LMFDB - Embedded newform 2025.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.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:	$119.478867762$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.0765073$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.57358 q^{2} +12.9176 q^{4} +20.1145 q^{7} -22.4912 q^{8} +O(q^{10})$	$q-4.57358 q^{2} +12.9176 q^{4} +20.1145 q^{7} -22.4912 q^{8} -66.3416 q^{11} +46.8006 q^{13} -91.9955 q^{14} -0.475708 q^{16} -47.6233 q^{17} -9.95276 q^{19} +303.418 q^{22} +9.59204 q^{23} -214.046 q^{26} +259.832 q^{28} +178.735 q^{29} +154.037 q^{31} +182.105 q^{32} +217.809 q^{34} -248.864 q^{37} +45.5197 q^{38} -249.664 q^{41} +212.245 q^{43} -856.976 q^{44} -43.8700 q^{46} +475.694 q^{47} +61.5946 q^{49} +604.553 q^{52} -546.314 q^{53} -452.400 q^{56} -817.459 q^{58} +419.296 q^{59} -545.210 q^{61} -704.502 q^{62} -829.068 q^{64} +447.877 q^{67} -615.181 q^{68} -409.542 q^{71} +358.548 q^{73} +1138.20 q^{74} -128.566 q^{76} -1334.43 q^{77} -651.552 q^{79} +1141.86 q^{82} -813.142 q^{83} -970.719 q^{86} +1492.10 q^{88} +201.000 q^{89} +941.373 q^{91} +123.906 q^{92} -2175.62 q^{94} -252.149 q^{97} -281.708 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8}+O(q^{10}) $ 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8	$ 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} - 14 q^{11} - 40 q^{13} + 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 376 q^{22} + 171 q^{23} - 434 q^{26} + 517 q^{28} + 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 402 q^{37} - 178 q^{38} - 187 q^{41} + 602 q^{43} - 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} + 456 q^{52} - 730 q^{53} - 705 q^{56} - 217 q^{58} + 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 1849 q^{67} - 710 q^{68} - 70 q^{71} + 368 q^{73} + 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} + 575 q^{82} - 831 q^{83} - 1580 q^{86} + 1428 q^{88} - 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} + 282 q^{97} + 2164 q^{98}+O(q^{100}) $ 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8 - 14 * q^11 - 40 * q^13 + 27 * q^14 - 13 * q^16 - 166 * q^17 - 164 * q^19 + 376 * q^22 + 171 * q^23 - 434 * q^26 + 517 * q^28 + 335 * q^29 - 352 * q^31 - 77 * q^32 - 52 * q^34 - 402 * q^37 - 178 * q^38 - 187 * q^41 + 602 * q^43 - 982 * q^44 - 201 * q^46 + 665 * q^47 + 430 * q^49 + 456 * q^52 - 730 * q^53 - 705 * q^56 - 217 * q^58 + 298 * q^59 - 1439 * q^61 - 1614 * q^62 - 1569 * q^64 + 1849 * q^67 - 710 * q^68 - 70 * q^71 + 368 * q^73 + 320 * q^74 + 204 * q^76 - 948 * q^77 - 382 * q^79 + 575 * q^82 - 831 * q^83 - 1580 * q^86 + 1428 * q^88 - 1719 * q^89 - 710 * q^91 - 1623 * q^92 - 2077 * q^94 + 282 * q^97 + 2164 * 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 $

$2$	−4.57358	−1.61700	−0.808502	−	0.588493i	$-0.799722\pi$
$2$	−4.57358	−1.61700	−0.808502	+	0.588493i	$0.799722\pi$
$3$	0	0
$3$	0	0
$4$	12.9176	1.61470
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.1145	1.08608	0.543041	−	0.839706i	$-0.317273\pi$
$7$	20.1145	1.08608	0.543041	+	0.839706i	$0.317273\pi$
$8$	−22.4912	−0.993981
$9$	0	0
$10$	0	0
$11$	−66.3416	−1.81843	−0.909215	−	0.416327i	$-0.863317\pi$
$11$	−66.3416	−1.81843	−0.909215	+	0.416327i	$0.863317\pi$
$12$	0	0
$13$	46.8006	0.998473	0.499237	−	0.866466i	$-0.333614\pi$
$13$	46.8006	0.998473	0.499237	+	0.866466i	$0.333614\pi$
$14$	−91.9955	−1.75620
$15$	0	0
$16$	−0.475708	−0.00743294
$17$	−47.6233	−0.679432	−0.339716	−	0.940528i	$-0.610331\pi$
$17$	−47.6233	−0.679432	−0.339716	+	0.940528i	$0.610331\pi$
$18$	0	0
$19$	−9.95276	−0.120175	−0.0600874	−	0.998193i	$-0.519138\pi$
$19$	−9.95276	−0.120175	−0.0600874	+	0.998193i	$0.519138\pi$
$20$	0	0
$21$	0	0
$22$	303.418	2.94041
$23$	9.59204	0.0869599	0.0434800	−	0.999054i	$-0.486156\pi$
$23$	9.59204	0.0869599	0.0434800	+	0.999054i	$0.486156\pi$
$24$	0	0
$25$	0	0
$26$	−214.046	−1.61454
$27$	0	0
$28$	259.832	1.75370
$29$	178.735	1.14449	0.572246	−	0.820082i	$-0.306072\pi$
$29$	178.735	1.14449	0.572246	+	0.820082i	$0.306072\pi$
$30$	0	0
$31$	154.037	0.892449	0.446224	−	0.894921i	$-0.352768\pi$
$31$	154.037	0.892449	0.446224	+	0.894921i	$0.352768\pi$
$32$	182.105	1.00600
$33$	0	0
$34$	217.809	1.09865
$35$	0	0
$36$	0	0
$37$	−248.864	−1.10576	−0.552878	−	0.833262i	$-0.686471\pi$
$37$	−248.864	−1.10576	−0.552878	+	0.833262i	$0.686471\pi$
$38$	45.5197	0.194323
$39$	0	0
$40$	0	0
$41$	−249.664	−0.951000	−0.475500	−	0.879716i	$-0.657733\pi$
$41$	−249.664	−0.951000	−0.475500	+	0.879716i	$0.657733\pi$
$42$	0	0
$43$	212.245	0.752722	0.376361	−	0.926473i	$-0.377175\pi$
$43$	212.245	0.752722	0.376361	+	0.926473i	$0.377175\pi$
$44$	−856.976	−2.93623
$45$	0	0
$46$	−43.8700	−0.140615
$47$	475.694	1.47632	0.738160	−	0.674626i	$-0.235695\pi$
$47$	475.694	1.47632	0.738160	+	0.674626i	$0.235695\pi$
$48$	0	0
$49$	61.5946	0.179576
$50$	0	0
$51$	0	0
$52$	604.553	1.61224
$53$	−546.314	−1.41589	−0.707944	−	0.706269i	$-0.750377\pi$
$53$	−546.314	−1.41589	−0.707944	+	0.706269i	$0.750377\pi$
$54$	0	0
$55$	0	0
$56$	−452.400	−1.07955
$57$	0	0
$58$	−817.459	−1.85065
$59$	419.296	0.925216	0.462608	−	0.886563i	$-0.346914\pi$
$59$	419.296	0.925216	0.462608	+	0.886563i	$0.346914\pi$
$60$	0	0
$61$	−545.210	−1.14438	−0.572188	−	0.820122i	$-0.693905\pi$
$61$	−545.210	−1.14438	−0.572188	+	0.820122i	$0.693905\pi$
$62$	−704.502	−1.44309
$63$	0	0
$64$	−829.068	−1.61927
$65$	0	0
$66$	0	0
$67$	447.877	0.816669	0.408335	−	0.912832i	$-0.366110\pi$
$67$	447.877	0.816669	0.408335	+	0.912832i	$0.366110\pi$
$68$	−615.181	−1.09708
$69$	0	0
$70$	0	0
$71$	−409.542	−0.684559	−0.342279	−	0.939598i	$-0.611199\pi$
$71$	−409.542	−0.684559	−0.342279	+	0.939598i	$0.611199\pi$
$72$	0	0
$73$	358.548	0.574861	0.287431	−	0.957801i	$-0.407199\pi$
$73$	358.548	0.574861	0.287431	+	0.957801i	$0.407199\pi$
$74$	1138.20	1.78801
$75$	0	0
$76$	−128.566	−0.194047
$77$	−1334.43	−1.97497
$78$	0	0
$79$	−651.552	−0.927915	−0.463958	−	0.885857i	$-0.653571\pi$
$79$	−651.552	−0.927915	−0.463958	+	0.885857i	$0.653571\pi$
$80$	0	0
$81$	0	0
$82$	1141.86	1.53777
$83$	−813.142	−1.07535	−0.537675	−	0.843152i	$-0.680697\pi$
$83$	−813.142	−1.07535	−0.537675	+	0.843152i	$0.680697\pi$
$84$	0	0
$85$	0	0
$86$	−970.719	−1.21715
$87$	0	0
$88$	1492.10	1.80749
$89$	201.000	0.239393	0.119696	−	0.992811i	$-0.461808\pi$
$89$	201.000	0.239393	0.119696	+	0.992811i	$0.461808\pi$
$90$	0	0
$91$	941.373	1.08442
$92$	123.906	0.140415
$93$	0	0
$94$	−2175.62	−2.38722
$95$	0	0
$96$	0	0
$97$	−252.149	−0.263936	−0.131968	−	0.991254i	$-0.542130\pi$
$97$	−252.149	−0.263936	−0.131968	+	0.991254i	$0.542130\pi$
$98$	−281.708	−0.290375
$99$	0	0
$100$	0	0
$101$	−43.6026	−0.0429567	−0.0214783	−	0.999769i	$-0.506837\pi$
$101$	−43.6026	−0.0429567	−0.0214783	+	0.999769i	$0.506837\pi$
$102$	0	0
$103$	1440.35	1.37788	0.688942	−	0.724816i	$-0.258075\pi$
$103$	1440.35	1.37788	0.688942	+	0.724816i	$0.258075\pi$
$104$	−1052.60	−0.992463
$105$	0	0
$106$	2498.61	2.28950
$107$	355.755	0.321422	0.160711	−	0.987002i	$-0.448621\pi$
$107$	355.755	0.321422	0.160711	+	0.987002i	$0.448621\pi$
$108$	0	0
$109$	−1522.51	−1.33789	−0.668946	−	0.743311i	$-0.733254\pi$
$109$	−1522.51	−1.33789	−0.668946	+	0.743311i	$0.733254\pi$
$110$	0	0
$111$	0	0
$112$	−9.56865	−0.00807279
$113$	−812.998	−0.676818	−0.338409	−	0.940999i	$-0.609889\pi$
$113$	−812.998	−0.676818	−0.338409	+	0.940999i	$0.609889\pi$
$114$	0	0
$115$	0	0
$116$	2308.83	1.84802
$117$	0	0
$118$	−1917.69	−1.49608
$119$	−957.921	−0.737920
$120$	0	0
$121$	3070.20	2.30669
$122$	2493.56	1.85046
$123$	0	0
$124$	1989.80	1.44104
$125$	0	0
$126$	0	0
$127$	864.662	0.604144	0.302072	−	0.953285i	$-0.402322\pi$
$127$	864.662	0.604144	0.302072	+	0.953285i	$0.402322\pi$
$128$	2334.97	1.61237
$129$	0	0
$130$	0	0
$131$	2178.52	1.45296	0.726482	−	0.687186i	$-0.241154\pi$
$131$	2178.52	1.45296	0.726482	+	0.687186i	$0.241154\pi$
$132$	0	0
$133$	−200.195	−0.130520
$134$	−2048.40	−1.32056
$135$	0	0
$136$	1071.11	0.675343
$137$	−2299.16	−1.43380	−0.716900	−	0.697176i	$-0.754440\pi$
$137$	−2299.16	−1.43380	−0.716900	+	0.697176i	$0.754440\pi$
$138$	0	0
$139$	−2133.95	−1.30215	−0.651077	−	0.759012i	$-0.725682\pi$
$139$	−2133.95	−1.30215	−0.651077	+	0.759012i	$0.725682\pi$
$140$	0	0
$141$	0	0
$142$	1873.07	1.10693
$143$	−3104.83	−1.81565
$144$	0	0
$145$	0	0
$146$	−1639.85	−0.929554
$147$	0	0
$148$	−3214.74	−1.78547
$149$	−1750.62	−0.962525	−0.481263	−	0.876576i	$-0.659822\pi$
$149$	−1750.62	−0.962525	−0.481263	+	0.876576i	$0.659822\pi$
$150$	0	0
$151$	875.954	0.472080	0.236040	−	0.971743i	$-0.424150\pi$
$151$	875.954	0.472080	0.236040	+	0.971743i	$0.424150\pi$
$152$	223.850	0.119451
$153$	0	0
$154$	6103.12	3.19353
$155$	0	0
$156$	0	0
$157$	259.395	0.131860	0.0659298	−	0.997824i	$-0.478999\pi$
$157$	259.395	0.131860	0.0659298	+	0.997824i	$0.478999\pi$
$158$	2979.92	1.50044
$159$	0	0
$160$	0	0
$161$	192.939	0.0944457
$162$	0	0
$163$	1201.80	0.577498	0.288749	−	0.957405i	$-0.406761\pi$
$163$	1201.80	0.577498	0.288749	+	0.957405i	$0.406761\pi$
$164$	−3225.07	−1.53558
$165$	0	0
$166$	3718.97	1.73884
$167$	−1678.90	−0.777949	−0.388975	−	0.921248i	$-0.627171\pi$
$167$	−1678.90	−0.777949	−0.388975	+	0.921248i	$0.627171\pi$
$168$	0	0
$169$	−6.70324	−0.00305109
$170$	0	0
$171$	0	0
$172$	2741.70	1.21542
$173$	931.798	0.409499	0.204749	−	0.978814i	$-0.434362\pi$
$173$	931.798	0.409499	0.204749	+	0.978814i	$0.434362\pi$
$174$	0	0
$175$	0	0
$176$	31.5592	0.0135163
$177$	0	0
$178$	−919.289	−0.387099
$179$	−1023.40	−0.427333	−0.213667	−	0.976907i	$-0.568541\pi$
$179$	−1023.40	−0.427333	−0.213667	+	0.976907i	$0.568541\pi$
$180$	0	0
$181$	2639.93	1.08411	0.542056	−	0.840342i	$-0.317646\pi$
$181$	2639.93	1.08411	0.542056	+	0.840342i	$0.317646\pi$
$182$	−4305.44	−1.75352
$183$	0	0
$184$	−215.737	−0.0864365
$185$	0	0
$186$	0	0
$187$	3159.41	1.23550
$188$	6144.84	2.38382
$189$	0	0
$190$	0	0
$191$	813.281	0.308099	0.154050	−	0.988063i	$-0.450768\pi$
$191$	813.281	0.308099	0.154050	+	0.988063i	$0.450768\pi$
$192$	0	0
$193$	814.242	0.303681	0.151840	−	0.988405i	$-0.451480\pi$
$193$	814.242	0.303681	0.151840	+	0.988405i	$0.451480\pi$
$194$	1153.22	0.426787
$195$	0	0
$196$	795.657	0.289962
$197$	4078.41	1.47500	0.737499	−	0.675348i	$-0.236006\pi$
$197$	4078.41	1.47500	0.737499	+	0.675348i	$0.236006\pi$
$198$	0	0
$199$	−1342.49	−0.478224	−0.239112	−	0.970992i	$-0.576856\pi$
$199$	−1342.49	−0.478224	−0.239112	+	0.970992i	$0.576856\pi$
$200$	0	0
$201$	0	0
$202$	199.420	0.0694611
$203$	3595.17	1.24301
$204$	0	0
$205$	0	0
$206$	−6587.56	−2.22805
$207$	0	0
$208$	−22.2634	−0.00742159
$209$	660.282	0.218529
$210$	0	0
$211$	−2954.97	−0.964118	−0.482059	−	0.876139i	$-0.660111\pi$
$211$	−2954.97	−0.964118	−0.482059	+	0.876139i	$0.660111\pi$
$212$	−7057.09	−2.28624
$213$	0	0
$214$	−1627.07	−0.519740
$215$	0	0
$216$	0	0
$217$	3098.39	0.969273
$218$	6963.33	2.16338
$219$	0	0
$220$	0	0
$221$	−2228.80	−0.678395
$222$	0	0
$223$	−3506.86	−1.05308	−0.526539	−	0.850151i	$-0.676511\pi$
$223$	−3506.86	−1.05308	−0.526539	+	0.850151i	$0.676511\pi$
$224$	3662.97	1.09260
$225$	0	0
$226$	3718.31	1.09442
$227$	652.826	0.190879	0.0954396	−	0.995435i	$-0.469574\pi$
$227$	652.826	0.190879	0.0954396	+	0.995435i	$0.469574\pi$
$228$	0	0
$229$	4583.55	1.32266	0.661331	−	0.750094i	$-0.269992\pi$
$229$	4583.55	1.32266	0.661331	+	0.750094i	$0.269992\pi$
$230$	0	0
$231$	0	0
$232$	−4019.97	−1.13760
$233$	317.527	0.0892785	0.0446392	−	0.999003i	$-0.485786\pi$
$233$	317.527	0.0892785	0.0446392	+	0.999003i	$0.485786\pi$
$234$	0	0
$235$	0	0
$236$	5416.32	1.49395
$237$	0	0
$238$	4381.13	1.19322
$239$	−1857.67	−0.502773	−0.251386	−	0.967887i	$-0.580887\pi$
$239$	−1857.67	−0.502773	−0.251386	+	0.967887i	$0.580887\pi$
$240$	0	0
$241$	−3266.94	−0.873204	−0.436602	−	0.899655i	$-0.643818\pi$
$241$	−3266.94	−0.873204	−0.436602	+	0.899655i	$0.643818\pi$
$242$	−14041.8	−3.72993
$243$	0	0
$244$	−7042.82	−1.84783
$245$	0	0
$246$	0	0
$247$	−465.795	−0.119991
$248$	−3464.49	−0.887077
$249$	0	0
$250$	0	0
$251$	5641.37	1.41865	0.709323	−	0.704884i	$-0.249001\pi$
$251$	5641.37	1.41865	0.709323	+	0.704884i	$0.249001\pi$
$252$	0	0
$253$	−636.351	−0.158131
$254$	−3954.60	−0.976904
$255$	0	0
$256$	−4046.61	−0.987943
$257$	−1173.46	−0.284819	−0.142410	−	0.989808i	$-0.545485\pi$
$257$	−1173.46	−0.284819	−0.142410	+	0.989808i	$0.545485\pi$
$258$	0	0
$259$	−5005.79	−1.20094
$260$	0	0
$261$	0	0
$262$	−9963.64	−2.34945
$263$	−2897.96	−0.679452	−0.339726	−	0.940524i	$-0.610334\pi$
$263$	−2897.96	−0.679452	−0.339726	+	0.940524i	$0.610334\pi$
$264$	0	0
$265$	0	0
$266$	915.609	0.211051
$267$	0	0
$268$	5785.51	1.31868
$269$	−2930.13	−0.664138	−0.332069	−	0.943255i	$-0.607747\pi$
$269$	−2930.13	−0.664138	−0.332069	+	0.943255i	$0.607747\pi$
$270$	0	0
$271$	−668.881	−0.149932	−0.0749661	−	0.997186i	$-0.523885\pi$
$271$	−668.881	−0.149932	−0.0749661	+	0.997186i	$0.523885\pi$
$272$	22.6548	0.00505018
$273$	0	0
$274$	10515.4	2.31846
$275$	0	0
$276$	0	0
$277$	632.630	0.137224	0.0686121	−	0.997643i	$-0.478143\pi$
$277$	632.630	0.137224	0.0686121	+	0.997643i	$0.478143\pi$
$278$	9759.80	2.10559
$279$	0	0
$280$	0	0
$281$	3595.30	0.763265	0.381633	−	0.924314i	$-0.375362\pi$
$281$	3595.30	0.763265	0.381633	+	0.924314i	$0.375362\pi$
$282$	0	0
$283$	−504.368	−0.105942	−0.0529710	−	0.998596i	$-0.516869\pi$
$283$	−504.368	−0.105942	−0.0529710	+	0.998596i	$0.516869\pi$
$284$	−5290.31	−1.10536
$285$	0	0
$286$	14200.2	2.93592
$287$	−5021.88	−1.03286
$288$	0	0
$289$	−2645.02	−0.538372
$290$	0	0
$291$	0	0
$292$	4631.60	0.928232
$293$	6198.50	1.23591	0.617953	−	0.786215i	$-0.287962\pi$
$293$	6198.50	1.23591	0.617953	+	0.786215i	$0.287962\pi$
$294$	0	0
$295$	0	0
$296$	5597.26	1.09910
$297$	0	0
$298$	8006.60	1.55641
$299$	448.913	0.0868271
$300$	0	0
$301$	4269.21	0.817518
$302$	−4006.25	−0.763356
$303$	0	0
$304$	4.73461	0.000893251 0
$305$	0	0
$306$	0	0
$307$	1966.79	0.365636	0.182818	−	0.983147i	$-0.441478\pi$
$307$	1966.79	0.365636	0.182818	+	0.983147i	$0.441478\pi$
$308$	−17237.7	−3.18899
$309$	0	0
$310$	0	0
$311$	2306.11	0.420474	0.210237	−	0.977650i	$-0.432576\pi$
$311$	2306.11	0.420474	0.210237	+	0.977650i	$0.432576\pi$
$312$	0	0
$313$	−10302.4	−1.86047	−0.930234	−	0.366968i	$-0.880396\pi$
$313$	−10302.4	−1.86047	−0.930234	+	0.366968i	$0.880396\pi$
$314$	−1186.36	−0.213218
$315$	0	0
$316$	−8416.51	−1.49831
$317$	−1701.83	−0.301528	−0.150764	−	0.988570i	$-0.548173\pi$
$317$	−1701.83	−0.301528	−0.150764	+	0.988570i	$0.548173\pi$
$318$	0	0
$319$	−11857.6	−2.08118
$320$	0	0
$321$	0	0
$322$	−882.424	−0.152719
$323$	473.983	0.0816506
$324$	0	0
$325$	0	0
$326$	−5496.52	−0.933817
$327$	0	0
$328$	5615.25	0.945276
$329$	9568.36	1.60341
$330$	0	0
$331$	−8351.49	−1.38683	−0.693413	−	0.720540i	$-0.743894\pi$
$331$	−8351.49	−1.38683	−0.693413	+	0.720540i	$0.743894\pi$
$332$	−10503.9	−1.73637
$333$	0	0
$334$	7678.61	1.25795
$335$	0	0
$336$	0	0
$337$	7857.42	1.27009	0.635046	−	0.772474i	$-0.280981\pi$
$337$	7857.42	1.27009	0.635046	+	0.772474i	$0.280981\pi$
$338$	30.6578	0.00493362
$339$	0	0
$340$	0	0
$341$	−10219.1	−1.62286
$342$	0	0
$343$	−5660.34	−0.891048
$344$	−4773.64	−0.748191
$345$	0	0
$346$	−4261.65	−0.662162
$347$	1212.18	0.187530	0.0937652	−	0.995594i	$-0.470110\pi$
$347$	1212.18	0.187530	0.0937652	+	0.995594i	$0.470110\pi$
$348$	0	0
$349$	1398.66	0.214524	0.107262	−	0.994231i	$-0.465792\pi$
$349$	1398.66	0.214524	0.107262	+	0.994231i	$0.465792\pi$
$350$	0	0
$351$	0	0
$352$	−12081.2	−1.82934
$353$	2628.43	0.396309	0.198154	−	0.980171i	$-0.436505\pi$
$353$	2628.43	0.396309	0.198154	+	0.980171i	$0.436505\pi$
$354$	0	0
$355$	0	0
$356$	2596.44	0.386548
$357$	0	0
$358$	4680.61	0.691000
$359$	−3677.48	−0.540640	−0.270320	−	0.962770i	$-0.587130\pi$
$359$	−3677.48	−0.540640	−0.270320	+	0.962770i	$0.587130\pi$
$360$	0	0
$361$	−6759.94	−0.985558
$362$	−12073.9	−1.75301
$363$	0	0
$364$	12160.3	1.75103
$365$	0	0
$366$	0	0
$367$	−11429.8	−1.62569	−0.812846	−	0.582479i	$-0.802083\pi$
$367$	−11429.8	−1.62569	−0.812846	+	0.582479i	$0.802083\pi$
$368$	−4.56301	−0.000646368 0
$369$	0	0
$370$	0	0
$371$	−10988.9	−1.53777
$372$	0	0
$373$	−2259.90	−0.313708	−0.156854	−	0.987622i	$-0.550135\pi$
$373$	−2259.90	−0.313708	−0.156854	+	0.987622i	$0.550135\pi$
$374$	−14449.8	−1.99781
$375$	0	0
$376$	−10698.9	−1.46743
$377$	8364.90	1.14274
$378$	0	0
$379$	−11815.8	−1.60142	−0.800709	−	0.599053i	$-0.795544\pi$
$379$	−11815.8	−1.60142	−0.800709	+	0.599053i	$0.795544\pi$
$380$	0	0
$381$	0	0
$382$	−3719.60	−0.498198
$383$	−8080.22	−1.07802	−0.539008	−	0.842301i	$-0.681201\pi$
$383$	−8080.22	−1.07802	−0.539008	+	0.842301i	$0.681201\pi$
$384$	0	0
$385$	0	0
$386$	−3724.00	−0.491054
$387$	0	0
$388$	−3257.17	−0.426180
$389$	−3100.43	−0.404108	−0.202054	−	0.979374i	$-0.564762\pi$
$389$	−3100.43	−0.404108	−0.202054	+	0.979374i	$0.564762\pi$
$390$	0	0
$391$	−456.805	−0.0590834
$392$	−1385.34	−0.178495
$393$	0	0
$394$	−18652.9	−2.38508
$395$	0	0
$396$	0	0
$397$	11990.1	1.51578	0.757890	−	0.652382i	$-0.226230\pi$
$397$	11990.1	1.51578	0.757890	+	0.652382i	$0.226230\pi$
$398$	6139.98	0.773291
$399$	0	0
$400$	0	0
$401$	−12853.3	−1.60065	−0.800326	−	0.599565i	$-0.795340\pi$
$401$	−12853.3	−1.60065	−0.800326	+	0.599565i	$0.795340\pi$
$402$	0	0
$403$	7209.04	0.891086
$404$	−563.243	−0.0693623
$405$	0	0
$406$	−16442.8	−2.00996
$407$	16510.0	2.01074
$408$	0	0
$409$	2225.09	0.269006	0.134503	−	0.990913i	$-0.457056\pi$
$409$	2225.09	0.269006	0.134503	+	0.990913i	$0.457056\pi$
$410$	0	0
$411$	0	0
$412$	18605.9	2.22488
$413$	8433.95	1.00486
$414$	0	0
$415$	0	0
$416$	8522.65	1.00446
$417$	0	0
$418$	−3019.85	−0.353363
$419$	−9676.65	−1.12825	−0.564123	−	0.825691i	$-0.690786\pi$
$419$	−9676.65	−1.12825	−0.564123	+	0.825691i	$0.690786\pi$
$420$	0	0
$421$	−9962.60	−1.15332	−0.576660	−	0.816984i	$-0.695644\pi$
$421$	−9962.60	−1.15332	−0.576660	+	0.816984i	$0.695644\pi$
$422$	13514.8	1.55898
$423$	0	0
$424$	12287.3	1.40737
$425$	0	0
$426$	0	0
$427$	−10966.6	−1.24289
$428$	4595.51	0.519001
$429$	0	0
$430$	0	0
$431$	2461.47	0.275092	0.137546	−	0.990495i	$-0.456078\pi$
$431$	2461.47	0.275092	0.137546	+	0.990495i	$0.456078\pi$
$432$	0	0
$433$	−7818.49	−0.867743	−0.433871	−	0.900975i	$-0.642853\pi$
$433$	−7818.49	−0.867743	−0.433871	+	0.900975i	$0.642853\pi$
$434$	−14170.7	−1.56732
$435$	0	0
$436$	−19667.3	−2.16030
$437$	−95.4672	−0.0104504
$438$	0	0
$439$	6211.83	0.675340	0.337670	−	0.941264i	$-0.390361\pi$
$439$	6211.83	0.675340	0.337670	+	0.941264i	$0.390361\pi$
$440$	0	0
$441$	0	0
$442$	10193.6	1.09697
$443$	2984.35	0.320069	0.160034	−	0.987111i	$-0.448839\pi$
$443$	2984.35	0.320069	0.160034	+	0.987111i	$0.448839\pi$
$444$	0	0
$445$	0	0
$446$	16038.9	1.70283
$447$	0	0
$448$	−16676.3	−1.75867
$449$	810.476	0.0851865	0.0425932	−	0.999092i	$-0.486438\pi$
$449$	810.476	0.0851865	0.0425932	+	0.999092i	$0.486438\pi$
$450$	0	0
$451$	16563.1	1.72933
$452$	−10502.0	−1.09286
$453$	0	0
$454$	−2985.75	−0.308653
$455$	0	0
$456$	0	0
$457$	−1571.77	−0.160885	−0.0804425	−	0.996759i	$-0.525633\pi$
$457$	−1571.77	−0.160885	−0.0804425	+	0.996759i	$0.525633\pi$
$458$	−20963.2	−2.13875
$459$	0	0
$460$	0	0
$461$	−2062.69	−0.208393	−0.104196	−	0.994557i	$-0.533227\pi$
$461$	−2062.69	−0.208393	−0.104196	+	0.994557i	$0.533227\pi$
$462$	0	0
$463$	−2783.23	−0.279369	−0.139684	−	0.990196i	$-0.544609\pi$
$463$	−2783.23	−0.279369	−0.139684	+	0.990196i	$0.544609\pi$
$464$	−85.0256	−0.00850693
$465$	0	0
$466$	−1452.24	−0.144364
$467$	10939.7	1.08400	0.541999	−	0.840379i	$-0.317668\pi$
$467$	10939.7	1.08400	0.541999	+	0.840379i	$0.317668\pi$
$468$	0	0
$469$	9008.83	0.886970
$470$	0	0
$471$	0	0
$472$	−9430.49	−0.919647
$473$	−14080.7	−1.36877
$474$	0	0
$475$	0	0
$476$	−12374.1	−1.19152
$477$	0	0
$478$	8496.21	0.812986
$479$	−14623.7	−1.39493	−0.697467	−	0.716617i	$-0.745690\pi$
$479$	−14623.7	−1.39493	−0.697467	+	0.716617i	$0.745690\pi$
$480$	0	0
$481$	−11647.0	−1.10407
$482$	14941.6	1.41197
$483$	0	0
$484$	39659.8	3.72462
$485$	0	0
$486$	0	0
$487$	−16473.6	−1.53284	−0.766419	−	0.642341i	$-0.777963\pi$
$487$	−16473.6	−1.53284	−0.766419	+	0.642341i	$0.777963\pi$
$488$	12262.4	1.13749
$489$	0	0
$490$	0	0
$491$	−20529.2	−1.88690	−0.943450	−	0.331514i	$-0.892441\pi$
$491$	−20529.2	−1.88690	−0.943450	+	0.331514i	$0.892441\pi$
$492$	0	0
$493$	−8511.95	−0.777604
$494$	2130.35	0.194026
$495$	0	0
$496$	−73.2768	−0.00663352
$497$	−8237.74	−0.743488
$498$	0	0
$499$	−14404.6	−1.29226	−0.646131	−	0.763226i	$-0.723614\pi$
$499$	−14404.6	−1.29226	−0.646131	+	0.763226i	$0.723614\pi$
$500$	0	0
$501$	0	0
$502$	−25801.3	−2.29396
$503$	−2953.63	−0.261821	−0.130910	−	0.991394i	$-0.541790\pi$
$503$	−2953.63	−0.261821	−0.130910	+	0.991394i	$0.541790\pi$
$504$	0	0
$505$	0	0
$506$	2910.40	0.255698
$507$	0	0
$508$	11169.4	0.975515
$509$	17368.9	1.51250	0.756250	−	0.654283i	$-0.227029\pi$
$509$	17368.9	1.51250	0.756250	+	0.654283i	$0.227029\pi$
$510$	0	0
$511$	7212.03	0.624347
$512$	−172.223	−0.0148657
$513$	0	0
$514$	5366.92	0.460554
$515$	0	0
$516$	0	0
$517$	−31558.3	−2.68459
$518$	22894.4	1.94193
$519$	0	0
$520$	0	0
$521$	6146.30	0.516841	0.258421	−	0.966033i	$-0.416798\pi$
$521$	6146.30	0.516841	0.258421	+	0.966033i	$0.416798\pi$
$522$	0	0
$523$	4554.68	0.380807	0.190404	−	0.981706i	$-0.439020\pi$
$523$	4554.68	0.380807	0.190404	+	0.981706i	$0.439020\pi$
$524$	28141.3	2.34611
$525$	0	0
$526$	13254.1	1.09868
$527$	−7335.77	−0.606359
$528$	0	0
$529$	−12075.0	−0.992438
$530$	0	0
$531$	0	0
$532$	−2586.05	−0.210751
$533$	−11684.4	−0.949548
$534$	0	0
$535$	0	0
$536$	−10073.3	−0.811753
$537$	0	0
$538$	13401.2	1.07391
$539$	−4086.28	−0.326547
$540$	0	0
$541$	18091.8	1.43776	0.718879	−	0.695135i	$-0.244655\pi$
$541$	18091.8	1.43776	0.718879	+	0.695135i	$0.244655\pi$
$542$	3059.18	0.242441
$543$	0	0
$544$	−8672.47	−0.683509
$545$	0	0
$546$	0	0
$547$	−15781.4	−1.23357	−0.616786	−	0.787131i	$-0.711565\pi$
$547$	−15781.4	−1.23357	−0.616786	+	0.787131i	$0.711565\pi$
$548$	−29699.7	−2.31516
$549$	0	0
$550$	0	0
$551$	−1778.91	−0.137539
$552$	0	0
$553$	−13105.7	−1.00779
$554$	−2893.39	−0.221892
$555$	0	0
$556$	−27565.6	−2.10259
$557$	−13954.5	−1.06153	−0.530766	−	0.847519i	$-0.678096\pi$
$557$	−13954.5	−1.06153	−0.530766	+	0.847519i	$0.678096\pi$
$558$	0	0
$559$	9933.18	0.751572
$560$	0	0
$561$	0	0
$562$	−16443.4	−1.23420
$563$	−13603.9	−1.01836	−0.509181	−	0.860659i	$-0.670052\pi$
$563$	−13603.9	−1.01836	−0.509181	+	0.860659i	$0.670052\pi$
$564$	0	0
$565$	0	0
$566$	2306.77	0.171309
$567$	0	0
$568$	9211.10	0.680438
$569$	12458.6	0.917912	0.458956	−	0.888459i	$-0.348224\pi$
$569$	12458.6	0.917912	0.458956	+	0.888459i	$0.348224\pi$
$570$	0	0
$571$	−13457.1	−0.986275	−0.493137	−	0.869951i	$-0.664150\pi$
$571$	−13457.1	−0.986275	−0.493137	+	0.869951i	$0.664150\pi$
$572$	−40107.0	−2.93175
$573$	0	0
$574$	22968.0	1.67015
$575$	0	0
$576$	0	0
$577$	−3722.70	−0.268592	−0.134296	−	0.990941i	$-0.542877\pi$
$577$	−3722.70	−0.268592	−0.134296	+	0.990941i	$0.542877\pi$
$578$	12097.2	0.870550
$579$	0	0
$580$	0	0
$581$	−16356.0	−1.16792
$582$	0	0
$583$	36243.4	2.57469
$584$	−8064.19	−0.571401
$585$	0	0
$586$	−28349.3	−1.99847
$587$	−17540.3	−1.23333	−0.616667	−	0.787224i	$-0.711517\pi$
$587$	−17540.3	−1.23333	−0.616667	+	0.787224i	$0.711517\pi$
$588$	0	0
$589$	−1533.10	−0.107250
$590$	0	0
$591$	0	0
$592$	118.387	0.00821902
$593$	−22350.6	−1.54777	−0.773886	−	0.633325i	$-0.781690\pi$
$593$	−22350.6	−1.54777	−0.773886	+	0.633325i	$0.781690\pi$
$594$	0	0
$595$	0	0
$596$	−22613.9	−1.55419
$597$	0	0
$598$	−2053.14	−0.140400
$599$	2561.62	0.174733	0.0873665	−	0.996176i	$-0.472155\pi$
$599$	2561.62	0.174733	0.0873665	+	0.996176i	$0.472155\pi$
$600$	0	0
$601$	13385.0	0.908463	0.454232	−	0.890884i	$-0.349914\pi$
$601$	13385.0	0.908463	0.454232	+	0.890884i	$0.349914\pi$
$602$	−19525.6	−1.32193
$603$	0	0
$604$	11315.3	0.762270
$605$	0	0
$606$	0	0
$607$	28628.6	1.91433	0.957166	−	0.289539i	$-0.0935021\pi$
$607$	28628.6	1.91433	0.957166	+	0.289539i	$0.0935021\pi$
$608$	−1812.45	−0.120896
$609$	0	0
$610$	0	0
$611$	22262.8	1.47407
$612$	0	0
$613$	−7188.12	−0.473614	−0.236807	−	0.971557i	$-0.576101\pi$
$613$	−7188.12	−0.473614	−0.236807	+	0.971557i	$0.576101\pi$
$614$	−8995.25	−0.591236
$615$	0	0
$616$	30013.0	1.96308
$617$	15533.8	1.01356	0.506779	−	0.862076i	$-0.330836\pi$
$617$	15533.8	1.01356	0.506779	+	0.862076i	$0.330836\pi$
$618$	0	0
$619$	−22159.8	−1.43890	−0.719450	−	0.694545i	$-0.755606\pi$
$619$	−22159.8	−1.43890	−0.719450	+	0.694545i	$0.755606\pi$
$620$	0	0
$621$	0	0
$622$	−10547.2	−0.679908
$623$	4043.02	0.260000
$624$	0	0
$625$	0	0
$626$	47118.9	3.00839
$627$	0	0
$628$	3350.77	0.212914
$629$	11851.7	0.751287
$630$	0	0
$631$	−25582.8	−1.61400	−0.807002	−	0.590549i	$-0.798911\pi$
$631$	−25582.8	−1.61400	−0.807002	+	0.590549i	$0.798911\pi$
$632$	14654.2	0.922330
$633$	0	0
$634$	7783.46	0.487572
$635$	0	0
$636$	0	0
$637$	2882.66	0.179302
$638$	54231.5	3.36527
$639$	0	0
$640$	0	0
$641$	−3810.68	−0.234809	−0.117405	−	0.993084i	$-0.537457\pi$
$641$	−3810.68	−0.234809	−0.117405	+	0.993084i	$0.537457\pi$
$642$	0	0
$643$	−26720.1	−1.63878	−0.819391	−	0.573235i	$-0.805688\pi$
$643$	−26720.1	−1.63878	−0.819391	+	0.573235i	$0.805688\pi$
$644$	2492.32	0.152502
$645$	0	0
$646$	−2167.80	−0.132029
$647$	5114.23	0.310759	0.155380	−	0.987855i	$-0.450340\pi$
$647$	5114.23	0.310759	0.155380	+	0.987855i	$0.450340\pi$
$648$	0	0
$649$	−27816.8	−1.68244
$650$	0	0
$651$	0	0
$652$	15524.4	0.932489
$653$	−8871.05	−0.531625	−0.265813	−	0.964025i	$-0.585640\pi$
$653$	−8871.05	−0.531625	−0.265813	+	0.964025i	$0.585640\pi$
$654$	0	0
$655$	0	0
$656$	118.767	0.00706872
$657$	0	0
$658$	−43761.7	−2.59272
$659$	24204.5	1.43076	0.715382	−	0.698734i	$-0.246253\pi$
$659$	24204.5	1.43076	0.715382	+	0.698734i	$0.246253\pi$
$660$	0	0
$661$	21379.5	1.25805	0.629023	−	0.777387i	$-0.283455\pi$
$661$	21379.5	1.25805	0.629023	+	0.777387i	$0.283455\pi$
$662$	38196.2	2.24250
$663$	0	0
$664$	18288.6	1.06888
$665$	0	0
$666$	0	0
$667$	1714.43	0.0995248
$668$	−21687.5	−1.25616
$669$	0	0
$670$	0	0
$671$	36170.1	2.08097
$672$	0	0
$673$	−29701.4	−1.70119	−0.850597	−	0.525818i	$-0.823759\pi$
$673$	−29701.4	−1.70119	−0.850597	+	0.525818i	$0.823759\pi$
$674$	−35936.6	−2.05375
$675$	0	0
$676$	−86.5900	−0.00492661
$677$	2661.48	0.151092	0.0755459	−	0.997142i	$-0.475930\pi$
$677$	2661.48	0.151092	0.0755459	+	0.997142i	$0.475930\pi$
$678$	0	0
$679$	−5071.86	−0.286657
$680$	0	0
$681$	0	0
$682$	46737.8	2.62417
$683$	28698.1	1.60776	0.803882	−	0.594789i	$-0.202765\pi$
$683$	28698.1	1.60776	0.803882	+	0.594789i	$0.202765\pi$
$684$	0	0
$685$	0	0
$686$	25888.0	1.44083
$687$	0	0
$688$	−100.967	−0.00559493
$689$	−25567.8	−1.41373
$690$	0	0
$691$	16825.2	0.926284	0.463142	−	0.886284i	$-0.346722\pi$
$691$	16825.2	0.926284	0.463142	+	0.886284i	$0.346722\pi$
$692$	12036.6	0.661220
$693$	0	0
$694$	−5543.99	−0.303238
$695$	0	0
$696$	0	0
$697$	11889.8	0.646140
$698$	−6396.90	−0.346886
$699$	0	0
$700$	0	0
$701$	998.795	0.0538145	0.0269073	−	0.999638i	$-0.491434\pi$
$701$	998.795	0.0538145	0.0269073	+	0.999638i	$0.491434\pi$
$702$	0	0
$703$	2476.88	0.132884
$704$	55001.7	2.94454
$705$	0	0
$706$	−12021.3	−0.640834
$707$	−877.047	−0.0466545
$708$	0	0
$709$	33253.8	1.76145	0.880727	−	0.473624i	$-0.157054\pi$
$709$	33253.8	1.76145	0.880727	+	0.473624i	$0.157054\pi$
$710$	0	0
$711$	0	0
$712$	−4520.73	−0.237952
$713$	1477.53	0.0776073
$714$	0	0
$715$	0	0
$716$	−13219.9	−0.690017
$717$	0	0
$718$	16819.2	0.874218
$719$	−1178.94	−0.0611503	−0.0305752	−	0.999532i	$-0.509734\pi$
$719$	−1178.94	−0.0611503	−0.0305752	+	0.999532i	$0.509734\pi$
$720$	0	0
$721$	28972.0	1.49650
$722$	30917.1	1.59365
$723$	0	0
$724$	34101.6	1.75052
$725$	0	0
$726$	0	0
$727$	−12676.7	−0.646703	−0.323351	−	0.946279i	$-0.604810\pi$
$727$	−12676.7	−0.646703	−0.323351	+	0.946279i	$0.604810\pi$
$728$	−21172.6	−1.07790
$729$	0	0
$730$	0	0
$731$	−10107.8	−0.511423
$732$	0	0
$733$	−9806.85	−0.494167	−0.247083	−	0.968994i	$-0.579472\pi$
$733$	−9806.85	−0.494167	−0.247083	+	0.968994i	$0.579472\pi$
$734$	52275.0	2.62875
$735$	0	0
$736$	1746.76	0.0874817
$737$	−29712.8	−1.48506
$738$	0	0
$739$	−29970.4	−1.49185	−0.745927	−	0.666028i	$-0.767993\pi$
$739$	−29970.4	−1.49185	−0.745927	+	0.666028i	$0.767993\pi$
$740$	0	0
$741$	0	0
$742$	50258.4	2.48658
$743$	21697.6	1.07134	0.535670	−	0.844427i	$-0.320059\pi$
$743$	21697.6	1.07134	0.535670	+	0.844427i	$0.320059\pi$
$744$	0	0
$745$	0	0
$746$	10335.8	0.507267
$747$	0	0
$748$	40812.1	1.99497
$749$	7155.84	0.349090
$750$	0	0
$751$	17024.2	0.827193	0.413596	−	0.910460i	$-0.364272\pi$
$751$	17024.2	0.827193	0.413596	+	0.910460i	$0.364272\pi$
$752$	−226.291	−0.0109734
$753$	0	0
$754$	−38257.6	−1.84782
$755$	0	0
$756$	0	0
$757$	−30745.2	−1.47616	−0.738080	−	0.674714i	$-0.764267\pi$
$757$	−30745.2	−1.47616	−0.738080	+	0.674714i	$0.764267\pi$
$758$	54040.6	2.58950
$759$	0	0
$760$	0	0
$761$	−21496.6	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$761$	−21496.6	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$762$	0	0
$763$	−30624.6	−1.45306
$764$	10505.7	0.497489
$765$	0	0
$766$	36955.5	1.74316
$767$	19623.3	0.923803
$768$	0	0
$769$	22057.5	1.03435	0.517174	−	0.855880i	$-0.326984\pi$
$769$	22057.5	1.03435	0.517174	+	0.855880i	$0.326984\pi$
$770$	0	0
$771$	0	0
$772$	10518.1	0.490355
$773$	30155.8	1.40314	0.701570	−	0.712601i	$-0.252483\pi$
$773$	30155.8	1.40314	0.701570	+	0.712601i	$0.252483\pi$
$774$	0	0
$775$	0	0
$776$	5671.14	0.262348
$777$	0	0
$778$	14180.1	0.653445
$779$	2484.85	0.114286
$780$	0	0
$781$	27169.6	1.24482
$782$	2089.23	0.0955381
$783$	0	0
$784$	−29.3010	−0.00133478
$785$	0	0
$786$	0	0
$787$	3248.72	0.147147	0.0735733	−	0.997290i	$-0.476560\pi$
$787$	3248.72	0.147147	0.0735733	+	0.997290i	$0.476560\pi$
$788$	52683.5	2.38169
$789$	0	0
$790$	0	0
$791$	−16353.1	−0.735081
$792$	0	0
$793$	−25516.1	−1.14263
$794$	−54837.6	−2.45103
$795$	0	0
$796$	−17341.8	−0.772191
$797$	−27710.2	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$797$	−27710.2	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$798$	0	0
$799$	−22654.1	−1.00306
$800$	0	0
$801$	0	0
$802$	58785.4	2.58826
$803$	−23786.6	−1.04535
$804$	0	0
$805$	0	0
$806$	−32971.1	−1.44089
$807$	0	0
$808$	980.676	0.0426981
$809$	2244.10	0.0975259	0.0487630	−	0.998810i	$-0.484472\pi$
$809$	2244.10	0.0975259	0.0487630	+	0.998810i	$0.484472\pi$
$810$	0	0
$811$	−2739.73	−0.118625	−0.0593126	−	0.998239i	$-0.518891\pi$
$811$	−2739.73	−0.118625	−0.0593126	+	0.998239i	$0.518891\pi$
$812$	46441.1	2.00710
$813$	0	0
$814$	−75510.0	−3.25138
$815$	0	0
$816$	0	0
$817$	−2112.42	−0.0904581
$818$	−10176.6	−0.434984
$819$	0	0
$820$	0	0
$821$	17232.1	0.732528	0.366264	−	0.930511i	$-0.380637\pi$
$821$	17232.1	0.732528	0.366264	+	0.930511i	$0.380637\pi$
$822$	0	0
$823$	−19285.9	−0.816845	−0.408422	−	0.912793i	$-0.633921\pi$
$823$	−19285.9	−0.816845	−0.408422	+	0.912793i	$0.633921\pi$
$824$	−32395.3	−1.36959
$825$	0	0
$826$	−38573.4	−1.62487
$827$	26379.4	1.10919	0.554595	−	0.832120i	$-0.312873\pi$
$827$	26379.4	1.10919	0.554595	+	0.832120i	$0.312873\pi$
$828$	0	0
$829$	−8718.15	−0.365252	−0.182626	−	0.983182i	$-0.558460\pi$
$829$	−8718.15	−0.365252	−0.182626	+	0.983182i	$0.558460\pi$
$830$	0	0
$831$	0	0
$832$	−38800.9	−1.61680
$833$	−2933.34	−0.122010
$834$	0	0
$835$	0	0
$836$	8529.28	0.352860
$837$	0	0
$838$	44256.9	1.82438
$839$	13476.2	0.554529	0.277265	−	0.960794i	$-0.410572\pi$
$839$	13476.2	0.554529	0.277265	+	0.960794i	$0.410572\pi$
$840$	0	0
$841$	7557.17	0.309860
$842$	45564.8	1.86492
$843$	0	0
$844$	−38171.3	−1.55677
$845$	0	0
$846$	0	0
$847$	61755.7	2.50526
$848$	259.886	0.0105242
$849$	0	0
$850$	0	0
$851$	−2387.11	−0.0961565
$852$	0	0
$853$	28236.6	1.13342	0.566708	−	0.823919i	$-0.308217\pi$
$853$	28236.6	1.13342	0.566708	+	0.823919i	$0.308217\pi$
$854$	50156.8	2.00975
$855$	0	0
$856$	−8001.36	−0.319487
$857$	8458.16	0.337136	0.168568	−	0.985690i	$-0.446086\pi$
$857$	8458.16	0.337136	0.168568	+	0.985690i	$0.446086\pi$
$858$	0	0
$859$	−23054.3	−0.915720	−0.457860	−	0.889024i	$-0.651384\pi$
$859$	−23054.3	−0.915720	−0.457860	+	0.889024i	$0.651384\pi$
$860$	0	0
$861$	0	0
$862$	−11257.7	−0.444826
$863$	−42314.4	−1.66906	−0.834529	−	0.550963i	$-0.814260\pi$
$863$	−42314.4	−1.66906	−0.834529	+	0.550963i	$0.814260\pi$
$864$	0	0
$865$	0	0
$866$	35758.5	1.40314
$867$	0	0
$868$	40023.9	1.56509
$869$	43225.0	1.68735
$870$	0	0
$871$	20960.9	0.815422
$872$	34243.1	1.32984
$873$	0	0
$874$	436.627	0.0168983
$875$	0	0
$876$	0	0
$877$	10180.0	0.391966	0.195983	−	0.980607i	$-0.437210\pi$
$877$	10180.0	0.391966	0.195983	+	0.980607i	$0.437210\pi$
$878$	−28410.3	−1.09203
$879$	0	0
$880$	0	0
$881$	−6030.64	−0.230621	−0.115311	−	0.993329i	$-0.536786\pi$
$881$	−6030.64	−0.230621	−0.115311	+	0.993329i	$0.536786\pi$
$882$	0	0
$883$	30238.0	1.15242	0.576211	−	0.817301i	$-0.304531\pi$
$883$	30238.0	1.15242	0.576211	+	0.817301i	$0.304531\pi$
$884$	−28790.8	−1.09541
$885$	0	0
$886$	−13649.1	−0.517553
$887$	18776.2	0.710758	0.355379	−	0.934722i	$-0.384352\pi$
$887$	18776.2	0.710758	0.355379	+	0.934722i	$0.384352\pi$
$888$	0	0
$889$	17392.3	0.656151
$890$	0	0
$891$	0	0
$892$	−45300.3	−1.70041
$893$	−4734.46	−0.177416
$894$	0	0
$895$	0	0
$896$	46966.8	1.75117
$897$	0	0
$898$	−3706.78	−0.137747
$899$	27531.8	1.02140
$900$	0	0
$901$	26017.3	0.962000
$902$	−75752.7	−2.79633
$903$	0	0
$904$	18285.3	0.672744
$905$	0	0
$906$	0	0
$907$	37652.5	1.37843	0.689213	−	0.724559i	$-0.257956\pi$
$907$	37652.5	1.37843	0.689213	+	0.724559i	$0.257956\pi$
$908$	8432.97	0.308214
$909$	0	0
$910$	0	0
$911$	47586.1	1.73062	0.865312	−	0.501233i	$-0.167120\pi$
$911$	47586.1	1.73062	0.865312	+	0.501233i	$0.167120\pi$
$912$	0	0
$913$	53945.1	1.95545
$914$	7188.63	0.260152
$915$	0	0
$916$	59208.7	2.13571
$917$	43819.9	1.57804
$918$	0	0
$919$	32177.3	1.15499	0.577493	−	0.816396i	$-0.304031\pi$
$919$	32177.3	1.15499	0.577493	+	0.816396i	$0.304031\pi$
$920$	0	0
$921$	0	0
$922$	9433.89	0.336972
$923$	−19166.8	−0.683514
$924$	0	0
$925$	0	0
$926$	12729.3	0.451741
$927$	0	0
$928$	32548.6	1.15136
$929$	−40643.7	−1.43539	−0.717694	−	0.696359i	$-0.754802\pi$
$929$	−40643.7	−1.43539	−0.717694	+	0.696359i	$0.754802\pi$
$930$	0	0
$931$	−613.036	−0.0215805
$932$	4101.70	0.144158
$933$	0	0
$934$	−50033.4	−1.75283
$935$	0	0
$936$	0	0
$937$	−6077.87	−0.211905	−0.105953	−	0.994371i	$-0.533789\pi$
$937$	−6077.87	−0.211905	−0.105953	+	0.994371i	$0.533789\pi$
$938$	−41202.6	−1.43424
$939$	0	0
$940$	0	0
$941$	−1464.59	−0.0507377	−0.0253688	−	0.999678i	$-0.508076\pi$
$941$	−1464.59	−0.0507377	−0.0253688	+	0.999678i	$0.508076\pi$
$942$	0	0
$943$	−2394.79	−0.0826989
$944$	−199.463	−0.00687707
$945$	0	0
$946$	64399.0	2.21331
$947$	−10608.7	−0.364031	−0.182015	−	0.983296i	$-0.558262\pi$
$947$	−10608.7	−0.364031	−0.182015	+	0.983296i	$0.558262\pi$
$948$	0	0
$949$	16780.3	0.573984
$950$	0	0
$951$	0	0
$952$	21544.8	0.733478
$953$	−43623.4	−1.48279	−0.741396	−	0.671068i	$-0.765836\pi$
$953$	−43623.4	−1.48279	−0.741396	+	0.671068i	$0.765836\pi$
$954$	0	0
$955$	0	0
$956$	−23996.7	−0.811830
$957$	0	0
$958$	66882.6	2.25562
$959$	−46246.6	−1.55723
$960$	0	0
$961$	−6063.52	−0.203535
$962$	53268.4	1.78528
$963$	0	0
$964$	−42201.2	−1.40997
$965$	0	0
$966$	0	0
$967$	−5080.37	−0.168949	−0.0844745	−	0.996426i	$-0.526921\pi$
$967$	−5080.37	−0.168949	−0.0844745	+	0.996426i	$0.526921\pi$
$968$	−69052.6	−2.29280
$969$	0	0
$970$	0	0
$971$	−9875.60	−0.326388	−0.163194	−	0.986594i	$-0.552180\pi$
$971$	−9875.60	−0.326388	−0.163194	+	0.986594i	$0.552180\pi$
$972$	0	0
$973$	−42923.4	−1.41425
$974$	75343.5	2.47861
$975$	0	0
$976$	259.361	0.00850608
$977$	24580.8	0.804923	0.402461	−	0.915437i	$-0.368155\pi$
$977$	24580.8	0.804923	0.402461	+	0.915437i	$0.368155\pi$
$978$	0	0
$979$	−13334.6	−0.435319
$980$	0	0
$981$	0	0
$982$	93891.8	3.05113
$983$	−31860.6	−1.03377	−0.516885	−	0.856055i	$-0.672908\pi$
$983$	−31860.6	−1.03377	−0.516885	+	0.856055i	$0.672908\pi$
$984$	0	0
$985$	0	0
$986$	38930.1	1.25739
$987$	0	0
$988$	−6016.97	−0.193750
$989$	2035.86	0.0654566
$990$	0	0
$991$	−36921.3	−1.18350	−0.591748	−	0.806123i	$-0.701562\pi$
$991$	−36921.3	−1.18350	−0.591748	+	0.806123i	$0.701562\pi$
$992$	28051.0	0.897803
$993$	0	0
$994$	37676.0	1.20222
$995$	0	0
$996$	0	0
$997$	3734.12	0.118617	0.0593083	−	0.998240i	$-0.481110\pi$
$997$	3734.12	0.118617	0.0593083	+	0.998240i	$0.481110\pi$
$998$	65880.7	2.08959
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.q.1.1		3
3.2	odd	2		2025.4.a.s.1.3		3
5.4	even	2		405.4.a.j.1.3		3
9.2	odd	6		225.4.e.c.76.1		6
9.5	odd	6		225.4.e.c.151.1		6
15.14	odd	2		405.4.a.h.1.1		3
45.2	even	12		225.4.k.c.49.6		12
45.4	even	6		135.4.e.b.46.1		6
45.14	odd	6		45.4.e.b.16.3	✓	6
45.23	even	12		225.4.k.c.124.6		12
45.29	odd	6		45.4.e.b.31.3	yes	6
45.32	even	12		225.4.k.c.124.1		12
45.34	even	6		135.4.e.b.91.1		6
45.38	even	12		225.4.k.c.49.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.b.16.3	✓	6	45.14	odd	6
45.4.e.b.31.3	yes	6	45.29	odd	6
135.4.e.b.46.1		6	45.4	even	6
135.4.e.b.91.1		6	45.34	even	6
225.4.e.c.76.1		6	9.2	odd	6
225.4.e.c.151.1		6	9.5	odd	6
225.4.k.c.49.1		12	45.38	even	12
225.4.k.c.49.6		12	45.2	even	12
225.4.k.c.124.1		12	45.32	even	12
225.4.k.c.124.6		12	45.23	even	12
405.4.a.h.1.1		3	15.14	odd	2
405.4.a.j.1.3		3	5.4	even	2
2025.4.a.q.1.1		3	1.1	even	1	trivial
2025.4.a.s.1.3		3	3.2	odd	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-4.57358 q^{2} +12.9176 q^{4} +20.1145 q^{7} -22.4912 q^{8} +O(q^{10})\)	\(q-4.57358 q^{2} +12.9176 q^{4} +20.1145 q^{7} -22.4912 q^{8} -66.3416 q^{11} +46.8006 q^{13} -91.9955 q^{14} -0.475708 q^{16} -47.6233 q^{17} -9.95276 q^{19} +303.418 q^{22} +9.59204 q^{23} -214.046 q^{26} +259.832 q^{28} +178.735 q^{29} +154.037 q^{31} +182.105 q^{32} +217.809 q^{34} -248.864 q^{37} +45.5197 q^{38} -249.664 q^{41} +212.245 q^{43} -856.976 q^{44} -43.8700 q^{46} +475.694 q^{47} +61.5946 q^{49} +604.553 q^{52} -546.314 q^{53} -452.400 q^{56} -817.459 q^{58} +419.296 q^{59} -545.210 q^{61} -704.502 q^{62} -829.068 q^{64} +447.877 q^{67} -615.181 q^{68} -409.542 q^{71} +358.548 q^{73} +1138.20 q^{74} -128.566 q^{76} -1334.43 q^{77} -651.552 q^{79} +1141.86 q^{82} -813.142 q^{83} -970.719 q^{86} +1492.10 q^{88} +201.000 q^{89} +941.373 q^{91} +123.906 q^{92} -2175.62 q^{94} -252.149 q^{97} -281.708 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8}+O(q^{10}) \) 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8	\( 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} - 14 q^{11} - 40 q^{13} + 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 376 q^{22} + 171 q^{23} - 434 q^{26} + 517 q^{28} + 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 402 q^{37} - 178 q^{38} - 187 q^{41} + 602 q^{43} - 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} + 456 q^{52} - 730 q^{53} - 705 q^{56} - 217 q^{58} + 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 1849 q^{67} - 710 q^{68} - 70 q^{71} + 368 q^{73} + 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} + 575 q^{82} - 831 q^{83} - 1580 q^{86} + 1428 q^{88} - 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} + 282 q^{97} + 2164 q^{98}+O(q^{100}) \) 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8 - 14 * q^11 - 40 * q^13 + 27 * q^14 - 13 * q^16 - 166 * q^17 - 164 * q^19 + 376 * q^22 + 171 * q^23 - 434 * q^26 + 517 * q^28 + 335 * q^29 - 352 * q^31 - 77 * q^32 - 52 * q^34 - 402 * q^37 - 178 * q^38 - 187 * q^41 + 602 * q^43 - 982 * q^44 - 201 * q^46 + 665 * q^47 + 430 * q^49 + 456 * q^52 - 730 * q^53 - 705 * q^56 - 217 * q^58 + 298 * q^59 - 1439 * q^61 - 1614 * q^62 - 1569 * q^64 + 1849 * q^67 - 710 * q^68 - 70 * q^71 + 368 * q^73 + 320 * q^74 + 204 * q^76 - 948 * q^77 - 382 * q^79 + 575 * q^82 - 831 * q^83 - 1580 * q^86 + 1428 * q^88 - 1719 * q^89 - 710 * q^91 - 1623 * q^92 - 2077 * q^94 + 282 * q^97 + 2164 * q^98

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