LMFDB - Embedded newform 2254.2.c.c.2253.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,2,Mod(2253,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2254.2253");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2254.c (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:	$17.9982806156$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 18x^{14} + 226x^{12} + 1434x^{10} + 6585x^{8} + 14406x^{6} + 22423x^{4} + 8085x^{2} + 2401 $ x^16 + 18x^14 + 226x^12 + 1434x^10 + 6585x^8 + 14406x^6 + 22423x^4 + 8085*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 322)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2253.12
Root			$0.306090 - 0.530164i$ of defining polynomial
Character	$\chi$	$=$	2254.2253
Dual form			2254.2.c.c.2253.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.21925i q^{3} +1.00000 q^{4} +2.47221 q^{5} +1.21925i q^{6} +1.00000 q^{8} +1.51344 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.21925i q^{3} +1.00000 q^{4} +2.47221 q^{5} +1.21925i q^{6} +1.00000 q^{8} +1.51344 q^{9} +2.47221 q^{10} -1.26776i q^{11} +1.21925i q^{12} -3.39445i q^{13} +3.01423i q^{15} +1.00000 q^{16} -4.29430 q^{17} +1.51344 q^{18} +2.47221 q^{19} +2.47221 q^{20} -1.26776i q^{22} +(4.62523 + 1.26776i) q^{23} +1.21925i q^{24} +1.11180 q^{25} -3.39445i q^{26} +5.50299i q^{27} +8.99116 q^{29} +3.01423i q^{30} +3.84060i q^{31} +1.00000 q^{32} +1.54571 q^{33} -4.29430 q^{34} +1.51344 q^{36} +1.43254i q^{37} +2.47221 q^{38} +4.13867 q^{39} +2.47221 q^{40} -3.39445i q^{41} -12.9877i q^{43} -1.26776i q^{44} +3.74153 q^{45} +(4.62523 + 1.26776i) q^{46} +1.66240i q^{47} +1.21925i q^{48} +1.11180 q^{50} -5.23581i q^{51} -3.39445i q^{52} -8.25004i q^{53} +5.50299i q^{54} -3.13415i q^{55} +3.01423i q^{57} +8.99116 q^{58} +9.74319i q^{59} +3.01423i q^{60} -9.23871 q^{61} +3.84060i q^{62} +1.00000 q^{64} -8.39177i q^{65} +1.54571 q^{66} +11.4060i q^{67} -4.29430 q^{68} +(-1.54571 + 5.63930i) q^{69} -10.1030 q^{71} +1.51344 q^{72} +3.84060i q^{73} +1.43254i q^{74} +1.35556i q^{75} +2.47221 q^{76} +4.13867 q^{78} +7.43795i q^{79} +2.47221 q^{80} -2.16920 q^{81} -3.39445i q^{82} +5.21176 q^{83} -10.6164 q^{85} -12.9877i q^{86} +10.9624i q^{87} -1.26776i q^{88} -6.86384 q^{89} +3.74153 q^{90} +(4.62523 + 1.26776i) q^{92} -4.68264 q^{93} +1.66240i q^{94} +6.11180 q^{95} +1.21925i q^{96} +6.49915 q^{97} -1.91867i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 20 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 16 * q^8 - 20 * q^9	$ 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 20 q^{9} + 16 q^{16} - 20 q^{18} + 8 q^{23} - 4 q^{25} + 16 q^{29} + 16 q^{32} - 20 q^{36} - 44 q^{39} + 8 q^{46} - 4 q^{50} + 16 q^{58} + 16 q^{64} - 12 q^{71} - 20 q^{72} - 44 q^{78} + 72 q^{81} + 24 q^{85} + 8 q^{92} + 76 q^{93} + 76 q^{95}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 16 * q^8 - 20 * q^9 + 16 * q^16 - 20 * q^18 + 8 * q^23 - 4 * q^25 + 16 * q^29 + 16 * q^32 - 20 * q^36 - 44 * q^39 + 8 * q^46 - 4 * q^50 + 16 * q^58 + 16 * q^64 - 12 * q^71 - 20 * q^72 - 44 * q^78 + 72 * q^81 + 24 * q^85 + 8 * q^92 + 76 * q^93 + 76 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2254\mathbb{Z}\right)^\times$.

$n$	$1473$	$1569$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$			1.21925i			0.703933i	0.936013	+	0.351966i	$0.114487\pi$
$3$			1.21925i			0.703933i	−0.936013	+	0.351966i	$0.885513\pi$
$4$	1.00000			0.500000
$5$	2.47221			1.10560			0.552802	−	0.833313i	$-0.313559\pi$
$5$	2.47221			1.10560			0.552802	+	0.833313i	$0.313559\pi$
$6$			1.21925i			0.497755i
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$	1.51344			0.504479
$10$	2.47221			0.781780
$11$		−	1.26776i		−	0.382243i	−0.981566	−	0.191121i	$-0.938788\pi$
$11$		−	1.26776i		−	0.382243i	0.981566	−	0.191121i	$-0.0612124\pi$
$12$			1.21925i			0.351966i
$13$		−	3.39445i		−	0.941451i	−0.882280	−	0.470725i	$-0.843992\pi$
$13$		−	3.39445i		−	0.941451i	0.882280	−	0.470725i	$-0.156008\pi$
$14$		0			0
$15$			3.01423i			0.778270i
$16$	1.00000			0.250000
$17$	−4.29430			−1.04152			−0.520760	−	0.853703i	$-0.674351\pi$
$17$	−4.29430			−1.04152			−0.520760	+	0.853703i	$0.674351\pi$
$18$	1.51344			0.356720
$19$	2.47221			0.567163			0.283581	−	0.958948i	$-0.408477\pi$
$19$	2.47221			0.567163			0.283581	+	0.958948i	$0.408477\pi$
$20$	2.47221			0.552802
$21$		0			0
$22$		−	1.26776i		−	0.270287i
$23$	4.62523	+	1.26776i	0.964428	+	0.264345i
$23$	4.62523	+	1.26776i	0.964428	+	0.264345i
$24$			1.21925i			0.248878i
$25$	1.11180			0.222360
$26$		−	3.39445i		−	0.665706i
$27$			5.50299i			1.05905i
$28$		0			0
$29$	8.99116			1.66962			0.834808	−	0.550541i	$-0.185579\pi$
$29$	8.99116			1.66962			0.834808	+	0.550541i	$0.185579\pi$
$30$			3.01423i			0.550320i
$31$			3.84060i			0.689791i	0.938641	+	0.344896i	$0.112086\pi$
$31$			3.84060i			0.689791i	−0.938641	+	0.344896i	$0.887914\pi$
$32$	1.00000			0.176777
$33$	1.54571			0.269073
$34$	−4.29430			−0.736466
$35$		0			0
$36$	1.51344			0.252239
$37$			1.43254i			0.235509i	0.993043	+	0.117754i	$0.0375695\pi$
$37$			1.43254i			0.235509i	−0.993043	+	0.117754i	$0.962430\pi$
$38$	2.47221			0.401045
$39$	4.13867			0.662718
$40$	2.47221			0.390890
$41$		−	3.39445i		−	0.530124i	−0.964231	−	0.265062i	$-0.914608\pi$
$41$		−	3.39445i		−	0.530124i	0.964231	−	0.265062i	$-0.0853924\pi$
$42$		0			0
$43$		−	12.9877i		−	1.98060i	−0.138933	−	0.990302i	$-0.544367\pi$
$43$		−	12.9877i		−	1.98060i	0.138933	−	0.990302i	$-0.455633\pi$
$44$		−	1.26776i		−	0.191121i
$45$	3.74153			0.557754
$46$	4.62523	+	1.26776i	0.681954	+	0.186920i
$47$			1.66240i			0.242486i	0.992623	+	0.121243i	$0.0386880\pi$
$47$			1.66240i			0.242486i	−0.992623	+	0.121243i	$0.961312\pi$
$48$			1.21925i			0.175983i
$49$		0			0
$50$	1.11180			0.157232
$51$		−	5.23581i		−	0.733160i
$52$		−	3.39445i		−	0.470725i
$53$		−	8.25004i		−	1.13323i	−0.823983	−	0.566615i	$-0.808253\pi$
$53$		−	8.25004i		−	1.13323i	0.823983	−	0.566615i	$-0.191747\pi$
$54$			5.50299i			0.748863i
$55$		−	3.13415i		−	0.422609i
$56$		0			0
$57$			3.01423i			0.399244i
$58$	8.99116			1.18060
$59$			9.74319i			1.26846i	0.773146	+	0.634228i	$0.218682\pi$
$59$			9.74319i			1.26846i	−0.773146	+	0.634228i	$0.781318\pi$
$60$			3.01423i			0.389135i
$61$	−9.23871			−1.18290			−0.591448	−	0.806343i	$-0.701443\pi$
$61$	−9.23871			−1.18290			−0.591448	+	0.806343i	$0.701443\pi$
$62$			3.84060i			0.487756i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	8.39177i		−	1.04087i
$66$	1.54571			0.190263
$67$			11.4060i			1.39346i	0.717331	+	0.696732i	$0.245364\pi$
$67$			11.4060i			1.39346i	−0.717331	+	0.696732i	$0.754636\pi$
$68$	−4.29430			−0.520760
$69$	−1.54571	+	5.63930i	−0.186081	+	0.678892i
$70$		0			0
$71$	−10.1030			−1.19900			−0.599500	−	0.800375i	$-0.704634\pi$
$71$	−10.1030			−1.19900			−0.599500	+	0.800375i	$0.704634\pi$
$72$	1.51344			0.178360
$73$			3.84060i			0.449508i	0.974416	+	0.224754i	$0.0721578\pi$
$73$			3.84060i			0.449508i	−0.974416	+	0.224754i	$0.927842\pi$
$74$			1.43254i			0.166530i
$75$			1.35556i			0.156526i
$76$	2.47221			0.283581
$77$		0			0
$78$	4.13867			0.468612
$79$			7.43795i			0.836834i	0.908255	+	0.418417i	$0.137415\pi$
$79$			7.43795i			0.836834i	−0.908255	+	0.418417i	$0.862585\pi$
$80$	2.47221			0.276401
$81$	−2.16920			−0.241022
$82$		−	3.39445i		−	0.374854i
$83$	5.21176			0.572065			0.286033	−	0.958220i	$-0.407663\pi$
$83$	5.21176			0.572065			0.286033	+	0.958220i	$0.407663\pi$
$84$		0			0
$85$	−10.6164			−1.15151
$86$		−	12.9877i		−	1.40050i
$87$			10.9624i			1.17530i
$88$		−	1.26776i		−	0.135143i
$89$	−6.86384			−0.727566			−0.363783	−	0.931484i	$-0.618515\pi$
$89$	−6.86384			−0.727566			−0.363783	+	0.931484i	$0.618515\pi$
$90$	3.74153			0.394391
$91$		0			0
$92$	4.62523	+	1.26776i	0.482214	+	0.132173i
$93$	−4.68264			−0.485567
$94$			1.66240i			0.171463i
$95$	6.11180			0.627057
$96$			1.21925i			0.124439i
$97$	6.49915			0.659889			0.329944	−	0.944000i	$-0.392970\pi$
$97$	6.49915			0.659889			0.329944	+	0.944000i	$0.392970\pi$
$98$		0			0
$99$		−	1.91867i		−	0.192833i
$100$	1.11180			0.111180
$101$			7.12184i			0.708650i	0.935122	+	0.354325i	$0.115289\pi$
$101$			7.12184i			0.708650i	−0.935122	+	0.354325i	$0.884711\pi$
$102$		−	5.23581i		−	0.518423i
$103$	−13.9067			−1.37027			−0.685136	−	0.728416i	$-0.740257\pi$
$103$	−13.9067			−1.37027			−0.685136	+	0.728416i	$0.740257\pi$
$104$		−	3.39445i		−	0.332853i
$105$		0			0
$106$		−	8.25004i		−	0.801315i
$107$		−	5.86367i		−	0.566862i	−0.958993	−	0.283431i	$-0.908527\pi$
$107$		−	5.86367i		−	0.566862i	0.958993	−	0.283431i	$-0.0914727\pi$
$108$			5.50299i			0.529526i
$109$		−	14.2554i		−	1.36542i	−0.730688	−	0.682712i	$-0.760800\pi$
$109$		−	14.2554i		−	1.36542i	0.730688	−	0.682712i	$-0.239200\pi$
$110$		−	3.13415i		−	0.298830i
$111$	−1.74662			−0.165782
$112$		0			0
$113$			15.6954i			1.47650i	0.674528	+	0.738249i	$0.264347\pi$
$113$			15.6954i			1.47650i	−0.674528	+	0.738249i	$0.735653\pi$
$114$			3.01423i			0.282308i
$115$	11.4345	+	3.13415i	1.06628	+	0.292261i
$116$	8.99116			0.834808
$117$		−	5.13728i		−	0.474942i
$118$			9.74319i			0.896934i
$119$		0			0
$120$			3.01423i			0.275160i
$121$	9.39279			0.853890
$122$	−9.23871			−0.836433
$123$	4.13867			0.373171
$124$			3.84060i			0.344896i
$125$	−9.61243			−0.859762
$126$		0			0
$127$	1.99116			0.176686			0.0883432	−	0.996090i	$-0.471843\pi$
$127$	1.99116			0.176686			0.0883432	+	0.996090i	$0.471843\pi$
$128$	1.00000			0.0883883
$129$	15.8352			1.39421
$130$		−	8.39177i		−	0.736007i
$131$		−	11.3389i		−	0.990687i	−0.868697	−	0.495344i	$-0.835042\pi$
$131$		−	11.3389i		−	0.990687i	0.868697	−	0.495344i	$-0.164958\pi$
$132$	1.54571			0.134537
$133$		0			0
$134$			11.4060i			0.985328i
$135$			13.6045i			1.17089i
$136$	−4.29430			−0.368233
$137$		−	1.58168i		−	0.135132i	−0.997715	−	0.0675662i	$-0.978477\pi$
$137$		−	1.58168i		−	0.135132i	0.997715	−	0.0675662i	$-0.0215234\pi$
$138$	−1.54571	+	5.63930i	−0.131579	+	0.480049i
$139$			14.1603i			1.20106i	0.799603	+	0.600528i	$0.205043\pi$
$139$			14.1603i			1.20106i	−0.799603	+	0.600528i	$0.794957\pi$
$140$		0			0
$141$	−2.02687			−0.170694
$142$	−10.1030			−0.847821
$143$	−4.30333			−0.359863
$144$	1.51344			0.126120
$145$	22.2280			1.84593
$146$			3.84060i			0.317850i
$147$		0			0
$148$			1.43254i			0.117754i
$149$			2.53551i			0.207717i	0.994592	+	0.103859i	$0.0331190\pi$
$149$			2.53551i			0.207717i	−0.994592	+	0.103859i	$0.966881\pi$
$150$			1.35556i			0.110681i
$151$	8.02687			0.653218			0.326609	−	0.945160i	$-0.394094\pi$
$151$	8.02687			0.653218			0.326609	+	0.945160i	$0.394094\pi$
$152$	2.47221			0.200522
$153$	−6.49915			−0.525425
$154$		0			0
$155$			9.49474i			0.762636i
$156$	4.13867			0.331359
$157$	−19.2030			−1.53257			−0.766283	−	0.642503i	$-0.777896\pi$
$157$	−19.2030			−1.53257			−0.766283	+	0.642503i	$0.777896\pi$
$158$			7.43795i			0.591731i
$159$	10.0588			0.797718
$160$	2.47221			0.195445
$161$		0			0
$162$	−2.16920			−0.170428
$163$	−2.48656			−0.194763			−0.0973813	−	0.995247i	$-0.531047\pi$
$163$	−2.48656			−0.194763			−0.0973813	+	0.995247i	$0.531047\pi$
$164$		−	3.39445i		−	0.265062i
$165$	3.82131			0.297488
$166$	5.21176			0.404511
$167$		−	8.20178i		−	0.634673i	−0.948313	−	0.317337i	$-0.897212\pi$
$167$		−	8.20178i		−	0.634673i	0.948313	−	0.317337i	$-0.102788\pi$
$168$		0			0
$169$	1.47772			0.113671
$170$	−10.6164			−0.814240
$171$	3.74153			0.286122
$172$		−	12.9877i		−	0.990302i
$173$		−	17.5715i		−	1.33593i	−0.744191	−	0.667967i	$-0.767165\pi$
$173$		−	17.5715i		−	1.33593i	0.744191	−	0.667967i	$-0.232835\pi$
$174$			10.9624i			0.831060i
$175$		0			0
$176$		−	1.26776i		−	0.0955607i
$177$	−11.8794			−0.892907
$178$	−6.86384			−0.514467
$179$	6.51344			0.486837			0.243419	−	0.969921i	$-0.421731\pi$
$179$	6.51344			0.486837			0.243419	+	0.969921i	$0.421731\pi$
$180$	3.74153			0.278877
$181$	−10.2407			−0.761184			−0.380592	−	0.924743i	$-0.624280\pi$
$181$	−10.2407			−0.761184			−0.380592	+	0.924743i	$0.624280\pi$
$182$		0			0
$183$		−	11.2643i		−	0.832679i
$184$	4.62523	+	1.26776i	0.340977	+	0.0934602i
$185$			3.54154i			0.260379i
$186$	−4.68264			−0.343347
$187$			5.44413i			0.398114i
$188$			1.66240i			0.121243i
$189$		0			0
$190$	6.11180			0.443396
$191$			9.83173i			0.711399i	0.934600	+	0.355699i	$0.115757\pi$
$191$			9.83173i			0.711399i	−0.934600	+	0.355699i	$0.884243\pi$
$192$			1.21925i			0.0879916i
$193$	−7.88455			−0.567542			−0.283771	−	0.958892i	$-0.591586\pi$
$193$	−7.88455			−0.567542			−0.283771	+	0.958892i	$0.591586\pi$
$194$	6.49915			0.466612
$195$	10.2316			0.732703
$196$		0			0
$197$	−5.96063			−0.424677			−0.212339	−	0.977196i	$-0.568108\pi$
$197$	−5.96063			−0.424677			−0.212339	+	0.977196i	$0.568108\pi$
$198$		−	1.91867i		−	0.136354i
$199$	23.2518			1.64828			0.824139	−	0.566388i	$-0.191660\pi$
$199$	23.2518			1.64828			0.824139	+	0.566388i	$0.191660\pi$
$200$	1.11180			0.0786160
$201$	−13.9067			−0.980905
$202$			7.12184i			0.501091i
$203$		0			0
$204$		−	5.23581i		−	0.366580i
$205$		−	8.39177i		−	0.586107i
$206$	−13.9067			−0.968928
$207$	7.00000	+	1.91867i	0.486534	+	0.133357i
$208$		−	3.39445i		−	0.235363i
$209$		−	3.13415i		−	0.216794i
$210$		0			0
$211$	−7.62889			−0.525194			−0.262597	−	0.964906i	$-0.584579\pi$
$211$	−7.62889			−0.525194			−0.262597	+	0.964906i	$0.584579\pi$
$212$		−	8.25004i		−	0.566615i
$213$		−	12.3180i		−	0.844015i
$214$		−	5.86367i		−	0.400832i
$215$		−	32.1082i		−	2.18976i
$216$			5.50299i			0.374431i
$217$		0			0
$218$		−	14.2554i		−	0.965500i
$219$	−4.68264			−0.316423
$220$		−	3.13415i		−	0.211305i
$221$			14.5768i			0.980540i
$222$	−1.74662			−0.117226
$223$			8.38464i			0.561477i	0.959784	+	0.280738i	$0.0905794\pi$
$223$			8.38464i			0.561477i	−0.959784	+	0.280738i	$0.909421\pi$
$224$		0			0
$225$	1.68264			0.112176
$226$			15.6954i			1.04404i
$227$	12.9893			0.862128			0.431064	−	0.902321i	$-0.358138\pi$
$227$	12.9893			0.862128			0.431064	+	0.902321i	$0.358138\pi$
$228$			3.01423i			0.199622i
$229$	−3.10045			−0.204883			−0.102442	−	0.994739i	$-0.532666\pi$
$229$	−3.10045			−0.204883			−0.102442	+	0.994739i	$0.532666\pi$
$230$	11.4345	+	3.13415i	0.753970	+	0.206660i
$231$		0			0
$232$	8.99116			0.590298
$233$	−24.8363			−1.62708			−0.813541	−	0.581507i	$-0.802463\pi$
$233$	−24.8363			−1.62708			−0.813541	+	0.581507i	$0.802463\pi$
$234$		−	5.13728i		−	0.335835i
$235$			4.10979i			0.268093i
$236$			9.74319i			0.634228i
$237$	−9.06869			−0.589075
$238$		0			0
$239$	−24.6701			−1.59578			−0.797889	−	0.602804i	$-0.794050\pi$
$239$	−24.6701			−1.59578			−0.797889	+	0.602804i	$0.794050\pi$
$240$			3.01423i			0.194568i
$241$	−20.1514			−1.29806			−0.649032	−	0.760761i	$-0.724826\pi$
$241$	−20.1514			−1.29806			−0.649032	+	0.760761i	$0.724826\pi$
$242$	9.39279			0.603792
$243$			13.8642i			0.889388i
$244$	−9.23871			−0.591448
$245$		0			0
$246$	4.13867			0.263872
$247$		−	8.39177i		−	0.533956i
$248$			3.84060i			0.243878i
$249$			6.35443i			0.402695i
$250$	−9.61243			−0.607944
$251$	29.3772			1.85427			0.927137	−	0.374721i	$-0.122262\pi$
$251$	29.3772			1.85427			0.927137	+	0.374721i	$0.122262\pi$
$252$		0			0
$253$	1.60721	−	5.86367i	0.101044	−	0.368646i
$254$	1.99116			0.124936
$255$		−	12.9440i		−	0.810585i
$256$	1.00000			0.0625000
$257$		−	4.54858i		−	0.283733i	−0.989886	−	0.141866i	$-0.954690\pi$
$257$		−	4.54858i		−	0.283733i	0.989886	−	0.141866i	$-0.0453103\pi$
$258$	15.8352			0.985856
$259$		0			0
$260$		−	8.39177i		−	0.520436i
$261$	13.6075			0.842286
$262$		−	11.3389i		−	0.700522i
$263$			5.71453i			0.352373i	0.984357	+	0.176186i	$0.0563762\pi$
$263$			5.71453i			0.352373i	−0.984357	+	0.176186i	$0.943624\pi$
$264$	1.54571			0.0951317
$265$		−	20.3958i		−	1.25290i
$266$		0			0
$267$		−	8.36872i		−	0.512157i
$268$			11.4060i			0.696732i
$269$		−	8.28376i		−	0.505070i	−0.967588	−	0.252535i	$-0.918736\pi$
$269$		−	8.28376i		−	0.505070i	0.967588	−	0.252535i	$-0.0812642\pi$
$270$			13.6045i			0.827945i
$271$		−	0.396604i		−	0.0240920i	−0.999927	−	0.0120460i	$-0.996166\pi$
$271$		−	0.396604i		−	0.0240920i	0.999927	−	0.0120460i	$-0.00383445\pi$
$272$	−4.29430			−0.260380
$273$		0			0
$274$		−	1.58168i		−	0.0955531i
$275$		−	1.40949i		−	0.0849954i
$276$	−1.54571	+	5.63930i	−0.0930407	+	0.339446i
$277$	−17.5190			−1.05261			−0.526306	−	0.850295i	$-0.676423\pi$
$277$	−17.5190			−1.05261			−0.526306	+	0.850295i	$0.676423\pi$
$278$			14.1603i			0.849275i
$279$			5.81250i			0.347985i
$280$		0			0
$281$			20.4451i			1.21965i	0.792536	+	0.609826i	$0.208761\pi$
$281$			20.4451i			1.21965i	−0.792536	+	0.609826i	$0.791239\pi$
$282$	−2.02687			−0.120699
$283$	−14.0131			−0.832992			−0.416496	−	0.909137i	$-0.636742\pi$
$283$	−14.0131			−0.832992			−0.416496	+	0.909137i	$0.636742\pi$
$284$	−10.1030			−0.599500
$285$			7.45179i			0.441406i
$286$	−4.30333			−0.254461
$287$		0			0
$288$	1.51344			0.0891801
$289$	1.44101			0.0847654
$290$	22.2280			1.30527
$291$			7.92407i			0.464517i
$292$			3.84060i			0.224754i
$293$	−4.28527			−0.250348			−0.125174	−	0.992135i	$-0.539949\pi$
$293$	−4.28527			−0.250348			−0.125174	+	0.992135i	$0.539949\pi$
$294$		0			0
$295$			24.0872i			1.40241i
$296$			1.43254i			0.0832649i
$297$	6.97645			0.404815
$298$			2.53551i			0.146878i
$299$	4.30333	−	15.7001i	0.248868	−	0.907961i
$300$			1.35556i			0.0782631i
$301$		0			0
$302$	8.02687			0.461895
$303$	−8.68329			−0.498842
$304$	2.47221			0.141791
$305$	−22.8400			−1.30781
$306$	−6.49915			−0.371532
$307$		−	29.9300i		−	1.70820i	−0.520111	−	0.854098i	$-0.674110\pi$
$307$		−	29.9300i		−	1.70820i	0.520111	−	0.854098i	$-0.325890\pi$
$308$		0			0
$309$		−	16.9557i		−	0.964578i
$310$			9.49474i			0.539265i
$311$			5.58497i			0.316695i	0.987383	+	0.158347i	$0.0506166\pi$
$311$			5.58497i			0.316695i	−0.987383	+	0.158347i	$0.949383\pi$
$312$	4.13867			0.234306
$313$	−8.85595			−0.500568			−0.250284	−	0.968172i	$-0.580524\pi$
$313$	−8.85595			−0.500568			−0.250284	+	0.968172i	$0.580524\pi$
$314$	−19.2030			−1.08369
$315$		0			0
$316$			7.43795i			0.418417i
$317$	19.0486			1.06987			0.534937	−	0.844892i	$-0.320335\pi$
$317$	19.0486			1.06987			0.534937	+	0.844892i	$0.320335\pi$
$318$	10.0588			0.564072
$319$		−	11.3986i		−	0.638199i
$320$	2.47221			0.138200
$321$	7.14926			0.399033
$322$		0			0
$323$	−10.6164			−0.590712
$324$	−2.16920			−0.120511
$325$		−	3.77394i		−	0.209341i
$326$	−2.48656			−0.137718
$327$	17.3809			0.961166
$328$		−	3.39445i		−	0.187427i
$329$		0			0
$330$	3.82131			0.210356
$331$	2.60201			0.143020			0.0715098	−	0.997440i	$-0.477218\pi$
$331$	2.60201			0.143020			0.0715098	+	0.997440i	$0.477218\pi$
$332$	5.21176			0.286033
$333$			2.16806i			0.118809i
$334$		−	8.20178i		−	0.448782i
$335$			28.1980i			1.54062i
$336$		0			0
$337$		−	5.37754i		−	0.292934i	−0.989216	−	0.146467i	$-0.953210\pi$
$337$		−	5.37754i		−	0.292934i	0.989216	−	0.146467i	$-0.0467901\pi$
$338$	1.47772			0.0803773
$339$	−19.1366			−1.03936
$340$	−10.6164			−0.575755
$341$	4.86894			0.263668
$342$	3.74153			0.202319
$343$		0			0
$344$		−	12.9877i		−	0.700249i
$345$	−3.82131	+	13.9415i	−0.205732	+	0.750586i
$346$		−	17.5715i		−	0.944648i
$347$	13.6925			0.735050			0.367525	−	0.930014i	$-0.380205\pi$
$347$	13.6925			0.735050			0.367525	+	0.930014i	$0.380205\pi$
$348$			10.9624i			0.587648i
$349$		−	23.2774i		−	1.24601i	−0.782217	−	0.623006i	$-0.785911\pi$
$349$		−	23.2774i		−	1.24601i	0.782217	−	0.623006i	$-0.214089\pi$
$350$		0			0
$351$	18.6796			0.997045
$352$		−	1.26776i		−	0.0675716i
$353$		−	31.6651i		−	1.68536i	−0.538413	−	0.842681i	$-0.680976\pi$
$353$		−	31.6651i		−	1.68536i	0.538413	−	0.842681i	$-0.319024\pi$
$354$	−11.8794			−0.631381
$355$	−24.9766			−1.32562
$356$	−6.86384			−0.363783
$357$		0			0
$358$	6.51344			0.344246
$359$			35.1792i			1.85669i	0.371721	+	0.928345i	$0.378768\pi$
$359$			35.1792i			1.85669i	−0.371721	+	0.928345i	$0.621232\pi$
$360$	3.74153			0.197196
$361$	−12.8882			−0.678326
$362$	−10.2407			−0.538238
$363$			11.4521i			0.601081i
$364$		0			0
$365$			9.49474i			0.496977i
$366$		−	11.2643i		−	0.588793i
$367$	−13.2476			−0.691519			−0.345759	−	0.938323i	$-0.612379\pi$
$367$	−13.2476			−0.691519			−0.345759	+	0.938323i	$0.612379\pi$
$368$	4.62523	+	1.26776i	0.241107	+	0.0660864i
$369$		−	5.13728i		−	0.267436i
$370$			3.54154i			0.184116i
$371$		0			0
$372$	−4.68264			−0.242783
$373$		−	36.5887i		−	1.89449i	−0.320507	−	0.947246i	$-0.603853\pi$
$373$		−	36.5887i		−	1.89449i	0.320507	−	0.947246i	$-0.396147\pi$
$374$			5.44413i			0.281509i
$375$		−	11.7199i		−	0.605215i
$376$			1.66240i			0.0857316i
$377$		−	30.5200i		−	1.57186i
$378$		0			0
$379$		−	1.29081i		−	0.0663045i	−0.999450	−	0.0331523i	$-0.989445\pi$
$379$		−	1.29081i		−	0.0663045i	0.999450	−	0.0331523i	$-0.0105546\pi$
$380$	6.11180			0.313529
$381$			2.42771i			0.124375i
$382$			9.83173i			0.503035i
$383$	4.84707			0.247674			0.123837	−	0.992303i	$-0.460480\pi$
$383$	4.84707			0.247674			0.123837	+	0.992303i	$0.460480\pi$
$384$			1.21925i			0.0622194i
$385$		0			0
$386$	−7.88455			−0.401313
$387$		−	19.6560i		−	0.999173i
$388$	6.49915			0.329944
$389$			14.2749i			0.723766i	0.932224	+	0.361883i	$0.117866\pi$
$389$			14.2749i			0.723766i	−0.932224	+	0.361883i	$0.882134\pi$
$390$	10.2316			0.518099
$391$	−19.8621	−	5.44413i	−1.00447	−	0.275321i
$392$		0			0
$393$	13.8250			0.697377
$394$	−5.96063			−0.300292
$395$			18.3881i			0.925207i
$396$		−	1.91867i		−	0.0964167i
$397$			31.7377i			1.59287i	0.604724	+	0.796435i	$0.293284\pi$
$397$			31.7377i			1.59287i	−0.604724	+	0.796435i	$0.706716\pi$
$398$	23.2518			1.16551
$399$		0			0
$400$	1.11180			0.0555899
$401$		−	4.25893i		−	0.212681i	−0.994330	−	0.106340i	$-0.966087\pi$
$401$		−	4.25893i		−	0.212681i	0.994330	−	0.106340i	$-0.0339133\pi$
$402$	−13.9067			−0.693605
$403$	13.0367			0.649405
$404$			7.12184i			0.354325i
$405$	−5.36270			−0.266475
$406$		0			0
$407$	1.81612			0.0900215
$408$		−	5.23581i		−	0.259211i
$409$		−	7.41636i		−	0.366715i	−0.983046	−	0.183358i	$-0.941303\pi$
$409$		−	7.41636i		−	0.366715i	0.983046	−	0.183358i	$-0.0586967\pi$
$410$		−	8.39177i		−	0.414440i
$411$	1.92846			0.0951241
$412$	−13.9067			−0.685136
$413$		0			0
$414$	7.00000	+	1.91867i	0.344031	+	0.0942974i
$415$	12.8845			0.632478
$416$		−	3.39445i		−	0.166427i
$417$	−17.2648			−0.845463
$418$		−	3.13415i		−	0.153296i
$419$	−33.1406			−1.61903			−0.809513	−	0.587102i	$-0.800269\pi$
$419$	−33.1406			−1.61903			−0.809513	+	0.587102i	$0.800269\pi$
$420$		0			0
$421$		−	28.6831i		−	1.39793i	−0.715157	−	0.698964i	$-0.753645\pi$
$421$		−	28.6831i		−	1.39793i	0.715157	−	0.698964i	$-0.246355\pi$
$422$	−7.62889			−0.371368
$423$			2.51593i			0.122329i
$424$		−	8.25004i		−	0.400657i
$425$	−4.77439			−0.231592
$426$		−	12.3180i		−	0.596809i
$427$		0			0
$428$		−	5.86367i		−	0.283431i
$429$		−	5.24683i		−	0.253319i
$430$		−	32.1082i		−	1.54840i
$431$		−	24.5733i		−	1.18365i	−0.806065	−	0.591827i	$-0.798407\pi$
$431$		−	24.5733i		−	1.18365i	0.806065	−	0.591827i	$-0.201593\pi$
$432$			5.50299i			0.264763i
$433$	24.3265			1.16906			0.584528	−	0.811374i	$-0.301280\pi$
$433$	24.3265			1.16906			0.584528	+	0.811374i	$0.301280\pi$
$434$		0			0
$435$			27.1014i			1.29941i
$436$		−	14.2554i		−	0.682712i
$437$	11.4345	+	3.13415i	0.546988	+	0.149927i
$438$	−4.68264			−0.223745
$439$		−	4.54404i		−	0.216875i	−0.994103	−	0.108438i	$-0.965415\pi$
$439$		−	4.54404i		−	0.216875i	0.994103	−	0.108438i	$-0.0345848\pi$
$440$		−	3.13415i		−	0.149415i
$441$		0			0
$442$			14.5768i			0.693347i
$443$	−20.4652			−0.972332			−0.486166	−	0.873866i	$-0.661605\pi$
$443$	−20.4652			−0.972332			−0.486166	+	0.873866i	$0.661605\pi$
$444$	−1.74662			−0.0828911
$445$	−16.9688			−0.804399
$446$			8.38464i			0.397024i
$447$	−3.09142			−0.146219
$448$		0			0
$449$	−31.4921			−1.48620			−0.743102	−	0.669178i	$-0.766646\pi$
$449$	−31.4921			−1.48620			−0.743102	+	0.669178i	$0.766646\pi$
$450$	1.68264			0.0793202
$451$	−4.30333			−0.202636
$452$			15.6954i			0.738249i
$453$			9.78674i			0.459821i
$454$	12.9893			0.609617
$455$		0			0
$456$			3.01423i			0.141154i
$457$			10.8086i			0.505605i	0.967518	+	0.252803i	$0.0813523\pi$
$457$			10.8086i			0.505605i	−0.967518	+	0.252803i	$0.918648\pi$
$458$	−3.10045			−0.144874
$459$		−	23.6315i		−	1.10302i
$460$	11.4345	+	3.13415i	0.533138	+	0.146131i
$461$		−	36.3796i		−	1.69437i	−0.531299	−	0.847185i	$-0.678296\pi$
$461$		−	36.3796i		−	1.69437i	0.531299	−	0.847185i	$-0.321704\pi$
$462$		0			0
$463$	25.6344			1.19133			0.595666	−	0.803232i	$-0.296888\pi$
$463$	25.6344			1.19133			0.595666	+	0.803232i	$0.296888\pi$
$464$	8.99116			0.417404
$465$	−11.5764			−0.536844
$466$	−24.8363			−1.15052
$467$	−36.4021			−1.68449			−0.842243	−	0.539098i	$-0.818765\pi$
$467$	−36.4021			−1.68449			−0.842243	+	0.539098i	$0.818765\pi$
$468$		−	5.13728i		−	0.237471i
$469$		0			0
$470$			4.10979i			0.189570i
$471$		−	23.4132i		−	1.07882i
$472$			9.74319i			0.448467i
$473$	−16.4652			−0.757072
$474$	−9.06869			−0.416539
$475$	2.74859			0.126114
$476$		0			0
$477$		−	12.4859i		−	0.571691i
$478$	−24.6701			−1.12839
$479$	−3.91154			−0.178723			−0.0893615	−	0.995999i	$-0.528483\pi$
$479$	−3.91154			−0.178723			−0.0893615	+	0.995999i	$0.528483\pi$
$480$			3.01423i			0.137580i
$481$	4.86270			0.221720
$482$	−20.1514			−0.917869
$483$		0			0
$484$	9.39279			0.426945
$485$	16.0672			0.729576
$486$			13.8642i			0.628893i
$487$	−30.3901			−1.37711			−0.688554	−	0.725185i	$-0.741754\pi$
$487$	−30.3901			−1.37711			−0.688554	+	0.725185i	$0.741754\pi$
$488$	−9.23871			−0.418217
$489$		−	3.03173i		−	0.137100i
$490$		0			0
$491$	−20.2990			−0.916082			−0.458041	−	0.888931i	$-0.651449\pi$
$491$	−20.2990			−0.916082			−0.458041	+	0.888931i	$0.651449\pi$
$492$	4.13867			0.186586
$493$	−38.6107			−1.73894
$494$		−	8.39177i		−	0.377564i
$495$		−	4.74334i		−	0.213197i
$496$			3.84060i			0.172448i
$497$		0			0
$498$			6.35443i			0.284749i
$499$	6.80946			0.304833			0.152417	−	0.988316i	$-0.451294\pi$
$499$	6.80946			0.304833			0.152417	+	0.988316i	$0.451294\pi$
$500$	−9.61243			−0.429881
$501$	10.0000			0.446767
$502$	29.3772			1.31117
$503$	−3.19779			−0.142582			−0.0712911	−	0.997456i	$-0.522712\pi$
$503$	−3.19779			−0.142582			−0.0712911	+	0.997456i	$0.522712\pi$
$504$		0			0
$505$			17.6067i			0.783486i
$506$	1.60721	−	5.86367i	0.0714490	−	0.260672i
$507$			1.80170i			0.0800165i
$508$	1.99116			0.0883432
$509$		−	8.13367i		−	0.360519i	−0.983619	−	0.180259i	$-0.942306\pi$
$509$		−	8.13367i		−	0.360519i	0.983619	−	0.180259i	$-0.0576937\pi$
$510$		−	12.9440i		−	0.573170i
$511$		0			0
$512$	1.00000			0.0441942
$513$			13.6045i			0.600655i
$514$		−	4.54858i		−	0.200629i
$515$	−34.3803			−1.51498
$516$	15.8352			0.697106
$517$	2.10752			0.0926884
$518$		0			0
$519$	21.4240			0.940408
$520$		−	8.39177i		−	0.368004i
$521$	39.7699			1.74235			0.871175	−	0.490973i	$-0.163359\pi$
$521$	39.7699			1.74235			0.871175	+	0.490973i	$0.163359\pi$
$522$	13.6075			0.595586
$523$	32.4506			1.41897			0.709483	−	0.704723i	$-0.248929\pi$
$523$	32.4506			1.41897			0.709483	+	0.704723i	$0.248929\pi$
$524$		−	11.3389i		−	0.495344i
$525$		0			0
$526$			5.71453i			0.249165i
$527$		−	16.4927i		−	0.718432i
$528$	1.54571			0.0672683
$529$	19.7856	+	11.7273i	0.860243	+	0.509884i
$530$		−	20.3958i		−	0.885937i
$531$			14.7457i			0.639909i
$532$		0			0
$533$	−11.5223			−0.499085
$534$		−	8.36872i		−	0.362150i
$535$		−	14.4962i		−	0.626725i
$536$			11.4060i			0.492664i
$537$			7.94149i			0.342701i
$538$		−	8.28376i		−	0.357138i
$539$		0			0
$540$			13.6045i			0.585446i
$541$	22.3229			0.959736			0.479868	−	0.877341i	$-0.340685\pi$
$541$	22.3229			0.959736			0.479868	+	0.877341i	$0.340685\pi$
$542$		−	0.396604i		−	0.0170356i
$543$		−	12.4859i		−	0.535822i
$544$	−4.29430			−0.184117
$545$		−	35.2424i		−	1.50962i
$546$		0			0
$547$	−12.9010			−0.551609			−0.275804	−	0.961214i	$-0.588944\pi$
$547$	−12.9010			−0.551609			−0.275804	+	0.961214i	$0.588944\pi$
$548$		−	1.58168i		−	0.0675662i
$549$	−13.9822			−0.596746
$550$		−	1.40949i		−	0.0601008i
$551$	22.2280			0.946944
$552$	−1.54571	+	5.63930i	−0.0657897	+	0.240025i
$553$		0			0
$554$	−17.5190			−0.744310
$555$	−4.31801			−0.183289
$556$			14.1603i			0.600528i
$557$		−	1.85774i		−	0.0787150i	−0.999225	−	0.0393575i	$-0.987469\pi$
$557$		−	1.85774i		−	0.0787150i	0.999225	−	0.0393575i	$-0.0125311\pi$
$558$			5.81250i			0.246063i
$559$	−44.0860			−1.86464
$560$		0			0
$561$	−6.63773			−0.280245
$562$			20.4451i			0.862424i
$563$	−5.76454			−0.242946			−0.121473	−	0.992595i	$-0.538762\pi$
$563$	−5.76454			−0.242946			−0.121473	+	0.992595i	$0.538762\pi$
$564$	−2.02687			−0.0853468
$565$			38.8022i			1.63242i
$566$	−14.0131			−0.589015
$567$		0			0
$568$	−10.1030			−0.423910
$569$			27.5340i			1.15429i	0.816643	+	0.577143i	$0.195832\pi$
$569$			27.5340i			1.15429i	−0.816643	+	0.577143i	$0.804168\pi$
$570$			7.45179i			0.312121i
$571$			30.8742i			1.29205i	0.763318	+	0.646023i	$0.223569\pi$
$571$			30.8742i			1.29205i	−0.763318	+	0.646023i	$0.776431\pi$
$572$	−4.30333			−0.179931
$573$	−11.9873			−0.500777
$574$		0			0
$575$	5.14233	+	1.40949i	0.214450	+	0.0587797i
$576$	1.51344			0.0630599
$577$		−	41.2735i		−	1.71824i	−0.511776	−	0.859119i	$-0.671012\pi$
$577$		−	41.2735i		−	1.71824i	0.511776	−	0.859119i	$-0.328988\pi$
$578$	1.44101			0.0599382
$579$		−	9.61321i		−	0.399511i
$580$	22.2280			0.922967
$581$		0			0
$582$			7.92407i			0.328463i
$583$	−10.4590			−0.433169
$584$			3.84060i			0.158925i
$585$		−	12.7004i		−	0.525098i
$586$	−4.28527			−0.177023
$587$		−	13.5868i		−	0.560787i	−0.959885	−	0.280393i	$-0.909535\pi$
$587$		−	13.5868i		−	0.560787i	0.959885	−	0.280393i	$-0.0904649\pi$
$588$		0			0
$589$			9.49474i			0.391224i
$590$			24.0872i			0.991653i
$591$		−	7.26748i		−	0.298944i
$592$			1.43254i			0.0588772i
$593$		−	14.2794i		−	0.586387i	−0.956053	−	0.293193i	$-0.905282\pi$
$593$		−	14.2794i		−	0.586387i	0.956053	−	0.293193i	$-0.0947180\pi$
$594$	6.97645			0.286247
$595$		0			0
$596$			2.53551i			0.103859i
$597$			28.3497i			1.16028i
$598$	4.30333	−	15.7001i	0.175976	−	0.642026i
$599$	23.3347			0.953432			0.476716	−	0.879057i	$-0.341827\pi$
$599$	23.3347			0.953432			0.476716	+	0.879057i	$0.341827\pi$
$600$			1.35556i			0.0553403i
$601$			41.8019i			1.70513i	0.522618	+	0.852567i	$0.324955\pi$
$601$			41.8019i			1.70513i	−0.522618	+	0.852567i	$0.675045\pi$
$602$		0			0
$603$			17.2623i			0.702974i
$604$	8.02687			0.326609
$605$	23.2209			0.944064
$606$	−8.68329			−0.352734
$607$			7.06897i			0.286921i	0.989656	+	0.143460i	$0.0458229\pi$
$607$			7.06897i			0.286921i	−0.989656	+	0.143460i	$0.954177\pi$
$608$	2.47221			0.100261
$609$		0			0
$610$	−22.8400			−0.924764
$611$	5.64292			0.228288
$612$	−6.49915			−0.262713
$613$			41.8550i			1.69051i	0.534365	+	0.845254i	$0.320551\pi$
$613$			41.8550i			1.69051i	−0.534365	+	0.845254i	$0.679449\pi$
$614$		−	29.9300i		−	1.20788i
$615$	10.2316			0.412580
$616$		0			0
$617$		−	44.8424i		−	1.80529i	−0.430390	−	0.902643i	$-0.641624\pi$
$617$		−	44.8424i		−	1.80529i	0.430390	−	0.902643i	$-0.358376\pi$
$618$		−	16.9557i		−	0.682060i
$619$	4.27623			0.171876			0.0859382	−	0.996300i	$-0.472611\pi$
$619$	4.27623			0.171876			0.0859382	+	0.996300i	$0.472611\pi$
$620$			9.49474i			0.381318i
$621$	−6.97645	+	25.4526i	−0.279955	+	1.02138i
$622$			5.58497i			0.223937i
$623$		0			0
$624$	4.13867			0.165679
$625$	−29.3229			−1.17292
$626$	−8.85595			−0.353955
$627$	3.82131			0.152608
$628$	−19.2030			−0.766283
$629$		−	6.15177i		−	0.245287i
$630$		0			0
$631$		−	34.7236i		−	1.38232i	−0.722700	−	0.691162i	$-0.757099\pi$
$631$		−	34.7236i		−	1.38232i	0.722700	−	0.691162i	$-0.242901\pi$
$632$			7.43795i			0.295866i
$633$		−	9.30150i		−	0.369701i
$634$	19.0486			0.756515
$635$	4.92254			0.195345
$636$	10.0588			0.398859
$637$		0			0
$638$		−	11.3986i		−	0.451275i
$639$	−15.2902			−0.604870
$640$	2.47221			0.0977225
$641$		−	7.77493i		−	0.307091i	−0.988142	−	0.153546i	$-0.950931\pi$
$641$		−	7.77493i		−	0.307091i	0.988142	−	0.153546i	$-0.0490692\pi$
$642$	7.14926			0.282159
$643$	29.0738			1.14656			0.573278	−	0.819361i	$-0.305671\pi$
$643$	29.0738			1.14656			0.573278	+	0.819361i	$0.305671\pi$
$644$		0			0
$645$	39.1479			1.54145
$646$	−10.6164			−0.417696
$647$			1.13248i			0.0445224i	0.999752	+	0.0222612i	$0.00708655\pi$
$647$			1.13248i			0.0445224i	−0.999752	+	0.0222612i	$0.992913\pi$
$648$	−2.16920			−0.0852142
$649$	12.3520			0.484858
$650$		−	3.77394i		−	0.148026i
$651$		0			0
$652$	−2.48656			−0.0973813
$653$	−23.0941			−0.903742			−0.451871	−	0.892083i	$-0.649243\pi$
$653$	−23.0941			−0.903742			−0.451871	+	0.892083i	$0.649243\pi$
$654$	17.3809			0.679647
$655$		−	28.0322i		−	1.09531i
$656$		−	3.39445i		−	0.132531i
$657$			5.81250i			0.226767i
$658$		0			0
$659$		−	4.27458i		−	0.166514i	−0.996528	−	0.0832569i	$-0.973468\pi$
$659$		−	4.27458i		−	0.166514i	0.996528	−	0.0832569i	$-0.0265322\pi$
$660$	3.82131			0.148744
$661$	36.9548			1.43738			0.718688	−	0.695332i	$-0.244743\pi$
$661$	36.9548			1.43738			0.718688	+	0.695332i	$0.244743\pi$
$662$	2.60201			0.101130
$663$	−17.7727			−0.690234
$664$	5.21176			0.202256
$665$		0			0
$666$			2.16806i			0.0840108i
$667$	41.5862	+	11.3986i	1.61022	+	0.441355i
$668$		−	8.20178i		−	0.317337i
$669$	−10.2229			−0.395242
$670$			28.1980i			1.08938i
$671$			11.7124i			0.452153i
$672$		0			0
$673$	16.0507			0.618711			0.309355	−	0.950947i	$-0.399887\pi$
$673$	16.0507			0.618711			0.309355	+	0.950947i	$0.399887\pi$
$674$		−	5.37754i		−	0.207135i
$675$			6.11822i			0.235490i
$676$	1.47772			0.0568353
$677$	−4.02695			−0.154768			−0.0773841	−	0.997001i	$-0.524657\pi$
$677$	−4.02695			−0.154768			−0.0773841	+	0.997001i	$0.524657\pi$
$678$	−19.1366			−0.734935
$679$		0			0
$680$	−10.6164			−0.407120
$681$			15.8371i			0.606880i
$682$	4.86894			0.186441
$683$	−23.3384			−0.893019			−0.446509	−	0.894779i	$-0.647333\pi$
$683$	−23.3384			−0.893019			−0.446509	+	0.894779i	$0.647333\pi$
$684$	3.74153			0.143061
$685$		−	3.91025i		−	0.149403i
$686$		0			0
$687$		−	3.78021i		−	0.144224i
$688$		−	12.9877i		−	0.495151i
$689$	−28.0043			−1.06688
$690$	−3.82131	+	13.9415i	−0.145475	+	0.530744i
$691$			37.8172i			1.43863i	0.694682	+	0.719317i	$0.255545\pi$
$691$			37.8172i			1.43863i	−0.694682	+	0.719317i	$0.744455\pi$
$692$		−	17.5715i		−	0.667967i
$693$		0			0
$694$	13.6925			0.519759
$695$			35.0070i			1.32789i
$696$			10.9624i			0.415530i
$697$			14.5768i			0.552135i
$698$		−	23.2774i		−	0.881064i
$699$		−	30.2816i		−	1.14536i
$700$		0			0
$701$		−	26.4771i		−	1.00003i	−0.866017	−	0.500014i	$-0.833328\pi$
$701$		−	26.4771i		−	1.00003i	0.866017	−	0.500014i	$-0.166672\pi$
$702$	18.6796			0.705017
$703$			3.54154i			0.133572i
$704$		−	1.26776i		−	0.0477804i
$705$	−5.01085			−0.188719
$706$		−	31.6651i		−	1.19173i
$707$		0			0
$708$	−11.8794			−0.446454
$709$		−	16.3427i		−	0.613763i	−0.951748	−	0.306882i	$-0.900714\pi$
$709$		−	16.3427i		−	0.613763i	0.951748	−	0.306882i	$-0.0992855\pi$
$710$	−24.9766			−0.937354
$711$			11.2569i			0.422165i
$712$	−6.86384			−0.257233
$713$	−4.86894	+	17.7637i	−0.182343	+	0.665254i
$714$		0			0
$715$	−10.6387			−0.397866
$716$	6.51344			0.243419
$717$		−	30.0790i		−	1.12332i
$718$			35.1792i			1.31288i
$719$			43.1466i			1.60910i	0.593886	+	0.804549i	$0.297593\pi$
$719$			43.1466i			1.60910i	−0.593886	+	0.804549i	$0.702407\pi$
$720$	3.74153			0.139438
$721$		0			0
$722$	−12.8882			−0.479649
$723$		−	24.5695i		−	0.913749i
$724$	−10.2407			−0.380592
$725$	9.99635			0.371255
$726$			11.4521i			0.425029i
$727$	13.4385			0.498405			0.249203	−	0.968451i	$-0.419832\pi$
$727$	13.4385			0.498405			0.249203	+	0.968451i	$0.419832\pi$
$728$		0			0
$729$	−23.4115			−0.867092
$730$			9.49474i			0.351416i
$731$			55.7730i			2.06284i
$732$		−	11.2643i		−	0.416339i
$733$	30.8412			1.13915			0.569573	−	0.821940i	$-0.307108\pi$
$733$	30.8412			1.13915			0.569573	+	0.821940i	$0.307108\pi$
$734$	−13.2476			−0.488977
$735$		0			0
$736$	4.62523	+	1.26776i	0.170488	+	0.0467301i
$737$	14.4600			0.532642
$738$		−	5.13728i		−	0.189106i
$739$	1.98651			0.0730750			0.0365375	−	0.999332i	$-0.488367\pi$
$739$	1.98651			0.0730750			0.0365375	+	0.999332i	$0.488367\pi$
$740$			3.54154i			0.130190i
$741$	10.2316			0.375869
$742$		0			0
$743$		−	1.57808i		−	0.0578941i	−0.999581	−	0.0289471i	$-0.990785\pi$
$743$		−	1.57808i		−	0.0578941i	0.999581	−	0.0289471i	$-0.00921543\pi$
$744$	−4.68264			−0.171674
$745$			6.26831i			0.229653i
$746$		−	36.5887i		−	1.33961i
$747$	7.88767			0.288595
$748$			5.44413i			0.199057i
$749$		0			0
$750$		−	11.7199i		−	0.427951i
$751$		−	17.3196i		−	0.632001i	−0.948759	−	0.316000i	$-0.897660\pi$
$751$		−	17.3196i		−	0.632001i	0.948759	−	0.316000i	$-0.102340\pi$
$752$			1.66240i			0.0606214i
$753$			35.8181i			1.30528i
$754$		−	30.5200i		−	1.11147i
$755$	19.8441			0.722200
$756$		0			0
$757$			29.1470i			1.05937i	0.848196	+	0.529683i	$0.177689\pi$
$757$			29.1470i			1.05937i	−0.848196	+	0.529683i	$0.822311\pi$
$758$		−	1.29081i		−	0.0468844i
$759$	7.14926	+	1.95958i	0.259502	+	0.0711283i
$760$	6.11180			0.221698
$761$			19.2216i			0.696784i	0.937349	+	0.348392i	$0.113272\pi$
$761$			19.2216i			0.696784i	−0.937349	+	0.348392i	$0.886728\pi$
$762$			2.42771i			0.0879467i
$763$		0			0
$764$			9.83173i			0.355699i
$765$	−16.0672			−0.580912
$766$	4.84707			0.175132
$767$	33.0728			1.19419
$768$			1.21925i			0.0439958i
$769$	32.3533			1.16669			0.583344	−	0.812225i	$-0.301744\pi$
$769$	32.3533			1.16669			0.583344	+	0.812225i	$0.301744\pi$
$770$		0			0
$771$	5.54584			0.199729
$772$	−7.88455			−0.283771
$773$	36.0283			1.29585			0.647925	−	0.761704i	$-0.275637\pi$
$773$	36.0283			1.29585			0.647925	+	0.761704i	$0.275637\pi$
$774$		−	19.6560i		−	0.706522i
$775$			4.26997i			0.153382i
$776$	6.49915			0.233306
$777$		0			0
$778$			14.2749i			0.511780i
$779$		−	8.39177i		−	0.300666i
$780$	10.2316			0.366352
$781$			12.8081i			0.458309i
$782$	−19.8621	−	5.44413i	−0.710269	−	0.194682i
$783$			49.4783i			1.76821i
$784$		0			0
$785$	−47.4738			−1.69441
$786$	13.8250			0.493120
$787$	29.3463			1.04608			0.523042	−	0.852307i	$-0.324797\pi$
$787$	29.3463			1.04608			0.523042	+	0.852307i	$0.324797\pi$
$788$	−5.96063			−0.212339
$789$	−6.96742			−0.248047
$790$			18.3881i			0.654220i
$791$		0			0
$792$		−	1.91867i		−	0.0681769i
$793$			31.3603i			1.11364i
$794$			31.7377i			1.12633i
$795$	24.8675			0.881960
$796$	23.2518			0.824139
$797$	23.8046			0.843202			0.421601	−	0.906782i	$-0.361468\pi$
$797$	23.8046			0.843202			0.421601	+	0.906782i	$0.361468\pi$
$798$		0			0
$799$		−	7.13883i		−	0.252554i
$800$	1.11180			0.0393080
$801$	−10.3880			−0.367042
$802$		−	4.25893i		−	0.150388i
$803$	4.86894			0.171821
$804$	−13.9067			−0.490453
$805$		0			0
$806$	13.0367			0.459198
$807$	10.0999			0.355535
$808$			7.12184i			0.250546i
$809$	−51.0474			−1.79473			−0.897366	−	0.441287i	$-0.854522\pi$
$809$	−51.0474			−1.79473			−0.897366	+	0.441287i	$0.854522\pi$
$810$	−5.36270			−0.188426
$811$			48.7844i			1.71305i	0.516104	+	0.856526i	$0.327382\pi$
$811$			48.7844i			1.71305i	−0.516104	+	0.856526i	$0.672618\pi$
$812$		0			0
$813$	0.483559			0.0169591
$814$	1.81612			0.0636548
$815$	−6.14729			−0.215330
$816$		−	5.23581i		−	0.183290i
$817$		−	32.1082i		−	1.12332i
$818$		−	7.41636i		−	0.259307i
$819$		0			0
$820$		−	8.39177i		−	0.293053i
$821$	0.737032			0.0257226			0.0128613	−	0.999917i	$-0.495906\pi$
$821$	0.737032			0.0257226			0.0128613	+	0.999917i	$0.495906\pi$
$822$	1.92846			0.0672629
$823$	50.5862			1.76332			0.881662	−	0.471881i	$-0.156425\pi$
$823$	50.5862			1.76332			0.881662	+	0.471881i	$0.156425\pi$
$824$	−13.9067			−0.484464
$825$	1.71851			0.0598310
$826$		0			0
$827$			18.2125i			0.633310i	0.948541	+	0.316655i	$0.102560\pi$
$827$			18.2125i			0.633310i	−0.948541	+	0.316655i	$0.897440\pi$
$828$	7.00000	+	1.91867i	0.243267	+	0.0666784i
$829$			24.4889i			0.850535i	0.905068	+	0.425267i	$0.139820\pi$
$829$			24.4889i			0.850535i	−0.905068	+	0.425267i	$0.860180\pi$
$830$	12.8845			0.447229
$831$		−	21.3599i		−	0.740968i
$832$		−	3.39445i		−	0.117681i
$833$		0			0
$834$	−17.2648			−0.597833
$835$		−	20.2765i		−	0.701697i
$836$		−	3.13415i		−	0.108397i
$837$	−21.1348			−0.730525
$838$	−33.1406			−1.14482
$839$	−20.0986			−0.693881			−0.346940	−	0.937887i	$-0.612779\pi$
$839$	−20.0986			−0.693881			−0.346940	+	0.937887i	$0.612779\pi$
$840$		0			0
$841$	51.8409			1.78762
$842$		−	28.6831i		−	0.988484i
$843$	−24.9276			−0.858552
$844$	−7.62889			−0.262597
$845$	3.65322			0.125675
$846$			2.51593i			0.0864996i
$847$		0			0
$848$		−	8.25004i		−	0.283308i
$849$		−	17.0854i		−	0.586371i
$850$	−4.77439			−0.163760
$851$	−1.81612	+	6.62585i	−0.0622557	+	0.227131i
$852$		−	12.3180i		−	0.422008i
$853$			28.1854i			0.965050i	0.875882	+	0.482525i	$0.160280\pi$
$853$			28.1854i			0.965050i	−0.875882	+	0.482525i	$0.839720\pi$
$854$		0			0
$855$	9.24982			0.316337
$856$		−	5.86367i		−	0.200416i
$857$		−	52.5163i		−	1.79392i	−0.442108	−	0.896962i	$-0.645769\pi$
$857$		−	52.5163i		−	1.79392i	0.442108	−	0.896962i	$-0.354231\pi$
$858$		−	5.24683i		−	0.179124i
$859$			9.90627i			0.337998i	0.985616	+	0.168999i	$0.0540534\pi$
$859$			9.90627i			0.337998i	−0.985616	+	0.168999i	$0.945947\pi$
$860$		−	32.1082i		−	1.09488i
$861$		0			0
$862$		−	24.5733i		−	0.836970i
$863$	6.97866			0.237556			0.118778	−	0.992921i	$-0.462102\pi$
$863$	6.97866			0.237556			0.118778	+	0.992921i	$0.462102\pi$
$864$			5.50299i			0.187216i
$865$		−	43.4403i		−	1.47701i
$866$	24.3265			0.826647
$867$			1.75695i			0.0596691i
$868$		0			0
$869$	9.42950			0.319874
$870$			27.1014i			0.918823i
$871$	38.7171			1.31188
$872$		−	14.2554i		−	0.482750i
$873$	9.83605			0.332900
$874$	11.4345	+	3.13415i	0.386779	+	0.106014i
$875$		0			0
$876$	−4.68264			−0.158212
$877$	−39.1338			−1.32146			−0.660728	−	0.750626i	$-0.729752\pi$
$877$	−39.1338			−1.32146			−0.660728	+	0.750626i	$0.729752\pi$
$878$		−	4.54404i		−	0.153354i
$879$		−	5.22480i		−	0.176228i
$880$		−	3.13415i		−	0.105652i
$881$	−16.1961			−0.545660			−0.272830	−	0.962062i	$-0.587960\pi$
$881$	−16.1961			−0.545660			−0.272830	+	0.962062i	$0.587960\pi$
$882$		0			0
$883$	3.29018			0.110723			0.0553617	−	0.998466i	$-0.482369\pi$
$883$	3.29018			0.110723			0.0553617	+	0.998466i	$0.482369\pi$
$884$			14.5768i			0.490270i
$885$	−29.3682			−0.987201
$886$	−20.4652			−0.687543
$887$			13.7913i			0.463066i	0.972827	+	0.231533i	$0.0743742\pi$
$887$			13.7913i			0.463066i	−0.972827	+	0.231533i	$0.925626\pi$
$888$	−1.74662			−0.0586129
$889$		0			0
$890$	−16.9688			−0.568796
$891$			2.75002i			0.0921290i
$892$			8.38464i			0.280738i
$893$			4.10979i			0.137529i
$894$	−3.09142			−0.103392
$895$	16.1026			0.538249
$896$		0			0
$897$	19.1423	+	5.24683i	0.639144	+	0.175186i
$898$	−31.4921			−1.05090
$899$			34.5314i			1.15169i
$900$	1.68264			0.0560879
$901$			35.4281i			1.18028i
$902$	−4.30333			−0.143285
$903$		0			0
$904$			15.6954i			0.522021i
$905$	−25.3171			−0.841567
$906$			9.78674i			0.325143i
$907$			11.5672i			0.384082i	0.981387	+	0.192041i	$0.0615107\pi$
$907$			11.5672i			0.384082i	−0.981387	+	0.192041i	$0.938489\pi$
$908$	12.9893			0.431064
$909$			10.7785i			0.357499i
$910$		0			0
$911$			52.9276i			1.75357i	0.480882	+	0.876785i	$0.340317\pi$
$911$			52.9276i			1.75357i	−0.480882	+	0.876785i	$0.659683\pi$
$912$			3.01423i			0.0998111i
$913$		−	6.60725i		−	0.218668i
$914$			10.8086i			0.357517i
$915$		−	27.8476i		−	0.920613i
$916$	−3.10045			−0.102442
$917$		0			0
$918$		−	23.6315i		−	0.779956i
$919$		−	1.39004i		−	0.0458531i	−0.999737	−	0.0229266i	$-0.992702\pi$
$919$		−	1.39004i		−	0.0458531i	0.999737	−	0.0229266i	$-0.00729839\pi$
$920$	11.4345	+	3.13415i	0.376985	+	0.103330i
$921$	36.4921			1.20246
$922$		−	36.3796i		−	1.19810i
$923$			34.2940i			1.12880i
$924$		0			0
$925$			1.59270i			0.0523676i
$926$	25.6344			0.842399
$927$	−21.0470			−0.691273
$928$	8.99116			0.295149
$929$		−	16.4808i		−	0.540716i	−0.962760	−	0.270358i	$-0.912858\pi$
$929$		−	16.4808i		−	0.540716i	0.962760	−	0.270358i	$-0.0871421\pi$
$930$	−11.5764			−0.379606
$931$		0			0
$932$	−24.8363			−0.813541
$933$	−6.80946			−0.222932
$934$	−36.4021			−1.19111
$935$			13.4590i			0.440156i
$936$		−	5.13728i		−	0.167917i
$937$	−41.9928			−1.37184			−0.685922	−	0.727675i	$-0.740601\pi$
$937$	−41.9928			−1.37184			−0.685922	+	0.727675i	$0.740601\pi$
$938$		0			0
$939$		−	10.7976i		−	0.352366i
$940$			4.10979i			0.134047i
$941$	−33.4498			−1.09043			−0.545216	−	0.838296i	$-0.683552\pi$
$941$	−33.4498			−1.09043			−0.545216	+	0.838296i	$0.683552\pi$
$942$		−	23.4132i		−	0.762843i
$943$	4.30333	−	15.7001i	0.140136	−	0.511266i
$944$			9.74319i			0.317114i
$945$		0			0
$946$	−16.4652			−0.535330
$947$	−10.0537			−0.326703			−0.163351	−	0.986568i	$-0.552230\pi$
$947$	−10.0537			−0.326703			−0.163351	+	0.986568i	$0.552230\pi$
$948$	−9.06869			−0.294537
$949$	13.0367			0.423189
$950$	2.74859			0.0891761
$951$			23.2249i			0.753119i
$952$		0			0
$953$			19.7821i			0.640806i	0.947281	+	0.320403i	$0.103818\pi$
$953$			19.7821i			0.640806i	−0.947281	+	0.320403i	$0.896182\pi$
$954$		−	12.4859i		−	0.404246i
$955$			24.3060i			0.786525i
$956$	−24.6701			−0.797889
$957$	13.8977			0.449249
$958$	−3.91154			−0.126376
$959$		0			0
$960$			3.01423i			0.0972838i
$961$	16.2498			0.524188
$962$	4.86270			0.156780
$963$		−	8.87429i		−	0.285970i
$964$	−20.1514			−0.649032
$965$	−19.4922			−0.627477
$966$		0			0
$967$	39.6200			1.27409			0.637047	−	0.770825i	$-0.280156\pi$
$967$	39.6200			1.27409			0.637047	+	0.770825i	$0.280156\pi$
$968$	9.39279			0.301896
$969$		−	12.9440i		−	0.415821i
$970$	16.0672			0.515888
$971$	−47.6130			−1.52797			−0.763987	−	0.645232i	$-0.776761\pi$
$971$	−47.6130			−1.52797			−0.763987	+	0.645232i	$0.776761\pi$
$972$			13.8642i			0.444694i
$973$		0			0
$974$	−30.3901			−0.973763
$975$	4.60137			0.147362
$976$	−9.23871			−0.295724
$977$		−	0.290874i		−	0.00930587i	−0.999989	−	0.00465293i	$-0.998519\pi$
$977$		−	0.290874i		−	0.00930587i	0.999989	−	0.00465293i	$-0.00148108\pi$
$978$		−	3.03173i		−	0.0969442i
$979$			8.70168i			0.278107i
$980$		0			0
$981$		−	21.5747i		−	0.688828i
$982$	−20.2990			−0.647768
$983$	−31.4358			−1.00265			−0.501323	−	0.865260i	$-0.667153\pi$
$983$	−31.4358			−1.00265			−0.501323	+	0.865260i	$0.667153\pi$
$984$	4.13867			0.131936
$985$	−14.7359			−0.469525
$986$	−38.6107			−1.22962
$987$		0			0
$988$		−	8.39177i		−	0.266978i
$989$	16.4652	−	60.0711i	0.523564	−	1.91015i
$990$		−	4.74334i		−	0.150753i
$991$	−13.6729			−0.434334			−0.217167	−	0.976134i	$-0.569682\pi$
$991$	−13.6729			−0.434334			−0.217167	+	0.976134i	$0.569682\pi$
$992$			3.84060i			0.121939i
$993$			3.17250i			0.100676i
$994$		0			0
$995$	57.4832			1.82234
$996$			6.35443i			0.201348i
$997$		−	9.95215i		−	0.315188i	−0.987504	−	0.157594i	$-0.949626\pi$
$997$		−	9.95215i		−	0.315188i	0.987504	−	0.157594i	$-0.0503737\pi$
$998$	6.80946			0.215550
$999$	−7.88328			−0.249416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.2.c.c.2253.12		16
7.4	even	3		322.2.g.a.229.5	yes	16
7.5	odd	6		322.2.g.a.45.6	yes	16
7.6	odd	2	inner	2254.2.c.c.2253.5		16
23.22	odd	2	inner	2254.2.c.c.2253.11		16
161.68	even	6		322.2.g.a.45.5	✓	16
161.137	odd	6		322.2.g.a.229.6	yes	16
161.160	even	2	inner	2254.2.c.c.2253.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.g.a.45.5	✓	16	161.68	even	6
322.2.g.a.45.6	yes	16	7.5	odd	6
322.2.g.a.229.5	yes	16	7.4	even	3
322.2.g.a.229.6	yes	16	161.137	odd	6
2254.2.c.c.2253.5		16	7.6	odd	2	inner
2254.2.c.c.2253.6		16	161.160	even	2	inner
2254.2.c.c.2253.11		16	23.22	odd	2	inner
2254.2.c.c.2253.12		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2254.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.21925i q^{3} +1.00000 q^{4} +2.47221 q^{5} +1.21925i q^{6} +1.00000 q^{8} +1.51344 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.21925i q^{3} +1.00000 q^{4} +2.47221 q^{5} +1.21925i q^{6} +1.00000 q^{8} +1.51344 q^{9} +2.47221 q^{10} -1.26776i q^{11} +1.21925i q^{12} -3.39445i q^{13} +3.01423i q^{15} +1.00000 q^{16} -4.29430 q^{17} +1.51344 q^{18} +2.47221 q^{19} +2.47221 q^{20} -1.26776i q^{22} +(4.62523 + 1.26776i) q^{23} +1.21925i q^{24} +1.11180 q^{25} -3.39445i q^{26} +5.50299i q^{27} +8.99116 q^{29} +3.01423i q^{30} +3.84060i q^{31} +1.00000 q^{32} +1.54571 q^{33} -4.29430 q^{34} +1.51344 q^{36} +1.43254i q^{37} +2.47221 q^{38} +4.13867 q^{39} +2.47221 q^{40} -3.39445i q^{41} -12.9877i q^{43} -1.26776i q^{44} +3.74153 q^{45} +(4.62523 + 1.26776i) q^{46} +1.66240i q^{47} +1.21925i q^{48} +1.11180 q^{50} -5.23581i q^{51} -3.39445i q^{52} -8.25004i q^{53} +5.50299i q^{54} -3.13415i q^{55} +3.01423i q^{57} +8.99116 q^{58} +9.74319i q^{59} +3.01423i q^{60} -9.23871 q^{61} +3.84060i q^{62} +1.00000 q^{64} -8.39177i q^{65} +1.54571 q^{66} +11.4060i q^{67} -4.29430 q^{68} +(-1.54571 + 5.63930i) q^{69} -10.1030 q^{71} +1.51344 q^{72} +3.84060i q^{73} +1.43254i q^{74} +1.35556i q^{75} +2.47221 q^{76} +4.13867 q^{78} +7.43795i q^{79} +2.47221 q^{80} -2.16920 q^{81} -3.39445i q^{82} +5.21176 q^{83} -10.6164 q^{85} -12.9877i q^{86} +10.9624i q^{87} -1.26776i q^{88} -6.86384 q^{89} +3.74153 q^{90} +(4.62523 + 1.26776i) q^{92} -4.68264 q^{93} +1.66240i q^{94} +6.11180 q^{95} +1.21925i q^{96} +6.49915 q^{97} -1.91867i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 20 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 16 * q^8 - 20 * q^9	\( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} - 20 q^{9} + 16 q^{16} - 20 q^{18} + 8 q^{23} - 4 q^{25} + 16 q^{29} + 16 q^{32} - 20 q^{36} - 44 q^{39} + 8 q^{46} - 4 q^{50} + 16 q^{58} + 16 q^{64} - 12 q^{71} - 20 q^{72} - 44 q^{78} + 72 q^{81} + 24 q^{85} + 8 q^{92} + 76 q^{93} + 76 q^{95}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 16 * q^8 - 20 * q^9 + 16 * q^16 - 20 * q^18 + 8 * q^23 - 4 * q^25 + 16 * q^29 + 16 * q^32 - 20 * q^36 - 44 * q^39 + 8 * q^46 - 4 * q^50 + 16 * q^58 + 16 * q^64 - 12 * q^71 - 20 * q^72 - 44 * q^78 + 72 * q^81 + 24 * q^85 + 8 * q^92 + 76 * q^93 + 76 * q^95

Introduction

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