LMFDB - Embedded newform 475.2.v.a.113.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(37,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([9, 10]))

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

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

chi := DirichletCharacter("475.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.v (of order $20$, degree $8$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	16.0.271737008656000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 9x^{14} + 56x^{12} + 279x^{10} + 1111x^{8} + 6975x^{6} + 35000x^{4} + 140625x^{2} + 390625 $ x^16 + 9x^14 + 56x^12 + 279x^10 + 1111x^8 + 6975x^6 + 35000x^4 + 140625*x^2 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			113.2
Root			$1.46932 + 1.68556i$ of defining polynomial
Character	$\chi$	$=$	475.113
Dual form			475.2.v.a.227.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.90211 - 0.618034i) q^{4} +(0.876540 + 2.05710i) q^{5} +(2.71241 - 2.71241i) q^{7} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})$	$q+(1.90211 - 0.618034i) q^{4} +(0.876540 + 2.05710i) q^{5} +(2.71241 - 2.71241i) q^{7} +(-1.76336 - 2.42705i) q^{9} +(0.475295 + 0.345322i) q^{11} +(3.23607 - 2.35114i) q^{16} +(0.320460 + 0.628938i) q^{17} +(-4.14556 - 1.34697i) q^{19} +(2.93864 + 3.37111i) q^{20} +(0.831374 + 5.24909i) q^{23} +(-3.46336 + 3.60627i) q^{25} +(3.48295 - 6.83568i) q^{28} +(7.95725 + 3.20218i) q^{35} +(-4.85410 - 3.52671i) q^{36} +(3.56784 + 3.56784i) q^{43} +(1.11749 + 0.363093i) q^{44} +(3.44705 - 5.75481i) q^{45} +(5.44421 + 2.77396i) q^{47} -7.71436i q^{49} +(-0.293749 + 1.28042i) q^{55} +(-12.6310 - 9.17699i) q^{61} +(-11.3661 - 1.80022i) q^{63} +(4.70228 - 6.47214i) q^{64} +(0.998256 + 0.998256i) q^{68} +(2.66326 + 16.8152i) q^{73} -8.71780 q^{76} +(2.22585 - 0.352540i) q^{77} +(7.67308 + 4.59606i) q^{80} +(-2.78115 + 8.55951i) q^{81} +(-16.2287 + 8.26894i) q^{83} +(-1.01290 + 1.21051i) q^{85} +(4.82548 + 9.47054i) q^{92} +(-0.862882 - 9.70852i) q^{95} -1.76249i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 16 * q - 2 * q^5 - 6 * q^7	$ 16 q - 2 q^{5} - 6 q^{7} + 16 q^{16} - 56 q^{17} - 8 q^{23} + 18 q^{25} + 12 q^{28} + 88 q^{35} - 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} + 8 q^{80} + 36 q^{81} - 128 q^{83} - 12 q^{85} - 16 q^{92} + 38 q^{95}+O(q^{100}) $ 16 * q - 2 * q^5 - 6 * q^7 + 16 * q^16 - 56 * q^17 - 8 * q^23 + 18 * q^25 + 12 * q^28 + 88 * q^35 - 24 * q^36 + 2 * q^43 - 26 * q^47 - 18 * q^63 - 28 * q^68 + 22 * q^73 + 38 * q^77 + 8 * q^80 + 36 * q^81 - 128 * q^83 - 12 * q^85 - 16 * q^92 + 38 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{19}{20}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$2$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$3$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$3$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$4$	1.90211	−	0.618034i	0.951057	−	0.309017i
$5$	0.876540	+	2.05710i	0.392000	+	0.919965i
$5$	0.876540	+	2.05710i	0.392000	+	0.919965i
$6$		0			0
$7$	2.71241	−	2.71241i	1.02520	−	1.02520i	0.0255212	−	0.999674i	$-0.491875\pi$
$7$	2.71241	−	2.71241i	1.02520	−	1.02520i	0.999674	−	0.0255212i	$-0.00812453\pi$
$8$		0			0
$9$	−1.76336	−	2.42705i	−0.587785	−	0.809017i
$10$		0			0
$11$	0.475295	+	0.345322i	0.143307	+	0.104119i	0.657129	−	0.753778i	$-0.271771\pi$
$11$	0.475295	+	0.345322i	0.143307	+	0.104119i	−0.513822	+	0.857897i	$0.671771\pi$
$12$		0			0
$13$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$13$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$14$		0			0
$15$		0			0
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$	0.320460	+	0.628938i	0.0777230	+	0.152540i	0.926598	−	0.376054i	$-0.122719\pi$
$17$	0.320460	+	0.628938i	0.0777230	+	0.152540i	−0.848875	+	0.528594i	$0.822719\pi$
$18$		0			0
$19$	−4.14556	−	1.34697i	−0.951057	−	0.309017i
$19$	−4.14556	−	1.34697i	−0.951057	−	0.309017i
$20$	2.93864	+	3.37111i	0.657099	+	0.753804i
$21$		0			0
$22$		0			0
$23$	0.831374	+	5.24909i	0.173353	+	1.09451i	0.908893	+	0.417029i	$0.136929\pi$
$23$	0.831374	+	5.24909i	0.173353	+	1.09451i	−0.735540	+	0.677481i	$0.763071\pi$
$24$		0			0
$25$	−3.46336	+	3.60627i	−0.692671	+	0.721253i
$26$		0			0
$27$		0			0
$28$	3.48295	−	6.83568i	0.658216	−	1.29182i
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.95725	+	3.20218i	1.34502	+	0.541267i
$36$	−4.85410	−	3.52671i	−0.809017	−	0.587785i
$37$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$37$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$38$		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
$43$	3.56784	+	3.56784i	0.544091	+	0.544091i	0.924725	−	0.380635i	$-0.124294\pi$
$43$	3.56784	+	3.56784i	0.544091	+	0.544091i	−0.380635	+	0.924725i	$0.624294\pi$
$44$	1.11749	+	0.363093i	0.168467	+	0.0547384i
$45$	3.44705	−	5.75481i	0.513855	−	0.857877i
$46$		0			0
$47$	5.44421	+	2.77396i	0.794120	+	0.404624i	0.803480	−	0.595332i	$-0.202979\pi$
$47$	5.44421	+	2.77396i	0.794120	+	0.404624i	−0.00936003	+	0.999956i	$0.502979\pi$
$48$		0			0
$49$		−	7.71436i		−	1.10205i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$53$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$54$		0			0
$55$	−0.293749	+	1.28042i	−0.0396090	+	0.172652i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$59$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	−12.6310	−	9.17699i	−1.61724	−	1.17499i	−0.827046	−	0.562134i	$-0.809981\pi$
$61$	−12.6310	−	9.17699i	−1.61724	−	1.17499i	−0.790193	−	0.612859i	$-0.790019\pi$
$62$		0			0
$63$	−11.3661	−	1.80022i	−1.43200	−	0.226806i
$64$	4.70228	−	6.47214i	0.587785	−	0.809017i
$65$		0			0
$66$		0			0
$67$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$67$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$68$	0.998256	+	0.998256i	0.121056	+	0.121056i
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$		0			0
$73$	2.66326	+	16.8152i	0.311712	+	1.96807i	0.240732	+	0.970592i	$0.422612\pi$
$73$	2.66326	+	16.8152i	0.311712	+	1.96807i	0.0709795	+	0.997478i	$0.477388\pi$
$74$		0			0
$75$		0			0
$76$	−8.71780			−1.00000
$77$	2.22585	−	0.352540i	0.253659	−	0.0401757i
$78$		0			0
$79$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$79$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$80$	7.67308	+	4.59606i	0.857877	+	0.513855i
$81$	−2.78115	+	8.55951i	−0.309017	+	0.951057i
$82$		0			0
$83$	−16.2287	+	8.26894i	−1.78133	+	0.907634i	−0.878114	+	0.478451i	$0.841198\pi$
$83$	−16.2287	+	8.26894i	−1.78133	+	0.907634i	−0.903218	+	0.429183i	$0.858802\pi$
$84$		0			0
$85$	−1.01290	+	1.21051i	−0.109864	+	0.131298i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$89$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$90$		0			0
$91$		0			0
$92$	4.82548	+	9.47054i	0.503091	+	0.987372i
$93$		0			0
$94$		0			0
$95$	−0.862882	−	9.70852i	−0.0885298	−	0.996074i
$96$		0			0
$97$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$97$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$98$		0			0
$99$		−	1.76249i		−	0.177137i
$100$	−4.35890	+	9.00000i	−0.435890	+	0.900000i
$101$	−14.8985			−1.48245			−0.741226	−	0.671255i	$-0.765755\pi$
$101$	−14.8985			−1.48245			−0.741226	+	0.671255i	$0.765755\pi$
$102$		0			0
$103$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$103$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$109$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$110$		0			0
$111$		0			0
$112$	2.40029	−	15.1548i	0.226806	−	1.43200i
$113$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$113$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$114$		0			0
$115$	−10.0692	+	6.31126i	−0.938957	+	0.588528i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.57516	+	0.836720i	0.236064	+	0.0767020i
$120$		0			0
$121$	−3.29253	−	10.1334i	−0.299321	−	0.921215i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.4542	−	3.96345i	−0.935055	−	0.354502i
$126$		0			0
$127$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$127$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.73680	+	17.6561i	−0.501227	+	1.54262i	0.305796	+	0.952097i	$0.401077\pi$
$131$	−5.73680	+	17.6561i	−0.501227	+	1.54262i	−0.807023	+	0.590520i	$0.798923\pi$
$132$		0			0
$133$	−14.8980	+	7.59092i	−1.29182	+	0.658216i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.44703	−	21.7637i	0.294499	−	1.85940i	−0.186203	−	0.982511i	$-0.559618\pi$
$137$	3.44703	−	21.7637i	0.294499	−	1.85940i	0.480702	−	0.876884i	$-0.340382\pi$
$138$		0			0
$139$	−3.25006	+	4.47332i	−0.275666	+	0.379422i	−0.924292	−	0.381685i	$-0.875344\pi$
$139$	−3.25006	+	4.47332i	−0.275666	+	0.379422i	0.648626	+	0.761107i	$0.275344\pi$
$140$	17.1146	+	1.17305i	1.44645	+	0.0991410i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−11.4127	−	3.70820i	−0.951057	−	0.309017i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	21.7097i		−	1.77853i	−0.457397	−	0.889263i	$-0.651218\pi$
$149$		−	21.7097i		−	1.77853i	0.457397	−	0.889263i	$-0.348782\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	0.961380	−	1.88681i	0.0777230	−	0.152540i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.74349	−	5.74349i	0.458380	−	0.458380i	−0.439743	−	0.898124i	$-0.644931\pi$
$157$	5.74349	−	5.74349i	0.458380	−	0.458380i	0.898124	+	0.439743i	$0.144931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.4927	+	11.9827i	1.29981	+	0.944366i
$162$		0			0
$163$	−22.8501	−	3.61911i	−1.78976	−	0.283470i	−0.828678	−	0.559726i	$-0.810906\pi$
$163$	−22.8501	−	3.61911i	−1.78976	−	0.283470i	−0.961084	+	0.276256i	$0.910906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$167$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$168$		0			0
$169$	12.3637	+	4.01722i	0.951057	+	0.309017i
$170$		0			0
$171$	4.04092	+	12.4367i	0.309017	+	0.951057i
$172$	8.99148	+	4.58139i	0.685594	+	0.349328i
$173$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$173$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$174$		0			0
$175$	0.387634	+	19.1757i	0.0293024	+	1.44955i
$176$	2.34999			0.177137
$177$		0			0
$178$		0			0
$179$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$179$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$180$	3.00000	−	13.0767i	0.223607	−	0.974679i
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.0648732	+	0.409593i	−0.00474400	+	0.0299524i
$188$	12.0699	+	1.91169i	0.880289	+	0.139424i
$189$		0			0
$190$		0			0
$191$	12.5191	−	9.09569i	0.905853	−	0.658141i	−0.0341095	−	0.999418i	$-0.510860\pi$
$191$	12.5191	−	9.09569i	0.905853	−	0.658141i	0.939963	+	0.341277i	$0.110860\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−4.76774	−	14.6736i	−0.340553	−	1.04811i
$197$	22.7412	+	11.5872i	1.62024	+	0.825554i	0.999124	+	0.0418369i	$0.0133210\pi$
$197$	22.7412	+	11.5872i	1.62024	+	0.825554i	0.621117	+	0.783718i	$0.286679\pi$
$198$		0			0
$199$			13.0767i			0.926982i	0.886102	+	0.463491i	$0.153403\pi$
$199$			13.0767i			0.926982i	−0.886102	+	0.463491i	$0.846597\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	11.2738	−	11.2738i	0.783583	−	0.783583i
$208$		0			0
$209$	−1.50522	−	2.07176i	−0.104119	−	0.143307i
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.21207	+	10.4668i	−0.287261	+	0.713828i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.232600	+	2.61705i	0.0156819	+	0.176442i
$221$		0			0
$222$		0			0
$223$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$223$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$224$		0			0
$225$	14.8597	+	2.04661i	0.990648	+	0.136441i
$226$		0			0
$227$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$227$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$228$		0			0
$229$	28.3413	−	9.20866i	1.87285	−	0.608525i	0.882426	−	0.470451i	$-0.155909\pi$
$229$	28.3413	−	9.20866i	1.87285	−	0.608525i	0.990422	−	0.138074i	$-0.0440911\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	24.6280	−	12.5486i	1.61343	−	0.822085i	0.613970	−	0.789329i	$-0.289572\pi$
$233$	24.6280	−	12.5486i	1.61343	−	0.822085i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$		0			0
$235$	−0.934266	+	13.6308i	−0.0609448	+	0.889175i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.33720	+	11.4752i	−0.539289	+	0.742268i	−0.988510	−	0.151153i	$-0.951701\pi$
$239$	−8.33720	+	11.4752i	−0.539289	+	0.742268i	0.449221	+	0.893420i	$0.351701\pi$
$240$		0			0
$241$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$241$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$242$		0			0
$243$		0			0
$244$	−29.6974	−	9.64925i	−1.90118	−	0.617730i
$245$	15.8692	−	6.76194i	1.01385	−	0.432005i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.8352			−1.75694			−0.878471	−	0.477796i	$-0.841436\pi$
$251$	−27.8352			−1.75694			−0.878471	+	0.477796i	$0.841436\pi$
$252$	−22.7322	+	3.60043i	−1.43200	+	0.226806i
$253$	−1.41748	+	2.78196i	−0.0891161	+	0.174900i
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.01640	−	0.952905i	−0.370987	−	0.0587586i	−0.0318427	−	0.999493i	$-0.510138\pi$
$263$	−6.01640	−	0.952905i	−0.370987	−	0.0587586i	−0.339145	+	0.940734i	$0.610138\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$269$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$270$		0			0
$271$	−10.1711	−	31.3033i	−0.617848	−	1.90154i	−0.332257	−	0.943189i	$-0.607810\pi$
$271$	−10.1711	−	31.3033i	−0.617848	−	1.90154i	−0.285591	−	0.958352i	$-0.592190\pi$
$272$	2.51575	+	1.28184i	0.152540	+	0.0777230i
$273$		0			0
$274$		0			0
$275$	−2.89144	+	0.518068i	−0.174360	+	0.0312407i
$276$		0			0
$277$	−30.9957	+	4.90924i	−1.86235	+	0.294968i	−0.983333	−	0.181815i	$-0.941803\pi$
$277$	−30.9957	+	4.90924i	−1.86235	+	0.294968i	−0.879021	+	0.476783i	$0.841803\pi$
$278$		0			0
$279$		0			0
$280$		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
$283$	29.1038	−	14.8291i	1.73004	−	0.881501i	0.756180	−	0.654364i	$-0.227063\pi$
$283$	29.1038	−	14.8291i	1.73004	−	0.881501i	0.973863	−	0.227137i	$-0.0729365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	9.69948	−	13.3502i	0.570558	−	0.785305i
$290$		0			0
$291$		0			0
$292$	15.4582	+	30.3384i	0.904622	+	1.77542i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	19.3549			1.11560
$302$		0			0
$303$		0			0
$304$	−16.5822	+	5.38790i	−0.951057	+	0.309017i
$305$	7.80641	−	34.0274i	0.446994	−	1.94840i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	4.01594	−	2.04622i	0.228829	−	0.116594i
$309$		0			0
$310$		0			0
$311$	25.8400	+	18.7739i	1.46525	+	1.06457i	0.981956	+	0.189112i	$0.0605608\pi$
$311$	25.8400	+	18.7739i	1.46525	+	1.06457i	0.483297	+	0.875457i	$0.339439\pi$
$312$		0			0
$313$	27.9975	+	4.43437i	1.58251	+	0.250646i	0.884883	−	0.465813i	$-0.154238\pi$
$313$	27.9975	+	4.43437i	1.58251	+	0.250646i	0.697630	+	0.716458i	$0.254238\pi$
$314$		0			0
$315$	−6.25962	−	24.9592i	−0.352690	−	1.40629i
$316$		0			0
$317$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$317$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$318$		0			0
$319$		0			0
$320$	17.4356	+	4.00000i	0.974679	+	0.223607i
$321$		0			0
$322$		0			0
$323$	−0.481323	−	3.03895i	−0.0267815	−	0.169092i
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.2911	−	7.24281i	1.22895	−	0.399309i
$330$		0			0
$331$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$331$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$332$	−25.7583	+	25.7583i	−1.41367	+	1.41367i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$337$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$338$		0			0
$339$		0			0
$340$	−1.17851	+	2.92853i	−0.0639135	+	0.158822i
$341$		0			0
$342$		0			0
$343$	−1.93764	−	1.93764i	−0.104623	−	0.104623i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.0576	−	13.2770i	−1.39885	−	0.712748i	−0.418167	−	0.908370i	$-0.637327\pi$
$347$	−26.0576	−	13.2770i	−1.39885	−	0.712748i	−0.980679	+	0.195622i	$0.937327\pi$
$348$		0			0
$349$		−	31.1518i		−	1.66752i	−0.552131	−	0.833758i	$-0.686185\pi$
$349$		−	31.1518i		−	1.66752i	0.552131	−	0.833758i	$-0.313815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.8503	+	25.2202i	−0.683954	+	1.34234i	0.244049	+	0.969763i	$0.421524\pi$
$353$	−12.8503	+	25.2202i	−0.683954	+	1.34234i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	22.2712	+	30.6537i	1.17543	+	1.61784i	0.600294	+	0.799780i	$0.295050\pi$
$359$	22.2712	+	30.6537i	1.17543	+	1.61784i	0.575135	+	0.818059i	$0.304950\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−32.2561	+	20.2178i	−1.68836	+	1.05825i
$366$		0			0
$367$	−0.592797	−	1.16343i	−0.0309438	−	0.0607305i	0.875017	−	0.484092i	$-0.160850\pi$
$367$	−0.592797	−	1.16343i	−0.0309438	−	0.0607305i	−0.905961	+	0.423362i	$0.860850\pi$
$368$	15.0317	+	15.0317i	0.783583	+	0.783583i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$373$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$379$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$380$	−7.64150	−	17.9334i	−0.392000	−	0.919965i
$381$		0			0
$382$		0			0
$383$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$383$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$384$		0			0
$385$	2.67626	+	4.26979i	0.136395	+	0.217609i
$386$		0			0
$387$	2.36796	−	14.9507i	0.120370	−	0.759987i
$388$		0			0
$389$	−17.9347	+	24.6850i	−0.909324	+	1.25158i	0.0580728	+	0.998312i	$0.481504\pi$
$389$	−17.9347	+	24.6850i	−0.909324	+	1.25158i	−0.967397	+	0.253265i	$0.918496\pi$
$390$		0			0
$391$	−3.03493	+	2.20501i	−0.153483	+	0.111512i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.08928	−	3.35246i	−0.0547384	−	0.168467i
$397$	0.692205	+	0.352696i	0.0347408	+	0.0177013i	0.471275	−	0.881987i	$-0.343794\pi$
$397$	0.692205	+	0.352696i	0.0347408	+	0.0177013i	−0.436534	+	0.899688i	$0.643794\pi$
$398$		0			0
$399$		0			0
$400$	−2.72881	+	19.8130i	−0.136441	+	0.990648i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−28.3386	+	9.20776i	−1.40990	+	0.458103i
$405$	−20.0456	+	1.78163i	−0.996074	+	0.0885298i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−31.2352	−	26.1361i	−1.53327	−	1.28297i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	29.0683	+	9.44487i	1.42008	+	0.461412i	0.915628	−	0.402027i	$-0.131694\pi$
$419$	29.0683	+	9.44487i	1.42008	+	0.461412i	0.504453	+	0.863439i	$0.331694\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$	−2.86753	−	18.1049i	−0.139424	−	0.880289i
$424$		0			0
$425$	−3.37799	−	1.02257i	−0.163856	−	0.0496021i
$426$		0			0
$427$	−59.1524	+	9.36881i	−2.86258	+	0.453389i
$428$		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	$0.600000\pi$
$432$		0			0
$433$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$433$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.62387	−	22.8802i	0.173353	−	1.09451i
$438$		0			0
$439$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$439$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$440$		0			0
$441$	−18.7231	+	13.6032i	−0.891578	+	0.647770i
$442$		0			0
$443$	−24.5177	−	24.5177i	−1.16487	−	1.16487i	−0.983396	−	0.181474i	$-0.941913\pi$
$443$	−24.5177	−	24.5177i	−1.16487	−	1.16487i	−0.181474	−	0.983396i	$-0.558087\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−4.80057	−	30.3096i	−0.226806	−	1.43200i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.1150	−	11.1150i	0.519940	−	0.519940i	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	11.1150	−	11.1150i	0.519940	−	0.519940i	0.917553	+	0.397613i	$0.130161\pi$
$458$		0			0
$459$		0			0
$460$	−15.2522	+	18.2278i	−0.711136	+	0.849877i
$461$	7.51914	+	5.46297i	0.350201	+	0.254436i	0.748953	−	0.662623i	$-0.230557\pi$
$461$	7.51914	+	5.46297i	0.350201	+	0.254436i	−0.398752	+	0.917059i	$0.630557\pi$
$462$		0			0
$463$	−19.0915	−	3.02380i	−0.887259	−	0.140528i	−0.303863	−	0.952716i	$-0.598276\pi$
$463$	−19.0915	−	3.02380i	−0.887259	−	0.140528i	−0.583396	+	0.812188i	$0.698276\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.6258	+	20.8544i	0.491705	+	0.965025i	0.994901	+	0.100853i	$0.0321571\pi$
$467$	10.6258	+	20.8544i	0.491705	+	0.965025i	−0.503197	+	0.864172i	$0.667843\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.463723	+	2.92783i	0.0213220	+	0.134622i
$474$		0			0
$475$	19.2151	−	10.2849i	0.881649	−	0.471906i
$476$	5.41537			0.248213
$477$		0			0
$478$		0			0
$479$	−38.2510	+	12.4285i	−1.74773	+	0.567873i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$	−38.2510	+	12.4285i	−1.74773	+	0.567873i	−0.751919	+	0.659256i	$0.770871\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−12.5255	−	17.2399i	−0.569342	−	0.783632i
$485$		0			0
$486$		0			0
$487$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$487$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.4917	−	11.2554i	0.699132	−	0.507949i	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	15.4917	−	11.2554i	0.699132	−	0.507949i	0.879649	+	0.475623i	$0.157777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	3.62563	−	1.54489i	0.162960	−	0.0694378i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			18.7412i			0.838970i	0.907762	+	0.419485i	$0.137789\pi$
$499$			18.7412i			0.838970i	−0.907762	+	0.419485i	$0.862211\pi$
$500$	−22.3347	−	1.07785i	−0.998838	−	0.0482030i
$501$		0			0
$502$		0			0
$503$	7.03394	−	13.8049i	0.313628	−	0.615529i	−0.679352	−	0.733813i	$-0.737739\pi$
$503$	7.03394	−	13.8049i	0.313628	−	0.615529i	0.992980	+	0.118283i	$0.0377391\pi$
$504$		0			0
$505$	−13.0591	−	30.6477i	−0.581122	−	1.36380i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		0			0
$511$	52.8336	+	38.3859i	2.33722	+	1.69809i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.62970	+	3.19846i	0.0716739	+	0.140668i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$523$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$524$			37.1294i			1.62200i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−4.98743	+	1.62051i	−0.216845	+	0.0704572i
$530$		0			0
$531$		0			0
$532$	−23.6463	+	23.6463i	−1.02520	+	1.02520i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.66394	−	3.66660i	0.114744	−	0.157932i
$540$		0			0
$541$	31.7378	−	23.0589i	1.36452	−	0.991378i	0.366372	−	0.930469i	$-0.380600\pi$
$541$	31.7378	−	23.0589i	1.36452	−	0.991378i	0.998143	−	0.0609096i	$-0.0194001\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$547$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$548$	−6.89405	−	43.5273i	−0.294499	−	1.85940i
$549$			46.8385i			1.99902i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−3.41731	+	10.5174i	−0.144926	+	0.446038i
$557$	8.24321	−	8.24321i	0.349276	−	0.349276i	−0.510564	−	0.859840i	$-0.670563\pi$
$557$	8.24321	−	8.24321i	0.349276	−	0.349276i	0.859840	+	0.510564i	$0.170563\pi$
$558$		0			0
$559$		0			0
$560$	33.2790	−	8.34616i	1.40629	−	0.352690i
$561$		0			0
$562$		0			0
$563$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$563$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.6733	+	30.7606i	0.658216	+	1.29182i
$568$		0			0
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		0			0
$571$	14.2531	+	43.8666i	0.596475	+	1.83576i	0.547243	+	0.836974i	$0.315677\pi$
$571$	14.2531	+	43.8666i	0.596475	+	1.83576i	0.0492323	+	0.998787i	$0.484323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.8090	−	15.1813i	−0.909496	−	0.633104i
$576$	−24.0000			−1.00000
$577$	−4.48177	+	0.709843i	−0.186579	+	0.0295511i	−0.249024	−	0.968497i	$-0.580110\pi$
$577$	−4.48177	+	0.709843i	−0.186579	+	0.0295511i	0.0624458	+	0.998048i	$0.480110\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.5902	+	66.4477i	−0.895711	+	2.75672i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.07811	−	38.3757i	0.250870	−	1.58393i	−0.464749	−	0.885443i	$-0.653855\pi$
$587$	6.07811	−	38.3757i	0.250870	−	1.58393i	0.715619	−	0.698491i	$-0.246145\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.8883	−	19.8883i	−0.816716	−	0.816716i	0.168915	−	0.985631i	$-0.445974\pi$
$593$	−19.8883	−	19.8883i	−0.816716	−	0.816716i	−0.985631	+	0.168915i	$0.945974\pi$
$594$		0			0
$595$	0.536009	+	6.03079i	0.0219742	+	0.247238i
$596$	−13.4173	−	41.2942i	−0.549595	−	1.69148i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.9593	−	15.6554i	0.730151	−	0.636481i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.662538	−	4.18310i	0.0267815	−	0.169092i
$613$	42.7600	+	6.77252i	1.72706	+	0.273540i	0.939463	−	0.342650i	$-0.111325\pi$
$613$	42.7600	+	6.77252i	1.72706	+	0.273540i	0.787598	+	0.616190i	$0.211325\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.1410	−	29.7159i	−0.609555	−	1.19632i	−0.965155	−	0.261680i	$-0.915723\pi$
$617$	−15.1410	−	29.7159i	−0.609555	−	1.19632i	0.355600	−	0.934638i	$-0.384277\pi$
$618$		0			0
$619$	32.3732	+	10.5187i	1.30119	+	0.422782i	0.875995	−	0.482321i	$-0.160206\pi$
$619$	32.3732	+	10.5187i	1.30119	+	0.422782i	0.425194	+	0.905102i	$0.360206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.01033	−	24.9796i	−0.0404132	−	0.999183i
$626$		0			0
$627$		0			0
$628$	7.37510	−	14.4744i	0.294298	−	0.577593i
$629$		0			0
$630$		0			0
$631$	8.98701	−	27.6592i	0.357767	−	1.10109i	−0.596621	−	0.802523i	$-0.703490\pi$
$631$	8.98701	−	27.6592i	0.357767	−	1.10109i	0.954388	−	0.298570i	$-0.0965097\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$641$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$642$		0			0
$643$	35.0697	+	35.0697i	1.38301	+	1.38301i	0.839218	+	0.543795i	$0.183013\pi$
$643$	35.0697	+	35.0697i	1.38301	+	1.38301i	0.543795	+	0.839218i	$0.316987\pi$
$644$	38.7767	+	12.5993i	1.52802	+	0.496483i
$645$		0			0
$646$		0			0
$647$	44.5654	+	22.7072i	1.75205	+	0.892713i	0.958929	+	0.283646i	$0.0915442\pi$
$647$	44.5654	+	22.7072i	1.75205	+	0.892713i	0.793119	+	0.609067i	$0.208456\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−45.7003	+	7.23822i	−1.78976	+	0.283470i
$653$	3.36676	−	6.60764i	0.131752	−	0.258577i	−0.815701	−	0.578474i	$-0.803648\pi$
$653$	3.36676	−	6.60764i	0.131752	−	0.258577i	0.947452	+	0.319897i	$0.103648\pi$
$654$		0			0
$655$	−41.3489	+	3.67504i	−1.61563	+	0.143596i
$656$		0			0
$657$	36.1150	−	36.1150i	1.40898	−	1.40898i
$658$		0			0
$659$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$659$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$660$		0			0
$661$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$661$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−28.6740	−	23.9930i	−1.11193	−	0.930410i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.83445	−	8.72355i	−0.109423	−	0.336769i
$672$		0			0
$673$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$673$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$674$		0			0
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$677$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$683$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$684$	15.3726	+	21.1585i	0.587785	+	0.809017i
$685$	47.7916	−	11.9858i	1.82602	−	0.457955i
$686$		0			0
$687$		0			0
$688$	19.9343	+	3.15728i	0.759987	+	0.120370i
$689$		0			0
$690$		0			0
$691$	36.7373	−	26.6912i	1.39755	−	1.01538i	0.402562	−	0.915393i	$-0.368120\pi$
$691$	36.7373	−	26.6912i	1.39755	−	1.01538i	0.994989	−	0.0999876i	$-0.0318803\pi$
$692$		0			0
$693$	−4.78060	−	4.78060i	−0.181600	−	0.181600i
$694$		0			0
$695$	−12.0509	−	2.76466i	−0.457116	−	0.104870i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	12.5886	+	36.2348i	0.475804	+	1.36955i
$701$	−17.4356			−0.658533			−0.329267	−	0.944237i	$-0.606802\pi$
$701$	−17.4356			−0.658533			−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$	4.46994	−	1.45237i	0.168467	−	0.0547384i
$705$		0			0
$706$		0			0
$707$	−40.4108	+	40.4108i	−1.51980	+	1.51980i
$708$		0			0
$709$	−21.4184	−	29.4800i	−0.804386	−	1.10714i	−0.992165	−	0.124932i	$-0.960129\pi$
$709$	−21.4184	−	29.4800i	−0.804386	−	1.10714i	0.187779	−	0.982211i	$-0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.5181	+	8.29134i	0.951666	+	0.309215i	0.743392	−	0.668856i	$-0.233216\pi$
$719$	25.5181	+	8.29134i	0.951666	+	0.309215i	0.208273	+	0.978071i	$0.433216\pi$
$720$	−2.37550	−	26.7275i	−0.0885298	−	0.996074i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−51.1148	+	8.09578i	−1.89574	+	0.300256i	−0.991840	−	0.127485i	$-0.959309\pi$
$727$	−51.1148	+	8.09578i	−1.89574	+	0.300256i	−0.903901	+	0.427741i	$0.859309\pi$
$728$		0			0
$729$	25.6785	−	8.34346i	0.951057	−	0.309017i
$730$		0			0
$731$	−1.10060	+	3.38730i	−0.0407072	+	0.125284i
$732$		0			0
$733$	−47.9833	+	24.4487i	−1.77230	+	0.903033i	−0.838822	+	0.544406i	$0.816755\pi$
$733$	−47.9833	+	24.4487i	−1.77230	+	0.903033i	−0.933481	+	0.358627i	$0.883245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	19.6136	−	26.9959i	0.721500	−	0.993059i	−0.277973	−	0.960589i	$-0.589663\pi$
$739$	19.6136	−	26.9959i	0.721500	−	0.993059i	0.999473	−	0.0324701i	$-0.0103374\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	44.6591	−	19.0294i	1.63618	−	0.697183i
$746$		0			0
$747$	48.6861	+	24.8068i	1.78133	+	0.907634i
$748$	0.129746	+	0.819186i	0.00474400	+	0.0299524i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	24.1398	−	3.82337i	0.880289	−	0.139424i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−21.5414	+	21.5414i	−0.782935	+	0.782935i	−0.980325	−	0.197390i	$-0.936754\pi$
$757$	−21.5414	+	21.5414i	−0.782935	+	0.782935i	0.197390	+	0.980325i	$0.436754\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−23.3011	−	16.9293i	−0.844665	−	0.613685i	0.0790050	−	0.996874i	$-0.474826\pi$
$761$	−23.3011	−	16.9293i	−0.844665	−	0.613685i	−0.923670	+	0.383189i	$0.874826\pi$
$762$		0			0
$763$		0			0
$764$	18.1914	−	25.0383i	0.658141	−	0.905853i
$765$	4.72406	+	0.323791i	0.170799	+	0.0117067i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−51.4240	−	16.7087i	−1.85440	−	0.602530i	−0.995980	−	0.0895741i	$-0.971449\pi$
$769$	−51.4240	−	16.7087i	−1.85440	−	0.602530i	−0.858415	−	0.512956i	$-0.828551\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$773$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−18.1376	−	24.9642i	−0.647770	−	0.891578i
$785$	16.8494	+	6.78056i	0.601379	+	0.242009i
$786$		0			0
$787$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$787$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$788$	50.4176	+	7.98536i	1.79605	+	0.284467i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	8.08184	+	24.8734i	0.286453	+	0.881613i
$797$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$797$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$798$		0			0
$799$			4.31302i			0.152584i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.54082	+	8.91186i	−0.160242	+	0.314493i
$804$		0			0
$805$	−10.1931	+	44.4305i	−0.359258	+	1.56597i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.49742	−	10.3193i	−0.263595	−	0.362808i	0.656619	−	0.754222i	$-0.271986\pi$
$809$	−7.49742	−	10.3193i	−0.263595	−	0.362808i	−0.920215	+	0.391414i	$0.871986\pi$
$810$		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
$815$	−12.5842	−	50.1774i	−0.440805	−	1.75764i
$816$		0			0
$817$	−9.98491	−	19.5965i	−0.349328	−	0.685594i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.4663	−	35.2896i	−0.400176	−	1.23162i	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−11.4663	−	35.2896i	−0.400176	−	1.23162i	0.524680	−	0.851299i	$-0.324185\pi$
$822$		0			0
$823$	8.53748	+	53.9035i	0.297598	+	1.87896i	0.453606	+	0.891202i	$0.350137\pi$
$823$	8.53748	+	53.9035i	0.297598	+	1.87896i	−0.156009	+	0.987756i	$0.549863\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$827$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$828$	14.4764	−	28.4116i	0.503091	−	0.987372i
$829$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$829$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.85186	−	2.47214i	0.168107	−	0.0856547i
$834$		0			0
$835$		0			0
$836$	−4.14353	−	3.01045i	−0.143307	−	0.104119i
$837$		0			0
$838$		0			0
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.57346	+	28.9547i	0.0885298	+	0.996074i
$846$		0			0
$847$	−36.4166	−	18.5552i	−1.25129	−	0.637563i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−7.94361	+	15.5902i	−0.271984	+	0.533799i	−0.986085	−	0.166244i	$-0.946836\pi$
$853$	−7.94361	+	15.5902i	−0.271984	+	0.533799i	0.714100	+	0.700043i	$0.246836\pi$
$854$		0			0
$855$	−22.0415	+	19.2138i	−0.753804	+	0.657099i
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$	21.7638	+	29.9552i	0.742570	+	1.02206i	0.998467	+	0.0553558i	$0.0176293\pi$
$859$	21.7638	+	29.9552i	0.742570	+	1.02206i	−0.255897	+	0.966704i	$0.582371\pi$
$860$	−1.54300	+	22.5122i	−0.0526160	+	0.767659i
$861$		0			0
$862$		0			0
$863$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$863$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−39.1067	+	17.6057i	−1.32205	+	0.595181i
$876$		0			0
$877$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$877$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$878$		0			0
$879$		0			0
$880$	2.05986	+	4.83417i	0.0694378	+	0.162960i
$881$	−5.70763	+	17.5663i	−0.192295	+	0.591823i	0.807703	+	0.589590i	$0.200711\pi$
$881$	−5.70763	+	17.5663i	−0.192295	+	0.591823i	−0.999998	+	0.00223258i	$0.999289\pi$
$882$		0			0
$883$	44.8534	−	22.8540i	1.50944	−	0.769097i	0.513410	−	0.858143i	$-0.328382\pi$
$883$	44.8534	−	22.8540i	1.50944	−	0.769097i	0.996027	+	0.0890464i	$0.0283819\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$887$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−4.27766	+	3.10790i	−0.143307	+	0.104119i
$892$		0			0
$893$	−18.8328	−	18.8328i	−0.630217	−	0.630217i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	29.5297	−	5.29093i	0.984325	−	0.176364i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	26.2713	+	36.1593i	0.871364	+	1.19933i
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$		0			0
$913$	−10.5689	−	1.67394i	−0.349779	−	0.0553995i
$914$		0			0
$915$		0			0
$916$	48.2172	−	35.0318i	1.59314	−	1.15748i
$917$	32.3299	+	63.4511i	1.06763	+	2.09534i
$918$		0			0
$919$	−26.8334	−	8.71869i	−0.885151	−	0.287603i	−0.169057	−	0.985606i	$-0.554072\pi$
$919$	−26.8334	−	8.71869i	−0.885151	−	0.287603i	−0.716094	+	0.698003i	$0.754072\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.1645	+	10.7758i	−1.08809	+	0.353542i	−0.797509	−	0.603308i	$-0.793849\pi$
$929$	−33.1645	+	10.7758i	−1.08809	+	0.353542i	−0.290583	+	0.956850i	$0.593849\pi$
$930$		0			0
$931$	−10.3910	+	31.9803i	−0.340553	+	1.04811i
$932$	39.0898	−	39.0898i	1.28043	−	1.28043i
$933$		0			0
$934$		0			0
$935$	−0.899440	+	0.225574i	−0.0294148	+	0.00737705i
$936$		0			0
$937$	9.33717	−	58.9525i	0.305032	−	1.92590i	−0.0671650	−	0.997742i	$-0.521395\pi$
$937$	9.33717	−	58.9525i	0.305032	−	1.92590i	0.372197	−	0.928154i	$-0.378605\pi$
$938$		0			0
$939$		0			0
$940$	6.64722	+	26.5047i	0.216808	+	0.864489i
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−54.8190	−	27.9317i	−1.78138	−	0.907657i	−0.902821	−	0.430016i	$-0.858508\pi$
$947$	−54.8190	−	27.9317i	−1.78138	−	0.907657i	−0.878555	−	0.477641i	$-0.841492\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$953$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$954$		0			0
$955$	29.6843	+	17.7804i	0.960561	+	0.575362i
$956$	−8.76626	+	26.9798i	−0.283521	+	0.872588i
$957$		0			0
$958$		0			0
$959$	−49.6823	−	68.3818i	−1.60432	−	2.20816i
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	17.4317	+	34.2117i	0.560566	+	1.10017i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	17.4317	+	34.2117i	0.560566	+	1.10017i	−0.420643	+	0.907226i	$0.638195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$		0			0
$973$	3.31799	+	20.9490i	0.106370	+	0.671594i
$974$		0			0
$975$		0			0
$976$	−62.4513			−1.99902
$977$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$977$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$978$		0			0
$979$		0			0
$980$	26.0060	−	22.6697i	0.830731	−	0.724157i
$981$		0			0
$982$		0			0
$983$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$983$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$984$		0			0
$985$	−3.90255	+	56.9376i	−0.124346	+	1.81418i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.7617	+	21.6941i	−0.501193	+	0.689833i
$990$		0			0
$991$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$991$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−26.9001	+	11.4622i	−0.852791	+	0.363378i
$996$		0			0
$997$	36.9460	+	18.8250i	1.17009	+	0.596192i	0.927459	−	0.373926i	$-0.121988\pi$
$997$	36.9460	+	18.8250i	1.17009	+	0.596192i	0.242634	+	0.970118i	$0.421988\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.a.113.2	✓	16
19.18	odd	2	CM	475.2.v.a.113.2	✓	16
25.2	odd	20	inner	475.2.v.a.227.2	yes	16
475.227	even	20	inner	475.2.v.a.227.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.a.113.2	✓	16	1.1	even	1	trivial
475.2.v.a.113.2	✓	16	19.18	odd	2	CM
475.2.v.a.227.2	yes	16	25.2	odd	20	inner
475.2.v.a.227.2	yes	16	475.227	even	20	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.v (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(1.90211 - 0.618034i) q^{4} +(0.876540 + 2.05710i) q^{5} +(2.71241 - 2.71241i) q^{7} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)	\(q+(1.90211 - 0.618034i) q^{4} +(0.876540 + 2.05710i) q^{5} +(2.71241 - 2.71241i) q^{7} +(-1.76336 - 2.42705i) q^{9} +(0.475295 + 0.345322i) q^{11} +(3.23607 - 2.35114i) q^{16} +(0.320460 + 0.628938i) q^{17} +(-4.14556 - 1.34697i) q^{19} +(2.93864 + 3.37111i) q^{20} +(0.831374 + 5.24909i) q^{23} +(-3.46336 + 3.60627i) q^{25} +(3.48295 - 6.83568i) q^{28} +(7.95725 + 3.20218i) q^{35} +(-4.85410 - 3.52671i) q^{36} +(3.56784 + 3.56784i) q^{43} +(1.11749 + 0.363093i) q^{44} +(3.44705 - 5.75481i) q^{45} +(5.44421 + 2.77396i) q^{47} -7.71436i q^{49} +(-0.293749 + 1.28042i) q^{55} +(-12.6310 - 9.17699i) q^{61} +(-11.3661 - 1.80022i) q^{63} +(4.70228 - 6.47214i) q^{64} +(0.998256 + 0.998256i) q^{68} +(2.66326 + 16.8152i) q^{73} -8.71780 q^{76} +(2.22585 - 0.352540i) q^{77} +(7.67308 + 4.59606i) q^{80} +(-2.78115 + 8.55951i) q^{81} +(-16.2287 + 8.26894i) q^{83} +(-1.01290 + 1.21051i) q^{85} +(4.82548 + 9.47054i) q^{92} +(-0.862882 - 9.70852i) q^{95} -1.76249i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 16 * q - 2 * q^5 - 6 * q^7	\( 16 q - 2 q^{5} - 6 q^{7} + 16 q^{16} - 56 q^{17} - 8 q^{23} + 18 q^{25} + 12 q^{28} + 88 q^{35} - 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} + 8 q^{80} + 36 q^{81} - 128 q^{83} - 12 q^{85} - 16 q^{92} + 38 q^{95}+O(q^{100}) \) 16 * q - 2 * q^5 - 6 * q^7 + 16 * q^16 - 56 * q^17 - 8 * q^23 + 18 * q^25 + 12 * q^28 + 88 * q^35 - 24 * q^36 + 2 * q^43 - 26 * q^47 - 18 * q^63 - 28 * q^68 + 22 * q^73 + 38 * q^77 + 8 * q^80 + 36 * q^81 - 128 * q^83 - 12 * q^85 - 16 * q^92 + 38 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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