LMFDB - Embedded newform 2175.2.f.b.724.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(724,2175)]

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, 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("2175.724");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.3674624396$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			724.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	2175.724
Dual form			2175.2.f.b.724.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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 $

$2$	1.61803			1.14412			0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803			1.14412			0.572061	+	0.820211i	$0.306144\pi$
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	0.618034			0.309017
$5$		0			0
$5$		0			0
$6$	1.61803			0.660560
$7$			4.23607i			1.60108i	0.599277	+	0.800542i	$0.295455\pi$
$7$			4.23607i			1.60108i	−0.599277	+	0.800542i	$0.704545\pi$
$8$	−2.23607			−0.790569
$9$	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	$-0.109446\pi$
$12$	0.618034			0.178411
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$			6.85410i			1.83184i
$15$		0			0
$16$	−4.85410			−1.21353
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$	1.61803			0.381374
$19$			7.70820i			1.76838i	0.467124	+	0.884192i	$0.345290\pi$
$19$			7.70820i			1.76838i	−0.467124	+	0.884192i	$0.654710\pi$
$20$		0			0
$21$			4.23607i			0.924386i
$22$		−	3.61803i		−	0.771367i
$23$			1.23607i			0.257738i	0.991662	+	0.128869i	$0.0411347\pi$
$23$			1.23607i			0.257738i	−0.991662	+	0.128869i	$0.958865\pi$
$24$	−2.23607			−0.456435
$25$		0			0
$26$		−	1.61803i		−	0.317323i
$27$	1.00000			0.192450
$28$			2.61803i			0.494762i
$29$	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	$0.774845\pi$
$32$	−3.38197			−0.597853
$33$		−	2.23607i		−	0.389249i
$34$	4.85410			0.832472
$35$		0			0
$36$	0.618034			0.103006
$37$	−4.76393			−0.783186			−0.391593	−	0.920139i	$-0.628076\pi$
$37$	−4.76393			−0.783186			−0.391593	+	0.920139i	$0.628076\pi$
$38$			12.4721i			2.02325i
$39$		−	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	$0.886448\pi$
$42$			6.85410i			1.05761i
$43$	−0.472136			−0.0720001			−0.0360000	−	0.999352i	$-0.511462\pi$
$43$	−0.472136			−0.0720001			−0.0360000	+	0.999352i	$0.511462\pi$
$44$		−	1.38197i		−	0.208339i
$45$		0			0
$46$			2.00000i			0.294884i
$47$	10.2361			1.49308			0.746542	−	0.665338i	$-0.231713\pi$
$47$	10.2361			1.49308			0.746542	+	0.665338i	$0.231713\pi$
$48$	−4.85410			−0.700629
$49$	−10.9443			−1.56347
$50$		0			0
$51$	3.00000			0.420084
$52$		−	0.618034i		−	0.0857059i
$53$			6.76393i			0.929098i	0.885548	+	0.464549i	$0.153783\pi$
$53$			6.76393i			0.929098i	−0.885548	+	0.464549i	$0.846217\pi$
$54$	1.61803			0.220187
$55$		0			0
$56$		−	9.47214i		−	1.26577i
$57$			7.70820i			1.02098i
$58$	7.23607	+	4.85410i	0.950142	+	0.637375i
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$		−	2.76393i		−	0.353885i	−0.984221	−	0.176943i	$-0.943379\pi$
$61$		−	2.76393i		−	0.353885i	0.984221	−	0.176943i	$-0.0566207\pi$
$62$			11.7082i			1.48694i
$63$			4.23607i			0.533694i
$64$	4.23607			0.529508
$65$		0			0
$66$		−	3.61803i		−	0.445349i
$67$		−	4.70820i		−	0.575199i	−0.957751	−	0.287599i	$-0.907143\pi$
$67$		−	4.70820i		−	0.575199i	0.957751	−	0.287599i	$-0.0928571\pi$
$68$	1.85410			0.224843
$69$			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	$0.408774\pi$
$72$	−2.23607			−0.263523
$73$	−7.70820			−0.902177			−0.451089	−	0.892479i	$-0.648964\pi$
$73$	−7.70820			−0.902177			−0.451089	+	0.892479i	$0.648964\pi$
$74$	−7.70820			−0.896061
$75$		0			0
$76$			4.76393i			0.546460i
$77$	9.47214			1.07945
$78$		−	1.61803i		−	0.183206i
$79$		−	2.29180i		−	0.257847i	−0.991655	−	0.128924i	$-0.958848\pi$
$79$		−	2.29180i		−	0.257847i	0.991655	−	0.128924i	$-0.0411522\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$			7.23607i			0.799090i
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$			2.61803i			0.285651i
$85$		0			0
$86$	−0.763932			−0.0823769
$87$	4.47214	+	3.00000i	0.479463	+	0.321634i
$88$			5.00000i			0.533002i
$89$		−	0.0557281i		−	0.00590717i	−0.999996	−	0.00295358i	$-0.999060\pi$
$89$		−	0.0557281i		−	0.00590717i	0.999996	−	0.00295358i	$-0.000940156\pi$
$90$		0			0
$91$	4.23607			0.444061
$92$			0.763932i			0.0796454i
$93$			7.23607i			0.750345i
$94$	16.5623			1.70827
$95$		0			0
$96$	−3.38197			−0.345170
$97$	−18.1803			−1.84593			−0.922967	−	0.384879i	$-0.874243\pi$
$97$	−18.1803			−1.84593			−0.922967	+	0.384879i	$0.874243\pi$
$98$	−17.7082			−1.78880
$99$		−	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	$0.937129\pi$
$102$	4.85410			0.480628
$103$		−	19.4164i		−	1.91316i	−0.291477	−	0.956578i	$-0.594147\pi$
$103$		−	19.4164i		−	1.91316i	0.291477	−	0.956578i	$-0.405853\pi$
$104$			2.23607i			0.219265i
$105$		0			0
$106$			10.9443i			1.06300i
$107$		−	0.763932i		−	0.0738521i	−0.999318	−	0.0369260i	$-0.988243\pi$
$107$		−	0.763932i		−	0.0738521i	0.999318	−	0.0369260i	$-0.0117566\pi$
$108$	0.618034			0.0594703
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$	−4.76393			−0.452172
$112$		−	20.5623i		−	1.94296i
$113$	4.52786			0.425946			0.212973	−	0.977058i	$-0.431685\pi$
$113$	4.52786			0.425946			0.212973	+	0.977058i	$0.431685\pi$
$114$			12.4721i			1.16812i
$115$		0			0
$116$	2.76393	+	1.85410i	0.256625	+	0.172149i
$117$		−	1.00000i		−	0.0924500i
$118$	−14.4721			−1.33227
$119$			12.7082i			1.16496i
$120$		0			0
$121$	6.00000			0.545455
$122$		−	4.47214i		−	0.404888i
$123$			4.47214i			0.403239i
$124$			4.47214i			0.401610i
$125$		0			0
$126$			6.85410i			0.610612i
$127$	12.4721			1.10672			0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721			1.10672			0.553362	+	0.832941i	$0.313345\pi$
$128$	13.6180			1.20368
$129$	−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	$-0.179513\pi$
$132$		−	1.38197i		−	0.120285i
$133$	−32.6525			−2.83133
$134$		−	7.61803i		−	0.658098i
$135$		0			0
$136$	−6.70820			−0.575224
$137$	3.52786			0.301406			0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786			0.301406			0.150703	+	0.988579i	$0.451846\pi$
$138$			2.00000i			0.170251i
$139$	6.70820			0.568982			0.284491	−	0.958679i	$-0.408175\pi$
$139$	6.70820			0.568982			0.284491	+	0.958679i	$0.408175\pi$
$140$		0			0
$141$	10.2361			0.862032
$142$	7.70820			0.646858
$143$	−2.23607			−0.186989
$144$	−4.85410			−0.404508
$145$		0			0
$146$	−12.4721			−1.03220
$147$	−10.9443			−0.902668
$148$	−2.94427			−0.242018
$149$	5.52786			0.452860			0.226430	−	0.974027i	$-0.427295\pi$
$149$	5.52786			0.452860			0.226430	+	0.974027i	$0.427295\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$		−	17.2361i		−	1.39803i
$153$	3.00000			0.242536
$154$	15.3262			1.23502
$155$		0			0
$156$		−	0.618034i		−	0.0494823i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	3.70820i		−	0.295009i
$159$			6.76393i			0.536415i
$160$		0			0
$161$	−5.23607			−0.412660
$162$	1.61803			0.127125
$163$	7.41641			0.580898			0.290449	−	0.956890i	$-0.406195\pi$
$163$	7.41641			0.580898			0.290449	+	0.956890i	$0.406195\pi$
$164$			2.76393i			0.215827i
$165$		0			0
$166$		−	9.70820i		−	0.753503i
$167$		−	15.8885i		−	1.22949i	−0.788725	−	0.614746i	$-0.789258\pi$
$167$		−	15.8885i		−	1.22949i	0.788725	−	0.614746i	$-0.210742\pi$
$168$		−	9.47214i		−	0.730791i
$169$	12.0000			0.923077
$170$		0			0
$171$			7.70820i			0.589461i
$172$	−0.291796			−0.0222492
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	7.23607	+	4.85410i	0.548565	+	0.367989i
$175$		0			0
$176$			10.8541i			0.818159i
$177$	−8.94427			−0.672293
$178$		−	0.0901699i		−	0.00675852i
$179$	17.2361			1.28828			0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361			1.28828			0.644142	+	0.764906i	$0.277215\pi$
$180$		0			0
$181$	20.4164			1.51754			0.758770	−	0.651359i	$-0.225801\pi$
$181$	20.4164			1.51754			0.758770	+	0.651359i	$0.225801\pi$
$182$	6.85410			0.508060
$183$		−	2.76393i		−	0.204316i
$184$		−	2.76393i		−	0.203760i
$185$		0			0
$186$			11.7082i			0.858487i
$187$		−	6.70820i		−	0.490552i
$188$	6.32624			0.461388
$189$			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	$-0.104891\pi$
$192$	4.23607			0.305712
$193$	17.4164			1.25366			0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164			1.25366			0.626830	+	0.779156i	$0.284352\pi$
$194$	−29.4164			−2.11198
$195$		0			0
$196$	−6.76393			−0.483138
$197$			3.70820i			0.264199i	0.991236	+	0.132099i	$0.0421718\pi$
$197$			3.70820i			0.264199i	−0.991236	+	0.132099i	$0.957828\pi$
$198$		−	3.61803i		−	0.257122i
$199$	20.1246			1.42660			0.713298	−	0.700861i	$-0.247201\pi$
$199$	20.1246			1.42660			0.713298	+	0.700861i	$0.247201\pi$
$200$		0			0
$201$		−	4.70820i		−	0.332091i
$202$			6.38197i			0.449034i
$203$	−12.7082	+	18.9443i	−0.891941	+	1.32963i
$204$	1.85410			0.129813
$205$		0			0
$206$		−	31.4164i		−	2.18888i
$207$			1.23607i			0.0859127i
$208$			4.85410i			0.336571i
$209$	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	$-0.111855\pi$
$212$			4.18034i			0.287107i
$213$	4.76393			0.326419
$214$		−	1.23607i		−	0.0844959i
$215$		0			0
$216$	−2.23607			−0.152145
$217$	−30.6525			−2.08083
$218$	−8.09017			−0.547935
$219$	−7.70820			−0.520872
$220$		0			0
$221$		−	3.00000i		−	0.201802i
$222$	−7.70820			−0.517341
$223$		−	12.7082i		−	0.851004i	−0.904957	−	0.425502i	$-0.860097\pi$
$223$		−	12.7082i		−	0.851004i	0.904957	−	0.425502i	$-0.139903\pi$
$224$		−	14.3262i		−	0.957212i
$225$		0			0
$226$	7.32624			0.487334
$227$		−	9.05573i		−	0.601050i	−0.953774	−	0.300525i	$-0.902838\pi$
$227$		−	9.05573i		−	0.601050i	0.953774	−	0.300525i	$-0.0971618\pi$
$228$			4.76393i			0.315499i
$229$			22.1803i			1.46572i	0.680380	+	0.732859i	$0.261815\pi$
$229$			22.1803i			1.46572i	−0.680380	+	0.732859i	$0.738185\pi$
$230$		0			0
$231$	9.47214			0.623221
$232$	−10.0000	−	6.70820i	−0.656532	−	0.440415i
$233$		−	10.4721i		−	0.686052i	−0.939326	−	0.343026i	$-0.888548\pi$
$233$		−	10.4721i		−	0.686052i	0.939326	−	0.343026i	$-0.111452\pi$
$234$		−	1.61803i		−	0.105774i
$235$		0			0
$236$	−5.52786			−0.359833
$237$		−	2.29180i		−	0.148868i
$238$			20.5623i			1.33286i
$239$	−26.1803			−1.69347			−0.846733	−	0.532019i	$-0.821434\pi$
$239$	−26.1803			−1.69347			−0.846733	+	0.532019i	$0.821434\pi$
$240$		0			0
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$	9.70820			0.624067
$243$	1.00000			0.0641500
$244$		−	1.70820i		−	0.109357i
$245$		0			0
$246$			7.23607i			0.461355i
$247$	7.70820			0.490461
$248$		−	16.1803i		−	1.02745i
$249$		−	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	$0.873799\pi$
$252$			2.61803i			0.164921i
$253$	2.76393			0.173767
$254$	20.1803			1.26623
$255$		0			0
$256$	13.5623			0.847644
$257$			26.4721i			1.65129i	0.564193	+	0.825643i	$0.309188\pi$
$257$			26.4721i			1.65129i	−0.564193	+	0.825643i	$0.690812\pi$
$258$	−0.763932			−0.0475603
$259$		−	20.1803i		−	1.25395i
$260$		0			0
$261$	4.47214	+	3.00000i	0.276818	+	0.185695i
$262$		−	19.7984i		−	1.22315i
$263$	−4.94427			−0.304877			−0.152438	−	0.988313i	$-0.548713\pi$
$263$	−4.94427			−0.304877			−0.152438	+	0.988313i	$0.548713\pi$
$264$			5.00000i			0.307729i
$265$		0			0
$266$	−52.8328			−3.23939
$267$		−	0.0557281i		−	0.00341050i
$268$		−	2.90983i		−	0.177746i
$269$		−	21.3607i		−	1.30238i	−0.758913	−	0.651192i	$-0.774269\pi$
$269$		−	21.3607i		−	1.30238i	0.758913	−	0.651192i	$-0.225731\pi$
$270$		0			0
$271$			23.4164i			1.42245i	0.702967	+	0.711223i	$0.251858\pi$
$271$			23.4164i			1.42245i	−0.702967	+	0.711223i	$0.748142\pi$
$272$	−14.5623			−0.882969
$273$	4.23607			0.256378
$274$	5.70820			0.344845
$275$		0			0
$276$			0.763932i			0.0459833i
$277$		−	6.41641i		−	0.385525i	−0.981245	−	0.192762i	$-0.938255\pi$
$277$		−	6.41641i		−	0.385525i	0.981245	−	0.192762i	$-0.0617446\pi$
$278$	10.8541			0.650986
$279$			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	$0.513452\pi$
$282$	16.5623			0.986271
$283$		−	13.8885i		−	0.825588i	−0.910824	−	0.412794i	$-0.864553\pi$
$283$		−	13.8885i		−	0.825588i	0.910824	−	0.412794i	$-0.135447\pi$
$284$	2.94427			0.174710
$285$		0			0
$286$	−3.61803			−0.213939
$287$	−18.9443			−1.11825
$288$	−3.38197			−0.199284
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−18.1803			−1.06575
$292$	−4.76393			−0.278788
$293$	−29.9443			−1.74936			−0.874682	−	0.484698i	$-0.838929\pi$
$293$	−29.9443			−1.74936			−0.874682	+	0.484698i	$0.838929\pi$
$294$	−17.7082			−1.03276
$295$		0			0
$296$	10.6525			0.619163
$297$		−	2.23607i		−	0.129750i
$298$	8.94427			0.518128
$299$	1.23607			0.0714837
$300$		0			0
$301$		−	2.00000i		−	0.115278i
$302$	19.4164			1.11729
$303$			3.94427i			0.226593i
$304$		−	37.4164i		−	2.14598i
$305$		0			0
$306$	4.85410			0.277491
$307$	−28.1803			−1.60834			−0.804168	−	0.594401i	$-0.797389\pi$
$307$	−28.1803			−1.60834			−0.804168	+	0.594401i	$0.797389\pi$
$308$	5.85410			0.333568
$309$		−	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	$0.853635\pi$
$312$			2.23607i			0.126592i
$313$		−	5.47214i		−	0.309303i	−0.987969	−	0.154652i	$-0.950574\pi$
$313$		−	5.47214i		−	0.309303i	0.987969	−	0.154652i	$-0.0494255\pi$
$314$	−3.23607			−0.182622
$315$		0			0
$316$		−	1.41641i		−	0.0796792i
$317$	−23.8328			−1.33858			−0.669292	−	0.742999i	$-0.733403\pi$
$317$	−23.8328			−1.33858			−0.669292	+	0.742999i	$0.733403\pi$
$318$			10.9443i			0.613724i
$319$	6.70820	−	10.0000i	0.375587	−	0.559893i
$320$		0			0
$321$		−	0.763932i		−	0.0426385i
$322$	−8.47214			−0.472134
$323$			23.1246i			1.28669i
$324$	0.618034			0.0343352
$325$		0			0
$326$	12.0000			0.664619
$327$	−5.00000			−0.276501
$328$		−	10.0000i		−	0.552158i
$329$			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	$0.926821\pi$
$332$		−	3.70820i		−	0.203514i
$333$	−4.76393			−0.261062
$334$		−	25.7082i		−	1.40669i
$335$		0			0
$336$		−	20.5623i		−	1.12177i
$337$	−19.2361			−1.04786			−0.523928	−	0.851763i	$-0.675534\pi$
$337$	−19.2361			−1.04786			−0.523928	+	0.851763i	$0.675534\pi$
$338$	19.4164			1.05611
$339$	4.52786			0.245920
$340$		0			0
$341$	16.1803			0.876215
$342$			12.4721i			0.674416i
$343$		−	16.7082i		−	0.902158i
$344$	1.05573			0.0569210
$345$		0			0
$346$		−	9.70820i		−	0.521916i
$347$			22.6525i			1.21605i	0.793918	+	0.608024i	$0.208038\pi$
$347$			22.6525i			1.21605i	−0.793918	+	0.608024i	$0.791962\pi$
$348$	2.76393	+	1.85410i	0.148162	+	0.0993903i
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		−	1.00000i		−	0.0533761i
$352$			7.56231i			0.403072i
$353$			4.65248i			0.247626i	0.992306	+	0.123813i	$0.0395123\pi$
$353$			4.65248i			0.247626i	−0.992306	+	0.123813i	$0.960488\pi$
$354$	−14.4721			−0.769185
$355$		0			0
$356$		−	0.0344419i		−	0.00182541i
$357$			12.7082i			0.672589i
$358$	27.8885			1.47396
$359$		−	25.3050i		−	1.33554i	−0.744366	−	0.667772i	$-0.767248\pi$
$359$		−	25.3050i		−	1.33554i	0.744366	−	0.667772i	$-0.232752\pi$
$360$		0			0
$361$	−40.4164			−2.12718
$362$	33.0344			1.73625
$363$	6.00000			0.314918
$364$	2.61803			0.137222
$365$		0			0
$366$		−	4.47214i		−	0.233762i
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$		−	6.00000i		−	0.312772i
$369$			4.47214i			0.232810i
$370$		0			0
$371$	−28.6525			−1.48756
$372$			4.47214i			0.231869i
$373$		−	3.88854i		−	0.201341i	−0.994920	−	0.100671i	$-0.967901\pi$
$373$		−	3.88854i		−	0.201341i	0.994920	−	0.100671i	$-0.0320988\pi$
$374$		−	10.8541i		−	0.561252i
$375$		0			0
$376$	−22.8885			−1.18039
$377$	3.00000	−	4.47214i	0.154508	−	0.230327i
$378$			6.85410i			0.352537i
$379$			14.2918i			0.734120i	0.930197	+	0.367060i	$0.119636\pi$
$379$			14.2918i			0.734120i	−0.930197	+	0.367060i	$0.880364\pi$
$380$		0			0
$381$	12.4721			0.638967
$382$		−	14.4721i		−	0.740459i
$383$		−	5.34752i		−	0.273246i	−0.990623	−	0.136623i	$-0.956375\pi$
$383$		−	5.34752i		−	0.273246i	0.990623	−	0.136623i	$-0.0436248\pi$
$384$	13.6180			0.694942
$385$		0			0
$386$	28.1803			1.43434
$387$	−0.472136			−0.0240000
$388$	−11.2361			−0.570425
$389$		−	13.4721i		−	0.683064i	−0.939870	−	0.341532i	$-0.889054\pi$
$389$		−	13.4721i		−	0.683064i	0.939870	−	0.341532i	$-0.110946\pi$
$390$		0			0
$391$			3.70820i			0.187532i
$392$	24.4721			1.23603
$393$		−	12.2361i		−	0.617228i
$394$			6.00000i			0.302276i
$395$		0			0
$396$		−	1.38197i		−	0.0694464i
$397$		−	6.94427i		−	0.348523i	−0.984699	−	0.174262i	$-0.944246\pi$
$397$		−	6.94427i		−	0.348523i	0.984699	−	0.174262i	$-0.0557538\pi$
$398$	32.5623			1.63220
$399$	−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	$0.181582\pi$
$402$		−	7.61803i		−	0.379953i
$403$	7.23607			0.360454
$404$			2.43769i			0.121280i
$405$		0			0
$406$	−20.5623	+	30.6525i	−1.02049	+	1.52126i
$407$			10.6525i			0.528024i
$408$	−6.70820			−0.332106
$409$		−	10.5836i		−	0.523325i	−0.965159	−	0.261662i	$-0.915729\pi$
$409$		−	10.5836i		−	0.523325i	0.965159	−	0.261662i	$-0.0842707\pi$
$410$		0			0
$411$	3.52786			0.174017
$412$		−	12.0000i		−	0.591198i
$413$		−	37.8885i		−	1.86437i
$414$			2.00000i			0.0982946i
$415$		0			0
$416$			3.38197i			0.165815i
$417$	6.70820			0.328502
$418$	27.8885			1.36407
$419$	−3.81966			−0.186603			−0.0933013	−	0.995638i	$-0.529742\pi$
$419$	−3.81966			−0.186603			−0.0933013	+	0.995638i	$0.529742\pi$
$420$		0			0
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$		−	16.1803i		−	0.787647i
$423$	10.2361			0.497695
$424$		−	15.1246i		−	0.734516i
$425$		0			0
$426$	7.70820			0.373464
$427$	11.7082			0.566600
$428$		−	0.472136i		−	0.0228216i
$429$	−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	$0.387586\pi$
$432$	−4.85410			−0.233543
$433$	−14.2918			−0.686820			−0.343410	−	0.939186i	$-0.611582\pi$
$433$	−14.2918			−0.686820			−0.343410	+	0.939186i	$0.611582\pi$
$434$	−49.5967			−2.38072
$435$		0			0
$436$	−3.09017			−0.147992
$437$	−9.52786			−0.455780
$438$	−12.4721			−0.595942
$439$	−13.2918			−0.634383			−0.317191	−	0.948362i	$-0.602740\pi$
$439$	−13.2918			−0.634383			−0.317191	+	0.948362i	$0.602740\pi$
$440$		0			0
$441$	−10.9443			−0.521156
$442$		−	4.85410i		−	0.230886i
$443$	9.65248			0.458603			0.229301	−	0.973355i	$-0.426356\pi$
$443$	9.65248			0.458603			0.229301	+	0.973355i	$0.426356\pi$
$444$	−2.94427			−0.139729
$445$		0			0
$446$		−	20.5623i		−	0.973653i
$447$	5.52786			0.261459
$448$			17.9443i			0.847787i
$449$			18.8885i			0.891405i	0.895181	+	0.445703i	$0.147046\pi$
$449$			18.8885i			0.891405i	−0.895181	+	0.445703i	$0.852954\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$	2.79837			0.131624
$453$	12.0000			0.563809
$454$		−	14.6525i		−	0.687675i
$455$		0			0
$456$		−	17.2361i		−	0.807153i
$457$		−	6.41641i		−	0.300147i	−0.988675	−	0.150073i	$-0.952049\pi$
$457$		−	6.41641i		−	0.300147i	0.988675	−	0.150073i	$-0.0479510\pi$
$458$			35.8885i			1.67696i
$459$	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	$0.854567\pi$
$462$	15.3262			0.713041
$463$			0.708204i			0.0329130i	0.999865	+	0.0164565i	$0.00523851\pi$
$463$			0.708204i			0.0329130i	−0.999865	+	0.0164565i	$0.994761\pi$
$464$	−21.7082	−	14.5623i	−1.00778	−	0.676038i
$465$		0			0
$466$		−	16.9443i		−	0.784928i
$467$	13.5279			0.625995			0.312997	−	0.949754i	$-0.398667\pi$
$467$	13.5279			0.625995			0.312997	+	0.949754i	$0.398667\pi$
$468$		−	0.618034i		−	0.0285686i
$469$	19.9443			0.920941
$470$		0			0
$471$	−2.00000			−0.0921551
$472$	20.0000			0.920575
$473$			1.05573i			0.0485424i
$474$		−	3.70820i		−	0.170323i
$475$		0			0
$476$			7.85410i			0.359992i
$477$			6.76393i			0.309699i
$478$	−42.3607			−1.93753
$479$			10.4721i			0.478484i	0.970960	+	0.239242i	$0.0768989\pi$
$479$			10.4721i			0.478484i	−0.970960	+	0.239242i	$0.923101\pi$
$480$		0			0
$481$			4.76393i			0.217217i
$482$	−4.85410			−0.221098
$483$	−5.23607			−0.238249
$484$	3.70820			0.168555
$485$		0			0
$486$	1.61803			0.0733955
$487$		−	20.3607i		−	0.922630i	−0.887236	−	0.461315i	$-0.847378\pi$
$487$		−	20.3607i		−	0.922630i	0.887236	−	0.461315i	$-0.152622\pi$
$488$			6.18034i			0.279771i
$489$	7.41641			0.335382
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$			2.76393i			0.124608i
$493$	13.4164	+	9.00000i	0.604245	+	0.405340i
$494$	12.4721			0.561148
$495$		0			0
$496$		−	35.1246i		−	1.57714i
$497$			20.1803i			0.905212i
$498$		−	9.70820i		−	0.435035i
$499$	−21.1803			−0.948162			−0.474081	−	0.880481i	$-0.657220\pi$
$499$	−21.1803			−0.948162			−0.474081	+	0.880481i	$0.657220\pi$
$500$		0			0
$501$		−	15.8885i		−	0.709848i
$502$			19.7984i			0.883645i
$503$	25.1803			1.12274			0.561368	−	0.827566i	$-0.310275\pi$
$503$	25.1803			1.12274			0.561368	+	0.827566i	$0.310275\pi$
$504$		−	9.47214i		−	0.421922i
$505$		0			0
$506$	4.47214			0.198811
$507$	12.0000			0.532939
$508$	7.70820			0.341996
$509$	26.8328			1.18934			0.594672	−	0.803969i	$-0.297282\pi$
$509$	26.8328			1.18934			0.594672	+	0.803969i	$0.297282\pi$
$510$		0			0
$511$		−	32.6525i		−	1.44446i
$512$	−5.29180			−0.233867
$513$			7.70820i			0.340326i
$514$			42.8328i			1.88927i
$515$		0			0
$516$	−0.291796			−0.0128456
$517$		−	22.8885i		−	1.00664i
$518$		−	32.6525i		−	1.43467i
$519$		−	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	$0.524623\pi$
$522$	7.23607	+	4.85410i	0.316714	+	0.212458i
$523$			33.0689i			1.44600i	0.690847	+	0.723001i	$0.257238\pi$
$523$			33.0689i			1.44600i	−0.690847	+	0.723001i	$0.742762\pi$
$524$		−	7.56231i		−	0.330361i
$525$		0			0
$526$	−8.00000			−0.348817
$527$			21.7082i			0.945624i
$528$			10.8541i			0.472364i
$529$	21.4721			0.933571
$530$		0			0
$531$	−8.94427			−0.388148
$532$	−20.1803			−0.874929
$533$	4.47214			0.193710
$534$		−	0.0901699i		−	0.00390204i
$535$		0			0
$536$			10.5279i			0.454734i
$537$	17.2361			0.743791
$538$		−	34.5623i		−	1.49009i
$539$			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	$0.926457\pi$
$542$			37.8885i			1.62745i
$543$	20.4164			0.876152
$544$	−10.1459			−0.435002
$545$		0			0
$546$	6.85410			0.293328
$547$			6.34752i			0.271401i	0.990750	+	0.135700i	$0.0433284\pi$
$547$			6.34752i			0.271401i	−0.990750	+	0.135700i	$0.956672\pi$
$548$	2.18034			0.0931395
$549$		−	2.76393i		−	0.117962i
$550$		0			0
$551$	−23.1246	+	34.4721i	−0.985142	+	1.46856i
$552$		−	2.76393i		−	0.117641i
$553$	9.70820			0.412835
$554$		−	10.3820i		−	0.441087i
$555$		0			0
$556$	4.14590			0.175825
$557$			16.4721i			0.697947i	0.937133	+	0.348973i	$0.113470\pi$
$557$			16.4721i			0.697947i	−0.937133	+	0.348973i	$0.886530\pi$
$558$			11.7082i			0.495648i
$559$			0.472136i			0.0199692i
$560$		0			0
$561$		−	6.70820i		−	0.283221i
$562$	−2.29180			−0.0966736
$563$	13.0689			0.550788			0.275394	−	0.961331i	$-0.411192\pi$
$563$	13.0689			0.550788			0.275394	+	0.961331i	$0.411192\pi$
$564$	6.32624			0.266383
$565$		0			0
$566$		−	22.4721i		−	0.944574i
$567$			4.23607i			0.177898i
$568$	−10.6525			−0.446968
$569$			21.0000i			0.880366i	0.897908	+	0.440183i	$0.145086\pi$
$569$			21.0000i			0.880366i	−0.897908	+	0.440183i	$0.854914\pi$
$570$		0			0
$571$	18.8328			0.788129			0.394064	−	0.919083i	$-0.371069\pi$
$571$	18.8328			0.788129			0.394064	+	0.919083i	$0.371069\pi$
$572$	−1.38197			−0.0577829
$573$		−	8.94427i		−	0.373652i
$574$	−30.6525			−1.27941
$575$		0			0
$576$	4.23607			0.176503
$577$	24.8328			1.03380			0.516902	−	0.856045i	$-0.327085\pi$
$577$	24.8328			1.03380			0.516902	+	0.856045i	$0.327085\pi$
$578$	−12.9443			−0.538411
$579$	17.4164			0.723801
$580$		0			0
$581$	25.4164			1.05445
$582$	−29.4164			−1.21935
$583$	15.1246			0.626397
$584$	17.2361			0.713234
$585$		0			0
$586$	−48.4508			−2.00149
$587$			0.944272i			0.0389743i	0.999810	+	0.0194871i	$0.00620334\pi$
$587$			0.944272i			0.0389743i	−0.999810	+	0.0194871i	$0.993797\pi$
$588$	−6.76393			−0.278940
$589$	−55.7771			−2.29825
$590$		0			0
$591$			3.70820i			0.152535i
$592$	23.1246			0.950416
$593$			40.1803i			1.65001i	0.565126	+	0.825004i	$0.308827\pi$
$593$			40.1803i			1.65001i	−0.565126	+	0.825004i	$0.691173\pi$
$594$		−	3.61803i		−	0.148450i
$595$		0			0
$596$	3.41641			0.139942
$597$	20.1246			0.823646
$598$	2.00000			0.0817861
$599$		−	1.76393i		−	0.0720723i	−0.999350	−	0.0360362i	$-0.988527\pi$
$599$		−	1.76393i		−	0.0720723i	0.999350	−	0.0360362i	$-0.0114731\pi$
$600$		0			0
$601$			18.5410i			0.756304i	0.925744	+	0.378152i	$0.123440\pi$
$601$			18.5410i			0.756304i	−0.925744	+	0.378152i	$0.876560\pi$
$602$		−	3.23607i		−	0.131892i
$603$		−	4.70820i		−	0.191733i
$604$	7.41641			0.301769
$605$		0			0
$606$			6.38197i			0.259250i
$607$	−25.4164			−1.03162			−0.515810	−	0.856703i	$-0.672509\pi$
$607$	−25.4164			−1.03162			−0.515810	+	0.856703i	$0.672509\pi$
$608$		−	26.0689i		−	1.05723i
$609$	−12.7082	+	18.9443i	−0.514962	+	0.767661i
$610$		0			0
$611$		−	10.2361i		−	0.414107i
$612$	1.85410			0.0749476
$613$		−	24.4164i		−	0.986169i	−0.869981	−	0.493085i	$-0.835869\pi$
$613$		−	24.4164i		−	0.986169i	0.869981	−	0.493085i	$-0.164131\pi$
$614$	−45.5967			−1.84013
$615$		0			0
$616$	−21.1803			−0.853380
$617$	−19.8885			−0.800683			−0.400341	−	0.916366i	$-0.631108\pi$
$617$	−19.8885			−0.800683			−0.400341	+	0.916366i	$0.631108\pi$
$618$		−	31.4164i		−	1.26375i
$619$		−	32.5410i		−	1.30793i	−0.756523	−	0.653967i	$-0.773104\pi$
$619$		−	32.5410i		−	1.30793i	0.756523	−	0.653967i	$-0.226896\pi$
$620$		0			0
$621$			1.23607i			0.0496017i
$622$			25.3262i			1.01549i
$623$	0.236068			0.00945786
$624$			4.85410i			0.194320i
$625$		0			0
$626$		−	8.85410i		−	0.353881i
$627$	17.2361			0.688342
$628$	−1.23607			−0.0493245
$629$	−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	$0.306390\pi$
$632$			5.12461i			0.203846i
$633$		−	10.0000i		−	0.397464i
$634$	−38.5623			−1.53150
$635$		0			0
$636$			4.18034i			0.165761i
$637$			10.9443i			0.433628i
$638$	10.8541	−	16.1803i	0.429718	−	0.640586i
$639$	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	$0.874356\pi$
$642$		−	1.23607i		−	0.0487837i
$643$			0.708204i			0.0279288i	0.999902	+	0.0139644i	$0.00444516\pi$
$643$			0.708204i			0.0279288i	−0.999902	+	0.0139644i	$0.995555\pi$
$644$	−3.23607			−0.127519
$645$		0			0
$646$			37.4164i			1.47213i
$647$		−	17.5967i		−	0.691800i	−0.938271	−	0.345900i	$-0.887574\pi$
$647$		−	17.5967i		−	0.691800i	0.938271	−	0.345900i	$-0.112426\pi$
$648$	−2.23607			−0.0878410
$649$			20.0000i			0.785069i
$650$		0			0
$651$	−30.6525			−1.20137
$652$	4.58359			0.179507
$653$	−32.3050			−1.26419			−0.632095	−	0.774891i	$-0.717805\pi$
$653$	−32.3050			−1.26419			−0.632095	+	0.774891i	$0.717805\pi$
$654$	−8.09017			−0.316351
$655$		0			0
$656$		−	21.7082i		−	0.847563i
$657$	−7.70820			−0.300726
$658$			70.1591i			2.73508i
$659$		−	26.2361i		−	1.02201i	−0.859577	−	0.511006i	$-0.829273\pi$
$659$		−	26.2361i		−	1.02201i	0.859577	−	0.511006i	$-0.170727\pi$
$660$		0			0
$661$	−29.8328			−1.16036			−0.580181	−	0.814488i	$-0.697018\pi$
$661$	−29.8328			−1.16036			−0.580181	+	0.814488i	$0.697018\pi$
$662$			13.4164i			0.521443i
$663$		−	3.00000i		−	0.116510i
$664$			13.4164i			0.520658i
$665$		0			0
$666$	−7.70820			−0.298687
$667$	−3.70820	+	5.52786i	−0.143582	+	0.214040i
$668$		−	9.81966i		−	0.379934i
$669$		−	12.7082i		−	0.491328i
$670$		0			0
$671$	−6.18034			−0.238589
$672$		−	14.3262i		−	0.552647i
$673$		−	4.41641i		−	0.170240i	−0.996371	−	0.0851200i	$-0.972873\pi$
$673$		−	4.41641i		−	0.170240i	0.996371	−	0.0851200i	$-0.0271274\pi$
$674$	−31.1246			−1.19888
$675$		0			0
$676$	7.41641			0.285246
$677$	−18.0557			−0.693938			−0.346969	−	0.937877i	$-0.612789\pi$
$677$	−18.0557			−0.693938			−0.346969	+	0.937877i	$0.612789\pi$
$678$	7.32624			0.281362
$679$		−	77.0132i		−	2.95549i
$680$		0			0
$681$		−	9.05573i		−	0.347016i
$682$	26.1803			1.00250
$683$			32.9443i			1.26058i	0.776361	+	0.630289i	$0.217064\pi$
$683$			32.9443i			1.26058i	−0.776361	+	0.630289i	$0.782936\pi$
$684$			4.76393i			0.182153i
$685$		0			0
$686$		−	27.0344i		−	1.03218i
$687$			22.1803i			0.846233i
$688$	2.29180			0.0873739
$689$	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	$0.695044\pi$
$692$		−	3.70820i		−	0.140965i
$693$	9.47214			0.359817
$694$			36.6525i			1.39131i
$695$		0			0
$696$	−10.0000	−	6.70820i	−0.379049	−	0.254274i
$697$			13.4164i			0.508183i
$698$	−16.1803			−0.612435
$699$		−	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	$0.450629\pi$
$702$		−	1.61803i		−	0.0610688i
$703$		−	36.7214i		−	1.38497i
$704$		−	9.47214i		−	0.356995i
$705$		0			0
$706$			7.52786i			0.283315i
$707$	−16.7082			−0.628377
$708$	−5.52786			−0.207750
$709$	13.4164			0.503864			0.251932	−	0.967745i	$-0.418934\pi$
$709$	13.4164			0.503864			0.251932	+	0.967745i	$0.418934\pi$
$710$		0			0
$711$		−	2.29180i		−	0.0859491i
$712$			0.124612i			0.00467002i
$713$	−8.94427			−0.334966
$714$			20.5623i			0.769525i
$715$		0			0
$716$	10.6525			0.398102
$717$	−26.1803			−0.977723
$718$		−	40.9443i		−	1.52803i
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	82.2492			3.06312
$722$	−65.3951			−2.43375
$723$	−3.00000			−0.111571
$724$	12.6180			0.468946
$725$		0			0
$726$	9.70820			0.360305
$727$	−5.81966			−0.215839			−0.107920	−	0.994160i	$-0.534419\pi$
$727$	−5.81966			−0.215839			−0.107920	+	0.994160i	$0.534419\pi$
$728$	−9.47214			−0.351061
$729$	1.00000			0.0370370
$730$		0			0
$731$	−1.41641			−0.0523877
$732$		−	1.70820i		−	0.0631370i
$733$	8.47214			0.312925			0.156463	−	0.987684i	$-0.449991\pi$
$733$	8.47214			0.312925			0.156463	+	0.987684i	$0.449991\pi$
$734$	29.1246			1.07501
$735$		0			0
$736$		−	4.18034i		−	0.154089i
$737$	−10.5279			−0.387799
$738$			7.23607i			0.266363i
$739$			12.1803i			0.448061i	0.974582	+	0.224031i	$0.0719215\pi$
$739$			12.1803i			0.448061i	−0.974582	+	0.224031i	$0.928078\pi$
$740$		0			0
$741$	7.70820			0.283168
$742$	−46.3607			−1.70195
$743$	43.0689			1.58004			0.790022	−	0.613078i	$-0.210069\pi$
$743$	43.0689			1.58004			0.790022	+	0.613078i	$0.210069\pi$
$744$		−	16.1803i		−	0.593200i
$745$		0			0
$746$		−	6.29180i		−	0.230359i
$747$		−	6.00000i		−	0.219529i
$748$		−	4.14590i		−	0.151589i
$749$	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	$-0.162847\pi$
$752$	−49.6869			−1.81190
$753$			12.2361i			0.445907i
$754$	4.85410	−	7.23607i	0.176776	−	0.263522i
$755$		0			0
$756$			2.61803i			0.0952170i
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$			23.1246i			0.839924i
$759$	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	$0.484691\pi$
$762$	20.1803			0.731057
$763$		−	21.1803i		−	0.766780i
$764$		−	5.52786i		−	0.199991i
$765$		0			0
$766$		−	8.65248i		−	0.312627i
$767$			8.94427i			0.322959i
$768$	13.5623			0.489388
$769$		−	40.1803i		−	1.44894i	−0.689306	−	0.724470i	$-0.742085\pi$
$769$		−	40.1803i		−	1.44894i	0.689306	−	0.724470i	$-0.257915\pi$
$770$		0			0
$771$			26.4721i			0.953371i
$772$	10.7639			0.387402
$773$	31.8885			1.14695			0.573476	−	0.819223i	$-0.305595\pi$
$773$	31.8885			1.14695			0.573476	+	0.819223i	$0.305595\pi$
$774$	−0.763932			−0.0274590
$775$		0			0
$776$	40.6525			1.45934
$777$		−	20.1803i		−	0.723966i
$778$		−	21.7984i		−	0.781510i
$779$	−34.4721			−1.23509
$780$		0			0
$781$		−	10.6525i		−	0.381176i
$782$			6.00000i			0.214560i
$783$	4.47214	+	3.00000i	0.159821	+	0.107211i
$784$	53.1246			1.89731
$785$		0			0
$786$		−	19.7984i		−	0.706185i
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$			2.29180i			0.0816419i
$789$	−4.94427			−0.176021
$790$		0			0
$791$			19.1803i			0.681974i
$792$			5.00000i			0.177667i
$793$	−2.76393			−0.0981501
$794$		−	11.2361i		−	0.398753i
$795$		0			0
$796$	12.4377			0.440842
$797$	44.8328			1.58806			0.794030	−	0.607879i	$-0.207979\pi$
$797$	44.8328			1.58806			0.794030	+	0.607879i	$0.207979\pi$
$798$	−52.8328			−1.87026
$799$	30.7082			1.08638
$800$		0			0
$801$		−	0.0557281i		−	0.00196906i
$802$	54.5410			1.92591
$803$			17.2361i			0.608248i
$804$		−	2.90983i		−	0.102622i
$805$		0			0
$806$	11.7082			0.412404
$807$		−	21.3607i		−	0.751932i
$808$		−	8.81966i		−	0.310275i
$809$			43.3607i			1.52448i	0.647294	+	0.762240i	$0.275900\pi$
$809$			43.3607i			1.52448i	−0.647294	+	0.762240i	$0.724100\pi$
$810$		0			0
$811$	46.5967			1.63623			0.818117	−	0.575052i	$-0.195018\pi$
$811$	46.5967			1.63623			0.818117	+	0.575052i	$0.195018\pi$
$812$	−7.85410	+	11.7082i	−0.275625	+	0.410877i
$813$			23.4164i			0.821249i
$814$			17.2361i			0.604124i
$815$		0			0
$816$	−14.5623			−0.509783
$817$		−	3.63932i		−	0.127324i
$818$		−	17.1246i		−	0.598748i
$819$	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	$0.569836\pi$
$822$	5.70820			0.199096
$823$	26.3607			0.918876			0.459438	−	0.888210i	$-0.348051\pi$
$823$	26.3607			0.918876			0.459438	+	0.888210i	$0.348051\pi$
$824$			43.4164i			1.51248i
$825$		0			0
$826$		−	61.3050i		−	2.13307i
$827$	−29.8885			−1.03933			−0.519663	−	0.854371i	$-0.673943\pi$
$827$	−29.8885			−1.03933			−0.519663	+	0.854371i	$0.673943\pi$
$828$			0.763932i			0.0265485i
$829$			27.3050i			0.948340i	0.880433	+	0.474170i	$0.157252\pi$
$829$			27.3050i			0.948340i	−0.880433	+	0.474170i	$0.842748\pi$
$830$		0			0
$831$		−	6.41641i		−	0.222583i
$832$		−	4.23607i		−	0.146859i
$833$	−32.8328			−1.13759
$834$	10.8541			0.375847
$835$		0			0
$836$	10.6525			0.368424
$837$			7.23607i			0.250115i
$838$	−6.18034			−0.213496
$839$		−	0.708204i		−	0.0244499i	−0.999925	−	0.0122250i	$-0.996109\pi$
$839$		−	0.708204i		−	0.0244499i	0.999925	−	0.0122250i	$-0.00389142\pi$
$840$		0			0
$841$	11.0000	+	26.8328i	0.379310	+	0.925270i
$842$		−	32.3607i		−	1.11522i
$843$	−1.41641			−0.0487837
$844$		−	6.18034i		−	0.212736i
$845$		0			0
$846$	16.5623			0.569424
$847$			25.4164i			0.873318i
$848$		−	32.8328i		−	1.12748i
$849$		−	13.8885i		−	0.476654i
$850$		0			0
$851$		−	5.88854i		−	0.201857i
$852$	2.94427			0.100869
$853$	−45.1935			−1.54740			−0.773698	−	0.633555i	$-0.781595\pi$
$853$	−45.1935			−1.54740			−0.773698	+	0.633555i	$0.781595\pi$
$854$	18.9443			0.648260
$855$		0			0
$856$			1.70820i			0.0583852i
$857$		−	37.5967i		−	1.28428i	−0.766587	−	0.642140i	$-0.778047\pi$
$857$		−	37.5967i		−	1.28428i	0.766587	−	0.642140i	$-0.221953\pi$
$858$	−3.61803			−0.123518
$859$		−	36.3607i		−	1.24061i	−0.784361	−	0.620305i	$-0.787009\pi$
$859$		−	36.3607i		−	1.24061i	0.784361	−	0.620305i	$-0.212991\pi$
$860$		0			0
$861$	−18.9443			−0.645619
$862$	23.2361			0.791424
$863$		−	46.6525i		−	1.58807i	−0.607873	−	0.794034i	$-0.707977\pi$
$863$		−	46.6525i		−	1.58807i	0.607873	−	0.794034i	$-0.292023\pi$
$864$	−3.38197			−0.115057
$865$		0			0
$866$	−23.1246			−0.785806
$867$	−8.00000			−0.271694
$868$	−18.9443			−0.643010
$869$	−5.12461			−0.173841
$870$		0			0
$871$	−4.70820			−0.159531
$872$	11.1803			0.378614
$873$	−18.1803			−0.615311
$874$	−15.4164			−0.521468
$875$		0			0
$876$	−4.76393			−0.160958
$877$		−	32.4721i		−	1.09651i	−0.836312	−	0.548253i	$-0.815293\pi$
$877$		−	32.4721i		−	1.09651i	0.836312	−	0.548253i	$-0.184707\pi$
$878$	−21.5066			−0.725812
$879$	−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	$0.619202\pi$
$882$	−17.7082			−0.596266
$883$			34.2492i			1.15258i	0.817246	+	0.576289i	$0.195500\pi$
$883$			34.2492i			1.15258i	−0.817246	+	0.576289i	$0.804500\pi$
$884$		−	1.85410i		−	0.0623602i
$885$		0			0
$886$	15.6180			0.524698
$887$	17.0689			0.573117			0.286559	−	0.958063i	$-0.407489\pi$
$887$	17.0689			0.573117			0.286559	+	0.958063i	$0.407489\pi$
$888$	10.6525			0.357474
$889$			52.8328i			1.77196i
$890$		0			0
$891$		−	2.23607i		−	0.0749111i
$892$		−	7.85410i		−	0.262975i
$893$			78.9017i			2.64034i
$894$	8.94427			0.299141
$895$		0			0
$896$			57.6869i			1.92718i
$897$	1.23607			0.0412711
$898$			30.5623i			1.01988i
$899$	−21.7082	+	32.3607i	−0.724009	+	1.07929i
$900$		0			0
$901$			20.2918i			0.676018i
$902$	16.1803			0.538746
$903$		−	2.00000i		−	0.0665558i
$904$	−10.1246			−0.336740
$905$		0			0
$906$	19.4164			0.645067
$907$	−26.4721			−0.878993			−0.439496	−	0.898244i	$-0.644843\pi$
$907$	−26.4721			−0.878993			−0.439496	+	0.898244i	$0.644843\pi$
$908$		−	5.59675i		−	0.185735i
$909$			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	$0.677903\pi$
$912$		−	37.4164i		−	1.23898i
$913$	−13.4164			−0.444018
$914$		−	10.3820i		−	0.343405i
$915$		0			0
$916$			13.7082i			0.452932i
$917$	51.8328			1.71167
$918$	4.85410			0.160209
$919$	25.6525			0.846197			0.423099	−	0.906084i	$-0.360942\pi$
$919$	25.6525			0.846197			0.423099	+	0.906084i	$0.360942\pi$
$920$		0			0
$921$	−28.1803			−0.928574
$922$			30.6525i			1.00949i
$923$		−	4.76393i		−	0.156807i
$924$	5.85410			0.192586
$925$		0			0
$926$			1.14590i			0.0376565i
$927$		−	19.4164i		−	0.637719i
$928$	−15.1246	−	10.1459i	−0.496490	−	0.333055i
$929$	32.3607			1.06172			0.530860	−	0.847460i	$-0.321869\pi$
$929$	32.3607			1.06172			0.530860	+	0.847460i	$0.321869\pi$
$930$		0			0
$931$		−	84.3607i		−	2.76481i
$932$		−	6.47214i		−	0.212002i
$933$			15.6525i			0.512439i
$934$	21.8885			0.716215
$935$		0			0
$936$			2.23607i			0.0730882i
$937$		−	33.0000i		−	1.07806i	−0.842286	−	0.539032i	$-0.818790\pi$
$937$		−	33.0000i		−	1.07806i	0.842286	−	0.539032i	$-0.181210\pi$
$938$	32.2705			1.05367
$939$		−	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	$0.760831\pi$
$942$	−3.23607			−0.105437
$943$	−5.52786			−0.180012
$944$	43.4164			1.41308
$945$		0			0
$946$			1.70820i			0.0555385i
$947$	16.8197			0.546566			0.273283	−	0.961934i	$-0.411891\pi$
$947$	16.8197			0.546566			0.273283	+	0.961934i	$0.411891\pi$
$948$		−	1.41641i		−	0.0460028i
$949$			7.70820i			0.250219i
$950$		0			0
$951$	−23.8328			−0.772832
$952$		−	28.4164i		−	0.920981i
$953$		−	59.6656i		−	1.93276i	−0.257119	−	0.966380i	$-0.582773\pi$
$953$		−	59.6656i		−	1.93276i	0.257119	−	0.966380i	$-0.417227\pi$
$954$			10.9443i			0.354334i
$955$		0			0
$956$	−16.1803			−0.523310
$957$	6.70820	−	10.0000i	0.216845	−	0.323254i
$958$			16.9443i			0.547445i
$959$			14.9443i			0.482576i
$960$		0			0
$961$	−21.3607			−0.689054
$962$			7.70820i			0.248522i
$963$		−	0.763932i		−	0.0246174i
$964$	−1.85410			−0.0597166
$965$		0			0
$966$	−8.47214			−0.272587
$967$	−27.1246			−0.872269			−0.436134	−	0.899882i	$-0.643653\pi$
$967$	−27.1246			−0.872269			−0.436134	+	0.899882i	$0.643653\pi$
$968$	−13.4164			−0.431220
$969$			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	$-0.0857340\pi$
$972$	0.618034			0.0198234
$973$			28.4164i			0.910988i
$974$		−	32.9443i		−	1.05560i
$975$		0			0
$976$			13.4164i			0.429449i
$977$		−	51.0132i		−	1.63206i	−0.578013	−	0.816028i	$-0.696172\pi$
$977$		−	51.0132i		−	1.63206i	0.578013	−	0.816028i	$-0.303828\pi$
$978$	12.0000			0.383718
$979$	−0.124612			−0.00398261
$980$		0			0
$981$	−5.00000			−0.159638
$982$		0			0
$983$	−13.8885			−0.442976			−0.221488	−	0.975163i	$-0.571091\pi$
$983$	−13.8885			−0.442976			−0.221488	+	0.975163i	$0.571091\pi$
$984$		−	10.0000i		−	0.318788i
$985$		0			0
$986$	21.7082	+	14.5623i	0.691330	+	0.463758i
$987$			43.3607i			1.38019i
$988$	4.76393			0.151561
$989$		−	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	$0.609811\pi$
$992$		−	24.4721i		−	0.776991i
$993$			8.29180i			0.263132i
$994$			32.6525i			1.03567i
$995$		0			0
$996$		−	3.70820i		−	0.117499i
$997$	24.5836			0.778570			0.389285	−	0.921117i	$-0.372722\pi$
$997$	24.5836			0.778570			0.389285	+	0.921117i	$0.372722\pi$
$998$	−34.2705			−1.08481
$999$	−4.76393			−0.150724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.f.b.724.4		4
5.2	odd	4		87.2.c.a.28.4	yes	4
5.3	odd	4		2175.2.d.e.376.1		4
5.4	even	2		2175.2.f.a.724.1		4
15.2	even	4		261.2.c.b.28.1		4
20.7	even	4		1392.2.o.i.289.4		4
29.28	even	2		2175.2.f.a.724.2		4
60.47	odd	4		4176.2.o.l.289.1		4
145.12	even	4		2523.2.a.d.1.1		2
145.17	even	4		2523.2.a.e.1.2		2
145.28	odd	4		2175.2.d.e.376.4		4
145.57	odd	4		87.2.c.a.28.1	✓	4
145.144	even	2	inner	2175.2.f.b.724.3		4
435.17	odd	4		7569.2.a.f.1.1		2
435.302	odd	4		7569.2.a.n.1.2		2
435.347	even	4		261.2.c.b.28.4		4
580.347	even	4		1392.2.o.i.289.2		4
1740.347	odd	4		4176.2.o.l.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.1	✓	4	145.57	odd	4
87.2.c.a.28.4	yes	4	5.2	odd	4
261.2.c.b.28.1		4	15.2	even	4
261.2.c.b.28.4		4	435.347	even	4
1392.2.o.i.289.2		4	580.347	even	4
1392.2.o.i.289.4		4	20.7	even	4
2175.2.d.e.376.1		4	5.3	odd	4
2175.2.d.e.376.4		4	145.28	odd	4
2175.2.f.a.724.1		4	5.4	even	2
2175.2.f.a.724.2		4	29.28	even	2
2175.2.f.b.724.3		4	145.144	even	2	inner
2175.2.f.b.724.4		4	1.1	even	1	trivial
2523.2.a.d.1.1		2	145.12	even	4
2523.2.a.e.1.2		2	145.17	even	4
4176.2.o.l.289.1		4	60.47	odd	4
4176.2.o.l.289.2		4	1740.347	odd	4
7569.2.a.f.1.1		2	435.17	odd	4
7569.2.a.n.1.2		2	435.302	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.f (of order \(2\), degree \(1\), not minimal)

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

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