LMFDB - Embedded newform 126.3.b.a.71.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,3,Mod(71,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("126.71");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	126.b (of order $2$, degree $1$, 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:	$3.43325133094$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			71.1
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	126.71
Dual form			126.3.b.a.71.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -6.06910i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} -6.06910i q^{5} -2.64575 q^{7} +2.82843i q^{8} -8.58301 q^{10} -12.1382i q^{11} -18.5830 q^{13} +3.74166i q^{14} +4.00000 q^{16} -10.9015i q^{17} +20.0000 q^{19} +12.1382i q^{20} -17.1660 q^{22} -12.1382i q^{23} -11.8340 q^{25} +26.2803i q^{26} +5.29150 q^{28} +41.8367i q^{29} +25.1660 q^{31} -5.65685i q^{32} -15.4170 q^{34} +16.0573i q^{35} +38.0000 q^{37} -28.2843i q^{38} +17.1660 q^{40} -60.6337i q^{41} +83.4980 q^{43} +24.2764i q^{44} -17.1660 q^{46} -16.9706i q^{47} +7.00000 q^{49} +16.7358i q^{50} +37.1660 q^{52} +94.0424i q^{53} -73.6680 q^{55} -7.48331i q^{56} +59.1660 q^{58} -58.2175i q^{59} +15.6680 q^{61} -35.5901i q^{62} -8.00000 q^{64} +112.782i q^{65} -132.664 q^{67} +21.8029i q^{68} +22.7085 q^{70} -12.1382i q^{71} -76.9150 q^{73} -53.7401i q^{74} -40.0000 q^{76} +32.1147i q^{77} +33.6680 q^{79} -24.2764i q^{80} -85.7490 q^{82} -60.5764i q^{83} -66.1621 q^{85} -118.084i q^{86} +34.3320 q^{88} -4.77506i q^{89} +49.1660 q^{91} +24.2764i q^{92} -24.0000 q^{94} -121.382i q^{95} -188.413 q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 8 q^{10} - 32 q^{13} + 16 q^{16} + 80 q^{19} + 16 q^{22} - 132 q^{25} + 16 q^{31} - 104 q^{34} + 152 q^{37} - 16 q^{40} + 80 q^{43} + 16 q^{46} + 28 q^{49} + 64 q^{52} - 464 q^{55} + 152 q^{58} + 232 q^{61} - 32 q^{64} - 192 q^{67} + 112 q^{70} - 96 q^{73} - 160 q^{76} + 304 q^{79} - 216 q^{82} + 328 q^{85} - 32 q^{88} + 112 q^{91} - 96 q^{94} - 288 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 8 * q^10 - 32 * q^13 + 16 * q^16 + 80 * q^19 + 16 * q^22 - 132 * q^25 + 16 * q^31 - 104 * q^34 + 152 * q^37 - 16 * q^40 + 80 * q^43 + 16 * q^46 + 28 * q^49 + 64 * q^52 - 464 * q^55 + 152 * q^58 + 232 * q^61 - 32 * q^64 - 192 * q^67 + 112 * q^70 - 96 * q^73 - 160 * q^76 + 304 * q^79 - 216 * q^82 + 328 * q^85 - 32 * q^88 + 112 * q^91 - 96 * q^94 - 288 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 6.06910i	− 1.21382i	−0.794770	−	0.606910i	$-0.792409\pi$
$5$	− 6.06910i	− 1.21382i	0.794770	−	0.606910i	$-0.207591\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−8.58301	−0.858301
$11$	− 12.1382i	− 1.10347i	−0.834019	−	0.551736i	$-0.813965\pi$
$11$	− 12.1382i	− 1.10347i	0.834019	−	0.551736i	$-0.186035\pi$
$12$	0	0
$13$	−18.5830	−1.42946	−0.714731	−	0.699399i	$-0.753451\pi$
$13$	−18.5830	−1.42946	−0.714731	+	0.699399i	$0.753451\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 10.9015i	− 0.641262i	−0.947204	−	0.320631i	$-0.896105\pi$
$17$	− 10.9015i	− 0.641262i	0.947204	−	0.320631i	$-0.103895\pi$
$18$	0	0
$19$	20.0000	1.05263	0.526316	−	0.850289i	$-0.323573\pi$
$19$	20.0000	1.05263	0.526316	+	0.850289i	$0.323573\pi$
$20$	12.1382i	0.606910i
$21$	0	0
$22$	−17.1660	−0.780273
$23$	− 12.1382i	− 0.527748i	−0.964557	−	0.263874i	$-0.915000\pi$
$23$	− 12.1382i	− 0.527748i	0.964557	−	0.263874i	$-0.0850003\pi$
$24$	0	0
$25$	−11.8340	−0.473360
$26$	26.2803i	1.01078i
$27$	0	0
$28$	5.29150	0.188982
$29$	41.8367i	1.44264i	0.692600	+	0.721322i	$0.256465\pi$
$29$	41.8367i	1.44264i	−0.692600	+	0.721322i	$0.743535\pi$
$30$	0	0
$31$	25.1660	0.811807	0.405903	−	0.913916i	$-0.366957\pi$
$31$	25.1660	0.811807	0.405903	+	0.913916i	$0.366957\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−15.4170	−0.453441
$35$	16.0573i	0.458781i
$36$	0	0
$37$	38.0000	1.02703	0.513514	−	0.858082i	$-0.328344\pi$
$37$	38.0000	1.02703	0.513514	+	0.858082i	$0.328344\pi$
$38$	− 28.2843i	− 0.744323i
$39$	0	0
$40$	17.1660	0.429150
$41$	− 60.6337i	− 1.47887i	−0.673227	−	0.739435i	$-0.735092\pi$
$41$	− 60.6337i	− 1.47887i	0.673227	−	0.739435i	$-0.264908\pi$
$42$	0	0
$43$	83.4980	1.94181	0.970907	−	0.239455i	$-0.0769689\pi$
$43$	83.4980	1.94181	0.970907	+	0.239455i	$0.0769689\pi$
$44$	24.2764i	0.551736i
$45$	0	0
$46$	−17.1660	−0.373174
$47$	− 16.9706i	− 0.361076i	−0.983568	−	0.180538i	$-0.942216\pi$
$47$	− 16.9706i	− 0.361076i	0.983568	−	0.180538i	$-0.0577838\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	16.7358i	0.334716i
$51$	0	0
$52$	37.1660	0.714731
$53$	94.0424i	1.77439i	0.461399	+	0.887193i	$0.347348\pi$
$53$	94.0424i	1.77439i	−0.461399	+	0.887193i	$0.652652\pi$
$54$	0	0
$55$	−73.6680	−1.33942
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	59.1660	1.02010
$59$	− 58.2175i	− 0.986738i	−0.869820	−	0.493369i	$-0.835765\pi$
$59$	− 58.2175i	− 0.986738i	0.869820	−	0.493369i	$-0.164235\pi$
$60$	0	0
$61$	15.6680	0.256852	0.128426	−	0.991719i	$-0.459008\pi$
$61$	15.6680	0.256852	0.128426	+	0.991719i	$0.459008\pi$
$62$	− 35.5901i	− 0.574034i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	112.782i	1.73511i
$66$	0	0
$67$	−132.664	−1.98006	−0.990030	−	0.140856i	$-0.955015\pi$
$67$	−132.664	−1.98006	−0.990030	+	0.140856i	$0.955015\pi$
$68$	21.8029i	0.320631i
$69$	0	0
$70$	22.7085	0.324407
$71$	− 12.1382i	− 0.170961i	−0.996340	−	0.0854803i	$-0.972758\pi$
$71$	− 12.1382i	− 0.170961i	0.996340	−	0.0854803i	$-0.0272425\pi$
$72$	0	0
$73$	−76.9150	−1.05363	−0.526815	−	0.849980i	$-0.676614\pi$
$73$	−76.9150	−1.05363	−0.526815	+	0.849980i	$0.676614\pi$
$74$	− 53.7401i	− 0.726218i
$75$	0	0
$76$	−40.0000	−0.526316
$77$	32.1147i	0.417074i
$78$	0	0
$79$	33.6680	0.426177	0.213088	−	0.977033i	$-0.431648\pi$
$79$	33.6680	0.426177	0.213088	+	0.977033i	$0.431648\pi$
$80$	− 24.2764i	− 0.303455i
$81$	0	0
$82$	−85.7490	−1.04572
$83$	− 60.5764i	− 0.729836i	−0.931040	−	0.364918i	$-0.881097\pi$
$83$	− 60.5764i	− 0.729836i	0.931040	−	0.364918i	$-0.118903\pi$
$84$	0	0
$85$	−66.1621	−0.778377
$86$	− 118.084i	− 1.37307i
$87$	0	0
$88$	34.3320	0.390137
$89$	− 4.77506i	− 0.0536523i	−0.999640	−	0.0268262i	$-0.991460\pi$
$89$	− 4.77506i	− 0.0536523i	0.999640	−	0.0268262i	$-0.00854006\pi$
$90$	0	0
$91$	49.1660	0.540286
$92$	24.2764i	0.263874i
$93$	0	0
$94$	−24.0000	−0.255319
$95$	− 121.382i	− 1.27771i
$96$	0	0
$97$	−188.413	−1.94240	−0.971201	−	0.238260i	$-0.923423\pi$
$97$	−188.413	−1.94240	−0.971201	+	0.238260i	$0.923423\pi$
$98$	− 9.89949i	− 0.101015i
$99$	0	0
$100$	23.6680	0.236680
$101$	106.713i	1.05656i	0.849069	+	0.528282i	$0.177164\pi$
$101$	106.713i	1.05656i	−0.849069	+	0.528282i	$0.822836\pi$
$102$	0	0
$103$	131.498	1.27668	0.638340	−	0.769755i	$-0.279621\pi$
$103$	131.498	1.27668	0.638340	+	0.769755i	$0.279621\pi$
$104$	− 52.5607i	− 0.505391i
$105$	0	0
$106$	132.996	1.25468
$107$	82.3793i	0.769900i	0.922937	+	0.384950i	$0.125781\pi$
$107$	82.3793i	0.769900i	−0.922937	+	0.384950i	$0.874219\pi$
$108$	0	0
$109$	33.8301	0.310367	0.155184	−	0.987886i	$-0.450403\pi$
$109$	33.8301	0.310367	0.155184	+	0.987886i	$0.450403\pi$
$110$	104.182i	0.947111i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	− 28.5190i	− 0.252381i	−0.992006	−	0.126190i	$-0.959725\pi$
$113$	− 28.5190i	− 0.252381i	0.992006	−	0.126190i	$-0.0402750\pi$
$114$	0	0
$115$	−73.6680	−0.640591
$116$	− 83.6734i	− 0.721322i
$117$	0	0
$118$	−82.3320	−0.697729
$119$	28.8426i	0.242374i
$120$	0	0
$121$	−26.3360	−0.217653
$122$	− 22.1579i	− 0.181622i
$123$	0	0
$124$	−50.3320	−0.405903
$125$	− 79.9059i	− 0.639247i
$126$	0	0
$127$	129.668	1.02101	0.510504	−	0.859875i	$-0.329459\pi$
$127$	129.668	1.02101	0.510504	+	0.859875i	$0.329459\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	159.498	1.22691
$131$	− 148.017i	− 1.12990i	−0.825124	−	0.564952i	$-0.808895\pi$
$131$	− 148.017i	− 1.12990i	0.825124	−	0.564952i	$-0.191105\pi$
$132$	0	0
$133$	−52.9150	−0.397857
$134$	187.615i	1.40011i
$135$	0	0
$136$	30.8340	0.226721
$137$	− 76.9573i	− 0.561732i	−0.959747	−	0.280866i	$-0.909378\pi$
$137$	− 76.9573i	− 0.561732i	0.959747	−	0.280866i	$-0.0906216\pi$
$138$	0	0
$139$	217.328	1.56351	0.781756	−	0.623585i	$-0.214324\pi$
$139$	217.328	1.56351	0.781756	+	0.623585i	$0.214324\pi$
$140$	− 32.1147i	− 0.229390i
$141$	0	0
$142$	−17.1660	−0.120887
$143$	225.564i	1.57737i
$144$	0	0
$145$	253.911	1.75111
$146$	108.774i	0.745029i
$147$	0	0
$148$	−76.0000	−0.513514
$149$	161.925i	1.08674i	0.839492	+	0.543371i	$0.182852\pi$
$149$	161.925i	1.08674i	−0.839492	+	0.543371i	$0.817148\pi$
$150$	0	0
$151$	−93.1660	−0.616993	−0.308497	−	0.951225i	$-0.599826\pi$
$151$	−93.1660	−0.616993	−0.308497	+	0.951225i	$0.599826\pi$
$152$	56.5685i	0.372161i
$153$	0	0
$154$	45.4170	0.294916
$155$	− 152.735i	− 0.985388i
$156$	0	0
$157$	−184.996	−1.17832	−0.589159	−	0.808017i	$-0.700541\pi$
$157$	−184.996	−1.17832	−0.589159	+	0.808017i	$0.700541\pi$
$158$	− 47.6137i	− 0.301353i
$159$	0	0
$160$	−34.3320	−0.214575
$161$	32.1147i	0.199470i
$162$	0	0
$163$	86.9961	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$163$	86.9961	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$164$	121.267i	0.739435i
$165$	0	0
$166$	−85.6680	−0.516072
$167$	60.5764i	0.362733i	0.983416	+	0.181366i	$0.0580520\pi$
$167$	60.5764i	0.362733i	−0.983416	+	0.181366i	$0.941948\pi$
$168$	0	0
$169$	176.328	1.04336
$170$	93.5673i	0.550396i
$171$	0	0
$172$	−166.996	−0.970907
$173$	162.572i	0.939721i	0.882741	+	0.469860i	$0.155696\pi$
$173$	162.572i	0.939721i	−0.882741	+	0.469860i	$0.844304\pi$
$174$	0	0
$175$	31.3098	0.178913
$176$	− 48.5528i	− 0.275868i
$177$	0	0
$178$	−6.75295	−0.0379379
$179$	223.091i	1.24632i	0.782095	+	0.623159i	$0.214151\pi$
$179$	223.091i	1.24632i	−0.782095	+	0.623159i	$0.785849\pi$
$180$	0	0
$181$	188.915	1.04373	0.521865	−	0.853028i	$-0.325237\pi$
$181$	188.915	1.04373	0.521865	+	0.853028i	$0.325237\pi$
$182$	− 69.5312i	− 0.382040i
$183$	0	0
$184$	34.3320	0.186587
$185$	− 230.626i	− 1.24663i
$186$	0	0
$187$	−132.324	−0.707616
$188$	33.9411i	0.180538i
$189$	0	0
$190$	−171.660	−0.903474
$191$	− 228.038i	− 1.19391i	−0.802273	−	0.596957i	$-0.796376\pi$
$191$	− 228.038i	− 1.19391i	0.802273	−	0.596957i	$-0.203624\pi$
$192$	0	0
$193$	134.000	0.694301	0.347150	−	0.937810i	$-0.387149\pi$
$193$	134.000	0.694301	0.347150	+	0.937810i	$0.387149\pi$
$194$	266.456i	1.37349i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	− 188.560i	− 0.957157i	−0.878045	−	0.478579i	$-0.841152\pi$
$197$	− 188.560i	− 0.957157i	0.878045	−	0.478579i	$-0.158848\pi$
$198$	0	0
$199$	102.494	0.515046	0.257523	−	0.966272i	$-0.417094\pi$
$199$	102.494	0.515046	0.257523	+	0.966272i	$0.417094\pi$
$200$	− 33.4716i	− 0.167358i
$201$	0	0
$202$	150.915	0.747104
$203$	− 110.689i	− 0.545268i
$204$	0	0
$205$	−367.992	−1.79508
$206$	− 185.966i	− 0.902749i
$207$	0	0
$208$	−74.3320	−0.357365
$209$	− 242.764i	− 1.16155i
$210$	0	0
$211$	−84.5020	−0.400483	−0.200242	−	0.979747i	$-0.564173\pi$
$211$	−84.5020	−0.400483	−0.200242	+	0.979747i	$0.564173\pi$
$212$	− 188.085i	− 0.887193i
$213$	0	0
$214$	116.502	0.544402
$215$	− 506.758i	− 2.35701i
$216$	0	0
$217$	−66.5830	−0.306834
$218$	− 47.8429i	− 0.219463i
$219$	0	0
$220$	147.336	0.669709
$221$	202.582i	0.916660i
$222$	0	0
$223$	−158.494	−0.710736	−0.355368	−	0.934727i	$-0.615644\pi$
$223$	−158.494	−0.710736	−0.355368	+	0.934727i	$0.615644\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−40.3320	−0.178460
$227$	101.823i	0.448561i	0.974525	+	0.224281i	$0.0720032\pi$
$227$	101.823i	0.448561i	−0.974525	+	0.224281i	$0.927997\pi$
$228$	0	0
$229$	−268.915	−1.17430	−0.587151	−	0.809478i	$-0.699750\pi$
$229$	−268.915	−1.17430	−0.587151	+	0.809478i	$0.699750\pi$
$230$	104.182i	0.452966i
$231$	0	0
$232$	−118.332	−0.510052
$233$	− 26.2748i	− 0.112767i	−0.998409	−	0.0563836i	$-0.982043\pi$
$233$	− 26.2748i	− 0.112767i	0.998409	−	0.0563836i	$-0.0179570\pi$
$234$	0	0
$235$	−102.996	−0.438281
$236$	116.435i	0.493369i
$237$	0	0
$238$	40.7895	0.171385
$239$	92.2733i	0.386081i	0.981191	+	0.193040i	$0.0618348\pi$
$239$	92.2733i	0.386081i	−0.981191	+	0.193040i	$0.938165\pi$
$240$	0	0
$241$	343.247	1.42426	0.712131	−	0.702047i	$-0.247730\pi$
$241$	343.247	1.42426	0.712131	+	0.702047i	$0.247730\pi$
$242$	37.2447i	0.153904i
$243$	0	0
$244$	−31.3360	−0.128426
$245$	− 42.4837i	− 0.173403i
$246$	0	0
$247$	−371.660	−1.50470
$248$	71.1802i	0.287017i
$249$	0	0
$250$	−113.004	−0.452016
$251$	356.382i	1.41985i	0.704278	+	0.709924i	$0.251271\pi$
$251$	356.382i	1.41985i	−0.704278	+	0.709924i	$0.748729\pi$
$252$	0	0
$253$	−147.336	−0.582356
$254$	− 183.378i	− 0.721961i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 254.730i	− 0.991169i	−0.868560	−	0.495584i	$-0.834954\pi$
$257$	− 254.730i	− 0.991169i	0.868560	−	0.495584i	$-0.165046\pi$
$258$	0	0
$259$	−100.539	−0.388180
$260$	− 225.564i	− 0.867555i
$261$	0	0
$262$	−209.328	−0.798962
$263$	261.979i	0.996117i	0.867143	+	0.498059i	$0.165954\pi$
$263$	261.979i	0.996117i	−0.867143	+	0.498059i	$0.834046\pi$
$264$	0	0
$265$	570.753	2.15378
$266$	74.8331i	0.281328i
$267$	0	0
$268$	265.328	0.990030
$269$	93.6246i	0.348047i	0.984742	+	0.174023i	$0.0556768\pi$
$269$	93.6246i	0.348047i	−0.984742	+	0.174023i	$0.944323\pi$
$270$	0	0
$271$	1.16601	0.00430262	0.00215131	−	0.999998i	$-0.499315\pi$
$271$	1.16601	0.00430262	0.00215131	+	0.999998i	$0.499315\pi$
$272$	− 43.6058i	− 0.160316i
$273$	0	0
$274$	−108.834	−0.397204
$275$	143.643i	0.522339i
$276$	0	0
$277$	32.0000	0.115523	0.0577617	−	0.998330i	$-0.481604\pi$
$277$	32.0000	0.115523	0.0577617	+	0.998330i	$0.481604\pi$
$278$	− 307.348i	− 1.10557i
$279$	0	0
$280$	−45.4170	−0.162204
$281$	166.757i	0.593441i	0.954964	+	0.296721i	$0.0958930\pi$
$281$	166.757i	0.593441i	−0.954964	+	0.296721i	$0.904107\pi$
$282$	0	0
$283$	16.3399	0.0577381	0.0288691	−	0.999583i	$-0.490809\pi$
$283$	16.3399	0.0577381	0.0288691	+	0.999583i	$0.490809\pi$
$284$	24.2764i	0.0854803i
$285$	0	0
$286$	318.996	1.11537
$287$	160.422i	0.558961i
$288$	0	0
$289$	170.158	0.588782
$290$	− 359.085i	− 1.23822i
$291$	0	0
$292$	153.830	0.526815
$293$	368.921i	1.25912i	0.776953	+	0.629558i	$0.216764\pi$
$293$	368.921i	1.25912i	−0.776953	+	0.629558i	$0.783236\pi$
$294$	0	0
$295$	−353.328	−1.19772
$296$	107.480i	0.363109i
$297$	0	0
$298$	228.996	0.768443
$299$	225.564i	0.754396i
$300$	0	0
$301$	−220.915	−0.733937
$302$	131.757i	0.436280i
$303$	0	0
$304$	80.0000	0.263158
$305$	− 95.0906i	− 0.311772i
$306$	0	0
$307$	−192.664	−0.627570	−0.313785	−	0.949494i	$-0.601597\pi$
$307$	−192.664	−0.627570	−0.313785	+	0.949494i	$0.601597\pi$
$308$	− 64.2293i	− 0.208537i
$309$	0	0
$310$	−216.000	−0.696774
$311$	131.276i	0.422109i	0.977474	+	0.211055i	$0.0676898\pi$
$311$	131.276i	0.422109i	−0.977474	+	0.211055i	$0.932310\pi$
$312$	0	0
$313$	43.3281	0.138428	0.0692142	−	0.997602i	$-0.477951\pi$
$313$	43.3281	0.138428	0.0692142	+	0.997602i	$0.477951\pi$
$314$	261.624i	0.833197i
$315$	0	0
$316$	−67.3360	−0.213088
$317$	− 251.724i	− 0.794083i	−0.917801	−	0.397042i	$-0.870037\pi$
$317$	− 251.724i	− 0.794083i	0.917801	−	0.397042i	$-0.129963\pi$
$318$	0	0
$319$	507.822	1.59192
$320$	48.5528i	0.151728i
$321$	0	0
$322$	45.4170	0.141047
$323$	− 218.029i	− 0.675013i
$324$	0	0
$325$	219.911	0.676650
$326$	− 123.031i	− 0.377396i
$327$	0	0
$328$	171.498	0.522860
$329$	44.8999i	0.136474i
$330$	0	0
$331$	361.490	1.09212	0.546058	−	0.837748i	$-0.316128\pi$
$331$	361.490	1.09212	0.546058	+	0.837748i	$0.316128\pi$
$332$	121.153i	0.364918i
$333$	0	0
$334$	85.6680	0.256491
$335$	805.151i	2.40344i
$336$	0	0
$337$	−298.834	−0.886748	−0.443374	−	0.896337i	$-0.646219\pi$
$337$	−298.834	−0.886748	−0.443374	+	0.896337i	$0.646219\pi$
$338$	− 249.366i	− 0.737768i
$339$	0	0
$340$	132.324	0.389189
$341$	− 305.470i	− 0.895807i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	236.168i	0.686535i
$345$	0	0
$346$	229.911	0.664483
$347$	206.120i	0.594006i	0.954876	+	0.297003i	$0.0959872\pi$
$347$	206.120i	0.594006i	−0.954876	+	0.297003i	$0.904013\pi$
$348$	0	0
$349$	434.324	1.24448	0.622241	−	0.782826i	$-0.286222\pi$
$349$	434.324	1.24448	0.622241	+	0.782826i	$0.286222\pi$
$350$	− 44.2787i	− 0.126511i
$351$	0	0
$352$	−68.6640	−0.195068
$353$	− 185.439i	− 0.525324i	−0.964888	−	0.262662i	$-0.915400\pi$
$353$	− 185.439i	− 0.525324i	0.964888	−	0.262662i	$-0.0846005\pi$
$354$	0	0
$355$	−73.6680	−0.207515
$356$	9.55012i	0.0268262i
$357$	0	0
$358$	315.498	0.881279
$359$	− 516.767i	− 1.43946i	−0.694254	−	0.719731i	$-0.744265\pi$
$359$	− 516.767i	− 1.43946i	0.694254	−	0.719731i	$-0.255735\pi$
$360$	0	0
$361$	39.0000	0.108033
$362$	− 267.166i	− 0.738028i
$363$	0	0
$364$	−98.3320	−0.270143
$365$	466.805i	1.27892i
$366$	0	0
$367$	−117.490	−0.320137	−0.160068	−	0.987106i	$-0.551171\pi$
$367$	−117.490	−0.320137	−0.160068	+	0.987106i	$0.551171\pi$
$368$	− 48.5528i	− 0.131937i
$369$	0	0
$370$	−326.154	−0.881498
$371$	− 248.813i	− 0.670655i
$372$	0	0
$373$	−402.664	−1.07953	−0.539764	−	0.841816i	$-0.681487\pi$
$373$	−402.664	−1.07953	−0.539764	+	0.841816i	$0.681487\pi$
$374$	187.135i	0.500360i
$375$	0	0
$376$	48.0000	0.127660
$377$	− 777.451i	− 2.06221i
$378$	0	0
$379$	398.834	1.05233	0.526166	−	0.850382i	$-0.323629\pi$
$379$	398.834	1.05233	0.526166	+	0.850382i	$0.323629\pi$
$380$	242.764i	0.638853i
$381$	0	0
$382$	−322.494	−0.844225
$383$	744.804i	1.94466i	0.233614	+	0.972329i	$0.424945\pi$
$383$	744.804i	1.94466i	−0.233614	+	0.972329i	$0.575055\pi$
$384$	0	0
$385$	194.907	0.506252
$386$	− 189.505i	− 0.490945i
$387$	0	0
$388$	376.826	0.971201
$389$	− 535.162i	− 1.37574i	−0.725834	−	0.687869i	$-0.758546\pi$
$389$	− 535.162i	− 1.37574i	0.725834	−	0.687869i	$-0.241454\pi$
$390$	0	0
$391$	−132.324	−0.338425
$392$	19.7990i	0.0505076i
$393$	0	0
$394$	−266.664	−0.676812
$395$	− 204.334i	− 0.517302i
$396$	0	0
$397$	−94.3241	−0.237592	−0.118796	−	0.992919i	$-0.537904\pi$
$397$	−94.3241	−0.237592	−0.118796	+	0.992919i	$0.537904\pi$
$398$	− 144.949i	− 0.364192i
$399$	0	0
$400$	−47.3360	−0.118340
$401$	− 103.593i	− 0.258335i	−0.991623	−	0.129168i	$-0.958769\pi$
$401$	− 103.593i	− 0.258335i	0.991623	−	0.129168i	$-0.0412306\pi$
$402$	0	0
$403$	−467.660	−1.16045
$404$	− 213.426i	− 0.528282i
$405$	0	0
$406$	−156.539	−0.385563
$407$	− 461.252i	− 1.13330i
$408$	0	0
$409$	−9.75689	−0.0238555	−0.0119277	−	0.999929i	$-0.503797\pi$
$409$	−9.75689	−0.0238555	−0.0119277	+	0.999929i	$0.503797\pi$
$410$	520.419i	1.26932i
$411$	0	0
$412$	−262.996	−0.638340
$413$	154.029i	0.372952i
$414$	0	0
$415$	−367.644	−0.885890
$416$	105.121i	0.252696i
$417$	0	0
$418$	−343.320	−0.821340
$419$	339.411i	0.810051i	0.914305	+	0.405025i	$0.132737\pi$
$419$	339.411i	0.810051i	−0.914305	+	0.405025i	$0.867263\pi$
$420$	0	0
$421$	−599.320	−1.42356	−0.711782	−	0.702401i	$-0.752111\pi$
$421$	−599.320	−1.42356	−0.711782	+	0.702401i	$0.752111\pi$
$422$	119.504i	0.283184i
$423$	0	0
$424$	−265.992	−0.627340
$425$	129.008i	0.303548i
$426$	0	0
$427$	−41.4536	−0.0970810
$428$	− 164.759i	− 0.384950i
$429$	0	0
$430$	−716.664	−1.66666
$431$	710.978i	1.64960i	0.565424	+	0.824800i	$0.308712\pi$
$431$	710.978i	1.64960i	−0.565424	+	0.824800i	$0.691288\pi$
$432$	0	0
$433$	377.984	0.872943	0.436471	−	0.899718i	$-0.356228\pi$
$433$	377.984	0.872943	0.436471	+	0.899718i	$0.356228\pi$
$434$	94.1626i	0.216964i
$435$	0	0
$436$	−67.6601	−0.155184
$437$	− 242.764i	− 0.555524i
$438$	0	0
$439$	528.146	1.20307	0.601533	−	0.798848i	$-0.294557\pi$
$439$	528.146	1.20307	0.601533	+	0.798848i	$0.294557\pi$
$440$	− 208.365i	− 0.473556i
$441$	0	0
$442$	286.494	0.648177
$443$	36.6438i	0.0827174i	0.999144	+	0.0413587i	$0.0131686\pi$
$443$	36.6438i	0.0827174i	−0.999144	+	0.0413587i	$0.986831\pi$
$444$	0	0
$445$	−28.9803	−0.0651243
$446$	224.144i	0.502566i
$447$	0	0
$448$	21.1660	0.0472456
$449$	− 397.612i	− 0.885550i	−0.896633	−	0.442775i	$-0.853994\pi$
$449$	− 397.612i	− 0.885550i	0.896633	−	0.442775i	$-0.146006\pi$
$450$	0	0
$451$	−735.984	−1.63189
$452$	57.0381i	0.126190i
$453$	0	0
$454$	144.000	0.317181
$455$	− 298.393i	− 0.655810i
$456$	0	0
$457$	344.324	0.753445	0.376722	−	0.926326i	$-0.377051\pi$
$457$	344.324	0.753445	0.376722	+	0.926326i	$0.377051\pi$
$458$	380.303i	0.830357i
$459$	0	0
$460$	147.336	0.320296
$461$	− 370.936i	− 0.804634i	−0.915500	−	0.402317i	$-0.868205\pi$
$461$	− 370.936i	− 0.804634i	0.915500	−	0.402317i	$-0.131795\pi$
$462$	0	0
$463$	78.3320	0.169184	0.0845918	−	0.996416i	$-0.473041\pi$
$463$	78.3320	0.169184	0.0845918	+	0.996416i	$0.473041\pi$
$464$	167.347i	0.360661i
$465$	0	0
$466$	−37.1581	−0.0797385
$467$	− 399.758i	− 0.856014i	−0.903775	−	0.428007i	$-0.859216\pi$
$467$	− 399.758i	− 0.856014i	0.903775	−	0.428007i	$-0.140784\pi$
$468$	0	0
$469$	350.996	0.748392
$470$	145.658i	0.309912i
$471$	0	0
$472$	164.664	0.348864
$473$	− 1013.52i	− 2.14274i
$474$	0	0
$475$	−236.680	−0.498273
$476$	− 57.6851i	− 0.121187i
$477$	0	0
$478$	130.494	0.273000
$479$	− 703.328i	− 1.46833i	−0.678973	−	0.734163i	$-0.737575\pi$
$479$	− 703.328i	− 1.46833i	0.678973	−	0.734163i	$-0.262425\pi$
$480$	0	0
$481$	−706.154	−1.46810
$482$	− 485.425i	− 1.00711i
$483$	0	0
$484$	52.6719	0.108826
$485$	1143.50i	2.35773i
$486$	0	0
$487$	−82.5098	−0.169425	−0.0847124	−	0.996405i	$-0.526997\pi$
$487$	−82.5098	−0.169425	−0.0847124	+	0.996405i	$0.526997\pi$
$488$	44.3157i	0.0908109i
$489$	0	0
$490$	−60.0810	−0.122614
$491$	− 184.203i	− 0.375158i	−0.982249	−	0.187579i	$-0.939936\pi$
$491$	− 184.203i	− 0.375158i	0.982249	−	0.187579i	$-0.0600641\pi$
$492$	0	0
$493$	456.081	0.925114
$494$	525.607i	1.06398i
$495$	0	0
$496$	100.664	0.202952
$497$	32.1147i	0.0646170i
$498$	0	0
$499$	752.810	1.50864	0.754319	−	0.656508i	$-0.227967\pi$
$499$	752.810	1.50864	0.754319	+	0.656508i	$0.227967\pi$
$500$	159.812i	0.319623i
$501$	0	0
$502$	504.000	1.00398
$503$	− 662.540i	− 1.31718i	−0.752504	−	0.658588i	$-0.771154\pi$
$503$	− 662.540i	− 1.31718i	0.752504	−	0.658588i	$-0.228846\pi$
$504$	0	0
$505$	647.652	1.28248
$506$	208.365i	0.411788i
$507$	0	0
$508$	−259.336	−0.510504
$509$	949.115i	1.86467i	0.361601	+	0.932333i	$0.382230\pi$
$509$	949.115i	1.86467i	−0.361601	+	0.932333i	$0.617770\pi$
$510$	0	0
$511$	203.498	0.398235
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−360.243	−0.700862
$515$	− 798.075i	− 1.54966i
$516$	0	0
$517$	−205.992	−0.398437
$518$	142.183i	0.274485i
$519$	0	0
$520$	−318.996	−0.613454
$521$	714.344i	1.37110i	0.728025	+	0.685551i	$0.240439\pi$
$521$	714.344i	1.37110i	−0.728025	+	0.685551i	$0.759561\pi$
$522$	0	0
$523$	−232.000	−0.443595	−0.221797	−	0.975093i	$-0.571192\pi$
$523$	−232.000	−0.443595	−0.221797	+	0.975093i	$0.571192\pi$
$524$	296.035i	0.564952i
$525$	0	0
$526$	370.494	0.704361
$527$	− 274.346i	− 0.520581i
$528$	0	0
$529$	381.664	0.721482
$530$	− 807.167i	− 1.52296i
$531$	0	0
$532$	105.830	0.198929
$533$	1126.76i	2.11399i
$534$	0	0
$535$	499.969	0.934521
$536$	− 375.231i	− 0.700057i
$537$	0	0
$538$	132.405	0.246106
$539$	− 84.9674i	− 0.157639i
$540$	0	0
$541$	165.668	0.306225	0.153113	−	0.988209i	$-0.451070\pi$
$541$	165.668	0.306225	0.153113	+	0.988209i	$0.451070\pi$
$542$	− 1.64899i	− 0.00304241i
$543$	0	0
$544$	−61.6680	−0.113360
$545$	− 205.318i	− 0.376730i
$546$	0	0
$547$	295.676	0.540541	0.270270	−	0.962784i	$-0.412887\pi$
$547$	295.676	0.540541	0.270270	+	0.962784i	$0.412887\pi$
$548$	153.915i	0.280866i
$549$	0	0
$550$	203.142	0.369350
$551$	836.734i	1.51857i
$552$	0	0
$553$	−89.0771	−0.161080
$554$	− 45.2548i	− 0.0816874i
$555$	0	0
$556$	−434.656	−0.781756
$557$	76.8426i	0.137958i	0.997618	+	0.0689790i	$0.0219742\pi$
$557$	76.8426i	0.137958i	−0.997618	+	0.0689790i	$0.978026\pi$
$558$	0	0
$559$	−1551.64	−2.77575
$560$	64.2293i	0.114695i
$561$	0	0
$562$	235.830	0.419626
$563$	1016.33i	1.80521i	0.430470	+	0.902605i	$0.358348\pi$
$563$	1016.33i	1.80521i	−0.430470	+	0.902605i	$0.641652\pi$
$564$	0	0
$565$	−173.085	−0.306345
$566$	− 23.1081i	− 0.0408270i
$567$	0	0
$568$	34.3320	0.0604437
$569$	− 586.533i	− 1.03081i	−0.856946	−	0.515406i	$-0.827641\pi$
$569$	− 586.533i	− 1.03081i	0.856946	−	0.515406i	$-0.172359\pi$
$570$	0	0
$571$	951.644	1.66663	0.833314	−	0.552800i	$-0.186441\pi$
$571$	951.644	1.66663	0.833314	+	0.552800i	$0.186441\pi$
$572$	− 451.129i	− 0.788686i
$573$	0	0
$574$	226.871	0.395245
$575$	143.643i	0.249815i
$576$	0	0
$577$	−148.672	−0.257664	−0.128832	−	0.991666i	$-0.541123\pi$
$577$	−148.672	−0.257664	−0.128832	+	0.991666i	$0.541123\pi$
$578$	− 240.640i	− 0.416332i
$579$	0	0
$580$	−507.822	−0.875555
$581$	160.270i	0.275852i
$582$	0	0
$583$	1141.51	1.95799
$584$	− 217.549i	− 0.372515i
$585$	0	0
$586$	521.733	0.890330
$587$	332.564i	0.566548i	0.959039	+	0.283274i	$0.0914206\pi$
$587$	332.564i	0.566548i	−0.959039	+	0.283274i	$0.908579\pi$
$588$	0	0
$589$	503.320	0.854533
$590$	499.681i	0.846918i
$591$	0	0
$592$	152.000	0.256757
$593$	217.251i	0.366359i	0.983079	+	0.183180i	$0.0586390\pi$
$593$	217.251i	0.366359i	−0.983079	+	0.183180i	$0.941361\pi$
$594$	0	0
$595$	175.048	0.294199
$596$	− 323.849i	− 0.543371i
$597$	0	0
$598$	318.996	0.533438
$599$	− 172.179i	− 0.287444i	−0.989618	−	0.143722i	$-0.954093\pi$
$599$	− 172.179i	− 0.287444i	0.989618	−	0.143722i	$-0.0459072\pi$
$600$	0	0
$601$	−418.000	−0.695507	−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000	−0.695507	−0.347754	+	0.937586i	$0.613055\pi$
$602$	312.421i	0.518972i
$603$	0	0
$604$	186.332	0.308497
$605$	159.836i	0.264191i
$606$	0	0
$607$	627.158	1.03321	0.516605	−	0.856224i	$-0.327196\pi$
$607$	627.158	1.03321	0.516605	+	0.856224i	$0.327196\pi$
$608$	− 113.137i	− 0.186081i
$609$	0	0
$610$	−134.478	−0.220456
$611$	315.364i	0.516144i
$612$	0	0
$613$	−279.328	−0.455674	−0.227837	−	0.973699i	$-0.573165\pi$
$613$	−279.328	−0.455674	−0.227837	+	0.973699i	$0.573165\pi$
$614$	272.468i	0.443759i
$615$	0	0
$616$	−90.8340	−0.147458
$617$	358.380i	0.580843i	0.956899	+	0.290422i	$0.0937955\pi$
$617$	358.380i	0.580843i	−0.956899	+	0.290422i	$0.906204\pi$
$618$	0	0
$619$	−983.644	−1.58909	−0.794543	−	0.607208i	$-0.792290\pi$
$619$	−983.644	−1.58909	−0.794543	+	0.607208i	$0.792290\pi$
$620$	305.470i	0.492694i
$621$	0	0
$622$	185.652	0.298476
$623$	12.6336i	0.0202787i
$624$	0	0
$625$	−780.806	−1.24929
$626$	− 61.2752i	− 0.0978836i
$627$	0	0
$628$	369.992	0.589159
$629$	− 414.256i	− 0.658594i
$630$	0	0
$631$	−298.996	−0.473845	−0.236922	−	0.971529i	$-0.576139\pi$
$631$	−298.996	−0.473845	−0.236922	+	0.971529i	$0.576139\pi$
$632$	95.2274i	0.150676i
$633$	0	0
$634$	−355.992	−0.561502
$635$	− 786.968i	− 1.23932i
$636$	0	0
$637$	−130.081	−0.204209
$638$	− 718.169i	− 1.12566i
$639$	0	0
$640$	68.6640	0.107288
$641$	− 311.957i	− 0.486672i	−0.969942	−	0.243336i	$-0.921758\pi$
$641$	− 311.957i	− 0.486672i	0.969942	−	0.243336i	$-0.0782418\pi$
$642$	0	0
$643$	−604.000	−0.939347	−0.469673	−	0.882840i	$-0.655628\pi$
$643$	−604.000	−0.939347	−0.469673	+	0.882840i	$0.655628\pi$
$644$	− 64.2293i	− 0.0997350i
$645$	0	0
$646$	−308.340	−0.477306
$647$	179.600i	0.277588i	0.990321	+	0.138794i	$0.0443226\pi$
$647$	179.600i	0.277588i	−0.990321	+	0.138794i	$0.955677\pi$
$648$	0	0
$649$	−706.656	−1.08884
$650$	− 311.001i	− 0.478463i
$651$	0	0
$652$	−173.992	−0.266859
$653$	481.892i	0.737966i	0.929436	+	0.368983i	$0.120294\pi$
$653$	481.892i	0.737966i	−0.929436	+	0.368983i	$0.879706\pi$
$654$	0	0
$655$	−898.332	−1.37150
$656$	− 242.535i	− 0.369718i
$657$	0	0
$658$	63.4980	0.0965016
$659$	877.408i	1.33142i	0.746209	+	0.665711i	$0.231872\pi$
$659$	877.408i	1.33142i	−0.746209	+	0.665711i	$0.768128\pi$
$660$	0	0
$661$	−521.644	−0.789175	−0.394587	−	0.918858i	$-0.629112\pi$
$661$	−521.644	−0.789175	−0.394587	+	0.918858i	$0.629112\pi$
$662$	− 511.224i	− 0.772242i
$663$	0	0
$664$	171.336	0.258036
$665$	321.147i	0.482927i
$666$	0	0
$667$	507.822	0.761353
$668$	− 121.153i	− 0.181366i
$669$	0	0
$670$	1138.66	1.69949
$671$	− 190.181i	− 0.283429i
$672$	0	0
$673$	−659.992	−0.980672	−0.490336	−	0.871534i	$-0.663126\pi$
$673$	−659.992	−0.980672	−0.490336	+	0.871534i	$0.663126\pi$
$674$	422.615i	0.627025i
$675$	0	0
$676$	−352.656	−0.521681
$677$	1016.28i	1.50115i	0.660787	+	0.750573i	$0.270222\pi$
$677$	1016.28i	1.50115i	−0.660787	+	0.750573i	$0.729778\pi$
$678$	0	0
$679$	498.494	0.734159
$680$	− 187.135i	− 0.275198i
$681$	0	0
$682$	−432.000	−0.633431
$683$	− 235.114i	− 0.344238i	−0.985076	−	0.172119i	$-0.944939\pi$
$683$	− 235.114i	− 0.344238i	0.985076	−	0.172119i	$-0.0550613\pi$
$684$	0	0
$685$	−467.061	−0.681841
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	333.992	0.485454
$689$	− 1747.59i	− 2.53642i
$690$	0	0
$691$	−50.9803	−0.0737776	−0.0368888	−	0.999319i	$-0.511745\pi$
$691$	−50.9803	−0.0737776	−0.0368888	+	0.999319i	$0.511745\pi$
$692$	− 325.143i	− 0.469860i
$693$	0	0
$694$	291.498	0.420026
$695$	− 1318.99i	− 1.89782i
$696$	0	0
$697$	−660.996	−0.948344
$698$	− 614.227i	− 0.879982i
$699$	0	0
$700$	−62.6196	−0.0894566
$701$	141.530i	0.201898i	0.994892	+	0.100949i	$0.0321879\pi$
$701$	141.530i	0.201898i	−0.994892	+	0.100949i	$0.967812\pi$
$702$	0	0
$703$	760.000	1.08108
$704$	97.1056i	0.137934i
$705$	0	0
$706$	−262.251	−0.371460
$707$	− 282.336i	− 0.399344i
$708$	0	0
$709$	−55.4980	−0.0782765	−0.0391382	−	0.999234i	$-0.512461\pi$
$709$	−55.4980	−0.0782765	−0.0391382	+	0.999234i	$0.512461\pi$
$710$	104.182i	0.146736i
$711$	0	0
$712$	13.5059	0.0189690
$713$	− 305.470i	− 0.428429i
$714$	0	0
$715$	1368.97	1.91465
$716$	− 446.182i	− 0.623159i
$717$	0	0
$718$	−730.818	−1.01785
$719$	− 1009.03i	− 1.40338i	−0.712484	−	0.701688i	$-0.752430\pi$
$719$	− 1009.03i	− 1.40338i	0.712484	−	0.701688i	$-0.247570\pi$
$720$	0	0
$721$	−347.911	−0.482540
$722$	− 55.1543i	− 0.0763910i
$723$	0	0
$724$	−377.830	−0.521865
$725$	− 495.095i	− 0.682890i
$726$	0	0
$727$	−365.182	−0.502313	−0.251157	−	0.967946i	$-0.580811\pi$
$727$	−365.182	−0.502313	−0.251157	+	0.967946i	$0.580811\pi$
$728$	139.062i	0.191020i
$729$	0	0
$730$	660.162	0.904332
$731$	− 910.251i	− 1.24521i
$732$	0	0
$733$	−353.077	−0.481688	−0.240844	−	0.970564i	$-0.577424\pi$
$733$	−353.077	−0.481688	−0.240844	+	0.970564i	$0.577424\pi$
$734$	166.156i	0.226371i
$735$	0	0
$736$	−68.6640	−0.0932935
$737$	1610.30i	2.18494i
$738$	0	0
$739$	329.684	0.446121	0.223061	−	0.974805i	$-0.428395\pi$
$739$	329.684	0.446121	0.223061	+	0.974805i	$0.428395\pi$
$740$	461.252i	0.623313i
$741$	0	0
$742$	−351.875	−0.474224
$743$	− 112.061i	− 0.150822i	−0.997153	−	0.0754112i	$-0.975973\pi$
$743$	− 112.061i	− 0.150822i	0.997153	−	0.0754112i	$-0.0240270\pi$
$744$	0	0
$745$	982.737	1.31911
$746$	569.453i	0.763342i
$747$	0	0
$748$	264.648	0.353808
$749$	− 217.955i	− 0.290995i
$750$	0	0
$751$	−144.826	−0.192844	−0.0964222	−	0.995341i	$-0.530740\pi$
$751$	−144.826	−0.192844	−0.0964222	+	0.995341i	$0.530740\pi$
$752$	− 67.8823i	− 0.0902690i
$753$	0	0
$754$	−1099.48	−1.45820
$755$	565.434i	0.748919i
$756$	0	0
$757$	78.1699	0.103263	0.0516314	−	0.998666i	$-0.483558\pi$
$757$	78.1699	0.103263	0.0516314	+	0.998666i	$0.483558\pi$
$758$	− 564.036i	− 0.744111i
$759$	0	0
$760$	343.320	0.451737
$761$	1465.50i	1.92576i	0.269928	+	0.962880i	$0.413000\pi$
$761$	1465.50i	1.92576i	−0.269928	+	0.962880i	$0.587000\pi$
$762$	0	0
$763$	−89.5059	−0.117308
$764$	456.076i	0.596957i
$765$	0	0
$766$	1053.31	1.37508
$767$	1081.86i	1.41050i
$768$	0	0
$769$	729.320	0.948401	0.474200	−	0.880417i	$-0.342737\pi$
$769$	729.320	0.948401	0.474200	+	0.880417i	$0.342737\pi$
$770$	− 275.640i	− 0.357974i
$771$	0	0
$772$	−268.000	−0.347150
$773$	− 434.559i	− 0.562172i	−0.959683	−	0.281086i	$-0.909305\pi$
$773$	− 434.559i	− 0.562172i	0.959683	−	0.281086i	$-0.0906947\pi$
$774$	0	0
$775$	−297.814	−0.384277
$776$	− 532.913i	− 0.686743i
$777$	0	0
$778$	−756.834	−0.972794
$779$	− 1212.67i	− 1.55671i
$780$	0	0
$781$	−147.336	−0.188650
$782$	187.135i	0.239303i
$783$	0	0
$784$	28.0000	0.0357143
$785$	1122.76i	1.43027i
$786$	0	0
$787$	−15.3517	−0.0195066	−0.00975331	−	0.999952i	$-0.503105\pi$
$787$	−15.3517	−0.0195066	−0.00975331	+	0.999952i	$0.503105\pi$
$788$	377.120i	0.478579i
$789$	0	0
$790$	−288.972	−0.365788
$791$	75.4543i	0.0953910i
$792$	0	0
$793$	−291.158	−0.367160
$794$	133.394i	0.168003i
$795$	0	0
$796$	−204.988	−0.257523
$797$	− 1043.48i	− 1.30927i	−0.755947	−	0.654633i	$-0.772823\pi$
$797$	− 1043.48i	− 1.30927i	0.755947	−	0.654633i	$-0.227177\pi$
$798$	0	0
$799$	−185.004	−0.231544
$800$	66.9432i	0.0836789i
$801$	0	0
$802$	−146.502	−0.182671
$803$	933.610i	1.16265i
$804$	0	0
$805$	194.907	0.242121
$806$	661.371i	0.820560i
$807$	0	0
$808$	−301.830	−0.373552
$809$	1041.31i	1.28716i	0.765378	+	0.643581i	$0.222552\pi$
$809$	1041.31i	1.28716i	−0.765378	+	0.643581i	$0.777448\pi$
$810$	0	0
$811$	502.316	0.619379	0.309689	−	0.950838i	$-0.399775\pi$
$811$	502.316	0.619379	0.309689	+	0.950838i	$0.399775\pi$
$812$	221.379i	0.272634i
$813$	0	0
$814$	−652.308	−0.801362
$815$	− 527.988i	− 0.647838i
$816$	0	0
$817$	1669.96	2.04402
$818$	13.7983i	0.0168684i
$819$	0	0
$820$	735.984	0.897542
$821$	− 23.1137i	− 0.0281531i	−0.999901	−	0.0140765i	$-0.995519\pi$
$821$	− 23.1137i	− 0.0281531i	0.999901	−	0.0140765i	$-0.00448085\pi$
$822$	0	0
$823$	−600.664	−0.729847	−0.364923	−	0.931038i	$-0.618905\pi$
$823$	−600.664	−0.729847	−0.364923	+	0.931038i	$0.618905\pi$
$824$	371.933i	0.451375i
$825$	0	0
$826$	217.830	0.263717
$827$	− 1309.21i	− 1.58308i	−0.611118	−	0.791540i	$-0.709280\pi$
$827$	− 1309.21i	− 1.58308i	0.611118	−	0.791540i	$-0.290720\pi$
$828$	0	0
$829$	−621.919	−0.750204	−0.375102	−	0.926984i	$-0.622392\pi$
$829$	−621.919	−0.750204	−0.375102	+	0.926984i	$0.622392\pi$
$830$	519.928i	0.626419i
$831$	0	0
$832$	148.664	0.178683
$833$	− 76.3102i	− 0.0916089i
$834$	0	0
$835$	367.644	0.440293
$836$	485.528i	0.580775i
$837$	0	0
$838$	480.000	0.572792
$839$	1190.30i	1.41871i	0.704851	+	0.709355i	$0.251014\pi$
$839$	1190.30i	1.41871i	−0.704851	+	0.709355i	$0.748986\pi$
$840$	0	0
$841$	−909.308	−1.08122
$842$	847.567i	1.00661i
$843$	0	0
$844$	169.004	0.200242
$845$	− 1070.15i	− 1.26645i
$846$	0	0
$847$	69.6784	0.0822649
$848$	376.170i	0.443596i
$849$	0	0
$850$	182.445	0.214641
$851$	− 461.252i	− 0.542011i
$852$	0	0
$853$	137.012	0.160623	0.0803117	−	0.996770i	$-0.474408\pi$
$853$	137.012	0.160623	0.0803117	+	0.996770i	$0.474408\pi$
$854$	58.6242i	0.0686466i
$855$	0	0
$856$	−233.004	−0.272201
$857$	466.141i	0.543922i	0.962308	+	0.271961i	$0.0876722\pi$
$857$	466.141i	0.543922i	−0.962308	+	0.271961i	$0.912328\pi$
$858$	0	0
$859$	23.9843	0.0279211	0.0139606	−	0.999903i	$-0.495556\pi$
$859$	23.9843	0.0279211	0.0139606	+	0.999903i	$0.495556\pi$
$860$	1013.52i	1.17851i
$861$	0	0
$862$	1005.47	1.16644
$863$	0.114603i	0 0.000132796i	1.00000		6.63982e-5i	$2.11352e-5\pi$
$863$	0.114603i	0 0.000132796i	−1.00000		6.63982e-5i	$0.999979\pi$
$864$	0	0
$865$	986.664	1.14065
$866$	− 534.550i	− 0.617264i
$867$	0	0
$868$	133.166	0.153417
$869$	− 408.669i	− 0.470275i
$870$	0	0
$871$	2465.30	2.83042
$872$	95.6858i	0.109731i
$873$	0	0
$874$	−343.320	−0.392815
$875$	211.411i	0.241613i
$876$	0	0
$877$	997.304	1.13718	0.568589	−	0.822622i	$-0.307490\pi$
$877$	997.304	1.13718	0.568589	+	0.822622i	$0.307490\pi$
$878$	− 746.912i	− 0.850697i
$879$	0	0
$880$	−294.672	−0.334854
$881$	935.649i	1.06203i	0.847362	+	0.531015i	$0.178189\pi$
$881$	935.649i	1.06203i	−0.847362	+	0.531015i	$0.821811\pi$
$882$	0	0
$883$	−1549.47	−1.75478	−0.877392	−	0.479774i	$-0.840719\pi$
$883$	−1549.47	−1.75478	−0.877392	+	0.479774i	$0.840719\pi$
$884$	− 405.164i	− 0.458330i
$885$	0	0
$886$	51.8222	0.0584900
$887$	− 894.493i	− 1.00845i	−0.863573	−	0.504224i	$-0.831779\pi$
$887$	− 894.493i	− 1.00845i	0.863573	−	0.504224i	$-0.168221\pi$
$888$	0	0
$889$	−343.069	−0.385905
$890$	40.9844i	0.0460498i
$891$	0	0
$892$	316.988	0.355368
$893$	− 339.411i	− 0.380080i
$894$	0	0
$895$	1353.96	1.51281
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−562.308	−0.626179
$899$	1052.86i	1.17115i
$900$	0	0
$901$	1025.20	1.13785
$902$	1040.84i	1.15392i
$903$	0	0
$904$	80.6640	0.0892301
$905$	− 1146.54i	− 1.26690i
$906$	0	0
$907$	−135.838	−0.149766	−0.0748831	−	0.997192i	$-0.523858\pi$
$907$	−135.838	−0.149766	−0.0748831	+	0.997192i	$0.523858\pi$
$908$	− 203.647i	− 0.224281i
$909$	0	0
$910$	−421.992	−0.463728
$911$	− 1242.01i	− 1.36335i	−0.731655	−	0.681675i	$-0.761252\pi$
$911$	− 1242.01i	− 1.36335i	0.731655	−	0.681675i	$-0.238748\pi$
$912$	0	0
$913$	−735.289	−0.805355
$914$	− 486.948i	− 0.532766i
$915$	0	0
$916$	537.830	0.587151
$917$	391.617i	0.427063i
$918$	0	0
$919$	−388.162	−0.422374	−0.211187	−	0.977446i	$-0.567733\pi$
$919$	−388.162	−0.422374	−0.211187	+	0.977446i	$0.567733\pi$
$920$	− 208.365i	− 0.226483i
$921$	0	0
$922$	−524.583	−0.568962
$923$	225.564i	0.244382i
$924$	0	0
$925$	−449.692	−0.486153
$926$	− 110.778i	− 0.119631i
$927$	0	0
$928$	236.664	0.255026
$929$	− 621.694i	− 0.669207i	−0.942359	−	0.334604i	$-0.891398\pi$
$929$	− 621.694i	− 0.669207i	0.942359	−	0.334604i	$-0.108602\pi$
$930$	0	0
$931$	140.000	0.150376
$932$	52.5495i	0.0563836i
$933$	0	0
$934$	−565.344	−0.605293
$935$	803.089i	0.858918i
$936$	0	0
$937$	1262.00	1.34685	0.673426	−	0.739255i	$-0.264822\pi$
$937$	1262.00	1.34685	0.673426	+	0.739255i	$0.264822\pi$
$938$	− 496.383i	− 0.529193i
$939$	0	0
$940$	205.992	0.219141
$941$	672.410i	0.714569i	0.933996	+	0.357285i	$0.116297\pi$
$941$	672.410i	0.714569i	−0.933996	+	0.357285i	$0.883703\pi$
$942$	0	0
$943$	−735.984	−0.780471
$944$	− 232.870i	− 0.246684i
$945$	0	0
$946$	−1433.33	−1.51515
$947$	1159.75i	1.22465i	0.790605	+	0.612327i	$0.209766\pi$
$947$	1159.75i	1.22465i	−0.790605	+	0.612327i	$0.790234\pi$
$948$	0	0
$949$	1429.31	1.50612
$950$	334.716i	0.352332i
$951$	0	0
$952$	−81.5791	−0.0856923
$953$	− 163.104i	− 0.171148i	−0.996332	−	0.0855740i	$-0.972728\pi$
$953$	− 163.104i	− 0.171148i	0.996332	−	0.0855740i	$-0.0272724\pi$
$954$	0	0
$955$	−1383.98	−1.44920
$956$	− 184.547i	− 0.193040i
$957$	0	0
$958$	−994.656	−1.03826
$959$	203.610i	0.212315i
$960$	0	0
$961$	−327.672	−0.340970
$962$	998.653i	1.03810i
$963$	0	0
$964$	−686.494	−0.712131
$965$	− 813.260i	− 0.842756i
$966$	0	0
$967$	887.012	0.917282	0.458641	−	0.888622i	$-0.348336\pi$
$967$	887.012	0.917282	0.458641	+	0.888622i	$0.348336\pi$
$968$	− 74.4893i	− 0.0769518i
$969$	0	0
$970$	1617.15	1.66717
$971$	− 1416.32i	− 1.45862i	−0.684183	−	0.729310i	$-0.739841\pi$
$971$	− 1416.32i	− 1.45862i	0.684183	−	0.729310i	$-0.260159\pi$
$972$	0	0
$973$	−574.996	−0.590952
$974$	116.687i	0.119801i
$975$	0	0
$976$	62.6719	0.0642130
$977$	339.051i	0.347032i	0.984831	+	0.173516i	$0.0555129\pi$
$977$	339.051i	0.347032i	−0.984831	+	0.173516i	$0.944487\pi$
$978$	0	0
$979$	−57.9606	−0.0592039
$980$	84.9674i	0.0867014i
$981$	0	0
$982$	−260.502	−0.265277
$983$	− 487.887i	− 0.496324i	−0.968718	−	0.248162i	$-0.920173\pi$
$983$	− 487.887i	− 0.496324i	0.968718	−	0.248162i	$-0.0798266\pi$
$984$	0	0
$985$	−1144.39	−1.16182
$986$	− 644.996i	− 0.654154i
$987$	0	0
$988$	743.320	0.752348
$989$	− 1013.52i	− 1.02479i
$990$	0	0
$991$	−937.474	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$991$	−937.474	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$992$	− 142.360i	− 0.143509i
$993$	0	0
$994$	45.4170	0.0456911
$995$	− 622.047i	− 0.625173i
$996$	0	0
$997$	461.012	0.462399	0.231200	−	0.972906i	$-0.425735\pi$
$997$	461.012	0.462399	0.231200	+	0.972906i	$0.425735\pi$
$998$	− 1064.63i	− 1.06677i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.b.a.71.1	✓	4
3.2	odd	2	inner	126.3.b.a.71.4	yes	4
4.3	odd	2		1008.3.d.a.449.2		4
5.2	odd	4		3150.3.c.b.449.6		8
5.3	odd	4		3150.3.c.b.449.4		8
5.4	even	2		3150.3.e.e.701.4		4
7.2	even	3		882.3.s.e.557.1		8
7.3	odd	6		882.3.s.i.863.3		8
7.4	even	3		882.3.s.e.863.4		8
7.5	odd	6		882.3.s.i.557.2		8
7.6	odd	2		882.3.b.f.197.2		4
8.3	odd	2		4032.3.d.j.449.3		4
8.5	even	2		4032.3.d.i.449.3		4
9.2	odd	6		1134.3.q.c.701.3		8
9.4	even	3		1134.3.q.c.1079.3		8
9.5	odd	6		1134.3.q.c.1079.2		8
9.7	even	3		1134.3.q.c.701.2		8
12.11	even	2		1008.3.d.a.449.3		4
15.2	even	4		3150.3.c.b.449.1		8
15.8	even	4		3150.3.c.b.449.7		8
15.14	odd	2		3150.3.e.e.701.2		4
21.2	odd	6		882.3.s.e.557.4		8
21.5	even	6		882.3.s.i.557.3		8
21.11	odd	6		882.3.s.e.863.1		8
21.17	even	6		882.3.s.i.863.2		8
21.20	even	2		882.3.b.f.197.3		4
24.5	odd	2		4032.3.d.i.449.2		4
24.11	even	2		4032.3.d.j.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.b.a.71.1	✓	4	1.1	even	1	trivial
126.3.b.a.71.4	yes	4	3.2	odd	2	inner
882.3.b.f.197.2		4	7.6	odd	2
882.3.b.f.197.3		4	21.20	even	2
882.3.s.e.557.1		8	7.2	even	3
882.3.s.e.557.4		8	21.2	odd	6
882.3.s.e.863.1		8	21.11	odd	6
882.3.s.e.863.4		8	7.4	even	3
882.3.s.i.557.2		8	7.5	odd	6
882.3.s.i.557.3		8	21.5	even	6
882.3.s.i.863.2		8	21.17	even	6
882.3.s.i.863.3		8	7.3	odd	6
1008.3.d.a.449.2		4	4.3	odd	2
1008.3.d.a.449.3		4	12.11	even	2
1134.3.q.c.701.2		8	9.7	even	3
1134.3.q.c.701.3		8	9.2	odd	6
1134.3.q.c.1079.2		8	9.5	odd	6
1134.3.q.c.1079.3		8	9.4	even	3
3150.3.c.b.449.1		8	15.2	even	4
3150.3.c.b.449.4		8	5.3	odd	4
3150.3.c.b.449.6		8	5.2	odd	4
3150.3.c.b.449.7		8	15.8	even	4
3150.3.e.e.701.2		4	15.14	odd	2
3150.3.e.e.701.4		4	5.4	even	2
4032.3.d.i.449.2		4	24.5	odd	2
4032.3.d.i.449.3		4	8.5	even	2
4032.3.d.j.449.2		4	24.11	even	2
4032.3.d.j.449.3		4	8.3	odd	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -6.06910i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -6.06910i q^{5} -2.64575 q^{7} +2.82843i q^{8} -8.58301 q^{10} -12.1382i q^{11} -18.5830 q^{13} +3.74166i q^{14} +4.00000 q^{16} -10.9015i q^{17} +20.0000 q^{19} +12.1382i q^{20} -17.1660 q^{22} -12.1382i q^{23} -11.8340 q^{25} +26.2803i q^{26} +5.29150 q^{28} +41.8367i q^{29} +25.1660 q^{31} -5.65685i q^{32} -15.4170 q^{34} +16.0573i q^{35} +38.0000 q^{37} -28.2843i q^{38} +17.1660 q^{40} -60.6337i q^{41} +83.4980 q^{43} +24.2764i q^{44} -17.1660 q^{46} -16.9706i q^{47} +7.00000 q^{49} +16.7358i q^{50} +37.1660 q^{52} +94.0424i q^{53} -73.6680 q^{55} -7.48331i q^{56} +59.1660 q^{58} -58.2175i q^{59} +15.6680 q^{61} -35.5901i q^{62} -8.00000 q^{64} +112.782i q^{65} -132.664 q^{67} +21.8029i q^{68} +22.7085 q^{70} -12.1382i q^{71} -76.9150 q^{73} -53.7401i q^{74} -40.0000 q^{76} +32.1147i q^{77} +33.6680 q^{79} -24.2764i q^{80} -85.7490 q^{82} -60.5764i q^{83} -66.1621 q^{85} -118.084i q^{86} +34.3320 q^{88} -4.77506i q^{89} +49.1660 q^{91} +24.2764i q^{92} -24.0000 q^{94} -121.382i q^{95} -188.413 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 8 q^{10} - 32 q^{13} + 16 q^{16} + 80 q^{19} + 16 q^{22} - 132 q^{25} + 16 q^{31} - 104 q^{34} + 152 q^{37} - 16 q^{40} + 80 q^{43} + 16 q^{46} + 28 q^{49} + 64 q^{52} - 464 q^{55} + 152 q^{58} + 232 q^{61} - 32 q^{64} - 192 q^{67} + 112 q^{70} - 96 q^{73} - 160 q^{76} + 304 q^{79} - 216 q^{82} + 328 q^{85} - 32 q^{88} + 112 q^{91} - 96 q^{94} - 288 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 8 * q^10 - 32 * q^13 + 16 * q^16 + 80 * q^19 + 16 * q^22 - 132 * q^25 + 16 * q^31 - 104 * q^34 + 152 * q^37 - 16 * q^40 + 80 * q^43 + 16 * q^46 + 28 * q^49 + 64 * q^52 - 464 * q^55 + 152 * q^58 + 232 * q^61 - 32 * q^64 - 192 * q^67 + 112 * q^70 - 96 * q^73 - 160 * q^76 + 304 * q^79 - 216 * q^82 + 328 * q^85 - 32 * q^88 + 112 * q^91 - 96 * q^94 - 288 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more