LMFDB - Embedded newform 441.4.c.b.440.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.8
Character	$\chi$	$=$	441.440
Dual form			441.4.c.b.440.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.70881i q^{2} -5.75528 q^{4} -4.54675 q^{5} +8.32523i q^{8} +O(q^{10})$	$q+3.70881i q^{2} -5.75528 q^{4} -4.54675 q^{5} +8.32523i q^{8} -16.8631i q^{10} -45.5164i q^{11} -11.6749i q^{13} -76.9190 q^{16} +91.5157 q^{17} -140.085i q^{19} +26.1679 q^{20} +168.812 q^{22} +45.0189i q^{23} -104.327 q^{25} +43.3000 q^{26} +47.7861i q^{29} -238.848i q^{31} -218.676i q^{32} +339.415i q^{34} -148.927 q^{37} +519.549 q^{38} -37.8528i q^{40} +393.755 q^{41} +412.265 q^{43} +261.960i q^{44} -166.967 q^{46} +408.757 q^{47} -386.929i q^{50} +67.1924i q^{52} +167.070i q^{53} +206.952i q^{55} -177.230 q^{58} +137.537 q^{59} -323.364i q^{61} +885.842 q^{62} +195.677 q^{64} +53.0829i q^{65} +424.152 q^{67} -526.699 q^{68} -727.190i q^{71} +1096.11i q^{73} -552.342i q^{74} +806.229i q^{76} -669.260 q^{79} +349.732 q^{80} +1460.36i q^{82} +199.014 q^{83} -416.100 q^{85} +1529.01i q^{86} +378.934 q^{88} -807.348 q^{89} -259.096i q^{92} +1516.00i q^{94} +636.932i q^{95} -701.053i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{4}+O(q^{10}) $ 24 * q - 96 * q^4	$ 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) $ 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.70881i	1.31126i	0.755081	+	0.655631i	$0.227597\pi$
$2$	3.70881i	1.31126i	−0.755081	+	0.655631i	$0.772403\pi$
$3$	0	0
$3$	0	0
$4$	−5.75528	−0.719410
$5$	−4.54675	−0.406674	−0.203337	−	0.979109i	$-0.565179\pi$
$5$	−4.54675	−0.406674	−0.203337	+	0.979109i	$0.565179\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.32523i	0.367927i
$9$	0	0
$10$	− 16.8631i	− 0.533257i
$11$	− 45.5164i	− 1.24761i	−0.781580	−	0.623805i	$-0.785586\pi$
$11$	− 45.5164i	− 1.24761i	0.781580	−	0.623805i	$-0.214414\pi$
$12$	0	0
$13$	− 11.6749i	− 0.249080i	−0.992215	−	0.124540i	$-0.960255\pi$
$13$	− 11.6749i	− 0.249080i	0.992215	−	0.124540i	$-0.0397455\pi$
$14$	0	0
$15$	0	0
$16$	−76.9190	−1.20186
$17$	91.5157	1.30564	0.652818	−	0.757514i	$-0.273586\pi$
$17$	91.5157	1.30564	0.652818	+	0.757514i	$0.273586\pi$
$18$	0	0
$19$	− 140.085i	− 1.69146i	−0.533612	−	0.845730i	$-0.679166\pi$
$19$	− 140.085i	− 1.69146i	0.533612	−	0.845730i	$-0.320834\pi$
$20$	26.1679	0.292566
$21$	0	0
$22$	168.812	1.63594
$23$	45.0189i	0.408134i	0.978957	+	0.204067i	$0.0654161\pi$
$23$	45.0189i	0.408134i	−0.978957	+	0.204067i	$0.934584\pi$
$24$	0	0
$25$	−104.327	−0.834616
$26$	43.3000	0.326609
$27$	0	0
$28$	0	0
$29$	47.7861i	0.305988i	0.988227	+	0.152994i	$0.0488916\pi$
$29$	47.7861i	0.305988i	−0.988227	+	0.152994i	$0.951108\pi$
$30$	0	0
$31$	− 238.848i	− 1.38382i	−0.721985	−	0.691909i	$-0.756770\pi$
$31$	− 238.848i	− 1.38382i	0.721985	−	0.691909i	$-0.243230\pi$
$32$	− 218.676i	− 1.20803i
$33$	0	0
$34$	339.415i	1.71203i
$35$	0	0
$36$	0	0
$37$	−148.927	−0.661715	−0.330858	−	0.943681i	$-0.607338\pi$
$37$	−148.927	−0.661715	−0.330858	+	0.943681i	$0.607338\pi$
$38$	519.549	2.21795
$39$	0	0
$40$	− 37.8528i	− 0.149626i
$41$	393.755	1.49986	0.749929	−	0.661518i	$-0.230088\pi$
$41$	393.755	1.49986	0.749929	+	0.661518i	$0.230088\pi$
$42$	0	0
$43$	412.265	1.46209	0.731045	−	0.682329i	$-0.239033\pi$
$43$	412.265	1.46209	0.731045	+	0.682329i	$0.239033\pi$
$44$	261.960i	0.897543i
$45$	0	0
$46$	−166.967	−0.535171
$47$	408.757	1.26858	0.634292	−	0.773094i	$-0.281292\pi$
$47$	408.757	1.26858	0.634292	+	0.773094i	$0.281292\pi$
$48$	0	0
$49$	0	0
$50$	− 386.929i	− 1.09440i
$51$	0	0
$52$	67.1924i	0.179191i
$53$	167.070i	0.432997i	0.976283	+	0.216498i	$0.0694636\pi$
$53$	167.070i	0.432997i	−0.976283	+	0.216498i	$0.930536\pi$
$54$	0	0
$55$	206.952i	0.507370i
$56$	0	0
$57$	0	0
$58$	−177.230	−0.401231
$59$	137.537	0.303489	0.151744	−	0.988420i	$-0.451511\pi$
$59$	137.537	0.303489	0.151744	+	0.988420i	$0.451511\pi$
$60$	0	0
$61$	− 323.364i	− 0.678730i	−0.940655	−	0.339365i	$-0.889788\pi$
$61$	− 323.364i	− 0.678730i	0.940655	−	0.339365i	$-0.110212\pi$
$62$	885.842	1.81455
$63$	0	0
$64$	195.677	0.382181
$65$	53.0829i	0.101294i
$66$	0	0
$67$	424.152	0.773409	0.386705	−	0.922204i	$-0.373613\pi$
$67$	424.152	0.773409	0.386705	+	0.922204i	$0.373613\pi$
$68$	−526.699	−0.939289
$69$	0	0
$70$	0	0
$71$	− 727.190i	− 1.21551i	−0.794123	−	0.607757i	$-0.792069\pi$
$71$	− 727.190i	− 1.21551i	0.794123	−	0.607757i	$-0.207931\pi$
$72$	0	0
$73$	1096.11i	1.75740i	0.477377	+	0.878699i	$0.341588\pi$
$73$	1096.11i	1.75740i	−0.477377	+	0.878699i	$0.658412\pi$
$74$	− 552.342i	− 0.867682i
$75$	0	0
$76$	806.229i	1.21685i
$77$	0	0
$78$	0	0
$79$	−669.260	−0.953135	−0.476568	−	0.879138i	$-0.658119\pi$
$79$	−669.260	−0.953135	−0.476568	+	0.879138i	$0.658119\pi$
$80$	349.732	0.488765
$81$	0	0
$82$	1460.36i	1.96671i
$83$	199.014	0.263189	0.131594	−	0.991304i	$-0.457990\pi$
$83$	199.014	0.263189	0.131594	+	0.991304i	$0.457990\pi$
$84$	0	0
$85$	−416.100	−0.530969
$86$	1529.01i	1.91718i
$87$	0	0
$88$	378.934	0.459029
$89$	−807.348	−0.961559	−0.480779	−	0.876842i	$-0.659646\pi$
$89$	−807.348	−0.961559	−0.480779	+	0.876842i	$0.659646\pi$
$90$	0	0
$91$	0	0
$92$	− 259.096i	− 0.293616i
$93$	0	0
$94$	1516.00i	1.66345i
$95$	636.932i	0.687872i
$96$	0	0
$97$	− 701.053i	− 0.733827i	−0.930255	−	0.366913i	$-0.880415\pi$
$97$	− 701.053i	− 0.733827i	0.930255	−	0.366913i	$-0.119585\pi$
$98$	0	0
$99$	0	0
$100$	600.432	0.600432
$101$	−1702.58	−1.67736	−0.838680	−	0.544625i	$-0.816672\pi$
$101$	−1702.58	−1.67736	−0.838680	+	0.544625i	$0.816672\pi$
$102$	0	0
$103$	− 1134.23i	− 1.08504i	−0.840042	−	0.542521i	$-0.817470\pi$
$103$	− 1134.23i	− 1.08504i	0.840042	−	0.542521i	$-0.182530\pi$
$104$	97.1963	0.0916431
$105$	0	0
$106$	−619.631	−0.567773
$107$	338.726i	0.306036i	0.988223	+	0.153018i	$0.0488993\pi$
$107$	338.726i	0.306036i	−0.988223	+	0.153018i	$0.951101\pi$
$108$	0	0
$109$	1975.54	1.73598	0.867991	−	0.496580i	$-0.165411\pi$
$109$	1975.54	1.73598	0.867991	+	0.496580i	$0.165411\pi$
$110$	−767.545	−0.665296
$111$	0	0
$112$	0	0
$113$	− 1685.74i	− 1.40337i	−0.712488	−	0.701685i	$-0.752432\pi$
$113$	− 1685.74i	− 1.40337i	0.712488	−	0.701685i	$-0.247568\pi$
$114$	0	0
$115$	− 204.690i	− 0.165978i
$116$	− 275.023i	− 0.220131i
$117$	0	0
$118$	510.100i	0.397954i
$119$	0	0
$120$	0	0
$121$	−740.739	−0.556529
$122$	1199.30	0.889993
$123$	0	0
$124$	1374.64i	0.995533i
$125$	1042.69	0.746091
$126$	0	0
$127$	−1155.98	−0.807691	−0.403846	−	0.914827i	$-0.632327\pi$
$127$	−1155.98	−0.807691	−0.403846	+	0.914827i	$0.632327\pi$
$128$	− 1023.68i	− 0.706886i
$129$	0	0
$130$	−196.875	−0.132823
$131$	635.067	0.423558	0.211779	−	0.977318i	$-0.432074\pi$
$131$	635.067	0.423558	0.211779	+	0.977318i	$0.432074\pi$
$132$	0	0
$133$	0	0
$134$	1573.10i	1.01414i
$135$	0	0
$136$	761.890i	0.480379i
$137$	2458.12i	1.53293i	0.642285	+	0.766466i	$0.277987\pi$
$137$	2458.12i	1.53293i	−0.642285	+	0.766466i	$0.722013\pi$
$138$	0	0
$139$	− 1094.81i	− 0.668062i	−0.942562	−	0.334031i	$-0.891591\pi$
$139$	− 1094.81i	− 0.668062i	0.942562	−	0.334031i	$-0.108409\pi$
$140$	0	0
$141$	0	0
$142$	2697.01	1.59386
$143$	−531.399	−0.310754
$144$	0	0
$145$	− 217.272i	− 0.124437i
$146$	−4065.27	−2.30441
$147$	0	0
$148$	857.117	0.476045
$149$	− 966.514i	− 0.531409i	−0.964055	−	0.265704i	$-0.914395\pi$
$149$	− 966.514i	− 0.531409i	0.964055	−	0.265704i	$-0.0856045\pi$
$150$	0	0
$151$	−2329.88	−1.25565	−0.627825	−	0.778354i	$-0.716055\pi$
$151$	−2329.88	−1.25565	−0.627825	+	0.778354i	$0.716055\pi$
$152$	1166.24	0.622333
$153$	0	0
$154$	0	0
$155$	1085.98i	0.562763i
$156$	0	0
$157$	− 3414.89i	− 1.73591i	−0.496641	−	0.867956i	$-0.665433\pi$
$157$	− 3414.89i	− 1.73591i	0.496641	−	0.867956i	$-0.334567\pi$
$158$	− 2482.16i	− 1.24981i
$159$	0	0
$160$	994.267i	0.491273i
$161$	0	0
$162$	0	0
$163$	−361.669	−0.173792	−0.0868960	−	0.996217i	$-0.527695\pi$
$163$	−361.669	−0.173792	−0.0868960	+	0.996217i	$0.527695\pi$
$164$	−2266.17	−1.07901
$165$	0	0
$166$	738.107i	0.345110i
$167$	1605.57	0.743969	0.371984	−	0.928239i	$-0.378678\pi$
$167$	1605.57	0.743969	0.371984	+	0.928239i	$0.378678\pi$
$168$	0	0
$169$	2060.70	0.937959
$170$	− 1543.23i	− 0.696239i
$171$	0	0
$172$	−2372.70	−1.05184
$173$	1466.87	0.644647	0.322324	−	0.946630i	$-0.395536\pi$
$173$	1466.87	0.644647	0.322324	+	0.946630i	$0.395536\pi$
$174$	0	0
$175$	0	0
$176$	3501.07i	1.49945i
$177$	0	0
$178$	− 2994.30i	− 1.26086i
$179$	981.846i	0.409981i	0.978764	+	0.204990i	$0.0657163\pi$
$179$	981.846i	0.409981i	−0.978764	+	0.204990i	$0.934284\pi$
$180$	0	0
$181$	− 1547.32i	− 0.635422i	−0.948188	−	0.317711i	$-0.897086\pi$
$181$	− 1547.32i	− 0.635422i	0.948188	−	0.317711i	$-0.102914\pi$
$182$	0	0
$183$	0	0
$184$	−374.793	−0.150163
$185$	677.135	0.269102
$186$	0	0
$187$	− 4165.46i	− 1.62892i
$188$	−2352.52	−0.912632
$189$	0	0
$190$	−2362.26	−0.901982
$191$	− 4784.63i	− 1.81259i	−0.422650	−	0.906293i	$-0.638900\pi$
$191$	− 4784.63i	− 1.81259i	0.422650	−	0.906293i	$-0.361100\pi$
$192$	0	0
$193$	−1276.37	−0.476037	−0.238019	−	0.971261i	$-0.576498\pi$
$193$	−1276.37	−0.476037	−0.238019	+	0.971261i	$0.576498\pi$
$194$	2600.08	0.962240
$195$	0	0
$196$	0	0
$197$	491.111i	0.177615i	0.996049	+	0.0888076i	$0.0283056\pi$
$197$	491.111i	0.177615i	−0.996049	+	0.0888076i	$0.971694\pi$
$198$	0	0
$199$	− 2988.44i	− 1.06455i	−0.846573	−	0.532273i	$-0.821338\pi$
$199$	− 2988.44i	− 1.06455i	0.846573	−	0.532273i	$-0.178662\pi$
$200$	− 868.547i	− 0.307078i
$201$	0	0
$202$	− 6314.56i	− 2.19946i
$203$	0	0
$204$	0	0
$205$	−1790.31	−0.609953
$206$	4206.66	1.42278
$207$	0	0
$208$	898.022i	0.299359i
$209$	−6376.16	−2.11028
$210$	0	0
$211$	978.467	0.319244	0.159622	−	0.987178i	$-0.448972\pi$
$211$	978.467	0.319244	0.159622	+	0.987178i	$0.448972\pi$
$212$	− 961.535i	− 0.311502i
$213$	0	0
$214$	−1256.27	−0.401294
$215$	−1874.47	−0.594594
$216$	0	0
$217$	0	0
$218$	7326.89i	2.27633i
$219$	0	0
$220$	− 1191.07i	− 0.365007i
$221$	− 1068.44i	− 0.325208i
$222$	0	0
$223$	727.150i	0.218357i	0.994022	+	0.109178i	$0.0348220\pi$
$223$	727.150i	0.218357i	−0.994022	+	0.109178i	$0.965178\pi$
$224$	0	0
$225$	0	0
$226$	6252.08	1.84019
$227$	−6230.08	−1.82161	−0.910803	−	0.412841i	$-0.864537\pi$
$227$	−6230.08	−1.82161	−0.910803	+	0.412841i	$0.864537\pi$
$228$	0	0
$229$	− 5670.06i	− 1.63619i	−0.575080	−	0.818097i	$-0.695029\pi$
$229$	− 5670.06i	− 1.63619i	0.575080	−	0.818097i	$-0.304971\pi$
$230$	759.156	0.217640
$231$	0	0
$232$	−397.830	−0.112581
$233$	3953.76i	1.11167i	0.831292	+	0.555836i	$0.187602\pi$
$233$	3953.76i	1.11167i	−0.831292	+	0.555836i	$0.812398\pi$
$234$	0	0
$235$	−1858.52	−0.515900
$236$	−791.567	−0.218333
$237$	0	0
$238$	0	0
$239$	− 3916.55i	− 1.06000i	−0.847997	−	0.530001i	$-0.822192\pi$
$239$	− 3916.55i	− 1.06000i	0.847997	−	0.530001i	$-0.177808\pi$
$240$	0	0
$241$	− 3271.69i	− 0.874474i	−0.899346	−	0.437237i	$-0.855957\pi$
$241$	− 3271.69i	− 0.874474i	0.899346	−	0.437237i	$-0.144043\pi$
$242$	− 2747.26i	− 0.729755i
$243$	0	0
$244$	1861.05i	0.488285i
$245$	0	0
$246$	0	0
$247$	−1635.48	−0.421308
$248$	1988.46	0.509144
$249$	0	0
$250$	3867.15i	0.978321i
$251$	−1401.75	−0.352502	−0.176251	−	0.984345i	$-0.556397\pi$
$251$	−1401.75	−0.352502	−0.176251	+	0.984345i	$0.556397\pi$
$252$	0	0
$253$	2049.10	0.509192
$254$	− 4287.32i	− 1.05910i
$255$	0	0
$256$	5362.05	1.30910
$257$	−4088.34	−0.992310	−0.496155	−	0.868234i	$-0.665255\pi$
$257$	−4088.34	−0.992310	−0.496155	+	0.868234i	$0.665255\pi$
$258$	0	0
$259$	0	0
$260$	− 305.507i	− 0.0728721i
$261$	0	0
$262$	2355.34i	0.555396i
$263$	4612.24i	1.08138i	0.841222	+	0.540690i	$0.181837\pi$
$263$	4612.24i	1.08138i	−0.841222	+	0.540690i	$0.818163\pi$
$264$	0	0
$265$	− 759.626i	− 0.176089i
$266$	0	0
$267$	0	0
$268$	−2441.12	−0.556399
$269$	−3678.31	−0.833720	−0.416860	−	0.908971i	$-0.636869\pi$
$269$	−3678.31	−0.833720	−0.416860	+	0.908971i	$0.636869\pi$
$270$	0	0
$271$	7159.95i	1.60493i	0.596699	+	0.802465i	$0.296479\pi$
$271$	7159.95i	1.60493i	−0.596699	+	0.802465i	$0.703521\pi$
$272$	−7039.30	−1.56919
$273$	0	0
$274$	−9116.72	−2.01008
$275$	4748.59i	1.04127i
$276$	0	0
$277$	1130.70	0.245260	0.122630	−	0.992452i	$-0.460867\pi$
$277$	1130.70	0.245260	0.122630	+	0.992452i	$0.460867\pi$
$278$	4060.45	0.876005
$279$	0	0
$280$	0	0
$281$	3919.67i	0.832128i	0.909335	+	0.416064i	$0.136591\pi$
$281$	3919.67i	0.832128i	−0.909335	+	0.416064i	$0.863409\pi$
$282$	0	0
$283$	4506.76i	0.946640i	0.880890	+	0.473320i	$0.156945\pi$
$283$	4506.76i	0.946640i	−0.880890	+	0.473320i	$0.843055\pi$
$284$	4185.18i	0.874454i
$285$	0	0
$286$	− 1970.86i	− 0.407480i
$287$	0	0
$288$	0	0
$289$	3462.13	0.704688
$290$	805.820	0.163170
$291$	0	0
$292$	− 6308.42i	− 1.26429i
$293$	8956.63	1.78584	0.892922	−	0.450212i	$-0.148652\pi$
$293$	8956.63	1.78584	0.892922	+	0.450212i	$0.148652\pi$
$294$	0	0
$295$	−625.349	−0.123421
$296$	− 1239.85i	− 0.243463i
$297$	0	0
$298$	3584.62	0.696817
$299$	525.591	0.101658
$300$	0	0
$301$	0	0
$302$	− 8641.10i	− 1.64649i
$303$	0	0
$304$	10775.2i	2.03290i
$305$	1470.26i	0.276022i
$306$	0	0
$307$	3522.69i	0.654888i	0.944871	+	0.327444i	$0.106187\pi$
$307$	3522.69i	0.654888i	−0.944871	+	0.327444i	$0.893813\pi$
$308$	0	0
$309$	0	0
$310$	−4027.71	−0.737930
$311$	−3555.82	−0.648335	−0.324168	−	0.946000i	$-0.605084\pi$
$311$	−3555.82	−0.648335	−0.324168	+	0.946000i	$0.605084\pi$
$312$	0	0
$313$	6339.77i	1.14487i	0.819949	+	0.572436i	$0.194001\pi$
$313$	6339.77i	1.14487i	−0.819949	+	0.572436i	$0.805999\pi$
$314$	12665.2	2.27624
$315$	0	0
$316$	3851.78	0.685695
$317$	− 2544.86i	− 0.450895i	−0.974255	−	0.225447i	$-0.927616\pi$
$317$	− 2544.86i	− 0.450895i	0.974255	−	0.225447i	$-0.0723844\pi$
$318$	0	0
$319$	2175.05	0.381754
$320$	−889.695	−0.155423
$321$	0	0
$322$	0	0
$323$	− 12820.0i	− 2.20843i
$324$	0	0
$325$	1218.01i	0.207886i
$326$	− 1341.36i	− 0.227887i
$327$	0	0
$328$	3278.10i	0.551838i
$329$	0	0
$330$	0	0
$331$	11398.4	1.89279	0.946396	−	0.323008i	$-0.104694\pi$
$331$	11398.4	1.89279	0.946396	+	0.323008i	$0.104694\pi$
$332$	−1145.38	−0.189341
$333$	0	0
$334$	5954.76i	0.975539i
$335$	−1928.52	−0.314525
$336$	0	0
$337$	−551.119	−0.0890842	−0.0445421	−	0.999008i	$-0.514183\pi$
$337$	−551.119	−0.0890842	−0.0445421	+	0.999008i	$0.514183\pi$
$338$	7642.74i	1.22991i
$339$	0	0
$340$	2394.77	0.381984
$341$	−10871.5	−1.72646
$342$	0	0
$343$	0	0
$344$	3432.20i	0.537942i
$345$	0	0
$346$	5440.34i	0.845302i
$347$	2887.43i	0.446702i	0.974738	+	0.223351i	$0.0716996\pi$
$347$	2887.43i	0.446702i	−0.974738	+	0.223351i	$0.928300\pi$
$348$	0	0
$349$	2290.34i	0.351286i	0.984454	+	0.175643i	$0.0562005\pi$
$349$	2290.34i	0.351286i	−0.984454	+	0.175643i	$0.943800\pi$
$350$	0	0
$351$	0	0
$352$	−9953.34	−1.50714
$353$	−1409.52	−0.212525	−0.106262	−	0.994338i	$-0.533888\pi$
$353$	−1409.52	−0.212525	−0.106262	+	0.994338i	$0.533888\pi$
$354$	0	0
$355$	3306.35i	0.494318i
$356$	4646.52	0.691755
$357$	0	0
$358$	−3641.48	−0.537593
$359$	6952.66i	1.02214i	0.859540	+	0.511068i	$0.170750\pi$
$359$	6952.66i	1.02214i	−0.859540	+	0.511068i	$0.829250\pi$
$360$	0	0
$361$	−12764.8	−1.86103
$362$	5738.72	0.833205
$363$	0	0
$364$	0	0
$365$	− 4983.74i	− 0.714688i
$366$	0	0
$367$	1502.10i	0.213648i	0.994278	+	0.106824i	$0.0340682\pi$
$367$	1502.10i	0.213648i	−0.994278	+	0.106824i	$0.965932\pi$
$368$	− 3462.81i	− 0.490520i
$369$	0	0
$370$	2511.37i	0.352864i
$371$	0	0
$372$	0	0
$373$	−7055.03	−0.979345	−0.489673	−	0.871906i	$-0.662884\pi$
$373$	−7055.03	−0.979345	−0.489673	+	0.871906i	$0.662884\pi$
$374$	15448.9	2.13595
$375$	0	0
$376$	3403.00i	0.466746i
$377$	557.898	0.0762155
$378$	0	0
$379$	5163.51	0.699820	0.349910	−	0.936783i	$-0.386212\pi$
$379$	5163.51	0.699820	0.349910	+	0.936783i	$0.386212\pi$
$380$	− 3665.73i	− 0.494863i
$381$	0	0
$382$	17745.3	2.37678
$383$	−2517.92	−0.335926	−0.167963	−	0.985793i	$-0.553719\pi$
$383$	−2517.92	−0.335926	−0.167963	+	0.985793i	$0.553719\pi$
$384$	0	0
$385$	0	0
$386$	− 4733.82i	− 0.624210i
$387$	0	0
$388$	4034.76i	0.527923i
$389$	− 4786.74i	− 0.623900i	−0.950099	−	0.311950i	$-0.899018\pi$
$389$	− 4786.74i	− 0.623900i	0.950099	−	0.311950i	$-0.100982\pi$
$390$	0	0
$391$	4119.94i	0.532875i
$392$	0	0
$393$	0	0
$394$	−1821.44	−0.232900
$395$	3042.96	0.387615
$396$	0	0
$397$	− 9594.49i	− 1.21293i	−0.795110	−	0.606466i	$-0.792587\pi$
$397$	− 9594.49i	− 1.21293i	0.795110	−	0.606466i	$-0.207413\pi$
$398$	11083.6	1.39590
$399$	0	0
$400$	8024.73	1.00309
$401$	13547.4i	1.68709i	0.537059	+	0.843544i	$0.319535\pi$
$401$	13547.4i	1.68709i	−0.537059	+	0.843544i	$0.680465\pi$
$402$	0	0
$403$	−2788.53	−0.344681
$404$	9798.85	1.20671
$405$	0	0
$406$	0	0
$407$	6778.62i	0.825562i
$408$	0	0
$409$	− 2162.24i	− 0.261408i	−0.991421	−	0.130704i	$-0.958276\pi$
$409$	− 2162.24i	− 0.261408i	0.991421	−	0.130704i	$-0.0417237\pi$
$410$	− 6639.91i	− 0.799809i
$411$	0	0
$412$	6527.83i	0.780590i
$413$	0	0
$414$	0	0
$415$	−904.870	−0.107032
$416$	−2553.02	−0.300895
$417$	0	0
$418$	− 23648.0i	− 2.76713i
$419$	4907.59	0.572199	0.286099	−	0.958200i	$-0.407641\pi$
$419$	4907.59	0.572199	0.286099	+	0.958200i	$0.407641\pi$
$420$	0	0
$421$	5719.49	0.662116	0.331058	−	0.943610i	$-0.392594\pi$
$421$	5719.49	0.662116	0.331058	+	0.943610i	$0.392594\pi$
$422$	3628.95i	0.418613i
$423$	0	0
$424$	−1390.90	−0.159311
$425$	−9547.56	−1.08971
$426$	0	0
$427$	0	0
$428$	− 1949.46i	− 0.220166i
$429$	0	0
$430$	− 6952.05i	− 0.779669i
$431$	− 3826.81i	− 0.427682i	−0.976868	−	0.213841i	$-0.931403\pi$
$431$	− 3826.81i	− 0.427682i	0.976868	−	0.213841i	$-0.0685975\pi$
$432$	0	0
$433$	9120.21i	1.01222i	0.862470	+	0.506108i	$0.168916\pi$
$433$	9120.21i	1.01222i	−0.862470	+	0.506108i	$0.831084\pi$
$434$	0	0
$435$	0	0
$436$	−11369.8	−1.24888
$437$	6306.47	0.690342
$438$	0	0
$439$	2944.49i	0.320120i	0.987107	+	0.160060i	$0.0511688\pi$
$439$	2944.49i	0.320120i	−0.987107	+	0.160060i	$0.948831\pi$
$440$	−1722.92	−0.186675
$441$	0	0
$442$	3962.63	0.426433
$443$	− 4263.66i	− 0.457275i	−0.973512	−	0.228637i	$-0.926573\pi$
$443$	− 4263.66i	− 0.457275i	0.973512	−	0.228637i	$-0.0734270\pi$
$444$	0	0
$445$	3670.81	0.391041
$446$	−2696.86	−0.286323
$447$	0	0
$448$	0	0
$449$	1608.94i	0.169110i	0.996419	+	0.0845550i	$0.0269469\pi$
$449$	1608.94i	0.169110i	−0.996419	+	0.0845550i	$0.973053\pi$
$450$	0	0
$451$	− 17922.3i	− 1.87124i
$452$	9701.89i	1.00960i
$453$	0	0
$454$	− 23106.2i	− 2.38861i
$455$	0	0
$456$	0	0
$457$	−5253.81	−0.537775	−0.268887	−	0.963172i	$-0.586656\pi$
$457$	−5253.81	−0.537775	−0.268887	+	0.963172i	$0.586656\pi$
$458$	21029.2	2.14548
$459$	0	0
$460$	1178.05i	0.119406i
$461$	−19229.1	−1.94271	−0.971356	−	0.237627i	$-0.923630\pi$
$461$	−19229.1	−1.94271	−0.971356	+	0.237627i	$0.923630\pi$
$462$	0	0
$463$	6539.63	0.656420	0.328210	−	0.944605i	$-0.393555\pi$
$463$	6539.63	0.656420	0.328210	+	0.944605i	$0.393555\pi$
$464$	− 3675.66i	− 0.367755i
$465$	0	0
$466$	−14663.8	−1.45769
$467$	9017.76	0.893559	0.446780	−	0.894644i	$-0.352571\pi$
$467$	9017.76	0.893559	0.446780	+	0.894644i	$0.352571\pi$
$468$	0	0
$469$	0	0
$470$	− 6892.90i	− 0.676480i
$471$	0	0
$472$	1145.03i	0.111662i
$473$	− 18764.8i	− 1.82412i
$474$	0	0
$475$	14614.7i	1.41172i
$476$	0	0
$477$	0	0
$478$	14525.7	1.38994
$479$	8225.75	0.784643	0.392322	−	0.919828i	$-0.371672\pi$
$479$	8225.75	0.784643	0.392322	+	0.919828i	$0.371672\pi$
$480$	0	0
$481$	1738.71i	0.164820i
$482$	12134.1	1.14666
$483$	0	0
$484$	4263.17	0.400372
$485$	3187.52i	0.298428i
$486$	0	0
$487$	−18034.8	−1.67810	−0.839048	−	0.544058i	$-0.816887\pi$
$487$	−18034.8	−1.67810	−0.839048	+	0.544058i	$0.816887\pi$
$488$	2692.08	0.249723
$489$	0	0
$490$	0	0
$491$	12148.2i	1.11658i	0.829646	+	0.558290i	$0.188542\pi$
$491$	12148.2i	1.11658i	−0.829646	+	0.558290i	$0.811458\pi$
$492$	0	0
$493$	4373.18i	0.399510i
$494$	− 6065.69i	− 0.552446i
$495$	0	0
$496$	18371.9i	1.66315i
$497$	0	0
$498$	0	0
$499$	−2946.55	−0.264340	−0.132170	−	0.991227i	$-0.542195\pi$
$499$	−2946.55	−0.264340	−0.132170	+	0.991227i	$0.542195\pi$
$500$	−6001.00	−0.536746
$501$	0	0
$502$	− 5198.84i	− 0.462222i
$503$	11290.2	1.00080	0.500402	−	0.865793i	$-0.333186\pi$
$503$	11290.2	1.00080	0.500402	+	0.865793i	$0.333186\pi$
$504$	0	0
$505$	7741.23	0.682139
$506$	7599.71i	0.667684i
$507$	0	0
$508$	6653.00	0.581062
$509$	6428.81	0.559827	0.279914	−	0.960025i	$-0.409694\pi$
$509$	6428.81	0.559827	0.279914	+	0.960025i	$0.409694\pi$
$510$	0	0
$511$	0	0
$512$	11697.4i	1.00968i
$513$	0	0
$514$	− 15162.9i	− 1.30118i
$515$	5157.08i	0.441258i
$516$	0	0
$517$	− 18605.2i	− 1.58270i
$518$	0	0
$519$	0	0
$520$	−441.928	−0.0372689
$521$	−6300.76	−0.529830	−0.264915	−	0.964272i	$-0.585344\pi$
$521$	−6300.76	−0.529830	−0.264915	+	0.964272i	$0.585344\pi$
$522$	0	0
$523$	13190.3i	1.10281i	0.834238	+	0.551405i	$0.185908\pi$
$523$	13190.3i	1.10281i	−0.834238	+	0.551405i	$0.814092\pi$
$524$	−3654.99	−0.304712
$525$	0	0
$526$	−17105.9	−1.41797
$527$	− 21858.3i	− 1.80676i
$528$	0	0
$529$	10140.3	0.833427
$530$	2817.31	0.230898
$531$	0	0
$532$	0	0
$533$	− 4597.05i	− 0.373584i
$534$	0	0
$535$	− 1540.10i	− 0.124457i
$536$	3531.16i	0.284558i
$537$	0	0
$538$	− 13642.2i	− 1.09323i
$539$	0	0
$540$	0	0
$541$	11893.0	0.945137	0.472568	−	0.881294i	$-0.343327\pi$
$541$	11893.0	0.945137	0.472568	+	0.881294i	$0.343327\pi$
$542$	−26554.9	−2.10449
$543$	0	0
$544$	− 20012.3i	− 1.57724i
$545$	−8982.27	−0.705979
$546$	0	0
$547$	1348.19	0.105383	0.0526915	−	0.998611i	$-0.483220\pi$
$547$	1348.19	0.105383	0.0526915	+	0.998611i	$0.483220\pi$
$548$	− 14147.2i	− 1.10281i
$549$	0	0
$550$	−17611.6	−1.36539
$551$	6694.12	0.517567
$552$	0	0
$553$	0	0
$554$	4193.54i	0.321600i
$555$	0	0
$556$	6300.95i	0.480611i
$557$	19008.3i	1.44598i	0.690861	+	0.722988i	$0.257232\pi$
$557$	19008.3i	1.44598i	−0.690861	+	0.722988i	$0.742768\pi$
$558$	0	0
$559$	− 4813.16i	− 0.364177i
$560$	0	0
$561$	0	0
$562$	−14537.3	−1.09114
$563$	2730.52	0.204401	0.102201	−	0.994764i	$-0.467412\pi$
$563$	2730.52	0.204401	0.102201	+	0.994764i	$0.467412\pi$
$564$	0	0
$565$	7664.63i	0.570714i
$566$	−16714.7	−1.24129
$567$	0	0
$568$	6054.02	0.447220
$569$	− 1541.26i	− 0.113556i	−0.998387	−	0.0567778i	$-0.981917\pi$
$569$	− 1541.26i	− 0.113556i	0.998387	−	0.0567778i	$-0.0180827\pi$
$570$	0	0
$571$	−3470.00	−0.254317	−0.127158	−	0.991882i	$-0.540586\pi$
$571$	−3470.00	−0.254317	−0.127158	+	0.991882i	$0.540586\pi$
$572$	3058.35	0.223560
$573$	0	0
$574$	0	0
$575$	− 4696.69i	− 0.340635i
$576$	0	0
$577$	4653.81i	0.335773i	0.985806	+	0.167886i	$0.0536941\pi$
$577$	4653.81i	0.335773i	−0.985806	+	0.167886i	$0.946306\pi$
$578$	12840.4i	0.924031i
$579$	0	0
$580$	1250.46i	0.0895216i
$581$	0	0
$582$	0	0
$583$	7604.42	0.540211
$584$	−9125.37	−0.646593
$585$	0	0
$586$	33218.5i	2.34171i
$587$	12056.1	0.847717	0.423858	−	0.905728i	$-0.360675\pi$
$587$	12056.1	0.847717	0.423858	+	0.905728i	$0.360675\pi$
$588$	0	0
$589$	−33459.0	−2.34067
$590$	− 2319.30i	− 0.161837i
$591$	0	0
$592$	11455.3	0.795288
$593$	−12900.0	−0.893324	−0.446662	−	0.894703i	$-0.647387\pi$
$593$	−12900.0	−0.893324	−0.446662	+	0.894703i	$0.647387\pi$
$594$	0	0
$595$	0	0
$596$	5562.56i	0.382301i
$597$	0	0
$598$	1949.32i	0.133300i
$599$	3507.81i	0.239274i	0.992818	+	0.119637i	$0.0381731\pi$
$599$	3507.81i	0.239274i	−0.992818	+	0.119637i	$0.961827\pi$
$600$	0	0
$601$	1767.94i	0.119993i	0.998199	+	0.0599966i	$0.0191090\pi$
$601$	1767.94i	0.119993i	−0.998199	+	0.0599966i	$0.980891\pi$
$602$	0	0
$603$	0	0
$604$	13409.1	0.903328
$605$	3367.96	0.226326
$606$	0	0
$607$	− 13887.7i	− 0.928639i	−0.885668	−	0.464320i	$-0.846299\pi$
$607$	− 13887.7i	− 0.928639i	0.885668	−	0.464320i	$-0.153701\pi$
$608$	−30633.3	−2.04333
$609$	0	0
$610$	−5452.90	−0.361937
$611$	− 4772.20i	− 0.315978i
$612$	0	0
$613$	22509.8	1.48314	0.741569	−	0.670877i	$-0.234082\pi$
$613$	22509.8	1.48314	0.741569	+	0.670877i	$0.234082\pi$
$614$	−13065.0	−0.858730
$615$	0	0
$616$	0	0
$617$	− 1591.00i	− 0.103811i	−0.998652	−	0.0519053i	$-0.983471\pi$
$617$	− 1591.00i	− 0.103811i	0.998652	−	0.0519053i	$-0.0165294\pi$
$618$	0	0
$619$	13273.5i	0.861886i	0.902379	+	0.430943i	$0.141819\pi$
$619$	13273.5i	0.861886i	−0.902379	+	0.430943i	$0.858181\pi$
$620$	− 6250.14i	− 0.404857i
$621$	0	0
$622$	− 13187.9i	− 0.850138i
$623$	0	0
$624$	0	0
$625$	8300.01	0.531200
$626$	−23513.0	−1.50123
$627$	0	0
$628$	19653.7i	1.24883i
$629$	−13629.2	−0.863960
$630$	0	0
$631$	−11933.2	−0.752860	−0.376430	−	0.926445i	$-0.622848\pi$
$631$	−11933.2	−0.752860	−0.376430	+	0.926445i	$0.622848\pi$
$632$	− 5571.75i	− 0.350684i
$633$	0	0
$634$	9438.41	0.591242
$635$	5255.96	0.328467
$636$	0	0
$637$	0	0
$638$	8066.85i	0.500580i
$639$	0	0
$640$	4654.42i	0.287472i
$641$	− 27542.5i	− 1.69713i	−0.529089	−	0.848566i	$-0.677466\pi$
$641$	− 27542.5i	− 1.69713i	0.529089	−	0.848566i	$-0.322534\pi$
$642$	0	0
$643$	− 19003.1i	− 1.16549i	−0.812655	−	0.582746i	$-0.801978\pi$
$643$	− 19003.1i	− 1.16549i	0.812655	−	0.582746i	$-0.198022\pi$
$644$	0	0
$645$	0	0
$646$	47546.9	2.89583
$647$	−25048.4	−1.52203	−0.761016	−	0.648733i	$-0.775299\pi$
$647$	−25048.4	−1.52203	−0.761016	+	0.648733i	$0.775299\pi$
$648$	0	0
$649$	− 6260.20i	− 0.378636i
$650$	−4517.36	−0.272593
$651$	0	0
$652$	2081.51	0.125028
$653$	13745.1i	0.823718i	0.911248	+	0.411859i	$0.135120\pi$
$653$	13745.1i	0.823718i	−0.911248	+	0.411859i	$0.864880\pi$
$654$	0	0
$655$	−2887.49	−0.172250
$656$	−30287.2	−1.80262
$657$	0	0
$658$	0	0
$659$	6965.25i	0.411726i	0.978581	+	0.205863i	$0.0660002\pi$
$659$	6965.25i	0.411726i	−0.978581	+	0.205863i	$0.934000\pi$
$660$	0	0
$661$	11853.6i	0.697504i	0.937215	+	0.348752i	$0.113394\pi$
$661$	11853.6i	0.697504i	−0.937215	+	0.348752i	$0.886606\pi$
$662$	42274.6i	2.48195i
$663$	0	0
$664$	1656.84i	0.0968342i
$665$	0	0
$666$	0	0
$667$	−2151.28	−0.124884
$668$	−9240.51	−0.535219
$669$	0	0
$670$	− 7152.50i	− 0.412426i
$671$	−14718.3	−0.846789
$672$	0	0
$673$	−30769.5	−1.76238	−0.881188	−	0.472766i	$-0.843255\pi$
$673$	−30769.5	−1.76238	−0.881188	+	0.472766i	$0.843255\pi$
$674$	− 2044.00i	− 0.116813i
$675$	0	0
$676$	−11859.9	−0.674778
$677$	1309.49	0.0743393	0.0371696	−	0.999309i	$-0.488166\pi$
$677$	1309.49	0.0743393	0.0371696	+	0.999309i	$0.488166\pi$
$678$	0	0
$679$	0	0
$680$	− 3464.12i	− 0.195358i
$681$	0	0
$682$	− 40320.3i	− 2.26385i
$683$	23134.7i	1.29608i	0.761606	+	0.648041i	$0.224411\pi$
$683$	23134.7i	1.29608i	−0.761606	+	0.648041i	$0.775589\pi$
$684$	0	0
$685$	− 11176.5i	− 0.623403i
$686$	0	0
$687$	0	0
$688$	−31711.0	−1.75723
$689$	1950.53	0.107851
$690$	0	0
$691$	24284.5i	1.33694i	0.743739	+	0.668471i	$0.233051\pi$
$691$	24284.5i	1.33694i	−0.743739	+	0.668471i	$0.766949\pi$
$692$	−8442.24	−0.463766
$693$	0	0
$694$	−10708.9	−0.585744
$695$	4977.84i	0.271684i
$696$	0	0
$697$	36034.8	1.95827
$698$	−8494.43	−0.460629
$699$	0	0
$700$	0	0
$701$	20398.8i	1.09907i	0.835469	+	0.549537i	$0.185196\pi$
$701$	20398.8i	1.09907i	−0.835469	+	0.549537i	$0.814804\pi$
$702$	0	0
$703$	20862.5i	1.11926i
$704$	− 8906.50i	− 0.476813i
$705$	0	0
$706$	− 5227.65i	− 0.278676i
$707$	0	0
$708$	0	0
$709$	−30857.0	−1.63450	−0.817250	−	0.576283i	$-0.804502\pi$
$709$	−30857.0	−1.63450	−0.817250	+	0.576283i	$0.804502\pi$
$710$	−12262.6	−0.648181
$711$	0	0
$712$	− 6721.36i	− 0.353783i
$713$	10752.7	0.564783
$714$	0	0
$715$	2416.14	0.126376
$716$	− 5650.80i	− 0.294945i
$717$	0	0
$718$	−25786.1	−1.34029
$719$	34407.2	1.78466	0.892332	−	0.451380i	$-0.149068\pi$
$719$	34407.2	1.78466	0.892332	+	0.451380i	$0.149068\pi$
$720$	0	0
$721$	0	0
$722$	− 47342.4i	− 2.44030i
$723$	0	0
$724$	8905.26i	0.457129i
$725$	− 4985.38i	− 0.255383i
$726$	0	0
$727$	− 11996.1i	− 0.611983i	−0.952034	−	0.305992i	$-0.901012\pi$
$727$	− 11996.1i	− 0.611983i	0.952034	−	0.305992i	$-0.0989880\pi$
$728$	0	0
$729$	0	0
$730$	18483.8	0.937144
$731$	37728.8	1.90896
$732$	0	0
$733$	− 32123.2i	− 1.61868i	−0.587337	−	0.809342i	$-0.699824\pi$
$733$	− 32123.2i	− 1.61868i	0.587337	−	0.809342i	$-0.300176\pi$
$734$	−5571.00	−0.280149
$735$	0	0
$736$	9844.55	0.493037
$737$	− 19305.9i	− 0.964913i
$738$	0	0
$739$	22871.9	1.13851	0.569254	−	0.822162i	$-0.307232\pi$
$739$	22871.9	1.13851	0.569254	+	0.822162i	$0.307232\pi$
$740$	−3897.10	−0.193595
$741$	0	0
$742$	0	0
$743$	20982.3i	1.03602i	0.855373	+	0.518012i	$0.173328\pi$
$743$	20982.3i	1.03602i	−0.855373	+	0.518012i	$0.826672\pi$
$744$	0	0
$745$	4394.50i	0.216110i
$746$	− 26165.8i	− 1.28418i
$747$	0	0
$748$	23973.4i	1.17187i
$749$	0	0
$750$	0	0
$751$	18642.5	0.905827	0.452914	−	0.891554i	$-0.350385\pi$
$751$	18642.5	0.905827	0.452914	+	0.891554i	$0.350385\pi$
$752$	−31441.2	−1.52466
$753$	0	0
$754$	2069.14i	0.0999385i
$755$	10593.4	0.510641
$756$	0	0
$757$	4135.46	0.198555	0.0992774	−	0.995060i	$-0.468347\pi$
$757$	4135.46	0.198555	0.0992774	+	0.995060i	$0.468347\pi$
$758$	19150.5i	0.917648i
$759$	0	0
$760$	−5302.61	−0.253087
$761$	−3581.69	−0.170612	−0.0853062	−	0.996355i	$-0.527187\pi$
$761$	−3581.69	−0.170612	−0.0853062	+	0.996355i	$0.527187\pi$
$762$	0	0
$763$	0	0
$764$	27536.9i	1.30399i
$765$	0	0
$766$	− 9338.50i	− 0.440488i
$767$	− 1605.74i	− 0.0755929i
$768$	0	0
$769$	− 11427.1i	− 0.535852i	−0.963439	−	0.267926i	$-0.913662\pi$
$769$	− 11427.1i	− 0.535852i	0.963439	−	0.267926i	$-0.0863383\pi$
$770$	0	0
$771$	0	0
$772$	7345.88	0.342466
$773$	1483.61	0.0690318	0.0345159	−	0.999404i	$-0.489011\pi$
$773$	1483.61	0.0690318	0.0345159	+	0.999404i	$0.489011\pi$
$774$	0	0
$775$	24918.3i	1.15496i
$776$	5836.43	0.269994
$777$	0	0
$778$	17753.1	0.818097
$779$	− 55159.2i	− 2.53695i
$780$	0	0
$781$	−33099.0	−1.51649
$782$	−15280.1	−0.698739
$783$	0	0
$784$	0	0
$785$	15526.7i	0.705951i
$786$	0	0
$787$	− 17520.2i	− 0.793557i	−0.917914	−	0.396778i	$-0.870128\pi$
$787$	− 17520.2i	− 0.793557i	0.917914	−	0.396778i	$-0.129872\pi$
$788$	− 2826.48i	− 0.127778i
$789$	0	0
$790$	11285.8i	0.508266i
$791$	0	0
$792$	0	0
$793$	−3775.24	−0.169058
$794$	35584.2	1.59047
$795$	0	0
$796$	17199.3i	0.765846i
$797$	−2687.85	−0.119459	−0.0597293	−	0.998215i	$-0.519024\pi$
$797$	−2687.85	−0.119459	−0.0597293	+	0.998215i	$0.519024\pi$
$798$	0	0
$799$	37407.7	1.65631
$800$	22813.8i	1.00824i
$801$	0	0
$802$	−50244.6	−2.21222
$803$	49890.9	2.19254
$804$	0	0
$805$	0	0
$806$	− 10342.1i	− 0.451967i
$807$	0	0
$808$	− 14174.4i	− 0.617145i
$809$	841.639i	0.0365766i	0.999833	+	0.0182883i	$0.00582167\pi$
$809$	841.639i	0.0365766i	−0.999833	+	0.0182883i	$0.994178\pi$
$810$	0	0
$811$	37115.6i	1.60704i	0.595281	+	0.803518i	$0.297041\pi$
$811$	37115.6i	1.60704i	−0.595281	+	0.803518i	$0.702959\pi$
$812$	0	0
$813$	0	0
$814$	−25140.6	−1.08253
$815$	1644.42	0.0706767
$816$	0	0
$817$	− 57752.2i	− 2.47306i
$818$	8019.33	0.342774
$819$	0	0
$820$	10303.7	0.438807
$821$	27601.6i	1.17333i	0.809831	+	0.586664i	$0.199559\pi$
$821$	27601.6i	1.17333i	−0.809831	+	0.586664i	$0.800441\pi$
$822$	0	0
$823$	7365.20	0.311950	0.155975	−	0.987761i	$-0.450148\pi$
$823$	7365.20	0.311950	0.155975	+	0.987761i	$0.450148\pi$
$824$	9442.75	0.399216
$825$	0	0
$826$	0	0
$827$	− 46801.3i	− 1.96789i	−0.178485	−	0.983943i	$-0.557120\pi$
$827$	− 46801.3i	− 1.96789i	0.178485	−	0.983943i	$-0.442880\pi$
$828$	0	0
$829$	10169.1i	0.426039i	0.977048	+	0.213020i	$0.0683298\pi$
$829$	10169.1i	0.426039i	−0.977048	+	0.213020i	$0.931670\pi$
$830$	− 3355.99i	− 0.140347i
$831$	0	0
$832$	− 2284.51i	− 0.0951936i
$833$	0	0
$834$	0	0
$835$	−7300.13	−0.302553
$836$	36696.6	1.51816
$837$	0	0
$838$	18201.3i	0.750303i
$839$	45477.6	1.87135	0.935674	−	0.352864i	$-0.114792\pi$
$839$	45477.6	1.87135	0.935674	+	0.352864i	$0.114792\pi$
$840$	0	0
$841$	22105.5	0.906371
$842$	21212.5i	0.868209i
$843$	0	0
$844$	−5631.36	−0.229667
$845$	−9369.48	−0.381444
$846$	0	0
$847$	0	0
$848$	− 12850.9i	− 0.520401i
$849$	0	0
$850$	− 35410.1i	− 1.42889i
$851$	− 6704.53i	− 0.270068i
$852$	0	0
$853$	3005.89i	0.120656i	0.998179	+	0.0603281i	$0.0192147\pi$
$853$	3005.89i	0.120656i	−0.998179	+	0.0603281i	$0.980785\pi$
$854$	0	0
$855$	0	0
$856$	−2819.97	−0.112599
$857$	44853.2	1.78781	0.893907	−	0.448252i	$-0.147953\pi$
$857$	44853.2	1.78781	0.893907	+	0.448252i	$0.147953\pi$
$858$	0	0
$859$	− 13090.8i	− 0.519968i	−0.965613	−	0.259984i	$-0.916283\pi$
$859$	− 13090.8i	− 0.519968i	0.965613	−	0.259984i	$-0.0837172\pi$
$860$	10788.1	0.427757
$861$	0	0
$862$	14192.9	0.560804
$863$	− 21219.2i	− 0.836977i	−0.908222	−	0.418488i	$-0.862560\pi$
$863$	− 21219.2i	− 0.836977i	0.908222	−	0.418488i	$-0.137440\pi$
$864$	0	0
$865$	−6669.49	−0.262161
$866$	−33825.2	−1.32728
$867$	0	0
$868$	0	0
$869$	30462.3i	1.18914i
$870$	0	0
$871$	− 4951.93i	− 0.192641i
$872$	16446.8i	0.638714i
$873$	0	0
$874$	23389.5i	0.905220i
$875$	0	0
$876$	0	0
$877$	−41876.4	−1.61239	−0.806195	−	0.591649i	$-0.798477\pi$
$877$	−41876.4	−1.61239	−0.806195	+	0.591649i	$0.798477\pi$
$878$	−10920.5	−0.419762
$879$	0	0
$880$	− 15918.5i	− 0.609788i
$881$	−16277.8	−0.622490	−0.311245	−	0.950330i	$-0.600746\pi$
$881$	−16277.8	−0.622490	−0.311245	+	0.950330i	$0.600746\pi$
$882$	0	0
$883$	−32041.9	−1.22117	−0.610586	−	0.791950i	$-0.709066\pi$
$883$	−32041.9	−1.22117	−0.610586	+	0.791950i	$0.709066\pi$
$884$	6149.16i	0.233958i
$885$	0	0
$886$	15813.1	0.599607
$887$	46153.6	1.74711	0.873555	−	0.486726i	$-0.161809\pi$
$887$	46153.6	1.74711	0.873555	+	0.486726i	$0.161809\pi$
$888$	0	0
$889$	0	0
$890$	13614.4i	0.512758i
$891$	0	0
$892$	− 4184.96i	− 0.157088i
$893$	− 57260.8i	− 2.14576i
$894$	0	0
$895$	− 4464.21i	− 0.166729i
$896$	0	0
$897$	0	0
$898$	−5967.24	−0.221748
$899$	11413.6	0.423432
$900$	0	0
$901$	15289.5i	0.565337i
$902$	66470.4	2.45368
$903$	0	0
$904$	14034.1	0.516337
$905$	7035.28i	0.258410i
$906$	0	0
$907$	27229.7	0.996854	0.498427	−	0.866932i	$-0.333911\pi$
$907$	27229.7	0.996854	0.498427	+	0.866932i	$0.333911\pi$
$908$	35855.9	1.31048
$909$	0	0
$910$	0	0
$911$	− 11012.2i	− 0.400494i	−0.979745	−	0.200247i	$-0.935826\pi$
$911$	− 11012.2i	− 0.400494i	0.979745	−	0.200247i	$-0.0641745\pi$
$912$	0	0
$913$	− 9058.41i	− 0.328357i
$914$	− 19485.4i	− 0.705164i
$915$	0	0
$916$	32632.8i	1.17709i
$917$	0	0
$918$	0	0
$919$	21777.5	0.781692	0.390846	−	0.920456i	$-0.372182\pi$
$919$	21777.5	0.781692	0.390846	+	0.920456i	$0.372182\pi$
$920$	1704.09	0.0610676
$921$	0	0
$922$	− 71317.3i	− 2.54741i
$923$	−8489.87	−0.302760
$924$	0	0
$925$	15537.1	0.552278
$926$	24254.3i	0.860739i
$927$	0	0
$928$	10449.7	0.369642
$929$	12372.3	0.436945	0.218472	−	0.975843i	$-0.429893\pi$
$929$	12372.3	0.436945	0.218472	+	0.975843i	$0.429893\pi$
$930$	0	0
$931$	0	0
$932$	− 22755.0i	− 0.799749i
$933$	0	0
$934$	33445.2i	1.17169i
$935$	18939.3i	0.662441i
$936$	0	0
$937$	− 41577.7i	− 1.44961i	−0.688954	−	0.724806i	$-0.741930\pi$
$937$	− 41577.7i	− 1.44961i	0.688954	−	0.724806i	$-0.258070\pi$
$938$	0	0
$939$	0	0
$940$	10696.3	0.371144
$941$	32490.8	1.12558	0.562789	−	0.826601i	$-0.309728\pi$
$941$	32490.8	1.12558	0.562789	+	0.826601i	$0.309728\pi$
$942$	0	0
$943$	17726.4i	0.612143i
$944$	−10579.2	−0.364751
$945$	0	0
$946$	69595.2	2.39190
$947$	50830.7i	1.74422i	0.489309	+	0.872110i	$0.337249\pi$
$947$	50830.7i	1.74422i	−0.489309	+	0.872110i	$0.662751\pi$
$948$	0	0
$949$	12797.0	0.437732
$950$	−54203.0	−1.85114
$951$	0	0
$952$	0	0
$953$	4892.40i	0.166296i	0.996537	+	0.0831482i	$0.0264975\pi$
$953$	4892.40i	0.166296i	−0.996537	+	0.0831482i	$0.973503\pi$
$954$	0	0
$955$	21754.5i	0.737132i
$956$	22540.9i	0.762577i
$957$	0	0
$958$	30507.8i	1.02887i
$959$	0	0
$960$	0	0
$961$	−27257.3	−0.914952
$962$	−6448.54	−0.216122
$963$	0	0
$964$	18829.5i	0.629105i
$965$	5803.35	0.193592
$966$	0	0
$967$	−41170.9	−1.36915	−0.684575	−	0.728943i	$-0.740012\pi$
$967$	−41170.9	−1.36915	−0.684575	+	0.728943i	$0.740012\pi$
$968$	− 6166.83i	− 0.204762i
$969$	0	0
$970$	−11821.9	−0.391318
$971$	−43287.6	−1.43065	−0.715327	−	0.698790i	$-0.753722\pi$
$971$	−43287.6	−1.43065	−0.715327	+	0.698790i	$0.753722\pi$
$972$	0	0
$973$	0	0
$974$	− 66887.5i	− 2.20042i
$975$	0	0
$976$	24872.8i	0.815737i
$977$	− 18073.1i	− 0.591821i	−0.955216	−	0.295911i	$-0.904377\pi$
$977$	− 18073.1i	− 0.591821i	0.955216	−	0.295911i	$-0.0956231\pi$
$978$	0	0
$979$	36747.5i	1.19965i
$980$	0	0
$981$	0	0
$982$	−45055.4	−1.46413
$983$	19730.6	0.640192	0.320096	−	0.947385i	$-0.396285\pi$
$983$	19730.6	0.640192	0.320096	+	0.947385i	$0.396285\pi$
$984$	0	0
$985$	− 2232.96i	− 0.0722315i
$986$	−16219.3	−0.523862
$987$	0	0
$988$	9412.65	0.303093
$989$	18559.7i	0.596729i
$990$	0	0
$991$	−13692.3	−0.438899	−0.219450	−	0.975624i	$-0.570426\pi$
$991$	−13692.3	−0.438899	−0.219450	+	0.975624i	$0.570426\pi$
$992$	−52230.4	−1.67169
$993$	0	0
$994$	0	0
$995$	13587.7i	0.432923i
$996$	0	0
$997$	− 3085.56i	− 0.0980148i	−0.998798	−	0.0490074i	$-0.984394\pi$
$997$	− 3085.56i	− 0.0980148i	0.998798	−	0.0490074i	$-0.0156058\pi$
$998$	− 10928.2i	− 0.346620i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.b.440.8	yes	24
3.2	odd	2	inner	441.4.c.b.440.17	yes	24
7.2	even	3		441.4.p.d.80.22		48
7.3	odd	6		441.4.p.d.215.3		48
7.4	even	3		441.4.p.d.215.4		48
7.5	odd	6		441.4.p.d.80.21		48
7.6	odd	2	inner	441.4.c.b.440.18	yes	24
21.2	odd	6		441.4.p.d.80.3		48
21.5	even	6		441.4.p.d.80.4		48
21.11	odd	6		441.4.p.d.215.21		48
21.17	even	6		441.4.p.d.215.22		48
21.20	even	2	inner	441.4.c.b.440.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.c.b.440.7	✓	24	21.20	even	2	inner
441.4.c.b.440.8	yes	24	1.1	even	1	trivial
441.4.c.b.440.17	yes	24	3.2	odd	2	inner
441.4.c.b.440.18	yes	24	7.6	odd	2	inner
441.4.p.d.80.3		48	21.2	odd	6
441.4.p.d.80.4		48	21.5	even	6
441.4.p.d.80.21		48	7.5	odd	6
441.4.p.d.80.22		48	7.2	even	3
441.4.p.d.215.3		48	7.3	odd	6
441.4.p.d.215.4		48	7.4	even	3
441.4.p.d.215.21		48	21.11	odd	6
441.4.p.d.215.22		48	21.17	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.70881i q^{2} -5.75528 q^{4} -4.54675 q^{5} +8.32523i q^{8} +O(q^{10})\)	\(q+3.70881i q^{2} -5.75528 q^{4} -4.54675 q^{5} +8.32523i q^{8} -16.8631i q^{10} -45.5164i q^{11} -11.6749i q^{13} -76.9190 q^{16} +91.5157 q^{17} -140.085i q^{19} +26.1679 q^{20} +168.812 q^{22} +45.0189i q^{23} -104.327 q^{25} +43.3000 q^{26} +47.7861i q^{29} -238.848i q^{31} -218.676i q^{32} +339.415i q^{34} -148.927 q^{37} +519.549 q^{38} -37.8528i q^{40} +393.755 q^{41} +412.265 q^{43} +261.960i q^{44} -166.967 q^{46} +408.757 q^{47} -386.929i q^{50} +67.1924i q^{52} +167.070i q^{53} +206.952i q^{55} -177.230 q^{58} +137.537 q^{59} -323.364i q^{61} +885.842 q^{62} +195.677 q^{64} +53.0829i q^{65} +424.152 q^{67} -526.699 q^{68} -727.190i q^{71} +1096.11i q^{73} -552.342i q^{74} +806.229i q^{76} -669.260 q^{79} +349.732 q^{80} +1460.36i q^{82} +199.014 q^{83} -416.100 q^{85} +1529.01i q^{86} +378.934 q^{88} -807.348 q^{89} -259.096i q^{92} +1516.00i q^{94} +636.932i q^{95} -701.053i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{4}+O(q^{10}) \) 24 * q - 96 * q^4	\( 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) \) 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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