LMFDB - Embedded newform 1078.2.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1078,2,Mod(1,1078)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,0,0,0,2,10,0,-2,0,0,0,0,2,0,10,0,0,0,-2,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} +2.82843 q^{12} +4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +5.00000 q^{18} -4.24264 q^{19} -1.00000 q^{22} +6.00000 q^{23} +2.82843 q^{24} -5.00000 q^{25} +4.24264 q^{26} +5.65685 q^{27} -4.00000 q^{29} -7.07107 q^{31} +1.00000 q^{32} -2.82843 q^{33} -2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} -4.24264 q^{38} +12.0000 q^{39} +2.82843 q^{41} +10.0000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -12.7279 q^{47} +2.82843 q^{48} -5.00000 q^{50} -8.00000 q^{51} +4.24264 q^{52} +2.00000 q^{53} +5.65685 q^{54} -12.0000 q^{57} -4.00000 q^{58} +11.3137 q^{59} -9.89949 q^{61} -7.07107 q^{62} +1.00000 q^{64} -2.82843 q^{66} +8.00000 q^{67} -2.82843 q^{68} +16.9706 q^{69} +16.0000 q^{71} +5.00000 q^{72} -8.48528 q^{73} +2.00000 q^{74} -14.1421 q^{75} -4.24264 q^{76} +12.0000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.82843 q^{82} -12.7279 q^{83} +10.0000 q^{86} -11.3137 q^{87} -1.00000 q^{88} +7.07107 q^{89} +6.00000 q^{92} -20.0000 q^{93} -12.7279 q^{94} +2.82843 q^{96} -7.07107 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} + 10 q^{18} - 2 q^{22} + 12 q^{23} - 10 q^{25} - 8 q^{29} + 2 q^{32} + 10 q^{36} + 4 q^{37} + 24 q^{39} + 20 q^{43} - 2 q^{44} + 12 q^{46}+ \cdots - 10 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9 - 2 * q^11 + 2 * q^16 + 10 * q^18 - 2 * q^22 + 12 * q^23 - 10 * q^25 - 8 * q^29 + 2 * q^32 + 10 * q^36 + 4 * q^37 + 24 * q^39 + 20 * q^43 - 2 * q^44 + 12 * q^46 - 10 * q^50 - 16 * q^51 + 4 * q^53 - 24 * q^57 - 8 * q^58 + 2 * q^64 + 16 * q^67 + 32 * q^71 + 10 * q^72 + 4 * q^74 + 24 * q^78 - 16 * q^79 + 2 * q^81 + 20 * q^86 - 2 * q^88 + 12 * q^92 - 40 * q^93 - 10 * 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
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.82843	1.15470
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	5.00000	1.66667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.82843	0.816497
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	5.00000	1.17851
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	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	$0.284877\pi$
$24$	2.82843	0.577350
$25$	−5.00000	−1.00000
$26$	4.24264	0.832050
$27$	5.65685	1.08866
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−7.07107	−1.27000	−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107	−1.27000	−0.635001	+	0.772512i	$0.719000\pi$
$32$	1.00000	0.176777
$33$	−2.82843	−0.492366
$34$	−2.82843	−0.485071
$35$	0	0
$36$	5.00000	0.833333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.24264	−0.688247
$39$	12.0000	1.92154
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	6.00000	0.884652
$47$	−12.7279	−1.85656	−0.928279	−	0.371884i	$-0.878712\pi$
$47$	−12.7279	−1.85656	−0.928279	+	0.371884i	$0.878712\pi$
$48$	2.82843	0.408248
$49$	0	0
$50$	−5.00000	−0.707107
$51$	−8.00000	−1.12022
$52$	4.24264	0.588348
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	−4.00000	−0.525226
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−9.89949	−1.26750	−0.633750	−	0.773538i	$-0.718485\pi$
$61$	−9.89949	−1.26750	−0.633750	+	0.773538i	$0.718485\pi$
$62$	−7.07107	−0.898027
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.82843	−0.348155
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−2.82843	−0.342997
$69$	16.9706	2.04302
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	5.00000	0.589256
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	2.00000	0.232495
$75$	−14.1421	−1.63299
$76$	−4.24264	−0.486664
$77$	0	0
$78$	12.0000	1.35873
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.82843	0.312348
$83$	−12.7279	−1.39707	−0.698535	−	0.715575i	$-0.746165\pi$
$83$	−12.7279	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	−11.3137	−1.21296
$88$	−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	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	−20.0000	−2.07390
$94$	−12.7279	−1.31278
$95$	0	0
$96$	2.82843	0.288675
$97$	−7.07107	−0.717958	−0.358979	−	0.933346i	$-0.616875\pi$
$97$	−7.07107	−0.717958	−0.358979	+	0.933346i	$0.616875\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	−5.00000	−0.500000
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	−8.00000	−0.792118
$103$	−1.41421	−0.139347	−0.0696733	−	0.997570i	$-0.522196\pi$
$103$	−1.41421	−0.139347	−0.0696733	+	0.997570i	$0.522196\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	2.00000	0.194257
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	5.65685	0.544331
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	−4.00000	−0.371391
$117$	21.2132	1.96116
$118$	11.3137	1.04151
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.89949	−0.896258
$123$	8.00000	0.721336
$124$	−7.07107	−0.635001
$125$	0	0
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	28.2843	2.49029
$130$	0	0
$131$	7.07107	0.617802	0.308901	−	0.951094i	$-0.400039\pi$
$131$	7.07107	0.617802	0.308901	+	0.951094i	$0.400039\pi$
$132$	−2.82843	−0.246183
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	−2.82843	−0.242536
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	16.9706	1.44463
$139$	18.3848	1.55938	0.779688	−	0.626168i	$-0.215378\pi$
$139$	18.3848	1.55938	0.779688	+	0.626168i	$0.215378\pi$
$140$	0	0
$141$	−36.0000	−3.03175
$142$	16.0000	1.34269
$143$	−4.24264	−0.354787
$144$	5.00000	0.416667
$145$	0	0
$146$	−8.48528	−0.702247
$147$	0	0
$148$	2.00000	0.164399
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	−14.1421	−1.15470
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−4.24264	−0.344124
$153$	−14.1421	−1.14332
$154$	0	0
$155$	0	0
$156$	12.0000	0.960769
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	−8.00000	−0.636446
$159$	5.65685	0.448618
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.82843	0.220863
$165$	0	0
$166$	−12.7279	−0.987878
$167$	−14.1421	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	−21.2132	−1.62221
$172$	10.0000	0.762493
$173$	7.07107	0.537603	0.268802	−	0.963196i	$-0.413372\pi$
$173$	7.07107	0.537603	0.268802	+	0.963196i	$0.413372\pi$
$174$	−11.3137	−0.857690
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	32.0000	2.40527
$178$	7.07107	0.529999
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	11.3137	0.840941	0.420471	−	0.907306i	$-0.361865\pi$
$181$	11.3137	0.840941	0.420471	+	0.907306i	$0.361865\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	6.00000	0.442326
$185$	0	0
$186$	−20.0000	−1.46647
$187$	2.82843	0.206835
$188$	−12.7279	−0.928279
$189$	0	0
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	2.82843	0.204124
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−7.07107	−0.507673
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−5.00000	−0.355335
$199$	9.89949	0.701757	0.350878	−	0.936421i	$-0.385883\pi$
$199$	9.89949	0.701757	0.350878	+	0.936421i	$0.385883\pi$
$200$	−5.00000	−0.353553
$201$	22.6274	1.59601
$202$	−1.41421	−0.0995037
$203$	0	0
$204$	−8.00000	−0.560112
$205$	0	0
$206$	−1.41421	−0.0985329
$207$	30.0000	2.08514
$208$	4.24264	0.294174
$209$	4.24264	0.293470
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	2.00000	0.137361
$213$	45.2548	3.10081
$214$	−14.0000	−0.957020
$215$	0	0
$216$	5.65685	0.384900
$217$	0	0
$218$	−14.0000	−0.948200
$219$	−24.0000	−1.62177
$220$	0	0
$221$	−12.0000	−0.807207
$222$	5.65685	0.379663
$223$	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
$225$	−25.0000	−1.66667
$226$	16.0000	1.06430
$227$	−7.07107	−0.469323	−0.234662	−	0.972077i	$-0.575398\pi$
$227$	−7.07107	−0.469323	−0.234662	+	0.972077i	$0.575398\pi$
$228$	−12.0000	−0.794719
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	0	0
$231$	0	0
$232$	−4.00000	−0.262613
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	21.2132	1.38675
$235$	0	0
$236$	11.3137	0.736460
$237$	−22.6274	−1.46981
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	1.00000	0.0642824
$243$	−14.1421	−0.907218
$244$	−9.89949	−0.633750
$245$	0	0
$246$	8.00000	0.510061
$247$	−18.0000	−1.14531
$248$	−7.07107	−0.449013
$249$	−36.0000	−2.28141
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.07107	0.441081	0.220541	−	0.975378i	$-0.429218\pi$
$257$	7.07107	0.441081	0.220541	+	0.975378i	$0.429218\pi$
$258$	28.2843	1.76090
$259$	0	0
$260$	0	0
$261$	−20.0000	−1.23797
$262$	7.07107	0.436852
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−2.82843	−0.174078
$265$	0	0
$266$	0	0
$267$	20.0000	1.22398
$268$	8.00000	0.488678
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	−6.00000	−0.362473
$275$	5.00000	0.301511
$276$	16.9706	1.02151
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	18.3848	1.10265
$279$	−35.3553	−2.11667
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	−36.0000	−2.14377
$283$	7.07107	0.420331	0.210166	−	0.977666i	$-0.432600\pi$
$283$	7.07107	0.420331	0.210166	+	0.977666i	$0.432600\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−4.24264	−0.250873
$287$	0	0
$288$	5.00000	0.294628
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−20.0000	−1.17242
$292$	−8.48528	−0.496564
$293$	−21.2132	−1.23929	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132	−1.23929	−0.619644	+	0.784883i	$0.712723\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	−5.65685	−0.328244
$298$	8.00000	0.463428
$299$	25.4558	1.47215
$300$	−14.1421	−0.816497
$301$	0	0
$302$	−8.00000	−0.460348
$303$	−4.00000	−0.229794
$304$	−4.24264	−0.243332
$305$	0	0
$306$	−14.1421	−0.808452
$307$	21.2132	1.21070	0.605351	−	0.795959i	$-0.293033\pi$
$307$	21.2132	1.21070	0.605351	+	0.795959i	$0.293033\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−1.41421	−0.0801927	−0.0400963	−	0.999196i	$-0.512766\pi$
$311$	−1.41421	−0.0801927	−0.0400963	+	0.999196i	$0.512766\pi$
$312$	12.0000	0.679366
$313$	−29.6985	−1.67866	−0.839329	−	0.543624i	$-0.817052\pi$
$313$	−29.6985	−1.67866	−0.839329	+	0.543624i	$0.817052\pi$
$314$	8.48528	0.478852
$315$	0	0
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	5.65685	0.317221
$319$	4.00000	0.223957
$320$	0	0
$321$	−39.5980	−2.21014
$322$	0	0
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	−21.2132	−1.17670
$326$	−4.00000	−0.221540
$327$	−39.5980	−2.18977
$328$	2.82843	0.156174
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−12.7279	−0.698535
$333$	10.0000	0.547997
$334$	−14.1421	−0.773823
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	5.00000	0.271964
$339$	45.2548	2.45791
$340$	0	0
$341$	7.07107	0.382920
$342$	−21.2132	−1.14708
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	7.07107	0.380143
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	−11.3137	−0.606478
$349$	−26.8701	−1.43832	−0.719161	−	0.694844i	$-0.755473\pi$
$349$	−26.8701	−1.43832	−0.719161	+	0.694844i	$0.755473\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	−1.00000	−0.0533002
$353$	15.5563	0.827981	0.413990	−	0.910281i	$-0.364135\pi$
$353$	15.5563	0.827981	0.413990	+	0.910281i	$0.364135\pi$
$354$	32.0000	1.70078
$355$	0	0
$356$	7.07107	0.374766
$357$	0	0
$358$	−12.0000	−0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	11.3137	0.594635
$363$	2.82843	0.148454
$364$	0	0
$365$	0	0
$366$	−28.0000	−1.46358
$367$	−21.2132	−1.10732	−0.553660	−	0.832743i	$-0.686769\pi$
$367$	−21.2132	−1.10732	−0.553660	+	0.832743i	$0.686769\pi$
$368$	6.00000	0.312772
$369$	14.1421	0.736210
$370$	0	0
$371$	0	0
$372$	−20.0000	−1.03695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	−12.7279	−0.656392
$377$	−16.9706	−0.874028
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	−56.5685	−2.89809
$382$	10.0000	0.511645
$383$	−15.5563	−0.794892	−0.397446	−	0.917625i	$-0.630103\pi$
$383$	−15.5563	−0.794892	−0.397446	+	0.917625i	$0.630103\pi$
$384$	2.82843	0.144338
$385$	0	0
$386$	6.00000	0.305392
$387$	50.0000	2.54164
$388$	−7.07107	−0.358979
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	20.0000	1.00887
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−5.00000	−0.251259
$397$	25.4558	1.27759	0.638796	−	0.769376i	$-0.279433\pi$
$397$	25.4558	1.27759	0.638796	+	0.769376i	$0.279433\pi$
$398$	9.89949	0.496217
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	22.6274	1.12855
$403$	−30.0000	−1.49441
$404$	−1.41421	−0.0703598
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	−8.00000	−0.396059
$409$	28.2843	1.39857	0.699284	−	0.714844i	$-0.253502\pi$
$409$	28.2843	1.39857	0.699284	+	0.714844i	$0.253502\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	−1.41421	−0.0696733
$413$	0	0
$414$	30.0000	1.47442
$415$	0	0
$416$	4.24264	0.208013
$417$	52.0000	2.54645
$418$	4.24264	0.207514
$419$	14.1421	0.690889	0.345444	−	0.938439i	$-0.387728\pi$
$419$	14.1421	0.690889	0.345444	+	0.938439i	$0.387728\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	26.0000	1.26566
$423$	−63.6396	−3.09426
$424$	2.00000	0.0971286
$425$	14.1421	0.685994
$426$	45.2548	2.19260
$427$	0	0
$428$	−14.0000	−0.676716
$429$	−12.0000	−0.579365
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	5.65685	0.272166
$433$	29.6985	1.42722	0.713609	−	0.700544i	$-0.247059\pi$
$433$	29.6985	1.42722	0.713609	+	0.700544i	$0.247059\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−25.4558	−1.21772
$438$	−24.0000	−1.14676
$439$	25.4558	1.21494	0.607471	−	0.794342i	$-0.292184\pi$
$439$	25.4558	1.21494	0.607471	+	0.794342i	$0.292184\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	5.65685	0.268462
$445$	0	0
$446$	21.2132	1.00447
$447$	22.6274	1.07024
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	−25.0000	−1.17851
$451$	−2.82843	−0.133185
$452$	16.0000	0.752577
$453$	−22.6274	−1.06313
$454$	−7.07107	−0.331862
$455$	0	0
$456$	−12.0000	−0.561951
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−22.6274	−1.05731
$459$	−16.0000	−0.746816
$460$	0	0
$461$	1.41421	0.0658665	0.0329332	−	0.999458i	$-0.489515\pi$
$461$	1.41421	0.0658665	0.0329332	+	0.999458i	$0.489515\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	26.0000	1.20443
$467$	25.4558	1.17796	0.588978	−	0.808149i	$-0.299530\pi$
$467$	25.4558	1.17796	0.588978	+	0.808149i	$0.299530\pi$
$468$	21.2132	0.980581
$469$	0	0
$470$	0	0
$471$	24.0000	1.10586
$472$	11.3137	0.520756
$473$	−10.0000	−0.459800
$474$	−22.6274	−1.03931
$475$	21.2132	0.973329
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	8.48528	0.386896
$482$	−8.48528	−0.386494
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−14.1421	−0.641500
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−9.89949	−0.448129
$489$	−11.3137	−0.511624
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	8.00000	0.360668
$493$	11.3137	0.509544
$494$	−18.0000	−0.809858
$495$	0	0
$496$	−7.07107	−0.317500
$497$	0	0
$498$	−36.0000	−1.61320
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	−40.0000	−1.78707
$502$	−5.65685	−0.252478
$503$	19.7990	0.882793	0.441397	−	0.897312i	$-0.354483\pi$
$503$	19.7990	0.882793	0.441397	+	0.897312i	$0.354483\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	14.1421	0.628074
$508$	−20.0000	−0.887357
$509$	16.9706	0.752207	0.376103	−	0.926578i	$-0.377264\pi$
$509$	16.9706	0.752207	0.376103	+	0.926578i	$0.377264\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−24.0000	−1.05963
$514$	7.07107	0.311891
$515$	0	0
$516$	28.2843	1.24515
$517$	12.7279	0.559773
$518$	0	0
$519$	20.0000	0.877903
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	−20.0000	−0.875376
$523$	4.24264	0.185518	0.0927589	−	0.995689i	$-0.470431\pi$
$523$	4.24264	0.185518	0.0927589	+	0.995689i	$0.470431\pi$
$524$	7.07107	0.308901
$525$	0	0
$526$	4.00000	0.174408
$527$	20.0000	0.871214
$528$	−2.82843	−0.123091
$529$	13.0000	0.565217
$530$	0	0
$531$	56.5685	2.45487
$532$	0	0
$533$	12.0000	0.519778
$534$	20.0000	0.865485
$535$	0	0
$536$	8.00000	0.345547
$537$	−33.9411	−1.46467
$538$	11.3137	0.487769
$539$	0	0
$540$	0	0
$541$	−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
$543$	32.0000	1.37325
$544$	−2.82843	−0.121268
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−6.00000	−0.256307
$549$	−49.4975	−2.11250
$550$	5.00000	0.213201
$551$	16.9706	0.722970
$552$	16.9706	0.722315
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	18.3848	0.779688
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−35.3553	−1.49671
$559$	42.4264	1.79445
$560$	0	0
$561$	8.00000	0.337760
$562$	22.0000	0.928014
$563$	41.0122	1.72846	0.864229	−	0.503099i	$-0.167807\pi$
$563$	41.0122	1.72846	0.864229	+	0.503099i	$0.167807\pi$
$564$	−36.0000	−1.51587
$565$	0	0
$566$	7.07107	0.297219
$567$	0	0
$568$	16.0000	0.671345
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−4.24264	−0.177394
$573$	28.2843	1.18159
$574$	0	0
$575$	−30.0000	−1.25109
$576$	5.00000	0.208333
$577$	−9.89949	−0.412121	−0.206061	−	0.978539i	$-0.566064\pi$
$577$	−9.89949	−0.412121	−0.206061	+	0.978539i	$0.566064\pi$
$578$	−9.00000	−0.374351
$579$	16.9706	0.705273
$580$	0	0
$581$	0	0
$582$	−20.0000	−0.829027
$583$	−2.00000	−0.0828315
$584$	−8.48528	−0.351123
$585$	0	0
$586$	−21.2132	−0.876309
$587$	5.65685	0.233483	0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685	0.233483	0.116742	+	0.993162i	$0.462755\pi$
$588$	0	0
$589$	30.0000	1.23613
$590$	0	0
$591$	−5.65685	−0.232692
$592$	2.00000	0.0821995
$593$	8.48528	0.348449	0.174224	−	0.984706i	$-0.444258\pi$
$593$	8.48528	0.348449	0.174224	+	0.984706i	$0.444258\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	8.00000	0.327693
$597$	28.0000	1.14596
$598$	25.4558	1.04097
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−14.1421	−0.577350
$601$	11.3137	0.461496	0.230748	−	0.973014i	$-0.425883\pi$
$601$	11.3137	0.461496	0.230748	+	0.973014i	$0.425883\pi$
$602$	0	0
$603$	40.0000	1.62893
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−4.00000	−0.162489
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	−4.24264	−0.172062
$609$	0	0
$610$	0	0
$611$	−54.0000	−2.18461
$612$	−14.1421	−0.571662
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	21.2132	0.856095
$615$	0	0
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	−4.00000	−0.160904
$619$	−14.1421	−0.568420	−0.284210	−	0.958762i	$-0.591731\pi$
$619$	−14.1421	−0.568420	−0.284210	+	0.958762i	$0.591731\pi$
$620$	0	0
$621$	33.9411	1.36201
$622$	−1.41421	−0.0567048
$623$	0	0
$624$	12.0000	0.480384
$625$	25.0000	1.00000
$626$	−29.6985	−1.18699
$627$	12.0000	0.479234
$628$	8.48528	0.338600
$629$	−5.65685	−0.225554
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	−8.00000	−0.318223
$633$	73.5391	2.92292
$634$	18.0000	0.714871
$635$	0	0
$636$	5.65685	0.224309
$637$	0	0
$638$	4.00000	0.158362
$639$	80.0000	3.16475
$640$	0	0
$641$	44.0000	1.73790	0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000	1.73790	0.868948	+	0.494904i	$0.164797\pi$
$642$	−39.5980	−1.56281
$643$	−33.9411	−1.33851	−0.669254	−	0.743034i	$-0.733386\pi$
$643$	−33.9411	−1.33851	−0.669254	+	0.743034i	$0.733386\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	32.5269	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$647$	32.5269	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$648$	1.00000	0.0392837
$649$	−11.3137	−0.444102
$650$	−21.2132	−0.832050
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−39.5980	−1.54840
$655$	0	0
$656$	2.82843	0.110432
$657$	−42.4264	−1.65521
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−16.9706	−0.660078	−0.330039	−	0.943967i	$-0.607062\pi$
$661$	−16.9706	−0.660078	−0.330039	+	0.943967i	$0.607062\pi$
$662$	8.00000	0.310929
$663$	−33.9411	−1.31816
$664$	−12.7279	−0.493939
$665$	0	0
$666$	10.0000	0.387492
$667$	−24.0000	−0.929284
$668$	−14.1421	−0.547176
$669$	60.0000	2.31973
$670$	0	0
$671$	9.89949	0.382166
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−22.0000	−0.847408
$675$	−28.2843	−1.08866
$676$	5.00000	0.192308
$677$	9.89949	0.380468	0.190234	−	0.981739i	$-0.439075\pi$
$677$	9.89949	0.380468	0.190234	+	0.981739i	$0.439075\pi$
$678$	45.2548	1.73800
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	7.07107	0.270765
$683$	32.0000	1.22445	0.612223	−	0.790685i	$-0.290275\pi$
$683$	32.0000	1.22445	0.612223	+	0.790685i	$0.290275\pi$
$684$	−21.2132	−0.811107
$685$	0	0
$686$	0	0
$687$	−64.0000	−2.44175
$688$	10.0000	0.381246
$689$	8.48528	0.323263
$690$	0	0
$691$	39.5980	1.50638	0.753189	−	0.657804i	$-0.228515\pi$
$691$	39.5980	1.50638	0.753189	+	0.657804i	$0.228515\pi$
$692$	7.07107	0.268802
$693$	0	0
$694$	26.0000	0.986947
$695$	0	0
$696$	−11.3137	−0.428845
$697$	−8.00000	−0.303022
$698$	−26.8701	−1.01705
$699$	73.5391	2.78150
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	24.0000	0.905822
$703$	−8.48528	−0.320028
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	15.5563	0.585471
$707$	0	0
$708$	32.0000	1.20263
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−40.0000	−1.50012
$712$	7.07107	0.264999
$713$	−42.4264	−1.58888
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	12.0000	0.447836
$719$	32.5269	1.21305	0.606525	−	0.795065i	$-0.292563\pi$
$719$	32.5269	1.21305	0.606525	+	0.795065i	$0.292563\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	−24.0000	−0.892570
$724$	11.3137	0.420471
$725$	20.0000	0.742781
$726$	2.82843	0.104973
$727$	−46.6690	−1.73086	−0.865430	−	0.501031i	$-0.832954\pi$
$727$	−46.6690	−1.73086	−0.865430	+	0.501031i	$0.832954\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−28.2843	−1.04613
$732$	−28.0000	−1.03491
$733$	7.07107	0.261176	0.130588	−	0.991437i	$-0.458314\pi$
$733$	7.07107	0.261176	0.130588	+	0.991437i	$0.458314\pi$
$734$	−21.2132	−0.782994
$735$	0	0
$736$	6.00000	0.221163
$737$	−8.00000	−0.294684
$738$	14.1421	0.520579
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−50.9117	−1.87029
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−20.0000	−0.733236
$745$	0	0
$746$	−10.0000	−0.366126
$747$	−63.6396	−2.32845
$748$	2.82843	0.103418
$749$	0	0
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	−12.7279	−0.464140
$753$	−16.0000	−0.583072
$754$	−16.9706	−0.618031
$755$	0	0
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	24.0000	0.871719
$759$	−16.9706	−0.615992
$760$	0	0
$761$	−33.9411	−1.23036	−0.615182	−	0.788385i	$-0.710918\pi$
$761$	−33.9411	−1.23036	−0.615182	+	0.788385i	$0.710918\pi$
$762$	−56.5685	−2.04926
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	−15.5563	−0.562074
$767$	48.0000	1.73318
$768$	2.82843	0.102062
$769$	5.65685	0.203991	0.101996	−	0.994785i	$-0.467477\pi$
$769$	5.65685	0.203991	0.101996	+	0.994785i	$0.467477\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	6.00000	0.215945
$773$	−48.0833	−1.72943	−0.864717	−	0.502259i	$-0.832502\pi$
$773$	−48.0833	−1.72943	−0.864717	+	0.502259i	$0.832502\pi$
$774$	50.0000	1.79721
$775$	35.3553	1.27000
$776$	−7.07107	−0.253837
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−16.9706	−0.606866
$783$	−22.6274	−0.808638
$784$	0	0
$785$	0	0
$786$	20.0000	0.713376
$787$	9.89949	0.352879	0.176439	−	0.984311i	$-0.443542\pi$
$787$	9.89949	0.352879	0.176439	+	0.984311i	$0.443542\pi$
$788$	−2.00000	−0.0712470
$789$	11.3137	0.402779
$790$	0	0
$791$	0	0
$792$	−5.00000	−0.177667
$793$	−42.0000	−1.49146
$794$	25.4558	0.903394
$795$	0	0
$796$	9.89949	0.350878
$797$	−2.82843	−0.100188	−0.0500940	−	0.998745i	$-0.515952\pi$
$797$	−2.82843	−0.100188	−0.0500940	+	0.998745i	$0.515952\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	−5.00000	−0.176777
$801$	35.3553	1.24922
$802$	−30.0000	−1.05934
$803$	8.48528	0.299439
$804$	22.6274	0.798007
$805$	0	0
$806$	−30.0000	−1.05670
$807$	32.0000	1.12645
$808$	−1.41421	−0.0497519
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−15.5563	−0.546257	−0.273129	−	0.961978i	$-0.588058\pi$
$811$	−15.5563	−0.546257	−0.273129	+	0.961978i	$0.588058\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	−8.00000	−0.280056
$817$	−42.4264	−1.48431
$818$	28.2843	0.988936
$819$	0	0
$820$	0	0
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	−16.9706	−0.591916
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−1.41421	−0.0492665
$825$	14.1421	0.492366
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	30.0000	1.04257
$829$	33.9411	1.17882	0.589412	−	0.807833i	$-0.299359\pi$
$829$	33.9411	1.17882	0.589412	+	0.807833i	$0.299359\pi$
$830$	0	0
$831$	−22.6274	−0.784936
$832$	4.24264	0.147087
$833$	0	0
$834$	52.0000	1.80061
$835$	0	0
$836$	4.24264	0.146735
$837$	−40.0000	−1.38260
$838$	14.1421	0.488532
$839$	−43.8406	−1.51355	−0.756773	−	0.653678i	$-0.773225\pi$
$839$	−43.8406	−1.51355	−0.756773	+	0.653678i	$0.773225\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−2.00000	−0.0689246
$843$	62.2254	2.14316
$844$	26.0000	0.894957
$845$	0	0
$846$	−63.6396	−2.18797
$847$	0	0
$848$	2.00000	0.0686803
$849$	20.0000	0.686398
$850$	14.1421	0.485071
$851$	12.0000	0.411355
$852$	45.2548	1.55041
$853$	−15.5563	−0.532639	−0.266320	−	0.963885i	$-0.585808\pi$
$853$	−15.5563	−0.532639	−0.266320	+	0.963885i	$0.585808\pi$
$854$	0	0
$855$	0	0
$856$	−14.0000	−0.478510
$857$	16.9706	0.579703	0.289852	−	0.957072i	$-0.406394\pi$
$857$	16.9706	0.579703	0.289852	+	0.957072i	$0.406394\pi$
$858$	−12.0000	−0.409673
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	29.6985	1.00920
$867$	−25.4558	−0.864526
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	33.9411	1.15005
$872$	−14.0000	−0.474100
$873$	−35.3553	−1.19660
$874$	−25.4558	−0.861057
$875$	0	0
$876$	−24.0000	−0.810885
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	25.4558	0.859093
$879$	−60.0000	−2.02375
$880$	0	0
$881$	−12.7279	−0.428815	−0.214407	−	0.976744i	$-0.568782\pi$
$881$	−12.7279	−0.428815	−0.214407	+	0.976744i	$0.568782\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	22.6274	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$887$	22.6274	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$888$	5.65685	0.189832
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	21.2132	0.710271
$893$	54.0000	1.80704
$894$	22.6274	0.756774
$895$	0	0
$896$	0	0
$897$	72.0000	2.40401
$898$	16.0000	0.533927
$899$	28.2843	0.943333
$900$	−25.0000	−0.833333
$901$	−5.65685	−0.188457
$902$	−2.82843	−0.0941763
$903$	0	0
$904$	16.0000	0.532152
$905$	0	0
$906$	−22.6274	−0.751746
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	−7.07107	−0.234662
$909$	−7.07107	−0.234533
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	−12.0000	−0.397360
$913$	12.7279	0.421233
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−22.6274	−0.747631
$917$	0	0
$918$	−16.0000	−0.528079
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	60.0000	1.97707
$922$	1.41421	0.0465746
$923$	67.8823	2.23437
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−16.0000	−0.525793
$927$	−7.07107	−0.232244
$928$	−4.00000	−0.131306
$929$	−26.8701	−0.881578	−0.440789	−	0.897611i	$-0.645301\pi$
$929$	−26.8701	−0.881578	−0.440789	+	0.897611i	$0.645301\pi$
$930$	0	0
$931$	0	0
$932$	26.0000	0.851658
$933$	−4.00000	−0.130954
$934$	25.4558	0.832941
$935$	0	0
$936$	21.2132	0.693375
$937$	−8.48528	−0.277202	−0.138601	−	0.990348i	$-0.544261\pi$
$937$	−8.48528	−0.277202	−0.138601	+	0.990348i	$0.544261\pi$
$938$	0	0
$939$	−84.0000	−2.74124
$940$	0	0
$941$	32.5269	1.06035	0.530174	−	0.847889i	$-0.322127\pi$
$941$	32.5269	1.06035	0.530174	+	0.847889i	$0.322127\pi$
$942$	24.0000	0.781962
$943$	16.9706	0.552638
$944$	11.3137	0.368230
$945$	0	0
$946$	−10.0000	−0.325128
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−22.6274	−0.734904
$949$	−36.0000	−1.16861
$950$	21.2132	0.688247
$951$	50.9117	1.65092
$952$	0	0
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	0	0
$957$	11.3137	0.365720
$958$	−28.2843	−0.913823
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	8.48528	0.273576
$963$	−70.0000	−2.25572
$964$	−8.48528	−0.273293
$965$	0	0
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	1.00000	0.0321412
$969$	33.9411	1.09035
$970$	0	0
$971$	−14.1421	−0.453843	−0.226921	−	0.973913i	$-0.572866\pi$
$971$	−14.1421	−0.453843	−0.226921	+	0.973913i	$0.572866\pi$
$972$	−14.1421	−0.453609
$973$	0	0
$974$	38.0000	1.21760
$975$	−60.0000	−1.92154
$976$	−9.89949	−0.316875
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	−11.3137	−0.361773
$979$	−7.07107	−0.225992
$980$	0	0
$981$	−70.0000	−2.23493
$982$	12.0000	0.382935
$983$	−26.8701	−0.857022	−0.428511	−	0.903537i	$-0.640962\pi$
$983$	−26.8701	−0.857022	−0.428511	+	0.903537i	$0.640962\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	11.3137	0.360302
$987$	0	0
$988$	−18.0000	−0.572656
$989$	60.0000	1.90789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−7.07107	−0.224507
$993$	22.6274	0.718059
$994$	0	0
$995$	0	0
$996$	−36.0000	−1.14070
$997$	−21.2132	−0.671829	−0.335914	−	0.941893i	$-0.609045\pi$
$997$	−21.2132	−0.671829	−0.335914	+	0.941893i	$0.609045\pi$
$998$	36.0000	1.13956
$999$	11.3137	0.357950

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.v.1.1	✓	2	7.6	odd	2	inner
1078.2.a.v.1.2	yes	2	1.1	even	1	trivial
1078.2.e.o.67.1		4	7.2	even	3
1078.2.e.o.67.2		4	7.5	odd	6
1078.2.e.o.177.1		4	7.4	even	3
1078.2.e.o.177.2		4	7.3	odd	6
8624.2.a.bz.1.1		2	4.3	odd	2
8624.2.a.bz.1.2		2	28.27	even	2
9702.2.a.co.1.1		2	21.20	even	2
9702.2.a.co.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} +2.82843 q^{12} +4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +5.00000 q^{18} -4.24264 q^{19} -1.00000 q^{22} +6.00000 q^{23} +2.82843 q^{24} -5.00000 q^{25} +4.24264 q^{26} +5.65685 q^{27} -4.00000 q^{29} -7.07107 q^{31} +1.00000 q^{32} -2.82843 q^{33} -2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} -4.24264 q^{38} +12.0000 q^{39} +2.82843 q^{41} +10.0000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -12.7279 q^{47} +2.82843 q^{48} -5.00000 q^{50} -8.00000 q^{51} +4.24264 q^{52} +2.00000 q^{53} +5.65685 q^{54} -12.0000 q^{57} -4.00000 q^{58} +11.3137 q^{59} -9.89949 q^{61} -7.07107 q^{62} +1.00000 q^{64} -2.82843 q^{66} +8.00000 q^{67} -2.82843 q^{68} +16.9706 q^{69} +16.0000 q^{71} +5.00000 q^{72} -8.48528 q^{73} +2.00000 q^{74} -14.1421 q^{75} -4.24264 q^{76} +12.0000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.82843 q^{82} -12.7279 q^{83} +10.0000 q^{86} -11.3137 q^{87} -1.00000 q^{88} +7.07107 q^{89} +6.00000 q^{92} -20.0000 q^{93} -12.7279 q^{94} +2.82843 q^{96} -7.07107 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} + 10 q^{18} - 2 q^{22} + 12 q^{23} - 10 q^{25} - 8 q^{29} + 2 q^{32} + 10 q^{36} + 4 q^{37} + 24 q^{39} + 20 q^{43} - 2 q^{44} + 12 q^{46}+ \cdots - 10 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 10 * q^9 - 2 * q^11 + 2 * q^16 + 10 * q^18 - 2 * q^22 + 12 * q^23 - 10 * q^25 - 8 * q^29 + 2 * q^32 + 10 * q^36 + 4 * q^37 + 24 * q^39 + 20 * q^43 - 2 * q^44 + 12 * q^46 - 10 * q^50 - 16 * q^51 + 4 * q^53 - 24 * q^57 - 8 * q^58 + 2 * q^64 + 16 * q^67 + 32 * q^71 + 10 * q^72 + 4 * q^74 + 24 * q^78 - 16 * q^79 + 2 * q^81 + 20 * q^86 - 2 * q^88 + 12 * q^92 - 40 * q^93 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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