LMFDB - Embedded newform 1344.4.c.f.673.5

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} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.5
Root			$-1.74348 - 0.467165i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +13.9871i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +13.9871i q^{5} -7.00000 q^{7} -9.00000 q^{9} +51.5791i q^{11} -30.7945i q^{13} +41.9613 q^{15} +0.246192 q^{17} -2.72473i q^{19} +21.0000i q^{21} -14.5441 q^{23} -70.6394 q^{25} +27.0000i q^{27} +167.061i q^{29} -198.718 q^{31} +154.737 q^{33} -97.9098i q^{35} +321.932i q^{37} -92.3834 q^{39} +448.055 q^{41} -86.5288i q^{43} -125.884i q^{45} +33.5268 q^{47} +49.0000 q^{49} -0.738577i q^{51} +299.811i q^{53} -721.443 q^{55} -8.17419 q^{57} -197.965i q^{59} -476.996i q^{61} +63.0000 q^{63} +430.726 q^{65} -824.500i q^{67} +43.6323i q^{69} -503.994 q^{71} -1127.61 q^{73} +211.918i q^{75} -361.054i q^{77} -684.567 q^{79} +81.0000 q^{81} -1119.55i q^{83} +3.44352i q^{85} +501.182 q^{87} -643.835 q^{89} +215.561i q^{91} +596.154i q^{93} +38.1111 q^{95} -168.589 q^{97} -464.212i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * 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
$5$	13.9871i	1.25105i	0.780206	+	0.625523i	$0.215114\pi$
$5$	13.9871i	1.25105i	−0.780206	+	0.625523i	$0.784886\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	51.5791i	1.41379i	0.707319	+	0.706895i	$0.249905\pi$
$11$	51.5791i	1.41379i	−0.707319	+	0.706895i	$0.750095\pi$
$12$	0	0
$13$	− 30.7945i	− 0.656988i	−0.944506	−	0.328494i	$-0.893459\pi$
$13$	− 30.7945i	− 0.656988i	0.944506	−	0.328494i	$-0.106541\pi$
$14$	0	0
$15$	41.9613	0.722292
$16$	0	0
$17$	0.246192	0.00351238	0.00175619	−	0.999998i	$-0.499441\pi$
$17$	0.246192	0.00351238	0.00175619	+	0.999998i	$0.499441\pi$
$18$	0	0
$19$	− 2.72473i	− 0.0328998i	−0.999865	−	0.0164499i	$-0.994764\pi$
$19$	− 2.72473i	− 0.0328998i	0.999865	−	0.0164499i	$-0.00523640\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−14.5441	−0.131855	−0.0659273	−	0.997824i	$-0.521001\pi$
$23$	−14.5441	−0.131855	−0.0659273	+	0.997824i	$0.521001\pi$
$24$	0	0
$25$	−70.6394	−0.565115
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	167.061i	1.06974i	0.844935	+	0.534868i	$0.179639\pi$
$29$	167.061i	1.06974i	−0.844935	+	0.534868i	$0.820361\pi$
$30$	0	0
$31$	−198.718	−1.15132	−0.575659	−	0.817690i	$-0.695254\pi$
$31$	−198.718	−1.15132	−0.575659	+	0.817690i	$0.695254\pi$
$32$	0	0
$33$	154.737	0.816252
$34$	0	0
$35$	− 97.9098i	− 0.472851i
$36$	0	0
$37$	321.932i	1.43041i	0.698914	+	0.715206i	$0.253667\pi$
$37$	321.932i	1.43041i	−0.698914	+	0.715206i	$0.746333\pi$
$38$	0	0
$39$	−92.3834	−0.379312
$40$	0	0
$41$	448.055	1.70669	0.853346	−	0.521345i	$-0.174569\pi$
$41$	448.055	1.70669	0.853346	+	0.521345i	$0.174569\pi$
$42$	0	0
$43$	− 86.5288i	− 0.306872i	−0.988159	−	0.153436i	$-0.950966\pi$
$43$	− 86.5288i	− 0.306872i	0.988159	−	0.153436i	$-0.0490340\pi$
$44$	0	0
$45$	− 125.884i	− 0.417015i
$46$	0	0
$47$	33.5268	0.104051	0.0520253	−	0.998646i	$-0.483432\pi$
$47$	33.5268	0.104051	0.0520253	+	0.998646i	$0.483432\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 0.738577i	− 0.00202787i
$52$	0	0
$53$	299.811i	0.777022i	0.921444	+	0.388511i	$0.127010\pi$
$53$	299.811i	0.777022i	−0.921444	+	0.388511i	$0.872990\pi$
$54$	0	0
$55$	−721.443	−1.76872
$56$	0	0
$57$	−8.17419	−0.0189947
$58$	0	0
$59$	− 197.965i	− 0.436827i	−0.975856	−	0.218414i	$-0.929912\pi$
$59$	− 197.965i	− 0.436827i	0.975856	−	0.218414i	$-0.0700882\pi$
$60$	0	0
$61$	− 476.996i	− 1.00120i	−0.865679	−	0.500599i	$-0.833113\pi$
$61$	− 476.996i	− 1.00120i	0.865679	−	0.500599i	$-0.166887\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	430.726	0.821922
$66$	0	0
$67$	− 824.500i	− 1.50341i	−0.659497	−	0.751707i	$-0.729231\pi$
$67$	− 824.500i	− 1.50341i	0.659497	−	0.751707i	$-0.270769\pi$
$68$	0	0
$69$	43.6323i	0.0761263i
$70$	0	0
$71$	−503.994	−0.842438	−0.421219	−	0.906959i	$-0.638398\pi$
$71$	−503.994	−0.842438	−0.421219	+	0.906959i	$0.638398\pi$
$72$	0	0
$73$	−1127.61	−1.80791	−0.903953	−	0.427631i	$-0.859348\pi$
$73$	−1127.61	−1.80791	−0.903953	+	0.427631i	$0.859348\pi$
$74$	0	0
$75$	211.918i	0.326269i
$76$	0	0
$77$	− 361.054i	− 0.534362i
$78$	0	0
$79$	−684.567	−0.974934	−0.487467	−	0.873141i	$-0.662079\pi$
$79$	−684.567	−0.974934	−0.487467	+	0.873141i	$0.662079\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1119.55i	− 1.48056i	−0.672297	−	0.740282i	$-0.734692\pi$
$83$	− 1119.55i	− 1.48056i	0.672297	−	0.740282i	$-0.265308\pi$
$84$	0	0
$85$	3.44352i	0.00439415i
$86$	0	0
$87$	501.182	0.617613
$88$	0	0
$89$	−643.835	−0.766814	−0.383407	−	0.923580i	$-0.625249\pi$
$89$	−643.835	−0.766814	−0.383407	+	0.923580i	$0.625249\pi$
$90$	0	0
$91$	215.561i	0.248318i
$92$	0	0
$93$	596.154i	0.664713i
$94$	0	0
$95$	38.1111	0.0411591
$96$	0	0
$97$	−168.589	−0.176470	−0.0882349	−	0.996100i	$-0.528123\pi$
$97$	−168.589	−0.176470	−0.0882349	+	0.996100i	$0.528123\pi$
$98$	0	0
$99$	− 464.212i	− 0.471263i
$100$	0	0
$101$	113.480i	0.111799i	0.998436	+	0.0558994i	$0.0178026\pi$
$101$	113.480i	0.111799i	−0.998436	+	0.0558994i	$0.982197\pi$
$102$	0	0
$103$	1124.28	1.07552	0.537760	−	0.843098i	$-0.319271\pi$
$103$	1124.28	1.07552	0.537760	+	0.843098i	$0.319271\pi$
$104$	0	0
$105$	−293.729	−0.273001
$106$	0	0
$107$	1073.84i	0.970210i	0.874456	+	0.485105i	$0.161219\pi$
$107$	1073.84i	0.970210i	−0.874456	+	0.485105i	$0.838781\pi$
$108$	0	0
$109$	− 1681.57i	− 1.47767i	−0.673888	−	0.738833i	$-0.735377\pi$
$109$	− 1681.57i	− 1.47767i	0.673888	−	0.738833i	$-0.264623\pi$
$110$	0	0
$111$	965.795	0.825849
$112$	0	0
$113$	−1711.39	−1.42473	−0.712363	−	0.701812i	$-0.752375\pi$
$113$	−1711.39	−1.42473	−0.712363	+	0.701812i	$0.752375\pi$
$114$	0	0
$115$	− 203.430i	− 0.164956i
$116$	0	0
$117$	277.150i	0.218996i
$118$	0	0
$119$	−1.72335	−0.00132755
$120$	0	0
$121$	−1329.40	−0.998801
$122$	0	0
$123$	− 1344.16i	− 0.985359i
$124$	0	0
$125$	760.348i	0.544061i
$126$	0	0
$127$	−1279.16	−0.893755	−0.446878	−	0.894595i	$-0.647464\pi$
$127$	−1279.16	−0.893755	−0.446878	+	0.894595i	$0.647464\pi$
$128$	0	0
$129$	−259.586	−0.177173
$130$	0	0
$131$	− 2380.09i	− 1.58740i	−0.608311	−	0.793699i	$-0.708153\pi$
$131$	− 2380.09i	− 1.58740i	0.608311	−	0.793699i	$-0.291847\pi$
$132$	0	0
$133$	19.0731i	0.0124349i
$134$	0	0
$135$	−377.652	−0.240764
$136$	0	0
$137$	1271.94	0.793206	0.396603	−	0.917990i	$-0.370189\pi$
$137$	1271.94	0.793206	0.396603	+	0.917990i	$0.370189\pi$
$138$	0	0
$139$	788.385i	0.481079i	0.970639	+	0.240539i	$0.0773243\pi$
$139$	788.385i	0.481079i	−0.970639	+	0.240539i	$0.922676\pi$
$140$	0	0
$141$	− 100.580i	− 0.0600737i
$142$	0	0
$143$	1588.35	0.928843
$144$	0	0
$145$	−2336.70	−1.33829
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	1976.18i	1.08655i	0.839556	+	0.543273i	$0.182815\pi$
$149$	1976.18i	1.08655i	−0.839556	+	0.543273i	$0.817185\pi$
$150$	0	0
$151$	−670.640	−0.361430	−0.180715	−	0.983536i	$-0.557841\pi$
$151$	−670.640	−0.361430	−0.180715	+	0.983536i	$0.557841\pi$
$152$	0	0
$153$	−2.21573	−0.00117079
$154$	0	0
$155$	− 2779.49i	− 1.44035i
$156$	0	0
$157$	337.739i	0.171685i	0.996309	+	0.0858423i	$0.0273581\pi$
$157$	337.739i	0.171685i	−0.996309	+	0.0858423i	$0.972642\pi$
$158$	0	0
$159$	899.432	0.448614
$160$	0	0
$161$	101.809	0.0498364
$162$	0	0
$163$	149.606i	0.0718899i	0.999354	+	0.0359450i	$0.0114441\pi$
$163$	149.606i	0.0718899i	−0.999354	+	0.0359450i	$0.988556\pi$
$164$	0	0
$165$	2164.33i	1.02117i
$166$	0	0
$167$	−3347.04	−1.55091	−0.775453	−	0.631405i	$-0.782479\pi$
$167$	−3347.04	−1.55091	−0.775453	+	0.631405i	$0.782479\pi$
$168$	0	0
$169$	1248.70	0.568367
$170$	0	0
$171$	24.5226i	0.0109666i
$172$	0	0
$173$	− 86.6015i	− 0.0380589i	−0.999819	−	0.0190295i	$-0.993942\pi$
$173$	− 86.6015i	− 0.0380589i	0.999819	−	0.0190295i	$-0.00605763\pi$
$174$	0	0
$175$	494.476	0.213593
$176$	0	0
$177$	−593.894	−0.252202
$178$	0	0
$179$	− 2306.58i	− 0.963140i	−0.876408	−	0.481570i	$-0.840067\pi$
$179$	− 2306.58i	− 0.963140i	0.876408	−	0.481570i	$-0.159933\pi$
$180$	0	0
$181$	− 2101.18i	− 0.862869i	−0.902144	−	0.431434i	$-0.858008\pi$
$181$	− 2101.18i	− 0.862869i	0.902144	−	0.431434i	$-0.141992\pi$
$182$	0	0
$183$	−1430.99	−0.578042
$184$	0	0
$185$	−4502.89	−1.78951
$186$	0	0
$187$	12.6984i	0.00496577i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	−1515.51	−0.574129	−0.287064	−	0.957911i	$-0.592679\pi$
$191$	−1515.51	−0.574129	−0.287064	+	0.957911i	$0.592679\pi$
$192$	0	0
$193$	−1380.38	−0.514830	−0.257415	−	0.966301i	$-0.582871\pi$
$193$	−1380.38	−0.514830	−0.257415	+	0.966301i	$0.582871\pi$
$194$	0	0
$195$	− 1292.18i	− 0.474537i
$196$	0	0
$197$	− 1057.44i	− 0.382435i	−0.981548	−	0.191217i	$-0.938756\pi$
$197$	− 1057.44i	− 0.382435i	0.981548	−	0.191217i	$-0.0612436\pi$
$198$	0	0
$199$	−982.569	−0.350012	−0.175006	−	0.984567i	$-0.555995\pi$
$199$	−982.569	−0.350012	−0.175006	+	0.984567i	$0.555995\pi$
$200$	0	0
$201$	−2473.50	−0.867996
$202$	0	0
$203$	− 1169.42i	− 0.404323i
$204$	0	0
$205$	6266.99i	2.13515i
$206$	0	0
$207$	130.897	0.0439516
$208$	0	0
$209$	140.539	0.0465134
$210$	0	0
$211$	− 1332.78i	− 0.434846i	−0.976078	−	0.217423i	$-0.930235\pi$
$211$	− 1332.78i	− 0.434846i	0.976078	−	0.217423i	$-0.0697651\pi$
$212$	0	0
$213$	1511.98i	0.486382i
$214$	0	0
$215$	1210.29	0.383911
$216$	0	0
$217$	1391.03	0.435157
$218$	0	0
$219$	3382.84i	1.04380i
$220$	0	0
$221$	− 7.58136i	− 0.00230759i
$222$	0	0
$223$	−2802.74	−0.841640	−0.420820	−	0.907144i	$-0.638258\pi$
$223$	−2802.74	−0.841640	−0.420820	+	0.907144i	$0.638258\pi$
$224$	0	0
$225$	635.755	0.188372
$226$	0	0
$227$	2933.73i	0.857792i	0.903354	+	0.428896i	$0.141097\pi$
$227$	2933.73i	0.857792i	−0.903354	+	0.428896i	$0.858903\pi$
$228$	0	0
$229$	3742.19i	1.07987i	0.841706	+	0.539936i	$0.181552\pi$
$229$	3742.19i	1.07987i	−0.841706	+	0.539936i	$0.818448\pi$
$230$	0	0
$231$	−1083.16	−0.308514
$232$	0	0
$233$	−1170.74	−0.329176	−0.164588	−	0.986362i	$-0.552629\pi$
$233$	−1170.74	−0.329176	−0.164588	+	0.986362i	$0.552629\pi$
$234$	0	0
$235$	468.943i	0.130172i
$236$	0	0
$237$	2053.70i	0.562878i
$238$	0	0
$239$	−1609.75	−0.435675	−0.217838	−	0.975985i	$-0.569900\pi$
$239$	−1609.75	−0.435675	−0.217838	+	0.975985i	$0.569900\pi$
$240$	0	0
$241$	−5408.12	−1.44551	−0.722755	−	0.691104i	$-0.757124\pi$
$241$	−5408.12	−1.44551	−0.722755	+	0.691104i	$0.757124\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	685.369i	0.178721i
$246$	0	0
$247$	−83.9066	−0.0216148
$248$	0	0
$249$	−3358.66	−0.854804
$250$	0	0
$251$	− 5299.79i	− 1.33275i	−0.745617	−	0.666374i	$-0.767845\pi$
$251$	− 5299.79i	− 1.33275i	0.745617	−	0.666374i	$-0.232155\pi$
$252$	0	0
$253$	− 750.173i	− 0.186415i
$254$	0	0
$255$	10.3306	0.00253696
$256$	0	0
$257$	42.9014	0.0104129	0.00520645	−	0.999986i	$-0.498343\pi$
$257$	42.9014	0.0104129	0.00520645	+	0.999986i	$0.498343\pi$
$258$	0	0
$259$	− 2253.52i	− 0.540645i
$260$	0	0
$261$	− 1503.55i	− 0.356579i
$262$	0	0
$263$	−3563.82	−0.835569	−0.417784	−	0.908546i	$-0.637193\pi$
$263$	−3563.82	−0.835569	−0.417784	+	0.908546i	$0.637193\pi$
$264$	0	0
$265$	−4193.49	−0.972090
$266$	0	0
$267$	1931.51i	0.442720i
$268$	0	0
$269$	5293.26i	1.19976i	0.800089	+	0.599881i	$0.204785\pi$
$269$	5293.26i	1.19976i	−0.800089	+	0.599881i	$0.795215\pi$
$270$	0	0
$271$	2544.94	0.570457	0.285229	−	0.958459i	$-0.407930\pi$
$271$	2544.94	0.570457	0.285229	+	0.958459i	$0.407930\pi$
$272$	0	0
$273$	646.684	0.143367
$274$	0	0
$275$	− 3643.52i	− 0.798954i
$276$	0	0
$277$	5360.35i	1.16272i	0.813648	+	0.581358i	$0.197478\pi$
$277$	5360.35i	1.16272i	−0.813648	+	0.581358i	$0.802522\pi$
$278$	0	0
$279$	1788.46	0.383772
$280$	0	0
$281$	3851.63	0.817684	0.408842	−	0.912605i	$-0.365933\pi$
$281$	3851.63	0.817684	0.408842	+	0.912605i	$0.365933\pi$
$282$	0	0
$283$	8884.40i	1.86616i	0.359670	+	0.933080i	$0.382889\pi$
$283$	8884.40i	1.86616i	−0.359670	+	0.933080i	$0.617111\pi$
$284$	0	0
$285$	− 114.333i	− 0.0237632i
$286$	0	0
$287$	−3136.38	−0.645069
$288$	0	0
$289$	−4912.94	−0.999988
$290$	0	0
$291$	505.766i	0.101885i
$292$	0	0
$293$	4308.01i	0.858965i	0.903075	+	0.429482i	$0.141304\pi$
$293$	4308.01i	0.858965i	−0.903075	+	0.429482i	$0.858696\pi$
$294$	0	0
$295$	2768.95	0.546491
$296$	0	0
$297$	−1392.64	−0.272084
$298$	0	0
$299$	447.878i	0.0866270i
$300$	0	0
$301$	605.701i	0.115987i
$302$	0	0
$303$	340.440	0.0645471
$304$	0	0
$305$	6671.80	1.25254
$306$	0	0
$307$	− 3920.97i	− 0.728930i	−0.931217	−	0.364465i	$-0.881252\pi$
$307$	− 3920.97i	− 0.728930i	0.931217	−	0.364465i	$-0.118748\pi$
$308$	0	0
$309$	− 3372.84i	− 0.620952i
$310$	0	0
$311$	−7152.81	−1.30418	−0.652088	−	0.758143i	$-0.726107\pi$
$311$	−7152.81	−1.30418	−0.652088	+	0.758143i	$0.726107\pi$
$312$	0	0
$313$	754.226	0.136202	0.0681012	−	0.997678i	$-0.478306\pi$
$313$	754.226	0.136202	0.0681012	+	0.997678i	$0.478306\pi$
$314$	0	0
$315$	881.188i	0.157617i
$316$	0	0
$317$	− 7954.23i	− 1.40932i	−0.709545	−	0.704660i	$-0.751100\pi$
$317$	− 7954.23i	− 1.40932i	0.709545	−	0.704660i	$-0.248900\pi$
$318$	0	0
$319$	−8616.83	−1.51238
$320$	0	0
$321$	3221.53	0.560151
$322$	0	0
$323$	− 0.670808i	0 0.000115557i
$324$	0	0
$325$	2175.30i	0.371274i
$326$	0	0
$327$	−5044.72	−0.853131
$328$	0	0
$329$	−234.687	−0.0393275
$330$	0	0
$331$	− 8919.36i	− 1.48113i	−0.671987	−	0.740563i	$-0.734559\pi$
$331$	− 8919.36i	− 1.48113i	0.671987	−	0.740563i	$-0.265441\pi$
$332$	0	0
$333$	− 2897.38i	− 0.476804i
$334$	0	0
$335$	11532.4	1.88084
$336$	0	0
$337$	−3689.51	−0.596381	−0.298191	−	0.954506i	$-0.596383\pi$
$337$	−3689.51	−0.596381	−0.298191	+	0.954506i	$0.596383\pi$
$338$	0	0
$339$	5134.17i	0.822566i
$340$	0	0
$341$	− 10249.7i	− 1.62772i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−610.291	−0.0952375
$346$	0	0
$347$	− 5809.87i	− 0.898819i	−0.893326	−	0.449410i	$-0.851634\pi$
$347$	− 5809.87i	− 0.898819i	0.893326	−	0.449410i	$-0.148366\pi$
$348$	0	0
$349$	− 5065.35i	− 0.776912i	−0.921467	−	0.388456i	$-0.873009\pi$
$349$	− 5065.35i	− 0.776912i	0.921467	−	0.388456i	$-0.126991\pi$
$350$	0	0
$351$	831.450	0.126437
$352$	0	0
$353$	−1513.12	−0.228145	−0.114072	−	0.993472i	$-0.536390\pi$
$353$	−1513.12	−0.228145	−0.114072	+	0.993472i	$0.536390\pi$
$354$	0	0
$355$	− 7049.42i	− 1.05393i
$356$	0	0
$357$	5.17004i	0 0.000766464i
$358$	0	0
$359$	1919.55	0.282200	0.141100	−	0.989995i	$-0.454936\pi$
$359$	1919.55	0.282200	0.141100	+	0.989995i	$0.454936\pi$
$360$	0	0
$361$	6851.58	0.998918
$362$	0	0
$363$	3988.21i	0.576658i
$364$	0	0
$365$	− 15772.1i	− 2.26177i
$366$	0	0
$367$	10414.2	1.48125	0.740625	−	0.671918i	$-0.234529\pi$
$367$	10414.2	1.48125	0.740625	+	0.671918i	$0.234529\pi$
$368$	0	0
$369$	−4032.49	−0.568897
$370$	0	0
$371$	− 2098.67i	− 0.293687i
$372$	0	0
$373$	9393.42i	1.30395i	0.758241	+	0.651975i	$0.226059\pi$
$373$	9393.42i	1.30395i	−0.758241	+	0.651975i	$0.773941\pi$
$374$	0	0
$375$	2281.04	0.314114
$376$	0	0
$377$	5144.54	0.702804
$378$	0	0
$379$	− 13916.5i	− 1.88613i	−0.332604	−	0.943067i	$-0.607927\pi$
$379$	− 13916.5i	− 1.88613i	0.332604	−	0.943067i	$-0.392073\pi$
$380$	0	0
$381$	3837.47i	0.516010i
$382$	0	0
$383$	1163.07	0.155170	0.0775852	−	0.996986i	$-0.475279\pi$
$383$	1163.07	0.155170	0.0775852	+	0.996986i	$0.475279\pi$
$384$	0	0
$385$	5050.10	0.668512
$386$	0	0
$387$	778.759i	0.102291i
$388$	0	0
$389$	− 4008.53i	− 0.522469i	−0.965275	−	0.261234i	$-0.915870\pi$
$389$	− 4008.53i	− 0.522469i	0.965275	−	0.261234i	$-0.0841295\pi$
$390$	0	0
$391$	−3.58065	−0.000463124 0
$392$	0	0
$393$	−7140.26	−0.916485
$394$	0	0
$395$	− 9575.11i	− 1.21969i
$396$	0	0
$397$	5436.43i	0.687271i	0.939103	+	0.343635i	$0.111658\pi$
$397$	5436.43i	0.687271i	−0.939103	+	0.343635i	$0.888342\pi$
$398$	0	0
$399$	57.2193	0.00717932
$400$	0	0
$401$	1337.79	0.166599	0.0832994	−	0.996525i	$-0.473454\pi$
$401$	1337.79	0.166599	0.0832994	+	0.996525i	$0.473454\pi$
$402$	0	0
$403$	6119.42i	0.756402i
$404$	0	0
$405$	1132.96i	0.139005i
$406$	0	0
$407$	−16604.9	−2.02230
$408$	0	0
$409$	5946.08	0.718863	0.359431	−	0.933172i	$-0.382971\pi$
$409$	5946.08	0.718863	0.359431	+	0.933172i	$0.382971\pi$
$410$	0	0
$411$	− 3815.82i	− 0.457958i
$412$	0	0
$413$	1385.75i	0.165105i
$414$	0	0
$415$	15659.3	1.85225
$416$	0	0
$417$	2365.15	0.277751
$418$	0	0
$419$	− 6475.08i	− 0.754960i	−0.926018	−	0.377480i	$-0.876791\pi$
$419$	− 6475.08i	− 0.754960i	0.926018	−	0.377480i	$-0.123209\pi$
$420$	0	0
$421$	148.460i	0.0171865i	0.999963	+	0.00859325i	$0.00273535\pi$
$421$	148.460i	0.0171865i	−0.999963	+	0.00859325i	$0.997265\pi$
$422$	0	0
$423$	−301.741	−0.0346836
$424$	0	0
$425$	−17.3909	−0.00198490
$426$	0	0
$427$	3338.97i	0.378417i
$428$	0	0
$429$	− 4765.05i	− 0.536268i
$430$	0	0
$431$	−3972.95	−0.444014	−0.222007	−	0.975045i	$-0.571261\pi$
$431$	−3972.95	−0.444014	−0.222007	+	0.975045i	$0.571261\pi$
$432$	0	0
$433$	−12727.5	−1.41257	−0.706287	−	0.707926i	$-0.749631\pi$
$433$	−12727.5	−1.41257	−0.706287	+	0.707926i	$0.749631\pi$
$434$	0	0
$435$	7010.09i	0.772662i
$436$	0	0
$437$	39.6288i	0.00433799i
$438$	0	0
$439$	13699.0	1.48933	0.744666	−	0.667437i	$-0.232609\pi$
$439$	13699.0	1.48933	0.744666	+	0.667437i	$0.232609\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	11556.0i	1.23938i	0.784848	+	0.619688i	$0.212741\pi$
$443$	11556.0i	1.23938i	−0.784848	+	0.619688i	$0.787259\pi$
$444$	0	0
$445$	− 9005.40i	− 0.959319i
$446$	0	0
$447$	5928.55	0.627317
$448$	0	0
$449$	13286.6	1.39651	0.698256	−	0.715848i	$-0.253960\pi$
$449$	13286.6	1.39651	0.698256	+	0.715848i	$0.253960\pi$
$450$	0	0
$451$	23110.3i	2.41290i
$452$	0	0
$453$	2011.92i	0.208672i
$454$	0	0
$455$	−3015.08	−0.310657
$456$	0	0
$457$	3989.44	0.408354	0.204177	−	0.978934i	$-0.434548\pi$
$457$	3989.44	0.408354	0.204177	+	0.978934i	$0.434548\pi$
$458$	0	0
$459$	6.64720i	0 0.000675958i
$460$	0	0
$461$	− 9542.23i	− 0.964048i	−0.876158	−	0.482024i	$-0.839902\pi$
$461$	− 9542.23i	− 0.964048i	0.876158	−	0.482024i	$-0.160098\pi$
$462$	0	0
$463$	10466.6	1.05059	0.525295	−	0.850920i	$-0.323955\pi$
$463$	10466.6	1.05059	0.525295	+	0.850920i	$0.323955\pi$
$464$	0	0
$465$	−8338.48	−0.831587
$466$	0	0
$467$	− 2048.14i	− 0.202947i	−0.994838	−	0.101474i	$-0.967644\pi$
$467$	− 2048.14i	− 0.202947i	0.994838	−	0.101474i	$-0.0323558\pi$
$468$	0	0
$469$	5771.50i	0.568237i
$470$	0	0
$471$	1013.22	0.0991222
$472$	0	0
$473$	4463.08	0.433853
$474$	0	0
$475$	192.473i	0.0185922i
$476$	0	0
$477$	− 2698.30i	− 0.259007i
$478$	0	0
$479$	−7566.57	−0.721765	−0.360883	−	0.932611i	$-0.617525\pi$
$479$	−7566.57	−0.721765	−0.360883	+	0.932611i	$0.617525\pi$
$480$	0	0
$481$	9913.71	0.939763
$482$	0	0
$483$	− 305.426i	− 0.0287731i
$484$	0	0
$485$	− 2358.07i	− 0.220772i
$486$	0	0
$487$	−7729.08	−0.719174	−0.359587	−	0.933111i	$-0.617082\pi$
$487$	−7729.08	−0.719174	−0.359587	+	0.933111i	$0.617082\pi$
$488$	0	0
$489$	448.818	0.0415057
$490$	0	0
$491$	− 14478.6i	− 1.33077i	−0.746500	−	0.665385i	$-0.768267\pi$
$491$	− 14478.6i	− 1.33077i	0.746500	−	0.665385i	$-0.231733\pi$
$492$	0	0
$493$	41.1291i	0.00375732i
$494$	0	0
$495$	6492.99	0.589572
$496$	0	0
$497$	3527.96	0.318412
$498$	0	0
$499$	833.888i	0.0748095i	0.999300	+	0.0374047i	$0.0119091\pi$
$499$	833.888i	0.0748095i	−0.999300	+	0.0374047i	$0.988091\pi$
$500$	0	0
$501$	10041.1i	0.895416i
$502$	0	0
$503$	−2969.77	−0.263251	−0.131626	−	0.991300i	$-0.542020\pi$
$503$	−2969.77	−0.263251	−0.131626	+	0.991300i	$0.542020\pi$
$504$	0	0
$505$	−1587.26	−0.139865
$506$	0	0
$507$	− 3746.10i	− 0.328147i
$508$	0	0
$509$	17527.8i	1.52634i	0.646198	+	0.763170i	$0.276358\pi$
$509$	17527.8i	1.52634i	−0.646198	+	0.763170i	$0.723642\pi$
$510$	0	0
$511$	7893.29	0.683324
$512$	0	0
$513$	73.5677	0.00633157
$514$	0	0
$515$	15725.4i	1.34552i
$516$	0	0
$517$	1729.28i	0.147106i
$518$	0	0
$519$	−259.805	−0.0219733
$520$	0	0
$521$	5661.35	0.476062	0.238031	−	0.971258i	$-0.423498\pi$
$521$	5661.35	0.476062	0.238031	+	0.971258i	$0.423498\pi$
$522$	0	0
$523$	21827.1i	1.82492i	0.409168	+	0.912459i	$0.365819\pi$
$523$	21827.1i	1.82492i	−0.409168	+	0.912459i	$0.634181\pi$
$524$	0	0
$525$	− 1483.43i	− 0.123318i
$526$	0	0
$527$	−48.9229	−0.00404386
$528$	0	0
$529$	−11955.5	−0.982614
$530$	0	0
$531$	1781.68i	0.145609i
$532$	0	0
$533$	− 13797.6i	− 1.12128i
$534$	0	0
$535$	−15020.0	−1.21378
$536$	0	0
$537$	−6919.75	−0.556069
$538$	0	0
$539$	2527.38i	0.201970i
$540$	0	0
$541$	− 729.984i	− 0.0580119i	−0.999579	−	0.0290060i	$-0.990766\pi$
$541$	− 729.984i	− 0.0580119i	0.999579	−	0.0290060i	$-0.00923418\pi$
$542$	0	0
$543$	−6303.53	−0.498178
$544$	0	0
$545$	23520.4	1.84863
$546$	0	0
$547$	23541.6i	1.84016i	0.391734	+	0.920078i	$0.371875\pi$
$547$	23541.6i	1.84016i	−0.391734	+	0.920078i	$0.628125\pi$
$548$	0	0
$549$	4292.96i	0.333733i
$550$	0	0
$551$	455.195	0.0351941
$552$	0	0
$553$	4791.97	0.368490
$554$	0	0
$555$	13508.7i	1.03317i
$556$	0	0
$557$	5392.39i	0.410203i	0.978741	+	0.205102i	$0.0657525\pi$
$557$	5392.39i	0.410203i	−0.978741	+	0.205102i	$0.934248\pi$
$558$	0	0
$559$	−2664.61	−0.201612
$560$	0	0
$561$	38.0952	0.00286699
$562$	0	0
$563$	− 8566.71i	− 0.641286i	−0.947200	−	0.320643i	$-0.896101\pi$
$563$	− 8566.71i	− 0.641286i	0.947200	−	0.320643i	$-0.103899\pi$
$564$	0	0
$565$	− 23937.4i	− 1.78240i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	25605.4	1.88653	0.943263	−	0.332045i	$-0.107739\pi$
$569$	25605.4	1.88653	0.943263	+	0.332045i	$0.107739\pi$
$570$	0	0
$571$	11694.6i	0.857102i	0.903518	+	0.428551i	$0.140976\pi$
$571$	11694.6i	0.857102i	−0.903518	+	0.428551i	$0.859024\pi$
$572$	0	0
$573$	4546.53i	0.331473i
$574$	0	0
$575$	1027.39	0.0745131
$576$	0	0
$577$	22162.0	1.59899	0.799495	−	0.600673i	$-0.205101\pi$
$577$	22162.0	1.59899	0.799495	+	0.600673i	$0.205101\pi$
$578$	0	0
$579$	4141.15i	0.297237i
$580$	0	0
$581$	7836.86i	0.559600i
$582$	0	0
$583$	−15464.0	−1.09855
$584$	0	0
$585$	−3876.53	−0.273974
$586$	0	0
$587$	310.769i	0.0218515i	0.999940	+	0.0109257i	$0.00347784\pi$
$587$	310.769i	0.0218515i	−0.999940	+	0.0109257i	$0.996522\pi$
$588$	0	0
$589$	541.453i	0.0378781i
$590$	0	0
$591$	−3172.33	−0.220799
$592$	0	0
$593$	10069.8	0.697328	0.348664	−	0.937248i	$-0.386635\pi$
$593$	10069.8	0.697328	0.348664	+	0.937248i	$0.386635\pi$
$594$	0	0
$595$	− 24.1047i	− 0.00166083i
$596$	0	0
$597$	2947.71i	0.202080i
$598$	0	0
$599$	−9888.41	−0.674507	−0.337253	−	0.941414i	$-0.609498\pi$
$599$	−9888.41	−0.674507	−0.337253	+	0.941414i	$0.609498\pi$
$600$	0	0
$601$	17962.8	1.21917	0.609584	−	0.792722i	$-0.291337\pi$
$601$	17962.8	1.21917	0.609584	+	0.792722i	$0.291337\pi$
$602$	0	0
$603$	7420.50i	0.501138i
$604$	0	0
$605$	− 18594.5i	− 1.24955i
$606$	0	0
$607$	26295.5	1.75832	0.879160	−	0.476527i	$-0.158105\pi$
$607$	26295.5	1.75832	0.879160	+	0.476527i	$0.158105\pi$
$608$	0	0
$609$	−3508.27	−0.233436
$610$	0	0
$611$	− 1032.44i	− 0.0683600i
$612$	0	0
$613$	− 3706.64i	− 0.244225i	−0.992516	−	0.122112i	$-0.961033\pi$
$613$	− 3706.64i	− 0.244225i	0.992516	−	0.122112i	$-0.0389669\pi$
$614$	0	0
$615$	18801.0	1.23273
$616$	0	0
$617$	−19879.0	−1.29708	−0.648539	−	0.761181i	$-0.724620\pi$
$617$	−19879.0	−1.29708	−0.648539	+	0.761181i	$0.724620\pi$
$618$	0	0
$619$	4047.25i	0.262799i	0.991330	+	0.131399i	$0.0419470\pi$
$619$	4047.25i	0.262799i	−0.991330	+	0.131399i	$0.958053\pi$
$620$	0	0
$621$	− 392.691i	− 0.0253754i
$622$	0	0
$623$	4506.85	0.289828
$624$	0	0
$625$	−19465.0	−1.24576
$626$	0	0
$627$	− 421.617i	− 0.0268545i
$628$	0	0
$629$	79.2571i	0.00502415i
$630$	0	0
$631$	−27715.5	−1.74855	−0.874277	−	0.485428i	$-0.838664\pi$
$631$	−27715.5	−1.74855	−0.874277	+	0.485428i	$0.838664\pi$
$632$	0	0
$633$	−3998.34	−0.251058
$634$	0	0
$635$	− 17891.7i	− 1.11813i
$636$	0	0
$637$	− 1508.93i	− 0.0938554i
$638$	0	0
$639$	4535.95	0.280813
$640$	0	0
$641$	18708.7	1.15281	0.576405	−	0.817164i	$-0.304455\pi$
$641$	18708.7	1.15281	0.576405	+	0.817164i	$0.304455\pi$
$642$	0	0
$643$	15544.7i	0.953379i	0.879072	+	0.476689i	$0.158163\pi$
$643$	15544.7i	0.953379i	−0.879072	+	0.476689i	$0.841837\pi$
$644$	0	0
$645$	− 3630.86i	− 0.221651i
$646$	0	0
$647$	7016.63	0.426356	0.213178	−	0.977013i	$-0.431619\pi$
$647$	7016.63	0.426356	0.213178	+	0.977013i	$0.431619\pi$
$648$	0	0
$649$	10210.8	0.617582
$650$	0	0
$651$	− 4173.08i	− 0.251238i
$652$	0	0
$653$	11732.4i	0.703102i	0.936169	+	0.351551i	$0.114346\pi$
$653$	11732.4i	0.703102i	−0.936169	+	0.351551i	$0.885654\pi$
$654$	0	0
$655$	33290.5	1.98591
$656$	0	0
$657$	10148.5	0.602636
$658$	0	0
$659$	30900.3i	1.82657i	0.407327	+	0.913283i	$0.366461\pi$
$659$	30900.3i	1.82657i	−0.407327	+	0.913283i	$0.633539\pi$
$660$	0	0
$661$	− 23522.3i	− 1.38413i	−0.721835	−	0.692065i	$-0.756701\pi$
$661$	− 23522.3i	− 1.38413i	0.721835	−	0.692065i	$-0.243299\pi$
$662$	0	0
$663$	−22.7441	−0.00133229
$664$	0	0
$665$	−266.778	−0.0155567
$666$	0	0
$667$	− 2429.75i	− 0.141050i
$668$	0	0
$669$	8408.23i	0.485921i
$670$	0	0
$671$	24603.0	1.41548
$672$	0	0
$673$	−31873.0	−1.82558	−0.912790	−	0.408429i	$-0.866077\pi$
$673$	−31873.0	−1.82558	−0.912790	+	0.408429i	$0.866077\pi$
$674$	0	0
$675$	− 1907.26i	− 0.108756i
$676$	0	0
$677$	− 28606.2i	− 1.62397i	−0.583681	−	0.811983i	$-0.698388\pi$
$677$	− 28606.2i	− 1.62397i	0.583681	−	0.811983i	$-0.301612\pi$
$678$	0	0
$679$	1180.12	0.0666993
$680$	0	0
$681$	8801.20	0.495246
$682$	0	0
$683$	− 18459.6i	− 1.03417i	−0.855935	−	0.517084i	$-0.827018\pi$
$683$	− 18459.6i	− 1.03417i	0.855935	−	0.517084i	$-0.172982\pi$
$684$	0	0
$685$	17790.8i	0.992337i
$686$	0	0
$687$	11226.6	0.623464
$688$	0	0
$689$	9232.51	0.510494
$690$	0	0
$691$	− 750.319i	− 0.0413075i	−0.999787	−	0.0206538i	$-0.993425\pi$
$691$	− 750.319i	− 0.0413075i	0.999787	−	0.0206538i	$-0.00657476\pi$
$692$	0	0
$693$	3249.48i	0.178121i
$694$	0	0
$695$	−11027.2	−0.601851
$696$	0	0
$697$	110.308	0.00599455
$698$	0	0
$699$	3512.23i	0.190050i
$700$	0	0
$701$	12365.7i	0.666257i	0.942881	+	0.333129i	$0.108104\pi$
$701$	12365.7i	0.666257i	−0.942881	+	0.333129i	$0.891896\pi$
$702$	0	0
$703$	877.177	0.0470602
$704$	0	0
$705$	1406.83	0.0751549
$706$	0	0
$707$	− 794.360i	− 0.0422560i
$708$	0	0
$709$	36658.2i	1.94179i	0.239511	+	0.970894i	$0.423013\pi$
$709$	36658.2i	1.94179i	−0.239511	+	0.970894i	$0.576987\pi$
$710$	0	0
$711$	6161.10	0.324978
$712$	0	0
$713$	2890.18	0.151807
$714$	0	0
$715$	22216.4i	1.16202i
$716$	0	0
$717$	4829.26i	0.251537i
$718$	0	0
$719$	22654.9	1.17508	0.587542	−	0.809194i	$-0.300096\pi$
$719$	22654.9	1.17508	0.587542	+	0.809194i	$0.300096\pi$
$720$	0	0
$721$	−7869.96	−0.406508
$722$	0	0
$723$	16224.4i	0.834565i
$724$	0	0
$725$	− 11801.1i	− 0.604524i
$726$	0	0
$727$	−2446.73	−0.124820	−0.0624100	−	0.998051i	$-0.519879\pi$
$727$	−2446.73	−0.124820	−0.0624100	+	0.998051i	$0.519879\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 21.3027i	− 0.00107785i
$732$	0	0
$733$	10555.2i	0.531874i	0.963990	+	0.265937i	$0.0856813\pi$
$733$	10555.2i	0.531874i	−0.963990	+	0.265937i	$0.914319\pi$
$734$	0	0
$735$	2056.11	0.103185
$736$	0	0
$737$	42527.0	2.12551
$738$	0	0
$739$	21756.1i	1.08296i	0.840713	+	0.541481i	$0.182136\pi$
$739$	21756.1i	1.08296i	−0.840713	+	0.541481i	$0.817864\pi$
$740$	0	0
$741$	251.720i	0.0124793i
$742$	0	0
$743$	4240.78	0.209393	0.104697	−	0.994504i	$-0.466613\pi$
$743$	4240.78	0.209393	0.104697	+	0.994504i	$0.466613\pi$
$744$	0	0
$745$	−27641.1	−1.35932
$746$	0	0
$747$	10076.0i	0.493521i
$748$	0	0
$749$	− 7516.91i	− 0.366705i
$750$	0	0
$751$	−17371.5	−0.844067	−0.422034	−	0.906580i	$-0.638684\pi$
$751$	−17371.5	−0.844067	−0.422034	+	0.906580i	$0.638684\pi$
$752$	0	0
$753$	−15899.4	−0.769463
$754$	0	0
$755$	− 9380.32i	− 0.452165i
$756$	0	0
$757$	− 33342.0i	− 1.60084i	−0.599439	−	0.800420i	$-0.704610\pi$
$757$	− 33342.0i	− 1.60084i	0.599439	−	0.800420i	$-0.295390\pi$
$758$	0	0
$759$	−2250.52	−0.107627
$760$	0	0
$761$	10831.6	0.515962	0.257981	−	0.966150i	$-0.416943\pi$
$761$	10831.6	0.515962	0.257981	+	0.966150i	$0.416943\pi$
$762$	0	0
$763$	11771.0i	0.558505i
$764$	0	0
$765$	− 30.9917i	− 0.00146472i
$766$	0	0
$767$	−6096.21	−0.286990
$768$	0	0
$769$	−19819.2	−0.929389	−0.464695	−	0.885471i	$-0.653836\pi$
$769$	−19819.2	−0.929389	−0.464695	+	0.885471i	$0.653836\pi$
$770$	0	0
$771$	− 128.704i	− 0.00601189i
$772$	0	0
$773$	− 19421.7i	− 0.903685i	−0.892098	−	0.451843i	$-0.850767\pi$
$773$	− 19421.7i	− 0.903685i	0.892098	−	0.451843i	$-0.149233\pi$
$774$	0	0
$775$	14037.3	0.650627
$776$	0	0
$777$	−6760.56	−0.312141
$778$	0	0
$779$	− 1220.83i	− 0.0561498i
$780$	0	0
$781$	− 25995.6i	− 1.19103i
$782$	0	0
$783$	−4510.64	−0.205871
$784$	0	0
$785$	−4723.99	−0.214785
$786$	0	0
$787$	634.033i	0.0287177i	0.999897	+	0.0143589i	$0.00457072\pi$
$787$	634.033i	0.0287177i	−0.999897	+	0.0143589i	$0.995429\pi$
$788$	0	0
$789$	10691.5i	0.482416i
$790$	0	0
$791$	11979.7	0.538496
$792$	0	0
$793$	−14688.8	−0.657775
$794$	0	0
$795$	12580.5i	0.561236i
$796$	0	0
$797$	26293.4i	1.16858i	0.811545	+	0.584290i	$0.198627\pi$
$797$	26293.4i	1.16858i	−0.811545	+	0.584290i	$0.801373\pi$
$798$	0	0
$799$	8.25404	0.000365465 0
$800$	0	0
$801$	5794.52	0.255605
$802$	0	0
$803$	− 58161.3i	− 2.55600i
$804$	0	0
$805$	1424.01i	0.0623476i
$806$	0	0
$807$	15879.8	0.692683
$808$	0	0
$809$	−10342.5	−0.449473	−0.224737	−	0.974420i	$-0.572152\pi$
$809$	−10342.5	−0.449473	−0.224737	+	0.974420i	$0.572152\pi$
$810$	0	0
$811$	− 16566.5i	− 0.717298i	−0.933472	−	0.358649i	$-0.883237\pi$
$811$	− 16566.5i	− 0.717298i	0.933472	−	0.358649i	$-0.116763\pi$
$812$	0	0
$813$	− 7634.81i	− 0.329354i
$814$	0	0
$815$	−2092.56	−0.0899376
$816$	0	0
$817$	−235.768	−0.0100960
$818$	0	0
$819$	− 1940.05i	− 0.0827727i
$820$	0	0
$821$	− 28950.0i	− 1.23065i	−0.788275	−	0.615323i	$-0.789025\pi$
$821$	− 28950.0i	− 1.23065i	0.788275	−	0.615323i	$-0.210975\pi$
$822$	0	0
$823$	−17682.2	−0.748922	−0.374461	−	0.927243i	$-0.622172\pi$
$823$	−17682.2	−0.748922	−0.374461	+	0.927243i	$0.622172\pi$
$824$	0	0
$825$	−10930.6	−0.461276
$826$	0	0
$827$	40381.1i	1.69793i	0.528449	+	0.848965i	$0.322774\pi$
$827$	40381.1i	1.69793i	−0.528449	+	0.848965i	$0.677226\pi$
$828$	0	0
$829$	− 37226.2i	− 1.55961i	−0.626022	−	0.779806i	$-0.715318\pi$
$829$	− 37226.2i	− 1.55961i	0.626022	−	0.779806i	$-0.284682\pi$
$830$	0	0
$831$	16081.0	0.671294
$832$	0	0
$833$	12.0634	0.000501768 0
$834$	0	0
$835$	− 46815.4i	− 1.94025i
$836$	0	0
$837$	− 5365.39i	− 0.221571i
$838$	0	0
$839$	−19414.0	−0.798862	−0.399431	−	0.916763i	$-0.630792\pi$
$839$	−19414.0	−0.798862	−0.399431	+	0.916763i	$0.630792\pi$
$840$	0	0
$841$	−3520.23	−0.144337
$842$	0	0
$843$	− 11554.9i	− 0.472090i
$844$	0	0
$845$	17465.7i	0.711053i
$846$	0	0
$847$	9305.83	0.377511
$848$	0	0
$849$	26653.2	1.07743
$850$	0	0
$851$	− 4682.21i	− 0.188606i
$852$	0	0
$853$	11641.9i	0.467304i	0.972320	+	0.233652i	$0.0750677\pi$
$853$	11641.9i	0.467304i	−0.972320	+	0.233652i	$0.924932\pi$
$854$	0	0
$855$	−343.000	−0.0137197
$856$	0	0
$857$	18755.4	0.747575	0.373788	−	0.927514i	$-0.378059\pi$
$857$	18755.4	0.747575	0.373788	+	0.927514i	$0.378059\pi$
$858$	0	0
$859$	− 5477.43i	− 0.217564i	−0.994066	−	0.108782i	$-0.965305\pi$
$859$	− 5477.43i	− 0.217564i	0.994066	−	0.108782i	$-0.0346951\pi$
$860$	0	0
$861$	9409.15i	0.372431i
$862$	0	0
$863$	44169.7	1.74224	0.871120	−	0.491071i	$-0.163394\pi$
$863$	44169.7	1.74224	0.871120	+	0.491071i	$0.163394\pi$
$864$	0	0
$865$	1211.31	0.0476134
$866$	0	0
$867$	14738.8i	0.577343i
$868$	0	0
$869$	− 35309.3i	− 1.37835i
$870$	0	0
$871$	−25390.0	−0.987725
$872$	0	0
$873$	1517.30	0.0588233
$874$	0	0
$875$	− 5322.44i	− 0.205636i
$876$	0	0
$877$	25235.0i	0.971636i	0.874060	+	0.485818i	$0.161478\pi$
$877$	25235.0i	0.971636i	−0.874060	+	0.485818i	$0.838522\pi$
$878$	0	0
$879$	12924.0	0.495924
$880$	0	0
$881$	14930.3	0.570957	0.285479	−	0.958385i	$-0.407847\pi$
$881$	14930.3	0.570957	0.285479	+	0.958385i	$0.407847\pi$
$882$	0	0
$883$	− 9325.73i	− 0.355420i	−0.984083	−	0.177710i	$-0.943131\pi$
$883$	− 9325.73i	− 0.355420i	0.984083	−	0.177710i	$-0.0568689\pi$
$884$	0	0
$885$	− 8306.86i	− 0.315517i
$886$	0	0
$887$	−13393.6	−0.507006	−0.253503	−	0.967335i	$-0.581583\pi$
$887$	−13393.6	−0.507006	−0.253503	+	0.967335i	$0.581583\pi$
$888$	0	0
$889$	8954.10	0.337808
$890$	0	0
$891$	4177.91i	0.157088i
$892$	0	0
$893$	− 91.3514i	− 0.00342324i
$894$	0	0
$895$	32262.4	1.20493
$896$	0	0
$897$	1343.63	0.0500141
$898$	0	0
$899$	− 33198.0i	− 1.23161i
$900$	0	0
$901$	73.8111i	0.00272920i
$902$	0	0
$903$	1817.10	0.0669651
$904$	0	0
$905$	29389.4	1.07949
$906$	0	0
$907$	34813.6i	1.27449i	0.770660	+	0.637247i	$0.219927\pi$
$907$	34813.6i	1.27449i	−0.770660	+	0.637247i	$0.780073\pi$
$908$	0	0
$909$	− 1021.32i	− 0.0372663i
$910$	0	0
$911$	−51367.2	−1.86814	−0.934068	−	0.357096i	$-0.883767\pi$
$911$	−51367.2	−1.86814	−0.934068	+	0.357096i	$0.883767\pi$
$912$	0	0
$913$	57745.5	2.09321
$914$	0	0
$915$	− 20015.4i	− 0.723157i
$916$	0	0
$917$	16660.6i	0.599980i
$918$	0	0
$919$	−25493.9	−0.915089	−0.457545	−	0.889187i	$-0.651271\pi$
$919$	−25493.9	−0.915089	−0.457545	+	0.889187i	$0.651271\pi$
$920$	0	0
$921$	−11762.9	−0.420848
$922$	0	0
$923$	15520.2i	0.553472i
$924$	0	0
$925$	− 22741.1i	− 0.808347i
$926$	0	0
$927$	−10118.5	−0.358507
$928$	0	0
$929$	−13198.9	−0.466139	−0.233069	−	0.972460i	$-0.574877\pi$
$929$	−13198.9	−0.466139	−0.233069	+	0.972460i	$0.574877\pi$
$930$	0	0
$931$	− 133.512i	− 0.00469997i
$932$	0	0
$933$	21458.4i	0.752966i
$934$	0	0
$935$	−177.614	−0.00621240
$936$	0	0
$937$	−25935.6	−0.904245	−0.452123	−	0.891956i	$-0.649333\pi$
$937$	−25935.6	−0.904245	−0.452123	+	0.891956i	$0.649333\pi$
$938$	0	0
$939$	− 2262.68i	− 0.0786365i
$940$	0	0
$941$	− 51952.9i	− 1.79980i	−0.436093	−	0.899901i	$-0.643638\pi$
$941$	− 51952.9i	− 1.79980i	0.436093	−	0.899901i	$-0.356362\pi$
$942$	0	0
$943$	−6516.56	−0.225035
$944$	0	0
$945$	2643.56	0.0910002
$946$	0	0
$947$	− 27316.9i	− 0.937359i	−0.883368	−	0.468680i	$-0.844730\pi$
$947$	− 27316.9i	− 0.937359i	0.883368	−	0.468680i	$-0.155270\pi$
$948$	0	0
$949$	34724.2i	1.18777i
$950$	0	0
$951$	−23862.7	−0.813671
$952$	0	0
$953$	2017.13	0.0685636	0.0342818	−	0.999412i	$-0.489086\pi$
$953$	2017.13	0.0685636	0.0342818	+	0.999412i	$0.489086\pi$
$954$	0	0
$955$	− 21197.6i	− 0.718261i
$956$	0	0
$957$	25850.5i	0.873175i
$958$	0	0
$959$	−8903.58	−0.299804
$960$	0	0
$961$	9697.89	0.325531
$962$	0	0
$963$	− 9664.60i	− 0.323403i
$964$	0	0
$965$	− 19307.6i	− 0.644075i
$966$	0	0
$967$	−35636.5	−1.18510	−0.592550	−	0.805534i	$-0.701879\pi$
$967$	−35636.5	−1.18510	−0.592550	+	0.805534i	$0.701879\pi$
$968$	0	0
$969$	−2.01242	−6.67166e−5 0
$970$	0	0
$971$	− 10179.9i	− 0.336445i	−0.985749	−	0.168222i	$-0.946197\pi$
$971$	− 10179.9i	− 0.336445i	0.985749	−	0.168222i	$-0.0538026\pi$
$972$	0	0
$973$	− 5518.69i	− 0.181831i
$974$	0	0
$975$	6525.90	0.214355
$976$	0	0
$977$	−60124.9	−1.96885	−0.984425	−	0.175806i	$-0.943747\pi$
$977$	−60124.9	−1.96885	−0.984425	+	0.175806i	$0.943747\pi$
$978$	0	0
$979$	− 33208.5i	− 1.08411i
$980$	0	0
$981$	15134.2i	0.492555i
$982$	0	0
$983$	−54363.0	−1.76390	−0.881948	−	0.471347i	$-0.843768\pi$
$983$	−54363.0	−1.76390	−0.881948	+	0.471347i	$0.843768\pi$
$984$	0	0
$985$	14790.6	0.478443
$986$	0	0
$987$	704.062i	0.0227057i
$988$	0	0
$989$	1258.48i	0.0404626i
$990$	0	0
$991$	32372.1	1.03767	0.518836	−	0.854874i	$-0.326365\pi$
$991$	32372.1	1.03767	0.518836	+	0.854874i	$0.326365\pi$
$992$	0	0
$993$	−26758.1	−0.855128
$994$	0	0
$995$	− 13743.3i	− 0.437882i
$996$	0	0
$997$	28233.4i	0.896851i	0.893820	+	0.448425i	$0.148015\pi$
$997$	28233.4i	0.896851i	−0.893820	+	0.448425i	$0.851985\pi$
$998$	0	0
$999$	−8692.15	−0.275283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.5	✓	12
4.3	odd	2		1344.4.c.g.673.11	yes	12
8.3	odd	2		1344.4.c.g.673.2	yes	12
8.5	even	2	inner	1344.4.c.f.673.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.5	✓	12	1.1	even	1	trivial
1344.4.c.f.673.8	yes	12	8.5	even	2	inner
1344.4.c.g.673.2	yes	12	8.3	odd	2
1344.4.c.g.673.11	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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +13.9871i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +13.9871i q^{5} -7.00000 q^{7} -9.00000 q^{9} +51.5791i q^{11} -30.7945i q^{13} +41.9613 q^{15} +0.246192 q^{17} -2.72473i q^{19} +21.0000i q^{21} -14.5441 q^{23} -70.6394 q^{25} +27.0000i q^{27} +167.061i q^{29} -198.718 q^{31} +154.737 q^{33} -97.9098i q^{35} +321.932i q^{37} -92.3834 q^{39} +448.055 q^{41} -86.5288i q^{43} -125.884i q^{45} +33.5268 q^{47} +49.0000 q^{49} -0.738577i q^{51} +299.811i q^{53} -721.443 q^{55} -8.17419 q^{57} -197.965i q^{59} -476.996i q^{61} +63.0000 q^{63} +430.726 q^{65} -824.500i q^{67} +43.6323i q^{69} -503.994 q^{71} -1127.61 q^{73} +211.918i q^{75} -361.054i q^{77} -684.567 q^{79} +81.0000 q^{81} -1119.55i q^{83} +3.44352i q^{85} +501.182 q^{87} -643.835 q^{89} +215.561i q^{91} +596.154i q^{93} +38.1111 q^{95} -168.589 q^{97} -464.212i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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