LMFDB - Embedded newform 5610.2.a.k.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -5.00000 q^{13} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +1.00000 q^{22} +7.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} -7.00000 q^{38} -5.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +3.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -7.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -5.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.00000 q^{56} +7.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +5.00000 q^{61} -3.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} +1.00000 q^{66} -7.00000 q^{67} +1.00000 q^{68} +7.00000 q^{69} -3.00000 q^{70} +14.0000 q^{71} -1.00000 q^{72} -12.0000 q^{73} -7.00000 q^{74} +1.00000 q^{75} +7.00000 q^{76} +3.00000 q^{77} +5.00000 q^{78} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +3.00000 q^{83} -3.00000 q^{84} -1.00000 q^{85} +8.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} +15.0000 q^{91} +7.00000 q^{92} +3.00000 q^{93} +6.00000 q^{94} -7.00000 q^{95} -1.00000 q^{96} -5.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$7$	−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
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.00000	0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	5.00000	0.980581
$27$	1.00000	0.192450
$28$	−3.00000	−0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	0.182574
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−7.00000	−1.13555
$39$	−5.00000	−0.800641
$40$	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	$0.549916\pi$
$42$	3.00000	0.462910
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	−7.00000	−1.03209
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	−5.00000	−0.693375
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	3.00000	0.400892
$57$	7.00000	0.927173
$58$	6.00000	0.787839
$59$	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
$61$	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
$65$	5.00000	0.620174
$66$	1.00000	0.123091
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	1.00000	0.121268
$69$	7.00000	0.842701
$70$	−3.00000	−0.358569
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	−1.00000	−0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−7.00000	−0.813733
$75$	1.00000	0.115470
$76$	7.00000	0.802955
$77$	3.00000	0.341882
$78$	5.00000	0.566139
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	−3.00000	−0.327327
$85$	−1.00000	−0.108465
$86$	8.00000	0.862662
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	−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
$91$	15.0000	1.57243
$92$	7.00000	0.729800
$93$	3.00000	0.311086
$94$	6.00000	0.618853
$95$	−7.00000	−0.718185
$96$	−1.00000	−0.102062
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−1.00000	−0.0990148
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	5.00000	0.490290
$105$	3.00000	0.292770
$106$	−14.0000	−1.35980
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	1.00000	0.0962250
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	−1.00000	−0.0953463
$111$	7.00000	0.664411
$112$	−3.00000	−0.283473
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−7.00000	−0.655610
$115$	−7.00000	−0.652753
$116$	−6.00000	−0.557086
$117$	−5.00000	−0.462250
$118$	−4.00000	−0.368230
$119$	−3.00000	−0.275010
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−5.00000	−0.452679
$123$	−2.00000	−0.180334
$124$	3.00000	0.269408
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−5.00000	−0.438529
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	−1.00000	−0.0870388
$133$	−21.0000	−1.82093
$134$	7.00000	0.604708
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−7.00000	−0.595880
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	3.00000	0.253546
$141$	−6.00000	−0.505291
$142$	−14.0000	−1.17485
$143$	5.00000	0.418121
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	12.0000	0.993127
$147$	2.00000	0.164957
$148$	7.00000	0.575396
$149$	23.0000	1.88423	0.942117	−	0.335285i	$-0.108833\pi$
$149$	23.0000	1.88423	0.942117	+	0.335285i	$0.108833\pi$
$150$	−1.00000	−0.0816497
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−7.00000	−0.567775
$153$	1.00000	0.0808452
$154$	−3.00000	−0.241747
$155$	−3.00000	−0.240966
$156$	−5.00000	−0.400320
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	2.00000	0.159111
$159$	14.0000	1.11027
$160$	1.00000	0.0790569
$161$	−21.0000	−1.65503
$162$	−1.00000	−0.0785674
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	−2.00000	−0.156174
$165$	1.00000	0.0778499
$166$	−3.00000	−0.232845
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	3.00000	0.231455
$169$	12.0000	0.923077
$170$	1.00000	0.0766965
$171$	7.00000	0.535303
$172$	−8.00000	−0.609994
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	6.00000	0.454859
$175$	−3.00000	−0.226779
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	18.0000	1.34916
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	−1.00000	−0.0745356
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	−15.0000	−1.11187
$183$	5.00000	0.369611
$184$	−7.00000	−0.516047
$185$	−7.00000	−0.514650
$186$	−3.00000	−0.219971
$187$	−1.00000	−0.0731272
$188$	−6.00000	−0.437595
$189$	−3.00000	−0.218218
$190$	7.00000	0.507833
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	5.00000	0.358979
$195$	5.00000	0.358057
$196$	2.00000	0.142857
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	1.00000	0.0710669
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	−1.00000	−0.0707107
$201$	−7.00000	−0.493742
$202$	6.00000	0.422159
$203$	18.0000	1.26335
$204$	1.00000	0.0700140
$205$	2.00000	0.139686
$206$	−1.00000	−0.0696733
$207$	7.00000	0.486534
$208$	−5.00000	−0.346688
$209$	−7.00000	−0.484200
$210$	−3.00000	−0.207020
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	14.0000	0.961524
$213$	14.0000	0.959264
$214$	18.0000	1.23045
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	−9.00000	−0.610960
$218$	17.0000	1.15139
$219$	−12.0000	−0.810885
$220$	1.00000	0.0674200
$221$	−5.00000	−0.336336
$222$	−7.00000	−0.469809
$223$	−25.0000	−1.67412	−0.837062	−	0.547108i	$-0.815729\pi$
$223$	−25.0000	−1.67412	−0.837062	+	0.547108i	$0.815729\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	7.00000	0.463586
$229$	3.00000	0.198246	0.0991228	−	0.995075i	$-0.468396\pi$
$229$	3.00000	0.198246	0.0991228	+	0.995075i	$0.468396\pi$
$230$	7.00000	0.461566
$231$	3.00000	0.197386
$232$	6.00000	0.393919
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	5.00000	0.326860
$235$	6.00000	0.391397
$236$	4.00000	0.260378
$237$	−2.00000	−0.129914
$238$	3.00000	0.194461
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	−1.00000	−0.0645497
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	5.00000	0.320092
$245$	−2.00000	−0.127775
$246$	2.00000	0.127515
$247$	−35.0000	−2.22700
$248$	−3.00000	−0.190500
$249$	3.00000	0.190117
$250$	1.00000	0.0632456
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	−3.00000	−0.188982
$253$	−7.00000	−0.440086
$254$	8.00000	0.501965
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	8.00000	0.498058
$259$	−21.0000	−1.30488
$260$	5.00000	0.310087
$261$	−6.00000	−0.371391
$262$	−1.00000	−0.0617802
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	1.00000	0.0615457
$265$	−14.0000	−0.860013
$266$	21.0000	1.28759
$267$	−18.0000	−1.10158
$268$	−7.00000	−0.427593
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	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
$273$	15.0000	0.907841
$274$	15.0000	0.906183
$275$	−1.00000	−0.0603023
$276$	7.00000	0.421350
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−18.0000	−1.07957
$279$	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	$0.481000\pi$
$282$	6.00000	0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	14.0000	0.830747
$285$	−7.00000	−0.414644
$286$	−5.00000	−0.295656
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	−5.00000	−0.293105
$292$	−12.0000	−0.702247
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	−2.00000	−0.116642
$295$	−4.00000	−0.232889
$296$	−7.00000	−0.406867
$297$	−1.00000	−0.0580259
$298$	−23.0000	−1.33235
$299$	−35.0000	−2.02410
$300$	1.00000	0.0577350
$301$	24.0000	1.38334
$302$	−5.00000	−0.287718
$303$	−6.00000	−0.344691
$304$	7.00000	0.401478
$305$	−5.00000	−0.286299
$306$	−1.00000	−0.0571662
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	3.00000	0.170941
$309$	1.00000	0.0568880
$310$	3.00000	0.170389
$311$	22.0000	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$311$	22.0000	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$312$	5.00000	0.283069
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	8.00000	0.451466
$315$	3.00000	0.169031
$316$	−2.00000	−0.112509
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−14.0000	−0.785081
$319$	6.00000	0.335936
$320$	−1.00000	−0.0559017
$321$	−18.0000	−1.00466
$322$	21.0000	1.17028
$323$	7.00000	0.389490
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	24.0000	1.32924
$327$	−17.0000	−0.940102
$328$	2.00000	0.110432
$329$	18.0000	0.992372
$330$	−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	$0.685239\pi$
$332$	3.00000	0.164646
$333$	7.00000	0.383598
$334$	−8.00000	−0.437741
$335$	7.00000	0.382451
$336$	−3.00000	−0.163663
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	−1.00000	−0.0542326
$341$	−3.00000	−0.162459
$342$	−7.00000	−0.378517
$343$	15.0000	0.809924
$344$	8.00000	0.431331
$345$	−7.00000	−0.376867
$346$	5.00000	0.268802
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	−6.00000	−0.321634
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	3.00000	0.160357
$351$	−5.00000	−0.266880
$352$	1.00000	0.0533002
$353$	−17.0000	−0.904819	−0.452409	−	0.891810i	$-0.649435\pi$
$353$	−17.0000	−0.904819	−0.452409	+	0.891810i	$0.649435\pi$
$354$	−4.00000	−0.212598
$355$	−14.0000	−0.743043
$356$	−18.0000	−0.953998
$357$	−3.00000	−0.158777
$358$	21.0000	1.10988
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	1.00000	0.0527046
$361$	30.0000	1.57895
$362$	20.0000	1.05118
$363$	1.00000	0.0524864
$364$	15.0000	0.786214
$365$	12.0000	0.628109
$366$	−5.00000	−0.261354
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	7.00000	0.364900
$369$	−2.00000	−0.104116
$370$	7.00000	0.363913
$371$	−42.0000	−2.18053
$372$	3.00000	0.155543
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	1.00000	0.0517088
$375$	−1.00000	−0.0516398
$376$	6.00000	0.309426
$377$	30.0000	1.54508
$378$	3.00000	0.154303
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	−7.00000	−0.359092
$381$	−8.00000	−0.409852
$382$	0	0
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	−1.00000	−0.0510310
$385$	−3.00000	−0.152894
$386$	8.00000	0.407189
$387$	−8.00000	−0.406663
$388$	−5.00000	−0.253837
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	−5.00000	−0.253185
$391$	7.00000	0.354005
$392$	−2.00000	−0.101015
$393$	1.00000	0.0504433
$394$	1.00000	0.0503793
$395$	2.00000	0.100631
$396$	−1.00000	−0.0502519
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	21.0000	1.05263
$399$	−21.0000	−1.05131
$400$	1.00000	0.0500000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	7.00000	0.349128
$403$	−15.0000	−0.747203
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	−18.0000	−0.893325
$407$	−7.00000	−0.346977
$408$	−1.00000	−0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−2.00000	−0.0987730
$411$	−15.0000	−0.739895
$412$	1.00000	0.0492665
$413$	−12.0000	−0.590481
$414$	−7.00000	−0.344031
$415$	−3.00000	−0.147264
$416$	5.00000	0.245145
$417$	18.0000	0.881464
$418$	7.00000	0.342381
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	3.00000	0.146385
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	−10.0000	−0.486792
$423$	−6.00000	−0.291730
$424$	−14.0000	−0.679900
$425$	1.00000	0.0485071
$426$	−14.0000	−0.678302
$427$	−15.0000	−0.725901
$428$	−18.0000	−0.870063
$429$	5.00000	0.241402
$430$	−8.00000	−0.385794
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	9.00000	0.432014
$435$	6.00000	0.287678
$436$	−17.0000	−0.814152
$437$	49.0000	2.34399
$438$	12.0000	0.573382
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	−1.00000	−0.0476731
$441$	2.00000	0.0952381
$442$	5.00000	0.237826
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	7.00000	0.332205
$445$	18.0000	0.853282
$446$	25.0000	1.18378
$447$	23.0000	1.08786
$448$	−3.00000	−0.141737
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	−1.00000	−0.0471405
$451$	2.00000	0.0941763
$452$	−6.00000	−0.282216
$453$	5.00000	0.234920
$454$	−6.00000	−0.281594
$455$	−15.0000	−0.703211
$456$	−7.00000	−0.327805
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−3.00000	−0.140181
$459$	1.00000	0.0466760
$460$	−7.00000	−0.326377
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	−3.00000	−0.139573
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	−6.00000	−0.278543
$465$	−3.00000	−0.139122
$466$	−14.0000	−0.648537
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−5.00000	−0.231125
$469$	21.0000	0.969690
$470$	−6.00000	−0.276759
$471$	−8.00000	−0.368621
$472$	−4.00000	−0.184115
$473$	8.00000	0.367840
$474$	2.00000	0.0918630
$475$	7.00000	0.321182
$476$	−3.00000	−0.137505
$477$	14.0000	0.641016
$478$	−18.0000	−0.823301
$479$	−29.0000	−1.32504	−0.662522	−	0.749043i	$-0.730514\pi$
$479$	−29.0000	−1.32504	−0.662522	+	0.749043i	$0.730514\pi$
$480$	1.00000	0.0456435
$481$	−35.0000	−1.59586
$482$	17.0000	0.774329
$483$	−21.0000	−0.955533
$484$	1.00000	0.0454545
$485$	5.00000	0.227038
$486$	−1.00000	−0.0453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−5.00000	−0.226339
$489$	−24.0000	−1.08532
$490$	2.00000	0.0903508
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	−2.00000	−0.0901670
$493$	−6.00000	−0.270226
$494$	35.0000	1.57472
$495$	1.00000	0.0449467
$496$	3.00000	0.134704
$497$	−42.0000	−1.88396
$498$	−3.00000	−0.134433
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	−23.0000	−1.02654
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	3.00000	0.133631
$505$	6.00000	0.266996
$506$	7.00000	0.311188
$507$	12.0000	0.532939
$508$	−8.00000	−0.354943
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	1.00000	0.0442807
$511$	36.0000	1.59255
$512$	−1.00000	−0.0441942
$513$	7.00000	0.309058
$514$	6.00000	0.264649
$515$	−1.00000	−0.0440653
$516$	−8.00000	−0.352180
$517$	6.00000	0.263880
$518$	21.0000	0.922687
$519$	−5.00000	−0.219476
$520$	−5.00000	−0.219265
$521$	29.0000	1.27051	0.635257	−	0.772301i	$-0.280894\pi$
$521$	29.0000	1.27051	0.635257	+	0.772301i	$0.280894\pi$
$522$	6.00000	0.262613
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	1.00000	0.0436852
$525$	−3.00000	−0.130931
$526$	9.00000	0.392419
$527$	3.00000	0.130682
$528$	−1.00000	−0.0435194
$529$	26.0000	1.13043
$530$	14.0000	0.608121
$531$	4.00000	0.173585
$532$	−21.0000	−0.910465
$533$	10.0000	0.433148
$534$	18.0000	0.778936
$535$	18.0000	0.778208
$536$	7.00000	0.302354
$537$	−21.0000	−0.906217
$538$	9.00000	0.388018
$539$	−2.00000	−0.0861461
$540$	−1.00000	−0.0430331
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	20.0000	0.859074
$543$	−20.0000	−0.858282
$544$	−1.00000	−0.0428746
$545$	17.0000	0.728200
$546$	−15.0000	−0.641941
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	−15.0000	−0.640768
$549$	5.00000	0.213395
$550$	1.00000	0.0426401
$551$	−42.0000	−1.78926
$552$	−7.00000	−0.297940
$553$	6.00000	0.255146
$554$	28.0000	1.18961
$555$	−7.00000	−0.297133
$556$	18.0000	0.763370
$557$	36.0000	1.52537	0.762684	−	0.646771i	$-0.223881\pi$
$557$	36.0000	1.52537	0.762684	+	0.646771i	$0.223881\pi$
$558$	−3.00000	−0.127000
$559$	40.0000	1.69182
$560$	3.00000	0.126773
$561$	−1.00000	−0.0422200
$562$	−2.00000	−0.0843649
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	−6.00000	−0.252646
$565$	6.00000	0.252422
$566$	−4.00000	−0.168133
$567$	−3.00000	−0.125988
$568$	−14.0000	−0.587427
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	7.00000	0.293198
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	5.00000	0.209061
$573$	0	0
$574$	−6.00000	−0.250435
$575$	7.00000	0.291920
$576$	1.00000	0.0416667
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−1.00000	−0.0415945
$579$	−8.00000	−0.332469
$580$	6.00000	0.249136
$581$	−9.00000	−0.373383
$582$	5.00000	0.207257
$583$	−14.0000	−0.579821
$584$	12.0000	0.496564
$585$	5.00000	0.206725
$586$	−12.0000	−0.495715
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	2.00000	0.0824786
$589$	21.0000	0.865290
$590$	4.00000	0.164677
$591$	−1.00000	−0.0411345
$592$	7.00000	0.287698
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	1.00000	0.0410305
$595$	3.00000	0.122988
$596$	23.0000	0.942117
$597$	−21.0000	−0.859473
$598$	35.0000	1.43126
$599$	17.0000	0.694601	0.347301	−	0.937754i	$-0.387098\pi$
$599$	17.0000	0.694601	0.347301	+	0.937754i	$0.387098\pi$
$600$	−1.00000	−0.0408248
$601$	23.0000	0.938190	0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000	0.938190	0.469095	+	0.883148i	$0.344580\pi$
$602$	−24.0000	−0.978167
$603$	−7.00000	−0.285062
$604$	5.00000	0.203447
$605$	−1.00000	−0.0406558
$606$	6.00000	0.243733
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−7.00000	−0.283887
$609$	18.0000	0.729397
$610$	5.00000	0.202444
$611$	30.0000	1.21367
$612$	1.00000	0.0404226
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	20.0000	0.807134
$615$	2.00000	0.0806478
$616$	−3.00000	−0.120873
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	−1.00000	−0.0402259
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	−3.00000	−0.120483
$621$	7.00000	0.280900
$622$	−22.0000	−0.882120
$623$	54.0000	2.16346
$624$	−5.00000	−0.200160
$625$	1.00000	0.0400000
$626$	−21.0000	−0.839329
$627$	−7.00000	−0.279553
$628$	−8.00000	−0.319235
$629$	7.00000	0.279108
$630$	−3.00000	−0.119523
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	2.00000	0.0795557
$633$	10.0000	0.397464
$634$	−18.0000	−0.714871
$635$	8.00000	0.317470
$636$	14.0000	0.555136
$637$	−10.0000	−0.396214
$638$	−6.00000	−0.237542
$639$	14.0000	0.553831
$640$	1.00000	0.0395285
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	18.0000	0.710403
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−21.0000	−0.827516
$645$	8.00000	0.315000
$646$	−7.00000	−0.275411
$647$	10.0000	0.393141	0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000	0.393141	0.196570	+	0.980490i	$0.437020\pi$
$648$	−1.00000	−0.0392837
$649$	−4.00000	−0.157014
$650$	5.00000	0.196116
$651$	−9.00000	−0.352738
$652$	−24.0000	−0.939913
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	17.0000	0.664753
$655$	−1.00000	−0.0390732
$656$	−2.00000	−0.0780869
$657$	−12.0000	−0.468165
$658$	−18.0000	−0.701713
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	1.00000	0.0389249
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	20.0000	0.777322
$663$	−5.00000	−0.194184
$664$	−3.00000	−0.116423
$665$	21.0000	0.814345
$666$	−7.00000	−0.271244
$667$	−42.0000	−1.62625
$668$	8.00000	0.309529
$669$	−25.0000	−0.966556
$670$	−7.00000	−0.270434
$671$	−5.00000	−0.193023
$672$	3.00000	0.115728
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	28.0000	1.07852
$675$	1.00000	0.0384900
$676$	12.0000	0.461538
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	6.00000	0.230429
$679$	15.0000	0.575647
$680$	1.00000	0.0383482
$681$	6.00000	0.229920
$682$	3.00000	0.114876
$683$	−37.0000	−1.41577	−0.707883	−	0.706330i	$-0.750350\pi$
$683$	−37.0000	−1.41577	−0.707883	+	0.706330i	$0.750350\pi$
$684$	7.00000	0.267652
$685$	15.0000	0.573121
$686$	−15.0000	−0.572703
$687$	3.00000	0.114457
$688$	−8.00000	−0.304997
$689$	−70.0000	−2.66679
$690$	7.00000	0.266485
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	−5.00000	−0.190071
$693$	3.00000	0.113961
$694$	14.0000	0.531433
$695$	−18.0000	−0.682779
$696$	6.00000	0.227429
$697$	−2.00000	−0.0757554
$698$	−18.0000	−0.681310
$699$	14.0000	0.529529
$700$	−3.00000	−0.113389
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	5.00000	0.188713
$703$	49.0000	1.84807
$704$	−1.00000	−0.0376889
$705$	6.00000	0.225973
$706$	17.0000	0.639803
$707$	18.0000	0.676960
$708$	4.00000	0.150329
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	14.0000	0.525411
$711$	−2.00000	−0.0750059
$712$	18.0000	0.674579
$713$	21.0000	0.786456
$714$	3.00000	0.112272
$715$	−5.00000	−0.186989
$716$	−21.0000	−0.784807
$717$	18.0000	0.672222
$718$	−10.0000	−0.373197
$719$	−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
$721$	−3.00000	−0.111726
$722$	−30.0000	−1.11648
$723$	−17.0000	−0.632237
$724$	−20.0000	−0.743294
$725$	−6.00000	−0.222834
$726$	−1.00000	−0.0371135
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−15.0000	−0.555937
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	−8.00000	−0.295891
$732$	5.00000	0.184805
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	−32.0000	−1.18114
$735$	−2.00000	−0.0737711
$736$	−7.00000	−0.258023
$737$	7.00000	0.257848
$738$	2.00000	0.0736210
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	−7.00000	−0.257325
$741$	−35.0000	−1.28576
$742$	42.0000	1.54187
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−3.00000	−0.109985
$745$	−23.0000	−0.842655
$746$	−18.0000	−0.659027
$747$	3.00000	0.109764
$748$	−1.00000	−0.0365636
$749$	54.0000	1.97312
$750$	1.00000	0.0365148
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−6.00000	−0.218797
$753$	23.0000	0.838167
$754$	−30.0000	−1.09254
$755$	−5.00000	−0.181969
$756$	−3.00000	−0.109109
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	21.0000	0.762754
$759$	−7.00000	−0.254084
$760$	7.00000	0.253917
$761$	−7.00000	−0.253750	−0.126875	−	0.991919i	$-0.540495\pi$
$761$	−7.00000	−0.253750	−0.126875	+	0.991919i	$0.540495\pi$
$762$	8.00000	0.289809
$763$	51.0000	1.84632
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	−26.0000	−0.939418
$767$	−20.0000	−0.722158
$768$	1.00000	0.0360844
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	3.00000	0.108112
$771$	−6.00000	−0.216085
$772$	−8.00000	−0.287926
$773$	−37.0000	−1.33080	−0.665399	−	0.746488i	$-0.731738\pi$
$773$	−37.0000	−1.33080	−0.665399	+	0.746488i	$0.731738\pi$
$774$	8.00000	0.287554
$775$	3.00000	0.107763
$776$	5.00000	0.179490
$777$	−21.0000	−0.753371
$778$	−16.0000	−0.573628
$779$	−14.0000	−0.501602
$780$	5.00000	0.179029
$781$	−14.0000	−0.500959
$782$	−7.00000	−0.250319
$783$	−6.00000	−0.214423
$784$	2.00000	0.0714286
$785$	8.00000	0.285532
$786$	−1.00000	−0.0356688
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	−1.00000	−0.0356235
$789$	−9.00000	−0.320408
$790$	−2.00000	−0.0711568
$791$	18.0000	0.640006
$792$	1.00000	0.0355335
$793$	−25.0000	−0.887776
$794$	−22.0000	−0.780751
$795$	−14.0000	−0.496529
$796$	−21.0000	−0.744325
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	21.0000	0.743392
$799$	−6.00000	−0.212265
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	27.0000	0.953403
$803$	12.0000	0.423471
$804$	−7.00000	−0.246871
$805$	21.0000	0.740153
$806$	15.0000	0.528352
$807$	−9.00000	−0.316815
$808$	6.00000	0.211079
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	1.00000	0.0351364
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	18.0000	0.631676
$813$	−20.0000	−0.701431
$814$	7.00000	0.245350
$815$	24.0000	0.840683
$816$	1.00000	0.0350070
$817$	−56.0000	−1.95919
$818$	−14.0000	−0.489499
$819$	15.0000	0.524142
$820$	2.00000	0.0698430
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	15.0000	0.523185
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−1.00000	−0.0348367
$825$	−1.00000	−0.0348155
$826$	12.0000	0.417533
$827$	−50.0000	−1.73867	−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000	−1.73867	−0.869335	+	0.494223i	$0.835453\pi$
$828$	7.00000	0.243267
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	3.00000	0.104132
$831$	−28.0000	−0.971309
$832$	−5.00000	−0.173344
$833$	2.00000	0.0692959
$834$	−18.0000	−0.623289
$835$	−8.00000	−0.276851
$836$	−7.00000	−0.242100
$837$	3.00000	0.103695
$838$	−28.0000	−0.967244
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	−3.00000	−0.103510
$841$	7.00000	0.241379
$842$	5.00000	0.172311
$843$	2.00000	0.0688837
$844$	10.0000	0.344214
$845$	−12.0000	−0.412813
$846$	6.00000	0.206284
$847$	−3.00000	−0.103081
$848$	14.0000	0.480762
$849$	4.00000	0.137280
$850$	−1.00000	−0.0342997
$851$	49.0000	1.67970
$852$	14.0000	0.479632
$853$	24.0000	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$853$	24.0000	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$854$	15.0000	0.513289
$855$	−7.00000	−0.239395
$856$	18.0000	0.615227
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	−5.00000	−0.170697
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	8.00000	0.272798
$861$	6.00000	0.204479
$862$	40.0000	1.36241
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	−1.00000	−0.0340207
$865$	5.00000	0.170005
$866$	2.00000	0.0679628
$867$	1.00000	0.0339618
$868$	−9.00000	−0.305480
$869$	2.00000	0.0678454
$870$	−6.00000	−0.203419
$871$	35.0000	1.18593
$872$	17.0000	0.575693
$873$	−5.00000	−0.169224
$874$	−49.0000	−1.65745
$875$	3.00000	0.101419
$876$	−12.0000	−0.405442
$877$	16.0000	0.540282	0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000	0.540282	0.270141	+	0.962821i	$0.412930\pi$
$878$	−10.0000	−0.337484
$879$	12.0000	0.404750
$880$	1.00000	0.0337100
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−2.00000	−0.0673435
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−5.00000	−0.168168
$885$	−4.00000	−0.134459
$886$	6.00000	0.201574
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−7.00000	−0.234905
$889$	24.0000	0.804934
$890$	−18.0000	−0.603361
$891$	−1.00000	−0.0335013
$892$	−25.0000	−0.837062
$893$	−42.0000	−1.40548
$894$	−23.0000	−0.769235
$895$	21.0000	0.701953
$896$	3.00000	0.100223
$897$	−35.0000	−1.16862
$898$	−39.0000	−1.30145
$899$	−18.0000	−0.600334
$900$	1.00000	0.0333333
$901$	14.0000	0.466408
$902$	−2.00000	−0.0665927
$903$	24.0000	0.798670
$904$	6.00000	0.199557
$905$	20.0000	0.664822
$906$	−5.00000	−0.166114
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	6.00000	0.199117
$909$	−6.00000	−0.199007
$910$	15.0000	0.497245
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	7.00000	0.231793
$913$	−3.00000	−0.0992855
$914$	11.0000	0.363848
$915$	−5.00000	−0.165295
$916$	3.00000	0.0991228
$917$	−3.00000	−0.0990687
$918$	−1.00000	−0.0330049
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	7.00000	0.230783
$921$	−20.0000	−0.659022
$922$	14.0000	0.461065
$923$	−70.0000	−2.30408
$924$	3.00000	0.0986928
$925$	7.00000	0.230159
$926$	5.00000	0.164310
$927$	1.00000	0.0328443
$928$	6.00000	0.196960
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	3.00000	0.0983739
$931$	14.0000	0.458831
$932$	14.0000	0.458585
$933$	22.0000	0.720248
$934$	8.00000	0.261768
$935$	1.00000	0.0327035
$936$	5.00000	0.163430
$937$	−19.0000	−0.620703	−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000	−0.620703	−0.310351	+	0.950622i	$0.600447\pi$
$938$	−21.0000	−0.685674
$939$	21.0000	0.685309
$940$	6.00000	0.195698
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	8.00000	0.260654
$943$	−14.0000	−0.455903
$944$	4.00000	0.130189
$945$	3.00000	0.0975900
$946$	−8.00000	−0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−2.00000	−0.0649570
$949$	60.0000	1.94768
$950$	−7.00000	−0.227110
$951$	18.0000	0.583690
$952$	3.00000	0.0972306
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	18.0000	0.582162
$957$	6.00000	0.193952
$958$	29.0000	0.936947
$959$	45.0000	1.45313
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	35.0000	1.12845
$963$	−18.0000	−0.580042
$964$	−17.0000	−0.547533
$965$	8.00000	0.257529
$966$	21.0000	0.675664
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−1.00000	−0.0321412
$969$	7.00000	0.224872
$970$	−5.00000	−0.160540
$971$	41.0000	1.31575	0.657876	−	0.753126i	$-0.271455\pi$
$971$	41.0000	1.31575	0.657876	+	0.753126i	$0.271455\pi$
$972$	1.00000	0.0320750
$973$	−54.0000	−1.73116
$974$	12.0000	0.384505
$975$	−5.00000	−0.160128
$976$	5.00000	0.160046
$977$	−53.0000	−1.69562	−0.847810	−	0.530300i	$-0.822079\pi$
$977$	−53.0000	−1.69562	−0.847810	+	0.530300i	$0.822079\pi$
$978$	24.0000	0.767435
$979$	18.0000	0.575282
$980$	−2.00000	−0.0638877
$981$	−17.0000	−0.542768
$982$	26.0000	0.829693
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	2.00000	0.0637577
$985$	1.00000	0.0318626
$986$	6.00000	0.191079
$987$	18.0000	0.572946
$988$	−35.0000	−1.11350
$989$	−56.0000	−1.78070
$990$	−1.00000	−0.0317821
$991$	41.0000	1.30241	0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000	1.30241	0.651204	+	0.758903i	$0.274264\pi$
$992$	−3.00000	−0.0952501
$993$	−20.0000	−0.634681
$994$	42.0000	1.33216
$995$	21.0000	0.665745
$996$	3.00000	0.0950586
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	20.0000	0.633089
$999$	7.00000	0.221470

Twists

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

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

Downloads

Learn more