LMFDB - Embedded newform 370.4.a.b.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:	$1$
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)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +2.00000 q^{6} -25.0000 q^{7} +8.00000 q^{8} -26.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +2.00000 q^{6} -25.0000 q^{7} +8.00000 q^{8} -26.0000 q^{9} +10.0000 q^{10} +9.00000 q^{11} +4.00000 q^{12} -76.0000 q^{13} -50.0000 q^{14} +5.00000 q^{15} +16.0000 q^{16} -24.0000 q^{17} -52.0000 q^{18} -40.0000 q^{19} +20.0000 q^{20} -25.0000 q^{21} +18.0000 q^{22} -72.0000 q^{23} +8.00000 q^{24} +25.0000 q^{25} -152.000 q^{26} -53.0000 q^{27} -100.000 q^{28} +60.0000 q^{29} +10.0000 q^{30} +26.0000 q^{31} +32.0000 q^{32} +9.00000 q^{33} -48.0000 q^{34} -125.000 q^{35} -104.000 q^{36} +37.0000 q^{37} -80.0000 q^{38} -76.0000 q^{39} +40.0000 q^{40} +267.000 q^{41} -50.0000 q^{42} -382.000 q^{43} +36.0000 q^{44} -130.000 q^{45} -144.000 q^{46} +267.000 q^{47} +16.0000 q^{48} +282.000 q^{49} +50.0000 q^{50} -24.0000 q^{51} -304.000 q^{52} +171.000 q^{53} -106.000 q^{54} +45.0000 q^{55} -200.000 q^{56} -40.0000 q^{57} +120.000 q^{58} +396.000 q^{59} +20.0000 q^{60} -898.000 q^{61} +52.0000 q^{62} +650.000 q^{63} +64.0000 q^{64} -380.000 q^{65} +18.0000 q^{66} -676.000 q^{67} -96.0000 q^{68} -72.0000 q^{69} -250.000 q^{70} -21.0000 q^{71} -208.000 q^{72} -691.000 q^{73} +74.0000 q^{74} +25.0000 q^{75} -160.000 q^{76} -225.000 q^{77} -152.000 q^{78} -394.000 q^{79} +80.0000 q^{80} +649.000 q^{81} +534.000 q^{82} +309.000 q^{83} -100.000 q^{84} -120.000 q^{85} -764.000 q^{86} +60.0000 q^{87} +72.0000 q^{88} -918.000 q^{89} -260.000 q^{90} +1900.00 q^{91} -288.000 q^{92} +26.0000 q^{93} +534.000 q^{94} -200.000 q^{95} +32.0000 q^{96} -766.000 q^{97} +564.000 q^{98} -234.000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.192450	0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000	0.192450	0.0962250	+	0.995360i	$0.469323\pi$
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	2.00000	0.136083
$7$	−25.0000	−1.34987	−0.674937	−	0.737876i	$-0.735829\pi$
$7$	−25.0000	−1.34987	−0.674937	+	0.737876i	$0.735829\pi$
$8$	8.00000	0.353553
$9$	−26.0000	−0.962963
$10$	10.0000	0.316228
$11$	9.00000	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$11$	9.00000	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$12$	4.00000	0.0962250
$13$	−76.0000	−1.62143	−0.810716	−	0.585440i	$-0.800922\pi$
$13$	−76.0000	−1.62143	−0.810716	+	0.585440i	$0.800922\pi$
$14$	−50.0000	−0.954504
$15$	5.00000	0.0860663
$16$	16.0000	0.250000
$17$	−24.0000	−0.342403	−0.171202	−	0.985236i	$-0.554765\pi$
$17$	−24.0000	−0.342403	−0.171202	+	0.985236i	$0.554765\pi$
$18$	−52.0000	−0.680918
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	20.0000	0.223607
$21$	−25.0000	−0.259783
$22$	18.0000	0.174437
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	8.00000	0.0680414
$25$	25.0000	0.200000
$26$	−152.000	−1.14653
$27$	−53.0000	−0.377772
$28$	−100.000	−0.674937
$29$	60.0000	0.384197	0.192099	−	0.981376i	$-0.438471\pi$
$29$	60.0000	0.384197	0.192099	+	0.981376i	$0.438471\pi$
$30$	10.0000	0.0608581
$31$	26.0000	0.150637	0.0753184	−	0.997160i	$-0.476003\pi$
$31$	26.0000	0.150637	0.0753184	+	0.997160i	$0.476003\pi$
$32$	32.0000	0.176777
$33$	9.00000	0.0474757
$34$	−48.0000	−0.242116
$35$	−125.000	−0.603682
$36$	−104.000	−0.481481
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	−80.0000	−0.341519
$39$	−76.0000	−0.312045
$40$	40.0000	0.158114
$41$	267.000	1.01703	0.508517	−	0.861052i	$-0.330194\pi$
$41$	267.000	1.01703	0.508517	+	0.861052i	$0.330194\pi$
$42$	−50.0000	−0.183694
$43$	−382.000	−1.35475	−0.677377	−	0.735636i	$-0.736884\pi$
$43$	−382.000	−1.35475	−0.677377	+	0.735636i	$0.736884\pi$
$44$	36.0000	0.123346
$45$	−130.000	−0.430650
$46$	−144.000	−0.461557
$47$	267.000	0.828637	0.414319	−	0.910132i	$-0.364020\pi$
$47$	267.000	0.828637	0.414319	+	0.910132i	$0.364020\pi$
$48$	16.0000	0.0481125
$49$	282.000	0.822157
$50$	50.0000	0.141421
$51$	−24.0000	−0.0658955
$52$	−304.000	−0.810716
$53$	171.000	0.443182	0.221591	−	0.975140i	$-0.428875\pi$
$53$	171.000	0.443182	0.221591	+	0.975140i	$0.428875\pi$
$54$	−106.000	−0.267125
$55$	45.0000	0.110324
$56$	−200.000	−0.477252
$57$	−40.0000	−0.0929496
$58$	120.000	0.271668
$59$	396.000	0.873810	0.436905	−	0.899508i	$-0.356075\pi$
$59$	396.000	0.873810	0.436905	+	0.899508i	$0.356075\pi$
$60$	20.0000	0.0430331
$61$	−898.000	−1.88487	−0.942435	−	0.334388i	$-0.891470\pi$
$61$	−898.000	−1.88487	−0.942435	+	0.334388i	$0.891470\pi$
$62$	52.0000	0.106516
$63$	650.000	1.29988
$64$	64.0000	0.125000
$65$	−380.000	−0.725126
$66$	18.0000	0.0335704
$67$	−676.000	−1.23263	−0.616317	−	0.787498i	$-0.711376\pi$
$67$	−676.000	−1.23263	−0.616317	+	0.787498i	$0.711376\pi$
$68$	−96.0000	−0.171202
$69$	−72.0000	−0.125620
$70$	−250.000	−0.426867
$71$	−21.0000	−0.0351020	−0.0175510	−	0.999846i	$-0.505587\pi$
$71$	−21.0000	−0.0351020	−0.0175510	+	0.999846i	$0.505587\pi$
$72$	−208.000	−0.340459
$73$	−691.000	−1.10788	−0.553941	−	0.832556i	$-0.686877\pi$
$73$	−691.000	−1.10788	−0.553941	+	0.832556i	$0.686877\pi$
$74$	74.0000	0.116248
$75$	25.0000	0.0384900
$76$	−160.000	−0.241490
$77$	−225.000	−0.333002
$78$	−152.000	−0.220649
$79$	−394.000	−0.561120	−0.280560	−	0.959837i	$-0.590520\pi$
$79$	−394.000	−0.561120	−0.280560	+	0.959837i	$0.590520\pi$
$80$	80.0000	0.111803
$81$	649.000	0.890261
$82$	534.000	0.719152
$83$	309.000	0.408640	0.204320	−	0.978904i	$-0.434502\pi$
$83$	309.000	0.408640	0.204320	+	0.978904i	$0.434502\pi$
$84$	−100.000	−0.129892
$85$	−120.000	−0.153127
$86$	−764.000	−0.957956
$87$	60.0000	0.0739388
$88$	72.0000	0.0872185
$89$	−918.000	−1.09335	−0.546673	−	0.837346i	$-0.684106\pi$
$89$	−918.000	−1.09335	−0.546673	+	0.837346i	$0.684106\pi$
$90$	−260.000	−0.304516
$91$	1900.00	2.18873
$92$	−288.000	−0.326370
$93$	26.0000	0.0289900
$94$	534.000	0.585935
$95$	−200.000	−0.215995
$96$	32.0000	0.0340207
$97$	−766.000	−0.801809	−0.400905	−	0.916120i	$-0.631304\pi$
$97$	−766.000	−0.801809	−0.400905	+	0.916120i	$0.631304\pi$
$98$	564.000	0.581353
$99$	−234.000	−0.237554
$100$	100.000	0.100000
$101$	1971.00	1.94180	0.970900	−	0.239485i	$-0.0769785\pi$
$101$	1971.00	1.94180	0.970900	+	0.239485i	$0.0769785\pi$
$102$	−48.0000	−0.0465952
$103$	−34.0000	−0.0325254	−0.0162627	−	0.999868i	$-0.505177\pi$
$103$	−34.0000	−0.0325254	−0.0162627	+	0.999868i	$0.505177\pi$
$104$	−608.000	−0.573263
$105$	−125.000	−0.116179
$106$	342.000	0.313377
$107$	2040.00	1.84312	0.921562	−	0.388231i	$-0.126914\pi$
$107$	2040.00	1.84312	0.921562	+	0.388231i	$0.126914\pi$
$108$	−212.000	−0.188886
$109$	−214.000	−0.188050	−0.0940251	−	0.995570i	$-0.529973\pi$
$109$	−214.000	−0.188050	−0.0940251	+	0.995570i	$0.529973\pi$
$110$	90.0000	0.0780106
$111$	37.0000	0.0316386
$112$	−400.000	−0.337468
$113$	−66.0000	−0.0549448	−0.0274724	−	0.999623i	$-0.508746\pi$
$113$	−66.0000	−0.0549448	−0.0274724	+	0.999623i	$0.508746\pi$
$114$	−80.0000	−0.0657253
$115$	−360.000	−0.291915
$116$	240.000	0.192099
$117$	1976.00	1.56138
$118$	792.000	0.617877
$119$	600.000	0.462201
$120$	40.0000	0.0304290
$121$	−1250.00	−0.939144
$122$	−1796.00	−1.33280
$123$	267.000	0.195728
$124$	104.000	0.0753184
$125$	125.000	0.0894427
$126$	1300.00	0.919152
$127$	1433.00	1.00125	0.500623	−	0.865666i	$-0.333104\pi$
$127$	1433.00	1.00125	0.500623	+	0.865666i	$0.333104\pi$
$128$	128.000	0.0883883
$129$	−382.000	−0.260723
$130$	−760.000	−0.512742
$131$	1278.00	0.852362	0.426181	−	0.904638i	$-0.359859\pi$
$131$	1278.00	0.852362	0.426181	+	0.904638i	$0.359859\pi$
$132$	36.0000	0.0237379
$133$	1000.00	0.651962
$134$	−1352.00	−0.871605
$135$	−265.000	−0.168945
$136$	−192.000	−0.121058
$137$	−1770.00	−1.10381	−0.551903	−	0.833909i	$-0.686098\pi$
$137$	−1770.00	−1.10381	−0.551903	+	0.833909i	$0.686098\pi$
$138$	−144.000	−0.0888268
$139$	−844.000	−0.515015	−0.257508	−	0.966276i	$-0.582901\pi$
$139$	−844.000	−0.515015	−0.257508	+	0.966276i	$0.582901\pi$
$140$	−500.000	−0.301841
$141$	267.000	0.159471
$142$	−42.0000	−0.0248209
$143$	−684.000	−0.399993
$144$	−416.000	−0.240741
$145$	300.000	0.171818
$146$	−1382.00	−0.783391
$147$	282.000	0.158224
$148$	148.000	0.0821995
$149$	−2553.00	−1.40369	−0.701845	−	0.712329i	$-0.747640\pi$
$149$	−2553.00	−1.40369	−0.701845	+	0.712329i	$0.747640\pi$
$150$	50.0000	0.0272166
$151$	3272.00	1.76339	0.881694	−	0.471822i	$-0.156403\pi$
$151$	3272.00	1.76339	0.881694	+	0.471822i	$0.156403\pi$
$152$	−320.000	−0.170759
$153$	624.000	0.329722
$154$	−450.000	−0.235468
$155$	130.000	0.0673668
$156$	−304.000	−0.156022
$157$	−331.000	−0.168259	−0.0841295	−	0.996455i	$-0.526811\pi$
$157$	−331.000	−0.168259	−0.0841295	+	0.996455i	$0.526811\pi$
$158$	−788.000	−0.396772
$159$	171.000	0.0852905
$160$	160.000	0.0790569
$161$	1800.00	0.881117
$162$	1298.00	0.629509
$163$	−1732.00	−0.832274	−0.416137	−	0.909302i	$-0.636616\pi$
$163$	−1732.00	−0.832274	−0.416137	+	0.909302i	$0.636616\pi$
$164$	1068.00	0.508517
$165$	45.0000	0.0212318
$166$	618.000	0.288952
$167$	3666.00	1.69870	0.849352	−	0.527827i	$-0.176993\pi$
$167$	3666.00	1.69870	0.849352	+	0.527827i	$0.176993\pi$
$168$	−200.000	−0.0918472
$169$	3579.00	1.62904
$170$	−240.000	−0.108277
$171$	1040.00	0.465092
$172$	−1528.00	−0.677377
$173$	2739.00	1.20371	0.601856	−	0.798604i	$-0.294428\pi$
$173$	2739.00	1.20371	0.601856	+	0.798604i	$0.294428\pi$
$174$	120.000	0.0522826
$175$	−625.000	−0.269975
$176$	144.000	0.0616728
$177$	396.000	0.168165
$178$	−1836.00	−0.773112
$179$	−2448.00	−1.02219	−0.511095	−	0.859524i	$-0.670760\pi$
$179$	−2448.00	−1.02219	−0.511095	+	0.859524i	$0.670760\pi$
$180$	−520.000	−0.215325
$181$	−307.000	−0.126073	−0.0630363	−	0.998011i	$-0.520078\pi$
$181$	−307.000	−0.126073	−0.0630363	+	0.998011i	$0.520078\pi$
$182$	3800.00	1.54766
$183$	−898.000	−0.362744
$184$	−576.000	−0.230779
$185$	185.000	0.0735215
$186$	52.0000	0.0204991
$187$	−216.000	−0.0844678
$188$	1068.00	0.414319
$189$	1325.00	0.509945
$190$	−400.000	−0.152732
$191$	−5004.00	−1.89569	−0.947845	−	0.318732i	$-0.896743\pi$
$191$	−5004.00	−1.89569	−0.947845	+	0.318732i	$0.896743\pi$
$192$	64.0000	0.0240563
$193$	−460.000	−0.171562	−0.0857812	−	0.996314i	$-0.527339\pi$
$193$	−460.000	−0.171562	−0.0857812	+	0.996314i	$0.527339\pi$
$194$	−1532.00	−0.566965
$195$	−380.000	−0.139551
$196$	1128.00	0.411079
$197$	69.0000	0.0249546	0.0124773	−	0.999922i	$-0.496028\pi$
$197$	69.0000	0.0249546	0.0124773	+	0.999922i	$0.496028\pi$
$198$	−468.000	−0.167976
$199$	−1006.00	−0.358359	−0.179180	−	0.983816i	$-0.557344\pi$
$199$	−1006.00	−0.358359	−0.179180	+	0.983816i	$0.557344\pi$
$200$	200.000	0.0707107
$201$	−676.000	−0.237221
$202$	3942.00	1.37306
$203$	−1500.00	−0.518618
$204$	−96.0000	−0.0329478
$205$	1335.00	0.454831
$206$	−68.0000	−0.0229990
$207$	1872.00	0.628565
$208$	−1216.00	−0.405358
$209$	−360.000	−0.119147
$210$	−250.000	−0.0821507
$211$	3353.00	1.09398	0.546991	−	0.837139i	$-0.315773\pi$
$211$	3353.00	1.09398	0.546991	+	0.837139i	$0.315773\pi$
$212$	684.000	0.221591
$213$	−21.0000	−0.00675538
$214$	4080.00	1.30329
$215$	−1910.00	−0.605865
$216$	−424.000	−0.133563
$217$	−650.000	−0.203340
$218$	−428.000	−0.132972
$219$	−691.000	−0.213212
$220$	180.000	0.0551618
$221$	1824.00	0.555183
$222$	74.0000	0.0223719
$223$	−5677.00	−1.70475	−0.852377	−	0.522928i	$-0.824840\pi$
$223$	−5677.00	−1.70475	−0.852377	+	0.522928i	$0.824840\pi$
$224$	−800.000	−0.238626
$225$	−650.000	−0.192593
$226$	−132.000	−0.0388518
$227$	2076.00	0.607000	0.303500	−	0.952831i	$-0.401845\pi$
$227$	2076.00	0.607000	0.303500	+	0.952831i	$0.401845\pi$
$228$	−160.000	−0.0464748
$229$	3671.00	1.05933	0.529665	−	0.848207i	$-0.322318\pi$
$229$	3671.00	1.05933	0.529665	+	0.848207i	$0.322318\pi$
$230$	−720.000	−0.206415
$231$	−225.000	−0.0640862
$232$	480.000	0.135834
$233$	−5742.00	−1.61447	−0.807234	−	0.590232i	$-0.799036\pi$
$233$	−5742.00	−1.61447	−0.807234	+	0.590232i	$0.799036\pi$
$234$	3952.00	1.10406
$235$	1335.00	0.370578
$236$	1584.00	0.436905
$237$	−394.000	−0.107988
$238$	1200.00	0.326825
$239$	3918.00	1.06039	0.530197	−	0.847874i	$-0.322118\pi$
$239$	3918.00	1.06039	0.530197	+	0.847874i	$0.322118\pi$
$240$	80.0000	0.0215166
$241$	2540.00	0.678904	0.339452	−	0.940623i	$-0.389758\pi$
$241$	2540.00	0.678904	0.339452	+	0.940623i	$0.389758\pi$
$242$	−2500.00	−0.664075
$243$	2080.00	0.549103
$244$	−3592.00	−0.942435
$245$	1410.00	0.367680
$246$	534.000	0.138401
$247$	3040.00	0.783120
$248$	208.000	0.0532581
$249$	309.000	0.0786429
$250$	250.000	0.0632456
$251$	444.000	0.111654	0.0558268	−	0.998440i	$-0.482221\pi$
$251$	444.000	0.111654	0.0558268	+	0.998440i	$0.482221\pi$
$252$	2600.00	0.649939
$253$	−648.000	−0.161025
$254$	2866.00	0.707988
$255$	−120.000	−0.0294694
$256$	256.000	0.0625000
$257$	−5034.00	−1.22184	−0.610919	−	0.791693i	$-0.709200\pi$
$257$	−5034.00	−1.22184	−0.610919	+	0.791693i	$0.709200\pi$
$258$	−764.000	−0.184359
$259$	−925.000	−0.221918
$260$	−1520.00	−0.362563
$261$	−1560.00	−0.369968
$262$	2556.00	0.602711
$263$	−393.000	−0.0921423	−0.0460711	−	0.998938i	$-0.514670\pi$
$263$	−393.000	−0.0921423	−0.0460711	+	0.998938i	$0.514670\pi$
$264$	72.0000	0.0167852
$265$	855.000	0.198197
$266$	2000.00	0.461007
$267$	−918.000	−0.210415
$268$	−2704.00	−0.616317
$269$	−1566.00	−0.354947	−0.177473	−	0.984126i	$-0.556792\pi$
$269$	−1566.00	−0.354947	−0.177473	+	0.984126i	$0.556792\pi$
$270$	−530.000	−0.119462
$271$	3431.00	0.769072	0.384536	−	0.923110i	$-0.374362\pi$
$271$	3431.00	0.769072	0.384536	+	0.923110i	$0.374362\pi$
$272$	−384.000	−0.0856008
$273$	1900.00	0.421221
$274$	−3540.00	−0.780508
$275$	225.000	0.0493382
$276$	−288.000	−0.0628100
$277$	−4450.00	−0.965251	−0.482626	−	0.875827i	$-0.660317\pi$
$277$	−4450.00	−0.965251	−0.482626	+	0.875827i	$0.660317\pi$
$278$	−1688.00	−0.364171
$279$	−676.000	−0.145058
$280$	−1000.00	−0.213434
$281$	318.000	0.0675099	0.0337550	−	0.999430i	$-0.489253\pi$
$281$	318.000	0.0675099	0.0337550	+	0.999430i	$0.489253\pi$
$282$	534.000	0.112763
$283$	−7936.00	−1.66695	−0.833474	−	0.552559i	$-0.813651\pi$
$283$	−7936.00	−1.66695	−0.833474	+	0.552559i	$0.813651\pi$
$284$	−84.0000	−0.0175510
$285$	−200.000	−0.0415683
$286$	−1368.00	−0.282838
$287$	−6675.00	−1.37287
$288$	−832.000	−0.170229
$289$	−4337.00	−0.882760
$290$	600.000	0.121494
$291$	−766.000	−0.154308
$292$	−2764.00	−0.553941
$293$	−1698.00	−0.338560	−0.169280	−	0.985568i	$-0.554144\pi$
$293$	−1698.00	−0.338560	−0.169280	+	0.985568i	$0.554144\pi$
$294$	564.000	0.111881
$295$	1980.00	0.390780
$296$	296.000	0.0581238
$297$	−477.000	−0.0931931
$298$	−5106.00	−0.992559
$299$	5472.00	1.05837
$300$	100.000	0.0192450
$301$	9550.00	1.82875
$302$	6544.00	1.24690
$303$	1971.00	0.373700
$304$	−640.000	−0.120745
$305$	−4490.00	−0.842940
$306$	1248.00	0.233148
$307$	−1189.00	−0.221042	−0.110521	−	0.993874i	$-0.535252\pi$
$307$	−1189.00	−0.221042	−0.110521	+	0.993874i	$0.535252\pi$
$308$	−900.000	−0.166501
$309$	−34.0000	−0.00625952
$310$	260.000	0.0476355
$311$	−3984.00	−0.726405	−0.363202	−	0.931710i	$-0.618317\pi$
$311$	−3984.00	−0.726405	−0.363202	+	0.931710i	$0.618317\pi$
$312$	−608.000	−0.110324
$313$	−6010.00	−1.08532	−0.542660	−	0.839952i	$-0.682583\pi$
$313$	−6010.00	−1.08532	−0.542660	+	0.839952i	$0.682583\pi$
$314$	−662.000	−0.118977
$315$	3250.00	0.581323
$316$	−1576.00	−0.280560
$317$	−6234.00	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$317$	−6234.00	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$318$	342.000	0.0603095
$319$	540.000	0.0947780
$320$	320.000	0.0559017
$321$	2040.00	0.354709
$322$	3600.00	0.623044
$323$	960.000	0.165374
$324$	2596.00	0.445130
$325$	−1900.00	−0.324286
$326$	−3464.00	−0.588507
$327$	−214.000	−0.0361903
$328$	2136.00	0.359576
$329$	−6675.00	−1.11856
$330$	90.0000	0.0150131
$331$	10316.0	1.71305	0.856524	−	0.516108i	$-0.172620\pi$
$331$	10316.0	1.71305	0.856524	+	0.516108i	$0.172620\pi$
$332$	1236.00	0.204320
$333$	−962.000	−0.158310
$334$	7332.00	1.20117
$335$	−3380.00	−0.551251
$336$	−400.000	−0.0649458
$337$	−4507.00	−0.728522	−0.364261	−	0.931297i	$-0.618678\pi$
$337$	−4507.00	−0.728522	−0.364261	+	0.931297i	$0.618678\pi$
$338$	7158.00	1.15190
$339$	−66.0000	−0.0105741
$340$	−480.000	−0.0765637
$341$	234.000	0.0371607
$342$	2080.00	0.328870
$343$	1525.00	0.240065
$344$	−3056.00	−0.478978
$345$	−360.000	−0.0561790
$346$	5478.00	0.851153
$347$	5136.00	0.794568	0.397284	−	0.917696i	$-0.369953\pi$
$347$	5136.00	0.794568	0.397284	+	0.917696i	$0.369953\pi$
$348$	240.000	0.0369694
$349$	−6658.00	−1.02119	−0.510594	−	0.859822i	$-0.670574\pi$
$349$	−6658.00	−1.02119	−0.510594	+	0.859822i	$0.670574\pi$
$350$	−1250.00	−0.190901
$351$	4028.00	0.612532
$352$	288.000	0.0436092
$353$	−1302.00	−0.196313	−0.0981565	−	0.995171i	$-0.531295\pi$
$353$	−1302.00	−0.196313	−0.0981565	+	0.995171i	$0.531295\pi$
$354$	792.000	0.118911
$355$	−105.000	−0.0156981
$356$	−3672.00	−0.546673
$357$	600.000	0.0889506
$358$	−4896.00	−0.722798
$359$	−2949.00	−0.433544	−0.216772	−	0.976222i	$-0.569553\pi$
$359$	−2949.00	−0.433544	−0.216772	+	0.976222i	$0.569553\pi$
$360$	−1040.00	−0.152258
$361$	−5259.00	−0.766730
$362$	−614.000	−0.0891468
$363$	−1250.00	−0.180738
$364$	7600.00	1.09436
$365$	−3455.00	−0.495460
$366$	−1796.00	−0.256498
$367$	−7672.00	−1.09121	−0.545606	−	0.838042i	$-0.683701\pi$
$367$	−7672.00	−1.09121	−0.545606	+	0.838042i	$0.683701\pi$
$368$	−1152.00	−0.163185
$369$	−6942.00	−0.979366
$370$	370.000	0.0519875
$371$	−4275.00	−0.598240
$372$	104.000	0.0144950
$373$	−9709.00	−1.34776	−0.673878	−	0.738843i	$-0.735373\pi$
$373$	−9709.00	−1.34776	−0.673878	+	0.738843i	$0.735373\pi$
$374$	−432.000	−0.0597278
$375$	125.000	0.0172133
$376$	2136.00	0.292968
$377$	−4560.00	−0.622949
$378$	2650.00	0.360585
$379$	−3991.00	−0.540907	−0.270454	−	0.962733i	$-0.587174\pi$
$379$	−3991.00	−0.540907	−0.270454	+	0.962733i	$0.587174\pi$
$380$	−800.000	−0.107998
$381$	1433.00	0.192690
$382$	−10008.0	−1.34046
$383$	−6042.00	−0.806088	−0.403044	−	0.915181i	$-0.632048\pi$
$383$	−6042.00	−0.806088	−0.403044	+	0.915181i	$0.632048\pi$
$384$	128.000	0.0170103
$385$	−1125.00	−0.148923
$386$	−920.000	−0.121313
$387$	9932.00	1.30458
$388$	−3064.00	−0.400905
$389$	4728.00	0.616245	0.308122	−	0.951347i	$-0.400299\pi$
$389$	4728.00	0.616245	0.308122	+	0.951347i	$0.400299\pi$
$390$	−760.000	−0.0986772
$391$	1728.00	0.223501
$392$	2256.00	0.290677
$393$	1278.00	0.164037
$394$	138.000	0.0176455
$395$	−1970.00	−0.250940
$396$	−936.000	−0.118777
$397$	−13051.0	−1.64990	−0.824951	−	0.565205i	$-0.808797\pi$
$397$	−13051.0	−1.64990	−0.824951	+	0.565205i	$0.808797\pi$
$398$	−2012.00	−0.253398
$399$	1000.00	0.125470
$400$	400.000	0.0500000
$401$	1098.00	0.136737	0.0683685	−	0.997660i	$-0.478221\pi$
$401$	1098.00	0.136737	0.0683685	+	0.997660i	$0.478221\pi$
$402$	−1352.00	−0.167740
$403$	−1976.00	−0.244247
$404$	7884.00	0.970900
$405$	3245.00	0.398137
$406$	−3000.00	−0.366718
$407$	333.000	0.0405558
$408$	−192.000	−0.0232976
$409$	−1996.00	−0.241310	−0.120655	−	0.992694i	$-0.538500\pi$
$409$	−1996.00	−0.241310	−0.120655	+	0.992694i	$0.538500\pi$
$410$	2670.00	0.321614
$411$	−1770.00	−0.212427
$412$	−136.000	−0.0162627
$413$	−9900.00	−1.17953
$414$	3744.00	0.444463
$415$	1545.00	0.182750
$416$	−2432.00	−0.286631
$417$	−844.000	−0.0991148
$418$	−720.000	−0.0842496
$419$	−7023.00	−0.818845	−0.409422	−	0.912345i	$-0.634270\pi$
$419$	−7023.00	−0.818845	−0.409422	+	0.912345i	$0.634270\pi$
$420$	−500.000	−0.0580893
$421$	3404.00	0.394064	0.197032	−	0.980397i	$-0.436870\pi$
$421$	3404.00	0.394064	0.197032	+	0.980397i	$0.436870\pi$
$422$	6706.00	0.773562
$423$	−6942.00	−0.797947
$424$	1368.00	0.156689
$425$	−600.000	−0.0684806
$426$	−42.0000	−0.00477678
$427$	22450.0	2.54434
$428$	8160.00	0.921562
$429$	−684.000	−0.0769786
$430$	−3820.00	−0.428411
$431$	2712.00	0.303092	0.151546	−	0.988450i	$-0.451575\pi$
$431$	2712.00	0.303092	0.151546	+	0.988450i	$0.451575\pi$
$432$	−848.000	−0.0944431
$433$	1043.00	0.115758	0.0578792	−	0.998324i	$-0.481566\pi$
$433$	1043.00	0.115758	0.0578792	+	0.998324i	$0.481566\pi$
$434$	−1300.00	−0.143783
$435$	300.000	0.0330664
$436$	−856.000	−0.0940251
$437$	2880.00	0.315261
$438$	−1382.00	−0.150764
$439$	8612.00	0.936283	0.468142	−	0.883653i	$-0.344924\pi$
$439$	8612.00	0.936283	0.468142	+	0.883653i	$0.344924\pi$
$440$	360.000	0.0390053
$441$	−7332.00	−0.791707
$442$	3648.00	0.392574
$443$	−4935.00	−0.529275	−0.264638	−	0.964348i	$-0.585252\pi$
$443$	−4935.00	−0.529275	−0.264638	+	0.964348i	$0.585252\pi$
$444$	148.000	0.0158193
$445$	−4590.00	−0.488959
$446$	−11354.0	−1.20544
$447$	−2553.00	−0.270140
$448$	−1600.00	−0.168734
$449$	1290.00	0.135588	0.0677938	−	0.997699i	$-0.478404\pi$
$449$	1290.00	0.135588	0.0677938	+	0.997699i	$0.478404\pi$
$450$	−1300.00	−0.136184
$451$	2403.00	0.250893
$452$	−264.000	−0.0274724
$453$	3272.00	0.339364
$454$	4152.00	0.429214
$455$	9500.00	0.978828
$456$	−320.000	−0.0328627
$457$	−5038.00	−0.515684	−0.257842	−	0.966187i	$-0.583011\pi$
$457$	−5038.00	−0.515684	−0.257842	+	0.966187i	$0.583011\pi$
$458$	7342.00	0.749059
$459$	1272.00	0.129350
$460$	−1440.00	−0.145957
$461$	1074.00	0.108506	0.0542529	−	0.998527i	$-0.482722\pi$
$461$	1074.00	0.108506	0.0542529	+	0.998527i	$0.482722\pi$
$462$	−450.000	−0.0453158
$463$	14762.0	1.48175	0.740873	−	0.671645i	$-0.234412\pi$
$463$	14762.0	1.48175	0.740873	+	0.671645i	$0.234412\pi$
$464$	960.000	0.0960493
$465$	130.000	0.0129647
$466$	−11484.0	−1.14160
$467$	9702.00	0.961360	0.480680	−	0.876896i	$-0.340390\pi$
$467$	9702.00	0.961360	0.480680	+	0.876896i	$0.340390\pi$
$468$	7904.00	0.780689
$469$	16900.0	1.66390
$470$	2670.00	0.262038
$471$	−331.000	−0.0323815
$472$	3168.00	0.308939
$473$	−3438.00	−0.334206
$474$	−788.000	−0.0763587
$475$	−1000.00	−0.0965961
$476$	2400.00	0.231100
$477$	−4446.00	−0.426768
$478$	7836.00	0.749812
$479$	5136.00	0.489916	0.244958	−	0.969534i	$-0.421226\pi$
$479$	5136.00	0.489916	0.244958	+	0.969534i	$0.421226\pi$
$480$	160.000	0.0152145
$481$	−2812.00	−0.266562
$482$	5080.00	0.480057
$483$	1800.00	0.169571
$484$	−5000.00	−0.469572
$485$	−3830.00	−0.358580
$486$	4160.00	0.388275
$487$	1772.00	0.164881	0.0824404	−	0.996596i	$-0.473729\pi$
$487$	1772.00	0.164881	0.0824404	+	0.996596i	$0.473729\pi$
$488$	−7184.00	−0.666402
$489$	−1732.00	−0.160171
$490$	2820.00	0.259989
$491$	12264.0	1.12722	0.563612	−	0.826040i	$-0.309411\pi$
$491$	12264.0	1.12722	0.563612	+	0.826040i	$0.309411\pi$
$492$	1068.00	0.0978641
$493$	−1440.00	−0.131550
$494$	6080.00	0.553749
$495$	−1170.00	−0.106238
$496$	416.000	0.0376592
$497$	525.000	0.0473832
$498$	618.000	0.0556089
$499$	−6586.00	−0.590841	−0.295421	−	0.955367i	$-0.595460\pi$
$499$	−6586.00	−0.590841	−0.295421	+	0.955367i	$0.595460\pi$
$500$	500.000	0.0447214
$501$	3666.00	0.326916
$502$	888.000	0.0789510
$503$	−6408.00	−0.568029	−0.284015	−	0.958820i	$-0.591666\pi$
$503$	−6408.00	−0.568029	−0.284015	+	0.958820i	$0.591666\pi$
$504$	5200.00	0.459576
$505$	9855.00	0.868399
$506$	−1296.00	−0.113862
$507$	3579.00	0.313509
$508$	5732.00	0.500623
$509$	6501.00	0.566113	0.283057	−	0.959103i	$-0.408652\pi$
$509$	6501.00	0.566113	0.283057	+	0.959103i	$0.408652\pi$
$510$	−240.000	−0.0208380
$511$	17275.0	1.49550
$512$	512.000	0.0441942
$513$	2120.00	0.182457
$514$	−10068.0	−0.863970
$515$	−170.000	−0.0145458
$516$	−1528.00	−0.130361
$517$	2403.00	0.204417
$518$	−1850.00	−0.156920
$519$	2739.00	0.231655
$520$	−3040.00	−0.256371
$521$	9615.00	0.808524	0.404262	−	0.914643i	$-0.367528\pi$
$521$	9615.00	0.808524	0.404262	+	0.914643i	$0.367528\pi$
$522$	−3120.00	−0.261607
$523$	590.000	0.0493287	0.0246643	−	0.999696i	$-0.492148\pi$
$523$	590.000	0.0493287	0.0246643	+	0.999696i	$0.492148\pi$
$524$	5112.00	0.426181
$525$	−625.000	−0.0519566
$526$	−786.000	−0.0651544
$527$	−624.000	−0.0515785
$528$	144.000	0.0118689
$529$	−6983.00	−0.573929
$530$	1710.00	0.140147
$531$	−10296.0	−0.841447
$532$	4000.00	0.325981
$533$	−20292.0	−1.64905
$534$	−1836.00	−0.148786
$535$	10200.0	0.824270
$536$	−5408.00	−0.435802
$537$	−2448.00	−0.196721
$538$	−3132.00	−0.250985
$539$	2538.00	0.202819
$540$	−1060.00	−0.0844725
$541$	13070.0	1.03868	0.519338	−	0.854569i	$-0.326179\pi$
$541$	13070.0	1.03868	0.519338	+	0.854569i	$0.326179\pi$
$542$	6862.00	0.543816
$543$	−307.000	−0.0242627
$544$	−768.000	−0.0605289
$545$	−1070.00	−0.0840986
$546$	3800.00	0.297848
$547$	−6928.00	−0.541535	−0.270768	−	0.962645i	$-0.587278\pi$
$547$	−6928.00	−0.541535	−0.270768	+	0.962645i	$0.587278\pi$
$548$	−7080.00	−0.551903
$549$	23348.0	1.81506
$550$	450.000	0.0348874
$551$	−2400.00	−0.185560
$552$	−576.000	−0.0444134
$553$	9850.00	0.757441
$554$	−8900.00	−0.682536
$555$	185.000	0.0141492
$556$	−3376.00	−0.257508
$557$	1860.00	0.141491	0.0707457	−	0.997494i	$-0.477462\pi$
$557$	1860.00	0.141491	0.0707457	+	0.997494i	$0.477462\pi$
$558$	−1352.00	−0.102571
$559$	29032.0	2.19664
$560$	−2000.00	−0.150920
$561$	−216.000	−0.0162558
$562$	636.000	0.0477367
$563$	1536.00	0.114982	0.0574908	−	0.998346i	$-0.481690\pi$
$563$	1536.00	0.114982	0.0574908	+	0.998346i	$0.481690\pi$
$564$	1068.00	0.0797357
$565$	−330.000	−0.0245720
$566$	−15872.0	−1.17871
$567$	−16225.0	−1.20174
$568$	−168.000	−0.0124104
$569$	−1254.00	−0.0923909	−0.0461954	−	0.998932i	$-0.514710\pi$
$569$	−1254.00	−0.0923909	−0.0461954	+	0.998932i	$0.514710\pi$
$570$	−400.000	−0.0293933
$571$	14789.0	1.08389	0.541944	−	0.840414i	$-0.317688\pi$
$571$	14789.0	1.08389	0.541944	+	0.840414i	$0.317688\pi$
$572$	−2736.00	−0.199996
$573$	−5004.00	−0.364826
$574$	−13350.0	−0.970763
$575$	−1800.00	−0.130548
$576$	−1664.00	−0.120370
$577$	15932.0	1.14949	0.574747	−	0.818331i	$-0.305101\pi$
$577$	15932.0	1.14949	0.574747	+	0.818331i	$0.305101\pi$
$578$	−8674.00	−0.624206
$579$	−460.000	−0.0330172
$580$	1200.00	0.0859091
$581$	−7725.00	−0.551613
$582$	−1532.00	−0.109112
$583$	1539.00	0.109329
$584$	−5528.00	−0.391696
$585$	9880.00	0.698270
$586$	−3396.00	−0.239398
$587$	−9546.00	−0.671219	−0.335610	−	0.942001i	$-0.608942\pi$
$587$	−9546.00	−0.671219	−0.335610	+	0.942001i	$0.608942\pi$
$588$	1128.00	0.0791121
$589$	−1040.00	−0.0727546
$590$	3960.00	0.276323
$591$	69.0000	0.00480251
$592$	592.000	0.0410997
$593$	8793.00	0.608913	0.304456	−	0.952526i	$-0.401525\pi$
$593$	8793.00	0.608913	0.304456	+	0.952526i	$0.401525\pi$
$594$	−954.000	−0.0658975
$595$	3000.00	0.206703
$596$	−10212.0	−0.701845
$597$	−1006.00	−0.0689662
$598$	10944.0	0.748384
$599$	25875.0	1.76498	0.882491	−	0.470330i	$-0.155865\pi$
$599$	25875.0	1.76498	0.882491	+	0.470330i	$0.155865\pi$
$600$	200.000	0.0136083
$601$	19478.0	1.32200	0.661002	−	0.750384i	$-0.270132\pi$
$601$	19478.0	1.32200	0.661002	+	0.750384i	$0.270132\pi$
$602$	19100.0	1.29312
$603$	17576.0	1.18698
$604$	13088.0	0.881694
$605$	−6250.00	−0.419998
$606$	3942.00	0.264246
$607$	−13696.0	−0.915821	−0.457911	−	0.888998i	$-0.651402\pi$
$607$	−13696.0	−0.915821	−0.457911	+	0.888998i	$0.651402\pi$
$608$	−1280.00	−0.0853797
$609$	−1500.00	−0.0998080
$610$	−8980.00	−0.596048
$611$	−20292.0	−1.34358
$612$	2496.00	0.164861
$613$	7049.00	0.464448	0.232224	−	0.972662i	$-0.425400\pi$
$613$	7049.00	0.464448	0.232224	+	0.972662i	$0.425400\pi$
$614$	−2378.00	−0.156300
$615$	1335.00	0.0875324
$616$	−1800.00	−0.117734
$617$	−12069.0	−0.787487	−0.393744	−	0.919220i	$-0.628820\pi$
$617$	−12069.0	−0.787487	−0.393744	+	0.919220i	$0.628820\pi$
$618$	−68.0000	−0.00442615
$619$	−18415.0	−1.19574	−0.597869	−	0.801594i	$-0.703986\pi$
$619$	−18415.0	−1.19574	−0.597869	+	0.801594i	$0.703986\pi$
$620$	520.000	0.0336834
$621$	3816.00	0.246587
$622$	−7968.00	−0.513646
$623$	22950.0	1.47588
$624$	−1216.00	−0.0780112
$625$	625.000	0.0400000
$626$	−12020.0	−0.767437
$627$	−360.000	−0.0229298
$628$	−1324.00	−0.0841295
$629$	−888.000	−0.0562907
$630$	6500.00	0.411057
$631$	10250.0	0.646666	0.323333	−	0.946285i	$-0.395197\pi$
$631$	10250.0	0.646666	0.323333	+	0.946285i	$0.395197\pi$
$632$	−3152.00	−0.198386
$633$	3353.00	0.210537
$634$	−12468.0	−0.781021
$635$	7165.00	0.447771
$636$	684.000	0.0426452
$637$	−21432.0	−1.33307
$638$	1080.00	0.0670182
$639$	546.000	0.0338019
$640$	640.000	0.0395285
$641$	−1569.00	−0.0966798	−0.0483399	−	0.998831i	$-0.515393\pi$
$641$	−1569.00	−0.0966798	−0.0483399	+	0.998831i	$0.515393\pi$
$642$	4080.00	0.250817
$643$	27596.0	1.69250	0.846252	−	0.532783i	$-0.178854\pi$
$643$	27596.0	1.69250	0.846252	+	0.532783i	$0.178854\pi$
$644$	7200.00	0.440559
$645$	−1910.00	−0.116599
$646$	1920.00	0.116937
$647$	−9126.00	−0.554529	−0.277264	−	0.960794i	$-0.589428\pi$
$647$	−9126.00	−0.554529	−0.277264	+	0.960794i	$0.589428\pi$
$648$	5192.00	0.314755
$649$	3564.00	0.215561
$650$	−3800.00	−0.229305
$651$	−650.000	−0.0391329
$652$	−6928.00	−0.416137
$653$	−14250.0	−0.853975	−0.426987	−	0.904258i	$-0.640425\pi$
$653$	−14250.0	−0.853975	−0.426987	+	0.904258i	$0.640425\pi$
$654$	−428.000	−0.0255904
$655$	6390.00	0.381188
$656$	4272.00	0.254259
$657$	17966.0	1.06685
$658$	−13350.0	−0.790938
$659$	−5985.00	−0.353782	−0.176891	−	0.984230i	$-0.556604\pi$
$659$	−5985.00	−0.353782	−0.176891	+	0.984230i	$0.556604\pi$
$660$	180.000	0.0106159
$661$	−16846.0	−0.991276	−0.495638	−	0.868529i	$-0.665066\pi$
$661$	−16846.0	−0.991276	−0.495638	+	0.868529i	$0.665066\pi$
$662$	20632.0	1.21131
$663$	1824.00	0.106845
$664$	2472.00	0.144476
$665$	5000.00	0.291566
$666$	−1924.00	−0.111942
$667$	−4320.00	−0.250781
$668$	14664.0	0.849352
$669$	−5677.00	−0.328080
$670$	−6760.00	−0.389793
$671$	−8082.00	−0.464981
$672$	−800.000	−0.0459236
$673$	2297.00	0.131564	0.0657822	−	0.997834i	$-0.479046\pi$
$673$	2297.00	0.131564	0.0657822	+	0.997834i	$0.479046\pi$
$674$	−9014.00	−0.515143
$675$	−1325.00	−0.0755545
$676$	14316.0	0.814520
$677$	27999.0	1.58950	0.794748	−	0.606939i	$-0.207603\pi$
$677$	27999.0	1.58950	0.794748	+	0.606939i	$0.207603\pi$
$678$	−132.000	−0.00747703
$679$	19150.0	1.08234
$680$	−960.000	−0.0541387
$681$	2076.00	0.116817
$682$	468.000	0.0262766
$683$	−28872.0	−1.61751	−0.808753	−	0.588149i	$-0.799857\pi$
$683$	−28872.0	−1.61751	−0.808753	+	0.588149i	$0.799857\pi$
$684$	4160.00	0.232546
$685$	−8850.00	−0.493637
$686$	3050.00	0.169752
$687$	3671.00	0.203868
$688$	−6112.00	−0.338689
$689$	−12996.0	−0.718590
$690$	−720.000	−0.0397245
$691$	−7720.00	−0.425011	−0.212506	−	0.977160i	$-0.568162\pi$
$691$	−7720.00	−0.425011	−0.212506	+	0.977160i	$0.568162\pi$
$692$	10956.0	0.601856
$693$	5850.00	0.320668
$694$	10272.0	0.561844
$695$	−4220.00	−0.230322
$696$	480.000	0.0261413
$697$	−6408.00	−0.348236
$698$	−13316.0	−0.722089
$699$	−5742.00	−0.310704
$700$	−2500.00	−0.134987
$701$	−31740.0	−1.71013	−0.855067	−	0.518518i	$-0.826484\pi$
$701$	−31740.0	−1.71013	−0.855067	+	0.518518i	$0.826484\pi$
$702$	8056.00	0.433126
$703$	−1480.00	−0.0794015
$704$	576.000	0.0308364
$705$	1335.00	0.0713177
$706$	−2604.00	−0.138814
$707$	−49275.0	−2.62118
$708$	1584.00	0.0840824
$709$	−33184.0	−1.75776	−0.878880	−	0.477044i	$-0.841708\pi$
$709$	−33184.0	−1.75776	−0.878880	+	0.477044i	$0.841708\pi$
$710$	−210.000	−0.0111002
$711$	10244.0	0.540338
$712$	−7344.00	−0.386556
$713$	−1872.00	−0.0983267
$714$	1200.00	0.0628976
$715$	−3420.00	−0.178882
$716$	−9792.00	−0.511095
$717$	3918.00	0.204073
$718$	−5898.00	−0.306562
$719$	−23703.0	−1.22945	−0.614724	−	0.788742i	$-0.710732\pi$
$719$	−23703.0	−1.22945	−0.614724	+	0.788742i	$0.710732\pi$
$720$	−2080.00	−0.107663
$721$	850.000	0.0439052
$722$	−10518.0	−0.542160
$723$	2540.00	0.130655
$724$	−1228.00	−0.0630363
$725$	1500.00	0.0768395
$726$	−2500.00	−0.127801
$727$	25730.0	1.31262	0.656309	−	0.754492i	$-0.272117\pi$
$727$	25730.0	1.31262	0.656309	+	0.754492i	$0.272117\pi$
$728$	15200.0	0.773832
$729$	−15443.0	−0.784586
$730$	−6910.00	−0.350343
$731$	9168.00	0.463872
$732$	−3592.00	−0.181372
$733$	26021.0	1.31120	0.655599	−	0.755110i	$-0.272416\pi$
$733$	26021.0	1.31120	0.655599	+	0.755110i	$0.272416\pi$
$734$	−15344.0	−0.771604
$735$	1410.00	0.0707600
$736$	−2304.00	−0.115389
$737$	−6084.00	−0.304080
$738$	−13884.0	−0.692516
$739$	−2563.00	−0.127580	−0.0637899	−	0.997963i	$-0.520319\pi$
$739$	−2563.00	−0.127580	−0.0637899	+	0.997963i	$0.520319\pi$
$740$	740.000	0.0367607
$741$	3040.00	0.150711
$742$	−8550.00	−0.423019
$743$	−3807.00	−0.187975	−0.0939874	−	0.995573i	$-0.529961\pi$
$743$	−3807.00	−0.187975	−0.0939874	+	0.995573i	$0.529961\pi$
$744$	208.000	0.0102495
$745$	−12765.0	−0.627750
$746$	−19418.0	−0.953007
$747$	−8034.00	−0.393506
$748$	−864.000	−0.0422339
$749$	−51000.0	−2.48798
$750$	250.000	0.0121716
$751$	−15109.0	−0.734135	−0.367067	−	0.930194i	$-0.619638\pi$
$751$	−15109.0	−0.734135	−0.367067	+	0.930194i	$0.619638\pi$
$752$	4272.00	0.207159
$753$	444.000	0.0214877
$754$	−9120.00	−0.440492
$755$	16360.0	0.788611
$756$	5300.00	0.254972
$757$	23804.0	1.14289	0.571447	−	0.820639i	$-0.306382\pi$
$757$	23804.0	1.14289	0.571447	+	0.820639i	$0.306382\pi$
$758$	−7982.00	−0.382479
$759$	−648.000	−0.0309893
$760$	−1600.00	−0.0763659
$761$	−35487.0	−1.69041	−0.845205	−	0.534441i	$-0.820522\pi$
$761$	−35487.0	−1.69041	−0.845205	+	0.534441i	$0.820522\pi$
$762$	2866.00	0.136252
$763$	5350.00	0.253844
$764$	−20016.0	−0.947845
$765$	3120.00	0.147456
$766$	−12084.0	−0.569990
$767$	−30096.0	−1.41682
$768$	256.000	0.0120281
$769$	11576.0	0.542836	0.271418	−	0.962462i	$-0.412507\pi$
$769$	11576.0	0.542836	0.271418	+	0.962462i	$0.412507\pi$
$770$	−2250.00	−0.105304
$771$	−5034.00	−0.235143
$772$	−1840.00	−0.0857812
$773$	36021.0	1.67605	0.838024	−	0.545633i	$-0.183711\pi$
$773$	36021.0	1.67605	0.838024	+	0.545633i	$0.183711\pi$
$774$	19864.0	0.922476
$775$	650.000	0.0301273
$776$	−6128.00	−0.283482
$777$	−925.000	−0.0427081
$778$	9456.00	0.435751
$779$	−10680.0	−0.491208
$780$	−1520.00	−0.0697753
$781$	−189.000	−0.00865935
$782$	3456.00	0.158039
$783$	−3180.00	−0.145139
$784$	4512.00	0.205539
$785$	−1655.00	−0.0752477
$786$	2556.00	0.115992
$787$	4847.00	0.219539	0.109769	−	0.993957i	$-0.464989\pi$
$787$	4847.00	0.219539	0.109769	+	0.993957i	$0.464989\pi$
$788$	276.000	0.0124773
$789$	−393.000	−0.0177328
$790$	−3940.00	−0.177442
$791$	1650.00	0.0741685
$792$	−1872.00	−0.0839882
$793$	68248.0	3.05619
$794$	−26102.0	−1.16666
$795$	855.000	0.0381431
$796$	−4024.00	−0.179180
$797$	−10176.0	−0.452261	−0.226131	−	0.974097i	$-0.572608\pi$
$797$	−10176.0	−0.452261	−0.226131	+	0.974097i	$0.572608\pi$
$798$	2000.00	0.0887208
$799$	−6408.00	−0.283728
$800$	800.000	0.0353553
$801$	23868.0	1.05285
$802$	2196.00	0.0966876
$803$	−6219.00	−0.273305
$804$	−2704.00	−0.118610
$805$	9000.00	0.394048
$806$	−3952.00	−0.172709
$807$	−1566.00	−0.0683096
$808$	15768.0	0.686530
$809$	−23568.0	−1.02424	−0.512118	−	0.858915i	$-0.671139\pi$
$809$	−23568.0	−1.02424	−0.512118	+	0.858915i	$0.671139\pi$
$810$	6490.00	0.281525
$811$	6629.00	0.287023	0.143512	−	0.989649i	$-0.454161\pi$
$811$	6629.00	0.287023	0.143512	+	0.989649i	$0.454161\pi$
$812$	−6000.00	−0.259309
$813$	3431.00	0.148008
$814$	666.000	0.0286773
$815$	−8660.00	−0.372204
$816$	−384.000	−0.0164739
$817$	15280.0	0.654320
$818$	−3992.00	−0.170632
$819$	−49400.0	−2.10766
$820$	5340.00	0.227416
$821$	37605.0	1.59857	0.799284	−	0.600954i	$-0.205213\pi$
$821$	37605.0	1.59857	0.799284	+	0.600954i	$0.205213\pi$
$822$	−3540.00	−0.150209
$823$	12632.0	0.535023	0.267511	−	0.963555i	$-0.413799\pi$
$823$	12632.0	0.535023	0.267511	+	0.963555i	$0.413799\pi$
$824$	−272.000	−0.0114995
$825$	225.000	0.00949514
$826$	−19800.0	−0.834056
$827$	−9744.00	−0.409712	−0.204856	−	0.978792i	$-0.565673\pi$
$827$	−9744.00	−0.409712	−0.204856	+	0.978792i	$0.565673\pi$
$828$	7488.00	0.314283
$829$	−22264.0	−0.932763	−0.466382	−	0.884584i	$-0.654443\pi$
$829$	−22264.0	−0.932763	−0.466382	+	0.884584i	$0.654443\pi$
$830$	3090.00	0.129223
$831$	−4450.00	−0.185763
$832$	−4864.00	−0.202679
$833$	−6768.00	−0.281509
$834$	−1688.00	−0.0700847
$835$	18330.0	0.759684
$836$	−1440.00	−0.0595735
$837$	−1378.00	−0.0569064
$838$	−14046.0	−0.579011
$839$	16416.0	0.675499	0.337749	−	0.941236i	$-0.390334\pi$
$839$	16416.0	0.675499	0.337749	+	0.941236i	$0.390334\pi$
$840$	−1000.00	−0.0410753
$841$	−20789.0	−0.852392
$842$	6808.00	0.278645
$843$	318.000	0.0129923
$844$	13412.0	0.546991
$845$	17895.0	0.728529
$846$	−13884.0	−0.564234
$847$	31250.0	1.26772
$848$	2736.00	0.110796
$849$	−7936.00	−0.320804
$850$	−1200.00	−0.0484231
$851$	−2664.00	−0.107310
$852$	−84.0000	−0.00337769
$853$	−7540.00	−0.302655	−0.151327	−	0.988484i	$-0.548355\pi$
$853$	−7540.00	−0.302655	−0.151327	+	0.988484i	$0.548355\pi$
$854$	44900.0	1.79912
$855$	5200.00	0.207996
$856$	16320.0	0.651643
$857$	−3282.00	−0.130818	−0.0654090	−	0.997859i	$-0.520835\pi$
$857$	−3282.00	−0.130818	−0.0654090	+	0.997859i	$0.520835\pi$
$858$	−1368.00	−0.0544321
$859$	12404.0	0.492688	0.246344	−	0.969182i	$-0.420771\pi$
$859$	12404.0	0.492688	0.246344	+	0.969182i	$0.420771\pi$
$860$	−7640.00	−0.302932
$861$	−6675.00	−0.264208
$862$	5424.00	0.214318
$863$	39528.0	1.55915	0.779576	−	0.626307i	$-0.215434\pi$
$863$	39528.0	1.55915	0.779576	+	0.626307i	$0.215434\pi$
$864$	−1696.00	−0.0667814
$865$	13695.0	0.538317
$866$	2086.00	0.0818535
$867$	−4337.00	−0.169887
$868$	−2600.00	−0.101670
$869$	−3546.00	−0.138423
$870$	600.000	0.0233815
$871$	51376.0	1.99863
$872$	−1712.00	−0.0664858
$873$	19916.0	0.772113
$874$	5760.00	0.222923
$875$	−3125.00	−0.120736
$876$	−2764.00	−0.106606
$877$	−6586.00	−0.253584	−0.126792	−	0.991929i	$-0.540468\pi$
$877$	−6586.00	−0.253584	−0.126792	+	0.991929i	$0.540468\pi$
$878$	17224.0	0.662052
$879$	−1698.00	−0.0651560
$880$	720.000	0.0275809
$881$	−41550.0	−1.58894	−0.794469	−	0.607304i	$-0.792251\pi$
$881$	−41550.0	−1.58894	−0.794469	+	0.607304i	$0.792251\pi$
$882$	−14664.0	−0.559822
$883$	−23944.0	−0.912548	−0.456274	−	0.889839i	$-0.650816\pi$
$883$	−23944.0	−0.912548	−0.456274	+	0.889839i	$0.650816\pi$
$884$	7296.00	0.277592
$885$	1980.00	0.0752056
$886$	−9870.00	−0.374254
$887$	6369.00	0.241094	0.120547	−	0.992708i	$-0.461535\pi$
$887$	6369.00	0.241094	0.120547	+	0.992708i	$0.461535\pi$
$888$	296.000	0.0111859
$889$	−35825.0	−1.35155
$890$	−9180.00	−0.345746
$891$	5841.00	0.219619
$892$	−22708.0	−0.852377
$893$	−10680.0	−0.400216
$894$	−5106.00	−0.191018
$895$	−12240.0	−0.457138
$896$	−3200.00	−0.119313
$897$	5472.00	0.203684
$898$	2580.00	0.0958750
$899$	1560.00	0.0578742
$900$	−2600.00	−0.0962963
$901$	−4104.00	−0.151747
$902$	4806.00	0.177408
$903$	9550.00	0.351943
$904$	−528.000	−0.0194259
$905$	−1535.00	−0.0563814
$906$	6544.00	0.239967
$907$	−22246.0	−0.814406	−0.407203	−	0.913338i	$-0.633496\pi$
$907$	−22246.0	−0.814406	−0.407203	+	0.913338i	$0.633496\pi$
$908$	8304.00	0.303500
$909$	−51246.0	−1.86988
$910$	19000.0	0.692136
$911$	−31770.0	−1.15542	−0.577710	−	0.816242i	$-0.696053\pi$
$911$	−31770.0	−1.15542	−0.577710	+	0.816242i	$0.696053\pi$
$912$	−640.000	−0.0232374
$913$	2781.00	0.100808
$914$	−10076.0	−0.364644
$915$	−4490.00	−0.162224
$916$	14684.0	0.529665
$917$	−31950.0	−1.15058
$918$	2544.00	0.0914646
$919$	21176.0	0.760100	0.380050	−	0.924966i	$-0.375907\pi$
$919$	21176.0	0.760100	0.380050	+	0.924966i	$0.375907\pi$
$920$	−2880.00	−0.103207
$921$	−1189.00	−0.0425395
$922$	2148.00	0.0767252
$923$	1596.00	0.0569155
$924$	−900.000	−0.0320431
$925$	925.000	0.0328798
$926$	29524.0	1.04775
$927$	884.000	0.0313208
$928$	1920.00	0.0679171
$929$	38766.0	1.36908	0.684538	−	0.728977i	$-0.260004\pi$
$929$	38766.0	1.36908	0.684538	+	0.728977i	$0.260004\pi$
$930$	260.000	0.00916746
$931$	−11280.0	−0.397086
$932$	−22968.0	−0.807234
$933$	−3984.00	−0.139797
$934$	19404.0	0.679784
$935$	−1080.00	−0.0377752
$936$	15808.0	0.552031
$937$	−37849.0	−1.31961	−0.659804	−	0.751438i	$-0.729361\pi$
$937$	−37849.0	−1.31961	−0.659804	+	0.751438i	$0.729361\pi$
$938$	33800.0	1.17656
$939$	−6010.00	−0.208870
$940$	5340.00	0.185289
$941$	−6114.00	−0.211807	−0.105904	−	0.994376i	$-0.533774\pi$
$941$	−6114.00	−0.211807	−0.105904	+	0.994376i	$0.533774\pi$
$942$	−662.000	−0.0228972
$943$	−19224.0	−0.663860
$944$	6336.00	0.218453
$945$	6625.00	0.228054
$946$	−6876.00	−0.236319
$947$	−6912.00	−0.237180	−0.118590	−	0.992943i	$-0.537837\pi$
$947$	−6912.00	−0.237180	−0.118590	+	0.992943i	$0.537837\pi$
$948$	−1576.00	−0.0539938
$949$	52516.0	1.79636
$950$	−2000.00	−0.0683038
$951$	−6234.00	−0.212567
$952$	4800.00	0.163413
$953$	50223.0	1.70712	0.853559	−	0.520997i	$-0.174440\pi$
$953$	50223.0	1.70712	0.853559	+	0.520997i	$0.174440\pi$
$954$	−8892.00	−0.301771
$955$	−25020.0	−0.847778
$956$	15672.0	0.530197
$957$	540.000	0.0182400
$958$	10272.0	0.346423
$959$	44250.0	1.49000
$960$	320.000	0.0107583
$961$	−29115.0	−0.977309
$962$	−5624.00	−0.188488
$963$	−53040.0	−1.77486
$964$	10160.0	0.339452
$965$	−2300.00	−0.0767250
$966$	3600.00	0.119905
$967$	−47266.0	−1.57184	−0.785921	−	0.618326i	$-0.787811\pi$
$967$	−47266.0	−1.57184	−0.785921	+	0.618326i	$0.787811\pi$
$968$	−10000.0	−0.332037
$969$	960.000	0.0318263
$970$	−7660.00	−0.253554
$971$	31860.0	1.05297	0.526486	−	0.850184i	$-0.323509\pi$
$971$	31860.0	1.05297	0.526486	+	0.850184i	$0.323509\pi$
$972$	8320.00	0.274552
$973$	21100.0	0.695205
$974$	3544.00	0.116588
$975$	−1900.00	−0.0624089
$976$	−14368.0	−0.471218
$977$	47004.0	1.53919	0.769596	−	0.638531i	$-0.220457\pi$
$977$	47004.0	1.53919	0.769596	+	0.638531i	$0.220457\pi$
$978$	−3464.00	−0.113258
$979$	−8262.00	−0.269719
$980$	5640.00	0.183840
$981$	5564.00	0.181085
$982$	24528.0	0.797067
$983$	2589.00	0.0840043	0.0420022	−	0.999118i	$-0.486626\pi$
$983$	2589.00	0.0840043	0.0420022	+	0.999118i	$0.486626\pi$
$984$	2136.00	0.0692004
$985$	345.000	0.0111600
$986$	−2880.00	−0.0930202
$987$	−6675.00	−0.215266
$988$	12160.0	0.391560
$989$	27504.0	0.884304
$990$	−2340.00	−0.0751213
$991$	4142.00	0.132770	0.0663849	−	0.997794i	$-0.478853\pi$
$991$	4142.00	0.132770	0.0663849	+	0.997794i	$0.478853\pi$
$992$	832.000	0.0266291
$993$	10316.0	0.329676
$994$	1050.00	0.0335050
$995$	−5030.00	−0.160263
$996$	1236.00	0.0393214
$997$	−13120.0	−0.416765	−0.208382	−	0.978047i	$-0.566820\pi$
$997$	−13120.0	−0.416765	−0.208382	+	0.978047i	$0.566820\pi$
$998$	−13172.0	−0.417788
$999$	−1961.00	−0.0621054

Twists

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

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

Downloads

Learn more