LMFDB - Embedded newform 49.7.d.a.31.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,7,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	49.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$11.2726500974$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	49.31
Dual form			49.7.d.a.19.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{2} +(-8.50000 + 14.7224i) q^{4} -423.000 q^{8} +(-364.500 - 631.333i) q^{9} +(-981.000 + 1699.14i) q^{11} +(2447.50 + 4239.19i) q^{16} +(-3280.50 + 5681.99i) q^{18} +17658.0 q^{22} +(11367.0 + 19688.2i) q^{23} +(-7812.50 + 13531.6i) q^{25} -21222.0 q^{29} +(8491.50 - 14707.7i) q^{32} +12393.0 q^{36} +(-50597.0 - 87636.6i) q^{37} -126614. q^{43} +(-16677.0 - 28885.4i) q^{44} +(102303. - 177194. i) q^{46} +140625. q^{50} +(-25173.0 + 43600.9i) q^{53} +(95499.0 + 165409. i) q^{58} +160433. q^{64} +(26963.0 - 46701.3i) q^{67} -242478. q^{71} +(154184. + 267054. i) q^{72} +(-455373. + 788729. i) q^{74} +(-464689. - 804865. i) q^{79} +(-265720. + 460241. i) q^{81} +(569763. + 986858. i) q^{86} +(414963. - 718737. i) q^{88} -386478. q^{92} +1.43030e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} - 17 q^{4} - 846 q^{8} - 729 q^{9} - 1962 q^{11} + 4895 q^{16} - 6561 q^{18} + 35316 q^{22} + 22734 q^{23} - 15625 q^{25} - 42444 q^{29} + 16983 q^{32} + 24786 q^{36} - 101194 q^{37} - 253228 q^{43}+ \cdots + 2860596 q^{99}+O(q^{100}) $ 2 * q - 9 * q^2 - 17 * q^4 - 846 * q^8 - 729 * q^9 - 1962 * q^11 + 4895 * q^16 - 6561 * q^18 + 35316 * q^22 + 22734 * q^23 - 15625 * q^25 - 42444 * q^29 + 16983 * q^32 + 24786 * q^36 - 101194 * q^37 - 253228 * q^43 - 33354 * q^44 + 204606 * q^46 + 281250 * q^50 - 50346 * q^53 + 190998 * q^58 + 320866 * q^64 + 53926 * q^67 - 484956 * q^71 + 308367 * q^72 - 910746 * q^74 - 929378 * q^79 - 531441 * q^81 + 1139526 * q^86 + 829926 * q^88 - 772956 * q^92 + 2860596 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−4.50000	−	7.79423i	−0.562500	−	0.974279i	−0.997277	−	0.0737406i	$-0.976506\pi$
$2$	−4.50000	−	7.79423i	−0.562500	−	0.974279i	0.434777	−	0.900538i	$-0.356827\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	−8.50000	+	14.7224i	−0.132812	+	0.230038i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−423.000			−0.826172
$9$	−364.500	−	631.333i	−0.500000	−	0.866025i
$10$		0			0
$11$	−981.000	+	1699.14i	−0.737040	+	1.27659i	0.216783	+	0.976220i	$0.430444\pi$
$11$	−981.000	+	1699.14i	−0.737040	+	1.27659i	−0.953823	+	0.300371i	$0.902890\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	2447.50	+	4239.19i	0.597534	+	1.03496i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−3280.50	+	5681.99i	−0.562500	+	0.974279i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$		0			0
$22$	17658.0			1.65834
$23$	11367.0	+	19688.2i	0.934248	+	1.61817i	0.775969	+	0.630771i	$0.217261\pi$
$23$	11367.0	+	19688.2i	0.934248	+	1.61817i	0.158280	+	0.987394i	$0.449405\pi$
$24$		0			0
$25$	−7812.50	+	13531.6i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−21222.0			−0.870146			−0.435073	−	0.900395i	$-0.643278\pi$
$29$	−21222.0			−0.870146			−0.435073	+	0.900395i	$0.643278\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	8491.50	−	14707.7i	0.259140	−	0.448844i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	12393.0			0.265625
$37$	−50597.0	−	87636.6i	−0.998894	−	1.73014i	−0.540159	−	0.841563i	$-0.681636\pi$
$37$	−50597.0	−	87636.6i	−0.998894	−	1.73014i	−0.458736	−	0.888573i	$-0.651698\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−126614.			−1.59249			−0.796244	−	0.604975i	$-0.793183\pi$
$43$	−126614.			−1.59249			−0.796244	+	0.604975i	$0.793183\pi$
$44$	−16677.0	−	28885.4i	−0.195776	−	0.339094i
$45$		0			0
$46$	102303.	−	177194.i	1.05103	−	1.82044i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$		0			0
$50$	140625.			1.12500
$51$		0			0
$52$		0			0
$53$	−25173.0	+	43600.9i	−0.169086	+	0.292865i	−0.938099	−	0.346368i	$-0.887415\pi$
$53$	−25173.0	+	43600.9i	−0.169086	+	0.292865i	0.769013	+	0.639233i	$0.220748\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	95499.0	+	165409.i	0.489457	+	0.847765i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	160433.			0.612003
$65$		0			0
$66$		0			0
$67$	26963.0	−	46701.3i	0.0896487	−	0.155276i	−0.817714	−	0.575625i	$-0.804759\pi$
$67$	26963.0	−	46701.3i	0.0896487	−	0.155276i	0.907363	+	0.420349i	$0.138092\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−242478.			−0.677481			−0.338741	−	0.940880i	$-0.610001\pi$
$71$	−242478.			−0.677481			−0.338741	+	0.940880i	$0.610001\pi$
$72$	154184.	+	267054.i	0.413086	+	0.715486i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−455373.	+	788729.i	−1.12376	+	1.94640i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−464689.	−	804865.i	−0.942499	−	1.63246i	−0.760682	−	0.649125i	$-0.775135\pi$
$79$	−464689.	−	804865.i	−0.942499	−	1.63246i	−0.181817	−	0.983332i	$-0.558198\pi$
$80$		0			0
$81$	−265720.	+	460241.i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	569763.	+	986858.i	0.895775	+	1.55153i
$87$		0			0
$88$	414963.	−	718737.i	0.608922	−	1.05468i
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$	−386478.			−0.496319
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	1.43030e6			1.47408
$100$	−132812.	−	230038.i	−0.132812	−	0.230038i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	453114.			0.380443
$107$	−23157.0	−	40109.1i	−0.0189030	−	0.0327410i	0.856419	−	0.516281i	$-0.172684\pi$
$107$	−23157.0	−	40109.1i	−0.0189030	−	0.0327410i	−0.875322	+	0.483540i	$0.839351\pi$
$108$		0			0
$109$	1.29357e6	−	2.24053e6i	0.998874	−	1.73010i	0.458354	−	0.888770i	$-0.348439\pi$
$109$	1.29357e6	−	2.24053e6i	0.998874	−	1.73010i	0.540520	−	0.841331i	$-0.318227\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.43689e6			−1.68889			−0.844445	−	0.535642i	$-0.820069\pi$
$113$	−2.43689e6			−1.68889			−0.844445	+	0.535642i	$0.820069\pi$
$114$		0			0
$115$		0			0
$116$	180387.	−	312439.i	0.115566	−	0.200167i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.03894e6	−	1.79950e6i	−0.586455	−	1.01577i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−96766.0			−0.0472402			−0.0236201	−	0.999721i	$-0.507519\pi$
$127$	−96766.0			−0.0472402			−0.0236201	+	0.999721i	$0.507519\pi$
$128$	−1.26540e6	−	2.19174e6i	−0.603392	−	1.04511i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$	−485334.			−0.201709
$135$		0			0
$136$		0			0
$137$	−2.26470e6	+	3.92257e6i	−0.880741	+	1.52549i	−0.0302232	+	0.999543i	$0.509622\pi$
$137$	−2.26470e6	+	3.92257e6i	−0.880741	+	1.52549i	−0.850518	+	0.525946i	$0.823712\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	1.09115e6	+	1.88993e6i	0.381083	+	0.660055i
$143$		0			0
$144$	1.78423e6	−	3.09037e6i	0.597534	−	1.03496i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.72030e6			0.530663
$149$	2.97665e6	+	5.15571e6i	0.899848	+	1.55858i	0.827688	+	0.561189i	$0.189656\pi$
$149$	2.97665e6	+	5.15571e6i	0.899848	+	1.55858i	0.0721604	+	0.997393i	$0.477011\pi$
$150$		0			0
$151$	−914201.	+	1.58344e6i	−0.265528	+	0.459909i	−0.967702	−	0.252097i	$-0.918880\pi$
$151$	−914201.	+	1.58344e6i	−0.265528	+	0.459909i	0.702174	+	0.712006i	$0.252213\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−4.18220e6	+	7.24378e6i	−1.06031	+	1.83651i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	4.78297e6			1.12500
$163$	2.74655e6	+	4.75716e6i	0.634197	+	1.09846i	0.986685	+	0.162645i	$0.0520025\pi$
$163$	2.74655e6	+	4.75716e6i	0.634197	+	1.09846i	−0.352488	+	0.935816i	$0.614664\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	4.82681e6			1.00000
$170$		0			0
$171$		0			0
$172$	1.07622e6	−	1.86407e6i	0.211502	−	0.366333i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	−9.60399e6			−1.76163
$177$		0			0
$178$		0			0
$179$	3.56381e6	−	6.17270e6i	0.621378	−	1.07626i	−0.367852	−	0.929884i	$-0.619907\pi$
$179$	3.56381e6	−	6.17270e6i	0.621378	−	1.07626i	0.989229	−	0.146373i	$-0.0467600\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−4.80824e6	−	8.32812e6i	−0.771850	−	1.33688i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.32272e6	−	2.29102e6i	−0.189831	−	0.328798i	0.755362	−	0.655307i	$-0.227461\pi$
$191$	−1.32272e6	−	2.29102e6i	−0.189831	−	0.328798i	−0.945194	+	0.326509i	$0.894127\pi$
$192$		0			0
$193$	−3.34499e6	+	5.79370e6i	−0.465290	+	0.805905i	−0.999215	−	0.0396267i	$-0.987383\pi$
$193$	−3.34499e6	+	5.79370e6i	−0.465290	+	0.805905i	0.533925	+	0.845532i	$0.320716\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.04066e7			−1.36117			−0.680585	−	0.732670i	$-0.738274\pi$
$197$	−1.04066e7			−1.36117			−0.680585	+	0.732670i	$0.738274\pi$
$198$	−6.43634e6	−	1.11481e7i	−0.829170	−	1.43616i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	3.30469e6	−	5.72389e6i	0.413086	−	0.715486i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.28654e6	−	1.43527e7i	0.934248	−	1.61817i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.56456e7			1.66550			0.832748	−	0.553652i	$-0.186766\pi$
$211$	1.56456e7			1.66550			0.832748	+	0.553652i	$0.186766\pi$
$212$	−427941.	−	741216.i	−0.0449134	−	0.0777924i
$213$		0			0
$214$	−208413.	+	360982.i	−0.0212659	+	0.0368336i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.32843e7			−2.24747
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.13906e7			1.00000
$226$	1.09660e7	+	1.89937e7i	0.950001	+	1.64545i
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	8.97691e6			0.718890
$233$	1.46289e6	+	2.53379e6i	0.115649	+	0.200310i	0.918039	−	0.396490i	$-0.129772\pi$
$233$	1.46289e6	+	2.53379e6i	0.115649	+	0.200310i	−0.802390	+	0.596800i	$0.796438\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.71015e7			1.98518			0.992591	−	0.121505i	$-0.0387721\pi$
$239$	2.71015e7			1.98518			0.992591	+	0.121505i	$0.0387721\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−9.35047e6	+	1.61955e7i	−0.659762	+	1.14274i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−4.46041e7			−2.75431
$254$	435447.	+	754216.i	0.0265726	+	0.0460251i
$255$		0			0
$256$	−6.25478e6	+	1.08336e7i	−0.372814	+	0.645733i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.73542e6	+	1.33981e7i	0.435073	+	0.753569i
$262$		0			0
$263$	−1.00838e7	+	1.74656e7i	−0.554313	+	0.960098i	0.443644	+	0.896203i	$0.353686\pi$
$263$	−1.00838e7	+	1.74656e7i	−0.554313	+	0.960098i	−0.997957	+	0.0638948i	$0.979648\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	458371.	+	793922.i	0.0238129	+	0.0412452i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	4.07645e7			1.98167
$275$	−1.53281e7	−	2.65491e7i	−0.737040	−	1.27659i
$276$		0			0
$277$	−5.46412e6	+	9.46413e6i	−0.257087	+	0.445288i	−0.965460	−	0.260550i	$-0.916096\pi$
$277$	−5.46412e6	+	9.46413e6i	−0.257087	+	0.445288i	0.708373	+	0.705838i	$0.249429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.55231e7			−1.15031			−0.575155	−	0.818045i	$-0.695058\pi$
$281$	−2.55231e7			−1.15031			−0.575155	+	0.818045i	$0.695058\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	2.06106e6	−	3.56987e6i	0.0899780	−	0.155846i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.23806e7			−0.518280
$289$	−1.20688e7	−	2.09037e7i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	2.14025e7	+	3.70703e7i	0.825258	+	1.42939i
$297$		0			0
$298$	2.67899e7	−	4.64014e7i	1.01233	−	1.75341i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	1.64556e7			0.597439
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	1.57994e7			0.500703
$317$	7.56456e6	+	1.31022e7i	0.237469	+	0.411308i	0.959987	−	0.280044i	$-0.0903490\pi$
$317$	7.56456e6	+	1.31022e7i	0.237469	+	0.411308i	−0.722519	+	0.691351i	$0.757016\pi$
$318$		0			0
$319$	2.08188e7	−	3.60592e7i	0.641333	−	1.11082i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−4.51725e6	−	7.82410e6i	−0.132812	−	0.230038i
$325$		0			0
$326$	2.47189e7	−	4.28144e7i	0.713472	−	1.23577i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.32275e7	−	4.02313e7i	−0.640500	−	1.10938i	−0.985321	−	0.170710i	$-0.945394\pi$
$331$	−2.32275e7	−	4.02313e7i	−0.640500	−	1.10938i	0.344821	−	0.938668i	$-0.387940\pi$
$332$		0			0
$333$	−3.68852e7	+	6.38871e7i	−0.998894	+	1.73014i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.54566e7			−1.71027			−0.855133	−	0.518408i	$-0.826525\pi$
$337$	−6.54566e7			−1.71027			−0.855133	+	0.518408i	$0.826525\pi$
$338$	−2.17206e7	−	3.76213e7i	−0.562500	−	0.974279i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	5.35577e7			1.31567
$345$		0			0
$346$		0			0
$347$	3.33769e7	−	5.78105e7i	0.798836	−	1.38363i	−0.121538	−	0.992587i	$-0.538783\pi$
$347$	3.33769e7	−	5.78105e7i	0.798836	−	1.38363i	0.920374	−	0.391039i	$-0.127884\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	1.66603e7	+	2.88565e7i	0.381993	+	0.661631i
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−6.41486e7			−1.39810
$359$	1.34309e7	+	2.32630e7i	0.290282	+	0.502784i	0.973876	−	0.227078i	$-0.0729174\pi$
$359$	1.34309e7	+	2.32630e7i	0.290282	+	0.502784i	−0.683594	+	0.729863i	$0.739584\pi$
$360$		0			0
$361$	−2.35229e7	+	4.07429e7i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−5.56415e7	+	9.63738e7i	−1.11649	+	1.93382i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.56853e7	−	7.91293e7i	−0.880340	−	1.52479i	−0.850964	−	0.525225i	$-0.823981\pi$
$373$	−4.56853e7	−	7.91293e7i	−0.880340	−	1.52479i	−0.0293761	−	0.999568i	$-0.509352\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.31112e7			0.608215			0.304107	−	0.952638i	$-0.401642\pi$
$379$	3.31112e7			0.608215			0.304107	+	0.952638i	$0.401642\pi$
$380$		0			0
$381$		0			0
$382$	−1.19045e7	+	2.06192e7i	−0.213560	+	0.369897i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$	6.02099e7			1.04690
$387$	4.61508e7	+	7.99355e7i	0.796244	+	1.37914i
$388$		0			0
$389$	3.46553e6	−	6.00248e6i	0.0588737	−	0.101972i	−0.835086	−	0.550119i	$-0.814582\pi$
$389$	3.46553e6	−	6.00248e6i	0.0588737	−	0.101972i	0.893960	+	0.448147i	$0.147916\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	4.68299e7	+	8.11118e7i	0.765658	+	1.32616i
$395$		0			0
$396$	−1.21575e7	+	2.10575e7i	−0.195776	+	0.339094i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	−7.64844e7			−1.19507
$401$	6.32044e7	+	1.09473e8i	0.980199	+	1.69775i	0.661586	+	0.749869i	$0.269884\pi$
$401$	6.32044e7	+	1.09473e8i	0.980199	+	1.69775i	0.318613	+	0.947885i	$0.396783\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.98543e8			2.94490
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.49158e8			−2.10206
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	8.36917e7			1.12160			0.560798	−	0.827953i	$-0.310495\pi$
$421$	8.36917e7			1.12160			0.560798	+	0.827953i	$0.310495\pi$
$422$	−7.04050e7	−	1.21945e8i	−0.936842	−	1.62266i
$423$		0			0
$424$	1.06482e7	−	1.84432e7i	0.139694	−	0.241957i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	787338.			0.0100422
$429$		0			0
$430$		0			0
$431$	4.30142e7	−	7.45027e7i	0.537254	−	0.930551i	−0.461797	−	0.886986i	$-0.652795\pi$
$431$	4.30142e7	−	7.45027e7i	0.537254	−	0.930551i	0.999051	−	0.0435653i	$-0.0138717\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	2.19907e7	+	3.80890e7i	0.265326	+	0.459558i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.56699e7	−	1.48385e8i	−0.985410	−	1.70678i	−0.640099	−	0.768292i	$-0.721107\pi$
$443$	−8.56699e7	−	1.48385e8i	−0.985410	−	1.70678i	−0.345311	−	0.938488i	$-0.612226\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.73823e8			−1.92030			−0.960149	−	0.279490i	$-0.909835\pi$
$449$	−1.73823e8			−1.92030			−0.960149	+	0.279490i	$0.909835\pi$
$450$	−5.12578e7	−	8.87811e7i	−0.562500	−	0.974279i
$451$		0			0
$452$	2.07136e7	−	3.58770e7i	0.224306	−	0.388509i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.33639e7	+	1.09750e8i	0.663886	+	1.14988i	0.979586	+	0.201025i	$0.0644274\pi$
$457$	6.33639e7	+	1.09750e8i	0.663886	+	1.14988i	−0.315700	+	0.948859i	$0.602239\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.27272e8			−1.28230			−0.641151	−	0.767415i	$-0.721543\pi$
$463$	−1.27272e8			−1.28230			−0.641151	+	0.767415i	$0.721543\pi$
$464$	−5.19408e7	−	8.99642e7i	−0.519942	−	0.900566i
$465$		0			0
$466$	1.31660e7	−	2.28042e7i	0.130105	−	0.225349i
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.24208e8	−	2.15135e8i	1.17373	−	2.03296i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.67022e7			0.338172
$478$	−1.21957e8	−	2.11236e8i	−1.11666	−	1.93412i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	3.53240e7			0.311554
$485$		0			0
$486$		0			0
$487$	−1.10067e8	+	1.90642e8i	−0.952955	+	1.65057i	−0.213974	+	0.976839i	$0.568641\pi$
$487$	−1.10067e8	+	1.90642e8i	−0.952955	+	1.65057i	−0.738981	+	0.673726i	$0.764693\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.78277e8			1.50609			0.753044	−	0.657970i	$-0.228585\pi$
$491$	1.78277e8			1.50609			0.753044	+	0.657970i	$0.228585\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	9.80717e7	+	1.69865e8i	0.789300	+	1.36711i	0.926397	+	0.376549i	$0.122889\pi$
$499$	9.80717e7	+	1.69865e8i	0.789300	+	1.36711i	−0.137097	+	0.990558i	$0.543777\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	2.00718e8	+	3.47655e8i	1.54930	+	2.68347i
$507$		0			0
$508$	822511.	−	1.42463e6i	0.00627409	−	0.0108670i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−4.93857e7			−0.367952
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	6.96188e7	−	1.20583e8i	0.489457	−	0.847765i
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	1.81508e8			1.24720
$527$		0			0
$528$		0			0
$529$	−1.84399e8	+	3.19389e8i	−1.24564	+	2.15751i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.14053e7	+	1.97546e7i	−0.0740652	+	0.128285i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.22850e7	+	5.59192e7i	0.203896	+	0.353158i	0.949780	−	0.312917i	$-0.101306\pi$
$541$	3.22850e7	+	5.59192e7i	0.203896	+	0.353158i	−0.745884	+	0.666075i	$0.767973\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.58854e8			−0.970592			−0.485296	−	0.874350i	$-0.661288\pi$
$547$	−1.58854e8			−0.970592			−0.485296	+	0.874350i	$0.661288\pi$
$548$	−3.84998e7	−	6.66837e7i	−0.233947	−	0.405208i
$549$		0			0
$550$	−1.37953e8	+	2.38942e8i	−0.829170	+	1.43616i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	9.83541e7			0.578447
$555$		0			0
$556$		0			0
$557$	−3.67018e7	+	6.35693e7i	−0.212384	+	0.367859i	−0.952460	−	0.304663i	$-0.901456\pi$
$557$	−3.67018e7	+	6.35693e7i	−0.212384	+	0.367859i	0.740076	+	0.672523i	$0.234789\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.14854e8	+	1.98933e8i	0.647049	+	1.12072i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.02568e8			0.559716
$569$	−1.77746e8	−	3.07866e8i	−0.964859	−	1.67119i	−0.709992	−	0.704209i	$-0.751302\pi$
$569$	−1.77746e8	−	3.07866e8i	−0.964859	−	1.67119i	−0.254867	−	0.966976i	$-0.582032\pi$
$570$		0			0
$571$	1.63131e8	−	2.82551e8i	0.876250	−	1.51771i	0.0208247	−	0.999783i	$-0.493371\pi$
$571$	1.63131e8	−	2.82551e8i	0.876250	−	1.51771i	0.855425	−	0.517926i	$-0.173296\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.55219e8			−1.86850
$576$	−5.84778e7	−	1.01287e8i	−0.306002	−	0.530010i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−1.08619e8	+	1.88134e8i	−0.562500	+	0.974279i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.93894e7	−	8.55450e7i	−0.249246	−	0.431707i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	2.47672e8	−	4.28981e8i	1.19375	−	2.06763i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	−1.01206e8			−0.478044
$597$		0			0
$598$		0			0
$599$	−9.10130e7	+	1.57639e8i	−0.423471	+	0.733472i	−0.996276	−	0.0862187i	$-0.972522\pi$
$599$	−9.10130e7	+	1.57639e8i	−0.423471	+	0.733472i	0.572806	+	0.819691i	$0.305855\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−3.93121e7			−0.179297
$604$	−1.55414e7	−	2.69185e7i	−0.0705310	−	0.122163i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.17696e8	−	2.03856e8i	0.510954	−	0.884998i	−0.488966	−	0.872303i	$-0.662626\pi$
$613$	1.17696e8	−	2.03856e8i	0.510954	−	0.884998i	0.999919	−	0.0126948i	$-0.00404098\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.63532e8			1.97344			0.986721	−	0.162423i	$-0.0519308\pi$
$617$	4.63532e8			1.97344			0.986721	+	0.162423i	$0.0519308\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.22070e8	−	2.11432e8i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.83542e8			0.730544			0.365272	−	0.930901i	$-0.380976\pi$
$631$	1.83542e8			0.730544			0.365272	+	0.930901i	$0.380976\pi$
$632$	1.96563e8	+	3.40458e8i	0.778667	+	1.34869i
$633$		0			0
$634$	6.80811e7	−	1.17920e8i	0.267152	−	0.462721i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−3.74738e8			−1.44300
$639$	8.83832e7	+	1.53084e8i	0.338741	+	0.586716i
$640$		0			0
$641$	2.23965e8	−	3.87919e8i	0.850367	−	1.47288i	−0.0305097	−	0.999534i	$-0.509713\pi$
$641$	2.23965e8	−	3.87919e8i	0.850367	−	1.47288i	0.880877	−	0.473345i	$-0.156954\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	1.12400e8	−	1.94682e8i	0.413086	−	0.715486i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−9.33826e7			−0.336917
$653$	−8.74062e7	−	1.51392e8i	−0.313908	−	0.543705i	0.665297	−	0.746579i	$-0.268305\pi$
$653$	−8.74062e7	−	1.51392e8i	−0.313908	−	0.543705i	−0.979205	+	0.202874i	$0.934972\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.69128e8			−1.98863			−0.994314	−	0.106485i	$-0.966040\pi$
$659$	−5.69128e8			−1.98863			−0.994314	+	0.106485i	$0.966040\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−2.09048e8	+	3.62082e8i	−0.720563	+	1.24805i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	6.63934e8			2.24751
$667$	−2.41230e8	−	4.17823e8i	−0.812933	−	1.40804i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	5.17302e8			1.69707			0.848534	−	0.529141i	$-0.177486\pi$
$673$	5.17302e8			1.69707			0.848534	+	0.529141i	$0.177486\pi$
$674$	2.94555e8	+	5.10184e8i	0.962025	+	1.66628i
$675$		0			0
$676$	−4.10279e7	+	7.10624e7i	−0.132812	+	0.230038i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.61430e8	+	4.52810e8i	−0.820527	+	1.42119i	0.0847629	+	0.996401i	$0.472987\pi$
$683$	−2.61430e8	+	4.52810e8i	−0.820527	+	1.42119i	−0.905290	+	0.424794i	$0.860347\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−3.09888e8	−	5.36741e8i	−0.951566	−	1.64816i
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$	−6.00785e8			−1.79738
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.71321e8			−1.94884			−0.974420	−	0.224735i	$-0.927848\pi$
$701$	−6.71321e8			−1.94884			−0.974420	+	0.224735i	$0.927848\pi$
$702$		0			0
$703$		0			0
$704$	−1.57385e8	+	2.72598e8i	−0.451071	+	0.781278i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.77700e8	+	4.80990e8i	0.779178	+	1.34958i	0.932416	+	0.361387i	$0.117697\pi$
$709$	2.77700e8	+	4.80990e8i	0.779178	+	1.34958i	−0.153238	+	0.988189i	$0.548970\pi$
$710$		0			0
$711$	−3.38758e8	+	5.86747e8i	−0.942499	+	1.63246i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	6.05848e7	+	1.04936e8i	0.165053	+	0.285881i
$717$		0			0
$718$	1.20878e8	−	2.09367e8i	0.326568	−	0.565632i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	4.23413e8			1.12500
$723$		0			0
$724$		0			0
$725$	1.65797e8	−	2.87169e8i	0.435073	−	0.753569i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	3.87420e8			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	3.86092e8			0.968405
$737$	5.29014e7	+	9.16279e7i	0.132149	+	0.228889i
$738$		0			0
$739$	8.29309e7	−	1.43641e8i	0.205486	−	0.355913i	−0.744801	−	0.667286i	$-0.767456\pi$
$739$	8.29309e7	−	1.43641e8i	0.205486	−	0.355913i	0.950288	+	0.311373i	$0.100789\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.87319e8			−0.456684			−0.228342	−	0.973581i	$-0.573330\pi$
$743$	−1.87319e8			−0.456684			−0.228342	+	0.973581i	$0.573330\pi$
$744$		0			0
$745$		0			0
$746$	−4.11168e8	+	7.12164e8i	−0.990382	+	1.71539i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.20568e8	+	7.28446e8i	0.992926	+	1.71980i	0.599291	+	0.800531i	$0.295449\pi$
$751$	4.20568e8	+	7.28446e8i	0.992926	+	1.71980i	0.393635	+	0.919267i	$0.371218\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	3.45533e8			0.796529			0.398264	−	0.917271i	$-0.369613\pi$
$757$	3.45533e8			0.796529			0.398264	+	0.917271i	$0.369613\pi$
$758$	−1.49000e8	−	2.58076e8i	−0.342121	−	0.592571i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	4.49725e7			0.100848
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−5.68649e7	−	9.84929e7i	−0.123593	−	0.214069i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$	4.15357e8	−	7.19420e8i	0.895775	−	1.55153i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−6.23796e7			−0.132466
$779$		0			0
$780$		0			0
$781$	2.37871e8	−	4.12005e8i	0.499331	−	0.864866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	8.84565e7	−	1.53211e8i	0.180780	−	0.313121i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−6.05016e8			−1.21784
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	1.32680e8	+	2.29808e8i	0.259140	+	0.448844i
$801$		0			0
$802$	5.68840e8	−	9.85259e8i	1.10272	−	1.90997i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.68186e8	+	2.91306e8i	−0.317646	+	0.550179i	−0.979996	−	0.199015i	$-0.936226\pi$
$809$	−1.68186e8	+	2.91306e8i	−0.317646	+	0.550179i	0.662351	+	0.749194i	$0.269559\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−8.93442e8	−	1.54749e9i	−1.65651	−	2.86915i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.94391e8	−	6.83105e8i	−0.712685	−	1.23441i	−0.963846	−	0.266461i	$-0.914146\pi$
$821$	−3.94391e8	−	6.83105e8i	−0.712685	−	1.23441i	0.251161	−	0.967945i	$-0.419188\pi$
$822$		0			0
$823$	−5.11627e8	+	8.86164e8i	−0.917813	+	1.58970i	−0.115082	+	0.993356i	$0.536713\pi$
$823$	−5.11627e8	+	8.86164e8i	−0.917813	+	1.58970i	−0.802731	+	0.596342i	$0.796620\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.56178e8			−0.983326			−0.491663	−	0.870786i	$-0.663611\pi$
$827$	−5.56178e8			−0.983326			−0.491663	+	0.870786i	$0.663611\pi$
$828$	1.40871e8	+	2.43996e8i	0.248160	+	0.429825i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−1.44450e8			−0.242845
$842$	−3.76613e8	−	6.52312e8i	−0.630897	−	1.09275i
$843$		0			0
$844$	−1.32987e8	+	2.30341e8i	−0.221199	+	0.383128i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−2.46444e8			−0.404138
$849$		0			0
$850$		0			0
$851$	1.15027e9	−	1.99233e9i	1.86643	−	3.23275i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	9.79541e6	+	1.69661e7i	0.0156171	+	0.0270497i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−7.74255e8			−1.20882
$863$	4.38716e8	+	7.59878e8i	0.682576	+	1.18226i	0.974192	+	0.225720i	$0.0724735\pi$
$863$	4.38716e8	+	7.59878e8i	0.682576	+	1.18226i	−0.291617	+	0.956535i	$0.594193\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.82344e9			2.77864
$870$		0			0
$871$		0			0
$872$	−5.47181e8	+	9.47744e8i	−0.825242	+	1.42936i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.53075e8	+	1.13116e9i	0.968198	+	1.67697i	0.700766	+	0.713392i	$0.252842\pi$
$877$	6.53075e8	+	1.13116e9i	0.968198	+	1.67697i	0.267432	+	0.963577i	$0.413825\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−4.73327e8			−0.687510			−0.343755	−	0.939059i	$-0.611699\pi$
$883$	−4.73327e8			−0.687510			−0.343755	+	0.939059i	$0.611699\pi$
$884$		0			0
$885$		0			0
$886$	−7.71029e8	+	1.33546e9i	−1.10859	+	1.92013i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−5.21344e8	−	9.02994e8i	−0.737040	−	1.27659i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	7.82204e8	+	1.35482e9i	1.08017	+	1.87090i
$899$		0			0
$900$	−9.68203e7	+	1.67698e8i	−0.132812	+	0.230038i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.03081e9			1.39531
$905$		0			0
$906$		0			0
$907$	−6.44611e8	+	1.11650e9i	−0.863925	+	1.49636i	0.00418597	+	0.999991i	$0.498668\pi$
$907$	−6.44611e8	+	1.11650e9i	−0.863925	+	1.49636i	−0.868111	+	0.496370i	$0.834666\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.85794e7			0.0774800			0.0387400	−	0.999249i	$-0.487666\pi$
$911$	5.85794e7			0.0774800			0.0387400	+	0.999249i	$0.487666\pi$
$912$		0			0
$913$		0			0
$914$	5.70275e8	−	9.87746e8i	0.746872	−	1.29362i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	5.39751e8	+	9.34876e8i	0.695419	+	1.20450i	0.970039	+	0.242949i	$0.0781147\pi$
$919$	5.39751e8	+	9.34876e8i	0.695419	+	1.20450i	−0.274620	+	0.961553i	$0.588552\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.58116e9			1.99779
$926$	5.72724e8	+	9.91988e8i	0.721295	+	1.24932i
$927$		0			0
$928$	−1.80207e8	+	3.12127e8i	−0.225490	+	0.390560i
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	−4.97382e7			−0.0614387
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−2.23575e9			−2.64089
$947$	−3.42418e8	−	5.93086e8i	−0.403187	−	0.698341i	0.590921	−	0.806729i	$-0.298764\pi$
$947$	−3.42418e8	−	5.93086e8i	−0.403187	−	0.698341i	−0.994109	+	0.108388i	$0.965431\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.73142e8			−1.00880			−0.504401	−	0.863469i	$-0.668287\pi$
$953$	−8.73142e8			−1.00880			−0.504401	+	0.863469i	$0.668287\pi$
$954$	−1.65160e8	−	2.86066e8i	−0.190222	−	0.329474i
$955$		0			0
$956$	−2.30363e8	+	3.99001e8i	−0.263657	+	0.456667i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−4.43752e8	−	7.68601e8i	−0.500000	−	0.866025i
$962$		0			0
$963$	−1.68815e7	+	2.92395e7i	−0.0189030	+	0.0327410i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	8.99699e8			0.994988			0.497494	−	0.867467i	$-0.334254\pi$
$967$	8.99699e8			0.994988			0.497494	+	0.867467i	$0.334254\pi$
$968$	4.39472e8	+	7.61188e8i	0.484513	+	0.839201i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$	1.98121e9			2.14415
$975$		0			0
$976$		0			0
$977$	2.29825e8	−	3.98068e8i	0.246441	−	0.426849i	−0.716095	−	0.698003i	$-0.754072\pi$
$977$	2.29825e8	−	3.98068e8i	0.246441	−	0.426849i	0.962536	+	0.271155i	$0.0874055\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.88603e9			−1.99775
$982$	−8.02246e8	−	1.38953e9i	−0.847175	−	1.46735i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.43922e9	−	2.49280e9i	−1.48778	−	2.57691i
$990$		0			0
$991$	−6.81493e8	+	1.18038e9i	−0.700230	+	1.21283i	0.268156	+	0.963376i	$0.413586\pi$
$991$	−6.81493e8	+	1.18038e9i	−0.700230	+	1.21283i	−0.968386	+	0.249458i	$0.919748\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	8.82645e8	−	1.52879e9i	0.887962	−	1.53800i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.7.d.a.31.1		2
7.2	even	3	inner	49.7.d.a.19.1		2
7.3	odd	6		7.7.b.a.6.1	✓	1
7.4	even	3		7.7.b.a.6.1	✓	1
7.5	odd	6	inner	49.7.d.a.19.1		2
7.6	odd	2	CM	49.7.d.a.31.1		2
21.11	odd	6		63.7.d.a.55.1		1
21.17	even	6		63.7.d.a.55.1		1
28.3	even	6		112.7.c.a.97.1		1
28.11	odd	6		112.7.c.a.97.1		1
35.3	even	12		175.7.c.a.174.1		2
35.4	even	6		175.7.d.a.76.1		1
35.17	even	12		175.7.c.a.174.2		2
35.18	odd	12		175.7.c.a.174.1		2
35.24	odd	6		175.7.d.a.76.1		1
35.32	odd	12		175.7.c.a.174.2		2
56.3	even	6		448.7.c.b.321.1		1
56.11	odd	6		448.7.c.b.321.1		1
56.45	odd	6		448.7.c.a.321.1		1
56.53	even	6		448.7.c.a.321.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.7.b.a.6.1	✓	1	7.3	odd	6
7.7.b.a.6.1	✓	1	7.4	even	3
49.7.d.a.19.1		2	7.2	even	3	inner
49.7.d.a.19.1		2	7.5	odd	6	inner
49.7.d.a.31.1		2	1.1	even	1	trivial
49.7.d.a.31.1		2	7.6	odd	2	CM
63.7.d.a.55.1		1	21.11	odd	6
63.7.d.a.55.1		1	21.17	even	6
112.7.c.a.97.1		1	28.3	even	6
112.7.c.a.97.1		1	28.11	odd	6
175.7.c.a.174.1		2	35.3	even	12
175.7.c.a.174.1		2	35.18	odd	12
175.7.c.a.174.2		2	35.17	even	12
175.7.c.a.174.2		2	35.32	odd	12
175.7.d.a.76.1		1	35.4	even	6
175.7.d.a.76.1		1	35.24	odd	6
448.7.c.a.321.1		1	56.45	odd	6
448.7.c.a.321.1		1	56.53	even	6
448.7.c.b.321.1		1	56.3	even	6
448.7.c.b.321.1		1	56.11	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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	49.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{2} +(-8.50000 + 14.7224i) q^{4} -423.000 q^{8} +(-364.500 - 631.333i) q^{9} +(-981.000 + 1699.14i) q^{11} +(2447.50 + 4239.19i) q^{16} +(-3280.50 + 5681.99i) q^{18} +17658.0 q^{22} +(11367.0 + 19688.2i) q^{23} +(-7812.50 + 13531.6i) q^{25} -21222.0 q^{29} +(8491.50 - 14707.7i) q^{32} +12393.0 q^{36} +(-50597.0 - 87636.6i) q^{37} -126614. q^{43} +(-16677.0 - 28885.4i) q^{44} +(102303. - 177194. i) q^{46} +140625. q^{50} +(-25173.0 + 43600.9i) q^{53} +(95499.0 + 165409. i) q^{58} +160433. q^{64} +(26963.0 - 46701.3i) q^{67} -242478. q^{71} +(154184. + 267054. i) q^{72} +(-455373. + 788729. i) q^{74} +(-464689. - 804865. i) q^{79} +(-265720. + 460241. i) q^{81} +(569763. + 986858. i) q^{86} +(414963. - 718737. i) q^{88} -386478. q^{92} +1.43030e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} - 17 q^{4} - 846 q^{8} - 729 q^{9} - 1962 q^{11} + 4895 q^{16} - 6561 q^{18} + 35316 q^{22} + 22734 q^{23} - 15625 q^{25} - 42444 q^{29} + 16983 q^{32} + 24786 q^{36} - 101194 q^{37} - 253228 q^{43}+ \cdots + 2860596 q^{99}+O(q^{100}) \) 2 * q - 9 * q^2 - 17 * q^4 - 846 * q^8 - 729 * q^9 - 1962 * q^11 + 4895 * q^16 - 6561 * q^18 + 35316 * q^22 + 22734 * q^23 - 15625 * q^25 - 42444 * q^29 + 16983 * q^32 + 24786 * q^36 - 101194 * q^37 - 253228 * q^43 - 33354 * q^44 + 204606 * q^46 + 281250 * q^50 - 50346 * q^53 + 190998 * q^58 + 320866 * q^64 + 53926 * q^67 - 484956 * q^71 + 308367 * q^72 - 910746 * q^74 - 929378 * q^79 - 531441 * q^81 + 1139526 * q^86 + 829926 * q^88 - 772956 * q^92 + 2860596 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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