LMFDB - Embedded newform 5610.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
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	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} -1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -4.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} -8.00000 q^{69} -4.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{84} -1.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +8.00000 q^{91} -8.00000 q^{92} -4.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	−1.00000	−0.213201
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	1.00000	0.171499
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	8.00000	1.17954
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	2.00000	0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	−4.00000	−0.534522
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	1.00000	0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−1.00000	−0.123091
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	−8.00000	−0.963087
$70$	−4.00000	−0.478091
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	4.00000	0.455842
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	4.00000	0.436436
$85$	−1.00000	−0.108465
$86$	−4.00000	−0.431331
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−1.00000	−0.105409
$91$	8.00000	0.838628
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−4.00000	−0.412568
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−9.00000	−0.909137
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	1.00000	0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	10.0000	0.971286
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	4.00000	0.377964
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	−4.00000	−0.366679
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	−4.00000	−0.356348
$127$	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
$129$	4.00000	0.352180
$130$	−2.00000	−0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	1.00000	0.0870388
$133$	16.0000	1.38738
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	8.00000	0.681005
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	0.338062
$141$	4.00000	0.336861
$142$	4.00000	0.335673
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−10.0000	−0.827606
$147$	9.00000	0.742307
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	−4.00000	−0.324443
$153$	−1.00000	−0.0808452
$154$	−4.00000	−0.322329
$155$	0	0
$156$	2.00000	0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	−1.00000	−0.0790569
$161$	−32.0000	−2.52195
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	1.00000	0.0778499
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	1.00000	0.0766965
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−2.00000	−0.151620
$175$	4.00000	0.302372
$176$	1.00000	0.0753778
$177$	4.00000	0.300658
$178$	−10.0000	−0.749532
$179$	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
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	−8.00000	−0.592999
$183$	2.00000	0.147844
$184$	8.00000	0.589768
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	4.00000	0.291730
$189$	4.00000	0.290957
$190$	−4.00000	−0.290191
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−2.00000	−0.143592
$195$	2.00000	0.143223
$196$	9.00000	0.642857
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−1.00000	−0.0710669
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	10.0000	0.703598
$203$	8.00000	0.561490
$204$	−1.00000	−0.0700140
$205$	6.00000	0.419058
$206$	8.00000	0.557386
$207$	−8.00000	−0.556038
$208$	2.00000	0.138675
$209$	4.00000	0.276686
$210$	−4.00000	−0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−10.0000	−0.686803
$213$	−4.00000	−0.274075
$214$	0	0
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	10.0000	0.675737
$220$	1.00000	0.0674200
$221$	−2.00000	−0.134535
$222$	2.00000	0.134231
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	4.00000	0.264906
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	8.00000	0.527504
$231$	4.00000	0.263181
$232$	−2.00000	−0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	4.00000	0.260378
$237$	0	0
$238$	4.00000	0.259281
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	9.00000	0.574989
$246$	−6.00000	−0.382546
$247$	8.00000	0.509028
$248$	0	0
$249$	4.00000	0.253490
$250$	−1.00000	−0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	4.00000	0.251976
$253$	−8.00000	−0.502956
$254$	−8.00000	−0.501965
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	−4.00000	−0.249029
$259$	−8.00000	−0.497096
$260$	2.00000	0.124035
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−1.00000	−0.0615457
$265$	−10.0000	−0.614295
$266$	−16.0000	−0.981023
$267$	10.0000	0.611990
$268$	8.00000	0.488678
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−1.00000	−0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−1.00000	−0.0606339
$273$	8.00000	0.484182
$274$	2.00000	0.120824
$275$	1.00000	0.0603023
$276$	−8.00000	−0.481543
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−4.00000	−0.239046
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−4.00000	−0.238197
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−4.00000	−0.237356
$285$	4.00000	0.236940
$286$	−2.00000	−0.118262
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−2.00000	−0.117444
$291$	2.00000	0.117242
$292$	10.0000	0.585206
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−9.00000	−0.524891
$295$	4.00000	0.232889
$296$	2.00000	0.116248
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	−16.0000	−0.925304
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	20.0000	1.15087
$303$	−10.0000	−0.574485
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	1.00000	0.0571662
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	4.00000	0.227921
$309$	−8.00000	−0.455104
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	−0.113228
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−10.0000	−0.564333
$315$	4.00000	0.225374
$316$	0	0
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	10.0000	0.560772
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	32.0000	1.78329
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	−14.0000	−0.774202
$328$	−6.00000	−0.331295
$329$	16.0000	0.882109
$330$	−1.00000	−0.0550482
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	8.00000	0.437741
$335$	8.00000	0.437087
$336$	4.00000	0.218218
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	9.00000	0.489535
$339$	10.0000	0.543125
$340$	−1.00000	−0.0542326
$341$	0	0
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	−8.00000	−0.430706
$346$	18.0000	0.967686
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	2.00000	0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−4.00000	−0.213809
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−4.00000	−0.212598
$355$	−4.00000	−0.212298
$356$	10.0000	0.529999
$357$	−4.00000	−0.211702
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−18.0000	−0.946059
$363$	1.00000	0.0524864
$364$	8.00000	0.419314
$365$	10.0000	0.523424
$366$	−2.00000	−0.104542
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−8.00000	−0.417029
$369$	6.00000	0.312348
$370$	2.00000	0.103975
$371$	−40.0000	−2.07670
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	1.00000	0.0517088
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	4.00000	0.206010
$378$	−4.00000	−0.205738
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	4.00000	0.205196
$381$	8.00000	0.409852
$382$	−24.0000	−1.22795
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	−1.00000	−0.0510310
$385$	4.00000	0.203859
$386$	6.00000	0.305392
$387$	4.00000	0.203331
$388$	2.00000	0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	8.00000	0.404577
$392$	−9.00000	−0.454569
$393$	−4.00000	−0.201773
$394$	2.00000	0.100759
$395$	0	0
$396$	1.00000	0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−8.00000	−0.401004
$399$	16.0000	0.801002
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	−2.00000	−0.0991363
$408$	1.00000	0.0495074
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	−2.00000	−0.0986527
$412$	−8.00000	−0.394132
$413$	16.0000	0.787309
$414$	8.00000	0.393179
$415$	4.00000	0.196352
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	−4.00000	−0.195646
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	4.00000	0.195180
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	12.0000	0.584151
$423$	4.00000	0.194487
$424$	10.0000	0.485643
$425$	−1.00000	−0.0485071
$426$	4.00000	0.193801
$427$	8.00000	0.387147
$428$	0	0
$429$	2.00000	0.0965609
$430$	−4.00000	−0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	−14.0000	−0.670478
$437$	−32.0000	−1.53077
$438$	−10.0000	−0.477818
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	−1.00000	−0.0476731
$441$	9.00000	0.428571
$442$	2.00000	0.0951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−2.00000	−0.0949158
$445$	10.0000	0.474045
$446$	8.00000	0.378811
$447$	6.00000	0.283790
$448$	4.00000	0.188982
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	10.0000	0.470360
$453$	−20.0000	−0.939682
$454$	0	0
$455$	8.00000	0.375046
$456$	−4.00000	−0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	18.0000	0.841085
$459$	−1.00000	−0.0466760
$460$	−8.00000	−0.373002
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−4.00000	−0.186097
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	32.0000	1.47762
$470$	−4.00000	−0.184506
$471$	10.0000	0.460776
$472$	−4.00000	−0.184115
$473$	4.00000	0.183920
$474$	0	0
$475$	4.00000	0.183533
$476$	−4.00000	−0.183340
$477$	−10.0000	−0.457869
$478$	24.0000	1.09773
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	−1.00000	−0.0456435
$481$	−4.00000	−0.182384
$482$	−14.0000	−0.637683
$483$	−32.0000	−1.45605
$484$	1.00000	0.0454545
$485$	2.00000	0.0908153
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−2.00000	−0.0905357
$489$	4.00000	0.180886
$490$	−9.00000	−0.406579
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	6.00000	0.270501
$493$	−2.00000	−0.0900755
$494$	−8.00000	−0.359937
$495$	1.00000	0.0449467
$496$	0	0
$497$	−16.0000	−0.717698
$498$	−4.00000	−0.179244
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	1.00000	0.0447214
$501$	−8.00000	−0.357414
$502$	−4.00000	−0.178529
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	−4.00000	−0.178174
$505$	−10.0000	−0.444994
$506$	8.00000	0.355643
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	1.00000	0.0442807
$511$	40.0000	1.76950
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	26.0000	1.14681
$515$	−8.00000	−0.352522
$516$	4.00000	0.176090
$517$	4.00000	0.175920
$518$	8.00000	0.351500
$519$	−18.0000	−0.790112
$520$	−2.00000	−0.0877058
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	−2.00000	−0.0875376
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−4.00000	−0.174741
$525$	4.00000	0.174574
$526$	24.0000	1.04645
$527$	0	0
$528$	1.00000	0.0435194
$529$	41.0000	1.78261
$530$	10.0000	0.434372
$531$	4.00000	0.173585
$532$	16.0000	0.693688
$533$	12.0000	0.519778
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−8.00000	−0.345547
$537$	12.0000	0.517838
$538$	−14.0000	−0.603583
$539$	9.00000	0.387657
$540$	1.00000	0.0430331
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	20.0000	0.859074
$543$	18.0000	0.772454
$544$	1.00000	0.0428746
$545$	−14.0000	−0.599694
$546$	−8.00000	−0.342368
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−2.00000	−0.0854358
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	8.00000	0.340811
$552$	8.00000	0.340503
$553$	0	0
$554$	22.0000	0.934690
$555$	−2.00000	−0.0848953
$556$	4.00000	0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	4.00000	0.169031
$561$	−1.00000	−0.0422200
$562$	−6.00000	−0.253095
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	4.00000	0.168430
$565$	10.0000	0.420703
$566$	−4.00000	−0.168133
$567$	4.00000	0.167984
$568$	4.00000	0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−4.00000	−0.167542
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	2.00000	0.0836242
$573$	24.0000	1.00261
$574$	−24.0000	−1.00174
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−1.00000	−0.0415945
$579$	−6.00000	−0.249351
$580$	2.00000	0.0830455
$581$	16.0000	0.663792
$582$	−2.00000	−0.0829027
$583$	−10.0000	−0.414158
$584$	−10.0000	−0.413803
$585$	2.00000	0.0826898
$586$	−26.0000	−1.07405
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	−4.00000	−0.164677
$591$	−2.00000	−0.0822690
$592$	−2.00000	−0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−1.00000	−0.0410305
$595$	−4.00000	−0.163984
$596$	6.00000	0.245770
$597$	8.00000	0.327418
$598$	16.0000	0.654289
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−1.00000	−0.0408248
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	−16.0000	−0.652111
$603$	8.00000	0.325785
$604$	−20.0000	−0.813788
$605$	1.00000	0.0406558
$606$	10.0000	0.406222
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−4.00000	−0.162221
$609$	8.00000	0.324176
$610$	−2.00000	−0.0809776
$611$	8.00000	0.323645
$612$	−1.00000	−0.0404226
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	4.00000	0.161427
$615$	6.00000	0.241943
$616$	−4.00000	−0.161165
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	8.00000	0.321807
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	−12.0000	−0.481156
$623$	40.0000	1.60257
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	4.00000	0.159745
$628$	10.0000	0.399043
$629$	2.00000	0.0797452
$630$	−4.00000	−0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	34.0000	1.35031
$635$	8.00000	0.317470
$636$	−10.0000	−0.396526
$637$	18.0000	0.713186
$638$	−2.00000	−0.0791808
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−32.0000	−1.26098
$645$	4.00000	0.157500
$646$	4.00000	0.157378
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	4.00000	0.157014
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	14.0000	0.547443
$655$	−4.00000	−0.156293
$656$	6.00000	0.234261
$657$	10.0000	0.390137
$658$	−16.0000	−0.623745
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	1.00000	0.0389249
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	28.0000	1.08825
$663$	−2.00000	−0.0776736
$664$	−4.00000	−0.155230
$665$	16.0000	0.620453
$666$	2.00000	0.0774984
$667$	−16.0000	−0.619522
$668$	−8.00000	−0.309529
$669$	−8.00000	−0.309298
$670$	−8.00000	−0.309067
$671$	2.00000	0.0772091
$672$	−4.00000	−0.154303
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−10.0000	−0.384048
$679$	8.00000	0.307012
$680$	1.00000	0.0383482
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	4.00000	0.152944
$685$	−2.00000	−0.0764161
$686$	−8.00000	−0.305441
$687$	−18.0000	−0.686743
$688$	4.00000	0.152499
$689$	−20.0000	−0.761939
$690$	8.00000	0.304555
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−18.0000	−0.684257
$693$	4.00000	0.151947
$694$	16.0000	0.607352
$695$	4.00000	0.151729
$696$	−2.00000	−0.0758098
$697$	−6.00000	−0.227266
$698$	−10.0000	−0.378506
$699$	6.00000	0.226941
$700$	4.00000	0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	−2.00000	−0.0754851
$703$	−8.00000	−0.301726
$704$	1.00000	0.0376889
$705$	4.00000	0.150649
$706$	18.0000	0.677439
$707$	−40.0000	−1.50435
$708$	4.00000	0.150329
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	4.00000	0.150117
$711$	0	0
$712$	−10.0000	−0.374766
$713$	0	0
$714$	4.00000	0.149696
$715$	2.00000	0.0747958
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000	0.0372678
$721$	−32.0000	−1.19174
$722$	3.00000	0.111648
$723$	14.0000	0.520666
$724$	18.0000	0.668965
$725$	2.00000	0.0742781
$726$	−1.00000	−0.0371135
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	−4.00000	−0.147945
$732$	2.00000	0.0739221
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	8.00000	0.295285
$735$	9.00000	0.331970
$736$	8.00000	0.294884
$737$	8.00000	0.294684
$738$	−6.00000	−0.220863
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−2.00000	−0.0735215
$741$	8.00000	0.293887
$742$	40.0000	1.46845
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−10.0000	−0.366126
$747$	4.00000	0.146352
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	4.00000	0.145865
$753$	4.00000	0.145768
$754$	−4.00000	−0.145671
$755$	−20.0000	−0.727875
$756$	4.00000	0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−32.0000	−1.16229
$759$	−8.00000	−0.290382
$760$	−4.00000	−0.145095
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−8.00000	−0.289809
$763$	−56.0000	−2.02734
$764$	24.0000	0.868290
$765$	−1.00000	−0.0361551
$766$	−12.0000	−0.433578
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−4.00000	−0.144150
$771$	−26.0000	−0.936367
$772$	−6.00000	−0.215945
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−8.00000	−0.286998
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	2.00000	0.0716115
$781$	−4.00000	−0.143131
$782$	−8.00000	−0.286079
$783$	2.00000	0.0714742
$784$	9.00000	0.321429
$785$	10.0000	0.356915
$786$	4.00000	0.142675
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−2.00000	−0.0712470
$789$	−24.0000	−0.854423
$790$	0	0
$791$	40.0000	1.42224
$792$	−1.00000	−0.0355335
$793$	4.00000	0.142044
$794$	2.00000	0.0709773
$795$	−10.0000	−0.354663
$796$	8.00000	0.283552
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−16.0000	−0.566394
$799$	−4.00000	−0.141510
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	−30.0000	−1.05934
$803$	10.0000	0.352892
$804$	8.00000	0.282138
$805$	−32.0000	−1.12785
$806$	0	0
$807$	14.0000	0.492823
$808$	10.0000	0.351799
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−1.00000	−0.0351364
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	8.00000	0.280745
$813$	−20.0000	−0.701431
$814$	2.00000	0.0701000
$815$	4.00000	0.140114
$816$	−1.00000	−0.0350070
$817$	16.0000	0.559769
$818$	−26.0000	−0.909069
$819$	8.00000	0.279543
$820$	6.00000	0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	2.00000	0.0697580
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	8.00000	0.278693
$825$	1.00000	0.0348155
$826$	−16.0000	−0.556711
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	−8.00000	−0.278019
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	−4.00000	−0.138842
$831$	−22.0000	−0.763172
$832$	2.00000	0.0693375
$833$	−9.00000	−0.311832
$834$	−4.00000	−0.138509
$835$	−8.00000	−0.276851
$836$	4.00000	0.138343
$837$	0	0
$838$	−20.0000	−0.690889
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	6.00000	0.206651
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	−4.00000	−0.137523
$847$	4.00000	0.137442
$848$	−10.0000	−0.343401
$849$	4.00000	0.137280
$850$	1.00000	0.0342997
$851$	16.0000	0.548473
$852$	−4.00000	−0.137038
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−8.00000	−0.273754
$855$	4.00000	0.136797
$856$	0	0
$857$	−58.0000	−1.98124	−0.990621	−	0.136637i	$-0.956370\pi$
$857$	−58.0000	−1.98124	−0.990621	+	0.136637i	$0.956370\pi$
$858$	−2.00000	−0.0682789
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	4.00000	0.136399
$861$	24.0000	0.817918
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	−2.00000	−0.0679628
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	16.0000	0.542139
$872$	14.0000	0.474100
$873$	2.00000	0.0676897
$874$	32.0000	1.08242
$875$	4.00000	0.135225
$876$	10.0000	0.337869
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	−16.0000	−0.539974
$879$	26.0000	0.876958
$880$	1.00000	0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−9.00000	−0.303046
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	−2.00000	−0.0672673
$885$	4.00000	0.134459
$886$	36.0000	1.20944
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	2.00000	0.0671156
$889$	32.0000	1.07325
$890$	−10.0000	−0.335201
$891$	1.00000	0.0335013
$892$	−8.00000	−0.267860
$893$	16.0000	0.535420
$894$	−6.00000	−0.200670
$895$	12.0000	0.401116
$896$	−4.00000	−0.133631
$897$	−16.0000	−0.534224
$898$	18.0000	0.600668
$899$	0	0
$900$	1.00000	0.0333333
$901$	10.0000	0.333148
$902$	−6.00000	−0.199778
$903$	16.0000	0.532447
$904$	−10.0000	−0.332595
$905$	18.0000	0.598340
$906$	20.0000	0.664455
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	−8.00000	−0.265197
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	4.00000	0.132453
$913$	4.00000	0.132381
$914$	10.0000	0.330771
$915$	2.00000	0.0661180
$916$	−18.0000	−0.594737
$917$	−16.0000	−0.528367
$918$	1.00000	0.0330049
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	8.00000	0.263752
$921$	−4.00000	−0.131804
$922$	−6.00000	−0.197599
$923$	−8.00000	−0.263323
$924$	4.00000	0.131590
$925$	−2.00000	−0.0657596
$926$	−8.00000	−0.262896
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	6.00000	0.196537
$933$	12.0000	0.392862
$934$	12.0000	0.392652
$935$	−1.00000	−0.0327035
$936$	−2.00000	−0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−32.0000	−1.04484
$939$	26.0000	0.848478
$940$	4.00000	0.130466
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−10.0000	−0.325818
$943$	−48.0000	−1.56310
$944$	4.00000	0.130189
$945$	4.00000	0.130120
$946$	−4.00000	−0.130051
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	−4.00000	−0.129777
$951$	−34.0000	−1.10253
$952$	4.00000	0.129641
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	10.0000	0.323762
$955$	24.0000	0.776622
$956$	−24.0000	−0.776215
$957$	2.00000	0.0646508
$958$	−16.0000	−0.516937
$959$	−8.00000	−0.258333
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	4.00000	0.128965
$963$	0	0
$964$	14.0000	0.450910
$965$	−6.00000	−0.193147
$966$	32.0000	1.02958
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	−1.00000	−0.0321412
$969$	−4.00000	−0.128499
$970$	−2.00000	−0.0642161
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	−16.0000	−0.512673
$975$	2.00000	0.0640513
$976$	2.00000	0.0640184
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	−4.00000	−0.127906
$979$	10.0000	0.319601
$980$	9.00000	0.287494
$981$	−14.0000	−0.446986
$982$	16.0000	0.510581
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−6.00000	−0.191273
$985$	−2.00000	−0.0637253
$986$	2.00000	0.0636930
$987$	16.0000	0.509286
$988$	8.00000	0.254514
$989$	−32.0000	−1.01754
$990$	−1.00000	−0.0317821
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	16.0000	0.507489
$995$	8.00000	0.253617
$996$	4.00000	0.126745
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	16.0000	0.506471
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.v.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.v.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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