LMFDB - Embedded newform 441.2.e.h.361.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,2,Mod(226,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.226"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,-10,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			361.2
Root			$1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	441.361
Dual form			441.2.e.h.226.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32288 + 2.29129i) q^{2} +(-2.50000 + 4.33013i) q^{4} -7.93725 q^{8} +(-2.64575 + 4.58258i) q^{11} +(-5.50000 - 9.52628i) q^{16} -14.0000 q^{22} +(2.64575 + 4.58258i) q^{23} +(2.50000 - 4.33013i) q^{25} +10.5830 q^{29} +(6.61438 - 11.4564i) q^{32} +(-3.00000 - 5.19615i) q^{37} +12.0000 q^{43} +(-13.2288 - 22.9129i) q^{44} +(-7.00000 + 12.1244i) q^{46} +13.2288 q^{50} +(-5.29150 + 9.16515i) q^{53} +(14.0000 + 24.2487i) q^{58} +13.0000 q^{64} +(-2.00000 + 3.46410i) q^{67} +5.29150 q^{71} +(7.93725 - 13.7477i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(15.8745 + 27.4955i) q^{86} +(21.0000 - 36.3731i) q^{88} -26.4575 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} - 22 q^{16} - 56 q^{22} + 10 q^{25} - 12 q^{37} + 48 q^{43} - 28 q^{46} + 56 q^{58} + 52 q^{64} - 8 q^{67} - 16 q^{79} + 84 q^{88}+O(q^{100}) $ 4 * q - 10 * q^4 - 22 * q^16 - 56 * q^22 + 10 * q^25 - 12 * q^37 + 48 * q^43 - 28 * q^46 + 56 * q^58 + 52 * q^64 - 8 * q^67 - 16 * q^79 + 84 * q^88

Character values

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

$n$	$199$	$344$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	1.32288	+	2.29129i	0.935414	+	1.62019i	0.773893	+	0.633316i	$0.218307\pi$
$2$	1.32288	+	2.29129i	0.935414	+	1.62019i	0.161521	+	0.986869i	$0.448360\pi$
$3$		0			0
$3$		0			0
$4$	−2.50000	+	4.33013i	−1.25000	+	2.16506i
$5$		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
$8$	−7.93725			−2.80624
$9$		0			0
$10$		0			0
$11$	−2.64575	+	4.58258i	−0.797724	+	1.38170i	0.123371	+	0.992361i	$0.460630\pi$
$11$	−2.64575	+	4.58258i	−0.797724	+	1.38170i	−0.921095	+	0.389338i	$0.872704\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−5.50000	−	9.52628i	−1.37500	−	2.38157i
$17$		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		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$	−14.0000			−2.98481
$23$	2.64575	+	4.58258i	0.551677	+	0.955533i	0.998154	+	0.0607377i	$0.0193453\pi$
$23$	2.64575	+	4.58258i	0.551677	+	0.955533i	−0.446476	+	0.894795i	$0.647321\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	10.5830			1.96521			0.982607	−	0.185695i	$-0.0594537\pi$
$29$	10.5830			1.96521			0.982607	+	0.185695i	$0.0594537\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	6.61438	−	11.4564i	1.16927	−	2.02523i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	0.506772	−	0.862080i	$-0.330838\pi$
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	−0.999969	+	0.00783774i	$0.997505\pi$
$38$		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
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$	−13.2288	−	22.9129i	−1.99431	−	3.45425i
$45$		0			0
$46$	−7.00000	+	12.1244i	−1.03209	+	1.78764i
$47$		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
$50$	13.2288			1.87083
$51$		0			0
$52$		0			0
$53$	−5.29150	+	9.16515i	−0.726844	+	1.25893i	0.231367	+	0.972867i	$0.425680\pi$
$53$	−5.29150	+	9.16515i	−0.726844	+	1.25893i	−0.958211	+	0.286064i	$0.907653\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	14.0000	+	24.2487i	1.83829	+	3.18401i
$59$		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		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$	13.0000			1.62500
$65$		0			0
$66$		0			0
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	0.717607	+	0.696449i	$0.245238\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.29150			0.627986			0.313993	−	0.949425i	$-0.398333\pi$
$71$	5.29150			0.627986			0.313993	+	0.949425i	$0.398333\pi$
$72$		0			0
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$	7.93725	−	13.7477i	0.922687	−	1.59814i
$75$		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
$86$	15.8745	+	27.4955i	1.71179	+	2.96491i
$87$		0			0
$88$	21.0000	−	36.3731i	2.23861	−	3.87738i
$89$		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
$92$	−26.4575			−2.75839
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	12.5000	+	21.6506i	1.25000	+	2.16506i
$101$		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		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$	−28.0000			−2.71960
$107$	−2.64575	−	4.58258i	−0.255774	−	0.443014i	0.709331	−	0.704875i	$-0.248997\pi$
$107$	−2.64575	−	4.58258i	−0.255774	−	0.443014i	−0.965106	+	0.261861i	$0.915664\pi$
$108$		0			0
$109$	9.00000	−	15.5885i	0.862044	−	1.49310i	−0.00790932	−	0.999969i	$-0.502518\pi$
$109$	9.00000	−	15.5885i	0.862044	−	1.49310i	0.869953	−	0.493135i	$-0.164149\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−21.1660			−1.99113			−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	−21.1660			−1.99113			−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$		0			0
$116$	−26.4575	+	45.8258i	−2.45652	+	4.25481i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.50000	−	14.7224i	−0.772727	−	1.33840i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$	3.96863	+	6.87386i	0.350780	+	0.607569i
$129$		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
$134$	−10.5830			−0.914232
$135$		0			0
$136$		0			0
$137$	10.5830	−	18.3303i	0.904167	−	1.56606i	0.0821359	−	0.996621i	$-0.473826\pi$
$137$	10.5830	−	18.3303i	0.904167	−	1.56606i	0.822031	−	0.569442i	$-0.192841\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	7.00000	+	12.1244i	0.587427	+	1.01745i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	30.0000			2.46598
$149$	5.29150	+	9.16515i	0.433497	+	0.750838i	0.997172	−	0.0751583i	$-0.0239462\pi$
$149$	5.29150	+	9.16515i	0.433497	+	0.750838i	−0.563675	+	0.825997i	$0.690613\pi$
$150$		0			0
$151$	−12.0000	+	20.7846i	−0.976546	+	1.69143i	−0.301811	+	0.953368i	$0.597591\pi$
$151$	−12.0000	+	20.7846i	−0.976546	+	1.69143i	−0.674735	+	0.738060i	$0.735742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$	10.5830	−	18.3303i	0.841939	−	1.45828i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	0.930033	+	0.367477i	$0.119778\pi$
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	−0.146772	+	0.989170i	$0.546888\pi$
$164$		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
$172$	−30.0000	+	51.9615i	−2.28748	+	3.96203i
$173$		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
$176$	58.2065			4.38748
$177$		0			0
$178$		0			0
$179$	13.2288	−	22.9129i	0.988764	−	1.71259i	0.364922	−	0.931038i	$-0.381096\pi$
$179$	13.2288	−	22.9129i	0.988764	−	1.71259i	0.623841	−	0.781551i	$-0.285571\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−21.0000	−	36.3731i	−1.54814	−	2.68146i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.2288	−	22.9129i	−0.957199	−	1.65792i	−0.729253	−	0.684244i	$-0.760132\pi$
$191$	−13.2288	−	22.9129i	−0.957199	−	1.65792i	−0.227946	−	0.973674i	$-0.573201\pi$
$192$		0			0
$193$	9.00000	−	15.5885i	0.647834	−	1.12208i	−0.335805	−	0.941932i	$-0.609008\pi$
$193$	9.00000	−	15.5885i	0.647834	−	1.12208i	0.983639	−	0.180150i	$-0.0576584\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.5830			−0.754008			−0.377004	−	0.926212i	$-0.623046\pi$
$197$	−10.5830			−0.754008			−0.377004	+	0.926212i	$0.623046\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	−19.8431	+	34.3693i	−1.40312	+	2.43028i
$201$		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
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−26.4575	−	45.8258i	−1.81711	−	3.14733i
$213$		0			0
$214$	7.00000	−	12.1244i	0.478510	−	0.828804i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	47.6235			3.22547
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−28.0000	−	48.4974i	−1.86253	−	3.22600i
$227$		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		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$	−84.0000			−5.51487
$233$	−10.5830	−	18.3303i	−0.693316	−	1.20086i	−0.970745	−	0.240112i	$-0.922816\pi$
$233$	−10.5830	−	18.3303i	−0.693316	−	1.20086i	0.277429	−	0.960746i	$-0.410518\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.4575			−1.71139			−0.855697	−	0.517477i	$-0.826871\pi$
$239$	−26.4575			−1.71139			−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	22.4889	−	38.9519i	1.44564	−	2.50392i
$243$		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
$253$	−28.0000			−1.76034
$254$	21.1660	+	36.6606i	1.32807	+	2.30029i
$255$		0			0
$256$	2.50000	−	4.33013i	0.156250	−	0.270633i
$257$		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
$263$	2.64575	−	4.58258i	0.163144	−	0.282574i	−0.772851	−	0.634588i	$-0.781170\pi$
$263$	2.64575	−	4.58258i	0.163144	−	0.282574i	0.935995	+	0.352014i	$0.114503\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−10.0000	−	17.3205i	−0.610847	−	1.05802i
$269$		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		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$	56.0000			3.38308
$275$	13.2288	+	22.9129i	0.797724	+	1.38170i
$276$		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
$281$	21.1660			1.26266			0.631329	−	0.775515i	$-0.282510\pi$
$281$	21.1660			1.26266			0.631329	+	0.775515i	$0.282510\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	−13.2288	+	22.9129i	−0.784982	+	1.35963i
$285$		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
$296$	23.8118	+	41.2432i	1.38403	+	2.39721i
$297$		0			0
$298$	−14.0000	+	24.2487i	−0.810998	+	1.40469i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−63.4980			−3.65390
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		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		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$	40.0000			2.25018
$317$	−5.29150	−	9.16515i	−0.297200	−	0.514766i	0.678294	−	0.734791i	$-0.262720\pi$
$317$	−5.29150	−	9.16515i	−0.297200	−	0.514766i	−0.975494	+	0.220024i	$0.929386\pi$
$318$		0			0
$319$	−28.0000	+	48.4974i	−1.56770	+	2.71533i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−26.4575	+	45.8258i	−1.46535	+	2.53805i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.0000	+	31.1769i	0.989369	+	1.71364i	0.620625	+	0.784107i	$0.286879\pi$
$331$	18.0000	+	31.1769i	0.989369	+	1.71364i	0.368744	+	0.929531i	$0.379788\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	−17.1974	−	29.7867i	−0.935414	−	1.62019i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−95.2470			−5.13538
$345$		0			0
$346$		0			0
$347$	−18.5203	+	32.0780i	−0.994220	+	1.72204i	−0.404128	+	0.914702i	$0.632425\pi$
$347$	−18.5203	+	32.0780i	−0.994220	+	1.72204i	−0.590091	+	0.807337i	$0.700908\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	35.0000	+	60.6218i	1.86551	+	3.23115i
$353$		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
$358$	70.0000			3.69961
$359$	18.5203	+	32.0780i	0.977462	+	1.69301i	0.671559	+	0.740951i	$0.265625\pi$
$359$	18.5203	+	32.0780i	0.977462	+	1.69301i	0.305903	+	0.952063i	$0.401042\pi$
$360$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	29.1033	−	50.4083i	1.51711	−	2.62772i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$	35.0000	−	60.6218i	1.79076	−	3.10168i
$383$		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
$386$	47.6235			2.42397
$387$		0			0
$388$		0			0
$389$	5.29150	−	9.16515i	0.268290	−	0.464692i	−0.700130	−	0.714015i	$-0.746875\pi$
$389$	5.29150	−	9.16515i	0.268290	−	0.464692i	0.968420	+	0.249323i	$0.0802082\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−14.0000	−	24.2487i	−0.705310	−	1.22163i
$395$		0			0
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−55.0000			−2.75000
$401$	10.5830	+	18.3303i	0.528490	+	0.915372i	0.999448	+	0.0332161i	$0.0105750\pi$
$401$	10.5830	+	18.3303i	0.528490	+	0.915372i	−0.470958	+	0.882156i	$0.656092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	31.7490			1.57374
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		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	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$	15.8745	+	27.4955i	0.772759	+	1.33846i
$423$		0			0
$424$	42.0000	−	72.7461i	2.03970	−	3.53286i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	26.4575			1.27887
$429$		0			0
$430$		0			0
$431$	−13.2288	+	22.9129i	−0.637207	+	1.10367i	0.348836	+	0.937184i	$0.386577\pi$
$431$	−13.2288	+	22.9129i	−0.637207	+	1.10367i	−0.986043	+	0.166491i	$0.946756\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	45.0000	+	77.9423i	2.15511	+	3.73276i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.5203	−	32.0780i	−0.879924	−	1.52407i	−0.851423	−	0.524479i	$-0.824260\pi$
$443$	−18.5203	−	32.0780i	−0.879924	−	1.52407i	−0.0285009	−	0.999594i	$-0.509073\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	42.3320			1.99777			0.998886	−	0.0471929i	$-0.0150276\pi$
$449$	42.3320			1.99777			0.998886	+	0.0471929i	$0.0150276\pi$
$450$		0			0
$451$		0			0
$452$	52.9150	−	91.6515i	2.48891	−	4.31092i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.00000	−	5.19615i	−0.140334	−	0.243066i	0.787288	−	0.616585i	$-0.211484\pi$
$457$	−3.00000	−	5.19615i	−0.140334	−	0.243066i	−0.927622	+	0.373519i	$0.878151\pi$
$458$		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	$0.879743\pi$
$464$	−58.2065	−	100.817i	−2.70217	−	4.68030i
$465$		0			0
$466$	28.0000	−	48.4974i	1.29707	−	2.24660i
$467$		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
$473$	−31.7490	+	54.9909i	−1.45982	+	2.52848i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−35.0000	−	60.6218i	−1.60086	−	2.77278i
$479$		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
$484$	85.0000			3.86364
$485$		0			0
$486$		0			0
$487$	−12.0000	+	20.7846i	−0.543772	+	0.941841i	0.454911	+	0.890537i	$0.349671\pi$
$487$	−12.0000	+	20.7846i	−0.543772	+	0.941841i	−0.998683	+	0.0513038i	$0.983662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.29150			−0.238802			−0.119401	−	0.992846i	$-0.538097\pi$
$491$	−5.29150			−0.238802			−0.119401	+	0.992846i	$0.538097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	0.915756	+	0.401735i	$0.131593\pi$
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	−0.109965	+	0.993935i	$0.535074\pi$
$500$		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
$506$	−37.0405	−	64.1561i	−1.64665	−	2.85208i
$507$		0			0
$508$	−40.0000	+	69.2820i	−1.77471	+	3.07389i
$509$		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
$512$	29.1033			1.28619
$513$		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		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$	14.0000			0.610429
$527$		0			0
$528$		0			0
$529$	−2.50000	+	4.33013i	−0.108696	+	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	15.8745	−	27.4955i	0.685674	−	1.18762i
$537$		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	$0.110215\pi$
$548$	52.9150	+	91.6515i	2.26042	+	3.91516i
$549$		0			0
$550$	−35.0000	+	60.6218i	−1.49241	+	2.58492i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	26.4575			1.12407
$555$		0			0
$556$		0			0
$557$	5.29150	−	9.16515i	0.224208	−	0.388340i	−0.731873	−	0.681441i	$-0.761354\pi$
$557$	5.29150	−	9.16515i	0.224208	−	0.388340i	0.956082	+	0.293101i	$0.0946871\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	28.0000	+	48.4974i	1.18111	+	2.04574i
$563$		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
$568$	−42.0000			−1.76228
$569$	21.1660	+	36.6606i	0.887325	+	1.53689i	0.843025	+	0.537874i	$0.180772\pi$
$569$	21.1660	+	36.6606i	0.887325	+	1.53689i	0.0443003	+	0.999018i	$0.485894\pi$
$570$		0			0
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	0.821138	+	0.570730i	$0.193340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26.4575			1.10335
$576$		0			0
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	−22.4889	+	38.9519i	−0.935414	+	1.62019i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−28.0000	−	48.4974i	−1.15964	−	2.00856i
$584$		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
$592$	−33.0000	+	57.1577i	−1.35629	+	2.34917i
$593$		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
$596$	−52.9150			−2.16748
$597$		0			0
$598$		0			0
$599$	18.5203	−	32.0780i	0.756717	−	1.31067i	−0.187799	−	0.982208i	$-0.560135\pi$
$599$	18.5203	−	32.0780i	0.756717	−	1.31067i	0.944516	−	0.328465i	$-0.106531\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$	−60.0000	−	103.923i	−2.44137	−	4.22857i
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		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	$-0.554882\pi$
$613$	19.0000	−	32.9090i	0.767403	−	1.32918i	0.938967	−	0.344008i	$-0.111785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.3320			−1.70422			−0.852111	−	0.523360i	$-0.824678\pi$
$617$	−42.3320			−1.70422			−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		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	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	31.7490	+	54.9909i	1.26291	+	2.18742i
$633$		0			0
$634$	14.0000	−	24.2487i	0.556011	−	0.963039i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−148.162			−5.86579
$639$		0			0
$640$		0			0
$641$	−10.5830	+	18.3303i	−0.418004	+	0.724003i	−0.995739	−	0.0922210i	$-0.970603\pi$
$641$	−10.5830	+	18.3303i	−0.418004	+	0.724003i	0.577735	+	0.816224i	$0.303937\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		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
$652$	−100.000			−3.91630
$653$	−5.29150	−	9.16515i	−0.207072	−	0.358660i	0.743719	−	0.668493i	$-0.233060\pi$
$653$	−5.29150	−	9.16515i	−0.207072	−	0.358660i	−0.950791	+	0.309833i	$0.899727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26.4575			1.03064			0.515319	−	0.856998i	$-0.327673\pi$
$659$	26.4575			1.03064			0.515319	+	0.856998i	$0.327673\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	−47.6235	+	82.4864i	−1.85094	+	3.20592i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	28.0000	+	48.4974i	1.08416	+	1.87783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−30.0000			−1.15642			−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000			−1.15642			−0.578208	+	0.815890i	$0.696248\pi$
$674$	−39.6863	−	68.7386i	−1.52866	−	2.64771i
$675$		0			0
$676$	32.5000	−	56.2917i	1.25000	−	2.16506i
$677$		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
$683$	2.64575	−	4.58258i	0.101237	−	0.175347i	−0.810958	−	0.585105i	$-0.801053\pi$
$683$	2.64575	−	4.58258i	0.101237	−	0.175347i	0.912194	+	0.409757i	$0.134387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−66.0000	−	114.315i	−2.51623	−	4.35823i
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	−98.0000			−3.72003
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−52.9150			−1.99857			−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	−52.9150			−1.99857			−0.999286	+	0.0377695i	$0.987975\pi$
$702$		0			0
$703$		0			0
$704$	−34.3948	+	59.5735i	−1.29630	+	2.24526i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	66.1438	+	114.564i	2.47191	+	4.28147i
$717$		0			0
$718$	−49.0000	+	84.8705i	−1.82866	+	3.16734i
$719$		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
$722$	50.2693			1.87083
$723$		0			0
$724$		0			0
$725$	26.4575	−	45.8258i	0.982607	−	1.70193i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$	70.0000			2.58023
$737$	−10.5830	−	18.3303i	−0.389830	−	0.675205i
$738$		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
$743$	−37.0405			−1.35888			−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405			−1.35888			−0.679442	+	0.733729i	$0.737778\pi$
$744$		0			0
$745$		0			0
$746$	29.1033	−	50.4083i	1.06555	−	1.84558i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.855938	−	0.517079i	$-0.827019\pi$
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.0198348	−	0.999803i	$-0.506314\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	54.0000			1.96266			0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000			1.96266			0.981332	+	0.192323i	$0.0616021\pi$
$758$	15.8745	+	27.4955i	0.576588	+	0.998680i
$759$		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
$764$	132.288			4.78600
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	45.0000	+	77.9423i	1.61959	+	2.80520i
$773$		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
$778$	28.0000			1.00385
$779$		0			0
$780$		0			0
$781$	−14.0000	+	24.2487i	−0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	26.4575	−	45.8258i	0.942510	−	1.63247i
$789$		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
$800$	−33.0719	−	57.2822i	−1.16927	−	2.02523i
$801$		0			0
$802$	−28.0000	+	48.4974i	−0.988714	+	1.71250i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21.1660	−	36.6606i	0.744157	−	1.28892i	−0.206430	−	0.978461i	$-0.566185\pi$
$809$	21.1660	−	36.6606i	0.744157	−	1.28892i	0.950587	−	0.310457i	$-0.100482\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	42.0000	+	72.7461i	1.47210	+	2.54975i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.4575	−	45.8258i	−0.923374	−	1.59933i	−0.794156	−	0.607714i	$-0.792087\pi$
$821$	−26.4575	−	45.8258i	−0.923374	−	1.59933i	−0.129217	−	0.991616i	$-0.541246\pi$
$822$		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
$827$	−37.0405			−1.28803			−0.644013	−	0.765015i	$-0.722732\pi$
$827$	−37.0405			−1.28803			−0.644013	+	0.765015i	$0.722732\pi$
$828$		0			0
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		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
$841$	83.0000			2.86207
$842$	−34.3948	−	59.5735i	−1.18532	−	2.05304i
$843$		0			0
$844$	−30.0000	+	51.9615i	−1.03264	+	1.78859i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	116.413			3.99764
$849$		0			0
$850$		0			0
$851$	15.8745	−	27.4955i	0.544171	−	0.942532i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	21.0000	+	36.3731i	0.717765	+	1.24321i
$857$		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		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	−70.0000			−2.38421
$863$	−29.1033	−	50.4083i	−0.990687	−	1.71592i	−0.613263	−	0.789879i	$-0.710143\pi$
$863$	−29.1033	−	50.4083i	−0.990687	−	1.71592i	−0.377424	−	0.926041i	$-0.623190\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	42.3320			1.43602
$870$		0			0
$871$		0			0
$872$	−71.4353	+	123.730i	−2.41910	+	4.19001i
$873$		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
$883$	12.0000			0.403832			0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000			0.403832			0.201916	+	0.979403i	$0.435283\pi$
$884$		0			0
$885$		0			0
$886$	49.0000	−	84.8705i	1.64619	−	2.85128i
$887$		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
$898$	56.0000	+	96.9948i	1.86874	+	3.23676i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	168.000			5.58760
$905$		0			0
$906$		0			0
$907$	30.0000	−	51.9615i	0.996134	−	1.72535i	0.421986	−	0.906602i	$-0.361333\pi$
$907$	30.0000	−	51.9615i	0.996134	−	1.72535i	0.574148	−	0.818752i	$-0.305333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−58.2065			−1.92847			−0.964234	−	0.265052i	$-0.914611\pi$
$911$	−58.2065			−1.92847			−0.964234	+	0.265052i	$0.914611\pi$
$912$		0			0
$913$		0			0
$914$	7.93725	−	13.7477i	0.262541	−	0.454734i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000	−	41.5692i	−0.791687	−	1.37124i	−0.924922	−	0.380158i	$-0.875870\pi$
$919$	−24.0000	−	41.5692i	−0.791687	−	1.37124i	0.133235	−	0.991084i	$-0.457464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−30.0000			−0.986394
$926$	−52.9150	−	91.6515i	−1.73890	−	3.01186i
$927$		0			0
$928$	70.0000	−	121.244i	2.29786	−	3.98001i
$929$		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
$932$	105.830			3.46658
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		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
$946$	−168.000			−5.46215
$947$	29.1033	+	50.4083i	0.945729	+	1.63805i	0.754285	+	0.656547i	$0.227984\pi$
$947$	29.1033	+	50.4083i	0.945729	+	1.63805i	0.191444	+	0.981504i	$0.438683\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.1660			0.685634			0.342817	−	0.939402i	$-0.388619\pi$
$953$	21.1660			0.685634			0.342817	+	0.939402i	$0.388619\pi$
$954$		0			0
$955$		0			0
$956$	66.1438	−	114.564i	2.13924	−	3.70528i
$957$		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	$0.722376\pi$
$968$	67.4667	+	116.856i	2.16846	+	3.75588i
$969$		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
$974$	−63.4980			−2.03461
$975$		0			0
$976$		0			0
$977$	21.1660	−	36.6606i	0.677161	−	1.17288i	−0.298672	−	0.954356i	$-0.596544\pi$
$977$	21.1660	−	36.6606i	0.677161	−	1.17288i	0.975832	−	0.218521i	$-0.0701231\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−7.00000	−	12.1244i	−0.223379	−	0.386904i
$983$		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
$989$	31.7490	+	54.9909i	1.00956	+	1.74861i
$990$		0			0
$991$	−12.0000	+	20.7846i	−0.381193	+	0.660245i	−0.991233	−	0.132125i	$-0.957820\pi$
$991$	−12.0000	+	20.7846i	−0.381193	+	0.660245i	0.610040	+	0.792370i	$0.291153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	−47.6235	+	82.4864i	−1.50750	+	2.61106i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.e.h.361.2		4
3.2	odd	2	inner	441.2.e.h.361.1		4
7.2	even	3	inner	441.2.e.h.226.2		4
7.3	odd	6		441.2.a.h.1.1	✓	2
7.4	even	3		441.2.a.h.1.1	✓	2
7.5	odd	6	inner	441.2.e.h.226.2		4
7.6	odd	2	CM	441.2.e.h.361.2		4
21.2	odd	6	inner	441.2.e.h.226.1		4
21.5	even	6	inner	441.2.e.h.226.1		4
21.11	odd	6		441.2.a.h.1.2	yes	2
21.17	even	6		441.2.a.h.1.2	yes	2
21.20	even	2	inner	441.2.e.h.361.1		4
28.3	even	6		7056.2.a.co.1.1		2
28.11	odd	6		7056.2.a.co.1.1		2
84.11	even	6		7056.2.a.co.1.2		2
84.59	odd	6		7056.2.a.co.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.a.h.1.1	✓	2	7.3	odd	6
441.2.a.h.1.1	✓	2	7.4	even	3
441.2.a.h.1.2	yes	2	21.11	odd	6
441.2.a.h.1.2	yes	2	21.17	even	6
441.2.e.h.226.1		4	21.2	odd	6	inner
441.2.e.h.226.1		4	21.5	even	6	inner
441.2.e.h.226.2		4	7.2	even	3	inner
441.2.e.h.226.2		4	7.5	odd	6	inner
441.2.e.h.361.1		4	3.2	odd	2	inner
441.2.e.h.361.1		4	21.20	even	2	inner
441.2.e.h.361.2		4	1.1	even	1	trivial
441.2.e.h.361.2		4	7.6	odd	2	CM
7056.2.a.co.1.1		2	28.3	even	6
7056.2.a.co.1.1		2	28.11	odd	6
7056.2.a.co.1.2		2	84.11	even	6
7056.2.a.co.1.2		2	84.59	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(1.32288 + 2.29129i) q^{2} +(-2.50000 + 4.33013i) q^{4} -7.93725 q^{8} +(-2.64575 + 4.58258i) q^{11} +(-5.50000 - 9.52628i) q^{16} -14.0000 q^{22} +(2.64575 + 4.58258i) q^{23} +(2.50000 - 4.33013i) q^{25} +10.5830 q^{29} +(6.61438 - 11.4564i) q^{32} +(-3.00000 - 5.19615i) q^{37} +12.0000 q^{43} +(-13.2288 - 22.9129i) q^{44} +(-7.00000 + 12.1244i) q^{46} +13.2288 q^{50} +(-5.29150 + 9.16515i) q^{53} +(14.0000 + 24.2487i) q^{58} +13.0000 q^{64} +(-2.00000 + 3.46410i) q^{67} +5.29150 q^{71} +(7.93725 - 13.7477i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(15.8745 + 27.4955i) q^{86} +(21.0000 - 36.3731i) q^{88} -26.4575 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 22 q^{16} - 56 q^{22} + 10 q^{25} - 12 q^{37} + 48 q^{43} - 28 q^{46} + 56 q^{58} + 52 q^{64} - 8 q^{67} - 16 q^{79} + 84 q^{88}+O(q^{100}) \) 4 * q - 10 * q^4 - 22 * q^16 - 56 * q^22 + 10 * q^25 - 12 * q^37 + 48 * q^43 - 28 * q^46 + 56 * q^58 + 52 * q^64 - 8 * q^67 - 16 * q^79 + 84 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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