LMFDB - Embedded newform 5625.2.a.n.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:	$0$
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			1.2
Root			$-1.12233$ 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)

$f(q)$	$=$	$q-1.12233 q^{2} -0.740367 q^{4} -1.11373 q^{7} +3.07561 q^{8} +O(q^{10})$	$q-1.12233 q^{2} -0.740367 q^{4} -1.11373 q^{7} +3.07561 q^{8} -3.67008 q^{11} +4.05204 q^{13} +1.24998 q^{14} -1.97112 q^{16} +2.12233 q^{17} -4.06064 q^{19} +4.11905 q^{22} +6.17438 q^{23} -4.54774 q^{26} +0.824573 q^{28} +2.25963 q^{29} +10.0520 q^{31} -3.93896 q^{32} -2.38197 q^{34} -7.37232 q^{37} +4.55739 q^{38} +7.47214 q^{41} -9.24998 q^{43} +2.71720 q^{44} -6.92971 q^{46} -3.12765 q^{47} -5.75960 q^{49} -3.00000 q^{52} -3.50961 q^{53} -3.42541 q^{56} -2.53606 q^{58} +6.59382 q^{59} -9.10408 q^{61} -11.2817 q^{62} +8.36307 q^{64} -2.62663 q^{67} -1.57131 q^{68} -0.660827 q^{71} -7.47542 q^{73} +8.27420 q^{74} +3.00637 q^{76} +4.08749 q^{77} +8.53711 q^{79} -8.38623 q^{82} +12.2639 q^{83} +10.3816 q^{86} -11.2877 q^{88} +15.2881 q^{89} -4.51290 q^{91} -4.57131 q^{92} +3.51026 q^{94} -13.5000 q^{97} +6.46419 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.12233	−0.793610	−0.396805	−	0.917903i	$-0.629881\pi$
$2$	−1.12233	−0.793610	−0.396805	+	0.917903i	$0.629881\pi$
$3$	0	0
$3$	0	0
$4$	−0.740367	−0.370184
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.11373	−0.420952	−0.210476	−	0.977599i	$-0.567501\pi$
$7$	−1.11373	−0.420952	−0.210476	+	0.977599i	$0.567501\pi$
$8$	3.07561	1.08739
$9$	0	0
$10$	0	0
$11$	−3.67008	−1.10657	−0.553285	−	0.832992i	$-0.686626\pi$
$11$	−3.67008	−1.10657	−0.553285	+	0.832992i	$0.686626\pi$
$12$	0	0
$13$	4.05204	1.12383	0.561917	−	0.827194i	$-0.310064\pi$
$13$	4.05204	1.12383	0.561917	+	0.827194i	$0.310064\pi$
$14$	1.24998	0.334072
$15$	0	0
$16$	−1.97112	−0.492780
$17$	2.12233	0.514741	0.257371	−	0.966313i	$-0.417144\pi$
$17$	2.12233	0.514741	0.257371	+	0.966313i	$0.417144\pi$
$18$	0	0
$19$	−4.06064	−0.931575	−0.465787	−	0.884897i	$-0.654229\pi$
$19$	−4.06064	−0.931575	−0.465787	+	0.884897i	$0.654229\pi$
$20$	0	0
$21$	0	0
$22$	4.11905	0.878184
$23$	6.17438	1.28745	0.643723	−	0.765258i	$-0.277389\pi$
$23$	6.17438	1.28745	0.643723	+	0.765258i	$0.277389\pi$
$24$	0	0
$25$	0	0
$26$	−4.54774	−0.891886
$27$	0	0
$28$	0.824573	0.155830
$29$	2.25963	0.419603	0.209802	−	0.977744i	$-0.432718\pi$
$29$	2.25963	0.419603	0.209802	+	0.977744i	$0.432718\pi$
$30$	0	0
$31$	10.0520	1.80540	0.902700	−	0.430271i	$-0.141582\pi$
$31$	10.0520	1.80540	0.902700	+	0.430271i	$0.141582\pi$
$32$	−3.93896	−0.696316
$33$	0	0
$34$	−2.38197	−0.408504
$35$	0	0
$36$	0	0
$37$	−7.37232	−1.21200	−0.606001	−	0.795464i	$-0.707227\pi$
$37$	−7.37232	−1.21200	−0.606001	+	0.795464i	$0.707227\pi$
$38$	4.55739	0.739307
$39$	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
$43$	−9.24998	−1.41061	−0.705304	−	0.708905i	$-0.749190\pi$
$43$	−9.24998	−1.41061	−0.705304	+	0.708905i	$0.749190\pi$
$44$	2.71720	0.409634
$45$	0	0
$46$	−6.92971	−1.02173
$47$	−3.12765	−0.456214	−0.228107	−	0.973636i	$-0.573254\pi$
$47$	−3.12765	−0.456214	−0.228107	+	0.973636i	$0.573254\pi$
$48$	0	0
$49$	−5.75960	−0.822799
$50$	0	0
$51$	0	0
$52$	−3.00000	−0.416025
$53$	−3.50961	−0.482083	−0.241041	−	0.970515i	$-0.577489\pi$
$53$	−3.50961	−0.482083	−0.241041	+	0.970515i	$0.577489\pi$
$54$	0	0
$55$	0	0
$56$	−3.42541	−0.457739
$57$	0	0
$58$	−2.53606	−0.333001
$59$	6.59382	0.858442	0.429221	−	0.903199i	$-0.358788\pi$
$59$	6.59382	0.858442	0.429221	+	0.903199i	$0.358788\pi$
$60$	0	0
$61$	−9.10408	−1.16566	−0.582829	−	0.812595i	$-0.698054\pi$
$61$	−9.10408	−1.16566	−0.582829	+	0.812595i	$0.698054\pi$
$62$	−11.2817	−1.43278
$63$	0	0
$64$	8.36307	1.04538
$65$	0	0
$66$	0	0
$67$	−2.62663	−0.320894	−0.160447	−	0.987044i	$-0.551294\pi$
$67$	−2.62663	−0.320894	−0.160447	+	0.987044i	$0.551294\pi$
$68$	−1.57131	−0.190549
$69$	0	0
$70$	0	0
$71$	−0.660827	−0.0784257	−0.0392128	−	0.999231i	$-0.512485\pi$
$71$	−0.660827	−0.0784257	−0.0392128	+	0.999231i	$0.512485\pi$
$72$	0	0
$73$	−7.47542	−0.874932	−0.437466	−	0.899235i	$-0.644124\pi$
$73$	−7.47542	−0.874932	−0.437466	+	0.899235i	$0.644124\pi$
$74$	8.27420	0.961856
$75$	0	0
$76$	3.00637	0.344854
$77$	4.08749	0.465813
$78$	0	0
$79$	8.53711	0.960500	0.480250	−	0.877132i	$-0.340546\pi$
$79$	8.53711	0.960500	0.480250	+	0.877132i	$0.340546\pi$
$80$	0	0
$81$	0	0
$82$	−8.38623	−0.926104
$83$	12.2639	1.34614	0.673069	−	0.739580i	$-0.264976\pi$
$83$	12.2639	1.34614	0.673069	+	0.739580i	$0.264976\pi$
$84$	0	0
$85$	0	0
$86$	10.3816	1.11947
$87$	0	0
$88$	−11.2877	−1.20327
$89$	15.2881	1.62054	0.810268	−	0.586059i	$-0.199321\pi$
$89$	15.2881	1.62054	0.810268	+	0.586059i	$0.199321\pi$
$90$	0	0
$91$	−4.51290	−0.473080
$92$	−4.57131	−0.476592
$93$	0	0
$94$	3.51026	0.362056
$95$	0	0
$96$	0	0
$97$	−13.5000	−1.37071	−0.685357	−	0.728207i	$-0.740354\pi$
$97$	−13.5000	−1.37071	−0.685357	+	0.728207i	$0.740354\pi$
$98$	6.46419	0.652982
$99$	0	0
$100$	0	0
$101$	−7.22642	−0.719055	−0.359528	−	0.933134i	$-0.617062\pi$
$101$	−7.22642	−0.719055	−0.359528	+	0.933134i	$0.617062\pi$
$102$	0	0
$103$	−5.27748	−0.520006	−0.260003	−	0.965608i	$-0.583723\pi$
$103$	−5.27748	−0.520006	−0.260003	+	0.965608i	$0.583723\pi$
$104$	12.4625	1.22205
$105$	0	0
$106$	3.93896	0.382585
$107$	5.46682	0.528498	0.264249	−	0.964455i	$-0.414876\pi$
$107$	5.46682	0.528498	0.264249	+	0.964455i	$0.414876\pi$
$108$	0	0
$109$	11.3395	1.08613	0.543064	−	0.839692i	$-0.317264\pi$
$109$	11.3395	1.08613	0.543064	+	0.839692i	$0.317264\pi$
$110$	0	0
$111$	0	0
$112$	2.19531	0.207437
$113$	4.83520	0.454858	0.227429	−	0.973795i	$-0.426968\pi$
$113$	4.83520	0.454858	0.227429	+	0.973795i	$0.426968\pi$
$114$	0	0
$115$	0	0
$116$	−1.67296	−0.155330
$117$	0	0
$118$	−7.40046	−0.681268
$119$	−2.36372	−0.216681
$120$	0	0
$121$	2.46946	0.224496
$122$	10.2178	0.925078
$123$	0	0
$124$	−7.44220	−0.668330
$125$	0	0
$126$	0	0
$127$	−0.759596	−0.0674032	−0.0337016	−	0.999432i	$-0.510730\pi$
$127$	−0.759596	−0.0674032	−0.0337016	+	0.999432i	$0.510730\pi$
$128$	−1.50823	−0.133310
$129$	0	0
$130$	0	0
$131$	−14.6551	−1.28042	−0.640211	−	0.768199i	$-0.721153\pi$
$131$	−14.6551	−1.28042	−0.640211	+	0.768199i	$0.721153\pi$
$132$	0	0
$133$	4.52248	0.392148
$134$	2.94796	0.254665
$135$	0	0
$136$	6.52746	0.559725
$137$	12.2914	1.05012	0.525062	−	0.851064i	$-0.324042\pi$
$137$	12.2914	1.05012	0.525062	+	0.851064i	$0.324042\pi$
$138$	0	0
$139$	14.7340	1.24972	0.624861	−	0.780736i	$-0.285156\pi$
$139$	14.7340	1.24972	0.624861	+	0.780736i	$0.285156\pi$
$140$	0	0
$141$	0	0
$142$	0.741668	0.0622394
$143$	−14.8713	−1.24360
$144$	0	0
$145$	0	0
$146$	8.38991	0.694354
$147$	0	0
$148$	5.45822	0.448663
$149$	−15.6498	−1.28208	−0.641041	−	0.767507i	$-0.721497\pi$
$149$	−15.6498	−1.28208	−0.641041	+	0.767507i	$0.721497\pi$
$150$	0	0
$151$	3.95819	0.322113	0.161056	−	0.986945i	$-0.448510\pi$
$151$	3.95819	0.322113	0.161056	+	0.986945i	$0.448510\pi$
$152$	−12.4889	−1.01299
$153$	0	0
$154$	−4.58753	−0.369673
$155$	0	0
$156$	0	0
$157$	4.50061	0.359188	0.179594	−	0.983741i	$-0.442522\pi$
$157$	4.50061	0.359188	0.179594	+	0.983741i	$0.442522\pi$
$158$	−9.58149	−0.762262
$159$	0	0
$160$	0	0
$161$	−6.87661	−0.541953
$162$	0	0
$163$	2.45389	0.192203	0.0961016	−	0.995372i	$-0.469363\pi$
$163$	2.45389	0.192203	0.0961016	+	0.995372i	$0.469363\pi$
$164$	−5.53213	−0.431987
$165$	0	0
$166$	−13.7642	−1.06831
$167$	−3.06328	−0.237043	−0.118522	−	0.992951i	$-0.537816\pi$
$167$	−3.06328	−0.237043	−0.118522	+	0.992951i	$0.537816\pi$
$168$	0	0
$169$	3.41904	0.263003
$170$	0	0
$171$	0	0
$172$	6.84839	0.522185
$173$	11.2596	0.856054	0.428027	−	0.903766i	$-0.359209\pi$
$173$	11.2596	0.856054	0.428027	+	0.903766i	$0.359209\pi$
$174$	0	0
$175$	0	0
$176$	7.23416	0.545296
$177$	0	0
$178$	−17.1584	−1.28607
$179$	−10.0574	−0.751722	−0.375861	−	0.926676i	$-0.622653\pi$
$179$	−10.0574	−0.751722	−0.375861	+	0.926676i	$0.622653\pi$
$180$	0	0
$181$	−19.4684	−1.44708	−0.723539	−	0.690283i	$-0.757486\pi$
$181$	−19.4684	−1.44708	−0.723539	+	0.690283i	$0.757486\pi$
$182$	5.06498	0.375441
$183$	0	0
$184$	18.9899	1.39996
$185$	0	0
$186$	0	0
$187$	−7.78912	−0.569597
$188$	2.31561	0.168883
$189$	0	0
$190$	0	0
$191$	11.6090	0.840000	0.420000	−	0.907524i	$-0.362030\pi$
$191$	11.6090	0.840000	0.420000	+	0.907524i	$0.362030\pi$
$192$	0	0
$193$	3.38156	0.243410	0.121705	−	0.992566i	$-0.461164\pi$
$193$	3.38156	0.243410	0.121705	+	0.992566i	$0.461164\pi$
$194$	15.1515	1.08781
$195$	0	0
$196$	4.26422	0.304587
$197$	−17.5881	−1.25310	−0.626550	−	0.779381i	$-0.715534\pi$
$197$	−17.5881	−1.25310	−0.626550	+	0.779381i	$0.715534\pi$
$198$	0	0
$199$	20.0102	1.41849	0.709244	−	0.704963i	$-0.249037\pi$
$199$	20.0102	1.41849	0.709244	+	0.704963i	$0.249037\pi$
$200$	0	0
$201$	0	0
$202$	8.11045	0.570649
$203$	−2.51663	−0.176633
$204$	0	0
$205$	0	0
$206$	5.92309	0.412681
$207$	0	0
$208$	−7.98706	−0.553803
$209$	14.9029	1.03085
$210$	0	0
$211$	2.44261	0.168156	0.0840780	−	0.996459i	$-0.473206\pi$
$211$	2.44261	0.168156	0.0840780	+	0.996459i	$0.473206\pi$
$212$	2.59840	0.178459
$213$	0	0
$214$	−6.13560	−0.419421
$215$	0	0
$216$	0	0
$217$	−11.1953	−0.759987
$218$	−12.7267	−0.861961
$219$	0	0
$220$	0	0
$221$	8.59978	0.578484
$222$	0	0
$223$	25.6369	1.71677	0.858386	−	0.513005i	$-0.171468\pi$
$223$	25.6369	1.71677	0.858386	+	0.513005i	$0.171468\pi$
$224$	4.38695	0.293116
$225$	0	0
$226$	−5.42671	−0.360979
$227$	6.56336	0.435625	0.217813	−	0.975991i	$-0.430108\pi$
$227$	6.56336	0.435625	0.217813	+	0.975991i	$0.430108\pi$
$228$	0	0
$229$	−7.96390	−0.526269	−0.263135	−	0.964759i	$-0.584756\pi$
$229$	−7.96390	−0.526269	−0.263135	+	0.964759i	$0.584756\pi$
$230$	0	0
$231$	0	0
$232$	6.94974	0.456273
$233$	−9.42107	−0.617195	−0.308597	−	0.951193i	$-0.599860\pi$
$233$	−9.42107	−0.617195	−0.308597	+	0.951193i	$0.599860\pi$
$234$	0	0
$235$	0	0
$236$	−4.88185	−0.317781
$237$	0	0
$238$	2.65288	0.171961
$239$	11.2231	0.725964	0.362982	−	0.931796i	$-0.381759\pi$
$239$	11.2231	0.725964	0.362982	+	0.931796i	$0.381759\pi$
$240$	0	0
$241$	22.6809	1.46101	0.730503	−	0.682910i	$-0.239286\pi$
$241$	22.6809	1.46101	0.730503	+	0.682910i	$0.239286\pi$
$242$	−2.77155	−0.178162
$243$	0	0
$244$	6.74037	0.431508
$245$	0	0
$246$	0	0
$247$	−16.4539	−1.04694
$248$	30.9161	1.96318
$249$	0	0
$250$	0	0
$251$	6.76819	0.427205	0.213602	−	0.976921i	$-0.431480\pi$
$251$	6.76819	0.427205	0.213602	+	0.976921i	$0.431480\pi$
$252$	0	0
$253$	−22.6604	−1.42465
$254$	0.852520	0.0534918
$255$	0	0
$256$	−15.0334	−0.939587
$257$	−14.7934	−0.922786	−0.461393	−	0.887196i	$-0.652650\pi$
$257$	−14.7934	−0.922786	−0.461393	+	0.887196i	$0.652650\pi$
$258$	0	0
$259$	8.21080	0.510194
$260$	0	0
$261$	0	0
$262$	16.4479	1.01616
$263$	19.9688	1.23133	0.615665	−	0.788008i	$-0.288887\pi$
$263$	19.9688	1.23133	0.615665	+	0.788008i	$0.288887\pi$
$264$	0	0
$265$	0	0
$266$	−5.07573	−0.311213
$267$	0	0
$268$	1.94467	0.118790
$269$	10.2381	0.624228	0.312114	−	0.950045i	$-0.398963\pi$
$269$	10.2381	0.624228	0.312114	+	0.950045i	$0.398963\pi$
$270$	0	0
$271$	12.3075	0.747630	0.373815	−	0.927503i	$-0.378050\pi$
$271$	12.3075	0.747630	0.373815	+	0.927503i	$0.378050\pi$
$272$	−4.18338	−0.253654
$273$	0	0
$274$	−13.7950	−0.833389
$275$	0	0
$276$	0	0
$277$	−31.5520	−1.89578	−0.947888	−	0.318603i	$-0.896786\pi$
$277$	−31.5520	−1.89578	−0.947888	+	0.318603i	$0.896786\pi$
$278$	−16.5365	−0.991791
$279$	0	0
$280$	0	0
$281$	21.3119	1.27136	0.635681	−	0.771952i	$-0.280719\pi$
$281$	21.3119	1.27136	0.635681	+	0.771952i	$0.280719\pi$
$282$	0	0
$283$	0.864403	0.0513834	0.0256917	−	0.999670i	$-0.491821\pi$
$283$	0.864403	0.0513834	0.0256917	+	0.999670i	$0.491821\pi$
$284$	0.489254	0.0290319
$285$	0	0
$286$	16.6906	0.986933
$287$	−8.32198	−0.491231
$288$	0	0
$289$	−12.4957	−0.735041
$290$	0	0
$291$	0	0
$292$	5.53456	0.323886
$293$	3.17701	0.185603	0.0928014	−	0.995685i	$-0.470418\pi$
$293$	3.17701	0.185603	0.0928014	+	0.995685i	$0.470418\pi$
$294$	0	0
$295$	0	0
$296$	−22.6743	−1.31792
$297$	0	0
$298$	17.5643	1.01747
$299$	25.0188	1.44688
$300$	0	0
$301$	10.3020	0.593799
$302$	−4.44240	−0.255632
$303$	0	0
$304$	8.00401	0.459062
$305$	0	0
$306$	0	0
$307$	0.507986	0.0289923	0.0144961	−	0.999895i	$-0.495386\pi$
$307$	0.507986	0.0289923	0.0144961	+	0.999895i	$0.495386\pi$
$308$	−3.02624	−0.172436
$309$	0	0
$310$	0	0
$311$	14.6438	0.830375	0.415188	−	0.909736i	$-0.363716\pi$
$311$	14.6438	0.830375	0.415188	+	0.909736i	$0.363716\pi$
$312$	0	0
$313$	14.9562	0.845372	0.422686	−	0.906276i	$-0.361087\pi$
$313$	14.9562	0.845372	0.422686	+	0.906276i	$0.361087\pi$
$314$	−5.05119	−0.285055
$315$	0	0
$316$	−6.32060	−0.355562
$317$	18.1190	1.01767	0.508834	−	0.860865i	$-0.330077\pi$
$317$	18.1190	1.01767	0.508834	+	0.860865i	$0.330077\pi$
$318$	0	0
$319$	−8.29302	−0.464320
$320$	0	0
$321$	0	0
$322$	7.71785	0.430099
$323$	−8.61803	−0.479520
$324$	0	0
$325$	0	0
$326$	−2.75408	−0.152534
$327$	0	0
$328$	22.9813	1.26893
$329$	3.48337	0.192044
$330$	0	0
$331$	26.3514	1.44841	0.724203	−	0.689587i	$-0.242208\pi$
$331$	26.3514	1.44841	0.724203	+	0.689587i	$0.242208\pi$
$332$	−9.07979	−0.498318
$333$	0	0
$334$	3.43802	0.188120
$335$	0	0
$336$	0	0
$337$	10.6948	0.582584	0.291292	−	0.956634i	$-0.405915\pi$
$337$	10.6948	0.582584	0.291292	+	0.956634i	$0.405915\pi$
$338$	−3.83731	−0.208722
$339$	0	0
$340$	0	0
$341$	−36.8918	−1.99780
$342$	0	0
$343$	14.2108	0.767311
$344$	−28.4493	−1.53388
$345$	0	0
$346$	−12.6371	−0.679373
$347$	16.1740	0.868264	0.434132	−	0.900849i	$-0.357055\pi$
$347$	16.1740	0.868264	0.434132	+	0.900849i	$0.357055\pi$
$348$	0	0
$349$	15.2383	0.815688	0.407844	−	0.913052i	$-0.366281\pi$
$349$	15.2383	0.815688	0.407844	+	0.913052i	$0.366281\pi$
$350$	0	0
$351$	0	0
$352$	14.4563	0.770522
$353$	−29.5383	−1.57217	−0.786084	−	0.618120i	$-0.787894\pi$
$353$	−29.5383	−1.57217	−0.786084	+	0.618120i	$0.787894\pi$
$354$	0	0
$355$	0	0
$356$	−11.3188	−0.599896
$357$	0	0
$358$	11.2877	0.596574
$359$	13.1668	0.694915	0.347457	−	0.937696i	$-0.387045\pi$
$359$	13.1668	0.694915	0.347457	+	0.937696i	$0.387045\pi$
$360$	0	0
$361$	−2.51120	−0.132168
$362$	21.8501	1.14842
$363$	0	0
$364$	3.34120	0.175127
$365$	0	0
$366$	0	0
$367$	3.88895	0.203001	0.101501	−	0.994835i	$-0.467636\pi$
$367$	3.88895	0.203001	0.101501	+	0.994835i	$0.467636\pi$
$368$	−12.1704	−0.634428
$369$	0	0
$370$	0	0
$371$	3.90878	0.202934
$372$	0	0
$373$	8.14326	0.421642	0.210821	−	0.977525i	$-0.432386\pi$
$373$	8.14326	0.421642	0.210821	+	0.977525i	$0.432386\pi$
$374$	8.74200	0.452038
$375$	0	0
$376$	−9.61941	−0.496083
$377$	9.15613	0.471564
$378$	0	0
$379$	9.90720	0.508898	0.254449	−	0.967086i	$-0.418106\pi$
$379$	9.90720	0.508898	0.254449	+	0.967086i	$0.418106\pi$
$380$	0	0
$381$	0	0
$382$	−13.0292	−0.666632
$383$	−5.22215	−0.266840	−0.133420	−	0.991060i	$-0.542596\pi$
$383$	−5.22215	−0.266840	−0.133420	+	0.991060i	$0.542596\pi$
$384$	0	0
$385$	0	0
$386$	−3.79524	−0.193173
$387$	0	0
$388$	9.99493	0.507416
$389$	3.72974	0.189105	0.0945526	−	0.995520i	$-0.469858\pi$
$389$	3.72974	0.189105	0.0945526	+	0.995520i	$0.469858\pi$
$390$	0	0
$391$	13.1041	0.662702
$392$	−17.7142	−0.894705
$393$	0	0
$394$	19.7397	0.994473
$395$	0	0
$396$	0	0
$397$	−4.01562	−0.201538	−0.100769	−	0.994910i	$-0.532130\pi$
$397$	−4.01562	−0.201538	−0.100769	+	0.994910i	$0.532130\pi$
$398$	−22.4581	−1.12573
$399$	0	0
$400$	0	0
$401$	24.9890	1.24789	0.623945	−	0.781468i	$-0.285529\pi$
$401$	24.9890	1.24789	0.623945	+	0.781468i	$0.285529\pi$
$402$	0	0
$403$	40.7313	2.02897
$404$	5.35020	0.266183
$405$	0	0
$406$	2.82450	0.140178
$407$	27.0570	1.34116
$408$	0	0
$409$	−25.3768	−1.25480	−0.627401	−	0.778697i	$-0.715881\pi$
$409$	−25.3768	−1.25480	−0.627401	+	0.778697i	$0.715881\pi$
$410$	0	0
$411$	0	0
$412$	3.90728	0.192498
$413$	−7.34376	−0.361363
$414$	0	0
$415$	0	0
$416$	−15.9608	−0.782544
$417$	0	0
$418$	−16.7260	−0.818094
$419$	30.8219	1.50575	0.752873	−	0.658165i	$-0.228667\pi$
$419$	30.8219	1.50575	0.752873	+	0.658165i	$0.228667\pi$
$420$	0	0
$421$	8.54649	0.416530	0.208265	−	0.978072i	$-0.433218\pi$
$421$	8.54649	0.416530	0.208265	+	0.978072i	$0.433218\pi$
$422$	−2.74142	−0.133450
$423$	0	0
$424$	−10.7942	−0.524212
$425$	0	0
$426$	0	0
$427$	10.1395	0.490686
$428$	−4.04746	−0.195641
$429$	0	0
$430$	0	0
$431$	26.3815	1.27075	0.635376	−	0.772203i	$-0.280845\pi$
$431$	26.3815	1.27075	0.635376	+	0.772203i	$0.280845\pi$
$432$	0	0
$433$	9.37272	0.450424	0.225212	−	0.974310i	$-0.427693\pi$
$433$	9.37272	0.450424	0.225212	+	0.974310i	$0.427693\pi$
$434$	12.5649	0.603133
$435$	0	0
$436$	−8.39540	−0.402067
$437$	−25.0719	−1.19935
$438$	0	0
$439$	−0.515980	−0.0246264	−0.0123132	−	0.999924i	$-0.503920\pi$
$439$	−0.515980	−0.0246264	−0.0123132	+	0.999924i	$0.503920\pi$
$440$	0	0
$441$	0	0
$442$	−9.65183	−0.459091
$443$	−17.8348	−0.847357	−0.423678	−	0.905813i	$-0.639261\pi$
$443$	−17.8348	−0.847357	−0.423678	+	0.905813i	$0.639261\pi$
$444$	0	0
$445$	0	0
$446$	−28.7731	−1.36245
$447$	0	0
$448$	−9.31423	−0.440056
$449$	4.16533	0.196574	0.0982870	−	0.995158i	$-0.468664\pi$
$449$	4.16533	0.196574	0.0982870	+	0.995158i	$0.468664\pi$
$450$	0	0
$451$	−27.4233	−1.29131
$452$	−3.57983	−0.168381
$453$	0	0
$454$	−7.36628	−0.345716
$455$	0	0
$456$	0	0
$457$	17.6734	0.826725	0.413362	−	0.910567i	$-0.364354\pi$
$457$	17.6734	0.826725	0.413362	+	0.910567i	$0.364354\pi$
$458$	8.93815	0.417652
$459$	0	0
$460$	0	0
$461$	0.204956	0.00954577	0.00477288	−	0.999989i	$-0.498481\pi$
$461$	0.204956	0.00954577	0.00477288	+	0.999989i	$0.498481\pi$
$462$	0	0
$463$	12.8176	0.595682	0.297841	−	0.954615i	$-0.403733\pi$
$463$	12.8176	0.595682	0.297841	+	0.954615i	$0.403733\pi$
$464$	−4.45401	−0.206772
$465$	0	0
$466$	10.5736	0.489812
$467$	1.11373	0.0515375	0.0257687	−	0.999668i	$-0.491797\pi$
$467$	1.11373	0.0515375	0.0257687	+	0.999668i	$0.491797\pi$
$468$	0	0
$469$	2.92537	0.135081
$470$	0	0
$471$	0	0
$472$	20.2800	0.933462
$473$	33.9481	1.56094
$474$	0	0
$475$	0	0
$476$	1.75002	0.0802120
$477$	0	0
$478$	−12.5961	−0.576132
$479$	25.1750	1.15027	0.575136	−	0.818057i	$-0.304949\pi$
$479$	25.1750	1.15027	0.575136	+	0.818057i	$0.304949\pi$
$480$	0	0
$481$	−29.8729	−1.36209
$482$	−25.4555	−1.15947
$483$	0	0
$484$	−1.82830	−0.0831048
$485$	0	0
$486$	0	0
$487$	−10.3598	−0.469447	−0.234723	−	0.972062i	$-0.575418\pi$
$487$	−10.3598	−0.469447	−0.234723	+	0.972062i	$0.575418\pi$
$488$	−28.0006	−1.26753
$489$	0	0
$490$	0	0
$491$	15.6571	0.706595	0.353297	−	0.935511i	$-0.385060\pi$
$491$	15.6571	0.706595	0.353297	+	0.935511i	$0.385060\pi$
$492$	0	0
$493$	4.79569	0.215987
$494$	18.4667	0.830858
$495$	0	0
$496$	−19.8138	−0.889665
$497$	0.735985	0.0330135
$498$	0	0
$499$	35.7864	1.60202	0.801010	−	0.598651i	$-0.204296\pi$
$499$	35.7864	1.60202	0.801010	+	0.598651i	$0.204296\pi$
$500$	0	0
$501$	0	0
$502$	−7.59617	−0.339034
$503$	−11.7791	−0.525203	−0.262601	−	0.964904i	$-0.584580\pi$
$503$	−11.7791	−0.525203	−0.262601	+	0.964904i	$0.584580\pi$
$504$	0	0
$505$	0	0
$506$	25.4326	1.13061
$507$	0	0
$508$	0.562380	0.0249516
$509$	−33.6507	−1.49154	−0.745771	−	0.666203i	$-0.767918\pi$
$509$	−33.6507	−1.49154	−0.745771	+	0.666203i	$0.767918\pi$
$510$	0	0
$511$	8.32563	0.368304
$512$	19.8889	0.878976
$513$	0	0
$514$	16.6031	0.732332
$515$	0	0
$516$	0	0
$517$	11.4787	0.504833
$518$	−9.21526	−0.404895
$519$	0	0
$520$	0	0
$521$	11.8448	0.518929	0.259465	−	0.965753i	$-0.416454\pi$
$521$	11.8448	0.518929	0.259465	+	0.965753i	$0.416454\pi$
$522$	0	0
$523$	−4.09694	−0.179147	−0.0895733	−	0.995980i	$-0.528550\pi$
$523$	−4.09694	−0.179147	−0.0895733	+	0.995980i	$0.528550\pi$
$524$	10.8502	0.473992
$525$	0	0
$526$	−22.4117	−0.977195
$527$	21.3338	0.929314
$528$	0	0
$529$	15.1229	0.657518
$530$	0	0
$531$	0	0
$532$	−3.34829	−0.145167
$533$	30.2774	1.31146
$534$	0	0
$535$	0	0
$536$	−8.07849	−0.348938
$537$	0	0
$538$	−11.4906	−0.495393
$539$	21.1382	0.910485
$540$	0	0
$541$	13.9582	0.600109	0.300055	−	0.953922i	$-0.402995\pi$
$541$	13.9582	0.600109	0.300055	+	0.953922i	$0.402995\pi$
$542$	−13.8132	−0.593326
$543$	0	0
$544$	−8.35978	−0.358423
$545$	0	0
$546$	0	0
$547$	26.5045	1.13325	0.566625	−	0.823976i	$-0.308249\pi$
$547$	26.5045	1.13325	0.566625	+	0.823976i	$0.308249\pi$
$548$	−9.10015	−0.388739
$549$	0	0
$550$	0	0
$551$	−9.17556	−0.390892
$552$	0	0
$553$	−9.50808	−0.404325
$554$	35.4119	1.50451
$555$	0	0
$556$	−10.9086	−0.462627
$557$	10.6860	0.452781	0.226391	−	0.974037i	$-0.427307\pi$
$557$	10.6860	0.452781	0.226391	+	0.974037i	$0.427307\pi$
$558$	0	0
$559$	−37.4813	−1.58529
$560$	0	0
$561$	0	0
$562$	−23.9191	−1.00897
$563$	−25.5750	−1.07786	−0.538928	−	0.842352i	$-0.681171\pi$
$563$	−25.5750	−1.07786	−0.538928	+	0.842352i	$0.681171\pi$
$564$	0	0
$565$	0	0
$566$	−0.970149	−0.0407784
$567$	0	0
$568$	−2.03244	−0.0852794
$569$	−6.10210	−0.255813	−0.127907	−	0.991786i	$-0.540826\pi$
$569$	−6.10210	−0.255813	−0.127907	+	0.991786i	$0.540826\pi$
$570$	0	0
$571$	23.7396	0.993473	0.496737	−	0.867901i	$-0.334532\pi$
$571$	23.7396	0.993473	0.496737	+	0.867901i	$0.334532\pi$
$572$	11.0102	0.460361
$573$	0	0
$574$	9.34003	0.389845
$575$	0	0
$576$	0	0
$577$	44.1639	1.83857	0.919284	−	0.393595i	$-0.128769\pi$
$577$	44.1639	1.83857	0.919284	+	0.393595i	$0.128769\pi$
$578$	14.0243	0.583336
$579$	0	0
$580$	0	0
$581$	−13.6587	−0.566659
$582$	0	0
$583$	12.8805	0.533458
$584$	−22.9915	−0.951393
$585$	0	0
$586$	−3.56566	−0.147296
$587$	0.511966	0.0211311	0.0105656	−	0.999944i	$-0.496637\pi$
$587$	0.511966	0.0211311	0.0105656	+	0.999944i	$0.496637\pi$
$588$	0	0
$589$	−40.8177	−1.68187
$590$	0	0
$591$	0	0
$592$	14.5317	0.597250
$593$	−18.8405	−0.773687	−0.386844	−	0.922145i	$-0.626435\pi$
$593$	−18.8405	−0.773687	−0.386844	+	0.922145i	$0.626435\pi$
$594$	0	0
$595$	0	0
$596$	11.5866	0.474606
$597$	0	0
$598$	−28.0795	−1.14825
$599$	−6.20712	−0.253616	−0.126808	−	0.991927i	$-0.540473\pi$
$599$	−6.20712	−0.253616	−0.126808	+	0.991927i	$0.540473\pi$
$600$	0	0
$601$	34.0303	1.38813	0.694063	−	0.719915i	$-0.255819\pi$
$601$	34.0303	1.38813	0.694063	+	0.719915i	$0.255819\pi$
$602$	−11.5623	−0.471244
$603$	0	0
$604$	−2.93051	−0.119241
$605$	0	0
$606$	0	0
$607$	−16.2488	−0.659518	−0.329759	−	0.944065i	$-0.606968\pi$
$607$	−16.2488	−0.659518	−0.329759	+	0.944065i	$0.606968\pi$
$608$	15.9947	0.648670
$609$	0	0
$610$	0	0
$611$	−12.6734	−0.512709
$612$	0	0
$613$	0.0679852	0.00274590	0.00137295	−	0.999999i	$-0.499563\pi$
$613$	0.0679852	0.00274590	0.00137295	+	0.999999i	$0.499563\pi$
$614$	−0.570130	−0.0230086
$615$	0	0
$616$	12.5715	0.506521
$617$	32.8108	1.32091	0.660456	−	0.750864i	$-0.270363\pi$
$617$	32.8108	1.32091	0.660456	+	0.750864i	$0.270363\pi$
$618$	0	0
$619$	−6.77712	−0.272395	−0.136198	−	0.990682i	$-0.543488\pi$
$619$	−6.77712	−0.272395	−0.136198	+	0.990682i	$0.543488\pi$
$620$	0	0
$621$	0	0
$622$	−16.4353	−0.658994
$623$	−17.0269	−0.682168
$624$	0	0
$625$	0	0
$626$	−16.7858	−0.670895
$627$	0	0
$628$	−3.33211	−0.132966
$629$	−15.6465	−0.623867
$630$	0	0
$631$	−7.66797	−0.305257	−0.152629	−	0.988284i	$-0.548774\pi$
$631$	−7.66797	−0.305257	−0.152629	+	0.988284i	$0.548774\pi$
$632$	26.2568	1.04444
$633$	0	0
$634$	−20.3356	−0.807630
$635$	0	0
$636$	0	0
$637$	−23.3381	−0.924690
$638$	9.30754	0.368489
$639$	0	0
$640$	0	0
$641$	23.6911	0.935744	0.467872	−	0.883796i	$-0.345021\pi$
$641$	23.6911	0.935744	0.467872	+	0.883796i	$0.345021\pi$
$642$	0	0
$643$	−2.16861	−0.0855218	−0.0427609	−	0.999085i	$-0.513615\pi$
$643$	−2.16861	−0.0855218	−0.0427609	+	0.999085i	$0.513615\pi$
$644$	5.09122	0.200622
$645$	0	0
$646$	9.67231	0.380552
$647$	2.55541	0.100463	0.0502317	−	0.998738i	$-0.484004\pi$
$647$	2.55541	0.100463	0.0502317	+	0.998738i	$0.484004\pi$
$648$	0	0
$649$	−24.1998	−0.949926
$650$	0	0
$651$	0	0
$652$	−1.81678	−0.0711505
$653$	32.4137	1.26845	0.634224	−	0.773150i	$-0.281320\pi$
$653$	32.4137	1.26845	0.634224	+	0.773150i	$0.281320\pi$
$654$	0	0
$655$	0	0
$656$	−14.7285	−0.575051
$657$	0	0
$658$	−3.90950	−0.152408
$659$	−0.449951	−0.0175276	−0.00876381	−	0.999962i	$-0.502790\pi$
$659$	−0.449951	−0.0175276	−0.00876381	+	0.999962i	$0.502790\pi$
$660$	0	0
$661$	−26.9827	−1.04951	−0.524753	−	0.851254i	$-0.675842\pi$
$661$	−26.9827	−1.04951	−0.524753	+	0.851254i	$0.675842\pi$
$662$	−29.5751	−1.14947
$663$	0	0
$664$	37.7189	1.46378
$665$	0	0
$666$	0	0
$667$	13.9518	0.540217
$668$	2.26795	0.0877496
$669$	0	0
$670$	0	0
$671$	33.4127	1.28988
$672$	0	0
$673$	−17.6224	−0.679292	−0.339646	−	0.940553i	$-0.610307\pi$
$673$	−17.6224	−0.679292	−0.339646	+	0.940553i	$0.610307\pi$
$674$	−12.0032	−0.462344
$675$	0	0
$676$	−2.53135	−0.0973595
$677$	−19.3484	−0.743621	−0.371810	−	0.928309i	$-0.621263\pi$
$677$	−19.3484	−0.743621	−0.371810	+	0.928309i	$0.621263\pi$
$678$	0	0
$679$	15.0354	0.577005
$680$	0	0
$681$	0	0
$682$	41.4049	1.58547
$683$	−29.9460	−1.14585	−0.572926	−	0.819607i	$-0.694192\pi$
$683$	−29.9460	−1.14585	−0.572926	+	0.819607i	$0.694192\pi$
$684$	0	0
$685$	0	0
$686$	−15.9493	−0.608946
$687$	0	0
$688$	18.2328	0.695120
$689$	−14.2211	−0.541781
$690$	0	0
$691$	−19.9349	−0.758361	−0.379181	−	0.925323i	$-0.623794\pi$
$691$	−19.9349	−0.758361	−0.379181	+	0.925323i	$0.623794\pi$
$692$	−8.33627	−0.316897
$693$	0	0
$694$	−18.1526	−0.689063
$695$	0	0
$696$	0	0
$697$	15.8584	0.600678
$698$	−17.1025	−0.647337
$699$	0	0
$700$	0	0
$701$	49.0150	1.85127	0.925636	−	0.378415i	$-0.123531\pi$
$701$	49.0150	1.85127	0.925636	+	0.378415i	$0.123531\pi$
$702$	0	0
$703$	29.9363	1.12907
$704$	−30.6931	−1.15679
$705$	0	0
$706$	33.1519	1.24769
$707$	8.04831	0.302688
$708$	0	0
$709$	−4.81347	−0.180774	−0.0903868	−	0.995907i	$-0.528810\pi$
$709$	−4.81347	−0.180774	−0.0903868	+	0.995907i	$0.528810\pi$
$710$	0	0
$711$	0	0
$712$	47.0202	1.76216
$713$	62.0651	2.32436
$714$	0	0
$715$	0	0
$716$	7.44614	0.278275
$717$	0	0
$718$	−14.7775	−0.551491
$719$	−32.2512	−1.20277	−0.601383	−	0.798961i	$-0.705383\pi$
$719$	−32.2512	−1.20277	−0.601383	+	0.798961i	$0.705383\pi$
$720$	0	0
$721$	5.87771	0.218897
$722$	2.81840	0.104890
$723$	0	0
$724$	14.4138	0.535685
$725$	0	0
$726$	0	0
$727$	29.1747	1.08203	0.541015	−	0.841013i	$-0.318040\pi$
$727$	29.1747	1.08203	0.541015	+	0.841013i	$0.318040\pi$
$728$	−13.8799	−0.514423
$729$	0	0
$730$	0	0
$731$	−19.6315	−0.726099
$732$	0	0
$733$	−5.20102	−0.192104	−0.0960521	−	0.995376i	$-0.530622\pi$
$733$	−5.20102	−0.192104	−0.0960521	+	0.995376i	$0.530622\pi$
$734$	−4.36469	−0.161104
$735$	0	0
$736$	−24.3206	−0.896469
$737$	9.63994	0.355092
$738$	0	0
$739$	−16.3206	−0.600363	−0.300182	−	0.953882i	$-0.597047\pi$
$739$	−16.3206	−0.600363	−0.300182	+	0.953882i	$0.597047\pi$
$740$	0	0
$741$	0	0
$742$	−4.38695	−0.161050
$743$	53.4487	1.96084	0.980421	−	0.196914i	$-0.0630919\pi$
$743$	53.4487	1.96084	0.980421	+	0.196914i	$0.0630919\pi$
$744$	0	0
$745$	0	0
$746$	−9.13946	−0.334619
$747$	0	0
$748$	5.76681	0.210856
$749$	−6.08859	−0.222472
$750$	0	0
$751$	−0.566970	−0.0206890	−0.0103445	−	0.999946i	$-0.503293\pi$
$751$	−0.566970	−0.0206890	−0.0103445	+	0.999946i	$0.503293\pi$
$752$	6.16497	0.224813
$753$	0	0
$754$	−10.2762	−0.374238
$755$	0	0
$756$	0	0
$757$	−53.0708	−1.92889	−0.964445	−	0.264282i	$-0.914865\pi$
$757$	−53.0708	−1.92889	−0.964445	+	0.264282i	$0.914865\pi$
$758$	−11.1192	−0.403867
$759$	0	0
$760$	0	0
$761$	27.5056	0.997077	0.498539	−	0.866867i	$-0.333870\pi$
$761$	27.5056	0.997077	0.498539	+	0.866867i	$0.333870\pi$
$762$	0	0
$763$	−12.6292	−0.457208
$764$	−8.59495	−0.310954
$765$	0	0
$766$	5.86100	0.211766
$767$	26.7184	0.964747
$768$	0	0
$769$	14.9716	0.539889	0.269944	−	0.962876i	$-0.412995\pi$
$769$	14.9716	0.539889	0.269944	+	0.962876i	$0.412995\pi$
$770$	0	0
$771$	0	0
$772$	−2.50360	−0.0901065
$773$	−2.05427	−0.0738871	−0.0369436	−	0.999317i	$-0.511762\pi$
$773$	−2.05427	−0.0738871	−0.0369436	+	0.999317i	$0.511762\pi$
$774$	0	0
$775$	0	0
$776$	−41.5206	−1.49050
$777$	0	0
$778$	−4.18601	−0.150076
$779$	−30.3417	−1.08710
$780$	0	0
$781$	2.42528	0.0867835
$782$	−14.7072	−0.525927
$783$	0	0
$784$	11.3529	0.405459
$785$	0	0
$786$	0	0
$787$	16.8269	0.599815	0.299908	−	0.953968i	$-0.403044\pi$
$787$	16.8269	0.599815	0.299908	+	0.953968i	$0.403044\pi$
$788$	13.0217	0.463877
$789$	0	0
$790$	0	0
$791$	−5.38513	−0.191473
$792$	0	0
$793$	−36.8901	−1.31001
$794$	4.50686	0.159942
$795$	0	0
$796$	−14.8149	−0.525101
$797$	−0.208891	−0.00739930	−0.00369965	−	0.999993i	$-0.501178\pi$
$797$	−0.208891	−0.00739930	−0.00369965	+	0.999993i	$0.501178\pi$
$798$	0	0
$799$	−6.63791	−0.234832
$800$	0	0
$801$	0	0
$802$	−28.0460	−0.990337
$803$	27.4354	0.968173
$804$	0	0
$805$	0	0
$806$	−45.7141	−1.61021
$807$	0	0
$808$	−22.2256	−0.781894
$809$	16.9655	0.596476	0.298238	−	0.954491i	$-0.403601\pi$
$809$	16.9655	0.596476	0.298238	+	0.954491i	$0.403601\pi$
$810$	0	0
$811$	−20.4684	−0.718742	−0.359371	−	0.933195i	$-0.617009\pi$
$811$	−20.4684	−0.718742	−0.359371	+	0.933195i	$0.617009\pi$
$812$	1.86323	0.0653866
$813$	0	0
$814$	−30.3669	−1.06436
$815$	0	0
$816$	0	0
$817$	37.5609	1.31409
$818$	28.4812	0.995822
$819$	0	0
$820$	0	0
$821$	−38.3636	−1.33890	−0.669449	−	0.742858i	$-0.733470\pi$
$821$	−38.3636	−1.33890	−0.669449	+	0.742858i	$0.733470\pi$
$822$	0	0
$823$	29.0098	1.01122	0.505609	−	0.862763i	$-0.331268\pi$
$823$	29.0098	1.01122	0.505609	+	0.862763i	$0.331268\pi$
$824$	−16.2315	−0.565449
$825$	0	0
$826$	8.24215	0.286781
$827$	14.3567	0.499233	0.249616	−	0.968345i	$-0.419695\pi$
$827$	14.3567	0.499233	0.249616	+	0.968345i	$0.419695\pi$
$828$	0	0
$829$	4.63563	0.161002	0.0805011	−	0.996755i	$-0.474348\pi$
$829$	4.63563	0.161002	0.0805011	+	0.996755i	$0.474348\pi$
$830$	0	0
$831$	0	0
$832$	33.8875	1.17484
$833$	−12.2238	−0.423529
$834$	0	0
$835$	0	0
$836$	−11.0336	−0.381605
$837$	0	0
$838$	−34.5924	−1.19498
$839$	−4.92642	−0.170079	−0.0850395	−	0.996378i	$-0.527102\pi$
$839$	−4.92642	−0.170079	−0.0850395	+	0.996378i	$0.527102\pi$
$840$	0	0
$841$	−23.8941	−0.823933
$842$	−9.59201	−0.330562
$843$	0	0
$844$	−1.80843	−0.0622486
$845$	0	0
$846$	0	0
$847$	−2.75032	−0.0945020
$848$	6.91787	0.237561
$849$	0	0
$850$	0	0
$851$	−45.5194	−1.56039
$852$	0	0
$853$	−53.6476	−1.83686	−0.918430	−	0.395584i	$-0.870542\pi$
$853$	−53.6476	−1.83686	−0.918430	+	0.395584i	$0.870542\pi$
$854$	−11.3799	−0.389413
$855$	0	0
$856$	16.8138	0.574684
$857$	−27.1144	−0.926210	−0.463105	−	0.886303i	$-0.653265\pi$
$857$	−27.1144	−0.926210	−0.463105	+	0.886303i	$0.653265\pi$
$858$	0	0
$859$	10.4190	0.355493	0.177747	−	0.984076i	$-0.443119\pi$
$859$	10.4190	0.355493	0.177747	+	0.984076i	$0.443119\pi$
$860$	0	0
$861$	0	0
$862$	−29.6088	−1.00848
$863$	−43.6432	−1.48563	−0.742816	−	0.669495i	$-0.766510\pi$
$863$	−43.6432	−1.48563	−0.742816	+	0.669495i	$0.766510\pi$
$864$	0	0
$865$	0	0
$866$	−10.5193	−0.357461
$867$	0	0
$868$	8.28864	0.281335
$869$	−31.3319	−1.06286
$870$	0	0
$871$	−10.6432	−0.360632
$872$	34.8758	1.18105
$873$	0	0
$874$	28.1391	0.951818
$875$	0	0
$876$	0	0
$877$	−53.6435	−1.81141	−0.905706	−	0.423907i	$-0.860658\pi$
$877$	−53.6435	−1.81141	−0.905706	+	0.423907i	$0.860658\pi$
$878$	0.579102	0.0195438
$879$	0	0
$880$	0	0
$881$	54.8950	1.84946	0.924730	−	0.380623i	$-0.124290\pi$
$881$	54.8950	1.84946	0.924730	+	0.380623i	$0.124290\pi$
$882$	0	0
$883$	19.0952	0.642603	0.321302	−	0.946977i	$-0.395880\pi$
$883$	19.0952	0.642603	0.321302	+	0.946977i	$0.395880\pi$
$884$	−6.36700	−0.214145
$885$	0	0
$886$	20.0166	0.672471
$887$	−11.2162	−0.376602	−0.188301	−	0.982111i	$-0.560298\pi$
$887$	−11.2162	−0.376602	−0.188301	+	0.982111i	$0.560298\pi$
$888$	0	0
$889$	0.845988	0.0283735
$890$	0	0
$891$	0	0
$892$	−18.9807	−0.635521
$893$	12.7003	0.424998
$894$	0	0
$895$	0	0
$896$	1.67977	0.0561173
$897$	0	0
$898$	−4.67489	−0.156003
$899$	22.7139	0.757552
$900$	0	0
$901$	−7.44857	−0.248148
$902$	30.7781	1.02480
$903$	0	0
$904$	14.8712	0.494608
$905$	0	0
$906$	0	0
$907$	36.4513	1.21034	0.605172	−	0.796095i	$-0.293104\pi$
$907$	36.4513	1.21034	0.605172	+	0.796095i	$0.293104\pi$
$908$	−4.85930	−0.161261
$909$	0	0
$910$	0	0
$911$	−1.16533	−0.0386091	−0.0193045	−	0.999814i	$-0.506145\pi$
$911$	−1.16533	−0.0386091	−0.0193045	+	0.999814i	$0.506145\pi$
$912$	0	0
$913$	−45.0094	−1.48959
$914$	−19.8354	−0.656097
$915$	0	0
$916$	5.89621	0.194816
$917$	16.3219	0.538997
$918$	0	0
$919$	43.9475	1.44969	0.724847	−	0.688910i	$-0.241911\pi$
$919$	43.9475	1.44969	0.724847	+	0.688910i	$0.241911\pi$
$920$	0	0
$921$	0	0
$922$	−0.230029	−0.00757561
$923$	−2.67770	−0.0881375
$924$	0	0
$925$	0	0
$926$	−14.3856	−0.472739
$927$	0	0
$928$	−8.90060	−0.292176
$929$	−31.2121	−1.02404	−0.512019	−	0.858974i	$-0.671102\pi$
$929$	−31.2121	−1.02404	−0.512019	+	0.858974i	$0.671102\pi$
$930$	0	0
$931$	23.3876	0.766499
$932$	6.97506	0.228476
$933$	0	0
$934$	−1.24998	−0.0409006
$935$	0	0
$936$	0	0
$937$	45.1060	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$937$	45.1060	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$938$	−3.28324	−0.107202
$939$	0	0
$940$	0	0
$941$	39.1341	1.27573	0.637867	−	0.770146i	$-0.279817\pi$
$941$	39.1341	1.27573	0.637867	+	0.770146i	$0.279817\pi$
$942$	0	0
$943$	46.1358	1.50239
$944$	−12.9972	−0.423023
$945$	0	0
$946$	−38.1011	−1.23877
$947$	−48.8537	−1.58753	−0.793766	−	0.608223i	$-0.791883\pi$
$947$	−48.8537	−1.58753	−0.793766	+	0.608223i	$0.791883\pi$
$948$	0	0
$949$	−30.2907	−0.983278
$950$	0	0
$951$	0	0
$952$	−7.26986	−0.235618
$953$	22.7824	0.737995	0.368998	−	0.929430i	$-0.379701\pi$
$953$	22.7824	0.737995	0.368998	+	0.929430i	$0.379701\pi$
$954$	0	0
$955$	0	0
$956$	−8.30924	−0.268740
$957$	0	0
$958$	−28.2547	−0.912868
$959$	−13.6893	−0.442052
$960$	0	0
$961$	70.0435	2.25947
$962$	33.5274	1.08097
$963$	0	0
$964$	−16.7922	−0.540841
$965$	0	0
$966$	0	0
$967$	−43.1927	−1.38898	−0.694492	−	0.719500i	$-0.744371\pi$
$967$	−43.1927	−1.38898	−0.694492	+	0.719500i	$0.744371\pi$
$968$	7.59507	0.244115
$969$	0	0
$970$	0	0
$971$	32.0252	1.02774	0.513869	−	0.857869i	$-0.328212\pi$
$971$	32.0252	1.02774	0.513869	+	0.857869i	$0.328212\pi$
$972$	0	0
$973$	−16.4098	−0.526073
$974$	11.6271	0.372557
$975$	0	0
$976$	17.9452	0.574413
$977$	−7.27011	−0.232591	−0.116296	−	0.993215i	$-0.537102\pi$
$977$	−7.27011	−0.232591	−0.116296	+	0.993215i	$0.537102\pi$
$978$	0	0
$979$	−56.1085	−1.79324
$980$	0	0
$981$	0	0
$982$	−17.5725	−0.560760
$983$	−5.09155	−0.162395	−0.0811976	−	0.996698i	$-0.525874\pi$
$983$	−5.09155	−0.162395	−0.0811976	+	0.996698i	$0.525874\pi$
$984$	0	0
$985$	0	0
$986$	−5.38237	−0.171410
$987$	0	0
$988$	12.1819	0.387559
$989$	−57.1129	−1.81608
$990$	0	0
$991$	−62.0762	−1.97192	−0.985958	−	0.166992i	$-0.946594\pi$
$991$	−62.0762	−1.97192	−0.985958	+	0.166992i	$0.946594\pi$
$992$	−39.5946	−1.25713
$993$	0	0
$994$	−0.826021	−0.0261998
$995$	0	0
$996$	0	0
$997$	−2.56895	−0.0813594	−0.0406797	−	0.999172i	$-0.512952\pi$
$997$	−2.56895	−0.0813594	−0.0406797	+	0.999172i	$0.512952\pi$
$998$	−40.1643	−1.27138
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.n.1.2		4
3.2	odd	2		1875.2.a.e.1.3		4
5.4	even	2		5625.2.a.i.1.3		4
15.2	even	4		1875.2.b.c.1249.5		8
15.8	even	4		1875.2.b.c.1249.4		8
15.14	odd	2		1875.2.a.h.1.2		4
25.9	even	10		225.2.h.c.181.2		8
25.14	even	10		225.2.h.c.46.2		8
75.2	even	20		375.2.i.b.274.3		16
75.11	odd	10		375.2.g.b.226.2		8
75.14	odd	10		75.2.g.b.46.1	yes	8
75.23	even	20		375.2.i.b.274.2		16
75.38	even	20		375.2.i.b.349.3		16
75.41	odd	10		375.2.g.b.151.2		8
75.59	odd	10		75.2.g.b.31.1	✓	8
75.62	even	20		375.2.i.b.349.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.31.1	✓	8	75.59	odd	10
75.2.g.b.46.1	yes	8	75.14	odd	10
225.2.h.c.46.2		8	25.14	even	10
225.2.h.c.181.2		8	25.9	even	10
375.2.g.b.151.2		8	75.41	odd	10
375.2.g.b.226.2		8	75.11	odd	10
375.2.i.b.274.2		16	75.23	even	20
375.2.i.b.274.3		16	75.2	even	20
375.2.i.b.349.2		16	75.62	even	20
375.2.i.b.349.3		16	75.38	even	20
1875.2.a.e.1.3		4	3.2	odd	2
1875.2.a.h.1.2		4	15.14	odd	2
1875.2.b.c.1249.4		8	15.8	even	4
1875.2.b.c.1249.5		8	15.2	even	4
5625.2.a.i.1.3		4	5.4	even	2
5625.2.a.n.1.2		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)

\(f(q)\)	\(=\)	\(q-1.12233 q^{2} -0.740367 q^{4} -1.11373 q^{7} +3.07561 q^{8} +O(q^{10})\)	\(q-1.12233 q^{2} -0.740367 q^{4} -1.11373 q^{7} +3.07561 q^{8} -3.67008 q^{11} +4.05204 q^{13} +1.24998 q^{14} -1.97112 q^{16} +2.12233 q^{17} -4.06064 q^{19} +4.11905 q^{22} +6.17438 q^{23} -4.54774 q^{26} +0.824573 q^{28} +2.25963 q^{29} +10.0520 q^{31} -3.93896 q^{32} -2.38197 q^{34} -7.37232 q^{37} +4.55739 q^{38} +7.47214 q^{41} -9.24998 q^{43} +2.71720 q^{44} -6.92971 q^{46} -3.12765 q^{47} -5.75960 q^{49} -3.00000 q^{52} -3.50961 q^{53} -3.42541 q^{56} -2.53606 q^{58} +6.59382 q^{59} -9.10408 q^{61} -11.2817 q^{62} +8.36307 q^{64} -2.62663 q^{67} -1.57131 q^{68} -0.660827 q^{71} -7.47542 q^{73} +8.27420 q^{74} +3.00637 q^{76} +4.08749 q^{77} +8.53711 q^{79} -8.38623 q^{82} +12.2639 q^{83} +10.3816 q^{86} -11.2877 q^{88} +15.2881 q^{89} -4.51290 q^{91} -4.57131 q^{92} +3.51026 q^{94} -13.5000 q^{97} +6.46419 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8}+O(q^{10}) \) 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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