LMFDB - Embedded newform 5610.2.a.e.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:	$1$
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	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -8.00000 q^{29} -1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{42} +1.00000 q^{44} -1.00000 q^{45} -2.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} -8.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -4.00000 q^{56} -4.00000 q^{57} +8.00000 q^{58} -8.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -2.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +8.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} -4.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -1.00000 q^{85} +8.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} -16.0000 q^{91} -2.00000 q^{93} +2.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−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
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−1.00000	−0.182574
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.00000	0.617213
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	−4.00000	−0.554700
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	−4.00000	−0.534522
$57$	−4.00000	−0.529813
$58$	8.00000	1.05045
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−2.00000	−0.254000
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	4.00000	0.478091
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	8.00000	0.929981
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	4.00000	0.455842
$78$	−4.00000	−0.452911
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	−1.00000	−0.108465
$86$	0	0
$87$	8.00000	0.857690
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	−16.0000	−1.67726
$92$	0	0
$93$	−2.00000	−0.207390
$94$	2.00000	0.206284
$95$	−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
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	1.00000	0.0990148
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	4.00000	0.392232
$105$	4.00000	0.390360
$106$	8.00000	0.777029
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.00000	−0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	1.00000	0.0953463
$111$	8.00000	0.759326
$112$	4.00000	0.377964
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−8.00000	−0.742781
$117$	−4.00000	−0.369800
$118$	8.00000	0.736460
$119$	4.00000	0.366679
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	2.00000	0.180334
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	16.0000	1.38738
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	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
$141$	2.00000	0.168430
$142$	−12.0000	−1.00702
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	−2.00000	−0.165521
$147$	−9.00000	−0.742307
$148$	−8.00000	−0.657596
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	1.00000	0.0816497
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	−4.00000	−0.322329
$155$	−2.00000	−0.160644
$156$	4.00000	0.320256
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−12.0000	−0.954669
$159$	8.00000	0.634441
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	1.00000	0.0778499
$166$	12.0000	0.931381
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	4.00000	0.308607
$169$	3.00000	0.230769
$170$	1.00000	0.0766965
$171$	4.00000	0.305888
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	−8.00000	−0.606478
$175$	4.00000	0.302372
$176$	1.00000	0.0753778
$177$	8.00000	0.601317
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	16.0000	1.18600
$183$	10.0000	0.739221
$184$	0	0
$185$	8.00000	0.588172
$186$	2.00000	0.146647
$187$	1.00000	0.0731272
$188$	−2.00000	−0.145865
$189$	−4.00000	−0.290957
$190$	4.00000	0.290191
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−2.00000	−0.143592
$195$	−4.00000	−0.286446
$196$	9.00000	0.642857
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	−1.00000	−0.0710669
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	−2.00000	−0.140720
$203$	−32.0000	−2.24596
$204$	−1.00000	−0.0700140
$205$	2.00000	0.139686
$206$	16.0000	1.11477
$207$	0	0
$208$	−4.00000	−0.277350
$209$	4.00000	0.276686
$210$	−4.00000	−0.276026
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−8.00000	−0.549442
$213$	−12.0000	−0.822226
$214$	−8.00000	−0.546869
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	18.0000	1.21911
$219$	−2.00000	−0.135147
$220$	−1.00000	−0.0674200
$221$	−4.00000	−0.269069
$222$	−8.00000	−0.536925
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	8.00000	0.525226
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	4.00000	0.261488
$235$	2.00000	0.130466
$236$	−8.00000	−0.520756
$237$	−12.0000	−0.779484
$238$	−4.00000	−0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	1.00000	0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−9.00000	−0.574989
$246$	−2.00000	−0.127515
$247$	−16.0000	−1.01806
$248$	−2.00000	−0.127000
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−14.0000	−0.878438
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	4.00000	0.248069
$261$	−8.00000	−0.495188
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	1.00000	0.0615457
$265$	8.00000	0.491436
$266$	−16.0000	−0.981023
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	1.00000	0.0606339
$273$	16.0000	0.968364
$274$	6.00000	0.362473
$275$	1.00000	0.0603023
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−4.00000	−0.239904
$279$	2.00000	0.119737
$280$	4.00000	0.239046
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−2.00000	−0.119098
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	12.0000	0.712069
$285$	4.00000	0.236940
$286$	4.00000	0.236525
$287$	−8.00000	−0.472225
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−8.00000	−0.469776
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	9.00000	0.524891
$295$	8.00000	0.465778
$296$	8.00000	0.464991
$297$	−1.00000	−0.0580259
$298$	14.0000	0.810998
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−2.00000	−0.115087
$303$	−2.00000	−0.114897
$304$	4.00000	0.229416
$305$	10.0000	0.572598
$306$	−1.00000	−0.0571662
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	4.00000	0.227921
$309$	16.0000	0.910208
$310$	2.00000	0.113592
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−4.00000	−0.226455
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−18.0000	−1.01580
$315$	−4.00000	−0.225374
$316$	12.0000	0.675053
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−8.00000	−0.448618
$319$	−8.00000	−0.447914
$320$	−1.00000	−0.0559017
$321$	−8.00000	−0.446516
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	4.00000	0.221540
$327$	18.0000	0.995402
$328$	2.00000	0.110432
$329$	−8.00000	−0.441054
$330$	−1.00000	−0.0550482
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	−8.00000	−0.438397
$334$	−22.0000	−1.20379
$335$	−4.00000	−0.218543
$336$	−4.00000	−0.218218
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−3.00000	−0.163178
$339$	18.0000	0.977626
$340$	−1.00000	−0.0542326
$341$	2.00000	0.108306
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	8.00000	0.428845
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	−4.00000	−0.213809
$351$	4.00000	0.213504
$352$	−1.00000	−0.0533002
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−8.00000	−0.425195
$355$	−12.0000	−0.636894
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	−20.0000	−1.05703
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−16.0000	−0.840941
$363$	−1.00000	−0.0524864
$364$	−16.0000	−0.838628
$365$	−2.00000	−0.104685
$366$	−10.0000	−0.522708
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	−8.00000	−0.415900
$371$	−32.0000	−1.66136
$372$	−2.00000	−0.103695
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	2.00000	0.103142
$377$	32.0000	1.64808
$378$	4.00000	0.205738
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−4.00000	−0.205196
$381$	−14.0000	−0.717242
$382$	6.00000	0.306987
$383$	10.0000	0.510976	0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000	0.510976	0.255488	+	0.966812i	$0.417764\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	2.00000	0.101797
$387$	0	0
$388$	2.00000	0.101535
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	−9.00000	−0.454569
$393$	0	0
$394$	24.0000	1.20910
$395$	−12.0000	−0.603786
$396$	1.00000	0.0502519
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	−10.0000	−0.501255
$399$	−16.0000	−0.801002
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	4.00000	0.199502
$403$	−8.00000	−0.398508
$404$	2.00000	0.0995037
$405$	−1.00000	−0.0496904
$406$	32.0000	1.58813
$407$	−8.00000	−0.396545
$408$	1.00000	0.0495074
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−2.00000	−0.0987730
$411$	6.00000	0.295958
$412$	−16.0000	−0.788263
$413$	−32.0000	−1.57462
$414$	0	0
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	−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
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	4.00000	0.194717
$423$	−2.00000	−0.0972433
$424$	8.00000	0.388514
$425$	1.00000	0.0485071
$426$	12.0000	0.581402
$427$	−40.0000	−1.93574
$428$	8.00000	0.386695
$429$	4.00000	0.193122
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	−8.00000	−0.384012
$435$	−8.00000	−0.383571
$436$	−18.0000	−0.862044
$437$	0	0
$438$	2.00000	0.0955637
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	1.00000	0.0476731
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	8.00000	0.379663
$445$	6.00000	0.284427
$446$	−4.00000	−0.189405
$447$	14.0000	0.662177
$448$	4.00000	0.188982
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−1.00000	−0.0471405
$451$	−2.00000	−0.0941763
$452$	−18.0000	−0.846649
$453$	−2.00000	−0.0939682
$454$	12.0000	0.563188
$455$	16.0000	0.750092
$456$	4.00000	0.187317
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	−10.0000	−0.467269
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	4.00000	0.186097
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−8.00000	−0.371391
$465$	2.00000	0.0927478
$466$	2.00000	0.0926482
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−4.00000	−0.184900
$469$	16.0000	0.738811
$470$	−2.00000	−0.0922531
$471$	−18.0000	−0.829396
$472$	8.00000	0.368230
$473$	0	0
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	4.00000	0.183340
$477$	−8.00000	−0.366295
$478$	−8.00000	−0.365911
$479$	26.0000	1.18797	0.593985	−	0.804476i	$-0.297554\pi$
$479$	26.0000	1.18797	0.593985	+	0.804476i	$0.297554\pi$
$480$	−1.00000	−0.0456435
$481$	32.0000	1.45907
$482$	18.0000	0.819878
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.00000	−0.0908153
$486$	1.00000	0.0453609
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	10.0000	0.452679
$489$	4.00000	0.180886
$490$	9.00000	0.406579
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	2.00000	0.0901670
$493$	−8.00000	−0.360302
$494$	16.0000	0.719874
$495$	−1.00000	−0.0449467
$496$	2.00000	0.0898027
$497$	48.0000	2.15309
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	−22.0000	−0.982888
$502$	−12.0000	−0.535586
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−4.00000	−0.178174
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	−3.00000	−0.133235
$508$	14.0000	0.621150
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	−1.00000	−0.0442807
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	2.00000	0.0882162
$515$	16.0000	0.705044
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	32.0000	1.40600
$519$	12.0000	0.526742
$520$	−4.00000	−0.175412
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	8.00000	0.350150
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	−4.00000	−0.174408
$527$	2.00000	0.0871214
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	−8.00000	−0.347498
$531$	−8.00000	−0.347170
$532$	16.0000	0.693688
$533$	8.00000	0.346518
$534$	−6.00000	−0.259645
$535$	−8.00000	−0.345870
$536$	−4.00000	−0.172774
$537$	−20.0000	−0.863064
$538$	−18.0000	−0.776035
$539$	9.00000	0.387657
$540$	1.00000	0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−14.0000	−0.601351
$543$	−16.0000	−0.686626
$544$	−1.00000	−0.0428746
$545$	18.0000	0.771035
$546$	−16.0000	−0.684737
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	−1.00000	−0.0426401
$551$	−32.0000	−1.36325
$552$	0	0
$553$	48.0000	2.04117
$554$	−18.0000	−0.764747
$555$	−8.00000	−0.339581
$556$	4.00000	0.169638
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−2.00000	−0.0846668
$559$	0	0
$560$	−4.00000	−0.169031
$561$	−1.00000	−0.0422200
$562$	−10.0000	−0.421825
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	2.00000	0.0842152
$565$	18.0000	0.757266
$566$	−12.0000	−0.504398
$567$	4.00000	0.167984
$568$	−12.0000	−0.503509
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	−4.00000	−0.167542
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−4.00000	−0.167248
$573$	6.00000	0.250654
$574$	8.00000	0.333914
$575$	0	0
$576$	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	$0.750276\pi$
$578$	−1.00000	−0.0415945
$579$	2.00000	0.0831172
$580$	8.00000	0.332182
$581$	−48.0000	−1.99138
$582$	2.00000	0.0829027
$583$	−8.00000	−0.331326
$584$	−2.00000	−0.0827606
$585$	4.00000	0.165380
$586$	14.0000	0.578335
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−9.00000	−0.371154
$589$	8.00000	0.329634
$590$	−8.00000	−0.329355
$591$	24.0000	0.987228
$592$	−8.00000	−0.328798
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	1.00000	0.0410305
$595$	−4.00000	−0.163984
$596$	−14.0000	−0.573462
$597$	−10.0000	−0.409273
$598$	0	0
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	1.00000	0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	2.00000	0.0813788
$605$	−1.00000	−0.0406558
$606$	2.00000	0.0812444
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	−4.00000	−0.162221
$609$	32.0000	1.29671
$610$	−10.0000	−0.404888
$611$	8.00000	0.323645
$612$	1.00000	0.0404226
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	28.0000	1.12999
$615$	−2.00000	−0.0806478
$616$	−4.00000	−0.161165
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−16.0000	−0.643614
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	24.0000	0.962312
$623$	−24.0000	−0.961540
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	−4.00000	−0.159745
$628$	18.0000	0.718278
$629$	−8.00000	−0.318981
$630$	4.00000	0.159364
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−12.0000	−0.477334
$633$	4.00000	0.158986
$634$	6.00000	0.238290
$635$	−14.0000	−0.555573
$636$	8.00000	0.317221
$637$	−36.0000	−1.42637
$638$	8.00000	0.316723
$639$	12.0000	0.474713
$640$	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	$0.512576\pi$
$642$	8.00000	0.315735
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−1.00000	−0.0392837
$649$	−8.00000	−0.314027
$650$	4.00000	0.156893
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	2.00000	0.0780274
$658$	8.00000	0.311872
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	1.00000	0.0389249
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	20.0000	0.777322
$663$	4.00000	0.155347
$664$	12.0000	0.465690
$665$	−16.0000	−0.620453
$666$	8.00000	0.309994
$667$	0	0
$668$	22.0000	0.851206
$669$	−4.00000	−0.154649
$670$	4.00000	0.154533
$671$	−10.0000	−0.386046
$672$	4.00000	0.154303
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	34.0000	1.30963
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	−16.0000	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$678$	−18.0000	−0.691286
$679$	8.00000	0.307012
$680$	1.00000	0.0383482
$681$	12.0000	0.459841
$682$	−2.00000	−0.0765840
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	4.00000	0.152944
$685$	6.00000	0.229248
$686$	−8.00000	−0.305441
$687$	−10.0000	−0.381524
$688$	0	0
$689$	32.0000	1.21910
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	−12.0000	−0.456172
$693$	4.00000	0.151947
$694$	32.0000	1.21470
$695$	−4.00000	−0.151729
$696$	−8.00000	−0.303239
$697$	−2.00000	−0.0757554
$698$	−8.00000	−0.302804
$699$	2.00000	0.0756469
$700$	4.00000	0.151186
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	−4.00000	−0.150970
$703$	−32.0000	−1.20690
$704$	1.00000	0.0376889
$705$	−2.00000	−0.0753244
$706$	34.0000	1.27961
$707$	8.00000	0.300871
$708$	8.00000	0.300658
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	12.0000	0.450352
$711$	12.0000	0.450035
$712$	6.00000	0.224860
$713$	0	0
$714$	4.00000	0.149696
$715$	4.00000	0.149592
$716$	20.0000	0.747435
$717$	−8.00000	−0.298765
$718$	12.0000	0.447836
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−1.00000	−0.0372678
$721$	−64.0000	−2.38348
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	16.0000	0.594635
$725$	−8.00000	−0.297113
$726$	1.00000	0.0371135
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	16.0000	0.592999
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	0	0
$732$	10.0000	0.369611
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	2.00000	0.0738213
$735$	9.00000	0.331970
$736$	0	0
$737$	4.00000	0.147342
$738$	2.00000	0.0736210
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	8.00000	0.294086
$741$	16.0000	0.587775
$742$	32.0000	1.17476
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	2.00000	0.0733236
$745$	14.0000	0.512920
$746$	20.0000	0.732252
$747$	−12.0000	−0.439057
$748$	1.00000	0.0365636
$749$	32.0000	1.16925
$750$	−1.00000	−0.0365148
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−2.00000	−0.0729325
$753$	−12.0000	−0.437304
$754$	−32.0000	−1.16537
$755$	−2.00000	−0.0727875
$756$	−4.00000	−0.145479
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	4.00000	0.145095
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	14.0000	0.507166
$763$	−72.0000	−2.60658
$764$	−6.00000	−0.217072
$765$	−1.00000	−0.0361551
$766$	−10.0000	−0.361315
$767$	32.0000	1.15545
$768$	−1.00000	−0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	4.00000	0.144150
$771$	2.00000	0.0720282
$772$	−2.00000	−0.0719816
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	−2.00000	−0.0717958
$777$	32.0000	1.14799
$778$	−8.00000	−0.286814
$779$	−8.00000	−0.286630
$780$	−4.00000	−0.143223
$781$	12.0000	0.429394
$782$	0	0
$783$	8.00000	0.285897
$784$	9.00000	0.321429
$785$	−18.0000	−0.642448
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−24.0000	−0.854965
$789$	−4.00000	−0.142404
$790$	12.0000	0.426941
$791$	−72.0000	−2.56003
$792$	−1.00000	−0.0355335
$793$	40.0000	1.42044
$794$	−12.0000	−0.425864
$795$	−8.00000	−0.283731
$796$	10.0000	0.354441
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	16.0000	0.566394
$799$	−2.00000	−0.0707549
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	30.0000	1.05934
$803$	2.00000	0.0705785
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	−18.0000	−0.633630
$808$	−2.00000	−0.0703598
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	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	$0.385792\pi$
$812$	−32.0000	−1.12298
$813$	−14.0000	−0.491001
$814$	8.00000	0.280400
$815$	4.00000	0.140114
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−6.00000	−0.209785
$819$	−16.0000	−0.559085
$820$	2.00000	0.0698430
$821$	28.0000	0.977207	0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000	0.977207	0.488603	+	0.872506i	$0.337507\pi$
$822$	−6.00000	−0.209274
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	16.0000	0.557386
$825$	−1.00000	−0.0348155
$826$	32.0000	1.11342
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	−12.0000	−0.416526
$831$	−18.0000	−0.624413
$832$	−4.00000	−0.138675
$833$	9.00000	0.311832
$834$	4.00000	0.138509
$835$	−22.0000	−0.761341
$836$	4.00000	0.138343
$837$	−2.00000	−0.0691301
$838$	−20.0000	−0.690889
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	−4.00000	−0.138013
$841$	35.0000	1.20690
$842$	6.00000	0.206774
$843$	−10.0000	−0.344418
$844$	−4.00000	−0.137686
$845$	−3.00000	−0.103203
$846$	2.00000	0.0687614
$847$	4.00000	0.137442
$848$	−8.00000	−0.274721
$849$	−12.0000	−0.411839
$850$	−1.00000	−0.0342997
$851$	0	0
$852$	−12.0000	−0.411113
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	40.0000	1.36877
$855$	−4.00000	−0.136797
$856$	−8.00000	−0.273434
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	−4.00000	−0.136558
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	10.0000	0.340601
$863$	34.0000	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$864$	1.00000	0.0340207
$865$	12.0000	0.408012
$866$	18.0000	0.611665
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	12.0000	0.407072
$870$	8.00000	0.271225
$871$	−16.0000	−0.542139
$872$	18.0000	0.609557
$873$	2.00000	0.0676897
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−2.00000	−0.0675737
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	32.0000	1.07995
$879$	14.0000	0.472208
$880$	−1.00000	−0.0337100
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−9.00000	−0.303046
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−4.00000	−0.134535
$885$	−8.00000	−0.268917
$886$	32.0000	1.07506
$887$	10.0000	0.335767	0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000	0.335767	0.167884	+	0.985807i	$0.446307\pi$
$888$	−8.00000	−0.268462
$889$	56.0000	1.87818
$890$	−6.00000	−0.201120
$891$	1.00000	0.0335013
$892$	4.00000	0.133930
$893$	−8.00000	−0.267710
$894$	−14.0000	−0.468230
$895$	−20.0000	−0.668526
$896$	−4.00000	−0.133631
$897$	0	0
$898$	6.00000	0.200223
$899$	−16.0000	−0.533630
$900$	1.00000	0.0333333
$901$	−8.00000	−0.266519
$902$	2.00000	0.0665927
$903$	0	0
$904$	18.0000	0.598671
$905$	−16.0000	−0.531858
$906$	2.00000	0.0664455
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−12.0000	−0.398234
$909$	2.00000	0.0663358
$910$	−16.0000	−0.530395
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−4.00000	−0.132453
$913$	−12.0000	−0.397142
$914$	30.0000	0.992312
$915$	−10.0000	−0.330590
$916$	10.0000	0.330409
$917$	0	0
$918$	1.00000	0.0330049
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	−30.0000	−0.987997
$923$	−48.0000	−1.57994
$924$	−4.00000	−0.131590
$925$	−8.00000	−0.263038
$926$	16.0000	0.525793
$927$	−16.0000	−0.525509
$928$	8.00000	0.262613
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	−2.00000	−0.0655826
$931$	36.0000	1.17985
$932$	−2.00000	−0.0655122
$933$	24.0000	0.785725
$934$	20.0000	0.654420
$935$	−1.00000	−0.0327035
$936$	4.00000	0.130744
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	−16.0000	−0.522419
$939$	14.0000	0.456873
$940$	2.00000	0.0652328
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	−8.00000	−0.260378
$945$	4.00000	0.130120
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−12.0000	−0.389742
$949$	−8.00000	−0.259691
$950$	−4.00000	−0.129777
$951$	6.00000	0.194563
$952$	−4.00000	−0.129641
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	8.00000	0.259010
$955$	6.00000	0.194155
$956$	8.00000	0.258738
$957$	8.00000	0.258603
$958$	−26.0000	−0.840022
$959$	−24.0000	−0.775000
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	−32.0000	−1.03172
$963$	8.00000	0.257796
$964$	−18.0000	−0.579741
$965$	2.00000	0.0643823
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	−1.00000	−0.0321412
$969$	−4.00000	−0.128499
$970$	2.00000	0.0642161
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	−34.0000	−1.08943
$975$	4.00000	0.128103
$976$	−10.0000	−0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	−4.00000	−0.127906
$979$	−6.00000	−0.191761
$980$	−9.00000	−0.287494
$981$	−18.0000	−0.574696
$982$	28.0000	0.893516
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	−2.00000	−0.0637577
$985$	24.0000	0.764704
$986$	8.00000	0.254772
$987$	8.00000	0.254643
$988$	−16.0000	−0.509028
$989$	0	0
$990$	1.00000	0.0317821
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	−2.00000	−0.0635001
$993$	20.0000	0.634681
$994$	−48.0000	−1.52247
$995$	−10.0000	−0.317021
$996$	12.0000	0.380235
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−4.00000	−0.126618
$999$	8.00000	0.253109

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.e.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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