LMFDB - Embedded newform 441.2.p.b.80.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(80,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("441.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			80.3
Root			$1.00781 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	441.80
Dual form			441.2.p.b.215.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00781 - 0.581861i) q^{2} +(-0.322876 + 0.559237i) q^{4} +3.07892i q^{8} +O(q^{10})$	$q+(1.00781 - 0.581861i) q^{2} +(-0.322876 + 0.559237i) q^{4} +3.07892i q^{8} +(5.68986 + 3.28504i) q^{11} +(1.14575 + 1.98450i) q^{16} +7.64575 q^{22} +(1.65861 - 0.957598i) q^{23} +(2.50000 - 4.33013i) q^{25} +8.89753i q^{29} +(-3.02344 - 1.74558i) q^{32} +(-5.29150 - 9.16515i) q^{37} -5.29150 q^{43} +(-3.67423 + 2.12132i) q^{44} +(1.11438 - 1.93016i) q^{46} -5.81861i q^{50} +(-0.357016 - 0.206123i) q^{53} +(5.17712 + 8.96704i) q^{58} -8.64575 q^{64} +(2.00000 - 3.46410i) q^{67} -15.0554i q^{71} +(-10.6657 - 6.15784i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-5.33284 + 3.07892i) q^{86} +(-10.1144 + 17.5186i) q^{88} +1.23674i q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4}+O(q^{10}) $ 8 * q + 8 * q^4	$ 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) $ 8 * q + 8 * q^4 - 12 * q^16 + 40 * q^22 + 20 * q^25 - 44 * q^46 + 52 * q^58 - 48 * q^64 + 16 * q^67 - 32 * q^79 - 28 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	1.00781	−	0.581861i	0.712631	−	0.411438i	−0.0994033	−	0.995047i	$-0.531693\pi$
$2$	1.00781	−	0.581861i	0.712631	−	0.411438i	0.812035	+	0.583609i	$0.198360\pi$
$3$		0			0
$3$		0			0
$4$	−0.322876	+	0.559237i	−0.161438	+	0.279619i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$			3.07892i			1.08856i
$9$		0			0
$10$		0			0
$11$	5.68986	+	3.28504i	1.71556	+	0.990478i	0.926616	+	0.376009i	$0.122704\pi$
$11$	5.68986	+	3.28504i	1.71556	+	0.990478i	0.788941	+	0.614468i	$0.210630\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	1.14575	+	1.98450i	0.286438	+	0.496125i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$		0			0
$22$	7.64575			1.63008
$23$	1.65861	−	0.957598i	0.345844	−	0.199673i	−0.317009	−	0.948422i	$-0.602679\pi$
$23$	1.65861	−	0.957598i	0.345844	−	0.199673i	0.662853	+	0.748749i	$0.269345\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.89753i			1.65223i	0.563502	+	0.826115i	$0.309454\pi$
$29$			8.89753i			1.65223i	−0.563502	+	0.826115i	$0.690546\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−3.02344	−	1.74558i	−0.534473	−	0.308578i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.29150	−	9.16515i	−0.869918	−	1.50674i	−0.862080	−	0.506772i	$-0.830838\pi$
$37$	−5.29150	−	9.16515i	−0.869918	−	1.50674i	−0.00783774	−	0.999969i	$-0.502495\pi$
$38$		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
$43$	−5.29150			−0.806947			−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150			−0.806947			−0.403473	+	0.914991i	$0.632197\pi$
$44$	−3.67423	+	2.12132i	−0.553912	+	0.319801i
$45$		0			0
$46$	1.11438	−	1.93016i	0.164306	−	0.284587i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		−	5.81861i		−	0.822876i
$51$		0			0
$52$		0			0
$53$	−0.357016	−	0.206123i	−0.0490400	−	0.0283132i	0.475280	−	0.879835i	$-0.342347\pi$
$53$	−0.357016	−	0.206123i	−0.0490400	−	0.0283132i	−0.524320	+	0.851522i	$0.675680\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	5.17712	+	8.96704i	0.679790	+	1.17743i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−8.64575			−1.08072
$65$		0			0
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	15.0554i		−	1.78674i	−0.449319	−	0.893372i	$-0.648333\pi$
$71$		−	15.0554i		−	1.78674i	0.449319	−	0.893372i	$-0.351667\pi$
$72$		0			0
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−10.6657	−	6.15784i	−1.23986	−	0.715834i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		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
$86$	−5.33284	+	3.07892i	−0.575055	+	0.332008i
$87$		0			0
$88$	−10.1144	+	17.5186i	−1.07820	+	1.86749i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$			1.23674i			0.128939i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	1.61438	+	2.79619i	0.161438	+	0.279619i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	−0.479741			−0.0465965
$107$	9.00708	−	5.20024i	0.870747	−	0.502726i	0.00315068	−	0.999995i	$-0.498997\pi$
$107$	9.00708	−	5.20024i	0.870747	−	0.502726i	0.867596	+	0.497269i	$0.165664\pi$
$108$		0			0
$109$	−5.29150	+	9.16515i	−0.506834	+	0.877862i	0.493135	+	0.869953i	$0.335851\pi$
$109$	−5.29150	+	9.16515i	−0.506834	+	0.877862i	−0.999969	+	0.00790932i	$0.997482\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	13.5524i		−	1.27490i	−0.770490	−	0.637452i	$-0.779988\pi$
$113$		−	13.5524i		−	1.27490i	0.770490	−	0.637452i	$-0.220012\pi$
$114$		0			0
$115$		0			0
$116$	−4.97583	−	2.87280i	−0.461994	−	0.266732i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	16.0830	+	27.8566i	1.46209	+	2.53242i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−2.66642	+	1.53946i	−0.235681	+	0.136070i
$129$		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
$133$		0			0
$134$		−	4.65489i		−	0.402121i
$135$		0			0
$136$		0			0
$137$	19.0852	+	11.0188i	1.63056	+	0.941404i	0.983920	+	0.178607i	$0.0571592\pi$
$137$	19.0852	+	11.0188i	1.63056	+	0.941404i	0.646639	+	0.762796i	$0.276174\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−8.76013	−	15.1730i	−0.735134	−	1.27329i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	6.83399			0.561750
$149$	−6.99145	+	4.03652i	−0.572762	+	0.330684i	−0.758252	−	0.651962i	$-0.773946\pi$
$149$	−6.99145	+	4.03652i	−0.572762	+	0.330684i	0.185490	+	0.982646i	$0.440613\pi$
$150$		0			0
$151$	2.64575	−	4.58258i	0.215308	−	0.372925i	−0.738060	−	0.674735i	$-0.764258\pi$
$151$	2.64575	−	4.58258i	0.215308	−	0.372925i	0.953368	+	0.301811i	$0.0975911\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−8.06250	−	4.65489i	−0.641418	−	0.370323i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	−0.930033	−	0.367477i	$-0.880222\pi$
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	0.146772	−	0.989170i	$-0.453112\pi$
$164$		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
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$	1.70850	−	2.95920i	0.130272	−	0.225637i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$			15.0554i			1.13484i
$177$		0			0
$178$		0			0
$179$	−13.7524	−	7.93993i	−1.02790	−	0.593458i	−0.111518	−	0.993762i	$-0.535571\pi$
$179$	−13.7524	−	7.93993i	−1.02790	−	0.593458i	−0.916382	+	0.400304i	$0.868904\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	2.94837	+	5.10672i	0.217357	+	0.376473i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.1008	−	12.1826i	1.52680	−	0.881500i	0.527309	−	0.849674i	$-0.323201\pi$
$191$	21.1008	−	12.1826i	1.52680	−	0.881500i	0.999493	−	0.0318264i	$-0.0101324\pi$
$192$		0			0
$193$	10.5830	−	18.3303i	0.761781	−	1.31944i	−0.180150	−	0.983639i	$-0.557658\pi$
$193$	10.5830	−	18.3303i	0.761781	−	1.31944i	0.941932	−	0.335805i	$-0.109008\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			25.8681i			1.84303i	0.388348	+	0.921513i	$0.373046\pi$
$197$			25.8681i			1.84303i	−0.388348	+	0.921513i	$0.626954\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	13.3321	+	7.69730i	0.942722	+	0.544281i
$201$		0			0
$202$		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
$211$	26.4575			1.82141			0.910705	−	0.413057i	$-0.135539\pi$
$211$	26.4575			1.82141			0.910705	+	0.413057i	$0.135539\pi$
$212$	0.230544	−	0.133105i	0.0158338	−	0.00914166i
$213$		0			0
$214$	6.05163	−	10.4817i	0.413681	−	0.716517i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			12.3157i			0.834123i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−7.88562	−	13.6583i	−0.524544	−	0.908536i
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	−27.3948			−1.79855
$233$	−26.4337	+	15.2615i	−1.73173	+	0.999813i	−0.856205	+	0.516636i	$0.827184\pi$
$233$	−26.4337	+	15.2615i	−1.73173	+	0.999813i	−0.875522	+	0.483178i	$0.839482\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			7.39458i			0.478316i	0.970981	+	0.239158i	$0.0768713\pi$
$239$			7.39458i			0.478316i	−0.970981	+	0.239158i	$0.923129\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	32.4173	+	18.7161i	2.08386	+	1.20312i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	12.5830			0.791087
$254$	−16.1250	+	9.30978i	−1.01177	+	0.584147i
$255$		0			0
$256$	6.85425	−	11.8719i	0.428391	−	0.741994i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.3555	−	9.44288i	−1.00853	−	0.582273i	−0.0977664	−	0.995209i	$-0.531170\pi$
$263$	−16.3555	−	9.44288i	−1.00853	−	0.582273i	−0.910760	+	0.412936i	$0.864503\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.29150	+	2.23695i	0.0788911	+	0.136643i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	25.6458			1.54932
$275$	28.4493	−	16.4252i	1.71556	−	0.990478i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.41815i			0.203910i	0.994789	+	0.101955i	$0.0325097\pi$
$281$			3.41815i			0.203910i	−0.994789	+	0.101955i	$0.967490\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	8.41952	+	4.86101i	0.499606	+	0.288448i
$285$		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
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	28.2188	−	16.2921i	1.64018	−	0.946959i
$297$		0			0
$298$	−4.69738	+	8.13611i	−0.272112	+	0.471312i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		−	6.15784i		−	0.354344i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	5.16601			0.290611
$317$	−14.3399	+	8.27916i	−0.805410	+	0.465004i	−0.845359	−	0.534198i	$-0.820614\pi$
$317$	−14.3399	+	8.27916i	−0.805410	+	0.465004i	0.0399492	+	0.999202i	$0.487280\pi$
$318$		0			0
$319$	−29.2288	+	50.6257i	−1.63650	+	2.83449i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−20.1563	−	11.6372i	−1.11635	−	0.644526i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.64575	+	4.58258i	0.145424	+	0.251881i	0.929531	−	0.368744i	$-0.120212\pi$
$331$	2.64575	+	4.58258i	0.145424	+	0.251881i	−0.784107	+	0.620625i	$0.786879\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−21.1660			−1.15299			−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660			−1.15299			−0.576493	+	0.817102i	$0.695579\pi$
$338$	13.1016	−	7.56419i	0.712631	−	0.411438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		−	16.2921i		−	0.878412i
$345$		0			0
$346$		0			0
$347$	25.1321	+	14.5100i	1.34916	+	0.778938i	0.988131	−	0.153616i	$-0.0490918\pi$
$347$	25.1321	+	14.5100i	1.34916	+	0.778938i	0.361030	+	0.932554i	$0.382425\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−11.4686	−	19.8642i	−0.611280	−	1.05877i
$353$		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
$358$	−18.4797			−0.976685
$359$	−17.7836	+	10.2674i	−0.938583	+	0.541891i	−0.889516	−	0.456904i	$-0.848958\pi$
$359$	−17.7836	+	10.2674i	−0.938583	+	0.541891i	−0.0490673	+	0.998795i	$0.515625\pi$
$360$		0			0
$361$	−9.50000	+	16.4545i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	3.80071	+	2.19434i	0.198126	+	0.114388i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−37.0405			−1.90264			−0.951322	−	0.308199i	$-0.900274\pi$
$379$	−37.0405			−1.90264			−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	14.1771	−	24.5555i	0.725365	−	1.25637i
$383$		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
$386$		−	24.6314i		−	1.25370i
$387$		0			0
$388$		0			0
$389$	−29.7509	−	17.1767i	−1.50843	−	0.870893i	−0.999952	−	0.00981780i	$-0.996875\pi$
$389$	−29.7509	−	17.1767i	−1.50843	−	0.870893i	−0.508478	−	0.861075i	$-0.669792\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	15.0516	+	26.0702i	0.758290	+	1.31340i
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	11.4575			0.572876
$401$	−33.7821	+	19.5041i	−1.68700	+	0.973990i	−0.730204	+	0.683229i	$0.760575\pi$
$401$	−33.7821	+	19.5041i	−1.68700	+	0.973990i	−0.956796	+	0.290760i	$0.906092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	69.5312i		−	3.44654i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		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
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$	26.6642	−	15.3946i	1.29799	−	0.749397i
$423$		0			0
$424$	0.634637	−	1.09922i	0.0308207	−	0.0533831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			6.71612i			0.324636i
$429$		0			0
$430$		0			0
$431$	−35.7978	−	20.6679i	−1.72432	−	0.995535i	−0.909353	−	0.416024i	$-0.863423\pi$
$431$	−35.7978	−	20.6679i	−1.72432	−	0.995535i	−0.814964	−	0.579511i	$-0.803244\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−3.41699	−	5.91841i	−0.163644	−	0.283440i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.4351	+	6.02473i	−0.495789	+	0.286244i	−0.726973	−	0.686666i	$-0.759073\pi$
$443$	−10.4351	+	6.02473i	−0.495789	+	0.286244i	0.231184	+	0.972910i	$0.425740\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.3475i			1.47938i	0.672948	+	0.739689i	$0.265028\pi$
$449$			31.3475i			1.47938i	−0.672948	+	0.739689i	$0.734972\pi$
$450$		0			0
$451$		0			0
$452$	7.57901	+	4.37575i	0.356487	+	0.205818i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−21.1660	−	36.6606i	−0.990104	−	1.71491i	−0.616585	−	0.787288i	$-0.711484\pi$
$457$	−21.1660	−	36.6606i	−0.990104	−	1.71491i	−0.373519	−	0.927622i	$-0.621849\pi$
$458$		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
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$	−17.6571	+	10.1944i	−0.819712	+	0.473261i
$465$		0			0
$466$	−17.7601	+	30.7614i	−0.822722	+	1.42500i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−30.1079	−	17.3828i	−1.38436	−	0.799262i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	4.30262	+	7.45235i	0.196797	+	0.340863i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−20.7712			−0.944147
$485$		0			0
$486$		0			0
$487$	18.5203	−	32.0780i	0.839233	−	1.45359i	−0.0513038	−	0.998683i	$-0.516338\pi$
$487$	18.5203	−	32.0780i	0.839233	−	1.45359i	0.890537	−	0.454911i	$-0.150329\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			27.3710i			1.23524i	0.786478	+	0.617619i	$0.211903\pi$
$491$			27.3710i			1.23524i	−0.786478	+	0.617619i	$0.788097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.2288	−	22.9129i	−0.592200	−	1.02572i	−0.993935	−	0.109965i	$-0.964926\pi$
$499$	−13.2288	−	22.9129i	−0.592200	−	1.02572i	0.401735	−	0.915756i	$-0.368407\pi$
$500$		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
$506$	12.6813	−	7.32156i	0.563753	−	0.325483i
$507$		0			0
$508$	5.16601	−	8.94779i	0.229205	−	0.396994i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		−	22.1107i		−	0.977165i
$513$		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
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	−21.9778			−0.958276
$527$		0			0
$528$		0			0
$529$	−9.66601	+	16.7420i	−0.420261	+	0.727914i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	10.6657	+	6.15784i	0.460688	+	0.265978i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$	−12.3243	+	7.11544i	−0.526468	+	0.303956i
$549$		0			0
$550$	19.1144	−	33.1071i	0.815040	−	1.41169i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	11.6372i		−	0.494418i
$555$		0			0
$556$		0			0
$557$	21.6884	+	12.5218i	0.918967	+	0.530566i	0.883305	−	0.468798i	$-0.155313\pi$
$557$	21.6884	+	12.5218i	0.918967	+	0.530566i	0.0356614	+	0.999364i	$0.488646\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.98889	+	3.44485i	0.0838961	+	0.145312i
$563$		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
$568$	46.3542			1.94498
$569$	12.4508	−	7.18845i	0.521963	−	0.301356i	−0.215774	−	0.976443i	$-0.569228\pi$
$569$	12.4508	−	7.18845i	0.521963	−	0.301356i	0.737738	+	0.675088i	$0.235894\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9.57598i		−	0.399346i
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	17.1328	+	9.89164i	0.712631	+	0.411438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.35425	−	2.34563i	−0.0560872	−	0.0971460i
$584$		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
$589$		0			0
$590$		0			0
$591$		0			0
$592$	12.1255	−	21.0020i	0.498355	−	0.863176i
$593$		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
$596$		−	5.21317i		−	0.213540i
$597$		0			0
$598$		0			0
$599$	3.08667	+	1.78209i	0.126118	+	0.0728143i	0.561732	−	0.827319i	$-0.310135\pi$
$599$	3.08667	+	1.78209i	0.126118	+	0.0728143i	−0.435614	+	0.900134i	$0.643469\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	1.70850	+	2.95920i	0.0695178	+	0.120408i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	0.171564	+	0.985173i	$0.445118\pi$
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	−0.938967	+	0.344008i	$0.888215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			48.3180i			1.94521i	0.232462	+	0.972605i	$0.425322\pi$
$617$			48.3180i			1.94521i	−0.232462	+	0.972605i	$0.574678\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	21.3314	−	12.3157i	0.848517	−	0.489891i
$633$		0			0
$634$	−9.63464	+	16.6877i	−0.382640	+	0.662753i
$635$		0			0
$636$		0			0
$637$		0			0
$638$			68.0283i			2.69327i
$639$		0			0
$640$		0			0
$641$	41.1306	+	23.7468i	1.62456	+	0.937941i	0.985678	+	0.168637i	$0.0539367\pi$
$641$	41.1306	+	23.7468i	1.62456	+	0.937941i	0.638883	+	0.769304i	$0.279397\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	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
$652$	12.9150			0.505791
$653$	37.0994	−	21.4193i	1.45181	−	0.838203i	0.453226	−	0.891396i	$-0.350273\pi$
$653$	37.0994	−	21.4193i	1.45181	−	0.838203i	0.998584	+	0.0531926i	$0.0169397\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			49.8210i			1.94075i	0.241604	+	0.970375i	$0.422327\pi$
$659$			49.8210i			1.94075i	−0.241604	+	0.970375i	$0.577673\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	5.33284	+	3.07892i	0.207267	+	0.119666i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.52026	+	14.7575i	0.329906	+	0.571414i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	42.3320			1.63178			0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320			1.63178			0.815890	+	0.578208i	$0.196248\pi$
$674$	−21.3314	+	12.3157i	−0.821654	+	0.474382i
$675$		0			0
$676$	−4.19738	+	7.27008i	−0.161438	+	0.279619i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.0837	+	20.2556i	1.34244	+	0.775059i	0.987165	−	0.159702i	$-0.0510533\pi$
$683$	35.0837	+	20.2556i	1.34244	+	0.775059i	0.355277	+	0.934761i	$0.384387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−6.06275	−	10.5010i	−0.231140	−	0.400346i
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$	33.7712			1.28194
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	36.0024i		−	1.35979i	−0.733309	−	0.679895i	$-0.762025\pi$
$701$		−	36.0024i		−	1.35979i	0.733309	−	0.679895i	$-0.237975\pi$
$702$		0			0
$703$		0			0
$704$	−49.1931	−	28.4017i	−1.85404	−	1.07043i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.4575	+	45.8258i	0.993633	+	1.72102i	0.594389	+	0.804178i	$0.297394\pi$
$709$	26.4575	+	45.8258i	0.993633	+	1.72102i	0.399244	+	0.916845i	$0.369273\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	8.88061	−	5.12722i	0.331884	−	0.191613i
$717$		0			0
$718$	−11.9484	+	20.6952i	−0.445909	+	0.772337i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$			22.1107i			0.822876i
$723$		0			0
$724$		0			0
$725$	38.5274	+	22.2438i	1.43087	+	0.826115i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	−6.68627			−0.246459
$737$	22.7594	−	13.1402i	0.838355	−	0.484024i
$738$		0			0
$739$	26.0000	−	45.0333i	0.956425	−	1.65658i	0.225354	−	0.974277i	$-0.427646\pi$
$739$	26.0000	−	45.0333i	0.956425	−	1.65658i	0.731072	−	0.682300i	$-0.239020\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	54.4759i		−	1.99853i	−0.0383863	−	0.999263i	$-0.512222\pi$
$743$		−	54.4759i		−	1.99853i	0.0383863	−	0.999263i	$-0.487778\pi$
$744$		0			0
$745$		0			0
$746$	22.1719	+	12.8009i	0.811770	+	0.468676i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−13.2288	−	22.9129i	−0.482724	−	0.836103i	0.517079	−	0.855938i	$-0.327019\pi$
$751$	−13.2288	−	22.9129i	−0.482724	−	0.836103i	−0.999803	+	0.0198348i	$0.993686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	10.5830			0.384646			0.192323	−	0.981332i	$-0.438398\pi$
$757$	10.5830			0.384646			0.192323	+	0.981332i	$0.438398\pi$
$758$	−37.3299	+	21.5524i	−1.35588	+	0.782820i
$759$		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
$763$		0			0
$764$			15.7338i			0.569230i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	6.83399	+	11.8368i	0.245961	+	0.426016i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−39.9778			−1.43327
$779$		0			0
$780$		0			0
$781$	49.4575	−	85.6629i	1.76973	−	3.06526i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−14.4664	−	8.35218i	−0.515344	−	0.297534i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		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
$800$	−15.1172	+	8.72791i	−0.534473	+	0.308578i
$801$		0			0
$802$	−22.6974	+	39.3130i	−0.801472	+	1.38819i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−49.1931	−	28.4017i	−1.72954	−	0.998549i	−0.891693	−	0.452641i	$-0.850482\pi$
$809$	−49.1931	−	28.4017i	−1.72954	−	0.998549i	−0.837845	−	0.545908i	$-0.816185\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−40.4575	−	70.0745i	−1.41804	−	2.45611i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−45.8759	+	26.4865i	−1.60108	+	0.924384i	−0.609808	+	0.792549i	$0.708754\pi$
$821$	−45.8759	+	26.4865i	−1.60108	+	0.924384i	−0.991272	+	0.131835i	$0.957913\pi$
$822$		0			0
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	0.439961	+	0.898017i	$0.354992\pi$
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.92110i			0.171123i	0.996333	+	0.0855616i	$0.0272685\pi$
$827$			4.92110i			0.171123i	−0.996333	+	0.0855616i	$0.972732\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−50.1660			−1.72986
$842$	26.2031	−	15.1284i	0.903020	−	0.521359i
$843$		0			0
$844$	−8.54249	+	14.7960i	−0.294045	+	0.509300i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		−	0.944665i		−	0.0324399i
$849$		0			0
$850$		0			0
$851$	−17.5531	−	10.1343i	−0.601711	−	0.347398i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	16.0111	+	27.7321i	0.547249	+	0.947862i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−48.1033			−1.63840
$863$	40.5431	−	23.4075i	1.38010	−	0.796802i	0.387931	−	0.921689i	$-0.373190\pi$
$863$	40.5431	−	23.4075i	1.38010	−	0.796802i	0.992171	+	0.124887i	$0.0398567\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	52.5607i		−	1.78300i
$870$		0			0
$871$		0			0
$872$	−28.2188	−	16.2921i	−0.955608	−	0.551720i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	−0.886323	−	0.463068i	$-0.846749\pi$
$877$	−25.0000	−	43.3013i	−0.844190	−	1.46218i	0.0421327	−	0.999112i	$-0.486585\pi$
$878$		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
$883$	58.2065			1.95881			0.979403	−	0.201916i	$-0.0647168\pi$
$883$	58.2065			1.95881			0.979403	+	0.201916i	$0.0647168\pi$
$884$		0			0
$885$		0			0
$886$	−7.01111	+	12.1436i	−0.235543	+	0.407972i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	18.2399	+	31.5924i	0.608672	+	1.05425i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	41.7268			1.38781
$905$		0			0
$906$		0			0
$907$	2.64575	−	4.58258i	0.0878507	−	0.152162i	−0.818752	−	0.574148i	$-0.805333\pi$
$907$	2.64575	−	4.58258i	0.0878507	−	0.152162i	0.906602	+	0.421986i	$0.138667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			29.8445i			0.988793i	0.869236	+	0.494397i	$0.164611\pi$
$911$			29.8445i			0.988793i	−0.869236	+	0.494397i	$0.835389\pi$
$912$		0			0
$913$		0			0
$914$	−42.6628	−	24.6314i	−1.41116	−	0.814733i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	18.5203	+	32.0780i	0.610927	+	1.05816i	0.991084	+	0.133235i	$0.0425364\pi$
$919$	18.5203	+	32.0780i	0.610927	+	1.05816i	−0.380158	+	0.924922i	$0.624130\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−52.9150			−1.73984
$926$	−40.3125	+	23.2744i	−1.32475	+	0.764845i
$927$		0			0
$928$	15.5314	−	26.9011i	0.509842	−	0.883073i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$		−	19.7103i		−	0.645631i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	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
$946$	−40.4575			−1.31539
$947$	47.8915	−	27.6502i	1.55627	−	0.898510i	0.558656	−	0.829399i	$-0.311317\pi$
$947$	47.8915	−	27.6502i	1.55627	−	0.898510i	0.997609	−	0.0691110i	$-0.0220163\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	55.9788i		−	1.81333i	−0.421849	−	0.906666i	$-0.638619\pi$
$953$		−	55.9788i		−	1.81333i	0.421849	−	0.906666i	$-0.361381\pi$
$954$		0			0
$955$		0			0
$956$	−4.13532	−	2.38753i	−0.133746	−	0.0772182i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−40.0000			−1.28631			−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000			−1.28631			−0.643157	+	0.765735i	$0.722376\pi$
$968$	−85.7682	+	49.5183i	−2.75669	+	1.59158i
$969$		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
$973$		0			0
$974$		−	43.1049i		−	1.38117i
$975$		0			0
$976$		0			0
$977$	2.24617	+	1.29683i	0.0718613	+	0.0414892i	0.535500	−	0.844535i	$-0.320123\pi$
$977$	2.24617	+	1.29683i	0.0718613	+	0.0414892i	−0.463639	+	0.886024i	$0.653456\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	15.9261	+	27.5849i	0.508224	+	0.880269i
$983$		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
$988$		0			0
$989$	−8.77653	+	5.06713i	−0.279078	+	0.161126i
$990$		0			0
$991$	−29.1033	+	50.4083i	−0.924496	+	1.60127i	−0.132125	+	0.991233i	$0.542180\pi$
$991$	−29.1033	+	50.4083i	−0.924496	+	1.60127i	−0.792370	+	0.610040i	$0.791153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−26.6642	−	15.3946i	−0.844041	−	0.487307i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.p.b.80.3		8
3.2	odd	2	inner	441.2.p.b.80.2		8
7.2	even	3	inner	441.2.p.b.215.2		8
7.3	odd	6		63.2.c.a.62.3	yes	4
7.4	even	3		63.2.c.a.62.3	yes	4
7.5	odd	6	inner	441.2.p.b.215.2		8
7.6	odd	2	CM	441.2.p.b.80.3		8
21.2	odd	6	inner	441.2.p.b.215.3		8
21.5	even	6	inner	441.2.p.b.215.3		8
21.11	odd	6		63.2.c.a.62.2	✓	4
21.17	even	6		63.2.c.a.62.2	✓	4
21.20	even	2	inner	441.2.p.b.80.2		8
28.3	even	6		1008.2.k.a.881.4		4
28.11	odd	6		1008.2.k.a.881.4		4
35.3	even	12		1575.2.g.d.1574.6		8
35.4	even	6		1575.2.b.a.251.2		4
35.17	even	12		1575.2.g.d.1574.3		8
35.18	odd	12		1575.2.g.d.1574.6		8
35.24	odd	6		1575.2.b.a.251.2		4
35.32	odd	12		1575.2.g.d.1574.3		8
56.3	even	6		4032.2.k.b.3905.3		4
56.11	odd	6		4032.2.k.b.3905.3		4
56.45	odd	6		4032.2.k.c.3905.2		4
56.53	even	6		4032.2.k.c.3905.2		4
63.4	even	3		567.2.o.f.188.2		8
63.11	odd	6		567.2.o.f.377.2		8
63.25	even	3		567.2.o.f.377.3		8
63.31	odd	6		567.2.o.f.188.2		8
63.32	odd	6		567.2.o.f.188.3		8
63.38	even	6		567.2.o.f.377.2		8
63.52	odd	6		567.2.o.f.377.3		8
63.59	even	6		567.2.o.f.188.3		8
84.11	even	6		1008.2.k.a.881.3		4
84.59	odd	6		1008.2.k.a.881.3		4
105.17	odd	12		1575.2.g.d.1574.5		8
105.32	even	12		1575.2.g.d.1574.5		8
105.38	odd	12		1575.2.g.d.1574.4		8
105.53	even	12		1575.2.g.d.1574.4		8
105.59	even	6		1575.2.b.a.251.3		4
105.74	odd	6		1575.2.b.a.251.3		4
168.11	even	6		4032.2.k.b.3905.4		4
168.53	odd	6		4032.2.k.c.3905.1		4
168.59	odd	6		4032.2.k.b.3905.4		4
168.101	even	6		4032.2.k.c.3905.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.c.a.62.2	✓	4	21.11	odd	6
63.2.c.a.62.2	✓	4	21.17	even	6
63.2.c.a.62.3	yes	4	7.3	odd	6
63.2.c.a.62.3	yes	4	7.4	even	3
441.2.p.b.80.2		8	3.2	odd	2	inner
441.2.p.b.80.2		8	21.20	even	2	inner
441.2.p.b.80.3		8	1.1	even	1	trivial
441.2.p.b.80.3		8	7.6	odd	2	CM
441.2.p.b.215.2		8	7.2	even	3	inner
441.2.p.b.215.2		8	7.5	odd	6	inner
441.2.p.b.215.3		8	21.2	odd	6	inner
441.2.p.b.215.3		8	21.5	even	6	inner
567.2.o.f.188.2		8	63.4	even	3
567.2.o.f.188.2		8	63.31	odd	6
567.2.o.f.188.3		8	63.32	odd	6
567.2.o.f.188.3		8	63.59	even	6
567.2.o.f.377.2		8	63.11	odd	6
567.2.o.f.377.2		8	63.38	even	6
567.2.o.f.377.3		8	63.25	even	3
567.2.o.f.377.3		8	63.52	odd	6
1008.2.k.a.881.3		4	84.11	even	6
1008.2.k.a.881.3		4	84.59	odd	6
1008.2.k.a.881.4		4	28.3	even	6
1008.2.k.a.881.4		4	28.11	odd	6
1575.2.b.a.251.2		4	35.4	even	6
1575.2.b.a.251.2		4	35.24	odd	6
1575.2.b.a.251.3		4	105.59	even	6
1575.2.b.a.251.3		4	105.74	odd	6
1575.2.g.d.1574.3		8	35.17	even	12
1575.2.g.d.1574.3		8	35.32	odd	12
1575.2.g.d.1574.4		8	105.38	odd	12
1575.2.g.d.1574.4		8	105.53	even	12
1575.2.g.d.1574.5		8	105.17	odd	12
1575.2.g.d.1574.5		8	105.32	even	12
1575.2.g.d.1574.6		8	35.3	even	12
1575.2.g.d.1574.6		8	35.18	odd	12
4032.2.k.b.3905.3		4	56.3	even	6
4032.2.k.b.3905.3		4	56.11	odd	6
4032.2.k.b.3905.4		4	168.11	even	6
4032.2.k.b.3905.4		4	168.59	odd	6
4032.2.k.c.3905.1		4	168.53	odd	6
4032.2.k.c.3905.1		4	168.101	even	6
4032.2.k.c.3905.2		4	56.45	odd	6
4032.2.k.c.3905.2		4	56.53	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00781 - 0.581861i) q^{2} +(-0.322876 + 0.559237i) q^{4} +3.07892i q^{8} +O(q^{10})\)	\(q+(1.00781 - 0.581861i) q^{2} +(-0.322876 + 0.559237i) q^{4} +3.07892i q^{8} +(5.68986 + 3.28504i) q^{11} +(1.14575 + 1.98450i) q^{16} +7.64575 q^{22} +(1.65861 - 0.957598i) q^{23} +(2.50000 - 4.33013i) q^{25} +8.89753i q^{29} +(-3.02344 - 1.74558i) q^{32} +(-5.29150 - 9.16515i) q^{37} -5.29150 q^{43} +(-3.67423 + 2.12132i) q^{44} +(1.11438 - 1.93016i) q^{46} -5.81861i q^{50} +(-0.357016 - 0.206123i) q^{53} +(5.17712 + 8.96704i) q^{58} -8.64575 q^{64} +(2.00000 - 3.46410i) q^{67} -15.0554i q^{71} +(-10.6657 - 6.15784i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-5.33284 + 3.07892i) q^{86} +(-10.1144 + 17.5186i) q^{88} +1.23674i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 8 * q + 8 * q^4	\( 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) 8 * q + 8 * q^4 - 12 * q^16 + 40 * q^22 + 20 * q^25 - 44 * q^46 + 52 * q^58 - 48 * q^64 + 16 * q^67 - 32 * q^79 - 28 * q^88

Introduction

L-functions

Modular forms

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