LMFDB - Embedded newform 1110.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.86339462436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -5.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} -5.00000 q^{21} -5.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -5.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -5.00000 q^{34} +5.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -3.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -5.00000 q^{42} +4.00000 q^{43} -5.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} +1.00000 q^{48} +18.0000 q^{49} +1.00000 q^{50} -5.00000 q^{51} -1.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} +5.00000 q^{55} -5.00000 q^{56} -3.00000 q^{57} +6.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} -6.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -5.00000 q^{66} -14.0000 q^{67} -5.00000 q^{68} +3.00000 q^{69} +5.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -9.00000 q^{73} -1.00000 q^{74} +1.00000 q^{75} -3.00000 q^{76} +25.0000 q^{77} -1.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +5.00000 q^{83} -5.00000 q^{84} +5.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -5.00000 q^{88} +13.0000 q^{89} -1.00000 q^{90} +5.00000 q^{91} +3.00000 q^{92} -6.00000 q^{93} +3.00000 q^{95} +1.00000 q^{96} -4.00000 q^{97} +18.0000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−5.00000	−1.33631
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−1.00000	−0.223607
$21$	−5.00000	−1.09109
$22$	−5.00000	−1.06600
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−5.00000	−0.944911
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−1.00000	−0.182574
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−5.00000	−0.857493
$35$	5.00000	0.845154
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−3.00000	−0.486664
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−5.00000	−0.771517
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−5.00000	−0.753778
$45$	−1.00000	−0.149071
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	18.0000	2.57143
$50$	1.00000	0.141421
$51$	−5.00000	−0.700140
$52$	−1.00000	−0.138675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	5.00000	0.674200
$56$	−5.00000	−0.668153
$57$	−3.00000	−0.397360
$58$	6.00000	0.787839
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.00000	−0.762001
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−5.00000	−0.615457
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−5.00000	−0.606339
$69$	3.00000	0.361158
$70$	5.00000	0.597614
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	−3.00000	−0.344124
$77$	25.0000	2.84901
$78$	−1.00000	−0.113228
$79$	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
$82$	0	0
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	−5.00000	−0.545545
$85$	5.00000	0.542326
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−5.00000	−0.533002
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	−1.00000	−0.105409
$91$	5.00000	0.524142
$92$	3.00000	0.312772
$93$	−6.00000	−0.622171
$94$	0	0
$95$	3.00000	0.307794
$96$	1.00000	0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	18.0000	1.81827
$99$	−5.00000	−0.502519
$100$	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	$0.596490\pi$
$102$	−5.00000	−0.495074
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−1.00000	−0.0980581
$105$	5.00000	0.487950
$106$	−3.00000	−0.291386
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	1.00000	0.0962250
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	5.00000	0.476731
$111$	−1.00000	−0.0949158
$112$	−5.00000	−0.472456
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−3.00000	−0.280976
$115$	−3.00000	−0.279751
$116$	6.00000	0.557086
$117$	−1.00000	−0.0924500
$118$	−10.0000	−0.920575
$119$	25.0000	2.29175
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	10.0000	0.905357
$123$	0	0
$124$	−6.00000	−0.538816
$125$	−1.00000	−0.0894427
$126$	−5.00000	−0.445435
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	1.00000	0.0877058
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	−5.00000	−0.435194
$133$	15.0000	1.30066
$134$	−14.0000	−1.20942
$135$	−1.00000	−0.0860663
$136$	−5.00000	−0.428746
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	3.00000	0.255377
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	5.00000	0.422577
$141$	0	0
$142$	6.00000	0.503509
$143$	5.00000	0.418121
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−9.00000	−0.744845
$147$	18.0000	1.48461
$148$	−1.00000	−0.0821995
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	1.00000	0.0816497
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	−3.00000	−0.243332
$153$	−5.00000	−0.404226
$154$	25.0000	2.01456
$155$	6.00000	0.481932
$156$	−1.00000	−0.0800641
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	−1.00000	−0.0790569
$161$	−15.0000	−1.18217
$162$	1.00000	0.0785674
$163$	−23.0000	−1.80150	−0.900750	−	0.434339i	$-0.856982\pi$
$163$	−23.0000	−1.80150	−0.900750	+	0.434339i	$0.856982\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	5.00000	0.388075
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	−5.00000	−0.385758
$169$	−12.0000	−0.923077
$170$	5.00000	0.383482
$171$	−3.00000	−0.229416
$172$	4.00000	0.304997
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	6.00000	0.454859
$175$	−5.00000	−0.377964
$176$	−5.00000	−0.376889
$177$	−10.0000	−0.751646
$178$	13.0000	0.974391
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	−1.00000	−0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	5.00000	0.370625
$183$	10.0000	0.739221
$184$	3.00000	0.221163
$185$	1.00000	0.0735215
$186$	−6.00000	−0.439941
$187$	25.0000	1.82818
$188$	0	0
$189$	−5.00000	−0.363696
$190$	3.00000	0.217643
$191$	−7.00000	−0.506502	−0.253251	−	0.967401i	$-0.581500\pi$
$191$	−7.00000	−0.506502	−0.253251	+	0.967401i	$0.581500\pi$
$192$	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	$0.522932\pi$
$194$	−4.00000	−0.287183
$195$	1.00000	0.0716115
$196$	18.0000	1.28571
$197$	23.0000	1.63868	0.819341	−	0.573306i	$-0.194340\pi$
$197$	23.0000	1.63868	0.819341	+	0.573306i	$0.194340\pi$
$198$	−5.00000	−0.355335
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000	0.0707107
$201$	−14.0000	−0.987484
$202$	−6.00000	−0.422159
$203$	−30.0000	−2.10559
$204$	−5.00000	−0.350070
$205$	0	0
$206$	−16.0000	−1.11477
$207$	3.00000	0.208514
$208$	−1.00000	−0.0693375
$209$	15.0000	1.03757
$210$	5.00000	0.345033
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	−3.00000	−0.206041
$213$	6.00000	0.411113
$214$	15.0000	1.02538
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	30.0000	2.03653
$218$	−5.00000	−0.338643
$219$	−9.00000	−0.608164
$220$	5.00000	0.337100
$221$	5.00000	0.336336
$222$	−1.00000	−0.0671156
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−5.00000	−0.334077
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−3.00000	−0.198680
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−3.00000	−0.197814
$231$	25.0000	1.64488
$232$	6.00000	0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	25.0000	1.62051
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−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	$0.696846\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	−18.0000	−1.14998
$246$	0	0
$247$	3.00000	0.190885
$248$	−6.00000	−0.381000
$249$	5.00000	0.316862
$250$	−1.00000	−0.0632456
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	−5.00000	−0.314970
$253$	−15.0000	−0.943042
$254$	−3.00000	−0.188237
$255$	5.00000	0.313112
$256$	1.00000	0.0625000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	4.00000	0.249029
$259$	5.00000	0.310685
$260$	1.00000	0.0620174
$261$	6.00000	0.371391
$262$	−20.0000	−1.23560
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−5.00000	−0.307729
$265$	3.00000	0.184289
$266$	15.0000	0.919709
$267$	13.0000	0.795587
$268$	−14.0000	−0.855186
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−1.00000	−0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−5.00000	−0.303170
$273$	5.00000	0.302614
$274$	−2.00000	−0.120824
$275$	−5.00000	−0.301511
$276$	3.00000	0.180579
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	2.00000	0.119952
$279$	−6.00000	−0.359211
$280$	5.00000	0.298807
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	6.00000	0.356034
$285$	3.00000	0.177705
$286$	5.00000	0.295656
$287$	0	0
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	−6.00000	−0.352332
$291$	−4.00000	−0.234484
$292$	−9.00000	−0.526685
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	18.0000	1.04978
$295$	10.0000	0.582223
$296$	−1.00000	−0.0581238
$297$	−5.00000	−0.290129
$298$	2.00000	0.115857
$299$	−3.00000	−0.173494
$300$	1.00000	0.0577350
$301$	−20.0000	−1.15278
$302$	15.0000	0.863153
$303$	−6.00000	−0.344691
$304$	−3.00000	−0.172062
$305$	−10.0000	−0.572598
$306$	−5.00000	−0.285831
$307$	−30.0000	−1.71219	−0.856095	−	0.516818i	$-0.827116\pi$
$307$	−30.0000	−1.71219	−0.856095	+	0.516818i	$0.827116\pi$
$308$	25.0000	1.42451
$309$	−16.0000	−0.910208
$310$	6.00000	0.340777
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−1.00000	−0.0566139
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	−20.0000	−1.12867
$315$	5.00000	0.281718
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−3.00000	−0.168232
$319$	−30.0000	−1.67968
$320$	−1.00000	−0.0559017
$321$	15.0000	0.837218
$322$	−15.0000	−0.835917
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−23.0000	−1.27385
$327$	−5.00000	−0.276501
$328$	0	0
$329$	0	0
$330$	5.00000	0.275241
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	5.00000	0.274411
$333$	−1.00000	−0.0547997
$334$	5.00000	0.273588
$335$	14.0000	0.764902
$336$	−5.00000	−0.272772
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	−12.0000	−0.652714
$339$	−10.0000	−0.543125
$340$	5.00000	0.271163
$341$	30.0000	1.62459
$342$	−3.00000	−0.162221
$343$	−55.0000	−2.96972
$344$	4.00000	0.215666
$345$	−3.00000	−0.161515
$346$	11.0000	0.591364
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	6.00000	0.321634
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	−5.00000	−0.267261
$351$	−1.00000	−0.0533761
$352$	−5.00000	−0.266501
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	−10.0000	−0.531494
$355$	−6.00000	−0.318447
$356$	13.0000	0.688999
$357$	25.0000	1.32314
$358$	8.00000	0.422813
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	−1.00000	−0.0527046
$361$	−10.0000	−0.526316
$362$	−10.0000	−0.525588
$363$	14.0000	0.734809
$364$	5.00000	0.262071
$365$	9.00000	0.471082
$366$	10.0000	0.522708
$367$	−5.00000	−0.260998	−0.130499	−	0.991448i	$-0.541658\pi$
$367$	−5.00000	−0.260998	−0.130499	+	0.991448i	$0.541658\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	1.00000	0.0519875
$371$	15.0000	0.778761
$372$	−6.00000	−0.311086
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	25.0000	1.29272
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−6.00000	−0.309016
$378$	−5.00000	−0.257172
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	3.00000	0.153897
$381$	−3.00000	−0.153695
$382$	−7.00000	−0.358151
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	1.00000	0.0510310
$385$	−25.0000	−1.27412
$386$	−2.00000	−0.101797
$387$	4.00000	0.203331
$388$	−4.00000	−0.203069
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	1.00000	0.0506370
$391$	−15.0000	−0.758583
$392$	18.0000	0.909137
$393$	−20.0000	−1.00887
$394$	23.0000	1.15872
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−36.0000	−1.80679	−0.903394	−	0.428811i	$-0.858933\pi$
$397$	−36.0000	−1.80679	−0.903394	+	0.428811i	$0.858933\pi$
$398$	−24.0000	−1.20301
$399$	15.0000	0.750939
$400$	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	$0.735493\pi$
$402$	−14.0000	−0.698257
$403$	6.00000	0.298881
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	−30.0000	−1.48888
$407$	5.00000	0.247841
$408$	−5.00000	−0.247537
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−16.0000	−0.788263
$413$	50.0000	2.46034
$414$	3.00000	0.147442
$415$	−5.00000	−0.245440
$416$	−1.00000	−0.0490290
$417$	2.00000	0.0979404
$418$	15.0000	0.733674
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	5.00000	0.243975
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−2.00000	−0.0973585
$423$	0	0
$424$	−3.00000	−0.145693
$425$	−5.00000	−0.242536
$426$	6.00000	0.290701
$427$	−50.0000	−2.41967
$428$	15.0000	0.725052
$429$	5.00000	0.241402
$430$	−4.00000	−0.192897
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	1.00000	0.0481125
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	30.0000	1.44005
$435$	−6.00000	−0.287678
$436$	−5.00000	−0.239457
$437$	−9.00000	−0.430528
$438$	−9.00000	−0.430037
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	5.00000	0.238366
$441$	18.0000	0.857143
$442$	5.00000	0.237826
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−1.00000	−0.0474579
$445$	−13.0000	−0.616259
$446$	−8.00000	−0.378811
$447$	2.00000	0.0945968
$448$	−5.00000	−0.236228
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−10.0000	−0.470360
$453$	15.0000	0.704761
$454$	8.00000	0.375459
$455$	−5.00000	−0.234404
$456$	−3.00000	−0.140488
$457$	4.00000	0.187112	0.0935561	−	0.995614i	$-0.470177\pi$
$457$	4.00000	0.187112	0.0935561	+	0.995614i	$0.470177\pi$
$458$	6.00000	0.280362
$459$	−5.00000	−0.233380
$460$	−3.00000	−0.139876
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	25.0000	1.16311
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	6.00000	0.278543
$465$	6.00000	0.278243
$466$	−26.0000	−1.20443
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−1.00000	−0.0462250
$469$	70.0000	3.23230
$470$	0	0
$471$	−20.0000	−0.921551
$472$	−10.0000	−0.460287
$473$	−20.0000	−0.919601
$474$	0	0
$475$	−3.00000	−0.137649
$476$	25.0000	1.14587
$477$	−3.00000	−0.137361
$478$	12.0000	0.548867
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	−1.00000	−0.0456435
$481$	1.00000	0.0455961
$482$	−18.0000	−0.819878
$483$	−15.0000	−0.682524
$484$	14.0000	0.636364
$485$	4.00000	0.181631
$486$	1.00000	0.0453609
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	10.0000	0.452679
$489$	−23.0000	−1.04010
$490$	−18.0000	−0.813157
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−30.0000	−1.35113
$494$	3.00000	0.134976
$495$	5.00000	0.224733
$496$	−6.00000	−0.269408
$497$	−30.0000	−1.34568
$498$	5.00000	0.224055
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	−1.00000	−0.0447214
$501$	5.00000	0.223384
$502$	30.0000	1.33897
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	−5.00000	−0.222718
$505$	6.00000	0.266996
$506$	−15.0000	−0.666831
$507$	−12.0000	−0.532939
$508$	−3.00000	−0.133103
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	5.00000	0.221404
$511$	45.0000	1.99068
$512$	1.00000	0.0441942
$513$	−3.00000	−0.132453
$514$	15.0000	0.661622
$515$	16.0000	0.705044
$516$	4.00000	0.176090
$517$	0	0
$518$	5.00000	0.219687
$519$	11.0000	0.482846
$520$	1.00000	0.0438529
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	6.00000	0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−20.0000	−0.873704
$525$	−5.00000	−0.218218
$526$	12.0000	0.523225
$527$	30.0000	1.30682
$528$	−5.00000	−0.217597
$529$	−14.0000	−0.608696
$530$	3.00000	0.130312
$531$	−10.0000	−0.433963
$532$	15.0000	0.650332
$533$	0	0
$534$	13.0000	0.562565
$535$	−15.0000	−0.648507
$536$	−14.0000	−0.604708
$537$	8.00000	0.345225
$538$	9.00000	0.388018
$539$	−90.0000	−3.87657
$540$	−1.00000	−0.0430331
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	20.0000	0.859074
$543$	−10.0000	−0.429141
$544$	−5.00000	−0.214373
$545$	5.00000	0.214176
$546$	5.00000	0.213980
$547$	−43.0000	−1.83855	−0.919274	−	0.393619i	$-0.871223\pi$
$547$	−43.0000	−1.83855	−0.919274	+	0.393619i	$0.871223\pi$
$548$	−2.00000	−0.0854358
$549$	10.0000	0.426790
$550$	−5.00000	−0.213201
$551$	−18.0000	−0.766826
$552$	3.00000	0.127688
$553$	0	0
$554$	25.0000	1.06215
$555$	1.00000	0.0424476
$556$	2.00000	0.0848189
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	−6.00000	−0.254000
$559$	−4.00000	−0.169182
$560$	5.00000	0.211289
$561$	25.0000	1.05550
$562$	1.00000	0.0421825
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	5.00000	0.210166
$567$	−5.00000	−0.209980
$568$	6.00000	0.251754
$569$	−13.0000	−0.544988	−0.272494	−	0.962157i	$-0.587849\pi$
$569$	−13.0000	−0.544988	−0.272494	+	0.962157i	$0.587849\pi$
$570$	3.00000	0.125656
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	5.00000	0.209061
$573$	−7.00000	−0.292429
$574$	0	0
$575$	3.00000	0.125109
$576$	1.00000	0.0416667
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	8.00000	0.332756
$579$	−2.00000	−0.0831172
$580$	−6.00000	−0.249136
$581$	−25.0000	−1.03717
$582$	−4.00000	−0.165805
$583$	15.0000	0.621237
$584$	−9.00000	−0.372423
$585$	1.00000	0.0413449
$586$	−11.0000	−0.454406
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	18.0000	0.742307
$589$	18.0000	0.741677
$590$	10.0000	0.411693
$591$	23.0000	0.946094
$592$	−1.00000	−0.0410997
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	−5.00000	−0.205152
$595$	−25.0000	−1.02490
$596$	2.00000	0.0819232
$597$	−24.0000	−0.982255
$598$	−3.00000	−0.122679
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	1.00000	0.0408248
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	−20.0000	−0.815139
$603$	−14.0000	−0.570124
$604$	15.0000	0.610341
$605$	−14.0000	−0.569181
$606$	−6.00000	−0.243733
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	−3.00000	−0.121666
$609$	−30.0000	−1.21566
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−5.00000	−0.202113
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−30.0000	−1.21070
$615$	0	0
$616$	25.0000	1.00728
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	−16.0000	−0.643614
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	6.00000	0.240966
$621$	3.00000	0.120386
$622$	8.00000	0.320771
$623$	−65.0000	−2.60417
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	24.0000	0.959233
$627$	15.0000	0.599042
$628$	−20.0000	−0.798087
$629$	5.00000	0.199363
$630$	5.00000	0.199205
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	−2.00000	−0.0794929
$634$	−10.0000	−0.397151
$635$	3.00000	0.119051
$636$	−3.00000	−0.118958
$637$	−18.0000	−0.713186
$638$	−30.0000	−1.18771
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	15.0000	0.592003
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	−15.0000	−0.591083
$645$	−4.00000	−0.157500
$646$	15.0000	0.590167
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	1.00000	0.0392837
$649$	50.0000	1.96267
$650$	−1.00000	−0.0392232
$651$	30.0000	1.17579
$652$	−23.0000	−0.900750
$653$	−48.0000	−1.87839	−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000	−1.87839	−0.939193	+	0.343391i	$0.888424\pi$
$654$	−5.00000	−0.195515
$655$	20.0000	0.781465
$656$	0	0
$657$	−9.00000	−0.351123
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	5.00000	0.194625
$661$	−47.0000	−1.82809	−0.914044	−	0.405615i	$-0.867057\pi$
$661$	−47.0000	−1.82809	−0.914044	+	0.405615i	$0.867057\pi$
$662$	4.00000	0.155464
$663$	5.00000	0.194184
$664$	5.00000	0.194038
$665$	−15.0000	−0.581675
$666$	−1.00000	−0.0387492
$667$	18.0000	0.696963
$668$	5.00000	0.193456
$669$	−8.00000	−0.309298
$670$	14.0000	0.540867
$671$	−50.0000	−1.93023
$672$	−5.00000	−0.192879
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−9.00000	−0.346667
$675$	1.00000	0.0384900
$676$	−12.0000	−0.461538
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	−10.0000	−0.384048
$679$	20.0000	0.767530
$680$	5.00000	0.191741
$681$	8.00000	0.306561
$682$	30.0000	1.14876
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	−3.00000	−0.114708
$685$	2.00000	0.0764161
$686$	−55.0000	−2.09991
$687$	6.00000	0.228914
$688$	4.00000	0.152499
$689$	3.00000	0.114291
$690$	−3.00000	−0.114208
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	11.0000	0.418157
$693$	25.0000	0.949671
$694$	−36.0000	−1.36654
$695$	−2.00000	−0.0758643
$696$	6.00000	0.227429
$697$	0	0
$698$	24.0000	0.908413
$699$	−26.0000	−0.983410
$700$	−5.00000	−0.188982
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−1.00000	−0.0377426
$703$	3.00000	0.113147
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−26.0000	−0.978523
$707$	30.0000	1.12827
$708$	−10.0000	−0.375823
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	13.0000	0.487196
$713$	−18.0000	−0.674105
$714$	25.0000	0.935601
$715$	−5.00000	−0.186989
$716$	8.00000	0.298974
$717$	12.0000	0.448148
$718$	36.0000	1.34351
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	−1.00000	−0.0372678
$721$	80.0000	2.97936
$722$	−10.0000	−0.372161
$723$	−18.0000	−0.669427
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	14.0000	0.519589
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	5.00000	0.185312
$729$	1.00000	0.0370370
$730$	9.00000	0.333105
$731$	−20.0000	−0.739727
$732$	10.0000	0.369611
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−5.00000	−0.184553
$735$	−18.0000	−0.663940
$736$	3.00000	0.110581
$737$	70.0000	2.57848
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	1.00000	0.0367607
$741$	3.00000	0.110208
$742$	15.0000	0.550667
$743$	10.0000	0.366864	0.183432	−	0.983032i	$-0.441279\pi$
$743$	10.0000	0.366864	0.183432	+	0.983032i	$0.441279\pi$
$744$	−6.00000	−0.219971
$745$	−2.00000	−0.0732743
$746$	14.0000	0.512576
$747$	5.00000	0.182940
$748$	25.0000	0.914091
$749$	−75.0000	−2.74044
$750$	−1.00000	−0.0365148
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	−6.00000	−0.218507
$755$	−15.0000	−0.545906
$756$	−5.00000	−0.181848
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	−26.0000	−0.944363
$759$	−15.0000	−0.544466
$760$	3.00000	0.108821
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−3.00000	−0.108679
$763$	25.0000	0.905061
$764$	−7.00000	−0.253251
$765$	5.00000	0.180775
$766$	9.00000	0.325183
$767$	10.0000	0.361079
$768$	1.00000	0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	−25.0000	−0.900937
$771$	15.0000	0.540212
$772$	−2.00000	−0.0719816
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	4.00000	0.143777
$775$	−6.00000	−0.215526
$776$	−4.00000	−0.143592
$777$	5.00000	0.179374
$778$	34.0000	1.21896
$779$	0	0
$780$	1.00000	0.0358057
$781$	−30.0000	−1.07348
$782$	−15.0000	−0.536399
$783$	6.00000	0.214423
$784$	18.0000	0.642857
$785$	20.0000	0.713831
$786$	−20.0000	−0.713376
$787$	42.0000	1.49714	0.748569	−	0.663057i	$-0.230741\pi$
$787$	42.0000	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$788$	23.0000	0.819341
$789$	12.0000	0.427211
$790$	0	0
$791$	50.0000	1.77780
$792$	−5.00000	−0.177667
$793$	−10.0000	−0.355110
$794$	−36.0000	−1.27759
$795$	3.00000	0.106399
$796$	−24.0000	−0.850657
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	15.0000	0.530994
$799$	0	0
$800$	1.00000	0.0353553
$801$	13.0000	0.459332
$802$	−27.0000	−0.953403
$803$	45.0000	1.58802
$804$	−14.0000	−0.493742
$805$	15.0000	0.528681
$806$	6.00000	0.211341
$807$	9.00000	0.316815
$808$	−6.00000	−0.211079
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	−1.00000	−0.0351364
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	−30.0000	−1.05279
$813$	20.0000	0.701431
$814$	5.00000	0.175250
$815$	23.0000	0.805655
$816$	−5.00000	−0.175035
$817$	−12.0000	−0.419827
$818$	24.0000	0.839140
$819$	5.00000	0.174714
$820$	0	0
$821$	39.0000	1.36111	0.680555	−	0.732697i	$-0.261739\pi$
$821$	39.0000	1.36111	0.680555	+	0.732697i	$0.261739\pi$
$822$	−2.00000	−0.0697580
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	−16.0000	−0.557386
$825$	−5.00000	−0.174078
$826$	50.0000	1.73972
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	3.00000	0.104257
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	−5.00000	−0.173553
$831$	25.0000	0.867240
$832$	−1.00000	−0.0346688
$833$	−90.0000	−3.11832
$834$	2.00000	0.0692543
$835$	−5.00000	−0.173032
$836$	15.0000	0.518786
$837$	−6.00000	−0.207390
$838$	−19.0000	−0.656344
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	5.00000	0.172516
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	1.00000	0.0344418
$844$	−2.00000	−0.0688428
$845$	12.0000	0.412813
$846$	0	0
$847$	−70.0000	−2.40523
$848$	−3.00000	−0.103020
$849$	5.00000	0.171600
$850$	−5.00000	−0.171499
$851$	−3.00000	−0.102839
$852$	6.00000	0.205557
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	−50.0000	−1.71096
$855$	3.00000	0.102598
$856$	15.0000	0.512689
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	5.00000	0.170697
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−33.0000	−1.12398
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	1.00000	0.0340207
$865$	−11.0000	−0.374011
$866$	21.0000	0.713609
$867$	8.00000	0.271694
$868$	30.0000	1.01827
$869$	0	0
$870$	−6.00000	−0.203419
$871$	14.0000	0.474372
$872$	−5.00000	−0.169321
$873$	−4.00000	−0.135379
$874$	−9.00000	−0.304430
$875$	5.00000	0.169031
$876$	−9.00000	−0.304082
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	36.0000	1.21494
$879$	−11.0000	−0.371021
$880$	5.00000	0.168550
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	18.0000	0.606092
$883$	21.0000	0.706706	0.353353	−	0.935490i	$-0.385041\pi$
$883$	21.0000	0.706706	0.353353	+	0.935490i	$0.385041\pi$
$884$	5.00000	0.168168
$885$	10.0000	0.336146
$886$	−28.0000	−0.940678
$887$	−38.0000	−1.27592	−0.637958	−	0.770072i	$-0.720220\pi$
$887$	−38.0000	−1.27592	−0.637958	+	0.770072i	$0.720220\pi$
$888$	−1.00000	−0.0335578
$889$	15.0000	0.503084
$890$	−13.0000	−0.435761
$891$	−5.00000	−0.167506
$892$	−8.00000	−0.267860
$893$	0	0
$894$	2.00000	0.0668900
$895$	−8.00000	−0.267411
$896$	−5.00000	−0.167038
$897$	−3.00000	−0.100167
$898$	30.0000	1.00111
$899$	−36.0000	−1.20067
$900$	1.00000	0.0333333
$901$	15.0000	0.499722
$902$	0	0
$903$	−20.0000	−0.665558
$904$	−10.0000	−0.332595
$905$	10.0000	0.332411
$906$	15.0000	0.498342
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	8.00000	0.265489
$909$	−6.00000	−0.199007
$910$	−5.00000	−0.165748
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	−3.00000	−0.0993399
$913$	−25.0000	−0.827379
$914$	4.00000	0.132308
$915$	−10.0000	−0.330590
$916$	6.00000	0.198246
$917$	100.000	3.30229
$918$	−5.00000	−0.165025
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−3.00000	−0.0989071
$921$	−30.0000	−0.988534
$922$	26.0000	0.856264
$923$	−6.00000	−0.197492
$924$	25.0000	0.822440
$925$	−1.00000	−0.0328798
$926$	−28.0000	−0.920137
$927$	−16.0000	−0.525509
$928$	6.00000	0.196960
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	6.00000	0.196748
$931$	−54.0000	−1.76978
$932$	−26.0000	−0.851658
$933$	8.00000	0.261908
$934$	−12.0000	−0.392652
$935$	−25.0000	−0.817587
$936$	−1.00000	−0.0326860
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	70.0000	2.28558
$939$	24.0000	0.783210
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	−10.0000	−0.325472
$945$	5.00000	0.162650
$946$	−20.0000	−0.650256
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	−3.00000	−0.0973329
$951$	−10.0000	−0.324272
$952$	25.0000	0.810255
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−3.00000	−0.0971286
$955$	7.00000	0.226515
$956$	12.0000	0.388108
$957$	−30.0000	−0.969762
$958$	−21.0000	−0.678479
$959$	10.0000	0.322917
$960$	−1.00000	−0.0322749
$961$	5.00000	0.161290
$962$	1.00000	0.0322413
$963$	15.0000	0.483368
$964$	−18.0000	−0.579741
$965$	2.00000	0.0643823
$966$	−15.0000	−0.482617
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	14.0000	0.449977
$969$	15.0000	0.481869
$970$	4.00000	0.128432
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	1.00000	0.0320750
$973$	−10.0000	−0.320585
$974$	26.0000	0.833094
$975$	−1.00000	−0.0320256
$976$	10.0000	0.320092
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	−23.0000	−0.735459
$979$	−65.0000	−2.07741
$980$	−18.0000	−0.574989
$981$	−5.00000	−0.159638
$982$	−9.00000	−0.287202
$983$	−58.0000	−1.84991	−0.924956	−	0.380073i	$-0.875899\pi$
$983$	−58.0000	−1.84991	−0.924956	+	0.380073i	$0.875899\pi$
$984$	0	0
$985$	−23.0000	−0.732841
$986$	−30.0000	−0.955395
$987$	0	0
$988$	3.00000	0.0954427
$989$	12.0000	0.381578
$990$	5.00000	0.158910
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−6.00000	−0.190500
$993$	4.00000	0.126936
$994$	−30.0000	−0.951542
$995$	24.0000	0.760851
$996$	5.00000	0.158431
$997$	−39.0000	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$997$	−39.0000	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$998$	−5.00000	−0.158272
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.m.1.1	✓	1
3.2	odd	2		3330.2.a.f.1.1		1
4.3	odd	2		8880.2.a.i.1.1		1
5.4	even	2		5550.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.m.1.1	✓	1	1.1	even	1	trivial
3330.2.a.f.1.1		1	3.2	odd	2
5550.2.a.k.1.1		1	5.4	even	2
8880.2.a.i.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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