LMFDB - Embedded newform 1521.2.b.m.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$1.80194i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.m.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80194i q^{2} -1.24698 q^{4} +1.44504i q^{5} +3.44504i q^{7} +1.35690i q^{8} -2.60388 q^{10} +5.18598i q^{11} -6.20775 q^{14} -4.93900 q^{16} -0.753020 q^{17} -7.96077i q^{19} -1.80194i q^{20} -9.34481 q^{22} -2.82908 q^{23} +2.91185 q^{25} -4.29590i q^{28} +3.91185 q^{29} +4.89977i q^{31} -6.18598i q^{32} -1.35690i q^{34} -4.97823 q^{35} +6.24698i q^{37} +14.3448 q^{38} -1.96077 q^{40} +1.80194i q^{41} +7.09783 q^{43} -6.46681i q^{44} -5.09783i q^{46} -10.5526i q^{47} -4.86831 q^{49} +5.24698i q^{50} +3.08815 q^{53} -7.49396 q^{55} -4.67456 q^{56} +7.04892i q^{58} -1.87800i q^{59} +3.34481 q^{61} -8.82908 q^{62} +1.26875 q^{64} +4.54288i q^{67} +0.939001 q^{68} -8.97046i q^{70} -9.11960i q^{71} +2.95108i q^{73} -11.2567 q^{74} +9.92692i q^{76} -17.8659 q^{77} -9.43296 q^{79} -7.13706i q^{80} -3.24698 q^{82} -6.46681i q^{83} -1.08815i q^{85} +12.7899i q^{86} -7.03684 q^{88} -1.15883i q^{89} +3.52781 q^{92} +19.0151 q^{94} +11.5036 q^{95} -8.65817i q^{97} -8.77240i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56}+ \cdots + 6 q^{95}+O(q^{100}) $ 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.80194i	1.27416i	0.770797	+	0.637081i	$0.219858\pi$
$2$	1.80194i	1.27416i	−0.770797	+	0.637081i	$0.780142\pi$
$3$	0	0
$3$	0	0
$4$	−1.24698	−0.623490
$5$	1.44504i	0.646242i	0.946358	+	0.323121i	$0.104732\pi$
$5$	1.44504i	0.646242i	−0.946358	+	0.323121i	$0.895268\pi$
$6$	0	0
$7$	3.44504i	1.30210i	0.759033	+	0.651052i	$0.225672\pi$
$7$	3.44504i	1.30210i	−0.759033	+	0.651052i	$0.774328\pi$
$8$	1.35690i	0.479735i
$9$	0	0
$10$	−2.60388	−0.823418
$11$	5.18598i	1.56363i	0.623509	+	0.781816i	$0.285706\pi$
$11$	5.18598i	1.56363i	−0.623509	+	0.781816i	$0.714294\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.20775	−1.65909
$15$	0	0
$16$	−4.93900	−1.23475
$17$	−0.753020	−0.182634	−0.0913171	−	0.995822i	$-0.529108\pi$
$17$	−0.753020	−0.182634	−0.0913171	+	0.995822i	$0.529108\pi$
$18$	0	0
$19$	− 7.96077i	− 1.82633i	−0.407594	−	0.913163i	$-0.633632\pi$
$19$	− 7.96077i	− 1.82633i	0.407594	−	0.913163i	$-0.366368\pi$
$20$	− 1.80194i	− 0.402926i
$21$	0	0
$22$	−9.34481	−1.99232
$23$	−2.82908	−0.589905	−0.294952	−	0.955512i	$-0.595304\pi$
$23$	−2.82908	−0.589905	−0.294952	+	0.955512i	$0.595304\pi$
$24$	0	0
$25$	2.91185	0.582371
$26$	0	0
$27$	0	0
$28$	− 4.29590i	− 0.811848i
$29$	3.91185	0.726413	0.363207	−	0.931709i	$-0.381682\pi$
$29$	3.91185	0.726413	0.363207	+	0.931709i	$0.381682\pi$
$30$	0	0
$31$	4.89977i	0.880025i	0.897992	+	0.440013i	$0.145026\pi$
$31$	4.89977i	0.880025i	−0.897992	+	0.440013i	$0.854974\pi$
$32$	− 6.18598i	− 1.09354i
$33$	0	0
$34$	− 1.35690i	− 0.232706i
$35$	−4.97823	−0.841474
$36$	0	0
$37$	6.24698i	1.02700i	0.858090	+	0.513499i	$0.171651\pi$
$37$	6.24698i	1.02700i	−0.858090	+	0.513499i	$0.828349\pi$
$38$	14.3448	2.32704
$39$	0	0
$40$	−1.96077	−0.310025
$41$	1.80194i	0.281415i	0.990051	+	0.140708i	$0.0449378\pi$
$41$	1.80194i	0.281415i	−0.990051	+	0.140708i	$0.955062\pi$
$42$	0	0
$43$	7.09783	1.08241	0.541205	−	0.840891i	$-0.317968\pi$
$43$	7.09783	1.08241	0.541205	+	0.840891i	$0.317968\pi$
$44$	− 6.46681i	− 0.974909i
$45$	0	0
$46$	− 5.09783i	− 0.751635i
$47$	− 10.5526i	− 1.53925i	−0.638496	−	0.769625i	$-0.720443\pi$
$47$	− 10.5526i	− 1.53925i	0.638496	−	0.769625i	$-0.279557\pi$
$48$	0	0
$49$	−4.86831	−0.695473
$50$	5.24698i	0.742035i
$51$	0	0
$52$	0	0
$53$	3.08815	0.424189	0.212095	−	0.977249i	$-0.431971\pi$
$53$	3.08815	0.424189	0.212095	+	0.977249i	$0.431971\pi$
$54$	0	0
$55$	−7.49396	−1.01049
$56$	−4.67456	−0.624665
$57$	0	0
$58$	7.04892i	0.925568i
$59$	− 1.87800i	− 0.244495i	−0.992500	−	0.122248i	$-0.960990\pi$
$59$	− 1.87800i	− 0.244495i	0.992500	−	0.122248i	$-0.0390102\pi$
$60$	0	0
$61$	3.34481	0.428260	0.214130	−	0.976805i	$-0.431308\pi$
$61$	3.34481	0.428260	0.214130	+	0.976805i	$0.431308\pi$
$62$	−8.82908	−1.12129
$63$	0	0
$64$	1.26875	0.158594
$65$	0	0
$66$	0	0
$67$	4.54288i	0.555001i	0.960726	+	0.277500i	$0.0895060\pi$
$67$	4.54288i	0.555001i	−0.960726	+	0.277500i	$0.910494\pi$
$68$	0.939001	0.113871
$69$	0	0
$70$	− 8.97046i	− 1.07218i
$71$	− 9.11960i	− 1.08230i	−0.840927	−	0.541149i	$-0.817989\pi$
$71$	− 9.11960i	− 1.08230i	0.840927	−	0.541149i	$-0.182011\pi$
$72$	0	0
$73$	2.95108i	0.345398i	0.984975	+	0.172699i	$0.0552488\pi$
$73$	2.95108i	0.345398i	−0.984975	+	0.172699i	$0.944751\pi$
$74$	−11.2567	−1.30856
$75$	0	0
$76$	9.92692i	1.13870i
$77$	−17.8659	−2.03601
$78$	0	0
$79$	−9.43296	−1.06129	−0.530645	−	0.847594i	$-0.678050\pi$
$79$	−9.43296	−1.06129	−0.530645	+	0.847594i	$0.678050\pi$
$80$	− 7.13706i	− 0.797948i
$81$	0	0
$82$	−3.24698	−0.358569
$83$	− 6.46681i	− 0.709825i	−0.934900	−	0.354912i	$-0.884511\pi$
$83$	− 6.46681i	− 0.709825i	0.934900	−	0.354912i	$-0.115489\pi$
$84$	0	0
$85$	− 1.08815i	− 0.118026i
$86$	12.7899i	1.37917i
$87$	0	0
$88$	−7.03684	−0.750129
$89$	− 1.15883i	− 0.122836i	−0.998112	−	0.0614181i	$-0.980438\pi$
$89$	− 1.15883i	− 0.122836i	0.998112	−	0.0614181i	$-0.0195623\pi$
$90$	0	0
$91$	0	0
$92$	3.52781	0.367800
$93$	0	0
$94$	19.0151	1.96125
$95$	11.5036	1.18025
$96$	0	0
$97$	− 8.65817i	− 0.879104i	−0.898217	−	0.439552i	$-0.855137\pi$
$97$	− 8.65817i	− 0.879104i	0.898217	−	0.439552i	$-0.144863\pi$
$98$	− 8.77240i	− 0.886146i
$99$	0	0
$100$	−3.63102	−0.363102
$101$	−8.47650	−0.843443	−0.421722	−	0.906725i	$-0.638574\pi$
$101$	−8.47650	−0.843443	−0.421722	+	0.906725i	$0.638574\pi$
$102$	0	0
$103$	−5.64742	−0.556456	−0.278228	−	0.960515i	$-0.589747\pi$
$103$	−5.64742	−0.556456	−0.278228	+	0.960515i	$0.589747\pi$
$104$	0	0
$105$	0	0
$106$	5.56465i	0.540486i
$107$	6.73556	0.651151	0.325576	−	0.945516i	$-0.394442\pi$
$107$	6.73556	0.651151	0.325576	+	0.945516i	$0.394442\pi$
$108$	0	0
$109$	2.07606i	0.198851i	0.995045	+	0.0994255i	$0.0317005\pi$
$109$	2.07606i	0.198851i	−0.995045	+	0.0994255i	$0.968300\pi$
$110$	− 13.5036i	− 1.28752i
$111$	0	0
$112$	− 17.0151i	− 1.60777i
$113$	−6.16852	−0.580286	−0.290143	−	0.956983i	$-0.593703\pi$
$113$	−6.16852	−0.580286	−0.290143	+	0.956983i	$0.593703\pi$
$114$	0	0
$115$	− 4.08815i	− 0.381222i
$116$	−4.87800	−0.452911
$117$	0	0
$118$	3.38404	0.311526
$119$	− 2.59419i	− 0.237809i
$120$	0	0
$121$	−15.8944	−1.44495
$122$	6.02715i	0.545672i
$123$	0	0
$124$	− 6.10992i	− 0.548687i
$125$	11.4330i	1.02260i
$126$	0	0
$127$	14.2620	1.26555	0.632776	−	0.774335i	$-0.281915\pi$
$127$	14.2620	1.26555	0.632776	+	0.774335i	$0.281915\pi$
$128$	− 10.0858i	− 0.891463i
$129$	0	0
$130$	0	0
$131$	−22.6015	−1.97470	−0.987350	−	0.158554i	$-0.949317\pi$
$131$	−22.6015	−1.97470	−0.987350	+	0.158554i	$0.949317\pi$
$132$	0	0
$133$	27.4252	2.37807
$134$	−8.18598	−0.707161
$135$	0	0
$136$	− 1.02177i	− 0.0876161i
$137$	13.6353i	1.16495i	0.812850	+	0.582473i	$0.197915\pi$
$137$	13.6353i	1.16495i	−0.812850	+	0.582473i	$0.802085\pi$
$138$	0	0
$139$	17.6015	1.49294	0.746469	−	0.665420i	$-0.231748\pi$
$139$	17.6015	1.49294	0.746469	+	0.665420i	$0.231748\pi$
$140$	6.20775	0.524651
$141$	0	0
$142$	16.4330	1.37902
$143$	0	0
$144$	0	0
$145$	5.65279i	0.469439i
$146$	−5.31767	−0.440093
$147$	0	0
$148$	− 7.78986i	− 0.640322i
$149$	12.7385i	1.04358i	0.853073	+	0.521791i	$0.174736\pi$
$149$	12.7385i	1.04358i	−0.853073	+	0.521791i	$0.825264\pi$
$150$	0	0
$151$	15.6407i	1.27282i	0.771350	+	0.636412i	$0.219582\pi$
$151$	15.6407i	1.27282i	−0.771350	+	0.636412i	$0.780418\pi$
$152$	10.8019	0.876153
$153$	0	0
$154$	− 32.1933i	− 2.59421i
$155$	−7.08038	−0.568710
$156$	0	0
$157$	−0.823708	−0.0657391	−0.0328695	−	0.999460i	$-0.510465\pi$
$157$	−0.823708	−0.0657391	−0.0328695	+	0.999460i	$0.510465\pi$
$158$	− 16.9976i	− 1.35226i
$159$	0	0
$160$	8.93900	0.706690
$161$	− 9.74632i	− 0.768117i
$162$	0	0
$163$	6.26875i	0.491006i	0.969396	+	0.245503i	$0.0789532\pi$
$163$	6.26875i	0.491006i	−0.969396	+	0.245503i	$0.921047\pi$
$164$	− 2.24698i	− 0.175460i
$165$	0	0
$166$	11.6528	0.904432
$167$	7.45042i	0.576531i	0.957551	+	0.288265i	$0.0930785\pi$
$167$	7.45042i	0.576531i	−0.957551	+	0.288265i	$0.906921\pi$
$168$	0	0
$169$	0	0
$170$	1.96077	0.150384
$171$	0	0
$172$	−8.85086	−0.674871
$173$	−2.00969	−0.152794	−0.0763969	−	0.997077i	$-0.524342\pi$
$173$	−2.00969	−0.152794	−0.0763969	+	0.997077i	$0.524342\pi$
$174$	0	0
$175$	10.0315i	0.758307i
$176$	− 25.6136i	− 1.93070i
$177$	0	0
$178$	2.08815	0.156513
$179$	20.0368	1.49762	0.748812	−	0.662783i	$-0.230625\pi$
$179$	20.0368	1.49762	0.748812	+	0.662783i	$0.230625\pi$
$180$	0	0
$181$	−24.1226	−1.79302	−0.896509	−	0.443026i	$-0.853905\pi$
$181$	−24.1226	−1.79302	−0.896509	+	0.443026i	$0.853905\pi$
$182$	0	0
$183$	0	0
$184$	− 3.83877i	− 0.282998i
$185$	−9.02715	−0.663689
$186$	0	0
$187$	− 3.90515i	− 0.285573i
$188$	13.1588i	0.959707i
$189$	0	0
$190$	20.7289i	1.50383i
$191$	7.08038	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$191$	7.08038	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$192$	0	0
$193$	9.76809i	0.703122i	0.936165	+	0.351561i	$0.114349\pi$
$193$	9.76809i	0.703122i	−0.936165	+	0.351561i	$0.885651\pi$
$194$	15.6015	1.12012
$195$	0	0
$196$	6.07069	0.433621
$197$	23.4112i	1.66798i	0.551781	+	0.833989i	$0.313948\pi$
$197$	23.4112i	1.66798i	−0.551781	+	0.833989i	$0.686052\pi$
$198$	0	0
$199$	−4.02475	−0.285307	−0.142654	−	0.989773i	$-0.545563\pi$
$199$	−4.02475	−0.285307	−0.142654	+	0.989773i	$0.545563\pi$
$200$	3.95108i	0.279384i
$201$	0	0
$202$	− 15.2741i	− 1.07468i
$203$	13.4765i	0.945865i
$204$	0	0
$205$	−2.60388	−0.181863
$206$	− 10.1763i	− 0.709016i
$207$	0	0
$208$	0	0
$209$	41.2844	2.85570
$210$	0	0
$211$	−3.91185	−0.269303	−0.134652	−	0.990893i	$-0.542992\pi$
$211$	−3.91185	−0.269303	−0.134652	+	0.990893i	$0.542992\pi$
$212$	−3.85086	−0.264478
$213$	0	0
$214$	12.1371i	0.829673i
$215$	10.2567i	0.699499i
$216$	0	0
$217$	−16.8799	−1.14588
$218$	−3.74094	−0.253368
$219$	0	0
$220$	9.34481	0.630027
$221$	0	0
$222$	0	0
$223$	− 7.44935i	− 0.498846i	−0.968395	−	0.249423i	$-0.919759\pi$
$223$	− 7.44935i	− 0.498846i	0.968395	−	0.249423i	$-0.0802409\pi$
$224$	21.3110	1.42390
$225$	0	0
$226$	− 11.1153i	− 0.739378i
$227$	− 21.2500i	− 1.41041i	−0.709004	−	0.705205i	$-0.750855\pi$
$227$	− 21.2500i	− 1.41041i	0.709004	−	0.705205i	$-0.249145\pi$
$228$	0	0
$229$	9.29590i	0.614290i	0.951663	+	0.307145i	$0.0993737\pi$
$229$	9.29590i	0.614290i	−0.951663	+	0.307145i	$0.900626\pi$
$230$	7.36658	0.485738
$231$	0	0
$232$	5.30798i	0.348486i
$233$	16.2107	1.06200	0.531000	−	0.847372i	$-0.321816\pi$
$233$	16.2107	1.06200	0.531000	+	0.847372i	$0.321816\pi$
$234$	0	0
$235$	15.2489	0.994728
$236$	2.34183i	0.152440i
$237$	0	0
$238$	4.67456	0.303007
$239$	− 13.5090i	− 0.873826i	−0.899504	−	0.436913i	$-0.856072\pi$
$239$	− 13.5090i	− 0.873826i	0.899504	−	0.436913i	$-0.143928\pi$
$240$	0	0
$241$	6.26875i	0.403806i	0.979406	+	0.201903i	$0.0647125\pi$
$241$	6.26875i	0.403806i	−0.979406	+	0.201903i	$0.935287\pi$
$242$	− 28.6407i	− 1.84109i
$243$	0	0
$244$	−4.17092	−0.267015
$245$	− 7.03492i	− 0.449444i
$246$	0	0
$247$	0	0
$248$	−6.64848	−0.422179
$249$	0	0
$250$	−20.6015	−1.30295
$251$	−0.753020	−0.0475302	−0.0237651	−	0.999718i	$-0.507565\pi$
$251$	−0.753020	−0.0475302	−0.0237651	+	0.999718i	$0.507565\pi$
$252$	0	0
$253$	− 14.6716i	− 0.922394i
$254$	25.6993i	1.61252i
$255$	0	0
$256$	20.7114	1.29446
$257$	19.7265	1.23050	0.615252	−	0.788331i	$-0.289054\pi$
$257$	19.7265	1.23050	0.615252	+	0.788331i	$0.289054\pi$
$258$	0	0
$259$	−21.5211	−1.33726
$260$	0	0
$261$	0	0
$262$	− 40.7265i	− 2.51609i
$263$	17.6093	1.08583	0.542917	−	0.839787i	$-0.317320\pi$
$263$	17.6093	1.08583	0.542917	+	0.839787i	$0.317320\pi$
$264$	0	0
$265$	4.46250i	0.274129i
$266$	49.4185i	3.03004i
$267$	0	0
$268$	− 5.66487i	− 0.346037i
$269$	16.3870	0.999135	0.499567	−	0.866275i	$-0.333492\pi$
$269$	16.3870	0.999135	0.499567	+	0.866275i	$0.333492\pi$
$270$	0	0
$271$	− 0.795233i	− 0.0483070i	−0.999708	−	0.0241535i	$-0.992311\pi$
$271$	− 0.795233i	− 0.0483070i	0.999708	−	0.0241535i	$-0.00768904\pi$
$272$	3.71917	0.225508
$273$	0	0
$274$	−24.5700	−1.48433
$275$	15.1008i	0.910614i
$276$	0	0
$277$	4.83340	0.290411	0.145205	−	0.989402i	$-0.453616\pi$
$277$	4.83340	0.290411	0.145205	+	0.989402i	$0.453616\pi$
$278$	31.7168i	1.90225i
$279$	0	0
$280$	− 6.75494i	− 0.403685i
$281$	− 18.7748i	− 1.12001i	−0.828489	−	0.560005i	$-0.810799\pi$
$281$	− 18.7748i	− 1.12001i	0.828489	−	0.560005i	$-0.189201\pi$
$282$	0	0
$283$	7.91723	0.470631	0.235315	−	0.971919i	$-0.424388\pi$
$283$	7.91723	0.470631	0.235315	+	0.971919i	$0.424388\pi$
$284$	11.3720i	0.674802i
$285$	0	0
$286$	0	0
$287$	−6.20775	−0.366432
$288$	0	0
$289$	−16.4330	−0.966645
$290$	−10.1860	−0.598141
$291$	0	0
$292$	− 3.67994i	− 0.215352i
$293$	− 6.57912i	− 0.384356i	−0.981360	−	0.192178i	$-0.938445\pi$
$293$	− 6.57912i	− 0.384356i	0.981360	−	0.192178i	$-0.0615552\pi$
$294$	0	0
$295$	2.71379	0.158003
$296$	−8.47650	−0.492687
$297$	0	0
$298$	−22.9541	−1.32969
$299$	0	0
$300$	0	0
$301$	24.4523i	1.40941i
$302$	−28.1836	−1.62178
$303$	0	0
$304$	39.3183i	2.25506i
$305$	4.83340i	0.276759i
$306$	0	0
$307$	− 24.8649i	− 1.41911i	−0.704649	−	0.709556i	$-0.748895\pi$
$307$	− 24.8649i	− 1.41911i	0.704649	−	0.709556i	$-0.251105\pi$
$308$	22.2784	1.26943
$309$	0	0
$310$	− 12.7584i	− 0.724628i
$311$	−17.0804	−0.968539	−0.484270	−	0.874919i	$-0.660915\pi$
$311$	−17.0804	−0.968539	−0.484270	+	0.874919i	$0.660915\pi$
$312$	0	0
$313$	15.6974	0.887269	0.443635	−	0.896208i	$-0.353689\pi$
$313$	15.6974	0.887269	0.443635	+	0.896208i	$0.353689\pi$
$314$	− 1.48427i	− 0.0837622i
$315$	0	0
$316$	11.7627	0.661704
$317$	32.7821i	1.84123i	0.390477	+	0.920613i	$0.372310\pi$
$317$	32.7821i	1.84123i	−0.390477	+	0.920613i	$0.627690\pi$
$318$	0	0
$319$	20.2868i	1.13584i
$320$	1.83340i	0.102490i
$321$	0	0
$322$	17.5623	0.978706
$323$	5.99462i	0.333550i
$324$	0	0
$325$	0	0
$326$	−11.2959	−0.625622
$327$	0	0
$328$	−2.44504	−0.135005
$329$	36.3540	2.00426
$330$	0	0
$331$	29.1618i	1.60288i	0.598076	+	0.801439i	$0.295932\pi$
$331$	29.1618i	1.60288i	−0.598076	+	0.801439i	$0.704068\pi$
$332$	8.06398i	0.442569i
$333$	0	0
$334$	−13.4252	−0.734594
$335$	−6.56465	−0.358665
$336$	0	0
$337$	33.2911	1.81348	0.906741	−	0.421688i	$-0.138562\pi$
$337$	33.2911	1.81348	0.906741	+	0.421688i	$0.138562\pi$
$338$	0	0
$339$	0	0
$340$	1.35690i	0.0735880i
$341$	−25.4101	−1.37604
$342$	0	0
$343$	7.34375i	0.396525i
$344$	9.63102i	0.519270i
$345$	0	0
$346$	− 3.62133i	− 0.194684i
$347$	0.873690	0.0469022	0.0234511	−	0.999725i	$-0.492535\pi$
$347$	0.873690	0.0469022	0.0234511	+	0.999725i	$0.492535\pi$
$348$	0	0
$349$	− 3.23191i	− 0.173000i	−0.996252	−	0.0865002i	$-0.972432\pi$
$349$	− 3.23191i	− 0.173000i	0.996252	−	0.0865002i	$-0.0275683\pi$
$350$	−18.0761	−0.966206
$351$	0	0
$352$	32.0804	1.70989
$353$	8.14675i	0.433608i	0.976215	+	0.216804i	$0.0695632\pi$
$353$	8.14675i	0.433608i	−0.976215	+	0.216804i	$0.930437\pi$
$354$	0	0
$355$	13.1782	0.699427
$356$	1.44504i	0.0765871i
$357$	0	0
$358$	36.1051i	1.90822i
$359$	− 2.64071i	− 0.139371i	−0.997569	−	0.0696857i	$-0.977800\pi$
$359$	− 2.64071i	− 0.139371i	0.997569	−	0.0696857i	$-0.0221996\pi$
$360$	0	0
$361$	−44.3739	−2.33547
$362$	− 43.4674i	− 2.28460i
$363$	0	0
$364$	0	0
$365$	−4.26444	−0.223211
$366$	0	0
$367$	−2.90408	−0.151592	−0.0757960	−	0.997123i	$-0.524150\pi$
$367$	−2.90408	−0.151592	−0.0757960	+	0.997123i	$0.524150\pi$
$368$	13.9729	0.728385
$369$	0	0
$370$	− 16.2664i	− 0.845648i
$371$	10.6388i	0.552339i
$372$	0	0
$373$	−8.39852	−0.434859	−0.217429	−	0.976076i	$-0.569767\pi$
$373$	−8.39852	−0.434859	−0.217429	+	0.976076i	$0.569767\pi$
$374$	7.03684	0.363866
$375$	0	0
$376$	14.3187	0.738432
$377$	0	0
$378$	0	0
$379$	− 15.7482i	− 0.808932i	−0.914553	−	0.404466i	$-0.867457\pi$
$379$	− 15.7482i	− 0.808932i	0.914553	−	0.404466i	$-0.132543\pi$
$380$	−14.3448	−0.735873
$381$	0	0
$382$	12.7584i	0.652776i
$383$	12.7385i	0.650909i	0.945558	+	0.325455i	$0.105517\pi$
$383$	12.7385i	0.650909i	−0.945558	+	0.325455i	$0.894483\pi$
$384$	0	0
$385$	− 25.8170i	− 1.31576i
$386$	−17.6015	−0.895892
$387$	0	0
$388$	10.7966i	0.548112i
$389$	−0.310371	−0.0157365	−0.00786823	−	0.999969i	$-0.502505\pi$
$389$	−0.310371	−0.0157365	−0.00786823	+	0.999969i	$0.502505\pi$
$390$	0	0
$391$	2.13036	0.107737
$392$	− 6.60579i	− 0.333643i
$393$	0	0
$394$	−42.1855	−2.12528
$395$	− 13.6310i	− 0.685851i
$396$	0	0
$397$	1.49098i	0.0748299i	0.999300	+	0.0374150i	$0.0119123\pi$
$397$	1.49098i	0.0748299i	−0.999300	+	0.0374150i	$0.988088\pi$
$398$	− 7.25236i	− 0.363528i
$399$	0	0
$400$	−14.3817	−0.719083
$401$	23.8334i	1.19018i	0.803658	+	0.595092i	$0.202884\pi$
$401$	23.8334i	1.19018i	−0.803658	+	0.595092i	$0.797116\pi$
$402$	0	0
$403$	0	0
$404$	10.5700	0.525878
$405$	0	0
$406$	−24.2838	−1.20519
$407$	−32.3967	−1.60585
$408$	0	0
$409$	− 4.26742i	− 0.211010i	−0.994419	−	0.105505i	$-0.966354\pi$
$409$	− 4.26742i	− 0.211010i	0.994419	−	0.105505i	$-0.0336460\pi$
$410$	− 4.69202i	− 0.231722i
$411$	0	0
$412$	7.04221	0.346945
$413$	6.46980	0.318358
$414$	0	0
$415$	9.34481	0.458719
$416$	0	0
$417$	0	0
$418$	74.3919i	3.63863i
$419$	−29.6896	−1.45043	−0.725217	−	0.688521i	$-0.758260\pi$
$419$	−29.6896	−1.45043	−0.725217	+	0.688521i	$0.758260\pi$
$420$	0	0
$421$	− 29.3991i	− 1.43282i	−0.697677	−	0.716412i	$-0.745783\pi$
$421$	− 29.3991i	− 1.43282i	0.697677	−	0.716412i	$-0.254217\pi$
$422$	− 7.04892i	− 0.343136i
$423$	0	0
$424$	4.19029i	0.203499i
$425$	−2.19269	−0.106361
$426$	0	0
$427$	11.5230i	0.557638i
$428$	−8.39911	−0.405986
$429$	0	0
$430$	−18.4819	−0.891275
$431$	− 33.0562i	− 1.59226i	−0.605124	−	0.796131i	$-0.706877\pi$
$431$	− 33.0562i	− 1.59226i	0.605124	−	0.796131i	$-0.293123\pi$
$432$	0	0
$433$	29.2664	1.40645	0.703226	−	0.710967i	$-0.251742\pi$
$433$	29.2664	1.40645	0.703226	+	0.710967i	$0.251742\pi$
$434$	− 30.4166i	− 1.46004i
$435$	0	0
$436$	− 2.58881i	− 0.123982i
$437$	22.5217i	1.07736i
$438$	0	0
$439$	2.13169	0.101740	0.0508699	−	0.998705i	$-0.483801\pi$
$439$	2.13169	0.101740	0.0508699	+	0.998705i	$0.483801\pi$
$440$	− 10.1685i	− 0.484765i
$441$	0	0
$442$	0	0
$443$	22.9922	1.09239	0.546197	−	0.837657i	$-0.316075\pi$
$443$	22.9922	1.09239	0.546197	+	0.837657i	$0.316075\pi$
$444$	0	0
$445$	1.67456	0.0793819
$446$	13.4233	0.635610
$447$	0	0
$448$	4.37090i	0.206505i
$449$	12.9379i	0.610579i	0.952260	+	0.305289i	$0.0987532\pi$
$449$	12.9379i	0.610579i	−0.952260	+	0.305289i	$0.901247\pi$
$450$	0	0
$451$	−9.34481	−0.440030
$452$	7.69202	0.361802
$453$	0	0
$454$	38.2911	1.79709
$455$	0	0
$456$	0	0
$457$	4.85325i	0.227025i	0.993537	+	0.113513i	$0.0362103\pi$
$457$	4.85325i	0.227025i	−0.993537	+	0.113513i	$0.963790\pi$
$458$	−16.7506	−0.782705
$459$	0	0
$460$	5.09783i	0.237688i
$461$	18.8345i	0.877208i	0.898680	+	0.438604i	$0.144527\pi$
$461$	18.8345i	0.877208i	−0.898680	+	0.438604i	$0.855473\pi$
$462$	0	0
$463$	− 22.8767i	− 1.06317i	−0.847005	−	0.531585i	$-0.821597\pi$
$463$	− 22.8767i	− 1.06317i	0.847005	−	0.531585i	$-0.178403\pi$
$464$	−19.3207	−0.896939
$465$	0	0
$466$	29.2107i	1.35316i
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	0	0
$469$	−15.6504	−0.722668
$470$	27.4776i	1.26745i
$471$	0	0
$472$	2.54825	0.117293
$473$	36.8092i	1.69249i
$474$	0	0
$475$	− 23.1806i	− 1.06360i
$476$	3.23490i	0.148271i
$477$	0	0
$478$	24.3424	1.11340
$479$	38.0901i	1.74038i	0.492717	+	0.870190i	$0.336004\pi$
$479$	38.0901i	1.74038i	−0.492717	+	0.870190i	$0.663996\pi$
$480$	0	0
$481$	0	0
$482$	−11.2959	−0.514514
$483$	0	0
$484$	19.8200	0.900909
$485$	12.5114	0.568114
$486$	0	0
$487$	21.2500i	0.962928i	0.876466	+	0.481464i	$0.159895\pi$
$487$	21.2500i	0.962928i	−0.876466	+	0.481464i	$0.840105\pi$
$488$	4.53856i	0.205451i
$489$	0	0
$490$	12.6765	0.572665
$491$	−6.35019	−0.286580	−0.143290	−	0.989681i	$-0.545768\pi$
$491$	−6.35019	−0.286580	−0.143290	+	0.989681i	$0.545768\pi$
$492$	0	0
$493$	−2.94571	−0.132668
$494$	0	0
$495$	0	0
$496$	− 24.2000i	− 1.08661i
$497$	31.4174	1.40926
$498$	0	0
$499$	4.65087i	0.208202i	0.994567	+	0.104101i	$0.0331965\pi$
$499$	4.65087i	0.208202i	−0.994567	+	0.104101i	$0.966804\pi$
$500$	− 14.2567i	− 0.637578i
$501$	0	0
$502$	− 1.35690i	− 0.0605612i
$503$	−15.4752	−0.690004	−0.345002	−	0.938602i	$-0.612122\pi$
$503$	−15.4752	−0.690004	−0.345002	+	0.938602i	$0.612122\pi$
$504$	0	0
$505$	− 12.2489i	− 0.545069i
$506$	26.4373	1.17528
$507$	0	0
$508$	−17.7845	−0.789059
$509$	− 20.5047i	− 0.908855i	−0.890784	−	0.454428i	$-0.849844\pi$
$509$	− 20.5047i	− 0.908855i	0.890784	−	0.454428i	$-0.150156\pi$
$510$	0	0
$511$	−10.1666	−0.449744
$512$	17.1491i	0.757892i
$513$	0	0
$514$	35.5459i	1.56786i
$515$	− 8.16075i	− 0.359606i
$516$	0	0
$517$	54.7254	2.40682
$518$	− 38.7797i	− 1.70388i
$519$	0	0
$520$	0	0
$521$	42.0267	1.84122	0.920611	−	0.390481i	$-0.127691\pi$
$521$	42.0267	1.84122	0.920611	+	0.390481i	$0.127691\pi$
$522$	0	0
$523$	−29.9885	−1.31131	−0.655653	−	0.755062i	$-0.727607\pi$
$523$	−29.9885	−1.31131	−0.655653	+	0.755062i	$0.727607\pi$
$524$	28.1836	1.23121
$525$	0	0
$526$	31.7308i	1.38353i
$527$	− 3.68963i	− 0.160723i
$528$	0	0
$529$	−14.9963	−0.652012
$530$	−8.04115	−0.349285
$531$	0	0
$532$	−34.1987	−1.48270
$533$	0	0
$534$	0	0
$535$	9.73317i	0.420802i
$536$	−6.16421	−0.266253
$537$	0	0
$538$	29.5284i	1.27306i
$539$	− 25.2470i	− 1.08746i
$540$	0	0
$541$	36.3803i	1.56411i	0.623208	+	0.782056i	$0.285829\pi$
$541$	36.3803i	1.56411i	−0.623208	+	0.782056i	$0.714171\pi$
$542$	1.43296	0.0615509
$543$	0	0
$544$	4.65817i	0.199717i
$545$	−3.00000	−0.128506
$546$	0	0
$547$	−25.8159	−1.10381	−0.551905	−	0.833907i	$-0.686099\pi$
$547$	−25.8159	−1.10381	−0.551905	+	0.833907i	$0.686099\pi$
$548$	− 17.0030i	− 0.726331i
$549$	0	0
$550$	−27.2107	−1.16027
$551$	− 31.1414i	− 1.32667i
$552$	0	0
$553$	− 32.4969i	− 1.38191i
$554$	8.70948i	0.370030i
$555$	0	0
$556$	−21.9487	−0.930832
$557$	− 17.9903i	− 0.762274i	−0.924519	−	0.381137i	$-0.875533\pi$
$557$	− 17.9903i	− 0.762274i	0.924519	−	0.381137i	$-0.124467\pi$
$558$	0	0
$559$	0	0
$560$	24.5875	1.03901
$561$	0	0
$562$	33.8310	1.42707
$563$	−39.1323	−1.64923	−0.824614	−	0.565695i	$-0.808608\pi$
$563$	−39.1323	−1.64923	−0.824614	+	0.565695i	$0.808608\pi$
$564$	0	0
$565$	− 8.91377i	− 0.375005i
$566$	14.2664i	0.599660i
$567$	0	0
$568$	12.3744	0.519216
$569$	30.6002	1.28283	0.641413	−	0.767196i	$-0.278349\pi$
$569$	30.6002	1.28283	0.641413	+	0.767196i	$0.278349\pi$
$570$	0	0
$571$	2.96184	0.123949	0.0619745	−	0.998078i	$-0.480260\pi$
$571$	2.96184	0.123949	0.0619745	+	0.998078i	$0.480260\pi$
$572$	0	0
$573$	0	0
$574$	− 11.1860i	− 0.466894i
$575$	−8.23788	−0.343543
$576$	0	0
$577$	0.819396i	0.0341119i	0.999855	+	0.0170560i	$0.00542934\pi$
$577$	0.819396i	0.0341119i	−0.999855	+	0.0170560i	$0.994571\pi$
$578$	− 29.6112i	− 1.23166i
$579$	0	0
$580$	− 7.04892i	− 0.292690i
$581$	22.2784	0.924265
$582$	0	0
$583$	16.0151i	0.663276i
$584$	−4.00431	−0.165700
$585$	0	0
$586$	11.8552	0.489732
$587$	− 31.7995i	− 1.31251i	−0.754540	−	0.656254i	$-0.772140\pi$
$587$	− 31.7995i	− 1.31251i	0.754540	−	0.656254i	$-0.227860\pi$
$588$	0	0
$589$	39.0060	1.60721
$590$	4.89008i	0.201322i
$591$	0	0
$592$	− 30.8538i	− 1.26808i
$593$	4.26337i	0.175076i	0.996161	+	0.0875379i	$0.0278999\pi$
$593$	4.26337i	0.175076i	−0.996161	+	0.0875379i	$0.972100\pi$
$594$	0	0
$595$	3.74871	0.153682
$596$	− 15.8847i	− 0.650663i
$597$	0	0
$598$	0	0
$599$	−24.7278	−1.01035	−0.505175	−	0.863017i	$-0.668572\pi$
$599$	−24.7278	−1.01035	−0.505175	+	0.863017i	$0.668572\pi$
$600$	0	0
$601$	6.82371	0.278345	0.139172	−	0.990268i	$-0.455556\pi$
$601$	6.82371	0.278345	0.139172	+	0.990268i	$0.455556\pi$
$602$	−44.0616	−1.79582
$603$	0	0
$604$	− 19.5036i	− 0.793592i
$605$	− 22.9681i	− 0.933785i
$606$	0	0
$607$	31.9963	1.29869	0.649344	−	0.760494i	$-0.275043\pi$
$607$	31.9963	1.29869	0.649344	+	0.760494i	$0.275043\pi$
$608$	−49.2452	−1.99716
$609$	0	0
$610$	−8.70948	−0.352637
$611$	0	0
$612$	0	0
$613$	− 33.5875i	− 1.35659i	−0.734792	−	0.678293i	$-0.762720\pi$
$613$	− 33.5875i	− 1.35659i	0.734792	−	0.678293i	$-0.237280\pi$
$614$	44.8049	1.80818
$615$	0	0
$616$	− 24.2422i	− 0.976746i
$617$	− 26.5870i	− 1.07035i	−0.844740	−	0.535176i	$-0.820245\pi$
$617$	− 26.5870i	− 1.07035i	0.844740	−	0.535176i	$-0.179755\pi$
$618$	0	0
$619$	− 9.17928i	− 0.368946i	−0.982838	−	0.184473i	$-0.940942\pi$
$619$	− 9.17928i	− 0.368946i	0.982838	−	0.184473i	$-0.0590579\pi$
$620$	8.82908	0.354585
$621$	0	0
$622$	− 30.7778i	− 1.23408i
$623$	3.99223	0.159945
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	28.2857i	1.13053i
$627$	0	0
$628$	1.02715	0.0409876
$629$	− 4.70410i	− 0.187565i
$630$	0	0
$631$	17.0043i	0.676931i	0.940979	+	0.338465i	$0.109908\pi$
$631$	17.0043i	0.676931i	−0.940979	+	0.338465i	$0.890092\pi$
$632$	− 12.7995i	− 0.509139i
$633$	0	0
$634$	−59.0713	−2.34602
$635$	20.6093i	0.817853i
$636$	0	0
$637$	0	0
$638$	−36.5555	−1.44725
$639$	0	0
$640$	14.5743	0.576101
$641$	−21.6649	−0.855711	−0.427856	−	0.903847i	$-0.640731\pi$
$641$	−21.6649	−0.855711	−0.427856	+	0.903847i	$0.640731\pi$
$642$	0	0
$643$	− 9.35557i	− 0.368948i	−0.982837	−	0.184474i	$-0.940942\pi$
$643$	− 9.35557i	− 0.368948i	0.982837	−	0.184474i	$-0.0590581\pi$
$644$	12.1535i	0.478913i
$645$	0	0
$646$	−10.8019	−0.424997
$647$	−0.702775	−0.0276289	−0.0138145	−	0.999905i	$-0.504397\pi$
$647$	−0.702775	−0.0276289	−0.0138145	+	0.999905i	$0.504397\pi$
$648$	0	0
$649$	9.73928	0.382300
$650$	0	0
$651$	0	0
$652$	− 7.81700i	− 0.306137i
$653$	37.3411	1.46127	0.730635	−	0.682768i	$-0.239224\pi$
$653$	37.3411	1.46127	0.730635	+	0.682768i	$0.239224\pi$
$654$	0	0
$655$	− 32.6601i	− 1.27614i
$656$	− 8.89977i	− 0.347478i
$657$	0	0
$658$	65.5077i	2.55376i
$659$	0.735562	0.0286534	0.0143267	−	0.999897i	$-0.495440\pi$
$659$	0.735562	0.0286534	0.0143267	+	0.999897i	$0.495440\pi$
$660$	0	0
$661$	13.8485i	0.538643i	0.963050	+	0.269321i	$0.0867994\pi$
$661$	13.8485i	0.538643i	−0.963050	+	0.269321i	$0.913201\pi$
$662$	−52.5478	−2.04233
$663$	0	0
$664$	8.77479	0.340528
$665$	39.6305i	1.53681i
$666$	0	0
$667$	−11.0670	−0.428515
$668$	− 9.29052i	− 0.359461i
$669$	0	0
$670$	− 11.8291i	− 0.456997i
$671$	17.3461i	0.669640i
$672$	0	0
$673$	6.35019	0.244782	0.122391	−	0.992482i	$-0.460944\pi$
$673$	6.35019	0.244782	0.122391	+	0.992482i	$0.460944\pi$
$674$	59.9885i	2.31067i
$675$	0	0
$676$	0	0
$677$	−33.7241	−1.29612	−0.648061	−	0.761589i	$-0.724420\pi$
$677$	−33.7241	−1.29612	−0.648061	+	0.761589i	$0.724420\pi$
$678$	0	0
$679$	29.8278	1.14468
$680$	1.47650	0.0566212
$681$	0	0
$682$	− 45.7875i	− 1.75329i
$683$	− 19.2687i	− 0.737298i	−0.929569	−	0.368649i	$-0.879820\pi$
$683$	− 19.2687i	− 0.737298i	0.929569	−	0.368649i	$-0.120180\pi$
$684$	0	0
$685$	−19.7036	−0.752837
$686$	−13.2330	−0.505237
$687$	0	0
$688$	−35.0562	−1.33651
$689$	0	0
$690$	0	0
$691$	39.4010i	1.49889i	0.662069	+	0.749443i	$0.269679\pi$
$691$	39.4010i	1.49889i	−0.662069	+	0.749443i	$0.730321\pi$
$692$	2.50604	0.0952654
$693$	0	0
$694$	1.57434i	0.0597610i
$695$	25.4349i	0.964800i
$696$	0	0
$697$	− 1.35690i	− 0.0513961i
$698$	5.82371	0.220431
$699$	0	0
$700$	− 12.5090i	− 0.472797i
$701$	18.3985	0.694902	0.347451	−	0.937698i	$-0.387047\pi$
$701$	18.3985	0.694902	0.347451	+	0.937698i	$0.387047\pi$
$702$	0	0
$703$	49.7308	1.87563
$704$	6.57971i	0.247982i
$705$	0	0
$706$	−14.6799	−0.552487
$707$	− 29.2019i	− 1.09825i
$708$	0	0
$709$	− 38.4553i	− 1.44422i	−0.691778	−	0.722110i	$-0.743172\pi$
$709$	− 38.4553i	− 1.44422i	0.691778	−	0.722110i	$-0.256828\pi$
$710$	23.7463i	0.891183i
$711$	0	0
$712$	1.57242	0.0589288
$713$	− 13.8619i	− 0.519131i
$714$	0	0
$715$	0	0
$716$	−24.9855	−0.933753
$717$	0	0
$718$	4.75840	0.177582
$719$	48.4999	1.80874	0.904371	−	0.426747i	$-0.140341\pi$
$719$	48.4999	1.80874	0.904371	+	0.426747i	$0.140341\pi$
$720$	0	0
$721$	− 19.4556i	− 0.724564i
$722$	− 79.9590i	− 2.97576i
$723$	0	0
$724$	30.0804	1.11793
$725$	11.3907	0.423042
$726$	0	0
$727$	−19.0344	−0.705948	−0.352974	−	0.935633i	$-0.614830\pi$
$727$	−19.0344	−0.705948	−0.352974	+	0.935633i	$0.614830\pi$
$728$	0	0
$729$	0	0
$730$	− 7.68425i	− 0.284407i
$731$	−5.34481	−0.197685
$732$	0	0
$733$	− 24.6213i	− 0.909410i	−0.890642	−	0.454705i	$-0.849745\pi$
$733$	− 24.6213i	− 0.909410i	0.890642	−	0.454705i	$-0.150255\pi$
$734$	− 5.23298i	− 0.193153i
$735$	0	0
$736$	17.5007i	0.645083i
$737$	−23.5593	−0.867817
$738$	0	0
$739$	− 44.5115i	− 1.63738i	−0.574233	−	0.818692i	$-0.694700\pi$
$739$	− 44.5115i	− 1.63738i	0.574233	−	0.818692i	$-0.305300\pi$
$740$	11.2567	0.413803
$741$	0	0
$742$	−19.1704	−0.703769
$743$	10.4112i	0.381950i	0.981595	+	0.190975i	$0.0611649\pi$
$743$	10.4112i	0.381950i	−0.981595	+	0.190975i	$0.938835\pi$
$744$	0	0
$745$	−18.4077	−0.674407
$746$	− 15.1336i	− 0.554081i
$747$	0	0
$748$	4.86964i	0.178052i
$749$	23.2043i	0.847866i
$750$	0	0
$751$	−1.69979	−0.0620263	−0.0310131	−	0.999519i	$-0.509873\pi$
$751$	−1.69979	−0.0620263	−0.0310131	+	0.999519i	$0.509873\pi$
$752$	52.1191i	1.90059i
$753$	0	0
$754$	0	0
$755$	−22.6015	−0.822552
$756$	0	0
$757$	27.4252	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$757$	27.4252	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$758$	28.3773	1.03071
$759$	0	0
$760$	15.6093i	0.566207i
$761$	5.02608i	0.182195i	0.995842	+	0.0910977i	$0.0290375\pi$
$761$	5.02608i	0.182195i	−0.995842	+	0.0910977i	$0.970962\pi$
$762$	0	0
$763$	−7.15213	−0.258924
$764$	−8.82908	−0.319425
$765$	0	0
$766$	−22.9541	−0.829364
$767$	0	0
$768$	0	0
$769$	− 42.4456i	− 1.53063i	−0.643657	−	0.765314i	$-0.722584\pi$
$769$	− 42.4456i	− 1.53063i	0.643657	−	0.765314i	$-0.277416\pi$
$770$	46.5206	1.67649
$771$	0	0
$772$	− 12.1806i	− 0.438390i
$773$	26.3593i	0.948078i	0.880504	+	0.474039i	$0.157204\pi$
$773$	26.3593i	0.948078i	−0.880504	+	0.474039i	$0.842796\pi$
$774$	0	0
$775$	14.2674i	0.512501i
$776$	11.7482	0.421737
$777$	0	0
$778$	− 0.559270i	− 0.0200508i
$779$	14.3448	0.513956
$780$	0	0
$781$	47.2941	1.69232
$782$	3.83877i	0.137274i
$783$	0	0
$784$	24.0446	0.858736
$785$	− 1.19029i	− 0.0424834i
$786$	0	0
$787$	17.1424i	0.611062i	0.952182	+	0.305531i	$0.0988340\pi$
$787$	17.1424i	0.611062i	−0.952182	+	0.305531i	$0.901166\pi$
$788$	− 29.1933i	− 1.03997i
$789$	0	0
$790$	24.5623	0.873886
$791$	− 21.2508i	− 0.755592i
$792$	0	0
$793$	0	0
$794$	−2.68664	−0.0953455
$795$	0	0
$796$	5.01879	0.177886
$797$	30.1629	1.06842	0.534212	−	0.845351i	$-0.320608\pi$
$797$	30.1629	1.06842	0.534212	+	0.845351i	$0.320608\pi$
$798$	0	0
$799$	7.94630i	0.281120i
$800$	− 18.0127i	− 0.636844i
$801$	0	0
$802$	−42.9463	−1.51649
$803$	−15.3043	−0.540076
$804$	0	0
$805$	14.0838	0.496390
$806$	0	0
$807$	0	0
$808$	− 11.5017i	− 0.404629i
$809$	29.8504	1.04948	0.524742	−	0.851261i	$-0.324162\pi$
$809$	29.8504	1.04948	0.524742	+	0.851261i	$0.324162\pi$
$810$	0	0
$811$	− 47.7362i	− 1.67624i	−0.545484	−	0.838122i	$-0.683654\pi$
$811$	− 47.7362i	− 1.67624i	0.545484	−	0.838122i	$-0.316346\pi$
$812$	− 16.8049i	− 0.589737i
$813$	0	0
$814$	− 58.3769i	− 2.04611i
$815$	−9.05861	−0.317309
$816$	0	0
$817$	− 56.5042i	− 1.97683i
$818$	7.68963	0.268862
$819$	0	0
$820$	3.24698	0.113389
$821$	− 17.9299i	− 0.625758i	−0.949793	−	0.312879i	$-0.898707\pi$
$821$	− 17.9299i	− 0.625758i	0.949793	−	0.312879i	$-0.101293\pi$
$822$	0	0
$823$	−54.3196	−1.89346	−0.946731	−	0.322026i	$-0.895636\pi$
$823$	−54.3196	−1.89346	−0.946731	+	0.322026i	$0.895636\pi$
$824$	− 7.66296i	− 0.266952i
$825$	0	0
$826$	11.6582i	0.405640i
$827$	− 49.1041i	− 1.70752i	−0.520670	−	0.853758i	$-0.674318\pi$
$827$	− 49.1041i	− 1.70752i	0.520670	−	0.853758i	$-0.325682\pi$
$828$	0	0
$829$	7.35796	0.255553	0.127776	−	0.991803i	$-0.459216\pi$
$829$	7.35796	0.255553	0.127776	+	0.991803i	$0.459216\pi$
$830$	16.8388i	0.584482i
$831$	0	0
$832$	0	0
$833$	3.66594	0.127017
$834$	0	0
$835$	−10.7662	−0.372579
$836$	−51.4808	−1.78050
$837$	0	0
$838$	− 53.4989i	− 1.84809i
$839$	36.5013i	1.26016i	0.776529	+	0.630082i	$0.216979\pi$
$839$	36.5013i	1.26016i	−0.776529	+	0.630082i	$0.783021\pi$
$840$	0	0
$841$	−13.6974	−0.472324
$842$	52.9754	1.82565
$843$	0	0
$844$	4.87800	0.167908
$845$	0	0
$846$	0	0
$847$	− 54.7569i	− 1.88147i
$848$	−15.2524	−0.523768
$849$	0	0
$850$	− 3.95108i	− 0.135521i
$851$	− 17.6732i	− 0.605831i
$852$	0	0
$853$	− 9.73855i	− 0.333441i	−0.986004	−	0.166721i	$-0.946682\pi$
$853$	− 9.73855i	− 0.333441i	0.986004	−	0.166721i	$-0.0533178\pi$
$854$	−20.7638	−0.710522
$855$	0	0
$856$	9.13946i	0.312380i
$857$	−15.2030	−0.519323	−0.259662	−	0.965700i	$-0.583611\pi$
$857$	−15.2030	−0.519323	−0.259662	+	0.965700i	$0.583611\pi$
$858$	0	0
$859$	−31.9885	−1.09143	−0.545717	−	0.837970i	$-0.683743\pi$
$859$	−31.9885	−1.09143	−0.545717	+	0.837970i	$0.683743\pi$
$860$	− 12.7899i	− 0.436130i
$861$	0	0
$862$	59.5652	2.02880
$863$	− 35.2905i	− 1.20130i	−0.799511	−	0.600652i	$-0.794908\pi$
$863$	− 35.2905i	− 1.20130i	0.799511	−	0.600652i	$-0.205092\pi$
$864$	0	0
$865$	− 2.90408i	− 0.0987418i
$866$	52.7362i	1.79205i
$867$	0	0
$868$	21.0489	0.714447
$869$	− 48.9191i	− 1.65947i
$870$	0	0
$871$	0	0
$872$	−2.81700	−0.0953958
$873$	0	0
$874$	−40.5827	−1.37273
$875$	−39.3870	−1.33152
$876$	0	0
$877$	15.3263i	0.517532i	0.965940	+	0.258766i	$0.0833159\pi$
$877$	15.3263i	0.517532i	−0.965940	+	0.258766i	$0.916684\pi$
$878$	3.84117i	0.129633i
$879$	0	0
$880$	37.0127	1.24770
$881$	−36.4306	−1.22738	−0.613689	−	0.789548i	$-0.710315\pi$
$881$	−36.4306	−1.22738	−0.613689	+	0.789548i	$0.710315\pi$
$882$	0	0
$883$	37.6819	1.26810	0.634048	−	0.773294i	$-0.281392\pi$
$883$	37.6819	1.26810	0.634048	+	0.773294i	$0.281392\pi$
$884$	0	0
$885$	0	0
$886$	41.4306i	1.39189i
$887$	−4.24890	−0.142664	−0.0713320	−	0.997453i	$-0.522725\pi$
$887$	−4.24890	−0.142664	−0.0713320	+	0.997453i	$0.522725\pi$
$888$	0	0
$889$	49.1333i	1.64788i
$890$	3.01746i	0.101145i
$891$	0	0
$892$	9.28919i	0.311025i
$893$	−84.0066	−2.81117
$894$	0	0
$895$	28.9541i	0.967828i
$896$	34.7458	1.16078
$897$	0	0
$898$	−23.3134	−0.777977
$899$	19.1672i	0.639262i
$900$	0	0
$901$	−2.32544	−0.0774715
$902$	− 16.8388i	− 0.560670i
$903$	0	0
$904$	− 8.37004i	− 0.278383i
$905$	− 34.8582i	− 1.15872i
$906$	0	0
$907$	12.6183	0.418985	0.209493	−	0.977810i	$-0.432819\pi$
$907$	12.6183	0.418985	0.209493	+	0.977810i	$0.432819\pi$
$908$	26.4983i	0.879376i
$909$	0	0
$910$	0	0
$911$	−6.77777	−0.224558	−0.112279	−	0.993677i	$-0.535815\pi$
$911$	−6.77777	−0.224558	−0.112279	+	0.993677i	$0.535815\pi$
$912$	0	0
$913$	33.5368	1.10990
$914$	−8.74525	−0.289267
$915$	0	0
$916$	− 11.5918i	− 0.383004i
$917$	− 77.8631i	− 2.57126i
$918$	0	0
$919$	−20.4674	−0.675157	−0.337579	−	0.941297i	$-0.609608\pi$
$919$	−20.4674	−0.675157	−0.337579	+	0.941297i	$0.609608\pi$
$920$	5.54719	0.182885
$921$	0	0
$922$	−33.9385	−1.11771
$923$	0	0
$924$	0	0
$925$	18.1903i	0.598093i
$926$	41.2223	1.35465
$927$	0	0
$928$	− 24.1987i	− 0.794360i
$929$	− 4.65220i	− 0.152634i	−0.997084	−	0.0763169i	$-0.975684\pi$
$929$	− 4.65220i	− 0.152634i	0.997084	−	0.0763169i	$-0.0243161\pi$
$930$	0	0
$931$	38.7555i	1.27016i
$932$	−20.2145	−0.662147
$933$	0	0
$934$	23.4252i	0.766496i
$935$	5.64310	0.184549
$936$	0	0
$937$	−41.8544	−1.36732	−0.683662	−	0.729798i	$-0.739614\pi$
$937$	−41.8544	−1.36732	−0.683662	+	0.729798i	$0.739614\pi$
$938$	− 28.2010i	− 0.920797i
$939$	0	0
$940$	−19.0151	−0.620203
$941$	30.3454i	0.989232i	0.869112	+	0.494616i	$0.164691\pi$
$941$	30.3454i	0.989232i	−0.869112	+	0.494616i	$0.835309\pi$
$942$	0	0
$943$	− 5.09783i	− 0.166008i
$944$	9.27545i	0.301890i
$945$	0	0
$946$	−66.3279	−2.15651
$947$	− 12.0325i	− 0.391004i	−0.980703	−	0.195502i	$-0.937366\pi$
$947$	− 12.0325i	− 0.391004i	0.980703	−	0.195502i	$-0.0626337\pi$
$948$	0	0
$949$	0	0
$950$	41.7700	1.35520
$951$	0	0
$952$	3.52004	0.114085
$953$	22.9825	0.744478	0.372239	−	0.928137i	$-0.378590\pi$
$953$	22.9825	0.744478	0.372239	+	0.928137i	$0.378590\pi$
$954$	0	0
$955$	10.2314i	0.331082i
$956$	16.8455i	0.544822i
$957$	0	0
$958$	−68.6359	−2.21753
$959$	−46.9743	−1.51688
$960$	0	0
$961$	6.99223	0.225556
$962$	0	0
$963$	0	0
$964$	− 7.81700i	− 0.251769i
$965$	−14.1153	−0.454387
$966$	0	0
$967$	38.8883i	1.25056i	0.780399	+	0.625281i	$0.215016\pi$
$967$	38.8883i	1.25056i	−0.780399	+	0.625281i	$0.784984\pi$
$968$	− 21.5670i	− 0.693191i
$969$	0	0
$970$	22.5448i	0.723870i
$971$	57.5133	1.84569	0.922845	−	0.385171i	$-0.125857\pi$
$971$	57.5133	1.84569	0.922845	+	0.385171i	$0.125857\pi$
$972$	0	0
$973$	60.6378i	1.94396i
$974$	−38.2911	−1.22693
$975$	0	0
$976$	−16.5200	−0.528794
$977$	16.3690i	0.523690i	0.965110	+	0.261845i	$0.0843309\pi$
$977$	16.3690i	0.523690i	−0.965110	+	0.261845i	$0.915669\pi$
$978$	0	0
$979$	6.00969	0.192070
$980$	8.77240i	0.280224i
$981$	0	0
$982$	− 11.4426i	− 0.365150i
$983$	15.6963i	0.500635i	0.968164	+	0.250318i	$0.0805351\pi$
$983$	15.6963i	0.500635i	−0.968164	+	0.250318i	$0.919465\pi$
$984$	0	0
$985$	−33.8301	−1.07792
$986$	− 5.30798i	− 0.169040i
$987$	0	0
$988$	0	0
$989$	−20.0804	−0.638519
$990$	0	0
$991$	−11.2644	−0.357827	−0.178913	−	0.983865i	$-0.557258\pi$
$991$	−11.2644	−0.357827	−0.178913	+	0.983865i	$0.557258\pi$
$992$	30.3099	0.962340
$993$	0	0
$994$	56.6122i	1.79563i
$995$	− 5.81594i	− 0.184378i
$996$	0	0
$997$	−7.70112	−0.243897	−0.121948	−	0.992536i	$-0.538914\pi$
$997$	−7.70112	−0.243897	−0.121948	+	0.992536i	$0.538914\pi$
$998$	−8.38059	−0.265283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.m.1351.6		6
3.2	odd	2		507.2.b.g.337.1		6
13.5	odd	4		1521.2.a.q.1.3		3
13.8	odd	4		1521.2.a.p.1.1		3
13.12	even	2	inner	1521.2.b.m.1351.1		6
39.2	even	12		507.2.e.k.22.3		6
39.5	even	4		507.2.a.j.1.1	✓	3
39.8	even	4		507.2.a.k.1.3	yes	3
39.11	even	12		507.2.e.j.22.1		6
39.17	odd	6		507.2.j.h.361.6		12
39.20	even	12		507.2.e.j.484.1		6
39.23	odd	6		507.2.j.h.316.1		12
39.29	odd	6		507.2.j.h.316.6		12
39.32	even	12		507.2.e.k.484.3		6
39.35	odd	6		507.2.j.h.361.1		12
39.38	odd	2		507.2.b.g.337.6		6
156.47	odd	4		8112.2.a.cf.1.2		3
156.83	odd	4		8112.2.a.by.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.1	✓	3	39.5	even	4
507.2.a.k.1.3	yes	3	39.8	even	4
507.2.b.g.337.1		6	3.2	odd	2
507.2.b.g.337.6		6	39.38	odd	2
507.2.e.j.22.1		6	39.11	even	12
507.2.e.j.484.1		6	39.20	even	12
507.2.e.k.22.3		6	39.2	even	12
507.2.e.k.484.3		6	39.32	even	12
507.2.j.h.316.1		12	39.23	odd	6
507.2.j.h.316.6		12	39.29	odd	6
507.2.j.h.361.1		12	39.35	odd	6
507.2.j.h.361.6		12	39.17	odd	6
1521.2.a.p.1.1		3	13.8	odd	4
1521.2.a.q.1.3		3	13.5	odd	4
1521.2.b.m.1351.1		6	13.12	even	2	inner
1521.2.b.m.1351.6		6	1.1	even	1	trivial
8112.2.a.by.1.2		3	156.83	odd	4
8112.2.a.cf.1.2		3	156.47	odd	4

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.80194i q^{2} -1.24698 q^{4} +1.44504i q^{5} +3.44504i q^{7} +1.35690i q^{8} -2.60388 q^{10} +5.18598i q^{11} -6.20775 q^{14} -4.93900 q^{16} -0.753020 q^{17} -7.96077i q^{19} -1.80194i q^{20} -9.34481 q^{22} -2.82908 q^{23} +2.91185 q^{25} -4.29590i q^{28} +3.91185 q^{29} +4.89977i q^{31} -6.18598i q^{32} -1.35690i q^{34} -4.97823 q^{35} +6.24698i q^{37} +14.3448 q^{38} -1.96077 q^{40} +1.80194i q^{41} +7.09783 q^{43} -6.46681i q^{44} -5.09783i q^{46} -10.5526i q^{47} -4.86831 q^{49} +5.24698i q^{50} +3.08815 q^{53} -7.49396 q^{55} -4.67456 q^{56} +7.04892i q^{58} -1.87800i q^{59} +3.34481 q^{61} -8.82908 q^{62} +1.26875 q^{64} +4.54288i q^{67} +0.939001 q^{68} -8.97046i q^{70} -9.11960i q^{71} +2.95108i q^{73} -11.2567 q^{74} +9.92692i q^{76} -17.8659 q^{77} -9.43296 q^{79} -7.13706i q^{80} -3.24698 q^{82} -6.46681i q^{83} -1.08815i q^{85} +12.7899i q^{86} -7.03684 q^{88} -1.15883i q^{89} +3.52781 q^{92} +19.0151 q^{94} +11.5036 q^{95} -8.65817i q^{97} -8.77240i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56}+ \cdots + 6 q^{95}+O(q^{100}) \) 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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