LMFDB - Embedded newform 1344.4.c.e.673.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 $ x^12 + 386x^10 + 54793x^8 + 3447408x^6 + 90154296x^4 + 707138208*x^2 + 525876624
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.4
Root			$11.2296i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.e.673.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +6.85537i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +6.85537i q^{5} -7.00000 q^{7} -9.00000 q^{9} +30.4375i q^{11} -45.1351i q^{13} +20.5661 q^{15} -99.4408 q^{17} -33.9662i q^{19} +21.0000i q^{21} -108.378 q^{23} +78.0039 q^{25} +27.0000i q^{27} +187.588i q^{29} +174.390 q^{31} +91.3124 q^{33} -47.9876i q^{35} -241.634i q^{37} -135.405 q^{39} +474.707 q^{41} -480.979i q^{43} -61.6983i q^{45} -516.624 q^{47} +49.0000 q^{49} +298.322i q^{51} -179.948i q^{53} -208.660 q^{55} -101.899 q^{57} +22.3098i q^{59} +932.621i q^{61} +63.0000 q^{63} +309.418 q^{65} +665.460i q^{67} +325.134i q^{69} +129.571 q^{71} +1089.47 q^{73} -234.012i q^{75} -213.062i q^{77} +739.469 q^{79} +81.0000 q^{81} +81.6835i q^{83} -681.703i q^{85} +562.765 q^{87} -1010.60 q^{89} +315.946i q^{91} -523.169i q^{93} +232.851 q^{95} +610.737 q^{97} -273.937i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	6.85537i	0.613163i	0.951844	+	0.306581i	$0.0991852\pi$
$5$	6.85537i	0.613163i	−0.951844	+	0.306581i	$0.900815\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	30.4375i	0.834295i	0.908839	+	0.417147i	$0.136970\pi$
$11$	30.4375i	0.834295i	−0.908839	+	0.417147i	$0.863030\pi$
$12$	0	0
$13$	− 45.1351i	− 0.962941i	−0.876462	−	0.481470i	$-0.840103\pi$
$13$	− 45.1351i	− 0.962941i	0.876462	−	0.481470i	$-0.159897\pi$
$14$	0	0
$15$	20.5661	0.354010
$16$	0	0
$17$	−99.4408	−1.41870	−0.709351	−	0.704855i	$-0.751012\pi$
$17$	−99.4408	−1.41870	−0.709351	+	0.704855i	$0.751012\pi$
$18$	0	0
$19$	− 33.9662i	− 0.410125i	−0.978749	−	0.205063i	$-0.934260\pi$
$19$	− 33.9662i	− 0.410125i	0.978749	−	0.205063i	$-0.0657398\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−108.378	−0.982539	−0.491270	−	0.871008i	$-0.663467\pi$
$23$	−108.378	−0.982539	−0.491270	+	0.871008i	$0.663467\pi$
$24$	0	0
$25$	78.0039	0.624031
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	187.588i	1.20118i	0.799557	+	0.600590i	$0.205068\pi$
$29$	187.588i	1.20118i	−0.799557	+	0.600590i	$0.794932\pi$
$30$	0	0
$31$	174.390	1.01037	0.505183	−	0.863012i	$-0.331425\pi$
$31$	174.390	1.01037	0.505183	+	0.863012i	$0.331425\pi$
$32$	0	0
$33$	91.3124	0.481680
$34$	0	0
$35$	− 47.9876i	− 0.231754i
$36$	0	0
$37$	− 241.634i	− 1.07363i	−0.843699	−	0.536817i	$-0.819627\pi$
$37$	− 241.634i	− 1.07363i	0.843699	−	0.536817i	$-0.180373\pi$
$38$	0	0
$39$	−135.405	−0.555954
$40$	0	0
$41$	474.707	1.80821	0.904107	−	0.427307i	$-0.140538\pi$
$41$	474.707	1.80821	0.904107	+	0.427307i	$0.140538\pi$
$42$	0	0
$43$	− 480.979i	− 1.70578i	−0.522090	−	0.852890i	$-0.674847\pi$
$43$	− 480.979i	− 1.70578i	0.522090	−	0.852890i	$-0.325153\pi$
$44$	0	0
$45$	− 61.6983i	− 0.204388i
$46$	0	0
$47$	−516.624	−1.60335	−0.801673	−	0.597762i	$-0.796057\pi$
$47$	−516.624	−1.60335	−0.801673	+	0.597762i	$0.796057\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	298.322i	0.819088i
$52$	0	0
$53$	− 179.948i	− 0.466374i	−0.972432	−	0.233187i	$-0.925085\pi$
$53$	− 179.948i	− 0.466374i	0.972432	−	0.233187i	$-0.0749154\pi$
$54$	0	0
$55$	−208.660	−0.511558
$56$	0	0
$57$	−101.899	−0.236786
$58$	0	0
$59$	22.3098i	0.0492287i	0.999697	+	0.0246143i	$0.00783578\pi$
$59$	22.3098i	0.0492287i	−0.999697	+	0.0246143i	$0.992164\pi$
$60$	0	0
$61$	932.621i	1.95754i	0.204963	+	0.978770i	$0.434292\pi$
$61$	932.621i	1.95754i	−0.204963	+	0.978770i	$0.565708\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	309.418	0.590439
$66$	0	0
$67$	665.460i	1.21342i	0.794925	+	0.606708i	$0.207510\pi$
$67$	665.460i	1.21342i	−0.794925	+	0.606708i	$0.792490\pi$
$68$	0	0
$69$	325.134i	0.567269i
$70$	0	0
$71$	129.571	0.216581	0.108291	−	0.994119i	$-0.465462\pi$
$71$	129.571	0.216581	0.108291	+	0.994119i	$0.465462\pi$
$72$	0	0
$73$	1089.47	1.74675	0.873377	−	0.487044i	$-0.161925\pi$
$73$	1089.47	1.74675	0.873377	+	0.487044i	$0.161925\pi$
$74$	0	0
$75$	− 234.012i	− 0.360285i
$76$	0	0
$77$	− 213.062i	− 0.315334i
$78$	0	0
$79$	739.469	1.05312	0.526562	−	0.850137i	$-0.323481\pi$
$79$	739.469	1.05312	0.526562	+	0.850137i	$0.323481\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	81.6835i	0.108023i	0.998540	+	0.0540116i	$0.0172008\pi$
$83$	81.6835i	0.108023i	−0.998540	+	0.0540116i	$0.982799\pi$
$84$	0	0
$85$	− 681.703i	− 0.869895i
$86$	0	0
$87$	562.765	0.693502
$88$	0	0
$89$	−1010.60	−1.20363	−0.601817	−	0.798634i	$-0.705556\pi$
$89$	−1010.60	−1.20363	−0.601817	+	0.798634i	$0.705556\pi$
$90$	0	0
$91$	315.946i	0.363957i
$92$	0	0
$93$	− 523.169i	− 0.583335i
$94$	0	0
$95$	232.851	0.251474
$96$	0	0
$97$	610.737	0.639288	0.319644	−	0.947538i	$-0.396437\pi$
$97$	610.737	0.639288	0.319644	+	0.947538i	$0.396437\pi$
$98$	0	0
$99$	− 273.937i	− 0.278098i
$100$	0	0
$101$	1306.93i	1.28757i	0.765206	+	0.643786i	$0.222637\pi$
$101$	1306.93i	1.28757i	−0.765206	+	0.643786i	$0.777363\pi$
$102$	0	0
$103$	1547.43	1.48032	0.740159	−	0.672432i	$-0.234750\pi$
$103$	1547.43	1.48032	0.740159	+	0.672432i	$0.234750\pi$
$104$	0	0
$105$	−143.963	−0.133803
$106$	0	0
$107$	− 1830.60i	− 1.65393i	−0.562251	−	0.826967i	$-0.690065\pi$
$107$	− 1830.60i	− 1.65393i	0.562251	−	0.826967i	$-0.309935\pi$
$108$	0	0
$109$	967.351i	0.850050i	0.905182	+	0.425025i	$0.139735\pi$
$109$	967.351i	0.850050i	−0.905182	+	0.425025i	$0.860265\pi$
$110$	0	0
$111$	−724.903	−0.619863
$112$	0	0
$113$	1885.99	1.57008	0.785039	−	0.619447i	$-0.212643\pi$
$113$	1885.99	1.57008	0.785039	+	0.619447i	$0.212643\pi$
$114$	0	0
$115$	− 742.972i	− 0.602456i
$116$	0	0
$117$	406.216i	0.320980i
$118$	0	0
$119$	696.086	0.536219
$120$	0	0
$121$	404.560	0.303952
$122$	0	0
$123$	− 1424.12i	− 1.04397i
$124$	0	0
$125$	1391.67i	0.995796i
$126$	0	0
$127$	−646.747	−0.451886	−0.225943	−	0.974141i	$-0.572546\pi$
$127$	−646.747	−0.451886	−0.225943	+	0.974141i	$0.572546\pi$
$128$	0	0
$129$	−1442.94	−0.984833
$130$	0	0
$131$	− 256.204i	− 0.170875i	−0.996344	−	0.0854375i	$-0.972771\pi$
$131$	− 256.204i	− 0.170875i	0.996344	−	0.0854375i	$-0.0272288\pi$
$132$	0	0
$133$	237.763i	0.155013i
$134$	0	0
$135$	−185.095	−0.118003
$136$	0	0
$137$	1886.18	1.17626	0.588130	−	0.808767i	$-0.299864\pi$
$137$	1886.18	1.17626	0.588130	+	0.808767i	$0.299864\pi$
$138$	0	0
$139$	611.918i	0.373397i	0.982417	+	0.186699i	$0.0597788\pi$
$139$	611.918i	0.373397i	−0.982417	+	0.186699i	$0.940221\pi$
$140$	0	0
$141$	1549.87i	0.925693i
$142$	0	0
$143$	1373.80	0.803377
$144$	0	0
$145$	−1285.99	−0.736519
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	− 1251.17i	− 0.687916i	−0.938985	−	0.343958i	$-0.888232\pi$
$149$	− 1251.17i	− 0.687916i	0.938985	−	0.343958i	$-0.111768\pi$
$150$	0	0
$151$	428.764	0.231075	0.115537	−	0.993303i	$-0.463141\pi$
$151$	428.764	0.231075	0.115537	+	0.993303i	$0.463141\pi$
$152$	0	0
$153$	894.967	0.472901
$154$	0	0
$155$	1195.51i	0.619519i
$156$	0	0
$157$	− 1699.44i	− 0.863887i	−0.901901	−	0.431944i	$-0.857828\pi$
$157$	− 1699.44i	− 0.863887i	0.901901	−	0.431944i	$-0.142172\pi$
$158$	0	0
$159$	−539.845	−0.269261
$160$	0	0
$161$	758.647	0.371365
$162$	0	0
$163$	− 2865.29i	− 1.37685i	−0.725308	−	0.688425i	$-0.758303\pi$
$163$	− 2865.29i	− 1.37685i	0.725308	−	0.688425i	$-0.241697\pi$
$164$	0	0
$165$	625.980i	0.295348i
$166$	0	0
$167$	−1.12650	−0.000521985 0	−0.000260992	−	1.00000i	$-0.500083\pi$
$167$	−1.12650	−0.000521985 0	−0.000260992		1.00000i	$0.500083\pi$
$168$	0	0
$169$	159.820	0.0727448
$170$	0	0
$171$	305.696i	0.136708i
$172$	0	0
$173$	140.487i	0.0617401i	0.999523	+	0.0308701i	$0.00982781\pi$
$173$	140.487i	0.0617401i	−0.999523	+	0.0308701i	$0.990172\pi$
$174$	0	0
$175$	−546.028	−0.235862
$176$	0	0
$177$	66.9295	0.0284222
$178$	0	0
$179$	− 120.665i	− 0.0503850i	−0.999683	−	0.0251925i	$-0.991980\pi$
$179$	− 120.665i	− 0.0503850i	0.999683	−	0.0251925i	$-0.00801987\pi$
$180$	0	0
$181$	− 1440.25i	− 0.591451i	−0.955273	−	0.295725i	$-0.904439\pi$
$181$	− 1440.25i	− 0.591451i	0.955273	−	0.295725i	$-0.0955613\pi$
$182$	0	0
$183$	2797.86	1.13019
$184$	0	0
$185$	1656.49	0.658312
$186$	0	0
$187$	− 3026.73i	− 1.18362i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	−2921.14	−1.10663	−0.553316	−	0.832972i	$-0.686638\pi$
$191$	−2921.14	−1.10663	−0.553316	+	0.832972i	$0.686638\pi$
$192$	0	0
$193$	2889.35	1.07761	0.538807	−	0.842429i	$-0.318875\pi$
$193$	2889.35	1.07761	0.538807	+	0.842429i	$0.318875\pi$
$194$	0	0
$195$	− 928.254i	− 0.340890i
$196$	0	0
$197$	1935.70i	0.700067i	0.936737	+	0.350033i	$0.113830\pi$
$197$	1935.70i	0.700067i	−0.936737	+	0.350033i	$0.886170\pi$
$198$	0	0
$199$	1431.80	0.510040	0.255020	−	0.966936i	$-0.417918\pi$
$199$	1431.80	0.510040	0.255020	+	0.966936i	$0.417918\pi$
$200$	0	0
$201$	1996.38	0.700566
$202$	0	0
$203$	− 1313.12i	− 0.454004i
$204$	0	0
$205$	3254.29i	1.10873i
$206$	0	0
$207$	975.403	0.327513
$208$	0	0
$209$	1033.85	0.342165
$210$	0	0
$211$	− 2117.07i	− 0.690735i	−0.938468	−	0.345367i	$-0.887754\pi$
$211$	− 2117.07i	− 0.690735i	0.938468	−	0.345367i	$-0.112246\pi$
$212$	0	0
$213$	− 388.713i	− 0.125043i
$214$	0	0
$215$	3297.29	1.04592
$216$	0	0
$217$	−1220.73	−0.381882
$218$	0	0
$219$	− 3268.42i	− 1.00849i
$220$	0	0
$221$	4488.27i	1.36613i
$222$	0	0
$223$	4992.28	1.49914	0.749570	−	0.661925i	$-0.230260\pi$
$223$	4992.28	1.49914	0.749570	+	0.661925i	$0.230260\pi$
$224$	0	0
$225$	−702.035	−0.208010
$226$	0	0
$227$	− 3430.04i	− 1.00291i	−0.865185	−	0.501453i	$-0.832799\pi$
$227$	− 3430.04i	− 1.00291i	0.865185	−	0.501453i	$-0.167201\pi$
$228$	0	0
$229$	2303.45i	0.664699i	0.943156	+	0.332349i	$0.107841\pi$
$229$	2303.45i	0.664699i	−0.943156	+	0.332349i	$0.892159\pi$
$230$	0	0
$231$	−639.187	−0.182058
$232$	0	0
$233$	−1087.30	−0.305713	−0.152857	−	0.988248i	$-0.548847\pi$
$233$	−1087.30	−0.305713	−0.152857	+	0.988248i	$0.548847\pi$
$234$	0	0
$235$	− 3541.64i	− 0.983112i
$236$	0	0
$237$	− 2218.41i	− 0.608021i
$238$	0	0
$239$	2738.86	0.741263	0.370632	−	0.928780i	$-0.379141\pi$
$239$	2738.86	0.741263	0.370632	+	0.928780i	$0.379141\pi$
$240$	0	0
$241$	−2115.29	−0.565386	−0.282693	−	0.959210i	$-0.591228\pi$
$241$	−2115.29	−0.565386	−0.282693	+	0.959210i	$0.591228\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	335.913i	0.0875947i
$246$	0	0
$247$	−1533.07	−0.394926
$248$	0	0
$249$	245.051	0.0623673
$250$	0	0
$251$	− 3874.74i	− 0.974389i	−0.873294	−	0.487194i	$-0.838020\pi$
$251$	− 3874.74i	− 0.974389i	0.873294	−	0.487194i	$-0.161980\pi$
$252$	0	0
$253$	− 3298.76i	− 0.819727i
$254$	0	0
$255$	−2045.11	−0.502234
$256$	0	0
$257$	7877.72	1.91206	0.956029	−	0.293272i	$-0.0947441\pi$
$257$	7877.72	1.91206	0.956029	+	0.293272i	$0.0947441\pi$
$258$	0	0
$259$	1691.44i	0.405795i
$260$	0	0
$261$	− 1688.29i	− 0.400394i
$262$	0	0
$263$	−4426.08	−1.03773	−0.518867	−	0.854855i	$-0.673646\pi$
$263$	−4426.08	−1.03773	−0.518867	+	0.854855i	$0.673646\pi$
$264$	0	0
$265$	1233.61	0.285963
$266$	0	0
$267$	3031.80i	0.694918i
$268$	0	0
$269$	− 3411.18i	− 0.773171i	−0.922254	−	0.386586i	$-0.873654\pi$
$269$	− 3411.18i	− 0.773171i	0.922254	−	0.386586i	$-0.126346\pi$
$270$	0	0
$271$	5417.43	1.21434	0.607169	−	0.794573i	$-0.292305\pi$
$271$	5417.43	1.21434	0.607169	+	0.794573i	$0.292305\pi$
$272$	0	0
$273$	947.838	0.210131
$274$	0	0
$275$	2374.24i	0.520626i
$276$	0	0
$277$	− 5249.20i	− 1.13860i	−0.822128	−	0.569302i	$-0.807213\pi$
$277$	− 5249.20i	− 1.13860i	0.822128	−	0.569302i	$-0.192787\pi$
$278$	0	0
$279$	−1569.51	−0.336789
$280$	0	0
$281$	4172.91	0.885889	0.442944	−	0.896549i	$-0.353934\pi$
$281$	4172.91	0.885889	0.442944	+	0.896549i	$0.353934\pi$
$282$	0	0
$283$	− 7600.93i	− 1.59657i	−0.602282	−	0.798284i	$-0.705742\pi$
$283$	− 7600.93i	− 1.59657i	0.602282	−	0.798284i	$-0.294258\pi$
$284$	0	0
$285$	− 698.552i	− 0.145188i
$286$	0	0
$287$	−3322.95	−0.683440
$288$	0	0
$289$	4975.47	1.01272
$290$	0	0
$291$	− 1832.21i	− 0.369093i
$292$	0	0
$293$	5751.98i	1.14687i	0.819250	+	0.573437i	$0.194390\pi$
$293$	5751.98i	1.14687i	−0.819250	+	0.573437i	$0.805610\pi$
$294$	0	0
$295$	−152.942	−0.0301852
$296$	0	0
$297$	−821.812	−0.160560
$298$	0	0
$299$	4891.66i	0.946127i
$300$	0	0
$301$	3366.85i	0.644725i
$302$	0	0
$303$	3920.80	0.743380
$304$	0	0
$305$	−6393.46	−1.20029
$306$	0	0
$307$	948.107i	0.176258i	0.996109	+	0.0881292i	$0.0280888\pi$
$307$	948.107i	0.176258i	−0.996109	+	0.0881292i	$0.971911\pi$
$308$	0	0
$309$	− 4642.29i	− 0.854662i
$310$	0	0
$311$	4726.33	0.861754	0.430877	−	0.902411i	$-0.358204\pi$
$311$	4726.33	0.861754	0.430877	+	0.902411i	$0.358204\pi$
$312$	0	0
$313$	−5594.30	−1.01025	−0.505126	−	0.863046i	$-0.668554\pi$
$313$	−5594.30	−1.01025	−0.505126	+	0.863046i	$0.668554\pi$
$314$	0	0
$315$	431.888i	0.0772512i
$316$	0	0
$317$	726.746i	0.128764i	0.997925	+	0.0643819i	$0.0205076\pi$
$317$	726.746i	0.128764i	−0.997925	+	0.0643819i	$0.979492\pi$
$318$	0	0
$319$	−5709.71	−1.00214
$320$	0	0
$321$	−5491.80	−0.954899
$322$	0	0
$323$	3377.63i	0.581846i
$324$	0	0
$325$	− 3520.72i	− 0.600905i
$326$	0	0
$327$	2902.05	0.490776
$328$	0	0
$329$	3616.37	0.606008
$330$	0	0
$331$	− 1398.69i	− 0.232263i	−0.993234	−	0.116132i	$-0.962951\pi$
$331$	− 1398.69i	− 0.232263i	0.993234	−	0.116132i	$-0.0370495\pi$
$332$	0	0
$333$	2174.71i	0.357878i
$334$	0	0
$335$	−4561.97	−0.744021
$336$	0	0
$337$	6562.34	1.06075	0.530376	−	0.847763i	$-0.322051\pi$
$337$	6562.34	1.06075	0.530376	+	0.847763i	$0.322051\pi$
$338$	0	0
$339$	− 5657.96i	− 0.906484i
$340$	0	0
$341$	5307.98i	0.842943i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2228.92	−0.347828
$346$	0	0
$347$	9993.55i	1.54606i	0.634371	+	0.773029i	$0.281259\pi$
$347$	9993.55i	1.54606i	−0.634371	+	0.773029i	$0.718741\pi$
$348$	0	0
$349$	8816.69i	1.35228i	0.736772	+	0.676141i	$0.236349\pi$
$349$	8816.69i	1.35228i	−0.736772	+	0.676141i	$0.763651\pi$
$350$	0	0
$351$	1218.65	0.185318
$352$	0	0
$353$	3670.85	0.553483	0.276742	−	0.960944i	$-0.410745\pi$
$353$	3670.85	0.553483	0.276742	+	0.960944i	$0.410745\pi$
$354$	0	0
$355$	888.258i	0.132800i
$356$	0	0
$357$	− 2088.26i	− 0.309586i
$358$	0	0
$359$	3848.87	0.565837	0.282919	−	0.959144i	$-0.408697\pi$
$359$	3848.87	0.565837	0.282919	+	0.959144i	$0.408697\pi$
$360$	0	0
$361$	5705.30	0.831797
$362$	0	0
$363$	− 1213.68i	− 0.175487i
$364$	0	0
$365$	7468.73i	1.07104i
$366$	0	0
$367$	−8050.78	−1.14509	−0.572544	−	0.819874i	$-0.694043\pi$
$367$	−8050.78	−1.14509	−0.572544	+	0.819874i	$0.694043\pi$
$368$	0	0
$369$	−4272.36	−0.602738
$370$	0	0
$371$	1259.64i	0.176273i
$372$	0	0
$373$	6135.41i	0.851687i	0.904797	+	0.425844i	$0.140023\pi$
$373$	6135.41i	0.851687i	−0.904797	+	0.425844i	$0.859977\pi$
$374$	0	0
$375$	4175.00	0.574923
$376$	0	0
$377$	8466.82	1.15667
$378$	0	0
$379$	6362.25i	0.862287i	0.902283	+	0.431143i	$0.141890\pi$
$379$	6362.25i	0.862287i	−0.902283	+	0.431143i	$0.858110\pi$
$380$	0	0
$381$	1940.24i	0.260897i
$382$	0	0
$383$	−7858.77	−1.04847	−0.524235	−	0.851573i	$-0.675649\pi$
$383$	−7858.77	−1.04847	−0.524235	+	0.851573i	$0.675649\pi$
$384$	0	0
$385$	1460.62	0.193351
$386$	0	0
$387$	4328.81i	0.568594i
$388$	0	0
$389$	− 12766.7i	− 1.66400i	−0.554775	−	0.832000i	$-0.687196\pi$
$389$	− 12766.7i	− 1.66400i	0.554775	−	0.832000i	$-0.312804\pi$
$390$	0	0
$391$	10777.2	1.39393
$392$	0	0
$393$	−768.611	−0.0986547
$394$	0	0
$395$	5069.33i	0.645736i
$396$	0	0
$397$	5851.29i	0.739718i	0.929088	+	0.369859i	$0.120594\pi$
$397$	5851.29i	0.739718i	−0.929088	+	0.369859i	$0.879406\pi$
$398$	0	0
$399$	713.290	0.0894967
$400$	0	0
$401$	6575.76	0.818897	0.409449	−	0.912333i	$-0.365721\pi$
$401$	6575.76	0.818897	0.409449	+	0.912333i	$0.365721\pi$
$402$	0	0
$403$	− 7871.11i	− 0.972922i
$404$	0	0
$405$	555.285i	0.0681292i
$406$	0	0
$407$	7354.74	0.895727
$408$	0	0
$409$	−6998.06	−0.846043	−0.423022	−	0.906120i	$-0.639031\pi$
$409$	−6998.06	−0.846043	−0.423022	+	0.906120i	$0.639031\pi$
$410$	0	0
$411$	− 5658.55i	− 0.679114i
$412$	0	0
$413$	− 156.169i	− 0.0186067i
$414$	0	0
$415$	−559.970	−0.0662358
$416$	0	0
$417$	1835.75	0.215581
$418$	0	0
$419$	2137.83i	0.249260i	0.992203	+	0.124630i	$0.0397744\pi$
$419$	2137.83i	0.249260i	−0.992203	+	0.124630i	$0.960226\pi$
$420$	0	0
$421$	8455.41i	0.978840i	0.872048	+	0.489420i	$0.162792\pi$
$421$	8455.41i	0.978840i	−0.872048	+	0.489420i	$0.837208\pi$
$422$	0	0
$423$	4649.61	0.534449
$424$	0	0
$425$	−7756.77	−0.885315
$426$	0	0
$427$	− 6528.35i	− 0.739880i
$428$	0	0
$429$	− 4121.40i	− 0.463830i
$430$	0	0
$431$	−5623.37	−0.628464	−0.314232	−	0.949346i	$-0.601747\pi$
$431$	−5623.37	−0.628464	−0.314232	+	0.949346i	$0.601747\pi$
$432$	0	0
$433$	−11800.5	−1.30969	−0.654846	−	0.755763i	$-0.727266\pi$
$433$	−11800.5	−1.30969	−0.654846	+	0.755763i	$0.727266\pi$
$434$	0	0
$435$	3857.96i	0.425230i
$436$	0	0
$437$	3681.19i	0.402964i
$438$	0	0
$439$	−13405.3	−1.45740	−0.728702	−	0.684831i	$-0.759876\pi$
$439$	−13405.3	−1.45740	−0.728702	+	0.684831i	$0.759876\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	9275.61i	0.994803i	0.867520	+	0.497401i	$0.165712\pi$
$443$	9275.61i	0.994803i	−0.867520	+	0.497401i	$0.834288\pi$
$444$	0	0
$445$	− 6928.04i	− 0.738024i
$446$	0	0
$447$	−3753.50	−0.397168
$448$	0	0
$449$	3033.99	0.318893	0.159446	−	0.987207i	$-0.449029\pi$
$449$	3033.99	0.318893	0.159446	+	0.987207i	$0.449029\pi$
$450$	0	0
$451$	14448.9i	1.50858i
$452$	0	0
$453$	− 1286.29i	− 0.133411i
$454$	0	0
$455$	−2165.93	−0.223165
$456$	0	0
$457$	−18937.3	−1.93841	−0.969203	−	0.246264i	$-0.920797\pi$
$457$	−18937.3	−1.93841	−0.969203	+	0.246264i	$0.920797\pi$
$458$	0	0
$459$	− 2684.90i	− 0.273029i
$460$	0	0
$461$	1583.51i	0.159981i	0.996796	+	0.0799906i	$0.0254890\pi$
$461$	1583.51i	0.159981i	−0.996796	+	0.0799906i	$0.974511\pi$
$462$	0	0
$463$	−1288.77	−0.129361	−0.0646807	−	0.997906i	$-0.520603\pi$
$463$	−1288.77	−0.129361	−0.0646807	+	0.997906i	$0.520603\pi$
$464$	0	0
$465$	3586.52	0.357679
$466$	0	0
$467$	5756.85i	0.570439i	0.958462	+	0.285220i	$0.0920666\pi$
$467$	5756.85i	0.570439i	−0.958462	+	0.285220i	$0.907933\pi$
$468$	0	0
$469$	− 4658.22i	− 0.458628i
$470$	0	0
$471$	−5098.33	−0.498766
$472$	0	0
$473$	14639.8	1.42312
$474$	0	0
$475$	− 2649.50i	− 0.255931i
$476$	0	0
$477$	1619.53i	0.155458i
$478$	0	0
$479$	−4549.67	−0.433987	−0.216993	−	0.976173i	$-0.569625\pi$
$479$	−4549.67	−0.433987	−0.216993	+	0.976173i	$0.569625\pi$
$480$	0	0
$481$	−10906.2	−1.03385
$482$	0	0
$483$	− 2275.94i	− 0.214408i
$484$	0	0
$485$	4186.82i	0.391987i
$486$	0	0
$487$	17106.0	1.59168	0.795840	−	0.605507i	$-0.207030\pi$
$487$	17106.0	1.59168	0.795840	+	0.605507i	$0.207030\pi$
$488$	0	0
$489$	−8595.86	−0.794925
$490$	0	0
$491$	13113.2i	1.20528i	0.798013	+	0.602640i	$0.205884\pi$
$491$	13113.2i	1.20528i	−0.798013	+	0.602640i	$0.794116\pi$
$492$	0	0
$493$	− 18653.9i	− 1.70412i
$494$	0	0
$495$	1877.94	0.170519
$496$	0	0
$497$	−906.998	−0.0818600
$498$	0	0
$499$	− 16356.0i	− 1.46733i	−0.679513	−	0.733663i	$-0.737809\pi$
$499$	− 16356.0i	− 1.46733i	0.679513	−	0.733663i	$-0.262191\pi$
$500$	0	0
$501$	3.37951i	0 0.000301368i
$502$	0	0
$503$	−9257.49	−0.820618	−0.410309	−	0.911947i	$-0.634579\pi$
$503$	−9257.49	−0.820618	−0.410309	+	0.911947i	$0.634579\pi$
$504$	0	0
$505$	−8959.50	−0.789491
$506$	0	0
$507$	− 479.461i	− 0.0419992i
$508$	0	0
$509$	7477.03i	0.651107i	0.945524	+	0.325554i	$0.105551\pi$
$509$	7477.03i	0.651107i	−0.945524	+	0.325554i	$0.894449\pi$
$510$	0	0
$511$	−7626.31	−0.660211
$512$	0	0
$513$	917.087	0.0789286
$514$	0	0
$515$	10608.2i	0.907676i
$516$	0	0
$517$	− 15724.7i	− 1.33766i
$518$	0	0
$519$	421.462	0.0356457
$520$	0	0
$521$	13165.9	1.10711	0.553557	−	0.832811i	$-0.313270\pi$
$521$	13165.9	1.10711	0.553557	+	0.832811i	$0.313270\pi$
$522$	0	0
$523$	6182.33i	0.516892i	0.966026	+	0.258446i	$0.0832103\pi$
$523$	6182.33i	0.516892i	−0.966026	+	0.258446i	$0.916790\pi$
$524$	0	0
$525$	1638.08i	0.136175i
$526$	0	0
$527$	−17341.5	−1.43341
$528$	0	0
$529$	−421.185	−0.0346170
$530$	0	0
$531$	− 200.788i	− 0.0164096i
$532$	0	0
$533$	− 21426.0i	− 1.74120i
$534$	0	0
$535$	12549.4	1.01413
$536$	0	0
$537$	−361.994	−0.0290898
$538$	0	0
$539$	1491.44i	0.119185i
$540$	0	0
$541$	− 16381.3i	− 1.30183i	−0.759152	−	0.650913i	$-0.774386\pi$
$541$	− 16381.3i	− 1.30183i	0.759152	−	0.650913i	$-0.225614\pi$
$542$	0	0
$543$	−4320.74	−0.341474
$544$	0	0
$545$	−6631.55	−0.521219
$546$	0	0
$547$	− 7397.09i	− 0.578203i	−0.957298	−	0.289101i	$-0.906644\pi$
$547$	− 7397.09i	− 0.578203i	0.957298	−	0.289101i	$-0.0933565\pi$
$548$	0	0
$549$	− 8393.59i	− 0.652513i
$550$	0	0
$551$	6371.66	0.492635
$552$	0	0
$553$	−5176.28	−0.398043
$554$	0	0
$555$	− 4969.48i	− 0.380077i
$556$	0	0
$557$	− 7579.32i	− 0.576564i	−0.957546	−	0.288282i	$-0.906916\pi$
$557$	− 7579.32i	− 0.576564i	0.957546	−	0.288282i	$-0.0930841\pi$
$558$	0	0
$559$	−21709.0	−1.64257
$560$	0	0
$561$	−9080.18	−0.683361
$562$	0	0
$563$	− 10639.0i	− 0.796412i	−0.917296	−	0.398206i	$-0.869633\pi$
$563$	− 10639.0i	− 0.796412i	0.917296	−	0.398206i	$-0.130367\pi$
$564$	0	0
$565$	12929.1i	0.962713i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	923.880	0.0680687	0.0340343	−	0.999421i	$-0.489164\pi$
$569$	923.880	0.0680687	0.0340343	+	0.999421i	$0.489164\pi$
$570$	0	0
$571$	25831.9i	1.89323i	0.322370	+	0.946614i	$0.395520\pi$
$571$	25831.9i	1.89323i	−0.322370	+	0.946614i	$0.604480\pi$
$572$	0	0
$573$	8763.43i	0.638914i
$574$	0	0
$575$	−8453.92	−0.613135
$576$	0	0
$577$	−6598.83	−0.476106	−0.238053	−	0.971252i	$-0.576509\pi$
$577$	−6598.83	−0.476106	−0.238053	+	0.971252i	$0.576509\pi$
$578$	0	0
$579$	− 8668.04i	− 0.622161i
$580$	0	0
$581$	− 571.785i	− 0.0408289i
$582$	0	0
$583$	5477.17	0.389093
$584$	0	0
$585$	−2784.76	−0.196813
$586$	0	0
$587$	21144.3i	1.48674i	0.668879	+	0.743371i	$0.266774\pi$
$587$	21144.3i	1.48674i	−0.668879	+	0.743371i	$0.733226\pi$
$588$	0	0
$589$	− 5923.36i	− 0.414376i
$590$	0	0
$591$	5807.11	0.404184
$592$	0	0
$593$	−12893.9	−0.892896	−0.446448	−	0.894810i	$-0.647311\pi$
$593$	−12893.9	−0.892896	−0.446448	+	0.894810i	$0.647311\pi$
$594$	0	0
$595$	4771.92i	0.328790i
$596$	0	0
$597$	− 4295.41i	− 0.294472i
$598$	0	0
$599$	−12097.1	−0.825165	−0.412582	−	0.910920i	$-0.635373\pi$
$599$	−12097.1	−0.825165	−0.412582	+	0.910920i	$0.635373\pi$
$600$	0	0
$601$	1802.44	0.122335	0.0611673	−	0.998128i	$-0.480518\pi$
$601$	1802.44	0.122335	0.0611673	+	0.998128i	$0.480518\pi$
$602$	0	0
$603$	− 5989.14i	− 0.404472i
$604$	0	0
$605$	2773.41i	0.186372i
$606$	0	0
$607$	−5092.87	−0.340549	−0.170275	−	0.985397i	$-0.554465\pi$
$607$	−5092.87	−0.340549	−0.170275	+	0.985397i	$0.554465\pi$
$608$	0	0
$609$	−3939.35	−0.262119
$610$	0	0
$611$	23317.9i	1.54393i
$612$	0	0
$613$	− 4763.25i	− 0.313843i	−0.987611	−	0.156922i	$-0.949843\pi$
$613$	− 4763.25i	− 0.313843i	0.987611	−	0.156922i	$-0.0501570\pi$
$614$	0	0
$615$	9762.87	0.640125
$616$	0	0
$617$	2643.13	0.172461	0.0862304	−	0.996275i	$-0.472518\pi$
$617$	2643.13	0.172461	0.0862304	+	0.996275i	$0.472518\pi$
$618$	0	0
$619$	13166.7i	0.854952i	0.904027	+	0.427476i	$0.140597\pi$
$619$	13166.7i	0.854952i	−0.904027	+	0.427476i	$0.859403\pi$
$620$	0	0
$621$	− 2926.21i	− 0.189090i
$622$	0	0
$623$	7074.20	0.454931
$624$	0	0
$625$	210.106	0.0134468
$626$	0	0
$627$	− 3101.54i	− 0.197549i
$628$	0	0
$629$	24028.3i	1.52317i
$630$	0	0
$631$	28012.9	1.76732	0.883659	−	0.468131i	$-0.155072\pi$
$631$	28012.9	1.76732	0.883659	+	0.468131i	$0.155072\pi$
$632$	0	0
$633$	−6351.21	−0.398796
$634$	0	0
$635$	− 4433.69i	− 0.277080i
$636$	0	0
$637$	− 2211.62i	− 0.137563i
$638$	0	0
$639$	−1166.14	−0.0721937
$640$	0	0
$641$	16283.5	1.00337	0.501685	−	0.865050i	$-0.332714\pi$
$641$	16283.5	1.00337	0.501685	+	0.865050i	$0.332714\pi$
$642$	0	0
$643$	− 1292.91i	− 0.0792961i	−0.999214	−	0.0396481i	$-0.987376\pi$
$643$	− 1292.91i	− 0.0792961i	0.999214	−	0.0396481i	$-0.0126237\pi$
$644$	0	0
$645$	− 9891.86i	− 0.603863i
$646$	0	0
$647$	−18335.5	−1.11413	−0.557067	−	0.830468i	$-0.688073\pi$
$647$	−18335.5	−1.11413	−0.557067	+	0.830468i	$0.688073\pi$
$648$	0	0
$649$	−679.055	−0.0410712
$650$	0	0
$651$	3662.19i	0.220480i
$652$	0	0
$653$	− 2508.06i	− 0.150303i	−0.997172	−	0.0751515i	$-0.976056\pi$
$653$	− 2508.06i	− 0.150303i	0.997172	−	0.0751515i	$-0.0239440\pi$
$654$	0	0
$655$	1756.37	0.104774
$656$	0	0
$657$	−9805.25	−0.582252
$658$	0	0
$659$	10689.3i	0.631859i	0.948783	+	0.315929i	$0.102316\pi$
$659$	10689.3i	0.631859i	−0.948783	+	0.315929i	$0.897684\pi$
$660$	0	0
$661$	− 18520.0i	− 1.08978i	−0.838508	−	0.544889i	$-0.816572\pi$
$661$	− 18520.0i	− 1.08978i	0.838508	−	0.544889i	$-0.183428\pi$
$662$	0	0
$663$	13464.8	0.788733
$664$	0	0
$665$	−1629.96	−0.0950481
$666$	0	0
$667$	− 20330.5i	− 1.18021i
$668$	0	0
$669$	− 14976.9i	− 0.865529i
$670$	0	0
$671$	−28386.6	−1.63316
$672$	0	0
$673$	−4841.15	−0.277285	−0.138642	−	0.990343i	$-0.544274\pi$
$673$	−4841.15	−0.277285	−0.138642	+	0.990343i	$0.544274\pi$
$674$	0	0
$675$	2106.11i	0.120095i
$676$	0	0
$677$	29003.8i	1.64654i	0.567652	+	0.823269i	$0.307852\pi$
$677$	29003.8i	1.64654i	−0.567652	+	0.823269i	$0.692148\pi$
$678$	0	0
$679$	−4275.16	−0.241628
$680$	0	0
$681$	−10290.1	−0.579028
$682$	0	0
$683$	− 23852.8i	− 1.33631i	−0.744021	−	0.668156i	$-0.767084\pi$
$683$	− 23852.8i	− 1.33631i	0.744021	−	0.668156i	$-0.232916\pi$
$684$	0	0
$685$	12930.5i	0.721238i
$686$	0	0
$687$	6910.34	0.383764
$688$	0	0
$689$	−8121.99	−0.449090
$690$	0	0
$691$	11568.5i	0.636883i	0.947943	+	0.318441i	$0.103159\pi$
$691$	11568.5i	0.636883i	−0.947943	+	0.318441i	$0.896841\pi$
$692$	0	0
$693$	1917.56i	0.105111i
$694$	0	0
$695$	−4194.92	−0.228953
$696$	0	0
$697$	−47205.2	−2.56532
$698$	0	0
$699$	3261.89i	0.176504i
$700$	0	0
$701$	− 30210.5i	− 1.62773i	−0.581057	−	0.813863i	$-0.697361\pi$
$701$	− 30210.5i	− 1.62773i	0.581057	−	0.813863i	$-0.302639\pi$
$702$	0	0
$703$	−8207.40	−0.440324
$704$	0	0
$705$	−10624.9	−0.567600
$706$	0	0
$707$	− 9148.53i	− 0.486656i
$708$	0	0
$709$	− 1281.57i	− 0.0678850i	−0.999424	−	0.0339425i	$-0.989194\pi$
$709$	− 1281.57i	− 0.0678850i	0.999424	−	0.0339425i	$-0.0108063\pi$
$710$	0	0
$711$	−6655.22	−0.351041
$712$	0	0
$713$	−18900.0	−0.992724
$714$	0	0
$715$	9417.90i	0.492601i
$716$	0	0
$717$	− 8216.57i	− 0.427969i
$718$	0	0
$719$	−9878.96	−0.512410	−0.256205	−	0.966622i	$-0.582472\pi$
$719$	−9878.96	−0.512410	−0.256205	+	0.966622i	$0.582472\pi$
$720$	0	0
$721$	−10832.0	−0.559508
$722$	0	0
$723$	6345.88i	0.326426i
$724$	0	0
$725$	14632.6i	0.749575i
$726$	0	0
$727$	−1362.29	−0.0694971	−0.0347485	−	0.999396i	$-0.511063\pi$
$727$	−1362.29	−0.0694971	−0.0347485	+	0.999396i	$0.511063\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	47828.9i	2.42000i
$732$	0	0
$733$	− 28003.3i	− 1.41109i	−0.708667	−	0.705543i	$-0.750703\pi$
$733$	− 28003.3i	− 1.41109i	0.708667	−	0.705543i	$-0.249297\pi$
$734$	0	0
$735$	1007.74	0.0505728
$736$	0	0
$737$	−20254.9	−1.01235
$738$	0	0
$739$	− 16192.5i	− 0.806023i	−0.915195	−	0.403012i	$-0.867963\pi$
$739$	− 16192.5i	− 0.806023i	0.915195	−	0.403012i	$-0.132037\pi$
$740$	0	0
$741$	4599.21i	0.228011i
$742$	0	0
$743$	35041.7	1.73022	0.865111	−	0.501580i	$-0.167248\pi$
$743$	35041.7	1.73022	0.865111	+	0.501580i	$0.167248\pi$
$744$	0	0
$745$	8577.20	0.421804
$746$	0	0
$747$	− 735.152i	− 0.0360077i
$748$	0	0
$749$	12814.2i	0.625128i
$750$	0	0
$751$	−36577.3	−1.77726	−0.888631	−	0.458624i	$-0.848343\pi$
$751$	−36577.3	−1.77726	−0.888631	+	0.458624i	$0.848343\pi$
$752$	0	0
$753$	−11624.2	−0.562564
$754$	0	0
$755$	2939.33i	0.141687i
$756$	0	0
$757$	− 24770.0i	− 1.18928i	−0.803994	−	0.594638i	$-0.797296\pi$
$757$	− 24770.0i	− 1.18928i	0.803994	−	0.594638i	$-0.202704\pi$
$758$	0	0
$759$	−9896.27	−0.473270
$760$	0	0
$761$	4300.59	0.204857	0.102429	−	0.994740i	$-0.467339\pi$
$761$	4300.59	0.204857	0.102429	+	0.994740i	$0.467339\pi$
$762$	0	0
$763$	− 6771.46i	− 0.321289i
$764$	0	0
$765$	6135.33i	0.289965i
$766$	0	0
$767$	1006.96	0.0474043
$768$	0	0
$769$	30850.0	1.44666	0.723328	−	0.690505i	$-0.242612\pi$
$769$	30850.0	1.44666	0.723328	+	0.690505i	$0.242612\pi$
$770$	0	0
$771$	− 23633.2i	− 1.10393i
$772$	0	0
$773$	23898.4i	1.11199i	0.831186	+	0.555994i	$0.187662\pi$
$773$	23898.4i	1.11199i	−0.831186	+	0.555994i	$0.812338\pi$
$774$	0	0
$775$	13603.1	0.630500
$776$	0	0
$777$	5074.32	0.234286
$778$	0	0
$779$	− 16124.0i	− 0.741594i
$780$	0	0
$781$	3943.82i	0.180693i
$782$	0	0
$783$	−5064.88	−0.231167
$784$	0	0
$785$	11650.3	0.529703
$786$	0	0
$787$	− 2774.72i	− 0.125678i	−0.998024	−	0.0628388i	$-0.979985\pi$
$787$	− 2774.72i	− 0.125678i	0.998024	−	0.0628388i	$-0.0200154\pi$
$788$	0	0
$789$	13278.2i	0.599135i
$790$	0	0
$791$	−13201.9	−0.593433
$792$	0	0
$793$	42094.0	1.88499
$794$	0	0
$795$	− 3700.84i	− 0.165101i
$796$	0	0
$797$	− 34888.8i	− 1.55060i	−0.631596	−	0.775298i	$-0.717600\pi$
$797$	− 34888.8i	− 1.55060i	0.631596	−	0.775298i	$-0.282400\pi$
$798$	0	0
$799$	51373.5	2.27467
$800$	0	0
$801$	9095.40	0.401211
$802$	0	0
$803$	33160.8i	1.45731i
$804$	0	0
$805$	5200.80i	0.227707i
$806$	0	0
$807$	−10233.5	−0.446391
$808$	0	0
$809$	647.431	0.0281366	0.0140683	−	0.999901i	$-0.495522\pi$
$809$	647.431	0.0281366	0.0140683	+	0.999901i	$0.495522\pi$
$810$	0	0
$811$	− 24867.7i	− 1.07672i	−0.842714	−	0.538362i	$-0.819044\pi$
$811$	− 24867.7i	− 1.07672i	0.842714	−	0.538362i	$-0.180956\pi$
$812$	0	0
$813$	− 16252.3i	− 0.701098i
$814$	0	0
$815$	19642.6	0.844233
$816$	0	0
$817$	−16337.0	−0.699584
$818$	0	0
$819$	− 2843.51i	− 0.121319i
$820$	0	0
$821$	− 5572.42i	− 0.236881i	−0.992961	−	0.118440i	$-0.962211\pi$
$821$	− 5572.42i	− 0.236881i	0.992961	−	0.118440i	$-0.0377894\pi$
$822$	0	0
$823$	8210.96	0.347772	0.173886	−	0.984766i	$-0.444368\pi$
$823$	8210.96	0.347772	0.173886	+	0.984766i	$0.444368\pi$
$824$	0	0
$825$	7122.73	0.300584
$826$	0	0
$827$	26176.4i	1.10065i	0.834949	+	0.550327i	$0.185497\pi$
$827$	26176.4i	1.10065i	−0.834949	+	0.550327i	$0.814503\pi$
$828$	0	0
$829$	31756.2i	1.33045i	0.746645	+	0.665223i	$0.231663\pi$
$829$	31756.2i	1.33045i	−0.746645	+	0.665223i	$0.768337\pi$
$830$	0	0
$831$	−15747.6	−0.657374
$832$	0	0
$833$	−4872.60	−0.202672
$834$	0	0
$835$	− 7.72260i	0 0.000320062i
$836$	0	0
$837$	4708.52i	0.194445i
$838$	0	0
$839$	23226.2	0.955728	0.477864	−	0.878434i	$-0.341411\pi$
$839$	23226.2	0.955728	0.477864	+	0.878434i	$0.341411\pi$
$840$	0	0
$841$	−10800.3	−0.442836
$842$	0	0
$843$	− 12518.7i	− 0.511468i
$844$	0	0
$845$	1095.63i	0.0446044i
$846$	0	0
$847$	−2831.92	−0.114883
$848$	0	0
$849$	−22802.8	−0.921779
$850$	0	0
$851$	26187.9i	1.05489i
$852$	0	0
$853$	1169.93i	0.0469609i	0.999724	+	0.0234804i	$0.00747474\pi$
$853$	1169.93i	0.0469609i	−0.999724	+	0.0234804i	$0.992525\pi$
$854$	0	0
$855$	−2095.66	−0.0838245
$856$	0	0
$857$	28011.2	1.11650	0.558252	−	0.829671i	$-0.311472\pi$
$857$	28011.2	1.11650	0.558252	+	0.829671i	$0.311472\pi$
$858$	0	0
$859$	− 22305.6i	− 0.885981i	−0.896526	−	0.442991i	$-0.853917\pi$
$859$	− 22305.6i	− 0.885981i	0.896526	−	0.442991i	$-0.146083\pi$
$860$	0	0
$861$	9968.84i	0.394585i
$862$	0	0
$863$	−32165.3	−1.26874	−0.634368	−	0.773031i	$-0.718739\pi$
$863$	−32165.3	−1.26874	−0.634368	+	0.773031i	$0.718739\pi$
$864$	0	0
$865$	−963.091	−0.0378567
$866$	0	0
$867$	− 14926.4i	− 0.584692i
$868$	0	0
$869$	22507.6i	0.878615i
$870$	0	0
$871$	30035.6	1.16845
$872$	0	0
$873$	−5496.63	−0.213096
$874$	0	0
$875$	− 9741.67i	− 0.376375i
$876$	0	0
$877$	35803.0i	1.37854i	0.724503	+	0.689271i	$0.242069\pi$
$877$	35803.0i	1.37854i	−0.724503	+	0.689271i	$0.757931\pi$
$878$	0	0
$879$	17255.9	0.662148
$880$	0	0
$881$	14860.3	0.568282	0.284141	−	0.958782i	$-0.408292\pi$
$881$	14860.3	0.568282	0.284141	+	0.958782i	$0.408292\pi$
$882$	0	0
$883$	52005.7i	1.98203i	0.133761	+	0.991014i	$0.457294\pi$
$883$	52005.7i	1.98203i	−0.133761	+	0.991014i	$0.542706\pi$
$884$	0	0
$885$	458.826i	0.0174274i
$886$	0	0
$887$	−18483.4	−0.699676	−0.349838	−	0.936810i	$-0.613763\pi$
$887$	−18483.4	−0.699676	−0.349838	+	0.936810i	$0.613763\pi$
$888$	0	0
$889$	4527.23	0.170797
$890$	0	0
$891$	2465.44i	0.0926994i
$892$	0	0
$893$	17547.7i	0.657573i
$894$	0	0
$895$	827.202	0.0308942
$896$	0	0
$897$	14675.0	0.546247
$898$	0	0
$899$	32713.5i	1.21363i
$900$	0	0
$901$	17894.2i	0.661645i
$902$	0	0
$903$	10100.6	0.372232
$904$	0	0
$905$	9873.41	0.362656
$906$	0	0
$907$	26109.9i	0.955861i	0.878398	+	0.477930i	$0.158613\pi$
$907$	26109.9i	0.955861i	−0.878398	+	0.477930i	$0.841387\pi$
$908$	0	0
$909$	− 11762.4i	− 0.429190i
$910$	0	0
$911$	−42877.1	−1.55937	−0.779683	−	0.626174i	$-0.784620\pi$
$911$	−42877.1	−1.55937	−0.779683	+	0.626174i	$0.784620\pi$
$912$	0	0
$913$	−2486.24	−0.0901232
$914$	0	0
$915$	19180.4i	0.692988i
$916$	0	0
$917$	1793.43i	0.0645847i
$918$	0	0
$919$	3299.71	0.118441	0.0592205	−	0.998245i	$-0.481138\pi$
$919$	3299.71	0.118441	0.0592205	+	0.998245i	$0.481138\pi$
$920$	0	0
$921$	2844.32	0.101763
$922$	0	0
$923$	− 5848.21i	− 0.208555i
$924$	0	0
$925$	− 18848.4i	− 0.669981i
$926$	0	0
$927$	−13926.9	−0.493440
$928$	0	0
$929$	8109.49	0.286398	0.143199	−	0.989694i	$-0.454261\pi$
$929$	8109.49	0.286398	0.143199	+	0.989694i	$0.454261\pi$
$930$	0	0
$931$	− 1664.34i	− 0.0585893i
$932$	0	0
$933$	− 14179.0i	− 0.497534i
$934$	0	0
$935$	20749.3	0.725749
$936$	0	0
$937$	−50079.2	−1.74601	−0.873007	−	0.487708i	$-0.837833\pi$
$937$	−50079.2	−1.74601	−0.873007	+	0.487708i	$0.837833\pi$
$938$	0	0
$939$	16782.9i	0.583269i
$940$	0	0
$941$	− 34701.5i	− 1.20216i	−0.799188	−	0.601082i	$-0.794737\pi$
$941$	− 34701.5i	− 1.20216i	0.799188	−	0.601082i	$-0.205263\pi$
$942$	0	0
$943$	−51447.8	−1.77664
$944$	0	0
$945$	1295.66	0.0446010
$946$	0	0
$947$	− 5170.49i	− 0.177422i	−0.996057	−	0.0887109i	$-0.971725\pi$
$947$	− 5170.49i	− 0.177422i	0.996057	−	0.0887109i	$-0.0282747\pi$
$948$	0	0
$949$	− 49173.5i	− 1.68202i
$950$	0	0
$951$	2180.24	0.0743418
$952$	0	0
$953$	53973.9	1.83461	0.917306	−	0.398183i	$-0.130359\pi$
$953$	53973.9	1.83461	0.917306	+	0.398183i	$0.130359\pi$
$954$	0	0
$955$	− 20025.5i	− 0.678545i
$956$	0	0
$957$	17129.1i	0.578585i
$958$	0	0
$959$	−13203.3	−0.444584
$960$	0	0
$961$	620.803	0.0208386
$962$	0	0
$963$	16475.4i	0.551311i
$964$	0	0
$965$	19807.5i	0.660753i
$966$	0	0
$967$	34514.5	1.14779	0.573894	−	0.818929i	$-0.305432\pi$
$967$	34514.5	1.14779	0.573894	+	0.818929i	$0.305432\pi$
$968$	0	0
$969$	10132.9	0.335929
$970$	0	0
$971$	− 38224.1i	− 1.26331i	−0.775251	−	0.631653i	$-0.782377\pi$
$971$	− 38224.1i	− 1.26331i	0.775251	−	0.631653i	$-0.217623\pi$
$972$	0	0
$973$	− 4283.43i	− 0.141131i
$974$	0	0
$975$	−10562.2	−0.346933
$976$	0	0
$977$	27748.4	0.908648	0.454324	−	0.890837i	$-0.349881\pi$
$977$	27748.4	0.908648	0.454324	+	0.890837i	$0.349881\pi$
$978$	0	0
$979$	− 30760.1i	− 1.00419i
$980$	0	0
$981$	− 8706.16i	− 0.283350i
$982$	0	0
$983$	37180.6	1.20639	0.603193	−	0.797595i	$-0.293895\pi$
$983$	37180.6	1.20639	0.603193	+	0.797595i	$0.293895\pi$
$984$	0	0
$985$	−13270.0	−0.429255
$986$	0	0
$987$	− 10849.1i	− 0.349879i
$988$	0	0
$989$	52127.6i	1.67600i
$990$	0	0
$991$	−52211.9	−1.67363	−0.836814	−	0.547487i	$-0.815585\pi$
$991$	−52211.9	−1.67363	−0.836814	+	0.547487i	$0.815585\pi$
$992$	0	0
$993$	−4196.08	−0.134097
$994$	0	0
$995$	9815.55i	0.312737i
$996$	0	0
$997$	− 11220.8i	− 0.356436i	−0.983991	−	0.178218i	$-0.942967\pi$
$997$	− 11220.8i	− 0.356436i	0.983991	−	0.178218i	$-0.0570332\pi$
$998$	0	0
$999$	6524.13	0.206621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.e.673.4	✓	12
4.3	odd	2		1344.4.c.h.673.10	yes	12
8.3	odd	2		1344.4.c.h.673.3	yes	12
8.5	even	2	inner	1344.4.c.e.673.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.e.673.4	✓	12	1.1	even	1	trivial
1344.4.c.e.673.9	yes	12	8.5	even	2	inner
1344.4.c.h.673.3	yes	12	8.3	odd	2
1344.4.c.h.673.10	yes	12	4.3	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +6.85537i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +6.85537i q^{5} -7.00000 q^{7} -9.00000 q^{9} +30.4375i q^{11} -45.1351i q^{13} +20.5661 q^{15} -99.4408 q^{17} -33.9662i q^{19} +21.0000i q^{21} -108.378 q^{23} +78.0039 q^{25} +27.0000i q^{27} +187.588i q^{29} +174.390 q^{31} +91.3124 q^{33} -47.9876i q^{35} -241.634i q^{37} -135.405 q^{39} +474.707 q^{41} -480.979i q^{43} -61.6983i q^{45} -516.624 q^{47} +49.0000 q^{49} +298.322i q^{51} -179.948i q^{53} -208.660 q^{55} -101.899 q^{57} +22.3098i q^{59} +932.621i q^{61} +63.0000 q^{63} +309.418 q^{65} +665.460i q^{67} +325.134i q^{69} +129.571 q^{71} +1089.47 q^{73} -234.012i q^{75} -213.062i q^{77} +739.469 q^{79} +81.0000 q^{81} +81.6835i q^{83} -681.703i q^{85} +562.765 q^{87} -1010.60 q^{89} +315.946i q^{91} -523.169i q^{93} +232.851 q^{95} +610.737 q^{97} -273.937i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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