LMFDB - Embedded newform 1920.2.h.l.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(1151,1920)]

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, 0, 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("1920.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.h (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:	$4$
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 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1920.1151
Dual form			1920.2.h.l.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.61803 + 0.618034i) q^{3} +1.00000i q^{5} -2.00000i q^{7} +(2.23607 + 2.00000i) q^{9} -6.47214 q^{11} +3.23607 q^{13} +(-0.618034 + 1.61803i) q^{15} +5.23607i q^{17} -0.763932i q^{19} +(1.23607 - 3.23607i) q^{21} +8.47214 q^{23} -1.00000 q^{25} +(2.38197 + 4.61803i) q^{27} +0.472136i q^{29} +9.23607i q^{31} +(-10.4721 - 4.00000i) q^{33} +2.00000 q^{35} +5.70820 q^{37} +(5.23607 + 2.00000i) q^{39} +10.4721i q^{41} +3.70820i q^{43} +(-2.00000 + 2.23607i) q^{45} +4.47214 q^{47} +3.00000 q^{49} +(-3.23607 + 8.47214i) q^{51} +4.47214i q^{53} -6.47214i q^{55} +(0.472136 - 1.23607i) q^{57} -2.47214 q^{59} -3.52786 q^{61} +(4.00000 - 4.47214i) q^{63} +3.23607i q^{65} -6.76393i q^{67} +(13.7082 + 5.23607i) q^{69} -4.00000 q^{71} +8.47214 q^{73} +(-1.61803 - 0.618034i) q^{75} +12.9443i q^{77} -14.1803i q^{79} +(1.00000 + 8.94427i) q^{81} +1.70820 q^{83} -5.23607 q^{85} +(-0.291796 + 0.763932i) q^{87} -10.4721i q^{89} -6.47214i q^{91} +(-5.70820 + 14.9443i) q^{93} +0.763932 q^{95} -3.52786 q^{97} +(-14.4721 - 12.9443i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 8 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{21} + 16 q^{23} - 4 q^{25} + 14 q^{27} - 24 q^{33} + 8 q^{35} - 4 q^{37} + 12 q^{39} - 8 q^{45} + 12 q^{49} - 4 q^{51} - 16 q^{57} + 8 q^{59} - 32 q^{61}+ \cdots - 40 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 8 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^21 + 16 * q^23 - 4 * q^25 + 14 * q^27 - 24 * q^33 + 8 * q^35 - 4 * q^37 + 12 * q^39 - 8 * q^45 + 12 * q^49 - 4 * q^51 - 16 * q^57 + 8 * q^59 - 32 * q^61 + 16 * q^63 + 28 * q^69 - 16 * q^71 + 16 * q^73 - 2 * q^75 + 4 * q^81 - 20 * q^83 - 12 * q^85 - 28 * q^87 + 4 * q^93 + 12 * q^95 - 32 * q^97 - 40 * 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
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	2.23607	+	2.00000i	0.745356	+	0.666667i
$10$		0			0
$11$	−6.47214			−1.95142			−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214			−1.95142			−0.975711	+	0.219061i	$0.929701\pi$
$12$		0			0
$13$	3.23607			0.897524			0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607			0.897524			0.448762	+	0.893651i	$0.351865\pi$
$14$		0			0
$15$	−0.618034	+	1.61803i	−0.159576	+	0.417775i
$16$		0			0
$17$			5.23607i			1.26993i	0.772540	+	0.634967i	$0.218986\pi$
$17$			5.23607i			1.26993i	−0.772540	+	0.634967i	$0.781014\pi$
$18$		0			0
$19$		−	0.763932i		−	0.175258i	−0.996153	−	0.0876290i	$-0.972071\pi$
$19$		−	0.763932i		−	0.175258i	0.996153	−	0.0876290i	$-0.0279290\pi$
$20$		0			0
$21$	1.23607	−	3.23607i	0.269732	−	0.706168i
$22$		0			0
$23$	8.47214			1.76656			0.883281	−	0.468844i	$-0.155329\pi$
$23$	8.47214			1.76656			0.883281	+	0.468844i	$0.155329\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	2.38197	+	4.61803i	0.458410	+	0.888741i
$28$		0			0
$29$			0.472136i			0.0876734i	0.999039	+	0.0438367i	$0.0139581\pi$
$29$			0.472136i			0.0876734i	−0.999039	+	0.0438367i	$0.986042\pi$
$30$		0			0
$31$			9.23607i			1.65885i	0.558620	+	0.829423i	$0.311331\pi$
$31$			9.23607i			1.65885i	−0.558620	+	0.829423i	$0.688669\pi$
$32$		0			0
$33$	−10.4721	−	4.00000i	−1.82296	−	0.696311i
$34$		0			0
$35$	2.00000			0.338062
$36$		0			0
$37$	5.70820			0.938423			0.469211	−	0.883086i	$-0.344538\pi$
$37$	5.70820			0.938423			0.469211	+	0.883086i	$0.344538\pi$
$38$		0			0
$39$	5.23607	+	2.00000i	0.838442	+	0.320256i
$40$		0			0
$41$			10.4721i			1.63547i	0.575593	+	0.817736i	$0.304771\pi$
$41$			10.4721i			1.63547i	−0.575593	+	0.817736i	$0.695229\pi$
$42$		0			0
$43$			3.70820i			0.565496i	0.959194	+	0.282748i	$0.0912460\pi$
$43$			3.70820i			0.565496i	−0.959194	+	0.282748i	$0.908754\pi$
$44$		0			0
$45$	−2.00000	+	2.23607i	−0.298142	+	0.333333i
$46$		0			0
$47$	4.47214			0.652328			0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214			0.652328			0.326164	+	0.945313i	$0.394244\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−3.23607	+	8.47214i	−0.453140	+	1.18634i
$52$		0			0
$53$			4.47214i			0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$			4.47214i			0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$		0			0
$55$		−	6.47214i		−	0.872703i
$56$		0			0
$57$	0.472136	−	1.23607i	0.0625359	−	0.163721i
$58$		0			0
$59$	−2.47214			−0.321845			−0.160922	−	0.986967i	$-0.551447\pi$
$59$	−2.47214			−0.321845			−0.160922	+	0.986967i	$0.551447\pi$
$60$		0			0
$61$	−3.52786			−0.451697			−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786			−0.451697			−0.225848	+	0.974162i	$0.572515\pi$
$62$		0			0
$63$	4.00000	−	4.47214i	0.503953	−	0.563436i
$64$		0			0
$65$			3.23607i			0.401385i
$66$		0			0
$67$		−	6.76393i		−	0.826346i	−0.910653	−	0.413173i	$-0.864421\pi$
$67$		−	6.76393i		−	0.826346i	0.910653	−	0.413173i	$-0.135579\pi$
$68$		0			0
$69$	13.7082	+	5.23607i	1.65027	+	0.630349i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	8.47214			0.991589			0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214			0.991589			0.495794	+	0.868440i	$0.334877\pi$
$74$		0			0
$75$	−1.61803	−	0.618034i	−0.186834	−	0.0713644i
$76$		0			0
$77$			12.9443i			1.47514i
$78$		0			0
$79$		−	14.1803i		−	1.59541i	−0.603046	−	0.797706i	$-0.706046\pi$
$79$		−	14.1803i		−	1.59541i	0.603046	−	0.797706i	$-0.293954\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	1.70820			0.187500			0.0937499	−	0.995596i	$-0.470115\pi$
$83$	1.70820			0.187500			0.0937499	+	0.995596i	$0.470115\pi$
$84$		0			0
$85$	−5.23607			−0.567931
$86$		0			0
$87$	−0.291796	+	0.763932i	−0.0312838	+	0.0819021i
$88$		0			0
$89$		−	10.4721i		−	1.11004i	−0.831836	−	0.555022i	$-0.812710\pi$
$89$		−	10.4721i		−	1.11004i	0.831836	−	0.555022i	$-0.187290\pi$
$90$		0			0
$91$		−	6.47214i		−	0.678464i
$92$		0			0
$93$	−5.70820	+	14.9443i	−0.591913	+	1.54965i
$94$		0			0
$95$	0.763932			0.0783778
$96$		0			0
$97$	−3.52786			−0.358200			−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786			−0.358200			−0.179100	+	0.983831i	$0.557319\pi$
$98$		0			0
$99$	−14.4721	−	12.9443i	−1.45450	−	1.30095i
$100$		0			0
$101$		−	9.41641i		−	0.936968i	−0.883472	−	0.468484i	$-0.844800\pi$
$101$		−	9.41641i		−	0.936968i	0.883472	−	0.468484i	$-0.155200\pi$
$102$		0			0
$103$		−	8.47214i		−	0.834784i	−0.908726	−	0.417392i	$-0.862944\pi$
$103$		−	8.47214i		−	0.834784i	0.908726	−	0.417392i	$-0.137056\pi$
$104$		0			0
$105$	3.23607	+	1.23607i	0.315808	+	0.120628i
$106$		0			0
$107$	−8.76393			−0.847241			−0.423621	−	0.905840i	$-0.639241\pi$
$107$	−8.76393			−0.847241			−0.423621	+	0.905840i	$0.639241\pi$
$108$		0			0
$109$	−18.9443			−1.81453			−0.907266	−	0.420557i	$-0.861835\pi$
$109$	−18.9443			−1.81453			−0.907266	+	0.420557i	$0.861835\pi$
$110$		0			0
$111$	9.23607	+	3.52786i	0.876649	+	0.334850i
$112$		0			0
$113$			19.1246i			1.79909i	0.436826	+	0.899546i	$0.356103\pi$
$113$			19.1246i			1.79909i	−0.436826	+	0.899546i	$0.643897\pi$
$114$		0			0
$115$			8.47214i			0.790031i
$116$		0			0
$117$	7.23607	+	6.47214i	0.668975	+	0.598349i
$118$		0			0
$119$	10.4721			0.959979
$120$		0			0
$121$	30.8885			2.80805
$122$		0			0
$123$	−6.47214	+	16.9443i	−0.583573	+	1.52781i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		0			0
$129$	−2.29180	+	6.00000i	−0.201781	+	0.528271i
$130$		0			0
$131$	−13.8885			−1.21345			−0.606724	−	0.794913i	$-0.707517\pi$
$131$	−13.8885			−1.21345			−0.606724	+	0.794913i	$0.707517\pi$
$132$		0			0
$133$	−1.52786			−0.132483
$134$		0			0
$135$	−4.61803	+	2.38197i	−0.397457	+	0.205007i
$136$		0			0
$137$		−	12.6525i		−	1.08097i	−0.841352	−	0.540487i	$-0.818240\pi$
$137$		−	12.6525i		−	1.08097i	0.841352	−	0.540487i	$-0.181760\pi$
$138$		0			0
$139$			8.18034i			0.693847i	0.937893	+	0.346924i	$0.112774\pi$
$139$			8.18034i			0.693847i	−0.937893	+	0.346924i	$0.887226\pi$
$140$		0			0
$141$	7.23607	+	2.76393i	0.609387	+	0.232765i
$142$		0			0
$143$	−20.9443			−1.75145
$144$		0			0
$145$	−0.472136			−0.0392088
$146$		0			0
$147$	4.85410	+	1.85410i	0.400360	+	0.152924i
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$			1.23607i			0.100590i	0.998734	+	0.0502949i	$0.0160161\pi$
$151$			1.23607i			0.100590i	−0.998734	+	0.0502949i	$0.983984\pi$
$152$		0			0
$153$	−10.4721	+	11.7082i	−0.846622	+	0.946552i
$154$		0			0
$155$	−9.23607			−0.741859
$156$		0			0
$157$	−1.70820			−0.136330			−0.0681648	−	0.997674i	$-0.521714\pi$
$157$	−1.70820			−0.136330			−0.0681648	+	0.997674i	$0.521714\pi$
$158$		0			0
$159$	−2.76393	+	7.23607i	−0.219194	+	0.573858i
$160$		0			0
$161$		−	16.9443i		−	1.33540i
$162$		0			0
$163$		−	1.23607i		−	0.0968163i	−0.998828	−	0.0484082i	$-0.984585\pi$
$163$		−	1.23607i		−	0.0968163i	0.998828	−	0.0484082i	$-0.0154148\pi$
$164$		0			0
$165$	4.00000	−	10.4721i	0.311400	−	0.815255i
$166$		0			0
$167$	14.9443			1.15642			0.578211	−	0.815887i	$-0.303751\pi$
$167$	14.9443			1.15642			0.578211	+	0.815887i	$0.303751\pi$
$168$		0			0
$169$	−2.52786			−0.194451
$170$		0			0
$171$	1.52786	−	1.70820i	0.116839	−	0.130630i
$172$		0			0
$173$		−	11.5279i		−	0.876447i	−0.898866	−	0.438224i	$-0.855608\pi$
$173$		−	11.5279i		−	0.876447i	0.898866	−	0.438224i	$-0.144392\pi$
$174$		0			0
$175$			2.00000i			0.151186i
$176$		0			0
$177$	−4.00000	−	1.52786i	−0.300658	−	0.114841i
$178$		0			0
$179$	10.4721			0.782724			0.391362	−	0.920237i	$-0.372004\pi$
$179$	10.4721			0.782724			0.391362	+	0.920237i	$0.372004\pi$
$180$		0			0
$181$	13.4164			0.997234			0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164			0.997234			0.498617	+	0.866822i	$0.333841\pi$
$182$		0			0
$183$	−5.70820	−	2.18034i	−0.421963	−	0.161175i
$184$		0			0
$185$			5.70820i			0.419675i
$186$		0			0
$187$		−	33.8885i		−	2.47818i
$188$		0			0
$189$	9.23607	−	4.76393i	0.671825	−	0.346525i
$190$		0			0
$191$	−5.52786			−0.399982			−0.199991	−	0.979798i	$-0.564091\pi$
$191$	−5.52786			−0.399982			−0.199991	+	0.979798i	$0.564091\pi$
$192$		0			0
$193$	−14.9443			−1.07571			−0.537856	−	0.843037i	$-0.680766\pi$
$193$	−14.9443			−1.07571			−0.537856	+	0.843037i	$0.680766\pi$
$194$		0			0
$195$	−2.00000	+	5.23607i	−0.143223	+	0.374963i
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$		−	4.29180i		−	0.304237i	−0.988362	−	0.152119i	$-0.951390\pi$
$199$		−	4.29180i		−	0.304237i	0.988362	−	0.152119i	$-0.0486096\pi$
$200$		0			0
$201$	4.18034	−	10.9443i	0.294858	−	0.771949i
$202$		0			0
$203$	0.944272			0.0662749
$204$		0			0
$205$	−10.4721			−0.731406
$206$		0			0
$207$	18.9443	+	16.9443i	1.31672	+	1.17771i
$208$		0			0
$209$			4.94427i			0.342002i
$210$		0			0
$211$		−	13.7082i		−	0.943712i	−0.881676	−	0.471856i	$-0.843584\pi$
$211$		−	13.7082i		−	0.943712i	0.881676	−	0.471856i	$-0.156416\pi$
$212$		0			0
$213$	−6.47214	−	2.47214i	−0.443463	−	0.169388i
$214$		0			0
$215$	−3.70820			−0.252897
$216$		0			0
$217$	18.4721			1.25397
$218$		0			0
$219$	13.7082	+	5.23607i	0.926315	+	0.353821i
$220$		0			0
$221$			16.9443i			1.13980i
$222$		0			0
$223$		−	14.9443i		−	1.00074i	−0.865811	−	0.500371i	$-0.833197\pi$
$223$		−	14.9443i		−	1.00074i	0.865811	−	0.500371i	$-0.166803\pi$
$224$		0			0
$225$	−2.23607	−	2.00000i	−0.149071	−	0.133333i
$226$		0			0
$227$	6.29180			0.417601			0.208801	−	0.977958i	$-0.433044\pi$
$227$	6.29180			0.417601			0.208801	+	0.977958i	$0.433044\pi$
$228$		0			0
$229$	2.94427			0.194563			0.0972815	−	0.995257i	$-0.468985\pi$
$229$	2.94427			0.194563			0.0972815	+	0.995257i	$0.468985\pi$
$230$		0			0
$231$	−8.00000	+	20.9443i	−0.526361	+	1.37803i
$232$		0			0
$233$		−	3.70820i		−	0.242933i	−0.992596	−	0.121466i	$-0.961240\pi$
$233$		−	3.70820i		−	0.242933i	0.992596	−	0.121466i	$-0.0387596\pi$
$234$		0			0
$235$			4.47214i			0.291730i
$236$		0			0
$237$	8.76393	−	22.9443i	0.569279	−	1.49039i
$238$		0			0
$239$	3.41641			0.220989			0.110495	−	0.993877i	$-0.464757\pi$
$239$	3.41641			0.220989			0.110495	+	0.993877i	$0.464757\pi$
$240$		0			0
$241$	5.05573			0.325668			0.162834	−	0.986653i	$-0.447936\pi$
$241$	5.05573			0.325668			0.162834	+	0.986653i	$0.447936\pi$
$242$		0			0
$243$	−3.90983	+	15.0902i	−0.250816	+	0.968035i
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$		−	2.47214i		−	0.157298i
$248$		0			0
$249$	2.76393	+	1.05573i	0.175157	+	0.0669040i
$250$		0			0
$251$	1.52786			0.0964379			0.0482190	−	0.998837i	$-0.484645\pi$
$251$	1.52786			0.0964379			0.0482190	+	0.998837i	$0.484645\pi$
$252$		0			0
$253$	−54.8328			−3.44731
$254$		0			0
$255$	−8.47214	−	3.23607i	−0.530546	−	0.202650i
$256$		0			0
$257$		−	1.23607i		−	0.0771038i	−0.999257	−	0.0385519i	$-0.987726\pi$
$257$		−	1.23607i		−	0.0771038i	0.999257	−	0.0385519i	$-0.0122745\pi$
$258$		0			0
$259$		−	11.4164i		−	0.709381i
$260$		0			0
$261$	−0.944272	+	1.05573i	−0.0584490	+	0.0653479i
$262$		0			0
$263$	−11.8885			−0.733079			−0.366540	−	0.930402i	$-0.619458\pi$
$263$	−11.8885			−0.733079			−0.366540	+	0.930402i	$0.619458\pi$
$264$		0			0
$265$	−4.47214			−0.274721
$266$		0			0
$267$	6.47214	−	16.9443i	0.396088	−	1.03697i
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$		−	5.81966i		−	0.353519i	−0.984254	−	0.176760i	$-0.943438\pi$
$271$		−	5.81966i		−	0.353519i	0.984254	−	0.176760i	$-0.0565615\pi$
$272$		0			0
$273$	4.00000	−	10.4721i	0.242091	−	0.633803i
$274$		0			0
$275$	6.47214			0.390284
$276$		0			0
$277$	18.6525			1.12072			0.560359	−	0.828250i	$-0.310663\pi$
$277$	18.6525			1.12072			0.560359	+	0.828250i	$0.310663\pi$
$278$		0			0
$279$	−18.4721	+	20.6525i	−1.10590	+	1.23643i
$280$		0			0
$281$			14.4721i			0.863335i	0.902033	+	0.431668i	$0.142075\pi$
$281$			14.4721i			0.863335i	−0.902033	+	0.431668i	$0.857925\pi$
$282$		0			0
$283$		−	6.76393i		−	0.402074i	−0.979584	−	0.201037i	$-0.935569\pi$
$283$		−	6.76393i		−	0.402074i	0.979584	−	0.201037i	$-0.0644312\pi$
$284$		0			0
$285$	1.23607	+	0.472136i	0.0732183	+	0.0279669i
$286$		0			0
$287$	20.9443			1.23630
$288$		0			0
$289$	−10.4164			−0.612730
$290$		0			0
$291$	−5.70820	−	2.18034i	−0.334621	−	0.127814i
$292$		0			0
$293$		−	2.58359i		−	0.150935i	−0.997148	−	0.0754675i	$-0.975955\pi$
$293$		−	2.58359i		−	0.150935i	0.997148	−	0.0754675i	$-0.0240449\pi$
$294$		0			0
$295$		−	2.47214i		−	0.143933i
$296$		0			0
$297$	−15.4164	−	29.8885i	−0.894551	−	1.73431i
$298$		0			0
$299$	27.4164			1.58553
$300$		0			0
$301$	7.41641			0.427475
$302$		0			0
$303$	5.81966	−	15.2361i	0.334331	−	0.875289i
$304$		0			0
$305$		−	3.52786i		−	0.202005i
$306$		0			0
$307$			23.1246i			1.31979i	0.751357	+	0.659896i	$0.229400\pi$
$307$			23.1246i			1.31979i	−0.751357	+	0.659896i	$0.770600\pi$
$308$		0			0
$309$	5.23607	−	13.7082i	0.297870	−	0.779832i
$310$		0			0
$311$	20.0000			1.13410			0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000			1.13410			0.567048	+	0.823685i	$0.308085\pi$
$312$		0			0
$313$	−11.8885			−0.671980			−0.335990	−	0.941866i	$-0.609071\pi$
$313$	−11.8885			−0.671980			−0.335990	+	0.941866i	$0.609071\pi$
$314$		0			0
$315$	4.47214	+	4.00000i	0.251976	+	0.225374i
$316$		0			0
$317$		−	27.8885i		−	1.56638i	−0.621785	−	0.783188i	$-0.713592\pi$
$317$		−	27.8885i		−	1.56638i	0.621785	−	0.783188i	$-0.286408\pi$
$318$		0			0
$319$		−	3.05573i		−	0.171088i
$320$		0			0
$321$	−14.1803	−	5.41641i	−0.791469	−	0.302314i
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	−3.23607			−0.179505
$326$		0			0
$327$	−30.6525	−	11.7082i	−1.69509	−	0.647465i
$328$		0			0
$329$		−	8.94427i		−	0.493114i
$330$		0			0
$331$			7.23607i			0.397730i	0.980027	+	0.198865i	$0.0637256\pi$
$331$			7.23607i			0.397730i	−0.980027	+	0.198865i	$0.936274\pi$
$332$		0			0
$333$	12.7639	+	11.4164i	0.699459	+	0.625615i
$334$		0			0
$335$	6.76393			0.369553
$336$		0			0
$337$	16.4721			0.897294			0.448647	−	0.893709i	$-0.351906\pi$
$337$	16.4721			0.897294			0.448647	+	0.893709i	$0.351906\pi$
$338$		0			0
$339$	−11.8197	+	30.9443i	−0.641956	+	1.68066i
$340$		0			0
$341$		−	59.7771i		−	3.23711i
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	−5.23607	+	13.7082i	−0.281900	+	0.738025i
$346$		0			0
$347$	−15.2361			−0.817915			−0.408957	−	0.912553i	$-0.634108\pi$
$347$	−15.2361			−0.817915			−0.408957	+	0.912553i	$0.634108\pi$
$348$		0			0
$349$	34.9443			1.87052			0.935262	−	0.353956i	$-0.115164\pi$
$349$	34.9443			1.87052			0.935262	+	0.353956i	$0.115164\pi$
$350$		0			0
$351$	7.70820	+	14.9443i	0.411433	+	0.797666i
$352$		0			0
$353$			9.23607i			0.491586i	0.969322	+	0.245793i	$0.0790484\pi$
$353$			9.23607i			0.491586i	−0.969322	+	0.245793i	$0.920952\pi$
$354$		0			0
$355$		−	4.00000i		−	0.212298i
$356$		0			0
$357$	16.9443	+	6.47214i	0.896786	+	0.342542i
$358$		0			0
$359$	4.58359			0.241913			0.120956	−	0.992658i	$-0.461404\pi$
$359$	4.58359			0.241913			0.120956	+	0.992658i	$0.461404\pi$
$360$		0			0
$361$	18.4164			0.969285
$362$		0			0
$363$	49.9787	+	19.0902i	2.62320	+	1.00197i
$364$		0			0
$365$			8.47214i			0.443452i
$366$		0			0
$367$			35.3050i			1.84290i	0.388493	+	0.921452i	$0.372996\pi$
$367$			35.3050i			1.84290i	−0.388493	+	0.921452i	$0.627004\pi$
$368$		0			0
$369$	−20.9443	+	23.4164i	−1.09032	+	1.21901i
$370$		0			0
$371$	8.94427			0.464363
$372$		0			0
$373$	−11.5967			−0.600457			−0.300228	−	0.953867i	$-0.597063\pi$
$373$	−11.5967			−0.600457			−0.300228	+	0.953867i	$0.597063\pi$
$374$		0			0
$375$	0.618034	−	1.61803i	0.0319151	−	0.0835549i
$376$		0			0
$377$			1.52786i			0.0786890i
$378$		0			0
$379$		−	21.7082i		−	1.11508i	−0.830152	−	0.557538i	$-0.811746\pi$
$379$		−	21.7082i		−	1.11508i	0.830152	−	0.557538i	$-0.188254\pi$
$380$		0			0
$381$	6.18034	−	16.1803i	0.316628	−	0.828944i
$382$		0			0
$383$	−14.9443			−0.763617			−0.381808	−	0.924242i	$-0.624699\pi$
$383$	−14.9443			−0.763617			−0.381808	+	0.924242i	$0.624699\pi$
$384$		0			0
$385$	−12.9443			−0.659701
$386$		0			0
$387$	−7.41641	+	8.29180i	−0.376997	+	0.421496i
$388$		0			0
$389$		−	14.9443i		−	0.757705i	−0.925457	−	0.378852i	$-0.876319\pi$
$389$		−	14.9443i		−	0.757705i	0.925457	−	0.378852i	$-0.123681\pi$
$390$		0			0
$391$			44.3607i			2.24342i
$392$		0			0
$393$	−22.4721	−	8.58359i	−1.13357	−	0.432985i
$394$		0			0
$395$	14.1803			0.713490
$396$		0			0
$397$	−20.1803			−1.01282			−0.506411	−	0.862292i	$-0.669028\pi$
$397$	−20.1803			−1.01282			−0.506411	+	0.862292i	$0.669028\pi$
$398$		0			0
$399$	−2.47214	−	0.944272i	−0.123762	−	0.0472727i
$400$		0			0
$401$		−	18.8328i		−	0.940466i	−0.882542	−	0.470233i	$-0.844170\pi$
$401$		−	18.8328i		−	0.940466i	0.882542	−	0.470233i	$-0.155830\pi$
$402$		0			0
$403$			29.8885i			1.48885i
$404$		0			0
$405$	−8.94427	+	1.00000i	−0.444444	+	0.0496904i
$406$		0			0
$407$	−36.9443			−1.83126
$408$		0			0
$409$	2.58359			0.127750			0.0638752	−	0.997958i	$-0.479654\pi$
$409$	2.58359			0.127750			0.0638752	+	0.997958i	$0.479654\pi$
$410$		0			0
$411$	7.81966	−	20.4721i	0.385715	−	1.00982i
$412$		0			0
$413$			4.94427i			0.243292i
$414$		0			0
$415$			1.70820i			0.0838524i
$416$		0			0
$417$	−5.05573	+	13.2361i	−0.247580	+	0.648173i
$418$		0			0
$419$	19.0557			0.930933			0.465467	−	0.885065i	$-0.345887\pi$
$419$	19.0557			0.930933			0.465467	+	0.885065i	$0.345887\pi$
$420$		0			0
$421$	−23.3050			−1.13581			−0.567907	−	0.823093i	$-0.692247\pi$
$421$	−23.3050			−1.13581			−0.567907	+	0.823093i	$0.692247\pi$
$422$		0			0
$423$	10.0000	+	8.94427i	0.486217	+	0.434885i
$424$		0			0
$425$		−	5.23607i		−	0.253987i
$426$		0			0
$427$			7.05573i			0.341451i
$428$		0			0
$429$	−33.8885	−	12.9443i	−1.63615	−	0.624955i
$430$		0			0
$431$	−28.0000			−1.34871			−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000			−1.34871			−0.674356	+	0.738406i	$0.735579\pi$
$432$		0			0
$433$	−26.3607			−1.26681			−0.633407	−	0.773819i	$-0.718344\pi$
$433$	−26.3607			−1.26681			−0.633407	+	0.773819i	$0.718344\pi$
$434$		0			0
$435$	−0.763932	−	0.291796i	−0.0366277	−	0.0139906i
$436$		0			0
$437$		−	6.47214i		−	0.309604i
$438$		0			0
$439$			24.6525i			1.17660i	0.808643	+	0.588299i	$0.200202\pi$
$439$			24.6525i			1.17660i	−0.808643	+	0.588299i	$0.799798\pi$
$440$		0			0
$441$	6.70820	+	6.00000i	0.319438	+	0.285714i
$442$		0			0
$443$	−1.34752			−0.0640228			−0.0320114	−	0.999488i	$-0.510191\pi$
$443$	−1.34752			−0.0640228			−0.0320114	+	0.999488i	$0.510191\pi$
$444$		0			0
$445$	10.4721			0.496427
$446$		0			0
$447$	−6.18034	+	16.1803i	−0.292320	+	0.765304i
$448$		0			0
$449$		−	36.9443i		−	1.74351i	−0.489944	−	0.871754i	$-0.662983\pi$
$449$		−	36.9443i		−	1.74351i	0.489944	−	0.871754i	$-0.337017\pi$
$450$		0			0
$451$		−	67.7771i		−	3.19150i
$452$		0			0
$453$	−0.763932	+	2.00000i	−0.0358927	+	0.0939682i
$454$		0			0
$455$	6.47214			0.303418
$456$		0			0
$457$	2.58359			0.120855			0.0604277	−	0.998173i	$-0.480754\pi$
$457$	2.58359			0.120855			0.0604277	+	0.998173i	$0.480754\pi$
$458$		0			0
$459$	−24.1803	+	12.4721i	−1.12864	+	0.582149i
$460$		0			0
$461$			0.472136i			0.0219896i	0.999940	+	0.0109948i	$0.00349982\pi$
$461$			0.472136i			0.0219896i	−0.999940	+	0.0109948i	$0.996500\pi$
$462$		0			0
$463$		−	26.3607i		−	1.22508i	−0.790438	−	0.612542i	$-0.790147\pi$
$463$		−	26.3607i		−	1.22508i	0.790438	−	0.612542i	$-0.209853\pi$
$464$		0			0
$465$	−14.9443	−	5.70820i	−0.693024	−	0.264712i
$466$		0			0
$467$	34.6525			1.60353			0.801763	−	0.597643i	$-0.203896\pi$
$467$	34.6525			1.60353			0.801763	+	0.597643i	$0.203896\pi$
$468$		0			0
$469$	−13.5279			−0.624659
$470$		0			0
$471$	−2.76393	−	1.05573i	−0.127355	−	0.0486454i
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$			0.763932i			0.0350516i
$476$		0			0
$477$	−8.94427	+	10.0000i	−0.409530	+	0.457869i
$478$		0			0
$479$	−32.3607			−1.47860			−0.739299	−	0.673378i	$-0.764843\pi$
$479$	−32.3607			−1.47860			−0.739299	+	0.673378i	$0.764843\pi$
$480$		0			0
$481$	18.4721			0.842257
$482$		0			0
$483$	10.4721	−	27.4164i	0.476499	−	1.24749i
$484$		0			0
$485$		−	3.52786i		−	0.160192i
$486$		0			0
$487$		−	16.4721i		−	0.746424i	−0.927746	−	0.373212i	$-0.878256\pi$
$487$		−	16.4721i		−	0.746424i	0.927746	−	0.373212i	$-0.121744\pi$
$488$		0			0
$489$	0.763932	−	2.00000i	0.0345462	−	0.0904431i
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	−2.47214			−0.111339
$494$		0			0
$495$	12.9443	−	14.4721i	0.581802	−	0.650474i
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$		−	17.7082i		−	0.792728i	−0.918093	−	0.396364i	$-0.870272\pi$
$499$		−	17.7082i		−	0.792728i	0.918093	−	0.396364i	$-0.129728\pi$
$500$		0			0
$501$	24.1803	+	9.23607i	1.08030	+	0.412637i
$502$		0			0
$503$	−11.8885			−0.530084			−0.265042	−	0.964237i	$-0.585386\pi$
$503$	−11.8885			−0.530084			−0.265042	+	0.964237i	$0.585386\pi$
$504$		0			0
$505$	9.41641			0.419025
$506$		0			0
$507$	−4.09017	−	1.56231i	−0.181651	−	0.0693844i
$508$		0			0
$509$		−	31.3050i		−	1.38757i	−0.720183	−	0.693784i	$-0.755942\pi$
$509$		−	31.3050i		−	1.38757i	0.720183	−	0.693784i	$-0.244058\pi$
$510$		0			0
$511$		−	16.9443i		−	0.749570i
$512$		0			0
$513$	3.52786	−	1.81966i	0.155759	−	0.0803400i
$514$		0			0
$515$	8.47214			0.373327
$516$		0			0
$517$	−28.9443			−1.27297
$518$		0			0
$519$	7.12461	−	18.6525i	0.312736	−	0.818753i
$520$		0			0
$521$			20.0000i			0.876216i	0.898922	+	0.438108i	$0.144351\pi$
$521$			20.0000i			0.876216i	−0.898922	+	0.438108i	$0.855649\pi$
$522$		0			0
$523$			12.6525i			0.553254i	0.960977	+	0.276627i	$0.0892167\pi$
$523$			12.6525i			0.553254i	−0.960977	+	0.276627i	$0.910783\pi$
$524$		0			0
$525$	−1.23607	+	3.23607i	−0.0539464	+	0.141234i
$526$		0			0
$527$	−48.3607			−2.10662
$528$		0			0
$529$	48.7771			2.12074
$530$		0			0
$531$	−5.52786	−	4.94427i	−0.239889	−	0.214563i
$532$		0			0
$533$			33.8885i			1.46788i
$534$		0			0
$535$		−	8.76393i		−	0.378898i
$536$		0			0
$537$	16.9443	+	6.47214i	0.731199	+	0.279293i
$538$		0			0
$539$	−19.4164			−0.836324
$540$		0			0
$541$	26.3607			1.13333			0.566667	−	0.823947i	$-0.308233\pi$
$541$	26.3607			1.13333			0.566667	+	0.823947i	$0.308233\pi$
$542$		0			0
$543$	21.7082	+	8.29180i	0.931588	+	0.355835i
$544$		0			0
$545$		−	18.9443i		−	0.811483i
$546$		0			0
$547$		−	32.0689i		−	1.37117i	−0.727994	−	0.685583i	$-0.759547\pi$
$547$		−	32.0689i		−	1.37117i	0.727994	−	0.685583i	$-0.240453\pi$
$548$		0			0
$549$	−7.88854	−	7.05573i	−0.336675	−	0.301131i
$550$		0			0
$551$	0.360680			0.0153655
$552$		0			0
$553$	−28.3607			−1.20602
$554$		0			0
$555$	−3.52786	+	9.23607i	−0.149749	+	0.392049i
$556$		0			0
$557$		−	36.2492i		−	1.53593i	−0.640492	−	0.767964i	$-0.721270\pi$
$557$		−	36.2492i		−	1.53593i	0.640492	−	0.767964i	$-0.278730\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$	20.9443	−	54.8328i	0.884268	−	2.31504i
$562$		0			0
$563$	14.0689			0.592933			0.296466	−	0.955043i	$-0.404192\pi$
$563$	14.0689			0.592933			0.296466	+	0.955043i	$0.404192\pi$
$564$		0			0
$565$	−19.1246			−0.804578
$566$		0			0
$567$	17.8885	−	2.00000i	0.751248	−	0.0839921i
$568$		0			0
$569$		−	36.9443i		−	1.54878i	−0.632706	−	0.774392i	$-0.718056\pi$
$569$		−	36.9443i		−	1.54878i	0.632706	−	0.774392i	$-0.281944\pi$
$570$		0			0
$571$		−	2.65248i		−	0.111003i	−0.998459	−	0.0555013i	$-0.982324\pi$
$571$		−	2.65248i		−	0.111003i	0.998459	−	0.0555013i	$-0.0176757\pi$
$572$		0			0
$573$	−8.94427	−	3.41641i	−0.373652	−	0.142722i
$574$		0			0
$575$	−8.47214			−0.353312
$576$		0			0
$577$	34.3607			1.43045			0.715227	−	0.698892i	$-0.246323\pi$
$577$	34.3607			1.43045			0.715227	+	0.698892i	$0.246323\pi$
$578$		0			0
$579$	−24.1803	−	9.23607i	−1.00490	−	0.383838i
$580$		0			0
$581$		−	3.41641i		−	0.141736i
$582$		0			0
$583$		−	28.9443i		−	1.19875i
$584$		0			0
$585$	−6.47214	+	7.23607i	−0.267590	+	0.299175i
$586$		0			0
$587$	−18.6525			−0.769870			−0.384935	−	0.922944i	$-0.625776\pi$
$587$	−18.6525			−0.769870			−0.384935	+	0.922944i	$0.625776\pi$
$588$		0			0
$589$	7.05573			0.290726
$590$		0			0
$591$	−6.18034	+	16.1803i	−0.254225	+	0.665570i
$592$		0			0
$593$			9.23607i			0.379280i	0.981854	+	0.189640i	$0.0607321\pi$
$593$			9.23607i			0.379280i	−0.981854	+	0.189640i	$0.939268\pi$
$594$		0			0
$595$			10.4721i			0.429316i
$596$		0			0
$597$	2.65248	−	6.94427i	0.108559	−	0.284210i
$598$		0			0
$599$	43.4164			1.77395			0.886973	−	0.461821i	$-0.152804\pi$
$599$	43.4164			1.77395			0.886973	+	0.461821i	$0.152804\pi$
$600$		0			0
$601$	−44.4721			−1.81406			−0.907028	−	0.421070i	$-0.861655\pi$
$601$	−44.4721			−1.81406			−0.907028	+	0.421070i	$0.861655\pi$
$602$		0			0
$603$	13.5279	−	15.1246i	0.550897	−	0.615922i
$604$		0			0
$605$			30.8885i			1.25580i
$606$		0			0
$607$			12.4721i			0.506228i	0.967436	+	0.253114i	$0.0814548\pi$
$607$			12.4721i			0.506228i	−0.967436	+	0.253114i	$0.918545\pi$
$608$		0			0
$609$	1.52786	+	0.583592i	0.0619122	+	0.0236483i
$610$		0			0
$611$	14.4721			0.585480
$612$		0			0
$613$	−15.2361			−0.615379			−0.307689	−	0.951487i	$-0.599556\pi$
$613$	−15.2361			−0.615379			−0.307689	+	0.951487i	$0.599556\pi$
$614$		0			0
$615$	−16.9443	−	6.47214i	−0.683259	−	0.260982i
$616$		0			0
$617$		−	1.23607i		−	0.0497622i	−0.999690	−	0.0248811i	$-0.992079\pi$
$617$		−	1.23607i		−	0.0497622i	0.999690	−	0.0248811i	$-0.00792072\pi$
$618$		0			0
$619$			1.34752i			0.0541616i	0.999633	+	0.0270808i	$0.00862114\pi$
$619$			1.34752i			0.0541616i	−0.999633	+	0.0270808i	$0.991379\pi$
$620$		0			0
$621$	20.1803	+	39.1246i	0.809809	+	1.57002i
$622$		0			0
$623$	−20.9443			−0.839115
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−3.05573	+	8.00000i	−0.122034	+	0.319489i
$628$		0			0
$629$			29.8885i			1.19173i
$630$		0			0
$631$		−	15.7082i		−	0.625334i	−0.949863	−	0.312667i	$-0.898778\pi$
$631$		−	15.7082i		−	0.625334i	0.949863	−	0.312667i	$-0.101222\pi$
$632$		0			0
$633$	8.47214	−	22.1803i	0.336737	−	0.881589i
$634$		0			0
$635$	10.0000			0.396838
$636$		0			0
$637$	9.70820			0.384653
$638$		0			0
$639$	−8.94427	−	8.00000i	−0.353830	−	0.316475i
$640$		0			0
$641$		−	25.3050i		−	0.999485i	−0.866174	−	0.499743i	$-0.833428\pi$
$641$		−	25.3050i		−	0.999485i	0.866174	−	0.499743i	$-0.166572\pi$
$642$		0			0
$643$			19.7082i			0.777216i	0.921403	+	0.388608i	$0.127044\pi$
$643$			19.7082i			0.777216i	−0.921403	+	0.388608i	$0.872956\pi$
$644$		0			0
$645$	−6.00000	−	2.29180i	−0.236250	−	0.0902394i
$646$		0			0
$647$	35.8885			1.41092			0.705462	−	0.708748i	$-0.250739\pi$
$647$	35.8885			1.41092			0.705462	+	0.708748i	$0.250739\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	29.8885	+	11.4164i	1.17142	+	0.447444i
$652$		0			0
$653$		−	36.4721i		−	1.42726i	−0.700520	−	0.713632i	$-0.747049\pi$
$653$		−	36.4721i		−	1.42726i	0.700520	−	0.713632i	$-0.252951\pi$
$654$		0			0
$655$		−	13.8885i		−	0.542670i
$656$		0			0
$657$	18.9443	+	16.9443i	0.739086	+	0.661059i
$658$		0			0
$659$	−5.52786			−0.215335			−0.107668	−	0.994187i	$-0.534338\pi$
$659$	−5.52786			−0.215335			−0.107668	+	0.994187i	$0.534338\pi$
$660$		0			0
$661$	31.5279			1.22629			0.613146	−	0.789970i	$-0.289904\pi$
$661$	31.5279			1.22629			0.613146	+	0.789970i	$0.289904\pi$
$662$		0			0
$663$	−10.4721	+	27.4164i	−0.406704	+	1.06477i
$664$		0			0
$665$		−	1.52786i		−	0.0592480i
$666$		0			0
$667$			4.00000i			0.154881i
$668$		0			0
$669$	9.23607	−	24.1803i	0.357087	−	0.934866i
$670$		0			0
$671$	22.8328			0.881451
$672$		0			0
$673$	9.41641			0.362976			0.181488	−	0.983393i	$-0.441909\pi$
$673$	9.41641			0.362976			0.181488	+	0.983393i	$0.441909\pi$
$674$		0			0
$675$	−2.38197	−	4.61803i	−0.0916819	−	0.177748i
$676$		0			0
$677$			30.3607i			1.16686i	0.812165	+	0.583428i	$0.198289\pi$
$677$			30.3607i			1.16686i	−0.812165	+	0.583428i	$0.801711\pi$
$678$		0			0
$679$			7.05573i			0.270774i
$680$		0			0
$681$	10.1803	+	3.88854i	0.390111	+	0.149009i
$682$		0			0
$683$	−32.7639			−1.25368			−0.626839	−	0.779149i	$-0.715651\pi$
$683$	−32.7639			−1.25368			−0.626839	+	0.779149i	$0.715651\pi$
$684$		0			0
$685$	12.6525			0.483426
$686$		0			0
$687$	4.76393	+	1.81966i	0.181755	+	0.0694244i
$688$		0			0
$689$			14.4721i			0.551344i
$690$		0			0
$691$			34.0689i			1.29604i	0.761623	+	0.648021i	$0.224403\pi$
$691$			34.0689i			1.29604i	−0.761623	+	0.648021i	$0.775597\pi$
$692$		0			0
$693$	−25.8885	+	28.9443i	−0.983424	+	1.09950i
$694$		0			0
$695$	−8.18034			−0.310298
$696$		0			0
$697$	−54.8328			−2.07694
$698$		0			0
$699$	2.29180	−	6.00000i	0.0866837	−	0.226941i
$700$		0			0
$701$			35.8885i			1.35549i	0.735296	+	0.677746i	$0.237043\pi$
$701$			35.8885i			1.35549i	−0.735296	+	0.677746i	$0.762957\pi$
$702$		0			0
$703$		−	4.36068i		−	0.164466i
$704$		0			0
$705$	−2.76393	+	7.23607i	−0.104096	+	0.272526i
$706$		0			0
$707$	−18.8328			−0.708281
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	28.3607	−	31.7082i	1.06361	−	1.18915i
$712$		0			0
$713$			78.2492i			2.93046i
$714$		0			0
$715$		−	20.9443i		−	0.783271i
$716$		0			0
$717$	5.52786	+	2.11146i	0.206442	+	0.0788538i
$718$		0			0
$719$	36.9443			1.37779			0.688894	−	0.724862i	$-0.258096\pi$
$719$	36.9443			1.37779			0.688894	+	0.724862i	$0.258096\pi$
$720$		0			0
$721$	−16.9443			−0.631038
$722$		0			0
$723$	8.18034	+	3.12461i	0.304230	+	0.116206i
$724$		0			0
$725$		−	0.472136i		−	0.0175347i
$726$		0			0
$727$			49.4164i			1.83275i	0.400317	+	0.916377i	$0.368900\pi$
$727$			49.4164i			1.83275i	−0.400317	+	0.916377i	$0.631100\pi$
$728$		0			0
$729$	−15.6525	+	22.0000i	−0.579721	+	0.814815i
$730$		0			0
$731$	−19.4164			−0.718142
$732$		0			0
$733$	8.76393			0.323703			0.161852	−	0.986815i	$-0.448253\pi$
$733$	8.76393			0.323703			0.161852	+	0.986815i	$0.448253\pi$
$734$		0			0
$735$	−1.85410	+	4.85410i	−0.0683896	+	0.179046i
$736$		0			0
$737$			43.7771i			1.61255i
$738$		0			0
$739$		−	30.6525i		−	1.12757i	−0.825922	−	0.563785i	$-0.809345\pi$
$739$		−	30.6525i		−	1.12757i	0.825922	−	0.563785i	$-0.190655\pi$
$740$		0			0
$741$	1.52786	−	4.00000i	0.0561275	−	0.146944i
$742$		0			0
$743$	−33.4164			−1.22593			−0.612965	−	0.790110i	$-0.710023\pi$
$743$	−33.4164			−1.22593			−0.612965	+	0.790110i	$0.710023\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	3.81966	+	3.41641i	0.139754	+	0.125000i
$748$		0			0
$749$			17.5279i			0.640454i
$750$		0			0
$751$			16.0689i			0.586362i	0.956057	+	0.293181i	$0.0947138\pi$
$751$			16.0689i			0.586362i	−0.956057	+	0.293181i	$0.905286\pi$
$752$		0			0
$753$	2.47214	+	0.944272i	0.0900896	+	0.0344112i
$754$		0			0
$755$	−1.23607			−0.0449851
$756$		0			0
$757$	7.59675			0.276108			0.138054	−	0.990425i	$-0.455915\pi$
$757$	7.59675			0.276108			0.138054	+	0.990425i	$0.455915\pi$
$758$		0			0
$759$	−88.7214	−	33.8885i	−3.22038	−	1.23008i
$760$		0			0
$761$		−	20.0000i		−	0.724999i	−0.931984	−	0.362500i	$-0.881923\pi$
$761$		−	20.0000i		−	0.724999i	0.931984	−	0.362500i	$-0.118077\pi$
$762$		0			0
$763$			37.8885i			1.37166i
$764$		0			0
$765$	−11.7082	−	10.4721i	−0.423311	−	0.378621i
$766$		0			0
$767$	−8.00000			−0.288863
$768$		0			0
$769$	14.3607			0.517859			0.258930	−	0.965896i	$-0.416630\pi$
$769$	14.3607			0.517859			0.258930	+	0.965896i	$0.416630\pi$
$770$		0			0
$771$	0.763932	−	2.00000i	0.0275123	−	0.0720282i
$772$		0			0
$773$		−	46.0000i		−	1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$		−	46.0000i		−	1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$		0			0
$775$		−	9.23607i		−	0.331769i
$776$		0			0
$777$	7.05573	−	18.4721i	0.253123	−	0.662684i
$778$		0			0
$779$	8.00000			0.286630
$780$		0			0
$781$	25.8885			0.926365
$782$		0			0
$783$	−2.18034	+	1.12461i	−0.0779190	+	0.0401903i
$784$		0			0
$785$		−	1.70820i		−	0.0609684i
$786$		0			0
$787$			7.70820i			0.274768i	0.990518	+	0.137384i	$0.0438694\pi$
$787$			7.70820i			0.274768i	−0.990518	+	0.137384i	$0.956131\pi$
$788$		0			0
$789$	−19.2361	−	7.34752i	−0.684822	−	0.261579i
$790$		0			0
$791$	38.2492			1.35999
$792$		0			0
$793$	−11.4164			−0.405409
$794$		0			0
$795$	−7.23607	−	2.76393i	−0.256637	−	0.0980266i
$796$		0			0
$797$		−	30.9443i		−	1.09610i	−0.836445	−	0.548051i	$-0.815370\pi$
$797$		−	30.9443i		−	1.09610i	0.836445	−	0.548051i	$-0.184630\pi$
$798$		0			0
$799$			23.4164i			0.828413i
$800$		0			0
$801$	20.9443	−	23.4164i	0.740029	−	0.827378i
$802$		0			0
$803$	−54.8328			−1.93501
$804$		0			0
$805$	16.9443			0.597207
$806$		0			0
$807$	−8.65248	+	22.6525i	−0.304582	+	0.797405i
$808$		0			0
$809$			45.3050i			1.59284i	0.604746	+	0.796419i	$0.293275\pi$
$809$			45.3050i			1.59284i	−0.604746	+	0.796419i	$0.706725\pi$
$810$		0			0
$811$		−	36.7639i		−	1.29096i	−0.763779	−	0.645478i	$-0.776658\pi$
$811$		−	36.7639i		−	1.29096i	0.763779	−	0.645478i	$-0.223342\pi$
$812$		0			0
$813$	3.59675	−	9.41641i	0.126143	−	0.330248i
$814$		0			0
$815$	1.23607			0.0432976
$816$		0			0
$817$	2.83282			0.0991077
$818$		0			0
$819$	12.9443	−	14.4721i	0.452309	−	0.505697i
$820$		0			0
$821$		−	12.1115i		−	0.422693i	−0.977411	−	0.211346i	$-0.932215\pi$
$821$		−	12.1115i		−	0.422693i	0.977411	−	0.211346i	$-0.0677848\pi$
$822$		0			0
$823$		−	26.3607i		−	0.918876i	−0.888210	−	0.459438i	$-0.848051\pi$
$823$		−	26.3607i		−	0.918876i	0.888210	−	0.459438i	$-0.151949\pi$
$824$		0			0
$825$	10.4721	+	4.00000i	0.364593	+	0.139262i
$826$		0			0
$827$	−13.1246			−0.456387			−0.228194	−	0.973616i	$-0.573282\pi$
$827$	−13.1246			−0.456387			−0.228194	+	0.973616i	$0.573282\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	30.1803	+	11.5279i	1.04694	+	0.399897i
$832$		0			0
$833$			15.7082i			0.544257i
$834$		0			0
$835$			14.9443i			0.517168i
$836$		0			0
$837$	−42.6525	+	22.0000i	−1.47429	+	0.760431i
$838$		0			0
$839$	30.4721			1.05201			0.526007	−	0.850480i	$-0.323688\pi$
$839$	30.4721			1.05201			0.526007	+	0.850480i	$0.323688\pi$
$840$		0			0
$841$	28.7771			0.992313
$842$		0			0
$843$	−8.94427	+	23.4164i	−0.308057	+	0.806504i
$844$		0			0
$845$		−	2.52786i		−	0.0869612i
$846$		0			0
$847$		−	61.7771i		−	2.12269i
$848$		0			0
$849$	4.18034	−	10.9443i	0.143469	−	0.375606i
$850$		0			0
$851$	48.3607			1.65778
$852$		0			0
$853$	−41.1246			−1.40808			−0.704040	−	0.710160i	$-0.748622\pi$
$853$	−41.1246			−1.40808			−0.704040	+	0.710160i	$0.748622\pi$
$854$		0			0
$855$	1.70820	+	1.52786i	0.0584193	+	0.0522518i
$856$		0			0
$857$		−	50.7639i		−	1.73406i	−0.498253	−	0.867031i	$-0.666025\pi$
$857$		−	50.7639i		−	1.73406i	0.498253	−	0.867031i	$-0.333975\pi$
$858$		0			0
$859$			36.1803i			1.23446i	0.786784	+	0.617229i	$0.211745\pi$
$859$			36.1803i			1.23446i	−0.786784	+	0.617229i	$0.788255\pi$
$860$		0			0
$861$	33.8885	+	12.9443i	1.15492	+	0.441140i
$862$		0			0
$863$	10.3607			0.352682			0.176341	−	0.984329i	$-0.443574\pi$
$863$	10.3607			0.352682			0.176341	+	0.984329i	$0.443574\pi$
$864$		0			0
$865$	11.5279			0.391959
$866$		0			0
$867$	−16.8541	−	6.43769i	−0.572395	−	0.218636i
$868$		0			0
$869$			91.7771i			3.11332i
$870$		0			0
$871$		−	21.8885i		−	0.741665i
$872$		0			0
$873$	−7.88854	−	7.05573i	−0.266987	−	0.238800i
$874$		0			0
$875$	−2.00000			−0.0676123
$876$		0			0
$877$	6.29180			0.212459			0.106229	−	0.994342i	$-0.466122\pi$
$877$	6.29180			0.212459			0.106229	+	0.994342i	$0.466122\pi$
$878$		0			0
$879$	1.59675	−	4.18034i	0.0538570	−	0.140999i
$880$		0			0
$881$		−	0.360680i		−	0.0121516i	−0.999982	−	0.00607581i	$-0.998066\pi$
$881$		−	0.360680i		−	0.0121516i	0.999982	−	0.00607581i	$-0.00193400\pi$
$882$		0			0
$883$			58.5410i			1.97006i	0.172377	+	0.985031i	$0.444855\pi$
$883$			58.5410i			1.97006i	−0.172377	+	0.985031i	$0.555145\pi$
$884$		0			0
$885$	1.52786	−	4.00000i	0.0513586	−	0.134459i
$886$		0			0
$887$	−4.83282			−0.162270			−0.0811350	−	0.996703i	$-0.525855\pi$
$887$	−4.83282			−0.162270			−0.0811350	+	0.996703i	$0.525855\pi$
$888$		0			0
$889$	−20.0000			−0.670778
$890$		0			0
$891$	−6.47214	−	57.8885i	−0.216825	−	1.93934i
$892$		0			0
$893$		−	3.41641i		−	0.114326i
$894$		0			0
$895$			10.4721i			0.350045i
$896$		0			0
$897$	44.3607	+	16.9443i	1.48116	+	0.565753i
$898$		0			0
$899$	−4.36068			−0.145437
$900$		0			0
$901$	−23.4164			−0.780114
$902$		0			0
$903$	12.0000	+	4.58359i	0.399335	+	0.152532i
$904$		0			0
$905$			13.4164i			0.445976i
$906$		0			0
$907$			49.5967i			1.64683i	0.567437	+	0.823416i	$0.307935\pi$
$907$			49.5967i			1.64683i	−0.567437	+	0.823416i	$0.692065\pi$
$908$		0			0
$909$	18.8328	−	21.0557i	0.624645	−	0.698374i
$910$		0			0
$911$	24.9443			0.826441			0.413220	−	0.910631i	$-0.364404\pi$
$911$	24.9443			0.826441			0.413220	+	0.910631i	$0.364404\pi$
$912$		0			0
$913$	−11.0557			−0.365891
$914$		0			0
$915$	2.18034	−	5.70820i	0.0720798	−	0.188707i
$916$		0			0
$917$			27.7771i			0.917280i
$918$		0			0
$919$		−	8.29180i		−	0.273521i	−0.990604	−	0.136761i	$-0.956331\pi$
$919$		−	8.29180i		−	0.273521i	0.990604	−	0.136761i	$-0.0436691\pi$
$920$		0			0
$921$	−14.2918	+	37.4164i	−0.470931	+	1.23291i
$922$		0			0
$923$	−12.9443			−0.426066
$924$		0			0
$925$	−5.70820			−0.187685
$926$		0			0
$927$	16.9443	−	18.9443i	0.556523	−	0.622212i
$928$		0			0
$929$			33.8885i			1.11185i	0.831233	+	0.555924i	$0.187635\pi$
$929$			33.8885i			1.11185i	−0.831233	+	0.555924i	$0.812365\pi$
$930$		0			0
$931$		−	2.29180i		−	0.0751106i
$932$		0			0
$933$	32.3607	+	12.3607i	1.05944	+	0.404670i
$934$		0			0
$935$	33.8885			1.10827
$936$		0			0
$937$	0.111456			0.00364111			0.00182056	−	0.999998i	$-0.499420\pi$
$937$	0.111456			0.00364111			0.00182056	+	0.999998i	$0.499420\pi$
$938$		0			0
$939$	−19.2361	−	7.34752i	−0.627745	−	0.239777i
$940$		0			0
$941$			15.5279i			0.506194i	0.967441	+	0.253097i	$0.0814492\pi$
$941$			15.5279i			0.506194i	−0.967441	+	0.253097i	$0.918551\pi$
$942$		0			0
$943$			88.7214i			2.88916i
$944$		0			0
$945$	4.76393	+	9.23607i	0.154971	+	0.300449i
$946$		0			0
$947$	−24.7639			−0.804720			−0.402360	−	0.915482i	$-0.631810\pi$
$947$	−24.7639			−0.804720			−0.402360	+	0.915482i	$0.631810\pi$
$948$		0			0
$949$	27.4164			0.889974
$950$		0			0
$951$	17.2361	−	45.1246i	0.558918	−	1.46327i
$952$		0			0
$953$			33.5967i			1.08831i	0.838986	+	0.544153i	$0.183149\pi$
$953$			33.5967i			1.08831i	−0.838986	+	0.544153i	$0.816851\pi$
$954$		0			0
$955$		−	5.52786i		−	0.178877i
$956$		0			0
$957$	1.88854	−	4.94427i	0.0610480	−	0.159826i
$958$		0			0
$959$	−25.3050			−0.817140
$960$		0			0
$961$	−54.3050			−1.75177
$962$		0			0
$963$	−19.5967	−	17.5279i	−0.631496	−	0.564828i
$964$		0			0
$965$		−	14.9443i		−	0.481073i
$966$		0			0
$967$			23.5279i			0.756605i	0.925682	+	0.378303i	$0.123492\pi$
$967$			23.5279i			0.756605i	−0.925682	+	0.378303i	$0.876508\pi$
$968$		0			0
$969$	6.47214	+	2.47214i	0.207915	+	0.0794164i
$970$		0			0
$971$	−9.52786			−0.305764			−0.152882	−	0.988244i	$-0.548855\pi$
$971$	−9.52786			−0.305764			−0.152882	+	0.988244i	$0.548855\pi$
$972$		0			0
$973$	16.3607			0.524499
$974$		0			0
$975$	−5.23607	−	2.00000i	−0.167688	−	0.0640513i
$976$		0			0
$977$			8.29180i			0.265278i	0.991164	+	0.132639i	$0.0423451\pi$
$977$			8.29180i			0.265278i	−0.991164	+	0.132639i	$0.957655\pi$
$978$		0			0
$979$			67.7771i			2.16617i
$980$		0			0
$981$	−42.3607	−	37.8885i	−1.35247	−	1.20969i
$982$		0			0
$983$	36.2492			1.15617			0.578085	−	0.815976i	$-0.303800\pi$
$983$	36.2492			1.15617			0.578085	+	0.815976i	$0.303800\pi$
$984$		0			0
$985$	−10.0000			−0.318626
$986$		0			0
$987$	5.52786	−	14.4721i	0.175954	−	0.460653i
$988$		0			0
$989$			31.4164i			0.998984i
$990$		0			0
$991$			5.23607i			0.166329i	0.996536	+	0.0831646i	$0.0265027\pi$
$991$			5.23607i			0.166329i	−0.996536	+	0.0831646i	$0.973497\pi$
$992$		0			0
$993$	−4.47214	+	11.7082i	−0.141919	+	0.371549i
$994$		0			0
$995$	4.29180			0.136059
$996$		0			0
$997$	56.1803			1.77925			0.889625	−	0.456693i	$-0.150966\pi$
$997$	56.1803			1.77925			0.889625	+	0.456693i	$0.150966\pi$
$998$		0			0
$999$	13.5967	+	26.3607i	0.430182	+	0.834015i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.h.l.1151.4	yes	4
3.2	odd	2		1920.2.h.f.1151.2	yes	4
4.3	odd	2		1920.2.h.f.1151.1	yes	4
8.3	odd	2		1920.2.h.k.1151.4	yes	4
8.5	even	2		1920.2.h.e.1151.1	✓	4
12.11	even	2	inner	1920.2.h.l.1151.3	yes	4
24.5	odd	2		1920.2.h.k.1151.3	yes	4
24.11	even	2		1920.2.h.e.1151.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.h.e.1151.1	✓	4	8.5	even	2
1920.2.h.e.1151.2	yes	4	24.11	even	2
1920.2.h.f.1151.1	yes	4	4.3	odd	2
1920.2.h.f.1151.2	yes	4	3.2	odd	2
1920.2.h.k.1151.3	yes	4	24.5	odd	2
1920.2.h.k.1151.4	yes	4	8.3	odd	2
1920.2.h.l.1151.3	yes	4	12.11	even	2	inner
1920.2.h.l.1151.4	yes	4	1.1	even	1	trivial

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 + 0.618034i) q^{3} +1.00000i q^{5} -2.00000i q^{7} +(2.23607 + 2.00000i) q^{9} -6.47214 q^{11} +3.23607 q^{13} +(-0.618034 + 1.61803i) q^{15} +5.23607i q^{17} -0.763932i q^{19} +(1.23607 - 3.23607i) q^{21} +8.47214 q^{23} -1.00000 q^{25} +(2.38197 + 4.61803i) q^{27} +0.472136i q^{29} +9.23607i q^{31} +(-10.4721 - 4.00000i) q^{33} +2.00000 q^{35} +5.70820 q^{37} +(5.23607 + 2.00000i) q^{39} +10.4721i q^{41} +3.70820i q^{43} +(-2.00000 + 2.23607i) q^{45} +4.47214 q^{47} +3.00000 q^{49} +(-3.23607 + 8.47214i) q^{51} +4.47214i q^{53} -6.47214i q^{55} +(0.472136 - 1.23607i) q^{57} -2.47214 q^{59} -3.52786 q^{61} +(4.00000 - 4.47214i) q^{63} +3.23607i q^{65} -6.76393i q^{67} +(13.7082 + 5.23607i) q^{69} -4.00000 q^{71} +8.47214 q^{73} +(-1.61803 - 0.618034i) q^{75} +12.9443i q^{77} -14.1803i q^{79} +(1.00000 + 8.94427i) q^{81} +1.70820 q^{83} -5.23607 q^{85} +(-0.291796 + 0.763932i) q^{87} -10.4721i q^{89} -6.47214i q^{91} +(-5.70820 + 14.9443i) q^{93} +0.763932 q^{95} -3.52786 q^{97} +(-14.4721 - 12.9443i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 8 q^{11} + 4 q^{13} + 2 q^{15} - 4 q^{21} + 16 q^{23} - 4 q^{25} + 14 q^{27} - 24 q^{33} + 8 q^{35} - 4 q^{37} + 12 q^{39} - 8 q^{45} + 12 q^{49} - 4 q^{51} - 16 q^{57} + 8 q^{59} - 32 q^{61}+ \cdots - 40 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 8 * q^11 + 4 * q^13 + 2 * q^15 - 4 * q^21 + 16 * q^23 - 4 * q^25 + 14 * q^27 - 24 * q^33 + 8 * q^35 - 4 * q^37 + 12 * q^39 - 8 * q^45 + 12 * q^49 - 4 * q^51 - 16 * q^57 + 8 * q^59 - 32 * q^61 + 16 * q^63 + 28 * q^69 - 16 * q^71 + 16 * q^73 - 2 * q^75 + 4 * q^81 - 20 * q^83 - 12 * q^85 - 28 * q^87 + 4 * q^93 + 12 * q^95 - 32 * q^97 - 40 * q^99

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