LMFDB - Embedded newform 84.5.p.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,5,Mod(53,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	84.p (of order $6$, degree $2$, 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:	$8.68307689904$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			53.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.53
Dual form			84.5.p.a.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{3} +(11.5000 + 47.6314i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	$q+(-4.50000 + 7.79423i) q^{3} +(11.5000 + 47.6314i) q^{7} +(-40.5000 - 70.1481i) q^{9} -337.000 q^{13} +(-323.500 - 560.318i) q^{19} +(-423.000 - 124.708i) q^{21} +(-312.500 + 541.266i) q^{25} +729.000 q^{27} +(-779.500 + 1350.13i) q^{31} +(264.500 + 458.127i) q^{37} +(1516.50 - 2626.66i) q^{39} +3191.00 q^{43} +(-2136.50 + 1095.52i) q^{49} +5823.00 q^{57} +(983.000 + 1702.61i) q^{61} +(2875.50 - 2735.77i) q^{63} +(-1451.50 + 2514.07i) q^{67} +(-4895.50 + 8479.25i) q^{73} +(-2812.50 - 4871.39i) q^{75} +(-2339.50 - 4052.13i) q^{79} +(-3280.50 + 5681.99i) q^{81} +(-3875.50 - 16051.8i) q^{91} +(-7015.50 - 12151.2i) q^{93} -18814.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} + 23 q^{7} - 81 q^{9}+O(q^{10}) $ 2 * q - 9 * q^3 + 23 * q^7 - 81 * q^9	$ 2 q - 9 q^{3} + 23 q^{7} - 81 q^{9} - 674 q^{13} - 647 q^{19} - 846 q^{21} - 625 q^{25} + 1458 q^{27} - 1559 q^{31} + 529 q^{37} + 3033 q^{39} + 6382 q^{43} - 4273 q^{49} + 11646 q^{57} + 1966 q^{61} + 5751 q^{63} - 2903 q^{67} - 9791 q^{73} - 5625 q^{75} - 4679 q^{79} - 6561 q^{81} - 7751 q^{91} - 14031 q^{93} - 37628 q^{97}+O(q^{100}) $ 2 * q - 9 * q^3 + 23 * q^7 - 81 * q^9 - 674 * q^13 - 647 * q^19 - 846 * q^21 - 625 * q^25 + 1458 * q^27 - 1559 * q^31 + 529 * q^37 + 3033 * q^39 + 6382 * q^43 - 4273 * q^49 + 11646 * q^57 + 1966 * q^61 + 5751 * q^63 - 2903 * q^67 - 9791 * q^73 - 5625 * q^75 - 4679 * q^79 - 6561 * q^81 - 7751 * q^91 - 14031 * q^93 - 37628 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$3$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	11.5000	+	47.6314i	0.234694	+	0.972069i
$7$	11.5000	+	47.6314i	0.234694	+	0.972069i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−337.000			−1.99408			−0.997041	−	0.0768662i	$-0.975509\pi$
$13$	−337.000			−1.99408			−0.997041	+	0.0768662i	$0.975509\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−323.500	−	560.318i	−0.896122	−	1.55213i	−0.832410	−	0.554160i	$-0.813039\pi$
$19$	−323.500	−	560.318i	−0.896122	−	1.55213i	−0.0637119	−	0.997968i	$-0.520294\pi$
$20$		0			0
$21$	−423.000	−	124.708i	−0.959184	−	0.282784i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−312.500	+	541.266i	−0.500000	+	0.866025i
$26$		0			0
$27$	729.000			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−779.500	+	1350.13i	−0.811134	+	1.40493i	0.100937	+	0.994893i	$0.467816\pi$
$31$	−779.500	+	1350.13i	−0.811134	+	1.40493i	−0.912071	+	0.410033i	$0.865517\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	264.500	+	458.127i	0.193207	+	0.334644i	0.946311	−	0.323257i	$-0.104778\pi$
$37$	264.500	+	458.127i	0.193207	+	0.334644i	−0.753104	+	0.657901i	$0.771445\pi$
$38$		0			0
$39$	1516.50	−	2626.66i	0.997041	−	1.72693i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	3191.00			1.72580			0.862899	−	0.505377i	$-0.168646\pi$
$43$	3191.00			1.72580			0.862899	+	0.505377i	$0.168646\pi$
$44$		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
$49$	−2136.50	+	1095.52i	−0.889838	+	0.456277i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5823.00			1.79224
$58$		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
$61$	983.000	+	1702.61i	0.264176	+	0.457567i	0.967347	−	0.253454i	$-0.0815666\pi$
$61$	983.000	+	1702.61i	0.264176	+	0.457567i	−0.703171	+	0.711021i	$0.748233\pi$
$62$		0			0
$63$	2875.50	−	2735.77i	0.724490	−	0.689286i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1451.50	+	2514.07i	−0.323346	+	0.560052i	−0.981176	−	0.193115i	$-0.938141\pi$
$67$	−1451.50	+	2514.07i	−0.323346	+	0.560052i	0.657830	+	0.753166i	$0.271474\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−4895.50	+	8479.25i	−0.918653	+	1.59115i	−0.117189	+	0.993110i	$0.537388\pi$
$73$	−4895.50	+	8479.25i	−0.918653	+	1.59115i	−0.801464	+	0.598043i	$0.795945\pi$
$74$		0			0
$75$	−2812.50	−	4871.39i	−0.500000	−	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2339.50	−	4052.13i	−0.374860	−	0.649276i	0.615446	−	0.788179i	$-0.288976\pi$
$79$	−2339.50	−	4052.13i	−0.374860	−	0.649276i	−0.990306	+	0.138903i	$0.955642\pi$
$80$		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
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	−3875.50	−	16051.8i	−0.467999	−	1.93839i
$92$		0			0
$93$	−7015.50	−	12151.2i	−0.811134	−	1.40493i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18814.0			−1.99957			−0.999787	−	0.0206175i	$-0.993437\pi$
$97$	−18814.0			−1.99957			−0.999787	+	0.0206175i	$0.993437\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	9924.50	+	17189.7i	0.935479	+	1.62030i	0.773777	+	0.633458i	$0.218365\pi$
$103$	9924.50	+	17189.7i	0.935479	+	1.62030i	0.161702	+	0.986840i	$0.448302\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	1656.50	−	2869.14i	0.139424	−	0.241490i	−0.787855	−	0.615861i	$-0.788808\pi$
$109$	1656.50	−	2869.14i	0.139424	−	0.241490i	0.927279	+	0.374371i	$0.122141\pi$
$110$		0			0
$111$	−4761.00			−0.386413
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	13648.5	+	23639.9i	0.997041	+	1.72693i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7320.50	−	12679.5i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	31751.0			1.96857			0.984283	−	0.176599i	$-0.0565094\pi$
$127$	31751.0			1.96857			0.984283	+	0.176599i	$0.0565094\pi$
$128$		0			0
$129$	−14359.5	+	24871.4i	−0.862899	+	1.49458i
$130$		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
$133$	22968.5	−	21852.4i	1.29846	−	1.23537i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	24359.0			1.26075			0.630376	−	0.776290i	$-0.282901\pi$
$139$	24359.0			1.26075			0.630376	+	0.776290i	$0.282901\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1075.50	−	21582.2i	0.0497709	−	0.998761i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−18097.0	+	31344.9i	−0.793693	+	1.37472i	0.129972	+	0.991518i	$0.458511\pi$
$151$	−18097.0	+	31344.9i	−0.793693	+	1.37472i	−0.923666	+	0.383199i	$0.874822\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17687.0	−	30634.8i	0.717554	−	1.24284i	−0.244412	−	0.969672i	$-0.578595\pi$
$157$	17687.0	−	30634.8i	0.717554	−	1.24284i	0.961966	−	0.273169i	$-0.0880719\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7753.00	−	13428.6i	−0.291806	−	0.505423i	0.682431	−	0.730950i	$-0.260923\pi$
$163$	−7753.00	−	13428.6i	−0.291806	−	0.505423i	−0.974237	+	0.225527i	$0.927590\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	85008.0			2.97637
$170$		0			0
$171$	−26203.5	+	45385.8i	−0.896122	+	1.55213i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−29375.0	−	8660.25i	−0.959184	−	0.282784i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	−65521.0			−1.99997			−0.999985	−	0.00552484i	$-0.998241\pi$
$181$	−65521.0			−1.99997			−0.999985	+	0.00552484i	$0.998241\pi$
$182$		0			0
$183$	−17694.0			−0.528353
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8383.50	+	34723.3i	0.234694	+	0.972069i
$190$		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
$193$	27024.5	−	46807.8i	0.725509	−	1.25662i	−0.233255	−	0.972416i	$-0.574938\pi$
$193$	27024.5	−	46807.8i	0.725509	−	1.25662i	0.958764	−	0.284203i	$-0.0917291\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−34897.0	+	60443.4i	−0.881215	+	1.52631i	−0.0312240	+	0.999512i	$0.509941\pi$
$199$	−34897.0	+	60443.4i	−0.881215	+	1.52631i	−0.849991	+	0.526797i	$0.823393\pi$
$200$		0			0
$201$	−13063.5	−	22626.6i	−0.323346	−	0.560052i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−61486.0			−1.38106			−0.690528	−	0.723306i	$-0.742622\pi$
$211$	−61486.0			−1.38106			−0.690528	+	0.723306i	$0.742622\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−73273.0	−	21602.1i	−1.55605	−	0.458751i
$218$		0			0
$219$	−44059.5	−	76313.3i	−0.918653	−	1.59115i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	14786.0			0.297332			0.148666	−	0.988888i	$-0.452502\pi$
$223$	14786.0			0.297332			0.148666	+	0.988888i	$0.452502\pi$
$224$		0			0
$225$	50625.0			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−20903.5	−	36205.9i	−0.398610	−	0.690413i	0.594945	−	0.803767i	$-0.297174\pi$
$229$	−20903.5	−	36205.9i	−0.398610	−	0.690413i	−0.993555	+	0.113354i	$0.963841\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	42111.0			0.749720
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	17183.0	−	29761.8i	0.295845	−	0.512419i	−0.679336	−	0.733828i	$-0.737732\pi$
$241$	17183.0	−	29761.8i	0.295845	−	0.512419i	0.975181	+	0.221408i	$0.0710653\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.500000	−	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	109020.	+	188827.i	1.78694	+	3.09507i
$248$		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
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	−18779.5	+	17867.0i	−0.279953	+	0.266349i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	44159.0	+	76485.6i	0.601285	+	1.04146i	0.992627	+	0.121211i	$0.0386778\pi$
$271$	44159.0	+	76485.6i	0.601285	+	1.04146i	−0.391341	+	0.920246i	$0.627989\pi$
$272$		0			0
$273$	142551.	+	42026.5i	1.91269	+	0.563894i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−63191.5	+	109451.i	−0.823567	+	1.42646i	0.0794419	+	0.996839i	$0.474686\pi$
$277$	−63191.5	+	109451.i	−0.823567	+	1.42646i	−0.903009	+	0.429621i	$0.858647\pi$
$278$		0			0
$279$	126279.			1.62227
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	78348.5	−	135704.i	0.978268	−	1.69441i	0.309568	−	0.950877i	$-0.399816\pi$
$283$	78348.5	−	135704.i	0.978268	−	1.69441i	0.668700	−	0.743532i	$-0.266851\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−41760.5	−	72331.3i	−0.500000	−	0.866025i
$290$		0			0
$291$	84663.0	−	146641.i	0.999787	−	1.73168i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	36696.5	+	151992.i	0.405034	+	1.67760i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−124489.			−1.32085			−0.660426	−	0.750891i	$-0.729624\pi$
$307$	−124489.			−1.32085			−0.660426	+	0.750891i	$0.729624\pi$
$308$		0			0
$309$	−178641.			−1.87096
$310$		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
$313$	−6455.50	−	11181.3i	−0.0658933	−	0.114131i	0.831197	−	0.555979i	$-0.187656\pi$
$313$	−6455.50	−	11181.3i	−0.0658933	−	0.114131i	−0.897090	+	0.441848i	$0.854323\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		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
$325$	105312.	−	182407.i	0.997041	−	1.72693i
$326$		0			0
$327$	14908.5	+	25822.3i	0.139424	+	0.241490i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	64860.5	+	112342.i	0.592004	+	1.02538i	0.993962	+	0.109722i	$0.0349962\pi$
$331$	64860.5	+	112342.i	0.592004	+	1.02538i	−0.401959	+	0.915658i	$0.631670\pi$
$332$		0			0
$333$	21424.5	−	37108.3i	0.193207	−	0.334644i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−199249.			−1.75443			−0.877216	−	0.480097i	$-0.840602\pi$
$337$	−199249.			−1.75443			−0.877216	+	0.480097i	$0.840602\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−76751.0	−	89166.0i	−0.652373	−	0.757898i
$344$		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
$349$	8402.00			0.0689814			0.0344907	−	0.999405i	$-0.489019\pi$
$349$	8402.00			0.0689814			0.0344907	+	0.999405i	$0.489019\pi$
$350$		0			0
$351$	−245673.			−1.99408
$352$		0			0
$353$		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
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−144144.	+	249665.i	−1.10607	+	1.91577i
$362$		0			0
$363$	131769.			1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$	108924.	−	188663.i	0.808711	−	1.40073i	−0.105046	−	0.994467i	$-0.533499\pi$
$367$	108924.	−	188663.i	0.808711	−	1.40073i	0.913757	−	0.406261i	$-0.133168\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	131808.	+	228299.i	0.947383	+	1.64092i	0.750907	+	0.660407i	$0.229616\pi$
$373$	131808.	+	228299.i	0.947383	+	1.64092i	0.196476	+	0.980509i	$0.437050\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−280393.			−1.95204			−0.976020	−	0.217681i	$-0.930151\pi$
$379$	−280393.			−1.95204			−0.976020	+	0.217681i	$0.930151\pi$
$380$		0			0
$381$	−142880.	+	247475.i	−0.984283	+	1.70483i
$382$		0			0
$383$		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
$386$		0			0
$387$	−129236.	−	223842.i	−0.862899	−	1.49458i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−156816.	−	271612.i	−0.994965	−	1.72333i	−0.584275	−	0.811556i	$-0.698621\pi$
$397$	−156816.	−	271612.i	−0.994965	−	1.72333i	−0.410690	−	0.911775i	$-0.634712\pi$
$398$		0			0
$399$	66964.5	+	277358.i	0.420629	+	1.74219i
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	262692.	−	454995.i	1.61747	−	2.80154i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	157056.	−	272030.i	0.938878	−	1.62618i	0.171311	−	0.985217i	$-0.445200\pi$
$409$	157056.	−	272030.i	0.938878	−	1.62618i	0.767568	−	0.640968i	$-0.221467\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−109616.	+	189860.i	−0.630376	+	1.09184i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	349439.			1.97155			0.985774	−	0.168079i	$-0.0537562\pi$
$421$	349439.			1.97155			0.985774	+	0.168079i	$0.0537562\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−69793.0	+	66401.6i	−0.382786	+	0.364186i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−121969.			−0.650539			−0.325270	−	0.945621i	$-0.605455\pi$
$433$	−121969.			−0.650539			−0.325270	+	0.945621i	$0.605455\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	188303.	+	326150.i	0.977076	+	1.69234i	0.672908	+	0.739726i	$0.265045\pi$
$439$	188303.	+	326150.i	0.977076	+	1.69234i	0.304168	+	0.952619i	$0.401622\pi$
$440$		0			0
$441$	163377.	+	105503.i	0.840067	+	0.542483i
$442$		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
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−162873.	−	282104.i	−0.793693	−	1.37472i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	207744.	+	359824.i	0.994711	+	1.72289i	0.586304	+	0.810091i	$0.300582\pi$
$457$	207744.	+	359824.i	0.994711	+	1.72289i	0.408408	+	0.912800i	$0.366084\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	423191.			1.97412			0.987062	−	0.160339i	$-0.0512587\pi$
$463$	423191.			1.97412			0.987062	+	0.160339i	$0.0512587\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	−136441.	−	40225.1i	−0.620296	−	0.182874i
$470$		0			0
$471$	159183.	+	275713.i	0.717554	+	1.24284i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	404375.			1.79224
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−89136.5	−	154389.i	−0.385270	−	0.667308i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	49284.5	−	85363.3i	0.207803	−	0.359926i	−0.743219	−	0.669048i	$-0.766702\pi$
$487$	49284.5	−	85363.3i	0.207803	−	0.359926i	0.951022	+	0.309122i	$0.100035\pi$
$488$		0			0
$489$	139554.			0.583612
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−135564.	−	234803.i	−0.544430	−	0.942980i	−0.998643	−	0.0520865i	$-0.983413\pi$
$499$	−135564.	−	234803.i	−0.544430	−	0.942980i	0.454213	−	0.890893i	$-0.349920\pi$
$500$		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
$507$	−382536.	+	662572.i	−1.48818	+	2.57761i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−460177.	−	135668.i	−1.76231	−	0.519560i
$512$		0			0
$513$	−235832.	−	408472.i	−0.896122	−	1.55213i
$514$		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	$0.666667\pi$
$522$		0			0
$523$	−48875.5	−	84654.8i	−0.178685	−	0.309491i	0.762745	−	0.646699i	$-0.223851\pi$
$523$	−48875.5	−	84654.8i	−0.178685	−	0.309491i	−0.941430	+	0.337208i	$0.890518\pi$
$524$		0			0
$525$	199688.	−	189984.i	0.724490	−	0.689286i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−139920.	+	242349.i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−21743.5	−	37660.8i	−0.0742908	−	0.128675i	0.826487	−	0.562956i	$-0.190336\pi$
$541$	−21743.5	−	37660.8i	−0.0742908	−	0.128675i	−0.900778	+	0.434281i	$0.857003\pi$
$542$		0			0
$543$	294844.	−	510686.i	0.999985	−	1.73202i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−342382.			−1.14429			−0.572145	−	0.820152i	$-0.693889\pi$
$547$	−342382.			−1.14429			−0.572145	+	0.820152i	$0.693889\pi$
$548$		0			0
$549$	79623.0	−	137911.i	0.264176	−	0.457567i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	166104.	−	158033.i	0.543164	−	0.516771i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	−1.07537e6			−3.44138
$560$		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
$567$	−308367.	−	90911.9i	−0.959184	−	0.282784i
$568$		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
$571$	−243203.	+	421241.i	−0.745929	+	1.29199i	0.203830	+	0.979006i	$0.434661\pi$
$571$	−243203.	+	421241.i	−0.745929	+	1.29199i	−0.949759	+	0.312981i	$0.898672\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−200616.	+	347476.i	−0.602577	+	1.04369i	0.389852	+	0.920878i	$0.372526\pi$
$577$	−200616.	+	347476.i	−0.602577	+	1.04369i	−0.992429	+	0.122817i	$0.960807\pi$
$578$		0			0
$579$	243220.	+	421270.i	0.725509	+	1.25662i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	1.00867e6			2.90750
$590$		0			0
$591$		0			0
$592$		0			0
$593$		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
$596$		0			0
$597$	−314073.	−	543990.i	−0.881215	−	1.52631i
$598$		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
$601$	−269473.			−0.746047			−0.373024	−	0.927822i	$-0.621679\pi$
$601$	−269473.			−0.746047			−0.373024	+	0.927822i	$0.621679\pi$
$602$		0			0
$603$	235143.			0.646692
$604$		0			0
$605$		0			0
$606$		0			0
$607$	298884.	+	517683.i	0.811196	+	1.40503i	0.912027	+	0.410130i	$0.134517\pi$
$607$	298884.	+	517683.i	0.811196	+	1.40503i	−0.100831	+	0.994904i	$0.532150\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−258169.	+	447162.i	−0.687042	+	1.18999i	0.285749	+	0.958305i	$0.407758\pi$
$613$	−258169.	+	447162.i	−0.687042	+	1.18999i	−0.972790	+	0.231687i	$0.925576\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−67139.5	+	116289.i	−0.175225	+	0.303499i	−0.940239	−	0.340515i	$-0.889399\pi$
$619$	−67139.5	+	116289.i	−0.175225	+	0.303499i	0.765014	+	0.644014i	$0.222732\pi$
$620$		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
$631$	−342046.			−0.859065			−0.429532	−	0.903052i	$-0.641322\pi$
$631$	−342046.			−0.859065			−0.429532	+	0.903052i	$0.641322\pi$
$632$		0			0
$633$	276687.	−	479236.i	0.690528	−	1.19603i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	720000.	−	369191.i	1.77441	−	0.909855i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	25031.0			0.0605419			0.0302710	−	0.999542i	$-0.490363\pi$
$643$	25031.0			0.0605419			0.0302710	+	0.999542i	$0.490363\pi$
$644$		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	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	498100.	−	473897.i	1.17532	−	1.11821i
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	793071.			1.83731
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−284280.	+	492387.i	−0.650643	+	1.12695i	0.332324	+	0.943165i	$0.392167\pi$
$661$	−284280.	+	492387.i	−0.650643	+	1.12695i	−0.982967	+	0.183781i	$0.941166\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−66537.0	+	115245.i	−0.148666	+	0.257497i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	479471.			1.05860			0.529300	−	0.848434i	$-0.322454\pi$
$673$	479471.			1.05860			0.529300	+	0.848434i	$0.322454\pi$
$674$		0			0
$675$	−227812.	+	394583.i	−0.500000	+	0.866025i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−216361.	−	896137.i	−0.469288	−	1.94373i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	376263.			0.797220
$688$		0			0
$689$		0			0
$690$		0			0
$691$	432780.	+	749598.i	0.906383	+	1.56990i	0.819050	+	0.573722i	$0.194501\pi$
$691$	432780.	+	749598.i	0.906383	+	1.56990i	0.0873323	+	0.996179i	$0.472166\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	171132.	−	296408.i	0.346274	−	0.599763i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	66503.0	+	115187.i	0.132297	+	0.229144i	0.924562	−	0.381033i	$-0.124432\pi$
$709$	66503.0	+	115187.i	0.132297	+	0.229144i	−0.792265	+	0.610177i	$0.791098\pi$
$710$		0			0
$711$	−189500.	+	328223.i	−0.374860	+	0.649276i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−704640.	+	670400.i	−1.35549	+	1.28962i
$722$		0			0
$723$	154647.	+	267856.i	0.295845	+	0.512419i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	984983.			1.86363			0.931815	−	0.362932i	$-0.118224\pi$
$727$	984983.			1.86363			0.931815	+	0.362932i	$0.118224\pi$
$728$		0			0
$729$	531441.			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	463945.	+	803575.i	0.863492	+	1.49561i	0.868537	+	0.495624i	$0.165061\pi$
$733$	463945.	+	803575.i	0.863492	+	1.49561i	−0.00504570	+	0.999987i	$0.501606\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	540180.	−	935620.i	0.989122	−	1.71321i	0.367173	−	0.930153i	$-0.380326\pi$
$739$	540180.	−	935620.i	0.989122	−	1.71321i	0.621949	−	0.783058i	$-0.286341\pi$
$740$		0			0
$741$	−1.96235e6			−3.57388
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−389580.	−	674771.i	−0.690743	−	1.19640i	−0.971595	−	0.236650i	$-0.923950\pi$
$751$	−389580.	−	674771.i	−0.690743	−	1.19640i	0.280852	−	0.959751i	$-0.409383\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−443854.			−0.774548			−0.387274	−	0.921965i	$-0.626583\pi$
$757$	−443854.			−0.774548			−0.387274	+	0.921965i	$0.626583\pi$
$758$		0			0
$759$		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
$763$	155711.	+	45906.3i	0.267467	+	0.0788539i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−732481.			−1.23864			−0.619318	−	0.785140i	$-0.712591\pi$
$769$	−732481.			−1.23864			−0.619318	+	0.785140i	$0.712591\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−487188.	−	843834.i	−0.811134	−	1.40493i
$776$		0			0
$777$	−54751.5	−	226773.i	−0.0906889	−	0.375621i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−600553.	+	1.04019e6i	−0.969621	+	1.67943i	−0.272969	+	0.962023i	$0.588006\pi$
$787$	−600553.	+	1.04019e6i	−0.969621	+	1.67943i	−0.696652	+	0.717409i	$0.745328\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−331271.	−	573778.i	−0.526789	−	0.912426i
$794$		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
$800$		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		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	976754.			1.48506			0.742529	−	0.669814i	$-0.233626\pi$
$811$	976754.			1.48506			0.742529	+	0.669814i	$0.233626\pi$
$812$		0			0
$813$	−794862.			−1.20257
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.03229e6	−	1.78798e6i	−1.54653	−	2.67866i
$818$		0			0
$819$	−969044.	+	921956.i	−1.44469	+	1.37449i
$820$		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
$823$	117647.	−	203771.i	0.173693	−	0.300844i	−0.766015	−	0.642822i	$-0.777763\pi$
$823$	117647.	−	203771.i	0.173693	−	0.300844i	0.939708	+	0.341978i	$0.111097\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	24840.5	−	43025.0i	0.0361453	−	0.0626054i	−0.847387	−	0.530976i	$-0.821825\pi$
$829$	24840.5	−	43025.0i	0.0361453	−	0.0626054i	0.883532	+	0.468371i	$0.155159\pi$
$830$		0			0
$831$	−568724.	−	985058.i	−0.823567	−	1.42646i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−568256.	+	984247.i	−0.811134	+	1.40493i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	707281.			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	519756.	−	494500.i	0.724490	−	0.689286i
$848$		0			0
$849$	705136.	+	1.22133e6i	0.978268	+	1.69441i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−69889.0			−0.0960530			−0.0480265	−	0.998846i	$-0.515293\pi$
$853$	−69889.0			−0.0960530			−0.0480265	+	0.998846i	$0.515293\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−267481.	−	463291.i	−0.362499	−	0.627866i	0.625873	−	0.779925i	$-0.284743\pi$
$859$	−267481.	−	463291.i	−0.362499	−	0.627866i	−0.988371	+	0.152059i	$0.951410\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	751689.			1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$	489156.	−	847242.i	0.644779	−	1.11679i
$872$		0			0
$873$	761967.	+	1.31977e6i	0.999787	+	1.73168i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	590327.	+	1.02248e6i	0.767527	+	1.32940i	0.938900	+	0.344189i	$0.111846\pi$
$877$	590327.	+	1.02248e6i	0.767527	+	1.32940i	−0.171374	+	0.985206i	$0.554821\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	25703.0			0.0329657			0.0164829	−	0.999864i	$-0.494753\pi$
$883$	25703.0			0.0329657			0.0164829	+	0.999864i	$0.494753\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	365136.	+	1.51234e6i	0.462010	+	1.91358i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		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
$903$	−1.34979e6	−	397942.i	−1.65536	−	0.488028i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−232956.	+	403491.i	−0.283177	+	0.490477i	−0.972166	−	0.234295i	$-0.924722\pi$
$907$	−232956.	+	403491.i	−0.283177	+	0.490477i	0.688988	+	0.724773i	$0.258055\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	373140.	+	646298.i	0.441816	+	0.765248i	0.997824	−	0.0659290i	$-0.0210011\pi$
$919$	373140.	+	646298.i	0.441816	+	0.765248i	−0.556008	+	0.831177i	$0.687668\pi$
$920$		0			0
$921$	560200.	−	970296.i	0.660426	−	1.14389i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−330625.			−0.386413
$926$		0			0
$927$	803884.	−	1.39237e6i	0.935479	−	1.62030i
$928$		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
$931$	1.30500e6	+	842719.i	1.50560	+	0.972262i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.65547e6			1.88557			0.942784	−	0.333403i	$-0.108197\pi$
$937$	1.65547e6			1.88557			0.942784	+	0.333403i	$0.108197\pi$
$938$		0			0
$939$	116199.			0.131787
$940$		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
$949$	1.64978e6	−	2.85751e6i	1.83187	−	3.17289i
$950$		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
$961$	−753480.	−	1.30507e6i	−0.815877	−	1.41314i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.80617e6			−1.93155			−0.965774	−	0.259386i	$-0.916480\pi$
$967$	−1.80617e6			−1.93155			−0.965774	+	0.259386i	$0.916480\pi$
$968$		0			0
$969$		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
$973$	280128.	+	1.16025e6i	0.295891	+	1.22554i
$974$		0			0
$975$	947812.	+	1.64166e6i	0.997041	+	1.72693i
$976$		0			0
$977$		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
$981$	−268353.			−0.278849
$982$		0			0
$983$		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
$989$		0			0
$990$		0			0
$991$	−529980.	+	917951.i	−0.539649	+	0.934700i	0.459273	+	0.888295i	$0.348110\pi$
$991$	−529980.	+	917951.i	−0.539649	+	0.934700i	−0.998923	+	0.0464053i	$0.985223\pi$
$992$		0			0
$993$	−1.16749e6			−1.18401
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−111576.	+	193254.i	−0.112248	+	0.194419i	−0.916676	−	0.399631i	$-0.869138\pi$
$997$	−111576.	+	193254.i	−0.112248	+	0.194419i	0.804428	+	0.594050i	$0.202472\pi$
$998$		0			0
$999$	192820.	+	333975.i	0.193207	+	0.334644i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.5.p.a.53.1	✓	2
3.2	odd	2	CM	84.5.p.a.53.1	✓	2
7.2	even	3	inner	84.5.p.a.65.1	yes	2
7.3	odd	6		588.5.c.b.197.1		1
7.4	even	3		588.5.c.c.197.1		1
21.2	odd	6	inner	84.5.p.a.65.1	yes	2
21.11	odd	6		588.5.c.c.197.1		1
21.17	even	6		588.5.c.b.197.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.5.p.a.53.1	✓	2	1.1	even	1	trivial
84.5.p.a.53.1	✓	2	3.2	odd	2	CM
84.5.p.a.65.1	yes	2	7.2	even	3	inner
84.5.p.a.65.1	yes	2	21.2	odd	6	inner
588.5.c.b.197.1		1	7.3	odd	6
588.5.c.b.197.1		1	21.17	even	6
588.5.c.c.197.1		1	7.4	even	3
588.5.c.c.197.1		1	21.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(11.5000 + 47.6314i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{3} +(11.5000 + 47.6314i) q^{7} +(-40.5000 - 70.1481i) q^{9} -337.000 q^{13} +(-323.500 - 560.318i) q^{19} +(-423.000 - 124.708i) q^{21} +(-312.500 + 541.266i) q^{25} +729.000 q^{27} +(-779.500 + 1350.13i) q^{31} +(264.500 + 458.127i) q^{37} +(1516.50 - 2626.66i) q^{39} +3191.00 q^{43} +(-2136.50 + 1095.52i) q^{49} +5823.00 q^{57} +(983.000 + 1702.61i) q^{61} +(2875.50 - 2735.77i) q^{63} +(-1451.50 + 2514.07i) q^{67} +(-4895.50 + 8479.25i) q^{73} +(-2812.50 - 4871.39i) q^{75} +(-2339.50 - 4052.13i) q^{79} +(-3280.50 + 5681.99i) q^{81} +(-3875.50 - 16051.8i) q^{91} +(-7015.50 - 12151.2i) q^{93} -18814.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} + 23 q^{7} - 81 q^{9}+O(q^{10}) \) 2 * q - 9 * q^3 + 23 * q^7 - 81 * q^9	\( 2 q - 9 q^{3} + 23 q^{7} - 81 q^{9} - 674 q^{13} - 647 q^{19} - 846 q^{21} - 625 q^{25} + 1458 q^{27} - 1559 q^{31} + 529 q^{37} + 3033 q^{39} + 6382 q^{43} - 4273 q^{49} + 11646 q^{57} + 1966 q^{61} + 5751 q^{63} - 2903 q^{67} - 9791 q^{73} - 5625 q^{75} - 4679 q^{79} - 6561 q^{81} - 7751 q^{91} - 14031 q^{93} - 37628 q^{97}+O(q^{100}) \) 2 * q - 9 * q^3 + 23 * q^7 - 81 * q^9 - 674 * q^13 - 647 * q^19 - 846 * q^21 - 625 * q^25 + 1458 * q^27 - 1559 * q^31 + 529 * q^37 + 3033 * q^39 + 6382 * q^43 - 4273 * q^49 + 11646 * q^57 + 1966 * q^61 + 5751 * q^63 - 2903 * q^67 - 9791 * q^73 - 5625 * q^75 - 4679 * q^79 - 6561 * q^81 - 7751 * q^91 - 14031 * q^93 - 37628 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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