LMFDB - Embedded newform 189.2.p.a.80.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(26,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("189.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.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:	$1.50917259820$
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			80.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.80
Dual form			189.2.p.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)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +6.92820i q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.50000 - 2.59808i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 - 1.73205i) q^{28} +(-1.50000 - 0.866025i) q^{31} +(-5.00000 - 8.66025i) q^{37} +13.0000 q^{43} +(-6.50000 + 2.59808i) q^{49} +(-12.0000 - 6.92820i) q^{52} +(-7.50000 + 4.33013i) q^{61} +8.00000 q^{64} +(8.00000 - 13.8564i) q^{67} +(13.5000 + 7.79423i) q^{73} +10.3923i q^{76} +(2.00000 + 3.46410i) q^{79} +(-18.0000 + 3.46410i) q^{91} +19.0526i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + q^{7}+O(q^{10}) $ 2 * q - 2 * q^4 + q^7	$ 2 q - 2 q^{4} + q^{7} - 4 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 3 q^{31} - 10 q^{37} + 26 q^{43} - 13 q^{49} - 24 q^{52} - 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} + 4 q^{79} - 36 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 + q^7 - 4 * q^16 + 9 * q^19 + 5 * q^25 - 10 * q^28 - 3 * q^31 - 10 * q^37 + 26 * q^43 - 13 * q^49 - 24 * q^52 - 15 * q^61 + 16 * q^64 + 16 * q^67 + 27 * q^73 + 4 * q^79 - 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		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
$13$			6.92820i			1.92154i	0.277350	+	0.960769i	$0.410544\pi$
$13$			6.92820i			1.92154i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		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
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	0.359211	−	0.933257i	$-0.383046\pi$
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$		0			0
$40$		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	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$	−12.0000	−	6.92820i	−1.66410	−	0.960769i
$53$		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
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	8.00000	−	13.8564i	0.977356	−	1.69283i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	8.00000	−	13.8564i	0.977356	−	1.69283i	0.671932	−	0.740613i	$-0.265465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	13.5000	+	7.79423i	1.58006	+	0.912245i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	13.5000	+	7.79423i	1.58006	+	0.912245i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$		0			0
$76$			10.3923i			1.19208i
$77$		0			0
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		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	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−18.0000	+	3.46410i	−1.88691	+	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.0526i			1.93449i	0.253837	+	0.967247i	$0.418307\pi$
$97$			19.0526i			1.93449i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	3.00000	−	1.73205i	0.295599	−	0.170664i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	3.00000	−	1.73205i	0.295599	−	0.170664i	0.640464	+	0.767988i	$0.278742\pi$
$104$		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
$109$	−9.50000	+	16.4545i	−0.909935	+	1.57605i	−0.0957826	+	0.995402i	$0.530535\pi$
$109$	−9.50000	+	16.4545i	−0.909935	+	1.57605i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$		0			0
$112$	8.00000	−	6.92820i	0.755929	−	0.654654i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$		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
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	9.00000	+	10.3923i	0.780399	+	0.901127i
$134$		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
$139$		−	17.3205i		−	1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$		−	17.3205i		−	1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	20.0000			1.64399
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	11.5000	−	19.9186i	0.935857	−	1.62095i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0000	−	10.3923i	−1.43656	−	0.829396i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−18.0000	−	10.3923i	−1.43656	−	0.829396i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	0.313304	−	0.949653i	$-0.398564\pi$
$164$		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
$169$	−35.0000			−2.69231
$170$		0			0
$171$		0			0
$172$	−13.0000	+	22.5167i	−0.991241	+	1.71688i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	12.5000	+	4.33013i	0.944911	+	0.327327i
$176$		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
$181$		−	25.9808i		−	1.93113i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$		−	25.9808i		−	1.93113i	0.260153	−	0.965567i	$-0.416227\pi$
$182$		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
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	2.00000	−	13.8564i	0.142857	−	0.989743i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	19.5000	+	11.2583i	1.38232	+	0.798082i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	19.5000	+	11.2583i	1.38232	+	0.798082i	0.389885	+	0.920864i	$0.372515\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	24.0000	−	13.8564i	1.66410	−	0.960769i
$209$		0			0
$210$		0			0
$211$	−29.0000			−1.99644			−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000			−1.99644			−0.998221	+	0.0596196i	$0.981011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.50000	−	4.33013i	0.101827	−	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			10.3923i			0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$			10.3923i			0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	0.231287	+	0.972886i	$0.425707\pi$
$230$		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
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.50000	−	0.866025i	−0.0966235	−	0.0557856i	0.450910	−	0.892570i	$-0.351100\pi$
$241$	−1.50000	−	0.866025i	−0.0966235	−	0.0557856i	−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$		0			0
$244$		−	17.3205i		−	1.10883i
$245$		0			0
$246$		0			0
$247$	18.0000	+	31.1769i	1.14531	+	1.98374i
$248$		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
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	20.0000	−	17.3205i	1.24274	−	1.07624i
$260$		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
$268$	16.0000	+	27.7128i	0.977356	+	1.69283i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−28.5000	+	16.4545i	−1.73125	+	0.999539i	−0.850439	+	0.526073i	$0.823664\pi$
$271$	−28.5000	+	16.4545i	−1.73125	+	0.999539i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−2.50000	+	4.33013i	−0.150210	+	0.260172i	0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$		0			0
$280$		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	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	19.5000	+	11.2583i	1.15915	+	0.669238i	0.208053	+	0.978117i	$0.433287\pi$
$284$		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
$291$		0			0
$292$	−27.0000	+	15.5885i	−1.58006	+	0.912245i
$293$		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
$301$	6.50000	+	33.7750i	0.374654	+	1.94676i
$302$		0			0
$303$		0			0
$304$	−18.0000	−	10.3923i	−1.03237	−	0.596040i
$305$		0			0
$306$		0			0
$307$		−	29.4449i		−	1.68051i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$		−	29.4449i		−	1.68051i	0.542194	−	0.840254i	$-0.317594\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	4.50000	−	2.59808i	0.254355	−	0.146852i	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	4.50000	−	2.59808i	0.254355	−	0.146852i	0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		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
$325$	30.0000	+	17.3205i	1.66410	+	0.960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	0.851957	−	0.523612i	$-0.175416\pi$
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		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	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		−	29.4449i		−	1.57615i	−0.615581	−	0.788074i	$-0.711079\pi$
$349$		−	29.4449i		−	1.57615i	0.615581	−	0.788074i	$-0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		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
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$		0			0
$364$	12.0000	−	34.6410i	0.628971	−	1.81568i
$365$		0			0
$366$		0			0
$367$	13.5000	+	7.79423i	0.704694	+	0.406855i	0.809093	−	0.587680i	$-0.199959\pi$
$367$	13.5000	+	7.79423i	0.704694	+	0.406855i	−0.104399	+	0.994535i	$0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−33.0000	−	19.0526i	−1.67532	−	0.967247i
$389$		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
$397$	34.5000	−	19.9186i	1.73151	−	0.999685i	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.5000	−	19.9186i	1.73151	−	0.999685i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	6.00000	−	10.3923i	0.298881	−	0.517678i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12.0000	−	6.92820i	−0.593362	−	0.342578i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	−12.0000	−	6.92820i	−0.593362	−	0.342578i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$			6.92820i			0.341328i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−15.0000	−	17.3205i	−0.725901	−	0.838198i
$428$		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
$433$			19.0526i			0.915608i	0.889053	+	0.457804i	$0.151364\pi$
$433$			19.0526i			0.915608i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$	−19.0000	−	32.9090i	−0.909935	−	1.57605i
$437$		0			0
$438$		0			0
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		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
$448$	4.00000	+	20.7846i	0.188982	+	0.981981i
$449$		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
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	0.233890	−	0.972263i	$-0.424854\pi$
$458$		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
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	40.0000	+	13.8564i	1.84703	+	0.639829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	25.9808i		−	1.19208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	60.0000	−	34.6410i	2.73576	−	1.57949i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	0.430486	+	0.902597i	$0.358342\pi$
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$		0			0
$490$		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
$496$			6.92820i			0.311086i
$497$		0			0
$498$		0			0
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		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
$507$		0			0
$508$	1.00000	−	1.73205i	0.0443678	−	0.0768473i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−13.5000	+	38.9711i	−0.597205	+	1.72398i
$512$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−39.0000	+	22.5167i	−1.70535	+	0.984585i	−0.765222	+	0.643767i	$0.777371\pi$
$523$	−39.0000	+	22.5167i	−1.70535	+	0.984585i	−0.940129	+	0.340818i	$0.889296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−27.0000	+	5.19615i	−1.17060	+	0.225282i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	0.365444	−	0.930834i	$-0.380917\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8.00000	+	6.92820i	−0.340195	+	0.294617i
$554$		0			0
$555$		0			0
$556$	30.0000	+	17.3205i	1.27228	+	0.734553i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$			90.0666i			3.80941i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		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
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.0000	−	6.92820i	−0.499567	−	0.288425i	0.228968	−	0.973434i	$-0.426465\pi$
$577$	−12.0000	−	6.92820i	−0.499567	−	0.288425i	−0.728535	+	0.685009i	$0.759798\pi$
$578$		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			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−9.00000			−0.370839
$590$		0			0
$591$		0			0
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		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
$601$			43.3013i			1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$			43.3013i			1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$	23.0000	+	39.8372i	0.935857	+	1.62095i
$605$		0			0
$606$		0			0
$607$	4.50000	−	2.59808i	0.182649	−	0.105453i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	4.50000	−	2.59808i	0.182649	−	0.105453i	0.588537	+	0.808470i	$0.299704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	18.5000	−	32.0429i	0.747208	−	1.29420i	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	18.5000	−	32.0429i	0.747208	−	1.29420i	0.949156	−	0.314806i	$-0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		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
$628$	36.0000	−	20.7846i	1.43656	−	0.829396i
$629$		0			0
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−18.0000	−	45.0333i	−0.713186	−	1.78429i
$638$		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
$643$			19.0526i			0.751360i	0.926750	+	0.375680i	$0.122591\pi$
$643$			19.0526i			0.751360i	−0.926750	+	0.375680i	$0.877409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	34.0000			1.33154
$653$		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
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	13.5000	+	7.79423i	0.525089	+	0.303160i	0.739014	−	0.673690i	$-0.235292\pi$
$661$	13.5000	+	7.79423i	0.525089	+	0.303160i	−0.213925	+	0.976850i	$0.568625\pi$
$662$		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
$673$	37.0000			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$674$		0			0
$675$		0			0
$676$	35.0000	−	60.6218i	1.34615	−	2.33161i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−49.5000	+	9.52628i	−1.89964	+	0.365585i
$680$		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
$687$		0			0
$688$	−26.0000	−	45.0333i	−0.991241	−	1.71688i
$689$		0			0
$690$		0			0
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−28.5000	+	16.4545i	−1.08419	+	0.625958i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−20.0000	+	17.3205i	−0.755929	+	0.654654i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−45.0000	−	25.9808i	−1.69721	−	0.979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	0.413114	−	0.910679i	$-0.364441\pi$
$710$		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		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	6.00000	+	6.92820i	0.223452	+	0.258020i
$722$		0			0
$723$		0			0
$724$	45.0000	+	25.9808i	1.67241	+	0.965567i
$725$		0			0
$726$		0			0
$727$		−	53.6936i		−	1.99138i	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$		−	53.6936i		−	1.99138i	0.0927199	−	0.995692i	$-0.470444\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.5000	+	16.4545i	−1.05267	+	0.607760i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−28.5000	+	16.4545i	−1.05267	+	0.607760i	−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		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
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	0.948753	+	0.316017i	$0.102346\pi$
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	55.0000			1.99901			0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	55.0000			1.99901			0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−47.5000	−	16.4545i	−1.71962	−	0.595692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	25.9808i		−	0.936890i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$		−	25.9808i		−	0.936890i	0.883493	−	0.468445i	$-0.155186\pi$
$770$		0			0
$771$		0			0
$772$	2.00000	+	3.46410i	0.0719816	+	0.124676i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−7.50000	+	4.33013i	−0.269408	+	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$		0			0
$786$		0			0
$787$	−43.5000	−	25.1147i	−1.55061	−	0.895244i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$	−43.5000	−	25.1147i	−1.55061	−	0.895244i	−0.552515	−	0.833503i	$-0.686332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−30.0000	−	51.9615i	−1.06533	−	1.84521i
$794$		0			0
$795$		0			0
$796$	−39.0000	+	22.5167i	−1.38232	+	0.798082i
$797$		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	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$			10.3923i			0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$			10.3923i			0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	58.5000	−	33.7750i	2.04666	−	1.18164i
$818$		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
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	0.0871445	+	0.996196i	$0.472226\pi$
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−49.5000	−	28.5788i	−1.71921	−	0.992584i	−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−49.5000	−	28.5788i	−1.71921	−	0.992584i	−0.798823	−	0.601566i	$-0.794544\pi$
$830$		0			0
$831$		0			0
$832$			55.4256i			1.92154i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		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
$844$	29.0000	−	50.2295i	0.998221	−	1.72897i
$845$		0			0
$846$		0			0
$847$	22.0000	−	19.0526i	0.755929	−	0.654654i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			46.7654i			1.60122i	0.599189	+	0.800608i	$0.295490\pi$
$853$			46.7654i			1.60122i	−0.599189	+	0.800608i	$0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.221777	−	0.975097i	$-0.428814\pi$
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		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
$867$		0			0
$868$	6.00000	+	6.92820i	0.203653	+	0.235159i
$869$		0			0
$870$		0			0
$871$	96.0000	+	55.4256i	3.25284	+	1.87803i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.5000	+	21.6506i	0.422095	+	0.731090i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	12.5000	+	21.6506i	0.422095	+	0.731090i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		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
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−0.500000	−	2.59808i	−0.0167695	−	0.0871367i
$890$		0			0
$891$		0			0
$892$	−18.0000	−	10.3923i	−0.602685	−	0.347960i
$893$		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
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	0.315442	+	0.948945i	$0.397847\pi$
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	−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
$916$		−	17.3205i		−	0.572286i
$917$		0			0
$918$		0			0
$919$	26.5000	+	45.8993i	0.874154	+	1.51408i	0.857661	+	0.514216i	$0.171917\pi$
$919$	26.5000	+	45.8993i	0.874154	+	1.51408i	0.0164935	+	0.999864i	$0.494750\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−50.0000			−1.64399
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−22.5000	+	28.5788i	−0.737408	+	0.936634i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			55.4256i			1.81068i	0.424691	+	0.905338i	$0.360383\pi$
$937$			55.4256i			1.81068i	−0.424691	+	0.905338i	$0.639617\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		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
$949$	−54.0000	+	93.5307i	−1.75291	+	3.03614i
$950$		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
$964$	3.00000	−	1.73205i	0.0966235	−	0.0557856i
$965$		0			0
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	45.0000	−	8.66025i	1.44263	−	0.277635i
$974$		0			0
$975$		0			0
$976$	30.0000	+	17.3205i	0.960277	+	0.554416i
$977$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−72.0000			−2.29063
$989$		0			0
$990$		0			0
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	0.968864	−	0.247592i	$-0.0796392\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	+	11.2583i	0.617571	+	0.356555i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	19.5000	+	11.2583i	0.617571	+	0.356555i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.p.a.80.1	yes	2
3.2	odd	2	CM	189.2.p.a.80.1	yes	2
7.3	odd	6		1323.2.c.a.1322.1		2
7.4	even	3		1323.2.c.a.1322.2		2
7.5	odd	6	inner	189.2.p.a.26.1	✓	2
9.2	odd	6		567.2.s.b.458.1		2
9.4	even	3		567.2.i.a.269.1		2
9.5	odd	6		567.2.i.a.269.1		2
9.7	even	3		567.2.s.b.458.1		2
21.5	even	6	inner	189.2.p.a.26.1	✓	2
21.11	odd	6		1323.2.c.a.1322.2		2
21.17	even	6		1323.2.c.a.1322.1		2
63.5	even	6		567.2.s.b.26.1		2
63.40	odd	6		567.2.s.b.26.1		2
63.47	even	6		567.2.i.a.215.1		2
63.61	odd	6		567.2.i.a.215.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.a.26.1	✓	2	7.5	odd	6	inner
189.2.p.a.26.1	✓	2	21.5	even	6	inner
189.2.p.a.80.1	yes	2	1.1	even	1	trivial
189.2.p.a.80.1	yes	2	3.2	odd	2	CM
567.2.i.a.215.1		2	63.47	even	6
567.2.i.a.215.1		2	63.61	odd	6
567.2.i.a.269.1		2	9.4	even	3
567.2.i.a.269.1		2	9.5	odd	6
567.2.s.b.26.1		2	63.5	even	6
567.2.s.b.26.1		2	63.40	odd	6
567.2.s.b.458.1		2	9.2	odd	6
567.2.s.b.458.1		2	9.7	even	3
1323.2.c.a.1322.1		2	7.3	odd	6
1323.2.c.a.1322.1		2	21.17	even	6
1323.2.c.a.1322.2		2	7.4	even	3
1323.2.c.a.1322.2		2	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +6.92820i q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.50000 - 2.59808i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 - 1.73205i) q^{28} +(-1.50000 - 0.866025i) q^{31} +(-5.00000 - 8.66025i) q^{37} +13.0000 q^{43} +(-6.50000 + 2.59808i) q^{49} +(-12.0000 - 6.92820i) q^{52} +(-7.50000 + 4.33013i) q^{61} +8.00000 q^{64} +(8.00000 - 13.8564i) q^{67} +(13.5000 + 7.79423i) q^{73} +10.3923i q^{76} +(2.00000 + 3.46410i) q^{79} +(-18.0000 + 3.46410i) q^{91} +19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + q^{7}+O(q^{10}) \) 2 * q - 2 * q^4 + q^7	\( 2 q - 2 q^{4} + q^{7} - 4 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 3 q^{31} - 10 q^{37} + 26 q^{43} - 13 q^{49} - 24 q^{52} - 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} + 4 q^{79} - 36 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 + q^7 - 4 * q^16 + 9 * q^19 + 5 * q^25 - 10 * q^28 - 3 * q^31 - 10 * q^37 + 26 * q^43 - 13 * q^49 - 24 * q^52 - 15 * q^61 + 16 * q^64 + 16 * q^67 + 27 * q^73 + 4 * q^79 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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