LMFDB - Embedded newform 5625.2.a.y.1.2

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.46980000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 $ x^8 - 15x^6 + 80x^4 - 180*x^2 + 145
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			1.2
Root			$-2.05160$ of defining polynomial
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)

$f(q)$	$=$	$q-2.05160 q^{2} +2.20906 q^{4} -1.27977 q^{7} -0.428901 q^{8} +O(q^{10})$	$q-2.05160 q^{2} +2.20906 q^{4} -1.27977 q^{7} -0.428901 q^{8} +3.17448 q^{11} +3.19236 q^{13} +2.62558 q^{14} -3.53818 q^{16} -1.35762 q^{17} +1.53818 q^{19} -6.51275 q^{22} -2.39083 q^{23} -6.54945 q^{26} -2.82709 q^{28} +6.80294 q^{29} -7.49606 q^{31} +8.11673 q^{32} +2.78530 q^{34} -5.24171 q^{37} -3.15573 q^{38} +6.41987 q^{41} -11.1216 q^{43} +7.01260 q^{44} +4.90503 q^{46} -12.4695 q^{47} -5.36218 q^{49} +7.05211 q^{52} -0.474735 q^{53} +0.548896 q^{56} -13.9569 q^{58} +8.04257 q^{59} +10.4813 q^{61} +15.3789 q^{62} -9.57591 q^{64} -6.45833 q^{67} -2.99907 q^{68} -14.8603 q^{71} +1.77583 q^{73} +10.7539 q^{74} +3.39793 q^{76} -4.06261 q^{77} -0.781806 q^{79} -13.1710 q^{82} +1.43379 q^{83} +22.8172 q^{86} -1.36154 q^{88} +7.89157 q^{89} -4.08550 q^{91} -5.28149 q^{92} +25.5824 q^{94} -13.7275 q^{97} +11.0010 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 - 10 * q^7	$ 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67} - 50 q^{73} - 34 q^{76} - 12 q^{79} - 60 q^{82} - 70 q^{88} - 10 q^{91} - 20 q^{94} - 50 q^{97}+O(q^{100}) $ 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * 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 $

$2$	−2.05160	−1.45070	−0.725350	−	0.688381i	$-0.758322\pi$
$2$	−2.05160	−1.45070	−0.725350	+	0.688381i	$0.758322\pi$
$3$	0	0
$3$	0	0
$4$	2.20906	1.10453
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.27977	−0.483709	−0.241854	−	0.970313i	$-0.577756\pi$
$7$	−1.27977	−0.483709	−0.241854	+	0.970313i	$0.577756\pi$
$8$	−0.428901	−0.151639
$9$	0	0
$10$	0	0
$11$	3.17448	0.957140	0.478570	−	0.878049i	$-0.341155\pi$
$11$	3.17448	0.957140	0.478570	+	0.878049i	$0.341155\pi$
$12$	0	0
$13$	3.19236	0.885402	0.442701	−	0.896669i	$-0.354020\pi$
$13$	3.19236	0.885402	0.442701	+	0.896669i	$0.354020\pi$
$14$	2.62558	0.701716
$15$	0	0
$16$	−3.53818	−0.884545
$17$	−1.35762	−0.329272	−0.164636	−	0.986354i	$-0.552645\pi$
$17$	−1.35762	−0.329272	−0.164636	+	0.986354i	$0.552645\pi$
$18$	0	0
$19$	1.53818	0.352883	0.176442	−	0.984311i	$-0.443541\pi$
$19$	1.53818	0.352883	0.176442	+	0.984311i	$0.443541\pi$
$20$	0	0
$21$	0	0
$22$	−6.51275	−1.38852
$23$	−2.39083	−0.498524	−0.249262	−	0.968436i	$-0.580188\pi$
$23$	−2.39083	−0.498524	−0.249262	+	0.968436i	$0.580188\pi$
$24$	0	0
$25$	0	0
$26$	−6.54945	−1.28445
$27$	0	0
$28$	−2.82709	−0.534270
$29$	6.80294	1.26327	0.631637	−	0.775264i	$-0.282383\pi$
$29$	6.80294	1.26327	0.631637	+	0.775264i	$0.282383\pi$
$30$	0	0
$31$	−7.49606	−1.34633	−0.673166	−	0.739492i	$-0.735066\pi$
$31$	−7.49606	−1.34633	−0.673166	+	0.739492i	$0.735066\pi$
$32$	8.11673	1.43485
$33$	0	0
$34$	2.78530	0.477674
$35$	0	0
$36$	0	0
$37$	−5.24171	−0.861733	−0.430866	−	0.902416i	$-0.641792\pi$
$37$	−5.24171	−0.861733	−0.430866	+	0.902416i	$0.641792\pi$
$38$	−3.15573	−0.511927
$39$	0	0
$40$	0	0
$41$	6.41987	1.00262	0.501308	−	0.865269i	$-0.332852\pi$
$41$	6.41987	1.00262	0.501308	+	0.865269i	$0.332852\pi$
$42$	0	0
$43$	−11.1216	−1.69604	−0.848018	−	0.529968i	$-0.822204\pi$
$43$	−11.1216	−1.69604	−0.848018	+	0.529968i	$0.822204\pi$
$44$	7.01260	1.05719
$45$	0	0
$46$	4.90503	0.723208
$47$	−12.4695	−1.81886	−0.909432	−	0.415853i	$-0.863483\pi$
$47$	−12.4695	−1.81886	−0.909432	+	0.415853i	$0.863483\pi$
$48$	0	0
$49$	−5.36218	−0.766026
$50$	0	0
$51$	0	0
$52$	7.05211	0.977952
$53$	−0.474735	−0.0652098	−0.0326049	−	0.999468i	$-0.510380\pi$
$53$	−0.474735	−0.0652098	−0.0326049	+	0.999468i	$0.510380\pi$
$54$	0	0
$55$	0	0
$56$	0.548896	0.0733493
$57$	0	0
$58$	−13.9569	−1.83263
$59$	8.04257	1.04705	0.523527	−	0.852009i	$-0.324616\pi$
$59$	8.04257	1.04705	0.523527	+	0.852009i	$0.324616\pi$
$60$	0	0
$61$	10.4813	1.34199	0.670995	−	0.741462i	$-0.265867\pi$
$61$	10.4813	1.34199	0.670995	+	0.741462i	$0.265867\pi$
$62$	15.3789	1.95312
$63$	0	0
$64$	−9.57591	−1.19699
$65$	0	0
$66$	0	0
$67$	−6.45833	−0.789010	−0.394505	−	0.918894i	$-0.629084\pi$
$67$	−6.45833	−0.789010	−0.394505	+	0.918894i	$0.629084\pi$
$68$	−2.99907	−0.363690
$69$	0	0
$70$	0	0
$71$	−14.8603	−1.76360	−0.881799	−	0.471626i	$-0.843667\pi$
$71$	−14.8603	−1.76360	−0.881799	+	0.471626i	$0.843667\pi$
$72$	0	0
$73$	1.77583	0.207845	0.103923	−	0.994585i	$-0.466861\pi$
$73$	1.77583	0.207845	0.103923	+	0.994585i	$0.466861\pi$
$74$	10.7539	1.25011
$75$	0	0
$76$	3.39793	0.389769
$77$	−4.06261	−0.462977
$78$	0	0
$79$	−0.781806	−0.0879601	−0.0439800	−	0.999032i	$-0.514004\pi$
$79$	−0.781806	−0.0879601	−0.0439800	+	0.999032i	$0.514004\pi$
$80$	0	0
$81$	0	0
$82$	−13.1710	−1.45449
$83$	1.43379	0.157379	0.0786893	−	0.996899i	$-0.474927\pi$
$83$	1.43379	0.157379	0.0786893	+	0.996899i	$0.474927\pi$
$84$	0	0
$85$	0	0
$86$	22.8172	2.46044
$87$	0	0
$88$	−1.36154	−0.145140
$89$	7.89157	0.836504	0.418252	−	0.908331i	$-0.362643\pi$
$89$	7.89157	0.836504	0.418252	+	0.908331i	$0.362643\pi$
$90$	0	0
$91$	−4.08550	−0.428277
$92$	−5.28149	−0.550633
$93$	0	0
$94$	25.5824	2.63862
$95$	0	0
$96$	0	0
$97$	−13.7275	−1.39381	−0.696906	−	0.717163i	$-0.745440\pi$
$97$	−13.7275	−1.39381	−0.696906	+	0.717163i	$0.745440\pi$
$98$	11.0010	1.11127
$99$	0	0
$100$	0	0
$101$	−19.3621	−1.92660	−0.963301	−	0.268425i	$-0.913497\pi$
$101$	−19.3621	−1.92660	−0.963301	+	0.268425i	$0.913497\pi$
$102$	0	0
$103$	−3.17193	−0.312540	−0.156270	−	0.987714i	$-0.549947\pi$
$103$	−3.17193	−0.312540	−0.156270	+	0.987714i	$0.549947\pi$
$104$	−1.36921	−0.134262
$105$	0	0
$106$	0.973965	0.0945999
$107$	12.0844	1.16825	0.584123	−	0.811665i	$-0.301439\pi$
$107$	12.0844	1.16825	0.584123	+	0.811665i	$0.301439\pi$
$108$	0	0
$109$	2.44512	0.234200	0.117100	−	0.993120i	$-0.462640\pi$
$109$	2.44512	0.234200	0.117100	+	0.993120i	$0.462640\pi$
$110$	0	0
$111$	0	0
$112$	4.52807	0.427862
$113$	15.9226	1.49787	0.748934	−	0.662645i	$-0.230566\pi$
$113$	15.9226	1.49787	0.748934	+	0.662645i	$0.230566\pi$
$114$	0	0
$115$	0	0
$116$	15.0281	1.39532
$117$	0	0
$118$	−16.5001	−1.51896
$119$	1.73745	0.159272
$120$	0	0
$121$	−0.922704	−0.0838821
$122$	−21.5034	−1.94682
$123$	0	0
$124$	−16.5592	−1.48706
$125$	0	0
$126$	0	0
$127$	2.44321	0.216800	0.108400	−	0.994107i	$-0.465427\pi$
$127$	2.44321	0.216800	0.108400	+	0.994107i	$0.465427\pi$
$128$	3.41246	0.301622
$129$	0	0
$130$	0	0
$131$	−4.16819	−0.364177	−0.182088	−	0.983282i	$-0.558286\pi$
$131$	−4.16819	−0.364177	−0.182088	+	0.983282i	$0.558286\pi$
$132$	0	0
$133$	−1.96852	−0.170693
$134$	13.2499	1.14462
$135$	0	0
$136$	0.582285	0.0499306
$137$	18.3154	1.56479	0.782395	−	0.622783i	$-0.213998\pi$
$137$	18.3154	1.56479	0.782395	+	0.622783i	$0.213998\pi$
$138$	0	0
$139$	−7.87953	−0.668333	−0.334167	−	0.942514i	$-0.608455\pi$
$139$	−7.87953	−0.668333	−0.334167	+	0.942514i	$0.608455\pi$
$140$	0	0
$141$	0	0
$142$	30.4874	2.55845
$143$	10.1341	0.847454
$144$	0	0
$145$	0	0
$146$	−3.64329	−0.301521
$147$	0	0
$148$	−11.5792	−0.951808
$149$	15.6658	1.28339	0.641697	−	0.766958i	$-0.278231\pi$
$149$	15.6658	1.28339	0.641697	+	0.766958i	$0.278231\pi$
$150$	0	0
$151$	6.21003	0.505365	0.252683	−	0.967549i	$-0.418687\pi$
$151$	6.21003	0.505365	0.252683	+	0.967549i	$0.418687\pi$
$152$	−0.659727	−0.0535110
$153$	0	0
$154$	8.33484	0.671641
$155$	0	0
$156$	0	0
$157$	−13.2210	−1.05515	−0.527574	−	0.849509i	$-0.676898\pi$
$157$	−13.2210	−1.05515	−0.527574	+	0.849509i	$0.676898\pi$
$158$	1.60395	0.127604
$159$	0	0
$160$	0	0
$161$	3.05973	0.241140
$162$	0	0
$163$	−17.6762	−1.38451	−0.692253	−	0.721655i	$-0.743382\pi$
$163$	−17.6762	−1.38451	−0.692253	+	0.721655i	$0.743382\pi$
$164$	14.1819	1.10742
$165$	0	0
$166$	−2.94155	−0.228309
$167$	−4.77843	−0.369766	−0.184883	−	0.982761i	$-0.559191\pi$
$167$	−4.77843	−0.369766	−0.184883	+	0.982761i	$0.559191\pi$
$168$	0	0
$169$	−2.80882	−0.216063
$170$	0	0
$171$	0	0
$172$	−24.5684	−1.87332
$173$	−15.1138	−1.14908	−0.574541	−	0.818476i	$-0.694820\pi$
$173$	−15.1138	−1.14908	−0.574541	+	0.818476i	$0.694820\pi$
$174$	0	0
$175$	0	0
$176$	−11.2319	−0.846634
$177$	0	0
$178$	−16.1903	−1.21352
$179$	19.3529	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$179$	19.3529	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$180$	0	0
$181$	3.81584	0.283629	0.141815	−	0.989893i	$-0.454706\pi$
$181$	3.81584	0.283629	0.141815	+	0.989893i	$0.454706\pi$
$182$	8.38181	0.621301
$183$	0	0
$184$	1.02543	0.0755958
$185$	0	0
$186$	0	0
$187$	−4.30974	−0.315159
$188$	−27.5458	−2.00899
$189$	0	0
$190$	0	0
$191$	0.690735	0.0499798	0.0249899	−	0.999688i	$-0.492045\pi$
$191$	0.690735	0.0499798	0.0249899	+	0.999688i	$0.492045\pi$
$192$	0	0
$193$	17.3564	1.24934	0.624671	−	0.780888i	$-0.285233\pi$
$193$	17.3564	1.24934	0.624671	+	0.780888i	$0.285233\pi$
$194$	28.1632	2.02200
$195$	0	0
$196$	−11.8454	−0.846097
$197$	−10.6985	−0.762235	−0.381118	−	0.924527i	$-0.624461\pi$
$197$	−10.6985	−0.762235	−0.381118	+	0.924527i	$0.624461\pi$
$198$	0	0
$199$	−10.9439	−0.775795	−0.387898	−	0.921702i	$-0.626799\pi$
$199$	−10.9439	−0.775795	−0.387898	+	0.921702i	$0.626799\pi$
$200$	0	0
$201$	0	0
$202$	39.7233	2.79492
$203$	−8.70622	−0.611057
$204$	0	0
$205$	0	0
$206$	6.50753	0.453401
$207$	0	0
$208$	−11.2952	−0.783178
$209$	4.88292	0.337759
$210$	0	0
$211$	−7.42883	−0.511422	−0.255711	−	0.966753i	$-0.582310\pi$
$211$	−7.42883	−0.511422	−0.255711	+	0.966753i	$0.582310\pi$
$212$	−1.04872	−0.0720261
$213$	0	0
$214$	−24.7924	−1.69477
$215$	0	0
$216$	0	0
$217$	9.59325	0.651232
$218$	−5.01641	−0.339754
$219$	0	0
$220$	0	0
$221$	−4.33402	−0.291538
$222$	0	0
$223$	−15.5938	−1.04424	−0.522118	−	0.852873i	$-0.674858\pi$
$223$	−15.5938	−1.04424	−0.522118	+	0.852873i	$0.674858\pi$
$224$	−10.3876	−0.694049
$225$	0	0
$226$	−32.6667	−2.17296
$227$	−11.5698	−0.767913	−0.383957	−	0.923351i	$-0.625439\pi$
$227$	−11.5698	−0.767913	−0.383957	+	0.923351i	$0.625439\pi$
$228$	0	0
$229$	16.0725	1.06210	0.531052	−	0.847339i	$-0.321797\pi$
$229$	16.0725	1.06210	0.531052	+	0.847339i	$0.321797\pi$
$230$	0	0
$231$	0	0
$232$	−2.91779	−0.191562
$233$	−6.13929	−0.402198	−0.201099	−	0.979571i	$-0.564451\pi$
$233$	−6.13929	−0.402198	−0.201099	+	0.979571i	$0.564451\pi$
$234$	0	0
$235$	0	0
$236$	17.7665	1.15650
$237$	0	0
$238$	−3.56455	−0.231055
$239$	−16.2825	−1.05323	−0.526615	−	0.850104i	$-0.676539\pi$
$239$	−16.2825	−1.05323	−0.526615	+	0.850104i	$0.676539\pi$
$240$	0	0
$241$	25.5758	1.64748	0.823741	−	0.566967i	$-0.191883\pi$
$241$	25.5758	1.64748	0.823741	+	0.566967i	$0.191883\pi$
$242$	1.89302	0.121688
$243$	0	0
$244$	23.1537	1.48227
$245$	0	0
$246$	0	0
$247$	4.91043	0.312443
$248$	3.21507	0.204157
$249$	0	0
$250$	0	0
$251$	−19.2234	−1.21337	−0.606684	−	0.794943i	$-0.707501\pi$
$251$	−19.2234	−1.21337	−0.606684	+	0.794943i	$0.707501\pi$
$252$	0	0
$253$	−7.58965	−0.477157
$254$	−5.01250	−0.314512
$255$	0	0
$256$	12.1508	0.759426
$257$	5.96388	0.372017	0.186008	−	0.982548i	$-0.440445\pi$
$257$	5.96388	0.372017	0.186008	+	0.982548i	$0.440445\pi$
$258$	0	0
$259$	6.70820	0.416828
$260$	0	0
$261$	0	0
$262$	8.55146	0.528311
$263$	15.4343	0.951721	0.475860	−	0.879521i	$-0.342137\pi$
$263$	15.4343	0.951721	0.475860	+	0.879521i	$0.342137\pi$
$264$	0	0
$265$	0	0
$266$	4.03862	0.247624
$267$	0	0
$268$	−14.2668	−0.871484
$269$	−27.0654	−1.65021	−0.825104	−	0.564980i	$-0.808884\pi$
$269$	−27.0654	−1.65021	−0.825104	+	0.564980i	$0.808884\pi$
$270$	0	0
$271$	−9.01669	−0.547725	−0.273863	−	0.961769i	$-0.588301\pi$
$271$	−9.01669	−0.547725	−0.273863	+	0.961769i	$0.588301\pi$
$272$	4.80351	0.291256
$273$	0	0
$274$	−37.5758	−2.27004
$275$	0	0
$276$	0	0
$277$	13.0034	0.781296	0.390648	−	0.920540i	$-0.372251\pi$
$277$	13.0034	0.781296	0.390648	+	0.920540i	$0.372251\pi$
$278$	16.1656	0.969551
$279$	0	0
$280$	0	0
$281$	−10.0922	−0.602048	−0.301024	−	0.953617i	$-0.597328\pi$
$281$	−10.0922	−0.602048	−0.301024	+	0.953617i	$0.597328\pi$
$282$	0	0
$283$	24.3549	1.44775	0.723873	−	0.689933i	$-0.242360\pi$
$283$	24.3549	1.44775	0.723873	+	0.689933i	$0.242360\pi$
$284$	−32.8273	−1.94794
$285$	0	0
$286$	−20.7911	−1.22940
$287$	−8.21598	−0.484974
$288$	0	0
$289$	−15.1569	−0.891580
$290$	0	0
$291$	0	0
$292$	3.92291	0.229571
$293$	−11.8059	−0.689705	−0.344853	−	0.938657i	$-0.612071\pi$
$293$	−11.8059	−0.689705	−0.344853	+	0.938657i	$0.612071\pi$
$294$	0	0
$295$	0	0
$296$	2.24818	0.130673
$297$	0	0
$298$	−32.1400	−1.86182
$299$	−7.63241	−0.441394
$300$	0	0
$301$	14.2332	0.820387
$302$	−12.7405	−0.733133
$303$	0	0
$304$	−5.44236	−0.312141
$305$	0	0
$306$	0	0
$307$	0.661409	0.0377486	0.0188743	−	0.999822i	$-0.493992\pi$
$307$	0.661409	0.0377486	0.0188743	+	0.999822i	$0.493992\pi$
$308$	−8.97453	−0.511371
$309$	0	0
$310$	0	0
$311$	−6.33221	−0.359067	−0.179533	−	0.983752i	$-0.557459\pi$
$311$	−6.33221	−0.359067	−0.179533	+	0.983752i	$0.557459\pi$
$312$	0	0
$313$	−32.5795	−1.84150	−0.920752	−	0.390149i	$-0.872423\pi$
$313$	−32.5795	−1.84150	−0.920752	+	0.390149i	$0.872423\pi$
$314$	27.1241	1.53070
$315$	0	0
$316$	−1.72705	−0.0971544
$317$	−2.63474	−0.147982	−0.0739910	−	0.997259i	$-0.523574\pi$
$317$	−2.63474	−0.147982	−0.0739910	+	0.997259i	$0.523574\pi$
$318$	0	0
$319$	21.5958	1.20913
$320$	0	0
$321$	0	0
$322$	−6.27733	−0.349822
$323$	−2.08827	−0.116194
$324$	0	0
$325$	0	0
$326$	36.2645	2.00850
$327$	0	0
$328$	−2.75349	−0.152036
$329$	15.9581	0.879800
$330$	0	0
$331$	7.37067	0.405129	0.202564	−	0.979269i	$-0.435072\pi$
$331$	7.37067	0.405129	0.202564	+	0.979269i	$0.435072\pi$
$332$	3.16732	0.173829
$333$	0	0
$334$	9.80342	0.536419
$335$	0	0
$336$	0	0
$337$	24.8563	1.35401	0.677005	−	0.735978i	$-0.263277\pi$
$337$	24.8563	1.35401	0.677005	+	0.735978i	$0.263277\pi$
$338$	5.76257	0.313442
$339$	0	0
$340$	0	0
$341$	−23.7960	−1.28863
$342$	0	0
$343$	15.8208	0.854242
$344$	4.77008	0.257186
$345$	0	0
$346$	31.0075	1.66697
$347$	22.2587	1.19491	0.597454	−	0.801903i	$-0.296179\pi$
$347$	22.2587	1.19491	0.597454	+	0.801903i	$0.296179\pi$
$348$	0	0
$349$	24.0115	1.28531	0.642653	−	0.766157i	$-0.277834\pi$
$349$	24.0115	1.28531	0.642653	+	0.766157i	$0.277834\pi$
$350$	0	0
$351$	0	0
$352$	25.7664	1.37335
$353$	−28.9964	−1.54332	−0.771660	−	0.636035i	$-0.780573\pi$
$353$	−28.9964	−1.54332	−0.771660	+	0.636035i	$0.780573\pi$
$354$	0	0
$355$	0	0
$356$	17.4329	0.923943
$357$	0	0
$358$	−39.7045	−2.09845
$359$	32.3192	1.70574	0.852870	−	0.522123i	$-0.174860\pi$
$359$	32.3192	1.70574	0.852870	+	0.522123i	$0.174860\pi$
$360$	0	0
$361$	−16.6340	−0.875474
$362$	−7.82857	−0.411461
$363$	0	0
$364$	−9.02510	−0.473044
$365$	0	0
$366$	0	0
$367$	21.0649	1.09958	0.549791	−	0.835303i	$-0.314708\pi$
$367$	21.0649	1.09958	0.549791	+	0.835303i	$0.314708\pi$
$368$	8.45921	0.440967
$369$	0	0
$370$	0	0
$371$	0.607553	0.0315426
$372$	0	0
$373$	4.99647	0.258707	0.129354	−	0.991599i	$-0.458710\pi$
$373$	4.99647	0.258707	0.129354	+	0.991599i	$0.458710\pi$
$374$	8.84186	0.457201
$375$	0	0
$376$	5.34818	0.275811
$377$	21.7174	1.11851
$378$	0	0
$379$	−5.19232	−0.266712	−0.133356	−	0.991068i	$-0.542575\pi$
$379$	−5.19232	−0.266712	−0.133356	+	0.991068i	$0.542575\pi$
$380$	0	0
$381$	0	0
$382$	−1.41711	−0.0725057
$383$	32.6926	1.67052	0.835258	−	0.549858i	$-0.185318\pi$
$383$	32.6926	1.67052	0.835258	+	0.549858i	$0.185318\pi$
$384$	0	0
$385$	0	0
$386$	−35.6084	−1.81242
$387$	0	0
$388$	−30.3247	−1.53950
$389$	−26.8706	−1.36239	−0.681197	−	0.732100i	$-0.738540\pi$
$389$	−26.8706	−1.36239	−0.681197	+	0.732100i	$0.738540\pi$
$390$	0	0
$391$	3.24585	0.164150
$392$	2.29984	0.116160
$393$	0	0
$394$	21.9490	1.10577
$395$	0	0
$396$	0	0
$397$	−21.0934	−1.05865	−0.529323	−	0.848420i	$-0.677554\pi$
$397$	−21.0934	−1.05865	−0.529323	+	0.848420i	$0.677554\pi$
$398$	22.4526	1.12545
$399$	0	0
$400$	0	0
$401$	26.1120	1.30397	0.651987	−	0.758230i	$-0.273936\pi$
$401$	26.1120	1.30397	0.651987	+	0.758230i	$0.273936\pi$
$402$	0	0
$403$	−23.9301	−1.19204
$404$	−42.7720	−2.12799
$405$	0	0
$406$	17.8617	0.886459
$407$	−16.6397	−0.824799
$408$	0	0
$409$	10.4994	0.519160	0.259580	−	0.965722i	$-0.416416\pi$
$409$	10.4994	0.519160	0.259580	+	0.965722i	$0.416416\pi$
$410$	0	0
$411$	0	0
$412$	−7.00698	−0.345209
$413$	−10.2927	−0.506469
$414$	0	0
$415$	0	0
$416$	25.9115	1.27042
$417$	0	0
$418$	−10.0178	−0.489986
$419$	−12.3587	−0.603760	−0.301880	−	0.953346i	$-0.597614\pi$
$419$	−12.3587	−0.603760	−0.301880	+	0.953346i	$0.597614\pi$
$420$	0	0
$421$	−29.0924	−1.41788	−0.708938	−	0.705270i	$-0.750826\pi$
$421$	−29.0924	−1.41788	−0.708938	+	0.705270i	$0.750826\pi$
$422$	15.2410	0.741919
$423$	0	0
$424$	0.203614	0.00988838
$425$	0	0
$426$	0	0
$427$	−13.4136	−0.649132
$428$	26.6952	1.29036
$429$	0	0
$430$	0	0
$431$	−19.2596	−0.927703	−0.463851	−	0.885913i	$-0.653533\pi$
$431$	−19.2596	−0.927703	−0.463851	+	0.885913i	$0.653533\pi$
$432$	0	0
$433$	15.5475	0.747166	0.373583	−	0.927597i	$-0.378129\pi$
$433$	15.5475	0.747166	0.373583	+	0.927597i	$0.378129\pi$
$434$	−19.6815	−0.944742
$435$	0	0
$436$	5.40142	0.258681
$437$	−3.67754	−0.175920
$438$	0	0
$439$	25.2432	1.20479	0.602395	−	0.798198i	$-0.294213\pi$
$439$	25.2432	1.20479	0.602395	+	0.798198i	$0.294213\pi$
$440$	0	0
$441$	0	0
$442$	8.89168	0.422934
$443$	2.35092	0.111696	0.0558479	−	0.998439i	$-0.482214\pi$
$443$	2.35092	0.111696	0.0558479	+	0.998439i	$0.482214\pi$
$444$	0	0
$445$	0	0
$446$	31.9922	1.51487
$447$	0	0
$448$	12.2550	0.578994
$449$	−27.1165	−1.27971	−0.639854	−	0.768497i	$-0.721005\pi$
$449$	−27.1165	−1.27971	−0.639854	+	0.768497i	$0.721005\pi$
$450$	0	0
$451$	20.3797	0.959644
$452$	35.1738	1.65444
$453$	0	0
$454$	23.7365	1.11401
$455$	0	0
$456$	0	0
$457$	2.68124	0.125423	0.0627115	−	0.998032i	$-0.480025\pi$
$457$	2.68124	0.125423	0.0627115	+	0.998032i	$0.480025\pi$
$458$	−32.9744	−1.54079
$459$	0	0
$460$	0	0
$461$	−25.4938	−1.18737	−0.593683	−	0.804699i	$-0.702327\pi$
$461$	−25.4938	−1.18737	−0.593683	+	0.804699i	$0.702327\pi$
$462$	0	0
$463$	−7.97430	−0.370597	−0.185298	−	0.982682i	$-0.559325\pi$
$463$	−7.97430	−0.370597	−0.185298	+	0.982682i	$0.559325\pi$
$464$	−24.0700	−1.11742
$465$	0	0
$466$	12.5954	0.583469
$467$	20.7835	0.961745	0.480872	−	0.876791i	$-0.340320\pi$
$467$	20.7835	0.961745	0.480872	+	0.876791i	$0.340320\pi$
$468$	0	0
$469$	8.26519	0.381651
$470$	0	0
$471$	0	0
$472$	−3.44947	−0.158775
$473$	−35.3054	−1.62334
$474$	0	0
$475$	0	0
$476$	3.83812	0.175920
$477$	0	0
$478$	33.4052	1.52792
$479$	−4.01153	−0.183291	−0.0916457	−	0.995792i	$-0.529213\pi$
$479$	−4.01153	−0.183291	−0.0916457	+	0.995792i	$0.529213\pi$
$480$	0	0
$481$	−16.7335	−0.762980
$482$	−52.4712	−2.39000
$483$	0	0
$484$	−2.03830	−0.0926502
$485$	0	0
$486$	0	0
$487$	−32.4746	−1.47156	−0.735782	−	0.677219i	$-0.763185\pi$
$487$	−32.4746	−1.47156	−0.735782	+	0.677219i	$0.763185\pi$
$488$	−4.49543	−0.203498
$489$	0	0
$490$	0	0
$491$	−27.7020	−1.25017	−0.625087	−	0.780555i	$-0.714936\pi$
$491$	−27.7020	−1.25017	−0.625087	+	0.780555i	$0.714936\pi$
$492$	0	0
$493$	−9.23582	−0.415960
$494$	−10.0742	−0.453261
$495$	0	0
$496$	26.5224	1.19089
$497$	19.0179	0.853067
$498$	0	0
$499$	−44.5471	−1.99420	−0.997100	−	0.0760989i	$-0.975754\pi$
$499$	−44.5471	−1.99420	−0.997100	+	0.0760989i	$0.975754\pi$
$500$	0	0
$501$	0	0
$502$	39.4386	1.76023
$503$	−11.7205	−0.522592	−0.261296	−	0.965259i	$-0.584150\pi$
$503$	−11.7205	−0.522592	−0.261296	+	0.965259i	$0.584150\pi$
$504$	0	0
$505$	0	0
$506$	15.5709	0.692211
$507$	0	0
$508$	5.39720	0.239462
$509$	−18.0471	−0.799922	−0.399961	−	0.916532i	$-0.630976\pi$
$509$	−18.0471	−0.799922	−0.399961	+	0.916532i	$0.630976\pi$
$510$	0	0
$511$	−2.27266	−0.100536
$512$	−31.7535	−1.40332
$513$	0	0
$514$	−12.2355	−0.539684
$515$	0	0
$516$	0	0
$517$	−39.5841	−1.74091
$518$	−13.7625	−0.604691
$519$	0	0
$520$	0	0
$521$	−7.84517	−0.343703	−0.171852	−	0.985123i	$-0.554975\pi$
$521$	−7.84517	−0.343703	−0.171852	+	0.985123i	$0.554975\pi$
$522$	0	0
$523$	−12.5448	−0.548547	−0.274274	−	0.961652i	$-0.588437\pi$
$523$	−12.5448	−0.548547	−0.274274	+	0.961652i	$0.588437\pi$
$524$	−9.20778	−0.402244
$525$	0	0
$526$	−31.6650	−1.38066
$527$	10.1768	0.443309
$528$	0	0
$529$	−17.2839	−0.751474
$530$	0	0
$531$	0	0
$532$	−4.34858	−0.188535
$533$	20.4946	0.887718
$534$	0	0
$535$	0	0
$536$	2.76998	0.119645
$537$	0	0
$538$	55.5274	2.39396
$539$	−17.0221	−0.733194
$540$	0	0
$541$	−21.2732	−0.914606	−0.457303	−	0.889311i	$-0.651185\pi$
$541$	−21.2732	−0.914606	−0.457303	+	0.889311i	$0.651185\pi$
$542$	18.4986	0.794584
$543$	0	0
$544$	−11.0195	−0.472455
$545$	0	0
$546$	0	0
$547$	−12.6304	−0.540036	−0.270018	−	0.962855i	$-0.587030\pi$
$547$	−12.6304	−0.540036	−0.270018	+	0.962855i	$0.587030\pi$
$548$	40.4597	1.72835
$549$	0	0
$550$	0	0
$551$	10.4642	0.445788
$552$	0	0
$553$	1.00053	0.0425471
$554$	−26.6777	−1.13343
$555$	0	0
$556$	−17.4063	−0.738193
$557$	−39.5221	−1.67461	−0.837303	−	0.546739i	$-0.815869\pi$
$557$	−39.5221	−1.67461	−0.837303	+	0.546739i	$0.815869\pi$
$558$	0	0
$559$	−35.5043	−1.50167
$560$	0	0
$561$	0	0
$562$	20.7051	0.873391
$563$	38.2284	1.61113	0.805567	−	0.592504i	$-0.201861\pi$
$563$	38.2284	1.61113	0.805567	+	0.592504i	$0.201861\pi$
$564$	0	0
$565$	0	0
$566$	−49.9664	−2.10025
$567$	0	0
$568$	6.37361	0.267431
$569$	−29.0712	−1.21873	−0.609364	−	0.792891i	$-0.708575\pi$
$569$	−29.0712	−1.21873	−0.609364	+	0.792891i	$0.708575\pi$
$570$	0	0
$571$	−9.97404	−0.417401	−0.208700	−	0.977980i	$-0.566923\pi$
$571$	−9.97404	−0.417401	−0.208700	+	0.977980i	$0.566923\pi$
$572$	22.3868	0.936037
$573$	0	0
$574$	16.8559	0.703551
$575$	0	0
$576$	0	0
$577$	−11.5868	−0.482366	−0.241183	−	0.970480i	$-0.577535\pi$
$577$	−11.5868	−0.482366	−0.241183	+	0.970480i	$0.577535\pi$
$578$	31.0958	1.29341
$579$	0	0
$580$	0	0
$581$	−1.83492	−0.0761253
$582$	0	0
$583$	−1.50703	−0.0624150
$584$	−0.761655	−0.0315175
$585$	0	0
$586$	24.2209	1.00055
$587$	−33.6580	−1.38922	−0.694608	−	0.719389i	$-0.744422\pi$
$587$	−33.6580	−1.38922	−0.694608	+	0.719389i	$0.744422\pi$
$588$	0	0
$589$	−11.5303	−0.475097
$590$	0	0
$591$	0	0
$592$	18.5461	0.762242
$593$	31.7271	1.30288	0.651438	−	0.758702i	$-0.274166\pi$
$593$	31.7271	1.30288	0.651438	+	0.758702i	$0.274166\pi$
$594$	0	0
$595$	0	0
$596$	34.6067	1.41755
$597$	0	0
$598$	15.6586	0.640330
$599$	24.0445	0.982433	0.491216	−	0.871038i	$-0.336552\pi$
$599$	24.0445	0.982433	0.491216	+	0.871038i	$0.336552\pi$
$600$	0	0
$601$	11.2832	0.460250	0.230125	−	0.973161i	$-0.426086\pi$
$601$	11.2832	0.460250	0.230125	+	0.973161i	$0.426086\pi$
$602$	−29.2008	−1.19013
$603$	0	0
$604$	13.7183	0.558190
$605$	0	0
$606$	0	0
$607$	−32.5290	−1.32031	−0.660155	−	0.751129i	$-0.729510\pi$
$607$	−32.5290	−1.32031	−0.660155	+	0.751129i	$0.729510\pi$
$608$	12.4850	0.506334
$609$	0	0
$610$	0	0
$611$	−39.8072	−1.61043
$612$	0	0
$613$	8.37368	0.338210	0.169105	−	0.985598i	$-0.445912\pi$
$613$	8.37368	0.338210	0.169105	+	0.985598i	$0.445912\pi$
$614$	−1.35694	−0.0547618
$615$	0	0
$616$	1.74246	0.0702056
$617$	−12.3024	−0.495277	−0.247639	−	0.968852i	$-0.579655\pi$
$617$	−12.3024	−0.495277	−0.247639	+	0.968852i	$0.579655\pi$
$618$	0	0
$619$	−19.5915	−0.787449	−0.393724	−	0.919229i	$-0.628814\pi$
$619$	−19.5915	−0.787449	−0.393724	+	0.919229i	$0.628814\pi$
$620$	0	0
$621$	0	0
$622$	12.9912	0.520898
$623$	−10.0994	−0.404624
$624$	0	0
$625$	0	0
$626$	66.8401	2.67147
$627$	0	0
$628$	−29.2058	−1.16544
$629$	7.11627	0.283744
$630$	0	0
$631$	−45.4577	−1.80964	−0.904821	−	0.425793i	$-0.859995\pi$
$631$	−45.4577	−1.80964	−0.904821	+	0.425793i	$0.859995\pi$
$632$	0.335317	0.0133382
$633$	0	0
$634$	5.40544	0.214677
$635$	0	0
$636$	0	0
$637$	−17.1180	−0.678241
$638$	−44.3058	−1.75408
$639$	0	0
$640$	0	0
$641$	−28.0923	−1.10958	−0.554790	−	0.831991i	$-0.687201\pi$
$641$	−28.0923	−1.10958	−0.554790	+	0.831991i	$0.687201\pi$
$642$	0	0
$643$	−4.03706	−0.159206	−0.0796030	−	0.996827i	$-0.525365\pi$
$643$	−4.03706	−0.159206	−0.0796030	+	0.996827i	$0.525365\pi$
$644$	6.75911	0.266346
$645$	0	0
$646$	4.28429	0.168563
$647$	33.8908	1.33238	0.666192	−	0.745780i	$-0.267923\pi$
$647$	33.8908	1.33238	0.666192	+	0.745780i	$0.267923\pi$
$648$	0	0
$649$	25.5309	1.00218
$650$	0	0
$651$	0	0
$652$	−39.0477	−1.52923
$653$	−32.9472	−1.28932	−0.644662	−	0.764467i	$-0.723002\pi$
$653$	−32.9472	−1.28932	−0.644662	+	0.764467i	$0.723002\pi$
$654$	0	0
$655$	0	0
$656$	−22.7147	−0.886859
$657$	0	0
$658$	−32.7397	−1.27633
$659$	−21.8704	−0.851949	−0.425974	−	0.904735i	$-0.640069\pi$
$659$	−21.8704	−0.851949	−0.425974	+	0.904735i	$0.640069\pi$
$660$	0	0
$661$	41.8645	1.62834	0.814171	−	0.580625i	$-0.197192\pi$
$661$	41.8645	1.62834	0.814171	+	0.580625i	$0.197192\pi$
$662$	−15.1217	−0.587720
$663$	0	0
$664$	−0.614952	−0.0238648
$665$	0	0
$666$	0	0
$667$	−16.2647	−0.629772
$668$	−10.5558	−0.408417
$669$	0	0
$670$	0	0
$671$	33.2725	1.28447
$672$	0	0
$673$	18.1375	0.699149	0.349574	−	0.936909i	$-0.386326\pi$
$673$	18.1375	0.699149	0.349574	+	0.936909i	$0.386326\pi$
$674$	−50.9952	−1.96426
$675$	0	0
$676$	−6.20484	−0.238648
$677$	−3.33506	−0.128177	−0.0640884	−	0.997944i	$-0.520414\pi$
$677$	−3.33506	−0.128177	−0.0640884	+	0.997944i	$0.520414\pi$
$678$	0	0
$679$	17.5680	0.674199
$680$	0	0
$681$	0	0
$682$	48.8199	1.86941
$683$	−37.3435	−1.42891	−0.714455	−	0.699682i	$-0.753325\pi$
$683$	−37.3435	−1.42891	−0.714455	+	0.699682i	$0.753325\pi$
$684$	0	0
$685$	0	0
$686$	−32.4579	−1.23925
$687$	0	0
$688$	39.3504	1.50022
$689$	−1.51553	−0.0577369
$690$	0	0
$691$	−19.7055	−0.749633	−0.374817	−	0.927099i	$-0.622294\pi$
$691$	−19.7055	−0.749633	−0.374817	+	0.927099i	$0.622294\pi$
$692$	−33.3873	−1.26919
$693$	0	0
$694$	−45.6659	−1.73345
$695$	0	0
$696$	0	0
$697$	−8.71576	−0.330133
$698$	−49.2620	−1.86459
$699$	0	0
$700$	0	0
$701$	−4.82618	−0.182282	−0.0911411	−	0.995838i	$-0.529051\pi$
$701$	−4.82618	−0.182282	−0.0911411	+	0.995838i	$0.529051\pi$
$702$	0	0
$703$	−8.06271	−0.304091
$704$	−30.3985	−1.14569
$705$	0	0
$706$	59.4889	2.23889
$707$	24.7791	0.931914
$708$	0	0
$709$	35.5151	1.33380	0.666898	−	0.745149i	$-0.267622\pi$
$709$	35.5151	1.33380	0.666898	+	0.745149i	$0.267622\pi$
$710$	0	0
$711$	0	0
$712$	−3.38470	−0.126847
$713$	17.9218	0.671178
$714$	0	0
$715$	0	0
$716$	42.7517	1.59771
$717$	0	0
$718$	−66.3059	−2.47452
$719$	−40.4979	−1.51032	−0.755158	−	0.655542i	$-0.772440\pi$
$719$	−40.4979	−1.51032	−0.755158	+	0.655542i	$0.772440\pi$
$720$	0	0
$721$	4.05935	0.151178
$722$	34.1263	1.27005
$723$	0	0
$724$	8.42941	0.313276
$725$	0	0
$726$	0	0
$727$	−39.0937	−1.44990	−0.724952	−	0.688800i	$-0.758138\pi$
$727$	−39.0937	−1.44990	−0.724952	+	0.688800i	$0.758138\pi$
$728$	1.75227	0.0649436
$729$	0	0
$730$	0	0
$731$	15.0990	0.558457
$732$	0	0
$733$	8.15011	0.301031	0.150516	−	0.988608i	$-0.451907\pi$
$733$	8.15011	0.301031	0.150516	+	0.988608i	$0.451907\pi$
$734$	−43.2168	−1.59516
$735$	0	0
$736$	−19.4058	−0.715306
$737$	−20.5018	−0.755194
$738$	0	0
$739$	12.4593	0.458324	0.229162	−	0.973388i	$-0.426401\pi$
$739$	12.4593	0.458324	0.229162	+	0.973388i	$0.426401\pi$
$740$	0	0
$741$	0	0
$742$	−1.24645	−0.0457588
$743$	8.37456	0.307233	0.153616	−	0.988131i	$-0.450908\pi$
$743$	8.37456	0.307233	0.153616	+	0.988131i	$0.450908\pi$
$744$	0	0
$745$	0	0
$746$	−10.2507	−0.375306
$747$	0	0
$748$	−9.52046	−0.348102
$749$	−15.4653	−0.565091
$750$	0	0
$751$	13.2093	0.482015	0.241008	−	0.970523i	$-0.422522\pi$
$751$	13.2093	0.482015	0.241008	+	0.970523i	$0.422522\pi$
$752$	44.1193	1.60887
$753$	0	0
$754$	−44.5555	−1.62262
$755$	0	0
$756$	0	0
$757$	−37.0385	−1.34619	−0.673093	−	0.739558i	$-0.735035\pi$
$757$	−37.0385	−1.34619	−0.673093	+	0.739558i	$0.735035\pi$
$758$	10.6526	0.386918
$759$	0	0
$760$	0	0
$761$	−21.2017	−0.768561	−0.384281	−	0.923216i	$-0.625550\pi$
$761$	−21.2017	−0.768561	−0.384281	+	0.923216i	$0.625550\pi$
$762$	0	0
$763$	−3.12920	−0.113285
$764$	1.52587	0.0552041
$765$	0	0
$766$	−67.0722	−2.42342
$767$	25.6748	0.927063
$768$	0	0
$769$	20.2439	0.730015	0.365008	−	0.931005i	$-0.381066\pi$
$769$	20.2439	0.730015	0.365008	+	0.931005i	$0.381066\pi$
$770$	0	0
$771$	0	0
$772$	38.3413	1.37993
$773$	41.8959	1.50689	0.753445	−	0.657511i	$-0.228391\pi$
$773$	41.8959	1.50689	0.753445	+	0.657511i	$0.228391\pi$
$774$	0	0
$775$	0	0
$776$	5.88772	0.211357
$777$	0	0
$778$	55.1277	1.97642
$779$	9.87493	0.353806
$780$	0	0
$781$	−47.1738	−1.68801
$782$	−6.65918	−0.238132
$783$	0	0
$784$	18.9724	0.677585
$785$	0	0
$786$	0	0
$787$	−18.3230	−0.653143	−0.326571	−	0.945173i	$-0.605893\pi$
$787$	−18.3230	−0.653143	−0.326571	+	0.945173i	$0.605893\pi$
$788$	−23.6335	−0.841910
$789$	0	0
$790$	0	0
$791$	−20.3772	−0.724531
$792$	0	0
$793$	33.4600	1.18820
$794$	43.2751	1.53578
$795$	0	0
$796$	−24.1758	−0.856888
$797$	10.8785	0.385336	0.192668	−	0.981264i	$-0.438286\pi$
$797$	10.8785	0.385336	0.192668	+	0.981264i	$0.438286\pi$
$798$	0	0
$799$	16.9289	0.598900
$800$	0	0
$801$	0	0
$802$	−53.5714	−1.89167
$803$	5.63733	0.198937
$804$	0	0
$805$	0	0
$806$	49.0950	1.72930
$807$	0	0
$808$	8.30442	0.292149
$809$	20.9195	0.735492	0.367746	−	0.929926i	$-0.380130\pi$
$809$	20.9195	0.735492	0.367746	+	0.929926i	$0.380130\pi$
$810$	0	0
$811$	36.7874	1.29178	0.645890	−	0.763430i	$-0.276486\pi$
$811$	36.7874	1.29178	0.645890	+	0.763430i	$0.276486\pi$
$812$	−19.2325	−0.674929
$813$	0	0
$814$	34.1380	1.19654
$815$	0	0
$816$	0	0
$817$	−17.1071	−0.598502
$818$	−21.5405	−0.753144
$819$	0	0
$820$	0	0
$821$	−11.8517	−0.413627	−0.206813	−	0.978380i	$-0.566309\pi$
$821$	−11.8517	−0.413627	−0.206813	+	0.978380i	$0.566309\pi$
$822$	0	0
$823$	21.1539	0.737380	0.368690	−	0.929552i	$-0.379806\pi$
$823$	21.1539	0.737380	0.368690	+	0.929552i	$0.379806\pi$
$824$	1.36044	0.0473933
$825$	0	0
$826$	21.1164	0.734734
$827$	2.92367	0.101666	0.0508329	−	0.998707i	$-0.483812\pi$
$827$	2.92367	0.101666	0.0508329	+	0.998707i	$0.483812\pi$
$828$	0	0
$829$	−23.9388	−0.831429	−0.415715	−	0.909495i	$-0.636469\pi$
$829$	−23.9388	−0.831429	−0.415715	+	0.909495i	$0.636469\pi$
$830$	0	0
$831$	0	0
$832$	−30.5698	−1.05982
$833$	7.27982	0.252231
$834$	0	0
$835$	0	0
$836$	10.7866	0.373064
$837$	0	0
$838$	25.3550	0.875875
$839$	−25.9386	−0.895501	−0.447751	−	0.894158i	$-0.647775\pi$
$839$	−25.9386	−0.895501	−0.447751	+	0.894158i	$0.647775\pi$
$840$	0	0
$841$	17.2800	0.595861
$842$	59.6859	2.05691
$843$	0	0
$844$	−16.4107	−0.564880
$845$	0	0
$846$	0	0
$847$	1.18085	0.0405745
$848$	1.67970	0.0576811
$849$	0	0
$850$	0	0
$851$	12.5321	0.429594
$852$	0	0
$853$	−22.8451	−0.782201	−0.391100	−	0.920348i	$-0.627905\pi$
$853$	−22.8451	−0.782201	−0.391100	+	0.920348i	$0.627905\pi$
$854$	27.5194	0.941695
$855$	0	0
$856$	−5.18302	−0.177152
$857$	−18.2986	−0.625067	−0.312533	−	0.949907i	$-0.601178\pi$
$857$	−18.2986	−0.625067	−0.312533	+	0.949907i	$0.601178\pi$
$858$	0	0
$859$	−52.0728	−1.77670	−0.888351	−	0.459165i	$-0.848149\pi$
$859$	−52.0728	−1.77670	−0.888351	+	0.459165i	$0.848149\pi$
$860$	0	0
$861$	0	0
$862$	39.5130	1.34582
$863$	23.5303	0.800981	0.400490	−	0.916301i	$-0.368840\pi$
$863$	23.5303	0.800981	0.400490	+	0.916301i	$0.368840\pi$
$864$	0	0
$865$	0	0
$866$	−31.8973	−1.08391
$867$	0	0
$868$	21.1920	0.719304
$869$	−2.48183	−0.0841902
$870$	0	0
$871$	−20.6173	−0.698592
$872$	−1.04872	−0.0355140
$873$	0	0
$874$	7.54483	0.255208
$875$	0	0
$876$	0	0
$877$	−14.7028	−0.496479	−0.248240	−	0.968699i	$-0.579852\pi$
$877$	−14.7028	−0.496479	−0.248240	+	0.968699i	$0.579852\pi$
$878$	−51.7888	−1.74779
$879$	0	0
$880$	0	0
$881$	−43.0118	−1.44911	−0.724553	−	0.689219i	$-0.757954\pi$
$881$	−43.0118	−1.44911	−0.724553	+	0.689219i	$0.757954\pi$
$882$	0	0
$883$	−2.70155	−0.0909142	−0.0454571	−	0.998966i	$-0.514474\pi$
$883$	−2.70155	−0.0909142	−0.0454571	+	0.998966i	$0.514474\pi$
$884$	−9.57411	−0.322012
$885$	0	0
$886$	−4.82315	−0.162037
$887$	16.8293	0.565072	0.282536	−	0.959257i	$-0.408824\pi$
$887$	16.8293	0.565072	0.282536	+	0.959257i	$0.408824\pi$
$888$	0	0
$889$	−3.12676	−0.104868
$890$	0	0
$891$	0	0
$892$	−34.4476	−1.15339
$893$	−19.1804	−0.641846
$894$	0	0
$895$	0	0
$896$	−4.36718	−0.145897
$897$	0	0
$898$	55.6322	1.85647
$899$	−50.9952	−1.70079
$900$	0	0
$901$	0.644511	0.0214718
$902$	−41.8110	−1.39215
$903$	0	0
$904$	−6.82920	−0.227136
$905$	0	0
$906$	0	0
$907$	42.8779	1.42374	0.711869	−	0.702312i	$-0.247849\pi$
$907$	42.8779	1.42374	0.711869	+	0.702312i	$0.247849\pi$
$908$	−25.5583	−0.848182
$909$	0	0
$910$	0	0
$911$	5.86788	0.194411	0.0972057	−	0.995264i	$-0.469010\pi$
$911$	5.86788	0.194411	0.0972057	+	0.995264i	$0.469010\pi$
$912$	0	0
$913$	4.55152	0.150633
$914$	−5.50082	−0.181951
$915$	0	0
$916$	35.5052	1.17312
$917$	5.33434	0.176156
$918$	0	0
$919$	−16.2621	−0.536439	−0.268219	−	0.963358i	$-0.586435\pi$
$919$	−16.2621	−0.536439	−0.268219	+	0.963358i	$0.586435\pi$
$920$	0	0
$921$	0	0
$922$	52.3031	1.72251
$923$	−47.4396	−1.56149
$924$	0	0
$925$	0	0
$926$	16.3601	0.537625
$927$	0	0
$928$	55.2176	1.81261
$929$	9.59435	0.314780	0.157390	−	0.987536i	$-0.449692\pi$
$929$	9.59435	0.314780	0.157390	+	0.987536i	$0.449692\pi$
$930$	0	0
$931$	−8.24801	−0.270318
$932$	−13.5620	−0.444240
$933$	0	0
$934$	−42.6394	−1.39520
$935$	0	0
$936$	0	0
$937$	−14.2369	−0.465098	−0.232549	−	0.972585i	$-0.574707\pi$
$937$	−14.2369	−0.465098	−0.232549	+	0.972585i	$0.574707\pi$
$938$	−16.9569	−0.553661
$939$	0	0
$940$	0	0
$941$	42.1039	1.37255	0.686273	−	0.727344i	$-0.259245\pi$
$941$	42.1039	1.37255	0.686273	+	0.727344i	$0.259245\pi$
$942$	0	0
$943$	−15.3489	−0.499827
$944$	−28.4561	−0.926166
$945$	0	0
$946$	72.4325	2.35498
$947$	−31.7355	−1.03126	−0.515632	−	0.856810i	$-0.672443\pi$
$947$	−31.7355	−1.03126	−0.515632	+	0.856810i	$0.672443\pi$
$948$	0	0
$949$	5.66909	0.184027
$950$	0	0
$951$	0	0
$952$	−0.745193	−0.0241518
$953$	46.9123	1.51964	0.759819	−	0.650135i	$-0.225288\pi$
$953$	46.9123	1.51964	0.759819	+	0.650135i	$0.225288\pi$
$954$	0	0
$955$	0	0
$956$	−35.9690	−1.16332
$957$	0	0
$958$	8.23005	0.265901
$959$	−23.4395	−0.756902
$960$	0	0
$961$	25.1909	0.812608
$962$	34.3303	1.10685
$963$	0	0
$964$	56.4984	1.81969
$965$	0	0
$966$	0	0
$967$	−34.9436	−1.12371	−0.561855	−	0.827235i	$-0.689912\pi$
$967$	−34.9436	−1.12371	−0.561855	+	0.827235i	$0.689912\pi$
$968$	0.395748	0.0127198
$969$	0	0
$970$	0	0
$971$	−25.9059	−0.831359	−0.415680	−	0.909511i	$-0.636456\pi$
$971$	−25.9059	−0.831359	−0.415680	+	0.909511i	$0.636456\pi$
$972$	0	0
$973$	10.0840	0.323279
$974$	66.6248	2.13480
$975$	0	0
$976$	−37.0846	−1.18705
$977$	−4.21429	−0.134827	−0.0674135	−	0.997725i	$-0.521475\pi$
$977$	−4.21429	−0.134827	−0.0674135	+	0.997725i	$0.521475\pi$
$978$	0	0
$979$	25.0516	0.800652
$980$	0	0
$981$	0	0
$982$	56.8334	1.81363
$983$	12.8262	0.409094	0.204547	−	0.978857i	$-0.434428\pi$
$983$	12.8262	0.409094	0.204547	+	0.978857i	$0.434428\pi$
$984$	0	0
$985$	0	0
$986$	18.9482	0.603434
$987$	0	0
$988$	10.8474	0.345103
$989$	26.5900	0.845514
$990$	0	0
$991$	48.3707	1.53655	0.768273	−	0.640122i	$-0.221116\pi$
$991$	48.3707	1.53655	0.768273	+	0.640122i	$0.221116\pi$
$992$	−60.8435	−1.93178
$993$	0	0
$994$	−39.0170	−1.23754
$995$	0	0
$996$	0	0
$997$	−12.1216	−0.383894	−0.191947	−	0.981405i	$-0.561480\pi$
$997$	−12.1216	−0.383894	−0.191947	+	0.981405i	$0.561480\pi$
$998$	91.3927	2.89299
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.y.1.2	✓	8
3.2	odd	2	inner	5625.2.a.y.1.7	yes	8
5.4	even	2		5625.2.a.ba.1.7	yes	8
15.14	odd	2		5625.2.a.ba.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.y.1.2	✓	8	1.1	even	1	trivial
5625.2.a.y.1.7	yes	8	3.2	odd	2	inner
5625.2.a.ba.1.2	yes	8	15.14	odd	2
5625.2.a.ba.1.7	yes	8	5.4	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05160 q^{2} +2.20906 q^{4} -1.27977 q^{7} -0.428901 q^{8} +O(q^{10})\)	\(q-2.05160 q^{2} +2.20906 q^{4} -1.27977 q^{7} -0.428901 q^{8} +3.17448 q^{11} +3.19236 q^{13} +2.62558 q^{14} -3.53818 q^{16} -1.35762 q^{17} +1.53818 q^{19} -6.51275 q^{22} -2.39083 q^{23} -6.54945 q^{26} -2.82709 q^{28} +6.80294 q^{29} -7.49606 q^{31} +8.11673 q^{32} +2.78530 q^{34} -5.24171 q^{37} -3.15573 q^{38} +6.41987 q^{41} -11.1216 q^{43} +7.01260 q^{44} +4.90503 q^{46} -12.4695 q^{47} -5.36218 q^{49} +7.05211 q^{52} -0.474735 q^{53} +0.548896 q^{56} -13.9569 q^{58} +8.04257 q^{59} +10.4813 q^{61} +15.3789 q^{62} -9.57591 q^{64} -6.45833 q^{67} -2.99907 q^{68} -14.8603 q^{71} +1.77583 q^{73} +10.7539 q^{74} +3.39793 q^{76} -4.06261 q^{77} -0.781806 q^{79} -13.1710 q^{82} +1.43379 q^{83} +22.8172 q^{86} -1.36154 q^{88} +7.89157 q^{89} -4.08550 q^{91} -5.28149 q^{92} +25.5824 q^{94} -13.7275 q^{97} +11.0010 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 - 10 * q^7	\( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} - 18 q^{16} + 2 q^{19} - 20 q^{22} - 10 q^{28} + 4 q^{31} + 50 q^{34} - 50 q^{43} - 30 q^{46} - 6 q^{49} - 30 q^{52} - 60 q^{58} + 46 q^{61} - 14 q^{64} - 40 q^{67} - 50 q^{73} - 34 q^{76} - 12 q^{79} - 60 q^{82} - 70 q^{88} - 10 q^{91} - 20 q^{94} - 50 q^{97}+O(q^{100}) \) 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 - 18 * q^16 + 2 * q^19 - 20 * q^22 - 10 * q^28 + 4 * q^31 + 50 * q^34 - 50 * q^43 - 30 * q^46 - 6 * q^49 - 30 * q^52 - 60 * q^58 + 46 * q^61 - 14 * q^64 - 40 * q^67 - 50 * q^73 - 34 * q^76 - 12 * q^79 - 60 * q^82 - 70 * q^88 - 10 * q^91 - 20 * q^94 - 50 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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