LMFDB - Embedded newform 8281.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8281 = 7^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8281.1

$q$ -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} -3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{6} -3.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} -3.00000 q^{11} +3.00000 q^{12} -9.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} +9.00000 q^{24} +4.00000 q^{25} -9.00000 q^{27} +7.00000 q^{29} -9.00000 q^{30} +3.00000 q^{31} +5.00000 q^{32} +9.00000 q^{33} -2.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} -9.00000 q^{40} +3.00000 q^{41} -7.00000 q^{43} +3.00000 q^{44} +18.0000 q^{45} +1.00000 q^{47} +3.00000 q^{48} +4.00000 q^{50} +6.00000 q^{51} +3.00000 q^{53} -9.00000 q^{54} -9.00000 q^{55} +3.00000 q^{57} +7.00000 q^{58} -4.00000 q^{59} +9.00000 q^{60} -13.0000 q^{61} +3.00000 q^{62} +7.00000 q^{64} +9.00000 q^{66} -3.00000 q^{67} +2.00000 q^{68} +13.0000 q^{71} -18.0000 q^{72} -13.0000 q^{73} +2.00000 q^{74} -12.0000 q^{75} +1.00000 q^{76} -3.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +3.00000 q^{82} -6.00000 q^{85} -7.00000 q^{86} -21.0000 q^{87} +9.00000 q^{88} +6.00000 q^{89} +18.0000 q^{90} -9.00000 q^{93} +1.00000 q^{94} -3.00000 q^{95} -15.0000 q^{96} -5.00000 q^{97} -18.0000 q^{99} +O(q^{100})

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	−0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−3.00000	−1.22474
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	6.00000	2.00000
$10$	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	$0.649385\pi$
$12$	3.00000	0.866025
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−9.00000	−2.32379
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	6.00000	1.41421
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	9.00000	1.83712
$25$	4.00000	0.800000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	−9.00000	−1.64317
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	5.00000	0.883883
$33$	9.00000	1.56670
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−6.00000	−1.00000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−9.00000	−1.42302
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	3.00000	0.452267
$45$	18.0000	2.68328
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	4.00000	0.565685
$51$	6.00000	0.840168
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−9.00000	−1.22474
$55$	−9.00000	−1.21356
$56$	0	0
$57$	3.00000	0.397360
$58$	7.00000	0.919145
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	9.00000	1.16190
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	9.00000	1.10782
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	−18.0000	−2.12132
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	2.00000	0.232495
$75$	−12.0000	−1.38564
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−3.00000	−0.335410
$81$	9.00000	1.00000
$82$	3.00000	0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−7.00000	−0.754829
$87$	−21.0000	−2.25144
$88$	9.00000	0.959403
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	18.0000	1.89737
$91$	0	0
$92$	0	0
$93$	−9.00000	−0.933257
$94$	1.00000	0.103142
$95$	−3.00000	−0.307794
$96$	−15.0000	−1.53093
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	−18.0000	−1.80907
$100$	−4.00000	−0.400000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	6.00000	0.594089
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	0	0
$106$	3.00000	0.291386
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	9.00000	0.866025
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−9.00000	−0.858116
$111$	−6.00000	−0.569495
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−7.00000	−0.649934
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	27.0000	2.46475
$121$	−2.00000	−0.181818
$122$	−13.0000	−1.17696
$123$	−9.00000	−0.811503
$124$	−3.00000	−0.269408
$125$	−3.00000	−0.268328
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	−3.00000	−0.265165
$129$	21.0000	1.84895
$130$	0	0
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	−9.00000	−0.783349
$133$	0	0
$134$	−3.00000	−0.259161
$135$	−27.0000	−2.32379
$136$	6.00000	0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	13.0000	1.09094
$143$	0	0
$144$	−6.00000	−0.500000
$145$	21.0000	1.74396
$146$	−13.0000	−1.07589
$147$	0	0
$148$	−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	$0.289388\pi$
$150$	−12.0000	−0.979796
$151$	−21.0000	−1.70896	−0.854478	−	0.519488i	$-0.826123\pi$
$151$	−21.0000	−1.70896	−0.854478	+	0.519488i	$0.826123\pi$
$152$	3.00000	0.243332
$153$	−12.0000	−0.970143
$154$	0	0
$155$	9.00000	0.722897
$156$	0	0
$157$	19.0000	1.51637	0.758183	−	0.652042i	$-0.226088\pi$
$157$	19.0000	1.51637	0.758183	+	0.652042i	$0.226088\pi$
$158$	−3.00000	−0.238667
$159$	−9.00000	−0.713746
$160$	15.0000	1.18585
$161$	0	0
$162$	9.00000	0.707107
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−3.00000	−0.234261
$165$	27.0000	2.10195
$166$	0	0
$167$	−13.0000	−1.00597	−0.502985	−	0.864295i	$-0.667765\pi$
$167$	−13.0000	−1.00597	−0.502985	+	0.864295i	$0.667765\pi$
$168$	0	0
$169$	0	0
$170$	−6.00000	−0.460179
$171$	−6.00000	−0.458831
$172$	7.00000	0.533745
$173$	19.0000	1.44454	0.722272	−	0.691609i	$-0.243098\pi$
$173$	19.0000	1.44454	0.722272	+	0.691609i	$0.243098\pi$
$174$	−21.0000	−1.59201
$175$	0	0
$176$	3.00000	0.226134
$177$	12.0000	0.901975
$178$	6.00000	0.449719
$179$	17.0000	1.27064	0.635320	−	0.772249i	$-0.280868\pi$
$179$	17.0000	1.27064	0.635320	+	0.772249i	$0.280868\pi$
$180$	−18.0000	−1.34164
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	39.0000	2.88296
$184$	0	0
$185$	6.00000	0.441129
$186$	−9.00000	−0.659912
$187$	6.00000	0.438763
$188$	−1.00000	−0.0729325
$189$	0	0
$190$	−3.00000	−0.217643
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	−21.0000	−1.51554
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	0	0
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	−18.0000	−1.27920
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−12.0000	−0.848528
$201$	9.00000	0.634811
$202$	−5.00000	−0.351799
$203$	0	0
$204$	−6.00000	−0.420084
$205$	9.00000	0.628587
$206$	5.00000	0.348367
$207$	0	0
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	−3.00000	−0.206041
$213$	−39.0000	−2.67224
$214$	8.00000	0.546869
$215$	−21.0000	−1.43219
$216$	27.0000	1.83712
$217$	0	0
$218$	7.00000	0.474100
$219$	39.0000	2.63538
$220$	9.00000	0.606780
$221$	0	0
$222$	−6.00000	−0.402694
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	0	0
$225$	24.0000	1.60000
$226$	15.0000	0.997785
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−3.00000	−0.198680
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	−21.0000	−1.37872
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	4.00000	0.260378
$237$	9.00000	0.584613
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	9.00000	0.580948
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	13.0000	0.832240
$245$	0	0
$246$	−9.00000	−0.573819
$247$	0	0
$248$	−9.00000	−0.571501
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	0	0
$254$	11.0000	0.690201
$255$	18.0000	1.12720
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	21.0000	1.30740
$259$	0	0
$260$	0	0
$261$	42.0000	2.59973
$262$	5.00000	0.308901
$263$	−27.0000	−1.66489	−0.832446	−	0.554107i	$-0.813060\pi$
$263$	−27.0000	−1.66489	−0.832446	+	0.554107i	$0.813060\pi$
$264$	−27.0000	−1.66174
$265$	9.00000	0.552866
$266$	0	0
$267$	−18.0000	−1.10158
$268$	3.00000	0.183254
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−27.0000	−1.64317
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	10.0000	0.604122
$275$	−12.0000	−0.723627
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−15.0000	−0.899640
$279$	18.0000	1.07763
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−3.00000	−0.178647
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−13.0000	−0.771408
$285$	9.00000	0.533114
$286$	0	0
$287$	0	0
$288$	30.0000	1.76777
$289$	−13.0000	−0.764706
$290$	21.0000	1.23316
$291$	15.0000	0.879316
$292$	13.0000	0.760767
$293$	11.0000	0.642627	0.321313	−	0.946973i	$-0.395876\pi$
$293$	11.0000	0.642627	0.321313	+	0.946973i	$0.395876\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	−6.00000	−0.348743
$297$	27.0000	1.56670
$298$	15.0000	0.868927
$299$	0	0
$300$	12.0000	0.692820
$301$	0	0
$302$	−21.0000	−1.20841
$303$	15.0000	0.861727
$304$	1.00000	0.0573539
$305$	−39.0000	−2.23313
$306$	−12.0000	−0.685994
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−15.0000	−0.853320
$310$	9.00000	0.511166
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	19.0000	1.07223
$315$	0	0
$316$	3.00000	0.168763
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	−9.00000	−0.504695
$319$	−21.0000	−1.17577
$320$	21.0000	1.17394
$321$	−24.0000	−1.33955
$322$	0	0
$323$	2.00000	0.111283
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−1.00000	−0.0553849
$327$	−21.0000	−1.16130
$328$	−9.00000	−0.496942
$329$	0	0
$330$	27.0000	1.48630
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	−13.0000	−0.711328
$335$	−9.00000	−0.491723
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−45.0000	−2.44406
$340$	6.00000	0.325396
$341$	−9.00000	−0.487377
$342$	−6.00000	−0.324443
$343$	0	0
$344$	21.0000	1.13224
$345$	0	0
$346$	19.0000	1.02145
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	21.0000	1.12572
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	0	0
$351$	0	0
$352$	−15.0000	−0.799503
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	12.0000	0.637793
$355$	39.0000	2.06991
$356$	−6.00000	−0.317999
$357$	0	0
$358$	17.0000	0.898478
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	−54.0000	−2.84605
$361$	−18.0000	−0.947368
$362$	−22.0000	−1.15629
$363$	6.00000	0.314918
$364$	0	0
$365$	−39.0000	−2.04135
$366$	39.0000	2.03856
$367$	−31.0000	−1.61819	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−31.0000	−1.61819	−0.809093	+	0.587680i	$0.800041\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	6.00000	0.311925
$371$	0	0
$372$	9.00000	0.466628
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	6.00000	0.310253
$375$	9.00000	0.464758
$376$	−3.00000	−0.154713
$377$	0	0
$378$	0	0
$379$	−33.0000	−1.69510	−0.847548	−	0.530719i	$-0.821922\pi$
$379$	−33.0000	−1.69510	−0.847548	+	0.530719i	$0.821922\pi$
$380$	3.00000	0.153897
$381$	−33.0000	−1.69064
$382$	−17.0000	−0.869796
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	9.00000	0.459279
$385$	0	0
$386$	7.00000	0.356291
$387$	−42.0000	−2.13498
$388$	5.00000	0.253837
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.0000	−0.756650
$394$	−1.00000	−0.0503793
$395$	−9.00000	−0.452839
$396$	18.0000	0.904534
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	9.00000	0.448879
$403$	0	0
$404$	5.00000	0.248759
$405$	27.0000	1.34164
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−18.0000	−0.891133
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	9.00000	0.444478
$411$	−30.0000	−1.47979
$412$	−5.00000	−0.246332
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	45.0000	2.20366
$418$	3.00000	0.146735
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	7.00000	0.340755
$423$	6.00000	0.291730
$424$	−9.00000	−0.437079
$425$	−8.00000	−0.388057
$426$	−39.0000	−1.88956
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−21.0000	−1.01271
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	9.00000	0.433013
$433$	27.0000	1.29754	0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000	1.29754	0.648769	+	0.760986i	$0.275284\pi$
$434$	0	0
$435$	−63.0000	−3.02062
$436$	−7.00000	−0.335239
$437$	0	0
$438$	39.0000	1.86349
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	27.0000	1.28717
$441$	0	0
$442$	0	0
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	6.00000	0.284747
$445$	18.0000	0.853282
$446$	−9.00000	−0.426162
$447$	−45.0000	−2.12843
$448$	0	0
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	24.0000	1.13137
$451$	−9.00000	−0.423793
$452$	−15.0000	−0.705541
$453$	63.0000	2.96000
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−9.00000	−0.421464
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−13.0000	−0.607450
$459$	18.0000	0.840168
$460$	0	0
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−7.00000	−0.324967
$465$	−27.0000	−1.25210
$466$	−21.0000	−0.972806
$467$	−7.00000	−0.323921	−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000	−0.323921	−0.161961	+	0.986797i	$0.551782\pi$
$468$	0	0
$469$	0	0
$470$	3.00000	0.138380
$471$	−57.0000	−2.62642
$472$	12.0000	0.552345
$473$	21.0000	0.965581
$474$	9.00000	0.413384
$475$	−4.00000	−0.183533
$476$	0	0
$477$	18.0000	0.824163
$478$	−4.00000	−0.182956
$479$	−35.0000	−1.59919	−0.799595	−	0.600539i	$-0.794953\pi$
$479$	−35.0000	−1.59919	−0.799595	+	0.600539i	$0.794953\pi$
$480$	−45.0000	−2.05396
$481$	0	0
$482$	−26.0000	−1.18427
$483$	0	0
$484$	2.00000	0.0909091
$485$	−15.0000	−0.681115
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	39.0000	1.76545
$489$	3.00000	0.135665
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	9.00000	0.405751
$493$	−14.0000	−0.630528
$494$	0	0
$495$	−54.0000	−2.42712
$496$	−3.00000	−0.134704
$497$	0	0
$498$	0	0
$499$	−31.0000	−1.38775	−0.693875	−	0.720095i	$-0.744098\pi$
$499$	−31.0000	−1.38775	−0.693875	+	0.720095i	$0.744098\pi$
$500$	3.00000	0.134164
$501$	39.0000	1.74239
$502$	−23.0000	−1.02654
$503$	−31.0000	−1.38222	−0.691111	−	0.722749i	$-0.742878\pi$
$503$	−31.0000	−1.38222	−0.691111	+	0.722749i	$0.742878\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	0	0
$507$	0	0
$508$	−11.0000	−0.488046
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	18.0000	0.797053
$511$	0	0
$512$	−11.0000	−0.486136
$513$	9.00000	0.397360
$514$	−2.00000	−0.0882162
$515$	15.0000	0.660979
$516$	−21.0000	−0.924473
$517$	−3.00000	−0.131940
$518$	0	0
$519$	−57.0000	−2.50202
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	42.0000	1.83829
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−5.00000	−0.218426
$525$	0	0
$526$	−27.0000	−1.17726
$527$	−6.00000	−0.261364
$528$	−9.00000	−0.391675
$529$	−23.0000	−1.00000
$530$	9.00000	0.390935
$531$	−24.0000	−1.04151
$532$	0	0
$533$	0	0
$534$	−18.0000	−0.778936
$535$	24.0000	1.03761
$536$	9.00000	0.388741
$537$	−51.0000	−2.20081
$538$	18.0000	0.776035
$539$	0	0
$540$	27.0000	1.16190
$541$	−37.0000	−1.59075	−0.795377	−	0.606115i	$-0.792727\pi$
$541$	−37.0000	−1.59075	−0.795377	+	0.606115i	$0.792727\pi$
$542$	−16.0000	−0.687259
$543$	66.0000	2.83233
$544$	−10.0000	−0.428746
$545$	21.0000	0.899541
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−10.0000	−0.427179
$549$	−78.0000	−3.32896
$550$	−12.0000	−0.511682
$551$	−7.00000	−0.298210
$552$	0	0
$553$	0	0
$554$	22.0000	0.934690
$555$	−18.0000	−0.764057
$556$	15.0000	0.636142
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	18.0000	0.762001
$559$	0	0
$560$	0	0
$561$	−18.0000	−0.759961
$562$	−18.0000	−0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	3.00000	0.126323
$565$	45.0000	1.89316
$566$	1.00000	0.0420331
$567$	0	0
$568$	−39.0000	−1.63640
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	9.00000	0.376969
$571$	43.0000	1.79949	0.899747	−	0.436412i	$-0.143751\pi$
$571$	43.0000	1.79949	0.899747	+	0.436412i	$0.143751\pi$
$572$	0	0
$573$	51.0000	2.13056
$574$	0	0
$575$	0	0
$576$	42.0000	1.75000
$577$	−1.00000	−0.0416305	−0.0208153	−	0.999783i	$-0.506626\pi$
$577$	−1.00000	−0.0416305	−0.0208153	+	0.999783i	$0.506626\pi$
$578$	−13.0000	−0.540729
$579$	−21.0000	−0.872730
$580$	−21.0000	−0.871978
$581$	0	0
$582$	15.0000	0.621770
$583$	−9.00000	−0.372742
$584$	39.0000	1.61383
$585$	0	0
$586$	11.0000	0.454406
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	−12.0000	−0.494032
$591$	3.00000	0.123404
$592$	−2.00000	−0.0821995
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	27.0000	1.10782
$595$	0	0
$596$	−15.0000	−0.614424
$597$	60.0000	2.45564
$598$	0	0
$599$	−25.0000	−1.02147	−0.510736	−	0.859738i	$-0.670627\pi$
$599$	−25.0000	−1.02147	−0.510736	+	0.859738i	$0.670627\pi$
$600$	36.0000	1.46969
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	−18.0000	−0.733017
$604$	21.0000	0.854478
$605$	−6.00000	−0.243935
$606$	15.0000	0.609333
$607$	11.0000	0.446476	0.223238	−	0.974764i	$-0.428337\pi$
$607$	11.0000	0.446476	0.223238	+	0.974764i	$0.428337\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	−39.0000	−1.57906
$611$	0	0
$612$	12.0000	0.485071
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	12.0000	0.484281
$615$	−27.0000	−1.08875
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	−15.0000	−0.603388
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	−9.00000	−0.361449
$621$	0	0
$622$	−9.00000	−0.360867
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	19.0000	0.759393
$627$	−9.00000	−0.359425
$628$	−19.0000	−0.758183
$629$	−4.00000	−0.159490
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	9.00000	0.358001
$633$	−21.0000	−0.834675
$634$	−9.00000	−0.357436
$635$	33.0000	1.30957
$636$	9.00000	0.356873
$637$	0	0
$638$	−21.0000	−0.831398
$639$	78.0000	3.08563
$640$	−9.00000	−0.355756
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−24.0000	−0.947204
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	63.0000	2.48062
$646$	2.00000	0.0786889
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	−27.0000	−1.06066
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	1.00000	0.0391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−21.0000	−0.821165
$655$	15.0000	0.586098
$656$	−3.00000	−0.117130
$657$	−78.0000	−3.04307
$658$	0	0
$659$	29.0000	1.12968	0.564840	−	0.825201i	$-0.308938\pi$
$659$	29.0000	1.12968	0.564840	+	0.825201i	$0.308938\pi$
$660$	−27.0000	−1.05097
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	−29.0000	−1.12712
$663$	0	0
$664$	0	0
$665$	0	0
$666$	12.0000	0.464991
$667$	0	0
$668$	13.0000	0.502985
$669$	27.0000	1.04388
$670$	−9.00000	−0.347700
$671$	39.0000	1.50558
$672$	0	0
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	14.0000	0.539260
$675$	−36.0000	−1.38564
$676$	0	0
$677$	7.00000	0.269032	0.134516	−	0.990911i	$-0.457052\pi$
$677$	7.00000	0.269032	0.134516	+	0.990911i	$0.457052\pi$
$678$	−45.0000	−1.72821
$679$	0	0
$680$	18.0000	0.690268
$681$	12.0000	0.459841
$682$	−9.00000	−0.344628
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	6.00000	0.229416
$685$	30.0000	1.14624
$686$	0	0
$687$	39.0000	1.48794
$688$	7.00000	0.266872
$689$	0	0
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−19.0000	−0.722272
$693$	0	0
$694$	−8.00000	−0.303676
$695$	−45.0000	−1.70695
$696$	63.0000	2.38801
$697$	−6.00000	−0.227266
$698$	23.0000	0.870563
$699$	63.0000	2.38288
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	−21.0000	−0.791467
$705$	−9.00000	−0.338960
$706$	−25.0000	−0.940887
$707$	0	0
$708$	−12.0000	−0.450988
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	39.0000	1.46364
$711$	−18.0000	−0.675053
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−17.0000	−0.635320
$717$	12.0000	0.448148
$718$	17.0000	0.634434
$719$	−9.00000	−0.335643	−0.167822	−	0.985817i	$-0.553673\pi$
$719$	−9.00000	−0.335643	−0.167822	+	0.985817i	$0.553673\pi$
$720$	−18.0000	−0.670820
$721$	0	0
$722$	−18.0000	−0.669891
$723$	78.0000	2.90085
$724$	22.0000	0.817624
$725$	28.0000	1.03989
$726$	6.00000	0.222681
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−39.0000	−1.44345
$731$	14.0000	0.517809
$732$	−39.0000	−1.44148
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	−31.0000	−1.14423
$735$	0	0
$736$	0	0
$737$	9.00000	0.331519
$738$	18.0000	0.662589
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	0	0
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	27.0000	0.989868
$745$	45.0000	1.64867
$746$	−9.00000	−0.329513
$747$	0	0
$748$	−6.00000	−0.219382
$749$	0	0
$750$	9.00000	0.328634
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	−1.00000	−0.0364662
$753$	69.0000	2.51450
$754$	0	0
$755$	−63.0000	−2.29280
$756$	0	0
$757$	3.00000	0.109037	0.0545184	−	0.998513i	$-0.482638\pi$
$757$	3.00000	0.109037	0.0545184	+	0.998513i	$0.482638\pi$
$758$	−33.0000	−1.19861
$759$	0	0
$760$	9.00000	0.326464
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	−33.0000	−1.19546
$763$	0	0
$764$	17.0000	0.615038
$765$	−36.0000	−1.30158
$766$	−21.0000	−0.758761
$767$	0	0
$768$	51.0000	1.84030
$769$	19.0000	0.685158	0.342579	−	0.939489i	$-0.388700\pi$
$769$	19.0000	0.685158	0.342579	+	0.939489i	$0.388700\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−7.00000	−0.251936
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−42.0000	−1.50966
$775$	12.0000	0.431053
$776$	15.0000	0.538469
$777$	0	0
$778$	−33.0000	−1.18311
$779$	−3.00000	−0.107486
$780$	0	0
$781$	−39.0000	−1.39553
$782$	0	0
$783$	−63.0000	−2.25144
$784$	0	0
$785$	57.0000	2.03442
$786$	−15.0000	−0.535032
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	1.00000	0.0356235
$789$	81.0000	2.88368
$790$	−9.00000	−0.320206
$791$	0	0
$792$	54.0000	1.91881
$793$	0	0
$794$	−1.00000	−0.0354887
$795$	−27.0000	−0.957591
$796$	20.0000	0.708881
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	0	0
$799$	−2.00000	−0.0707549
$800$	20.0000	0.707107
$801$	36.0000	1.27200
$802$	−2.00000	−0.0706225
$803$	39.0000	1.37628
$804$	−9.00000	−0.317406
$805$	0	0
$806$	0	0
$807$	−54.0000	−1.90089
$808$	15.0000	0.527698
$809$	11.0000	0.386739	0.193370	−	0.981126i	$-0.438058\pi$
$809$	11.0000	0.386739	0.193370	+	0.981126i	$0.438058\pi$
$810$	27.0000	0.948683
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	48.0000	1.68343
$814$	−6.00000	−0.210300
$815$	−3.00000	−0.105085
$816$	−6.00000	−0.210042
$817$	7.00000	0.244899
$818$	14.0000	0.489499
$819$	0	0
$820$	−9.00000	−0.314294
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	−30.0000	−1.04637
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−15.0000	−0.522550
$825$	36.0000	1.25336
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−66.0000	−2.28951
$832$	0	0
$833$	0	0
$834$	45.0000	1.55822
$835$	−39.0000	−1.34965
$836$	−3.00000	−0.103757
$837$	−27.0000	−0.933257
$838$	−25.0000	−0.863611
$839$	37.0000	1.27738	0.638691	−	0.769463i	$-0.279476\pi$
$839$	37.0000	1.27738	0.638691	+	0.769463i	$0.279476\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	18.0000	0.620321
$843$	54.0000	1.85986
$844$	−7.00000	−0.240950
$845$	0	0
$846$	6.00000	0.206284
$847$	0	0
$848$	−3.00000	−0.103020
$849$	−3.00000	−0.102960
$850$	−8.00000	−0.274398
$851$	0	0
$852$	39.0000	1.33612
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	−24.0000	−0.820303
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	21.0000	0.716094
$861$	0	0
$862$	9.00000	0.306541
$863$	37.0000	1.25949	0.629747	−	0.776800i	$-0.283158\pi$
$863$	37.0000	1.25949	0.629747	+	0.776800i	$0.283158\pi$
$864$	−45.0000	−1.53093
$865$	57.0000	1.93806
$866$	27.0000	0.917497
$867$	39.0000	1.32451
$868$	0	0
$869$	9.00000	0.305304
$870$	−63.0000	−2.13590
$871$	0	0
$872$	−21.0000	−0.711150
$873$	−30.0000	−1.01535
$874$	0	0
$875$	0	0
$876$	−39.0000	−1.31769
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	−16.0000	−0.539974
$879$	−33.0000	−1.11306
$880$	9.00000	0.303390
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	−11.0000	−0.369552
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	18.0000	0.603361
$891$	−27.0000	−0.904534
$892$	9.00000	0.301342
$893$	−1.00000	−0.0334637
$894$	−45.0000	−1.50503
$895$	51.0000	1.70474
$896$	0	0
$897$	0	0
$898$	15.0000	0.500556
$899$	21.0000	0.700389
$900$	−24.0000	−0.800000
$901$	−6.00000	−0.199889
$902$	−9.00000	−0.299667
$903$	0	0
$904$	−45.0000	−1.49668
$905$	−66.0000	−2.19391
$906$	63.0000	2.09303
$907$	−47.0000	−1.56061	−0.780305	−	0.625400i	$-0.784936\pi$
$907$	−47.0000	−1.56061	−0.780305	+	0.625400i	$0.784936\pi$
$908$	4.00000	0.132745
$909$	−30.0000	−0.995037
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−3.00000	−0.0993399
$913$	0	0
$914$	−18.0000	−0.595387
$915$	117.000	3.86790
$916$	13.0000	0.429532
$917$	0	0
$918$	18.0000	0.594089
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	−36.0000	−1.18624
$922$	35.0000	1.15266
$923$	0	0
$924$	0	0
$925$	8.00000	0.263038
$926$	−8.00000	−0.262896
$927$	30.0000	0.985329
$928$	35.0000	1.14893
$929$	−13.0000	−0.426516	−0.213258	−	0.976996i	$-0.568408\pi$
$929$	−13.0000	−0.426516	−0.213258	+	0.976996i	$0.568408\pi$
$930$	−27.0000	−0.885365
$931$	0	0
$932$	21.0000	0.687878
$933$	27.0000	0.883940
$934$	−7.00000	−0.229047
$935$	18.0000	0.588663
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−57.0000	−1.86012
$940$	−3.00000	−0.0978492
$941$	−17.0000	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$942$	−57.0000	−1.85716
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	21.0000	0.682769
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−9.00000	−0.292306
$949$	0	0
$950$	−4.00000	−0.129777
$951$	27.0000	0.875535
$952$	0	0
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	18.0000	0.582772
$955$	−51.0000	−1.65032
$956$	4.00000	0.129369
$957$	63.0000	2.03650
$958$	−35.0000	−1.13080
$959$	0	0
$960$	−63.0000	−2.03332
$961$	−22.0000	−0.709677
$962$	0	0
$963$	48.0000	1.54678
$964$	26.0000	0.837404
$965$	21.0000	0.676014
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	6.00000	0.192847
$969$	−6.00000	−0.192748
$970$	−15.0000	−0.481621
$971$	−1.00000	−0.0320915	−0.0160458	−	0.999871i	$-0.505108\pi$
$971$	−1.00000	−0.0320915	−0.0160458	+	0.999871i	$0.505108\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	13.0000	0.416120
$977$	−5.00000	−0.159964	−0.0799821	−	0.996796i	$-0.525486\pi$
$977$	−5.00000	−0.159964	−0.0799821	+	0.996796i	$0.525486\pi$
$978$	3.00000	0.0959294
$979$	−18.0000	−0.575282
$980$	0	0
$981$	42.0000	1.34096
$982$	15.0000	0.478669
$983$	47.0000	1.49907	0.749534	−	0.661966i	$-0.230278\pi$
$983$	47.0000	1.49907	0.749534	+	0.661966i	$0.230278\pi$
$984$	27.0000	0.860729
$985$	−3.00000	−0.0955879
$986$	−14.0000	−0.445851
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−54.0000	−1.71623
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	15.0000	0.476250
$993$	87.0000	2.76086
$994$	0	0
$995$	−60.0000	−1.90213
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−31.0000	−0.981288
$999$	−18.0000	−0.569495

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.i.1.1		1
7.2	even	3		1183.2.e.a.508.1		2
7.4	even	3		1183.2.e.a.170.1		2
7.6	odd	2		8281.2.a.j.1.1		1
13.3	even	3		637.2.f.b.295.1		2
13.9	even	3		637.2.f.b.393.1		2
13.12	even	2		8281.2.a.c.1.1		1
91.3	odd	6		637.2.g.a.373.1		2
91.9	even	3		91.2.g.a.81.1	yes	2
91.16	even	3		91.2.h.a.74.1	yes	2
91.25	even	6		1183.2.e.c.170.1		2
91.48	odd	6		637.2.f.a.393.1		2
91.51	even	6		1183.2.e.c.508.1		2
91.55	odd	6		637.2.f.a.295.1		2
91.61	odd	6		637.2.g.a.263.1		2
91.68	odd	6		637.2.h.a.165.1		2
91.74	even	3		91.2.h.a.16.1	yes	2
91.81	even	3		91.2.g.a.9.1	✓	2
91.87	odd	6		637.2.h.a.471.1		2
91.90	odd	2		8281.2.a.g.1.1		1
273.74	odd	6		819.2.s.a.289.1		2
273.107	odd	6		819.2.s.a.802.1		2
273.191	odd	6		819.2.n.c.172.1		2
273.263	odd	6		819.2.n.c.100.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.a.9.1	✓	2	91.81	even	3
91.2.g.a.81.1	yes	2	91.9	even	3
91.2.h.a.16.1	yes	2	91.74	even	3
91.2.h.a.74.1	yes	2	91.16	even	3
637.2.f.a.295.1		2	91.55	odd	6
637.2.f.a.393.1		2	91.48	odd	6
637.2.f.b.295.1		2	13.3	even	3
637.2.f.b.393.1		2	13.9	even	3
637.2.g.a.263.1		2	91.61	odd	6
637.2.g.a.373.1		2	91.3	odd	6
637.2.h.a.165.1		2	91.68	odd	6
637.2.h.a.471.1		2	91.87	odd	6
819.2.n.c.100.1		2	273.263	odd	6
819.2.n.c.172.1		2	273.191	odd	6
819.2.s.a.289.1		2	273.74	odd	6
819.2.s.a.802.1		2	273.107	odd	6
1183.2.e.a.170.1		2	7.4	even	3
1183.2.e.a.508.1		2	7.2	even	3
1183.2.e.c.170.1		2	91.25	even	6
1183.2.e.c.508.1		2	91.51	even	6
8281.2.a.c.1.1		1	13.12	even	2
8281.2.a.g.1.1		1	91.90	odd	2
8281.2.a.i.1.1		1	1.1	even	1	trivial
8281.2.a.j.1.1		1	7.6	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8281.2.a.i.1.1

Level	$8281$
Weight	$2$
Character	8281.1
Self dual	yes
Analytic conductor	$66.124$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$