LMFDB - Embedded newform 49.5.d.a.31.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,5,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, 5); 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, 5, names="a")

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	49.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$5.06512819111$
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:	$\Z[a_1, a_2]$
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			49.5.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)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(7.50000 - 12.9904i) q^{4} -31.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +(103.000 - 178.401i) q^{11} +(-104.500 - 180.999i) q^{16} +(-40.5000 + 70.1481i) q^{18} -206.000 q^{22} +(367.000 + 635.663i) q^{23} +(-312.500 + 541.266i) q^{25} +1234.00 q^{29} +(-352.500 + 610.548i) q^{32} -1215.00 q^{36} +(647.000 + 1120.64i) q^{37} -334.000 q^{43} +(-1545.00 - 2676.02i) q^{44} +(367.000 - 635.663i) q^{46} +625.000 q^{50} +(2791.00 - 4834.15i) q^{53} +(-617.000 - 1068.68i) q^{58} -2639.00 q^{64} +(-2473.00 + 4283.36i) q^{67} +2914.00 q^{71} +(1255.50 + 2174.59i) q^{72} +(647.000 - 1120.64i) q^{74} +(1823.00 + 3157.53i) q^{79} +(-3280.50 + 5681.99i) q^{81} +(167.000 + 289.252i) q^{86} +(-3193.00 + 5530.44i) q^{88} +11010.0 q^{92} -16686.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 15 q^{4} - 62 q^{8} - 81 q^{9} + 206 q^{11} - 209 q^{16} - 81 q^{18} - 412 q^{22} + 734 q^{23} - 625 q^{25} + 2468 q^{29} - 705 q^{32} - 2430 q^{36} + 1294 q^{37} - 668 q^{43} - 3090 q^{44}+ \cdots - 33372 q^{99}+O(q^{100}) $ 2 * q - q^2 + 15 * q^4 - 62 * q^8 - 81 * q^9 + 206 * q^11 - 209 * q^16 - 81 * q^18 - 412 * q^22 + 734 * q^23 - 625 * q^25 + 2468 * q^29 - 705 * q^32 - 2430 * q^36 + 1294 * q^37 - 668 * q^43 - 3090 * q^44 + 734 * q^46 + 1250 * q^50 + 5582 * q^53 - 1234 * q^58 - 5278 * q^64 - 4946 * q^67 + 5828 * q^71 + 2511 * q^72 + 1294 * q^74 + 3646 * q^79 - 6561 * q^81 + 334 * q^86 - 6386 * q^88 + 22020 * q^92 - 33372 * 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 $

$2$	−0.500000	−	0.866025i	−0.125000	−	0.216506i	0.796733	−	0.604332i	$-0.206560\pi$
$2$	−0.500000	−	0.866025i	−0.125000	−	0.216506i	−0.921733	+	0.387825i	$0.873226\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	7.50000	−	12.9904i	0.468750	−	0.811899i
$5$		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
$8$	−31.0000			−0.484375
$9$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$10$		0			0
$11$	103.000	−	178.401i	0.851240	−	1.47439i	−0.0288505	−	0.999584i	$-0.509185\pi$
$11$	103.000	−	178.401i	0.851240	−	1.47439i	0.880090	−	0.474807i	$-0.157482\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−104.500	−	180.999i	−0.408203	−	0.707029i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−40.5000	+	70.1481i	−0.125000	+	0.216506i
$19$		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
$22$	−206.000			−0.425620
$23$	367.000	+	635.663i	0.693762	+	1.20163i	0.970596	+	0.240713i	$0.0773813\pi$
$23$	367.000	+	635.663i	0.693762	+	1.20163i	−0.276834	+	0.960918i	$0.589285\pi$
$24$		0			0
$25$	−312.500	+	541.266i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1234.00			1.46730			0.733650	−	0.679527i	$-0.237815\pi$
$29$	1234.00			1.46730			0.733650	+	0.679527i	$0.237815\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−352.500	+	610.548i	−0.344238	+	0.596238i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1215.00			−0.937500
$37$	647.000	+	1120.64i	0.472608	+	0.818581i	0.999509	−	0.0313461i	$-0.00997942\pi$
$37$	647.000	+	1120.64i	0.472608	+	0.818581i	−0.526901	+	0.849927i	$0.676646\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−334.000			−0.180638			−0.0903191	−	0.995913i	$-0.528789\pi$
$43$	−334.000			−0.180638			−0.0903191	+	0.995913i	$0.528789\pi$
$44$	−1545.00	−	2676.02i	−0.798037	−	1.38224i
$45$		0			0
$46$	367.000	−	635.663i	0.173440	−	0.300408i
$47$		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
$50$	625.000			0.250000
$51$		0			0
$52$		0			0
$53$	2791.00	−	4834.15i	0.993592	−	1.72095i	0.398913	−	0.916989i	$-0.369388\pi$
$53$	2791.00	−	4834.15i	0.993592	−	1.72095i	0.594679	−	0.803963i	$-0.297279\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−617.000	−	1068.68i	−0.183413	−	0.317680i
$59$		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
$64$	−2639.00			−0.644287
$65$		0			0
$66$		0			0
$67$	−2473.00	+	4283.36i	−0.550902	+	0.954191i	0.447308	+	0.894380i	$0.352383\pi$
$67$	−2473.00	+	4283.36i	−0.550902	+	0.954191i	−0.998210	+	0.0598104i	$0.980950\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2914.00			0.578060			0.289030	−	0.957320i	$-0.406667\pi$
$71$	2914.00			0.578060			0.289030	+	0.957320i	$0.406667\pi$
$72$	1255.50	+	2174.59i	0.242188	+	0.419481i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	647.000	−	1120.64i	0.118152	−	0.204645i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1823.00	+	3157.53i	0.292101	+	0.505933i	0.974306	−	0.225227i	$-0.0723124\pi$
$79$	1823.00	+	3157.53i	0.292101	+	0.505933i	−0.682206	+	0.731160i	$0.738979\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−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
$86$	167.000	+	289.252i	0.0225798	+	0.0391093i
$87$		0			0
$88$	−3193.00	+	5530.44i	−0.412319	+	0.714158i
$89$		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
$92$	11010.0			1.30080
$93$		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
$99$	−16686.0			−1.70248
$100$	4687.50	+	8118.99i	0.468750	+	0.811899i
$101$		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
$106$	−5582.00			−0.496796
$107$	−5849.00	−	10130.8i	−0.510874	−	0.884860i	−0.999921	−	0.0126024i	$-0.995988\pi$
$107$	−5849.00	−	10130.8i	−0.510874	−	0.884860i	0.489046	−	0.872258i	$-0.337345\pi$
$108$		0			0
$109$	6263.00	−	10847.8i	0.527144	−	0.913041i	−0.472355	−	0.881408i	$-0.656596\pi$
$109$	6263.00	−	10847.8i	0.527144	−	0.913041i	0.999500	−	0.0316323i	$-0.0100706\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	23746.0			1.85966			0.929830	−	0.367989i	$-0.119954\pi$
$113$	23746.0			1.85966			0.929830	+	0.367989i	$0.119954\pi$
$114$		0			0
$115$		0			0
$116$	9255.00	−	16030.1i	0.687797	−	1.19130i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−13897.5	−	24071.2i	−0.949218	−	1.64409i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−32254.0			−1.99975			−0.999876	−	0.0157475i	$-0.994987\pi$
$127$	−32254.0			−1.99975			−0.999876	+	0.0157475i	$0.994987\pi$
$128$	6959.50	+	12054.2i	0.424774	+	0.735730i
$129$		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
$134$	4946.00			0.275451
$135$		0			0
$136$		0			0
$137$	3631.00	−	6289.08i	0.193457	−	0.335078i	−0.752936	−	0.658093i	$-0.771363\pi$
$137$	3631.00	−	6289.08i	0.193457	−	0.335078i	0.946394	+	0.323015i	$0.104697\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−1457.00	−	2523.60i	−0.0722575	−	0.125154i
$143$		0			0
$144$	−8464.50	+	14660.9i	−0.408203	+	0.707029i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	19410.0			0.886140
$149$	4903.00	+	8492.25i	0.220846	+	0.382516i	0.955065	−	0.296396i	$-0.0957849\pi$
$149$	4903.00	+	8492.25i	0.220846	+	0.382516i	−0.734219	+	0.678913i	$0.762452\pi$
$150$		0			0
$151$	−14737.0	+	25525.2i	−0.646331	+	1.11948i	0.337661	+	0.941268i	$0.390364\pi$
$151$	−14737.0	+	25525.2i	−0.646331	+	1.11948i	−0.983992	+	0.178211i	$0.942969\pi$
$152$		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	$0.666667\pi$
$158$	1823.00	−	3157.53i	0.0730252	−	0.126483i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	6561.00			0.250000
$163$	23831.0	+	41276.5i	0.896948	+	1.55356i	0.831376	+	0.555711i	$0.187554\pi$
$163$	23831.0	+	41276.5i	0.896948	+	1.55356i	0.0655720	+	0.997848i	$0.479113\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	28561.0			1.00000
$170$		0			0
$171$		0			0
$172$	−2505.00	+	4338.79i	−0.0846741	+	0.146660i
$173$		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
$176$	−43054.0			−1.38991
$177$		0			0
$178$		0			0
$179$	−26441.0	+	45797.2i	−0.825224	+	1.42933i	0.0765241	+	0.997068i	$0.475618\pi$
$179$	−26441.0	+	45797.2i	−0.825224	+	1.42933i	−0.901748	+	0.432262i	$0.857716\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−11377.0	−	19705.5i	−0.336041	−	0.582040i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14081.0	−	24389.0i	−0.385982	−	0.668540i	0.605923	−	0.795523i	$-0.292804\pi$
$191$	−14081.0	−	24389.0i	−0.385982	−	0.668540i	−0.991905	+	0.126983i	$0.959471\pi$
$192$		0			0
$193$	35327.0	−	61188.2i	0.948401	−	1.64268i	0.199608	−	0.979876i	$-0.436033\pi$
$193$	35327.0	−	61188.2i	0.948401	−	1.64268i	0.748793	−	0.662803i	$-0.230633\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1906.00			0.0491123			0.0245562	−	0.999698i	$-0.492183\pi$
$197$	1906.00			0.0491123			0.0245562	+	0.999698i	$0.492183\pi$
$198$	8343.00	+	14450.5i	0.212810	+	0.368598i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	9687.50	−	16779.2i	0.242188	−	0.419481i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	29727.0	−	51488.7i	0.693762	−	1.20163i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−11758.0			−0.264100			−0.132050	−	0.991243i	$-0.542156\pi$
$211$	−11758.0			−0.264100			−0.132050	+	0.991243i	$0.542156\pi$
$212$	−41865.0	−	72512.3i	−0.931493	−	1.61339i
$213$		0			0
$214$	−5849.00	+	10130.8i	−0.127719	+	0.221215i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−12526.0			−0.263572
$219$		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
$225$	50625.0			1.00000
$226$	−11873.0	−	20564.6i	−0.232458	−	0.402628i
$227$		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
$232$	−38254.0			−0.710724
$233$	54127.0	+	93750.7i	0.997016	+	1.72688i	0.565361	+	0.824843i	$0.308737\pi$
$233$	54127.0	+	93750.7i	0.997016	+	1.72688i	0.431655	+	0.902039i	$0.357930\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−64958.0			−1.13720			−0.568600	−	0.822614i	$-0.692515\pi$
$239$	−64958.0			−1.13720			−0.568600	+	0.822614i	$0.692515\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−13897.5	+	24071.2i	−0.237304	+	0.411023i
$243$		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
$253$	151204.			2.36223
$254$	16127.0	+	27932.8i	0.249969	+	0.432959i
$255$		0			0
$256$	−14152.5	+	24512.8i	−0.215950	+	0.374036i
$257$		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
$261$	−49977.0	−	86562.7i	−0.733650	−	1.27072i
$262$		0			0
$263$	−54833.0	+	94973.5i	−0.792740	+	1.37307i	0.131525	+	0.991313i	$0.458013\pi$
$263$	−54833.0	+	94973.5i	−0.792740	+	1.37307i	−0.924265	+	0.381752i	$0.875321\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	37095.0	+	64250.4i	0.516471	+	0.894554i
$269$		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
$274$	−7262.00			−0.0967286
$275$	64375.0	+	111501.i	0.851240	+	1.47439i
$276$		0			0
$277$	−26329.0	+	45603.2i	−0.343143	+	0.594341i	−0.985015	−	0.172471i	$-0.944825\pi$
$277$	−26329.0	+	45603.2i	−0.343143	+	0.594341i	0.641872	+	0.766812i	$0.278158\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−144926.			−1.83541			−0.917706	−	0.397260i	$-0.869961\pi$
$281$	−144926.			−1.83541			−0.917706	+	0.397260i	$0.869961\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	21855.0	−	37854.0i	0.270966	−	0.469326i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	57105.0			0.688477
$289$	−41760.5	−	72331.3i	−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
$296$	−20057.0	−	34739.7i	−0.228919	−	0.396500i
$297$		0			0
$298$	4903.00	−	8492.25i	0.0552115	−	0.0956291i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	29474.0			0.323166
$303$		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
$316$	54690.0			0.547689
$317$	−35753.0	−	61926.0i	−0.355790	−	0.616247i	0.631463	−	0.775406i	$-0.282455\pi$
$317$	−35753.0	−	61926.0i	−0.355790	−	0.616247i	−0.987253	+	0.159160i	$0.949122\pi$
$318$		0			0
$319$	127102.	−	220147.i	1.24902	−	2.16337i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	49207.5	+	85229.9i	0.468750	+	0.811899i
$325$		0			0
$326$	23831.0	−	41276.5i	0.224237	−	0.388390i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−91417.0	−	158339.i	−0.834394	−	1.44521i	−0.894523	−	0.447022i	$-0.852485\pi$
$331$	−91417.0	−	158339.i	−0.834394	−	1.44521i	0.0601296	−	0.998191i	$-0.480849\pi$
$332$		0			0
$333$	52407.0	−	90771.6i	0.472608	−	0.818581i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−176062.			−1.55026			−0.775132	−	0.631799i	$-0.782317\pi$
$337$	−176062.			−1.55026			−0.775132	+	0.631799i	$0.782317\pi$
$338$	−14280.5	−	24734.6i	−0.125000	−	0.216506i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	10354.0			0.0874966
$345$		0			0
$346$		0			0
$347$	−109433.	+	189544.i	−0.908844	+	1.57416i	−0.0931715	+	0.995650i	$0.529700\pi$
$347$	−109433.	+	189544.i	−0.908844	+	1.57416i	−0.815672	+	0.578514i	$0.803633\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	72615.0	+	125773.i	0.586059	+	1.01508i
$353$		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
$358$	52882.0			0.412612
$359$	−84977.0	−	147184.i	−0.659345	−	1.14202i	−0.980786	−	0.195089i	$-0.937500\pi$
$359$	−84977.0	−	147184.i	−0.659345	−	1.14202i	0.321441	−	0.946930i	$-0.395833\pi$
$360$		0			0
$361$	−65160.5	+	112861.i	−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	$0.666667\pi$
$368$	76703.0	−	132853.i	0.566391	−	0.981019i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	104807.	+	181531.i	0.753308	+	1.30477i	0.946211	+	0.323550i	$0.104876\pi$
$373$	104807.	+	181531.i	0.753308	+	1.30477i	−0.192903	+	0.981218i	$0.561790\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	89714.0			0.624571			0.312285	−	0.949988i	$-0.398905\pi$
$379$	89714.0			0.624571			0.312285	+	0.949988i	$0.398905\pi$
$380$		0			0
$381$		0			0
$382$	−14081.0	+	24389.0i	−0.0964954	+	0.167135i
$383$		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
$386$	−70654.0			−0.474201
$387$	13527.0	+	23429.5i	0.0903191	+	0.156437i
$388$		0			0
$389$	−70457.0	+	122035.i	−0.465613	+	0.806465i	−0.999229	−	0.0392617i	$-0.987499\pi$
$389$	−70457.0	+	122035.i	−0.465613	+	0.806465i	0.533616	+	0.845727i	$0.320833\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−953.000	−	1650.64i	−0.00613904	−	0.0106331i
$395$		0			0
$396$	−125145.	+	216757.i	−0.798037	+	1.38224i
$397$		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
$400$	130625.			0.816406
$401$	98143.0	+	169989.i	0.610338	+	1.05714i	0.991183	+	0.132498i	$0.0422998\pi$
$401$	98143.0	+	169989.i	0.610338	+	1.05714i	−0.380845	+	0.924639i	$0.624367\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	266564.			1.60921
$408$		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
$414$	−59454.0			−0.346881
$415$		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
$421$	−326926.			−1.84453			−0.922264	−	0.386561i	$-0.873663\pi$
$421$	−326926.			−1.84453			−0.922264	+	0.386561i	$0.873663\pi$
$422$	5879.00	+	10182.7i	0.0330125	+	0.0571793i
$423$		0			0
$424$	−86521.0	+	149859.i	−0.481271	+	0.833586i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−175470.			−0.957889
$429$		0			0
$430$		0			0
$431$	172639.	−	299020.i	0.929361	−	1.60970i	0.144968	−	0.989436i	$-0.453692\pi$
$431$	172639.	−	299020.i	0.929361	−	1.60970i	0.784393	−	0.620264i	$-0.212974\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−93945.0	−	162718.i	−0.494198	−	0.855975i
$437$		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
$443$	78151.0	+	135362.i	0.398224	+	0.689744i	0.993507	−	0.113772i	$-0.0362934\pi$
$443$	78151.0	+	135362.i	0.398224	+	0.689744i	−0.595283	+	0.803516i	$0.702960\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	396034.			1.96444			0.982222	−	0.187721i	$-0.0601102\pi$
$449$	396034.			1.96444			0.982222	+	0.187721i	$0.0601102\pi$
$450$	−25312.5	−	43842.5i	−0.125000	−	0.216506i
$451$		0			0
$452$	178095.	−	308470.i	0.871716	−	1.50986i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−176593.	−	305868.i	−0.845553	−	1.46454i	−0.885140	−	0.465325i	$-0.845937\pi$
$457$	−176593.	−	305868.i	−0.845553	−	1.46454i	0.0395862	−	0.999216i	$-0.487396\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	25538.0			0.119131			0.0595655	−	0.998224i	$-0.481028\pi$
$463$	25538.0			0.119131			0.0595655	+	0.998224i	$0.481028\pi$
$464$	−128953.	−	223353.i	−0.598957	−	1.03742i
$465$		0			0
$466$	54127.0	−	93750.7i	0.249254	−	0.431721i
$467$		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
$473$	−34402.0	+	59586.0i	−0.153766	+	0.266331i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−452142.			−1.98718
$478$	32479.0	+	56255.3i	0.142150	+	0.246211i
$479$		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
$484$	−416925.			−1.77978
$485$		0			0
$486$		0			0
$487$	157967.	−	273607.i	0.666052	−	1.15364i	−0.312946	−	0.949771i	$-0.601316\pi$
$487$	157967.	−	273607.i	0.666052	−	1.15364i	0.978999	−	0.203866i	$-0.0653507\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	427954.			1.77515			0.887573	−	0.460667i	$-0.152390\pi$
$491$	427954.			1.77515			0.887573	+	0.460667i	$0.152390\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	204599.	+	354376.i	0.821679	+	1.42319i	0.904431	+	0.426620i	$0.140296\pi$
$499$	204599.	+	354376.i	0.821679	+	1.42319i	−0.0827514	+	0.996570i	$0.526371\pi$
$500$		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
$506$	−75602.0	−	130947.i	−0.295279	−	0.511438i
$507$		0			0
$508$	−241905.	+	418992.i	−0.937384	+	1.62360i
$509$		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
$512$	251009.			0.957523
$513$		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	$0.666667\pi$
$522$	−49977.0	+	86562.7i	−0.183413	+	0.317680i
$523$		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
$526$	109666.			0.396370
$527$		0			0
$528$		0			0
$529$	−129458.	+	224227.i	−0.462611	+	0.801266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	76663.0	−	132784.i	0.266843	−	0.462186i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	289943.	+	502196.i	0.990645	+	1.71585i	0.613503	+	0.789692i	$0.289760\pi$
$541$	289943.	+	502196.i	0.990645	+	1.71585i	0.377142	+	0.926156i	$0.376907\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	110546.			0.369461			0.184730	−	0.982789i	$-0.440859\pi$
$547$	110546.			0.369461			0.184730	+	0.982789i	$0.440859\pi$
$548$	−54465.0	−	94336.1i	−0.181366	−	0.314136i
$549$		0			0
$550$	64375.0	−	111501.i	0.212810	−	0.368598i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	52658.0			0.171571
$555$		0			0
$556$		0			0
$557$	−191753.	+	332126.i	−0.618062	+	1.07051i	0.371777	+	0.928322i	$0.378748\pi$
$557$	−191753.	+	332126.i	−0.618062	+	1.07051i	−0.989839	+	0.142192i	$0.954585\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	72463.0	+	125510.i	0.229427	+	0.397378i
$563$		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
$568$	−90334.0			−0.279998
$569$	109903.	+	190358.i	0.339457	+	0.587957i	0.984331	−	0.176332i	$-0.0564233\pi$
$569$	109903.	+	190358.i	0.339457	+	0.587957i	−0.644874	+	0.764289i	$0.723090\pi$
$570$		0			0
$571$	−307897.	+	533293.i	−0.944351	+	1.63566i	−0.187304	+	0.982302i	$0.559975\pi$
$571$	−307897.	+	533293.i	−0.944351	+	1.63566i	−0.757046	+	0.653361i	$0.773358\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−458750.			−1.38752
$576$	106880.	+	185121.i	0.322144	+	0.557969i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−41760.5	+	72331.3i	−0.125000	+	0.216506i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−574946.	−	995836.i	−1.69157	−	2.92989i
$584$		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
$592$	135223.	−	234213.i	0.385840	−	0.668294i
$593$		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
$596$	147090.			0.414086
$597$		0			0
$598$		0			0
$599$	343663.	−	595242.i	0.957809	−	1.65897i	0.230006	−	0.973189i	$-0.426126\pi$
$599$	343663.	−	595242.i	0.957809	−	1.65897i	0.727804	−	0.685785i	$-0.240541\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	400626.			1.10180
$604$	221055.	+	382878.i	0.605936	+	1.04951i
$605$		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
$613$	352007.	−	609694.i	0.936764	−	1.62252i	0.165307	−	0.986242i	$-0.447138\pi$
$613$	352007.	−	609694.i	0.936764	−	1.62252i	0.771457	−	0.636281i	$-0.219528\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−450014.			−1.18210			−0.591052	−	0.806633i	$-0.701287\pi$
$617$	−450014.			−1.18210			−0.591052	+	0.806633i	$0.701287\pi$
$618$		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
$625$	−195312.	−	338291.i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	215714.			0.541776			0.270888	−	0.962611i	$-0.412683\pi$
$631$	215714.			0.541776			0.270888	+	0.962611i	$0.412683\pi$
$632$	−56513.0	−	97883.4i	−0.141486	−	0.245061i
$633$		0			0
$634$	−35753.0	+	61926.0i	−0.0889475	+	0.154062i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−254204.			−0.624512
$639$	−118017.	−	204411.i	−0.289030	−	0.500615i
$640$		0			0
$641$	63103.0	−	109298.i	0.153580	−	0.266008i	−0.778961	−	0.627072i	$-0.784253\pi$
$641$	63103.0	−	109298.i	0.153580	−	0.266008i	0.932541	+	0.361064i	$0.117586\pi$
$642$		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	$0.666667\pi$
$648$	101696.	−	176142.i	0.242188	−	0.419481i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	714930.			1.68178
$653$	−286409.	−	496075.i	−0.671677	−	1.16338i	−0.977428	−	0.211267i	$-0.932241\pi$
$653$	−286409.	−	496075.i	−0.671677	−	1.16338i	0.305752	−	0.952111i	$-0.401092\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−486638.			−1.12056			−0.560280	−	0.828303i	$-0.689307\pi$
$659$	−486638.			−1.12056			−0.560280	+	0.828303i	$0.689307\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−91417.0	+	158339.i	−0.208598	+	0.361303i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−104814.			−0.236304
$667$	452878.	+	784408.i	1.01796	+	1.76315i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−706942.			−1.56082			−0.780411	−	0.625266i	$-0.784990\pi$
$673$	−706942.			−1.56082			−0.780411	+	0.625266i	$0.784990\pi$
$674$	88031.0	+	152474.i	0.193783	+	0.335642i
$675$		0			0
$676$	214208.	−	371018.i	0.468750	−	0.811899i
$677$		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
$683$	−428633.	+	742414.i	−0.918849	+	1.59149i	−0.117683	+	0.993051i	$0.537547\pi$
$683$	−428633.	+	742414.i	−0.918849	+	1.59149i	−0.801166	+	0.598442i	$0.795787\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	34903.0	+	60453.8i	0.0737371	+	0.127716i
$689$		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
$694$	218866.			0.454422
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	971602.			1.97721			0.988604	−	0.150539i	$-0.0481010\pi$
$701$	971602.			1.97721			0.988604	+	0.150539i	$0.0481010\pi$
$702$		0			0
$703$		0			0
$704$	−271817.	+	470801.i	−0.548443	+	0.949931i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−452281.	−	783374.i	−0.899738	−	1.55839i	−0.827829	−	0.560980i	$-0.810424\pi$
$709$	−452281.	−	783374.i	−0.899738	−	1.55839i	−0.0719083	−	0.997411i	$-0.522909\pi$
$710$		0			0
$711$	147663.	−	255760.i	0.292101	−	0.505933i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	396615.	+	686957.i	0.773647	+	1.34000i
$717$		0			0
$718$	−84977.0	+	147184.i	−0.164836	+	0.285505i
$719$		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
$722$	130321.			0.250000
$723$		0			0
$724$		0			0
$725$	−385625.	+	667922.i	−0.733650	+	1.27072i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	531441.			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
$736$	−517470.			−0.955277
$737$	509438.	+	882372.i	0.937900	+	1.62449i
$738$		0			0
$739$	−205417.	+	355793.i	−0.376138	+	0.651491i	−0.990497	−	0.137536i	$-0.956082\pi$
$739$	−205417.	+	355793.i	−0.376138	+	0.651491i	0.614358	+	0.789027i	$0.289415\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.09110e6			−1.97646			−0.988229	−	0.152980i	$-0.951113\pi$
$743$	−1.09110e6			−1.97646			−0.988229	+	0.152980i	$0.951113\pi$
$744$		0			0
$745$		0			0
$746$	104807.	−	181531.i	0.188327	−	0.326192i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	242399.	+	419847.i	0.429785	+	0.744409i	0.996854	−	0.0792612i	$-0.0252561\pi$
$751$	242399.	+	419847.i	0.429785	+	0.744409i	−0.567069	+	0.823670i	$0.691923\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	819506.			1.43008			0.715040	−	0.699083i	$-0.246408\pi$
$757$	819506.			1.43008			0.715040	+	0.699083i	$0.246408\pi$
$758$	−44857.0	−	77694.6i	−0.0780714	−	0.135224i
$759$		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
$764$	−422430.			−0.723716
$765$		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
$772$	−529905.	−	917822.i	−0.889126	−	1.54001i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$	13527.0	−	23429.5i	0.0225798	−	0.0391093i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	140914.			0.232806
$779$		0			0
$780$		0			0
$781$	300142.	−	519861.i	0.492068	−	0.852286i
$782$		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	$0.666667\pi$
$788$	14295.0	−	24759.7i	0.0230214	−	0.0398742i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	517266.			0.824638
$793$		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
$800$	−220312.	−	381592.i	−0.344238	−	0.596238i
$801$		0			0
$802$	98143.0	−	169989.i	0.152585	−	0.264284i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	639343.	−	1.10737e6i	0.976870	−	1.69199i	0.303250	−	0.952911i	$-0.401928\pi$
$809$	639343.	−	1.10737e6i	0.976870	−	1.69199i	0.673620	−	0.739078i	$-0.264738\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−133282.	−	230851.i	−0.201151	−	0.348404i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3559.00	+	6164.37i	0.00528009	+	0.00914539i	0.868653	−	0.495420i	$-0.164986\pi$
$821$	3559.00	+	6164.37i	0.00528009	+	0.00914539i	−0.863373	+	0.504566i	$0.831653\pi$
$822$		0			0
$823$	483887.	−	838117.i	0.714405	−	1.23739i	−0.248784	−	0.968559i	$-0.580031\pi$
$823$	483887.	−	838117.i	0.714405	−	1.23739i	0.963189	−	0.268826i	$-0.0866357\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.28833e6			−1.88372			−0.941862	−	0.335999i	$-0.890926\pi$
$827$	−1.28833e6			−1.88372			−0.941862	+	0.335999i	$0.890926\pi$
$828$	−445905.	−	772330.i	−0.650402	−	1.12653i
$829$		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
$841$	815475.			1.15297
$842$	163463.	+	283126.i	0.230566	+	0.399352i
$843$		0			0
$844$	−88185.0	+	152741.i	−0.123797	+	0.214423i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.16664e6			−1.62235
$849$		0			0
$850$		0			0
$851$	−474898.	+	822547.i	−0.655754	+	1.13580i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	181319.	+	314054.i	0.247455	+	0.428604i
$857$		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
$862$	−345278.			−0.464680
$863$	−636353.	−	1.10220e6i	−0.854430	−	1.47992i	−0.877173	−	0.480175i	$-0.840573\pi$
$863$	−636353.	−	1.10220e6i	−0.854430	−	1.47992i	0.0227428	−	0.999741i	$-0.492760\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	751076.			0.994591
$870$		0			0
$871$		0			0
$872$	−194153.	+	336283.i	−0.255335	+	0.442254i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	490871.	+	850214.i	0.638217	+	1.10542i	0.985824	+	0.167784i	$0.0536610\pi$
$877$	490871.	+	850214.i	0.638217	+	1.10542i	−0.347607	+	0.937640i	$0.613006\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1.07151e6			1.37427			0.687137	−	0.726528i	$-0.258867\pi$
$883$	1.07151e6			1.37427			0.687137	+	0.726528i	$0.258867\pi$
$884$		0			0
$885$		0			0
$886$	78151.0	−	135362.i	0.0995559	−	0.172436i
$887$		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
$891$	675783.	+	1.17049e6i	0.851240	+	1.47439i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−198017.	−	342976.i	−0.245556	−	0.425315i
$899$		0			0
$900$	379688.	−	657638.i	0.468750	−	0.811899i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−736126.			−0.900773
$905$		0			0
$906$		0			0
$907$	−772249.	+	1.33757e6i	−0.938735	+	1.62594i	−0.170899	+	0.985289i	$0.554667\pi$
$907$	−772249.	+	1.33757e6i	−0.938735	+	1.62594i	−0.767835	+	0.640647i	$0.778666\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	792514.			0.954927			0.477464	−	0.878652i	$-0.341556\pi$
$911$	792514.			0.954927			0.477464	+	0.878652i	$0.341556\pi$
$912$		0			0
$913$		0			0
$914$	−176593.	+	305868.i	−0.211388	+	0.366135i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	735983.	+	1.27476e6i	0.871439	+	1.50938i	0.860509	+	0.509436i	$0.170146\pi$
$919$	735983.	+	1.27476e6i	0.871439	+	1.50938i	0.0109297	+	0.999940i	$0.496521\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−808750.			−0.945215
$926$	−12769.0	−	22116.6i	−0.0148914	−	0.0257926i
$927$		0			0
$928$	−434985.	+	753416.i	−0.505101	+	0.874861i
$929$		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
$932$	1.62381e6			1.86940
$933$		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
$946$	68804.0			0.0768832
$947$	−219209.	−	379681.i	−0.244432	−	0.423369i	0.717540	−	0.696518i	$-0.245268\pi$
$947$	−219209.	−	379681.i	−0.244432	−	0.423369i	−0.961972	+	0.273149i	$0.911935\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	309346.			0.340611			0.170306	−	0.985391i	$-0.445525\pi$
$953$	309346.			0.340611			0.170306	+	0.985391i	$0.445525\pi$
$954$	226071.	+	391566.i	0.248398	+	0.430238i
$955$		0			0
$956$	−487185.	+	843829.i	−0.533062	+	0.923291i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−461760.	−	799793.i	−0.500000	−	0.866025i
$962$		0			0
$963$	−473769.	+	820592.i	−0.510874	+	0.884860i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.75862e6			−1.88070			−0.940350	−	0.340208i	$-0.889502\pi$
$967$	−1.75862e6			−1.88070			−0.940350	+	0.340208i	$0.889502\pi$
$968$	430822.	+	746206.i	0.459777	+	0.796358i
$969$		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
$974$	−315934.			−0.333026
$975$		0			0
$976$		0			0
$977$	941407.	−	1.63056e6i	0.986253	−	1.70824i	0.350022	−	0.936742i	$-0.386174\pi$
$977$	941407.	−	1.63056e6i	0.986253	−	1.70824i	0.636231	−	0.771499i	$-0.280493\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.01461e6			−1.05429
$982$	−213977.	−	370619.i	−0.221893	−	0.384330i
$983$		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
$989$	−122578.	−	212311.i	−0.125320	−	0.217060i
$990$		0			0
$991$	−6337.00	+	10976.0i	−0.00645262	+	0.0111763i	−0.869234	−	0.494401i	$-0.835387\pi$
$991$	−6337.00	+	10976.0i	−0.00645262	+	0.0111763i	0.862781	+	0.505578i	$0.168721\pi$
$992$		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	$0.666667\pi$
$998$	204599.	−	354376.i	0.205420	−	0.355798i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.5.d.a.31.1		2
7.2	even	3	inner	49.5.d.a.19.1		2
7.3	odd	6		7.5.b.a.6.1	✓	1
7.4	even	3		7.5.b.a.6.1	✓	1
7.5	odd	6	inner	49.5.d.a.19.1		2
7.6	odd	2	CM	49.5.d.a.31.1		2
21.11	odd	6		63.5.d.a.55.1		1
21.17	even	6		63.5.d.a.55.1		1
28.3	even	6		112.5.c.a.97.1		1
28.11	odd	6		112.5.c.a.97.1		1
35.3	even	12		175.5.c.a.174.1		2
35.4	even	6		175.5.d.a.76.1		1
35.17	even	12		175.5.c.a.174.2		2
35.18	odd	12		175.5.c.a.174.1		2
35.24	odd	6		175.5.d.a.76.1		1
35.32	odd	12		175.5.c.a.174.2		2
56.3	even	6		448.5.c.a.321.1		1
56.11	odd	6		448.5.c.a.321.1		1
56.45	odd	6		448.5.c.b.321.1		1
56.53	even	6		448.5.c.b.321.1		1
84.11	even	6		1008.5.f.a.433.1		1
84.59	odd	6		1008.5.f.a.433.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.5.b.a.6.1	✓	1	7.3	odd	6
7.5.b.a.6.1	✓	1	7.4	even	3
49.5.d.a.19.1		2	7.2	even	3	inner
49.5.d.a.19.1		2	7.5	odd	6	inner
49.5.d.a.31.1		2	1.1	even	1	trivial
49.5.d.a.31.1		2	7.6	odd	2	CM
63.5.d.a.55.1		1	21.11	odd	6
63.5.d.a.55.1		1	21.17	even	6
112.5.c.a.97.1		1	28.3	even	6
112.5.c.a.97.1		1	28.11	odd	6
175.5.c.a.174.1		2	35.3	even	12
175.5.c.a.174.1		2	35.18	odd	12
175.5.c.a.174.2		2	35.17	even	12
175.5.c.a.174.2		2	35.32	odd	12
175.5.d.a.76.1		1	35.4	even	6
175.5.d.a.76.1		1	35.24	odd	6
448.5.c.a.321.1		1	56.3	even	6
448.5.c.a.321.1		1	56.11	odd	6
448.5.c.b.321.1		1	56.45	odd	6
448.5.c.b.321.1		1	56.53	even	6
1008.5.f.a.433.1		1	84.11	even	6
1008.5.f.a.433.1		1	84.59	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	49.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(7.50000 - 12.9904i) q^{4} -31.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +(103.000 - 178.401i) q^{11} +(-104.500 - 180.999i) q^{16} +(-40.5000 + 70.1481i) q^{18} -206.000 q^{22} +(367.000 + 635.663i) q^{23} +(-312.500 + 541.266i) q^{25} +1234.00 q^{29} +(-352.500 + 610.548i) q^{32} -1215.00 q^{36} +(647.000 + 1120.64i) q^{37} -334.000 q^{43} +(-1545.00 - 2676.02i) q^{44} +(367.000 - 635.663i) q^{46} +625.000 q^{50} +(2791.00 - 4834.15i) q^{53} +(-617.000 - 1068.68i) q^{58} -2639.00 q^{64} +(-2473.00 + 4283.36i) q^{67} +2914.00 q^{71} +(1255.50 + 2174.59i) q^{72} +(647.000 - 1120.64i) q^{74} +(1823.00 + 3157.53i) q^{79} +(-3280.50 + 5681.99i) q^{81} +(167.000 + 289.252i) q^{86} +(-3193.00 + 5530.44i) q^{88} +11010.0 q^{92} -16686.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 15 q^{4} - 62 q^{8} - 81 q^{9} + 206 q^{11} - 209 q^{16} - 81 q^{18} - 412 q^{22} + 734 q^{23} - 625 q^{25} + 2468 q^{29} - 705 q^{32} - 2430 q^{36} + 1294 q^{37} - 668 q^{43} - 3090 q^{44}+ \cdots - 33372 q^{99}+O(q^{100}) \) 2 * q - q^2 + 15 * q^4 - 62 * q^8 - 81 * q^9 + 206 * q^11 - 209 * q^16 - 81 * q^18 - 412 * q^22 + 734 * q^23 - 625 * q^25 + 2468 * q^29 - 705 * q^32 - 2430 * q^36 + 1294 * q^37 - 668 * q^43 - 3090 * q^44 + 734 * q^46 + 1250 * q^50 + 5582 * q^53 - 1234 * q^58 - 5278 * q^64 - 4946 * q^67 + 5828 * q^71 + 2511 * q^72 + 1294 * q^74 + 3646 * q^79 - 6561 * q^81 + 334 * q^86 - 6386 * q^88 + 22020 * q^92 - 33372 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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