LMFDB - Embedded newform 76.7.h.a.69.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,7,Mod(65,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	76.h (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.4841103551$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 $ x^20 + 9460x^18 + 36670708x^16 + 75655761912x^14 + 90488614064544x^12 + 64290781142478432x^10 + 26741969359067432592x^8 + 6092313787108272225600x^6 + 632563499034568748983104x^4 + 15152233769669432690883072*x^2 + 103372598905826219735270400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8}\cdot 19^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			69.7
Root			$-17.5071i$ of defining polynomial
Character	$\chi$	$=$	76.69
Dual form			76.7.h.a.65.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.6616 - 7.88751i) q^{3} +(93.3237 + 161.641i) q^{5} -434.577 q^{7} +(-240.074 + 415.821i) q^{9} +O(q^{10})$	\(q+(13.6616 - 7.88751i) q^{3} +(93.3237 + 161.641i) q^{5} -434.577 q^{7} +(-240.074 + 415.821i) q^{9} -255.032 q^{11} +(-2469.86 - 1425.97i) q^{13} +(2549.90 + 1472.18i) q^{15} +(-3363.84 - 5826.34i) q^{17} +(3749.31 + 5743.56i) q^{19} +(-5937.00 + 3427.73i) q^{21} +(-5566.10 + 9640.78i) q^{23} +(-9606.11 + 16638.3i) q^{25} +19074.3i q^{27} +(-16545.6 - 9552.59i) q^{29} -6024.94i q^{31} +(-3484.13 + 2011.57i) q^{33} +(-40556.3 - 70245.6i) q^{35} +36269.0i q^{37} -44989.5 q^{39} +(-76636.3 + 44246.0i) q^{41} +(71053.3 + 123068. i) q^{43} -89618.4 q^{45} +(17200.2 - 29791.6i) q^{47} +71208.1 q^{49} +(-91910.6 - 53064.6i) q^{51} +(173772. + 100327. i) q^{53} +(-23800.5 - 41223.7i) q^{55} +(96524.0 + 48893.4i) q^{57} +(93693.5 - 54094.0i) q^{59} +(176836. - 306288. i) q^{61} +(104331. - 180706. i) q^{63} -532308. i q^{65} +(297793. + 171931. i) q^{67} +175611. i q^{69} +(-272809. + 157506. i) q^{71} +(11611.0 + 20110.9i) q^{73} +303073. i q^{75} +110831. q^{77} +(-110948. + 64055.7i) q^{79} +(-24565.0 - 42547.8i) q^{81} +433643. q^{83} +(627851. - 1.08747e6i) q^{85} -301385. q^{87} +(-490924. - 283435. i) q^{89} +(1.07334e6 + 619695. i) q^{91} +(-47521.8 - 82310.2i) q^{93} +(-578498. + 1.14205e6i) q^{95} +(-1.34892e6 + 778798. i) q^{97} +(61226.6 - 106048. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	13.6616	−	7.88751i	0.505984	−	0.292130i	−0.225197	−	0.974313i	$-0.572303\pi$
$3$	13.6616	−	7.88751i	0.505984	−	0.292130i	0.731181	+	0.682183i	$0.238969\pi$
$4$		0			0
$5$	93.3237	+	161.641i	0.746589	+	1.29313i	0.949449	+	0.313923i	$0.101643\pi$
$5$	93.3237	+	161.641i	0.746589	+	1.29313i	−0.202859	+	0.979208i	$0.565023\pi$
$6$		0			0
$7$	−434.577			−1.26699			−0.633494	−	0.773748i	$-0.718380\pi$
$7$	−434.577			−1.26699			−0.633494	+	0.773748i	$0.718380\pi$
$8$		0			0
$9$	−240.074	+	415.821i	−0.329320	+	0.570399i
$10$		0			0
$11$	−255.032			−0.191609			−0.0958045	−	0.995400i	$-0.530542\pi$
$11$	−255.032			−0.191609			−0.0958045	+	0.995400i	$0.530542\pi$
$12$		0			0
$13$	−2469.86	−	1425.97i	−1.12420	−	0.649055i	−0.181727	−	0.983349i	$-0.558169\pi$
$13$	−2469.86	−	1425.97i	−1.12420	−	0.649055i	−0.942469	+	0.334294i	$0.891502\pi$
$14$		0			0
$15$	2549.90	+	1472.18i	0.755525	+	0.436202i
$16$		0			0
$17$	−3363.84	−	5826.34i	−0.684681	−	1.18590i	−0.973537	−	0.228530i	$-0.926608\pi$
$17$	−3363.84	−	5826.34i	−0.684681	−	1.18590i	0.288856	−	0.957373i	$-0.406725\pi$
$18$		0			0
$19$	3749.31	+	5743.56i	0.546627	+	0.837376i
$19$	3749.31	+	5743.56i	0.546627	+	0.837376i
$20$		0			0
$21$	−5937.00	+	3427.73i	−0.641076	+	0.370125i
$22$		0			0
$23$	−5566.10	+	9640.78i	−0.457475	+	0.792371i	−0.998827	−	0.0484257i	$-0.984580\pi$
$23$	−5566.10	+	9640.78i	−0.457475	+	0.792371i	0.541351	+	0.840797i	$0.317913\pi$
$24$		0			0
$25$	−9606.11	+	16638.3i	−0.614791	+	1.06485i
$26$		0			0
$27$			19074.3i			0.969077i
$28$		0			0
$29$	−16545.6	−	9552.59i	−0.678403	−	0.391676i	0.120850	−	0.992671i	$-0.461438\pi$
$29$	−16545.6	−	9552.59i	−0.678403	−	0.391676i	−0.799253	+	0.600995i	$0.794771\pi$
$30$		0			0
$31$		−	6024.94i		−	0.202240i	−0.994874	−	0.101120i	$-0.967757\pi$
$31$		−	6024.94i		−	0.202240i	0.994874	−	0.101120i	$-0.0322427\pi$
$32$		0			0
$33$	−3484.13	+	2011.57i	−0.0969512	+	0.0559748i
$34$		0			0
$35$	−40556.3	−	70245.6i	−0.945920	−	1.63838i
$36$		0			0
$37$			36269.0i			0.716028i	0.933716	+	0.358014i	$0.116546\pi$
$37$			36269.0i			0.716028i	−0.933716	+	0.358014i	$0.883454\pi$
$38$		0			0
$39$	−44989.5			−0.758434
$40$		0			0
$41$	−76636.3	+	44246.0i	−1.11194	+	0.641981i	−0.939332	−	0.343009i	$-0.888554\pi$
$41$	−76636.3	+	44246.0i	−1.11194	+	0.641981i	−0.172611	+	0.984990i	$0.555220\pi$
$42$		0			0
$43$	71053.3	+	123068.i	0.893673	+	1.54789i	0.835438	+	0.549585i	$0.185214\pi$
$43$	71053.3	+	123068.i	0.893673	+	1.54789i	0.0582353	+	0.998303i	$0.481453\pi$
$44$		0			0
$45$	−89618.4			−0.983467
$46$		0			0
$47$	17200.2	−	29791.6i	0.165669	−	0.286946i	−0.771224	−	0.636564i	$-0.780355\pi$
$47$	17200.2	−	29791.6i	0.165669	−	0.286946i	0.936892	+	0.349618i	$0.113688\pi$
$48$		0			0
$49$	71208.1			0.605259
$50$		0			0
$51$	−91910.6	−	53064.6i	−0.692875	−	0.400032i
$52$		0			0
$53$	173772.	+	100327.i	1.16722	+	0.673895i	0.953024	−	0.302896i	$-0.0979535\pi$
$53$	173772.	+	100327.i	1.16722	+	0.673895i	0.214196	+	0.976791i	$0.431287\pi$
$54$		0			0
$55$	−23800.5	−	41223.7i	−0.143053	−	0.247776i
$56$		0			0
$57$	96524.0	+	48893.4i	0.521207	+	0.264013i
$58$		0			0
$59$	93693.5	−	54094.0i	0.456198	−	0.263386i	−0.254246	−	0.967140i	$-0.581827\pi$
$59$	93693.5	−	54094.0i	0.456198	−	0.263386i	0.710444	+	0.703753i	$0.248494\pi$
$60$		0			0
$61$	176836.	−	306288.i	0.779077	−	1.34940i	−0.153397	−	0.988165i	$-0.549021\pi$
$61$	176836.	−	306288.i	0.779077	−	1.34940i	0.932474	−	0.361237i	$-0.117645\pi$
$62$		0			0
$63$	104331.	−	180706.i	0.417245	−	0.722689i
$64$		0			0
$65$		−	532308.i		−	1.93831i
$66$		0			0
$67$	297793.	+	171931.i	0.990126	+	0.571650i	0.905312	−	0.424747i	$-0.139637\pi$
$67$	297793.	+	171931.i	0.990126	+	0.571650i	0.0848142	+	0.996397i	$0.472970\pi$
$68$		0			0
$69$			175611.i			0.534569i
$70$		0			0
$71$	−272809.	+	157506.i	−0.762225	+	0.440071i	−0.830094	−	0.557623i	$-0.811713\pi$
$71$	−272809.	+	157506.i	−0.762225	+	0.440071i	0.0678689	+	0.997694i	$0.478380\pi$
$72$		0			0
$73$	11611.0	+	20110.9i	0.0298471	+	0.0516967i	0.880563	−	0.473929i	$-0.157165\pi$
$73$	11611.0	+	20110.9i	0.0298471	+	0.0516967i	−0.850716	+	0.525626i	$0.823831\pi$
$74$		0			0
$75$			303073.i			0.718396i
$76$		0			0
$77$	110831.			0.242766
$78$		0			0
$79$	−110948.	+	64055.7i	−0.225028	+	0.129920i	−0.608276	−	0.793725i	$-0.708139\pi$
$79$	−110948.	+	64055.7i	−0.225028	+	0.129920i	0.383248	+	0.923645i	$0.374805\pi$
$80$		0			0
$81$	−24565.0	−	42547.8i	−0.0462233	−	0.0800611i
$82$		0			0
$83$	433643.			0.758399			0.379199	−	0.925315i	$-0.376199\pi$
$83$	433643.			0.758399			0.379199	+	0.925315i	$0.376199\pi$
$84$		0			0
$85$	627851.	−	1.08747e6i	1.02235	−	1.77076i
$86$		0			0
$87$	−301385.			−0.457682
$88$		0			0
$89$	−490924.	−	283435.i	−0.696377	−	0.402054i	0.109619	−	0.993974i	$-0.465037\pi$
$89$	−490924.	−	283435.i	−0.696377	−	0.402054i	−0.805997	+	0.591920i	$0.798370\pi$
$90$		0			0
$91$	1.07334e6	+	619695.i	1.42434	+	0.822345i
$92$		0			0
$93$	−47521.8	−	82310.2i	−0.0590805	−	0.102330i
$94$		0			0
$95$	−578498.	+	1.14205e6i	−0.674731	+	1.33204i
$96$		0			0
$97$	−1.34892e6	+	778798.i	−1.47799	+	0.853316i	−0.999690	−	0.0248827i	$-0.992079\pi$
$97$	−1.34892e6	+	778798.i	−1.47799	+	0.853316i	−0.478296	+	0.878199i	$0.658745\pi$
$98$		0			0
$99$	61226.6	−	106048.i	0.0631007	−	0.109294i
$100$		0			0
$101$	874889.	−	1.51535e6i	0.849159	−	1.47079i	−0.0328007	−	0.999462i	$-0.510443\pi$
$101$	874889.	−	1.51535e6i	0.849159	−	1.47079i	0.881960	−	0.471325i	$-0.156224\pi$
$102$		0			0
$103$		−	1.93228e6i		−	1.76831i	−0.467194	−	0.884155i	$-0.654735\pi$
$103$		−	1.93228e6i		−	1.76831i	0.467194	−	0.884155i	$-0.345265\pi$
$104$		0			0
$105$	−1.10813e6	−	639777.i	−0.957241	−	0.552663i
$106$		0			0
$107$			2.08169e6i			1.69928i	0.527363	+	0.849640i	$0.323181\pi$
$107$			2.08169e6i			1.69928i	−0.527363	+	0.849640i	$0.676819\pi$
$108$		0			0
$109$	−385219.	+	222406.i	−0.297460	+	0.171738i	−0.641301	−	0.767289i	$-0.721605\pi$
$109$	−385219.	+	222406.i	−0.297460	+	0.171738i	0.343841	+	0.939028i	$0.388272\pi$
$110$		0			0
$111$	286072.	+	495491.i	0.209173	+	0.362299i
$112$		0			0
$113$			1.88569e6i			1.30688i	0.756979	+	0.653439i	$0.226674\pi$
$113$			1.88569e6i			1.30688i	−0.756979	+	0.653439i	$0.773326\pi$
$114$		0			0
$115$	−2.07780e6			−1.36619
$116$		0			0
$117$	1.18590e6	−	684679.i	0.740440	−	0.427493i
$118$		0			0
$119$	1.46185e6	+	2.53199e6i	0.867482	+	1.50252i
$120$		0			0
$121$	−1.70652e6			−0.963286
$122$		0			0
$123$	−697981.	+	1.20894e6i	−0.375084	+	0.649664i
$124$		0			0
$125$	−669545.			−0.342807
$126$		0			0
$127$	−689988.	−	398365.i	−0.336845	−	0.194478i	0.322031	−	0.946729i	$-0.395635\pi$
$127$	−689988.	−	398365.i	−0.336845	−	0.194478i	−0.658876	+	0.752251i	$0.728968\pi$
$128$		0			0
$129$	1.94140e6	+	1.12087e6i	0.904369	+	0.522138i
$130$		0			0
$131$	−1.10680e6	−	1.91704e6i	−0.492331	−	0.852742i	0.507630	−	0.861575i	$-0.330522\pi$
$131$	−1.10680e6	−	1.91704e6i	−0.492331	−	0.852742i	−0.999961	+	0.00883334i	$0.997188\pi$
$132$		0			0
$133$	−1.62936e6	−	2.49602e6i	−0.692570	−	1.06095i
$134$		0			0
$135$	−3.08320e6	+	1.78009e6i	−1.25314	+	0.723503i
$136$		0			0
$137$	495427.	−	858105.i	0.192672	−	0.333717i	−0.753463	−	0.657490i	$-0.771618\pi$
$137$	495427.	−	858105.i	0.192672	−	0.333717i	0.946135	+	0.323773i	$0.104951\pi$
$138$		0			0
$139$	−1.59951e6	+	2.77043e6i	−0.595583	+	1.03158i	0.397882	+	0.917437i	$0.369746\pi$
$139$	−1.59951e6	+	2.77043e6i	−0.595583	+	1.03158i	−0.993464	+	0.114143i	$0.963588\pi$
$140$		0			0
$141$		−	542667.i		−	0.193587i
$142$		0			0
$143$	629892.	+	363668.i	0.215406	+	0.124365i
$144$		0			0
$145$		−	3.56593e6i		−	1.16968i
$146$		0			0
$147$	972815.	−	561655.i	0.306251	−	0.176814i
$148$		0			0
$149$	−957224.	−	1.65796e6i	−0.289371	−	0.501205i	0.684289	−	0.729211i	$-0.260113\pi$
$149$	−957224.	−	1.65796e6i	−0.289371	−	0.501205i	−0.973660	+	0.228006i	$0.926779\pi$
$150$		0			0
$151$		−	890825.i		−	0.258739i	−0.991596	−	0.129369i	$-0.958705\pi$
$151$		−	890825.i		−	0.258739i	0.991596	−	0.129369i	$-0.0412953\pi$
$152$		0			0
$153$	3.23028e6			0.901916
$154$		0			0
$155$	973880.	−	562270.i	0.261523	−	0.150990i
$156$		0			0
$157$	2.52568e6	+	4.37461e6i	0.652650	+	1.13042i	0.982477	+	0.186381i	$0.0596760\pi$
$157$	2.52568e6	+	4.37461e6i	0.652650	+	1.13042i	−0.329828	+	0.944041i	$0.606991\pi$
$158$		0			0
$159$	3.16534e6			0.787460
$160$		0			0
$161$	2.41890e6	−	4.18966e6i	0.579616	−	1.00392i
$162$		0			0
$163$	6.07226e6			1.40213			0.701064	−	0.713098i	$-0.252709\pi$
$163$	6.07226e6			1.40213			0.701064	+	0.713098i	$0.252709\pi$
$164$		0			0
$165$	−650304.	−	375453.i	−0.144765	−	0.0835803i
$166$		0			0
$167$	2.66450e6	+	1.53835e6i	0.572093	+	0.330298i	0.757985	−	0.652272i	$-0.226184\pi$
$167$	2.66450e6	+	1.53835e6i	0.572093	+	0.330298i	−0.185892	+	0.982570i	$0.559517\pi$
$168$		0			0
$169$	1.65339e6	+	2.86376e6i	0.342544	+	0.593304i
$170$		0			0
$171$	−3.28841e6	+	180161.i	−0.657654	+	0.0360306i
$172$		0			0
$173$	−4.67794e6	+	2.70081e6i	−0.903475	+	0.521622i	−0.878326	−	0.478062i	$-0.841339\pi$
$173$	−4.67794e6	+	2.70081e6i	−0.903475	+	0.521622i	−0.0251490	+	0.999684i	$0.508006\pi$
$174$		0			0
$175$	4.17459e6	−	7.23061e6i	0.778933	−	1.34915i
$176$		0			0
$177$	853334.	−	1.47802e6i	0.153886	−	0.266538i
$178$		0			0
$179$			3.82333e6i			0.666627i	0.942816	+	0.333314i	$0.108167\pi$
$179$			3.82333e6i			0.666627i	−0.942816	+	0.333314i	$0.891833\pi$
$180$		0			0
$181$	432946.	+	249961.i	0.0730126	+	0.0421539i	0.536062	−	0.844179i	$-0.319911\pi$
$181$	432946.	+	249961.i	0.0730126	+	0.0421539i	−0.463049	+	0.886333i	$0.653245\pi$
$182$		0			0
$183$		−	5.57918e6i		−	0.910368i
$184$		0			0
$185$	−5.86256e6	+	3.38475e6i	−0.925918	+	0.534579i
$186$		0			0
$187$	857885.	+	1.48590e6i	0.131191	+	0.227230i
$188$		0			0
$189$		−	8.28927e6i		−	1.22781i
$190$		0			0
$191$	4.86151e6			0.697704			0.348852	−	0.937178i	$-0.386572\pi$
$191$	4.86151e6			0.697704			0.348852	+	0.937178i	$0.386572\pi$
$192$		0			0
$193$	5.05578e6	−	2.91896e6i	0.703260	−	0.406028i	−0.105300	−	0.994440i	$-0.533580\pi$
$193$	5.05578e6	−	2.91896e6i	0.703260	−	0.406028i	0.808561	+	0.588413i	$0.200247\pi$
$194$		0			0
$195$	−4.19859e6	−	7.27217e6i	−0.566238	−	0.980754i
$196$		0			0
$197$	5.09057e6			0.665837			0.332919	−	0.942956i	$-0.391967\pi$
$197$	5.09057e6			0.665837			0.332919	+	0.942956i	$0.391967\pi$
$198$		0			0
$199$	−1.77612e6	+	3.07633e6i	−0.225379	+	0.390367i	−0.956433	−	0.291952i	$-0.905695\pi$
$199$	−1.77612e6	+	3.07633e6i	−0.225379	+	0.390367i	0.731054	+	0.682319i	$0.239029\pi$
$200$		0			0
$201$	5.42443e6			0.667984
$202$		0			0
$203$	7.19032e6	+	4.15133e6i	0.859528	+	0.496249i
$204$		0			0
$205$	−1.43040e7	−	8.25839e6i	−1.66033	−	0.958592i
$206$		0			0
$207$	−2.67256e6	−	4.62900e6i	−0.301312	−	0.521887i
$208$		0			0
$209$	−956194.	−	1.46479e6i	−0.104739	−	0.160449i
$210$		0			0
$211$	4.77345e6	−	2.75595e6i	0.508142	−	0.293376i	−0.223928	−	0.974606i	$-0.571888\pi$
$211$	4.77345e6	−	2.75595e6i	0.508142	−	0.293376i	0.732070	+	0.681230i	$0.238555\pi$
$212$		0			0
$213$	−2.48466e6	+	4.30357e6i	−0.257116	+	0.445338i
$214$		0			0
$215$	−1.32619e7	+	2.29703e7i	−1.33441	+	2.31127i
$216$		0			0
$217$			2.61830e6i			0.256236i
$218$		0			0
$219$	317250.	+	183164.i	0.0302043	+	0.0174385i
$220$		0			0
$221$			1.91870e7i			1.77758i
$222$		0			0
$223$	−6.28438e6	+	3.62829e6i	−0.566693	+	0.327181i	−0.755828	−	0.654771i	$-0.772765\pi$
$223$	−6.28438e6	+	3.62829e6i	−0.566693	+	0.327181i	0.189134	+	0.981951i	$0.439432\pi$
$224$		0			0
$225$	−4.61236e6	−	7.98884e6i	−0.404926	−	0.701352i
$226$		0			0
$227$			6.70391e6i			0.573127i	0.958061	+	0.286563i	$0.0925129\pi$
$227$			6.70391e6i			0.573127i	−0.958061	+	0.286563i	$0.907487\pi$
$228$		0			0
$229$	−2.75501e6			−0.229413			−0.114706	−	0.993399i	$-0.536593\pi$
$229$	−2.75501e6			−0.229413			−0.114706	+	0.993399i	$0.536593\pi$
$230$		0			0
$231$	1.51412e6	−	874180.i	0.122836	−	0.0709194i
$232$		0			0
$233$	−3.62219e6	−	6.27382e6i	−0.286354	−	0.495980i	0.686582	−	0.727052i	$-0.259110\pi$
$233$	−3.62219e6	−	6.27382e6i	−0.286354	−	0.495980i	−0.972937	+	0.231072i	$0.925777\pi$
$234$		0			0
$235$	6.42074e6			0.494745
$236$		0			0
$237$	−1.01048e6	+	1.75020e6i	−0.0759072	+	0.131475i
$238$		0			0
$239$	−1.32361e7			−0.969542			−0.484771	−	0.874641i	$-0.661097\pi$
$239$	−1.32361e7			−0.969542			−0.484771	+	0.874641i	$0.661097\pi$
$240$		0			0
$241$	1.58330e7	+	9.14121e6i	1.13113	+	0.653059i	0.944219	−	0.329317i	$-0.106819\pi$
$241$	1.58330e7	+	9.14121e6i	1.13113	+	0.653059i	0.186913	+	0.982377i	$0.440152\pi$
$242$		0			0
$243$	−1.27134e7	−	7.34011e6i	−0.886022	−	0.511545i
$244$		0			0
$245$	6.64540e6	+	1.15102e7i	0.451880	+	0.782679i
$246$		0			0
$247$	−1.07010e6	−	1.95322e7i	−0.0710124	−	1.29617i
$248$		0			0
$249$	5.92424e6	−	3.42036e6i	0.383738	−	0.221551i
$250$		0			0
$251$	−1.19440e7	+	2.06875e7i	−0.755313	+	1.30824i	0.189905	+	0.981802i	$0.439182\pi$
$251$	−1.19440e7	+	2.06875e7i	−0.755313	+	1.30824i	−0.945219	+	0.326438i	$0.894151\pi$
$252$		0			0
$253$	1.41953e6	−	2.45870e6i	0.0876565	−	0.151825i
$254$		0			0
$255$		−	1.98087e7i		−	1.19464i
$256$		0			0
$257$	−8.40872e6	−	4.85477e6i	−0.495371	−	0.286002i	0.231429	−	0.972852i	$-0.425660\pi$
$257$	−8.40872e6	−	4.85477e6i	−0.495371	−	0.286002i	−0.726800	+	0.686849i	$0.758993\pi$
$258$		0			0
$259$		−	1.57617e7i		−	0.907199i
$260$		0			0
$261$	7.94433e6	−	4.58666e6i	0.446823	−	0.257974i
$262$		0			0
$263$	−860313.	−	1.49011e6i	−0.0472922	−	0.0819125i	0.841410	−	0.540397i	$-0.181726\pi$
$263$	−860313.	−	1.49011e6i	−0.0472922	−	0.0819125i	−0.888702	+	0.458484i	$0.848393\pi$
$264$		0			0
$265$			3.74517e7i			2.01249i
$266$		0			0
$267$	−8.94240e6			−0.469808
$268$		0			0
$269$	−2.45089e7	+	1.41502e7i	−1.25912	+	0.726952i	−0.972903	−	0.231213i	$-0.925731\pi$
$269$	−2.45089e7	+	1.41502e7i	−1.25912	+	0.726952i	−0.286215	+	0.958165i	$0.592397\pi$
$270$		0			0
$271$	−8.02548e6	−	1.39005e7i	−0.403240	−	0.698432i	0.590875	−	0.806763i	$-0.298783\pi$
$271$	−8.02548e6	−	1.39005e7i	−0.403240	−	0.698432i	−0.994115	+	0.108331i	$0.965449\pi$
$272$		0			0
$273$	1.95514e7			0.960926
$274$		0			0
$275$	2.44986e6	−	4.24329e6i	0.117800	−	0.204035i
$276$		0			0
$277$	−3.25237e7			−1.53024			−0.765122	−	0.643885i	$-0.777322\pi$
$277$	−3.25237e7			−1.53024			−0.765122	+	0.643885i	$0.777322\pi$
$278$		0			0
$279$	2.50530e6	+	1.44643e6i	0.115358	+	0.0666018i
$280$		0			0
$281$	1.95761e7	+	1.13023e7i	0.882284	+	0.509387i	0.871411	−	0.490554i	$-0.163206\pi$
$281$	1.95761e7	+	1.13023e7i	0.882284	+	0.509387i	0.0108730	+	0.999941i	$0.496539\pi$
$282$		0			0
$283$	−4.01226e6	−	6.94943e6i	−0.177023	−	0.306613i	0.763837	−	0.645410i	$-0.223313\pi$
$283$	−4.01226e6	−	6.94943e6i	−0.177023	−	0.306613i	−0.940859	+	0.338797i	$0.889980\pi$
$284$		0			0
$285$	1.10478e6	+	2.01652e7i	0.0477245	+	0.871098i
$286$		0			0
$287$	3.33043e7	−	1.92283e7i	1.40882	−	0.813382i
$288$		0			0
$289$	−1.05620e7	+	1.82939e7i	−0.437576	+	0.757903i
$290$		0			0
$291$	−1.22856e7	+	2.12792e7i	−0.498559	+	0.863529i
$292$		0			0
$293$			2.92619e7i			1.16332i	0.813431	+	0.581661i	$0.197597\pi$
$293$			2.92619e7i			1.16332i	−0.813431	+	0.581661i	$0.802403\pi$
$294$		0			0
$295$	1.74876e7	+	1.00965e7i	0.681185	+	0.393282i
$296$		0			0
$297$		−	4.86456e6i		−	0.185684i
$298$		0			0
$299$	2.74950e7	−	1.58742e7i	1.02858	−	0.593853i
$300$		0			0
$301$	−3.08781e7	−	5.34825e7i	−1.13227	−	1.96116i
$302$		0			0
$303$		−	2.76028e7i		−	0.992260i
$304$		0			0
$305$	6.60118e7			2.32660
$306$		0			0
$307$	2.55956e7	−	1.47776e7i	0.884607	−	0.510728i	0.0124322	−	0.999923i	$-0.496043\pi$
$307$	2.55956e7	−	1.47776e7i	0.884607	−	0.510728i	0.872175	+	0.489195i	$0.162709\pi$
$308$		0			0
$309$	−1.52409e7	−	2.63980e7i	−0.516577	−	0.894737i
$310$		0			0
$311$	−4.45907e7			−1.48239			−0.741196	−	0.671289i	$-0.765741\pi$
$311$	−4.45907e7			−1.48239			−0.741196	+	0.671289i	$0.765741\pi$
$312$		0			0
$313$	1.96053e7	−	3.39574e7i	0.639353	−	1.10739i	−0.346222	−	0.938152i	$-0.612536\pi$
$313$	1.96053e7	−	3.39574e7i	0.639353	−	1.10739i	0.985575	−	0.169239i	$-0.0541309\pi$
$314$		0			0
$315$	3.89461e7			1.24604
$316$		0			0
$317$	−686537.	−	396372.i	−0.0215519	−	0.0124430i	0.489185	−	0.872180i	$-0.337294\pi$
$317$	−686537.	−	396372.i	−0.0215519	−	0.0124430i	−0.510737	+	0.859737i	$0.670628\pi$
$318$		0			0
$319$	4.21964e6	+	2.43621e6i	0.129988	+	0.0750487i
$320$		0			0
$321$	1.64194e7	+	2.84392e7i	0.496411	+	0.859809i
$322$		0			0
$323$	2.08519e7	−	4.11652e7i	0.618781	−	1.22158i
$324$		0			0
$325$	4.74515e7	−	2.73961e7i	1.38229	−	0.798066i
$326$		0			0
$327$	−3.50846e6	+	6.07684e6i	−0.100340	+	0.173794i
$328$		0			0
$329$	−7.47481e6	+	1.29468e7i	−0.209900	+	0.363558i
$330$		0			0
$331$		−	5.39124e7i		−	1.48664i	−0.668938	−	0.743318i	$-0.733251\pi$
$331$		−	5.39124e7i		−	1.48664i	0.668938	−	0.743318i	$-0.266749\pi$
$332$		0			0
$333$	−1.50814e7	−	8.70725e6i	−0.408422	−	0.235802i
$334$		0			0
$335$			6.41809e7i			1.70715i
$336$		0			0
$337$	−5.89576e7	+	3.40392e7i	−1.54046	+	0.889384i	−0.541650	+	0.840604i	$0.682200\pi$
$337$	−5.89576e7	+	3.40392e7i	−1.54046	+	0.889384i	−0.998810	+	0.0487801i	$0.984467\pi$
$338$		0			0
$339$	1.48734e7	+	2.57615e7i	0.381778	+	0.661260i
$340$		0			0
$341$			1.53655e6i			0.0387511i
$342$		0			0
$343$	2.01821e7			0.500132
$344$		0			0
$345$	−2.83860e7	+	1.63886e7i	−0.691268	+	0.399104i
$346$		0			0
$347$	1.56502e7	+	2.71070e7i	0.374569	+	0.648773i	0.990262	−	0.139213i	$-0.0444573\pi$
$347$	1.56502e7	+	2.71070e7i	0.374569	+	0.648773i	−0.615693	+	0.787986i	$0.711124\pi$
$348$		0			0
$349$	−9.20941e6			−0.216648			−0.108324	−	0.994116i	$-0.534548\pi$
$349$	−9.20941e6			−0.216648			−0.108324	+	0.994116i	$0.534548\pi$
$350$		0			0
$351$	2.71995e7	−	4.71109e7i	0.628984	−	1.08943i
$352$		0			0
$353$	6.21034e6			0.141186			0.0705929	−	0.997505i	$-0.477511\pi$
$353$	6.21034e6			0.141186			0.0705929	+	0.997505i	$0.477511\pi$
$354$		0			0
$355$	−5.09190e7	−	2.93981e7i	−1.13814	−	0.657104i
$356$		0			0
$357$	3.99422e7	+	2.30607e7i	0.877865	+	0.506836i
$358$		0			0
$359$	2.46475e7	+	4.26908e7i	0.532709	+	0.922679i	0.999270	+	0.0381903i	$0.0121593\pi$
$359$	2.46475e7	+	4.26908e7i	0.532709	+	0.922679i	−0.466561	+	0.884489i	$0.654507\pi$
$360$		0			0
$361$	−1.89312e7	+	4.30688e7i	−0.402398	+	0.915465i
$362$		0			0
$363$	−2.33137e7	+	1.34602e7i	−0.487407	+	0.281405i
$364$		0			0
$365$	−2.16717e6	+	3.75364e6i	−0.0445670	+	0.0771923i
$366$		0			0
$367$	−7.21105e6	+	1.24899e7i	−0.145882	+	0.252674i	−0.929702	−	0.368314i	$-0.879935\pi$
$367$	−7.21105e6	+	1.24899e7i	−0.145882	+	0.252674i	0.783820	+	0.620988i	$0.213269\pi$
$368$		0			0
$369$		−	4.24893e7i		−	0.845669i
$370$		0			0
$371$	−7.55174e7	−	4.36000e7i	−1.47885	−	0.853816i
$372$		0			0
$373$		−	9.50012e7i		−	1.83064i	−0.402728	−	0.915319i	$-0.631938\pi$
$373$		−	9.50012e7i		−	1.83064i	0.402728	−	0.915319i	$-0.368062\pi$
$374$		0			0
$375$	−9.14703e6	+	5.28104e6i	−0.173455	+	0.100144i
$376$		0			0
$377$	2.72435e7	+	4.71871e7i	0.508438	+	0.880641i
$378$		0			0
$379$			3.19628e7i			0.587121i	0.955941	+	0.293560i	$0.0948402\pi$
$379$			3.19628e7i			0.587121i	−0.955941	+	0.293560i	$0.905160\pi$
$380$		0			0
$381$	−1.25684e7			−0.227251
$382$		0			0
$383$	−4.76741e7	+	2.75247e7i	−0.848567	+	0.489921i	−0.860167	−	0.510012i	$-0.829641\pi$
$383$	−4.76741e7	+	2.75247e7i	−0.848567	+	0.489921i	0.0115998	+	0.999933i	$0.496308\pi$
$384$		0			0
$385$	1.03431e7	+	1.79149e7i	0.181247	+	0.313929i
$386$		0			0
$387$	−6.82323e7			−1.17722
$388$		0			0
$389$	−6.39537e6	+	1.10771e7i	−0.108647	+	0.188182i	−0.915222	−	0.402949i	$-0.867985\pi$
$389$	−6.39537e6	+	1.10771e7i	−0.108647	+	0.188182i	0.806575	+	0.591131i	$0.201318\pi$
$390$		0			0
$391$	7.48939e7			1.25290
$392$		0			0
$393$	−3.02414e7	−	1.74599e7i	−0.498223	−	0.287649i
$394$		0			0
$395$	−2.07081e7	−	1.19558e7i	−0.336008	−	0.193994i
$396$		0			0
$397$	2.34442e7	+	4.06066e7i	0.374683	+	0.648970i	0.990280	−	0.139091i	$-0.0444182\pi$
$397$	2.34442e7	+	4.06066e7i	0.374683	+	0.648970i	−0.615596	+	0.788062i	$0.711085\pi$
$398$		0			0
$399$	−4.19471e7	−	2.12479e7i	−0.660364	−	0.334501i
$400$		0			0
$401$	−3.25668e7	+	1.88025e7i	−0.505059	+	0.291596i	−0.730800	−	0.682591i	$-0.760853\pi$
$401$	−3.25668e7	+	1.88025e7i	−0.505059	+	0.291596i	0.225741	+	0.974187i	$0.427520\pi$
$402$		0			0
$403$	−8.59141e6	+	1.48808e7i	−0.131265	+	0.227358i
$404$		0			0
$405$	4.58498e6	−	7.94143e6i	0.0690197	−	0.119546i
$406$		0			0
$407$		−	9.24974e6i		−	0.137197i
$408$		0			0
$409$	−1.01262e8	−	5.84636e7i	−1.48005	−	0.854506i	−0.480304	−	0.877102i	$-0.659474\pi$
$409$	−1.01262e8	−	5.84636e7i	−1.48005	−	0.854506i	−0.999745	+	0.0225956i	$0.992807\pi$
$410$		0			0
$411$		−	1.56308e7i		−	0.225141i
$412$		0			0
$413$	−4.07170e7	+	2.35080e7i	−0.577998	+	0.333707i
$414$		0			0
$415$	4.04691e7	+	7.00946e7i	0.566212	+	0.980709i
$416$		0			0
$417$			5.04646e7i			0.695951i
$418$		0			0
$419$	−7.74454e6			−0.105282			−0.0526409	−	0.998614i	$-0.516764\pi$
$419$	−7.74454e6			−0.105282			−0.0526409	+	0.998614i	$0.516764\pi$
$420$		0			0
$421$	6.83733e7	−	3.94753e7i	0.916305	−	0.529029i	0.0338507	−	0.999427i	$-0.489223\pi$
$421$	6.83733e7	−	3.94753e7i	0.916305	−	0.529029i	0.882454	+	0.470398i	$0.155890\pi$
$422$		0			0
$423$	8.25865e6	+	1.43044e7i	0.109116	+	0.188994i
$424$		0			0
$425$	1.29254e8			1.68374
$426$		0			0
$427$	−7.68487e7	+	1.33106e8i	−0.987082	+	1.70968i
$428$		0			0
$429$	1.14738e7			0.145323
$430$		0			0
$431$	−1.62876e7	−	9.40365e6i	−0.203435	−	0.117453i	0.394822	−	0.918758i	$-0.370806\pi$
$431$	−1.62876e7	−	9.40365e6i	−0.203435	−	0.117453i	−0.598257	+	0.801305i	$0.704140\pi$
$432$		0			0
$433$	9.99598e7	+	5.77118e7i	1.23129	+	0.710888i	0.967300	−	0.253636i	$-0.0816266\pi$
$433$	9.99598e7	+	5.77118e7i	1.23129	+	0.710888i	0.263994	+	0.964524i	$0.414960\pi$
$434$		0			0
$435$	−2.81263e7	−	4.87162e7i	−0.341700	−	0.591842i
$436$		0			0
$437$	−7.62415e7	+	4.17701e6i	−0.913581	+	0.0500519i
$438$		0			0
$439$	4.63182e7	−	2.67418e7i	0.547467	−	0.316080i	−0.200633	−	0.979667i	$-0.564300\pi$
$439$	4.63182e7	−	2.67418e7i	0.547467	−	0.316080i	0.748100	+	0.663586i	$0.230966\pi$
$440$		0			0
$441$	−1.70952e7	+	2.96098e7i	−0.199324	+	0.345239i
$442$		0			0
$443$	−4.59316e7	+	7.95559e7i	−0.528324	+	0.915085i	0.471130	+	0.882064i	$0.343846\pi$
$443$	−4.59316e7	+	7.95559e7i	−0.528324	+	0.915085i	−0.999455	+	0.0330210i	$0.989487\pi$
$444$		0			0
$445$		−	1.05805e8i		−	1.20068i
$446$		0			0
$447$	−2.61544e7	−	1.51002e7i	−0.292834	−	0.169068i
$448$		0			0
$449$		−	1.24474e8i		−	1.37511i	−0.726130	−	0.687557i	$-0.758683\pi$
$449$		−	1.24474e8i		−	1.37511i	0.726130	−	0.687557i	$-0.241317\pi$
$450$		0			0
$451$	1.95447e7	−	1.12841e7i	0.213058	−	0.123009i
$452$		0			0
$453$	−7.02639e6	−	1.21701e7i	−0.0755854	−	0.130918i
$454$		0			0
$455$			2.31329e8i			2.45581i
$456$		0			0
$457$	1.15601e8			1.21120			0.605598	−	0.795771i	$-0.292934\pi$
$457$	1.15601e8			1.21120			0.605598	+	0.795771i	$0.292934\pi$
$458$		0			0
$459$	1.11134e8	−	6.41630e7i	1.14923	−	0.663509i
$460$		0			0
$461$	−7.74014e7	−	1.34063e8i	−0.790035	−	1.36838i	−0.925945	−	0.377659i	$-0.876729\pi$
$461$	−7.74014e7	−	1.34063e8i	−0.790035	−	1.36838i	0.135910	−	0.990721i	$-0.456604\pi$
$462$		0			0
$463$	−2.96494e7			−0.298726			−0.149363	−	0.988782i	$-0.547722\pi$
$463$	−2.96494e7			−0.298726			−0.149363	+	0.988782i	$0.547722\pi$
$464$		0			0
$465$	8.86982e6	−	1.53630e7i	0.0882177	−	0.152798i
$466$		0			0
$467$	1.63514e8			1.60548			0.802738	−	0.596332i	$-0.203376\pi$
$467$	1.63514e8			1.60548			0.802738	+	0.596332i	$0.203376\pi$
$468$		0			0
$469$	−1.29414e8	−	7.47173e7i	−1.25448	−	0.724273i
$470$		0			0
$471$	6.90097e7	+	3.98427e7i	0.660461	+	0.381317i
$472$		0			0
$473$	−1.81208e7	−	3.13862e7i	−0.171236	−	0.296589i
$474$		0			0
$475$	−1.31579e8	+	7.20878e6i	−1.22774	+	0.0672637i
$476$		0			0
$477$	−8.34365e7	+	4.81721e7i	−0.768778	+	0.443854i
$478$		0			0
$479$	6.37760e7	−	1.10463e8i	0.580298	−	1.00511i	−0.415146	−	0.909755i	$-0.636269\pi$
$479$	6.37760e7	−	1.10463e8i	0.580298	−	1.00511i	0.995444	−	0.0953505i	$-0.0303972\pi$
$480$		0			0
$481$	5.17186e7	−	8.95792e7i	0.464741	−	0.804956i
$482$		0			0
$483$		−	7.63164e7i		−	0.677293i
$484$		0			0
$485$	−2.51772e8	−	1.45361e8i	−2.20690	−	1.27415i
$486$		0			0
$487$			3.43397e7i			0.297310i	0.988889	+	0.148655i	$0.0474944\pi$
$487$			3.43397e7i			0.297310i	−0.988889	+	0.148655i	$0.952506\pi$
$488$		0			0
$489$	8.29567e7	−	4.78951e7i	0.709455	−	0.409604i
$490$		0			0
$491$	6.70061e7	+	1.16058e8i	0.566069	+	0.980461i	0.996949	+	0.0780518i	$0.0248699\pi$
$491$	6.70061e7	+	1.16058e8i	0.566069	+	0.980461i	−0.430880	+	0.902409i	$0.641797\pi$
$492$		0			0
$493$			1.28533e8i			1.07269i
$494$		0			0
$495$	2.28555e7			0.188441
$496$		0			0
$497$	1.18556e8	−	6.84486e7i	0.965730	−	0.557565i
$498$		0			0
$499$	9.06362e7	+	1.56986e8i	0.729457	+	1.26346i	0.957113	+	0.289715i	$0.0935606\pi$
$499$	9.06362e7	+	1.56986e8i	0.729457	+	1.26346i	−0.227656	+	0.973742i	$0.573106\pi$
$500$		0			0
$501$	4.85351e7			0.385960
$502$		0			0
$503$	5.62620e7	−	9.74487e7i	0.442091	−	0.765723i	−0.555754	−	0.831347i	$-0.687570\pi$
$503$	5.62620e7	−	9.74487e7i	0.442091	−	0.765723i	0.997844	+	0.0656235i	$0.0209036\pi$
$504$		0			0
$505$	3.26592e8			2.53589
$506$		0			0
$507$	4.51759e7	+	2.60823e7i	0.346644	+	0.200135i
$508$		0			0
$509$	−1.42761e8	−	8.24233e7i	−1.08257	−	0.625024i	−0.150983	−	0.988536i	$-0.548244\pi$
$509$	−1.42761e8	−	8.24233e7i	−1.08257	−	0.625024i	−0.931589	+	0.363513i	$0.881577\pi$
$510$		0			0
$511$	−5.04588e6	−	8.73972e6i	−0.0378159	−	0.0654990i
$512$		0			0
$513$	−1.09555e8	+	7.15157e7i	−0.811482	+	0.529724i
$514$		0			0
$515$	3.12336e8	−	1.80327e8i	2.28666	−	1.32020i
$516$		0			0
$517$	−4.38660e6	+	7.59781e6i	−0.0317436	+	0.0549815i
$518$		0			0
$519$	−4.26053e7	+	7.37946e7i	−0.304763	+	0.527865i
$520$		0			0
$521$			1.29458e8i			0.915407i	0.889105	+	0.457704i	$0.151328\pi$
$521$			1.29458e8i			0.915407i	−0.889105	+	0.457704i	$0.848672\pi$
$522$		0			0
$523$	−3.68805e7	−	2.12929e7i	−0.257805	−	0.148844i	0.365528	−	0.930800i	$-0.380888\pi$
$523$	−3.68805e7	−	2.12929e7i	−0.257805	−	0.148844i	−0.623333	+	0.781957i	$0.714222\pi$
$524$		0			0
$525$		−	1.31709e8i		−	0.910199i
$526$		0			0
$527$	−3.51033e7	+	2.02669e7i	−0.239837	+	0.138470i
$528$		0			0
$529$	1.20549e7	+	2.08797e7i	0.0814324	+	0.141045i
$530$		0			0
$531$			5.19463e7i			0.346953i
$532$		0			0
$533$	2.52374e8			1.66672
$534$		0			0
$535$	−3.36487e8	+	1.94271e8i	−2.19739	+	1.26866i
$536$		0			0
$537$	3.01566e7	+	5.22327e7i	0.194742	+	0.337303i
$538$		0			0
$539$	−1.81603e7			−0.115973
$540$		0			0
$541$	2.66258e7	−	4.61173e7i	0.168156	−	0.291254i	−0.769616	−	0.638507i	$-0.779552\pi$
$541$	2.66258e7	−	4.61173e7i	0.168156	−	0.291254i	0.937771	+	0.347253i	$0.112886\pi$
$542$		0			0
$543$	7.88630e6			0.0492577
$544$		0			0
$545$	−7.19001e7	−	4.15115e7i	−0.444160	−	0.256436i
$546$		0			0
$547$	−1.13824e8	−	6.57162e7i	−0.695458	−	0.401523i	0.110195	−	0.993910i	$-0.464852\pi$
$547$	−1.13824e8	−	6.57162e7i	−0.695458	−	0.401523i	−0.805654	+	0.592387i	$0.798186\pi$
$548$		0			0
$549$	8.49074e7	+	1.47064e8i	0.513131	+	0.888770i
$550$		0			0
$551$	−7.16861e6	−	1.30846e8i	−0.0428529	−	0.782179i
$552$		0			0
$553$	4.82154e7	−	2.78371e7i	0.285108	−	0.164607i
$554$		0			0
$555$	−5.33946e7	+	9.24821e7i	−0.312333	+	0.540977i
$556$		0			0
$557$	−1.48316e8	+	2.56890e8i	−0.858265	+	1.48656i	0.0153176	+	0.999883i	$0.495124\pi$
$557$	−1.48316e8	+	2.56890e8i	−0.858265	+	1.48656i	−0.873583	+	0.486676i	$0.838209\pi$
$558$		0			0
$559$		−	4.05280e8i		−	2.32017i
$560$		0			0
$561$	2.34401e7	+	1.35332e7i	0.132761	+	0.0766497i
$562$		0			0
$563$		−	1.41586e8i		−	0.793407i	−0.917947	−	0.396704i	$-0.870154\pi$
$563$		−	1.41586e8i		−	0.793407i	0.917947	−	0.396704i	$-0.129846\pi$
$564$		0			0
$565$	−3.04806e8	+	1.75980e8i	−1.68996	+	0.975701i
$566$		0			0
$567$	1.06754e7	+	1.84903e7i	0.0585644	+	0.101436i
$568$		0			0
$569$		−	2.27347e8i		−	1.23411i	−0.786921	−	0.617054i	$-0.788326\pi$
$569$		−	2.27347e8i		−	1.23411i	0.786921	−	0.617054i	$-0.211674\pi$
$570$		0			0
$571$	−9.60082e6			−0.0515704			−0.0257852	−	0.999668i	$-0.508209\pi$
$571$	−9.60082e6			−0.0515704			−0.0257852	+	0.999668i	$0.508209\pi$
$572$		0			0
$573$	6.64159e7	−	3.83452e7i	0.353027	−	0.203820i
$574$		0			0
$575$	−1.06937e8	−	1.85221e8i	−0.562504	−	0.974285i
$576$		0			0
$577$	7.05802e7			0.367414			0.183707	−	0.982981i	$-0.441190\pi$
$577$	7.05802e7			0.367414			0.183707	+	0.982981i	$0.441190\pi$
$578$		0			0
$579$	4.60466e7	−	7.97550e7i	0.237226	−	0.410887i
$580$		0			0
$581$	−1.88451e8			−0.960882
$582$		0			0
$583$	−4.43174e7	−	2.55867e7i	−0.223650	−	0.129124i
$584$		0			0
$585$	2.21345e8	+	1.27793e8i	1.10561	+	0.638324i
$586$		0			0
$587$	1.28398e8	+	2.22392e8i	0.634812	+	1.09953i	0.986555	+	0.163430i	$0.0522558\pi$
$587$	1.28398e8	+	2.22392e8i	0.634812	+	1.09953i	−0.351743	+	0.936097i	$0.614411\pi$
$588$		0			0
$589$	3.46046e7	−	2.25894e7i	0.169351	−	0.110550i
$590$		0			0
$591$	6.95453e7	−	4.01520e7i	0.336903	−	0.194511i
$592$		0			0
$593$	−1.75619e8	+	3.04181e8i	−0.842186	+	1.45871i	0.0458574	+	0.998948i	$0.485398\pi$
$593$	−1.75619e8	+	3.04181e8i	−0.842186	+	1.45871i	−0.888043	+	0.459760i	$0.847935\pi$
$594$		0			0
$595$	−2.72850e8	+	4.72589e8i	−1.29531	+	2.24354i
$596$		0			0
$597$			5.60366e7i			0.263359i
$598$		0			0
$599$	−1.61224e8	−	9.30828e7i	−0.750152	−	0.433101i	0.0755966	−	0.997138i	$-0.475914\pi$
$599$	−1.61224e8	−	9.30828e7i	−0.750152	−	0.433101i	−0.825749	+	0.564038i	$0.809247\pi$
$600$		0			0
$601$			3.14603e8i			1.44924i	0.689150	+	0.724619i	$0.257984\pi$
$601$			3.14603e8i			1.44924i	−0.689150	+	0.724619i	$0.742016\pi$
$602$		0			0
$603$	−1.42985e8	+	8.25525e7i	−0.652137	+	0.376511i
$604$		0			0
$605$	−1.59259e8	−	2.75844e8i	−0.719179	−	1.24565i
$606$		0			0
$607$			3.10604e8i			1.38880i	0.719587	+	0.694402i	$0.244331\pi$
$607$			3.10604e8i			1.38880i	−0.719587	+	0.694402i	$0.755669\pi$
$608$		0			0
$609$	1.30975e8			0.579877
$610$		0			0
$611$	−8.49641e7	+	4.90541e7i	−0.372488	+	0.215056i
$612$		0			0
$613$	−1.60234e8	−	2.77534e8i	−0.695623	−	1.20486i	−0.969970	−	0.243224i	$-0.921795\pi$
$613$	−1.60234e8	−	2.77534e8i	−0.695623	−	1.20486i	0.274347	−	0.961631i	$-0.411538\pi$
$614$		0			0
$615$	−2.60553e8			−1.12013
$616$		0			0
$617$	−2.99213e7	+	5.18252e7i	−0.127387	+	0.220641i	−0.922663	−	0.385606i	$-0.873992\pi$
$617$	−2.99213e7	+	5.18252e7i	−0.127387	+	0.220641i	0.795277	+	0.606247i	$0.207326\pi$
$618$		0			0
$619$	1.52099e8			0.641290			0.320645	−	0.947200i	$-0.396100\pi$
$619$	1.52099e8			0.641290			0.320645	+	0.947200i	$0.396100\pi$
$620$		0			0
$621$	−1.83892e8	−	1.06170e8i	−0.767869	−	0.443329i
$622$		0			0
$623$	2.13344e8	+	1.23174e8i	0.882302	+	0.509397i
$624$		0			0
$625$	8.76111e7	+	1.51747e8i	0.358855	+	0.621555i
$626$		0			0
$627$	−2.46167e7	−	1.24694e7i	−0.0998681	−	0.0505873i
$628$		0			0
$629$	2.11315e8	−	1.22003e8i	0.849139	−	0.490251i
$630$		0			0
$631$	−1.10061e8	+	1.90631e8i	−0.438071	+	0.758761i	−0.997541	−	0.0700894i	$-0.977672\pi$
$631$	−1.10061e8	+	1.90631e8i	−0.438071	+	0.758761i	0.559470	+	0.828851i	$0.311005\pi$
$632$		0			0
$633$	4.34752e7	−	7.53013e7i	0.171408	−	0.296887i
$634$		0			0
$635$		−	1.48707e8i		−	0.580780i
$636$		0			0
$637$	−1.75874e8	−	1.01541e8i	−0.680429	−	0.392846i
$638$		0			0
$639$		−	1.51253e8i		−	0.579697i
$640$		0			0
$641$	−1.30176e8	+	7.51571e7i	−0.494261	+	0.285362i	−0.726341	−	0.687335i	$-0.758780\pi$
$641$	−1.30176e8	+	7.51571e7i	−0.494261	+	0.285362i	0.232079	+	0.972697i	$0.425447\pi$
$642$		0			0
$643$	−2.05308e8	−	3.55603e8i	−0.772275	−	1.33762i	−0.936313	−	0.351166i	$-0.885785\pi$
$643$	−2.05308e8	−	3.55603e8i	−0.772275	−	1.33762i	0.164038	−	0.986454i	$-0.447548\pi$
$644$		0			0
$645$			4.18414e8i			1.55929i
$646$		0			0
$647$	3.71980e8			1.37343			0.686716	−	0.726926i	$-0.259052\pi$
$647$	3.71980e8			1.37343			0.686716	+	0.726926i	$0.259052\pi$
$648$		0			0
$649$	−2.38948e7	+	1.37957e7i	−0.0874117	+	0.0504672i
$650$		0			0
$651$	2.06519e7	+	3.57701e7i	0.0748543	+	0.129651i
$652$		0			0
$653$	−3.92895e7			−0.141103			−0.0705516	−	0.997508i	$-0.522476\pi$
$653$	−3.92895e7			−0.141103			−0.0705516	+	0.997508i	$0.522476\pi$
$654$		0			0
$655$	2.06582e8	−	3.57810e8i	0.735137	−	1.27330i
$656$		0			0
$657$	−1.11500e7			−0.0393170
$658$		0			0
$659$	3.38065e8	+	1.95182e8i	1.18126	+	0.681999i	0.956305	−	0.292370i	$-0.0944440\pi$
$659$	3.38065e8	+	1.95182e8i	1.18126	+	0.681999i	0.224952	+	0.974370i	$0.427777\pi$
$660$		0			0
$661$	3.25385e7	+	1.87861e7i	0.112666	+	0.0650479i	0.555274	−	0.831667i	$-0.312613\pi$
$661$	3.25385e7	+	1.87861e7i	0.112666	+	0.0650479i	−0.442608	+	0.896715i	$0.645947\pi$
$662$		0			0
$663$	1.51337e8	+	2.62124e8i	0.519285	+	0.899428i
$664$		0			0
$665$	2.51402e8	−	4.96310e8i	0.854876	−	1.68767i
$666$		0			0
$667$	1.84189e8	−	1.06341e8i	0.620705	−	0.358364i
$668$		0			0
$669$	−5.72364e7	+	9.91363e7i	−0.191159	+	0.331096i
$670$		0			0
$671$	−4.50987e7	+	7.81133e7i	−0.149278	+	0.258558i
$672$		0			0
$673$		−	1.04527e8i		−	0.342914i	−0.985192	−	0.171457i	$-0.945153\pi$
$673$		−	1.04527e8i		−	0.342914i	0.985192	−	0.171457i	$-0.0548475\pi$
$674$		0			0
$675$	−3.17364e8	−	1.83230e8i	−1.03192	−	0.595780i
$676$		0			0
$677$		−	9.38180e6i		−	0.0302357i	−0.999886	−	0.0151179i	$-0.995188\pi$
$677$		−	9.38180e6i		−	0.0302357i	0.999886	−	0.0151179i	$-0.00481235\pi$
$678$		0			0
$679$	5.86209e8	−	3.38448e8i	1.87259	−	1.08114i
$680$		0			0
$681$	5.28772e7	+	9.15860e7i	0.167428	+	0.289993i
$682$		0			0
$683$		−	5.12614e8i		−	1.60890i	−0.594023	−	0.804448i	$-0.702461\pi$
$683$		−	5.12614e8i		−	1.60890i	0.594023	−	0.804448i	$-0.297539\pi$
$684$		0			0
$685$	1.84940e8			0.575387
$686$		0			0
$687$	−3.76378e7	+	2.17302e7i	−0.116079	+	0.0670183i
$688$		0			0
$689$	−2.86128e8	−	4.95589e8i	−0.874789	−	1.51518i
$690$		0			0
$691$	4.24447e8			1.28644			0.643219	−	0.765682i	$-0.277598\pi$
$691$	4.24447e8			1.28644			0.643219	+	0.765682i	$0.277598\pi$
$692$		0			0
$693$	−2.66076e7	+	4.60858e7i	−0.0799478	+	0.138474i
$694$		0			0
$695$	−5.97088e8			−1.77862
$696$		0			0
$697$	5.15584e8	+	2.97672e8i	1.52265	+	0.879104i
$698$		0			0
$699$	−9.89697e7	−	5.71402e7i	−0.289781	−	0.167305i
$700$		0			0
$701$	1.85294e8	+	3.20939e8i	0.537908	+	0.931684i	0.999016	+	0.0443403i	$0.0141186\pi$
$701$	1.85294e8	+	3.20939e8i	0.537908	+	0.931684i	−0.461108	+	0.887344i	$0.652548\pi$
$702$		0			0
$703$	−2.08313e8	+	1.35984e8i	−0.599585	+	0.391400i
$704$		0			0
$705$	8.77174e7	−	5.06437e7i	0.250333	−	0.144530i
$706$		0			0
$707$	−3.80207e8	+	6.58537e8i	−1.07587	+	1.86347i
$708$		0			0
$709$	−2.34846e8	+	4.06765e8i	−0.658938	+	1.14131i	0.321953	+	0.946756i	$0.395661\pi$
$709$	−2.34846e8	+	4.06765e8i	−0.658938	+	1.14131i	−0.980891	+	0.194558i	$0.937673\pi$
$710$		0			0
$711$		−	6.15126e7i		−	0.171141i
$712$		0			0
$713$	5.80851e7	+	3.35355e7i	0.160249	+	0.0925200i
$714$		0			0
$715$			1.35755e8i			0.371398i
$716$		0			0
$717$	−1.80826e8	+	1.04400e8i	−0.490573	+	0.283233i
$718$		0			0
$719$	−1.86660e8	−	3.23305e8i	−0.502187	−	0.869813i	−0.999997	−	0.00252720i	$-0.999196\pi$
$719$	−1.86660e8	−	3.23305e8i	−0.502187	−	0.869813i	0.497810	−	0.867286i	$-0.334138\pi$
$720$		0			0
$721$			8.39724e8i			2.24043i
$722$		0			0
$723$	2.88406e8			0.763113
$724$		0			0
$725$	3.17877e8	−	1.83526e8i	0.834152	−	0.481598i
$726$		0			0
$727$	−7.09528e7	−	1.22894e8i	−0.184657	−	0.319836i	0.758804	−	0.651319i	$-0.225784\pi$
$727$	−7.09528e7	−	1.22894e8i	−0.184657	−	0.319836i	−0.943461	+	0.331484i	$0.892451\pi$
$728$		0			0
$729$	−1.95765e8			−0.505304
$730$		0			0
$731$	4.78023e8	−	8.27961e8i	1.22376	−	2.11962i
$732$		0			0
$733$	4.12879e8			1.04836			0.524181	−	0.851607i	$-0.324372\pi$
$733$	4.12879e8			1.04836			0.524181	+	0.851607i	$0.324372\pi$
$734$		0			0
$735$	1.81573e8	+	1.04831e8i	0.457288	+	0.264015i
$736$		0			0
$737$	−7.59467e7	−	4.38479e7i	−0.189717	−	0.109533i
$738$		0			0
$739$	−3.16380e8	−	5.47986e8i	−0.783926	−	1.35780i	−0.929638	−	0.368473i	$-0.879881\pi$
$739$	−3.16380e8	−	5.47986e8i	−0.783926	−	1.35780i	0.145712	−	0.989327i	$-0.453453\pi$
$740$		0			0
$741$	−1.68680e8	−	2.58400e8i	−0.414580	−	0.635094i
$742$		0			0
$743$	4.32236e8	−	2.49551e8i	1.05379	−	0.608406i	0.130082	−	0.991503i	$-0.458476\pi$
$743$	4.32236e8	−	2.49551e8i	1.05379	−	0.608406i	0.923708	+	0.383097i	$0.125143\pi$
$744$		0			0
$745$	1.78663e8	−	3.09454e8i	0.432082	−	0.748389i
$746$		0			0
$747$	−1.04106e8	+	1.80318e8i	−0.249756	+	0.432590i
$748$		0			0
$749$		−	9.04655e8i		−	2.15297i
$750$		0			0
$751$	8.80887e7	+	5.08580e7i	0.207970	+	0.120071i	0.600368	−	0.799724i	$-0.295021\pi$
$751$	8.80887e7	+	5.08580e7i	0.207970	+	0.120071i	−0.392398	+	0.919796i	$0.628354\pi$
$752$		0			0
$753$			3.76832e8i			0.882599i
$754$		0			0
$755$	1.43994e8	−	8.31350e7i	0.334583	−	0.193172i
$756$		0			0
$757$	1.49966e8	+	2.59748e8i	0.345704	+	0.598777i	0.985481	−	0.169783i	$-0.0543067\pi$
$757$	1.49966e8	+	2.59748e8i	0.345704	+	0.598777i	−0.639777	+	0.768560i	$0.720973\pi$
$758$		0			0
$759$		−	4.47863e7i		−	0.102428i
$760$		0			0
$761$	1.38038e8			0.313216			0.156608	−	0.987661i	$-0.449944\pi$
$761$	1.38038e8			0.313216			0.156608	+	0.987661i	$0.449944\pi$
$762$		0			0
$763$	1.67407e8	−	9.66526e7i	0.376878	−	0.217591i
$764$		0			0
$765$	3.01462e8	+	5.22147e8i	0.673361	+	1.16630i
$766$		0			0
$767$	−3.08546e8			−0.683808
$768$		0			0
$769$	2.12384e7	−	3.67860e7i	0.0467028	−	0.0808916i	−0.841729	−	0.539900i	$-0.818462\pi$
$769$	2.12384e7	−	3.67860e7i	0.0467028	−	0.0808916i	0.888432	+	0.459009i	$0.151795\pi$
$770$		0			0
$771$	−1.53168e8			−0.334200
$772$		0			0
$773$	−2.34552e8	−	1.35419e8i	−0.507809	−	0.293184i	0.224123	−	0.974561i	$-0.428048\pi$
$773$	−2.34552e8	−	1.35419e8i	−0.507809	−	0.293184i	−0.731933	+	0.681377i	$0.761381\pi$
$774$		0			0
$775$	1.00245e8	+	5.78763e7i	0.215355	+	0.124336i
$776$		0			0
$777$	−1.24320e8	−	2.15329e8i	−0.265020	−	0.459028i
$778$		0			0
$779$	−5.41463e8	−	2.74273e8i	−1.14540	−	0.580191i
$780$		0			0
$781$	6.95749e7	−	4.01691e7i	0.146049	−	0.0843216i
$782$		0			0
$783$	1.82209e8	−	3.15596e8i	0.379564	−	0.657425i
$784$		0			0
$785$	−4.71412e8	+	8.16510e8i	−0.974523	+	1.68792i
$786$		0			0
$787$			5.51738e8i			1.13190i	0.824439	+	0.565951i	$0.191491\pi$
$787$			5.51738e8i			1.13190i	−0.824439	+	0.565951i	$0.808509\pi$
$788$		0			0
$789$	−2.35065e7	−	1.35715e7i	−0.0478582	−	0.0276309i
$790$		0			0
$791$		−	8.19478e8i		−	1.65580i
$792$		0			0
$793$	−8.73518e8	+	5.04326e8i	−1.75167	+	1.01133i
$794$		0			0
$795$	2.95401e8	+	5.11649e8i	0.587909	+	1.01829i
$796$		0			0
$797$			4.12939e6i			0.00815663i	0.999992	+	0.00407832i	$0.00129817\pi$
$797$			4.12939e6i			0.00815663i	−0.999992	+	0.00407832i	$0.998702\pi$
$798$		0			0
$799$	−2.31435e8			−0.453720
$800$		0			0
$801$	2.35717e8	−	1.36091e8i	0.458662	−	0.264809i
$802$		0			0
$803$	−2.96118e6	−	5.12891e6i	−0.00571897	−	0.00990555i
$804$		0			0
$805$	9.02962e8			1.73094
$806$		0			0
$807$	−2.23220e8	+	3.86628e8i	−0.424729	+	0.735653i
$808$		0			0
$809$	5.62237e8			1.06188			0.530938	−	0.847411i	$-0.321840\pi$
$809$	5.62237e8			1.06188			0.530938	+	0.847411i	$0.321840\pi$
$810$		0			0
$811$	−9.01498e8	−	5.20480e8i	−1.69006	−	0.975757i	−0.954459	−	0.298344i	$-0.903566\pi$
$811$	−9.01498e8	−	5.20480e8i	−1.69006	−	0.975757i	−0.735602	−	0.677413i	$-0.763101\pi$
$812$		0			0
$813$	−2.19281e8	−	1.26602e8i	−0.408066	−	0.235597i
$814$		0			0
$815$	5.66686e8	+	9.81529e8i	1.04681	+	1.81314i
$816$		0			0
$817$	−4.40447e8	+	8.69519e8i	−0.807659	+	1.59446i
$818$		0			0
$819$	−5.15364e8	+	2.97546e8i	−0.938129	+	0.541629i
$820$		0			0
$821$	−1.23413e8	+	2.13758e8i	−0.223014	+	0.386272i	−0.955722	−	0.294272i	$-0.904923\pi$
$821$	−1.23413e8	+	2.13758e8i	−0.223014	+	0.386272i	0.732708	+	0.680544i	$0.238256\pi$
$822$		0			0
$823$	1.03483e7	−	1.79238e7i	0.0185640	−	0.0321537i	−0.856594	−	0.515991i	$-0.827424\pi$
$823$	1.03483e7	−	1.79238e7i	0.0185640	−	0.0321537i	0.875158	+	0.483837i	$0.160757\pi$
$824$		0			0
$825$		−	7.72933e7i		−	0.137651i
$826$		0			0
$827$	2.02693e8	+	1.17025e8i	0.358362	+	0.206900i	0.668362	−	0.743836i	$-0.266996\pi$
$827$	2.02693e8	+	1.17025e8i	0.358362	+	0.206900i	−0.310000	+	0.950737i	$0.600329\pi$
$828$		0			0
$829$		−	1.75521e8i		−	0.308081i	−0.988065	−	0.154041i	$-0.950771\pi$
$829$		−	1.75521e8i		−	0.308081i	0.988065	−	0.154041i	$-0.0492286\pi$
$830$		0			0
$831$	−4.44325e8	+	2.56531e8i	−0.774280	+	0.447031i
$832$		0			0
$833$	−2.39532e8	−	4.14882e8i	−0.414409	−	0.717778i
$834$		0			0
$835$			5.74258e8i			0.986388i
$836$		0			0
$837$	1.14922e8			0.195987
$838$		0			0
$839$	−4.34017e8	+	2.50580e8i	−0.734888	+	0.424288i	−0.820208	−	0.572066i	$-0.806142\pi$
$839$	−4.34017e8	+	2.50580e8i	−0.734888	+	0.424288i	0.0853197	+	0.996354i	$0.472809\pi$
$840$		0			0
$841$	−1.14908e8	−	1.99026e8i	−0.193180	−	0.334597i
$842$		0			0
$843$	3.56588e8			0.595229
$844$		0			0
$845$	−3.08602e8	+	5.34514e8i	−0.511479	+	0.885908i
$846$		0			0
$847$	7.41614e8			1.22047
$848$		0			0
$849$	−1.09627e8	−	6.32935e7i	−0.179142	−	0.103427i
$850$		0			0
$851$	−3.49661e8	−	2.01877e8i	−0.567360	−	0.327565i
$852$		0			0
$853$	−3.96916e8	−	6.87478e8i	−0.639516	−	1.10767i	−0.985539	−	0.169448i	$-0.945802\pi$
$853$	−3.96916e8	−	6.87478e8i	−0.639516	−	1.10767i	0.346023	−	0.938226i	$-0.387532\pi$
$854$		0			0
$855$	−3.36008e8	−	5.14729e8i	−0.537589	−	0.823532i
$856$		0			0
$857$	−5.75670e8	+	3.32363e8i	−0.914600	+	0.528045i	−0.881908	−	0.471421i	$-0.843741\pi$
$857$	−5.75670e8	+	3.32363e8i	−0.914600	+	0.528045i	−0.0326916	+	0.999465i	$0.510408\pi$
$858$		0			0
$859$	4.61906e8	−	8.00044e8i	0.728742	−	1.26222i	−0.228673	−	0.973503i	$-0.573439\pi$
$859$	4.61906e8	−	8.00044e8i	0.728742	−	1.26222i	0.957415	−	0.288715i	$-0.0932281\pi$
$860$		0			0
$861$	3.03327e8	−	5.25377e8i	0.475227	−	0.823117i
$862$		0			0
$863$			1.01000e9i			1.57140i	0.618607	+	0.785700i	$0.287697\pi$
$863$			1.01000e9i			1.57140i	−0.618607	+	0.785700i	$0.712303\pi$
$864$		0			0
$865$	−8.73125e8	−	5.04099e8i	−1.34905	−	0.778874i
$866$		0			0
$867$			3.33232e8i			0.511316i
$868$		0			0
$869$	2.82952e7	−	1.63362e7i	0.0431175	−	0.0248939i
$870$		0			0
$871$	−4.90338e8	−	8.49291e8i	−0.742064	−	1.28529i
$872$		0			0
$873$		−	7.47878e8i		−	1.12406i
$874$		0			0
$875$	2.90969e8			0.434332
$876$		0			0
$877$	−5.82966e8	+	3.36576e8i	−0.864261	+	0.498981i	−0.865437	−	0.501018i	$-0.832959\pi$
$877$	−5.82966e8	+	3.36576e8i	−0.864261	+	0.498981i	0.00117605	+	0.999999i	$0.499626\pi$
$878$		0			0
$879$	2.30804e8	+	3.99764e8i	0.339842	+	0.588623i
$880$		0			0
$881$	−3.78834e8			−0.554015			−0.277008	−	0.960868i	$-0.589343\pi$
$881$	−3.78834e8			−0.554015			−0.277008	+	0.960868i	$0.589343\pi$
$882$		0			0
$883$	−2.25640e8	+	3.90819e8i	−0.327743	+	0.567667i	−0.982064	−	0.188550i	$-0.939621\pi$
$883$	−2.25640e8	+	3.90819e8i	−0.327743	+	0.567667i	0.654321	+	0.756217i	$0.272955\pi$
$884$		0			0
$885$	3.18545e8			0.459559
$886$		0			0
$887$	9.82622e8	+	5.67317e8i	1.40804	+	0.812933i	0.995199	−	0.0978689i	$-0.0312026\pi$
$887$	9.82622e8	+	5.67317e8i	1.40804	+	0.812933i	0.412843	+	0.910802i	$0.364536\pi$
$888$		0			0
$889$	2.99853e8	+	1.73120e8i	0.426779	+	0.246401i
$890$		0			0
$891$	6.26485e6	+	1.08510e7i	0.00885681	+	0.0153404i
$892$		0			0
$893$	2.35599e8	−	1.29077e7i	0.330841	−	0.0181256i
$894$		0			0
$895$	−6.18008e8	+	3.56807e8i	−0.862036	+	0.497697i
$896$		0			0
$897$	2.50416e8	−	4.33734e8i	0.346965	−	0.600961i
$898$		0			0
$899$	−5.75538e7	+	9.96861e7i	−0.0792127	+	0.137200i
$900$		0			0
$901$		−	1.34994e9i		−	1.84561i
$902$		0			0
$903$	−8.43687e8	−	4.87103e8i	−1.14582	−	0.661542i
$904$		0			0
$905$			9.33093e7i			0.125886i
$906$		0			0
$907$	−4.53334e8	+	2.61733e8i	−0.607570	+	0.350781i	−0.772014	−	0.635606i	$-0.780750\pi$
$907$	−4.53334e8	+	2.61733e8i	−0.607570	+	0.350781i	0.164444	+	0.986386i	$0.447417\pi$
$908$		0			0
$909$	4.20077e8	+	7.27595e8i	0.559290	+	0.968719i
$910$		0			0
$911$			7.90767e8i			1.04591i	0.852361	+	0.522954i	$0.175170\pi$
$911$			7.90767e8i			1.04591i	−0.852361	+	0.522954i	$0.824830\pi$
$912$		0			0
$913$	−1.10593e8			−0.145316
$914$		0			0
$915$	9.01825e8	−	5.20669e8i	1.17722	−	0.679671i
$916$		0			0
$917$	4.80991e8	+	8.33102e8i	0.623777	+	1.08041i
$918$		0			0
$919$	5.38251e8			0.693487			0.346744	−	0.937960i	$-0.387287\pi$
$919$	5.38251e8			0.693487			0.346744	+	0.937960i	$0.387287\pi$
$920$		0			0
$921$	2.33117e8	−	4.03771e8i	0.298398	−	0.516841i
$922$		0			0
$923$	8.98399e8			1.14252
$924$		0			0
$925$	−6.03453e8	−	3.48404e8i	−0.762462	−	0.440208i
$926$		0			0
$927$	8.03482e8	+	4.63891e8i	1.00864	+	0.582340i
$928$		0			0
$929$	−3.40869e8	−	5.90402e8i	−0.425148	−	0.736378i	0.571286	−	0.820751i	$-0.306445\pi$
$929$	−3.40869e8	−	5.90402e8i	−0.425148	−	0.736378i	−0.996434	+	0.0843729i	$0.973111\pi$
$930$		0			0
$931$	2.66981e8	+	4.08988e8i	0.330851	+	0.506829i
$932$		0			0
$933$	−6.09179e8	+	3.51710e8i	−0.750067	+	0.433051i
$934$		0			0
$935$	−1.60122e8	+	2.77339e8i	−0.195892	+	0.339294i
$936$		0			0
$937$	−6.22146e7	+	1.07759e8i	−0.0756265	+	0.130989i	−0.901359	−	0.433074i	$-0.857429\pi$
$937$	−6.22146e7	+	1.07759e8i	−0.0756265	+	0.130989i	0.825732	+	0.564063i	$0.190762\pi$
$938$		0			0
$939$		−	6.18548e8i		−	0.747097i
$940$		0			0
$941$	8.21054e8	+	4.74036e8i	0.985378	+	0.568908i	0.903889	−	0.427766i	$-0.140699\pi$
$941$	8.21054e8	+	4.74036e8i	0.985378	+	0.568908i	0.0814882	+	0.996674i	$0.474033\pi$
$942$		0			0
$943$		−	9.85111e8i		−	1.17476i
$944$		0			0
$945$	1.33989e9	−	7.73585e8i	1.58772	−	0.916669i
$946$		0			0
$947$	2.69898e8	+	4.67477e8i	0.317797	+	0.550440i	0.980028	−	0.198859i	$-0.0637236\pi$
$947$	2.69898e8	+	4.67477e8i	0.317797	+	0.550440i	−0.662231	+	0.749300i	$0.730390\pi$
$948$		0			0
$949$		−	6.62280e7i		−	0.0774896i
$950$		0			0
$951$	−1.25056e7			−0.0145399
$952$		0			0
$953$	2.88809e8	−	1.66744e8i	0.333682	−	0.192651i	−0.323793	−	0.946128i	$-0.604958\pi$
$953$	2.88809e8	−	1.66744e8i	0.333682	−	0.192651i	0.657475	+	0.753477i	$0.271625\pi$
$954$		0			0
$955$	4.53694e8	+	7.85821e8i	0.520898	+	0.902223i
$956$		0			0
$957$	7.68626e7			0.0876959
$958$		0			0
$959$	−2.15301e8	+	3.72913e8i	−0.244113	+	0.422816i
$960$		0			0
$961$	8.51204e8			0.959099
$962$		0			0
$963$	−8.65610e8	−	4.99760e8i	−0.969267	−	0.559607i
$964$		0			0
$965$	9.43647e8	+	5.44815e8i	1.05009	+	0.606272i
$966$		0			0
$967$	−4.71120e8	−	8.16005e8i	−0.521018	−	0.902429i	−0.999701	−	0.0244417i	$-0.992219\pi$
$967$	−4.71120e8	−	8.16005e8i	−0.521018	−	0.902429i	0.478683	−	0.877988i	$-0.341114\pi$
$968$		0			0
$969$	−3.98216e7	−	7.26850e8i	−0.0437671	−	0.798866i
$970$		0			0
$971$	1.89205e8	−	1.09238e8i	0.206669	−	0.119321i	−0.393093	−	0.919499i	$-0.628595\pi$
$971$	1.89205e8	−	1.09238e8i	0.206669	−	0.119321i	0.599763	+	0.800178i	$0.295262\pi$
$972$		0			0
$973$	6.95109e8	−	1.20396e9i	0.754596	−	1.30700i
$974$		0			0
$975$	4.32174e8	−	7.48548e8i	0.466278	−	0.807618i
$976$		0			0
$977$		−	9.42883e8i		−	1.01105i	−0.862811	−	0.505527i	$-0.831298\pi$
$977$		−	9.42883e8i		−	1.01105i	0.862811	−	0.505527i	$-0.168702\pi$
$978$		0			0
$979$	1.25201e8	+	7.22850e7i	0.133432	+	0.0770371i
$980$		0			0
$981$		−	2.13576e8i		−	0.226228i
$982$		0			0
$983$	2.29638e8	−	1.32582e8i	0.241759	−	0.139580i	−0.374226	−	0.927338i	$-0.622091\pi$
$983$	2.29638e8	−	1.32582e8i	0.241759	−	0.139580i	0.615985	+	0.787758i	$0.288758\pi$
$984$		0			0
$985$	4.75071e8	+	8.22847e8i	0.497107	+	0.861014i
$986$		0			0
$987$			2.35831e8i			0.245272i
$988$		0			0
$989$	−1.58196e9			−1.63533
$990$		0			0
$991$	2.21400e8	−	1.27826e8i	0.227488	−	0.131340i	−0.381925	−	0.924193i	$-0.624739\pi$
$991$	2.21400e8	−	1.27826e8i	0.227488	−	0.131340i	0.609412	+	0.792853i	$0.291405\pi$
$992$		0			0
$993$	−4.25235e8	−	7.36529e8i	−0.434291	−	0.752215i
$994$		0			0
$995$	−6.63015e8			−0.673061
$996$		0			0
$997$	5.58185e7	−	9.66804e7i	0.0563239	−	0.0975558i	−0.836489	−	0.547984i	$-0.815395\pi$
$997$	5.58185e7	−	9.66804e7i	0.0563239	−	0.0975558i	0.892813	+	0.450428i	$0.148729\pi$
$998$		0			0
$999$	−6.91807e8			−0.693887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.h.a.69.7	yes	20
3.2	odd	2		684.7.y.c.145.1		20
19.8	odd	6	inner	76.7.h.a.65.7	✓	20
57.8	even	6		684.7.y.c.217.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.h.a.65.7	✓	20	19.8	odd	6	inner
76.7.h.a.69.7	yes	20	1.1	even	1	trivial
684.7.y.c.145.1		20	3.2	odd	2
684.7.y.c.217.1		20	57.8	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(13.6616 - 7.88751i) q^{3} +(93.3237 + 161.641i) q^{5} -434.577 q^{7} +(-240.074 + 415.821i) q^{9} +O(q^{10})\)	\(q+(13.6616 - 7.88751i) q^{3} +(93.3237 + 161.641i) q^{5} -434.577 q^{7} +(-240.074 + 415.821i) q^{9} -255.032 q^{11} +(-2469.86 - 1425.97i) q^{13} +(2549.90 + 1472.18i) q^{15} +(-3363.84 - 5826.34i) q^{17} +(3749.31 + 5743.56i) q^{19} +(-5937.00 + 3427.73i) q^{21} +(-5566.10 + 9640.78i) q^{23} +(-9606.11 + 16638.3i) q^{25} +19074.3i q^{27} +(-16545.6 - 9552.59i) q^{29} -6024.94i q^{31} +(-3484.13 + 2011.57i) q^{33} +(-40556.3 - 70245.6i) q^{35} +36269.0i q^{37} -44989.5 q^{39} +(-76636.3 + 44246.0i) q^{41} +(71053.3 + 123068. i) q^{43} -89618.4 q^{45} +(17200.2 - 29791.6i) q^{47} +71208.1 q^{49} +(-91910.6 - 53064.6i) q^{51} +(173772. + 100327. i) q^{53} +(-23800.5 - 41223.7i) q^{55} +(96524.0 + 48893.4i) q^{57} +(93693.5 - 54094.0i) q^{59} +(176836. - 306288. i) q^{61} +(104331. - 180706. i) q^{63} -532308. i q^{65} +(297793. + 171931. i) q^{67} +175611. i q^{69} +(-272809. + 157506. i) q^{71} +(11611.0 + 20110.9i) q^{73} +303073. i q^{75} +110831. q^{77} +(-110948. + 64055.7i) q^{79} +(-24565.0 - 42547.8i) q^{81} +433643. q^{83} +(627851. - 1.08747e6i) q^{85} -301385. q^{87} +(-490924. - 283435. i) q^{89} +(1.07334e6 + 619695. i) q^{91} +(-47521.8 - 82310.2i) q^{93} +(-578498. + 1.14205e6i) q^{95} +(-1.34892e6 + 778798. i) q^{97} +(61226.6 - 106048. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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