LMFDB - Embedded newform 2025.4.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,-1,0,1,0,0,-9,-9,0,0,37,0,112,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q-3.37228 q^{2} +3.37228 q^{4} -1.62772 q^{7} +15.6060 q^{8} +32.8614 q^{11} +33.0217 q^{13} +5.48913 q^{14} -79.6060 q^{16} +110.307 q^{17} -54.3070 q^{19} -110.818 q^{22} -67.4375 q^{23} -111.359 q^{26} -5.48913 q^{28} -274.519 q^{29} -6.00000 q^{31} +143.606 q^{32} -371.986 q^{34} +347.723 q^{37} +183.139 q^{38} -291.337 q^{41} -201.388 q^{43} +110.818 q^{44} +227.418 q^{46} -482.095 q^{47} -340.351 q^{49} +111.359 q^{52} -175.228 q^{53} -25.4021 q^{56} +925.755 q^{58} +183.242 q^{59} +436.595 q^{61} +20.2337 q^{62} +152.568 q^{64} +831.500 q^{67} +371.986 q^{68} +118.951 q^{71} -183.318 q^{73} -1172.62 q^{74} -183.139 q^{76} -53.4891 q^{77} -638.293 q^{79} +982.470 q^{82} -1494.64 q^{83} +679.139 q^{86} +512.834 q^{88} +1437.27 q^{89} -53.7501 q^{91} -227.418 q^{92} +1625.76 q^{94} +891.769 q^{97} +1147.76 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + q^{4} - 9 q^{7} - 9 q^{8} + 37 q^{11} + 112 q^{13} - 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} - 101 q^{22} - 267 q^{23} + 76 q^{26} + 12 q^{28} - 325 q^{29} - 12 q^{31} + 247 q^{32}+ \cdots + 463 q^{98}+O(q^{100}) $ 2 * q - q^2 + q^4 - 9 * q^7 - 9 * q^8 + 37 * q^11 + 112 * q^13 - 12 * q^14 - 119 * q^16 + 77 * q^17 + 35 * q^19 - 101 * q^22 - 267 * q^23 + 76 * q^26 + 12 * q^28 - 325 * q^29 - 12 * q^31 + 247 * q^32 - 451 * q^34 + 638 * q^37 + 395 * q^38 - 238 * q^41 + 97 * q^43 + 101 * q^44 - 246 * q^46 - 901 * q^47 - 629 * q^49 - 76 * q^52 + 224 * q^53 + 156 * q^56 + 806 * q^58 + 85 * q^59 - 247 * q^61 + 6 * q^62 + 713 * q^64 + 606 * q^67 + 451 * q^68 - 394 * q^71 + 811 * q^73 - 484 * q^74 - 395 * q^76 - 84 * q^77 - 840 * q^79 + 1109 * q^82 - 387 * q^83 + 1387 * q^86 + 411 * q^88 + 1065 * q^89 - 636 * q^91 + 246 * q^92 + 632 * q^94 + 1031 * q^97 + 463 * q^98

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 $

$2$	−3.37228	−1.19228	−0.596141	−	0.802880i	$-0.703300\pi$
$2$	−3.37228	−1.19228	−0.596141	+	0.802880i	$0.703300\pi$
$3$	0	0
$3$	0	0
$4$	3.37228	0.421535
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.62772	−0.0878885	−0.0439443	−	0.999034i	$-0.513992\pi$
$7$	−1.62772	−0.0878885	−0.0439443	+	0.999034i	$0.513992\pi$
$8$	15.6060	0.689693
$9$	0	0
$10$	0	0
$11$	32.8614	0.900735	0.450368	−	0.892843i	$-0.351293\pi$
$11$	32.8614	0.900735	0.450368	+	0.892843i	$0.351293\pi$
$12$	0	0
$13$	33.0217	0.704507	0.352253	−	0.935905i	$-0.385416\pi$
$13$	33.0217	0.704507	0.352253	+	0.935905i	$0.385416\pi$
$14$	5.48913	0.104788
$15$	0	0
$16$	−79.6060	−1.24384
$17$	110.307	1.57373	0.786864	−	0.617126i	$-0.211703\pi$
$17$	110.307	1.57373	0.786864	+	0.617126i	$0.211703\pi$
$18$	0	0
$19$	−54.3070	−0.655731	−0.327865	−	0.944724i	$-0.606329\pi$
$19$	−54.3070	−0.655731	−0.327865	+	0.944724i	$0.606329\pi$
$20$	0	0
$21$	0	0
$22$	−110.818	−1.07393
$23$	−67.4375	−0.611378	−0.305689	−	0.952131i	$-0.598887\pi$
$23$	−67.4375	−0.611378	−0.305689	+	0.952131i	$0.598887\pi$
$24$	0	0
$25$	0	0
$26$	−111.359	−0.839970
$27$	0	0
$28$	−5.48913	−0.0370481
$29$	−274.519	−1.75782	−0.878912	−	0.476984i	$-0.841730\pi$
$29$	−274.519	−1.75782	−0.878912	+	0.476984i	$0.841730\pi$
$30$	0	0
$31$	−6.00000	−0.0347623	−0.0173812	−	0.999849i	$-0.505533\pi$
$31$	−6.00000	−0.0347623	−0.0173812	+	0.999849i	$0.505533\pi$
$32$	143.606	0.793318
$33$	0	0
$34$	−371.986	−1.87633
$35$	0	0
$36$	0	0
$37$	347.723	1.54501	0.772504	−	0.635010i	$-0.219004\pi$
$37$	347.723	1.54501	0.772504	+	0.635010i	$0.219004\pi$
$38$	183.139	0.781816
$39$	0	0
$40$	0	0
$41$	−291.337	−1.10974	−0.554868	−	0.831938i	$-0.687231\pi$
$41$	−291.337	−1.10974	−0.554868	+	0.831938i	$0.687231\pi$
$42$	0	0
$43$	−201.388	−0.714220	−0.357110	−	0.934062i	$-0.616238\pi$
$43$	−201.388	−0.714220	−0.357110	+	0.934062i	$0.616238\pi$
$44$	110.818	0.379692
$45$	0	0
$46$	227.418	0.728935
$47$	−482.095	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$47$	−482.095	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$48$	0	0
$49$	−340.351	−0.992276
$50$	0	0
$51$	0	0
$52$	111.359	0.296974
$53$	−175.228	−0.454140	−0.227070	−	0.973878i	$-0.572915\pi$
$53$	−175.228	−0.454140	−0.227070	+	0.973878i	$0.572915\pi$
$54$	0	0
$55$	0	0
$56$	−25.4021	−0.0606161
$57$	0	0
$58$	925.755	2.09582
$59$	183.242	0.404340	0.202170	−	0.979350i	$-0.435201\pi$
$59$	183.242	0.404340	0.202170	+	0.979350i	$0.435201\pi$
$60$	0	0
$61$	436.595	0.916397	0.458199	−	0.888850i	$-0.348495\pi$
$61$	436.595	0.916397	0.458199	+	0.888850i	$0.348495\pi$
$62$	20.2337	0.0414465
$63$	0	0
$64$	152.568	0.297984
$65$	0	0
$66$	0	0
$67$	831.500	1.51618	0.758089	−	0.652152i	$-0.226133\pi$
$67$	831.500	1.51618	0.758089	+	0.652152i	$0.226133\pi$
$68$	371.986	0.663382
$69$	0	0
$70$	0	0
$71$	118.951	0.198829	0.0994146	−	0.995046i	$-0.468303\pi$
$71$	118.951	0.198829	0.0994146	+	0.995046i	$0.468303\pi$
$72$	0	0
$73$	−183.318	−0.293914	−0.146957	−	0.989143i	$-0.546948\pi$
$73$	−183.318	−0.293914	−0.146957	+	0.989143i	$0.546948\pi$
$74$	−1172.62	−1.84208
$75$	0	0
$76$	−183.139	−0.276414
$77$	−53.4891	−0.0791643
$78$	0	0
$79$	−638.293	−0.909033	−0.454517	−	0.890738i	$-0.650188\pi$
$79$	−638.293	−0.909033	−0.454517	+	0.890738i	$0.650188\pi$
$80$	0	0
$81$	0	0
$82$	982.470	1.32312
$83$	−1494.64	−1.97661	−0.988304	−	0.152498i	$-0.951268\pi$
$83$	−1494.64	−1.97661	−0.988304	+	0.152498i	$0.951268\pi$
$84$	0	0
$85$	0	0
$86$	679.139	0.851551
$87$	0	0
$88$	512.834	0.621231
$89$	1437.27	1.71180	0.855900	−	0.517142i	$-0.173004\pi$
$89$	1437.27	1.71180	0.855900	+	0.517142i	$0.173004\pi$
$90$	0	0
$91$	−53.7501	−0.0619181
$92$	−227.418	−0.257717
$93$	0	0
$94$	1625.76	1.78388
$95$	0	0
$96$	0	0
$97$	891.769	0.933458	0.466729	−	0.884400i	$-0.345432\pi$
$97$	891.769	0.933458	0.466729	+	0.884400i	$0.345432\pi$
$98$	1147.76	1.18307
$99$	0	0
$100$	0	0
$101$	153.744	0.151466	0.0757332	−	0.997128i	$-0.475870\pi$
$101$	153.744	0.151466	0.0757332	+	0.997128i	$0.475870\pi$
$102$	0	0
$103$	−135.294	−0.129426	−0.0647131	−	0.997904i	$-0.520613\pi$
$103$	−135.294	−0.129426	−0.0647131	+	0.997904i	$0.520613\pi$
$104$	515.336	0.485893
$105$	0	0
$106$	590.919	0.541463
$107$	−718.783	−0.649414	−0.324707	−	0.945815i	$-0.605266\pi$
$107$	−718.783	−0.649414	−0.324707	+	0.945815i	$0.605266\pi$
$108$	0	0
$109$	−2010.56	−1.76676	−0.883378	−	0.468661i	$-0.844736\pi$
$109$	−2010.56	−1.76676	−0.883378	+	0.468661i	$0.844736\pi$
$110$	0	0
$111$	0	0
$112$	129.576	0.109320
$113$	−226.745	−0.188764	−0.0943820	−	0.995536i	$-0.530088\pi$
$113$	−226.745	−0.188764	−0.0943820	+	0.995536i	$0.530088\pi$
$114$	0	0
$115$	0	0
$116$	−925.755	−0.740985
$117$	0	0
$118$	−617.943	−0.482087
$119$	−179.549	−0.138313
$120$	0	0
$121$	−251.128	−0.188676
$122$	−1472.32	−1.09260
$123$	0	0
$124$	−20.2337	−0.0146535
$125$	0	0
$126$	0	0
$127$	−1132.24	−0.791100	−0.395550	−	0.918444i	$-0.629446\pi$
$127$	−1132.24	−0.791100	−0.395550	+	0.918444i	$0.629446\pi$
$128$	−1663.35	−1.14860
$129$	0	0
$130$	0	0
$131$	−777.505	−0.518557	−0.259278	−	0.965803i	$-0.583485\pi$
$131$	−777.505	−0.518557	−0.259278	+	0.965803i	$0.583485\pi$
$132$	0	0
$133$	88.3966	0.0576312
$134$	−2804.05	−1.80771
$135$	0	0
$136$	1721.45	1.08539
$137$	623.638	0.388913	0.194456	−	0.980911i	$-0.437706\pi$
$137$	623.638	0.388913	0.194456	+	0.980911i	$0.437706\pi$
$138$	0	0
$139$	1282.68	0.782702	0.391351	−	0.920241i	$-0.372008\pi$
$139$	1282.68	0.782702	0.391351	+	0.920241i	$0.372008\pi$
$140$	0	0
$141$	0	0
$142$	−401.136	−0.237060
$143$	1085.14	0.634574
$144$	0	0
$145$	0	0
$146$	618.199	0.350428
$147$	0	0
$148$	1172.62	0.651275
$149$	−1524.31	−0.838098	−0.419049	−	0.907964i	$-0.637636\pi$
$149$	−1524.31	−0.838098	−0.419049	+	0.907964i	$0.637636\pi$
$150$	0	0
$151$	3162.56	1.70441	0.852203	−	0.523211i	$-0.175266\pi$
$151$	3162.56	1.70441	0.852203	+	0.523211i	$0.175266\pi$
$152$	−847.514	−0.452253
$153$	0	0
$154$	180.380	0.0943861
$155$	0	0
$156$	0	0
$157$	2388.70	1.21426	0.607131	−	0.794602i	$-0.292320\pi$
$157$	2388.70	1.21426	0.607131	+	0.794602i	$0.292320\pi$
$158$	2152.50	1.08382
$159$	0	0
$160$	0	0
$161$	109.769	0.0537331
$162$	0	0
$163$	2544.79	1.22284	0.611422	−	0.791305i	$-0.290598\pi$
$163$	2544.79	1.22284	0.611422	+	0.791305i	$0.290598\pi$
$164$	−982.470	−0.467793
$165$	0	0
$166$	5040.36	2.35667
$167$	−1374.56	−0.636925	−0.318462	−	0.947935i	$-0.603166\pi$
$167$	−1374.56	−0.636925	−0.318462	+	0.947935i	$0.603166\pi$
$168$	0	0
$169$	−1106.56	−0.503670
$170$	0	0
$171$	0	0
$172$	−679.139	−0.301069
$173$	2360.45	1.03735	0.518675	−	0.854971i	$-0.326425\pi$
$173$	2360.45	1.03735	0.518675	+	0.854971i	$0.326425\pi$
$174$	0	0
$175$	0	0
$176$	−2615.96	−1.12037
$177$	0	0
$178$	−4846.87	−2.04095
$179$	−1305.11	−0.544963	−0.272482	−	0.962161i	$-0.587844\pi$
$179$	−1305.11	−0.544963	−0.272482	+	0.962161i	$0.587844\pi$
$180$	0	0
$181$	3099.43	1.27281	0.636406	−	0.771355i	$-0.280420\pi$
$181$	3099.43	1.27281	0.636406	+	0.771355i	$0.280420\pi$
$182$	181.261	0.0738238
$183$	0	0
$184$	−1052.43	−0.421663
$185$	0	0
$186$	0	0
$187$	3624.84	1.41751
$188$	−1625.76	−0.630696
$189$	0	0
$190$	0	0
$191$	−380.712	−0.144227	−0.0721135	−	0.997396i	$-0.522974\pi$
$191$	−380.712	−0.144227	−0.0721135	+	0.997396i	$0.522974\pi$
$192$	0	0
$193$	−1546.04	−0.576613	−0.288307	−	0.957538i	$-0.593092\pi$
$193$	−1546.04	−0.576613	−0.288307	+	0.957538i	$0.593092\pi$
$194$	−3007.30	−1.11294
$195$	0	0
$196$	−1147.76	−0.418279
$197$	−4284.60	−1.54957	−0.774784	−	0.632226i	$-0.782141\pi$
$197$	−4284.60	−1.54957	−0.774784	+	0.632226i	$0.782141\pi$
$198$	0	0
$199$	−1402.85	−0.499727	−0.249863	−	0.968281i	$-0.580386\pi$
$199$	−1402.85	−0.499727	−0.249863	+	0.968281i	$0.580386\pi$
$200$	0	0
$201$	0	0
$202$	−518.468	−0.180591
$203$	446.840	0.154493
$204$	0	0
$205$	0	0
$206$	456.249	0.154312
$207$	0	0
$208$	−2628.73	−0.876296
$209$	−1784.61	−0.590640
$210$	0	0
$211$	−2150.46	−0.701628	−0.350814	−	0.936445i	$-0.614095\pi$
$211$	−2150.46	−0.701628	−0.350814	+	0.936445i	$0.614095\pi$
$212$	−590.919	−0.191436
$213$	0	0
$214$	2423.94	0.774285
$215$	0	0
$216$	0	0
$217$	9.76631	0.00305521
$218$	6780.16	2.10647
$219$	0	0
$220$	0	0
$221$	3642.53	1.10870
$222$	0	0
$223$	2558.15	0.768190	0.384095	−	0.923294i	$-0.374514\pi$
$223$	2558.15	0.768190	0.384095	+	0.923294i	$0.374514\pi$
$224$	−233.750	−0.0697236
$225$	0	0
$226$	764.646	0.225060
$227$	−4084.54	−1.19428	−0.597138	−	0.802139i	$-0.703695\pi$
$227$	−4084.54	−1.19428	−0.597138	+	0.802139i	$0.703695\pi$
$228$	0	0
$229$	−2741.91	−0.791225	−0.395612	−	0.918418i	$-0.629468\pi$
$229$	−2741.91	−0.791225	−0.395612	+	0.918418i	$0.629468\pi$
$230$	0	0
$231$	0	0
$232$	−4284.13	−1.21236
$233$	5084.70	1.42966	0.714828	−	0.699300i	$-0.246505\pi$
$233$	5084.70	1.42966	0.714828	+	0.699300i	$0.246505\pi$
$234$	0	0
$235$	0	0
$236$	617.943	0.170443
$237$	0	0
$238$	605.489	0.164908
$239$	−1476.47	−0.399603	−0.199801	−	0.979836i	$-0.564030\pi$
$239$	−1476.47	−0.399603	−0.199801	+	0.979836i	$0.564030\pi$
$240$	0	0
$241$	1763.46	0.471345	0.235673	−	0.971833i	$-0.424271\pi$
$241$	1763.46	0.471345	0.235673	+	0.971833i	$0.424271\pi$
$242$	846.874	0.224955
$243$	0	0
$244$	1472.32	0.386294
$245$	0	0
$246$	0	0
$247$	−1793.31	−0.461967
$248$	−93.6358	−0.0239753
$249$	0	0
$250$	0	0
$251$	−1705.16	−0.428801	−0.214400	−	0.976746i	$-0.568780\pi$
$251$	−1705.16	−0.428801	−0.214400	+	0.976746i	$0.568780\pi$
$252$	0	0
$253$	−2216.09	−0.550690
$254$	3818.22	0.943214
$255$	0	0
$256$	4388.74	1.07147
$257$	228.596	0.0554842	0.0277421	−	0.999615i	$-0.491168\pi$
$257$	228.596	0.0554842	0.0277421	+	0.999615i	$0.491168\pi$
$258$	0	0
$259$	−565.995	−0.135788
$260$	0	0
$261$	0	0
$262$	2621.97	0.618266
$263$	−1288.76	−0.302161	−0.151081	−	0.988521i	$-0.548275\pi$
$263$	−1288.76	−0.302161	−0.151081	+	0.988521i	$0.548275\pi$
$264$	0	0
$265$	0	0
$266$	−298.098	−0.0687127
$267$	0	0
$268$	2804.05	0.639122
$269$	−973.981	−0.220761	−0.110380	−	0.993889i	$-0.535207\pi$
$269$	−973.981	−0.220761	−0.110380	+	0.993889i	$0.535207\pi$
$270$	0	0
$271$	−4021.83	−0.901508	−0.450754	−	0.892648i	$-0.648845\pi$
$271$	−4021.83	−0.901508	−0.450754	+	0.892648i	$0.648845\pi$
$272$	−8781.10	−1.95747
$273$	0	0
$274$	−2103.08	−0.463693
$275$	0	0
$276$	0	0
$277$	3379.17	0.732977	0.366489	−	0.930422i	$-0.380560\pi$
$277$	3379.17	0.732977	0.366489	+	0.930422i	$0.380560\pi$
$278$	−4325.56	−0.933201
$279$	0	0
$280$	0	0
$281$	−733.307	−0.155678	−0.0778388	−	0.996966i	$-0.524802\pi$
$281$	−733.307	−0.155678	−0.0778388	+	0.996966i	$0.524802\pi$
$282$	0	0
$283$	−6909.03	−1.45123	−0.725617	−	0.688099i	$-0.758445\pi$
$283$	−6909.03	−1.45123	−0.725617	+	0.688099i	$0.758445\pi$
$284$	401.136	0.0838135
$285$	0	0
$286$	−3659.40	−0.756591
$287$	474.214	0.0975331
$288$	0	0
$289$	7254.64	1.47662
$290$	0	0
$291$	0	0
$292$	−618.199	−0.123895
$293$	−4649.49	−0.927051	−0.463525	−	0.886084i	$-0.653416\pi$
$293$	−4649.49	−0.927051	−0.463525	+	0.886084i	$0.653416\pi$
$294$	0	0
$295$	0	0
$296$	5426.55	1.06558
$297$	0	0
$298$	5140.41	0.999248
$299$	−2226.91	−0.430720
$300$	0	0
$301$	327.804	0.0627717
$302$	−10665.0	−2.03213
$303$	0	0
$304$	4323.16	0.815627
$305$	0	0
$306$	0	0
$307$	7361.42	1.36853	0.684264	−	0.729234i	$-0.260123\pi$
$307$	7361.42	1.36853	0.684264	+	0.729234i	$0.260123\pi$
$308$	−180.380	−0.0333705
$309$	0	0
$310$	0	0
$311$	2709.21	0.493971	0.246986	−	0.969019i	$-0.420560\pi$
$311$	2709.21	0.493971	0.246986	+	0.969019i	$0.420560\pi$
$312$	0	0
$313$	6243.72	1.12753	0.563763	−	0.825936i	$-0.309353\pi$
$313$	6243.72	1.12753	0.563763	+	0.825936i	$0.309353\pi$
$314$	−8055.37	−1.44774
$315$	0	0
$316$	−2152.50	−0.383189
$317$	2084.03	0.369245	0.184623	−	0.982809i	$-0.440894\pi$
$317$	2084.03	0.369245	0.184623	+	0.982809i	$0.440894\pi$
$318$	0	0
$319$	−9021.08	−1.58333
$320$	0	0
$321$	0	0
$322$	−370.173	−0.0640650
$323$	−5990.45	−1.03194
$324$	0	0
$325$	0	0
$326$	−8581.76	−1.45797
$327$	0	0
$328$	−4546.59	−0.765377
$329$	784.715	0.131498
$330$	0	0
$331$	−227.876	−0.0378406	−0.0189203	−	0.999821i	$-0.506023\pi$
$331$	−227.876	−0.0378406	−0.0189203	+	0.999821i	$0.506023\pi$
$332$	−5040.36	−0.833210
$333$	0	0
$334$	4635.39	0.759394
$335$	0	0
$336$	0	0
$337$	−3417.48	−0.552409	−0.276205	−	0.961099i	$-0.589077\pi$
$337$	−3417.48	−0.552409	−0.276205	+	0.961099i	$0.589077\pi$
$338$	3731.65	0.600517
$339$	0	0
$340$	0	0
$341$	−197.168	−0.0313116
$342$	0	0
$343$	1112.30	0.175098
$344$	−3142.86	−0.492592
$345$	0	0
$346$	−7960.10	−1.23681
$347$	3609.60	0.558425	0.279212	−	0.960229i	$-0.409927\pi$
$347$	3609.60	0.558425	0.279212	+	0.960229i	$0.409927\pi$
$348$	0	0
$349$	−8800.83	−1.34985	−0.674925	−	0.737886i	$-0.735824\pi$
$349$	−8800.83	−1.34985	−0.674925	+	0.737886i	$0.735824\pi$
$350$	0	0
$351$	0	0
$352$	4719.09	0.714570
$353$	8510.49	1.28319	0.641597	−	0.767042i	$-0.278272\pi$
$353$	8510.49	1.28319	0.641597	+	0.767042i	$0.278272\pi$
$354$	0	0
$355$	0	0
$356$	4846.87	0.721584
$357$	0	0
$358$	4401.19	0.649750
$359$	−4320.35	−0.635152	−0.317576	−	0.948233i	$-0.602869\pi$
$359$	−4320.35	−0.635152	−0.317576	+	0.948233i	$0.602869\pi$
$360$	0	0
$361$	−3909.75	−0.570017
$362$	−10452.2	−1.51755
$363$	0	0
$364$	−181.261	−0.0261006
$365$	0	0
$366$	0	0
$367$	−6600.39	−0.938794	−0.469397	−	0.882987i	$-0.655529\pi$
$367$	−6600.39	−0.938794	−0.469397	+	0.882987i	$0.655529\pi$
$368$	5368.43	0.760459
$369$	0	0
$370$	0	0
$371$	285.222	0.0399137
$372$	0	0
$373$	8608.45	1.19498	0.597492	−	0.801875i	$-0.296164\pi$
$373$	8608.45	1.19498	0.597492	+	0.801875i	$0.296164\pi$
$374$	−12224.0	−1.69007
$375$	0	0
$376$	−7523.56	−1.03191
$377$	−9065.10	−1.23840
$378$	0	0
$379$	−7129.80	−0.966314	−0.483157	−	0.875534i	$-0.660510\pi$
$379$	−7129.80	−0.966314	−0.483157	+	0.875534i	$0.660510\pi$
$380$	0	0
$381$	0	0
$382$	1283.87	0.171959
$383$	5927.46	0.790807	0.395404	−	0.918507i	$-0.370605\pi$
$383$	5927.46	0.790807	0.395404	+	0.918507i	$0.370605\pi$
$384$	0	0
$385$	0	0
$386$	5213.68	0.687486
$387$	0	0
$388$	3007.30	0.393485
$389$	−1038.86	−0.135405	−0.0677024	−	0.997706i	$-0.521567\pi$
$389$	−1038.86	−0.135405	−0.0677024	+	0.997706i	$0.521567\pi$
$390$	0	0
$391$	−7438.83	−0.962143
$392$	−5311.50	−0.684365
$393$	0	0
$394$	14448.9	1.84752
$395$	0	0
$396$	0	0
$397$	−13441.4	−1.69926	−0.849628	−	0.527382i	$-0.823174\pi$
$397$	−13441.4	−1.69926	−0.849628	+	0.527382i	$0.823174\pi$
$398$	4730.81	0.595815
$399$	0	0
$400$	0	0
$401$	−13075.4	−1.62831	−0.814157	−	0.580644i	$-0.802801\pi$
$401$	−13075.4	−1.62831	−0.814157	+	0.580644i	$0.802801\pi$
$402$	0	0
$403$	−198.130	−0.0244903
$404$	518.468	0.0638484
$405$	0	0
$406$	−1506.87	−0.184199
$407$	11426.7	1.39164
$408$	0	0
$409$	−5273.19	−0.637512	−0.318756	−	0.947837i	$-0.603265\pi$
$409$	−5273.19	−0.637512	−0.318756	+	0.947837i	$0.603265\pi$
$410$	0	0
$411$	0	0
$412$	−456.249	−0.0545577
$413$	−298.266	−0.0355368
$414$	0	0
$415$	0	0
$416$	4742.12	0.558898
$417$	0	0
$418$	6018.19	0.704209
$419$	−13846.1	−1.61438	−0.807192	−	0.590289i	$-0.799014\pi$
$419$	−13846.1	−1.61438	−0.807192	+	0.590289i	$0.799014\pi$
$420$	0	0
$421$	1097.37	0.127037	0.0635184	−	0.997981i	$-0.479768\pi$
$421$	1097.37	0.127037	0.0635184	+	0.997981i	$0.479768\pi$
$422$	7251.94	0.836538
$423$	0	0
$424$	−2734.60	−0.313217
$425$	0	0
$426$	0	0
$427$	−710.654	−0.0805408
$428$	−2423.94	−0.273751
$429$	0	0
$430$	0	0
$431$	−15912.8	−1.77841	−0.889205	−	0.457509i	$-0.848742\pi$
$431$	−15912.8	−1.77841	−0.889205	+	0.457509i	$0.848742\pi$
$432$	0	0
$433$	−3566.31	−0.395810	−0.197905	−	0.980221i	$-0.563414\pi$
$433$	−3566.31	−0.395810	−0.197905	+	0.980221i	$0.563414\pi$
$434$	−32.9348	−0.00364267
$435$	0	0
$436$	−6780.16	−0.744750
$437$	3662.33	0.400900
$438$	0	0
$439$	−580.822	−0.0631461	−0.0315730	−	0.999501i	$-0.510052\pi$
$439$	−580.822	−0.0631461	−0.0315730	+	0.999501i	$0.510052\pi$
$440$	0	0
$441$	0	0
$442$	−12283.6	−1.32188
$443$	−10792.8	−1.15752	−0.578759	−	0.815499i	$-0.696463\pi$
$443$	−10792.8	−1.15752	−0.578759	+	0.815499i	$0.696463\pi$
$444$	0	0
$445$	0	0
$446$	−8626.80	−0.915898
$447$	0	0
$448$	−248.338	−0.0261894
$449$	−2894.01	−0.304180	−0.152090	−	0.988367i	$-0.548600\pi$
$449$	−2894.01	−0.304180	−0.152090	+	0.988367i	$0.548600\pi$
$450$	0	0
$451$	−9573.74	−0.999578
$452$	−764.646	−0.0795707
$453$	0	0
$454$	13774.2	1.42391
$455$	0	0
$456$	0	0
$457$	−3199.92	−0.327540	−0.163770	−	0.986499i	$-0.552365\pi$
$457$	−3199.92	−0.327540	−0.163770	+	0.986499i	$0.552365\pi$
$458$	9246.49	0.943363
$459$	0	0
$460$	0	0
$461$	−7316.38	−0.739171	−0.369585	−	0.929197i	$-0.620500\pi$
$461$	−7316.38	−0.739171	−0.369585	+	0.929197i	$0.620500\pi$
$462$	0	0
$463$	−7011.98	−0.703832	−0.351916	−	0.936032i	$-0.614470\pi$
$463$	−7011.98	−0.703832	−0.351916	+	0.936032i	$0.614470\pi$
$464$	21853.3	2.18646
$465$	0	0
$466$	−17147.1	−1.70455
$467$	−8002.63	−0.792971	−0.396485	−	0.918041i	$-0.629770\pi$
$467$	−8002.63	−0.792971	−0.396485	+	0.918041i	$0.629770\pi$
$468$	0	0
$469$	−1353.45	−0.133255
$470$	0	0
$471$	0	0
$472$	2859.67	0.278870
$473$	−6617.91	−0.643323
$474$	0	0
$475$	0	0
$476$	−605.489	−0.0583037
$477$	0	0
$478$	4979.08	0.476439
$479$	3178.84	0.303225	0.151613	−	0.988440i	$-0.451553\pi$
$479$	3178.84	0.303225	0.151613	+	0.988440i	$0.451553\pi$
$480$	0	0
$481$	11482.4	1.08847
$482$	−5946.87	−0.561976
$483$	0	0
$484$	−846.874	−0.0795336
$485$	0	0
$486$	0	0
$487$	−13060.3	−1.21523	−0.607615	−	0.794232i	$-0.707874\pi$
$487$	−13060.3	−1.21523	−0.607615	+	0.794232i	$0.707874\pi$
$488$	6813.49	0.632033
$489$	0	0
$490$	0	0
$491$	4401.61	0.404566	0.202283	−	0.979327i	$-0.435164\pi$
$491$	4401.61	0.404566	0.202283	+	0.979327i	$0.435164\pi$
$492$	0	0
$493$	−30281.4	−2.76634
$494$	6047.56	0.550794
$495$	0	0
$496$	477.636	0.0432389
$497$	−193.619	−0.0174748
$498$	0	0
$499$	18725.2	1.67987	0.839935	−	0.542687i	$-0.182593\pi$
$499$	18725.2	1.67987	0.839935	+	0.542687i	$0.182593\pi$
$500$	0	0
$501$	0	0
$502$	5750.29	0.511251
$503$	−3811.68	−0.337882	−0.168941	−	0.985626i	$-0.554035\pi$
$503$	−3811.68	−0.337882	−0.168941	+	0.985626i	$0.554035\pi$
$504$	0	0
$505$	0	0
$506$	7473.29	0.656577
$507$	0	0
$508$	−3818.22	−0.333477
$509$	−4495.48	−0.391471	−0.195735	−	0.980657i	$-0.562709\pi$
$509$	−4495.48	−0.391471	−0.195735	+	0.980657i	$0.562709\pi$
$510$	0	0
$511$	298.390	0.0258317
$512$	−1493.27	−0.128894
$513$	0	0
$514$	−770.891	−0.0661528
$515$	0	0
$516$	0	0
$517$	−15842.3	−1.34767
$518$	1908.69	0.161898
$519$	0	0
$520$	0	0
$521$	−12095.0	−1.01706	−0.508531	−	0.861043i	$-0.669811\pi$
$521$	−12095.0	−1.01706	−0.508531	+	0.861043i	$0.669811\pi$
$522$	0	0
$523$	−7385.38	−0.617476	−0.308738	−	0.951147i	$-0.599907\pi$
$523$	−7385.38	−0.617476	−0.308738	+	0.951147i	$0.599907\pi$
$524$	−2621.97	−0.218590
$525$	0	0
$526$	4346.06	0.360261
$527$	−661.842	−0.0547064
$528$	0	0
$529$	−7619.18	−0.626217
$530$	0	0
$531$	0	0
$532$	298.098	0.0242936
$533$	−9620.45	−0.781816
$534$	0	0
$535$	0	0
$536$	12976.4	1.04570
$537$	0	0
$538$	3284.54	0.263209
$539$	−11184.4	−0.893778
$540$	0	0
$541$	−5935.19	−0.471670	−0.235835	−	0.971793i	$-0.575783\pi$
$541$	−5935.19	−0.471670	−0.235835	+	0.971793i	$0.575783\pi$
$542$	13562.7	1.07485
$543$	0	0
$544$	15840.7	1.24847
$545$	0	0
$546$	0	0
$547$	10157.2	0.793951	0.396976	−	0.917829i	$-0.370060\pi$
$547$	10157.2	0.793951	0.396976	+	0.917829i	$0.370060\pi$
$548$	2103.08	0.163940
$549$	0	0
$550$	0	0
$551$	14908.3	1.15266
$552$	0	0
$553$	1038.96	0.0798936
$554$	−11395.5	−0.873915
$555$	0	0
$556$	4325.56	0.329937
$557$	5709.62	0.434334	0.217167	−	0.976134i	$-0.430318\pi$
$557$	5709.62	0.434334	0.217167	+	0.976134i	$0.430318\pi$
$558$	0	0
$559$	−6650.20	−0.503173
$560$	0	0
$561$	0	0
$562$	2472.92	0.185612
$563$	12939.1	0.968595	0.484297	−	0.874903i	$-0.339075\pi$
$563$	12939.1	0.968595	0.484297	+	0.874903i	$0.339075\pi$
$564$	0	0
$565$	0	0
$566$	23299.2	1.73028
$567$	0	0
$568$	1856.34	0.137131
$569$	6125.42	0.451302	0.225651	−	0.974208i	$-0.427549\pi$
$569$	6125.42	0.451302	0.225651	+	0.974208i	$0.427549\pi$
$570$	0	0
$571$	−19282.4	−1.41321	−0.706605	−	0.707608i	$-0.749774\pi$
$571$	−19282.4	−1.41321	−0.706605	+	0.707608i	$0.749774\pi$
$572$	3659.40	0.267495
$573$	0	0
$574$	−1599.18	−0.116287
$575$	0	0
$576$	0	0
$577$	4988.14	0.359894	0.179947	−	0.983676i	$-0.442407\pi$
$577$	4988.14	0.359894	0.179947	+	0.983676i	$0.442407\pi$
$578$	−24464.7	−1.76055
$579$	0	0
$580$	0	0
$581$	2432.86	0.173721
$582$	0	0
$583$	−5758.24	−0.409060
$584$	−2860.85	−0.202710
$585$	0	0
$586$	15679.4	1.10531
$587$	−15294.0	−1.07539	−0.537693	−	0.843141i	$-0.680704\pi$
$587$	−15294.0	−1.07539	−0.537693	+	0.843141i	$0.680704\pi$
$588$	0	0
$589$	325.842	0.0227947
$590$	0	0
$591$	0	0
$592$	−27680.8	−1.92175
$593$	−11090.9	−0.768040	−0.384020	−	0.923325i	$-0.625461\pi$
$593$	−11090.9	−0.768040	−0.384020	+	0.923325i	$0.625461\pi$
$594$	0	0
$595$	0	0
$596$	−5140.41	−0.353288
$597$	0	0
$598$	7509.75	0.513539
$599$	28201.6	1.92368	0.961840	−	0.273612i	$-0.0882184\pi$
$599$	28201.6	1.92368	0.961840	+	0.273612i	$0.0882184\pi$
$600$	0	0
$601$	21144.0	1.43508	0.717539	−	0.696518i	$-0.245268\pi$
$601$	21144.0	1.43508	0.717539	+	0.696518i	$0.245268\pi$
$602$	−1105.45	−0.0748416
$603$	0	0
$604$	10665.0	0.718467
$605$	0	0
$606$	0	0
$607$	8798.98	0.588369	0.294184	−	0.955749i	$-0.404952\pi$
$607$	8798.98	0.588369	0.294184	+	0.955749i	$0.404952\pi$
$608$	−7798.81	−0.520203
$609$	0	0
$610$	0	0
$611$	−15919.6	−1.05407
$612$	0	0
$613$	−19539.8	−1.28745	−0.643724	−	0.765257i	$-0.722612\pi$
$613$	−19539.8	−1.28745	−0.643724	+	0.765257i	$0.722612\pi$
$614$	−24824.8	−1.63167
$615$	0	0
$616$	−834.750	−0.0545991
$617$	−3273.87	−0.213616	−0.106808	−	0.994280i	$-0.534063\pi$
$617$	−3273.87	−0.213616	−0.106808	+	0.994280i	$0.534063\pi$
$618$	0	0
$619$	9386.47	0.609490	0.304745	−	0.952434i	$-0.401429\pi$
$619$	9386.47	0.609490	0.304745	+	0.952434i	$0.401429\pi$
$620$	0	0
$621$	0	0
$622$	−9136.21	−0.588953
$623$	−2339.47	−0.150448
$624$	0	0
$625$	0	0
$626$	−21055.6	−1.34433
$627$	0	0
$628$	8055.37	0.511854
$629$	38356.3	2.43142
$630$	0	0
$631$	9647.08	0.608628	0.304314	−	0.952572i	$-0.401573\pi$
$631$	9647.08	0.608628	0.304314	+	0.952572i	$0.401573\pi$
$632$	−9961.19	−0.626954
$633$	0	0
$634$	−7027.94	−0.440245
$635$	0	0
$636$	0	0
$637$	−11239.0	−0.699065
$638$	30421.6	1.88778
$639$	0	0
$640$	0	0
$641$	7003.77	0.431563	0.215782	−	0.976442i	$-0.430770\pi$
$641$	7003.77	0.431563	0.215782	+	0.976442i	$0.430770\pi$
$642$	0	0
$643$	6288.89	0.385707	0.192853	−	0.981228i	$-0.438226\pi$
$643$	6288.89	0.385707	0.192853	+	0.981228i	$0.438226\pi$
$644$	370.173	0.0226504
$645$	0	0
$646$	20201.5	1.23037
$647$	−5900.85	−0.358557	−0.179279	−	0.983798i	$-0.557376\pi$
$647$	−5900.85	−0.358557	−0.179279	+	0.983798i	$0.557376\pi$
$648$	0	0
$649$	6021.58	0.364203
$650$	0	0
$651$	0	0
$652$	8581.76	0.515472
$653$	−7885.15	−0.472542	−0.236271	−	0.971687i	$-0.575925\pi$
$653$	−7885.15	−0.472542	−0.236271	+	0.971687i	$0.575925\pi$
$654$	0	0
$655$	0	0
$656$	23192.2	1.38034
$657$	0	0
$658$	−2646.28	−0.156782
$659$	−28757.8	−1.69992	−0.849959	−	0.526848i	$-0.823374\pi$
$659$	−28757.8	−1.69992	−0.849959	+	0.526848i	$0.823374\pi$
$660$	0	0
$661$	8260.23	0.486060	0.243030	−	0.970019i	$-0.421859\pi$
$661$	8260.23	0.486060	0.243030	+	0.970019i	$0.421859\pi$
$662$	768.464	0.0451166
$663$	0	0
$664$	−23325.4	−1.36325
$665$	0	0
$666$	0	0
$667$	18512.9	1.07470
$668$	−4635.39	−0.268486
$669$	0	0
$670$	0	0
$671$	14347.1	0.825431
$672$	0	0
$673$	−804.962	−0.0461055	−0.0230528	−	0.999734i	$-0.507339\pi$
$673$	−804.962	−0.0461055	−0.0230528	+	0.999734i	$0.507339\pi$
$674$	11524.7	0.658627
$675$	0	0
$676$	−3731.65	−0.212315
$677$	1363.27	0.0773926	0.0386963	−	0.999251i	$-0.487680\pi$
$677$	1363.27	0.0773926	0.0386963	+	0.999251i	$0.487680\pi$
$678$	0	0
$679$	−1451.55	−0.0820403
$680$	0	0
$681$	0	0
$682$	664.907	0.0373323
$683$	−11434.6	−0.640604	−0.320302	−	0.947315i	$-0.603784\pi$
$683$	−11434.6	−0.640604	−0.320302	+	0.947315i	$0.603784\pi$
$684$	0	0
$685$	0	0
$686$	−3751.00	−0.208766
$687$	0	0
$688$	16031.7	0.888378
$689$	−5786.34	−0.319945
$690$	0	0
$691$	−19530.6	−1.07522	−0.537611	−	0.843193i	$-0.680673\pi$
$691$	−19530.6	−1.07522	−0.537611	+	0.843193i	$0.680673\pi$
$692$	7960.10	0.437280
$693$	0	0
$694$	−12172.6	−0.665799
$695$	0	0
$696$	0	0
$697$	−32136.5	−1.74642
$698$	29678.9	1.60940
$699$	0	0
$700$	0	0
$701$	11041.4	0.594903	0.297452	−	0.954737i	$-0.403863\pi$
$701$	11041.4	0.594903	0.297452	+	0.954737i	$0.403863\pi$
$702$	0	0
$703$	−18883.8	−1.01311
$704$	5013.60	0.268405
$705$	0	0
$706$	−28699.8	−1.52993
$707$	−250.252	−0.0133122
$708$	0	0
$709$	3939.08	0.208653	0.104327	−	0.994543i	$-0.466731\pi$
$709$	3939.08	0.208653	0.104327	+	0.994543i	$0.466731\pi$
$710$	0	0
$711$	0	0
$712$	22430.0	1.18062
$713$	404.625	0.0212529
$714$	0	0
$715$	0	0
$716$	−4401.19	−0.229721
$717$	0	0
$718$	14569.4	0.757280
$719$	23298.2	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$719$	23298.2	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$720$	0	0
$721$	220.220	0.0113751
$722$	13184.8	0.679621
$723$	0	0
$724$	10452.2	0.536535
$725$	0	0
$726$	0	0
$727$	9504.48	0.484872	0.242436	−	0.970167i	$-0.422054\pi$
$727$	9504.48	0.484872	0.242436	+	0.970167i	$0.422054\pi$
$728$	−838.823	−0.0427044
$729$	0	0
$730$	0	0
$731$	−22214.6	−1.12399
$732$	0	0
$733$	−7320.81	−0.368895	−0.184448	−	0.982842i	$-0.559050\pi$
$733$	−7320.81	−0.368895	−0.184448	+	0.982842i	$0.559050\pi$
$734$	22258.4	1.11931
$735$	0	0
$736$	−9684.43	−0.485018
$737$	27324.3	1.36567
$738$	0	0
$739$	−1274.52	−0.0634424	−0.0317212	−	0.999497i	$-0.510099\pi$
$739$	−1274.52	−0.0634424	−0.0317212	+	0.999497i	$0.510099\pi$
$740$	0	0
$741$	0	0
$742$	−961.849	−0.0475884
$743$	1941.15	0.0958465	0.0479232	−	0.998851i	$-0.484740\pi$
$743$	1941.15	0.0958465	0.0479232	+	0.998851i	$0.484740\pi$
$744$	0	0
$745$	0	0
$746$	−29030.1	−1.42476
$747$	0	0
$748$	12224.0	0.597531
$749$	1169.98	0.0570761
$750$	0	0
$751$	16540.0	0.803667	0.401834	−	0.915713i	$-0.368373\pi$
$751$	16540.0	0.803667	0.401834	+	0.915713i	$0.368373\pi$
$752$	38377.6	1.86102
$753$	0	0
$754$	30570.1	1.47652
$755$	0	0
$756$	0	0
$757$	−21145.7	−1.01526	−0.507631	−	0.861575i	$-0.669479\pi$
$757$	−21145.7	−1.01526	−0.507631	+	0.861575i	$0.669479\pi$
$758$	24043.7	1.15212
$759$	0	0
$760$	0	0
$761$	20134.2	0.959087	0.479543	−	0.877518i	$-0.340802\pi$
$761$	20134.2	0.959087	0.479543	+	0.877518i	$0.340802\pi$
$762$	0	0
$763$	3272.62	0.155278
$764$	−1283.87	−0.0607968
$765$	0	0
$766$	−19989.1	−0.942865
$767$	6050.96	0.284860
$768$	0	0
$769$	2194.72	0.102917	0.0514587	−	0.998675i	$-0.483613\pi$
$769$	2194.72	0.102917	0.0514587	+	0.998675i	$0.483613\pi$
$770$	0	0
$771$	0	0
$772$	−5213.68	−0.243063
$773$	−8327.70	−0.387486	−0.193743	−	0.981052i	$-0.562063\pi$
$773$	−8327.70	−0.387486	−0.193743	+	0.981052i	$0.562063\pi$
$774$	0	0
$775$	0	0
$776$	13916.9	0.643799
$777$	0	0
$778$	3503.34	0.161441
$779$	15821.6	0.727688
$780$	0	0
$781$	3908.90	0.179093
$782$	25085.8	1.14715
$783$	0	0
$784$	27093.9	1.23424
$785$	0	0
$786$	0	0
$787$	18963.3	0.858918	0.429459	−	0.903086i	$-0.358704\pi$
$787$	18963.3	0.858918	0.429459	+	0.903086i	$0.358704\pi$
$788$	−14448.9	−0.653197
$789$	0	0
$790$	0	0
$791$	369.076	0.0165902
$792$	0	0
$793$	14417.1	0.645608
$794$	45328.2	2.02599
$795$	0	0
$796$	−4730.81	−0.210652
$797$	34265.1	1.52288	0.761439	−	0.648237i	$-0.224493\pi$
$797$	34265.1	1.52288	0.761439	+	0.648237i	$0.224493\pi$
$798$	0	0
$799$	−53178.5	−2.35459
$800$	0	0
$801$	0	0
$802$	44093.9	1.94141
$803$	−6024.08	−0.264739
$804$	0	0
$805$	0	0
$806$	668.152	0.0291993
$807$	0	0
$808$	2399.33	0.104465
$809$	−36425.5	−1.58300	−0.791502	−	0.611166i	$-0.790701\pi$
$809$	−36425.5	−1.58300	−0.791502	+	0.611166i	$0.790701\pi$
$810$	0	0
$811$	45174.0	1.95595	0.977973	−	0.208732i	$-0.0669335\pi$
$811$	45174.0	1.95595	0.977973	+	0.208732i	$0.0669335\pi$
$812$	1506.87	0.0651241
$813$	0	0
$814$	−38533.9	−1.65923
$815$	0	0
$816$	0	0
$817$	10936.8	0.468336
$818$	17782.7	0.760093
$819$	0	0
$820$	0	0
$821$	6919.30	0.294136	0.147068	−	0.989126i	$-0.453016\pi$
$821$	6919.30	0.294136	0.147068	+	0.989126i	$0.453016\pi$
$822$	0	0
$823$	11398.5	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$823$	11398.5	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$824$	−2111.39	−0.0892643
$825$	0	0
$826$	1005.84	0.0423699
$827$	34712.0	1.45956	0.729779	−	0.683683i	$-0.239623\pi$
$827$	34712.0	1.45956	0.729779	+	0.683683i	$0.239623\pi$
$828$	0	0
$829$	2732.97	0.114500	0.0572498	−	0.998360i	$-0.481767\pi$
$829$	2732.97	0.114500	0.0572498	+	0.998360i	$0.481767\pi$
$830$	0	0
$831$	0	0
$832$	5038.06	0.209932
$833$	−37543.1	−1.56157
$834$	0	0
$835$	0	0
$836$	−6018.19	−0.248975
$837$	0	0
$838$	46693.0	1.92480
$839$	9575.79	0.394032	0.197016	−	0.980400i	$-0.436875\pi$
$839$	9575.79	0.394032	0.197016	+	0.980400i	$0.436875\pi$
$840$	0	0
$841$	50971.7	2.08994
$842$	−3700.64	−0.151464
$843$	0	0
$844$	−7251.94	−0.295761
$845$	0	0
$846$	0	0
$847$	408.766	0.0165825
$848$	13949.2	0.564879
$849$	0	0
$850$	0	0
$851$	−23449.6	−0.944584
$852$	0	0
$853$	−9041.89	−0.362941	−0.181470	−	0.983396i	$-0.558086\pi$
$853$	−9041.89	−0.362941	−0.181470	+	0.983396i	$0.558086\pi$
$854$	2396.52	0.0960274
$855$	0	0
$856$	−11217.3	−0.447896
$857$	−2315.28	−0.0922851	−0.0461426	−	0.998935i	$-0.514693\pi$
$857$	−2315.28	−0.0922851	−0.0461426	+	0.998935i	$0.514693\pi$
$858$	0	0
$859$	−31403.1	−1.24733	−0.623667	−	0.781690i	$-0.714358\pi$
$859$	−31403.1	−1.24733	−0.623667	+	0.781690i	$0.714358\pi$
$860$	0	0
$861$	0	0
$862$	53662.6	2.12036
$863$	−40883.5	−1.61262	−0.806310	−	0.591493i	$-0.798539\pi$
$863$	−40883.5	−1.61262	−0.806310	+	0.591493i	$0.798539\pi$
$864$	0	0
$865$	0	0
$866$	12026.6	0.471917
$867$	0	0
$868$	32.9348	0.00128788
$869$	−20975.2	−0.818798
$870$	0	0
$871$	27457.6	1.06816
$872$	−31376.7	−1.21852
$873$	0	0
$874$	−12350.4	−0.477985
$875$	0	0
$876$	0	0
$877$	28314.7	1.09021	0.545107	−	0.838366i	$-0.316489\pi$
$877$	28314.7	1.09021	0.545107	+	0.838366i	$0.316489\pi$
$878$	1958.70	0.0752879
$879$	0	0
$880$	0	0
$881$	−6479.51	−0.247787	−0.123893	−	0.992296i	$-0.539538\pi$
$881$	−6479.51	−0.247787	−0.123893	+	0.992296i	$0.539538\pi$
$882$	0	0
$883$	17769.0	0.677208	0.338604	−	0.940929i	$-0.390045\pi$
$883$	17769.0	0.677208	0.338604	+	0.940929i	$0.390045\pi$
$884$	12283.6	0.467357
$885$	0	0
$886$	36396.3	1.38009
$887$	−25888.1	−0.979976	−0.489988	−	0.871729i	$-0.662999\pi$
$887$	−25888.1	−0.979976	−0.489988	+	0.871729i	$0.662999\pi$
$888$	0	0
$889$	1842.96	0.0695287
$890$	0	0
$891$	0	0
$892$	8626.80	0.323819
$893$	26181.2	0.981096
$894$	0	0
$895$	0	0
$896$	2707.47	0.100949
$897$	0	0
$898$	9759.41	0.362668
$899$	1647.11	0.0611060
$900$	0	0
$901$	−19328.9	−0.714694
$902$	32285.3	1.19178
$903$	0	0
$904$	−3538.57	−0.130189
$905$	0	0
$906$	0	0
$907$	45188.2	1.65430	0.827150	−	0.561981i	$-0.189960\pi$
$907$	45188.2	1.65430	0.827150	+	0.561981i	$0.189960\pi$
$908$	−13774.2	−0.503429
$909$	0	0
$910$	0	0
$911$	−13174.2	−0.479121	−0.239560	−	0.970881i	$-0.577003\pi$
$911$	−13174.2	−0.479121	−0.239560	+	0.970881i	$0.577003\pi$
$912$	0	0
$913$	−49116.1	−1.78040
$914$	10791.0	0.390520
$915$	0	0
$916$	−9246.49	−0.333529
$917$	1265.56	0.0455752
$918$	0	0
$919$	−54078.5	−1.94112	−0.970558	−	0.240867i	$-0.922568\pi$
$919$	−54078.5	−1.94112	−0.970558	+	0.240867i	$0.922568\pi$
$920$	0	0
$921$	0	0
$922$	24672.9	0.881299
$923$	3927.97	0.140077
$924$	0	0
$925$	0	0
$926$	23646.4	0.839166
$927$	0	0
$928$	−39422.6	−1.39451
$929$	45717.8	1.61459	0.807294	−	0.590150i	$-0.200931\pi$
$929$	45717.8	1.61459	0.807294	+	0.590150i	$0.200931\pi$
$930$	0	0
$931$	18483.4	0.650666
$932$	17147.1	0.602651
$933$	0	0
$934$	26987.1	0.945445
$935$	0	0
$936$	0	0
$937$	5055.54	0.176262	0.0881309	−	0.996109i	$-0.471911\pi$
$937$	5055.54	0.176262	0.0881309	+	0.996109i	$0.471911\pi$
$938$	4564.21	0.158877
$939$	0	0
$940$	0	0
$941$	−28112.0	−0.973883	−0.486942	−	0.873435i	$-0.661888\pi$
$941$	−28112.0	−0.973883	−0.486942	+	0.873435i	$0.661888\pi$
$942$	0	0
$943$	19647.0	0.678468
$944$	−14587.1	−0.502935
$945$	0	0
$946$	22317.4	0.767022
$947$	30120.4	1.03356	0.516780	−	0.856118i	$-0.327130\pi$
$947$	30120.4	1.03356	0.516780	+	0.856118i	$0.327130\pi$
$948$	0	0
$949$	−6053.47	−0.207064
$950$	0	0
$951$	0	0
$952$	−2802.03	−0.0953933
$953$	19362.7	0.658154	0.329077	−	0.944303i	$-0.393262\pi$
$953$	19362.7	0.658154	0.329077	+	0.944303i	$0.393262\pi$
$954$	0	0
$955$	0	0
$956$	−4979.08	−0.168447
$957$	0	0
$958$	−10719.9	−0.361530
$959$	−1015.11	−0.0341810
$960$	0	0
$961$	−29755.0	−0.998792
$962$	−38721.9	−1.29776
$963$	0	0
$964$	5946.87	0.198689
$965$	0	0
$966$	0	0
$967$	30724.5	1.02175	0.510876	−	0.859654i	$-0.329321\pi$
$967$	30724.5	1.02175	0.510876	+	0.859654i	$0.329321\pi$
$968$	−3919.10	−0.130129
$969$	0	0
$970$	0	0
$971$	49791.0	1.64559	0.822796	−	0.568336i	$-0.192413\pi$
$971$	49791.0	1.64559	0.822796	+	0.568336i	$0.192413\pi$
$972$	0	0
$973$	−2087.84	−0.0687906
$974$	44042.9	1.44890
$975$	0	0
$976$	−34755.6	−1.13985
$977$	−22391.9	−0.733244	−0.366622	−	0.930370i	$-0.619486\pi$
$977$	−22391.9	−0.733244	−0.366622	+	0.930370i	$0.619486\pi$
$978$	0	0
$979$	47230.7	1.54188
$980$	0	0
$981$	0	0
$982$	−14843.5	−0.482357
$983$	−24619.3	−0.798815	−0.399408	−	0.916773i	$-0.630784\pi$
$983$	−24619.3	−0.798815	−0.399408	+	0.916773i	$0.630784\pi$
$984$	0	0
$985$	0	0
$986$	102117.	3.29825
$987$	0	0
$988$	−6047.56	−0.194735
$989$	13581.1	0.436658
$990$	0	0
$991$	24540.3	0.786629	0.393314	−	0.919404i	$-0.371328\pi$
$991$	24540.3	0.786629	0.393314	+	0.919404i	$0.371328\pi$
$992$	−861.636	−0.0275776
$993$	0	0
$994$	652.937	0.0208349
$995$	0	0
$996$	0	0
$997$	−45956.1	−1.45982	−0.729912	−	0.683542i	$-0.760439\pi$
$997$	−45956.1	−1.45982	−0.729912	+	0.683542i	$0.760439\pi$
$998$	−63146.7	−2.00288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.j.1.1		2
3.2	odd	2		2025.4.a.l.1.2		2
5.4	even	2		405.4.a.e.1.2		2
9.2	odd	6		225.4.e.a.76.1		4
9.5	odd	6		225.4.e.a.151.1		4
15.14	odd	2		405.4.a.d.1.1		2
45.2	even	12		225.4.k.a.49.4		8
45.4	even	6		135.4.e.a.46.1		4
45.14	odd	6		45.4.e.a.16.2	✓	4
45.23	even	12		225.4.k.a.124.4		8
45.29	odd	6		45.4.e.a.31.2	yes	4
45.32	even	12		225.4.k.a.124.1		8
45.34	even	6		135.4.e.a.91.1		4
45.38	even	12		225.4.k.a.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.a.16.2	✓	4	45.14	odd	6
45.4.e.a.31.2	yes	4	45.29	odd	6
135.4.e.a.46.1		4	45.4	even	6
135.4.e.a.91.1		4	45.34	even	6
225.4.e.a.76.1		4	9.2	odd	6
225.4.e.a.151.1		4	9.5	odd	6
225.4.k.a.49.1		8	45.38	even	12
225.4.k.a.49.4		8	45.2	even	12
225.4.k.a.124.1		8	45.32	even	12
225.4.k.a.124.4		8	45.23	even	12
405.4.a.d.1.1		2	15.14	odd	2
405.4.a.e.1.2		2	5.4	even	2
2025.4.a.j.1.1		2	1.1	even	1	trivial
2025.4.a.l.1.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-3.37228 q^{2} +3.37228 q^{4} -1.62772 q^{7} +15.6060 q^{8} +32.8614 q^{11} +33.0217 q^{13} +5.48913 q^{14} -79.6060 q^{16} +110.307 q^{17} -54.3070 q^{19} -110.818 q^{22} -67.4375 q^{23} -111.359 q^{26} -5.48913 q^{28} -274.519 q^{29} -6.00000 q^{31} +143.606 q^{32} -371.986 q^{34} +347.723 q^{37} +183.139 q^{38} -291.337 q^{41} -201.388 q^{43} +110.818 q^{44} +227.418 q^{46} -482.095 q^{47} -340.351 q^{49} +111.359 q^{52} -175.228 q^{53} -25.4021 q^{56} +925.755 q^{58} +183.242 q^{59} +436.595 q^{61} +20.2337 q^{62} +152.568 q^{64} +831.500 q^{67} +371.986 q^{68} +118.951 q^{71} -183.318 q^{73} -1172.62 q^{74} -183.139 q^{76} -53.4891 q^{77} -638.293 q^{79} +982.470 q^{82} -1494.64 q^{83} +679.139 q^{86} +512.834 q^{88} +1437.27 q^{89} -53.7501 q^{91} -227.418 q^{92} +1625.76 q^{94} +891.769 q^{97} +1147.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{4} - 9 q^{7} - 9 q^{8} + 37 q^{11} + 112 q^{13} - 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} - 101 q^{22} - 267 q^{23} + 76 q^{26} + 12 q^{28} - 325 q^{29} - 12 q^{31} + 247 q^{32}+ \cdots + 463 q^{98}+O(q^{100}) \) 2 * q - q^2 + q^4 - 9 * q^7 - 9 * q^8 + 37 * q^11 + 112 * q^13 - 12 * q^14 - 119 * q^16 + 77 * q^17 + 35 * q^19 - 101 * q^22 - 267 * q^23 + 76 * q^26 + 12 * q^28 - 325 * q^29 - 12 * q^31 + 247 * q^32 - 451 * q^34 + 638 * q^37 + 395 * q^38 - 238 * q^41 + 97 * q^43 + 101 * q^44 - 246 * q^46 - 901 * q^47 - 629 * q^49 - 76 * q^52 + 224 * q^53 + 156 * q^56 + 806 * q^58 + 85 * q^59 - 247 * q^61 + 6 * q^62 + 713 * q^64 + 606 * q^67 + 451 * q^68 - 394 * q^71 + 811 * q^73 - 484 * q^74 - 395 * q^76 - 84 * q^77 - 840 * q^79 + 1109 * q^82 - 387 * q^83 + 1387 * q^86 + 411 * q^88 + 1065 * q^89 - 636 * q^91 + 246 * q^92 + 632 * q^94 + 1031 * q^97 + 463 * q^98

Introduction

L-functions

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Database

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