LMFDB - Embedded newform 605.2.m.a.112.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(112,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([5, 2]))

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

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

chi := DirichletCharacter("605.112");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.m (of order $20$, degree $8$, 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:	$4.83094932229$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	16.0.3429742096000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 $ x^16 + 5x^14 + 16x^12 + 35x^10 + 31x^8 + 315x^6 + 1296x^4 + 3645*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			112.1
Root			$-1.04771 + 1.37924i$ of defining polynomial
Character	$\chi$	$=$	605.112
Dual form			605.2.m.a.578.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.256255 + 1.61793i) q^{3} +(1.17557 - 1.61803i) q^{4} +(0.914138 - 2.04067i) q^{5} +(0.301128 + 0.0978424i) q^{9} +(2.31662 + 2.31662i) q^{12} +(3.06742 + 2.00195i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(-2.22725 - 3.87806i) q^{20} +(6.15831 - 6.15831i) q^{23} +(-3.32870 - 3.73092i) q^{25} +(-2.46652 + 4.84081i) q^{27} +(3.07468 - 9.46289i) q^{31} +(0.512310 - 0.372215i) q^{36} +(1.87493 + 11.8378i) q^{37} +(0.474937 - 0.525063i) q^{45} +(3.74814 + 0.593648i) q^{47} +(6.47173 - 1.02502i) q^{48} +(-6.65740 + 2.16312i) q^{49} +(6.18493 + 12.1386i) q^{53} +(1.94946 - 2.68321i) q^{59} +(6.84519 - 2.60976i) q^{60} +(-7.60845 - 2.47214i) q^{64} +(1.52506 + 1.52506i) q^{67} +(8.38564 + 11.5418i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(6.88937 - 4.42955i) q^{75} +(-8.89312 - 0.955176i) q^{80} +(-6.43158 - 4.67282i) q^{81} +9.00000i q^{89} +(-2.72483 - 17.2039i) q^{92} +(14.5224 + 7.39955i) q^{93} +(-16.9794 + 8.65144i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) $ 16 * q + 2 * q^3 - 16 * q^12 + 8 * q^15 + 16 * q^16 - 12 * q^20 + 72 * q^23 - 2 * q^25 - 22 * q^27 - 24 * q^36 + 14 * q^37 - 72 * q^45 + 24 * q^47 - 8 * q^48 + 12 * q^53 + 28 * q^60 + 104 * q^67 + 12 * q^71 - 32 * q^75 + 8 * q^81 - 36 * q^92 - 66 * q^93 + 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{10}\right)$

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		0.891007	−	0.453990i	$-0.150000\pi$
$2$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$3$	−0.256255	+	1.61793i	−0.147949	+	0.934114i	0.796305	+	0.604896i	$0.206785\pi$
$3$	−0.256255	+	1.61793i	−0.147949	+	0.934114i	−0.944254	+	0.329218i	$0.893215\pi$
$4$	1.17557	−	1.61803i	0.587785	−	0.809017i
$5$	0.914138	−	2.04067i	0.408815	−	0.912617i
$5$	0.914138	−	2.04067i	0.408815	−	0.912617i
$6$		0			0
$7$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$7$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$8$		0			0
$9$	0.301128	+	0.0978424i	0.100376	+	0.0326141i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	2.31662	+	2.31662i	0.668752	+	0.668752i
$13$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$13$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$14$		0			0
$15$	3.06742	+	2.00195i	0.792005	+	0.516901i
$16$	−1.23607	−	3.80423i	−0.309017	−	0.951057i
$17$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$17$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	−2.22725	−	3.87806i	−0.498027	−	0.867161i
$21$		0			0
$22$		0			0
$23$	6.15831	−	6.15831i	1.28410	−	1.28410i	0.345782	−	0.938315i	$-0.387614\pi$
$23$	6.15831	−	6.15831i	1.28410	−	1.28410i	0.938315	−	0.345782i	$-0.112386\pi$
$24$		0			0
$25$	−3.32870	−	3.73092i	−0.665741	−	0.746183i
$26$		0			0
$27$	−2.46652	+	4.84081i	−0.474681	+	0.931614i
$28$		0			0
$29$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$29$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$	3.07468	−	9.46289i	0.552229	−	1.69959i	−0.150923	−	0.988546i	$-0.548224\pi$
$31$	3.07468	−	9.46289i	0.552229	−	1.69959i	0.703151	−	0.711040i	$-0.251776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.512310	−	0.372215i	0.0853849	−	0.0620358i
$37$	1.87493	+	11.8378i	0.308236	+	1.94612i	0.323465	+	0.946240i	$0.395152\pi$
$37$	1.87493	+	11.8378i	0.308236	+	1.94612i	−0.0152291	+	0.999884i	$0.504848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	0.474937	−	0.525063i	0.0707995	−	0.0782717i
$46$		0			0
$47$	3.74814	+	0.593648i	0.546723	+	0.0865924i	0.423685	−	0.905810i	$-0.360736\pi$
$47$	3.74814	+	0.593648i	0.546723	+	0.0865924i	0.123038	+	0.992402i	$0.460736\pi$
$48$	6.47173	−	1.02502i	0.934114	−	0.147949i
$49$	−6.65740	+	2.16312i	−0.951057	+	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.18493	+	12.1386i	0.849565	+	1.66737i	0.739212	+	0.673473i	$0.235198\pi$
$53$	6.18493	+	12.1386i	0.849565	+	1.66737i	0.110353	+	0.993892i	$0.464802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.94946	−	2.68321i	0.253798	−	0.349324i	−0.663039	−	0.748585i	$-0.730733\pi$
$59$	1.94946	−	2.68321i	0.253798	−	0.349324i	0.916837	+	0.399262i	$0.130733\pi$
$60$	6.84519	−	2.60976i	0.883710	−	0.336919i
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$		0			0
$64$	−7.60845	−	2.47214i	−0.951057	−	0.309017i
$65$		0			0
$66$		0			0
$67$	1.52506	+	1.52506i	0.186316	+	0.186316i	0.794101	−	0.607785i	$-0.207942\pi$
$67$	1.52506	+	1.52506i	0.186316	+	0.186316i	−0.607785	+	0.794101i	$0.707942\pi$
$68$		0			0
$69$	8.38564	+	11.5418i	1.00951	+	1.38947i
$70$		0			0
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	0.880855	−	0.473386i	$-0.156968\pi$
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	−0.990876	+	0.134777i	$0.956968\pi$
$72$		0			0
$73$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$73$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$74$		0			0
$75$	6.88937	−	4.42955i	0.795516	−	0.511481i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$	−8.89312	−	0.955176i	−0.994281	−	0.106792i
$81$	−6.43158	−	4.67282i	−0.714620	−	0.519202i
$82$		0			0
$83$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$83$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$		0			0
$92$	−2.72483	−	17.2039i	−0.284083	−	1.79363i
$93$	14.5224	+	7.39955i	1.50591	+	0.767297i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.9794	+	8.65144i	−1.72400	+	0.878421i	−0.747101	+	0.664711i	$0.768555\pi$
$97$	−16.9794	+	8.65144i	−1.72400	+	0.878421i	−0.976897	+	0.213710i	$0.931445\pi$
$98$		0			0
$99$		0			0
$100$	−9.94987	+	1.00000i	−0.994987	+	0.100000i
$101$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$101$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$102$		0			0
$103$	−16.6916	+	2.64369i	−1.64467	+	0.260491i	−0.908985	−	0.416829i	$-0.863141\pi$
$103$	−16.6916	+	2.64369i	−1.64467	+	0.260491i	−0.735689	+	0.677320i	$0.763141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$107$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$108$	4.93303	+	9.68162i	0.474681	+	0.931614i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−19.6332			−1.86351
$112$		0			0
$113$	1.95606	−	12.3501i	0.184011	−	1.16180i	−0.706796	−	0.707417i	$-0.749860\pi$
$113$	1.95606	−	12.3501i	0.184011	−	1.16180i	0.890807	−	0.454382i	$-0.150140\pi$
$114$		0			0
$115$	−6.93756	−	18.1967i	−0.646931	−	1.69685i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−11.6968	−	16.0992i	−1.05040	−	1.44575i
$125$	−10.6565	+	3.38223i	−0.953144	+	0.302516i
$126$		0			0
$127$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$127$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	7.62378	+	9.45852i	0.656150	+	0.814060i
$136$		0			0
$137$	−8.41597	+	16.5173i	−0.719025	+	1.41117i	0.184588	+	0.982816i	$0.440905\pi$
$137$	−8.41597	+	16.5173i	−0.719025	+	1.41117i	−0.903613	+	0.428350i	$0.859095\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$	−1.92097	+	5.91212i	−0.161774	+	0.497891i
$142$		0			0
$143$		0			0
$144$		−	1.26650i		−	0.105542i
$145$		0			0
$146$		0			0
$147$	−1.79379	−	11.3255i	−0.147949	−	0.934114i
$148$	21.3581	+	10.8825i	1.75562	+	0.894535i
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$		0			0
$151$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$151$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5000	−	14.9248i	−1.32531	−	1.19879i
$156$		0			0
$157$	−9.11422	−	1.44355i	−0.727394	−	0.115208i	−0.218255	−	0.975892i	$-0.570036\pi$
$157$	−9.11422	−	1.44355i	−0.727394	−	0.115208i	−0.509140	+	0.860684i	$0.670036\pi$
$158$		0			0
$159$	−21.2244	+	6.89622i	−1.68320	+	0.546906i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.25190	−	2.45699i	−0.0980561	−	0.192446i	0.836768	−	0.547558i	$-0.184443\pi$
$163$	−1.25190	−	2.45699i	−0.0980561	−	0.192446i	−0.934824	+	0.355112i	$0.884443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$167$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$168$		0			0
$169$	−7.64121	+	10.5172i	−0.587785	+	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$173$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.84169	+	3.84169i	0.288759	+	0.288759i
$178$		0			0
$179$	12.3435	+	16.9894i	0.922596	+	1.26984i	0.962678	+	0.270648i	$0.0872379\pi$
$179$	12.3435	+	16.9894i	0.922596	+	1.26984i	−0.0400827	+	0.999196i	$0.512762\pi$
$180$	−0.291247	−	1.38571i	−0.0217083	−	0.103285i
$181$	3.07468	+	9.46289i	0.228539	+	0.703371i	0.997913	+	0.0645725i	$0.0205684\pi$
$181$	3.07468	+	9.46289i	0.228539	+	0.703371i	−0.769374	+	0.638799i	$0.779432\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	25.8711	+	6.99528i	1.90208	+	0.514303i
$186$		0			0
$187$		0			0
$188$	5.36675	−	5.36675i	0.391411	−	0.391411i
$189$		0			0
$190$		0			0
$191$	18.7824	+	13.6462i	1.35905	+	0.987407i	0.998505	+	0.0546656i	$0.0174093\pi$
$191$	18.7824	+	13.6462i	1.35905	+	0.987407i	0.360545	+	0.932742i	$0.382591\pi$
$192$	5.94946	−	11.6765i	0.429365	−	0.842677i
$193$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$193$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$194$		0			0
$195$		0			0
$196$	−4.32624	+	13.3148i	−0.309017	+	0.951057i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			19.8997i			1.41066i	0.708881	+	0.705328i	$0.249200\pi$
$199$			19.8997i			1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−2.85826	+	2.07664i	−0.201606	+	0.146475i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.45699	−	1.25190i	0.170772	−	0.0870128i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$	26.9115	+	4.26236i	1.84829	+	0.292740i
$213$	4.85380	−	0.768766i	0.332577	−	0.0526750i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.41246	−	21.5454i	0.228515	−	1.44279i	−0.560368	−	0.828244i	$-0.689340\pi$
$223$	3.41246	−	21.5454i	0.228515	−	1.44279i	0.788883	−	0.614544i	$-0.210660\pi$
$224$		0			0
$225$	−0.637324	−	1.44917i	−0.0424883	−	0.0966115i
$226$		0			0
$227$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$227$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$228$		0			0
$229$	−28.3887	−	9.22404i	−1.87598	−	0.609542i	−0.989039	−	0.147652i	$-0.952828\pi$
$229$	−28.3887	−	9.22404i	−1.87598	−	0.609542i	−0.886937	−	0.461890i	$-0.847172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$233$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$234$		0			0
$235$	4.63776	−	7.10607i	0.302534	−	0.463549i
$236$	−2.04979	−	6.30860i	−0.133430	−	0.410655i
$237$		0			0
$238$		0			0
$239$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$239$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$240$	3.82432	−	14.1437i	0.246859	−	0.912973i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−2.31662	+	2.31662i	−0.148612	+	0.148612i
$244$		0			0
$245$	−1.67156	+	15.5630i	−0.106792	+	0.994281i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.34346	−	25.6785i	0.526634	−	1.62081i	−0.234427	−	0.972134i	$-0.575321\pi$
$251$	8.34346	−	25.6785i	0.526634	−	1.62081i	0.761061	−	0.648680i	$-0.224679\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	0.943885	+	5.95946i	0.0588779	+	0.371741i	0.999480	+	0.0322308i	$0.0102612\pi$
$257$	0.943885	+	5.95946i	0.0588779	+	0.371741i	−0.940603	+	0.339510i	$0.889739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	30.4248	−	1.52506i	1.86898	−	0.0936839i
$266$		0			0
$267$	−14.5614	−	2.30630i	−0.891143	−	0.141143i
$268$	4.26042	−	0.674785i	0.260247	−	0.0412190i
$269$	12.6172	−	4.09957i	0.769284	−	0.249955i	0.102025	−	0.994782i	$-0.467468\pi$
$269$	12.6172	−	4.09957i	0.769284	−	0.249955i	0.667258	+	0.744826i	$0.267468\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	28.5330			1.71748
$277$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$277$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$278$		0			0
$279$	1.85175	−	2.54871i	0.110861	−	0.152587i
$280$		0			0
$281$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$281$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$283$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$284$	−5.70634	−	1.85410i	−0.338609	−	0.110021i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.99235	−	13.7533i	−0.587785	−	0.809017i
$290$		0			0
$291$	−9.64639	−	29.6885i	−0.565481	−	1.74037i
$292$		0			0
$293$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$293$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$294$		0			0
$295$	−3.69347	−	6.43104i	−0.215042	−	0.374430i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.931776	−	16.3545i	0.0537961	−	0.944227i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		−	27.6834i		−	1.57485i
$310$		0			0
$311$	−9.70820	+	7.05342i	−0.550502	+	0.399963i	−0.827970	−	0.560772i	$-0.810505\pi$
$311$	−9.70820	+	7.05342i	−0.550502	+	0.399963i	0.277469	+	0.960735i	$0.410505\pi$
$312$		0			0
$313$	−6.83566	−	3.48294i	−0.386374	−	0.196868i	0.250004	−	0.968245i	$-0.419568\pi$
$313$	−6.83566	−	3.48294i	−0.386374	−	0.196868i	−0.636378	+	0.771377i	$0.719568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.38382	+	1.21462i	−0.133889	+	0.0682198i	−0.519652	−	0.854378i	$-0.673938\pi$
$317$	−2.38382	+	1.21462i	−0.133889	+	0.0682198i	0.385763	+	0.922598i	$0.373938\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	+	13.2665i	−0.670820	+	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−15.1216	+	4.91329i	−0.840087	+	0.272961i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.94987			0.546895			0.273447	−	0.961887i	$-0.411836\pi$
$331$	9.94987			0.546895			0.273447	+	0.961887i	$0.411836\pi$
$332$		0			0
$333$	−0.593648	+	3.74814i	−0.0325317	+	0.205397i
$334$		0			0
$335$	4.50627	−	1.71804i	0.246204	−	0.0938665i
$336$		0			0
$337$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$337$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$338$		0			0
$339$	19.4804	+	6.32956i	1.05803	+	0.343774i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	31.2188	−	6.56152i	1.68076	−	0.353260i
$346$		0			0
$347$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$347$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.7414	−	22.7414i	1.21040	−	1.21040i	0.239511	−	0.970894i	$-0.423013\pi$
$353$	22.7414	−	22.7414i	1.21040	−	1.21040i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$		0			0
$355$	−6.66984	−	0.716382i	−0.353998	−	0.0380216i
$356$	14.5623	+	10.5801i	0.771801	+	0.560746i
$357$		0			0
$358$		0			0
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−5.87132	+	18.0701i	−0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	5.19340	+	32.7898i	0.271093	+	1.71162i	0.628620	+	0.777713i	$0.283620\pi$
$367$	5.19340	+	32.7898i	0.271093	+	1.71162i	−0.357526	+	0.933903i	$0.616380\pi$
$368$	−31.0397	−	15.8155i	−1.61806	−	0.824441i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	29.0449	−	14.7991i	1.50591	−	0.767297i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	−2.74144	−	18.1082i	−0.141567	−	0.935103i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	28.3887	−	9.22404i	1.45823	−	0.473807i	0.530700	−	0.847560i	$-0.321929\pi$
$379$	28.3887	−	9.22404i	1.45823	−	0.473807i	0.927528	+	0.373753i	$0.121929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.5845	−	26.6610i	−0.694134	−	1.36231i	−0.921451	−	0.388494i	$-0.872995\pi$
$383$	−13.5845	−	26.6610i	−0.694134	−	1.36231i	0.227317	−	0.973821i	$-0.427005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−5.96217	+	37.6436i	−0.302683	+	1.91107i
$389$	21.4441	−	29.5153i	1.08726	−	1.49648i	0.235988	−	0.971756i	$-0.424167\pi$
$389$	21.4441	−	29.5153i	1.08726	−	1.49648i	0.851270	−	0.524727i	$-0.175833\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.8997	+	18.8997i	0.948551	+	0.948551i	0.998740	−	0.0501886i	$-0.0159822\pi$
$397$	18.8997	+	18.8997i	0.948551	+	0.948551i	−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$		0			0
$400$	−10.0787	+	17.2748i	−0.503937	+	0.863740i
$401$	8.19915	+	25.2344i	0.409446	+	1.26014i	0.917125	+	0.398599i	$0.130503\pi$
$401$	8.19915	+	25.2344i	0.409446	+	1.26014i	−0.507679	+	0.861546i	$0.669497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−15.4151	+	8.85317i	−0.765980	+	0.439917i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$	−24.5672	−	17.8491i	−1.21181	−	0.880433i
$412$	−15.3446	+	30.1154i	−0.755973	+	1.48368i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	32.1985	−	23.3936i	1.56926	−	1.14013i	0.641394	−	0.767211i	$-0.278356\pi$
$421$	32.1985	−	23.3936i	1.56926	−	1.14013i	0.927863	−	0.372921i	$-0.121644\pi$
$422$		0			0
$423$	1.07059	+	0.545492i	0.0520537	+	0.0265227i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$431$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$432$	21.4643	+	3.39961i	1.03270	+	0.163564i
$433$	0.593378	−	0.0939818i	0.0285159	−	0.00451648i	−0.142160	−	0.989844i	$-0.545405\pi$
$433$	0.593378	−	0.0939818i	0.0285159	−	0.00451648i	0.170676	+	0.985327i	$0.445405\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−2.21637			−0.105542
$442$		0			0
$443$	−1.71265	+	10.8133i	−0.0813705	+	0.513753i	0.913014	+	0.407928i	$0.133748\pi$
$443$	−1.71265	+	10.8133i	−0.0813705	+	0.513753i	−0.994385	+	0.105825i	$0.966252\pi$
$444$	−23.0803	+	31.7673i	−1.09534	+	1.50761i
$445$	18.3661	+	8.22724i	0.870635	+	0.390009i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	37.0912	+	12.0517i	1.75044	+	0.568753i	0.996140	−	0.0877747i	$-0.0279756\pi$
$449$	37.0912	+	12.0517i	1.75044	+	0.568753i	0.754302	+	0.656528i	$0.227976\pi$
$450$		0			0
$451$		0			0
$452$	−17.6834	−	17.6834i	−0.831756	−	0.831756i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$457$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$458$		0			0
$459$		0			0
$460$	−37.5984	−	10.1662i	−1.75303	−	0.474004i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−30.4248	+	30.4248i	−1.41396	+	1.41396i	−0.693615	+	0.720346i	$0.743983\pi$
$463$	−30.4248	+	30.4248i	−1.41396	+	1.41396i	−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	28.3756	−	22.8713i	1.31589	−	1.06063i
$466$		0			0
$467$	12.8781	−	25.2746i	0.595926	−	1.16957i	−0.374286	−	0.927313i	$-0.622112\pi$
$467$	12.8781	−	25.2746i	0.595926	−	1.16957i	0.970212	−	0.242257i	$-0.0778878\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4.67114	−	14.3763i	0.215235	−	0.662424i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.674785	+	4.26042i	0.0308963	+	0.195071i
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.13325	+	42.5581i	0.0968659	+	1.93246i
$486$		0			0
$487$	−23.0822	−	3.65587i	−1.04596	−	0.165663i	−0.390273	−	0.920699i	$-0.627619\pi$
$487$	−23.0822	−	3.65587i	−1.04596	−	0.165663i	−0.655683	+	0.755036i	$0.727619\pi$
$488$		0			0
$489$	4.29604	−	1.39587i	0.194274	−	0.0631234i
$490$		0			0
$491$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$491$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−39.7995			−1.78705
$497$		0			0
$498$		0			0
$499$	11.6968	−	16.0992i	0.523620	−	0.720701i	−0.462522	−	0.886608i	$-0.653055\pi$
$499$	11.6968	−	16.0992i	0.523620	−	0.720701i	0.986141	+	0.165907i	$0.0530552\pi$
$500$	−7.05488	+	21.2186i	−0.315504	+	0.948924i
$501$		0			0
$502$		0			0
$503$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$503$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.0581	−	15.0581i	−0.668752	−	0.668752i
$508$		0			0
$509$	−1.94946	−	2.68321i	−0.0864084	−	0.118931i	0.763624	−	0.645661i	$-0.223418\pi$
$509$	−1.94946	−	2.68321i	−0.0864084	−	0.118931i	−0.850033	+	0.526730i	$0.823418\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.86352	+	36.4788i	−0.434639	+	1.60745i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.8817	−	25.3430i	−1.52819	−	1.11030i	−0.957232	−	0.289321i	$-0.906570\pi$
$521$	−34.8817	−	25.3430i	−1.52819	−	1.11030i	−0.570962	−	0.820977i	$-0.693430\pi$
$522$		0			0
$523$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$523$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	52.8496i		−	2.29781i
$530$		0			0
$531$	0.849569	−	0.617248i	0.0368682	−	0.0267863i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−30.6507	+	15.6173i	−1.32268	+	0.673937i
$538$		0			0
$539$		0			0
$540$	24.2665	−	1.21637i	1.04426	−	0.0523444i
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$		0			0
$543$	−16.0982	+	2.54971i	−0.690841	+	0.109419i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$547$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$548$	16.8319	+	33.0346i	0.719025	+	1.41117i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−17.9475	+	40.0651i	−0.761829	+	1.70067i
$556$		0			0
$557$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$557$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$563$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$564$	7.30779	+	10.0583i	0.307713	+	0.423531i
$565$	−23.4144	−	15.2814i	−0.985051	−	0.642892i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$569$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−26.8918	+	26.8918i	−1.12342	+	1.12342i
$574$		0			0
$575$	−43.4753	−	2.47695i	−1.81305	−	0.103296i
$576$	−2.04924	−	1.48886i	−0.0853849	−	0.0620358i
$577$	18.2820	−	35.8805i	0.761091	−	1.49373i	−0.105344	−	0.994436i	$-0.533594\pi$
$577$	18.2820	−	35.8805i	0.761091	−	1.49373i	0.866435	−	0.499290i	$-0.166406\pi$
$578$		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
$587$	−6.04330	−	38.1559i	−0.249434	−	1.57486i	−0.720938	−	0.693000i	$-0.756289\pi$
$587$	−6.04330	−	38.1559i	−0.249434	−	1.57486i	0.471504	−	0.881864i	$-0.343711\pi$
$588$	−20.4338	−	10.4116i	−0.842677	−	0.429365i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	42.7162	−	21.7650i	1.75562	−	0.894535i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−32.1965	−	5.09942i	−1.31771	−	0.208705i
$598$		0			0
$599$	34.2380	−	11.1246i	1.39893	−	0.454539i	0.490084	−	0.871675i	$-0.336966\pi$
$599$	34.2380	−	11.1246i	1.39893	−	0.454539i	0.908844	+	0.417136i	$0.136966\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$	0.310023	+	0.608455i	0.0126251	+	0.0247782i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$607$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$613$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.2665	−	34.2665i	−1.37952	−	1.37952i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−34.2665	−	34.2665i	−1.37952	−	1.37952i	−0.534089	−	0.845428i	$-0.679345\pi$
$618$		0			0
$619$	0.587785	+	0.809017i	0.0236251	+	0.0325171i	0.820666	−	0.571408i	$-0.193603\pi$
$619$	0.587785	+	0.809017i	0.0236251	+	0.0325171i	−0.797041	+	0.603925i	$0.793603\pi$
$620$	−43.5458	+	9.15239i	−1.74884	+	0.367569i
$621$	14.6216	+	45.0008i	0.586746	+	1.80582i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.83947	+	24.8382i	−0.113579	+	0.993529i
$626$		0			0
$627$		0			0
$628$	−13.0501	+	13.0501i	−0.520757	+	0.520757i
$629$		0			0
$630$		0			0
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	0.469329	−	0.883023i	$-0.344496\pi$
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	−0.694774	+	0.719228i	$0.744496\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−13.7924	+	42.4487i	−0.546906	+	1.68320i
$637$		0			0
$638$		0			0
$639$		−	0.949874i		−	0.0375764i
$640$		0			0
$641$	18.7824	−	13.6462i	0.741862	−	0.538994i	−0.151432	−	0.988468i	$-0.548388\pi$
$641$	18.7824	−	13.6462i	0.741862	−	0.538994i	0.893294	+	0.449474i	$0.148388\pi$
$642$		0			0
$643$	−44.6379	−	22.7441i	−1.76035	−	0.896941i	−0.950753	−	0.309948i	$-0.899688\pi$
$643$	−44.6379	−	22.7441i	−1.76035	−	0.896941i	−0.809592	−	0.586993i	$-0.800312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−44.1757	+	22.5087i	−1.73673	+	0.884907i	−0.766666	+	0.642046i	$0.778086\pi$
$647$	−44.1757	+	22.5087i	−1.73673	+	0.884907i	−0.970061	+	0.242861i	$0.921914\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−5.44718	−	0.862748i	−0.213328	−	0.0337878i
$653$	33.3021	−	5.27454i	1.30321	−	0.206409i	0.534045	−	0.845456i	$-0.320671\pi$
$653$	33.3021	−	5.27454i	1.30321	−	0.206409i	0.769167	+	0.639047i	$0.220671\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	33.9846	+	11.0423i	1.31392	+	0.426918i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$673$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$674$		0			0
$675$	26.2709	−	6.91125i	1.01117	−	0.266014i
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$677$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.2164	+	11.2164i	−0.429183	+	0.429183i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	−11.2164	+	11.2164i	−0.429183	+	0.429183i	0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	26.0130	+	32.2733i	0.993907	+	1.23310i
$686$		0			0
$687$	22.1986	−	43.5673i	0.846931	−	1.66220i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	−0.800041	−	0.599945i	$-0.795189\pi$
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	0.999886	−	0.0151132i	$-0.00481087\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	10.3087	+	9.32456i	0.388248	+	0.351183i
$706$		0			0
$707$		0			0
$708$	10.7322	−	1.69981i	0.403339	−	0.0638826i
$709$	−18.0701	+	5.87132i	−0.678636	+	0.220502i	−0.627998	−	0.778215i	$-0.716125\pi$
$709$	−18.0701	+	5.87132i	−0.678636	+	0.220502i	−0.0506378	+	0.998717i	$0.516125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−39.3406	−	77.2103i	−1.47332	−	2.89155i
$714$		0			0
$715$		0			0
$716$	42.0000			1.56961
$717$		0			0
$718$		0			0
$719$	−29.9770	+	41.2599i	−1.11796	+	1.53873i	−0.308811	+	0.951123i	$0.599931\pi$
$719$	−29.9770	+	41.2599i	−1.11796	+	1.53873i	−0.809144	+	0.587610i	$0.800069\pi$
$720$	−2.58451	−	1.15776i	−0.0963191	−	0.0431470i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	18.9258	+	6.14936i	0.703371	+	0.228539i
$725$		0			0
$726$		0			0
$727$	21.5251	+	21.5251i	0.798320	+	0.798320i	0.982831	−	0.184510i	$-0.0590699\pi$
$727$	21.5251	+	21.5251i	0.798320	+	0.798320i	−0.184510	+	0.982831i	$0.559070\pi$
$728$		0			0
$729$	−17.1730	−	23.6365i	−0.636035	−	0.875428i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$733$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$734$		0			0
$735$	−24.7515	−	6.69256i	−0.912973	−	0.246859i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	41.7319	−	33.6368i	1.53409	−	1.23651i
$741$		0			0
$742$		0			0
$743$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$743$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	−0.194030	−	0.980996i	$-0.562156\pi$
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	0.873024	+	0.487678i	$0.162156\pi$
$752$	−2.37459	−	14.9926i	−0.0865924	−	0.546723i
$753$	39.4081	+	20.0794i	1.43611	+	0.731735i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	49.0165	−	24.9752i	1.78154	−	0.907738i	0.880270	−	0.474473i	$-0.157361\pi$
$757$	49.0165	−	24.9752i	1.78154	−	0.907738i	0.901266	−	0.433266i	$-0.142639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$761$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$762$		0			0
$763$		0			0
$764$	44.1602	−	14.3485i	1.59766	−	0.519111i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−11.8989	−	23.3529i	−0.429365	−	0.842677i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−9.88388			−0.355959
$772$		0			0
$773$	−4.50577	+	28.4483i	−0.162061	+	1.02322i	0.763828	+	0.645420i	$0.223318\pi$
$773$	−4.50577	+	28.4483i	−0.162061	+	1.02322i	−0.925889	+	0.377795i	$0.876682\pi$
$774$		0			0
$775$	−45.5400	+	20.0278i	−1.63584	+	0.719419i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	16.4580	+	22.6525i	0.587785	+	0.809017i
$785$	−11.2775	+	17.2796i	−0.402510	+	0.616734i
$786$		0			0
$787$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$787$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−5.32907	+	49.6161i	−0.189003	+	1.75970i
$796$	32.1985	+	23.3936i	1.14124	+	0.829163i
$797$	−19.0630	+	37.4132i	−0.675246	+	1.32524i	0.258048	+	0.966132i	$0.416921\pi$
$797$	−19.0630	+	37.4132i	−0.675246	+	1.32524i	−0.933294	+	0.359113i	$0.883079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.880582	+	2.71015i	−0.0311138	+	0.0957585i
$802$		0			0
$803$		0			0
$804$			7.06600i			0.249198i
$805$		0			0
$806$		0			0
$807$	3.39961	+	21.4643i	0.119672	+	0.755580i
$808$		0			0
$809$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$809$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$810$		0			0
$811$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$811$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.15831	+	0.308689i	−0.215716	+	0.0108129i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$821$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$822$		0			0
$823$	−6.14765	−	12.0654i	−0.214294	−	0.420575i	0.758689	−	0.651453i	$-0.225840\pi$
$823$	−6.14765	−	12.0654i	−0.214294	−	0.420575i	−0.972983	+	0.230878i	$0.925840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$827$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$828$	0.862748	−	5.44718i	0.0299826	−	0.189303i
$829$	17.0458	−	23.4615i	0.592024	−	0.814851i	−0.402925	−	0.915233i	$-0.632006\pi$
$829$	17.0458	−	23.4615i	0.592024	−	0.814851i	0.994949	+	0.100382i	$0.0320064\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	38.2243	+	38.2243i	1.32123	+	1.32123i
$838$		0			0
$839$	−21.4441	−	29.5153i	−0.740332	−	1.01898i	−0.998599	−	0.0529065i	$-0.983151\pi$
$839$	−21.4441	−	29.5153i	−0.740332	−	1.01898i	0.258267	−	0.966074i	$-0.416849\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.4771	+	25.2074i	0.498027	+	0.867161i
$846$		0			0
$847$		0			0
$848$	38.5330	−	38.5330i	1.32323	−	1.32323i
$849$		0			0
$850$		0			0
$851$	84.4473	+	61.3546i	2.89482	+	2.10321i
$852$	4.46209	−	8.75735i	0.152869	−	0.300022i
$853$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$853$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−51.9357	−	26.4625i	−1.76791	−	0.900795i	−0.940958	−	0.338524i	$-0.890072\pi$
$863$	−51.9357	−	26.4625i	−1.76791	−	0.900795i	−0.826953	−	0.562272i	$-0.809928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	24.8125	−	12.6426i	0.842677	−	0.429365i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−5.95946	+	0.943885i	−0.201697	+	0.0319457i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$877$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$	3.99326	−	25.2125i	0.134384	−	0.848467i	−0.824747	−	0.565503i	$-0.808682\pi$
$883$	3.99326	−	25.2125i	0.134384	−	0.848467i	0.959130	−	0.282964i	$-0.0913178\pi$
$884$		0			0
$885$	11.3515	−	4.32780i	0.381575	−	0.145477i
$886$		0			0
$887$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$887$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−30.8496	−	30.8496i	−1.03292	−	1.03292i
$893$		0			0
$894$		0			0
$895$	45.9534	−	9.65843i	1.53605	−	0.322846i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−3.09403	−	0.672392i	−0.103134	−	0.0224131i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.1214	+	2.37597i	0.735339	+	0.0789799i
$906$		0			0
$907$	16.5965	−	32.5724i	0.551077	−	1.08155i	−0.432597	−	0.901588i	$-0.642403\pi$
$907$	16.5965	−	32.5724i	0.551077	−	1.08155i	0.983674	−	0.179962i	$-0.0575975\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.04979	+	6.30860i	−0.0679125	+	0.209013i	−0.979253	−	0.202639i	$-0.935048\pi$
$911$	−2.04979	+	6.30860i	−0.0679125	+	0.209013i	0.911341	+	0.411652i	$0.135048\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−48.2977	+	35.0903i	−1.59580	+	1.15942i
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	37.9248	−	46.3997i	1.24696	−	1.52561i
$926$		0			0
$927$	−5.28498	−	0.837058i	−0.173581	−	0.0274926i
$928$		0			0
$929$	−50.4688	+	16.3983i	−1.65583	+	0.538011i	−0.979991	−	0.199042i	$-0.936217\pi$
$929$	−50.4688	+	16.3983i	−1.65583	+	0.538011i	−0.675835	+	0.737053i	$0.736217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−8.92419	−	17.5147i	−0.292165	−	0.573406i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$937$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$938$		0			0
$939$	7.38685	−	10.1671i	0.241061	−	0.331791i
$940$	−6.04584	−	15.8577i	−0.197194	−	0.517222i
$941$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$941$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$	−12.6172	−	4.09957i	−0.410655	−	0.133430i
$945$		0			0
$946$		0			0
$947$	−16.8918	−	16.8918i	−0.548910	−	0.548910i	0.377215	−	0.926126i	$-0.376882\pi$
$947$	−16.8918	−	16.8918i	−0.548910	−	0.548910i	−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−1.35430	−	4.16812i	−0.0439163	−	0.135161i
$952$		0			0
$953$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$953$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$954$		0			0
$955$	45.0173	−	25.8543i	1.45672	−	0.836625i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−18.3892	−	22.8148i	−0.593510	−	0.736345i
$961$	−55.0132	−	39.9694i	−1.77462	−	1.28934i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−34.8817	+	25.3430i	−1.11941	+	0.813296i	−0.984119	−	0.177510i	$-0.943196\pi$
$971$	−34.8817	+	25.3430i	−1.11941	+	0.813296i	−0.135287	+	0.990806i	$0.543196\pi$
$972$	1.02502	+	6.47173i	0.0328776	+	0.207581i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.5121	−	9.43240i	0.592256	−	0.301769i	−0.132056	−	0.991242i	$-0.542158\pi$
$977$	18.5121	−	9.43240i	0.592256	−	0.301769i	0.724311	+	0.689473i	$0.242158\pi$
$978$		0			0
$979$		0			0
$980$	23.2164	+	21.0000i	0.741620	+	0.670820i
$981$		0			0
$982$		0			0
$983$	10.1388	−	1.60582i	0.323377	−	0.0512179i	0.00736431	−	0.999973i	$-0.497656\pi$
$983$	10.1388	−	1.60582i	0.323377	−	0.0512179i	0.316012	+	0.948755i	$0.397656\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	59.6992			1.89641			0.948205	−	0.317660i	$-0.102897\pi$
$991$	59.6992			1.89641			0.948205	+	0.317660i	$0.102897\pi$
$992$		0			0
$993$	−2.54971	+	16.0982i	−0.0809126	+	0.510862i
$994$		0			0
$995$	40.6089	+	18.1911i	1.28739	+	0.576697i
$996$		0			0
$997$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$997$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$998$		0			0
$999$	−61.9291	−	20.1220i	−1.95935	−	0.636632i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.a.112.1		16
5.3	odd	4	inner	605.2.m.a.233.1		16
11.2	odd	10	inner	605.2.m.a.602.2		16
11.3	even	5	inner	605.2.m.a.282.1		16
11.4	even	5		55.2.e.b.32.2	✓	4
11.5	even	5	inner	605.2.m.a.457.1		16
11.6	odd	10	inner	605.2.m.a.457.1		16
11.7	odd	10		55.2.e.b.32.2	✓	4
11.8	odd	10	inner	605.2.m.a.282.1		16
11.9	even	5	inner	605.2.m.a.602.2		16
11.10	odd	2	CM	605.2.m.a.112.1		16
33.26	odd	10		495.2.k.a.307.2		4
33.29	even	10		495.2.k.a.307.2		4
44.7	even	10		880.2.bd.d.417.1		4
44.15	odd	10		880.2.bd.d.417.1		4
55.3	odd	20	inner	605.2.m.a.403.2		16
55.4	even	10		275.2.e.a.32.1		4
55.7	even	20		275.2.e.a.43.1		4
55.8	even	20	inner	605.2.m.a.403.2		16
55.13	even	20	inner	605.2.m.a.118.1		16
55.18	even	20		55.2.e.b.43.2	yes	4
55.28	even	20	inner	605.2.m.a.578.1		16
55.29	odd	10		275.2.e.a.32.1		4
55.37	odd	20		275.2.e.a.43.1		4
55.38	odd	20	inner	605.2.m.a.578.1		16
55.43	even	4	inner	605.2.m.a.233.1		16
55.48	odd	20		55.2.e.b.43.2	yes	4
55.53	odd	20	inner	605.2.m.a.118.1		16
165.128	odd	20		495.2.k.a.208.2		4
165.158	even	20		495.2.k.a.208.2		4
220.103	even	20		880.2.bd.d.593.1		4
220.183	odd	20		880.2.bd.d.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.2	✓	4	11.4	even	5
55.2.e.b.32.2	✓	4	11.7	odd	10
55.2.e.b.43.2	yes	4	55.18	even	20
55.2.e.b.43.2	yes	4	55.48	odd	20
275.2.e.a.32.1		4	55.4	even	10
275.2.e.a.32.1		4	55.29	odd	10
275.2.e.a.43.1		4	55.7	even	20
275.2.e.a.43.1		4	55.37	odd	20
495.2.k.a.208.2		4	165.128	odd	20
495.2.k.a.208.2		4	165.158	even	20
495.2.k.a.307.2		4	33.26	odd	10
495.2.k.a.307.2		4	33.29	even	10
605.2.m.a.112.1		16	1.1	even	1	trivial
605.2.m.a.112.1		16	11.10	odd	2	CM
605.2.m.a.118.1		16	55.13	even	20	inner
605.2.m.a.118.1		16	55.53	odd	20	inner
605.2.m.a.233.1		16	5.3	odd	4	inner
605.2.m.a.233.1		16	55.43	even	4	inner
605.2.m.a.282.1		16	11.3	even	5	inner
605.2.m.a.282.1		16	11.8	odd	10	inner
605.2.m.a.403.2		16	55.3	odd	20	inner
605.2.m.a.403.2		16	55.8	even	20	inner
605.2.m.a.457.1		16	11.5	even	5	inner
605.2.m.a.457.1		16	11.6	odd	10	inner
605.2.m.a.578.1		16	55.28	even	20	inner
605.2.m.a.578.1		16	55.38	odd	20	inner
605.2.m.a.602.2		16	11.2	odd	10	inner
605.2.m.a.602.2		16	11.9	even	5	inner
880.2.bd.d.417.1		4	44.7	even	10
880.2.bd.d.417.1		4	44.15	odd	10
880.2.bd.d.593.1		4	220.103	even	20
880.2.bd.d.593.1		4	220.183	odd	20

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:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.256255 + 1.61793i) q^{3} +(1.17557 - 1.61803i) q^{4} +(0.914138 - 2.04067i) q^{5} +(0.301128 + 0.0978424i) q^{9} +(2.31662 + 2.31662i) q^{12} +(3.06742 + 2.00195i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(-2.22725 - 3.87806i) q^{20} +(6.15831 - 6.15831i) q^{23} +(-3.32870 - 3.73092i) q^{25} +(-2.46652 + 4.84081i) q^{27} +(3.07468 - 9.46289i) q^{31} +(0.512310 - 0.372215i) q^{36} +(1.87493 + 11.8378i) q^{37} +(0.474937 - 0.525063i) q^{45} +(3.74814 + 0.593648i) q^{47} +(6.47173 - 1.02502i) q^{48} +(-6.65740 + 2.16312i) q^{49} +(6.18493 + 12.1386i) q^{53} +(1.94946 - 2.68321i) q^{59} +(6.84519 - 2.60976i) q^{60} +(-7.60845 - 2.47214i) q^{64} +(1.52506 + 1.52506i) q^{67} +(8.38564 + 11.5418i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(6.88937 - 4.42955i) q^{75} +(-8.89312 - 0.955176i) q^{80} +(-6.43158 - 4.67282i) q^{81} +9.00000i q^{89} +(-2.72483 - 17.2039i) q^{92} +(14.5224 + 7.39955i) q^{93} +(-16.9794 + 8.65144i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 16 * q + 2 * q^3 - 16 * q^12 + 8 * q^15 + 16 * q^16 - 12 * q^20 + 72 * q^23 - 2 * q^25 - 22 * q^27 - 24 * q^36 + 14 * q^37 - 72 * q^45 + 24 * q^47 - 8 * q^48 + 12 * q^53 + 28 * q^60 + 104 * q^67 + 12 * q^71 - 32 * q^75 + 8 * q^81 - 36 * q^92 - 66 * q^93 + 34 * q^97

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