LMFDB - Embedded newform 3136.2.f.e.3135.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(3135,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("3136.3135");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3135.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3136.3135
Dual form			3136.2.f.e.3135.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -1.73205i q^{5} +O(q^{10})$	$q+1.73205 q^{3} -1.73205i q^{5} -1.00000i q^{11} -3.46410i q^{13} -3.00000i q^{15} -1.73205i q^{17} -5.19615 q^{19} -1.00000i q^{23} +2.00000 q^{25} -5.19615 q^{27} -4.00000 q^{29} +1.73205 q^{31} -1.73205i q^{33} -3.00000 q^{37} -6.00000i q^{39} -3.46410i q^{41} +2.00000i q^{43} -8.66025 q^{47} -3.00000i q^{51} +1.00000 q^{53} -1.73205 q^{55} -9.00000 q^{57} -5.19615 q^{59} -5.19615i q^{61} -6.00000 q^{65} -3.00000i q^{67} -1.73205i q^{69} +14.0000i q^{71} +8.66025i q^{73} +3.46410 q^{75} -9.00000i q^{79} -9.00000 q^{81} +13.8564 q^{83} -3.00000 q^{85} -6.92820 q^{87} -15.5885i q^{89} +3.00000 q^{93} +9.00000i q^{95} -17.3205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{25} - 16 q^{29} - 12 q^{37} + 4 q^{53} - 36 q^{57} - 24 q^{65} - 36 q^{81} - 12 q^{85} + 12 q^{93}+O(q^{100}) $ 4 * q + 8 * q^25 - 16 * q^29 - 12 * q^37 + 4 * q^53 - 36 * q^57 - 24 * q^65 - 36 * q^81 - 12 * q^85 + 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	0	0
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i	−0.988571	−	0.150756i	$-0.951829\pi$
$11$	− 1.00000i	− 0.301511i	0.988571	−	0.150756i	$-0.0481707\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	− 3.00000i	− 0.774597i
$16$	0	0
$17$	− 1.73205i	− 0.420084i	−0.977692	−	0.210042i	$-0.932640\pi$
$17$	− 1.73205i	− 0.420084i	0.977692	−	0.210042i	$-0.0673601\pi$
$18$	0	0
$19$	−5.19615	−1.19208	−0.596040	−	0.802955i	$-0.703260\pi$
$19$	−5.19615	−1.19208	−0.596040	+	0.802955i	$0.703260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	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
$31$	1.73205	0.311086	0.155543	−	0.987829i	$-0.450287\pi$
$31$	1.73205	0.311086	0.155543	+	0.987829i	$0.450287\pi$
$32$	0	0
$33$	− 1.73205i	− 0.301511i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	− 6.00000i	− 0.960769i
$40$	0	0
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.66025	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−1.73205	−0.233550
$56$	0	0
$57$	−9.00000	−1.19208
$58$	0	0
$59$	−5.19615	−0.676481	−0.338241	−	0.941060i	$-0.609832\pi$
$59$	−5.19615	−0.676481	−0.338241	+	0.941060i	$0.609832\pi$
$60$	0	0
$61$	− 5.19615i	− 0.665299i	−0.943051	−	0.332650i	$-0.892057\pi$
$61$	− 5.19615i	− 0.665299i	0.943051	−	0.332650i	$-0.107943\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	0	0
$69$	− 1.73205i	− 0.208514i
$70$	0	0
$71$	14.0000i	1.66149i	0.556650	+	0.830747i	$0.312086\pi$
$71$	14.0000i	1.66149i	−0.556650	+	0.830747i	$0.687914\pi$
$72$	0	0
$73$	8.66025i	1.01361i	0.862062	+	0.506803i	$0.169173\pi$
$73$	8.66025i	1.01361i	−0.862062	+	0.506803i	$0.830827\pi$
$74$	0	0
$75$	3.46410	0.400000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 9.00000i	− 1.01258i	−0.862364	−	0.506290i	$-0.831017\pi$
$79$	− 9.00000i	− 1.01258i	0.862364	−	0.506290i	$-0.168983\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−6.92820	−0.742781
$88$	0	0
$89$	− 15.5885i	− 1.65237i	−0.563397	−	0.826187i	$-0.690506\pi$
$89$	− 15.5885i	− 1.65237i	0.563397	−	0.826187i	$-0.309494\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	9.00000i	0.923381i
$96$	0	0
$97$	− 17.3205i	− 1.75863i	−0.476240	−	0.879316i	$-0.658000\pi$
$97$	− 17.3205i	− 1.75863i	0.476240	−	0.879316i	$-0.342000\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.66025i	0.861727i	0.902417	+	0.430864i	$0.141791\pi$
$101$	8.66025i	0.861727i	−0.902417	+	0.430864i	$0.858209\pi$
$102$	0	0
$103$	8.66025	0.853320	0.426660	−	0.904412i	$-0.359690\pi$
$103$	8.66025	0.853320	0.426660	+	0.904412i	$0.359690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.0000i	− 1.25676i	−0.777908	−	0.628379i	$-0.783719\pi$
$107$	− 13.0000i	− 1.25676i	0.777908	−	0.628379i	$-0.216281\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	−5.19615	−0.493197
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	−1.73205	−0.161515
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.0000	0.909091
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	3.46410i	0.304997i
$130$	0	0
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.00000i	0.774597i
$136$	0	0
$137$	1.00000	0.0854358	0.0427179	−	0.999087i	$-0.486398\pi$
$137$	1.00000	0.0854358	0.0427179	+	0.999087i	$0.486398\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	−15.0000	−1.26323
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	6.92820i	0.575356i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	− 7.00000i	− 0.569652i	−0.958579	−	0.284826i	$-0.908064\pi$
$151$	− 7.00000i	− 0.569652i	0.958579	−	0.284826i	$-0.0919358\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.00000i	− 0.240966i
$156$	0	0
$157$	− 1.73205i	− 0.138233i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	− 1.73205i	− 0.138233i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$	0	0
$159$	1.73205	0.137361
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.0000i	1.64485i	0.568876	+	0.822423i	$0.307379\pi$
$163$	21.0000i	1.64485i	−0.568876	+	0.822423i	$0.692621\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.1244i	− 0.921798i	−0.887453	−	0.460899i	$-0.847527\pi$
$173$	− 12.1244i	− 0.921798i	0.887453	−	0.460899i	$-0.152473\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	19.0000i	1.42013i	0.704138	+	0.710063i	$0.251334\pi$
$179$	19.0000i	1.42013i	−0.704138	+	0.710063i	$0.748666\pi$
$180$	0	0
$181$	− 6.92820i	− 0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	− 6.92820i	− 0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$	0	0
$183$	− 9.00000i	− 0.665299i
$184$	0	0
$185$	5.19615i	0.382029i
$186$	0	0
$187$	−1.73205	−0.126660
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.00000i	0.0723575i	0.999345	+	0.0361787i	$0.0115186\pi$
$191$	1.00000i	0.0723575i	−0.999345	+	0.0361787i	$0.988481\pi$
$192$	0	0
$193$	−15.0000	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$193$	−15.0000	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$194$	0	0
$195$	−10.3923	−0.744208
$196$	0	0
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	22.5167	1.59616	0.798082	−	0.602549i	$-0.205848\pi$
$199$	22.5167	1.59616	0.798082	+	0.602549i	$0.205848\pi$
$200$	0	0
$201$	− 5.19615i	− 0.366508i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.19615i	0.359425i
$210$	0	0
$211$	10.0000i	0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$	10.0000i	0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	0	0
$213$	24.2487i	1.66149i
$214$	0	0
$215$	3.46410	0.236250
$216$	0	0
$217$	0	0
$218$	0	0
$219$	15.0000i	1.01361i
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.0526	1.26456	0.632281	−	0.774739i	$-0.282119\pi$
$227$	19.0526	1.26456	0.632281	+	0.774739i	$0.282119\pi$
$228$	0	0
$229$	− 15.5885i	− 1.03011i	−0.857156	−	0.515057i	$-0.827771\pi$
$229$	− 15.5885i	− 1.03011i	0.857156	−	0.515057i	$-0.172229\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.00000	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$234$	0	0
$235$	15.0000i	0.978492i
$236$	0	0
$237$	− 15.5885i	− 1.01258i
$238$	0	0
$239$	− 20.0000i	− 1.29369i	−0.762620	−	0.646846i	$-0.776088\pi$
$239$	− 20.0000i	− 1.29369i	0.762620	−	0.646846i	$-0.223912\pi$
$240$	0	0
$241$	− 5.19615i	− 0.334714i	−0.985896	−	0.167357i	$-0.946477\pi$
$241$	− 5.19615i	− 0.334714i	0.985896	−	0.167357i	$-0.0535232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.0000i	1.14531i
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	−3.46410	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−5.19615	−0.325396
$256$	0	0
$257$	− 5.19615i	− 0.324127i	−0.986780	−	0.162064i	$-0.948185\pi$
$257$	− 5.19615i	− 0.324127i	0.986780	−	0.162064i	$-0.0518150\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.0000i	1.41824i	0.705087	+	0.709120i	$0.250908\pi$
$263$	23.0000i	1.41824i	−0.705087	+	0.709120i	$0.749092\pi$
$264$	0	0
$265$	− 1.73205i	− 0.106399i
$266$	0	0
$267$	− 27.0000i	− 1.65237i
$268$	0	0
$269$	− 22.5167i	− 1.37287i	−0.727194	−	0.686433i	$-0.759176\pi$
$269$	− 22.5167i	− 1.37287i	0.727194	−	0.686433i	$-0.240824\pi$
$270$	0	0
$271$	15.5885	0.946931	0.473466	−	0.880812i	$-0.343003\pi$
$271$	15.5885	0.946931	0.473466	+	0.880812i	$0.343003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	12.1244	0.720718	0.360359	−	0.932814i	$-0.382654\pi$
$283$	12.1244	0.720718	0.360359	+	0.932814i	$0.382654\pi$
$284$	0	0
$285$	15.5885i	0.923381i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	14.0000	0.823529
$290$	0	0
$291$	− 30.0000i	− 1.75863i
$292$	0	0
$293$	20.7846i	1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$	20.7846i	1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$	0	0
$295$	9.00000i	0.524000i
$296$	0	0
$297$	5.19615i	0.301511i
$298$	0	0
$299$	−3.46410	−0.200334
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.0000i	0.861727i
$304$	0	0
$305$	−9.00000	−0.515339
$306$	0	0
$307$	20.7846	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$307$	20.7846	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$308$	0	0
$309$	15.0000	0.853320
$310$	0	0
$311$	−8.66025	−0.491078	−0.245539	−	0.969387i	$-0.578965\pi$
$311$	−8.66025	−0.491078	−0.245539	+	0.969387i	$0.578965\pi$
$312$	0	0
$313$	− 1.73205i	− 0.0979013i	−0.998801	−	0.0489506i	$-0.984412\pi$
$313$	− 1.73205i	− 0.0979013i	0.998801	−	0.0489506i	$-0.0155877\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	0	0
$319$	4.00000i	0.223957i
$320$	0	0
$321$	− 22.5167i	− 1.25676i
$322$	0	0
$323$	9.00000i	0.500773i
$324$	0	0
$325$	− 6.92820i	− 0.384308i
$326$	0	0
$327$	−15.5885	−0.862044
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 7.00000i	− 0.384755i	−0.981321	−	0.192377i	$-0.938380\pi$
$331$	− 7.00000i	− 0.384755i	0.981321	−	0.192377i	$-0.0616198\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.19615	−0.283896
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	−27.7128	−1.50515
$340$	0	0
$341$	− 1.73205i	− 0.0937958i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	13.0000i	0.697877i	0.937146	+	0.348938i	$0.113458\pi$
$347$	13.0000i	0.697877i	−0.937146	+	0.348938i	$0.886542\pi$
$348$	0	0
$349$	− 10.3923i	− 0.556287i	−0.960539	−	0.278144i	$-0.910281\pi$
$349$	− 10.3923i	− 0.556287i	0.960539	−	0.278144i	$-0.0897191\pi$
$350$	0	0
$351$	18.0000i	0.960769i
$352$	0	0
$353$	− 29.4449i	− 1.56719i	−0.621271	−	0.783596i	$-0.713383\pi$
$353$	− 29.4449i	− 1.56719i	0.621271	−	0.783596i	$-0.286617\pi$
$354$	0	0
$355$	24.2487	1.28699
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 23.0000i	− 1.21389i	−0.794742	−	0.606947i	$-0.792394\pi$
$359$	− 23.0000i	− 1.21389i	0.794742	−	0.606947i	$-0.207606\pi$
$360$	0	0
$361$	8.00000	0.421053
$362$	0	0
$363$	17.3205	0.909091
$364$	0	0
$365$	15.0000	0.785136
$366$	0	0
$367$	−1.73205	−0.0904123	−0.0452062	−	0.998978i	$-0.514394\pi$
$367$	−1.73205	−0.0904123	−0.0452062	+	0.998978i	$0.514394\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	0	0
$375$	− 21.0000i	− 1.08444i
$376$	0	0
$377$	13.8564i	0.713641i
$378$	0	0
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	0	0
$381$	− 10.3923i	− 0.532414i
$382$	0	0
$383$	5.19615	0.265511	0.132755	−	0.991149i	$-0.457617\pi$
$383$	5.19615	0.265511	0.132755	+	0.991149i	$0.457617\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	−1.73205	−0.0875936
$392$	0	0
$393$	9.00000	0.453990
$394$	0	0
$395$	−15.5885	−0.784340
$396$	0	0
$397$	19.0526i	0.956221i	0.878300	+	0.478110i	$0.158678\pi$
$397$	19.0526i	0.956221i	−0.878300	+	0.478110i	$0.841322\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	0	0
$405$	15.5885i	0.774597i
$406$	0	0
$407$	3.00000i	0.148704i
$408$	0	0
$409$	25.9808i	1.28467i	0.766426	+	0.642333i	$0.222033\pi$
$409$	25.9808i	1.28467i	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	1.73205	0.0854358
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 24.0000i	− 1.17811i
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−20.7846	−1.01539	−0.507697	−	0.861536i	$-0.669503\pi$
$419$	−20.7846	−1.01539	−0.507697	+	0.861536i	$0.669503\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.46410i	− 0.168034i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	− 23.0000i	− 1.10787i	−0.832560	−	0.553936i	$-0.813125\pi$
$431$	− 23.0000i	− 1.10787i	0.832560	−	0.553936i	$-0.186875\pi$
$432$	0	0
$433$	− 10.3923i	− 0.499422i	−0.968320	−	0.249711i	$-0.919664\pi$
$433$	− 10.3923i	− 0.499422i	0.968320	−	0.249711i	$-0.0803357\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	5.19615i	0.248566i
$438$	0	0
$439$	−22.5167	−1.07466	−0.537331	−	0.843372i	$-0.680567\pi$
$439$	−22.5167	−1.07466	−0.537331	+	0.843372i	$0.680567\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17.0000i	− 0.807694i	−0.914826	−	0.403847i	$-0.867673\pi$
$443$	− 17.0000i	− 0.807694i	0.914826	−	0.403847i	$-0.132327\pi$
$444$	0	0
$445$	−27.0000	−1.27992
$446$	0	0
$447$	−1.73205	−0.0819232
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−3.46410	−0.163118
$452$	0	0
$453$	− 12.1244i	− 0.569652i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0000	0.701670	0.350835	−	0.936437i	$-0.385898\pi$
$457$	15.0000	0.701670	0.350835	+	0.936437i	$0.385898\pi$
$458$	0	0
$459$	9.00000i	0.420084i
$460$	0	0
$461$	17.3205i	0.806696i	0.915047	+	0.403348i	$0.132154\pi$
$461$	17.3205i	0.806696i	−0.915047	+	0.403348i	$0.867846\pi$
$462$	0	0
$463$	30.0000i	1.39422i	0.716965	+	0.697109i	$0.245531\pi$
$463$	30.0000i	1.39422i	−0.716965	+	0.697109i	$0.754469\pi$
$464$	0	0
$465$	− 5.19615i	− 0.240966i
$466$	0	0
$467$	−8.66025	−0.400749	−0.200374	−	0.979719i	$-0.564216\pi$
$467$	−8.66025	−0.400749	−0.200374	+	0.979719i	$0.564216\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 3.00000i	− 0.138233i
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−10.3923	−0.476832
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.1244	0.553976	0.276988	−	0.960873i	$-0.410664\pi$
$479$	12.1244	0.553976	0.276988	+	0.960873i	$0.410664\pi$
$480$	0	0
$481$	10.3923i	0.473848i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30.0000	−1.36223
$486$	0	0
$487$	31.0000i	1.40474i	0.711810	+	0.702372i	$0.247876\pi$
$487$	31.0000i	1.40474i	−0.711810	+	0.702372i	$0.752124\pi$
$488$	0	0
$489$	36.3731i	1.64485i
$490$	0	0
$491$	32.0000i	1.44414i	0.691820	+	0.722070i	$0.256809\pi$
$491$	32.0000i	1.44414i	−0.691820	+	0.722070i	$0.743191\pi$
$492$	0	0
$493$	6.92820i	0.312031i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 35.0000i	− 1.56682i	−0.621508	−	0.783408i	$-0.713480\pi$
$499$	− 35.0000i	− 1.56682i	0.621508	−	0.783408i	$-0.286520\pi$
$500$	0	0
$501$	30.0000	1.34030
$502$	0	0
$503$	6.92820	0.308913	0.154457	−	0.988000i	$-0.450637\pi$
$503$	6.92820	0.308913	0.154457	+	0.988000i	$0.450637\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	1.73205	0.0769231
$508$	0	0
$509$	12.1244i	0.537403i	0.963224	+	0.268701i	$0.0865945\pi$
$509$	12.1244i	0.537403i	−0.963224	+	0.268701i	$0.913406\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	27.0000	1.19208
$514$	0	0
$515$	− 15.0000i	− 0.660979i
$516$	0	0
$517$	8.66025i	0.380878i
$518$	0	0
$519$	− 21.0000i	− 0.921798i
$520$	0	0
$521$	1.73205i	0.0758825i	0.999280	+	0.0379413i	$0.0120800\pi$
$521$	1.73205i	0.0758825i	−0.999280	+	0.0379413i	$0.987920\pi$
$522$	0	0
$523$	−25.9808	−1.13606	−0.568030	−	0.823008i	$-0.692294\pi$
$523$	−25.9808	−1.13606	−0.568030	+	0.823008i	$0.692294\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.00000i	− 0.130682i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−22.5167	−0.973480
$536$	0	0
$537$	32.9090i	1.42013i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.0000	0.816874	0.408437	−	0.912787i	$-0.366074\pi$
$541$	19.0000	0.816874	0.408437	+	0.912787i	$0.366074\pi$
$542$	0	0
$543$	− 12.0000i	− 0.514969i
$544$	0	0
$545$	15.5885i	0.667736i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	0	0
$554$	0	0
$555$	9.00000i	0.382029i
$556$	0	0
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	0	0
$559$	6.92820	0.293032
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	22.5167	0.948964	0.474482	−	0.880265i	$-0.342635\pi$
$563$	22.5167	0.948964	0.474482	+	0.880265i	$0.342635\pi$
$564$	0	0
$565$	27.7128i	1.16589i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.0000	−0.544988	−0.272494	−	0.962157i	$-0.587849\pi$
$569$	−13.0000	−0.544988	−0.272494	+	0.962157i	$0.587849\pi$
$570$	0	0
$571$	− 21.0000i	− 0.878823i	−0.898286	−	0.439411i	$-0.855187\pi$
$571$	− 21.0000i	− 0.878823i	0.898286	−	0.439411i	$-0.144813\pi$
$572$	0	0
$573$	1.73205i	0.0723575i
$574$	0	0
$575$	− 2.00000i	− 0.0834058i
$576$	0	0
$577$	− 32.9090i	− 1.37002i	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	− 32.9090i	− 1.37002i	0.728535	−	0.685009i	$-0.240202\pi$
$578$	0	0
$579$	−25.9808	−1.07972
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 1.00000i	− 0.0414158i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	−27.7128	−1.13995
$592$	0	0
$593$	15.5885i	0.640141i	0.947394	+	0.320071i	$0.103707\pi$
$593$	15.5885i	0.640141i	−0.947394	+	0.320071i	$0.896293\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	39.0000	1.59616
$598$	0	0
$599$	− 17.0000i	− 0.694601i	−0.937754	−	0.347301i	$-0.887098\pi$
$599$	− 17.0000i	− 0.694601i	0.937754	−	0.347301i	$-0.112902\pi$
$600$	0	0
$601$	38.1051i	1.55434i	0.629291	+	0.777170i	$0.283346\pi$
$601$	38.1051i	1.55434i	−0.629291	+	0.777170i	$0.716654\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 17.3205i	− 0.704179i
$606$	0	0
$607$	−15.5885	−0.632716	−0.316358	−	0.948640i	$-0.602460\pi$
$607$	−15.5885	−0.632716	−0.316358	+	0.948640i	$0.602460\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0000i	1.21367i
$612$	0	0
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	0	0
$615$	−10.3923	−0.419058
$616$	0	0
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	0	0
$619$	15.5885	0.626553	0.313276	−	0.949662i	$-0.398573\pi$
$619$	15.5885	0.626553	0.313276	+	0.949662i	$0.398573\pi$
$620$	0	0
$621$	5.19615i	0.208514i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	9.00000i	0.359425i
$628$	0	0
$629$	5.19615i	0.207184i
$630$	0	0
$631$	− 30.0000i	− 1.19428i	−0.802137	−	0.597141i	$-0.796303\pi$
$631$	− 30.0000i	− 1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$632$	0	0
$633$	17.3205i	0.688428i
$634$	0	0
$635$	−10.3923	−0.412406
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.0000	−0.513469	−0.256735	−	0.966482i	$-0.582647\pi$
$641$	−13.0000	−0.513469	−0.256735	+	0.966482i	$0.582647\pi$
$642$	0	0
$643$	−13.8564	−0.546443	−0.273222	−	0.961951i	$-0.588089\pi$
$643$	−13.8564	−0.546443	−0.273222	+	0.961951i	$0.588089\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	32.9090	1.29378	0.646892	−	0.762581i	$-0.276068\pi$
$647$	32.9090	1.29378	0.646892	+	0.762581i	$0.276068\pi$
$648$	0	0
$649$	5.19615i	0.203967i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.0000	−1.21312	−0.606562	−	0.795036i	$-0.707452\pi$
$653$	−31.0000	−1.21312	−0.606562	+	0.795036i	$0.707452\pi$
$654$	0	0
$655$	− 9.00000i	− 0.351659i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 38.0000i	− 1.48027i	−0.672458	−	0.740135i	$-0.734762\pi$
$659$	− 38.0000i	− 1.48027i	0.672458	−	0.740135i	$-0.265238\pi$
$660$	0	0
$661$	39.8372i	1.54949i	0.632276	+	0.774743i	$0.282121\pi$
$661$	39.8372i	1.54949i	−0.632276	+	0.774743i	$0.717879\pi$
$662$	0	0
$663$	−10.3923	−0.403604
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000i	0.154881i
$668$	0	0
$669$	−12.0000	−0.463947
$670$	0	0
$671$	−5.19615	−0.200595
$672$	0	0
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	0	0
$675$	−10.3923	−0.400000
$676$	0	0
$677$	− 43.3013i	− 1.66420i	−0.554623	−	0.832102i	$-0.687138\pi$
$677$	− 43.3013i	− 1.66420i	0.554623	−	0.832102i	$-0.312862\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	33.0000	1.26456
$682$	0	0
$683$	− 25.0000i	− 0.956598i	−0.878197	−	0.478299i	$-0.841253\pi$
$683$	− 25.0000i	− 0.956598i	0.878197	−	0.478299i	$-0.158747\pi$
$684$	0	0
$685$	− 1.73205i	− 0.0661783i
$686$	0	0
$687$	− 27.0000i	− 1.03011i
$688$	0	0
$689$	− 3.46410i	− 0.131972i
$690$	0	0
$691$	12.1244	0.461232	0.230616	−	0.973045i	$-0.425926\pi$
$691$	12.1244	0.461232	0.230616	+	0.973045i	$0.425926\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000i	0.455186i
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	−12.1244	−0.458585
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	15.5885	0.587930
$704$	0	0
$705$	25.9808i	0.978492i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	9.00000	0.338002	0.169001	−	0.985616i	$-0.445946\pi$
$709$	9.00000	0.338002	0.169001	+	0.985616i	$0.445946\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.73205i	− 0.0648658i
$714$	0	0
$715$	6.00000i	0.224387i
$716$	0	0
$717$	− 34.6410i	− 1.29369i
$718$	0	0
$719$	−25.9808	−0.968919	−0.484459	−	0.874814i	$-0.660984\pi$
$719$	−25.9808	−0.968919	−0.484459	+	0.874814i	$0.660984\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 9.00000i	− 0.334714i
$724$	0	0
$725$	−8.00000	−0.297113
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	3.46410	0.128124
$732$	0	0
$733$	43.3013i	1.59937i	0.600420	+	0.799684i	$0.295000\pi$
$733$	43.3013i	1.59937i	−0.600420	+	0.799684i	$0.705000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.00000	−0.110506
$738$	0	0
$739$	51.0000i	1.87607i	0.346547	+	0.938033i	$0.387354\pi$
$739$	51.0000i	1.87607i	−0.346547	+	0.938033i	$0.612646\pi$
$740$	0	0
$741$	31.1769i	1.14531i
$742$	0	0
$743$	34.0000i	1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$	34.0000i	1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$	0	0
$745$	1.73205i	0.0634574i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.0000i	0.912263i	0.889912	+	0.456131i	$0.150765\pi$
$751$	25.0000i	0.912263i	−0.889912	+	0.456131i	$0.849235\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−12.1244	−0.441250
$756$	0	0
$757$	48.0000	1.74459	0.872295	−	0.488980i	$-0.162631\pi$
$757$	48.0000	1.74459	0.872295	+	0.488980i	$0.162631\pi$
$758$	0	0
$759$	−1.73205	−0.0628695
$760$	0	0
$761$	− 19.0526i	− 0.690655i	−0.938482	−	0.345327i	$-0.887768\pi$
$761$	− 19.0526i	− 0.690655i	0.938482	−	0.345327i	$-0.112232\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0000i	0.649942i
$768$	0	0
$769$	− 3.46410i	− 0.124919i	−0.998048	−	0.0624593i	$-0.980106\pi$
$769$	− 3.46410i	− 0.124919i	0.998048	−	0.0624593i	$-0.0198944\pi$
$770$	0	0
$771$	− 9.00000i	− 0.324127i
$772$	0	0
$773$	− 25.9808i	− 0.934463i	−0.884135	−	0.467232i	$-0.845251\pi$
$773$	− 25.9808i	− 0.934463i	0.884135	−	0.467232i	$-0.154749\pi$
$774$	0	0
$775$	3.46410	0.124434
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000i	0.644917i
$780$	0	0
$781$	14.0000	0.500959
$782$	0	0
$783$	20.7846	0.742781
$784$	0	0
$785$	−3.00000	−0.107075
$786$	0	0
$787$	−5.19615	−0.185223	−0.0926114	−	0.995702i	$-0.529521\pi$
$787$	−5.19615	−0.185223	−0.0926114	+	0.995702i	$0.529521\pi$
$788$	0	0
$789$	39.8372i	1.41824i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	− 3.00000i	− 0.106399i
$796$	0	0
$797$	10.3923i	0.368114i	0.982916	+	0.184057i	$0.0589232\pi$
$797$	10.3923i	0.368114i	−0.982916	+	0.184057i	$0.941077\pi$
$798$	0	0
$799$	15.0000i	0.530662i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.66025	0.305614
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 39.0000i	− 1.37287i
$808$	0	0
$809$	43.0000	1.51180	0.755900	−	0.654687i	$-0.227200\pi$
$809$	43.0000	1.51180	0.755900	+	0.654687i	$0.227200\pi$
$810$	0	0
$811$	13.8564	0.486564	0.243282	−	0.969956i	$-0.421776\pi$
$811$	13.8564	0.486564	0.243282	+	0.969956i	$0.421776\pi$
$812$	0	0
$813$	27.0000	0.946931
$814$	0	0
$815$	36.3731	1.27409
$816$	0	0
$817$	− 10.3923i	− 0.363581i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.0000	0.383903	0.191951	−	0.981404i	$-0.438518\pi$
$821$	11.0000	0.383903	0.191951	+	0.981404i	$0.438518\pi$
$822$	0	0
$823$	− 9.00000i	− 0.313720i	−0.987621	−	0.156860i	$-0.949863\pi$
$823$	− 9.00000i	− 0.313720i	0.987621	−	0.156860i	$-0.0501372\pi$
$824$	0	0
$825$	− 3.46410i	− 0.120605i
$826$	0	0
$827$	22.0000i	0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$	22.0000i	0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$	0	0
$829$	8.66025i	0.300783i	0.988627	+	0.150392i	$0.0480534\pi$
$829$	8.66025i	0.300783i	−0.988627	+	0.150392i	$0.951947\pi$
$830$	0	0
$831$	22.5167	0.781094
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 30.0000i	− 1.03819i
$836$	0	0
$837$	−9.00000	−0.311086
$838$	0	0
$839$	−48.4974	−1.67432	−0.837158	−	0.546960i	$-0.815785\pi$
$839$	−48.4974	−1.67432	−0.837158	+	0.546960i	$0.815785\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−6.92820	−0.238620
$844$	0	0
$845$	− 1.73205i	− 0.0595844i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	3.00000i	0.102839i
$852$	0	0
$853$	24.2487i	0.830260i	0.909762	+	0.415130i	$0.136264\pi$
$853$	24.2487i	0.830260i	−0.909762	+	0.415130i	$0.863736\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.9808i	− 0.887486i	−0.896154	−	0.443743i	$-0.853650\pi$
$857$	− 25.9808i	− 0.887486i	0.896154	−	0.443743i	$-0.146350\pi$
$858$	0	0
$859$	50.2295	1.71381	0.856904	−	0.515476i	$-0.172385\pi$
$859$	50.2295	1.71381	0.856904	+	0.515476i	$0.172385\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.0000i	1.19141i	0.803202	+	0.595707i	$0.203128\pi$
$863$	35.0000i	1.19141i	−0.803202	+	0.595707i	$0.796872\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	24.2487	0.823529
$868$	0	0
$869$	−9.00000	−0.305304
$870$	0	0
$871$	−10.3923	−0.352130
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.00000	−0.0337676	−0.0168838	−	0.999857i	$-0.505375\pi$
$877$	−1.00000	−0.0337676	−0.0168838	+	0.999857i	$0.505375\pi$
$878$	0	0
$879$	36.0000i	1.21425i
$880$	0	0
$881$	13.8564i	0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$	13.8564i	0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$	0	0
$883$	− 10.0000i	− 0.336527i	−0.985742	−	0.168263i	$-0.946184\pi$
$883$	− 10.0000i	− 0.336527i	0.985742	−	0.168263i	$-0.0538159\pi$
$884$	0	0
$885$	15.5885i	0.524000i
$886$	0	0
$887$	25.9808	0.872349	0.436174	−	0.899862i	$-0.356333\pi$
$887$	25.9808	0.872349	0.436174	+	0.899862i	$0.356333\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000i	0.301511i
$892$	0	0
$893$	45.0000	1.50587
$894$	0	0
$895$	32.9090	1.10003
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	−6.92820	−0.231069
$900$	0	0
$901$	− 1.73205i	− 0.0577030i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	− 7.00000i	− 0.232431i	−0.993224	−	0.116216i	$-0.962924\pi$
$907$	− 7.00000i	− 0.232431i	0.993224	−	0.116216i	$-0.0370764\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.0000i	0.861418i	0.902491	+	0.430709i	$0.141737\pi$
$911$	26.0000i	0.861418i	−0.902491	+	0.430709i	$0.858263\pi$
$912$	0	0
$913$	− 13.8564i	− 0.458580i
$914$	0	0
$915$	−15.5885	−0.515339
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.00000i	0.0329870i	0.999864	+	0.0164935i	$0.00525028\pi$
$919$	1.00000i	0.0329870i	−0.999864	+	0.0164935i	$0.994750\pi$
$920$	0	0
$921$	36.0000	1.18624
$922$	0	0
$923$	48.4974	1.59631
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.66025i	− 0.284134i	−0.989857	−	0.142067i	$-0.954625\pi$
$929$	− 8.66025i	− 0.284134i	0.989857	−	0.142067i	$-0.0453748\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.0000	−0.491078
$934$	0	0
$935$	3.00000i	0.0981105i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	− 3.00000i	− 0.0979013i
$940$	0	0
$941$	− 57.1577i	− 1.86329i	−0.363374	−	0.931644i	$-0.618375\pi$
$941$	− 57.1577i	− 1.86329i	0.363374	−	0.931644i	$-0.381625\pi$
$942$	0	0
$943$	−3.46410	−0.112807
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.0000i	0.942373i	0.882034	+	0.471187i	$0.156174\pi$
$947$	29.0000i	0.942373i	−0.882034	+	0.471187i	$0.843826\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	−19.0526	−0.617822
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	1.73205	0.0560478
$956$	0	0
$957$	6.92820i	0.223957i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.0000	−0.903226
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.9808i	0.836350i
$966$	0	0
$967$	6.00000i	0.192947i	0.995336	+	0.0964735i	$0.0307563\pi$
$967$	6.00000i	0.192947i	−0.995336	+	0.0964735i	$0.969244\pi$
$968$	0	0
$969$	15.5885i	0.500773i
$970$	0	0
$971$	60.6218	1.94545	0.972723	−	0.231971i	$-0.0745174\pi$
$971$	60.6218	1.94545	0.972723	+	0.231971i	$0.0745174\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 12.0000i	− 0.384308i
$976$	0	0
$977$	−31.0000	−0.991778	−0.495889	−	0.868386i	$-0.665158\pi$
$977$	−31.0000	−0.991778	−0.495889	+	0.868386i	$0.665158\pi$
$978$	0	0
$979$	−15.5885	−0.498209
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−60.6218	−1.93353	−0.966767	−	0.255658i	$-0.917708\pi$
$983$	−60.6218	−1.93353	−0.966767	+	0.255658i	$0.917708\pi$
$984$	0	0
$985$	27.7128i	0.883004i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	23.0000i	0.730619i	0.930886	+	0.365310i	$0.119037\pi$
$991$	23.0000i	0.730619i	−0.930886	+	0.365310i	$0.880963\pi$
$992$	0	0
$993$	− 12.1244i	− 0.384755i
$994$	0	0
$995$	− 39.0000i	− 1.23638i
$996$	0	0
$997$	− 22.5167i	− 0.713110i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	− 22.5167i	− 0.713110i	0.934274	−	0.356555i	$-0.116049\pi$
$998$	0	0
$999$	15.5885	0.493197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.f.e.3135.3		4
4.3	odd	2	inner	3136.2.f.e.3135.1		4
7.4	even	3		448.2.p.d.383.1		4
7.5	odd	6		448.2.p.d.255.2		4
7.6	odd	2	inner	3136.2.f.e.3135.2		4
8.3	odd	2		196.2.d.b.195.2		4
8.5	even	2		196.2.d.b.195.3		4
24.5	odd	2		1764.2.b.a.1567.1		4
24.11	even	2		1764.2.b.a.1567.3		4
28.11	odd	6		448.2.p.d.383.2		4
28.19	even	6		448.2.p.d.255.1		4
28.27	even	2	inner	3136.2.f.e.3135.4		4
56.3	even	6		196.2.f.a.19.2		4
56.5	odd	6		28.2.f.a.3.2	yes	4
56.11	odd	6		28.2.f.a.19.2	yes	4
56.13	odd	2		196.2.d.b.195.4		4
56.19	even	6		28.2.f.a.3.1	✓	4
56.27	even	2		196.2.d.b.195.1		4
56.37	even	6		196.2.f.a.31.2		4
56.45	odd	6		196.2.f.a.19.1		4
56.51	odd	6		196.2.f.a.31.1		4
56.53	even	6		28.2.f.a.19.1	yes	4
168.5	even	6		252.2.bf.e.199.1		4
168.11	even	6		252.2.bf.e.19.1		4
168.53	odd	6		252.2.bf.e.19.2		4
168.83	odd	2		1764.2.b.a.1567.4		4
168.125	even	2		1764.2.b.a.1567.2		4
168.131	odd	6		252.2.bf.e.199.2		4
280.19	even	6		700.2.p.a.451.2		4
280.53	odd	12		700.2.t.a.299.2		4
280.67	even	12		700.2.t.a.299.1		4
280.109	even	6		700.2.p.a.551.2		4
280.117	even	12		700.2.t.b.199.2		4
280.123	even	12		700.2.t.b.299.2		4
280.173	even	12		700.2.t.a.199.1		4
280.179	odd	6		700.2.p.a.551.1		4
280.187	odd	12		700.2.t.a.199.2		4
280.229	odd	6		700.2.p.a.451.1		4
280.243	odd	12		700.2.t.b.199.1		4
280.277	odd	12		700.2.t.b.299.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.f.a.3.1	✓	4	56.19	even	6
28.2.f.a.3.2	yes	4	56.5	odd	6
28.2.f.a.19.1	yes	4	56.53	even	6
28.2.f.a.19.2	yes	4	56.11	odd	6
196.2.d.b.195.1		4	56.27	even	2
196.2.d.b.195.2		4	8.3	odd	2
196.2.d.b.195.3		4	8.5	even	2
196.2.d.b.195.4		4	56.13	odd	2
196.2.f.a.19.1		4	56.45	odd	6
196.2.f.a.19.2		4	56.3	even	6
196.2.f.a.31.1		4	56.51	odd	6
196.2.f.a.31.2		4	56.37	even	6
252.2.bf.e.19.1		4	168.11	even	6
252.2.bf.e.19.2		4	168.53	odd	6
252.2.bf.e.199.1		4	168.5	even	6
252.2.bf.e.199.2		4	168.131	odd	6
448.2.p.d.255.1		4	28.19	even	6
448.2.p.d.255.2		4	7.5	odd	6
448.2.p.d.383.1		4	7.4	even	3
448.2.p.d.383.2		4	28.11	odd	6
700.2.p.a.451.1		4	280.229	odd	6
700.2.p.a.451.2		4	280.19	even	6
700.2.p.a.551.1		4	280.179	odd	6
700.2.p.a.551.2		4	280.109	even	6
700.2.t.a.199.1		4	280.173	even	12
700.2.t.a.199.2		4	280.187	odd	12
700.2.t.a.299.1		4	280.67	even	12
700.2.t.a.299.2		4	280.53	odd	12
700.2.t.b.199.1		4	280.243	odd	12
700.2.t.b.199.2		4	280.117	even	12
700.2.t.b.299.1		4	280.277	odd	12
700.2.t.b.299.2		4	280.123	even	12
1764.2.b.a.1567.1		4	24.5	odd	2
1764.2.b.a.1567.2		4	168.125	even	2
1764.2.b.a.1567.3		4	24.11	even	2
1764.2.b.a.1567.4		4	168.83	odd	2
3136.2.f.e.3135.1		4	4.3	odd	2	inner
3136.2.f.e.3135.2		4	7.6	odd	2	inner
3136.2.f.e.3135.3		4	1.1	even	1	trivial
3136.2.f.e.3135.4		4	28.27	even	2	inner

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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -1.73205i q^{5} +O(q^{10})\)	\(q+1.73205 q^{3} -1.73205i q^{5} -1.00000i q^{11} -3.46410i q^{13} -3.00000i q^{15} -1.73205i q^{17} -5.19615 q^{19} -1.00000i q^{23} +2.00000 q^{25} -5.19615 q^{27} -4.00000 q^{29} +1.73205 q^{31} -1.73205i q^{33} -3.00000 q^{37} -6.00000i q^{39} -3.46410i q^{41} +2.00000i q^{43} -8.66025 q^{47} -3.00000i q^{51} +1.00000 q^{53} -1.73205 q^{55} -9.00000 q^{57} -5.19615 q^{59} -5.19615i q^{61} -6.00000 q^{65} -3.00000i q^{67} -1.73205i q^{69} +14.0000i q^{71} +8.66025i q^{73} +3.46410 q^{75} -9.00000i q^{79} -9.00000 q^{81} +13.8564 q^{83} -3.00000 q^{85} -6.92820 q^{87} -15.5885i q^{89} +3.00000 q^{93} +9.00000i q^{95} -17.3205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{25} - 16 q^{29} - 12 q^{37} + 4 q^{53} - 36 q^{57} - 24 q^{65} - 36 q^{81} - 12 q^{85} + 12 q^{93}+O(q^{100}) \) 4 * q + 8 * q^25 - 16 * q^29 - 12 * q^37 + 4 * q^53 - 36 * q^57 - 24 * q^65 - 36 * q^81 - 12 * q^85 + 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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