LMFDB - Embedded newform 2601.2.a.bj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.45769536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 72x^{2} - 109 $ x^6 - 15x^4 + 72x^2 - 109
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.31242$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.31242 q^{2} +3.34730 q^{4} -0.803096 q^{5} -2.65270 q^{7} +3.11552 q^{8} +O(q^{10})$	$q+2.31242 q^{2} +3.34730 q^{4} -0.803096 q^{5} -2.65270 q^{7} +3.11552 q^{8} -1.85710 q^{10} -5.85526 q^{11} +3.94356 q^{13} -6.13417 q^{14} +0.509800 q^{16} -7.10607 q^{19} -2.68820 q^{20} -13.5398 q^{22} +6.37944 q^{23} -4.35504 q^{25} +9.11918 q^{26} -8.87939 q^{28} -5.14903 q^{29} -4.46791 q^{31} -5.05216 q^{32} +2.13038 q^{35} -1.71688 q^{37} -16.4322 q^{38} -2.50206 q^{40} +7.18254 q^{41} +1.30541 q^{43} -19.5993 q^{44} +14.7520 q^{46} -1.08201 q^{47} +0.0368366 q^{49} -10.0707 q^{50} +13.2003 q^{52} -9.67701 q^{53} +4.70233 q^{55} -8.26455 q^{56} -11.9067 q^{58} +12.0863 q^{59} -6.18479 q^{61} -10.3317 q^{62} -12.7023 q^{64} -3.16706 q^{65} +5.43376 q^{67} +4.92633 q^{70} +5.95212 q^{71} +2.53209 q^{73} -3.97015 q^{74} -23.7861 q^{76} +15.5323 q^{77} -6.30541 q^{79} -0.409418 q^{80} +16.6091 q^{82} -15.2534 q^{83} +3.01865 q^{86} -18.2422 q^{88} -3.63970 q^{89} -10.4611 q^{91} +21.3539 q^{92} -2.50206 q^{94} +5.70685 q^{95} -8.19253 q^{97} +0.0851818 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{4} - 18 q^{7}+O(q^{10}) $ 6 * q + 18 * q^4 - 18 * q^7	$ 6 q + 18 q^{4} - 18 q^{7} - 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} - 24 q^{22} + 24 q^{25} - 42 q^{28} - 36 q^{31} + 6 q^{37} - 66 q^{40} + 12 q^{43} - 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} - 18 q^{58} - 30 q^{61} - 24 q^{64} - 18 q^{70} + 6 q^{73} - 78 q^{76} - 42 q^{79} - 6 q^{82} + 12 q^{91} - 66 q^{94} + 6 q^{97}+O(q^{100}) $ 6 * q + 18 * q^4 - 18 * q^7 - 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 - 24 * q^22 + 24 * q^25 - 42 * q^28 - 36 * q^31 + 6 * q^37 - 66 * q^40 + 12 * q^43 - 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 - 18 * q^58 - 30 * q^61 - 24 * q^64 - 18 * q^70 + 6 * q^73 - 78 * q^76 - 42 * q^79 - 6 * q^82 + 12 * q^91 - 66 * q^94 + 6 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	2.31242	1.63513	0.817565	−	0.575837i	$-0.195324\pi$
$2$	2.31242	1.63513	0.817565	+	0.575837i	$0.195324\pi$
$3$	0	0
$3$	0	0
$4$	3.34730	1.67365
$5$	−0.803096	−0.359155	−0.179578	−	0.983744i	$-0.557473\pi$
$5$	−0.803096	−0.359155	−0.179578	+	0.983744i	$0.557473\pi$
$6$	0	0
$7$	−2.65270	−1.00263	−0.501314	−	0.865266i	$-0.667150\pi$
$7$	−2.65270	−1.00263	−0.501314	+	0.865266i	$0.667150\pi$
$8$	3.11552	1.10150
$9$	0	0
$10$	−1.85710	−0.587265
$11$	−5.85526	−1.76543	−0.882713	−	0.469912i	$-0.844286\pi$
$11$	−5.85526	−1.76543	−0.882713	+	0.469912i	$0.844286\pi$
$12$	0	0
$13$	3.94356	1.09375	0.546874	−	0.837215i	$-0.315818\pi$
$13$	3.94356	1.09375	0.546874	+	0.837215i	$0.315818\pi$
$14$	−6.13417	−1.63943
$15$	0	0
$16$	0.509800	0.127450
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−7.10607	−1.63024	−0.815122	−	0.579290i	$-0.803330\pi$
$19$	−7.10607	−1.63024	−0.815122	+	0.579290i	$0.803330\pi$
$20$	−2.68820	−0.601100
$21$	0	0
$22$	−13.5398	−2.88670
$23$	6.37944	1.33021	0.665103	−	0.746752i	$-0.268388\pi$
$23$	6.37944	1.33021	0.665103	+	0.746752i	$0.268388\pi$
$24$	0	0
$25$	−4.35504	−0.871007
$26$	9.11918	1.78842
$27$	0	0
$28$	−8.87939	−1.67805
$29$	−5.14903	−0.956150	−0.478075	−	0.878319i	$-0.658665\pi$
$29$	−5.14903	−0.956150	−0.478075	+	0.878319i	$0.658665\pi$
$30$	0	0
$31$	−4.46791	−0.802461	−0.401230	−	0.915977i	$-0.631417\pi$
$31$	−4.46791	−0.802461	−0.401230	+	0.915977i	$0.631417\pi$
$32$	−5.05216	−0.893105
$33$	0	0
$34$	0	0
$35$	2.13038	0.360099
$36$	0	0
$37$	−1.71688	−0.282254	−0.141127	−	0.989992i	$-0.545073\pi$
$37$	−1.71688	−0.282254	−0.141127	+	0.989992i	$0.545073\pi$
$38$	−16.4322	−2.66566
$39$	0	0
$40$	−2.50206	−0.395610
$41$	7.18254	1.12172	0.560862	−	0.827909i	$-0.310470\pi$
$41$	7.18254	1.12172	0.560862	+	0.827909i	$0.310470\pi$
$42$	0	0
$43$	1.30541	0.199073	0.0995364	−	0.995034i	$-0.468264\pi$
$43$	1.30541	0.199073	0.0995364	+	0.995034i	$0.468264\pi$
$44$	−19.5993	−2.95470
$45$	0	0
$46$	14.7520	2.17506
$47$	−1.08201	−0.157827	−0.0789135	−	0.996881i	$-0.525145\pi$
$47$	−1.08201	−0.157827	−0.0789135	+	0.996881i	$0.525145\pi$
$48$	0	0
$49$	0.0368366	0.00526238
$50$	−10.0707	−1.42421
$51$	0	0
$52$	13.2003	1.83055
$53$	−9.67701	−1.32924	−0.664620	−	0.747182i	$-0.731406\pi$
$53$	−9.67701	−1.32924	−0.664620	+	0.747182i	$0.731406\pi$
$54$	0	0
$55$	4.70233	0.634063
$56$	−8.26455	−1.10440
$57$	0	0
$58$	−11.9067	−1.56343
$59$	12.0863	1.57350	0.786751	−	0.617271i	$-0.211762\pi$
$59$	12.0863	1.57350	0.786751	+	0.617271i	$0.211762\pi$
$60$	0	0
$61$	−6.18479	−0.791882	−0.395941	−	0.918276i	$-0.629581\pi$
$61$	−6.18479	−0.791882	−0.395941	+	0.918276i	$0.629581\pi$
$62$	−10.3317	−1.31213
$63$	0	0
$64$	−12.7023	−1.58779
$65$	−3.16706	−0.392825
$66$	0	0
$67$	5.43376	0.663840	0.331920	−	0.943308i	$-0.392304\pi$
$67$	5.43376	0.663840	0.331920	+	0.943308i	$0.392304\pi$
$68$	0	0
$69$	0	0
$70$	4.92633	0.588809
$71$	5.95212	0.706387	0.353194	−	0.935550i	$-0.385096\pi$
$71$	5.95212	0.706387	0.353194	+	0.935550i	$0.385096\pi$
$72$	0	0
$73$	2.53209	0.296359	0.148179	−	0.988961i	$-0.452659\pi$
$73$	2.53209	0.296359	0.148179	+	0.988961i	$0.452659\pi$
$74$	−3.97015	−0.461521
$75$	0	0
$76$	−23.7861	−2.72845
$77$	15.5323	1.77007
$78$	0	0
$79$	−6.30541	−0.709414	−0.354707	−	0.934978i	$-0.615419\pi$
$79$	−6.30541	−0.709414	−0.354707	+	0.934978i	$0.615419\pi$
$80$	−0.409418	−0.0457744
$81$	0	0
$82$	16.6091	1.83416
$83$	−15.2534	−1.67427	−0.837137	−	0.546994i	$-0.815772\pi$
$83$	−15.2534	−1.67427	−0.837137	+	0.546994i	$0.815772\pi$
$84$	0	0
$85$	0	0
$86$	3.01865	0.325510
$87$	0	0
$88$	−18.2422	−1.94462
$89$	−3.63970	−0.385808	−0.192904	−	0.981218i	$-0.561791\pi$
$89$	−3.63970	−0.385808	−0.192904	+	0.981218i	$0.561791\pi$
$90$	0	0
$91$	−10.4611	−1.09662
$92$	21.3539	2.22630
$93$	0	0
$94$	−2.50206	−0.258068
$95$	5.70685	0.585511
$96$	0	0
$97$	−8.19253	−0.831826	−0.415913	−	0.909404i	$-0.636538\pi$
$97$	−8.19253	−0.831826	−0.415913	+	0.909404i	$0.636538\pi$
$98$	0.0851818	0.00860466
$99$	0	0
$100$	−14.5776	−1.45776
$101$	17.8447	1.77561	0.887806	−	0.460217i	$-0.152228\pi$
$101$	17.8447	1.77561	0.887806	+	0.460217i	$0.152228\pi$
$102$	0	0
$103$	−5.39693	−0.531775	−0.265887	−	0.964004i	$-0.585665\pi$
$103$	−5.39693	−0.531775	−0.265887	+	0.964004i	$0.585665\pi$
$104$	12.2862	1.20477
$105$	0	0
$106$	−22.3773	−2.17348
$107$	1.50933	0.145912	0.0729560	−	0.997335i	$-0.476757\pi$
$107$	1.50933	0.145912	0.0729560	+	0.997335i	$0.476757\pi$
$108$	0	0
$109$	16.0719	1.53941	0.769705	−	0.638399i	$-0.220403\pi$
$109$	16.0719	1.53941	0.769705	+	0.638399i	$0.220403\pi$
$110$	10.8738	1.03677
$111$	0	0
$112$	−1.35235	−0.127785
$113$	−11.6769	−1.09847	−0.549234	−	0.835669i	$-0.685080\pi$
$113$	−11.6769	−1.09847	−0.549234	+	0.835669i	$0.685080\pi$
$114$	0	0
$115$	−5.12330	−0.477750
$116$	−17.2353	−1.60026
$117$	0	0
$118$	27.9486	2.57288
$119$	0	0
$120$	0	0
$121$	23.2841	2.11673
$122$	−14.3019	−1.29483
$123$	0	0
$124$	−14.9554	−1.34304
$125$	7.51299	0.671982
$126$	0	0
$127$	5.29860	0.470175	0.235087	−	0.971974i	$-0.424462\pi$
$127$	5.29860	0.470175	0.235087	+	0.971974i	$0.424462\pi$
$128$	−19.2688	−1.70314
$129$	0	0
$130$	−7.32358	−0.642320
$131$	9.24969	0.808149	0.404075	−	0.914726i	$-0.367594\pi$
$131$	9.24969	0.808149	0.404075	+	0.914726i	$0.367594\pi$
$132$	0	0
$133$	18.8503	1.63453
$134$	12.5652	1.08546
$135$	0	0
$136$	0	0
$137$	−4.77325	−0.407806	−0.203903	−	0.978991i	$-0.565363\pi$
$137$	−4.77325	−0.407806	−0.203903	+	0.978991i	$0.565363\pi$
$138$	0	0
$139$	−18.6382	−1.58087	−0.790434	−	0.612547i	$-0.790145\pi$
$139$	−18.6382	−1.58087	−0.790434	+	0.612547i	$0.790145\pi$
$140$	7.13100	0.602679
$141$	0	0
$142$	13.7638	1.15503
$143$	−23.0906	−1.93093
$144$	0	0
$145$	4.13516	0.343407
$146$	5.85526	0.484585
$147$	0	0
$148$	−5.74691	−0.472393
$149$	10.4465	0.855808	0.427904	−	0.903824i	$-0.359252\pi$
$149$	10.4465	0.855808	0.427904	+	0.903824i	$0.359252\pi$
$150$	0	0
$151$	−3.67499	−0.299067	−0.149533	−	0.988757i	$-0.547777\pi$
$151$	−3.67499	−0.299067	−0.149533	+	0.988757i	$0.547777\pi$
$152$	−22.1391	−1.79572
$153$	0	0
$154$	35.9172	2.89429
$155$	3.58816	0.288208
$156$	0	0
$157$	5.07873	0.405326	0.202663	−	0.979248i	$-0.435040\pi$
$157$	5.07873	0.405326	0.202663	+	0.979248i	$0.435040\pi$
$158$	−14.5808	−1.15998
$159$	0	0
$160$	4.05737	0.320763
$161$	−16.9228	−1.33370
$162$	0	0
$163$	−6.87258	−0.538302	−0.269151	−	0.963098i	$-0.586743\pi$
$163$	−6.87258	−0.538302	−0.269151	+	0.963098i	$0.586743\pi$
$164$	24.0421	1.87737
$165$	0	0
$166$	−35.2722	−2.73765
$167$	13.2198	1.02298	0.511491	−	0.859289i	$-0.329093\pi$
$167$	13.2198	1.02298	0.511491	+	0.859289i	$0.329093\pi$
$168$	0	0
$169$	2.55169	0.196284
$170$	0	0
$171$	0	0
$172$	4.36959	0.333178
$173$	1.36092	0.103469	0.0517344	−	0.998661i	$-0.483525\pi$
$173$	1.36092	0.103469	0.0517344	+	0.998661i	$0.483525\pi$
$174$	0	0
$175$	11.5526	0.873296
$176$	−2.98501	−0.225004
$177$	0	0
$178$	−8.41653	−0.630845
$179$	−1.65773	−0.123905	−0.0619524	−	0.998079i	$-0.519733\pi$
$179$	−1.65773	−0.123905	−0.0619524	+	0.998079i	$0.519733\pi$
$180$	0	0
$181$	13.3259	0.990509	0.495255	−	0.868748i	$-0.335075\pi$
$181$	13.3259	0.990509	0.495255	+	0.868748i	$0.335075\pi$
$182$	−24.1905	−1.79312
$183$	0	0
$184$	19.8753	1.46522
$185$	1.37882	0.101373
$186$	0	0
$187$	0	0
$188$	−3.62180	−0.264147
$189$	0	0
$190$	13.1967	0.957386
$191$	−11.4316	−0.827162	−0.413581	−	0.910467i	$-0.635722\pi$
$191$	−11.4316	−0.827162	−0.413581	+	0.910467i	$0.635722\pi$
$192$	0	0
$193$	15.5057	1.11612	0.558062	−	0.829799i	$-0.311545\pi$
$193$	15.5057	1.11612	0.558062	+	0.829799i	$0.311545\pi$
$194$	−18.9446	−1.36014
$195$	0	0
$196$	0.123303	0.00880736
$197$	19.8446	1.41387	0.706933	−	0.707280i	$-0.250078\pi$
$197$	19.8446	1.41387	0.706933	+	0.707280i	$0.250078\pi$
$198$	0	0
$199$	−2.81521	−0.199565	−0.0997824	−	0.995009i	$-0.531815\pi$
$199$	−2.81521	−0.199565	−0.0997824	+	0.995009i	$0.531815\pi$
$200$	−13.5682	−0.959416
$201$	0	0
$202$	41.2645	2.90336
$203$	13.6588	0.958663
$204$	0	0
$205$	−5.76827	−0.402873
$206$	−12.4800	−0.869521
$207$	0	0
$208$	2.01043	0.139398
$209$	41.6079	2.87808
$210$	0	0
$211$	−17.2121	−1.18493	−0.592466	−	0.805595i	$-0.701846\pi$
$211$	−17.2121	−1.18493	−0.592466	+	0.805595i	$0.701846\pi$
$212$	−32.3918	−2.22468
$213$	0	0
$214$	3.49020	0.238585
$215$	−1.04837	−0.0714980
$216$	0	0
$217$	11.8520	0.804569
$218$	37.1651	2.51714
$219$	0	0
$220$	15.7401	1.06120
$221$	0	0
$222$	0	0
$223$	−16.7324	−1.12048	−0.560241	−	0.828330i	$-0.689291\pi$
$223$	−16.7324	−1.12048	−0.560241	+	0.828330i	$0.689291\pi$
$224$	13.4019	0.895451
$225$	0	0
$226$	−27.0019	−1.79614
$227$	0.706231	0.0468742	0.0234371	−	0.999725i	$-0.492539\pi$
$227$	0.706231	0.0468742	0.0234371	+	0.999725i	$0.492539\pi$
$228$	0	0
$229$	−1.65270	−0.109214	−0.0546069	−	0.998508i	$-0.517391\pi$
$229$	−1.65270	−0.109214	−0.0546069	+	0.998508i	$0.517391\pi$
$230$	−11.8472	−0.781184
$231$	0	0
$232$	−16.0419	−1.05320
$233$	22.3844	1.46645	0.733224	−	0.679987i	$-0.238015\pi$
$233$	22.3844	1.46645	0.733224	+	0.679987i	$0.238015\pi$
$234$	0	0
$235$	0.868956	0.0566844
$236$	40.4564	2.63349
$237$	0	0
$238$	0	0
$239$	−12.3831	−0.800997	−0.400498	−	0.916298i	$-0.631163\pi$
$239$	−12.3831	−0.800997	−0.400498	+	0.916298i	$0.631163\pi$
$240$	0	0
$241$	−16.5790	−1.06795	−0.533975	−	0.845501i	$-0.679302\pi$
$241$	−16.5790	−1.06795	−0.533975	+	0.845501i	$0.679302\pi$
$242$	53.8426	3.46113
$243$	0	0
$244$	−20.7023	−1.32533
$245$	−0.0295833	−0.00189001
$246$	0	0
$247$	−28.0232	−1.78307
$248$	−13.9199	−0.883912
$249$	0	0
$250$	17.3732	1.09878
$251$	5.01852	0.316766	0.158383	−	0.987378i	$-0.449372\pi$
$251$	5.01852	0.316766	0.158383	+	0.987378i	$0.449372\pi$
$252$	0	0
$253$	−37.3533	−2.34838
$254$	12.2526	0.768797
$255$	0	0
$256$	−19.1530	−1.19706
$257$	−19.5141	−1.21726	−0.608628	−	0.793456i	$-0.708280\pi$
$257$	−19.5141	−1.21726	−0.608628	+	0.793456i	$0.708280\pi$
$258$	0	0
$259$	4.55438	0.282995
$260$	−10.6011	−0.657451
$261$	0	0
$262$	21.3892	1.32143
$263$	−9.28333	−0.572435	−0.286217	−	0.958165i	$-0.592398\pi$
$263$	−9.28333	−0.572435	−0.286217	+	0.958165i	$0.592398\pi$
$264$	0	0
$265$	7.77156	0.477403
$266$	43.5898	2.67266
$267$	0	0
$268$	18.1884	1.11103
$269$	−0.0515409	−0.00314250	−0.00157125	−	0.999999i	$-0.500500\pi$
$269$	−0.0515409	−0.00314250	−0.00157125	+	0.999999i	$0.500500\pi$
$270$	0	0
$271$	−18.1506	−1.10257	−0.551287	−	0.834316i	$-0.685863\pi$
$271$	−18.1506	−1.10257	−0.551287	+	0.834316i	$0.685863\pi$
$272$	0	0
$273$	0	0
$274$	−11.0378	−0.666816
$275$	25.4999	1.53770
$276$	0	0
$277$	−5.55438	−0.333730	−0.166865	−	0.985980i	$-0.553364\pi$
$277$	−5.55438	−0.333730	−0.166865	+	0.985980i	$0.553364\pi$
$278$	−43.0993	−2.58492
$279$	0	0
$280$	6.63722	0.396650
$281$	−2.16402	−0.129094	−0.0645472	−	0.997915i	$-0.520560\pi$
$281$	−2.16402	−0.129094	−0.0645472	+	0.997915i	$0.520560\pi$
$282$	0	0
$283$	−25.0401	−1.48848	−0.744241	−	0.667911i	$-0.767188\pi$
$283$	−25.0401	−1.48848	−0.744241	+	0.667911i	$0.767188\pi$
$284$	19.9235	1.18224
$285$	0	0
$286$	−53.3952	−3.15732
$287$	−19.0531	−1.12467
$288$	0	0
$289$	0	0
$290$	9.56224	0.561514
$291$	0	0
$292$	8.47565	0.496000
$293$	28.9005	1.68839	0.844193	−	0.536039i	$-0.180080\pi$
$293$	28.9005	1.68839	0.844193	+	0.536039i	$0.180080\pi$
$294$	0	0
$295$	−9.70645	−0.565132
$296$	−5.34897	−0.310903
$297$	0	0
$298$	24.1566	1.39936
$299$	25.1577	1.45491
$300$	0	0
$301$	−3.46286	−0.199596
$302$	−8.49813	−0.489013
$303$	0	0
$304$	−3.62267	−0.207775
$305$	4.96698	0.284408
$306$	0	0
$307$	7.09152	0.404734	0.202367	−	0.979310i	$-0.435137\pi$
$307$	7.09152	0.404734	0.202367	+	0.979310i	$0.435137\pi$
$308$	51.9911	2.96247
$309$	0	0
$310$	8.29734	0.471257
$311$	9.38019	0.531902	0.265951	−	0.963987i	$-0.414314\pi$
$311$	9.38019	0.531902	0.265951	+	0.963987i	$0.414314\pi$
$312$	0	0
$313$	−29.1857	−1.64967	−0.824837	−	0.565370i	$-0.808733\pi$
$313$	−29.1857	−1.64967	−0.824837	+	0.565370i	$0.808733\pi$
$314$	11.7442	0.662761
$315$	0	0
$316$	−21.1061	−1.18731
$317$	15.6291	0.877819	0.438910	−	0.898531i	$-0.355365\pi$
$317$	15.6291	0.877819	0.438910	+	0.898531i	$0.355365\pi$
$318$	0	0
$319$	30.1489	1.68801
$320$	10.2012	0.570264
$321$	0	0
$322$	−39.1326	−2.18077
$323$	0	0
$324$	0	0
$325$	−17.1744	−0.952662
$326$	−15.8923	−0.880193
$327$	0	0
$328$	22.3773	1.23558
$329$	2.87025	0.158242
$330$	0	0
$331$	4.08883	0.224742	0.112371	−	0.993666i	$-0.464155\pi$
$331$	4.08883	0.224742	0.112371	+	0.993666i	$0.464155\pi$
$332$	−51.0575	−2.80214
$333$	0	0
$334$	30.5699	1.67271
$335$	−4.36383	−0.238422
$336$	0	0
$337$	−4.07604	−0.222036	−0.111018	−	0.993818i	$-0.535411\pi$
$337$	−4.07604	−0.222036	−0.111018	+	0.993818i	$0.535411\pi$
$338$	5.90058	0.320949
$339$	0	0
$340$	0	0
$341$	26.1608	1.41669
$342$	0	0
$343$	18.4712	0.997352
$344$	4.06702	0.219279
$345$	0	0
$346$	3.14702	0.169185
$347$	−11.2832	−0.605714	−0.302857	−	0.953036i	$-0.597940\pi$
$347$	−11.2832	−0.605714	−0.302857	+	0.953036i	$0.597940\pi$
$348$	0	0
$349$	27.2814	1.46034	0.730169	−	0.683267i	$-0.239441\pi$
$349$	27.2814	1.46034	0.730169	+	0.683267i	$0.239441\pi$
$350$	26.7145	1.42795
$351$	0	0
$352$	29.5817	1.57671
$353$	−21.0117	−1.11834	−0.559171	−	0.829052i	$-0.688881\pi$
$353$	−21.0117	−1.11834	−0.559171	+	0.829052i	$0.688881\pi$
$354$	0	0
$355$	−4.78013	−0.253703
$356$	−12.1832	−0.645706
$357$	0	0
$358$	−3.83338	−0.202600
$359$	−30.8488	−1.62814	−0.814070	−	0.580767i	$-0.802753\pi$
$359$	−30.8488	−1.62814	−0.814070	+	0.580767i	$0.802753\pi$
$360$	0	0
$361$	31.4962	1.65769
$362$	30.8152	1.61961
$363$	0	0
$364$	−35.0164	−1.83536
$365$	−2.03351	−0.106439
$366$	0	0
$367$	−26.7219	−1.39487	−0.697437	−	0.716646i	$-0.745676\pi$
$367$	−26.7219	−1.39487	−0.697437	+	0.716646i	$0.745676\pi$
$368$	3.25224	0.169535
$369$	0	0
$370$	3.18841	0.165758
$371$	25.6702	1.33273
$372$	0	0
$373$	−5.24392	−0.271520	−0.135760	−	0.990742i	$-0.543348\pi$
$373$	−5.24392	−0.271520	−0.135760	+	0.990742i	$0.543348\pi$
$374$	0	0
$375$	0	0
$376$	−3.37102	−0.173847
$377$	−20.3055	−1.04579
$378$	0	0
$379$	8.06923	0.414489	0.207244	−	0.978289i	$-0.433551\pi$
$379$	8.06923	0.414489	0.207244	+	0.978289i	$0.433551\pi$
$380$	19.1025	0.979939
$381$	0	0
$382$	−26.4347	−1.35252
$383$	−3.11552	−0.159196	−0.0795978	−	0.996827i	$-0.525364\pi$
$383$	−3.11552	−0.159196	−0.0795978	+	0.996827i	$0.525364\pi$
$384$	0	0
$385$	−12.4739	−0.635729
$386$	35.8557	1.82501
$387$	0	0
$388$	−27.4228	−1.39218
$389$	−10.1833	−0.516313	−0.258157	−	0.966103i	$-0.583115\pi$
$389$	−10.1833	−0.516313	−0.258157	+	0.966103i	$0.583115\pi$
$390$	0	0
$391$	0	0
$392$	0.114765	0.00579652
$393$	0	0
$394$	45.8890	2.31185
$395$	5.06385	0.254790
$396$	0	0
$397$	−26.3756	−1.32375	−0.661876	−	0.749613i	$-0.730239\pi$
$397$	−26.3756	−1.32375	−0.661876	+	0.749613i	$0.730239\pi$
$398$	−6.50995	−0.326314
$399$	0	0
$400$	−2.22020	−0.111010
$401$	26.0639	1.30157	0.650785	−	0.759262i	$-0.274440\pi$
$401$	26.0639	1.30157	0.650785	+	0.759262i	$0.274440\pi$
$402$	0	0
$403$	−17.6195	−0.877689
$404$	59.7315	2.97175
$405$	0	0
$406$	31.5850	1.56754
$407$	10.0528	0.498298
$408$	0	0
$409$	−29.4097	−1.45422	−0.727109	−	0.686523i	$-0.759136\pi$
$409$	−29.4097	−1.45422	−0.727109	+	0.686523i	$0.759136\pi$
$410$	−13.3387	−0.658750
$411$	0	0
$412$	−18.0651	−0.890004
$413$	−32.0614	−1.57764
$414$	0	0
$415$	12.2499	0.601324
$416$	−19.9235	−0.976831
$417$	0	0
$418$	96.2149	4.70603
$419$	−3.77021	−0.184187	−0.0920933	−	0.995750i	$-0.529356\pi$
$419$	−3.77021	−0.184187	−0.0920933	+	0.995750i	$0.529356\pi$
$420$	0	0
$421$	4.50475	0.219548	0.109774	−	0.993957i	$-0.464987\pi$
$421$	4.50475	0.219548	0.109774	+	0.993957i	$0.464987\pi$
$422$	−39.8017	−1.93752
$423$	0	0
$424$	−30.1489	−1.46416
$425$	0	0
$426$	0	0
$427$	16.4064	0.793962
$428$	5.05216	0.244206
$429$	0	0
$430$	−2.42427	−0.116909
$431$	18.7625	0.903760	0.451880	−	0.892079i	$-0.350753\pi$
$431$	18.7625	0.903760	0.451880	+	0.892079i	$0.350753\pi$
$432$	0	0
$433$	16.3455	0.785517	0.392758	−	0.919642i	$-0.371521\pi$
$433$	16.3455	0.785517	0.392758	+	0.919642i	$0.371521\pi$
$434$	27.4069	1.31557
$435$	0	0
$436$	53.7975	2.57643
$437$	−45.3327	−2.16856
$438$	0	0
$439$	−15.3200	−0.731182	−0.365591	−	0.930776i	$-0.619133\pi$
$439$	−15.3200	−0.731182	−0.365591	+	0.930776i	$0.619133\pi$
$440$	14.6502	0.698421
$441$	0	0
$442$	0	0
$443$	−3.77021	−0.179128	−0.0895640	−	0.995981i	$-0.528547\pi$
$443$	−3.77021	−0.179128	−0.0895640	+	0.995981i	$0.528547\pi$
$444$	0	0
$445$	2.92303	0.138565
$446$	−38.6923	−1.83213
$447$	0	0
$448$	33.6955	1.59196
$449$	−11.8926	−0.561245	−0.280622	−	0.959818i	$-0.590541\pi$
$449$	−11.8926	−0.561245	−0.280622	+	0.959818i	$0.590541\pi$
$450$	0	0
$451$	−42.0556	−1.98032
$452$	−39.0860	−1.83845
$453$	0	0
$454$	1.63310	0.0766453
$455$	8.40127	0.393858
$456$	0	0
$457$	−13.2472	−0.619679	−0.309839	−	0.950789i	$-0.600275\pi$
$457$	−13.2472	−0.619679	−0.309839	+	0.950789i	$0.600275\pi$
$458$	−3.82175	−0.178579
$459$	0	0
$460$	−17.1492	−0.799586
$461$	23.7632	1.10676	0.553381	−	0.832929i	$-0.313338\pi$
$461$	23.7632	1.10676	0.553381	+	0.832929i	$0.313338\pi$
$462$	0	0
$463$	0.576666	0.0268000	0.0134000	−	0.999910i	$-0.495735\pi$
$463$	0.576666	0.0268000	0.0134000	+	0.999910i	$0.495735\pi$
$464$	−2.62498	−0.121861
$465$	0	0
$466$	51.7621	2.39783
$467$	−9.67701	−0.447798	−0.223899	−	0.974612i	$-0.571879\pi$
$467$	−9.67701	−0.447798	−0.223899	+	0.974612i	$0.571879\pi$
$468$	0	0
$469$	−14.4142	−0.665584
$470$	2.00939	0.0926864
$471$	0	0
$472$	37.6551	1.73321
$473$	−7.64350	−0.351448
$474$	0	0
$475$	30.9472	1.41995
$476$	0	0
$477$	0	0
$478$	−28.6350	−1.30973
$479$	−10.6442	−0.486348	−0.243174	−	0.969983i	$-0.578189\pi$
$479$	−10.6442	−0.486348	−0.243174	+	0.969983i	$0.578189\pi$
$480$	0	0
$481$	−6.77063	−0.308714
$482$	−38.3377	−1.74624
$483$	0	0
$484$	77.9386	3.54266
$485$	6.57939	0.298755
$486$	0	0
$487$	−1.15982	−0.0525563	−0.0262781	−	0.999655i	$-0.508366\pi$
$487$	−1.15982	−0.0525563	−0.0262781	+	0.999655i	$0.508366\pi$
$488$	−19.2688	−0.872259
$489$	0	0
$490$	−0.0684092	−0.00309041
$491$	31.8340	1.43665	0.718324	−	0.695709i	$-0.244910\pi$
$491$	31.8340	1.43665	0.718324	+	0.695709i	$0.244910\pi$
$492$	0	0
$493$	0	0
$494$	−64.8015	−2.91556
$495$	0	0
$496$	−2.27774	−0.102274
$497$	−15.7892	−0.708243
$498$	0	0
$499$	−16.0378	−0.717949	−0.358975	−	0.933347i	$-0.616874\pi$
$499$	−16.0378	−0.717949	−0.358975	+	0.933347i	$0.616874\pi$
$500$	25.1482	1.12466
$501$	0	0
$502$	11.6049	0.517954
$503$	−28.1091	−1.25332	−0.626661	−	0.779292i	$-0.715579\pi$
$503$	−28.1091	−1.25332	−0.626661	+	0.779292i	$0.715579\pi$
$504$	0	0
$505$	−14.3310	−0.637721
$506$	−86.3766	−3.83991
$507$	0	0
$508$	17.7360	0.786907
$509$	−7.13100	−0.316076	−0.158038	−	0.987433i	$-0.550517\pi$
$509$	−7.13100	−0.316076	−0.158038	+	0.987433i	$0.550517\pi$
$510$	0	0
$511$	−6.71688	−0.297137
$512$	−5.75218	−0.254213
$513$	0	0
$514$	−45.1248	−1.99037
$515$	4.33425	0.190990
$516$	0	0
$517$	6.33544	0.278632
$518$	10.5316	0.462734
$519$	0	0
$520$	−9.86703	−0.432698
$521$	−37.1135	−1.62597	−0.812986	−	0.582283i	$-0.802160\pi$
$521$	−37.1135	−1.62597	−0.812986	+	0.582283i	$0.802160\pi$
$522$	0	0
$523$	−21.3182	−0.932180	−0.466090	−	0.884737i	$-0.654338\pi$
$523$	−21.3182	−0.932180	−0.466090	+	0.884737i	$0.654338\pi$
$524$	30.9614	1.35256
$525$	0	0
$526$	−21.4670	−0.936005
$527$	0	0
$528$	0	0
$529$	17.6973	0.769447
$530$	17.9711	0.780616
$531$	0	0
$532$	63.0975	2.73562
$533$	28.3248	1.22688
$534$	0	0
$535$	−1.21213	−0.0524051
$536$	16.9290	0.731221
$537$	0	0
$538$	−0.119184	−0.00513840
$539$	−0.215688	−0.00929034
$540$	0	0
$541$	32.8780	1.41353	0.706767	−	0.707447i	$-0.250153\pi$
$541$	32.8780	1.41353	0.706767	+	0.707447i	$0.250153\pi$
$542$	−41.9720	−1.80285
$543$	0	0
$544$	0	0
$545$	−12.9073	−0.552888
$546$	0	0
$547$	−12.8571	−0.549730	−0.274865	−	0.961483i	$-0.588633\pi$
$547$	−12.8571	−0.549730	−0.274865	+	0.961483i	$0.588633\pi$
$548$	−15.9775	−0.682524
$549$	0	0
$550$	58.9665	2.51434
$551$	36.5893	1.55876
$552$	0	0
$553$	16.7264	0.711278
$554$	−12.8441	−0.545692
$555$	0	0
$556$	−62.3874	−2.64582
$557$	30.8193	1.30585	0.652927	−	0.757421i	$-0.273541\pi$
$557$	30.8193	1.30585	0.652927	+	0.757421i	$0.273541\pi$
$558$	0	0
$559$	5.14796	0.217735
$560$	1.08607	0.0458946
$561$	0	0
$562$	−5.00412	−0.211086
$563$	−9.88864	−0.416756	−0.208378	−	0.978048i	$-0.566818\pi$
$563$	−9.88864	−0.416756	−0.208378	+	0.978048i	$0.566818\pi$
$564$	0	0
$565$	9.37765	0.394521
$566$	−57.9034	−2.43386
$567$	0	0
$568$	18.5439	0.778087
$569$	41.5742	1.74288	0.871441	−	0.490500i	$-0.163186\pi$
$569$	41.5742	1.74288	0.871441	+	0.490500i	$0.163186\pi$
$570$	0	0
$571$	−31.4219	−1.31497	−0.657483	−	0.753469i	$-0.728379\pi$
$571$	−31.4219	−1.31497	−0.657483	+	0.753469i	$0.728379\pi$
$572$	−77.2910	−3.23170
$573$	0	0
$574$	−44.0589	−1.83898
$575$	−27.7827	−1.15862
$576$	0	0
$577$	−27.9837	−1.16498	−0.582488	−	0.812839i	$-0.697921\pi$
$577$	−27.9837	−1.16498	−0.582488	+	0.812839i	$0.697921\pi$
$578$	0	0
$579$	0	0
$580$	13.8416	0.574742
$581$	40.4626	1.67867
$582$	0	0
$583$	56.6614	2.34667
$584$	7.88877	0.326440
$585$	0	0
$586$	66.8302	2.76073
$587$	21.1244	0.871895	0.435948	−	0.899972i	$-0.356413\pi$
$587$	21.1244	0.871895	0.435948	+	0.899972i	$0.356413\pi$
$588$	0	0
$589$	31.7493	1.30821
$590$	−22.4454	−0.924063
$591$	0	0
$592$	−0.875266	−0.0359732
$593$	−13.1230	−0.538896	−0.269448	−	0.963015i	$-0.586841\pi$
$593$	−13.1230	−0.538896	−0.269448	+	0.963015i	$0.586841\pi$
$594$	0	0
$595$	0	0
$596$	34.9674	1.43232
$597$	0	0
$598$	58.1753	2.37896
$599$	−7.55832	−0.308824	−0.154412	−	0.988007i	$-0.549348\pi$
$599$	−7.55832	−0.308824	−0.154412	+	0.988007i	$0.549348\pi$
$600$	0	0
$601$	17.7861	0.725511	0.362755	−	0.931884i	$-0.381836\pi$
$601$	17.7861	0.725511	0.362755	+	0.931884i	$0.381836\pi$
$602$	−8.00759	−0.326365
$603$	0	0
$604$	−12.3013	−0.500532
$605$	−18.6993	−0.760236
$606$	0	0
$607$	−37.8999	−1.53831	−0.769155	−	0.639062i	$-0.779323\pi$
$607$	−37.8999	−1.53831	−0.769155	+	0.639062i	$0.779323\pi$
$608$	35.9010	1.45598
$609$	0	0
$610$	11.4858	0.465045
$611$	−4.26697	−0.172623
$612$	0	0
$613$	−16.9240	−0.683552	−0.341776	−	0.939781i	$-0.611028\pi$
$613$	−16.9240	−0.683552	−0.341776	+	0.939781i	$0.611028\pi$
$614$	16.3986	0.661793
$615$	0	0
$616$	48.3911	1.94973
$617$	−0.375777	−0.0151282	−0.00756412	−	0.999971i	$-0.502408\pi$
$617$	−0.375777	−0.0151282	−0.00756412	+	0.999971i	$0.502408\pi$
$618$	0	0
$619$	22.4074	0.900628	0.450314	−	0.892870i	$-0.351312\pi$
$619$	22.4074	0.900628	0.450314	+	0.892870i	$0.351312\pi$
$620$	12.0106	0.482359
$621$	0	0
$622$	21.6910	0.869729
$623$	9.65505	0.386821
$624$	0	0
$625$	15.7415	0.629661
$626$	−67.4897	−2.69743
$627$	0	0
$628$	17.0000	0.678374
$629$	0	0
$630$	0	0
$631$	12.4929	0.497334	0.248667	−	0.968589i	$-0.420008\pi$
$631$	12.4929	0.497334	0.248667	+	0.968589i	$0.420008\pi$
$632$	−19.6446	−0.781421
$633$	0	0
$634$	36.1411	1.43535
$635$	−4.25528	−0.168866
$636$	0	0
$637$	0.145268	0.00575571
$638$	69.7170	2.76012
$639$	0	0
$640$	15.4747	0.611692
$641$	28.1943	1.11361	0.556804	−	0.830644i	$-0.312028\pi$
$641$	28.1943	1.11361	0.556804	+	0.830644i	$0.312028\pi$
$642$	0	0
$643$	38.3773	1.51345	0.756727	−	0.653731i	$-0.226797\pi$
$643$	38.3773	1.51345	0.756727	+	0.653731i	$0.226797\pi$
$644$	−56.6455	−2.23215
$645$	0	0
$646$	0	0
$647$	22.0718	0.867732	0.433866	−	0.900977i	$-0.357149\pi$
$647$	22.0718	0.867732	0.433866	+	0.900977i	$0.357149\pi$
$648$	0	0
$649$	−70.7684	−2.77790
$650$	−39.7144	−1.55773
$651$	0	0
$652$	−23.0046	−0.900928
$653$	9.83710	0.384955	0.192478	−	0.981301i	$-0.438348\pi$
$653$	9.83710	0.384955	0.192478	+	0.981301i	$0.438348\pi$
$654$	0	0
$655$	−7.42839	−0.290251
$656$	3.66166	0.142964
$657$	0	0
$658$	6.63722	0.258746
$659$	0.0968652	0.00377333	0.00188667	−	0.999998i	$-0.499399\pi$
$659$	0.0968652	0.00377333	0.00188667	+	0.999998i	$0.499399\pi$
$660$	0	0
$661$	16.3523	0.636033	0.318016	−	0.948085i	$-0.396983\pi$
$661$	16.3523	0.636033	0.318016	+	0.948085i	$0.396983\pi$
$662$	9.45510	0.367483
$663$	0	0
$664$	−47.5221	−1.84422
$665$	−15.1386	−0.587049
$666$	0	0
$667$	−32.8479	−1.27188
$668$	44.2507	1.71211
$669$	0	0
$670$	−10.0910	−0.389850
$671$	36.2136	1.39801
$672$	0	0
$673$	36.6705	1.41355	0.706773	−	0.707441i	$-0.250150\pi$
$673$	36.6705	1.41355	0.706773	+	0.707441i	$0.250150\pi$
$674$	−9.42552	−0.363057
$675$	0	0
$676$	8.54126	0.328510
$677$	18.1016	0.695703	0.347851	−	0.937550i	$-0.386911\pi$
$677$	18.1016	0.695703	0.347851	+	0.937550i	$0.386911\pi$
$678$	0	0
$679$	21.7324	0.834012
$680$	0	0
$681$	0	0
$682$	60.4948	2.31646
$683$	−51.1028	−1.95540	−0.977698	−	0.210018i	$-0.932648\pi$
$683$	−51.1028	−1.95540	−0.977698	+	0.210018i	$0.932648\pi$
$684$	0	0
$685$	3.83338	0.146466
$686$	42.7132	1.63080
$687$	0	0
$688$	0.665497	0.0253718
$689$	−38.1619	−1.45385
$690$	0	0
$691$	16.2131	0.616774	0.308387	−	0.951261i	$-0.400211\pi$
$691$	16.2131	0.616774	0.308387	+	0.951261i	$0.400211\pi$
$692$	4.55540	0.173170
$693$	0	0
$694$	−26.0915	−0.990421
$695$	14.9682	0.567777
$696$	0	0
$697$	0	0
$698$	63.0860	2.38784
$699$	0	0
$700$	38.6701	1.46159
$701$	−32.6371	−1.23269	−0.616343	−	0.787478i	$-0.711386\pi$
$701$	−32.6371	−1.23269	−0.616343	+	0.787478i	$0.711386\pi$
$702$	0	0
$703$	12.2003	0.460142
$704$	74.3754	2.80313
$705$	0	0
$706$	−48.5880	−1.82863
$707$	−47.3367	−1.78028
$708$	0	0
$709$	21.7014	0.815013	0.407507	−	0.913202i	$-0.366398\pi$
$709$	21.7014	0.815013	0.407507	+	0.913202i	$0.366398\pi$
$710$	−11.0537	−0.414837
$711$	0	0
$712$	−11.3396	−0.424968
$713$	−28.5028	−1.06744
$714$	0	0
$715$	18.5439	0.693504
$716$	−5.54892	−0.207373
$717$	0	0
$718$	−71.3355	−2.66222
$719$	−16.9963	−0.633854	−0.316927	−	0.948450i	$-0.602651\pi$
$719$	−16.9963	−0.633854	−0.316927	+	0.948450i	$0.602651\pi$
$720$	0	0
$721$	14.3164	0.533172
$722$	72.8325	2.71054
$723$	0	0
$724$	44.6059	1.65776
$725$	22.4242	0.832814
$726$	0	0
$727$	23.8212	0.883479	0.441740	−	0.897143i	$-0.354362\pi$
$727$	23.8212	0.883479	0.441740	+	0.897143i	$0.354362\pi$
$728$	−32.5918	−1.20793
$729$	0	0
$730$	−4.70233	−0.174041
$731$	0	0
$732$	0	0
$733$	−44.1661	−1.63131	−0.815657	−	0.578536i	$-0.803624\pi$
$733$	−44.1661	−1.63131	−0.815657	+	0.578536i	$0.803624\pi$
$734$	−61.7924	−2.28080
$735$	0	0
$736$	−32.2300	−1.18801
$737$	−31.8161	−1.17196
$738$	0	0
$739$	−27.7219	−1.01977	−0.509884	−	0.860243i	$-0.670312\pi$
$739$	−27.7219	−1.01977	−0.509884	+	0.860243i	$0.670312\pi$
$740$	4.61532	0.169663
$741$	0	0
$742$	59.3604	2.17919
$743$	21.4727	0.787757	0.393879	−	0.919162i	$-0.371133\pi$
$743$	21.4727	0.787757	0.393879	+	0.919162i	$0.371133\pi$
$744$	0	0
$745$	−8.38951	−0.307368
$746$	−12.1262	−0.443970
$747$	0	0
$748$	0	0
$749$	−4.00380	−0.146296
$750$	0	0
$751$	1.40467	0.0512570	0.0256285	−	0.999672i	$-0.491841\pi$
$751$	1.40467	0.0512570	0.0256285	+	0.999672i	$0.491841\pi$
$752$	−0.551608	−0.0201151
$753$	0	0
$754$	−46.9549	−1.71000
$755$	2.95137	0.107411
$756$	0	0
$757$	30.8262	1.12040	0.560199	−	0.828358i	$-0.310724\pi$
$757$	30.8262	1.12040	0.560199	+	0.828358i	$0.310724\pi$
$758$	18.6595	0.677742
$759$	0	0
$760$	17.7798	0.644941
$761$	−35.1432	−1.27394	−0.636971	−	0.770888i	$-0.719813\pi$
$761$	−35.1432	−1.27394	−0.636971	+	0.770888i	$0.719813\pi$
$762$	0	0
$763$	−42.6340	−1.54346
$764$	−38.2650	−1.38438
$765$	0	0
$766$	−7.20439	−0.260305
$767$	47.6631	1.72101
$768$	0	0
$769$	24.5449	0.885111	0.442556	−	0.896741i	$-0.354072\pi$
$769$	24.5449	0.885111	0.442556	+	0.896741i	$0.354072\pi$
$770$	−28.8449	−1.03950
$771$	0	0
$772$	51.9021	1.86800
$773$	−19.6625	−0.707211	−0.353606	−	0.935395i	$-0.615045\pi$
$773$	−19.6625	−0.707211	−0.353606	+	0.935395i	$0.615045\pi$
$774$	0	0
$775$	19.4579	0.698949
$776$	−25.5240	−0.916258
$777$	0	0
$778$	−23.5481	−0.844239
$779$	−51.0396	−1.82868
$780$	0	0
$781$	−34.8512	−1.24707
$782$	0	0
$783$	0	0
$784$	0.0187793	0.000670690 0
$785$	−4.07870	−0.145575
$786$	0	0
$787$	18.5354	0.660715	0.330358	−	0.943856i	$-0.392831\pi$
$787$	18.5354	0.660715	0.330358	+	0.943856i	$0.392831\pi$
$788$	66.4256	2.36631
$789$	0	0
$790$	11.7098	0.416614
$791$	30.9753	1.10135
$792$	0	0
$793$	−24.3901	−0.866119
$794$	−60.9915	−2.16451
$795$	0	0
$796$	−9.42333	−0.334001
$797$	−39.3923	−1.39535	−0.697673	−	0.716416i	$-0.745781\pi$
$797$	−39.3923	−1.39535	−0.697673	+	0.716416i	$0.745781\pi$
$798$	0	0
$799$	0	0
$800$	22.0024	0.777901
$801$	0	0
$802$	60.2708	2.12823
$803$	−14.8260	−0.523199
$804$	0	0
$805$	13.5906	0.479006
$806$	−40.7437	−1.43514
$807$	0	0
$808$	55.5954	1.95584
$809$	−53.6426	−1.88597	−0.942987	−	0.332830i	$-0.891997\pi$
$809$	−53.6426	−1.88597	−0.942987	+	0.332830i	$0.891997\pi$
$810$	0	0
$811$	−42.3327	−1.48650	−0.743252	−	0.669012i	$-0.766718\pi$
$811$	−42.3327	−1.48650	−0.743252	+	0.669012i	$0.766718\pi$
$812$	45.7202	1.60446
$813$	0	0
$814$	23.2463	0.814782
$815$	5.51934	0.193334
$816$	0	0
$817$	−9.27631	−0.324537
$818$	−68.0077	−2.37783
$819$	0	0
$820$	−19.3081	−0.674268
$821$	−33.3433	−1.16369	−0.581845	−	0.813300i	$-0.697669\pi$
$821$	−33.3433	−1.16369	−0.581845	+	0.813300i	$0.697669\pi$
$822$	0	0
$823$	−30.9469	−1.07874	−0.539370	−	0.842069i	$-0.681338\pi$
$823$	−30.9469	−1.07874	−0.539370	+	0.842069i	$0.681338\pi$
$824$	−16.8142	−0.585751
$825$	0	0
$826$	−74.1394	−2.57964
$827$	−7.02245	−0.244194	−0.122097	−	0.992518i	$-0.538962\pi$
$827$	−7.02245	−0.244194	−0.122097	+	0.992518i	$0.538962\pi$
$828$	0	0
$829$	13.2959	0.461786	0.230893	−	0.972979i	$-0.425835\pi$
$829$	13.2959	0.461786	0.230893	+	0.972979i	$0.425835\pi$
$830$	28.3269	0.983243
$831$	0	0
$832$	−50.0925	−1.73664
$833$	0	0
$834$	0	0
$835$	−10.6168	−0.367410
$836$	139.274	4.81689
$837$	0	0
$838$	−8.71831	−0.301169
$839$	−32.9613	−1.13795	−0.568976	−	0.822354i	$-0.692660\pi$
$839$	−32.9613	−1.13795	−0.568976	+	0.822354i	$0.692660\pi$
$840$	0	0
$841$	−2.48751	−0.0857763
$842$	10.4169	0.358989
$843$	0	0
$844$	−57.6141	−1.98316
$845$	−2.04925	−0.0704964
$846$	0	0
$847$	−61.7657	−2.12229
$848$	−4.93334	−0.169412
$849$	0	0
$850$	0	0
$851$	−10.9527	−0.375455
$852$	0	0
$853$	6.33511	0.216910	0.108455	−	0.994101i	$-0.465410\pi$
$853$	6.33511	0.216910	0.108455	+	0.994101i	$0.465410\pi$
$854$	37.9386	1.29823
$855$	0	0
$856$	4.70233	0.160722
$857$	−11.4769	−0.392044	−0.196022	−	0.980599i	$-0.562802\pi$
$857$	−11.4769	−0.392044	−0.196022	+	0.980599i	$0.562802\pi$
$858$	0	0
$859$	13.8817	0.473639	0.236820	−	0.971554i	$-0.423895\pi$
$859$	13.8817	0.473639	0.236820	+	0.971554i	$0.423895\pi$
$860$	−3.50920	−0.119663
$861$	0	0
$862$	43.3869	1.47776
$863$	−14.3534	−0.488595	−0.244298	−	0.969700i	$-0.578557\pi$
$863$	−14.3534	−0.488595	−0.244298	+	0.969700i	$0.578557\pi$
$864$	0	0
$865$	−1.09295	−0.0371614
$866$	37.7978	1.28442
$867$	0	0
$868$	39.6723	1.34657
$869$	36.9198	1.25242
$870$	0	0
$871$	21.4284	0.726073
$872$	50.0724	1.69566
$873$	0	0
$874$	−104.828	−3.54587
$875$	−19.9297	−0.673748
$876$	0	0
$877$	−49.3201	−1.66542	−0.832710	−	0.553709i	$-0.813212\pi$
$877$	−49.3201	−1.66542	−0.832710	+	0.553709i	$0.813212\pi$
$878$	−35.4262	−1.19558
$879$	0	0
$880$	2.39725	0.0808113
$881$	16.2049	0.545955	0.272978	−	0.962020i	$-0.411991\pi$
$881$	16.2049	0.545955	0.272978	+	0.962020i	$0.411991\pi$
$882$	0	0
$883$	45.1293	1.51872	0.759361	−	0.650670i	$-0.225512\pi$
$883$	45.1293	1.51872	0.759361	+	0.650670i	$0.225512\pi$
$884$	0	0
$885$	0	0
$886$	−8.71831	−0.292897
$887$	−13.9199	−0.467383	−0.233692	−	0.972311i	$-0.575081\pi$
$887$	−13.9199	−0.467383	−0.233692	+	0.972311i	$0.575081\pi$
$888$	0	0
$889$	−14.0556	−0.471410
$890$	6.75928	0.226571
$891$	0	0
$892$	−56.0082	−1.87529
$893$	7.68882	0.257297
$894$	0	0
$895$	1.33132	0.0445011
$896$	51.1145	1.70762
$897$	0	0
$898$	−27.5006	−0.917708
$899$	23.0054	0.767273
$900$	0	0
$901$	0	0
$902$	−97.2503	−3.23808
$903$	0	0
$904$	−36.3795	−1.20996
$905$	−10.7020	−0.355747
$906$	0	0
$907$	−9.87164	−0.327783	−0.163891	−	0.986478i	$-0.552405\pi$
$907$	−9.87164	−0.327783	−0.163891	+	0.986478i	$0.552405\pi$
$908$	2.36396	0.0784509
$909$	0	0
$910$	19.4273	0.644008
$911$	38.5596	1.27754	0.638769	−	0.769399i	$-0.279444\pi$
$911$	38.5596	1.27754	0.638769	+	0.769399i	$0.279444\pi$
$912$	0	0
$913$	89.3123	2.95581
$914$	−30.6332	−1.01325
$915$	0	0
$916$	−5.53209	−0.182785
$917$	−24.5367	−0.810273
$918$	0	0
$919$	0.885371	0.0292057	0.0146029	−	0.999893i	$-0.495352\pi$
$919$	0.885371	0.0292057	0.0146029	+	0.999893i	$0.495352\pi$
$920$	−15.9617	−0.526243
$921$	0	0
$922$	54.9505	1.80970
$923$	23.4726	0.772609
$924$	0	0
$925$	7.47708	0.245845
$926$	1.33350	0.0438214
$927$	0	0
$928$	26.0137	0.853942
$929$	−8.34973	−0.273946	−0.136973	−	0.990575i	$-0.543737\pi$
$929$	−8.34973	−0.273946	−0.136973	+	0.990575i	$0.543737\pi$
$930$	0	0
$931$	−0.261764	−0.00857895
$932$	74.9271	2.45432
$933$	0	0
$934$	−22.3773	−0.732208
$935$	0	0
$936$	0	0
$937$	27.7374	0.906142	0.453071	−	0.891474i	$-0.350328\pi$
$937$	27.7374	0.906142	0.453071	+	0.891474i	$0.350328\pi$
$938$	−33.3316	−1.08832
$939$	0	0
$940$	2.90865	0.0948698
$941$	−40.6962	−1.32666	−0.663329	−	0.748328i	$-0.730857\pi$
$941$	−40.6962	−1.32666	−0.663329	+	0.748328i	$0.730857\pi$
$942$	0	0
$943$	45.8206	1.49212
$944$	6.16160	0.200543
$945$	0	0
$946$	−17.6750	−0.574664
$947$	−39.2775	−1.27635	−0.638174	−	0.769892i	$-0.720310\pi$
$947$	−39.2775	−1.27635	−0.638174	+	0.769892i	$0.720310\pi$
$948$	0	0
$949$	9.98545	0.324141
$950$	71.5630	2.32181
$951$	0	0
$952$	0	0
$953$	−29.7943	−0.965131	−0.482565	−	0.875860i	$-0.660295\pi$
$953$	−29.7943	−0.965131	−0.482565	+	0.875860i	$0.660295\pi$
$954$	0	0
$955$	9.18067	0.297080
$956$	−41.4499	−1.34059
$957$	0	0
$958$	−24.6140	−0.795242
$959$	12.6620	0.408878
$960$	0	0
$961$	−11.0378	−0.356057
$962$	−15.6566	−0.504788
$963$	0	0
$964$	−55.4949	−1.78737
$965$	−12.4525	−0.400862
$966$	0	0
$967$	1.53384	0.0493251	0.0246625	−	0.999696i	$-0.492149\pi$
$967$	1.53384	0.0493251	0.0246625	+	0.999696i	$0.492149\pi$
$968$	72.5419	2.33158
$969$	0	0
$970$	15.2143	0.488503
$971$	32.2214	1.03404	0.517018	−	0.855975i	$-0.327042\pi$
$971$	32.2214	1.03404	0.517018	+	0.855975i	$0.327042\pi$
$972$	0	0
$973$	49.4415	1.58502
$974$	−2.68198	−0.0859363
$975$	0	0
$976$	−3.15301	−0.100925
$977$	−5.18889	−0.166007	−0.0830036	−	0.996549i	$-0.526451\pi$
$977$	−5.18889	−0.166007	−0.0830036	+	0.996549i	$0.526451\pi$
$978$	0	0
$979$	21.3114	0.681115
$980$	−0.0990242	−0.00316321
$981$	0	0
$982$	73.6136	2.34910
$983$	−34.8307	−1.11093	−0.555463	−	0.831541i	$-0.687459\pi$
$983$	−34.8307	−1.11093	−0.555463	+	0.831541i	$0.687459\pi$
$984$	0	0
$985$	−15.9371	−0.507798
$986$	0	0
$987$	0	0
$988$	−93.8020	−2.98424
$989$	8.32777	0.264808
$990$	0	0
$991$	−48.5580	−1.54250	−0.771248	−	0.636535i	$-0.780367\pi$
$991$	−48.5580	−1.54250	−0.771248	+	0.636535i	$0.780367\pi$
$992$	22.5726	0.716681
$993$	0	0
$994$	−36.5113	−1.15807
$995$	2.26088	0.0716748
$996$	0	0
$997$	−56.2576	−1.78170	−0.890849	−	0.454300i	$-0.849889\pi$
$997$	−56.2576	−1.78170	−0.890849	+	0.454300i	$0.849889\pi$
$998$	−37.0861	−1.17394
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bj.1.5	yes	6
3.2	odd	2	inner	2601.2.a.bj.1.2	✓	6
17.16	even	2		2601.2.a.bk.1.5	yes	6
51.50	odd	2		2601.2.a.bk.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2601.2.a.bj.1.2	✓	6	3.2	odd	2	inner
2601.2.a.bj.1.5	yes	6	1.1	even	1	trivial
2601.2.a.bk.1.2	yes	6	51.50	odd	2
2601.2.a.bk.1.5	yes	6	17.16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+2.31242 q^{2} +3.34730 q^{4} -0.803096 q^{5} -2.65270 q^{7} +3.11552 q^{8} +O(q^{10})\)	\(q+2.31242 q^{2} +3.34730 q^{4} -0.803096 q^{5} -2.65270 q^{7} +3.11552 q^{8} -1.85710 q^{10} -5.85526 q^{11} +3.94356 q^{13} -6.13417 q^{14} +0.509800 q^{16} -7.10607 q^{19} -2.68820 q^{20} -13.5398 q^{22} +6.37944 q^{23} -4.35504 q^{25} +9.11918 q^{26} -8.87939 q^{28} -5.14903 q^{29} -4.46791 q^{31} -5.05216 q^{32} +2.13038 q^{35} -1.71688 q^{37} -16.4322 q^{38} -2.50206 q^{40} +7.18254 q^{41} +1.30541 q^{43} -19.5993 q^{44} +14.7520 q^{46} -1.08201 q^{47} +0.0368366 q^{49} -10.0707 q^{50} +13.2003 q^{52} -9.67701 q^{53} +4.70233 q^{55} -8.26455 q^{56} -11.9067 q^{58} +12.0863 q^{59} -6.18479 q^{61} -10.3317 q^{62} -12.7023 q^{64} -3.16706 q^{65} +5.43376 q^{67} +4.92633 q^{70} +5.95212 q^{71} +2.53209 q^{73} -3.97015 q^{74} -23.7861 q^{76} +15.5323 q^{77} -6.30541 q^{79} -0.409418 q^{80} +16.6091 q^{82} -15.2534 q^{83} +3.01865 q^{86} -18.2422 q^{88} -3.63970 q^{89} -10.4611 q^{91} +21.3539 q^{92} -2.50206 q^{94} +5.70685 q^{95} -8.19253 q^{97} +0.0851818 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{4} - 18 q^{7}+O(q^{10}) \) 6 * q + 18 * q^4 - 18 * q^7	\( 6 q + 18 q^{4} - 18 q^{7} - 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} - 24 q^{22} + 24 q^{25} - 42 q^{28} - 36 q^{31} + 6 q^{37} - 66 q^{40} + 12 q^{43} - 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} - 18 q^{58} - 30 q^{61} - 24 q^{64} - 18 q^{70} + 6 q^{73} - 78 q^{76} - 42 q^{79} - 6 q^{82} + 12 q^{91} - 66 q^{94} + 6 q^{97}+O(q^{100}) \) 6 * q + 18 * q^4 - 18 * q^7 - 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 - 24 * q^22 + 24 * q^25 - 42 * q^28 - 36 * q^31 + 6 * q^37 - 66 * q^40 + 12 * q^43 - 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 - 18 * q^58 - 30 * q^61 - 24 * q^64 - 18 * q^70 + 6 * q^73 - 78 * q^76 - 42 * q^79 - 6 * q^82 + 12 * q^91 - 66 * q^94 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more