LMFDB - Embedded newform 2025.4.a.bl.1.14

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			1.14
Root			$4.07626$ 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)

$f(q)$	$=$	$q+4.07626 q^{2} +8.61587 q^{4} +13.3430 q^{7} +2.51043 q^{8} +O(q^{10})$	$q+4.07626 q^{2} +8.61587 q^{4} +13.3430 q^{7} +2.51043 q^{8} -11.5617 q^{11} -40.0796 q^{13} +54.3893 q^{14} -58.6938 q^{16} -93.3392 q^{17} +75.1002 q^{19} -47.1284 q^{22} +142.001 q^{23} -163.375 q^{26} +114.961 q^{28} +174.047 q^{29} +248.868 q^{31} -259.334 q^{32} -380.475 q^{34} +82.9086 q^{37} +306.128 q^{38} +449.454 q^{41} +279.850 q^{43} -99.6140 q^{44} +578.831 q^{46} -33.8764 q^{47} -164.966 q^{49} -345.320 q^{52} +423.321 q^{53} +33.4966 q^{56} +709.461 q^{58} +615.628 q^{59} -502.608 q^{61} +1014.45 q^{62} -587.563 q^{64} -57.7537 q^{67} -804.198 q^{68} +252.598 q^{71} +823.934 q^{73} +337.957 q^{74} +647.054 q^{76} -154.267 q^{77} -205.565 q^{79} +1832.09 q^{82} -646.559 q^{83} +1140.74 q^{86} -29.0248 q^{88} +1324.94 q^{89} -534.780 q^{91} +1223.46 q^{92} -138.089 q^{94} -563.768 q^{97} -672.442 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.07626	1.44117	0.720587	−	0.693364i	$-0.243872\pi$
$2$	4.07626	1.44117	0.720587	+	0.693364i	$0.243872\pi$
$3$	0	0
$3$	0	0
$4$	8.61587	1.07698
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.3430	0.720452	0.360226	−	0.932865i	$-0.382700\pi$
$7$	13.3430	0.720452	0.360226	+	0.932865i	$0.382700\pi$
$8$	2.51043	0.110946
$9$	0	0
$10$	0	0
$11$	−11.5617	−0.316907	−0.158454	−	0.987366i	$-0.550651\pi$
$11$	−11.5617	−0.316907	−0.158454	+	0.987366i	$0.550651\pi$
$12$	0	0
$13$	−40.0796	−0.855083	−0.427541	−	0.903996i	$-0.640620\pi$
$13$	−40.0796	−0.855083	−0.427541	+	0.903996i	$0.640620\pi$
$14$	54.3893	1.03830
$15$	0	0
$16$	−58.6938	−0.917090
$17$	−93.3392	−1.33165	−0.665826	−	0.746107i	$-0.731921\pi$
$17$	−93.3392	−1.33165	−0.665826	+	0.746107i	$0.731921\pi$
$18$	0	0
$19$	75.1002	0.906799	0.453399	−	0.891307i	$-0.350211\pi$
$19$	75.1002	0.906799	0.453399	+	0.891307i	$0.350211\pi$
$20$	0	0
$21$	0	0
$22$	−47.1284	−0.456719
$23$	142.001	1.28735	0.643677	−	0.765297i	$-0.277408\pi$
$23$	142.001	1.28735	0.643677	+	0.765297i	$0.277408\pi$
$24$	0	0
$25$	0	0
$26$	−163.375	−1.23232
$27$	0	0
$28$	114.961	0.775915
$29$	174.047	1.11447	0.557237	−	0.830354i	$-0.311862\pi$
$29$	174.047	1.11447	0.557237	+	0.830354i	$0.311862\pi$
$30$	0	0
$31$	248.868	1.44187	0.720937	−	0.693001i	$-0.243712\pi$
$31$	248.868	1.44187	0.720937	+	0.693001i	$0.243712\pi$
$32$	−259.334	−1.43263
$33$	0	0
$34$	−380.475	−1.91914
$35$	0	0
$36$	0	0
$37$	82.9086	0.368381	0.184190	−	0.982891i	$-0.441034\pi$
$37$	82.9086	0.368381	0.184190	+	0.982891i	$0.441034\pi$
$38$	306.128	1.30686
$39$	0	0
$40$	0	0
$41$	449.454	1.71202	0.856011	−	0.516957i	$-0.172935\pi$
$41$	449.454	1.71202	0.856011	+	0.516957i	$0.172935\pi$
$42$	0	0
$43$	279.850	0.992481	0.496241	−	0.868185i	$-0.334713\pi$
$43$	279.850	0.992481	0.496241	+	0.868185i	$0.334713\pi$
$44$	−99.6140	−0.341304
$45$	0	0
$46$	578.831	1.85530
$47$	−33.8764	−0.105136	−0.0525678	−	0.998617i	$-0.516741\pi$
$47$	−33.8764	−0.105136	−0.0525678	+	0.998617i	$0.516741\pi$
$48$	0	0
$49$	−164.966	−0.480949
$50$	0	0
$51$	0	0
$52$	−345.320	−0.920910
$53$	423.321	1.09712	0.548562	−	0.836110i	$-0.315175\pi$
$53$	423.321	1.09712	0.548562	+	0.836110i	$0.315175\pi$
$54$	0	0
$55$	0	0
$56$	33.4966	0.0799316
$57$	0	0
$58$	709.461	1.60615
$59$	615.628	1.35844	0.679220	−	0.733935i	$-0.262318\pi$
$59$	615.628	1.35844	0.679220	+	0.733935i	$0.262318\pi$
$60$	0	0
$61$	−502.608	−1.05496	−0.527479	−	0.849568i	$-0.676863\pi$
$61$	−502.608	−1.05496	−0.527479	+	0.849568i	$0.676863\pi$
$62$	1014.45	2.07799
$63$	0	0
$64$	−587.563	−1.14758
$65$	0	0
$66$	0	0
$67$	−57.7537	−0.105310	−0.0526548	−	0.998613i	$-0.516768\pi$
$67$	−57.7537	−0.105310	−0.0526548	+	0.998613i	$0.516768\pi$
$68$	−804.198	−1.43417
$69$	0	0
$70$	0	0
$71$	252.598	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$71$	252.598	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$72$	0	0
$73$	823.934	1.32102	0.660508	−	0.750819i	$-0.270341\pi$
$73$	823.934	1.32102	0.660508	+	0.750819i	$0.270341\pi$
$74$	337.957	0.530901
$75$	0	0
$76$	647.054	0.976607
$77$	−154.267	−0.228317
$78$	0	0
$79$	−205.565	−0.292758	−0.146379	−	0.989229i	$-0.546762\pi$
$79$	−205.565	−0.292758	−0.146379	+	0.989229i	$0.546762\pi$
$80$	0	0
$81$	0	0
$82$	1832.09	2.46732
$83$	−646.559	−0.855049	−0.427525	−	0.904004i	$-0.640614\pi$
$83$	−646.559	−0.855049	−0.427525	+	0.904004i	$0.640614\pi$
$84$	0	0
$85$	0	0
$86$	1140.74	1.43034
$87$	0	0
$88$	−29.0248	−0.0351598
$89$	1324.94	1.57802	0.789010	−	0.614380i	$-0.210594\pi$
$89$	1324.94	1.57802	0.789010	+	0.614380i	$0.210594\pi$
$90$	0	0
$91$	−534.780	−0.616046
$92$	1223.46	1.38646
$93$	0	0
$94$	−138.089	−0.151519
$95$	0	0
$96$	0	0
$97$	−563.768	−0.590124	−0.295062	−	0.955478i	$-0.595340\pi$
$97$	−563.768	−0.590124	−0.295062	+	0.955478i	$0.595340\pi$
$98$	−672.442	−0.693132
$99$	0	0
$100$	0	0
$101$	311.591	0.306975	0.153487	−	0.988151i	$-0.450950\pi$
$101$	311.591	0.306975	0.153487	+	0.988151i	$0.450950\pi$
$102$	0	0
$103$	81.5226	0.0779870	0.0389935	−	0.999239i	$-0.487585\pi$
$103$	81.5226	0.0779870	0.0389935	+	0.999239i	$0.487585\pi$
$104$	−100.617	−0.0948684
$105$	0	0
$106$	1725.56	1.58115
$107$	−145.138	−0.131131	−0.0655655	−	0.997848i	$-0.520885\pi$
$107$	−145.138	−0.131131	−0.0655655	+	0.997848i	$0.520885\pi$
$108$	0	0
$109$	326.484	0.286894	0.143447	−	0.989658i	$-0.454181\pi$
$109$	326.484	0.286894	0.143447	+	0.989658i	$0.454181\pi$
$110$	0	0
$111$	0	0
$112$	−783.148	−0.660719
$113$	−52.0212	−0.0433074	−0.0216537	−	0.999766i	$-0.506893\pi$
$113$	−52.0212	−0.0433074	−0.0216537	+	0.999766i	$0.506893\pi$
$114$	0	0
$115$	0	0
$116$	1499.57	1.20027
$117$	0	0
$118$	2509.46	1.95775
$119$	−1245.42	−0.959391
$120$	0	0
$121$	−1197.33	−0.899570
$122$	−2048.76	−1.52038
$123$	0	0
$124$	2144.22	1.55287
$125$	0	0
$126$	0	0
$127$	−266.008	−0.185862	−0.0929309	−	0.995673i	$-0.529624\pi$
$127$	−266.008	−0.185862	−0.0929309	+	0.995673i	$0.529624\pi$
$128$	−320.383	−0.221235
$129$	0	0
$130$	0	0
$131$	−2542.39	−1.69565	−0.847825	−	0.530277i	$-0.822088\pi$
$131$	−2542.39	−1.69565	−0.847825	+	0.530277i	$0.822088\pi$
$132$	0	0
$133$	1002.06	0.653305
$134$	−235.419	−0.151769
$135$	0	0
$136$	−234.322	−0.147742
$137$	−123.495	−0.0770140	−0.0385070	−	0.999258i	$-0.512260\pi$
$137$	−123.495	−0.0770140	−0.0385070	+	0.999258i	$0.512260\pi$
$138$	0	0
$139$	215.123	0.131270	0.0656348	−	0.997844i	$-0.479093\pi$
$139$	215.123	0.131270	0.0656348	+	0.997844i	$0.479093\pi$
$140$	0	0
$141$	0	0
$142$	1029.66	0.608499
$143$	463.388	0.270982
$144$	0	0
$145$	0	0
$146$	3358.57	1.90382
$147$	0	0
$148$	714.329	0.396740
$149$	221.877	0.121993	0.0609963	−	0.998138i	$-0.480572\pi$
$149$	221.877	0.121993	0.0609963	+	0.998138i	$0.480572\pi$
$150$	0	0
$151$	495.348	0.266959	0.133480	−	0.991052i	$-0.457385\pi$
$151$	495.348	0.266959	0.133480	+	0.991052i	$0.457385\pi$
$152$	188.534	0.100606
$153$	0	0
$154$	−628.832	−0.329044
$155$	0	0
$156$	0	0
$157$	2206.41	1.12160	0.560798	−	0.827952i	$-0.310494\pi$
$157$	2206.41	1.12160	0.560798	+	0.827952i	$0.310494\pi$
$158$	−837.935	−0.421915
$159$	0	0
$160$	0	0
$161$	1894.71	0.927477
$162$	0	0
$163$	−931.108	−0.447423	−0.223712	−	0.974655i	$-0.571817\pi$
$163$	−931.108	−0.447423	−0.223712	+	0.974655i	$0.571817\pi$
$164$	3872.44	1.84382
$165$	0	0
$166$	−2635.54	−1.23227
$167$	−1724.43	−0.799045	−0.399522	−	0.916723i	$-0.630824\pi$
$167$	−1724.43	−0.799045	−0.399522	+	0.916723i	$0.630824\pi$
$168$	0	0
$169$	−590.628	−0.268834
$170$	0	0
$171$	0	0
$172$	2411.15	1.06889
$173$	2451.99	1.07758	0.538790	−	0.842440i	$-0.318881\pi$
$173$	2451.99	1.07758	0.538790	+	0.842440i	$0.318881\pi$
$174$	0	0
$175$	0	0
$176$	678.599	0.290633
$177$	0	0
$178$	5400.81	2.27420
$179$	1051.07	0.438885	0.219443	−	0.975625i	$-0.429576\pi$
$179$	1051.07	0.438885	0.219443	+	0.975625i	$0.429576\pi$
$180$	0	0
$181$	−2663.06	−1.09361	−0.546805	−	0.837260i	$-0.684156\pi$
$181$	−2663.06	−1.09361	−0.546805	+	0.837260i	$0.684156\pi$
$182$	−2179.90	−0.887829
$183$	0	0
$184$	356.483	0.142827
$185$	0	0
$186$	0	0
$187$	1079.16	0.422010
$188$	−291.874	−0.113229
$189$	0	0
$190$	0	0
$191$	−2223.59	−0.842374	−0.421187	−	0.906974i	$-0.638386\pi$
$191$	−2223.59	−0.842374	−0.421187	+	0.906974i	$0.638386\pi$
$192$	0	0
$193$	4850.19	1.80894	0.904468	−	0.426542i	$-0.140268\pi$
$193$	4850.19	1.80894	0.904468	+	0.426542i	$0.140268\pi$
$194$	−2298.06	−0.850471
$195$	0	0
$196$	−1421.32	−0.517974
$197$	372.662	0.134777	0.0673884	−	0.997727i	$-0.478533\pi$
$197$	372.662	0.134777	0.0673884	+	0.997727i	$0.478533\pi$
$198$	0	0
$199$	4996.59	1.77990	0.889948	−	0.456063i	$-0.150741\pi$
$199$	4996.59	1.77990	0.889948	+	0.456063i	$0.150741\pi$
$200$	0	0
$201$	0	0
$202$	1270.12	0.442404
$203$	2322.30	0.802925
$204$	0	0
$205$	0	0
$206$	332.307	0.112393
$207$	0	0
$208$	2352.42	0.784188
$209$	−868.286	−0.287371
$210$	0	0
$211$	−563.402	−0.183821	−0.0919104	−	0.995767i	$-0.529297\pi$
$211$	−563.402	−0.183821	−0.0919104	+	0.995767i	$0.529297\pi$
$212$	3647.28	1.18158
$213$	0	0
$214$	−591.619	−0.188982
$215$	0	0
$216$	0	0
$217$	3320.64	1.03880
$218$	1330.83	0.413465
$219$	0	0
$220$	0	0
$221$	3741.00	1.13867
$222$	0	0
$223$	−5374.35	−1.61387	−0.806936	−	0.590639i	$-0.798876\pi$
$223$	−5374.35	−1.61387	−0.806936	+	0.590639i	$0.798876\pi$
$224$	−3460.29	−1.03214
$225$	0	0
$226$	−212.052	−0.0624136
$227$	−1111.67	−0.325040	−0.162520	−	0.986705i	$-0.551962\pi$
$227$	−1111.67	−0.325040	−0.162520	+	0.986705i	$0.551962\pi$
$228$	0	0
$229$	2037.75	0.588027	0.294013	−	0.955801i	$-0.405009\pi$
$229$	2037.75	0.588027	0.294013	+	0.955801i	$0.405009\pi$
$230$	0	0
$231$	0	0
$232$	436.933	0.123647
$233$	2127.08	0.598066	0.299033	−	0.954243i	$-0.403336\pi$
$233$	2127.08	0.598066	0.299033	+	0.954243i	$0.403336\pi$
$234$	0	0
$235$	0	0
$236$	5304.17	1.46302
$237$	0	0
$238$	−5076.66	−1.38265
$239$	1982.88	0.536659	0.268330	−	0.963327i	$-0.413528\pi$
$239$	1982.88	0.536659	0.268330	+	0.963327i	$0.413528\pi$
$240$	0	0
$241$	−4219.68	−1.12786	−0.563929	−	0.825823i	$-0.690711\pi$
$241$	−4219.68	−1.12786	−0.563929	+	0.825823i	$0.690711\pi$
$242$	−4880.61	−1.29644
$243$	0	0
$244$	−4330.41	−1.13617
$245$	0	0
$246$	0	0
$247$	−3009.99	−0.775388
$248$	624.767	0.159971
$249$	0	0
$250$	0	0
$251$	1638.58	0.412057	0.206028	−	0.978546i	$-0.433946\pi$
$251$	1638.58	0.412057	0.206028	+	0.978546i	$0.433946\pi$
$252$	0	0
$253$	−1641.77	−0.407972
$254$	−1084.32	−0.267859
$255$	0	0
$256$	3394.54	0.828745
$257$	6167.31	1.49691	0.748456	−	0.663184i	$-0.230795\pi$
$257$	6167.31	1.49691	0.748456	+	0.663184i	$0.230795\pi$
$258$	0	0
$259$	1106.25	0.265401
$260$	0	0
$261$	0	0
$262$	−10363.5	−2.44373
$263$	−6361.88	−1.49160	−0.745799	−	0.666171i	$-0.767932\pi$
$263$	−6361.88	−1.49160	−0.745799	+	0.666171i	$0.767932\pi$
$264$	0	0
$265$	0	0
$266$	4084.65	0.941526
$267$	0	0
$268$	−497.598	−0.113417
$269$	−2404.84	−0.545077	−0.272539	−	0.962145i	$-0.587863\pi$
$269$	−2404.84	−0.545077	−0.272539	+	0.962145i	$0.587863\pi$
$270$	0	0
$271$	6986.46	1.56604	0.783021	−	0.621995i	$-0.213678\pi$
$271$	6986.46	1.56604	0.783021	+	0.621995i	$0.213678\pi$
$272$	5478.43	1.22125
$273$	0	0
$274$	−503.398	−0.110991
$275$	0	0
$276$	0	0
$277$	1502.01	0.325803	0.162901	−	0.986642i	$-0.447915\pi$
$277$	1502.01	0.325803	0.162901	+	0.986642i	$0.447915\pi$
$278$	876.896	0.189182
$279$	0	0
$280$	0	0
$281$	1443.06	0.306355	0.153178	−	0.988199i	$-0.451049\pi$
$281$	1443.06	0.306355	0.153178	+	0.988199i	$0.451049\pi$
$282$	0	0
$283$	4662.08	0.979264	0.489632	−	0.871929i	$-0.337131\pi$
$283$	4662.08	0.979264	0.489632	+	0.871929i	$0.337131\pi$
$284$	2176.35	0.454728
$285$	0	0
$286$	1888.89	0.390532
$287$	5997.04	1.23343
$288$	0	0
$289$	3799.21	0.773297
$290$	0	0
$291$	0	0
$292$	7098.91	1.42271
$293$	1105.95	0.220513	0.110256	−	0.993903i	$-0.464833\pi$
$293$	1105.95	0.220513	0.110256	+	0.993903i	$0.464833\pi$
$294$	0	0
$295$	0	0
$296$	208.136	0.0408705
$297$	0	0
$298$	904.429	0.175813
$299$	−5691.32	−1.10079
$300$	0	0
$301$	3734.02	0.715035
$302$	2019.16	0.384734
$303$	0	0
$304$	−4407.92	−0.831616
$305$	0	0
$306$	0	0
$307$	−8585.39	−1.59607	−0.798036	−	0.602609i	$-0.794128\pi$
$307$	−8585.39	−1.59607	−0.798036	+	0.602609i	$0.794128\pi$
$308$	−1329.15	−0.245893
$309$	0	0
$310$	0	0
$311$	7039.00	1.28342	0.641712	−	0.766945i	$-0.278224\pi$
$311$	7039.00	1.28342	0.641712	+	0.766945i	$0.278224\pi$
$312$	0	0
$313$	5196.42	0.938399	0.469199	−	0.883092i	$-0.344543\pi$
$313$	5196.42	0.938399	0.469199	+	0.883092i	$0.344543\pi$
$314$	8993.89	1.61642
$315$	0	0
$316$	−1771.12	−0.315295
$317$	−556.011	−0.0985133	−0.0492566	−	0.998786i	$-0.515685\pi$
$317$	−556.011	−0.0985133	−0.0492566	+	0.998786i	$0.515685\pi$
$318$	0	0
$319$	−2012.28	−0.353185
$320$	0	0
$321$	0	0
$322$	7723.31	1.33666
$323$	−7009.80	−1.20754
$324$	0	0
$325$	0	0
$326$	−3795.43	−0.644815
$327$	0	0
$328$	1128.32	0.189943
$329$	−452.011	−0.0757452
$330$	0	0
$331$	−509.573	−0.0846184	−0.0423092	−	0.999105i	$-0.513471\pi$
$331$	−509.573	−0.0846184	−0.0423092	+	0.999105i	$0.513471\pi$
$332$	−5570.67	−0.920874
$333$	0	0
$334$	−7029.22	−1.15156
$335$	0	0
$336$	0	0
$337$	783.515	0.126649	0.0633246	−	0.997993i	$-0.479830\pi$
$337$	783.515	0.126649	0.0633246	+	0.997993i	$0.479830\pi$
$338$	−2407.55	−0.387436
$339$	0	0
$340$	0	0
$341$	−2877.34	−0.456940
$342$	0	0
$343$	−6777.76	−1.06695
$344$	702.544	0.110112
$345$	0	0
$346$	9994.95	1.55298
$347$	630.674	0.0975688	0.0487844	−	0.998809i	$-0.484465\pi$
$347$	630.674	0.0975688	0.0487844	+	0.998809i	$0.484465\pi$
$348$	0	0
$349$	387.854	0.0594882	0.0297441	−	0.999558i	$-0.490531\pi$
$349$	387.854	0.0594882	0.0297441	+	0.999558i	$0.490531\pi$
$350$	0	0
$351$	0	0
$352$	2998.34	0.454012
$353$	−2140.74	−0.322776	−0.161388	−	0.986891i	$-0.551597\pi$
$353$	−2140.74	−0.322776	−0.161388	+	0.986891i	$0.551597\pi$
$354$	0	0
$355$	0	0
$356$	11415.5	1.69950
$357$	0	0
$358$	4284.42	0.632510
$359$	−1180.69	−0.173578	−0.0867889	−	0.996227i	$-0.527661\pi$
$359$	−1180.69	−0.173578	−0.0867889	+	0.996227i	$0.527661\pi$
$360$	0	0
$361$	−1218.95	−0.177716
$362$	−10855.3	−1.57608
$363$	0	0
$364$	−4607.59	−0.663471
$365$	0	0
$366$	0	0
$367$	−8058.99	−1.14626	−0.573128	−	0.819466i	$-0.694270\pi$
$367$	−8058.99	−1.14626	−0.573128	+	0.819466i	$0.694270\pi$
$368$	−8334.55	−1.18062
$369$	0	0
$370$	0	0
$371$	5648.35	0.790425
$372$	0	0
$373$	−9866.56	−1.36963	−0.684813	−	0.728718i	$-0.740116\pi$
$373$	−9866.56	−1.36963	−0.684813	+	0.728718i	$0.740116\pi$
$374$	4398.93	0.608191
$375$	0	0
$376$	−85.0443	−0.0116644
$377$	−6975.73	−0.952967
$378$	0	0
$379$	−4799.17	−0.650440	−0.325220	−	0.945638i	$-0.605438\pi$
$379$	−4799.17	−0.650440	−0.325220	+	0.945638i	$0.605438\pi$
$380$	0	0
$381$	0	0
$382$	−9063.93	−1.21401
$383$	5701.76	0.760695	0.380347	−	0.924844i	$-0.375804\pi$
$383$	5701.76	0.760695	0.380347	+	0.924844i	$0.375804\pi$
$384$	0	0
$385$	0	0
$386$	19770.6	2.60699
$387$	0	0
$388$	−4857.35	−0.635553
$389$	8912.17	1.16161	0.580804	−	0.814044i	$-0.302738\pi$
$389$	8912.17	1.16161	0.580804	+	0.814044i	$0.302738\pi$
$390$	0	0
$391$	−13254.2	−1.71431
$392$	−414.135	−0.0533596
$393$	0	0
$394$	1519.06	0.194237
$395$	0	0
$396$	0	0
$397$	−9971.00	−1.26053	−0.630264	−	0.776381i	$-0.717054\pi$
$397$	−9971.00	−1.26053	−0.630264	+	0.776381i	$0.717054\pi$
$398$	20367.4	2.56514
$399$	0	0
$400$	0	0
$401$	710.271	0.0884520	0.0442260	−	0.999022i	$-0.485918\pi$
$401$	710.271	0.0884520	0.0442260	+	0.999022i	$0.485918\pi$
$402$	0	0
$403$	−9974.54	−1.23292
$404$	2684.62	0.330606
$405$	0	0
$406$	9466.30	1.15715
$407$	−958.563	−0.116743
$408$	0	0
$409$	6162.68	0.745048	0.372524	−	0.928023i	$-0.378492\pi$
$409$	6162.68	0.745048	0.372524	+	0.928023i	$0.378492\pi$
$410$	0	0
$411$	0	0
$412$	702.388	0.0839907
$413$	8214.30	0.978691
$414$	0	0
$415$	0	0
$416$	10394.0	1.22502
$417$	0	0
$418$	−3539.36	−0.414152
$419$	13217.7	1.54111	0.770556	−	0.637372i	$-0.219978\pi$
$419$	13217.7	1.54111	0.770556	+	0.637372i	$0.219978\pi$
$420$	0	0
$421$	2760.80	0.319603	0.159802	−	0.987149i	$-0.448915\pi$
$421$	2760.80	0.319603	0.159802	+	0.987149i	$0.448915\pi$
$422$	−2296.57	−0.264918
$423$	0	0
$424$	1062.72	0.121722
$425$	0	0
$426$	0	0
$427$	−6706.28	−0.760046
$428$	−1250.49	−0.141226
$429$	0	0
$430$	0	0
$431$	3057.57	0.341712	0.170856	−	0.985296i	$-0.445347\pi$
$431$	3057.57	0.341712	0.170856	+	0.985296i	$0.445347\pi$
$432$	0	0
$433$	−14533.5	−1.61302	−0.806510	−	0.591221i	$-0.798646\pi$
$433$	−14533.5	−1.61302	−0.806510	+	0.591221i	$0.798646\pi$
$434$	13535.8	1.49709
$435$	0	0
$436$	2812.94	0.308980
$437$	10664.3	1.16737
$438$	0	0
$439$	−1497.89	−0.162848	−0.0814240	−	0.996680i	$-0.525947\pi$
$439$	−1497.89	−0.162848	−0.0814240	+	0.996680i	$0.525947\pi$
$440$	0	0
$441$	0	0
$442$	15249.3	1.64103
$443$	−13363.3	−1.43320	−0.716600	−	0.697484i	$-0.754303\pi$
$443$	−13363.3	−1.43320	−0.716600	+	0.697484i	$0.754303\pi$
$444$	0	0
$445$	0	0
$446$	−21907.2	−2.32587
$447$	0	0
$448$	−7839.83	−0.826779
$449$	15023.0	1.57902	0.789509	−	0.613739i	$-0.210335\pi$
$449$	15023.0	1.57902	0.789509	+	0.613739i	$0.210335\pi$
$450$	0	0
$451$	−5196.45	−0.542553
$452$	−448.207	−0.0466414
$453$	0	0
$454$	−4531.44	−0.468439
$455$	0	0
$456$	0	0
$457$	−14460.7	−1.48018	−0.740091	−	0.672507i	$-0.765218\pi$
$457$	−14460.7	−1.48018	−0.740091	+	0.672507i	$0.765218\pi$
$458$	8306.38	0.847449
$459$	0	0
$460$	0	0
$461$	6397.52	0.646339	0.323170	−	0.946341i	$-0.395252\pi$
$461$	6397.52	0.646339	0.323170	+	0.946341i	$0.395252\pi$
$462$	0	0
$463$	−6228.94	−0.625234	−0.312617	−	0.949879i	$-0.601206\pi$
$463$	−6228.94	−0.625234	−0.312617	+	0.949879i	$0.601206\pi$
$464$	−10215.5	−1.02207
$465$	0	0
$466$	8670.51	0.861917
$467$	1434.12	0.142105	0.0710526	−	0.997473i	$-0.477364\pi$
$467$	1434.12	0.142105	0.0710526	+	0.997473i	$0.477364\pi$
$468$	0	0
$469$	−770.605	−0.0758705
$470$	0	0
$471$	0	0
$472$	1545.49	0.150714
$473$	−3235.54	−0.314525
$474$	0	0
$475$	0	0
$476$	−10730.4	−1.03325
$477$	0	0
$478$	8082.71	0.773419
$479$	12095.5	1.15378	0.576888	−	0.816823i	$-0.304267\pi$
$479$	12095.5	1.15378	0.576888	+	0.816823i	$0.304267\pi$
$480$	0	0
$481$	−3322.94	−0.314996
$482$	−17200.5	−1.62544
$483$	0	0
$484$	−10316.0	−0.968822
$485$	0	0
$486$	0	0
$487$	−9600.57	−0.893313	−0.446656	−	0.894706i	$-0.647385\pi$
$487$	−9600.57	−0.893313	−0.446656	+	0.894706i	$0.647385\pi$
$488$	−1261.76	−0.117044
$489$	0	0
$490$	0	0
$491$	2091.41	0.192228	0.0961141	−	0.995370i	$-0.469359\pi$
$491$	2091.41	0.192228	0.0961141	+	0.995370i	$0.469359\pi$
$492$	0	0
$493$	−16245.4	−1.48409
$494$	−12269.5	−1.11747
$495$	0	0
$496$	−14607.0	−1.32233
$497$	3370.41	0.304192
$498$	0	0
$499$	−18595.9	−1.66827	−0.834133	−	0.551563i	$-0.814032\pi$
$499$	−18595.9	−1.66827	−0.834133	+	0.551563i	$0.814032\pi$
$500$	0	0
$501$	0	0
$502$	6679.27	0.593845
$503$	5038.79	0.446657	0.223329	−	0.974743i	$-0.428308\pi$
$503$	5038.79	0.446657	0.223329	+	0.974743i	$0.428308\pi$
$504$	0	0
$505$	0	0
$506$	−6692.26	−0.587959
$507$	0	0
$508$	−2291.89	−0.200170
$509$	−6501.99	−0.566200	−0.283100	−	0.959090i	$-0.591363\pi$
$509$	−6501.99	−0.566200	−0.283100	+	0.959090i	$0.591363\pi$
$510$	0	0
$511$	10993.7	0.951729
$512$	16400.1	1.41560
$513$	0	0
$514$	25139.6	2.15731
$515$	0	0
$516$	0	0
$517$	391.668	0.0333183
$518$	4509.34	0.382488
$519$	0	0
$520$	0	0
$521$	3689.43	0.310244	0.155122	−	0.987895i	$-0.450423\pi$
$521$	3689.43	0.310244	0.155122	+	0.987895i	$0.450423\pi$
$522$	0	0
$523$	5040.68	0.421441	0.210720	−	0.977546i	$-0.432419\pi$
$523$	5040.68	0.421441	0.210720	+	0.977546i	$0.432419\pi$
$524$	−21904.9	−1.82619
$525$	0	0
$526$	−25932.7	−2.14965
$527$	−23229.2	−1.92007
$528$	0	0
$529$	7997.15	0.657282
$530$	0	0
$531$	0	0
$532$	8633.61	0.703599
$533$	−18013.9	−1.46392
$534$	0	0
$535$	0	0
$536$	−144.987	−0.0116837
$537$	0	0
$538$	−9802.75	−0.785552
$539$	1907.28	0.152416
$540$	0	0
$541$	744.536	0.0591684	0.0295842	−	0.999562i	$-0.490582\pi$
$541$	744.536	0.0591684	0.0295842	+	0.999562i	$0.490582\pi$
$542$	28478.6	2.25694
$543$	0	0
$544$	24206.1	1.90777
$545$	0	0
$546$	0	0
$547$	−6401.69	−0.500396	−0.250198	−	0.968195i	$-0.580496\pi$
$547$	−6401.69	−0.500396	−0.250198	+	0.968195i	$0.580496\pi$
$548$	−1064.02	−0.0829428
$549$	0	0
$550$	0	0
$551$	13071.0	1.01060
$552$	0	0
$553$	−2742.84	−0.210918
$554$	6122.60	0.469538
$555$	0	0
$556$	1853.47	0.141375
$557$	6623.77	0.503874	0.251937	−	0.967744i	$-0.418932\pi$
$557$	6623.77	0.503874	0.251937	+	0.967744i	$0.418932\pi$
$558$	0	0
$559$	−11216.3	−0.848653
$560$	0	0
$561$	0	0
$562$	5882.29	0.441512
$563$	−13355.7	−0.999778	−0.499889	−	0.866089i	$-0.666626\pi$
$563$	−13355.7	−0.999778	−0.499889	+	0.866089i	$0.666626\pi$
$564$	0	0
$565$	0	0
$566$	19003.8	1.41129
$567$	0	0
$568$	634.131	0.0468443
$569$	20926.1	1.54177	0.770887	−	0.636972i	$-0.219813\pi$
$569$	20926.1	1.54177	0.770887	+	0.636972i	$0.219813\pi$
$570$	0	0
$571$	25369.8	1.85936	0.929679	−	0.368370i	$-0.120084\pi$
$571$	25369.8	1.85936	0.929679	+	0.368370i	$0.120084\pi$
$572$	3992.49	0.291843
$573$	0	0
$574$	24445.5	1.77759
$575$	0	0
$576$	0	0
$577$	−2474.30	−0.178521	−0.0892605	−	0.996008i	$-0.528450\pi$
$577$	−2474.30	−0.178521	−0.0892605	+	0.996008i	$0.528450\pi$
$578$	15486.6	1.11446
$579$	0	0
$580$	0	0
$581$	−8627.01	−0.616022
$582$	0	0
$583$	−4894.30	−0.347687
$584$	2068.43	0.146562
$585$	0	0
$586$	4508.13	0.317797
$587$	−15752.6	−1.10763	−0.553817	−	0.832639i	$-0.686829\pi$
$587$	−15752.6	−1.10763	−0.553817	+	0.832639i	$0.686829\pi$
$588$	0	0
$589$	18690.1	1.30749
$590$	0	0
$591$	0	0
$592$	−4866.22	−0.337838
$593$	−6162.03	−0.426719	−0.213360	−	0.976974i	$-0.568441\pi$
$593$	−6162.03	−0.426719	−0.213360	+	0.976974i	$0.568441\pi$
$594$	0	0
$595$	0	0
$596$	1911.67	0.131384
$597$	0	0
$598$	−23199.3	−1.58644
$599$	−7814.07	−0.533012	−0.266506	−	0.963833i	$-0.585869\pi$
$599$	−7814.07	−0.533012	−0.266506	+	0.963833i	$0.585869\pi$
$600$	0	0
$601$	−24315.4	−1.65033	−0.825164	−	0.564893i	$-0.808917\pi$
$601$	−24315.4	−1.65033	−0.825164	+	0.564893i	$0.808917\pi$
$602$	15220.8	1.03049
$603$	0	0
$604$	4267.85	0.287510
$605$	0	0
$606$	0	0
$607$	−1586.65	−0.106096	−0.0530478	−	0.998592i	$-0.516894\pi$
$607$	−1586.65	−0.106096	−0.0530478	+	0.998592i	$0.516894\pi$
$608$	−19476.1	−1.29911
$609$	0	0
$610$	0	0
$611$	1357.75	0.0898997
$612$	0	0
$613$	−12856.4	−0.847089	−0.423545	−	0.905875i	$-0.639214\pi$
$613$	−12856.4	−0.847089	−0.423545	+	0.905875i	$0.639214\pi$
$614$	−34996.3	−2.30022
$615$	0	0
$616$	−387.277	−0.0253309
$617$	3854.46	0.251499	0.125749	−	0.992062i	$-0.459867\pi$
$617$	3854.46	0.251499	0.125749	+	0.992062i	$0.459867\pi$
$618$	0	0
$619$	5644.03	0.366482	0.183241	−	0.983068i	$-0.441341\pi$
$619$	5644.03	0.366482	0.183241	+	0.983068i	$0.441341\pi$
$620$	0	0
$621$	0	0
$622$	28692.8	1.84964
$623$	17678.7	1.13689
$624$	0	0
$625$	0	0
$626$	21181.9	1.35240
$627$	0	0
$628$	19010.1	1.20794
$629$	−7738.62	−0.490555
$630$	0	0
$631$	−6379.75	−0.402494	−0.201247	−	0.979541i	$-0.564499\pi$
$631$	−6379.75	−0.402494	−0.201247	+	0.979541i	$0.564499\pi$
$632$	−516.057	−0.0324804
$633$	0	0
$634$	−2266.44	−0.141975
$635$	0	0
$636$	0	0
$637$	6611.75	0.411251
$638$	−8202.57	−0.509001
$639$	0	0
$640$	0	0
$641$	−19543.9	−1.20427	−0.602135	−	0.798395i	$-0.705683\pi$
$641$	−19543.9	−1.20427	−0.602135	+	0.798395i	$0.705683\pi$
$642$	0	0
$643$	26175.3	1.60537	0.802686	−	0.596401i	$-0.203403\pi$
$643$	26175.3	1.60537	0.802686	+	0.596401i	$0.203403\pi$
$644$	16324.5	0.998877
$645$	0	0
$646$	−28573.7	−1.74028
$647$	−20349.8	−1.23653	−0.618264	−	0.785971i	$-0.712164\pi$
$647$	−20349.8	−1.23653	−0.618264	+	0.785971i	$0.712164\pi$
$648$	0	0
$649$	−7117.71	−0.430500
$650$	0	0
$651$	0	0
$652$	−8022.30	−0.481867
$653$	21661.4	1.29813	0.649064	−	0.760734i	$-0.275161\pi$
$653$	21661.4	1.29813	0.649064	+	0.760734i	$0.275161\pi$
$654$	0	0
$655$	0	0
$656$	−26380.2	−1.57008
$657$	0	0
$658$	−1842.51	−0.109162
$659$	−9447.97	−0.558484	−0.279242	−	0.960221i	$-0.590083\pi$
$659$	−9447.97	−0.558484	−0.279242	+	0.960221i	$0.590083\pi$
$660$	0	0
$661$	−14301.8	−0.841569	−0.420784	−	0.907161i	$-0.638245\pi$
$661$	−14301.8	−0.841569	−0.420784	+	0.907161i	$0.638245\pi$
$662$	−2077.15	−0.121950
$663$	0	0
$664$	−1623.14	−0.0948647
$665$	0	0
$666$	0	0
$667$	24714.8	1.43472
$668$	−14857.5	−0.860558
$669$	0	0
$670$	0	0
$671$	5811.00	0.334324
$672$	0	0
$673$	13078.6	0.749096	0.374548	−	0.927208i	$-0.377798\pi$
$673$	13078.6	0.749096	0.374548	+	0.927208i	$0.377798\pi$
$674$	3193.81	0.182524
$675$	0	0
$676$	−5088.77	−0.289529
$677$	15622.6	0.886891	0.443446	−	0.896301i	$-0.353756\pi$
$677$	15622.6	0.886891	0.443446	+	0.896301i	$0.353756\pi$
$678$	0	0
$679$	−7522.33	−0.425156
$680$	0	0
$681$	0	0
$682$	−11728.8	−0.658531
$683$	3176.18	0.177940	0.0889701	−	0.996034i	$-0.471642\pi$
$683$	3176.18	0.177940	0.0889701	+	0.996034i	$0.471642\pi$
$684$	0	0
$685$	0	0
$686$	−27627.9	−1.53766
$687$	0	0
$688$	−16425.4	−0.910195
$689$	−16966.5	−0.938132
$690$	0	0
$691$	30731.4	1.69186	0.845931	−	0.533293i	$-0.179046\pi$
$691$	30731.4	1.69186	0.845931	+	0.533293i	$0.179046\pi$
$692$	21126.0	1.16054
$693$	0	0
$694$	2570.79	0.140614
$695$	0	0
$696$	0	0
$697$	−41951.7	−2.27982
$698$	1580.99	0.0857328
$699$	0	0
$700$	0	0
$701$	−18536.2	−0.998718	−0.499359	−	0.866395i	$-0.666431\pi$
$701$	−18536.2	−0.998718	−0.499359	+	0.866395i	$0.666431\pi$
$702$	0	0
$703$	6226.45	0.334047
$704$	6793.22	0.363678
$705$	0	0
$706$	−8726.19	−0.465176
$707$	4157.54	0.221160
$708$	0	0
$709$	−16619.3	−0.880325	−0.440162	−	0.897918i	$-0.645079\pi$
$709$	−16619.3	−0.880325	−0.440162	+	0.897918i	$0.645079\pi$
$710$	0	0
$711$	0	0
$712$	3326.18	0.175076
$713$	35339.4	1.85620
$714$	0	0
$715$	0	0
$716$	9055.86	0.472672
$717$	0	0
$718$	−4812.80	−0.250156
$719$	−10245.8	−0.531441	−0.265720	−	0.964050i	$-0.585610\pi$
$719$	−10245.8	−0.531441	−0.265720	+	0.964050i	$0.585610\pi$
$720$	0	0
$721$	1087.75	0.0561859
$722$	−4968.77	−0.256120
$723$	0	0
$724$	−22944.6	−1.17780
$725$	0	0
$726$	0	0
$727$	−30936.5	−1.57823	−0.789114	−	0.614247i	$-0.789460\pi$
$727$	−30936.5	−1.57823	−0.789114	+	0.614247i	$0.789460\pi$
$728$	−1342.53	−0.0683481
$729$	0	0
$730$	0	0
$731$	−26121.0	−1.32164
$732$	0	0
$733$	−3336.77	−0.168140	−0.0840698	−	0.996460i	$-0.526792\pi$
$733$	−3336.77	−0.168140	−0.0840698	+	0.996460i	$0.526792\pi$
$734$	−32850.5	−1.65195
$735$	0	0
$736$	−36825.6	−1.84431
$737$	667.731	0.0333734
$738$	0	0
$739$	−23494.6	−1.16950	−0.584752	−	0.811212i	$-0.698808\pi$
$739$	−23494.6	−1.16950	−0.584752	+	0.811212i	$0.698808\pi$
$740$	0	0
$741$	0	0
$742$	23024.1	1.13914
$743$	−2905.75	−0.143475	−0.0717373	−	0.997424i	$-0.522854\pi$
$743$	−2905.75	−0.143475	−0.0717373	+	0.997424i	$0.522854\pi$
$744$	0	0
$745$	0	0
$746$	−40218.6	−1.97387
$747$	0	0
$748$	9297.89	0.454498
$749$	−1936.57	−0.0944735
$750$	0	0
$751$	30691.2	1.49126	0.745632	−	0.666358i	$-0.232148\pi$
$751$	30691.2	1.49126	0.745632	+	0.666358i	$0.232148\pi$
$752$	1988.33	0.0964189
$753$	0	0
$754$	−28434.9	−1.37339
$755$	0	0
$756$	0	0
$757$	−35875.7	−1.72249	−0.861244	−	0.508192i	$-0.830314\pi$
$757$	−35875.7	−1.72249	−0.861244	+	0.508192i	$0.830314\pi$
$758$	−19562.6	−0.937397
$759$	0	0
$760$	0	0
$761$	−26842.1	−1.27861	−0.639307	−	0.768951i	$-0.720779\pi$
$761$	−26842.1	−1.27861	−0.639307	+	0.768951i	$0.720779\pi$
$762$	0	0
$763$	4356.26	0.206694
$764$	−19158.2	−0.907223
$765$	0	0
$766$	23241.8	1.09629
$767$	−24674.1	−1.16158
$768$	0	0
$769$	12755.2	0.598133	0.299067	−	0.954232i	$-0.403325\pi$
$769$	12755.2	0.598133	0.299067	+	0.954232i	$0.403325\pi$
$770$	0	0
$771$	0	0
$772$	41788.6	1.94819
$773$	9984.62	0.464582	0.232291	−	0.972646i	$-0.425378\pi$
$773$	9984.62	0.464582	0.232291	+	0.972646i	$0.425378\pi$
$774$	0	0
$775$	0	0
$776$	−1415.30	−0.0654721
$777$	0	0
$778$	36328.3	1.67408
$779$	33754.1	1.55246
$780$	0	0
$781$	−2920.46	−0.133806
$782$	−54027.6	−2.47062
$783$	0	0
$784$	9682.45	0.441074
$785$	0	0
$786$	0	0
$787$	35676.2	1.61591	0.807954	−	0.589245i	$-0.200575\pi$
$787$	35676.2	1.61591	0.807954	+	0.589245i	$0.200575\pi$
$788$	3210.80	0.145152
$789$	0	0
$790$	0	0
$791$	−694.116	−0.0312009
$792$	0	0
$793$	20144.3	0.902076
$794$	−40644.3	−1.81664
$795$	0	0
$796$	43050.0	1.91692
$797$	−41560.5	−1.84711	−0.923555	−	0.383465i	$-0.874731\pi$
$797$	−41560.5	−1.84711	−0.923555	+	0.383465i	$0.874731\pi$
$798$	0	0
$799$	3161.99	0.140004
$800$	0	0
$801$	0	0
$802$	2895.25	0.127475
$803$	−9526.08	−0.418640
$804$	0	0
$805$	0	0
$806$	−40658.8	−1.77685
$807$	0	0
$808$	782.227	0.0340577
$809$	−12100.2	−0.525861	−0.262931	−	0.964815i	$-0.584689\pi$
$809$	−12100.2	−0.525861	−0.262931	+	0.964815i	$0.584689\pi$
$810$	0	0
$811$	30794.9	1.33336	0.666680	−	0.745344i	$-0.267715\pi$
$811$	30794.9	1.33336	0.666680	+	0.745344i	$0.267715\pi$
$812$	20008.7	0.864736
$813$	0	0
$814$	−3907.35	−0.168246
$815$	0	0
$816$	0	0
$817$	21016.8	0.899981
$818$	25120.7	1.07374
$819$	0	0
$820$	0	0
$821$	32746.1	1.39202	0.696009	−	0.718033i	$-0.254958\pi$
$821$	32746.1	1.39202	0.696009	+	0.718033i	$0.254958\pi$
$822$	0	0
$823$	19816.1	0.839303	0.419652	−	0.907685i	$-0.362152\pi$
$823$	19816.1	0.839303	0.419652	+	0.907685i	$0.362152\pi$
$824$	204.657	0.00865238
$825$	0	0
$826$	33483.6	1.41046
$827$	37093.9	1.55971	0.779855	−	0.625960i	$-0.215293\pi$
$827$	37093.9	1.55971	0.779855	+	0.625960i	$0.215293\pi$
$828$	0	0
$829$	−2960.66	−0.124039	−0.0620193	−	0.998075i	$-0.519754\pi$
$829$	−2960.66	−0.124039	−0.0620193	+	0.998075i	$0.519754\pi$
$830$	0	0
$831$	0	0
$832$	23549.3	0.981279
$833$	15397.8	0.640457
$834$	0	0
$835$	0	0
$836$	−7481.04	−0.309494
$837$	0	0
$838$	53878.7	2.22101
$839$	8121.13	0.334175	0.167087	−	0.985942i	$-0.446564\pi$
$839$	8121.13	0.334175	0.167087	+	0.985942i	$0.446564\pi$
$840$	0	0
$841$	5903.40	0.242052
$842$	11253.7	0.460604
$843$	0	0
$844$	−4854.20	−0.197972
$845$	0	0
$846$	0	0
$847$	−15975.9	−0.648097
$848$	−24846.3	−1.00616
$849$	0	0
$850$	0	0
$851$	11773.1	0.474237
$852$	0	0
$853$	−13277.5	−0.532957	−0.266478	−	0.963841i	$-0.585860\pi$
$853$	−13277.5	−0.532957	−0.266478	+	0.963841i	$0.585860\pi$
$854$	−27336.5	−1.09536
$855$	0	0
$856$	−364.359	−0.0145485
$857$	−17722.7	−0.706411	−0.353206	−	0.935546i	$-0.614908\pi$
$857$	−17722.7	−0.706411	−0.353206	+	0.935546i	$0.614908\pi$
$858$	0	0
$859$	9026.72	0.358542	0.179271	−	0.983800i	$-0.442626\pi$
$859$	9026.72	0.358542	0.179271	+	0.983800i	$0.442626\pi$
$860$	0	0
$861$	0	0
$862$	12463.4	0.492467
$863$	10197.6	0.402235	0.201117	−	0.979567i	$-0.435543\pi$
$863$	10197.6	0.402235	0.201117	+	0.979567i	$0.435543\pi$
$864$	0	0
$865$	0	0
$866$	−59242.4	−2.32464
$867$	0	0
$868$	28610.2	1.11877
$869$	2376.68	0.0927771
$870$	0	0
$871$	2314.74	0.0900484
$872$	819.616	0.0318299
$873$	0	0
$874$	43470.3	1.68239
$875$	0	0
$876$	0	0
$877$	−13326.7	−0.513127	−0.256563	−	0.966527i	$-0.582590\pi$
$877$	−13326.7	−0.513127	−0.256563	+	0.966527i	$0.582590\pi$
$878$	−6105.78	−0.234692
$879$	0	0
$880$	0	0
$881$	15016.4	0.574252	0.287126	−	0.957893i	$-0.407300\pi$
$881$	15016.4	0.574252	0.287126	+	0.957893i	$0.407300\pi$
$882$	0	0
$883$	6981.59	0.266081	0.133040	−	0.991111i	$-0.457526\pi$
$883$	6981.59	0.266081	0.133040	+	0.991111i	$0.457526\pi$
$884$	32231.9	1.22633
$885$	0	0
$886$	−54472.1	−2.06549
$887$	25728.2	0.973920	0.486960	−	0.873424i	$-0.338106\pi$
$887$	25728.2	0.973920	0.486960	+	0.873424i	$0.338106\pi$
$888$	0	0
$889$	−3549.34	−0.133904
$890$	0	0
$891$	0	0
$892$	−46304.7	−1.73811
$893$	−2544.12	−0.0953369
$894$	0	0
$895$	0	0
$896$	−4274.86	−0.159389
$897$	0	0
$898$	61237.6	2.27564
$899$	43314.8	1.60693
$900$	0	0
$901$	−39512.4	−1.46099
$902$	−21182.1	−0.781913
$903$	0	0
$904$	−130.596	−0.00480481
$905$	0	0
$906$	0	0
$907$	−17952.3	−0.657218	−0.328609	−	0.944466i	$-0.606580\pi$
$907$	−17952.3	−0.657218	−0.328609	+	0.944466i	$0.606580\pi$
$908$	−9577.98	−0.350062
$909$	0	0
$910$	0	0
$911$	12037.2	0.437773	0.218886	−	0.975750i	$-0.429758\pi$
$911$	12037.2	0.437773	0.218886	+	0.975750i	$0.429758\pi$
$912$	0	0
$913$	7475.32	0.270971
$914$	−58945.5	−2.13320
$915$	0	0
$916$	17557.0	0.633295
$917$	−33923.1	−1.22163
$918$	0	0
$919$	−14182.6	−0.509075	−0.254537	−	0.967063i	$-0.581923\pi$
$919$	−14182.6	−0.509075	−0.254537	+	0.967063i	$0.581923\pi$
$920$	0	0
$921$	0	0
$922$	26077.9	0.931487
$923$	−10124.0	−0.361037
$924$	0	0
$925$	0	0
$926$	−25390.8	−0.901071
$927$	0	0
$928$	−45136.4	−1.59663
$929$	30537.9	1.07849	0.539244	−	0.842150i	$-0.318710\pi$
$929$	30537.9	1.07849	0.539244	+	0.842150i	$0.318710\pi$
$930$	0	0
$931$	−12389.0	−0.436124
$932$	18326.6	0.644107
$933$	0	0
$934$	5845.84	0.204798
$935$	0	0
$936$	0	0
$937$	26958.0	0.939892	0.469946	−	0.882695i	$-0.344273\pi$
$937$	26958.0	0.939892	0.469946	+	0.882695i	$0.344273\pi$
$938$	−3141.18	−0.109343
$939$	0	0
$940$	0	0
$941$	−16063.4	−0.556486	−0.278243	−	0.960511i	$-0.589752\pi$
$941$	−16063.4	−0.556486	−0.278243	+	0.960511i	$0.589752\pi$
$942$	0	0
$943$	63822.7	2.20398
$944$	−36133.6	−1.24581
$945$	0	0
$946$	−13188.9	−0.453285
$947$	38504.8	1.32127	0.660633	−	0.750709i	$-0.270288\pi$
$947$	38504.8	1.32127	0.660633	+	0.750709i	$0.270288\pi$
$948$	0	0
$949$	−33022.9	−1.12958
$950$	0	0
$951$	0	0
$952$	−3126.55	−0.106441
$953$	−48509.6	−1.64888	−0.824438	−	0.565952i	$-0.808509\pi$
$953$	−48509.6	−1.64888	−0.824438	+	0.565952i	$0.808509\pi$
$954$	0	0
$955$	0	0
$956$	17084.2	0.577973
$957$	0	0
$958$	49304.5	1.66279
$959$	−1647.79	−0.0554849
$960$	0	0
$961$	32144.5	1.07900
$962$	−13545.2	−0.453964
$963$	0	0
$964$	−36356.2	−1.21468
$965$	0	0
$966$	0	0
$967$	−30364.0	−1.00976	−0.504882	−	0.863188i	$-0.668464\pi$
$967$	−30364.0	−1.00976	−0.504882	+	0.863188i	$0.668464\pi$
$968$	−3005.81	−0.0998041
$969$	0	0
$970$	0	0
$971$	15199.6	0.502346	0.251173	−	0.967942i	$-0.419184\pi$
$971$	15199.6	0.502346	0.251173	+	0.967942i	$0.419184\pi$
$972$	0	0
$973$	2870.37	0.0945734
$974$	−39134.4	−1.28742
$975$	0	0
$976$	29500.0	0.967491
$977$	59956.5	1.96333	0.981667	−	0.190605i	$-0.0610450\pi$
$977$	59956.5	1.96333	0.981667	+	0.190605i	$0.0610450\pi$
$978$	0	0
$979$	−15318.6	−0.500086
$980$	0	0
$981$	0	0
$982$	8525.13	0.277034
$983$	−10733.0	−0.348250	−0.174125	−	0.984724i	$-0.555710\pi$
$983$	−10733.0	−0.348250	−0.174125	+	0.984724i	$0.555710\pi$
$984$	0	0
$985$	0	0
$986$	−66220.5	−2.13883
$987$	0	0
$988$	−25933.6	−0.835080
$989$	39738.8	1.27768
$990$	0	0
$991$	30040.3	0.962928	0.481464	−	0.876466i	$-0.340105\pi$
$991$	30040.3	0.962928	0.481464	+	0.876466i	$0.340105\pi$
$992$	−64540.1	−2.06568
$993$	0	0
$994$	13738.7	0.438394
$995$	0	0
$996$	0	0
$997$	59325.1	1.88450	0.942249	−	0.334912i	$-0.108707\pi$
$997$	59325.1	1.88450	0.942249	+	0.334912i	$0.108707\pi$
$998$	−75801.5	−2.40426
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bl.1.14		16
3.2	odd	2		2025.4.a.bk.1.3		16
5.2	odd	4		405.4.b.f.244.14		16
5.3	odd	4		405.4.b.f.244.3		16
5.4	even	2	inner	2025.4.a.bl.1.3		16
9.2	odd	6		225.4.e.g.76.14		32
9.5	odd	6		225.4.e.g.151.14		32
15.2	even	4		405.4.b.e.244.3		16
15.8	even	4		405.4.b.e.244.14		16
15.14	odd	2		2025.4.a.bk.1.14		16
45.2	even	12		45.4.j.a.4.3	✓	32
45.7	odd	12		135.4.j.a.64.14		32
45.13	odd	12		135.4.j.a.19.14		32
45.14	odd	6		225.4.e.g.151.3		32
45.22	odd	12		135.4.j.a.19.3		32
45.23	even	12		45.4.j.a.34.3	yes	32
45.29	odd	6		225.4.e.g.76.3		32
45.32	even	12		45.4.j.a.34.14	yes	32
45.38	even	12		45.4.j.a.4.14	yes	32
45.43	odd	12		135.4.j.a.64.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.3	✓	32	45.2	even	12
45.4.j.a.4.14	yes	32	45.38	even	12
45.4.j.a.34.3	yes	32	45.23	even	12
45.4.j.a.34.14	yes	32	45.32	even	12
135.4.j.a.19.3		32	45.22	odd	12
135.4.j.a.19.14		32	45.13	odd	12
135.4.j.a.64.3		32	45.43	odd	12
135.4.j.a.64.14		32	45.7	odd	12
225.4.e.g.76.3		32	45.29	odd	6
225.4.e.g.76.14		32	9.2	odd	6
225.4.e.g.151.3		32	45.14	odd	6
225.4.e.g.151.14		32	9.5	odd	6
405.4.b.e.244.3		16	15.2	even	4
405.4.b.e.244.14		16	15.8	even	4
405.4.b.f.244.3		16	5.3	odd	4
405.4.b.f.244.14		16	5.2	odd	4
2025.4.a.bk.1.3		16	3.2	odd	2
2025.4.a.bk.1.14		16	15.14	odd	2
2025.4.a.bl.1.3		16	5.4	even	2	inner
2025.4.a.bl.1.14		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)

\(f(q)\)	\(=\)	\(q+4.07626 q^{2} +8.61587 q^{4} +13.3430 q^{7} +2.51043 q^{8} +O(q^{10})\)	\(q+4.07626 q^{2} +8.61587 q^{4} +13.3430 q^{7} +2.51043 q^{8} -11.5617 q^{11} -40.0796 q^{13} +54.3893 q^{14} -58.6938 q^{16} -93.3392 q^{17} +75.1002 q^{19} -47.1284 q^{22} +142.001 q^{23} -163.375 q^{26} +114.961 q^{28} +174.047 q^{29} +248.868 q^{31} -259.334 q^{32} -380.475 q^{34} +82.9086 q^{37} +306.128 q^{38} +449.454 q^{41} +279.850 q^{43} -99.6140 q^{44} +578.831 q^{46} -33.8764 q^{47} -164.966 q^{49} -345.320 q^{52} +423.321 q^{53} +33.4966 q^{56} +709.461 q^{58} +615.628 q^{59} -502.608 q^{61} +1014.45 q^{62} -587.563 q^{64} -57.7537 q^{67} -804.198 q^{68} +252.598 q^{71} +823.934 q^{73} +337.957 q^{74} +647.054 q^{76} -154.267 q^{77} -205.565 q^{79} +1832.09 q^{82} -646.559 q^{83} +1140.74 q^{86} -29.0248 q^{88} +1324.94 q^{89} -534.780 q^{91} +1223.46 q^{92} -138.089 q^{94} -563.768 q^{97} -672.442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 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

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Database

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