LMFDB - Embedded newform 81.7.d.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,7,Mod(26,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("81.26");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	81.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.6343807732$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			53.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.53
Dual form			81.7.d.a.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-32.0000 + 55.4256i) q^{4} +(143.000 + 247.683i) q^{7} +O(q^{10})$	$q+(-32.0000 + 55.4256i) q^{4} +(143.000 + 247.683i) q^{7} +(-253.000 + 438.209i) q^{13} +(-2048.00 - 3547.24i) q^{16} -10582.0 q^{19} +(-7812.50 - 13531.6i) q^{25} -18304.0 q^{28} +(-17641.0 + 30555.1i) q^{31} -89206.0 q^{37} +(-55693.0 - 96463.1i) q^{43} +(17926.5 - 31049.6i) q^{49} +(-16192.0 - 28045.4i) q^{52} +(210419. + 364456. i) q^{61} +262144. q^{64} +(-86437.0 + 149713. i) q^{67} +638066. q^{73} +(338624. - 586514. i) q^{76} +(102311. + 177208. i) q^{79} -144716. q^{91} +(28223.0 + 48883.7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{4} + 286 q^{7}+O(q^{10}) $ 2 * q - 64 * q^4 + 286 * q^7	$ 2 q - 64 q^{4} + 286 q^{7} - 506 q^{13} - 4096 q^{16} - 21164 q^{19} - 15625 q^{25} - 36608 q^{28} - 35282 q^{31} - 178412 q^{37} - 111386 q^{43} + 35853 q^{49} - 32384 q^{52} + 420838 q^{61} + 524288 q^{64} - 172874 q^{67} + 1276132 q^{73} + 677248 q^{76} + 204622 q^{79} - 289432 q^{91} + 56446 q^{97}+O(q^{100}) $ 2 * q - 64 * q^4 + 286 * q^7 - 506 * q^13 - 4096 * q^16 - 21164 * q^19 - 15625 * q^25 - 36608 * q^28 - 35282 * q^31 - 178412 * q^37 - 111386 * q^43 + 35853 * q^49 - 32384 * q^52 + 420838 * q^61 + 524288 * q^64 - 172874 * q^67 + 1276132 * q^73 + 677248 * q^76 + 204622 * q^79 - 289432 * q^91 + 56446 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		0			0
$3$		0			0
$4$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	143.000	+	247.683i	0.416910	+	0.722109i	0.995627	−	0.0934196i	$-0.0297798\pi$
$7$	143.000	+	247.683i	0.416910	+	0.722109i	−0.578717	+	0.815528i	$0.696446\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−253.000	+	438.209i	−0.115157	+	0.199458i	−0.917843	−	0.396945i	$-0.870070\pi$
$13$	−253.000	+	438.209i	−0.115157	+	0.199458i	0.802685	+	0.596403i	$0.203404\pi$
$14$		0			0
$15$		0			0
$16$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−10582.0			−1.54279			−0.771395	−	0.636356i	$-0.780441\pi$
$19$	−10582.0			−1.54279			−0.771395	+	0.636356i	$0.780441\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−18304.0			−0.833819
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−17641.0	+	30555.1i	−0.592159	+	1.02565i	0.401782	+	0.915735i	$0.368391\pi$
$31$	−17641.0	+	30555.1i	−0.592159	+	1.02565i	−0.993941	+	0.109914i	$0.964943\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−89206.0			−1.76112			−0.880560	−	0.473935i	$-0.842833\pi$
$37$	−89206.0			−1.76112			−0.880560	+	0.473935i	$0.842833\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−55693.0	−	96463.1i	−0.700479	−	1.21327i	−0.968298	−	0.249796i	$-0.919636\pi$
$43$	−55693.0	−	96463.1i	−0.700479	−	1.21327i	0.267819	−	0.963469i	$-0.413697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	17926.5	−	31049.6i	0.152373	−	0.263917i
$50$		0			0
$51$		0			0
$52$	−16192.0	−	28045.4i	−0.115157	−	0.199458i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	210419.	+	364456.i	0.927034	+	1.60567i	0.788258	+	0.615345i	$0.210983\pi$
$61$	210419.	+	364456.i	0.927034	+	1.60567i	0.138776	+	0.990324i	$0.455683\pi$
$62$		0			0
$63$		0			0
$64$	262144.			1.00000
$65$		0			0
$66$		0			0
$67$	−86437.0	+	149713.i	−0.287392	+	0.497778i	−0.973187	−	0.230017i	$-0.926122\pi$
$67$	−86437.0	+	149713.i	−0.287392	+	0.497778i	0.685794	+	0.727796i	$0.259455\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	638066.			1.64020			0.820100	−	0.572220i	$-0.193918\pi$
$73$	638066.			1.64020			0.820100	+	0.572220i	$0.193918\pi$
$74$		0			0
$75$		0			0
$76$	338624.	−	586514.i	0.771395	−	1.33610i
$77$		0			0
$78$		0			0
$79$	102311.	+	177208.i	0.207511	+	0.359420i	0.950930	−	0.309407i	$-0.100130\pi$
$79$	102311.	+	177208.i	0.207511	+	0.359420i	−0.743419	+	0.668826i	$0.766797\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−144716.			−0.192040
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	28223.0	+	48883.7i	0.0309235	+	0.0535610i	0.881073	−	0.472980i	$-0.156822\pi$
$97$	28223.0	+	48883.7i	0.0309235	+	0.0535610i	−0.850150	+	0.526541i	$0.823489\pi$
$98$		0			0
$99$		0			0
$100$	1.00000e6			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−563473.	+	975964.i	−0.515658	+	0.893145i	0.484177	+	0.874970i	$0.339119\pi$
$103$	−563473.	+	975964.i	−0.515658	+	0.893145i	−0.999835	+	0.0181752i	$0.994214\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−2.17274e6			−1.67776			−0.838878	−	0.544320i	$-0.816788\pi$
$109$	−2.17274e6			−1.67776			−0.838878	+	0.544320i	$0.816788\pi$
$110$		0			0
$111$		0			0
$112$	585728.	−	1.01451e6i	0.416910	−	0.722109i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−885780.	+	1.53422e6i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.12902e6	−	1.95553e6i	−0.592159	−	1.02565i
$125$		0			0
$126$		0			0
$127$	−3.95237e6			−1.92951			−0.964753	−	0.263158i	$-0.915236\pi$
$127$	−3.95237e6			−1.92951			−0.964753	+	0.263158i	$0.915236\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−1.51323e6	−	2.62098e6i	−0.643204	−	1.11406i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−632269.	+	1.09512e6i	−0.235428	+	0.407773i	−0.959397	−	0.282060i	$-0.908982\pi$
$139$	−632269.	+	1.09512e6i	−0.235428	+	0.407773i	0.723969	+	0.689832i	$0.242316\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	2.85459e6	−	4.94430e6i	0.880560	−	1.52517i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	1.91520e6	+	3.31722e6i	0.556267	+	0.963482i	0.997804	+	0.0662392i	$0.0211000\pi$
$151$	1.91520e6	+	3.31722e6i	0.556267	+	0.963482i	−0.441537	+	0.897243i	$0.645567\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.54136e6	+	6.13381e6i	−0.915105	+	1.58501i	−0.108357	+	0.994112i	$0.534559\pi$
$157$	−3.54136e6	+	6.13381e6i	−0.915105	+	1.58501i	−0.806748	+	0.590896i	$0.798774\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.89851e6			0.669285			0.334643	−	0.942345i	$-0.391384\pi$
$163$	2.89851e6			0.669285			0.334643	+	0.942345i	$0.391384\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	2.28539e6	+	3.95841e6i	0.473478	+	0.820087i
$170$		0			0
$171$		0			0
$172$	7.12870e6			1.40096
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	2.23438e6	−	3.87005e6i	0.416910	−	0.722109i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	98282.0			0.0165744			0.00828721	−	0.999966i	$-0.497362\pi$
$181$	98282.0			0.0165744			0.00828721	+	0.999966i	$0.497362\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	6.52781e6	−	1.13065e7i	0.908020	−	1.57274i	0.0912086	−	0.995832i	$-0.470927\pi$
$193$	6.52781e6	−	1.13065e7i	0.908020	−	1.57274i	0.816811	−	0.576905i	$-0.195740\pi$
$194$		0			0
$195$		0			0
$196$	1.14730e6	+	1.98717e6i	0.152373	+	0.263917i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.16545e7			1.47888			0.739442	−	0.673220i	$-0.235089\pi$
$199$	1.16545e7			1.47888			0.739442	+	0.673220i	$0.235089\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2.07258e6			0.230314
$209$		0			0
$210$		0			0
$211$	−8.79858e6	+	1.52396e7i	−0.936624	+	1.62228i	−0.164912	+	0.986308i	$0.552734\pi$
$211$	−8.79858e6	+	1.52396e7i	−0.936624	+	1.62228i	−0.771712	+	0.635972i	$0.780599\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.00907e7			−0.987507
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	5.90537e6	+	1.02284e7i	0.532516	+	0.922344i	0.999279	+	0.0379619i	$0.0120866\pi$
$223$	5.90537e6	+	1.02284e7i	0.532516	+	0.922344i	−0.466764	+	0.884382i	$0.654580\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	2.03641e6	−	3.52717e6i	0.169574	−	0.293711i	−0.768696	−	0.639614i	$-0.779094\pi$
$229$	2.03641e6	−	3.52717e6i	0.169574	−	0.293711i	0.938270	+	0.345903i	$0.112428\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−1.32199e7	−	2.28976e7i	−0.944447	−	1.63583i	−0.756854	−	0.653584i	$-0.773265\pi$
$241$	−1.32199e7	−	2.28976e7i	−0.944447	−	1.63583i	−0.187593	−	0.982247i	$-0.560069\pi$
$242$		0			0
$243$		0			0
$244$	−2.69336e7			−1.85407
$245$		0			0
$246$		0			0
$247$	2.67725e6	−	4.63713e6i	0.177663	−	0.307722i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.38861e6	+	1.45295e7i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−1.27565e7	−	2.20948e7i	−0.734228	−	1.27172i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−5.53197e6	−	9.58165e6i	−0.287392	−	0.497778i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−3.91457e7			−1.96687			−0.983436	−	0.181258i	$-0.941983\pi$
$271$	−3.91457e7			−1.96687			−0.983436	+	0.181258i	$0.941983\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.31335e7	+	2.27479e7i	0.617932	+	1.07029i	0.989863	+	0.142029i	$0.0453626\pi$
$277$	1.31335e7	+	2.27479e7i	0.617932	+	1.07029i	−0.371931	+	0.928261i	$0.621304\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	6.98519e6	−	1.20987e7i	0.308190	−	0.533801i	−0.669776	−	0.742563i	$-0.733610\pi$
$283$	6.98519e6	−	1.20987e7i	0.308190	−	0.533801i	0.977966	+	0.208762i	$0.0669434\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.41376e7			1.00000
$290$		0			0
$291$		0			0
$292$	−2.04181e7	+	3.53652e7i	−0.820100	+	1.42046i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.59282e7	−	2.75884e7i	0.584073	−	1.01164i
$302$		0			0
$303$		0			0
$304$	2.16719e7	+	3.75369e7i	0.771395	+	1.33610i
$305$		0			0
$306$		0			0
$307$	5.53407e7			1.91262			0.956312	−	0.292348i	$-0.0944365\pi$
$307$	5.53407e7			1.91262			0.956312	+	0.292348i	$0.0944365\pi$
$308$		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
$313$	−1.94358e7	−	3.36637e7i	−0.633824	−	1.09781i	−0.986763	−	0.162168i	$-0.948151\pi$
$313$	−1.94358e7	−	3.36637e7i	−0.633824	−	1.09781i	0.352940	−	0.935646i	$-0.385182\pi$
$314$		0			0
$315$		0			0
$316$	−1.30958e7			−0.415022
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	7.90625e6			0.230314
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.57727e7	+	6.19601e7i	0.986432	+	1.70855i	0.635391	+	0.772191i	$0.280839\pi$
$331$	3.57727e7	+	6.19601e7i	0.986432	+	1.70855i	0.351041	+	0.936360i	$0.385828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.61203e7	−	4.52417e7i	0.682478	−	1.18209i	−0.291745	−	0.956496i	$-0.594236\pi$
$337$	2.61203e7	−	4.52417e7i	0.682478	−	1.18209i	0.974222	−	0.225590i	$-0.0724309\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	4.39016e7			1.08792
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−2.84631e7	−	4.92996e7i	−0.669586	−	1.15976i	−0.978020	−	0.208512i	$-0.933138\pi$
$349$	−2.84631e7	−	4.92996e7i	−0.669586	−	1.15976i	0.308433	−	0.951246i	$-0.400195\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	6.49328e7			1.38020
$362$		0			0
$363$		0			0
$364$	4.63091e6	−	8.02097e6i	0.0960201	−	0.166312i
$365$		0			0
$366$		0			0
$367$	4.00131e7	+	6.93047e7i	0.809475	+	1.40205i	0.913228	+	0.407449i	$0.133582\pi$
$367$	4.00131e7	+	6.93047e7i	0.809475	+	1.40205i	−0.103753	+	0.994603i	$0.533085\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.43661e7	+	4.22034e7i	−0.469527	+	0.813244i	−0.999393	−	0.0348373i	$-0.988909\pi$
$373$	−2.43661e7	+	4.22034e7i	−0.469527	+	0.813244i	0.529866	+	0.848081i	$0.322242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.51948e7			−0.646489			−0.323245	−	0.946315i	$-0.604774\pi$
$379$	−3.51948e7			−0.646489			−0.323245	+	0.946315i	$0.604774\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−3.61254e6			−0.0618469
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.23725e8			−1.97737			−0.988684	−	0.150013i	$-0.952068\pi$
$397$	−1.23725e8			−1.97737			−0.988684	+	0.150013i	$0.952068\pi$
$398$		0			0
$399$		0			0
$400$	−3.20000e7	+	5.54256e7i	−0.500000	+	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−8.92635e6	−	1.54609e7i	−0.136382	−	0.236221i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.44197e7	−	5.96166e7i	0.503080	−	0.871360i	−0.496914	−	0.867800i	$-0.665534\pi$
$409$	3.44197e7	−	5.96166e7i	0.503080	−	0.871360i	0.999994	−	0.00355979i	$-0.00113312\pi$
$410$		0			0
$411$		0			0
$412$	−3.60623e7	−	6.24617e7i	−0.515658	−	0.893145i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−7.22371e7	−	1.25118e8i	−0.968086	−	1.67677i	−0.701086	−	0.713077i	$-0.747301\pi$
$421$	−7.22371e7	−	1.25118e8i	−0.968086	−	1.67677i	−0.267000	−	0.963697i	$-0.586032\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.01798e7	+	1.04235e8i	−0.772978	+	1.33884i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−1.55657e8			−1.91737			−0.958685	−	0.284469i	$-0.908183\pi$
$433$	−1.55657e8			−1.91737			−0.958685	+	0.284469i	$0.908183\pi$
$434$		0			0
$435$		0			0
$436$	6.95277e7	−	1.20426e8i	0.838878	−	1.45298i
$437$		0			0
$438$		0			0
$439$	−2.67584e7	−	4.63469e7i	−0.316276	−	0.547806i	0.663432	−	0.748237i	$-0.269099\pi$
$439$	−2.67584e7	−	4.63469e7i	−0.316276	−	0.547806i	−0.979708	+	0.200431i	$0.935766\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.74866e7	+	6.49287e7i	0.416910	+	0.722109i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.46719e7	+	2.54126e7i	0.153723	+	0.266256i	0.932593	−	0.360929i	$-0.117540\pi$
$457$	1.46719e7	+	2.54126e7i	0.153723	+	0.266256i	−0.778870	+	0.627185i	$0.784207\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	−9.63714e7	+	1.66920e8i	−0.970968	+	1.68177i	−0.278323	+	0.960487i	$0.589779\pi$
$463$	−9.63714e7	+	1.66920e8i	−0.970968	+	1.68177i	−0.692645	+	0.721279i	$0.743555\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−4.94420e7			−0.479267
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	8.26719e7	+	1.43192e8i	0.771395	+	1.33610i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	2.25691e7	−	3.90909e7i	0.202805	−	0.351269i
$482$		0			0
$483$		0			0
$484$	−5.66900e7	−	9.81899e7i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	2.05807e8			1.78186			0.890931	−	0.454139i	$-0.150053\pi$
$487$	2.05807e8			1.78186			0.890931	+	0.454139i	$0.150053\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.44515e8			1.18432
$497$		0			0
$498$		0			0
$499$	9.70225e6	−	1.68048e7i	0.0780856	−	0.135248i	−0.824338	−	0.566097i	$-0.808453\pi$
$499$	9.70225e6	−	1.68048e7i	0.0780856	−	0.135248i	0.902424	+	0.430849i	$0.141786\pi$
$500$		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
$506$		0			0
$507$		0			0
$508$	1.26476e8	−	2.19062e8i	0.964753	−	1.67100i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	9.12434e7	+	1.58038e8i	0.683815	+	1.18440i
$512$		0			0
$513$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−1.41150e8			−0.986677			−0.493338	−	0.869837i	$-0.664224\pi$
$523$	−1.41150e8			−0.986677			−0.493338	+	0.869837i	$0.664224\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.40179e7	−	1.28203e8i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	1.93693e8			1.28641
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.97611e8			1.24801			0.624006	−	0.781419i	$-0.285504\pi$
$541$	1.97611e8			1.24801			0.624006	+	0.781419i	$0.285504\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.62323e8	+	2.81151e8i	0.991783	+	1.71782i	0.606681	+	0.794945i	$0.292500\pi$
$547$	1.62323e8	+	2.81151e8i	0.991783	+	1.71782i	0.385102	+	0.922874i	$0.374166\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.92609e7	+	5.06814e7i	−0.173027	+	0.299691i
$554$		0			0
$555$		0			0
$556$	−4.04652e7	−	7.00878e7i	−0.235428	−	0.407773i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	5.63613e7			0.322660
$560$		0			0
$561$		0			0
$562$		0			0
$563$		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
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−8.55553e7	+	1.48186e8i	−0.459556	+	0.795974i	−0.998937	−	0.0460873i	$-0.985325\pi$
$571$	−8.55553e7	+	1.48186e8i	−0.459556	+	0.795974i	0.539381	+	0.842062i	$0.318658\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.05560e7			−0.367288			−0.183644	−	0.982993i	$-0.558789\pi$
$577$	−7.05560e7			−0.367288			−0.183644	+	0.982993i	$0.558789\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	1.86677e8	−	3.23334e8i	0.913577	−	1.58236i
$590$		0			0
$591$		0			0
$592$	1.82694e8	+	3.16435e8i	0.880560	+	1.52517i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−2.12222e8	−	3.67579e8i	−0.977612	−	1.69327i	−0.671031	−	0.741429i	$-0.734148\pi$
$601$	−2.12222e8	−	3.67579e8i	−0.977612	−	1.69327i	−0.306581	−	0.951845i	$-0.599185\pi$
$602$		0			0
$603$		0			0
$604$	−2.45145e8			−1.11253
$605$		0			0
$606$		0			0
$607$	−1.80200e8	+	3.12115e8i	−0.805727	+	1.39556i	0.110072	+	0.993924i	$0.464892\pi$
$607$	−1.80200e8	+	3.12115e8i	−0.805727	+	1.39556i	−0.915799	+	0.401636i	$0.868442\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.58281e8			−0.687142			−0.343571	−	0.939127i	$-0.611637\pi$
$613$	−1.58281e8			−0.687142			−0.343571	+	0.939127i	$0.611637\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−1.18095e8	−	2.04546e8i	−0.497918	−	0.862419i	0.502079	−	0.864822i	$-0.332569\pi$
$619$	−1.18095e8	−	2.04546e8i	−0.497918	−	0.862419i	−0.999997	+	0.00240251i	$0.999235\pi$
$620$		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
$628$	−2.26647e8	−	3.92564e8i	−0.915105	−	1.58501i
$629$		0			0
$630$		0			0
$631$	−4.98900e8			−1.98575			−0.992876	−	0.119152i	$-0.961982\pi$
$631$	−4.98900e8			−1.98575			−0.992876	+	0.119152i	$0.961982\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	9.07081e6	+	1.57111e7i	0.0350936	+	0.0607839i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	1.79255e8	−	3.10479e8i	0.674277	−	1.16788i	−0.302403	−	0.953180i	$-0.597789\pi$
$643$	1.79255e8	−	3.10479e8i	0.674277	−	1.16788i	0.976680	−	0.214701i	$-0.0688778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−9.27522e7	+	1.60652e8i	−0.334643	+	0.579618i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	7.90486e7	−	1.36916e8i	0.273710	−	0.474079i	−0.696099	−	0.717946i	$-0.745083\pi$
$661$	7.90486e7	−	1.36916e8i	0.273710	−	0.474079i	0.969809	+	0.243867i	$0.0784160\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.56200e7	+	2.70546e7i	0.0512430	+	0.0887556i	0.890509	−	0.454965i	$-0.150348\pi$
$673$	1.56200e7	+	2.70546e7i	0.0512430	+	0.0887556i	−0.839266	+	0.543721i	$0.817015\pi$
$674$		0			0
$675$		0			0
$676$	−2.92529e8			−0.946955
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−8.07178e6	+	1.39807e7i	−0.0257846	+	0.0446602i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−2.28119e8	+	3.95113e8i	−0.700479	+	1.21327i
$689$		0			0
$690$		0			0
$691$	2.00785e8	+	3.47769e8i	0.608551	+	1.05404i	0.991480	+	0.130263i	$0.0415821\pi$
$691$	2.00785e8	+	3.47769e8i	0.608551	+	1.05404i	−0.382929	+	0.923778i	$0.625085\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.43000e8	+	2.47683e8i	0.416910	+	0.722109i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	9.43978e8			2.71704
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−2.96866e8	−	5.14187e8i	−0.832955	−	1.44272i	−0.895685	−	0.444690i	$-0.853314\pi$
$709$	−2.96866e8	−	5.14187e8i	−0.832955	−	1.44272i	0.0627298	−	0.998031i	$-0.480019\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−3.22307e8			−0.859930
$722$		0			0
$723$		0			0
$724$	−3.14502e6	+	5.44734e6i	−0.00828721	+	0.0143539i
$725$		0			0
$726$		0			0
$727$	3.26136e8	+	5.64885e8i	0.848782	+	1.47013i	0.882296	+	0.470695i	$0.155997\pi$
$727$	3.26136e8	+	5.64885e8i	0.848782	+	1.47013i	−0.0335144	+	0.999438i	$0.510670\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2.80582e8	−	4.85982e8i	0.712438	−	1.23398i	−0.251501	−	0.967857i	$-0.580924\pi$
$733$	2.80582e8	−	4.85982e8i	0.712438	−	1.23398i	0.963939	−	0.266122i	$-0.0857425\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.77287e8			0.439281			0.219641	−	0.975581i	$-0.429512\pi$
$739$	1.77287e8			0.439281			0.219641	+	0.975581i	$0.429512\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.48567e8	−	2.57325e8i	0.350753	−	0.607522i	−0.635629	−	0.771995i	$-0.719259\pi$
$751$	1.48567e8	−	2.57325e8i	0.350753	−	0.607522i	0.986382	+	0.164473i	$0.0525923\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	8.52165e8			1.96443			0.982214	−	0.187767i	$-0.0601251\pi$
$757$	8.52165e8			1.96443			0.982214	+	0.187767i	$0.0601251\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−3.10702e8	−	5.38152e8i	−0.699472	−	1.21152i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	4.44149e8	−	7.69289e8i	0.976674	−	1.69165i	0.302378	−	0.953188i	$-0.402220\pi$
$769$	4.44149e8	−	7.69289e8i	0.976674	−	1.69165i	0.674296	−	0.738461i	$-0.264447\pi$
$770$		0			0
$771$		0			0
$772$	4.17780e8	+	7.23616e8i	0.908020	+	1.57274i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	5.51281e8			1.18432
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.46854e8			−0.304745
$785$		0			0
$786$		0			0
$787$	−4.54337e8	+	7.86934e8i	−0.932081	+	1.61441i	−0.152322	+	0.988331i	$0.548675\pi$
$787$	−4.54337e8	+	7.86934e8i	−0.932081	+	1.61441i	−0.779759	+	0.626080i	$0.784658\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.12944e8			−0.427018
$794$		0			0
$795$		0			0
$796$	−3.72944e8	+	6.45958e8i	−0.739442	+	1.28075i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	4.83016e8			0.905522			0.452761	−	0.891632i	$-0.350439\pi$
$811$	4.83016e8			0.905522			0.452761	+	0.891632i	$0.350439\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	5.89343e8	+	1.02077e9i	1.08069	+	1.87181i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	4.82778e8	−	8.36195e8i	0.866059	−	1.50006i	6.75828e−5	−	1.00000i	$-0.499978\pi$
$823$	4.82778e8	−	8.36195e8i	0.866059	−	1.50006i	0.865992	−	0.500059i	$-0.166688\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−8.48197e8			−1.48879			−0.744395	−	0.667740i	$-0.767262\pi$
$829$	−8.48197e8			−1.48879			−0.744395	+	0.667740i	$0.767262\pi$
$830$		0			0
$831$		0			0
$832$	−6.63224e7	+	1.14874e8i	−0.115157	+	0.199458i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−2.97412e8	+	5.15132e8i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−5.63109e8	−	9.75334e8i	−0.936624	−	1.62228i
$845$		0			0
$846$		0			0
$847$	−5.06666e8			−0.833819
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−4.69328e8	−	8.12899e8i	−0.756187	−	1.30975i	−0.944782	−	0.327699i	$-0.893727\pi$
$853$	−4.69328e8	−	8.12899e8i	−0.756187	−	1.30975i	0.188596	−	0.982055i	$-0.439606\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	1.43870e8	−	2.49189e8i	0.226981	−	0.393143i	−0.729931	−	0.683521i	$-0.760448\pi$
$859$	1.43870e8	−	2.49189e8i	0.226981	−	0.393143i	0.956912	+	0.290378i	$0.0937811\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	3.22901e8	−	5.59281e8i	0.493753	−	0.855206i
$869$		0			0
$870$		0			0
$871$	−4.37371e7	−	7.57549e7i	−0.0661905	−	0.114645i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.82985e8	+	1.00976e9i	−0.864288	+	1.49699i	0.00346360	+	0.999994i	$0.498897\pi$
$877$	−5.82985e8	+	1.00976e9i	−0.864288	+	1.49699i	−0.867752	+	0.496997i	$0.834436\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−9.49268e8			−1.37882			−0.689409	−	0.724372i	$-0.742130\pi$
$883$	−9.49268e8			−1.37882			−0.689409	+	0.724372i	$0.742130\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−5.65188e8	−	9.78935e8i	−0.804429	−	1.39331i
$890$		0			0
$891$		0			0
$892$	−7.55887e8			−1.06503
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.59170e8	−	4.48896e8i	−0.347347	−	0.601622i	0.638431	−	0.769679i	$-0.279584\pi$
$907$	−2.59170e8	−	4.48896e8i	−0.347347	−	0.601622i	−0.985777	+	0.168057i	$0.946251\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.30330e8	+	2.25739e8i	0.169574	+	0.293711i
$917$		0			0
$918$		0			0
$919$	−1.54471e9			−1.99021			−0.995107	−	0.0988039i	$-0.968498\pi$
$919$	−1.54471e9			−1.99021			−0.995107	+	0.0988039i	$0.968498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	6.96922e8	+	1.20710e9i	0.880560	+	1.52517i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−1.89698e8	+	3.28567e8i	−0.235079	+	0.407169i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.43605e9			1.74562			0.872810	−	0.488060i	$-0.162295\pi$
$937$	1.43605e9			1.74562			0.872810	+	0.488060i	$0.162295\pi$
$938$		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
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−1.61431e8	+	2.79606e8i	−0.188881	+	0.327151i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.78658e8	−	3.09445e8i	−0.201304	−	0.348669i
$962$		0			0
$963$		0			0
$964$	1.69215e9			1.88889
$965$		0			0
$966$		0			0
$967$	−3.46769e8	+	6.00622e8i	−0.383496	+	0.664235i	−0.991559	−	0.129654i	$-0.958613\pi$
$967$	−3.46769e8	+	6.00622e8i	−0.383496	+	0.664235i	0.608063	+	0.793889i	$0.291947\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−3.61658e8			−0.392608
$974$		0			0
$975$		0			0
$976$	8.61876e8	−	1.49281e9i	0.927034	−	1.60567i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.71344e8	+	2.96776e8i	0.177663	+	0.307722i
$989$		0			0
$990$		0			0
$991$	1.35430e8			0.139153			0.0695766	−	0.997577i	$-0.477835\pi$
$991$	1.35430e8			0.139153			0.0695766	+	0.997577i	$0.477835\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.73133e8	+	9.92695e8i	0.578322	+	1.00168i	0.995672	+	0.0929375i	$0.0296257\pi$
$997$	5.73133e8	+	9.92695e8i	0.578322	+	1.00168i	−0.417350	+	0.908746i	$0.637041\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.7.d.a.53.1		2
3.2	odd	2	CM	81.7.d.a.53.1		2
9.2	odd	6	inner	81.7.d.a.26.1		2
9.4	even	3		3.7.b.a.2.1	✓	1
9.5	odd	6		3.7.b.a.2.1	✓	1
9.7	even	3	inner	81.7.d.a.26.1		2
36.23	even	6		48.7.e.a.17.1		1
36.31	odd	6		48.7.e.a.17.1		1
45.4	even	6		75.7.c.a.26.1		1
45.13	odd	12		75.7.d.a.74.1		2
45.14	odd	6		75.7.c.a.26.1		1
45.22	odd	12		75.7.d.a.74.2		2
45.23	even	12		75.7.d.a.74.1		2
45.32	even	12		75.7.d.a.74.2		2
63.13	odd	6		147.7.b.a.50.1		1
63.41	even	6		147.7.b.a.50.1		1
72.5	odd	6		192.7.e.b.65.1		1
72.13	even	6		192.7.e.b.65.1		1
72.59	even	6		192.7.e.a.65.1		1
72.67	odd	6		192.7.e.a.65.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.7.b.a.2.1	✓	1	9.4	even	3
3.7.b.a.2.1	✓	1	9.5	odd	6
48.7.e.a.17.1		1	36.23	even	6
48.7.e.a.17.1		1	36.31	odd	6
75.7.c.a.26.1		1	45.4	even	6
75.7.c.a.26.1		1	45.14	odd	6
75.7.d.a.74.1		2	45.13	odd	12
75.7.d.a.74.1		2	45.23	even	12
75.7.d.a.74.2		2	45.22	odd	12
75.7.d.a.74.2		2	45.32	even	12
81.7.d.a.26.1		2	9.2	odd	6	inner
81.7.d.a.26.1		2	9.7	even	3	inner
81.7.d.a.53.1		2	1.1	even	1	trivial
81.7.d.a.53.1		2	3.2	odd	2	CM
147.7.b.a.50.1		1	63.13	odd	6
147.7.b.a.50.1		1	63.41	even	6
192.7.e.a.65.1		1	72.59	even	6
192.7.e.a.65.1		1	72.67	odd	6
192.7.e.b.65.1		1	72.5	odd	6
192.7.e.b.65.1		1	72.13	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-32.0000 + 55.4256i) q^{4} +(143.000 + 247.683i) q^{7} +O(q^{10})\)	\(q+(-32.0000 + 55.4256i) q^{4} +(143.000 + 247.683i) q^{7} +(-253.000 + 438.209i) q^{13} +(-2048.00 - 3547.24i) q^{16} -10582.0 q^{19} +(-7812.50 - 13531.6i) q^{25} -18304.0 q^{28} +(-17641.0 + 30555.1i) q^{31} -89206.0 q^{37} +(-55693.0 - 96463.1i) q^{43} +(17926.5 - 31049.6i) q^{49} +(-16192.0 - 28045.4i) q^{52} +(210419. + 364456. i) q^{61} +262144. q^{64} +(-86437.0 + 149713. i) q^{67} +638066. q^{73} +(338624. - 586514. i) q^{76} +(102311. + 177208. i) q^{79} -144716. q^{91} +(28223.0 + 48883.7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{4} + 286 q^{7}+O(q^{10}) \) 2 * q - 64 * q^4 + 286 * q^7	\( 2 q - 64 q^{4} + 286 q^{7} - 506 q^{13} - 4096 q^{16} - 21164 q^{19} - 15625 q^{25} - 36608 q^{28} - 35282 q^{31} - 178412 q^{37} - 111386 q^{43} + 35853 q^{49} - 32384 q^{52} + 420838 q^{61} + 524288 q^{64} - 172874 q^{67} + 1276132 q^{73} + 677248 q^{76} + 204622 q^{79} - 289432 q^{91} + 56446 q^{97}+O(q^{100}) \) 2 * q - 64 * q^4 + 286 * q^7 - 506 * q^13 - 4096 * q^16 - 21164 * q^19 - 15625 * q^25 - 36608 * q^28 - 35282 * q^31 - 178412 * q^37 - 111386 * q^43 + 35853 * q^49 - 32384 * q^52 + 420838 * q^61 + 524288 * q^64 - 172874 * q^67 + 1276132 * q^73 + 677248 * q^76 + 204622 * q^79 - 289432 * q^91 + 56446 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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