LMFDB - Embedded newform 1078.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} -4.24264 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} -4.24264 q^{10} -1.00000 q^{11} -1.41421 q^{12} +6.00000 q^{15} +1.00000 q^{16} +5.65685 q^{17} -1.00000 q^{18} -4.24264 q^{20} -1.00000 q^{22} +6.00000 q^{23} -1.41421 q^{24} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +6.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} +1.41421 q^{33} +5.65685 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.24264 q^{40} +11.3137 q^{41} -8.00000 q^{43} -1.00000 q^{44} +4.24264 q^{45} +6.00000 q^{46} -4.24264 q^{47} -1.41421 q^{48} +13.0000 q^{50} -8.00000 q^{51} +8.00000 q^{53} +5.65685 q^{54} +4.24264 q^{55} +2.00000 q^{58} -1.41421 q^{59} +6.00000 q^{60} +2.82843 q^{61} +1.41421 q^{62} +1.00000 q^{64} +1.41421 q^{66} +2.00000 q^{67} +5.65685 q^{68} -8.48528 q^{69} -2.00000 q^{71} -1.00000 q^{72} -8.48528 q^{73} -10.0000 q^{74} -18.3848 q^{75} +16.0000 q^{79} -4.24264 q^{80} -5.00000 q^{81} +11.3137 q^{82} +16.9706 q^{83} -24.0000 q^{85} -8.00000 q^{86} -2.82843 q^{87} -1.00000 q^{88} +7.07107 q^{89} +4.24264 q^{90} +6.00000 q^{92} -2.00000 q^{93} -4.24264 q^{94} -1.41421 q^{96} +9.89949 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{11} + 12 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{22} + 12 q^{23} + 26 q^{25} + 4 q^{29} + 12 q^{30} + 2 q^{32} - 2 q^{36} - 20 q^{37} - 16 q^{43} - 2 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^11 + 12 * q^15 + 2 * q^16 - 2 * q^18 - 2 * q^22 + 12 * q^23 + 26 * q^25 + 4 * q^29 + 12 * q^30 + 2 * q^32 - 2 * q^36 - 20 * q^37 - 16 * q^43 - 2 * q^44 + 12 * q^46 + 26 * q^50 - 16 * q^51 + 16 * q^53 + 4 * q^58 + 12 * q^60 + 2 * q^64 + 4 * q^67 - 4 * q^71 - 2 * q^72 - 20 * q^74 + 32 * q^79 - 10 * q^81 - 48 * q^85 - 16 * q^86 - 2 * q^88 + 12 * q^92 - 4 * q^93 + 2 * q^99

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	−4.24264	−1.89737	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−4.24264	−1.89737	−0.948683	+	0.316228i	$0.897584\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	−4.24264	−1.34164
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.41421	−0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	6.00000	1.54919
$16$	1.00000	0.250000
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−4.24264	−0.948683
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.41421	−0.288675
$25$	13.0000	2.60000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	6.00000	1.09545
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	1.00000	0.176777
$33$	1.41421	0.246183
$34$	5.65685	0.970143
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	−4.24264	−0.670820
$41$	11.3137	1.76690	0.883452	−	0.468521i	$-0.155213\pi$
$41$	11.3137	1.76690	0.883452	+	0.468521i	$0.155213\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	4.24264	0.632456
$46$	6.00000	0.884652
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	13.0000	1.83848
$51$	−8.00000	−1.12022
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	5.65685	0.769800
$55$	4.24264	0.572078
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	6.00000	0.774597
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	1.41421	0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.41421	0.174078
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	5.65685	0.685994
$69$	−8.48528	−1.02151
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	−10.0000	−1.16248
$75$	−18.3848	−2.12289
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−4.24264	−0.474342
$81$	−5.00000	−0.555556
$82$	11.3137	1.24939
$83$	16.9706	1.86276	0.931381	−	0.364047i	$-0.118605\pi$
$83$	16.9706	1.86276	0.931381	+	0.364047i	$0.118605\pi$
$84$	0	0
$85$	−24.0000	−2.60317
$86$	−8.00000	−0.862662
$87$	−2.82843	−0.303239
$88$	−1.00000	−0.106600
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	4.24264	0.447214
$91$	0	0
$92$	6.00000	0.625543
$93$	−2.00000	−0.207390
$94$	−4.24264	−0.437595
$95$	0	0
$96$	−1.41421	−0.144338
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	13.0000	1.30000
$101$	−5.65685	−0.562878	−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685	−0.562878	−0.281439	+	0.959579i	$0.590812\pi$
$102$	−8.00000	−0.792118
$103$	−18.3848	−1.81151	−0.905753	−	0.423806i	$-0.860694\pi$
$103$	−18.3848	−1.81151	−0.905753	+	0.423806i	$0.860694\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	5.65685	0.544331
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	4.24264	0.404520
$111$	14.1421	1.34231
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−25.4558	−2.37377
$116$	2.00000	0.185695
$117$	0	0
$118$	−1.41421	−0.130189
$119$	0	0
$120$	6.00000	0.547723
$121$	1.00000	0.0909091
$122$	2.82843	0.256074
$123$	−16.0000	−1.44267
$124$	1.41421	0.127000
$125$	−33.9411	−3.03579
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	11.3137	0.996116
$130$	0	0
$131$	19.7990	1.72985	0.864923	−	0.501905i	$-0.167367\pi$
$131$	19.7990	1.72985	0.864923	+	0.501905i	$0.167367\pi$
$132$	1.41421	0.123091
$133$	0	0
$134$	2.00000	0.172774
$135$	−24.0000	−2.06559
$136$	5.65685	0.485071
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−8.48528	−0.722315
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−2.00000	−0.167836
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−8.48528	−0.704664
$146$	−8.48528	−0.702247
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−18.3848	−1.50111
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	−5.65685	−0.457330
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−4.24264	−0.338600	−0.169300	−	0.985565i	$-0.554151\pi$
$157$	−4.24264	−0.338600	−0.169300	+	0.985565i	$0.554151\pi$
$158$	16.0000	1.27289
$159$	−11.3137	−0.897235
$160$	−4.24264	−0.335410
$161$	0	0
$162$	−5.00000	−0.392837
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	11.3137	0.883452
$165$	−6.00000	−0.467099
$166$	16.9706	1.31717
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−24.0000	−1.84072
$171$	0	0
$172$	−8.00000	−0.609994
$173$	11.3137	0.860165	0.430083	−	0.902790i	$-0.358484\pi$
$173$	11.3137	0.860165	0.430083	+	0.902790i	$0.358484\pi$
$174$	−2.82843	−0.214423
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	2.00000	0.150329
$178$	7.07107	0.529999
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	4.24264	0.316228
$181$	7.07107	0.525588	0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107	0.525588	0.262794	+	0.964852i	$0.415356\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	6.00000	0.442326
$185$	42.4264	3.11925
$186$	−2.00000	−0.146647
$187$	−5.65685	−0.413670
$188$	−4.24264	−0.309426
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−1.41421	−0.102062
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	9.89949	0.710742
$195$	0	0
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	1.00000	0.0710669
$199$	1.41421	0.100251	0.0501255	−	0.998743i	$-0.484038\pi$
$199$	1.41421	0.100251	0.0501255	+	0.998743i	$0.484038\pi$
$200$	13.0000	0.919239
$201$	−2.82843	−0.199502
$202$	−5.65685	−0.398015
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−48.0000	−3.35247
$206$	−18.3848	−1.28093
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	8.00000	0.549442
$213$	2.82843	0.193801
$214$	16.0000	1.09374
$215$	33.9411	2.31477
$216$	5.65685	0.384900
$217$	0	0
$218$	−2.00000	−0.135457
$219$	12.0000	0.810885
$220$	4.24264	0.286039
$221$	0	0
$222$	14.1421	0.949158
$223$	21.2132	1.42054	0.710271	−	0.703929i	$-0.248573\pi$
$223$	21.2132	1.42054	0.710271	+	0.703929i	$0.248573\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	−2.00000	−0.133038
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	−9.89949	−0.654177	−0.327089	−	0.944994i	$-0.606068\pi$
$229$	−9.89949	−0.654177	−0.327089	+	0.944994i	$0.606068\pi$
$230$	−25.4558	−1.67851
$231$	0	0
$232$	2.00000	0.131306
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	−1.41421	−0.0920575
$237$	−22.6274	−1.46981
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	6.00000	0.387298
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	1.00000	0.0642824
$243$	−9.89949	−0.635053
$244$	2.82843	0.181071
$245$	0	0
$246$	−16.0000	−1.02012
$247$	0	0
$248$	1.41421	0.0898027
$249$	−24.0000	−1.52094
$250$	−33.9411	−2.14663
$251$	−18.3848	−1.16044	−0.580218	−	0.814461i	$-0.697033\pi$
$251$	−18.3848	−1.16044	−0.580218	+	0.814461i	$0.697033\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	16.0000	1.00393
$255$	33.9411	2.12548
$256$	1.00000	0.0625000
$257$	−1.41421	−0.0882162	−0.0441081	−	0.999027i	$-0.514045\pi$
$257$	−1.41421	−0.0882162	−0.0441081	+	0.999027i	$0.514045\pi$
$258$	11.3137	0.704361
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	19.7990	1.22319
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	1.41421	0.0870388
$265$	−33.9411	−2.08499
$266$	0	0
$267$	−10.0000	−0.611990
$268$	2.00000	0.122169
$269$	−18.3848	−1.12094	−0.560470	−	0.828175i	$-0.689379\pi$
$269$	−18.3848	−1.12094	−0.560470	+	0.828175i	$0.689379\pi$
$270$	−24.0000	−1.46059
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−13.0000	−0.783929
$276$	−8.48528	−0.510754
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−11.3137	−0.678551
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	6.00000	0.357295
$283$	19.7990	1.17693	0.588464	−	0.808523i	$-0.299733\pi$
$283$	19.7990	1.17693	0.588464	+	0.808523i	$0.299733\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	15.0000	0.882353
$290$	−8.48528	−0.498273
$291$	−14.0000	−0.820695
$292$	−8.48528	−0.496564
$293$	8.48528	0.495715	0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528	0.495715	0.247858	+	0.968796i	$0.420273\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	−10.0000	−0.581238
$297$	−5.65685	−0.328244
$298$	−10.0000	−0.579284
$299$	0	0
$300$	−18.3848	−1.06145
$301$	0	0
$302$	4.00000	0.230174
$303$	8.00000	0.459588
$304$	0	0
$305$	−12.0000	−0.687118
$306$	−5.65685	−0.323381
$307$	25.4558	1.45284	0.726421	−	0.687250i	$-0.241182\pi$
$307$	25.4558	1.45284	0.726421	+	0.687250i	$0.241182\pi$
$308$	0	0
$309$	26.0000	1.47909
$310$	−6.00000	−0.340777
$311$	−18.3848	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$311$	−18.3848	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	−4.24264	−0.239426
$315$	0	0
$316$	16.0000	0.900070
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−11.3137	−0.634441
$319$	−2.00000	−0.111979
$320$	−4.24264	−0.237171
$321$	−22.6274	−1.26294
$322$	0	0
$323$	0	0
$324$	−5.00000	−0.277778
$325$	0	0
$326$	−10.0000	−0.553849
$327$	2.82843	0.156412
$328$	11.3137	0.624695
$329$	0	0
$330$	−6.00000	−0.330289
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	16.9706	0.931381
$333$	10.0000	0.547997
$334$	−5.65685	−0.309529
$335$	−8.48528	−0.463600
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−13.0000	−0.707107
$339$	2.82843	0.153619
$340$	−24.0000	−1.30158
$341$	−1.41421	−0.0765840
$342$	0	0
$343$	0	0
$344$	−8.00000	−0.431331
$345$	36.0000	1.93817
$346$	11.3137	0.608229
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−2.82843	−0.151620
$349$	−14.1421	−0.757011	−0.378506	−	0.925599i	$-0.623562\pi$
$349$	−14.1421	−0.757011	−0.378506	+	0.925599i	$0.623562\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−1.41421	−0.0752710	−0.0376355	−	0.999292i	$-0.511983\pi$
$353$	−1.41421	−0.0752710	−0.0376355	+	0.999292i	$0.511983\pi$
$354$	2.00000	0.106299
$355$	8.48528	0.450352
$356$	7.07107	0.374766
$357$	0	0
$358$	12.0000	0.634220
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	4.24264	0.223607
$361$	−19.0000	−1.00000
$362$	7.07107	0.371647
$363$	−1.41421	−0.0742270
$364$	0	0
$365$	36.0000	1.88433
$366$	−4.00000	−0.209083
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	6.00000	0.312772
$369$	−11.3137	−0.588968
$370$	42.4264	2.20564
$371$	0	0
$372$	−2.00000	−0.103695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−5.65685	−0.292509
$375$	48.0000	2.47871
$376$	−4.24264	−0.218797
$377$	0	0
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−22.6274	−1.15924
$382$	16.0000	0.818631
$383$	−15.5563	−0.794892	−0.397446	−	0.917625i	$-0.630103\pi$
$383$	−15.5563	−0.794892	−0.397446	+	0.917625i	$0.630103\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	−6.00000	−0.305392
$387$	8.00000	0.406663
$388$	9.89949	0.502571
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	33.9411	1.71648
$392$	0	0
$393$	−28.0000	−1.41241
$394$	22.0000	1.10834
$395$	−67.8823	−3.41553
$396$	1.00000	0.0502519
$397$	−12.7279	−0.638796	−0.319398	−	0.947621i	$-0.603481\pi$
$397$	−12.7279	−0.638796	−0.319398	+	0.947621i	$0.603481\pi$
$398$	1.41421	0.0708881
$399$	0	0
$400$	13.0000	0.650000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−2.82843	−0.141069
$403$	0	0
$404$	−5.65685	−0.281439
$405$	21.2132	1.05409
$406$	0	0
$407$	10.0000	0.495682
$408$	−8.00000	−0.396059
$409$	2.82843	0.139857	0.0699284	−	0.997552i	$-0.477723\pi$
$409$	2.82843	0.139857	0.0699284	+	0.997552i	$0.477723\pi$
$410$	−48.0000	−2.37055
$411$	25.4558	1.25564
$412$	−18.3848	−0.905753
$413$	0	0
$414$	−6.00000	−0.294884
$415$	−72.0000	−3.53434
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	8.00000	0.389434
$423$	4.24264	0.206284
$424$	8.00000	0.388514
$425$	73.5391	3.56717
$426$	2.82843	0.137038
$427$	0	0
$428$	16.0000	0.773389
$429$	0	0
$430$	33.9411	1.63679
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	5.65685	0.272166
$433$	12.7279	0.611665	0.305832	−	0.952085i	$-0.401065\pi$
$433$	12.7279	0.611665	0.305832	+	0.952085i	$0.401065\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	12.0000	0.573382
$439$	25.4558	1.21494	0.607471	−	0.794342i	$-0.292184\pi$
$439$	25.4558	1.21494	0.607471	+	0.794342i	$0.292184\pi$
$440$	4.24264	0.202260
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	14.1421	0.671156
$445$	−30.0000	−1.42214
$446$	21.2132	1.00447
$447$	14.1421	0.668900
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	−13.0000	−0.612826
$451$	−11.3137	−0.532742
$452$	−2.00000	−0.0940721
$453$	−5.65685	−0.265782
$454$	14.1421	0.663723
$455$	0	0
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−9.89949	−0.462573
$459$	32.0000	1.49363
$460$	−25.4558	−1.18688
$461$	−2.82843	−0.131733	−0.0658665	−	0.997828i	$-0.520981\pi$
$461$	−2.82843	−0.131733	−0.0658665	+	0.997828i	$0.520981\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	2.00000	0.0928477
$465$	8.48528	0.393496
$466$	14.0000	0.648537
$467$	4.24264	0.196326	0.0981630	−	0.995170i	$-0.468703\pi$
$467$	4.24264	0.196326	0.0981630	+	0.995170i	$0.468703\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	6.00000	0.276465
$472$	−1.41421	−0.0650945
$473$	8.00000	0.367840
$474$	−22.6274	−1.03931
$475$	0	0
$476$	0	0
$477$	−8.00000	−0.366295
$478$	−12.0000	−0.548867
$479$	−2.82843	−0.129234	−0.0646171	−	0.997910i	$-0.520583\pi$
$479$	−2.82843	−0.129234	−0.0646171	+	0.997910i	$0.520583\pi$
$480$	6.00000	0.273861
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.00000	0.0454545
$485$	−42.0000	−1.90712
$486$	−9.89949	−0.449050
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	2.82843	0.128037
$489$	14.1421	0.639529
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−16.0000	−0.721336
$493$	11.3137	0.509544
$494$	0	0
$495$	−4.24264	−0.190693
$496$	1.41421	0.0635001
$497$	0	0
$498$	−24.0000	−1.07547
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	−33.9411	−1.51789
$501$	8.00000	0.357414
$502$	−18.3848	−0.820553
$503$	−31.1127	−1.38725	−0.693623	−	0.720338i	$-0.743987\pi$
$503$	−31.1127	−1.38725	−0.693623	+	0.720338i	$0.743987\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	−6.00000	−0.266733
$507$	18.3848	0.816497
$508$	16.0000	0.709885
$509$	−12.7279	−0.564155	−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279	−0.564155	−0.282078	+	0.959392i	$0.591024\pi$
$510$	33.9411	1.50294
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−1.41421	−0.0623783
$515$	78.0000	3.43709
$516$	11.3137	0.498058
$517$	4.24264	0.186591
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	−2.00000	−0.0875376
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	19.7990	0.864923
$525$	0	0
$526$	−8.00000	−0.348817
$527$	8.00000	0.348485
$528$	1.41421	0.0615457
$529$	13.0000	0.565217
$530$	−33.9411	−1.47431
$531$	1.41421	0.0613716
$532$	0	0
$533$	0	0
$534$	−10.0000	−0.432742
$535$	−67.8823	−2.93481
$536$	2.00000	0.0863868
$537$	−16.9706	−0.732334
$538$	−18.3848	−0.792624
$539$	0	0
$540$	−24.0000	−1.03280
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	8.48528	0.364474
$543$	−10.0000	−0.429141
$544$	5.65685	0.242536
$545$	8.48528	0.363470
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−18.0000	−0.768922
$549$	−2.82843	−0.120714
$550$	−13.0000	−0.554322
$551$	0	0
$552$	−8.48528	−0.361158
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	−60.0000	−2.54686
$556$	−11.3137	−0.479808
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−1.41421	−0.0598684
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	−14.0000	−0.590554
$563$	−22.6274	−0.953632	−0.476816	−	0.879003i	$-0.658209\pi$
$563$	−22.6274	−0.953632	−0.476816	+	0.879003i	$0.658209\pi$
$564$	6.00000	0.252646
$565$	8.48528	0.356978
$566$	19.7990	0.832214
$567$	0	0
$568$	−2.00000	−0.0839181
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	−22.6274	−0.945274
$574$	0	0
$575$	78.0000	3.25282
$576$	−1.00000	−0.0416667
$577$	7.07107	0.294372	0.147186	−	0.989109i	$-0.452978\pi$
$577$	7.07107	0.294372	0.147186	+	0.989109i	$0.452978\pi$
$578$	15.0000	0.623918
$579$	8.48528	0.352636
$580$	−8.48528	−0.352332
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−8.00000	−0.331326
$584$	−8.48528	−0.351123
$585$	0	0
$586$	8.48528	0.350524
$587$	26.8701	1.10905	0.554523	−	0.832168i	$-0.312901\pi$
$587$	26.8701	1.10905	0.554523	+	0.832168i	$0.312901\pi$
$588$	0	0
$589$	0	0
$590$	6.00000	0.247016
$591$	−31.1127	−1.27981
$592$	−10.0000	−0.410997
$593$	−42.4264	−1.74224	−0.871122	−	0.491067i	$-0.836607\pi$
$593$	−42.4264	−1.74224	−0.871122	+	0.491067i	$0.836607\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	−18.3848	−0.750555
$601$	−5.65685	−0.230748	−0.115374	−	0.993322i	$-0.536807\pi$
$601$	−5.65685	−0.230748	−0.115374	+	0.993322i	$0.536807\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	4.00000	0.162758
$605$	−4.24264	−0.172488
$606$	8.00000	0.324978
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	0	0
$610$	−12.0000	−0.485866
$611$	0	0
$612$	−5.65685	−0.228665
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	25.4558	1.02731
$615$	67.8823	2.73728
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	26.0000	1.04587
$619$	−9.89949	−0.397894	−0.198947	−	0.980010i	$-0.563752\pi$
$619$	−9.89949	−0.397894	−0.198947	+	0.980010i	$0.563752\pi$
$620$	−6.00000	−0.240966
$621$	33.9411	1.36201
$622$	−18.3848	−0.737162
$623$	0	0
$624$	0	0
$625$	79.0000	3.16000
$626$	12.7279	0.508710
$627$	0	0
$628$	−4.24264	−0.169300
$629$	−56.5685	−2.25554
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	16.0000	0.636446
$633$	−11.3137	−0.449680
$634$	30.0000	1.19145
$635$	−67.8823	−2.69382
$636$	−11.3137	−0.448618
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	2.00000	0.0791188
$640$	−4.24264	−0.167705
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	−22.6274	−0.893033
$643$	38.1838	1.50582	0.752910	−	0.658123i	$-0.228649\pi$
$643$	38.1838	1.50582	0.752910	+	0.658123i	$0.228649\pi$
$644$	0	0
$645$	−48.0000	−1.89000
$646$	0	0
$647$	15.5563	0.611583	0.305792	−	0.952098i	$-0.401079\pi$
$647$	15.5563	0.611583	0.305792	+	0.952098i	$0.401079\pi$
$648$	−5.00000	−0.196419
$649$	1.41421	0.0555127
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	2.82843	0.110600
$655$	−84.0000	−3.28215
$656$	11.3137	0.441726
$657$	8.48528	0.331042
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−6.00000	−0.233550
$661$	−12.7279	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$661$	−12.7279	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	16.9706	0.658586
$665$	0	0
$666$	10.0000	0.387492
$667$	12.0000	0.464642
$668$	−5.65685	−0.218870
$669$	−30.0000	−1.15987
$670$	−8.48528	−0.327815
$671$	−2.82843	−0.109190
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	2.00000	0.0770371
$675$	73.5391	2.83052
$676$	−13.0000	−0.500000
$677$	39.5980	1.52187	0.760937	−	0.648826i	$-0.224740\pi$
$677$	39.5980	1.52187	0.760937	+	0.648826i	$0.224740\pi$
$678$	2.82843	0.108625
$679$	0	0
$680$	−24.0000	−0.920358
$681$	−20.0000	−0.766402
$682$	−1.41421	−0.0541530
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	76.3675	2.91785
$686$	0	0
$687$	14.0000	0.534133
$688$	−8.00000	−0.304997
$689$	0	0
$690$	36.0000	1.37050
$691$	−15.5563	−0.591791	−0.295896	−	0.955220i	$-0.595618\pi$
$691$	−15.5563	−0.591791	−0.295896	+	0.955220i	$0.595618\pi$
$692$	11.3137	0.430083
$693$	0	0
$694$	20.0000	0.759190
$695$	48.0000	1.82074
$696$	−2.82843	−0.107211
$697$	64.0000	2.42417
$698$	−14.1421	−0.535288
$699$	−19.7990	−0.748867
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−25.4558	−0.958723
$706$	−1.41421	−0.0532246
$707$	0	0
$708$	2.00000	0.0751646
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	8.48528	0.318447
$711$	−16.0000	−0.600047
$712$	7.07107	0.264999
$713$	8.48528	0.317776
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	16.9706	0.633777
$718$	−12.0000	−0.447836
$719$	−18.3848	−0.685636	−0.342818	−	0.939402i	$-0.611381\pi$
$719$	−18.3848	−0.685636	−0.342818	+	0.939402i	$0.611381\pi$
$720$	4.24264	0.158114
$721$	0	0
$722$	−19.0000	−0.707107
$723$	0	0
$724$	7.07107	0.262794
$725$	26.0000	0.965616
$726$	−1.41421	−0.0524864
$727$	−12.7279	−0.472052	−0.236026	−	0.971747i	$-0.575845\pi$
$727$	−12.7279	−0.472052	−0.236026	+	0.971747i	$0.575845\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	36.0000	1.33242
$731$	−45.2548	−1.67381
$732$	−4.00000	−0.147844
$733$	−14.1421	−0.522352	−0.261176	−	0.965291i	$-0.584110\pi$
$733$	−14.1421	−0.522352	−0.261176	+	0.965291i	$0.584110\pi$
$734$	21.2132	0.782994
$735$	0	0
$736$	6.00000	0.221163
$737$	−2.00000	−0.0736709
$738$	−11.3137	−0.416463
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	42.4264	1.55963
$741$	0	0
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	−2.00000	−0.0733236
$745$	42.4264	1.55438
$746$	−10.0000	−0.366126
$747$	−16.9706	−0.620920
$748$	−5.65685	−0.206835
$749$	0	0
$750$	48.0000	1.75271
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	−4.24264	−0.154713
$753$	26.0000	0.947493
$754$	0	0
$755$	−16.9706	−0.617622
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	−6.00000	−0.217930
$759$	8.48528	0.307996
$760$	0	0
$761$	42.4264	1.53796	0.768978	−	0.639275i	$-0.220766\pi$
$761$	42.4264	1.53796	0.768978	+	0.639275i	$0.220766\pi$
$762$	−22.6274	−0.819705
$763$	0	0
$764$	16.0000	0.578860
$765$	24.0000	0.867722
$766$	−15.5563	−0.562074
$767$	0	0
$768$	−1.41421	−0.0510310
$769$	−19.7990	−0.713970	−0.356985	−	0.934110i	$-0.616195\pi$
$769$	−19.7990	−0.713970	−0.356985	+	0.934110i	$0.616195\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	−6.00000	−0.215945
$773$	−1.41421	−0.0508657	−0.0254329	−	0.999677i	$-0.508096\pi$
$773$	−1.41421	−0.0508657	−0.0254329	+	0.999677i	$0.508096\pi$
$774$	8.00000	0.287554
$775$	18.3848	0.660401
$776$	9.89949	0.355371
$777$	0	0
$778$	−24.0000	−0.860442
$779$	0	0
$780$	0	0
$781$	2.00000	0.0715656
$782$	33.9411	1.21373
$783$	11.3137	0.404319
$784$	0	0
$785$	18.0000	0.642448
$786$	−28.0000	−0.998727
$787$	5.65685	0.201645	0.100823	−	0.994904i	$-0.467853\pi$
$787$	5.65685	0.201645	0.100823	+	0.994904i	$0.467853\pi$
$788$	22.0000	0.783718
$789$	11.3137	0.402779
$790$	−67.8823	−2.41514
$791$	0	0
$792$	1.00000	0.0355335
$793$	0	0
$794$	−12.7279	−0.451697
$795$	48.0000	1.70238
$796$	1.41421	0.0501255
$797$	−7.07107	−0.250470	−0.125235	−	0.992127i	$-0.539968\pi$
$797$	−7.07107	−0.250470	−0.125235	+	0.992127i	$0.539968\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	13.0000	0.459619
$801$	−7.07107	−0.249844
$802$	12.0000	0.423735
$803$	8.48528	0.299439
$804$	−2.82843	−0.0997509
$805$	0	0
$806$	0	0
$807$	26.0000	0.915243
$808$	−5.65685	−0.199007
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	21.2132	0.745356
$811$	−36.7696	−1.29115	−0.645577	−	0.763695i	$-0.723383\pi$
$811$	−36.7696	−1.29115	−0.645577	+	0.763695i	$0.723383\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	10.0000	0.350500
$815$	42.4264	1.48613
$816$	−8.00000	−0.280056
$817$	0	0
$818$	2.82843	0.0988936
$819$	0	0
$820$	−48.0000	−1.67623
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	25.4558	0.887875
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−18.3848	−0.640464
$825$	18.3848	0.640076
$826$	0	0
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	−6.00000	−0.208514
$829$	4.24264	0.147353	0.0736765	−	0.997282i	$-0.476527\pi$
$829$	4.24264	0.147353	0.0736765	+	0.997282i	$0.476527\pi$
$830$	−72.0000	−2.49916
$831$	2.82843	0.0981170
$832$	0	0
$833$	0	0
$834$	16.0000	0.554035
$835$	24.0000	0.830554
$836$	0	0
$837$	8.00000	0.276520
$838$	−24.0416	−0.830504
$839$	−1.41421	−0.0488241	−0.0244120	−	0.999702i	$-0.507771\pi$
$839$	−1.41421	−0.0488241	−0.0244120	+	0.999702i	$0.507771\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−20.0000	−0.689246
$843$	19.7990	0.681913
$844$	8.00000	0.275371
$845$	55.1543	1.89737
$846$	4.24264	0.145865
$847$	0	0
$848$	8.00000	0.274721
$849$	−28.0000	−0.960958
$850$	73.5391	2.52237
$851$	−60.0000	−2.05677
$852$	2.82843	0.0969003
$853$	−28.2843	−0.968435	−0.484218	−	0.874948i	$-0.660896\pi$
$853$	−28.2843	−0.968435	−0.484218	+	0.874948i	$0.660896\pi$
$854$	0	0
$855$	0	0
$856$	16.0000	0.546869
$857$	25.4558	0.869555	0.434778	−	0.900538i	$-0.356827\pi$
$857$	25.4558	0.869555	0.434778	+	0.900538i	$0.356827\pi$
$858$	0	0
$859$	−12.7279	−0.434271	−0.217136	−	0.976141i	$-0.569671\pi$
$859$	−12.7279	−0.434271	−0.217136	+	0.976141i	$0.569671\pi$
$860$	33.9411	1.15738
$861$	0	0
$862$	32.0000	1.08992
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	5.65685	0.192450
$865$	−48.0000	−1.63205
$866$	12.7279	0.432512
$867$	−21.2132	−0.720438
$868$	0	0
$869$	−16.0000	−0.542763
$870$	12.0000	0.406838
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	−9.89949	−0.335047
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	25.4558	0.859093
$879$	−12.0000	−0.404750
$880$	4.24264	0.143019
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	36.0000	1.20944
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	14.1421	0.474579
$889$	0	0
$890$	−30.0000	−1.00560
$891$	5.00000	0.167506
$892$	21.2132	0.710271
$893$	0	0
$894$	14.1421	0.472984
$895$	−50.9117	−1.70179
$896$	0	0
$897$	0	0
$898$	−20.0000	−0.667409
$899$	2.82843	0.0943333
$900$	−13.0000	−0.433333
$901$	45.2548	1.50766
$902$	−11.3137	−0.376705
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	−30.0000	−0.997234
$906$	−5.65685	−0.187936
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	14.1421	0.469323
$909$	5.65685	0.187626
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	−16.9706	−0.561644
$914$	14.0000	0.463079
$915$	16.9706	0.561029
$916$	−9.89949	−0.327089
$917$	0	0
$918$	32.0000	1.05616
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−25.4558	−0.839254
$921$	−36.0000	−1.18624
$922$	−2.82843	−0.0931493
$923$	0	0
$924$	0	0
$925$	−130.000	−4.27437
$926$	26.0000	0.854413
$927$	18.3848	0.603835
$928$	2.00000	0.0656532
$929$	32.5269	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$929$	32.5269	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$930$	8.48528	0.278243
$931$	0	0
$932$	14.0000	0.458585
$933$	26.0000	0.851202
$934$	4.24264	0.138823
$935$	24.0000	0.784884
$936$	0	0
$937$	−25.4558	−0.831606	−0.415803	−	0.909455i	$-0.636499\pi$
$937$	−25.4558	−0.831606	−0.415803	+	0.909455i	$0.636499\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	18.0000	0.587095
$941$	−31.1127	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$941$	−31.1127	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$942$	6.00000	0.195491
$943$	67.8823	2.21055
$944$	−1.41421	−0.0460287
$945$	0	0
$946$	8.00000	0.260102
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	−22.6274	−0.734904
$949$	0	0
$950$	0	0
$951$	−42.4264	−1.37577
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−8.00000	−0.259010
$955$	−67.8823	−2.19662
$956$	−12.0000	−0.388108
$957$	2.82843	0.0914301
$958$	−2.82843	−0.0913823
$959$	0	0
$960$	6.00000	0.193649
$961$	−29.0000	−0.935484
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	25.4558	0.819453
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−42.0000	−1.34854
$971$	−52.3259	−1.67922	−0.839609	−	0.543191i	$-0.817216\pi$
$971$	−52.3259	−1.67922	−0.839609	+	0.543191i	$0.817216\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	2.82843	0.0905357
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	14.1421	0.452216
$979$	−7.07107	−0.225992
$980$	0	0
$981$	2.00000	0.0638551
$982$	−12.0000	−0.382935
$983$	−1.41421	−0.0451064	−0.0225532	−	0.999746i	$-0.507180\pi$
$983$	−1.41421	−0.0451064	−0.0225532	+	0.999746i	$0.507180\pi$
$984$	−16.0000	−0.510061
$985$	−93.3381	−2.97400
$986$	11.3137	0.360302
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	−4.24264	−0.134840
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	1.41421	0.0449013
$993$	−28.2843	−0.897574
$994$	0	0
$995$	−6.00000	−0.190213
$996$	−24.0000	−0.760469
$997$	−16.9706	−0.537463	−0.268732	−	0.963215i	$-0.586604\pi$
$997$	−16.9706	−0.537463	−0.268732	+	0.963215i	$0.586604\pi$
$998$	6.00000	0.189927
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.u.1.1	✓	2
3.2	odd	2		9702.2.a.cp.1.2		2
4.3	odd	2		8624.2.a.bs.1.2		2
7.2	even	3		1078.2.e.p.67.2		4
7.3	odd	6		1078.2.e.p.177.1		4
7.4	even	3		1078.2.e.p.177.2		4
7.5	odd	6		1078.2.e.p.67.1		4
7.6	odd	2	inner	1078.2.a.u.1.2	yes	2
21.20	even	2		9702.2.a.cp.1.1		2
28.27	even	2		8624.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.u.1.1	✓	2	1.1	even	1	trivial
1078.2.a.u.1.2	yes	2	7.6	odd	2	inner
1078.2.e.p.67.1		4	7.5	odd	6
1078.2.e.p.67.2		4	7.2	even	3
1078.2.e.p.177.1		4	7.3	odd	6
1078.2.e.p.177.2		4	7.4	even	3
8624.2.a.bs.1.1		2	28.27	even	2
8624.2.a.bs.1.2		2	4.3	odd	2
9702.2.a.cp.1.1		2	21.20	even	2
9702.2.a.cp.1.2		2	3.2	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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} -4.24264 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} -4.24264 q^{10} -1.00000 q^{11} -1.41421 q^{12} +6.00000 q^{15} +1.00000 q^{16} +5.65685 q^{17} -1.00000 q^{18} -4.24264 q^{20} -1.00000 q^{22} +6.00000 q^{23} -1.41421 q^{24} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +6.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} +1.41421 q^{33} +5.65685 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.24264 q^{40} +11.3137 q^{41} -8.00000 q^{43} -1.00000 q^{44} +4.24264 q^{45} +6.00000 q^{46} -4.24264 q^{47} -1.41421 q^{48} +13.0000 q^{50} -8.00000 q^{51} +8.00000 q^{53} +5.65685 q^{54} +4.24264 q^{55} +2.00000 q^{58} -1.41421 q^{59} +6.00000 q^{60} +2.82843 q^{61} +1.41421 q^{62} +1.00000 q^{64} +1.41421 q^{66} +2.00000 q^{67} +5.65685 q^{68} -8.48528 q^{69} -2.00000 q^{71} -1.00000 q^{72} -8.48528 q^{73} -10.0000 q^{74} -18.3848 q^{75} +16.0000 q^{79} -4.24264 q^{80} -5.00000 q^{81} +11.3137 q^{82} +16.9706 q^{83} -24.0000 q^{85} -8.00000 q^{86} -2.82843 q^{87} -1.00000 q^{88} +7.07107 q^{89} +4.24264 q^{90} +6.00000 q^{92} -2.00000 q^{93} -4.24264 q^{94} -1.41421 q^{96} +9.89949 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{11} + 12 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{22} + 12 q^{23} + 26 q^{25} + 4 q^{29} + 12 q^{30} + 2 q^{32} - 2 q^{36} - 20 q^{37} - 16 q^{43} - 2 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 2 * q^11 + 12 * q^15 + 2 * q^16 - 2 * q^18 - 2 * q^22 + 12 * q^23 + 26 * q^25 + 4 * q^29 + 12 * q^30 + 2 * q^32 - 2 * q^36 - 20 * q^37 - 16 * q^43 - 2 * q^44 + 12 * q^46 + 26 * q^50 - 16 * q^51 + 16 * q^53 + 4 * q^58 + 12 * q^60 + 2 * q^64 + 4 * q^67 - 4 * q^71 - 2 * q^72 - 20 * q^74 + 32 * q^79 - 10 * q^81 - 48 * q^85 - 16 * q^86 - 2 * q^88 + 12 * q^92 - 4 * q^93 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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