LMFDB - Embedded newform 975.2.h.f.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(649,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("975.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.h (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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	975.649
Dual form			975.2.h.f.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.73205 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} -3.46410i q^{11} +1.00000i q^{12} +(3.46410 + 1.00000i) q^{13} -6.00000 q^{14} -5.00000 q^{16} -6.00000i q^{17} +1.73205 q^{18} -3.46410i q^{19} +3.46410i q^{21} +6.00000i q^{22} +1.73205i q^{24} +(-6.00000 - 1.73205i) q^{26} -1.00000i q^{27} +3.46410 q^{28} -6.00000 q^{29} -3.46410i q^{31} +5.19615 q^{32} +3.46410 q^{33} +10.3923i q^{34} -1.00000 q^{36} -6.92820 q^{37} +6.00000i q^{38} +(-1.00000 + 3.46410i) q^{39} -6.92820i q^{41} -6.00000i q^{42} +4.00000i q^{43} -3.46410i q^{44} -3.46410 q^{47} -5.00000i q^{48} +5.00000 q^{49} +6.00000 q^{51} +(3.46410 + 1.00000i) q^{52} -6.00000i q^{53} +1.73205i q^{54} +6.00000 q^{56} +3.46410 q^{57} +10.3923 q^{58} -10.3923i q^{59} -2.00000 q^{61} +6.00000i q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} -6.00000i q^{68} +3.46410i q^{71} -1.73205 q^{72} +12.0000 q^{74} -3.46410i q^{76} -12.0000i q^{77} +(1.73205 - 6.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} -3.46410 q^{83} +3.46410i q^{84} -6.92820i q^{86} -6.00000i q^{87} -6.00000i q^{88} +6.92820i q^{89} +(12.0000 + 3.46410i) q^{91} +3.46410 q^{93} +6.00000 q^{94} +5.19615i q^{96} +13.8564 q^{97} -8.66025 q^{98} +3.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^4 - 4 * q^9	$ 4 q + 4 q^{4} - 4 q^{9} - 24 q^{14} - 20 q^{16} - 24 q^{26} - 24 q^{29} - 4 q^{36} - 4 q^{39} + 20 q^{49} + 24 q^{51} + 24 q^{56} - 8 q^{61} + 4 q^{64} - 24 q^{66} + 48 q^{74} + 32 q^{79} + 4 q^{81} + 48 q^{91} + 24 q^{94}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^9 - 24 * q^14 - 20 * q^16 - 24 * q^26 - 24 * q^29 - 4 * q^36 - 4 * q^39 + 20 * q^49 + 24 * q^51 + 24 * q^56 - 8 * q^61 + 4 * q^64 - 24 * q^66 + 48 * q^74 + 32 * q^79 + 4 * q^81 + 48 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.73205			−1.22474			−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205			−1.22474			−0.612372	+	0.790569i	$0.709785\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		−	1.73205i		−	0.707107i
$7$	3.46410			1.30931			0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410			1.30931			0.654654	+	0.755929i	$0.272814\pi$
$8$	1.73205			0.612372
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$			1.00000i			0.288675i
$13$	3.46410	+	1.00000i	0.960769	+	0.277350i
$13$	3.46410	+	1.00000i	0.960769	+	0.277350i
$14$	−6.00000			−1.60357
$15$		0			0
$16$	−5.00000			−1.25000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.73205			0.408248
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$			3.46410i			0.755929i
$22$			6.00000i			1.27920i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$			1.73205i			0.353553i
$25$		0			0
$26$	−6.00000	−	1.73205i	−1.17670	−	0.339683i
$27$		−	1.00000i		−	0.192450i
$28$	3.46410			0.654654
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	5.19615			0.918559
$33$	3.46410			0.603023
$34$			10.3923i			1.78227i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	−6.92820			−1.13899			−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820			−1.13899			−0.569495	+	0.821995i	$0.692861\pi$
$38$			6.00000i			0.973329i
$39$	−1.00000	+	3.46410i	−0.160128	+	0.554700i
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		−	6.00000i		−	0.925820i
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		−	3.46410i		−	0.522233i
$45$		0			0
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		−	5.00000i		−	0.721688i
$49$	5.00000			0.714286
$50$		0			0
$51$	6.00000			0.840168
$52$	3.46410	+	1.00000i	0.480384	+	0.138675i
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$			1.73205i			0.235702i
$55$		0			0
$56$	6.00000			0.801784
$57$	3.46410			0.458831
$58$	10.3923			1.36458
$59$		−	10.3923i		−	1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$		−	10.3923i		−	1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$			6.00000i			0.762001i
$63$	−3.46410			−0.436436
$64$	1.00000			0.125000
$65$		0			0
$66$	−6.00000			−0.738549
$67$	10.3923			1.26962			0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923			1.26962			0.634811	+	0.772667i	$0.281078\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$	−1.73205			−0.204124
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	12.0000			1.39497
$75$		0			0
$76$		−	3.46410i		−	0.397360i
$77$		−	12.0000i		−	1.36753i
$78$	1.73205	−	6.00000i	0.196116	−	0.679366i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$			12.0000i			1.32518i
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$			3.46410i			0.377964i
$85$		0			0
$86$		−	6.92820i		−	0.747087i
$87$		−	6.00000i		−	0.643268i
$88$		−	6.00000i		−	0.639602i
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$	12.0000	+	3.46410i	1.25794	+	0.363137i
$92$		0			0
$93$	3.46410			0.359211
$94$	6.00000			0.618853
$95$		0			0
$96$			5.19615i			0.530330i
$97$	13.8564			1.40690			0.703452	−	0.710742i	$-0.251641\pi$
$97$	13.8564			1.40690			0.703452	+	0.710742i	$0.251641\pi$
$98$	−8.66025			−0.874818
$99$			3.46410i			0.348155i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$	−10.3923			−1.02899
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	6.00000	+	1.73205i	0.588348	+	0.169842i
$105$		0			0
$106$			10.3923i			1.00939i
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		−	1.00000i		−	0.0962250i
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		−	6.92820i		−	0.657596i
$112$	−17.3205			−1.63663
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	−6.00000			−0.561951
$115$		0			0
$116$	−6.00000			−0.557086
$117$	−3.46410	−	1.00000i	−0.320256	−	0.0924500i
$118$			18.0000i			1.65703i
$119$		−	20.7846i		−	1.90532i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	3.46410			0.313625
$123$	6.92820			0.624695
$124$		−	3.46410i		−	0.311086i
$125$		0			0
$126$	6.00000			0.534522
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	−12.1244			−1.07165
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.46410			0.301511
$133$		−	12.0000i		−	1.04053i
$134$	−18.0000			−1.55496
$135$		0			0
$136$		−	10.3923i		−	0.891133i
$137$	20.7846			1.77575			0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846			1.77575			0.887875	+	0.460086i	$0.152181\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		−	3.46410i		−	0.291730i
$142$		−	6.00000i		−	0.503509i
$143$	3.46410	−	12.0000i	0.289683	−	1.00349i
$144$	5.00000			0.416667
$145$		0			0
$146$		0			0
$147$			5.00000i			0.412393i
$148$	−6.92820			−0.569495
$149$		−	13.8564i		−	1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$149$		−	13.8564i		−	1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		−	6.00000i		−	0.486664i
$153$			6.00000i			0.485071i
$154$			20.7846i			1.67487i
$155$		0			0
$156$	−1.00000	+	3.46410i	−0.0800641	+	0.277350i
$157$			14.0000i			1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$			14.0000i			1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	−13.8564			−1.10236
$159$	6.00000			0.475831
$160$		0			0
$161$		0			0
$162$	−1.73205			−0.136083
$163$	3.46410			0.271329			0.135665	−	0.990755i	$-0.456683\pi$
$163$	3.46410			0.271329			0.135665	+	0.990755i	$0.456683\pi$
$164$		−	6.92820i		−	0.541002i
$165$		0			0
$166$	6.00000			0.465690
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$			6.00000i			0.462910i
$169$	11.0000	+	6.92820i	0.846154	+	0.532939i
$170$		0			0
$171$			3.46410i			0.264906i
$172$			4.00000i			0.304997i
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$			10.3923i			0.787839i
$175$		0			0
$176$			17.3205i			1.30558i
$177$	10.3923			0.781133
$178$		−	12.0000i		−	0.899438i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−20.7846	−	6.00000i	−1.54066	−	0.444750i
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$		0			0
$186$	−6.00000			−0.439941
$187$	−20.7846			−1.51992
$188$	−3.46410			−0.252646
$189$		−	3.46410i		−	0.251976i
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$			1.00000i			0.0721688i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$	−24.0000			−1.72310
$195$		0			0
$196$	5.00000			0.357143
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		−	6.00000i		−	0.426401i
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$			10.3923i			0.733017i
$202$	−10.3923			−0.731200
$203$	−20.7846			−1.45879
$204$	6.00000			0.420084
$205$		0			0
$206$			13.8564i			0.965422i
$207$		0			0
$208$	−17.3205	−	5.00000i	−1.20096	−	0.346688i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		−	6.00000i		−	0.412082i
$213$	−3.46410			−0.237356
$214$		−	20.7846i		−	1.42081i
$215$		0			0
$216$		−	1.73205i		−	0.117851i
$217$		−	12.0000i		−	0.814613i
$218$			12.0000i			0.812743i
$219$		0			0
$220$		0			0
$221$	6.00000	−	20.7846i	0.403604	−	1.39812i
$222$			12.0000i			0.805387i
$223$	−3.46410			−0.231973			−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410			−0.231973			−0.115987	+	0.993251i	$0.537003\pi$
$224$	18.0000			1.20268
$225$		0			0
$226$		−	10.3923i		−	0.691286i
$227$	17.3205			1.14960			0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205			1.14960			0.574801	+	0.818293i	$0.305079\pi$
$228$	3.46410			0.229416
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$	−10.3923			−0.682288
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	6.00000	+	1.73205i	0.392232	+	0.113228i
$235$		0			0
$236$		−	10.3923i		−	0.676481i
$237$			8.00000i			0.519656i
$238$			36.0000i			2.33353i
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$	1.73205			0.111340
$243$			1.00000i			0.0641500i
$244$	−2.00000			−0.128037
$245$		0			0
$246$	−12.0000			−0.765092
$247$	3.46410	−	12.0000i	0.220416	−	0.763542i
$248$		−	6.00000i		−	0.381000i
$249$		−	3.46410i		−	0.219529i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.46410			−0.218218
$253$		0			0
$254$			13.8564i			0.869428i
$255$		0			0
$256$	19.0000			1.18750
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	6.92820			0.431331
$259$	−24.0000			−1.49129
$260$		0			0
$261$	6.00000			0.371391
$262$	20.7846			1.28408
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	6.00000			0.369274
$265$		0			0
$266$			20.7846i			1.27439i
$267$	−6.92820			−0.423999
$268$	10.3923			0.634811
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$			30.0000i			1.81902i
$273$	−3.46410	+	12.0000i	−0.209657	+	0.726273i
$274$	−36.0000			−2.17484
$275$		0			0
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	−6.92820			−0.415526
$279$			3.46410i			0.207390i
$280$		0			0
$281$			6.92820i			0.413302i	0.978415	+	0.206651i	$0.0662565\pi$
$281$			6.92820i			0.413302i	−0.978415	+	0.206651i	$0.933744\pi$
$282$			6.00000i			0.357295i
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			3.46410i			0.205557i
$285$		0			0
$286$	−6.00000	+	20.7846i	−0.354787	+	1.22902i
$287$		−	24.0000i		−	1.41668i
$288$	−5.19615			−0.306186
$289$	−19.0000			−1.11765
$290$		0			0
$291$			13.8564i			0.812277i
$292$		0			0
$293$	27.7128			1.61900			0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128			1.61900			0.809500	+	0.587120i	$0.199738\pi$
$294$		−	8.66025i		−	0.505076i
$295$		0			0
$296$	−12.0000			−0.697486
$297$	−3.46410			−0.201008
$298$			24.0000i			1.39028i
$299$		0			0
$300$		0			0
$301$			13.8564i			0.798670i
$302$		−	18.0000i		−	1.03578i
$303$			6.00000i			0.344691i
$304$			17.3205i			0.993399i
$305$		0			0
$306$		−	10.3923i		−	0.594089i
$307$	10.3923			0.593120			0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923			0.593120			0.296560	+	0.955014i	$0.404160\pi$
$308$		−	12.0000i		−	0.683763i
$309$	8.00000			0.455104
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−1.73205	+	6.00000i	−0.0980581	+	0.339683i
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		−	24.2487i		−	1.36843i
$315$		0			0
$316$	8.00000			0.450035
$317$	−13.8564			−0.778253			−0.389127	−	0.921184i	$-0.627223\pi$
$317$	−13.8564			−0.778253			−0.389127	+	0.921184i	$0.627223\pi$
$318$	−10.3923			−0.582772
$319$			20.7846i			1.16371i
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$	−20.7846			−1.15649
$324$	1.00000			0.0555556
$325$		0			0
$326$	−6.00000			−0.332309
$327$	6.92820			0.383131
$328$		−	12.0000i		−	0.662589i
$329$	−12.0000			−0.661581
$330$		0			0
$331$			3.46410i			0.190404i	0.995458	+	0.0952021i	$0.0303497\pi$
$331$			3.46410i			0.190404i	−0.995458	+	0.0952021i	$0.969650\pi$
$332$	−3.46410			−0.190117
$333$	6.92820			0.379663
$334$	30.0000			1.64153
$335$		0			0
$336$		−	17.3205i		−	0.944911i
$337$		−	14.0000i		−	0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$		−	14.0000i		−	0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	−19.0526	−	12.0000i	−1.03632	−	0.652714i
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−12.0000			−0.649836
$342$		−	6.00000i		−	0.324443i
$343$	−6.92820			−0.374088
$344$			6.92820i			0.373544i
$345$		0			0
$346$		−	31.1769i		−	1.67608i
$347$			36.0000i			1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$			36.0000i			1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$		−	6.00000i		−	0.321634i
$349$			6.92820i			0.370858i	0.982658	+	0.185429i	$0.0593675\pi$
$349$			6.92820i			0.370858i	−0.982658	+	0.185429i	$0.940632\pi$
$350$		0			0
$351$	1.00000	−	3.46410i	0.0533761	−	0.184900i
$352$		−	18.0000i		−	0.959403i
$353$	34.6410			1.84376			0.921878	−	0.387481i	$-0.126655\pi$
$353$	34.6410			1.84376			0.921878	+	0.387481i	$0.126655\pi$
$354$	−18.0000			−0.956689
$355$		0			0
$356$			6.92820i			0.367194i
$357$	20.7846			1.10004
$358$	−20.7846			−1.09850
$359$		−	17.3205i		−	0.914141i	−0.889430	−	0.457071i	$-0.848899\pi$
$359$		−	17.3205i		−	0.914141i	0.889430	−	0.457071i	$-0.151101\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$	17.3205			0.910346
$363$		−	1.00000i		−	0.0524864i
$364$	12.0000	+	3.46410i	0.628971	+	0.181568i
$365$		0			0
$366$			3.46410i			0.181071i
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$		0			0
$369$			6.92820i			0.360668i
$370$		0			0
$371$		−	20.7846i		−	1.07908i
$372$	3.46410			0.179605
$373$		−	22.0000i		−	1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$		−	22.0000i		−	1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	36.0000			1.86152
$375$		0			0
$376$	−6.00000			−0.309426
$377$	−20.7846	−	6.00000i	−1.07046	−	0.309016i
$378$			6.00000i			0.308607i
$379$		−	17.3205i		−	0.889695i	−0.895606	−	0.444847i	$-0.853258\pi$
$379$		−	17.3205i		−	0.889695i	0.895606	−	0.444847i	$-0.146742\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$	−41.5692			−2.12687
$383$	3.46410			0.177007			0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410			0.177007			0.0885037	+	0.996076i	$0.471792\pi$
$384$		−	12.1244i		−	0.618718i
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$	13.8564			0.703452
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		0			0
$392$	8.66025			0.437409
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$		0			0
$396$			3.46410i			0.174078i
$397$	34.6410			1.73858			0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410			1.73858			0.869291	+	0.494300i	$0.164576\pi$
$398$	27.7128			1.38912
$399$	12.0000			0.600751
$400$		0			0
$401$			6.92820i			0.345978i	0.984924	+	0.172989i	$0.0553425\pi$
$401$			6.92820i			0.345978i	−0.984924	+	0.172989i	$0.944657\pi$
$402$		−	18.0000i		−	0.897758i
$403$	3.46410	−	12.0000i	0.172559	−	0.597763i
$404$	6.00000			0.298511
$405$		0			0
$406$	36.0000			1.78665
$407$			24.0000i			1.18964i
$408$	10.3923			0.514496
$409$		−	27.7128i		−	1.37031i	−0.728397	−	0.685155i	$-0.759734\pi$
$409$		−	27.7128i		−	1.37031i	0.728397	−	0.685155i	$-0.240266\pi$
$410$		0			0
$411$			20.7846i			1.02523i
$412$		−	8.00000i		−	0.394132i
$413$		−	36.0000i		−	1.77144i
$414$		0			0
$415$		0			0
$416$	18.0000	+	5.19615i	0.882523	+	0.254762i
$417$			4.00000i			0.195881i
$418$	20.7846			1.01661
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$		−	34.6410i		−	1.68830i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$		−	34.6410i		−	1.68830i	0.536107	−	0.844150i	$-0.319894\pi$
$422$	34.6410			1.68630
$423$	3.46410			0.168430
$424$		−	10.3923i		−	0.504695i
$425$		0			0
$426$	6.00000			0.290701
$427$	−6.92820			−0.335279
$428$			12.0000i			0.580042i
$429$	12.0000	+	3.46410i	0.579365	+	0.167248i
$430$		0			0
$431$		−	24.2487i		−	1.16802i	−0.811747	−	0.584010i	$-0.801483\pi$
$431$		−	24.2487i		−	1.16802i	0.811747	−	0.584010i	$-0.198517\pi$
$432$			5.00000i			0.240563i
$433$		−	34.0000i		−	1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$		−	34.0000i		−	1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$			20.7846i			0.997693i
$435$		0			0
$436$		−	6.92820i		−	0.331801i
$437$		0			0
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$	−10.3923	+	36.0000i	−0.494312	+	1.71235i
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		−	6.92820i		−	0.328798i
$445$		0			0
$446$	6.00000			0.284108
$447$	13.8564			0.655386
$448$	3.46410			0.163663
$449$		−	6.92820i		−	0.326962i	−0.986546	−	0.163481i	$-0.947728\pi$
$449$		−	6.92820i		−	0.326962i	0.986546	−	0.163481i	$-0.0522723\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$			6.00000i			0.282216i
$453$	−10.3923			−0.488273
$454$	−30.0000			−1.40797
$455$		0			0
$456$	6.00000			0.280976
$457$	27.7128			1.29635			0.648175	−	0.761491i	$-0.275532\pi$
$457$	27.7128			1.29635			0.648175	+	0.761491i	$0.275532\pi$
$458$		−	12.0000i		−	0.560723i
$459$	−6.00000			−0.280056
$460$		0			0
$461$		−	13.8564i		−	0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$		−	13.8564i		−	0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$	−20.7846			−0.966988
$463$	−17.3205			−0.804952			−0.402476	−	0.915430i	$-0.631850\pi$
$463$	−17.3205			−0.804952			−0.402476	+	0.915430i	$0.631850\pi$
$464$	30.0000			1.39272
$465$		0			0
$466$		−	10.3923i		−	0.481414i
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	−3.46410	−	1.00000i	−0.160128	−	0.0462250i
$469$	36.0000			1.66233
$470$		0			0
$471$	−14.0000			−0.645086
$472$		−	18.0000i		−	0.828517i
$473$	13.8564			0.637118
$474$		−	13.8564i		−	0.636446i
$475$		0			0
$476$		−	20.7846i		−	0.952661i
$477$			6.00000i			0.274721i
$478$		−	18.0000i		−	0.823301i
$479$			10.3923i			0.474837i	0.971408	+	0.237418i	$0.0763012\pi$
$479$			10.3923i			0.474837i	−0.971408	+	0.237418i	$0.923699\pi$
$480$		0			0
$481$	−24.0000	−	6.92820i	−1.09431	−	0.315899i
$482$		−	24.0000i		−	1.09317i
$483$		0			0
$484$	−1.00000			−0.0454545
$485$		0			0
$486$		−	1.73205i		−	0.0785674i
$487$	−38.1051			−1.72671			−0.863354	−	0.504599i	$-0.831640\pi$
$487$	−38.1051			−1.72671			−0.863354	+	0.504599i	$0.831640\pi$
$488$	−3.46410			−0.156813
$489$			3.46410i			0.156652i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.92820			0.312348
$493$			36.0000i			1.62136i
$494$	−6.00000	+	20.7846i	−0.269953	+	0.935144i
$495$		0			0
$496$			17.3205i			0.777714i
$497$			12.0000i			0.538274i
$498$			6.00000i			0.268866i
$499$			10.3923i			0.465223i	0.972570	+	0.232612i	$0.0747271\pi$
$499$			10.3923i			0.465223i	−0.972570	+	0.232612i	$0.925273\pi$
$500$		0			0
$501$		−	17.3205i		−	0.773823i
$502$	20.7846			0.927663
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	−6.00000			−0.267261
$505$		0			0
$506$		0			0
$507$	−6.92820	+	11.0000i	−0.307692	+	0.488527i
$508$		−	8.00000i		−	0.354943i
$509$			41.5692i			1.84252i	0.388943	+	0.921262i	$0.372840\pi$
$509$			41.5692i			1.84252i	−0.388943	+	0.921262i	$0.627160\pi$
$510$		0			0
$511$		0			0
$512$	−8.66025			−0.382733
$513$	−3.46410			−0.152944
$514$		−	31.1769i		−	1.37515i
$515$		0			0
$516$	−4.00000			−0.176090
$517$			12.0000i			0.527759i
$518$	41.5692			1.82645
$519$	−18.0000			−0.790112
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$	−10.3923			−0.454859
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$		−	41.5692i		−	1.81250i
$527$	−20.7846			−0.905392
$528$	−17.3205			−0.753778
$529$	23.0000			1.00000
$530$		0			0
$531$			10.3923i			0.450988i
$532$		−	12.0000i		−	0.520266i
$533$	6.92820	−	24.0000i	0.300094	−	1.03956i
$534$	12.0000			0.519291
$535$		0			0
$536$	18.0000			0.777482
$537$			12.0000i			0.517838i
$538$	10.3923			0.448044
$539$		−	17.3205i		−	0.746047i
$540$		0			0
$541$			6.92820i			0.297867i	0.988847	+	0.148933i	$0.0475840\pi$
$541$			6.92820i			0.297867i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		−	18.0000i		−	0.773166i
$543$		−	10.0000i		−	0.429141i
$544$		−	31.1769i		−	1.33670i
$545$		0			0
$546$	6.00000	−	20.7846i	0.256776	−	0.889499i
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	20.7846			0.887875
$549$	2.00000			0.0853579
$550$		0			0
$551$			20.7846i			0.885454i
$552$		0			0
$553$	27.7128			1.17847
$554$			17.3205i			0.735878i
$555$		0			0
$556$	4.00000			0.169638
$557$	−13.8564			−0.587115			−0.293557	−	0.955941i	$-0.594839\pi$
$557$	−13.8564			−0.587115			−0.293557	+	0.955941i	$0.594839\pi$
$558$		−	6.00000i		−	0.254000i
$559$	−4.00000	+	13.8564i	−0.169182	+	0.586064i
$560$		0			0
$561$		−	20.7846i		−	0.877527i
$562$		−	12.0000i		−	0.506189i
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		−	3.46410i		−	0.145865i
$565$		0			0
$566$			6.92820i			0.291214i
$567$	3.46410			0.145479
$568$			6.00000i			0.251754i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	3.46410	−	12.0000i	0.144841	−	0.501745i
$573$			24.0000i			1.00261i
$574$			41.5692i			1.73507i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	32.9090			1.36883
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		−	24.0000i		−	0.994832i
$583$	−20.7846			−0.860811
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$	−10.3923			−0.428936			−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923			−0.428936			−0.214468	+	0.976731i	$0.568802\pi$
$588$			5.00000i			0.206197i
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$	34.6410			1.42374
$593$	6.92820			0.284507			0.142254	−	0.989830i	$-0.454565\pi$
$593$	6.92820			0.284507			0.142254	+	0.989830i	$0.454565\pi$
$594$	6.00000			0.246183
$595$		0			0
$596$		−	13.8564i		−	0.567581i
$597$		−	16.0000i		−	0.654836i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		−	24.0000i		−	0.978167i
$603$	−10.3923			−0.423207
$604$			10.3923i			0.422857i
$605$		0			0
$606$		−	10.3923i		−	0.422159i
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		−	18.0000i		−	0.729996i
$609$		−	20.7846i		−	0.842235i
$610$		0			0
$611$	−12.0000	−	3.46410i	−0.485468	−	0.140143i
$612$			6.00000i			0.242536i
$613$	−20.7846			−0.839482			−0.419741	−	0.907644i	$-0.637879\pi$
$613$	−20.7846			−0.839482			−0.419741	+	0.907644i	$0.637879\pi$
$614$	−18.0000			−0.726421
$615$		0			0
$616$		−	20.7846i		−	0.837436i
$617$	6.92820			0.278919			0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820			0.278919			0.139459	+	0.990228i	$0.455464\pi$
$618$	−13.8564			−0.557386
$619$		−	31.1769i		−	1.25311i	−0.779379	−	0.626553i	$-0.784465\pi$
$619$		−	31.1769i		−	1.25311i	0.779379	−	0.626553i	$-0.215535\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			24.0000i			0.961540i
$624$	5.00000	−	17.3205i	0.200160	−	0.693375i
$625$		0			0
$626$			17.3205i			0.692267i
$627$		−	12.0000i		−	0.479234i
$628$			14.0000i			0.558661i
$629$			41.5692i			1.65747i
$630$		0			0
$631$			38.1051i			1.51694i	0.651707	+	0.758470i	$0.274053\pi$
$631$			38.1051i			1.51694i	−0.651707	+	0.758470i	$0.725947\pi$
$632$	13.8564			0.551178
$633$		−	20.0000i		−	0.794929i
$634$	24.0000			0.953162
$635$		0			0
$636$	6.00000			0.237915
$637$	17.3205	+	5.00000i	0.686264	+	0.198107i
$638$		−	36.0000i		−	1.42525i
$639$		−	3.46410i		−	0.137038i
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$	20.7846			0.820303
$643$	−10.3923			−0.409832			−0.204916	−	0.978780i	$-0.565692\pi$
$643$	−10.3923			−0.409832			−0.204916	+	0.978780i	$0.565692\pi$
$644$		0			0
$645$		0			0
$646$	36.0000			1.41640
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.73205			0.0680414
$649$	−36.0000			−1.41312
$650$		0			0
$651$	12.0000			0.470317
$652$	3.46410			0.135665
$653$		−	6.00000i		−	0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$		−	6.00000i		−	0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	−12.0000			−0.469237
$655$		0			0
$656$			34.6410i			1.35250i
$657$		0			0
$658$	20.7846			0.810268
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$		−	20.7846i		−	0.808428i	−0.914665	−	0.404214i	$-0.867545\pi$
$661$		−	20.7846i		−	0.808428i	0.914665	−	0.404214i	$-0.132455\pi$
$662$		−	6.00000i		−	0.233197i
$663$	20.7846	+	6.00000i	0.807207	+	0.233021i
$664$	−6.00000			−0.232845
$665$		0			0
$666$	−12.0000			−0.464991
$667$		0			0
$668$	−17.3205			−0.670151
$669$		−	3.46410i		−	0.133930i
$670$		0			0
$671$			6.92820i			0.267460i
$672$			18.0000i			0.694365i
$673$			46.0000i			1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$			46.0000i			1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$			24.2487i			0.934025i
$675$		0			0
$676$	11.0000	+	6.92820i	0.423077	+	0.266469i
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	10.3923			0.399114
$679$	48.0000			1.84207
$680$		0			0
$681$			17.3205i			0.663723i
$682$	20.7846			0.795884
$683$	−31.1769			−1.19295			−0.596476	−	0.802631i	$-0.703433\pi$
$683$	−31.1769			−1.19295			−0.596476	+	0.802631i	$0.703433\pi$
$684$			3.46410i			0.132453i
$685$		0			0
$686$	12.0000			0.458162
$687$	−6.92820			−0.264327
$688$		−	20.0000i		−	0.762493i
$689$	6.00000	−	20.7846i	0.228582	−	0.791831i
$690$		0			0
$691$			45.0333i			1.71315i	0.516024	+	0.856574i	$0.327412\pi$
$691$			45.0333i			1.71315i	−0.516024	+	0.856574i	$0.672588\pi$
$692$			18.0000i			0.684257i
$693$			12.0000i			0.455842i
$694$		−	62.3538i		−	2.36692i
$695$		0			0
$696$		−	10.3923i		−	0.393919i
$697$	−41.5692			−1.57455
$698$		−	12.0000i		−	0.454207i
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$	−1.73205	+	6.00000i	−0.0653720	+	0.226455i
$703$			24.0000i			0.905177i
$704$		−	3.46410i		−	0.130558i
$705$		0			0
$706$	−60.0000			−2.25813
$707$	20.7846			0.781686
$708$	10.3923			0.390567
$709$			6.92820i			0.260194i	0.991501	+	0.130097i	$0.0415289\pi$
$709$			6.92820i			0.260194i	−0.991501	+	0.130097i	$0.958471\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$			12.0000i			0.449719i
$713$		0			0
$714$	−36.0000			−1.34727
$715$		0			0
$716$	12.0000			0.448461
$717$	−10.3923			−0.388108
$718$			30.0000i			1.11959i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		−	27.7128i		−	1.03208i
$722$	−12.1244			−0.451222
$723$	−13.8564			−0.515325
$724$	−10.0000			−0.371647
$725$		0			0
$726$			1.73205i			0.0642824i
$727$			16.0000i			0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$			16.0000i			0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	20.7846	+	6.00000i	0.770329	+	0.222375i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	24.0000			0.887672
$732$		−	2.00000i		−	0.0739221i
$733$	34.6410			1.27950			0.639748	−	0.768585i	$-0.279039\pi$
$733$	34.6410			1.27950			0.639748	+	0.768585i	$0.279039\pi$
$734$			27.7128i			1.02290i
$735$		0			0
$736$		0			0
$737$		−	36.0000i		−	1.32608i
$738$		−	12.0000i		−	0.441726i
$739$			38.1051i			1.40172i	0.713299	+	0.700860i	$0.247200\pi$
$739$			38.1051i			1.40172i	−0.713299	+	0.700860i	$0.752800\pi$
$740$		0			0
$741$	12.0000	+	3.46410i	0.440831	+	0.127257i
$742$			36.0000i			1.32160i
$743$	3.46410			0.127086			0.0635428	−	0.997979i	$-0.479760\pi$
$743$	3.46410			0.127086			0.0635428	+	0.997979i	$0.479760\pi$
$744$	6.00000			0.219971
$745$		0			0
$746$			38.1051i			1.39513i
$747$	3.46410			0.126745
$748$	−20.7846			−0.759961
$749$			41.5692i			1.51891i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$	17.3205			0.631614
$753$		−	12.0000i		−	0.437304i
$754$	36.0000	+	10.3923i	1.31104	+	0.378465i
$755$		0			0
$756$		−	3.46410i		−	0.125988i
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$			30.0000i			1.08965i
$759$		0			0
$760$		0			0
$761$		−	48.4974i		−	1.75803i	−0.476794	−	0.879015i	$-0.658201\pi$
$761$		−	48.4974i		−	1.75803i	0.476794	−	0.879015i	$-0.341799\pi$
$762$	−13.8564			−0.501965
$763$		−	24.0000i		−	0.868858i
$764$	24.0000			0.868290
$765$		0			0
$766$	−6.00000			−0.216789
$767$	10.3923	−	36.0000i	0.375244	−	1.29988i
$768$			19.0000i			0.685603i
$769$		−	27.7128i		−	0.999350i	−0.866213	−	0.499675i	$-0.833453\pi$
$769$		−	27.7128i		−	0.999350i	0.866213	−	0.499675i	$-0.166547\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	−13.8564			−0.498380			−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564			−0.498380			−0.249190	+	0.968455i	$0.580164\pi$
$774$			6.92820i			0.249029i
$775$		0			0
$776$	24.0000			0.861550
$777$		−	24.0000i		−	0.860995i
$778$	−31.1769			−1.11775
$779$	−24.0000			−0.859889
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$			6.00000i			0.214423i
$784$	−25.0000			−0.892857
$785$		0			0
$786$			20.7846i			0.741362i
$787$	10.3923			0.370446			0.185223	−	0.982697i	$-0.440699\pi$
$787$	10.3923			0.370446			0.185223	+	0.982697i	$0.440699\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$			20.7846i			0.739016i
$792$			6.00000i			0.213201i
$793$	−6.92820	−	2.00000i	−0.246028	−	0.0710221i
$794$	−60.0000			−2.12932
$795$		0			0
$796$	−16.0000			−0.567105
$797$		−	42.0000i		−	1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$		−	42.0000i		−	1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	−20.7846			−0.735767
$799$			20.7846i			0.735307i
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		−	12.0000i		−	0.423735i
$803$		0			0
$804$			10.3923i			0.366508i
$805$		0			0
$806$	−6.00000	+	20.7846i	−0.211341	+	0.732107i
$807$		−	6.00000i		−	0.211210i
$808$	10.3923			0.365600
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$		−	38.1051i		−	1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$		−	38.1051i		−	1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	−20.7846			−0.729397
$813$	−10.3923			−0.364474
$814$		−	41.5692i		−	1.45700i
$815$		0			0
$816$	−30.0000			−1.05021
$817$	13.8564			0.484774
$818$			48.0000i			1.67828i
$819$	−12.0000	−	3.46410i	−0.419314	−	0.121046i
$820$		0			0
$821$			13.8564i			0.483592i	0.970327	+	0.241796i	$0.0777365\pi$
$821$			13.8564i			0.483592i	−0.970327	+	0.241796i	$0.922264\pi$
$822$		−	36.0000i		−	1.25564i
$823$			40.0000i			1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$			40.0000i			1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$		−	13.8564i		−	0.482711i
$825$		0			0
$826$			62.3538i			2.16957i
$827$	−24.2487			−0.843210			−0.421605	−	0.906780i	$-0.638533\pi$
$827$	−24.2487			−0.843210			−0.421605	+	0.906780i	$0.638533\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$	3.46410	+	1.00000i	0.120096	+	0.0346688i
$833$		−	30.0000i		−	1.03944i
$834$		−	6.92820i		−	0.239904i
$835$		0			0
$836$	−12.0000			−0.415029
$837$	−3.46410			−0.119737
$838$	−20.7846			−0.717992
$839$		−	3.46410i		−	0.119594i	−0.998211	−	0.0597970i	$-0.980955\pi$
$839$		−	3.46410i		−	0.119594i	0.998211	−	0.0597970i	$-0.0190453\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$			60.0000i			2.06774i
$843$	−6.92820			−0.238620
$844$	−20.0000			−0.688428
$845$		0			0
$846$	−6.00000			−0.206284
$847$	−3.46410			−0.119028
$848$			30.0000i			1.03020i
$849$	4.00000			0.137280
$850$		0			0
$851$		0			0
$852$	−3.46410			−0.118678
$853$	−20.7846			−0.711651			−0.355826	−	0.934552i	$-0.615800\pi$
$853$	−20.7846			−0.711651			−0.355826	+	0.934552i	$0.615800\pi$
$854$	12.0000			0.410632
$855$		0			0
$856$			20.7846i			0.710403i
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	−20.7846	−	6.00000i	−0.709575	−	0.204837i
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$			42.0000i			1.43053i
$863$	31.1769			1.06127			0.530637	−	0.847599i	$-0.321953\pi$
$863$	31.1769			1.06127			0.530637	+	0.847599i	$0.321953\pi$
$864$		−	5.19615i		−	0.176777i
$865$		0			0
$866$			58.8897i			2.00115i
$867$		−	19.0000i		−	0.645274i
$868$		−	12.0000i		−	0.407307i
$869$		−	27.7128i		−	0.940093i
$870$		0			0
$871$	36.0000	+	10.3923i	1.21981	+	0.352130i
$872$		−	12.0000i		−	0.406371i
$873$	−13.8564			−0.468968
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−48.4974			−1.63764			−0.818821	−	0.574049i	$-0.805372\pi$
$877$	−48.4974			−1.63764			−0.818821	+	0.574049i	$0.805372\pi$
$878$	13.8564			0.467631
$879$			27.7128i			0.934730i
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$	8.66025			0.291606
$883$			20.0000i			0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$			20.0000i			0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	6.00000	−	20.7846i	0.201802	−	0.699062i
$885$		0			0
$886$		−	62.3538i		−	2.09482i
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		−	12.0000i		−	0.402694i
$889$		−	27.7128i		−	0.929458i
$890$		0			0
$891$		−	3.46410i		−	0.116052i
$892$	−3.46410			−0.115987
$893$			12.0000i			0.401565i
$894$	−24.0000			−0.802680
$895$		0			0
$896$	−42.0000			−1.40312
$897$		0			0
$898$			12.0000i			0.400445i
$899$			20.7846i			0.693206i
$900$		0			0
$901$	−36.0000			−1.19933
$902$	41.5692			1.38410
$903$	−13.8564			−0.461112
$904$			10.3923i			0.345643i
$905$		0			0
$906$	18.0000			0.598010
$907$		−	44.0000i		−	1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$		−	44.0000i		−	1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	17.3205			0.574801
$909$	−6.00000			−0.199007
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$	−17.3205			−0.573539
$913$			12.0000i			0.397142i
$914$	−48.0000			−1.58770
$915$		0			0
$916$			6.92820i			0.228914i
$917$	−41.5692			−1.37274
$918$	10.3923			0.342997
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$			10.3923i			0.342438i
$922$			24.0000i			0.790398i
$923$	−3.46410	+	12.0000i	−0.114022	+	0.394985i
$924$	12.0000			0.394771
$925$		0			0
$926$	30.0000			0.985861
$927$			8.00000i			0.262754i
$928$	−31.1769			−1.02343
$929$			20.7846i			0.681921i	0.940078	+	0.340960i	$0.110752\pi$
$929$			20.7846i			0.681921i	−0.940078	+	0.340960i	$0.889248\pi$
$930$		0			0
$931$		−	17.3205i		−	0.567657i
$932$			6.00000i			0.196537i
$933$		0			0
$934$		−	20.7846i		−	0.680093i
$935$		0			0
$936$	−6.00000	−	1.73205i	−0.196116	−	0.0566139i
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	−62.3538			−2.03592
$939$	10.0000			0.326338
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$	24.2487			0.790066
$943$		0			0
$944$			51.9615i			1.69120i
$945$		0			0
$946$	−24.0000			−0.780307
$947$	−51.9615			−1.68852			−0.844261	−	0.535932i	$-0.819960\pi$
$947$	−51.9615			−1.68852			−0.844261	+	0.535932i	$0.819960\pi$
$948$			8.00000i			0.259828i
$949$		0			0
$950$		0			0
$951$		−	13.8564i		−	0.449325i
$952$		−	36.0000i		−	1.16677i
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		−	10.3923i		−	0.336463i
$955$		0			0
$956$			10.3923i			0.336111i
$957$	−20.7846			−0.671871
$958$		−	18.0000i		−	0.581554i
$959$	72.0000			2.32500
$960$		0			0
$961$	19.0000			0.612903
$962$	41.5692	+	12.0000i	1.34025	+	0.386896i
$963$		−	12.0000i		−	0.386695i
$964$			13.8564i			0.446285i
$965$		0			0
$966$		0			0
$967$	−10.3923			−0.334194			−0.167097	−	0.985940i	$-0.553439\pi$
$967$	−10.3923			−0.334194			−0.167097	+	0.985940i	$0.553439\pi$
$968$	−1.73205			−0.0556702
$969$		−	20.7846i		−	0.667698i
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$			1.00000i			0.0320750i
$973$	13.8564			0.444216
$974$	66.0000			2.11478
$975$		0			0
$976$	10.0000			0.320092
$977$	−48.4974			−1.55157			−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974			−1.55157			−0.775785	+	0.630997i	$0.782646\pi$
$978$		−	6.00000i		−	0.191859i
$979$	24.0000			0.767043
$980$		0			0
$981$			6.92820i			0.221201i
$982$	20.7846			0.663264
$983$	−51.9615			−1.65732			−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615			−1.65732			−0.828658	+	0.559756i	$0.810895\pi$
$984$	12.0000			0.382546
$985$		0			0
$986$		−	62.3538i		−	1.98575i
$987$		−	12.0000i		−	0.381964i
$988$	3.46410	−	12.0000i	0.110208	−	0.381771i
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		−	18.0000i		−	0.571501i
$993$	−3.46410			−0.109930
$994$		−	20.7846i		−	0.659248i
$995$		0			0
$996$		−	3.46410i		−	0.109764i
$997$			38.0000i			1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$			38.0000i			1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$		−	18.0000i		−	0.569780i
$999$			6.92820i			0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.h.f.649.2		4
5.2	odd	4		975.2.b.d.376.1		2
5.3	odd	4		39.2.b.a.25.2	yes	2
5.4	even	2	inner	975.2.h.f.649.3		4
13.12	even	2	inner	975.2.h.f.649.4		4
15.8	even	4		117.2.b.a.64.1		2
20.3	even	4		624.2.c.e.337.2		2
35.13	even	4		1911.2.c.d.883.2		2
40.3	even	4		2496.2.c.d.961.2		2
40.13	odd	4		2496.2.c.k.961.1		2
60.23	odd	4		1872.2.c.e.1585.2		2
65.3	odd	12		507.2.j.a.316.1		2
65.8	even	4		507.2.a.f.1.1		2
65.12	odd	4		975.2.b.d.376.2		2
65.18	even	4		507.2.a.f.1.2		2
65.23	odd	12		507.2.j.c.316.1		2
65.28	even	12		507.2.e.e.22.1		4
65.33	even	12		507.2.e.e.484.2		4
65.38	odd	4		39.2.b.a.25.1	✓	2
65.43	odd	12		507.2.j.a.361.1		2
65.48	odd	12		507.2.j.c.361.1		2
65.58	even	12		507.2.e.e.484.1		4
65.63	even	12		507.2.e.e.22.2		4
65.64	even	2	inner	975.2.h.f.649.1		4
195.8	odd	4		1521.2.a.l.1.2		2
195.38	even	4		117.2.b.a.64.2		2
195.83	odd	4		1521.2.a.l.1.1		2
260.83	odd	4		8112.2.a.bv.1.1		2
260.103	even	4		624.2.c.e.337.1		2
260.203	odd	4		8112.2.a.bv.1.2		2
455.363	even	4		1911.2.c.d.883.1		2
520.363	even	4		2496.2.c.d.961.1		2
520.493	odd	4		2496.2.c.k.961.2		2
780.623	odd	4		1872.2.c.e.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.b.a.25.1	✓	2	65.38	odd	4
39.2.b.a.25.2	yes	2	5.3	odd	4
117.2.b.a.64.1		2	15.8	even	4
117.2.b.a.64.2		2	195.38	even	4
507.2.a.f.1.1		2	65.8	even	4
507.2.a.f.1.2		2	65.18	even	4
507.2.e.e.22.1		4	65.28	even	12
507.2.e.e.22.2		4	65.63	even	12
507.2.e.e.484.1		4	65.58	even	12
507.2.e.e.484.2		4	65.33	even	12
507.2.j.a.316.1		2	65.3	odd	12
507.2.j.a.361.1		2	65.43	odd	12
507.2.j.c.316.1		2	65.23	odd	12
507.2.j.c.361.1		2	65.48	odd	12
624.2.c.e.337.1		2	260.103	even	4
624.2.c.e.337.2		2	20.3	even	4
975.2.b.d.376.1		2	5.2	odd	4
975.2.b.d.376.2		2	65.12	odd	4
975.2.h.f.649.1		4	65.64	even	2	inner
975.2.h.f.649.2		4	1.1	even	1	trivial
975.2.h.f.649.3		4	5.4	even	2	inner
975.2.h.f.649.4		4	13.12	even	2	inner
1521.2.a.l.1.1		2	195.83	odd	4
1521.2.a.l.1.2		2	195.8	odd	4
1872.2.c.e.1585.1		2	780.623	odd	4
1872.2.c.e.1585.2		2	60.23	odd	4
1911.2.c.d.883.1		2	455.363	even	4
1911.2.c.d.883.2		2	35.13	even	4
2496.2.c.d.961.1		2	520.363	even	4
2496.2.c.d.961.2		2	40.3	even	4
2496.2.c.k.961.1		2	40.13	odd	4
2496.2.c.k.961.2		2	520.493	odd	4
8112.2.a.bv.1.1		2	260.83	odd	4
8112.2.a.bv.1.2		2	260.203	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} -3.46410i q^{11} +1.00000i q^{12} +(3.46410 + 1.00000i) q^{13} -6.00000 q^{14} -5.00000 q^{16} -6.00000i q^{17} +1.73205 q^{18} -3.46410i q^{19} +3.46410i q^{21} +6.00000i q^{22} +1.73205i q^{24} +(-6.00000 - 1.73205i) q^{26} -1.00000i q^{27} +3.46410 q^{28} -6.00000 q^{29} -3.46410i q^{31} +5.19615 q^{32} +3.46410 q^{33} +10.3923i q^{34} -1.00000 q^{36} -6.92820 q^{37} +6.00000i q^{38} +(-1.00000 + 3.46410i) q^{39} -6.92820i q^{41} -6.00000i q^{42} +4.00000i q^{43} -3.46410i q^{44} -3.46410 q^{47} -5.00000i q^{48} +5.00000 q^{49} +6.00000 q^{51} +(3.46410 + 1.00000i) q^{52} -6.00000i q^{53} +1.73205i q^{54} +6.00000 q^{56} +3.46410 q^{57} +10.3923 q^{58} -10.3923i q^{59} -2.00000 q^{61} +6.00000i q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} -6.00000i q^{68} +3.46410i q^{71} -1.73205 q^{72} +12.0000 q^{74} -3.46410i q^{76} -12.0000i q^{77} +(1.73205 - 6.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{82} -3.46410 q^{83} +3.46410i q^{84} -6.92820i q^{86} -6.00000i q^{87} -6.00000i q^{88} +6.92820i q^{89} +(12.0000 + 3.46410i) q^{91} +3.46410 q^{93} +6.00000 q^{94} +5.19615i q^{96} +13.8564 q^{97} -8.66025 q^{98} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^4 - 4 * q^9	\( 4 q + 4 q^{4} - 4 q^{9} - 24 q^{14} - 20 q^{16} - 24 q^{26} - 24 q^{29} - 4 q^{36} - 4 q^{39} + 20 q^{49} + 24 q^{51} + 24 q^{56} - 8 q^{61} + 4 q^{64} - 24 q^{66} + 48 q^{74} + 32 q^{79} + 4 q^{81} + 48 q^{91} + 24 q^{94}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^9 - 24 * q^14 - 20 * q^16 - 24 * q^26 - 24 * q^29 - 4 * q^36 - 4 * q^39 + 20 * q^49 + 24 * q^51 + 24 * q^56 - 8 * q^61 + 4 * q^64 - 24 * q^66 + 48 * q^74 + 32 * q^79 + 4 * q^81 + 48 * q^91 + 24 * q^94

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