LMFDB - Embedded newform 1334.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +4.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{18} +8.00000 q^{19} -3.00000 q^{20} -4.00000 q^{21} +3.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} -4.00000 q^{28} -1.00000 q^{29} +3.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +6.00000 q^{34} +12.0000 q^{35} -2.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} +5.00000 q^{39} +3.00000 q^{40} +12.0000 q^{41} +4.00000 q^{42} +5.00000 q^{43} -3.00000 q^{44} +6.00000 q^{45} -1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} -6.00000 q^{51} +5.00000 q^{52} +9.00000 q^{53} +5.00000 q^{54} +9.00000 q^{55} +4.00000 q^{56} +8.00000 q^{57} +1.00000 q^{58} -3.00000 q^{60} -10.0000 q^{61} -5.00000 q^{62} +8.00000 q^{63} +1.00000 q^{64} -15.0000 q^{65} +3.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -12.0000 q^{70} +6.00000 q^{71} +2.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +4.00000 q^{75} +8.00000 q^{76} +12.0000 q^{77} -5.00000 q^{78} +5.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -12.0000 q^{83} -4.00000 q^{84} +18.0000 q^{85} -5.00000 q^{86} -1.00000 q^{87} +3.00000 q^{88} +12.0000 q^{89} -6.00000 q^{90} -20.0000 q^{91} +1.00000 q^{92} +5.00000 q^{93} +3.00000 q^{94} -24.0000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	4.00000	1.06904
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	2.00000	0.471405
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−3.00000	−0.670820
$21$	−4.00000	−0.872872
$22$	3.00000	0.639602
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−5.00000	−0.980581
$27$	−5.00000	−0.962250
$28$	−4.00000	−0.755929
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.00000	0.547723
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	6.00000	1.02899
$35$	12.0000	2.02837
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−8.00000	−1.29777
$39$	5.00000	0.800641
$40$	3.00000	0.474342
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	4.00000	0.617213
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−3.00000	−0.452267
$45$	6.00000	0.894427
$46$	−1.00000	−0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	−6.00000	−0.840168
$52$	5.00000	0.693375
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	5.00000	0.680414
$55$	9.00000	1.21356
$56$	4.00000	0.534522
$57$	8.00000	1.05963
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−3.00000	−0.387298
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−5.00000	−0.635001
$63$	8.00000	1.00791
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	3.00000	0.369274
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	−12.0000	−1.43427
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.00000	0.235702
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	4.00000	0.461880
$76$	8.00000	0.917663
$77$	12.0000	1.36753
$78$	−5.00000	−0.566139
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	−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
$85$	18.0000	1.95237
$86$	−5.00000	−0.539164
$87$	−1.00000	−0.107211
$88$	3.00000	0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−6.00000	−0.632456
$91$	−20.0000	−2.09657
$92$	1.00000	0.104257
$93$	5.00000	0.518476
$94$	3.00000	0.309426
$95$	−24.0000	−2.46235
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	6.00000	0.603023
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−5.00000	−0.490290
$105$	12.0000	1.17108
$106$	−9.00000	−0.874157
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	−5.00000	−0.481125
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−9.00000	−0.858116
$111$	2.00000	0.189832
$112$	−4.00000	−0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−8.00000	−0.749269
$115$	−3.00000	−0.279751
$116$	−1.00000	−0.0928477
$117$	−10.0000	−0.924500
$118$	0	0
$119$	24.0000	2.20008
$120$	3.00000	0.273861
$121$	−2.00000	−0.181818
$122$	10.0000	0.905357
$123$	12.0000	1.08200
$124$	5.00000	0.449013
$125$	3.00000	0.268328
$126$	−8.00000	−0.712697
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	5.00000	0.440225
$130$	15.0000	1.31559
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−3.00000	−0.261116
$133$	−32.0000	−2.77475
$134$	10.0000	0.863868
$135$	15.0000	1.29099
$136$	6.00000	0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−1.00000	−0.0851257
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	12.0000	1.01419
$141$	−3.00000	−0.252646
$142$	−6.00000	−0.503509
$143$	−15.0000	−1.25436
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	−2.00000	−0.165521
$147$	9.00000	0.742307
$148$	2.00000	0.164399
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	−4.00000	−0.326599
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−8.00000	−0.648886
$153$	12.0000	0.970143
$154$	−12.0000	−0.966988
$155$	−15.0000	−1.20483
$156$	5.00000	0.400320
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−5.00000	−0.397779
$159$	9.00000	0.713746
$160$	3.00000	0.237171
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	12.0000	0.937043
$165$	9.00000	0.700649
$166$	12.0000	0.931381
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	4.00000	0.308607
$169$	12.0000	0.923077
$170$	−18.0000	−1.38054
$171$	−16.0000	−1.22355
$172$	5.00000	0.381246
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	1.00000	0.0758098
$175$	−16.0000	−1.20949
$176$	−3.00000	−0.226134
$177$	0	0
$178$	−12.0000	−0.899438
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	6.00000	0.447214
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	20.0000	1.48250
$183$	−10.0000	−0.739221
$184$	−1.00000	−0.0737210
$185$	−6.00000	−0.441129
$186$	−5.00000	−0.366618
$187$	18.0000	1.31629
$188$	−3.00000	−0.218797
$189$	20.0000	1.45479
$190$	24.0000	1.74114
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	10.0000	0.717958
$195$	−15.0000	−1.07417
$196$	9.00000	0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−4.00000	−0.282843
$201$	−10.0000	−0.705346
$202$	6.00000	0.422159
$203$	4.00000	0.280745
$204$	−6.00000	−0.420084
$205$	−36.0000	−2.51435
$206$	−14.0000	−0.975426
$207$	−2.00000	−0.139010
$208$	5.00000	0.346688
$209$	−24.0000	−1.66011
$210$	−12.0000	−0.828079
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	9.00000	0.618123
$213$	6.00000	0.411113
$214$	−6.00000	−0.410152
$215$	−15.0000	−1.02299
$216$	5.00000	0.340207
$217$	−20.0000	−1.35769
$218$	7.00000	0.474100
$219$	2.00000	0.135147
$220$	9.00000	0.606780
$221$	−30.0000	−2.01802
$222$	−2.00000	−0.134231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	4.00000	0.267261
$225$	−8.00000	−0.533333
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	8.00000	0.529813
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	3.00000	0.197814
$231$	12.0000	0.789542
$232$	1.00000	0.0656532
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	10.0000	0.653720
$235$	9.00000	0.587095
$236$	0	0
$237$	5.00000	0.324785
$238$	−24.0000	−1.55569
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−3.00000	−0.193649
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	2.00000	0.128565
$243$	16.0000	1.02640
$244$	−10.0000	−0.640184
$245$	−27.0000	−1.72497
$246$	−12.0000	−0.765092
$247$	40.0000	2.54514
$248$	−5.00000	−0.317500
$249$	−12.0000	−0.760469
$250$	−3.00000	−0.189737
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	8.00000	0.503953
$253$	−3.00000	−0.188608
$254$	−8.00000	−0.501965
$255$	18.0000	1.12720
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−5.00000	−0.311286
$259$	−8.00000	−0.497096
$260$	−15.0000	−0.930261
$261$	2.00000	0.123797
$262$	12.0000	0.741362
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	3.00000	0.184637
$265$	−27.0000	−1.65860
$266$	32.0000	1.96205
$267$	12.0000	0.734388
$268$	−10.0000	−0.610847
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−15.0000	−0.912871
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−6.00000	−0.363803
$273$	−20.0000	−1.21046
$274$	18.0000	1.08742
$275$	−12.0000	−0.723627
$276$	1.00000	0.0601929
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	22.0000	1.31947
$279$	−10.0000	−0.598684
$280$	−12.0000	−0.717137
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	3.00000	0.178647
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	6.00000	0.356034
$285$	−24.0000	−1.42164
$286$	15.0000	0.886969
$287$	−48.0000	−2.83335
$288$	2.00000	0.117851
$289$	19.0000	1.11765
$290$	−3.00000	−0.176166
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	15.0000	0.870388
$298$	−15.0000	−0.868927
$299$	5.00000	0.289157
$300$	4.00000	0.230940
$301$	−20.0000	−1.15278
$302$	−14.0000	−0.805609
$303$	−6.00000	−0.344691
$304$	8.00000	0.458831
$305$	30.0000	1.71780
$306$	−12.0000	−0.685994
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	12.0000	0.683763
$309$	14.0000	0.796432
$310$	15.0000	0.851943
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−5.00000	−0.283069
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	4.00000	0.225733
$315$	−24.0000	−1.35225
$316$	5.00000	0.281272
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−9.00000	−0.504695
$319$	3.00000	0.167968
$320$	−3.00000	−0.167705
$321$	6.00000	0.334887
$322$	4.00000	0.222911
$323$	−48.0000	−2.67079
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	−11.0000	−0.609234
$327$	−7.00000	−0.387101
$328$	−12.0000	−0.662589
$329$	12.0000	0.661581
$330$	−9.00000	−0.495434
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	−12.0000	−0.658586
$333$	−4.00000	−0.219199
$334$	−6.00000	−0.328305
$335$	30.0000	1.63908
$336$	−4.00000	−0.218218
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−12.0000	−0.652714
$339$	6.00000	0.325875
$340$	18.0000	0.976187
$341$	−15.0000	−0.812296
$342$	16.0000	0.865181
$343$	−8.00000	−0.431959
$344$	−5.00000	−0.269582
$345$	−3.00000	−0.161515
$346$	−18.0000	−0.967686
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−1.00000	−0.0536056
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	16.0000	0.855236
$351$	−25.0000	−1.33440
$352$	3.00000	0.159901
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	12.0000	0.635999
$357$	24.0000	1.27021
$358$	−24.0000	−1.26844
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	−6.00000	−0.316228
$361$	45.0000	2.36842
$362$	−23.0000	−1.20885
$363$	−2.00000	−0.104973
$364$	−20.0000	−1.04828
$365$	−6.00000	−0.314054
$366$	10.0000	0.522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.00000	0.0521286
$369$	−24.0000	−1.24939
$370$	6.00000	0.311925
$371$	−36.0000	−1.86903
$372$	5.00000	0.259238
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	−18.0000	−0.930758
$375$	3.00000	0.154919
$376$	3.00000	0.154713
$377$	−5.00000	−0.257513
$378$	−20.0000	−1.02869
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−24.0000	−1.23117
$381$	8.00000	0.409852
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	−36.0000	−1.83473
$386$	−14.0000	−0.712581
$387$	−10.0000	−0.508329
$388$	−10.0000	−0.507673
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	15.0000	0.759555
$391$	−6.00000	−0.303433
$392$	−9.00000	−0.454569
$393$	−12.0000	−0.605320
$394$	−6.00000	−0.302276
$395$	−15.0000	−0.754732
$396$	6.00000	0.301511
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	−2.00000	−0.100251
$399$	−32.0000	−1.60200
$400$	4.00000	0.200000
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	10.0000	0.498755
$403$	25.0000	1.24534
$404$	−6.00000	−0.298511
$405$	−3.00000	−0.149071
$406$	−4.00000	−0.198517
$407$	−6.00000	−0.297409
$408$	6.00000	0.297044
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	36.0000	1.77791
$411$	−18.0000	−0.887875
$412$	14.0000	0.689730
$413$	0	0
$414$	2.00000	0.0982946
$415$	36.0000	1.76717
$416$	−5.00000	−0.245145
$417$	−22.0000	−1.07734
$418$	24.0000	1.17388
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	12.0000	0.585540
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	13.0000	0.632830
$423$	6.00000	0.291730
$424$	−9.00000	−0.437079
$425$	−24.0000	−1.16417
$426$	−6.00000	−0.290701
$427$	40.0000	1.93574
$428$	6.00000	0.290021
$429$	−15.0000	−0.724207
$430$	15.0000	0.723364
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−5.00000	−0.240563
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	20.0000	0.960031
$435$	3.00000	0.143839
$436$	−7.00000	−0.335239
$437$	8.00000	0.382692
$438$	−2.00000	−0.0955637
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	−9.00000	−0.429058
$441$	−18.0000	−0.857143
$442$	30.0000	1.42695
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	2.00000	0.0949158
$445$	−36.0000	−1.70656
$446$	−14.0000	−0.662919
$447$	15.0000	0.709476
$448$	−4.00000	−0.188982
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	8.00000	0.377124
$451$	−36.0000	−1.69517
$452$	6.00000	0.282216
$453$	14.0000	0.657777
$454$	−12.0000	−0.563188
$455$	60.0000	2.81284
$456$	−8.00000	−0.374634
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−8.00000	−0.373815
$459$	30.0000	1.40028
$460$	−3.00000	−0.139876
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	−12.0000	−0.558291
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−1.00000	−0.0464238
$465$	−15.0000	−0.695608
$466$	−3.00000	−0.138972
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	−10.0000	−0.462250
$469$	40.0000	1.84703
$470$	−9.00000	−0.415139
$471$	−4.00000	−0.184310
$472$	0	0
$473$	−15.0000	−0.689701
$474$	−5.00000	−0.229658
$475$	32.0000	1.46826
$476$	24.0000	1.10004
$477$	−18.0000	−0.824163
$478$	6.00000	0.274434
$479$	9.00000	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$479$	9.00000	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$480$	3.00000	0.136931
$481$	10.0000	0.455961
$482$	1.00000	0.0455488
$483$	−4.00000	−0.182006
$484$	−2.00000	−0.0909091
$485$	30.0000	1.36223
$486$	−16.0000	−0.725775
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	10.0000	0.452679
$489$	11.0000	0.497437
$490$	27.0000	1.21974
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	12.0000	0.541002
$493$	6.00000	0.270226
$494$	−40.0000	−1.79969
$495$	−18.0000	−0.809040
$496$	5.00000	0.224507
$497$	−24.0000	−1.07655
$498$	12.0000	0.537733
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	3.00000	0.134164
$501$	6.00000	0.268060
$502$	21.0000	0.937276
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	−8.00000	−0.356348
$505$	18.0000	0.800989
$506$	3.00000	0.133366
$507$	12.0000	0.532939
$508$	8.00000	0.354943
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	−18.0000	−0.797053
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	−40.0000	−1.76604
$514$	3.00000	0.132324
$515$	−42.0000	−1.85074
$516$	5.00000	0.220113
$517$	9.00000	0.395820
$518$	8.00000	0.351500
$519$	18.0000	0.790112
$520$	15.0000	0.657794
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	−2.00000	−0.0875376
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−12.0000	−0.524222
$525$	−16.0000	−0.698297
$526$	21.0000	0.915644
$527$	−30.0000	−1.30682
$528$	−3.00000	−0.130558
$529$	1.00000	0.0434783
$530$	27.0000	1.17281
$531$	0	0
$532$	−32.0000	−1.38738
$533$	60.0000	2.59889
$534$	−12.0000	−0.519291
$535$	−18.0000	−0.778208
$536$	10.0000	0.431934
$537$	24.0000	1.03568
$538$	6.00000	0.258678
$539$	−27.0000	−1.16297
$540$	15.0000	0.645497
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−11.0000	−0.472490
$543$	23.0000	0.987024
$544$	6.00000	0.257248
$545$	21.0000	0.899541
$546$	20.0000	0.855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−18.0000	−0.768922
$549$	20.0000	0.853579
$550$	12.0000	0.511682
$551$	−8.00000	−0.340811
$552$	−1.00000	−0.0425628
$553$	−20.0000	−0.850487
$554$	22.0000	0.934690
$555$	−6.00000	−0.254686
$556$	−22.0000	−0.933008
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	10.0000	0.423334
$559$	25.0000	1.05739
$560$	12.0000	0.507093
$561$	18.0000	0.759961
$562$	−21.0000	−0.885832
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	−3.00000	−0.126323
$565$	−18.0000	−0.757266
$566$	22.0000	0.924729
$567$	−4.00000	−0.167984
$568$	−6.00000	−0.251754
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	24.0000	1.00525
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−15.0000	−0.627182
$573$	0	0
$574$	48.0000	2.00348
$575$	4.00000	0.166812
$576$	−2.00000	−0.0833333
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−19.0000	−0.790296
$579$	14.0000	0.581820
$580$	3.00000	0.124568
$581$	48.0000	1.99138
$582$	10.0000	0.414513
$583$	−27.0000	−1.11823
$584$	−2.00000	−0.0827606
$585$	30.0000	1.24035
$586$	−24.0000	−0.991431
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	9.00000	0.371154
$589$	40.0000	1.64817
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	−15.0000	−0.615457
$595$	−72.0000	−2.95171
$596$	15.0000	0.614424
$597$	2.00000	0.0818546
$598$	−5.00000	−0.204465
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	−4.00000	−0.163299
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	20.0000	0.815139
$603$	20.0000	0.814463
$604$	14.0000	0.569652
$605$	6.00000	0.243935
$606$	6.00000	0.243733
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	−8.00000	−0.324443
$609$	4.00000	0.162088
$610$	−30.0000	−1.21466
$611$	−15.0000	−0.606835
$612$	12.0000	0.485071
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	−23.0000	−0.928204
$615$	−36.0000	−1.45166
$616$	−12.0000	−0.483494
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	−14.0000	−0.563163
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−15.0000	−0.602414
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−48.0000	−1.92308
$624$	5.00000	0.200160
$625$	−29.0000	−1.16000
$626$	1.00000	0.0399680
$627$	−24.0000	−0.958468
$628$	−4.00000	−0.159617
$629$	−12.0000	−0.478471
$630$	24.0000	0.956183
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−5.00000	−0.198889
$633$	−13.0000	−0.516704
$634$	18.0000	0.714871
$635$	−24.0000	−0.952411
$636$	9.00000	0.356873
$637$	45.0000	1.78296
$638$	−3.00000	−0.118771
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	−6.00000	−0.236801
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−4.00000	−0.157622
$645$	−15.0000	−0.590624
$646$	48.0000	1.88853
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−20.0000	−0.784465
$651$	−20.0000	−0.783862
$652$	11.0000	0.430793
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	7.00000	0.273722
$655$	36.0000	1.40664
$656$	12.0000	0.468521
$657$	−4.00000	−0.156055
$658$	−12.0000	−0.467809
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	9.00000	0.350325
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−35.0000	−1.36031
$663$	−30.0000	−1.16510
$664$	12.0000	0.465690
$665$	96.0000	3.72272
$666$	4.00000	0.154997
$667$	−1.00000	−0.0387202
$668$	6.00000	0.232147
$669$	14.0000	0.541271
$670$	−30.0000	−1.15900
$671$	30.0000	1.15814
$672$	4.00000	0.154303
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−8.00000	−0.308148
$675$	−20.0000	−0.769800
$676$	12.0000	0.461538
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	−6.00000	−0.230429
$679$	40.0000	1.53506
$680$	−18.0000	−0.690268
$681$	12.0000	0.459841
$682$	15.0000	0.574380
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−16.0000	−0.611775
$685$	54.0000	2.06323
$686$	8.00000	0.305441
$687$	8.00000	0.305219
$688$	5.00000	0.190623
$689$	45.0000	1.71436
$690$	3.00000	0.114208
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	18.0000	0.684257
$693$	−24.0000	−0.911685
$694$	18.0000	0.683271
$695$	66.0000	2.50352
$696$	1.00000	0.0379049
$697$	−72.0000	−2.72719
$698$	−11.0000	−0.416356
$699$	3.00000	0.113470
$700$	−16.0000	−0.604743
$701$	3.00000	0.113308	0.0566542	−	0.998394i	$-0.481957\pi$
$701$	3.00000	0.113308	0.0566542	+	0.998394i	$0.481957\pi$
$702$	25.0000	0.943564
$703$	16.0000	0.603451
$704$	−3.00000	−0.113067
$705$	9.00000	0.338960
$706$	18.0000	0.677439
$707$	24.0000	0.902613
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	18.0000	0.675528
$711$	−10.0000	−0.375029
$712$	−12.0000	−0.449719
$713$	5.00000	0.187251
$714$	−24.0000	−0.898177
$715$	45.0000	1.68290
$716$	24.0000	0.896922
$717$	−6.00000	−0.224074
$718$	15.0000	0.559795
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	6.00000	0.223607
$721$	−56.0000	−2.08555
$722$	−45.0000	−1.67473
$723$	−1.00000	−0.0371904
$724$	23.0000	0.854788
$725$	−4.00000	−0.148556
$726$	2.00000	0.0742270
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	20.0000	0.741249
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	−30.0000	−1.10959
$732$	−10.0000	−0.369611
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	16.0000	0.590571
$735$	−27.0000	−0.995910
$736$	−1.00000	−0.0368605
$737$	30.0000	1.10506
$738$	24.0000	0.883452
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	−6.00000	−0.220564
$741$	40.0000	1.46944
$742$	36.0000	1.32160
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	−5.00000	−0.183309
$745$	−45.0000	−1.64867
$746$	−23.0000	−0.842090
$747$	24.0000	0.878114
$748$	18.0000	0.658145
$749$	−24.0000	−0.876941
$750$	−3.00000	−0.109545
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−3.00000	−0.109399
$753$	−21.0000	−0.765283
$754$	5.00000	0.182089
$755$	−42.0000	−1.52854
$756$	20.0000	0.727393
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	16.0000	0.581146
$759$	−3.00000	−0.108893
$760$	24.0000	0.870572
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−8.00000	−0.289809
$763$	28.0000	1.01367
$764$	0	0
$765$	−36.0000	−1.30158
$766$	24.0000	0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	36.0000	1.29735
$771$	−3.00000	−0.108042
$772$	14.0000	0.503871
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	10.0000	0.359443
$775$	20.0000	0.718421
$776$	10.0000	0.358979
$777$	−8.00000	−0.286998
$778$	−12.0000	−0.430221
$779$	96.0000	3.43956
$780$	−15.0000	−0.537086
$781$	−18.0000	−0.644091
$782$	6.00000	0.214560
$783$	5.00000	0.178685
$784$	9.00000	0.321429
$785$	12.0000	0.428298
$786$	12.0000	0.428026
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	6.00000	0.213741
$789$	−21.0000	−0.747620
$790$	15.0000	0.533676
$791$	−24.0000	−0.853342
$792$	−6.00000	−0.213201
$793$	−50.0000	−1.77555
$794$	−5.00000	−0.177443
$795$	−27.0000	−0.957591
$796$	2.00000	0.0708881
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	32.0000	1.13279
$799$	18.0000	0.636794
$800$	−4.00000	−0.141421
$801$	−24.0000	−0.847998
$802$	−33.0000	−1.16527
$803$	−6.00000	−0.211735
$804$	−10.0000	−0.352673
$805$	12.0000	0.422944
$806$	−25.0000	−0.880587
$807$	−6.00000	−0.211210
$808$	6.00000	0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	3.00000	0.105409
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	4.00000	0.140372
$813$	11.0000	0.385787
$814$	6.00000	0.210300
$815$	−33.0000	−1.15594
$816$	−6.00000	−0.210042
$817$	40.0000	1.39942
$818$	−8.00000	−0.279713
$819$	40.0000	1.39771
$820$	−36.0000	−1.25717
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	18.0000	0.627822
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−14.0000	−0.487713
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	−2.00000	−0.0695048
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	−36.0000	−1.24958
$831$	−22.0000	−0.763172
$832$	5.00000	0.173344
$833$	−54.0000	−1.87099
$834$	22.0000	0.761798
$835$	−18.0000	−0.622916
$836$	−24.0000	−0.830057
$837$	−25.0000	−0.864126
$838$	0	0
$839$	39.0000	1.34643	0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000	1.34643	0.673215	+	0.739447i	$0.264913\pi$
$840$	−12.0000	−0.414039
$841$	1.00000	0.0344828
$842$	−32.0000	−1.10279
$843$	21.0000	0.723278
$844$	−13.0000	−0.447478
$845$	−36.0000	−1.23844
$846$	−6.00000	−0.206284
$847$	8.00000	0.274883
$848$	9.00000	0.309061
$849$	−22.0000	−0.755038
$850$	24.0000	0.823193
$851$	2.00000	0.0685591
$852$	6.00000	0.205557
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−40.0000	−1.36877
$855$	48.0000	1.64157
$856$	−6.00000	−0.205076
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	15.0000	0.512092
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	−15.0000	−0.511496
$861$	−48.0000	−1.63584
$862$	0	0
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	5.00000	0.170103
$865$	−54.0000	−1.83606
$866$	10.0000	0.339814
$867$	19.0000	0.645274
$868$	−20.0000	−0.678844
$869$	−15.0000	−0.508840
$870$	−3.00000	−0.101710
$871$	−50.0000	−1.69419
$872$	7.00000	0.237050
$873$	20.0000	0.676897
$874$	−8.00000	−0.270604
$875$	−12.0000	−0.405674
$876$	2.00000	0.0675737
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	34.0000	1.14744
$879$	24.0000	0.809500
$880$	9.00000	0.303390
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	18.0000	0.606092
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	−2.00000	−0.0671156
$889$	−32.0000	−1.07325
$890$	36.0000	1.20672
$891$	−3.00000	−0.100504
$892$	14.0000	0.468755
$893$	−24.0000	−0.803129
$894$	−15.0000	−0.501675
$895$	−72.0000	−2.40669
$896$	4.00000	0.133631
$897$	5.00000	0.166945
$898$	0	0
$899$	−5.00000	−0.166759
$900$	−8.00000	−0.266667
$901$	−54.0000	−1.79900
$902$	36.0000	1.19867
$903$	−20.0000	−0.665558
$904$	−6.00000	−0.199557
$905$	−69.0000	−2.29364
$906$	−14.0000	−0.465119
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	12.0000	0.398234
$909$	12.0000	0.398015
$910$	−60.0000	−1.98898
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	8.00000	0.264906
$913$	36.0000	1.19143
$914$	22.0000	0.727695
$915$	30.0000	0.991769
$916$	8.00000	0.264327
$917$	48.0000	1.58510
$918$	−30.0000	−0.990148
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	3.00000	0.0989071
$921$	23.0000	0.757876
$922$	−36.0000	−1.18560
$923$	30.0000	0.987462
$924$	12.0000	0.394771
$925$	8.00000	0.263038
$926$	40.0000	1.31448
$927$	−28.0000	−0.919641
$928$	1.00000	0.0328266
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	15.0000	0.491869
$931$	72.0000	2.35970
$932$	3.00000	0.0982683
$933$	0	0
$934$	15.0000	0.490815
$935$	−54.0000	−1.76599
$936$	10.0000	0.326860
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−40.0000	−1.30605
$939$	−1.00000	−0.0326338
$940$	9.00000	0.293548
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	4.00000	0.130327
$943$	12.0000	0.390774
$944$	0	0
$945$	−60.0000	−1.95180
$946$	15.0000	0.487692
$947$	39.0000	1.26733	0.633665	−	0.773608i	$-0.281550\pi$
$947$	39.0000	1.26733	0.633665	+	0.773608i	$0.281550\pi$
$948$	5.00000	0.162392
$949$	10.0000	0.324614
$950$	−32.0000	−1.03822
$951$	−18.0000	−0.583690
$952$	−24.0000	−0.777844
$953$	−27.0000	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	−6.00000	−0.194054
$957$	3.00000	0.0969762
$958$	−9.00000	−0.290777
$959$	72.0000	2.32500
$960$	−3.00000	−0.0968246
$961$	−6.00000	−0.193548
$962$	−10.0000	−0.322413
$963$	−12.0000	−0.386695
$964$	−1.00000	−0.0322078
$965$	−42.0000	−1.35203
$966$	4.00000	0.128698
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	2.00000	0.0642824
$969$	−48.0000	−1.54198
$970$	−30.0000	−0.963242
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	16.0000	0.513200
$973$	88.0000	2.82115
$974$	28.0000	0.897178
$975$	20.0000	0.640513
$976$	−10.0000	−0.320092
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	−11.0000	−0.351741
$979$	−36.0000	−1.15056
$980$	−27.0000	−0.862483
$981$	14.0000	0.446986
$982$	−33.0000	−1.05307
$983$	15.0000	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$983$	15.0000	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$984$	−12.0000	−0.382546
$985$	−18.0000	−0.573528
$986$	−6.00000	−0.191079
$987$	12.0000	0.381964
$988$	40.0000	1.27257
$989$	5.00000	0.158991
$990$	18.0000	0.572078
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−5.00000	−0.158750
$993$	35.0000	1.11069
$994$	24.0000	0.761234
$995$	−6.00000	−0.190213
$996$	−12.0000	−0.380235
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−8.00000	−0.253236
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.b.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:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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