LMFDB - Embedded newform 605.2.m.a.457.1

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			457.1
Root			$-1.63550 - 0.570223i$ of defining polynomial
Character	$\chi$	$=$	605.457
Dual form			605.2.m.a.233.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61793 + 0.256255i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(-1.93903 + 1.11362i) q^{5} +(-0.301128 + 0.0978424i) q^{9} +(2.31662 + 2.31662i) q^{12} +(2.85185 - 2.29866i) q^{15} +(-1.23607 + 3.80423i) q^{16} +(4.08135 + 1.82828i) q^{20} +(6.15831 - 6.15831i) q^{23} +(2.51969 - 4.31870i) q^{25} +(4.84081 - 2.46652i) q^{27} +(3.07468 + 9.46289i) q^{31} +(0.512310 + 0.372215i) q^{36} +(11.8378 + 1.87493i) q^{37} +(0.474937 - 0.525063i) q^{45} +(0.593648 + 3.74814i) q^{47} +(1.02502 - 6.47173i) q^{48} +(6.65740 + 2.16312i) q^{49} +(-12.1386 - 6.18493i) q^{53} +(-1.94946 - 2.68321i) q^{59} +(-7.07186 - 1.91216i) q^{60} +(7.60845 - 2.47214i) q^{64} +(1.52506 + 1.52506i) q^{67} +(-8.38564 + 11.5418i) q^{69} +(-0.927051 + 2.85317i) q^{71} +(-2.96999 + 7.63305i) q^{75} +(-1.83970 - 8.75303i) q^{80} +(-6.43158 + 4.67282i) q^{81} +9.00000i q^{89} +(-17.2039 - 2.72483i) q^{92} +(-7.39955 - 14.5224i) q^{93} +(8.65144 - 16.9794i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{9}{10}\right)$

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		0.453990	−	0.891007i	$-0.350000\pi$
$2$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$3$	−1.61793	+	0.256255i	−0.934114	+	0.147949i	−0.604896	−	0.796305i	$-0.706785\pi$
$3$	−1.61793	+	0.256255i	−0.934114	+	0.147949i	−0.329218	+	0.944254i	$0.606785\pi$
$4$	−1.17557	−	1.61803i	−0.587785	−	0.809017i
$5$	−1.93903	+	1.11362i	−0.867161	+	0.498027i
$5$	−1.93903	+	1.11362i	−0.867161	+	0.498027i
$6$		0			0
$7$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$7$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$8$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$13$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$14$		0			0
$15$	2.85185	−	2.29866i	0.736345	−	0.593510i
$16$	−1.23607	+	3.80423i	−0.309017	+	0.951057i
$17$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$17$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$	4.08135	+	1.82828i	0.912617	+	0.408815i
$21$		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
$25$	2.51969	−	4.31870i	0.503937	−	0.863740i
$26$		0			0
$27$	4.84081	−	2.46652i	0.931614	−	0.474681i
$28$		0			0
$29$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$29$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$30$		0			0
$31$	3.07468	+	9.46289i	0.552229	+	1.69959i	0.703151	+	0.711040i	$0.251776\pi$
$31$	3.07468	+	9.46289i	0.552229	+	1.69959i	−0.150923	+	0.988546i	$0.548224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.512310	+	0.372215i	0.0853849	+	0.0620358i
$37$	11.8378	+	1.87493i	1.94612	+	0.308236i	0.999884	−	0.0152291i	$-0.00484777\pi$
$37$	11.8378	+	1.87493i	1.94612	+	0.308236i	0.946240	+	0.323465i	$0.104848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$41$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$42$		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
$47$	0.593648	+	3.74814i	0.0865924	+	0.546723i	0.992402	+	0.123038i	$0.0392637\pi$
$47$	0.593648	+	3.74814i	0.0865924	+	0.546723i	−0.905810	+	0.423685i	$0.860736\pi$
$48$	1.02502	−	6.47173i	0.147949	−	0.934114i
$49$	6.65740	+	2.16312i	0.951057	+	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−12.1386	−	6.18493i	−1.66737	−	0.849565i	−0.993892	−	0.110353i	$-0.964802\pi$
$53$	−12.1386	−	6.18493i	−1.66737	−	0.849565i	−0.673473	−	0.739212i	$-0.735198\pi$
$54$		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	$-0.269267\pi$
$59$	−1.94946	−	2.68321i	−0.253798	−	0.349324i	−0.916837	+	0.399262i	$0.869267\pi$
$60$	−7.07186	−	1.91216i	−0.912973	−	0.246859i
$61$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$61$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$62$		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	−0.990876	−	0.134777i	$-0.956968\pi$
$71$	−0.927051	+	2.85317i	−0.110021	+	0.338609i	0.880855	+	0.473386i	$0.156968\pi$
$72$		0			0
$73$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$73$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$74$		0			0
$75$	−2.96999	+	7.63305i	−0.342945	+	0.881389i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$79$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$80$	−1.83970	−	8.75303i	−0.205685	−	0.978618i
$81$	−6.43158	+	4.67282i	−0.714620	+	0.519202i
$82$		0			0
$83$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$83$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$84$		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
$92$	−17.2039	−	2.72483i	−1.79363	−	0.284083i
$93$	−7.39955	−	14.5224i	−0.767297	−	1.50591i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.65144	−	16.9794i	0.878421	−	1.72400i	0.213710	−	0.976897i	$-0.431445\pi$
$97$	8.65144	−	16.9794i	0.878421	−	1.72400i	0.664711	−	0.747101i	$-0.268555\pi$
$98$		0			0
$99$		0			0
$100$	−9.94987	+	1.00000i	−0.994987	+	0.100000i
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	−2.64369	+	16.6916i	−0.260491	+	1.64467i	0.416829	+	0.908985i	$0.363141\pi$
$103$	−2.64369	+	16.6916i	−0.260491	+	1.64467i	−0.677320	+	0.735689i	$0.736859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$107$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$108$	−9.68162	−	4.93303i	−0.931614	−	0.474681i
$109$		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
$113$	12.3501	−	1.95606i	1.16180	−	0.184011i	0.454382	−	0.890807i	$-0.349860\pi$
$113$	12.3501	−	1.95606i	1.16180	−	0.184011i	0.707417	+	0.706796i	$0.249860\pi$
$114$		0			0
$115$	−5.08312	+	18.7992i	−0.474004	+	1.75303i
$116$		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
$125$	−0.0763444	+	11.1801i	−0.00682845	+	0.999977i
$126$		0			0
$127$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$127$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$128$		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
$135$	−6.63971	+	10.1735i	−0.571455	+	0.875594i
$136$		0			0
$137$	16.5173	−	8.41597i	1.41117	−	0.719025i	0.428350	−	0.903613i	$-0.359095\pi$
$137$	16.5173	−	8.41597i	1.41117	−	0.719025i	0.982816	+	0.184588i	$0.0590949\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		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
$147$	−11.3255	−	1.79379i	−0.934114	−	0.147949i
$148$	−10.8825	−	21.3581i	−0.894535	−	1.75562i
$149$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$149$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$150$		0			0
$151$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$151$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5000	−	14.9248i	−1.32531	−	1.19879i
$156$		0			0
$157$	−1.44355	−	9.11422i	−0.115208	−	0.727394i	−0.975892	−	0.218255i	$-0.929964\pi$
$157$	−1.44355	−	9.11422i	−0.115208	−	0.727394i	0.860684	−	0.509140i	$-0.170036\pi$
$158$		0			0
$159$	21.2244	+	6.89622i	1.68320	+	0.546906i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.45699	+	1.25190i	0.192446	+	0.0980561i	0.547558	−	0.836768i	$-0.315557\pi$
$163$	2.45699	+	1.25190i	0.192446	+	0.0980561i	−0.355112	+	0.934824i	$0.615557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$167$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$168$		0			0
$169$	7.64121	+	10.5172i	0.587785	+	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$173$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$174$		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	0.0400827	+	0.999196i	$0.487238\pi$
$179$	−12.3435	+	16.9894i	−0.922596	+	1.26984i	−0.962678	+	0.270648i	$0.912762\pi$
$180$	−1.40789	−	0.151216i	−0.104938	−	0.0112710i
$181$	3.07468	−	9.46289i	0.228539	−	0.703371i	−0.769374	−	0.638799i	$-0.779432\pi$
$181$	3.07468	−	9.46289i	0.228539	−	0.703371i	0.997913	−	0.0645725i	$-0.0205684\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−25.0418	+	9.54732i	−1.84111	+	0.701933i
$186$		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	0.360545	−	0.932742i	$-0.382591\pi$
$191$	18.7824	−	13.6462i	1.35905	−	0.987407i	0.998505	−	0.0546656i	$-0.0174093\pi$
$192$	−11.6765	+	5.94946i	−0.842677	+	0.429365i
$193$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$193$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$194$		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
$207$	−1.25190	+	2.45699i	−0.0870128	+	0.170772i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$211$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$212$	4.26236	+	26.9115i	0.292740	+	1.84829i
$213$	0.768766	−	4.85380i	0.0526750	−	0.332577i
$214$		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
$223$	21.5454	−	3.41246i	1.44279	−	0.228515i	0.614544	−	0.788883i	$-0.289340\pi$
$223$	21.5454	−	3.41246i	1.44279	−	0.228515i	0.828244	+	0.560368i	$0.189340\pi$
$224$		0			0
$225$	−0.336196	+	1.54701i	−0.0224131	+	0.103134i
$226$		0			0
$227$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$227$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$228$		0			0
$229$	28.3887	−	9.22404i	1.87598	−	0.609542i	0.886937	−	0.461890i	$-0.152828\pi$
$229$	28.3887	−	9.22404i	1.87598	−	0.609542i	0.989039	−	0.147652i	$-0.0471715\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$233$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$234$		0			0
$235$	−5.32512	−	6.60667i	−0.347373	−	0.430972i
$236$	−2.04979	+	6.30860i	−0.133430	+	0.410655i
$237$		0			0
$238$		0			0
$239$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$239$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$240$	5.21952	+	13.6904i	0.336919	+	0.883710i
$241$		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
$245$	−15.3178	+	3.21948i	−0.978618	+	0.205685i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.34346	+	25.6785i	0.526634	+	1.62081i	0.761061	+	0.648680i	$0.224679\pi$
$251$	8.34346	+	25.6785i	0.526634	+	1.62081i	−0.234427	+	0.972134i	$0.575321\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	−	9.40456i	−0.809017	−	0.587785i
$257$	5.95946	+	0.943885i	0.371741	+	0.0588779i	0.339510	−	0.940603i	$-0.389739\pi$
$257$	5.95946	+	0.943885i	0.371741	+	0.0588779i	0.0322308	+	0.999480i	$0.489739\pi$
$258$		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
$267$	−2.30630	−	14.5614i	−0.141143	−	0.891143i
$268$	0.674785	−	4.26042i	0.0412190	−	0.260247i
$269$	−12.6172	−	4.09957i	−0.769284	−	0.249955i	−0.102025	−	0.994782i	$-0.532532\pi$
$269$	−12.6172	−	4.09957i	−0.769284	−	0.249955i	−0.667258	+	0.744826i	$0.732532\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	28.5330			1.71748
$277$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$277$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$278$		0			0
$279$	−1.85175	−	2.54871i	−0.110861	−	0.152587i
$280$		0			0
$281$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$281$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$283$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$284$	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		0.156434	−	0.987688i	$-0.450000\pi$
$293$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$294$		0			0
$295$	6.76815	+	3.03185i	0.394057	+	0.176521i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	15.8420	−	4.16764i	0.914637	−	0.240619i
$301$		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	0.277469	−	0.960735i	$-0.410505\pi$
$311$	−9.70820	−	7.05342i	−0.550502	−	0.399963i	−0.827970	+	0.560772i	$0.810505\pi$
$312$		0			0
$313$	3.48294	+	6.83566i	0.196868	+	0.386374i	0.968245	−	0.250004i	$-0.0804318\pi$
$313$	3.48294	+	6.83566i	0.196868	+	0.386374i	−0.771377	+	0.636378i	$0.780432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.21462	−	2.38382i	0.0682198	−	0.133889i	−0.854378	−	0.519652i	$-0.826062\pi$
$317$	1.21462	−	2.38382i	0.0682198	−	0.133889i	0.922598	+	0.385763i	$0.126062\pi$
$318$		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
$333$	−3.74814	+	0.593648i	−0.205397	+	0.0325317i
$334$		0			0
$335$	−4.65549	−	1.25880i	−0.254357	−	0.0687756i
$336$		0			0
$337$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$337$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$338$		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
$345$	3.40675	−	31.7184i	0.183413	−	1.70766i
$346$		0			0
$347$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$347$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$		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
$355$	−1.37978	−	6.56477i	−0.0732309	−	0.348422i
$356$	14.5623	−	10.5801i	0.771801	−	0.560746i
$357$		0			0
$358$		0			0
$359$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$359$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$360$		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
$367$	32.7898	+	5.19340i	1.71162	+	0.271093i	0.933903	−	0.357526i	$-0.116380\pi$
$367$	32.7898	+	5.19340i	1.71162	+	0.271093i	0.777713	+	0.628620i	$0.216380\pi$
$368$	15.8155	+	31.0397i	0.824441	+	1.61806i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−14.7991	+	29.0449i	−0.767297	+	1.50591i
$373$		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	$-0.678071\pi$
$379$	−28.3887	−	9.22404i	−1.45823	−	0.473807i	−0.927528	+	0.373753i	$0.878071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.6610	+	13.5845i	1.36231	+	0.694134i	0.973821	−	0.227317i	$-0.0729954\pi$
$383$	26.6610	+	13.5845i	1.36231	+	0.694134i	0.388494	+	0.921451i	$0.372995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−37.6436	+	5.96217i	−1.91107	+	0.302683i
$389$	−21.4441	−	29.5153i	−1.08726	−	1.49648i	−0.851270	−	0.524727i	$-0.824167\pi$
$389$	−21.4441	−	29.5153i	−1.08726	−	1.49648i	−0.235988	−	0.971756i	$-0.575833\pi$
$390$		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
$400$	13.3148	+	14.9237i	0.665741	+	0.746183i
$401$	8.19915	−	25.2344i	0.409446	−	1.26014i	−0.507679	−	0.861546i	$-0.669497\pi$
$401$	8.19915	−	25.2344i	0.409446	−	1.26014i	0.917125	−	0.398599i	$-0.130503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	7.26728	−	16.2231i	0.361114	−	0.806133i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$409$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$410$		0			0
$411$	−24.5672	+	17.8491i	−1.21181	+	0.880433i
$412$	30.1154	−	15.3446i	1.48368	−	0.755973i
$413$		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	0.927863	+	0.372921i	$0.121644\pi$
$421$	32.1985	+	23.3936i	1.56926	+	1.14013i	0.641394	+	0.767211i	$0.278356\pi$
$422$		0			0
$423$	−0.545492	−	1.07059i	−0.0265227	−	0.0520537i
$424$		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	$-0.600000\pi$
$431$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$432$	3.39961	+	21.4643i	0.163564	+	1.03270i
$433$	0.0939818	−	0.593378i	0.00451648	−	0.0285159i	−0.985327	−	0.170676i	$-0.945405\pi$
$433$	0.0939818	−	0.593378i	0.00451648	−	0.0285159i	0.989844	+	0.142160i	$0.0454048\pi$
$434$		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
$443$	−10.8133	+	1.71265i	−0.513753	+	0.0813705i	−0.407928	−	0.913014i	$-0.633748\pi$
$443$	−10.8133	+	1.71265i	−0.513753	+	0.0813705i	−0.105825	+	0.994385i	$0.533748\pi$
$444$	23.0803	+	31.7673i	1.09534	+	1.50761i
$445$	−10.0226	−	17.4513i	−0.475117	−	0.827270i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−37.0912	+	12.0517i	−1.75044	+	0.568753i	−0.996140	−	0.0877747i	$-0.972024\pi$
$449$	−37.0912	+	12.0517i	−1.75044	+	0.568753i	−0.754302	+	0.656528i	$0.772024\pi$
$450$		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		0.891007	−	0.453990i	$-0.150000\pi$
$457$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$458$		0			0
$459$		0			0
$460$	36.3933	−	13.8751i	1.69685	−	0.646931i
$461$		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
$465$	30.5205	+	19.9191i	1.41535	+	0.923728i
$466$		0			0
$467$	−25.2746	+	12.8781i	−1.16957	+	0.595926i	−0.927313	−	0.374286i	$-0.877888\pi$
$467$	−25.2746	+	12.8781i	−1.16957	+	0.595926i	−0.242257	+	0.970212i	$0.577888\pi$
$468$		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
$477$	4.26042	+	0.674785i	0.195071	+	0.0308963i
$478$		0			0
$479$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$479$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$480$		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
$487$	−3.65587	−	23.0822i	−0.165663	−	1.04596i	−0.920699	−	0.390273i	$-0.872381\pi$
$487$	−3.65587	−	23.0822i	−0.165663	−	1.04596i	0.755036	−	0.655683i	$-0.227619\pi$
$488$		0			0
$489$	−4.29604	−	1.39587i	−0.194274	−	0.0631234i
$490$		0			0
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		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	$-0.346945\pi$
$499$	−11.6968	−	16.0992i	−0.523620	−	0.720701i	−0.986141	+	0.165907i	$0.946945\pi$
$500$	18.1795	−	13.0194i	0.813012	−	0.582247i
$501$		0			0
$502$		0			0
$503$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$503$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$504$		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	$-0.776582\pi$
$509$	1.94946	−	2.68321i	0.0864084	−	0.118931i	0.850033	+	0.526730i	$0.176582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.4620	−	35.3096i	−0.593205	−	1.55593i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.8817	+	25.3430i	−1.52819	+	1.11030i	−0.570962	+	0.820977i	$0.693430\pi$
$521$	−34.8817	+	25.3430i	−1.52819	+	1.11030i	−0.957232	+	0.289321i	$0.906570\pi$
$522$		0			0
$523$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$523$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$524$		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
$537$	15.6173	−	30.6507i	0.673937	−	1.32268i
$538$		0			0
$539$		0			0
$540$	24.2665	−	1.21637i	1.04426	−	0.0523444i
$541$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$541$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$542$		0			0
$543$	−2.54971	+	16.0982i	−0.109419	+	0.690841i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$547$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$548$	−33.0346	−	16.8319i	−1.41117	−	0.719025i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	38.0695	−	21.8640i	1.61596	−	0.928077i
$556$		0			0
$557$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$557$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$563$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$564$	−7.30779	+	10.0583i	−0.307713	+	0.423531i
$565$	−21.7689	+	17.5462i	−0.915824	+	0.738175i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$569$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$570$		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
$575$	−11.0789	−	42.1129i	−0.462022	−	1.75623i
$576$	−2.04924	+	1.48886i	−0.0853849	+	0.0620358i
$577$	−35.8805	+	18.2820i	−1.49373	+	0.761091i	−0.994436	−	0.105344i	$-0.966406\pi$
$577$	−35.8805	+	18.2820i	−1.49373	+	0.761091i	−0.499290	+	0.866435i	$0.666406\pi$
$578$		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
$587$	−38.1559	−	6.04330i	−1.57486	−	0.249434i	−0.693000	−	0.720938i	$-0.743711\pi$
$587$	−38.1559	−	6.04330i	−1.57486	−	0.249434i	−0.881864	+	0.471504i	$0.843711\pi$
$588$	10.4116	+	20.4338i	0.429365	+	0.842677i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−21.7650	+	42.7162i	−0.894535	+	1.75562i
$593$		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
$597$	−5.09942	−	32.1965i	−0.208705	−	1.31771i
$598$		0			0
$599$	−34.2380	−	11.1246i	−1.39893	−	0.454539i	−0.490084	−	0.871675i	$-0.663034\pi$
$599$	−34.2380	−	11.1246i	−1.39893	−	0.454539i	−0.908844	+	0.417136i	$0.863034\pi$
$600$		0			0
$601$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$601$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$	−0.608455	−	0.310023i	−0.0247782	−	0.0126251i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$607$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$613$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$614$		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	$-0.806397\pi$
$619$	−0.587785	+	0.809017i	−0.0236251	+	0.0325171i	0.797041	+	0.603925i	$0.206397\pi$
$620$	−4.75194	+	44.2427i	−0.190842	+	1.77683i
$621$	14.6216	−	45.0008i	0.586746	−	1.80582i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.3024	−	21.7635i	−0.492095	−	0.870542i
$626$		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	−0.694774	−	0.719228i	$-0.744496\pi$
$631$	−5.66312	+	4.11450i	−0.225445	+	0.163796i	0.469329	+	0.883023i	$0.344496\pi$
$632$		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	0.893294	−	0.449474i	$-0.148388\pi$
$641$	18.7824	+	13.6462i	0.741862	+	0.538994i	−0.151432	+	0.988468i	$0.548388\pi$
$642$		0			0
$643$	22.7441	+	44.6379i	0.896941	+	1.76035i	0.586993	+	0.809592i	$0.300312\pi$
$643$	22.7441	+	44.6379i	0.896941	+	1.76035i	0.309948	+	0.950753i	$0.399688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.5087	−	44.1757i	0.884907	−	1.73673i	0.242861	−	0.970061i	$-0.421914\pi$
$647$	22.5087	−	44.1757i	0.884907	−	1.73673i	0.642046	−	0.766666i	$-0.278086\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.862748	−	5.44718i	−0.0337878	−	0.213328i
$653$	5.27454	−	33.3021i	0.206409	−	1.30321i	−0.639047	−	0.769167i	$-0.720671\pi$
$653$	5.27454	−	33.3021i	0.206409	−	1.30321i	0.845456	−	0.534045i	$-0.179329\pi$
$654$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$673$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$674$		0			0
$675$	1.54518	−	27.1209i	0.0594739	−	1.04388i
$676$	8.03444	−	24.7275i	0.309017	−	0.951057i
$677$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$677$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$678$		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
$685$	−22.6553	+	34.7129i	−0.865614	+	1.32631i
$686$		0			0
$687$	−43.5673	+	22.1986i	−1.66220	+	0.846931i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.25329	+	16.1680i	0.199845	+	0.615058i	0.999886	+	0.0151132i	$0.00481087\pi$
$691$	5.25329	+	16.1680i	0.199845	+	0.615058i	−0.800041	+	0.599945i	$0.795189\pi$
$692$		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	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	10.3087	+	9.32456i	0.388248	+	0.351183i
$706$		0			0
$707$		0			0
$708$	1.69981	−	10.7322i	0.0638826	−	0.403339i
$709$	18.0701	+	5.87132i	0.678636	+	0.220502i	0.627998	−	0.778215i	$-0.283875\pi$
$709$	18.0701	+	5.87132i	0.678636	+	0.220502i	0.0506378	+	0.998717i	$0.483875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	77.2103	+	39.3406i	2.89155	+	1.47332i
$714$		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	0.809144	+	0.587610i	$0.199931\pi$
$719$	29.9770	+	41.2599i	1.11796	+	1.53873i	0.308811	+	0.951123i	$0.400069\pi$
$720$	1.41040	+	2.45578i	0.0525626	+	0.0915216i
$721$		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		0.156434	−	0.987688i	$-0.450000\pi$
$733$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$734$		0			0
$735$	23.9582	−	9.13417i	0.883710	−	0.336919i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$	44.8864	+	29.2950i	1.65006	+	1.07691i
$741$		0			0
$742$		0			0
$743$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$743$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$744$		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	0.873024	−	0.487678i	$-0.162156\pi$
$751$	18.6074	+	13.5191i	0.678993	+	0.493318i	−0.194030	+	0.980996i	$0.562156\pi$
$752$	−14.9926	−	2.37459i	−0.546723	−	0.0865924i
$753$	−20.0794	−	39.4081i	−0.731735	−	1.43611i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−24.9752	+	49.0165i	−0.907738	+	1.78154i	−0.433266	+	0.901266i	$0.642639\pi$
$757$	−24.9752	+	49.0165i	−0.907738	+	1.78154i	−0.474473	+	0.880270i	$0.657361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$		0			0
$764$	−44.1602	−	14.3485i	−1.59766	−	0.519111i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	23.3529	+	11.8989i	0.842677	+	0.429365i
$769$		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
$773$	−28.4483	+	4.50577i	−1.02322	+	0.162061i	−0.645420	−	0.763828i	$-0.723318\pi$
$773$	−28.4483	+	4.50577i	−1.02322	+	0.162061i	−0.377795	+	0.925889i	$0.623318\pi$
$774$		0			0
$775$	48.6146	+	10.5649i	1.74629	+	0.379502i
$776$		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
$785$	12.9489	+	16.0652i	0.462166	+	0.573391i
$786$		0			0
$787$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$787$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−48.8345	+	10.2640i	−1.73198	+	0.364026i
$796$	32.1985	−	23.3936i	1.14124	−	0.829163i
$797$	37.4132	−	19.0630i	1.32524	−	0.675246i	0.359113	−	0.933294i	$-0.383079\pi$
$797$	37.4132	−	19.0630i	1.32524	−	0.675246i	0.966132	+	0.258048i	$0.0830794\pi$
$798$		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
$807$	21.4643	+	3.39961i	0.755580	+	0.119672i
$808$		0			0
$809$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$809$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$810$		0			0
$811$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$811$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$812$		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	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	12.0654	+	6.14765i	0.420575	+	0.214294i	0.651453	−	0.758689i	$-0.274160\pi$
$823$	12.0654	+	6.14765i	0.420575	+	0.214294i	−0.230878	+	0.972983i	$0.574160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$827$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$828$	5.44718	−	0.862748i	0.189303	−	0.0299826i
$829$	−17.0458	−	23.4615i	−0.592024	−	0.814851i	0.402925	−	0.915233i	$-0.367994\pi$
$829$	−17.0458	−	23.4615i	−0.592024	−	0.814851i	−0.994949	+	0.100382i	$0.967994\pi$
$830$		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	−0.258267	−	0.966074i	$-0.583151\pi$
$839$	21.4441	−	29.5153i	0.740332	−	1.01898i	0.998599	−	0.0529065i	$-0.0168485\pi$
$840$		0			0
$841$	−8.96149	+	27.5806i	−0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.5288	−	11.8838i	−0.912617	−	0.408815i
$846$		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
$852$	−8.75735	+	4.46209i	−0.300022	+	0.152869i
$853$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$853$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$854$		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
$863$	26.4625	+	51.9357i	0.900795	+	1.76791i	0.562272	+	0.826953i	$0.309928\pi$
$863$	26.4625	+	51.9357i	0.900795	+	1.76791i	0.338524	+	0.940958i	$0.390072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−12.6426	+	24.8125i	−0.429365	+	0.842677i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.943885	+	5.95946i	−0.0319457	+	0.201697i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$877$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$878$		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
$883$	25.2125	−	3.99326i	0.848467	−	0.134384i	0.282964	−	0.959130i	$-0.408682\pi$
$883$	25.2125	−	3.99326i	0.848467	−	0.134384i	0.565503	+	0.824747i	$0.308682\pi$
$884$		0			0
$885$	−11.7273	−	3.17096i	−0.394210	−	0.106591i
$886$		0			0
$887$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$887$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$888$		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
$895$	5.01467	−	46.6889i	0.167622	−	1.56064i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2.89834	−	1.27465i	0.0966115	−	0.0424883i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.57620	+	21.7729i	0.152118	+	0.723755i
$906$		0			0
$907$	−32.5724	+	16.5965i	−1.08155	+	0.551077i	−0.901588	−	0.432597i	$-0.857597\pi$
$907$	−32.5724	+	16.5965i	−1.08155	+	0.551077i	−0.179962	+	0.983674i	$0.557597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.04979	−	6.30860i	−0.0679125	−	0.209013i	0.911341	−	0.411652i	$-0.135048\pi$
$911$	−2.04979	−	6.30860i	−0.0679125	−	0.209013i	−0.979253	+	0.202639i	$0.935048\pi$
$912$		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	$-0.900000\pi$
$919$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$920$		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
$927$	−0.837058	−	5.28498i	−0.0274926	−	0.173581i
$928$		0			0
$929$	50.4688	+	16.3983i	1.65583	+	0.538011i	0.979991	−	0.199042i	$-0.0637830\pi$
$929$	50.4688	+	16.3983i	1.65583	+	0.538011i	0.675835	+	0.737053i	$0.263783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	17.5147	+	8.92419i	0.573406	+	0.292165i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$937$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$938$		0			0
$939$	−7.38685	−	10.1671i	−0.241061	−	0.331791i
$940$	−4.42976	+	16.3828i	−0.144483	+	0.534349i
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		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		0.156434	−	0.987688i	$-0.450000\pi$
$953$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$954$		0			0
$955$	−21.2230	+	47.3771i	−0.686759	+	1.53309i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	16.0156	−	24.5394i	0.516901	−	0.792005i
$961$	−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	−0.135287	−	0.990806i	$-0.543196\pi$
$971$	−34.8817	−	25.3430i	−1.11941	−	0.813296i	−0.984119	+	0.177510i	$0.943196\pi$
$972$	6.47173	+	1.02502i	0.207581	+	0.0328776i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.43240	+	18.5121i	−0.301769	+	0.592256i	−0.991242	−	0.132056i	$-0.957842\pi$
$977$	−9.43240	+	18.5121i	−0.301769	+	0.592256i	0.689473	+	0.724311i	$0.257842\pi$
$978$		0			0
$979$		0			0
$980$	23.2164	+	21.0000i	0.741620	+	0.670820i
$981$		0			0
$982$		0			0
$983$	1.60582	−	10.1388i	0.0512179	−	0.323377i	−0.948755	−	0.316012i	$-0.897656\pi$
$983$	1.60582	−	10.1388i	0.0512179	−	0.323377i	0.999973	−	0.00736431i	$-0.00234415\pi$
$984$		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
$993$	−16.0982	+	2.54971i	−0.510862	+	0.0809126i
$994$		0			0
$995$	−22.1608	−	38.5862i	−0.702545	−	1.22327i
$996$		0			0
$997$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$997$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$998$		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.457.1		16
5.3	odd	4	inner	605.2.m.a.578.1		16
11.2	odd	10	inner	605.2.m.a.112.1		16
11.3	even	5		55.2.e.b.32.2	✓	4
11.4	even	5	inner	605.2.m.a.602.2		16
11.5	even	5	inner	605.2.m.a.282.1		16
11.6	odd	10	inner	605.2.m.a.282.1		16
11.7	odd	10	inner	605.2.m.a.602.2		16
11.8	odd	10		55.2.e.b.32.2	✓	4
11.9	even	5	inner	605.2.m.a.112.1		16
11.10	odd	2	CM	605.2.m.a.457.1		16
33.8	even	10		495.2.k.a.307.2		4
33.14	odd	10		495.2.k.a.307.2		4
44.3	odd	10		880.2.bd.d.417.1		4
44.19	even	10		880.2.bd.d.417.1		4
55.3	odd	20		55.2.e.b.43.2	yes	4
55.8	even	20		55.2.e.b.43.2	yes	4
55.13	even	20	inner	605.2.m.a.233.1		16
55.14	even	10		275.2.e.a.32.1		4
55.18	even	20	inner	605.2.m.a.118.1		16
55.19	odd	10		275.2.e.a.32.1		4
55.28	even	20	inner	605.2.m.a.403.2		16
55.38	odd	20	inner	605.2.m.a.403.2		16
55.43	even	4	inner	605.2.m.a.578.1		16
55.47	odd	20		275.2.e.a.43.1		4
55.48	odd	20	inner	605.2.m.a.118.1		16
55.52	even	20		275.2.e.a.43.1		4
55.53	odd	20	inner	605.2.m.a.233.1		16
165.8	odd	20		495.2.k.a.208.2		4
165.113	even	20		495.2.k.a.208.2		4
220.3	even	20		880.2.bd.d.593.1		4
220.63	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	11.3	even	5
55.2.e.b.32.2	✓	4	11.8	odd	10
55.2.e.b.43.2	yes	4	55.3	odd	20
55.2.e.b.43.2	yes	4	55.8	even	20
275.2.e.a.32.1		4	55.14	even	10
275.2.e.a.32.1		4	55.19	odd	10
275.2.e.a.43.1		4	55.47	odd	20
275.2.e.a.43.1		4	55.52	even	20
495.2.k.a.208.2		4	165.8	odd	20
495.2.k.a.208.2		4	165.113	even	20
495.2.k.a.307.2		4	33.8	even	10
495.2.k.a.307.2		4	33.14	odd	10
605.2.m.a.112.1		16	11.2	odd	10	inner
605.2.m.a.112.1		16	11.9	even	5	inner
605.2.m.a.118.1		16	55.18	even	20	inner
605.2.m.a.118.1		16	55.48	odd	20	inner
605.2.m.a.233.1		16	55.13	even	20	inner
605.2.m.a.233.1		16	55.53	odd	20	inner
605.2.m.a.282.1		16	11.5	even	5	inner
605.2.m.a.282.1		16	11.6	odd	10	inner
605.2.m.a.403.2		16	55.28	even	20	inner
605.2.m.a.403.2		16	55.38	odd	20	inner
605.2.m.a.457.1		16	1.1	even	1	trivial
605.2.m.a.457.1		16	11.10	odd	2	CM
605.2.m.a.578.1		16	5.3	odd	4	inner
605.2.m.a.578.1		16	55.43	even	4	inner
605.2.m.a.602.2		16	11.4	even	5	inner
605.2.m.a.602.2		16	11.7	odd	10	inner
880.2.bd.d.417.1		4	44.3	odd	10
880.2.bd.d.417.1		4	44.19	even	10
880.2.bd.d.593.1		4	220.3	even	20
880.2.bd.d.593.1		4	220.63	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)

\(f(q)\)	\(=\)	\(q+(-1.61793 + 0.256255i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(-1.93903 + 1.11362i) q^{5} +(-0.301128 + 0.0978424i) q^{9} +(2.31662 + 2.31662i) q^{12} +(2.85185 - 2.29866i) q^{15} +(-1.23607 + 3.80423i) q^{16} +(4.08135 + 1.82828i) q^{20} +(6.15831 - 6.15831i) q^{23} +(2.51969 - 4.31870i) q^{25} +(4.84081 - 2.46652i) q^{27} +(3.07468 + 9.46289i) q^{31} +(0.512310 + 0.372215i) q^{36} +(11.8378 + 1.87493i) q^{37} +(0.474937 - 0.525063i) q^{45} +(0.593648 + 3.74814i) q^{47} +(1.02502 - 6.47173i) q^{48} +(6.65740 + 2.16312i) q^{49} +(-12.1386 - 6.18493i) q^{53} +(-1.94946 - 2.68321i) q^{59} +(-7.07186 - 1.91216i) q^{60} +(7.60845 - 2.47214i) q^{64} +(1.52506 + 1.52506i) q^{67} +(-8.38564 + 11.5418i) q^{69} +(-0.927051 + 2.85317i) q^{71} +(-2.96999 + 7.63305i) q^{75} +(-1.83970 - 8.75303i) q^{80} +(-6.43158 + 4.67282i) q^{81} +9.00000i q^{89} +(-17.2039 - 2.72483i) q^{92} +(-7.39955 - 14.5224i) q^{93} +(8.65144 - 16.9794i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more