LMFDB - Embedded newform 370.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,4,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, 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("370.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	370.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:	$21.8307067021$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	370.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +6.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +12.0000 q^{6} +3.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +6.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +12.0000 q^{6} +3.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +5.00000 q^{11} +24.0000 q^{12} -16.0000 q^{13} +6.00000 q^{14} +30.0000 q^{15} +16.0000 q^{16} +115.000 q^{17} +18.0000 q^{18} +110.000 q^{19} +20.0000 q^{20} +18.0000 q^{21} +10.0000 q^{22} +6.00000 q^{23} +48.0000 q^{24} +25.0000 q^{25} -32.0000 q^{26} -108.000 q^{27} +12.0000 q^{28} -111.000 q^{29} +60.0000 q^{30} -79.0000 q^{31} +32.0000 q^{32} +30.0000 q^{33} +230.000 q^{34} +15.0000 q^{35} +36.0000 q^{36} -37.0000 q^{37} +220.000 q^{38} -96.0000 q^{39} +40.0000 q^{40} +171.000 q^{41} +36.0000 q^{42} +361.000 q^{43} +20.0000 q^{44} +45.0000 q^{45} +12.0000 q^{46} -428.000 q^{47} +96.0000 q^{48} -334.000 q^{49} +50.0000 q^{50} +690.000 q^{51} -64.0000 q^{52} -527.000 q^{53} -216.000 q^{54} +25.0000 q^{55} +24.0000 q^{56} +660.000 q^{57} -222.000 q^{58} +112.000 q^{59} +120.000 q^{60} -323.000 q^{61} -158.000 q^{62} +27.0000 q^{63} +64.0000 q^{64} -80.0000 q^{65} +60.0000 q^{66} -464.000 q^{67} +460.000 q^{68} +36.0000 q^{69} +30.0000 q^{70} -366.000 q^{71} +72.0000 q^{72} +712.000 q^{73} -74.0000 q^{74} +150.000 q^{75} +440.000 q^{76} +15.0000 q^{77} -192.000 q^{78} +176.000 q^{79} +80.0000 q^{80} -891.000 q^{81} +342.000 q^{82} -180.000 q^{83} +72.0000 q^{84} +575.000 q^{85} +722.000 q^{86} -666.000 q^{87} +40.0000 q^{88} +446.000 q^{89} +90.0000 q^{90} -48.0000 q^{91} +24.0000 q^{92} -474.000 q^{93} -856.000 q^{94} +550.000 q^{95} +192.000 q^{96} -1407.00 q^{97} -668.000 q^{98} +45.0000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	6.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	6.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	12.0000	0.816497
$7$	3.00000	0.161985	0.0809924	−	0.996715i	$-0.474191\pi$
$7$	3.00000	0.161985	0.0809924	+	0.996715i	$0.474191\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	5.00000	0.137051	0.0685253	−	0.997649i	$-0.478171\pi$
$11$	5.00000	0.137051	0.0685253	+	0.997649i	$0.478171\pi$
$12$	24.0000	0.577350
$13$	−16.0000	−0.341354	−0.170677	−	0.985327i	$-0.554595\pi$
$13$	−16.0000	−0.341354	−0.170677	+	0.985327i	$0.554595\pi$
$14$	6.00000	0.114541
$15$	30.0000	0.516398
$16$	16.0000	0.250000
$17$	115.000	1.64068	0.820341	−	0.571875i	$-0.193784\pi$
$17$	115.000	1.64068	0.820341	+	0.571875i	$0.193784\pi$
$18$	18.0000	0.235702
$19$	110.000	1.32820	0.664098	−	0.747645i	$-0.268816\pi$
$19$	110.000	1.32820	0.664098	+	0.747645i	$0.268816\pi$
$20$	20.0000	0.223607
$21$	18.0000	0.187044
$22$	10.0000	0.0969094
$23$	6.00000	0.0543951	0.0271975	−	0.999630i	$-0.491342\pi$
$23$	6.00000	0.0543951	0.0271975	+	0.999630i	$0.491342\pi$
$24$	48.0000	0.408248
$25$	25.0000	0.200000
$26$	−32.0000	−0.241374
$27$	−108.000	−0.769800
$28$	12.0000	0.0809924
$29$	−111.000	−0.710765	−0.355382	−	0.934721i	$-0.615649\pi$
$29$	−111.000	−0.710765	−0.355382	+	0.934721i	$0.615649\pi$
$30$	60.0000	0.365148
$31$	−79.0000	−0.457704	−0.228852	−	0.973461i	$-0.573497\pi$
$31$	−79.0000	−0.457704	−0.228852	+	0.973461i	$0.573497\pi$
$32$	32.0000	0.176777
$33$	30.0000	0.158252
$34$	230.000	1.16014
$35$	15.0000	0.0724418
$36$	36.0000	0.166667
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	220.000	0.939177
$39$	−96.0000	−0.394162
$40$	40.0000	0.158114
$41$	171.000	0.651359	0.325679	−	0.945480i	$-0.394407\pi$
$41$	171.000	0.651359	0.325679	+	0.945480i	$0.394407\pi$
$42$	36.0000	0.132260
$43$	361.000	1.28028	0.640139	−	0.768259i	$-0.278877\pi$
$43$	361.000	1.28028	0.640139	+	0.768259i	$0.278877\pi$
$44$	20.0000	0.0685253
$45$	45.0000	0.149071
$46$	12.0000	0.0384631
$47$	−428.000	−1.32830	−0.664151	−	0.747598i	$-0.731207\pi$
$47$	−428.000	−1.32830	−0.664151	+	0.747598i	$0.731207\pi$
$48$	96.0000	0.288675
$49$	−334.000	−0.973761
$50$	50.0000	0.141421
$51$	690.000	1.89450
$52$	−64.0000	−0.170677
$53$	−527.000	−1.36583	−0.682915	−	0.730498i	$-0.739288\pi$
$53$	−527.000	−1.36583	−0.682915	+	0.730498i	$0.739288\pi$
$54$	−216.000	−0.544331
$55$	25.0000	0.0612909
$56$	24.0000	0.0572703
$57$	660.000	1.53367
$58$	−222.000	−0.502587
$59$	112.000	0.247138	0.123569	−	0.992336i	$-0.460566\pi$
$59$	112.000	0.247138	0.123569	+	0.992336i	$0.460566\pi$
$60$	120.000	0.258199
$61$	−323.000	−0.677966	−0.338983	−	0.940793i	$-0.610083\pi$
$61$	−323.000	−0.677966	−0.338983	+	0.940793i	$0.610083\pi$
$62$	−158.000	−0.323645
$63$	27.0000	0.0539949
$64$	64.0000	0.125000
$65$	−80.0000	−0.152658
$66$	60.0000	0.111901
$67$	−464.000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$67$	−464.000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$68$	460.000	0.820341
$69$	36.0000	0.0628100
$70$	30.0000	0.0512241
$71$	−366.000	−0.611778	−0.305889	−	0.952067i	$-0.598954\pi$
$71$	−366.000	−0.611778	−0.305889	+	0.952067i	$0.598954\pi$
$72$	72.0000	0.117851
$73$	712.000	1.14155	0.570776	−	0.821106i	$-0.306642\pi$
$73$	712.000	1.14155	0.570776	+	0.821106i	$0.306642\pi$
$74$	−74.0000	−0.116248
$75$	150.000	0.230940
$76$	440.000	0.664098
$77$	15.0000	0.0222001
$78$	−192.000	−0.278714
$79$	176.000	0.250652	0.125326	−	0.992116i	$-0.460002\pi$
$79$	176.000	0.250652	0.125326	+	0.992116i	$0.460002\pi$
$80$	80.0000	0.111803
$81$	−891.000	−1.22222
$82$	342.000	0.460580
$83$	−180.000	−0.238043	−0.119021	−	0.992892i	$-0.537976\pi$
$83$	−180.000	−0.238043	−0.119021	+	0.992892i	$0.537976\pi$
$84$	72.0000	0.0935220
$85$	575.000	0.733735
$86$	722.000	0.905294
$87$	−666.000	−0.820721
$88$	40.0000	0.0484547
$89$	446.000	0.531190	0.265595	−	0.964085i	$-0.414432\pi$
$89$	446.000	0.531190	0.265595	+	0.964085i	$0.414432\pi$
$90$	90.0000	0.105409
$91$	−48.0000	−0.0552941
$92$	24.0000	0.0271975
$93$	−474.000	−0.528511
$94$	−856.000	−0.939252
$95$	550.000	0.593987
$96$	192.000	0.204124
$97$	−1407.00	−1.47278	−0.736388	−	0.676560i	$-0.763470\pi$
$97$	−1407.00	−1.47278	−0.736388	+	0.676560i	$0.763470\pi$
$98$	−668.000	−0.688553
$99$	45.0000	0.0456835
$100$	100.000	0.100000
$101$	1758.00	1.73196	0.865978	−	0.500082i	$-0.166697\pi$
$101$	1758.00	1.73196	0.865978	+	0.500082i	$0.166697\pi$
$102$	1380.00	1.33961
$103$	384.000	0.367346	0.183673	−	0.982987i	$-0.441201\pi$
$103$	384.000	0.367346	0.183673	+	0.982987i	$0.441201\pi$
$104$	−128.000	−0.120687
$105$	90.0000	0.0836486
$106$	−1054.00	−0.965788
$107$	−794.000	−0.717373	−0.358686	−	0.933458i	$-0.616775\pi$
$107$	−794.000	−0.717373	−0.358686	+	0.933458i	$0.616775\pi$
$108$	−432.000	−0.384900
$109$	201.000	0.176627	0.0883133	−	0.996093i	$-0.471852\pi$
$109$	201.000	0.176627	0.0883133	+	0.996093i	$0.471852\pi$
$110$	50.0000	0.0433392
$111$	−222.000	−0.189832
$112$	48.0000	0.0404962
$113$	−1759.00	−1.46436	−0.732181	−	0.681111i	$-0.761497\pi$
$113$	−1759.00	−1.46436	−0.732181	+	0.681111i	$0.761497\pi$
$114$	1320.00	1.08447
$115$	30.0000	0.0243262
$116$	−444.000	−0.355382
$117$	−144.000	−0.113785
$118$	224.000	0.174753
$119$	345.000	0.265766
$120$	240.000	0.182574
$121$	−1306.00	−0.981217
$122$	−646.000	−0.479394
$123$	1026.00	0.752124
$124$	−316.000	−0.228852
$125$	125.000	0.0894427
$126$	54.0000	0.0381802
$127$	1052.00	0.735039	0.367519	−	0.930016i	$-0.380207\pi$
$127$	1052.00	0.735039	0.367519	+	0.930016i	$0.380207\pi$
$128$	128.000	0.0883883
$129$	2166.00	1.47834
$130$	−160.000	−0.107946
$131$	468.000	0.312132	0.156066	−	0.987747i	$-0.450119\pi$
$131$	468.000	0.312132	0.156066	+	0.987747i	$0.450119\pi$
$132$	120.000	0.0791262
$133$	330.000	0.215148
$134$	−928.000	−0.598261
$135$	−540.000	−0.344265
$136$	920.000	0.580069
$137$	48.0000	0.0299337	0.0149668	−	0.999888i	$-0.495236\pi$
$137$	48.0000	0.0299337	0.0149668	+	0.999888i	$0.495236\pi$
$138$	72.0000	0.0444134
$139$	941.000	0.574206	0.287103	−	0.957900i	$-0.407308\pi$
$139$	941.000	0.574206	0.287103	+	0.957900i	$0.407308\pi$
$140$	60.0000	0.0362209
$141$	−2568.00	−1.53379
$142$	−732.000	−0.432592
$143$	−80.0000	−0.0467828
$144$	144.000	0.0833333
$145$	−555.000	−0.317864
$146$	1424.00	0.807199
$147$	−2004.00	−1.12440
$148$	−148.000	−0.0821995
$149$	−790.000	−0.434358	−0.217179	−	0.976132i	$-0.569686\pi$
$149$	−790.000	−0.434358	−0.217179	+	0.976132i	$0.569686\pi$
$150$	300.000	0.163299
$151$	−504.000	−0.271622	−0.135811	−	0.990735i	$-0.543364\pi$
$151$	−504.000	−0.271622	−0.135811	+	0.990735i	$0.543364\pi$
$152$	880.000	0.469588
$153$	1035.00	0.546894
$154$	30.0000	0.0156978
$155$	−395.000	−0.204691
$156$	−384.000	−0.197081
$157$	−2199.00	−1.11783	−0.558915	−	0.829225i	$-0.688782\pi$
$157$	−2199.00	−1.11783	−0.558915	+	0.829225i	$0.688782\pi$
$158$	352.000	0.177238
$159$	−3162.00	−1.57713
$160$	160.000	0.0790569
$161$	18.0000	0.00881117
$162$	−1782.00	−0.864242
$163$	−379.000	−0.182120	−0.0910600	−	0.995845i	$-0.529026\pi$
$163$	−379.000	−0.182120	−0.0910600	+	0.995845i	$0.529026\pi$
$164$	684.000	0.325679
$165$	150.000	0.0707726
$166$	−360.000	−0.168322
$167$	−2644.00	−1.22514	−0.612571	−	0.790415i	$-0.709865\pi$
$167$	−2644.00	−1.22514	−0.612571	+	0.790415i	$0.709865\pi$
$168$	144.000	0.0661300
$169$	−1941.00	−0.883477
$170$	1150.00	0.518829
$171$	990.000	0.442732
$172$	1444.00	0.640139
$173$	517.000	0.227207	0.113603	−	0.993526i	$-0.463761\pi$
$173$	517.000	0.227207	0.113603	+	0.993526i	$0.463761\pi$
$174$	−1332.00	−0.580337
$175$	75.0000	0.0323970
$176$	80.0000	0.0342627
$177$	672.000	0.285371
$178$	892.000	0.375608
$179$	2872.00	1.19924	0.599618	−	0.800286i	$-0.295319\pi$
$179$	2872.00	1.19924	0.599618	+	0.800286i	$0.295319\pi$
$180$	180.000	0.0745356
$181$	−110.000	−0.0451726	−0.0225863	−	0.999745i	$-0.507190\pi$
$181$	−110.000	−0.0451726	−0.0225863	+	0.999745i	$0.507190\pi$
$182$	−96.0000	−0.0390989
$183$	−1938.00	−0.782847
$184$	48.0000	0.0192316
$185$	−185.000	−0.0735215
$186$	−948.000	−0.373714
$187$	575.000	0.224856
$188$	−1712.00	−0.664151
$189$	−324.000	−0.124696
$190$	1100.00	0.420013
$191$	−3609.00	−1.36722	−0.683608	−	0.729850i	$-0.739590\pi$
$191$	−3609.00	−1.36722	−0.683608	+	0.729850i	$0.739590\pi$
$192$	384.000	0.144338
$193$	−58.0000	−0.0216318	−0.0108159	−	0.999942i	$-0.503443\pi$
$193$	−58.0000	−0.0216318	−0.0108159	+	0.999942i	$0.503443\pi$
$194$	−2814.00	−1.04141
$195$	−480.000	−0.176274
$196$	−1336.00	−0.486880
$197$	−1354.00	−0.489688	−0.244844	−	0.969563i	$-0.578737\pi$
$197$	−1354.00	−0.489688	−0.244844	+	0.969563i	$0.578737\pi$
$198$	90.0000	0.0323031
$199$	1888.00	0.672547	0.336273	−	0.941764i	$-0.390833\pi$
$199$	1888.00	0.672547	0.336273	+	0.941764i	$0.390833\pi$
$200$	200.000	0.0707107
$201$	−2784.00	−0.976956
$202$	3516.00	1.22468
$203$	−333.000	−0.115133
$204$	2760.00	0.947248
$205$	855.000	0.291297
$206$	768.000	0.259753
$207$	54.0000	0.0181317
$208$	−256.000	−0.0853385
$209$	550.000	0.182030
$210$	180.000	0.0591485
$211$	−3893.00	−1.27017	−0.635083	−	0.772444i	$-0.719034\pi$
$211$	−3893.00	−1.27017	−0.635083	+	0.772444i	$0.719034\pi$
$212$	−2108.00	−0.682915
$213$	−2196.00	−0.706420
$214$	−1588.00	−0.507259
$215$	1805.00	0.572558
$216$	−864.000	−0.272166
$217$	−237.000	−0.0741411
$218$	402.000	0.124894
$219$	4272.00	1.31815
$220$	100.000	0.0306454
$221$	−1840.00	−0.560053
$222$	−444.000	−0.134231
$223$	5909.00	1.77442	0.887211	−	0.461365i	$-0.152640\pi$
$223$	5909.00	1.77442	0.887211	+	0.461365i	$0.152640\pi$
$224$	96.0000	0.0286351
$225$	225.000	0.0666667
$226$	−3518.00	−1.03546
$227$	−2917.00	−0.852899	−0.426450	−	0.904511i	$-0.640236\pi$
$227$	−2917.00	−0.852899	−0.426450	+	0.904511i	$0.640236\pi$
$228$	2640.00	0.766835
$229$	−2142.00	−0.618111	−0.309055	−	0.951044i	$-0.600013\pi$
$229$	−2142.00	−0.618111	−0.309055	+	0.951044i	$0.600013\pi$
$230$	60.0000	0.0172012
$231$	90.0000	0.0256345
$232$	−888.000	−0.251293
$233$	5588.00	1.57117	0.785584	−	0.618755i	$-0.212363\pi$
$233$	5588.00	1.57117	0.785584	+	0.618755i	$0.212363\pi$
$234$	−288.000	−0.0804579
$235$	−2140.00	−0.594035
$236$	448.000	0.123569
$237$	1056.00	0.289429
$238$	690.000	0.187925
$239$	1371.00	0.371057	0.185528	−	0.982639i	$-0.440600\pi$
$239$	1371.00	0.371057	0.185528	+	0.982639i	$0.440600\pi$
$240$	480.000	0.129099
$241$	4184.00	1.11832	0.559160	−	0.829060i	$-0.311124\pi$
$241$	4184.00	1.11832	0.559160	+	0.829060i	$0.311124\pi$
$242$	−2612.00	−0.693825
$243$	−2430.00	−0.641500
$244$	−1292.00	−0.338983
$245$	−1670.00	−0.435479
$246$	2052.00	0.531832
$247$	−1760.00	−0.453385
$248$	−632.000	−0.161823
$249$	−1080.00	−0.274868
$250$	250.000	0.0632456
$251$	−5062.00	−1.27295	−0.636475	−	0.771297i	$-0.719608\pi$
$251$	−5062.00	−1.27295	−0.636475	+	0.771297i	$0.719608\pi$
$252$	108.000	0.0269975
$253$	30.0000	0.00745488
$254$	2104.00	0.519751
$255$	3450.00	0.847245
$256$	256.000	0.0625000
$257$	5450.00	1.32281	0.661404	−	0.750030i	$-0.269961\pi$
$257$	5450.00	1.32281	0.661404	+	0.750030i	$0.269961\pi$
$258$	4332.00	1.04534
$259$	−111.000	−0.0266301
$260$	−320.000	−0.0763291
$261$	−999.000	−0.236922
$262$	936.000	0.220711
$263$	6517.00	1.52797	0.763984	−	0.645236i	$-0.223241\pi$
$263$	6517.00	1.52797	0.763984	+	0.645236i	$0.223241\pi$
$264$	240.000	0.0559507
$265$	−2635.00	−0.610818
$266$	660.000	0.152132
$267$	2676.00	0.613365
$268$	−1856.00	−0.423034
$269$	6686.00	1.51544	0.757719	−	0.652581i	$-0.226314\pi$
$269$	6686.00	1.51544	0.757719	+	0.652581i	$0.226314\pi$
$270$	−1080.00	−0.243432
$271$	−7046.00	−1.57939	−0.789694	−	0.613501i	$-0.789761\pi$
$271$	−7046.00	−1.57939	−0.789694	+	0.613501i	$0.789761\pi$
$272$	1840.00	0.410171
$273$	−288.000	−0.0638482
$274$	96.0000	0.0211663
$275$	125.000	0.0274101
$276$	144.000	0.0314050
$277$	−5116.00	−1.10971	−0.554857	−	0.831946i	$-0.687227\pi$
$277$	−5116.00	−1.10971	−0.554857	+	0.831946i	$0.687227\pi$
$278$	1882.00	0.406025
$279$	−711.000	−0.152568
$280$	120.000	0.0256120
$281$	−808.000	−0.171535	−0.0857673	−	0.996315i	$-0.527334\pi$
$281$	−808.000	−0.171535	−0.0857673	+	0.996315i	$0.527334\pi$
$282$	−5136.00	−1.08455
$283$	2092.00	0.439422	0.219711	−	0.975565i	$-0.429489\pi$
$283$	2092.00	0.439422	0.219711	+	0.975565i	$0.429489\pi$
$284$	−1464.00	−0.305889
$285$	3300.00	0.685878
$286$	−160.000	−0.0330804
$287$	513.000	0.105510
$288$	288.000	0.0589256
$289$	8312.00	1.69184
$290$	−1110.00	−0.224764
$291$	−8442.00	−1.70061
$292$	2848.00	0.570776
$293$	2199.00	0.438454	0.219227	−	0.975674i	$-0.429647\pi$
$293$	2199.00	0.438454	0.219227	+	0.975674i	$0.429647\pi$
$294$	−4008.00	−0.795072
$295$	560.000	0.110524
$296$	−296.000	−0.0581238
$297$	−540.000	−0.105502
$298$	−1580.00	−0.307137
$299$	−96.0000	−0.0185680
$300$	600.000	0.115470
$301$	1083.00	0.207386
$302$	−1008.00	−0.192066
$303$	10548.0	1.99989
$304$	1760.00	0.332049
$305$	−1615.00	−0.303196
$306$	2070.00	0.386712
$307$	3166.00	0.588577	0.294289	−	0.955717i	$-0.404917\pi$
$307$	3166.00	0.588577	0.294289	+	0.955717i	$0.404917\pi$
$308$	60.0000	0.0111001
$309$	2304.00	0.424175
$310$	−790.000	−0.144739
$311$	−9045.00	−1.64918	−0.824590	−	0.565731i	$-0.808594\pi$
$311$	−9045.00	−1.64918	−0.824590	+	0.565731i	$0.808594\pi$
$312$	−768.000	−0.139357
$313$	8906.00	1.60830	0.804148	−	0.594429i	$-0.202622\pi$
$313$	8906.00	1.60830	0.804148	+	0.594429i	$0.202622\pi$
$314$	−4398.00	−0.790425
$315$	135.000	0.0241473
$316$	704.000	0.125326
$317$	9689.00	1.71668	0.858341	−	0.513079i	$-0.171495\pi$
$317$	9689.00	1.71668	0.858341	+	0.513079i	$0.171495\pi$
$318$	−6324.00	−1.11520
$319$	−555.000	−0.0974108
$320$	320.000	0.0559017
$321$	−4764.00	−0.828351
$322$	36.0000	0.00623044
$323$	12650.0	2.17915
$324$	−3564.00	−0.611111
$325$	−400.000	−0.0682708
$326$	−758.000	−0.128778
$327$	1206.00	0.203951
$328$	1368.00	0.230290
$329$	−1284.00	−0.215165
$330$	300.000	0.0500438
$331$	98.0000	0.0162736	0.00813681	−	0.999967i	$-0.497410\pi$
$331$	98.0000	0.0162736	0.00813681	+	0.999967i	$0.497410\pi$
$332$	−720.000	−0.119021
$333$	−333.000	−0.0547997
$334$	−5288.00	−0.866307
$335$	−2320.00	−0.378374
$336$	288.000	0.0467610
$337$	−6020.00	−0.973087	−0.486543	−	0.873656i	$-0.661742\pi$
$337$	−6020.00	−0.973087	−0.486543	+	0.873656i	$0.661742\pi$
$338$	−3882.00	−0.624713
$339$	−10554.0	−1.69090
$340$	2300.00	0.366868
$341$	−395.000	−0.0627286
$342$	1980.00	0.313059
$343$	−2031.00	−0.319719
$344$	2888.00	0.452647
$345$	180.000	0.0280895
$346$	1034.00	0.160659
$347$	9700.00	1.50064	0.750322	−	0.661073i	$-0.229899\pi$
$347$	9700.00	1.50064	0.750322	+	0.661073i	$0.229899\pi$
$348$	−2664.00	−0.410360
$349$	6234.00	0.956156	0.478078	−	0.878317i	$-0.341334\pi$
$349$	6234.00	0.956156	0.478078	+	0.878317i	$0.341334\pi$
$350$	150.000	0.0229081
$351$	1728.00	0.262774
$352$	160.000	0.0242274
$353$	9939.00	1.49858	0.749291	−	0.662241i	$-0.230394\pi$
$353$	9939.00	1.49858	0.749291	+	0.662241i	$0.230394\pi$
$354$	1344.00	0.201788
$355$	−1830.00	−0.273595
$356$	1784.00	0.265595
$357$	2070.00	0.306880
$358$	5744.00	0.847988
$359$	−1920.00	−0.282267	−0.141133	−	0.989991i	$-0.545075\pi$
$359$	−1920.00	−0.282267	−0.141133	+	0.989991i	$0.545075\pi$
$360$	360.000	0.0527046
$361$	5241.00	0.764106
$362$	−220.000	−0.0319418
$363$	−7836.00	−1.13301
$364$	−192.000	−0.0276471
$365$	3560.00	0.510518
$366$	−3876.00	−0.553557
$367$	7031.00	1.00004	0.500021	−	0.866013i	$-0.333326\pi$
$367$	7031.00	1.00004	0.500021	+	0.866013i	$0.333326\pi$
$368$	96.0000	0.0135988
$369$	1539.00	0.217120
$370$	−370.000	−0.0519875
$371$	−1581.00	−0.221244
$372$	−1896.00	−0.264255
$373$	2158.00	0.299563	0.149782	−	0.988719i	$-0.452143\pi$
$373$	2158.00	0.299563	0.149782	+	0.988719i	$0.452143\pi$
$374$	1150.00	0.158998
$375$	750.000	0.103280
$376$	−3424.00	−0.469626
$377$	1776.00	0.242622
$378$	−648.000	−0.0881733
$379$	−2440.00	−0.330698	−0.165349	−	0.986235i	$-0.552875\pi$
$379$	−2440.00	−0.330698	−0.165349	+	0.986235i	$0.552875\pi$
$380$	2200.00	0.296994
$381$	6312.00	0.848749
$382$	−7218.00	−0.966767
$383$	−1860.00	−0.248150	−0.124075	−	0.992273i	$-0.539596\pi$
$383$	−1860.00	−0.248150	−0.124075	+	0.992273i	$0.539596\pi$
$384$	768.000	0.102062
$385$	75.0000	0.00992819
$386$	−116.000	−0.0152960
$387$	3249.00	0.426760
$388$	−5628.00	−0.736388
$389$	2011.00	0.262112	0.131056	−	0.991375i	$-0.458163\pi$
$389$	2011.00	0.262112	0.131056	+	0.991375i	$0.458163\pi$
$390$	−960.000	−0.124645
$391$	690.000	0.0892450
$392$	−2672.00	−0.344276
$393$	2808.00	0.360419
$394$	−2708.00	−0.346262
$395$	880.000	0.112095
$396$	180.000	0.0228418
$397$	6010.00	0.759781	0.379891	−	0.925031i	$-0.375962\pi$
$397$	6010.00	0.759781	0.379891	+	0.925031i	$0.375962\pi$
$398$	3776.00	0.475562
$399$	1980.00	0.248431
$400$	400.000	0.0500000
$401$	9338.00	1.16289	0.581443	−	0.813587i	$-0.302488\pi$
$401$	9338.00	1.16289	0.581443	+	0.813587i	$0.302488\pi$
$402$	−5568.00	−0.690812
$403$	1264.00	0.156239
$404$	7032.00	0.865978
$405$	−4455.00	−0.546594
$406$	−666.000	−0.0814114
$407$	−185.000	−0.0225310
$408$	5520.00	0.669806
$409$	−760.000	−0.0918816	−0.0459408	−	0.998944i	$-0.514629\pi$
$409$	−760.000	−0.0918816	−0.0459408	+	0.998944i	$0.514629\pi$
$410$	1710.00	0.205978
$411$	288.000	0.0345645
$412$	1536.00	0.183673
$413$	336.000	0.0400326
$414$	108.000	0.0128210
$415$	−900.000	−0.106456
$416$	−512.000	−0.0603434
$417$	5646.00	0.663036
$418$	1100.00	0.128715
$419$	−10772.0	−1.25596	−0.627979	−	0.778230i	$-0.716118\pi$
$419$	−10772.0	−1.25596	−0.627979	+	0.778230i	$0.716118\pi$
$420$	360.000	0.0418243
$421$	−1014.00	−0.117386	−0.0586928	−	0.998276i	$-0.518693\pi$
$421$	−1014.00	−0.117386	−0.0586928	+	0.998276i	$0.518693\pi$
$422$	−7786.00	−0.898143
$423$	−3852.00	−0.442767
$424$	−4216.00	−0.482894
$425$	2875.00	0.328136
$426$	−4392.00	−0.499514
$427$	−969.000	−0.109820
$428$	−3176.00	−0.358686
$429$	−480.000	−0.0540201
$430$	3610.00	0.404860
$431$	−4445.00	−0.496771	−0.248385	−	0.968661i	$-0.579900\pi$
$431$	−4445.00	−0.496771	−0.248385	+	0.968661i	$0.579900\pi$
$432$	−1728.00	−0.192450
$433$	8482.00	0.941383	0.470692	−	0.882298i	$-0.344004\pi$
$433$	8482.00	0.941383	0.470692	+	0.882298i	$0.344004\pi$
$434$	−474.000	−0.0524256
$435$	−3330.00	−0.367037
$436$	804.000	0.0883133
$437$	660.000	0.0722473
$438$	8544.00	0.932073
$439$	−7433.00	−0.808104	−0.404052	−	0.914736i	$-0.632399\pi$
$439$	−7433.00	−0.808104	−0.404052	+	0.914736i	$0.632399\pi$
$440$	200.000	0.0216696
$441$	−3006.00	−0.324587
$442$	−3680.00	−0.396018
$443$	−5382.00	−0.577216	−0.288608	−	0.957447i	$-0.593192\pi$
$443$	−5382.00	−0.577216	−0.288608	+	0.957447i	$0.593192\pi$
$444$	−888.000	−0.0949158
$445$	2230.00	0.237555
$446$	11818.0	1.25471
$447$	−4740.00	−0.501553
$448$	192.000	0.0202481
$449$	4974.00	0.522801	0.261400	−	0.965230i	$-0.415816\pi$
$449$	4974.00	0.522801	0.261400	+	0.965230i	$0.415816\pi$
$450$	450.000	0.0471405
$451$	855.000	0.0892691
$452$	−7036.00	−0.732181
$453$	−3024.00	−0.313642
$454$	−5834.00	−0.603091
$455$	−240.000	−0.0247283
$456$	5280.00	0.542234
$457$	2721.00	0.278519	0.139259	−	0.990256i	$-0.455528\pi$
$457$	2721.00	0.278519	0.139259	+	0.990256i	$0.455528\pi$
$458$	−4284.00	−0.437070
$459$	−12420.0	−1.26300
$460$	120.000	0.0121631
$461$	13075.0	1.32096	0.660481	−	0.750843i	$-0.270352\pi$
$461$	13075.0	1.32096	0.660481	+	0.750843i	$0.270352\pi$
$462$	180.000	0.0181263
$463$	5084.00	0.510310	0.255155	−	0.966900i	$-0.417874\pi$
$463$	5084.00	0.510310	0.255155	+	0.966900i	$0.417874\pi$
$464$	−1776.00	−0.177691
$465$	−2370.00	−0.236357
$466$	11176.0	1.11098
$467$	11941.0	1.18322	0.591610	−	0.806224i	$-0.298493\pi$
$467$	11941.0	1.18322	0.591610	+	0.806224i	$0.298493\pi$
$468$	−576.000	−0.0568923
$469$	−1392.00	−0.137050
$470$	−4280.00	−0.420046
$471$	−13194.0	−1.29076
$472$	896.000	0.0873766
$473$	1805.00	0.175463
$474$	2112.00	0.204657
$475$	2750.00	0.265639
$476$	1380.00	0.132883
$477$	−4743.00	−0.455277
$478$	2742.00	0.262377
$479$	1216.00	0.115993	0.0579963	−	0.998317i	$-0.481529\pi$
$479$	1216.00	0.115993	0.0579963	+	0.998317i	$0.481529\pi$
$480$	960.000	0.0912871
$481$	592.000	0.0561182
$482$	8368.00	0.790772
$483$	108.000	0.0101743
$484$	−5224.00	−0.490609
$485$	−7035.00	−0.658645
$486$	−4860.00	−0.453609
$487$	−1618.00	−0.150551	−0.0752757	−	0.997163i	$-0.523984\pi$
$487$	−1618.00	−0.150551	−0.0752757	+	0.997163i	$0.523984\pi$
$488$	−2584.00	−0.239697
$489$	−2274.00	−0.210294
$490$	−3340.00	−0.307930
$491$	11132.0	1.02318	0.511589	−	0.859230i	$-0.329057\pi$
$491$	11132.0	1.02318	0.511589	+	0.859230i	$0.329057\pi$
$492$	4104.00	0.376062
$493$	−12765.0	−1.16614
$494$	−3520.00	−0.320592
$495$	225.000	0.0204303
$496$	−1264.00	−0.114426
$497$	−1098.00	−0.0990987
$498$	−2160.00	−0.194361
$499$	−16562.0	−1.48581	−0.742903	−	0.669399i	$-0.766552\pi$
$499$	−16562.0	−1.48581	−0.742903	+	0.669399i	$0.766552\pi$
$500$	500.000	0.0447214
$501$	−15864.0	−1.41467
$502$	−10124.0	−0.900112
$503$	−11232.0	−0.995646	−0.497823	−	0.867279i	$-0.665867\pi$
$503$	−11232.0	−0.995646	−0.497823	+	0.867279i	$0.665867\pi$
$504$	216.000	0.0190901
$505$	8790.00	0.774554
$506$	60.0000	0.00527139
$507$	−11646.0	−1.02015
$508$	4208.00	0.367519
$509$	3036.00	0.264378	0.132189	−	0.991225i	$-0.457799\pi$
$509$	3036.00	0.264378	0.132189	+	0.991225i	$0.457799\pi$
$510$	6900.00	0.599092
$511$	2136.00	0.184914
$512$	512.000	0.0441942
$513$	−11880.0	−1.02245
$514$	10900.0	0.935367
$515$	1920.00	0.164282
$516$	8664.00	0.739169
$517$	−2140.00	−0.182045
$518$	−222.000	−0.0188303
$519$	3102.00	0.262356
$520$	−640.000	−0.0539728
$521$	19417.0	1.63277	0.816386	−	0.577507i	$-0.195974\pi$
$521$	19417.0	1.63277	0.816386	+	0.577507i	$0.195974\pi$
$522$	−1998.00	−0.167529
$523$	−156.000	−0.0130428	−0.00652142	−	0.999979i	$-0.502076\pi$
$523$	−156.000	−0.0130428	−0.00652142	+	0.999979i	$0.502076\pi$
$524$	1872.00	0.156066
$525$	450.000	0.0374088
$526$	13034.0	1.08044
$527$	−9085.00	−0.750947
$528$	480.000	0.0395631
$529$	−12131.0	−0.997041
$530$	−5270.00	−0.431914
$531$	1008.00	0.0823794
$532$	1320.00	0.107574
$533$	−2736.00	−0.222344
$534$	5352.00	0.433715
$535$	−3970.00	−0.320819
$536$	−3712.00	−0.299131
$537$	17232.0	1.38476
$538$	13372.0	1.07158
$539$	−1670.00	−0.133455
$540$	−2160.00	−0.172133
$541$	−19290.0	−1.53298	−0.766490	−	0.642257i	$-0.777998\pi$
$541$	−19290.0	−1.53298	−0.766490	+	0.642257i	$0.777998\pi$
$542$	−14092.0	−1.11680
$543$	−660.000	−0.0521608
$544$	3680.00	0.290034
$545$	1005.00	0.0789899
$546$	−576.000	−0.0451475
$547$	−24479.0	−1.91343	−0.956715	−	0.291026i	$-0.906003\pi$
$547$	−24479.0	−1.91343	−0.956715	+	0.291026i	$0.906003\pi$
$548$	192.000	0.0149668
$549$	−2907.00	−0.225989
$550$	250.000	0.0193819
$551$	−12210.0	−0.944035
$552$	288.000	0.0222067
$553$	528.000	0.0406019
$554$	−10232.0	−0.784686
$555$	−1110.00	−0.0848953
$556$	3764.00	0.287103
$557$	−3496.00	−0.265943	−0.132972	−	0.991120i	$-0.542452\pi$
$557$	−3496.00	−0.265943	−0.132972	+	0.991120i	$0.542452\pi$
$558$	−1422.00	−0.107882
$559$	−5776.00	−0.437028
$560$	240.000	0.0181104
$561$	3450.00	0.259642
$562$	−1616.00	−0.121293
$563$	8475.00	0.634420	0.317210	−	0.948355i	$-0.397254\pi$
$563$	8475.00	0.634420	0.317210	+	0.948355i	$0.397254\pi$
$564$	−10272.0	−0.766896
$565$	−8795.00	−0.654882
$566$	4184.00	0.310718
$567$	−2673.00	−0.197981
$568$	−2928.00	−0.216296
$569$	12594.0	0.927887	0.463944	−	0.885865i	$-0.346434\pi$
$569$	12594.0	0.927887	0.463944	+	0.885865i	$0.346434\pi$
$570$	6600.00	0.484989
$571$	−9993.00	−0.732389	−0.366194	−	0.930538i	$-0.619339\pi$
$571$	−9993.00	−0.732389	−0.366194	+	0.930538i	$0.619339\pi$
$572$	−320.000	−0.0233914
$573$	−21654.0	−1.57872
$574$	1026.00	0.0746070
$575$	150.000	0.0108790
$576$	576.000	0.0416667
$577$	8830.00	0.637084	0.318542	−	0.947909i	$-0.396807\pi$
$577$	8830.00	0.637084	0.318542	+	0.947909i	$0.396807\pi$
$578$	16624.0	1.19631
$579$	−348.000	−0.0249782
$580$	−2220.00	−0.158932
$581$	−540.000	−0.0385593
$582$	−16884.0	−1.20252
$583$	−2635.00	−0.187188
$584$	5696.00	0.403600
$585$	−720.000	−0.0508860
$586$	4398.00	0.310034
$587$	−18699.0	−1.31480	−0.657402	−	0.753540i	$-0.728345\pi$
$587$	−18699.0	−1.31480	−0.657402	+	0.753540i	$0.728345\pi$
$588$	−8016.00	−0.562201
$589$	−8690.00	−0.607921
$590$	1120.00	0.0781520
$591$	−8124.00	−0.565443
$592$	−592.000	−0.0410997
$593$	9700.00	0.671722	0.335861	−	0.941912i	$-0.390973\pi$
$593$	9700.00	0.671722	0.335861	+	0.941912i	$0.390973\pi$
$594$	−1080.00	−0.0746009
$595$	1725.00	0.118854
$596$	−3160.00	−0.217179
$597$	11328.0	0.776590
$598$	−192.000	−0.0131295
$599$	−7896.00	−0.538601	−0.269300	−	0.963056i	$-0.586792\pi$
$599$	−7896.00	−0.538601	−0.269300	+	0.963056i	$0.586792\pi$
$600$	1200.00	0.0816497
$601$	−8867.00	−0.601818	−0.300909	−	0.953653i	$-0.597290\pi$
$601$	−8867.00	−0.601818	−0.300909	+	0.953653i	$0.597290\pi$
$602$	2166.00	0.146644
$603$	−4176.00	−0.282023
$604$	−2016.00	−0.135811
$605$	−6530.00	−0.438814
$606$	21096.0	1.41414
$607$	−3174.00	−0.212238	−0.106119	−	0.994353i	$-0.533843\pi$
$607$	−3174.00	−0.212238	−0.106119	+	0.994353i	$0.533843\pi$
$608$	3520.00	0.234794
$609$	−1998.00	−0.132944
$610$	−3230.00	−0.214392
$611$	6848.00	0.453421
$612$	4140.00	0.273447
$613$	−16185.0	−1.06640	−0.533202	−	0.845988i	$-0.679012\pi$
$613$	−16185.0	−1.06640	−0.533202	+	0.845988i	$0.679012\pi$
$614$	6332.00	0.416187
$615$	5130.00	0.336360
$616$	120.000	0.00784892
$617$	204.000	0.0133107	0.00665537	−	0.999978i	$-0.497882\pi$
$617$	204.000	0.0133107	0.00665537	+	0.999978i	$0.497882\pi$
$618$	4608.00	0.299937
$619$	−2699.00	−0.175254	−0.0876268	−	0.996153i	$-0.527928\pi$
$619$	−2699.00	−0.175254	−0.0876268	+	0.996153i	$0.527928\pi$
$620$	−1580.00	−0.102346
$621$	−648.000	−0.0418733
$622$	−18090.0	−1.16615
$623$	1338.00	0.0860447
$624$	−1536.00	−0.0985404
$625$	625.000	0.0400000
$626$	17812.0	1.13724
$627$	3300.00	0.210190
$628$	−8796.00	−0.558915
$629$	−4255.00	−0.269726
$630$	270.000	0.0170747
$631$	−23523.0	−1.48405	−0.742025	−	0.670372i	$-0.766135\pi$
$631$	−23523.0	−1.48405	−0.742025	+	0.670372i	$0.766135\pi$
$632$	1408.00	0.0886190
$633$	−23358.0	−1.46666
$634$	19378.0	1.21388
$635$	5260.00	0.328719
$636$	−12648.0	−0.788563
$637$	5344.00	0.332397
$638$	−1110.00	−0.0688798
$639$	−3294.00	−0.203926
$640$	640.000	0.0395285
$641$	16319.0	1.00556	0.502778	−	0.864415i	$-0.332311\pi$
$641$	16319.0	1.00556	0.502778	+	0.864415i	$0.332311\pi$
$642$	−9528.00	−0.585732
$643$	−7403.00	−0.454037	−0.227019	−	0.973890i	$-0.572898\pi$
$643$	−7403.00	−0.454037	−0.227019	+	0.973890i	$0.572898\pi$
$644$	72.0000	0.00440559
$645$	10830.0	0.661133
$646$	25300.0	1.54089
$647$	15344.0	0.932357	0.466178	−	0.884691i	$-0.345630\pi$
$647$	15344.0	0.932357	0.466178	+	0.884691i	$0.345630\pi$
$648$	−7128.00	−0.432121
$649$	560.000	0.0338705
$650$	−800.000	−0.0482747
$651$	−1422.00	−0.0856107
$652$	−1516.00	−0.0910600
$653$	6118.00	0.366640	0.183320	−	0.983053i	$-0.441316\pi$
$653$	6118.00	0.366640	0.183320	+	0.983053i	$0.441316\pi$
$654$	2412.00	0.144215
$655$	2340.00	0.139590
$656$	2736.00	0.162840
$657$	6408.00	0.380517
$658$	−2568.00	−0.152144
$659$	5540.00	0.327478	0.163739	−	0.986504i	$-0.447645\pi$
$659$	5540.00	0.327478	0.163739	+	0.986504i	$0.447645\pi$
$660$	600.000	0.0353863
$661$	1247.00	0.0733777	0.0366889	−	0.999327i	$-0.488319\pi$
$661$	1247.00	0.0733777	0.0366889	+	0.999327i	$0.488319\pi$
$662$	196.000	0.0115072
$663$	−11040.0	−0.646694
$664$	−1440.00	−0.0841609
$665$	1650.00	0.0962169
$666$	−666.000	−0.0387492
$667$	−666.000	−0.0386621
$668$	−10576.0	−0.612571
$669$	35454.0	2.04893
$670$	−4640.00	−0.267551
$671$	−1615.00	−0.0929156
$672$	576.000	0.0330650
$673$	19006.0	1.08860	0.544300	−	0.838891i	$-0.316795\pi$
$673$	19006.0	1.08860	0.544300	+	0.838891i	$0.316795\pi$
$674$	−12040.0	−0.688076
$675$	−2700.00	−0.153960
$676$	−7764.00	−0.441739
$677$	6938.00	0.393869	0.196934	−	0.980417i	$-0.436901\pi$
$677$	6938.00	0.393869	0.196934	+	0.980417i	$0.436901\pi$
$678$	−21108.0	−1.19565
$679$	−4221.00	−0.238567
$680$	4600.00	0.259415
$681$	−17502.0	−0.984843
$682$	−790.000	−0.0443558
$683$	4719.00	0.264374	0.132187	−	0.991225i	$-0.457800\pi$
$683$	4719.00	0.264374	0.132187	+	0.991225i	$0.457800\pi$
$684$	3960.00	0.221366
$685$	240.000	0.0133868
$686$	−4062.00	−0.226076
$687$	−12852.0	−0.713733
$688$	5776.00	0.320070
$689$	8432.00	0.466232
$690$	360.000	0.0198623
$691$	−22805.0	−1.25549	−0.627745	−	0.778419i	$-0.716022\pi$
$691$	−22805.0	−1.25549	−0.627745	+	0.778419i	$0.716022\pi$
$692$	2068.00	0.113603
$693$	135.000	0.00740004
$694$	19400.0	1.06112
$695$	4705.00	0.256793
$696$	−5328.00	−0.290169
$697$	19665.0	1.06867
$698$	12468.0	0.676104
$699$	33528.0	1.81423
$700$	300.000	0.0161985
$701$	17378.0	0.936317	0.468158	−	0.883645i	$-0.344918\pi$
$701$	17378.0	0.936317	0.468158	+	0.883645i	$0.344918\pi$
$702$	3456.00	0.185810
$703$	−4070.00	−0.218354
$704$	320.000	0.0171313
$705$	−12840.0	−0.685932
$706$	19878.0	1.05966
$707$	5274.00	0.280550
$708$	2688.00	0.142685
$709$	−15015.0	−0.795346	−0.397673	−	0.917527i	$-0.630182\pi$
$709$	−15015.0	−0.795346	−0.397673	+	0.917527i	$0.630182\pi$
$710$	−3660.00	−0.193461
$711$	1584.00	0.0835508
$712$	3568.00	0.187804
$713$	−474.000	−0.0248968
$714$	4140.00	0.216997
$715$	−400.000	−0.0209219
$716$	11488.0	0.599618
$717$	8226.00	0.428460
$718$	−3840.00	−0.199593
$719$	14306.0	0.742036	0.371018	−	0.928626i	$-0.379009\pi$
$719$	14306.0	0.742036	0.371018	+	0.928626i	$0.379009\pi$
$720$	720.000	0.0372678
$721$	1152.00	0.0595045
$722$	10482.0	0.540304
$723$	25104.0	1.29132
$724$	−440.000	−0.0225863
$725$	−2775.00	−0.142153
$726$	−15672.0	−0.801160
$727$	−16988.0	−0.866644	−0.433322	−	0.901239i	$-0.642659\pi$
$727$	−16988.0	−0.866644	−0.433322	+	0.901239i	$0.642659\pi$
$728$	−384.000	−0.0195494
$729$	9477.00	0.481481
$730$	7120.00	0.360990
$731$	41515.0	2.10053
$732$	−7752.00	−0.391424
$733$	−2909.00	−0.146584	−0.0732922	−	0.997311i	$-0.523351\pi$
$733$	−2909.00	−0.146584	−0.0732922	+	0.997311i	$0.523351\pi$
$734$	14062.0	0.707136
$735$	−10020.0	−0.502848
$736$	192.000	0.00961578
$737$	−2320.00	−0.115954
$738$	3078.00	0.153527
$739$	10149.0	0.505192	0.252596	−	0.967572i	$-0.418716\pi$
$739$	10149.0	0.505192	0.252596	+	0.967572i	$0.418716\pi$
$740$	−740.000	−0.0367607
$741$	−10560.0	−0.523524
$742$	−3162.00	−0.156443
$743$	37473.0	1.85027	0.925135	−	0.379638i	$-0.123951\pi$
$743$	37473.0	1.85027	0.925135	+	0.379638i	$0.123951\pi$
$744$	−3792.00	−0.186857
$745$	−3950.00	−0.194251
$746$	4316.00	0.211823
$747$	−1620.00	−0.0793477
$748$	2300.00	0.112428
$749$	−2382.00	−0.116203
$750$	1500.00	0.0730297
$751$	9950.00	0.483463	0.241731	−	0.970343i	$-0.422285\pi$
$751$	9950.00	0.483463	0.241731	+	0.970343i	$0.422285\pi$
$752$	−6848.00	−0.332076
$753$	−30372.0	−1.46988
$754$	3552.00	0.171560
$755$	−2520.00	−0.121473
$756$	−1296.00	−0.0623480
$757$	20830.0	1.00010	0.500052	−	0.865995i	$-0.333314\pi$
$757$	20830.0	1.00010	0.500052	+	0.865995i	$0.333314\pi$
$758$	−4880.00	−0.233838
$759$	180.000	0.00860815
$760$	4400.00	0.210006
$761$	22515.0	1.07249	0.536247	−	0.844061i	$-0.319842\pi$
$761$	22515.0	1.07249	0.536247	+	0.844061i	$0.319842\pi$
$762$	12624.0	0.600157
$763$	603.000	0.0286108
$764$	−14436.0	−0.683608
$765$	5175.00	0.244578
$766$	−3720.00	−0.175469
$767$	−1792.00	−0.0843616
$768$	1536.00	0.0721688
$769$	29064.0	1.36291	0.681453	−	0.731862i	$-0.261348\pi$
$769$	29064.0	1.36291	0.681453	+	0.731862i	$0.261348\pi$
$770$	150.000	0.00702029
$771$	32700.0	1.52745
$772$	−232.000	−0.0108159
$773$	8843.00	0.411463	0.205731	−	0.978609i	$-0.434043\pi$
$773$	8843.00	0.411463	0.205731	+	0.978609i	$0.434043\pi$
$774$	6498.00	0.301765
$775$	−1975.00	−0.0915408
$776$	−11256.0	−0.520705
$777$	−666.000	−0.0307498
$778$	4022.00	0.185341
$779$	18810.0	0.865132
$780$	−1920.00	−0.0881372
$781$	−1830.00	−0.0838445
$782$	1380.00	0.0631058
$783$	11988.0	0.547147
$784$	−5344.00	−0.243440
$785$	−10995.0	−0.499909
$786$	5616.00	0.254855
$787$	33450.0	1.51507	0.757537	−	0.652792i	$-0.226402\pi$
$787$	33450.0	1.51507	0.757537	+	0.652792i	$0.226402\pi$
$788$	−5416.00	−0.244844
$789$	39102.0	1.76434
$790$	1760.00	0.0792633
$791$	−5277.00	−0.237204
$792$	360.000	0.0161516
$793$	5168.00	0.231426
$794$	12020.0	0.537247
$795$	−15810.0	−0.705312
$796$	7552.00	0.336273
$797$	−10866.0	−0.482928	−0.241464	−	0.970410i	$-0.577628\pi$
$797$	−10866.0	−0.482928	−0.241464	+	0.970410i	$0.577628\pi$
$798$	3960.00	0.175667
$799$	−49220.0	−2.17932
$800$	800.000	0.0353553
$801$	4014.00	0.177063
$802$	18676.0	0.822285
$803$	3560.00	0.156450
$804$	−11136.0	−0.488478
$805$	90.0000	0.00394048
$806$	2528.00	0.110478
$807$	40116.0	1.74988
$808$	14064.0	0.612339
$809$	−22342.0	−0.970955	−0.485478	−	0.874249i	$-0.661354\pi$
$809$	−22342.0	−0.970955	−0.485478	+	0.874249i	$0.661354\pi$
$810$	−8910.00	−0.386501
$811$	7432.00	0.321791	0.160896	−	0.986971i	$-0.448562\pi$
$811$	7432.00	0.321791	0.160896	+	0.986971i	$0.448562\pi$
$812$	−1332.00	−0.0575665
$813$	−42276.0	−1.82372
$814$	−370.000	−0.0159318
$815$	−1895.00	−0.0814466
$816$	11040.0	0.473624
$817$	39710.0	1.70046
$818$	−1520.00	−0.0649701
$819$	−432.000	−0.0184314
$820$	3420.00	0.145648
$821$	20208.0	0.859031	0.429515	−	0.903060i	$-0.358684\pi$
$821$	20208.0	0.859031	0.429515	+	0.903060i	$0.358684\pi$
$822$	576.000	0.0244408
$823$	−36448.0	−1.54374	−0.771870	−	0.635781i	$-0.780678\pi$
$823$	−36448.0	−1.54374	−0.771870	+	0.635781i	$0.780678\pi$
$824$	3072.00	0.129876
$825$	750.000	0.0316505
$826$	672.000	0.0283073
$827$	12819.0	0.539009	0.269504	−	0.962999i	$-0.413140\pi$
$827$	12819.0	0.539009	0.269504	+	0.962999i	$0.413140\pi$
$828$	216.000	0.00906584
$829$	−18851.0	−0.789774	−0.394887	−	0.918730i	$-0.629216\pi$
$829$	−18851.0	−0.789774	−0.394887	+	0.918730i	$0.629216\pi$
$830$	−1800.00	−0.0752758
$831$	−30696.0	−1.28139
$832$	−1024.00	−0.0426692
$833$	−38410.0	−1.59763
$834$	11292.0	0.468837
$835$	−13220.0	−0.547901
$836$	2200.00	0.0910151
$837$	8532.00	0.352341
$838$	−21544.0	−0.888097
$839$	−5844.00	−0.240474	−0.120237	−	0.992745i	$-0.538365\pi$
$839$	−5844.00	−0.240474	−0.120237	+	0.992745i	$0.538365\pi$
$840$	720.000	0.0295742
$841$	−12068.0	−0.494813
$842$	−2028.00	−0.0830042
$843$	−4848.00	−0.198071
$844$	−15572.0	−0.635083
$845$	−9705.00	−0.395103
$846$	−7704.00	−0.313084
$847$	−3918.00	−0.158942
$848$	−8432.00	−0.341458
$849$	12552.0	0.507401
$850$	5750.00	0.232027
$851$	−222.000	−0.00894249
$852$	−8784.00	−0.353210
$853$	−27602.0	−1.10794	−0.553971	−	0.832536i	$-0.686888\pi$
$853$	−27602.0	−1.10794	−0.553971	+	0.832536i	$0.686888\pi$
$854$	−1938.00	−0.0776546
$855$	4950.00	0.197996
$856$	−6352.00	−0.253630
$857$	2889.00	0.115153	0.0575766	−	0.998341i	$-0.481663\pi$
$857$	2889.00	0.115153	0.0575766	+	0.998341i	$0.481663\pi$
$858$	−960.000	−0.0381980
$859$	−2260.00	−0.0897674	−0.0448837	−	0.998992i	$-0.514292\pi$
$859$	−2260.00	−0.0897674	−0.0448837	+	0.998992i	$0.514292\pi$
$860$	7220.00	0.286279
$861$	3078.00	0.121833
$862$	−8890.00	−0.351270
$863$	49323.0	1.94551	0.972755	−	0.231837i	$-0.0744737\pi$
$863$	49323.0	1.94551	0.972755	+	0.231837i	$0.0744737\pi$
$864$	−3456.00	−0.136083
$865$	2585.00	0.101610
$866$	16964.0	0.665658
$867$	49872.0	1.95357
$868$	−948.000	−0.0370705
$869$	880.000	0.0343521
$870$	−6660.00	−0.259535
$871$	7424.00	0.288809
$872$	1608.00	0.0624470
$873$	−12663.0	−0.490925
$874$	1320.00	0.0510866
$875$	375.000	0.0144884
$876$	17088.0	0.659075
$877$	−24255.0	−0.933903	−0.466952	−	0.884283i	$-0.654648\pi$
$877$	−24255.0	−0.933903	−0.466952	+	0.884283i	$0.654648\pi$
$878$	−14866.0	−0.571416
$879$	13194.0	0.506283
$880$	400.000	0.0153227
$881$	−42849.0	−1.63861	−0.819307	−	0.573355i	$-0.805642\pi$
$881$	−42849.0	−1.63861	−0.819307	+	0.573355i	$0.805642\pi$
$882$	−6012.00	−0.229518
$883$	−22021.0	−0.839259	−0.419629	−	0.907695i	$-0.637840\pi$
$883$	−22021.0	−0.839259	−0.419629	+	0.907695i	$0.637840\pi$
$884$	−7360.00	−0.280027
$885$	3360.00	0.127622
$886$	−10764.0	−0.408153
$887$	−9955.00	−0.376839	−0.188419	−	0.982089i	$-0.560336\pi$
$887$	−9955.00	−0.376839	−0.188419	+	0.982089i	$0.560336\pi$
$888$	−1776.00	−0.0671156
$889$	3156.00	0.119065
$890$	4460.00	0.167977
$891$	−4455.00	−0.167506
$892$	23636.0	0.887211
$893$	−47080.0	−1.76425
$894$	−9480.00	−0.354652
$895$	14360.0	0.536315
$896$	384.000	0.0143176
$897$	−576.000	−0.0214404
$898$	9948.00	0.369676
$899$	8769.00	0.325320
$900$	900.000	0.0333333
$901$	−60605.0	−2.24089
$902$	1710.00	0.0631228
$903$	6498.00	0.239468
$904$	−14072.0	−0.517730
$905$	−550.000	−0.0202018
$906$	−6048.00	−0.221779
$907$	26644.0	0.975413	0.487706	−	0.873008i	$-0.337834\pi$
$907$	26644.0	0.975413	0.487706	+	0.873008i	$0.337834\pi$
$908$	−11668.0	−0.426450
$909$	15822.0	0.577319
$910$	−480.000	−0.0174855
$911$	12696.0	0.461731	0.230866	−	0.972986i	$-0.425844\pi$
$911$	12696.0	0.461731	0.230866	+	0.972986i	$0.425844\pi$
$912$	10560.0	0.383417
$913$	−900.000	−0.0326239
$914$	5442.00	0.196942
$915$	−9690.00	−0.350100
$916$	−8568.00	−0.309055
$917$	1404.00	0.0505607
$918$	−24840.0	−0.893074
$919$	1448.00	0.0519751	0.0259875	−	0.999662i	$-0.491727\pi$
$919$	1448.00	0.0519751	0.0259875	+	0.999662i	$0.491727\pi$
$920$	240.000	0.00860061
$921$	18996.0	0.679630
$922$	26150.0	0.934061
$923$	5856.00	0.208833
$924$	360.000	0.0128172
$925$	−925.000	−0.0328798
$926$	10168.0	0.360844
$927$	3456.00	0.122449
$928$	−3552.00	−0.125647
$929$	−20283.0	−0.716323	−0.358161	−	0.933660i	$-0.616596\pi$
$929$	−20283.0	−0.716323	−0.358161	+	0.933660i	$0.616596\pi$
$930$	−4740.00	−0.167130
$931$	−36740.0	−1.29335
$932$	22352.0	0.785584
$933$	−54270.0	−1.90431
$934$	23882.0	0.836663
$935$	2875.00	0.100559
$936$	−1152.00	−0.0402290
$937$	−54206.0	−1.88990	−0.944948	−	0.327220i	$-0.893888\pi$
$937$	−54206.0	−1.88990	−0.944948	+	0.327220i	$0.893888\pi$
$938$	−2784.00	−0.0969092
$939$	53436.0	1.85710
$940$	−8560.00	−0.297017
$941$	29370.0	1.01747	0.508733	−	0.860925i	$-0.330114\pi$
$941$	29370.0	1.01747	0.508733	+	0.860925i	$0.330114\pi$
$942$	−26388.0	−0.912704
$943$	1026.00	0.0354307
$944$	1792.00	0.0617846
$945$	−1620.00	−0.0557657
$946$	3610.00	0.124071
$947$	−18721.0	−0.642398	−0.321199	−	0.947012i	$-0.604086\pi$
$947$	−18721.0	−0.642398	−0.321199	+	0.947012i	$0.604086\pi$
$948$	4224.00	0.144714
$949$	−11392.0	−0.389673
$950$	5500.00	0.187835
$951$	58134.0	1.98225
$952$	2760.00	0.0939623
$953$	−53772.0	−1.82775	−0.913875	−	0.405995i	$-0.866925\pi$
$953$	−53772.0	−1.82775	−0.913875	+	0.405995i	$0.866925\pi$
$954$	−9486.00	−0.321929
$955$	−18045.0	−0.611437
$956$	5484.00	0.185528
$957$	−3330.00	−0.112480
$958$	2432.00	0.0820192
$959$	144.000	0.00484880
$960$	1920.00	0.0645497
$961$	−23550.0	−0.790507
$962$	1184.00	0.0396816
$963$	−7146.00	−0.239124
$964$	16736.0	0.559160
$965$	−290.000	−0.00967402
$966$	216.000	0.00719429
$967$	−25448.0	−0.846280	−0.423140	−	0.906064i	$-0.639072\pi$
$967$	−25448.0	−0.846280	−0.423140	+	0.906064i	$0.639072\pi$
$968$	−10448.0	−0.346913
$969$	75900.0	2.51626
$970$	−14070.0	−0.465732
$971$	31407.0	1.03800	0.519000	−	0.854774i	$-0.326304\pi$
$971$	31407.0	1.03800	0.519000	+	0.854774i	$0.326304\pi$
$972$	−9720.00	−0.320750
$973$	2823.00	0.0930126
$974$	−3236.00	−0.106456
$975$	−2400.00	−0.0788323
$976$	−5168.00	−0.169491
$977$	−7589.00	−0.248509	−0.124255	−	0.992250i	$-0.539654\pi$
$977$	−7589.00	−0.248509	−0.124255	+	0.992250i	$0.539654\pi$
$978$	−4548.00	−0.148700
$979$	2230.00	0.0727999
$980$	−6680.00	−0.217740
$981$	1809.00	0.0588756
$982$	22264.0	0.723496
$983$	−48513.0	−1.57408	−0.787042	−	0.616900i	$-0.788388\pi$
$983$	−48513.0	−1.57408	−0.787042	+	0.616900i	$0.788388\pi$
$984$	8208.00	0.265916
$985$	−6770.00	−0.218995
$986$	−25530.0	−0.824585
$987$	−7704.00	−0.248451
$988$	−7040.00	−0.226693
$989$	2166.00	0.0696408
$990$	450.000	0.0144464
$991$	−13213.0	−0.423537	−0.211768	−	0.977320i	$-0.567922\pi$
$991$	−13213.0	−0.423537	−0.211768	+	0.977320i	$0.567922\pi$
$992$	−2528.00	−0.0809114
$993$	588.000	0.0187912
$994$	−2196.00	−0.0700733
$995$	9440.00	0.300772
$996$	−4320.00	−0.137434
$997$	50596.0	1.60721	0.803607	−	0.595161i	$-0.202912\pi$
$997$	50596.0	1.60721	0.803607	+	0.595161i	$0.202912\pi$
$998$	−33124.0	−1.05062
$999$	3996.00	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.4.a.c.1.1	✓	1
5.4	even	2		1850.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.4.a.c.1.1	✓	1	1.1	even	1	trivial
1850.4.a.a.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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