LMFDB - Embedded newform 414.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,4,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.a (trivial)

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:	yes
Analytic conductor:	$24.4267907424$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 $ x^4 - 2x^3 - 219x^2 - 468*x + 3240
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.93708$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +19.0098 q^{5} -26.4014 q^{7} -8.00000 q^{8} -38.0197 q^{10} -27.2657 q^{11} +69.8938 q^{13} +52.8028 q^{14} +16.0000 q^{16} +15.2407 q^{17} +65.6587 q^{19} +76.0393 q^{20} +54.5314 q^{22} -23.0000 q^{23} +236.374 q^{25} -139.788 q^{26} -105.606 q^{28} -124.619 q^{29} +302.744 q^{31} -32.0000 q^{32} -30.4815 q^{34} -501.886 q^{35} +5.93992 q^{37} -131.317 q^{38} -152.079 q^{40} +166.526 q^{41} -318.295 q^{43} -109.063 q^{44} +46.0000 q^{46} -144.643 q^{47} +354.033 q^{49} -472.747 q^{50} +279.575 q^{52} +502.487 q^{53} -518.317 q^{55} +211.211 q^{56} +249.237 q^{58} +600.540 q^{59} +636.071 q^{61} -605.489 q^{62} +64.0000 q^{64} +1328.67 q^{65} +206.163 q^{67} +60.9630 q^{68} +1003.77 q^{70} -440.948 q^{71} +390.246 q^{73} -11.8798 q^{74} +262.635 q^{76} +719.853 q^{77} +681.317 q^{79} +304.157 q^{80} -333.051 q^{82} +695.269 q^{83} +289.724 q^{85} +636.589 q^{86} +218.126 q^{88} -948.900 q^{89} -1845.29 q^{91} -92.0000 q^{92} +289.286 q^{94} +1248.16 q^{95} +746.861 q^{97} -708.067 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 18 q^{7} - 32 q^{8} - 42 q^{11} + 108 q^{13} - 36 q^{14} + 64 q^{16} - 26 q^{17} + 132 q^{19} + 84 q^{22} - 92 q^{23} + 260 q^{25} - 216 q^{26} + 72 q^{28} - 252 q^{29} + 428 q^{31}+ \cdots - 2152 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 18 * q^7 - 32 * q^8 - 42 * q^11 + 108 * q^13 - 36 * q^14 + 64 * q^16 - 26 * q^17 + 132 * q^19 + 84 * q^22 - 92 * q^23 + 260 * q^25 - 216 * q^26 + 72 * q^28 - 252 * q^29 + 428 * q^31 - 128 * q^32 + 52 * q^34 - 348 * q^35 + 278 * q^37 - 264 * q^38 + 4 * q^41 + 216 * q^43 - 168 * q^44 + 184 * q^46 + 4 * q^47 + 1076 * q^49 - 520 * q^50 + 432 * q^52 + 460 * q^53 + 512 * q^55 - 144 * q^56 + 504 * q^58 + 504 * q^59 + 1086 * q^61 - 856 * q^62 + 256 * q^64 + 1420 * q^65 + 1484 * q^67 - 104 * q^68 + 696 * q^70 + 876 * q^71 + 1072 * q^73 - 556 * q^74 + 528 * q^76 + 1580 * q^77 + 1634 * q^79 - 8 * q^82 + 2246 * q^83 + 872 * q^85 - 432 * q^86 + 336 * q^88 + 474 * q^89 + 40 * q^91 - 368 * q^92 - 8 * q^94 + 2780 * q^95 + 1044 * q^97 - 2152 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	19.0098	1.70029	0.850145	−	0.526548i	$-0.176514\pi$
$5$	19.0098	1.70029	0.850145	+	0.526548i	$0.176514\pi$
$6$	0	0
$7$	−26.4014	−1.42554	−0.712770	−	0.701397i	$-0.752560\pi$
$7$	−26.4014	−1.42554	−0.712770	+	0.701397i	$0.752560\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−38.0197	−1.20229
$11$	−27.2657	−0.747357	−0.373678	−	0.927558i	$-0.621904\pi$
$11$	−27.2657	−0.747357	−0.373678	+	0.927558i	$0.621904\pi$
$12$	0	0
$13$	69.8938	1.49116	0.745579	−	0.666417i	$-0.232173\pi$
$13$	69.8938	1.49116	0.745579	+	0.666417i	$0.232173\pi$
$14$	52.8028	1.00801
$15$	0	0
$16$	16.0000	0.250000
$17$	15.2407	0.217437	0.108718	−	0.994073i	$-0.465325\pi$
$17$	15.2407	0.217437	0.108718	+	0.994073i	$0.465325\pi$
$18$	0	0
$19$	65.6587	0.792797	0.396398	−	0.918079i	$-0.370260\pi$
$19$	65.6587	0.792797	0.396398	+	0.918079i	$0.370260\pi$
$20$	76.0393	0.850145
$21$	0	0
$22$	54.5314	0.528461
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	236.374	1.89099
$26$	−139.788	−1.05441
$27$	0	0
$28$	−105.606	−0.712770
$29$	−124.619	−0.797968	−0.398984	−	0.916958i	$-0.630637\pi$
$29$	−124.619	−0.797968	−0.398984	+	0.916958i	$0.630637\pi$
$30$	0	0
$31$	302.744	1.75402	0.877008	−	0.480476i	$-0.159536\pi$
$31$	302.744	1.75402	0.877008	+	0.480476i	$0.159536\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−30.4815	−0.153751
$35$	−501.886	−2.42383
$36$	0	0
$37$	5.93992	0.0263924	0.0131962	−	0.999913i	$-0.495799\pi$
$37$	5.93992	0.0263924	0.0131962	+	0.999913i	$0.495799\pi$
$38$	−131.317	−0.560592
$39$	0	0
$40$	−152.079	−0.601144
$41$	166.526	0.634316	0.317158	−	0.948373i	$-0.397271\pi$
$41$	166.526	0.634316	0.317158	+	0.948373i	$0.397271\pi$
$42$	0	0
$43$	−318.295	−1.12882	−0.564412	−	0.825493i	$-0.690897\pi$
$43$	−318.295	−1.12882	−0.564412	+	0.825493i	$0.690897\pi$
$44$	−109.063	−0.373678
$45$	0	0
$46$	46.0000	0.147442
$47$	−144.643	−0.448902	−0.224451	−	0.974485i	$-0.572059\pi$
$47$	−144.643	−0.448902	−0.224451	+	0.974485i	$0.572059\pi$
$48$	0	0
$49$	354.033	1.03217
$50$	−472.747	−1.33713
$51$	0	0
$52$	279.575	0.745579
$53$	502.487	1.30230	0.651150	−	0.758949i	$-0.274287\pi$
$53$	502.487	1.30230	0.651150	+	0.758949i	$0.274287\pi$
$54$	0	0
$55$	−518.317	−1.27072
$56$	211.211	0.504005
$57$	0	0
$58$	249.237	0.564249
$59$	600.540	1.32515	0.662573	−	0.748997i	$-0.269464\pi$
$59$	600.540	1.32515	0.662573	+	0.748997i	$0.269464\pi$
$60$	0	0
$61$	636.071	1.33509	0.667546	−	0.744569i	$-0.267345\pi$
$61$	636.071	1.33509	0.667546	+	0.744569i	$0.267345\pi$
$62$	−605.489	−1.24028
$63$	0	0
$64$	64.0000	0.125000
$65$	1328.67	2.53540
$66$	0	0
$67$	206.163	0.375922	0.187961	−	0.982176i	$-0.439812\pi$
$67$	206.163	0.375922	0.187961	+	0.982176i	$0.439812\pi$
$68$	60.9630	0.108718
$69$	0	0
$70$	1003.77	1.71391
$71$	−440.948	−0.737055	−0.368527	−	0.929617i	$-0.620138\pi$
$71$	−440.948	−0.737055	−0.368527	+	0.929617i	$0.620138\pi$
$72$	0	0
$73$	390.246	0.625683	0.312841	−	0.949805i	$-0.398719\pi$
$73$	390.246	0.625683	0.312841	+	0.949805i	$0.398719\pi$
$74$	−11.8798	−0.0186622
$75$	0	0
$76$	262.635	0.396398
$77$	719.853	1.06539
$78$	0	0
$79$	681.317	0.970306	0.485153	−	0.874429i	$-0.338764\pi$
$79$	681.317	0.970306	0.485153	+	0.874429i	$0.338764\pi$
$80$	304.157	0.425073
$81$	0	0
$82$	−333.051	−0.448529
$83$	695.269	0.919466	0.459733	−	0.888057i	$-0.347945\pi$
$83$	695.269	0.919466	0.459733	+	0.888057i	$0.347945\pi$
$84$	0	0
$85$	289.724	0.369705
$86$	636.589	0.798200
$87$	0	0
$88$	218.126	0.264230
$89$	−948.900	−1.13015	−0.565074	−	0.825040i	$-0.691152\pi$
$89$	−948.900	−1.13015	−0.565074	+	0.825040i	$0.691152\pi$
$90$	0	0
$91$	−1845.29	−2.12571
$92$	−92.0000	−0.104257
$93$	0	0
$94$	289.286	0.317421
$95$	1248.16	1.34798
$96$	0	0
$97$	746.861	0.781775	0.390888	−	0.920438i	$-0.372168\pi$
$97$	746.861	0.781775	0.390888	+	0.920438i	$0.372168\pi$
$98$	−708.067	−0.729852
$99$	0	0
$100$	945.494	0.945494
$101$	−1168.00	−1.15070	−0.575349	−	0.817908i	$-0.695134\pi$
$101$	−1168.00	−1.15070	−0.575349	+	0.817908i	$0.695134\pi$
$102$	0	0
$103$	608.376	0.581991	0.290995	−	0.956724i	$-0.406014\pi$
$103$	608.376	0.581991	0.290995	+	0.956724i	$0.406014\pi$
$104$	−559.150	−0.527204
$105$	0	0
$106$	−1004.97	−0.920866
$107$	−1134.79	−1.02528	−0.512638	−	0.858605i	$-0.671332\pi$
$107$	−1134.79	−1.02528	−0.512638	+	0.858605i	$0.671332\pi$
$108$	0	0
$109$	1602.61	1.40828	0.704140	−	0.710061i	$-0.251333\pi$
$109$	1602.61	1.40828	0.704140	+	0.710061i	$0.251333\pi$
$110$	1036.63	0.898537
$111$	0	0
$112$	−422.422	−0.356385
$113$	2049.37	1.70609	0.853046	−	0.521836i	$-0.174753\pi$
$113$	2049.37	1.70609	0.853046	+	0.521836i	$0.174753\pi$
$114$	0	0
$115$	−437.226	−0.354535
$116$	−498.474	−0.398984
$117$	0	0
$118$	−1201.08	−0.937020
$119$	−402.377	−0.309965
$120$	0	0
$121$	−587.581	−0.441458
$122$	−1272.14	−0.944052
$123$	0	0
$124$	1210.98	0.877008
$125$	2117.19	1.51494
$126$	0	0
$127$	1322.84	0.924278	0.462139	−	0.886807i	$-0.347082\pi$
$127$	1322.84	0.924278	0.462139	+	0.886807i	$0.347082\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−2657.34	−1.79280
$131$	−353.591	−0.235827	−0.117914	−	0.993024i	$-0.537621\pi$
$131$	−353.591	−0.235827	−0.117914	+	0.993024i	$0.537621\pi$
$132$	0	0
$133$	−1733.48	−1.13016
$134$	−412.326	−0.265817
$135$	0	0
$136$	−121.926	−0.0768755
$137$	−3171.79	−1.97799	−0.988995	−	0.147948i	$-0.952733\pi$
$137$	−3171.79	−1.97799	−0.988995	+	0.147948i	$0.952733\pi$
$138$	0	0
$139$	508.055	0.310019	0.155010	−	0.987913i	$-0.450459\pi$
$139$	508.055	0.310019	0.155010	+	0.987913i	$0.450459\pi$
$140$	−2007.54	−1.21192
$141$	0	0
$142$	881.896	0.521176
$143$	−1905.71	−1.11443
$144$	0	0
$145$	−2368.98	−1.35678
$146$	−780.492	−0.442424
$147$	0	0
$148$	23.7597	0.0131962
$149$	2954.62	1.62451	0.812255	−	0.583303i	$-0.198240\pi$
$149$	2954.62	1.62451	0.812255	+	0.583303i	$0.198240\pi$
$150$	0	0
$151$	−3383.54	−1.82350	−0.911749	−	0.410747i	$-0.865268\pi$
$151$	−3383.54	−1.82350	−0.911749	+	0.410747i	$0.865268\pi$
$152$	−525.270	−0.280296
$153$	0	0
$154$	−1439.71	−0.753343
$155$	5755.12	2.98234
$156$	0	0
$157$	−293.879	−0.149389	−0.0746945	−	0.997206i	$-0.523798\pi$
$157$	−293.879	−0.149389	−0.0746945	+	0.997206i	$0.523798\pi$
$158$	−1362.63	−0.686110
$159$	0	0
$160$	−608.314	−0.300572
$161$	607.232	0.297246
$162$	0	0
$163$	1277.17	0.613717	0.306859	−	0.951755i	$-0.400722\pi$
$163$	1277.17	0.613717	0.306859	+	0.951755i	$0.400722\pi$
$164$	666.103	0.317158
$165$	0	0
$166$	−1390.54	−0.650161
$167$	−1509.84	−0.699609	−0.349805	−	0.936823i	$-0.613752\pi$
$167$	−1509.84	−0.699609	−0.349805	+	0.936823i	$0.613752\pi$
$168$	0	0
$169$	2688.15	1.22355
$170$	−579.448	−0.261421
$171$	0	0
$172$	−1273.18	−0.564412
$173$	−137.777	−0.0605490	−0.0302745	−	0.999542i	$-0.509638\pi$
$173$	−137.777	−0.0605490	−0.0302745	+	0.999542i	$0.509638\pi$
$174$	0	0
$175$	−6240.59	−2.69568
$176$	−436.252	−0.186839
$177$	0	0
$178$	1897.80	0.799135
$179$	3043.97	1.27105	0.635523	−	0.772082i	$-0.280784\pi$
$179$	3043.97	1.27105	0.635523	+	0.772082i	$0.280784\pi$
$180$	0	0
$181$	3906.90	1.60441	0.802204	−	0.597051i	$-0.203661\pi$
$181$	3906.90	1.60441	0.802204	+	0.597051i	$0.203661\pi$
$182$	3690.59	1.50310
$183$	0	0
$184$	184.000	0.0737210
$185$	112.917	0.0448747
$186$	0	0
$187$	−415.550	−0.162503
$188$	−578.573	−0.224451
$189$	0	0
$190$	−2496.32	−0.953169
$191$	−1018.93	−0.386007	−0.193004	−	0.981198i	$-0.561823\pi$
$191$	−1018.93	−0.386007	−0.193004	+	0.981198i	$0.561823\pi$
$192$	0	0
$193$	−4302.92	−1.60482	−0.802412	−	0.596771i	$-0.796450\pi$
$193$	−4302.92	−1.60482	−0.802412	+	0.596771i	$0.796450\pi$
$194$	−1493.72	−0.552799
$195$	0	0
$196$	1416.13	0.516083
$197$	−1439.36	−0.520558	−0.260279	−	0.965533i	$-0.583815\pi$
$197$	−1439.36	−0.520558	−0.260279	+	0.965533i	$0.583815\pi$
$198$	0	0
$199$	5262.93	1.87477	0.937385	−	0.348295i	$-0.113239\pi$
$199$	5262.93	1.87477	0.937385	+	0.348295i	$0.113239\pi$
$200$	−1890.99	−0.668565
$201$	0	0
$202$	2336.00	0.813666
$203$	3290.10	1.13754
$204$	0	0
$205$	3165.63	1.07852
$206$	−1216.75	−0.411530
$207$	0	0
$208$	1118.30	0.372790
$209$	−1790.23	−0.592502
$210$	0	0
$211$	353.694	0.115399	0.0576997	−	0.998334i	$-0.481623\pi$
$211$	353.694	0.115399	0.0576997	+	0.998334i	$0.481623\pi$
$212$	2009.95	0.651150
$213$	0	0
$214$	2269.58	0.724979
$215$	−6050.73	−1.91933
$216$	0	0
$217$	−7992.87	−2.50042
$218$	−3205.23	−0.995804
$219$	0	0
$220$	−2073.27	−0.635362
$221$	1065.23	0.324232
$222$	0	0
$223$	−5336.31	−1.60245	−0.801223	−	0.598366i	$-0.795817\pi$
$223$	−5336.31	−1.60245	−0.801223	+	0.598366i	$0.795817\pi$
$224$	844.844	0.252002
$225$	0	0
$226$	−4098.74	−1.20639
$227$	−2213.30	−0.647144	−0.323572	−	0.946204i	$-0.604884\pi$
$227$	−2213.30	−0.647144	−0.323572	+	0.946204i	$0.604884\pi$
$228$	0	0
$229$	−5141.09	−1.48355	−0.741774	−	0.670650i	$-0.766015\pi$
$229$	−5141.09	−1.48355	−0.741774	+	0.670650i	$0.766015\pi$
$230$	874.452	0.250694
$231$	0	0
$232$	996.948	0.282124
$233$	−6582.34	−1.85074	−0.925372	−	0.379061i	$-0.876247\pi$
$233$	−6582.34	−1.85074	−0.925372	+	0.379061i	$0.876247\pi$
$234$	0	0
$235$	−2749.64	−0.763263
$236$	2402.16	0.662573
$237$	0	0
$238$	804.753	0.219178
$239$	−6205.96	−1.67962	−0.839812	−	0.542877i	$-0.817335\pi$
$239$	−6205.96	−1.67962	−0.839812	+	0.542877i	$0.817335\pi$
$240$	0	0
$241$	−6363.48	−1.70086	−0.850432	−	0.526086i	$-0.823659\pi$
$241$	−6363.48	−1.70086	−0.850432	+	0.526086i	$0.823659\pi$
$242$	1175.16	0.312158
$243$	0	0
$244$	2544.28	0.667546
$245$	6730.11	1.75498
$246$	0	0
$247$	4589.14	1.18219
$248$	−2421.95	−0.620138
$249$	0	0
$250$	−4234.38	−1.07122
$251$	−1761.97	−0.443087	−0.221544	−	0.975150i	$-0.571109\pi$
$251$	−1761.97	−0.443087	−0.221544	+	0.975150i	$0.571109\pi$
$252$	0	0
$253$	627.112	0.155835
$254$	−2645.69	−0.653563
$255$	0	0
$256$	256.000	0.0625000
$257$	446.547	0.108385	0.0541923	−	0.998531i	$-0.482742\pi$
$257$	446.547	0.108385	0.0541923	+	0.998531i	$0.482742\pi$
$258$	0	0
$259$	−156.822	−0.0376234
$260$	5314.68	1.26770
$261$	0	0
$262$	707.182	0.166755
$263$	8390.91	1.96732	0.983661	−	0.180033i	$-0.0576205\pi$
$263$	8390.91	1.96732	0.983661	+	0.180033i	$0.0576205\pi$
$264$	0	0
$265$	9552.20	2.21429
$266$	3466.96	0.799147
$267$	0	0
$268$	824.651	0.187961
$269$	4219.17	0.956310	0.478155	−	0.878275i	$-0.341306\pi$
$269$	4219.17	0.956310	0.478155	+	0.878275i	$0.341306\pi$
$270$	0	0
$271$	4711.39	1.05608	0.528038	−	0.849221i	$-0.322928\pi$
$271$	4711.39	1.05608	0.528038	+	0.849221i	$0.322928\pi$
$272$	243.852	0.0543592
$273$	0	0
$274$	6343.59	1.39865
$275$	−6444.89	−1.41324
$276$	0	0
$277$	−6017.91	−1.30535	−0.652674	−	0.757639i	$-0.726353\pi$
$277$	−6017.91	−1.30535	−0.652674	+	0.757639i	$0.726353\pi$
$278$	−1016.11	−0.219217
$279$	0	0
$280$	4015.09	0.856955
$281$	5210.98	1.10627	0.553134	−	0.833092i	$-0.313432\pi$
$281$	5210.98	1.10627	0.553134	+	0.833092i	$0.313432\pi$
$282$	0	0
$283$	−5156.61	−1.08314	−0.541570	−	0.840655i	$-0.682170\pi$
$283$	−5156.61	−1.08314	−0.541570	+	0.840655i	$0.682170\pi$
$284$	−1763.79	−0.368527
$285$	0	0
$286$	3811.41	0.788019
$287$	−4396.51	−0.904243
$288$	0	0
$289$	−4680.72	−0.952721
$290$	4737.95	0.959387
$291$	0	0
$292$	1560.98	0.312841
$293$	−4978.19	−0.992590	−0.496295	−	0.868154i	$-0.665307\pi$
$293$	−4978.19	−0.992590	−0.496295	+	0.868154i	$0.665307\pi$
$294$	0	0
$295$	11416.2	2.25313
$296$	−47.5194	−0.00933111
$297$	0	0
$298$	−5909.24	−1.14870
$299$	−1607.56	−0.310928
$300$	0	0
$301$	8403.42	1.60919
$302$	6767.07	1.28941
$303$	0	0
$304$	1050.54	0.198199
$305$	12091.6	2.27004
$306$	0	0
$307$	3035.93	0.564396	0.282198	−	0.959356i	$-0.408937\pi$
$307$	3035.93	0.564396	0.282198	+	0.959356i	$0.408937\pi$
$308$	2879.41	0.532694
$309$	0	0
$310$	−11510.2	−2.10883
$311$	−4811.62	−0.877305	−0.438652	−	0.898657i	$-0.644544\pi$
$311$	−4811.62	−0.877305	−0.438652	+	0.898657i	$0.644544\pi$
$312$	0	0
$313$	−6959.90	−1.25686	−0.628429	−	0.777867i	$-0.716302\pi$
$313$	−6959.90	−1.25686	−0.628429	+	0.777867i	$0.716302\pi$
$314$	587.758	0.105634
$315$	0	0
$316$	2725.27	0.485153
$317$	2817.16	0.499140	0.249570	−	0.968357i	$-0.419711\pi$
$317$	2817.16	0.499140	0.249570	+	0.968357i	$0.419711\pi$
$318$	0	0
$319$	3397.81	0.596367
$320$	1216.63	0.212536
$321$	0	0
$322$	−1214.46	−0.210185
$323$	1000.69	0.172383
$324$	0	0
$325$	16521.0	2.81976
$326$	−2554.35	−0.433964
$327$	0	0
$328$	−1332.21	−0.224265
$329$	3818.78	0.639928
$330$	0	0
$331$	8262.04	1.37197	0.685986	−	0.727614i	$-0.259371\pi$
$331$	8262.04	1.37197	0.685986	+	0.727614i	$0.259371\pi$
$332$	2781.08	0.459733
$333$	0	0
$334$	3019.68	0.494699
$335$	3919.12	0.639177
$336$	0	0
$337$	−8303.24	−1.34215	−0.671077	−	0.741387i	$-0.734168\pi$
$337$	−8303.24	−1.34215	−0.671077	+	0.741387i	$0.734168\pi$
$338$	−5376.29	−0.865182
$339$	0	0
$340$	1158.90	0.184853
$341$	−8254.54	−1.31088
$342$	0	0
$343$	−291.293	−0.0458553
$344$	2546.36	0.399100
$345$	0	0
$346$	275.554	0.0428146
$347$	307.537	0.0475776	0.0237888	−	0.999717i	$-0.492427\pi$
$347$	307.537	0.0475776	0.0237888	+	0.999717i	$0.492427\pi$
$348$	0	0
$349$	−10.9250	−0.00167564	−0.000837822	−	1.00000i	$-0.500267\pi$
$349$	−10.9250	−0.00167564	−0.000837822		1.00000i	$0.500267\pi$
$350$	12481.2	1.90613
$351$	0	0
$352$	872.503	0.132115
$353$	525.894	0.0792932	0.0396466	−	0.999214i	$-0.487377\pi$
$353$	525.894	0.0792932	0.0396466	+	0.999214i	$0.487377\pi$
$354$	0	0
$355$	−8382.34	−1.25321
$356$	−3795.60	−0.565074
$357$	0	0
$358$	−6087.95	−0.898766
$359$	1726.99	0.253891	0.126946	−	0.991910i	$-0.459483\pi$
$359$	1726.99	0.253891	0.126946	+	0.991910i	$0.459483\pi$
$360$	0	0
$361$	−2547.94	−0.371473
$362$	−7813.80	−1.13449
$363$	0	0
$364$	−7381.17	−1.06285
$365$	7418.51	1.06384
$366$	0	0
$367$	−537.816	−0.0764953	−0.0382477	−	0.999268i	$-0.512178\pi$
$367$	−537.816	−0.0764953	−0.0382477	+	0.999268i	$0.512178\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	−225.834	−0.0317312
$371$	−13266.4	−1.85648
$372$	0	0
$373$	3074.20	0.426745	0.213373	−	0.976971i	$-0.431555\pi$
$373$	3074.20	0.426745	0.213373	+	0.976971i	$0.431555\pi$
$374$	831.100	0.114907
$375$	0	0
$376$	1157.15	0.158711
$377$	−8710.07	−1.18990
$378$	0	0
$379$	9245.15	1.25301	0.626506	−	0.779417i	$-0.284485\pi$
$379$	9245.15	1.25301	0.626506	+	0.779417i	$0.284485\pi$
$380$	4992.64	0.673992
$381$	0	0
$382$	2037.87	0.272948
$383$	−2029.45	−0.270758	−0.135379	−	0.990794i	$-0.543225\pi$
$383$	−2029.45	−0.270758	−0.135379	+	0.990794i	$0.543225\pi$
$384$	0	0
$385$	13684.3	1.81147
$386$	8605.84	1.13478
$387$	0	0
$388$	2987.44	0.390888
$389$	−7081.67	−0.923020	−0.461510	−	0.887135i	$-0.652692\pi$
$389$	−7081.67	−0.923020	−0.461510	+	0.887135i	$0.652692\pi$
$390$	0	0
$391$	−350.537	−0.0453387
$392$	−2832.27	−0.364926
$393$	0	0
$394$	2878.71	0.368090
$395$	12951.7	1.64980
$396$	0	0
$397$	3974.11	0.502405	0.251203	−	0.967935i	$-0.419174\pi$
$397$	3974.11	0.502405	0.251203	+	0.967935i	$0.419174\pi$
$398$	−10525.9	−1.32566
$399$	0	0
$400$	3781.98	0.472747
$401$	1228.32	0.152966	0.0764830	−	0.997071i	$-0.475631\pi$
$401$	1228.32	0.152966	0.0764830	+	0.997071i	$0.475631\pi$
$402$	0	0
$403$	21160.0	2.61551
$404$	−4672.01	−0.575349
$405$	0	0
$406$	−6580.21	−0.804360
$407$	−161.956	−0.0197245
$408$	0	0
$409$	−4559.87	−0.551274	−0.275637	−	0.961262i	$-0.588889\pi$
$409$	−4559.87	−0.551274	−0.275637	+	0.961262i	$0.588889\pi$
$410$	−6331.25	−0.762630
$411$	0	0
$412$	2433.50	0.290995
$413$	−15855.1	−1.88905
$414$	0	0
$415$	13216.9	1.56336
$416$	−2236.60	−0.263602
$417$	0	0
$418$	3580.46	0.418962
$419$	−16364.6	−1.90803	−0.954013	−	0.299766i	$-0.903092\pi$
$419$	−16364.6	−1.90803	−0.954013	+	0.299766i	$0.903092\pi$
$420$	0	0
$421$	8467.50	0.980239	0.490120	−	0.871655i	$-0.336953\pi$
$421$	8467.50	0.980239	0.490120	+	0.871655i	$0.336953\pi$
$422$	−707.387	−0.0815997
$423$	0	0
$424$	−4019.90	−0.460433
$425$	3602.51	0.411170
$426$	0	0
$427$	−16793.2	−1.90323
$428$	−4539.17	−0.512638
$429$	0	0
$430$	12101.5	1.35717
$431$	−12440.5	−1.39034	−0.695172	−	0.718843i	$-0.744672\pi$
$431$	−12440.5	−1.39034	−0.695172	+	0.718843i	$0.744672\pi$
$432$	0	0
$433$	4079.23	0.452737	0.226369	−	0.974042i	$-0.427315\pi$
$433$	4079.23	0.452737	0.226369	+	0.974042i	$0.427315\pi$
$434$	15985.7	1.76806
$435$	0	0
$436$	6410.45	0.704140
$437$	−1510.15	−0.165310
$438$	0	0
$439$	3486.22	0.379017	0.189508	−	0.981879i	$-0.439311\pi$
$439$	3486.22	0.379017	0.189508	+	0.981879i	$0.439311\pi$
$440$	4146.53	0.449269
$441$	0	0
$442$	−2130.47	−0.229267
$443$	−11320.3	−1.21409	−0.607047	−	0.794666i	$-0.707646\pi$
$443$	−11320.3	−1.21409	−0.607047	+	0.794666i	$0.707646\pi$
$444$	0	0
$445$	−18038.4	−1.92158
$446$	10672.6	1.13310
$447$	0	0
$448$	−1689.69	−0.178193
$449$	16854.6	1.77154	0.885768	−	0.464129i	$-0.153633\pi$
$449$	16854.6	1.77154	0.885768	+	0.464129i	$0.153633\pi$
$450$	0	0
$451$	−4540.44	−0.474060
$452$	8197.47	0.853046
$453$	0	0
$454$	4426.59	0.457600
$455$	−35078.7	−3.61432
$456$	0	0
$457$	−5329.53	−0.545525	−0.272762	−	0.962081i	$-0.587937\pi$
$457$	−5329.53	−0.545525	−0.272762	+	0.962081i	$0.587937\pi$
$458$	10282.2	1.04903
$459$	0	0
$460$	−1748.90	−0.177268
$461$	−8612.31	−0.870098	−0.435049	−	0.900407i	$-0.643269\pi$
$461$	−8612.31	−0.870098	−0.435049	+	0.900407i	$0.643269\pi$
$462$	0	0
$463$	−11553.9	−1.15973	−0.579867	−	0.814711i	$-0.696896\pi$
$463$	−11553.9	−1.15973	−0.579867	+	0.814711i	$0.696896\pi$
$464$	−1993.90	−0.199492
$465$	0	0
$466$	13164.7	1.30867
$467$	−6425.53	−0.636698	−0.318349	−	0.947974i	$-0.603128\pi$
$467$	−6425.53	−0.636698	−0.318349	+	0.947974i	$0.603128\pi$
$468$	0	0
$469$	−5442.98	−0.535893
$470$	5499.28	0.539709
$471$	0	0
$472$	−4804.32	−0.468510
$473$	8678.53	0.843635
$474$	0	0
$475$	15520.0	1.49917
$476$	−1609.51	−0.154982
$477$	0	0
$478$	12411.9	1.18767
$479$	−8666.10	−0.826647	−0.413324	−	0.910584i	$-0.635632\pi$
$479$	−8666.10	−0.826647	−0.413324	+	0.910584i	$0.635632\pi$
$480$	0	0
$481$	415.164	0.0393552
$482$	12727.0	1.20269
$483$	0	0
$484$	−2350.32	−0.220729
$485$	14197.7	1.32925
$486$	0	0
$487$	−18563.6	−1.72730	−0.863652	−	0.504088i	$-0.831829\pi$
$487$	−18563.6	−1.72730	−0.863652	+	0.504088i	$0.831829\pi$
$488$	−5088.57	−0.472026
$489$	0	0
$490$	−13460.2	−1.24096
$491$	11797.2	1.08432	0.542161	−	0.840274i	$-0.317606\pi$
$491$	11797.2	1.08432	0.542161	+	0.840274i	$0.317606\pi$
$492$	0	0
$493$	−1899.28	−0.173508
$494$	−9178.27	−0.835931
$495$	0	0
$496$	4843.91	0.438504
$497$	11641.6	1.05070
$498$	0	0
$499$	−3453.29	−0.309801	−0.154900	−	0.987930i	$-0.549506\pi$
$499$	−3453.29	−0.309801	−0.154900	+	0.987930i	$0.549506\pi$
$500$	8468.77	0.757469
$501$	0	0
$502$	3523.95	0.313310
$503$	−9326.34	−0.826721	−0.413361	−	0.910567i	$-0.635645\pi$
$503$	−9326.34	−0.826721	−0.413361	+	0.910567i	$0.635645\pi$
$504$	0	0
$505$	−22203.5	−1.95652
$506$	−1254.22	−0.110192
$507$	0	0
$508$	5291.37	0.462139
$509$	12603.6	1.09754	0.548768	−	0.835975i	$-0.315097\pi$
$509$	12603.6	1.09754	0.548768	+	0.835975i	$0.315097\pi$
$510$	0	0
$511$	−10303.0	−0.891936
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−893.094	−0.0766394
$515$	11565.1	0.989554
$516$	0	0
$517$	3943.80	0.335490
$518$	313.645	0.0266038
$519$	0	0
$520$	−10629.4	−0.896400
$521$	17821.9	1.49864	0.749322	−	0.662206i	$-0.230380\pi$
$521$	17821.9	1.49864	0.749322	+	0.662206i	$0.230380\pi$
$522$	0	0
$523$	−3816.68	−0.319104	−0.159552	−	0.987189i	$-0.551005\pi$
$523$	−3816.68	−0.319104	−0.159552	+	0.987189i	$0.551005\pi$
$524$	−1414.36	−0.117914
$525$	0	0
$526$	−16781.8	−1.39111
$527$	4614.05	0.381387
$528$	0	0
$529$	529.000	0.0434783
$530$	−19104.4	−1.56574
$531$	0	0
$532$	−6933.92	−0.565082
$533$	11639.1	0.945865
$534$	0	0
$535$	−21572.2	−1.74327
$536$	−1649.30	−0.132909
$537$	0	0
$538$	−8438.34	−0.676213
$539$	−9652.97	−0.771397
$540$	0	0
$541$	12730.5	1.01169	0.505846	−	0.862624i	$-0.331180\pi$
$541$	12730.5	1.01169	0.505846	+	0.862624i	$0.331180\pi$
$542$	−9422.78	−0.746759
$543$	0	0
$544$	−487.704	−0.0384377
$545$	30465.4	2.39449
$546$	0	0
$547$	175.554	0.0137224	0.00686119	−	0.999976i	$-0.497816\pi$
$547$	175.554	0.0137224	0.00686119	+	0.999976i	$0.497816\pi$
$548$	−12687.2	−0.988995
$549$	0	0
$550$	12889.8	0.999313
$551$	−8182.29	−0.632627
$552$	0	0
$553$	−17987.7	−1.38321
$554$	12035.8	0.923020
$555$	0	0
$556$	2032.22	0.155010
$557$	−4603.34	−0.350179	−0.175089	−	0.984553i	$-0.556021\pi$
$557$	−4603.34	−0.350179	−0.175089	+	0.984553i	$0.556021\pi$
$558$	0	0
$559$	−22246.8	−1.68326
$560$	−8030.17	−0.605958
$561$	0	0
$562$	−10422.0	−0.782249
$563$	−3425.06	−0.256393	−0.128196	−	0.991749i	$-0.540919\pi$
$563$	−3425.06	−0.256393	−0.128196	+	0.991749i	$0.540919\pi$
$564$	0	0
$565$	38958.1	2.90085
$566$	10313.2	0.765896
$567$	0	0
$568$	3527.58	0.260588
$569$	23899.8	1.76086	0.880431	−	0.474175i	$-0.157254\pi$
$569$	23899.8	1.76086	0.880431	+	0.474175i	$0.157254\pi$
$570$	0	0
$571$	−12192.1	−0.893563	−0.446781	−	0.894643i	$-0.647430\pi$
$571$	−12192.1	−0.893563	−0.446781	+	0.894643i	$0.647430\pi$
$572$	−7622.82	−0.557214
$573$	0	0
$574$	8793.02	0.639397
$575$	−5436.59	−0.394298
$576$	0	0
$577$	16729.8	1.20706	0.603529	−	0.797341i	$-0.293761\pi$
$577$	16729.8	1.20706	0.603529	+	0.797341i	$0.293761\pi$
$578$	9361.44	0.673676
$579$	0	0
$580$	−9475.91	−0.678389
$581$	−18356.1	−1.31074
$582$	0	0
$583$	−13700.7	−0.973283
$584$	−3121.97	−0.221212
$585$	0	0
$586$	9956.38	0.701867
$587$	−10502.3	−0.738463	−0.369231	−	0.929338i	$-0.620379\pi$
$587$	−10502.3	−0.738463	−0.369231	+	0.929338i	$0.620379\pi$
$588$	0	0
$589$	19877.8	1.39058
$590$	−22832.3	−1.59321
$591$	0	0
$592$	95.0388	0.00659809
$593$	−19802.9	−1.37134	−0.685671	−	0.727912i	$-0.740491\pi$
$593$	−19802.9	−1.37134	−0.685671	+	0.727912i	$0.740491\pi$
$594$	0	0
$595$	−7649.11	−0.527030
$596$	11818.5	0.812255
$597$	0	0
$598$	3215.12	0.219859
$599$	26702.7	1.82144	0.910720	−	0.413024i	$-0.135528\pi$
$599$	26702.7	1.82144	0.910720	+	0.413024i	$0.135528\pi$
$600$	0	0
$601$	−18863.6	−1.28030	−0.640150	−	0.768250i	$-0.721128\pi$
$601$	−18863.6	−1.28030	−0.640150	+	0.768250i	$0.721128\pi$
$602$	−16806.8	−1.13787
$603$	0	0
$604$	−13534.1	−0.911749
$605$	−11169.8	−0.750607
$606$	0	0
$607$	10095.7	0.675079	0.337539	−	0.941311i	$-0.390405\pi$
$607$	10095.7	0.675079	0.337539	+	0.941311i	$0.390405\pi$
$608$	−2101.08	−0.140148
$609$	0	0
$610$	−24183.2	−1.60516
$611$	−10109.7	−0.669383
$612$	0	0
$613$	770.460	0.0507644	0.0253822	−	0.999678i	$-0.491920\pi$
$613$	770.460	0.0507644	0.0253822	+	0.999678i	$0.491920\pi$
$614$	−6071.85	−0.399088
$615$	0	0
$616$	−5758.82	−0.376671
$617$	6297.91	0.410931	0.205465	−	0.978664i	$-0.434129\pi$
$617$	6297.91	0.410931	0.205465	+	0.978664i	$0.434129\pi$
$618$	0	0
$619$	893.359	0.0580083	0.0290041	−	0.999579i	$-0.490766\pi$
$619$	893.359	0.0580083	0.0290041	+	0.999579i	$0.490766\pi$
$620$	23020.5	1.49117
$621$	0	0
$622$	9623.23	0.620348
$623$	25052.3	1.61107
$624$	0	0
$625$	10700.8	0.684848
$626$	13919.8	0.888733
$627$	0	0
$628$	−1175.52	−0.0746945
$629$	90.5289	0.00573867
$630$	0	0
$631$	16031.2	1.01140	0.505698	−	0.862711i	$-0.331235\pi$
$631$	16031.2	1.01140	0.505698	+	0.862711i	$0.331235\pi$
$632$	−5450.54	−0.343055
$633$	0	0
$634$	−5634.31	−0.352945
$635$	25147.0	1.57154
$636$	0	0
$637$	24744.7	1.53912
$638$	−6795.63	−0.421695
$639$	0	0
$640$	−2433.26	−0.150286
$641$	−11366.2	−0.700370	−0.350185	−	0.936681i	$-0.613881\pi$
$641$	−11366.2	−0.700370	−0.350185	+	0.936681i	$0.613881\pi$
$642$	0	0
$643$	−267.056	−0.0163789	−0.00818946	−	0.999966i	$-0.502607\pi$
$643$	−267.056	−0.0163789	−0.00818946	+	0.999966i	$0.502607\pi$
$644$	2428.93	0.148623
$645$	0	0
$646$	−2001.37	−0.121893
$647$	10111.1	0.614389	0.307194	−	0.951647i	$-0.400610\pi$
$647$	10111.1	0.614389	0.307194	+	0.951647i	$0.400610\pi$
$648$	0	0
$649$	−16374.1	−0.990357
$650$	−33042.1	−1.99387
$651$	0	0
$652$	5108.69	0.306859
$653$	−12953.9	−0.776299	−0.388150	−	0.921596i	$-0.626886\pi$
$653$	−12953.9	−0.776299	−0.388150	+	0.921596i	$0.626886\pi$
$654$	0	0
$655$	−6721.70	−0.400975
$656$	2664.41	0.158579
$657$	0	0
$658$	−7637.56	−0.452497
$659$	920.513	0.0544129	0.0272064	−	0.999630i	$-0.491339\pi$
$659$	920.513	0.0544129	0.0272064	+	0.999630i	$0.491339\pi$
$660$	0	0
$661$	6224.81	0.366289	0.183144	−	0.983086i	$-0.441372\pi$
$661$	6224.81	0.366289	0.183144	+	0.983086i	$0.441372\pi$
$662$	−16524.1	−0.970131
$663$	0	0
$664$	−5562.15	−0.325080
$665$	−32953.2	−1.92161
$666$	0	0
$667$	2866.23	0.166388
$668$	−6039.35	−0.349805
$669$	0	0
$670$	−7838.24	−0.451966
$671$	−17342.9	−0.997789
$672$	0	0
$673$	7195.10	0.412111	0.206056	−	0.978540i	$-0.433937\pi$
$673$	7195.10	0.412111	0.206056	+	0.978540i	$0.433937\pi$
$674$	16606.5	0.949047
$675$	0	0
$676$	10752.6	0.611776
$677$	2584.01	0.146694	0.0733468	−	0.997306i	$-0.476632\pi$
$677$	2584.01	0.146694	0.0733468	+	0.997306i	$0.476632\pi$
$678$	0	0
$679$	−19718.2	−1.11445
$680$	−2317.79	−0.130711
$681$	0	0
$682$	16509.1	0.926929
$683$	15445.0	0.865280	0.432640	−	0.901567i	$-0.357582\pi$
$683$	15445.0	0.865280	0.432640	+	0.901567i	$0.357582\pi$
$684$	0	0
$685$	−60295.3	−3.36316
$686$	582.587	0.0324246
$687$	0	0
$688$	−5092.71	−0.282206
$689$	35120.7	1.94194
$690$	0	0
$691$	28002.8	1.54164	0.770822	−	0.637050i	$-0.219846\pi$
$691$	28002.8	1.54164	0.770822	+	0.637050i	$0.219846\pi$
$692$	−551.107	−0.0302745
$693$	0	0
$694$	−615.074	−0.0336425
$695$	9658.05	0.527123
$696$	0	0
$697$	2537.98	0.137924
$698$	21.8499	0.00118486
$699$	0	0
$700$	−24962.4	−1.34784
$701$	31379.7	1.69072	0.845360	−	0.534198i	$-0.179386\pi$
$701$	31379.7	1.69072	0.845360	+	0.534198i	$0.179386\pi$
$702$	0	0
$703$	390.008	0.0209238
$704$	−1745.01	−0.0934196
$705$	0	0
$706$	−1051.79	−0.0560688
$707$	30836.9	1.64037
$708$	0	0
$709$	−12612.6	−0.668092	−0.334046	−	0.942557i	$-0.608414\pi$
$709$	−12612.6	−0.668092	−0.334046	+	0.942557i	$0.608414\pi$
$710$	16764.7	0.886151
$711$	0	0
$712$	7591.20	0.399568
$713$	−6963.12	−0.365738
$714$	0	0
$715$	−36227.1	−1.89485
$716$	12175.9	0.635523
$717$	0	0
$718$	−3453.98	−0.179528
$719$	8522.72	0.442064	0.221032	−	0.975267i	$-0.429058\pi$
$719$	8522.72	0.442064	0.221032	+	0.975267i	$0.429058\pi$
$720$	0	0
$721$	−16062.0	−0.829652
$722$	5095.87	0.262671
$723$	0	0
$724$	15627.6	0.802204
$725$	−29456.5	−1.50895
$726$	0	0
$727$	9853.76	0.502690	0.251345	−	0.967898i	$-0.419127\pi$
$727$	9853.76	0.502690	0.251345	+	0.967898i	$0.419127\pi$
$728$	14762.3	0.751551
$729$	0	0
$730$	−14837.0	−0.752250
$731$	−4851.05	−0.245448
$732$	0	0
$733$	−21851.1	−1.10107	−0.550537	−	0.834810i	$-0.685577\pi$
$733$	−21851.1	−1.10107	−0.550537	+	0.834810i	$0.685577\pi$
$734$	1075.63	0.0540903
$735$	0	0
$736$	736.000	0.0368605
$737$	−5621.18	−0.280948
$738$	0	0
$739$	9049.47	0.450460	0.225230	−	0.974306i	$-0.427687\pi$
$739$	9049.47	0.450460	0.225230	+	0.974306i	$0.427687\pi$
$740$	451.668	0.0224373
$741$	0	0
$742$	26532.7	1.31273
$743$	10650.0	0.525855	0.262927	−	0.964816i	$-0.415312\pi$
$743$	10650.0	0.525855	0.262927	+	0.964816i	$0.415312\pi$
$744$	0	0
$745$	56166.8	2.76214
$746$	−6148.39	−0.301754
$747$	0	0
$748$	−1662.20	−0.0812514
$749$	29960.1	1.46157
$750$	0	0
$751$	−9340.20	−0.453833	−0.226917	−	0.973914i	$-0.572864\pi$
$751$	−9340.20	−0.453833	−0.226917	+	0.973914i	$0.572864\pi$
$752$	−2314.29	−0.112225
$753$	0	0
$754$	17420.1	0.841384
$755$	−64320.4	−3.10048
$756$	0	0
$757$	−12383.4	−0.594562	−0.297281	−	0.954790i	$-0.596080\pi$
$757$	−12383.4	−0.594562	−0.297281	+	0.954790i	$0.596080\pi$
$758$	−18490.3	−0.886013
$759$	0	0
$760$	−9985.28	−0.476585
$761$	−5906.69	−0.281363	−0.140682	−	0.990055i	$-0.544929\pi$
$761$	−5906.69	−0.281363	−0.140682	+	0.990055i	$0.544929\pi$
$762$	0	0
$763$	−42311.2	−2.00756
$764$	−4075.73	−0.193004
$765$	0	0
$766$	4058.91	0.191455
$767$	41974.0	1.97600
$768$	0	0
$769$	17056.5	0.799834	0.399917	−	0.916551i	$-0.369039\pi$
$769$	17056.5	0.799834	0.399917	+	0.916551i	$0.369039\pi$
$770$	−27368.6	−1.28090
$771$	0	0
$772$	−17211.7	−0.802412
$773$	28498.6	1.32603	0.663017	−	0.748604i	$-0.269276\pi$
$773$	28498.6	1.32603	0.663017	+	0.748604i	$0.269276\pi$
$774$	0	0
$775$	71560.7	3.31682
$776$	−5974.89	−0.276399
$777$	0	0
$778$	14163.3	0.652674
$779$	10933.9	0.502884
$780$	0	0
$781$	12022.8	0.550843
$782$	701.074	0.0320593
$783$	0	0
$784$	5664.53	0.258042
$785$	−5586.59	−0.254005
$786$	0	0
$787$	30401.9	1.37701	0.688507	−	0.725230i	$-0.258266\pi$
$787$	30401.9	1.37701	0.688507	+	0.725230i	$0.258266\pi$
$788$	−5757.43	−0.260279
$789$	0	0
$790$	−25903.4	−1.16659
$791$	−54106.2	−2.43210
$792$	0	0
$793$	44457.4	1.99083
$794$	−7948.22	−0.355254
$795$	0	0
$796$	21051.7	0.937385
$797$	3453.59	0.153491	0.0767456	−	0.997051i	$-0.475547\pi$
$797$	3453.59	0.153491	0.0767456	+	0.997051i	$0.475547\pi$
$798$	0	0
$799$	−2204.47	−0.0976077
$800$	−7563.95	−0.334283
$801$	0	0
$802$	−2456.64	−0.108163
$803$	−10640.3	−0.467608
$804$	0	0
$805$	11543.4	0.505404
$806$	−42319.9	−1.84945
$807$	0	0
$808$	9344.01	0.406833
$809$	−31655.0	−1.37569	−0.687844	−	0.725859i	$-0.741443\pi$
$809$	−31655.0	−1.37569	−0.687844	+	0.725859i	$0.741443\pi$
$810$	0	0
$811$	−2071.19	−0.0896786	−0.0448393	−	0.998994i	$-0.514278\pi$
$811$	−2071.19	−0.0896786	−0.0448393	+	0.998994i	$0.514278\pi$
$812$	13160.4	0.568768
$813$	0	0
$814$	323.913	0.0139473
$815$	24278.8	1.04350
$816$	0	0
$817$	−20898.8	−0.894929
$818$	9119.74	0.389809
$819$	0	0
$820$	12662.5	0.539261
$821$	−3082.16	−0.131021	−0.0655105	−	0.997852i	$-0.520868\pi$
$821$	−3082.16	−0.131021	−0.0655105	+	0.997852i	$0.520868\pi$
$822$	0	0
$823$	−16572.0	−0.701901	−0.350951	−	0.936394i	$-0.614142\pi$
$823$	−16572.0	−0.701901	−0.350951	+	0.936394i	$0.614142\pi$
$824$	−4867.01	−0.205765
$825$	0	0
$826$	31710.2	1.33576
$827$	−5374.02	−0.225965	−0.112983	−	0.993597i	$-0.536040\pi$
$827$	−5374.02	−0.225965	−0.112983	+	0.993597i	$0.536040\pi$
$828$	0	0
$829$	−19312.7	−0.809117	−0.404559	−	0.914512i	$-0.632575\pi$
$829$	−19312.7	−0.809117	−0.404559	+	0.914512i	$0.632575\pi$
$830$	−26433.9	−1.10546
$831$	0	0
$832$	4473.20	0.186395
$833$	5395.73	0.224431
$834$	0	0
$835$	−28701.8	−1.18954
$836$	−7160.93	−0.296251
$837$	0	0
$838$	32729.2	1.34918
$839$	−8072.94	−0.332192	−0.166096	−	0.986110i	$-0.553116\pi$
$839$	−8072.94	−0.332192	−0.166096	+	0.986110i	$0.553116\pi$
$840$	0	0
$841$	−8859.21	−0.363246
$842$	−16935.0	−0.693134
$843$	0	0
$844$	1414.77	0.0576997
$845$	51101.2	2.08039
$846$	0	0
$847$	15512.9	0.629316
$848$	8039.80	0.325575
$849$	0	0
$850$	−7205.02	−0.290741
$851$	−136.618	−0.00550319
$852$	0	0
$853$	−3676.62	−0.147579	−0.0737896	−	0.997274i	$-0.523509\pi$
$853$	−3676.62	−0.147579	−0.0737896	+	0.997274i	$0.523509\pi$
$854$	33586.3	1.34578
$855$	0	0
$856$	9078.34	0.362490
$857$	−33806.9	−1.34752	−0.673758	−	0.738952i	$-0.735321\pi$
$857$	−33806.9	−1.34752	−0.673758	+	0.738952i	$0.735321\pi$
$858$	0	0
$859$	−16752.0	−0.665391	−0.332696	−	0.943034i	$-0.607958\pi$
$859$	−16752.0	−0.665391	−0.332696	+	0.943034i	$0.607958\pi$
$860$	−24202.9	−0.959665
$861$	0	0
$862$	24881.0	0.983122
$863$	26052.3	1.02761	0.513806	−	0.857906i	$-0.328235\pi$
$863$	26052.3	1.02761	0.513806	+	0.857906i	$0.328235\pi$
$864$	0	0
$865$	−2619.11	−0.102951
$866$	−8158.46	−0.320134
$867$	0	0
$868$	−31971.5	−1.25021
$869$	−18576.6	−0.725165
$870$	0	0
$871$	14409.5	0.560560
$872$	−12820.9	−0.497902
$873$	0	0
$874$	3020.30	0.116891
$875$	−55896.8	−2.15961
$876$	0	0
$877$	−7214.34	−0.277778	−0.138889	−	0.990308i	$-0.544353\pi$
$877$	−7214.34	−0.277778	−0.138889	+	0.990308i	$0.544353\pi$
$878$	−6972.45	−0.268005
$879$	0	0
$880$	−8293.07	−0.317681
$881$	30172.3	1.15384	0.576919	−	0.816801i	$-0.304255\pi$
$881$	30172.3	1.15384	0.576919	+	0.816801i	$0.304255\pi$
$882$	0	0
$883$	37498.4	1.42913	0.714564	−	0.699570i	$-0.246625\pi$
$883$	37498.4	1.42913	0.714564	+	0.699570i	$0.246625\pi$
$884$	4260.93	0.162116
$885$	0	0
$886$	22640.6	0.858494
$887$	−17991.8	−0.681065	−0.340533	−	0.940233i	$-0.610607\pi$
$887$	−17991.8	−0.681065	−0.340533	+	0.940233i	$0.610607\pi$
$888$	0	0
$889$	−34924.9	−1.31760
$890$	36076.8	1.35876
$891$	0	0
$892$	−21345.2	−0.801223
$893$	−9497.08	−0.355888
$894$	0	0
$895$	57865.4	2.16115
$896$	3379.38	0.126001
$897$	0	0
$898$	−33709.3	−1.25266
$899$	−37727.6	−1.39965
$900$	0	0
$901$	7658.28	0.283168
$902$	9080.89	0.335211
$903$	0	0
$904$	−16394.9	−0.603194
$905$	74269.5	2.72796
$906$	0	0
$907$	44123.2	1.61531	0.807655	−	0.589655i	$-0.200736\pi$
$907$	44123.2	1.61531	0.807655	+	0.589655i	$0.200736\pi$
$908$	−8853.19	−0.323572
$909$	0	0
$910$	70157.4	2.55571
$911$	10096.6	0.367196	0.183598	−	0.983001i	$-0.441226\pi$
$911$	10096.6	0.367196	0.183598	+	0.983001i	$0.441226\pi$
$912$	0	0
$913$	−18957.0	−0.687169
$914$	10659.1	0.385744
$915$	0	0
$916$	−20564.4	−0.741774
$917$	9335.29	0.336181
$918$	0	0
$919$	40900.3	1.46809	0.734045	−	0.679100i	$-0.237630\pi$
$919$	40900.3	1.46809	0.734045	+	0.679100i	$0.237630\pi$
$920$	3497.81	0.125347
$921$	0	0
$922$	17224.6	0.615252
$923$	−30819.5	−1.09907
$924$	0	0
$925$	1404.04	0.0499077
$926$	23107.9	0.820056
$927$	0	0
$928$	3987.79	0.141062
$929$	−22417.0	−0.791688	−0.395844	−	0.918318i	$-0.629548\pi$
$929$	−22417.0	−0.791688	−0.395844	+	0.918318i	$0.629548\pi$
$930$	0	0
$931$	23245.4	0.818298
$932$	−26329.3	−0.925372
$933$	0	0
$934$	12851.1	0.450214
$935$	−7899.53	−0.276302
$936$	0	0
$937$	3106.56	0.108310	0.0541552	−	0.998533i	$-0.482753\pi$
$937$	3106.56	0.108310	0.0541552	+	0.998533i	$0.482753\pi$
$938$	10886.0	0.378933
$939$	0	0
$940$	−10998.6	−0.381632
$941$	19428.6	0.673066	0.336533	−	0.941672i	$-0.390746\pi$
$941$	19428.6	0.673066	0.336533	+	0.941672i	$0.390746\pi$
$942$	0	0
$943$	−3830.09	−0.132264
$944$	9608.63	0.331286
$945$	0	0
$946$	−17357.1	−0.596540
$947$	26080.2	0.894924	0.447462	−	0.894303i	$-0.352328\pi$
$947$	26080.2	0.894924	0.447462	+	0.894303i	$0.352328\pi$
$948$	0	0
$949$	27275.8	0.932992
$950$	−31040.0	−1.06007
$951$	0	0
$952$	3219.01	0.109589
$953$	12821.6	0.435815	0.217907	−	0.975969i	$-0.430077\pi$
$953$	12821.6	0.435815	0.217907	+	0.975969i	$0.430077\pi$
$954$	0	0
$955$	−19369.7	−0.656325
$956$	−24823.8	−0.839812
$957$	0	0
$958$	17332.2	0.584528
$959$	83739.8	2.81971
$960$	0	0
$961$	61863.1	2.07657
$962$	−830.328	−0.0278283
$963$	0	0
$964$	−25453.9	−0.850432
$965$	−81797.8	−2.72867
$966$	0	0
$967$	−4778.68	−0.158916	−0.0794582	−	0.996838i	$-0.525319\pi$
$967$	−4778.68	−0.158916	−0.0794582	+	0.996838i	$0.525319\pi$
$968$	4700.64	0.156079
$969$	0	0
$970$	−28395.4	−0.939918
$971$	7439.58	0.245878	0.122939	−	0.992414i	$-0.460768\pi$
$971$	7439.58	0.245878	0.122939	+	0.992414i	$0.460768\pi$
$972$	0	0
$973$	−13413.4	−0.441945
$974$	37127.2	1.22139
$975$	0	0
$976$	10177.1	0.333773
$977$	−9975.97	−0.326673	−0.163336	−	0.986570i	$-0.552226\pi$
$977$	−9975.97	−0.326673	−0.163336	+	0.986570i	$0.552226\pi$
$978$	0	0
$979$	25872.4	0.844624
$980$	26920.4	0.877492
$981$	0	0
$982$	−23594.5	−0.766732
$983$	−21419.6	−0.694995	−0.347497	−	0.937681i	$-0.612968\pi$
$983$	−21419.6	−0.694995	−0.347497	+	0.937681i	$0.612968\pi$
$984$	0	0
$985$	−27361.9	−0.885100
$986$	3798.56	0.122688
$987$	0	0
$988$	18356.5	0.591093
$989$	7320.78	0.235376
$990$	0	0
$991$	−43927.4	−1.40807	−0.704037	−	0.710164i	$-0.748621\pi$
$991$	−43927.4	−1.40807	−0.704037	+	0.710164i	$0.748621\pi$
$992$	−9687.82	−0.310069
$993$	0	0
$994$	−23283.3	−0.742958
$995$	100047.	3.18765
$996$	0	0
$997$	−39862.3	−1.26625	−0.633125	−	0.774049i	$-0.718228\pi$
$997$	−39862.3	−1.26625	−0.633125	+	0.774049i	$0.718228\pi$
$998$	6906.58	0.219062
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.m.1.4	✓	4
3.2	odd	2		414.4.a.n.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.4.a.m.1.4	✓	4	1.1	even	1	trivial
414.4.a.n.1.1	yes	4	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +19.0098 q^{5} -26.4014 q^{7} -8.00000 q^{8} -38.0197 q^{10} -27.2657 q^{11} +69.8938 q^{13} +52.8028 q^{14} +16.0000 q^{16} +15.2407 q^{17} +65.6587 q^{19} +76.0393 q^{20} +54.5314 q^{22} -23.0000 q^{23} +236.374 q^{25} -139.788 q^{26} -105.606 q^{28} -124.619 q^{29} +302.744 q^{31} -32.0000 q^{32} -30.4815 q^{34} -501.886 q^{35} +5.93992 q^{37} -131.317 q^{38} -152.079 q^{40} +166.526 q^{41} -318.295 q^{43} -109.063 q^{44} +46.0000 q^{46} -144.643 q^{47} +354.033 q^{49} -472.747 q^{50} +279.575 q^{52} +502.487 q^{53} -518.317 q^{55} +211.211 q^{56} +249.237 q^{58} +600.540 q^{59} +636.071 q^{61} -605.489 q^{62} +64.0000 q^{64} +1328.67 q^{65} +206.163 q^{67} +60.9630 q^{68} +1003.77 q^{70} -440.948 q^{71} +390.246 q^{73} -11.8798 q^{74} +262.635 q^{76} +719.853 q^{77} +681.317 q^{79} +304.157 q^{80} -333.051 q^{82} +695.269 q^{83} +289.724 q^{85} +636.589 q^{86} +218.126 q^{88} -948.900 q^{89} -1845.29 q^{91} -92.0000 q^{92} +289.286 q^{94} +1248.16 q^{95} +746.861 q^{97} -708.067 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 18 q^{7} - 32 q^{8} - 42 q^{11} + 108 q^{13} - 36 q^{14} + 64 q^{16} - 26 q^{17} + 132 q^{19} + 84 q^{22} - 92 q^{23} + 260 q^{25} - 216 q^{26} + 72 q^{28} - 252 q^{29} + 428 q^{31}+ \cdots - 2152 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 18 * q^7 - 32 * q^8 - 42 * q^11 + 108 * q^13 - 36 * q^14 + 64 * q^16 - 26 * q^17 + 132 * q^19 + 84 * q^22 - 92 * q^23 + 260 * q^25 - 216 * q^26 + 72 * q^28 - 252 * q^29 + 428 * q^31 - 128 * q^32 + 52 * q^34 - 348 * q^35 + 278 * q^37 - 264 * q^38 + 4 * q^41 + 216 * q^43 - 168 * q^44 + 184 * q^46 + 4 * q^47 + 1076 * q^49 - 520 * q^50 + 432 * q^52 + 460 * q^53 + 512 * q^55 - 144 * q^56 + 504 * q^58 + 504 * q^59 + 1086 * q^61 - 856 * q^62 + 256 * q^64 + 1420 * q^65 + 1484 * q^67 - 104 * q^68 + 696 * q^70 + 876 * q^71 + 1072 * q^73 - 556 * q^74 + 528 * q^76 + 1580 * q^77 + 1634 * q^79 - 8 * q^82 + 2246 * q^83 + 872 * q^85 - 432 * q^86 + 336 * q^88 + 474 * q^89 + 40 * q^91 - 368 * q^92 - 8 * q^94 + 2780 * q^95 + 1044 * q^97 - 2152 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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