LMFDB - Embedded newform 1920.2.b.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(191,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("1920.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.b (of order $2$, degree $1$, 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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$-0.252709 + 0.252709i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.72585 - 0.146426i) q^{3} +1.00000 q^{5} +1.45170i q^{7} +(2.95712 + 0.505418i) q^{9} +1.01084i q^{11} -3.74455i q^{13} +(-1.72585 - 0.146426i) q^{15} -4.75539i q^{17} -2.16969 q^{19} +(0.212567 - 2.50542i) q^{21} +3.30369 q^{23} +1.00000 q^{25} +(-5.02954 - 1.30527i) q^{27} -3.59654 q^{29} +2.75539i q^{31} +(0.148013 - 1.74455i) q^{33} +1.45170i q^{35} -6.06225i q^{37} +(-0.548299 + 6.46254i) q^{39} +7.32853i q^{41} -0.0374041 q^{43} +(2.95712 + 0.505418i) q^{45} +5.62139 q^{47} +4.89257 q^{49} +(-0.696312 + 8.20709i) q^{51} +4.31770 q^{53} +1.01084i q^{55} +(3.74455 + 0.317698i) q^{57} -11.9142i q^{59} -5.30686i q^{61} +(-0.733716 + 4.29285i) q^{63} -3.74455i q^{65} +8.03740 q^{67} +(-5.70167 - 0.483745i) q^{69} -12.4142 q^{71} +10.3394 q^{73} +(-1.72585 - 0.146426i) q^{75} -1.46743 q^{77} -11.3411i q^{79} +(8.48910 + 2.98916i) q^{81} +11.1963i q^{83} -4.75539i q^{85} +(6.20709 + 0.526626i) q^{87} +6.31770i q^{89} +5.43597 q^{91} +(0.403460 - 4.75539i) q^{93} -2.16969 q^{95} +9.48910 q^{97} +(-0.510895 + 2.98916i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 8 q^{5} - 4 q^{9} - 2 q^{15} + 8 q^{19} + 4 q^{21} + 12 q^{23} + 8 q^{25} - 14 q^{27} - 8 q^{29} - 8 q^{33} - 28 q^{39} + 36 q^{43} - 4 q^{45} - 4 q^{47} - 20 q^{51} + 16 q^{63} + 28 q^{67}+ \cdots - 64 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 + 8 * q^5 - 4 * q^9 - 2 * q^15 + 8 * q^19 + 4 * q^21 + 12 * q^23 + 8 * q^25 - 14 * q^27 - 8 * q^29 - 8 * q^33 - 28 * q^39 + 36 * q^43 - 4 * q^45 - 4 * q^47 - 20 * q^51 + 16 * q^63 + 28 * q^67 + 12 * q^69 + 24 * q^71 + 32 * q^73 - 2 * q^75 + 32 * q^77 + 8 * q^81 - 12 * q^87 + 40 * q^91 + 24 * q^93 + 8 * q^95 + 16 * q^97 - 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−1.72585	−	0.146426i	−0.996420	−	0.0845390i
$3$	−1.72585	−	0.146426i	−0.996420	−	0.0845390i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			1.45170i			0.548691i	0.961631	+	0.274346i	$0.0884612\pi$
$7$			1.45170i			0.548691i	−0.961631	+	0.274346i	$0.911539\pi$
$8$		0			0
$9$	2.95712	+	0.505418i	0.985706	+	0.168473i
$10$		0			0
$11$			1.01084i			0.304779i	0.988321	+	0.152389i	$0.0486968\pi$
$11$			1.01084i			0.304779i	−0.988321	+	0.152389i	$0.951303\pi$
$12$		0			0
$13$		−	3.74455i		−	1.03855i	−0.854607	−	0.519276i	$-0.826202\pi$
$13$		−	3.74455i		−	1.03855i	0.854607	−	0.519276i	$-0.173798\pi$
$14$		0			0
$15$	−1.72585	−	0.146426i	−0.445613	−	0.0378070i
$16$		0			0
$17$		−	4.75539i		−	1.15335i	−0.816973	−	0.576676i	$-0.804350\pi$
$17$		−	4.75539i		−	1.15335i	0.816973	−	0.576676i	$-0.195650\pi$
$18$		0			0
$19$	−2.16969			−0.497760			−0.248880	−	0.968534i	$-0.580062\pi$
$19$	−2.16969			−0.497760			−0.248880	+	0.968534i	$0.580062\pi$
$20$		0			0
$21$	0.212567	−	2.50542i	0.0463858	−	0.546727i
$22$		0			0
$23$	3.30369			0.688867			0.344433	−	0.938811i	$-0.388071\pi$
$23$	3.30369			0.688867			0.344433	+	0.938811i	$0.388071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.02954	−	1.30527i	−0.967935	−	0.251200i
$28$		0			0
$29$	−3.59654			−0.667861			−0.333930	−	0.942598i	$-0.608375\pi$
$29$	−3.59654			−0.667861			−0.333930	+	0.942598i	$0.608375\pi$
$30$		0			0
$31$			2.75539i			0.494882i	0.968903	+	0.247441i	$0.0795897\pi$
$31$			2.75539i			0.494882i	−0.968903	+	0.247441i	$0.920410\pi$
$32$		0			0
$33$	0.148013	−	1.74455i	0.0257657	−	0.303688i
$34$		0			0
$35$			1.45170i			0.245382i
$36$		0			0
$37$		−	6.06225i		−	0.996628i	−0.866997	−	0.498314i	$-0.833953\pi$
$37$		−	6.06225i		−	0.996628i	0.866997	−	0.498314i	$-0.166047\pi$
$38$		0			0
$39$	−0.548299	+	6.46254i	−0.0877982	+	1.03483i
$40$		0			0
$41$			7.32853i			1.14452i	0.820071	+	0.572262i	$0.193934\pi$
$41$			7.32853i			1.14452i	−0.820071	+	0.572262i	$0.806066\pi$
$42$		0			0
$43$	−0.0374041			−0.00570408			−0.00285204	−	0.999996i	$-0.500908\pi$
$43$	−0.0374041			−0.00570408			−0.00285204	+	0.999996i	$0.500908\pi$
$44$		0			0
$45$	2.95712	+	0.505418i	0.440821	+	0.0753433i
$46$		0			0
$47$	5.62139			0.819963			0.409982	−	0.912094i	$-0.365535\pi$
$47$	5.62139			0.819963			0.409982	+	0.912094i	$0.365535\pi$
$48$		0			0
$49$	4.89257			0.698938
$50$		0			0
$51$	−0.696312	+	8.20709i	−0.0975032	+	1.14922i
$52$		0			0
$53$	4.31770			0.593081			0.296541	−	0.955020i	$-0.404167\pi$
$53$	4.31770			0.593081			0.296541	+	0.955020i	$0.404167\pi$
$54$		0			0
$55$			1.01084i			0.136301i
$56$		0			0
$57$	3.74455	+	0.317698i	0.495978	+	0.0420801i
$58$		0			0
$59$		−	11.9142i		−	1.55110i	−0.631285	−	0.775551i	$-0.717472\pi$
$59$		−	11.9142i		−	1.55110i	0.631285	−	0.775551i	$-0.282528\pi$
$60$		0			0
$61$		−	5.30686i		−	0.679474i	−0.940520	−	0.339737i	$-0.889662\pi$
$61$		−	5.30686i		−	0.679474i	0.940520	−	0.339737i	$-0.110338\pi$
$62$		0			0
$63$	−0.733716	+	4.29285i	−0.0924395	+	0.540848i
$64$		0			0
$65$		−	3.74455i		−	0.464455i
$66$		0			0
$67$	8.03740			0.981925			0.490963	−	0.871181i	$-0.336645\pi$
$67$	8.03740			0.981925			0.490963	+	0.871181i	$0.336645\pi$
$68$		0			0
$69$	−5.70167	−	0.483745i	−0.686401	−	0.0582361i
$70$		0			0
$71$	−12.4142			−1.47329			−0.736646	−	0.676279i	$-0.763592\pi$
$71$	−12.4142			−1.47329			−0.736646	+	0.676279i	$0.763592\pi$
$72$		0			0
$73$	10.3394			1.21013			0.605066	−	0.796175i	$-0.293147\pi$
$73$	10.3394			1.21013			0.605066	+	0.796175i	$0.293147\pi$
$74$		0			0
$75$	−1.72585	−	0.146426i	−0.199284	−	0.0169078i
$76$		0			0
$77$	−1.46743			−0.167229
$78$		0			0
$79$		−	11.3411i		−	1.27597i	−0.770048	−	0.637986i	$-0.779768\pi$
$79$		−	11.3411i		−	1.27597i	0.770048	−	0.637986i	$-0.220232\pi$
$80$		0			0
$81$	8.48910	+	2.98916i	0.943234	+	0.332129i
$82$		0			0
$83$			11.1963i			1.22895i	0.788937	+	0.614474i	$0.210632\pi$
$83$			11.1963i			1.22895i	−0.788937	+	0.614474i	$0.789368\pi$
$84$		0			0
$85$		−	4.75539i		−	0.515794i
$86$		0			0
$87$	6.20709	+	0.526626i	0.665470	+	0.0564603i
$88$		0			0
$89$			6.31770i			0.669675i	0.942276	+	0.334837i	$0.108681\pi$
$89$			6.31770i			0.669675i	−0.942276	+	0.334837i	$0.891319\pi$
$90$		0			0
$91$	5.43597			0.569844
$92$		0			0
$93$	0.403460	−	4.75539i	0.0418369	−	0.493111i
$94$		0			0
$95$	−2.16969			−0.222605
$96$		0			0
$97$	9.48910			0.963473			0.481736	−	0.876316i	$-0.340006\pi$
$97$	9.48910			0.963473			0.481736	+	0.876316i	$0.340006\pi$
$98$		0			0
$99$	−0.510895	+	2.98916i	−0.0513469	+	0.300422i
$100$		0			0
$101$	−13.4033			−1.33368			−0.666841	−	0.745200i	$-0.732354\pi$
$101$	−13.4033			−1.33368			−0.666841	+	0.745200i	$0.732354\pi$
$102$		0			0
$103$		−	11.4734i		−	1.13051i	−0.824918	−	0.565253i	$-0.808779\pi$
$103$		−	11.4734i		−	1.13051i	0.824918	−	0.565253i	$-0.191221\pi$
$104$		0			0
$105$	0.212567	−	2.50542i	0.0207444	−	0.244504i
$106$		0			0
$107$		−	19.0247i		−	1.83919i	−0.392868	−	0.919595i	$-0.628517\pi$
$107$		−	19.0247i		−	1.83919i	0.392868	−	0.919595i	$-0.371483\pi$
$108$		0			0
$109$		−	16.8393i		−	1.61291i	−0.591293	−	0.806457i	$-0.701382\pi$
$109$		−	16.8393i		−	1.61291i	0.591293	−	0.806457i	$-0.298618\pi$
$110$		0			0
$111$	−0.887670	+	10.4625i	−0.0842539	+	0.993060i
$112$		0			0
$113$		−	14.2228i		−	1.33797i	−0.743276	−	0.668985i	$-0.766729\pi$
$113$		−	14.2228i		−	1.33797i	0.743276	−	0.668985i	$-0.233271\pi$
$114$		0			0
$115$	3.30369			0.308071
$116$		0			0
$117$	1.89257	−	11.0731i	0.174968	−	1.02371i
$118$		0			0
$119$	6.90340			0.632834
$120$		0			0
$121$	9.97821			0.907110
$122$		0			0
$123$	1.07309	−	12.6480i	0.0967570	−	1.14043i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	13.7911i		−	1.22376i	−0.790951	−	0.611880i	$-0.790414\pi$
$127$		−	13.7911i		−	1.22376i	0.790951	−	0.611880i	$-0.209586\pi$
$128$		0			0
$129$	0.0645539	+	0.00547693i	0.00568366	+	0.000482217i
$130$		0			0
$131$			17.1353i			1.49712i	0.663066	+	0.748561i	$0.269255\pi$
$131$			17.1353i			1.49712i	−0.663066	+	0.748561i	$0.730745\pi$
$132$		0			0
$133$		−	3.14973i		−	0.273117i
$134$		0			0
$135$	−5.02954	−	1.30527i	−0.432874	−	0.112340i
$136$		0			0
$137$			9.26628i			0.791672i	0.918321	+	0.395836i	$0.129545\pi$
$137$			9.26628i			0.791672i	−0.918321	+	0.395836i	$0.870455\pi$
$138$		0			0
$139$	10.4657			0.887690			0.443845	−	0.896104i	$-0.353614\pi$
$139$	10.4657			0.887690			0.443845	+	0.896104i	$0.353614\pi$
$140$		0			0
$141$	−9.70167	−	0.823116i	−0.817028	−	0.0693189i
$142$		0			0
$143$	3.78513			0.316528
$144$		0			0
$145$	−3.59654			−0.298676
$146$		0			0
$147$	−8.44384	−	0.716398i	−0.696436	−	0.0590875i
$148$		0			0
$149$	11.8501			0.970802			0.485401	−	0.874292i	$-0.338674\pi$
$149$	11.8501			0.970802			0.485401	+	0.874292i	$0.338674\pi$
$150$		0			0
$151$		−	22.5405i		−	1.83432i	−0.398518	−	0.917161i	$-0.630475\pi$
$151$		−	22.5405i		−	1.83432i	0.398518	−	0.917161i	$-0.369525\pi$
$152$		0			0
$153$	2.40346	−	14.0623i	0.194308	−	1.13687i
$154$		0			0
$155$			2.75539i			0.221318i
$156$		0			0
$157$			7.57303i			0.604393i	0.953246	+	0.302197i	$0.0977200\pi$
$157$			7.57303i			0.604393i	−0.953246	+	0.302197i	$0.902280\pi$
$158$		0			0
$159$	−7.45170	−	0.632223i	−0.590958	−	0.0501385i
$160$		0			0
$161$			4.79597i			0.377975i
$162$		0			0
$163$	11.5265			0.902826			0.451413	−	0.892315i	$-0.350920\pi$
$163$	11.5265			0.902826			0.451413	+	0.892315i	$0.350920\pi$
$164$		0			0
$165$	0.148013	−	1.74455i	0.0115228	−	0.135813i
$166$		0			0
$167$	6.79279			0.525642			0.262821	−	0.964845i	$-0.415347\pi$
$167$	6.79279			0.525642			0.262821	+	0.964845i	$0.415347\pi$
$168$		0			0
$169$	−1.02167			−0.0785902
$170$		0			0
$171$	−6.41602	−	1.09660i	−0.490645	−	0.0838590i
$172$		0			0
$173$	25.6136			1.94737			0.973683	−	0.227906i	$-0.0731878\pi$
$173$	25.6136			1.94737			0.973683	+	0.227906i	$0.0731878\pi$
$174$		0			0
$175$			1.45170i			0.109738i
$176$		0			0
$177$	−1.74455	+	20.5622i	−0.131129	+	1.54555i
$178$		0			0
$179$			1.89257i			0.141457i	0.997496	+	0.0707285i	$0.0225324\pi$
$179$			1.89257i			0.141457i	−0.997496	+	0.0707285i	$0.977468\pi$
$180$		0			0
$181$			5.17141i			0.384388i	0.981357	+	0.192194i	$0.0615602\pi$
$181$			5.17141i			0.384388i	−0.981357	+	0.192194i	$0.938440\pi$
$182$		0			0
$183$	−0.777062	+	9.15885i	−0.0574421	+	0.677042i
$184$		0			0
$185$		−	6.06225i		−	0.445706i
$186$		0			0
$187$	4.80692			0.351517
$188$		0			0
$189$	1.89487	−	7.30138i	0.137831	−	0.531098i
$190$		0			0
$191$	−19.5388			−1.41378			−0.706889	−	0.707325i	$-0.749902\pi$
$191$	−19.5388			−1.41378			−0.706889	+	0.707325i	$0.749902\pi$
$192$		0			0
$193$	−0.853709			−0.0614513			−0.0307257	−	0.999528i	$-0.509782\pi$
$193$	−0.853709			−0.0614513			−0.0307257	+	0.999528i	$0.509782\pi$
$194$		0			0
$195$	−0.548299	+	6.46254i	−0.0392645	+	0.462792i
$196$		0			0
$197$	−19.3393			−1.37786			−0.688932	−	0.724826i	$-0.741920\pi$
$197$	−19.3393			−1.37786			−0.688932	+	0.724826i	$0.741920\pi$
$198$		0			0
$199$			1.31942i			0.0935312i	0.998906	+	0.0467656i	$0.0148914\pi$
$199$			1.31942i			0.0935312i	−0.998906	+	0.0467656i	$0.985109\pi$
$200$		0			0
$201$	−13.8714	−	1.17688i	−0.978410	−	0.0830110i
$202$		0			0
$203$		−	5.22110i		−	0.366449i
$204$		0			0
$205$			7.32853i			0.511847i
$206$		0			0
$207$	9.76940	+	1.66974i	0.679020	+	0.116055i
$208$		0			0
$209$		−	2.19320i		−	0.151707i
$210$		0			0
$211$	17.9485			1.23562			0.617812	−	0.786326i	$-0.288019\pi$
$211$	17.9485			1.23562			0.617812	+	0.786326i	$0.288019\pi$
$212$		0			0
$213$	21.4250	+	1.81776i	1.46802	+	0.124551i
$214$		0			0
$215$	−0.0374041			−0.00255094
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	−17.8442	−	1.51395i	−1.20580	−	0.102303i
$220$		0			0
$221$	−17.8068			−1.19782
$222$		0			0
$223$		−	3.94092i		−	0.263904i	−0.991256	−	0.131952i	$-0.957876\pi$
$223$		−	3.94092i		−	0.263904i	0.991256	−	0.131952i	$-0.0421244\pi$
$224$		0			0
$225$	2.95712	+	0.505418i	0.197141	+	0.0336945i
$226$		0			0
$227$			6.27118i			0.416233i	0.978104	+	0.208116i	$0.0667333\pi$
$227$			6.27118i			0.416233i	−0.978104	+	0.208116i	$0.933267\pi$
$228$		0			0
$229$			19.3176i			1.27654i	0.769812	+	0.638271i	$0.220350\pi$
$229$			19.3176i			1.27654i	−0.769812	+	0.638271i	$0.779650\pi$
$230$		0			0
$231$	2.53257	+	0.214870i	0.166631	+	0.0141374i
$232$		0			0
$233$		−	18.9016i		−	1.23828i	−0.785280	−	0.619141i	$-0.787481\pi$
$233$		−	18.9016i		−	1.23828i	0.785280	−	0.619141i	$-0.212519\pi$
$234$		0			0
$235$	5.62139			0.366699
$236$		0			0
$237$	−1.66063	+	19.5730i	−0.107869	+	1.27140i
$238$		0			0
$239$	20.3394			1.31564			0.657822	−	0.753173i	$-0.271478\pi$
$239$	20.3394			1.31564			0.657822	+	0.753173i	$0.271478\pi$
$240$		0			0
$241$	−0.699485			−0.0450578			−0.0225289	−	0.999746i	$-0.507172\pi$
$241$	−0.699485			−0.0450578			−0.0225289	+	0.999746i	$0.507172\pi$
$242$		0			0
$243$	−14.2132	−	6.40187i	−0.911779	−	0.410680i
$244$		0			0
$245$	4.89257			0.312575
$246$		0			0
$247$			8.12450i			0.516950i
$248$		0			0
$249$	1.63942	−	19.3231i	0.103894	−	1.22455i
$250$		0			0
$251$			1.01084i			0.0638034i	0.999491	+	0.0319017i	$0.0101564\pi$
$251$			1.01084i			0.0638034i	−0.999491	+	0.0319017i	$0.989844\pi$
$252$		0			0
$253$			3.33949i			0.209952i
$254$		0			0
$255$	−0.696312	+	8.20709i	−0.0436047	+	0.513948i
$256$		0			0
$257$			10.2662i			0.640386i	0.947352	+	0.320193i	$0.103748\pi$
$257$			10.2662i			0.640386i	−0.947352	+	0.320193i	$0.896252\pi$
$258$		0			0
$259$	8.80057			0.546841
$260$		0			0
$261$	−10.6354	−	1.81776i	−0.658314	−	0.112516i
$262$		0			0
$263$	−17.2820			−1.06565			−0.532827	−	0.846224i	$-0.678870\pi$
$263$	−17.2820			−1.06565			−0.532827	+	0.846224i	$0.678870\pi$
$264$		0			0
$265$	4.31770			0.265234
$266$		0			0
$267$	0.925074	−	10.9034i	0.0566136	−	0.667277i
$268$		0			0
$269$	−0.871939			−0.0531631			−0.0265815	−	0.999647i	$-0.508462\pi$
$269$	−0.871939			−0.0531631			−0.0265815	+	0.999647i	$0.508462\pi$
$270$		0			0
$271$			17.7085i			1.07571i	0.843036	+	0.537857i	$0.180766\pi$
$271$			17.7085i			1.07571i	−0.843036	+	0.537857i	$0.819234\pi$
$272$		0			0
$273$	−9.38167	−	0.795966i	−0.567804	−	0.0481741i
$274$		0			0
$275$			1.01084i			0.0609557i
$276$		0			0
$277$		−	12.1273i		−	0.728657i	−0.931271	−	0.364328i	$-0.881299\pi$
$277$		−	12.1273i		−	0.728657i	0.931271	−	0.364328i	$-0.118701\pi$
$278$		0			0
$279$	−1.39262	+	8.14801i	−0.0833742	+	0.487809i
$280$		0			0
$281$			20.4566i			1.22034i	0.792271	+	0.610169i	$0.208899\pi$
$281$			20.4566i			1.22034i	−0.792271	+	0.610169i	$0.791101\pi$
$282$		0			0
$283$	3.62322			0.215378			0.107689	−	0.994185i	$-0.465655\pi$
$283$	3.62322			0.215378			0.107689	+	0.994185i	$0.465655\pi$
$284$		0			0
$285$	3.74455	+	0.317698i	0.221808	+	0.0188188i
$286$		0			0
$287$	−10.6388			−0.627991
$288$		0			0
$289$	−5.61372			−0.330219
$290$		0			0
$291$	−16.3768	−	1.38945i	−0.960024	−	0.0814510i
$292$		0			0
$293$	20.9782			1.22556			0.612780	−	0.790254i	$-0.290051\pi$
$293$	20.9782			1.22556			0.612780	+	0.790254i	$0.290051\pi$
$294$		0			0
$295$		−	11.9142i		−	0.693674i
$296$		0			0
$297$	1.31942	−	5.08404i	0.0765605	−	0.295006i
$298$		0			0
$299$		−	12.3708i		−	0.715424i
$300$		0			0
$301$		−	0.0542996i		−	0.00312978i
$302$		0			0
$303$	23.1322	+	1.96260i	1.32891	+	0.112748i
$304$		0			0
$305$		−	5.30686i		−	0.303870i
$306$		0			0
$307$	−14.3984			−0.821763			−0.410881	−	0.911689i	$-0.634779\pi$
$307$	−14.3984			−0.821763			−0.410881	+	0.911689i	$0.634779\pi$
$308$		0			0
$309$	−1.68000	+	19.8013i	−0.0955718	+	1.12646i
$310$		0			0
$311$	−1.97198			−0.111821			−0.0559104	−	0.998436i	$-0.517806\pi$
$311$	−1.97198			−0.111821			−0.0559104	+	0.998436i	$0.517806\pi$
$312$		0			0
$313$	−31.9998			−1.80873			−0.904367	−	0.426756i	$-0.859656\pi$
$313$	−31.9998			−1.80873			−0.904367	+	0.426756i	$0.859656\pi$
$314$		0			0
$315$	−0.733716	+	4.29285i	−0.0413402	+	0.241875i
$316$		0			0
$317$	−20.9531			−1.17684			−0.588422	−	0.808554i	$-0.700250\pi$
$317$	−20.9531			−1.17684			−0.588422	+	0.808554i	$0.700250\pi$
$318$		0			0
$319$		−	3.63551i		−	0.203550i
$320$		0			0
$321$	−2.78571	+	32.8338i	−0.155483	+	1.83261i
$322$		0			0
$323$			10.3177i			0.574092i
$324$		0			0
$325$		−	3.74455i		−	0.207710i
$326$		0			0
$327$	−2.46571	+	29.0621i	−0.136354	+	1.60714i
$328$		0			0
$329$			8.16057i			0.449907i
$330$		0			0
$331$	−6.58386			−0.361882			−0.180941	−	0.983494i	$-0.557914\pi$
$331$	−6.58386			−0.361882			−0.180941	+	0.983494i	$0.557914\pi$
$332$		0			0
$333$	3.06397	−	17.9268i	0.167905	−	0.982382i
$334$		0			0
$335$	8.03740			0.439130
$336$		0			0
$337$	−2.80692			−0.152903			−0.0764513	−	0.997073i	$-0.524359\pi$
$337$	−2.80692			−0.152903			−0.0764513	+	0.997073i	$0.524359\pi$
$338$		0			0
$339$	−2.08259	+	24.5465i	−0.113111	+	1.33318i
$340$		0			0
$341$	−2.78525			−0.150830
$342$		0			0
$343$			17.2644i			0.932192i
$344$		0			0
$345$	−5.70167	−	0.483745i	−0.306968	−	0.0260440i
$346$		0			0
$347$			8.12133i			0.435976i	0.975951	+	0.217988i	$0.0699493\pi$
$347$			8.12133i			0.435976i	−0.975951	+	0.217988i	$0.930051\pi$
$348$		0			0
$349$			5.68230i			0.304167i	0.988368	+	0.152083i	$0.0485982\pi$
$349$			5.68230i			0.304167i	−0.988368	+	0.152083i	$0.951402\pi$
$350$		0			0
$351$	−4.88767	+	18.8334i	−0.260885	+	1.00525i
$352$		0			0
$353$		−	29.9017i		−	1.59151i	−0.605621	−	0.795753i	$-0.707075\pi$
$353$		−	29.9017i		−	1.59151i	0.605621	−	0.795753i	$-0.292925\pi$
$354$		0			0
$355$	−12.4142			−0.658876
$356$		0			0
$357$	−11.9142	−	1.01084i	−0.630568	−	0.0534991i
$358$		0			0
$359$	16.8754			0.890649			0.445324	−	0.895369i	$-0.353088\pi$
$359$	16.8754			0.890649			0.445324	+	0.895369i	$0.353088\pi$
$360$		0			0
$361$	−14.2925			−0.752235
$362$		0			0
$363$	−17.2209	−	1.46107i	−0.903863	−	0.0766862i
$364$		0			0
$365$	10.3394			0.541187
$366$		0			0
$367$		−	35.3333i		−	1.84438i	−0.386733	−	0.922192i	$-0.626396\pi$
$367$		−	35.3333i		−	1.84438i	0.386733	−	0.922192i	$-0.373604\pi$
$368$		0			0
$369$	−3.70397	+	21.6713i	−0.192821	+	1.12817i
$370$		0			0
$371$			6.26801i			0.325419i
$372$		0			0
$373$			10.6976i			0.553903i	0.960884	+	0.276952i	$0.0893242\pi$
$373$			10.6976i			0.553903i	−0.960884	+	0.276952i	$0.910676\pi$
$374$		0			0
$375$	−1.72585	−	0.146426i	−0.0891225	−	0.00756140i
$376$		0			0
$377$			13.4674i			0.693608i
$378$		0			0
$379$	−5.83031			−0.299483			−0.149742	−	0.988725i	$-0.547844\pi$
$379$	−5.83031			−0.299483			−0.149742	+	0.988725i	$0.547844\pi$
$380$		0			0
$381$	−2.01937	+	23.8013i	−0.103455	+	1.21938i
$382$		0			0
$383$	−37.8207			−1.93255			−0.966274	−	0.257518i	$-0.917095\pi$
$383$	−37.8207			−1.93255			−0.966274	+	0.257518i	$0.917095\pi$
$384$		0			0
$385$	−1.46743			−0.0747873
$386$		0			0
$387$	−0.110608	−	0.0189047i	−0.00562254	−	0.000960981i
$388$		0			0
$389$	−6.63540			−0.336428			−0.168214	−	0.985751i	$-0.553800\pi$
$389$	−6.63540			−0.336428			−0.168214	+	0.985751i	$0.553800\pi$
$390$		0			0
$391$		−	15.7103i		−	0.794505i
$392$		0			0
$393$	2.50906	−	29.5730i	0.126565	−	1.49176i
$394$		0			0
$395$		−	11.3411i		−	0.570632i
$396$		0			0
$397$			32.7878i			1.64557i	0.568352	+	0.822786i	$0.307581\pi$
$397$			32.7878i			1.64557i	−0.568352	+	0.822786i	$0.692419\pi$
$398$		0			0
$399$	−0.461202	+	5.43597i	−0.0230890	+	0.272139i
$400$		0			0
$401$			22.8067i			1.13891i	0.822022	+	0.569456i	$0.192846\pi$
$401$			22.8067i			1.13891i	−0.822022	+	0.569456i	$0.807154\pi$
$402$		0			0
$403$	10.3177			0.513961
$404$		0			0
$405$	8.48910	+	2.98916i	0.421827	+	0.148533i
$406$		0			0
$407$	6.12794			0.303751
$408$		0			0
$409$	−9.87089			−0.488084			−0.244042	−	0.969765i	$-0.578474\pi$
$409$	−9.87089			−0.488084			−0.244042	+	0.969765i	$0.578474\pi$
$410$		0			0
$411$	1.35682	−	15.9922i	0.0669272	−	0.788838i
$412$		0			0
$413$	17.2959			0.851076
$414$		0			0
$415$			11.1963i			0.549602i
$416$		0			0
$417$	−18.0623	−	1.53245i	−0.884512	−	0.0750444i
$418$		0			0
$419$		−	12.3818i		−	0.604890i	−0.953167	−	0.302445i	$-0.902197\pi$
$419$		−	12.3818i		−	0.604890i	0.953167	−	0.302445i	$-0.0978028\pi$
$420$		0			0
$421$			19.6930i			0.959779i	0.877329	+	0.479890i	$0.159323\pi$
$421$			19.6930i			0.959779i	−0.877329	+	0.479890i	$0.840677\pi$
$422$		0			0
$423$	16.6231	+	2.84115i	0.808243	+	0.138141i
$424$		0			0
$425$		−	4.75539i		−	0.230670i
$426$		0			0
$427$	7.70397			0.372821
$428$		0			0
$429$	−6.53257	−	0.554241i	−0.315395	−	0.0267590i
$430$		0			0
$431$	28.5638			1.37587			0.687935	−	0.725772i	$-0.258517\pi$
$431$	28.5638			1.37587			0.687935	+	0.725772i	$0.258517\pi$
$432$		0			0
$433$	−12.4673			−0.599141			−0.299570	−	0.954074i	$-0.596843\pi$
$433$	−12.4673			−0.599141			−0.299570	+	0.954074i	$0.596843\pi$
$434$		0			0
$435$	6.20709	+	0.526626i	0.297607	+	0.0252498i
$436$		0			0
$437$	−7.16796			−0.342890
$438$		0			0
$439$		−	3.21659i		−	0.153520i	−0.997050	−	0.0767598i	$-0.975543\pi$
$439$		−	3.21659i		−	0.153520i	0.997050	−	0.0767598i	$-0.0244574\pi$
$440$		0			0
$441$	14.4679	+	2.47279i	0.688947	+	0.117752i
$442$		0			0
$443$			3.92837i			0.186642i	0.995636	+	0.0933211i	$0.0297483\pi$
$443$			3.92837i			0.186642i	−0.995636	+	0.0933211i	$0.970252\pi$
$444$		0			0
$445$			6.31770i			0.299488i
$446$		0			0
$447$	−20.4516	−	1.73517i	−0.967327	−	0.0820706i
$448$		0			0
$449$		−	6.98916i		−	0.329839i	−0.986307	−	0.164920i	$-0.947264\pi$
$449$		−	6.98916i		−	0.329839i	0.986307	−	0.164920i	$-0.0527364\pi$
$450$		0			0
$451$	−7.40795			−0.348827
$452$		0			0
$453$	−3.30051	+	38.9016i	−0.155072	+	1.82775i
$454$		0			0
$455$	5.43597			0.254842
$456$		0			0
$457$	−18.4639			−0.863703			−0.431852	−	0.901945i	$-0.642140\pi$
$457$	−18.4639			−0.863703			−0.431852	+	0.901945i	$0.642140\pi$
$458$		0			0
$459$	−6.20709	+	23.9174i	−0.289722	+	1.11637i
$460$		0			0
$461$	−12.0171			−0.559691			−0.279845	−	0.960045i	$-0.590283\pi$
$461$	−12.0171			−0.559691			−0.279845	+	0.960045i	$0.590283\pi$
$462$		0			0
$463$		−	30.2304i		−	1.40492i	−0.711721	−	0.702462i	$-0.752084\pi$
$463$		−	30.2304i		−	1.40492i	0.711721	−	0.702462i	$-0.247916\pi$
$464$		0			0
$465$	0.403460	−	4.75539i	0.0187100	−	0.220526i
$466$		0			0
$467$			6.06819i			0.280802i	0.990095	+	0.140401i	$0.0448392\pi$
$467$			6.06819i			0.280802i	−0.990095	+	0.140401i	$0.955161\pi$
$468$		0			0
$469$			11.6679i			0.538774i
$470$		0			0
$471$	1.10889	−	13.0699i	0.0510948	−	0.602230i
$472$		0			0
$473$		−	0.0378095i		−	0.00173848i
$474$		0			0
$475$	−2.16969			−0.0995520
$476$		0			0
$477$	12.7679	+	2.18224i	0.584604	+	0.0999181i
$478$		0			0
$479$	−12.4890			−0.570636			−0.285318	−	0.958433i	$-0.592099\pi$
$479$	−12.4890			−0.570636			−0.285318	+	0.958433i	$0.592099\pi$
$480$		0			0
$481$	−22.7004			−1.03505
$482$		0			0
$483$	0.702253	−	8.27712i	0.0319536	−	0.376622i
$484$		0			0
$485$	9.48910			0.430878
$486$		0			0
$487$			16.0157i			0.725742i	0.931839	+	0.362871i	$0.118203\pi$
$487$			16.0157i			0.725742i	−0.931839	+	0.362871i	$0.881797\pi$
$488$		0			0
$489$	−19.8930	−	1.68778i	−0.899594	−	0.0763240i
$490$		0			0
$491$			2.64979i			0.119583i	0.998211	+	0.0597917i	$0.0190437\pi$
$491$			2.64979i			0.119583i	−0.998211	+	0.0597917i	$0.980956\pi$
$492$		0			0
$493$			17.1029i			0.770278i
$494$		0			0
$495$	−0.510895	+	2.98916i	−0.0229630	+	0.134353i
$496$		0			0
$497$		−	18.0217i		−	0.808383i
$498$		0			0
$499$	10.1697			0.455258			0.227629	−	0.973748i	$-0.426903\pi$
$499$	10.1697			0.455258			0.227629	+	0.973748i	$0.426903\pi$
$500$		0			0
$501$	−11.7233	−	0.994640i	−0.523760	−	0.0444373i
$502$		0			0
$503$	4.22876			0.188551			0.0942756	−	0.995546i	$-0.469947\pi$
$503$	4.22876			0.188551			0.0942756	+	0.995546i	$0.469947\pi$
$504$		0			0
$505$	−13.4033			−0.596441
$506$		0			0
$507$	1.76325	+	0.149599i	0.0783089	+	0.00664394i
$508$		0			0
$509$	−13.5749			−0.601695			−0.300848	−	0.953672i	$-0.597270\pi$
$509$	−13.5749			−0.601695			−0.300848	+	0.953672i	$0.597270\pi$
$510$		0			0
$511$			15.0097i			0.663989i
$512$		0			0
$513$	10.9125	+	2.83204i	0.481799	+	0.125037i
$514$		0			0
$515$		−	11.4734i		−	0.505577i
$516$		0			0
$517$			5.68230i			0.249907i
$518$		0			0
$519$	−44.2053	−	3.75049i	−1.94040	−	0.164628i
$520$		0			0
$521$			41.9998i			1.84004i	0.391868	+	0.920022i	$0.371829\pi$
$521$			41.9998i			1.84004i	−0.391868	+	0.920022i	$0.628171\pi$
$522$		0			0
$523$	−7.70438			−0.336889			−0.168445	−	0.985711i	$-0.553874\pi$
$523$	−7.70438			−0.336889			−0.168445	+	0.985711i	$0.553874\pi$
$524$		0			0
$525$	0.212567	−	2.50542i	0.00927716	−	0.109345i
$526$		0			0
$527$	13.1029			0.570773
$528$		0			0
$529$	−12.0856			−0.525463
$530$		0			0
$531$	6.02167	−	35.2318i	0.261318	−	1.52893i
$532$		0			0
$533$	27.4421			1.18865
$534$		0			0
$535$		−	19.0247i		−	0.822511i
$536$		0			0
$537$	0.277120	−	3.26628i	0.0119586	−	0.140951i
$538$		0			0
$539$			4.94558i			0.213021i
$540$		0			0
$541$		−	4.85027i		−	0.208529i	−0.994550	−	0.104265i	$-0.966751\pi$
$541$		−	4.85027i		−	0.208529i	0.994550	−	0.104265i	$-0.0332489\pi$
$542$		0			0
$543$	0.757228	−	8.92507i	0.0324957	−	0.383012i
$544$		0			0
$545$		−	16.8393i		−	0.721317i
$546$		0			0
$547$	24.4201			1.04413			0.522064	−	0.852906i	$-0.325162\pi$
$547$	24.4201			1.04413			0.522064	+	0.852906i	$0.325162\pi$
$548$		0			0
$549$	2.68218	−	15.6930i	0.114473	−	0.669762i
$550$		0			0
$551$	7.80336			0.332434
$552$		0			0
$553$	16.4639			0.700115
$554$		0			0
$555$	−0.887670	+	10.4625i	−0.0376795	+	0.444110i
$556$		0			0
$557$	25.8718			1.09622			0.548112	−	0.836405i	$-0.315347\pi$
$557$	25.8718			1.09622			0.548112	+	0.836405i	$0.315347\pi$
$558$		0			0
$559$			0.140062i			0.00592398i
$560$		0			0
$561$	−8.29603	−	0.703857i	−0.350258	−	0.0297169i
$562$		0			0
$563$		−	11.0247i		−	0.464637i	−0.972640	−	0.232318i	$-0.925369\pi$
$563$		−	11.0247i		−	0.464637i	0.972640	−	0.232318i	$-0.0746312\pi$
$564$		0			0
$565$		−	14.2228i		−	0.598359i
$566$		0			0
$567$	−4.33937	+	12.3236i	−0.182236	+	0.517544i
$568$		0			0
$569$			28.2039i			1.18237i	0.806536	+	0.591185i	$0.201340\pi$
$569$			28.2039i			1.18237i	−0.806536	+	0.591185i	$0.798660\pi$
$570$		0			0
$571$	24.0513			1.00652			0.503258	−	0.864136i	$-0.332135\pi$
$571$	24.0513			1.00652			0.503258	+	0.864136i	$0.332135\pi$
$572$		0			0
$573$	33.7210	+	2.86099i	1.40872	+	0.119519i
$574$		0			0
$575$	3.30369			0.137773
$576$		0			0
$577$	−21.6570			−0.901591			−0.450795	−	0.892627i	$-0.648860\pi$
$577$	−21.6570			−0.901591			−0.450795	+	0.892627i	$0.648860\pi$
$578$		0			0
$579$	1.47337	+	0.125005i	0.0612313	+	0.00519503i
$580$		0			0
$581$	−16.2536			−0.674313
$582$		0			0
$583$			4.36449i			0.180759i
$584$		0			0
$585$	1.89257	−	11.0731i	0.0782479	−	0.457816i
$586$		0			0
$587$			39.0995i			1.61381i	0.590681	+	0.806905i	$0.298859\pi$
$587$			39.0995i			1.61381i	−0.590681	+	0.806905i	$0.701141\pi$
$588$		0			0
$589$		−	5.97833i		−	0.246333i
$590$		0			0
$591$	33.3767	+	2.83177i	1.37293	+	0.116483i
$592$		0			0
$593$		−	33.5153i		−	1.37631i	−0.725565	−	0.688154i	$-0.758421\pi$
$593$		−	33.5153i		−	1.37631i	0.725565	−	0.688154i	$-0.241579\pi$
$594$		0			0
$595$	6.90340			0.283012
$596$		0			0
$597$	0.193197	−	2.27712i	0.00790703	−	0.0931963i
$598$		0			0
$599$	7.61728			0.311234			0.155617	−	0.987817i	$-0.450263\pi$
$599$	7.61728			0.311234			0.155617	+	0.987817i	$0.450263\pi$
$600$		0			0
$601$	17.7861			0.725508			0.362754	−	0.931885i	$-0.381836\pi$
$601$	17.7861			0.725508			0.362754	+	0.931885i	$0.381836\pi$
$602$		0			0
$603$	23.7676	+	4.06225i	0.967890	+	0.165428i
$604$		0			0
$605$	9.97821			0.405672
$606$		0			0
$607$		−	18.6295i		−	0.756146i	−0.925776	−	0.378073i	$-0.876587\pi$
$607$		−	18.6295i		−	0.756146i	0.925776	−	0.378073i	$-0.123413\pi$
$608$		0			0
$609$	−0.764504	+	9.01084i	−0.0309793	+	0.365138i
$610$		0			0
$611$		−	21.0496i		−	0.851575i
$612$		0			0
$613$		−	34.1307i		−	1.37853i	−0.724511	−	0.689263i	$-0.757934\pi$
$613$		−	34.1307i		−	1.37853i	0.724511	−	0.689263i	$-0.242066\pi$
$614$		0			0
$615$	1.07309	−	12.6480i	0.0432710	−	0.510015i
$616$		0			0
$617$			14.8582i			0.598169i	0.954227	+	0.299085i	$0.0966813\pi$
$617$			14.8582i			0.598169i	−0.954227	+	0.299085i	$0.903319\pi$
$618$		0			0
$619$	45.6369			1.83430			0.917151	−	0.398541i	$-0.130483\pi$
$619$	45.6369			1.83430			0.917151	+	0.398541i	$0.130483\pi$
$620$		0			0
$621$	−16.6160	−	4.31222i	−0.666778	−	0.173043i
$622$		0			0
$623$	−9.17141			−0.367445
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.321141	+	3.78513i	−0.0128251	+	0.151164i
$628$		0			0
$629$	−28.8284			−1.14946
$630$		0			0
$631$		−	43.5367i		−	1.73317i	−0.499030	−	0.866585i	$-0.666310\pi$
$631$		−	43.5367i		−	1.73317i	0.499030	−	0.866585i	$-0.333690\pi$
$632$		0			0
$633$	−30.9764	−	2.62812i	−1.23120	−	0.104458i
$634$		0			0
$635$		−	13.7911i		−	0.547282i
$636$		0			0
$637$		−	18.3205i		−	0.725883i
$638$		0			0
$639$	−36.7102	−	6.27435i	−1.45223	−	0.248210i
$640$		0			0
$641$		−	31.4782i		−	1.24331i	−0.783290	−	0.621656i	$-0.786460\pi$
$641$		−	31.4782i		−	1.24331i	0.783290	−	0.621656i	$-0.213540\pi$
$642$		0			0
$643$	−4.32708			−0.170643			−0.0853217	−	0.996353i	$-0.527192\pi$
$643$	−4.32708			−0.170643			−0.0853217	+	0.996353i	$0.527192\pi$
$644$		0			0
$645$	0.0645539	+	0.00547693i	0.00254181	+	0.000215654i
$646$		0			0
$647$	−48.8171			−1.91920			−0.959600	−	0.281368i	$-0.909212\pi$
$647$	−48.8171			−1.91920			−0.959600	+	0.281368i	$0.909212\pi$
$648$		0			0
$649$	12.0433			0.472743
$650$		0			0
$651$	6.90340	+	0.585703i	0.270566	+	0.0229555i
$652$		0			0
$653$	−4.18952			−0.163949			−0.0819743	−	0.996634i	$-0.526123\pi$
$653$	−4.18952			−0.163949			−0.0819743	+	0.996634i	$0.526123\pi$
$654$		0			0
$655$			17.1353i			0.669533i
$656$		0			0
$657$	30.5747	+	5.22571i	1.19283	+	0.203874i
$658$		0			0
$659$		−	42.3004i		−	1.64779i	−0.566743	−	0.823895i	$-0.691797\pi$
$659$		−	42.3004i		−	1.64779i	0.566743	−	0.823895i	$-0.308203\pi$
$660$		0			0
$661$			21.6280i			0.841232i	0.907239	+	0.420616i	$0.138186\pi$
$661$			21.6280i			0.841232i	−0.907239	+	0.420616i	$0.861814\pi$
$662$		0			0
$663$	30.7319	+	2.60738i	1.19353	+	0.101262i
$664$		0			0
$665$		−	3.14973i		−	0.122141i
$666$		0			0
$667$	−11.8818			−0.460067
$668$		0			0
$669$	−0.577053	+	6.80144i	−0.0223102	+	0.262959i
$670$		0			0
$671$	5.36437			0.207089
$672$		0			0
$673$	22.1245			0.852837			0.426418	−	0.904526i	$-0.359775\pi$
$673$	22.1245			0.852837			0.426418	+	0.904526i	$0.359775\pi$
$674$		0			0
$675$	−5.02954	−	1.30527i	−0.193587	−	0.0502401i
$676$		0			0
$677$	24.1895			0.929679			0.464839	−	0.885395i	$-0.346112\pi$
$677$	24.1895			0.929679			0.464839	+	0.885395i	$0.346112\pi$
$678$		0			0
$679$			13.7753i			0.528649i
$680$		0			0
$681$	0.918263	−	10.8231i	0.0351879	−	0.414743i
$682$		0			0
$683$			49.7069i			1.90198i	0.309218	+	0.950991i	$0.399933\pi$
$683$			49.7069i			1.90198i	−0.309218	+	0.950991i	$0.600067\pi$
$684$		0			0
$685$			9.26628i			0.354047i
$686$		0			0
$687$	2.82859	−	33.3393i	0.107918	−	1.27197i
$688$		0			0
$689$		−	16.1678i		−	0.615946i
$690$		0			0
$691$	−1.94847			−0.0741232			−0.0370616	−	0.999313i	$-0.511800\pi$
$691$	−1.94847			−0.0741232			−0.0370616	+	0.999313i	$0.511800\pi$
$692$		0			0
$693$	−4.33937	−	0.741667i	−0.164839	−	0.0281736i
$694$		0			0
$695$	10.4657			0.396987
$696$		0			0
$697$	34.8500			1.32004
$698$		0			0
$699$	−2.76768	+	32.6213i	−0.104683	+	1.23385i
$700$		0			0
$701$	−37.0216			−1.39828			−0.699142	−	0.714982i	$-0.746435\pi$
$701$	−37.0216			−1.39828			−0.699142	+	0.714982i	$0.746435\pi$
$702$		0			0
$703$			13.1532i			0.496081i
$704$		0			0
$705$	−9.70167	−	0.823116i	−0.365386	−	0.0310004i
$706$		0			0
$707$		−	19.4576i		−	0.731780i
$708$		0			0
$709$			4.66051i			0.175029i	0.996163	+	0.0875146i	$0.0278924\pi$
$709$			4.66051i			0.175029i	−0.996163	+	0.0875146i	$0.972108\pi$
$710$		0			0
$711$	5.73199	−	33.5370i	0.214967	−	1.25773i
$712$		0			0
$713$			9.10295i			0.340908i
$714$		0			0
$715$	3.78513			0.141556
$716$		0			0
$717$	−35.1027	−	2.97821i	−1.31093	−	0.111223i
$718$		0			0
$719$	45.3771			1.69228			0.846139	−	0.532962i	$-0.178921\pi$
$719$	45.3771			1.69228			0.846139	+	0.532962i	$0.178921\pi$
$720$		0			0
$721$	16.6559			0.620298
$722$		0			0
$723$	1.20721	+	0.102423i	0.0448965	+	0.00380914i
$724$		0			0
$725$	−3.59654			−0.133572
$726$		0			0
$727$			37.4795i			1.39004i	0.718992	+	0.695019i	$0.244604\pi$
$727$			37.4795i			1.39004i	−0.718992	+	0.695019i	$0.755396\pi$
$728$		0			0
$729$	23.5925	+	13.1299i	0.873797	+	0.486291i
$730$		0			0
$731$			0.177871i			0.00657880i
$732$		0			0
$733$			7.08404i			0.261655i	0.991405	+	0.130828i	$0.0417634\pi$
$733$			7.08404i			0.261655i	−0.991405	+	0.130828i	$0.958237\pi$
$734$		0			0
$735$	−8.44384	−	0.716398i	−0.311456	−	0.0264247i
$736$		0			0
$737$			8.12450i			0.299270i
$738$		0			0
$739$	−46.6587			−1.71637			−0.858184	−	0.513343i	$-0.828407\pi$
$739$	−46.6587			−1.71637			−0.858184	+	0.513343i	$0.828407\pi$
$740$		0			0
$741$	1.18964	−	14.0217i	0.0437024	−	0.515099i
$742$		0			0
$743$	−22.2504			−0.816289			−0.408145	−	0.912917i	$-0.633824\pi$
$743$	−22.2504			−0.816289			−0.408145	+	0.912917i	$0.633824\pi$
$744$		0			0
$745$	11.8501			0.434156
$746$		0			0
$747$	−5.65879	+	33.1087i	−0.207044	+	1.21138i
$748$		0			0
$749$	27.6182			1.00915
$750$		0			0
$751$			38.9610i			1.42171i	0.703339	+	0.710854i	$0.251692\pi$
$751$			38.9610i			1.42171i	−0.703339	+	0.710854i	$0.748308\pi$
$752$		0			0
$753$	0.148013	−	1.74455i	0.00539388	−	0.0635750i
$754$		0			0
$755$		−	22.5405i		−	0.820333i
$756$		0			0
$757$			6.98098i			0.253728i	0.991920	+	0.126864i	$0.0404912\pi$
$757$			6.98098i			0.253728i	−0.991920	+	0.126864i	$0.959509\pi$
$758$		0			0
$759$	0.488987	−	5.76346i	0.0177491	−	0.209200i
$760$		0			0
$761$			3.20577i			0.116209i	0.998311	+	0.0581046i	$0.0185057\pi$
$761$			3.20577i			0.116209i	−0.998311	+	0.0581046i	$0.981494\pi$
$762$		0			0
$763$	24.4456			0.884992
$764$		0			0
$765$	2.40346	−	14.0623i	0.0868973	−	0.508422i
$766$		0			0
$767$	−44.6135			−1.61090
$768$		0			0
$769$	15.6353			0.563823			0.281911	−	0.959440i	$-0.409032\pi$
$769$	15.6353			0.563823			0.281911	+	0.959440i	$0.409032\pi$
$770$		0			0
$771$	1.50323	−	17.7179i	0.0541376	−	0.638093i
$772$		0			0
$773$	24.3610			0.876206			0.438103	−	0.898925i	$-0.355651\pi$
$773$	24.3610			0.876206			0.438103	+	0.898925i	$0.355651\pi$
$774$		0			0
$775$			2.75539i			0.0989765i
$776$		0			0
$777$	−15.1885	−	1.28863i	−0.544883	−	0.0462294i
$778$		0			0
$779$		−	15.9006i		−	0.569699i
$780$		0			0
$781$		−	12.5487i		−	0.449028i
$782$		0			0
$783$	18.0889	+	4.69447i	0.646446	+	0.167767i
$784$		0			0
$785$			7.57303i			0.270293i
$786$		0			0
$787$	10.9590			0.390647			0.195324	−	0.980739i	$-0.437424\pi$
$787$	10.9590			0.390647			0.195324	+	0.980739i	$0.437424\pi$
$788$		0			0
$789$	29.8262	+	2.53053i	1.06184	+	0.0900894i
$790$		0			0
$791$	20.6473			0.734133
$792$		0			0
$793$	−19.8718			−0.705669
$794$		0			0
$795$	−7.45170	−	0.632223i	−0.264285	−	0.0224226i
$796$		0			0
$797$	−11.0903			−0.392837			−0.196418	−	0.980520i	$-0.562931\pi$
$797$	−11.0903			−0.392837			−0.196418	+	0.980520i	$0.562931\pi$
$798$		0			0
$799$		−	26.7319i		−	0.945706i
$800$		0			0
$801$	−3.19308	+	18.6822i	−0.112822	+	0.660103i
$802$		0			0
$803$			10.4514i			0.368822i
$804$		0			0
$805$			4.79597i			0.169036i
$806$		0			0
$807$	1.50484	+	0.127674i	0.0529727	+	0.00449435i
$808$		0			0
$809$			7.16796i			0.252012i	0.992029	+	0.126006i	$0.0402159\pi$
$809$			7.16796i			0.252012i	−0.992029	+	0.126006i	$0.959784\pi$
$810$		0			0
$811$	−55.5586			−1.95093			−0.975464	−	0.220160i	$-0.929342\pi$
$811$	−55.5586			−1.95093			−0.975464	+	0.220160i	$0.929342\pi$
$812$		0			0
$813$	2.59298	−	30.5622i	0.0909398	−	1.07186i
$814$		0			0
$815$	11.5265			0.403756
$816$		0			0
$817$	0.0811552			0.00283926
$818$		0			0
$819$	16.0748	+	2.74744i	0.561699	+	0.0960033i
$820$		0			0
$821$	−42.4420			−1.48123			−0.740617	−	0.671927i	$-0.765467\pi$
$821$	−42.4420			−1.48123			−0.740617	+	0.671927i	$0.765467\pi$
$822$		0			0
$823$		−	16.6881i		−	0.581712i	−0.956767	−	0.290856i	$-0.906060\pi$
$823$		−	16.6881i		−	0.581712i	0.956767	−	0.290856i	$-0.0939400\pi$
$824$		0			0
$825$	0.148013	−	1.74455i	0.00515314	−	0.0607375i
$826$		0			0
$827$			20.8968i			0.726652i	0.931662	+	0.363326i	$0.118359\pi$
$827$			20.8968i			0.726652i	−0.931662	+	0.363326i	$0.881641\pi$
$828$		0			0
$829$			8.83931i			0.307002i	0.988148	+	0.153501i	$0.0490548\pi$
$829$			8.83931i			0.307002i	−0.988148	+	0.153501i	$0.950945\pi$
$830$		0			0
$831$	−1.77575	+	20.9299i	−0.0615999	+	0.726049i
$832$		0			0
$833$		−	23.2660i		−	0.806121i
$834$		0			0
$835$	6.79279			0.235074
$836$		0			0
$837$	3.59654	−	13.8583i	0.124315	−	0.479014i
$838$		0			0
$839$	−6.10283			−0.210693			−0.105347	−	0.994436i	$-0.533595\pi$
$839$	−6.10283			−0.210693			−0.105347	+	0.994436i	$0.533595\pi$
$840$		0			0
$841$	−16.0649			−0.553962
$842$		0			0
$843$	2.99537	−	35.3050i	0.103166	−	1.21597i
$844$		0			0
$845$	−1.02167			−0.0351466
$846$		0			0
$847$			14.4854i			0.497723i
$848$		0			0
$849$	−6.25314	−	0.530534i	−0.214607	−	0.0182079i
$850$		0			0
$851$		−	20.0278i		−	0.686544i
$852$		0			0
$853$		−	32.3799i		−	1.10867i	−0.832294	−	0.554334i	$-0.812973\pi$
$853$		−	32.3799i		−	1.10867i	0.832294	−	0.554334i	$-0.187027\pi$
$854$		0			0
$855$	−6.41602	−	1.09660i	−0.219423	−	0.0375029i
$856$		0			0
$857$			32.2445i			1.10145i	0.834686	+	0.550725i	$0.185649\pi$
$857$			32.2445i			1.10145i	−0.834686	+	0.550725i	$0.814351\pi$
$858$		0			0
$859$	−47.9295			−1.63533			−0.817666	−	0.575693i	$-0.804732\pi$
$859$	−47.9295			−1.63533			−0.817666	+	0.575693i	$0.804732\pi$
$860$		0			0
$861$	18.3610	+	1.55780i	0.625743	+	0.0530897i
$862$		0			0
$863$	−5.96443			−0.203032			−0.101516	−	0.994834i	$-0.532369\pi$
$863$	−5.96443			−0.203032			−0.101516	+	0.994834i	$0.532369\pi$
$864$		0			0
$865$	25.6136			0.870889
$866$		0			0
$867$	9.68845	+	0.821994i	0.329037	+	0.0279164i
$868$		0			0
$869$	11.4640			0.388889
$870$		0			0
$871$		−	30.0965i		−	1.01978i
$872$		0			0
$873$	28.0604	+	4.79597i	0.949701	+	0.162319i
$874$		0			0
$875$			1.45170i			0.0490764i
$876$		0			0
$877$		−	39.0621i		−	1.31903i	−0.751689	−	0.659517i	$-0.770761\pi$
$877$		−	39.0621i		−	1.31903i	0.751689	−	0.659517i	$-0.229239\pi$
$878$		0			0
$879$	−36.2053	−	3.07175i	−1.22117	−	0.103608i
$880$		0			0
$881$			11.9639i			0.403075i	0.979481	+	0.201538i	$0.0645938\pi$
$881$			11.9639i			0.403075i	−0.979481	+	0.201538i	$0.935406\pi$
$882$		0			0
$883$	−32.5798			−1.09640			−0.548198	−	0.836348i	$-0.684686\pi$
$883$	−32.5798			−1.09640			−0.548198	+	0.836348i	$0.684686\pi$
$884$		0			0
$885$	−1.74455	+	20.5622i	−0.0586425	+	0.691191i
$886$		0			0
$887$	27.7115			0.930462			0.465231	−	0.885189i	$-0.345971\pi$
$887$	27.7115			0.930462			0.465231	+	0.885189i	$0.345971\pi$
$888$		0			0
$889$	20.0205			0.671466
$890$		0			0
$891$	−3.02156	+	8.58110i	−0.101226	+	0.287478i
$892$		0			0
$893$	−12.1966			−0.408145
$894$		0			0
$895$			1.89257i			0.0632615i
$896$		0			0
$897$	−1.81141	+	21.3502i	−0.0604812	+	0.712863i
$898$		0			0
$899$		−	9.90987i		−	0.330513i
$900$		0			0
$901$		−	20.5323i		−	0.684031i
$902$		0			0
$903$	−0.00795087	+	0.0937130i	−0.000264588	+	0.00311857i
$904$		0			0
$905$			5.17141i			0.171903i
$906$		0			0
$907$	−8.64134			−0.286931			−0.143465	−	0.989655i	$-0.545825\pi$
$907$	−8.64134			−0.286931			−0.143465	+	0.989655i	$0.545825\pi$
$908$		0			0
$909$	−39.6353	−	6.77429i	−1.31462	−	0.224689i
$910$		0			0
$911$	15.3490			0.508536			0.254268	−	0.967134i	$-0.418165\pi$
$911$	15.3490			0.508536			0.254268	+	0.967134i	$0.418165\pi$
$912$		0			0
$913$	−11.3176			−0.374557
$914$		0			0
$915$	−0.777062	+	9.15885i	−0.0256889	+	0.302782i
$916$		0			0
$917$	−24.8754			−0.821457
$918$		0			0
$919$			53.4873i			1.76438i	0.470892	+	0.882191i	$0.343932\pi$
$919$			53.4873i			1.76438i	−0.470892	+	0.882191i	$0.656068\pi$
$920$		0			0
$921$	24.8496	+	2.10830i	0.818821	+	0.0694710i
$922$		0			0
$923$			46.4855i			1.53009i
$924$		0			0
$925$		−	6.06225i		−	0.199326i
$926$		0			0
$927$	5.79885	−	33.9281i	0.190459	−	1.11435i
$928$		0			0
$929$		−	19.9457i		−	0.654397i	−0.944956	−	0.327199i	$-0.893895\pi$
$929$		−	19.9457i		−	0.654397i	0.944956	−	0.327199i	$-0.106105\pi$
$930$		0			0
$931$	−10.6153			−0.347903
$932$		0			0
$933$	3.40334	+	0.288749i	0.111420	+	0.00945321i
$934$		0			0
$935$	4.80692			0.157203
$936$		0			0
$937$	17.9096			0.585082			0.292541	−	0.956253i	$-0.405499\pi$
$937$	17.9096			0.585082			0.292541	+	0.956253i	$0.405499\pi$
$938$		0			0
$939$	55.2268	+	4.68559i	1.80226	+	0.152909i
$940$		0			0
$941$	37.1885			1.21231			0.606155	−	0.795347i	$-0.292711\pi$
$941$	37.1885			1.21231			0.606155	+	0.795347i	$0.292711\pi$
$942$		0			0
$943$			24.2112i			0.788425i
$944$		0			0
$945$	1.89487	−	7.30138i	0.0616401	−	0.237514i
$946$		0			0
$947$			22.7720i			0.739992i	0.929033	+	0.369996i	$0.120641\pi$
$947$			22.7720i			0.739992i	−0.929033	+	0.369996i	$0.879359\pi$
$948$		0			0
$949$		−	38.7163i		−	1.25678i
$950$		0			0
$951$	36.1619	+	3.06807i	1.17263	+	0.0994891i
$952$		0			0
$953$		−	21.2010i		−	0.686769i	−0.939195	−	0.343384i	$-0.888427\pi$
$953$		−	21.2010i		−	0.686769i	0.939195	−	0.343384i	$-0.111573\pi$
$954$		0			0
$955$	−19.5388			−0.632261
$956$		0			0
$957$	−0.532333	+	6.27435i	−0.0172079	+	0.202821i
$958$		0			0
$959$	−13.4519			−0.434384
$960$		0			0
$961$	23.4078			0.755091
$962$		0			0
$963$	9.61544	−	56.2584i	0.309853	−	1.81290i
$964$		0			0
$965$	−0.853709			−0.0274819
$966$		0			0
$967$			36.7629i			1.18222i	0.806592	+	0.591108i	$0.201309\pi$
$967$			36.7629i			1.18222i	−0.806592	+	0.591108i	$0.798691\pi$
$968$		0			0
$969$	1.51078	−	17.8068i	0.0485332	−	0.572037i
$970$		0			0
$971$			15.9890i			0.513113i	0.966529	+	0.256556i	$0.0825880\pi$
$971$			15.9890i			0.513113i	−0.966529	+	0.256556i	$0.917412\pi$
$972$		0			0
$973$			15.1931i			0.487068i
$974$		0			0
$975$	−0.548299	+	6.46254i	−0.0175596	+	0.206967i
$976$		0			0
$977$			5.85833i			0.187425i	0.995599	+	0.0937124i	$0.0298734\pi$
$977$			5.85833i			0.187425i	−0.995599	+	0.0937124i	$0.970127\pi$
$978$		0			0
$979$	−6.38616			−0.204103
$980$		0			0
$981$	8.51090	−	49.7958i	0.271732	−	1.58986i
$982$		0			0
$983$	−29.0790			−0.927477			−0.463738	−	0.885972i	$-0.653492\pi$
$983$	−29.0790			−0.927477			−0.463738	+	0.885972i	$0.653492\pi$
$984$		0			0
$985$	−19.3393			−0.616200
$986$		0			0
$987$	1.19492	−	14.0839i	0.0380347	−	0.448296i
$988$		0			0
$989$	−0.123572			−0.00392935
$990$		0			0
$991$			58.8799i			1.87038i	0.354145	+	0.935191i	$0.384772\pi$
$991$			58.8799i			1.87038i	−0.354145	+	0.935191i	$0.615228\pi$
$992$		0			0
$993$	11.3628	+	0.964048i	0.360586	+	0.0305931i
$994$		0			0
$995$			1.31942i			0.0418284i
$996$		0			0
$997$			24.3799i			0.772121i	0.922474	+	0.386060i	$0.126164\pi$
$997$			24.3799i			0.772121i	−0.922474	+	0.386060i	$0.873836\pi$
$998$		0			0
$999$	−7.91290	+	30.4903i	−0.250353	+	0.964671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.b.191.1	yes	8
3.2	odd	2		1920.2.b.a.191.2	yes	8
4.3	odd	2		1920.2.b.h.191.8	yes	8
8.3	odd	2		1920.2.b.a.191.1	✓	8
8.5	even	2		1920.2.b.g.191.8	yes	8
12.11	even	2		1920.2.b.g.191.7	yes	8
24.5	odd	2		1920.2.b.h.191.7	yes	8
24.11	even	2	inner	1920.2.b.b.191.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.a.191.1	✓	8	8.3	odd	2
1920.2.b.a.191.2	yes	8	3.2	odd	2
1920.2.b.b.191.1	yes	8	1.1	even	1	trivial
1920.2.b.b.191.2	yes	8	24.11	even	2	inner
1920.2.b.g.191.7	yes	8	12.11	even	2
1920.2.b.g.191.8	yes	8	8.5	even	2
1920.2.b.h.191.7	yes	8	24.5	odd	2
1920.2.b.h.191.8	yes	8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72585 - 0.146426i) q^{3} +1.00000 q^{5} +1.45170i q^{7} +(2.95712 + 0.505418i) q^{9} +1.01084i q^{11} -3.74455i q^{13} +(-1.72585 - 0.146426i) q^{15} -4.75539i q^{17} -2.16969 q^{19} +(0.212567 - 2.50542i) q^{21} +3.30369 q^{23} +1.00000 q^{25} +(-5.02954 - 1.30527i) q^{27} -3.59654 q^{29} +2.75539i q^{31} +(0.148013 - 1.74455i) q^{33} +1.45170i q^{35} -6.06225i q^{37} +(-0.548299 + 6.46254i) q^{39} +7.32853i q^{41} -0.0374041 q^{43} +(2.95712 + 0.505418i) q^{45} +5.62139 q^{47} +4.89257 q^{49} +(-0.696312 + 8.20709i) q^{51} +4.31770 q^{53} +1.01084i q^{55} +(3.74455 + 0.317698i) q^{57} -11.9142i q^{59} -5.30686i q^{61} +(-0.733716 + 4.29285i) q^{63} -3.74455i q^{65} +8.03740 q^{67} +(-5.70167 - 0.483745i) q^{69} -12.4142 q^{71} +10.3394 q^{73} +(-1.72585 - 0.146426i) q^{75} -1.46743 q^{77} -11.3411i q^{79} +(8.48910 + 2.98916i) q^{81} +11.1963i q^{83} -4.75539i q^{85} +(6.20709 + 0.526626i) q^{87} +6.31770i q^{89} +5.43597 q^{91} +(0.403460 - 4.75539i) q^{93} -2.16969 q^{95} +9.48910 q^{97} +(-0.510895 + 2.98916i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 8 q^{5} - 4 q^{9} - 2 q^{15} + 8 q^{19} + 4 q^{21} + 12 q^{23} + 8 q^{25} - 14 q^{27} - 8 q^{29} - 8 q^{33} - 28 q^{39} + 36 q^{43} - 4 q^{45} - 4 q^{47} - 20 q^{51} + 16 q^{63} + 28 q^{67}+ \cdots - 64 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 + 8 * q^5 - 4 * q^9 - 2 * q^15 + 8 * q^19 + 4 * q^21 + 12 * q^23 + 8 * q^25 - 14 * q^27 - 8 * q^29 - 8 * q^33 - 28 * q^39 + 36 * q^43 - 4 * q^45 - 4 * q^47 - 20 * q^51 + 16 * q^63 + 28 * q^67 + 12 * q^69 + 24 * q^71 + 32 * q^73 - 2 * q^75 + 32 * q^77 + 8 * q^81 - 12 * q^87 + 40 * q^91 + 24 * q^93 + 8 * q^95 + 16 * q^97 - 64 * q^99

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