LMFDB - Embedded newform 5625.2.a.bf.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.305545 q^{2} -1.90664 q^{4} -1.87098 q^{7} +1.19366 q^{8} +O(q^{10})$	$q-0.305545 q^{2} -1.90664 q^{4} -1.87098 q^{7} +1.19366 q^{8} -5.53904 q^{11} -3.38583 q^{13} +0.571667 q^{14} +3.44857 q^{16} +6.94371 q^{17} -3.10611 q^{19} +1.69243 q^{22} +4.02357 q^{23} +1.03452 q^{26} +3.56728 q^{28} -4.73587 q^{29} +3.79640 q^{31} -3.44100 q^{32} -2.12162 q^{34} -7.24043 q^{37} +0.949055 q^{38} -4.81115 q^{41} -8.46556 q^{43} +10.5610 q^{44} -1.22938 q^{46} -6.19480 q^{47} -3.49945 q^{49} +6.45556 q^{52} -10.7247 q^{53} -2.23330 q^{56} +1.44702 q^{58} -5.75740 q^{59} +2.85563 q^{61} -1.15997 q^{62} -5.84576 q^{64} -3.56340 q^{67} -13.2392 q^{68} +8.39069 q^{71} -1.51245 q^{73} +2.21228 q^{74} +5.92223 q^{76} +10.3634 q^{77} -0.629411 q^{79} +1.47002 q^{82} -0.927819 q^{83} +2.58661 q^{86} -6.61171 q^{88} +0.959243 q^{89} +6.33480 q^{91} -7.67151 q^{92} +1.89279 q^{94} +15.8588 q^{97} +1.06924 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4}+O(q^{10}) $ 24 * q + 32 * q^4	$ 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) $ 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

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 $

$2$	−0.305545	−0.216053	−0.108026	−	0.994148i	$-0.534453\pi$
$2$	−0.305545	−0.216053	−0.108026	+	0.994148i	$0.534453\pi$
$3$	0	0
$3$	0	0
$4$	−1.90664	−0.953321
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.87098	−0.707162	−0.353581	−	0.935404i	$-0.615036\pi$
$7$	−1.87098	−0.707162	−0.353581	+	0.935404i	$0.615036\pi$
$8$	1.19366	0.422021
$9$	0	0
$10$	0	0
$11$	−5.53904	−1.67008	−0.835042	−	0.550186i	$-0.814557\pi$
$11$	−5.53904	−1.67008	−0.835042	+	0.550186i	$0.814557\pi$
$12$	0	0
$13$	−3.38583	−0.939059	−0.469530	−	0.882917i	$-0.655576\pi$
$13$	−3.38583	−0.939059	−0.469530	+	0.882917i	$0.655576\pi$
$14$	0.571667	0.152785
$15$	0	0
$16$	3.44857	0.862142
$17$	6.94371	1.68410	0.842048	−	0.539402i	$-0.181350\pi$
$17$	6.94371	1.68410	0.842048	+	0.539402i	$0.181350\pi$
$18$	0	0
$19$	−3.10611	−0.712590	−0.356295	−	0.934374i	$-0.615960\pi$
$19$	−3.10611	−0.712590	−0.356295	+	0.934374i	$0.615960\pi$
$20$	0	0
$21$	0	0
$22$	1.69243	0.360827
$23$	4.02357	0.838972	0.419486	−	0.907762i	$-0.362210\pi$
$23$	4.02357	0.838972	0.419486	+	0.907762i	$0.362210\pi$
$24$	0	0
$25$	0	0
$26$	1.03452	0.202887
$27$	0	0
$28$	3.56728	0.674153
$29$	−4.73587	−0.879429	−0.439714	−	0.898138i	$-0.644920\pi$
$29$	−4.73587	−0.879429	−0.439714	+	0.898138i	$0.644920\pi$
$30$	0	0
$31$	3.79640	0.681854	0.340927	−	0.940090i	$-0.389259\pi$
$31$	3.79640	0.681854	0.340927	+	0.940090i	$0.389259\pi$
$32$	−3.44100	−0.608289
$33$	0	0
$34$	−2.12162	−0.363854
$35$	0	0
$36$	0	0
$37$	−7.24043	−1.19032	−0.595160	−	0.803607i	$-0.702911\pi$
$37$	−7.24043	−1.19032	−0.595160	+	0.803607i	$0.702911\pi$
$38$	0.949055	0.153957
$39$	0	0
$40$	0	0
$41$	−4.81115	−0.751375	−0.375688	−	0.926746i	$-0.622593\pi$
$41$	−4.81115	−0.751375	−0.375688	+	0.926746i	$0.622593\pi$
$42$	0	0
$43$	−8.46556	−1.29099	−0.645493	−	0.763766i	$-0.723348\pi$
$43$	−8.46556	−1.29099	−0.645493	+	0.763766i	$0.723348\pi$
$44$	10.5610	1.59213
$45$	0	0
$46$	−1.22938	−0.181262
$47$	−6.19480	−0.903605	−0.451803	−	0.892118i	$-0.649219\pi$
$47$	−6.19480	−0.903605	−0.451803	+	0.892118i	$0.649219\pi$
$48$	0	0
$49$	−3.49945	−0.499921
$50$	0	0
$51$	0	0
$52$	6.45556	0.895225
$53$	−10.7247	−1.47315	−0.736577	−	0.676354i	$-0.763559\pi$
$53$	−10.7247	−1.47315	−0.736577	+	0.676354i	$0.763559\pi$
$54$	0	0
$55$	0	0
$56$	−2.23330	−0.298437
$57$	0	0
$58$	1.44702	0.190003
$59$	−5.75740	−0.749550	−0.374775	−	0.927116i	$-0.622280\pi$
$59$	−5.75740	−0.749550	−0.374775	+	0.927116i	$0.622280\pi$
$60$	0	0
$61$	2.85563	0.365626	0.182813	−	0.983148i	$-0.441480\pi$
$61$	2.85563	0.365626	0.182813	+	0.983148i	$0.441480\pi$
$62$	−1.15997	−0.147317
$63$	0	0
$64$	−5.84576	−0.730720
$65$	0	0
$66$	0	0
$67$	−3.56340	−0.435339	−0.217669	−	0.976023i	$-0.569845\pi$
$67$	−3.56340	−0.435339	−0.217669	+	0.976023i	$0.569845\pi$
$68$	−13.2392	−1.60548
$69$	0	0
$70$	0	0
$71$	8.39069	0.995792	0.497896	−	0.867237i	$-0.334106\pi$
$71$	8.39069	0.995792	0.497896	+	0.867237i	$0.334106\pi$
$72$	0	0
$73$	−1.51245	−0.177019	−0.0885096	−	0.996075i	$-0.528210\pi$
$73$	−1.51245	−0.177019	−0.0885096	+	0.996075i	$0.528210\pi$
$74$	2.21228	0.257172
$75$	0	0
$76$	5.92223	0.679327
$77$	10.3634	1.18102
$78$	0	0
$79$	−0.629411	−0.0708143	−0.0354072	−	0.999373i	$-0.511273\pi$
$79$	−0.629411	−0.0708143	−0.0354072	+	0.999373i	$0.511273\pi$
$80$	0	0
$81$	0	0
$82$	1.47002	0.162337
$83$	−0.927819	−0.101841	−0.0509207	−	0.998703i	$-0.516216\pi$
$83$	−0.927819	−0.101841	−0.0509207	+	0.998703i	$0.516216\pi$
$84$	0	0
$85$	0	0
$86$	2.58661	0.278921
$87$	0	0
$88$	−6.61171	−0.704810
$89$	0.959243	0.101680	0.0508398	−	0.998707i	$-0.483810\pi$
$89$	0.959243	0.101680	0.0508398	+	0.998707i	$0.483810\pi$
$90$	0	0
$91$	6.33480	0.664067
$92$	−7.67151	−0.799810
$93$	0	0
$94$	1.89279	0.195227
$95$	0	0
$96$	0	0
$97$	15.8588	1.61021	0.805107	−	0.593129i	$-0.202108\pi$
$97$	15.8588	1.61021	0.805107	+	0.593129i	$0.202108\pi$
$98$	1.06924	0.108010
$99$	0	0
$100$	0	0
$101$	9.33669	0.929035	0.464518	−	0.885564i	$-0.346228\pi$
$101$	9.33669	0.929035	0.464518	+	0.885564i	$0.346228\pi$
$102$	0	0
$103$	6.88108	0.678013	0.339007	−	0.940784i	$-0.389909\pi$
$103$	6.88108	0.678013	0.339007	+	0.940784i	$0.389909\pi$
$104$	−4.04151	−0.396303
$105$	0	0
$106$	3.27689	0.318279
$107$	10.3647	1.00199	0.500995	−	0.865450i	$-0.332967\pi$
$107$	10.3647	1.00199	0.500995	+	0.865450i	$0.332967\pi$
$108$	0	0
$109$	−5.48241	−0.525120	−0.262560	−	0.964916i	$-0.584567\pi$
$109$	−5.48241	−0.525120	−0.262560	+	0.964916i	$0.584567\pi$
$110$	0	0
$111$	0	0
$112$	−6.45219	−0.609675
$113$	−6.96825	−0.655518	−0.327759	−	0.944761i	$-0.606293\pi$
$113$	−6.96825	−0.655518	−0.327759	+	0.944761i	$0.606293\pi$
$114$	0	0
$115$	0	0
$116$	9.02961	0.838378
$117$	0	0
$118$	1.75914	0.161942
$119$	−12.9915	−1.19093
$120$	0	0
$121$	19.6810	1.78918
$122$	−0.872524	−0.0789946
$123$	0	0
$124$	−7.23838	−0.650026
$125$	0	0
$126$	0	0
$127$	18.0457	1.60130	0.800651	−	0.599131i	$-0.204487\pi$
$127$	18.0457	1.60130	0.800651	+	0.599131i	$0.204487\pi$
$128$	8.66815	0.766163
$129$	0	0
$130$	0	0
$131$	15.4011	1.34560	0.672801	−	0.739823i	$-0.265091\pi$
$131$	15.4011	1.34560	0.672801	+	0.739823i	$0.265091\pi$
$132$	0	0
$133$	5.81145	0.503917
$134$	1.08878	0.0940563
$135$	0	0
$136$	8.28839	0.710724
$137$	−10.8190	−0.924333	−0.462166	−	0.886793i	$-0.652928\pi$
$137$	−10.8190	−0.924333	−0.462166	+	0.886793i	$0.652928\pi$
$138$	0	0
$139$	2.40719	0.204175	0.102088	−	0.994775i	$-0.467448\pi$
$139$	2.40719	0.204175	0.102088	+	0.994775i	$0.467448\pi$
$140$	0	0
$141$	0	0
$142$	−2.56373	−0.215144
$143$	18.7542	1.56831
$144$	0	0
$145$	0	0
$146$	0.462122	0.0382455
$147$	0	0
$148$	13.8049	1.13476
$149$	−4.53337	−0.371388	−0.185694	−	0.982608i	$-0.559453\pi$
$149$	−4.53337	−0.371388	−0.185694	+	0.982608i	$0.559453\pi$
$150$	0	0
$151$	10.2345	0.832875	0.416438	−	0.909164i	$-0.363278\pi$
$151$	10.2345	0.832875	0.416438	+	0.909164i	$0.363278\pi$
$152$	−3.70762	−0.300728
$153$	0	0
$154$	−3.16649	−0.255163
$155$	0	0
$156$	0	0
$157$	−16.1453	−1.28853	−0.644267	−	0.764801i	$-0.722837\pi$
$157$	−16.1453	−1.28853	−0.644267	+	0.764801i	$0.722837\pi$
$158$	0.192314	0.0152996
$159$	0	0
$160$	0	0
$161$	−7.52800	−0.593290
$162$	0	0
$163$	11.3134	0.886134	0.443067	−	0.896489i	$-0.353890\pi$
$163$	11.3134	0.886134	0.443067	+	0.896489i	$0.353890\pi$
$164$	9.17314	0.716302
$165$	0	0
$166$	0.283491	0.0220031
$167$	−5.10743	−0.395225	−0.197612	−	0.980280i	$-0.563319\pi$
$167$	−5.10743	−0.395225	−0.197612	+	0.980280i	$0.563319\pi$
$168$	0	0
$169$	−1.53618	−0.118168
$170$	0	0
$171$	0	0
$172$	16.1408	1.23072
$173$	−15.2699	−1.16095	−0.580475	−	0.814278i	$-0.697133\pi$
$173$	−15.2699	−1.16095	−0.580475	+	0.814278i	$0.697133\pi$
$174$	0	0
$175$	0	0
$176$	−19.1018	−1.43985
$177$	0	0
$178$	−0.293092	−0.0219682
$179$	21.7896	1.62863	0.814316	−	0.580422i	$-0.197112\pi$
$179$	21.7896	1.62863	0.814316	+	0.580422i	$0.197112\pi$
$180$	0	0
$181$	13.3973	0.995815	0.497907	−	0.867230i	$-0.334102\pi$
$181$	13.3973	0.995815	0.497907	+	0.867230i	$0.334102\pi$
$182$	−1.93557	−0.143474
$183$	0	0
$184$	4.80275	0.354064
$185$	0	0
$186$	0	0
$187$	−38.4615	−2.81258
$188$	11.8113	0.861426
$189$	0	0
$190$	0	0
$191$	5.40409	0.391026	0.195513	−	0.980701i	$-0.437363\pi$
$191$	5.40409	0.391026	0.195513	+	0.980701i	$0.437363\pi$
$192$	0	0
$193$	8.72818	0.628268	0.314134	−	0.949379i	$-0.398286\pi$
$193$	8.72818	0.628268	0.314134	+	0.949379i	$0.398286\pi$
$194$	−4.84557	−0.347892
$195$	0	0
$196$	6.67220	0.476586
$197$	9.25006	0.659040	0.329520	−	0.944149i	$-0.393113\pi$
$197$	9.25006	0.659040	0.329520	+	0.944149i	$0.393113\pi$
$198$	0	0
$199$	19.1122	1.35483	0.677413	−	0.735603i	$-0.263101\pi$
$199$	19.1122	1.35483	0.677413	+	0.735603i	$0.263101\pi$
$200$	0	0
$201$	0	0
$202$	−2.85278	−0.200721
$203$	8.86070	0.621899
$204$	0	0
$205$	0	0
$206$	−2.10248	−0.146487
$207$	0	0
$208$	−11.6763	−0.809602
$209$	17.2049	1.19008
$210$	0	0
$211$	28.4132	1.95604	0.978022	−	0.208502i	$-0.0668587\pi$
$211$	28.4132	1.95604	0.978022	+	0.208502i	$0.0668587\pi$
$212$	20.4482	1.40439
$213$	0	0
$214$	−3.16688	−0.216483
$215$	0	0
$216$	0	0
$217$	−7.10298	−0.482182
$218$	1.67512	0.113454
$219$	0	0
$220$	0	0
$221$	−23.5102	−1.58147
$222$	0	0
$223$	−15.5268	−1.03975	−0.519877	−	0.854241i	$-0.674022\pi$
$223$	−15.5268	−1.03975	−0.519877	+	0.854241i	$0.674022\pi$
$224$	6.43803	0.430159
$225$	0	0
$226$	2.12911	0.141627
$227$	−27.4429	−1.82145	−0.910726	−	0.413011i	$-0.864477\pi$
$227$	−27.4429	−1.82145	−0.910726	+	0.413011i	$0.864477\pi$
$228$	0	0
$229$	12.4428	0.822245	0.411123	−	0.911580i	$-0.365137\pi$
$229$	12.4428	0.822245	0.411123	+	0.911580i	$0.365137\pi$
$230$	0	0
$231$	0	0
$232$	−5.65299	−0.371137
$233$	−7.04038	−0.461231	−0.230615	−	0.973045i	$-0.574074\pi$
$233$	−7.04038	−0.461231	−0.230615	+	0.973045i	$0.574074\pi$
$234$	0	0
$235$	0	0
$236$	10.9773	0.714561
$237$	0	0
$238$	3.96949	0.257304
$239$	−25.4313	−1.64501	−0.822507	−	0.568756i	$-0.807425\pi$
$239$	−25.4313	−1.64501	−0.822507	+	0.568756i	$0.807425\pi$
$240$	0	0
$241$	13.7218	0.883899	0.441949	−	0.897040i	$-0.354287\pi$
$241$	13.7218	0.883899	0.441949	+	0.897040i	$0.354287\pi$
$242$	−6.01343	−0.386558
$243$	0	0
$244$	−5.44467	−0.348559
$245$	0	0
$246$	0	0
$247$	10.5167	0.669164
$248$	4.53160	0.287757
$249$	0	0
$250$	0	0
$251$	−17.7275	−1.11895	−0.559474	−	0.828848i	$-0.688997\pi$
$251$	−17.7275	−1.11895	−0.559474	+	0.828848i	$0.688997\pi$
$252$	0	0
$253$	−22.2867	−1.40115
$254$	−5.51379	−0.345966
$255$	0	0
$256$	9.04300	0.565188
$257$	−21.3592	−1.33235	−0.666176	−	0.745795i	$-0.732070\pi$
$257$	−21.3592	−1.33235	−0.666176	+	0.745795i	$0.732070\pi$
$258$	0	0
$259$	13.5467	0.841749
$260$	0	0
$261$	0	0
$262$	−4.70574	−0.290721
$263$	−16.8729	−1.04043	−0.520214	−	0.854036i	$-0.674148\pi$
$263$	−16.8729	−1.04043	−0.520214	+	0.854036i	$0.674148\pi$
$264$	0	0
$265$	0	0
$266$	−1.77566	−0.108873
$267$	0	0
$268$	6.79413	0.415018
$269$	−3.69307	−0.225170	−0.112585	−	0.993642i	$-0.535913\pi$
$269$	−3.69307	−0.225170	−0.112585	+	0.993642i	$0.535913\pi$
$270$	0	0
$271$	9.77446	0.593756	0.296878	−	0.954915i	$-0.404054\pi$
$271$	9.77446	0.593756	0.296878	+	0.954915i	$0.404054\pi$
$272$	23.9459	1.45193
$273$	0	0
$274$	3.30570	0.199705
$275$	0	0
$276$	0	0
$277$	28.0822	1.68730	0.843648	−	0.536896i	$-0.180403\pi$
$277$	28.0822	1.68730	0.843648	+	0.536896i	$0.180403\pi$
$278$	−0.735506	−0.0441127
$279$	0	0
$280$	0	0
$281$	−15.9029	−0.948689	−0.474345	−	0.880339i	$-0.657315\pi$
$281$	−15.9029	−0.948689	−0.474345	+	0.880339i	$0.657315\pi$
$282$	0	0
$283$	7.54078	0.448253	0.224126	−	0.974560i	$-0.428047\pi$
$283$	7.54078	0.448253	0.224126	+	0.974560i	$0.428047\pi$
$284$	−15.9980	−0.949309
$285$	0	0
$286$	−5.73026	−0.338838
$287$	9.00154	0.531344
$288$	0	0
$289$	31.2151	1.83618
$290$	0	0
$291$	0	0
$292$	2.88371	0.168756
$293$	−12.3332	−0.720515	−0.360258	−	0.932853i	$-0.617311\pi$
$293$	−12.3332	−0.720515	−0.360258	+	0.932853i	$0.617311\pi$
$294$	0	0
$295$	0	0
$296$	−8.64258	−0.502340
$297$	0	0
$298$	1.38515	0.0802395
$299$	−13.6231	−0.787844
$300$	0	0
$301$	15.8389	0.912937
$302$	−3.12711	−0.179945
$303$	0	0
$304$	−10.7116	−0.614354
$305$	0	0
$306$	0	0
$307$	−2.01051	−0.114746	−0.0573730	−	0.998353i	$-0.518272\pi$
$307$	−2.01051	−0.114746	−0.0573730	+	0.998353i	$0.518272\pi$
$308$	−19.7593	−1.12589
$309$	0	0
$310$	0	0
$311$	24.0066	1.36129	0.680644	−	0.732614i	$-0.261700\pi$
$311$	24.0066	1.36129	0.680644	+	0.732614i	$0.261700\pi$
$312$	0	0
$313$	21.7210	1.22774	0.613871	−	0.789406i	$-0.289611\pi$
$313$	21.7210	1.22774	0.613871	+	0.789406i	$0.289611\pi$
$314$	4.93311	0.278391
$315$	0	0
$316$	1.20006	0.0675088
$317$	15.2323	0.855529	0.427765	−	0.903890i	$-0.359301\pi$
$317$	15.2323	0.855529	0.427765	+	0.903890i	$0.359301\pi$
$318$	0	0
$319$	26.2322	1.46872
$320$	0	0
$321$	0	0
$322$	2.30014	0.128182
$323$	−21.5679	−1.20007
$324$	0	0
$325$	0	0
$326$	−3.45675	−0.191452
$327$	0	0
$328$	−5.74285	−0.317096
$329$	11.5903	0.638996
$330$	0	0
$331$	4.48751	0.246656	0.123328	−	0.992366i	$-0.460643\pi$
$331$	4.48751	0.246656	0.123328	+	0.992366i	$0.460643\pi$
$332$	1.76902	0.0970876
$333$	0	0
$334$	1.56055	0.0853895
$335$	0	0
$336$	0	0
$337$	−8.93226	−0.486571	−0.243286	−	0.969955i	$-0.578225\pi$
$337$	−8.93226	−0.486571	−0.243286	+	0.969955i	$0.578225\pi$
$338$	0.469374	0.0255306
$339$	0	0
$340$	0	0
$341$	−21.0284	−1.13875
$342$	0	0
$343$	19.6442	1.06069
$344$	−10.1050	−0.544823
$345$	0	0
$346$	4.66565	0.250827
$347$	−5.82511	−0.312708	−0.156354	−	0.987701i	$-0.549974\pi$
$347$	−5.82511	−0.312708	−0.156354	+	0.987701i	$0.549974\pi$
$348$	0	0
$349$	−2.21387	−0.118506	−0.0592529	−	0.998243i	$-0.518872\pi$
$349$	−2.21387	−0.118506	−0.0592529	+	0.998243i	$0.518872\pi$
$350$	0	0
$351$	0	0
$352$	19.0599	1.01589
$353$	10.9314	0.581819	0.290909	−	0.956751i	$-0.406042\pi$
$353$	10.9314	0.581819	0.290909	+	0.956751i	$0.406042\pi$
$354$	0	0
$355$	0	0
$356$	−1.82893	−0.0969333
$357$	0	0
$358$	−6.65771	−0.351871
$359$	25.4919	1.34541	0.672707	−	0.739909i	$-0.265132\pi$
$359$	25.4919	1.34541	0.672707	+	0.739909i	$0.265132\pi$
$360$	0	0
$361$	−9.35210	−0.492216
$362$	−4.09348	−0.215149
$363$	0	0
$364$	−12.0782	−0.633069
$365$	0	0
$366$	0	0
$367$	−21.6948	−1.13246	−0.566229	−	0.824248i	$-0.691598\pi$
$367$	−21.6948	−1.13246	−0.566229	+	0.824248i	$0.691598\pi$
$368$	13.8756	0.723313
$369$	0	0
$370$	0	0
$371$	20.0657	1.04176
$372$	0	0
$373$	−16.2827	−0.843088	−0.421544	−	0.906808i	$-0.638512\pi$
$373$	−16.2827	−0.843088	−0.421544	+	0.906808i	$0.638512\pi$
$374$	11.7517	0.607667
$375$	0	0
$376$	−7.39446	−0.381340
$377$	16.0348	0.825836
$378$	0	0
$379$	19.3168	0.992238	0.496119	−	0.868254i	$-0.334758\pi$
$379$	19.3168	0.992238	0.496119	+	0.868254i	$0.334758\pi$
$380$	0	0
$381$	0	0
$382$	−1.65119	−0.0844824
$383$	28.3611	1.44919	0.724593	−	0.689177i	$-0.242028\pi$
$383$	28.3611	1.44919	0.724593	+	0.689177i	$0.242028\pi$
$384$	0	0
$385$	0	0
$386$	−2.66685	−0.135739
$387$	0	0
$388$	−30.2370	−1.53505
$389$	10.2487	0.519628	0.259814	−	0.965659i	$-0.416339\pi$
$389$	10.2487	0.519628	0.259814	+	0.965659i	$0.416339\pi$
$390$	0	0
$391$	27.9385	1.41291
$392$	−4.17714	−0.210977
$393$	0	0
$394$	−2.82631	−0.142387
$395$	0	0
$396$	0	0
$397$	29.3897	1.47503	0.737513	−	0.675333i	$-0.236000\pi$
$397$	29.3897	1.47503	0.737513	+	0.675333i	$0.236000\pi$
$398$	−5.83963	−0.292714
$399$	0	0
$400$	0	0
$401$	22.6657	1.13187	0.565936	−	0.824449i	$-0.308515\pi$
$401$	22.6657	1.13187	0.565936	+	0.824449i	$0.308515\pi$
$402$	0	0
$403$	−12.8540	−0.640301
$404$	−17.8017	−0.885669
$405$	0	0
$406$	−2.70734	−0.134363
$407$	40.1051	1.98793
$408$	0	0
$409$	7.98590	0.394877	0.197439	−	0.980315i	$-0.436738\pi$
$409$	7.98590	0.394877	0.197439	+	0.980315i	$0.436738\pi$
$410$	0	0
$411$	0	0
$412$	−13.1198	−0.646365
$413$	10.7720	0.530053
$414$	0	0
$415$	0	0
$416$	11.6506	0.571220
$417$	0	0
$418$	−5.25686	−0.257121
$419$	17.5503	0.857388	0.428694	−	0.903450i	$-0.358974\pi$
$419$	17.5503	0.857388	0.428694	+	0.903450i	$0.358974\pi$
$420$	0	0
$421$	−23.2752	−1.13436	−0.567181	−	0.823593i	$-0.691966\pi$
$421$	−23.2752	−1.13436	−0.567181	+	0.823593i	$0.691966\pi$
$422$	−8.68151	−0.422609
$423$	0	0
$424$	−12.8016	−0.621702
$425$	0	0
$426$	0	0
$427$	−5.34282	−0.258557
$428$	−19.7617	−0.955219
$429$	0	0
$430$	0	0
$431$	12.5466	0.604349	0.302174	−	0.953253i	$-0.402287\pi$
$431$	12.5466	0.604349	0.302174	+	0.953253i	$0.402287\pi$
$432$	0	0
$433$	−8.21647	−0.394858	−0.197429	−	0.980317i	$-0.563259\pi$
$433$	−8.21647	−0.394858	−0.197429	+	0.980317i	$0.563259\pi$
$434$	2.17028	0.104177
$435$	0	0
$436$	10.4530	0.500608
$437$	−12.4976	−0.597843
$438$	0	0
$439$	−13.2810	−0.633868	−0.316934	−	0.948448i	$-0.602653\pi$
$439$	−13.2810	−0.633868	−0.316934	+	0.948448i	$0.602653\pi$
$440$	0	0
$441$	0	0
$442$	7.18342	0.341681
$443$	7.03665	0.334322	0.167161	−	0.985930i	$-0.446540\pi$
$443$	7.03665	0.334322	0.167161	+	0.985930i	$0.446540\pi$
$444$	0	0
$445$	0	0
$446$	4.74415	0.224642
$447$	0	0
$448$	10.9373	0.516737
$449$	1.17018	0.0552242	0.0276121	−	0.999619i	$-0.491210\pi$
$449$	1.17018	0.0552242	0.0276121	+	0.999619i	$0.491210\pi$
$450$	0	0
$451$	26.6492	1.25486
$452$	13.2860	0.624919
$453$	0	0
$454$	8.38505	0.393530
$455$	0	0
$456$	0	0
$457$	5.45077	0.254976	0.127488	−	0.991840i	$-0.459309\pi$
$457$	5.45077	0.254976	0.127488	+	0.991840i	$0.459309\pi$
$458$	−3.80184	−0.177649
$459$	0	0
$460$	0	0
$461$	−37.1747	−1.73140	−0.865700	−	0.500564i	$-0.833126\pi$
$461$	−37.1747	−1.73140	−0.865700	+	0.500564i	$0.833126\pi$
$462$	0	0
$463$	15.7015	0.729711	0.364856	−	0.931064i	$-0.381118\pi$
$463$	15.7015	0.729711	0.364856	+	0.931064i	$0.381118\pi$
$464$	−16.3320	−0.758193
$465$	0	0
$466$	2.15115	0.0996503
$467$	−11.4062	−0.527815	−0.263907	−	0.964548i	$-0.585011\pi$
$467$	−11.4062	−0.527815	−0.263907	+	0.964548i	$0.585011\pi$
$468$	0	0
$469$	6.66704	0.307855
$470$	0	0
$471$	0	0
$472$	−6.87235	−0.316326
$473$	46.8911	2.15605
$474$	0	0
$475$	0	0
$476$	24.7702	1.13534
$477$	0	0
$478$	7.77040	0.355410
$479$	3.50849	0.160307	0.0801536	−	0.996783i	$-0.474459\pi$
$479$	3.50849	0.160307	0.0801536	+	0.996783i	$0.474459\pi$
$480$	0	0
$481$	24.5148	1.11778
$482$	−4.19263	−0.190969
$483$	0	0
$484$	−37.5246	−1.70566
$485$	0	0
$486$	0	0
$487$	−7.87359	−0.356786	−0.178393	−	0.983959i	$-0.557090\pi$
$487$	−7.87359	−0.356786	−0.178393	+	0.983959i	$0.557090\pi$
$488$	3.40864	0.154302
$489$	0	0
$490$	0	0
$491$	25.4698	1.14944	0.574718	−	0.818352i	$-0.305112\pi$
$491$	25.4698	1.14944	0.574718	+	0.818352i	$0.305112\pi$
$492$	0	0
$493$	−32.8845	−1.48104
$494$	−3.21334	−0.144575
$495$	0	0
$496$	13.0922	0.587855
$497$	−15.6988	−0.704186
$498$	0	0
$499$	5.04661	0.225917	0.112959	−	0.993600i	$-0.463967\pi$
$499$	5.04661	0.225917	0.112959	+	0.993600i	$0.463967\pi$
$500$	0	0
$501$	0	0
$502$	5.41654	0.241752
$503$	11.5862	0.516604	0.258302	−	0.966064i	$-0.416837\pi$
$503$	11.5862	0.516604	0.258302	+	0.966064i	$0.416837\pi$
$504$	0	0
$505$	0	0
$506$	6.80960	0.302723
$507$	0	0
$508$	−34.4068	−1.52655
$509$	−36.3623	−1.61173	−0.805866	−	0.592099i	$-0.798300\pi$
$509$	−36.3623	−1.61173	−0.805866	+	0.592099i	$0.798300\pi$
$510$	0	0
$511$	2.82976	0.125181
$512$	−20.0993	−0.888274
$513$	0	0
$514$	6.52620	0.287858
$515$	0	0
$516$	0	0
$517$	34.3133	1.50910
$518$	−4.13912	−0.181862
$519$	0	0
$520$	0	0
$521$	12.9450	0.567130	0.283565	−	0.958953i	$-0.408483\pi$
$521$	12.9450	0.567130	0.283565	+	0.958953i	$0.408483\pi$
$522$	0	0
$523$	10.8031	0.472385	0.236193	−	0.971706i	$-0.424100\pi$
$523$	10.8031	0.472385	0.236193	+	0.971706i	$0.424100\pi$
$524$	−29.3644	−1.28279
$525$	0	0
$526$	5.15543	0.224788
$527$	26.3611	1.14831
$528$	0	0
$529$	−6.81089	−0.296126
$530$	0	0
$531$	0	0
$532$	−11.0804	−0.480394
$533$	16.2897	0.705586
$534$	0	0
$535$	0	0
$536$	−4.25347	−0.183722
$537$	0	0
$538$	1.12840	0.0486487
$539$	19.3836	0.834911
$540$	0	0
$541$	−5.90345	−0.253809	−0.126905	−	0.991915i	$-0.540504\pi$
$541$	−5.90345	−0.253809	−0.126905	+	0.991915i	$0.540504\pi$
$542$	−2.98654	−0.128283
$543$	0	0
$544$	−23.8933	−1.02442
$545$	0	0
$546$	0	0
$547$	12.9396	0.553256	0.276628	−	0.960977i	$-0.410783\pi$
$547$	12.9396	0.553256	0.276628	+	0.960977i	$0.410783\pi$
$548$	20.6280	0.881186
$549$	0	0
$550$	0	0
$551$	14.7101	0.626672
$552$	0	0
$553$	1.17761	0.0500772
$554$	−8.58038	−0.364545
$555$	0	0
$556$	−4.58966	−0.194645
$557$	−20.9324	−0.886933	−0.443466	−	0.896291i	$-0.646251\pi$
$557$	−20.9324	−0.886933	−0.443466	+	0.896291i	$0.646251\pi$
$558$	0	0
$559$	28.6629	1.21231
$560$	0	0
$561$	0	0
$562$	4.85906	0.204967
$563$	10.7269	0.452086	0.226043	−	0.974117i	$-0.427421\pi$
$563$	10.7269	0.452086	0.226043	+	0.974117i	$0.427421\pi$
$564$	0	0
$565$	0	0
$566$	−2.30405	−0.0968463
$567$	0	0
$568$	10.0156	0.420245
$569$	41.1358	1.72450	0.862252	−	0.506479i	$-0.169053\pi$
$569$	41.1358	1.72450	0.862252	+	0.506479i	$0.169053\pi$
$570$	0	0
$571$	16.0169	0.670286	0.335143	−	0.942167i	$-0.391215\pi$
$571$	16.0169	0.670286	0.335143	+	0.942167i	$0.391215\pi$
$572$	−35.7576	−1.49510
$573$	0	0
$574$	−2.75038	−0.114799
$575$	0	0
$576$	0	0
$577$	−16.6101	−0.691486	−0.345743	−	0.938329i	$-0.612373\pi$
$577$	−16.6101	−0.691486	−0.345743	+	0.938329i	$0.612373\pi$
$578$	−9.53762	−0.396713
$579$	0	0
$580$	0	0
$581$	1.73593	0.0720184
$582$	0	0
$583$	59.4047	2.46029
$584$	−1.80535	−0.0747058
$585$	0	0
$586$	3.76836	0.155669
$587$	3.97530	0.164078	0.0820392	−	0.996629i	$-0.473857\pi$
$587$	3.97530	0.164078	0.0820392	+	0.996629i	$0.473857\pi$
$588$	0	0
$589$	−11.7920	−0.485882
$590$	0	0
$591$	0	0
$592$	−24.9691	−1.02622
$593$	−43.3755	−1.78122	−0.890609	−	0.454769i	$-0.849722\pi$
$593$	−43.3755	−1.78122	−0.890609	+	0.454769i	$0.849722\pi$
$594$	0	0
$595$	0	0
$596$	8.64351	0.354052
$597$	0	0
$598$	4.16247	0.170216
$599$	45.0977	1.84264	0.921321	−	0.388802i	$-0.127111\pi$
$599$	45.0977	1.84264	0.921321	+	0.388802i	$0.127111\pi$
$600$	0	0
$601$	−30.6180	−1.24893	−0.624467	−	0.781051i	$-0.714684\pi$
$601$	−30.6180	−1.24893	−0.624467	+	0.781051i	$0.714684\pi$
$602$	−4.83948	−0.197243
$603$	0	0
$604$	−19.5136	−0.793997
$605$	0	0
$606$	0	0
$607$	−7.94687	−0.322553	−0.161277	−	0.986909i	$-0.551561\pi$
$607$	−7.94687	−0.322553	−0.161277	+	0.986909i	$0.551561\pi$
$608$	10.6881	0.433461
$609$	0	0
$610$	0	0
$611$	20.9745	0.848539
$612$	0	0
$613$	−10.9745	−0.443255	−0.221628	−	0.975131i	$-0.571137\pi$
$613$	−10.9745	−0.443255	−0.221628	+	0.975131i	$0.571137\pi$
$614$	0.614302	0.0247912
$615$	0	0
$616$	12.3703	0.498415
$617$	16.1269	0.649243	0.324622	−	0.945844i	$-0.394763\pi$
$617$	16.1269	0.649243	0.324622	+	0.945844i	$0.394763\pi$
$618$	0	0
$619$	−10.4095	−0.418393	−0.209197	−	0.977874i	$-0.567085\pi$
$619$	−10.4095	−0.418393	−0.209197	+	0.977874i	$0.567085\pi$
$620$	0	0
$621$	0	0
$622$	−7.33509	−0.294110
$623$	−1.79472	−0.0719040
$624$	0	0
$625$	0	0
$626$	−6.63674	−0.265257
$627$	0	0
$628$	30.7833	1.22839
$629$	−50.2755	−2.00461
$630$	0	0
$631$	−15.8644	−0.631550	−0.315775	−	0.948834i	$-0.602264\pi$
$631$	−15.8644	−0.631550	−0.315775	+	0.948834i	$0.602264\pi$
$632$	−0.751300	−0.0298851
$633$	0	0
$634$	−4.65414	−0.184840
$635$	0	0
$636$	0	0
$637$	11.8485	0.469456
$638$	−8.01511	−0.317321
$639$	0	0
$640$	0	0
$641$	−34.8092	−1.37488	−0.687441	−	0.726240i	$-0.741266\pi$
$641$	−34.8092	−1.37488	−0.687441	+	0.726240i	$0.741266\pi$
$642$	0	0
$643$	−9.19973	−0.362802	−0.181401	−	0.983409i	$-0.558063\pi$
$643$	−9.19973	−0.362802	−0.181401	+	0.983409i	$0.558063\pi$
$644$	14.3532	0.565595
$645$	0	0
$646$	6.58996	0.259279
$647$	8.06144	0.316928	0.158464	−	0.987365i	$-0.449346\pi$
$647$	8.06144	0.316928	0.158464	+	0.987365i	$0.449346\pi$
$648$	0	0
$649$	31.8905	1.25181
$650$	0	0
$651$	0	0
$652$	−21.5706	−0.844770
$653$	−36.7787	−1.43926	−0.719630	−	0.694358i	$-0.755689\pi$
$653$	−36.7787	−1.43926	−0.719630	+	0.694358i	$0.755689\pi$
$654$	0	0
$655$	0	0
$656$	−16.5916	−0.647792
$657$	0	0
$658$	−3.54137	−0.138057
$659$	−24.1456	−0.940578	−0.470289	−	0.882512i	$-0.655850\pi$
$659$	−24.1456	−0.940578	−0.470289	+	0.882512i	$0.655850\pi$
$660$	0	0
$661$	−15.6031	−0.606891	−0.303446	−	0.952849i	$-0.598137\pi$
$661$	−15.6031	−0.606891	−0.303446	+	0.952849i	$0.598137\pi$
$662$	−1.37114	−0.0532908
$663$	0	0
$664$	−1.10750	−0.0429792
$665$	0	0
$666$	0	0
$667$	−19.0551	−0.737816
$668$	9.73804	0.376776
$669$	0	0
$670$	0	0
$671$	−15.8175	−0.610626
$672$	0	0
$673$	−46.2089	−1.78122	−0.890612	−	0.454764i	$-0.849724\pi$
$673$	−46.2089	−1.78122	−0.890612	+	0.454764i	$0.849724\pi$
$674$	2.72921	0.105125
$675$	0	0
$676$	2.92896	0.112652
$677$	4.72177	0.181472	0.0907362	−	0.995875i	$-0.471078\pi$
$677$	4.72177	0.181472	0.0907362	+	0.995875i	$0.471078\pi$
$678$	0	0
$679$	−29.6714	−1.13868
$680$	0	0
$681$	0	0
$682$	6.42514	0.246031
$683$	−8.86671	−0.339275	−0.169638	−	0.985506i	$-0.554260\pi$
$683$	−8.86671	−0.339275	−0.169638	+	0.985506i	$0.554260\pi$
$684$	0	0
$685$	0	0
$686$	−6.00219	−0.229165
$687$	0	0
$688$	−29.1941	−1.11301
$689$	36.3120	1.38338
$690$	0	0
$691$	−51.7396	−1.96827	−0.984133	−	0.177433i	$-0.943221\pi$
$691$	−51.7396	−1.96827	−0.984133	+	0.177433i	$0.943221\pi$
$692$	29.1143	1.10676
$693$	0	0
$694$	1.77983	0.0675616
$695$	0	0
$696$	0	0
$697$	−33.4072	−1.26539
$698$	0.676438	0.0256035
$699$	0	0
$700$	0	0
$701$	0.824350	0.0311353	0.0155676	−	0.999879i	$-0.495044\pi$
$701$	0.824350	0.0311353	0.0155676	+	0.999879i	$0.495044\pi$
$702$	0	0
$703$	22.4896	0.848210
$704$	32.3799	1.22036
$705$	0	0
$706$	−3.34003	−0.125704
$707$	−17.4687	−0.656979
$708$	0	0
$709$	−14.4315	−0.541986	−0.270993	−	0.962581i	$-0.587352\pi$
$709$	−14.4315	−0.541986	−0.270993	+	0.962581i	$0.587352\pi$
$710$	0	0
$711$	0	0
$712$	1.14501	0.0429109
$713$	15.2751	0.572057
$714$	0	0
$715$	0	0
$716$	−41.5450	−1.55261
$717$	0	0
$718$	−7.78894	−0.290681
$719$	−36.2869	−1.35327	−0.676636	−	0.736318i	$-0.736563\pi$
$719$	−36.2869	−1.35327	−0.676636	+	0.736318i	$0.736563\pi$
$720$	0	0
$721$	−12.8743	−0.479466
$722$	2.85749	0.106345
$723$	0	0
$724$	−25.5439	−0.949331
$725$	0	0
$726$	0	0
$727$	21.3585	0.792144	0.396072	−	0.918219i	$-0.370373\pi$
$727$	21.3585	0.792144	0.396072	+	0.918219i	$0.370373\pi$
$728$	7.56156	0.280250
$729$	0	0
$730$	0	0
$731$	−58.7824	−2.17414
$732$	0	0
$733$	14.1185	0.521479	0.260740	−	0.965409i	$-0.416034\pi$
$733$	14.1185	0.521479	0.260740	+	0.965409i	$0.416034\pi$
$734$	6.62873	0.244671
$735$	0	0
$736$	−13.8451	−0.510338
$737$	19.7378	0.727053
$738$	0	0
$739$	15.6183	0.574527	0.287264	−	0.957852i	$-0.407254\pi$
$739$	15.6183	0.574527	0.287264	+	0.957852i	$0.407254\pi$
$740$	0	0
$741$	0	0
$742$	−6.13097	−0.225075
$743$	9.33091	0.342318	0.171159	−	0.985243i	$-0.445249\pi$
$743$	9.33091	0.342318	0.171159	+	0.985243i	$0.445249\pi$
$744$	0	0
$745$	0	0
$746$	4.97511	0.182152
$747$	0	0
$748$	73.3323	2.68129
$749$	−19.3921	−0.708570
$750$	0	0
$751$	−4.05895	−0.148113	−0.0740565	−	0.997254i	$-0.523595\pi$
$751$	−4.05895	−0.148113	−0.0740565	+	0.997254i	$0.523595\pi$
$752$	−21.3632	−0.779036
$753$	0	0
$754$	−4.89936	−0.178424
$755$	0	0
$756$	0	0
$757$	10.3385	0.375760	0.187880	−	0.982192i	$-0.439838\pi$
$757$	10.3385	0.375760	0.187880	+	0.982192i	$0.439838\pi$
$758$	−5.90216	−0.214376
$759$	0	0
$760$	0	0
$761$	5.44584	0.197411	0.0987057	−	0.995117i	$-0.468530\pi$
$761$	5.44584	0.197411	0.0987057	+	0.995117i	$0.468530\pi$
$762$	0	0
$763$	10.2575	0.371345
$764$	−10.3037	−0.372774
$765$	0	0
$766$	−8.66561	−0.313101
$767$	19.4935	0.703871
$768$	0	0
$769$	9.82697	0.354370	0.177185	−	0.984178i	$-0.443301\pi$
$769$	9.82697	0.354370	0.177185	+	0.984178i	$0.443301\pi$
$770$	0	0
$771$	0	0
$772$	−16.6415	−0.598941
$773$	13.0635	0.469862	0.234931	−	0.972012i	$-0.424514\pi$
$773$	13.0635	0.469862	0.234931	+	0.972012i	$0.424514\pi$
$774$	0	0
$775$	0	0
$776$	18.9299	0.679544
$777$	0	0
$778$	−3.13143	−0.112267
$779$	14.9439	0.535422
$780$	0	0
$781$	−46.4764	−1.66306
$782$	−8.53647	−0.305263
$783$	0	0
$784$	−12.0681	−0.431003
$785$	0	0
$786$	0	0
$787$	46.7276	1.66566	0.832830	−	0.553529i	$-0.186719\pi$
$787$	46.7276	1.66566	0.832830	+	0.553529i	$0.186719\pi$
$788$	−17.6366	−0.628276
$789$	0	0
$790$	0	0
$791$	13.0374	0.463558
$792$	0	0
$793$	−9.66867	−0.343344
$794$	−8.97988	−0.318684
$795$	0	0
$796$	−36.4401	−1.29158
$797$	15.1941	0.538203	0.269101	−	0.963112i	$-0.413273\pi$
$797$	15.1941	0.538203	0.269101	+	0.963112i	$0.413273\pi$
$798$	0	0
$799$	−43.0149	−1.52176
$800$	0	0
$801$	0	0
$802$	−6.92539	−0.244544
$803$	8.37754	0.295637
$804$	0	0
$805$	0	0
$806$	3.92746	0.138339
$807$	0	0
$808$	11.1448	0.392072
$809$	−4.47083	−0.157186	−0.0785930	−	0.996907i	$-0.525043\pi$
$809$	−4.47083	−0.157186	−0.0785930	+	0.996907i	$0.525043\pi$
$810$	0	0
$811$	22.3921	0.786294	0.393147	−	0.919476i	$-0.371386\pi$
$811$	22.3921	0.786294	0.393147	+	0.919476i	$0.371386\pi$
$812$	−16.8942	−0.592869
$813$	0	0
$814$	−12.2539	−0.429499
$815$	0	0
$816$	0	0
$817$	26.2949	0.919943
$818$	−2.44005	−0.0853145
$819$	0	0
$820$	0	0
$821$	7.46571	0.260555	0.130278	−	0.991478i	$-0.458413\pi$
$821$	7.46571	0.260555	0.130278	+	0.991478i	$0.458413\pi$
$822$	0	0
$823$	−12.0203	−0.419002	−0.209501	−	0.977808i	$-0.567184\pi$
$823$	−12.0203	−0.419002	−0.209501	+	0.977808i	$0.567184\pi$
$824$	8.21364	0.286136
$825$	0	0
$826$	−3.29132	−0.114520
$827$	−43.0718	−1.49775	−0.748876	−	0.662710i	$-0.769406\pi$
$827$	−43.0718	−1.49775	−0.748876	+	0.662710i	$0.769406\pi$
$828$	0	0
$829$	36.5021	1.26777	0.633885	−	0.773427i	$-0.281459\pi$
$829$	36.5021	1.26777	0.633885	+	0.773427i	$0.281459\pi$
$830$	0	0
$831$	0	0
$832$	19.7927	0.686189
$833$	−24.2992	−0.841916
$834$	0	0
$835$	0	0
$836$	−32.8035	−1.13453
$837$	0	0
$838$	−5.36240	−0.185241
$839$	−20.6669	−0.713502	−0.356751	−	0.934199i	$-0.616116\pi$
$839$	−20.6669	−0.713502	−0.356751	+	0.934199i	$0.616116\pi$
$840$	0	0
$841$	−6.57155	−0.226605
$842$	7.11161	0.245082
$843$	0	0
$844$	−54.1738	−1.86474
$845$	0	0
$846$	0	0
$847$	−36.8226	−1.26524
$848$	−36.9849	−1.27007
$849$	0	0
$850$	0	0
$851$	−29.1324	−0.998645
$852$	0	0
$853$	30.1301	1.03163	0.515817	−	0.856699i	$-0.327488\pi$
$853$	30.1301	1.03163	0.515817	+	0.856699i	$0.327488\pi$
$854$	1.63247	0.0558620
$855$	0	0
$856$	12.3718	0.422861
$857$	−2.03243	−0.0694266	−0.0347133	−	0.999397i	$-0.511052\pi$
$857$	−2.03243	−0.0694266	−0.0347133	+	0.999397i	$0.511052\pi$
$858$	0	0
$859$	−14.1565	−0.483015	−0.241507	−	0.970399i	$-0.577642\pi$
$859$	−14.1565	−0.483015	−0.241507	+	0.970399i	$0.577642\pi$
$860$	0	0
$861$	0	0
$862$	−3.83355	−0.130571
$863$	−15.4225	−0.524989	−0.262495	−	0.964933i	$-0.584545\pi$
$863$	−15.4225	−0.524989	−0.262495	+	0.964933i	$0.584545\pi$
$864$	0	0
$865$	0	0
$866$	2.51050	0.0853103
$867$	0	0
$868$	13.5428	0.459674
$869$	3.48634	0.118266
$870$	0	0
$871$	12.0651	0.408809
$872$	−6.54411	−0.221612
$873$	0	0
$874$	3.81859	0.129166
$875$	0	0
$876$	0	0
$877$	−51.2442	−1.73039	−0.865197	−	0.501433i	$-0.832806\pi$
$877$	−51.2442	−1.73039	−0.865197	+	0.501433i	$0.832806\pi$
$878$	4.05794	0.136949
$879$	0	0
$880$	0	0
$881$	16.0748	0.541572	0.270786	−	0.962640i	$-0.412716\pi$
$881$	16.0748	0.541572	0.270786	+	0.962640i	$0.412716\pi$
$882$	0	0
$883$	−30.4342	−1.02419	−0.512096	−	0.858928i	$-0.671131\pi$
$883$	−30.4342	−1.02419	−0.512096	+	0.858928i	$0.671131\pi$
$884$	44.8255	1.50765
$885$	0	0
$886$	−2.15002	−0.0722312
$887$	14.9790	0.502945	0.251473	−	0.967864i	$-0.419085\pi$
$887$	14.9790	0.502945	0.251473	+	0.967864i	$0.419085\pi$
$888$	0	0
$889$	−33.7632	−1.13238
$890$	0	0
$891$	0	0
$892$	29.6041	0.991220
$893$	19.2417	0.643900
$894$	0	0
$895$	0	0
$896$	−16.2179	−0.541802
$897$	0	0
$898$	−0.357543	−0.0119313
$899$	−17.9793	−0.599642
$900$	0	0
$901$	−74.4693	−2.48093
$902$	−8.14252	−0.271116
$903$	0	0
$904$	−8.31769	−0.276642
$905$	0	0
$906$	0	0
$907$	34.9255	1.15968	0.579841	−	0.814730i	$-0.303115\pi$
$907$	34.9255	1.15968	0.579841	+	0.814730i	$0.303115\pi$
$908$	52.3239	1.73643
$909$	0	0
$910$	0	0
$911$	15.1544	0.502087	0.251043	−	0.967976i	$-0.419226\pi$
$911$	15.1544	0.502087	0.251043	+	0.967976i	$0.419226\pi$
$912$	0	0
$913$	5.13923	0.170084
$914$	−1.66546	−0.0550884
$915$	0	0
$916$	−23.7240	−0.783864
$917$	−28.8151	−0.951559
$918$	0	0
$919$	35.1591	1.15979	0.579896	−	0.814690i	$-0.303093\pi$
$919$	35.1591	1.15979	0.579896	+	0.814690i	$0.303093\pi$
$920$	0	0
$921$	0	0
$922$	11.3586	0.374074
$923$	−28.4094	−0.935107
$924$	0	0
$925$	0	0
$926$	−4.79752	−0.157656
$927$	0	0
$928$	16.2961	0.534947
$929$	−7.75919	−0.254571	−0.127285	−	0.991866i	$-0.540626\pi$
$929$	−7.75919	−0.254571	−0.127285	+	0.991866i	$0.540626\pi$
$930$	0	0
$931$	10.8697	0.356239
$932$	13.4235	0.439701
$933$	0	0
$934$	3.48510	0.114036
$935$	0	0
$936$	0	0
$937$	31.5637	1.03114	0.515571	−	0.856847i	$-0.327580\pi$
$937$	31.5637	1.03114	0.515571	+	0.856847i	$0.327580\pi$
$938$	−2.03708	−0.0665131
$939$	0	0
$940$	0	0
$941$	32.3923	1.05596	0.527979	−	0.849257i	$-0.322950\pi$
$941$	32.3923	1.05596	0.527979	+	0.849257i	$0.322950\pi$
$942$	0	0
$943$	−19.3580	−0.630383
$944$	−19.8548	−0.646218
$945$	0	0
$946$	−14.3273	−0.465822
$947$	40.4439	1.31425	0.657125	−	0.753782i	$-0.271773\pi$
$947$	40.4439	1.31425	0.657125	+	0.753782i	$0.271773\pi$
$948$	0	0
$949$	5.12090	0.166231
$950$	0	0
$951$	0	0
$952$	−15.5074	−0.502597
$953$	46.2149	1.49705	0.748523	−	0.663108i	$-0.230763\pi$
$953$	46.2149	1.49705	0.748523	+	0.663108i	$0.230763\pi$
$954$	0	0
$955$	0	0
$956$	48.4884	1.56823
$957$	0	0
$958$	−1.07200	−0.0346348
$959$	20.2422	0.653653
$960$	0	0
$961$	−16.5873	−0.535075
$962$	−7.49039	−0.241500
$963$	0	0
$964$	−26.1626	−0.842639
$965$	0	0
$966$	0	0
$967$	13.5029	0.434225	0.217112	−	0.976147i	$-0.430336\pi$
$967$	13.5029	0.434225	0.217112	+	0.976147i	$0.430336\pi$
$968$	23.4923	0.755071
$969$	0	0
$970$	0	0
$971$	−42.6730	−1.36944	−0.684721	−	0.728805i	$-0.740076\pi$
$971$	−42.6730	−1.36944	−0.684721	+	0.728805i	$0.740076\pi$
$972$	0	0
$973$	−4.50380	−0.144385
$974$	2.40574	0.0770848
$975$	0	0
$976$	9.84784	0.315222
$977$	25.2622	0.808208	0.404104	−	0.914713i	$-0.367583\pi$
$977$	25.2622	0.808208	0.404104	+	0.914713i	$0.367583\pi$
$978$	0	0
$979$	−5.31329	−0.169813
$980$	0	0
$981$	0	0
$982$	−7.78217	−0.248339
$983$	−26.7259	−0.852425	−0.426212	−	0.904623i	$-0.640152\pi$
$983$	−26.7259	−0.852425	−0.426212	+	0.904623i	$0.640152\pi$
$984$	0	0
$985$	0	0
$986$	10.0477	0.319984
$987$	0	0
$988$	−20.0516	−0.637928
$989$	−34.0618	−1.08310
$990$	0	0
$991$	7.54636	0.239718	0.119859	−	0.992791i	$-0.461756\pi$
$991$	7.54636	0.239718	0.119859	+	0.992791i	$0.461756\pi$
$992$	−13.0634	−0.414765
$993$	0	0
$994$	4.79668	0.152142
$995$	0	0
$996$	0	0
$997$	−20.7403	−0.656851	−0.328425	−	0.944530i	$-0.606518\pi$
$997$	−20.7403	−0.656851	−0.328425	+	0.944530i	$0.606518\pi$
$998$	−1.54197	−0.0488101
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bf.1.11		24
3.2	odd	2	inner	5625.2.a.bf.1.13		24
5.4	even	2	inner	5625.2.a.bf.1.14		24
15.14	odd	2	inner	5625.2.a.bf.1.12		24
25.3	odd	20		225.2.m.c.109.4	yes	24
25.17	odd	20		225.2.m.c.64.4	yes	24
75.17	even	20		225.2.m.c.64.3	✓	24
75.53	even	20		225.2.m.c.109.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.64.3	✓	24	75.17	even	20
225.2.m.c.64.4	yes	24	25.17	odd	20
225.2.m.c.109.3	yes	24	75.53	even	20
225.2.m.c.109.4	yes	24	25.3	odd	20
5625.2.a.bf.1.11		24	1.1	even	1	trivial
5625.2.a.bf.1.12		24	15.14	odd	2	inner
5625.2.a.bf.1.13		24	3.2	odd	2	inner
5625.2.a.bf.1.14		24	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.305545 q^{2} -1.90664 q^{4} -1.87098 q^{7} +1.19366 q^{8} +O(q^{10})\)	\(q-0.305545 q^{2} -1.90664 q^{4} -1.87098 q^{7} +1.19366 q^{8} -5.53904 q^{11} -3.38583 q^{13} +0.571667 q^{14} +3.44857 q^{16} +6.94371 q^{17} -3.10611 q^{19} +1.69243 q^{22} +4.02357 q^{23} +1.03452 q^{26} +3.56728 q^{28} -4.73587 q^{29} +3.79640 q^{31} -3.44100 q^{32} -2.12162 q^{34} -7.24043 q^{37} +0.949055 q^{38} -4.81115 q^{41} -8.46556 q^{43} +10.5610 q^{44} -1.22938 q^{46} -6.19480 q^{47} -3.49945 q^{49} +6.45556 q^{52} -10.7247 q^{53} -2.23330 q^{56} +1.44702 q^{58} -5.75740 q^{59} +2.85563 q^{61} -1.15997 q^{62} -5.84576 q^{64} -3.56340 q^{67} -13.2392 q^{68} +8.39069 q^{71} -1.51245 q^{73} +2.21228 q^{74} +5.92223 q^{76} +10.3634 q^{77} -0.629411 q^{79} +1.47002 q^{82} -0.927819 q^{83} +2.58661 q^{86} -6.61171 q^{88} +0.959243 q^{89} +6.33480 q^{91} -7.67151 q^{92} +1.89279 q^{94} +15.8588 q^{97} +1.06924 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4}+O(q^{10}) \) 24 * q + 32 * q^4	\( 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) \) 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

Introduction

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