LMFDB - Embedded newform 1600.2.s.e.207.9

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			207.9
Character	$\chi$	$=$	1600.207
Dual form			1600.2.s.e.943.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86755 q^{3} +(-0.719989 - 0.719989i) q^{7} +0.487737 q^{9} +O(q^{10})$	$q+1.86755 q^{3} +(-0.719989 - 0.719989i) q^{7} +0.487737 q^{9} +(0.805654 - 0.805654i) q^{11} -5.90473i q^{13} +(5.17145 + 5.17145i) q^{17} +(1.16370 - 1.16370i) q^{19} +(-1.34461 - 1.34461i) q^{21} +(2.30177 - 2.30177i) q^{23} -4.69177 q^{27} +(3.71953 + 3.71953i) q^{29} -9.82775i q^{31} +(1.50460 - 1.50460i) q^{33} +1.71983i q^{37} -11.0274i q^{39} +3.93637i q^{41} -8.82362i q^{43} +(3.21130 - 3.21130i) q^{47} -5.96323i q^{49} +(9.65793 + 9.65793i) q^{51} +8.60748 q^{53} +(2.17326 - 2.17326i) q^{57} +(-5.24522 - 5.24522i) q^{59} +(1.59176 - 1.59176i) q^{61} +(-0.351165 - 0.351165i) q^{63} +9.29532i q^{67} +(4.29867 - 4.29867i) q^{69} -9.33581 q^{71} +(8.57821 + 8.57821i) q^{73} -1.16012 q^{77} +1.70231 q^{79} -10.2253 q^{81} +13.8974 q^{83} +(6.94640 + 6.94640i) q^{87} -4.48540 q^{89} +(-4.25134 + 4.25134i) q^{91} -18.3538i q^{93} +(4.46476 + 4.46476i) q^{97} +(0.392947 - 0.392947i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 40 q^{9}+O(q^{10}) $ 24 * q + 40 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	1.86755			1.07823			0.539115	−	0.842232i	$-0.318759\pi$
$3$	1.86755			1.07823			0.539115	+	0.842232i	$0.318759\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.719989	−	0.719989i	−0.272130	−	0.272130i	0.557827	−	0.829957i	$-0.311635\pi$
$7$	−0.719989	−	0.719989i	−0.272130	−	0.272130i	−0.829957	+	0.557827i	$0.811635\pi$
$8$		0			0
$9$	0.487737			0.162579
$10$		0			0
$11$	0.805654	−	0.805654i	0.242914	−	0.242914i	−0.575141	−	0.818055i	$-0.695053\pi$
$11$	0.805654	−	0.805654i	0.242914	−	0.242914i	0.818055	+	0.575141i	$0.195053\pi$
$12$		0			0
$13$		−	5.90473i		−	1.63768i	−0.574023	−	0.818839i	$-0.694618\pi$
$13$		−	5.90473i		−	1.63768i	0.574023	−	0.818839i	$-0.305382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.17145	+	5.17145i	1.25426	+	1.25426i	0.953792	+	0.300469i	$0.0971430\pi$
$17$	5.17145	+	5.17145i	1.25426	+	1.25426i	0.300469	+	0.953792i	$0.402857\pi$
$18$		0			0
$19$	1.16370	−	1.16370i	0.266971	−	0.266971i	−0.560908	−	0.827878i	$-0.689548\pi$
$19$	1.16370	−	1.16370i	0.266971	−	0.266971i	0.827878	+	0.560908i	$0.189548\pi$
$20$		0			0
$21$	−1.34461	−	1.34461i	−0.293419	−	0.293419i
$22$		0			0
$23$	2.30177	−	2.30177i	0.479953	−	0.479953i	−0.425164	−	0.905116i	$-0.639783\pi$
$23$	2.30177	−	2.30177i	0.479953	−	0.479953i	0.905116	+	0.425164i	$0.139783\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.69177			−0.902932
$28$		0			0
$29$	3.71953	+	3.71953i	0.690699	+	0.690699i	0.962386	−	0.271687i	$-0.0875814\pi$
$29$	3.71953	+	3.71953i	0.690699	+	0.690699i	−0.271687	+	0.962386i	$0.587581\pi$
$30$		0			0
$31$		−	9.82775i		−	1.76512i	−0.470204	−	0.882558i	$-0.655820\pi$
$31$		−	9.82775i		−	1.76512i	0.470204	−	0.882558i	$-0.344180\pi$
$32$		0			0
$33$	1.50460	−	1.50460i	0.261917	−	0.261917i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.71983i			0.282739i	0.989957	+	0.141369i	$0.0451505\pi$
$37$			1.71983i			0.282739i	−0.989957	+	0.141369i	$0.954850\pi$
$38$		0			0
$39$		−	11.0274i		−	1.76579i
$40$		0			0
$41$			3.93637i			0.614758i	0.951587	+	0.307379i	$0.0994519\pi$
$41$			3.93637i			0.614758i	−0.951587	+	0.307379i	$0.900548\pi$
$42$		0			0
$43$		−	8.82362i		−	1.34559i	−0.739829	−	0.672794i	$-0.765094\pi$
$43$		−	8.82362i		−	1.34559i	0.739829	−	0.672794i	$-0.234906\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.21130	−	3.21130i	0.468417	−	0.468417i	−0.432985	−	0.901401i	$-0.642540\pi$
$47$	3.21130	−	3.21130i	0.468417	−	0.468417i	0.901401	+	0.432985i	$0.142540\pi$
$48$		0			0
$49$		−	5.96323i		−	0.851890i
$50$		0			0
$51$	9.65793	+	9.65793i	1.35238	+	1.35238i
$52$		0			0
$53$	8.60748			1.18233			0.591164	−	0.806551i	$-0.298668\pi$
$53$	8.60748			1.18233			0.591164	+	0.806551i	$0.298668\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.17326	−	2.17326i	0.287856	−	0.287856i
$58$		0			0
$59$	−5.24522	−	5.24522i	−0.682870	−	0.682870i	0.277776	−	0.960646i	$-0.410403\pi$
$59$	−5.24522	−	5.24522i	−0.682870	−	0.682870i	−0.960646	+	0.277776i	$0.910403\pi$
$60$		0			0
$61$	1.59176	−	1.59176i	0.203804	−	0.203804i	−0.597824	−	0.801628i	$-0.703968\pi$
$61$	1.59176	−	1.59176i	0.203804	−	0.203804i	0.801628	+	0.597824i	$0.203968\pi$
$62$		0			0
$63$	−0.351165	−	0.351165i	−0.0442426	−	0.0442426i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			9.29532i			1.13560i	0.823165	+	0.567802i	$0.192206\pi$
$67$			9.29532i			1.13560i	−0.823165	+	0.567802i	$0.807794\pi$
$68$		0			0
$69$	4.29867	−	4.29867i	0.517499	−	0.517499i
$70$		0			0
$71$	−9.33581			−1.10796			−0.553979	−	0.832531i	$-0.686891\pi$
$71$	−9.33581			−1.10796			−0.553979	+	0.832531i	$0.686891\pi$
$72$		0			0
$73$	8.57821	+	8.57821i	1.00400	+	1.00400i	0.999992	+	0.00401200i	$0.00127706\pi$
$73$	8.57821	+	8.57821i	1.00400	+	1.00400i	0.00401200	+	0.999992i	$0.498723\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.16012			−0.132208
$78$		0			0
$79$	1.70231			0.191525			0.0957625	−	0.995404i	$-0.469471\pi$
$79$	1.70231			0.191525			0.0957625	+	0.995404i	$0.469471\pi$
$80$		0			0
$81$	−10.2253			−1.13615
$82$		0			0
$83$	13.8974			1.52544			0.762718	−	0.646732i	$-0.223865\pi$
$83$	13.8974			1.52544			0.762718	+	0.646732i	$0.223865\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	6.94640	+	6.94640i	0.744732	+	0.744732i
$88$		0			0
$89$	−4.48540			−0.475452			−0.237726	−	0.971332i	$-0.576402\pi$
$89$	−4.48540			−0.475452			−0.237726	+	0.971332i	$0.576402\pi$
$90$		0			0
$91$	−4.25134	+	4.25134i	−0.445662	+	0.445662i
$92$		0			0
$93$		−	18.3538i		−	1.90320i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.46476	+	4.46476i	0.453327	+	0.453327i	0.896457	−	0.443130i	$-0.146132\pi$
$97$	4.46476	+	4.46476i	0.453327	+	0.453327i	−0.443130	+	0.896457i	$0.646132\pi$
$98$		0			0
$99$	0.392947	−	0.392947i	0.0394927	−	0.0394927i
$100$		0			0
$101$	−9.04273	−	9.04273i	−0.899785	−	0.899785i	0.0956319	−	0.995417i	$-0.469513\pi$
$101$	−9.04273	−	9.04273i	−0.899785	−	0.899785i	−0.995417	+	0.0956319i	$0.969513\pi$
$102$		0			0
$103$	−8.89360	+	8.89360i	−0.876312	+	0.876312i	−0.993151	−	0.116839i	$-0.962724\pi$
$103$	−8.89360	+	8.89360i	−0.876312	+	0.876312i	0.116839	+	0.993151i	$0.462724\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.29532			0.898612			0.449306	−	0.893378i	$-0.351671\pi$
$107$	9.29532			0.898612			0.449306	+	0.893378i	$0.351671\pi$
$108$		0			0
$109$	−4.10635	−	4.10635i	−0.393317	−	0.393317i	0.482551	−	0.875868i	$-0.339710\pi$
$109$	−4.10635	−	4.10635i	−0.393317	−	0.393317i	−0.875868	+	0.482551i	$0.839710\pi$
$110$		0			0
$111$			3.21187i			0.304857i
$112$		0			0
$113$	−7.51147	+	7.51147i	−0.706619	+	0.706619i	−0.965823	−	0.259203i	$-0.916540\pi$
$113$	−7.51147	+	7.51147i	−0.706619	+	0.706619i	0.259203	+	0.965823i	$0.416540\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	2.87996i		−	0.266252i
$118$		0			0
$119$		−	7.44677i		−	0.682644i
$120$		0			0
$121$			9.70184i			0.881986i
$122$		0			0
$123$			7.35136i			0.662850i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.66639	+	7.66639i	−0.680282	+	0.680282i	−0.960064	−	0.279782i	$-0.909738\pi$
$127$	−7.66639	+	7.66639i	−0.680282	+	0.680282i	0.279782	+	0.960064i	$0.409738\pi$
$128$		0			0
$129$		−	16.4785i		−	1.45085i
$130$		0			0
$131$	1.70610	+	1.70610i	0.149062	+	0.149062i	0.777699	−	0.628637i	$-0.216387\pi$
$131$	1.70610	+	1.70610i	0.149062	+	0.149062i	−0.628637	+	0.777699i	$0.716387\pi$
$132$		0			0
$133$	−1.67570			−0.145302
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.69414	−	1.69414i	0.144740	−	0.144740i	−0.631024	−	0.775764i	$-0.717365\pi$
$137$	1.69414	−	1.69414i	0.144740	−	0.144740i	0.775764	+	0.631024i	$0.217365\pi$
$138$		0			0
$139$	2.38206	+	2.38206i	0.202044	+	0.202044i	0.800875	−	0.598831i	$-0.204368\pi$
$139$	2.38206	+	2.38206i	0.202044	+	0.202044i	−0.598831	+	0.800875i	$0.704368\pi$
$140$		0			0
$141$	5.99726	−	5.99726i	0.505061	−	0.505061i
$142$		0			0
$143$	−4.75717	−	4.75717i	−0.397815	−	0.397815i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	11.1366i		−	0.918533i
$148$		0			0
$149$	2.49691	−	2.49691i	0.204555	−	0.204555i	−0.597393	−	0.801948i	$-0.703797\pi$
$149$	2.49691	−	2.49691i	0.204555	−	0.204555i	0.801948	+	0.597393i	$0.203797\pi$
$150$		0			0
$151$	−16.5505			−1.34686			−0.673431	−	0.739250i	$-0.735180\pi$
$151$	−16.5505			−1.34686			−0.673431	+	0.739250i	$0.735180\pi$
$152$		0			0
$153$	2.52231	+	2.52231i	0.203916	+	0.203916i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.07087			−0.404699			−0.202350	−	0.979313i	$-0.564858\pi$
$157$	−5.07087			−0.404699			−0.202350	+	0.979313i	$0.564858\pi$
$158$		0			0
$159$	16.0749			1.27482
$160$		0			0
$161$	−3.31450			−0.261219
$162$		0			0
$163$	15.3065			1.19890			0.599450	−	0.800412i	$-0.295386\pi$
$163$	15.3065			1.19890			0.599450	+	0.800412i	$0.295386\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.78800	+	4.78800i	0.370506	+	0.370506i	0.867662	−	0.497155i	$-0.165622\pi$
$167$	4.78800	+	4.78800i	0.370506	+	0.370506i	−0.497155	+	0.867662i	$0.665622\pi$
$168$		0			0
$169$	−21.8659			−1.68199
$170$		0			0
$171$	0.567579	−	0.567579i	0.0434038	−	0.0434038i
$172$		0			0
$173$			9.32156i			0.708705i	0.935112	+	0.354353i	$0.115299\pi$
$173$			9.32156i			0.708705i	−0.935112	+	0.354353i	$0.884701\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.79570	−	9.79570i	−0.736290	−	0.736290i
$178$		0			0
$179$	−15.6273	+	15.6273i	−1.16804	+	1.16804i	−0.185369	+	0.982669i	$0.559348\pi$
$179$	−15.6273	+	15.6273i	−1.16804	+	1.16804i	−0.982669	+	0.185369i	$0.940652\pi$
$180$		0			0
$181$	17.0056	+	17.0056i	1.26401	+	1.26401i	0.949131	+	0.314882i	$0.101965\pi$
$181$	17.0056	+	17.0056i	1.26401	+	1.26401i	0.314882	+	0.949131i	$0.398035\pi$
$182$		0			0
$183$	2.97268	−	2.97268i	0.219747	−	0.219747i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.33280			0.609355
$188$		0			0
$189$	3.37802	+	3.37802i	0.245715	+	0.245715i
$190$		0			0
$191$		−	3.88531i		−	0.281131i	−0.990071	−	0.140566i	$-0.955108\pi$
$191$		−	3.88531i		−	0.281131i	0.990071	−	0.140566i	$-0.0448921\pi$
$192$		0			0
$193$	3.68299	−	3.68299i	0.265108	−	0.265108i	−0.562018	−	0.827125i	$-0.689975\pi$
$193$	3.68299	−	3.68299i	0.265108	−	0.265108i	0.827125	+	0.562018i	$0.189975\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.2901i			0.875633i	0.899064	+	0.437816i	$0.144248\pi$
$197$			12.2901i			0.875633i	−0.899064	+	0.437816i	$0.855752\pi$
$198$		0			0
$199$			12.6548i			0.897075i	0.893764	+	0.448537i	$0.148055\pi$
$199$			12.6548i			0.897075i	−0.893764	+	0.448537i	$0.851945\pi$
$200$		0			0
$201$			17.3595i			1.22444i
$202$		0			0
$203$		−	5.35604i		−	0.375920i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.12266	−	1.12266i	0.0780302	−	0.0780302i
$208$		0			0
$209$		−	1.87508i		−	0.129702i
$210$		0			0
$211$	−2.57346	−	2.57346i	−0.177164	−	0.177164i	0.612954	−	0.790118i	$-0.289981\pi$
$211$	−2.57346	−	2.57346i	−0.177164	−	0.177164i	−0.790118	+	0.612954i	$0.789981\pi$
$212$		0			0
$213$	−17.4351			−1.19463
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−7.07587	+	7.07587i	−0.480341	+	0.480341i
$218$		0			0
$219$	16.0202	+	16.0202i	1.08255	+	1.08255i
$220$		0			0
$221$	30.5360	−	30.5360i	2.05407	−	2.05407i
$222$		0			0
$223$	−9.55375	−	9.55375i	−0.639766	−	0.639766i	0.310731	−	0.950498i	$-0.399426\pi$
$223$	−9.55375	−	9.55375i	−0.639766	−	0.639766i	−0.950498	+	0.310731i	$0.899426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	12.8481i		−	0.852759i	−0.904544	−	0.426380i	$-0.859789\pi$
$227$		−	12.8481i		−	0.852759i	0.904544	−	0.426380i	$-0.140211\pi$
$228$		0			0
$229$	1.96090	−	1.96090i	0.129580	−	0.129580i	−0.639342	−	0.768922i	$-0.720793\pi$
$229$	1.96090	−	1.96090i	0.129580	−	0.129580i	0.768922	+	0.639342i	$0.220793\pi$
$230$		0			0
$231$	−2.16659			−0.142551
$232$		0			0
$233$	7.39089	+	7.39089i	0.484193	+	0.484193i	0.906468	−	0.422275i	$-0.138768\pi$
$233$	7.39089	+	7.39089i	0.484193	+	0.484193i	−0.422275	+	0.906468i	$0.638768\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.17915			0.206508
$238$		0			0
$239$	−10.7765			−0.697074			−0.348537	−	0.937295i	$-0.613321\pi$
$239$	−10.7765			−0.697074			−0.348537	+	0.937295i	$0.613321\pi$
$240$		0			0
$241$	−14.5670			−0.938344			−0.469172	−	0.883107i	$-0.655448\pi$
$241$	−14.5670			−0.938344			−0.469172	+	0.883107i	$0.655448\pi$
$242$		0			0
$243$	−5.02097			−0.322095
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.87133	−	6.87133i	−0.437212	−	0.437212i
$248$		0			0
$249$	25.9540			1.64477
$250$		0			0
$251$	12.1001	−	12.1001i	0.763750	−	0.763750i	−0.213248	−	0.976998i	$-0.568404\pi$
$251$	12.1001	−	12.1001i	0.763750	−	0.763750i	0.976998	+	0.213248i	$0.0684043\pi$
$252$		0			0
$253$		−	3.70887i		−	0.233174i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.6083	−	17.6083i	−1.09838	−	1.09838i	−0.994601	−	0.103775i	$-0.966908\pi$
$257$	−17.6083	−	17.6083i	−1.09838	−	1.09838i	−0.103775	−	0.994601i	$-0.533092\pi$
$258$		0			0
$259$	1.23826	−	1.23826i	0.0769417	−	0.0769417i
$260$		0			0
$261$	1.81415	+	1.81415i	0.112293	+	0.112293i
$262$		0			0
$263$	−12.1083	+	12.1083i	−0.746630	+	0.746630i	−0.973845	−	0.227214i	$-0.927038\pi$
$263$	−12.1083	+	12.1083i	−0.746630	+	0.746630i	0.227214	+	0.973845i	$0.427038\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−8.37671			−0.512646
$268$		0			0
$269$	6.20149	+	6.20149i	0.378112	+	0.378112i	0.870421	−	0.492309i	$-0.163847\pi$
$269$	6.20149	+	6.20149i	0.378112	+	0.378112i	−0.492309	+	0.870421i	$0.663847\pi$
$270$		0			0
$271$		−	5.11166i		−	0.310511i	−0.987874	−	0.155256i	$-0.950380\pi$
$271$		−	5.11166i		−	0.310511i	0.987874	−	0.155256i	$-0.0496201\pi$
$272$		0			0
$273$	−7.93959	+	7.93959i	−0.480526	+	0.480526i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	4.65720i		−	0.279824i	−0.990164	−	0.139912i	$-0.955318\pi$
$277$		−	4.65720i		−	0.279824i	0.990164	−	0.139912i	$-0.0446820\pi$
$278$		0			0
$279$		−	4.79335i		−	0.286971i
$280$		0			0
$281$		−	26.7402i		−	1.59519i	−0.603194	−	0.797595i	$-0.706105\pi$
$281$		−	26.7402i		−	1.59519i	0.603194	−	0.797595i	$-0.293895\pi$
$282$		0			0
$283$			2.12034i			0.126041i	0.998012	+	0.0630205i	$0.0200733\pi$
$283$			2.12034i			0.126041i	−0.998012	+	0.0630205i	$0.979927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.83414	−	2.83414i	0.167294	−	0.167294i
$288$		0			0
$289$			36.4877i			2.14634i
$290$		0			0
$291$	8.33815	+	8.33815i	0.488791	+	0.488791i
$292$		0			0
$293$	−15.7244			−0.918632			−0.459316	−	0.888273i	$-0.651905\pi$
$293$	−15.7244			−0.918632			−0.459316	+	0.888273i	$0.651905\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.77995	+	3.77995i	−0.219335	+	0.219335i
$298$		0			0
$299$	−13.5914	−	13.5914i	−0.786008	−	0.786008i
$300$		0			0
$301$	−6.35291	+	6.35291i	−0.366175	+	0.366175i
$302$		0			0
$303$	−16.8877	−	16.8877i	−0.970175	−	0.970175i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	5.32655i		−	0.304002i	−0.988380	−	0.152001i	$-0.951428\pi$
$307$		−	5.32655i		−	0.304002i	0.988380	−	0.152001i	$-0.0485717\pi$
$308$		0			0
$309$	−16.6092	+	16.6092i	−0.944866	+	0.944866i
$310$		0			0
$311$	15.8269			0.897459			0.448730	−	0.893668i	$-0.351877\pi$
$311$	15.8269			0.897459			0.448730	+	0.893668i	$0.351877\pi$
$312$		0			0
$313$	−14.4637	−	14.4637i	−0.817539	−	0.817539i	0.168212	−	0.985751i	$-0.446201\pi$
$313$	−14.4637	−	14.4637i	−0.817539	−	0.817539i	−0.985751	+	0.168212i	$0.946201\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.38384			0.302387			0.151193	−	0.988504i	$-0.451688\pi$
$317$	5.38384			0.302387			0.151193	+	0.988504i	$0.451688\pi$
$318$		0			0
$319$	5.99331			0.335561
$320$		0			0
$321$	17.3595			0.968910
$322$		0			0
$323$	12.0360			0.669702
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−7.66882	−	7.66882i	−0.424086	−	0.424086i
$328$		0			0
$329$	−4.62420			−0.254941
$330$		0			0
$331$	5.04895	−	5.04895i	0.277516	−	0.277516i	−0.554601	−	0.832116i	$-0.687129\pi$
$331$	5.04895	−	5.04895i	0.277516	−	0.277516i	0.832116	+	0.554601i	$0.187129\pi$
$332$		0			0
$333$			0.838825i			0.0459673i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.10233	+	8.10233i	0.441362	+	0.441362i	0.892470	−	0.451107i	$-0.148971\pi$
$337$	8.10233	+	8.10233i	0.441362	+	0.441362i	−0.451107	+	0.892470i	$0.648971\pi$
$338$		0			0
$339$	−14.0280	+	14.0280i	−0.761898	+	0.761898i
$340$		0			0
$341$	−7.91777	−	7.91777i	−0.428771	−	0.428771i
$342$		0			0
$343$	−9.33338	+	9.33338i	−0.503955	+	0.503955i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.87019			−0.422494			−0.211247	−	0.977433i	$-0.567752\pi$
$347$	−7.87019			−0.422494			−0.211247	+	0.977433i	$0.567752\pi$
$348$		0			0
$349$	18.7492	+	18.7492i	1.00362	+	1.00362i	0.999993	+	0.00363019i	$0.00115553\pi$
$349$	18.7492	+	18.7492i	1.00362	+	1.00362i	0.00363019	+	0.999993i	$0.498844\pi$
$350$		0			0
$351$			27.7037i			1.47871i
$352$		0			0
$353$	−0.0830593	+	0.0830593i	−0.00442080	+	0.00442080i	−0.709314	−	0.704893i	$-0.750995\pi$
$353$	−0.0830593	+	0.0830593i	−0.00442080	+	0.00442080i	0.704893	+	0.709314i	$0.250995\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	13.9072i		−	0.736047i
$358$		0			0
$359$			35.9409i			1.89689i	0.316941	+	0.948445i	$0.397344\pi$
$359$			35.9409i			1.89689i	−0.316941	+	0.948445i	$0.602656\pi$
$360$		0			0
$361$			16.2916i			0.857453i
$362$		0			0
$363$			18.1187i			0.950983i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−23.6922	+	23.6922i	−1.23672	+	1.23672i	−0.275389	+	0.961333i	$0.588807\pi$
$367$	−23.6922	+	23.6922i	−1.23672	+	1.23672i	−0.961333	+	0.275389i	$0.911193\pi$
$368$		0			0
$369$			1.91991i			0.0999467i
$370$		0			0
$371$	−6.19729	−	6.19729i	−0.321747	−	0.321747i
$372$		0			0
$373$	27.8655			1.44282			0.721410	−	0.692508i	$-0.243494\pi$
$373$	27.8655			1.44282			0.721410	+	0.692508i	$0.243494\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.9628	−	21.9628i	1.13114	−	1.13114i
$378$		0			0
$379$	18.4005	+	18.4005i	0.945168	+	0.945168i	0.998573	−	0.0534045i	$-0.0170073\pi$
$379$	18.4005	+	18.4005i	0.945168	+	0.945168i	−0.0534045	+	0.998573i	$0.517007\pi$
$380$		0			0
$381$	−14.3173	+	14.3173i	−0.733500	+	0.733500i
$382$		0			0
$383$	27.3966	+	27.3966i	1.39990	+	1.39990i	0.800298	+	0.599603i	$0.204675\pi$
$383$	27.3966	+	27.3966i	1.39990	+	1.39990i	0.599603	+	0.800298i	$0.295325\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.30360i		−	0.218764i
$388$		0			0
$389$	−9.05190	+	9.05190i	−0.458950	+	0.458950i	−0.898311	−	0.439361i	$-0.855205\pi$
$389$	−9.05190	+	9.05190i	−0.458950	+	0.458950i	0.439361	+	0.898311i	$0.355205\pi$
$390$		0			0
$391$	23.8070			1.20397
$392$		0			0
$393$	3.18622	+	3.18622i	0.160724	+	0.160724i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−11.6052			−0.582448			−0.291224	−	0.956655i	$-0.594063\pi$
$397$	−11.6052			−0.582448			−0.291224	+	0.956655i	$0.594063\pi$
$398$		0			0
$399$	−3.12945			−0.156668
$400$		0			0
$401$	−9.94759			−0.496759			−0.248379	−	0.968663i	$-0.579898\pi$
$401$	−9.94759			−0.496759			−0.248379	+	0.968663i	$0.579898\pi$
$402$		0			0
$403$	−58.0302			−2.89069
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.38559	+	1.38559i	0.0686811	+	0.0686811i
$408$		0			0
$409$	−24.7129			−1.22198			−0.610988	−	0.791640i	$-0.709228\pi$
$409$	−24.7129			−1.22198			−0.610988	+	0.791640i	$0.709228\pi$
$410$		0			0
$411$	3.16389	−	3.16389i	0.156063	−	0.156063i
$412$		0			0
$413$			7.55300i			0.371659i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.44861	+	4.44861i	0.217849	+	0.217849i
$418$		0			0
$419$	12.9537	−	12.9537i	0.632829	−	0.632829i	−0.315947	−	0.948777i	$-0.602322\pi$
$419$	12.9537	−	12.9537i	0.632829	−	0.632829i	0.948777	+	0.315947i	$0.102322\pi$
$420$		0			0
$421$	−8.99009	−	8.99009i	−0.438150	−	0.438150i	0.453239	−	0.891389i	$-0.350268\pi$
$421$	−8.99009	−	8.99009i	−0.438150	−	0.438150i	−0.891389	+	0.453239i	$0.850268\pi$
$422$		0			0
$423$	1.56627	−	1.56627i	0.0761547	−	0.0761547i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.29210			−0.110922
$428$		0			0
$429$	−8.88425	−	8.88425i	−0.428936	−	0.428936i
$430$		0			0
$431$		−	20.8177i		−	1.00275i	−0.865230	−	0.501376i	$-0.832828\pi$
$431$		−	20.8177i		−	1.00275i	0.865230	−	0.501376i	$-0.167172\pi$
$432$		0			0
$433$	−7.70002	+	7.70002i	−0.370039	+	0.370039i	−0.867491	−	0.497452i	$-0.834269\pi$
$433$	−7.70002	+	7.70002i	−0.370039	+	0.370039i	0.497452	+	0.867491i	$0.334269\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5.35714i		−	0.256267i
$438$		0			0
$439$		−	38.4535i		−	1.83529i	−0.397407	−	0.917643i	$-0.630090\pi$
$439$		−	38.4535i		−	1.83529i	0.397407	−	0.917643i	$-0.369910\pi$
$440$		0			0
$441$		−	2.90849i		−	0.138499i
$442$		0			0
$443$			17.1242i			0.813595i	0.913518	+	0.406797i	$0.133355\pi$
$443$			17.1242i			0.813595i	−0.913518	+	0.406797i	$0.866645\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	4.66310	−	4.66310i	0.220557	−	0.220557i
$448$		0			0
$449$			12.1296i			0.572431i	0.958165	+	0.286215i	$0.0923973\pi$
$449$			12.1296i			0.572431i	−0.958165	+	0.286215i	$0.907603\pi$
$450$		0			0
$451$	3.17135	+	3.17135i	0.149333	+	0.149333i
$452$		0			0
$453$	−30.9089			−1.45223
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.96532	−	3.96532i	0.185490	−	0.185490i	−0.608253	−	0.793743i	$-0.708129\pi$
$457$	3.96532	−	3.96532i	0.185490	−	0.185490i	0.793743	+	0.608253i	$0.208129\pi$
$458$		0			0
$459$	−24.2633	−	24.2633i	−1.13251	−	1.13251i
$460$		0			0
$461$	−25.6620	+	25.6620i	−1.19520	+	1.19520i	−0.219614	+	0.975587i	$0.570480\pi$
$461$	−25.6620	+	25.6620i	−1.19520	+	1.19520i	−0.975587	+	0.219614i	$0.929520\pi$
$462$		0			0
$463$	−3.98789	−	3.98789i	−0.185333	−	0.185333i	0.608342	−	0.793675i	$-0.291835\pi$
$463$	−3.98789	−	3.98789i	−0.185333	−	0.185333i	−0.793675	+	0.608342i	$0.791835\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.8576i			0.502429i	0.967931	+	0.251215i	$0.0808300\pi$
$467$			10.8576i			0.502429i	−0.967931	+	0.251215i	$0.919170\pi$
$468$		0			0
$469$	6.69253	−	6.69253i	0.309032	−	0.309032i
$470$		0			0
$471$	−9.47009			−0.436359
$472$		0			0
$473$	−7.10878	−	7.10878i	−0.326862	−	0.326862i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.19819			0.192222
$478$		0			0
$479$	−0.144583			−0.00660618			−0.00330309	−	0.999995i	$-0.501051\pi$
$479$	−0.144583			−0.00660618			−0.00330309	+	0.999995i	$0.501051\pi$
$480$		0			0
$481$	10.1551			0.463035
$482$		0			0
$483$	−6.18999			−0.281654
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−24.9905	−	24.9905i	−1.13243	−	1.13243i	−0.989772	−	0.142655i	$-0.954436\pi$
$487$	−24.9905	−	24.9905i	−1.13243	−	1.13243i	−0.142655	−	0.989772i	$-0.545564\pi$
$488$		0			0
$489$	28.5857			1.29269
$490$		0			0
$491$	−16.8603	+	16.8603i	−0.760893	+	0.760893i	−0.976484	−	0.215591i	$-0.930832\pi$
$491$	−16.8603	+	16.8603i	−0.760893	+	0.760893i	0.215591	+	0.976484i	$0.430832\pi$
$492$		0			0
$493$			38.4707i			1.73263i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.72168	+	6.72168i	0.301509	+	0.301509i
$498$		0			0
$499$	12.2949	−	12.2949i	0.550397	−	0.550397i	−0.376159	−	0.926555i	$-0.622755\pi$
$499$	12.2949	−	12.2949i	0.550397	−	0.550397i	0.926555	+	0.376159i	$0.122755\pi$
$500$		0			0
$501$	8.94182	+	8.94182i	0.399491	+	0.399491i
$502$		0			0
$503$	−2.55961	+	2.55961i	−0.114128	+	0.114128i	−0.761864	−	0.647737i	$-0.775716\pi$
$503$	−2.55961	+	2.55961i	−0.114128	+	0.114128i	0.647737	+	0.761864i	$0.275716\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−40.8356			−1.81357
$508$		0			0
$509$	−6.59987	−	6.59987i	−0.292534	−	0.292534i	0.545547	−	0.838080i	$-0.316322\pi$
$509$	−6.59987	−	6.59987i	−0.292534	−	0.292534i	−0.838080	+	0.545547i	$0.816322\pi$
$510$		0			0
$511$		−	12.3524i		−	0.546440i
$512$		0			0
$513$	−5.45981	+	5.45981i	−0.241056	+	0.241056i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	5.17440i		−	0.227570i
$518$		0			0
$519$			17.4085i			0.764147i
$520$		0			0
$521$		−	13.9510i		−	0.611204i	−0.952159	−	0.305602i	$-0.901142\pi$
$521$		−	13.9510i		−	0.611204i	0.952159	−	0.305602i	$-0.0988577\pi$
$522$		0			0
$523$		−	24.6076i		−	1.07601i	−0.842941	−	0.538007i	$-0.819178\pi$
$523$		−	24.6076i		−	1.07601i	0.842941	−	0.538007i	$-0.180822\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	50.8237	−	50.8237i	2.21391	−	2.21391i
$528$		0			0
$529$			12.4037i			0.539291i
$530$		0			0
$531$	−2.55829	−	2.55829i	−0.111020	−	0.111020i
$532$		0			0
$533$	23.2432			1.00678
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−29.1847	+	29.1847i	−1.25941	+	1.25941i
$538$		0			0
$539$	−4.80430	−	4.80430i	−0.206936	−	0.206936i
$540$		0			0
$541$	−14.4785	+	14.4785i	−0.622481	+	0.622481i	−0.946165	−	0.323684i	$-0.895078\pi$
$541$	−14.4785	+	14.4785i	−0.622481	+	0.622481i	0.323684	+	0.946165i	$0.395078\pi$
$542$		0			0
$543$	31.7587	+	31.7587i	1.36290	+	1.36290i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			27.5523i			1.17805i	0.808114	+	0.589026i	$0.200488\pi$
$547$			27.5523i			1.17805i	−0.808114	+	0.589026i	$0.799512\pi$
$548$		0			0
$549$	0.776359	−	0.776359i	0.0331342	−	0.0331342i
$550$		0			0
$551$	8.65682			0.368793
$552$		0			0
$553$	−1.22565	−	1.22565i	−0.0521197	−	0.0521197i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.5208			0.996608			0.498304	−	0.867002i	$-0.333956\pi$
$557$	23.5208			0.996608			0.498304	+	0.867002i	$0.333956\pi$
$558$		0			0
$559$	−52.1011			−2.20364
$560$		0			0
$561$	15.5619			0.657024
$562$		0			0
$563$	−11.3970			−0.480327			−0.240163	−	0.970732i	$-0.577201\pi$
$563$	−11.3970			−0.480327			−0.240163	+	0.970732i	$0.577201\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.36212	+	7.36212i	0.309180	+	0.309180i
$568$		0			0
$569$	21.2444			0.890613			0.445307	−	0.895378i	$-0.353095\pi$
$569$	21.2444			0.890613			0.445307	+	0.895378i	$0.353095\pi$
$570$		0			0
$571$	−23.0980	+	23.0980i	−0.966622	+	0.966622i	−0.999461	−	0.0328390i	$-0.989545\pi$
$571$	−23.0980	+	23.0980i	−0.966622	+	0.966622i	0.0328390	+	0.999461i	$0.489545\pi$
$572$		0			0
$573$		−	7.25601i		−	0.303124i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.54721	+	8.54721i	0.355825	+	0.355825i	0.862271	−	0.506446i	$-0.169041\pi$
$577$	8.54721	+	8.54721i	0.355825	+	0.355825i	−0.506446	+	0.862271i	$0.669041\pi$
$578$		0			0
$579$	6.87817	−	6.87817i	0.285847	−	0.285847i
$580$		0			0
$581$	−10.0060	−	10.0060i	−0.415117	−	0.415117i
$582$		0			0
$583$	6.93466	−	6.93466i	0.287204	−	0.287204i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.9980			1.19687			0.598437	−	0.801170i	$-0.295789\pi$
$587$	28.9980			1.19687			0.598437	+	0.801170i	$0.295789\pi$
$588$		0			0
$589$	−11.4365	−	11.4365i	−0.471234	−	0.471234i
$590$		0			0
$591$			22.9523i			0.944133i
$592$		0			0
$593$	−1.73827	+	1.73827i	−0.0713822	+	0.0713822i	−0.741897	−	0.670514i	$-0.766073\pi$
$593$	−1.73827	+	1.73827i	−0.0713822	+	0.0713822i	0.670514	+	0.741897i	$0.266073\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			23.6334i			0.967252i
$598$		0			0
$599$			23.3429i			0.953764i	0.878967	+	0.476882i	$0.158233\pi$
$599$			23.3429i			0.953764i	−0.878967	+	0.476882i	$0.841767\pi$
$600$		0			0
$601$		−	14.2850i		−	0.582697i	−0.956617	−	0.291348i	$-0.905896\pi$
$601$		−	14.2850i		−	0.582697i	0.956617	−	0.291348i	$-0.0941039\pi$
$602$		0			0
$603$			4.53367i			0.184625i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.84325	+	4.84325i	−0.196581	+	0.196581i	−0.798533	−	0.601951i	$-0.794390\pi$
$607$	−4.84325	+	4.84325i	−0.196581	+	0.196581i	0.601951	+	0.798533i	$0.294390\pi$
$608$		0			0
$609$		−	10.0027i		−	0.405328i
$610$		0			0
$611$	−18.9619	−	18.9619i	−0.767116	−	0.767116i
$612$		0			0
$613$	6.37122			0.257331			0.128666	−	0.991688i	$-0.458931\pi$
$613$	6.37122			0.257331			0.128666	+	0.991688i	$0.458931\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.7251	−	13.7251i	0.552553	−	0.552553i	−0.374624	−	0.927177i	$-0.622228\pi$
$617$	13.7251	−	13.7251i	0.552553	−	0.552553i	0.927177	+	0.374624i	$0.122228\pi$
$618$		0			0
$619$	6.92352	+	6.92352i	0.278280	+	0.278280i	0.832422	−	0.554142i	$-0.186954\pi$
$619$	6.92352	+	6.92352i	0.278280	+	0.278280i	−0.554142	+	0.832422i	$0.686954\pi$
$620$		0			0
$621$	−10.7994	+	10.7994i	−0.433365	+	0.433365i
$622$		0			0
$623$	3.22944	+	3.22944i	0.129385	+	0.129385i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	3.50180i		−	0.139848i
$628$		0			0
$629$	−8.89402	+	8.89402i	−0.354628	+	0.354628i
$630$		0			0
$631$	−0.299394			−0.0119187			−0.00595935	−	0.999982i	$-0.501897\pi$
$631$	−0.299394			−0.0119187			−0.00595935	+	0.999982i	$0.501897\pi$
$632$		0			0
$633$	−4.80606	−	4.80606i	−0.191024	−	0.191024i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−35.2113			−1.39512
$638$		0			0
$639$	−4.55342			−0.180130
$640$		0			0
$641$	45.7708			1.80784			0.903920	−	0.427702i	$-0.140677\pi$
$641$	45.7708			1.80784			0.903920	+	0.427702i	$0.140677\pi$
$642$		0			0
$643$	9.26732			0.365467			0.182734	−	0.983162i	$-0.441505\pi$
$643$	9.26732			0.365467			0.182734	+	0.983162i	$0.441505\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.284672	+	0.284672i	0.0111916	+	0.0111916i	0.712680	−	0.701489i	$-0.247481\pi$
$647$	0.284672	+	0.284672i	0.0111916	+	0.0111916i	−0.701489	+	0.712680i	$0.747481\pi$
$648$		0			0
$649$	−8.45167			−0.331757
$650$		0			0
$651$	−13.2145	+	13.2145i	−0.517918	+	0.517918i
$652$		0			0
$653$			11.1970i			0.438173i	0.975705	+	0.219087i	$0.0703078\pi$
$653$			11.1970i			0.438173i	−0.975705	+	0.219087i	$0.929692\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.18391	+	4.18391i	0.163230	+	0.163230i
$658$		0			0
$659$	5.66498	−	5.66498i	0.220676	−	0.220676i	−0.588107	−	0.808783i	$-0.700127\pi$
$659$	5.66498	−	5.66498i	0.220676	−	0.220676i	0.808783	+	0.588107i	$0.200127\pi$
$660$		0			0
$661$	23.3785	+	23.3785i	0.909320	+	0.909320i	0.996217	−	0.0868975i	$-0.0276953\pi$
$661$	23.3785	+	23.3785i	0.909320	+	0.909320i	−0.0868975	+	0.996217i	$0.527695\pi$
$662$		0			0
$663$	57.0275	−	57.0275i	2.21476	−	2.21476i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.1230			0.663006
$668$		0			0
$669$	−17.8421	−	17.8421i	−0.689815	−	0.689815i
$670$		0			0
$671$		−	2.56481i		−	0.0990135i
$672$		0			0
$673$	26.6324	−	26.6324i	1.02660	−	1.02660i	0.0269656	−	0.999636i	$-0.491416\pi$
$673$	26.6324	−	26.6324i	1.02660	−	1.02660i	0.999636	−	0.0269656i	$-0.00858447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	13.5539i		−	0.520918i	−0.965485	−	0.260459i	$-0.916126\pi$
$677$		−	13.5539i		−	0.520918i	0.965485	−	0.260459i	$-0.0838739\pi$
$678$		0			0
$679$		−	6.42915i		−	0.246728i
$680$		0			0
$681$		−	23.9945i		−	0.919470i
$682$		0			0
$683$		−	15.3467i		−	0.587225i	−0.955925	−	0.293613i	$-0.905142\pi$
$683$		−	15.3467i		−	0.587225i	0.955925	−	0.293613i	$-0.0948575\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	3.66207	−	3.66207i	0.139717	−	0.139717i
$688$		0			0
$689$		−	50.8249i		−	1.93627i
$690$		0			0
$691$	−10.6170	−	10.6170i	−0.403891	−	0.403891i	0.475710	−	0.879602i	$-0.342191\pi$
$691$	−10.6170	−	10.6170i	−0.403891	−	0.403891i	−0.879602	+	0.475710i	$0.842191\pi$
$692$		0			0
$693$	−0.565835			−0.0214943
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−20.3567	+	20.3567i	−0.771067	+	0.771067i
$698$		0			0
$699$	13.8028	+	13.8028i	0.522072	+	0.522072i
$700$		0			0
$701$	−5.96737	+	5.96737i	−0.225384	+	0.225384i	−0.810761	−	0.585377i	$-0.800947\pi$
$701$	−5.96737	+	5.96737i	−0.225384	+	0.225384i	0.585377	+	0.810761i	$0.300947\pi$
$702$		0			0
$703$	2.00137	+	2.00137i	0.0754829	+	0.0754829i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			13.0213i			0.489717i
$708$		0			0
$709$	35.3379	−	35.3379i	1.32714	−	1.32714i	0.419287	−	0.907854i	$-0.362280\pi$
$709$	35.3379	−	35.3379i	1.32714	−	1.32714i	0.907854	−	0.419287i	$-0.137720\pi$
$710$		0			0
$711$	0.830280			0.0311379
$712$		0			0
$713$	−22.6212	−	22.6212i	−0.847172	−	0.847172i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−20.1256			−0.751606
$718$		0			0
$719$	37.8418			1.41126			0.705631	−	0.708580i	$-0.250664\pi$
$719$	37.8418			1.41126			0.705631	+	0.708580i	$0.250664\pi$
$720$		0			0
$721$	12.8066			0.476942
$722$		0			0
$723$	−27.2046			−1.01175
$724$		0			0
$725$		0			0
$726$		0			0
$727$	32.3722	+	32.3722i	1.20062	+	1.20062i	0.973978	+	0.226641i	$0.0727745\pi$
$727$	32.3722	+	32.3722i	1.20062	+	1.20062i	0.226641	+	0.973978i	$0.427226\pi$
$728$		0			0
$729$	21.2991			0.788855
$730$		0			0
$731$	45.6309	−	45.6309i	1.68772	−	1.68772i
$732$		0			0
$733$			1.96701i			0.0726532i	0.999340	+	0.0363266i	$0.0115657\pi$
$733$			1.96701i			0.0726532i	−0.999340	+	0.0363266i	$0.988434\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.48881	+	7.48881i	0.275854	+	0.275854i
$738$		0			0
$739$	14.3605	−	14.3605i	0.528261	−	0.528261i	−0.391793	−	0.920054i	$-0.628145\pi$
$739$	14.3605	−	14.3605i	0.528261	−	0.528261i	0.920054	+	0.391793i	$0.128145\pi$
$740$		0			0
$741$	−12.8325	−	12.8325i	−0.471415	−	0.471415i
$742$		0			0
$743$	2.28846	−	2.28846i	0.0839556	−	0.0839556i	−0.663882	−	0.747837i	$-0.731092\pi$
$743$	2.28846	−	2.28846i	0.0839556	−	0.0839556i	0.747837	+	0.663882i	$0.231092\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.77826			0.248004
$748$		0			0
$749$	−6.69253	−	6.69253i	−0.244540	−	0.244540i
$750$		0			0
$751$			23.7064i			0.865058i	0.901620	+	0.432529i	$0.142379\pi$
$751$			23.7064i			0.865058i	−0.901620	+	0.432529i	$0.857621\pi$
$752$		0			0
$753$	22.5975	−	22.5975i	0.823497	−	0.823497i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	30.5469i		−	1.11025i	−0.831768	−	0.555124i	$-0.812671\pi$
$757$		−	30.5469i		−	1.11025i	0.831768	−	0.555124i	$-0.187329\pi$
$758$		0			0
$759$		−	6.92649i		−	0.251416i
$760$		0			0
$761$		−	23.6988i		−	0.859080i	−0.903048	−	0.429540i	$-0.858676\pi$
$761$		−	23.6988i		−	0.859080i	0.903048	−	0.429540i	$-0.141324\pi$
$762$		0			0
$763$			5.91306i			0.214067i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.9716	+	30.9716i	−1.11832	+	1.11832i
$768$		0			0
$769$			2.76629i			0.0997548i	0.998755	+	0.0498774i	$0.0158831\pi$
$769$			2.76629i			0.0997548i	−0.998755	+	0.0498774i	$0.984117\pi$
$770$		0			0
$771$	−32.8844	−	32.8844i	−1.18430	−	1.18430i
$772$		0			0
$773$	36.0726			1.29744			0.648720	−	0.761027i	$-0.275304\pi$
$773$	36.0726			1.29744			0.648720	+	0.761027i	$0.275304\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.31251	−	2.31251i	0.0829608	−	0.0829608i
$778$		0			0
$779$	4.58075	+	4.58075i	0.164122	+	0.164122i
$780$		0			0
$781$	−7.52144	+	7.52144i	−0.269138	+	0.269138i
$782$		0			0
$783$	−17.4512	−	17.4512i	−0.623654	−	0.623654i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	12.1024i		−	0.431402i	−0.976459	−	0.215701i	$-0.930796\pi$
$787$		−	12.1024i		−	0.431402i	0.976459	−	0.215701i	$-0.0692037\pi$
$788$		0			0
$789$	−22.6129	+	22.6129i	−0.805039	+	0.805039i
$790$		0			0
$791$	10.8163			0.384585
$792$		0			0
$793$	−9.39890	−	9.39890i	−0.333765	−	0.333765i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.6614			0.838130			0.419065	−	0.907956i	$-0.362358\pi$
$797$	23.6614			0.838130			0.419065	+	0.907956i	$0.362358\pi$
$798$		0			0
$799$	33.2142			1.17503
$800$		0			0
$801$	−2.18770			−0.0772984
$802$		0			0
$803$	13.8221			0.487773
$804$		0			0
$805$		0			0
$806$		0			0
$807$	11.5816	+	11.5816i	0.407691	+	0.407691i
$808$		0			0
$809$	−23.5574			−0.828235			−0.414117	−	0.910223i	$-0.635910\pi$
$809$	−23.5574			−0.828235			−0.414117	+	0.910223i	$0.635910\pi$
$810$		0			0
$811$	−14.0698	+	14.0698i	−0.494056	+	0.494056i	−0.909581	−	0.415526i	$-0.863598\pi$
$811$	−14.0698	+	14.0698i	−0.494056	+	0.494056i	0.415526	+	0.909581i	$0.363598\pi$
$812$		0			0
$813$		−	9.54627i		−	0.334802i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.2680	−	10.2680i	−0.359233	−	0.359233i
$818$		0			0
$819$	−2.07354	+	2.07354i	−0.0724552	+	0.0724552i
$820$		0			0
$821$	−28.1332	−	28.1332i	−0.981855	−	0.981855i	0.0179834	−	0.999838i	$-0.494275\pi$
$821$	−28.1332	−	28.1332i	−0.981855	−	0.981855i	−0.999838	+	0.0179834i	$0.994275\pi$
$822$		0			0
$823$	27.6068	−	27.6068i	0.962313	−	0.962313i	−0.0370020	−	0.999315i	$-0.511781\pi$
$823$	27.6068	−	27.6068i	0.962313	−	0.962313i	0.999315	+	0.0370020i	$0.0117808\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.79320			−0.236223			−0.118111	−	0.993000i	$-0.537684\pi$
$827$	−6.79320			−0.236223			−0.118111	+	0.993000i	$0.537684\pi$
$828$		0			0
$829$	−25.7474	−	25.7474i	−0.894245	−	0.894245i	0.100675	−	0.994919i	$-0.467900\pi$
$829$	−25.7474	−	25.7474i	−0.894245	−	0.894245i	−0.994919	+	0.100675i	$0.967900\pi$
$830$		0			0
$831$		−	8.69755i		−	0.301715i
$832$		0			0
$833$	30.8385	−	30.8385i	1.06849	−	1.06849i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			46.1096i			1.59378i
$838$		0			0
$839$			11.4280i			0.394538i	0.980349	+	0.197269i	$0.0632073\pi$
$839$			11.4280i			0.394538i	−0.980349	+	0.197269i	$0.936793\pi$
$840$		0			0
$841$		−	1.33022i		−	0.0458696i
$842$		0			0
$843$		−	49.9387i		−	1.71998i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.98522	−	6.98522i	0.240015	−	0.240015i
$848$		0			0
$849$			3.95983i			0.135901i
$850$		0			0
$851$	3.95866	+	3.95866i	0.135701	+	0.135701i
$852$		0			0
$853$	−9.67475			−0.331257			−0.165629	−	0.986188i	$-0.552965\pi$
$853$	−9.67475			−0.331257			−0.165629	+	0.986188i	$0.552965\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.5831	−	26.5831i	0.908061	−	0.908061i	−0.0880544	−	0.996116i	$-0.528065\pi$
$857$	26.5831	−	26.5831i	0.908061	−	0.908061i	0.996116	+	0.0880544i	$0.0280649\pi$
$858$		0			0
$859$	1.58572	+	1.58572i	0.0541040	+	0.0541040i	0.733641	−	0.679537i	$-0.237819\pi$
$859$	1.58572	+	1.58572i	0.0541040	+	0.0541040i	−0.679537	+	0.733641i	$0.737819\pi$
$860$		0			0
$861$	5.29290	−	5.29290i	0.180382	−	0.180382i
$862$		0			0
$863$	9.59115	+	9.59115i	0.326486	+	0.326486i	0.851249	−	0.524762i	$-0.175846\pi$
$863$	9.59115	+	9.59115i	0.326486	+	0.326486i	−0.524762	+	0.851249i	$0.675846\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			68.1426i			2.31424i
$868$		0			0
$869$	1.37147	−	1.37147i	0.0465241	−	0.0465241i
$870$		0			0
$871$	54.8864			1.85975
$872$		0			0
$873$	2.17763	+	2.17763i	0.0737015	+	0.0737015i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	32.7116			1.10459			0.552295	−	0.833649i	$-0.313752\pi$
$877$	32.7116			1.10459			0.552295	+	0.833649i	$0.313752\pi$
$878$		0			0
$879$	−29.3661			−0.990496
$880$		0			0
$881$	9.42337			0.317481			0.158741	−	0.987320i	$-0.449257\pi$
$881$	9.42337			0.317481			0.158741	+	0.987320i	$0.449257\pi$
$882$		0			0
$883$	13.1729			0.443304			0.221652	−	0.975126i	$-0.428855\pi$
$883$	13.1729			0.443304			0.221652	+	0.975126i	$0.428855\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.04550	−	7.04550i	−0.236565	−	0.236565i	0.578861	−	0.815426i	$-0.303497\pi$
$887$	−7.04550	−	7.04550i	−0.236565	−	0.236565i	−0.815426	+	0.578861i	$0.803497\pi$
$888$		0			0
$889$	11.0394			0.370251
$890$		0			0
$891$	−8.23808	+	8.23808i	−0.275986	+	0.275986i
$892$		0			0
$893$		−	7.47397i		−	0.250107i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−25.3825	−	25.3825i	−0.847497	−	0.847497i
$898$		0			0
$899$	36.5546	−	36.5546i	1.21916	−	1.21916i
$900$		0			0
$901$	44.5131	+	44.5131i	1.48295	+	1.48295i
$902$		0			0
$903$	−11.8644	+	11.8644i	−0.394821	+	0.394821i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	45.4995			1.51079			0.755394	−	0.655271i	$-0.227446\pi$
$907$	45.4995			1.51079			0.755394	+	0.655271i	$0.227446\pi$
$908$		0			0
$909$	−4.41047	−	4.41047i	−0.146286	−	0.146286i
$910$		0			0
$911$			41.2904i			1.36801i	0.729477	+	0.684005i	$0.239763\pi$
$911$			41.2904i			1.36801i	−0.729477	+	0.684005i	$0.760237\pi$
$912$		0			0
$913$	11.1965	−	11.1965i	0.370549	−	0.370549i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.45674i		−	0.0811288i
$918$		0			0
$919$			20.1715i			0.665397i	0.943033	+	0.332698i	$0.107959\pi$
$919$			20.1715i			0.665397i	−0.943033	+	0.332698i	$0.892041\pi$
$920$		0			0
$921$		−	9.94759i		−	0.327784i
$922$		0			0
$923$			55.1255i			1.81448i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−4.33774	+	4.33774i	−0.142470	+	0.142470i
$928$		0			0
$929$		−	8.18969i		−	0.268695i	−0.990934	−	0.134348i	$-0.957106\pi$
$929$		−	8.18969i		−	0.268695i	0.990934	−	0.134348i	$-0.0428939\pi$
$930$		0			0
$931$	−6.93940	−	6.93940i	−0.227430	−	0.227430i
$932$		0			0
$933$	29.5574			0.967667
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−33.0511	+	33.0511i	−1.07973	+	1.07973i	−0.0831985	+	0.996533i	$0.526514\pi$
$937$	−33.0511	+	33.0511i	−1.07973	+	1.07973i	−0.996533	+	0.0831985i	$0.973486\pi$
$938$		0			0
$939$	−27.0117	−	27.0117i	−0.881495	−	0.881495i
$940$		0			0
$941$	26.6371	−	26.6371i	0.868345	−	0.868345i	−0.123944	−	0.992289i	$-0.539554\pi$
$941$	26.6371	−	26.6371i	0.868345	−	0.868345i	0.992289	+	0.123944i	$0.0395544\pi$
$942$		0			0
$943$	9.06063	+	9.06063i	0.295055	+	0.295055i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	37.5777i		−	1.22111i	−0.791973	−	0.610556i	$-0.790946\pi$
$947$		−	37.5777i		−	1.22111i	0.791973	−	0.610556i	$-0.209054\pi$
$948$		0			0
$949$	50.6521	−	50.6521i	1.64424	−	1.64424i
$950$		0			0
$951$	10.0546			0.326042
$952$		0			0
$953$	7.96284	+	7.96284i	0.257942	+	0.257942i	0.824217	−	0.566275i	$-0.191616\pi$
$953$	7.96284	+	7.96284i	0.257942	+	0.257942i	−0.566275	+	0.824217i	$0.691616\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	11.1928			0.361812
$958$		0			0
$959$	−2.43952			−0.0787763
$960$		0			0
$961$	−65.5846			−2.11563
$962$		0			0
$963$	4.53367			0.146095
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−16.6144	−	16.6144i	−0.534282	−	0.534282i	0.387562	−	0.921844i	$-0.373317\pi$
$967$	−16.6144	−	16.6144i	−0.534282	−	0.534282i	−0.921844	+	0.387562i	$0.873317\pi$
$968$		0			0
$969$	22.4778			0.722092
$970$		0			0
$971$	17.9269	−	17.9269i	0.575301	−	0.575301i	−0.358304	−	0.933605i	$-0.616645\pi$
$971$	17.9269	−	17.9269i	0.575301	−	0.575301i	0.933605	+	0.358304i	$0.116645\pi$
$972$		0			0
$973$		−	3.43011i		−	0.109964i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−35.0721	−	35.0721i	−1.12206	−	1.12206i	−0.991432	−	0.130625i	$-0.958302\pi$
$977$	−35.0721	−	35.0721i	−1.12206	−	1.12206i	−0.130625	−	0.991432i	$-0.541698\pi$
$978$		0			0
$979$	−3.61368	+	3.61368i	−0.115494	+	0.115494i
$980$		0			0
$981$	−2.00282	−	2.00282i	−0.0639451	−	0.0639451i
$982$		0			0
$983$	24.9265	−	24.9265i	0.795033	−	0.795033i	−0.187274	−	0.982308i	$-0.559965\pi$
$983$	24.9265	−	24.9265i	0.795033	−	0.795033i	0.982308	+	0.187274i	$0.0599653\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−8.63592			−0.274884
$988$		0			0
$989$	−20.3100	−	20.3100i	−0.645819	−	0.645819i
$990$		0			0
$991$			10.7686i			0.342076i	0.985264	+	0.171038i	$0.0547121\pi$
$991$			10.7686i			0.342076i	−0.985264	+	0.171038i	$0.945288\pi$
$992$		0			0
$993$	9.42916	−	9.42916i	0.299226	−	0.299226i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	10.3858i		−	0.328922i	−0.986384	−	0.164461i	$-0.947412\pi$
$997$		−	10.3858i		−	0.328922i	0.986384	−	0.164461i	$-0.0525885\pi$
$998$		0			0
$999$		−	8.06906i		−	0.255294i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.s.e.207.9		24
4.3	odd	2		400.2.s.e.107.2	yes	24
5.2	odd	4		1600.2.j.e.143.4		24
5.3	odd	4		1600.2.j.e.143.9		24
5.4	even	2	inner	1600.2.s.e.207.4		24
16.3	odd	4		1600.2.j.e.1007.4		24
16.13	even	4		400.2.j.e.307.8	yes	24
20.3	even	4		400.2.j.e.43.8	yes	24
20.7	even	4		400.2.j.e.43.5	✓	24
20.19	odd	2		400.2.s.e.107.11	yes	24
80.3	even	4	inner	1600.2.s.e.943.9		24
80.13	odd	4		400.2.s.e.243.2	yes	24
80.19	odd	4		1600.2.j.e.1007.9		24
80.29	even	4		400.2.j.e.307.5	yes	24
80.67	even	4	inner	1600.2.s.e.943.4		24
80.77	odd	4		400.2.s.e.243.11	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.5	✓	24	20.7	even	4
400.2.j.e.43.8	yes	24	20.3	even	4
400.2.j.e.307.5	yes	24	80.29	even	4
400.2.j.e.307.8	yes	24	16.13	even	4
400.2.s.e.107.2	yes	24	4.3	odd	2
400.2.s.e.107.11	yes	24	20.19	odd	2
400.2.s.e.243.2	yes	24	80.13	odd	4
400.2.s.e.243.11	yes	24	80.77	odd	4
1600.2.j.e.143.4		24	5.2	odd	4
1600.2.j.e.143.9		24	5.3	odd	4
1600.2.j.e.1007.4		24	16.3	odd	4
1600.2.j.e.1007.9		24	80.19	odd	4
1600.2.s.e.207.4		24	5.4	even	2	inner
1600.2.s.e.207.9		24	1.1	even	1	trivial
1600.2.s.e.943.4		24	80.67	even	4	inner
1600.2.s.e.943.9		24	80.3	even	4	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)

\(f(q)\)	\(=\)	\(q+1.86755 q^{3} +(-0.719989 - 0.719989i) q^{7} +0.487737 q^{9} +O(q^{10})\)	\(q+1.86755 q^{3} +(-0.719989 - 0.719989i) q^{7} +0.487737 q^{9} +(0.805654 - 0.805654i) q^{11} -5.90473i q^{13} +(5.17145 + 5.17145i) q^{17} +(1.16370 - 1.16370i) q^{19} +(-1.34461 - 1.34461i) q^{21} +(2.30177 - 2.30177i) q^{23} -4.69177 q^{27} +(3.71953 + 3.71953i) q^{29} -9.82775i q^{31} +(1.50460 - 1.50460i) q^{33} +1.71983i q^{37} -11.0274i q^{39} +3.93637i q^{41} -8.82362i q^{43} +(3.21130 - 3.21130i) q^{47} -5.96323i q^{49} +(9.65793 + 9.65793i) q^{51} +8.60748 q^{53} +(2.17326 - 2.17326i) q^{57} +(-5.24522 - 5.24522i) q^{59} +(1.59176 - 1.59176i) q^{61} +(-0.351165 - 0.351165i) q^{63} +9.29532i q^{67} +(4.29867 - 4.29867i) q^{69} -9.33581 q^{71} +(8.57821 + 8.57821i) q^{73} -1.16012 q^{77} +1.70231 q^{79} -10.2253 q^{81} +13.8974 q^{83} +(6.94640 + 6.94640i) q^{87} -4.48540 q^{89} +(-4.25134 + 4.25134i) q^{91} -18.3538i q^{93} +(4.46476 + 4.46476i) q^{97} +(0.392947 - 0.392947i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{9}+O(q^{10}) \) 24 * q + 40 * q^9	\( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \) 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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