LMFDB - Embedded newform 2175.2.d.e.376.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(376,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2175.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			376.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2175.376
Dual form			2175.2.d.e.376.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +4.23607 q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +4.23607 q^{7} -2.23607i q^{8} -1.00000 q^{9} -2.23607i q^{11} -0.618034i q^{12} +1.00000 q^{13} -6.85410i q^{14} -4.85410 q^{16} -3.00000i q^{17} +1.61803i q^{18} -7.70820i q^{19} +4.23607i q^{21} -3.61803 q^{22} -1.23607 q^{23} +2.23607 q^{24} -1.61803i q^{26} -1.00000i q^{27} -2.61803 q^{28} +(-4.47214 - 3.00000i) q^{29} +7.23607i q^{31} +3.38197i q^{32} +2.23607 q^{33} -4.85410 q^{34} +0.618034 q^{36} +4.76393i q^{37} -12.4721 q^{38} +1.00000i q^{39} +4.47214i q^{41} +6.85410 q^{42} -0.472136i q^{43} +1.38197i q^{44} +2.00000i q^{46} -10.2361i q^{47} -4.85410i q^{48} +10.9443 q^{49} +3.00000 q^{51} -0.618034 q^{52} -6.76393 q^{53} -1.61803 q^{54} -9.47214i q^{56} +7.70820 q^{57} +(-4.85410 + 7.23607i) q^{58} +8.94427 q^{59} -2.76393i q^{61} +11.7082 q^{62} -4.23607 q^{63} -4.23607 q^{64} -3.61803i q^{66} -4.70820 q^{67} +1.85410i q^{68} -1.23607i q^{69} +4.76393 q^{71} +2.23607i q^{72} -7.70820i q^{73} +7.70820 q^{74} +4.76393i q^{76} -9.47214i q^{77} +1.61803 q^{78} +2.29180i q^{79} +1.00000 q^{81} +7.23607 q^{82} +6.00000 q^{83} -2.61803i q^{84} -0.763932 q^{86} +(3.00000 - 4.47214i) q^{87} -5.00000 q^{88} +0.0557281i q^{89} +4.23607 q^{91} +0.763932 q^{92} -7.23607 q^{93} -16.5623 q^{94} -3.38197 q^{96} +18.1803i q^{97} -17.7082i q^{98} +2.23607i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 2 q^{6} + 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 2 * q^6 + 8 * q^7 - 4 * q^9	$ 4 q + 2 q^{4} + 2 q^{6} + 8 q^{7} - 4 q^{9} + 4 q^{13} - 6 q^{16} - 10 q^{22} + 4 q^{23} - 6 q^{28} - 6 q^{34} - 2 q^{36} - 32 q^{38} + 14 q^{42} + 8 q^{49} + 12 q^{51} + 2 q^{52} - 36 q^{53} - 2 q^{54} + 4 q^{57} - 6 q^{58} + 20 q^{62} - 8 q^{63} - 8 q^{64} + 8 q^{67} + 28 q^{71} + 4 q^{74} + 2 q^{78} + 4 q^{81} + 20 q^{82} + 24 q^{83} - 12 q^{86} + 12 q^{87} - 20 q^{88} + 8 q^{91} + 12 q^{92} - 20 q^{93} - 26 q^{94} - 18 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 + 2 * q^6 + 8 * q^7 - 4 * q^9 + 4 * q^13 - 6 * q^16 - 10 * q^22 + 4 * q^23 - 6 * q^28 - 6 * q^34 - 2 * q^36 - 32 * q^38 + 14 * q^42 + 8 * q^49 + 12 * q^51 + 2 * q^52 - 36 * q^53 - 2 * q^54 + 4 * q^57 - 6 * q^58 + 20 * q^62 - 8 * q^63 - 8 * q^64 + 8 * q^67 + 28 * q^71 + 4 * q^74 + 2 * q^78 + 4 * q^81 + 20 * q^82 + 24 * q^83 - 12 * q^86 + 12 * q^87 - 20 * q^88 + 8 * q^91 + 12 * q^92 - 20 * q^93 - 26 * q^94 - 18 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.61803i		−	1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$		−	1.61803i		−	1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−0.618034			−0.309017
$5$		0			0
$5$		0			0
$6$	1.61803			0.660560
$7$	4.23607			1.60108			0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607			1.60108			0.800542	+	0.599277i	$0.204545\pi$
$8$		−	2.23607i		−	0.790569i
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		−	0.618034i		−	0.178411i
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		−	6.85410i		−	1.83184i
$15$		0			0
$16$	−4.85410			−1.21353
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$			1.61803i			0.381374i
$19$		−	7.70820i		−	1.76838i	−0.467124	−	0.884192i	$-0.654710\pi$
$19$		−	7.70820i		−	1.76838i	0.467124	−	0.884192i	$-0.345290\pi$
$20$		0			0
$21$			4.23607i			0.924386i
$22$	−3.61803			−0.771367
$23$	−1.23607			−0.257738			−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607			−0.257738			−0.128869	+	0.991662i	$0.541135\pi$
$24$	2.23607			0.456435
$25$		0			0
$26$		−	1.61803i		−	0.317323i
$27$		−	1.00000i		−	0.192450i
$28$	−2.61803			−0.494762
$29$	−4.47214	−	3.00000i	−0.830455	−	0.557086i
$29$	−4.47214	−	3.00000i	−0.830455	−	0.557086i
$30$		0			0
$31$			7.23607i			1.29964i	0.760090	+	0.649818i	$0.225155\pi$
$31$			7.23607i			1.29964i	−0.760090	+	0.649818i	$0.774845\pi$
$32$			3.38197i			0.597853i
$33$	2.23607			0.389249
$34$	−4.85410			−0.832472
$35$		0			0
$36$	0.618034			0.103006
$37$			4.76393i			0.783186i	0.920139	+	0.391593i	$0.128076\pi$
$37$			4.76393i			0.783186i	−0.920139	+	0.391593i	$0.871924\pi$
$38$	−12.4721			−2.02325
$39$			1.00000i			0.160128i
$40$		0			0
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$	6.85410			1.05761
$43$		−	0.472136i		−	0.0720001i	−0.999352	−	0.0360000i	$-0.988538\pi$
$43$		−	0.472136i		−	0.0720001i	0.999352	−	0.0360000i	$-0.0114616\pi$
$44$			1.38197i			0.208339i
$45$		0			0
$46$			2.00000i			0.294884i
$47$		−	10.2361i		−	1.49308i	−0.665338	−	0.746542i	$-0.731713\pi$
$47$		−	10.2361i		−	1.49308i	0.665338	−	0.746542i	$-0.268287\pi$
$48$		−	4.85410i		−	0.700629i
$49$	10.9443			1.56347
$50$		0			0
$51$	3.00000			0.420084
$52$	−0.618034			−0.0857059
$53$	−6.76393			−0.929098			−0.464549	−	0.885548i	$-0.653783\pi$
$53$	−6.76393			−0.929098			−0.464549	+	0.885548i	$0.653783\pi$
$54$	−1.61803			−0.220187
$55$		0			0
$56$		−	9.47214i		−	1.26577i
$57$	7.70820			1.02098
$58$	−4.85410	+	7.23607i	−0.637375	+	0.950142i
$59$	8.94427			1.16445			0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427			1.16445			0.582223	+	0.813029i	$0.302183\pi$
$60$		0			0
$61$		−	2.76393i		−	0.353885i	−0.984221	−	0.176943i	$-0.943379\pi$
$61$		−	2.76393i		−	0.353885i	0.984221	−	0.176943i	$-0.0566207\pi$
$62$	11.7082			1.48694
$63$	−4.23607			−0.533694
$64$	−4.23607			−0.529508
$65$		0			0
$66$		−	3.61803i		−	0.445349i
$67$	−4.70820			−0.575199			−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820			−0.575199			−0.287599	+	0.957751i	$0.592857\pi$
$68$			1.85410i			0.224843i
$69$		−	1.23607i		−	0.148805i
$70$		0			0
$71$	4.76393			0.565375			0.282687	−	0.959212i	$-0.408774\pi$
$71$	4.76393			0.565375			0.282687	+	0.959212i	$0.408774\pi$
$72$			2.23607i			0.263523i
$73$		−	7.70820i		−	0.902177i	−0.892479	−	0.451089i	$-0.851036\pi$
$73$		−	7.70820i		−	0.902177i	0.892479	−	0.451089i	$-0.148964\pi$
$74$	7.70820			0.896061
$75$		0			0
$76$			4.76393i			0.546460i
$77$		−	9.47214i		−	1.07945i
$78$	1.61803			0.183206
$79$			2.29180i			0.257847i	0.991655	+	0.128924i	$0.0411522\pi$
$79$			2.29180i			0.257847i	−0.991655	+	0.128924i	$0.958848\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	7.23607			0.799090
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		−	2.61803i		−	0.285651i
$85$		0			0
$86$	−0.763932			−0.0823769
$87$	3.00000	−	4.47214i	0.321634	−	0.479463i
$88$	−5.00000			−0.533002
$89$			0.0557281i			0.00590717i	0.999996	+	0.00295358i	$0.000940156\pi$
$89$			0.0557281i			0.00590717i	−0.999996	+	0.00295358i	$0.999060\pi$
$90$		0			0
$91$	4.23607			0.444061
$92$	0.763932			0.0796454
$93$	−7.23607			−0.750345
$94$	−16.5623			−1.70827
$95$		0			0
$96$	−3.38197			−0.345170
$97$			18.1803i			1.84593i	0.384879	+	0.922967i	$0.374243\pi$
$97$			18.1803i			1.84593i	−0.384879	+	0.922967i	$0.625757\pi$
$98$		−	17.7082i		−	1.78880i
$99$			2.23607i			0.224733i
$100$		0			0
$101$			3.94427i			0.392470i	0.980557	+	0.196235i	$0.0628715\pi$
$101$			3.94427i			0.392470i	−0.980557	+	0.196235i	$0.937129\pi$
$102$		−	4.85410i		−	0.480628i
$103$	19.4164			1.91316			0.956578	−	0.291477i	$-0.0941468\pi$
$103$	19.4164			1.91316			0.956578	+	0.291477i	$0.0941468\pi$
$104$		−	2.23607i		−	0.219265i
$105$		0			0
$106$			10.9443i			1.06300i
$107$	−0.763932			−0.0738521			−0.0369260	−	0.999318i	$-0.511757\pi$
$107$	−0.763932			−0.0738521			−0.0369260	+	0.999318i	$0.511757\pi$
$108$			0.618034i			0.0594703i
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$	−4.76393			−0.452172
$112$	−20.5623			−1.94296
$113$			4.52786i			0.425946i	0.977058	+	0.212973i	$0.0683146\pi$
$113$			4.52786i			0.425946i	−0.977058	+	0.212973i	$0.931685\pi$
$114$		−	12.4721i		−	1.16812i
$115$		0			0
$116$	2.76393	+	1.85410i	0.256625	+	0.172149i
$117$	−1.00000			−0.0924500
$118$		−	14.4721i		−	1.33227i
$119$		−	12.7082i		−	1.16496i
$120$		0			0
$121$	6.00000			0.545455
$122$	−4.47214			−0.404888
$123$	−4.47214			−0.403239
$124$		−	4.47214i		−	0.401610i
$125$		0			0
$126$			6.85410i			0.610612i
$127$		−	12.4721i		−	1.10672i	−0.832941	−	0.553362i	$-0.813345\pi$
$127$		−	12.4721i		−	1.10672i	0.832941	−	0.553362i	$-0.186655\pi$
$128$			13.6180i			1.20368i
$129$	0.472136			0.0415693
$130$		0			0
$131$		−	12.2361i		−	1.06907i	−0.845146	−	0.534535i	$-0.820487\pi$
$131$		−	12.2361i		−	1.06907i	0.845146	−	0.534535i	$-0.179513\pi$
$132$	−1.38197			−0.120285
$133$		−	32.6525i		−	2.83133i
$134$			7.61803i			0.658098i
$135$		0			0
$136$	−6.70820			−0.575224
$137$		−	3.52786i		−	0.301406i	−0.988579	−	0.150703i	$-0.951846\pi$
$137$		−	3.52786i		−	0.301406i	0.988579	−	0.150703i	$-0.0481537\pi$
$138$	−2.00000			−0.170251
$139$	−6.70820			−0.568982			−0.284491	−	0.958679i	$-0.591825\pi$
$139$	−6.70820			−0.568982			−0.284491	+	0.958679i	$0.591825\pi$
$140$		0			0
$141$	10.2361			0.862032
$142$		−	7.70820i		−	0.646858i
$143$		−	2.23607i		−	0.186989i
$144$	4.85410			0.404508
$145$		0			0
$146$	−12.4721			−1.03220
$147$			10.9443i			0.902668i
$148$		−	2.94427i		−	0.242018i
$149$	−5.52786			−0.452860			−0.226430	−	0.974027i	$-0.572705\pi$
$149$	−5.52786			−0.452860			−0.226430	+	0.974027i	$0.572705\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	−17.2361			−1.39803
$153$			3.00000i			0.242536i
$154$	−15.3262			−1.23502
$155$		0			0
$156$		−	0.618034i		−	0.0494823i
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	3.70820			0.295009
$159$		−	6.76393i		−	0.536415i
$160$		0			0
$161$	−5.23607			−0.412660
$162$		−	1.61803i		−	0.127125i
$163$			7.41641i			0.580898i	0.956890	+	0.290449i	$0.0938046\pi$
$163$			7.41641i			0.580898i	−0.956890	+	0.290449i	$0.906195\pi$
$164$		−	2.76393i		−	0.215827i
$165$		0			0
$166$		−	9.70820i		−	0.753503i
$167$	−15.8885			−1.22949			−0.614746	−	0.788725i	$-0.710742\pi$
$167$	−15.8885			−1.22949			−0.614746	+	0.788725i	$0.710742\pi$
$168$	9.47214			0.730791
$169$	−12.0000			−0.923077
$170$		0			0
$171$			7.70820i			0.589461i
$172$			0.291796i			0.0222492i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$	−7.23607	−	4.85410i	−0.548565	−	0.367989i
$175$		0			0
$176$			10.8541i			0.818159i
$177$			8.94427i			0.672293i
$178$	0.0901699			0.00675852
$179$	−17.2361			−1.28828			−0.644142	−	0.764906i	$-0.722785\pi$
$179$	−17.2361			−1.28828			−0.644142	+	0.764906i	$0.722785\pi$
$180$		0			0
$181$	20.4164			1.51754			0.758770	−	0.651359i	$-0.225801\pi$
$181$	20.4164			1.51754			0.758770	+	0.651359i	$0.225801\pi$
$182$		−	6.85410i		−	0.508060i
$183$	2.76393			0.204316
$184$			2.76393i			0.203760i
$185$		0			0
$186$			11.7082i			0.858487i
$187$	−6.70820			−0.490552
$188$			6.32624i			0.461388i
$189$		−	4.23607i		−	0.308129i
$190$		0			0
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$		−	4.23607i		−	0.305712i
$193$			17.4164i			1.25366i	0.779156	+	0.626830i	$0.215648\pi$
$193$			17.4164i			1.25366i	−0.779156	+	0.626830i	$0.784352\pi$
$194$	29.4164			2.11198
$195$		0			0
$196$	−6.76393			−0.483138
$197$	3.70820			0.264199			0.132099	−	0.991236i	$-0.457828\pi$
$197$	3.70820			0.264199			0.132099	+	0.991236i	$0.457828\pi$
$198$	3.61803			0.257122
$199$	−20.1246			−1.42660			−0.713298	−	0.700861i	$-0.752799\pi$
$199$	−20.1246			−1.42660			−0.713298	+	0.700861i	$0.752799\pi$
$200$		0			0
$201$		−	4.70820i		−	0.332091i
$202$	6.38197			0.449034
$203$	−18.9443	−	12.7082i	−1.32963	−	0.891941i
$204$	−1.85410			−0.129813
$205$		0			0
$206$		−	31.4164i		−	2.18888i
$207$	1.23607			0.0859127
$208$	−4.85410			−0.336571
$209$	−17.2361			−1.19224
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$	4.18034			0.287107
$213$			4.76393i			0.326419i
$214$			1.23607i			0.0844959i
$215$		0			0
$216$	−2.23607			−0.152145
$217$			30.6525i			2.08083i
$218$		−	8.09017i		−	0.547935i
$219$	7.70820			0.520872
$220$		0			0
$221$		−	3.00000i		−	0.201802i
$222$			7.70820i			0.517341i
$223$	12.7082			0.851004			0.425502	−	0.904957i	$-0.360097\pi$
$223$	12.7082			0.851004			0.425502	+	0.904957i	$0.360097\pi$
$224$			14.3262i			0.957212i
$225$		0			0
$226$	7.32624			0.487334
$227$	−9.05573			−0.601050			−0.300525	−	0.953774i	$-0.597162\pi$
$227$	−9.05573			−0.601050			−0.300525	+	0.953774i	$0.597162\pi$
$228$	−4.76393			−0.315499
$229$		−	22.1803i		−	1.46572i	−0.680380	−	0.732859i	$-0.738185\pi$
$229$		−	22.1803i		−	1.46572i	0.680380	−	0.732859i	$-0.261815\pi$
$230$		0			0
$231$	9.47214			0.623221
$232$	−6.70820	+	10.0000i	−0.440415	+	0.656532i
$233$	10.4721			0.686052			0.343026	−	0.939326i	$-0.388548\pi$
$233$	10.4721			0.686052			0.343026	+	0.939326i	$0.388548\pi$
$234$			1.61803i			0.105774i
$235$		0			0
$236$	−5.52786			−0.359833
$237$	−2.29180			−0.148868
$238$	−20.5623			−1.33286
$239$	26.1803			1.69347			0.846733	−	0.532019i	$-0.178566\pi$
$239$	26.1803			1.69347			0.846733	+	0.532019i	$0.178566\pi$
$240$		0			0
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$		−	9.70820i		−	0.624067i
$243$			1.00000i			0.0641500i
$244$			1.70820i			0.109357i
$245$		0			0
$246$			7.23607i			0.461355i
$247$		−	7.70820i		−	0.490461i
$248$	16.1803			1.02745
$249$			6.00000i			0.380235i
$250$		0			0
$251$			12.2361i			0.772334i	0.922429	+	0.386167i	$0.126201\pi$
$251$			12.2361i			0.772334i	−0.922429	+	0.386167i	$0.873799\pi$
$252$	2.61803			0.164921
$253$			2.76393i			0.173767i
$254$	−20.1803			−1.26623
$255$		0			0
$256$	13.5623			0.847644
$257$	26.4721			1.65129			0.825643	−	0.564193i	$-0.190812\pi$
$257$	26.4721			1.65129			0.825643	+	0.564193i	$0.190812\pi$
$258$		−	0.763932i		−	0.0475603i
$259$			20.1803i			1.25395i
$260$		0			0
$261$	4.47214	+	3.00000i	0.276818	+	0.185695i
$262$	−19.7984			−1.22315
$263$		−	4.94427i		−	0.304877i	−0.988313	−	0.152438i	$-0.951287\pi$
$263$		−	4.94427i		−	0.304877i	0.988313	−	0.152438i	$-0.0487126\pi$
$264$		−	5.00000i		−	0.307729i
$265$		0			0
$266$	−52.8328			−3.23939
$267$	−0.0557281			−0.00341050
$268$	2.90983			0.177746
$269$			21.3607i			1.30238i	0.758913	+	0.651192i	$0.225731\pi$
$269$			21.3607i			1.30238i	−0.758913	+	0.651192i	$0.774269\pi$
$270$		0			0
$271$			23.4164i			1.42245i	0.702967	+	0.711223i	$0.251858\pi$
$271$			23.4164i			1.42245i	−0.702967	+	0.711223i	$0.748142\pi$
$272$			14.5623i			0.882969i
$273$			4.23607i			0.256378i
$274$	−5.70820			−0.344845
$275$		0			0
$276$			0.763932i			0.0459833i
$277$	−6.41641			−0.385525			−0.192762	−	0.981245i	$-0.561745\pi$
$277$	−6.41641			−0.385525			−0.192762	+	0.981245i	$0.561745\pi$
$278$			10.8541i			0.650986i
$279$		−	7.23607i		−	0.433212i
$280$		0			0
$281$	−1.41641			−0.0844958			−0.0422479	−	0.999107i	$-0.513452\pi$
$281$	−1.41641			−0.0844958			−0.0422479	+	0.999107i	$0.513452\pi$
$282$		−	16.5623i		−	0.986271i
$283$	13.8885			0.825588			0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885			0.825588			0.412794	+	0.910824i	$0.364553\pi$
$284$	−2.94427			−0.174710
$285$		0			0
$286$	−3.61803			−0.213939
$287$			18.9443i			1.11825i
$288$		−	3.38197i		−	0.199284i
$289$	8.00000			0.470588
$290$		0			0
$291$	−18.1803			−1.06575
$292$			4.76393i			0.278788i
$293$		−	29.9443i		−	1.74936i	−0.484698	−	0.874682i	$-0.661071\pi$
$293$		−	29.9443i		−	1.74936i	0.484698	−	0.874682i	$-0.338929\pi$
$294$	17.7082			1.03276
$295$		0			0
$296$	10.6525			0.619163
$297$	−2.23607			−0.129750
$298$			8.94427i			0.518128i
$299$	−1.23607			−0.0714837
$300$		0			0
$301$		−	2.00000i		−	0.115278i
$302$		−	19.4164i		−	1.11729i
$303$	−3.94427			−0.226593
$304$			37.4164i			2.14598i
$305$		0			0
$306$	4.85410			0.277491
$307$			28.1803i			1.60834i	0.594401	+	0.804168i	$0.297389\pi$
$307$			28.1803i			1.60834i	−0.594401	+	0.804168i	$0.702611\pi$
$308$			5.85410i			0.333568i
$309$			19.4164i			1.10456i
$310$		0			0
$311$			15.6525i			0.887570i	0.896133	+	0.443785i	$0.146365\pi$
$311$			15.6525i			0.887570i	−0.896133	+	0.443785i	$0.853635\pi$
$312$	2.23607			0.126592
$313$	5.47214			0.309303			0.154652	−	0.987969i	$-0.450574\pi$
$313$	5.47214			0.309303			0.154652	+	0.987969i	$0.450574\pi$
$314$	3.23607			0.182622
$315$		0			0
$316$		−	1.41641i		−	0.0796792i
$317$			23.8328i			1.33858i	0.742999	+	0.669292i	$0.233403\pi$
$317$			23.8328i			1.33858i	−0.742999	+	0.669292i	$0.766597\pi$
$318$	−10.9443			−0.613724
$319$	−6.70820	+	10.0000i	−0.375587	+	0.559893i
$320$		0			0
$321$		−	0.763932i		−	0.0426385i
$322$			8.47214i			0.472134i
$323$	−23.1246			−1.28669
$324$	−0.618034			−0.0343352
$325$		0			0
$326$	12.0000			0.664619
$327$			5.00000i			0.276501i
$328$	10.0000			0.552158
$329$		−	43.3607i		−	2.39055i
$330$		0			0
$331$			8.29180i			0.455758i	0.973689	+	0.227879i	$0.0731791\pi$
$331$			8.29180i			0.455758i	−0.973689	+	0.227879i	$0.926821\pi$
$332$	−3.70820			−0.203514
$333$		−	4.76393i		−	0.261062i
$334$			25.7082i			1.40669i
$335$		0			0
$336$		−	20.5623i		−	1.12177i
$337$			19.2361i			1.04786i	0.851763	+	0.523928i	$0.175534\pi$
$337$			19.2361i			1.04786i	−0.851763	+	0.523928i	$0.824466\pi$
$338$			19.4164i			1.05611i
$339$	−4.52786			−0.245920
$340$		0			0
$341$	16.1803			0.876215
$342$	12.4721			0.674416
$343$	16.7082			0.902158
$344$	−1.05573			−0.0569210
$345$		0			0
$346$		−	9.70820i		−	0.521916i
$347$	22.6525			1.21605			0.608024	−	0.793918i	$-0.291962\pi$
$347$	22.6525			1.21605			0.608024	+	0.793918i	$0.291962\pi$
$348$	−1.85410	+	2.76393i	−0.0993903	+	0.148162i
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		−	1.00000i		−	0.0533761i
$352$	7.56231			0.403072
$353$	−4.65248			−0.247626			−0.123813	−	0.992306i	$-0.539512\pi$
$353$	−4.65248			−0.247626			−0.123813	+	0.992306i	$0.539512\pi$
$354$	14.4721			0.769185
$355$		0			0
$356$		−	0.0344419i		−	0.00182541i
$357$	12.7082			0.672589
$358$			27.8885i			1.47396i
$359$			25.3050i			1.33554i	0.744366	+	0.667772i	$0.232752\pi$
$359$			25.3050i			1.33554i	−0.744366	+	0.667772i	$0.767248\pi$
$360$		0			0
$361$	−40.4164			−2.12718
$362$		−	33.0344i		−	1.73625i
$363$			6.00000i			0.314918i
$364$	−2.61803			−0.137222
$365$		0			0
$366$		−	4.47214i		−	0.233762i
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	6.00000			0.312772
$369$		−	4.47214i		−	0.232810i
$370$		0			0
$371$	−28.6525			−1.48756
$372$	4.47214			0.231869
$373$	3.88854			0.201341			0.100671	−	0.994920i	$-0.467901\pi$
$373$	3.88854			0.201341			0.100671	+	0.994920i	$0.467901\pi$
$374$			10.8541i			0.561252i
$375$		0			0
$376$	−22.8885			−1.18039
$377$	−4.47214	−	3.00000i	−0.230327	−	0.154508i
$378$	−6.85410			−0.352537
$379$		−	14.2918i		−	0.734120i	−0.930197	−	0.367060i	$-0.880364\pi$
$379$		−	14.2918i		−	0.734120i	0.930197	−	0.367060i	$-0.119636\pi$
$380$		0			0
$381$	12.4721			0.638967
$382$	−14.4721			−0.740459
$383$	5.34752			0.273246			0.136623	−	0.990623i	$-0.456375\pi$
$383$	5.34752			0.273246			0.136623	+	0.990623i	$0.456375\pi$
$384$	−13.6180			−0.694942
$385$		0			0
$386$	28.1803			1.43434
$387$			0.472136i			0.0240000i
$388$		−	11.2361i		−	0.570425i
$389$			13.4721i			0.683064i	0.939870	+	0.341532i	$0.110946\pi$
$389$			13.4721i			0.683064i	−0.939870	+	0.341532i	$0.889054\pi$
$390$		0			0
$391$			3.70820i			0.187532i
$392$		−	24.4721i		−	1.23603i
$393$	12.2361			0.617228
$394$		−	6.00000i		−	0.302276i
$395$		0			0
$396$		−	1.38197i		−	0.0694464i
$397$	−6.94427			−0.348523			−0.174262	−	0.984699i	$-0.555754\pi$
$397$	−6.94427			−0.348523			−0.174262	+	0.984699i	$0.555754\pi$
$398$			32.5623i			1.63220i
$399$	32.6525			1.63467
$400$		0			0
$401$	33.7082			1.68331			0.841654	−	0.540018i	$-0.181582\pi$
$401$	33.7082			1.68331			0.841654	+	0.540018i	$0.181582\pi$
$402$	−7.61803			−0.379953
$403$			7.23607i			0.360454i
$404$		−	2.43769i		−	0.121280i
$405$		0			0
$406$	−20.5623	+	30.6525i	−1.02049	+	1.52126i
$407$	10.6525			0.528024
$408$		−	6.70820i		−	0.332106i
$409$			10.5836i			0.523325i	0.965159	+	0.261662i	$0.0842707\pi$
$409$			10.5836i			0.523325i	−0.965159	+	0.261662i	$0.915729\pi$
$410$		0			0
$411$	3.52786			0.174017
$412$	−12.0000			−0.591198
$413$	37.8885			1.86437
$414$		−	2.00000i		−	0.0982946i
$415$		0			0
$416$			3.38197i			0.165815i
$417$		−	6.70820i		−	0.328502i
$418$			27.8885i			1.36407i
$419$	3.81966			0.186603			0.0933013	−	0.995638i	$-0.470258\pi$
$419$	3.81966			0.186603			0.0933013	+	0.995638i	$0.470258\pi$
$420$		0			0
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	−16.1803			−0.787647
$423$			10.2361i			0.497695i
$424$			15.1246i			0.734516i
$425$		0			0
$426$	7.70820			0.373464
$427$		−	11.7082i		−	0.566600i
$428$	0.472136			0.0228216
$429$	2.23607			0.107958
$430$		0			0
$431$	14.3607			0.691730			0.345865	−	0.938284i	$-0.387586\pi$
$431$	14.3607			0.691730			0.345865	+	0.938284i	$0.387586\pi$
$432$			4.85410i			0.233543i
$433$		−	14.2918i		−	0.686820i	−0.939186	−	0.343410i	$-0.888418\pi$
$433$		−	14.2918i		−	0.686820i	0.939186	−	0.343410i	$-0.111582\pi$
$434$	49.5967			2.38072
$435$		0			0
$436$	−3.09017			−0.147992
$437$			9.52786i			0.455780i
$438$		−	12.4721i		−	0.595942i
$439$	13.2918			0.634383			0.317191	−	0.948362i	$-0.397260\pi$
$439$	13.2918			0.634383			0.317191	+	0.948362i	$0.397260\pi$
$440$		0			0
$441$	−10.9443			−0.521156
$442$	−4.85410			−0.230886
$443$			9.65248i			0.458603i	0.973355	+	0.229301i	$0.0736442\pi$
$443$			9.65248i			0.458603i	−0.973355	+	0.229301i	$0.926356\pi$
$444$	2.94427			0.139729
$445$		0			0
$446$		−	20.5623i		−	0.973653i
$447$		−	5.52786i		−	0.261459i
$448$	−17.9443			−0.847787
$449$		−	18.8885i		−	0.891405i	−0.895181	−	0.445703i	$-0.852954\pi$
$449$		−	18.8885i		−	0.891405i	0.895181	−	0.445703i	$-0.147046\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		−	2.79837i		−	0.131624i
$453$			12.0000i			0.563809i
$454$			14.6525i			0.687675i
$455$		0			0
$456$		−	17.2361i		−	0.807153i
$457$	−6.41641			−0.300147			−0.150073	−	0.988675i	$-0.547951\pi$
$457$	−6.41641			−0.300147			−0.150073	+	0.988675i	$0.547951\pi$
$458$	−35.8885			−1.67696
$459$	−3.00000			−0.140028
$460$		0			0
$461$			18.9443i			0.882323i	0.897428	+	0.441161i	$0.145433\pi$
$461$			18.9443i			0.882323i	−0.897428	+	0.441161i	$0.854567\pi$
$462$		−	15.3262i		−	0.713041i
$463$	−0.708204			−0.0329130			−0.0164565	−	0.999865i	$-0.505239\pi$
$463$	−0.708204			−0.0329130			−0.0164565	+	0.999865i	$0.505239\pi$
$464$	21.7082	+	14.5623i	1.00778	+	0.676038i
$465$		0			0
$466$		−	16.9443i		−	0.784928i
$467$		−	13.5279i		−	0.625995i	−0.949754	−	0.312997i	$-0.898667\pi$
$467$		−	13.5279i		−	0.625995i	0.949754	−	0.312997i	$-0.101333\pi$
$468$	0.618034			0.0285686
$469$	−19.9443			−0.920941
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		−	20.0000i		−	0.920575i
$473$	−1.05573			−0.0485424
$474$			3.70820i			0.170323i
$475$		0			0
$476$			7.85410i			0.359992i
$477$	6.76393			0.309699
$478$		−	42.3607i		−	1.93753i
$479$		−	10.4721i		−	0.478484i	−0.970960	−	0.239242i	$-0.923101\pi$
$479$		−	10.4721i		−	0.478484i	0.970960	−	0.239242i	$-0.0768989\pi$
$480$		0			0
$481$			4.76393i			0.217217i
$482$			4.85410i			0.221098i
$483$		−	5.23607i		−	0.238249i
$484$	−3.70820			−0.168555
$485$		0			0
$486$	1.61803			0.0733955
$487$	−20.3607			−0.922630			−0.461315	−	0.887236i	$-0.652622\pi$
$487$	−20.3607			−0.922630			−0.461315	+	0.887236i	$0.652622\pi$
$488$	−6.18034			−0.279771
$489$	−7.41641			−0.335382
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	2.76393			0.124608
$493$	−9.00000	+	13.4164i	−0.405340	+	0.604245i
$494$	−12.4721			−0.561148
$495$		0			0
$496$		−	35.1246i		−	1.57714i
$497$	20.1803			0.905212
$498$	9.70820			0.435035
$499$	21.1803			0.948162			0.474081	−	0.880481i	$-0.342780\pi$
$499$	21.1803			0.948162			0.474081	+	0.880481i	$0.342780\pi$
$500$		0			0
$501$		−	15.8885i		−	0.709848i
$502$	19.7984			0.883645
$503$			25.1803i			1.12274i	0.827566	+	0.561368i	$0.189725\pi$
$503$			25.1803i			1.12274i	−0.827566	+	0.561368i	$0.810275\pi$
$504$			9.47214i			0.421922i
$505$		0			0
$506$	4.47214			0.198811
$507$		−	12.0000i		−	0.532939i
$508$			7.70820i			0.341996i
$509$	−26.8328			−1.18934			−0.594672	−	0.803969i	$-0.702718\pi$
$509$	−26.8328			−1.18934			−0.594672	+	0.803969i	$0.702718\pi$
$510$		0			0
$511$		−	32.6525i		−	1.44446i
$512$			5.29180i			0.233867i
$513$	−7.70820			−0.340326
$514$		−	42.8328i		−	1.88927i
$515$		0			0
$516$	−0.291796			−0.0128456
$517$	−22.8885			−1.00664
$518$	32.6525			1.43467
$519$			6.00000i			0.263371i
$520$		0			0
$521$	−3.52786			−0.154559			−0.0772793	−	0.997009i	$-0.524623\pi$
$521$	−3.52786			−0.154559			−0.0772793	+	0.997009i	$0.524623\pi$
$522$	4.85410	−	7.23607i	0.212458	−	0.316714i
$523$	−33.0689			−1.44600			−0.723001	−	0.690847i	$-0.757238\pi$
$523$	−33.0689			−1.44600			−0.723001	+	0.690847i	$0.757238\pi$
$524$			7.56231i			0.330361i
$525$		0			0
$526$	−8.00000			−0.348817
$527$	21.7082			0.945624
$528$	−10.8541			−0.472364
$529$	−21.4721			−0.933571
$530$		0			0
$531$	−8.94427			−0.388148
$532$			20.1803i			0.874929i
$533$			4.47214i			0.193710i
$534$			0.0901699i			0.00390204i
$535$		0			0
$536$			10.5279i			0.454734i
$537$		−	17.2361i		−	0.743791i
$538$	34.5623			1.49009
$539$		−	24.4721i		−	1.05409i
$540$		0			0
$541$			10.6525i			0.457986i	0.973428	+	0.228993i	$0.0735432\pi$
$541$			10.6525i			0.457986i	−0.973428	+	0.228993i	$0.926457\pi$
$542$	37.8885			1.62745
$543$			20.4164i			0.876152i
$544$	10.1459			0.435002
$545$		0			0
$546$	6.85410			0.293328
$547$	6.34752			0.271401			0.135700	−	0.990750i	$-0.456672\pi$
$547$	6.34752			0.271401			0.135700	+	0.990750i	$0.456672\pi$
$548$			2.18034i			0.0931395i
$549$			2.76393i			0.117962i
$550$		0			0
$551$	−23.1246	+	34.4721i	−0.985142	+	1.46856i
$552$	−2.76393			−0.117641
$553$			9.70820i			0.412835i
$554$			10.3820i			0.441087i
$555$		0			0
$556$	4.14590			0.175825
$557$	16.4721			0.697947			0.348973	−	0.937133i	$-0.386530\pi$
$557$	16.4721			0.697947			0.348973	+	0.937133i	$0.386530\pi$
$558$	−11.7082			−0.495648
$559$		−	0.472136i		−	0.0199692i
$560$		0			0
$561$		−	6.70820i		−	0.283221i
$562$			2.29180i			0.0966736i
$563$			13.0689i			0.550788i	0.961331	+	0.275394i	$0.0888083\pi$
$563$			13.0689i			0.550788i	−0.961331	+	0.275394i	$0.911192\pi$
$564$	−6.32624			−0.266383
$565$		0			0
$566$		−	22.4721i		−	0.944574i
$567$	4.23607			0.177898
$568$		−	10.6525i		−	0.446968i
$569$		−	21.0000i		−	0.880366i	−0.897908	−	0.440183i	$-0.854914\pi$
$569$		−	21.0000i		−	0.880366i	0.897908	−	0.440183i	$-0.145086\pi$
$570$		0			0
$571$	18.8328			0.788129			0.394064	−	0.919083i	$-0.371069\pi$
$571$	18.8328			0.788129			0.394064	+	0.919083i	$0.371069\pi$
$572$			1.38197i			0.0577829i
$573$	8.94427			0.373652
$574$	30.6525			1.27941
$575$		0			0
$576$	4.23607			0.176503
$577$		−	24.8328i		−	1.03380i	−0.856045	−	0.516902i	$-0.827085\pi$
$577$		−	24.8328i		−	1.03380i	0.856045	−	0.516902i	$-0.172915\pi$
$578$		−	12.9443i		−	0.538411i
$579$	−17.4164			−0.723801
$580$		0			0
$581$	25.4164			1.05445
$582$			29.4164i			1.21935i
$583$			15.1246i			0.626397i
$584$	−17.2361			−0.713234
$585$		0			0
$586$	−48.4508			−2.00149
$587$	0.944272			0.0389743			0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272			0.0389743			0.0194871	+	0.999810i	$0.493797\pi$
$588$		−	6.76393i		−	0.278940i
$589$	55.7771			2.29825
$590$		0			0
$591$			3.70820i			0.152535i
$592$		−	23.1246i		−	0.950416i
$593$	−40.1803			−1.65001			−0.825004	−	0.565126i	$-0.808827\pi$
$593$	−40.1803			−1.65001			−0.825004	+	0.565126i	$0.808827\pi$
$594$			3.61803i			0.148450i
$595$		0			0
$596$	3.41641			0.139942
$597$		−	20.1246i		−	0.823646i
$598$			2.00000i			0.0817861i
$599$			1.76393i			0.0720723i	0.999350	+	0.0360362i	$0.0114731\pi$
$599$			1.76393i			0.0720723i	−0.999350	+	0.0360362i	$0.988527\pi$
$600$		0			0
$601$			18.5410i			0.756304i	0.925744	+	0.378152i	$0.123440\pi$
$601$			18.5410i			0.756304i	−0.925744	+	0.378152i	$0.876560\pi$
$602$	−3.23607			−0.131892
$603$	4.70820			0.191733
$604$	−7.41641			−0.301769
$605$		0			0
$606$			6.38197i			0.259250i
$607$			25.4164i			1.03162i	0.856703	+	0.515810i	$0.172509\pi$
$607$			25.4164i			1.03162i	−0.856703	+	0.515810i	$0.827491\pi$
$608$	26.0689			1.05723
$609$	12.7082	−	18.9443i	0.514962	−	0.767661i
$610$		0			0
$611$		−	10.2361i		−	0.414107i
$612$		−	1.85410i		−	0.0749476i
$613$	24.4164			0.986169			0.493085	−	0.869981i	$-0.335869\pi$
$613$	24.4164			0.986169			0.493085	+	0.869981i	$0.335869\pi$
$614$	45.5967			1.84013
$615$		0			0
$616$	−21.1803			−0.853380
$617$			19.8885i			0.800683i	0.916366	+	0.400341i	$0.131108\pi$
$617$			19.8885i			0.800683i	−0.916366	+	0.400341i	$0.868892\pi$
$618$	31.4164			1.26375
$619$			32.5410i			1.30793i	0.756523	+	0.653967i	$0.226896\pi$
$619$			32.5410i			1.30793i	−0.756523	+	0.653967i	$0.773104\pi$
$620$		0			0
$621$			1.23607i			0.0496017i
$622$	25.3262			1.01549
$623$			0.236068i			0.00945786i
$624$		−	4.85410i		−	0.194320i
$625$		0			0
$626$		−	8.85410i		−	0.353881i
$627$		−	17.2361i		−	0.688342i
$628$		−	1.23607i		−	0.0493245i
$629$	14.2918			0.569851
$630$		0			0
$631$	28.7082			1.14286			0.571428	−	0.820652i	$-0.306390\pi$
$631$	28.7082			1.14286			0.571428	+	0.820652i	$0.306390\pi$
$632$	5.12461			0.203846
$633$	10.0000			0.397464
$634$	38.5623			1.53150
$635$		0			0
$636$			4.18034i			0.165761i
$637$	10.9443			0.433628
$638$	16.1803	+	10.8541i	0.640586	+	0.429718i
$639$	−4.76393			−0.188458
$640$		0			0
$641$			19.4721i			0.769103i	0.923104	+	0.384552i	$0.125644\pi$
$641$			19.4721i			0.769103i	−0.923104	+	0.384552i	$0.874356\pi$
$642$	−1.23607			−0.0487837
$643$	−0.708204			−0.0279288			−0.0139644	−	0.999902i	$-0.504445\pi$
$643$	−0.708204			−0.0279288			−0.0139644	+	0.999902i	$0.504445\pi$
$644$	3.23607			0.127519
$645$		0			0
$646$			37.4164i			1.47213i
$647$	−17.5967			−0.691800			−0.345900	−	0.938271i	$-0.612426\pi$
$647$	−17.5967			−0.691800			−0.345900	+	0.938271i	$0.612426\pi$
$648$		−	2.23607i		−	0.0878410i
$649$		−	20.0000i		−	0.785069i
$650$		0			0
$651$	−30.6525			−1.20137
$652$		−	4.58359i		−	0.179507i
$653$		−	32.3050i		−	1.26419i	−0.774891	−	0.632095i	$-0.782195\pi$
$653$		−	32.3050i		−	1.26419i	0.774891	−	0.632095i	$-0.217805\pi$
$654$	8.09017			0.316351
$655$		0			0
$656$		−	21.7082i		−	0.847563i
$657$			7.70820i			0.300726i
$658$	−70.1591			−2.73508
$659$			26.2361i			1.02201i	0.859577	+	0.511006i	$0.170727\pi$
$659$			26.2361i			1.02201i	−0.859577	+	0.511006i	$0.829273\pi$
$660$		0			0
$661$	−29.8328			−1.16036			−0.580181	−	0.814488i	$-0.697018\pi$
$661$	−29.8328			−1.16036			−0.580181	+	0.814488i	$0.697018\pi$
$662$	13.4164			0.521443
$663$	3.00000			0.116510
$664$		−	13.4164i		−	0.520658i
$665$		0			0
$666$	−7.70820			−0.298687
$667$	5.52786	+	3.70820i	0.214040	+	0.143582i
$668$	9.81966			0.379934
$669$			12.7082i			0.491328i
$670$		0			0
$671$	−6.18034			−0.238589
$672$	−14.3262			−0.552647
$673$	4.41641			0.170240			0.0851200	−	0.996371i	$-0.472873\pi$
$673$	4.41641			0.170240			0.0851200	+	0.996371i	$0.472873\pi$
$674$	31.1246			1.19888
$675$		0			0
$676$	7.41641			0.285246
$677$			18.0557i			0.693938i	0.937877	+	0.346969i	$0.112789\pi$
$677$			18.0557i			0.693938i	−0.937877	+	0.346969i	$0.887211\pi$
$678$			7.32624i			0.281362i
$679$			77.0132i			2.95549i
$680$		0			0
$681$		−	9.05573i		−	0.347016i
$682$		−	26.1803i		−	1.00250i
$683$	−32.9443			−1.26058			−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443			−1.26058			−0.630289	+	0.776361i	$0.717064\pi$
$684$		−	4.76393i		−	0.182153i
$685$		0			0
$686$		−	27.0344i		−	1.03218i
$687$	22.1803			0.846233
$688$			2.29180i			0.0873739i
$689$	−6.76393			−0.257685
$690$		0			0
$691$	−30.2361			−1.15023			−0.575117	−	0.818071i	$-0.695044\pi$
$691$	−30.2361			−1.15023			−0.575117	+	0.818071i	$0.695044\pi$
$692$	−3.70820			−0.140965
$693$			9.47214i			0.359817i
$694$		−	36.6525i		−	1.39131i
$695$		0			0
$696$	−10.0000	−	6.70820i	−0.379049	−	0.254274i
$697$	13.4164			0.508183
$698$		−	16.1803i		−	0.612435i
$699$			10.4721i			0.396093i
$700$		0			0
$701$	8.18034			0.308967			0.154484	−	0.987995i	$-0.450629\pi$
$701$	8.18034			0.308967			0.154484	+	0.987995i	$0.450629\pi$
$702$	−1.61803			−0.0610688
$703$	36.7214			1.38497
$704$			9.47214i			0.356995i
$705$		0			0
$706$			7.52786i			0.283315i
$707$			16.7082i			0.628377i
$708$		−	5.52786i		−	0.207750i
$709$	−13.4164			−0.503864			−0.251932	−	0.967745i	$-0.581066\pi$
$709$	−13.4164			−0.503864			−0.251932	+	0.967745i	$0.581066\pi$
$710$		0			0
$711$		−	2.29180i		−	0.0859491i
$712$	0.124612			0.00467002
$713$		−	8.94427i		−	0.334966i
$714$		−	20.5623i		−	0.769525i
$715$		0			0
$716$	10.6525			0.398102
$717$			26.1803i			0.977723i
$718$	40.9443			1.52803
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	82.2492			3.06312
$722$			65.3951i			2.43375i
$723$		−	3.00000i		−	0.111571i
$724$	−12.6180			−0.468946
$725$		0			0
$726$	9.70820			0.360305
$727$			5.81966i			0.215839i	0.994160	+	0.107920i	$0.0344189\pi$
$727$			5.81966i			0.215839i	−0.994160	+	0.107920i	$0.965581\pi$
$728$		−	9.47214i		−	0.351061i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−1.41641			−0.0523877
$732$	−1.70820			−0.0631370
$733$			8.47214i			0.312925i	0.987684	+	0.156463i	$0.0500091\pi$
$733$			8.47214i			0.312925i	−0.987684	+	0.156463i	$0.949991\pi$
$734$	−29.1246			−1.07501
$735$		0			0
$736$		−	4.18034i		−	0.154089i
$737$			10.5279i			0.387799i
$738$	−7.23607			−0.266363
$739$		−	12.1803i		−	0.448061i	−0.974582	−	0.224031i	$-0.928078\pi$
$739$		−	12.1803i		−	0.448061i	0.974582	−	0.224031i	$-0.0719215\pi$
$740$		0			0
$741$	7.70820			0.283168
$742$			46.3607i			1.70195i
$743$			43.0689i			1.58004i	0.613078	+	0.790022i	$0.289931\pi$
$743$			43.0689i			1.58004i	−0.613078	+	0.790022i	$0.710069\pi$
$744$			16.1803i			0.593200i
$745$		0			0
$746$		−	6.29180i		−	0.230359i
$747$	−6.00000			−0.219529
$748$	4.14590			0.151589
$749$	−3.23607			−0.118243
$750$		0			0
$751$		−	26.8328i		−	0.979143i	−0.871963	−	0.489572i	$-0.837153\pi$
$751$		−	26.8328i		−	0.979143i	0.871963	−	0.489572i	$-0.162847\pi$
$752$			49.6869i			1.81190i
$753$	−12.2361			−0.445907
$754$	−4.85410	+	7.23607i	−0.176776	+	0.263522i
$755$		0			0
$756$			2.61803i			0.0952170i
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	−23.1246			−0.839924
$759$	−2.76393			−0.100324
$760$		0			0
$761$	2.65248			0.0961522			0.0480761	−	0.998844i	$-0.484691\pi$
$761$	2.65248			0.0961522			0.0480761	+	0.998844i	$0.484691\pi$
$762$		−	20.1803i		−	0.731057i
$763$	21.1803			0.766780
$764$			5.52786i			0.199991i
$765$		0			0
$766$		−	8.65248i		−	0.312627i
$767$	8.94427			0.322959
$768$			13.5623i			0.489388i
$769$			40.1803i			1.44894i	0.689306	+	0.724470i	$0.257915\pi$
$769$			40.1803i			1.44894i	−0.689306	+	0.724470i	$0.742085\pi$
$770$		0			0
$771$			26.4721i			0.953371i
$772$		−	10.7639i		−	0.387402i
$773$			31.8885i			1.14695i	0.819223	+	0.573476i	$0.194405\pi$
$773$			31.8885i			1.14695i	−0.819223	+	0.573476i	$0.805595\pi$
$774$	0.763932			0.0274590
$775$		0			0
$776$	40.6525			1.45934
$777$	−20.1803			−0.723966
$778$	21.7984			0.781510
$779$	34.4721			1.23509
$780$		0			0
$781$		−	10.6525i		−	0.381176i
$782$	6.00000			0.214560
$783$	−3.00000	+	4.47214i	−0.107211	+	0.159821i
$784$	−53.1246			−1.89731
$785$		0			0
$786$		−	19.7984i		−	0.706185i
$787$	12.0000			0.427754			0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000			0.427754			0.213877	+	0.976861i	$0.431391\pi$
$788$	−2.29180			−0.0816419
$789$	4.94427			0.176021
$790$		0			0
$791$			19.1803i			0.681974i
$792$	5.00000			0.177667
$793$		−	2.76393i		−	0.0981501i
$794$			11.2361i			0.398753i
$795$		0			0
$796$	12.4377			0.440842
$797$		−	44.8328i		−	1.58806i	−0.607879	−	0.794030i	$-0.707979\pi$
$797$		−	44.8328i		−	1.58806i	0.607879	−	0.794030i	$-0.292021\pi$
$798$		−	52.8328i		−	1.87026i
$799$	−30.7082			−1.08638
$800$		0			0
$801$		−	0.0557281i		−	0.00196906i
$802$		−	54.5410i		−	1.92591i
$803$	−17.2361			−0.608248
$804$			2.90983i			0.102622i
$805$		0			0
$806$	11.7082			0.412404
$807$	−21.3607			−0.751932
$808$	8.81966			0.310275
$809$		−	43.3607i		−	1.52448i	−0.647294	−	0.762240i	$-0.724100\pi$
$809$		−	43.3607i		−	1.52448i	0.647294	−	0.762240i	$-0.275900\pi$
$810$		0			0
$811$	46.5967			1.63623			0.818117	−	0.575052i	$-0.195018\pi$
$811$	46.5967			1.63623			0.818117	+	0.575052i	$0.195018\pi$
$812$	11.7082	+	7.85410i	0.410877	+	0.275625i
$813$	−23.4164			−0.821249
$814$		−	17.2361i		−	0.604124i
$815$		0			0
$816$	−14.5623			−0.509783
$817$	−3.63932			−0.127324
$818$	17.1246			0.598748
$819$	−4.23607			−0.148020
$820$		0			0
$821$	−12.4721			−0.435281			−0.217640	−	0.976029i	$-0.569836\pi$
$821$	−12.4721			−0.435281			−0.217640	+	0.976029i	$0.569836\pi$
$822$		−	5.70820i		−	0.199096i
$823$			26.3607i			0.918876i	0.888210	+	0.459438i	$0.151949\pi$
$823$			26.3607i			0.918876i	−0.888210	+	0.459438i	$0.848051\pi$
$824$		−	43.4164i		−	1.51248i
$825$		0			0
$826$		−	61.3050i		−	2.13307i
$827$			29.8885i			1.03933i	0.854371	+	0.519663i	$0.173943\pi$
$827$			29.8885i			1.03933i	−0.854371	+	0.519663i	$0.826057\pi$
$828$	−0.763932			−0.0265485
$829$		−	27.3050i		−	0.948340i	−0.880433	−	0.474170i	$-0.842748\pi$
$829$		−	27.3050i		−	0.948340i	0.880433	−	0.474170i	$-0.157252\pi$
$830$		0			0
$831$		−	6.41641i		−	0.222583i
$832$	−4.23607			−0.146859
$833$		−	32.8328i		−	1.13759i
$834$	−10.8541			−0.375847
$835$		0			0
$836$	10.6525			0.368424
$837$	7.23607			0.250115
$838$		−	6.18034i		−	0.213496i
$839$			0.708204i			0.0244499i	0.999925	+	0.0122250i	$0.00389142\pi$
$839$			0.708204i			0.0244499i	−0.999925	+	0.0122250i	$0.996109\pi$
$840$		0			0
$841$	11.0000	+	26.8328i	0.379310	+	0.925270i
$842$	−32.3607			−1.11522
$843$		−	1.41641i		−	0.0487837i
$844$			6.18034i			0.212736i
$845$		0			0
$846$	16.5623			0.569424
$847$	25.4164			0.873318
$848$	32.8328			1.12748
$849$			13.8885i			0.476654i
$850$		0			0
$851$		−	5.88854i		−	0.201857i
$852$		−	2.94427i		−	0.100869i
$853$		−	45.1935i		−	1.54740i	−0.633555	−	0.773698i	$-0.718405\pi$
$853$		−	45.1935i		−	1.54740i	0.633555	−	0.773698i	$-0.281595\pi$
$854$	−18.9443			−0.648260
$855$		0			0
$856$			1.70820i			0.0583852i
$857$	−37.5967			−1.28428			−0.642140	−	0.766587i	$-0.721953\pi$
$857$	−37.5967			−1.28428			−0.642140	+	0.766587i	$0.721953\pi$
$858$		−	3.61803i		−	0.123518i
$859$			36.3607i			1.24061i	0.784361	+	0.620305i	$0.212991\pi$
$859$			36.3607i			1.24061i	−0.784361	+	0.620305i	$0.787009\pi$
$860$		0			0
$861$	−18.9443			−0.645619
$862$		−	23.2361i		−	0.791424i
$863$	46.6525			1.58807			0.794034	−	0.607873i	$-0.207977\pi$
$863$	46.6525			1.58807			0.794034	+	0.607873i	$0.207977\pi$
$864$	3.38197			0.115057
$865$		0			0
$866$	−23.1246			−0.785806
$867$			8.00000i			0.271694i
$868$		−	18.9443i		−	0.643010i
$869$	5.12461			0.173841
$870$		0			0
$871$	−4.70820			−0.159531
$872$		−	11.1803i		−	0.378614i
$873$		−	18.1803i		−	0.615311i
$874$	15.4164			0.521468
$875$		0			0
$876$	−4.76393			−0.160958
$877$	−32.4721			−1.09651			−0.548253	−	0.836312i	$-0.684707\pi$
$877$	−32.4721			−1.09651			−0.548253	+	0.836312i	$0.684707\pi$
$878$		−	21.5066i		−	0.725812i
$879$	29.9443			1.01000
$880$		0			0
$881$			55.2492i			1.86139i	0.365792	+	0.930697i	$0.380798\pi$
$881$			55.2492i			1.86139i	−0.365792	+	0.930697i	$0.619202\pi$
$882$			17.7082i			0.596266i
$883$	−34.2492			−1.15258			−0.576289	−	0.817246i	$-0.695500\pi$
$883$	−34.2492			−1.15258			−0.576289	+	0.817246i	$0.695500\pi$
$884$			1.85410i			0.0623602i
$885$		0			0
$886$	15.6180			0.524698
$887$		−	17.0689i		−	0.573117i	−0.958063	−	0.286559i	$-0.907489\pi$
$887$		−	17.0689i		−	0.573117i	0.958063	−	0.286559i	$-0.0925113\pi$
$888$			10.6525i			0.357474i
$889$		−	52.8328i		−	1.77196i
$890$		0			0
$891$		−	2.23607i		−	0.0749111i
$892$	−7.85410			−0.262975
$893$	−78.9017			−2.64034
$894$	−8.94427			−0.299141
$895$		0			0
$896$			57.6869i			1.92718i
$897$		−	1.23607i		−	0.0412711i
$898$	−30.5623			−1.01988
$899$	21.7082	−	32.3607i	0.724009	−	1.07929i
$900$		0			0
$901$			20.2918i			0.676018i
$902$		−	16.1803i		−	0.538746i
$903$	2.00000			0.0665558
$904$	10.1246			0.336740
$905$		0			0
$906$	19.4164			0.645067
$907$			26.4721i			0.878993i	0.898244	+	0.439496i	$0.144843\pi$
$907$			26.4721i			0.878993i	−0.898244	+	0.439496i	$0.855157\pi$
$908$	5.59675			0.185735
$909$		−	3.94427i		−	0.130823i
$910$		0			0
$911$			51.1803i			1.69568i	0.530252	+	0.847840i	$0.322097\pi$
$911$			51.1803i			1.69568i	−0.530252	+	0.847840i	$0.677903\pi$
$912$	−37.4164			−1.23898
$913$		−	13.4164i		−	0.444018i
$914$			10.3820i			0.343405i
$915$		0			0
$916$			13.7082i			0.452932i
$917$		−	51.8328i		−	1.71167i
$918$			4.85410i			0.160209i
$919$	−25.6525			−0.846197			−0.423099	−	0.906084i	$-0.639058\pi$
$919$	−25.6525			−0.846197			−0.423099	+	0.906084i	$0.639058\pi$
$920$		0			0
$921$	−28.1803			−0.928574
$922$	30.6525			1.00949
$923$	4.76393			0.156807
$924$	−5.85410			−0.192586
$925$		0			0
$926$			1.14590i			0.0376565i
$927$	−19.4164			−0.637719
$928$	10.1459	−	15.1246i	0.333055	−	0.496490i
$929$	−32.3607			−1.06172			−0.530860	−	0.847460i	$-0.678131\pi$
$929$	−32.3607			−1.06172			−0.530860	+	0.847460i	$0.678131\pi$
$930$		0			0
$931$		−	84.3607i		−	2.76481i
$932$	−6.47214			−0.212002
$933$	−15.6525			−0.512439
$934$	−21.8885			−0.716215
$935$		0			0
$936$			2.23607i			0.0730882i
$937$	−33.0000			−1.07806			−0.539032	−	0.842286i	$-0.681210\pi$
$937$	−33.0000			−1.07806			−0.539032	+	0.842286i	$0.681210\pi$
$938$			32.2705i			1.05367i
$939$			5.47214i			0.178576i
$940$		0			0
$941$	−44.8328			−1.46151			−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328			−1.46151			−0.730754	+	0.682641i	$0.760831\pi$
$942$			3.23607i			0.105437i
$943$		−	5.52786i		−	0.180012i
$944$	−43.4164			−1.41308
$945$		0			0
$946$			1.70820i			0.0555385i
$947$		−	16.8197i		−	0.546566i	−0.961934	−	0.273283i	$-0.911891\pi$
$947$		−	16.8197i		−	0.546566i	0.961934	−	0.273283i	$-0.0881095\pi$
$948$	1.41641			0.0460028
$949$		−	7.70820i		−	0.250219i
$950$		0			0
$951$	−23.8328			−0.772832
$952$	−28.4164			−0.920981
$953$	59.6656			1.93276			0.966380	−	0.257119i	$-0.0827733\pi$
$953$	59.6656			1.93276			0.966380	+	0.257119i	$0.0827733\pi$
$954$		−	10.9443i		−	0.354334i
$955$		0			0
$956$	−16.1803			−0.523310
$957$	−10.0000	−	6.70820i	−0.323254	−	0.216845i
$958$	−16.9443			−0.547445
$959$		−	14.9443i		−	0.482576i
$960$		0			0
$961$	−21.3607			−0.689054
$962$	7.70820			0.248522
$963$	0.763932			0.0246174
$964$	1.85410			0.0597166
$965$		0			0
$966$	−8.47214			−0.272587
$967$			27.1246i			0.872269i	0.899882	+	0.436134i	$0.143653\pi$
$967$			27.1246i			0.872269i	−0.899882	+	0.436134i	$0.856347\pi$
$968$		−	13.4164i		−	0.431220i
$969$		−	23.1246i		−	0.742870i
$970$		0			0
$971$		−	16.5836i		−	0.532193i	−0.963946	−	0.266096i	$-0.914266\pi$
$971$		−	16.5836i		−	0.532193i	0.963946	−	0.266096i	$-0.0857340\pi$
$972$		−	0.618034i		−	0.0198234i
$973$	−28.4164			−0.910988
$974$			32.9443i			1.05560i
$975$		0			0
$976$			13.4164i			0.429449i
$977$	−51.0132			−1.63206			−0.816028	−	0.578013i	$-0.803828\pi$
$977$	−51.0132			−1.63206			−0.816028	+	0.578013i	$0.803828\pi$
$978$			12.0000i			0.383718i
$979$	0.124612			0.00398261
$980$		0			0
$981$	−5.00000			−0.159638
$982$		0			0
$983$		−	13.8885i		−	0.442976i	−0.975163	−	0.221488i	$-0.928909\pi$
$983$		−	13.8885i		−	0.442976i	0.975163	−	0.221488i	$-0.0710913\pi$
$984$			10.0000i			0.318788i
$985$		0			0
$986$	21.7082	+	14.5623i	0.691330	+	0.463758i
$987$	43.3607			1.38019
$988$			4.76393i			0.151561i
$989$			0.583592i			0.0185572i
$990$		0			0
$991$	−21.2918			−0.676356			−0.338178	−	0.941082i	$-0.609811\pi$
$991$	−21.2918			−0.676356			−0.338178	+	0.941082i	$0.609811\pi$
$992$	−24.4721			−0.776991
$993$	−8.29180			−0.263132
$994$		−	32.6525i		−	1.03567i
$995$		0			0
$996$		−	3.70820i		−	0.117499i
$997$		−	24.5836i		−	0.778570i	−0.921117	−	0.389285i	$-0.872722\pi$
$997$		−	24.5836i		−	0.778570i	0.921117	−	0.389285i	$-0.127278\pi$
$998$		−	34.2705i		−	1.08481i
$999$	4.76393			0.150724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.d.e.376.1		4
5.2	odd	4		2175.2.f.b.724.4		4
5.3	odd	4		2175.2.f.a.724.1		4
5.4	even	2		87.2.c.a.28.4	yes	4
15.14	odd	2		261.2.c.b.28.1		4
20.19	odd	2		1392.2.o.i.289.4		4
29.28	even	2	inner	2175.2.d.e.376.4		4
60.59	even	2		4176.2.o.l.289.1		4
145.28	odd	4		2175.2.f.b.724.3		4
145.57	odd	4		2175.2.f.a.724.2		4
145.99	odd	4		2523.2.a.d.1.1		2
145.104	odd	4		2523.2.a.e.1.2		2
145.144	even	2		87.2.c.a.28.1	✓	4
435.104	even	4		7569.2.a.f.1.1		2
435.389	even	4		7569.2.a.n.1.2		2
435.434	odd	2		261.2.c.b.28.4		4
580.579	odd	2		1392.2.o.i.289.2		4
1740.1739	even	2		4176.2.o.l.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.1	✓	4	145.144	even	2
87.2.c.a.28.4	yes	4	5.4	even	2
261.2.c.b.28.1		4	15.14	odd	2
261.2.c.b.28.4		4	435.434	odd	2
1392.2.o.i.289.2		4	580.579	odd	2
1392.2.o.i.289.4		4	20.19	odd	2
2175.2.d.e.376.1		4	1.1	even	1	trivial
2175.2.d.e.376.4		4	29.28	even	2	inner
2175.2.f.a.724.1		4	5.3	odd	4
2175.2.f.a.724.2		4	145.57	odd	4
2175.2.f.b.724.3		4	145.28	odd	4
2175.2.f.b.724.4		4	5.2	odd	4
2523.2.a.d.1.1		2	145.99	odd	4
2523.2.a.e.1.2		2	145.104	odd	4
4176.2.o.l.289.1		4	60.59	even	2
4176.2.o.l.289.2		4	1740.1739	even	2
7569.2.a.f.1.1		2	435.104	even	4
7569.2.a.n.1.2		2	435.389	even	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +4.23607 q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.61803i q^{2} +1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} +4.23607 q^{7} -2.23607i q^{8} -1.00000 q^{9} -2.23607i q^{11} -0.618034i q^{12} +1.00000 q^{13} -6.85410i q^{14} -4.85410 q^{16} -3.00000i q^{17} +1.61803i q^{18} -7.70820i q^{19} +4.23607i q^{21} -3.61803 q^{22} -1.23607 q^{23} +2.23607 q^{24} -1.61803i q^{26} -1.00000i q^{27} -2.61803 q^{28} +(-4.47214 - 3.00000i) q^{29} +7.23607i q^{31} +3.38197i q^{32} +2.23607 q^{33} -4.85410 q^{34} +0.618034 q^{36} +4.76393i q^{37} -12.4721 q^{38} +1.00000i q^{39} +4.47214i q^{41} +6.85410 q^{42} -0.472136i q^{43} +1.38197i q^{44} +2.00000i q^{46} -10.2361i q^{47} -4.85410i q^{48} +10.9443 q^{49} +3.00000 q^{51} -0.618034 q^{52} -6.76393 q^{53} -1.61803 q^{54} -9.47214i q^{56} +7.70820 q^{57} +(-4.85410 + 7.23607i) q^{58} +8.94427 q^{59} -2.76393i q^{61} +11.7082 q^{62} -4.23607 q^{63} -4.23607 q^{64} -3.61803i q^{66} -4.70820 q^{67} +1.85410i q^{68} -1.23607i q^{69} +4.76393 q^{71} +2.23607i q^{72} -7.70820i q^{73} +7.70820 q^{74} +4.76393i q^{76} -9.47214i q^{77} +1.61803 q^{78} +2.29180i q^{79} +1.00000 q^{81} +7.23607 q^{82} +6.00000 q^{83} -2.61803i q^{84} -0.763932 q^{86} +(3.00000 - 4.47214i) q^{87} -5.00000 q^{88} +0.0557281i q^{89} +4.23607 q^{91} +0.763932 q^{92} -7.23607 q^{93} -16.5623 q^{94} -3.38197 q^{96} +18.1803i q^{97} -17.7082i q^{98} +2.23607i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 2 q^{6} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 2 * q^6 + 8 * q^7 - 4 * q^9	\( 4 q + 2 q^{4} + 2 q^{6} + 8 q^{7} - 4 q^{9} + 4 q^{13} - 6 q^{16} - 10 q^{22} + 4 q^{23} - 6 q^{28} - 6 q^{34} - 2 q^{36} - 32 q^{38} + 14 q^{42} + 8 q^{49} + 12 q^{51} + 2 q^{52} - 36 q^{53} - 2 q^{54} + 4 q^{57} - 6 q^{58} + 20 q^{62} - 8 q^{63} - 8 q^{64} + 8 q^{67} + 28 q^{71} + 4 q^{74} + 2 q^{78} + 4 q^{81} + 20 q^{82} + 24 q^{83} - 12 q^{86} + 12 q^{87} - 20 q^{88} + 8 q^{91} + 12 q^{92} - 20 q^{93} - 26 q^{94} - 18 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 + 2 * q^6 + 8 * q^7 - 4 * q^9 + 4 * q^13 - 6 * q^16 - 10 * q^22 + 4 * q^23 - 6 * q^28 - 6 * q^34 - 2 * q^36 - 32 * q^38 + 14 * q^42 + 8 * q^49 + 12 * q^51 + 2 * q^52 - 36 * q^53 - 2 * q^54 + 4 * q^57 - 6 * q^58 + 20 * q^62 - 8 * q^63 - 8 * q^64 + 8 * q^67 + 28 * q^71 + 4 * q^74 + 2 * q^78 + 4 * q^81 + 20 * q^82 + 24 * q^83 - 12 * q^86 + 12 * q^87 - 20 * q^88 + 8 * q^91 + 12 * q^92 - 20 * q^93 - 26 * q^94 - 18 * q^96

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