LMFDB - Embedded newform 2736.2.k.a.2431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.a.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{5} +1.73205i q^{7} +O(q^{10})$	$q-3.00000 q^{5} +1.73205i q^{7} +5.19615i q^{11} -6.92820i q^{13} -3.00000 q^{17} +(-4.00000 - 1.73205i) q^{19} -3.46410i q^{23} +4.00000 q^{25} +4.00000 q^{31} -5.19615i q^{35} +6.92820i q^{37} -6.92820i q^{41} +8.66025i q^{43} -8.66025i q^{47} +4.00000 q^{49} +6.92820i q^{53} -15.5885i q^{55} +12.0000 q^{59} +7.00000 q^{61} +20.7846i q^{65} -8.00000 q^{67} +12.0000 q^{71} -5.00000 q^{73} -9.00000 q^{77} +8.00000 q^{79} +3.46410i q^{83} +9.00000 q^{85} -6.92820i q^{89} +12.0000 q^{91} +(12.0000 + 5.19615i) q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5}+O(q^{10}) $ 2 * q - 6 * q^5	$ 2 q - 6 q^{5} - 6 q^{17} - 8 q^{19} + 8 q^{25} + 8 q^{31} + 8 q^{49} + 24 q^{59} + 14 q^{61} - 16 q^{67} + 24 q^{71} - 10 q^{73} - 18 q^{77} + 16 q^{79} + 18 q^{85} + 24 q^{91} + 24 q^{95}+O(q^{100}) $ 2 * q - 6 * q^5 - 6 * q^17 - 8 * q^19 + 8 * q^25 + 8 * q^31 + 8 * q^49 + 24 * q^59 + 14 * q^61 - 16 * q^67 + 24 * q^71 - 10 * q^73 - 18 * q^77 + 16 * q^79 + 18 * q^85 + 24 * q^91 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.00000			−1.34164			−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000			−1.34164			−0.670820	+	0.741620i	$0.734058\pi$
$6$		0			0
$7$			1.73205i			0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$			1.73205i			0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$		−	6.92820i		−	1.92154i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$		−	6.92820i		−	1.92154i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	5.19615i		−	0.878310i
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$			8.66025i			1.32068i	0.750968	+	0.660338i	$0.229587\pi$
$43$			8.66025i			1.32068i	−0.750968	+	0.660338i	$0.770413\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.66025i		−	1.26323i	−0.775283	−	0.631614i	$-0.782393\pi$
$47$		−	8.66025i		−	1.26323i	0.775283	−	0.631614i	$-0.217607\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.92820i			0.951662i	0.879537	+	0.475831i	$0.157853\pi$
$53$			6.92820i			0.951662i	−0.879537	+	0.475831i	$0.842147\pi$
$54$		0			0
$55$		−	15.5885i		−	2.10195i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	7.00000			0.896258			0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000			0.896258			0.448129	+	0.893969i	$0.352090\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			20.7846i			2.57801i
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−5.00000			−0.585206			−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000			−0.585206			−0.292603	+	0.956234i	$0.594521\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000			−1.02565
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	9.00000			0.976187
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000	+	5.19615i	1.23117	+	0.533114i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	13.8564i		−	1.30350i	−0.758433	−	0.651751i	$-0.774035\pi$
$113$		−	13.8564i		−	1.30350i	0.758433	−	0.651751i	$-0.225965\pi$
$114$		0			0
$115$			10.3923i			0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.19615i		−	0.476331i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	15.5885i		−	1.36197i	−0.732297	−	0.680985i	$-0.761552\pi$
$131$		−	15.5885i		−	1.36197i	0.732297	−	0.680985i	$-0.238448\pi$
$132$		0			0
$133$	3.00000	−	6.92820i	0.260133	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	21.0000			1.79415			0.897076	−	0.441877i	$-0.145687\pi$
$137$	21.0000			1.79415			0.897076	+	0.441877i	$0.145687\pi$
$138$		0			0
$139$		−	19.0526i		−	1.61602i	−0.589171	−	0.808008i	$-0.700546\pi$
$139$		−	19.0526i		−	1.61602i	0.589171	−	0.808008i	$-0.299454\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	36.0000			3.01047
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000			0.737309			0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000			0.737309			0.368654	+	0.929567i	$0.379819\pi$
$150$		0			0
$151$	20.0000			1.62758			0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000			1.62758			0.813788	+	0.581161i	$0.197401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−35.0000			−2.69231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$			6.92820i			0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	20.7846i		−	1.52811i
$186$		0			0
$187$		−	15.5885i		−	1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			12.1244i			0.877288i	0.898661	+	0.438644i	$0.144541\pi$
$191$			12.1244i			0.877288i	−0.898661	+	0.438644i	$0.855459\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$		−	19.0526i		−	1.35060i	−0.737543	−	0.675300i	$-0.764014\pi$
$199$		−	19.0526i		−	1.35060i	0.737543	−	0.675300i	$-0.235986\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			20.7846i			1.45166i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.00000	−	20.7846i	0.622543	−	1.43770i
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	25.9808i		−	1.77187i
$216$		0			0
$217$			6.92820i			0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			20.7846i			1.39812i
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	−13.0000			−0.859064			−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000			−0.859064			−0.429532	+	0.903052i	$0.641321\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.0000			1.37576			0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000			1.37576			0.687878	+	0.725826i	$0.258542\pi$
$234$		0			0
$235$			25.9808i			1.69480i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	22.5167i		−	1.45648i	−0.685321	−	0.728241i	$-0.740338\pi$
$239$		−	22.5167i		−	1.45648i	0.685321	−	0.728241i	$-0.259662\pi$
$240$		0			0
$241$			6.92820i			0.446285i	0.974786	+	0.223142i	$0.0716315\pi$
$241$			6.92820i			0.446285i	−0.974786	+	0.223142i	$0.928369\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.0000			−0.766652
$246$		0			0
$247$	−12.0000	+	27.7128i	−0.763542	+	1.76332i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.73205i		−	0.109326i	−0.998505	−	0.0546630i	$-0.982592\pi$
$251$		−	1.73205i		−	0.109326i	0.998505	−	0.0546630i	$-0.0174085\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$			27.7128i			1.72868i	0.502910	+	0.864339i	$0.332263\pi$
$257$			27.7128i			1.72868i	−0.502910	+	0.864339i	$0.667737\pi$
$258$		0			0
$259$	−12.0000			−0.745644
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.0526i			1.17483i	0.809285	+	0.587416i	$0.199855\pi$
$263$			19.0526i			1.17483i	−0.809285	+	0.587416i	$0.800145\pi$
$264$		0			0
$265$		−	20.7846i		−	1.27679i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.92820i		−	0.422420i	−0.977441	−	0.211210i	$-0.932260\pi$
$269$		−	6.92820i		−	0.422420i	0.977441	−	0.211210i	$-0.0677404\pi$
$270$		0			0
$271$		−	10.3923i		−	0.631288i	−0.948878	−	0.315644i	$-0.897780\pi$
$271$		−	10.3923i		−	0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			20.7846i			1.25336i
$276$		0			0
$277$	23.0000			1.38194			0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000			1.38194			0.690968	+	0.722885i	$0.257185\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			13.8564i			0.826604i	0.910594	+	0.413302i	$0.135625\pi$
$281$			13.8564i			0.826604i	−0.910594	+	0.413302i	$0.864375\pi$
$282$		0			0
$283$			8.66025i			0.514799i	0.966305	+	0.257399i	$0.0828656\pi$
$283$			8.66025i			0.514799i	−0.966305	+	0.257399i	$0.917134\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.8564i		−	0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$		−	13.8564i		−	0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$		0			0
$295$	−36.0000			−2.09600
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.0000			−1.38796
$300$		0			0
$301$	−15.0000			−0.864586
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.0000			−1.20246
$306$		0			0
$307$	32.0000			1.82634			0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000			1.82634			0.913168	+	0.407583i	$0.133628\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			19.0526i			1.08037i	0.841546	+	0.540186i	$0.181646\pi$
$311$			19.0526i			1.08037i	−0.841546	+	0.540186i	$0.818354\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	34.6410i		−	1.94563i	−0.231577	−	0.972817i	$-0.574388\pi$
$317$		−	34.6410i		−	1.94563i	0.231577	−	0.972817i	$-0.425612\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.0000	+	5.19615i	0.667698	+	0.289122i
$324$		0			0
$325$		−	27.7128i		−	1.53723i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.0000			0.826977
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000			1.31126
$336$		0			0
$337$			6.92820i			0.377403i	0.982034	+	0.188702i	$0.0604279\pi$
$337$			6.92820i			0.377403i	−0.982034	+	0.188702i	$0.939572\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			20.7846i			1.12555i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			19.0526i			1.02279i	0.859344	+	0.511397i	$0.170872\pi$
$347$			19.0526i			1.02279i	−0.859344	+	0.511397i	$0.829128\pi$
$348$		0			0
$349$	−13.0000			−0.695874			−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000			−0.695874			−0.347937	+	0.937518i	$0.613118\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$		0			0
$355$	−36.0000			−1.91068
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.5885i		−	0.822727i	−0.911471	−	0.411364i	$-0.865053\pi$
$359$		−	15.5885i		−	0.822727i	0.911471	−	0.411364i	$-0.134947\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.0000			0.785136
$366$		0			0
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000			−0.623009
$372$		0			0
$373$			13.8564i			0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$			13.8564i			0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$	27.0000			1.37605
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000			−0.152106			−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000			−0.152106			−0.0760530	+	0.997104i	$0.524232\pi$
$390$		0			0
$391$			10.3923i			0.525561i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.0000			−1.20757
$396$		0			0
$397$	−25.0000			−1.25471			−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000			−1.25471			−0.627357	+	0.778732i	$0.715863\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			13.8564i			0.691956i	0.938243	+	0.345978i	$0.112453\pi$
$401$			13.8564i			0.691956i	−0.938243	+	0.345978i	$0.887547\pi$
$402$		0			0
$403$		−	27.7128i		−	1.38047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−36.0000			−1.78445
$408$		0			0
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			20.7846i			1.02274i
$414$		0			0
$415$		−	10.3923i		−	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.2487i		−	1.18463i	−0.805708	−	0.592314i	$-0.798215\pi$
$419$		−	24.2487i		−	1.18463i	0.805708	−	0.592314i	$-0.201785\pi$
$420$		0			0
$421$		−	34.6410i		−	1.68830i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$		−	34.6410i		−	1.68830i	0.536107	−	0.844150i	$-0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$			12.1244i			0.586739i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	13.8564i		−	0.665896i	−0.942945	−	0.332948i	$-0.891957\pi$
$433$		−	13.8564i		−	0.665896i	0.942945	−	0.332948i	$-0.108043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.00000	+	13.8564i	−0.287019	+	0.662842i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.19615i			0.246877i	0.992352	+	0.123438i	$0.0393921\pi$
$443$			5.19615i			0.246877i	−0.992352	+	0.123438i	$0.960608\pi$
$444$		0			0
$445$			20.7846i			0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.8564i		−	0.653924i	−0.945037	−	0.326962i	$-0.893975\pi$
$449$		−	13.8564i		−	0.653924i	0.945037	−	0.326962i	$-0.106025\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−36.0000			−1.68771
$456$		0			0
$457$	−17.0000			−0.795226			−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000			−0.795226			−0.397613	+	0.917553i	$0.630161\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.00000			0.419172			0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000			0.419172			0.209586	+	0.977790i	$0.432788\pi$
$462$		0			0
$463$		−	19.0526i		−	0.885448i	−0.896658	−	0.442724i	$-0.854012\pi$
$463$		−	19.0526i		−	0.885448i	0.896658	−	0.442724i	$-0.145988\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			32.9090i			1.52285i	0.648256	+	0.761423i	$0.275499\pi$
$467$			32.9090i			1.52285i	−0.648256	+	0.761423i	$0.724501\pi$
$468$		0			0
$469$		−	13.8564i		−	0.639829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−45.0000			−2.06910
$474$		0			0
$475$	−16.0000	−	6.92820i	−0.734130	−	0.317888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	17.3205i		−	0.791394i	−0.918381	−	0.395697i	$-0.870503\pi$
$479$		−	17.3205i		−	0.791394i	0.918381	−	0.395697i	$-0.129497\pi$
$480$		0			0
$481$	48.0000			2.18861
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	20.7846i		−	0.943781i
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	10.3923i		−	0.468998i	−0.972116	−	0.234499i	$-0.924655\pi$
$491$		−	10.3923i		−	0.468998i	0.972116	−	0.234499i	$-0.0753450\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			20.7846i			0.932317i
$498$		0			0
$499$			36.3731i			1.62828i	0.580667	+	0.814141i	$0.302792\pi$
$499$			36.3731i			1.62828i	−0.580667	+	0.814141i	$0.697208\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	3.46410i		−	0.154457i	−0.997013	−	0.0772283i	$-0.975393\pi$
$503$		−	3.46410i		−	0.154457i	0.997013	−	0.0772283i	$-0.0246070\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	6.92820i		−	0.307087i	−0.988142	−	0.153544i	$-0.950931\pi$
$509$		−	6.92820i		−	0.307087i	0.988142	−	0.153544i	$-0.0490686\pi$
$510$		0			0
$511$		−	8.66025i		−	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000			−1.05757
$516$		0			0
$517$	45.0000			1.97910
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.8564i		−	0.607060i	−0.952822	−	0.303530i	$-0.901835\pi$
$521$		−	13.8564i		−	0.607060i	0.952822	−	0.303530i	$-0.0981653\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−48.0000			−2.07911
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			20.7846i			0.895257i
$540$		0			0
$541$	7.00000			0.300954			0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000			0.300954			0.150477	+	0.988614i	$0.451919\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			41.5692i			1.78063i
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			13.8564i			0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00000			−0.127114			−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000			−0.127114			−0.0635570	+	0.997978i	$0.520244\pi$
$558$		0			0
$559$	60.0000			2.53773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$			41.5692i			1.74883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			6.92820i			0.290445i	0.989399	+	0.145223i	$0.0463899\pi$
$569$			6.92820i			0.290445i	−0.989399	+	0.145223i	$0.953610\pi$
$570$		0			0
$571$		−	45.0333i		−	1.88459i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$		−	45.0333i		−	1.88459i	0.334790	−	0.942293i	$-0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	13.8564i		−	0.577852i
$576$		0			0
$577$	−5.00000			−0.208153			−0.104076	−	0.994569i	$-0.533189\pi$
$577$	−5.00000			−0.208153			−0.104076	+	0.994569i	$0.533189\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000			−0.248922
$582$		0			0
$583$	−36.0000			−1.49097
$584$		0			0
$585$		0			0
$586$		0			0
$587$			25.9808i			1.07234i	0.844110	+	0.536170i	$0.180130\pi$
$587$			25.9808i			1.07234i	−0.844110	+	0.536170i	$0.819870\pi$
$588$		0			0
$589$	−16.0000	−	6.92820i	−0.659269	−	0.285472i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$			15.5885i			0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$		−	34.6410i		−	1.41304i	−0.707695	−	0.706518i	$-0.750265\pi$
$601$		−	34.6410i		−	1.41304i	0.707695	−	0.706518i	$-0.249735\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	48.0000			1.95148
$606$		0			0
$607$	4.00000			0.162355			0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000			0.162355			0.0811775	+	0.996700i	$0.474132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−60.0000			−2.42734
$612$		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
$615$		0			0
$616$		0			0
$617$	9.00000			0.362326			0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000			0.362326			0.181163	+	0.983453i	$0.442014\pi$
$618$		0			0
$619$			10.3923i			0.417702i	0.977947	+	0.208851i	$0.0669724\pi$
$619$			10.3923i			0.417702i	−0.977947	+	0.208851i	$0.933028\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000			0.480770
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	20.7846i		−	0.828737i
$630$		0			0
$631$			8.66025i			0.344759i	0.985031	+	0.172380i	$0.0551456\pi$
$631$			8.66025i			0.344759i	−0.985031	+	0.172380i	$0.944854\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−48.0000			−1.90482
$636$		0			0
$637$		−	27.7128i		−	1.09802i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	20.7846i		−	0.820943i	−0.911873	−	0.410471i	$-0.865364\pi$
$641$		−	20.7846i		−	0.820943i	0.911873	−	0.410471i	$-0.134636\pi$
$642$		0			0
$643$			36.3731i			1.43441i	0.696860	+	0.717207i	$0.254580\pi$
$643$			36.3731i			1.43441i	−0.696860	+	0.717207i	$0.745420\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.9808i			1.02141i	0.859756	+	0.510705i	$0.170615\pi$
$647$			25.9808i			1.02141i	−0.859756	+	0.510705i	$0.829385\pi$
$648$		0			0
$649$			62.3538i			2.44760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.0000			0.821794			0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000			0.821794			0.410897	+	0.911682i	$0.365216\pi$
$654$		0			0
$655$			46.7654i			1.82727i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	48.0000			1.86981			0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000			1.86981			0.934907	+	0.354892i	$0.115482\pi$
$660$		0			0
$661$		−	41.5692i		−	1.61686i	−0.588596	−	0.808428i	$-0.700319\pi$
$661$		−	41.5692i		−	1.61686i	0.588596	−	0.808428i	$-0.299681\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.00000	+	20.7846i	−0.349005	+	0.805993i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$			36.3731i			1.40417i
$672$		0			0
$673$		−	34.6410i		−	1.33531i	−0.744469	−	0.667657i	$-0.767297\pi$
$673$		−	34.6410i		−	1.33531i	0.744469	−	0.667657i	$-0.232703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	27.7128i		−	1.06509i	−0.846402	−	0.532545i	$-0.821236\pi$
$677$		−	27.7128i		−	1.06509i	0.846402	−	0.532545i	$-0.178764\pi$
$678$		0			0
$679$	−12.0000			−0.460518
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−63.0000			−2.40711
$686$		0			0
$687$		0			0
$688$		0			0
$689$	48.0000			1.82865
$690$		0			0
$691$			1.73205i			0.0658903i	0.999457	+	0.0329452i	$0.0104887\pi$
$691$			1.73205i			0.0658903i	−0.999457	+	0.0329452i	$0.989511\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			57.1577i			2.16811i
$696$		0			0
$697$			20.7846i			0.787273i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	12.0000	−	27.7128i	0.452589	−	1.04521i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	10.3923i		−	0.390843i
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	13.8564i		−	0.518927i
$714$		0			0
$715$	−108.000			−4.03897
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.73205i		−	0.0645946i	−0.999478	−	0.0322973i	$-0.989718\pi$
$719$		−	1.73205i		−	0.0645946i	0.999478	−	0.0322973i	$-0.0102823\pi$
$720$		0			0
$721$			13.8564i			0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	25.9808i		−	0.963573i	−0.876289	−	0.481787i	$-0.839988\pi$
$727$		−	25.9808i		−	0.963573i	0.876289	−	0.481787i	$-0.160012\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	25.9808i		−	0.960933i
$732$		0			0
$733$	−2.00000			−0.0738717			−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000			−0.0738717			−0.0369358	+	0.999318i	$0.511760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	41.5692i		−	1.53122i
$738$		0			0
$739$		−	12.1244i		−	0.446002i	−0.974818	−	0.223001i	$-0.928415\pi$
$739$		−	12.1244i		−	0.446002i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0000			−0.880475			−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000			−0.880475			−0.440237	+	0.897881i	$0.645106\pi$
$744$		0			0
$745$	−27.0000			−0.989203
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−60.0000			−2.18362
$756$		0			0
$757$	−13.0000			−0.472493			−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000			−0.472493			−0.236247	+	0.971693i	$0.575917\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000			−0.108750			−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000			−0.108750			−0.0543750	+	0.998521i	$0.517317\pi$
$762$		0			0
$763$	24.0000			0.868858
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	83.1384i		−	3.00196i
$768$		0			0
$769$	−5.00000			−0.180305			−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000			−0.180305			−0.0901523	+	0.995928i	$0.528735\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			34.6410i			1.24595i	0.782241	+	0.622975i	$0.214076\pi$
$773$			34.6410i			1.24595i	−0.782241	+	0.622975i	$0.785924\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	+	27.7128i	−0.429945	+	0.992915i
$780$		0			0
$781$			62.3538i			2.23120i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.00000			0.214149
$786$		0			0
$787$	−8.00000			−0.285169			−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000			−0.285169			−0.142585	+	0.989783i	$0.545541\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$		−	48.4974i		−	1.72219i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$			25.9808i			0.919133i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	25.9808i		−	0.916841i
$804$		0			0
$805$	−18.0000			−0.634417
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.00000			0.316423			0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000			0.316423			0.158212	+	0.987405i	$0.449427\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			10.3923i			0.364027i
$816$		0			0
$817$	15.0000	−	34.6410i	0.524784	−	1.21194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.0000			1.15171			0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000			1.15171			0.575854	+	0.817553i	$0.304670\pi$
$822$		0			0
$823$			22.5167i			0.784881i	0.919777	+	0.392441i	$0.128369\pi$
$823$			22.5167i			0.784881i	−0.919777	+	0.392441i	$0.871631\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−48.0000			−1.66912			−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000			−1.66912			−0.834562	+	0.550914i	$0.814279\pi$
$828$		0			0
$829$			27.7128i			0.962506i	0.876582	+	0.481253i	$0.159818\pi$
$829$			27.7128i			0.962506i	−0.876582	+	0.481253i	$0.840182\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.0000			−0.415775
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	105.000			3.61211
$846$		0			0
$847$		−	27.7128i		−	0.952224i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.92820i		−	0.236663i	−0.992974	−	0.118331i	$-0.962245\pi$
$857$		−	6.92820i		−	0.236663i	0.992974	−	0.118331i	$-0.0377545\pi$
$858$		0			0
$859$			22.5167i			0.768259i	0.923279	+	0.384129i	$0.125498\pi$
$859$			22.5167i			0.768259i	−0.923279	+	0.384129i	$0.874502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			41.5692i			1.41014i
$870$		0			0
$871$			55.4256i			1.87803i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			5.19615i			0.175662i
$876$		0			0
$877$			13.8564i			0.467898i	0.972249	+	0.233949i	$0.0751648\pi$
$877$			13.8564i			0.467898i	−0.972249	+	0.233949i	$0.924835\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.0000			−1.31394			−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000			−1.31394			−0.656972	+	0.753915i	$0.728163\pi$
$882$		0			0
$883$		−	12.1244i		−	0.408017i	−0.978969	−	0.204009i	$-0.934603\pi$
$883$		−	12.1244i		−	0.408017i	0.978969	−	0.204009i	$-0.0653970\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$			27.7128i			0.929458i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.0000	+	34.6410i	−0.501956	+	1.15922i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	20.7846i		−	0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	41.5692i		−	1.38181i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−18.0000			−0.595713
$914$		0			0
$915$		0			0
$916$		0			0
$917$	27.0000			0.891619
$918$		0			0
$919$		−	51.9615i		−	1.71405i	−0.515273	−	0.857026i	$-0.672309\pi$
$919$		−	51.9615i		−	1.71405i	0.515273	−	0.857026i	$-0.327691\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	83.1384i		−	2.73654i
$924$		0			0
$925$			27.7128i			0.911192i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	−16.0000	−	6.92820i	−0.524379	−	0.227063i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			46.7654i			1.52939i
$936$		0			0
$937$	−1.00000			−0.0326686			−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000			−0.0326686			−0.0163343	+	0.999867i	$0.505200\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	55.4256i		−	1.80682i	−0.428774	−	0.903412i	$-0.641054\pi$
$941$		−	55.4256i		−	1.80682i	0.428774	−	0.903412i	$-0.358946\pi$
$942$		0			0
$943$	−24.0000			−0.781548
$944$		0			0
$945$		0			0
$946$		0			0
$947$			58.8897i			1.91366i	0.290650	+	0.956830i	$0.406129\pi$
$947$			58.8897i			1.91366i	−0.290650	+	0.956830i	$0.593871\pi$
$948$		0			0
$949$			34.6410i			1.12449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			27.7128i			0.897706i	0.893606	+	0.448853i	$0.148167\pi$
$953$			27.7128i			0.897706i	−0.893606	+	0.448853i	$0.851833\pi$
$954$		0			0
$955$		−	36.3731i		−	1.17700i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			36.3731i			1.17455i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			31.1769i			1.00258i	0.865279	+	0.501291i	$0.167141\pi$
$967$			31.1769i			1.00258i	−0.865279	+	0.501291i	$0.832859\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	33.0000			1.05793
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	20.7846i		−	0.664959i	−0.943111	−	0.332479i	$-0.892115\pi$
$977$		−	20.7846i		−	0.664959i	0.943111	−	0.332479i	$-0.107885\pi$
$978$		0			0
$979$	36.0000			1.15056
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000			0.765481			0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000			0.765481			0.382741	+	0.923856i	$0.374980\pi$
$984$		0			0
$985$	18.0000			0.573528
$986$		0			0
$987$		0			0
$988$		0			0
$989$	30.0000			0.953945
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			57.1577i			1.81202i
$996$		0			0
$997$	7.00000			0.221692			0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000			0.221692			0.110846	+	0.993838i	$0.464644\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.k.a.2431.2		2
3.2	odd	2		912.2.k.f.607.2	yes	2
4.3	odd	2		2736.2.k.b.2431.1		2
12.11	even	2		912.2.k.c.607.1	✓	2
19.18	odd	2		2736.2.k.b.2431.2		2
24.5	odd	2		3648.2.k.a.2431.2		2
24.11	even	2		3648.2.k.d.2431.1		2
57.56	even	2		912.2.k.c.607.2	yes	2
76.75	even	2	inner	2736.2.k.a.2431.1		2
228.227	odd	2		912.2.k.f.607.1	yes	2
456.227	odd	2		3648.2.k.a.2431.1		2
456.341	even	2		3648.2.k.d.2431.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.k.c.607.1	✓	2	12.11	even	2
912.2.k.c.607.2	yes	2	57.56	even	2
912.2.k.f.607.1	yes	2	228.227	odd	2
912.2.k.f.607.2	yes	2	3.2	odd	2
2736.2.k.a.2431.1		2	76.75	even	2	inner
2736.2.k.a.2431.2		2	1.1	even	1	trivial
2736.2.k.b.2431.1		2	4.3	odd	2
2736.2.k.b.2431.2		2	19.18	odd	2
3648.2.k.a.2431.1		2	456.227	odd	2
3648.2.k.a.2431.2		2	24.5	odd	2
3648.2.k.d.2431.1		2	24.11	even	2
3648.2.k.d.2431.2		2	456.341	even	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +1.73205i q^{7} +O(q^{10})\)	\(q-3.00000 q^{5} +1.73205i q^{7} +5.19615i q^{11} -6.92820i q^{13} -3.00000 q^{17} +(-4.00000 - 1.73205i) q^{19} -3.46410i q^{23} +4.00000 q^{25} +4.00000 q^{31} -5.19615i q^{35} +6.92820i q^{37} -6.92820i q^{41} +8.66025i q^{43} -8.66025i q^{47} +4.00000 q^{49} +6.92820i q^{53} -15.5885i q^{55} +12.0000 q^{59} +7.00000 q^{61} +20.7846i q^{65} -8.00000 q^{67} +12.0000 q^{71} -5.00000 q^{73} -9.00000 q^{77} +8.00000 q^{79} +3.46410i q^{83} +9.00000 q^{85} -6.92820i q^{89} +12.0000 q^{91} +(12.0000 + 5.19615i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5}+O(q^{10}) \) 2 * q - 6 * q^5	\( 2 q - 6 q^{5} - 6 q^{17} - 8 q^{19} + 8 q^{25} + 8 q^{31} + 8 q^{49} + 24 q^{59} + 14 q^{61} - 16 q^{67} + 24 q^{71} - 10 q^{73} - 18 q^{77} + 16 q^{79} + 18 q^{85} + 24 q^{91} + 24 q^{95}+O(q^{100}) \) 2 * q - 6 * q^5 - 6 * q^17 - 8 * q^19 + 8 * q^25 + 8 * q^31 + 8 * q^49 + 24 * q^59 + 14 * q^61 - 16 * q^67 + 24 * q^71 - 10 * q^73 - 18 * q^77 + 16 * q^79 + 18 * q^85 + 24 * q^91 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more