LMFDB - Embedded newform 1600.2.s.e.943.5

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			943.5
Character	$\chi$	$=$	1600.943
Dual form			1600.2.s.e.207.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.790153 q^{3} +(-0.139907 + 0.139907i) q^{7} -2.37566 q^{9} +(-2.94816 - 2.94816i) q^{11} -0.235568i q^{13} +(2.06145 - 2.06145i) q^{17} +(2.55293 + 2.55293i) q^{19} +(0.110548 - 0.110548i) q^{21} +(4.62421 + 4.62421i) q^{23} +4.24759 q^{27} +(-6.66417 + 6.66417i) q^{29} -3.43202i q^{31} +(2.32950 + 2.32950i) q^{33} -1.38457i q^{37} +0.186135i q^{39} +8.26242i q^{41} +5.40057i q^{43} +(6.84602 + 6.84602i) q^{47} +6.96085i q^{49} +(-1.62886 + 1.62886i) q^{51} +8.19252 q^{53} +(-2.01720 - 2.01720i) q^{57} +(-4.32313 + 4.32313i) q^{59} +(-9.15188 - 9.15188i) q^{61} +(0.332372 - 0.332372i) q^{63} -5.00083i q^{67} +(-3.65383 - 3.65383i) q^{69} +6.06473 q^{71} +(-11.3646 + 11.3646i) q^{73} +0.824938 q^{77} -4.44776 q^{79} +3.77073 q^{81} -11.4778 q^{83} +(5.26571 - 5.26571i) q^{87} +5.84762 q^{89} +(0.0329576 + 0.0329576i) q^{91} +2.71182i q^{93} +(-0.515382 + 0.515382i) q^{97} +(7.00383 + 7.00383i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	−0.790153			−0.456195			−0.228097	−	0.973638i	$-0.573250\pi$
$3$	−0.790153			−0.456195			−0.228097	+	0.973638i	$0.573250\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.139907	+	0.139907i	−0.0528800	+	0.0528800i	−0.733052	−	0.680172i	$-0.761905\pi$
$7$	−0.139907	+	0.139907i	−0.0528800	+	0.0528800i	0.680172	+	0.733052i	$0.261905\pi$
$8$		0			0
$9$	−2.37566			−0.791886
$10$		0			0
$11$	−2.94816	−	2.94816i	−0.888904	−	0.888904i	0.105514	−	0.994418i	$-0.466351\pi$
$11$	−2.94816	−	2.94816i	−0.888904	−	0.888904i	−0.994418	+	0.105514i	$0.966351\pi$
$12$		0			0
$13$		−	0.235568i		−	0.0653348i	−0.999466	−	0.0326674i	$-0.989600\pi$
$13$		−	0.235568i		−	0.0653348i	0.999466	−	0.0326674i	$-0.0104002\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.06145	−	2.06145i	0.499975	−	0.499975i	−0.411455	−	0.911430i	$-0.634979\pi$
$17$	2.06145	−	2.06145i	0.499975	−	0.499975i	0.911430	+	0.411455i	$0.134979\pi$
$18$		0			0
$19$	2.55293	+	2.55293i	0.585682	+	0.585682i	0.936459	−	0.350777i	$-0.114083\pi$
$19$	2.55293	+	2.55293i	0.585682	+	0.585682i	−0.350777	+	0.936459i	$0.614083\pi$
$20$		0			0
$21$	0.110548	−	0.110548i	0.0241236	−	0.0241236i
$22$		0			0
$23$	4.62421	+	4.62421i	0.964214	+	0.964214i	0.999381	−	0.0351672i	$-0.0111964\pi$
$23$	4.62421	+	4.62421i	0.964214	+	0.964214i	−0.0351672	+	0.999381i	$0.511196\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.24759			0.817449
$28$		0			0
$29$	−6.66417	+	6.66417i	−1.23751	+	1.23751i	−0.276488	+	0.961017i	$0.589171\pi$
$29$	−6.66417	+	6.66417i	−1.23751	+	1.23751i	−0.961017	+	0.276488i	$0.910829\pi$
$30$		0			0
$31$		−	3.43202i		−	0.616408i	−0.951320	−	0.308204i	$-0.900272\pi$
$31$		−	3.43202i		−	0.616408i	0.951320	−	0.308204i	$-0.0997280\pi$
$32$		0			0
$33$	2.32950	+	2.32950i	0.405513	+	0.405513i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	1.38457i		−	0.227621i	−0.993502	−	0.113811i	$-0.963694\pi$
$37$		−	1.38457i		−	0.227621i	0.993502	−	0.113811i	$-0.0363057\pi$
$38$		0			0
$39$			0.186135i			0.0298054i
$40$		0			0
$41$			8.26242i			1.29037i	0.764025	+	0.645187i	$0.223220\pi$
$41$			8.26242i			1.29037i	−0.764025	+	0.645187i	$0.776780\pi$
$42$		0			0
$43$			5.40057i			0.823580i	0.911279	+	0.411790i	$0.135096\pi$
$43$			5.40057i			0.823580i	−0.911279	+	0.411790i	$0.864904\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.84602	+	6.84602i	0.998594	+	0.998594i	0.999999	−	0.00140497i	$-0.000447216\pi$
$47$	6.84602	+	6.84602i	0.998594	+	0.998594i	−0.00140497	+	0.999999i	$0.500447\pi$
$48$		0			0
$49$			6.96085i			0.994407i
$50$		0			0
$51$	−1.62886	+	1.62886i	−0.228086	+	0.228086i
$52$		0			0
$53$	8.19252			1.12533			0.562665	−	0.826685i	$-0.309776\pi$
$53$	8.19252			1.12533			0.562665	+	0.826685i	$0.309776\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.01720	−	2.01720i	−0.267185	−	0.267185i
$58$		0			0
$59$	−4.32313	+	4.32313i	−0.562823	+	0.562823i	−0.930108	−	0.367285i	$-0.880287\pi$
$59$	−4.32313	+	4.32313i	−0.562823	+	0.562823i	0.367285	+	0.930108i	$0.380287\pi$
$60$		0			0
$61$	−9.15188	−	9.15188i	−1.17178	−	1.17178i	−0.981786	−	0.189992i	$-0.939154\pi$
$61$	−9.15188	−	9.15188i	−1.17178	−	1.17178i	−0.189992	−	0.981786i	$-0.560846\pi$
$62$		0			0
$63$	0.332372	−	0.332372i	0.0418749	−	0.0418749i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	5.00083i		−	0.610948i	−0.952200	−	0.305474i	$-0.901185\pi$
$67$		−	5.00083i		−	0.610948i	0.952200	−	0.305474i	$-0.0988150\pi$
$68$		0			0
$69$	−3.65383	−	3.65383i	−0.439870	−	0.439870i
$70$		0			0
$71$	6.06473			0.719751			0.359875	−	0.933000i	$-0.382819\pi$
$71$	6.06473			0.719751			0.359875	+	0.933000i	$0.382819\pi$
$72$		0			0
$73$	−11.3646	+	11.3646i	−1.33012	+	1.33012i	−0.424863	+	0.905258i	$0.639678\pi$
$73$	−11.3646	+	11.3646i	−1.33012	+	1.33012i	−0.905258	+	0.424863i	$0.860322\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.824938			0.0940104
$78$		0			0
$79$	−4.44776			−0.500413			−0.250206	−	0.968193i	$-0.580498\pi$
$79$	−4.44776			−0.500413			−0.250206	+	0.968193i	$0.580498\pi$
$80$		0			0
$81$	3.77073			0.418970
$82$		0			0
$83$	−11.4778			−1.25986			−0.629928	−	0.776654i	$-0.716915\pi$
$83$	−11.4778			−1.25986			−0.629928	+	0.776654i	$0.716915\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5.26571	−	5.26571i	0.564543	−	0.564543i
$88$		0			0
$89$	5.84762			0.619846			0.309923	−	0.950762i	$-0.399697\pi$
$89$	5.84762			0.619846			0.309923	+	0.950762i	$0.399697\pi$
$90$		0			0
$91$	0.0329576	+	0.0329576i	0.00345490	+	0.00345490i
$92$		0			0
$93$			2.71182i			0.281202i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.515382	+	0.515382i	−0.0523291	+	0.0523291i	−0.732787	−	0.680458i	$-0.761781\pi$
$97$	−0.515382	+	0.515382i	−0.0523291	+	0.0523291i	0.680458	+	0.732787i	$0.261781\pi$
$98$		0			0
$99$	7.00383	+	7.00383i	0.703911	+	0.703911i
$100$		0			0
$101$	3.56668	−	3.56668i	0.354898	−	0.354898i	−0.507030	−	0.861928i	$-0.669257\pi$
$101$	3.56668	−	3.56668i	0.354898	−	0.354898i	0.861928	+	0.507030i	$0.169257\pi$
$102$		0			0
$103$	11.6666	+	11.6666i	1.14954	+	1.14954i	0.986642	+	0.162901i	$0.0520851\pi$
$103$	11.6666	+	11.6666i	1.14954	+	1.14954i	0.162901	+	0.986642i	$0.447915\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.00083			0.483448			0.241724	−	0.970345i	$-0.422287\pi$
$107$	5.00083			0.483448			0.241724	+	0.970345i	$0.422287\pi$
$108$		0			0
$109$	−3.69574	+	3.69574i	−0.353988	+	0.353988i	−0.861591	−	0.507603i	$-0.830532\pi$
$109$	−3.69574	+	3.69574i	−0.353988	+	0.353988i	0.507603	+	0.861591i	$0.330532\pi$
$110$		0			0
$111$			1.09402i			0.103840i
$112$		0			0
$113$	11.6416	+	11.6416i	1.09515	+	1.09515i	0.994969	+	0.100184i	$0.0319430\pi$
$113$	11.6416	+	11.6416i	1.09515	+	1.09515i	0.100184	+	0.994969i	$0.468057\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			0.559629i			0.0517377i
$118$		0			0
$119$			0.576823i			0.0528773i
$120$		0			0
$121$			6.38331i			0.580301i
$122$		0			0
$123$		−	6.52858i		−	0.588662i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.40562	−	5.40562i	−0.479671	−	0.479671i	0.425355	−	0.905026i	$-0.360149\pi$
$127$	−5.40562	−	5.40562i	−0.479671	−	0.479671i	−0.905026	+	0.425355i	$0.860149\pi$
$128$		0			0
$129$		−	4.26728i		−	0.375713i
$130$		0			0
$131$	−15.2758	+	15.2758i	−1.33465	+	1.33465i	−0.433500	+	0.901154i	$0.642722\pi$
$131$	−15.2758	+	15.2758i	−1.33465	+	1.33465i	−0.901154	+	0.433500i	$0.857278\pi$
$132$		0			0
$133$	−0.714346			−0.0619417
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.41140	+	4.41140i	0.376891	+	0.376891i	0.869979	−	0.493088i	$-0.164132\pi$
$137$	4.41140	+	4.41140i	0.376891	+	0.376891i	−0.493088	+	0.869979i	$0.664132\pi$
$138$		0			0
$139$	−10.3472	+	10.3472i	−0.877640	+	0.877640i	−0.993290	−	0.115650i	$-0.963105\pi$
$139$	−10.3472	+	10.3472i	−0.877640	+	0.877640i	0.115650	+	0.993290i	$0.463105\pi$
$140$		0			0
$141$	−5.40940	−	5.40940i	−0.455553	−	0.455553i
$142$		0			0
$143$	−0.694492	+	0.694492i	−0.0580763	+	0.0580763i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	5.50014i		−	0.453644i
$148$		0			0
$149$	7.12848	+	7.12848i	0.583987	+	0.583987i	0.935997	−	0.352009i	$-0.114501\pi$
$149$	7.12848	+	7.12848i	0.583987	+	0.583987i	−0.352009	+	0.935997i	$0.614501\pi$
$150$		0			0
$151$	19.7239			1.60511			0.802555	−	0.596578i	$-0.203473\pi$
$151$	19.7239			1.60511			0.802555	+	0.596578i	$0.203473\pi$
$152$		0			0
$153$	−4.89730	+	4.89730i	−0.395923	+	0.395923i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.0585			1.28161			0.640804	−	0.767704i	$-0.278601\pi$
$157$	16.0585			1.28161			0.640804	+	0.767704i	$0.278601\pi$
$158$		0			0
$159$	−6.47334			−0.513370
$160$		0			0
$161$	−1.29392			−0.101975
$162$		0			0
$163$	1.10043			0.0861920			0.0430960	−	0.999071i	$-0.486278\pi$
$163$	1.10043			0.0861920			0.0430960	+	0.999071i	$0.486278\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.6039	+	11.6039i	−0.897940	+	0.897940i	−0.995254	−	0.0973136i	$-0.968975\pi$
$167$	−11.6039	+	11.6039i	−0.897940	+	0.897940i	0.0973136	+	0.995254i	$0.468975\pi$
$168$		0			0
$169$	12.9445			0.995731
$170$		0			0
$171$	−6.06489	−	6.06489i	−0.463794	−	0.463794i
$172$		0			0
$173$		−	14.7116i		−	1.11851i	−0.828997	−	0.559253i	$-0.811088\pi$
$173$		−	14.7116i		−	1.11851i	0.828997	−	0.559253i	$-0.188912\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.41593	−	3.41593i	0.256757	−	0.256757i
$178$		0			0
$179$	−1.97591	−	1.97591i	−0.147686	−	0.147686i	0.629397	−	0.777084i	$-0.283302\pi$
$179$	−1.97591	−	1.97591i	−0.147686	−	0.147686i	−0.777084	+	0.629397i	$0.783302\pi$
$180$		0			0
$181$	1.45673	−	1.45673i	0.108278	−	0.108278i	−0.650892	−	0.759170i	$-0.725605\pi$
$181$	1.45673	−	1.45673i	0.108278	−	0.108278i	0.759170	+	0.650892i	$0.225605\pi$
$182$		0			0
$183$	7.23138	+	7.23138i	0.534559	+	0.534559i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.1550			−0.888860
$188$		0			0
$189$	−0.594269	+	0.594269i	−0.0432267	+	0.0432267i
$190$		0			0
$191$			0.285625i			0.0206671i	0.999947	+	0.0103336i	$0.00328933\pi$
$191$			0.285625i			0.0206671i	−0.999947	+	0.0103336i	$0.996711\pi$
$192$		0			0
$193$	−8.95931	−	8.95931i	−0.644905	−	0.644905i	0.306852	−	0.951757i	$-0.400724\pi$
$193$	−8.95931	−	8.95931i	−0.644905	−	0.644905i	−0.951757	+	0.306852i	$0.900724\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.8094i			0.912632i	0.889818	+	0.456316i	$0.150831\pi$
$197$			12.8094i			0.912632i	−0.889818	+	0.456316i	$0.849169\pi$
$198$		0			0
$199$		−	18.2117i		−	1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$199$		−	18.2117i		−	1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$200$		0			0
$201$			3.95142i			0.278711i
$202$		0			0
$203$		−	1.86473i		−	0.130878i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.9855	−	10.9855i	−0.763548	−	0.763548i
$208$		0			0
$209$		−	15.0529i		−	1.04123i
$210$		0			0
$211$	−2.40291	+	2.40291i	−0.165423	+	0.165423i	−0.784964	−	0.619541i	$-0.787319\pi$
$211$	−2.40291	+	2.40291i	−0.165423	+	0.165423i	0.619541	+	0.784964i	$0.287319\pi$
$212$		0			0
$213$	−4.79206			−0.328347
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.480164	+	0.480164i	0.0325956	+	0.0325956i
$218$		0			0
$219$	8.97973	−	8.97973i	0.606794	−	0.606794i
$220$		0			0
$221$	−0.485611	−	0.485611i	−0.0326657	−	0.0326657i
$222$		0			0
$223$	8.26331	−	8.26331i	0.553352	−	0.553352i	−0.374054	−	0.927407i	$-0.622033\pi$
$223$	8.26331	−	8.26331i	0.553352	−	0.553352i	0.927407	+	0.374054i	$0.122033\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			10.5895i			0.702847i	0.936217	+	0.351423i	$0.114302\pi$
$227$			10.5895i			0.702847i	−0.936217	+	0.351423i	$0.885698\pi$
$228$		0			0
$229$	−4.51111	−	4.51111i	−0.298102	−	0.298102i	0.542168	−	0.840270i	$-0.317604\pi$
$229$	−4.51111	−	4.51111i	−0.298102	−	0.298102i	−0.840270	+	0.542168i	$0.817604\pi$
$230$		0			0
$231$	−0.651827			−0.0428871
$232$		0			0
$233$	−1.60312	+	1.60312i	−0.105024	+	0.105024i	−0.757666	−	0.652642i	$-0.773660\pi$
$233$	−1.60312	+	1.60312i	−0.105024	+	0.105024i	0.652642	+	0.757666i	$0.273660\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.51441			0.228286
$238$		0			0
$239$	−14.1546			−0.915587			−0.457794	−	0.889058i	$-0.651360\pi$
$239$	−14.1546			−0.915587			−0.457794	+	0.889058i	$0.651360\pi$
$240$		0			0
$241$	4.25207			0.273900			0.136950	−	0.990578i	$-0.456270\pi$
$241$	4.25207			0.273900			0.136950	+	0.990578i	$0.456270\pi$
$242$		0			0
$243$	−15.7222			−1.00858
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.601388	−	0.601388i	0.0382654	−	0.0382654i
$248$		0			0
$249$	9.06923			0.574739
$250$		0			0
$251$	1.29050	+	1.29050i	0.0814559	+	0.0814559i	0.746661	−	0.665205i	$-0.231656\pi$
$251$	1.29050	+	1.29050i	0.0814559	+	0.0814559i	−0.665205	+	0.746661i	$0.731656\pi$
$252$		0			0
$253$		−	27.2658i		−	1.71419i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.72463	+	5.72463i	−0.357093	+	0.357093i	−0.862740	−	0.505647i	$-0.831254\pi$
$257$	−5.72463	+	5.72463i	−0.357093	+	0.357093i	0.505647	+	0.862740i	$0.331254\pi$
$258$		0			0
$259$	0.193711	+	0.193711i	0.0120366	+	0.0120366i
$260$		0			0
$261$	15.8318	−	15.8318i	0.979963	−	0.979963i
$262$		0			0
$263$	−17.0683	−	17.0683i	−1.05248	−	1.05248i	−0.998545	−	0.0539323i	$-0.982824\pi$
$263$	−17.0683	−	17.0683i	−1.05248	−	1.05248i	−0.0539323	−	0.998545i	$-0.517176\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.62051			−0.282771
$268$		0			0
$269$	6.15456	−	6.15456i	0.375250	−	0.375250i	−0.494135	−	0.869385i	$-0.664515\pi$
$269$	6.15456	−	6.15456i	0.375250	−	0.375250i	0.869385	+	0.494135i	$0.164515\pi$
$270$		0			0
$271$		−	18.4342i		−	1.11980i	−0.828561	−	0.559899i	$-0.810840\pi$
$271$		−	18.4342i		−	1.11980i	0.828561	−	0.559899i	$-0.189160\pi$
$272$		0			0
$273$	−0.0260416	−	0.0260416i	−0.00157611	−	0.00157611i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	15.2281i		−	0.914967i	−0.889218	−	0.457484i	$-0.848751\pi$
$277$		−	15.2281i		−	0.914967i	0.889218	−	0.457484i	$-0.151249\pi$
$278$		0			0
$279$			8.15330i			0.488125i
$280$		0			0
$281$			7.07835i			0.422259i	0.977458	+	0.211129i	$0.0677142\pi$
$281$			7.07835i			0.422259i	−0.977458	+	0.211129i	$0.932286\pi$
$282$		0			0
$283$		−	19.9173i		−	1.18396i	−0.805953	−	0.591979i	$-0.798347\pi$
$283$		−	19.9173i		−	1.18396i	0.805953	−	0.591979i	$-0.201653\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.15597	−	1.15597i	−0.0682349	−	0.0682349i
$288$		0			0
$289$			8.50085i			0.500050i
$290$		0			0
$291$	0.407231	−	0.407231i	0.0238723	−	0.0238723i
$292$		0			0
$293$	23.1293			1.35123			0.675614	−	0.737256i	$-0.263879\pi$
$293$	23.1293			1.35123			0.675614	+	0.737256i	$0.263879\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.5226	−	12.5226i	−0.726634	−	0.726634i
$298$		0			0
$299$	1.08931	−	1.08931i	0.0629967	−	0.0629967i
$300$		0			0
$301$	−0.755579	−	0.755579i	−0.0435509	−	0.0435509i
$302$		0			0
$303$	−2.81822	+	2.81822i	−0.161903	+	0.161903i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			3.82901i			0.218533i	0.994012	+	0.109267i	$0.0348502\pi$
$307$			3.82901i			0.218533i	−0.994012	+	0.109267i	$0.965150\pi$
$308$		0			0
$309$	−9.21839	−	9.21839i	−0.524416	−	0.524416i
$310$		0			0
$311$	−9.07002			−0.514314			−0.257157	−	0.966370i	$-0.582786\pi$
$311$	−9.07002			−0.514314			−0.257157	+	0.966370i	$0.582786\pi$
$312$		0			0
$313$	2.78399	−	2.78399i	0.157361	−	0.157361i	−0.624035	−	0.781396i	$-0.714508\pi$
$313$	2.78399	−	2.78399i	0.157361	−	0.157361i	0.781396	+	0.624035i	$0.214508\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.8639			−0.834838			−0.417419	−	0.908714i	$-0.637065\pi$
$317$	−14.8639			−0.834838			−0.417419	+	0.908714i	$0.637065\pi$
$318$		0			0
$319$	39.2941			2.20005
$320$		0			0
$321$	−3.95142			−0.220547
$322$		0			0
$323$	10.5255			0.585653
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.92020	−	2.92020i	0.161487	−	0.161487i
$328$		0			0
$329$	−1.91561			−0.105611
$330$		0			0
$331$	8.73942	+	8.73942i	0.480362	+	0.480362i	0.905247	−	0.424885i	$-0.139686\pi$
$331$	8.73942	+	8.73942i	0.480362	+	0.480362i	−0.424885	+	0.905247i	$0.639686\pi$
$332$		0			0
$333$			3.28926i			0.180250i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.3405	+	20.3405i	−1.10802	+	1.10802i	−0.114607	+	0.993411i	$0.536561\pi$
$337$	−20.3405	+	20.3405i	−1.10802	+	1.10802i	−0.993411	+	0.114607i	$0.963439\pi$
$338$		0			0
$339$	−9.19867	−	9.19867i	−0.499603	−	0.499603i
$340$		0			0
$341$	−10.1181	+	10.1181i	−0.547928	+	0.547928i
$342$		0			0
$343$	−1.95322	−	1.95322i	−0.105464	−	0.105464i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.33669			0.393854			0.196927	−	0.980418i	$-0.436904\pi$
$347$	7.33669			0.393854			0.196927	+	0.980418i	$0.436904\pi$
$348$		0			0
$349$	4.99392	−	4.99392i	0.267319	−	0.267319i	−0.560700	−	0.828019i	$-0.689468\pi$
$349$	4.99392	−	4.99392i	0.267319	−	0.267319i	0.828019	+	0.560700i	$0.189468\pi$
$350$		0			0
$351$		−	1.00060i		−	0.0534078i
$352$		0			0
$353$	−5.74673	−	5.74673i	−0.305868	−	0.305868i	0.537436	−	0.843304i	$-0.319393\pi$
$353$	−5.74673	−	5.74673i	−0.305868	−	0.305868i	−0.843304	+	0.537436i	$0.819393\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	0.455778i		−	0.0241224i
$358$		0			0
$359$		−	22.2959i		−	1.17673i	−0.808594	−	0.588366i	$-0.799771\pi$
$359$		−	22.2959i		−	1.17673i	0.808594	−	0.588366i	$-0.200229\pi$
$360$		0			0
$361$		−	5.96511i		−	0.313953i
$362$		0			0
$363$		−	5.04379i		−	0.264730i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	22.1923	+	22.1923i	1.15843	+	1.15843i	0.984814	+	0.173615i	$0.0555448\pi$
$367$	22.1923	+	22.1923i	1.15843	+	1.15843i	0.173615	+	0.984814i	$0.444455\pi$
$368$		0			0
$369$		−	19.6287i		−	1.02183i
$370$		0			0
$371$	−1.14619	+	1.14619i	−0.0595074	+	0.0595074i
$372$		0			0
$373$	−27.1593			−1.40625			−0.703127	−	0.711064i	$-0.748214\pi$
$373$	−27.1593			−1.40625			−0.703127	+	0.711064i	$0.748214\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.56986	+	1.56986i	0.0808521	+	0.0808521i
$378$		0			0
$379$	14.7602	−	14.7602i	0.758180	−	0.758180i	−0.217811	−	0.975991i	$-0.569892\pi$
$379$	14.7602	−	14.7602i	0.758180	−	0.758180i	0.975991	+	0.217811i	$0.0698917\pi$
$380$		0			0
$381$	4.27126	+	4.27126i	0.218823	+	0.218823i
$382$		0			0
$383$	14.5976	−	14.5976i	0.745901	−	0.745901i	−0.227805	−	0.973707i	$-0.573155\pi$
$383$	14.5976	−	14.5976i	0.745901	−	0.745901i	0.973707	+	0.227805i	$0.0731550\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	12.8299i		−	0.652182i
$388$		0			0
$389$	−3.93745	−	3.93745i	−0.199637	−	0.199637i	0.600208	−	0.799844i	$-0.295085\pi$
$389$	−3.93745	−	3.93745i	−0.199637	−	0.199637i	−0.799844	+	0.600208i	$0.795085\pi$
$390$		0			0
$391$	19.0651			0.964166
$392$		0			0
$393$	12.0702	−	12.0702i	0.608862	−	0.608862i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−27.0000			−1.35509			−0.677545	−	0.735481i	$-0.736956\pi$
$397$	−27.0000			−1.35509			−0.677545	+	0.735481i	$0.736956\pi$
$398$		0			0
$399$	0.564442			0.0282575
$400$		0			0
$401$	3.02550			0.151086			0.0755432	−	0.997143i	$-0.475931\pi$
$401$	3.02550			0.151086			0.0755432	+	0.997143i	$0.475931\pi$
$402$		0			0
$403$	−0.808473			−0.0402729
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.08193	+	4.08193i	−0.202334	+	0.202334i
$408$		0			0
$409$	−19.6553			−0.971894			−0.485947	−	0.873988i	$-0.661525\pi$
$409$	−19.6553			−0.971894			−0.485947	+	0.873988i	$0.661525\pi$
$410$		0			0
$411$	−3.48568	−	3.48568i	−0.171936	−	0.171936i
$412$		0			0
$413$		−	1.20967i		−	0.0595241i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8.17588	−	8.17588i	0.400375	−	0.400375i
$418$		0			0
$419$	16.1834	+	16.1834i	0.790609	+	0.790609i	0.981593	−	0.190984i	$-0.0611679\pi$
$419$	16.1834	+	16.1834i	0.790609	+	0.790609i	−0.190984	+	0.981593i	$0.561168\pi$
$420$		0			0
$421$	−23.1841	+	23.1841i	−1.12993	+	1.12993i	−0.139737	+	0.990189i	$0.544626\pi$
$421$	−23.1841	+	23.1841i	−1.12993	+	1.12993i	−0.990189	+	0.139737i	$0.955374\pi$
$422$		0			0
$423$	−16.2638	−	16.2638i	−0.790773	−	0.790773i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.56083			0.123927
$428$		0			0
$429$	0.548755	−	0.548755i	0.0264941	−	0.0264941i
$430$		0			0
$431$			29.7907i			1.43497i	0.696575	+	0.717484i	$0.254706\pi$
$431$			29.7907i			1.43497i	−0.696575	+	0.717484i	$0.745294\pi$
$432$		0			0
$433$	−19.0587	−	19.0587i	−0.915903	−	0.915903i	0.0808251	−	0.996728i	$-0.474244\pi$
$433$	−19.0587	−	19.0587i	−0.915903	−	0.915903i	−0.996728	+	0.0808251i	$0.974244\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			23.6105i			1.12945i
$438$		0			0
$439$		−	6.10665i		−	0.291454i	−0.989325	−	0.145727i	$-0.953448\pi$
$439$		−	6.10665i		−	0.291454i	0.989325	−	0.145727i	$-0.0465522\pi$
$440$		0			0
$441$		−	16.5366i		−	0.787458i
$442$		0			0
$443$		−	16.1163i		−	0.765708i	−0.923809	−	0.382854i	$-0.874941\pi$
$443$		−	16.1163i		−	0.765708i	0.923809	−	0.382854i	$-0.125059\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−5.63258	−	5.63258i	−0.266412	−	0.266412i
$448$		0			0
$449$		−	1.87161i		−	0.0883268i	−0.999024	−	0.0441634i	$-0.985938\pi$
$449$		−	1.87161i		−	0.0883268i	0.999024	−	0.0441634i	$-0.0140622\pi$
$450$		0			0
$451$	24.3590	−	24.3590i	1.14702	−	1.14702i
$452$		0			0
$453$	−15.5849			−0.732243
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.7459	−	10.7459i	−0.502673	−	0.502673i	0.409594	−	0.912268i	$-0.365670\pi$
$457$	−10.7459	−	10.7459i	−0.502673	−	0.502673i	−0.912268	+	0.409594i	$0.865670\pi$
$458$		0			0
$459$	8.75619	−	8.75619i	0.408704	−	0.408704i
$460$		0			0
$461$	16.5710	+	16.5710i	0.771790	+	0.771790i	0.978419	−	0.206630i	$-0.0662495\pi$
$461$	16.5710	+	16.5710i	0.771790	+	0.771790i	−0.206630	+	0.978419i	$0.566249\pi$
$462$		0			0
$463$	−19.1271	+	19.1271i	−0.888912	+	0.888912i	−0.994419	−	0.105507i	$-0.966354\pi$
$463$	−19.1271	+	19.1271i	−0.888912	+	0.888912i	0.105507	+	0.994419i	$0.466354\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			35.9184i			1.66210i	0.556195	+	0.831052i	$0.312261\pi$
$467$			35.9184i			1.66210i	−0.556195	+	0.831052i	$0.687739\pi$
$468$		0			0
$469$	0.699652	+	0.699652i	0.0323069	+	0.0323069i
$470$		0			0
$471$	−12.6887			−0.584663
$472$		0			0
$473$	15.9218	−	15.9218i	0.732083	−	0.732083i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−19.4626			−0.891133
$478$		0			0
$479$	7.66614			0.350275			0.175137	−	0.984544i	$-0.443963\pi$
$479$	7.66614			0.350275			0.175137	+	0.984544i	$0.443963\pi$
$480$		0			0
$481$	−0.326159			−0.0148716
$482$		0			0
$483$	1.02239			0.0465206
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.0452267	−	0.0452267i	0.00204942	−	0.00204942i	−0.706081	−	0.708131i	$-0.749539\pi$
$487$	0.0452267	−	0.0452267i	0.00204942	−	0.00204942i	0.708131	+	0.706081i	$0.249539\pi$
$488$		0			0
$489$	−0.869504			−0.0393203
$490$		0			0
$491$	−3.49963	−	3.49963i	−0.157936	−	0.157936i	0.623715	−	0.781651i	$-0.285622\pi$
$491$	−3.49963	−	3.49963i	−0.157936	−	0.157936i	−0.781651	+	0.623715i	$0.785622\pi$
$492$		0			0
$493$			27.4757i			1.23744i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.848499	+	0.848499i	−0.0380604	+	0.0380604i
$498$		0			0
$499$	−18.7985	−	18.7985i	−0.841535	−	0.841535i	0.147523	−	0.989059i	$-0.452870\pi$
$499$	−18.7985	−	18.7985i	−0.841535	−	0.841535i	−0.989059	+	0.147523i	$0.952870\pi$
$500$		0			0
$501$	9.16889	−	9.16889i	0.409636	−	0.409636i
$502$		0			0
$503$	11.4064	+	11.4064i	0.508588	+	0.508588i	0.914093	−	0.405505i	$-0.132904\pi$
$503$	11.4064	+	11.4064i	0.508588	+	0.508588i	−0.405505	+	0.914093i	$0.632904\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.2281			−0.454247
$508$		0			0
$509$	9.87431	−	9.87431i	0.437671	−	0.437671i	−0.453556	−	0.891228i	$-0.649845\pi$
$509$	9.87431	−	9.87431i	0.437671	−	0.437671i	0.891228	+	0.453556i	$0.149845\pi$
$510$		0			0
$511$		−	3.17997i		−	0.140673i
$512$		0			0
$513$	10.8438	+	10.8438i	0.478765	+	0.478765i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	40.3663i		−	1.77531i
$518$		0			0
$519$			11.6244i			0.510256i
$520$		0			0
$521$			21.6730i			0.949512i	0.880117	+	0.474756i	$0.157464\pi$
$521$			21.6730i			0.949512i	−0.880117	+	0.474756i	$0.842536\pi$
$522$		0			0
$523$			40.3785i			1.76563i	0.469724	+	0.882813i	$0.344353\pi$
$523$			40.3785i			1.76563i	−0.469724	+	0.882813i	$0.655647\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.07493	−	7.07493i	−0.308189	−	0.308189i
$528$		0			0
$529$			19.7666i			0.859418i
$530$		0			0
$531$	10.2703	−	10.2703i	0.445692	−	0.445692i
$532$		0			0
$533$	1.94636			0.0843063
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.56127	+	1.56127i	0.0673737	+	0.0673737i
$538$		0			0
$539$	20.5217	−	20.5217i	0.883933	−	0.883933i
$540$		0			0
$541$	6.26728	+	6.26728i	0.269451	+	0.269451i	0.828879	−	0.559428i	$-0.188979\pi$
$541$	6.26728	+	6.26728i	0.269451	+	0.269451i	−0.559428	+	0.828879i	$0.688979\pi$
$542$		0			0
$543$	−1.15104	+	1.15104i	−0.0493958	+	0.0493958i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	34.5372i		−	1.47671i	−0.674415	−	0.738353i	$-0.735604\pi$
$547$		−	34.5372i		−	1.47671i	0.674415	−	0.738353i	$-0.264396\pi$
$548$		0			0
$549$	21.7417	+	21.7417i	0.927915	+	0.927915i
$550$		0			0
$551$	−34.0263			−1.44957
$552$		0			0
$553$	0.622274	−	0.622274i	0.0264618	−	0.0264618i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.4740			0.909881			0.454940	−	0.890522i	$-0.349661\pi$
$557$	21.4740			0.909881			0.454940	+	0.890522i	$0.349661\pi$
$558$		0			0
$559$	1.27220			0.0538084
$560$		0			0
$561$	9.60428			0.405493
$562$		0			0
$563$	6.10126			0.257138			0.128569	−	0.991701i	$-0.458962\pi$
$563$	6.10126			0.257138			0.128569	+	0.991701i	$0.458962\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.527553	+	0.527553i	−0.0221551	+	0.0221551i
$568$		0			0
$569$	−31.1884			−1.30749			−0.653744	−	0.756716i	$-0.726802\pi$
$569$	−31.1884			−1.30749			−0.653744	+	0.756716i	$0.726802\pi$
$570$		0			0
$571$	9.39471	+	9.39471i	0.393156	+	0.393156i	0.875811	−	0.482654i	$-0.160327\pi$
$571$	9.39471	+	9.39471i	0.393156	+	0.393156i	−0.482654	+	0.875811i	$0.660327\pi$
$572$		0			0
$573$		−	0.225688i		−	0.00942824i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.2136	−	16.2136i	0.674980	−	0.674980i	−0.283880	−	0.958860i	$-0.591622\pi$
$577$	16.2136	−	16.2136i	0.674980	−	0.674980i	0.958860	+	0.283880i	$0.0916218\pi$
$578$		0			0
$579$	7.07922	+	7.07922i	0.294203	+	0.294203i
$580$		0			0
$581$	1.60583	−	1.60583i	0.0666211	−	0.0666211i
$582$		0			0
$583$	−24.1529	−	24.1529i	−1.00031	−	1.00031i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.3084			0.920765			0.460382	−	0.887721i	$-0.347712\pi$
$587$	22.3084			0.920765			0.460382	+	0.887721i	$0.347712\pi$
$588$		0			0
$589$	8.76169	−	8.76169i	0.361019	−	0.361019i
$590$		0			0
$591$		−	10.1214i		−	0.416338i
$592$		0			0
$593$	−5.08162	−	5.08162i	−0.208677	−	0.208677i	0.595028	−	0.803705i	$-0.297141\pi$
$593$	−5.08162	−	5.08162i	−0.208677	−	0.208677i	−0.803705	+	0.595028i	$0.797141\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			14.3900i			0.588945i
$598$		0			0
$599$			45.7467i			1.86916i	0.355752	+	0.934580i	$0.384225\pi$
$599$			45.7467i			1.86916i	−0.355752	+	0.934580i	$0.615775\pi$
$600$		0			0
$601$		−	34.5280i		−	1.40843i	−0.709989	−	0.704213i	$-0.751300\pi$
$601$		−	34.5280i		−	1.40843i	0.709989	−	0.704213i	$-0.248700\pi$
$602$		0			0
$603$			11.8803i			0.483802i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.881912	−	0.881912i	−0.0357957	−	0.0357957i	0.688982	−	0.724778i	$-0.258058\pi$
$607$	−0.881912	−	0.881912i	−0.0357957	−	0.0357957i	−0.724778	+	0.688982i	$0.758058\pi$
$608$		0			0
$609$			1.47342i			0.0597061i
$610$		0			0
$611$	1.61270	−	1.61270i	0.0652429	−	0.0652429i
$612$		0			0
$613$	19.7457			0.797521			0.398760	−	0.917055i	$-0.369440\pi$
$613$	19.7457			0.797521			0.398760	+	0.917055i	$0.369440\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.0643	+	24.0643i	0.968793	+	0.968793i	0.999528	−	0.0307346i	$-0.00978467\pi$
$617$	24.0643	+	24.0643i	0.968793	+	0.968793i	−0.0307346	+	0.999528i	$0.509785\pi$
$618$		0			0
$619$	−29.9131	+	29.9131i	−1.20231	+	1.20231i	−0.228846	+	0.973463i	$0.573495\pi$
$619$	−29.9131	+	29.9131i	−1.20231	+	1.20231i	−0.973463	+	0.228846i	$0.926505\pi$
$620$		0			0
$621$	19.6417	+	19.6417i	0.788196	+	0.788196i
$622$		0			0
$623$	−0.818124	+	0.818124i	−0.0327774	+	0.0327774i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			11.8941i			0.475004i
$628$		0			0
$629$	−2.85421	−	2.85421i	−0.113805	−	0.113805i
$630$		0			0
$631$	−49.7586			−1.98086			−0.990429	−	0.138022i	$-0.955926\pi$
$631$	−49.7586			−1.98086			−0.990429	+	0.138022i	$0.955926\pi$
$632$		0			0
$633$	1.89867	−	1.89867i	0.0754652	−	0.0754652i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.63975			0.0649694
$638$		0			0
$639$	−14.4077			−0.569961
$640$		0			0
$641$	−17.3779			−0.686386			−0.343193	−	0.939265i	$-0.611509\pi$
$641$	−17.3779			−0.686386			−0.343193	+	0.939265i	$0.611509\pi$
$642$		0			0
$643$	36.4216			1.43633			0.718165	−	0.695873i	$-0.244982\pi$
$643$	36.4216			1.43633			0.718165	+	0.695873i	$0.244982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.55333	+	3.55333i	−0.139696	+	0.139696i	−0.773496	−	0.633801i	$-0.781494\pi$
$647$	−3.55333	+	3.55333i	−0.139696	+	0.139696i	0.633801	+	0.773496i	$0.281494\pi$
$648$		0			0
$649$	25.4906			1.00059
$650$		0			0
$651$	−0.379403	−	0.379403i	−0.0148700	−	0.0148700i
$652$		0			0
$653$			38.4729i			1.50556i	0.658271	+	0.752781i	$0.271288\pi$
$653$			38.4729i			1.50556i	−0.658271	+	0.752781i	$0.728712\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	26.9983	−	26.9983i	1.05330	−	1.05330i
$658$		0			0
$659$	−17.1857	−	17.1857i	−0.669459	−	0.669459i	0.288132	−	0.957591i	$-0.406966\pi$
$659$	−17.1857	−	17.1857i	−0.669459	−	0.669459i	−0.957591	+	0.288132i	$0.906966\pi$
$660$		0			0
$661$	−4.63141	+	4.63141i	−0.180141	+	0.180141i	−0.791417	−	0.611276i	$-0.790656\pi$
$661$	−4.63141	+	4.63141i	−0.180141	+	0.180141i	0.611276	+	0.791417i	$0.290656\pi$
$662$		0			0
$663$	0.383707	+	0.383707i	0.0149019	+	0.0149019i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−61.6330			−2.38644
$668$		0			0
$669$	−6.52928	+	6.52928i	−0.252436	+	0.252436i
$670$		0			0
$671$			53.9624i			2.08320i
$672$		0			0
$673$	−3.70786	−	3.70786i	−0.142928	−	0.142928i	0.632022	−	0.774950i	$-0.282225\pi$
$673$	−3.70786	−	3.70786i	−0.142928	−	0.142928i	−0.774950	+	0.632022i	$0.782225\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			35.7948i			1.37570i	0.725851	+	0.687852i	$0.241446\pi$
$677$			35.7948i			1.37570i	−0.725851	+	0.687852i	$0.758554\pi$
$678$		0			0
$679$		−	0.144211i		−	0.00553432i
$680$		0			0
$681$		−	8.36729i		−	0.320635i
$682$		0			0
$683$			30.5276i			1.16810i	0.811716	+	0.584052i	$0.198534\pi$
$683$			30.5276i			1.16810i	−0.811716	+	0.584052i	$0.801466\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	3.56446	+	3.56446i	0.135993	+	0.135993i
$688$		0			0
$689$		−	1.92989i		−	0.0735231i
$690$		0			0
$691$	16.3626	−	16.3626i	0.622463	−	0.622463i	−0.323698	−	0.946161i	$-0.604926\pi$
$691$	16.3626	−	16.3626i	0.622463	−	0.622463i	0.946161	+	0.323698i	$0.104926\pi$
$692$		0			0
$693$	−1.95977			−0.0744456
$694$		0			0
$695$		0			0
$696$		0			0
$697$	17.0326	+	17.0326i	0.645155	+	0.645155i
$698$		0			0
$699$	1.26671	−	1.26671i	0.0479114	−	0.0479114i
$700$		0			0
$701$	−5.97112	−	5.97112i	−0.225526	−	0.225526i	0.585295	−	0.810821i	$-0.300979\pi$
$701$	−5.97112	−	5.97112i	−0.225526	−	0.225526i	−0.810821	+	0.585295i	$0.800979\pi$
$702$		0			0
$703$	3.53470	−	3.53470i	0.133314	−	0.133314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			0.998009i			0.0375340i
$708$		0			0
$709$	−4.50479	−	4.50479i	−0.169181	−	0.169181i	0.617438	−	0.786619i	$-0.288170\pi$
$709$	−4.50479	−	4.50479i	−0.169181	−	0.169181i	−0.786619	+	0.617438i	$0.788170\pi$
$710$		0			0
$711$	10.5664			0.396270
$712$		0			0
$713$	15.8704	−	15.8704i	0.594350	−	0.594350i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	11.1843			0.417686
$718$		0			0
$719$	−34.8855			−1.30101			−0.650504	−	0.759503i	$-0.725442\pi$
$719$	−34.8855			−1.30101			−0.650504	+	0.759503i	$0.725442\pi$
$720$		0			0
$721$	−3.26448			−0.121576
$722$		0			0
$723$	−3.35979			−0.124952
$724$		0			0
$725$		0			0
$726$		0			0
$727$	20.1354	−	20.1354i	0.746781	−	0.746781i	−0.227092	−	0.973873i	$-0.572922\pi$
$727$	20.1354	−	20.1354i	0.746781	−	0.746781i	0.973873	+	0.227092i	$0.0729219\pi$
$728$		0			0
$729$	1.11076			0.0411393
$730$		0			0
$731$	11.1330	+	11.1330i	0.411769	+	0.411769i
$732$		0			0
$733$			29.8111i			1.10110i	0.834803	+	0.550549i	$0.185581\pi$
$733$			29.8111i			1.10110i	−0.834803	+	0.550549i	$0.814419\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.7432	+	14.7432i	−0.543074	+	0.543074i
$738$		0			0
$739$	−18.8014	−	18.8014i	−0.691620	−	0.691620i	0.270969	−	0.962588i	$-0.412656\pi$
$739$	−18.8014	−	18.8014i	−0.691620	−	0.691620i	−0.962588	+	0.270969i	$0.912656\pi$
$740$		0			0
$741$	−0.475188	+	0.475188i	−0.0174565	+	0.0174565i
$742$		0			0
$743$	−8.34445	−	8.34445i	−0.306128	−	0.306128i	0.537277	−	0.843406i	$-0.319453\pi$
$743$	−8.34445	−	8.34445i	−0.306128	−	0.306128i	−0.843406	+	0.537277i	$0.819453\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	27.2674			0.997662
$748$		0			0
$749$	−0.699652	+	0.699652i	−0.0255647	+	0.0255647i
$750$		0			0
$751$		−	40.1477i		−	1.46501i	−0.680761	−	0.732505i	$-0.738351\pi$
$751$		−	40.1477i		−	1.46501i	0.680761	−	0.732505i	$-0.261649\pi$
$752$		0			0
$753$	−1.01970	−	1.01970i	−0.0371598	−	0.0371598i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	31.5795i		−	1.14778i	−0.818934	−	0.573888i	$-0.805435\pi$
$757$		−	31.5795i		−	1.14778i	0.818934	−	0.573888i	$-0.194565\pi$
$758$		0			0
$759$			21.5442i			0.782004i
$760$		0			0
$761$		−	29.0804i		−	1.05416i	−0.849815	−	0.527081i	$-0.823286\pi$
$761$		−	29.0804i		−	1.05416i	0.849815	−	0.527081i	$-0.176714\pi$
$762$		0			0
$763$		−	1.03412i		−	0.0374377i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.01839	+	1.01839i	0.0367719	+	0.0367719i
$768$		0			0
$769$			44.9984i			1.62268i	0.584573	+	0.811341i	$0.301262\pi$
$769$			44.9984i			1.62268i	−0.584573	+	0.811341i	$0.698738\pi$
$770$		0			0
$771$	4.52333	−	4.52333i	0.162904	−	0.162904i
$772$		0			0
$773$	−35.9788			−1.29407			−0.647035	−	0.762461i	$-0.723991\pi$
$773$	−35.9788			−1.29407			−0.647035	+	0.762461i	$0.723991\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.153061	−	0.153061i	−0.00549104	−	0.00549104i
$778$		0			0
$779$	−21.0934	+	21.0934i	−0.755749	+	0.755749i
$780$		0			0
$781$	−17.8798	−	17.8798i	−0.639789	−	0.639789i
$782$		0			0
$783$	−28.3067	+	28.3067i	−1.01160	+	1.01160i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	18.5853i		−	0.662496i	−0.943544	−	0.331248i	$-0.892530\pi$
$787$		−	18.5853i		−	0.662496i	0.943544	−	0.331248i	$-0.107470\pi$
$788$		0			0
$789$	13.4866	+	13.4866i	0.480135	+	0.480135i
$790$		0			0
$791$	−3.25750			−0.115823
$792$		0			0
$793$	−2.15589	+	2.15589i	−0.0765578	+	0.0765578i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.5437			0.975649			0.487824	−	0.872942i	$-0.337791\pi$
$797$	27.5437			0.975649			0.487824	+	0.872942i	$0.337791\pi$
$798$		0			0
$799$	28.2254			0.998544
$800$		0			0
$801$	−13.8919			−0.490848
$802$		0			0
$803$	67.0091			2.36470
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.86304	+	4.86304i	−0.171187	+	0.171187i
$808$		0			0
$809$	−45.0587			−1.58418			−0.792090	−	0.610404i	$-0.791007\pi$
$809$	−45.0587			−1.58418			−0.792090	+	0.610404i	$0.791007\pi$
$810$		0			0
$811$	19.2189	+	19.2189i	0.674865	+	0.674865i	0.958834	−	0.283968i	$-0.0916511\pi$
$811$	19.2189	+	19.2189i	0.674865	+	0.674865i	−0.283968	+	0.958834i	$0.591651\pi$
$812$		0			0
$813$			14.5658i			0.510846i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−13.7873	+	13.7873i	−0.482356	+	0.482356i
$818$		0			0
$819$	−0.0782961	−	0.0782961i	−0.00273589	−	0.00273589i
$820$		0			0
$821$	−30.5315	+	30.5315i	−1.06556	+	1.06556i	−0.0678632	+	0.997695i	$0.521618\pi$
$821$	−30.5315	+	30.5315i	−1.06556	+	1.06556i	−0.997695	+	0.0678632i	$0.978382\pi$
$822$		0			0
$823$	18.1848	+	18.1848i	0.633884	+	0.633884i	0.949040	−	0.315156i	$-0.102057\pi$
$823$	18.1848	+	18.1848i	0.633884	+	0.633884i	−0.315156	+	0.949040i	$0.602057\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.7073			0.720063			0.360032	−	0.932940i	$-0.382766\pi$
$827$	20.7073			0.720063			0.360032	+	0.932940i	$0.382766\pi$
$828$		0			0
$829$	19.5152	−	19.5152i	0.677790	−	0.677790i	−0.281710	−	0.959500i	$-0.590901\pi$
$829$	19.5152	−	19.5152i	0.677790	−	0.677790i	0.959500	+	0.281710i	$0.0909015\pi$
$830$		0			0
$831$			12.0325i			0.417403i
$832$		0			0
$833$	14.3494	+	14.3494i	0.497179	+	0.497179i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	14.5778i		−	0.503882i
$838$		0			0
$839$			10.8185i			0.373496i	0.982408	+	0.186748i	$0.0597948\pi$
$839$			10.8185i			0.373496i	−0.982408	+	0.186748i	$0.940205\pi$
$840$		0			0
$841$		−	59.8223i		−	2.06284i
$842$		0			0
$843$		−	5.59298i		−	0.192632i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.893071	−	0.893071i	−0.0306863	−	0.0306863i
$848$		0			0
$849$			15.7377i			0.540116i
$850$		0			0
$851$	6.40253	−	6.40253i	0.219476	−	0.219476i
$852$		0			0
$853$	5.42003			0.185578			0.0927892	−	0.995686i	$-0.470422\pi$
$853$	5.42003			0.185578			0.0927892	+	0.995686i	$0.470422\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.89352	+	4.89352i	0.167160	+	0.167160i	0.785730	−	0.618570i	$-0.212288\pi$
$857$	4.89352	+	4.89352i	0.167160	+	0.167160i	−0.618570	+	0.785730i	$0.712288\pi$
$858$		0			0
$859$	−33.4966	+	33.4966i	−1.14289	+	1.14289i	−0.154969	+	0.987919i	$0.549528\pi$
$859$	−33.4966	+	33.4966i	−1.14289	+	1.14289i	−0.987919	+	0.154969i	$0.950472\pi$
$860$		0			0
$861$	0.913395	+	0.913395i	0.0311284	+	0.0311284i
$862$		0			0
$863$	28.0996	−	28.0996i	0.956521	−	0.956521i	−0.0425724	−	0.999093i	$-0.513555\pi$
$863$	28.0996	−	28.0996i	0.956521	−	0.956521i	0.999093	+	0.0425724i	$0.0135553\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	6.71697i		−	0.228120i
$868$		0			0
$869$	13.1127	+	13.1127i	0.444819	+	0.444819i
$870$		0			0
$871$	−1.17803			−0.0399162
$872$		0			0
$873$	1.22437	−	1.22437i	0.0414387	−	0.0414387i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	21.3398			0.720593			0.360296	−	0.932838i	$-0.382676\pi$
$877$	21.3398			0.720593			0.360296	+	0.932838i	$0.382676\pi$
$878$		0			0
$879$	−18.2757			−0.616423
$880$		0			0
$881$	−34.8632			−1.17457			−0.587285	−	0.809380i	$-0.699803\pi$
$881$	−34.8632			−1.17457			−0.587285	+	0.809380i	$0.699803\pi$
$882$		0			0
$883$	−16.9490			−0.570381			−0.285190	−	0.958471i	$-0.592057\pi$
$883$	−16.9490			−0.570381			−0.285190	+	0.958471i	$0.592057\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	33.7217	−	33.7217i	1.13226	−	1.13226i	0.142464	−	0.989800i	$-0.454498\pi$
$887$	33.7217	−	33.7217i	1.13226	−	1.13226i	0.989800	−	0.142464i	$-0.0455025\pi$
$888$		0			0
$889$	1.51257			0.0507300
$890$		0			0
$891$	−11.1167	−	11.1167i	−0.372424	−	0.372424i
$892$		0			0
$893$			34.9548i			1.16972i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.860725	+	0.860725i	−0.0287388	+	0.0287388i
$898$		0			0
$899$	22.8715	+	22.8715i	0.762808	+	0.762808i
$900$		0			0
$901$	16.8885	−	16.8885i	0.562637	−	0.562637i
$902$		0			0
$903$	0.597023	+	0.597023i	0.0198677	+	0.0198677i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	18.9797			0.630210			0.315105	−	0.949057i	$-0.397960\pi$
$907$	18.9797			0.630210			0.315105	+	0.949057i	$0.397960\pi$
$908$		0			0
$909$	−8.47322	+	8.47322i	−0.281039	+	0.281039i
$910$		0			0
$911$			15.0326i			0.498053i	0.968497	+	0.249027i	$0.0801107\pi$
$911$			15.0326i			0.498053i	−0.968497	+	0.249027i	$0.919889\pi$
$912$		0			0
$913$	33.8385	+	33.8385i	1.11989	+	1.11989i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.27439i		−	0.141153i
$918$		0			0
$919$			37.9183i			1.25081i	0.780301	+	0.625404i	$0.215066\pi$
$919$			37.9183i			1.25081i	−0.780301	+	0.625404i	$0.784934\pi$
$920$		0			0
$921$		−	3.02550i		−	0.0996938i
$922$		0			0
$923$		−	1.42865i		−	0.0470247i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−27.7158	−	27.7158i	−0.910308	−	0.910308i
$928$		0			0
$929$		−	22.5607i		−	0.740193i	−0.928993	−	0.370096i	$-0.879325\pi$
$929$		−	22.5607i		−	0.740193i	0.928993	−	0.370096i	$-0.120675\pi$
$930$		0			0
$931$	−17.7706	+	17.7706i	−0.582406	+	0.582406i
$932$		0			0
$933$	7.16670			0.234627
$934$		0			0
$935$		0			0
$936$		0			0
$937$	7.98622	+	7.98622i	0.260898	+	0.260898i	0.825419	−	0.564521i	$-0.190939\pi$
$937$	7.98622	+	7.98622i	0.260898	+	0.260898i	−0.564521	+	0.825419i	$0.690939\pi$
$938$		0			0
$939$	−2.19978	+	2.19978i	−0.0717871	+	0.0717871i
$940$		0			0
$941$	26.4926	+	26.4926i	0.863633	+	0.863633i	0.991758	−	0.128125i	$-0.0408958\pi$
$941$	26.4926	+	26.4926i	0.863633	+	0.863633i	−0.128125	+	0.991758i	$0.540896\pi$
$942$		0			0
$943$	−38.2072	+	38.2072i	−1.24420	+	1.24420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			25.8742i			0.840797i	0.907339	+	0.420399i	$0.138110\pi$
$947$			25.8742i			0.840797i	−0.907339	+	0.420399i	$0.861890\pi$
$948$		0			0
$949$	2.67712	+	2.67712i	0.0869031	+	0.0869031i
$950$		0			0
$951$	11.7447			0.380849
$952$		0			0
$953$	−0.934991	+	0.934991i	−0.0302873	+	0.0302873i	−0.722088	−	0.691801i	$-0.756818\pi$
$953$	−0.934991	+	0.934991i	−0.0302873	+	0.0302873i	0.691801	+	0.722088i	$0.256818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−31.0483			−1.00365
$958$		0			0
$959$	−1.23437			−0.0398600
$960$		0			0
$961$	19.2213			0.620041
$962$		0			0
$963$	−11.8803			−0.382836
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.89923	−	1.89923i	0.0610752	−	0.0610752i	−0.675909	−	0.736985i	$-0.736249\pi$
$967$	1.89923	−	1.89923i	0.0610752	−	0.0610752i	0.736985	+	0.675909i	$0.236249\pi$
$968$		0			0
$969$	−8.31672			−0.267172
$970$		0			0
$971$	−11.8787	−	11.8787i	−0.381204	−	0.381204i	0.490332	−	0.871536i	$-0.336876\pi$
$971$	−11.8787	−	11.8787i	−0.381204	−	0.381204i	−0.871536	+	0.490332i	$0.836876\pi$
$972$		0			0
$973$		−	2.89530i		−	0.0928191i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.4462	−	23.4462i	0.750109	−	0.750109i	−0.224390	−	0.974499i	$-0.572039\pi$
$977$	23.4462	−	23.4462i	0.750109	−	0.750109i	0.974499	+	0.224390i	$0.0720390\pi$
$978$		0			0
$979$	−17.2397	−	17.2397i	−0.550984	−	0.550984i
$980$		0			0
$981$	8.77982	−	8.77982i	0.280318	−	0.280318i
$982$		0			0
$983$	5.90331	+	5.90331i	0.188286	+	0.188286i	0.794955	−	0.606669i	$-0.207494\pi$
$983$	5.90331	+	5.90331i	0.188286	+	0.188286i	−0.606669	+	0.794955i	$0.707494\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.51363			0.0481793
$988$		0			0
$989$	−24.9734	+	24.9734i	−0.794107	+	0.794107i
$990$		0			0
$991$		−	8.28808i		−	0.263280i	−0.991298	−	0.131640i	$-0.957976\pi$
$991$		−	8.28808i		−	0.263280i	0.991298	−	0.131640i	$-0.0420242\pi$
$992$		0			0
$993$	−6.90548	−	6.90548i	−0.219139	−	0.219139i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			12.9184i			0.409130i	0.978853	+	0.204565i	$0.0655780\pi$
$997$			12.9184i			0.409130i	−0.978853	+	0.204565i	$0.934422\pi$
$998$		0			0
$999$		−	5.88107i		−	0.186069i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.s.e.943.5		24
4.3	odd	2		400.2.s.e.243.12	yes	24
5.2	odd	4		1600.2.j.e.1007.8		24
5.3	odd	4		1600.2.j.e.1007.5		24
5.4	even	2	inner	1600.2.s.e.943.8		24
16.5	even	4		400.2.j.e.43.7	yes	24
16.11	odd	4		1600.2.j.e.143.5		24
20.3	even	4		400.2.j.e.307.6	yes	24
20.7	even	4		400.2.j.e.307.7	yes	24
20.19	odd	2		400.2.s.e.243.1	yes	24
80.27	even	4	inner	1600.2.s.e.207.5		24
80.37	odd	4		400.2.s.e.107.12	yes	24
80.43	even	4	inner	1600.2.s.e.207.8		24
80.53	odd	4		400.2.s.e.107.1	yes	24
80.59	odd	4		1600.2.j.e.143.8		24
80.69	even	4		400.2.j.e.43.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.6	✓	24	80.69	even	4
400.2.j.e.43.7	yes	24	16.5	even	4
400.2.j.e.307.6	yes	24	20.3	even	4
400.2.j.e.307.7	yes	24	20.7	even	4
400.2.s.e.107.1	yes	24	80.53	odd	4
400.2.s.e.107.12	yes	24	80.37	odd	4
400.2.s.e.243.1	yes	24	20.19	odd	2
400.2.s.e.243.12	yes	24	4.3	odd	2
1600.2.j.e.143.5		24	16.11	odd	4
1600.2.j.e.143.8		24	80.59	odd	4
1600.2.j.e.1007.5		24	5.3	odd	4
1600.2.j.e.1007.8		24	5.2	odd	4
1600.2.s.e.207.5		24	80.27	even	4	inner
1600.2.s.e.207.8		24	80.43	even	4	inner
1600.2.s.e.943.5		24	1.1	even	1	trivial
1600.2.s.e.943.8		24	5.4	even	2	inner

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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)

\(f(q)\)	\(=\)	\(q-0.790153 q^{3} +(-0.139907 + 0.139907i) q^{7} -2.37566 q^{9} +(-2.94816 - 2.94816i) q^{11} -0.235568i q^{13} +(2.06145 - 2.06145i) q^{17} +(2.55293 + 2.55293i) q^{19} +(0.110548 - 0.110548i) q^{21} +(4.62421 + 4.62421i) q^{23} +4.24759 q^{27} +(-6.66417 + 6.66417i) q^{29} -3.43202i q^{31} +(2.32950 + 2.32950i) q^{33} -1.38457i q^{37} +0.186135i q^{39} +8.26242i q^{41} +5.40057i q^{43} +(6.84602 + 6.84602i) q^{47} +6.96085i q^{49} +(-1.62886 + 1.62886i) q^{51} +8.19252 q^{53} +(-2.01720 - 2.01720i) q^{57} +(-4.32313 + 4.32313i) q^{59} +(-9.15188 - 9.15188i) q^{61} +(0.332372 - 0.332372i) q^{63} -5.00083i q^{67} +(-3.65383 - 3.65383i) q^{69} +6.06473 q^{71} +(-11.3646 + 11.3646i) q^{73} +0.824938 q^{77} -4.44776 q^{79} +3.77073 q^{81} -11.4778 q^{83} +(5.26571 - 5.26571i) q^{87} +5.84762 q^{89} +(0.0329576 + 0.0329576i) q^{91} +2.71182i q^{93} +(-0.515382 + 0.515382i) q^{97} +(7.00383 + 7.00383i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \) 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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