LMFDB - Embedded newform 1920.2.m.r.959.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(959,1920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1920.959"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,12,0,0,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			959.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.r.959.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61803 - 0.618034i) q^{3} +2.23607 q^{5} +0.763932 q^{7} +(2.23607 - 2.00000i) q^{9} +(3.61803 - 1.38197i) q^{15} +(1.23607 - 0.472136i) q^{21} +7.70820i q^{23} +5.00000 q^{25} +(2.38197 - 4.61803i) q^{27} +6.00000 q^{29} +1.70820 q^{35} -12.0000i q^{41} -6.76393i q^{43} +(5.00000 - 4.47214i) q^{45} +0.291796i q^{47} -6.41641 q^{49} +8.00000i q^{61} +(1.70820 - 1.52786i) q^{63} +14.1803i q^{67} +(4.76393 + 12.4721i) q^{69} +(8.09017 - 3.09017i) q^{75} +(1.00000 - 8.94427i) q^{81} -17.7082 q^{83} +(9.70820 - 3.70820i) q^{87} -17.8885i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 12 q^{7} + 10 q^{15} - 4 q^{21} + 20 q^{25} + 14 q^{27} + 24 q^{29} - 20 q^{35} + 20 q^{45} + 28 q^{49} - 20 q^{63} + 28 q^{69} + 10 q^{75} + 4 q^{81} - 44 q^{83} + 12 q^{87}+O(q^{100}) $ 4 * q + 2 * q^3 + 12 * q^7 + 10 * q^15 - 4 * q^21 + 20 * q^25 + 14 * q^27 + 24 * q^29 - 20 * q^35 + 20 * q^45 + 28 * q^49 - 20 * q^63 + 28 * q^69 + 10 * q^75 + 4 * q^81 - 44 * q^83 + 12 * q^87

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
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$4$		0			0
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$		0			0
$7$	0.763932			0.288739			0.144370	−	0.989524i	$-0.453885\pi$
$7$	0.763932			0.288739			0.144370	+	0.989524i	$0.453885\pi$
$8$		0			0
$9$	2.23607	−	2.00000i	0.745356	−	0.666667i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	3.61803	−	1.38197i	0.934172	−	0.356822i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	1.23607	−	0.472136i	0.269732	−	0.103029i
$22$		0			0
$23$			7.70820i			1.60727i	0.595121	+	0.803636i	$0.297104\pi$
$23$			7.70820i			1.60727i	−0.595121	+	0.803636i	$0.702896\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	2.38197	−	4.61803i	0.458410	−	0.888741i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.70820			0.288739
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	12.0000i		−	1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$		−	12.0000i		−	1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$		0			0
$43$		−	6.76393i		−	1.03149i	−0.856742	−	0.515745i	$-0.827515\pi$
$43$		−	6.76393i		−	1.03149i	0.856742	−	0.515745i	$-0.172485\pi$
$44$		0			0
$45$	5.00000	−	4.47214i	0.745356	−	0.666667i
$46$		0			0
$47$			0.291796i			0.0425628i	0.999774	+	0.0212814i	$0.00677460\pi$
$47$			0.291796i			0.0425628i	−0.999774	+	0.0212814i	$0.993225\pi$
$48$		0			0
$49$	−6.41641			−0.916630
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$	1.70820	−	1.52786i	0.215213	−	0.192493i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			14.1803i			1.73240i	0.499694	+	0.866202i	$0.333446\pi$
$67$			14.1803i			1.73240i	−0.499694	+	0.866202i	$0.666554\pi$
$68$		0			0
$69$	4.76393	+	12.4721i	0.573510	+	1.50147i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	8.09017	−	3.09017i	0.934172	−	0.356822i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−17.7082			−1.94373			−0.971864	−	0.235543i	$-0.924313\pi$
$83$	−17.7082			−1.94373			−0.971864	+	0.235543i	$0.924313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.70820	−	3.70820i	1.04083	−	0.397561i
$88$		0			0
$89$		−	17.8885i		−	1.89618i	−0.317999	−	0.948091i	$-0.603011\pi$
$89$		−	17.8885i		−	1.89618i	0.317999	−	0.948091i	$-0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−20.1803			−1.98843			−0.994214	−	0.107418i	$-0.965742\pi$
$103$	−20.1803			−1.98843			−0.994214	+	0.107418i	$0.965742\pi$
$104$		0			0
$105$	2.76393	−	1.05573i	0.269732	−	0.103029i
$106$		0			0
$107$	6.29180			0.608251			0.304125	−	0.952632i	$-0.401636\pi$
$107$	6.29180			0.608251			0.304125	+	0.952632i	$0.401636\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$			17.2361i			1.60727i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−7.41641	−	19.4164i	−0.668715	−	1.75072i
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$	−18.6525			−1.65514			−0.827570	−	0.561363i	$-0.810277\pi$
$127$	−18.6525			−1.65514			−0.827570	+	0.561363i	$0.810277\pi$
$128$		0			0
$129$	−4.18034	−	10.9443i	−0.368058	−	0.963589i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	5.32624	−	10.3262i	0.458410	−	0.888741i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	0.180340	+	0.472136i	0.0151874	+	0.0397610i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	13.4164			1.11417
$146$		0			0
$147$	−10.3820	+	3.96556i	−0.856290	+	0.327074i
$148$		0			0
$149$	−4.47214			−0.366372			−0.183186	−	0.983078i	$-0.558641\pi$
$149$	−4.47214			−0.366372			−0.183186	+	0.983078i	$0.558641\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			5.88854i			0.464082i
$162$		0			0
$163$			24.6525i			1.93093i	0.260531	+	0.965465i	$0.416102\pi$
$163$			24.6525i			1.93093i	−0.260531	+	0.965465i	$0.583898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	23.7082i		−	1.83460i	−0.398202	−	0.917298i	$-0.630366\pi$
$167$		−	23.7082i		−	1.83460i	0.398202	−	0.917298i	$-0.369634\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	3.81966			0.288739
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	26.8328i		−	1.99447i	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$		−	26.8328i		−	1.99447i	0.0743294	−	0.997234i	$-0.476318\pi$
$182$		0			0
$183$	4.94427	+	12.9443i	0.365491	+	0.956868i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.81966	−	3.52786i	0.132361	−	0.256614i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	8.76393	+	22.9443i	0.618160	+	1.61836i
$202$		0			0
$203$	4.58359			0.321705
$204$		0			0
$205$		−	26.8328i		−	1.87409i
$206$		0			0
$207$	15.4164	+	17.2361i	1.07151	+	1.19799i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	15.1246i		−	1.03149i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−23.2361			−1.55600			−0.778001	−	0.628263i	$-0.783766\pi$
$223$	−23.2361			−1.55600			−0.778001	+	0.628263i	$0.783766\pi$
$224$		0			0
$225$	11.1803	−	10.0000i	0.745356	−	0.666667i
$226$		0			0
$227$	13.1246			0.871111			0.435556	−	0.900162i	$-0.356552\pi$
$227$	13.1246			0.871111			0.435556	+	0.900162i	$0.356552\pi$
$228$		0			0
$229$			26.8328i			1.77316i	0.462573	+	0.886581i	$0.346926\pi$
$229$			26.8328i			1.77316i	−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			0.652476i			0.0425628i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	13.4164			0.864227			0.432113	−	0.901819i	$-0.357768\pi$
$241$	13.4164			0.864227			0.432113	+	0.901819i	$0.357768\pi$
$242$		0			0
$243$	−3.90983	−	15.0902i	−0.250816	−	0.968035i
$244$		0			0
$245$	−14.3475			−0.916630
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−28.6525	+	10.9443i	−1.81578	+	0.693565i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	13.4164	−	12.0000i	0.830455	−	0.742781i
$262$		0			0
$263$			31.1246i			1.91923i	0.281324	+	0.959613i	$0.409226\pi$
$263$			31.1246i			1.91923i	−0.281324	+	0.959613i	$0.590774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−11.0557	−	28.9443i	−0.676600	−	1.77136i
$268$		0			0
$269$	−22.3607			−1.36335			−0.681677	−	0.731653i	$-0.738749\pi$
$269$	−22.3607			−1.36335			−0.681677	+	0.731653i	$0.738749\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$		−	9.81966i		−	0.583718i	−0.956461	−	0.291859i	$-0.905726\pi$
$283$		−	9.81966i		−	0.583718i	0.956461	−	0.291859i	$-0.0942738\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	9.16718i		−	0.541122i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	5.16718i		−	0.297832i
$302$		0			0
$303$	29.1246	−	11.1246i	1.67317	−	0.639092i
$304$		0			0
$305$			17.8885i			1.02430i
$306$		0			0
$307$			21.5967i			1.23259i	0.787515	+	0.616296i	$0.211367\pi$
$307$			21.5967i			1.23259i	−0.787515	+	0.616296i	$0.788633\pi$
$308$		0			0
$309$	−32.6525	+	12.4721i	−1.85753	+	0.709515i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	3.81966	−	3.41641i	0.215213	−	0.192493i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	10.1803	−	3.88854i	0.568211	−	0.217037i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	9.88854	+	25.8885i	0.546838	+	1.43164i
$328$		0			0
$329$			0.222912i			0.0122896i
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			31.7082i			1.73240i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.2492			−0.553406
$344$		0			0
$345$	10.6525	+	27.8885i	0.573510	+	1.50147i
$346$		0			0
$347$	37.1246			1.99295			0.996477	−	0.0838690i	$-0.0267277\pi$
$347$	37.1246			1.99295			0.996477	+	0.0838690i	$0.0267277\pi$
$348$		0			0
$349$			26.8328i			1.43633i	0.695874	+	0.718164i	$0.255017\pi$
$349$			26.8328i			1.43633i	−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	17.7984	−	6.79837i	0.934172	−	0.356822i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	24.7639			1.29267			0.646333	−	0.763055i	$-0.276302\pi$
$367$	24.7639			1.29267			0.646333	+	0.763055i	$0.276302\pi$
$368$		0			0
$369$	−24.0000	−	26.8328i	−1.24939	−	1.39686i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	18.0902	−	6.90983i	0.934172	−	0.356822i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−30.1803	+	11.5279i	−1.54619	+	0.590590i
$382$		0			0
$383$		−	39.1246i		−	1.99917i	−0.0287325	−	0.999587i	$-0.509147\pi$
$383$		−	39.1246i		−	1.99917i	0.0287325	−	0.999587i	$-0.490853\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.5279	−	15.1246i	−0.687660	−	0.768827i
$388$		0			0
$389$	−31.3050			−1.58722			−0.793612	−	0.608424i	$-0.791802\pi$
$389$	−31.3050			−1.58722			−0.793612	+	0.608424i	$0.791802\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			35.7771i			1.78662i	0.449439	+	0.893311i	$0.351624\pi$
$401$			35.7771i			1.78662i	−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.23607	−	20.0000i	0.111111	−	0.993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−40.2492			−1.99020			−0.995098	−	0.0988936i	$-0.968470\pi$
$409$	−40.2492			−1.99020			−0.995098	+	0.0988936i	$0.968470\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−39.5967			−1.94373
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	8.00000i		−	0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$421$		−	8.00000i		−	0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$		0			0
$423$	0.583592	+	0.652476i	0.0283752	+	0.0317245i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			6.11146i			0.295754i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	21.7082	−	8.29180i	1.04083	−	0.397561i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−14.3475	+	12.8328i	−0.683215	+	0.611086i
$442$		0			0
$443$	−22.2918			−1.05912			−0.529558	−	0.848274i	$-0.677642\pi$
$443$	−22.2918			−1.05912			−0.529558	+	0.848274i	$0.677642\pi$
$444$		0			0
$445$		−	40.0000i		−	1.89618i
$446$		0			0
$447$	−7.23607	+	2.76393i	−0.342254	+	0.130729i
$448$		0			0
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	38.0689			1.76921			0.884606	−	0.466340i	$-0.154428\pi$
$463$	38.0689			1.76921			0.884606	+	0.466340i	$0.154428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.87539			0.133057			0.0665285	−	0.997785i	$-0.478808\pi$
$467$	2.87539			0.133057			0.0665285	+	0.997785i	$0.478808\pi$
$468$		0			0
$469$			10.8328i			0.500213i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	3.63932	+	9.52786i	0.165595	+	0.433533i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−42.6525			−1.93277			−0.966384	−	0.257103i	$-0.917232\pi$
$487$	−42.6525			−1.93277			−0.966384	+	0.257103i	$0.917232\pi$
$488$		0			0
$489$	15.2361	+	39.8885i	0.688999	+	1.80382i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−14.6525	−	38.3607i	−0.654624	−	1.71383i
$502$		0			0
$503$			24.2918i			1.08312i	0.840663	+	0.541559i	$0.182166\pi$
$503$			24.2918i			1.08312i	−0.840663	+	0.541559i	$0.817834\pi$
$504$		0			0
$505$	40.2492			1.79107
$506$		0			0
$507$	−21.0344	+	8.03444i	−0.934172	+	0.356822i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−45.1246			−1.98843
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	17.8885i		−	0.783711i	−0.920027	−	0.391856i	$-0.871833\pi$
$521$		−	17.8885i		−	0.783711i	0.920027	−	0.391856i	$-0.128167\pi$
$522$		0			0
$523$			45.5967i			1.99381i	0.0786374	+	0.996903i	$0.474943\pi$
$523$			45.5967i			1.99381i	−0.0786374	+	0.996903i	$0.525057\pi$
$524$		0			0
$525$	6.18034	−	2.36068i	0.269732	−	0.103029i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−36.4164			−1.58332
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	14.0689			0.608251
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	26.8328i		−	1.15363i	−0.816874	−	0.576816i	$-0.804295\pi$
$541$		−	26.8328i		−	1.15363i	0.816874	−	0.576816i	$-0.195705\pi$
$542$		0			0
$543$	−16.5836	−	43.4164i	−0.711670	−	1.86318i
$544$		0			0
$545$			35.7771i			1.53252i
$546$		0			0
$547$		−	30.7639i		−	1.31537i	−0.753293	−	0.657685i	$-0.771536\pi$
$547$		−	30.7639i		−	1.31537i	0.753293	−	0.657685i	$-0.228464\pi$
$548$		0			0
$549$	16.0000	+	17.8885i	0.682863	+	0.763464i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	32.5410			1.37144			0.685720	−	0.727865i	$-0.259487\pi$
$563$	32.5410			1.37144			0.685720	+	0.727865i	$0.259487\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.763932	−	6.83282i	0.0320821	−	0.286951i
$568$		0			0
$569$		−	36.0000i		−	1.50920i	−0.656186	−	0.754599i	$-0.727831\pi$
$569$		−	36.0000i		−	1.50920i	0.656186	−	0.754599i	$-0.272169\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			38.5410i			1.60727i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.5279			−0.561230
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−40.5410			−1.67331			−0.836653	−	0.547733i	$-0.815491\pi$
$587$	−40.5410			−1.67331			−0.836653	+	0.547733i	$0.815491\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	40.2492			1.64180			0.820900	−	0.571072i	$-0.193472\pi$
$601$	40.2492			1.64180			0.820900	+	0.571072i	$0.193472\pi$
$602$		0			0
$603$	28.3607	+	31.7082i	1.15494	+	1.29126i
$604$		0			0
$605$	24.5967			1.00000
$606$		0			0
$607$	−44.1803			−1.79322			−0.896612	−	0.442816i	$-0.853979\pi$
$607$	−44.1803			−1.79322			−0.896612	+	0.442816i	$0.853979\pi$
$608$		0			0
$609$	7.41641	−	2.83282i	0.300528	−	0.114791i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$	−16.5836	−	43.4164i	−0.668715	−	1.75072i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	35.5967	+	18.3607i	1.42845	+	0.736789i
$622$		0			0
$623$		−	13.6656i		−	0.547502i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−41.7082			−1.65514
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12.0000i			0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$			12.0000i			0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$		0			0
$643$		−	8.06888i		−	0.318206i	−0.987262	−	0.159103i	$-0.949140\pi$
$643$		−	8.06888i		−	0.318206i	0.987262	−	0.159103i	$-0.0508601\pi$
$644$		0			0
$645$	−9.34752	−	24.4721i	−0.368058	−	0.963589i
$646$		0			0
$647$		−	46.5410i		−	1.82972i	−0.403775	−	0.914858i	$-0.632302\pi$
$647$		−	46.5410i		−	1.82972i	0.403775	−	0.914858i	$-0.367698\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			32.0000i			1.24466i	0.782757	+	0.622328i	$0.213813\pi$
$661$			32.0000i			1.24466i	−0.782757	+	0.622328i	$0.786187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			46.2492i			1.79078i
$668$		0			0
$669$	−37.5967	+	14.3607i	−1.45357	+	0.555216i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	11.9098	−	23.0902i	0.458410	−	0.888741i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	21.2361	−	8.11146i	0.813768	−	0.310832i
$682$		0			0
$683$	−10.8754			−0.416135			−0.208068	−	0.978114i	$-0.566717\pi$
$683$	−10.8754			−0.416135			−0.208068	+	0.978114i	$0.566717\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	16.5836	+	43.4164i	0.632704	+	1.65644i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.3607			0.844551			0.422276	−	0.906467i	$-0.361231\pi$
$701$	22.3607			0.844551			0.422276	+	0.906467i	$0.361231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.403252	+	1.05573i	0.0151874	+	0.0397610i
$706$		0			0
$707$	13.7508			0.517151
$708$		0			0
$709$		−	26.8328i		−	1.00773i	−0.863783	−	0.503864i	$-0.831911\pi$
$709$		−	26.8328i		−	1.00773i	0.863783	−	0.503864i	$-0.168089\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−15.4164			−0.574137
$722$		0			0
$723$	21.7082	−	8.29180i	0.807337	−	0.308375i
$724$		0			0
$725$	30.0000			1.11417
$726$		0			0
$727$	35.0132			1.29857			0.649283	−	0.760547i	$-0.275069\pi$
$727$	35.0132			1.29857			0.649283	+	0.760547i	$0.275069\pi$
$728$		0			0
$729$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−23.2148	+	8.86726i	−0.856290	+	0.327074i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			14.5410i			0.533458i	0.963772	+	0.266729i	$0.0859429\pi$
$743$			14.5410i			0.533458i	−0.963772	+	0.266729i	$0.914057\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	−39.5967	+	35.4164i	−1.44877	+	1.29582i
$748$		0			0
$749$	4.80650			0.175626
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	35.7771i		−	1.29692i	−0.761249	−	0.648459i	$-0.775414\pi$
$761$		−	35.7771i		−	1.29692i	0.761249	−	0.648459i	$-0.224586\pi$
$762$		0			0
$763$			12.2229i			0.442499i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	14.2918	−	27.7082i	0.510747	−	0.990210i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	56.0689i		−	1.99864i	−0.0368739	−	0.999320i	$-0.511740\pi$
$787$		−	56.0689i		−	1.99864i	0.0368739	−	0.999320i	$-0.488260\pi$
$788$		0			0
$789$	19.2361	+	50.3607i	0.684822	+	1.79289i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−35.7771	−	40.0000i	−1.26412	−	1.41333i
$802$		0			0
$803$		0			0
$804$		0			0
$805$			13.1672i			0.464082i
$806$		0			0
$807$	−36.1803	+	13.8197i	−1.27361	+	0.486475i
$808$		0			0
$809$		−	17.8885i		−	0.628928i	−0.949269	−	0.314464i	$-0.898175\pi$
$809$		−	17.8885i		−	0.628928i	0.949269	−	0.314464i	$-0.101825\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			55.1246i			1.93093i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	31.3050			1.09255			0.546275	−	0.837606i	$-0.316045\pi$
$821$	31.3050			1.09255			0.546275	+	0.837606i	$0.316045\pi$
$822$		0			0
$823$	27.8197			0.969732			0.484866	−	0.874588i	$-0.338868\pi$
$823$	27.8197			0.969732			0.484866	+	0.874588i	$0.338868\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	56.5410			1.96612			0.983062	−	0.183274i	$-0.0586694\pi$
$827$	56.5410			1.96612			0.983062	+	0.183274i	$0.0586694\pi$
$828$		0			0
$829$			56.0000i			1.94496i	0.232986	+	0.972480i	$0.425151\pi$
$829$			56.0000i			1.94496i	−0.232986	+	0.972480i	$0.574849\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	53.0132i		−	1.83460i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	7.41641	+	19.4164i	0.255435	+	0.668737i
$844$		0			0
$845$	−29.0689			−1.00000
$846$		0			0
$847$	8.40325			0.288739
$848$		0			0
$849$	−6.06888	−	15.8885i	−0.208284	−	0.545293i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	−5.66563	−	14.8328i	−0.193084	−	0.505501i
$862$		0			0
$863$		−	47.7082i		−	1.62401i	−0.583653	−	0.812003i	$-0.698377\pi$
$863$		−	47.7082i		−	1.62401i	0.583653	−	0.812003i	$-0.301623\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−27.5066	+	10.5066i	−0.934172	+	0.356822i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.54102			0.288739
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.0000i			0.404290i	0.979356	+	0.202145i	$0.0647913\pi$
$881$			12.0000i			0.404290i	−0.979356	+	0.202145i	$0.935209\pi$
$882$		0			0
$883$		−	23.3475i		−	0.785707i	−0.919601	−	0.392853i	$-0.871488\pi$
$883$		−	23.3475i		−	0.785707i	0.919601	−	0.392853i	$-0.128512\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.8754i		−	0.566620i	−0.959028	−	0.283310i	$-0.908567\pi$
$887$		−	16.8754i		−	0.566620i	0.959028	−	0.283310i	$-0.0914325\pi$
$888$		0			0
$889$	−14.2492			−0.477904
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−3.19350	−	8.36068i	−0.106273	−	0.278226i
$904$		0			0
$905$		−	60.0000i		−	1.99447i
$906$		0			0
$907$		−	39.4853i		−	1.31109i	−0.755157	−	0.655544i	$-0.772439\pi$
$907$		−	39.4853i		−	1.31109i	0.755157	−	0.655544i	$-0.227561\pi$
$908$		0			0
$909$	40.2492	−	36.0000i	1.33498	−	1.19404i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	11.0557	+	28.9443i	0.365491	+	0.956868i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	13.3475	+	34.9443i	0.439816	+	1.15145i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−45.1246	+	40.3607i	−1.48209	+	1.32562i
$928$		0			0
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	42.0000			1.36916			0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000			1.36916			0.684580	+	0.728937i	$0.259985\pi$
$942$		0			0
$943$	92.4984			3.01216
$944$		0			0
$945$	4.06888	−	7.88854i	0.132361	−	0.256614i
$946$		0			0
$947$	49.7082			1.61530			0.807650	−	0.589662i	$-0.200739\pi$
$947$	49.7082			1.61530			0.807650	+	0.589662i	$0.200739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	14.0689	−	12.5836i	0.453363	−	0.405501i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	62.0689			1.99600			0.998000	−	0.0632081i	$-0.0201332\pi$
$967$	62.0689			1.99600			0.998000	+	0.0632081i	$0.0201332\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	32.0000	+	35.7771i	1.02168	+	1.14227i
$982$		0			0
$983$			62.5410i			1.99475i	0.0724180	+	0.997374i	$0.476928\pi$
$983$			62.5410i			1.99475i	−0.0724180	+	0.997374i	$0.523072\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.137767	+	0.360680i	0.00438519	+	0.0114806i
$988$		0			0
$989$	52.1378			1.65788
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.m.r.959.3	yes	4
3.2	odd	2		1920.2.m.f.959.1	yes	4
4.3	odd	2		1920.2.m.e.959.2	yes	4
5.4	even	2		1920.2.m.e.959.2	yes	4
8.3	odd	2		1920.2.m.q.959.3	yes	4
8.5	even	2		1920.2.m.f.959.2	yes	4
12.11	even	2		1920.2.m.q.959.4	yes	4
15.14	odd	2		1920.2.m.q.959.4	yes	4
20.19	odd	2	CM	1920.2.m.r.959.3	yes	4
24.5	odd	2	inner	1920.2.m.r.959.4	yes	4
24.11	even	2		1920.2.m.e.959.1	✓	4
40.19	odd	2		1920.2.m.f.959.2	yes	4
40.29	even	2		1920.2.m.q.959.3	yes	4
60.59	even	2		1920.2.m.f.959.1	yes	4
120.29	odd	2		1920.2.m.e.959.1	✓	4
120.59	even	2	inner	1920.2.m.r.959.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.m.e.959.1	✓	4	24.11	even	2
1920.2.m.e.959.1	✓	4	120.29	odd	2
1920.2.m.e.959.2	yes	4	4.3	odd	2
1920.2.m.e.959.2	yes	4	5.4	even	2
1920.2.m.f.959.1	yes	4	3.2	odd	2
1920.2.m.f.959.1	yes	4	60.59	even	2
1920.2.m.f.959.2	yes	4	8.5	even	2
1920.2.m.f.959.2	yes	4	40.19	odd	2
1920.2.m.q.959.3	yes	4	8.3	odd	2
1920.2.m.q.959.3	yes	4	40.29	even	2
1920.2.m.q.959.4	yes	4	12.11	even	2
1920.2.m.q.959.4	yes	4	15.14	odd	2
1920.2.m.r.959.3	yes	4	1.1	even	1	trivial
1920.2.m.r.959.3	yes	4	20.19	odd	2	CM
1920.2.m.r.959.4	yes	4	24.5	odd	2	inner
1920.2.m.r.959.4	yes	4	120.59	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.61803 - 0.618034i) q^{3} +2.23607 q^{5} +0.763932 q^{7} +(2.23607 - 2.00000i) q^{9} +(3.61803 - 1.38197i) q^{15} +(1.23607 - 0.472136i) q^{21} +7.70820i q^{23} +5.00000 q^{25} +(2.38197 - 4.61803i) q^{27} +6.00000 q^{29} +1.70820 q^{35} -12.0000i q^{41} -6.76393i q^{43} +(5.00000 - 4.47214i) q^{45} +0.291796i q^{47} -6.41641 q^{49} +8.00000i q^{61} +(1.70820 - 1.52786i) q^{63} +14.1803i q^{67} +(4.76393 + 12.4721i) q^{69} +(8.09017 - 3.09017i) q^{75} +(1.00000 - 8.94427i) q^{81} -17.7082 q^{83} +(9.70820 - 3.70820i) q^{87} -17.8885i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 12 q^{7} + 10 q^{15} - 4 q^{21} + 20 q^{25} + 14 q^{27} + 24 q^{29} - 20 q^{35} + 20 q^{45} + 28 q^{49} - 20 q^{63} + 28 q^{69} + 10 q^{75} + 4 q^{81} - 44 q^{83} + 12 q^{87}+O(q^{100}) \) 4 * q + 2 * q^3 + 12 * q^7 + 10 * q^15 - 4 * q^21 + 20 * q^25 + 14 * q^27 + 24 * q^29 - 20 * q^35 + 20 * q^45 + 28 * q^49 - 20 * q^63 + 28 * q^69 + 10 * q^75 + 4 * q^81 - 44 * q^83 + 12 * q^87

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