LMFDB - Embedded newform 900.3.f.e.199.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(199,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.199");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.8
Root			$-0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	900.199
Dual form			900.3.f.e.199.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.90211 + 0.618034i) q^{2} +(3.23607 + 2.35114i) q^{4} +5.25731 q^{7} +(4.70228 + 6.47214i) q^{8} +O(q^{10})$	\(q+(1.90211 + 0.618034i) q^{2} +(3.23607 + 2.35114i) q^{4} +5.25731 q^{7} +(4.70228 + 6.47214i) q^{8} +19.9192i q^{11} -8.47214i q^{13} +(10.0000 + 3.24920i) q^{14} +(4.94427 + 15.2169i) q^{16} +11.8885i q^{17} -15.2169i q^{19} +(-12.3107 + 37.8885i) q^{22} +0.555029 q^{23} +(5.23607 - 16.1150i) q^{26} +(17.0130 + 12.3607i) q^{28} -10.9443 q^{29} +8.29451i q^{31} +32.0000i q^{32} +(-7.34752 + 22.6134i) q^{34} +18.3607i q^{37} +(9.40456 - 28.9443i) q^{38} +14.5836 q^{41} +22.2703 q^{43} +(-46.8328 + 64.4598i) q^{44} +(1.05573 + 0.343027i) q^{46} +53.3902 q^{47} -21.3607 q^{49} +(19.9192 - 27.4164i) q^{52} +66.3607i q^{53} +(24.7214 + 34.0260i) q^{56} +(-20.8172 - 6.76393i) q^{58} -17.4370i q^{59} +90.1378 q^{61} +(-5.12629 + 15.7771i) q^{62} +(-19.7771 + 60.8676i) q^{64} -50.2220 q^{67} +(-27.9516 + 38.4721i) q^{68} -80.7868i q^{71} -5.55418i q^{73} +(-11.3475 + 34.9241i) q^{74} +(35.7771 - 49.2429i) q^{76} +104.721i q^{77} -13.8448i q^{79} +(27.7396 + 9.01316i) q^{82} +76.2155 q^{83} +(42.3607 + 13.7638i) q^{86} +(-128.920 + 93.6656i) q^{88} -111.443 q^{89} -44.5407i q^{91} +(1.79611 + 1.30495i) q^{92} +(101.554 + 32.9970i) q^{94} +92.8328i q^{97} +(-40.6304 - 13.2016i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4}+O(q^{10}) $ 8 * q + 8 * q^4	$ 8 q + 8 q^{4} + 80 q^{14} - 32 q^{16} + 24 q^{26} - 16 q^{29} - 184 q^{34} + 224 q^{41} - 160 q^{44} + 80 q^{46} + 8 q^{49} - 160 q^{56} + 256 q^{61} + 128 q^{64} - 216 q^{74} + 160 q^{86} - 176 q^{89} + 240 q^{94}+O(q^{100}) $ 8 * q + 8 * q^4 + 80 * q^14 - 32 * q^16 + 24 * q^26 - 16 * q^29 - 184 * q^34 + 224 * q^41 - 160 * q^44 + 80 * q^46 + 8 * q^49 - 160 * q^56 + 256 * q^61 + 128 * q^64 - 216 * q^74 + 160 * q^86 - 176 * q^89 + 240 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.90211	+	0.618034i	0.951057	+	0.309017i
$2$	1.90211	+	0.618034i	0.951057	+	0.309017i
$3$		0			0
$3$		0			0
$4$	3.23607	+	2.35114i	0.809017	+	0.587785i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.25731			0.751044			0.375522	−	0.926813i	$-0.377463\pi$
$7$	5.25731			0.751044			0.375522	+	0.926813i	$0.377463\pi$
$8$	4.70228	+	6.47214i	0.587785	+	0.809017i
$9$		0			0
$10$		0			0
$11$			19.9192i			1.81084i	0.424522	+	0.905418i	$0.360442\pi$
$11$			19.9192i			1.81084i	−0.424522	+	0.905418i	$0.639558\pi$
$12$		0			0
$13$		−	8.47214i		−	0.651703i	−0.945421	−	0.325851i	$-0.894349\pi$
$13$		−	8.47214i		−	0.651703i	0.945421	−	0.325851i	$-0.105651\pi$
$14$	10.0000	+	3.24920i	0.714286	+	0.232085i
$15$		0			0
$16$	4.94427	+	15.2169i	0.309017	+	0.951057i
$17$			11.8885i			0.699326i	0.936876	+	0.349663i	$0.113704\pi$
$17$			11.8885i			0.699326i	−0.936876	+	0.349663i	$0.886296\pi$
$18$		0			0
$19$		−	15.2169i		−	0.800890i	−0.916321	−	0.400445i	$-0.868856\pi$
$19$		−	15.2169i		−	0.800890i	0.916321	−	0.400445i	$-0.131144\pi$
$20$		0			0
$21$		0			0
$22$	−12.3107	+	37.8885i	−0.559579	+	1.72221i
$23$	0.555029			0.0241317			0.0120659	−	0.999927i	$-0.496159\pi$
$23$	0.555029			0.0241317			0.0120659	+	0.999927i	$0.496159\pi$
$24$		0			0
$25$		0			0
$26$	5.23607	−	16.1150i	0.201387	−	0.619806i
$27$		0			0
$28$	17.0130	+	12.3607i	0.607608	+	0.441453i
$29$	−10.9443			−0.377389			−0.188694	−	0.982036i	$-0.560426\pi$
$29$	−10.9443			−0.377389			−0.188694	+	0.982036i	$0.560426\pi$
$30$		0			0
$31$			8.29451i			0.267565i	0.991011	+	0.133782i	$0.0427123\pi$
$31$			8.29451i			0.267565i	−0.991011	+	0.133782i	$0.957288\pi$
$32$			32.0000i			1.00000i
$33$		0			0
$34$	−7.34752	+	22.6134i	−0.216104	+	0.665099i
$35$		0			0
$36$		0			0
$37$			18.3607i			0.496235i	0.968730	+	0.248117i	$0.0798118\pi$
$37$			18.3607i			0.496235i	−0.968730	+	0.248117i	$0.920188\pi$
$38$	9.40456	−	28.9443i	0.247489	−	0.761691i
$39$		0			0
$40$		0			0
$41$	14.5836			0.355697			0.177849	−	0.984058i	$-0.443086\pi$
$41$	14.5836			0.355697			0.177849	+	0.984058i	$0.443086\pi$
$42$		0			0
$43$	22.2703			0.517915			0.258957	−	0.965889i	$-0.416621\pi$
$43$	22.2703			0.517915			0.258957	+	0.965889i	$0.416621\pi$
$44$	−46.8328	+	64.4598i	−1.06438	+	1.46500i
$45$		0			0
$46$	1.05573	+	0.343027i	0.0229506	+	0.00745711i
$47$	53.3902			1.13596			0.567981	−	0.823042i	$-0.307725\pi$
$47$	53.3902			1.13596			0.567981	+	0.823042i	$0.307725\pi$
$48$		0			0
$49$	−21.3607			−0.435932
$50$		0			0
$51$		0			0
$52$	19.9192	−	27.4164i	0.383061	−	0.527239i
$53$			66.3607i			1.25209i	0.779788	+	0.626044i	$0.215327\pi$
$53$			66.3607i			1.25209i	−0.779788	+	0.626044i	$0.784673\pi$
$54$		0			0
$55$		0			0
$56$	24.7214	+	34.0260i	0.441453	+	0.607608i
$57$		0			0
$58$	−20.8172	−	6.76393i	−0.358918	−	0.116620i
$59$		−	17.4370i		−	0.295543i	−0.989022	−	0.147771i	$-0.952790\pi$
$59$		−	17.4370i		−	0.295543i	0.989022	−	0.147771i	$-0.0472100\pi$
$60$		0			0
$61$	90.1378			1.47767			0.738834	−	0.673887i	$-0.235377\pi$
$61$	90.1378			1.47767			0.738834	+	0.673887i	$0.235377\pi$
$62$	−5.12629	+	15.7771i	−0.0826820	+	0.254469i
$63$		0			0
$64$	−19.7771	+	60.8676i	−0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	−50.2220			−0.749582			−0.374791	−	0.927109i	$-0.622285\pi$
$67$	−50.2220			−0.749582			−0.374791	+	0.927109i	$0.622285\pi$
$68$	−27.9516	+	38.4721i	−0.411054	+	0.565767i
$69$		0			0
$70$		0			0
$71$		−	80.7868i		−	1.13784i	−0.822392	−	0.568921i	$-0.807361\pi$
$71$		−	80.7868i		−	1.13784i	0.822392	−	0.568921i	$-0.192639\pi$
$72$		0			0
$73$		−	5.55418i		−	0.0760846i	−0.999276	−	0.0380423i	$-0.987888\pi$
$73$		−	5.55418i		−	0.0760846i	0.999276	−	0.0380423i	$-0.0121122\pi$
$74$	−11.3475	+	34.9241i	−0.153345	+	0.471947i
$75$		0			0
$76$	35.7771	−	49.2429i	0.470751	−	0.647933i
$77$			104.721i			1.36002i
$78$		0			0
$79$		−	13.8448i		−	0.175251i	−0.996154	−	0.0876253i	$-0.972072\pi$
$79$		−	13.8448i		−	0.175251i	0.996154	−	0.0876253i	$-0.0279278\pi$
$80$		0			0
$81$		0			0
$82$	27.7396	+	9.01316i	0.338288	+	0.109917i
$83$	76.2155			0.918260			0.459130	−	0.888369i	$-0.348161\pi$
$83$	76.2155			0.918260			0.459130	+	0.888369i	$0.348161\pi$
$84$		0			0
$85$		0			0
$86$	42.3607	+	13.7638i	0.492566	+	0.160044i
$87$		0			0
$88$	−128.920	+	93.6656i	−1.46500	+	1.06438i
$89$	−111.443			−1.25217			−0.626083	−	0.779757i	$-0.715343\pi$
$89$	−111.443			−1.25217			−0.626083	+	0.779757i	$0.715343\pi$
$90$		0			0
$91$		−	44.5407i		−	0.489458i
$92$	1.79611	+	1.30495i	0.0195230	+	0.0141843i
$93$		0			0
$94$	101.554	+	32.9970i	1.08036	+	0.351031i
$95$		0			0
$96$		0			0
$97$			92.8328i			0.957039i	0.878077	+	0.478520i	$0.158826\pi$
$97$			92.8328i			0.957039i	−0.878077	+	0.478520i	$0.841174\pi$
$98$	−40.6304	−	13.2016i	−0.414596	−	0.134710i
$99$		0			0
$100$		0			0
$101$	−64.1115			−0.634767			−0.317383	−	0.948297i	$-0.602804\pi$
$101$	−64.1115			−0.634767			−0.317383	+	0.948297i	$0.602804\pi$
$102$		0			0
$103$	137.769			1.33757			0.668783	−	0.743458i	$-0.266816\pi$
$103$	137.769			1.33757			0.668783	+	0.743458i	$0.266816\pi$
$104$	54.8328	−	39.8384i	0.527239	−	0.383061i
$105$		0			0
$106$	−41.0132	+	126.226i	−0.386917	+	1.19081i
$107$	−51.3320			−0.479739			−0.239869	−	0.970805i	$-0.577105\pi$
$107$	−51.3320			−0.479739			−0.239869	+	0.970805i	$0.577105\pi$
$108$		0			0
$109$	−133.469			−1.22449			−0.612243	−	0.790669i	$-0.709733\pi$
$109$	−133.469			−1.22449			−0.612243	+	0.790669i	$0.709733\pi$
$110$		0			0
$111$		0			0
$112$	25.9936	+	80.0000i	0.232085	+	0.714286i
$113$		−	170.721i		−	1.51081i	−0.655259	−	0.755404i	$-0.727441\pi$
$113$		−	170.721i		−	1.51081i	0.655259	−	0.755404i	$-0.272559\pi$
$114$		0			0
$115$		0			0
$116$	−35.4164	−	25.7315i	−0.305314	−	0.221824i
$117$		0			0
$118$	10.7767	−	33.1672i	0.0913277	−	0.281078i
$119$			62.5018i			0.525225i
$120$		0			0
$121$	−275.774			−2.27912
$122$	171.452	+	55.7082i	1.40535	+	0.456625i
$123$		0			0
$124$	−19.5016	+	26.8416i	−0.157271	+	0.216464i
$125$		0			0
$126$		0			0
$127$	198.637			1.56407			0.782035	−	0.623235i	$-0.214182\pi$
$127$	198.637			1.56407			0.782035	+	0.623235i	$0.214182\pi$
$128$	−75.2365	+	103.554i	−0.587785	+	0.809017i
$129$		0			0
$130$		0			0
$131$			7.77041i			0.0593161i	0.999560	+	0.0296580i	$0.00944183\pi$
$131$			7.77041i			0.0593161i	−0.999560	+	0.0296580i	$0.990558\pi$
$132$		0			0
$133$		−	80.0000i		−	0.601504i
$134$	−95.5279	−	31.0389i	−0.712895	−	0.231633i
$135$		0			0
$136$	−76.9443	+	55.9033i	−0.565767	+	0.411054i
$137$			0.832816i			0.00607895i	0.999995	+	0.00303947i	$0.000967496\pi$
$137$			0.832816i			0.00607895i	−0.999995	+	0.00303947i	$0.999033\pi$
$138$		0			0
$139$		−	237.658i		−	1.70977i	−0.518817	−	0.854885i	$-0.673627\pi$
$139$		−	237.658i		−	1.70977i	0.518817	−	0.854885i	$-0.326373\pi$
$140$		0			0
$141$		0			0
$142$	49.9290	−	153.666i	0.351613	−	1.08215i
$143$	168.758			1.18013
$144$		0			0
$145$		0			0
$146$	3.43267	−	10.5647i	0.0235114	−	0.0723607i
$147$		0			0
$148$	−43.1685	+	59.4164i	−0.291679	+	0.401462i
$149$	−36.9706			−0.248125			−0.124062	−	0.992274i	$-0.539592\pi$
$149$	−36.9706			−0.248125			−0.124062	+	0.992274i	$0.539592\pi$
$150$		0			0
$151$		−	282.723i		−	1.87234i	−0.351552	−	0.936168i	$-0.614346\pi$
$151$		−	282.723i		−	1.87234i	0.351552	−	0.936168i	$-0.385654\pi$
$152$	98.4859	−	71.5542i	0.647933	−	0.470751i
$153$		0			0
$154$	−64.7214	+	199.192i	−0.420269	+	1.29345i
$155$		0			0
$156$		0			0
$157$		−	204.748i		−	1.30413i	−0.758165	−	0.652063i	$-0.773904\pi$
$157$		−	204.748i		−	1.30413i	0.758165	−	0.652063i	$-0.226096\pi$
$158$	8.55656	−	26.3344i	0.0541554	−	0.166673i
$159$		0			0
$160$		0			0
$161$	2.91796			0.0181240
$162$		0			0
$163$	107.235			0.657885			0.328943	−	0.944350i	$-0.393308\pi$
$163$	107.235			0.657885			0.328943	+	0.944350i	$0.393308\pi$
$164$	47.1935	+	34.2881i	0.287765	+	0.209074i
$165$		0			0
$166$	144.971	+	47.1038i	0.873317	+	0.283758i
$167$	33.2090			0.198856			0.0994280	−	0.995045i	$-0.468299\pi$
$167$	33.2090			0.198856			0.0994280	+	0.995045i	$0.468299\pi$
$168$		0			0
$169$	97.2229			0.575284
$170$		0			0
$171$		0			0
$172$	72.0683	+	52.3607i	0.419002	+	0.304423i
$173$			226.361i			1.30844i	0.756303	+	0.654222i	$0.227004\pi$
$173$			226.361i			1.30844i	−0.756303	+	0.654222i	$0.772996\pi$
$174$		0			0
$175$		0			0
$176$	−303.108	+	98.4859i	−1.72221	+	0.559579i
$177$		0			0
$178$	−211.977	−	68.8754i	−1.19088	−	0.386940i
$179$		−	224.337i		−	1.25328i	−0.779308	−	0.626641i	$-0.784429\pi$
$179$		−	224.337i		−	1.25328i	0.779308	−	0.626641i	$-0.215571\pi$
$180$		0			0
$181$	86.2229			0.476370			0.238185	−	0.971220i	$-0.423448\pi$
$181$	86.2229			0.476370			0.238185	+	0.971220i	$0.423448\pi$
$182$	27.5276	−	84.7214i	0.151251	−	0.465502i
$183$		0			0
$184$	2.60990	+	3.59222i	0.0141843	+	0.0195230i
$185$		0			0
$186$		0			0
$187$	−236.810			−1.26636
$188$	172.774	+	125.528i	0.919012	+	0.667701i
$189$		0			0
$190$		0			0
$191$		−	31.0198i		−	0.162407i	−0.996698	−	0.0812036i	$-0.974124\pi$
$191$		−	31.0198i		−	0.162407i	0.996698	−	0.0812036i	$-0.0258764\pi$
$192$		0			0
$193$			110.223i			0.571103i	0.958363	+	0.285552i	$0.0921768\pi$
$193$			110.223i			0.571103i	−0.958363	+	0.285552i	$0.907823\pi$
$194$	−57.3738	+	176.579i	−0.295741	+	0.910198i
$195$		0			0
$196$	−69.1246	−	50.2220i	−0.352677	−	0.256235i
$197$		−	172.525i		−	0.875760i	−0.899033	−	0.437880i	$-0.855729\pi$
$197$		−	172.525i		−	0.875760i	0.899033	−	0.437880i	$-0.144271\pi$
$198$		0			0
$199$		−	272.208i		−	1.36788i	−0.729538	−	0.683940i	$-0.760265\pi$
$199$		−	272.208i		−	1.36788i	0.729538	−	0.683940i	$-0.239735\pi$
$200$		0			0
$201$		0			0
$202$	−121.947	−	39.6231i	−0.603699	−	0.196154i
$203$	−57.5374			−0.283436
$204$		0			0
$205$		0			0
$206$	262.053	+	85.1461i	1.27210	+	0.413330i
$207$		0			0
$208$	128.920	−	41.8885i	0.619806	−	0.201387i
$209$	303.108			1.45028
$210$		0			0
$211$			205.266i			0.972826i	0.873729	+	0.486413i	$0.161695\pi$
$211$			205.266i			0.972826i	−0.873729	+	0.486413i	$0.838305\pi$
$212$	−156.023	+	214.748i	−0.735959	+	1.01296i
$213$		0			0
$214$	−97.6393	−	31.7249i	−0.456259	−	0.148247i
$215$		0			0
$216$		0			0
$217$			43.6068i			0.200953i
$218$	−253.873	−	82.4884i	−1.16456	−	0.378387i
$219$		0			0
$220$		0			0
$221$	100.721			0.455753
$222$		0			0
$223$	235.731			1.05709			0.528545	−	0.848905i	$-0.322738\pi$
$223$	235.731			1.05709			0.528545	+	0.848905i	$0.322738\pi$
$224$			168.234i			0.751044i
$225$		0			0
$226$	105.512	−	324.731i	0.466865	−	1.43686i
$227$	−58.5165			−0.257782			−0.128891	−	0.991659i	$-0.541142\pi$
$227$	−58.5165			−0.257782			−0.128891	+	0.991659i	$0.541142\pi$
$228$		0			0
$229$	−162.721			−0.710574			−0.355287	−	0.934757i	$-0.615617\pi$
$229$	−162.721			−0.710574			−0.355287	+	0.934757i	$0.615617\pi$
$230$		0			0
$231$		0			0
$232$	−51.4631	−	70.8328i	−0.221824	−	0.305314i
$233$		−	319.050i		−	1.36931i	−0.728867	−	0.684656i	$-0.759953\pi$
$233$		−	319.050i		−	1.36931i	0.728867	−	0.684656i	$-0.240047\pi$
$234$		0			0
$235$		0			0
$236$	40.9969	−	56.4274i	0.173716	−	0.239099i
$237$		0			0
$238$	−38.6282	+	118.885i	−0.162303	+	0.499519i
$239$		−	236.810i		−	0.990837i	−0.868654	−	0.495419i	$-0.835015\pi$
$239$		−	236.810i		−	0.990837i	0.868654	−	0.495419i	$-0.164985\pi$
$240$		0			0
$241$	−0.917961			−0.00380897			−0.00190448	−	0.999998i	$-0.500606\pi$
$241$	−0.917961			−0.00380897			−0.00190448	+	0.999998i	$0.500606\pi$
$242$	−524.553	−	170.438i	−2.16758	−	0.704288i
$243$		0			0
$244$	291.692	+	211.927i	1.19546	+	0.868552i
$245$		0			0
$246$		0			0
$247$	−128.920			−0.521942
$248$	−53.6832	+	39.0031i	−0.216464	+	0.157271i
$249$		0			0
$250$		0			0
$251$		−	136.690i		−	0.544582i	−0.962215	−	0.272291i	$-0.912219\pi$
$251$		−	136.690i		−	0.544582i	0.962215	−	0.272291i	$-0.0877813\pi$
$252$		0			0
$253$			11.0557i			0.0436985i
$254$	377.830	+	122.764i	1.48752	+	0.483324i
$255$		0			0
$256$	−207.108	+	150.473i	−0.809017	+	0.587785i
$257$		−	274.944i		−	1.06982i	−0.844908	−	0.534911i	$-0.820345\pi$
$257$		−	274.944i		−	1.06982i	0.844908	−	0.534911i	$-0.179655\pi$
$258$		0			0
$259$			96.5278i			0.372694i
$260$		0			0
$261$		0			0
$262$	−4.80238	+	14.7802i	−0.0183297	+	0.0564130i
$263$	−406.385			−1.54519			−0.772596	−	0.634899i	$-0.781042\pi$
$263$	−406.385			−1.54519			−0.772596	+	0.634899i	$0.781042\pi$
$264$		0			0
$265$		0			0
$266$	49.4427	−	152.169i	0.185875	−	0.572064i
$267$		0			0
$268$	−162.522	−	118.079i	−0.606424	−	0.440593i
$269$	−348.525			−1.29563			−0.647816	−	0.761797i	$-0.724317\pi$
$269$	−348.525			−1.29563			−0.647816	+	0.761797i	$0.724317\pi$
$270$		0			0
$271$			247.849i			0.914571i	0.889320	+	0.457286i	$0.151178\pi$
$271$			247.849i			0.914571i	−0.889320	+	0.457286i	$0.848822\pi$
$272$	−180.907	+	58.7802i	−0.665099	+	0.216104i
$273$		0			0
$274$	−0.514708	+	1.58411i	−0.00187850	+	0.00578142i
$275$		0			0
$276$		0			0
$277$			54.7539i			0.197667i	0.995104	+	0.0988337i	$0.0315112\pi$
$277$			54.7539i			0.197667i	−0.995104	+	0.0988337i	$0.968489\pi$
$278$	146.881	−	452.053i	0.528348	−	1.62609i
$279$		0			0
$280$		0			0
$281$	50.3607			0.179220			0.0896098	−	0.995977i	$-0.471438\pi$
$281$	50.3607			0.179220			0.0896098	+	0.995977i	$0.471438\pi$
$282$		0			0
$283$	−147.336			−0.520621			−0.260310	−	0.965525i	$-0.583825\pi$
$283$	−147.336			−0.520621			−0.260310	+	0.965525i	$0.583825\pi$
$284$	189.941	−	261.432i	0.668807	−	0.920534i
$285$		0			0
$286$	320.997	+	104.298i	1.12237	+	0.364679i
$287$	76.6705			0.267145
$288$		0			0
$289$	147.663			0.510943
$290$		0			0
$291$		0			0
$292$	13.0586	−	17.9737i	0.0447214	−	0.0615537i
$293$		−	178.859i		−	0.610441i	−0.952282	−	0.305220i	$-0.901270\pi$
$293$		−	178.859i		−	0.610441i	0.952282	−	0.305220i	$-0.0987301\pi$
$294$		0			0
$295$		0			0
$296$	−118.833	+	86.3371i	−0.401462	+	0.291679i
$297$		0			0
$298$	−70.3222	−	22.8491i	−0.235981	−	0.0766748i
$299$		−	4.70228i		−	0.0157267i
$300$		0			0
$301$	117.082			0.388977
$302$	174.732	−	537.771i	0.578584	−	1.78070i
$303$		0			0
$304$	231.554	−	75.2365i	0.761691	−	0.247489i
$305$		0			0
$306$		0			0
$307$	284.550			0.926873			0.463436	−	0.886130i	$-0.346616\pi$
$307$	284.550			0.926873			0.463436	+	0.886130i	$0.346616\pi$
$308$	−246.215	+	338.885i	−0.799398	+	1.10028i
$309$		0			0
$310$		0			0
$311$			282.199i			0.907392i	0.891157	+	0.453696i	$0.149895\pi$
$311$			282.199i			0.907392i	−0.891157	+	0.453696i	$0.850105\pi$
$312$		0			0
$313$		−	567.548i		−	1.81325i	−0.421935	−	0.906626i	$-0.638649\pi$
$313$		−	567.548i		−	1.81325i	0.421935	−	0.906626i	$-0.361351\pi$
$314$	126.541	−	389.453i	0.402997	−	1.24030i
$315$		0			0
$316$	32.5511	−	44.8027i	0.103010	−	0.141781i
$317$			161.141i			0.508331i	0.967161	+	0.254165i	$0.0818008\pi$
$317$			161.141i			0.508331i	−0.967161	+	0.254165i	$0.918199\pi$
$318$		0			0
$319$		−	218.001i		−	0.683389i
$320$		0			0
$321$		0			0
$322$	5.55029	+	1.80340i	0.0172369	+	0.00560062i
$323$	180.907			0.560083
$324$		0			0
$325$		0			0
$326$	203.974	+	66.2751i	0.625686	+	0.203298i
$327$		0			0
$328$	68.5762	+	94.3870i	0.209074	+	0.287765i
$329$	280.689			0.853158
$330$		0			0
$331$		−	331.966i		−	1.00292i	−0.865181	−	0.501459i	$-0.832797\pi$
$331$		−	331.966i		−	1.00292i	0.865181	−	0.501459i	$-0.167203\pi$
$332$	246.639	+	179.193i	0.742888	+	0.539739i
$333$		0			0
$334$	63.1672	+	20.5243i	0.189123	+	0.0614499i
$335$		0			0
$336$		0			0
$337$			269.108i			0.798541i	0.916833	+	0.399271i	$0.130737\pi$
$337$			269.108i			0.798541i	−0.916833	+	0.399271i	$0.869263\pi$
$338$	184.929	+	60.0871i	0.547127	+	0.177772i
$339$		0			0
$340$		0			0
$341$	−165.220			−0.484516
$342$		0			0
$343$	−369.908			−1.07845
$344$	104.721	+	144.137i	0.304423	+	0.419002i
$345$		0			0
$346$	−139.899	+	430.564i	−0.404331	+	1.24440i
$347$	−503.075			−1.44978			−0.724892	−	0.688863i	$-0.758110\pi$
$347$	−503.075			−1.44978			−0.724892	+	0.688863i	$0.758110\pi$
$348$		0			0
$349$	0.504658			0.00144601			0.000723006	−	1.00000i	$-0.499770\pi$
$349$	0.504658			0.00144601			0.000723006		1.00000i	$0.499770\pi$
$350$		0			0
$351$		0			0
$352$	−637.414			−1.81084
$353$			335.994i			0.951824i	0.879493	+	0.475912i	$0.157882\pi$
$353$			335.994i			0.951824i	−0.879493	+	0.475912i	$0.842118\pi$
$354$		0			0
$355$		0			0
$356$	−360.636	−	262.018i	−1.01302	−	0.736004i
$357$		0			0
$358$	138.648	−	426.715i	0.387285	−	1.19194i
$359$		−	98.4859i		−	0.274334i	−0.990548	−	0.137167i	$-0.956200\pi$
$359$		−	98.4859i		−	0.274334i	0.990548	−	0.137167i	$-0.0437997\pi$
$360$		0			0
$361$	129.446			0.358576
$362$	164.006	+	53.2887i	0.453054	+	0.147206i
$363$		0			0
$364$	104.721	−	144.137i	0.287696	−	0.395980i
$365$		0			0
$366$		0			0
$367$	−498.473			−1.35824			−0.679118	−	0.734029i	$-0.737638\pi$
$367$	−498.473			−1.35824			−0.679118	+	0.734029i	$0.737638\pi$
$368$	2.74421	+	8.44582i	0.00745711	+	0.0229506i
$369$		0			0
$370$		0			0
$371$			348.879i			0.940374i
$372$		0			0
$373$			600.354i			1.60953i	0.593594	+	0.804765i	$0.297709\pi$
$373$			600.354i			1.60953i	−0.593594	+	0.804765i	$0.702291\pi$
$374$	−450.440	−	146.357i	−1.20438	−	0.391328i
$375$		0			0
$376$	251.056	+	345.549i	0.667701	+	0.919012i
$377$			92.7214i			0.245945i
$378$		0			0
$379$			303.490i			0.800765i	0.916348	+	0.400383i	$0.131123\pi$
$379$			303.490i			0.800765i	−0.916348	+	0.400383i	$0.868877\pi$
$380$		0			0
$381$		0			0
$382$	19.1713	−	59.0031i	0.0501866	−	0.154458i
$383$	332.583			0.868362			0.434181	−	0.900826i	$-0.357038\pi$
$383$	332.583			0.868362			0.434181	+	0.900826i	$0.357038\pi$
$384$		0			0
$385$		0			0
$386$	−68.1215	+	209.656i	−0.176481	+	0.543151i
$387$		0			0
$388$	−218.263	+	300.413i	−0.562534	+	0.774261i
$389$	392.354			1.00862			0.504312	−	0.863522i	$-0.331746\pi$
$389$	392.354			1.00862			0.504312	+	0.863522i	$0.331746\pi$
$390$		0			0
$391$			6.59849i			0.0168759i
$392$	−100.444	−	138.249i	−0.256235	−	0.352677i
$393$		0			0
$394$	106.626	−	328.162i	0.270625	−	0.832897i
$395$		0			0
$396$		0			0
$397$		−	334.190i		−	0.841789i	−0.907110	−	0.420895i	$-0.861716\pi$
$397$		−	334.190i		−	0.841789i	0.907110	−	0.420895i	$-0.138284\pi$
$398$	168.234	−	517.771i	0.422698	−	1.30093i
$399$		0			0
$400$		0			0
$401$	−121.003			−0.301753			−0.150877	−	0.988553i	$-0.548210\pi$
$401$	−121.003			−0.301753			−0.150877	+	0.988553i	$0.548210\pi$
$402$		0			0
$403$	70.2722			0.174373
$404$	−207.469	−	150.735i	−0.513537	−	0.373107i
$405$		0			0
$406$	−109.443	−	35.5601i	−0.269563	−	0.0875864i
$407$	−365.730			−0.898599
$408$		0			0
$409$	607.410			1.48511			0.742555	−	0.669785i	$-0.233614\pi$
$409$	607.410			1.48511			0.742555	+	0.669785i	$0.233614\pi$
$410$		0			0
$411$		0			0
$412$	445.831	+	323.915i	1.08211	+	0.786201i
$413$		−	91.6718i		−	0.221966i
$414$		0			0
$415$		0			0
$416$	271.108			0.651703
$417$		0			0
$418$	576.546	+	187.331i	1.37930	+	0.448161i
$419$			466.760i			1.11398i	0.830518	+	0.556992i	$0.188045\pi$
$419$			466.760i			1.11398i	−0.830518	+	0.556992i	$0.811955\pi$
$420$		0			0
$421$	−73.0883			−0.173606			−0.0868031	−	0.996225i	$-0.527665\pi$
$421$	−73.0883			−0.173606			−0.0868031	+	0.996225i	$0.527665\pi$
$422$	−126.862	+	390.440i	−0.300620	+	0.925212i
$423$		0			0
$424$	−429.495	+	312.047i	−1.01296	+	0.735959i
$425$		0			0
$426$		0			0
$427$	473.882			1.10979
$428$	−166.114	−	120.689i	−0.388117	−	0.281983i
$429$		0			0
$430$		0			0
$431$		−	463.630i		−	1.07571i	−0.843038	−	0.537853i	$-0.819235\pi$
$431$		−	463.630i		−	1.07571i	0.843038	−	0.537853i	$-0.180765\pi$
$432$		0			0
$433$		−	99.8359i		−	0.230568i	−0.993333	−	0.115284i	$-0.963222\pi$
$433$		−	99.8359i		−	0.230568i	0.993333	−	0.115284i	$-0.0367778\pi$
$434$	−26.9505	+	82.9451i	−0.0620979	+	0.191118i
$435$		0			0
$436$	−431.915	−	313.805i	−0.990630	−	0.719735i
$437$		−	8.44582i		−	0.0193268i
$438$		0			0
$439$			374.086i			0.852133i	0.904692	+	0.426066i	$0.140101\pi$
$439$			374.086i			0.852133i	−0.904692	+	0.426066i	$0.859899\pi$
$440$		0			0
$441$		0			0
$442$	191.583	+	62.2492i	0.433447	+	0.140835i
$443$	290.100			0.654854			0.327427	−	0.944877i	$-0.393818\pi$
$443$	290.100			0.654854			0.327427	+	0.944877i	$0.393818\pi$
$444$		0			0
$445$		0			0
$446$	448.387	+	145.690i	1.00535	+	0.326659i
$447$		0			0
$448$	−103.974	+	320.000i	−0.232085	+	0.714286i
$449$	299.921			0.667976			0.333988	−	0.942577i	$-0.391606\pi$
$449$	299.921			0.667976			0.333988	+	0.942577i	$0.391606\pi$
$450$		0			0
$451$			290.493i			0.644109i
$452$	401.390	−	552.466i	0.888031	−	1.22227i
$453$		0			0
$454$	−111.305	−	36.1652i	−0.245165	−	0.0796590i
$455$		0			0
$456$		0			0
$457$		−	822.328i		−	1.79941i	−0.436503	−	0.899703i	$-0.643783\pi$
$457$		−	822.328i		−	1.79941i	0.436503	−	0.899703i	$-0.356217\pi$
$458$	−309.514	−	100.567i	−0.675796	−	0.219579i
$459$		0			0
$460$		0			0
$461$	456.885			0.991075			0.495537	−	0.868587i	$-0.334971\pi$
$461$	456.885			0.991075			0.495537	+	0.868587i	$0.334971\pi$
$462$		0			0
$463$	400.249			0.864469			0.432234	−	0.901761i	$-0.357725\pi$
$463$	400.249			0.864469			0.432234	+	0.901761i	$0.357725\pi$
$464$	−54.1115	−	166.538i	−0.116620	−	0.358918i
$465$		0			0
$466$	197.183	−	606.868i	0.423140	−	1.30229i
$467$	913.145			1.95534			0.977672	−	0.210139i	$-0.0673915\pi$
$467$	913.145			1.95534			0.977672	+	0.210139i	$0.0673915\pi$
$468$		0			0
$469$	−264.033			−0.562969
$470$		0			0
$471$		0			0
$472$	112.855	−	81.9938i	0.239099	−	0.173716i
$473$			443.607i			0.937858i
$474$		0			0
$475$		0			0
$476$	−146.950	+	202.260i	−0.308720	+	0.424916i
$477$		0			0
$478$	146.357	−	450.440i	0.306186	−	0.942342i
$479$			526.131i			1.09840i	0.835692	+	0.549198i	$0.185067\pi$
$479$			526.131i			1.09840i	−0.835692	+	0.549198i	$0.814933\pi$
$480$		0			0
$481$	155.554			0.323397
$482$	−1.74606	−	0.567331i	−0.00362254	−	0.00117704i
$483$		0			0
$484$	−892.423	−	648.384i	−1.84385	−	1.33964i
$485$		0			0
$486$		0			0
$487$	−443.541			−0.910762			−0.455381	−	0.890297i	$-0.650497\pi$
$487$	−443.541			−0.910762			−0.455381	+	0.890297i	$0.650497\pi$
$488$	423.853	+	583.384i	0.868552	+	1.19546i
$489$		0			0
$490$		0			0
$491$		−	287.163i		−	0.584854i	−0.956288	−	0.292427i	$-0.905537\pi$
$491$		−	287.163i		−	0.584854i	0.956288	−	0.292427i	$-0.0944628\pi$
$492$		0			0
$493$		−	130.111i		−	0.263918i
$494$	−245.220	−	79.6767i	−0.496396	−	0.161289i
$495$		0			0
$496$	−126.217	+	41.0103i	−0.254469	+	0.0826820i
$497$		−	424.721i		−	0.854570i
$498$		0			0
$499$			810.936i			1.62512i	0.582876	+	0.812561i	$0.301927\pi$
$499$			810.936i			1.62512i	−0.582876	+	0.812561i	$0.698073\pi$
$500$		0			0
$501$		0			0
$502$	84.4791	−	260.000i	0.168285	−	0.517928i
$503$	−642.471			−1.27728			−0.638639	−	0.769506i	$-0.720502\pi$
$503$	−642.471			−1.27728			−0.638639	+	0.769506i	$0.720502\pi$
$504$		0			0
$505$		0			0
$506$	−6.83282	+	21.0292i	−0.0135036	+	0.0415598i
$507$		0			0
$508$	642.802	+	467.023i	1.26536	+	0.919337i
$509$	−915.050			−1.79774			−0.898870	−	0.438216i	$-0.855611\pi$
$509$	−915.050			−1.79774			−0.898870	+	0.438216i	$0.855611\pi$
$510$		0			0
$511$		−	29.2000i		−	0.0571429i
$512$	−486.941	+	158.217i	−0.951057	+	0.309017i
$513$		0			0
$514$	169.925	−	522.975i	0.330593	−	1.01746i
$515$		0			0
$516$		0			0
$517$			1063.49i			2.05704i
$518$	−59.6575	+	183.607i	−0.115169	+	0.354453i
$519$		0			0
$520$		0			0
$521$	−1006.98			−1.93279			−0.966396	−	0.257058i	$-0.917247\pi$
$521$	−1006.98			−1.93279			−0.966396	+	0.257058i	$0.917247\pi$
$522$		0			0
$523$	−774.173			−1.48025			−0.740127	−	0.672467i	$-0.765235\pi$
$523$	−774.173			−1.48025			−0.740127	+	0.672467i	$0.765235\pi$
$524$	−18.2693	+	25.1456i	−0.0348651	+	0.0479877i
$525$		0			0
$526$	−772.991	−	251.160i	−1.46956	−	0.477490i
$527$	−98.6096			−0.187115
$528$		0			0
$529$	−528.692			−0.999418
$530$		0			0
$531$		0			0
$532$	188.091	−	258.885i	0.353555	−	0.486627i
$533$		−	123.554i		−	0.231809i
$534$		0			0
$535$		0			0
$536$	−236.158	−	325.043i	−0.440593	−	0.606424i
$537$		0			0
$538$	−662.933	−	215.400i	−1.23222	−	0.400372i
$539$		−	425.487i		−	0.789401i
$540$		0			0
$541$	−259.115			−0.478955			−0.239477	−	0.970902i	$-0.576976\pi$
$541$	−259.115			−0.478955			−0.239477	+	0.970902i	$0.576976\pi$
$542$	−153.179	+	471.437i	−0.282618	+	0.869809i
$543$		0			0
$544$	−380.433			−0.699326
$545$		0			0
$546$		0			0
$547$	149.818			0.273890			0.136945	−	0.990579i	$-0.456272\pi$
$547$	149.818			0.273890			0.136945	+	0.990579i	$0.456272\pi$
$548$	−1.95807	+	2.69505i	−0.00357312	+	0.00491797i
$549$		0			0
$550$		0			0
$551$			166.538i			0.302247i
$552$		0			0
$553$		−	72.7864i		−	0.131621i
$554$	−33.8398	+	104.148i	−0.0610826	+	0.187993i
$555$		0			0
$556$	558.768	−	769.078i	1.00498	−	1.38323i
$557$			511.698i			0.918668i	0.888264	+	0.459334i	$0.151912\pi$
$557$			511.698i			0.918668i	−0.888264	+	0.459334i	$0.848088\pi$
$558$		0			0
$559$		−	188.677i		−	0.337526i
$560$		0			0
$561$		0			0
$562$	95.7917	+	31.1246i	0.170448	+	0.0553819i
$563$	−490.726			−0.871627			−0.435814	−	0.900037i	$-0.643539\pi$
$563$	−490.726			−0.871627			−0.435814	+	0.900037i	$0.643539\pi$
$564$		0			0
$565$		0			0
$566$	−280.249	−	91.0585i	−0.495140	−	0.160881i
$567$		0			0
$568$	522.863	−	379.882i	0.920534	−	0.668807i
$569$	−232.748			−0.409047			−0.204523	−	0.978862i	$-0.565564\pi$
$569$	−232.748			−0.409047			−0.204523	+	0.978862i	$0.565564\pi$
$570$		0			0
$571$			210.755i			0.369098i	0.982823	+	0.184549i	$0.0590824\pi$
$571$			210.755i			0.369098i	−0.982823	+	0.184549i	$0.940918\pi$
$572$	546.113	+	396.774i	0.954742	+	0.693661i
$573$		0			0
$574$	145.836	+	47.3850i	0.254070	+	0.0825522i
$575$		0			0
$576$		0			0
$577$		−	341.712i		−	0.592222i	−0.955154	−	0.296111i	$-0.904310\pi$
$577$		−	341.712i		−	0.592222i	0.955154	−	0.296111i	$-0.0956898\pi$
$578$	280.871	+	91.2605i	0.485936	+	0.157890i
$579$		0			0
$580$		0			0
$581$	400.689			0.689654
$582$		0			0
$583$	−1321.85			−2.26733
$584$	35.9474	−	26.1173i	0.0615537	−	0.0447214i
$585$		0			0
$586$	110.541	−	340.210i	0.188637	−	0.580564i
$587$	618.412			1.05351			0.526756	−	0.850016i	$-0.323408\pi$
$587$	618.412			1.05351			0.526756	+	0.850016i	$0.323408\pi$
$588$		0			0
$589$	126.217			0.214290
$590$		0			0
$591$		0			0
$592$	−279.393	+	90.7802i	−0.471947	+	0.153345i
$593$		−	120.663i		−	0.203478i	−0.994811	−	0.101739i	$-0.967559\pi$
$593$		−	120.663i		−	0.203478i	0.994811	−	0.101739i	$-0.0324407\pi$
$594$		0			0
$595$		0			0
$596$	−119.639	−	86.9231i	−0.200737	−	0.145844i
$597$		0			0
$598$	2.90617	−	8.94427i	0.00485982	−	0.0149570i
$599$			849.927i			1.41891i	0.704751	+	0.709455i	$0.251059\pi$
$599$			849.927i			1.41891i	−0.704751	+	0.709455i	$0.748941\pi$
$600$		0			0
$601$	−11.3576			−0.0188978			−0.00944890	−	0.999955i	$-0.503008\pi$
$601$	−11.3576			−0.0188978			−0.00944890	+	0.999955i	$0.503008\pi$
$602$	222.703	+	72.3607i	0.369939	+	0.120200i
$603$		0			0
$604$	664.721	−	914.910i	1.10053	−	1.51475i
$605$		0			0
$606$		0			0
$607$	1115.12			1.83710			0.918550	−	0.395305i	$-0.129361\pi$
$607$	1115.12			1.83710			0.918550	+	0.395305i	$0.129361\pi$
$608$	486.941			0.800890
$609$		0			0
$610$		0			0
$611$		−	452.329i		−	0.740309i
$612$		0			0
$613$			499.475i			0.814805i	0.913249	+	0.407402i	$0.133565\pi$
$613$			499.475i			0.814805i	−0.913249	+	0.407402i	$0.866435\pi$
$614$	541.246	+	175.862i	0.881508	+	0.286419i
$615$		0			0
$616$	−677.771	+	492.429i	−1.10028	+	0.799398i
$617$			545.935i			0.884822i	0.896813	+	0.442411i	$0.145877\pi$
$617$			545.935i			0.884822i	−0.896813	+	0.442411i	$0.854123\pi$
$618$		0			0
$619$			455.011i			0.735075i	0.930009	+	0.367537i	$0.119799\pi$
$619$			455.011i			0.735075i	−0.930009	+	0.367537i	$0.880201\pi$
$620$		0			0
$621$		0			0
$622$	−174.408	+	536.774i	−0.280399	+	0.862981i
$623$	−585.889			−0.940432
$624$		0			0
$625$		0			0
$626$	350.764	−	1079.54i	0.560326	−	1.72451i
$627$		0			0
$628$	481.391	−	662.577i	0.766546	−	1.05506i
$629$	−218.282			−0.347030
$630$		0			0
$631$		−	267.706i		−	0.424257i	−0.977242	−	0.212128i	$-0.931960\pi$
$631$		−	267.706i		−	0.424257i	0.977242	−	0.212128i	$-0.0680395\pi$
$632$	89.6054	−	65.1021i	0.141781	−	0.103010i
$633$		0			0
$634$	−99.5905	+	306.508i	−0.157083	+	0.483451i
$635$		0			0
$636$		0			0
$637$			180.971i			0.284098i
$638$	134.732	−	414.663i	0.211179	−	0.649941i
$639$		0			0
$640$		0			0
$641$	418.571			0.652997			0.326499	−	0.945198i	$-0.394131\pi$
$641$	418.571			0.652997			0.326499	+	0.945198i	$0.394131\pi$
$642$		0			0
$643$	439.339			0.683265			0.341633	−	0.939834i	$-0.389020\pi$
$643$	439.339			0.683265			0.341633	+	0.939834i	$0.389020\pi$
$644$	9.44272	+	6.86054i	0.0146626	+	0.0106530i
$645$		0			0
$646$	344.105	+	111.807i	0.532671	+	0.173075i
$647$	419.644			0.648600			0.324300	−	0.945954i	$-0.394871\pi$
$647$	419.644			0.648600			0.324300	+	0.945954i	$0.394871\pi$
$648$		0			0
$649$	347.331			0.535179
$650$		0			0
$651$		0			0
$652$	347.021	+	252.125i	0.532240	+	0.386695i
$653$			370.085i			0.566746i	0.959010	+	0.283373i	$0.0914534\pi$
$653$			370.085i			0.566746i	−0.959010	+	0.283373i	$0.908547\pi$
$654$		0			0
$655$		0			0
$656$	72.1052	+	221.917i	0.109917	+	0.338288i
$657$		0			0
$658$	533.902	+	173.475i	0.811401	+	0.263640i
$659$			322.823i			0.489868i	0.969540	+	0.244934i	$0.0787664\pi$
$659$			322.823i			0.489868i	−0.969540	+	0.244934i	$0.921234\pi$
$660$		0			0
$661$	−812.735			−1.22955			−0.614777	−	0.788701i	$-0.710754\pi$
$661$	−812.735			−1.22955			−0.614777	+	0.788701i	$0.710754\pi$
$662$	205.166	−	631.437i	0.309919	−	0.953832i
$663$		0			0
$664$	358.387	+	493.277i	0.539739	+	0.742888i
$665$		0			0
$666$		0			0
$667$	−6.07439			−0.00910703
$668$	107.466	+	78.0789i	0.160878	+	0.116885i
$669$		0			0
$670$		0			0
$671$			1795.47i			2.67581i
$672$		0			0
$673$			467.378i			0.694469i	0.937778	+	0.347235i	$0.112879\pi$
$673$			467.378i			0.694469i	−0.937778	+	0.347235i	$0.887121\pi$
$674$	−166.318	+	511.875i	−0.246763	+	0.759458i
$675$		0			0
$676$	314.620	+	228.585i	0.465414	+	0.338143i
$677$			548.237i			0.809803i	0.914360	+	0.404902i	$0.132694\pi$
$677$			548.237i			0.809803i	−0.914360	+	0.404902i	$0.867306\pi$
$678$		0			0
$679$			488.051i			0.718779i
$680$		0			0
$681$		0			0
$682$	−314.267	−	102.111i	−0.460802	−	0.149724i
$683$	−23.9663			−0.0350898			−0.0175449	−	0.999846i	$-0.505585\pi$
$683$	−23.9663			−0.0350898			−0.0175449	+	0.999846i	$0.505585\pi$
$684$		0			0
$685$		0			0
$686$	−703.607	−	228.616i	−1.02567	−	0.333259i
$687$		0			0
$688$	110.111	+	338.885i	0.160044	+	0.492566i
$689$	562.217			0.815989
$690$		0			0
$691$			186.981i			0.270595i	0.990805	+	0.135298i	$0.0431990\pi$
$691$			186.981i			0.270595i	−0.990805	+	0.135298i	$0.956801\pi$
$692$	−532.206	+	732.519i	−0.769084	+	1.05855i
$693$		0			0
$694$	−956.906	−	310.917i	−1.37883	−	0.448008i
$695$		0			0
$696$		0			0
$697$			173.378i			0.248748i
$698$	0.959917	+	0.311896i	0.00137524	+	0.000446842i
$699$		0			0
$700$		0			0
$701$	706.636			1.00804			0.504020	−	0.863692i	$-0.331854\pi$
$701$	706.636			1.00804			0.504020	+	0.863692i	$0.331854\pi$
$702$		0			0
$703$	279.393			0.397429
$704$	−1212.43	−	393.943i	−1.72221	−	0.559579i
$705$		0			0
$706$	−207.656	+	639.098i	−0.294130	+	0.905238i
$707$	−337.054			−0.476738
$708$		0			0
$709$	188.597			0.266005			0.133002	−	0.991116i	$-0.457538\pi$
$709$	188.597			0.266005			0.133002	+	0.991116i	$0.457538\pi$
$710$		0			0
$711$		0			0
$712$	−524.035	−	721.272i	−0.736004	−	1.01302i
$713$			4.60369i			0.00645679i
$714$		0			0
$715$		0			0
$716$	527.449	−	725.971i	0.736661	−	1.01393i
$717$		0			0
$718$	60.8676	−	187.331i	0.0847738	−	0.260907i
$719$		−	156.085i		−	0.217086i	−0.994092	−	0.108543i	$-0.965381\pi$
$719$		−	156.085i		−	0.217086i	0.994092	−	0.108543i	$-0.0346186\pi$
$720$		0			0
$721$	724.296			1.00457
$722$	246.221	+	80.0019i	0.341026	+	0.110806i
$723$		0			0
$724$	279.023	+	202.722i	0.385391	+	0.280003i
$725$		0			0
$726$		0			0
$727$	715.164			0.983719			0.491859	−	0.870675i	$-0.336317\pi$
$727$	715.164			0.983719			0.491859	+	0.870675i	$0.336317\pi$
$728$	288.273	−	209.443i	0.395980	−	0.287696i
$729$		0			0
$730$		0			0
$731$			264.762i			0.362191i
$732$		0			0
$733$			1233.29i			1.68252i	0.540632	+	0.841259i	$0.318185\pi$
$733$			1233.29i			1.68252i	−0.540632	+	0.841259i	$0.681815\pi$
$734$	−948.152	−	308.073i	−1.29176	−	0.419718i
$735$		0			0
$736$			17.7609i			0.0241317i
$737$		−	1000.38i		−	1.35737i
$738$		0			0
$739$			8.55656i			0.0115786i	0.999983	+	0.00578928i	$0.00184280\pi$
$739$			8.55656i			0.0115786i	−0.999983	+	0.00578928i	$0.998157\pi$
$740$		0			0
$741$		0			0
$742$	−215.619	+	663.607i	−0.290592	+	0.894349i
$743$	1010.56			1.36011			0.680053	−	0.733163i	$-0.261957\pi$
$743$	1010.56			1.36011			0.680053	+	0.733163i	$0.261957\pi$
$744$		0			0
$745$		0			0
$746$	−371.039	+	1141.94i	−0.497372	+	1.53075i
$747$		0			0
$748$	−766.334	−	556.774i	−1.02451	−	0.744350i
$749$	−269.868			−0.360305
$750$		0			0
$751$			1104.31i			1.47046i	0.677820	+	0.735228i	$0.262925\pi$
$751$			1104.31i			1.47046i	−0.677820	+	0.735228i	$0.737075\pi$
$752$	263.976	+	812.433i	0.351031	+	1.08036i
$753$		0			0
$754$	−57.3050	+	176.367i	−0.0760013	+	0.233908i
$755$		0			0
$756$		0			0
$757$			875.633i			1.15671i	0.815783	+	0.578357i	$0.196306\pi$
$757$			875.633i			1.15671i	−0.815783	+	0.578357i	$0.803694\pi$
$758$	−187.567	+	577.272i	−0.247450	+	0.761573i
$759$		0			0
$760$		0			0
$761$	647.207			0.850470			0.425235	−	0.905083i	$-0.360192\pi$
$761$	647.207			0.850470			0.425235	+	0.905083i	$0.360192\pi$
$762$		0			0
$763$	−701.688			−0.919644
$764$	72.9318	−	100.382i	0.0954605	−	0.131390i
$765$		0			0
$766$	632.610	+	205.547i	0.825861	+	0.268339i
$767$	−147.729			−0.192606
$768$		0			0
$769$	−631.430			−0.821106			−0.410553	−	0.911837i	$-0.634664\pi$
$769$	−631.430			−0.821106			−0.410553	+	0.911837i	$0.634664\pi$
$770$		0			0
$771$		0			0
$772$	−259.150	+	356.689i	−0.335686	+	0.462032i
$773$			421.522i			0.545306i	0.962112	+	0.272653i	$0.0879011\pi$
$773$			421.522i			0.545306i	−0.962112	+	0.272653i	$0.912099\pi$
$774$		0			0
$775$		0			0
$776$	−600.827	+	436.526i	−0.774261	+	0.562534i
$777$		0			0
$778$	746.303	+	242.488i	0.959258	+	0.311682i
$779$		−	221.917i		−	0.284874i
$780$		0			0
$781$	1609.21			2.06044
$782$	−4.07809	+	12.5511i	−0.00521495	+	0.0160500i
$783$		0			0
$784$	−105.613	−	325.043i	−0.134710	−	0.414596i
$785$		0			0
$786$		0			0
$787$	−838.633			−1.06561			−0.532804	−	0.846239i	$-0.678862\pi$
$787$	−838.633			−1.06561			−0.532804	+	0.846239i	$0.678862\pi$
$788$	405.630	−	558.302i	0.514759	−	0.708505i
$789$		0			0
$790$		0			0
$791$		−	897.535i		−	1.13468i
$792$		0			0
$793$		−	763.659i		−	0.963001i
$794$	206.541	−	635.668i	0.260127	−	0.800589i
$795$		0			0
$796$	640.000	−	880.884i	0.804020	−	1.10664i
$797$			1213.57i			1.52268i	0.648354	+	0.761339i	$0.275458\pi$
$797$			1213.57i			1.52268i	−0.648354	+	0.761339i	$0.724542\pi$
$798$		0			0
$799$			634.732i			0.794408i
$800$		0			0
$801$		0			0
$802$	−230.162	−	74.7840i	−0.286985	−	0.0932469i
$803$	110.635			0.137777
$804$		0			0
$805$		0			0
$806$	133.666	+	43.4306i	0.165838	+	0.0538841i
$807$		0			0
$808$	−301.470	−	414.938i	−0.373107	−	0.513537i
$809$	−229.214			−0.283330			−0.141665	−	0.989915i	$-0.545246\pi$
$809$	−229.214			−0.283330			−0.141665	+	0.989915i	$0.545246\pi$
$810$		0			0
$811$		−	454.225i		−	0.560080i	−0.959988	−	0.280040i	$-0.909652\pi$
$811$		−	454.225i		−	0.560080i	0.959988	−	0.280040i	$-0.0903478\pi$
$812$	−186.195	−	135.279i	−0.229304	−	0.166599i
$813$		0			0
$814$	−695.659	−	226.033i	−0.854618	−	0.277682i
$815$		0			0
$816$		0			0
$817$		−	338.885i		−	0.414792i
$818$	1155.36	+	375.400i	1.41242	+	0.458924i
$819$		0			0
$820$		0			0
$821$	−1130.90			−1.37747			−0.688733	−	0.725015i	$-0.741833\pi$
$821$	−1130.90			−1.37747			−0.688733	+	0.725015i	$0.741833\pi$
$822$		0			0
$823$	780.148			0.947931			0.473966	−	0.880543i	$-0.342822\pi$
$823$	780.148			0.947931			0.473966	+	0.880543i	$0.342822\pi$
$824$	647.830	+	891.661i	0.786201	+	1.08211i
$825$		0			0
$826$	56.6563	−	174.370i	0.0685912	−	0.211102i
$827$	209.175			0.252932			0.126466	−	0.991971i	$-0.459636\pi$
$827$	209.175			0.252932			0.126466	+	0.991971i	$0.459636\pi$
$828$		0			0
$829$	508.525			0.613419			0.306710	−	0.951803i	$-0.400772\pi$
$829$	508.525			0.613419			0.306710	+	0.951803i	$0.400772\pi$
$830$		0			0
$831$		0			0
$832$	515.679	+	167.554i	0.619806	+	0.201387i
$833$		−	253.947i		−	0.304859i
$834$		0			0
$835$		0			0
$836$	980.879	+	712.650i	1.17330	+	0.852453i
$837$		0			0
$838$	−288.473	+	887.830i	−0.344240	+	1.05946i
$839$			274.028i			0.326613i	0.986575	+	0.163306i	$0.0522159\pi$
$839$			274.028i			0.326613i	−0.986575	+	0.163306i	$0.947784\pi$
$840$		0			0
$841$	−721.223			−0.857578
$842$	−139.022	−	45.1710i	−0.165109	−	0.0536473i
$843$		0			0
$844$	−482.610	+	664.256i	−0.571813	+	0.787033i
$845$		0			0
$846$		0			0
$847$	−1449.83			−1.71172
$848$	−1009.80	+	328.105i	−1.19081	+	0.386917i
$849$		0			0
$850$		0			0
$851$			10.1907i			0.0119750i
$852$		0			0
$853$		−	1583.28i		−	1.85613i	−0.372416	−	0.928066i	$-0.621471\pi$
$853$		−	1583.28i		−	1.85613i	0.372416	−	0.928066i	$-0.378529\pi$
$854$	901.378	+	292.875i	1.05548	+	0.342945i
$855$		0			0
$856$	−241.378	−	332.228i	−0.281983	−	0.388117i
$857$		−	1007.38i		−	1.17547i	−0.809054	−	0.587735i	$-0.800020\pi$
$857$		−	1007.38i		−	1.17547i	0.809054	−	0.587735i	$-0.199980\pi$
$858$		0			0
$859$		−	76.6086i		−	0.0891835i	−0.999005	−	0.0445917i	$-0.985801\pi$
$859$		−	76.6086i		−	0.0891835i	0.999005	−	0.0445917i	$-0.0141987\pi$
$860$		0			0
$861$		0			0
$862$	286.539	−	881.876i	0.332412	−	1.02306i
$863$	255.450			0.296002			0.148001	−	0.988987i	$-0.452716\pi$
$863$	255.450			0.296002			0.148001	+	0.988987i	$0.452716\pi$
$864$		0			0
$865$		0			0
$866$	61.7020	−	189.899i	0.0712494	−	0.219283i
$867$		0			0
$868$	−102.526	+	141.115i	−0.118117	+	0.162574i
$869$	275.777			0.317350
$870$		0			0
$871$			425.487i			0.488504i
$872$	−627.609	−	863.830i	−0.719735	−	0.990630i
$873$		0			0
$874$	5.21981	−	16.0649i	0.00597232	−	0.0183809i
$875$		0			0
$876$		0			0
$877$		−	601.522i		−	0.685886i	−0.939356	−	0.342943i	$-0.888576\pi$
$877$		−	601.522i		−	0.685886i	0.939356	−	0.342943i	$-0.111424\pi$
$878$	−231.198	+	711.554i	−0.263323	+	0.810426i
$879$		0			0
$880$		0			0
$881$	−237.850			−0.269977			−0.134989	−	0.990847i	$-0.543100\pi$
$881$	−237.850			−0.269977			−0.134989	+	0.990847i	$0.543100\pi$
$882$		0			0
$883$	−1.30294			−0.00147559			−0.000737794	−	1.00000i	$-0.500235\pi$
$883$	−1.30294			−0.00147559			−0.000737794		1.00000i	$0.500235\pi$
$884$	325.941	+	236.810i	0.368712	+	0.267885i
$885$		0			0
$886$	551.803	+	179.292i	0.622803	+	0.202361i
$887$	−536.353			−0.604682			−0.302341	−	0.953200i	$-0.597768\pi$
$887$	−536.353			−0.604682			−0.302341	+	0.953200i	$0.597768\pi$
$888$		0			0
$889$	1044.30			1.17469
$890$		0			0
$891$		0			0
$892$	762.842	+	554.237i	0.855203	+	0.621342i
$893$		−	812.433i		−	0.909780i
$894$		0			0
$895$		0			0
$896$	−395.542	+	544.417i	−0.441453	+	0.607608i
$897$		0			0
$898$	570.484	+	185.361i	0.635283	+	0.206416i
$899$		−	90.7773i		−	0.100976i
$900$		0			0
$901$	−788.932			−0.875618
$902$	−179.535	+	552.551i	−0.199041	+	0.612584i
$903$		0			0
$904$	1104.93	−	802.780i	1.22227	−	0.888031i
$905$		0			0
$906$		0			0
$907$	−332.159			−0.366217			−0.183108	−	0.983093i	$-0.558616\pi$
$907$	−332.159			−0.366217			−0.183108	+	0.983093i	$0.558616\pi$
$908$	−189.363	−	137.580i	−0.208550	−	0.151520i
$909$		0			0
$910$		0			0
$911$		−	1450.06i		−	1.59172i	−0.605478	−	0.795862i	$-0.707018\pi$
$911$		−	1450.06i		−	1.59172i	0.605478	−	0.795862i	$-0.292982\pi$
$912$		0			0
$913$			1518.15i			1.66282i
$914$	508.227	−	1564.16i	0.556047	−	1.71134i
$915$		0			0
$916$	−526.577	−	382.581i	−0.574866	−	0.417665i
$917$			40.8514i			0.0445490i
$918$		0			0
$919$			814.405i			0.886186i	0.896476	+	0.443093i	$0.146119\pi$
$919$			814.405i			0.886186i	−0.896476	+	0.443093i	$0.853881\pi$
$920$		0			0
$921$		0			0
$922$	869.048	+	282.371i	0.942568	+	0.306259i
$923$	−684.437			−0.741535
$924$		0			0
$925$		0			0
$926$	761.319	+	247.367i	0.822159	+	0.267136i
$927$		0			0
$928$		−	350.217i		−	0.377389i
$929$	400.039			0.430612			0.215306	−	0.976547i	$-0.430925\pi$
$929$	400.039			0.430612			0.215306	+	0.976547i	$0.430925\pi$
$930$		0			0
$931$			325.043i			0.349134i
$932$	750.130	−	1032.47i	0.804861	−	1.10780i
$933$		0			0
$934$	1736.91	+	564.355i	1.85964	+	0.604234i
$935$		0			0
$936$		0			0
$937$		−	249.279i		−	0.266039i	−0.991113	−	0.133020i	$-0.957533\pi$
$937$		−	249.279i		−	0.266039i	0.991113	−	0.133020i	$-0.0424673\pi$
$938$	−502.220	−	163.181i	−0.535415	−	0.173967i
$939$		0			0
$940$		0			0
$941$	−724.229			−0.769638			−0.384819	−	0.922992i	$-0.625736\pi$
$941$	−724.229			−0.769638			−0.384819	+	0.922992i	$0.625736\pi$
$942$		0			0
$943$	8.09432			0.00858358
$944$	265.337	−	86.2134i	0.281078	−	0.0913277i
$945$		0			0
$946$	−274.164	+	843.790i	−0.289814	+	0.891956i
$947$	−1141.54			−1.20542			−0.602712	−	0.797959i	$-0.705913\pi$
$947$	−1141.54			−1.20542			−0.602712	+	0.797959i	$0.705913\pi$
$948$		0			0
$949$	−47.0557			−0.0495845
$950$		0			0
$951$		0			0
$952$	−404.520	+	293.901i	−0.424916	+	0.308720i
$953$		−	1295.33i		−	1.35921i	−0.733579	−	0.679604i	$-0.762152\pi$
$953$		−	1295.33i		−	1.35921i	0.733579	−	0.679604i	$-0.237848\pi$
$954$		0			0
$955$		0			0
$956$	556.774	−	766.334i	0.582400	−	0.801604i
$957$		0			0
$958$	−325.167	+	1000.76i	−0.339423	+	1.04464i
$959$			4.37837i			0.00456556i
$960$		0			0
$961$	892.201			0.928409
$962$	295.882	+	96.1378i	0.307569	+	0.0999353i
$963$		0			0
$964$	−2.97058	−	2.15825i	−0.00308152	−	0.00223885i
$965$		0			0
$966$		0			0
$967$	398.477			0.412075			0.206037	−	0.978544i	$-0.433943\pi$
$967$	398.477			0.412075			0.206037	+	0.978544i	$0.433943\pi$
$968$	−1296.77	−	1784.85i	−1.33964	−	1.84385i
$969$		0			0
$970$		0			0
$971$		−	928.093i		−	0.955811i	−0.878411	−	0.477906i	$-0.841396\pi$
$971$		−	928.093i		−	0.955811i	0.878411	−	0.477906i	$-0.158604\pi$
$972$		0			0
$973$		−	1249.44i		−	1.28411i
$974$	−843.666	−	274.124i	−0.866186	−	0.281441i
$975$		0			0
$976$	445.666	+	1371.62i	0.456625	+	1.40535i
$977$		−	1378.05i		−	1.41049i	−0.708962	−	0.705247i	$-0.750836\pi$
$977$		−	1378.05i		−	1.41049i	0.708962	−	0.705247i	$-0.249164\pi$
$978$		0			0
$979$		−	2219.85i		−	2.26747i
$980$		0			0
$981$		0			0
$982$	177.477	−	546.217i	0.180730	−	0.556229i
$983$	−311.291			−0.316675			−0.158337	−	0.987385i	$-0.550613\pi$
$983$	−311.291			−0.316675			−0.158337	+	0.987385i	$0.550613\pi$
$984$		0			0
$985$		0			0
$986$	80.4133	−	247.487i	0.0815551	−	0.251001i
$987$		0			0
$988$	−417.193	−	303.108i	−0.422260	−	0.306790i
$989$	12.3607			0.0124982
$990$		0			0
$991$		−	961.147i		−	0.969876i	−0.874549	−	0.484938i	$-0.838842\pi$
$991$		−	961.147i		−	0.969876i	0.874549	−	0.484938i	$-0.161158\pi$
$992$	−265.424			−0.267565
$993$		0			0
$994$	262.492	−	807.868i	0.264077	−	0.812745i
$995$		0			0
$996$		0			0
$997$			1089.68i			1.09296i	0.837473	+	0.546479i	$0.184032\pi$
$997$			1089.68i			1.09296i	−0.837473	+	0.546479i	$0.815968\pi$
$998$	−501.186	+	1542.49i	−0.502190	+	1.54558i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.f.e.199.8		8
3.2	odd	2		100.3.d.b.99.1		8
4.3	odd	2	inner	900.3.f.e.199.2		8
5.2	odd	4		180.3.c.a.91.2		4
5.3	odd	4		900.3.c.k.451.3		4
5.4	even	2	inner	900.3.f.e.199.1		8
12.11	even	2		100.3.d.b.99.7		8
15.2	even	4		20.3.b.a.11.3	✓	4
15.8	even	4		100.3.b.f.51.2		4
15.14	odd	2		100.3.d.b.99.8		8
20.3	even	4		900.3.c.k.451.4		4
20.7	even	4		180.3.c.a.91.1		4
20.19	odd	2	inner	900.3.f.e.199.7		8
24.5	odd	2		1600.3.h.n.1599.6		8
24.11	even	2		1600.3.h.n.1599.3		8
40.27	even	4		2880.3.e.e.2431.1		4
40.37	odd	4		2880.3.e.e.2431.2		4
60.23	odd	4		100.3.b.f.51.1		4
60.47	odd	4		20.3.b.a.11.4	yes	4
60.59	even	2		100.3.d.b.99.2		8
120.29	odd	2		1600.3.h.n.1599.4		8
120.53	even	4		1600.3.b.s.1151.3		4
120.59	even	2		1600.3.h.n.1599.5		8
120.77	even	4		320.3.b.c.191.2		4
120.83	odd	4		1600.3.b.s.1151.2		4
120.107	odd	4		320.3.b.c.191.3		4
240.77	even	4		1280.3.g.e.1151.4		8
240.107	odd	4		1280.3.g.e.1151.3		8
240.197	even	4		1280.3.g.e.1151.5		8
240.227	odd	4		1280.3.g.e.1151.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.3	✓	4	15.2	even	4
20.3.b.a.11.4	yes	4	60.47	odd	4
100.3.b.f.51.1		4	60.23	odd	4
100.3.b.f.51.2		4	15.8	even	4
100.3.d.b.99.1		8	3.2	odd	2
100.3.d.b.99.2		8	60.59	even	2
100.3.d.b.99.7		8	12.11	even	2
100.3.d.b.99.8		8	15.14	odd	2
180.3.c.a.91.1		4	20.7	even	4
180.3.c.a.91.2		4	5.2	odd	4
320.3.b.c.191.2		4	120.77	even	4
320.3.b.c.191.3		4	120.107	odd	4
900.3.c.k.451.3		4	5.3	odd	4
900.3.c.k.451.4		4	20.3	even	4
900.3.f.e.199.1		8	5.4	even	2	inner
900.3.f.e.199.2		8	4.3	odd	2	inner
900.3.f.e.199.7		8	20.19	odd	2	inner
900.3.f.e.199.8		8	1.1	even	1	trivial
1280.3.g.e.1151.3		8	240.107	odd	4
1280.3.g.e.1151.4		8	240.77	even	4
1280.3.g.e.1151.5		8	240.197	even	4
1280.3.g.e.1151.6		8	240.227	odd	4
1600.3.b.s.1151.2		4	120.83	odd	4
1600.3.b.s.1151.3		4	120.53	even	4
1600.3.h.n.1599.3		8	24.11	even	2
1600.3.h.n.1599.4		8	120.29	odd	2
1600.3.h.n.1599.5		8	120.59	even	2
1600.3.h.n.1599.6		8	24.5	odd	2
2880.3.e.e.2431.1		4	40.27	even	4
2880.3.e.e.2431.2		4	40.37	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.90211 + 0.618034i) q^{2} +(3.23607 + 2.35114i) q^{4} +5.25731 q^{7} +(4.70228 + 6.47214i) q^{8} +O(q^{10})\)	\(q+(1.90211 + 0.618034i) q^{2} +(3.23607 + 2.35114i) q^{4} +5.25731 q^{7} +(4.70228 + 6.47214i) q^{8} +19.9192i q^{11} -8.47214i q^{13} +(10.0000 + 3.24920i) q^{14} +(4.94427 + 15.2169i) q^{16} +11.8885i q^{17} -15.2169i q^{19} +(-12.3107 + 37.8885i) q^{22} +0.555029 q^{23} +(5.23607 - 16.1150i) q^{26} +(17.0130 + 12.3607i) q^{28} -10.9443 q^{29} +8.29451i q^{31} +32.0000i q^{32} +(-7.34752 + 22.6134i) q^{34} +18.3607i q^{37} +(9.40456 - 28.9443i) q^{38} +14.5836 q^{41} +22.2703 q^{43} +(-46.8328 + 64.4598i) q^{44} +(1.05573 + 0.343027i) q^{46} +53.3902 q^{47} -21.3607 q^{49} +(19.9192 - 27.4164i) q^{52} +66.3607i q^{53} +(24.7214 + 34.0260i) q^{56} +(-20.8172 - 6.76393i) q^{58} -17.4370i q^{59} +90.1378 q^{61} +(-5.12629 + 15.7771i) q^{62} +(-19.7771 + 60.8676i) q^{64} -50.2220 q^{67} +(-27.9516 + 38.4721i) q^{68} -80.7868i q^{71} -5.55418i q^{73} +(-11.3475 + 34.9241i) q^{74} +(35.7771 - 49.2429i) q^{76} +104.721i q^{77} -13.8448i q^{79} +(27.7396 + 9.01316i) q^{82} +76.2155 q^{83} +(42.3607 + 13.7638i) q^{86} +(-128.920 + 93.6656i) q^{88} -111.443 q^{89} -44.5407i q^{91} +(1.79611 + 1.30495i) q^{92} +(101.554 + 32.9970i) q^{94} +92.8328i q^{97} +(-40.6304 - 13.2016i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 8 * q + 8 * q^4	\( 8 q + 8 q^{4} + 80 q^{14} - 32 q^{16} + 24 q^{26} - 16 q^{29} - 184 q^{34} + 224 q^{41} - 160 q^{44} + 80 q^{46} + 8 q^{49} - 160 q^{56} + 256 q^{61} + 128 q^{64} - 216 q^{74} + 160 q^{86} - 176 q^{89} + 240 q^{94}+O(q^{100}) \) 8 * q + 8 * q^4 + 80 * q^14 - 32 * q^16 + 24 * q^26 - 16 * q^29 - 184 * q^34 + 224 * q^41 - 160 * q^44 + 80 * q^46 + 8 * q^49 - 160 * q^56 + 256 * q^61 + 128 * q^64 - 216 * q^74 + 160 * q^86 - 176 * q^89 + 240 * q^94

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