LMFDB - Embedded newform 414.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [414,4,Mod(1,414)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,-4,0,8,-8,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.4267907424$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +7.31371 q^{5} +17.3137 q^{7} -8.00000 q^{8} -14.6274 q^{10} -69.2548 q^{11} -3.25483 q^{13} -34.6274 q^{14} +16.0000 q^{16} -85.9411 q^{17} -19.9411 q^{19} +29.2548 q^{20} +138.510 q^{22} -23.0000 q^{23} -71.5097 q^{25} +6.50967 q^{26} +69.2548 q^{28} +47.1371 q^{29} +109.137 q^{31} -32.0000 q^{32} +171.882 q^{34} +126.627 q^{35} +86.9016 q^{37} +39.8823 q^{38} -58.5097 q^{40} -65.7645 q^{41} -398.451 q^{43} -277.019 q^{44} +46.0000 q^{46} +164.118 q^{47} -43.2355 q^{49} +143.019 q^{50} -13.0193 q^{52} -631.862 q^{53} -506.510 q^{55} -138.510 q^{56} -94.2742 q^{58} +665.450 q^{59} -490.431 q^{61} -218.274 q^{62} +64.0000 q^{64} -23.8049 q^{65} +83.4701 q^{67} -343.765 q^{68} -253.255 q^{70} -969.568 q^{71} +462.313 q^{73} -173.803 q^{74} -79.7645 q^{76} -1199.06 q^{77} -857.314 q^{79} +117.019 q^{80} +131.529 q^{82} +1195.61 q^{83} -628.548 q^{85} +796.902 q^{86} +554.039 q^{88} +382.960 q^{89} -56.3532 q^{91} -92.0000 q^{92} -328.235 q^{94} -145.844 q^{95} -817.724 q^{97} +86.4710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 8 q^{5} + 12 q^{7} - 16 q^{8} + 16 q^{10} - 48 q^{11} + 84 q^{13} - 24 q^{14} + 32 q^{16} - 104 q^{17} + 28 q^{19} - 32 q^{20} + 96 q^{22} - 46 q^{23} + 38 q^{25} - 168 q^{26}+ \cdots + 716 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 8 * q^5 + 12 * q^7 - 16 * q^8 + 16 * q^10 - 48 * q^11 + 84 * q^13 - 24 * q^14 + 32 * q^16 - 104 * q^17 + 28 * q^19 - 32 * q^20 + 96 * q^22 - 46 * q^23 + 38 * q^25 - 168 * q^26 + 48 * q^28 - 132 * q^29 - 8 * q^31 - 64 * q^32 + 208 * q^34 + 208 * q^35 - 324 * q^37 - 56 * q^38 + 64 * q^40 + 140 * q^41 - 548 * q^43 - 192 * q^44 + 92 * q^46 + 464 * q^47 - 358 * q^49 - 76 * q^50 + 336 * q^52 - 336 * q^53 - 832 * q^55 - 96 * q^56 + 264 * q^58 - 72 * q^59 + 60 * q^61 + 16 * q^62 + 128 * q^64 - 1360 * q^65 - 444 * q^67 - 416 * q^68 - 416 * q^70 - 672 * q^71 - 252 * q^73 + 648 * q^74 + 112 * q^76 - 1312 * q^77 - 1692 * q^79 - 128 * q^80 - 280 * q^82 + 400 * q^83 - 352 * q^85 + 1096 * q^86 + 384 * q^88 + 336 * q^89 - 520 * q^91 - 184 * q^92 - 928 * q^94 - 880 * q^95 + 220 * q^97 + 716 * 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
$5$	7.31371	0.654158	0.327079	−	0.944997i	$-0.393936\pi$
$5$	7.31371	0.654158	0.327079	+	0.944997i	$0.393936\pi$
$6$	0	0
$7$	17.3137	0.934852	0.467426	−	0.884032i	$-0.345181\pi$
$7$	17.3137	0.934852	0.467426	+	0.884032i	$0.345181\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−14.6274	−0.462560
$11$	−69.2548	−1.89828	−0.949142	−	0.314849i	$-0.898046\pi$
$11$	−69.2548	−1.89828	−0.949142	+	0.314849i	$0.898046\pi$
$12$	0	0
$13$	−3.25483	−0.0694407	−0.0347203	−	0.999397i	$-0.511054\pi$
$13$	−3.25483	−0.0694407	−0.0347203	+	0.999397i	$0.511054\pi$
$14$	−34.6274	−0.661040
$15$	0	0
$16$	16.0000	0.250000
$17$	−85.9411	−1.22610	−0.613052	−	0.790042i	$-0.710059\pi$
$17$	−85.9411	−1.22610	−0.613052	+	0.790042i	$0.710059\pi$
$18$	0	0
$19$	−19.9411	−0.240779	−0.120390	−	0.992727i	$-0.538414\pi$
$19$	−19.9411	−0.240779	−0.120390	+	0.992727i	$0.538414\pi$
$20$	29.2548	0.327079
$21$	0	0
$22$	138.510	1.34229
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−71.5097	−0.572077
$26$	6.50967	0.0491020
$27$	0	0
$28$	69.2548	0.467426
$29$	47.1371	0.301832	0.150916	−	0.988547i	$-0.451778\pi$
$29$	47.1371	0.301832	0.150916	+	0.988547i	$0.451778\pi$
$30$	0	0
$31$	109.137	0.632310	0.316155	−	0.948708i	$-0.397608\pi$
$31$	109.137	0.632310	0.316155	+	0.948708i	$0.397608\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	171.882	0.866987
$35$	126.627	0.611541
$36$	0	0
$37$	86.9016	0.386123	0.193061	−	0.981187i	$-0.438158\pi$
$37$	86.9016	0.386123	0.193061	+	0.981187i	$0.438158\pi$
$38$	39.8823	0.170257
$39$	0	0
$40$	−58.5097	−0.231280
$41$	−65.7645	−0.250505	−0.125252	−	0.992125i	$-0.539974\pi$
$41$	−65.7645	−0.250505	−0.125252	+	0.992125i	$0.539974\pi$
$42$	0	0
$43$	−398.451	−1.41310	−0.706549	−	0.707665i	$-0.749749\pi$
$43$	−398.451	−1.41310	−0.706549	+	0.707665i	$0.749749\pi$
$44$	−277.019	−0.949142
$45$	0	0
$46$	46.0000	0.147442
$47$	164.118	0.509341	0.254671	−	0.967028i	$-0.418033\pi$
$47$	164.118	0.509341	0.254671	+	0.967028i	$0.418033\pi$
$48$	0	0
$49$	−43.2355	−0.126051
$50$	143.019	0.404520
$51$	0	0
$52$	−13.0193	−0.0347203
$53$	−631.862	−1.63760	−0.818801	−	0.574077i	$-0.805361\pi$
$53$	−631.862	−1.63760	−0.818801	+	0.574077i	$0.805361\pi$
$54$	0	0
$55$	−506.510	−1.24178
$56$	−138.510	−0.330520
$57$	0	0
$58$	−94.2742	−0.213428
$59$	665.450	1.46838	0.734188	−	0.678946i	$-0.237563\pi$
$59$	665.450	1.46838	0.734188	+	0.678946i	$0.237563\pi$
$60$	0	0
$61$	−490.431	−1.02940	−0.514698	−	0.857371i	$-0.672096\pi$
$61$	−490.431	−1.02940	−0.514698	+	0.857371i	$0.672096\pi$
$62$	−218.274	−0.447110
$63$	0	0
$64$	64.0000	0.125000
$65$	−23.8049	−0.0454252
$66$	0	0
$67$	83.4701	0.152201	0.0761007	−	0.997100i	$-0.475753\pi$
$67$	83.4701	0.152201	0.0761007	+	0.997100i	$0.475753\pi$
$68$	−343.765	−0.613052
$69$	0	0
$70$	−253.255	−0.432425
$71$	−969.568	−1.62066	−0.810328	−	0.585977i	$-0.800711\pi$
$71$	−969.568	−1.62066	−0.810328	+	0.585977i	$0.800711\pi$
$72$	0	0
$73$	462.313	0.741228	0.370614	−	0.928787i	$-0.379147\pi$
$73$	462.313	0.741228	0.370614	+	0.928787i	$0.379147\pi$
$74$	−173.803	−0.273030
$75$	0	0
$76$	−79.7645	−0.120390
$77$	−1199.06	−1.77461
$78$	0	0
$79$	−857.314	−1.22095	−0.610477	−	0.792034i	$-0.709022\pi$
$79$	−857.314	−1.22095	−0.610477	+	0.792034i	$0.709022\pi$
$80$	117.019	0.163539
$81$	0	0
$82$	131.529	0.177134
$83$	1195.61	1.58114	0.790571	−	0.612370i	$-0.209784\pi$
$83$	1195.61	1.58114	0.790571	+	0.612370i	$0.209784\pi$
$84$	0	0
$85$	−628.548	−0.802066
$86$	796.902	0.999211
$87$	0	0
$88$	554.039	0.671145
$89$	382.960	0.456109	0.228055	−	0.973648i	$-0.426764\pi$
$89$	382.960	0.456109	0.228055	+	0.973648i	$0.426764\pi$
$90$	0	0
$91$	−56.3532	−0.0649168
$92$	−92.0000	−0.104257
$93$	0	0
$94$	−328.235	−0.360159
$95$	−145.844	−0.157508
$96$	0	0
$97$	−817.724	−0.855952	−0.427976	−	0.903790i	$-0.640773\pi$
$97$	−817.724	−0.855952	−0.427976	+	0.903790i	$0.640773\pi$
$98$	86.4710	0.0891315
$99$	0	0
$100$	−286.039	−0.286039
$101$	−530.548	−0.522688	−0.261344	−	0.965246i	$-0.584166\pi$
$101$	−530.548	−0.522688	−0.261344	+	0.965246i	$0.584166\pi$
$102$	0	0
$103$	−1427.12	−1.36522	−0.682612	−	0.730781i	$-0.739156\pi$
$103$	−1427.12	−1.36522	−0.682612	+	0.730781i	$0.739156\pi$
$104$	26.0387	0.0245510
$105$	0	0
$106$	1263.72	1.15796
$107$	−1085.49	−0.980732	−0.490366	−	0.871517i	$-0.663137\pi$
$107$	−1085.49	−0.980732	−0.490366	+	0.871517i	$0.663137\pi$
$108$	0	0
$109$	−100.432	−0.0882539	−0.0441269	−	0.999026i	$-0.514051\pi$
$109$	−100.432	−0.0882539	−0.0441269	+	0.999026i	$0.514051\pi$
$110$	1013.02	0.878069
$111$	0	0
$112$	277.019	0.233713
$113$	−2289.43	−1.90594	−0.952971	−	0.303062i	$-0.901991\pi$
$113$	−2289.43	−1.90594	−0.952971	+	0.303062i	$0.901991\pi$
$114$	0	0
$115$	−168.215	−0.136401
$116$	188.548	0.150916
$117$	0	0
$118$	−1330.90	−1.03830
$119$	−1487.96	−1.14623
$120$	0	0
$121$	3465.23	2.60348
$122$	980.861	0.727893
$123$	0	0
$124$	436.548	0.316155
$125$	−1437.21	−1.02839
$126$	0	0
$127$	313.531	0.219066	0.109533	−	0.993983i	$-0.465065\pi$
$127$	313.531	0.219066	0.109533	+	0.993983i	$0.465065\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	47.6098	0.0321204
$131$	517.568	0.345192	0.172596	−	0.984993i	$-0.444785\pi$
$131$	517.568	0.345192	0.172596	+	0.984993i	$0.444785\pi$
$132$	0	0
$133$	−345.255	−0.225093
$134$	−166.940	−0.107623
$135$	0	0
$136$	687.529	0.433494
$137$	435.038	0.271298	0.135649	−	0.990757i	$-0.456688\pi$
$137$	435.038	0.271298	0.135649	+	0.990757i	$0.456688\pi$
$138$	0	0
$139$	124.626	0.0760476	0.0380238	−	0.999277i	$-0.487894\pi$
$139$	124.626	0.0760476	0.0380238	+	0.999277i	$0.487894\pi$
$140$	506.510	0.305771
$141$	0	0
$142$	1939.14	1.14598
$143$	225.413	0.131818
$144$	0	0
$145$	344.747	0.197446
$146$	−924.626	−0.524127
$147$	0	0
$148$	347.606	0.193061
$149$	2790.29	1.53416	0.767079	−	0.641553i	$-0.221709\pi$
$149$	2790.29	1.53416	0.767079	+	0.641553i	$0.221709\pi$
$150$	0	0
$151$	2883.14	1.55382	0.776908	−	0.629614i	$-0.216787\pi$
$151$	2883.14	1.55382	0.776908	+	0.629614i	$0.216787\pi$
$152$	159.529	0.0851284
$153$	0	0
$154$	2398.12	1.25484
$155$	798.197	0.413630
$156$	0	0
$157$	1524.86	0.775142	0.387571	−	0.921840i	$-0.373314\pi$
$157$	1524.86	0.775142	0.387571	+	0.921840i	$0.373314\pi$
$158$	1714.63	0.863345
$159$	0	0
$160$	−234.039	−0.115640
$161$	−398.215	−0.194930
$162$	0	0
$163$	−1796.74	−0.863385	−0.431693	−	0.902021i	$-0.642083\pi$
$163$	−1796.74	−0.863385	−0.431693	+	0.902021i	$0.642083\pi$
$164$	−263.058	−0.125252
$165$	0	0
$166$	−2391.21	−1.11804
$167$	1339.61	0.620730	0.310365	−	0.950618i	$-0.399549\pi$
$167$	1339.61	0.620730	0.310365	+	0.950618i	$0.399549\pi$
$168$	0	0
$169$	−2186.41	−0.995178
$170$	1257.10	0.567147
$171$	0	0
$172$	−1593.80	−0.706549
$173$	1753.65	0.770678	0.385339	−	0.922775i	$-0.374085\pi$
$173$	1753.65	0.770678	0.385339	+	0.922775i	$0.374085\pi$
$174$	0	0
$175$	−1238.10	−0.534808
$176$	−1108.08	−0.474571
$177$	0	0
$178$	−765.921	−0.322518
$179$	4375.18	1.82690	0.913452	−	0.406945i	$-0.133406\pi$
$179$	4375.18	1.82690	0.913452	+	0.406945i	$0.133406\pi$
$180$	0	0
$181$	−3619.02	−1.48619	−0.743093	−	0.669188i	$-0.766642\pi$
$181$	−3619.02	−1.48619	−0.743093	+	0.669188i	$0.766642\pi$
$182$	112.706	0.0459031
$183$	0	0
$184$	184.000	0.0737210
$185$	635.573	0.252585
$186$	0	0
$187$	5951.84	2.32749
$188$	656.471	0.254671
$189$	0	0
$190$	291.687	0.111375
$191$	−537.369	−0.203574	−0.101787	−	0.994806i	$-0.532456\pi$
$191$	−537.369	−0.203574	−0.101787	+	0.994806i	$0.532456\pi$
$192$	0	0
$193$	−334.784	−0.124861	−0.0624307	−	0.998049i	$-0.519885\pi$
$193$	−334.784	−0.124861	−0.0624307	+	0.998049i	$0.519885\pi$
$194$	1635.45	0.605249
$195$	0	0
$196$	−172.942	−0.0630255
$197$	1674.19	0.605488	0.302744	−	0.953072i	$-0.402097\pi$
$197$	1674.19	0.605488	0.302744	+	0.953072i	$0.402097\pi$
$198$	0	0
$199$	767.231	0.273304	0.136652	−	0.990619i	$-0.456366\pi$
$199$	767.231	0.273304	0.136652	+	0.990619i	$0.456366\pi$
$200$	572.077	0.202260
$201$	0	0
$202$	1061.10	0.369597
$203$	816.118	0.282169
$204$	0	0
$205$	−480.982	−0.163870
$206$	2854.23	0.965359
$207$	0	0
$208$	−52.0773	−0.0173602
$209$	1381.02	0.457067
$210$	0	0
$211$	−4779.40	−1.55937	−0.779687	−	0.626170i	$-0.784622\pi$
$211$	−4779.40	−1.55937	−0.779687	+	0.626170i	$0.784622\pi$
$212$	−2527.45	−0.818801
$213$	0	0
$214$	2170.98	0.693482
$215$	−2914.15	−0.924389
$216$	0	0
$217$	1889.57	0.591116
$218$	200.865	0.0624049
$219$	0	0
$220$	−2026.04	−0.620889
$221$	279.724	0.0851415
$222$	0	0
$223$	−797.410	−0.239455	−0.119728	−	0.992807i	$-0.538202\pi$
$223$	−797.410	−0.239455	−0.119728	+	0.992807i	$0.538202\pi$
$224$	−554.039	−0.165260
$225$	0	0
$226$	4578.86	1.34770
$227$	5898.63	1.72469	0.862347	−	0.506317i	$-0.168994\pi$
$227$	5898.63	1.72469	0.862347	+	0.506317i	$0.168994\pi$
$228$	0	0
$229$	−168.826	−0.0487176	−0.0243588	−	0.999703i	$-0.507754\pi$
$229$	−168.826	−0.0487176	−0.0243588	+	0.999703i	$0.507754\pi$
$230$	336.431	0.0964503
$231$	0	0
$232$	−377.097	−0.106714
$233$	−4184.67	−1.17660	−0.588298	−	0.808644i	$-0.700202\pi$
$233$	−4184.67	−1.17660	−0.588298	+	0.808644i	$0.700202\pi$
$234$	0	0
$235$	1200.31	0.333190
$236$	2661.80	0.734188
$237$	0	0
$238$	2975.92	0.810505
$239$	−2573.96	−0.696635	−0.348317	−	0.937377i	$-0.613247\pi$
$239$	−2573.96	−0.696635	−0.348317	+	0.937377i	$0.613247\pi$
$240$	0	0
$241$	−1092.40	−0.291981	−0.145990	−	0.989286i	$-0.546637\pi$
$241$	−1092.40	−0.291981	−0.145990	+	0.989286i	$0.546637\pi$
$242$	−6930.46	−1.84094
$243$	0	0
$244$	−1961.72	−0.514698
$245$	−316.212	−0.0824573
$246$	0	0
$247$	64.9051	0.0167199
$248$	−873.097	−0.223555
$249$	0	0
$250$	2874.43	0.727179
$251$	5504.11	1.38413	0.692065	−	0.721836i	$-0.256701\pi$
$251$	5504.11	1.38413	0.692065	+	0.721836i	$0.256701\pi$
$252$	0	0
$253$	1592.86	0.395819
$254$	−627.061	−0.154903
$255$	0	0
$256$	256.000	0.0625000
$257$	−3654.39	−0.886984	−0.443492	−	0.896278i	$-0.646260\pi$
$257$	−3654.39	−0.886984	−0.443492	+	0.896278i	$0.646260\pi$
$258$	0	0
$259$	1504.59	0.360968
$260$	−95.2196	−0.0227126
$261$	0	0
$262$	−1035.14	−0.244087
$263$	5352.43	1.25492	0.627462	−	0.778648i	$-0.284094\pi$
$263$	5352.43	1.25492	0.627462	+	0.778648i	$0.284094\pi$
$264$	0	0
$265$	−4621.25	−1.07125
$266$	690.510	0.159165
$267$	0	0
$268$	333.881	0.0761007
$269$	−4423.21	−1.00256	−0.501279	−	0.865286i	$-0.667137\pi$
$269$	−4423.21	−1.00256	−0.501279	+	0.865286i	$0.667137\pi$
$270$	0	0
$271$	−3179.72	−0.712747	−0.356374	−	0.934344i	$-0.615987\pi$
$271$	−3179.72	−0.712747	−0.356374	+	0.934344i	$0.615987\pi$
$272$	−1375.06	−0.306526
$273$	0	0
$274$	−870.076	−0.191836
$275$	4952.39	1.08596
$276$	0	0
$277$	1135.26	0.246249	0.123125	−	0.992391i	$-0.460708\pi$
$277$	1135.26	0.246249	0.123125	+	0.992391i	$0.460708\pi$
$278$	−249.251	−0.0537738
$279$	0	0
$280$	−1013.02	−0.216212
$281$	2869.86	0.609257	0.304629	−	0.952471i	$-0.401468\pi$
$281$	2869.86	0.609257	0.304629	+	0.952471i	$0.401468\pi$
$282$	0	0
$283$	857.505	0.180118	0.0900590	−	0.995936i	$-0.471294\pi$
$283$	857.505	0.180118	0.0900590	+	0.995936i	$0.471294\pi$
$284$	−3878.27	−0.810328
$285$	0	0
$286$	−450.826	−0.0932094
$287$	−1138.63	−0.234185
$288$	0	0
$289$	2472.88	0.503333
$290$	−689.494	−0.139615
$291$	0	0
$292$	1849.25	0.370614
$293$	−8823.46	−1.75929	−0.879645	−	0.475631i	$-0.842220\pi$
$293$	−8823.46	−1.75929	−0.879645	+	0.475631i	$0.842220\pi$
$294$	0	0
$295$	4866.91	0.960550
$296$	−695.213	−0.136515
$297$	0	0
$298$	−5580.58	−1.08481
$299$	74.8612	0.0144794
$300$	0	0
$301$	−6898.66	−1.32104
$302$	−5766.27	−1.09871
$303$	0	0
$304$	−319.058	−0.0601948
$305$	−3586.87	−0.673388
$306$	0	0
$307$	8123.41	1.51019	0.755094	−	0.655617i	$-0.227591\pi$
$307$	8123.41	1.51019	0.755094	+	0.655617i	$0.227591\pi$
$308$	−4796.23	−0.887307
$309$	0	0
$310$	−1596.39	−0.292481
$311$	−3416.66	−0.622961	−0.311481	−	0.950252i	$-0.600825\pi$
$311$	−3416.66	−0.622961	−0.311481	+	0.950252i	$0.600825\pi$
$312$	0	0
$313$	5325.14	0.961644	0.480822	−	0.876818i	$-0.340338\pi$
$313$	5325.14	0.961644	0.480822	+	0.876818i	$0.340338\pi$
$314$	−3049.73	−0.548108
$315$	0	0
$316$	−3429.25	−0.610477
$317$	952.315	0.168730	0.0843649	−	0.996435i	$-0.473114\pi$
$317$	952.315	0.168730	0.0843649	+	0.996435i	$0.473114\pi$
$318$	0	0
$319$	−3264.47	−0.572963
$320$	468.077	0.0817697
$321$	0	0
$322$	796.431	0.137836
$323$	1713.76	0.295221
$324$	0	0
$325$	232.752	0.0397254
$326$	3593.49	0.610506
$327$	0	0
$328$	526.116	0.0885668
$329$	2841.49	0.476159
$330$	0	0
$331$	5728.43	0.951248	0.475624	−	0.879649i	$-0.342222\pi$
$331$	5728.43	0.951248	0.475624	+	0.879649i	$0.342222\pi$
$332$	4782.43	0.790571
$333$	0	0
$334$	−2679.21	−0.438922
$335$	610.476	0.0995638
$336$	0	0
$337$	−6304.39	−1.01906	−0.509528	−	0.860454i	$-0.670180\pi$
$337$	−6304.39	−1.01906	−0.509528	+	0.860454i	$0.670180\pi$
$338$	4372.81	0.703697
$339$	0	0
$340$	−2514.19	−0.401033
$341$	−7558.27	−1.20030
$342$	0	0
$343$	−6687.17	−1.05269
$344$	3187.61	0.499605
$345$	0	0
$346$	−3507.29	−0.544952
$347$	−9338.30	−1.44469	−0.722344	−	0.691534i	$-0.756935\pi$
$347$	−9338.30	−1.44469	−0.722344	+	0.691534i	$0.756935\pi$
$348$	0	0
$349$	4224.27	0.647908	0.323954	−	0.946073i	$-0.394988\pi$
$349$	4224.27	0.647908	0.323954	+	0.946073i	$0.394988\pi$
$350$	2476.20	0.378166
$351$	0	0
$352$	2216.15	0.335572
$353$	6291.13	0.948564	0.474282	−	0.880373i	$-0.342708\pi$
$353$	6291.13	0.948564	0.474282	+	0.880373i	$0.342708\pi$
$354$	0	0
$355$	−7091.14	−1.06016
$356$	1531.84	0.228055
$357$	0	0
$358$	−8750.35	−1.29182
$359$	−12068.0	−1.77416	−0.887079	−	0.461618i	$-0.847269\pi$
$359$	−12068.0	−1.77416	−0.887079	+	0.461618i	$0.847269\pi$
$360$	0	0
$361$	−6461.35	−0.942025
$362$	7238.04	1.05089
$363$	0	0
$364$	−225.413	−0.0324584
$365$	3381.22	0.484880
$366$	0	0
$367$	8119.00	1.15479	0.577395	−	0.816465i	$-0.304069\pi$
$367$	8119.00	1.15479	0.577395	+	0.816465i	$0.304069\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	−1271.15	−0.178605
$371$	−10939.9	−1.53092
$372$	0	0
$373$	5256.30	0.729654	0.364827	−	0.931075i	$-0.381128\pi$
$373$	5256.30	0.729654	0.364827	+	0.931075i	$0.381128\pi$
$374$	−11903.7	−1.64579
$375$	0	0
$376$	−1312.94	−0.180079
$377$	−153.423	−0.0209594
$378$	0	0
$379$	3375.35	0.457467	0.228734	−	0.973489i	$-0.426542\pi$
$379$	3375.35	0.457467	0.228734	+	0.973489i	$0.426542\pi$
$380$	−583.374	−0.0787539
$381$	0	0
$382$	1074.74	0.143949
$383$	8034.61	1.07193	0.535966	−	0.844240i	$-0.319948\pi$
$383$	8034.61	1.07193	0.535966	+	0.844240i	$0.319948\pi$
$384$	0	0
$385$	−8769.56	−1.16088
$386$	669.568	0.0882904
$387$	0	0
$388$	−3270.90	−0.427976
$389$	−26.5264	−0.00345743	−0.00172872	−	0.999999i	$-0.500550\pi$
$389$	−26.5264	−0.00345743	−0.00172872	+	0.999999i	$0.500550\pi$
$390$	0	0
$391$	1976.65	0.255661
$392$	345.884	0.0445658
$393$	0	0
$394$	−3348.38	−0.428145
$395$	−6270.14	−0.798696
$396$	0	0
$397$	9160.19	1.15803	0.579014	−	0.815318i	$-0.303438\pi$
$397$	9160.19	1.15803	0.579014	+	0.815318i	$0.303438\pi$
$398$	−1534.46	−0.193255
$399$	0	0
$400$	−1144.15	−0.143019
$401$	−4494.80	−0.559750	−0.279875	−	0.960036i	$-0.590293\pi$
$401$	−4494.80	−0.559750	−0.279875	+	0.960036i	$0.590293\pi$
$402$	0	0
$403$	−355.223	−0.0439080
$404$	−2122.19	−0.261344
$405$	0	0
$406$	−1632.24	−0.199523
$407$	−6018.35	−0.732970
$408$	0	0
$409$	−7988.04	−0.965729	−0.482864	−	0.875695i	$-0.660404\pi$
$409$	−7988.04	−0.965729	−0.482864	+	0.875695i	$0.660404\pi$
$410$	961.965	0.115873
$411$	0	0
$412$	−5708.47	−0.682612
$413$	11521.4	1.37272
$414$	0	0
$415$	8744.32	1.03432
$416$	104.155	0.0122755
$417$	0	0
$418$	−2762.04	−0.323196
$419$	12558.9	1.46430	0.732150	−	0.681143i	$-0.238517\pi$
$419$	12558.9	1.46430	0.732150	+	0.681143i	$0.238517\pi$
$420$	0	0
$421$	−12609.7	−1.45976	−0.729879	−	0.683577i	$-0.760423\pi$
$421$	−12609.7	−1.45976	−0.729879	+	0.683577i	$0.760423\pi$
$422$	9558.81	1.10264
$423$	0	0
$424$	5054.90	0.578980
$425$	6145.62	0.701427
$426$	0	0
$427$	−8491.17	−0.962334
$428$	−4341.96	−0.490366
$429$	0	0
$430$	5828.31	0.653642
$431$	1715.05	0.191673	0.0958364	−	0.995397i	$-0.469447\pi$
$431$	1715.05	0.191673	0.0958364	+	0.995397i	$0.469447\pi$
$432$	0	0
$433$	16686.2	1.85194	0.925969	−	0.377601i	$-0.123251\pi$
$433$	16686.2	1.85194	0.925969	+	0.377601i	$0.123251\pi$
$434$	−3779.14	−0.417982
$435$	0	0
$436$	−401.729	−0.0441269
$437$	458.646	0.0502060
$438$	0	0
$439$	11447.0	1.24450	0.622249	−	0.782819i	$-0.286219\pi$
$439$	11447.0	1.24450	0.622249	+	0.782819i	$0.286219\pi$
$440$	4052.08	0.439035
$441$	0	0
$442$	−559.448	−0.0602042
$443$	−12008.1	−1.28786	−0.643932	−	0.765082i	$-0.722698\pi$
$443$	−12008.1	−1.28786	−0.643932	+	0.765082i	$0.722698\pi$
$444$	0	0
$445$	2800.86	0.298368
$446$	1594.82	0.169320
$447$	0	0
$448$	1108.08	0.116857
$449$	−3225.83	−0.339057	−0.169528	−	0.985525i	$-0.554224\pi$
$449$	−3225.83	−0.339057	−0.169528	+	0.985525i	$0.554224\pi$
$450$	0	0
$451$	4554.51	0.475529
$452$	−9157.72	−0.952971
$453$	0	0
$454$	−11797.3	−1.21954
$455$	−412.151	−0.0424658
$456$	0	0
$457$	1318.20	0.134929	0.0674645	−	0.997722i	$-0.478509\pi$
$457$	1318.20	0.134929	0.0674645	+	0.997722i	$0.478509\pi$
$458$	337.652	0.0344486
$459$	0	0
$460$	−672.861	−0.0682007
$461$	7837.18	0.791787	0.395894	−	0.918296i	$-0.370435\pi$
$461$	7837.18	0.791787	0.395894	+	0.918296i	$0.370435\pi$
$462$	0	0
$463$	15812.5	1.58719	0.793594	−	0.608448i	$-0.208208\pi$
$463$	15812.5	1.58719	0.793594	+	0.608448i	$0.208208\pi$
$464$	754.193	0.0754581
$465$	0	0
$466$	8369.34	0.831979
$467$	8142.16	0.806797	0.403399	−	0.915024i	$-0.367829\pi$
$467$	8142.16	0.806797	0.403399	+	0.915024i	$0.367829\pi$
$468$	0	0
$469$	1445.18	0.142286
$470$	−2400.62	−0.235601
$471$	0	0
$472$	−5323.60	−0.519149
$473$	27594.6	2.68246
$474$	0	0
$475$	1425.98	0.137744
$476$	−5951.84	−0.573114
$477$	0	0
$478$	5147.92	0.492595
$479$	2216.85	0.211463	0.105731	−	0.994395i	$-0.466282\pi$
$479$	2216.85	0.211463	0.105731	+	0.994395i	$0.466282\pi$
$480$	0	0
$481$	−282.850	−0.0268126
$482$	2184.79	0.206462
$483$	0	0
$484$	13860.9	1.30174
$485$	−5980.60	−0.559928
$486$	0	0
$487$	−85.0055	−0.00790958	−0.00395479	−	0.999992i	$-0.501259\pi$
$487$	−85.0055	−0.00790958	−0.00395479	+	0.999992i	$0.501259\pi$
$488$	3923.44	0.363947
$489$	0	0
$490$	632.424	0.0583061
$491$	1200.29	0.110322	0.0551611	−	0.998477i	$-0.482433\pi$
$491$	1200.29	0.110322	0.0551611	+	0.998477i	$0.482433\pi$
$492$	0	0
$493$	−4051.01	−0.370078
$494$	−129.810	−0.0118227
$495$	0	0
$496$	1746.19	0.158077
$497$	−16786.8	−1.51507
$498$	0	0
$499$	−18846.0	−1.69071	−0.845353	−	0.534208i	$-0.820610\pi$
$499$	−18846.0	−1.69071	−0.845353	+	0.534208i	$0.820610\pi$
$500$	−5748.86	−0.514193
$501$	0	0
$502$	−11008.2	−0.978727
$503$	8489.56	0.752546	0.376273	−	0.926509i	$-0.377205\pi$
$503$	8489.56	0.752546	0.376273	+	0.926509i	$0.377205\pi$
$504$	0	0
$505$	−3880.28	−0.341921
$506$	−3185.72	−0.279887
$507$	0	0
$508$	1254.12	0.109533
$509$	−7158.36	−0.623357	−0.311679	−	0.950188i	$-0.600891\pi$
$509$	−7158.36	−0.623357	−0.311679	+	0.950188i	$0.600891\pi$
$510$	0	0
$511$	8004.35	0.692939
$512$	−512.000	−0.0441942
$513$	0	0
$514$	7308.79	0.627192
$515$	−10437.5	−0.893072
$516$	0	0
$517$	−11365.9	−0.966874
$518$	−3009.18	−0.255243
$519$	0	0
$520$	190.439	0.0160602
$521$	4283.35	0.360186	0.180093	−	0.983650i	$-0.442360\pi$
$521$	4283.35	0.360186	0.180093	+	0.983650i	$0.442360\pi$
$522$	0	0
$523$	19831.0	1.65803	0.829014	−	0.559228i	$-0.188902\pi$
$523$	19831.0	1.65803	0.829014	+	0.559228i	$0.188902\pi$
$524$	2070.27	0.172596
$525$	0	0
$526$	−10704.9	−0.887365
$527$	−9379.36	−0.775278
$528$	0	0
$529$	529.000	0.0434783
$530$	9242.51	0.757489
$531$	0	0
$532$	−1381.02	−0.112547
$533$	214.053	0.0173952
$534$	0	0
$535$	−7938.96	−0.641554
$536$	−667.761	−0.0538114
$537$	0	0
$538$	8846.43	0.708916
$539$	2994.27	0.239281
$540$	0	0
$541$	−15713.7	−1.24877	−0.624384	−	0.781117i	$-0.714650\pi$
$541$	−15713.7	−1.24877	−0.624384	+	0.781117i	$0.714650\pi$
$542$	6359.45	0.503988
$543$	0	0
$544$	2750.12	0.216747
$545$	−734.533	−0.0577320
$546$	0	0
$547$	21143.1	1.65267	0.826337	−	0.563176i	$-0.190421\pi$
$547$	21143.1	1.65267	0.826337	+	0.563176i	$0.190421\pi$
$548$	1740.15	0.135649
$549$	0	0
$550$	−9904.78	−0.767893
$551$	−939.967	−0.0726750
$552$	0	0
$553$	−14843.3	−1.14141
$554$	−2270.52	−0.174125
$555$	0	0
$556$	498.503	0.0380238
$557$	−3746.75	−0.285018	−0.142509	−	0.989794i	$-0.545517\pi$
$557$	−3746.75	−0.285018	−0.142509	+	0.989794i	$0.545517\pi$
$558$	0	0
$559$	1296.89	0.0981264
$560$	2026.04	0.152885
$561$	0	0
$562$	−5739.71	−0.430810
$563$	16060.2	1.20223	0.601116	−	0.799162i	$-0.294723\pi$
$563$	16060.2	1.20223	0.601116	+	0.799162i	$0.294723\pi$
$564$	0	0
$565$	−16744.2	−1.24679
$566$	−1715.01	−0.127363
$567$	0	0
$568$	7756.54	0.572988
$569$	−17547.1	−1.29282	−0.646409	−	0.762991i	$-0.723730\pi$
$569$	−17547.1	−1.29282	−0.646409	+	0.762991i	$0.723730\pi$
$570$	0	0
$571$	−5551.81	−0.406893	−0.203446	−	0.979086i	$-0.565214\pi$
$571$	−5551.81	−0.406893	−0.203446	+	0.979086i	$0.565214\pi$
$572$	901.652	0.0659090
$573$	0	0
$574$	2277.25	0.165594
$575$	1644.72	0.119286
$576$	0	0
$577$	1394.14	0.100587	0.0502937	−	0.998734i	$-0.483984\pi$
$577$	1394.14	0.100587	0.0502937	+	0.998734i	$0.483984\pi$
$578$	−4945.75	−0.355910
$579$	0	0
$580$	1378.99	0.0987230
$581$	20700.4	1.47814
$582$	0	0
$583$	43759.5	3.10863
$584$	−3698.50	−0.262064
$585$	0	0
$586$	17646.9	1.24401
$587$	26307.6	1.84980	0.924900	−	0.380210i	$-0.124148\pi$
$587$	26307.6	1.84980	0.924900	+	0.380210i	$0.124148\pi$
$588$	0	0
$589$	−2176.32	−0.152247
$590$	−9733.81	−0.679211
$591$	0	0
$592$	1390.43	0.0965306
$593$	−10657.9	−0.738056	−0.369028	−	0.929418i	$-0.620309\pi$
$593$	−10657.9	−0.738056	−0.369028	+	0.929418i	$0.620309\pi$
$594$	0	0
$595$	−10882.5	−0.749814
$596$	11161.2	0.767079
$597$	0	0
$598$	−149.722	−0.0102385
$599$	−5018.67	−0.342333	−0.171166	−	0.985242i	$-0.554754\pi$
$599$	−5018.67	−0.342333	−0.171166	+	0.985242i	$0.554754\pi$
$600$	0	0
$601$	7715.55	0.523667	0.261833	−	0.965113i	$-0.415673\pi$
$601$	7715.55	0.523667	0.261833	+	0.965113i	$0.415673\pi$
$602$	13797.3	0.934114
$603$	0	0
$604$	11532.5	0.776908
$605$	25343.7	1.70309
$606$	0	0
$607$	26998.5	1.80533	0.902664	−	0.430346i	$-0.141608\pi$
$607$	26998.5	1.80533	0.902664	+	0.430346i	$0.141608\pi$
$608$	638.116	0.0425642
$609$	0	0
$610$	7173.73	0.476157
$611$	−534.176	−0.0353690
$612$	0	0
$613$	3756.96	0.247540	0.123770	−	0.992311i	$-0.460501\pi$
$613$	3756.96	0.247540	0.123770	+	0.992311i	$0.460501\pi$
$614$	−16246.8	−1.06786
$615$	0	0
$616$	9592.46	0.627421
$617$	−14186.2	−0.925632	−0.462816	−	0.886454i	$-0.653161\pi$
$617$	−14186.2	−0.925632	−0.462816	+	0.886454i	$0.653161\pi$
$618$	0	0
$619$	−25670.2	−1.66684	−0.833418	−	0.552643i	$-0.813619\pi$
$619$	−25670.2	−1.66684	−0.833418	+	0.552643i	$0.813619\pi$
$620$	3192.79	0.206815
$621$	0	0
$622$	6833.32	0.440500
$623$	6630.47	0.426395
$624$	0	0
$625$	−1572.66	−0.100650
$626$	−10650.3	−0.679985
$627$	0	0
$628$	6099.45	0.387571
$629$	−7468.42	−0.473427
$630$	0	0
$631$	−948.530	−0.0598421	−0.0299211	−	0.999552i	$-0.509526\pi$
$631$	−948.530	−0.0598421	−0.0299211	+	0.999552i	$0.509526\pi$
$632$	6858.51	0.431672
$633$	0	0
$634$	−1904.63	−0.119310
$635$	2293.07	0.143304
$636$	0	0
$637$	140.724	0.00875307
$638$	6528.94	0.405146
$639$	0	0
$640$	−936.155	−0.0578199
$641$	28678.3	1.76712	0.883560	−	0.468318i	$-0.155140\pi$
$641$	28678.3	1.76712	0.883560	+	0.468318i	$0.155140\pi$
$642$	0	0
$643$	7606.41	0.466513	0.233256	−	0.972415i	$-0.425062\pi$
$643$	7606.41	0.466513	0.233256	+	0.972415i	$0.425062\pi$
$644$	−1592.86	−0.0974651
$645$	0	0
$646$	−3427.53	−0.208753
$647$	−546.880	−0.0332304	−0.0166152	−	0.999862i	$-0.505289\pi$
$647$	−546.880	−0.0332304	−0.0166152	+	0.999862i	$0.505289\pi$
$648$	0	0
$649$	−46085.6	−2.78739
$650$	−465.504	−0.0280901
$651$	0	0
$652$	−7186.97	−0.431693
$653$	−2615.66	−0.156751	−0.0783756	−	0.996924i	$-0.524973\pi$
$653$	−2615.66	−0.156751	−0.0783756	+	0.996924i	$0.524973\pi$
$654$	0	0
$655$	3785.34	0.225810
$656$	−1052.23	−0.0626262
$657$	0	0
$658$	−5682.97	−0.336695
$659$	−5101.12	−0.301535	−0.150767	−	0.988569i	$-0.548174\pi$
$659$	−5101.12	−0.301535	−0.150767	+	0.988569i	$0.548174\pi$
$660$	0	0
$661$	−33184.1	−1.95267	−0.976334	−	0.216268i	$-0.930611\pi$
$661$	−33184.1	−1.95267	−0.976334	+	0.216268i	$0.930611\pi$
$662$	−11456.9	−0.672634
$663$	0	0
$664$	−9564.85	−0.559018
$665$	−2525.09	−0.147246
$666$	0	0
$667$	−1084.15	−0.0629364
$668$	5358.43	0.310365
$669$	0	0
$670$	−1220.95	−0.0704022
$671$	33964.7	1.95409
$672$	0	0
$673$	−22540.9	−1.29107	−0.645533	−	0.763733i	$-0.723365\pi$
$673$	−22540.9	−1.29107	−0.645533	+	0.763733i	$0.723365\pi$
$674$	12608.8	0.720582
$675$	0	0
$676$	−8745.62	−0.497589
$677$	−30165.0	−1.71246	−0.856229	−	0.516596i	$-0.827199\pi$
$677$	−30165.0	−1.71246	−0.856229	+	0.516596i	$0.827199\pi$
$678$	0	0
$679$	−14157.8	−0.800188
$680$	5028.39	0.283573
$681$	0	0
$682$	15116.5	0.848742
$683$	−417.726	−0.0234024	−0.0117012	−	0.999932i	$-0.503725\pi$
$683$	−417.726	−0.0234024	−0.0117012	+	0.999932i	$0.503725\pi$
$684$	0	0
$685$	3181.74	0.177472
$686$	13374.3	0.744365
$687$	0	0
$688$	−6375.21	−0.353274
$689$	2056.61	0.113716
$690$	0	0
$691$	−5421.98	−0.298498	−0.149249	−	0.988800i	$-0.547686\pi$
$691$	−5421.98	−0.298498	−0.149249	+	0.988800i	$0.547686\pi$
$692$	7014.59	0.385339
$693$	0	0
$694$	18676.6	1.02155
$695$	911.476	0.0497471
$696$	0	0
$697$	5651.88	0.307145
$698$	−8448.54	−0.458140
$699$	0	0
$700$	−4952.39	−0.267404
$701$	−14840.0	−0.799572	−0.399786	−	0.916608i	$-0.630916\pi$
$701$	−14840.0	−0.799572	−0.399786	+	0.916608i	$0.630916\pi$
$702$	0	0
$703$	−1732.92	−0.0929703
$704$	−4432.31	−0.237285
$705$	0	0
$706$	−12582.3	−0.670736
$707$	−9185.76	−0.488637
$708$	0	0
$709$	15302.4	0.810569	0.405284	−	0.914191i	$-0.367173\pi$
$709$	15302.4	0.810569	0.405284	+	0.914191i	$0.367173\pi$
$710$	14182.3	0.749649
$711$	0	0
$712$	−3063.68	−0.161259
$713$	−2510.15	−0.131846
$714$	0	0
$715$	1648.60	0.0862298
$716$	17500.7	0.913452
$717$	0	0
$718$	24135.9	1.25452
$719$	−26125.2	−1.35508	−0.677542	−	0.735484i	$-0.736955\pi$
$719$	−26125.2	−1.35508	−0.677542	+	0.735484i	$0.736955\pi$
$720$	0	0
$721$	−24708.7	−1.27628
$722$	12922.7	0.666112
$723$	0	0
$724$	−14476.1	−0.743093
$725$	−3370.76	−0.172671
$726$	0	0
$727$	−4928.32	−0.251419	−0.125709	−	0.992067i	$-0.540121\pi$
$727$	−4928.32	−0.251419	−0.125709	+	0.992067i	$0.540121\pi$
$728$	450.826	0.0229515
$729$	0	0
$730$	−6762.44	−0.342862
$731$	34243.3	1.73261
$732$	0	0
$733$	16142.2	0.813405	0.406703	−	0.913561i	$-0.366679\pi$
$733$	16142.2	0.813405	0.406703	+	0.913561i	$0.366679\pi$
$734$	−16238.0	−0.816560
$735$	0	0
$736$	736.000	0.0368605
$737$	−5780.71	−0.288922
$738$	0	0
$739$	−1612.09	−0.0802457	−0.0401229	−	0.999195i	$-0.512775\pi$
$739$	−1612.09	−0.0802457	−0.0401229	+	0.999195i	$0.512775\pi$
$740$	2542.29	0.126293
$741$	0	0
$742$	21879.8	1.08252
$743$	33768.1	1.66734	0.833668	−	0.552265i	$-0.186236\pi$
$743$	33768.1	1.66734	0.833668	+	0.552265i	$0.186236\pi$
$744$	0	0
$745$	20407.4	1.00358
$746$	−10512.6	−0.515943
$747$	0	0
$748$	23807.4	1.16375
$749$	−18793.9	−0.916840
$750$	0	0
$751$	−12656.6	−0.614977	−0.307488	−	0.951552i	$-0.599489\pi$
$751$	−12656.6	−0.614977	−0.307488	+	0.951552i	$0.599489\pi$
$752$	2625.88	0.127335
$753$	0	0
$754$	306.847	0.0148206
$755$	21086.4	1.01644
$756$	0	0
$757$	−35451.3	−1.70211	−0.851056	−	0.525075i	$-0.824037\pi$
$757$	−35451.3	−1.70211	−0.851056	+	0.525075i	$0.824037\pi$
$758$	−6750.70	−0.323478
$759$	0	0
$760$	1166.75	0.0556874
$761$	−4234.99	−0.201732	−0.100866	−	0.994900i	$-0.532161\pi$
$761$	−4234.99	−0.201732	−0.100866	+	0.994900i	$0.532161\pi$
$762$	0	0
$763$	−1738.86	−0.0825043
$764$	−2149.48	−0.101787
$765$	0	0
$766$	−16069.2	−0.757970
$767$	−2165.93	−0.101965
$768$	0	0
$769$	23381.4	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$769$	23381.4	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$770$	17539.1	0.820865
$771$	0	0
$772$	−1339.14	−0.0624307
$773$	−8976.44	−0.417672	−0.208836	−	0.977951i	$-0.566967\pi$
$773$	−8976.44	−0.417672	−0.208836	+	0.977951i	$0.566967\pi$
$774$	0	0
$775$	−7804.36	−0.361730
$776$	6541.79	0.302625
$777$	0	0
$778$	53.0528	0.00244478
$779$	1311.42	0.0603163
$780$	0	0
$781$	67147.2	3.07646
$782$	−3953.29	−0.180779
$783$	0	0
$784$	−691.768	−0.0315128
$785$	11152.4	0.507065
$786$	0	0
$787$	−33632.8	−1.52335	−0.761677	−	0.647957i	$-0.775624\pi$
$787$	−33632.8	−1.52335	−0.761677	+	0.647957i	$0.775624\pi$
$788$	6696.77	0.302744
$789$	0	0
$790$	12540.3	0.564764
$791$	−39638.5	−1.78177
$792$	0	0
$793$	1596.27	0.0714820
$794$	−18320.4	−0.818849
$795$	0	0
$796$	3068.92	0.136652
$797$	−38371.3	−1.70537	−0.852686	−	0.522424i	$-0.825028\pi$
$797$	−38371.3	−1.70537	−0.852686	+	0.522424i	$0.825028\pi$
$798$	0	0
$799$	−14104.5	−0.624506
$800$	2288.31	0.101130
$801$	0	0
$802$	8989.60	0.395803
$803$	−32017.4	−1.40706
$804$	0	0
$805$	−2912.43	−0.127515
$806$	710.446	0.0310476
$807$	0	0
$808$	4244.39	0.184798
$809$	15249.5	0.662722	0.331361	−	0.943504i	$-0.392492\pi$
$809$	15249.5	0.662722	0.331361	+	0.943504i	$0.392492\pi$
$810$	0	0
$811$	9708.02	0.420339	0.210169	−	0.977665i	$-0.432598\pi$
$811$	9708.02	0.420339	0.210169	+	0.977665i	$0.432598\pi$
$812$	3264.47	0.141084
$813$	0	0
$814$	12036.7	0.518288
$815$	−13140.9	−0.564790
$816$	0	0
$817$	7945.56	0.340245
$818$	15976.1	0.682873
$819$	0	0
$820$	−1923.93	−0.0819348
$821$	−9028.52	−0.383797	−0.191899	−	0.981415i	$-0.561464\pi$
$821$	−9028.52	−0.383797	−0.191899	+	0.981415i	$0.561464\pi$
$822$	0	0
$823$	42539.1	1.80172	0.900862	−	0.434105i	$-0.142935\pi$
$823$	42539.1	1.80172	0.900862	+	0.434105i	$0.142935\pi$
$824$	11416.9	0.482679
$825$	0	0
$826$	−23042.8	−0.970656
$827$	−21526.4	−0.905134	−0.452567	−	0.891730i	$-0.649492\pi$
$827$	−21526.4	−0.905134	−0.452567	+	0.891730i	$0.649492\pi$
$828$	0	0
$829$	38550.4	1.61509	0.807546	−	0.589804i	$-0.200795\pi$
$829$	38550.4	1.61509	0.807546	+	0.589804i	$0.200795\pi$
$830$	−17488.6	−0.731373
$831$	0	0
$832$	−208.309	−0.00868008
$833$	3715.71	0.154552
$834$	0	0
$835$	9797.49	0.406055
$836$	5524.08	0.228534
$837$	0	0
$838$	−25117.8	−1.03542
$839$	−30566.4	−1.25777	−0.628884	−	0.777499i	$-0.716488\pi$
$839$	−30566.4	−1.25777	−0.628884	+	0.777499i	$0.716488\pi$
$840$	0	0
$841$	−22167.1	−0.908897
$842$	25219.3	1.03220
$843$	0	0
$844$	−19117.6	−0.779687
$845$	−15990.7	−0.651004
$846$	0	0
$847$	59996.0	2.43387
$848$	−10109.8	−0.409401
$849$	0	0
$850$	−12291.2	−0.495984
$851$	−1998.74	−0.0805121
$852$	0	0
$853$	20617.3	0.827576	0.413788	−	0.910373i	$-0.364205\pi$
$853$	20617.3	0.827576	0.413788	+	0.910373i	$0.364205\pi$
$854$	16982.3	0.680473
$855$	0	0
$856$	8683.92	0.346741
$857$	−1553.09	−0.0619048	−0.0309524	−	0.999521i	$-0.509854\pi$
$857$	−1553.09	−0.0619048	−0.0309524	+	0.999521i	$0.509854\pi$
$858$	0	0
$859$	−34732.9	−1.37959	−0.689797	−	0.724003i	$-0.742300\pi$
$859$	−34732.9	−1.37959	−0.689797	+	0.724003i	$0.742300\pi$
$860$	−11656.6	−0.462194
$861$	0	0
$862$	−3430.10	−0.135533
$863$	−9162.01	−0.361389	−0.180694	−	0.983539i	$-0.557834\pi$
$863$	−9162.01	−0.361389	−0.180694	+	0.983539i	$0.557834\pi$
$864$	0	0
$865$	12825.7	0.504145
$866$	−33372.5	−1.30952
$867$	0	0
$868$	7558.27	0.295558
$869$	59373.1	2.31772
$870$	0	0
$871$	−271.681	−0.0105690
$872$	803.459	0.0312025
$873$	0	0
$874$	−917.292	−0.0355010
$875$	−24883.5	−0.961390
$876$	0	0
$877$	−5772.11	−0.222247	−0.111123	−	0.993807i	$-0.535445\pi$
$877$	−5772.11	−0.222247	−0.111123	+	0.993807i	$0.535445\pi$
$878$	−22894.0	−0.879993
$879$	0	0
$880$	−8104.15	−0.310444
$881$	−9412.65	−0.359955	−0.179977	−	0.983671i	$-0.557602\pi$
$881$	−9412.65	−0.359955	−0.179977	+	0.983671i	$0.557602\pi$
$882$	0	0
$883$	−45232.2	−1.72388	−0.861940	−	0.507011i	$-0.830750\pi$
$883$	−45232.2	−1.72388	−0.861940	+	0.507011i	$0.830750\pi$
$884$	1118.90	0.0425708
$885$	0	0
$886$	24016.3	0.910658
$887$	−31353.8	−1.18687	−0.593437	−	0.804880i	$-0.702229\pi$
$887$	−31353.8	−1.18687	−0.593437	+	0.804880i	$0.702229\pi$
$888$	0	0
$889$	5428.38	0.204794
$890$	−5601.72	−0.210978
$891$	0	0
$892$	−3189.64	−0.119728
$893$	−3272.69	−0.122639
$894$	0	0
$895$	31998.8	1.19508
$896$	−2216.15	−0.0826301
$897$	0	0
$898$	6451.66	0.239749
$899$	5144.40	0.190851
$900$	0	0
$901$	54302.9	2.00787
$902$	−9109.02	−0.336250
$903$	0	0
$904$	18315.4	0.673852
$905$	−26468.5	−0.972200
$906$	0	0
$907$	−8652.83	−0.316772	−0.158386	−	0.987377i	$-0.550629\pi$
$907$	−8652.83	−0.316772	−0.158386	+	0.987377i	$0.550629\pi$
$908$	23594.5	0.862347
$909$	0	0
$910$	824.302	0.0300279
$911$	−23107.3	−0.840370	−0.420185	−	0.907438i	$-0.638035\pi$
$911$	−23107.3	−0.840370	−0.420185	+	0.907438i	$0.638035\pi$
$912$	0	0
$913$	−82801.5	−3.00146
$914$	−2636.39	−0.0954092
$915$	0	0
$916$	−675.304	−0.0243588
$917$	8961.02	0.322703
$918$	0	0
$919$	−42056.1	−1.50958	−0.754790	−	0.655966i	$-0.772261\pi$
$919$	−42056.1	−1.50958	−0.754790	+	0.655966i	$0.772261\pi$
$920$	1345.72	0.0482252
$921$	0	0
$922$	−15674.4	−0.559878
$923$	3155.78	0.112539
$924$	0	0
$925$	−6214.30	−0.220892
$926$	−31624.9	−1.12231
$927$	0	0
$928$	−1508.39	−0.0533569
$929$	−1263.92	−0.0446369	−0.0223185	−	0.999751i	$-0.507105\pi$
$929$	−1263.92	−0.0446369	−0.0223185	+	0.999751i	$0.507105\pi$
$930$	0	0
$931$	862.164	0.0303505
$932$	−16738.7	−0.588298
$933$	0	0
$934$	−16284.3	−0.570492
$935$	43530.0	1.52255
$936$	0	0
$937$	−27756.8	−0.967743	−0.483871	−	0.875139i	$-0.660770\pi$
$937$	−27756.8	−0.967743	−0.483871	+	0.875139i	$0.660770\pi$
$938$	−2890.35	−0.100611
$939$	0	0
$940$	4801.24	0.166595
$941$	7081.47	0.245323	0.122662	−	0.992449i	$-0.460857\pi$
$941$	7081.47	0.245323	0.122662	+	0.992449i	$0.460857\pi$
$942$	0	0
$943$	1512.58	0.0522338
$944$	10647.2	0.367094
$945$	0	0
$946$	−55189.3	−1.89678
$947$	34048.7	1.16836	0.584179	−	0.811625i	$-0.301417\pi$
$947$	34048.7	1.16836	0.584179	+	0.811625i	$0.301417\pi$
$948$	0	0
$949$	−1504.75	−0.0514713
$950$	−2851.97	−0.0974000
$951$	0	0
$952$	11903.7	0.405252
$953$	10453.3	0.355314	0.177657	−	0.984092i	$-0.443148\pi$
$953$	10453.3	0.355314	0.177657	+	0.984092i	$0.443148\pi$
$954$	0	0
$955$	−3930.16	−0.133170
$956$	−10295.8	−0.348317
$957$	0	0
$958$	−4433.71	−0.149527
$959$	7532.12	0.253623
$960$	0	0
$961$	−17880.1	−0.600185
$962$	565.700	0.0189594
$963$	0	0
$964$	−4369.58	−0.145990
$965$	−2448.51	−0.0816791
$966$	0	0
$967$	−22280.5	−0.740945	−0.370473	−	0.928843i	$-0.620804\pi$
$967$	−22280.5	−0.740945	−0.370473	+	0.928843i	$0.620804\pi$
$968$	−27721.9	−0.920469
$969$	0	0
$970$	11961.2	0.395929
$971$	39430.9	1.30319	0.651595	−	0.758567i	$-0.274100\pi$
$971$	39430.9	1.30319	0.651595	+	0.758567i	$0.274100\pi$
$972$	0	0
$973$	2157.73	0.0710933
$974$	170.011	0.00559292
$975$	0	0
$976$	−7846.89	−0.257349
$977$	−56586.0	−1.85297	−0.926483	−	0.376337i	$-0.877183\pi$
$977$	−56586.0	−1.85297	−0.926483	+	0.376337i	$0.877183\pi$
$978$	0	0
$979$	−26521.9	−0.865825
$980$	−1264.85	−0.0412286
$981$	0	0
$982$	−2400.57	−0.0780095
$983$	−42422.5	−1.37647	−0.688235	−	0.725488i	$-0.741614\pi$
$983$	−42422.5	−1.37647	−0.688235	+	0.725488i	$0.741614\pi$
$984$	0	0
$985$	12244.5	0.396085
$986$	8102.03	0.261685
$987$	0	0
$988$	259.620	0.00835994
$989$	9164.37	0.294651
$990$	0	0
$991$	−23798.6	−0.762853	−0.381426	−	0.924399i	$-0.624567\pi$
$991$	−23798.6	−0.762853	−0.381426	+	0.924399i	$0.624567\pi$
$992$	−3492.39	−0.111778
$993$	0	0
$994$	33573.6	1.07132
$995$	5611.31	0.178784
$996$	0	0
$997$	−35708.9	−1.13431	−0.567157	−	0.823610i	$-0.691957\pi$
$997$	−35708.9	−1.13431	−0.567157	+	0.823610i	$0.691957\pi$
$998$	37692.0	1.19551
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.g.1.2		2
3.2	odd	2		138.4.a.e.1.1	✓	2
12.11	even	2		1104.4.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.a.e.1.1	✓	2	3.2	odd	2
414.4.a.g.1.2		2	1.1	even	1	trivial
1104.4.a.p.1.1		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +7.31371 q^{5} +17.3137 q^{7} -8.00000 q^{8} -14.6274 q^{10} -69.2548 q^{11} -3.25483 q^{13} -34.6274 q^{14} +16.0000 q^{16} -85.9411 q^{17} -19.9411 q^{19} +29.2548 q^{20} +138.510 q^{22} -23.0000 q^{23} -71.5097 q^{25} +6.50967 q^{26} +69.2548 q^{28} +47.1371 q^{29} +109.137 q^{31} -32.0000 q^{32} +171.882 q^{34} +126.627 q^{35} +86.9016 q^{37} +39.8823 q^{38} -58.5097 q^{40} -65.7645 q^{41} -398.451 q^{43} -277.019 q^{44} +46.0000 q^{46} +164.118 q^{47} -43.2355 q^{49} +143.019 q^{50} -13.0193 q^{52} -631.862 q^{53} -506.510 q^{55} -138.510 q^{56} -94.2742 q^{58} +665.450 q^{59} -490.431 q^{61} -218.274 q^{62} +64.0000 q^{64} -23.8049 q^{65} +83.4701 q^{67} -343.765 q^{68} -253.255 q^{70} -969.568 q^{71} +462.313 q^{73} -173.803 q^{74} -79.7645 q^{76} -1199.06 q^{77} -857.314 q^{79} +117.019 q^{80} +131.529 q^{82} +1195.61 q^{83} -628.548 q^{85} +796.902 q^{86} +554.039 q^{88} +382.960 q^{89} -56.3532 q^{91} -92.0000 q^{92} -328.235 q^{94} -145.844 q^{95} -817.724 q^{97} +86.4710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 8 q^{5} + 12 q^{7} - 16 q^{8} + 16 q^{10} - 48 q^{11} + 84 q^{13} - 24 q^{14} + 32 q^{16} - 104 q^{17} + 28 q^{19} - 32 q^{20} + 96 q^{22} - 46 q^{23} + 38 q^{25} - 168 q^{26}+ \cdots + 716 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 8 * q^5 + 12 * q^7 - 16 * q^8 + 16 * q^10 - 48 * q^11 + 84 * q^13 - 24 * q^14 + 32 * q^16 - 104 * q^17 + 28 * q^19 - 32 * q^20 + 96 * q^22 - 46 * q^23 + 38 * q^25 - 168 * q^26 + 48 * q^28 - 132 * q^29 - 8 * q^31 - 64 * q^32 + 208 * q^34 + 208 * q^35 - 324 * q^37 - 56 * q^38 + 64 * q^40 + 140 * q^41 - 548 * q^43 - 192 * q^44 + 92 * q^46 + 464 * q^47 - 358 * q^49 - 76 * q^50 + 336 * q^52 - 336 * q^53 - 832 * q^55 - 96 * q^56 + 264 * q^58 - 72 * q^59 + 60 * q^61 + 16 * q^62 + 128 * q^64 - 1360 * q^65 - 444 * q^67 - 416 * q^68 - 416 * q^70 - 672 * q^71 - 252 * q^73 + 648 * q^74 + 112 * q^76 - 1312 * q^77 - 1692 * q^79 - 128 * q^80 - 280 * q^82 + 400 * q^83 - 352 * q^85 + 1096 * q^86 + 384 * q^88 + 336 * q^89 - 520 * q^91 - 184 * q^92 - 928 * q^94 - 880 * q^95 + 220 * q^97 + 716 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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