LMFDB - Embedded newform 1617.2.c.b.538.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(538,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.538");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.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:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			538.5
Character	$\chi$	$=$	1617.538
Dual form			1617.2.c.b.538.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34878i q^{2} -1.00000i q^{3} -3.51679 q^{4} +3.84365i q^{5} -2.34878 q^{6} +3.56261i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.34878i q^{2} -1.00000i q^{3} -3.51679 q^{4} +3.84365i q^{5} -2.34878 q^{6} +3.56261i q^{8} -1.00000 q^{9} +9.02791 q^{10} +(0.648688 - 3.25257i) q^{11} +3.51679i q^{12} +2.56122 q^{13} +3.84365 q^{15} +1.33423 q^{16} +0.436269 q^{17} +2.34878i q^{18} -5.74787 q^{19} -13.5173i q^{20} +(-7.63958 - 1.52363i) q^{22} -5.71576 q^{23} +3.56261 q^{24} -9.77365 q^{25} -6.01575i q^{26} +1.00000i q^{27} +1.22751i q^{29} -9.02791i q^{30} +5.39790i q^{31} +3.99141i q^{32} +(-3.25257 - 0.648688i) q^{33} -1.02470i q^{34} +3.51679 q^{36} -9.95670 q^{37} +13.5005i q^{38} -2.56122i q^{39} -13.6934 q^{40} -12.3106 q^{41} -8.05674i q^{43} +(-2.28130 + 11.4386i) q^{44} -3.84365i q^{45} +13.4251i q^{46} -5.43927i q^{47} -1.33423i q^{48} +22.9562i q^{50} -0.436269i q^{51} -9.00726 q^{52} +6.51185 q^{53} +2.34878 q^{54} +(12.5017 + 2.49333i) q^{55} +5.74787i q^{57} +2.88315 q^{58} -3.88092i q^{59} -13.5173 q^{60} +4.47453 q^{61} +12.6785 q^{62} +12.0434 q^{64} +9.84442i q^{65} +(-1.52363 + 7.63958i) q^{66} -4.08459 q^{67} -1.53427 q^{68} +5.71576i q^{69} -5.05430 q^{71} -3.56261i q^{72} +3.52022 q^{73} +23.3861i q^{74} +9.77365i q^{75} +20.2140 q^{76} -6.01575 q^{78} +16.6372i q^{79} +5.12831i q^{80} +1.00000 q^{81} +28.9150i q^{82} +4.69573 q^{83} +1.67686i q^{85} -18.9235 q^{86} +1.22751 q^{87} +(11.5876 + 2.31102i) q^{88} -2.88525i q^{89} -9.02791 q^{90} +20.1011 q^{92} +5.39790 q^{93} -12.7757 q^{94} -22.0928i q^{95} +3.99141 q^{96} -11.2534i q^{97} +(-0.648688 + 3.25257i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) $ 48 * q - 64 * q^4 - 48 * q^9	$ 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) $ 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1617\mathbb{Z}\right)^\times$.

$n$	$199$	$442$	$1079$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	2.34878i		−	1.66084i	−0.557137	−	0.830421i	$-0.688100\pi$
$2$		−	2.34878i		−	1.66084i	0.557137	−	0.830421i	$-0.311900\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−3.51679			−1.75839
$5$			3.84365i			1.71893i	0.511192	+	0.859466i	$0.329204\pi$
$5$			3.84365i			1.71893i	−0.511192	+	0.859466i	$0.670796\pi$
$6$	−2.34878			−0.958887
$7$		0			0
$7$		0			0
$8$			3.56261i			1.25957i
$9$	−1.00000			−0.333333
$10$	9.02791			2.85487
$11$	0.648688	−	3.25257i	0.195587	−	0.980686i
$11$	0.648688	−	3.25257i	0.195587	−	0.980686i
$12$			3.51679i			1.01521i
$13$	2.56122			0.710354			0.355177	−	0.934799i	$-0.384421\pi$
$13$	2.56122			0.710354			0.355177	+	0.934799i	$0.384421\pi$
$14$		0			0
$15$	3.84365			0.992426
$16$	1.33423			0.333557
$17$	0.436269			0.105811			0.0529054	−	0.998600i	$-0.483152\pi$
$17$	0.436269			0.105811			0.0529054	+	0.998600i	$0.483152\pi$
$18$			2.34878i			0.553614i
$19$	−5.74787			−1.31865			−0.659325	−	0.751858i	$-0.729158\pi$
$19$	−5.74787			−1.31865			−0.659325	+	0.751858i	$0.729158\pi$
$20$		−	13.5173i		−	3.02256i
$21$		0			0
$22$	−7.63958	−	1.52363i	−1.62876	−	0.324838i
$23$	−5.71576			−1.19182			−0.595909	−	0.803052i	$-0.703208\pi$
$23$	−5.71576			−1.19182			−0.595909	+	0.803052i	$0.703208\pi$
$24$	3.56261			0.727215
$25$	−9.77365			−1.95473
$26$		−	6.01575i		−	1.17978i
$27$			1.00000i			0.192450i
$28$		0			0
$29$			1.22751i			0.227942i	0.993484	+	0.113971i	$0.0363571\pi$
$29$			1.22751i			0.227942i	−0.993484	+	0.113971i	$0.963643\pi$
$30$		−	9.02791i		−	1.64826i
$31$			5.39790i			0.969492i	0.874655	+	0.484746i	$0.161088\pi$
$31$			5.39790i			0.969492i	−0.874655	+	0.484746i	$0.838912\pi$
$32$			3.99141i			0.705588i
$33$	−3.25257	−	0.648688i	−0.566200	−	0.112922i
$34$		−	1.02470i		−	0.175735i
$35$		0			0
$36$	3.51679			0.586132
$37$	−9.95670			−1.63687			−0.818436	−	0.574598i	$-0.805158\pi$
$37$	−9.95670			−1.63687			−0.818436	+	0.574598i	$0.805158\pi$
$38$			13.5005i			2.19007i
$39$		−	2.56122i		−	0.410123i
$40$	−13.6934			−2.16512
$41$	−12.3106			−1.92260			−0.961299	−	0.275506i	$-0.911155\pi$
$41$	−12.3106			−1.92260			−0.961299	+	0.275506i	$0.911155\pi$
$42$		0			0
$43$		−	8.05674i		−	1.22864i	−0.789057	−	0.614320i	$-0.789430\pi$
$43$		−	8.05674i		−	1.22864i	0.789057	−	0.614320i	$-0.210570\pi$
$44$	−2.28130	+	11.4386i	−0.343919	+	1.72443i
$45$		−	3.84365i		−	0.572978i
$46$			13.4251i			1.97942i
$47$		−	5.43927i		−	0.793400i	−0.917948	−	0.396700i	$-0.870155\pi$
$47$		−	5.43927i		−	0.793400i	0.917948	−	0.396700i	$-0.129845\pi$
$48$		−	1.33423i		−	0.192579i
$49$		0			0
$50$			22.9562i			3.24650i
$51$		−	0.436269i		−	0.0610899i
$52$	−9.00726			−1.24908
$53$	6.51185			0.894472			0.447236	−	0.894416i	$-0.352408\pi$
$53$	6.51185			0.894472			0.447236	+	0.894416i	$0.352408\pi$
$54$	2.34878			0.319629
$55$	12.5017	+	2.49333i	1.68573	+	0.336200i
$56$		0			0
$57$			5.74787i			0.761323i
$58$	2.88315			0.378576
$59$		−	3.88092i		−	0.505253i	−0.967564	−	0.252626i	$-0.918706\pi$
$59$		−	3.88092i		−	0.505253i	0.967564	−	0.252626i	$-0.0812943\pi$
$60$	−13.5173			−1.74508
$61$	4.47453			0.572904			0.286452	−	0.958095i	$-0.407524\pi$
$61$	4.47453			0.572904			0.286452	+	0.958095i	$0.407524\pi$
$62$	12.6785			1.61017
$63$		0			0
$64$	12.0434			1.50543
$65$			9.84442i			1.22105i
$66$	−1.52363	+	7.63958i	−0.187546	+	0.940368i
$67$	−4.08459			−0.499013			−0.249506	−	0.968373i	$-0.580268\pi$
$67$	−4.08459			−0.499013			−0.249506	+	0.968373i	$0.580268\pi$
$68$	−1.53427			−0.186057
$69$			5.71576i			0.688097i
$70$		0			0
$71$	−5.05430			−0.599835			−0.299918	−	0.953965i	$-0.596959\pi$
$71$	−5.05430			−0.599835			−0.299918	+	0.953965i	$0.596959\pi$
$72$		−	3.56261i		−	0.419858i
$73$	3.52022			0.412011			0.206005	−	0.978551i	$-0.433954\pi$
$73$	3.52022			0.412011			0.206005	+	0.978551i	$0.433954\pi$
$74$			23.3861i			2.71858i
$75$			9.77365i			1.12856i
$76$	20.2140			2.31871
$77$		0			0
$78$	−6.01575			−0.681149
$79$			16.6372i			1.87183i	0.352227	+	0.935914i	$0.385424\pi$
$79$			16.6372i			1.87183i	−0.352227	+	0.935914i	$0.614576\pi$
$80$			5.12831i			0.573363i
$81$	1.00000			0.111111
$82$			28.9150i			3.19313i
$83$	4.69573			0.515424			0.257712	−	0.966222i	$-0.417031\pi$
$83$	4.69573			0.515424			0.257712	+	0.966222i	$0.417031\pi$
$84$		0			0
$85$			1.67686i			0.181882i
$86$	−18.9235			−2.04058
$87$	1.22751			0.131602
$88$	11.5876	+	2.31102i	1.23525	+	0.246356i
$89$		−	2.88525i		−	0.305836i	−0.988239	−	0.152918i	$-0.951133\pi$
$89$		−	2.88525i		−	0.305836i	0.988239	−	0.152918i	$-0.0488670\pi$
$90$	−9.02791			−0.951625
$91$		0			0
$92$	20.1011			2.09569
$93$	5.39790			0.559736
$94$	−12.7757			−1.31771
$95$		−	22.0928i		−	2.26667i
$96$	3.99141			0.407371
$97$		−	11.2534i		−	1.14261i	−0.820737	−	0.571305i	$-0.806437\pi$
$97$		−	11.2534i		−	1.14261i	0.820737	−	0.571305i	$-0.193563\pi$
$98$		0			0
$99$	−0.648688	+	3.25257i	−0.0651956	+	0.326895i
$100$	34.3719			3.43719
$101$	−17.3219			−1.72360			−0.861799	−	0.507250i	$-0.830662\pi$
$101$	−17.3219			−1.72360			−0.861799	+	0.507250i	$0.830662\pi$
$102$	−1.02470			−0.101461
$103$			18.0178i			1.77534i	0.460476	+	0.887672i	$0.347679\pi$
$103$			18.0178i			1.77534i	−0.460476	+	0.887672i	$0.652321\pi$
$104$			9.12462i			0.894743i
$105$		0			0
$106$		−	15.2949i		−	1.48558i
$107$		−	9.65458i		−	0.933343i	−0.884431	−	0.466672i	$-0.845453\pi$
$107$		−	9.65458i		−	0.933343i	0.884431	−	0.466672i	$-0.154547\pi$
$108$		−	3.51679i		−	0.338403i
$109$			8.52361i			0.816413i	0.912890	+	0.408207i	$0.133846\pi$
$109$			8.52361i			0.816413i	−0.912890	+	0.408207i	$0.866154\pi$
$110$	5.85629	−	29.3639i	0.558375	−	2.79974i
$111$			9.95670i			0.945048i
$112$		0			0
$113$	−13.6546			−1.28452			−0.642259	−	0.766488i	$-0.722002\pi$
$113$	−13.6546			−1.28452			−0.642259	+	0.766488i	$0.722002\pi$
$114$	13.5005			1.26444
$115$		−	21.9694i		−	2.04865i
$116$		−	4.31688i		−	0.400812i
$117$	−2.56122			−0.236785
$118$	−9.11544			−0.839144
$119$		0			0
$120$			13.6934i			1.25003i
$121$	−10.1584	−	4.21980i	−0.923492	−	0.383618i
$122$		−	10.5097i		−	0.951503i
$123$			12.3106i			1.11001i
$124$		−	18.9833i		−	1.70475i
$125$		−	18.3482i		−	1.64112i
$126$		0			0
$127$			17.6262i			1.56407i	0.623235	+	0.782035i	$0.285818\pi$
$127$			17.6262i			1.56407i	−0.623235	+	0.782035i	$0.714182\pi$
$128$		−	20.3046i		−	1.79469i
$129$	−8.05674			−0.709356
$130$	23.1224			2.02797
$131$	−1.57624			−0.137717			−0.0688583	−	0.997626i	$-0.521936\pi$
$131$	−1.57624			−0.137717			−0.0688583	+	0.997626i	$0.521936\pi$
$132$	11.4386	+	2.28130i	0.995602	+	0.198561i
$133$		0			0
$134$			9.59383i			0.828781i
$135$	−3.84365			−0.330809
$136$			1.55426i			0.133276i
$137$	0.722567			0.0617331			0.0308665	−	0.999524i	$-0.490173\pi$
$137$	0.722567			0.0617331			0.0308665	+	0.999524i	$0.490173\pi$
$138$	13.4251			1.14282
$139$	2.37513			0.201456			0.100728	−	0.994914i	$-0.467883\pi$
$139$	2.37513			0.201456			0.100728	+	0.994914i	$0.467883\pi$
$140$		0			0
$141$	−5.43927			−0.458070
$142$			11.8715i			0.996232i
$143$	1.66143	−	8.33053i	0.138936	−	0.696634i
$144$	−1.33423			−0.111186
$145$	−4.71810			−0.391817
$146$		−	8.26824i		−	0.684284i
$147$		0			0
$148$	35.0156			2.87827
$149$		−	2.66454i		−	0.218287i	−0.994026	−	0.109144i	$-0.965189\pi$
$149$		−	2.66454i		−	0.218287i	0.994026	−	0.109144i	$-0.0348109\pi$
$150$	22.9562			1.87437
$151$		−	9.57465i		−	0.779174i	−0.920990	−	0.389587i	$-0.872618\pi$
$151$		−	9.57465i		−	0.779174i	0.920990	−	0.389587i	$-0.127382\pi$
$152$		−	20.4774i		−	1.66094i
$153$	−0.436269			−0.0352702
$154$		0			0
$155$	−20.7476			−1.66649
$156$			9.00726i			0.721158i
$157$			17.2160i			1.37399i	0.726662	+	0.686995i	$0.241071\pi$
$157$			17.2160i			1.37399i	−0.726662	+	0.686995i	$0.758929\pi$
$158$	39.0772			3.10881
$159$		−	6.51185i		−	0.516423i
$160$	−15.3416			−1.21286
$161$		0			0
$162$		−	2.34878i		−	0.184538i
$163$	10.6669			0.835497			0.417748	−	0.908563i	$-0.362819\pi$
$163$	10.6669			0.835497			0.417748	+	0.908563i	$0.362819\pi$
$164$	43.2939			3.38069
$165$	2.49333	−	12.5017i	0.194105	−	0.973259i
$166$		−	11.0293i		−	0.856037i
$167$	15.8643			1.22761			0.613807	−	0.789456i	$-0.289637\pi$
$167$	15.8643			1.22761			0.613807	+	0.789456i	$0.289637\pi$
$168$		0			0
$169$	−6.44017			−0.495398
$170$	3.93859			0.302076
$171$	5.74787			0.439550
$172$			28.3338i			2.16044i
$173$	−14.4467			−1.09837			−0.549183	−	0.835702i	$-0.685061\pi$
$173$	−14.4467			−1.09837			−0.549183	+	0.835702i	$0.685061\pi$
$174$		−	2.88315i		−	0.218571i
$175$		0			0
$176$	0.865498	−	4.33967i	0.0652394	−	0.327115i
$177$	−3.88092			−0.291708
$178$	−6.77683			−0.507945
$179$	−2.87300			−0.214738			−0.107369	−	0.994219i	$-0.534243\pi$
$179$	−2.87300			−0.214738			−0.107369	+	0.994219i	$0.534243\pi$
$180$			13.5173i			1.00752i
$181$			4.96366i			0.368946i	0.982838	+	0.184473i	$0.0590578\pi$
$181$			4.96366i			0.368946i	−0.982838	+	0.184473i	$0.940942\pi$
$182$		0			0
$183$		−	4.47453i		−	0.330766i
$184$		−	20.3630i		−	1.50118i
$185$		−	38.2701i		−	2.81367i
$186$		−	12.6785i		−	0.929633i
$187$	0.283002	−	1.41899i	0.0206952	−	0.103767i
$188$			19.1288i			1.39511i
$189$		0			0
$190$	−51.8912			−3.76458
$191$	9.09190			0.657867			0.328934	−	0.944353i	$-0.393311\pi$
$191$	9.09190			0.657867			0.328934	+	0.944353i	$0.393311\pi$
$192$		−	12.0434i		−	0.869158i
$193$		−	9.48779i		−	0.682946i	−0.939892	−	0.341473i	$-0.889074\pi$
$193$		−	9.48779i		−	0.682946i	0.939892	−	0.341473i	$-0.110926\pi$
$194$	−26.4318			−1.89770
$195$	9.84442			0.704973
$196$		0			0
$197$		−	8.29213i		−	0.590790i	−0.955375	−	0.295395i	$-0.904549\pi$
$197$		−	8.29213i		−	0.590790i	0.955375	−	0.295395i	$-0.0954513\pi$
$198$	7.63958	+	1.52363i	0.542922	+	0.108279i
$199$			5.84519i			0.414355i	0.978303	+	0.207177i	$0.0664277\pi$
$199$			5.84519i			0.414355i	−0.978303	+	0.207177i	$0.933572\pi$
$200$		−	34.8197i		−	2.46213i
$201$			4.08459i			0.288105i
$202$			40.6855i			2.86262i
$203$		0			0
$204$			1.53427i			0.107420i
$205$		−	47.3178i		−	3.30482i
$206$	42.3199			2.94857
$207$	5.71576			0.397273
$208$	3.41725			0.236944
$209$	−3.72857	+	18.6953i	−0.257911	+	1.29318i
$210$		0			0
$211$		−	19.3323i		−	1.33089i	−0.746446	−	0.665446i	$-0.768241\pi$
$211$		−	19.3323i		−	1.33089i	0.746446	−	0.665446i	$-0.231759\pi$
$212$	−22.9008			−1.57283
$213$			5.05430i			0.346315i
$214$	−22.6765			−1.55014
$215$	30.9673			2.11195
$216$	−3.56261			−0.242405
$217$		0			0
$218$	20.0201			1.35593
$219$		−	3.52022i		−	0.237874i
$220$	−43.9660	−	8.76851i	−2.96419	−	0.591173i
$221$	1.11738			0.0751630
$222$	23.3861			1.56957
$223$			10.3611i			0.693828i	0.937897	+	0.346914i	$0.112770\pi$
$223$			10.3611i			0.693828i	−0.937897	+	0.346914i	$0.887230\pi$
$224$		0			0
$225$	9.77365			0.651576
$226$			32.0717i			2.13338i
$227$	10.1930			0.676534			0.338267	−	0.941050i	$-0.390159\pi$
$227$	10.1930			0.676534			0.338267	+	0.941050i	$0.390159\pi$
$228$		−	20.2140i		−	1.33871i
$229$			11.7876i			0.778950i	0.921037	+	0.389475i	$0.127343\pi$
$229$			11.7876i			0.778950i	−0.921037	+	0.389475i	$0.872657\pi$
$230$	−51.6013			−3.40249
$231$		0			0
$232$	−4.37312			−0.287110
$233$		−	22.5934i		−	1.48015i	−0.672527	−	0.740073i	$-0.734791\pi$
$233$		−	22.5934i		−	1.48015i	0.672527	−	0.740073i	$-0.265209\pi$
$234$			6.01575i			0.393262i
$235$	20.9067			1.36380
$236$			13.6484i			0.888433i
$237$	16.6372			1.08070
$238$		0			0
$239$		−	9.15208i		−	0.591999i	−0.955188	−	0.295999i	$-0.904347\pi$
$239$		−	9.15208i		−	0.591999i	0.955188	−	0.295999i	$-0.0956526\pi$
$240$	5.12831			0.331031
$241$	−23.4348			−1.50957			−0.754783	−	0.655975i	$-0.772258\pi$
$241$	−23.4348			−1.50957			−0.754783	+	0.655975i	$0.772258\pi$
$242$	−9.91141	+	23.8599i	−0.637129	+	1.53377i
$243$		−	1.00000i		−	0.0641500i
$244$	−15.7360			−1.00739
$245$		0			0
$246$	28.9150			1.84356
$247$	−14.7215			−0.936708
$248$	−19.2306			−1.22115
$249$		−	4.69573i		−	0.297580i
$250$	−43.0960			−2.72563
$251$		−	28.1432i		−	1.77638i	−0.459476	−	0.888190i	$-0.651963\pi$
$251$		−	28.1432i		−	1.77638i	0.459476	−	0.888190i	$-0.348037\pi$
$252$		0			0
$253$	−3.70774	+	18.5909i	−0.233104	+	1.16880i
$254$	41.4001			2.59767
$255$	1.67686			0.105009
$256$	−23.6042			−1.47527
$257$			19.8203i			1.23635i	0.786039	+	0.618177i	$0.212129\pi$
$257$			19.8203i			1.23635i	−0.786039	+	0.618177i	$0.787871\pi$
$258$			18.9235i			1.17813i
$259$		0			0
$260$		−	34.6207i		−	2.14709i
$261$		−	1.22751i		−	0.0759807i
$262$			3.70224i			0.228725i
$263$		−	11.5493i		−	0.712161i	−0.934455	−	0.356081i	$-0.884113\pi$
$263$		−	11.5493i		−	0.712161i	0.934455	−	0.356081i	$-0.115887\pi$
$264$	2.31102	−	11.5876i	0.142234	−	0.713170i
$265$			25.0293i			1.53754i
$266$		0			0
$267$	−2.88525			−0.176575
$268$	14.3647			0.877461
$269$			21.1556i			1.28988i	0.764233	+	0.644940i	$0.223118\pi$
$269$			21.1556i			1.28988i	−0.764233	+	0.644940i	$0.776882\pi$
$270$			9.02791i			0.549421i
$271$	26.9307			1.63593			0.817963	−	0.575271i	$-0.195103\pi$
$271$	26.9307			1.63593			0.817963	+	0.575271i	$0.195103\pi$
$272$	0.582083			0.0352940
$273$		0			0
$274$		−	1.69715i		−	0.102529i
$275$	−6.34004	+	31.7895i	−0.382319	+	1.91698i
$276$		−	20.1011i		−	1.20995i
$277$			16.2156i			0.974299i	0.873319	+	0.487149i	$0.161963\pi$
$277$			16.2156i			0.974299i	−0.873319	+	0.487149i	$0.838037\pi$
$278$		−	5.57868i		−	0.334587i
$279$		−	5.39790i		−	0.323164i
$280$		0			0
$281$		−	23.2290i		−	1.38573i	−0.721068	−	0.692864i	$-0.756349\pi$
$281$		−	23.2290i		−	1.38573i	0.721068	−	0.692864i	$-0.243651\pi$
$282$			12.7757i			0.760781i
$283$	1.16316			0.0691429			0.0345714	−	0.999402i	$-0.488993\pi$
$283$	1.16316			0.0691429			0.0345714	+	0.999402i	$0.488993\pi$
$284$	17.7749			1.05475
$285$	−22.0928			−1.30866
$286$	−19.5666	−	3.90234i	−1.15700	−	0.230750i
$287$		0			0
$288$		−	3.99141i		−	0.235196i
$289$	−16.8097			−0.988804
$290$			11.0818i			0.650746i
$291$	−11.2534			−0.659687
$292$	−12.3799			−0.724477
$293$	−6.73723			−0.393593			−0.196797	−	0.980444i	$-0.563054\pi$
$293$	−6.73723			−0.393593			−0.196797	+	0.980444i	$0.563054\pi$
$294$		0			0
$295$	14.9169			0.868495
$296$		−	35.4719i		−	2.06176i
$297$	3.25257	+	0.648688i	0.188733	+	0.0376407i
$298$	−6.25842			−0.362541
$299$	−14.6393			−0.846612
$300$		−	34.3719i		−	1.98446i
$301$		0			0
$302$	−22.4888			−1.29408
$303$			17.3219i			0.995120i
$304$	−7.66897			−0.439846
$305$			17.1985i			0.984784i
$306$			1.02470i			0.0585783i
$307$	12.3115			0.702655			0.351328	−	0.936253i	$-0.385730\pi$
$307$	12.3115			0.702655			0.351328	+	0.936253i	$0.385730\pi$
$308$		0			0
$309$	18.0178			1.02500
$310$			48.7318i			2.76778i
$311$			6.72951i			0.381596i	0.981629	+	0.190798i	$0.0611075\pi$
$311$			6.72951i			0.381596i	−0.981629	+	0.190798i	$0.938892\pi$
$312$	9.12462			0.516580
$313$		−	8.88544i		−	0.502235i	−0.967957	−	0.251117i	$-0.919202\pi$
$313$		−	8.88544i		−	0.502235i	0.967957	−	0.251117i	$-0.0807980\pi$
$314$	40.4368			2.28198
$315$		0			0
$316$		−	58.5095i		−	3.29141i
$317$	−10.1489			−0.570020			−0.285010	−	0.958525i	$-0.591997\pi$
$317$	−10.1489			−0.570020			−0.285010	+	0.958525i	$0.591997\pi$
$318$	−15.2949			−0.857697
$319$	3.99255	+	0.796267i	0.223540	+	0.0445824i
$320$			46.2907i			2.58773i
$321$	−9.65458			−0.538866
$322$		0			0
$323$	−2.50761			−0.139527
$324$	−3.51679			−0.195377
$325$	−25.0324			−1.38855
$326$		−	25.0543i		−	1.38763i
$327$	8.52361			0.471356
$328$		−	43.8580i		−	2.42165i
$329$		0			0
$330$	−29.3639	−	5.85629i	−1.61643	−	0.322378i
$331$	−10.3967			−0.571454			−0.285727	−	0.958311i	$-0.592235\pi$
$331$	−10.3967			−0.571454			−0.285727	+	0.958311i	$0.592235\pi$
$332$	−16.5139			−0.906319
$333$	9.95670			0.545624
$334$		−	37.2617i		−	2.03887i
$335$		−	15.6998i		−	0.857769i
$336$		0			0
$337$			12.7142i			0.692584i	0.938127	+	0.346292i	$0.112559\pi$
$337$			12.7142i			0.692584i	−0.938127	+	0.346292i	$0.887441\pi$
$338$			15.1266i			0.822777i
$339$			13.6546i			0.741616i
$340$		−	5.89718i		−	0.319819i
$341$	17.5570	+	3.50155i	0.950768	+	0.189620i
$342$		−	13.5005i		−	0.730023i
$343$		0			0
$344$	28.7030			1.54756
$345$	−21.9694			−1.18279
$346$			33.9323i			1.82421i
$347$		−	26.7361i		−	1.43527i	−0.696420	−	0.717635i	$-0.745225\pi$
$347$		−	26.7361i		−	1.43527i	0.696420	−	0.717635i	$-0.254775\pi$
$348$	−4.31688			−0.231409
$349$	12.4372			0.665748			0.332874	−	0.942971i	$-0.391982\pi$
$349$	12.4372			0.665748			0.332874	+	0.942971i	$0.391982\pi$
$350$		0			0
$351$			2.56122i			0.136708i
$352$	12.9823	+	2.58918i	0.691960	+	0.138004i
$353$			18.3442i			0.976363i	0.872742	+	0.488181i	$0.162340\pi$
$353$			18.3442i			0.976363i	−0.872742	+	0.488181i	$0.837660\pi$
$354$			9.11544i			0.484480i
$355$		−	19.4270i		−	1.03108i
$356$			10.1468i			0.537780i
$357$		0			0
$358$			6.74806i			0.356646i
$359$		−	15.0354i		−	0.793537i	−0.917919	−	0.396768i	$-0.870132\pi$
$359$		−	15.0354i		−	0.793537i	0.917919	−	0.396768i	$-0.129868\pi$
$360$	13.6934			0.721707
$361$	14.0380			0.738840
$362$	11.6586			0.612760
$363$	−4.21980	+	10.1584i	−0.221482	+	0.533178i
$364$		0			0
$365$			13.5305i			0.708219i
$366$	−10.5097			−0.549351
$367$		−	19.6825i		−	1.02742i	−0.857964	−	0.513709i	$-0.828271\pi$
$367$		−	19.6825i		−	1.02742i	0.857964	−	0.513709i	$-0.171729\pi$
$368$	−7.62613			−0.397540
$369$	12.3106			0.640866
$370$	−89.8881			−4.67306
$371$		0			0
$372$	−18.9833			−0.984238
$373$		−	32.3966i		−	1.67743i	−0.544570	−	0.838715i	$-0.683307\pi$
$373$		−	32.3966i		−	1.67743i	0.544570	−	0.838715i	$-0.316693\pi$
$374$	−3.33291	−	0.664711i	−0.172341	−	0.0343714i
$375$	−18.3482			−0.947498
$376$	19.3780			0.999345
$377$			3.14391i			0.161919i
$378$		0			0
$379$	9.99877			0.513602			0.256801	−	0.966464i	$-0.417331\pi$
$379$	9.99877			0.513602			0.256801	+	0.966464i	$0.417331\pi$
$380$			77.6957i			3.98570i
$381$	17.6262			0.903016
$382$		−	21.3549i		−	1.09261i
$383$		−	25.0805i		−	1.28155i	−0.767727	−	0.640777i	$-0.778612\pi$
$383$		−	25.0805i		−	1.28155i	0.767727	−	0.640777i	$-0.221388\pi$
$384$	−20.3046			−1.03616
$385$		0			0
$386$	−22.2848			−1.13427
$387$			8.05674i			0.409547i
$388$			39.5759i			2.00916i
$389$	−27.5757			−1.39814			−0.699070	−	0.715053i	$-0.746403\pi$
$389$	−27.5757			−1.39814			−0.699070	+	0.715053i	$0.746403\pi$
$390$		−	23.1224i		−	1.17085i
$391$	−2.49361			−0.126107
$392$		0			0
$393$			1.57624i			0.0795107i
$394$	−19.4764			−0.981209
$395$	−63.9475			−3.21755
$396$	2.28130	−	11.4386i	0.114640	−	0.574811i
$397$			0.744268i			0.0373537i	0.999826	+	0.0186769i	$0.00594538\pi$
$397$			0.744268i			0.0373537i	−0.999826	+	0.0186769i	$0.994055\pi$
$398$	13.7291			0.688178
$399$		0			0
$400$	−13.0403			−0.652014
$401$	16.1488			0.806433			0.403217	−	0.915105i	$-0.367892\pi$
$401$	16.1488			0.806433			0.403217	+	0.915105i	$0.367892\pi$
$402$	9.59383			0.478497
$403$			13.8252i			0.688682i
$404$	60.9176			3.03077
$405$			3.84365i			0.190993i
$406$		0			0
$407$	−6.45879	+	32.3848i	−0.320150	+	1.60526i
$408$	1.55426			0.0769472
$409$	−3.92547			−0.194102			−0.0970510	−	0.995279i	$-0.530941\pi$
$409$	−3.92547			−0.194102			−0.0970510	+	0.995279i	$0.530941\pi$
$410$	−111.139			−5.48878
$411$		−	0.722567i		−	0.0356416i
$412$		−	63.3647i		−	3.12176i
$413$		0			0
$414$		−	13.4251i		−	0.659807i
$415$			18.0488i			0.885979i
$416$			10.2229i			0.501217i
$417$		−	2.37513i		−	0.116311i
$418$	43.9113	+	8.75761i	2.14777	+	0.428349i
$419$		−	18.8112i		−	0.918989i	−0.888181	−	0.459494i	$-0.848031\pi$
$419$		−	18.8112i		−	0.918989i	0.888181	−	0.459494i	$-0.151969\pi$
$420$		0			0
$421$	11.0047			0.536337			0.268169	−	0.963372i	$-0.413582\pi$
$421$	11.0047			0.536337			0.268169	+	0.963372i	$0.413582\pi$
$422$	−45.4075			−2.21040
$423$			5.43927i			0.264467i
$424$			23.1992i			1.12665i
$425$	−4.26394			−0.206831
$426$	11.8715			0.575175
$427$		0			0
$428$			33.9531i			1.64119i
$429$	−8.33053	−	1.66143i	−0.402202	−	0.0802146i
$430$		−	72.7355i		−	3.50762i
$431$		−	13.3953i		−	0.645228i	−0.946531	−	0.322614i	$-0.895438\pi$
$431$		−	13.3953i		−	0.645228i	0.946531	−	0.322614i	$-0.104562\pi$
$432$			1.33423i			0.0641932i
$433$			2.06098i			0.0990446i	0.998773	+	0.0495223i	$0.0157699\pi$
$433$			2.06098i			0.0990446i	−0.998773	+	0.0495223i	$0.984230\pi$
$434$		0			0
$435$			4.71810i			0.226216i
$436$		−	29.9757i		−	1.43558i
$437$	32.8534			1.57159
$438$	−8.26824			−0.395072
$439$	14.7525			0.704098			0.352049	−	0.935982i	$-0.385485\pi$
$439$	14.7525			0.704098			0.352049	+	0.935982i	$0.385485\pi$
$440$	−8.88276	+	44.5388i	−0.423469	+	2.12331i
$441$		0			0
$442$		−	2.62448i		−	0.124834i
$443$	8.66961			0.411905			0.205953	−	0.978562i	$-0.433971\pi$
$443$	8.66961			0.411905			0.205953	+	0.978562i	$0.433971\pi$
$444$		−	35.0156i		−	1.66177i
$445$	11.0899			0.525712
$446$	24.3359			1.15234
$447$	−2.66454			−0.126028
$448$		0			0
$449$	19.7486			0.931994			0.465997	−	0.884786i	$-0.345696\pi$
$449$	19.7486			0.931994			0.465997	+	0.884786i	$0.345696\pi$
$450$		−	22.9562i		−	1.08217i
$451$	−7.98576	+	40.0412i	−0.376035	+	1.88547i
$452$	48.0204			2.25869
$453$	−9.57465			−0.449856
$454$		−	23.9412i		−	1.12362i
$455$		0			0
$456$	−20.4774			−0.958943
$457$			22.1378i			1.03556i	0.855513	+	0.517782i	$0.173242\pi$
$457$			22.1378i			1.03556i	−0.855513	+	0.517782i	$0.826758\pi$
$458$	27.6866			1.29371
$459$			0.436269i			0.0203633i
$460$			77.2617i			3.60234i
$461$	−9.81168			−0.456976			−0.228488	−	0.973547i	$-0.573378\pi$
$461$	−9.81168			−0.456976			−0.228488	+	0.973547i	$0.573378\pi$
$462$		0			0
$463$	−10.4489			−0.485603			−0.242802	−	0.970076i	$-0.578066\pi$
$463$	−10.4489			−0.485603			−0.242802	+	0.970076i	$0.578066\pi$
$464$			1.63777i			0.0760317i
$465$			20.7476i			0.962149i
$466$	−53.0671			−2.45829
$467$			31.8510i			1.47389i	0.675953	+	0.736944i	$0.263732\pi$
$467$			31.8510i			1.47389i	−0.675953	+	0.736944i	$0.736268\pi$
$468$	9.00726			0.416361
$469$		0			0
$470$		−	49.1053i		−	2.26506i
$471$	17.2160			0.793273
$472$	13.8262			0.636403
$473$	−26.2051	−	5.22631i	−1.20491	−	0.240306i
$474$		−	39.0772i		−	1.79487i
$475$	56.1776			2.57761
$476$		0			0
$477$	−6.51185			−0.298157
$478$	−21.4963			−0.983216
$479$	9.53225			0.435540			0.217770	−	0.976000i	$-0.430122\pi$
$479$	9.53225			0.435540			0.217770	+	0.976000i	$0.430122\pi$
$480$			15.3416i			0.700244i
$481$	−25.5013			−1.16276
$482$			55.0432i			2.50715i
$483$		0			0
$484$	35.7250	+	14.8402i	1.62386	+	0.674553i
$485$	43.2542			1.96407
$486$	−2.34878			−0.106543
$487$	2.44181			0.110649			0.0553244	−	0.998468i	$-0.482381\pi$
$487$	2.44181			0.110649			0.0553244	+	0.998468i	$0.482381\pi$
$488$			15.9410i			0.721615i
$489$		−	10.6669i		−	0.482374i
$490$		0			0
$491$			7.87199i			0.355258i	0.984098	+	0.177629i	$0.0568427\pi$
$491$			7.87199i			0.355258i	−0.984098	+	0.177629i	$0.943157\pi$
$492$		−	43.2939i		−	1.95184i
$493$			0.535522i			0.0241187i
$494$			34.5777i			1.55572i
$495$	−12.5017	−	2.49333i	−0.561911	−	0.112067i
$496$			7.20204i			0.323381i
$497$		0			0
$498$	−11.0293			−0.494233
$499$	24.0027			1.07451			0.537254	−	0.843421i	$-0.319462\pi$
$499$	24.0027			1.07451			0.537254	+	0.843421i	$0.319462\pi$
$500$			64.5268i			2.88573i
$501$		−	15.8643i		−	0.708763i
$502$	−66.1022			−2.95029
$503$	−1.30790			−0.0583163			−0.0291582	−	0.999575i	$-0.509283\pi$
$503$	−1.30790			−0.0583163			−0.0291582	+	0.999575i	$0.509283\pi$
$504$		0			0
$505$		−	66.5795i		−	2.96275i
$506$	43.6660	+	8.70869i	1.94119	+	0.387148i
$507$			6.44017i			0.286018i
$508$		−	61.9875i		−	2.75025i
$509$			8.29130i			0.367505i	0.982973	+	0.183753i	$0.0588246\pi$
$509$			8.29130i			0.367505i	−0.982973	+	0.183753i	$0.941175\pi$
$510$		−	3.93859i		−	0.174404i
$511$		0			0
$512$			14.8321i			0.655494i
$513$		−	5.74787i		−	0.253774i
$514$	46.5536			2.05339
$515$	−69.2540			−3.05170
$516$	28.3338			1.24733
$517$	−17.6916	−	3.52839i	−0.778076	−	0.155178i
$518$		0			0
$519$			14.4467i			0.634141i
$520$	−35.0718			−1.53800
$521$		−	23.7610i		−	1.04099i	−0.853866	−	0.520493i	$-0.825748\pi$
$521$		−	23.7610i		−	1.04099i	0.853866	−	0.520493i	$-0.174252\pi$
$522$	−2.88315			−0.126192
$523$	26.5352			1.16030			0.580151	−	0.814509i	$-0.302994\pi$
$523$	26.5352			1.16030			0.580151	+	0.814509i	$0.302994\pi$
$524$	5.54330			0.242160
$525$		0			0
$526$	−27.1269			−1.18279
$527$			2.35494i			0.102583i
$528$	−4.33967	−	0.865498i	−0.188860	−	0.0376660i
$529$	9.66990			0.420430
$530$	58.7884			2.55360
$531$			3.88092i			0.168418i
$532$		0			0
$533$	−31.5302			−1.36572
$534$			6.77683i			0.293262i
$535$	37.1088			1.60435
$536$		−	14.5518i		−	0.628543i
$537$			2.87300i			0.123979i
$538$	49.6900			2.14229
$539$		0			0
$540$	13.5173			0.581692
$541$		−	15.2823i		−	0.657038i	−0.944497	−	0.328519i	$-0.893450\pi$
$541$		−	15.2823i		−	0.657038i	0.944497	−	0.328519i	$-0.106550\pi$
$542$		−	63.2545i		−	2.71701i
$543$	4.96366			0.213011
$544$			1.74133i			0.0746587i
$545$	−32.7618			−1.40336
$546$		0			0
$547$			16.8133i			0.718884i	0.933167	+	0.359442i	$0.117033\pi$
$547$			16.8133i			0.718884i	−0.933167	+	0.359442i	$0.882967\pi$
$548$	−2.54112			−0.108551
$549$	−4.47453			−0.190968
$550$	74.6666	+	14.8914i	3.18379	+	0.634971i
$551$		−	7.05554i		−	0.300576i
$552$	−20.3630			−0.866708
$553$		0			0
$554$	38.0869			1.61816
$555$	−38.2701			−1.62447
$556$	−8.35285			−0.354240
$557$			38.4503i			1.62919i	0.580030	+	0.814595i	$0.303041\pi$
$557$			38.4503i			1.62919i	−0.580030	+	0.814595i	$0.696959\pi$
$558$	−12.6785			−0.536724
$559$		−	20.6350i		−	0.872769i
$560$		0			0
$561$	−1.41899	−	0.283002i	−0.0599100	−	0.0119484i
$562$	−54.5600			−2.30147
$563$	−36.9144			−1.55576			−0.777879	−	0.628414i	$-0.783704\pi$
$563$	−36.9144			−1.55576			−0.777879	+	0.628414i	$0.783704\pi$
$564$	19.1288			0.805467
$565$		−	52.4835i		−	2.20800i
$566$		−	2.73202i		−	0.114835i
$567$		0			0
$568$		−	18.0065i		−	0.755537i
$569$		−	36.3839i		−	1.52529i	−0.646815	−	0.762647i	$-0.723899\pi$
$569$		−	36.3839i		−	1.52529i	0.646815	−	0.762647i	$-0.276101\pi$
$570$			51.8912i			2.17348i
$571$		−	9.68732i		−	0.405402i	−0.979241	−	0.202701i	$-0.935028\pi$
$571$		−	9.68732i		−	0.405402i	0.979241	−	0.202701i	$-0.0649719\pi$
$572$	−5.84290	+	29.2967i	−0.244304	+	1.22496i
$573$		−	9.09190i		−	0.379820i
$574$		0			0
$575$	55.8638			2.32968
$576$	−12.0434			−0.501809
$577$		−	4.98652i		−	0.207592i	−0.994599	−	0.103796i	$-0.966901\pi$
$577$		−	4.98652i		−	0.207592i	0.994599	−	0.103796i	$-0.0330988\pi$
$578$			39.4823i			1.64225i
$579$	−9.48779			−0.394299
$580$	16.5926			0.688969
$581$		0			0
$582$			26.4318i			1.09564i
$583$	4.22416	−	21.1802i	0.174947	−	0.877196i
$584$			12.5412i			0.518958i
$585$		−	9.84442i		−	0.407017i
$586$			15.8243i			0.653696i
$587$			1.52709i			0.0630297i	0.999503	+	0.0315148i	$0.0100331\pi$
$587$			1.52709i			0.0630297i	−0.999503	+	0.0315148i	$0.989967\pi$
$588$		0			0
$589$		−	31.0264i		−	1.27842i
$590$		−	35.0366i		−	1.44243i
$591$	−8.29213			−0.341093
$592$	−13.2845			−0.545990
$593$	−24.6736			−1.01322			−0.506612	−	0.862174i	$-0.669102\pi$
$593$	−24.6736			−1.01322			−0.506612	+	0.862174i	$0.669102\pi$
$594$	1.52363	−	7.63958i	0.0625152	−	0.313456i
$595$		0			0
$596$			9.37061i			0.383835i
$597$	5.84519			0.239228
$598$			34.3846i			1.40609i
$599$	−14.3324			−0.585607			−0.292803	−	0.956173i	$-0.594588\pi$
$599$	−14.3324			−0.585607			−0.292803	+	0.956173i	$0.594588\pi$
$600$	−34.8197			−1.42151
$601$	−14.3873			−0.586870			−0.293435	−	0.955979i	$-0.594798\pi$
$601$	−14.3873			−0.586870			−0.293435	+	0.955979i	$0.594798\pi$
$602$		0			0
$603$	4.08459			0.166338
$604$			33.6720i			1.37009i
$605$	16.2194	−	39.0454i	0.659414	−	1.58742i
$606$	40.6855			1.65274
$607$	11.7773			0.478025			0.239013	−	0.971016i	$-0.423176\pi$
$607$	11.7773			0.478025			0.239013	+	0.971016i	$0.423176\pi$
$608$		−	22.9421i		−	0.930424i
$609$		0			0
$610$	40.3956			1.63557
$611$		−	13.9312i		−	0.563594i
$612$	1.53427			0.0620190
$613$		−	29.0239i		−	1.17226i	−0.810216	−	0.586131i	$-0.800650\pi$
$613$		−	29.0239i		−	1.17226i	0.810216	−	0.586131i	$-0.199350\pi$
$614$		−	28.9171i		−	1.16700i
$615$	−47.3178			−1.90804
$616$		0			0
$617$	14.4008			0.579755			0.289878	−	0.957064i	$-0.406385\pi$
$617$	14.4008			0.579755			0.289878	+	0.957064i	$0.406385\pi$
$618$		−	42.3199i		−	1.70236i
$619$			11.9495i			0.480290i	0.970737	+	0.240145i	$0.0771950\pi$
$619$			11.9495i			0.480290i	−0.970737	+	0.240145i	$0.922805\pi$
$620$	72.9651			2.93035
$621$		−	5.71576i		−	0.229366i
$622$	15.8062			0.633770
$623$		0			0
$624$		−	3.41725i		−	0.136799i
$625$	21.6559			0.866237
$626$	−20.8700			−0.834132
$627$	18.6953	+	3.72857i	0.746620	+	0.148905i
$628$		−	60.5452i		−	2.41602i
$629$	−4.34380			−0.173199
$630$		0			0
$631$	−24.9661			−0.993886			−0.496943	−	0.867783i	$-0.665544\pi$
$631$	−24.9661			−0.993886			−0.496943	+	0.867783i	$0.665544\pi$
$632$	−59.2718			−2.35771
$633$	−19.3323			−0.768391
$634$			23.8376i			0.946712i
$635$	−67.7488			−2.68853
$636$			22.9008i			0.908076i
$637$		0			0
$638$	1.87026	−	9.37763i	0.0740443	−	0.371264i
$639$	5.05430			0.199945
$640$	78.0437			3.08495
$641$	23.1524			0.914463			0.457232	−	0.889348i	$-0.348841\pi$
$641$	23.1524			0.914463			0.457232	+	0.889348i	$0.348841\pi$
$642$			22.6765i			0.894971i
$643$			14.2436i			0.561713i	0.959750	+	0.280857i	$0.0906186\pi$
$643$			14.2436i			0.561713i	−0.959750	+	0.280857i	$0.909381\pi$
$644$		0			0
$645$		−	30.9673i		−	1.21934i
$646$			5.88985i			0.231733i
$647$		−	18.4493i		−	0.725316i	−0.931922	−	0.362658i	$-0.881869\pi$
$647$		−	18.4493i		−	0.725316i	0.931922	−	0.362658i	$-0.118131\pi$
$648$			3.56261i			0.139953i
$649$	−12.6230	−	2.51750i	−0.495494	−	0.0988207i
$650$			58.7958i			2.30616i
$651$		0			0
$652$	−37.5133			−1.46913
$653$	33.3924			1.30674			0.653372	−	0.757037i	$-0.273354\pi$
$653$	33.3924			1.30674			0.653372	+	0.757037i	$0.273354\pi$
$654$		−	20.0201i		−	0.782848i
$655$		−	6.05851i		−	0.236725i
$656$	−16.4252			−0.641297
$657$	−3.52022			−0.137337
$658$		0			0
$659$			32.3153i			1.25882i	0.777072	+	0.629412i	$0.216704\pi$
$659$			32.3153i			1.25882i	−0.777072	+	0.629412i	$0.783296\pi$
$660$	−8.76851	+	43.9660i	−0.341314	+	1.71137i
$661$		−	16.7523i		−	0.651588i	−0.945441	−	0.325794i	$-0.894368\pi$
$661$		−	16.7523i		−	0.651588i	0.945441	−	0.325794i	$-0.105632\pi$
$662$			24.4196i			0.949094i
$663$		−	1.11738i		−	0.0433954i
$664$			16.7291i			0.649214i
$665$		0			0
$666$		−	23.3861i		−	0.906194i
$667$		−	7.01612i		−	0.271665i
$668$	−55.7913			−2.15863
$669$	10.3611			0.400582
$670$	−36.8753			−1.42462
$671$	2.90257	−	14.5537i	0.112052	−	0.561840i
$672$		0			0
$673$			27.6753i			1.06680i	0.845862	+	0.533402i	$0.179087\pi$
$673$			27.6753i			1.06680i	−0.845862	+	0.533402i	$0.820913\pi$
$674$	29.8628			1.15027
$675$		−	9.77365i		−	0.376188i
$676$	22.6487			0.871105
$677$	−17.1160			−0.657821			−0.328911	−	0.944361i	$-0.606682\pi$
$677$	−17.1160			−0.657821			−0.328911	+	0.944361i	$0.606682\pi$
$678$	32.0717			1.23171
$679$		0			0
$680$	−5.97402			−0.229093
$681$		−	10.1930i		−	0.390597i
$682$	8.22439	−	41.2377i	0.314928	−	1.57907i
$683$	41.9563			1.60541			0.802706	−	0.596375i	$-0.203393\pi$
$683$	41.9563			1.60541			0.802706	+	0.596375i	$0.203393\pi$
$684$	−20.2140			−0.772903
$685$			2.77730i			0.106115i
$686$		0			0
$687$	11.7876			0.449727
$688$		−	10.7495i		−	0.409822i
$689$	16.6783			0.635391
$690$			51.6013i			1.96443i
$691$		−	4.57118i		−	0.173896i	−0.996213	−	0.0869480i	$-0.972289\pi$
$691$		−	4.57118i		−	0.173896i	0.996213	−	0.0869480i	$-0.0277114\pi$
$692$	50.8061			1.93136
$693$		0			0
$694$	−62.7974			−2.38376
$695$			9.12919i			0.346290i
$696$			4.37312i			0.165763i
$697$	−5.37075			−0.203432
$698$		−	29.2123i		−	1.10570i
$699$	−22.5934			−0.854563
$700$		0			0
$701$		−	8.58375i		−	0.324204i	−0.986774	−	0.162102i	$-0.948173\pi$
$701$		−	8.58375i		−	0.324204i	0.986774	−	0.162102i	$-0.0518273\pi$
$702$	6.01575			0.227050
$703$	57.2298			2.15846
$704$	7.81241	−	39.1720i	0.294441	−	1.47635i
$705$		−	20.9067i		−	0.787391i
$706$	43.0866			1.62158
$707$		0			0
$708$	13.6484			0.512937
$709$	22.0275			0.827261			0.413631	−	0.910445i	$-0.364261\pi$
$709$	22.0275			0.827261			0.413631	+	0.910445i	$0.364261\pi$
$710$	−45.6298			−1.71245
$711$		−	16.6372i		−	0.623943i
$712$	10.2790			0.385223
$713$		−	30.8531i		−	1.15546i
$714$		0			0
$715$	32.0197	+	6.38595i	1.19747	+	0.238821i
$716$	10.1037			0.377594
$717$	−9.15208			−0.341791
$718$	−35.3148			−1.31794
$719$			15.2177i			0.567525i	0.958895	+	0.283762i	$0.0915826\pi$
$719$			15.2177i			0.567525i	−0.958895	+	0.283762i	$0.908417\pi$
$720$		−	5.12831i		−	0.191121i
$721$		0			0
$722$		−	32.9721i		−	1.22710i
$723$			23.4348i			0.871548i
$724$		−	17.4561i		−	0.648752i
$725$		−	11.9972i		−	0.445565i
$726$	23.8599	+	9.91141i	0.885525	+	0.367847i
$727$			4.87985i			0.180983i	0.995897	+	0.0904917i	$0.0288439\pi$
$727$			4.87985i			0.180983i	−0.995897	+	0.0904917i	$0.971156\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	31.7802			1.17624
$731$		−	3.51490i		−	0.130003i
$732$			15.7360i			0.581618i
$733$	−7.00427			−0.258709			−0.129354	−	0.991598i	$-0.541290\pi$
$733$	−7.00427			−0.258709			−0.129354	+	0.991598i	$0.541290\pi$
$734$	−46.2300			−1.70638
$735$		0			0
$736$		−	22.8139i		−	0.840932i
$737$	−2.64963	+	13.2854i	−0.0976002	+	0.489375i
$738$		−	28.9150i		−	1.06438i
$739$			35.4668i			1.30467i	0.757932	+	0.652334i	$0.226210\pi$
$739$			35.4668i			1.30467i	−0.757932	+	0.652334i	$0.773790\pi$
$740$			134.588i			4.94754i
$741$			14.7215i			0.540809i
$742$		0			0
$743$			7.21238i			0.264596i	0.991210	+	0.132298i	$0.0422357\pi$
$743$			7.21238i			0.264596i	−0.991210	+	0.132298i	$0.957764\pi$
$744$			19.2306i			0.705029i
$745$	10.2415			0.375221
$746$	−76.0926			−2.78595
$747$	−4.69573			−0.171808
$748$	−0.995259	+	4.99030i	−0.0363903	+	0.182464i
$749$		0			0
$750$			43.0960i			1.57364i
$751$	−12.7527			−0.465352			−0.232676	−	0.972554i	$-0.574748\pi$
$751$	−12.7527			−0.465352			−0.232676	+	0.972554i	$0.574748\pi$
$752$		−	7.25724i		−	0.264644i
$753$	−28.1432			−1.02559
$754$	7.38436			0.268922
$755$	36.8016			1.33935
$756$		0			0
$757$	−40.7104			−1.47965			−0.739823	−	0.672802i	$-0.765091\pi$
$757$	−40.7104			−1.47965			−0.739823	+	0.672802i	$0.765091\pi$
$758$		−	23.4850i		−	0.853012i
$759$	18.5909	+	3.70774i	0.674807	+	0.134582i
$760$	78.7080			2.85504
$761$	4.36635			0.158280			0.0791401	−	0.996864i	$-0.474783\pi$
$761$	4.36635			0.158280			0.0791401	+	0.996864i	$0.474783\pi$
$762$		−	41.4001i		−	1.49977i
$763$		0			0
$764$	−31.9743			−1.15679
$765$		−	1.67686i		−	0.0606272i
$766$	−58.9087			−2.12846
$767$		−	9.93987i		−	0.358908i
$768$			23.6042i			0.851745i
$769$	29.9317			1.07937			0.539683	−	0.841868i	$-0.318544\pi$
$769$	29.9317			1.07937			0.539683	+	0.841868i	$0.318544\pi$
$770$		0			0
$771$	19.8203			0.713810
$772$			33.3666i			1.20089i
$773$			19.7675i			0.710989i	0.934678	+	0.355495i	$0.115688\pi$
$773$			19.7675i			0.710989i	−0.934678	+	0.355495i	$0.884312\pi$
$774$	18.9235			0.680193
$775$		−	52.7572i		−	1.89509i
$776$	40.0915			1.43920
$777$		0			0
$778$			64.7693i			2.32209i
$779$	70.7599			2.53524
$780$	−34.6207			−1.23962
$781$	−3.27866	+	16.4395i	−0.117320	+	0.588250i
$782$			5.85695i			0.209444i
$783$	−1.22751			−0.0438675
$784$		0			0
$785$	−66.1725			−2.36180
$786$	3.70224			0.132055
$787$	−51.5141			−1.83628			−0.918140	−	0.396256i	$-0.870309\pi$
$787$	−51.5141			−1.83628			−0.918140	+	0.396256i	$0.870309\pi$
$788$			29.1617i			1.03884i
$789$	−11.5493			−0.411167
$790$			150.199i			5.34384i
$791$		0			0
$792$	−11.5876	−	2.31102i	−0.411749	−	0.0821186i
$793$	11.4602			0.406965
$794$	1.74812			0.0620386
$795$	25.0293			0.887697
$796$		−	20.5563i		−	0.728599i
$797$			6.04239i			0.214033i	0.994257	+	0.107016i	$0.0341297\pi$
$797$			6.04239i			0.214033i	−0.994257	+	0.107016i	$0.965870\pi$
$798$		0			0
$799$		−	2.37299i		−	0.0839502i
$800$		−	39.0106i		−	1.37923i
$801$			2.88525i			0.101945i
$802$		−	37.9301i		−	1.33936i
$803$	2.28352	−	11.4498i	0.0805838	−	0.404053i
$804$		−	14.3647i		−	0.506603i
$805$		0			0
$806$	32.4724			1.14379
$807$	21.1556			0.744713
$808$		−	61.7114i		−	2.17100i
$809$			9.07112i			0.318924i	0.987204	+	0.159462i	$0.0509759\pi$
$809$			9.07112i			0.318924i	−0.987204	+	0.159462i	$0.949024\pi$
$810$	9.02791			0.317208
$811$	−15.6199			−0.548488			−0.274244	−	0.961660i	$-0.588428\pi$
$811$	−15.6199			−0.548488			−0.274244	+	0.961660i	$0.588428\pi$
$812$		0			0
$813$		−	26.9307i		−	0.944502i
$814$	76.0650	+	15.1703i	2.66608	+	0.531719i
$815$			40.9999i			1.43616i
$816$		−	0.582083i		−	0.0203770i
$817$			46.3090i			1.62015i
$818$			9.22009i			0.322373i
$819$		0			0
$820$			166.407i			5.81117i
$821$			56.5452i			1.97344i	0.162431	+	0.986720i	$0.448067\pi$
$821$			56.5452i			1.97344i	−0.162431	+	0.986720i	$0.551933\pi$
$822$	−1.69715			−0.0591951
$823$	−47.6964			−1.66259			−0.831297	−	0.555829i	$-0.812401\pi$
$823$	−47.6964			−1.66259			−0.831297	+	0.555829i	$0.812401\pi$
$824$	−64.1904			−2.23618
$825$	31.7895	+	6.34004i	1.10677	+	0.220732i
$826$		0			0
$827$			36.5468i			1.27086i	0.772160	+	0.635428i	$0.219176\pi$
$827$			36.5468i			1.27086i	−0.772160	+	0.635428i	$0.780824\pi$
$828$	−20.1011			−0.698562
$829$			44.4395i			1.54345i	0.635958	+	0.771723i	$0.280605\pi$
$829$			44.4395i			1.54345i	−0.635958	+	0.771723i	$0.719395\pi$
$830$	42.3927			1.47147
$831$	16.2156			0.562512
$832$	30.8458			1.06939
$833$		0			0
$834$	−5.57868			−0.193174
$835$			60.9767i			2.11019i
$836$	13.1126	−	65.7475i	0.453508	−	2.27393i
$837$	−5.39790			−0.186579
$838$	−44.1835			−1.52629
$839$			53.0313i			1.83084i	0.402495	+	0.915422i	$0.368143\pi$
$839$			53.0313i			1.83084i	−0.402495	+	0.915422i	$0.631857\pi$
$840$		0			0
$841$	27.4932			0.948042
$842$		−	25.8477i		−	0.890772i
$843$	−23.2290			−0.800050
$844$			67.9877i			2.34023i
$845$		−	24.7538i		−	0.851555i
$846$	12.7757			0.439237
$847$		0			0
$848$	8.68830			0.298358
$849$		−	1.16316i		−	0.0399197i
$850$			10.0151i			0.343514i
$851$	56.9101			1.95085
$852$		−	17.7749i		−	0.608959i
$853$	−28.8055			−0.986282			−0.493141	−	0.869949i	$-0.664151\pi$
$853$	−28.8055			−0.986282			−0.493141	+	0.869949i	$0.664151\pi$
$854$		0			0
$855$			22.0928i			0.755557i
$856$	34.3955			1.17561
$857$	−1.93896			−0.0662336			−0.0331168	−	0.999451i	$-0.510543\pi$
$857$	−1.93896			−0.0662336			−0.0331168	+	0.999451i	$0.510543\pi$
$858$	−3.90234	+	19.5666i	−0.133224	+	0.667994i
$859$		−	22.5764i		−	0.770297i	−0.922855	−	0.385148i	$-0.874150\pi$
$859$		−	22.5764i		−	0.770297i	0.922855	−	0.385148i	$-0.125850\pi$
$860$	−108.905			−3.71364
$861$		0			0
$862$	−31.4626			−1.07162
$863$	−10.3518			−0.352380			−0.176190	−	0.984356i	$-0.556377\pi$
$863$	−10.3518			−0.352380			−0.176190	+	0.984356i	$0.556377\pi$
$864$	−3.99141			−0.135790
$865$		−	55.5282i		−	1.88802i
$866$	4.84081			0.164497
$867$			16.8097i			0.570886i
$868$		0			0
$869$	54.1136	+	10.7923i	1.83568	+	0.366105i
$870$	11.0818			0.375708
$871$	−10.4615			−0.354475
$872$	−30.3663			−1.02833
$873$			11.2534i			0.380870i
$874$		−	77.1656i		−	2.61017i
$875$		0			0
$876$			12.3799i			0.418277i
$877$		−	29.5021i		−	0.996215i	−0.867115	−	0.498108i	$-0.834028\pi$
$877$		−	29.5021i		−	0.996215i	0.867115	−	0.498108i	$-0.165972\pi$
$878$		−	34.6504i		−	1.16939i
$879$			6.73723i			0.227241i
$880$	16.6802	+	3.32667i	0.562289	+	0.112142i
$881$		−	32.3910i		−	1.09128i	−0.838019	−	0.545640i	$-0.816286\pi$
$881$		−	32.3910i		−	1.09128i	0.838019	−	0.545640i	$-0.183714\pi$
$882$		0			0
$883$	35.2639			1.18672			0.593362	−	0.804936i	$-0.297800\pi$
$883$	35.2639			1.18672			0.593362	+	0.804936i	$0.297800\pi$
$884$	−3.92959			−0.132166
$885$		−	14.9169i		−	0.501426i
$886$		−	20.3630i		−	0.684110i
$887$	−6.25834			−0.210135			−0.105067	−	0.994465i	$-0.533506\pi$
$887$	−6.25834			−0.210135			−0.105067	+	0.994465i	$0.533506\pi$
$888$	−35.4719			−1.19036
$889$		0			0
$890$		−	26.0478i		−	0.873124i
$891$	0.648688	−	3.25257i	0.0217319	−	0.108965i
$892$		−	36.4377i		−	1.22002i
$893$			31.2642i			1.04622i
$894$			6.25842i			0.209313i
$895$		−	11.0428i		−	0.369120i
$896$		0			0
$897$			14.6393i			0.488792i
$898$		−	46.3852i		−	1.54789i
$899$	−6.62595			−0.220988
$900$	−34.3719			−1.14573
$901$	2.84092			0.0946447
$902$	94.0482	+	18.7568i	3.13146	+	0.624534i
$903$		0			0
$904$		−	48.6461i		−	1.61794i
$905$	−19.0786			−0.634193
$906$			22.4888i			0.747140i
$907$	−33.6871			−1.11856			−0.559281	−	0.828978i	$-0.688923\pi$
$907$	−33.6871			−1.11856			−0.559281	+	0.828978i	$0.688923\pi$
$908$	−35.8467			−1.18961
$909$	17.3219			0.574533
$910$		0			0
$911$	−26.1080			−0.864997			−0.432499	−	0.901635i	$-0.642368\pi$
$911$	−26.1080			−0.864997			−0.432499	+	0.901635i	$0.642368\pi$
$912$			7.66897i			0.253945i
$913$	3.04606	−	15.2732i	0.100810	−	0.505469i
$914$	51.9970			1.71991
$915$	17.1985			0.568565
$916$		−	41.4547i		−	1.36970i
$917$		0			0
$918$	1.02470			0.0338202
$919$			6.38229i			0.210532i	0.994444	+	0.105266i	$0.0335695\pi$
$919$			6.38229i			0.210532i	−0.994444	+	0.105266i	$0.966431\pi$
$920$	78.2684			2.58043
$921$		−	12.3115i		−	0.405678i
$922$			23.0455i			0.758964i
$923$	−12.9452			−0.426095
$924$		0			0
$925$	97.3132			3.19964
$926$			24.5423i			0.806510i
$927$		−	18.0178i		−	0.591782i
$928$	−4.89947			−0.160833
$929$			9.48788i			0.311287i	0.987813	+	0.155644i	$0.0497451\pi$
$929$			9.48788i			0.311287i	−0.987813	+	0.155644i	$0.950255\pi$
$930$	48.7318			1.59798
$931$		0			0
$932$			79.4564i			2.60268i
$933$	6.72951			0.220314
$934$	74.8112			2.44790
$935$	5.45412	+	1.08776i	0.178369	+	0.0355736i
$936$		−	9.12462i		−	0.298248i
$937$	20.6699			0.675257			0.337629	−	0.941279i	$-0.390375\pi$
$937$	20.6699			0.675257			0.337629	+	0.941279i	$0.390375\pi$
$938$		0			0
$939$	−8.88544			−0.289965
$940$	−73.5243			−2.39810
$941$	29.5014			0.961719			0.480860	−	0.876798i	$-0.340325\pi$
$941$	29.5014			0.961719			0.480860	+	0.876798i	$0.340325\pi$
$942$		−	40.4368i		−	1.31750i
$943$	70.3646			2.29139
$944$		−	5.17804i		−	0.168531i
$945$		0			0
$946$	−12.2755	+	61.5501i	−0.399110	+	2.00117i
$947$	3.50144			0.113782			0.0568908	−	0.998380i	$-0.481881\pi$
$947$	3.50144			0.113782			0.0568908	+	0.998380i	$0.481881\pi$
$948$	−58.5095			−1.90030
$949$	9.01605			0.292673
$950$		−	131.949i		−	4.28099i
$951$			10.1489i			0.329101i
$952$		0			0
$953$		−	35.0364i		−	1.13494i	−0.823394	−	0.567471i	$-0.807922\pi$
$953$		−	35.0364i		−	1.13494i	0.823394	−	0.567471i	$-0.192078\pi$
$954$			15.2949i			0.495192i
$955$			34.9461i			1.13083i
$956$			32.1859i			1.04097i
$957$	0.796267	−	3.99255i	0.0257397	−	0.129061i
$958$		−	22.3892i		−	0.723362i
$959$		0			0
$960$	46.2907			1.49402
$961$	1.86265			0.0600855
$962$			59.8970i			1.93116i
$963$			9.65458i			0.311114i
$964$	82.4151			2.65441
$965$	36.4677			1.17394
$966$		0			0
$967$		−	29.3774i		−	0.944713i	−0.881407	−	0.472357i	$-0.843403\pi$
$967$		−	29.3774i		−	0.944713i	0.881407	−	0.472357i	$-0.156597\pi$
$968$	15.0335	−	36.1905i	0.483196	−	1.16321i
$969$			2.50761i			0.0805562i
$970$		−	101.595i		−	3.26201i
$971$			0.709513i			0.0227694i	0.999935	+	0.0113847i	$0.00362393\pi$
$971$			0.709513i			0.0227694i	−0.999935	+	0.0113847i	$0.996376\pi$
$972$			3.51679i			0.112801i
$973$		0			0
$974$		−	5.73528i		−	0.183770i
$975$			25.0324i			0.801679i
$976$	5.97005			0.191096
$977$	59.5119			1.90396			0.951978	−	0.306167i	$-0.0990466\pi$
$977$	59.5119			1.90396			0.951978	+	0.306167i	$0.0990466\pi$
$978$	−25.0543			−0.801147
$979$	−9.38448	−	1.87163i	−0.299929	−	0.0598174i
$980$		0			0
$981$		−	8.52361i		−	0.272138i
$982$	18.4896			0.590027
$983$			41.2626i			1.31607i	0.752987	+	0.658036i	$0.228612\pi$
$983$			41.2626i			1.31607i	−0.752987	+	0.658036i	$0.771388\pi$
$984$	−43.8580			−1.39814
$985$	31.8721			1.01553
$986$	1.25783			0.0400574
$987$		0			0
$988$	51.7725			1.64710
$989$			46.0504i			1.46432i
$990$	−5.85629	+	29.3639i	−0.186125	+	0.933246i
$991$	58.7002			1.86467			0.932336	−	0.361593i	$-0.117767\pi$
$991$	58.7002			1.86467			0.932336	+	0.361593i	$0.117767\pi$
$992$	−21.5452			−0.684061
$993$			10.3967i			0.329929i
$994$		0			0
$995$	−22.4669			−0.712248
$996$			16.5139i			0.523263i
$997$	−21.2611			−0.673345			−0.336673	−	0.941622i	$-0.609302\pi$
$997$	−21.2611			−0.673345			−0.336673	+	0.941622i	$0.609302\pi$
$998$		−	56.3771i		−	1.78459i
$999$		−	9.95670i		−	0.315016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.c.b.538.5	✓	48
7.6	odd	2	inner	1617.2.c.b.538.6	yes	48
11.10	odd	2	inner	1617.2.c.b.538.43	yes	48
77.76	even	2	inner	1617.2.c.b.538.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.c.b.538.5	✓	48	1.1	even	1	trivial
1617.2.c.b.538.6	yes	48	7.6	odd	2	inner
1617.2.c.b.538.43	yes	48	11.10	odd	2	inner
1617.2.c.b.538.44	yes	48	77.76	even	2	inner

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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.34878i q^{2} -1.00000i q^{3} -3.51679 q^{4} +3.84365i q^{5} -2.34878 q^{6} +3.56261i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.34878i q^{2} -1.00000i q^{3} -3.51679 q^{4} +3.84365i q^{5} -2.34878 q^{6} +3.56261i q^{8} -1.00000 q^{9} +9.02791 q^{10} +(0.648688 - 3.25257i) q^{11} +3.51679i q^{12} +2.56122 q^{13} +3.84365 q^{15} +1.33423 q^{16} +0.436269 q^{17} +2.34878i q^{18} -5.74787 q^{19} -13.5173i q^{20} +(-7.63958 - 1.52363i) q^{22} -5.71576 q^{23} +3.56261 q^{24} -9.77365 q^{25} -6.01575i q^{26} +1.00000i q^{27} +1.22751i q^{29} -9.02791i q^{30} +5.39790i q^{31} +3.99141i q^{32} +(-3.25257 - 0.648688i) q^{33} -1.02470i q^{34} +3.51679 q^{36} -9.95670 q^{37} +13.5005i q^{38} -2.56122i q^{39} -13.6934 q^{40} -12.3106 q^{41} -8.05674i q^{43} +(-2.28130 + 11.4386i) q^{44} -3.84365i q^{45} +13.4251i q^{46} -5.43927i q^{47} -1.33423i q^{48} +22.9562i q^{50} -0.436269i q^{51} -9.00726 q^{52} +6.51185 q^{53} +2.34878 q^{54} +(12.5017 + 2.49333i) q^{55} +5.74787i q^{57} +2.88315 q^{58} -3.88092i q^{59} -13.5173 q^{60} +4.47453 q^{61} +12.6785 q^{62} +12.0434 q^{64} +9.84442i q^{65} +(-1.52363 + 7.63958i) q^{66} -4.08459 q^{67} -1.53427 q^{68} +5.71576i q^{69} -5.05430 q^{71} -3.56261i q^{72} +3.52022 q^{73} +23.3861i q^{74} +9.77365i q^{75} +20.2140 q^{76} -6.01575 q^{78} +16.6372i q^{79} +5.12831i q^{80} +1.00000 q^{81} +28.9150i q^{82} +4.69573 q^{83} +1.67686i q^{85} -18.9235 q^{86} +1.22751 q^{87} +(11.5876 + 2.31102i) q^{88} -2.88525i q^{89} -9.02791 q^{90} +20.1011 q^{92} +5.39790 q^{93} -12.7757 q^{94} -22.0928i q^{95} +3.99141 q^{96} -11.2534i q^{97} +(-0.648688 + 3.25257i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) \) 48 * q - 64 * q^4 - 48 * q^9	\( 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) \) 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Introduction

L-functions

Modular forms

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