LMFDB - Embedded newform 5625.2.a.p.1.1

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:	$6$
Coefficient field:	6.6.44400625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 $ x^6 - 11x^4 - x^3 + 29x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44028$ 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)

$f(q)$	$=$	$q-2.44028 q^{2} +3.95498 q^{4} -3.44028 q^{7} -4.77071 q^{8} +O(q^{10})$	$q-2.44028 q^{2} +3.95498 q^{4} -3.44028 q^{7} -4.77071 q^{8} +3.26656 q^{11} -3.23204 q^{13} +8.39527 q^{14} +3.73192 q^{16} +5.05337 q^{17} -3.08119 q^{19} -7.97134 q^{22} -1.54765 q^{23} +7.88709 q^{26} -13.6063 q^{28} +3.12218 q^{29} +7.44212 q^{31} +0.434479 q^{32} -12.3317 q^{34} -5.75838 q^{37} +7.51899 q^{38} -5.41962 q^{41} +2.53106 q^{43} +12.9192 q^{44} +3.77670 q^{46} -7.07162 q^{47} +4.83555 q^{49} -12.7827 q^{52} +10.1515 q^{53} +16.4126 q^{56} -7.61901 q^{58} -1.73056 q^{59} +7.83611 q^{61} -18.1609 q^{62} -8.52409 q^{64} -1.84910 q^{67} +19.9860 q^{68} +0.713969 q^{71} -1.88027 q^{73} +14.0521 q^{74} -12.1861 q^{76} -11.2379 q^{77} -13.3332 q^{79} +13.2254 q^{82} -3.95747 q^{83} -6.17649 q^{86} -15.5838 q^{88} -8.53392 q^{89} +11.1191 q^{91} -6.12092 q^{92} +17.2567 q^{94} -10.6528 q^{97} -11.8001 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8	$ 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

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 $

$2$	−2.44028	−1.72554	−0.862770	−	0.505596i	$-0.831273\pi$
$2$	−2.44028	−1.72554	−0.862770	+	0.505596i	$0.831273\pi$
$3$	0	0
$3$	0	0
$4$	3.95498	1.97749
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.44028	−1.30030	−0.650152	−	0.759804i	$-0.725295\pi$
$7$	−3.44028	−1.30030	−0.650152	+	0.759804i	$0.725295\pi$
$8$	−4.77071	−1.68670
$9$	0	0
$10$	0	0
$11$	3.26656	0.984906	0.492453	−	0.870339i	$-0.336100\pi$
$11$	3.26656	0.984906	0.492453	+	0.870339i	$0.336100\pi$
$12$	0	0
$13$	−3.23204	−0.896406	−0.448203	−	0.893932i	$-0.647936\pi$
$13$	−3.23204	−0.896406	−0.448203	+	0.893932i	$0.647936\pi$
$14$	8.39527	2.24373
$15$	0	0
$16$	3.73192	0.932980
$17$	5.05337	1.22562	0.612811	−	0.790229i	$-0.290039\pi$
$17$	5.05337	1.22562	0.612811	+	0.790229i	$0.290039\pi$
$18$	0	0
$19$	−3.08119	−0.706874	−0.353437	−	0.935458i	$-0.614987\pi$
$19$	−3.08119	−0.706874	−0.353437	+	0.935458i	$0.614987\pi$
$20$	0	0
$21$	0	0
$22$	−7.97134	−1.69950
$23$	−1.54765	−0.322707	−0.161354	−	0.986897i	$-0.551586\pi$
$23$	−1.54765	−0.322707	−0.161354	+	0.986897i	$0.551586\pi$
$24$	0	0
$25$	0	0
$26$	7.88709	1.54679
$27$	0	0
$28$	−13.6063	−2.57134
$29$	3.12218	0.579775	0.289887	−	0.957061i	$-0.406382\pi$
$29$	3.12218	0.579775	0.289887	+	0.957061i	$0.406382\pi$
$30$	0	0
$31$	7.44212	1.33664	0.668322	−	0.743872i	$-0.267013\pi$
$31$	7.44212	1.33664	0.668322	+	0.743872i	$0.267013\pi$
$32$	0.434479	0.0768057
$33$	0	0
$34$	−12.3317	−2.11486
$35$	0	0
$36$	0	0
$37$	−5.75838	−0.946673	−0.473336	−	0.880882i	$-0.656950\pi$
$37$	−5.75838	−0.946673	−0.473336	+	0.880882i	$0.656950\pi$
$38$	7.51899	1.21974
$39$	0	0
$40$	0	0
$41$	−5.41962	−0.846403	−0.423201	−	0.906036i	$-0.639094\pi$
$41$	−5.41962	−0.846403	−0.423201	+	0.906036i	$0.639094\pi$
$42$	0	0
$43$	2.53106	0.385982	0.192991	−	0.981201i	$-0.438181\pi$
$43$	2.53106	0.385982	0.192991	+	0.981201i	$0.438181\pi$
$44$	12.9192	1.94764
$45$	0	0
$46$	3.77670	0.556844
$47$	−7.07162	−1.03150	−0.515751	−	0.856739i	$-0.672487\pi$
$47$	−7.07162	−1.03150	−0.515751	+	0.856739i	$0.672487\pi$
$48$	0	0
$49$	4.83555	0.690793
$50$	0	0
$51$	0	0
$52$	−12.7827	−1.77263
$53$	10.1515	1.39442	0.697211	−	0.716866i	$-0.254424\pi$
$53$	10.1515	1.39442	0.697211	+	0.716866i	$0.254424\pi$
$54$	0	0
$55$	0	0
$56$	16.4126	2.19323
$57$	0	0
$58$	−7.61901	−1.00042
$59$	−1.73056	−0.225300	−0.112650	−	0.993635i	$-0.535934\pi$
$59$	−1.73056	−0.225300	−0.112650	+	0.993635i	$0.535934\pi$
$60$	0	0
$61$	7.83611	1.00331	0.501656	−	0.865067i	$-0.332724\pi$
$61$	7.83611	1.00331	0.501656	+	0.865067i	$0.332724\pi$
$62$	−18.1609	−2.30643
$63$	0	0
$64$	−8.52409	−1.06551
$65$	0	0
$66$	0	0
$67$	−1.84910	−0.225903	−0.112952	−	0.993600i	$-0.536031\pi$
$67$	−1.84910	−0.225903	−0.112952	+	0.993600i	$0.536031\pi$
$68$	19.9860	2.42366
$69$	0	0
$70$	0	0
$71$	0.713969	0.0847325	0.0423663	−	0.999102i	$-0.486510\pi$
$71$	0.713969	0.0847325	0.0423663	+	0.999102i	$0.486510\pi$
$72$	0	0
$73$	−1.88027	−0.220069	−0.110035	−	0.993928i	$-0.535096\pi$
$73$	−1.88027	−0.220069	−0.110035	+	0.993928i	$0.535096\pi$
$74$	14.0521	1.63352
$75$	0	0
$76$	−12.1861	−1.39784
$77$	−11.2379	−1.28068
$78$	0	0
$79$	−13.3332	−1.50011	−0.750053	−	0.661378i	$-0.769972\pi$
$79$	−13.3332	−1.50011	−0.750053	+	0.661378i	$0.769972\pi$
$80$	0	0
$81$	0	0
$82$	13.2254	1.46050
$83$	−3.95747	−0.434389	−0.217195	−	0.976128i	$-0.569691\pi$
$83$	−3.95747	−0.434389	−0.217195	+	0.976128i	$0.569691\pi$
$84$	0	0
$85$	0	0
$86$	−6.17649	−0.666028
$87$	0	0
$88$	−15.5838	−1.66124
$89$	−8.53392	−0.904593	−0.452297	−	0.891868i	$-0.649395\pi$
$89$	−8.53392	−0.904593	−0.452297	+	0.891868i	$0.649395\pi$
$90$	0	0
$91$	11.1191	1.16560
$92$	−6.12092	−0.638150
$93$	0	0
$94$	17.2567	1.77990
$95$	0	0
$96$	0	0
$97$	−10.6528	−1.08163	−0.540815	−	0.841142i	$-0.681884\pi$
$97$	−10.6528	−1.08163	−0.540815	+	0.841142i	$0.681884\pi$
$98$	−11.8001	−1.19199
$99$	0	0
$100$	0	0
$101$	−1.76173	−0.175299	−0.0876496	−	0.996151i	$-0.527936\pi$
$101$	−1.76173	−0.175299	−0.0876496	+	0.996151i	$0.527936\pi$
$102$	0	0
$103$	15.9143	1.56808	0.784039	−	0.620712i	$-0.213156\pi$
$103$	15.9143	1.56808	0.784039	+	0.620712i	$0.213156\pi$
$104$	15.4191	1.51197
$105$	0	0
$106$	−24.7726	−2.40613
$107$	15.7807	1.52558	0.762788	−	0.646649i	$-0.223830\pi$
$107$	15.7807	1.52558	0.762788	+	0.646649i	$0.223830\pi$
$108$	0	0
$109$	−9.91980	−0.950144	−0.475072	−	0.879947i	$-0.657578\pi$
$109$	−9.91980	−0.950144	−0.475072	+	0.879947i	$0.657578\pi$
$110$	0	0
$111$	0	0
$112$	−12.8389	−1.21316
$113$	−5.51386	−0.518700	−0.259350	−	0.965783i	$-0.583508\pi$
$113$	−5.51386	−0.518700	−0.259350	+	0.965783i	$0.583508\pi$
$114$	0	0
$115$	0	0
$116$	12.3482	1.14650
$117$	0	0
$118$	4.22306	0.388764
$119$	−17.3850	−1.59368
$120$	0	0
$121$	−0.329568	−0.0299607
$122$	−19.1223	−1.73126
$123$	0	0
$124$	29.4334	2.64320
$125$	0	0
$126$	0	0
$127$	−0.992089	−0.0880336	−0.0440168	−	0.999031i	$-0.514016\pi$
$127$	−0.992089	−0.0880336	−0.0440168	+	0.999031i	$0.514016\pi$
$128$	19.9322	1.76178
$129$	0	0
$130$	0	0
$131$	12.6115	1.10187	0.550937	−	0.834547i	$-0.314271\pi$
$131$	12.6115	1.10187	0.550937	+	0.834547i	$0.314271\pi$
$132$	0	0
$133$	10.6002	0.919152
$134$	4.51232	0.389805
$135$	0	0
$136$	−24.1082	−2.06726
$137$	−8.24065	−0.704046	−0.352023	−	0.935991i	$-0.614506\pi$
$137$	−8.24065	−0.704046	−0.352023	+	0.935991i	$0.614506\pi$
$138$	0	0
$139$	8.06371	0.683955	0.341977	−	0.939708i	$-0.388903\pi$
$139$	8.06371	0.683955	0.341977	+	0.939708i	$0.388903\pi$
$140$	0	0
$141$	0	0
$142$	−1.74229	−0.146209
$143$	−10.5577	−0.882875
$144$	0	0
$145$	0	0
$146$	4.58840	0.379739
$147$	0	0
$148$	−22.7743	−1.87204
$149$	19.1101	1.56556	0.782781	−	0.622298i	$-0.213801\pi$
$149$	19.1101	1.56556	0.782781	+	0.622298i	$0.213801\pi$
$150$	0	0
$151$	1.58550	0.129026	0.0645132	−	0.997917i	$-0.479451\pi$
$151$	1.58550	0.129026	0.0645132	+	0.997917i	$0.479451\pi$
$152$	14.6995	1.19229
$153$	0	0
$154$	27.4237	2.20986
$155$	0	0
$156$	0	0
$157$	−21.8510	−1.74390	−0.871948	−	0.489599i	$-0.837143\pi$
$157$	−21.8510	−1.74390	−0.871948	+	0.489599i	$0.837143\pi$
$158$	32.5369	2.58849
$159$	0	0
$160$	0	0
$161$	5.32435	0.419617
$162$	0	0
$163$	10.1338	0.793744	0.396872	−	0.917874i	$-0.370096\pi$
$163$	10.1338	0.793744	0.396872	+	0.917874i	$0.370096\pi$
$164$	−21.4345	−1.67375
$165$	0	0
$166$	9.65736	0.749556
$167$	1.51109	0.116931	0.0584657	−	0.998289i	$-0.481379\pi$
$167$	1.51109	0.116931	0.0584657	+	0.998289i	$0.481379\pi$
$168$	0	0
$169$	−2.55393	−0.196456
$170$	0	0
$171$	0	0
$172$	10.0103	0.763277
$173$	13.8553	1.05340	0.526700	−	0.850051i	$-0.323429\pi$
$173$	13.8553	1.05340	0.526700	+	0.850051i	$0.323429\pi$
$174$	0	0
$175$	0	0
$176$	12.1906	0.918897
$177$	0	0
$178$	20.8252	1.56091
$179$	−14.4557	−1.08047	−0.540235	−	0.841514i	$-0.681665\pi$
$179$	−14.4557	−1.08047	−0.540235	+	0.841514i	$0.681665\pi$
$180$	0	0
$181$	17.1898	1.27771	0.638854	−	0.769328i	$-0.279409\pi$
$181$	17.1898	1.27771	0.638854	+	0.769328i	$0.279409\pi$
$182$	−27.1338	−2.01129
$183$	0	0
$184$	7.38338	0.544310
$185$	0	0
$186$	0	0
$187$	16.5072	1.20712
$188$	−27.9681	−2.03978
$189$	0	0
$190$	0	0
$191$	−10.5424	−0.762821	−0.381410	−	0.924406i	$-0.624562\pi$
$191$	−10.5424	−0.762821	−0.381410	+	0.924406i	$0.624562\pi$
$192$	0	0
$193$	−0.682908	−0.0491568	−0.0245784	−	0.999698i	$-0.507824\pi$
$193$	−0.682908	−0.0491568	−0.0245784	+	0.999698i	$0.507824\pi$
$194$	25.9959	1.86640
$195$	0	0
$196$	19.1245	1.36604
$197$	23.2972	1.65986	0.829928	−	0.557870i	$-0.188381\pi$
$197$	23.2972	1.65986	0.829928	+	0.557870i	$0.188381\pi$
$198$	0	0
$199$	10.1946	0.722679	0.361339	−	0.932434i	$-0.382320\pi$
$199$	10.1946	0.722679	0.361339	+	0.932434i	$0.382320\pi$
$200$	0	0
$201$	0	0
$202$	4.29913	0.302486
$203$	−10.7412	−0.753884
$204$	0	0
$205$	0	0
$206$	−38.8353	−2.70578
$207$	0	0
$208$	−12.0617	−0.836329
$209$	−10.0649	−0.696205
$210$	0	0
$211$	18.2570	1.25686	0.628431	−	0.777865i	$-0.283697\pi$
$211$	18.2570	1.25686	0.628431	+	0.777865i	$0.283697\pi$
$212$	40.1492	2.75746
$213$	0	0
$214$	−38.5093	−2.63244
$215$	0	0
$216$	0	0
$217$	−25.6030	−1.73804
$218$	24.2071	1.63951
$219$	0	0
$220$	0	0
$221$	−16.3327	−1.09866
$222$	0	0
$223$	17.2244	1.15343	0.576714	−	0.816946i	$-0.304335\pi$
$223$	17.2244	1.15343	0.576714	+	0.816946i	$0.304335\pi$
$224$	−1.49473	−0.0998708
$225$	0	0
$226$	13.4554	0.895039
$227$	4.32879	0.287312	0.143656	−	0.989628i	$-0.454114\pi$
$227$	4.32879	0.287312	0.143656	+	0.989628i	$0.454114\pi$
$228$	0	0
$229$	20.3147	1.34243	0.671216	−	0.741262i	$-0.265772\pi$
$229$	20.3147	1.34243	0.671216	+	0.741262i	$0.265772\pi$
$230$	0	0
$231$	0	0
$232$	−14.8950	−0.977906
$233$	0.340956	0.0223368	0.0111684	−	0.999938i	$-0.496445\pi$
$233$	0.340956	0.0223368	0.0111684	+	0.999938i	$0.496445\pi$
$234$	0	0
$235$	0	0
$236$	−6.84434	−0.445529
$237$	0	0
$238$	42.4244	2.74997
$239$	7.16488	0.463458	0.231729	−	0.972780i	$-0.425562\pi$
$239$	7.16488	0.463458	0.231729	+	0.972780i	$0.425562\pi$
$240$	0	0
$241$	−12.5622	−0.809204	−0.404602	−	0.914493i	$-0.632590\pi$
$241$	−12.5622	−0.809204	−0.404602	+	0.914493i	$0.632590\pi$
$242$	0.804239	0.0516985
$243$	0	0
$244$	30.9917	1.98404
$245$	0	0
$246$	0	0
$247$	9.95854	0.633646
$248$	−35.5042	−2.25452
$249$	0	0
$250$	0	0
$251$	−17.0160	−1.07404	−0.537022	−	0.843568i	$-0.680451\pi$
$251$	−17.0160	−1.07404	−0.537022	+	0.843568i	$0.680451\pi$
$252$	0	0
$253$	−5.05549	−0.317836
$254$	2.42098	0.151906
$255$	0	0
$256$	−31.5921	−1.97451
$257$	−4.13200	−0.257747	−0.128874	−	0.991661i	$-0.541136\pi$
$257$	−4.13200	−0.257747	−0.128874	+	0.991661i	$0.541136\pi$
$258$	0	0
$259$	19.8105	1.23096
$260$	0	0
$261$	0	0
$262$	−30.7757	−1.90133
$263$	1.03773	0.0639892	0.0319946	−	0.999488i	$-0.489814\pi$
$263$	1.03773	0.0639892	0.0319946	+	0.999488i	$0.489814\pi$
$264$	0	0
$265$	0	0
$266$	−25.8674	−1.58603
$267$	0	0
$268$	−7.31315	−0.446722
$269$	15.7897	0.962713	0.481357	−	0.876525i	$-0.340144\pi$
$269$	15.7897	0.962713	0.481357	+	0.876525i	$0.340144\pi$
$270$	0	0
$271$	−13.7003	−0.832237	−0.416118	−	0.909310i	$-0.636610\pi$
$271$	−13.7003	−0.832237	−0.416118	+	0.909310i	$0.636610\pi$
$272$	18.8588	1.14348
$273$	0	0
$274$	20.1095	1.21486
$275$	0	0
$276$	0	0
$277$	6.97989	0.419381	0.209691	−	0.977768i	$-0.432754\pi$
$277$	6.97989	0.419381	0.209691	+	0.977768i	$0.432754\pi$
$278$	−19.6777	−1.18019
$279$	0	0
$280$	0	0
$281$	−7.95823	−0.474748	−0.237374	−	0.971418i	$-0.576287\pi$
$281$	−7.95823	−0.474748	−0.237374	+	0.971418i	$0.576287\pi$
$282$	0	0
$283$	−27.7952	−1.65225	−0.826126	−	0.563486i	$-0.809460\pi$
$283$	−27.7952	−1.65225	−0.826126	+	0.563486i	$0.809460\pi$
$284$	2.82373	0.167558
$285$	0	0
$286$	25.7637	1.52344
$287$	18.6450	1.10058
$288$	0	0
$289$	8.53657	0.502151
$290$	0	0
$291$	0	0
$292$	−7.43645	−0.435185
$293$	14.2098	0.830146	0.415073	−	0.909788i	$-0.363756\pi$
$293$	14.2098	0.830146	0.415073	+	0.909788i	$0.363756\pi$
$294$	0	0
$295$	0	0
$296$	27.4716	1.59675
$297$	0	0
$298$	−46.6341	−2.70144
$299$	5.00206	0.289276
$300$	0	0
$301$	−8.70755	−0.501895
$302$	−3.86908	−0.222640
$303$	0	0
$304$	−11.4988	−0.659500
$305$	0	0
$306$	0	0
$307$	−23.2911	−1.32930	−0.664648	−	0.747157i	$-0.731418\pi$
$307$	−23.2911	−1.32930	−0.664648	+	0.747157i	$0.731418\pi$
$308$	−44.4457	−2.53253
$309$	0	0
$310$	0	0
$311$	7.54924	0.428078	0.214039	−	0.976825i	$-0.431338\pi$
$311$	7.54924	0.428078	0.214039	+	0.976825i	$0.431338\pi$
$312$	0	0
$313$	−31.6565	−1.78933	−0.894667	−	0.446734i	$-0.852587\pi$
$313$	−31.6565	−1.78933	−0.894667	+	0.446734i	$0.852587\pi$
$314$	53.3225	3.00916
$315$	0	0
$316$	−52.7327	−2.96645
$317$	−25.8362	−1.45110	−0.725552	−	0.688167i	$-0.758416\pi$
$317$	−25.8362	−1.45110	−0.725552	+	0.688167i	$0.758416\pi$
$318$	0	0
$319$	10.1988	0.571023
$320$	0	0
$321$	0	0
$322$	−12.9929	−0.724067
$323$	−15.5704	−0.866361
$324$	0	0
$325$	0	0
$326$	−24.7295	−1.36964
$327$	0	0
$328$	25.8555	1.42763
$329$	24.3284	1.34127
$330$	0	0
$331$	20.1083	1.10525	0.552625	−	0.833430i	$-0.313626\pi$
$331$	20.1083	1.10525	0.552625	+	0.833430i	$0.313626\pi$
$332$	−15.6517	−0.859001
$333$	0	0
$334$	−3.68748	−0.201770
$335$	0	0
$336$	0	0
$337$	3.38378	0.184326	0.0921630	−	0.995744i	$-0.470622\pi$
$337$	3.38378	0.184326	0.0921630	+	0.995744i	$0.470622\pi$
$338$	6.23232	0.338993
$339$	0	0
$340$	0	0
$341$	24.3101	1.31647
$342$	0	0
$343$	7.44633	0.402064
$344$	−12.0749	−0.651037
$345$	0	0
$346$	−33.8109	−1.81768
$347$	29.1964	1.56734	0.783672	−	0.621175i	$-0.213344\pi$
$347$	29.1964	1.56734	0.783672	+	0.621175i	$0.213344\pi$
$348$	0	0
$349$	−20.3979	−1.09187	−0.545937	−	0.837826i	$-0.683826\pi$
$349$	−20.3979	−1.09187	−0.545937	+	0.837826i	$0.683826\pi$
$350$	0	0
$351$	0	0
$352$	1.41925	0.0756464
$353$	1.86346	0.0991821	0.0495910	−	0.998770i	$-0.484208\pi$
$353$	1.86346	0.0991821	0.0495910	+	0.998770i	$0.484208\pi$
$354$	0	0
$355$	0	0
$356$	−33.7515	−1.78883
$357$	0	0
$358$	35.2760	1.86440
$359$	−3.83795	−0.202559	−0.101280	−	0.994858i	$-0.532294\pi$
$359$	−3.83795	−0.202559	−0.101280	+	0.994858i	$0.532294\pi$
$360$	0	0
$361$	−9.50624	−0.500329
$362$	−41.9480	−2.20474
$363$	0	0
$364$	43.9759	2.30497
$365$	0	0
$366$	0	0
$367$	25.6281	1.33777	0.668887	−	0.743364i	$-0.266771\pi$
$367$	25.6281	1.33777	0.668887	+	0.743364i	$0.266771\pi$
$368$	−5.77570	−0.301079
$369$	0	0
$370$	0	0
$371$	−34.9242	−1.81317
$372$	0	0
$373$	21.4369	1.10996	0.554980	−	0.831863i	$-0.312726\pi$
$373$	21.4369	1.10996	0.554980	+	0.831863i	$0.312726\pi$
$374$	−40.2821	−2.08294
$375$	0	0
$376$	33.7366	1.73983
$377$	−10.0910	−0.519713
$378$	0	0
$379$	−25.8713	−1.32892	−0.664461	−	0.747323i	$-0.731339\pi$
$379$	−25.8713	−1.32892	−0.664461	+	0.747323i	$0.731339\pi$
$380$	0	0
$381$	0	0
$382$	25.7264	1.31628
$383$	14.5336	0.742632	0.371316	−	0.928507i	$-0.378907\pi$
$383$	14.5336	0.742632	0.371316	+	0.928507i	$0.378907\pi$
$384$	0	0
$385$	0	0
$386$	1.66649	0.0848221
$387$	0	0
$388$	−42.1317	−2.13891
$389$	−23.1047	−1.17145	−0.585726	−	0.810509i	$-0.699191\pi$
$389$	−23.1047	−1.17145	−0.585726	+	0.810509i	$0.699191\pi$
$390$	0	0
$391$	−7.82084	−0.395517
$392$	−23.0690	−1.16516
$393$	0	0
$394$	−56.8517	−2.86415
$395$	0	0
$396$	0	0
$397$	13.9345	0.699355	0.349677	−	0.936870i	$-0.386291\pi$
$397$	13.9345	0.699355	0.349677	+	0.936870i	$0.386291\pi$
$398$	−24.8778	−1.24701
$399$	0	0
$400$	0	0
$401$	27.5822	1.37739	0.688694	−	0.725052i	$-0.258184\pi$
$401$	27.5822	1.37739	0.688694	+	0.725052i	$0.258184\pi$
$402$	0	0
$403$	−24.0532	−1.19818
$404$	−6.96763	−0.346653
$405$	0	0
$406$	26.2115	1.30086
$407$	−18.8101	−0.932383
$408$	0	0
$409$	13.2125	0.653315	0.326658	−	0.945143i	$-0.394078\pi$
$409$	13.2125	0.653315	0.326658	+	0.945143i	$0.394078\pi$
$410$	0	0
$411$	0	0
$412$	62.9406	3.10086
$413$	5.95362	0.292959
$414$	0	0
$415$	0	0
$416$	−1.40425	−0.0688491
$417$	0	0
$418$	24.5612	1.20133
$419$	8.84774	0.432240	0.216120	−	0.976367i	$-0.430660\pi$
$419$	8.84774	0.432240	0.216120	+	0.976367i	$0.430660\pi$
$420$	0	0
$421$	−37.4766	−1.82650	−0.913248	−	0.407404i	$-0.866434\pi$
$421$	−37.4766	−1.82650	−0.913248	+	0.407404i	$0.866434\pi$
$422$	−44.5522	−2.16877
$423$	0	0
$424$	−48.4301	−2.35197
$425$	0	0
$426$	0	0
$427$	−26.9585	−1.30461
$428$	62.4123	3.01681
$429$	0	0
$430$	0	0
$431$	26.6245	1.28246	0.641228	−	0.767350i	$-0.278425\pi$
$431$	26.6245	1.28246	0.641228	+	0.767350i	$0.278425\pi$
$432$	0	0
$433$	34.9972	1.68186	0.840928	−	0.541146i	$-0.182010\pi$
$433$	34.9972	1.68186	0.840928	+	0.541146i	$0.182010\pi$
$434$	62.4785	2.99907
$435$	0	0
$436$	−39.2326	−1.87890
$437$	4.76861	0.228113
$438$	0	0
$439$	20.0539	0.957122	0.478561	−	0.878054i	$-0.341159\pi$
$439$	20.0539	0.957122	0.478561	+	0.878054i	$0.341159\pi$
$440$	0	0
$441$	0	0
$442$	39.8564	1.89577
$443$	−4.14871	−0.197111	−0.0985556	−	0.995132i	$-0.531422\pi$
$443$	−4.14871	−0.197111	−0.0985556	+	0.995132i	$0.531422\pi$
$444$	0	0
$445$	0	0
$446$	−42.0323	−1.99029
$447$	0	0
$448$	29.3253	1.38549
$449$	8.34804	0.393969	0.196984	−	0.980407i	$-0.436885\pi$
$449$	8.34804	0.393969	0.196984	+	0.980407i	$0.436885\pi$
$450$	0	0
$451$	−17.7035	−0.833627
$452$	−21.8072	−1.02573
$453$	0	0
$454$	−10.5635	−0.495768
$455$	0	0
$456$	0	0
$457$	20.5774	0.962571	0.481285	−	0.876564i	$-0.340170\pi$
$457$	20.5774	0.962571	0.481285	+	0.876564i	$0.340170\pi$
$458$	−49.5736	−2.31642
$459$	0	0
$460$	0	0
$461$	−28.6891	−1.33618	−0.668092	−	0.744079i	$-0.732889\pi$
$461$	−28.6891	−1.33618	−0.668092	+	0.744079i	$0.732889\pi$
$462$	0	0
$463$	40.0797	1.86266	0.931331	−	0.364173i	$-0.118648\pi$
$463$	40.0797	1.86266	0.931331	+	0.364173i	$0.118648\pi$
$464$	11.6517	0.540918
$465$	0	0
$466$	−0.832030	−0.0385430
$467$	8.14191	0.376763	0.188381	−	0.982096i	$-0.439676\pi$
$467$	8.14191	0.376763	0.188381	+	0.982096i	$0.439676\pi$
$468$	0	0
$469$	6.36142	0.293743
$470$	0	0
$471$	0	0
$472$	8.25601	0.380014
$473$	8.26785	0.380156
$474$	0	0
$475$	0	0
$476$	−68.7575	−3.15149
$477$	0	0
$478$	−17.4843	−0.799715
$479$	28.0621	1.28219	0.641094	−	0.767462i	$-0.278481\pi$
$479$	28.0621	1.28219	0.641094	+	0.767462i	$0.278481\pi$
$480$	0	0
$481$	18.6113	0.848603
$482$	30.6554	1.39631
$483$	0	0
$484$	−1.30344	−0.0592471
$485$	0	0
$486$	0	0
$487$	−15.2173	−0.689560	−0.344780	−	0.938684i	$-0.612046\pi$
$487$	−15.2173	−0.689560	−0.344780	+	0.938684i	$0.612046\pi$
$488$	−37.3838	−1.69229
$489$	0	0
$490$	0	0
$491$	−8.29230	−0.374226	−0.187113	−	0.982338i	$-0.559913\pi$
$491$	−8.29230	−0.374226	−0.187113	+	0.982338i	$0.559913\pi$
$492$	0	0
$493$	15.7775	0.710585
$494$	−24.3016	−1.09338
$495$	0	0
$496$	27.7734	1.24706
$497$	−2.45625	−0.110178
$498$	0	0
$499$	−24.4006	−1.09232	−0.546160	−	0.837681i	$-0.683911\pi$
$499$	−24.4006	−1.09232	−0.546160	+	0.837681i	$0.683911\pi$
$500$	0	0
$501$	0	0
$502$	41.5240	1.85331
$503$	−10.4731	−0.466974	−0.233487	−	0.972360i	$-0.575014\pi$
$503$	−10.4731	−0.466974	−0.233487	+	0.972360i	$0.575014\pi$
$504$	0	0
$505$	0	0
$506$	12.3368	0.548439
$507$	0	0
$508$	−3.92369	−0.174086
$509$	11.0599	0.490220	0.245110	−	0.969495i	$-0.421176\pi$
$509$	11.0599	0.490220	0.245110	+	0.969495i	$0.421176\pi$
$510$	0	0
$511$	6.46867	0.286157
$512$	37.2293	1.64532
$513$	0	0
$514$	10.0832	0.444753
$515$	0	0
$516$	0	0
$517$	−23.0999	−1.01593
$518$	−48.3432	−2.12408
$519$	0	0
$520$	0	0
$521$	41.4212	1.81470	0.907348	−	0.420380i	$-0.138103\pi$
$521$	41.4212	1.81470	0.907348	+	0.420380i	$0.138103\pi$
$522$	0	0
$523$	13.6425	0.596544	0.298272	−	0.954481i	$-0.403590\pi$
$523$	13.6425	0.596544	0.298272	+	0.954481i	$0.403590\pi$
$524$	49.8783	2.17894
$525$	0	0
$526$	−2.53235	−0.110416
$527$	37.6078	1.63822
$528$	0	0
$529$	−20.6048	−0.895860
$530$	0	0
$531$	0	0
$532$	41.9235	1.81762
$533$	17.5164	0.758721
$534$	0	0
$535$	0	0
$536$	8.82151	0.381031
$537$	0	0
$538$	−38.5313	−1.66120
$539$	15.7956	0.680366
$540$	0	0
$541$	14.6598	0.630272	0.315136	−	0.949046i	$-0.397950\pi$
$541$	14.6598	0.630272	0.315136	+	0.949046i	$0.397950\pi$
$542$	33.4327	1.43606
$543$	0	0
$544$	2.19558	0.0941348
$545$	0	0
$546$	0	0
$547$	26.6123	1.13786	0.568931	−	0.822385i	$-0.307357\pi$
$547$	26.6123	1.13786	0.568931	+	0.822385i	$0.307357\pi$
$548$	−32.5916	−1.39225
$549$	0	0
$550$	0	0
$551$	−9.62005	−0.409828
$552$	0	0
$553$	45.8701	1.95060
$554$	−17.0329	−0.723659
$555$	0	0
$556$	31.8918	1.35251
$557$	10.3141	0.437020	0.218510	−	0.975835i	$-0.429880\pi$
$557$	10.3141	0.437020	0.218510	+	0.975835i	$0.429880\pi$
$558$	0	0
$559$	−8.18047	−0.345997
$560$	0	0
$561$	0	0
$562$	19.4203	0.819197
$563$	34.4031	1.44992	0.724958	−	0.688793i	$-0.241859\pi$
$563$	34.4031	1.44992	0.724958	+	0.688793i	$0.241859\pi$
$564$	0	0
$565$	0	0
$566$	67.8281	2.85103
$567$	0	0
$568$	−3.40614	−0.142918
$569$	12.2048	0.511651	0.255825	−	0.966723i	$-0.417653\pi$
$569$	12.2048	0.511651	0.255825	+	0.966723i	$0.417653\pi$
$570$	0	0
$571$	42.0644	1.76034	0.880170	−	0.474658i	$-0.157428\pi$
$571$	42.0644	1.76034	0.880170	+	0.474658i	$0.157428\pi$
$572$	−41.7553	−1.74588
$573$	0	0
$574$	−45.4992	−1.89910
$575$	0	0
$576$	0	0
$577$	17.5351	0.729994	0.364997	−	0.931009i	$-0.381070\pi$
$577$	17.5351	0.729994	0.364997	+	0.931009i	$0.381070\pi$
$578$	−20.8316	−0.866482
$579$	0	0
$580$	0	0
$581$	13.6148	0.564838
$582$	0	0
$583$	33.1607	1.37337
$584$	8.97024	0.371191
$585$	0	0
$586$	−34.6759	−1.43245
$587$	21.9119	0.904402	0.452201	−	0.891916i	$-0.350639\pi$
$587$	21.9119	0.904402	0.452201	+	0.891916i	$0.350639\pi$
$588$	0	0
$589$	−22.9306	−0.944839
$590$	0	0
$591$	0	0
$592$	−21.4898	−0.883227
$593$	38.0061	1.56072	0.780361	−	0.625330i	$-0.215035\pi$
$593$	38.0061	1.56072	0.780361	+	0.625330i	$0.215035\pi$
$594$	0	0
$595$	0	0
$596$	75.5802	3.09588
$597$	0	0
$598$	−12.2064	−0.499158
$599$	−16.3154	−0.666629	−0.333314	−	0.942816i	$-0.608167\pi$
$599$	−16.3154	−0.666629	−0.333314	+	0.942816i	$0.608167\pi$
$600$	0	0
$601$	2.31871	0.0945822	0.0472911	−	0.998881i	$-0.484941\pi$
$601$	2.31871	0.0945822	0.0472911	+	0.998881i	$0.484941\pi$
$602$	21.2489	0.866040
$603$	0	0
$604$	6.27064	0.255149
$605$	0	0
$606$	0	0
$607$	32.2134	1.30750	0.653752	−	0.756709i	$-0.273194\pi$
$607$	32.2134	1.30750	0.653752	+	0.756709i	$0.273194\pi$
$608$	−1.33871	−0.0542920
$609$	0	0
$610$	0	0
$611$	22.8557	0.924644
$612$	0	0
$613$	25.9482	1.04804	0.524018	−	0.851707i	$-0.324432\pi$
$613$	25.9482	1.04804	0.524018	+	0.851707i	$0.324432\pi$
$614$	56.8370	2.29375
$615$	0	0
$616$	53.6128	2.16012
$617$	−1.90374	−0.0766416	−0.0383208	−	0.999265i	$-0.512201\pi$
$617$	−1.90374	−0.0766416	−0.0383208	+	0.999265i	$0.512201\pi$
$618$	0	0
$619$	−19.8795	−0.799025	−0.399513	−	0.916728i	$-0.630821\pi$
$619$	−19.8795	−0.799025	−0.399513	+	0.916728i	$0.630821\pi$
$620$	0	0
$621$	0	0
$622$	−18.4223	−0.738666
$623$	29.3591	1.17625
$624$	0	0
$625$	0	0
$626$	77.2509	3.08757
$627$	0	0
$628$	−86.4201	−3.44854
$629$	−29.0993	−1.16026
$630$	0	0
$631$	−32.8982	−1.30966	−0.654828	−	0.755778i	$-0.727259\pi$
$631$	−32.8982	−1.30966	−0.654828	+	0.755778i	$0.727259\pi$
$632$	63.6090	2.53023
$633$	0	0
$634$	63.0476	2.50394
$635$	0	0
$636$	0	0
$637$	−15.6287	−0.619231
$638$	−24.8880	−0.985324
$639$	0	0
$640$	0	0
$641$	40.7624	1.61002	0.805009	−	0.593263i	$-0.202161\pi$
$641$	40.7624	1.61002	0.805009	+	0.593263i	$0.202161\pi$
$642$	0	0
$643$	24.9947	0.985695	0.492847	−	0.870116i	$-0.335956\pi$
$643$	24.9947	0.985695	0.492847	+	0.870116i	$0.335956\pi$
$644$	21.0577	0.829790
$645$	0	0
$646$	37.9962	1.49494
$647$	32.0232	1.25896	0.629481	−	0.777016i	$-0.283268\pi$
$647$	32.0232	1.25896	0.629481	+	0.777016i	$0.283268\pi$
$648$	0	0
$649$	−5.65299	−0.221899
$650$	0	0
$651$	0	0
$652$	40.0792	1.56962
$653$	44.6772	1.74835	0.874177	−	0.485607i	$-0.161401\pi$
$653$	44.6772	1.74835	0.874177	+	0.485607i	$0.161401\pi$
$654$	0	0
$655$	0	0
$656$	−20.2256	−0.789677
$657$	0	0
$658$	−59.3681	−2.31441
$659$	24.0215	0.935747	0.467873	−	0.883796i	$-0.345020\pi$
$659$	24.0215	0.935747	0.467873	+	0.883796i	$0.345020\pi$
$660$	0	0
$661$	37.8928	1.47386	0.736929	−	0.675970i	$-0.236275\pi$
$661$	37.8928	1.47386	0.736929	+	0.675970i	$0.236275\pi$
$662$	−49.0699	−1.90715
$663$	0	0
$664$	18.8800	0.732684
$665$	0	0
$666$	0	0
$667$	−4.83204	−0.187097
$668$	5.97633	0.231231
$669$	0	0
$670$	0	0
$671$	25.5972	0.988167
$672$	0	0
$673$	−18.4267	−0.710299	−0.355149	−	0.934810i	$-0.615570\pi$
$673$	−18.4267	−0.710299	−0.355149	+	0.934810i	$0.615570\pi$
$674$	−8.25737	−0.318062
$675$	0	0
$676$	−10.1008	−0.388491
$677$	22.8164	0.876904	0.438452	−	0.898755i	$-0.355527\pi$
$677$	22.8164	0.876904	0.438452	+	0.898755i	$0.355527\pi$
$678$	0	0
$679$	36.6487	1.40645
$680$	0	0
$681$	0	0
$682$	−59.3236	−2.27162
$683$	−32.2064	−1.23234	−0.616172	−	0.787612i	$-0.711317\pi$
$683$	−32.2064	−1.23234	−0.616172	+	0.787612i	$0.711317\pi$
$684$	0	0
$685$	0	0
$686$	−18.1711	−0.693777
$687$	0	0
$688$	9.44570	0.360114
$689$	−32.8102	−1.24997
$690$	0	0
$691$	24.5769	0.934947	0.467474	−	0.884007i	$-0.345164\pi$
$691$	24.5769	0.934947	0.467474	+	0.884007i	$0.345164\pi$
$692$	54.7975	2.08309
$693$	0	0
$694$	−71.2475	−2.70452
$695$	0	0
$696$	0	0
$697$	−27.3874	−1.03737
$698$	49.7766	1.88407
$699$	0	0
$700$	0	0
$701$	3.66355	0.138370	0.0691852	−	0.997604i	$-0.477960\pi$
$701$	3.66355	0.138370	0.0691852	+	0.997604i	$0.477960\pi$
$702$	0	0
$703$	17.7427	0.669179
$704$	−27.8445	−1.04943
$705$	0	0
$706$	−4.54737	−0.171143
$707$	6.06087	0.227942
$708$	0	0
$709$	−26.6969	−1.00262	−0.501312	−	0.865267i	$-0.667149\pi$
$709$	−26.6969	−1.00262	−0.501312	+	0.865267i	$0.667149\pi$
$710$	0	0
$711$	0	0
$712$	40.7129	1.52578
$713$	−11.5178	−0.431344
$714$	0	0
$715$	0	0
$716$	−57.1721	−2.13662
$717$	0	0
$718$	9.36569	0.349524
$719$	−17.8699	−0.666433	−0.333217	−	0.942850i	$-0.608134\pi$
$719$	−17.8699	−0.666433	−0.333217	+	0.942850i	$0.608134\pi$
$720$	0	0
$721$	−54.7495	−2.03898
$722$	23.1979	0.863337
$723$	0	0
$724$	67.9853	2.52665
$725$	0	0
$726$	0	0
$727$	−34.0727	−1.26369	−0.631843	−	0.775097i	$-0.717701\pi$
$727$	−34.0727	−1.26369	−0.631843	+	0.775097i	$0.717701\pi$
$728$	−53.0461	−1.96602
$729$	0	0
$730$	0	0
$731$	12.7904	0.473069
$732$	0	0
$733$	−40.1806	−1.48410	−0.742052	−	0.670342i	$-0.766147\pi$
$733$	−40.1806	−1.48410	−0.742052	+	0.670342i	$0.766147\pi$
$734$	−62.5398	−2.30838
$735$	0	0
$736$	−0.672420	−0.0247857
$737$	−6.04019	−0.222493
$738$	0	0
$739$	15.4750	0.569256	0.284628	−	0.958638i	$-0.408130\pi$
$739$	15.4750	0.569256	0.284628	+	0.958638i	$0.408130\pi$
$740$	0	0
$741$	0	0
$742$	85.2249	3.12870
$743$	17.1140	0.627853	0.313926	−	0.949447i	$-0.398355\pi$
$743$	17.1140	0.627853	0.313926	+	0.949447i	$0.398355\pi$
$744$	0	0
$745$	0	0
$746$	−52.3121	−1.91528
$747$	0	0
$748$	65.2855	2.38707
$749$	−54.2900	−1.98371
$750$	0	0
$751$	−11.1559	−0.407086	−0.203543	−	0.979066i	$-0.565246\pi$
$751$	−11.1559	−0.407086	−0.203543	+	0.979066i	$0.565246\pi$
$752$	−26.3907	−0.962370
$753$	0	0
$754$	24.6249	0.896787
$755$	0	0
$756$	0	0
$757$	24.6773	0.896911	0.448456	−	0.893805i	$-0.351974\pi$
$757$	24.6773	0.896911	0.448456	+	0.893805i	$0.351974\pi$
$758$	63.1334	2.29311
$759$	0	0
$760$	0	0
$761$	40.2478	1.45898	0.729490	−	0.683991i	$-0.239757\pi$
$761$	40.2478	1.45898	0.729490	+	0.683991i	$0.239757\pi$
$762$	0	0
$763$	34.1269	1.23548
$764$	−41.6950	−1.50847
$765$	0	0
$766$	−35.4661	−1.28144
$767$	5.59324	0.201960
$768$	0	0
$769$	22.1010	0.796983	0.398492	−	0.917172i	$-0.369534\pi$
$769$	22.1010	0.796983	0.398492	+	0.917172i	$0.369534\pi$
$770$	0	0
$771$	0	0
$772$	−2.70089	−0.0972072
$773$	−4.90033	−0.176253	−0.0881264	−	0.996109i	$-0.528088\pi$
$773$	−4.90033	−0.176253	−0.0881264	+	0.996109i	$0.528088\pi$
$774$	0	0
$775$	0	0
$776$	50.8215	1.82439
$777$	0	0
$778$	56.3819	2.02139
$779$	16.6989	0.598301
$780$	0	0
$781$	2.33222	0.0834535
$782$	19.0851	0.682481
$783$	0	0
$784$	18.0459	0.644496
$785$	0	0
$786$	0	0
$787$	−3.73477	−0.133130	−0.0665652	−	0.997782i	$-0.521204\pi$
$787$	−3.73477	−0.133130	−0.0665652	+	0.997782i	$0.521204\pi$
$788$	92.1400	3.28235
$789$	0	0
$790$	0	0
$791$	18.9692	0.674469
$792$	0	0
$793$	−25.3266	−0.899375
$794$	−34.0042	−1.20677
$795$	0	0
$796$	40.3196	1.42909
$797$	−3.34404	−0.118452	−0.0592259	−	0.998245i	$-0.518863\pi$
$797$	−3.34404	−0.118452	−0.0592259	+	0.998245i	$0.518863\pi$
$798$	0	0
$799$	−35.7355	−1.26423
$800$	0	0
$801$	0	0
$802$	−67.3083	−2.37674
$803$	−6.14203	−0.216748
$804$	0	0
$805$	0	0
$806$	58.6966	2.06750
$807$	0	0
$808$	8.40473	0.295677
$809$	−34.2875	−1.20549	−0.602743	−	0.797935i	$-0.705926\pi$
$809$	−34.2875	−1.20549	−0.602743	+	0.797935i	$0.705926\pi$
$810$	0	0
$811$	−3.46750	−0.121760	−0.0608801	−	0.998145i	$-0.519391\pi$
$811$	−3.46750	−0.121760	−0.0608801	+	0.998145i	$0.519391\pi$
$812$	−42.4812	−1.49080
$813$	0	0
$814$	45.9020	1.60887
$815$	0	0
$816$	0	0
$817$	−7.79867	−0.272841
$818$	−32.2422	−1.12732
$819$	0	0
$820$	0	0
$821$	14.6809	0.512367	0.256184	−	0.966628i	$-0.417535\pi$
$821$	14.6809	0.512367	0.256184	+	0.966628i	$0.417535\pi$
$822$	0	0
$823$	−20.8698	−0.727475	−0.363737	−	0.931502i	$-0.618499\pi$
$823$	−20.8698	−0.727475	−0.363737	+	0.931502i	$0.618499\pi$
$824$	−75.9223	−2.64488
$825$	0	0
$826$	−14.5285	−0.505512
$827$	−45.2876	−1.57480	−0.787401	−	0.616441i	$-0.788574\pi$
$827$	−45.2876	−1.57480	−0.787401	+	0.616441i	$0.788574\pi$
$828$	0	0
$829$	−9.65729	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$829$	−9.65729	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$830$	0	0
$831$	0	0
$832$	27.5502	0.955131
$833$	24.4358	0.846651
$834$	0	0
$835$	0	0
$836$	−39.8066	−1.37674
$837$	0	0
$838$	−21.5910	−0.745848
$839$	25.1717	0.869023	0.434511	−	0.900666i	$-0.356921\pi$
$839$	25.1717	0.869023	0.434511	+	0.900666i	$0.356921\pi$
$840$	0	0
$841$	−19.2520	−0.663861
$842$	91.4534	3.15169
$843$	0	0
$844$	72.2060	2.48543
$845$	0	0
$846$	0	0
$847$	1.13381	0.0389581
$848$	37.8848	1.30097
$849$	0	0
$850$	0	0
$851$	8.91196	0.305498
$852$	0	0
$853$	−3.20847	−0.109856	−0.0549280	−	0.998490i	$-0.517493\pi$
$853$	−3.20847	−0.109856	−0.0549280	+	0.998490i	$0.517493\pi$
$854$	65.7863	2.25116
$855$	0	0
$856$	−75.2851	−2.57319
$857$	−19.4569	−0.664634	−0.332317	−	0.943168i	$-0.607830\pi$
$857$	−19.4569	−0.664634	−0.332317	+	0.943168i	$0.607830\pi$
$858$	0	0
$859$	16.8332	0.574340	0.287170	−	0.957880i	$-0.407286\pi$
$859$	16.8332	0.574340	0.287170	+	0.957880i	$0.407286\pi$
$860$	0	0
$861$	0	0
$862$	−64.9713	−2.21293
$863$	−3.81218	−0.129768	−0.0648841	−	0.997893i	$-0.520668\pi$
$863$	−3.81218	−0.129768	−0.0648841	+	0.997893i	$0.520668\pi$
$864$	0	0
$865$	0	0
$866$	−85.4030	−2.90211
$867$	0	0
$868$	−101.259	−3.43697
$869$	−43.5538	−1.47746
$870$	0	0
$871$	5.97635	0.202501
$872$	47.3245	1.60261
$873$	0	0
$874$	−11.6367	−0.393619
$875$	0	0
$876$	0	0
$877$	−25.7451	−0.869352	−0.434676	−	0.900587i	$-0.643137\pi$
$877$	−25.7451	−0.869352	−0.434676	+	0.900587i	$0.643137\pi$
$878$	−48.9373	−1.65155
$879$	0	0
$880$	0	0
$881$	−45.2537	−1.52464	−0.762318	−	0.647202i	$-0.775939\pi$
$881$	−45.2537	−1.52464	−0.762318	+	0.647202i	$0.775939\pi$
$882$	0	0
$883$	44.3978	1.49410	0.747052	−	0.664765i	$-0.231469\pi$
$883$	44.3978	1.49410	0.747052	+	0.664765i	$0.231469\pi$
$884$	−64.5955	−2.17258
$885$	0	0
$886$	10.1240	0.340124
$887$	38.9928	1.30925	0.654626	−	0.755953i	$-0.272826\pi$
$887$	38.9928	1.30925	0.654626	+	0.755953i	$0.272826\pi$
$888$	0	0
$889$	3.41307	0.114471
$890$	0	0
$891$	0	0
$892$	68.1220	2.28089
$893$	21.7890	0.729142
$894$	0	0
$895$	0	0
$896$	−68.5726	−2.29085
$897$	0	0
$898$	−20.3716	−0.679809
$899$	23.2356	0.774952
$900$	0	0
$901$	51.2995	1.70903
$902$	43.2017	1.43846
$903$	0	0
$904$	26.3050	0.874892
$905$	0	0
$906$	0	0
$907$	11.0201	0.365915	0.182958	−	0.983121i	$-0.441433\pi$
$907$	11.0201	0.365915	0.182958	+	0.983121i	$0.441433\pi$
$908$	17.1203	0.568156
$909$	0	0
$910$	0	0
$911$	52.9227	1.75341	0.876704	−	0.481031i	$-0.159737\pi$
$911$	52.9227	1.75341	0.876704	+	0.481031i	$0.159737\pi$
$912$	0	0
$913$	−12.9273	−0.427832
$914$	−50.2147	−1.66096
$915$	0	0
$916$	80.3442	2.65465
$917$	−43.3872	−1.43277
$918$	0	0
$919$	−47.6045	−1.57033	−0.785164	−	0.619288i	$-0.787421\pi$
$919$	−47.6045	−1.57033	−0.785164	+	0.619288i	$0.787421\pi$
$920$	0	0
$921$	0	0
$922$	70.0095	2.30564
$923$	−2.30757	−0.0759547
$924$	0	0
$925$	0	0
$926$	−97.8059	−3.21410
$927$	0	0
$928$	1.35652	0.0445300
$929$	−21.6887	−0.711582	−0.355791	−	0.934566i	$-0.615789\pi$
$929$	−21.6887	−0.711582	−0.355791	+	0.934566i	$0.615789\pi$
$930$	0	0
$931$	−14.8993	−0.488304
$932$	1.34848	0.0441708
$933$	0	0
$934$	−19.8686	−0.650119
$935$	0	0
$936$	0	0
$937$	2.20089	0.0718999	0.0359500	−	0.999354i	$-0.488554\pi$
$937$	2.20089	0.0718999	0.0359500	+	0.999354i	$0.488554\pi$
$938$	−15.5237	−0.506866
$939$	0	0
$940$	0	0
$941$	16.7898	0.547331	0.273666	−	0.961825i	$-0.411764\pi$
$941$	16.7898	0.547331	0.273666	+	0.961825i	$0.411764\pi$
$942$	0	0
$943$	8.38767	0.273140
$944$	−6.45832	−0.210200
$945$	0	0
$946$	−20.1759	−0.655975
$947$	−36.6132	−1.18977	−0.594885	−	0.803811i	$-0.702802\pi$
$947$	−36.6132	−1.18977	−0.594885	+	0.803811i	$0.702802\pi$
$948$	0	0
$949$	6.07711	0.197271
$950$	0	0
$951$	0	0
$952$	82.9389	2.68807
$953$	−43.9277	−1.42296	−0.711479	−	0.702707i	$-0.751974\pi$
$953$	−43.9277	−1.42296	−0.711479	+	0.702707i	$0.751974\pi$
$954$	0	0
$955$	0	0
$956$	28.3370	0.916483
$957$	0	0
$958$	−68.4794	−2.21247
$959$	28.3502	0.915475
$960$	0	0
$961$	24.3851	0.786616
$962$	−45.4169	−1.46430
$963$	0	0
$964$	−49.6834	−1.60019
$965$	0	0
$966$	0	0
$967$	24.4815	0.787272	0.393636	−	0.919266i	$-0.371217\pi$
$967$	24.4815	0.787272	0.393636	+	0.919266i	$0.371217\pi$
$968$	1.57227	0.0505348
$969$	0	0
$970$	0	0
$971$	19.6254	0.629808	0.314904	−	0.949123i	$-0.398028\pi$
$971$	19.6254	0.629808	0.314904	+	0.949123i	$0.398028\pi$
$972$	0	0
$973$	−27.7414	−0.889349
$974$	37.1344	1.18986
$975$	0	0
$976$	29.2438	0.936070
$977$	−30.5238	−0.976541	−0.488271	−	0.872692i	$-0.662372\pi$
$977$	−30.5238	−0.976541	−0.488271	+	0.872692i	$0.662372\pi$
$978$	0	0
$979$	−27.8766	−0.890939
$980$	0	0
$981$	0	0
$982$	20.2356	0.645743
$983$	−10.3655	−0.330609	−0.165305	−	0.986243i	$-0.552861\pi$
$983$	−10.3655	−0.330609	−0.165305	+	0.986243i	$0.552861\pi$
$984$	0	0
$985$	0	0
$986$	−38.5017	−1.22614
$987$	0	0
$988$	39.3858	1.25303
$989$	−3.91718	−0.124559
$990$	0	0
$991$	−10.1087	−0.321113	−0.160557	−	0.987027i	$-0.551329\pi$
$991$	−10.1087	−0.321113	−0.160557	+	0.987027i	$0.551329\pi$
$992$	3.23344	0.102662
$993$	0	0
$994$	5.99396	0.190117
$995$	0	0
$996$	0	0
$997$	21.3639	0.676602	0.338301	−	0.941038i	$-0.390148\pi$
$997$	21.3639	0.676602	0.338301	+	0.941038i	$0.390148\pi$
$998$	59.5443	1.88484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.p.1.1		6
3.2	odd	2		1875.2.a.j.1.6		6
5.4	even	2		5625.2.a.q.1.6		6
15.2	even	4		1875.2.b.f.1249.11		12
15.8	even	4		1875.2.b.f.1249.2		12
15.14	odd	2		1875.2.a.k.1.1		6
25.11	even	5		225.2.h.d.46.1		12
25.16	even	5		225.2.h.d.181.1		12
75.2	even	20		375.2.i.d.274.5		24
75.11	odd	10		75.2.g.c.46.3	yes	12
75.14	odd	10		375.2.g.c.226.1		12
75.23	even	20		375.2.i.d.274.2		24
75.38	even	20		375.2.i.d.349.5		24
75.41	odd	10		75.2.g.c.31.3	✓	12
75.59	odd	10		375.2.g.c.151.1		12
75.62	even	20		375.2.i.d.349.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.31.3	✓	12	75.41	odd	10
75.2.g.c.46.3	yes	12	75.11	odd	10
225.2.h.d.46.1		12	25.11	even	5
225.2.h.d.181.1		12	25.16	even	5
375.2.g.c.151.1		12	75.59	odd	10
375.2.g.c.226.1		12	75.14	odd	10
375.2.i.d.274.2		24	75.23	even	20
375.2.i.d.274.5		24	75.2	even	20
375.2.i.d.349.2		24	75.62	even	20
375.2.i.d.349.5		24	75.38	even	20
1875.2.a.j.1.6		6	3.2	odd	2
1875.2.a.k.1.1		6	15.14	odd	2
1875.2.b.f.1249.2		12	15.8	even	4
1875.2.b.f.1249.11		12	15.2	even	4
5625.2.a.p.1.1		6	1.1	even	1	trivial
5625.2.a.q.1.6		6	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)

\(f(q)\)	\(=\)	\(q-2.44028 q^{2} +3.95498 q^{4} -3.44028 q^{7} -4.77071 q^{8} +O(q^{10})\)	\(q-2.44028 q^{2} +3.95498 q^{4} -3.44028 q^{7} -4.77071 q^{8} +3.26656 q^{11} -3.23204 q^{13} +8.39527 q^{14} +3.73192 q^{16} +5.05337 q^{17} -3.08119 q^{19} -7.97134 q^{22} -1.54765 q^{23} +7.88709 q^{26} -13.6063 q^{28} +3.12218 q^{29} +7.44212 q^{31} +0.434479 q^{32} -12.3317 q^{34} -5.75838 q^{37} +7.51899 q^{38} -5.41962 q^{41} +2.53106 q^{43} +12.9192 q^{44} +3.77670 q^{46} -7.07162 q^{47} +4.83555 q^{49} -12.7827 q^{52} +10.1515 q^{53} +16.4126 q^{56} -7.61901 q^{58} -1.73056 q^{59} +7.83611 q^{61} -18.1609 q^{62} -8.52409 q^{64} -1.84910 q^{67} +19.9860 q^{68} +0.713969 q^{71} -1.88027 q^{73} +14.0521 q^{74} -12.1861 q^{76} -11.2379 q^{77} -13.3332 q^{79} +13.2254 q^{82} -3.95747 q^{83} -6.17649 q^{86} -15.5838 q^{88} -8.53392 q^{89} +11.1191 q^{91} -6.12092 q^{92} +17.2567 q^{94} -10.6528 q^{97} -11.8001 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8	\( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more