LMFDB - Embedded newform 304.5.e.e.113.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,5,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not 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:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 $ x^8 + 450x^6 + 68229x^4 + 4001228*x^2 + 77475204
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.8
Root			$13.9305i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.e.113.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.3447i q^{3} +41.6240 q^{5} +62.4342 q^{7} -154.460 q^{9} +O(q^{10})$	$q+15.3447i q^{3} +41.6240 q^{5} +62.4342 q^{7} -154.460 q^{9} -122.938 q^{11} +68.9610i q^{13} +638.707i q^{15} +297.375 q^{17} +(195.642 + 303.390i) q^{19} +958.034i q^{21} -268.685 q^{23} +1107.56 q^{25} -1127.21i q^{27} -561.405i q^{29} +252.423i q^{31} -1886.44i q^{33} +2598.76 q^{35} +2407.33i q^{37} -1058.18 q^{39} -690.246i q^{41} -218.905 q^{43} -6429.22 q^{45} +83.1049 q^{47} +1497.03 q^{49} +4563.13i q^{51} +4389.68i q^{53} -5117.16 q^{55} +(-4655.42 + 3002.06i) q^{57} +476.981i q^{59} +3965.09 q^{61} -9643.57 q^{63} +2870.43i q^{65} -5111.96i q^{67} -4122.89i q^{69} -8182.93i q^{71} -7345.23 q^{73} +16995.1i q^{75} -7675.53 q^{77} +1485.61i q^{79} +4785.54 q^{81} -5347.30 q^{83} +12377.9 q^{85} +8614.59 q^{87} -11172.3i q^{89} +4305.53i q^{91} -3873.35 q^{93} +(8143.39 + 12628.3i) q^{95} +1078.13i q^{97} +18988.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			15.3447i			1.70497i	0.522755	+	0.852483i	$0.324904\pi$
$3$			15.3447i			1.70497i	−0.522755	+	0.852483i	$0.675096\pi$
$4$		0			0
$5$	41.6240			1.66496			0.832480	−	0.554055i	$-0.186920\pi$
$5$	41.6240			1.66496			0.832480	+	0.554055i	$0.186920\pi$
$6$		0			0
$7$	62.4342			1.27417			0.637084	−	0.770794i	$-0.280140\pi$
$7$	62.4342			1.27417			0.637084	+	0.770794i	$0.280140\pi$
$8$		0			0
$9$	−154.460			−1.90691
$10$		0			0
$11$	−122.938			−1.01601			−0.508007	−	0.861353i	$-0.669618\pi$
$11$	−122.938			−1.01601			−0.508007	+	0.861353i	$0.669618\pi$
$12$		0			0
$13$			68.9610i			0.408053i	0.978965	+	0.204027i	$0.0654029\pi$
$13$			68.9610i			0.408053i	−0.978965	+	0.204027i	$0.934597\pi$
$14$		0			0
$15$			638.707i			2.83870i
$16$		0			0
$17$	297.375			1.02898			0.514490	−	0.857496i	$-0.327981\pi$
$17$	297.375			1.02898			0.514490	+	0.857496i	$0.327981\pi$
$18$		0			0
$19$	195.642	+	303.390i	0.541944	+	0.840415i
$19$	195.642	+	303.390i	0.541944	+	0.840415i
$20$		0			0
$21$			958.034i			2.17241i
$22$		0			0
$23$	−268.685			−0.507911			−0.253956	−	0.967216i	$-0.581732\pi$
$23$	−268.685			−0.507911			−0.253956	+	0.967216i	$0.581732\pi$
$24$		0			0
$25$	1107.56			1.77209
$26$		0			0
$27$		−	1127.21i		−	1.54625i
$28$		0			0
$29$		−	561.405i		−	0.667545i	−0.942654	−	0.333772i	$-0.891678\pi$
$29$		−	561.405i		−	0.667545i	0.942654	−	0.333772i	$-0.108322\pi$
$30$		0			0
$31$			252.423i			0.262667i	0.991338	+	0.131333i	$0.0419259\pi$
$31$			252.423i			0.262667i	−0.991338	+	0.131333i	$0.958074\pi$
$32$		0			0
$33$		−	1886.44i		−	1.73227i
$34$		0			0
$35$	2598.76			2.12144
$36$		0			0
$37$			2407.33i			1.75846i	0.476400	+	0.879228i	$0.341941\pi$
$37$			2407.33i			1.75846i	−0.476400	+	0.879228i	$0.658059\pi$
$38$		0			0
$39$	−1058.18			−0.695717
$40$		0			0
$41$		−	690.246i		−	0.410616i	−0.978697	−	0.205308i	$-0.934180\pi$
$41$		−	690.246i		−	0.410616i	0.978697	−	0.205308i	$-0.0658197\pi$
$42$		0			0
$43$	−218.905			−0.118391			−0.0591954	−	0.998246i	$-0.518854\pi$
$43$	−218.905			−0.118391			−0.0591954	+	0.998246i	$0.518854\pi$
$44$		0			0
$45$	−6429.22			−3.17493
$46$		0			0
$47$	83.1049			0.0376211			0.0188105	−	0.999823i	$-0.494012\pi$
$47$	83.1049			0.0376211			0.0188105	+	0.999823i	$0.494012\pi$
$48$		0			0
$49$	1497.03			0.623505
$50$		0			0
$51$			4563.13i			1.75438i
$52$		0			0
$53$			4389.68i			1.56272i	0.624080	+	0.781360i	$0.285474\pi$
$53$			4389.68i			1.56272i	−0.624080	+	0.781360i	$0.714526\pi$
$54$		0			0
$55$	−5117.16			−1.69162
$56$		0			0
$57$	−4655.42	+	3002.06i	−1.43288	+	0.923996i
$58$		0			0
$59$			476.981i			0.137024i	0.997650	+	0.0685120i	$0.0218252\pi$
$59$			476.981i			0.137024i	−0.997650	+	0.0685120i	$0.978175\pi$
$60$		0			0
$61$	3965.09			1.06560			0.532799	−	0.846242i	$-0.321140\pi$
$61$	3965.09			1.06560			0.532799	+	0.846242i	$0.321140\pi$
$62$		0			0
$63$	−9643.57			−2.42972
$64$		0			0
$65$			2870.43i			0.679392i
$66$		0			0
$67$		−	5111.96i		−	1.13878i	−0.822069	−	0.569388i	$-0.807180\pi$
$67$		−	5111.96i		−	1.13878i	0.822069	−	0.569388i	$-0.192820\pi$
$68$		0			0
$69$		−	4122.89i		−	0.865972i
$70$		0			0
$71$		−	8182.93i		−	1.62327i	−0.584161	−	0.811637i	$-0.698577\pi$
$71$		−	8182.93i		−	1.62327i	0.584161	−	0.811637i	$-0.301423\pi$
$72$		0			0
$73$	−7345.23			−1.37835			−0.689175	−	0.724595i	$-0.742027\pi$
$73$	−7345.23			−1.37835			−0.689175	+	0.724595i	$0.742027\pi$
$74$		0			0
$75$			16995.1i			3.02135i
$76$		0			0
$77$	−7675.53			−1.29457
$78$		0			0
$79$			1485.61i			0.238041i	0.992892	+	0.119021i	$0.0379754\pi$
$79$			1485.61i			0.238041i	−0.992892	+	0.119021i	$0.962025\pi$
$80$		0			0
$81$	4785.54			0.729391
$82$		0			0
$83$	−5347.30			−0.776209			−0.388104	−	0.921615i	$-0.626870\pi$
$83$	−5347.30			−0.776209			−0.388104	+	0.921615i	$0.626870\pi$
$84$		0			0
$85$	12377.9			1.71321
$86$		0			0
$87$	8614.59			1.13814
$88$		0			0
$89$		−	11172.3i		−	1.41047i	−0.708975	−	0.705234i	$-0.750842\pi$
$89$		−	11172.3i		−	1.41047i	0.708975	−	0.705234i	$-0.249158\pi$
$90$		0			0
$91$			4305.53i			0.519928i
$92$		0			0
$93$	−3873.35			−0.447838
$94$		0			0
$95$	8143.39	+	12628.3i	0.902315	+	1.39926i
$96$		0			0
$97$			1078.13i			0.114585i	0.998357	+	0.0572927i	$0.0182468\pi$
$97$			1078.13i			0.114585i	−0.998357	+	0.0572927i	$0.981753\pi$
$98$		0			0
$99$	18988.9			1.93745
$100$		0			0
$101$	1393.71			0.136625			0.0683126	−	0.997664i	$-0.478238\pi$
$101$	1393.71			0.136625			0.0683126	+	0.997664i	$0.478238\pi$
$102$		0			0
$103$		−	11163.2i		−	1.05224i	−0.850411	−	0.526119i	$-0.823647\pi$
$103$		−	11163.2i		−	1.05224i	0.850411	−	0.526119i	$-0.176353\pi$
$104$		0			0
$105$			39877.2i			3.61698i
$106$		0			0
$107$		−	502.078i		−	0.0438534i	−0.999760	−	0.0219267i	$-0.993020\pi$
$107$		−	502.078i		−	0.0438534i	0.999760	−	0.0219267i	$-0.00698005\pi$
$108$		0			0
$109$		−	11387.7i		−	0.958477i	−0.877685	−	0.479238i	$-0.840913\pi$
$109$		−	11387.7i		−	0.958477i	0.877685	−	0.479238i	$-0.159087\pi$
$110$		0			0
$111$	−36939.7			−2.99811
$112$		0			0
$113$			4779.02i			0.374267i	0.982334	+	0.187133i	$0.0599197\pi$
$113$			4779.02i			0.374267i	−0.982334	+	0.187133i	$0.940080\pi$
$114$		0			0
$115$	−11183.7			−0.845652
$116$		0			0
$117$		−	10651.7i		−	0.778120i
$118$		0			0
$119$	18566.4			1.31109
$120$		0			0
$121$	472.695			0.0322857
$122$		0			0
$123$	10591.6			0.700087
$124$		0			0
$125$	20085.9			1.28550
$126$		0			0
$127$			17552.6i			1.08826i	0.839001	+	0.544130i	$0.183140\pi$
$127$			17552.6i			1.08826i	−0.839001	+	0.544130i	$0.816860\pi$
$128$		0			0
$129$		−	3359.02i		−	0.201852i
$130$		0			0
$131$	7401.84			0.431317			0.215659	−	0.976469i	$-0.430810\pi$
$131$	7401.84			0.431317			0.215659	+	0.976469i	$0.430810\pi$
$132$		0			0
$133$	12214.7	+	18941.9i	0.690528	+	1.07083i
$134$		0			0
$135$		−	46919.2i		−	2.57444i
$136$		0			0
$137$	1692.77			0.0901896			0.0450948	−	0.998983i	$-0.485641\pi$
$137$	1692.77			0.0901896			0.0450948	+	0.998983i	$0.485641\pi$
$138$		0			0
$139$	37366.7			1.93399			0.966996	−	0.254791i	$-0.0820066\pi$
$139$	37366.7			1.93399			0.966996	+	0.254791i	$0.0820066\pi$
$140$		0			0
$141$			1275.22i			0.0641426i
$142$		0			0
$143$		−	8477.91i		−	0.414588i
$144$		0			0
$145$		−	23367.9i		−	1.11143i
$146$		0			0
$147$			22971.5i			1.06305i
$148$		0			0
$149$	−6037.22			−0.271935			−0.135967	−	0.990713i	$-0.543414\pi$
$149$	−6037.22			−0.271935			−0.135967	+	0.990713i	$0.543414\pi$
$150$		0			0
$151$		−	20619.1i		−	0.904307i	−0.891940	−	0.452153i	$-0.850656\pi$
$151$		−	20619.1i		−	0.904307i	0.891940	−	0.452153i	$-0.149344\pi$
$152$		0			0
$153$	−45932.4			−1.96217
$154$		0			0
$155$			10506.8i			0.437330i
$156$		0			0
$157$	−1043.55			−0.0423362			−0.0211681	−	0.999776i	$-0.506739\pi$
$157$	−1043.55			−0.0423362			−0.0211681	+	0.999776i	$0.506739\pi$
$158$		0			0
$159$	−67358.3			−2.66439
$160$		0			0
$161$	−16775.2			−0.647165
$162$		0			0
$163$	−1587.92			−0.0597658			−0.0298829	−	0.999553i	$-0.509513\pi$
$163$	−1587.92			−0.0597658			−0.0298829	+	0.999553i	$0.509513\pi$
$164$		0			0
$165$		−	78521.3i		−	2.88416i
$166$		0			0
$167$			21277.9i			0.762948i	0.924380	+	0.381474i	$0.124583\pi$
$167$			21277.9i			0.762948i	−0.924380	+	0.381474i	$0.875417\pi$
$168$		0			0
$169$	23805.4			0.833493
$170$		0			0
$171$	−30218.7	−	46861.4i	−1.03344	−	1.60259i
$172$		0			0
$173$			23629.3i			0.789513i	0.918786	+	0.394756i	$0.129171\pi$
$173$			23629.3i			0.789513i	−0.918786	+	0.394756i	$0.870829\pi$
$174$		0			0
$175$	69149.5			2.25794
$176$		0			0
$177$	−7319.12			−0.233621
$178$		0			0
$179$		−	54580.1i		−	1.70345i	−0.523993	−	0.851723i	$-0.675558\pi$
$179$		−	54580.1i		−	1.70345i	0.523993	−	0.851723i	$-0.324442\pi$
$180$		0			0
$181$		−	48461.8i		−	1.47925i	−0.673017	−	0.739627i	$-0.735002\pi$
$181$		−	48461.8i		−	1.47925i	0.673017	−	0.739627i	$-0.264998\pi$
$182$		0			0
$183$			60843.0i			1.81681i
$184$		0			0
$185$			100203.i			2.92776i
$186$		0			0
$187$	−36558.6			−1.04546
$188$		0			0
$189$		−	70376.8i		−	1.97018i
$190$		0			0
$191$	−10718.8			−0.293819			−0.146909	−	0.989150i	$-0.546933\pi$
$191$	−10718.8			−0.293819			−0.146909	+	0.989150i	$0.546933\pi$
$192$		0			0
$193$		−	25358.2i		−	0.680776i	−0.940285	−	0.340388i	$-0.889442\pi$
$193$		−	25358.2i		−	0.680776i	0.940285	−	0.340388i	$-0.110558\pi$
$194$		0			0
$195$	−44045.9			−1.15834
$196$		0			0
$197$	−4536.94			−0.116904			−0.0584521	−	0.998290i	$-0.518617\pi$
$197$	−4536.94			−0.116904			−0.0584521	+	0.998290i	$0.518617\pi$
$198$		0			0
$199$	67698.1			1.70951			0.854753	−	0.519035i	$-0.173709\pi$
$199$	67698.1			1.70951			0.854753	+	0.519035i	$0.173709\pi$
$200$		0			0
$201$	78441.5			1.94157
$202$		0			0
$203$		−	35050.9i		−	0.850564i
$204$		0			0
$205$		−	28730.8i		−	0.683660i
$206$		0			0
$207$	41501.0			0.968540
$208$		0			0
$209$	−24051.8	−	37298.1i	−0.550623	−	0.853874i
$210$		0			0
$211$			2816.81i			0.0632691i	0.999500	+	0.0316346i	$0.0100713\pi$
$211$			2816.81i			0.0632691i	−0.999500	+	0.0316346i	$0.989929\pi$
$212$		0			0
$213$	125565.			2.76763
$214$		0			0
$215$	−9111.68			−0.197116
$216$		0			0
$217$			15759.8i			0.334682i
$218$		0			0
$219$		−	112710.i		−	2.35004i
$220$		0			0
$221$			20507.3i			0.419878i
$222$		0			0
$223$			36870.4i			0.741426i	0.928747	+	0.370713i	$0.120887\pi$
$223$			36870.4i			0.741426i	−0.928747	+	0.370713i	$0.879113\pi$
$224$		0			0
$225$	−171073.			−3.37921
$226$		0			0
$227$			56826.9i			1.10281i	0.834236	+	0.551407i	$0.185909\pi$
$227$			56826.9i			1.10281i	−0.834236	+	0.551407i	$0.814091\pi$
$228$		0			0
$229$	7632.64			0.145547			0.0727736	−	0.997348i	$-0.476815\pi$
$229$	7632.64			0.145547			0.0727736	+	0.997348i	$0.476815\pi$
$230$		0			0
$231$		−	117779.i		−	2.20720i
$232$		0			0
$233$	91249.2			1.68080			0.840402	−	0.541964i	$-0.182319\pi$
$233$	91249.2			1.68080			0.840402	+	0.541964i	$0.182319\pi$
$234$		0			0
$235$	3459.16			0.0626375
$236$		0			0
$237$	−22796.3			−0.405852
$238$		0			0
$239$	110161.			1.92856			0.964280	−	0.264884i	$-0.0853337\pi$
$239$	110161.			1.92856			0.964280	+	0.264884i	$0.0853337\pi$
$240$		0			0
$241$			58423.0i			1.00589i	0.864319	+	0.502944i	$0.167750\pi$
$241$			58423.0i			1.00589i	−0.864319	+	0.502944i	$0.832250\pi$
$242$		0			0
$243$		−	17871.8i		−	0.302661i
$244$		0			0
$245$	62312.6			1.03811
$246$		0			0
$247$	−20922.0	+	13491.6i	−0.342934	+	0.221142i
$248$		0			0
$249$		−	82052.7i		−	1.32341i
$250$		0			0
$251$	17258.3			0.273936			0.136968	−	0.990575i	$-0.456264\pi$
$251$	17258.3			0.273936			0.136968	+	0.990575i	$0.456264\pi$
$252$		0			0
$253$	33031.5			0.516045
$254$		0			0
$255$			189936.i			2.92096i
$256$		0			0
$257$			41195.7i			0.623714i	0.950129	+	0.311857i	$0.100951\pi$
$257$			41195.7i			0.623714i	−0.950129	+	0.311857i	$0.899049\pi$
$258$		0			0
$259$			150300.i			2.24057i
$260$		0			0
$261$			86714.4i			1.27295i
$262$		0			0
$263$	41179.7			0.595349			0.297675	−	0.954667i	$-0.403789\pi$
$263$	41179.7			0.595349			0.297675	+	0.954667i	$0.403789\pi$
$264$		0			0
$265$			182716.i			2.60187i
$266$		0			0
$267$	171436.			2.40480
$268$		0			0
$269$		−	19858.8i		−	0.274440i	−0.990541	−	0.137220i	$-0.956183\pi$
$269$		−	19858.8i		−	0.274440i	0.990541	−	0.137220i	$-0.0438168\pi$
$270$		0			0
$271$	−110976.			−1.51110			−0.755549	−	0.655093i	$-0.772630\pi$
$271$	−110976.			−1.51110			−0.755549	+	0.655093i	$0.772630\pi$
$272$		0			0
$273$	−66067.0			−0.886460
$274$		0			0
$275$	−136161.			−1.80047
$276$		0			0
$277$	−87534.8			−1.14083			−0.570415	−	0.821357i	$-0.693218\pi$
$277$	−87534.8			−1.14083			−0.570415	+	0.821357i	$0.693218\pi$
$278$		0			0
$279$		−	38989.1i		−	0.500882i
$280$		0			0
$281$		−	133298.i		−	1.68815i	−0.536228	−	0.844073i	$-0.680151\pi$
$281$		−	133298.i		−	1.68815i	0.536228	−	0.844073i	$-0.319849\pi$
$282$		0			0
$283$	−29905.8			−0.373407			−0.186704	−	0.982416i	$-0.559780\pi$
$283$	−29905.8			−0.373407			−0.186704	+	0.982416i	$0.559780\pi$
$284$		0			0
$285$	−193777.	+	124958.i	−2.38568	+	1.53842i
$286$		0			0
$287$		−	43095.0i		−	0.523194i
$288$		0			0
$289$	4910.99			0.0587994
$290$		0			0
$291$	−16543.6			−0.195364
$292$		0			0
$293$		−	42625.4i		−	0.496516i	−0.968694	−	0.248258i	$-0.920142\pi$
$293$		−	42625.4i		−	0.496516i	0.968694	−	0.248258i	$-0.0798580\pi$
$294$		0			0
$295$			19853.8i			0.228140i
$296$		0			0
$297$			138577.i			1.57101i
$298$		0			0
$299$		−	18528.8i		−	0.207255i
$300$		0			0
$301$	−13667.1			−0.150850
$302$		0			0
$303$			21386.1i			0.232941i
$304$		0			0
$305$	165043.			1.77418
$306$		0			0
$307$		−	39047.5i		−	0.414302i	−0.978309	−	0.207151i	$-0.933581\pi$
$307$		−	39047.5i		−	0.414302i	0.978309	−	0.207151i	$-0.0664191\pi$
$308$		0			0
$309$	171296.			1.79403
$310$		0			0
$311$	−79232.0			−0.819180			−0.409590	−	0.912270i	$-0.634328\pi$
$311$	−79232.0			−0.819180			−0.409590	+	0.912270i	$0.634328\pi$
$312$		0			0
$313$	−107079.			−1.09298			−0.546492	−	0.837464i	$-0.684037\pi$
$313$	−107079.			−1.09298			−0.546492	+	0.837464i	$0.684037\pi$
$314$		0			0
$315$	−401404.			−4.04539
$316$		0			0
$317$		−	132924.i		−	1.32277i	−0.750046	−	0.661385i	$-0.769969\pi$
$317$		−	132924.i		−	1.32277i	0.750046	−	0.661385i	$-0.230031\pi$
$318$		0			0
$319$			69017.9i			0.678235i
$320$		0			0
$321$	7704.23			0.0747686
$322$		0			0
$323$	58179.0	+	90220.6i	0.557649	+	0.864770i
$324$		0			0
$325$			76378.2i			0.723107i
$326$		0			0
$327$	174740.			1.63417
$328$		0			0
$329$	5188.59			0.0479356
$330$		0			0
$331$			44623.1i			0.407290i	0.979045	+	0.203645i	$0.0652788\pi$
$331$			44623.1i			0.407290i	−0.979045	+	0.203645i	$0.934721\pi$
$332$		0			0
$333$		−	371835.i		−	3.35322i
$334$		0			0
$335$		−	212780.i		−	1.89602i
$336$		0			0
$337$			5142.51i			0.0452810i	0.999744	+	0.0226405i	$0.00720730\pi$
$337$			5142.51i			0.0452810i	−0.999744	+	0.0226405i	$0.992793\pi$
$338$		0			0
$339$	−73332.5			−0.638112
$340$		0			0
$341$		−	31032.3i		−	0.266873i
$342$		0			0
$343$	−56438.4			−0.479718
$344$		0			0
$345$		−	171611.i		−	1.44181i
$346$		0			0
$347$	175661.			1.45887			0.729436	−	0.684049i	$-0.239782\pi$
$347$	175661.			1.45887			0.729436	+	0.684049i	$0.239782\pi$
$348$		0			0
$349$	137125.			1.12581			0.562906	−	0.826521i	$-0.309683\pi$
$349$	137125.			1.12581			0.562906	+	0.826521i	$0.309683\pi$
$350$		0			0
$351$	77733.8			0.630951
$352$		0			0
$353$	−199391.			−1.60013			−0.800065	−	0.599913i	$-0.795202\pi$
$353$	−199391.			−1.60013			−0.800065	+	0.599913i	$0.795202\pi$
$354$		0			0
$355$		−	340606.i		−	2.70269i
$356$		0			0
$357$			284896.i			2.23537i
$358$		0			0
$359$	−6307.95			−0.0489440			−0.0244720	−	0.999701i	$-0.507790\pi$
$359$	−6307.95			−0.0489440			−0.0244720	+	0.999701i	$0.507790\pi$
$360$		0			0
$361$	−53769.6	+	118711.i	−0.412594	+	0.910915i
$362$		0			0
$363$			7253.36i			0.0550460i
$364$		0			0
$365$	−305738.			−2.29490
$366$		0			0
$367$	−52946.3			−0.393100			−0.196550	−	0.980494i	$-0.562974\pi$
$367$	−52946.3			−0.393100			−0.196550	+	0.980494i	$0.562974\pi$
$368$		0			0
$369$			106615.i			0.783008i
$370$		0			0
$371$			274067.i			1.99117i
$372$		0			0
$373$		−	9312.99i		−	0.0669378i	−0.999440	−	0.0334689i	$-0.989345\pi$
$373$		−	9312.99i		−	0.0669378i	0.999440	−	0.0334689i	$-0.0106555\pi$
$374$		0			0
$375$			308212.i			2.19173i
$376$		0			0
$377$	38715.0			0.272394
$378$		0			0
$379$			178291.i			1.24123i	0.784117	+	0.620613i	$0.213116\pi$
$379$			178291.i			1.24123i	−0.784117	+	0.620613i	$0.786884\pi$
$380$		0			0
$381$	−269339.			−1.85545
$382$		0			0
$383$		−	27764.8i		−	0.189276i	−0.995512	−	0.0946382i	$-0.969831\pi$
$383$		−	27764.8i		−	0.189276i	0.995512	−	0.0946382i	$-0.0301694\pi$
$384$		0			0
$385$	−319486.			−2.15541
$386$		0			0
$387$	33811.9			0.225760
$388$		0			0
$389$	241135.			1.59353			0.796765	−	0.604289i	$-0.206543\pi$
$389$	241135.			1.59353			0.796765	+	0.604289i	$0.206543\pi$
$390$		0			0
$391$	−79900.3			−0.522631
$392$		0			0
$393$			113579.i			0.735381i
$394$		0			0
$395$			61837.2i			0.396329i
$396$		0			0
$397$	234920.			1.49052			0.745261	−	0.666772i	$-0.232325\pi$
$397$	234920.			1.49052			0.745261	+	0.666772i	$0.232325\pi$
$398$		0			0
$399$	−290658.	+	187432.i	−1.82573	+	1.17733i
$400$		0			0
$401$		−	311500.i		−	1.93718i	−0.248667	−	0.968589i	$-0.579993\pi$
$401$		−	311500.i		−	1.93718i	0.248667	−	0.968589i	$-0.420007\pi$
$402$		0			0
$403$	−17407.3			−0.107182
$404$		0			0
$405$	199193.			1.21441
$406$		0			0
$407$		−	295951.i		−	1.78662i
$408$		0			0
$409$			243636.i			1.45645i	0.685338	+	0.728225i	$0.259655\pi$
$409$			243636.i			1.45645i	−0.685338	+	0.728225i	$0.740345\pi$
$410$		0			0
$411$			25975.0i			0.153770i
$412$		0			0
$413$			29779.9i			0.174592i
$414$		0			0
$415$	−222576.			−1.29236
$416$		0			0
$417$			573380.i			3.29739i
$418$		0			0
$419$	193805.			1.10392			0.551960	−	0.833871i	$-0.313880\pi$
$419$	193805.			1.10392			0.551960	+	0.833871i	$0.313880\pi$
$420$		0			0
$421$			136536.i			0.770339i	0.922846	+	0.385169i	$0.125857\pi$
$421$			136536.i			0.770339i	−0.922846	+	0.385169i	$0.874143\pi$
$422$		0			0
$423$	−12836.3			−0.0717399
$424$		0			0
$425$	329360.			1.82345
$426$		0			0
$427$	247557.			1.35775
$428$		0			0
$429$	130091.			0.706858
$430$		0			0
$431$		−	323841.i		−	1.74332i	−0.490113	−	0.871659i	$-0.663044\pi$
$431$		−	323841.i		−	1.74332i	0.490113	−	0.871659i	$-0.336956\pi$
$432$		0			0
$433$			286129.i			1.52611i	0.646333	+	0.763056i	$0.276302\pi$
$433$			286129.i			1.52611i	−0.646333	+	0.763056i	$0.723698\pi$
$434$		0			0
$435$	358574.			1.89496
$436$		0			0
$437$	−52566.0	−	81516.3i	−0.275259	−	0.426856i
$438$		0			0
$439$			26283.3i			0.136380i	0.997672	+	0.0681901i	$0.0217224\pi$
$439$			26283.3i			0.136380i	−0.997672	+	0.0681901i	$0.978278\pi$
$440$		0			0
$441$	−231231.			−1.18897
$442$		0			0
$443$	317887.			1.61981			0.809907	−	0.586558i	$-0.199518\pi$
$443$	317887.			1.61981			0.809907	+	0.586558i	$0.199518\pi$
$444$		0			0
$445$		−	465036.i		−	2.34837i
$446$		0			0
$447$		−	92639.3i		−	0.463639i
$448$		0			0
$449$		−	68728.7i		−	0.340915i	−0.985365	−	0.170457i	$-0.945476\pi$
$449$		−	68728.7i		−	0.340915i	0.985365	−	0.170457i	$-0.0545245\pi$
$450$		0			0
$451$			84857.3i			0.417192i
$452$		0			0
$453$	316394.			1.54181
$454$		0			0
$455$			179213.i			0.865660i
$456$		0			0
$457$	182859.			0.875555			0.437778	−	0.899083i	$-0.355766\pi$
$457$	182859.			0.875555			0.437778	+	0.899083i	$0.355766\pi$
$458$		0			0
$459$		−	335206.i		−	1.59106i
$460$		0			0
$461$	142804.			0.671951			0.335976	−	0.941871i	$-0.390934\pi$
$461$	142804.			0.671951			0.335976	+	0.941871i	$0.390934\pi$
$462$		0			0
$463$	−96212.9			−0.448819			−0.224409	−	0.974495i	$-0.572045\pi$
$463$	−96212.9			−0.448819			−0.224409	+	0.974495i	$0.572045\pi$
$464$		0			0
$465$	−161224.			−0.745632
$466$		0			0
$467$	−287321.			−1.31745			−0.658724	−	0.752384i	$-0.728904\pi$
$467$	−287321.			−1.31745			−0.658724	+	0.752384i	$0.728904\pi$
$468$		0			0
$469$		−	319162.i		−	1.45099i
$470$		0			0
$471$		−	16012.9i		−	0.0721818i
$472$		0			0
$473$	26911.6			0.120287
$474$		0			0
$475$	216684.	+	336021.i	0.960374	+	1.48929i
$476$		0			0
$477$		−	678029.i		−	2.97997i
$478$		0			0
$479$	−101492.			−0.442346			−0.221173	−	0.975235i	$-0.570989\pi$
$479$	−101492.			−0.442346			−0.221173	+	0.975235i	$0.570989\pi$
$480$		0			0
$481$	−166012.			−0.717544
$482$		0			0
$483$		−	257410.i		−	1.10339i
$484$		0			0
$485$			44876.2i			0.190780i
$486$		0			0
$487$		−	203340.i		−	0.857363i	−0.903456	−	0.428682i	$-0.858978\pi$
$487$		−	203340.i		−	0.857363i	0.903456	−	0.428682i	$-0.141022\pi$
$488$		0			0
$489$		−	24366.1i		−	0.101899i
$490$		0			0
$491$	227758.			0.944735			0.472368	−	0.881402i	$-0.343399\pi$
$491$	227758.			0.944735			0.472368	+	0.881402i	$0.343399\pi$
$492$		0			0
$493$		−	166948.i		−	0.686890i
$494$		0			0
$495$	790394.			3.22577
$496$		0			0
$497$		−	510895.i		−	2.06833i
$498$		0			0
$499$	132829.			0.533447			0.266724	−	0.963773i	$-0.414059\pi$
$499$	132829.			0.533447			0.266724	+	0.963773i	$0.414059\pi$
$500$		0			0
$501$	−326502.			−1.30080
$502$		0			0
$503$	−279294.			−1.10389			−0.551945	−	0.833880i	$-0.686114\pi$
$503$	−279294.			−1.10389			−0.551945	+	0.833880i	$0.686114\pi$
$504$		0			0
$505$	58011.9			0.227475
$506$		0			0
$507$			365286.i			1.42108i
$508$		0			0
$509$			304788.i			1.17642i	0.808709	+	0.588209i	$0.200167\pi$
$509$			304788.i			1.17642i	−0.808709	+	0.588209i	$0.799833\pi$
$510$		0			0
$511$	−458594.			−1.75625
$512$		0			0
$513$	341985.	−	220530.i	1.29949	−	0.837980i
$514$		0			0
$515$		−	464656.i		−	1.75193i
$516$		0			0
$517$	−10216.7			−0.0382235
$518$		0			0
$519$	−362585.			−1.34609
$520$		0			0
$521$		−	181301.i		−	0.667919i	−0.942587	−	0.333959i	$-0.891615\pi$
$521$		−	181301.i		−	0.667919i	0.942587	−	0.333959i	$-0.108385\pi$
$522$		0			0
$523$		−	5966.86i		−	0.0218144i	−0.999941	−	0.0109072i	$-0.996528\pi$
$523$		−	5966.86i		−	0.0218144i	0.999941	−	0.0109072i	$-0.00347193\pi$
$524$		0			0
$525$			1.06108e6i			3.84971i
$526$		0			0
$527$			75064.3i			0.270279i
$528$		0			0
$529$	−207649.			−0.742026
$530$		0			0
$531$		−	73674.2i		−	0.261292i
$532$		0			0
$533$	47600.1			0.167553
$534$		0			0
$535$		−	20898.5i		−	0.0730142i
$536$		0			0
$537$	837515.			2.90432
$538$		0			0
$539$	−184042.			−0.633490
$540$		0			0
$541$	−121383.			−0.414727			−0.207364	−	0.978264i	$-0.566488\pi$
$541$	−121383.			−0.414727			−0.207364	+	0.978264i	$0.566488\pi$
$542$		0			0
$543$	743632.			2.52208
$544$		0			0
$545$		−	474000.i		−	1.59583i
$546$		0			0
$547$		−	297533.i		−	0.994399i	−0.867636	−	0.497200i	$-0.834362\pi$
$547$		−	297533.i		−	0.994399i	0.867636	−	0.497200i	$-0.165638\pi$
$548$		0			0
$549$	−612446.			−2.03200
$550$		0			0
$551$	170325.	−	109834.i	0.561014	−	0.361772i
$552$		0			0
$553$			92753.2i			0.303304i
$554$		0			0
$555$	−1.53758e6			−4.99173
$556$		0			0
$557$	192568.			0.620687			0.310344	−	0.950624i	$-0.399556\pi$
$557$	192568.			0.620687			0.310344	+	0.950624i	$0.399556\pi$
$558$		0			0
$559$		−	15095.9i		−	0.0483097i
$560$		0			0
$561$		−	560981.i		−	1.78247i
$562$		0			0
$563$		−	21470.7i		−	0.0677376i	−0.999426	−	0.0338688i	$-0.989217\pi$
$563$		−	21470.7i		−	0.0677376i	0.999426	−	0.0338688i	$-0.0107828\pi$
$564$		0			0
$565$			198922.i			0.623139i
$566$		0			0
$567$	298781.			0.929367
$568$		0			0
$569$			51249.7i			0.158295i	0.996863	+	0.0791474i	$0.0252198\pi$
$569$			51249.7i			0.158295i	−0.996863	+	0.0791474i	$0.974780\pi$
$570$		0			0
$571$	−515089.			−1.57983			−0.789914	−	0.613218i	$-0.789875\pi$
$571$	−515089.			−1.57983			−0.789914	+	0.613218i	$0.789875\pi$
$572$		0			0
$573$		−	164477.i		−	0.500951i
$574$		0			0
$575$	−297584.			−0.900065
$576$		0			0
$577$	−419849.			−1.26108			−0.630538	−	0.776159i	$-0.717166\pi$
$577$	−419849.			−1.26108			−0.630538	+	0.776159i	$0.717166\pi$
$578$		0			0
$579$	389114.			1.16070
$580$		0			0
$581$	−333855.			−0.989020
$582$		0			0
$583$		−	539658.i		−	1.58775i
$584$		0			0
$585$		−	443366.i		−	1.29554i
$586$		0			0
$587$	−311469.			−0.903939			−0.451970	−	0.892033i	$-0.649278\pi$
$587$	−311469.			−0.903939			−0.451970	+	0.892033i	$0.649278\pi$
$588$		0			0
$589$	−76582.5	+	49384.5i	−0.220749	+	0.142351i
$590$		0			0
$591$		−	69617.9i		−	0.199318i
$592$		0			0
$593$	−131604.			−0.374247			−0.187124	−	0.982336i	$-0.559917\pi$
$593$	−131604.			−0.374247			−0.187124	+	0.982336i	$0.559917\pi$
$594$		0			0
$595$	772807.			2.18292
$596$		0			0
$597$			1.03881e6i			2.91465i
$598$		0			0
$599$		−	199776.i		−	0.556788i	−0.960467	−	0.278394i	$-0.910198\pi$
$599$		−	199776.i		−	0.556788i	0.960467	−	0.278394i	$-0.0898021\pi$
$600$		0			0
$601$		−	328476.i		−	0.909399i	−0.890645	−	0.454700i	$-0.849747\pi$
$601$		−	328476.i		−	0.909399i	0.890645	−	0.454700i	$-0.150253\pi$
$602$		0			0
$603$			789592.i			2.17154i
$604$		0			0
$605$	19675.5			0.0537544
$606$		0			0
$607$			275312.i			0.747219i	0.927586	+	0.373610i	$0.121880\pi$
$607$			275312.i			0.747219i	−0.927586	+	0.373610i	$0.878120\pi$
$608$		0			0
$609$	537845.			1.45018
$610$		0			0
$611$			5731.00i			0.0153514i
$612$		0			0
$613$	−473140.			−1.25912			−0.629562	−	0.776950i	$-0.716766\pi$
$613$	−473140.			−1.25912			−0.629562	+	0.776950i	$0.716766\pi$
$614$		0			0
$615$	440865.			1.16562
$616$		0			0
$617$	10864.2			0.0285382			0.0142691	−	0.999898i	$-0.495458\pi$
$617$	10864.2			0.0285382			0.0142691	+	0.999898i	$0.495458\pi$
$618$		0			0
$619$	−8963.62			−0.0233939			−0.0116969	−	0.999932i	$-0.503723\pi$
$619$	−8963.62			−0.0233939			−0.0116969	+	0.999932i	$0.503723\pi$
$620$		0			0
$621$			302866.i			0.785357i
$622$		0			0
$623$		−	697535.i		−	1.79717i
$624$		0			0
$625$	143834.			0.368214
$626$		0			0
$627$	572327.	−	369067.i	1.45583	−	0.938793i
$628$		0			0
$629$			715879.i			1.80942i
$630$		0			0
$631$	439268.			1.10324			0.551621	−	0.834095i	$-0.314009\pi$
$631$	439268.			1.10324			0.551621	+	0.834095i	$0.314009\pi$
$632$		0			0
$633$	−43223.0			−0.107872
$634$		0			0
$635$			730608.i			1.81191i
$636$		0			0
$637$			103237.i			0.254423i
$638$		0			0
$639$			1.26393e6i			3.09544i
$640$		0			0
$641$		−	351470.i		−	0.855406i	−0.903919	−	0.427703i	$-0.859323\pi$
$641$		−	351470.i		−	0.855406i	0.903919	−	0.427703i	$-0.140677\pi$
$642$		0			0
$643$	257336.			0.622412			0.311206	−	0.950342i	$-0.399267\pi$
$643$	257336.			0.622412			0.311206	+	0.950342i	$0.399267\pi$
$644$		0			0
$645$		−	139816.i		−	0.336076i
$646$		0			0
$647$	−352433.			−0.841915			−0.420957	−	0.907080i	$-0.638306\pi$
$647$	−352433.			−0.841915			−0.420957	+	0.907080i	$0.638306\pi$
$648$		0			0
$649$		−	58639.0i		−	0.139218i
$650$		0			0
$651$	−241830.			−0.570621
$652$		0			0
$653$	−51680.8			−0.121200			−0.0606001	−	0.998162i	$-0.519301\pi$
$653$	−51680.8			−0.121200			−0.0606001	+	0.998162i	$0.519301\pi$
$654$		0			0
$655$	308094.			0.718126
$656$		0			0
$657$	1.13454e6			2.62839
$658$		0			0
$659$		−	144227.i		−	0.332105i	−0.986117	−	0.166053i	$-0.946898\pi$
$659$		−	144227.i		−	0.332105i	0.986117	−	0.166053i	$-0.0531021\pi$
$660$		0			0
$661$		−	615181.i		−	1.40799i	−0.710205	−	0.703995i	$-0.751398\pi$
$661$		−	615181.i		−	1.40799i	0.710205	−	0.703995i	$-0.248602\pi$
$662$		0			0
$663$	−314678.			−0.715878
$664$		0			0
$665$	508426.	+	788438.i	1.14970	+	1.78289i
$666$		0			0
$667$			150841.i			0.339054i
$668$		0			0
$669$	−565765.			−1.26411
$670$		0			0
$671$	−487459.			−1.08266
$672$		0			0
$673$			334936.i			0.739489i	0.929133	+	0.369745i	$0.120555\pi$
$673$			334936.i			0.739489i	−0.929133	+	0.369745i	$0.879445\pi$
$674$		0			0
$675$		−	1.24845e6i		−	2.74009i
$676$		0			0
$677$		−	504808.i		−	1.10141i	−0.834700	−	0.550705i	$-0.814359\pi$
$677$		−	504808.i		−	1.10141i	0.834700	−	0.550705i	$-0.185641\pi$
$678$		0			0
$679$			67312.5i			0.146001i
$680$		0			0
$681$	−871992.			−1.88026
$682$		0			0
$683$		−	690506.i		−	1.48022i	−0.672486	−	0.740110i	$-0.734773\pi$
$683$		−	690506.i		−	1.48022i	0.672486	−	0.740110i	$-0.265227\pi$
$684$		0			0
$685$	70459.8			0.150162
$686$		0			0
$687$			117121.i			0.248153i
$688$		0			0
$689$	−302717.			−0.637673
$690$		0			0
$691$	−461905.			−0.967379			−0.483690	−	0.875240i	$-0.660704\pi$
$691$	−461905.			−0.967379			−0.483690	+	0.875240i	$0.660704\pi$
$692$		0			0
$693$	1.18556e6			2.46863
$694$		0			0
$695$	1.55535e6			3.22002
$696$		0			0
$697$		−	205262.i		−	0.422516i
$698$		0			0
$699$			1.40019e6i			2.86571i
$700$		0			0
$701$	−410170.			−0.834694			−0.417347	−	0.908747i	$-0.637040\pi$
$701$	−410170.			−0.834694			−0.417347	+	0.908747i	$0.637040\pi$
$702$		0			0
$703$	−730358.	+	470974.i	−1.47783	+	0.952985i
$704$		0			0
$705$			53079.7i			0.106795i
$706$		0			0
$707$	87015.4			0.174083
$708$		0			0
$709$	368245.			0.732562			0.366281	−	0.930504i	$-0.380631\pi$
$709$	368245.			0.732562			0.366281	+	0.930504i	$0.380631\pi$
$710$		0			0
$711$		−	229467.i		−	0.453923i
$712$		0			0
$713$		−	67822.3i		−	0.133411i
$714$		0			0
$715$		−	352884.i		−	0.690272i
$716$		0			0
$717$			1.69039e6i			3.28813i
$718$		0			0
$719$	460066.			0.889943			0.444971	−	0.895545i	$-0.353214\pi$
$719$	460066.			0.889943			0.444971	+	0.895545i	$0.353214\pi$
$720$		0			0
$721$		−	696965.i		−	1.34073i
$722$		0			0
$723$	−896482.			−1.71500
$724$		0			0
$725$		−	621788.i		−	1.18295i
$726$		0			0
$727$	−196263.			−0.371339			−0.185670	−	0.982612i	$-0.559445\pi$
$727$	−196263.			−0.371339			−0.185670	+	0.982612i	$0.559445\pi$
$728$		0			0
$729$	661866.			1.24542
$730$		0			0
$731$	−65096.8			−0.121822
$732$		0			0
$733$	578892.			1.07743			0.538716	−	0.842487i	$-0.318910\pi$
$733$	578892.			1.07743			0.538716	+	0.842487i	$0.318910\pi$
$734$		0			0
$735$			956167.i			1.76994i
$736$		0			0
$737$			628453.i			1.15701i
$738$		0			0
$739$	−721624.			−1.32136			−0.660681	−	0.750667i	$-0.729732\pi$
$739$	−721624.			−1.32136			−0.660681	+	0.750667i	$0.729732\pi$
$740$		0			0
$741$	−207025.	−	321042.i	−0.377039	−	0.584690i
$742$		0			0
$743$			963975.i			1.74618i	0.487562	+	0.873088i	$0.337886\pi$
$743$			963975.i			1.74618i	−0.487562	+	0.873088i	$0.662114\pi$
$744$		0			0
$745$	−251293.			−0.452760
$746$		0			0
$747$	825942.			1.48016
$748$		0			0
$749$		−	31346.8i		−	0.0558766i
$750$		0			0
$751$			632486.i			1.12143i	0.828010	+	0.560713i	$0.189473\pi$
$751$			632486.i			1.12143i	−0.828010	+	0.560713i	$0.810527\pi$
$752$		0			0
$753$			264823.i			0.467052i
$754$		0			0
$755$		−	858249.i		−	1.50563i
$756$		0			0
$757$	684418.			1.19434			0.597172	−	0.802113i	$-0.296291\pi$
$757$	684418.			1.19434			0.597172	+	0.802113i	$0.296291\pi$
$758$		0			0
$759$			506859.i			0.879840i
$760$		0			0
$761$	−201444.			−0.347845			−0.173923	−	0.984759i	$-0.555644\pi$
$761$	−201444.			−0.347845			−0.173923	+	0.984759i	$0.555644\pi$
$762$		0			0
$763$		−	710980.i		−	1.22126i
$764$		0			0
$765$	−1.91189e6			−3.26693
$766$		0			0
$767$	−32893.1			−0.0559131
$768$		0			0
$769$	343992.			0.581695			0.290847	−	0.956769i	$-0.406063\pi$
$769$	343992.			0.581695			0.290847	+	0.956769i	$0.406063\pi$
$770$		0			0
$771$	−632135.			−1.06341
$772$		0			0
$773$		−	323326.i		−	0.541105i	−0.962705	−	0.270553i	$-0.912794\pi$
$773$		−	323326.i		−	0.541105i	0.962705	−	0.270553i	$-0.0872064\pi$
$774$		0			0
$775$			279573.i			0.465469i
$776$		0			0
$777$	−2.30630e6			−3.82010
$778$		0			0
$779$	209414.	−	135041.i	0.345088	−	0.222531i
$780$		0			0
$781$			1.00599e6i			1.64927i
$782$		0			0
$783$	−632824.			−1.03219
$784$		0			0
$785$	−43436.5			−0.0704881
$786$		0			0
$787$			526771.i			0.850496i	0.905077	+	0.425248i	$0.139813\pi$
$787$			526771.i			0.850496i	−0.905077	+	0.425248i	$0.860187\pi$
$788$		0			0
$789$			631890.i			1.01505i
$790$		0			0
$791$			298374.i			0.476879i
$792$		0			0
$793$			273436.i			0.434820i
$794$		0			0
$795$	−2.80372e6			−4.43609
$796$		0			0
$797$			937353.i			1.47566i	0.674986	+	0.737831i	$0.264150\pi$
$797$			937353.i			1.47566i	−0.674986	+	0.737831i	$0.735850\pi$
$798$		0			0
$799$	24713.3			0.0387113
$800$		0			0
$801$			1.72567e6i			2.68963i
$802$		0			0
$803$	903006.			1.40042
$804$		0			0
$805$	−698249.			−1.07750
$806$		0			0
$807$	304727.			0.467911
$808$		0			0
$809$	−339941.			−0.519405			−0.259702	−	0.965689i	$-0.583624\pi$
$809$	−339941.			−0.519405			−0.259702	+	0.965689i	$0.583624\pi$
$810$		0			0
$811$			421572.i			0.640958i	0.947256	+	0.320479i	$0.103844\pi$
$811$			421572.i			0.640958i	−0.947256	+	0.320479i	$0.896156\pi$
$812$		0			0
$813$		−	1.70290e6i		−	2.57637i
$814$		0			0
$815$	−66095.4			−0.0995076
$816$		0			0
$817$	−42826.9	−	66413.4i	−0.0641612	−	0.0994974i
$818$		0			0
$819$		−	665030.i		−	0.991456i
$820$		0			0
$821$	−832671.			−1.23534			−0.617671	−	0.786437i	$-0.711923\pi$
$821$	−832671.			−1.23534			−0.617671	+	0.786437i	$0.711923\pi$
$822$		0			0
$823$	−193463.			−0.285627			−0.142813	−	0.989750i	$-0.545615\pi$
$823$	−193463.			−0.285627			−0.142813	+	0.989750i	$0.545615\pi$
$824$		0			0
$825$		−	2.08934e6i		−	3.06974i
$826$		0			0
$827$			414119.i			0.605500i	0.953070	+	0.302750i	$0.0979047\pi$
$827$			414119.i			0.605500i	−0.953070	+	0.302750i	$0.902095\pi$
$828$		0			0
$829$		−	1.10281e6i		−	1.60469i	−0.596858	−	0.802347i	$-0.703585\pi$
$829$		−	1.10281e6i		−	1.60469i	0.596858	−	0.802347i	$-0.296415\pi$
$830$		0			0
$831$		−	1.34319e6i		−	1.94508i
$832$		0			0
$833$	445181.			0.641574
$834$		0			0
$835$			885669.i			1.27028i
$836$		0			0
$837$	284535.			0.406148
$838$		0			0
$839$		−	420379.i		−	0.597196i	−0.954379	−	0.298598i	$-0.903481\pi$
$839$		−	420379.i		−	0.597196i	0.954379	−	0.298598i	$-0.0965189\pi$
$840$		0			0
$841$	392105.			0.554384
$842$		0			0
$843$	2.04541e6			2.87823
$844$		0			0
$845$	990875.			1.38773
$846$		0			0
$847$	29512.4			0.0411374
$848$		0			0
$849$		−	458895.i		−	0.636646i
$850$		0			0
$851$		−	646813.i		−	0.893140i
$852$		0			0
$853$	−1.00746e6			−1.38462			−0.692310	−	0.721600i	$-0.743407\pi$
$853$	−1.00746e6			−1.38462			−0.692310	+	0.721600i	$0.743407\pi$
$854$		0			0
$855$	−1.25782e6	−	1.95056e6i	−1.72063	−	2.66825i
$856$		0			0
$857$		−	425334.i		−	0.579119i	−0.957160	−	0.289560i	$-0.906491\pi$
$857$		−	425334.i		−	0.579119i	0.957160	−	0.289560i	$-0.0935089\pi$
$858$		0			0
$859$	101991.			0.138221			0.0691105	−	0.997609i	$-0.477984\pi$
$859$	101991.			0.138221			0.0691105	+	0.997609i	$0.477984\pi$
$860$		0			0
$861$	661280.			0.892029
$862$		0			0
$863$			295578.i			0.396872i	0.980114	+	0.198436i	$0.0635862\pi$
$863$			295578.i			0.396872i	−0.980114	+	0.198436i	$0.936414\pi$
$864$		0			0
$865$			983547.i			1.31451i
$866$		0			0
$867$			75357.6i			0.100251i
$868$		0			0
$869$		−	182638.i		−	0.241853i
$870$		0			0
$871$	352526.			0.464681
$872$		0			0
$873$		−	166528.i		−	0.218504i
$874$		0			0
$875$	1.25405e6			1.63794
$876$		0			0
$877$			267600.i			0.347926i	0.984752	+	0.173963i	$0.0556573\pi$
$877$			267600.i			0.347926i	−0.984752	+	0.173963i	$0.944343\pi$
$878$		0			0
$879$	654073.			0.846542
$880$		0			0
$881$	1.32998e6			1.71353			0.856765	−	0.515707i	$-0.172471\pi$
$881$	1.32998e6			1.71353			0.856765	+	0.515707i	$0.172471\pi$
$882$		0			0
$883$	229460.			0.294296			0.147148	−	0.989114i	$-0.452991\pi$
$883$	229460.			0.294296			0.147148	+	0.989114i	$0.452991\pi$
$884$		0			0
$885$	−304651.			−0.388970
$886$		0			0
$887$		−	161937.i		−	0.205825i	−0.994690	−	0.102913i	$-0.967184\pi$
$887$		−	161937.i		−	0.205825i	0.994690	−	0.102913i	$-0.0328162\pi$
$888$		0			0
$889$			1.09588e6i			1.38663i
$890$		0			0
$891$	−588323.			−0.741072
$892$		0			0
$893$	16258.8	+	25213.2i	0.0203885	+	0.0316173i
$894$		0			0
$895$		−	2.27184e6i		−	2.83617i
$896$		0			0
$897$	284319.			0.353362
$898$		0			0
$899$	141711.			0.175342
$900$		0			0
$901$			1.30538e6i			1.60801i
$902$		0			0
$903$		−	209718.i		−	0.257194i
$904$		0			0
$905$		−	2.01717e6i		−	2.46290i
$906$		0			0
$907$			530320.i			0.644649i	0.946629	+	0.322325i	$0.104464\pi$
$907$			530320.i			0.644649i	−0.946629	+	0.322325i	$0.895536\pi$
$908$		0			0
$909$	−215272.			−0.260532
$910$		0			0
$911$		−	969623.i		−	1.16833i	−0.811634	−	0.584166i	$-0.801422\pi$
$911$		−	969623.i		−	1.16833i	0.811634	−	0.584166i	$-0.198578\pi$
$912$		0			0
$913$	657385.			0.788639
$914$		0			0
$915$			2.53253e6i			3.02491i
$916$		0			0
$917$	462128.			0.549571
$918$		0			0
$919$	−853445.			−1.01052			−0.505260	−	0.862967i	$-0.668603\pi$
$919$	−853445.			−1.01052			−0.505260	+	0.862967i	$0.668603\pi$
$920$		0			0
$921$	599172.			0.706370
$922$		0			0
$923$	564303.			0.662382
$924$		0			0
$925$			2.66625e6i			3.11614i
$926$		0			0
$927$			1.72426e6i			2.00652i
$928$		0			0
$929$	580185.			0.672257			0.336128	−	0.941816i	$-0.390882\pi$
$929$	580185.			0.672257			0.336128	+	0.941816i	$0.390882\pi$
$930$		0			0
$931$	292882.	+	454185.i	0.337905	+	0.524002i
$932$		0			0
$933$		−	1.21579e6i		−	1.39667i
$934$		0			0
$935$	−1.52172e6			−1.74065
$936$		0			0
$937$	−173996.			−0.198180			−0.0990901	−	0.995078i	$-0.531593\pi$
$937$	−173996.			−0.198180			−0.0990901	+	0.995078i	$0.531593\pi$
$938$		0			0
$939$		−	1.64309e6i		−	1.86350i
$940$		0			0
$941$			288487.i			0.325797i	0.986643	+	0.162899i	$0.0520844\pi$
$941$			288487.i			0.325797i	−0.986643	+	0.162899i	$0.947916\pi$
$942$		0			0
$943$			185459.i			0.208557i
$944$		0			0
$945$		−	2.92936e6i		−	3.28027i
$946$		0			0
$947$	−484204.			−0.539919			−0.269959	−	0.962872i	$-0.587010\pi$
$947$	−484204.			−0.539919			−0.269959	+	0.962872i	$0.587010\pi$
$948$		0			0
$949$		−	506534.i		−	0.562440i
$950$		0			0
$951$	2.03968e6			2.25528
$952$		0			0
$953$			834271.i			0.918590i	0.888284	+	0.459295i	$0.151898\pi$
$953$			834271.i			0.918590i	−0.888284	+	0.459295i	$0.848102\pi$
$954$		0			0
$955$	−446159.			−0.489196
$956$		0			0
$957$	−1.05906e6			−1.15637
$958$		0			0
$959$	105687.			0.114917
$960$		0			0
$961$	859804.			0.931006
$962$		0			0
$963$			77550.7i			0.0836244i
$964$		0			0
$965$		−	1.05551e6i		−	1.13346i
$966$		0			0
$967$	268509.			0.287148			0.143574	−	0.989640i	$-0.454141\pi$
$967$	268509.			0.287148			0.143574	+	0.989640i	$0.454141\pi$
$968$		0			0
$969$	−1.38441e6	+	892739.i	−1.47440	+	0.950773i
$970$		0			0
$971$			513847.i			0.544999i	0.962156	+	0.272499i	$0.0878503\pi$
$971$			513847.i			0.544999i	−0.962156	+	0.272499i	$0.912150\pi$
$972$		0			0
$973$	2.33296e6			2.46423
$974$		0			0
$975$	−1.17200e6			−1.23287
$976$		0			0
$977$		−	880047.i		−	0.921970i	−0.887408	−	0.460985i	$-0.847496\pi$
$977$		−	880047.i		−	0.921970i	0.887408	−	0.460985i	$-0.152504\pi$
$978$		0			0
$979$			1.37350e6i			1.43306i
$980$		0			0
$981$			1.75893e6i			1.82773i
$982$		0			0
$983$		−	1.15115e6i		−	1.19131i	−0.803239	−	0.595656i	$-0.796892\pi$
$983$		−	1.15115e6i		−	1.19131i	0.803239	−	0.595656i	$-0.203108\pi$
$984$		0			0
$985$	−188845.			−0.194641
$986$		0			0
$987$			79617.3i			0.0817285i
$988$		0			0
$989$	58816.4			0.0601320
$990$		0			0
$991$			1.55344e6i			1.58178i	0.611956	+	0.790892i	$0.290383\pi$
$991$			1.55344e6i			1.58178i	−0.611956	+	0.790892i	$0.709617\pi$
$992$		0			0
$993$	−684727.			−0.694415
$994$		0			0
$995$	2.81787e6			2.84626
$996$		0			0
$997$	−1.19520e6			−1.20240			−0.601200	−	0.799099i	$-0.705310\pi$
$997$	−1.19520e6			−1.20240			−0.601200	+	0.799099i	$0.705310\pi$
$998$		0			0
$999$	2.71357e6			2.71901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.e.113.8		8
4.3	odd	2		38.5.b.a.37.1	✓	8
12.11	even	2		342.5.d.a.37.5		8
19.18	odd	2	inner	304.5.e.e.113.1		8
76.75	even	2		38.5.b.a.37.8	yes	8
228.227	odd	2		342.5.d.a.37.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.1	✓	8	4.3	odd	2
38.5.b.a.37.8	yes	8	76.75	even	2
304.5.e.e.113.1		8	19.18	odd	2	inner
304.5.e.e.113.8		8	1.1	even	1	trivial
342.5.d.a.37.1		8	228.227	odd	2
342.5.d.a.37.5		8	12.11	even	2

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.3447i q^{3} +41.6240 q^{5} +62.4342 q^{7} -154.460 q^{9} +O(q^{10})\)	\(q+15.3447i q^{3} +41.6240 q^{5} +62.4342 q^{7} -154.460 q^{9} -122.938 q^{11} +68.9610i q^{13} +638.707i q^{15} +297.375 q^{17} +(195.642 + 303.390i) q^{19} +958.034i q^{21} -268.685 q^{23} +1107.56 q^{25} -1127.21i q^{27} -561.405i q^{29} +252.423i q^{31} -1886.44i q^{33} +2598.76 q^{35} +2407.33i q^{37} -1058.18 q^{39} -690.246i q^{41} -218.905 q^{43} -6429.22 q^{45} +83.1049 q^{47} +1497.03 q^{49} +4563.13i q^{51} +4389.68i q^{53} -5117.16 q^{55} +(-4655.42 + 3002.06i) q^{57} +476.981i q^{59} +3965.09 q^{61} -9643.57 q^{63} +2870.43i q^{65} -5111.96i q^{67} -4122.89i q^{69} -8182.93i q^{71} -7345.23 q^{73} +16995.1i q^{75} -7675.53 q^{77} +1485.61i q^{79} +4785.54 q^{81} -5347.30 q^{83} +12377.9 q^{85} +8614.59 q^{87} -11172.3i q^{89} +4305.53i q^{91} -3873.35 q^{93} +(8143.39 + 12628.3i) q^{95} +1078.13i q^{97} +18988.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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