LMFDB - Embedded newform 1680.2.f.i.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(881,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.4148675396$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.i.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.618034 - 1.61803i) q^{3} +1.00000 q^{5} +(-0.381966 - 2.61803i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	\(q+(-0.618034 - 1.61803i) q^{3} +1.00000 q^{5} +(-0.381966 - 2.61803i) q^{7} +(-2.23607 + 2.00000i) q^{9} +4.47214i q^{11} +3.23607i q^{13} +(-0.618034 - 1.61803i) q^{15} +0.763932 q^{17} +0.472136i q^{19} +(-4.00000 + 2.23607i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +(4.61803 + 2.38197i) q^{27} +5.70820i q^{29} +7.23607i q^{31} +(7.23607 - 2.76393i) q^{33} +(-0.381966 - 2.61803i) q^{35} +5.23607 q^{37} +(5.23607 - 2.00000i) q^{39} +6.47214 q^{41} -12.9443 q^{43} +(-2.23607 + 2.00000i) q^{45} -2.47214 q^{47} +(-6.70820 + 2.00000i) q^{49} +(-0.472136 - 1.23607i) q^{51} -8.47214i q^{53} +4.47214i q^{55} +(0.763932 - 0.291796i) q^{57} +4.47214 q^{59} +2.76393i q^{61} +(6.09017 + 5.09017i) q^{63} +3.23607i q^{65} +12.0000 q^{67} +(6.47214 - 2.47214i) q^{69} +2.76393i q^{71} -6.76393i q^{73} +(-0.618034 - 1.61803i) q^{75} +(11.7082 - 1.70820i) q^{77} -8.94427 q^{79} +(1.00000 - 8.94427i) q^{81} +16.6525 q^{83} +0.763932 q^{85} +(9.23607 - 3.52786i) q^{87} -14.4721 q^{89} +(8.47214 - 1.23607i) q^{91} +(11.7082 - 4.47214i) q^{93} +0.472136i q^{95} +5.23607i q^{97} +(-8.94427 - 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} - 6 q^{7}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 - 6 * q^7	$ 4 q + 2 q^{3} + 4 q^{5} - 6 q^{7} + 2 q^{15} + 12 q^{17} - 16 q^{21} + 4 q^{25} + 14 q^{27} + 20 q^{33} - 6 q^{35} + 12 q^{37} + 12 q^{39} + 8 q^{41} - 16 q^{43} + 8 q^{47} + 16 q^{51} + 12 q^{57} + 2 q^{63} + 48 q^{67} + 8 q^{69} + 2 q^{75} + 20 q^{77} + 4 q^{81} + 4 q^{83} + 12 q^{85} + 28 q^{87} - 40 q^{89} + 16 q^{91} + 20 q^{93}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 - 6 * q^7 + 2 * q^15 + 12 * q^17 - 16 * q^21 + 4 * q^25 + 14 * q^27 + 20 * q^33 - 6 * q^35 + 12 * q^37 + 12 * q^39 + 8 * q^41 - 16 * q^43 + 8 * q^47 + 16 * q^51 + 12 * q^57 + 2 * q^63 + 48 * q^67 + 8 * q^69 + 2 * q^75 + 20 * q^77 + 4 * q^81 + 4 * q^83 + 12 * q^85 + 28 * q^87 - 40 * q^89 + 16 * q^91 + 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$-1$	$1$

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
$3$	−0.618034	−	1.61803i	−0.356822	−	0.934172i
$3$	−0.618034	−	1.61803i	−0.356822	−	0.934172i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−0.381966	−	2.61803i	−0.144370	−	0.989524i
$7$	−0.381966	−	2.61803i	−0.144370	−	0.989524i
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$			4.47214i			1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$			4.47214i			1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$		0			0
$13$			3.23607i			0.897524i	0.893651	+	0.448762i	$0.148135\pi$
$13$			3.23607i			0.897524i	−0.893651	+	0.448762i	$0.851865\pi$
$14$		0			0
$15$	−0.618034	−	1.61803i	−0.159576	−	0.417775i
$16$		0			0
$17$	0.763932			0.185281			0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932			0.185281			0.0926404	+	0.995700i	$0.470469\pi$
$18$		0			0
$19$			0.472136i			0.108315i	0.998532	+	0.0541577i	$0.0172474\pi$
$19$			0.472136i			0.108315i	−0.998532	+	0.0541577i	$0.982753\pi$
$20$		0			0
$21$	−4.00000	+	2.23607i	−0.872872	+	0.487950i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.61803	+	2.38197i	0.888741	+	0.458410i
$28$		0			0
$29$			5.70820i			1.05999i	0.848002	+	0.529993i	$0.177806\pi$
$29$			5.70820i			1.05999i	−0.848002	+	0.529993i	$0.822194\pi$
$30$		0			0
$31$			7.23607i			1.29964i	0.760090	+	0.649818i	$0.225155\pi$
$31$			7.23607i			1.29964i	−0.760090	+	0.649818i	$0.774845\pi$
$32$		0			0
$33$	7.23607	−	2.76393i	1.25964	−	0.481139i
$34$		0			0
$35$	−0.381966	−	2.61803i	−0.0645640	−	0.442529i
$36$		0			0
$37$	5.23607			0.860804			0.430402	−	0.902637i	$-0.358372\pi$
$37$	5.23607			0.860804			0.430402	+	0.902637i	$0.358372\pi$
$38$		0			0
$39$	5.23607	−	2.00000i	0.838442	−	0.320256i
$40$		0			0
$41$	6.47214			1.01078			0.505389	−	0.862892i	$-0.331349\pi$
$41$	6.47214			1.01078			0.505389	+	0.862892i	$0.331349\pi$
$42$		0			0
$43$	−12.9443			−1.97398			−0.986991	−	0.160773i	$-0.948601\pi$
$43$	−12.9443			−1.97398			−0.986991	+	0.160773i	$0.948601\pi$
$44$		0			0
$45$	−2.23607	+	2.00000i	−0.333333	+	0.298142i
$46$		0			0
$47$	−2.47214			−0.360598			−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214			−0.360598			−0.180299	+	0.983612i	$0.557707\pi$
$48$		0			0
$49$	−6.70820	+	2.00000i	−0.958315	+	0.285714i
$50$		0			0
$51$	−0.472136	−	1.23607i	−0.0661123	−	0.173084i
$52$		0			0
$53$		−	8.47214i		−	1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$		−	8.47214i		−	1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$		0			0
$55$			4.47214i			0.603023i
$56$		0			0
$57$	0.763932	−	0.291796i	0.101185	−	0.0386493i
$58$		0			0
$59$	4.47214			0.582223			0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214			0.582223			0.291111	+	0.956689i	$0.405975\pi$
$60$		0			0
$61$			2.76393i			0.353885i	0.984221	+	0.176943i	$0.0566207\pi$
$61$			2.76393i			0.353885i	−0.984221	+	0.176943i	$0.943379\pi$
$62$		0			0
$63$	6.09017	+	5.09017i	0.767289	+	0.641301i
$64$		0			0
$65$			3.23607i			0.401385i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$	6.47214	−	2.47214i	0.779154	−	0.297610i
$70$		0			0
$71$			2.76393i			0.328018i	0.986459	+	0.164009i	$0.0524427\pi$
$71$			2.76393i			0.328018i	−0.986459	+	0.164009i	$0.947557\pi$
$72$		0			0
$73$		−	6.76393i		−	0.791658i	−0.918324	−	0.395829i	$-0.870457\pi$
$73$		−	6.76393i		−	0.791658i	0.918324	−	0.395829i	$-0.129543\pi$
$74$		0			0
$75$	−0.618034	−	1.61803i	−0.0713644	−	0.186834i
$76$		0			0
$77$	11.7082	−	1.70820i	1.33427	−	0.194668i
$78$		0			0
$79$	−8.94427			−1.00631			−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427			−1.00631			−0.503155	+	0.864196i	$0.667827\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	16.6525			1.82785			0.913923	−	0.405887i	$-0.133037\pi$
$83$	16.6525			1.82785			0.913923	+	0.405887i	$0.133037\pi$
$84$		0			0
$85$	0.763932			0.0828601
$86$		0			0
$87$	9.23607	−	3.52786i	0.990210	−	0.378227i
$88$		0			0
$89$	−14.4721			−1.53404			−0.767022	−	0.641621i	$-0.778262\pi$
$89$	−14.4721			−1.53404			−0.767022	+	0.641621i	$0.778262\pi$
$90$		0			0
$91$	8.47214	−	1.23607i	0.888121	−	0.129575i
$92$		0			0
$93$	11.7082	−	4.47214i	1.21408	−	0.463739i
$94$		0			0
$95$			0.472136i			0.0484401i
$96$		0			0
$97$			5.23607i			0.531642i	0.964022	+	0.265821i	$0.0856430\pi$
$97$			5.23607i			0.531642i	−0.964022	+	0.265821i	$0.914357\pi$
$98$		0			0
$99$	−8.94427	−	10.0000i	−0.898933	−	1.00504i
$100$		0			0
$101$	−3.52786			−0.351036			−0.175518	−	0.984476i	$-0.556160\pi$
$101$	−3.52786			−0.351036			−0.175518	+	0.984476i	$0.556160\pi$
$102$		0			0
$103$		−	16.6525i		−	1.64082i	−0.571778	−	0.820409i	$-0.693746\pi$
$103$		−	16.6525i		−	1.64082i	0.571778	−	0.820409i	$-0.306254\pi$
$104$		0			0
$105$	−4.00000	+	2.23607i	−0.390360	+	0.218218i
$106$		0			0
$107$			15.4164i			1.49036i	0.666863	+	0.745180i	$0.267637\pi$
$107$			15.4164i			1.49036i	−0.666863	+	0.745180i	$0.732363\pi$
$108$		0			0
$109$	−4.47214			−0.428353			−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214			−0.428353			−0.214176	+	0.976795i	$0.568707\pi$
$110$		0			0
$111$	−3.23607	−	8.47214i	−0.307154	−	0.804140i
$112$		0			0
$113$			14.9443i			1.40584i	0.711269	+	0.702919i	$0.248121\pi$
$113$			14.9443i			1.40584i	−0.711269	+	0.702919i	$0.751879\pi$
$114$		0			0
$115$			4.00000i			0.373002i
$116$		0			0
$117$	−6.47214	−	7.23607i	−0.598349	−	0.668975i
$118$		0			0
$119$	−0.291796	−	2.00000i	−0.0267489	−	0.183340i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	−4.00000	−	10.4721i	−0.360668	−	0.944241i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	13.7082			1.21641			0.608203	−	0.793781i	$-0.291891\pi$
$127$	13.7082			1.21641			0.608203	+	0.793781i	$0.291891\pi$
$128$		0			0
$129$	8.00000	+	20.9443i	0.704361	+	1.84404i
$130$		0			0
$131$	21.4164			1.87116			0.935580	−	0.353114i	$-0.114877\pi$
$131$	21.4164			1.87116			0.935580	+	0.353114i	$0.114877\pi$
$132$		0			0
$133$	1.23607	−	0.180340i	0.107181	−	0.0156375i
$134$		0			0
$135$	4.61803	+	2.38197i	0.397457	+	0.205007i
$136$		0			0
$137$			12.4721i			1.06557i	0.846252	+	0.532783i	$0.178854\pi$
$137$			12.4721i			1.06557i	−0.846252	+	0.532783i	$0.821146\pi$
$138$		0			0
$139$			20.4721i			1.73642i	0.496194	+	0.868212i	$0.334731\pi$
$139$			20.4721i			1.73642i	−0.496194	+	0.868212i	$0.665269\pi$
$140$		0			0
$141$	1.52786	+	4.00000i	0.128669	+	0.336861i
$142$		0			0
$143$	−14.4721			−1.21022
$144$		0			0
$145$			5.70820i			0.474041i
$146$		0			0
$147$	7.38197	+	9.61803i	0.608854	+	0.793282i
$148$		0			0
$149$		−	17.7082i		−	1.45071i	−0.688374	−	0.725356i	$-0.741675\pi$
$149$		−	17.7082i		−	1.45071i	0.688374	−	0.725356i	$-0.258325\pi$
$150$		0			0
$151$	−3.05573			−0.248672			−0.124336	−	0.992240i	$-0.539680\pi$
$151$	−3.05573			−0.248672			−0.124336	+	0.992240i	$0.539680\pi$
$152$		0			0
$153$	−1.70820	+	1.52786i	−0.138100	+	0.123520i
$154$		0			0
$155$			7.23607i			0.581215i
$156$		0			0
$157$		−	4.76393i		−	0.380203i	−0.981764	−	0.190102i	$-0.939118\pi$
$157$		−	4.76393i		−	0.380203i	0.981764	−	0.190102i	$-0.0608817\pi$
$158$		0			0
$159$	−13.7082	+	5.23607i	−1.08713	+	0.415247i
$160$		0			0
$161$	10.4721	−	1.52786i	0.825320	−	0.120413i
$162$		0			0
$163$	1.52786			0.119672			0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786			0.119672			0.0598358	+	0.998208i	$0.480942\pi$
$164$		0			0
$165$	7.23607	−	2.76393i	0.563327	−	0.215172i
$166$		0			0
$167$	−16.9443			−1.31119			−0.655594	−	0.755114i	$-0.727582\pi$
$167$	−16.9443			−1.31119			−0.655594	+	0.755114i	$0.727582\pi$
$168$		0			0
$169$	2.52786			0.194451
$170$		0			0
$171$	−0.944272	−	1.05573i	−0.0722103	−	0.0807335i
$172$		0			0
$173$	17.4164			1.32414			0.662072	−	0.749440i	$-0.269677\pi$
$173$	17.4164			1.32414			0.662072	+	0.749440i	$0.269677\pi$
$174$		0			0
$175$	−0.381966	−	2.61803i	−0.0288739	−	0.197905i
$176$		0			0
$177$	−2.76393	−	7.23607i	−0.207750	−	0.543896i
$178$		0			0
$179$		−	2.94427i		−	0.220065i	−0.993928	−	0.110033i	$-0.964904\pi$
$179$		−	2.94427i		−	0.220065i	0.993928	−	0.110033i	$-0.0350955\pi$
$180$		0			0
$181$			6.18034i			0.459381i	0.973264	+	0.229691i	$0.0737714\pi$
$181$			6.18034i			0.459381i	−0.973264	+	0.229691i	$0.926229\pi$
$182$		0			0
$183$	4.47214	−	1.70820i	0.330590	−	0.126274i
$184$		0			0
$185$	5.23607			0.384963
$186$		0			0
$187$			3.41641i			0.249832i
$188$		0			0
$189$	4.47214	−	13.0000i	0.325300	−	0.945611i
$190$		0			0
$191$		−	2.76393i		−	0.199991i	−0.994988	−	0.0999956i	$-0.968117\pi$
$191$		−	2.76393i		−	0.199991i	0.994988	−	0.0999956i	$-0.0318829\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$		0			0
$195$	5.23607	−	2.00000i	0.374963	−	0.143223i
$196$		0			0
$197$			12.4721i			0.888603i	0.895877	+	0.444301i	$0.146548\pi$
$197$			12.4721i			0.888603i	−0.895877	+	0.444301i	$0.853452\pi$
$198$		0			0
$199$		−	0.180340i		−	0.0127840i	−0.999980	−	0.00639198i	$-0.997965\pi$
$199$		−	0.180340i		−	0.0127840i	0.999980	−	0.00639198i	$-0.00203464\pi$
$200$		0			0
$201$	−7.41641	−	19.4164i	−0.523113	−	1.36953i
$202$		0			0
$203$	14.9443	−	2.18034i	1.04888	−	0.153030i
$204$		0			0
$205$	6.47214			0.452034
$206$		0			0
$207$	−8.00000	−	8.94427i	−0.556038	−	0.621670i
$208$		0			0
$209$	−2.11146			−0.146052
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$	4.47214	−	1.70820i	0.306426	−	0.117044i
$214$		0			0
$215$	−12.9443			−0.882792
$216$		0			0
$217$	18.9443	−	2.76393i	1.28602	−	0.187628i
$218$		0			0
$219$	−10.9443	+	4.18034i	−0.739545	+	0.282481i
$220$		0			0
$221$			2.47214i			0.166294i
$222$		0			0
$223$		−	4.29180i		−	0.287400i	−0.989621	−	0.143700i	$-0.954100\pi$
$223$		−	4.29180i		−	0.287400i	0.989621	−	0.143700i	$-0.0459000\pi$
$224$		0			0
$225$	−2.23607	+	2.00000i	−0.149071	+	0.133333i
$226$		0			0
$227$	−5.23607			−0.347530			−0.173765	−	0.984787i	$-0.555593\pi$
$227$	−5.23607			−0.347530			−0.173765	+	0.984787i	$0.555593\pi$
$228$		0			0
$229$		−	13.2361i		−	0.874664i	−0.899300	−	0.437332i	$-0.855923\pi$
$229$		−	13.2361i		−	0.874664i	0.899300	−	0.437332i	$-0.144077\pi$
$230$		0			0
$231$	−10.0000	−	17.8885i	−0.657952	−	1.17698i
$232$		0			0
$233$			20.4721i			1.34117i	0.741831	+	0.670587i	$0.233958\pi$
$233$			20.4721i			1.34117i	−0.741831	+	0.670587i	$0.766042\pi$
$234$		0			0
$235$	−2.47214			−0.161264
$236$		0			0
$237$	5.52786	+	14.4721i	0.359073	+	0.940066i
$238$		0			0
$239$			22.1803i			1.43473i	0.696699	+	0.717363i	$0.254651\pi$
$239$			22.1803i			1.43473i	−0.696699	+	0.717363i	$0.745349\pi$
$240$		0			0
$241$		−	17.8885i		−	1.15230i	−0.817343	−	0.576151i	$-0.804554\pi$
$241$		−	17.8885i		−	1.15230i	0.817343	−	0.576151i	$-0.195446\pi$
$242$		0			0
$243$	−15.0902	+	3.90983i	−0.968035	+	0.250816i
$244$		0			0
$245$	−6.70820	+	2.00000i	−0.428571	+	0.127775i
$246$		0			0
$247$	−1.52786			−0.0972157
$248$		0			0
$249$	−10.2918	−	26.9443i	−0.652216	−	1.70752i
$250$		0			0
$251$	3.52786			0.222677			0.111338	−	0.993783i	$-0.464486\pi$
$251$	3.52786			0.222677			0.111338	+	0.993783i	$0.464486\pi$
$252$		0			0
$253$	−17.8885			−1.12464
$254$		0			0
$255$	−0.472136	−	1.23607i	−0.0295663	−	0.0774056i
$256$		0			0
$257$	−8.18034			−0.510276			−0.255138	−	0.966905i	$-0.582121\pi$
$257$	−8.18034			−0.510276			−0.255138	+	0.966905i	$0.582121\pi$
$258$		0			0
$259$	−2.00000	−	13.7082i	−0.124274	−	0.851786i
$260$		0			0
$261$	−11.4164	−	12.7639i	−0.706658	−	0.790068i
$262$		0			0
$263$		−	4.94427i		−	0.304877i	−0.988313	−	0.152438i	$-0.951287\pi$
$263$		−	4.94427i		−	0.304877i	0.988313	−	0.152438i	$-0.0487126\pi$
$264$		0			0
$265$		−	8.47214i		−	0.520439i
$266$		0			0
$267$	8.94427	+	23.4164i	0.547381	+	1.43306i
$268$		0			0
$269$	−4.47214			−0.272671			−0.136335	−	0.990663i	$-0.543533\pi$
$269$	−4.47214			−0.272671			−0.136335	+	0.990663i	$0.543533\pi$
$270$		0			0
$271$		−	18.2918i		−	1.11115i	−0.831467	−	0.555574i	$-0.812499\pi$
$271$		−	18.2918i		−	1.11115i	0.831467	−	0.555574i	$-0.187501\pi$
$272$		0			0
$273$	−7.23607	−	12.9443i	−0.437947	−	0.783423i
$274$		0			0
$275$			4.47214i			0.269680i
$276$		0			0
$277$	23.1246			1.38942			0.694712	−	0.719288i	$-0.255532\pi$
$277$	23.1246			1.38942			0.694712	+	0.719288i	$0.255532\pi$
$278$		0			0
$279$	−14.4721	−	16.1803i	−0.866424	−	0.968692i
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$		−	8.76393i		−	0.520962i	−0.965479	−	0.260481i	$-0.916119\pi$
$283$		−	8.76393i		−	0.520962i	0.965479	−	0.260481i	$-0.0838811\pi$
$284$		0			0
$285$	0.763932	−	0.291796i	0.0452514	−	0.0172845i
$286$		0			0
$287$	−2.47214	−	16.9443i	−0.145926	−	1.00019i
$288$		0			0
$289$	−16.4164			−0.965671
$290$		0			0
$291$	8.47214	−	3.23607i	0.496645	−	0.189702i
$292$		0			0
$293$	−29.4164			−1.71852			−0.859262	−	0.511535i	$-0.829077\pi$
$293$	−29.4164			−1.71852			−0.859262	+	0.511535i	$0.829077\pi$
$294$		0			0
$295$	4.47214			0.260378
$296$		0			0
$297$	−10.6525	+	20.6525i	−0.618119	+	1.19838i
$298$		0			0
$299$	−12.9443			−0.748587
$300$		0			0
$301$	4.94427	+	33.8885i	0.284983	+	1.95330i
$302$		0			0
$303$	2.18034	+	5.70820i	0.125257	+	0.327928i
$304$		0			0
$305$			2.76393i			0.158262i
$306$		0			0
$307$		−	4.18034i		−	0.238585i	−0.992859	−	0.119292i	$-0.961937\pi$
$307$		−	4.18034i		−	0.238585i	0.992859	−	0.119292i	$-0.0380626\pi$
$308$		0			0
$309$	−26.9443	+	10.2918i	−1.53281	+	0.585480i
$310$		0			0
$311$	−26.4721			−1.50110			−0.750549	−	0.660815i	$-0.770211\pi$
$311$	−26.4721			−1.50110			−0.750549	+	0.660815i	$0.770211\pi$
$312$		0			0
$313$			7.70820i			0.435693i	0.975983	+	0.217847i	$0.0699033\pi$
$313$			7.70820i			0.435693i	−0.975983	+	0.217847i	$0.930097\pi$
$314$		0			0
$315$	6.09017	+	5.09017i	0.343142	+	0.286799i
$316$		0			0
$317$			6.94427i			0.390029i	0.980800	+	0.195015i	$0.0624754\pi$
$317$			6.94427i			0.390029i	−0.980800	+	0.195015i	$0.937525\pi$
$318$		0			0
$319$	−25.5279			−1.42929
$320$		0			0
$321$	24.9443	−	9.52786i	1.39225	−	0.531794i
$322$		0			0
$323$			0.360680i			0.0200688i
$324$		0			0
$325$			3.23607i			0.179505i
$326$		0			0
$327$	2.76393	+	7.23607i	0.152846	+	0.400155i
$328$		0			0
$329$	0.944272	+	6.47214i	0.0520594	+	0.356820i
$330$		0			0
$331$	−20.9443			−1.15120			−0.575601	−	0.817731i	$-0.695232\pi$
$331$	−20.9443			−1.15120			−0.575601	+	0.817731i	$0.695232\pi$
$332$		0			0
$333$	−11.7082	+	10.4721i	−0.641606	+	0.573870i
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	−5.41641			−0.295051			−0.147525	−	0.989058i	$-0.547131\pi$
$337$	−5.41641			−0.295051			−0.147525	+	0.989058i	$0.547131\pi$
$338$		0			0
$339$	24.1803	−	9.23607i	1.31330	−	0.501634i
$340$		0			0
$341$	−32.3607			−1.75243
$342$		0			0
$343$	7.79837	+	16.7984i	0.421073	+	0.907027i
$344$		0			0
$345$	6.47214	−	2.47214i	0.348448	−	0.133095i
$346$		0			0
$347$			6.47214i			0.347442i	0.984795	+	0.173721i	$0.0555792\pi$
$347$			6.47214i			0.347442i	−0.984795	+	0.173721i	$0.944421\pi$
$348$		0			0
$349$		−	2.18034i		−	0.116711i	−0.998296	−	0.0583555i	$-0.981414\pi$
$349$		−	2.18034i		−	0.116711i	0.998296	−	0.0583555i	$-0.0185857\pi$
$350$		0			0
$351$	−7.70820	+	14.9443i	−0.411433	+	0.797666i
$352$		0			0
$353$	−14.2918			−0.760676			−0.380338	−	0.924848i	$-0.624192\pi$
$353$	−14.2918			−0.760676			−0.380338	+	0.924848i	$0.624192\pi$
$354$		0			0
$355$			2.76393i			0.146694i
$356$		0			0
$357$	−3.05573	+	1.70820i	−0.161726	+	0.0904077i
$358$		0			0
$359$		−	10.1803i		−	0.537298i	−0.963238	−	0.268649i	$-0.913423\pi$
$359$		−	10.1803i		−	0.537298i	0.963238	−	0.268649i	$-0.0865771\pi$
$360$		0			0
$361$	18.7771			0.988268
$362$		0			0
$363$	5.56231	+	14.5623i	0.291945	+	0.764323i
$364$		0			0
$365$		−	6.76393i		−	0.354040i
$366$		0			0
$367$		−	14.1803i		−	0.740208i	−0.928990	−	0.370104i	$-0.879322\pi$
$367$		−	14.1803i		−	0.740208i	0.928990	−	0.370104i	$-0.120678\pi$
$368$		0			0
$369$	−14.4721	+	12.9443i	−0.753389	+	0.673852i
$370$		0			0
$371$	−22.1803	+	3.23607i	−1.15155	+	0.168008i
$372$		0			0
$373$	−9.81966			−0.508443			−0.254221	−	0.967146i	$-0.581819\pi$
$373$	−9.81966			−0.508443			−0.254221	+	0.967146i	$0.581819\pi$
$374$		0			0
$375$	−0.618034	−	1.61803i	−0.0319151	−	0.0835549i
$376$		0			0
$377$	−18.4721			−0.951363
$378$		0			0
$379$	−17.8885			−0.918873			−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885			−0.918873			−0.459436	+	0.888211i	$0.651949\pi$
$380$		0			0
$381$	−8.47214	−	22.1803i	−0.434041	−	1.13633i
$382$		0			0
$383$	−21.8885			−1.11845			−0.559226	−	0.829015i	$-0.688902\pi$
$383$	−21.8885			−1.11845			−0.559226	+	0.829015i	$0.688902\pi$
$384$		0			0
$385$	11.7082	−	1.70820i	0.596705	−	0.0870581i
$386$		0			0
$387$	28.9443	−	25.8885i	1.47132	−	1.31599i
$388$		0			0
$389$			7.81966i			0.396473i	0.980154	+	0.198236i	$0.0635213\pi$
$389$			7.81966i			0.396473i	−0.980154	+	0.198236i	$0.936479\pi$
$390$		0			0
$391$			3.05573i			0.154535i
$392$		0			0
$393$	−13.2361	−	34.6525i	−0.667671	−	1.74799i
$394$		0			0
$395$	−8.94427			−0.450035
$396$		0			0
$397$		−	17.1246i		−	0.859460i	−0.902958	−	0.429730i	$-0.858609\pi$
$397$		−	17.1246i		−	0.859460i	0.902958	−	0.429730i	$-0.141391\pi$
$398$		0			0
$399$	−1.05573	−	1.88854i	−0.0528525	−	0.0945454i
$400$		0			0
$401$			5.52786i			0.276048i	0.990429	+	0.138024i	$0.0440752\pi$
$401$			5.52786i			0.276048i	−0.990429	+	0.138024i	$0.955925\pi$
$402$		0			0
$403$	−23.4164			−1.16645
$404$		0			0
$405$	1.00000	−	8.94427i	0.0496904	−	0.444444i
$406$		0			0
$407$			23.4164i			1.16071i
$408$		0			0
$409$		−	19.4164i		−	0.960080i	−0.877247	−	0.480040i	$-0.840622\pi$
$409$		−	19.4164i		−	0.960080i	0.877247	−	0.480040i	$-0.159378\pi$
$410$		0			0
$411$	20.1803	−	7.70820i	0.995423	−	0.380218i
$412$		0			0
$413$	−1.70820	−	11.7082i	−0.0840552	−	0.576123i
$414$		0			0
$415$	16.6525			0.817438
$416$		0			0
$417$	33.1246	−	12.6525i	1.62212	−	0.619594i
$418$		0			0
$419$	16.8328			0.822337			0.411168	−	0.911559i	$-0.365121\pi$
$419$	16.8328			0.822337			0.411168	+	0.911559i	$0.365121\pi$
$420$		0			0
$421$	−12.4721			−0.607855			−0.303927	−	0.952695i	$-0.598298\pi$
$421$	−12.4721			−0.607855			−0.303927	+	0.952695i	$0.598298\pi$
$422$		0			0
$423$	5.52786	−	4.94427i	0.268774	−	0.240399i
$424$		0			0
$425$	0.763932			0.0370561
$426$		0			0
$427$	7.23607	−	1.05573i	0.350178	−	0.0510903i
$428$		0			0
$429$	8.94427	+	23.4164i	0.431834	+	1.13055i
$430$		0			0
$431$		−	9.59675i		−	0.462259i	−0.972923	−	0.231130i	$-0.925758\pi$
$431$		−	9.59675i		−	0.462259i	0.972923	−	0.231130i	$-0.0742421\pi$
$432$		0			0
$433$		−	4.65248i		−	0.223584i	−0.993732	−	0.111792i	$-0.964341\pi$
$433$		−	4.65248i		−	0.223584i	0.993732	−	0.111792i	$-0.0356590\pi$
$434$		0			0
$435$	9.23607	−	3.52786i	0.442836	−	0.169148i
$436$		0			0
$437$	−1.88854			−0.0903413
$438$		0			0
$439$			12.1803i			0.581336i	0.956824	+	0.290668i	$0.0938775\pi$
$439$			12.1803i			0.581336i	−0.956824	+	0.290668i	$0.906123\pi$
$440$		0			0
$441$	11.0000	−	17.8885i	0.523810	−	0.851835i
$442$		0			0
$443$			9.52786i			0.452682i	0.974048	+	0.226341i	$0.0726764\pi$
$443$			9.52786i			0.452682i	−0.974048	+	0.226341i	$0.927324\pi$
$444$		0			0
$445$	−14.4721			−0.686045
$446$		0			0
$447$	−28.6525	+	10.9443i	−1.35522	+	0.517646i
$448$		0			0
$449$		−	1.52786i		−	0.0721044i	−0.999350	−	0.0360522i	$-0.988522\pi$
$449$		−	1.52786i		−	0.0721044i	0.999350	−	0.0360522i	$-0.0114783\pi$
$450$		0			0
$451$			28.9443i			1.36293i
$452$		0			0
$453$	1.88854	+	4.94427i	0.0887315	+	0.232302i
$454$		0			0
$455$	8.47214	−	1.23607i	0.397180	−	0.0579478i
$456$		0			0
$457$	9.05573			0.423609			0.211805	−	0.977312i	$-0.432066\pi$
$457$	9.05573			0.423609			0.211805	+	0.977312i	$0.432066\pi$
$458$		0			0
$459$	3.52786	+	1.81966i	0.164667	+	0.0849345i
$460$		0			0
$461$	10.9443			0.509726			0.254863	−	0.966977i	$-0.417970\pi$
$461$	10.9443			0.509726			0.254863	+	0.966977i	$0.417970\pi$
$462$		0			0
$463$	−0.180340			−0.00838111			−0.00419055	−	0.999991i	$-0.501334\pi$
$463$	−0.180340			−0.00838111			−0.00419055	+	0.999991i	$0.501334\pi$
$464$		0			0
$465$	11.7082	−	4.47214i	0.542955	−	0.207390i
$466$		0			0
$467$	20.2918			0.938992			0.469496	−	0.882935i	$-0.344436\pi$
$467$	20.2918			0.938992			0.469496	+	0.882935i	$0.344436\pi$
$468$		0			0
$469$	−4.58359	−	31.4164i	−0.211651	−	1.45067i
$470$		0			0
$471$	−7.70820	+	2.94427i	−0.355175	+	0.135665i
$472$		0			0
$473$		−	57.8885i		−	2.66172i
$474$		0			0
$475$			0.472136i			0.0216631i
$476$		0			0
$477$	16.9443	+	18.9443i	0.775825	+	0.867399i
$478$		0			0
$479$	−2.11146			−0.0964749			−0.0482374	−	0.998836i	$-0.515360\pi$
$479$	−2.11146			−0.0964749			−0.0482374	+	0.998836i	$0.515360\pi$
$480$		0			0
$481$			16.9443i			0.772592i
$482$		0			0
$483$	−8.94427	−	16.0000i	−0.406978	−	0.728025i
$484$		0			0
$485$			5.23607i			0.237758i
$486$		0			0
$487$	31.5967			1.43179			0.715893	−	0.698210i	$-0.246020\pi$
$487$	31.5967			1.43179			0.715893	+	0.698210i	$0.246020\pi$
$488$		0			0
$489$	−0.944272	−	2.47214i	−0.0427015	−	0.111794i
$490$		0			0
$491$			13.4164i			0.605474i	0.953074	+	0.302737i	$0.0979004\pi$
$491$			13.4164i			0.605474i	−0.953074	+	0.302737i	$0.902100\pi$
$492$		0			0
$493$			4.36068i			0.196395i
$494$		0			0
$495$	−8.94427	−	10.0000i	−0.402015	−	0.449467i
$496$		0			0
$497$	7.23607	−	1.05573i	0.324582	−	0.0473559i
$498$		0			0
$499$	17.8885			0.800801			0.400401	−	0.916340i	$-0.368871\pi$
$499$	17.8885			0.800801			0.400401	+	0.916340i	$0.368871\pi$
$500$		0			0
$501$	10.4721	+	27.4164i	0.467861	+	1.22487i
$502$		0			0
$503$	−16.3607			−0.729487			−0.364743	−	0.931108i	$-0.618843\pi$
$503$	−16.3607			−0.729487			−0.364743	+	0.931108i	$0.618843\pi$
$504$		0			0
$505$	−3.52786			−0.156988
$506$		0			0
$507$	−1.56231	−	4.09017i	−0.0693844	−	0.181651i
$508$		0			0
$509$	24.4721			1.08471			0.542354	−	0.840150i	$-0.317533\pi$
$509$	24.4721			1.08471			0.542354	+	0.840150i	$0.317533\pi$
$510$		0			0
$511$	−17.7082	+	2.58359i	−0.783365	+	0.114291i
$512$		0			0
$513$	−1.12461	+	2.18034i	−0.0496528	+	0.0962644i
$514$		0			0
$515$		−	16.6525i		−	0.733796i
$516$		0			0
$517$		−	11.0557i		−	0.486230i
$518$		0			0
$519$	−10.7639	−	28.1803i	−0.472484	−	1.23698i
$520$		0			0
$521$	−19.0557			−0.834847			−0.417423	−	0.908712i	$-0.637067\pi$
$521$	−19.0557			−0.834847			−0.417423	+	0.908712i	$0.637067\pi$
$522$		0			0
$523$		−	24.5410i		−	1.07310i	−0.843867	−	0.536552i	$-0.819727\pi$
$523$		−	24.5410i		−	1.07310i	0.843867	−	0.536552i	$-0.180273\pi$
$524$		0			0
$525$	−4.00000	+	2.23607i	−0.174574	+	0.0975900i
$526$		0			0
$527$			5.52786i			0.240798i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	−10.0000	+	8.94427i	−0.433963	+	0.388148i
$532$		0			0
$533$			20.9443i			0.907197i
$534$		0			0
$535$			15.4164i			0.666509i
$536$		0			0
$537$	−4.76393	+	1.81966i	−0.205579	+	0.0785241i
$538$		0			0
$539$	−8.94427	−	30.0000i	−0.385257	−	1.29219i
$540$		0			0
$541$	13.0557			0.561310			0.280655	−	0.959809i	$-0.409448\pi$
$541$	13.0557			0.561310			0.280655	+	0.959809i	$0.409448\pi$
$542$		0			0
$543$	10.0000	−	3.81966i	0.429141	−	0.163917i
$544$		0			0
$545$	−4.47214			−0.191565
$546$		0			0
$547$	8.58359			0.367008			0.183504	−	0.983019i	$-0.441256\pi$
$547$	8.58359			0.367008			0.183504	+	0.983019i	$0.441256\pi$
$548$		0			0
$549$	−5.52786	−	6.18034i	−0.235923	−	0.263770i
$550$		0			0
$551$	−2.69505			−0.114813
$552$		0			0
$553$	3.41641	+	23.4164i	0.145280	+	0.995767i
$554$		0			0
$555$	−3.23607	−	8.47214i	−0.137363	−	0.359622i
$556$		0			0
$557$		−	16.4721i		−	0.697947i	−0.937133	−	0.348973i	$-0.886530\pi$
$557$		−	16.4721i		−	0.697947i	0.937133	−	0.348973i	$-0.113470\pi$
$558$		0			0
$559$		−	41.8885i		−	1.77170i
$560$		0			0
$561$	5.52786	−	2.11146i	0.233387	−	0.0891457i
$562$		0			0
$563$	−17.8197			−0.751009			−0.375505	−	0.926821i	$-0.622531\pi$
$563$	−17.8197			−0.751009			−0.375505	+	0.926821i	$0.622531\pi$
$564$		0			0
$565$			14.9443i			0.628710i
$566$		0			0
$567$	−23.7984	+	0.798374i	−0.999438	+	0.0335286i
$568$		0			0
$569$			18.4721i			0.774392i	0.921997	+	0.387196i	$0.126556\pi$
$569$			18.4721i			0.774392i	−0.921997	+	0.387196i	$0.873444\pi$
$570$		0			0
$571$	34.8328			1.45771			0.728854	−	0.684669i	$-0.240053\pi$
$571$	34.8328			1.45771			0.728854	+	0.684669i	$0.240053\pi$
$572$		0			0
$573$	−4.47214	+	1.70820i	−0.186826	+	0.0713612i
$574$		0			0
$575$			4.00000i			0.166812i
$576$		0			0
$577$			14.1803i			0.590335i	0.955445	+	0.295168i	$0.0953755\pi$
$577$			14.1803i			0.590335i	−0.955445	+	0.295168i	$0.904625\pi$
$578$		0			0
$579$	3.70820	+	9.70820i	0.154108	+	0.403459i
$580$		0			0
$581$	−6.36068	−	43.5967i	−0.263885	−	1.80870i
$582$		0			0
$583$	37.8885			1.56918
$584$		0			0
$585$	−6.47214	−	7.23607i	−0.267590	−	0.299175i
$586$		0			0
$587$	23.7082			0.978542			0.489271	−	0.872132i	$-0.337263\pi$
$587$	23.7082			0.978542			0.489271	+	0.872132i	$0.337263\pi$
$588$		0			0
$589$	−3.41641			−0.140771
$590$		0			0
$591$	20.1803	−	7.70820i	0.830108	−	0.317073i
$592$		0			0
$593$	47.0132			1.93060			0.965299	−	0.261145i	$-0.0841002\pi$
$593$	47.0132			1.93060			0.965299	+	0.261145i	$0.0841002\pi$
$594$		0			0
$595$	−0.291796	−	2.00000i	−0.0119625	−	0.0819920i
$596$		0			0
$597$	−0.291796	+	0.111456i	−0.0119424	+	0.00456160i
$598$		0			0
$599$		−	41.2361i		−	1.68486i	−0.538806	−	0.842430i	$-0.681124\pi$
$599$		−	41.2361i		−	1.68486i	0.538806	−	0.842430i	$-0.318876\pi$
$600$		0			0
$601$			14.4721i			0.590331i	0.955446	+	0.295165i	$0.0953747\pi$
$601$			14.4721i			0.590331i	−0.955446	+	0.295165i	$0.904625\pi$
$602$		0			0
$603$	−26.8328	+	24.0000i	−1.09272	+	0.977356i
$604$		0			0
$605$	−9.00000			−0.365902
$606$		0			0
$607$			14.7639i			0.599250i	0.954057	+	0.299625i	$0.0968615\pi$
$607$			14.7639i			0.599250i	−0.954057	+	0.299625i	$0.903139\pi$
$608$		0			0
$609$	−12.7639	−	22.8328i	−0.517221	−	0.925232i
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$	−20.0689			−0.810575			−0.405287	−	0.914189i	$-0.632829\pi$
$613$	−20.0689			−0.810575			−0.405287	+	0.914189i	$0.632829\pi$
$614$		0			0
$615$	−4.00000	−	10.4721i	−0.161296	−	0.422277i
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$	−9.52786	+	18.4721i	−0.382340	+	0.741261i
$622$		0			0
$623$	5.52786	+	37.8885i	0.221469	+	1.51797i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.30495	+	3.41641i	0.0521148	+	0.136438i
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	−29.8885			−1.18984			−0.594922	−	0.803783i	$-0.702817\pi$
$631$	−29.8885			−1.18984			−0.594922	+	0.803783i	$0.702817\pi$
$632$		0			0
$633$	−4.94427	−	12.9443i	−0.196517	−	0.514489i
$634$		0			0
$635$	13.7082			0.543993
$636$		0			0
$637$	−6.47214	−	21.7082i	−0.256435	−	0.860110i
$638$		0			0
$639$	−5.52786	−	6.18034i	−0.218679	−	0.244490i
$640$		0			0
$641$		−	3.41641i		−	0.134940i	−0.997721	−	0.0674700i	$-0.978507\pi$
$641$		−	3.41641i		−	0.134940i	0.997721	−	0.0674700i	$-0.0214927\pi$
$642$		0			0
$643$			25.7082i			1.01383i	0.861995	+	0.506916i	$0.169215\pi$
$643$			25.7082i			1.01383i	−0.861995	+	0.506916i	$0.830785\pi$
$644$		0			0
$645$	8.00000	+	20.9443i	0.315000	+	0.824680i
$646$		0			0
$647$	18.8328			0.740394			0.370197	−	0.928953i	$-0.379290\pi$
$647$	18.8328			0.740394			0.370197	+	0.928953i	$0.379290\pi$
$648$		0			0
$649$			20.0000i			0.785069i
$650$		0			0
$651$	−16.1803	−	28.9443i	−0.634158	−	1.13442i
$652$		0			0
$653$		−	11.8885i		−	0.465235i	−0.972568	−	0.232617i	$-0.925271\pi$
$653$		−	11.8885i		−	0.465235i	0.972568	−	0.232617i	$-0.0747290\pi$
$654$		0			0
$655$	21.4164			0.836808
$656$		0			0
$657$	13.5279	+	15.1246i	0.527772	+	0.590067i
$658$		0			0
$659$			40.4721i			1.57657i	0.615310	+	0.788285i	$0.289031\pi$
$659$			40.4721i			1.57657i	−0.615310	+	0.788285i	$0.710969\pi$
$660$		0			0
$661$			31.7082i			1.23331i	0.787235	+	0.616653i	$0.211512\pi$
$661$			31.7082i			1.23331i	−0.787235	+	0.616653i	$0.788488\pi$
$662$		0			0
$663$	4.00000	−	1.52786i	0.155347	−	0.0593373i
$664$		0			0
$665$	1.23607	−	0.180340i	0.0479327	−	0.00699328i
$666$		0			0
$667$	−22.8328			−0.884090
$668$		0			0
$669$	−6.94427	+	2.65248i	−0.268481	+	0.102551i
$670$		0			0
$671$	−12.3607			−0.477179
$672$		0			0
$673$	28.4721			1.09752			0.548760	−	0.835980i	$-0.315100\pi$
$673$	28.4721			1.09752			0.548760	+	0.835980i	$0.315100\pi$
$674$		0			0
$675$	4.61803	+	2.38197i	0.177748	+	0.0916819i
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	13.7082	−	2.00000i	0.526073	−	0.0767530i
$680$		0			0
$681$	3.23607	+	8.47214i	0.124006	+	0.324653i
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$			12.4721i			0.476536i
$686$		0			0
$687$	−21.4164	+	8.18034i	−0.817087	+	0.312099i
$688$		0			0
$689$	27.4164			1.04448
$690$		0			0
$691$			38.9443i			1.48151i	0.671775	+	0.740755i	$0.265532\pi$
$691$			38.9443i			1.48151i	−0.671775	+	0.740755i	$0.734468\pi$
$692$		0			0
$693$	−22.7639	+	27.2361i	−0.864730	+	1.03461i
$694$		0			0
$695$			20.4721i			0.776552i
$696$		0			0
$697$	4.94427			0.187278
$698$		0			0
$699$	33.1246	−	12.6525i	1.25289	−	0.478561i
$700$		0			0
$701$			14.0689i			0.531374i	0.964059	+	0.265687i	$0.0855988\pi$
$701$			14.0689i			0.531374i	−0.964059	+	0.265687i	$0.914401\pi$
$702$		0			0
$703$			2.47214i			0.0932384i
$704$		0			0
$705$	1.52786	+	4.00000i	0.0575427	+	0.150649i
$706$		0			0
$707$	1.34752	+	9.23607i	0.0506789	+	0.347358i
$708$		0			0
$709$	−24.4721			−0.919070			−0.459535	−	0.888160i	$-0.651984\pi$
$709$	−24.4721			−0.919070			−0.459535	+	0.888160i	$0.651984\pi$
$710$		0			0
$711$	20.0000	−	17.8885i	0.750059	−	0.670873i
$712$		0			0
$713$	−28.9443			−1.08397
$714$		0			0
$715$	−14.4721			−0.541227
$716$		0			0
$717$	35.8885	−	13.7082i	1.34028	−	0.511942i
$718$		0			0
$719$	25.5279			0.952029			0.476014	−	0.879438i	$-0.342081\pi$
$719$	25.5279			0.952029			0.476014	+	0.879438i	$0.342081\pi$
$720$		0			0
$721$	−43.5967	+	6.36068i	−1.62363	+	0.236884i
$722$		0			0
$723$	−28.9443	+	11.0557i	−1.07645	+	0.411167i
$724$		0			0
$725$			5.70820i			0.211997i
$726$		0			0
$727$		−	39.7082i		−	1.47270i	−0.676603	−	0.736348i	$-0.736549\pi$
$727$		−	39.7082i		−	1.47270i	0.676603	−	0.736348i	$-0.263451\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	−9.88854			−0.365741
$732$		0			0
$733$		−	38.0689i		−	1.40611i	−0.711137	−	0.703053i	$-0.751820\pi$
$733$		−	38.0689i		−	1.40611i	0.711137	−	0.703053i	$-0.248180\pi$
$734$		0			0
$735$	7.38197	+	9.61803i	0.272288	+	0.354767i
$736$		0			0
$737$			53.6656i			1.97680i
$738$		0			0
$739$	8.94427			0.329020			0.164510	−	0.986375i	$-0.447396\pi$
$739$	8.94427			0.329020			0.164510	+	0.986375i	$0.447396\pi$
$740$		0			0
$741$	0.944272	+	2.47214i	0.0346887	+	0.0908162i
$742$		0			0
$743$		−	1.52786i		−	0.0560519i	−0.999607	−	0.0280259i	$-0.991078\pi$
$743$		−	1.52786i		−	0.0560519i	0.999607	−	0.0280259i	$-0.00892210\pi$
$744$		0			0
$745$		−	17.7082i		−	0.648778i
$746$		0			0
$747$	−37.2361	+	33.3050i	−1.36240	+	1.21856i
$748$		0			0
$749$	40.3607	−	5.88854i	1.47475	−	0.215163i
$750$		0			0
$751$	−35.4164			−1.29236			−0.646182	−	0.763184i	$-0.723635\pi$
$751$	−35.4164			−1.29236			−0.646182	+	0.763184i	$0.723635\pi$
$752$		0			0
$753$	−2.18034	−	5.70820i	−0.0794560	−	0.208019i
$754$		0			0
$755$	−3.05573			−0.111209
$756$		0			0
$757$	28.6525			1.04139			0.520696	−	0.853742i	$-0.325673\pi$
$757$	28.6525			1.04139			0.520696	+	0.853742i	$0.325673\pi$
$758$		0			0
$759$	11.0557	+	28.9443i	0.401298	+	1.05061i
$760$		0			0
$761$	29.8885			1.08346			0.541729	−	0.840553i	$-0.317770\pi$
$761$	29.8885			1.08346			0.541729	+	0.840553i	$0.317770\pi$
$762$		0			0
$763$	1.70820	+	11.7082i	0.0618411	+	0.423865i
$764$		0			0
$765$	−1.70820	+	1.52786i	−0.0617602	+	0.0552400i
$766$		0			0
$767$			14.4721i			0.522559i
$768$		0			0
$769$		−	36.0000i		−	1.29819i	−0.760706	−	0.649097i	$-0.775147\pi$
$769$		−	36.0000i		−	1.29819i	0.760706	−	0.649097i	$-0.224853\pi$
$770$		0			0
$771$	5.05573	+	13.2361i	0.182078	+	0.476685i
$772$		0			0
$773$	37.4164			1.34577			0.672887	−	0.739745i	$-0.265054\pi$
$773$	37.4164			1.34577			0.672887	+	0.739745i	$0.265054\pi$
$774$		0			0
$775$			7.23607i			0.259927i
$776$		0			0
$777$	−20.9443	+	11.7082i	−0.751372	+	0.420029i
$778$		0			0
$779$			3.05573i			0.109483i
$780$		0			0
$781$	−12.3607			−0.442300
$782$		0			0
$783$	−13.5967	+	26.3607i	−0.485908	+	0.942054i
$784$		0			0
$785$		−	4.76393i		−	0.170032i
$786$		0			0
$787$		−	49.7082i		−	1.77191i	−0.463775	−	0.885953i	$-0.653505\pi$
$787$		−	49.7082i		−	1.77191i	0.463775	−	0.885953i	$-0.346495\pi$
$788$		0			0
$789$	−8.00000	+	3.05573i	−0.284808	+	0.108787i
$790$		0			0
$791$	39.1246	−	5.70820i	1.39111	−	0.202960i
$792$		0			0
$793$	−8.94427			−0.317620
$794$		0			0
$795$	−13.7082	+	5.23607i	−0.486180	+	0.185704i
$796$		0			0
$797$	−5.41641			−0.191859			−0.0959295	−	0.995388i	$-0.530582\pi$
$797$	−5.41641			−0.191859			−0.0959295	+	0.995388i	$0.530582\pi$
$798$		0			0
$799$	−1.88854			−0.0668119
$800$		0			0
$801$	32.3607	−	28.9443i	1.14341	−	1.02270i
$802$		0			0
$803$	30.2492			1.06747
$804$		0			0
$805$	10.4721	−	1.52786i	0.369094	−	0.0538501i
$806$		0			0
$807$	2.76393	+	7.23607i	0.0972950	+	0.254722i
$808$		0			0
$809$		−	8.36068i		−	0.293946i	−0.989141	−	0.146973i	$-0.953047\pi$
$809$		−	8.36068i		−	0.293946i	0.989141	−	0.146973i	$-0.0469530\pi$
$810$		0			0
$811$			16.8328i			0.591080i	0.955330	+	0.295540i	$0.0954996\pi$
$811$			16.8328i			0.591080i	−0.955330	+	0.295540i	$0.904500\pi$
$812$		0			0
$813$	−29.5967	+	11.3050i	−1.03800	+	0.396482i
$814$		0			0
$815$	1.52786			0.0535187
$816$		0			0
$817$		−	6.11146i		−	0.213813i
$818$		0			0
$819$	−16.4721	+	19.7082i	−0.575583	+	0.688660i
$820$		0			0
$821$		−	18.2918i		−	0.638388i	−0.947689	−	0.319194i	$-0.896588\pi$
$821$		−	18.2918i		−	0.638388i	0.947689	−	0.319194i	$-0.103412\pi$
$822$		0			0
$823$	−0.180340			−0.00628625			−0.00314313	−	0.999995i	$-0.501000\pi$
$823$	−0.180340			−0.00628625			−0.00314313	+	0.999995i	$0.501000\pi$
$824$		0			0
$825$	7.23607	−	2.76393i	0.251928	−	0.0962278i
$826$		0			0
$827$		−	14.8328i		−	0.515788i	−0.966173	−	0.257894i	$-0.916972\pi$
$827$		−	14.8328i		−	0.515788i	0.966173	−	0.257894i	$-0.0830285\pi$
$828$		0			0
$829$			30.1803i			1.04821i	0.851655	+	0.524103i	$0.175599\pi$
$829$			30.1803i			1.04821i	−0.851655	+	0.524103i	$0.824401\pi$
$830$		0			0
$831$	−14.2918	−	37.4164i	−0.495777	−	1.29796i
$832$		0			0
$833$	−5.12461	+	1.52786i	−0.177557	+	0.0529374i
$834$		0			0
$835$	−16.9443			−0.586381
$836$		0			0
$837$	−17.2361	+	33.4164i	−0.595766	+	1.15504i
$838$		0			0
$839$	−14.4721			−0.499634			−0.249817	−	0.968293i	$-0.580370\pi$
$839$	−14.4721			−0.499634			−0.249817	+	0.968293i	$0.580370\pi$
$840$		0			0
$841$	−3.58359			−0.123572
$842$		0			0
$843$	32.3607	−	12.3607i	1.11456	−	0.425724i
$844$		0			0
$845$	2.52786			0.0869612
$846$		0			0
$847$	3.43769	+	23.5623i	0.118121	+	0.809610i
$848$		0			0
$849$	−14.1803	+	5.41641i	−0.486668	+	0.185891i
$850$		0			0
$851$			20.9443i			0.717960i
$852$		0			0
$853$		−	5.70820i		−	0.195445i	−0.995214	−	0.0977226i	$-0.968844\pi$
$853$		−	5.70820i		−	0.195445i	0.995214	−	0.0977226i	$-0.0311558\pi$
$854$		0			0
$855$	−0.944272	−	1.05573i	−0.0322934	−	0.0361051i
$856$		0			0
$857$	−42.6525			−1.45698			−0.728490	−	0.685056i	$-0.759778\pi$
$857$	−42.6525			−1.45698			−0.728490	+	0.685056i	$0.759778\pi$
$858$		0			0
$859$			34.9443i			1.19228i	0.802879	+	0.596142i	$0.203300\pi$
$859$			34.9443i			1.19228i	−0.802879	+	0.596142i	$0.796700\pi$
$860$		0			0
$861$	−25.8885	+	14.4721i	−0.882279	+	0.493209i
$862$		0			0
$863$		−	21.5279i		−	0.732817i	−0.930454	−	0.366409i	$-0.880587\pi$
$863$		−	21.5279i		−	0.732817i	0.930454	−	0.366409i	$-0.119413\pi$
$864$		0			0
$865$	17.4164			0.592176
$866$		0			0
$867$	10.1459	+	26.5623i	0.344573	+	0.902103i
$868$		0			0
$869$		−	40.0000i		−	1.35691i
$870$		0			0
$871$			38.8328i			1.31580i
$872$		0			0
$873$	−10.4721	−	11.7082i	−0.354428	−	0.396263i
$874$		0			0
$875$	−0.381966	−	2.61803i	−0.0129128	−	0.0885057i
$876$		0			0
$877$	7.34752			0.248108			0.124054	−	0.992275i	$-0.460410\pi$
$877$	7.34752			0.248108			0.124054	+	0.992275i	$0.460410\pi$
$878$		0			0
$879$	18.1803	+	47.5967i	0.613208	+	1.60540i
$880$		0			0
$881$	−11.4164			−0.384629			−0.192314	−	0.981333i	$-0.561599\pi$
$881$	−11.4164			−0.384629			−0.192314	+	0.981333i	$0.561599\pi$
$882$		0			0
$883$	−21.8885			−0.736608			−0.368304	−	0.929705i	$-0.620061\pi$
$883$	−21.8885			−0.736608			−0.368304	+	0.929705i	$0.620061\pi$
$884$		0			0
$885$	−2.76393	−	7.23607i	−0.0929086	−	0.243238i
$886$		0			0
$887$	26.4721			0.888847			0.444424	−	0.895817i	$-0.353408\pi$
$887$	26.4721			0.888847			0.444424	+	0.895817i	$0.353408\pi$
$888$		0			0
$889$	−5.23607	−	35.8885i	−0.175612	−	1.20366i
$890$		0			0
$891$	40.0000	+	4.47214i	1.34005	+	0.149822i
$892$		0			0
$893$		−	1.16718i		−	0.0390583i
$894$		0			0
$895$		−	2.94427i		−	0.0984162i
$896$		0			0
$897$	8.00000	+	20.9443i	0.267112	+	0.699309i
$898$		0			0
$899$	−41.3050			−1.37760
$900$		0			0
$901$		−	6.47214i		−	0.215618i
$902$		0			0
$903$	51.7771	−	28.9443i	1.72303	−	0.963205i
$904$		0			0
$905$			6.18034i			0.205441i
$906$		0			0
$907$	26.4721			0.878993			0.439496	−	0.898244i	$-0.355157\pi$
$907$	26.4721			0.878993			0.439496	+	0.898244i	$0.355157\pi$
$908$		0			0
$909$	7.88854	−	7.05573i	0.261646	−	0.234024i
$910$		0			0
$911$		−	19.3475i		−	0.641012i	−0.947246	−	0.320506i	$-0.896147\pi$
$911$		−	19.3475i		−	0.641012i	0.947246	−	0.320506i	$-0.103853\pi$
$912$		0			0
$913$			74.4721i			2.46467i
$914$		0			0
$915$	4.47214	−	1.70820i	0.147844	−	0.0564715i
$916$		0			0
$917$	−8.18034	−	56.0689i	−0.270139	−	1.85156i
$918$		0			0
$919$	−44.7214			−1.47522			−0.737611	−	0.675226i	$-0.764046\pi$
$919$	−44.7214			−1.47522			−0.737611	+	0.675226i	$0.764046\pi$
$920$		0			0
$921$	−6.76393	+	2.58359i	−0.222879	+	0.0851323i
$922$		0			0
$923$	−8.94427			−0.294404
$924$		0			0
$925$	5.23607			0.172161
$926$		0			0
$927$	33.3050	+	37.2361i	1.09388	+	1.22299i
$928$		0			0
$929$	2.11146			0.0692746			0.0346373	−	0.999400i	$-0.488972\pi$
$929$	2.11146			0.0692746			0.0346373	+	0.999400i	$0.488972\pi$
$930$		0			0
$931$	−0.944272	−	3.16718i	−0.0309473	−	0.103800i
$932$		0			0
$933$	16.3607	+	42.8328i	0.535625	+	1.40228i
$934$		0			0
$935$			3.41641i			0.111728i
$936$		0			0
$937$		−	16.0689i		−	0.524948i	−0.964939	−	0.262474i	$-0.915462\pi$
$937$		−	16.0689i		−	0.524948i	0.964939	−	0.262474i	$-0.0845383\pi$
$938$		0			0
$939$	12.4721	−	4.76393i	0.407013	−	0.155465i
$940$		0			0
$941$	22.0000			0.717180			0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000			0.717180			0.358590	+	0.933495i	$0.383258\pi$
$942$		0			0
$943$			25.8885i			0.843047i
$944$		0			0
$945$	4.47214	−	13.0000i	0.145479	−	0.422890i
$946$		0			0
$947$		−	15.6393i		−	0.508210i	−0.967177	−	0.254105i	$-0.918219\pi$
$947$		−	15.6393i		−	0.508210i	0.967177	−	0.254105i	$-0.0817808\pi$
$948$		0			0
$949$	21.8885			0.710532
$950$		0			0
$951$	11.2361	−	4.29180i	0.364354	−	0.139171i
$952$		0			0
$953$			49.4164i			1.60075i	0.599497	+	0.800377i	$0.295367\pi$
$953$			49.4164i			1.60075i	−0.599497	+	0.800377i	$0.704633\pi$
$954$		0			0
$955$		−	2.76393i		−	0.0894387i
$956$		0			0
$957$	15.7771	+	41.3050i	0.510001	+	1.33520i
$958$		0			0
$959$	32.6525	−	4.76393i	1.05440	−	0.153835i
$960$		0			0
$961$	−21.3607			−0.689054
$962$		0			0
$963$	−30.8328	−	34.4721i	−0.993574	−	1.11085i
$964$		0			0
$965$	−6.00000			−0.193147
$966$		0			0
$967$	−45.4853			−1.46271			−0.731354	−	0.681998i	$-0.761111\pi$
$967$	−45.4853			−1.46271			−0.731354	+	0.681998i	$0.761111\pi$
$968$		0			0
$969$	0.583592	−	0.222912i	0.0187477	−	0.00716098i
$970$		0			0
$971$	58.0000			1.86131			0.930654	−	0.365900i	$-0.119239\pi$
$971$	58.0000			1.86131			0.930654	+	0.365900i	$0.119239\pi$
$972$		0			0
$973$	53.5967	−	7.81966i	1.71823	−	0.250687i
$974$		0			0
$975$	5.23607	−	2.00000i	0.167688	−	0.0640513i
$976$		0			0
$977$		−	25.4164i		−	0.813143i	−0.913619	−	0.406571i	$-0.866724\pi$
$977$		−	25.4164i		−	0.813143i	0.913619	−	0.406571i	$-0.133276\pi$
$978$		0			0
$979$		−	64.7214i		−	2.06850i
$980$		0			0
$981$	10.0000	−	8.94427i	0.319275	−	0.285569i
$982$		0			0
$983$	39.4164			1.25719			0.628594	−	0.777734i	$-0.283631\pi$
$983$	39.4164			1.25719			0.628594	+	0.777734i	$0.283631\pi$
$984$		0			0
$985$			12.4721i			0.397395i
$986$		0			0
$987$	9.88854	−	5.52786i	0.314756	−	0.175954i
$988$		0			0
$989$		−	51.7771i		−	1.64642i
$990$		0			0
$991$	−26.4721			−0.840915			−0.420458	−	0.907312i	$-0.638130\pi$
$991$	−26.4721			−0.840915			−0.420458	+	0.907312i	$0.638130\pi$
$992$		0			0
$993$	12.9443	+	33.8885i	0.410774	+	1.07542i
$994$		0			0
$995$		−	0.180340i		−	0.00571716i
$996$		0			0
$997$			20.7639i			0.657600i	0.944400	+	0.328800i	$0.106644\pi$
$997$			20.7639i			0.657600i	−0.944400	+	0.328800i	$0.893356\pi$
$998$		0			0
$999$	24.1803	+	12.4721i	0.765032	+	0.394601i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.i.881.1		4
3.2	odd	2		1680.2.f.e.881.3		4
4.3	odd	2		210.2.b.a.41.4	yes	4
7.6	odd	2		1680.2.f.e.881.4		4
12.11	even	2		210.2.b.b.41.1	yes	4
20.3	even	4		1050.2.d.d.1049.2		4
20.7	even	4		1050.2.d.c.1049.3		4
20.19	odd	2		1050.2.b.c.251.1		4
21.20	even	2	inner	1680.2.f.i.881.2		4
28.27	even	2		210.2.b.b.41.3	yes	4
60.23	odd	4		1050.2.d.a.1049.1		4
60.47	odd	4		1050.2.d.f.1049.4		4
60.59	even	2		1050.2.b.a.251.4		4
84.83	odd	2		210.2.b.a.41.2	✓	4
140.27	odd	4		1050.2.d.a.1049.2		4
140.83	odd	4		1050.2.d.f.1049.3		4
140.139	even	2		1050.2.b.a.251.2		4
420.83	even	4		1050.2.d.c.1049.4		4
420.167	even	4		1050.2.d.d.1049.1		4
420.419	odd	2		1050.2.b.c.251.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.b.a.41.2	✓	4	84.83	odd	2
210.2.b.a.41.4	yes	4	4.3	odd	2
210.2.b.b.41.1	yes	4	12.11	even	2
210.2.b.b.41.3	yes	4	28.27	even	2
1050.2.b.a.251.2		4	140.139	even	2
1050.2.b.a.251.4		4	60.59	even	2
1050.2.b.c.251.1		4	20.19	odd	2
1050.2.b.c.251.3		4	420.419	odd	2
1050.2.d.a.1049.1		4	60.23	odd	4
1050.2.d.a.1049.2		4	140.27	odd	4
1050.2.d.c.1049.3		4	20.7	even	4
1050.2.d.c.1049.4		4	420.83	even	4
1050.2.d.d.1049.1		4	420.167	even	4
1050.2.d.d.1049.2		4	20.3	even	4
1050.2.d.f.1049.3		4	140.83	odd	4
1050.2.d.f.1049.4		4	60.47	odd	4
1680.2.f.e.881.3		4	3.2	odd	2
1680.2.f.e.881.4		4	7.6	odd	2
1680.2.f.i.881.1		4	1.1	even	1	trivial
1680.2.f.i.881.2		4	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.618034 - 1.61803i) q^{3} +1.00000 q^{5} +(-0.381966 - 2.61803i) q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(-0.618034 - 1.61803i) q^{3} +1.00000 q^{5} +(-0.381966 - 2.61803i) q^{7} +(-2.23607 + 2.00000i) q^{9} +4.47214i q^{11} +3.23607i q^{13} +(-0.618034 - 1.61803i) q^{15} +0.763932 q^{17} +0.472136i q^{19} +(-4.00000 + 2.23607i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +(4.61803 + 2.38197i) q^{27} +5.70820i q^{29} +7.23607i q^{31} +(7.23607 - 2.76393i) q^{33} +(-0.381966 - 2.61803i) q^{35} +5.23607 q^{37} +(5.23607 - 2.00000i) q^{39} +6.47214 q^{41} -12.9443 q^{43} +(-2.23607 + 2.00000i) q^{45} -2.47214 q^{47} +(-6.70820 + 2.00000i) q^{49} +(-0.472136 - 1.23607i) q^{51} -8.47214i q^{53} +4.47214i q^{55} +(0.763932 - 0.291796i) q^{57} +4.47214 q^{59} +2.76393i q^{61} +(6.09017 + 5.09017i) q^{63} +3.23607i q^{65} +12.0000 q^{67} +(6.47214 - 2.47214i) q^{69} +2.76393i q^{71} -6.76393i q^{73} +(-0.618034 - 1.61803i) q^{75} +(11.7082 - 1.70820i) q^{77} -8.94427 q^{79} +(1.00000 - 8.94427i) q^{81} +16.6525 q^{83} +0.763932 q^{85} +(9.23607 - 3.52786i) q^{87} -14.4721 q^{89} +(8.47214 - 1.23607i) q^{91} +(11.7082 - 4.47214i) q^{93} +0.472136i q^{95} +5.23607i q^{97} +(-8.94427 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} - 6 q^{7}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 - 6 * q^7	\( 4 q + 2 q^{3} + 4 q^{5} - 6 q^{7} + 2 q^{15} + 12 q^{17} - 16 q^{21} + 4 q^{25} + 14 q^{27} + 20 q^{33} - 6 q^{35} + 12 q^{37} + 12 q^{39} + 8 q^{41} - 16 q^{43} + 8 q^{47} + 16 q^{51} + 12 q^{57} + 2 q^{63} + 48 q^{67} + 8 q^{69} + 2 q^{75} + 20 q^{77} + 4 q^{81} + 4 q^{83} + 12 q^{85} + 28 q^{87} - 40 q^{89} + 16 q^{91} + 20 q^{93}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 - 6 * q^7 + 2 * q^15 + 12 * q^17 - 16 * q^21 + 4 * q^25 + 14 * q^27 + 20 * q^33 - 6 * q^35 + 12 * q^37 + 12 * q^39 + 8 * q^41 - 16 * q^43 + 8 * q^47 + 16 * q^51 + 12 * q^57 + 2 * q^63 + 48 * q^67 + 8 * q^69 + 2 * q^75 + 20 * q^77 + 4 * q^81 + 4 * q^83 + 12 * q^85 + 28 * q^87 - 40 * q^89 + 16 * q^91 + 20 * q^93

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