LMFDB - Embedded newform 1600.2.s.e.207.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 1]))

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.s (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.1
Character	$\chi$	$=$	1600.207
Dual form			1600.2.s.e.943.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25766 q^{3} +(2.54012 + 2.54012i) q^{7} +7.61238 q^{9} +(0.462406 - 0.462406i) q^{11} +1.33473i q^{13} +(2.37342 + 2.37342i) q^{17} +(2.69776 - 2.69776i) q^{19} +(-8.27486 - 8.27486i) q^{21} +(-2.10420 + 2.10420i) q^{23} -15.0256 q^{27} +(1.97767 + 1.97767i) q^{29} -7.03054i q^{31} +(-1.50636 + 1.50636i) q^{33} -7.81135i q^{37} -4.34811i q^{39} +2.17459i q^{41} -3.10070i q^{43} +(0.0727309 - 0.0727309i) q^{47} +5.90443i q^{49} +(-7.73182 - 7.73182i) q^{51} +0.719718 q^{53} +(-8.78840 + 8.78840i) q^{57} +(8.67421 + 8.67421i) q^{59} +(-7.10027 + 7.10027i) q^{61} +(19.3364 + 19.3364i) q^{63} +10.8172i q^{67} +(6.85476 - 6.85476i) q^{69} +15.3474 q^{71} +(-0.905052 - 0.905052i) q^{73} +2.34913 q^{77} -3.90167 q^{79} +26.1112 q^{81} +6.02206 q^{83} +(-6.44260 - 6.44260i) q^{87} -7.46664 q^{89} +(-3.39038 + 3.39038i) q^{91} +22.9031i q^{93} +(3.74551 + 3.74551i) q^{97} +(3.52001 - 3.52001i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) $ 24 * q + 40 * q^9 + 20 * q^11 - 12 * q^19 - 8 * q^29 - 20 * q^51 + 8 * q^59 - 48 * q^61 + 64 * q^69 + 16 * q^71 - 104 * q^79 + 48 * q^81 - 96 * q^89 - 64 * q^91 + 128 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{4}\right)$	$-1$

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
$2$		0			0
$3$	−3.25766			−1.88081			−0.940407	−	0.340052i	$-0.889555\pi$
$3$	−3.25766			−1.88081			−0.940407	+	0.340052i	$0.889555\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.54012	+	2.54012i	0.960075	+	0.960075i	0.999233	−	0.0391576i	$-0.0124674\pi$
$7$	2.54012	+	2.54012i	0.960075	+	0.960075i	−0.0391576	+	0.999233i	$0.512467\pi$
$8$		0			0
$9$	7.61238			2.53746
$10$		0			0
$11$	0.462406	−	0.462406i	0.139421	−	0.139421i	−0.633952	−	0.773372i	$-0.718568\pi$
$11$	0.462406	−	0.462406i	0.139421	−	0.139421i	0.773372	+	0.633952i	$0.218568\pi$
$12$		0			0
$13$			1.33473i			0.370188i	0.982721	+	0.185094i	$0.0592590\pi$
$13$			1.33473i			0.370188i	−0.982721	+	0.185094i	$0.940741\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.37342	+	2.37342i	0.575640	+	0.575640i	0.933699	−	0.358059i	$-0.116561\pi$
$17$	2.37342	+	2.37342i	0.575640	+	0.575640i	−0.358059	+	0.933699i	$0.616561\pi$
$18$		0			0
$19$	2.69776	−	2.69776i	0.618909	−	0.618909i	−0.326343	−	0.945251i	$-0.605816\pi$
$19$	2.69776	−	2.69776i	0.618909	−	0.618909i	0.945251	+	0.326343i	$0.105816\pi$
$20$		0			0
$21$	−8.27486	−	8.27486i	−1.80572	−	1.80572i
$22$		0			0
$23$	−2.10420	+	2.10420i	−0.438755	+	0.438755i	−0.891593	−	0.452838i	$-0.850412\pi$
$23$	−2.10420	+	2.10420i	−0.438755	+	0.438755i	0.452838	+	0.891593i	$0.350412\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−15.0256			−2.89168
$28$		0			0
$29$	1.97767	+	1.97767i	0.367245	+	0.367245i	0.866471	−	0.499227i	$-0.166382\pi$
$29$	1.97767	+	1.97767i	0.367245	+	0.367245i	−0.499227	+	0.866471i	$0.666382\pi$
$30$		0			0
$31$		−	7.03054i		−	1.26272i	−0.775489	−	0.631361i	$-0.782497\pi$
$31$		−	7.03054i		−	1.26272i	0.775489	−	0.631361i	$-0.217503\pi$
$32$		0			0
$33$	−1.50636	+	1.50636i	−0.262224	+	0.262224i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	7.81135i		−	1.28418i	−0.766630	−	0.642089i	$-0.778068\pi$
$37$		−	7.81135i		−	1.28418i	0.766630	−	0.642089i	$-0.221932\pi$
$38$		0			0
$39$		−	4.34811i		−	0.696255i
$40$		0			0
$41$			2.17459i			0.339614i	0.985477	+	0.169807i	$0.0543144\pi$
$41$			2.17459i			0.339614i	−0.985477	+	0.169807i	$0.945686\pi$
$42$		0			0
$43$		−	3.10070i		−	0.472852i	−0.971650	−	0.236426i	$-0.924024\pi$
$43$		−	3.10070i		−	0.472852i	0.971650	−	0.236426i	$-0.0759761\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.0727309	−	0.0727309i	0.0106089	−	0.0106089i	−0.701782	−	0.712391i	$-0.747612\pi$
$47$	0.0727309	−	0.0727309i	0.0106089	−	0.0106089i	0.712391	+	0.701782i	$0.247612\pi$
$48$		0			0
$49$			5.90443i			0.843490i
$50$		0			0
$51$	−7.73182	−	7.73182i	−1.08267	−	1.08267i
$52$		0			0
$53$	0.719718			0.0988609			0.0494304	−	0.998778i	$-0.484259\pi$
$53$	0.719718			0.0988609			0.0494304	+	0.998778i	$0.484259\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.78840	+	8.78840i	−1.16405	+	1.16405i
$58$		0			0
$59$	8.67421	+	8.67421i	1.12929	+	1.12929i	0.990293	+	0.138993i	$0.0443866\pi$
$59$	8.67421	+	8.67421i	1.12929	+	1.12929i	0.138993	+	0.990293i	$0.455613\pi$
$60$		0			0
$61$	−7.10027	+	7.10027i	−0.909097	+	0.909097i	−0.996199	−	0.0871025i	$-0.972239\pi$
$61$	−7.10027	+	7.10027i	−0.909097	+	0.909097i	0.0871025	+	0.996199i	$0.472239\pi$
$62$		0			0
$63$	19.3364	+	19.3364i	2.43615	+	2.43615i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			10.8172i			1.32153i	0.750593	+	0.660765i	$0.229768\pi$
$67$			10.8172i			1.32153i	−0.750593	+	0.660765i	$0.770232\pi$
$68$		0			0
$69$	6.85476	−	6.85476i	0.825217	−	0.825217i
$70$		0			0
$71$	15.3474			1.82140			0.910698	−	0.413072i	$-0.135544\pi$
$71$	15.3474			1.82140			0.910698	+	0.413072i	$0.135544\pi$
$72$		0			0
$73$	−0.905052	−	0.905052i	−0.105928	−	0.105928i	0.652156	−	0.758085i	$-0.273865\pi$
$73$	−0.905052	−	0.905052i	−0.105928	−	0.105928i	−0.758085	+	0.652156i	$0.773865\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.34913			0.267709
$78$		0			0
$79$	−3.90167			−0.438972			−0.219486	−	0.975616i	$-0.570438\pi$
$79$	−3.90167			−0.438972			−0.219486	+	0.975616i	$0.570438\pi$
$80$		0			0
$81$	26.1112			2.90124
$82$		0			0
$83$	6.02206			0.661007			0.330503	−	0.943805i	$-0.392782\pi$
$83$	6.02206			0.661007			0.330503	+	0.943805i	$0.392782\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.44260	−	6.44260i	−0.690719	−	0.690719i
$88$		0			0
$89$	−7.46664			−0.791462			−0.395731	−	0.918366i	$-0.629509\pi$
$89$	−7.46664			−0.791462			−0.395731	+	0.918366i	$0.629509\pi$
$90$		0			0
$91$	−3.39038	+	3.39038i	−0.355408	+	0.355408i
$92$		0			0
$93$			22.9031i			2.37494i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.74551	+	3.74551i	0.380299	+	0.380299i	0.871210	−	0.490911i	$-0.163336\pi$
$97$	3.74551	+	3.74551i	0.380299	+	0.380299i	−0.490911	+	0.871210i	$0.663336\pi$
$98$		0			0
$99$	3.52001	−	3.52001i	0.353774	−	0.353774i
$100$		0			0
$101$	4.39232	+	4.39232i	0.437052	+	0.437052i	0.891019	−	0.453966i	$-0.149991\pi$
$101$	4.39232	+	4.39232i	0.437052	+	0.437052i	−0.453966	+	0.891019i	$0.649991\pi$
$102$		0			0
$103$	−6.36518	+	6.36518i	−0.627179	+	0.627179i	−0.947357	−	0.320178i	$-0.896257\pi$
$103$	−6.36518	+	6.36518i	−0.627179	+	0.627179i	0.320178	+	0.947357i	$0.396257\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.8172			1.04574			0.522869	−	0.852413i	$-0.324862\pi$
$107$	10.8172			1.04574			0.522869	+	0.852413i	$0.324862\pi$
$108$		0			0
$109$	7.56691	+	7.56691i	0.724779	+	0.724779i	0.969575	−	0.244796i	$-0.0787210\pi$
$109$	7.56691	+	7.56691i	0.724779	+	0.724779i	−0.244796	+	0.969575i	$0.578721\pi$
$110$		0			0
$111$			25.4468i			2.41530i
$112$		0			0
$113$	12.2142	−	12.2142i	1.14902	−	1.14902i	0.162270	−	0.986746i	$-0.448118\pi$
$113$	12.2142	−	12.2142i	1.14902	−	1.14902i	0.986746	−	0.162270i	$-0.0518816\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			10.1605i			0.939337i
$118$		0			0
$119$			12.0576i			1.10532i
$120$		0			0
$121$			10.5724i			0.961124i
$122$		0			0
$123$		−	7.08408i		−	0.638750i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.98272	−	8.98272i	0.797087	−	0.797087i	−0.185548	−	0.982635i	$-0.559406\pi$
$127$	8.98272	−	8.98272i	0.797087	−	0.797087i	0.982635	+	0.185548i	$0.0594059\pi$
$128$		0			0
$129$			10.1010i			0.889346i
$130$		0			0
$131$	5.01718	+	5.01718i	0.438353	+	0.438353i	0.891458	−	0.453104i	$-0.149683\pi$
$131$	5.01718	+	5.01718i	0.438353	+	0.438353i	−0.453104	+	0.891458i	$0.649683\pi$
$132$		0			0
$133$	13.7053			1.18840
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.21068	−	3.21068i	0.274307	−	0.274307i	−0.556524	−	0.830831i	$-0.687865\pi$
$137$	3.21068	−	3.21068i	0.274307	−	0.274307i	0.830831	+	0.556524i	$0.187865\pi$
$138$		0			0
$139$	0.102562	+	0.102562i	0.00869923	+	0.00869923i	0.711443	−	0.702744i	$-0.248042\pi$
$139$	0.102562	+	0.102562i	0.00869923	+	0.00869923i	−0.702744	+	0.711443i	$0.748042\pi$
$140$		0			0
$141$	−0.236933	+	0.236933i	−0.0199533	+	0.0199533i
$142$		0			0
$143$	0.617188	+	0.617188i	0.0516118	+	0.0516118i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	19.2346i		−	1.58645i
$148$		0			0
$149$	2.12805	−	2.12805i	0.174337	−	0.174337i	−0.614545	−	0.788882i	$-0.710660\pi$
$149$	2.12805	−	2.12805i	0.174337	−	0.174337i	0.788882	+	0.614545i	$0.210660\pi$
$150$		0			0
$151$	−21.6723			−1.76366			−0.881832	−	0.471564i	$-0.843690\pi$
$151$	−21.6723			−1.76366			−0.881832	+	0.471564i	$0.843690\pi$
$152$		0			0
$153$	18.0674	+	18.0674i	1.46066	+	1.46066i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.3539			−1.38499			−0.692496	−	0.721421i	$-0.743489\pi$
$157$	−17.3539			−1.38499			−0.692496	+	0.721421i	$0.743489\pi$
$158$		0			0
$159$	−2.34460			−0.185939
$160$		0			0
$161$	−10.6898			−0.842476
$162$		0			0
$163$	−6.29032			−0.492696			−0.246348	−	0.969181i	$-0.579231\pi$
$163$	−6.29032			−0.492696			−0.246348	+	0.969181i	$0.579231\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.61495	+	6.61495i	0.511880	+	0.511880i	0.915102	−	0.403222i	$-0.132110\pi$
$167$	6.61495	+	6.61495i	0.511880	+	0.511880i	−0.403222	+	0.915102i	$0.632110\pi$
$168$		0			0
$169$	11.2185			0.862961
$170$		0			0
$171$	20.5364	−	20.5364i	1.57046	−	1.57046i
$172$		0			0
$173$		−	0.857921i		−	0.0652265i	−0.999468	−	0.0326133i	$-0.989617\pi$
$173$		−	0.857921i		−	0.0652265i	0.999468	−	0.0326133i	$-0.0103830\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−28.2577	−	28.2577i	−2.12398	−	2.12398i
$178$		0			0
$179$	0.571652	−	0.571652i	0.0427273	−	0.0427273i	−0.685420	−	0.728148i	$-0.740382\pi$
$179$	0.571652	−	0.571652i	0.0427273	−	0.0427273i	0.728148	+	0.685420i	$0.240382\pi$
$180$		0			0
$181$	−10.3879	−	10.3879i	−0.772130	−	0.772130i	0.206349	−	0.978478i	$-0.433842\pi$
$181$	−10.3879	−	10.3879i	−0.772130	−	0.772130i	−0.978478	+	0.206349i	$0.933842\pi$
$182$		0			0
$183$	23.1303	−	23.1303i	1.70984	−	1.70984i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.19497			0.160512
$188$		0			0
$189$	−38.1668	−	38.1668i	−2.77623	−	2.77623i
$190$		0			0
$191$			22.5293i			1.63017i	0.579344	+	0.815083i	$0.303309\pi$
$191$			22.5293i			1.63017i	−0.579344	+	0.815083i	$0.696691\pi$
$192$		0			0
$193$	7.95941	−	7.95941i	0.572931	−	0.572931i	−0.360015	−	0.932946i	$-0.617229\pi$
$193$	7.95941	−	7.95941i	0.572931	−	0.572931i	0.932946	+	0.360015i	$0.117229\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.5546i			1.25071i	0.780338	+	0.625357i	$0.215047\pi$
$197$			17.5546i			1.25071i	−0.780338	+	0.625357i	$0.784953\pi$
$198$		0			0
$199$			18.7910i			1.33206i	0.745924	+	0.666031i	$0.232008\pi$
$199$			18.7910i			1.33206i	−0.745924	+	0.666031i	$0.767992\pi$
$200$		0			0
$201$		−	35.2388i		−	2.48555i
$202$		0			0
$203$			10.0471i			0.705165i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−16.0179	+	16.0179i	−1.11332	+	1.11332i
$208$		0			0
$209$		−	2.49492i		−	0.172577i
$210$		0			0
$211$	−9.00592	−	9.00592i	−0.619993	−	0.619993i	0.325536	−	0.945530i	$-0.394455\pi$
$211$	−9.00592	−	9.00592i	−0.619993	−	0.619993i	−0.945530	+	0.325536i	$0.894455\pi$
$212$		0			0
$213$	−49.9965			−3.42571
$214$		0			0
$215$		0			0
$216$		0			0
$217$	17.8584	−	17.8584i	1.21231	−	1.21231i
$218$		0			0
$219$	2.94836	+	2.94836i	0.199232	+	0.199232i
$220$		0			0
$221$	−3.16788	+	3.16788i	−0.213095	+	0.213095i
$222$		0			0
$223$	−9.16630	−	9.16630i	−0.613821	−	0.613821i	0.330119	−	0.943939i	$-0.392911\pi$
$223$	−9.16630	−	9.16630i	−0.613821	−	0.613821i	−0.943939	+	0.330119i	$0.892911\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			21.2474i			1.41024i	0.709088	+	0.705120i	$0.249107\pi$
$227$			21.2474i			1.41024i	−0.709088	+	0.705120i	$0.750893\pi$
$228$		0			0
$229$	−14.0502	+	14.0502i	−0.928461	+	0.928461i	−0.997607	−	0.0691452i	$-0.977973\pi$
$229$	−14.0502	+	14.0502i	−0.928461	+	0.928461i	0.0691452	+	0.997607i	$0.477973\pi$
$230$		0			0
$231$	−7.65269			−0.503510
$232$		0			0
$233$	−19.7571	−	19.7571i	−1.29433	−	1.29433i	−0.932083	−	0.362244i	$-0.882011\pi$
$233$	−19.7571	−	19.7571i	−1.29433	−	1.29433i	−0.362244	−	0.932083i	$-0.617989\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.7103			0.825625
$238$		0			0
$239$	1.51630			0.0980814			0.0490407	−	0.998797i	$-0.484384\pi$
$239$	1.51630			0.0980814			0.0490407	+	0.998797i	$0.484384\pi$
$240$		0			0
$241$	5.91616			0.381094			0.190547	−	0.981678i	$-0.438974\pi$
$241$	5.91616			0.381094			0.190547	+	0.981678i	$0.438974\pi$
$242$		0			0
$243$	−39.9847			−2.56502
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.60079	+	3.60079i	0.229112	+	0.229112i
$248$		0			0
$249$	−19.6178			−1.24323
$250$		0			0
$251$	11.8723	−	11.8723i	0.749376	−	0.749376i	−0.224986	−	0.974362i	$-0.572234\pi$
$251$	11.8723	−	11.8723i	0.749376	−	0.749376i	0.974362	+	0.224986i	$0.0722336\pi$
$252$		0			0
$253$			1.94598i			0.122343i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.39452	+	8.39452i	0.523636	+	0.523636i	0.918667	−	0.395032i	$-0.129266\pi$
$257$	8.39452	+	8.39452i	0.523636	+	0.523636i	−0.395032	+	0.918667i	$0.629266\pi$
$258$		0			0
$259$	19.8418	−	19.8418i	1.23291	−	1.23291i
$260$		0			0
$261$	15.0548	+	15.0548i	0.931869	+	0.931869i
$262$		0			0
$263$	−10.8623	+	10.8623i	−0.669798	+	0.669798i	−0.957669	−	0.287871i	$-0.907053\pi$
$263$	−10.8623	+	10.8623i	−0.669798	+	0.669798i	0.287871	+	0.957669i	$0.407053\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	24.3238			1.48859
$268$		0			0
$269$	12.9373	+	12.9373i	0.788804	+	0.788804i	0.981298	−	0.192494i	$-0.0616577\pi$
$269$	12.9373	+	12.9373i	0.788804	+	0.788804i	−0.192494	+	0.981298i	$0.561658\pi$
$270$		0			0
$271$			1.44017i			0.0874842i	0.999043	+	0.0437421i	$0.0139280\pi$
$271$			1.44017i			0.0874842i	−0.999043	+	0.0437421i	$0.986072\pi$
$272$		0			0
$273$	11.0447	−	11.0447i	0.668457	−	0.668457i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			13.0758i			0.785647i	0.919614	+	0.392823i	$0.128502\pi$
$277$			13.0758i			0.785647i	−0.919614	+	0.392823i	$0.871498\pi$
$278$		0			0
$279$		−	53.5191i		−	3.20411i
$280$		0			0
$281$		−	6.00639i		−	0.358311i	−0.983821	−	0.179156i	$-0.942663\pi$
$281$		−	6.00639i		−	0.358311i	0.983821	−	0.179156i	$-0.0573366\pi$
$282$		0			0
$283$			0.542533i			0.0322502i	0.999870	+	0.0161251i	$0.00513301\pi$
$283$			0.542533i			0.0322502i	−0.999870	+	0.0161251i	$0.994867\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.52372	+	5.52372i	−0.326055	+	0.326055i
$288$		0			0
$289$		−	5.73372i		−	0.337278i
$290$		0			0
$291$	−12.2016	−	12.2016i	−0.715271	−	0.715271i
$292$		0			0
$293$	−26.6018			−1.55409			−0.777046	−	0.629443i	$-0.783283\pi$
$293$	−26.6018			−1.55409			−0.777046	+	0.629443i	$0.783283\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.94792	+	6.94792i	−0.403159	+	0.403159i
$298$		0			0
$299$	−2.80854	−	2.80854i	−0.162422	−	0.162422i
$300$		0			0
$301$	7.87615	−	7.87615i	0.453973	−	0.453973i
$302$		0			0
$303$	−14.3087	−	14.3087i	−0.822014	−	0.822014i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	6.72156i		−	0.383619i	−0.981432	−	0.191810i	$-0.938564\pi$
$307$		−	6.72156i		−	0.383619i	0.981432	−	0.191810i	$-0.0614357\pi$
$308$		0			0
$309$	20.7356	−	20.7356i	1.17961	−	1.17961i
$310$		0			0
$311$	−0.636143			−0.0360723			−0.0180362	−	0.999837i	$-0.505741\pi$
$311$	−0.636143			−0.0360723			−0.0180362	+	0.999837i	$0.505741\pi$
$312$		0			0
$313$	0.109268	+	0.109268i	0.00617619	+	0.00617619i	0.710188	−	0.704012i	$-0.248610\pi$
$313$	0.109268	+	0.109268i	0.00617619	+	0.00617619i	−0.704012	+	0.710188i	$0.748610\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	34.6913			1.94846			0.974228	−	0.225566i	$-0.0724231\pi$
$317$	34.6913			1.94846			0.974228	+	0.225566i	$0.0724231\pi$
$318$		0			0
$319$	1.82898			0.102403
$320$		0			0
$321$	−35.2388			−1.96684
$322$		0			0
$323$	12.8059			0.712537
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−24.6505	−	24.6505i	−1.36317	−	1.36317i
$328$		0			0
$329$	0.369490			0.0203707
$330$		0			0
$331$	−7.51089	+	7.51089i	−0.412836	+	0.412836i	−0.882725	−	0.469890i	$-0.844294\pi$
$331$	−7.51089	+	7.51089i	−0.412836	+	0.412836i	0.469890	+	0.882725i	$0.344294\pi$
$332$		0			0
$333$		−	59.4630i		−	3.25855i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.00053	+	6.00053i	0.326870	+	0.326870i	0.851395	−	0.524525i	$-0.175757\pi$
$337$	6.00053	+	6.00053i	0.326870	+	0.326870i	−0.524525	+	0.851395i	$0.675757\pi$
$338$		0			0
$339$	−39.7898	+	39.7898i	−2.16109	+	2.16109i
$340$		0			0
$341$	−3.25096	−	3.25096i	−0.176049	−	0.176049i
$342$		0			0
$343$	2.78289	−	2.78289i	0.150262	−	0.150262i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.5337			1.26336			0.631678	−	0.775231i	$-0.282366\pi$
$347$	23.5337			1.26336			0.631678	+	0.775231i	$0.282366\pi$
$348$		0			0
$349$	14.4715	+	14.4715i	0.774643	+	0.774643i	0.978914	−	0.204271i	$-0.0654825\pi$
$349$	14.4715	+	14.4715i	0.774643	+	0.774643i	−0.204271	+	0.978914i	$0.565482\pi$
$350$		0			0
$351$		−	20.0551i		−	1.07046i
$352$		0			0
$353$	−20.4898	+	20.4898i	−1.09056	+	1.09056i	−0.0950957	+	0.995468i	$0.530316\pi$
$353$	−20.4898	+	20.4898i	−1.09056	+	1.09056i	−0.995468	+	0.0950957i	$0.969684\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	39.2795i		−	2.07889i
$358$		0			0
$359$			9.83355i			0.518995i	0.965744	+	0.259498i	$0.0835570\pi$
$359$			9.83355i			0.518995i	−0.965744	+	0.259498i	$0.916443\pi$
$360$		0			0
$361$			4.44419i			0.233905i
$362$		0			0
$363$		−	34.4412i		−	1.80769i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	25.2123	−	25.2123i	1.31607	−	1.31607i	0.399214	−	0.916858i	$-0.369283\pi$
$367$	25.2123	−	25.2123i	1.31607	−	1.31607i	0.916858	−	0.399214i	$-0.130717\pi$
$368$		0			0
$369$			16.5538i			0.861756i
$370$		0			0
$371$	1.82817	+	1.82817i	0.0949139	+	0.0949139i
$372$		0			0
$373$	15.6020			0.807839			0.403920	−	0.914794i	$-0.367648\pi$
$373$	15.6020			0.807839			0.403920	+	0.914794i	$0.367648\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.63966	+	2.63966i	−0.135950	+	0.135950i
$378$		0			0
$379$	−11.1868	−	11.1868i	−0.574627	−	0.574627i	0.358791	−	0.933418i	$-0.383189\pi$
$379$	−11.1868	−	11.1868i	−0.574627	−	0.574627i	−0.933418	+	0.358791i	$0.883189\pi$
$380$		0			0
$381$	−29.2627	+	29.2627i	−1.49917	+	1.49917i
$382$		0			0
$383$	−11.3975	−	11.3975i	−0.582386	−	0.582386i	0.353172	−	0.935558i	$-0.385103\pi$
$383$	−11.3975	−	11.3975i	−0.582386	−	0.582386i	−0.935558	+	0.353172i	$0.885103\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	23.6037i		−	1.19984i
$388$		0			0
$389$	11.8767	−	11.8767i	0.602170	−	0.602170i	−0.338718	−	0.940888i	$-0.609993\pi$
$389$	11.8767	−	11.8767i	0.602170	−	0.602170i	0.940888	+	0.338718i	$0.109993\pi$
$390$		0			0
$391$	−9.98829			−0.505130
$392$		0			0
$393$	−16.3443	−	16.3443i	−0.824461	−	0.824461i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.0768			−0.806870			−0.403435	−	0.915008i	$-0.632184\pi$
$397$	−16.0768			−0.806870			−0.403435	+	0.915008i	$0.632184\pi$
$398$		0			0
$399$	−44.6472			−2.23515
$400$		0			0
$401$	21.8966			1.09346			0.546731	−	0.837308i	$-0.315872\pi$
$401$	21.8966			1.09346			0.546731	+	0.837308i	$0.315872\pi$
$402$		0			0
$403$	9.38388			0.467444
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.61201	−	3.61201i	−0.179041	−	0.179041i
$408$		0			0
$409$	−20.1426			−0.995987			−0.497993	−	0.867181i	$-0.665930\pi$
$409$	−20.1426			−0.995987			−0.497993	+	0.867181i	$0.665930\pi$
$410$		0			0
$411$	−10.4593	+	10.4593i	−0.515921	+	0.515921i
$412$		0			0
$413$			44.0671i			2.16840i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.334114	−	0.334114i	−0.0163616	−	0.0163616i
$418$		0			0
$419$	2.87077	−	2.87077i	0.140246	−	0.140246i	−0.633498	−	0.773744i	$-0.718381\pi$
$419$	2.87077	−	2.87077i	0.140246	−	0.140246i	0.773744	+	0.633498i	$0.218381\pi$
$420$		0			0
$421$	12.9834	+	12.9834i	0.632774	+	0.632774i	0.948763	−	0.315989i	$-0.102336\pi$
$421$	12.9834	+	12.9834i	0.632774	+	0.632774i	−0.315989	+	0.948763i	$0.602336\pi$
$422$		0			0
$423$	0.553655	−	0.553655i	0.0269196	−	0.0269196i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−36.0711			−1.74560
$428$		0			0
$429$	−2.01059	−	2.01059i	−0.0970722	−	0.0970722i
$430$		0			0
$431$		−	11.2042i		−	0.539686i	−0.962904	−	0.269843i	$-0.913028\pi$
$431$		−	11.2042i		−	0.539686i	0.962904	−	0.269843i	$-0.0869719\pi$
$432$		0			0
$433$	20.8634	−	20.8634i	1.00263	−	1.00263i	0.00263391	−	0.999997i	$-0.499162\pi$
$433$	20.8634	−	20.8634i	1.00263	−	1.00263i	0.999997	−	0.00263391i	$-0.000838399\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			11.3532i			0.543099i
$438$		0			0
$439$			26.6495i			1.27191i	0.771726	+	0.635956i	$0.219394\pi$
$439$			26.6495i			1.27191i	−0.771726	+	0.635956i	$0.780606\pi$
$440$		0			0
$441$			44.9467i			2.14032i
$442$		0			0
$443$			5.97706i			0.283979i	0.989868	+	0.141989i	$0.0453499\pi$
$443$			5.97706i			0.283979i	−0.989868	+	0.141989i	$0.954650\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−6.93247	+	6.93247i	−0.327895	+	0.327895i
$448$		0			0
$449$		−	29.5495i		−	1.39453i	−0.716814	−	0.697264i	$-0.754400\pi$
$449$		−	29.5495i		−	1.39453i	0.716814	−	0.697264i	$-0.245600\pi$
$450$		0			0
$451$	1.00554	+	1.00554i	0.0473492	+	0.0473492i
$452$		0			0
$453$	70.6010			3.31712
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.1813	−	19.1813i	0.897262	−	0.897262i	−0.0979311	−	0.995193i	$-0.531222\pi$
$457$	19.1813	−	19.1813i	0.897262	−	0.897262i	0.995193	+	0.0979311i	$0.0312225\pi$
$458$		0			0
$459$	−35.6621	−	35.6621i	−1.66456	−	1.66456i
$460$		0			0
$461$	18.3016	−	18.3016i	0.852390	−	0.852390i	−0.138037	−	0.990427i	$-0.544079\pi$
$461$	18.3016	−	18.3016i	0.852390	−	0.852390i	0.990427	+	0.138037i	$0.0440794\pi$
$462$		0			0
$463$	2.71513	+	2.71513i	0.126183	+	0.126183i	0.767378	−	0.641195i	$-0.221561\pi$
$463$	2.71513	+	2.71513i	0.126183	+	0.126183i	−0.641195	+	0.767378i	$0.721561\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			27.1096i			1.25448i	0.778826	+	0.627241i	$0.215816\pi$
$467$			27.1096i			1.25448i	−0.778826	+	0.627241i	$0.784184\pi$
$468$		0			0
$469$	−27.4770	+	27.4770i	−1.26877	+	1.26877i
$470$		0			0
$471$	56.5332			2.60491
$472$		0			0
$473$	−1.43378	−	1.43378i	−0.0659253	−	0.0659253i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.47877			0.250855
$478$		0			0
$479$	−8.86610			−0.405102			−0.202551	−	0.979272i	$-0.564923\pi$
$479$	−8.86610			−0.405102			−0.202551	+	0.979272i	$0.564923\pi$
$480$		0			0
$481$	10.4261			0.475387
$482$		0			0
$483$	34.8239			1.58454
$484$		0			0
$485$		0			0
$486$		0			0
$487$	8.24646	+	8.24646i	0.373683	+	0.373683i	0.868817	−	0.495134i	$-0.164881\pi$
$487$	8.24646	+	8.24646i	0.373683	+	0.373683i	−0.495134	+	0.868817i	$0.664881\pi$
$488$		0			0
$489$	20.4918			0.926670
$490$		0			0
$491$	−23.4901	+	23.4901i	−1.06010	+	1.06010i	−0.0620205	+	0.998075i	$0.519754\pi$
$491$	−23.4901	+	23.4901i	−1.06010	+	1.06010i	−0.998075	+	0.0620205i	$0.980246\pi$
$492$		0			0
$493$			9.38771i			0.422801i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	38.9841	+	38.9841i	1.74868	+	1.74868i
$498$		0			0
$499$	15.8371	−	15.8371i	0.708965	−	0.708965i	−0.257353	−	0.966317i	$-0.582850\pi$
$499$	15.8371	−	15.8371i	0.708965	−	0.708965i	0.966317	+	0.257353i	$0.0828503\pi$
$500$		0			0
$501$	−21.5493	−	21.5493i	−0.962751	−	0.962751i
$502$		0			0
$503$	−16.4080	+	16.4080i	−0.731597	+	0.731597i	−0.970936	−	0.239339i	$-0.923069\pi$
$503$	−16.4080	+	16.4080i	−0.731597	+	0.731597i	0.239339	+	0.970936i	$0.423069\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−36.5461			−1.62307
$508$		0			0
$509$	−0.698178	−	0.698178i	−0.0309462	−	0.0309462i	0.691464	−	0.722411i	$-0.256966\pi$
$509$	−0.698178	−	0.698178i	−0.0309462	−	0.0309462i	−0.722411	+	0.691464i	$0.756966\pi$
$510$		0			0
$511$		−	4.59789i		−	0.203398i
$512$		0			0
$513$	−40.5354	+	40.5354i	−1.78968	+	1.78968i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	0.0672624i		−	0.00295819i
$518$		0			0
$519$			2.79482i			0.122679i
$520$		0			0
$521$			24.0336i			1.05293i	0.850196	+	0.526466i	$0.176483\pi$
$521$			24.0336i			1.05293i	−0.850196	+	0.526466i	$0.823517\pi$
$522$		0			0
$523$		−	21.7876i		−	0.952705i	−0.879254	−	0.476353i	$-0.841959\pi$
$523$		−	21.7876i		−	0.952705i	0.879254	−	0.476353i	$-0.158041\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.6864	−	16.6864i	0.726873	−	0.726873i
$528$		0			0
$529$			14.1447i			0.614988i
$530$		0			0
$531$	66.0314	+	66.0314i	2.86552	+	2.86552i
$532$		0			0
$533$	−2.90249			−0.125721
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.86225	+	1.86225i	−0.0803620	+	0.0803620i
$538$		0			0
$539$	2.73024	+	2.73024i	0.117600	+	0.117600i
$540$		0			0
$541$	12.1010	−	12.1010i	0.520264	−	0.520264i	−0.397387	−	0.917651i	$-0.630083\pi$
$541$	12.1010	−	12.1010i	0.520264	−	0.520264i	0.917651	+	0.397387i	$0.130083\pi$
$542$		0			0
$543$	33.8404	+	33.8404i	1.45223	+	1.45223i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	27.3827i		−	1.17080i	−0.810745	−	0.585399i	$-0.800938\pi$
$547$		−	27.3827i		−	1.17080i	0.810745	−	0.585399i	$-0.199062\pi$
$548$		0			0
$549$	−54.0500	+	54.0500i	−2.30680	+	2.30680i
$550$		0			0
$551$	10.6706			0.454582
$552$		0			0
$553$	−9.91071	−	9.91071i	−0.421446	−	0.421446i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.8103			−0.797018			−0.398509	−	0.917164i	$-0.630472\pi$
$557$	−18.8103			−0.797018			−0.398509	+	0.917164i	$0.630472\pi$
$558$		0			0
$559$	4.13860			0.175044
$560$		0			0
$561$	−7.15048			−0.301893
$562$		0			0
$563$	−4.43590			−0.186951			−0.0934755	−	0.995622i	$-0.529798\pi$
$563$	−4.43590			−0.186951			−0.0934755	+	0.995622i	$0.529798\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	66.3256	+	66.3256i	2.78541	+	2.78541i
$568$		0			0
$569$	−8.76043			−0.367256			−0.183628	−	0.982996i	$-0.558784\pi$
$569$	−8.76043			−0.367256			−0.183628	+	0.982996i	$0.558784\pi$
$570$		0			0
$571$	4.44131	−	4.44131i	0.185863	−	0.185863i	−0.608042	−	0.793905i	$-0.708045\pi$
$571$	4.44131	−	4.44131i	0.185863	−	0.185863i	0.793905	+	0.608042i	$0.208045\pi$
$572$		0			0
$573$		−	73.3930i		−	3.06604i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.25383	−	9.25383i	−0.385242	−	0.385242i	0.487745	−	0.872986i	$-0.337820\pi$
$577$	−9.25383	−	9.25383i	−0.385242	−	0.385242i	−0.872986	+	0.487745i	$0.837820\pi$
$578$		0			0
$579$	−25.9291	+	25.9291i	−1.07758	+	1.07758i
$580$		0			0
$581$	15.2968	+	15.2968i	0.634616	+	0.634616i
$582$		0			0
$583$	0.332802	−	0.332802i	0.0137832	−	0.0137832i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.9676			−0.906700			−0.453350	−	0.891333i	$-0.649771\pi$
$587$	−21.9676			−0.906700			−0.453350	+	0.891333i	$0.649771\pi$
$588$		0			0
$589$	−18.9667	−	18.9667i	−0.781509	−	0.781509i
$590$		0			0
$591$		−	57.1871i		−	2.35236i
$592$		0			0
$593$	−9.10461	+	9.10461i	−0.373881	+	0.373881i	−0.868889	−	0.495007i	$-0.835165\pi$
$593$	−9.10461	+	9.10461i	−0.373881	+	0.373881i	0.495007	+	0.868889i	$0.335165\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	61.2149i		−	2.50536i
$598$		0			0
$599$		−	28.3551i		−	1.15856i	−0.815130	−	0.579279i	$-0.803334\pi$
$599$		−	28.3551i		−	1.15856i	0.815130	−	0.579279i	$-0.196666\pi$
$600$		0			0
$601$			40.3072i			1.64417i	0.569368	+	0.822083i	$0.307188\pi$
$601$			40.3072i			1.64417i	−0.569368	+	0.822083i	$0.692812\pi$
$602$		0			0
$603$			82.3446i			3.35333i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−33.3427	+	33.3427i	−1.35334	+	1.35334i	−0.471442	+	0.881897i	$0.656266\pi$
$607$	−33.3427	+	33.3427i	−1.35334	+	1.35334i	−0.881897	+	0.471442i	$0.843734\pi$
$608$		0			0
$609$		−	32.7300i		−	1.32628i
$610$		0			0
$611$	0.0970762	+	0.0970762i	0.00392728	+	0.00392728i
$612$		0			0
$613$	22.1895			0.896227			0.448113	−	0.893977i	$-0.352096\pi$
$613$	22.1895			0.896227			0.448113	+	0.893977i	$0.352096\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.7719	+	15.7719i	−0.634954	+	0.634954i	−0.949306	−	0.314353i	$-0.898213\pi$
$617$	−15.7719	+	15.7719i	−0.634954	+	0.634954i	0.314353	+	0.949306i	$0.398213\pi$
$618$		0			0
$619$	5.07600	+	5.07600i	0.204022	+	0.204022i	0.801721	−	0.597699i	$-0.203918\pi$
$619$	5.07600	+	5.07600i	0.204022	+	0.204022i	−0.597699	+	0.801721i	$0.703918\pi$
$620$		0			0
$621$	31.6168	−	31.6168i	1.26874	−	1.26874i
$622$		0			0
$623$	−18.9662	−	18.9662i	−0.759863	−	0.759863i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			8.12761i			0.324586i
$628$		0			0
$629$	18.5396	−	18.5396i	0.739224	−	0.739224i
$630$		0			0
$631$	−31.2617			−1.24451			−0.622254	−	0.782816i	$-0.713783\pi$
$631$	−31.2617			−1.24451			−0.622254	+	0.782816i	$0.713783\pi$
$632$		0			0
$633$	29.3383	+	29.3383i	1.16609	+	1.16609i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−7.88083			−0.312250
$638$		0			0
$639$	116.830			4.62172
$640$		0			0
$641$	−4.38681			−0.173269			−0.0866343	−	0.996240i	$-0.527611\pi$
$641$	−4.38681			−0.173269			−0.0866343	+	0.996240i	$0.527611\pi$
$642$		0			0
$643$	28.1726			1.11102			0.555509	−	0.831511i	$-0.312524\pi$
$643$	28.1726			1.11102			0.555509	+	0.831511i	$0.312524\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.3260	−	13.3260i	−0.523900	−	0.523900i	0.394847	−	0.918747i	$-0.370798\pi$
$647$	−13.3260	−	13.3260i	−0.523900	−	0.523900i	−0.918747	+	0.394847i	$0.870798\pi$
$648$		0			0
$649$	8.02201			0.314892
$650$		0			0
$651$	−58.1767	+	58.1767i	−2.28013	+	2.28013i
$652$		0			0
$653$			3.61001i			0.141271i	0.997502	+	0.0706354i	$0.0225027\pi$
$653$			3.61001i			0.141271i	−0.997502	+	0.0706354i	$0.977497\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.88960	−	6.88960i	−0.268789	−	0.268789i
$658$		0			0
$659$	10.6066	−	10.6066i	0.413173	−	0.413173i	−0.469669	−	0.882842i	$-0.655627\pi$
$659$	10.6066	−	10.6066i	0.413173	−	0.413173i	0.882842	+	0.469669i	$0.155627\pi$
$660$		0			0
$661$	−14.7397	−	14.7397i	−0.573307	−	0.573307i	0.359744	−	0.933051i	$-0.382864\pi$
$661$	−14.7397	−	14.7397i	−0.573307	−	0.573307i	−0.933051	+	0.359744i	$0.882864\pi$
$662$		0			0
$663$	10.3199	−	10.3199i	0.400792	−	0.400792i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.32282			−0.322261
$668$		0			0
$669$	29.8607	+	29.8607i	1.15448	+	1.15448i
$670$		0			0
$671$			6.56642i			0.253494i
$672$		0			0
$673$	1.21655	−	1.21655i	0.0468946	−	0.0468946i	−0.683271	−	0.730165i	$-0.739443\pi$
$673$	1.21655	−	1.21655i	0.0468946	−	0.0468946i	0.730165	+	0.683271i	$0.239443\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	29.4713i		−	1.13267i	−0.824174	−	0.566337i	$-0.808360\pi$
$677$		−	29.4713i		−	1.13267i	0.824174	−	0.566337i	$-0.191640\pi$
$678$		0			0
$679$			19.0281i			0.730231i
$680$		0			0
$681$		−	69.2169i		−	2.65240i
$682$		0			0
$683$		−	45.5003i		−	1.74102i	−0.492151	−	0.870510i	$-0.663789\pi$
$683$		−	45.5003i		−	1.74102i	0.492151	−	0.870510i	$-0.336211\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	45.7707	−	45.7707i	1.74626	−	1.74626i
$688$		0			0
$689$			0.960630i			0.0365971i
$690$		0			0
$691$	−21.4526	−	21.4526i	−0.816096	−	0.816096i	0.169444	−	0.985540i	$-0.445803\pi$
$691$	−21.4526	−	21.4526i	−0.816096	−	0.816096i	−0.985540	+	0.169444i	$0.945803\pi$
$692$		0			0
$693$	17.8825			0.679300
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5.16122	+	5.16122i	−0.195495	+	0.195495i
$698$		0			0
$699$	64.3618	+	64.3618i	2.43439	+	2.43439i
$700$		0			0
$701$	−17.4263	+	17.4263i	−0.658183	+	0.658183i	−0.954950	−	0.296767i	$-0.904091\pi$
$701$	−17.4263	+	17.4263i	−0.658183	+	0.658183i	0.296767	+	0.954950i	$0.404091\pi$
$702$		0			0
$703$	−21.0731	−	21.0731i	−0.794789	−	0.794789i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			22.3141i			0.839207i
$708$		0			0
$709$	14.0431	−	14.0431i	0.527401	−	0.527401i	−0.392396	−	0.919796i	$-0.628354\pi$
$709$	14.0431	−	14.0431i	0.527401	−	0.527401i	0.919796	+	0.392396i	$0.128354\pi$
$710$		0			0
$711$	−29.7010			−1.11387
$712$		0			0
$713$	14.7936	+	14.7936i	0.554025	+	0.554025i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−4.93960			−0.184473
$718$		0			0
$719$	−36.8139			−1.37293			−0.686463	−	0.727165i	$-0.740838\pi$
$719$	−36.8139			−1.37293			−0.686463	+	0.727165i	$0.740838\pi$
$720$		0			0
$721$	−32.3366			−1.20428
$722$		0			0
$723$	−19.2729			−0.716766
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−21.9475	−	21.9475i	−0.813989	−	0.813989i	0.171240	−	0.985229i	$-0.445222\pi$
$727$	−21.9475	−	21.9475i	−0.813989	−	0.813989i	−0.985229	+	0.171240i	$0.945222\pi$
$728$		0			0
$729$	51.9233			1.92308
$730$		0			0
$731$	7.35927	−	7.35927i	0.272192	−	0.272192i
$732$		0			0
$733$			34.2145i			1.26374i	0.775074	+	0.631870i	$0.217712\pi$
$733$			34.2145i			1.26374i	−0.775074	+	0.631870i	$0.782288\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.00193	+	5.00193i	0.184249	+	0.184249i
$738$		0			0
$739$	5.64086	−	5.64086i	0.207502	−	0.207502i	−0.595703	−	0.803205i	$-0.703126\pi$
$739$	5.64086	−	5.64086i	0.207502	−	0.207502i	0.803205	+	0.595703i	$0.203126\pi$
$740$		0			0
$741$	−11.7302	−	11.7302i	−0.430918	−	0.430918i
$742$		0			0
$743$	−0.765980	+	0.765980i	−0.0281011	+	0.0281011i	−0.721018	−	0.692917i	$-0.756325\pi$
$743$	−0.765980	+	0.765980i	−0.0281011	+	0.0281011i	0.692917	+	0.721018i	$0.256325\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	45.8422			1.67728
$748$		0			0
$749$	27.4770	+	27.4770i	1.00399	+	1.00399i
$750$		0			0
$751$		−	9.66983i		−	0.352857i	−0.984313	−	0.176429i	$-0.943546\pi$
$751$		−	9.66983i		−	0.352857i	0.984313	−	0.176429i	$-0.0564544\pi$
$752$		0			0
$753$	−38.6761	+	38.6761i	−1.40944	+	1.40944i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			4.33702i			0.157632i	0.996889	+	0.0788158i	$0.0251139\pi$
$757$			4.33702i			0.157632i	−0.996889	+	0.0788158i	$0.974886\pi$
$758$		0			0
$759$		−	6.33937i		−	0.230104i
$760$		0			0
$761$		−	49.2936i		−	1.78689i	−0.449170	−	0.893446i	$-0.648280\pi$
$761$		−	49.2936i		−	1.78689i	0.449170	−	0.893446i	$-0.351720\pi$
$762$		0			0
$763$			38.4417i			1.39168i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.5777	+	11.5777i	−0.418048	+	0.418048i
$768$		0			0
$769$		−	30.9736i		−	1.11694i	−0.829526	−	0.558468i	$-0.811389\pi$
$769$		−	30.9736i		−	1.11694i	0.829526	−	0.558468i	$-0.188611\pi$
$770$		0			0
$771$	−27.3465	−	27.3465i	−0.984861	−	0.984861i
$772$		0			0
$773$	−15.1058			−0.543319			−0.271660	−	0.962393i	$-0.587573\pi$
$773$	−15.1058			−0.543319			−0.271660	+	0.962393i	$0.587573\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−64.6378	+	64.6378i	−2.31887	+	2.31887i
$778$		0			0
$779$	5.86652	+	5.86652i	0.210190	+	0.210190i
$780$		0			0
$781$	7.09671	−	7.09671i	0.253940	−	0.253940i
$782$		0			0
$783$	−29.7157	−	29.7157i	−1.06195	−	1.06195i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	30.2095i		−	1.07685i	−0.842673	−	0.538426i	$-0.819019\pi$
$787$		−	30.2095i		−	1.07685i	0.842673	−	0.538426i	$-0.180981\pi$
$788$		0			0
$789$	35.3857	−	35.3857i	1.25977	−	1.25977i
$790$		0			0
$791$	62.0512			2.20629
$792$		0			0
$793$	−9.47696	−	9.47696i	−0.336537	−	0.336537i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.39226			−0.332691			−0.166346	−	0.986068i	$-0.553197\pi$
$797$	−9.39226			−0.332691			−0.166346	+	0.986068i	$0.553197\pi$
$798$		0			0
$799$	0.345242			0.0122138
$800$		0			0
$801$	−56.8389			−2.00830
$802$		0			0
$803$	−0.837003			−0.0295372
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−42.1455	−	42.1455i	−1.48359	−	1.48359i
$808$		0			0
$809$	−49.0307			−1.72383			−0.861914	−	0.507054i	$-0.830734\pi$
$809$	−49.0307			−1.72383			−0.861914	+	0.507054i	$0.830734\pi$
$810$		0			0
$811$	20.4561	−	20.4561i	0.718311	−	0.718311i	−0.249948	−	0.968259i	$-0.580414\pi$
$811$	20.4561	−	20.4561i	0.718311	−	0.718311i	0.968259	+	0.249948i	$0.0804137\pi$
$812$		0			0
$813$		−	4.69160i		−	0.164541i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.36494	−	8.36494i	−0.292652	−	0.292652i
$818$		0			0
$819$	−25.8089	+	25.8089i	−0.901834	+	0.901834i
$820$		0			0
$821$	34.8437	+	34.8437i	1.21605	+	1.21605i	0.969003	+	0.247050i	$0.0794610\pi$
$821$	34.8437	+	34.8437i	1.21605	+	1.21605i	0.247050	+	0.969003i	$0.420539\pi$
$822$		0			0
$823$	−8.05406	+	8.05406i	−0.280747	+	0.280747i	−0.833407	−	0.552660i	$-0.813613\pi$
$823$	−8.05406	+	8.05406i	−0.280747	+	0.280747i	0.552660	+	0.833407i	$0.313613\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.7148			1.38102			0.690510	−	0.723323i	$-0.257386\pi$
$827$	39.7148			1.38102			0.690510	+	0.723323i	$0.257386\pi$
$828$		0			0
$829$	−20.7529	−	20.7529i	−0.720778	−	0.720778i	0.247985	−	0.968764i	$-0.420231\pi$
$829$	−20.7529	−	20.7529i	−0.720778	−	0.720778i	−0.968764	+	0.247985i	$0.920231\pi$
$830$		0			0
$831$		−	42.5965i		−	1.47765i
$832$		0			0
$833$	−14.0137	+	14.0137i	−0.485546	+	0.485546i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			105.638i			3.65138i
$838$		0			0
$839$		−	44.3056i		−	1.52960i	−0.644267	−	0.764800i	$-0.722838\pi$
$839$		−	44.3056i		−	1.52960i	0.644267	−	0.764800i	$-0.277162\pi$
$840$		0			0
$841$		−	21.1776i		−	0.730262i
$842$		0			0
$843$			19.5668i			0.673917i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−26.8551	+	26.8551i	−0.922751	+	0.922751i
$848$		0			0
$849$		−	1.76739i		−	0.0606567i
$850$		0			0
$851$	16.4366	+	16.4366i	0.563440	+	0.563440i
$852$		0			0
$853$	−29.5903			−1.01315			−0.506577	−	0.862195i	$-0.669089\pi$
$853$	−29.5903			−1.01315			−0.506577	+	0.862195i	$0.669089\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.9315	+	22.9315i	−0.783326	+	0.783326i	−0.980391	−	0.197064i	$-0.936859\pi$
$857$	−22.9315	+	22.9315i	−0.783326	+	0.783326i	0.197064	+	0.980391i	$0.436859\pi$
$858$		0			0
$859$	−23.1388	−	23.1388i	−0.789486	−	0.789486i	0.191924	−	0.981410i	$-0.438527\pi$
$859$	−23.1388	−	23.1388i	−0.789486	−	0.789486i	−0.981410	+	0.191924i	$0.938527\pi$
$860$		0			0
$861$	17.9944	−	17.9944i	0.613248	−	0.613248i
$862$		0			0
$863$	10.8789	+	10.8789i	0.370322	+	0.370322i	0.867594	−	0.497273i	$-0.165665\pi$
$863$	10.8789	+	10.8789i	0.370322	+	0.370322i	−0.497273	+	0.867594i	$0.665665\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			18.6786i			0.634357i
$868$		0			0
$869$	−1.80415	+	1.80415i	−0.0612018	+	0.0612018i
$870$		0			0
$871$	−14.4381			−0.489215
$872$		0			0
$873$	28.5122	+	28.5122i	0.964993	+	0.964993i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.3146			−0.719742			−0.359871	−	0.933002i	$-0.617179\pi$
$877$	−21.3146			−0.719742			−0.359871	+	0.933002i	$0.617179\pi$
$878$		0			0
$879$	86.6597			2.92296
$880$		0			0
$881$	43.6540			1.47074			0.735370	−	0.677666i	$-0.237009\pi$
$881$	43.6540			1.47074			0.735370	+	0.677666i	$0.237009\pi$
$882$		0			0
$883$	−12.9408			−0.435492			−0.217746	−	0.976006i	$-0.569870\pi$
$883$	−12.9408			−0.435492			−0.217746	+	0.976006i	$0.569870\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.9454	+	28.9454i	0.971892	+	0.971892i	0.999616	−	0.0277234i	$-0.00882577\pi$
$887$	28.9454	+	28.9454i	0.971892	+	0.971892i	−0.0277234	+	0.999616i	$0.508826\pi$
$888$		0			0
$889$	45.6344			1.53053
$890$		0			0
$891$	12.0740	−	12.0740i	0.404493	−	0.404493i
$892$		0			0
$893$		−	0.392421i		−	0.0131319i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	9.14927	+	9.14927i	0.305485	+	0.305485i
$898$		0			0
$899$	13.9041	−	13.9041i	0.463728	−	0.463728i
$900$		0			0
$901$	1.70820	+	1.70820i	0.0569082	+	0.0569082i
$902$		0			0
$903$	−25.6578	+	25.6578i	−0.853840	+	0.853840i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.83334			0.226897			0.113449	−	0.993544i	$-0.463810\pi$
$907$	6.83334			0.226897			0.113449	+	0.993544i	$0.463810\pi$
$908$		0			0
$909$	33.4360	+	33.4360i	1.10900	+	1.10900i
$910$		0			0
$911$		−	25.5381i		−	0.846114i	−0.906103	−	0.423057i	$-0.860957\pi$
$911$		−	25.5381i		−	0.846114i	0.906103	−	0.423057i	$-0.139043\pi$
$912$		0			0
$913$	2.78463	−	2.78463i	0.0921579	−	0.0921579i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			25.4885i			0.841704i
$918$		0			0
$919$			2.49036i			0.0821493i	0.999156	+	0.0410746i	$0.0130781\pi$
$919$			2.49036i			0.0821493i	−0.999156	+	0.0410746i	$0.986922\pi$
$920$		0			0
$921$			21.8966i			0.721517i
$922$		0			0
$923$			20.4846i			0.674259i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−48.4541	+	48.4541i	−1.59144	+	1.59144i
$928$		0			0
$929$			8.45665i			0.277454i	0.990331	+	0.138727i	$0.0443010\pi$
$929$			8.45665i			0.277454i	−0.990331	+	0.138727i	$0.955699\pi$
$930$		0			0
$931$	15.9287	+	15.9287i	0.522043	+	0.522043i
$932$		0			0
$933$	2.07234			0.0678453
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40.1339	−	40.1339i	1.31112	−	1.31112i	0.390523	−	0.920593i	$-0.372294\pi$
$937$	40.1339	−	40.1339i	1.31112	−	1.31112i	0.920593	−	0.390523i	$-0.127706\pi$
$938$		0			0
$939$	−0.355959	−	0.355959i	−0.0116163	−	0.0116163i
$940$		0			0
$941$	35.3857	−	35.3857i	1.15354	−	1.15354i	0.167704	−	0.985837i	$-0.446365\pi$
$941$	35.3857	−	35.3857i	1.15354	−	1.15354i	0.985837	−	0.167704i	$-0.0536354\pi$
$942$		0			0
$943$	−4.57576	−	4.57576i	−0.149007	−	0.149007i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			10.4999i			0.341200i	0.985340	+	0.170600i	$0.0545706\pi$
$947$			10.4999i			0.341200i	−0.985340	+	0.170600i	$0.945429\pi$
$948$		0			0
$949$	1.20800	−	1.20800i	0.0392134	−	0.0392134i
$950$		0			0
$951$	−113.013			−3.66468
$952$		0			0
$953$	−4.72367	−	4.72367i	−0.153015	−	0.153015i	0.626448	−	0.779463i	$-0.284508\pi$
$953$	−4.72367	−	4.72367i	−0.153015	−	0.153015i	−0.779463	+	0.626448i	$0.784508\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−5.95819			−0.192601
$958$		0			0
$959$	16.3111			0.526711
$960$		0			0
$961$	−18.4284			−0.594466
$962$		0			0
$963$	82.3446			2.65352
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.08577	+	4.08577i	0.131390	+	0.131390i	0.769743	−	0.638354i	$-0.220384\pi$
$967$	4.08577	+	4.08577i	0.131390	+	0.131390i	−0.638354	+	0.769743i	$0.720384\pi$
$968$		0			0
$969$	−41.7172			−1.34015
$970$		0			0
$971$	13.5569	−	13.5569i	0.435062	−	0.435062i	−0.455284	−	0.890346i	$-0.650462\pi$
$971$	13.5569	−	13.5569i	0.435062	−	0.435062i	0.890346	+	0.455284i	$0.150462\pi$
$972$		0			0
$973$			0.521042i			0.0167038i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.7102	−	24.7102i	−0.790550	−	0.790550i	0.191034	−	0.981584i	$-0.438816\pi$
$977$	−24.7102	−	24.7102i	−0.790550	−	0.790550i	−0.981584	+	0.191034i	$0.938816\pi$
$978$		0			0
$979$	−3.45262	+	3.45262i	−0.110346	+	0.110346i
$980$		0			0
$981$	57.6022	+	57.6022i	1.83910	+	1.83910i
$982$		0			0
$983$	15.4292	−	15.4292i	0.492115	−	0.492115i	−0.416857	−	0.908972i	$-0.636868\pi$
$983$	15.4292	−	15.4292i	0.492115	−	0.492115i	0.908972	+	0.416857i	$0.136868\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.20368			−0.0383134
$988$		0			0
$989$	6.52447	+	6.52447i	0.207466	+	0.207466i
$990$		0			0
$991$			29.6537i			0.941981i	0.882138	+	0.470990i	$0.156103\pi$
$991$			29.6537i			0.941981i	−0.882138	+	0.470990i	$0.843897\pi$
$992$		0			0
$993$	24.4680	−	24.4680i	0.776467	−	0.776467i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			36.4336i			1.15386i	0.816792	+	0.576932i	$0.195750\pi$
$997$			36.4336i			1.15386i	−0.816792	+	0.576932i	$0.804250\pi$
$998$		0			0
$999$			117.370i			3.71343i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.s.e.207.1		24
4.3	odd	2		400.2.s.e.107.8	yes	24
5.2	odd	4		1600.2.j.e.143.12		24
5.3	odd	4		1600.2.j.e.143.1		24
5.4	even	2	inner	1600.2.s.e.207.12		24
16.3	odd	4		1600.2.j.e.1007.12		24
16.13	even	4		400.2.j.e.307.2	yes	24
20.3	even	4		400.2.j.e.43.2	✓	24
20.7	even	4		400.2.j.e.43.11	yes	24
20.19	odd	2		400.2.s.e.107.5	yes	24
80.3	even	4	inner	1600.2.s.e.943.1		24
80.13	odd	4		400.2.s.e.243.8	yes	24
80.19	odd	4		1600.2.j.e.1007.1		24
80.29	even	4		400.2.j.e.307.11	yes	24
80.67	even	4	inner	1600.2.s.e.943.12		24
80.77	odd	4		400.2.s.e.243.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.2	✓	24	20.3	even	4
400.2.j.e.43.11	yes	24	20.7	even	4
400.2.j.e.307.2	yes	24	16.13	even	4
400.2.j.e.307.11	yes	24	80.29	even	4
400.2.s.e.107.5	yes	24	20.19	odd	2
400.2.s.e.107.8	yes	24	4.3	odd	2
400.2.s.e.243.5	yes	24	80.77	odd	4
400.2.s.e.243.8	yes	24	80.13	odd	4
1600.2.j.e.143.1		24	5.3	odd	4
1600.2.j.e.143.12		24	5.2	odd	4
1600.2.j.e.1007.1		24	80.19	odd	4
1600.2.j.e.1007.12		24	16.3	odd	4
1600.2.s.e.207.1		24	1.1	even	1	trivial
1600.2.s.e.207.12		24	5.4	even	2	inner
1600.2.s.e.943.1		24	80.3	even	4	inner
1600.2.s.e.943.12		24	80.67	even	4	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.s (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.25766 q^{3} +(2.54012 + 2.54012i) q^{7} +7.61238 q^{9} +(0.462406 - 0.462406i) q^{11} +1.33473i q^{13} +(2.37342 + 2.37342i) q^{17} +(2.69776 - 2.69776i) q^{19} +(-8.27486 - 8.27486i) q^{21} +(-2.10420 + 2.10420i) q^{23} -15.0256 q^{27} +(1.97767 + 1.97767i) q^{29} -7.03054i q^{31} +(-1.50636 + 1.50636i) q^{33} -7.81135i q^{37} -4.34811i q^{39} +2.17459i q^{41} -3.10070i q^{43} +(0.0727309 - 0.0727309i) q^{47} +5.90443i q^{49} +(-7.73182 - 7.73182i) q^{51} +0.719718 q^{53} +(-8.78840 + 8.78840i) q^{57} +(8.67421 + 8.67421i) q^{59} +(-7.10027 + 7.10027i) q^{61} +(19.3364 + 19.3364i) q^{63} +10.8172i q^{67} +(6.85476 - 6.85476i) q^{69} +15.3474 q^{71} +(-0.905052 - 0.905052i) q^{73} +2.34913 q^{77} -3.90167 q^{79} +26.1112 q^{81} +6.02206 q^{83} +(-6.44260 - 6.44260i) q^{87} -7.46664 q^{89} +(-3.39038 + 3.39038i) q^{91} +22.9031i q^{93} +(3.74551 + 3.74551i) q^{97} +(3.52001 - 3.52001i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \) 24 * q + 40 * q^9 + 20 * q^11 - 12 * q^19 - 8 * q^29 - 20 * q^51 + 8 * q^59 - 48 * q^61 + 64 * q^69 + 16 * q^71 - 104 * q^79 + 48 * q^81 - 96 * q^89 - 64 * q^91 + 128 * q^99

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