LMFDB - Embedded newform 1110.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
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	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -5.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} +3.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -7.00000 q^{41} +3.00000 q^{42} +3.00000 q^{43} -5.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} -3.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} -5.00000 q^{55} -3.00000 q^{56} +6.00000 q^{57} -1.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +5.00000 q^{61} -3.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} +4.00000 q^{69} -3.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -1.00000 q^{74} -1.00000 q^{75} -6.00000 q^{76} +15.0000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -7.00000 q^{82} +6.00000 q^{83} +3.00000 q^{84} +3.00000 q^{85} +3.00000 q^{86} +1.00000 q^{87} -5.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} +6.00000 q^{91} -4.00000 q^{92} +3.00000 q^{93} -6.00000 q^{95} -1.00000 q^{96} -13.0000 q^{97} +2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	3.00000	0.654654
$22$	−5.00000	−1.06600
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−1.00000	−0.182574
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000	0.176777
$33$	5.00000	0.870388
$34$	3.00000	0.514496
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−6.00000	−0.973329
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	3.00000	0.462910
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	−5.00000	−0.753778
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	−3.00000	−0.420084
$52$	−2.00000	−0.277350
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	−1.00000	−0.136083
$55$	−5.00000	−0.674200
$56$	−3.00000	−0.400892
$57$	6.00000	0.794719
$58$	−1.00000	−0.131306
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−3.00000	−0.381000
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	5.00000	0.615457
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	0.363803
$69$	4.00000	0.481543
$70$	−3.00000	−0.358569
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−1.00000	−0.116248
$75$	−1.00000	−0.115470
$76$	−6.00000	−0.688247
$77$	15.0000	1.70941
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	3.00000	0.327327
$85$	3.00000	0.325396
$86$	3.00000	0.323498
$87$	1.00000	0.107211
$88$	−5.00000	−0.533002
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	−4.00000	−0.417029
$93$	3.00000	0.311086
$94$	0	0
$95$	−6.00000	−0.615587
$96$	−1.00000	−0.102062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	2.00000	0.202031
$99$	−5.00000	−0.502519
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−3.00000	−0.297044
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	3.00000	0.292770
$106$	5.00000	0.485643
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	−5.00000	−0.476731
$111$	1.00000	0.0949158
$112$	−3.00000	−0.283473
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	6.00000	0.561951
$115$	−4.00000	−0.373002
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	6.00000	0.552345
$119$	−9.00000	−0.825029
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	5.00000	0.452679
$123$	7.00000	0.631169
$124$	−3.00000	−0.269408
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−3.00000	−0.264135
$130$	−2.00000	−0.175412
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	5.00000	0.435194
$133$	18.0000	1.56080
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	3.00000	0.257248
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	4.00000	0.340503
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−12.0000	−1.00702
$143$	10.0000	0.836242
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−2.00000	−0.164957
$148$	−1.00000	−0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−6.00000	−0.486664
$153$	3.00000	0.242536
$154$	15.0000	1.20873
$155$	−3.00000	−0.240966
$156$	2.00000	0.160128
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−4.00000	−0.318223
$159$	−5.00000	−0.396526
$160$	1.00000	0.0790569
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−7.00000	−0.546608
$165$	5.00000	0.389249
$166$	6.00000	0.465690
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	3.00000	0.230089
$171$	−6.00000	−0.458831
$172$	3.00000	0.228748
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	1.00000	0.0758098
$175$	−3.00000	−0.226779
$176$	−5.00000	−0.376889
$177$	−6.00000	−0.450988
$178$	−18.0000	−1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	1.00000	0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	6.00000	0.444750
$183$	−5.00000	−0.369611
$184$	−4.00000	−0.294884
$185$	−1.00000	−0.0735215
$186$	3.00000	0.219971
$187$	−15.0000	−1.09691
$188$	0	0
$189$	3.00000	0.218218
$190$	−6.00000	−0.435286
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−1.00000	−0.0721688
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	−13.0000	−0.933346
$195$	2.00000	0.143223
$196$	2.00000	0.142857
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−5.00000	−0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	−12.0000	−0.844317
$203$	3.00000	0.210559
$204$	−3.00000	−0.210042
$205$	−7.00000	−0.488901
$206$	16.0000	1.11477
$207$	−4.00000	−0.278019
$208$	−2.00000	−0.138675
$209$	30.0000	2.07514
$210$	3.00000	0.207020
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	5.00000	0.343401
$213$	12.0000	0.822226
$214$	2.00000	0.136717
$215$	3.00000	0.204598
$216$	−1.00000	−0.0680414
$217$	9.00000	0.610960
$218$	5.00000	0.338643
$219$	0	0
$220$	−5.00000	−0.337100
$221$	−6.00000	−0.403604
$222$	1.00000	0.0671156
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	9.00000	0.598671
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	6.00000	0.397360
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	−4.00000	−0.263752
$231$	−15.0000	−0.986928
$232$	−1.00000	−0.0656532
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	4.00000	0.259828
$238$	−9.00000	−0.583383
$239$	29.0000	1.87585	0.937927	−	0.346833i	$-0.112743\pi$
$239$	29.0000	1.87585	0.937927	+	0.346833i	$0.112743\pi$
$240$	−1.00000	−0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	5.00000	0.320092
$245$	2.00000	0.127775
$246$	7.00000	0.446304
$247$	12.0000	0.763542
$248$	−3.00000	−0.190500
$249$	−6.00000	−0.380235
$250$	1.00000	0.0632456
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	−3.00000	−0.188982
$253$	20.0000	1.25739
$254$	8.00000	0.501965
$255$	−3.00000	−0.187867
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	−3.00000	−0.186772
$259$	3.00000	0.186411
$260$	−2.00000	−0.124035
$261$	−1.00000	−0.0618984
$262$	10.0000	0.617802
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	5.00000	0.307729
$265$	5.00000	0.307148
$266$	18.0000	1.10365
$267$	18.0000	1.10158
$268$	−4.00000	−0.244339
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	3.00000	0.181902
$273$	−6.00000	−0.363137
$274$	−10.0000	−0.604122
$275$	−5.00000	−0.301511
$276$	4.00000	0.240772
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	5.00000	0.299880
$279$	−3.00000	−0.179605
$280$	−3.00000	−0.179284
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−12.0000	−0.712069
$285$	6.00000	0.355409
$286$	10.0000	0.591312
$287$	21.0000	1.23959
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−1.00000	−0.0587220
$291$	13.0000	0.762073
$292$	0	0
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	−2.00000	−0.116642
$295$	6.00000	0.349334
$296$	−1.00000	−0.0581238
$297$	5.00000	0.290129
$298$	−18.0000	−1.04271
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	−9.00000	−0.518751
$302$	10.0000	0.575435
$303$	12.0000	0.689382
$304$	−6.00000	−0.344124
$305$	5.00000	0.286299
$306$	3.00000	0.171499
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	15.0000	0.854704
$309$	−16.0000	−0.910208
$310$	−3.00000	−0.170389
$311$	−31.0000	−1.75785	−0.878924	−	0.476961i	$-0.841738\pi$
$311$	−31.0000	−1.75785	−0.878924	+	0.476961i	$0.841738\pi$
$312$	2.00000	0.113228
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	11.0000	0.620766
$315$	−3.00000	−0.169031
$316$	−4.00000	−0.225018
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	−5.00000	−0.280386
$319$	5.00000	0.279946
$320$	1.00000	0.0559017
$321$	−2.00000	−0.111629
$322$	12.0000	0.668734
$323$	−18.0000	−1.00155
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	11.0000	0.609234
$327$	−5.00000	−0.276501
$328$	−7.00000	−0.386510
$329$	0	0
$330$	5.00000	0.275241
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	6.00000	0.329293
$333$	−1.00000	−0.0547997
$334$	−24.0000	−1.31322
$335$	−4.00000	−0.218543
$336$	3.00000	0.163663
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−9.00000	−0.489535
$339$	−9.00000	−0.488813
$340$	3.00000	0.162698
$341$	15.0000	0.812296
$342$	−6.00000	−0.324443
$343$	15.0000	0.809924
$344$	3.00000	0.161749
$345$	4.00000	0.215353
$346$	13.0000	0.698884
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	1.00000	0.0536056
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−3.00000	−0.160357
$351$	2.00000	0.106752
$352$	−5.00000	−0.266501
$353$	−37.0000	−1.96931	−0.984656	−	0.174509i	$-0.944166\pi$
$353$	−37.0000	−1.96931	−0.984656	+	0.174509i	$0.944166\pi$
$354$	−6.00000	−0.318896
$355$	−12.0000	−0.636894
$356$	−18.0000	−0.953998
$357$	9.00000	0.476331
$358$	12.0000	0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	1.00000	0.0527046
$361$	17.0000	0.894737
$362$	10.0000	0.525588
$363$	−14.0000	−0.734809
$364$	6.00000	0.314485
$365$	0	0
$366$	−5.00000	−0.261354
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−4.00000	−0.208514
$369$	−7.00000	−0.364405
$370$	−1.00000	−0.0519875
$371$	−15.0000	−0.778761
$372$	3.00000	0.155543
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−15.0000	−0.775632
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	2.00000	0.103005
$378$	3.00000	0.154303
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	−6.00000	−0.307794
$381$	−8.00000	−0.409852
$382$	−3.00000	−0.153493
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−1.00000	−0.0510310
$385$	15.0000	0.764471
$386$	18.0000	0.916176
$387$	3.00000	0.152499
$388$	−13.0000	−0.659975
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	2.00000	0.101274
$391$	−12.0000	−0.606866
$392$	2.00000	0.101015
$393$	−10.0000	−0.504433
$394$	−2.00000	−0.100759
$395$	−4.00000	−0.201262
$396$	−5.00000	−0.251259
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−4.00000	−0.200502
$399$	−18.0000	−0.901127
$400$	1.00000	0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	4.00000	0.199502
$403$	6.00000	0.298881
$404$	−12.0000	−0.597022
$405$	1.00000	0.0496904
$406$	3.00000	0.148888
$407$	5.00000	0.247841
$408$	−3.00000	−0.148522
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−7.00000	−0.345705
$411$	10.0000	0.493264
$412$	16.0000	0.788263
$413$	−18.0000	−0.885722
$414$	−4.00000	−0.196589
$415$	6.00000	0.294528
$416$	−2.00000	−0.0980581
$417$	−5.00000	−0.244851
$418$	30.0000	1.46735
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	3.00000	0.146385
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	19.0000	0.924906
$423$	0	0
$424$	5.00000	0.242821
$425$	3.00000	0.145521
$426$	12.0000	0.581402
$427$	−15.0000	−0.725901
$428$	2.00000	0.0966736
$429$	−10.0000	−0.482805
$430$	3.00000	0.144673
$431$	−7.00000	−0.337178	−0.168589	−	0.985686i	$-0.553921\pi$
$431$	−7.00000	−0.337178	−0.168589	+	0.985686i	$0.553921\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	9.00000	0.432014
$435$	1.00000	0.0479463
$436$	5.00000	0.239457
$437$	24.0000	1.14808
$438$	0	0
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	−5.00000	−0.238366
$441$	2.00000	0.0952381
$442$	−6.00000	−0.285391
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	1.00000	0.0474579
$445$	−18.0000	−0.853282
$446$	7.00000	0.331460
$447$	18.0000	0.851371
$448$	−3.00000	−0.141737
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	1.00000	0.0471405
$451$	35.0000	1.64809
$452$	9.00000	0.423324
$453$	−10.0000	−0.469841
$454$	−21.0000	−0.985579
$455$	6.00000	0.281284
$456$	6.00000	0.280976
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	12.0000	0.560723
$459$	−3.00000	−0.140028
$460$	−4.00000	−0.186501
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	−15.0000	−0.697863
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−1.00000	−0.0464238
$465$	3.00000	0.139122
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	−2.00000	−0.0924500
$469$	12.0000	0.554109
$470$	0	0
$471$	−11.0000	−0.506853
$472$	6.00000	0.276172
$473$	−15.0000	−0.689701
$474$	4.00000	0.183726
$475$	−6.00000	−0.275299
$476$	−9.00000	−0.412514
$477$	5.00000	0.228934
$478$	29.0000	1.32643
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−1.00000	−0.0456435
$481$	2.00000	0.0911922
$482$	−10.0000	−0.455488
$483$	−12.0000	−0.546019
$484$	14.0000	0.636364
$485$	−13.0000	−0.590300
$486$	−1.00000	−0.0453609
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	5.00000	0.226339
$489$	−11.0000	−0.497437
$490$	2.00000	0.0903508
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	7.00000	0.315584
$493$	−3.00000	−0.135113
$494$	12.0000	0.539906
$495$	−5.00000	−0.224733
$496$	−3.00000	−0.134704
$497$	36.0000	1.61482
$498$	−6.00000	−0.268866
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	1.00000	0.0447214
$501$	24.0000	1.07224
$502$	−24.0000	−1.07117
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−3.00000	−0.133631
$505$	−12.0000	−0.533993
$506$	20.0000	0.889108
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	−3.00000	−0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	26.0000	1.14681
$515$	16.0000	0.705044
$516$	−3.00000	−0.132068
$517$	0	0
$518$	3.00000	0.131812
$519$	−13.0000	−0.570637
$520$	−2.00000	−0.0877058
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	−1.00000	−0.0437688
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	10.0000	0.436852
$525$	3.00000	0.130931
$526$	5.00000	0.218010
$527$	−9.00000	−0.392046
$528$	5.00000	0.217597
$529$	−7.00000	−0.304348
$530$	5.00000	0.217186
$531$	6.00000	0.260378
$532$	18.0000	0.780399
$533$	14.0000	0.606407
$534$	18.0000	0.778936
$535$	2.00000	0.0864675
$536$	−4.00000	−0.172774
$537$	−12.0000	−0.517838
$538$	−16.0000	−0.689809
$539$	−10.0000	−0.430730
$540$	−1.00000	−0.0430331
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	8.00000	0.343629
$543$	−10.0000	−0.429141
$544$	3.00000	0.128624
$545$	5.00000	0.214176
$546$	−6.00000	−0.256776
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	−10.0000	−0.427179
$549$	5.00000	0.213395
$550$	−5.00000	−0.213201
$551$	6.00000	0.255609
$552$	4.00000	0.170251
$553$	12.0000	0.510292
$554$	−16.0000	−0.679775
$555$	1.00000	0.0424476
$556$	5.00000	0.212047
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−3.00000	−0.127000
$559$	−6.00000	−0.253773
$560$	−3.00000	−0.126773
$561$	15.0000	0.633300
$562$	2.00000	0.0843649
$563$	−17.0000	−0.716465	−0.358232	−	0.933632i	$-0.616620\pi$
$563$	−17.0000	−0.716465	−0.358232	+	0.933632i	$0.616620\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	4.00000	0.168133
$567$	−3.00000	−0.125988
$568$	−12.0000	−0.503509
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	6.00000	0.251312
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	10.0000	0.418121
$573$	3.00000	0.125327
$574$	21.0000	0.876523
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−8.00000	−0.332756
$579$	−18.0000	−0.748054
$580$	−1.00000	−0.0415227
$581$	−18.0000	−0.746766
$582$	13.0000	0.538867
$583$	−25.0000	−1.03539
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	7.00000	0.289167
$587$	1.00000	0.0412744	0.0206372	−	0.999787i	$-0.493431\pi$
$587$	1.00000	0.0412744	0.0206372	+	0.999787i	$0.493431\pi$
$588$	−2.00000	−0.0824786
$589$	18.0000	0.741677
$590$	6.00000	0.247016
$591$	2.00000	0.0822690
$592$	−1.00000	−0.0410997
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	5.00000	0.205152
$595$	−9.00000	−0.368964
$596$	−18.0000	−0.737309
$597$	4.00000	0.163709
$598$	8.00000	0.327144
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	−1.00000	−0.0408248
$601$	−31.0000	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$602$	−9.00000	−0.366813
$603$	−4.00000	−0.162893
$604$	10.0000	0.406894
$605$	14.0000	0.569181
$606$	12.0000	0.487467
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−6.00000	−0.243332
$609$	−3.00000	−0.121566
$610$	5.00000	0.202444
$611$	0	0
$612$	3.00000	0.121268
$613$	−43.0000	−1.73675	−0.868377	−	0.495905i	$-0.834836\pi$
$613$	−43.0000	−1.73675	−0.868377	+	0.495905i	$0.834836\pi$
$614$	−2.00000	−0.0807134
$615$	7.00000	0.282267
$616$	15.0000	0.604367
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	−16.0000	−0.643614
$619$	−27.0000	−1.08522	−0.542611	−	0.839984i	$-0.682564\pi$
$619$	−27.0000	−1.08522	−0.542611	+	0.839984i	$0.682564\pi$
$620$	−3.00000	−0.120483
$621$	4.00000	0.160514
$622$	−31.0000	−1.24299
$623$	54.0000	2.16346
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−30.0000	−1.19904
$627$	−30.0000	−1.19808
$628$	11.0000	0.438948
$629$	−3.00000	−0.119618
$630$	−3.00000	−0.119523
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	−4.00000	−0.159111
$633$	−19.0000	−0.755182
$634$	−27.0000	−1.07231
$635$	8.00000	0.317470
$636$	−5.00000	−0.198263
$637$	−4.00000	−0.158486
$638$	5.00000	0.197952
$639$	−12.0000	−0.474713
$640$	1.00000	0.0395285
$641$	−23.0000	−0.908445	−0.454223	−	0.890888i	$-0.650083\pi$
$641$	−23.0000	−0.908445	−0.454223	+	0.890888i	$0.650083\pi$
$642$	−2.00000	−0.0789337
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	12.0000	0.472866
$645$	−3.00000	−0.118125
$646$	−18.0000	−0.708201
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	1.00000	0.0392837
$649$	−30.0000	−1.17760
$650$	−2.00000	−0.0784465
$651$	−9.00000	−0.352738
$652$	11.0000	0.430793
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	−5.00000	−0.195515
$655$	10.0000	0.390732
$656$	−7.00000	−0.273304
$657$	0	0
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	5.00000	0.194625
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	−20.0000	−0.777322
$663$	6.00000	0.233021
$664$	6.00000	0.232845
$665$	18.0000	0.698010
$666$	−1.00000	−0.0387492
$667$	4.00000	0.154881
$668$	−24.0000	−0.928588
$669$	−7.00000	−0.270636
$670$	−4.00000	−0.154533
$671$	−25.0000	−0.965114
$672$	3.00000	0.115728
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	−6.00000	−0.231111
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	10.0000	0.384331	0.192166	−	0.981363i	$-0.438449\pi$
$677$	10.0000	0.384331	0.192166	+	0.981363i	$0.438449\pi$
$678$	−9.00000	−0.345643
$679$	39.0000	1.49668
$680$	3.00000	0.115045
$681$	21.0000	0.804722
$682$	15.0000	0.574380
$683$	−29.0000	−1.10965	−0.554827	−	0.831966i	$-0.687216\pi$
$683$	−29.0000	−1.10965	−0.554827	+	0.831966i	$0.687216\pi$
$684$	−6.00000	−0.229416
$685$	−10.0000	−0.382080
$686$	15.0000	0.572703
$687$	−12.0000	−0.457829
$688$	3.00000	0.114374
$689$	−10.0000	−0.380970
$690$	4.00000	0.152277
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	13.0000	0.494186
$693$	15.0000	0.569803
$694$	−20.0000	−0.759190
$695$	5.00000	0.189661
$696$	1.00000	0.0379049
$697$	−21.0000	−0.795432
$698$	−20.0000	−0.757011
$699$	0	0
$700$	−3.00000	−0.113389
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	2.00000	0.0754851
$703$	6.00000	0.226294
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−37.0000	−1.39251
$707$	36.0000	1.35392
$708$	−6.00000	−0.225494
$709$	49.0000	1.84023	0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000	1.84023	0.920117	+	0.391644i	$0.128094\pi$
$710$	−12.0000	−0.450352
$711$	−4.00000	−0.150012
$712$	−18.0000	−0.674579
$713$	12.0000	0.449404
$714$	9.00000	0.336817
$715$	10.0000	0.373979
$716$	12.0000	0.448461
$717$	−29.0000	−1.08302
$718$	−30.0000	−1.11959
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	1.00000	0.0372678
$721$	−48.0000	−1.78761
$722$	17.0000	0.632674
$723$	10.0000	0.371904
$724$	10.0000	0.371647
$725$	−1.00000	−0.0371391
$726$	−14.0000	−0.519589
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.00000	0.332877
$732$	−5.00000	−0.184805
$733$	−51.0000	−1.88373	−0.941864	−	0.335994i	$-0.890928\pi$
$733$	−51.0000	−1.88373	−0.941864	+	0.335994i	$0.890928\pi$
$734$	−3.00000	−0.110732
$735$	−2.00000	−0.0737711
$736$	−4.00000	−0.147442
$737$	20.0000	0.736709
$738$	−7.00000	−0.257674
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	−1.00000	−0.0367607
$741$	−12.0000	−0.440831
$742$	−15.0000	−0.550667
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	3.00000	0.109985
$745$	−18.0000	−0.659469
$746$	26.0000	0.951928
$747$	6.00000	0.219529
$748$	−15.0000	−0.548454
$749$	−6.00000	−0.219235
$750$	−1.00000	−0.0365148
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	2.00000	0.0728357
$755$	10.0000	0.363937
$756$	3.00000	0.109109
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	32.0000	1.16229
$759$	−20.0000	−0.725954
$760$	−6.00000	−0.217643
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	−8.00000	−0.289809
$763$	−15.0000	−0.543036
$764$	−3.00000	−0.108536
$765$	3.00000	0.108465
$766$	−20.0000	−0.722629
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	15.0000	0.540562
$771$	−26.0000	−0.936367
$772$	18.0000	0.647834
$773$	−17.0000	−0.611448	−0.305724	−	0.952120i	$-0.598898\pi$
$773$	−17.0000	−0.611448	−0.305724	+	0.952120i	$0.598898\pi$
$774$	3.00000	0.107833
$775$	−3.00000	−0.107763
$776$	−13.0000	−0.466673
$777$	−3.00000	−0.107624
$778$	−3.00000	−0.107555
$779$	42.0000	1.50481
$780$	2.00000	0.0716115
$781$	60.0000	2.14697
$782$	−12.0000	−0.429119
$783$	1.00000	0.0357371
$784$	2.00000	0.0714286
$785$	11.0000	0.392607
$786$	−10.0000	−0.356688
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−2.00000	−0.0712470
$789$	−5.00000	−0.178005
$790$	−4.00000	−0.142314
$791$	−27.0000	−0.960009
$792$	−5.00000	−0.177667
$793$	−10.0000	−0.355110
$794$	18.0000	0.638796
$795$	−5.00000	−0.177332
$796$	−4.00000	−0.141776
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	−18.0000	−0.637193
$799$	0	0
$800$	1.00000	0.0353553
$801$	−18.0000	−0.635999
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	4.00000	0.141069
$805$	12.0000	0.422944
$806$	6.00000	0.211341
$807$	16.0000	0.563227
$808$	−12.0000	−0.422159
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	1.00000	0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	3.00000	0.105279
$813$	−8.00000	−0.280572
$814$	5.00000	0.175250
$815$	11.0000	0.385313
$816$	−3.00000	−0.105021
$817$	−18.0000	−0.629740
$818$	6.00000	0.209785
$819$	6.00000	0.209657
$820$	−7.00000	−0.244451
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	10.0000	0.348790
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	16.0000	0.557386
$825$	5.00000	0.174078
$826$	−18.0000	−0.626300
$827$	23.0000	0.799788	0.399894	−	0.916561i	$-0.369047\pi$
$827$	23.0000	0.799788	0.399894	+	0.916561i	$0.369047\pi$
$828$	−4.00000	−0.139010
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	6.00000	0.208263
$831$	16.0000	0.555034
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	−5.00000	−0.173136
$835$	−24.0000	−0.830554
$836$	30.0000	1.03757
$837$	3.00000	0.103695
$838$	−12.0000	−0.414533
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	3.00000	0.103510
$841$	−28.0000	−0.965517
$842$	2.00000	0.0689246
$843$	−2.00000	−0.0688837
$844$	19.0000	0.654007
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−42.0000	−1.44314
$848$	5.00000	0.171701
$849$	−4.00000	−0.137280
$850$	3.00000	0.102899
$851$	4.00000	0.137118
$852$	12.0000	0.411113
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−15.0000	−0.513289
$855$	−6.00000	−0.205196
$856$	2.00000	0.0683586
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	−10.0000	−0.341394
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	3.00000	0.102299
$861$	−21.0000	−0.715678
$862$	−7.00000	−0.238421
$863$	27.0000	0.919091	0.459545	−	0.888154i	$-0.348012\pi$
$863$	27.0000	0.919091	0.459545	+	0.888154i	$0.348012\pi$
$864$	−1.00000	−0.0340207
$865$	13.0000	0.442013
$866$	−8.00000	−0.271851
$867$	8.00000	0.271694
$868$	9.00000	0.305480
$869$	20.0000	0.678454
$870$	1.00000	0.0339032
$871$	8.00000	0.271070
$872$	5.00000	0.169321
$873$	−13.0000	−0.439983
$874$	24.0000	0.811812
$875$	−3.00000	−0.101419
$876$	0	0
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	−29.0000	−0.978703
$879$	−7.00000	−0.236104
$880$	−5.00000	−0.168550
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	2.00000	0.0673435
$883$	−3.00000	−0.100958	−0.0504790	−	0.998725i	$-0.516075\pi$
$883$	−3.00000	−0.100958	−0.0504790	+	0.998725i	$0.516075\pi$
$884$	−6.00000	−0.201802
$885$	−6.00000	−0.201688
$886$	14.0000	0.470339
$887$	−7.00000	−0.235037	−0.117518	−	0.993071i	$-0.537494\pi$
$887$	−7.00000	−0.235037	−0.117518	+	0.993071i	$0.537494\pi$
$888$	1.00000	0.0335578
$889$	−24.0000	−0.804934
$890$	−18.0000	−0.603361
$891$	−5.00000	−0.167506
$892$	7.00000	0.234377
$893$	0	0
$894$	18.0000	0.602010
$895$	12.0000	0.401116
$896$	−3.00000	−0.100223
$897$	−8.00000	−0.267112
$898$	−36.0000	−1.20134
$899$	3.00000	0.100056
$900$	1.00000	0.0333333
$901$	15.0000	0.499722
$902$	35.0000	1.16537
$903$	9.00000	0.299501
$904$	9.00000	0.299336
$905$	10.0000	0.332411
$906$	−10.0000	−0.332228
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−21.0000	−0.696909
$909$	−12.0000	−0.398015
$910$	6.00000	0.198898
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	6.00000	0.198680
$913$	−30.0000	−0.992855
$914$	−5.00000	−0.165385
$915$	−5.00000	−0.165295
$916$	12.0000	0.396491
$917$	−30.0000	−0.990687
$918$	−3.00000	−0.0990148
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	−4.00000	−0.131876
$921$	2.00000	0.0659022
$922$	21.0000	0.691598
$923$	24.0000	0.789970
$924$	−15.0000	−0.493464
$925$	−1.00000	−0.0328798
$926$	34.0000	1.11731
$927$	16.0000	0.525509
$928$	−1.00000	−0.0328266
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	3.00000	0.0983739
$931$	−12.0000	−0.393284
$932$	0	0
$933$	31.0000	1.01489
$934$	−15.0000	−0.490815
$935$	−15.0000	−0.490552
$936$	−2.00000	−0.0653720
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	12.0000	0.391814
$939$	30.0000	0.979013
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	−11.0000	−0.358399
$943$	28.0000	0.911805
$944$	6.00000	0.195283
$945$	3.00000	0.0975900
$946$	−15.0000	−0.487692
$947$	−29.0000	−0.942373	−0.471187	−	0.882034i	$-0.656174\pi$
$947$	−29.0000	−0.942373	−0.471187	+	0.882034i	$0.656174\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	−6.00000	−0.194666
$951$	27.0000	0.875535
$952$	−9.00000	−0.291692
$953$	−40.0000	−1.29573	−0.647864	−	0.761756i	$-0.724337\pi$
$953$	−40.0000	−1.29573	−0.647864	+	0.761756i	$0.724337\pi$
$954$	5.00000	0.161881
$955$	−3.00000	−0.0970777
$956$	29.0000	0.937927
$957$	−5.00000	−0.161627
$958$	−16.0000	−0.516937
$959$	30.0000	0.968751
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	2.00000	0.0644826
$963$	2.00000	0.0644491
$964$	−10.0000	−0.322078
$965$	18.0000	0.579441
$966$	−12.0000	−0.386094
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	14.0000	0.449977
$969$	18.0000	0.578243
$970$	−13.0000	−0.417405
$971$	−35.0000	−1.12320	−0.561602	−	0.827408i	$-0.689815\pi$
$971$	−35.0000	−1.12320	−0.561602	+	0.827408i	$0.689815\pi$
$972$	−1.00000	−0.0320750
$973$	−15.0000	−0.480878
$974$	−6.00000	−0.192252
$975$	2.00000	0.0640513
$976$	5.00000	0.160046
$977$	−25.0000	−0.799821	−0.399910	−	0.916554i	$-0.630959\pi$
$977$	−25.0000	−0.799821	−0.399910	+	0.916554i	$0.630959\pi$
$978$	−11.0000	−0.351741
$979$	90.0000	2.87641
$980$	2.00000	0.0638877
$981$	5.00000	0.159638
$982$	4.00000	0.127645
$983$	−29.0000	−0.924956	−0.462478	−	0.886631i	$-0.653040\pi$
$983$	−29.0000	−0.924956	−0.462478	+	0.886631i	$0.653040\pi$
$984$	7.00000	0.223152
$985$	−2.00000	−0.0637253
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	12.0000	0.381771
$989$	−12.0000	−0.381578
$990$	−5.00000	−0.158910
$991$	7.00000	0.222362	0.111181	−	0.993800i	$-0.464537\pi$
$991$	7.00000	0.222362	0.111181	+	0.993800i	$0.464537\pi$
$992$	−3.00000	−0.0952501
$993$	20.0000	0.634681
$994$	36.0000	1.14185
$995$	−4.00000	−0.126809
$996$	−6.00000	−0.190117
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.j.1.1	✓	1
3.2	odd	2		3330.2.a.b.1.1		1
4.3	odd	2		8880.2.a.bc.1.1		1
5.4	even	2		5550.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.j.1.1	✓	1	1.1	even	1	trivial
3330.2.a.b.1.1		1	3.2	odd	2
5550.2.a.t.1.1		1	5.4	even	2
8880.2.a.bc.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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