LMFDB - Embedded newform 84.2.k.c.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,2,Mod(5,84)]

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, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("84.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	84.k (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:	$0.670743376979$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			5.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.5
Dual form			84.2.k.c.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 0.866025i) q^{3} +(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +5.19615i q^{13} +(-7.50000 - 4.33013i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 + 4.33013i) q^{25} -5.19615i q^{27} +(-1.50000 + 0.866025i) q^{31} +(5.50000 - 9.52628i) q^{37} +(4.50000 + 7.79423i) q^{39} +13.0000 q^{43} +(5.50000 - 4.33013i) q^{49} -15.0000 q^{57} +(-6.00000 - 3.46410i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-2.50000 - 4.33013i) q^{67} +(-1.50000 + 0.866025i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-8.50000 + 14.7224i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 - 12.9904i) q^{91} +(-1.50000 + 2.59808i) q^{93} +13.8564i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 5 q^{7} + 3 q^{9} - 15 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{31} + 11 q^{37} + 9 q^{39} + 26 q^{43} + 11 q^{49} - 30 q^{57} - 12 q^{61} - 3 q^{63} - 5 q^{67} - 3 q^{73} + 15 q^{75} - 17 q^{79}+ \cdots - 3 q^{93}+O(q^{100}) $ 2 * q + 3 * q^3 - 5 * q^7 + 3 * q^9 - 15 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^31 + 11 * q^37 + 9 * q^39 + 26 * q^43 + 11 * q^49 - 30 * q^57 - 12 * q^61 - 3 * q^63 - 5 * q^67 - 3 * q^73 + 15 * q^75 - 17 * q^79 - 9 * q^81 - 9 * q^91 - 3 * q^93

Character values

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

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$2$		0			0
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		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
$13$			5.19615i			1.44115i	0.693375	+	0.720577i	$0.256123\pi$
$13$			5.19615i			1.44115i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−7.50000	−	4.33013i	−1.72062	−	0.993399i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−7.50000	−	4.33013i	−1.72062	−	0.993399i	−0.802955	−	0.596040i	$-0.796740\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		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
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$	4.50000	+	7.79423i	0.720577	+	1.24808i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	13.0000			1.98248			0.991241	−	0.132068i	$-0.0421616\pi$
$43$	13.0000			1.98248			0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−15.0000			−1.98680
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−6.00000	−	3.46410i	−0.768221	−	0.443533i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	−6.00000	−	3.46410i	−0.768221	−	0.443533i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	−1.50000	+	7.79423i	−0.188982	+	0.981981i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.50000	+	0.866025i	−0.175562	+	0.101361i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−1.50000	+	0.866025i	−0.175562	+	0.101361i	0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−4.50000	−	12.9904i	−0.471728	−	1.36176i
$92$		0			0
$93$	−1.50000	+	2.59808i	−0.155543	+	0.269408i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	13.5000	+	7.79423i	1.33019	+	0.767988i	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	13.5000	+	7.79423i	1.33019	+	0.767988i	0.344865	+	0.938652i	$0.387925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.0957826	−	0.995402i	$-0.530535\pi$
$110$		0			0
$111$		−	19.0526i		−	1.80839i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	13.5000	+	7.79423i	1.24808	+	0.720577i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$	19.5000	−	11.2583i	1.71688	−	0.991241i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	22.5000	+	4.33013i	1.95100	+	0.375470i
$134$		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
$139$			5.19615i			0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$			5.19615i			0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.50000	−	11.2583i	0.371154	−	0.928571i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−2.00000	−	3.46410i	−0.162758	−	0.281905i	0.773099	−	0.634285i	$-0.218706\pi$
$151$	−2.00000	−	3.46410i	−0.162758	−	0.281905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0000	+	10.3923i	−1.43656	+	0.829396i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−18.0000	+	10.3923i	−1.43656	+	0.829396i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		0			0
$171$	−22.5000	+	12.9904i	−1.72062	+	0.993399i
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	19.0526i		−	1.41617i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$		−	19.0526i		−	1.41617i	0.706129	−	0.708083i	$-0.250440\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.50000	+	12.9904i	0.327327	+	0.944911i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	0.899770	+	0.436365i	$0.143734\pi$
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	3.00000	−	1.73205i	0.212664	−	0.122782i	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	3.00000	−	1.73205i	0.212664	−	0.122782i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$	−7.50000	−	4.33013i	−0.529009	−	0.305424i
$202$		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
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.00000	−	3.46410i	0.203653	−	0.235159i
$218$		0			0
$219$	−1.50000	+	2.59808i	−0.101361	+	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−7.50000	−	4.33013i	−0.495614	−	0.286143i	0.231287	−	0.972886i	$-0.425707\pi$
$229$	−7.50000	−	4.33013i	−0.495614	−	0.286143i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			29.4449i			1.91265i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	24.0000	−	13.8564i	1.54598	−	0.892570i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	24.0000	−	13.8564i	1.54598	−	0.892570i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	22.5000	−	38.9711i	1.43164	−	2.47967i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−5.50000	+	28.5788i	−0.341753	+	1.77580i
$260$		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
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$	−18.0000	−	15.5885i	−1.08941	−	0.943456i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$			5.19615i			0.311086i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	19.5000	−	11.2583i	1.15915	−	0.669238i	0.208053	−	0.978117i	$-0.433287\pi$
$283$	19.5000	−	11.2583i	1.15915	−	0.669238i	0.951101	+	0.308879i	$0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	12.0000	+	20.7846i	0.703452	+	1.21842i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−32.5000	+	11.2583i	−1.87327	+	0.648919i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			29.4449i			1.68051i	0.542194	+	0.840254i	$0.317594\pi$
$307$			29.4449i			1.68051i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$	27.0000			1.53598
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−28.5000	−	16.4545i	−1.61092	−	0.930062i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−28.5000	−	16.4545i	−1.61092	−	0.930062i	−0.621757	−	0.783210i	$-0.713581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−22.5000	+	12.9904i	−1.24808	+	0.720577i
$326$		0			0
$327$	−28.5000	−	16.4545i	−1.57605	−	0.909935i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	0.0274825	+	0.999622i	$0.491251\pi$
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−16.5000	−	28.5788i	−0.904194	−	1.56611i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−29.0000			−1.57973			−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000			−1.57973			−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		−	34.6410i		−	1.85429i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$		−	34.6410i		−	1.85429i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$	27.0000			1.44115
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	28.0000	+	48.4974i	1.47368	+	2.55250i
$362$		0			0
$363$			19.0526i			1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.5000	−	11.2583i	1.01789	−	0.587680i	0.104399	−	0.994535i	$-0.466708\pi$
$367$	19.5000	−	11.2583i	1.01789	−	0.587680i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$	−1.50000	+	0.866025i	−0.0768473	+	0.0443678i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	19.5000	−	33.7750i	0.991241	−	1.71688i
$388$		0			0
$389$		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
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.878300	+	0.478110i	$0.158678\pi$
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$	37.5000	−	12.9904i	1.87735	−	0.650332i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−4.50000	−	7.79423i	−0.224161	−	0.388258i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.50000	+	7.79423i	0.220366	+	0.381685i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	41.0000			1.99822			0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	41.0000			1.99822			0.999109	+	0.0422075i	$0.0134391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	18.0000	+	3.46410i	0.871081	+	0.167640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		−	19.0526i		−	0.915608i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$		−	19.0526i		−	0.915608i	0.889053	−	0.457804i	$-0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		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
$453$	−6.00000	−	3.46410i	−0.281905	−	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−43.0000			−1.99838			−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−43.0000			−1.99838			−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	10.0000	+	8.66025i	0.461757	+	0.399893i
$470$		0			0
$471$	−18.0000	+	31.1769i	−0.829396	+	1.43656i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	43.3013i		−	1.98680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	49.5000	+	28.5788i	2.25701	+	1.30308i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$			13.8564i			0.626608i
$490$		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
$499$	5.50000	−	9.52628i	0.246214	−	0.426455i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	5.50000	−	9.52628i	0.246214	−	0.426455i	0.962472	+	0.271380i	$0.0874801\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.0000	+	12.1244i	−0.932643	+	0.538462i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	3.00000	−	3.46410i	0.132712	−	0.153243i
$512$		0			0
$513$	−22.5000	+	38.9711i	−0.993399	+	1.72062i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	13.5000	+	7.79423i	0.590314	+	0.340818i	0.765222	−	0.643767i	$-0.222629\pi$
$523$	13.5000	+	7.79423i	0.590314	+	0.340818i	−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	−22.5000	−	4.33013i	−0.981981	−	0.188982i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−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
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	0.623404	+	0.781900i	$0.285749\pi$
$542$		0			0
$543$	−16.5000	−	28.5788i	−0.708083	−	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	−18.0000	+	10.3923i	−0.768221	+	0.443533i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	8.50000	−	44.1673i	0.361457	−	1.87818i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			67.5500i			2.85706i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−23.5000	−	40.7032i	−0.983444	−	1.70338i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−23.5000	−	40.7032i	−0.983444	−	1.70338i	−0.334790	−	0.942293i	$-0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	40.5000	−	23.3827i	1.68604	−	0.973434i	0.728535	−	0.685009i	$-0.240202\pi$
$577$	40.5000	−	23.3827i	1.68604	−	0.973434i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	34.5000	+	19.9186i	1.43377	+	0.827788i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	15.0000			0.618064
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.00000	−	5.19615i	0.122782	−	0.212664i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		−	43.3013i		−	1.76630i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$		−	43.3013i		−	1.76630i	0.469095	−	0.883148i	$-0.344580\pi$
$602$		0			0
$603$	−15.0000			−0.610847
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.984738	−	0.174042i	$-0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$	24.0000	−	13.8564i	0.953914	−	0.550743i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.5000	+	28.5788i	0.891482	+	1.13233i
$638$		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
$643$		−	19.0526i		−	0.751360i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$		−	19.0526i		−	0.751360i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.50000	−	7.79423i	0.0587896	−	0.305480i
$652$		0			0
$653$		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
$657$			5.19615i			0.202721i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−43.5000	+	25.1147i	−1.69195	+	0.976850i	−0.739014	+	0.673690i	$0.764708\pi$
$661$	−43.5000	+	25.1147i	−1.69195	+	0.976850i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−9.00000	−	15.5885i	−0.347960	−	0.602685i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	13.0000			0.501113			0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000			0.501113			0.250557	+	0.968102i	$0.419386\pi$
$674$		0			0
$675$	22.5000	−	12.9904i	0.866025	−	0.500000i
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−12.0000	−	34.6410i	−0.460518	−	1.32940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−15.0000			−0.572286
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		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
$703$	−82.5000	+	47.6314i	−3.11155	+	1.79645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$	25.5000	+	44.1673i	0.956325	+	1.65640i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−40.5000	−	7.79423i	−1.50830	−	0.290272i
$722$		0			0
$723$	24.0000	−	41.5692i	0.892570	−	1.54598i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			53.6936i			1.99138i	0.0927199	+	0.995692i	$0.470444\pi$
$727$			53.6936i			1.99138i	−0.0927199	+	0.995692i	$0.529556\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		−	77.9423i		−	2.86328i
$742$		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
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	38.0000	+	32.9090i	1.37569	+	1.19138i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			29.4449i			1.06181i	0.847432	+	0.530904i	$0.178148\pi$
$769$			29.4449i			1.06181i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−7.50000	−	4.33013i	−0.269408	−	0.155543i
$776$		0			0
$777$	16.5000	+	47.6314i	0.591934	+	1.70877i
$778$		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
$787$	3.00000	−	1.73205i	0.106938	−	0.0617409i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	3.00000	−	1.73205i	0.106938	−	0.0617409i	0.552515	+	0.833503i	$0.313668\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18.0000	−	31.1769i	0.639199	−	1.10712i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		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	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$	30.0000			1.05215
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−97.5000	−	56.2917i	−3.41109	−	1.96940i
$818$		0			0
$819$	−40.5000	−	7.79423i	−1.41518	−	0.272352i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.819159	+	0.573567i	$0.194441\pi$
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	19.5000	−	11.2583i	0.677263	−	0.391018i	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	19.5000	−	11.2583i	0.677263	−	0.391018i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	−7.50000	−	4.33013i	−0.260172	−	0.150210i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.50000	+	7.79423i	0.155543	+	0.269408i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.50000	−	28.5788i	0.188982	−	0.981981i
$848$		0			0
$849$	19.5000	−	33.7750i	0.669238	−	1.15915i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			53.6936i			1.83843i	0.393753	+	0.919216i	$0.371177\pi$
$853$			53.6936i			1.83843i	−0.393753	+	0.919216i	$0.628823\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	15.0000	+	8.66025i	0.511793	+	0.295484i	0.733571	−	0.679613i	$-0.237852\pi$
$859$	15.0000	+	8.66025i	0.511793	+	0.295484i	−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	22.5000	−	12.9904i	0.762383	−	0.440162i
$872$		0			0
$873$	36.0000	+	20.7846i	1.21842	+	0.703452i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17.0000	−	29.4449i	0.574049	−	0.994282i	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	17.0000	−	29.4449i	0.574049	−	0.994282i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	55.0000			1.85090			0.925449	−	0.378873i	$-0.123688\pi$
$883$	55.0000			1.85090			0.925449	+	0.378873i	$0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	2.50000	−	0.866025i	0.0838473	−	0.0290456i
$890$		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
$903$	−39.0000	+	45.0333i	−1.29784	+	1.49862i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.50000	−	16.4545i	−0.315442	−	0.546362i	0.664089	−	0.747653i	$-0.268820\pi$
$907$	−9.50000	−	16.4545i	−0.315442	−	0.546362i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		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
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$	25.5000	+	44.1673i	0.840254	+	1.45536i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	55.0000			1.80839
$926$		0			0
$927$	40.5000	−	23.3827i	1.33019	−	0.767988i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−60.0000	+	8.66025i	−1.96642	+	0.283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			5.19615i			0.169751i	0.996392	+	0.0848755i	$0.0270492\pi$
$937$			5.19615i			0.169751i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$	−57.0000			−1.86012
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	−4.50000	−	7.79423i	−0.146076	−	0.253011i
$950$		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
$961$	−14.0000	+	24.2487i	−0.451613	+	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	41.0000			1.31847			0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000			1.31847			0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	−4.50000	−	12.9904i	−0.144263	−	0.416452i
$974$		0			0
$975$	−22.5000	+	38.9711i	−0.720577	+	1.24808i
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−57.0000			−1.81987
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−30.5000	−	52.8275i	−0.968864	−	1.67812i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−30.5000	−	52.8275i	−0.968864	−	1.67812i	−0.270011	−	0.962857i	$-0.587027\pi$
$992$		0			0
$993$			53.6936i			1.70391i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	−49.5000	−	28.5788i	−1.56611	−	0.904194i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.2.k.c.5.1	✓	2
3.2	odd	2	CM	84.2.k.c.5.1	✓	2
4.3	odd	2		336.2.bc.a.257.1		2
5.2	odd	4		2100.2.bo.e.1349.1		4
5.3	odd	4		2100.2.bo.e.1349.2		4
5.4	even	2		2100.2.bi.d.1601.1		2
7.2	even	3		588.2.f.b.293.2		2
7.3	odd	6	inner	84.2.k.c.17.1	yes	2
7.4	even	3		588.2.k.b.521.1		2
7.5	odd	6		588.2.f.b.293.1		2
7.6	odd	2		588.2.k.b.509.1		2
9.2	odd	6		2268.2.w.e.1349.1		2
9.4	even	3		2268.2.bm.d.593.1		2
9.5	odd	6		2268.2.bm.d.593.1		2
9.7	even	3		2268.2.w.e.1349.1		2
12.11	even	2		336.2.bc.a.257.1		2
15.2	even	4		2100.2.bo.e.1349.1		4
15.8	even	4		2100.2.bo.e.1349.2		4
15.14	odd	2		2100.2.bi.d.1601.1		2
21.2	odd	6		588.2.f.b.293.2		2
21.5	even	6		588.2.f.b.293.1		2
21.11	odd	6		588.2.k.b.521.1		2
21.17	even	6	inner	84.2.k.c.17.1	yes	2
21.20	even	2		588.2.k.b.509.1		2
28.3	even	6		336.2.bc.a.17.1		2
28.19	even	6		2352.2.k.b.881.2		2
28.23	odd	6		2352.2.k.b.881.1		2
35.3	even	12		2100.2.bo.e.1949.1		4
35.17	even	12		2100.2.bo.e.1949.2		4
35.24	odd	6		2100.2.bi.d.101.1		2
63.31	odd	6		2268.2.w.e.269.1		2
63.38	even	6		2268.2.bm.d.1025.1		2
63.52	odd	6		2268.2.bm.d.1025.1		2
63.59	even	6		2268.2.w.e.269.1		2
84.23	even	6		2352.2.k.b.881.1		2
84.47	odd	6		2352.2.k.b.881.2		2
84.59	odd	6		336.2.bc.a.17.1		2
105.17	odd	12		2100.2.bo.e.1949.2		4
105.38	odd	12		2100.2.bo.e.1949.1		4
105.59	even	6		2100.2.bi.d.101.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.k.c.5.1	✓	2	1.1	even	1	trivial
84.2.k.c.5.1	✓	2	3.2	odd	2	CM
84.2.k.c.17.1	yes	2	7.3	odd	6	inner
84.2.k.c.17.1	yes	2	21.17	even	6	inner
336.2.bc.a.17.1		2	28.3	even	6
336.2.bc.a.17.1		2	84.59	odd	6
336.2.bc.a.257.1		2	4.3	odd	2
336.2.bc.a.257.1		2	12.11	even	2
588.2.f.b.293.1		2	7.5	odd	6
588.2.f.b.293.1		2	21.5	even	6
588.2.f.b.293.2		2	7.2	even	3
588.2.f.b.293.2		2	21.2	odd	6
588.2.k.b.509.1		2	7.6	odd	2
588.2.k.b.509.1		2	21.20	even	2
588.2.k.b.521.1		2	7.4	even	3
588.2.k.b.521.1		2	21.11	odd	6
2100.2.bi.d.101.1		2	35.24	odd	6
2100.2.bi.d.101.1		2	105.59	even	6
2100.2.bi.d.1601.1		2	5.4	even	2
2100.2.bi.d.1601.1		2	15.14	odd	2
2100.2.bo.e.1349.1		4	5.2	odd	4
2100.2.bo.e.1349.1		4	15.2	even	4
2100.2.bo.e.1349.2		4	5.3	odd	4
2100.2.bo.e.1349.2		4	15.8	even	4
2100.2.bo.e.1949.1		4	35.3	even	12
2100.2.bo.e.1949.1		4	105.38	odd	12
2100.2.bo.e.1949.2		4	35.17	even	12
2100.2.bo.e.1949.2		4	105.17	odd	12
2268.2.w.e.269.1		2	63.31	odd	6
2268.2.w.e.269.1		2	63.59	even	6
2268.2.w.e.1349.1		2	9.2	odd	6
2268.2.w.e.1349.1		2	9.7	even	3
2268.2.bm.d.593.1		2	9.4	even	3
2268.2.bm.d.593.1		2	9.5	odd	6
2268.2.bm.d.1025.1		2	63.38	even	6
2268.2.bm.d.1025.1		2	63.52	odd	6
2352.2.k.b.881.1		2	28.23	odd	6
2352.2.k.b.881.1		2	84.23	even	6
2352.2.k.b.881.2		2	28.19	even	6
2352.2.k.b.881.2		2	84.47	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	84.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{3} +(-2.50000 + 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +5.19615i q^{13} +(-7.50000 - 4.33013i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 + 4.33013i) q^{25} -5.19615i q^{27} +(-1.50000 + 0.866025i) q^{31} +(5.50000 - 9.52628i) q^{37} +(4.50000 + 7.79423i) q^{39} +13.0000 q^{43} +(5.50000 - 4.33013i) q^{49} -15.0000 q^{57} +(-6.00000 - 3.46410i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-2.50000 - 4.33013i) q^{67} +(-1.50000 + 0.866025i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-8.50000 + 14.7224i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 - 12.9904i) q^{91} +(-1.50000 + 2.59808i) q^{93} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 5 q^{7} + 3 q^{9} - 15 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{31} + 11 q^{37} + 9 q^{39} + 26 q^{43} + 11 q^{49} - 30 q^{57} - 12 q^{61} - 3 q^{63} - 5 q^{67} - 3 q^{73} + 15 q^{75} - 17 q^{79}+ \cdots - 3 q^{93}+O(q^{100}) \) 2 * q + 3 * q^3 - 5 * q^7 + 3 * q^9 - 15 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^31 + 11 * q^37 + 9 * q^39 + 26 * q^43 + 11 * q^49 - 30 * q^57 - 12 * q^61 - 3 * q^63 - 5 * q^67 - 3 * q^73 + 15 * q^75 - 17 * q^79 - 9 * q^81 - 9 * q^91 - 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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