LMFDB - Embedded newform 480.2.b.b.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(431,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("480.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.b (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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1649659456.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 $ x^8 - x^7 - 2x^5 + 4x^4 - 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$0.814732 + 1.15595i$ of defining polynomial
Character	$\chi$	$=$	480.431
Dual form			480.2.b.b.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.48716 + 0.887900i) q^{3} +1.00000 q^{5} +0.797253i q^{7} +(1.42327 - 2.64089i) q^{9} +O(q^{10})$	\(q+(-1.48716 + 0.887900i) q^{3} +1.00000 q^{5} +0.797253i q^{7} +(1.42327 - 2.64089i) q^{9} +0.320548i q^{11} +4.30324i q^{13} +(-1.48716 + 0.887900i) q^{15} +2.57305i q^{17} +6.10546 q^{19} +(-0.707881 - 1.18564i) q^{21} -3.13115 q^{23} +1.00000 q^{25} +(0.228229 + 5.19114i) q^{27} -8.79516 q^{29} +9.90557i q^{31} +(-0.284615 - 0.476705i) q^{33} +0.797253i q^{35} +8.49593i q^{37} +(-3.82085 - 6.39959i) q^{39} +5.28178i q^{41} +2.97431 q^{43} +(1.42327 - 2.64089i) q^{45} +6.56192 q^{47} +6.36439 q^{49} +(-2.28461 - 3.82653i) q^{51} +3.94862 q^{53} +0.320548i q^{55} +(-9.07977 + 5.42104i) q^{57} -12.4786i q^{59} -8.83339i q^{61} +(2.10546 + 1.13470i) q^{63} +4.30324i q^{65} -4.66738 q^{67} +(4.65651 - 2.78015i) q^{69} -3.43077 q^{71} -1.43077 q^{73} +(-1.48716 + 0.887900i) q^{75} -0.255558 q^{77} -2.89360i q^{79} +(-4.94862 - 7.51739i) q^{81} -3.37031i q^{83} +2.57305i q^{85} +(13.0798 - 7.80922i) q^{87} -13.7526i q^{89} -3.43077 q^{91} +(-8.79516 - 14.7311i) q^{93} +6.10546 q^{95} -4.26230 q^{97} +(0.846533 + 0.456225i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 12 q^{27} - 4 q^{33} + 16 q^{39} + 28 q^{47} - 16 q^{49} - 20 q^{51} - 16 q^{53} - 4 q^{57} - 28 q^{63} + 32 q^{67} - 20 q^{69} - 24 q^{71} - 8 q^{73} + 8 q^{81} + 36 q^{87} - 24 q^{91} + 4 q^{95} + 8 q^{97} - 16 q^{99}+O(q^{100}) $ 8 * q + 8 * q^5 + 4 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 + 12 * q^27 - 4 * q^33 + 16 * q^39 + 28 * q^47 - 16 * q^49 - 20 * q^51 - 16 * q^53 - 4 * q^57 - 28 * q^63 + 32 * q^67 - 20 * q^69 - 24 * q^71 - 8 * q^73 + 8 * q^81 + 36 * q^87 - 24 * q^91 + 4 * q^95 + 8 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.48716	+	0.887900i	−0.858610	+	0.512629i
$3$	−1.48716	+	0.887900i	−0.858610	+	0.512629i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			0.797253i			0.301333i	0.988585	+	0.150667i	$0.0481420\pi$
$7$			0.797253i			0.301333i	−0.988585	+	0.150667i	$0.951858\pi$
$8$		0			0
$9$	1.42327	−	2.64089i	0.474422	−	0.880297i
$10$		0			0
$11$			0.320548i			0.0966489i	0.998832	+	0.0483245i	$0.0153881\pi$
$11$			0.320548i			0.0966489i	−0.998832	+	0.0483245i	$0.984612\pi$
$12$		0			0
$13$			4.30324i			1.19350i	0.802426	+	0.596752i	$0.203542\pi$
$13$			4.30324i			1.19350i	−0.802426	+	0.596752i	$0.796458\pi$
$14$		0			0
$15$	−1.48716	+	0.887900i	−0.383982	+	0.229255i
$16$		0			0
$17$			2.57305i			0.624057i	0.950073	+	0.312029i	$0.101008\pi$
$17$			2.57305i			0.624057i	−0.950073	+	0.312029i	$0.898992\pi$
$18$		0			0
$19$	6.10546			1.40069			0.700344	−	0.713805i	$-0.253030\pi$
$19$	6.10546			1.40069			0.700344	+	0.713805i	$0.253030\pi$
$20$		0			0
$21$	−0.707881	−	1.18564i	−0.154472	−	0.258728i
$22$		0			0
$23$	−3.13115			−0.652889			−0.326445	−	0.945216i	$-0.605851\pi$
$23$	−3.13115			−0.652889			−0.326445	+	0.945216i	$0.605851\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	0.228229	+	5.19114i	0.0439227	+	0.999035i
$28$		0			0
$29$	−8.79516			−1.63322			−0.816610	−	0.577190i	$-0.804149\pi$
$29$	−8.79516			−1.63322			−0.816610	+	0.577190i	$0.804149\pi$
$30$		0			0
$31$			9.90557i			1.77909i	0.456845	+	0.889546i	$0.348980\pi$
$31$			9.90557i			1.77909i	−0.456845	+	0.889546i	$0.651020\pi$
$32$		0			0
$33$	−0.284615	−	0.476705i	−0.0495451	−	0.0829837i
$34$		0			0
$35$			0.797253i			0.134760i
$36$		0			0
$37$			8.49593i			1.39672i	0.715745	+	0.698362i	$0.246087\pi$
$37$			8.49593i			1.39672i	−0.715745	+	0.698362i	$0.753913\pi$
$38$		0			0
$39$	−3.82085	−	6.39959i	−0.611825	−	1.02475i
$40$		0			0
$41$			5.28178i			0.824876i	0.910986	+	0.412438i	$0.135323\pi$
$41$			5.28178i			0.824876i	−0.910986	+	0.412438i	$0.864677\pi$
$42$		0			0
$43$	2.97431			0.453578			0.226789	−	0.973944i	$-0.427177\pi$
$43$	2.97431			0.453578			0.226789	+	0.973944i	$0.427177\pi$
$44$		0			0
$45$	1.42327	−	2.64089i	0.212168	−	0.393681i
$46$		0			0
$47$	6.56192			0.957154			0.478577	−	0.878046i	$-0.341153\pi$
$47$	6.56192			0.957154			0.478577	+	0.878046i	$0.341153\pi$
$48$		0			0
$49$	6.36439			0.909198
$50$		0			0
$51$	−2.28461	−	3.82653i	−0.319910	−	0.535822i
$52$		0			0
$53$	3.94862			0.542385			0.271193	−	0.962525i	$-0.412582\pi$
$53$	3.94862			0.542385			0.271193	+	0.962525i	$0.412582\pi$
$54$		0			0
$55$			0.320548i			0.0432227i
$56$		0			0
$57$	−9.07977	+	5.42104i	−1.20265	+	0.718034i
$58$		0			0
$59$		−	12.4786i		−	1.62458i	−0.583255	−	0.812289i	$-0.698221\pi$
$59$		−	12.4786i		−	1.62458i	0.583255	−	0.812289i	$-0.301779\pi$
$60$		0			0
$61$		−	8.83339i		−	1.13100i	−0.824749	−	0.565500i	$-0.808683\pi$
$61$		−	8.83339i		−	1.13100i	0.824749	−	0.565500i	$-0.191317\pi$
$62$		0			0
$63$	2.10546	+	1.13470i	0.265263	+	0.142959i
$64$		0			0
$65$			4.30324i			0.533751i
$66$		0			0
$67$	−4.66738			−0.570211			−0.285106	−	0.958496i	$-0.592029\pi$
$67$	−4.66738			−0.570211			−0.285106	+	0.958496i	$0.592029\pi$
$68$		0			0
$69$	4.65651	−	2.78015i	0.560577	−	0.334690i
$70$		0			0
$71$	−3.43077			−0.407158			−0.203579	−	0.979059i	$-0.565257\pi$
$71$	−3.43077			−0.407158			−0.203579	+	0.979059i	$0.565257\pi$
$72$		0			0
$73$	−1.43077			−0.167459			−0.0837295	−	0.996489i	$-0.526683\pi$
$73$	−1.43077			−0.167459			−0.0837295	+	0.996489i	$0.526683\pi$
$74$		0			0
$75$	−1.48716	+	0.887900i	−0.171722	+	0.102526i
$76$		0			0
$77$	−0.255558			−0.0291235
$78$		0			0
$79$		−	2.89360i		−	0.325556i	−0.986663	−	0.162778i	$-0.947955\pi$
$79$		−	2.89360i		−	0.325556i	0.986663	−	0.162778i	$-0.0520454\pi$
$80$		0			0
$81$	−4.94862	−	7.51739i	−0.549847	−	0.835265i
$82$		0			0
$83$		−	3.37031i		−	0.369939i	−0.982744	−	0.184970i	$-0.940781\pi$
$83$		−	3.37031i		−	0.369939i	0.982744	−	0.184970i	$-0.0592187\pi$
$84$		0			0
$85$			2.57305i			0.279087i
$86$		0			0
$87$	13.0798	−	7.80922i	1.40230	−	0.837236i
$88$		0			0
$89$		−	13.7526i		−	1.45777i	−0.684636	−	0.728885i	$-0.740039\pi$
$89$		−	13.7526i		−	1.45777i	0.684636	−	0.728885i	$-0.259961\pi$
$90$		0			0
$91$	−3.43077			−0.359642
$92$		0			0
$93$	−8.79516	−	14.7311i	−0.912015	−	1.52755i
$94$		0			0
$95$	6.10546			0.626407
$96$		0			0
$97$	−4.26230			−0.432771			−0.216385	−	0.976308i	$-0.569427\pi$
$97$	−4.26230			−0.432771			−0.216385	+	0.976308i	$0.569427\pi$
$98$		0			0
$99$	0.846533	+	0.456225i	0.0850798	+	0.0458524i
$100$		0			0
$101$	15.3130			1.52370			0.761851	−	0.647753i	$-0.224291\pi$
$101$	15.3130			1.52370			0.761851	+	0.647753i	$0.224291\pi$
$102$		0			0
$103$		−	7.25936i		−	0.715286i	−0.933858	−	0.357643i	$-0.883581\pi$
$103$		−	7.25936i		−	0.715286i	0.933858	−	0.357643i	$-0.116419\pi$
$104$		0			0
$105$	−0.707881	−	1.18564i	−0.0690821	−	0.115707i
$106$		0			0
$107$			13.2928i			1.28506i	0.766260	+	0.642531i	$0.222116\pi$
$107$			13.2928i			1.28506i	−0.766260	+	0.642531i	$0.777884\pi$
$108$		0			0
$109$		−	3.41592i		−	0.327186i	−0.986528	−	0.163593i	$-0.947692\pi$
$109$		−	3.41592i		−	0.327186i	0.986528	−	0.163593i	$-0.0523084\pi$
$110$		0			0
$111$	−7.54354	−	12.6348i	−0.716001	−	1.19924i
$112$		0			0
$113$		−	10.2261i		−	0.961992i	−0.876723	−	0.480996i	$-0.840275\pi$
$113$		−	10.2261i		−	0.961992i	0.876723	−	0.480996i	$-0.159725\pi$
$114$		0			0
$115$	−3.13115			−0.291981
$116$		0			0
$117$	11.3644	+	6.12465i	1.05064	+	0.566224i
$118$		0			0
$119$	−2.05138			−0.188049
$120$		0			0
$121$	10.8972			0.990659
$122$		0			0
$123$	−4.68970	−	7.85484i	−0.422856	−	0.708247i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			4.98995i			0.442786i	0.975185	+	0.221393i	$0.0710604\pi$
$127$			4.98995i			0.442786i	−0.975185	+	0.221393i	$0.928940\pi$
$128$		0			0
$129$	−4.42327	+	2.64089i	−0.389447	+	0.232518i
$130$		0			0
$131$			8.92702i			0.779958i	0.920824	+	0.389979i	$0.127518\pi$
$131$			8.92702i			0.779958i	−0.920824	+	0.389979i	$0.872482\pi$
$132$		0			0
$133$			4.86760i			0.422074i
$134$		0			0
$135$	0.228229	+	5.19114i	0.0196428	+	0.446782i
$136$		0			0
$137$		−	1.61964i		−	0.138375i	−0.997604	−	0.0691877i	$-0.977959\pi$
$137$		−	1.61964i		−	0.138375i	0.997604	−	0.0691877i	$-0.0220407\pi$
$138$		0			0
$139$	−3.58761			−0.304297			−0.152148	−	0.988358i	$-0.548619\pi$
$139$	−3.58761			−0.304297			−0.152148	+	0.988358i	$0.548619\pi$
$140$		0			0
$141$	−9.75860	+	5.82633i	−0.821822	+	0.490665i
$142$		0			0
$143$	−1.37939			−0.115351
$144$		0			0
$145$	−8.79516			−0.730398
$146$		0			0
$147$	−9.46484	+	5.65094i	−0.780647	+	0.466082i
$148$		0			0
$149$	−2.31367			−0.189543			−0.0947717	−	0.995499i	$-0.530212\pi$
$149$	−2.31367			−0.189543			−0.0947717	+	0.995499i	$0.530212\pi$
$150$		0			0
$151$		−	3.44347i		−	0.280225i	−0.990136	−	0.140113i	$-0.955254\pi$
$151$		−	3.44347i		−	0.280225i	0.990136	−	0.140113i	$-0.0447465\pi$
$152$		0			0
$153$	6.79516	+	3.66214i	0.549356	+	0.296067i
$154$		0			0
$155$			9.90557i			0.795635i
$156$		0			0
$157$		−	9.17084i		−	0.731912i	−0.930632	−	0.365956i	$-0.880742\pi$
$157$		−	9.17084i		−	0.731912i	0.930632	−	0.365956i	$-0.119258\pi$
$158$		0			0
$159$	−5.87222	+	3.50598i	−0.465697	+	0.278043i
$160$		0			0
$161$		−	2.49632i		−	0.196737i
$162$		0			0
$163$	10.6160			0.831510			0.415755	−	0.909477i	$-0.363517\pi$
$163$	10.6160			0.831510			0.415755	+	0.909477i	$0.363517\pi$
$164$		0			0
$165$	−0.284615	−	0.476705i	−0.0221572	−	0.0371114i
$166$		0			0
$167$	−13.3353			−1.03192			−0.515959	−	0.856613i	$-0.672564\pi$
$167$	−13.3353			−1.03192			−0.515959	+	0.856613i	$0.672564\pi$
$168$		0			0
$169$	−5.51785			−0.424450
$170$		0			0
$171$	8.68970	−	16.1239i	0.664518	−	1.23302i
$172$		0			0
$173$	−13.8972			−1.05659			−0.528294	−	0.849061i	$-0.677168\pi$
$173$	−13.8972			−1.05659			−0.528294	+	0.849061i	$0.677168\pi$
$174$		0			0
$175$			0.797253i			0.0602667i
$176$		0			0
$177$	11.0798	+	18.5577i	0.832807	+	1.39488i
$178$		0			0
$179$			5.18815i			0.387780i	0.981023	+	0.193890i	$0.0621105\pi$
$179$			5.18815i			0.387780i	−0.981023	+	0.193890i	$0.937889\pi$
$180$		0			0
$181$			2.59819i			0.193122i	0.995327	+	0.0965610i	$0.0307843\pi$
$181$			2.59819i			0.193122i	−0.995327	+	0.0965610i	$0.969216\pi$
$182$		0			0
$183$	7.84316	+	13.1366i	0.579783	+	0.971087i
$184$		0			0
$185$			8.49593i			0.624634i
$186$		0			0
$187$	−0.824788			−0.0603144
$188$		0			0
$189$	−4.13865	+	0.181956i	−0.301043	+	0.0132354i
$190$		0			0
$191$	12.2556			0.886781			0.443391	−	0.896329i	$-0.353775\pi$
$191$	12.2556			0.886781			0.443391	+	0.896329i	$0.353775\pi$
$192$		0			0
$193$	8.26230			0.594733			0.297367	−	0.954763i	$-0.403892\pi$
$193$	8.26230			0.594733			0.297367	+	0.954763i	$0.403892\pi$
$194$		0			0
$195$	−3.82085	−	6.39959i	−0.273616	−	0.458284i
$196$		0			0
$197$	10.8102			0.770192			0.385096	−	0.922876i	$-0.374168\pi$
$197$	10.8102			0.770192			0.385096	+	0.922876i	$0.374168\pi$
$198$		0			0
$199$			5.71287i			0.404975i	0.979285	+	0.202487i	$0.0649025\pi$
$199$			5.71287i			0.404975i	−0.979285	+	0.202487i	$0.935097\pi$
$200$		0			0
$201$	6.94112	−	4.14417i	0.489589	−	0.292307i
$202$		0			0
$203$		−	7.01197i		−	0.492144i
$204$		0			0
$205$			5.28178i			0.368896i
$206$		0			0
$207$	−4.45646	+	8.26902i	−0.309745	+	0.574737i
$208$		0			0
$209$			1.95709i			0.135375i
$210$		0			0
$211$	8.15684			0.561540			0.280770	−	0.959775i	$-0.409410\pi$
$211$	8.15684			0.561540			0.280770	+	0.959775i	$0.409410\pi$
$212$		0			0
$213$	5.10209	−	3.04618i	0.349590	−	0.208721i
$214$		0			0
$215$	2.97431			0.202846
$216$		0			0
$217$	−7.89725			−0.536100
$218$		0			0
$219$	2.12778	−	1.27038i	0.143782	−	0.0858444i
$220$		0			0
$221$	−11.0725			−0.744814
$222$		0			0
$223$		−	20.6084i		−	1.38004i	−0.723790	−	0.690020i	$-0.757602\pi$
$223$		−	20.6084i		−	1.38004i	0.723790	−	0.690020i	$-0.242398\pi$
$224$		0			0
$225$	1.42327	−	2.64089i	0.0948844	−	0.176059i
$226$		0			0
$227$		−	27.0044i		−	1.79235i	−0.443706	−	0.896173i	$-0.646336\pi$
$227$		−	27.0044i		−	1.79235i	0.443706	−	0.896173i	$-0.353664\pi$
$228$		0			0
$229$			9.65112i			0.637764i	0.947794	+	0.318882i	$0.103307\pi$
$229$			9.65112i			0.637764i	−0.947794	+	0.318882i	$0.896693\pi$
$230$		0			0
$231$	0.380055	−	0.226910i	0.0250058	−	0.0149296i
$232$		0			0
$233$		−	1.29086i		−	0.0845671i	−0.999106	−	0.0422836i	$-0.986537\pi$
$233$		−	1.29086i		−	0.0845671i	0.999106	−	0.0422836i	$-0.0134633\pi$
$234$		0			0
$235$	6.56192			0.428052
$236$		0			0
$237$	2.56923	+	4.30324i	0.166889	+	0.279525i
$238$		0			0
$239$	−4.21092			−0.272382			−0.136191	−	0.990683i	$-0.543486\pi$
$239$	−4.21092			−0.272382			−0.136191	+	0.990683i	$0.543486\pi$
$240$		0			0
$241$	−19.5686			−1.26052			−0.630261	−	0.776383i	$-0.717052\pi$
$241$	−19.5686			−1.26052			−0.630261	+	0.776383i	$0.717052\pi$
$242$		0			0
$243$	14.0341	+	6.78564i	0.900286	+	0.435299i
$244$		0			0
$245$	6.36439			0.406606
$246$		0			0
$247$			26.2732i			1.67173i
$248$		0			0
$249$	2.99250	+	5.01217i	0.189642	+	0.317634i
$250$		0			0
$251$			17.5335i			1.10670i	0.832947	+	0.553352i	$0.186652\pi$
$251$			17.5335i			1.10670i	−0.832947	+	0.553352i	$0.813348\pi$
$252$		0			0
$253$		−	1.00368i		−	0.0631011i
$254$		0			0
$255$	−2.28461	−	3.82653i	−0.143068	−	0.239627i
$256$		0			0
$257$		−	1.16582i		−	0.0727221i	−0.999339	−	0.0363610i	$-0.988423\pi$
$257$		−	1.16582i		−	0.0727221i	0.999339	−	0.0363610i	$-0.0115766\pi$
$258$		0			0
$259$	−6.77341			−0.420879
$260$		0			0
$261$	−12.5179	+	23.2271i	−0.774836	+	1.43772i
$262$		0			0
$263$	−15.3867			−0.948785			−0.474392	−	0.880313i	$-0.657332\pi$
$263$	−15.3867			−0.948785			−0.474392	+	0.880313i	$0.657332\pi$
$264$		0			0
$265$	3.94862			0.242562
$266$		0			0
$267$	12.2109	+	20.4522i	0.747296	+	1.25166i
$268$		0			0
$269$	−2.82479			−0.172230			−0.0861152	−	0.996285i	$-0.527445\pi$
$269$	−2.82479			−0.172230			−0.0861152	+	0.996285i	$0.527445\pi$
$270$		0			0
$271$		−	3.89729i		−	0.236743i	−0.992969	−	0.118372i	$-0.962233\pi$
$271$		−	3.89729i		−	0.236743i	0.992969	−	0.118372i	$-0.0377674\pi$
$272$		0			0
$273$	5.10209	−	3.04618i	0.308793	−	0.184363i
$274$		0			0
$275$			0.320548i			0.0193298i
$276$		0			0
$277$		−	21.8450i		−	1.31254i	−0.754527	−	0.656269i	$-0.772134\pi$
$277$		−	21.8450i		−	1.31254i	0.754527	−	0.656269i	$-0.227866\pi$
$278$		0			0
$279$	26.1595	+	14.0983i	1.56613	+	0.844041i
$280$		0			0
$281$			20.7201i			1.23606i	0.786155	+	0.618029i	$0.212069\pi$
$281$			20.7201i			1.23606i	−0.786155	+	0.618029i	$0.787931\pi$
$282$		0			0
$283$	−1.23661			−0.0735087			−0.0367544	−	0.999324i	$-0.511702\pi$
$283$	−1.23661			−0.0735087			−0.0367544	+	0.999324i	$0.511702\pi$
$284$		0			0
$285$	−9.07977	+	5.42104i	−0.537839	+	0.321115i
$286$		0			0
$287$	−4.21092			−0.248563
$288$		0			0
$289$	10.3794			0.610553
$290$		0			0
$291$	6.33870	−	3.78449i	0.371581	−	0.221851i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		−	12.4786i		−	0.726534i
$296$		0			0
$297$	−1.66401	+	0.0731584i	−0.0965556	+	0.00424508i
$298$		0			0
$299$		−	13.4741i		−	0.779226i
$300$		0			0
$301$			2.37128i			0.136678i
$302$		0			0
$303$	−22.7728	+	13.5964i	−1.30827	+	0.781094i
$304$		0			0
$305$		−	8.83339i		−	0.505798i
$306$		0			0
$307$	−6.71875			−0.383460			−0.191730	−	0.981448i	$-0.561410\pi$
$307$	−6.71875			−0.383460			−0.191730	+	0.981448i	$0.561410\pi$
$308$		0			0
$309$	6.44559	+	10.7958i	0.366677	+	0.614152i
$310$		0			0
$311$	22.0568			1.25073			0.625363	−	0.780334i	$-0.284951\pi$
$311$	22.0568			1.25073			0.625363	+	0.780334i	$0.284951\pi$
$312$		0			0
$313$	−11.0357			−0.623775			−0.311888	−	0.950119i	$-0.600961\pi$
$313$	−11.0357			−0.623775			−0.311888	+	0.950119i	$0.600961\pi$
$314$		0			0
$315$	2.10546	+	1.13470i	0.118629	+	0.0639333i
$316$		0			0
$317$	−24.3705			−1.36878			−0.684391	−	0.729115i	$-0.739932\pi$
$317$	−24.3705			−1.36878			−0.684391	+	0.729115i	$0.739932\pi$
$318$		0			0
$319$		−	2.81927i		−	0.157849i
$320$		0			0
$321$	−11.8027	−	19.7684i	−0.658760	−	1.10337i
$322$		0			0
$323$			15.7097i			0.874110i
$324$		0			0
$325$			4.30324i			0.238701i
$326$		0			0
$327$	3.03300	+	5.08001i	0.167725	+	0.280925i
$328$		0			0
$329$			5.23151i			0.288423i
$330$		0			0
$331$	−13.1925			−0.725128			−0.362564	−	0.931959i	$-0.618099\pi$
$331$	−13.1925			−0.725128			−0.362564	+	0.931959i	$0.618099\pi$
$332$		0			0
$333$	22.4368	+	12.0920i	1.22953	+	0.662636i
$334$		0			0
$335$	−4.66738			−0.255006
$336$		0			0
$337$	27.4876			1.49734			0.748672	−	0.662941i	$-0.230692\pi$
$337$	27.4876			1.49734			0.748672	+	0.662941i	$0.230692\pi$
$338$		0			0
$339$	9.07977	+	15.2078i	0.493146	+	0.825976i
$340$		0			0
$341$	−3.17521			−0.171947
$342$		0			0
$343$			10.6548i			0.575305i
$344$		0			0
$345$	4.65651	−	2.78015i	0.250698	−	0.149678i
$346$		0			0
$347$		−	10.9731i		−	0.589067i	−0.955641	−	0.294533i	$-0.904836\pi$
$347$		−	10.9731i		−	0.589067i	0.955641	−	0.294533i	$-0.0951642\pi$
$348$		0			0
$349$		−	31.6066i		−	1.69186i	−0.533291	−	0.845932i	$-0.679045\pi$
$349$		−	31.6066i		−	1.69186i	0.533291	−	0.845932i	$-0.320955\pi$
$350$		0			0
$351$	−22.3387	+	0.982124i	−1.19235	+	0.0524219i
$352$		0			0
$353$			21.4646i			1.14244i	0.820795	+	0.571222i	$0.193531\pi$
$353$			21.4646i			1.14244i	−0.820795	+	0.571222i	$0.806469\pi$
$354$		0			0
$355$	−3.43077			−0.182086
$356$		0			0
$357$	3.05072	−	1.82142i	0.161461	−	0.0963996i
$358$		0			0
$359$	−34.3124			−1.81094			−0.905468	−	0.424414i	$-0.860480\pi$
$359$	−34.3124			−1.81094			−0.905468	+	0.424414i	$0.860480\pi$
$360$		0			0
$361$	18.2766			0.961929
$362$		0			0
$363$	−16.2059	+	9.67567i	−0.850590	+	0.507841i
$364$		0			0
$365$	−1.43077			−0.0748899
$366$		0			0
$367$		−	27.9901i		−	1.46107i	−0.682874	−	0.730536i	$-0.739270\pi$
$367$		−	27.9901i		−	1.46107i	0.682874	−	0.730536i	$-0.260730\pi$
$368$		0			0
$369$	13.9486	+	7.51739i	0.726136	+	0.391340i
$370$		0			0
$371$			3.14805i			0.163439i
$372$		0			0
$373$			13.9134i			0.720408i	0.932873	+	0.360204i	$0.117293\pi$
$373$			13.9134i			0.720408i	−0.932873	+	0.360204i	$0.882707\pi$
$374$		0			0
$375$	−1.48716	+	0.887900i	−0.0767964	+	0.0458510i
$376$		0			0
$377$		−	37.8477i		−	1.94925i
$378$		0			0
$379$	−7.79853			−0.400583			−0.200292	−	0.979736i	$-0.564189\pi$
$379$	−7.79853			−0.400583			−0.200292	+	0.979736i	$0.564189\pi$
$380$		0			0
$381$	−4.43058	−	7.42084i	−0.226985	−	0.380181i
$382$		0			0
$383$	27.8386			1.42248			0.711242	−	0.702947i	$-0.248133\pi$
$383$	27.8386			1.42248			0.711242	+	0.702947i	$0.248133\pi$
$384$		0			0
$385$	−0.255558			−0.0130244
$386$		0			0
$387$	4.23324	−	7.85484i	0.215188	−	0.399284i
$388$		0			0
$389$	18.5246			0.939234			0.469617	−	0.882870i	$-0.344392\pi$
$389$	18.5246			0.939234			0.469617	+	0.882870i	$0.344392\pi$
$390$		0			0
$391$		−	8.05661i		−	0.407440i
$392$		0			0
$393$	−7.92631	−	13.2759i	−0.399829	−	0.669679i
$394$		0			0
$395$		−	2.89360i		−	0.145593i
$396$		0			0
$397$			4.97814i			0.249846i	0.992166	+	0.124923i	$0.0398683\pi$
$397$			4.97814i			0.249846i	−0.992166	+	0.124923i	$0.960132\pi$
$398$		0			0
$399$	−4.32194	−	7.23888i	−0.216368	−	0.362397i
$400$		0			0
$401$			16.8094i			0.839422i	0.907658	+	0.419711i	$0.137868\pi$
$401$			16.8094i			0.839422i	−0.907658	+	0.419711i	$0.862132\pi$
$402$		0			0
$403$	−42.6260			−2.12335
$404$		0			0
$405$	−4.94862	−	7.51739i	−0.245899	−	0.373542i
$406$		0			0
$407$	−2.72336			−0.134992
$408$		0			0
$409$	−17.6053			−0.870527			−0.435264	−	0.900303i	$-0.643345\pi$
$409$	−17.6053			−0.870527			−0.435264	+	0.900303i	$0.643345\pi$
$410$		0			0
$411$	1.43808	+	2.40866i	0.0709353	+	0.118811i
$412$		0			0
$413$	9.94862			0.489540
$414$		0			0
$415$		−	3.37031i		−	0.165442i
$416$		0			0
$417$	5.33533	−	3.18544i	0.261272	−	0.155991i
$418$		0			0
$419$			13.3408i			0.651741i	0.945414	+	0.325870i	$0.105657\pi$
$419$			13.3408i			0.651741i	−0.945414	+	0.325870i	$0.894343\pi$
$420$		0			0
$421$		−	16.7650i		−	0.817074i	−0.912742	−	0.408537i	$-0.866039\pi$
$421$		−	16.7650i		−	0.817074i	0.912742	−	0.408537i	$-0.133961\pi$
$422$		0			0
$423$	9.33936	−	17.3293i	0.454095	−	0.842580i
$424$		0			0
$425$			2.57305i			0.124811i
$426$		0			0
$427$	7.04245			0.340808
$428$		0			0
$429$	2.05138	−	1.22476i	0.0990413	−	0.0591322i
$430$		0			0
$431$	28.9911			1.39645			0.698225	−	0.715878i	$-0.253973\pi$
$431$	28.9911			1.39645			0.698225	+	0.715878i	$0.253973\pi$
$432$		0			0
$433$	−23.6484			−1.13647			−0.568235	−	0.822866i	$-0.692374\pi$
$433$	−23.6484			−1.13647			−0.568235	+	0.822866i	$0.692374\pi$
$434$		0			0
$435$	13.0798	−	7.80922i	0.627127	−	0.374424i
$436$		0			0
$437$	−19.1171			−0.914495
$438$		0			0
$439$		−	7.85724i		−	0.375006i	−0.982264	−	0.187503i	$-0.939961\pi$
$439$		−	7.85724i		−	0.375006i	0.982264	−	0.187503i	$-0.0600394\pi$
$440$		0			0
$441$	9.05822	−	16.8077i	0.431344	−	0.800365i
$442$		0			0
$443$		−	12.9805i		−	0.616721i	−0.951270	−	0.308360i	$-0.900220\pi$
$443$		−	12.9805i		−	0.616721i	0.951270	−	0.308360i	$-0.0997802\pi$
$444$		0			0
$445$		−	13.7526i		−	0.651935i
$446$		0			0
$447$	3.44079	−	2.05431i	0.162744	−	0.0971655i
$448$		0			0
$449$		−	13.4847i		−	0.636383i	−0.948026	−	0.318191i	$-0.896925\pi$
$449$		−	13.4847i		−	0.636383i	0.948026	−	0.318191i	$-0.103075\pi$
$450$		0			0
$451$	−1.69307			−0.0797234
$452$		0			0
$453$	3.05745	+	5.12097i	0.143652	+	0.240604i
$454$		0			0
$455$	−3.43077			−0.160837
$456$		0			0
$457$	−28.1014			−1.31453			−0.657265	−	0.753660i	$-0.728287\pi$
$457$	−28.1014			−1.31453			−0.657265	+	0.753660i	$0.728287\pi$
$458$		0			0
$459$	−13.3571	+	0.587246i	−0.623455	+	0.0274103i
$460$		0			0
$461$	29.2170			1.36077			0.680386	−	0.732854i	$-0.261812\pi$
$461$	29.2170			1.36077			0.680386	+	0.732854i	$0.261812\pi$
$462$		0			0
$463$			14.1463i			0.657434i	0.944429	+	0.328717i	$0.106616\pi$
$463$			14.1463i			0.657434i	−0.944429	+	0.328717i	$0.893384\pi$
$464$		0			0
$465$	−8.79516	−	14.7311i	−0.407866	−	0.683140i
$466$		0			0
$467$		−	28.9687i		−	1.34051i	−0.742131	−	0.670255i	$-0.766185\pi$
$467$		−	28.9687i		−	1.34051i	0.742131	−	0.670255i	$-0.233815\pi$
$468$		0			0
$469$		−	3.72108i		−	0.171824i
$470$		0			0
$471$	8.14279	+	13.6385i	0.375200	+	0.628427i
$472$		0			0
$473$			0.953410i			0.0438379i
$474$		0			0
$475$	6.10546			0.280138
$476$		0			0
$477$	5.61995	−	10.4279i	0.257320	−	0.477460i
$478$		0			0
$479$	37.9040			1.73188			0.865939	−	0.500150i	$-0.166722\pi$
$479$	37.9040			1.73188			0.865939	+	0.500150i	$0.166722\pi$
$480$		0			0
$481$	−36.5600			−1.66699
$482$		0			0
$483$	2.21648	+	3.71241i	0.100853	+	0.168921i
$484$		0			0
$485$	−4.26230			−0.193541
$486$		0			0
$487$		−	41.9180i		−	1.89949i	−0.313031	−	0.949743i	$-0.601344\pi$
$487$		−	41.9180i		−	1.89949i	0.313031	−	0.949743i	$-0.398656\pi$
$488$		0			0
$489$	−15.7877	+	9.42595i	−0.713942	+	0.426256i
$490$		0			0
$491$		−	5.09691i		−	0.230020i	−0.993364	−	0.115010i	$-0.963310\pi$
$491$		−	5.09691i		−	0.230020i	0.993364	−	0.115010i	$-0.0366901\pi$
$492$		0			0
$493$		−	22.6304i		−	1.01922i
$494$		0			0
$495$	0.846533	+	0.456225i	0.0380488	+	0.0205058i
$496$		0			0
$497$		−	2.73519i		−	0.122690i
$498$		0			0
$499$	27.3821			1.22579			0.612896	−	0.790164i	$-0.290005\pi$
$499$	27.3821			1.22579			0.612896	+	0.790164i	$0.290005\pi$
$500$		0			0
$501$	19.8317	−	11.8404i	0.886016	−	0.528992i
$502$		0			0
$503$	21.7572			0.970104			0.485052	−	0.874485i	$-0.338801\pi$
$503$	21.7572			0.970104			0.485052	+	0.874485i	$0.338801\pi$
$504$		0			0
$505$	15.3130			0.681420
$506$		0			0
$507$	8.20591	−	4.89930i	0.364437	−	0.217586i
$508$		0			0
$509$	25.0061			1.10837			0.554187	−	0.832392i	$-0.313029\pi$
$509$	25.0061			1.10837			0.554187	+	0.832392i	$0.313029\pi$
$510$		0			0
$511$		−	1.14069i		−	0.0504610i
$512$		0			0
$513$	1.39344	+	31.6943i	0.0615220	+	1.39934i
$514$		0			0
$515$		−	7.25936i		−	0.319886i
$516$		0			0
$517$			2.10341i			0.0925079i
$518$		0			0
$519$	20.6674	−	12.3394i	0.907197	−	0.541638i
$520$		0			0
$521$			26.3235i			1.15325i	0.817007	+	0.576627i	$0.195631\pi$
$521$			26.3235i			1.15325i	−0.817007	+	0.576627i	$0.804369\pi$
$522$		0			0
$523$	23.5435			1.02949			0.514744	−	0.857344i	$-0.327887\pi$
$523$	23.5435			1.02949			0.514744	+	0.857344i	$0.327887\pi$
$524$		0			0
$525$	−0.707881	−	1.18564i	−0.0308945	−	0.0517456i
$526$		0			0
$527$	−25.4876			−1.11026
$528$		0			0
$529$	−13.1959			−0.573735
$530$		0			0
$531$	−32.9547	−	17.7604i	−1.43011	−	0.770736i
$532$		0			0
$533$	−22.7288			−0.984492
$534$		0			0
$535$			13.2928i			0.574697i
$536$		0			0
$537$	−4.60656	−	7.71558i	−0.198788	−	0.332952i
$538$		0			0
$539$			2.04009i			0.0878730i
$540$		0			0
$541$			18.2576i			0.784955i	0.919762	+	0.392478i	$0.128382\pi$
$541$			18.2576i			0.784955i	−0.919762	+	0.392478i	$0.871618\pi$
$542$		0			0
$543$	−2.30693	−	3.86391i	−0.0990000	−	0.165816i
$544$		0			0
$545$		−	3.41592i		−	0.146322i
$546$		0			0
$547$	6.40508			0.273862			0.136931	−	0.990581i	$-0.456276\pi$
$547$	6.40508			0.273862			0.136931	+	0.990581i	$0.456276\pi$
$548$		0			0
$549$	−23.3280	−	12.5723i	−0.995616	−	0.536571i
$550$		0			0
$551$	−53.6985			−2.28763
$552$		0			0
$553$	2.30693			0.0981008
$554$		0			0
$555$	−7.54354	−	12.6348i	−0.320206	−	0.536317i
$556$		0			0
$557$	−7.79582			−0.330319			−0.165160	−	0.986267i	$-0.552814\pi$
$557$	−7.79582			−0.330319			−0.165160	+	0.986267i	$0.552814\pi$
$558$		0			0
$559$			12.7992i			0.541347i
$560$		0			0
$561$	1.22659	−	0.732329i	0.0517866	−	0.0309190i
$562$		0			0
$563$		−	4.37399i		−	0.184342i	−0.995743	−	0.0921709i	$-0.970619\pi$
$563$		−	4.37399i		−	0.184342i	0.995743	−	0.0921709i	$-0.0293806\pi$
$564$		0			0
$565$		−	10.2261i		−	0.430216i
$566$		0			0
$567$	5.99326	−	3.94531i	0.251693	−	0.165687i
$568$		0			0
$569$			21.8198i			0.914735i	0.889278	+	0.457368i	$0.151208\pi$
$569$			21.8198i			0.914735i	−0.889278	+	0.457368i	$0.848792\pi$
$570$		0			0
$571$	−1.42806			−0.0597625			−0.0298813	−	0.999553i	$-0.509513\pi$
$571$	−1.42806			−0.0597625			−0.0298813	+	0.999553i	$0.509513\pi$
$572$		0			0
$573$	−18.2259	+	10.8817i	−0.761399	+	0.454590i
$574$		0			0
$575$	−3.13115			−0.130578
$576$		0			0
$577$	43.3548			1.80488			0.902442	−	0.430812i	$-0.141773\pi$
$577$	43.3548			1.80488			0.902442	+	0.430812i	$0.141773\pi$
$578$		0			0
$579$	−12.2873	+	7.33609i	−0.510644	+	0.304878i
$580$		0			0
$581$	2.68699			0.111475
$582$		0			0
$583$			1.26572i			0.0524209i
$584$		0			0
$585$	11.3644	+	6.12465i	0.469860	+	0.253223i
$586$		0			0
$587$		−	6.88810i		−	0.284302i	−0.989845	−	0.142151i	$-0.954598\pi$
$587$		−	6.88810i		−	0.284302i	0.989845	−	0.142151i	$-0.0454019\pi$
$588$		0			0
$589$			60.4781i			2.49196i
$590$		0			0
$591$	−16.0764	+	9.59835i	−0.661295	+	0.394823i
$592$		0			0
$593$			0.894469i			0.0367314i	0.999831	+	0.0183657i	$0.00584632\pi$
$593$			0.894469i			0.0367314i	−0.999831	+	0.0183657i	$0.994154\pi$
$594$		0			0
$595$	−2.05138			−0.0840982
$596$		0			0
$597$	−5.07246	−	8.49593i	−0.207602	−	0.347715i
$598$		0			0
$599$	11.5836			0.473292			0.236646	−	0.971596i	$-0.423952\pi$
$599$	11.5836			0.473292			0.236646	+	0.971596i	$0.423952\pi$
$600$		0			0
$601$	24.9480			1.01765			0.508824	−	0.860870i	$-0.330080\pi$
$601$	24.9480			1.01765			0.508824	+	0.860870i	$0.330080\pi$
$602$		0			0
$603$	−6.64292	+	12.3260i	−0.270521	+	0.501955i
$604$		0			0
$605$	10.8972			0.443036
$606$		0			0
$607$			21.8741i			0.887843i	0.896066	+	0.443922i	$0.146413\pi$
$607$			21.8741i			0.887843i	−0.896066	+	0.443922i	$0.853587\pi$
$608$		0			0
$609$	6.22593	+	10.4279i	0.252287	+	0.422560i
$610$		0			0
$611$			28.2375i			1.14237i
$612$		0			0
$613$		−	25.8339i		−	1.04342i	−0.853122	−	0.521711i	$-0.825294\pi$
$613$		−	25.8339i		−	1.04342i	0.853122	−	0.521711i	$-0.174706\pi$
$614$		0			0
$615$	−4.68970	−	7.85484i	−0.189107	−	0.316738i
$616$		0			0
$617$		−	27.5641i		−	1.10969i	−0.831954	−	0.554845i	$-0.812778\pi$
$617$		−	27.5641i		−	1.10969i	0.831954	−	0.554845i	$-0.187222\pi$
$618$		0			0
$619$	−22.2136			−0.892841			−0.446421	−	0.894823i	$-0.647301\pi$
$619$	−22.2136			−0.892841			−0.446421	+	0.894823i	$0.647301\pi$
$620$		0			0
$621$	−0.714619	−	16.2542i	−0.0286767	−	0.652259i
$622$		0			0
$623$	10.9643			0.439275
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−1.73770	−	2.91050i	−0.0693972	−	0.116234i
$628$		0			0
$629$	−21.8605			−0.871635
$630$		0			0
$631$			26.2225i			1.04390i	0.852975	+	0.521951i	$0.174796\pi$
$631$			26.2225i			1.04390i	−0.852975	+	0.521951i	$0.825204\pi$
$632$		0			0
$633$	−12.1305	+	7.24246i	−0.482144	+	0.287862i
$634$		0			0
$635$			4.98995i			0.198020i
$636$		0			0
$637$			27.3875i			1.08513i
$638$		0			0
$639$	−4.88290	+	9.06030i	−0.193165	+	0.358420i
$640$		0			0
$641$		−	47.0436i		−	1.85811i	−0.369940	−	0.929056i	$-0.620622\pi$
$641$		−	47.0436i		−	1.85811i	0.369940	−	0.929056i	$-0.379378\pi$
$642$		0			0
$643$	18.1696			0.716538			0.358269	−	0.933618i	$-0.383367\pi$
$643$	18.1696			0.716538			0.358269	+	0.933618i	$0.383367\pi$
$644$		0			0
$645$	−4.42327	+	2.64089i	−0.174166	+	0.103985i
$646$		0			0
$647$	−36.9324			−1.45196			−0.725981	−	0.687715i	$-0.758614\pi$
$647$	−36.9324			−1.45196			−0.725981	+	0.687715i	$0.758614\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	11.7444	−	7.01197i	0.460301	−	0.274821i
$652$		0			0
$653$	1.11710			0.0437155			0.0218577	−	0.999761i	$-0.493042\pi$
$653$	1.11710			0.0437155			0.0218577	+	0.999761i	$0.493042\pi$
$654$		0			0
$655$			8.92702i			0.348808i
$656$		0			0
$657$	−2.03637	+	3.77851i	−0.0794463	+	0.147414i
$658$		0			0
$659$			30.7865i			1.19927i	0.800273	+	0.599636i	$0.204688\pi$
$659$			30.7865i			1.19927i	−0.800273	+	0.599636i	$0.795312\pi$
$660$		0			0
$661$			11.5686i			0.449966i	0.974363	+	0.224983i	$0.0722326\pi$
$661$			11.5686i			0.449966i	−0.974363	+	0.224983i	$0.927767\pi$
$662$		0			0
$663$	16.4665	−	9.83124i	0.639505	−	0.381814i
$664$		0			0
$665$			4.86760i			0.188757i
$666$		0			0
$667$	27.5389			1.06631
$668$		0			0
$669$	18.2982	+	30.6479i	0.707449	+	1.18492i
$670$		0			0
$671$	2.83153			0.109310
$672$		0			0
$673$	30.4072			1.17211			0.586056	−	0.810271i	$-0.300680\pi$
$673$	30.4072			1.17211			0.586056	+	0.810271i	$0.300680\pi$
$674$		0			0
$675$	0.228229	+	5.19114i	0.00878454	+	0.199807i
$676$		0			0
$677$	−29.9608			−1.15149			−0.575743	−	0.817631i	$-0.695287\pi$
$677$	−29.9608			−1.15149			−0.575743	+	0.817631i	$0.695287\pi$
$678$		0			0
$679$		−	3.39813i		−	0.130408i
$680$		0			0
$681$	23.9772	+	40.1598i	0.918809	+	1.53893i
$682$		0			0
$683$			30.8345i			1.17985i	0.807458	+	0.589925i	$0.200843\pi$
$683$			30.8345i			1.17985i	−0.807458	+	0.589925i	$0.799157\pi$
$684$		0			0
$685$		−	1.61964i		−	0.0618834i
$686$		0			0
$687$	−8.56923	−	14.3527i	−0.326936	−	0.547590i
$688$		0			0
$689$			16.9919i			0.647339i
$690$		0			0
$691$	19.2293			0.731517			0.365758	−	0.930710i	$-0.380810\pi$
$691$	19.2293			0.731517			0.365758	+	0.930710i	$0.380810\pi$
$692$		0			0
$693$	−0.363727	+	0.674901i	−0.0138169	+	0.0256374i
$694$		0			0
$695$	−3.58761			−0.136086
$696$		0			0
$697$	−13.5903			−0.514770
$698$		0			0
$699$	1.14616	+	1.91971i	0.0433516	+	0.0726102i
$700$		0			0
$701$	25.8972			0.978126			0.489063	−	0.872249i	$-0.337339\pi$
$701$	25.8972			0.978126			0.489063	+	0.872249i	$0.337339\pi$
$702$		0			0
$703$			51.8716i			1.95637i
$704$		0			0
$705$	−9.75860	+	5.82633i	−0.367530	+	0.219432i
$706$		0			0
$707$			12.2083i			0.459142i
$708$		0			0
$709$			6.00828i			0.225646i	0.993615	+	0.112823i	$0.0359893\pi$
$709$			6.00828i			0.225646i	−0.993615	+	0.112823i	$0.964011\pi$
$710$		0			0
$711$	−7.64169	−	4.11837i	−0.286586	−	0.154451i
$712$		0			0
$713$		−	31.0158i		−	1.16155i
$714$		0			0
$715$	−1.37939			−0.0515864
$716$		0			0
$717$	6.26230	−	3.73888i	0.233870	−	0.139631i
$718$		0			0
$719$	−3.34264			−0.124659			−0.0623297	−	0.998056i	$-0.519853\pi$
$719$	−3.34264			−0.124659			−0.0623297	+	0.998056i	$0.519853\pi$
$720$		0			0
$721$	5.78755			0.215540
$722$		0			0
$723$	29.1015	−	17.3749i	1.08230	−	0.646181i
$724$		0			0
$725$	−8.79516			−0.326644
$726$		0			0
$727$		−	15.4120i		−	0.571600i	−0.958289	−	0.285800i	$-0.907741\pi$
$727$		−	15.4120i		−	0.571600i	0.958289	−	0.285800i	$-0.0922593\pi$
$728$		0			0
$729$	−26.8958	+	2.36954i	−0.996142	+	0.0877607i
$730$		0			0
$731$			7.65306i			0.283059i
$732$		0			0
$733$			42.7008i			1.57719i	0.614914	+	0.788594i	$0.289191\pi$
$733$			42.7008i			1.57719i	−0.614914	+	0.788594i	$0.710809\pi$
$734$		0			0
$735$	−9.46484	+	5.65094i	−0.349116	+	0.208438i
$736$		0			0
$737$		−	1.49612i		−	0.0551103i
$738$		0			0
$739$	−30.4546			−1.12029			−0.560145	−	0.828395i	$-0.689254\pi$
$739$	−30.4546			−1.12029			−0.560145	+	0.828395i	$0.689254\pi$
$740$		0			0
$741$	−23.3280	−	39.0724i	−0.856976	−	1.43536i
$742$		0			0
$743$	−49.8954			−1.83048			−0.915242	−	0.402906i	$-0.868000\pi$
$743$	−49.8954			−1.83048			−0.915242	+	0.402906i	$0.868000\pi$
$744$		0			0
$745$	−2.31367			−0.0847664
$746$		0			0
$747$	−8.90062	−	4.79685i	−0.325657	−	0.175507i
$748$		0			0
$749$	−10.5977			−0.387232
$750$		0			0
$751$			3.81321i			0.139146i	0.997577	+	0.0695730i	$0.0221637\pi$
$751$			3.81321i			0.139146i	−0.997577	+	0.0695730i	$0.977836\pi$
$752$		0			0
$753$	−15.5680	−	26.0750i	−0.567329	−	0.950228i
$754$		0			0
$755$		−	3.44347i		−	0.125321i
$756$		0			0
$757$			38.1793i			1.38765i	0.720144	+	0.693825i	$0.244076\pi$
$757$			38.1793i			1.38765i	−0.720144	+	0.693825i	$0.755924\pi$
$758$		0			0
$759$	0.891171	+	1.49263i	0.0323475	+	0.0541792i
$760$		0			0
$761$			10.7460i			0.389543i	0.980849	+	0.194772i	$0.0623966\pi$
$761$			10.7460i			0.389543i	−0.980849	+	0.194772i	$0.937603\pi$
$762$		0			0
$763$	2.72336			0.0985921
$764$		0			0
$765$	6.79516	+	3.66214i	0.245679	+	0.132405i
$766$		0			0
$767$	53.6985			1.93894
$768$		0			0
$769$	1.13172			0.0408109			0.0204055	−	0.999792i	$-0.493504\pi$
$769$	1.13172			0.0408109			0.0204055	+	0.999792i	$0.493504\pi$
$770$		0			0
$771$	1.03513	+	1.73376i	0.0372795	+	0.0624399i
$772$		0			0
$773$	13.7144			0.493274			0.246637	−	0.969108i	$-0.420675\pi$
$773$	13.7144			0.493274			0.246637	+	0.969108i	$0.420675\pi$
$774$		0			0
$775$			9.90557i			0.355819i
$776$		0			0
$777$	10.0731	−	6.01411i	0.361371	−	0.215755i
$778$		0			0
$779$			32.2477i			1.15539i
$780$		0			0
$781$		−	1.09973i		−	0.0393513i
$782$		0			0
$783$	−2.00731	−	45.6569i	−0.0717354	−	1.63164i
$784$		0			0
$785$		−	9.17084i		−	0.327321i
$786$		0			0
$787$	32.6374			1.16340			0.581698	−	0.813405i	$-0.302388\pi$
$787$	32.6374			1.16340			0.581698	+	0.813405i	$0.302388\pi$
$788$		0			0
$789$	22.8824	−	13.6619i	0.814636	−	0.486375i
$790$		0			0
$791$	8.15281			0.289880
$792$		0			0
$793$	38.0122			1.34985
$794$		0			0
$795$	−5.87222	+	3.50598i	−0.208266	+	0.124344i
$796$		0			0
$797$	39.1293			1.38603			0.693015	−	0.720923i	$-0.256282\pi$
$797$	39.1293			1.38603			0.693015	+	0.720923i	$0.256282\pi$
$798$		0			0
$799$			16.8842i			0.597319i
$800$		0			0
$801$	−36.3191	−	19.5736i	−1.28327	−	0.691599i
$802$		0			0
$803$		−	0.458631i		−	0.0161847i
$804$		0			0
$805$		−	2.49632i		−	0.0879837i
$806$		0			0
$807$	4.20090	−	2.50813i	0.147879	−	0.0882903i
$808$		0			0
$809$		−	25.5481i		−	0.898222i	−0.893476	−	0.449111i	$-0.851741\pi$
$809$		−	25.5481i		−	0.898222i	0.893476	−	0.449111i	$-0.148259\pi$
$810$		0			0
$811$	−5.63898			−0.198011			−0.0990057	−	0.995087i	$-0.531566\pi$
$811$	−5.63898			−0.198011			−0.0990057	+	0.995087i	$0.531566\pi$
$812$		0			0
$813$	3.46040	+	5.79587i	0.121362	+	0.203270i
$814$		0			0
$815$	10.6160			0.371862
$816$		0			0
$817$	18.1595			0.635322
$818$		0			0
$819$	−4.88290	+	9.06030i	−0.170622	+	0.316592i
$820$		0			0
$821$	−26.8248			−0.936192			−0.468096	−	0.883678i	$-0.655060\pi$
$821$	−26.8248			−0.936192			−0.468096	+	0.883678i	$0.655060\pi$
$822$		0			0
$823$		−	8.35909i		−	0.291379i	−0.989330	−	0.145690i	$-0.953460\pi$
$823$		−	8.35909i		−	0.291379i	0.989330	−	0.145690i	$-0.0465401\pi$
$824$		0			0
$825$	−0.284615	−	0.476705i	−0.00990901	−	0.0165967i
$826$		0			0
$827$		−	6.27366i		−	0.218156i	−0.994033	−	0.109078i	$-0.965210\pi$
$827$		−	6.27366i		−	0.218156i	0.994033	−	0.109078i	$-0.0347899\pi$
$828$		0			0
$829$			8.61230i			0.299118i	0.988753	+	0.149559i	$0.0477853\pi$
$829$			8.61230i			0.299118i	−0.988753	+	0.149559i	$0.952215\pi$
$830$		0			0
$831$	19.3962	+	32.4869i	0.672845	+	1.12696i
$832$		0			0
$833$			16.3759i			0.567392i
$834$		0			0
$835$	−13.3353			−0.461488
$836$		0			0
$837$	−51.4212	+	2.26074i	−1.77738	+	0.0781426i
$838$		0			0
$839$	2.93969			0.101490			0.0507448	−	0.998712i	$-0.483840\pi$
$839$	2.93969			0.101490			0.0507448	+	0.998712i	$0.483840\pi$
$840$		0			0
$841$	48.3548			1.66741
$842$		0			0
$843$	−18.3974	−	30.8140i	−0.633640	−	1.06129i
$844$		0			0
$845$	−5.51785			−0.189820
$846$		0			0
$847$			8.68787i			0.298519i
$848$		0			0
$849$	1.83903	−	1.09798i	0.0631153	−	0.0376827i
$850$		0			0
$851$		−	26.6020i		−	0.911906i
$852$		0			0
$853$		−	34.2193i		−	1.17165i	−0.810439	−	0.585824i	$-0.800771\pi$
$853$		−	34.2193i		−	1.17165i	0.810439	−	0.585824i	$-0.199229\pi$
$854$		0			0
$855$	8.68970	−	16.1239i	0.297181	−	0.551424i
$856$		0			0
$857$		−	0.519916i		−	0.0177600i	−0.999961	−	0.00888000i	$-0.997173\pi$
$857$		−	0.519916i		−	0.0177600i	0.999961	−	0.00888000i	$-0.00282663\pi$
$858$		0			0
$859$	32.9724			1.12500			0.562502	−	0.826796i	$-0.309839\pi$
$859$	32.9724			1.12500			0.562502	+	0.826796i	$0.309839\pi$
$860$		0			0
$861$	6.26230	−	3.73888i	0.213418	−	0.127421i
$862$		0			0
$863$	−21.8453			−0.743623			−0.371811	−	0.928308i	$-0.621263\pi$
$863$	−21.8453			−0.743623			−0.371811	+	0.928308i	$0.621263\pi$
$864$		0			0
$865$	−13.8972			−0.472521
$866$		0			0
$867$	−15.4358	+	9.21587i	−0.524227	+	0.312987i
$868$		0			0
$869$	0.927539			0.0314646
$870$		0			0
$871$		−	20.0848i		−	0.680549i
$872$		0			0
$873$	−6.06638	+	11.2563i	−0.205316	+	0.380967i
$874$		0			0
$875$			0.797253i			0.0269521i
$876$		0			0
$877$		−	37.2187i		−	1.25679i	−0.777896	−	0.628393i	$-0.783713\pi$
$877$		−	37.2187i		−	1.25679i	0.777896	−	0.628393i	$-0.216287\pi$
$878$		0			0
$879$	−8.92294	+	5.32740i	−0.300963	+	0.179689i
$880$		0			0
$881$			17.2984i			0.582796i	0.956602	+	0.291398i	$0.0941205\pi$
$881$			17.2984i			0.582796i	−0.956602	+	0.291398i	$0.905880\pi$
$882$		0			0
$883$	−51.3234			−1.72717			−0.863585	−	0.504203i	$-0.831786\pi$
$883$	−51.3234			−1.72717			−0.863585	+	0.504203i	$0.831786\pi$
$884$		0			0
$885$	11.0798	+	18.5577i	0.372442	+	0.623809i
$886$		0			0
$887$	15.3867			0.516635			0.258318	−	0.966060i	$-0.416832\pi$
$887$	15.3867			0.516635			0.258318	+	0.966060i	$0.416832\pi$
$888$		0			0
$889$	−3.97825			−0.133426
$890$		0			0
$891$	2.40968	−	1.58627i	0.0807275	−	0.0531421i
$892$		0			0
$893$	40.0635			1.34067
$894$		0			0
$895$			5.18815i			0.173421i
$896$		0			0
$897$	11.9636	+	20.0381i	0.399454	+	0.669051i
$898$		0			0
$899$		−	87.1211i		−	2.90565i
$900$		0			0
$901$			10.1600i			0.338479i
$902$		0			0
$903$	−2.10546	−	3.52646i	−0.0700653	−	0.117353i
$904$		0			0
$905$			2.59819i			0.0863668i
$906$		0			0
$907$	−34.3304			−1.13992			−0.569962	−	0.821671i	$-0.693042\pi$
$907$	−34.3304			−1.13992			−0.569962	+	0.821671i	$0.693042\pi$
$908$		0			0
$909$	21.7945	−	40.4400i	0.722878	−	1.34131i
$910$		0			0
$911$	−20.7856			−0.688657			−0.344328	−	0.938849i	$-0.611893\pi$
$911$	−20.7856			−0.688657			−0.344328	+	0.938849i	$0.611893\pi$
$912$		0			0
$913$	1.08035			0.0357542
$914$		0			0
$915$	7.84316	+	13.1366i	0.259287	+	0.434283i
$916$		0			0
$917$	−7.11710			−0.235027
$918$		0			0
$919$			35.6290i			1.17529i	0.809119	+	0.587645i	$0.199945\pi$
$919$			35.6290i			1.17529i	−0.809119	+	0.587645i	$0.800055\pi$
$920$		0			0
$921$	9.99184	−	5.96558i	0.329242	−	0.196573i
$922$		0			0
$923$		−	14.7634i		−	0.485944i
$924$		0			0
$925$			8.49593i			0.279345i
$926$		0			0
$927$	−19.1712	−	10.3320i	−0.629664	−	0.339347i
$928$		0			0
$929$			44.9041i			1.47325i	0.676299	+	0.736627i	$0.263583\pi$
$929$			44.9041i			1.47325i	−0.676299	+	0.736627i	$0.736417\pi$
$930$		0			0
$931$	38.8575			1.27350
$932$		0			0
$933$	−32.8019	+	19.5842i	−1.07389	+	0.641159i
$934$		0			0
$935$	−0.824788			−0.0269734
$936$		0			0
$937$	−56.5086			−1.84606			−0.923029	−	0.384731i	$-0.874294\pi$
$937$	−56.5086			−1.84606			−0.923029	+	0.384731i	$0.874294\pi$
$938$		0			0
$939$	16.4118	−	9.79861i	0.535580	−	0.319765i
$940$		0			0
$941$	−29.6334			−0.966022			−0.483011	−	0.875614i	$-0.660457\pi$
$941$	−29.6334			−0.966022			−0.483011	+	0.875614i	$0.660457\pi$
$942$		0			0
$943$		−	16.5380i		−	0.538553i
$944$		0			0
$945$	−4.13865	+	0.181956i	−0.134630	+	0.00591904i
$946$		0			0
$947$			29.2810i			0.951504i	0.879580	+	0.475752i	$0.157824\pi$
$947$			29.2810i			0.951504i	−0.879580	+	0.475752i	$0.842176\pi$
$948$		0			0
$949$		−	6.15695i		−	0.199863i
$950$		0			0
$951$	36.2427	−	21.6385i	1.17525	−	0.701678i
$952$		0			0
$953$			40.2311i			1.30321i	0.758557	+	0.651606i	$0.225905\pi$
$953$			40.2311i			1.30321i	−0.758557	+	0.651606i	$0.774095\pi$
$954$		0			0
$955$	12.2556			0.396581
$956$		0			0
$957$	2.50323	+	4.19270i	0.0809180	+	0.135531i
$958$		0			0
$959$	1.29127			0.0416972
$960$		0			0
$961$	−67.1203			−2.16517
$962$		0			0
$963$	35.1048	+	18.9192i	1.13124	+	0.609662i
$964$		0			0
$965$	8.26230			0.265973
$966$		0			0
$967$			34.4522i			1.10791i	0.832547	+	0.553954i	$0.186882\pi$
$967$			34.4522i			1.10791i	−0.832547	+	0.553954i	$0.813118\pi$
$968$		0			0
$969$	−13.9486	−	23.3627i	−0.448094	−	0.750519i
$970$		0			0
$971$		−	26.9133i		−	0.863688i	−0.901948	−	0.431844i	$-0.857863\pi$
$971$		−	26.9133i		−	0.863688i	0.901948	−	0.431844i	$-0.142137\pi$
$972$		0			0
$973$		−	2.86023i		−	0.0916948i
$974$		0			0
$975$	−3.82085	−	6.39959i	−0.122365	−	0.204951i
$976$		0			0
$977$			41.9916i			1.34343i	0.740810	+	0.671715i	$0.234442\pi$
$977$			41.9916i			1.34343i	−0.740810	+	0.671715i	$0.765558\pi$
$978$		0			0
$979$	4.40837			0.140892
$980$		0			0
$981$	−9.02109	−	4.86177i	−0.288021	−	0.155224i
$982$		0			0
$983$	3.83856			0.122431			0.0612156	−	0.998125i	$-0.480502\pi$
$983$	3.83856			0.122431			0.0612156	+	0.998125i	$0.480502\pi$
$984$		0			0
$985$	10.8102			0.344441
$986$		0			0
$987$	−4.64506	−	7.78007i	−0.147854	−	0.247642i
$988$		0			0
$989$	−9.31301			−0.296137
$990$		0			0
$991$		−	48.3440i		−	1.53570i	−0.640630	−	0.767850i	$-0.721327\pi$
$991$		−	48.3440i		−	1.53570i	0.640630	−	0.767850i	$-0.278673\pi$
$992$		0			0
$993$	19.6194	−	11.7137i	0.622602	−	0.371722i
$994$		0			0
$995$			5.71287i			0.181110i
$996$		0			0
$997$		−	23.5236i		−	0.744998i	−0.928032	−	0.372499i	$-0.878501\pi$
$997$		−	23.5236i		−	0.744998i	0.928032	−	0.372499i	$-0.121499\pi$
$998$		0			0
$999$	−44.1036	+	1.93902i	−1.39538	+	0.0613479i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.b.b.431.2		8
3.2	odd	2		480.2.b.a.431.1		8
4.3	odd	2		120.2.b.a.11.4	yes	8
5.2	odd	4		2400.2.m.d.1199.12		16
5.3	odd	4		2400.2.m.d.1199.5		16
5.4	even	2		2400.2.b.f.2351.7		8
8.3	odd	2		480.2.b.a.431.2		8
8.5	even	2		120.2.b.b.11.6	yes	8
12.11	even	2		120.2.b.b.11.5	yes	8
15.2	even	4		2400.2.m.c.1199.6		16
15.8	even	4		2400.2.m.c.1199.11		16
15.14	odd	2		2400.2.b.e.2351.8		8
20.3	even	4		600.2.m.c.299.14		16
20.7	even	4		600.2.m.c.299.3		16
20.19	odd	2		600.2.b.f.251.5		8
24.5	odd	2		120.2.b.a.11.3	✓	8
24.11	even	2	inner	480.2.b.b.431.1		8
40.3	even	4		2400.2.m.c.1199.5		16
40.13	odd	4		600.2.m.d.299.13		16
40.19	odd	2		2400.2.b.e.2351.7		8
40.27	even	4		2400.2.m.c.1199.12		16
40.29	even	2		600.2.b.e.251.3		8
40.37	odd	4		600.2.m.d.299.4		16
60.23	odd	4		600.2.m.d.299.3		16
60.47	odd	4		600.2.m.d.299.14		16
60.59	even	2		600.2.b.e.251.4		8
120.29	odd	2		600.2.b.f.251.6		8
120.53	even	4		600.2.m.c.299.4		16
120.59	even	2		2400.2.b.f.2351.8		8
120.77	even	4		600.2.m.c.299.13		16
120.83	odd	4		2400.2.m.d.1199.11		16
120.107	odd	4		2400.2.m.d.1199.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.b.a.11.3	✓	8	24.5	odd	2
120.2.b.a.11.4	yes	8	4.3	odd	2
120.2.b.b.11.5	yes	8	12.11	even	2
120.2.b.b.11.6	yes	8	8.5	even	2
480.2.b.a.431.1		8	3.2	odd	2
480.2.b.a.431.2		8	8.3	odd	2
480.2.b.b.431.1		8	24.11	even	2	inner
480.2.b.b.431.2		8	1.1	even	1	trivial
600.2.b.e.251.3		8	40.29	even	2
600.2.b.e.251.4		8	60.59	even	2
600.2.b.f.251.5		8	20.19	odd	2
600.2.b.f.251.6		8	120.29	odd	2
600.2.m.c.299.3		16	20.7	even	4
600.2.m.c.299.4		16	120.53	even	4
600.2.m.c.299.13		16	120.77	even	4
600.2.m.c.299.14		16	20.3	even	4
600.2.m.d.299.3		16	60.23	odd	4
600.2.m.d.299.4		16	40.37	odd	4
600.2.m.d.299.13		16	40.13	odd	4
600.2.m.d.299.14		16	60.47	odd	4
2400.2.b.e.2351.7		8	40.19	odd	2
2400.2.b.e.2351.8		8	15.14	odd	2
2400.2.b.f.2351.7		8	5.4	even	2
2400.2.b.f.2351.8		8	120.59	even	2
2400.2.m.c.1199.5		16	40.3	even	4
2400.2.m.c.1199.6		16	15.2	even	4
2400.2.m.c.1199.11		16	15.8	even	4
2400.2.m.c.1199.12		16	40.27	even	4
2400.2.m.d.1199.5		16	5.3	odd	4
2400.2.m.d.1199.6		16	120.107	odd	4
2400.2.m.d.1199.11		16	120.83	odd	4
2400.2.m.d.1199.12		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.48716 + 0.887900i) q^{3} +1.00000 q^{5} +0.797253i q^{7} +(1.42327 - 2.64089i) q^{9} +O(q^{10})\)	\(q+(-1.48716 + 0.887900i) q^{3} +1.00000 q^{5} +0.797253i q^{7} +(1.42327 - 2.64089i) q^{9} +0.320548i q^{11} +4.30324i q^{13} +(-1.48716 + 0.887900i) q^{15} +2.57305i q^{17} +6.10546 q^{19} +(-0.707881 - 1.18564i) q^{21} -3.13115 q^{23} +1.00000 q^{25} +(0.228229 + 5.19114i) q^{27} -8.79516 q^{29} +9.90557i q^{31} +(-0.284615 - 0.476705i) q^{33} +0.797253i q^{35} +8.49593i q^{37} +(-3.82085 - 6.39959i) q^{39} +5.28178i q^{41} +2.97431 q^{43} +(1.42327 - 2.64089i) q^{45} +6.56192 q^{47} +6.36439 q^{49} +(-2.28461 - 3.82653i) q^{51} +3.94862 q^{53} +0.320548i q^{55} +(-9.07977 + 5.42104i) q^{57} -12.4786i q^{59} -8.83339i q^{61} +(2.10546 + 1.13470i) q^{63} +4.30324i q^{65} -4.66738 q^{67} +(4.65651 - 2.78015i) q^{69} -3.43077 q^{71} -1.43077 q^{73} +(-1.48716 + 0.887900i) q^{75} -0.255558 q^{77} -2.89360i q^{79} +(-4.94862 - 7.51739i) q^{81} -3.37031i q^{83} +2.57305i q^{85} +(13.0798 - 7.80922i) q^{87} -13.7526i q^{89} -3.43077 q^{91} +(-8.79516 - 14.7311i) q^{93} +6.10546 q^{95} -4.26230 q^{97} +(0.846533 + 0.456225i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 12 q^{27} - 4 q^{33} + 16 q^{39} + 28 q^{47} - 16 q^{49} - 20 q^{51} - 16 q^{53} - 4 q^{57} - 28 q^{63} + 32 q^{67} - 20 q^{69} - 24 q^{71} - 8 q^{73} + 8 q^{81} + 36 q^{87} - 24 q^{91} + 4 q^{95} + 8 q^{97} - 16 q^{99}+O(q^{100}) \) 8 * q + 8 * q^5 + 4 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 + 12 * q^27 - 4 * q^33 + 16 * q^39 + 28 * q^47 - 16 * q^49 - 20 * q^51 - 16 * q^53 - 4 * q^57 - 28 * q^63 + 32 * q^67 - 20 * q^69 - 24 * q^71 - 8 * q^73 + 8 * q^81 + 36 * q^87 - 24 * q^91 + 4 * q^95 + 8 * q^97 - 16 * q^99

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