LMFDB - Embedded newform 1250.4.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,4,Mod(1,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1250.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1250.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:	$73.7523875072$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 145x^{6} - 180x^{5} + 5585x^{4} + 8550x^{3} - 49600x^{2} - 23400x + 24400 $ x^8 - 145x^6 - 180x^5 + 5585x^4 + 8550x^3 - 49600x^2 - 23400x + 24400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-7.69551$ of defining polynomial
Character	$\chi$	$=$	1250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +8.31354 q^{3} +4.00000 q^{4} +16.6271 q^{6} -26.4146 q^{7} +8.00000 q^{8} +42.1150 q^{9} -62.6568 q^{11} +33.2542 q^{12} +0.482265 q^{13} -52.8293 q^{14} +16.0000 q^{16} -105.983 q^{17} +84.2300 q^{18} +76.0259 q^{19} -219.599 q^{21} -125.314 q^{22} -95.6651 q^{23} +66.5084 q^{24} +0.964531 q^{26} +125.659 q^{27} -105.659 q^{28} +3.73342 q^{29} -2.31872 q^{31} +32.0000 q^{32} -520.900 q^{33} -211.967 q^{34} +168.460 q^{36} -11.8767 q^{37} +152.052 q^{38} +4.00934 q^{39} -52.6278 q^{41} -439.199 q^{42} -262.484 q^{43} -250.627 q^{44} -191.330 q^{46} -271.270 q^{47} +133.017 q^{48} +354.734 q^{49} -881.098 q^{51} +1.92906 q^{52} +359.541 q^{53} +251.319 q^{54} -211.317 q^{56} +632.045 q^{57} +7.46684 q^{58} +62.0750 q^{59} -364.767 q^{61} -4.63745 q^{62} -1112.45 q^{63} +64.0000 q^{64} -1041.80 q^{66} -221.351 q^{67} -423.934 q^{68} -795.316 q^{69} +714.680 q^{71} +336.920 q^{72} +128.238 q^{73} -23.7534 q^{74} +304.104 q^{76} +1655.06 q^{77} +8.01867 q^{78} -1116.77 q^{79} -92.4304 q^{81} -105.256 q^{82} +500.570 q^{83} -878.397 q^{84} -524.968 q^{86} +31.0379 q^{87} -501.255 q^{88} -1138.03 q^{89} -12.7389 q^{91} -382.660 q^{92} -19.2768 q^{93} -542.541 q^{94} +266.033 q^{96} -735.557 q^{97} +709.467 q^{98} -2638.79 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} - 4 q^{3} + 32 q^{4} - 8 q^{6} - 27 q^{7} + 64 q^{8} + 76 q^{9} - 39 q^{11} - 16 q^{12} - 179 q^{13} - 54 q^{14} + 128 q^{16} - 247 q^{17} + 152 q^{18} - 25 q^{19} - 104 q^{21} - 78 q^{22}+ \cdots - 893 q^{99}+O(q^{100}) $ 8 * q + 16 * q^2 - 4 * q^3 + 32 * q^4 - 8 * q^6 - 27 * q^7 + 64 * q^8 + 76 * q^9 - 39 * q^11 - 16 * q^12 - 179 * q^13 - 54 * q^14 + 128 * q^16 - 247 * q^17 + 152 * q^18 - 25 * q^19 - 104 * q^21 - 78 * q^22 - 179 * q^23 - 32 * q^24 - 358 * q^26 - 610 * q^27 - 108 * q^28 - 565 * q^29 - 89 * q^31 + 256 * q^32 - 563 * q^33 - 494 * q^34 + 304 * q^36 - 287 * q^37 - 50 * q^38 + 722 * q^39 - 394 * q^41 - 208 * q^42 - 529 * q^43 - 156 * q^44 - 358 * q^46 + 758 * q^47 - 64 * q^48 + 849 * q^49 - 1804 * q^51 - 716 * q^52 + 56 * q^53 - 1220 * q^54 - 216 * q^56 - 760 * q^57 - 1130 * q^58 - 1220 * q^59 - 489 * q^61 - 178 * q^62 - 3819 * q^63 + 512 * q^64 - 1126 * q^66 - 2547 * q^67 - 988 * q^68 + 17 * q^69 + 1576 * q^71 + 608 * q^72 + 166 * q^73 - 574 * q^74 - 100 * q^76 - 954 * q^77 + 1444 * q^78 - 1540 * q^79 + 1868 * q^81 - 788 * q^82 - 929 * q^83 - 416 * q^84 - 1058 * q^86 - 1755 * q^87 - 312 * q^88 - 1150 * q^89 - 1939 * q^91 - 716 * q^92 - 128 * q^93 + 1516 * q^94 - 128 * q^96 - 632 * q^97 + 1698 * q^98 - 893 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.31354	1.59994	0.799971	−	0.600038i	$-0.204848\pi$
$3$	8.31354	1.59994	0.799971	+	0.600038i	$0.204848\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	16.6271	1.13133
$7$	−26.4146	−1.42626	−0.713128	−	0.701033i	$-0.752722\pi$
$7$	−26.4146	−1.42626	−0.713128	+	0.701033i	$0.752722\pi$
$8$	8.00000	0.353553
$9$	42.1150	1.55982
$10$	0	0
$11$	−62.6568	−1.71743	−0.858716	−	0.512452i	$-0.828737\pi$
$11$	−62.6568	−1.71743	−0.858716	+	0.512452i	$0.828737\pi$
$12$	33.2542	0.799971
$13$	0.482265	0.0102890	0.00514448	−	0.999987i	$-0.498362\pi$
$13$	0.482265	0.0102890	0.00514448	+	0.999987i	$0.498362\pi$
$14$	−52.8293	−1.00852
$15$	0	0
$16$	16.0000	0.250000
$17$	−105.983	−1.51204	−0.756022	−	0.654546i	$-0.772860\pi$
$17$	−105.983	−1.51204	−0.756022	+	0.654546i	$0.772860\pi$
$18$	84.2300	1.10296
$19$	76.0259	0.917976	0.458988	−	0.888442i	$-0.348212\pi$
$19$	76.0259	0.917976	0.458988	+	0.888442i	$0.348212\pi$
$20$	0	0
$21$	−219.599	−2.28193
$22$	−125.314	−1.21441
$23$	−95.6651	−0.867285	−0.433642	−	0.901085i	$-0.642772\pi$
$23$	−95.6651	−0.867285	−0.433642	+	0.901085i	$0.642772\pi$
$24$	66.5084	0.565665
$25$	0	0
$26$	0.964531	0.00727539
$27$	125.659	0.895673
$28$	−105.659	−0.713128
$29$	3.73342	0.0239061	0.0119531	−	0.999929i	$-0.496195\pi$
$29$	3.73342	0.0239061	0.0119531	+	0.999929i	$0.496195\pi$
$30$	0	0
$31$	−2.31872	−0.0134340	−0.00671702	−	0.999977i	$-0.502138\pi$
$31$	−2.31872	−0.0134340	−0.00671702	+	0.999977i	$0.502138\pi$
$32$	32.0000	0.176777
$33$	−520.900	−2.74779
$34$	−211.967	−1.06918
$35$	0	0
$36$	168.460	0.779908
$37$	−11.8767	−0.0527708	−0.0263854	−	0.999652i	$-0.508400\pi$
$37$	−11.8767	−0.0527708	−0.0263854	+	0.999652i	$0.508400\pi$
$38$	152.052	0.649107
$39$	4.00934	0.0164617
$40$	0	0
$41$	−52.6278	−0.200465	−0.100233	−	0.994964i	$-0.531959\pi$
$41$	−52.6278	−0.200465	−0.100233	+	0.994964i	$0.531959\pi$
$42$	−439.199	−1.61357
$43$	−262.484	−0.930895	−0.465447	−	0.885076i	$-0.654107\pi$
$43$	−262.484	−0.930895	−0.465447	+	0.885076i	$0.654107\pi$
$44$	−250.627	−0.858716
$45$	0	0
$46$	−191.330	−0.613263
$47$	−271.270	−0.841891	−0.420945	−	0.907086i	$-0.638302\pi$
$47$	−271.270	−0.841891	−0.420945	+	0.907086i	$0.638302\pi$
$48$	133.017	0.399986
$49$	354.734	1.03421
$50$	0	0
$51$	−881.098	−2.41918
$52$	1.92906	0.00514448
$53$	359.541	0.931824	0.465912	−	0.884831i	$-0.345726\pi$
$53$	359.541	0.931824	0.465912	+	0.884831i	$0.345726\pi$
$54$	251.319	0.633336
$55$	0	0
$56$	−211.317	−0.504258
$57$	632.045	1.46871
$58$	7.46684	0.0169042
$59$	62.0750	0.136974	0.0684871	−	0.997652i	$-0.478183\pi$
$59$	62.0750	0.136974	0.0684871	+	0.997652i	$0.478183\pi$
$60$	0	0
$61$	−364.767	−0.765634	−0.382817	−	0.923824i	$-0.625046\pi$
$61$	−364.767	−0.765634	−0.382817	+	0.923824i	$0.625046\pi$
$62$	−4.63745	−0.00949930
$63$	−1112.45	−2.22470
$64$	64.0000	0.125000
$65$	0	0
$66$	−1041.80	−1.94298
$67$	−221.351	−0.403617	−0.201809	−	0.979425i	$-0.564682\pi$
$67$	−221.351	−0.403617	−0.201809	+	0.979425i	$0.564682\pi$
$68$	−423.934	−0.756022
$69$	−795.316	−1.38761
$70$	0	0
$71$	714.680	1.19460	0.597302	−	0.802016i	$-0.296239\pi$
$71$	714.680	1.19460	0.597302	+	0.802016i	$0.296239\pi$
$72$	336.920	0.551478
$73$	128.238	0.205604	0.102802	−	0.994702i	$-0.467219\pi$
$73$	128.238	0.205604	0.102802	+	0.994702i	$0.467219\pi$
$74$	−23.7534	−0.0373146
$75$	0	0
$76$	304.104	0.458988
$77$	1655.06	2.44950
$78$	8.01867	0.0116402
$79$	−1116.77	−1.59045	−0.795227	−	0.606311i	$-0.792648\pi$
$79$	−1116.77	−1.59045	−0.795227	+	0.606311i	$0.792648\pi$
$80$	0	0
$81$	−92.4304	−0.126791
$82$	−105.256	−0.141750
$83$	500.570	0.661985	0.330992	−	0.943633i	$-0.392617\pi$
$83$	500.570	0.661985	0.330992	+	0.943633i	$0.392617\pi$
$84$	−878.397	−1.14096
$85$	0	0
$86$	−524.968	−0.658242
$87$	31.0379	0.0382485
$88$	−501.255	−0.607204
$89$	−1138.03	−1.35541	−0.677703	−	0.735336i	$-0.737025\pi$
$89$	−1138.03	−1.35541	−0.677703	+	0.735336i	$0.737025\pi$
$90$	0	0
$91$	−12.7389	−0.0146747
$92$	−382.660	−0.433642
$93$	−19.2768	−0.0214937
$94$	−542.541	−0.595307
$95$	0	0
$96$	266.033	0.282833
$97$	−735.557	−0.769944	−0.384972	−	0.922928i	$-0.625789\pi$
$97$	−735.557	−0.769944	−0.384972	+	0.922928i	$0.625789\pi$
$98$	709.467	0.731296
$99$	−2638.79	−2.67888
$100$	0	0
$101$	−1249.44	−1.23093	−0.615463	−	0.788166i	$-0.711031\pi$
$101$	−1249.44	−1.23093	−0.615463	+	0.788166i	$0.711031\pi$
$102$	−1762.20	−1.71062
$103$	332.831	0.318397	0.159198	−	0.987247i	$-0.449109\pi$
$103$	332.831	0.318397	0.159198	+	0.987247i	$0.449109\pi$
$104$	3.85812	0.00363769
$105$	0	0
$106$	719.081	0.658899
$107$	2079.52	1.87883	0.939414	−	0.342786i	$-0.111370\pi$
$107$	2079.52	1.87883	0.939414	+	0.342786i	$0.111370\pi$
$108$	502.638	0.447836
$109$	704.965	0.619481	0.309740	−	0.950821i	$-0.399758\pi$
$109$	704.965	0.619481	0.309740	+	0.950821i	$0.399758\pi$
$110$	0	0
$111$	−98.7376	−0.0844303
$112$	−422.634	−0.356564
$113$	379.678	0.316080	0.158040	−	0.987433i	$-0.449482\pi$
$113$	379.678	0.316080	0.158040	+	0.987433i	$0.449482\pi$
$114$	1264.09	1.03853
$115$	0	0
$116$	14.9337	0.0119531
$117$	20.3106	0.0160489
$118$	124.150	0.0968554
$119$	2799.51	2.15656
$120$	0	0
$121$	2594.88	1.94957
$122$	−729.535	−0.541385
$123$	−437.524	−0.320733
$124$	−9.27490	−0.00671702
$125$	0	0
$126$	−2224.91	−1.57310
$127$	−1403.00	−0.980286	−0.490143	−	0.871642i	$-0.663055\pi$
$127$	−1403.00	−0.980286	−0.490143	+	0.871642i	$0.663055\pi$
$128$	128.000	0.0883883
$129$	−2182.17	−1.48938
$130$	0	0
$131$	1460.80	0.974279	0.487139	−	0.873324i	$-0.338040\pi$
$131$	1460.80	0.974279	0.487139	+	0.873324i	$0.338040\pi$
$132$	−2083.60	−1.37390
$133$	−2008.20	−1.30927
$134$	−442.702	−0.285401
$135$	0	0
$136$	−847.867	−0.534588
$137$	−1486.96	−0.927294	−0.463647	−	0.886020i	$-0.653459\pi$
$137$	−1486.96	−0.927294	−0.463647	+	0.886020i	$0.653459\pi$
$138$	−1590.63	−0.981186
$139$	129.580	0.0790705	0.0395353	−	0.999218i	$-0.487412\pi$
$139$	129.580	0.0790705	0.0395353	+	0.999218i	$0.487412\pi$
$140$	0	0
$141$	−2255.22	−1.34698
$142$	1429.36	0.844713
$143$	−30.2172	−0.0176706
$144$	673.840	0.389954
$145$	0	0
$146$	256.476	0.145384
$147$	2949.09	1.65467
$148$	−47.5069	−0.0263854
$149$	1055.54	0.580357	0.290179	−	0.956973i	$-0.406285\pi$
$149$	1055.54	0.580357	0.290179	+	0.956973i	$0.406285\pi$
$150$	0	0
$151$	2197.85	1.18449	0.592246	−	0.805757i	$-0.298241\pi$
$151$	2197.85	1.18449	0.592246	+	0.805757i	$0.298241\pi$
$152$	608.207	0.324554
$153$	−4463.49	−2.35851
$154$	3310.12	1.73206
$155$	0	0
$156$	16.0373	0.00823087
$157$	−1392.21	−0.707712	−0.353856	−	0.935300i	$-0.615130\pi$
$157$	−1392.21	−0.707712	−0.353856	+	0.935300i	$0.615130\pi$
$158$	−2233.53	−1.12462
$159$	2989.06	1.49087
$160$	0	0
$161$	2526.96	1.23697
$162$	−184.861	−0.0896545
$163$	−3494.56	−1.67923	−0.839617	−	0.543178i	$-0.817221\pi$
$163$	−3494.56	−1.67923	−0.839617	+	0.543178i	$0.817221\pi$
$164$	−210.511	−0.100233
$165$	0	0
$166$	1001.14	0.468094
$167$	3166.47	1.46724	0.733620	−	0.679560i	$-0.237829\pi$
$167$	3166.47	1.46724	0.733620	+	0.679560i	$0.237829\pi$
$168$	−1756.79	−0.806784
$169$	−2196.77	−0.999894
$170$	0	0
$171$	3201.83	1.43187
$172$	−1049.94	−0.465447
$173$	−309.384	−0.135966	−0.0679828	−	0.997686i	$-0.521656\pi$
$173$	−309.384	−0.135966	−0.0679828	+	0.997686i	$0.521656\pi$
$174$	62.0759	0.0270457
$175$	0	0
$176$	−1002.51	−0.429358
$177$	516.064	0.219151
$178$	−2276.06	−0.958417
$179$	3711.31	1.54970	0.774850	−	0.632145i	$-0.217826\pi$
$179$	3711.31	1.54970	0.774850	+	0.632145i	$0.217826\pi$
$180$	0	0
$181$	943.659	0.387523	0.193761	−	0.981049i	$-0.437931\pi$
$181$	943.659	0.387523	0.193761	+	0.981049i	$0.437931\pi$
$182$	−25.4777	−0.0103766
$183$	−3032.51	−1.22497
$184$	−765.321	−0.306632
$185$	0	0
$186$	−38.5536	−0.0151983
$187$	6640.59	2.59683
$188$	−1085.08	−0.420945
$189$	−3319.25	−1.27746
$190$	0	0
$191$	2477.22	0.938456	0.469228	−	0.883077i	$-0.344532\pi$
$191$	2477.22	0.938456	0.469228	+	0.883077i	$0.344532\pi$
$192$	532.067	0.199993
$193$	−2345.06	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$193$	−2345.06	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$194$	−1471.11	−0.544432
$195$	0	0
$196$	1418.93	0.517104
$197$	3803.71	1.37565	0.687824	−	0.725877i	$-0.258566\pi$
$197$	3803.71	1.37565	0.687824	+	0.725877i	$0.258566\pi$
$198$	−5277.59	−1.89425
$199$	2972.61	1.05891	0.529454	−	0.848339i	$-0.322397\pi$
$199$	2972.61	1.05891	0.529454	+	0.848339i	$0.322397\pi$
$200$	0	0
$201$	−1840.21	−0.645764
$202$	−2498.87	−0.870396
$203$	−98.6169	−0.0340963
$204$	−3524.39	−1.20959
$205$	0	0
$206$	665.663	0.225140
$207$	−4028.94	−1.35280
$208$	7.71625	0.00257224
$209$	−4763.55	−1.57656
$210$	0	0
$211$	2055.82	0.670751	0.335375	−	0.942085i	$-0.391137\pi$
$211$	2055.82	0.670751	0.335375	+	0.942085i	$0.391137\pi$
$212$	1438.16	0.465912
$213$	5941.53	1.91130
$214$	4159.03	1.32853
$215$	0	0
$216$	1005.28	0.316668
$217$	61.2483	0.0191604
$218$	1409.93	0.438039
$219$	1066.11	0.328955
$220$	0	0
$221$	−51.1121	−0.0155574
$222$	−197.475	−0.0597012
$223$	−3154.19	−0.947175	−0.473588	−	0.880747i	$-0.657041\pi$
$223$	−3154.19	−0.947175	−0.473588	+	0.880747i	$0.657041\pi$
$224$	−845.269	−0.252129
$225$	0	0
$226$	759.355	0.223502
$227$	4452.14	1.30176	0.650879	−	0.759181i	$-0.274400\pi$
$227$	4452.14	1.30176	0.650879	+	0.759181i	$0.274400\pi$
$228$	2528.18	0.734354
$229$	813.711	0.234810	0.117405	−	0.993084i	$-0.462542\pi$
$229$	813.711	0.234810	0.117405	+	0.993084i	$0.462542\pi$
$230$	0	0
$231$	13759.4	3.91906
$232$	29.8673	0.00845210
$233$	5120.62	1.43975	0.719877	−	0.694101i	$-0.244198\pi$
$233$	5120.62	1.43975	0.719877	+	0.694101i	$0.244198\pi$
$234$	40.6212	0.0113483
$235$	0	0
$236$	248.300	0.0684871
$237$	−9284.28	−2.54464
$238$	5599.03	1.52492
$239$	−320.787	−0.0868201	−0.0434101	−	0.999057i	$-0.513822\pi$
$239$	−320.787	−0.0868201	−0.0434101	+	0.999057i	$0.513822\pi$
$240$	0	0
$241$	−89.5597	−0.0239379	−0.0119690	−	0.999928i	$-0.503810\pi$
$241$	−89.5597	−0.0239379	−0.0119690	+	0.999928i	$0.503810\pi$
$242$	5189.76	1.37856
$243$	−4161.23	−1.09853
$244$	−1459.07	−0.382817
$245$	0	0
$246$	−875.047	−0.226793
$247$	36.6647	0.00944501
$248$	−18.5498	−0.00474965
$249$	4161.51	1.05914
$250$	0	0
$251$	−7071.96	−1.77840	−0.889199	−	0.457520i	$-0.848738\pi$
$251$	−7071.96	−1.77840	−0.889199	+	0.457520i	$0.848738\pi$
$252$	−4449.81	−1.11235
$253$	5994.07	1.48950
$254$	−2806.00	−0.693167
$255$	0	0
$256$	256.000	0.0625000
$257$	−2518.68	−0.611327	−0.305663	−	0.952140i	$-0.598878\pi$
$257$	−2518.68	−0.611327	−0.305663	+	0.952140i	$0.598878\pi$
$258$	−4364.35	−1.05315
$259$	313.719	0.0752647
$260$	0	0
$261$	157.233	0.0372892
$262$	2921.60	0.688919
$263$	−8135.10	−1.90734	−0.953672	−	0.300847i	$-0.902731\pi$
$263$	−8135.10	−1.90734	−0.953672	+	0.300847i	$0.902731\pi$
$264$	−4167.20	−0.971491
$265$	0	0
$266$	−4016.40	−0.925794
$267$	−9461.08	−2.16857
$268$	−885.405	−0.201809
$269$	649.925	0.147311	0.0736554	−	0.997284i	$-0.476533\pi$
$269$	649.925	0.147311	0.0736554	+	0.997284i	$0.476533\pi$
$270$	0	0
$271$	−291.643	−0.0653728	−0.0326864	−	0.999466i	$-0.510406\pi$
$271$	−291.643	−0.0653728	−0.0326864	+	0.999466i	$0.510406\pi$
$272$	−1695.73	−0.378011
$273$	−105.905	−0.0234787
$274$	−2973.91	−0.655696
$275$	0	0
$276$	−3181.26	−0.693803
$277$	−3635.51	−0.788580	−0.394290	−	0.918986i	$-0.629009\pi$
$277$	−3635.51	−0.788580	−0.394290	+	0.918986i	$0.629009\pi$
$278$	259.159	0.0559113
$279$	−97.6531	−0.0209546
$280$	0	0
$281$	−8408.24	−1.78503	−0.892516	−	0.451016i	$-0.851062\pi$
$281$	−8408.24	−1.78503	−0.892516	+	0.451016i	$0.851062\pi$
$282$	−4510.44	−0.952456
$283$	−929.858	−0.195316	−0.0976578	−	0.995220i	$-0.531135\pi$
$283$	−929.858	−0.195316	−0.0976578	+	0.995220i	$0.531135\pi$
$284$	2858.72	0.597302
$285$	0	0
$286$	−60.4345	−0.0124950
$287$	1390.15	0.285915
$288$	1347.68	0.275739
$289$	6319.49	1.28628
$290$	0	0
$291$	−6115.09	−1.23187
$292$	512.952	0.102802
$293$	−413.694	−0.0824856	−0.0412428	−	0.999149i	$-0.513132\pi$
$293$	−413.694	−0.0824856	−0.0412428	+	0.999149i	$0.513132\pi$
$294$	5898.19	1.17003
$295$	0	0
$296$	−95.0137	−0.0186573
$297$	−7873.42	−1.53826
$298$	2111.08	0.410375
$299$	−46.1360	−0.00892345
$300$	0	0
$301$	6933.43	1.32769
$302$	4395.69	0.837562
$303$	−10387.2	−1.96941
$304$	1216.41	0.229494
$305$	0	0
$306$	−8926.99	−1.66772
$307$	5657.76	1.05181	0.525905	−	0.850544i	$-0.323727\pi$
$307$	5657.76	1.05181	0.525905	+	0.850544i	$0.323727\pi$
$308$	6620.23	1.22475
$309$	2767.01	0.509416
$310$	0	0
$311$	−1685.09	−0.307244	−0.153622	−	0.988130i	$-0.549094\pi$
$311$	−1685.09	−0.307244	−0.153622	+	0.988130i	$0.549094\pi$
$312$	32.0747	0.00582010
$313$	−3655.64	−0.660156	−0.330078	−	0.943954i	$-0.607075\pi$
$313$	−3655.64	−0.660156	−0.330078	+	0.943954i	$0.607075\pi$
$314$	−2784.43	−0.500428
$315$	0	0
$316$	−4467.06	−0.795227
$317$	−10257.8	−1.81747	−0.908735	−	0.417374i	$-0.862950\pi$
$317$	−10257.8	−1.81747	−0.908735	+	0.417374i	$0.862950\pi$
$318$	5978.11	1.05420
$319$	−233.924	−0.0410572
$320$	0	0
$321$	17288.2	3.00602
$322$	5053.92	0.874671
$323$	−8057.49	−1.38802
$324$	−369.722	−0.0633953
$325$	0	0
$326$	−6989.13	−1.18740
$327$	5860.76	0.991134
$328$	−421.023	−0.0708752
$329$	7165.51	1.20075
$330$	0	0
$331$	8118.03	1.34806	0.674029	−	0.738705i	$-0.264562\pi$
$331$	8118.03	1.34806	0.674029	+	0.738705i	$0.264562\pi$
$332$	2002.28	0.330992
$333$	−500.188	−0.0823127
$334$	6332.95	1.03750
$335$	0	0
$336$	−3513.59	−0.570482
$337$	−2334.38	−0.377335	−0.188668	−	0.982041i	$-0.560417\pi$
$337$	−2334.38	−0.377335	−0.188668	+	0.982041i	$0.560417\pi$
$338$	−4393.53	−0.707032
$339$	3156.47	0.505710
$340$	0	0
$341$	145.284	0.0230720
$342$	6403.67	1.01249
$343$	−309.940	−0.0487906
$344$	−2099.87	−0.329121
$345$	0	0
$346$	−618.769	−0.0961422
$347$	−7436.64	−1.15049	−0.575244	−	0.817982i	$-0.695093\pi$
$347$	−7436.64	−1.15049	−0.575244	+	0.817982i	$0.695093\pi$
$348$	124.152	0.0191242
$349$	−2740.81	−0.420379	−0.210190	−	0.977661i	$-0.567408\pi$
$349$	−2740.81	−0.420379	−0.210190	+	0.977661i	$0.567408\pi$
$350$	0	0
$351$	60.6012	0.00921554
$352$	−2005.02	−0.303602
$353$	−8340.03	−1.25749	−0.628747	−	0.777610i	$-0.716432\pi$
$353$	−8340.03	−1.25749	−0.628747	+	0.777610i	$0.716432\pi$
$354$	1032.13	0.154963
$355$	0	0
$356$	−4552.13	−0.677703
$357$	23273.9	3.45038
$358$	7422.62	1.09580
$359$	2000.28	0.294069	0.147034	−	0.989131i	$-0.453027\pi$
$359$	2000.28	0.294069	0.147034	+	0.989131i	$0.453027\pi$
$360$	0	0
$361$	−1079.06	−0.157320
$362$	1887.32	0.274020
$363$	21572.7	3.11920
$364$	−50.9555	−0.00733734
$365$	0	0
$366$	−6065.02	−0.866185
$367$	−4855.63	−0.690631	−0.345316	−	0.938487i	$-0.612228\pi$
$367$	−4855.63	−0.690631	−0.345316	+	0.938487i	$0.612228\pi$
$368$	−1530.64	−0.216821
$369$	−2216.42	−0.312689
$370$	0	0
$371$	−9497.14	−1.32902
$372$	−77.1073	−0.0107468
$373$	10746.2	1.49174	0.745868	−	0.666094i	$-0.232035\pi$
$373$	10746.2	1.49174	0.745868	+	0.666094i	$0.232035\pi$
$374$	13281.2	1.83624
$375$	0	0
$376$	−2170.16	−0.297653
$377$	1.80050	0.000245969 0
$378$	−6638.50	−0.903300
$379$	−8006.27	−1.08510	−0.542552	−	0.840022i	$-0.682542\pi$
$379$	−8006.27	−1.08510	−0.542552	+	0.840022i	$0.682542\pi$
$380$	0	0
$381$	−11663.9	−1.56840
$382$	4954.44	0.663589
$383$	156.601	0.0208928	0.0104464	−	0.999945i	$-0.496675\pi$
$383$	156.601	0.0208928	0.0104464	+	0.999945i	$0.496675\pi$
$384$	1064.13	0.141416
$385$	0	0
$386$	−4690.11	−0.618447
$387$	−11054.5	−1.45202
$388$	−2942.23	−0.384972
$389$	11362.5	1.48098	0.740491	−	0.672066i	$-0.234593\pi$
$389$	11362.5	1.48098	0.740491	+	0.672066i	$0.234593\pi$
$390$	0	0
$391$	10138.9	1.31137
$392$	2837.87	0.365648
$393$	12144.4	1.55879
$394$	7607.41	0.972731
$395$	0	0
$396$	−10555.2	−1.33944
$397$	630.607	0.0797210	0.0398605	−	0.999205i	$-0.487309\pi$
$397$	630.607	0.0797210	0.0398605	+	0.999205i	$0.487309\pi$
$398$	5945.22	0.748761
$399$	−16695.2	−2.09476
$400$	0	0
$401$	−2493.80	−0.310560	−0.155280	−	0.987871i	$-0.549628\pi$
$401$	−2493.80	−0.310560	−0.155280	+	0.987871i	$0.549628\pi$
$402$	−3680.43	−0.456624
$403$	−1.11824	−0.000138222 0
$404$	−4997.74	−0.615463
$405$	0	0
$406$	−197.234	−0.0241097
$407$	744.157	0.0906303
$408$	−7048.78	−0.855311
$409$	−11742.3	−1.41960	−0.709801	−	0.704402i	$-0.751215\pi$
$409$	−11742.3	−1.41960	−0.709801	+	0.704402i	$0.751215\pi$
$410$	0	0
$411$	−12361.9	−1.48362
$412$	1331.33	0.159198
$413$	−1639.69	−0.195361
$414$	−8057.88	−0.956577
$415$	0	0
$416$	15.4325	0.00181885
$417$	1077.27	0.126508
$418$	−9527.09	−1.11480
$419$	−7553.08	−0.880650	−0.440325	−	0.897839i	$-0.645137\pi$
$419$	−7553.08	−0.880650	−0.440325	+	0.897839i	$0.645137\pi$
$420$	0	0
$421$	−11542.8	−1.33626	−0.668128	−	0.744046i	$-0.732904\pi$
$421$	−11542.8	−1.33626	−0.668128	+	0.744046i	$0.732904\pi$
$422$	4111.64	0.474293
$423$	−11424.6	−1.31319
$424$	2876.32	0.329450
$425$	0	0
$426$	11883.1	1.35149
$427$	9635.20	1.09199
$428$	8318.07	0.939414
$429$	−251.212	−0.0282719
$430$	0	0
$431$	−7597.04	−0.849040	−0.424520	−	0.905418i	$-0.639557\pi$
$431$	−7597.04	−0.849040	−0.424520	+	0.905418i	$0.639557\pi$
$432$	2010.55	0.223918
$433$	−8689.29	−0.964390	−0.482195	−	0.876064i	$-0.660160\pi$
$433$	−8689.29	−0.964390	−0.482195	+	0.876064i	$0.660160\pi$
$434$	122.497	0.0135484
$435$	0	0
$436$	2819.86	0.309740
$437$	−7273.03	−0.796147
$438$	2132.22	0.232606
$439$	15366.3	1.67060	0.835300	−	0.549794i	$-0.185294\pi$
$439$	15366.3	1.67060	0.835300	+	0.549794i	$0.185294\pi$
$440$	0	0
$441$	14939.6	1.61318
$442$	−102.224	−0.0110007
$443$	2948.65	0.316240	0.158120	−	0.987420i	$-0.449457\pi$
$443$	2948.65	0.316240	0.158120	+	0.987420i	$0.449457\pi$
$444$	−394.950	−0.0422151
$445$	0	0
$446$	−6308.38	−0.669754
$447$	8775.28	0.928538
$448$	−1690.54	−0.178282
$449$	12471.0	1.31079	0.655394	−	0.755287i	$-0.272503\pi$
$449$	12471.0	1.31079	0.655394	+	0.755287i	$0.272503\pi$
$450$	0	0
$451$	3297.49	0.344286
$452$	1518.71	0.158040
$453$	18271.9	1.89512
$454$	8904.29	0.920482
$455$	0	0
$456$	5056.36	0.519267
$457$	−13767.5	−1.40922	−0.704612	−	0.709593i	$-0.748879\pi$
$457$	−13767.5	−1.40922	−0.704612	+	0.709593i	$0.748879\pi$
$458$	1627.42	0.166036
$459$	−13317.8	−1.35430
$460$	0	0
$461$	13896.0	1.40390	0.701952	−	0.712224i	$-0.252312\pi$
$461$	13896.0	1.40390	0.701952	+	0.712224i	$0.252312\pi$
$462$	27518.8	2.77119
$463$	−15358.0	−1.54157	−0.770783	−	0.637098i	$-0.780135\pi$
$463$	−15358.0	−1.54157	−0.770783	+	0.637098i	$0.780135\pi$
$464$	59.7347	0.00597654
$465$	0	0
$466$	10241.2	1.01806
$467$	−11489.4	−1.13847	−0.569236	−	0.822174i	$-0.692761\pi$
$467$	−11489.4	−1.13847	−0.569236	+	0.822174i	$0.692761\pi$
$468$	81.2425	0.00802443
$469$	5846.92	0.575662
$470$	0	0
$471$	−11574.2	−1.13230
$472$	496.600	0.0484277
$473$	16446.4	1.59875
$474$	−18568.6	−1.79933
$475$	0	0
$476$	11198.1	1.07828
$477$	15142.1	1.45347
$478$	−641.575	−0.0613911
$479$	11566.2	1.10328	0.551640	−	0.834082i	$-0.314002\pi$
$479$	11566.2	1.10328	0.551640	+	0.834082i	$0.314002\pi$
$480$	0	0
$481$	−5.72773	−0.000542956 0
$482$	−179.119	−0.0169267
$483$	21008.0	1.97908
$484$	10379.5	0.974786
$485$	0	0
$486$	−8322.46	−0.776778
$487$	−9203.06	−0.856326	−0.428163	−	0.903702i	$-0.640839\pi$
$487$	−9203.06	−0.856326	−0.428163	+	0.903702i	$0.640839\pi$
$488$	−2918.14	−0.270693
$489$	−29052.2	−2.68668
$490$	0	0
$491$	2016.76	0.185367	0.0926833	−	0.995696i	$-0.470456\pi$
$491$	2016.76	0.185367	0.0926833	+	0.995696i	$0.470456\pi$
$492$	−1750.09	−0.160367
$493$	−395.680	−0.0361472
$494$	73.3294	0.00667863
$495$	0	0
$496$	−37.0996	−0.00335851
$497$	−18878.0	−1.70381
$498$	8323.03	0.748923
$499$	−1509.37	−0.135408	−0.0677041	−	0.997705i	$-0.521567\pi$
$499$	−1509.37	−0.135408	−0.0677041	+	0.997705i	$0.521567\pi$
$500$	0	0
$501$	26324.6	2.34750
$502$	−14143.9	−1.25752
$503$	16226.0	1.43833	0.719165	−	0.694839i	$-0.244524\pi$
$503$	16226.0	1.43833	0.719165	+	0.694839i	$0.244524\pi$
$504$	−8899.63	−0.786550
$505$	0	0
$506$	11988.1	1.05324
$507$	−18262.9	−1.59977
$508$	−5612.01	−0.490143
$509$	13391.3	1.16613	0.583063	−	0.812427i	$-0.301854\pi$
$509$	13391.3	1.16613	0.583063	+	0.812427i	$0.301854\pi$
$510$	0	0
$511$	−3387.36	−0.293245
$512$	512.000	0.0441942
$513$	9553.38	0.822206
$514$	−5037.36	−0.432273
$515$	0	0
$516$	−8728.70	−0.744689
$517$	16996.9	1.44589
$518$	627.438	0.0532202
$519$	−2572.08	−0.217537
$520$	0	0
$521$	1311.86	0.110314	0.0551570	−	0.998478i	$-0.482434\pi$
$521$	1311.86	0.110314	0.0551570	+	0.998478i	$0.482434\pi$
$522$	314.466	0.0263674
$523$	−7029.07	−0.587686	−0.293843	−	0.955854i	$-0.594934\pi$
$523$	−7029.07	−0.587686	−0.293843	+	0.955854i	$0.594934\pi$
$524$	5843.19	0.487139
$525$	0	0
$526$	−16270.2	−1.34870
$527$	245.746	0.0203129
$528$	−8334.41	−0.686948
$529$	−3015.19	−0.247817
$530$	0	0
$531$	2614.29	0.213655
$532$	−8032.79	−0.654635
$533$	−25.3806	−0.00206258
$534$	−18922.2	−1.53341
$535$	0	0
$536$	−1770.81	−0.142700
$537$	30854.1	2.47943
$538$	1299.85	0.104165
$539$	−22226.5	−1.77618
$540$	0	0
$541$	−15540.8	−1.23503	−0.617515	−	0.786559i	$-0.711861\pi$
$541$	−15540.8	−1.23503	−0.617515	+	0.786559i	$0.711861\pi$
$542$	−583.286	−0.0462256
$543$	7845.15	0.620014
$544$	−3391.47	−0.267294
$545$	0	0
$546$	−211.810	−0.0166019
$547$	12014.1	0.939098	0.469549	−	0.882906i	$-0.344417\pi$
$547$	12014.1	0.939098	0.469549	+	0.882906i	$0.344417\pi$
$548$	−5947.83	−0.463647
$549$	−15362.2	−1.19425
$550$	0	0
$551$	283.837	0.0219453
$552$	−6362.53	−0.490593
$553$	29499.0	2.26840
$554$	−7271.02	−0.557610
$555$	0	0
$556$	518.319	0.0395353
$557$	−2730.35	−0.207700	−0.103850	−	0.994593i	$-0.533116\pi$
$557$	−2730.35	−0.207700	−0.103850	+	0.994593i	$0.533116\pi$
$558$	−195.306	−0.0148172
$559$	−126.587	−0.00957793
$560$	0	0
$561$	55206.8	4.15478
$562$	−16816.5	−1.26221
$563$	−14430.0	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$563$	−14430.0	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$564$	−9020.87	−0.673488
$565$	0	0
$566$	−1859.72	−0.138109
$567$	2441.52	0.180836
$568$	5717.44	0.422357
$569$	−575.994	−0.0424375	−0.0212187	−	0.999775i	$-0.506755\pi$
$569$	−575.994	−0.0424375	−0.0212187	+	0.999775i	$0.506755\pi$
$570$	0	0
$571$	5000.10	0.366458	0.183229	−	0.983070i	$-0.441345\pi$
$571$	5000.10	0.366458	0.183229	+	0.983070i	$0.441345\pi$
$572$	−120.869	−0.00883529
$573$	20594.5	1.50148
$574$	2780.29	0.202173
$575$	0	0
$576$	2695.36	0.194977
$577$	−11691.7	−0.843555	−0.421777	−	0.906699i	$-0.638594\pi$
$577$	−11691.7	−0.843555	−0.421777	+	0.906699i	$0.638594\pi$
$578$	12639.0	0.909536
$579$	−19495.7	−1.39933
$580$	0	0
$581$	−13222.4	−0.944160
$582$	−12230.2	−0.871061
$583$	−22527.7	−1.60034
$584$	1025.90	0.0726921
$585$	0	0
$586$	−827.389	−0.0583261
$587$	−13087.0	−0.920205	−0.460103	−	0.887866i	$-0.652187\pi$
$587$	−13087.0	−0.920205	−0.460103	+	0.887866i	$0.652187\pi$
$588$	11796.4	0.827337
$589$	−176.283	−0.0123321
$590$	0	0
$591$	31622.3	2.20096
$592$	−190.027	−0.0131927
$593$	−5702.64	−0.394906	−0.197453	−	0.980312i	$-0.563267\pi$
$593$	−5702.64	−0.394906	−0.197453	+	0.980312i	$0.563267\pi$
$594$	−15746.8	−1.08771
$595$	0	0
$596$	4222.16	0.290179
$597$	24712.9	1.69419
$598$	−92.2720	−0.00630983
$599$	16685.8	1.13817	0.569084	−	0.822279i	$-0.307298\pi$
$599$	16685.8	1.13817	0.569084	+	0.822279i	$0.307298\pi$
$600$	0	0
$601$	27123.2	1.84089	0.920446	−	0.390869i	$-0.127826\pi$
$601$	27123.2	1.84089	0.920446	+	0.390869i	$0.127826\pi$
$602$	13866.9	0.938822
$603$	−9322.21	−0.629569
$604$	8791.39	0.592246
$605$	0	0
$606$	−20774.5	−1.39258
$607$	1749.87	0.117010	0.0585050	−	0.998287i	$-0.481367\pi$
$607$	1749.87	0.117010	0.0585050	+	0.998287i	$0.481367\pi$
$608$	2432.83	0.162277
$609$	−819.856	−0.0545521
$610$	0	0
$611$	−130.824	−0.00866217
$612$	−17854.0	−1.17926
$613$	27965.3	1.84259	0.921295	−	0.388865i	$-0.127133\pi$
$613$	27965.3	1.84259	0.921295	+	0.388865i	$0.127133\pi$
$614$	11315.5	0.743741
$615$	0	0
$616$	13240.5	0.866029
$617$	8833.05	0.576346	0.288173	−	0.957578i	$-0.406952\pi$
$617$	8833.05	0.576346	0.288173	+	0.957578i	$0.406952\pi$
$618$	5534.02	0.360212
$619$	10011.3	0.650062	0.325031	−	0.945703i	$-0.394625\pi$
$619$	10011.3	0.650062	0.325031	+	0.945703i	$0.394625\pi$
$620$	0	0
$621$	−12021.2	−0.776804
$622$	−3370.19	−0.217254
$623$	30060.7	1.93316
$624$	64.1494	0.00411543
$625$	0	0
$626$	−7311.28	−0.466801
$627$	−39601.9	−2.52241
$628$	−5568.85	−0.353856
$629$	1258.73	0.0797918
$630$	0	0
$631$	−5847.76	−0.368932	−0.184466	−	0.982839i	$-0.559055\pi$
$631$	−5847.76	−0.368932	−0.184466	+	0.982839i	$0.559055\pi$
$632$	−8934.12	−0.562311
$633$	17091.1	1.07316
$634$	−20515.7	−1.28514
$635$	0	0
$636$	11956.2	0.745433
$637$	171.076	0.0106409
$638$	−467.848	−0.0290318
$639$	30098.8	1.86336
$640$	0	0
$641$	−3414.57	−0.210402	−0.105201	−	0.994451i	$-0.533549\pi$
$641$	−3414.57	−0.210402	−0.105201	+	0.994451i	$0.533549\pi$
$642$	34576.3	2.12557
$643$	−31025.3	−1.90283	−0.951413	−	0.307917i	$-0.900368\pi$
$643$	−31025.3	−1.90283	−0.951413	+	0.307917i	$0.900368\pi$
$644$	10107.8	0.618486
$645$	0	0
$646$	−16115.0	−0.981479
$647$	−11289.4	−0.685982	−0.342991	−	0.939339i	$-0.611440\pi$
$647$	−11289.4	−0.685982	−0.342991	+	0.939339i	$0.611440\pi$
$648$	−739.443	−0.0448273
$649$	−3889.43	−0.235244
$650$	0	0
$651$	509.190	0.0306555
$652$	−13978.3	−0.839617
$653$	−1742.21	−0.104407	−0.0522037	−	0.998636i	$-0.516624\pi$
$653$	−1742.21	−0.104407	−0.0522037	+	0.998636i	$0.516624\pi$
$654$	11721.5	0.700837
$655$	0	0
$656$	−842.045	−0.0501164
$657$	5400.75	0.320705
$658$	14331.0	0.849060
$659$	9358.55	0.553198	0.276599	−	0.960985i	$-0.410793\pi$
$659$	9358.55	0.553198	0.276599	+	0.960985i	$0.410793\pi$
$660$	0	0
$661$	−16056.7	−0.944834	−0.472417	−	0.881375i	$-0.656618\pi$
$661$	−16056.7	−0.944834	−0.472417	+	0.881375i	$0.656618\pi$
$662$	16236.1	0.953221
$663$	−424.923	−0.0248909
$664$	4004.56	0.234047
$665$	0	0
$666$	−1000.38	−0.0582039
$667$	−357.158	−0.0207334
$668$	12665.9	0.733620
$669$	−26222.5	−1.51543
$670$	0	0
$671$	22855.2	1.31492
$672$	−7027.18	−0.403392
$673$	2727.45	0.156219	0.0781095	−	0.996945i	$-0.475112\pi$
$673$	2727.45	0.156219	0.0781095	+	0.996945i	$0.475112\pi$
$674$	−4668.76	−0.266816
$675$	0	0
$676$	−8787.07	−0.499947
$677$	13552.5	0.769373	0.384686	−	0.923047i	$-0.374310\pi$
$677$	13552.5	0.769373	0.384686	+	0.923047i	$0.374310\pi$
$678$	6312.93	0.357591
$679$	19429.5	1.09814
$680$	0	0
$681$	37013.1	2.08274
$682$	290.568	0.0163144
$683$	12869.8	0.721009	0.360504	−	0.932757i	$-0.382605\pi$
$683$	12869.8	0.721009	0.360504	+	0.932757i	$0.382605\pi$
$684$	12807.3	0.715937
$685$	0	0
$686$	−619.879	−0.0345001
$687$	6764.83	0.375683
$688$	−4199.75	−0.232724
$689$	173.394	0.00958750
$690$	0	0
$691$	7750.81	0.426707	0.213354	−	0.976975i	$-0.431561\pi$
$691$	7750.81	0.426707	0.213354	+	0.976975i	$0.431561\pi$
$692$	−1237.54	−0.0679828
$693$	69702.8	3.82077
$694$	−14873.3	−0.813518
$695$	0	0
$696$	248.303	0.0135229
$697$	5577.68	0.303113
$698$	−5481.63	−0.297253
$699$	42570.5	2.30352
$700$	0	0
$701$	18533.5	0.998574	0.499287	−	0.866437i	$-0.333595\pi$
$701$	18533.5	0.998574	0.499287	+	0.866437i	$0.333595\pi$
$702$	121.202	0.00651637
$703$	−902.938	−0.0484423
$704$	−4010.04	−0.214679
$705$	0	0
$706$	−16680.1	−0.889182
$707$	33003.4	1.75562
$708$	2064.25	0.109575
$709$	−12499.8	−0.662113	−0.331056	−	0.943611i	$-0.607405\pi$
$709$	−12499.8	−0.662113	−0.331056	+	0.943611i	$0.607405\pi$
$710$	0	0
$711$	−47032.6	−2.48082
$712$	−9104.26	−0.479209
$713$	221.821	0.0116511
$714$	46547.8	2.43979
$715$	0	0
$716$	14845.2	0.774850
$717$	−2666.88	−0.138907
$718$	4000.55	0.207938
$719$	17144.5	0.889266	0.444633	−	0.895713i	$-0.353334\pi$
$719$	17144.5	0.889266	0.444633	+	0.895713i	$0.353334\pi$
$720$	0	0
$721$	−8791.62	−0.454115
$722$	−2158.11	−0.111242
$723$	−744.558	−0.0382993
$724$	3774.64	0.193761
$725$	0	0
$726$	43145.3	2.20561
$727$	33920.5	1.73045	0.865227	−	0.501380i	$-0.167174\pi$
$727$	33920.5	1.73045	0.865227	+	0.501380i	$0.167174\pi$
$728$	−101.911	−0.00518829
$729$	−32098.9	−1.63080
$730$	0	0
$731$	27819.0	1.40755
$732$	−12130.0	−0.612485
$733$	18018.5	0.907954	0.453977	−	0.891013i	$-0.350005\pi$
$733$	18018.5	0.907954	0.453977	+	0.891013i	$0.350005\pi$
$734$	−9711.25	−0.488350
$735$	0	0
$736$	−3061.28	−0.153316
$737$	13869.2	0.693185
$738$	−4432.84	−0.221105
$739$	5764.33	0.286934	0.143467	−	0.989655i	$-0.454175\pi$
$739$	5764.33	0.286934	0.143467	+	0.989655i	$0.454175\pi$
$740$	0	0
$741$	304.813	0.0151115
$742$	−18994.3	−0.939760
$743$	−3472.42	−0.171454	−0.0857272	−	0.996319i	$-0.527321\pi$
$743$	−3472.42	−0.171454	−0.0857272	+	0.996319i	$0.527321\pi$
$744$	−154.215	−0.00759917
$745$	0	0
$746$	21492.4	1.05482
$747$	21081.5	1.03257
$748$	26562.3	1.29842
$749$	−54929.7	−2.67969
$750$	0	0
$751$	14372.4	0.698344	0.349172	−	0.937059i	$-0.386463\pi$
$751$	14372.4	0.698344	0.349172	+	0.937059i	$0.386463\pi$
$752$	−4340.33	−0.210473
$753$	−58793.1	−2.84534
$754$	3.60100	0.000173926 0
$755$	0	0
$756$	−13277.0	−0.638730
$757$	−18932.8	−0.909017	−0.454509	−	0.890742i	$-0.650185\pi$
$757$	−18932.8	−0.909017	−0.454509	+	0.890742i	$0.650185\pi$
$758$	−16012.5	−0.767285
$759$	49832.0	2.38312
$760$	0	0
$761$	−30942.4	−1.47393	−0.736965	−	0.675931i	$-0.763742\pi$
$761$	−30942.4	−1.47393	−0.736965	+	0.675931i	$0.763742\pi$
$762$	−23327.8	−1.10903
$763$	−18621.4	−0.883539
$764$	9908.87	0.469228
$765$	0	0
$766$	313.203	0.0147735
$767$	29.9366	0.00140932
$768$	2128.27	0.0999964
$769$	10199.3	0.478277	0.239139	−	0.970985i	$-0.423135\pi$
$769$	10199.3	0.478277	0.239139	+	0.970985i	$0.423135\pi$
$770$	0	0
$771$	−20939.2	−0.978087
$772$	−9380.22	−0.437308
$773$	−16453.8	−0.765589	−0.382795	−	0.923833i	$-0.625038\pi$
$773$	−16453.8	−0.765589	−0.382795	+	0.923833i	$0.625038\pi$
$774$	−22109.1	−1.02674
$775$	0	0
$776$	−5884.46	−0.272216
$777$	2608.12	0.120419
$778$	22725.0	1.04721
$779$	−4001.08	−0.184023
$780$	0	0
$781$	−44779.6	−2.05165
$782$	20277.8	0.927281
$783$	469.139	0.0214121
$784$	5675.74	0.258552
$785$	0	0
$786$	24288.8	1.10223
$787$	−7714.34	−0.349411	−0.174706	−	0.984621i	$-0.555897\pi$
$787$	−7714.34	−0.349411	−0.174706	+	0.984621i	$0.555897\pi$
$788$	15214.8	0.687824
$789$	−67631.5	−3.05164
$790$	0	0
$791$	−10029.1	−0.450812
$792$	−21110.4	−0.947126
$793$	−175.915	−0.00787757
$794$	1261.21	0.0563713
$795$	0	0
$796$	11890.4	0.529454
$797$	23818.7	1.05860	0.529299	−	0.848435i	$-0.322455\pi$
$797$	23818.7	1.05860	0.529299	+	0.848435i	$0.322455\pi$
$798$	−33390.5	−1.48122
$799$	28750.2	1.27298
$800$	0	0
$801$	−47928.3	−2.11418
$802$	−4987.60	−0.219599
$803$	−8034.99	−0.353112
$804$	−7360.85	−0.322882
$805$	0	0
$806$	−2.23648	−9.77378e−5 0
$807$	5403.18	0.235689
$808$	−9995.48	−0.435198
$809$	10209.0	0.443670	0.221835	−	0.975084i	$-0.428795\pi$
$809$	10209.0	0.443670	0.221835	+	0.975084i	$0.428795\pi$
$810$	0	0
$811$	28815.1	1.24764	0.623819	−	0.781569i	$-0.285580\pi$
$811$	28815.1	1.24764	0.623819	+	0.781569i	$0.285580\pi$
$812$	−394.468	−0.0170482
$813$	−2424.59	−0.104593
$814$	1488.31	0.0640853
$815$	0	0
$816$	−14097.6	−0.604796
$817$	−19955.6	−0.854539
$818$	−23484.5	−1.00381
$819$	−536.498	−0.0228898
$820$	0	0
$821$	−38057.4	−1.61780	−0.808900	−	0.587946i	$-0.799937\pi$
$821$	−38057.4	−1.61780	−0.808900	+	0.587946i	$0.799937\pi$
$822$	−24723.8	−1.04908
$823$	−33325.6	−1.41149	−0.705746	−	0.708464i	$-0.749388\pi$
$823$	−33325.6	−1.41149	−0.705746	+	0.708464i	$0.749388\pi$
$824$	2662.65	0.112570
$825$	0	0
$826$	−3279.38	−0.138141
$827$	−17041.4	−0.716551	−0.358275	−	0.933616i	$-0.616635\pi$
$827$	−17041.4	−0.716551	−0.358275	+	0.933616i	$0.616635\pi$
$828$	−16115.8	−0.676402
$829$	9485.19	0.397388	0.198694	−	0.980062i	$-0.436330\pi$
$829$	9485.19	0.397388	0.198694	+	0.980062i	$0.436330\pi$
$830$	0	0
$831$	−30224.0	−1.26168
$832$	30.8650	0.00128612
$833$	−37595.9	−1.56377
$834$	2154.53	0.0894549
$835$	0	0
$836$	−19054.2	−0.788281
$837$	−291.370	−0.0120325
$838$	−15106.2	−0.622713
$839$	−14950.4	−0.615192	−0.307596	−	0.951517i	$-0.599525\pi$
$839$	−14950.4	−0.615192	−0.307596	+	0.951517i	$0.599525\pi$
$840$	0	0
$841$	−24375.1	−0.999428
$842$	−23085.7	−0.944876
$843$	−69902.3	−2.85595
$844$	8223.28	0.335375
$845$	0	0
$846$	−22849.1	−0.928568
$847$	−68542.8	−2.78059
$848$	5752.65	0.232956
$849$	−7730.42	−0.312494
$850$	0	0
$851$	1136.19	0.0457673
$852$	23766.1	0.955650
$853$	−23751.3	−0.953374	−0.476687	−	0.879073i	$-0.658162\pi$
$853$	−23751.3	−0.953374	−0.476687	+	0.879073i	$0.658162\pi$
$854$	19270.4	0.772154
$855$	0	0
$856$	16636.1	0.664266
$857$	37144.3	1.48054	0.740272	−	0.672307i	$-0.234697\pi$
$857$	37144.3	1.48054	0.740272	+	0.672307i	$0.234697\pi$
$858$	−502.425	−0.0199913
$859$	−14064.3	−0.558635	−0.279318	−	0.960199i	$-0.590108\pi$
$859$	−14064.3	−0.558635	−0.279318	+	0.960199i	$0.590108\pi$
$860$	0	0
$861$	11557.0	0.457448
$862$	−15194.1	−0.600362
$863$	16051.4	0.633134	0.316567	−	0.948570i	$-0.397470\pi$
$863$	16051.4	0.633134	0.316567	+	0.948570i	$0.397470\pi$
$864$	4021.10	0.158334
$865$	0	0
$866$	−17378.6	−0.681927
$867$	52537.3	2.05797
$868$	244.993	0.00958019
$869$	69973.0	2.73150
$870$	0	0
$871$	−106.750	−0.00415280
$872$	5639.72	0.219020
$873$	−30978.0	−1.20097
$874$	−14546.1	−0.562961
$875$	0	0
$876$	4264.45	0.164478
$877$	4872.25	0.187599	0.0937995	−	0.995591i	$-0.470099\pi$
$877$	4872.25	0.187599	0.0937995	+	0.995591i	$0.470099\pi$
$878$	30732.6	1.18129
$879$	−3439.27	−0.131972
$880$	0	0
$881$	−22564.5	−0.862903	−0.431451	−	0.902136i	$-0.641998\pi$
$881$	−22564.5	−0.862903	−0.431451	+	0.902136i	$0.641998\pi$
$882$	29879.2	1.14069
$883$	38276.9	1.45880	0.729400	−	0.684088i	$-0.239799\pi$
$883$	38276.9	1.45880	0.729400	+	0.684088i	$0.239799\pi$
$884$	−204.449	−0.00777868
$885$	0	0
$886$	5897.29	0.223616
$887$	20986.8	0.794439	0.397220	−	0.917724i	$-0.369975\pi$
$887$	20986.8	0.794439	0.397220	+	0.917724i	$0.369975\pi$
$888$	−789.901	−0.0298506
$889$	37059.8	1.39814
$890$	0	0
$891$	5791.40	0.217754
$892$	−12616.8	−0.473588
$893$	−20623.6	−0.772835
$894$	17550.6	0.656576
$895$	0	0
$896$	−3381.07	−0.126064
$897$	−383.554	−0.0142770
$898$	24942.0	0.926867
$899$	−8.65677	−0.000321156 0
$900$	0	0
$901$	−38105.3	−1.40896
$902$	6594.99	0.243447
$903$	57641.4	2.12424
$904$	3037.42	0.111751
$905$	0	0
$906$	36543.8	1.34005
$907$	−29670.2	−1.08620	−0.543100	−	0.839668i	$-0.682749\pi$
$907$	−29670.2	−1.08620	−0.543100	+	0.839668i	$0.682749\pi$
$908$	17808.6	0.650879
$909$	−52620.0	−1.92002
$910$	0	0
$911$	−3225.70	−0.117313	−0.0586565	−	0.998278i	$-0.518682\pi$
$911$	−3225.70	−0.117313	−0.0586565	+	0.998278i	$0.518682\pi$
$912$	10112.7	0.367177
$913$	−31364.2	−1.13691
$914$	−27534.9	−0.996471
$915$	0	0
$916$	3254.85	0.117405
$917$	−38586.5	−1.38957
$918$	−26635.6	−0.957633
$919$	−37518.0	−1.34669	−0.673344	−	0.739330i	$-0.735143\pi$
$919$	−37518.0	−1.34669	−0.673344	+	0.739330i	$0.735143\pi$
$920$	0	0
$921$	47036.0	1.68283
$922$	27791.9	0.992710
$923$	344.666	0.0122912
$924$	55037.6	1.95953
$925$	0	0
$926$	−30715.9	−1.09005
$927$	14017.2	0.496640
$928$	119.469	0.00422605
$929$	−6931.01	−0.244778	−0.122389	−	0.992482i	$-0.539056\pi$
$929$	−6931.01	−0.244778	−0.122389	+	0.992482i	$0.539056\pi$
$930$	0	0
$931$	26969.0	0.949379
$932$	20482.5	0.719877
$933$	−14009.1	−0.491573
$934$	−22978.8	−0.805021
$935$	0	0
$936$	162.485	0.00567413
$937$	−27841.6	−0.970700	−0.485350	−	0.874320i	$-0.661308\pi$
$937$	−27841.6	−0.970700	−0.485350	+	0.874320i	$0.661308\pi$
$938$	11693.8	0.407054
$939$	−30391.3	−1.05621
$940$	0	0
$941$	−19925.2	−0.690268	−0.345134	−	0.938553i	$-0.612167\pi$
$941$	−19925.2	−0.690268	−0.345134	+	0.938553i	$0.612167\pi$
$942$	−23148.5	−0.800655
$943$	5034.65	0.173861
$944$	993.201	0.0342436
$945$	0	0
$946$	32892.9	1.13049
$947$	14762.2	0.506553	0.253276	−	0.967394i	$-0.418492\pi$
$947$	14762.2	0.506553	0.253276	+	0.967394i	$0.418492\pi$
$948$	−37137.1	−1.27232
$949$	61.8448	0.00211545
$950$	0	0
$951$	−85279.0	−2.90785
$952$	22396.1	0.762460
$953$	4072.58	0.138430	0.0692150	−	0.997602i	$-0.477951\pi$
$953$	4072.58	0.138430	0.0692150	+	0.997602i	$0.477951\pi$
$954$	30284.1	1.02776
$955$	0	0
$956$	−1283.15	−0.0434101
$957$	−1944.74	−0.0656891
$958$	23132.3	0.780137
$959$	39277.4	1.32256
$960$	0	0
$961$	−29785.6	−0.999820
$962$	−11.4555	−0.000383928 0
$963$	87578.9	2.93062
$964$	−358.239	−0.0119690
$965$	0	0
$966$	42016.0	1.39942
$967$	−25273.5	−0.840477	−0.420239	−	0.907414i	$-0.638054\pi$
$967$	−25273.5	−0.840477	−0.420239	+	0.907414i	$0.638054\pi$
$968$	20759.0	0.689278
$969$	−66986.3	−2.22075
$970$	0	0
$971$	−37609.3	−1.24299	−0.621494	−	0.783419i	$-0.713474\pi$
$971$	−37609.3	−1.24299	−0.621494	+	0.783419i	$0.713474\pi$
$972$	−16644.9	−0.549265
$973$	−3422.80	−0.112775
$974$	−18406.1	−0.605514
$975$	0	0
$976$	−5836.28	−0.191409
$977$	−12802.6	−0.419233	−0.209616	−	0.977784i	$-0.567221\pi$
$977$	−12802.6	−0.419233	−0.209616	+	0.977784i	$0.567221\pi$
$978$	−58104.4	−1.89977
$979$	71305.5	2.32782
$980$	0	0
$981$	29689.6	0.966276
$982$	4033.52	0.131074
$983$	−21311.3	−0.691482	−0.345741	−	0.938330i	$-0.612372\pi$
$983$	−21311.3	−0.691482	−0.345741	+	0.938330i	$0.612372\pi$
$984$	−3500.19	−0.113396
$985$	0	0
$986$	−791.361	−0.0255599
$987$	59570.8	1.92113
$988$	146.659	0.00472251
$989$	25110.6	0.807351
$990$	0	0
$991$	−44868.1	−1.43823	−0.719113	−	0.694893i	$-0.755452\pi$
$991$	−44868.1	−1.43823	−0.719113	+	0.694893i	$0.755452\pi$
$992$	−74.1992	−0.00237482
$993$	67489.6	2.15681
$994$	−37756.1	−1.20478
$995$	0	0
$996$	16646.1	0.529569
$997$	−8039.98	−0.255395	−0.127697	−	0.991813i	$-0.540759\pi$
$997$	−8039.98	−0.255395	−0.127697	+	0.991813i	$0.540759\pi$
$998$	−3018.74	−0.0957480
$999$	−1492.42	−0.0472654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.4.a.j.1.8		8
5.4	even	2		1250.4.a.g.1.1		8
25.3	odd	20		250.4.e.c.49.1		32
25.4	even	10		250.4.d.b.201.1		16
25.6	even	5		50.4.d.b.11.4	✓	16
25.8	odd	20		250.4.e.c.199.8		32
25.17	odd	20		250.4.e.c.199.1		32
25.19	even	10		250.4.d.b.51.1		16
25.21	even	5		50.4.d.b.41.4	yes	16
25.22	odd	20		250.4.e.c.49.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.d.b.11.4	✓	16	25.6	even	5
50.4.d.b.41.4	yes	16	25.21	even	5
250.4.d.b.51.1		16	25.19	even	10
250.4.d.b.201.1		16	25.4	even	10
250.4.e.c.49.1		32	25.3	odd	20
250.4.e.c.49.8		32	25.22	odd	20
250.4.e.c.199.1		32	25.17	odd	20
250.4.e.c.199.8		32	25.8	odd	20
1250.4.a.g.1.1		8	5.4	even	2
1250.4.a.j.1.8		8	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:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.31354 q^{3} +4.00000 q^{4} +16.6271 q^{6} -26.4146 q^{7} +8.00000 q^{8} +42.1150 q^{9} -62.6568 q^{11} +33.2542 q^{12} +0.482265 q^{13} -52.8293 q^{14} +16.0000 q^{16} -105.983 q^{17} +84.2300 q^{18} +76.0259 q^{19} -219.599 q^{21} -125.314 q^{22} -95.6651 q^{23} +66.5084 q^{24} +0.964531 q^{26} +125.659 q^{27} -105.659 q^{28} +3.73342 q^{29} -2.31872 q^{31} +32.0000 q^{32} -520.900 q^{33} -211.967 q^{34} +168.460 q^{36} -11.8767 q^{37} +152.052 q^{38} +4.00934 q^{39} -52.6278 q^{41} -439.199 q^{42} -262.484 q^{43} -250.627 q^{44} -191.330 q^{46} -271.270 q^{47} +133.017 q^{48} +354.734 q^{49} -881.098 q^{51} +1.92906 q^{52} +359.541 q^{53} +251.319 q^{54} -211.317 q^{56} +632.045 q^{57} +7.46684 q^{58} +62.0750 q^{59} -364.767 q^{61} -4.63745 q^{62} -1112.45 q^{63} +64.0000 q^{64} -1041.80 q^{66} -221.351 q^{67} -423.934 q^{68} -795.316 q^{69} +714.680 q^{71} +336.920 q^{72} +128.238 q^{73} -23.7534 q^{74} +304.104 q^{76} +1655.06 q^{77} +8.01867 q^{78} -1116.77 q^{79} -92.4304 q^{81} -105.256 q^{82} +500.570 q^{83} -878.397 q^{84} -524.968 q^{86} +31.0379 q^{87} -501.255 q^{88} -1138.03 q^{89} -12.7389 q^{91} -382.660 q^{92} -19.2768 q^{93} -542.541 q^{94} +266.033 q^{96} -735.557 q^{97} +709.467 q^{98} -2638.79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} - 4 q^{3} + 32 q^{4} - 8 q^{6} - 27 q^{7} + 64 q^{8} + 76 q^{9} - 39 q^{11} - 16 q^{12} - 179 q^{13} - 54 q^{14} + 128 q^{16} - 247 q^{17} + 152 q^{18} - 25 q^{19} - 104 q^{21} - 78 q^{22}+ \cdots - 893 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 - 4 * q^3 + 32 * q^4 - 8 * q^6 - 27 * q^7 + 64 * q^8 + 76 * q^9 - 39 * q^11 - 16 * q^12 - 179 * q^13 - 54 * q^14 + 128 * q^16 - 247 * q^17 + 152 * q^18 - 25 * q^19 - 104 * q^21 - 78 * q^22 - 179 * q^23 - 32 * q^24 - 358 * q^26 - 610 * q^27 - 108 * q^28 - 565 * q^29 - 89 * q^31 + 256 * q^32 - 563 * q^33 - 494 * q^34 + 304 * q^36 - 287 * q^37 - 50 * q^38 + 722 * q^39 - 394 * q^41 - 208 * q^42 - 529 * q^43 - 156 * q^44 - 358 * q^46 + 758 * q^47 - 64 * q^48 + 849 * q^49 - 1804 * q^51 - 716 * q^52 + 56 * q^53 - 1220 * q^54 - 216 * q^56 - 760 * q^57 - 1130 * q^58 - 1220 * q^59 - 489 * q^61 - 178 * q^62 - 3819 * q^63 + 512 * q^64 - 1126 * q^66 - 2547 * q^67 - 988 * q^68 + 17 * q^69 + 1576 * q^71 + 608 * q^72 + 166 * q^73 - 574 * q^74 - 100 * q^76 - 954 * q^77 + 1444 * q^78 - 1540 * q^79 + 1868 * q^81 - 788 * q^82 - 929 * q^83 - 416 * q^84 - 1058 * q^86 - 1755 * q^87 - 312 * q^88 - 1150 * q^89 - 1939 * q^91 - 716 * q^92 - 128 * q^93 + 1516 * q^94 - 128 * q^96 - 632 * q^97 + 1698 * q^98 - 893 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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