LMFDB - Embedded newform 1456.2.r.g.625.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,2,Mod(417,1456)] 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("1456.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.6262185343$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			625.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1456.625
Dual form			1456.2.r.g.417.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(-3.50000 - 6.06218i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{29} +(-4.00000 + 6.92820i) q^{37} -2.00000 q^{43} +(3.50000 - 6.06218i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(1.50000 + 2.59808i) q^{53} +(-3.50000 - 6.06218i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-7.50000 - 2.59808i) q^{63} +(-1.50000 - 2.59808i) q^{67} +5.00000 q^{71} +(-7.00000 - 12.1244i) q^{73} +(-6.00000 + 5.19615i) q^{77} +(-3.00000 + 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(0.500000 + 2.59808i) q^{91} -14.0000 q^{97} -9.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{17} - 7 q^{19} - 6 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{37} - 4 q^{43} + 7 q^{47} - 13 q^{49} + 3 q^{53} - 7 q^{59} + 7 q^{61} - 15 q^{63} - 3 q^{67}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 7 * q^17 - 7 * q^19 - 6 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^37 - 4 * q^43 + 7 * q^47 - 13 * q^49 + 3 * q^53 - 7 * q^59 + 7 * q^61 - 15 * q^63 - 3 * q^67 + 10 * q^71 - 14 * q^73 - 12 * q^77 - 6 * q^79 - 9 * q^81 + q^91 - 28 * q^97 - 18 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	0.0333386	−	0.999444i	$-0.489386\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.50000	−	6.06218i	0.510527	−	0.884260i	−0.489398	−	0.872060i	$-0.662783\pi$
$47$	3.50000	−	6.06218i	0.510527	−	0.884260i	0.999926	−	0.0121990i	$-0.00388317\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.50000	−	6.06218i	−0.455661	−	0.789228i	0.543065	−	0.839691i	$-0.317264\pi$
$59$	−3.50000	−	6.06218i	−0.455661	−	0.789228i	−0.998726	+	0.0504625i	$0.983930\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$	−7.50000	−	2.59808i	−0.944911	−	0.327327i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.00000			0.593391			0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000			0.593391			0.296695	+	0.954972i	$0.404115\pi$
$72$		0			0
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	−0.906208	−	0.422833i	$-0.861036\pi$
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	0.0869195	−	0.996215i	$-0.472298\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	5.19615i	−0.683763	+	0.592157i
$78$		0			0
$79$	−3.00000	+	5.19615i	−0.337526	+	0.584613i	−0.983967	−	0.178352i	$-0.942924\pi$
$79$	−3.00000	+	5.19615i	−0.337526	+	0.584613i	0.646440	+	0.762964i	$0.276257\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	0.500000	+	2.59808i	0.0524142	+	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−9.00000			−0.904534
$100$		0			0
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	0.969664	+	0.244443i	$0.0786053\pi$
$101$	7.00000	+	12.1244i	0.696526	+	1.20642i	−0.273138	+	0.961975i	$0.588061\pi$
$102$		0			0
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	−0.282194	−	0.959357i	$-0.591062\pi$
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	0.971925	−	0.235291i	$-0.0756043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	−0.605308	−	0.795991i	$-0.706950\pi$
$107$	4.00000	−	6.92820i	0.386695	−	0.669775i	0.992003	+	0.126217i	$0.0402834\pi$
$108$		0			0
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	0.754204	−	0.656640i	$-0.228023\pi$
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	−0.945769	+	0.324840i	$0.894690\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.00000			0.846649			0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000			0.846649			0.423324	+	0.905978i	$0.360863\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.50000	+	2.59808i	−0.138675	+	0.240192i
$118$		0			0
$119$	−14.0000	+	12.1244i	−1.28338	+	1.11144i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.00000	−	12.1244i	0.611593	−	1.05931i	−0.379379	−	0.925241i	$-0.623862\pi$
$131$	7.00000	−	12.1244i	0.611593	−	1.05931i	0.990972	−	0.134069i	$-0.0428042\pi$
$132$		0			0
$133$	17.5000	+	6.06218i	1.51744	+	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000	−	3.46410i	−0.170872	−	0.295958i	0.767853	−	0.640626i	$-0.221325\pi$
$137$	−2.00000	−	3.46410i	−0.170872	−	0.295958i	−0.938725	+	0.344668i	$0.887992\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.50000	+	2.59808i	0.125436	+	0.217262i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	0.798515	−	0.601975i	$-0.205619\pi$
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	−0.920583	+	0.390547i	$0.872286\pi$
$152$		0			0
$153$	−21.0000			−1.69775
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.0000	+	5.19615i	1.18217	+	0.409514i
$162$		0			0
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	0.490825	+	0.871258i	$0.336695\pi$
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	−0.999944	+	0.0105623i	$0.996638\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.00000			0.541676			0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000			0.541676			0.270838	+	0.962625i	$0.412699\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	10.5000	+	18.1865i	0.802955	+	1.39076i
$172$		0			0
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	−0.967851	−	0.251523i	$-0.919068\pi$
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	0.701751	+	0.712422i	$0.252402\pi$
$174$		0			0
$175$	10.0000	−	8.66025i	0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.00000	−	8.66025i	−0.373718	−	0.647298i	0.616417	−	0.787420i	$-0.288584\pi$
$179$	−5.00000	−	8.66025i	−0.373718	−	0.647298i	−0.990134	+	0.140122i	$0.955250\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.5000	+	18.1865i	−0.767836	+	1.32993i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	0.235983	+	0.971757i	$0.424169\pi$
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	−0.959558	+	0.281511i	$0.909164\pi$
$192$		0			0
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	0.785022	−	0.619467i	$-0.212651\pi$
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	−0.928986	+	0.370116i	$0.879318\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	0.503774	−	0.863836i	$-0.331945\pi$
$199$	−7.00000	−	12.1244i	−0.496217	−	0.859473i	−0.999990	+	0.00436292i	$0.998611\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.50000	+	12.9904i	0.175466	+	0.911746i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	9.00000	+	15.5885i	0.625543	+	1.08347i
$208$		0			0
$209$	21.0000			1.45260
$210$		0			0
$211$	26.0000			1.78991			0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000			1.78991			0.894957	+	0.446153i	$0.147206\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.50000	+	6.06218i	0.235435	+	0.407786i
$222$		0			0
$223$	−21.0000			−1.40626			−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000			−1.40626			−0.703132	+	0.711059i	$0.748216\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		0			0
$227$	−14.0000	−	24.2487i	−0.929213	−	1.60944i	−0.784642	−	0.619949i	$-0.787153\pi$
$227$	−14.0000	−	24.2487i	−0.929213	−	1.60944i	−0.144571	−	0.989494i	$-0.546180\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.0380310	−	0.999277i	$-0.487891\pi$
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.846383	−	0.532574i	$-0.178775\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.0000			1.22901			0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000			1.22901			0.614504	+	0.788914i	$0.289356\pi$
$240$		0			0
$241$	−14.0000	−	24.2487i	−0.901819	−	1.56200i	−0.825131	−	0.564942i	$-0.808899\pi$
$241$	−14.0000	−	24.2487i	−0.901819	−	1.56200i	−0.0766885	−	0.997055i	$-0.524435\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.50000	−	6.06218i	0.222700	−	0.385727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.0000			0.883672			0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000			0.883672			0.441836	+	0.897096i	$0.354327\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	−0.560781	−	0.827964i	$-0.689499\pi$
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	0.997429	+	0.0716680i	$0.0228322\pi$
$258$		0			0
$259$	20.0000	+	6.92820i	1.24274	+	0.430498i
$260$		0			0
$261$	−7.50000	+	12.9904i	−0.464238	+	0.804084i
$262$		0			0
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	−0.952517	−	0.304487i	$-0.901515\pi$
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	0.212565	−	0.977147i	$-0.431818\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	0.985389	+	0.170321i	$0.0544803\pi$
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	−0.345192	+	0.938532i	$0.612186\pi$
$270$		0			0
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	−0.952531	−	0.304443i	$-0.901530\pi$
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	0.739921	+	0.672694i	$0.234863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.50000	−	12.9904i	0.452267	−	0.783349i
$276$		0			0
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.0000	+	27.7128i	−0.941176	+	1.63017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	1.00000	+	5.19615i	0.0576390	+	0.299501i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−21.0000			−1.19853			−0.599267	−	0.800549i	$-0.704541\pi$
$307$	−21.0000			−1.19853			−0.599267	+	0.800549i	$0.704541\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	−0.597522	−	0.801852i	$-0.703848\pi$
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	0.993186	+	0.116543i	$0.0371814\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$	7.50000	+	12.9904i	0.419919	+	0.727322i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	49.0000			2.72643
$324$		0			0
$325$	−2.50000	−	4.33013i	−0.138675	−	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−17.5000	−	6.06218i	−0.964806	−	0.334219i
$330$		0			0
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	0.448649	+	0.893708i	$0.351905\pi$
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	−0.998298	+	0.0583130i	$0.981428\pi$
$332$		0			0
$333$	12.0000	+	20.7846i	0.657596	+	1.13899i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	−0.807337	+	0.590091i	$0.799092\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	0.617388	−	0.786659i	$-0.288191\pi$
$353$	−7.00000	−	12.1244i	−0.372572	−	0.645314i	−0.989960	+	0.141344i	$0.954858\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.00000	−	6.92820i	0.211112	−	0.365657i	−0.740951	−	0.671559i	$-0.765625\pi$
$359$	4.00000	−	6.92820i	0.211112	−	0.365657i	0.952063	+	0.305903i	$0.0989582\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.00000	+	12.1244i	0.365397	+	0.632886i	0.988840	−	0.148983i	$-0.0475999\pi$
$367$	7.00000	+	12.1244i	0.365397	+	0.632886i	−0.623443	+	0.781869i	$0.714267\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000	−	5.19615i	0.311504	−	0.269771i
$372$		0			0
$373$	−7.50000	+	12.9904i	−0.388335	+	0.672616i	−0.992226	−	0.124451i	$-0.960283\pi$
$373$	−7.50000	+	12.9904i	−0.388335	+	0.672616i	0.603890	+	0.797067i	$0.293616\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.00000			0.257513
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.00000	+	5.19615i	−0.152499	+	0.264135i
$388$		0			0
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	0.901544	−	0.432688i	$-0.142435\pi$
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	−0.825491	+	0.564416i	$0.809102\pi$
$390$		0			0
$391$	42.0000			2.12403
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	−0.986481	−	0.163876i	$-0.947600\pi$
$397$	−7.00000	+	12.1244i	−0.351320	+	0.608504i	0.635161	+	0.772380i	$0.280934\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	0.449008	+	0.893528i	$0.351777\pi$
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	−0.998322	+	0.0579116i	$0.981556\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	14.0000	+	24.2487i	0.692255	+	1.19902i	0.971097	+	0.238685i	$0.0767162\pi$
$409$	14.0000	+	24.2487i	0.692255	+	1.19902i	−0.278842	+	0.960337i	$0.589950\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.0000	+	12.1244i	−0.688895	+	0.596601i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.0000			−0.683945			−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000			−0.683945			−0.341972	+	0.939710i	$0.611095\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$	−10.5000	−	18.1865i	−0.510527	−	0.884260i
$424$		0			0
$425$	17.5000	−	30.3109i	0.848875	−	1.47029i
$426$		0			0
$427$	−17.5000	−	6.06218i	−0.846884	−	0.293369i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000	−	20.7846i	−0.578020	−	1.00116i	−0.995706	−	0.0925683i	$-0.970492\pi$
$431$	−12.0000	−	20.7846i	−0.578020	−	1.00116i	0.417687	−	0.908591i	$-0.362841\pi$
$432$		0			0
$433$	21.0000			1.00920			0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000			1.00920			0.504598	+	0.863355i	$0.331641\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0000	−	36.3731i	−1.00457	−	1.73996i
$438$		0			0
$439$	7.00000	−	12.1244i	0.334092	−	0.578664i	−0.649218	−	0.760602i	$-0.724904\pi$
$439$	7.00000	−	12.1244i	0.334092	−	0.578664i	0.983310	+	0.181938i	$0.0582371\pi$
$440$		0			0
$441$	−3.00000	+	20.7846i	−0.142857	+	0.989743i
$442$		0			0
$443$	−10.0000	+	17.3205i	−0.475114	+	0.822922i	−0.999594	−	0.0285009i	$-0.990927\pi$
$443$	−10.0000	+	17.3205i	−0.475114	+	0.822922i	0.524479	+	0.851423i	$0.324260\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.00000	−	5.19615i	0.140334	−	0.243066i	−0.787288	−	0.616585i	$-0.788516\pi$
$457$	3.00000	−	5.19615i	0.140334	−	0.243066i	0.927622	+	0.373519i	$0.121849\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−28.0000			−1.30409			−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000			−1.30409			−0.652045	+	0.758180i	$0.726089\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.00000	−	12.1244i	0.323921	−	0.561048i	−0.657372	−	0.753566i	$-0.728332\pi$
$467$	7.00000	−	12.1244i	0.323921	−	0.561048i	0.981293	+	0.192518i	$0.0616653\pi$
$468$		0			0
$469$	−6.00000	+	5.19615i	−0.277054	+	0.239936i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.00000	+	5.19615i	0.137940	+	0.238919i
$474$		0			0
$475$	−35.0000			−1.60591
$476$		0			0
$477$	9.00000			0.412082
$478$		0			0
$479$	3.50000	+	6.06218i	0.159919	+	0.276988i	0.934839	−	0.355071i	$-0.115543\pi$
$479$	3.50000	+	6.06218i	0.159919	+	0.276988i	−0.774920	+	0.632059i	$0.782210\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.0000			−1.35388			−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000			−1.35388			−0.676941	+	0.736038i	$0.736695\pi$
$492$		0			0
$493$	17.5000	+	30.3109i	0.788160	+	1.36513i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.50000	−	12.9904i	−0.112140	−	0.582698i
$498$		0			0
$499$	4.00000	−	6.92820i	0.179065	−	0.310149i	−0.762496	−	0.646993i	$-0.776026\pi$
$499$	4.00000	−	6.92820i	0.179065	−	0.310149i	0.941560	+	0.336844i	$0.109360\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	28.0000			1.24846			0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000			1.24846			0.624229	+	0.781241i	$0.285413\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.0000	−	24.2487i	0.620539	−	1.07481i	−0.368846	−	0.929490i	$-0.620247\pi$
$509$	14.0000	−	24.2487i	0.620539	−	1.07481i	0.989385	−	0.145315i	$-0.0464195\pi$
$510$		0			0
$511$	−28.0000	+	24.2487i	−1.23865	+	1.07270i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−21.0000			−0.923579
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	0.977633	−	0.210318i	$-0.0674500\pi$
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	−0.670957	+	0.741496i	$0.734117\pi$
$522$		0			0
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	−0.977503	−	0.210921i	$-0.932354\pi$
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	0.671414	+	0.741082i	$0.265687\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	−21.0000			−0.911322
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.5000	+	12.9904i	0.710705	+	0.559535i
$540$		0			0
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	−0.939110	−	0.343617i	$-0.888348\pi$
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	0.767136	+	0.641484i	$0.221681\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.00000			−0.0855138			−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000			−0.0855138			−0.0427569	+	0.999086i	$0.513614\pi$
$548$		0			0
$549$	−10.5000	−	18.1865i	−0.448129	−	0.776182i
$550$		0			0
$551$	17.5000	−	30.3109i	0.745525	−	1.29129i
$552$		0			0
$553$	15.0000	+	5.19615i	0.637865	+	0.220963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	0.609912	−	0.792469i	$-0.291205\pi$
$557$	−9.00000	−	15.5885i	−0.381342	−	0.660504i	−0.991254	+	0.131965i	$0.957871\pi$
$558$		0			0
$559$	2.00000			0.0845910
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.0000	+	24.2487i	0.590030	+	1.02196i	0.994228	+	0.107290i	$0.0342173\pi$
$563$	14.0000	+	24.2487i	0.590030	+	1.02196i	−0.404198	+	0.914671i	$0.632449\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−18.0000	+	15.5885i	−0.755929	+	0.654654i
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	0.682730	−	0.730670i	$-0.260792\pi$
$577$	−7.00000	−	12.1244i	−0.291414	−	0.504744i	−0.974144	+	0.225927i	$0.927459\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.50000	−	7.79423i	0.186371	−	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.0000			−0.866763			−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000			−0.866763			−0.433381	+	0.901211i	$0.642680\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000	+	27.7128i	0.653742	+	1.13231i	0.982208	+	0.187799i	$0.0601353\pi$
$599$	16.0000	+	27.7128i	0.653742	+	1.13231i	−0.328465	+	0.944516i	$0.606531\pi$
$600$		0			0
$601$	−7.00000			−0.285536			−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000			−0.285536			−0.142768	+	0.989756i	$0.545600\pi$
$602$		0			0
$603$	−9.00000			−0.366508
$604$		0			0
$605$		0			0
$606$		0			0
$607$	7.00000	−	12.1244i	0.284121	−	0.492112i	−0.688274	−	0.725450i	$-0.741632\pi$
$607$	7.00000	−	12.1244i	0.284121	−	0.492112i	0.972396	+	0.233338i	$0.0749648\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.50000	+	6.06218i	−0.141595	+	0.245249i
$612$		0			0
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	−0.984015	−	0.178085i	$-0.943010\pi$
$613$	−16.0000	−	27.7128i	−0.646234	−	1.11931i	0.337781	−	0.941225i	$-0.390324\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	−14.0000	−	24.2487i	−0.562708	−	0.974638i	−0.997259	−	0.0739910i	$-0.976426\pi$
$619$	−14.0000	−	24.2487i	−0.562708	−	0.974638i	0.434551	−	0.900647i	$-0.356907\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	56.0000			2.23287
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.50000	−	2.59808i	0.257539	−	0.102940i
$638$		0			0
$639$	7.50000	−	12.9904i	0.296695	−	0.513892i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	7.00000			0.276053			0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000			0.276053			0.138027	+	0.990429i	$0.455924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	−0.901464	−	0.432855i	$-0.857506\pi$
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	0.0758684	−	0.997118i	$-0.475827\pi$
$648$		0			0
$649$	−10.5000	+	18.1865i	−0.412161	+	0.713884i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−42.0000			−1.63858
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	15.0000	−	25.9808i	0.580802	−	1.00598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.0000			−0.810696
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.5000	−	30.3109i	0.672580	−	1.16494i	−0.304590	−	0.952483i	$-0.598520\pi$
$677$	17.5000	−	30.3109i	0.672580	−	1.16494i	0.977170	−	0.212459i	$-0.0681471\pi$
$678$		0			0
$679$	7.00000	+	36.3731i	0.268635	+	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	0.539750	−	0.841825i	$-0.318519\pi$
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	−0.998917	+	0.0465244i	$0.985185\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.50000	−	2.59808i	−0.0571454	−	0.0989788i
$690$		0			0
$691$	17.5000	−	30.3109i	0.665731	−	1.15308i	−0.313355	−	0.949636i	$-0.601453\pi$
$691$	17.5000	−	30.3109i	0.665731	−	1.15308i	0.979086	−	0.203445i	$-0.0652137\pi$
$692$		0			0
$693$	4.50000	+	23.3827i	0.170941	+	0.888235i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−28.0000	−	48.4974i	−1.05604	−	1.82911i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	28.0000	−	24.2487i	1.05305	−	0.911967i
$708$		0			0
$709$	−25.0000	+	43.3013i	−0.938895	+	1.62621i	−0.171358	+	0.985209i	$0.554815\pi$
$709$	−25.0000	+	43.3013i	−0.938895	+	1.62621i	−0.767537	+	0.641004i	$0.778518\pi$
$710$		0			0
$711$	9.00000	+	15.5885i	0.337526	+	0.584613i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.0000	+	36.3731i	−0.783168	+	1.35649i	0.146920	+	0.989148i	$0.453064\pi$
$719$	−21.0000	+	36.3731i	−0.783168	+	1.35649i	−0.930087	+	0.367338i	$0.880269\pi$
$720$		0			0
$721$	−35.0000	−	12.1244i	−1.30347	−	0.451535i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.5000	−	21.6506i	−0.464238	−	0.804084i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	7.00000	+	12.1244i	0.258904	+	0.448435i
$732$		0			0
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	0.158774	+	0.987315i	$0.449246\pi$
$733$	−21.0000	+	36.3731i	−0.775653	+	1.34347i	−0.934427	+	0.356155i	$0.884088\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.50000	+	7.79423i	−0.165760	+	0.287104i
$738$		0			0
$739$	2.00000	+	3.46410i	0.0735712	+	0.127429i	0.900464	−	0.434930i	$-0.143227\pi$
$739$	2.00000	+	3.46410i	0.0735712	+	0.127429i	−0.826893	+	0.562360i	$0.809894\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.00000			−0.330178			−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000			−0.330178			−0.165089	+	0.986279i	$0.552791\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−20.0000	−	6.92820i	−0.730784	−	0.253151i
$750$		0			0
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	−0.988761	−	0.149505i	$-0.952232\pi$
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	0.623856	+	0.781540i	$0.285565\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.00000			0.327111			0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000			0.327111			0.163555	+	0.986534i	$0.447704\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−8.00000	+	6.92820i	−0.289619	+	0.250818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.50000	+	6.06218i	0.126378	+	0.218893i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21.0000	−	36.3731i	−0.755318	−	1.30825i	−0.945216	−	0.326445i	$-0.894149\pi$
$773$	−21.0000	−	36.3731i	−0.755318	−	1.30825i	0.189899	−	0.981804i	$-0.439184\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−7.50000	−	12.9904i	−0.268371	−	0.464832i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	3.50000	+	6.06218i	0.124762	+	0.216093i	0.921640	−	0.388047i	$-0.126850\pi$
$787$	3.50000	+	6.06218i	0.124762	+	0.216093i	−0.796878	+	0.604140i	$0.793517\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.50000	−	23.3827i	−0.160002	−	0.831393i
$792$		0			0
$793$	−3.50000	+	6.06218i	−0.124289	+	0.215274i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$	−49.0000			−1.73350
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.0000	+	36.3731i	−0.741074	+	1.28358i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.5000	+	26.8468i	0.544951	+	0.943883i	0.998610	+	0.0527074i	$0.0167851\pi$
$809$	15.5000	+	26.8468i	0.544951	+	0.943883i	−0.453659	+	0.891175i	$0.649882\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.00000	−	12.1244i	0.244899	−	0.424178i
$818$		0			0
$819$	7.50000	+	2.59808i	0.262071	+	0.0907841i
$820$		0			0
$821$	−18.0000	+	31.1769i	−0.628204	+	1.08808i	0.359708	+	0.933065i	$0.382876\pi$
$821$	−18.0000	+	31.1769i	−0.628204	+	1.08808i	−0.987912	+	0.155017i	$0.950457\pi$
$822$		0			0
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	0.898776	−	0.438408i	$-0.144457\pi$
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	−0.829060	+	0.559159i	$0.811124\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.00000			0.173867			0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000			0.173867			0.0869335	+	0.996214i	$0.472293\pi$
$828$		0			0
$829$	−17.5000	−	30.3109i	−0.607800	−	1.05274i	−0.991602	−	0.129325i	$-0.958719\pi$
$829$	−17.5000	−	30.3109i	−0.607800	−	1.05274i	0.383802	−	0.923415i	$-0.374614\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	38.5000	+	30.3109i	1.33395	+	1.05021i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.00000			−0.241667			−0.120833	−	0.992673i	$-0.538557\pi$
$839$	−7.00000			−0.241667			−0.120833	+	0.992673i	$0.538557\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.00000	−	1.73205i	−0.171802	−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.0000	−	41.5692i	−0.822709	−	1.42497i
$852$		0			0
$853$	−42.0000			−1.43805			−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000			−1.43805			−0.719026	+	0.694983i	$0.755412\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	0.629066	−	0.777352i	$-0.283437\pi$
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	−0.987739	+	0.156112i	$0.950104\pi$
$858$		0			0
$859$	−28.0000	+	48.4974i	−0.955348	+	1.65471i	−0.221777	+	0.975097i	$0.571186\pi$
$859$	−28.0000	+	48.4974i	−0.955348	+	1.65471i	−0.733571	+	0.679613i	$0.762148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.0000	+	17.3205i	−0.340404	+	0.589597i	−0.984508	−	0.175341i	$-0.943897\pi$
$863$	−10.0000	+	17.3205i	−0.340404	+	0.589597i	0.644104	+	0.764938i	$0.277230\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.0000			0.610608
$870$		0			0
$871$	1.50000	+	2.59808i	0.0508256	+	0.0880325i
$872$		0			0
$873$	−21.0000	+	36.3731i	−0.710742	+	1.23104i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	24.0000	−	41.5692i	0.810422	−	1.40369i	−0.102146	−	0.994769i	$-0.532571\pi$
$877$	24.0000	−	41.5692i	0.810422	−	1.40369i	0.912569	−	0.408923i	$-0.134096\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	40.0000			1.34611			0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000			1.34611			0.673054	+	0.739594i	$0.264982\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	−0.261540	−	0.965193i	$-0.584230\pi$
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	0.966651	−	0.256096i	$-0.0824362\pi$
$888$		0			0
$889$	1.00000	+	5.19615i	0.0335389	+	0.174273i
$890$		0			0
$891$	−13.5000	+	23.3827i	−0.452267	+	0.783349i
$892$		0			0
$893$	24.5000	+	42.4352i	0.819861	+	1.42004i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	10.5000	−	18.1865i	0.349806	−	0.605881i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	−0.987371	−	0.158424i	$-0.949359\pi$
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	0.356487	−	0.934300i	$-0.383975\pi$
$908$		0			0
$909$	42.0000			1.39305
$910$		0			0
$911$	54.0000			1.78910			0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000			1.78910			0.894550	+	0.446968i	$0.147496\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35.0000	−	12.1244i	−1.15580	−	0.400381i
$918$		0			0
$919$	25.0000	−	43.3013i	0.824674	−	1.42838i	−0.0774944	−	0.996993i	$-0.524692\pi$
$919$	25.0000	−	43.3013i	0.824674	−	1.42838i	0.902168	−	0.431384i	$-0.141975\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.00000			−0.164577
$924$		0			0
$925$	−40.0000			−1.31519
$926$		0			0
$927$	−21.0000	−	36.3731i	−0.689730	−	1.19465i
$928$		0			0
$929$	−7.00000	+	12.1244i	−0.229663	+	0.397787i	−0.957708	−	0.287742i	$-0.907096\pi$
$929$	−7.00000	+	12.1244i	−0.229663	+	0.397787i	0.728046	+	0.685529i	$0.240429\pi$
$930$		0			0
$931$	7.00000	−	48.4974i	0.229416	−	1.58944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	7.00000			0.228680			0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000			0.228680			0.114340	+	0.993442i	$0.463525\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0000	−	24.2487i	−0.456387	−	0.790485i	0.542380	−	0.840133i	$-0.317523\pi$
$941$	−14.0000	−	24.2487i	−0.456387	−	0.790485i	−0.998767	+	0.0496480i	$0.984190\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	−0.997589	−	0.0694014i	$-0.977891\pi$
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	0.558898	+	0.829237i	$0.311224\pi$
$948$		0			0
$949$	7.00000	+	12.1244i	0.227230	+	0.393573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000	+	6.92820i	−0.258333	+	0.223723i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$	−12.0000	−	20.7846i	−0.386695	−	0.669775i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	5.00000			0.160789			0.0803946	−	0.996763i	$-0.474382\pi$
$967$	5.00000			0.160789			0.0803946	+	0.996763i	$0.474382\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.0000	+	24.2487i	−0.449281	+	0.778178i	−0.998339	−	0.0576061i	$-0.981653\pi$
$971$	−14.0000	+	24.2487i	−0.449281	+	0.778178i	0.549058	+	0.835784i	$0.314987\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.0000	+	32.9090i	0.607864	+	1.05285i	0.991592	+	0.129405i	$0.0413067\pi$
$977$	19.0000	+	32.9090i	0.607864	+	1.05285i	−0.383728	+	0.923446i	$0.625360\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−12.0000			−0.383131
$982$		0			0
$983$	−3.50000	−	6.06218i	−0.111633	−	0.193353i	0.804796	−	0.593551i	$-0.202275\pi$
$983$	−3.50000	−	6.06218i	−0.111633	−	0.193353i	−0.916429	+	0.400198i	$0.868941\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.00000	−	10.3923i	0.190789	−	0.330456i
$990$		0			0
$991$	2.00000	+	3.46410i	0.0635321	+	0.110041i	0.896042	−	0.443969i	$-0.146430\pi$
$991$	2.00000	+	3.46410i	0.0635321	+	0.110041i	−0.832510	+	0.554010i	$0.813097\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.50000	+	6.06218i	0.110846	+	0.191991i	0.916112	−	0.400923i	$-0.131311\pi$
$997$	3.50000	+	6.06218i	0.110846	+	0.191991i	−0.805266	+	0.592914i	$0.797977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.r.g.625.1		2
4.3	odd	2		91.2.e.a.79.1	yes	2
7.4	even	3	inner	1456.2.r.g.417.1		2
12.11	even	2		819.2.j.b.352.1		2
28.3	even	6		637.2.e.a.508.1		2
28.11	odd	6		91.2.e.a.53.1	✓	2
28.19	even	6		637.2.a.d.1.1		1
28.23	odd	6		637.2.a.c.1.1		1
28.27	even	2		637.2.e.a.79.1		2
52.51	odd	2		1183.2.e.b.170.1		2
84.11	even	6		819.2.j.b.235.1		2
84.23	even	6		5733.2.a.c.1.1		1
84.47	odd	6		5733.2.a.d.1.1		1
364.51	odd	6		8281.2.a.f.1.1		1
364.103	even	6		8281.2.a.e.1.1		1
364.207	odd	6		1183.2.e.b.508.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.a.53.1	✓	2	28.11	odd	6
91.2.e.a.79.1	yes	2	4.3	odd	2
637.2.a.c.1.1		1	28.23	odd	6
637.2.a.d.1.1		1	28.19	even	6
637.2.e.a.79.1		2	28.27	even	2
637.2.e.a.508.1		2	28.3	even	6
819.2.j.b.235.1		2	84.11	even	6
819.2.j.b.352.1		2	12.11	even	2
1183.2.e.b.170.1		2	52.51	odd	2
1183.2.e.b.508.1		2	364.207	odd	6
1456.2.r.g.417.1		2	7.4	even	3	inner
1456.2.r.g.625.1		2	1.1	even	1	trivial
5733.2.a.c.1.1		1	84.23	even	6
5733.2.a.d.1.1		1	84.47	odd	6
8281.2.a.e.1.1		1	364.103	even	6
8281.2.a.f.1.1		1	364.51	odd	6

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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(-3.50000 - 6.06218i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{29} +(-4.00000 + 6.92820i) q^{37} -2.00000 q^{43} +(3.50000 - 6.06218i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(1.50000 + 2.59808i) q^{53} +(-3.50000 - 6.06218i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-7.50000 - 2.59808i) q^{63} +(-1.50000 - 2.59808i) q^{67} +5.00000 q^{71} +(-7.00000 - 12.1244i) q^{73} +(-6.00000 + 5.19615i) q^{77} +(-3.00000 + 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(0.500000 + 2.59808i) q^{91} -14.0000 q^{97} -9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{17} - 7 q^{19} - 6 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{37} - 4 q^{43} + 7 q^{47} - 13 q^{49} + 3 q^{53} - 7 q^{59} + 7 q^{61} - 15 q^{63} - 3 q^{67}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 7 * q^17 - 7 * q^19 - 6 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^37 - 4 * q^43 + 7 * q^47 - 13 * q^49 + 3 * q^53 - 7 * q^59 + 7 * q^61 - 15 * q^63 - 3 * q^67 + 10 * q^71 - 14 * q^73 - 12 * q^77 - 6 * q^79 - 9 * q^81 + q^91 - 28 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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