LMFDB - Embedded newform 30.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [30,4,Mod(1,30)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(30, 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("30.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 30 = 2 \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	30.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:	$1.77005730017$
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	$\chi$	$=$	30.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -4.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -48.0000 q^{11} +12.0000 q^{12} +2.00000 q^{13} -8.00000 q^{14} -15.0000 q^{15} +16.0000 q^{16} -114.000 q^{17} +18.0000 q^{18} +140.000 q^{19} -20.0000 q^{20} -12.0000 q^{21} -96.0000 q^{22} +72.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +4.00000 q^{26} +27.0000 q^{27} -16.0000 q^{28} +210.000 q^{29} -30.0000 q^{30} +272.000 q^{31} +32.0000 q^{32} -144.000 q^{33} -228.000 q^{34} +20.0000 q^{35} +36.0000 q^{36} -334.000 q^{37} +280.000 q^{38} +6.00000 q^{39} -40.0000 q^{40} -198.000 q^{41} -24.0000 q^{42} -268.000 q^{43} -192.000 q^{44} -45.0000 q^{45} +144.000 q^{46} +216.000 q^{47} +48.0000 q^{48} -327.000 q^{49} +50.0000 q^{50} -342.000 q^{51} +8.00000 q^{52} -78.0000 q^{53} +54.0000 q^{54} +240.000 q^{55} -32.0000 q^{56} +420.000 q^{57} +420.000 q^{58} +240.000 q^{59} -60.0000 q^{60} +302.000 q^{61} +544.000 q^{62} -36.0000 q^{63} +64.0000 q^{64} -10.0000 q^{65} -288.000 q^{66} +596.000 q^{67} -456.000 q^{68} +216.000 q^{69} +40.0000 q^{70} -768.000 q^{71} +72.0000 q^{72} -478.000 q^{73} -668.000 q^{74} +75.0000 q^{75} +560.000 q^{76} +192.000 q^{77} +12.0000 q^{78} -640.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -396.000 q^{82} -348.000 q^{83} -48.0000 q^{84} +570.000 q^{85} -536.000 q^{86} +630.000 q^{87} -384.000 q^{88} +210.000 q^{89} -90.0000 q^{90} -8.00000 q^{91} +288.000 q^{92} +816.000 q^{93} +432.000 q^{94} -700.000 q^{95} +96.0000 q^{96} -1534.00 q^{97} -654.000 q^{98} -432.000 q^{99} +O(q^{100})\)

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.00000	0.408248
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	12.0000	0.288675
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	−8.00000	−0.152721
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	18.0000	0.235702
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	−20.0000	−0.223607
$21$	−12.0000	−0.124696
$22$	−96.0000	−0.930330
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	4.00000	0.0301717
$27$	27.0000	0.192450
$28$	−16.0000	−0.107990
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	−30.0000	−0.182574
$31$	272.000	1.57589	0.787946	−	0.615745i	$-0.211145\pi$
$31$	272.000	1.57589	0.787946	+	0.615745i	$0.211145\pi$
$32$	32.0000	0.176777
$33$	−144.000	−0.759612
$34$	−228.000	−1.15005
$35$	20.0000	0.0965891
$36$	36.0000	0.166667
$37$	−334.000	−1.48403	−0.742017	−	0.670381i	$-0.766131\pi$
$37$	−334.000	−1.48403	−0.742017	+	0.670381i	$0.766131\pi$
$38$	280.000	1.19532
$39$	6.00000	0.0246351
$40$	−40.0000	−0.158114
$41$	−198.000	−0.754205	−0.377102	−	0.926172i	$-0.623080\pi$
$41$	−198.000	−0.754205	−0.377102	+	0.926172i	$0.623080\pi$
$42$	−24.0000	−0.0881733
$43$	−268.000	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−268.000	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	−192.000	−0.657843
$45$	−45.0000	−0.149071
$46$	144.000	0.461557
$47$	216.000	0.670358	0.335179	−	0.942154i	$-0.391203\pi$
$47$	216.000	0.670358	0.335179	+	0.942154i	$0.391203\pi$
$48$	48.0000	0.144338
$49$	−327.000	−0.953353
$50$	50.0000	0.141421
$51$	−342.000	−0.939011
$52$	8.00000	0.0213346
$53$	−78.0000	−0.202153	−0.101077	−	0.994879i	$-0.532229\pi$
$53$	−78.0000	−0.202153	−0.101077	+	0.994879i	$0.532229\pi$
$54$	54.0000	0.136083
$55$	240.000	0.588393
$56$	−32.0000	−0.0763604
$57$	420.000	0.975971
$58$	420.000	0.950840
$59$	240.000	0.529582	0.264791	−	0.964306i	$-0.414697\pi$
$59$	240.000	0.529582	0.264791	+	0.964306i	$0.414697\pi$
$60$	−60.0000	−0.129099
$61$	302.000	0.633888	0.316944	−	0.948444i	$-0.397343\pi$
$61$	302.000	0.633888	0.316944	+	0.948444i	$0.397343\pi$
$62$	544.000	1.11432
$63$	−36.0000	−0.0719932
$64$	64.0000	0.125000
$65$	−10.0000	−0.0190823
$66$	−288.000	−0.537127
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	−456.000	−0.813208
$69$	216.000	0.376860
$70$	40.0000	0.0682988
$71$	−768.000	−1.28373	−0.641865	−	0.766818i	$-0.721839\pi$
$71$	−768.000	−1.28373	−0.641865	+	0.766818i	$0.721839\pi$
$72$	72.0000	0.117851
$73$	−478.000	−0.766379	−0.383190	−	0.923670i	$-0.625174\pi$
$73$	−478.000	−0.766379	−0.383190	+	0.923670i	$0.625174\pi$
$74$	−668.000	−1.04937
$75$	75.0000	0.115470
$76$	560.000	0.845216
$77$	192.000	0.284161
$78$	12.0000	0.0174196
$79$	−640.000	−0.911464	−0.455732	−	0.890117i	$-0.650622\pi$
$79$	−640.000	−0.911464	−0.455732	+	0.890117i	$0.650622\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−396.000	−0.533303
$83$	−348.000	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$83$	−348.000	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$84$	−48.0000	−0.0623480
$85$	570.000	0.727355
$86$	−536.000	−0.672074
$87$	630.000	0.776357
$88$	−384.000	−0.465165
$89$	210.000	0.250112	0.125056	−	0.992150i	$-0.460089\pi$
$89$	210.000	0.250112	0.125056	+	0.992150i	$0.460089\pi$
$90$	−90.0000	−0.105409
$91$	−8.00000	−0.00921569
$92$	288.000	0.326370
$93$	816.000	0.909841
$94$	432.000	0.474015
$95$	−700.000	−0.755984
$96$	96.0000	0.102062
$97$	−1534.00	−1.60571	−0.802856	−	0.596173i	$-0.796687\pi$
$97$	−1534.00	−1.60571	−0.802856	+	0.596173i	$0.796687\pi$
$98$	−654.000	−0.674122
$99$	−432.000	−0.438562
$100$	100.000	0.100000
$101$	1722.00	1.69649	0.848245	−	0.529605i	$-0.177660\pi$
$101$	1722.00	1.69649	0.848245	+	0.529605i	$0.177660\pi$
$102$	−684.000	−0.663981
$103$	1052.00	1.00638	0.503188	−	0.864177i	$-0.332160\pi$
$103$	1052.00	1.00638	0.503188	+	0.864177i	$0.332160\pi$
$104$	16.0000	0.0150859
$105$	60.0000	0.0557657
$106$	−156.000	−0.142944
$107$	−564.000	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$108$	108.000	0.0962250
$109$	−610.000	−0.536031	−0.268016	−	0.963415i	$-0.586368\pi$
$109$	−610.000	−0.536031	−0.268016	+	0.963415i	$0.586368\pi$
$110$	480.000	0.416056
$111$	−1002.00	−0.856807
$112$	−64.0000	−0.0539949
$113$	1302.00	1.08391	0.541955	−	0.840407i	$-0.317684\pi$
$113$	1302.00	1.08391	0.541955	+	0.840407i	$0.317684\pi$
$114$	840.000	0.690116
$115$	−360.000	−0.291915
$116$	840.000	0.672345
$117$	18.0000	0.0142231
$118$	480.000	0.374471
$119$	456.000	0.351273
$120$	−120.000	−0.0912871
$121$	973.000	0.731029
$122$	604.000	0.448226
$123$	−594.000	−0.435440
$124$	1088.00	0.787946
$125$	−125.000	−0.0894427
$126$	−72.0000	−0.0509069
$127$	−124.000	−0.0866395	−0.0433198	−	0.999061i	$-0.513793\pi$
$127$	−124.000	−0.0866395	−0.0433198	+	0.999061i	$0.513793\pi$
$128$	128.000	0.0883883
$129$	−804.000	−0.548746
$130$	−20.0000	−0.0134932
$131$	192.000	0.128054	0.0640272	−	0.997948i	$-0.479606\pi$
$131$	192.000	0.128054	0.0640272	+	0.997948i	$0.479606\pi$
$132$	−576.000	−0.379806
$133$	−560.000	−0.365099
$134$	1192.00	0.768456
$135$	−135.000	−0.0860663
$136$	−912.000	−0.575025
$137$	−2514.00	−1.56778	−0.783889	−	0.620901i	$-0.786767\pi$
$137$	−2514.00	−1.56778	−0.783889	+	0.620901i	$0.786767\pi$
$138$	432.000	0.266480
$139$	1340.00	0.817679	0.408839	−	0.912606i	$-0.365934\pi$
$139$	1340.00	0.817679	0.408839	+	0.912606i	$0.365934\pi$
$140$	80.0000	0.0482945
$141$	648.000	0.387032
$142$	−1536.00	−0.907734
$143$	−96.0000	−0.0561393
$144$	144.000	0.0833333
$145$	−1050.00	−0.601364
$146$	−956.000	−0.541912
$147$	−981.000	−0.550418
$148$	−1336.00	−0.742017
$149$	1410.00	0.775246	0.387623	−	0.921818i	$-0.373296\pi$
$149$	1410.00	0.775246	0.387623	+	0.921818i	$0.373296\pi$
$150$	150.000	0.0816497
$151$	−2128.00	−1.14685	−0.573424	−	0.819258i	$-0.694385\pi$
$151$	−2128.00	−1.14685	−0.573424	+	0.819258i	$0.694385\pi$
$152$	1120.00	0.597658
$153$	−1026.00	−0.542138
$154$	384.000	0.200932
$155$	−1360.00	−0.704760
$156$	24.0000	0.0123176
$157$	3026.00	1.53822	0.769112	−	0.639114i	$-0.220699\pi$
$157$	3026.00	1.53822	0.769112	+	0.639114i	$0.220699\pi$
$158$	−1280.00	−0.644502
$159$	−234.000	−0.116713
$160$	−160.000	−0.0790569
$161$	−288.000	−0.140979
$162$	162.000	0.0785674
$163$	2612.00	1.25514	0.627569	−	0.778561i	$-0.284050\pi$
$163$	2612.00	1.25514	0.627569	+	0.778561i	$0.284050\pi$
$164$	−792.000	−0.377102
$165$	720.000	0.339709
$166$	−696.000	−0.325422
$167$	−24.0000	−0.0111208	−0.00556041	−	0.999985i	$-0.501770\pi$
$167$	−24.0000	−0.0111208	−0.00556041	+	0.999985i	$0.501770\pi$
$168$	−96.0000	−0.0440867
$169$	−2193.00	−0.998179
$170$	1140.00	0.514318
$171$	1260.00	0.563477
$172$	−1072.00	−0.475228
$173$	1962.00	0.862243	0.431122	−	0.902294i	$-0.358118\pi$
$173$	1962.00	0.862243	0.431122	+	0.902294i	$0.358118\pi$
$174$	1260.00	0.548968
$175$	−100.000	−0.0431959
$176$	−768.000	−0.328921
$177$	720.000	0.305754
$178$	420.000	0.176856
$179$	−120.000	−0.0501074	−0.0250537	−	0.999686i	$-0.507976\pi$
$179$	−120.000	−0.0501074	−0.0250537	+	0.999686i	$0.507976\pi$
$180$	−180.000	−0.0745356
$181$	902.000	0.370415	0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000	0.370415	0.185208	+	0.982699i	$0.440704\pi$
$182$	−16.0000	−0.00651648
$183$	906.000	0.365975
$184$	576.000	0.230779
$185$	1670.00	0.663680
$186$	1632.00	0.643355
$187$	5472.00	2.13985
$188$	864.000	0.335179
$189$	−108.000	−0.0415653
$190$	−1400.00	−0.534561
$191$	−168.000	−0.0636443	−0.0318221	−	0.999494i	$-0.510131\pi$
$191$	−168.000	−0.0636443	−0.0318221	+	0.999494i	$0.510131\pi$
$192$	192.000	0.0721688
$193$	−1318.00	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$193$	−1318.00	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$194$	−3068.00	−1.13541
$195$	−30.0000	−0.0110172
$196$	−1308.00	−0.476676
$197$	−4014.00	−1.45170	−0.725852	−	0.687851i	$-0.758554\pi$
$197$	−4014.00	−1.45170	−0.725852	+	0.687851i	$0.758554\pi$
$198$	−864.000	−0.310110
$199$	2000.00	0.712443	0.356222	−	0.934401i	$-0.384065\pi$
$199$	2000.00	0.712443	0.356222	+	0.934401i	$0.384065\pi$
$200$	200.000	0.0707107
$201$	1788.00	0.627442
$202$	3444.00	1.19960
$203$	−840.000	−0.290426
$204$	−1368.00	−0.469506
$205$	990.000	0.337291
$206$	2104.00	0.711615
$207$	648.000	0.217580
$208$	32.0000	0.0106673
$209$	−6720.00	−2.22408
$210$	120.000	0.0394323
$211$	−3868.00	−1.26201	−0.631005	−	0.775779i	$-0.717357\pi$
$211$	−3868.00	−1.26201	−0.631005	+	0.775779i	$0.717357\pi$
$212$	−312.000	−0.101077
$213$	−2304.00	−0.741162
$214$	−1128.00	−0.360320
$215$	1340.00	0.425057
$216$	216.000	0.0680414
$217$	−1088.00	−0.340361
$218$	−1220.00	−0.379031
$219$	−1434.00	−0.442469
$220$	960.000	0.294196
$221$	−228.000	−0.0693979
$222$	−2004.00	−0.605854
$223$	−3148.00	−0.945317	−0.472658	−	0.881246i	$-0.656706\pi$
$223$	−3148.00	−0.945317	−0.472658	+	0.881246i	$0.656706\pi$
$224$	−128.000	−0.0381802
$225$	225.000	0.0666667
$226$	2604.00	0.766440
$227$	2556.00	0.747347	0.373673	−	0.927560i	$-0.378098\pi$
$227$	2556.00	0.747347	0.373673	+	0.927560i	$0.378098\pi$
$228$	1680.00	0.487986
$229$	−610.000	−0.176026	−0.0880130	−	0.996119i	$-0.528052\pi$
$229$	−610.000	−0.176026	−0.0880130	+	0.996119i	$0.528052\pi$
$230$	−720.000	−0.206415
$231$	576.000	0.164061
$232$	1680.00	0.475420
$233$	−2058.00	−0.578644	−0.289322	−	0.957232i	$-0.593430\pi$
$233$	−2058.00	−0.578644	−0.289322	+	0.957232i	$0.593430\pi$
$234$	36.0000	0.0100572
$235$	−1080.00	−0.299793
$236$	960.000	0.264791
$237$	−1920.00	−0.526234
$238$	912.000	0.248387
$239$	4920.00	1.33158	0.665792	−	0.746138i	$-0.268094\pi$
$239$	4920.00	1.33158	0.665792	+	0.746138i	$0.268094\pi$
$240$	−240.000	−0.0645497
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	1946.00	0.516916
$243$	243.000	0.0641500
$244$	1208.00	0.316944
$245$	1635.00	0.426352
$246$	−1188.00	−0.307903
$247$	280.000	0.0721294
$248$	2176.00	0.557162
$249$	−1044.00	−0.265706
$250$	−250.000	−0.0632456
$251$	792.000	0.199166	0.0995829	−	0.995029i	$-0.468249\pi$
$251$	792.000	0.199166	0.0995829	+	0.995029i	$0.468249\pi$
$252$	−144.000	−0.0359966
$253$	−3456.00	−0.858802
$254$	−248.000	−0.0612634
$255$	1710.00	0.419939
$256$	256.000	0.0625000
$257$	2166.00	0.525725	0.262863	−	0.964833i	$-0.415333\pi$
$257$	2166.00	0.525725	0.262863	+	0.964833i	$0.415333\pi$
$258$	−1608.00	−0.388022
$259$	1336.00	0.320521
$260$	−40.0000	−0.00954113
$261$	1890.00	0.448230
$262$	384.000	0.0905481
$263$	3192.00	0.748392	0.374196	−	0.927350i	$-0.377919\pi$
$263$	3192.00	0.748392	0.374196	+	0.927350i	$0.377919\pi$
$264$	−1152.00	−0.268563
$265$	390.000	0.0904057
$266$	−1120.00	−0.258164
$267$	630.000	0.144402
$268$	2384.00	0.543381
$269$	5490.00	1.24435	0.622177	−	0.782877i	$-0.286248\pi$
$269$	5490.00	1.24435	0.622177	+	0.782877i	$0.286248\pi$
$270$	−270.000	−0.0608581
$271$	−6328.00	−1.41845	−0.709223	−	0.704985i	$-0.750954\pi$
$271$	−6328.00	−1.41845	−0.709223	+	0.704985i	$0.750954\pi$
$272$	−1824.00	−0.406604
$273$	−24.0000	−0.00532068
$274$	−5028.00	−1.10859
$275$	−1200.00	−0.263137
$276$	864.000	0.188430
$277$	−574.000	−0.124507	−0.0622533	−	0.998060i	$-0.519829\pi$
$277$	−574.000	−0.124507	−0.0622533	+	0.998060i	$0.519829\pi$
$278$	2680.00	0.578186
$279$	2448.00	0.525297
$280$	160.000	0.0341494
$281$	4242.00	0.900557	0.450278	−	0.892888i	$-0.351325\pi$
$281$	4242.00	0.900557	0.450278	+	0.892888i	$0.351325\pi$
$282$	1296.00	0.273673
$283$	−628.000	−0.131911	−0.0659553	−	0.997823i	$-0.521009\pi$
$283$	−628.000	−0.131911	−0.0659553	+	0.997823i	$0.521009\pi$
$284$	−3072.00	−0.641865
$285$	−2100.00	−0.436468
$286$	−192.000	−0.0396965
$287$	792.000	0.162893
$288$	288.000	0.0589256
$289$	8083.00	1.64523
$290$	−2100.00	−0.425228
$291$	−4602.00	−0.927058
$292$	−1912.00	−0.383190
$293$	−558.000	−0.111258	−0.0556292	−	0.998451i	$-0.517716\pi$
$293$	−558.000	−0.111258	−0.0556292	+	0.998451i	$0.517716\pi$
$294$	−1962.00	−0.389205
$295$	−1200.00	−0.236836
$296$	−2672.00	−0.524685
$297$	−1296.00	−0.253204
$298$	2820.00	0.548182
$299$	144.000	0.0278520
$300$	300.000	0.0577350
$301$	1072.00	0.205279
$302$	−4256.00	−0.810945
$303$	5166.00	0.979468
$304$	2240.00	0.422608
$305$	−1510.00	−0.283483
$306$	−2052.00	−0.383350
$307$	−6964.00	−1.29465	−0.647323	−	0.762216i	$-0.724112\pi$
$307$	−6964.00	−1.29465	−0.647323	+	0.762216i	$0.724112\pi$
$308$	768.000	0.142081
$309$	3156.00	0.581031
$310$	−2720.00	−0.498341
$311$	2832.00	0.516360	0.258180	−	0.966097i	$-0.416877\pi$
$311$	2832.00	0.516360	0.258180	+	0.966097i	$0.416877\pi$
$312$	48.0000	0.00870982
$313$	8642.00	1.56062	0.780311	−	0.625392i	$-0.215061\pi$
$313$	8642.00	1.56062	0.780311	+	0.625392i	$0.215061\pi$
$314$	6052.00	1.08769
$315$	180.000	0.0321964
$316$	−2560.00	−0.455732
$317$	−2214.00	−0.392273	−0.196137	−	0.980577i	$-0.562840\pi$
$317$	−2214.00	−0.392273	−0.196137	+	0.980577i	$0.562840\pi$
$318$	−468.000	−0.0825287
$319$	−10080.0	−1.76919
$320$	−320.000	−0.0559017
$321$	−1692.00	−0.294200
$322$	−576.000	−0.0996870
$323$	−15960.0	−2.74934
$324$	324.000	0.0555556
$325$	50.0000	0.00853385
$326$	5224.00	0.887517
$327$	−1830.00	−0.309478
$328$	−1584.00	−0.266652
$329$	−864.000	−0.144784
$330$	1440.00	0.240210
$331$	10772.0	1.78877	0.894385	−	0.447299i	$-0.147614\pi$
$331$	10772.0	1.78877	0.894385	+	0.447299i	$0.147614\pi$
$332$	−1392.00	−0.230108
$333$	−3006.00	−0.494678
$334$	−48.0000	−0.00786360
$335$	−2980.00	−0.486014
$336$	−192.000	−0.0311740
$337$	−1654.00	−0.267356	−0.133678	−	0.991025i	$-0.542679\pi$
$337$	−1654.00	−0.267356	−0.133678	+	0.991025i	$0.542679\pi$
$338$	−4386.00	−0.705819
$339$	3906.00	0.625796
$340$	2280.00	0.363678
$341$	−13056.0	−2.07338
$342$	2520.00	0.398439
$343$	2680.00	0.421885
$344$	−2144.00	−0.336037
$345$	−1080.00	−0.168537
$346$	3924.00	0.609698
$347$	2196.00	0.339733	0.169867	−	0.985467i	$-0.445666\pi$
$347$	2196.00	0.339733	0.169867	+	0.985467i	$0.445666\pi$
$348$	2520.00	0.388179
$349$	8270.00	1.26843	0.634216	−	0.773156i	$-0.281323\pi$
$349$	8270.00	1.26843	0.634216	+	0.773156i	$0.281323\pi$
$350$	−200.000	−0.0305441
$351$	54.0000	0.00821170
$352$	−1536.00	−0.232583
$353$	10302.0	1.55331	0.776657	−	0.629923i	$-0.216914\pi$
$353$	10302.0	1.55331	0.776657	+	0.629923i	$0.216914\pi$
$354$	1440.00	0.216201
$355$	3840.00	0.574102
$356$	840.000	0.125056
$357$	1368.00	0.202807
$358$	−240.000	−0.0354313
$359$	−2280.00	−0.335192	−0.167596	−	0.985856i	$-0.553600\pi$
$359$	−2280.00	−0.335192	−0.167596	+	0.985856i	$0.553600\pi$
$360$	−360.000	−0.0527046
$361$	12741.0	1.85756
$362$	1804.00	0.261923
$363$	2919.00	0.422060
$364$	−32.0000	−0.00460785
$365$	2390.00	0.342735
$366$	1812.00	0.258783
$367$	−8764.00	−1.24653	−0.623266	−	0.782010i	$-0.714195\pi$
$367$	−8764.00	−1.24653	−0.623266	+	0.782010i	$0.714195\pi$
$368$	1152.00	0.163185
$369$	−1782.00	−0.251402
$370$	3340.00	0.469293
$371$	312.000	0.0436610
$372$	3264.00	0.454921
$373$	−1318.00	−0.182958	−0.0914792	−	0.995807i	$-0.529159\pi$
$373$	−1318.00	−0.182958	−0.0914792	+	0.995807i	$0.529159\pi$
$374$	10944.0	1.51310
$375$	−375.000	−0.0516398
$376$	1728.00	0.237007
$377$	420.000	0.0573769
$378$	−216.000	−0.0293911
$379$	1100.00	0.149085	0.0745425	−	0.997218i	$-0.476250\pi$
$379$	1100.00	0.149085	0.0745425	+	0.997218i	$0.476250\pi$
$380$	−2800.00	−0.377992
$381$	−372.000	−0.0500214
$382$	−336.000	−0.0450033
$383$	−3528.00	−0.470685	−0.235343	−	0.971912i	$-0.575621\pi$
$383$	−3528.00	−0.470685	−0.235343	+	0.971912i	$0.575621\pi$
$384$	384.000	0.0510310
$385$	−960.000	−0.127081
$386$	−2636.00	−0.347588
$387$	−2412.00	−0.316819
$388$	−6136.00	−0.802856
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	−60.0000	−0.00779030
$391$	−8208.00	−1.06163
$392$	−2616.00	−0.337061
$393$	576.000	0.0739322
$394$	−8028.00	−1.02651
$395$	3200.00	0.407619
$396$	−1728.00	−0.219281
$397$	−3094.00	−0.391142	−0.195571	−	0.980690i	$-0.562656\pi$
$397$	−3094.00	−0.391142	−0.195571	+	0.980690i	$0.562656\pi$
$398$	4000.00	0.503774
$399$	−1680.00	−0.210790
$400$	400.000	0.0500000
$401$	−1638.00	−0.203985	−0.101992	−	0.994785i	$-0.532522\pi$
$401$	−1638.00	−0.203985	−0.101992	+	0.994785i	$0.532522\pi$
$402$	3576.00	0.443668
$403$	544.000	0.0672421
$404$	6888.00	0.848245
$405$	−405.000	−0.0496904
$406$	−1680.00	−0.205362
$407$	16032.0	1.95252
$408$	−2736.00	−0.331991
$409$	−13750.0	−1.66233	−0.831166	−	0.556024i	$-0.812326\pi$
$409$	−13750.0	−1.66233	−0.831166	+	0.556024i	$0.812326\pi$
$410$	1980.00	0.238501
$411$	−7542.00	−0.905157
$412$	4208.00	0.503188
$413$	−960.000	−0.114379
$414$	1296.00	0.153852
$415$	1740.00	0.205815
$416$	64.0000	0.00754293
$417$	4020.00	0.472087
$418$	−13440.0	−1.57266
$419$	−12480.0	−1.45510	−0.727551	−	0.686053i	$-0.759342\pi$
$419$	−12480.0	−1.45510	−0.727551	+	0.686053i	$0.759342\pi$
$420$	240.000	0.0278829
$421$	7262.00	0.840685	0.420342	−	0.907366i	$-0.361910\pi$
$421$	7262.00	0.840685	0.420342	+	0.907366i	$0.361910\pi$
$422$	−7736.00	−0.892376
$423$	1944.00	0.223453
$424$	−624.000	−0.0714720
$425$	−2850.00	−0.325283
$426$	−4608.00	−0.524081
$427$	−1208.00	−0.136907
$428$	−2256.00	−0.254785
$429$	−288.000	−0.0324121
$430$	2680.00	0.300561
$431$	9792.00	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$431$	9792.00	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$432$	432.000	0.0481125
$433$	1802.00	0.199997	0.0999984	−	0.994988i	$-0.468116\pi$
$433$	1802.00	0.199997	0.0999984	+	0.994988i	$0.468116\pi$
$434$	−2176.00	−0.240671
$435$	−3150.00	−0.347198
$436$	−2440.00	−0.268016
$437$	10080.0	1.10341
$438$	−2868.00	−0.312873
$439$	−2320.00	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$439$	−2320.00	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$440$	1920.00	0.208028
$441$	−2943.00	−0.317784
$442$	−456.000	−0.0490717
$443$	11172.0	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$443$	11172.0	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$444$	−4008.00	−0.428404
$445$	−1050.00	−0.111853
$446$	−6296.00	−0.668440
$447$	4230.00	0.447589
$448$	−256.000	−0.0269975
$449$	6810.00	0.715777	0.357888	−	0.933764i	$-0.383497\pi$
$449$	6810.00	0.715777	0.357888	+	0.933764i	$0.383497\pi$
$450$	450.000	0.0471405
$451$	9504.00	0.992297
$452$	5208.00	0.541955
$453$	−6384.00	−0.662134
$454$	5112.00	0.528454
$455$	40.0000	0.00412138
$456$	3360.00	0.345058
$457$	17066.0	1.74686	0.873429	−	0.486952i	$-0.161891\pi$
$457$	17066.0	1.74686	0.873429	+	0.486952i	$0.161891\pi$
$458$	−1220.00	−0.124469
$459$	−3078.00	−0.313004
$460$	−1440.00	−0.145957
$461$	−18918.0	−1.91128	−0.955639	−	0.294541i	$-0.904833\pi$
$461$	−18918.0	−1.91128	−0.955639	+	0.294541i	$0.904833\pi$
$462$	1152.00	0.116008
$463$	1052.00	0.105595	0.0527976	−	0.998605i	$-0.483186\pi$
$463$	1052.00	0.105595	0.0527976	+	0.998605i	$0.483186\pi$
$464$	3360.00	0.336173
$465$	−4080.00	−0.406893
$466$	−4116.00	−0.409163
$467$	11076.0	1.09751	0.548754	−	0.835984i	$-0.315102\pi$
$467$	11076.0	1.09751	0.548754	+	0.835984i	$0.315102\pi$
$468$	72.0000	0.00711154
$469$	−2384.00	−0.234718
$470$	−2160.00	−0.211986
$471$	9078.00	0.888094
$472$	1920.00	0.187236
$473$	12864.0	1.25050
$474$	−3840.00	−0.372103
$475$	3500.00	0.338086
$476$	1824.00	0.175636
$477$	−702.000	−0.0673844
$478$	9840.00	0.941571
$479$	−9000.00	−0.858498	−0.429249	−	0.903186i	$-0.641222\pi$
$479$	−9000.00	−0.858498	−0.429249	+	0.903186i	$0.641222\pi$
$480$	−480.000	−0.0456435
$481$	−668.000	−0.0633226
$482$	−2876.00	−0.271781
$483$	−864.000	−0.0813941
$484$	3892.00	0.365515
$485$	7670.00	0.718096
$486$	486.000	0.0453609
$487$	−8764.00	−0.815472	−0.407736	−	0.913100i	$-0.633682\pi$
$487$	−8764.00	−0.815472	−0.407736	+	0.913100i	$0.633682\pi$
$488$	2416.00	0.224113
$489$	7836.00	0.724655
$490$	3270.00	0.301477
$491$	5592.00	0.513978	0.256989	−	0.966414i	$-0.417270\pi$
$491$	5592.00	0.513978	0.256989	+	0.966414i	$0.417270\pi$
$492$	−2376.00	−0.217720
$493$	−23940.0	−2.18703
$494$	560.000	0.0510032
$495$	2160.00	0.196131
$496$	4352.00	0.393973
$497$	3072.00	0.277260
$498$	−2088.00	−0.187883
$499$	4700.00	0.421645	0.210823	−	0.977524i	$-0.432386\pi$
$499$	4700.00	0.421645	0.210823	+	0.977524i	$0.432386\pi$
$500$	−500.000	−0.0447214
$501$	−72.0000	−0.00642060
$502$	1584.00	0.140831
$503$	−11808.0	−1.04671	−0.523353	−	0.852116i	$-0.675319\pi$
$503$	−11808.0	−1.04671	−0.523353	+	0.852116i	$0.675319\pi$
$504$	−288.000	−0.0254535
$505$	−8610.00	−0.758693
$506$	−6912.00	−0.607265
$507$	−6579.00	−0.576299
$508$	−496.000	−0.0433198
$509$	1170.00	0.101885	0.0509424	−	0.998702i	$-0.483778\pi$
$509$	1170.00	0.101885	0.0509424	+	0.998702i	$0.483778\pi$
$510$	3420.00	0.296941
$511$	1912.00	0.165522
$512$	512.000	0.0441942
$513$	3780.00	0.325324
$514$	4332.00	0.371744
$515$	−5260.00	−0.450065
$516$	−3216.00	−0.274373
$517$	−10368.0	−0.881981
$518$	2672.00	0.226643
$519$	5886.00	0.497816
$520$	−80.0000	−0.00674660
$521$	−16638.0	−1.39909	−0.699543	−	0.714590i	$-0.746613\pi$
$521$	−16638.0	−1.39909	−0.699543	+	0.714590i	$0.746613\pi$
$522$	3780.00	0.316947
$523$	15692.0	1.31198	0.655988	−	0.754771i	$-0.272252\pi$
$523$	15692.0	1.31198	0.655988	+	0.754771i	$0.272252\pi$
$524$	768.000	0.0640272
$525$	−300.000	−0.0249392
$526$	6384.00	0.529193
$527$	−31008.0	−2.56305
$528$	−2304.00	−0.189903
$529$	−6983.00	−0.573929
$530$	780.000	0.0639265
$531$	2160.00	0.176527
$532$	−2240.00	−0.182549
$533$	−396.000	−0.0321814
$534$	1260.00	0.102108
$535$	2820.00	0.227886
$536$	4768.00	0.384228
$537$	−360.000	−0.0289295
$538$	10980.0	0.879891
$539$	15696.0	1.25431
$540$	−540.000	−0.0430331
$541$	−22018.0	−1.74977	−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0	−1.74977	−0.874887	+	0.484327i	$0.839064\pi$
$542$	−12656.0	−1.00299
$543$	2706.00	0.213859
$544$	−3648.00	−0.287512
$545$	3050.00	0.239720
$546$	−48.0000	−0.00376229
$547$	−4564.00	−0.356751	−0.178375	−	0.983963i	$-0.557084\pi$
$547$	−4564.00	−0.356751	−0.178375	+	0.983963i	$0.557084\pi$
$548$	−10056.0	−0.783889
$549$	2718.00	0.211296
$550$	−2400.00	−0.186066
$551$	29400.0	2.27311
$552$	1728.00	0.133240
$553$	2560.00	0.196858
$554$	−1148.00	−0.0880394
$555$	5010.00	0.383176
$556$	5360.00	0.408839
$557$	−7734.00	−0.588331	−0.294165	−	0.955755i	$-0.595042\pi$
$557$	−7734.00	−0.588331	−0.294165	+	0.955755i	$0.595042\pi$
$558$	4896.00	0.371441
$559$	−536.000	−0.0405552
$560$	320.000	0.0241473
$561$	16416.0	1.23544
$562$	8484.00	0.636790
$563$	−20148.0	−1.50824	−0.754118	−	0.656739i	$-0.771935\pi$
$563$	−20148.0	−1.50824	−0.754118	+	0.656739i	$0.771935\pi$
$564$	2592.00	0.193516
$565$	−6510.00	−0.484739
$566$	−1256.00	−0.0932749
$567$	−324.000	−0.0239977
$568$	−6144.00	−0.453867
$569$	−24030.0	−1.77046	−0.885228	−	0.465156i	$-0.845998\pi$
$569$	−24030.0	−1.77046	−0.885228	+	0.465156i	$0.845998\pi$
$570$	−4200.00	−0.308629
$571$	2372.00	0.173844	0.0869222	−	0.996215i	$-0.472297\pi$
$571$	2372.00	0.173844	0.0869222	+	0.996215i	$0.472297\pi$
$572$	−384.000	−0.0280697
$573$	−504.000	−0.0367450
$574$	1584.00	0.115183
$575$	1800.00	0.130548
$576$	576.000	0.0416667
$577$	8546.00	0.616594	0.308297	−	0.951290i	$-0.400241\pi$
$577$	8546.00	0.616594	0.308297	+	0.951290i	$0.400241\pi$
$578$	16166.0	1.16335
$579$	−3954.00	−0.283804
$580$	−4200.00	−0.300682
$581$	1392.00	0.0993974
$582$	−9204.00	−0.655529
$583$	3744.00	0.265970
$584$	−3824.00	−0.270956
$585$	−90.0000	−0.00636076
$586$	−1116.00	−0.0786716
$587$	−15444.0	−1.08593	−0.542966	−	0.839755i	$-0.682699\pi$
$587$	−15444.0	−1.08593	−0.542966	+	0.839755i	$0.682699\pi$
$588$	−3924.00	−0.275209
$589$	38080.0	2.66394
$590$	−2400.00	−0.167469
$591$	−12042.0	−0.838142
$592$	−5344.00	−0.371009
$593$	18342.0	1.27018	0.635089	−	0.772439i	$-0.280963\pi$
$593$	18342.0	1.27018	0.635089	+	0.772439i	$0.280963\pi$
$594$	−2592.00	−0.179042
$595$	−2280.00	−0.157094
$596$	5640.00	0.387623
$597$	6000.00	0.411329
$598$	288.000	0.0196943
$599$	24600.0	1.67801	0.839006	−	0.544123i	$-0.183137\pi$
$599$	24600.0	1.67801	0.839006	+	0.544123i	$0.183137\pi$
$600$	600.000	0.0408248
$601$	−8998.00	−0.610709	−0.305354	−	0.952239i	$-0.598775\pi$
$601$	−8998.00	−0.610709	−0.305354	+	0.952239i	$0.598775\pi$
$602$	2144.00	0.145154
$603$	5364.00	0.362254
$604$	−8512.00	−0.573424
$605$	−4865.00	−0.326926
$606$	10332.0	0.692589
$607$	4076.00	0.272553	0.136277	−	0.990671i	$-0.456486\pi$
$607$	4076.00	0.272553	0.136277	+	0.990671i	$0.456486\pi$
$608$	4480.00	0.298829
$609$	−2520.00	−0.167677
$610$	−3020.00	−0.200453
$611$	432.000	0.0286037
$612$	−4104.00	−0.271069
$613$	−4078.00	−0.268693	−0.134347	−	0.990934i	$-0.542894\pi$
$613$	−4078.00	−0.268693	−0.134347	+	0.990934i	$0.542894\pi$
$614$	−13928.0	−0.915453
$615$	2970.00	0.194735
$616$	1536.00	0.100466
$617$	10086.0	0.658099	0.329049	−	0.944313i	$-0.393272\pi$
$617$	10086.0	0.658099	0.329049	+	0.944313i	$0.393272\pi$
$618$	6312.00	0.410851
$619$	8780.00	0.570110	0.285055	−	0.958511i	$-0.407988\pi$
$619$	8780.00	0.570110	0.285055	+	0.958511i	$0.407988\pi$
$620$	−5440.00	−0.352380
$621$	1944.00	0.125620
$622$	5664.00	0.365122
$623$	−840.000	−0.0540191
$624$	96.0000	0.00615878
$625$	625.000	0.0400000
$626$	17284.0	1.10353
$627$	−20160.0	−1.28407
$628$	12104.0	0.769112
$629$	38076.0	2.41366
$630$	360.000	0.0227663
$631$	2792.00	0.176145	0.0880727	−	0.996114i	$-0.471929\pi$
$631$	2792.00	0.176145	0.0880727	+	0.996114i	$0.471929\pi$
$632$	−5120.00	−0.322251
$633$	−11604.0	−0.728622
$634$	−4428.00	−0.277379
$635$	620.000	0.0387464
$636$	−936.000	−0.0583566
$637$	−654.000	−0.0406788
$638$	−20160.0	−1.25101
$639$	−6912.00	−0.427910
$640$	−640.000	−0.0395285
$641$	7602.00	0.468426	0.234213	−	0.972185i	$-0.424749\pi$
$641$	7602.00	0.468426	0.234213	+	0.972185i	$0.424749\pi$
$642$	−3384.00	−0.208031
$643$	24212.0	1.48496	0.742479	−	0.669869i	$-0.233650\pi$
$643$	24212.0	1.48496	0.742479	+	0.669869i	$0.233650\pi$
$644$	−1152.00	−0.0704894
$645$	4020.00	0.245407
$646$	−31920.0	−1.94408
$647$	9456.00	0.574581	0.287290	−	0.957844i	$-0.407246\pi$
$647$	9456.00	0.574581	0.287290	+	0.957844i	$0.407246\pi$
$648$	648.000	0.0392837
$649$	−11520.0	−0.696764
$650$	100.000	0.00603434
$651$	−3264.00	−0.196507
$652$	10448.0	0.627569
$653$	−9558.00	−0.572792	−0.286396	−	0.958111i	$-0.592457\pi$
$653$	−9558.00	−0.572792	−0.286396	+	0.958111i	$0.592457\pi$
$654$	−3660.00	−0.218834
$655$	−960.000	−0.0572676
$656$	−3168.00	−0.188551
$657$	−4302.00	−0.255460
$658$	−1728.00	−0.102378
$659$	−29280.0	−1.73078	−0.865392	−	0.501095i	$-0.832931\pi$
$659$	−29280.0	−1.73078	−0.865392	+	0.501095i	$0.832931\pi$
$660$	2880.00	0.169854
$661$	−29098.0	−1.71223	−0.856113	−	0.516789i	$-0.827127\pi$
$661$	−29098.0	−1.71223	−0.856113	+	0.516789i	$0.827127\pi$
$662$	21544.0	1.26485
$663$	−684.000	−0.0400669
$664$	−2784.00	−0.162711
$665$	2800.00	0.163277
$666$	−6012.00	−0.349790
$667$	15120.0	0.877734
$668$	−96.0000	−0.00556041
$669$	−9444.00	−0.545779
$670$	−5960.00	−0.343664
$671$	−14496.0	−0.833997
$672$	−384.000	−0.0220433
$673$	−11638.0	−0.666585	−0.333293	−	0.942823i	$-0.608160\pi$
$673$	−11638.0	−0.666585	−0.333293	+	0.942823i	$0.608160\pi$
$674$	−3308.00	−0.189050
$675$	675.000	0.0384900
$676$	−8772.00	−0.499090
$677$	3426.00	0.194493	0.0972466	−	0.995260i	$-0.468996\pi$
$677$	3426.00	0.194493	0.0972466	+	0.995260i	$0.468996\pi$
$678$	7812.00	0.442505
$679$	6136.00	0.346801
$680$	4560.00	0.257159
$681$	7668.00	0.431481
$682$	−26112.0	−1.46610
$683$	−20148.0	−1.12876	−0.564379	−	0.825516i	$-0.690884\pi$
$683$	−20148.0	−1.12876	−0.564379	+	0.825516i	$0.690884\pi$
$684$	5040.00	0.281739
$685$	12570.0	0.701131
$686$	5360.00	0.298317
$687$	−1830.00	−0.101629
$688$	−4288.00	−0.237614
$689$	−156.000	−0.00862573
$690$	−2160.00	−0.119174
$691$	−29428.0	−1.62011	−0.810053	−	0.586356i	$-0.800562\pi$
$691$	−29428.0	−1.62011	−0.810053	+	0.586356i	$0.800562\pi$
$692$	7848.00	0.431122
$693$	1728.00	0.0947205
$694$	4392.00	0.240228
$695$	−6700.00	−0.365677
$696$	5040.00	0.274484
$697$	22572.0	1.22665
$698$	16540.0	0.896917
$699$	−6174.00	−0.334080
$700$	−400.000	−0.0215980
$701$	16242.0	0.875110	0.437555	−	0.899192i	$-0.355845\pi$
$701$	16242.0	0.875110	0.437555	+	0.899192i	$0.355845\pi$
$702$	108.000	0.00580655
$703$	−46760.0	−2.50866
$704$	−3072.00	−0.164461
$705$	−3240.00	−0.173086
$706$	20604.0	1.09836
$707$	−6888.00	−0.366407
$708$	2880.00	0.152877
$709$	2030.00	0.107529	0.0537646	−	0.998554i	$-0.482878\pi$
$709$	2030.00	0.107529	0.0537646	+	0.998554i	$0.482878\pi$
$710$	7680.00	0.405951
$711$	−5760.00	−0.303821
$712$	1680.00	0.0884279
$713$	19584.0	1.02865
$714$	2736.00	0.143406
$715$	480.000	0.0251063
$716$	−480.000	−0.0250537
$717$	14760.0	0.768790
$718$	−4560.00	−0.237016
$719$	6960.00	0.361007	0.180504	−	0.983574i	$-0.442227\pi$
$719$	6960.00	0.361007	0.180504	+	0.983574i	$0.442227\pi$
$720$	−720.000	−0.0372678
$721$	−4208.00	−0.217357
$722$	25482.0	1.31349
$723$	−4314.00	−0.221908
$724$	3608.00	0.185208
$725$	5250.00	0.268938
$726$	5838.00	0.298441
$727$	18596.0	0.948676	0.474338	−	0.880343i	$-0.342687\pi$
$727$	18596.0	0.948676	0.474338	+	0.880343i	$0.342687\pi$
$728$	−64.0000	−0.00325824
$729$	729.000	0.0370370
$730$	4780.00	0.242350
$731$	30552.0	1.54584
$732$	3624.00	0.182988
$733$	21242.0	1.07038	0.535192	−	0.844731i	$-0.320239\pi$
$733$	21242.0	1.07038	0.535192	+	0.844731i	$0.320239\pi$
$734$	−17528.0	−0.881431
$735$	4905.00	0.246155
$736$	2304.00	0.115389
$737$	−28608.0	−1.42984
$738$	−3564.00	−0.177768
$739$	−340.000	−0.0169244	−0.00846218	−	0.999964i	$-0.502694\pi$
$739$	−340.000	−0.0169244	−0.00846218	+	0.999964i	$0.502694\pi$
$740$	6680.00	0.331840
$741$	840.000	0.0416440
$742$	624.000	0.0308730
$743$	−21888.0	−1.08074	−0.540372	−	0.841426i	$-0.681716\pi$
$743$	−21888.0	−1.08074	−0.540372	+	0.841426i	$0.681716\pi$
$744$	6528.00	0.321678
$745$	−7050.00	−0.346701
$746$	−2636.00	−0.129371
$747$	−3132.00	−0.153405
$748$	21888.0	1.06993
$749$	2256.00	0.110057
$750$	−750.000	−0.0365148
$751$	17792.0	0.864500	0.432250	−	0.901754i	$-0.357720\pi$
$751$	17792.0	0.864500	0.432250	+	0.901754i	$0.357720\pi$
$752$	3456.00	0.167590
$753$	2376.00	0.114988
$754$	840.000	0.0405716
$755$	10640.0	0.512886
$756$	−432.000	−0.0207827
$757$	37346.0	1.79308	0.896541	−	0.442960i	$-0.146072\pi$
$757$	37346.0	1.79308	0.896541	+	0.442960i	$0.146072\pi$
$758$	2200.00	0.105419
$759$	−10368.0	−0.495829
$760$	−5600.00	−0.267281
$761$	−11358.0	−0.541034	−0.270517	−	0.962715i	$-0.587195\pi$
$761$	−11358.0	−0.541034	−0.270517	+	0.962715i	$0.587195\pi$
$762$	−744.000	−0.0353704
$763$	2440.00	0.115772
$764$	−672.000	−0.0318221
$765$	5130.00	0.242452
$766$	−7056.00	−0.332825
$767$	480.000	0.0225969
$768$	768.000	0.0360844
$769$	−34270.0	−1.60703	−0.803516	−	0.595283i	$-0.797040\pi$
$769$	−34270.0	−1.60703	−0.803516	+	0.595283i	$0.797040\pi$
$770$	−1920.00	−0.0898597
$771$	6498.00	0.303528
$772$	−5272.00	−0.245782
$773$	−13278.0	−0.617822	−0.308911	−	0.951091i	$-0.599965\pi$
$773$	−13278.0	−0.617822	−0.308911	+	0.951091i	$0.599965\pi$
$774$	−4824.00	−0.224025
$775$	6800.00	0.315178
$776$	−12272.0	−0.567705
$777$	4008.00	0.185053
$778$	−19260.0	−0.887538
$779$	−27720.0	−1.27493
$780$	−120.000	−0.00550858
$781$	36864.0	1.68899
$782$	−16416.0	−0.750684
$783$	5670.00	0.258786
$784$	−5232.00	−0.238338
$785$	−15130.0	−0.687914
$786$	1152.00	0.0522780
$787$	−11164.0	−0.505659	−0.252829	−	0.967511i	$-0.581361\pi$
$787$	−11164.0	−0.505659	−0.252829	+	0.967511i	$0.581361\pi$
$788$	−16056.0	−0.725852
$789$	9576.00	0.432084
$790$	6400.00	0.288230
$791$	−5208.00	−0.234103
$792$	−3456.00	−0.155055
$793$	604.000	0.0270475
$794$	−6188.00	−0.276579
$795$	1170.00	0.0521958
$796$	8000.00	0.356222
$797$	−5094.00	−0.226397	−0.113199	−	0.993572i	$-0.536110\pi$
$797$	−5094.00	−0.226397	−0.113199	+	0.993572i	$0.536110\pi$
$798$	−3360.00	−0.149051
$799$	−24624.0	−1.09028
$800$	800.000	0.0353553
$801$	1890.00	0.0833706
$802$	−3276.00	−0.144239
$803$	22944.0	1.00831
$804$	7152.00	0.313721
$805$	1440.00	0.0630476
$806$	1088.00	0.0475474
$807$	16470.0	0.718428
$808$	13776.0	0.599799
$809$	−8790.00	−0.382002	−0.191001	−	0.981590i	$-0.561173\pi$
$809$	−8790.00	−0.382002	−0.191001	+	0.981590i	$0.561173\pi$
$810$	−810.000	−0.0351364
$811$	5852.00	0.253380	0.126690	−	0.991942i	$-0.459565\pi$
$811$	5852.00	0.253380	0.126690	+	0.991942i	$0.459565\pi$
$812$	−3360.00	−0.145213
$813$	−18984.0	−0.818940
$814$	32064.0	1.38064
$815$	−13060.0	−0.561315
$816$	−5472.00	−0.234753
$817$	−37520.0	−1.60668
$818$	−27500.0	−1.17545
$819$	−72.0000	−0.00307190
$820$	3960.00	0.168645
$821$	−29478.0	−1.25309	−0.626546	−	0.779384i	$-0.715532\pi$
$821$	−29478.0	−1.25309	−0.626546	+	0.779384i	$0.715532\pi$
$822$	−15084.0	−0.640042
$823$	39332.0	1.66589	0.832945	−	0.553356i	$-0.186653\pi$
$823$	39332.0	1.66589	0.832945	+	0.553356i	$0.186653\pi$
$824$	8416.00	0.355807
$825$	−3600.00	−0.151922
$826$	−1920.00	−0.0808781
$827$	6756.00	0.284074	0.142037	−	0.989861i	$-0.454635\pi$
$827$	6756.00	0.284074	0.142037	+	0.989861i	$0.454635\pi$
$828$	2592.00	0.108790
$829$	3950.00	0.165488	0.0827438	−	0.996571i	$-0.473632\pi$
$829$	3950.00	0.165488	0.0827438	+	0.996571i	$0.473632\pi$
$830$	3480.00	0.145533
$831$	−1722.00	−0.0718839
$832$	128.000	0.00533366
$833$	37278.0	1.55055
$834$	8040.00	0.333816
$835$	120.000	0.00497338
$836$	−26880.0	−1.11204
$837$	7344.00	0.303280
$838$	−24960.0	−1.02891
$839$	12360.0	0.508599	0.254300	−	0.967126i	$-0.418155\pi$
$839$	12360.0	0.508599	0.254300	+	0.967126i	$0.418155\pi$
$840$	480.000	0.0197162
$841$	19711.0	0.808192
$842$	14524.0	0.594454
$843$	12726.0	0.519937
$844$	−15472.0	−0.631005
$845$	10965.0	0.446399
$846$	3888.00	0.158005
$847$	−3892.00	−0.157887
$848$	−1248.00	−0.0505383
$849$	−1884.00	−0.0761587
$850$	−5700.00	−0.230010
$851$	−24048.0	−0.968690
$852$	−9216.00	−0.370581
$853$	−35998.0	−1.44496	−0.722478	−	0.691394i	$-0.756997\pi$
$853$	−35998.0	−1.44496	−0.722478	+	0.691394i	$0.756997\pi$
$854$	−2416.00	−0.0968077
$855$	−6300.00	−0.251995
$856$	−4512.00	−0.180160
$857$	−21594.0	−0.860720	−0.430360	−	0.902657i	$-0.641613\pi$
$857$	−21594.0	−0.860720	−0.430360	+	0.902657i	$0.641613\pi$
$858$	−576.000	−0.0229188
$859$	9260.00	0.367808	0.183904	−	0.982944i	$-0.441126\pi$
$859$	9260.00	0.367808	0.183904	+	0.982944i	$0.441126\pi$
$860$	5360.00	0.212528
$861$	2376.00	0.0940463
$862$	19584.0	0.773821
$863$	31632.0	1.24770	0.623850	−	0.781544i	$-0.285567\pi$
$863$	31632.0	1.24770	0.623850	+	0.781544i	$0.285567\pi$
$864$	864.000	0.0340207
$865$	−9810.00	−0.385607
$866$	3604.00	0.141419
$867$	24249.0	0.949872
$868$	−4352.00	−0.170180
$869$	30720.0	1.19920
$870$	−6300.00	−0.245506
$871$	1192.00	0.0463713
$872$	−4880.00	−0.189516
$873$	−13806.0	−0.535237
$874$	20160.0	0.780231
$875$	500.000	0.0193178
$876$	−5736.00	−0.221235
$877$	−39694.0	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$877$	−39694.0	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$878$	−4640.00	−0.178351
$879$	−1674.00	−0.0642351
$880$	3840.00	0.147098
$881$	1242.00	0.0474961	0.0237480	−	0.999718i	$-0.492440\pi$
$881$	1242.00	0.0474961	0.0237480	+	0.999718i	$0.492440\pi$
$882$	−5886.00	−0.224707
$883$	−2668.00	−0.101682	−0.0508411	−	0.998707i	$-0.516190\pi$
$883$	−2668.00	−0.101682	−0.0508411	+	0.998707i	$0.516190\pi$
$884$	−912.000	−0.0346990
$885$	−3600.00	−0.136737
$886$	22344.0	0.847248
$887$	−4344.00	−0.164439	−0.0822194	−	0.996614i	$-0.526201\pi$
$887$	−4344.00	−0.164439	−0.0822194	+	0.996614i	$0.526201\pi$
$888$	−8016.00	−0.302927
$889$	496.000	0.0187124
$890$	−2100.00	−0.0790923
$891$	−3888.00	−0.146187
$892$	−12592.0	−0.472658
$893$	30240.0	1.13319
$894$	8460.00	0.316493
$895$	600.000	0.0224087
$896$	−512.000	−0.0190901
$897$	432.000	0.0160803
$898$	13620.0	0.506131
$899$	57120.0	2.11909
$900$	900.000	0.0333333
$901$	8892.00	0.328785
$902$	19008.0	0.701660
$903$	3216.00	0.118518
$904$	10416.0	0.383220
$905$	−4510.00	−0.165655
$906$	−12768.0	−0.468199
$907$	4436.00	0.162398	0.0811990	−	0.996698i	$-0.474125\pi$
$907$	4436.00	0.162398	0.0811990	+	0.996698i	$0.474125\pi$
$908$	10224.0	0.373673
$909$	15498.0	0.565496
$910$	80.0000	0.00291426
$911$	22752.0	0.827450	0.413725	−	0.910402i	$-0.364227\pi$
$911$	22752.0	0.827450	0.413725	+	0.910402i	$0.364227\pi$
$912$	6720.00	0.243993
$913$	16704.0	0.605500
$914$	34132.0	1.23521
$915$	−4530.00	−0.163669
$916$	−2440.00	−0.0880130
$917$	−768.000	−0.0276571
$918$	−6156.00	−0.221327
$919$	−27160.0	−0.974892	−0.487446	−	0.873153i	$-0.662071\pi$
$919$	−27160.0	−0.974892	−0.487446	+	0.873153i	$0.662071\pi$
$920$	−2880.00	−0.103207
$921$	−20892.0	−0.747465
$922$	−37836.0	−1.35148
$923$	−1536.00	−0.0547758
$924$	2304.00	0.0820303
$925$	−8350.00	−0.296807
$926$	2104.00	0.0746671
$927$	9468.00	0.335458
$928$	6720.00	0.237710
$929$	−33030.0	−1.16650	−0.583250	−	0.812292i	$-0.698219\pi$
$929$	−33030.0	−1.16650	−0.583250	+	0.812292i	$0.698219\pi$
$930$	−8160.00	−0.287717
$931$	−45780.0	−1.61158
$932$	−8232.00	−0.289322
$933$	8496.00	0.298121
$934$	22152.0	0.776055
$935$	−27360.0	−0.956971
$936$	144.000	0.00502862
$937$	−29974.0	−1.04505	−0.522523	−	0.852625i	$-0.675009\pi$
$937$	−29974.0	−1.04505	−0.522523	+	0.852625i	$0.675009\pi$
$938$	−4768.00	−0.165971
$939$	25926.0	0.901026
$940$	−4320.00	−0.149897
$941$	13962.0	0.483686	0.241843	−	0.970315i	$-0.422248\pi$
$941$	13962.0	0.483686	0.241843	+	0.970315i	$0.422248\pi$
$942$	18156.0	0.627977
$943$	−14256.0	−0.492300
$944$	3840.00	0.132396
$945$	540.000	0.0185886
$946$	25728.0	0.884238
$947$	35196.0	1.20773	0.603863	−	0.797088i	$-0.293627\pi$
$947$	35196.0	1.20773	0.603863	+	0.797088i	$0.293627\pi$
$948$	−7680.00	−0.263117
$949$	−956.000	−0.0327008
$950$	7000.00	0.239063
$951$	−6642.00	−0.226479
$952$	3648.00	0.124194
$953$	−28338.0	−0.963230	−0.481615	−	0.876383i	$-0.659950\pi$
$953$	−28338.0	−0.963230	−0.481615	+	0.876383i	$0.659950\pi$
$954$	−1404.00	−0.0476480
$955$	840.000	0.0284626
$956$	19680.0	0.665792
$957$	−30240.0	−1.02144
$958$	−18000.0	−0.607050
$959$	10056.0	0.338608
$960$	−960.000	−0.0322749
$961$	44193.0	1.48343
$962$	−1336.00	−0.0447759
$963$	−5076.00	−0.169857
$964$	−5752.00	−0.192178
$965$	6590.00	0.219834
$966$	−1728.00	−0.0575543
$967$	−17524.0	−0.582765	−0.291383	−	0.956607i	$-0.594115\pi$
$967$	−17524.0	−0.582765	−0.291383	+	0.956607i	$0.594115\pi$
$968$	7784.00	0.258458
$969$	−47880.0	−1.58733
$970$	15340.0	0.507771
$971$	−26808.0	−0.886004	−0.443002	−	0.896521i	$-0.646087\pi$
$971$	−26808.0	−0.886004	−0.443002	+	0.896521i	$0.646087\pi$
$972$	972.000	0.0320750
$973$	−5360.00	−0.176602
$974$	−17528.0	−0.576626
$975$	150.000	0.00492702
$976$	4832.00	0.158472
$977$	−10914.0	−0.357390	−0.178695	−	0.983905i	$-0.557187\pi$
$977$	−10914.0	−0.357390	−0.178695	+	0.983905i	$0.557187\pi$
$978$	15672.0	0.512408
$979$	−10080.0	−0.329069
$980$	6540.00	0.213176
$981$	−5490.00	−0.178677
$982$	11184.0	0.363438
$983$	22272.0	0.722652	0.361326	−	0.932440i	$-0.382324\pi$
$983$	22272.0	0.722652	0.361326	+	0.932440i	$0.382324\pi$
$984$	−4752.00	−0.153951
$985$	20070.0	0.649222
$986$	−47880.0	−1.54646
$987$	−2592.00	−0.0835910
$988$	1120.00	0.0360647
$989$	−19296.0	−0.620402
$990$	4320.00	0.138685
$991$	14072.0	0.451071	0.225536	−	0.974235i	$-0.427587\pi$
$991$	14072.0	0.451071	0.225536	+	0.974235i	$0.427587\pi$
$992$	8704.00	0.278581
$993$	32316.0	1.03275
$994$	6144.00	0.196052
$995$	−10000.0	−0.318614
$996$	−4176.00	−0.132853
$997$	4826.00	0.153301	0.0766504	−	0.997058i	$-0.475577\pi$
$997$	4826.00	0.153301	0.0766504	+	0.997058i	$0.475577\pi$
$998$	9400.00	0.298148
$999$	−9018.00	−0.285602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	30.4.a.b.1.1	✓	1
3.2	odd	2		90.4.a.c.1.1		1
4.3	odd	2		240.4.a.b.1.1		1
5.2	odd	4		150.4.c.c.49.2		2
5.3	odd	4		150.4.c.c.49.1		2
5.4	even	2		150.4.a.b.1.1		1
7.6	odd	2		1470.4.a.r.1.1		1
8.3	odd	2		960.4.a.bg.1.1		1
8.5	even	2		960.4.a.n.1.1		1
9.2	odd	6		810.4.e.p.271.1		2
9.4	even	3		810.4.e.i.541.1		2
9.5	odd	6		810.4.e.p.541.1		2
9.7	even	3		810.4.e.i.271.1		2
12.11	even	2		720.4.a.y.1.1		1
15.2	even	4		450.4.c.j.199.1		2
15.8	even	4		450.4.c.j.199.2		2
15.14	odd	2		450.4.a.r.1.1		1
20.3	even	4		1200.4.f.r.49.1		2
20.7	even	4		1200.4.f.r.49.2		2
20.19	odd	2		1200.4.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	1.1	even	1	trivial
90.4.a.c.1.1		1	3.2	odd	2
150.4.a.b.1.1		1	5.4	even	2
150.4.c.c.49.1		2	5.3	odd	4
150.4.c.c.49.2		2	5.2	odd	4
240.4.a.b.1.1		1	4.3	odd	2
450.4.a.r.1.1		1	15.14	odd	2
450.4.c.j.199.1		2	15.2	even	4
450.4.c.j.199.2		2	15.8	even	4
720.4.a.y.1.1		1	12.11	even	2
810.4.e.i.271.1		2	9.7	even	3
810.4.e.i.541.1		2	9.4	even	3
810.4.e.p.271.1		2	9.2	odd	6
810.4.e.p.541.1		2	9.5	odd	6
960.4.a.n.1.1		1	8.5	even	2
960.4.a.bg.1.1		1	8.3	odd	2
1200.4.a.ba.1.1		1	20.19	odd	2
1200.4.f.r.49.1		2	20.3	even	4
1200.4.f.r.49.2		2	20.7	even	4
1470.4.a.r.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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