LMFDB - Embedded newform 2025.4.a.bl.1.10

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.785333$ 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)

$f(q)$	$=$	$q+0.785333 q^{2} -7.38325 q^{4} +20.9136 q^{7} -12.0810 q^{8} +O(q^{10})$	$q+0.785333 q^{2} -7.38325 q^{4} +20.9136 q^{7} -12.0810 q^{8} +59.3838 q^{11} -45.9320 q^{13} +16.4241 q^{14} +49.5784 q^{16} -43.0872 q^{17} +140.178 q^{19} +46.6360 q^{22} +101.756 q^{23} -36.0719 q^{26} -154.410 q^{28} -12.3918 q^{29} +71.9790 q^{31} +135.583 q^{32} -33.8378 q^{34} -150.734 q^{37} +110.087 q^{38} -51.0821 q^{41} -34.4716 q^{43} -438.445 q^{44} +79.9121 q^{46} -117.976 q^{47} +94.3781 q^{49} +339.128 q^{52} +137.196 q^{53} -252.657 q^{56} -9.73173 q^{58} -496.314 q^{59} -247.700 q^{61} +56.5275 q^{62} -290.149 q^{64} +826.052 q^{67} +318.124 q^{68} +260.049 q^{71} -372.227 q^{73} -118.376 q^{74} -1034.97 q^{76} +1241.93 q^{77} +476.404 q^{79} -40.1164 q^{82} -313.173 q^{83} -27.0717 q^{86} -717.414 q^{88} +817.628 q^{89} -960.603 q^{91} -751.288 q^{92} -92.6502 q^{94} +941.296 q^{97} +74.1183 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4}+O(q^{10}) $ 16 * q + 54 * q^4	$ 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

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 $

$2$	0.785333	0.277657	0.138829	−	0.990316i	$-0.455666\pi$
$2$	0.785333	0.277657	0.138829	+	0.990316i	$0.455666\pi$
$3$	0	0
$3$	0	0
$4$	−7.38325	−0.922907
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.9136	1.12923	0.564614	−	0.825355i	$-0.309025\pi$
$7$	20.9136	1.12923	0.564614	+	0.825355i	$0.309025\pi$
$8$	−12.0810	−0.533909
$9$	0	0
$10$	0	0
$11$	59.3838	1.62772	0.813858	−	0.581064i	$-0.197363\pi$
$11$	59.3838	1.62772	0.813858	+	0.581064i	$0.197363\pi$
$12$	0	0
$13$	−45.9320	−0.979942	−0.489971	−	0.871739i	$-0.662993\pi$
$13$	−45.9320	−0.979942	−0.489971	+	0.871739i	$0.662993\pi$
$14$	16.4241	0.313538
$15$	0	0
$16$	49.5784	0.774663
$17$	−43.0872	−0.614717	−0.307359	−	0.951594i	$-0.599445\pi$
$17$	−43.0872	−0.614717	−0.307359	+	0.951594i	$0.599445\pi$
$18$	0	0
$19$	140.178	1.69258	0.846292	−	0.532719i	$-0.178830\pi$
$19$	140.178	1.69258	0.846292	+	0.532719i	$0.178830\pi$
$20$	0	0
$21$	0	0
$22$	46.6360	0.451947
$23$	101.756	0.922501	0.461251	−	0.887270i	$-0.347401\pi$
$23$	101.756	0.922501	0.461251	+	0.887270i	$0.347401\pi$
$24$	0	0
$25$	0	0
$26$	−36.0719	−0.272088
$27$	0	0
$28$	−154.410	−1.04217
$29$	−12.3918	−0.0793486	−0.0396743	−	0.999213i	$-0.512632\pi$
$29$	−12.3918	−0.0793486	−0.0396743	+	0.999213i	$0.512632\pi$
$30$	0	0
$31$	71.9790	0.417026	0.208513	−	0.978020i	$-0.433138\pi$
$31$	71.9790	0.417026	0.208513	+	0.978020i	$0.433138\pi$
$32$	135.583	0.748999
$33$	0	0
$34$	−33.8378	−0.170681
$35$	0	0
$36$	0	0
$37$	−150.734	−0.669742	−0.334871	−	0.942264i	$-0.608693\pi$
$37$	−150.734	−0.669742	−0.334871	+	0.942264i	$0.608693\pi$
$38$	110.087	0.469958
$39$	0	0
$40$	0	0
$41$	−51.0821	−0.194577	−0.0972887	−	0.995256i	$-0.531017\pi$
$41$	−51.0821	−0.194577	−0.0972887	+	0.995256i	$0.531017\pi$
$42$	0	0
$43$	−34.4716	−0.122253	−0.0611264	−	0.998130i	$-0.519469\pi$
$43$	−34.4716	−0.122253	−0.0611264	+	0.998130i	$0.519469\pi$
$44$	−438.445	−1.50223
$45$	0	0
$46$	79.9121	0.256139
$47$	−117.976	−0.366139	−0.183069	−	0.983100i	$-0.558603\pi$
$47$	−117.976	−0.366139	−0.183069	+	0.983100i	$0.558603\pi$
$48$	0	0
$49$	94.3781	0.275155
$50$	0	0
$51$	0	0
$52$	339.128	0.904395
$53$	137.196	0.355573	0.177787	−	0.984069i	$-0.443106\pi$
$53$	137.196	0.355573	0.177787	+	0.984069i	$0.443106\pi$
$54$	0	0
$55$	0	0
$56$	−252.657	−0.602904
$57$	0	0
$58$	−9.73173	−0.0220317
$59$	−496.314	−1.09516	−0.547581	−	0.836753i	$-0.684451\pi$
$59$	−496.314	−1.09516	−0.547581	+	0.836753i	$0.684451\pi$
$60$	0	0
$61$	−247.700	−0.519914	−0.259957	−	0.965620i	$-0.583708\pi$
$61$	−247.700	−0.519914	−0.259957	+	0.965620i	$0.583708\pi$
$62$	56.5275	0.115790
$63$	0	0
$64$	−290.149	−0.566698
$65$	0	0
$66$	0	0
$67$	826.052	1.50624	0.753121	−	0.657881i	$-0.228547\pi$
$67$	826.052	1.50624	0.753121	+	0.657881i	$0.228547\pi$
$68$	318.124	0.567326
$69$	0	0
$70$	0	0
$71$	260.049	0.434677	0.217339	−	0.976096i	$-0.430262\pi$
$71$	260.049	0.434677	0.217339	+	0.976096i	$0.430262\pi$
$72$	0	0
$73$	−372.227	−0.596793	−0.298396	−	0.954442i	$-0.596452\pi$
$73$	−372.227	−0.596793	−0.298396	+	0.954442i	$0.596452\pi$
$74$	−118.376	−0.185959
$75$	0	0
$76$	−1034.97	−1.56210
$77$	1241.93	1.83806
$78$	0	0
$79$	476.404	0.678476	0.339238	−	0.940701i	$-0.389831\pi$
$79$	476.404	0.678476	0.339238	+	0.940701i	$0.389831\pi$
$80$	0	0
$81$	0	0
$82$	−40.1164	−0.0540258
$83$	−313.173	−0.414160	−0.207080	−	0.978324i	$-0.566396\pi$
$83$	−313.173	−0.414160	−0.207080	+	0.978324i	$0.566396\pi$
$84$	0	0
$85$	0	0
$86$	−27.0717	−0.0339444
$87$	0	0
$88$	−717.414	−0.869052
$89$	817.628	0.973802	0.486901	−	0.873457i	$-0.338127\pi$
$89$	817.628	0.973802	0.486901	+	0.873457i	$0.338127\pi$
$90$	0	0
$91$	−960.603	−1.10658
$92$	−751.288	−0.851382
$93$	0	0
$94$	−92.6502	−0.101661
$95$	0	0
$96$	0	0
$97$	941.296	0.985300	0.492650	−	0.870227i	$-0.336028\pi$
$97$	941.296	0.985300	0.492650	+	0.870227i	$0.336028\pi$
$98$	74.1183	0.0763987
$99$	0	0
$100$	0	0
$101$	1002.87	0.988017	0.494008	−	0.869457i	$-0.335531\pi$
$101$	1002.87	0.988017	0.494008	+	0.869457i	$0.335531\pi$
$102$	0	0
$103$	298.834	0.285873	0.142937	−	0.989732i	$-0.454345\pi$
$103$	298.834	0.285873	0.142937	+	0.989732i	$0.454345\pi$
$104$	554.903	0.523199
$105$	0	0
$106$	107.745	0.0987275
$107$	−984.533	−0.889518	−0.444759	−	0.895650i	$-0.646711\pi$
$107$	−984.533	−0.889518	−0.444759	+	0.895650i	$0.646711\pi$
$108$	0	0
$109$	−224.770	−0.197514	−0.0987572	−	0.995112i	$-0.531487\pi$
$109$	−224.770	−0.197514	−0.0987572	+	0.995112i	$0.531487\pi$
$110$	0	0
$111$	0	0
$112$	1036.86	0.874771
$113$	−137.764	−0.114688	−0.0573440	−	0.998354i	$-0.518263\pi$
$113$	−137.764	−0.114688	−0.0573440	+	0.998354i	$0.518263\pi$
$114$	0	0
$115$	0	0
$116$	91.4921	0.0732313
$117$	0	0
$118$	−389.772	−0.304080
$119$	−901.109	−0.694155
$120$	0	0
$121$	2195.43	1.64946
$122$	−194.527	−0.144358
$123$	0	0
$124$	−531.439	−0.384876
$125$	0	0
$126$	0	0
$127$	−2742.19	−1.91598	−0.957991	−	0.286800i	$-0.907409\pi$
$127$	−2742.19	−1.91598	−0.957991	+	0.286800i	$0.907409\pi$
$128$	−1312.53	−0.906347
$129$	0	0
$130$	0	0
$131$	960.401	0.640539	0.320270	−	0.947326i	$-0.396226\pi$
$131$	960.401	0.640539	0.320270	+	0.947326i	$0.396226\pi$
$132$	0	0
$133$	2931.63	1.91131
$134$	648.726	0.418219
$135$	0	0
$136$	520.536	0.328203
$137$	−726.517	−0.453070	−0.226535	−	0.974003i	$-0.572740\pi$
$137$	−726.517	−0.453070	−0.226535	+	0.974003i	$0.572740\pi$
$138$	0	0
$139$	499.893	0.305039	0.152519	−	0.988300i	$-0.451261\pi$
$139$	499.893	0.305039	0.152519	+	0.988300i	$0.451261\pi$
$140$	0	0
$141$	0	0
$142$	204.225	0.120691
$143$	−2727.61	−1.59507
$144$	0	0
$145$	0	0
$146$	−292.322	−0.165704
$147$	0	0
$148$	1112.90	0.618109
$149$	19.6970	0.0108298	0.00541492	−	0.999985i	$-0.498276\pi$
$149$	19.6970	0.0108298	0.00541492	+	0.999985i	$0.498276\pi$
$150$	0	0
$151$	655.417	0.353226	0.176613	−	0.984280i	$-0.443486\pi$
$151$	655.417	0.353226	0.176613	+	0.984280i	$0.443486\pi$
$152$	−1693.49	−0.903686
$153$	0	0
$154$	975.326	0.510351
$155$	0	0
$156$	0	0
$157$	−1703.54	−0.865969	−0.432984	−	0.901401i	$-0.642540\pi$
$157$	−1703.54	−0.865969	−0.432984	+	0.901401i	$0.642540\pi$
$158$	374.136	0.188384
$159$	0	0
$160$	0	0
$161$	2128.08	1.04171
$162$	0	0
$163$	2606.84	1.25266	0.626329	−	0.779559i	$-0.284557\pi$
$163$	2606.84	1.25266	0.626329	+	0.779559i	$0.284557\pi$
$164$	377.152	0.179577
$165$	0	0
$166$	−245.945	−0.114994
$167$	1639.42	0.759651	0.379826	−	0.925058i	$-0.375984\pi$
$167$	1639.42	0.759651	0.379826	+	0.925058i	$0.375984\pi$
$168$	0	0
$169$	−87.2514	−0.0397139
$170$	0	0
$171$	0	0
$172$	254.513	0.112828
$173$	1174.44	0.516132	0.258066	−	0.966127i	$-0.416915\pi$
$173$	1174.44	0.516132	0.258066	+	0.966127i	$0.416915\pi$
$174$	0	0
$175$	0	0
$176$	2944.15	1.26093
$177$	0	0
$178$	642.110	0.270383
$179$	−2512.58	−1.04916	−0.524578	−	0.851362i	$-0.675777\pi$
$179$	−2512.58	−1.04916	−0.524578	+	0.851362i	$0.675777\pi$
$180$	0	0
$181$	3685.91	1.51366	0.756828	−	0.653614i	$-0.226748\pi$
$181$	3685.91	1.51366	0.756828	+	0.653614i	$0.226748\pi$
$182$	−754.393	−0.307249
$183$	0	0
$184$	−1229.31	−0.492531
$185$	0	0
$186$	0	0
$187$	−2558.68	−1.00058
$188$	871.044	0.337912
$189$	0	0
$190$	0	0
$191$	2716.35	1.02905	0.514524	−	0.857476i	$-0.327969\pi$
$191$	2716.35	1.02905	0.514524	+	0.857476i	$0.327969\pi$
$192$	0	0
$193$	1972.75	0.735762	0.367881	−	0.929873i	$-0.380083\pi$
$193$	1972.75	0.735762	0.367881	+	0.929873i	$0.380083\pi$
$194$	739.231	0.273576
$195$	0	0
$196$	−696.818	−0.253942
$197$	3406.43	1.23197	0.615985	−	0.787758i	$-0.288758\pi$
$197$	3406.43	1.23197	0.615985	+	0.787758i	$0.288758\pi$
$198$	0	0
$199$	−981.722	−0.349711	−0.174855	−	0.984594i	$-0.555946\pi$
$199$	−981.722	−0.349711	−0.174855	+	0.984594i	$0.555946\pi$
$200$	0	0
$201$	0	0
$202$	787.590	0.274330
$203$	−259.158	−0.0896026
$204$	0	0
$205$	0	0
$206$	234.684	0.0793748
$207$	0	0
$208$	−2277.24	−0.759125
$209$	8324.31	2.75505
$210$	0	0
$211$	−2017.52	−0.658254	−0.329127	−	0.944286i	$-0.606754\pi$
$211$	−2017.52	−0.658254	−0.329127	+	0.944286i	$0.606754\pi$
$212$	−1012.96	−0.328161
$213$	0	0
$214$	−773.186	−0.246981
$215$	0	0
$216$	0	0
$217$	1505.34	0.470918
$218$	−176.519	−0.0548413
$219$	0	0
$220$	0	0
$221$	1979.08	0.602387
$222$	0	0
$223$	−265.712	−0.0797911	−0.0398956	−	0.999204i	$-0.512703\pi$
$223$	−265.712	−0.0797911	−0.0398956	+	0.999204i	$0.512703\pi$
$224$	2835.53	0.845791
$225$	0	0
$226$	−108.191	−0.0318440
$227$	5124.95	1.49848	0.749239	−	0.662300i	$-0.230419\pi$
$227$	5124.95	1.49848	0.749239	+	0.662300i	$0.230419\pi$
$228$	0	0
$229$	90.7643	0.0261916	0.0130958	−	0.999914i	$-0.495831\pi$
$229$	90.7643	0.0261916	0.0130958	+	0.999914i	$0.495831\pi$
$230$	0	0
$231$	0	0
$232$	149.706	0.0423649
$233$	5856.17	1.64657	0.823284	−	0.567629i	$-0.192139\pi$
$233$	5856.17	1.64657	0.823284	+	0.567629i	$0.192139\pi$
$234$	0	0
$235$	0	0
$236$	3664.41	1.01073
$237$	0	0
$238$	−707.670	−0.192737
$239$	6868.82	1.85903	0.929513	−	0.368790i	$-0.120228\pi$
$239$	6868.82	1.85903	0.929513	+	0.368790i	$0.120228\pi$
$240$	0	0
$241$	−25.1374	−0.00671886	−0.00335943	−	0.999994i	$-0.501069\pi$
$241$	−25.1374	−0.00671886	−0.00335943	+	0.999994i	$0.501069\pi$
$242$	1724.14	0.457984
$243$	0	0
$244$	1828.83	0.479832
$245$	0	0
$246$	0	0
$247$	−6438.67	−1.65863
$248$	−869.577	−0.222654
$249$	0	0
$250$	0	0
$251$	5765.12	1.44976	0.724882	−	0.688873i	$-0.241894\pi$
$251$	5765.12	1.44976	0.724882	+	0.688873i	$0.241894\pi$
$252$	0	0
$253$	6042.63	1.50157
$254$	−2153.53	−0.531986
$255$	0	0
$256$	1290.42	0.315044
$257$	−5609.90	−1.36162	−0.680810	−	0.732460i	$-0.738372\pi$
$257$	−5609.90	−1.36162	−0.680810	+	0.732460i	$0.738372\pi$
$258$	0	0
$259$	−3152.38	−0.756291
$260$	0	0
$261$	0	0
$262$	754.235	0.177850
$263$	5803.41	1.36066	0.680330	−	0.732906i	$-0.261837\pi$
$263$	5803.41	1.36066	0.680330	+	0.732906i	$0.261837\pi$
$264$	0	0
$265$	0	0
$266$	2302.31	0.530690
$267$	0	0
$268$	−6098.95	−1.39012
$269$	7317.07	1.65847	0.829237	−	0.558897i	$-0.188775\pi$
$269$	7317.07	1.65847	0.829237	+	0.558897i	$0.188775\pi$
$270$	0	0
$271$	−8201.78	−1.83846	−0.919230	−	0.393722i	$-0.871187\pi$
$271$	−8201.78	−1.83846	−0.919230	+	0.393722i	$0.871187\pi$
$272$	−2136.20	−0.476199
$273$	0	0
$274$	−570.558	−0.125798
$275$	0	0
$276$	0	0
$277$	−4598.40	−0.997441	−0.498720	−	0.866763i	$-0.666197\pi$
$277$	−4598.40	−0.997441	−0.498720	+	0.866763i	$0.666197\pi$
$278$	392.582	0.0846961
$279$	0	0
$280$	0	0
$281$	3811.87	0.809243	0.404621	−	0.914484i	$-0.367403\pi$
$281$	3811.87	0.809243	0.404621	+	0.914484i	$0.367403\pi$
$282$	0	0
$283$	−4099.09	−0.861009	−0.430504	−	0.902588i	$-0.641664\pi$
$283$	−4099.09	−0.861009	−0.430504	+	0.902588i	$0.641664\pi$
$284$	−1920.00	−0.401167
$285$	0	0
$286$	−2142.09	−0.442882
$287$	−1068.31	−0.219722
$288$	0	0
$289$	−3056.49	−0.622123
$290$	0	0
$291$	0	0
$292$	2748.25	0.550784
$293$	573.453	0.114340	0.0571698	−	0.998364i	$-0.481792\pi$
$293$	573.453	0.114340	0.0571698	+	0.998364i	$0.481792\pi$
$294$	0	0
$295$	0	0
$296$	1821.01	0.357581
$297$	0	0
$298$	15.4687	0.00300698
$299$	−4673.84	−0.903997
$300$	0	0
$301$	−720.925	−0.138051
$302$	514.721	0.0980757
$303$	0	0
$304$	6949.82	1.31118
$305$	0	0
$306$	0	0
$307$	8610.98	1.60083	0.800414	−	0.599447i	$-0.204613\pi$
$307$	8610.98	1.60083	0.800414	+	0.599447i	$0.204613\pi$
$308$	−9169.46	−1.69636
$309$	0	0
$310$	0	0
$311$	3049.69	0.556052	0.278026	−	0.960574i	$-0.410320\pi$
$311$	3049.69	0.556052	0.278026	+	0.960574i	$0.410320\pi$
$312$	0	0
$313$	4105.22	0.741344	0.370672	−	0.928764i	$-0.379127\pi$
$313$	4105.22	0.741344	0.370672	+	0.928764i	$0.379127\pi$
$314$	−1337.84	−0.240442
$315$	0	0
$316$	−3517.41	−0.626170
$317$	5420.67	0.960427	0.480214	−	0.877152i	$-0.340559\pi$
$317$	5420.67	0.960427	0.480214	+	0.877152i	$0.340559\pi$
$318$	0	0
$319$	−735.874	−0.129157
$320$	0	0
$321$	0	0
$322$	1671.25	0.289239
$323$	−6039.90	−1.04046
$324$	0	0
$325$	0	0
$326$	2047.23	0.347809
$327$	0	0
$328$	617.121	0.103887
$329$	−2467.29	−0.413454
$330$	0	0
$331$	5909.23	0.981272	0.490636	−	0.871365i	$-0.336765\pi$
$331$	5909.23	0.981272	0.490636	+	0.871365i	$0.336765\pi$
$332$	2312.24	0.382231
$333$	0	0
$334$	1287.49	0.210923
$335$	0	0
$336$	0	0
$337$	1324.01	0.214016	0.107008	−	0.994258i	$-0.465873\pi$
$337$	1324.01	0.214016	0.107008	+	0.994258i	$0.465873\pi$
$338$	−68.5214	−0.0110268
$339$	0	0
$340$	0	0
$341$	4274.39	0.678800
$342$	0	0
$343$	−5199.58	−0.818515
$344$	416.451	0.0652718
$345$	0	0
$346$	922.324	0.143308
$347$	−11227.7	−1.73699	−0.868495	−	0.495697i	$-0.834913\pi$
$347$	−11227.7	−1.73699	−0.868495	+	0.495697i	$0.834913\pi$
$348$	0	0
$349$	1042.88	0.159954	0.0799772	−	0.996797i	$-0.474515\pi$
$349$	1042.88	0.159954	0.0799772	+	0.996797i	$0.474515\pi$
$350$	0	0
$351$	0	0
$352$	8051.45	1.21916
$353$	11522.2	1.73730	0.868650	−	0.495426i	$-0.164988\pi$
$353$	11522.2	1.73730	0.868650	+	0.495426i	$0.164988\pi$
$354$	0	0
$355$	0	0
$356$	−6036.75	−0.898728
$357$	0	0
$358$	−1973.21	−0.291306
$359$	4751.13	0.698482	0.349241	−	0.937033i	$-0.386440\pi$
$359$	4751.13	0.698482	0.349241	+	0.937033i	$0.386440\pi$
$360$	0	0
$361$	12791.0	1.86484
$362$	2894.67	0.420277
$363$	0	0
$364$	7092.37	1.02127
$365$	0	0
$366$	0	0
$367$	11120.8	1.58174	0.790872	−	0.611981i	$-0.209627\pi$
$367$	11120.8	1.58174	0.790872	+	0.611981i	$0.209627\pi$
$368$	5044.89	0.714627
$369$	0	0
$370$	0	0
$371$	2869.27	0.401523
$372$	0	0
$373$	−9512.54	−1.32048	−0.660242	−	0.751053i	$-0.729546\pi$
$373$	−9512.54	−1.32048	−0.660242	+	0.751053i	$0.729546\pi$
$374$	−2009.42	−0.277819
$375$	0	0
$376$	1425.26	0.195485
$377$	569.182	0.0777570
$378$	0	0
$379$	−3029.06	−0.410534	−0.205267	−	0.978706i	$-0.565806\pi$
$379$	−3029.06	−0.410534	−0.205267	+	0.978706i	$0.565806\pi$
$380$	0	0
$381$	0	0
$382$	2133.24	0.285722
$383$	−10405.3	−1.38822	−0.694108	−	0.719871i	$-0.744201\pi$
$383$	−10405.3	−1.38822	−0.694108	+	0.719871i	$0.744201\pi$
$384$	0	0
$385$	0	0
$386$	1549.27	0.204289
$387$	0	0
$388$	−6949.83	−0.909340
$389$	9049.43	1.17950	0.589748	−	0.807587i	$-0.299227\pi$
$389$	9049.43	1.17950	0.589748	+	0.807587i	$0.299227\pi$
$390$	0	0
$391$	−4384.37	−0.567077
$392$	−1140.18	−0.146908
$393$	0	0
$394$	2675.18	0.342065
$395$	0	0
$396$	0	0
$397$	−276.228	−0.0349206	−0.0174603	−	0.999848i	$-0.505558\pi$
$397$	−276.228	−0.0349206	−0.0174603	+	0.999848i	$0.505558\pi$
$398$	−770.979	−0.0970997
$399$	0	0
$400$	0	0
$401$	14902.5	1.85585	0.927925	−	0.372767i	$-0.121591\pi$
$401$	14902.5	1.85585	0.927925	+	0.372767i	$0.121591\pi$
$402$	0	0
$403$	−3306.14	−0.408662
$404$	−7404.47	−0.911847
$405$	0	0
$406$	−203.525	−0.0248788
$407$	−8951.13	−1.09015
$408$	0	0
$409$	−1116.83	−0.135021	−0.0675104	−	0.997719i	$-0.521506\pi$
$409$	−1116.83	−0.135021	−0.0675104	+	0.997719i	$0.521506\pi$
$410$	0	0
$411$	0	0
$412$	−2206.36	−0.263834
$413$	−10379.7	−1.23669
$414$	0	0
$415$	0	0
$416$	−6227.62	−0.733976
$417$	0	0
$418$	6537.36	0.764958
$419$	5086.23	0.593027	0.296514	−	0.955029i	$-0.404176\pi$
$419$	5086.23	0.593027	0.296514	+	0.955029i	$0.404176\pi$
$420$	0	0
$421$	−9640.06	−1.11598	−0.557990	−	0.829848i	$-0.688427\pi$
$421$	−9640.06	−1.11598	−0.557990	+	0.829848i	$0.688427\pi$
$422$	−1584.42	−0.182769
$423$	0	0
$424$	−1657.47	−0.189844
$425$	0	0
$426$	0	0
$427$	−5180.30	−0.587101
$428$	7269.06	0.820942
$429$	0	0
$430$	0	0
$431$	857.490	0.0958326	0.0479163	−	0.998851i	$-0.484742\pi$
$431$	857.490	0.0958326	0.0479163	+	0.998851i	$0.484742\pi$
$432$	0	0
$433$	2306.40	0.255978	0.127989	−	0.991776i	$-0.459148\pi$
$433$	2306.40	0.255978	0.127989	+	0.991776i	$0.459148\pi$
$434$	1182.19	0.130754
$435$	0	0
$436$	1659.53	0.182287
$437$	14263.9	1.56141
$438$	0	0
$439$	−711.237	−0.0773246	−0.0386623	−	0.999252i	$-0.512310\pi$
$439$	−711.237	−0.0773246	−0.0386623	+	0.999252i	$0.512310\pi$
$440$	0	0
$441$	0	0
$442$	1554.24	0.167257
$443$	−15639.9	−1.67737	−0.838684	−	0.544618i	$-0.816675\pi$
$443$	−15639.9	−1.67737	−0.838684	+	0.544618i	$0.816675\pi$
$444$	0	0
$445$	0	0
$446$	−208.673	−0.0221546
$447$	0	0
$448$	−6068.06	−0.639931
$449$	−8156.47	−0.857299	−0.428650	−	0.903471i	$-0.641011\pi$
$449$	−8156.47	−0.857299	−0.428650	+	0.903471i	$0.641011\pi$
$450$	0	0
$451$	−3033.44	−0.316717
$452$	1017.15	0.105846
$453$	0	0
$454$	4024.79	0.416063
$455$	0	0
$456$	0	0
$457$	−14785.2	−1.51339	−0.756697	−	0.653766i	$-0.773188\pi$
$457$	−14785.2	−1.51339	−0.756697	+	0.653766i	$0.773188\pi$
$458$	71.2802	0.00727228
$459$	0	0
$460$	0	0
$461$	8293.15	0.837853	0.418927	−	0.908020i	$-0.362406\pi$
$461$	8293.15	0.837853	0.418927	+	0.908020i	$0.362406\pi$
$462$	0	0
$463$	−14357.0	−1.44109	−0.720546	−	0.693407i	$-0.756109\pi$
$463$	−14357.0	−1.44109	−0.720546	+	0.693407i	$0.756109\pi$
$464$	−614.368	−0.0614684
$465$	0	0
$466$	4599.04	0.457181
$467$	−4815.25	−0.477138	−0.238569	−	0.971126i	$-0.576678\pi$
$467$	−4815.25	−0.477138	−0.238569	+	0.971126i	$0.576678\pi$
$468$	0	0
$469$	17275.7	1.70089
$470$	0	0
$471$	0	0
$472$	5995.96	0.584717
$473$	−2047.05	−0.198993
$474$	0	0
$475$	0	0
$476$	6653.11	0.640641
$477$	0	0
$478$	5394.31	0.516172
$479$	−11827.7	−1.12823	−0.564116	−	0.825695i	$-0.690783\pi$
$479$	−11827.7	−1.12823	−0.564116	+	0.825695i	$0.690783\pi$
$480$	0	0
$481$	6923.50	0.656308
$482$	−19.7413	−0.00186554
$483$	0	0
$484$	−16209.4	−1.52230
$485$	0	0
$486$	0	0
$487$	−13087.8	−1.21779	−0.608894	−	0.793251i	$-0.708387\pi$
$487$	−13087.8	−1.21779	−0.608894	+	0.793251i	$0.708387\pi$
$488$	2992.46	0.277586
$489$	0	0
$490$	0	0
$491$	6593.39	0.606019	0.303010	−	0.952988i	$-0.402009\pi$
$491$	6593.39	0.606019	0.303010	+	0.952988i	$0.402009\pi$
$492$	0	0
$493$	533.930	0.0487769
$494$	−5056.50	−0.460532
$495$	0	0
$496$	3568.61	0.323055
$497$	5438.55	0.490850
$498$	0	0
$499$	−16184.8	−1.45196	−0.725982	−	0.687713i	$-0.758615\pi$
$499$	−16184.8	−1.45196	−0.725982	+	0.687713i	$0.758615\pi$
$500$	0	0
$501$	0	0
$502$	4527.54	0.402537
$503$	10342.9	0.916833	0.458416	−	0.888738i	$-0.348417\pi$
$503$	10342.9	0.916833	0.458416	+	0.888738i	$0.348417\pi$
$504$	0	0
$505$	0	0
$506$	4745.48	0.416921
$507$	0	0
$508$	20246.2	1.76827
$509$	−5648.61	−0.491887	−0.245943	−	0.969284i	$-0.579098\pi$
$509$	−5648.61	−0.491887	−0.245943	+	0.969284i	$0.579098\pi$
$510$	0	0
$511$	−7784.60	−0.673915
$512$	11513.7	0.993821
$513$	0	0
$514$	−4405.64	−0.378063
$515$	0	0
$516$	0	0
$517$	−7005.84	−0.595970
$518$	−2475.67	−0.209990
$519$	0	0
$520$	0	0
$521$	−15511.7	−1.30438	−0.652188	−	0.758058i	$-0.726149\pi$
$521$	−15511.7	−1.30438	−0.652188	+	0.758058i	$0.726149\pi$
$522$	0	0
$523$	−613.845	−0.0513223	−0.0256612	−	0.999671i	$-0.508169\pi$
$523$	−613.845	−0.0513223	−0.0256612	+	0.999671i	$0.508169\pi$
$524$	−7090.89	−0.591158
$525$	0	0
$526$	4557.61	0.377797
$527$	−3101.38	−0.256353
$528$	0	0
$529$	−1812.78	−0.148992
$530$	0	0
$531$	0	0
$532$	−21645.0	−1.76396
$533$	2346.30	0.190675
$534$	0	0
$535$	0	0
$536$	−9979.51	−0.804196
$537$	0	0
$538$	5746.34	0.460487
$539$	5604.53	0.447874
$540$	0	0
$541$	−12701.1	−1.00936	−0.504678	−	0.863308i	$-0.668389\pi$
$541$	−12701.1	−1.00936	−0.504678	+	0.863308i	$0.668389\pi$
$542$	−6441.12	−0.510461
$543$	0	0
$544$	−5841.91	−0.460423
$545$	0	0
$546$	0	0
$547$	9413.79	0.735840	0.367920	−	0.929857i	$-0.380070\pi$
$547$	9413.79	0.735840	0.367920	+	0.929857i	$0.380070\pi$
$548$	5364.06	0.418141
$549$	0	0
$550$	0	0
$551$	−1737.07	−0.134304
$552$	0	0
$553$	9963.31	0.766154
$554$	−3611.27	−0.276947
$555$	0	0
$556$	−3690.83	−0.281522
$557$	−14085.5	−1.07149	−0.535746	−	0.844379i	$-0.679969\pi$
$557$	−14085.5	−1.07149	−0.535746	+	0.844379i	$0.679969\pi$
$558$	0	0
$559$	1583.35	0.119801
$560$	0	0
$561$	0	0
$562$	2993.59	0.224692
$563$	8442.04	0.631953	0.315977	−	0.948767i	$-0.397668\pi$
$563$	8442.04	0.631953	0.315977	+	0.948767i	$0.397668\pi$
$564$	0	0
$565$	0	0
$566$	−3219.15	−0.239065
$567$	0	0
$568$	−3141.64	−0.232078
$569$	3528.48	0.259968	0.129984	−	0.991516i	$-0.458507\pi$
$569$	3528.48	0.259968	0.129984	+	0.991516i	$0.458507\pi$
$570$	0	0
$571$	−14738.1	−1.08016	−0.540078	−	0.841615i	$-0.681605\pi$
$571$	−14738.1	−1.08016	−0.540078	+	0.841615i	$0.681605\pi$
$572$	20138.7	1.47210
$573$	0	0
$574$	−838.978	−0.0610074
$575$	0	0
$576$	0	0
$577$	−17485.6	−1.26159	−0.630793	−	0.775951i	$-0.717270\pi$
$577$	−17485.6	−1.26159	−0.630793	+	0.775951i	$0.717270\pi$
$578$	−2400.36	−0.172737
$579$	0	0
$580$	0	0
$581$	−6549.58	−0.467681
$582$	0	0
$583$	8147.24	0.578772
$584$	4496.87	0.318633
$585$	0	0
$586$	450.352	0.0317472
$587$	−12494.4	−0.878536	−0.439268	−	0.898356i	$-0.644762\pi$
$587$	−12494.4	−0.878536	−0.439268	+	0.898356i	$0.644762\pi$
$588$	0	0
$589$	10089.9	0.705853
$590$	0	0
$591$	0	0
$592$	−7473.14	−0.518824
$593$	−4044.65	−0.280091	−0.140046	−	0.990145i	$-0.544725\pi$
$593$	−4044.65	−0.280091	−0.140046	+	0.990145i	$0.544725\pi$
$594$	0	0
$595$	0	0
$596$	−145.428	−0.00999492
$597$	0	0
$598$	−3670.52	−0.251001
$599$	21563.1	1.47086	0.735431	−	0.677600i	$-0.236980\pi$
$599$	21563.1	1.47086	0.735431	+	0.677600i	$0.236980\pi$
$600$	0	0
$601$	20115.9	1.36530	0.682648	−	0.730747i	$-0.260828\pi$
$601$	20115.9	1.36530	0.682648	+	0.730747i	$0.260828\pi$
$602$	−566.166	−0.0383309
$603$	0	0
$604$	−4839.11	−0.325994
$605$	0	0
$606$	0	0
$607$	17200.6	1.15017	0.575084	−	0.818094i	$-0.304969\pi$
$607$	17200.6	1.15017	0.575084	+	0.818094i	$0.304969\pi$
$608$	19005.8	1.26774
$609$	0	0
$610$	0	0
$611$	5418.86	0.358795
$612$	0	0
$613$	−894.343	−0.0589269	−0.0294634	−	0.999566i	$-0.509380\pi$
$613$	−894.343	−0.0589269	−0.0294634	+	0.999566i	$0.509380\pi$
$614$	6762.48	0.444481
$615$	0	0
$616$	−15003.7	−0.981357
$617$	6856.71	0.447392	0.223696	−	0.974659i	$-0.428188\pi$
$617$	6856.71	0.447392	0.223696	+	0.974659i	$0.428188\pi$
$618$	0	0
$619$	21165.2	1.37431	0.687157	−	0.726509i	$-0.258858\pi$
$619$	21165.2	1.37431	0.687157	+	0.726509i	$0.258858\pi$
$620$	0	0
$621$	0	0
$622$	2395.02	0.154392
$623$	17099.5	1.09964
$624$	0	0
$625$	0	0
$626$	3223.96	0.205839
$627$	0	0
$628$	12577.6	0.799208
$629$	6494.70	0.411702
$630$	0	0
$631$	14889.2	0.939352	0.469676	−	0.882839i	$-0.344371\pi$
$631$	14889.2	0.939352	0.469676	+	0.882839i	$0.344371\pi$
$632$	−5755.42	−0.362244
$633$	0	0
$634$	4257.03	0.266669
$635$	0	0
$636$	0	0
$637$	−4334.98	−0.269636
$638$	−577.906	−0.0358613
$639$	0	0
$640$	0	0
$641$	−22428.3	−1.38200	−0.691001	−	0.722854i	$-0.742830\pi$
$641$	−22428.3	−1.38200	−0.691001	+	0.722854i	$0.742830\pi$
$642$	0	0
$643$	−4262.51	−0.261426	−0.130713	−	0.991420i	$-0.541727\pi$
$643$	−4262.51	−0.261426	−0.130713	+	0.991420i	$0.541727\pi$
$644$	−15712.1	−0.961404
$645$	0	0
$646$	−4743.33	−0.288891
$647$	−19619.7	−1.19216	−0.596082	−	0.802924i	$-0.703277\pi$
$647$	−19619.7	−1.19216	−0.596082	+	0.802924i	$0.703277\pi$
$648$	0	0
$649$	−29473.0	−1.78261
$650$	0	0
$651$	0	0
$652$	−19246.9	−1.15609
$653$	4708.94	0.282198	0.141099	−	0.989996i	$-0.454936\pi$
$653$	4708.94	0.282198	0.141099	+	0.989996i	$0.454936\pi$
$654$	0	0
$655$	0	0
$656$	−2532.57	−0.150732
$657$	0	0
$658$	−1937.65	−0.114798
$659$	24138.8	1.42688	0.713441	−	0.700715i	$-0.247135\pi$
$659$	24138.8	1.42688	0.713441	+	0.700715i	$0.247135\pi$
$660$	0	0
$661$	31706.5	1.86572	0.932859	−	0.360241i	$-0.117305\pi$
$661$	31706.5	1.86572	0.932859	+	0.360241i	$0.117305\pi$
$662$	4640.72	0.272457
$663$	0	0
$664$	3783.44	0.221123
$665$	0	0
$666$	0	0
$667$	−1260.94	−0.0731991
$668$	−12104.2	−0.701087
$669$	0	0
$670$	0	0
$671$	−14709.4	−0.846272
$672$	0	0
$673$	277.641	0.0159023	0.00795116	−	0.999968i	$-0.497469\pi$
$673$	277.641	0.0159023	0.00795116	+	0.999968i	$0.497469\pi$
$674$	1039.79	0.0594231
$675$	0	0
$676$	644.199	0.0366522
$677$	−8744.57	−0.496427	−0.248213	−	0.968705i	$-0.579843\pi$
$677$	−8744.57	−0.496427	−0.248213	+	0.968705i	$0.579843\pi$
$678$	0	0
$679$	19685.9	1.11263
$680$	0	0
$681$	0	0
$682$	3356.82	0.188474
$683$	−34097.3	−1.91025	−0.955123	−	0.296210i	$-0.904277\pi$
$683$	−34097.3	−1.91025	−0.955123	+	0.296210i	$0.904277\pi$
$684$	0	0
$685$	0	0
$686$	−4083.40	−0.227267
$687$	0	0
$688$	−1709.05	−0.0947047
$689$	−6301.71	−0.348441
$690$	0	0
$691$	−4982.20	−0.274286	−0.137143	−	0.990551i	$-0.543792\pi$
$691$	−4982.20	−0.274286	−0.137143	+	0.990551i	$0.543792\pi$
$692$	−8671.17	−0.476341
$693$	0	0
$694$	−8817.50	−0.482288
$695$	0	0
$696$	0	0
$697$	2200.98	0.119610
$698$	819.008	0.0444125
$699$	0	0
$700$	0	0
$701$	16450.0	0.886315	0.443158	−	0.896444i	$-0.353858\pi$
$701$	16450.0	0.886315	0.443158	+	0.896444i	$0.353858\pi$
$702$	0	0
$703$	−21129.6	−1.13360
$704$	−17230.2	−0.922423
$705$	0	0
$706$	9048.79	0.482374
$707$	20973.7	1.11570
$708$	0	0
$709$	−9139.71	−0.484131	−0.242066	−	0.970260i	$-0.577825\pi$
$709$	−9139.71	−0.484131	−0.242066	+	0.970260i	$0.577825\pi$
$710$	0	0
$711$	0	0
$712$	−9877.74	−0.519921
$713$	7324.28	0.384707
$714$	0	0
$715$	0	0
$716$	18551.0	0.968273
$717$	0	0
$718$	3731.22	0.193938
$719$	28268.2	1.46624	0.733121	−	0.680098i	$-0.238063\pi$
$719$	28268.2	1.46624	0.733121	+	0.680098i	$0.238063\pi$
$720$	0	0
$721$	6249.68	0.322816
$722$	10045.2	0.517787
$723$	0	0
$724$	−27214.0	−1.39696
$725$	0	0
$726$	0	0
$727$	32221.9	1.64380	0.821900	−	0.569632i	$-0.192914\pi$
$727$	32221.9	1.64380	0.821900	+	0.569632i	$0.192914\pi$
$728$	11605.0	0.590811
$729$	0	0
$730$	0	0
$731$	1485.29	0.0751509
$732$	0	0
$733$	10764.2	0.542408	0.271204	−	0.962522i	$-0.412578\pi$
$733$	10764.2	0.542408	0.271204	+	0.962522i	$0.412578\pi$
$734$	8733.52	0.439183
$735$	0	0
$736$	13796.4	0.690953
$737$	49054.0	2.45174
$738$	0	0
$739$	21955.9	1.09291	0.546456	−	0.837488i	$-0.315977\pi$
$739$	21955.9	1.09291	0.546456	+	0.837488i	$0.315977\pi$
$740$	0	0
$741$	0	0
$742$	2253.33	0.111486
$743$	−5929.17	−0.292759	−0.146380	−	0.989228i	$-0.546762\pi$
$743$	−5929.17	−0.292759	−0.146380	+	0.989228i	$0.546762\pi$
$744$	0	0
$745$	0	0
$746$	−7470.51	−0.366642
$747$	0	0
$748$	18891.4	0.923446
$749$	−20590.1	−1.00447
$750$	0	0
$751$	−14894.6	−0.723716	−0.361858	−	0.932233i	$-0.617857\pi$
$751$	−14894.6	−0.723716	−0.361858	+	0.932233i	$0.617857\pi$
$752$	−5849.05	−0.283634
$753$	0	0
$754$	446.998	0.0215898
$755$	0	0
$756$	0	0
$757$	11935.5	0.573058	0.286529	−	0.958072i	$-0.407499\pi$
$757$	11935.5	0.573058	0.286529	+	0.958072i	$0.407499\pi$
$758$	−2378.82	−0.113988
$759$	0	0
$760$	0	0
$761$	−28482.5	−1.35675	−0.678377	−	0.734714i	$-0.737317\pi$
$761$	−28482.5	−1.35675	−0.678377	+	0.734714i	$0.737317\pi$
$762$	0	0
$763$	−4700.75	−0.223039
$764$	−20055.5	−0.949715
$765$	0	0
$766$	−8171.63	−0.385448
$767$	22796.7	1.07320
$768$	0	0
$769$	13392.3	0.628009	0.314005	−	0.949421i	$-0.398329\pi$
$769$	13392.3	0.628009	0.314005	+	0.949421i	$0.398329\pi$
$770$	0	0
$771$	0	0
$772$	−14565.3	−0.679039
$773$	37031.6	1.72307	0.861536	−	0.507696i	$-0.169503\pi$
$773$	37031.6	1.72307	0.861536	+	0.507696i	$0.169503\pi$
$774$	0	0
$775$	0	0
$776$	−11371.8	−0.526060
$777$	0	0
$778$	7106.81	0.327496
$779$	−7160.60	−0.329339
$780$	0	0
$781$	15442.7	0.707531
$782$	−3443.19	−0.157453
$783$	0	0
$784$	4679.12	0.213152
$785$	0	0
$786$	0	0
$787$	29263.5	1.32545	0.662726	−	0.748862i	$-0.269399\pi$
$787$	29263.5	1.32545	0.662726	+	0.748862i	$0.269399\pi$
$788$	−25150.5	−1.13699
$789$	0	0
$790$	0	0
$791$	−2881.14	−0.129509
$792$	0	0
$793$	11377.4	0.509485
$794$	−216.931	−0.00969596
$795$	0	0
$796$	7248.30	0.322750
$797$	−35700.2	−1.58666	−0.793329	−	0.608793i	$-0.791654\pi$
$797$	−35700.2	−1.58666	−0.793329	+	0.608793i	$0.791654\pi$
$798$	0	0
$799$	5083.24	0.225072
$800$	0	0
$801$	0	0
$802$	11703.4	0.515290
$803$	−22104.2	−0.971409
$804$	0	0
$805$	0	0
$806$	−2596.42	−0.113468
$807$	0	0
$808$	−12115.7	−0.527511
$809$	−24027.7	−1.04421	−0.522106	−	0.852881i	$-0.674853\pi$
$809$	−24027.7	−1.04421	−0.522106	+	0.852881i	$0.674853\pi$
$810$	0	0
$811$	−26426.3	−1.14421	−0.572104	−	0.820181i	$-0.693873\pi$
$811$	−26426.3	−1.14421	−0.572104	+	0.820181i	$0.693873\pi$
$812$	1913.43	0.0826948
$813$	0	0
$814$	−7029.62	−0.302688
$815$	0	0
$816$	0	0
$817$	−4832.17	−0.206923
$818$	−877.080	−0.0374895
$819$	0	0
$820$	0	0
$821$	24496.7	1.04134	0.520669	−	0.853758i	$-0.325682\pi$
$821$	24496.7	1.04134	0.520669	+	0.853758i	$0.325682\pi$
$822$	0	0
$823$	−14743.0	−0.624434	−0.312217	−	0.950011i	$-0.601072\pi$
$823$	−14743.0	−0.624434	−0.312217	+	0.950011i	$0.601072\pi$
$824$	−3610.20	−0.152630
$825$	0	0
$826$	−8151.52	−0.343375
$827$	−27985.3	−1.17672	−0.588358	−	0.808601i	$-0.700225\pi$
$827$	−27985.3	−1.17672	−0.588358	+	0.808601i	$0.700225\pi$
$828$	0	0
$829$	39350.4	1.64861	0.824304	−	0.566147i	$-0.191567\pi$
$829$	39350.4	1.64861	0.824304	+	0.566147i	$0.191567\pi$
$830$	0	0
$831$	0	0
$832$	13327.1	0.555331
$833$	−4066.49	−0.169142
$834$	0	0
$835$	0	0
$836$	−61460.5	−2.54265
$837$	0	0
$838$	3994.38	0.164658
$839$	22700.9	0.934113	0.467057	−	0.884227i	$-0.345314\pi$
$839$	22700.9	0.934113	0.467057	+	0.884227i	$0.345314\pi$
$840$	0	0
$841$	−24235.4	−0.993704
$842$	−7570.65	−0.309860
$843$	0	0
$844$	14895.8	0.607507
$845$	0	0
$846$	0	0
$847$	45914.3	1.86261
$848$	6801.99	0.275449
$849$	0	0
$850$	0	0
$851$	−15338.0	−0.617838
$852$	0	0
$853$	−34055.9	−1.36700	−0.683500	−	0.729951i	$-0.739543\pi$
$853$	−34055.9	−1.36700	−0.683500	+	0.729951i	$0.739543\pi$
$854$	−4068.26	−0.163013
$855$	0	0
$856$	11894.1	0.474921
$857$	−22647.0	−0.902693	−0.451347	−	0.892349i	$-0.649056\pi$
$857$	−22647.0	−0.902693	−0.451347	+	0.892349i	$0.649056\pi$
$858$	0	0
$859$	−12157.1	−0.482882	−0.241441	−	0.970416i	$-0.577620\pi$
$859$	−12157.1	−0.482882	−0.241441	+	0.970416i	$0.577620\pi$
$860$	0	0
$861$	0	0
$862$	673.415	0.0266086
$863$	−11517.7	−0.454305	−0.227153	−	0.973859i	$-0.572942\pi$
$863$	−11517.7	−0.454305	−0.227153	+	0.973859i	$0.572942\pi$
$864$	0	0
$865$	0	0
$866$	1811.29	0.0710742
$867$	0	0
$868$	−11114.3	−0.434613
$869$	28290.6	1.10437
$870$	0	0
$871$	−37942.2	−1.47603
$872$	2715.44	0.105455
$873$	0	0
$874$	11201.9	0.433537
$875$	0	0
$876$	0	0
$877$	−27266.1	−1.04984	−0.524921	−	0.851151i	$-0.675905\pi$
$877$	−27266.1	−1.04984	−0.524921	+	0.851151i	$0.675905\pi$
$878$	−558.558	−0.0214697
$879$	0	0
$880$	0	0
$881$	−9173.42	−0.350806	−0.175403	−	0.984497i	$-0.556123\pi$
$881$	−9173.42	−0.350806	−0.175403	+	0.984497i	$0.556123\pi$
$882$	0	0
$883$	−8150.59	−0.310633	−0.155317	−	0.987865i	$-0.549640\pi$
$883$	−8150.59	−0.310633	−0.155317	+	0.987865i	$0.549640\pi$
$884$	−14612.1	−0.555947
$885$	0	0
$886$	−12282.5	−0.465733
$887$	1442.34	0.0545986	0.0272993	−	0.999627i	$-0.491309\pi$
$887$	1442.34	0.0545986	0.0272993	+	0.999627i	$0.491309\pi$
$888$	0	0
$889$	−57348.9	−2.16358
$890$	0	0
$891$	0	0
$892$	1961.82	0.0736398
$893$	−16537.6	−0.619721
$894$	0	0
$895$	0	0
$896$	−27449.7	−1.02347
$897$	0	0
$898$	−6405.54	−0.238035
$899$	−891.953	−0.0330904
$900$	0	0
$901$	−5911.42	−0.218577
$902$	−2382.26	−0.0879387
$903$	0	0
$904$	1664.32	0.0612329
$905$	0	0
$906$	0	0
$907$	5472.41	0.200340	0.100170	−	0.994970i	$-0.468061\pi$
$907$	5472.41	0.200340	0.100170	+	0.994970i	$0.468061\pi$
$908$	−37838.8	−1.38296
$909$	0	0
$910$	0	0
$911$	−23045.6	−0.838130	−0.419065	−	0.907956i	$-0.637642\pi$
$911$	−23045.6	−0.838130	−0.419065	+	0.907956i	$0.637642\pi$
$912$	0	0
$913$	−18597.4	−0.674134
$914$	−11611.3	−0.420204
$915$	0	0
$916$	−670.136	−0.0241724
$917$	20085.4	0.723315
$918$	0	0
$919$	−23130.3	−0.830247	−0.415124	−	0.909765i	$-0.636262\pi$
$919$	−23130.3	−0.830247	−0.415124	+	0.909765i	$0.636262\pi$
$920$	0	0
$921$	0	0
$922$	6512.88	0.232636
$923$	−11944.6	−0.425959
$924$	0	0
$925$	0	0
$926$	−11275.0	−0.400130
$927$	0	0
$928$	−1680.13	−0.0594320
$929$	12258.9	0.432942	0.216471	−	0.976289i	$-0.430545\pi$
$929$	12258.9	0.432942	0.216471	+	0.976289i	$0.430545\pi$
$930$	0	0
$931$	13229.8	0.465723
$932$	−43237.6	−1.51963
$933$	0	0
$934$	−3781.57	−0.132481
$935$	0	0
$936$	0	0
$937$	37083.2	1.29291	0.646454	−	0.762953i	$-0.276251\pi$
$937$	37083.2	1.29291	0.646454	+	0.762953i	$0.276251\pi$
$938$	13567.2	0.472264
$939$	0	0
$940$	0	0
$941$	2612.96	0.0905206	0.0452603	−	0.998975i	$-0.485588\pi$
$941$	2612.96	0.0905206	0.0452603	+	0.998975i	$0.485588\pi$
$942$	0	0
$943$	−5197.89	−0.179498
$944$	−24606.5	−0.848382
$945$	0	0
$946$	−1607.62	−0.0552518
$947$	34322.4	1.17775	0.588875	−	0.808224i	$-0.299571\pi$
$947$	34322.4	1.17775	0.588875	+	0.808224i	$0.299571\pi$
$948$	0	0
$949$	17097.1	0.584822
$950$	0	0
$951$	0	0
$952$	10886.3	0.370616
$953$	33734.8	1.14667	0.573335	−	0.819321i	$-0.305649\pi$
$953$	33734.8	1.14667	0.573335	+	0.819321i	$0.305649\pi$
$954$	0	0
$955$	0	0
$956$	−50714.2	−1.71571
$957$	0	0
$958$	−9288.72	−0.313262
$959$	−15194.1	−0.511619
$960$	0	0
$961$	−24610.0	−0.826089
$962$	5437.25	0.182229
$963$	0	0
$964$	185.596	0.00620088
$965$	0	0
$966$	0	0
$967$	−27695.9	−0.921033	−0.460517	−	0.887651i	$-0.652336\pi$
$967$	−27695.9	−0.921033	−0.460517	+	0.887651i	$0.652336\pi$
$968$	−26522.9	−0.880660
$969$	0	0
$970$	0	0
$971$	3160.23	0.104446	0.0522228	−	0.998635i	$-0.483369\pi$
$971$	3160.23	0.104446	0.0522228	+	0.998635i	$0.483369\pi$
$972$	0	0
$973$	10454.6	0.344458
$974$	−10278.3	−0.338128
$975$	0	0
$976$	−12280.6	−0.402758
$977$	−31041.9	−1.01650	−0.508248	−	0.861211i	$-0.669707\pi$
$977$	−31041.9	−1.01650	−0.508248	+	0.861211i	$0.669707\pi$
$978$	0	0
$979$	48553.8	1.58507
$980$	0	0
$981$	0	0
$982$	5178.00	0.168265
$983$	7417.84	0.240684	0.120342	−	0.992732i	$-0.461601\pi$
$983$	7417.84	0.240684	0.120342	+	0.992732i	$0.461601\pi$
$984$	0	0
$985$	0	0
$986$	419.313	0.0135433
$987$	0	0
$988$	47538.3	1.53076
$989$	−3507.68	−0.112778
$990$	0	0
$991$	13109.6	0.420221	0.210110	−	0.977678i	$-0.432618\pi$
$991$	13109.6	0.420221	0.210110	+	0.977678i	$0.432618\pi$
$992$	9759.16	0.312352
$993$	0	0
$994$	4271.07	0.136288
$995$	0	0
$996$	0	0
$997$	−35703.9	−1.13416	−0.567079	−	0.823664i	$-0.691926\pi$
$997$	−35703.9	−1.13416	−0.567079	+	0.823664i	$0.691926\pi$
$998$	−12710.4	−0.403148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bl.1.10		16
3.2	odd	2		2025.4.a.bk.1.7		16
5.2	odd	4		405.4.b.f.244.10		16
5.3	odd	4		405.4.b.f.244.7		16
5.4	even	2	inner	2025.4.a.bl.1.7		16
9.2	odd	6		225.4.e.g.76.10		32
9.5	odd	6		225.4.e.g.151.10		32
15.2	even	4		405.4.b.e.244.7		16
15.8	even	4		405.4.b.e.244.10		16
15.14	odd	2		2025.4.a.bk.1.10		16
45.2	even	12		45.4.j.a.4.7	✓	32
45.7	odd	12		135.4.j.a.64.10		32
45.13	odd	12		135.4.j.a.19.10		32
45.14	odd	6		225.4.e.g.151.7		32
45.22	odd	12		135.4.j.a.19.7		32
45.23	even	12		45.4.j.a.34.7	yes	32
45.29	odd	6		225.4.e.g.76.7		32
45.32	even	12		45.4.j.a.34.10	yes	32
45.38	even	12		45.4.j.a.4.10	yes	32
45.43	odd	12		135.4.j.a.64.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.7	✓	32	45.2	even	12
45.4.j.a.4.10	yes	32	45.38	even	12
45.4.j.a.34.7	yes	32	45.23	even	12
45.4.j.a.34.10	yes	32	45.32	even	12
135.4.j.a.19.7		32	45.22	odd	12
135.4.j.a.19.10		32	45.13	odd	12
135.4.j.a.64.7		32	45.43	odd	12
135.4.j.a.64.10		32	45.7	odd	12
225.4.e.g.76.7		32	45.29	odd	6
225.4.e.g.76.10		32	9.2	odd	6
225.4.e.g.151.7		32	45.14	odd	6
225.4.e.g.151.10		32	9.5	odd	6
405.4.b.e.244.7		16	15.2	even	4
405.4.b.e.244.10		16	15.8	even	4
405.4.b.f.244.7		16	5.3	odd	4
405.4.b.f.244.10		16	5.2	odd	4
2025.4.a.bk.1.7		16	3.2	odd	2
2025.4.a.bk.1.10		16	15.14	odd	2
2025.4.a.bl.1.7		16	5.4	even	2	inner
2025.4.a.bl.1.10		16	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.785333 q^{2} -7.38325 q^{4} +20.9136 q^{7} -12.0810 q^{8} +O(q^{10})\)	\(q+0.785333 q^{2} -7.38325 q^{4} +20.9136 q^{7} -12.0810 q^{8} +59.3838 q^{11} -45.9320 q^{13} +16.4241 q^{14} +49.5784 q^{16} -43.0872 q^{17} +140.178 q^{19} +46.6360 q^{22} +101.756 q^{23} -36.0719 q^{26} -154.410 q^{28} -12.3918 q^{29} +71.9790 q^{31} +135.583 q^{32} -33.8378 q^{34} -150.734 q^{37} +110.087 q^{38} -51.0821 q^{41} -34.4716 q^{43} -438.445 q^{44} +79.9121 q^{46} -117.976 q^{47} +94.3781 q^{49} +339.128 q^{52} +137.196 q^{53} -252.657 q^{56} -9.73173 q^{58} -496.314 q^{59} -247.700 q^{61} +56.5275 q^{62} -290.149 q^{64} +826.052 q^{67} +318.124 q^{68} +260.049 q^{71} -372.227 q^{73} -118.376 q^{74} -1034.97 q^{76} +1241.93 q^{77} +476.404 q^{79} -40.1164 q^{82} -313.173 q^{83} -27.0717 q^{86} -717.414 q^{88} +817.628 q^{89} -960.603 q^{91} -751.288 q^{92} -92.6502 q^{94} +941.296 q^{97} +74.1183 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 16 * q + 54 * q^4	\( 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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