LMFDB - Embedded newform 5625.2.a.n.1.3

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:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
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			1.3
Root			$2.12233$ 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)

$f(q)$	$=$	$q+2.12233 q^{2} +2.50430 q^{4} -4.35840 q^{7} +1.07029 q^{8} +O(q^{10})$	$q+2.12233 q^{2} +2.50430 q^{4} -4.35840 q^{7} +1.07029 q^{8} +1.57991 q^{11} -1.19794 q^{13} -9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +7.67867 q^{19} +3.35309 q^{22} -2.32027 q^{23} -2.54243 q^{26} -10.9147 q^{28} +5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -2.38197 q^{34} +6.37232 q^{37} +16.2967 q^{38} +7.47214 q^{41} +1.24998 q^{43} +3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} +11.9957 q^{49} -3.00000 q^{52} +3.74568 q^{53} -4.66476 q^{56} +11.6820 q^{58} -9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} -11.3975 q^{64} +3.86270 q^{67} -2.81066 q^{68} +10.6051 q^{71} -4.99672 q^{73} +13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +14.5531 q^{79} +15.8584 q^{82} -8.73603 q^{83} +2.65288 q^{86} +1.69096 q^{88} +10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +8.76025 q^{94} +7.49996 q^{97} +25.4588 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 - 2 * q^7 + 15 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8} + 7 q^{11} - q^{13} - 16 q^{14} + 4 q^{16} + 2 q^{17} + 5 q^{19} + 6 q^{22} + q^{23} - 3 q^{26} - 9 q^{28} + 20 q^{29} + 23 q^{31} + 12 q^{32} - 14 q^{34} - 2 q^{37} + 35 q^{38} + 12 q^{41} - 16 q^{43} + 29 q^{44} - 17 q^{46} + 2 q^{47} + 8 q^{49} - 12 q^{52} - 4 q^{53} - 5 q^{56} + 25 q^{58} + 15 q^{59} - 2 q^{61} + 9 q^{62} + 23 q^{64} - 2 q^{67} - 11 q^{68} + 2 q^{71} - 16 q^{73} + 19 q^{74} + 40 q^{76} + 19 q^{77} + 35 q^{79} + 6 q^{82} + 16 q^{83} - 3 q^{86} + 30 q^{88} + 35 q^{89} - 12 q^{91} - 23 q^{92} - 9 q^{94} - 12 q^{97} - q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 - 2 * q^7 + 15 * q^8 + 7 * q^11 - q^13 - 16 * q^14 + 4 * q^16 + 2 * q^17 + 5 * q^19 + 6 * q^22 + q^23 - 3 * q^26 - 9 * q^28 + 20 * q^29 + 23 * q^31 + 12 * q^32 - 14 * q^34 - 2 * q^37 + 35 * q^38 + 12 * q^41 - 16 * q^43 + 29 * q^44 - 17 * q^46 + 2 * q^47 + 8 * q^49 - 12 * q^52 - 4 * q^53 - 5 * q^56 + 25 * q^58 + 15 * q^59 - 2 * q^61 + 9 * q^62 + 23 * q^64 - 2 * q^67 - 11 * q^68 + 2 * q^71 - 16 * q^73 + 19 * q^74 + 40 * q^76 + 19 * q^77 + 35 * q^79 + 6 * q^82 + 16 * q^83 - 3 * q^86 + 30 * q^88 + 35 * q^89 - 12 * q^91 - 23 * q^92 - 9 * q^94 - 12 * q^97 - 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 $

$2$	2.12233	1.50072	0.750358	−	0.661031i	$-0.229881\pi$
$2$	2.12233	1.50072	0.750358	+	0.661031i	$0.229881\pi$
$3$	0	0
$3$	0	0
$4$	2.50430	1.25215
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.35840	−1.64732	−0.823660	−	0.567083i	$-0.808072\pi$
$7$	−4.35840	−1.64732	−0.823660	+	0.567083i	$0.808072\pi$
$8$	1.07029	0.378405
$9$	0	0
$10$	0	0
$11$	1.57991	0.476360	0.238180	−	0.971221i	$-0.423449\pi$
$11$	1.57991	0.476360	0.238180	+	0.971221i	$0.423449\pi$
$12$	0	0
$13$	−1.19794	−0.332249	−0.166124	−	0.986105i	$-0.553125\pi$
$13$	−1.19794	−0.332249	−0.166124	+	0.986105i	$0.553125\pi$
$14$	−9.24998	−2.47216
$15$	0	0
$16$	−2.73708	−0.684271
$17$	−1.12233	−0.272206	−0.136103	−	0.990695i	$-0.543458\pi$
$17$	−1.12233	−0.272206	−0.136103	+	0.990695i	$0.543458\pi$
$18$	0	0
$19$	7.67867	1.76161	0.880804	−	0.473480i	$-0.157002\pi$
$19$	7.67867	1.76161	0.880804	+	0.473480i	$0.157002\pi$
$20$	0	0
$21$	0	0
$22$	3.35309	0.714881
$23$	−2.32027	−0.483810	−0.241905	−	0.970300i	$-0.577772\pi$
$23$	−2.32027	−0.483810	−0.241905	+	0.970300i	$0.577772\pi$
$24$	0	0
$25$	0	0
$26$	−2.54243	−0.498611
$27$	0	0
$28$	−10.9147	−2.06269
$29$	5.50430	1.02212	0.511061	−	0.859544i	$-0.329252\pi$
$29$	5.50430	1.02212	0.511061	+	0.859544i	$0.329252\pi$
$30$	0	0
$31$	4.80206	0.862475	0.431238	−	0.902238i	$-0.358077\pi$
$31$	4.80206	0.862475	0.431238	+	0.902238i	$0.358077\pi$
$32$	−7.94959	−1.40530
$33$	0	0
$34$	−2.38197	−0.408504
$35$	0	0
$36$	0	0
$37$	6.37232	1.04760	0.523801	−	0.851841i	$-0.324514\pi$
$37$	6.37232	1.04760	0.523801	+	0.851841i	$0.324514\pi$
$38$	16.2967	2.64368
$39$	0	0
$40$	0	0
$41$	7.47214	1.16695	0.583476	−	0.812131i	$-0.301692\pi$
$41$	7.47214	1.16695	0.583476	+	0.812131i	$0.301692\pi$
$42$	0	0
$43$	1.24998	0.190620	0.0953102	−	0.995448i	$-0.469616\pi$
$43$	1.24998	0.190620	0.0953102	+	0.995448i	$0.469616\pi$
$44$	3.95656	0.596473
$45$	0	0
$46$	−4.92439	−0.726062
$47$	4.12765	0.602079	0.301040	−	0.953612i	$-0.402666\pi$
$47$	4.12765	0.602079	0.301040	+	0.953612i	$0.402666\pi$
$48$	0	0
$49$	11.9957	1.71367
$50$	0	0
$51$	0	0
$52$	−3.00000	−0.416025
$53$	3.74568	0.514509	0.257254	−	0.966344i	$-0.417182\pi$
$53$	3.74568	0.514509	0.257254	+	0.966344i	$0.417182\pi$
$54$	0	0
$55$	0	0
$56$	−4.66476	−0.623355
$57$	0	0
$58$	11.6820	1.53392
$59$	−9.15613	−1.19203	−0.596013	−	0.802975i	$-0.703249\pi$
$59$	−9.15613	−1.19203	−0.596013	+	0.802975i	$0.703249\pi$
$60$	0	0
$61$	1.39588	0.178724	0.0893620	−	0.995999i	$-0.471517\pi$
$61$	1.39588	0.178724	0.0893620	+	0.995999i	$0.471517\pi$
$62$	10.1916	1.29433
$63$	0	0
$64$	−11.3975	−1.42469
$65$	0	0
$66$	0	0
$67$	3.86270	0.471904	0.235952	−	0.971765i	$-0.424179\pi$
$67$	3.86270	0.471904	0.235952	+	0.971765i	$0.424179\pi$
$68$	−2.81066	−0.340843
$69$	0	0
$70$	0	0
$71$	10.6051	1.25859	0.629297	−	0.777165i	$-0.283343\pi$
$71$	10.6051	1.25859	0.629297	+	0.777165i	$0.283343\pi$
$72$	0	0
$73$	−4.99672	−0.584821	−0.292411	−	0.956293i	$-0.594457\pi$
$73$	−4.99672	−0.584821	−0.292411	+	0.956293i	$0.594457\pi$
$74$	13.5242	1.57215
$75$	0	0
$76$	19.2297	2.20580
$77$	−6.88586	−0.784717
$78$	0	0
$79$	14.5531	1.63735	0.818673	−	0.574259i	$-0.194710\pi$
$79$	14.5531	1.63735	0.818673	+	0.574259i	$0.194710\pi$
$80$	0	0
$81$	0	0
$82$	15.8584	1.75126
$83$	−8.73603	−0.958904	−0.479452	−	0.877568i	$-0.659165\pi$
$83$	−8.73603	−0.958904	−0.479452	+	0.877568i	$0.659165\pi$
$84$	0	0
$85$	0	0
$86$	2.65288	0.286067
$87$	0	0
$88$	1.69096	0.180257
$89$	10.0381	1.06404	0.532020	−	0.846732i	$-0.321433\pi$
$89$	10.0381	1.06404	0.532020	+	0.846732i	$0.321433\pi$
$90$	0	0
$91$	5.22110	0.547320
$92$	−5.81066	−0.605803
$93$	0	0
$94$	8.76025	0.903550
$95$	0	0
$96$	0	0
$97$	7.49996	0.761506	0.380753	−	0.924677i	$-0.375665\pi$
$97$	7.49996	0.761506	0.380753	+	0.924677i	$0.375665\pi$
$98$	25.4588	2.57173
$99$	0	0
$100$	0	0
$101$	6.51821	0.648586	0.324293	−	0.945957i	$-0.394874\pi$
$101$	6.51821	0.648586	0.324293	+	0.945957i	$0.394874\pi$
$102$	0	0
$103$	−8.04876	−0.793068	−0.396534	−	0.918020i	$-0.629787\pi$
$103$	−8.04876	−0.793068	−0.396534	+	0.918020i	$0.629787\pi$
$104$	−1.28215	−0.125725
$105$	0	0
$106$	7.94959	0.772132
$107$	9.47745	0.916220	0.458110	−	0.888896i	$-0.348527\pi$
$107$	9.47745	0.916220	0.458110	+	0.888896i	$0.348527\pi$
$108$	0	0
$109$	−11.6657	−1.11738	−0.558688	−	0.829378i	$-0.688695\pi$
$109$	−11.6657	−1.11738	−0.558688	+	0.829378i	$0.688695\pi$
$110$	0	0
$111$	0	0
$112$	11.9293	1.12721
$113$	−14.9254	−1.40406	−0.702030	−	0.712147i	$-0.747723\pi$
$113$	−14.9254	−1.40406	−0.702030	+	0.712147i	$0.747723\pi$
$114$	0	0
$115$	0	0
$116$	13.7844	1.27985
$117$	0	0
$118$	−19.4324	−1.78889
$119$	4.89158	0.448410
$120$	0	0
$121$	−8.50390	−0.773082
$122$	2.96252	0.268214
$123$	0	0
$124$	12.0258	1.07995
$125$	0	0
$126$	0	0
$127$	16.9957	1.50812	0.754061	−	0.656805i	$-0.228092\pi$
$127$	16.9957	1.50812	0.754061	+	0.656805i	$0.228092\pi$
$128$	−8.29014	−0.732752
$129$	0	0
$130$	0	0
$131$	0.328872	0.0287337	0.0143669	−	0.999897i	$-0.495427\pi$
$131$	0.328872	0.0287337	0.0143669	+	0.999897i	$0.495427\pi$
$132$	0	0
$133$	−33.4667	−2.90194
$134$	8.19794	0.708194
$135$	0	0
$136$	−1.20122	−0.103004
$137$	4.56271	0.389818	0.194909	−	0.980821i	$-0.437559\pi$
$137$	4.56271	0.389818	0.194909	+	0.980821i	$0.437559\pi$
$138$	0	0
$139$	−4.73400	−0.401533	−0.200766	−	0.979639i	$-0.564343\pi$
$139$	−4.73400	−0.401533	−0.200766	+	0.979639i	$0.564343\pi$
$140$	0	0
$141$	0	0
$142$	22.5076	1.88879
$143$	−1.89263	−0.158270
$144$	0	0
$145$	0	0
$146$	−10.6047	−0.877651
$147$	0	0
$148$	15.9582	1.31175
$149$	−4.67644	−0.383109	−0.191555	−	0.981482i	$-0.561353\pi$
$149$	−4.67644	−0.383109	−0.191555	+	0.981482i	$0.561353\pi$
$150$	0	0
$151$	−6.54178	−0.532362	−0.266181	−	0.963923i	$-0.585762\pi$
$151$	−6.54178	−0.532362	−0.266181	+	0.963923i	$0.585762\pi$
$152$	8.21842	0.666602
$153$	0	0
$154$	−14.6141	−1.17764
$155$	0	0
$156$	0	0
$157$	−3.99404	−0.318759	−0.159379	−	0.987217i	$-0.550949\pi$
$157$	−3.99404	−0.318759	−0.159379	+	0.987217i	$0.550949\pi$
$158$	30.8864	2.45719
$159$	0	0
$160$	0	0
$161$	10.1127	0.796991
$162$	0	0
$163$	−4.80141	−0.376075	−0.188038	−	0.982162i	$-0.560213\pi$
$163$	−4.80141	−0.376075	−0.188038	+	0.982162i	$0.560213\pi$
$164$	18.7125	1.46120
$165$	0	0
$166$	−18.5408	−1.43904
$167$	23.6600	1.83087	0.915434	−	0.402469i	$-0.131848\pi$
$167$	23.6600	1.83087	0.915434	+	0.402469i	$0.131848\pi$
$168$	0	0
$169$	−11.5649	−0.889611
$170$	0	0
$171$	0	0
$172$	3.13033	0.238685
$173$	14.5043	1.10274	0.551371	−	0.834260i	$-0.314105\pi$
$173$	14.5043	1.10274	0.551371	+	0.834260i	$0.314105\pi$
$174$	0	0
$175$	0	0
$176$	−4.32433	−0.325959
$177$	0	0
$178$	21.3043	1.59682
$179$	−0.796746	−0.0595516	−0.0297758	−	0.999557i	$-0.509479\pi$
$179$	−0.796746	−0.0595516	−0.0297758	+	0.999557i	$0.509479\pi$
$180$	0	0
$181$	−14.2185	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$181$	−14.2185	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$182$	11.0809	0.821372
$183$	0	0
$184$	−2.48337	−0.183076
$185$	0	0
$186$	0	0
$187$	−1.77318	−0.129668
$188$	10.3369	0.753894
$189$	0	0
$190$	0	0
$191$	10.3697	0.750324	0.375162	−	0.926959i	$-0.377587\pi$
$191$	10.3697	0.750324	0.375162	+	0.926959i	$0.377587\pi$
$192$	0	0
$193$	−4.34712	−0.312913	−0.156456	−	0.987685i	$-0.550007\pi$
$193$	−4.34712	−0.312913	−0.156456	+	0.987685i	$0.550007\pi$
$194$	15.9174	1.14280
$195$	0	0
$196$	30.0407	2.14577
$197$	1.87990	0.133937	0.0669686	−	0.997755i	$-0.478667\pi$
$197$	1.87990	0.133937	0.0669686	+	0.997755i	$0.478667\pi$
$198$	0	0
$199$	4.26028	0.302003	0.151002	−	0.988534i	$-0.451750\pi$
$199$	4.26028	0.302003	0.151002	+	0.988534i	$0.451750\pi$
$200$	0	0
$201$	0	0
$202$	13.8338	0.973344
$203$	−23.9899	−1.68376
$204$	0	0
$205$	0	0
$206$	−17.0821	−1.19017
$207$	0	0
$208$	3.27886	0.227348
$209$	12.1316	0.839159
$210$	0	0
$211$	−9.29671	−0.640012	−0.320006	−	0.947416i	$-0.603685\pi$
$211$	−9.29671	−0.640012	−0.320006	+	0.947416i	$0.603685\pi$
$212$	9.38031	0.644242
$213$	0	0
$214$	20.1143	1.37499
$215$	0	0
$216$	0	0
$217$	−20.9293	−1.42077
$218$	−24.7586	−1.67686
$219$	0	0
$220$	0	0
$221$	1.34449	0.0904400
$222$	0	0
$223$	3.39758	0.227519	0.113759	−	0.993508i	$-0.463711\pi$
$223$	3.39758	0.227519	0.113759	+	0.993508i	$0.463711\pi$
$224$	34.6475	2.31498
$225$	0	0
$226$	−31.6766	−2.10710
$227$	26.7973	1.77860	0.889300	−	0.457324i	$-0.151192\pi$
$227$	26.7973	1.77860	0.889300	+	0.457324i	$0.151192\pi$
$228$	0	0
$229$	−1.18200	−0.0781085	−0.0390543	−	0.999237i	$-0.512435\pi$
$229$	−1.18200	−0.0781085	−0.0390543	+	0.999237i	$0.512435\pi$
$230$	0	0
$231$	0	0
$232$	5.89121	0.386777
$233$	7.09483	0.464798	0.232399	−	0.972621i	$-0.425342\pi$
$233$	7.09483	0.464798	0.232399	+	0.972621i	$0.425342\pi$
$234$	0	0
$235$	0	0
$236$	−22.9297	−1.49260
$237$	0	0
$238$	10.3816	0.672937
$239$	−0.0427926	−0.00276802	−0.00138401	−	0.999999i	$-0.500441\pi$
$239$	−0.0427926	−0.00276802	−0.00138401	+	0.999999i	$0.500441\pi$
$240$	0	0
$241$	−11.7711	−0.758242	−0.379121	−	0.925347i	$-0.623774\pi$
$241$	−11.7711	−0.758242	−0.379121	+	0.925347i	$0.623774\pi$
$242$	−18.0481	−1.16018
$243$	0	0
$244$	3.49570	0.223789
$245$	0	0
$246$	0	0
$247$	−9.19859	−0.585292
$248$	5.13961	0.326365
$249$	0	0
$250$	0	0
$251$	−17.4764	−1.10310	−0.551550	−	0.834142i	$-0.685964\pi$
$251$	−17.4764	−1.10310	−0.551550	+	0.834142i	$0.685964\pi$
$252$	0	0
$253$	−3.66581	−0.230468
$254$	36.0705	2.26326
$255$	0	0
$256$	5.20057	0.325036
$257$	15.4671	0.964814	0.482407	−	0.875947i	$-0.339763\pi$
$257$	15.4671	0.964814	0.482407	+	0.875947i	$0.339763\pi$
$258$	0	0
$259$	−27.7731	−1.72574
$260$	0	0
$261$	0	0
$262$	0.697977	0.0431211
$263$	1.44759	0.0892624	0.0446312	−	0.999004i	$-0.485789\pi$
$263$	1.44759	0.0892624	0.0446312	+	0.999004i	$0.485789\pi$
$264$	0	0
$265$	0	0
$266$	−71.0276	−4.35498
$267$	0	0
$268$	9.67336	0.590895
$269$	8.70617	0.530825	0.265412	−	0.964135i	$-0.414492\pi$
$269$	8.70617	0.530825	0.265412	+	0.964135i	$0.414492\pi$
$270$	0	0
$271$	−28.6338	−1.73938	−0.869689	−	0.493600i	$-0.835681\pi$
$271$	−28.6338	−1.73938	−0.869689	+	0.493600i	$0.835681\pi$
$272$	3.07192	0.186263
$273$	0	0
$274$	9.68359	0.585007
$275$	0	0
$276$	0	0
$277$	−5.30210	−0.318572	−0.159286	−	0.987232i	$-0.550919\pi$
$277$	−5.30210	−0.318572	−0.159286	+	0.987232i	$0.550919\pi$
$278$	−10.0471	−0.602587
$279$	0	0
$280$	0	0
$281$	24.0832	1.43668	0.718342	−	0.695691i	$-0.244902\pi$
$281$	24.0832	1.43668	0.718342	+	0.695691i	$0.244902\pi$
$282$	0	0
$283$	27.1143	1.61178	0.805889	−	0.592066i	$-0.201688\pi$
$283$	27.1143	1.61178	0.805889	+	0.592066i	$0.201688\pi$
$284$	26.5583	1.57595
$285$	0	0
$286$	−4.01680	−0.237518
$287$	−32.5666	−1.92234
$288$	0	0
$289$	−15.7404	−0.925904
$290$	0	0
$291$	0	0
$292$	−12.5133	−0.732284
$293$	−20.3016	−1.18603	−0.593017	−	0.805190i	$-0.702063\pi$
$293$	−20.3016	−1.18603	−0.593017	+	0.805190i	$0.702063\pi$
$294$	0	0
$295$	0	0
$296$	6.82024	0.396418
$297$	0	0
$298$	−9.92497	−0.574938
$299$	2.77955	0.160745
$300$	0	0
$301$	−5.44792	−0.314013
$302$	−13.8838	−0.798925
$303$	0	0
$304$	−21.0172	−1.20542
$305$	0	0
$306$	0	0
$307$	2.51330	0.143442	0.0717208	−	0.997425i	$-0.477151\pi$
$307$	2.51330	0.143442	0.0717208	+	0.997425i	$0.477151\pi$
$308$	−17.2443	−0.982583
$309$	0	0
$310$	0	0
$311$	−4.82417	−0.273554	−0.136777	−	0.990602i	$-0.543674\pi$
$311$	−4.82417	−0.273554	−0.136777	+	0.990602i	$0.543674\pi$
$312$	0	0
$313$	5.98812	0.338468	0.169234	−	0.985576i	$-0.445871\pi$
$313$	5.98812	0.338468	0.169234	+	0.985576i	$0.445871\pi$
$314$	−8.47668	−0.478366
$315$	0	0
$316$	36.4452	2.05020
$317$	17.3531	0.974646	0.487323	−	0.873222i	$-0.337973\pi$
$317$	17.3531	0.974646	0.487323	+	0.873222i	$0.337973\pi$
$318$	0	0
$319$	8.69627	0.486898
$320$	0	0
$321$	0	0
$322$	21.4625	1.19606
$323$	−8.61803	−0.479520
$324$	0	0
$325$	0	0
$326$	−10.1902	−0.564383
$327$	0	0
$328$	7.99737	0.441581
$329$	−17.9899	−0.991818
$330$	0	0
$331$	20.3355	1.11774	0.558870	−	0.829255i	$-0.311235\pi$
$331$	20.3355	1.11774	0.558870	+	0.829255i	$0.311235\pi$
$332$	−21.8776	−1.20069
$333$	0	0
$334$	50.2145	2.74761
$335$	0	0
$336$	0	0
$337$	−34.2571	−1.86610	−0.933052	−	0.359741i	$-0.882865\pi$
$337$	−34.2571	−1.86610	−0.933052	+	0.359741i	$0.882865\pi$
$338$	−24.5447	−1.33505
$339$	0	0
$340$	0	0
$341$	7.58680	0.410848
$342$	0	0
$343$	−21.7731	−1.17564
$344$	1.33784	0.0721318
$345$	0	0
$346$	30.7830	1.65490
$347$	−0.0493616	−0.00264987	−0.00132494	−	0.999999i	$-0.500422\pi$
$347$	−0.0493616	−0.00264987	−0.00132494	+	0.999999i	$0.500422\pi$
$348$	0	0
$349$	−7.47437	−0.400094	−0.200047	−	0.979786i	$-0.564109\pi$
$349$	−7.47437	−0.400094	−0.200047	+	0.979786i	$0.564109\pi$
$350$	0	0
$351$	0	0
$352$	−12.5596	−0.669429
$353$	−18.5650	−0.988115	−0.494057	−	0.869429i	$-0.664487\pi$
$353$	−18.5650	−0.988115	−0.494057	+	0.869429i	$0.664487\pi$
$354$	0	0
$355$	0	0
$356$	25.1385	1.33234
$357$	0	0
$358$	−1.69096	−0.0893700
$359$	−10.6044	−0.559681	−0.279841	−	0.960046i	$-0.590282\pi$
$359$	−10.6044	−0.559681	−0.279841	+	0.960046i	$0.590282\pi$
$360$	0	0
$361$	39.9620	2.10327
$362$	−30.1763	−1.58603
$363$	0	0
$364$	13.0752	0.685327
$365$	0	0
$366$	0	0
$367$	11.6176	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$367$	11.6176	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$368$	6.35078	0.331057
$369$	0	0
$370$	0	0
$371$	−16.3252	−0.847561
$372$	0	0
$373$	23.1272	1.19748	0.598742	−	0.800942i	$-0.295668\pi$
$373$	23.1272	1.19748	0.598742	+	0.800942i	$0.295668\pi$
$374$	−3.76328	−0.194595
$375$	0	0
$376$	4.41779	0.227830
$377$	−6.59382	−0.339599
$378$	0	0
$379$	24.8912	1.27857	0.639287	−	0.768968i	$-0.279230\pi$
$379$	24.8912	1.27857	0.639287	+	0.768968i	$0.279230\pi$
$380$	0	0
$381$	0	0
$382$	22.0079	1.12602
$383$	−15.7221	−0.803363	−0.401681	−	0.915780i	$-0.631574\pi$
$383$	−15.7221	−0.803363	−0.401681	+	0.915780i	$0.631574\pi$
$384$	0	0
$385$	0	0
$386$	−9.22604	−0.469593
$387$	0	0
$388$	18.7822	0.953519
$389$	8.50633	0.431288	0.215644	−	0.976472i	$-0.430815\pi$
$389$	8.50633	0.431288	0.215644	+	0.976472i	$0.430815\pi$
$390$	0	0
$391$	2.60412	0.131696
$392$	12.8389	0.648460
$393$	0	0
$394$	3.98977	0.201002
$395$	0	0
$396$	0	0
$397$	−26.2549	−1.31770	−0.658848	−	0.752276i	$-0.728956\pi$
$397$	−26.2549	−1.31770	−0.658848	+	0.752276i	$0.728956\pi$
$398$	9.04174	0.453222
$399$	0	0
$400$	0	0
$401$	25.2815	1.26250	0.631250	−	0.775579i	$-0.282542\pi$
$401$	25.2815	1.26250	0.631250	+	0.775579i	$0.282542\pi$
$402$	0	0
$403$	−5.75258	−0.286556
$404$	16.3236	0.812127
$405$	0	0
$406$	−50.9147	−2.52685
$407$	10.0677	0.499035
$408$	0	0
$409$	33.7932	1.67097	0.835483	−	0.549517i	$-0.185188\pi$
$409$	33.7932	1.67097	0.835483	+	0.549517i	$0.185188\pi$
$410$	0	0
$411$	0	0
$412$	−20.1565	−0.993039
$413$	39.9061	1.96365
$414$	0	0
$415$	0	0
$416$	9.52313	0.466910
$417$	0	0
$418$	25.7473	1.25934
$419$	−7.93332	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$419$	−7.93332	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$420$	0	0
$421$	7.48795	0.364941	0.182470	−	0.983211i	$-0.441591\pi$
$421$	7.48795	0.364941	0.182470	+	0.983211i	$0.441591\pi$
$422$	−19.7307	−0.960476
$423$	0	0
$424$	4.00897	0.194693
$425$	0	0
$426$	0	0
$427$	−6.08380	−0.294416
$428$	23.7344	1.14724
$429$	0	0
$430$	0	0
$431$	−23.3470	−1.12459	−0.562294	−	0.826937i	$-0.690081\pi$
$431$	−23.3470	−1.12459	−0.562294	+	0.826937i	$0.690081\pi$
$432$	0	0
$433$	3.35677	0.161316	0.0806581	−	0.996742i	$-0.474298\pi$
$433$	3.35677	0.161316	0.0806581	+	0.996742i	$0.474298\pi$
$434$	−44.4190	−2.13218
$435$	0	0
$436$	−29.2145	−1.39912
$437$	−17.8166	−0.852285
$438$	0	0
$439$	−9.48402	−0.452648	−0.226324	−	0.974052i	$-0.572671\pi$
$439$	−9.48402	−0.452648	−0.226324	+	0.974052i	$0.572671\pi$
$440$	0	0
$441$	0	0
$442$	2.85345	0.135725
$443$	9.65446	0.458697	0.229349	−	0.973344i	$-0.426340\pi$
$443$	9.65446	0.458697	0.229349	+	0.973344i	$0.426340\pi$
$444$	0	0
$445$	0	0
$446$	7.21080	0.341441
$447$	0	0
$448$	49.6749	2.34692
$449$	−31.5260	−1.48780	−0.743902	−	0.668289i	$-0.767027\pi$
$449$	−31.5260	−1.48780	−0.743902	+	0.668289i	$0.767027\pi$
$450$	0	0
$451$	11.8053	0.555888
$452$	−37.3776	−1.75809
$453$	0	0
$454$	56.8729	2.66918
$455$	0	0
$456$	0	0
$457$	9.94467	0.465192	0.232596	−	0.972573i	$-0.425278\pi$
$457$	9.94467	0.465192	0.232596	+	0.972573i	$0.425278\pi$
$458$	−2.50859	−0.117219
$459$	0	0
$460$	0	0
$461$	23.6836	1.10305	0.551527	−	0.834157i	$-0.314045\pi$
$461$	23.6836	1.10305	0.551527	+	0.834157i	$0.314045\pi$
$462$	0	0
$463$	−27.6504	−1.28502	−0.642511	−	0.766277i	$-0.722107\pi$
$463$	−27.6504	−1.28502	−0.642511	+	0.766277i	$0.722107\pi$
$464$	−15.0657	−0.699409
$465$	0	0
$466$	15.0576	0.697530
$467$	4.35840	0.201683	0.100841	−	0.994903i	$-0.467847\pi$
$467$	4.35840	0.201683	0.100841	+	0.994903i	$0.467847\pi$
$468$	0	0
$469$	−16.8352	−0.777377
$470$	0	0
$471$	0	0
$472$	−9.79973	−0.451069
$473$	1.97485	0.0908038
$474$	0	0
$475$	0	0
$476$	12.2500	0.561477
$477$	0	0
$478$	−0.0908201	−0.00415401
$479$	−12.8143	−0.585499	−0.292750	−	0.956189i	$-0.594570\pi$
$479$	−12.8143	−0.585499	−0.292750	+	0.956189i	$0.594570\pi$
$480$	0	0
$481$	−7.63365	−0.348064
$482$	−24.9822	−1.13791
$483$	0	0
$484$	−21.2963	−0.968014
$485$	0	0
$486$	0	0
$487$	6.92209	0.313670	0.156835	−	0.987625i	$-0.449871\pi$
$487$	6.92209	0.313670	0.156835	+	0.987625i	$0.449871\pi$
$488$	1.49400	0.0676301
$489$	0	0
$490$	0	0
$491$	−26.8161	−1.21020	−0.605098	−	0.796151i	$-0.706866\pi$
$491$	−26.8161	−1.21020	−0.605098	+	0.796151i	$0.706866\pi$
$492$	0	0
$493$	−6.17766	−0.278228
$494$	−19.5225	−0.878358
$495$	0	0
$496$	−13.1436	−0.590167
$497$	−46.2213	−2.07331
$498$	0	0
$499$	2.75460	0.123313	0.0616565	−	0.998097i	$-0.480362\pi$
$499$	2.75460	0.123313	0.0616565	+	0.998097i	$0.480362\pi$
$500$	0	0
$501$	0	0
$502$	−37.0907	−1.65544
$503$	15.7102	0.700483	0.350241	−	0.936659i	$-0.386100\pi$
$503$	15.7102	0.700483	0.350241	+	0.936659i	$0.386100\pi$
$504$	0	0
$505$	0	0
$506$	−7.78008	−0.345867
$507$	0	0
$508$	42.5622	1.88839
$509$	41.0884	1.82121	0.910606	−	0.413277i	$-0.135616\pi$
$509$	41.0884	1.82121	0.910606	+	0.413277i	$0.135616\pi$
$510$	0	0
$511$	21.7777	0.963388
$512$	27.6176	1.22054
$513$	0	0
$514$	32.8264	1.44791
$515$	0	0
$516$	0	0
$517$	6.52129	0.286806
$518$	−58.9438	−2.58984
$519$	0	0
$520$	0	0
$521$	−36.1710	−1.58468	−0.792341	−	0.610079i	$-0.791138\pi$
$521$	−36.1710	−1.58468	−0.792341	+	0.610079i	$0.791138\pi$
$522$	0	0
$523$	−28.0490	−1.22650	−0.613248	−	0.789891i	$-0.710137\pi$
$523$	−28.0490	−1.22650	−0.613248	+	0.789891i	$0.710137\pi$
$524$	0.823595	0.0359789
$525$	0	0
$526$	3.07228	0.133958
$527$	−5.38951	−0.234771
$528$	0	0
$529$	−17.6163	−0.765927
$530$	0	0
$531$	0	0
$532$	−83.8108	−3.63366
$533$	−8.95117	−0.387718
$534$	0	0
$535$	0	0
$536$	4.13422	0.178571
$537$	0	0
$538$	18.4774	0.796617
$539$	18.9520	0.816321
$540$	0	0
$541$	3.45822	0.148681	0.0743403	−	0.997233i	$-0.476315\pi$
$541$	3.45822	0.148681	0.0743403	+	0.997233i	$0.476315\pi$
$542$	−60.7704	−2.61031
$543$	0	0
$544$	8.92209	0.382531
$545$	0	0
$546$	0	0
$547$	−13.9635	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$547$	−13.9635	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$548$	11.4264	0.488111
$549$	0	0
$550$	0	0
$551$	42.2657	1.80058
$552$	0	0
$553$	−63.4281	−2.69724
$554$	−11.2528	−0.478086
$555$	0	0
$556$	−11.8554	−0.502779
$557$	−6.59585	−0.279475	−0.139738	−	0.990189i	$-0.544626\pi$
$557$	−6.59585	−0.279475	−0.139738	+	0.990189i	$0.544626\pi$
$558$	0	0
$559$	−1.49740	−0.0633334
$560$	0	0
$561$	0	0
$562$	51.1126	2.15605
$563$	−16.0218	−0.675238	−0.337619	−	0.941283i	$-0.609621\pi$
$563$	−16.0218	−0.675238	−0.337619	+	0.941283i	$0.609621\pi$
$564$	0	0
$565$	0	0
$566$	57.5456	2.41882
$567$	0	0
$568$	11.3505	0.476258
$569$	−23.0914	−0.968042	−0.484021	−	0.875056i	$-0.660824\pi$
$569$	−23.0914	−0.968042	−0.484021	+	0.875056i	$0.660824\pi$
$570$	0	0
$571$	−34.0102	−1.42328	−0.711640	−	0.702544i	$-0.752047\pi$
$571$	−34.0102	−1.42328	−0.711640	+	0.702544i	$0.752047\pi$
$572$	−4.73972	−0.198178
$573$	0	0
$574$	−69.1171	−2.88489
$575$	0	0
$576$	0	0
$577$	6.46727	0.269236	0.134618	−	0.990898i	$-0.457019\pi$
$577$	6.46727	0.269236	0.134618	+	0.990898i	$0.457019\pi$
$578$	−33.4063	−1.38952
$579$	0	0
$580$	0	0
$581$	38.0751	1.57962
$582$	0	0
$583$	5.91782	0.245091
$584$	−5.34794	−0.221299
$585$	0	0
$586$	−43.0868	−1.77990
$587$	38.5012	1.58911	0.794557	−	0.607189i	$-0.207703\pi$
$587$	38.5012	1.58911	0.794557	+	0.607189i	$0.207703\pi$
$588$	0	0
$589$	36.8735	1.51934
$590$	0	0
$591$	0	0
$592$	−17.4416	−0.716843
$593$	4.93069	0.202479	0.101240	−	0.994862i	$-0.467719\pi$
$593$	4.93069	0.202479	0.101240	+	0.994862i	$0.467719\pi$
$594$	0	0
$595$	0	0
$596$	−11.7112	−0.479710
$597$	0	0
$598$	5.89913	0.241233
$599$	35.0268	1.43116	0.715578	−	0.698533i	$-0.246164\pi$
$599$	35.0268	1.43116	0.715578	+	0.698533i	$0.246164\pi$
$600$	0	0
$601$	−4.90570	−0.200108	−0.100054	−	0.994982i	$-0.531901\pi$
$601$	−4.90570	−0.200108	−0.100054	+	0.994982i	$0.531901\pi$
$602$	−11.5623	−0.471244
$603$	0	0
$604$	−16.3826	−0.666597
$605$	0	0
$606$	0	0
$607$	−48.6955	−1.97649	−0.988244	−	0.152884i	$-0.951144\pi$
$607$	−48.6955	−1.97649	−0.988244	+	0.152884i	$0.951144\pi$
$608$	−61.0423	−2.47559
$609$	0	0
$610$	0	0
$611$	−4.94467	−0.200040
$612$	0	0
$613$	−17.2139	−0.695262	−0.347631	−	0.937631i	$-0.613014\pi$
$613$	−17.2139	−0.695262	−0.347631	+	0.937631i	$0.613014\pi$
$614$	5.33406	0.215265
$615$	0	0
$616$	−7.36988	−0.296941
$617$	−31.6092	−1.27254	−0.636268	−	0.771468i	$-0.719523\pi$
$617$	−31.6092	−1.27254	−0.636268	+	0.771468i	$0.719523\pi$
$618$	0	0
$619$	−22.8196	−0.917198	−0.458599	−	0.888643i	$-0.651649\pi$
$619$	−22.8196	−0.917198	−0.458599	+	0.888643i	$0.651649\pi$
$620$	0	0
$621$	0	0
$622$	−10.2385	−0.410526
$623$	−43.7502	−1.75281
$624$	0	0
$625$	0	0
$626$	12.7088	0.507945
$627$	0	0
$628$	−10.0023	−0.399134
$629$	−7.15186	−0.285163
$630$	0	0
$631$	38.0499	1.51474	0.757372	−	0.652984i	$-0.226483\pi$
$631$	38.0499	1.51474	0.757372	+	0.652984i	$0.226483\pi$
$632$	15.5760	0.619581
$633$	0	0
$634$	36.8290	1.46267
$635$	0	0
$636$	0	0
$637$	−14.3701	−0.569363
$638$	18.4564	0.730696
$639$	0	0
$640$	0	0
$641$	−26.5108	−1.04711	−0.523557	−	0.851991i	$-0.675395\pi$
$641$	−26.5108	−1.04711	−0.523557	+	0.851991i	$0.675395\pi$
$642$	0	0
$643$	36.0014	1.41976	0.709879	−	0.704324i	$-0.248750\pi$
$643$	36.0014	1.41976	0.709879	+	0.704324i	$0.248750\pi$
$644$	25.3252	0.997952
$645$	0	0
$646$	−18.2903	−0.719624
$647$	41.7840	1.64270	0.821349	−	0.570426i	$-0.193222\pi$
$647$	41.7840	1.64270	0.821349	+	0.570426i	$0.193222\pi$
$648$	0	0
$649$	−14.4658	−0.567833
$650$	0	0
$651$	0	0
$652$	−12.0242	−0.470903
$653$	21.4404	0.839027	0.419513	−	0.907749i	$-0.362201\pi$
$653$	21.4404	0.839027	0.419513	+	0.907749i	$0.362201\pi$
$654$	0	0
$655$	0	0
$656$	−20.4519	−0.798511
$657$	0	0
$658$	−38.1807	−1.48844
$659$	16.8319	0.655678	0.327839	−	0.944734i	$-0.393680\pi$
$659$	16.8319	0.655678	0.327839	+	0.944734i	$0.393680\pi$
$660$	0	0
$661$	2.03846	0.0792868	0.0396434	−	0.999214i	$-0.487378\pi$
$661$	2.03846	0.0792868	0.0396434	+	0.999214i	$0.487378\pi$
$662$	43.1587	1.67741
$663$	0	0
$664$	−9.35010	−0.362854
$665$	0	0
$666$	0	0
$667$	−12.7715	−0.494514
$668$	59.2518	2.29252
$669$	0	0
$670$	0	0
$671$	2.20536	0.0851369
$672$	0	0
$673$	−35.3776	−1.36371	−0.681854	−	0.731489i	$-0.738826\pi$
$673$	−35.3776	−1.36371	−0.681854	+	0.731489i	$0.738826\pi$
$674$	−72.7050	−2.80049
$675$	0	0
$676$	−28.9621	−1.11393
$677$	44.4173	1.70710	0.853548	−	0.521014i	$-0.174446\pi$
$677$	44.4173	1.70710	0.853548	+	0.521014i	$0.174446\pi$
$678$	0	0
$679$	−32.6879	−1.25444
$680$	0	0
$681$	0	0
$682$	16.1017	0.616567
$683$	42.4952	1.62603	0.813016	−	0.582241i	$-0.197824\pi$
$683$	42.4952	1.62603	0.813016	+	0.582241i	$0.197824\pi$
$684$	0	0
$685$	0	0
$686$	−46.2098	−1.76430
$687$	0	0
$688$	−3.42130	−0.130436
$689$	−4.48710	−0.170945
$690$	0	0
$691$	28.0808	1.06825	0.534123	−	0.845407i	$-0.320642\pi$
$691$	28.0808	1.06825	0.534123	+	0.845407i	$0.320642\pi$
$692$	36.3231	1.38080
$693$	0	0
$694$	−0.104762	−0.00397671
$695$	0	0
$696$	0	0
$697$	−8.38623	−0.317651
$698$	−15.8631	−0.600427
$699$	0	0
$700$	0	0
$701$	−46.4314	−1.75369	−0.876845	−	0.480772i	$-0.840356\pi$
$701$	−46.4314	−1.75369	−0.876845	+	0.480772i	$0.840356\pi$
$702$	0	0
$703$	48.9309	1.84547
$704$	−18.0070	−0.678664
$705$	0	0
$706$	−39.4011	−1.48288
$707$	−28.4090	−1.06843
$708$	0	0
$709$	48.6331	1.82646	0.913228	−	0.407450i	$-0.133582\pi$
$709$	48.6331	1.82646	0.913228	+	0.407450i	$0.133582\pi$
$710$	0	0
$711$	0	0
$712$	10.7437	0.402638
$713$	−11.1421	−0.417275
$714$	0	0
$715$	0	0
$716$	−1.99529	−0.0745675
$717$	0	0
$718$	−22.5062	−0.839923
$719$	21.1954	0.790456	0.395228	−	0.918583i	$-0.370666\pi$
$719$	21.1954	0.790456	0.395228	+	0.918583i	$0.370666\pi$
$720$	0	0
$721$	35.0797	1.30644
$722$	84.8128	3.15641
$723$	0	0
$724$	−35.6073	−1.32333
$725$	0	0
$726$	0	0
$727$	−13.5911	−0.504066	−0.252033	−	0.967719i	$-0.581099\pi$
$727$	−13.5911	−0.504066	−0.252033	+	0.967719i	$0.581099\pi$
$728$	5.58810	0.207109
$729$	0	0
$730$	0	0
$731$	−1.40290	−0.0518880
$732$	0	0
$733$	−18.6531	−0.688967	−0.344484	−	0.938792i	$-0.611946\pi$
$733$	−18.6531	−0.688967	−0.344484	+	0.938792i	$0.611946\pi$
$734$	24.6565	0.910088
$735$	0	0
$736$	18.4452	0.679900
$737$	6.10270	0.224796
$738$	0	0
$739$	26.4452	0.972803	0.486401	−	0.873735i	$-0.338309\pi$
$739$	26.4452	0.972803	0.486401	+	0.873735i	$0.338309\pi$
$740$	0	0
$741$	0	0
$742$	−34.6475	−1.27195
$743$	−29.0191	−1.06461	−0.532304	−	0.846553i	$-0.678674\pi$
$743$	−29.0191	−1.06461	−0.532304	+	0.846553i	$0.678674\pi$
$744$	0	0
$745$	0	0
$746$	49.0837	1.79708
$747$	0	0
$748$	−4.44058	−0.162364
$749$	−41.3065	−1.50931
$750$	0	0
$751$	15.9489	0.581985	0.290992	−	0.956725i	$-0.406015\pi$
$751$	15.9489	0.581985	0.290992	+	0.956725i	$0.406015\pi$
$752$	−11.2977	−0.411985
$753$	0	0
$754$	−13.9943	−0.509642
$755$	0	0
$756$	0	0
$757$	16.4183	0.596734	0.298367	−	0.954451i	$-0.403558\pi$
$757$	16.4183	0.596734	0.298367	+	0.954451i	$0.403558\pi$
$758$	52.8274	1.91878
$759$	0	0
$760$	0	0
$761$	49.2715	1.78609	0.893045	−	0.449967i	$-0.148564\pi$
$761$	49.2715	1.78609	0.893045	+	0.449967i	$0.148564\pi$
$762$	0	0
$763$	50.8440	1.84068
$764$	25.9688	0.939518
$765$	0	0
$766$	−33.3676	−1.20562
$767$	10.9685	0.396049
$768$	0	0
$769$	49.4236	1.78226	0.891129	−	0.453749i	$-0.149914\pi$
$769$	49.4236	1.78226	0.891129	+	0.453749i	$0.149914\pi$
$770$	0	0
$771$	0	0
$772$	−10.8865	−0.391814
$773$	25.9084	0.931860	0.465930	−	0.884822i	$-0.345720\pi$
$773$	25.9084	0.931860	0.465930	+	0.884822i	$0.345720\pi$
$774$	0	0
$775$	0	0
$776$	8.02715	0.288158
$777$	0	0
$778$	18.0533	0.647241
$779$	57.3761	2.05571
$780$	0	0
$781$	16.7551	0.599543
$782$	5.52681	0.197638
$783$	0	0
$784$	−32.8331	−1.17261
$785$	0	0
$786$	0	0
$787$	−30.7155	−1.09489	−0.547444	−	0.836842i	$-0.684399\pi$
$787$	−30.7155	−1.09489	−0.547444	+	0.836842i	$0.684399\pi$
$788$	4.70783	0.167710
$789$	0	0
$790$	0	0
$791$	65.0508	2.31294
$792$	0	0
$793$	−1.67218	−0.0593808
$794$	−55.7216	−1.97749
$795$	0	0
$796$	10.6690	0.378154
$797$	−33.7141	−1.19421	−0.597107	−	0.802161i	$-0.703683\pi$
$797$	−33.7141	−1.19421	−0.597107	+	0.802161i	$0.703683\pi$
$798$	0	0
$799$	−4.63260	−0.163890
$800$	0	0
$801$	0	0
$802$	53.6559	1.89465
$803$	−7.89434	−0.278585
$804$	0	0
$805$	0	0
$806$	−12.2089	−0.430040
$807$	0	0
$808$	6.97639	0.245429
$809$	0.923014	0.0324514	0.0162257	−	0.999868i	$-0.494835\pi$
$809$	0.923014	0.0324514	0.0162257	+	0.999868i	$0.494835\pi$
$810$	0	0
$811$	26.7815	0.940424	0.470212	−	0.882553i	$-0.344177\pi$
$811$	26.7815	0.940424	0.470212	+	0.882553i	$0.344177\pi$
$812$	−60.0780	−2.10832
$813$	0	0
$814$	21.3669	0.748910
$815$	0	0
$816$	0	0
$817$	9.59820	0.335799
$818$	71.7204	2.50765
$819$	0	0
$820$	0	0
$821$	36.8489	1.28604	0.643018	−	0.765851i	$-0.277682\pi$
$821$	36.8489	1.28604	0.643018	+	0.765851i	$0.277682\pi$
$822$	0	0
$823$	−20.4262	−0.712012	−0.356006	−	0.934484i	$-0.615862\pi$
$823$	−20.4262	−0.712012	−0.356006	+	0.934484i	$0.615862\pi$
$824$	−8.61452	−0.300101
$825$	0	0
$826$	84.6940	2.94688
$827$	4.33017	0.150575	0.0752874	−	0.997162i	$-0.476013\pi$
$827$	4.33017	0.150575	0.0752874	+	0.997162i	$0.476013\pi$
$828$	0	0
$829$	−0.614348	−0.0213372	−0.0106686	−	0.999943i	$-0.503396\pi$
$829$	−0.614348	−0.0213372	−0.0106686	+	0.999943i	$0.503396\pi$
$830$	0	0
$831$	0	0
$832$	13.6535	0.473351
$833$	−13.4631	−0.466470
$834$	0	0
$835$	0	0
$836$	30.3811	1.05075
$837$	0	0
$838$	−16.8372	−0.581630
$839$	−5.39981	−0.186422	−0.0932111	−	0.995646i	$-0.529713\pi$
$839$	−5.39981	−0.186422	−0.0932111	+	0.995646i	$0.529713\pi$
$840$	0	0
$841$	1.29731	0.0447349
$842$	15.8919	0.547672
$843$	0	0
$844$	−23.2817	−0.801391
$845$	0	0
$846$	0	0
$847$	37.0634	1.27351
$848$	−10.2522	−0.352063
$849$	0	0
$850$	0	0
$851$	−14.7855	−0.506841
$852$	0	0
$853$	39.7935	1.36250	0.681252	−	0.732049i	$-0.261436\pi$
$853$	39.7935	1.36250	0.681252	+	0.732049i	$0.261436\pi$
$854$	−12.9119	−0.441835
$855$	0	0
$856$	10.1436	0.346702
$857$	−42.8643	−1.46422	−0.732109	−	0.681188i	$-0.761464\pi$
$857$	−42.8643	−1.46422	−0.732109	+	0.681188i	$0.761464\pi$
$858$	0	0
$859$	−4.56494	−0.155754	−0.0778769	−	0.996963i	$-0.524814\pi$
$859$	−4.56494	−0.155754	−0.0778769	+	0.996963i	$0.524814\pi$
$860$	0	0
$861$	0	0
$862$	−49.5502	−1.68769
$863$	−37.6273	−1.28085	−0.640424	−	0.768022i	$-0.721241\pi$
$863$	−37.6273	−1.28085	−0.640424	+	0.768022i	$0.721241\pi$
$864$	0	0
$865$	0	0
$866$	7.12419	0.242090
$867$	0	0
$868$	−52.4133	−1.77902
$869$	22.9925	0.779966
$870$	0	0
$871$	−4.62728	−0.156790
$872$	−12.4857	−0.422821
$873$	0	0
$874$	−37.8128	−1.27904
$875$	0	0
$876$	0	0
$877$	−52.4041	−1.76956	−0.884781	−	0.466008i	$-0.845692\pi$
$877$	−52.4041	−1.76956	−0.884781	+	0.466008i	$0.845692\pi$
$878$	−20.1283	−0.679296
$879$	0	0
$880$	0	0
$881$	7.93778	0.267430	0.133715	−	0.991020i	$-0.457309\pi$
$881$	7.93778	0.267430	0.133715	+	0.991020i	$0.457309\pi$
$882$	0	0
$883$	−31.6919	−1.06652	−0.533259	−	0.845952i	$-0.679033\pi$
$883$	−31.6919	−1.06652	−0.533259	+	0.845952i	$0.679033\pi$
$884$	3.36700	0.113244
$885$	0	0
$886$	20.4900	0.688374
$887$	2.82104	0.0947213	0.0473606	−	0.998878i	$-0.484919\pi$
$887$	2.82104	0.0947213	0.0473606	+	0.998878i	$0.484919\pi$
$888$	0	0
$889$	−74.0739	−2.48436
$890$	0	0
$891$	0	0
$892$	8.50856	0.284888
$893$	31.6949	1.06063
$894$	0	0
$895$	0	0
$896$	36.1318	1.20708
$897$	0	0
$898$	−66.9087	−2.23277
$899$	26.4320	0.881556
$900$	0	0
$901$	−4.20390	−0.140052
$902$	25.0547	0.834231
$903$	0	0
$904$	−15.9745	−0.531304
$905$	0	0
$906$	0	0
$907$	44.1799	1.46697	0.733485	−	0.679705i	$-0.237892\pi$
$907$	44.1799	1.46697	0.733485	+	0.679705i	$0.237892\pi$
$908$	67.1085	2.22707
$909$	0	0
$910$	0	0
$911$	34.5260	1.14390	0.571949	−	0.820289i	$-0.306188\pi$
$911$	34.5260	1.14390	0.571949	+	0.820289i	$0.306188\pi$
$912$	0	0
$913$	−13.8021	−0.456783
$914$	21.1059	0.698122
$915$	0	0
$916$	−2.96007	−0.0978036
$917$	−1.43336	−0.0473336
$918$	0	0
$919$	−0.531075	−0.0175185	−0.00875927	−	0.999962i	$-0.502788\pi$
$919$	−0.531075	−0.0175185	−0.00875927	+	0.999962i	$0.502788\pi$
$920$	0	0
$921$	0	0
$922$	50.2645	1.65537
$923$	−12.7043	−0.418166
$924$	0	0
$925$	0	0
$926$	−58.6833	−1.92845
$927$	0	0
$928$	−43.7569	−1.43639
$929$	−46.1961	−1.51565	−0.757823	−	0.652460i	$-0.773737\pi$
$929$	−46.1961	−1.51565	−0.757823	+	0.652460i	$0.773737\pi$
$930$	0	0
$931$	92.1108	3.01881
$932$	17.7676	0.581997
$933$	0	0
$934$	9.24998	0.302669
$935$	0	0
$936$	0	0
$937$	30.1220	0.984042	0.492021	−	0.870583i	$-0.336258\pi$
$937$	30.1220	0.984042	0.492021	+	0.870583i	$0.336258\pi$
$938$	−35.7299	−1.16662
$939$	0	0
$940$	0	0
$941$	−57.0783	−1.86070	−0.930350	−	0.366672i	$-0.880497\pi$
$941$	−57.0783	−1.86070	−0.930350	+	0.366672i	$0.880497\pi$
$942$	0	0
$943$	−17.3374	−0.564583
$944$	25.0611	0.815668
$945$	0	0
$946$	4.19130	0.136271
$947$	−41.1250	−1.33638	−0.668192	−	0.743989i	$-0.732931\pi$
$947$	−41.1250	−1.33638	−0.668192	+	0.743989i	$0.732931\pi$
$948$	0	0
$949$	5.98576	0.194306
$950$	0	0
$951$	0	0
$952$	5.23542	0.169681
$953$	34.8143	1.12775	0.563873	−	0.825861i	$-0.309311\pi$
$953$	34.8143	1.12775	0.563873	+	0.825861i	$0.309311\pi$
$954$	0	0
$955$	0	0
$956$	−0.107165	−0.00346598
$957$	0	0
$958$	−27.1962	−0.878668
$959$	−19.8861	−0.642156
$960$	0	0
$961$	−7.94022	−0.256136
$962$	−16.2012	−0.522346
$963$	0	0
$964$	−29.4783	−0.949432
$965$	0	0
$966$	0	0
$967$	37.2697	1.19851	0.599257	−	0.800557i	$-0.295463\pi$
$967$	37.2697	1.19851	0.599257	+	0.800557i	$0.295463\pi$
$968$	−9.10165	−0.292538
$969$	0	0
$970$	0	0
$971$	26.0092	0.834677	0.417338	−	0.908751i	$-0.362963\pi$
$971$	26.0092	0.834677	0.417338	+	0.908751i	$0.362963\pi$
$972$	0	0
$973$	20.6327	0.661453
$974$	14.6910	0.470729
$975$	0	0
$976$	−3.82064	−0.122296
$977$	0.458579	0.0146712	0.00733561	−	0.999973i	$-0.497665\pi$
$977$	0.458579	0.0146712	0.00733561	+	0.999973i	$0.497665\pi$
$978$	0	0
$979$	15.8593	0.506865
$980$	0	0
$981$	0	0
$982$	−56.9128	−1.81616
$983$	8.94565	0.285322	0.142661	−	0.989772i	$-0.454434\pi$
$983$	8.94565	0.285322	0.142661	+	0.989772i	$0.454434\pi$
$984$	0	0
$985$	0	0
$986$	−13.1111	−0.417541
$987$	0	0
$988$	−23.0360	−0.732874
$989$	−2.90030	−0.0922241
$990$	0	0
$991$	−56.5337	−1.79585	−0.897926	−	0.440147i	$-0.854926\pi$
$991$	−56.5337	−1.79585	−0.897926	+	0.440147i	$0.854926\pi$
$992$	−38.1744	−1.21204
$993$	0	0
$994$	−98.0970	−3.11145
$995$	0	0
$996$	0	0
$997$	−39.6114	−1.25451	−0.627253	−	0.778816i	$-0.715821\pi$
$997$	−39.6114	−1.25451	−0.627253	+	0.778816i	$0.715821\pi$
$998$	5.84619	0.185058
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.n.1.3		4
3.2	odd	2		1875.2.a.e.1.2		4
5.4	even	2		5625.2.a.i.1.2		4
15.2	even	4		1875.2.b.c.1249.2		8
15.8	even	4		1875.2.b.c.1249.7		8
15.14	odd	2		1875.2.a.h.1.3		4
25.9	even	10		225.2.h.c.181.1		8
25.14	even	10		225.2.h.c.46.1		8
75.2	even	20		375.2.i.b.274.1		16
75.11	odd	10		375.2.g.b.226.1		8
75.14	odd	10		75.2.g.b.46.2	yes	8
75.23	even	20		375.2.i.b.274.4		16
75.38	even	20		375.2.i.b.349.1		16
75.41	odd	10		375.2.g.b.151.1		8
75.59	odd	10		75.2.g.b.31.2	✓	8
75.62	even	20		375.2.i.b.349.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.31.2	✓	8	75.59	odd	10
75.2.g.b.46.2	yes	8	75.14	odd	10
225.2.h.c.46.1		8	25.14	even	10
225.2.h.c.181.1		8	25.9	even	10
375.2.g.b.151.1		8	75.41	odd	10
375.2.g.b.226.1		8	75.11	odd	10
375.2.i.b.274.1		16	75.2	even	20
375.2.i.b.274.4		16	75.23	even	20
375.2.i.b.349.1		16	75.38	even	20
375.2.i.b.349.4		16	75.62	even	20
1875.2.a.e.1.2		4	3.2	odd	2
1875.2.a.h.1.3		4	15.14	odd	2
1875.2.b.c.1249.2		8	15.2	even	4
1875.2.b.c.1249.7		8	15.8	even	4
5625.2.a.i.1.2		4	5.4	even	2
5625.2.a.n.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.12233 q^{2} +2.50430 q^{4} -4.35840 q^{7} +1.07029 q^{8} +O(q^{10})\)	\(q+2.12233 q^{2} +2.50430 q^{4} -4.35840 q^{7} +1.07029 q^{8} +1.57991 q^{11} -1.19794 q^{13} -9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +7.67867 q^{19} +3.35309 q^{22} -2.32027 q^{23} -2.54243 q^{26} -10.9147 q^{28} +5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -2.38197 q^{34} +6.37232 q^{37} +16.2967 q^{38} +7.47214 q^{41} +1.24998 q^{43} +3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} +11.9957 q^{49} -3.00000 q^{52} +3.74568 q^{53} -4.66476 q^{56} +11.6820 q^{58} -9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} -11.3975 q^{64} +3.86270 q^{67} -2.81066 q^{68} +10.6051 q^{71} -4.99672 q^{73} +13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +14.5531 q^{79} +15.8584 q^{82} -8.73603 q^{83} +2.65288 q^{86} +1.69096 q^{88} +10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +8.76025 q^{94} +7.49996 q^{97} +25.4588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 - 2 * q^7 + 15 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8} + 7 q^{11} - q^{13} - 16 q^{14} + 4 q^{16} + 2 q^{17} + 5 q^{19} + 6 q^{22} + q^{23} - 3 q^{26} - 9 q^{28} + 20 q^{29} + 23 q^{31} + 12 q^{32} - 14 q^{34} - 2 q^{37} + 35 q^{38} + 12 q^{41} - 16 q^{43} + 29 q^{44} - 17 q^{46} + 2 q^{47} + 8 q^{49} - 12 q^{52} - 4 q^{53} - 5 q^{56} + 25 q^{58} + 15 q^{59} - 2 q^{61} + 9 q^{62} + 23 q^{64} - 2 q^{67} - 11 q^{68} + 2 q^{71} - 16 q^{73} + 19 q^{74} + 40 q^{76} + 19 q^{77} + 35 q^{79} + 6 q^{82} + 16 q^{83} - 3 q^{86} + 30 q^{88} + 35 q^{89} - 12 q^{91} - 23 q^{92} - 9 q^{94} - 12 q^{97} - q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 - 2 * q^7 + 15 * q^8 + 7 * q^11 - q^13 - 16 * q^14 + 4 * q^16 + 2 * q^17 + 5 * q^19 + 6 * q^22 + q^23 - 3 * q^26 - 9 * q^28 + 20 * q^29 + 23 * q^31 + 12 * q^32 - 14 * q^34 - 2 * q^37 + 35 * q^38 + 12 * q^41 - 16 * q^43 + 29 * q^44 - 17 * q^46 + 2 * q^47 + 8 * q^49 - 12 * q^52 - 4 * q^53 - 5 * q^56 + 25 * q^58 + 15 * q^59 - 2 * q^61 + 9 * q^62 + 23 * q^64 - 2 * q^67 - 11 * q^68 + 2 * q^71 - 16 * q^73 + 19 * q^74 + 40 * q^76 + 19 * q^77 + 35 * q^79 + 6 * q^82 + 16 * q^83 - 3 * q^86 + 30 * q^88 + 35 * q^89 - 12 * q^91 - 23 * q^92 - 9 * q^94 - 12 * q^97 - q^98

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