LMFDB - Embedded newform 3822.2.c.i.883.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3822,2,Mod(883,3822)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3822.883"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3822, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,-6,0,0,0,0,6,-4,0,6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.3
Root			$0.254102i$ of defining polynomial
Character	$\chi$	$=$	3822.883
Dual form			3822.2.c.i.883.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.68133i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +1.68133 q^{10} +2.18953i q^{11} +1.00000 q^{12} +(1.68133 - 3.18953i) q^{13} -1.68133i q^{15} +1.00000 q^{16} -4.18953 q^{17} -1.00000i q^{18} +1.17313i q^{19} -1.68133i q^{20} +2.18953 q^{22} -5.68133 q^{23} -1.00000i q^{24} +2.17313 q^{25} +(-3.18953 - 1.68133i) q^{26} -1.00000 q^{27} +2.69774 q^{29} -1.68133 q^{30} -7.36266i q^{31} -1.00000i q^{32} -2.18953i q^{33} +4.18953i q^{34} -1.00000 q^{36} -3.68133i q^{37} +1.17313 q^{38} +(-1.68133 + 3.18953i) q^{39} -1.68133 q^{40} -9.36266i q^{41} +12.9149 q^{43} -2.18953i q^{44} +1.68133i q^{45} +5.68133i q^{46} +2.37907i q^{47} -1.00000 q^{48} -2.17313i q^{50} +4.18953 q^{51} +(-1.68133 + 3.18953i) q^{52} -3.01641 q^{53} +1.00000i q^{54} -3.68133 q^{55} -1.17313i q^{57} -2.69774i q^{58} +5.36266i q^{59} +1.68133i q^{60} -0.697737 q^{61} -7.36266 q^{62} -1.00000 q^{64} +(5.36266 + 2.82687i) q^{65} -2.18953 q^{66} +15.1044i q^{67} +4.18953 q^{68} +5.68133 q^{69} +5.01641i q^{71} +1.00000i q^{72} +3.81047i q^{73} -3.68133 q^{74} -2.17313 q^{75} -1.17313i q^{76} +(3.18953 + 1.68133i) q^{78} +8.37907 q^{79} +1.68133i q^{80} +1.00000 q^{81} -9.36266 q^{82} -3.01641i q^{83} -7.04399i q^{85} -12.9149i q^{86} -2.69774 q^{87} -2.18953 q^{88} +3.01641i q^{89} +1.68133 q^{90} +5.68133 q^{92} +7.36266i q^{93} +2.37907 q^{94} -1.97241 q^{95} +1.00000i q^{96} +2.00000i q^{97} +2.18953i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 4 q^{10} + 6 q^{12} - 4 q^{13} + 6 q^{16} - 8 q^{17} - 4 q^{22} - 20 q^{23} + 2 q^{25} - 2 q^{26} - 6 q^{27} - 4 q^{29} + 4 q^{30} - 6 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40}+ \cdots - 48 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 4 * q^10 + 6 * q^12 - 4 * q^13 + 6 * q^16 - 8 * q^17 - 4 * q^22 - 20 * q^23 + 2 * q^25 - 2 * q^26 - 6 * q^27 - 4 * q^29 + 4 * q^30 - 6 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^40 + 4 * q^43 - 6 * q^48 + 8 * q^51 + 4 * q^52 - 12 * q^53 - 8 * q^55 + 16 * q^61 - 16 * q^62 - 6 * q^64 + 4 * q^65 + 4 * q^66 + 8 * q^68 + 20 * q^69 - 8 * q^74 - 2 * q^75 + 2 * q^78 + 16 * q^79 + 6 * q^81 - 28 * q^82 + 4 * q^87 + 4 * q^88 - 4 * q^90 + 20 * q^92 - 20 * q^94 - 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$.

$n$	$1471$	$2549$	$3433$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$			1.68133i			0.751914i	0.926637	+	0.375957i	$0.122686\pi$
$5$			1.68133i			0.751914i	−0.926637	+	0.375957i	$0.877314\pi$
$6$			1.00000i			0.408248i
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$	1.00000			0.333333
$10$	1.68133			0.531683
$11$			2.18953i			0.660169i	0.943951	+	0.330085i	$0.107077\pi$
$11$			2.18953i			0.660169i	−0.943951	+	0.330085i	$0.892923\pi$
$12$	1.00000			0.288675
$13$	1.68133	−	3.18953i	0.466317	−	0.884618i
$13$	1.68133	−	3.18953i	0.466317	−	0.884618i
$14$		0			0
$15$		−	1.68133i		−	0.434118i
$16$	1.00000			0.250000
$17$	−4.18953			−1.01611			−0.508056	−	0.861324i	$-0.669636\pi$
$17$	−4.18953			−1.01611			−0.508056	+	0.861324i	$0.669636\pi$
$18$		−	1.00000i		−	0.235702i
$19$			1.17313i			0.269134i	0.990905	+	0.134567i	$0.0429643\pi$
$19$			1.17313i			0.269134i	−0.990905	+	0.134567i	$0.957036\pi$
$20$		−	1.68133i		−	0.375957i
$21$		0			0
$22$	2.18953			0.466810
$23$	−5.68133			−1.18464			−0.592320	−	0.805703i	$-0.701788\pi$
$23$	−5.68133			−1.18464			−0.592320	+	0.805703i	$0.701788\pi$
$24$		−	1.00000i		−	0.204124i
$25$	2.17313			0.434625
$26$	−3.18953	−	1.68133i	−0.625519	−	0.329736i
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.69774			0.500957			0.250479	−	0.968122i	$-0.419412\pi$
$29$	2.69774			0.500957			0.250479	+	0.968122i	$0.419412\pi$
$30$	−1.68133			−0.306968
$31$		−	7.36266i		−	1.32237i	−0.750222	−	0.661187i	$-0.770053\pi$
$31$		−	7.36266i		−	1.32237i	0.750222	−	0.661187i	$-0.229947\pi$
$32$		−	1.00000i		−	0.176777i
$33$		−	2.18953i		−	0.381149i
$34$			4.18953i			0.718499i
$35$		0			0
$36$	−1.00000			−0.166667
$37$		−	3.68133i		−	0.605207i	−0.953117	−	0.302604i	$-0.902144\pi$
$37$		−	3.68133i		−	0.605207i	0.953117	−	0.302604i	$-0.0978558\pi$
$38$	1.17313			0.190306
$39$	−1.68133	+	3.18953i	−0.269228	+	0.510734i
$40$	−1.68133			−0.265842
$41$		−	9.36266i		−	1.46220i	−0.682269	−	0.731101i	$-0.739007\pi$
$41$		−	9.36266i		−	1.46220i	0.682269	−	0.731101i	$-0.260993\pi$
$42$		0			0
$43$	12.9149			1.96950			0.984749	−	0.173983i	$-0.0556639\pi$
$43$	12.9149			1.96950			0.984749	+	0.173983i	$0.0556639\pi$
$44$		−	2.18953i		−	0.330085i
$45$			1.68133i			0.250638i
$46$			5.68133i			0.837667i
$47$			2.37907i			0.347023i	0.984832	+	0.173511i	$0.0555113\pi$
$47$			2.37907i			0.347023i	−0.984832	+	0.173511i	$0.944489\pi$
$48$	−1.00000			−0.144338
$49$		0			0
$50$		−	2.17313i		−	0.307327i
$51$	4.18953			0.586652
$52$	−1.68133	+	3.18953i	−0.233159	+	0.442309i
$53$	−3.01641			−0.414335			−0.207168	−	0.978305i	$-0.566425\pi$
$53$	−3.01641			−0.414335			−0.207168	+	0.978305i	$0.566425\pi$
$54$			1.00000i			0.136083i
$55$	−3.68133			−0.496391
$56$		0			0
$57$		−	1.17313i		−	0.155385i
$58$		−	2.69774i		−	0.354230i
$59$			5.36266i			0.698159i	0.937093	+	0.349080i	$0.113506\pi$
$59$			5.36266i			0.698159i	−0.937093	+	0.349080i	$0.886494\pi$
$60$			1.68133i			0.217059i
$61$	−0.697737			−0.0893361			−0.0446681	−	0.999002i	$-0.514223\pi$
$61$	−0.697737			−0.0893361			−0.0446681	+	0.999002i	$0.514223\pi$
$62$	−7.36266			−0.935059
$63$		0			0
$64$	−1.00000			−0.125000
$65$	5.36266	+	2.82687i	0.665156	+	0.350630i
$66$	−2.18953			−0.269513
$67$			15.1044i			1.84530i	0.385644	+	0.922648i	$0.373979\pi$
$67$			15.1044i			1.84530i	−0.385644	+	0.922648i	$0.626021\pi$
$68$	4.18953			0.508056
$69$	5.68133			0.683952
$70$		0			0
$71$			5.01641i			0.595338i	0.954669	+	0.297669i	$0.0962092\pi$
$71$			5.01641i			0.595338i	−0.954669	+	0.297669i	$0.903791\pi$
$72$			1.00000i			0.117851i
$73$			3.81047i			0.445981i	0.974821	+	0.222991i	$0.0715819\pi$
$73$			3.81047i			0.445981i	−0.974821	+	0.222991i	$0.928418\pi$
$74$	−3.68133			−0.427946
$75$	−2.17313			−0.250931
$76$		−	1.17313i		−	0.134567i
$77$		0			0
$78$	3.18953	+	1.68133i	0.361144	+	0.190373i
$79$	8.37907			0.942719			0.471359	−	0.881941i	$-0.343763\pi$
$79$	8.37907			0.942719			0.471359	+	0.881941i	$0.343763\pi$
$80$			1.68133i			0.187978i
$81$	1.00000			0.111111
$82$	−9.36266			−1.03393
$83$		−	3.01641i		−	0.331094i	−0.986202	−	0.165547i	$-0.947061\pi$
$83$		−	3.01641i		−	0.331094i	0.986202	−	0.165547i	$-0.0529389\pi$
$84$		0			0
$85$		−	7.04399i		−	0.764028i
$86$		−	12.9149i		−	1.39264i
$87$	−2.69774			−0.289228
$88$	−2.18953			−0.233405
$89$			3.01641i			0.319738i	0.987138	+	0.159869i	$0.0511073\pi$
$89$			3.01641i			0.319738i	−0.987138	+	0.159869i	$0.948893\pi$
$90$	1.68133			0.177228
$91$		0			0
$92$	5.68133			0.592320
$93$			7.36266i			0.763472i
$94$	2.37907			0.245382
$95$	−1.97241			−0.202365
$96$			1.00000i			0.102062i
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$			2.18953i			0.220056i
$100$	−2.17313			−0.217313
$101$	−9.36266			−0.931620			−0.465810	−	0.884885i	$-0.654237\pi$
$101$	−9.36266			−0.931620			−0.465810	+	0.884885i	$0.654237\pi$
$102$		−	4.18953i		−	0.414826i
$103$	−12.4395			−1.22570			−0.612849	−	0.790200i	$-0.709976\pi$
$103$	−12.4395			−1.22570			−0.612849	+	0.790200i	$0.709976\pi$
$104$	3.18953	+	1.68133i	0.312760	+	0.164868i
$105$		0			0
$106$			3.01641i			0.292979i
$107$	13.1044			1.26685			0.633425	−	0.773804i	$-0.281649\pi$
$107$	13.1044			1.26685			0.633425	+	0.773804i	$0.281649\pi$
$108$	1.00000			0.0962250
$109$			7.04399i			0.674692i	0.941381	+	0.337346i	$0.109529\pi$
$109$			7.04399i			0.674692i	−0.941381	+	0.337346i	$0.890471\pi$
$110$			3.68133i			0.351001i
$111$			3.68133i			0.349416i
$112$		0			0
$113$	7.39547			0.695708			0.347854	−	0.937549i	$-0.386911\pi$
$113$	7.39547			0.695708			0.347854	+	0.937549i	$0.386911\pi$
$114$	−1.17313			−0.109873
$115$		−	9.55220i		−	0.890747i
$116$	−2.69774			−0.250479
$117$	1.68133	−	3.18953i	0.155439	−	0.294873i
$118$	5.36266			0.493673
$119$		0			0
$120$	1.68133			0.153484
$121$	6.20594			0.564176
$122$			0.697737i			0.0631702i
$123$			9.36266i			0.844203i
$124$			7.36266i			0.661187i
$125$			12.0604i			1.07871i
$126$		0			0
$127$	12.1208			1.07555			0.537773	−	0.843089i	$-0.319266\pi$
$127$	12.1208			1.07555			0.537773	+	0.843089i	$0.319266\pi$
$128$			1.00000i			0.0883883i
$129$	−12.9149			−1.13709
$130$	2.82687	−	5.36266i	0.247933	−	0.470337i
$131$	12.6566			1.10581			0.552906	−	0.833244i	$-0.313519\pi$
$131$	12.6566			1.10581			0.552906	+	0.833244i	$0.313519\pi$
$132$			2.18953i			0.190574i
$133$		0			0
$134$	15.1044			1.30482
$135$		−	1.68133i		−	0.144706i
$136$		−	4.18953i		−	0.359250i
$137$			9.58501i			0.818903i	0.912332	+	0.409451i	$0.134280\pi$
$137$			9.58501i			0.818903i	−0.912332	+	0.409451i	$0.865720\pi$
$138$		−	5.68133i		−	0.483627i
$139$	19.7417			1.67447			0.837236	−	0.546842i	$-0.184170\pi$
$139$	19.7417			1.67447			0.837236	+	0.546842i	$0.184170\pi$
$140$		0			0
$141$		−	2.37907i		−	0.200354i
$142$	5.01641			0.420968
$143$	6.98359	+	3.68133i	0.583997	+	0.307848i
$144$	1.00000			0.0833333
$145$			4.53579i			0.376677i
$146$	3.81047			0.315356
$147$		0			0
$148$			3.68133i			0.302604i
$149$		−	23.4506i		−	1.92115i	−0.278018	−	0.960576i	$-0.589678\pi$
$149$		−	23.4506i		−	1.92115i	0.278018	−	0.960576i	$-0.410322\pi$
$150$			2.17313i			0.177435i
$151$		−	10.0276i		−	0.816033i	−0.912974	−	0.408017i	$-0.866221\pi$
$151$		−	10.0276i		−	0.816033i	0.912974	−	0.408017i	$-0.133779\pi$
$152$	−1.17313			−0.0951532
$153$	−4.18953			−0.338704
$154$		0			0
$155$	12.3791			0.994311
$156$	1.68133	−	3.18953i	0.134614	−	0.255367i
$157$	18.0932			1.44400			0.721998	−	0.691895i	$-0.243224\pi$
$157$	18.0932			1.44400			0.721998	+	0.691895i	$0.243224\pi$
$158$		−	8.37907i		−	0.666603i
$159$	3.01641			0.239217
$160$	1.68133			0.132921
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$			10.0328i			0.785831i	0.919575	+	0.392915i	$0.128533\pi$
$163$			10.0328i			0.785831i	−0.919575	+	0.392915i	$0.871467\pi$
$164$			9.36266i			0.731101i
$165$	3.68133			0.286591
$166$	−3.01641			−0.234119
$167$			4.66492i			0.360983i	0.983577	+	0.180491i	$0.0577688\pi$
$167$			4.66492i			0.360983i	−0.983577	+	0.180491i	$0.942231\pi$
$168$		0			0
$169$	−7.34625	−	10.7253i	−0.565097	−	0.825025i
$170$	−7.04399			−0.540250
$171$			1.17313i			0.0897113i
$172$	−12.9149			−0.984749
$173$	15.0716			1.14587			0.572935	−	0.819601i	$-0.305805\pi$
$173$	15.0716			1.14587			0.572935	+	0.819601i	$0.305805\pi$
$174$			2.69774i			0.204515i
$175$		0			0
$176$			2.18953i			0.165042i
$177$		−	5.36266i		−	0.403082i
$178$	3.01641			0.226089
$179$	9.10439			0.680494			0.340247	−	0.940336i	$-0.389489\pi$
$179$	9.10439			0.680494			0.340247	+	0.940336i	$0.389489\pi$
$180$		−	1.68133i		−	0.125319i
$181$	−9.32985			−0.693482			−0.346741	−	0.937961i	$-0.612712\pi$
$181$	−9.32985			−0.693482			−0.346741	+	0.937961i	$0.612712\pi$
$182$		0			0
$183$	0.697737			0.0515782
$184$		−	5.68133i		−	0.418833i
$185$	6.18953			0.455064
$186$	7.36266			0.539857
$187$		−	9.17313i		−	0.670806i
$188$		−	2.37907i		−	0.173511i
$189$		0			0
$190$			1.97241i			0.143094i
$191$	6.43947			0.465943			0.232972	−	0.972484i	$-0.425155\pi$
$191$	6.43947			0.465943			0.232972	+	0.972484i	$0.425155\pi$
$192$	1.00000			0.0721688
$193$			3.10439i			0.223459i	0.993739	+	0.111729i	$0.0356390\pi$
$193$			3.10439i			0.223459i	−0.993739	+	0.111729i	$0.964361\pi$
$194$	2.00000			0.143592
$195$	−5.36266	−	2.82687i	−0.384028	−	0.202437i
$196$		0			0
$197$		−	6.12080i		−	0.436089i	−0.975939	−	0.218044i	$-0.930032\pi$
$197$		−	6.12080i		−	0.436089i	0.975939	−	0.218044i	$-0.0699678\pi$
$198$	2.18953			0.155603
$199$	13.7141			0.972170			0.486085	−	0.873912i	$-0.338425\pi$
$199$	13.7141			0.972170			0.486085	+	0.873912i	$0.338425\pi$
$200$			2.17313i			0.153663i
$201$		−	15.1044i		−	1.06538i
$202$			9.36266i			0.658755i
$203$		0			0
$204$	−4.18953			−0.293326
$205$	15.7417			1.09945
$206$			12.4395i			0.866699i
$207$	−5.68133			−0.394880
$208$	1.68133	−	3.18953i	0.116579	−	0.221154i
$209$	−2.56860			−0.177674
$210$		0			0
$211$	3.46421			0.238486			0.119243	−	0.992865i	$-0.461953\pi$
$211$	3.46421			0.238486			0.119243	+	0.992865i	$0.461953\pi$
$212$	3.01641			0.207168
$213$		−	5.01641i		−	0.343719i
$214$		−	13.1044i		−	0.895798i
$215$			21.7141i			1.48089i
$216$		−	1.00000i		−	0.0680414i
$217$		0			0
$218$	7.04399			0.477079
$219$		−	3.81047i		−	0.257487i
$220$	3.68133			0.248195
$221$	−7.04399	+	13.3627i	−0.473830	+	0.898870i
$222$	3.68133			0.247075
$223$		−	0.258271i		−	0.0172951i	−0.999963	−	0.00864754i	$-0.997247\pi$
$223$		−	0.258271i		−	0.0172951i	0.999963	−	0.00864754i	$-0.00275263\pi$
$224$		0			0
$225$	2.17313			0.144875
$226$		−	7.39547i		−	0.491940i
$227$		−	2.25827i		−	0.149887i	−0.997188	−	0.0749433i	$-0.976122\pi$
$227$		−	2.25827i		−	0.149887i	0.997188	−	0.0749433i	$-0.0238776\pi$
$228$			1.17313i			0.0776923i
$229$		−	18.3791i		−	1.21452i	−0.794502	−	0.607262i	$-0.792268\pi$
$229$		−	18.3791i		−	1.21452i	0.794502	−	0.607262i	$-0.207732\pi$
$230$	−9.55220			−0.629853
$231$		0			0
$232$			2.69774i			0.177115i
$233$	17.7417			1.16230			0.581150	−	0.813797i	$-0.302603\pi$
$233$	17.7417			1.16230			0.581150	+	0.813797i	$0.302603\pi$
$234$	−3.18953	−	1.68133i	−0.208506	−	0.109912i
$235$	−4.00000			−0.260931
$236$		−	5.36266i		−	0.349080i
$237$	−8.37907			−0.544279
$238$		0			0
$239$		−	17.0716i		−	1.10427i	−0.833755	−	0.552134i	$-0.813814\pi$
$239$		−	17.0716i		−	1.10427i	0.833755	−	0.552134i	$-0.186186\pi$
$240$		−	1.68133i		−	0.108529i
$241$		−	11.4506i		−	0.737601i	−0.929509	−	0.368800i	$-0.879769\pi$
$241$		−	11.4506i		−	0.737601i	0.929509	−	0.368800i	$-0.120231\pi$
$242$		−	6.20594i		−	0.398933i
$243$	−1.00000			−0.0641500
$244$	0.697737			0.0446681
$245$		0			0
$246$	9.36266			0.596941
$247$	3.74173	+	1.97241i	0.238081	+	0.125502i
$248$	7.36266			0.467529
$249$			3.01641i			0.191157i
$250$	12.0604			0.762767
$251$	2.82687			0.178431			0.0892153	−	0.996012i	$-0.471564\pi$
$251$	2.82687			0.178431			0.0892153	+	0.996012i	$0.471564\pi$
$252$		0			0
$253$		−	12.4395i		−	0.782063i
$254$		−	12.1208i		−	0.760526i
$255$			7.04399i			0.441112i
$256$	1.00000			0.0625000
$257$	4.03281			0.251560			0.125780	−	0.992058i	$-0.459857\pi$
$257$	4.03281			0.251560			0.125780	+	0.992058i	$0.459857\pi$
$258$			12.9149i			0.804044i
$259$		0			0
$260$	−5.36266	−	2.82687i	−0.332578	−	0.175315i
$261$	2.69774			0.166986
$262$		−	12.6566i		−	0.781926i
$263$	−15.8297			−0.976102			−0.488051	−	0.872815i	$-0.662292\pi$
$263$	−15.8297			−0.976102			−0.488051	+	0.872815i	$0.662292\pi$
$264$	2.18953			0.134757
$265$		−	5.07158i		−	0.311545i
$266$		0			0
$267$		−	3.01641i		−	0.184601i
$268$		−	15.1044i		−	0.922648i
$269$	−18.7581			−1.14370			−0.571852	−	0.820357i	$-0.693775\pi$
$269$	−18.7581			−1.14370			−0.571852	+	0.820357i	$0.693775\pi$
$270$	−1.68133			−0.102323
$271$			7.62093i			0.462939i	0.972842	+	0.231469i	$0.0743533\pi$
$271$			7.62093i			0.462939i	−0.972842	+	0.231469i	$0.925647\pi$
$272$	−4.18953			−0.254028
$273$		0			0
$274$	9.58501			0.579052
$275$			4.75814i			0.286926i
$276$	−5.68133			−0.341976
$277$	18.4342			1.10761			0.553803	−	0.832648i	$-0.313176\pi$
$277$	18.4342			1.10761			0.553803	+	0.832648i	$0.313176\pi$
$278$		−	19.7417i		−	1.18403i
$279$		−	7.36266i		−	0.440791i
$280$		0			0
$281$		−	8.03281i		−	0.479197i	−0.970872	−	0.239599i	$-0.922984\pi$
$281$		−	8.03281i		−	0.479197i	0.970872	−	0.239599i	$-0.0770159\pi$
$282$	−2.37907			−0.141671
$283$	14.8461			0.882510			0.441255	−	0.897382i	$-0.354533\pi$
$283$	14.8461			0.882510			0.441255	+	0.897382i	$0.354533\pi$
$284$		−	5.01641i		−	0.297669i
$285$	1.97241			0.116836
$286$	3.68133	−	6.98359i	0.217682	−	0.412949i
$287$		0			0
$288$		−	1.00000i		−	0.0589256i
$289$	0.552195			0.0324821
$290$	4.53579			0.266351
$291$		−	2.00000i		−	0.117242i
$292$		−	3.81047i		−	0.222991i
$293$			21.7417i			1.27017i	0.772444	+	0.635083i	$0.219034\pi$
$293$			21.7417i			1.27017i	−0.772444	+	0.635083i	$0.780966\pi$
$294$		0			0
$295$	−9.01641			−0.524955
$296$	3.68133			0.213973
$297$		−	2.18953i		−	0.127050i
$298$	−23.4506			−1.35846
$299$	−9.55220	+	18.1208i	−0.552418	+	1.04795i
$300$	2.17313			0.125466
$301$		0			0
$302$	−10.0276			−0.577023
$303$	9.36266			0.537871
$304$			1.17313i			0.0672835i
$305$		−	1.17313i		−	0.0671731i
$306$			4.18953i			0.239500i
$307$		−	27.4835i		−	1.56856i	−0.620405	−	0.784282i	$-0.713032\pi$
$307$		−	27.4835i		−	1.56856i	0.620405	−	0.784282i	$-0.286968\pi$
$308$		0			0
$309$	12.4395			0.707657
$310$		−	12.3791i		−	0.703084i
$311$	−6.29108			−0.356735			−0.178367	−	0.983964i	$-0.557082\pi$
$311$	−6.29108			−0.356735			−0.178367	+	0.983964i	$0.557082\pi$
$312$	−3.18953	−	1.68133i	−0.180572	−	0.0951866i
$313$	−32.4342			−1.83329			−0.916646	−	0.399701i	$-0.869114\pi$
$313$	−32.4342			−1.83329			−0.916646	+	0.399701i	$0.869114\pi$
$314$		−	18.0932i		−	1.02106i
$315$		0			0
$316$	−8.37907			−0.471359
$317$			29.7417i			1.67046i	0.549899	+	0.835231i	$0.314666\pi$
$317$			29.7417i			1.67046i	−0.549899	+	0.835231i	$0.685334\pi$
$318$		−	3.01641i		−	0.169152i
$319$			5.90679i			0.330717i
$320$		−	1.68133i		−	0.0939892i
$321$	−13.1044			−0.731416
$322$		0			0
$323$		−	4.91486i		−	0.273470i
$324$	−1.00000			−0.0555556
$325$	3.65375	−	6.93126i	0.202673	−	0.384477i
$326$	10.0328			0.555666
$327$		−	7.04399i		−	0.389534i
$328$	9.36266			0.516966
$329$		0			0
$330$		−	3.68133i		−	0.202651i
$331$			25.3955i			1.39586i	0.716165	+	0.697931i	$0.245896\pi$
$331$			25.3955i			1.39586i	−0.716165	+	0.697931i	$0.754104\pi$
$332$			3.01641i			0.165547i
$333$		−	3.68133i		−	0.201736i
$334$	4.66492			0.255253
$335$	−25.3955			−1.38750
$336$		0			0
$337$	10.4806			0.570916			0.285458	−	0.958391i	$-0.407854\pi$
$337$	10.4806			0.570916			0.285458	+	0.958391i	$0.407854\pi$
$338$	−10.7253	+	7.34625i	−0.583381	+	0.399584i
$339$	−7.39547			−0.401667
$340$			7.04399i			0.382014i
$341$	16.1208			0.872990
$342$	1.17313			0.0634355
$343$		0			0
$344$			12.9149i			0.696322i
$345$			9.55220i			0.514273i
$346$		−	15.0716i		−	0.810253i
$347$	19.7089			1.05803			0.529015	−	0.848612i	$-0.322562\pi$
$347$	19.7089			1.05803			0.529015	+	0.848612i	$0.322562\pi$
$348$	2.69774			0.144614
$349$		−	0.467052i		−	0.0250007i	−0.999922	−	0.0125004i	$-0.996021\pi$
$349$		−	0.467052i		−	0.0250007i	0.999922	−	0.0125004i	$-0.00397909\pi$
$350$		0			0
$351$	−1.68133	+	3.18953i	−0.0897428	+	0.170245i
$352$	2.18953			0.116703
$353$			31.1372i			1.65727i	0.559792	+	0.828633i	$0.310881\pi$
$353$			31.1372i			1.65727i	−0.559792	+	0.828633i	$0.689119\pi$
$354$	−5.36266			−0.285022
$355$	−8.43424			−0.447643
$356$		−	3.01641i		−	0.159869i
$357$		0			0
$358$		−	9.10439i		−	0.481182i
$359$			8.12080i			0.428599i	0.976768	+	0.214300i	$0.0687469\pi$
$359$			8.12080i			0.428599i	−0.976768	+	0.214300i	$0.931253\pi$
$360$	−1.68133			−0.0886139
$361$	17.6238			0.927567
$362$			9.32985i			0.490366i
$363$	−6.20594			−0.325727
$364$		0			0
$365$	−6.40665			−0.335340
$366$		−	0.697737i		−	0.0364713i
$367$	11.4835			0.599432			0.299716	−	0.954029i	$-0.403108\pi$
$367$	11.4835			0.599432			0.299716	+	0.954029i	$0.403108\pi$
$368$	−5.68133			−0.296160
$369$		−	9.36266i		−	0.487401i
$370$		−	6.18953i		−	0.321779i
$371$		0			0
$372$		−	7.36266i		−	0.381736i
$373$	−11.3955			−0.590035			−0.295018	−	0.955492i	$-0.595326\pi$
$373$	−11.3955			−0.590035			−0.295018	+	0.955492i	$0.595326\pi$
$374$	−9.17313			−0.474331
$375$		−	12.0604i		−	0.622796i
$376$	−2.37907			−0.122691
$377$	4.53579	−	8.60453i	0.233605	−	0.443156i
$378$		0			0
$379$		−	14.3463i		−	0.736918i	−0.929644	−	0.368459i	$-0.879886\pi$
$379$		−	14.3463i		−	0.736918i	0.929644	−	0.368459i	$-0.120114\pi$
$380$	1.97241			0.101183
$381$	−12.1208			−0.620967
$382$		−	6.43947i		−	0.329472i
$383$			5.36789i			0.274286i	0.990551	+	0.137143i	$0.0437920\pi$
$383$			5.36789i			0.274286i	−0.990551	+	0.137143i	$0.956208\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	3.10439			0.158009
$387$	12.9149			0.656499
$388$		−	2.00000i		−	0.101535i
$389$	−14.4342			−0.731845			−0.365922	−	0.930645i	$-0.619246\pi$
$389$	−14.4342			−0.731845			−0.365922	+	0.930645i	$0.619246\pi$
$390$	−2.82687	+	5.36266i	−0.143144	+	0.271549i
$391$	23.8021			1.20373
$392$		0			0
$393$	−12.6566			−0.638440
$394$	−6.12080			−0.308361
$395$			14.0880i			0.708843i
$396$		−	2.18953i		−	0.110028i
$397$			12.6045i			0.632603i	0.948659	+	0.316301i	$0.102441\pi$
$397$			12.6045i			0.632603i	−0.948659	+	0.316301i	$0.897559\pi$
$398$		−	13.7141i		−	0.687428i
$399$		0			0
$400$	2.17313			0.108656
$401$			4.72532i			0.235971i	0.993015	+	0.117986i	$0.0376437\pi$
$401$			4.72532i			0.235971i	−0.993015	+	0.117986i	$0.962356\pi$
$402$	−15.1044			−0.753339
$403$	−23.4835	−	12.3791i	−1.16979	−	0.616645i
$404$	9.36266			0.465810
$405$			1.68133i			0.0835460i
$406$		0			0
$407$	8.06040			0.399539
$408$			4.18953i			0.207413i
$409$		−	31.6074i		−	1.56288i	−0.623978	−	0.781442i	$-0.714485\pi$
$409$		−	31.6074i		−	1.56288i	0.623978	−	0.781442i	$-0.285515\pi$
$410$		−	15.7417i		−	0.777429i
$411$		−	9.58501i		−	0.472794i
$412$	12.4395			0.612849
$413$		0			0
$414$			5.68133i			0.279222i
$415$	5.07158			0.248954
$416$	−3.18953	−	1.68133i	−0.156380	−	0.0824340i
$417$	−19.7417			−0.966757
$418$			2.56860i			0.125634i
$419$	−21.9313			−1.07141			−0.535706	−	0.844404i	$-0.679955\pi$
$419$	−21.9313			−1.07141			−0.535706	+	0.844404i	$0.679955\pi$
$420$		0			0
$421$		−	15.4283i		−	0.751929i	−0.926634	−	0.375964i	$-0.877311\pi$
$421$		−	15.4283i		−	0.751929i	0.926634	−	0.375964i	$-0.122689\pi$
$422$		−	3.46421i		−	0.168635i
$423$			2.37907i			0.115674i
$424$		−	3.01641i		−	0.146490i
$425$	−9.10439			−0.441628
$426$	−5.01641			−0.243046
$427$		0			0
$428$	−13.1044			−0.633425
$429$	−6.98359	−	3.68133i	−0.337171	−	0.177736i
$430$	21.7141			1.04715
$431$		−	5.65375i		−	0.272331i	−0.990686	−	0.136166i	$-0.956522\pi$
$431$		−	5.65375i		−	0.272331i	0.990686	−	0.136166i	$-0.0434779\pi$
$432$	−1.00000			−0.0481125
$433$	19.7417			0.948727			0.474363	−	0.880329i	$-0.342678\pi$
$433$	19.7417			0.948727			0.474363	+	0.880329i	$0.342678\pi$
$434$		0			0
$435$		−	4.53579i		−	0.217474i
$436$		−	7.04399i		−	0.337346i
$437$		−	6.66492i		−	0.318827i
$438$	−3.81047			−0.182071
$439$	19.7365			0.941972			0.470986	−	0.882141i	$-0.343898\pi$
$439$	19.7365			0.941972			0.470986	+	0.882141i	$0.343898\pi$
$440$		−	3.68133i		−	0.175501i
$441$		0			0
$442$	13.3627	+	7.04399i	0.635597	+	0.335049i
$443$	22.4342			1.06588			0.532941	−	0.846152i	$-0.321087\pi$
$443$	22.4342			1.06588			0.532941	+	0.846152i	$0.321087\pi$
$444$		−	3.68133i		−	0.174708i
$445$	−5.07158			−0.240416
$446$	−0.258271			−0.0122295
$447$			23.4506i			1.10918i
$448$		0			0
$449$		−	0.447805i		−	0.0211332i	−0.999944	−	0.0105666i	$-0.996636\pi$
$449$		−	0.447805i		−	0.0211332i	0.999944	−	0.0105666i	$-0.00336352\pi$
$450$		−	2.17313i		−	0.102442i
$451$	20.4999			0.965301
$452$	−7.39547			−0.347854
$453$			10.0276i			0.471137i
$454$	−2.25827			−0.105986
$455$		0			0
$456$	1.17313			0.0549367
$457$		−	12.6373i		−	0.591150i	−0.955320	−	0.295575i	$-0.904489\pi$
$457$		−	12.6373i		−	0.591150i	0.955320	−	0.295575i	$-0.0955112\pi$
$458$	−18.3791			−0.858797
$459$	4.18953			0.195551
$460$			9.55220i			0.445373i
$461$			38.5603i			1.79593i	0.440066	+	0.897965i	$0.354955\pi$
$461$			38.5603i			1.79593i	−0.440066	+	0.897965i	$0.645045\pi$
$462$		0			0
$463$			40.1156i			1.86433i	0.362036	+	0.932164i	$0.382082\pi$
$463$			40.1156i			1.86433i	−0.362036	+	0.932164i	$0.617918\pi$
$464$	2.69774			0.125239
$465$	−12.3791			−0.574066
$466$		−	17.7417i		−	0.821870i
$467$	37.7282			1.74585			0.872926	−	0.487853i	$-0.162220\pi$
$467$	37.7282			1.74585			0.872926	+	0.487853i	$0.162220\pi$
$468$	−1.68133	+	3.18953i	−0.0777195	+	0.147436i
$469$		0			0
$470$			4.00000i			0.184506i
$471$	−18.0932			−0.833691
$472$	−5.36266			−0.246836
$473$			28.2775i			1.30020i
$474$			8.37907i			0.384863i
$475$			2.54935i			0.116972i
$476$		0			0
$477$	−3.01641			−0.138112
$478$	−17.0716			−0.780836
$479$		−	23.4559i		−	1.07173i	−0.844305	−	0.535863i	$-0.819986\pi$
$479$		−	23.4559i		−	1.07173i	0.844305	−	0.535863i	$-0.180014\pi$
$480$	−1.68133			−0.0767419
$481$	−11.7417	−	6.18953i	−0.535377	−	0.282218i
$482$	−11.4506			−0.521563
$483$		0			0
$484$	−6.20594			−0.282088
$485$	−3.36266			−0.152691
$486$			1.00000i			0.0453609i
$487$			11.4835i			0.520365i	0.965559	+	0.260183i	$0.0837828\pi$
$487$			11.4835i			0.520365i	−0.965559	+	0.260183i	$0.916217\pi$
$488$		−	0.697737i		−	0.0315851i
$489$		−	10.0328i		−	0.453700i
$490$		0			0
$491$	−17.4283			−0.786528			−0.393264	−	0.919426i	$-0.628654\pi$
$491$	−17.4283			−0.786528			−0.393264	+	0.919426i	$0.628654\pi$
$492$		−	9.36266i		−	0.422101i
$493$	−11.3023			−0.509028
$494$	1.97241	−	3.74173i	0.0887431	−	0.168348i
$495$	−3.68133			−0.165464
$496$		−	7.36266i		−	0.330593i
$497$		0			0
$498$	3.01641			0.135168
$499$			32.2416i			1.44333i	0.692241	+	0.721666i	$0.256623\pi$
$499$			32.2416i			1.44333i	−0.692241	+	0.721666i	$0.743377\pi$
$500$		−	12.0604i		−	0.539357i
$501$		−	4.66492i		−	0.208413i
$502$		−	2.82687i		−	0.126170i
$503$	24.4342			1.08947			0.544734	−	0.838609i	$-0.316630\pi$
$503$	24.4342			1.08947			0.544734	+	0.838609i	$0.316630\pi$
$504$		0			0
$505$		−	15.7417i		−	0.700498i
$506$	−12.4395			−0.553002
$507$	7.34625	+	10.7253i	0.326259	+	0.476328i
$508$	−12.1208			−0.537773
$509$		−	28.4067i		−	1.25910i	−0.776959	−	0.629551i	$-0.783239\pi$
$509$		−	28.4067i		−	1.25910i	0.776959	−	0.629551i	$-0.216761\pi$
$510$	7.04399			0.311913
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		−	1.17313i		−	0.0517948i
$514$		−	4.03281i		−	0.177880i
$515$		−	20.9149i		−	0.921619i
$516$	12.9149			0.568545
$517$	−5.20905			−0.229094
$518$		0			0
$519$	−15.0716			−0.661569
$520$	−2.82687	+	5.36266i	−0.123967	+	0.235168i
$521$	9.77765			0.428367			0.214183	−	0.976793i	$-0.431291\pi$
$521$	9.77765			0.428367			0.214183	+	0.976793i	$0.431291\pi$
$522$		−	2.69774i		−	0.118077i
$523$	−13.1372			−0.574450			−0.287225	−	0.957863i	$-0.592733\pi$
$523$	−13.1372			−0.574450			−0.287225	+	0.957863i	$0.592733\pi$
$524$	−12.6566			−0.552906
$525$		0			0
$526$			15.8297i			0.690208i
$527$			30.8461i			1.34368i
$528$		−	2.18953i		−	0.0952872i
$529$	9.27752			0.403370
$530$	−5.07158			−0.220295
$531$			5.36266i			0.232720i
$532$		0			0
$533$	−29.8625	−	15.7417i	−1.29349	−	0.681850i
$534$	−3.01641			−0.130533
$535$			22.0328i			0.952562i
$536$	−15.1044			−0.652410
$537$	−9.10439			−0.392883
$538$			18.7581i			0.808721i
$539$		0			0
$540$			1.68133i			0.0723530i
$541$		−	9.59335i		−	0.412450i	−0.978505	−	0.206225i	$-0.933882\pi$
$541$		−	9.59335i		−	0.412450i	0.978505	−	0.206225i	$-0.0661179\pi$
$542$	7.62093			0.327347
$543$	9.32985			0.400382
$544$			4.18953i			0.179625i
$545$	−11.8433			−0.507310
$546$		0			0
$547$	−6.03281			−0.257944			−0.128972	−	0.991648i	$-0.541168\pi$
$547$	−6.03281			−0.257944			−0.128972	+	0.991648i	$0.541168\pi$
$548$		−	9.58501i		−	0.409451i
$549$	−0.697737			−0.0297787
$550$	4.75814			0.202888
$551$			3.16479i			0.134825i
$552$			5.68133i			0.241813i
$553$		0			0
$554$		−	18.4342i		−	0.783196i
$555$	−6.18953			−0.262731
$556$	−19.7417			−0.837236
$557$			5.74173i			0.243285i	0.992574	+	0.121642i	$0.0388161\pi$
$557$			5.74173i			0.243285i	−0.992574	+	0.121642i	$0.961184\pi$
$558$	−7.36266			−0.311686
$559$	21.7141	−	41.1924i	0.918410	−	1.74225i
$560$		0			0
$561$			9.17313i			0.387290i
$562$	−8.03281			−0.338844
$563$	30.2447			1.27466			0.637331	−	0.770590i	$-0.280038\pi$
$563$	30.2447			1.27466			0.637331	+	0.770590i	$0.280038\pi$
$564$			2.37907i			0.100177i
$565$			12.4342i			0.523112i
$566$		−	14.8461i		−	0.624029i
$567$		0			0
$568$	−5.01641			−0.210484
$569$	−43.9505			−1.84250			−0.921251	−	0.388970i	$-0.872831\pi$
$569$	−43.9505			−1.84250			−0.921251	+	0.388970i	$0.872831\pi$
$570$		−	1.97241i		−	0.0826154i
$571$	−4.69251			−0.196375			−0.0981877	−	0.995168i	$-0.531305\pi$
$571$	−4.69251			−0.196375			−0.0981877	+	0.995168i	$0.531305\pi$
$572$	−6.98359	−	3.68133i	−0.291999	−	0.153924i
$573$	−6.43947			−0.269013
$574$		0			0
$575$	−12.3463			−0.514874
$576$	−1.00000			−0.0416667
$577$		−	17.4835i		−	0.727846i	−0.931429	−	0.363923i	$-0.881437\pi$
$577$		−	17.4835i		−	0.727846i	0.931429	−	0.363923i	$-0.118563\pi$
$578$		−	0.552195i		−	0.0229683i
$579$		−	3.10439i		−	0.129014i
$580$		−	4.53579i		−	0.188338i
$581$		0			0
$582$	−2.00000			−0.0829027
$583$		−	6.60453i		−	0.273531i
$584$	−3.81047			−0.157678
$585$	5.36266	+	2.82687i	0.221719	+	0.116877i
$586$	21.7417			0.898143
$587$		−	21.6209i		−	0.892391i	−0.894935	−	0.446196i	$-0.852779\pi$
$587$		−	21.6209i		−	0.892391i	0.894935	−	0.446196i	$-0.147221\pi$
$588$		0			0
$589$	8.63734			0.355895
$590$			9.01641i			0.371200i
$591$			6.12080i			0.251776i
$592$		−	3.68133i		−	0.151302i
$593$			9.48346i			0.389439i	0.980859	+	0.194719i	$0.0623797\pi$
$593$			9.48346i			0.389439i	−0.980859	+	0.194719i	$0.937620\pi$
$594$	−2.18953			−0.0898377
$595$		0			0
$596$			23.4506i			0.960576i
$597$	−13.7141			−0.561283
$598$	18.1208	+	9.55220i	0.741015	+	0.390618i
$599$	12.3515			0.504668			0.252334	−	0.967640i	$-0.418802\pi$
$599$	12.3515			0.504668			0.252334	+	0.967640i	$0.418802\pi$
$600$		−	2.17313i		−	0.0887175i
$601$	29.2580			1.19346			0.596729	−	0.802443i	$-0.296467\pi$
$601$	29.2580			1.19346			0.596729	+	0.802443i	$0.296467\pi$
$602$		0			0
$603$			15.1044i			0.615098i
$604$			10.0276i			0.408017i
$605$			10.4342i			0.424212i
$606$		−	9.36266i		−	0.380332i
$607$	0.439467			0.0178374			0.00891870	−	0.999960i	$-0.497161\pi$
$607$	0.439467			0.0178374			0.00891870	+	0.999960i	$0.497161\pi$
$608$	1.17313			0.0475766
$609$		0			0
$610$	−1.17313			−0.0474985
$611$	7.58812	+	4.00000i	0.306982	+	0.161823i
$612$	4.18953			0.169352
$613$		−	41.8901i		−	1.69193i	−0.533242	−	0.845963i	$-0.679026\pi$
$613$		−	41.8901i		−	1.69193i	0.533242	−	0.845963i	$-0.320974\pi$
$614$	−27.4835			−1.10914
$615$	−15.7417			−0.634768
$616$		0			0
$617$		−	35.4147i		−	1.42574i	−0.701295	−	0.712872i	$-0.747394\pi$
$617$		−	35.4147i		−	1.42574i	0.701295	−	0.712872i	$-0.252606\pi$
$618$		−	12.4395i		−	0.500389i
$619$			5.55220i			0.223162i	0.993755	+	0.111581i	$0.0355914\pi$
$619$			5.55220i			0.223162i	−0.993755	+	0.111581i	$0.964409\pi$
$620$	−12.3791			−0.497155
$621$	5.68133			0.227984
$622$			6.29108i			0.252249i
$623$		0			0
$624$	−1.68133	+	3.18953i	−0.0673071	+	0.127684i
$625$	−9.41188			−0.376475
$626$			32.4342i			1.29633i
$627$	2.56860			0.102580
$628$	−18.0932			−0.721998
$629$			15.4231i			0.614958i
$630$		0			0
$631$			28.8737i			1.14944i	0.818349	+	0.574722i	$0.194890\pi$
$631$			28.8737i			1.14944i	−0.818349	+	0.574722i	$0.805110\pi$
$632$			8.37907i			0.333301i
$633$	−3.46421			−0.137690
$634$	29.7417			1.18119
$635$			20.3791i			0.808719i
$636$	−3.01641			−0.119608
$637$		0			0
$638$	5.90679			0.233852
$639$			5.01641i			0.198446i
$640$	−1.68133			−0.0664604
$641$	5.22519			0.206383			0.103191	−	0.994662i	$-0.467095\pi$
$641$	5.22519			0.206383			0.103191	+	0.994662i	$0.467095\pi$
$642$			13.1044i			0.517189i
$643$			32.2775i			1.27290i	0.771318	+	0.636451i	$0.219598\pi$
$643$			32.2775i			1.27290i	−0.771318	+	0.636451i	$0.780402\pi$
$644$		0			0
$645$		−	21.7141i		−	0.854994i
$646$	−4.91486			−0.193372
$647$	−42.5878			−1.67430			−0.837151	−	0.546973i	$-0.815780\pi$
$647$	−42.5878			−1.67430			−0.837151	+	0.546973i	$0.815780\pi$
$648$			1.00000i			0.0392837i
$649$	−11.7417			−0.460903
$650$	−6.93126	−	3.65375i	−0.271867	−	0.143312i
$651$		0			0
$652$		−	10.0328i		−	0.392915i
$653$	18.0604			0.706758			0.353379	−	0.935480i	$-0.385033\pi$
$653$	18.0604			0.706758			0.353379	+	0.935480i	$0.385033\pi$
$654$	−7.04399			−0.275442
$655$			21.2799i			0.831475i
$656$		−	9.36266i		−	0.365551i
$657$			3.81047i			0.148660i
$658$		0			0
$659$	34.1760			1.33131			0.665653	−	0.746261i	$-0.268153\pi$
$659$	34.1760			1.33131			0.665653	+	0.746261i	$0.268153\pi$
$660$	−3.68133			−0.143296
$661$			12.4671i			0.484912i	0.970162	+	0.242456i	$0.0779530\pi$
$661$			12.4671i			0.484912i	−0.970162	+	0.242456i	$0.922047\pi$
$662$	25.3955			0.987023
$663$	7.04399	−	13.3627i	0.273566	−	0.518963i
$664$	3.01641			0.117059
$665$		0			0
$666$	−3.68133			−0.142649
$667$	−15.3267			−0.593454
$668$		−	4.66492i		−	0.180491i
$669$			0.258271i			0.00998532i
$670$			25.3955i			0.981113i
$671$		−	1.52772i		−	0.0589770i
$672$		0			0
$673$	−33.5850			−1.29461			−0.647303	−	0.762232i	$-0.724103\pi$
$673$	−33.5850			−1.29461			−0.647303	+	0.762232i	$0.724103\pi$
$674$		−	10.4806i		−	0.403698i
$675$	−2.17313			−0.0836437
$676$	7.34625	+	10.7253i	0.282548	+	0.412512i
$677$	−41.1700			−1.58229			−0.791146	−	0.611627i	$-0.790515\pi$
$677$	−41.1700			−1.58229			−0.791146	+	0.611627i	$0.790515\pi$
$678$			7.39547i			0.284021i
$679$		0			0
$680$	7.04399			0.270125
$681$			2.25827i			0.0865371i
$682$		−	16.1208i		−	0.617297i
$683$		−	4.22235i		−	0.161564i	−0.996732	−	0.0807818i	$-0.974258\pi$
$683$		−	4.22235i		−	0.161564i	0.996732	−	0.0807818i	$-0.0257417\pi$
$684$		−	1.17313i		−	0.0448556i
$685$	−16.1156			−0.615744
$686$		0			0
$687$			18.3791i			0.701205i
$688$	12.9149			0.492374
$689$	−5.07158	+	9.62093i	−0.193212	+	0.366528i
$690$	9.55220			0.363646
$691$		−	9.27468i		−	0.352825i	−0.984316	−	0.176413i	$-0.943551\pi$
$691$		−	9.27468i		−	0.352825i	0.984316	−	0.176413i	$-0.0564493\pi$
$692$	−15.0716			−0.572935
$693$		0			0
$694$		−	19.7089i		−	0.748140i
$695$			33.1924i			1.25906i
$696$		−	2.69774i		−	0.102257i
$697$			39.2252i			1.48576i
$698$	−0.467052			−0.0176782
$699$	−17.7417			−0.671054
$700$		0			0
$701$	−39.9505			−1.50891			−0.754455	−	0.656352i	$-0.772099\pi$
$701$	−39.9505			−1.50891			−0.754455	+	0.656352i	$0.772099\pi$
$702$	3.18953	+	1.68133i	0.120381	+	0.0634577i
$703$	4.31867			0.162882
$704$		−	2.18953i		−	0.0825212i
$705$	4.00000			0.150649
$706$	31.1372			1.17186
$707$		0			0
$708$			5.36266i			0.201541i
$709$		−	2.73578i		−	0.102744i	−0.998680	−	0.0513722i	$-0.983641\pi$
$709$		−	2.73578i		−	0.102744i	0.998680	−	0.0513722i	$-0.0163595\pi$
$710$			8.43424i			0.316531i
$711$	8.37907			0.314240
$712$	−3.01641			−0.113045
$713$			41.8297i			1.56654i
$714$		0			0
$715$	−6.18953	+	11.7417i	−0.231475	+	0.439116i
$716$	−9.10439			−0.340247
$717$			17.0716i			0.637550i
$718$	8.12080			0.303065
$719$	6.84612			0.255317			0.127659	−	0.991818i	$-0.459254\pi$
$719$	6.84612			0.255317			0.127659	+	0.991818i	$0.459254\pi$
$720$			1.68133i			0.0626595i
$721$		0			0
$722$		−	17.6238i		−	0.655889i
$723$			11.4506i			0.425854i
$724$	9.32985			0.346741
$725$	5.86253			0.217729
$726$			6.20594i			0.230324i
$727$	45.2528			1.67833			0.839166	−	0.543875i	$-0.183043\pi$
$727$	45.2528			1.67833			0.839166	+	0.543875i	$0.183043\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$			6.40665i			0.237121i
$731$	−54.1072			−2.00123
$732$	−0.697737			−0.0257891
$733$		−	32.7909i		−	1.21116i	−0.795784	−	0.605581i	$-0.792941\pi$
$733$		−	32.7909i		−	1.21116i	0.795784	−	0.605581i	$-0.207059\pi$
$734$		−	11.4835i		−	0.423862i
$735$		0			0
$736$			5.68133i			0.209417i
$737$	−33.0716			−1.21821
$738$	−9.36266			−0.344644
$739$		−	40.1760i		−	1.47790i	−0.673762	−	0.738948i	$-0.735323\pi$
$739$		−	40.1760i		−	1.47790i	0.673762	−	0.738948i	$-0.264677\pi$
$740$	−6.18953			−0.227532
$741$	−3.74173	−	1.97241i	−0.137456	−	0.0724585i
$742$		0			0
$743$			48.6758i			1.78574i	0.450311	+	0.892872i	$0.351313\pi$
$743$			48.6758i			1.78574i	−0.450311	+	0.892872i	$0.648687\pi$
$744$	−7.36266			−0.269928
$745$	39.4283			1.44454
$746$			11.3955i			0.417218i
$747$		−	3.01641i		−	0.110365i
$748$			9.17313i			0.335403i
$749$		0			0
$750$	−12.0604			−0.440383
$751$	−23.1044			−0.843091			−0.421546	−	0.906807i	$-0.638512\pi$
$751$	−23.1044			−0.843091			−0.421546	+	0.906807i	$0.638512\pi$
$752$			2.37907i			0.0867557i
$753$	−2.82687			−0.103017
$754$	−8.60453	−	4.53579i	−0.313358	−	0.165184i
$755$	16.8597			0.613587
$756$		0			0
$757$	−38.2968			−1.39192			−0.695960	−	0.718081i	$-0.745021\pi$
$757$	−38.2968			−1.39192			−0.695960	+	0.718081i	$0.745021\pi$
$758$	−14.3463			−0.521079
$759$			12.4395i			0.451524i
$760$		−	1.97241i		−	0.0715470i
$761$			30.2583i			1.09686i	0.836196	+	0.548431i	$0.184775\pi$
$761$			30.2583i			1.09686i	−0.836196	+	0.548431i	$0.815225\pi$
$762$			12.1208i			0.439090i
$763$		0			0
$764$	−6.43947			−0.232972
$765$		−	7.04399i		−	0.254676i
$766$	5.36789			0.193950
$767$	17.1044	+	9.01641i	0.617604	+	0.325564i
$768$	−1.00000			−0.0360844
$769$		−	40.1895i		−	1.44927i	−0.689132	−	0.724636i	$-0.742008\pi$
$769$		−	40.1895i		−	1.44927i	0.689132	−	0.724636i	$-0.257992\pi$
$770$		0			0
$771$	−4.03281			−0.145238
$772$		−	3.10439i		−	0.111729i
$773$			5.73650i			0.206328i	0.994664	+	0.103164i	$0.0328966\pi$
$773$			5.73650i			0.206328i	−0.994664	+	0.103164i	$0.967103\pi$
$774$		−	12.9149i		−	0.464215i
$775$		−	16.0000i		−	0.574737i
$776$	−2.00000			−0.0717958
$777$		0			0
$778$			14.4342i			0.517493i
$779$	10.9836			0.393528
$780$	5.36266	+	2.82687i	0.192014	+	0.101218i
$781$	−10.9836			−0.393024
$782$		−	23.8021i		−	0.851162i
$783$	−2.69774			−0.0964093
$784$		0			0
$785$			30.4207i			1.08576i
$786$			12.6566i			0.451445i
$787$			9.61782i			0.342838i	0.985198	+	0.171419i	$0.0548352\pi$
$787$			9.61782i			0.342838i	−0.985198	+	0.171419i	$0.945165\pi$
$788$			6.12080i			0.218044i
$789$	15.8297			0.563553
$790$	14.0880			0.501228
$791$		0			0
$792$	−2.18953			−0.0778017
$793$	−1.17313	+	2.22546i	−0.0416590	+	0.0790283i
$794$	12.6045			0.447318
$795$			5.07158i			0.179870i
$796$	−13.7141			−0.486085
$797$	3.58812			0.127098			0.0635488	−	0.997979i	$-0.479758\pi$
$797$	3.58812			0.127098			0.0635488	+	0.997979i	$0.479758\pi$
$798$		0			0
$799$		−	9.96719i		−	0.352614i
$800$		−	2.17313i		−	0.0768317i
$801$			3.01641i			0.106579i
$802$	4.72532			0.166857
$803$	−8.34314			−0.294423
$804$			15.1044i			0.532691i
$805$		0			0
$806$	−12.3791	+	23.4835i	−0.436034	+	0.827170i
$807$	18.7581			0.660318
$808$		−	9.36266i		−	0.329377i
$809$	48.4671			1.70401			0.852005	−	0.523533i	$-0.175386\pi$
$809$	48.4671			1.70401			0.852005	+	0.523533i	$0.175386\pi$
$810$	1.68133			0.0590759
$811$		−	33.7938i		−	1.18666i	−0.804959	−	0.593330i	$-0.797813\pi$
$811$		−	33.7938i		−	1.18666i	0.804959	−	0.593330i	$-0.202187\pi$
$812$		0			0
$813$		−	7.62093i		−	0.267278i
$814$		−	8.06040i		−	0.282517i
$815$	−16.8685			−0.590877
$816$	4.18953			0.146663
$817$			15.1508i			0.530058i
$818$	−31.6074			−1.10513
$819$		0			0
$820$	−15.7417			−0.549725
$821$		−	15.1372i		−	0.528292i	−0.964483	−	0.264146i	$-0.914910\pi$
$821$		−	15.1372i		−	0.528292i	0.964483	−	0.264146i	$-0.0850901\pi$
$822$	−9.58501			−0.334316
$823$	25.3955			0.885231			0.442615	−	0.896712i	$-0.354051\pi$
$823$	25.3955			0.885231			0.442615	+	0.896712i	$0.354051\pi$
$824$		−	12.4395i		−	0.433349i
$825$		−	4.75814i		−	0.165657i
$826$		0			0
$827$			26.4311i			0.919100i	0.888152	+	0.459550i	$0.151989\pi$
$827$			26.4311i			0.919100i	−0.888152	+	0.459550i	$0.848011\pi$
$828$	5.68133			0.197440
$829$	−4.57694			−0.158964			−0.0794818	−	0.996836i	$-0.525327\pi$
$829$	−4.57694			−0.158964			−0.0794818	+	0.996836i	$0.525327\pi$
$830$		−	5.07158i		−	0.176037i
$831$	−18.4342			−0.639477
$832$	−1.68133	+	3.18953i	−0.0582897	+	0.110577i
$833$		0			0
$834$			19.7417i			0.683600i
$835$	−7.84328			−0.271428
$836$	2.56860			0.0888370
$837$			7.36266i			0.254491i
$838$			21.9313i			0.757603i
$839$			56.5222i			1.95136i	0.219191	+	0.975682i	$0.429658\pi$
$839$			56.5222i			1.95136i	−0.219191	+	0.975682i	$0.570342\pi$
$840$		0			0
$841$	−21.7222			−0.749042
$842$	−15.4283			−0.531694
$843$			8.03281i			0.276665i
$844$	−3.46421			−0.119243
$845$	18.0328	−	12.3515i	0.620348	−	0.424904i
$846$	2.37907			0.0817940
$847$		0			0
$848$	−3.01641			−0.103584
$849$	−14.8461			−0.509518
$850$			9.10439i			0.312278i
$851$			20.9149i			0.716952i
$852$			5.01641i			0.171859i
$853$		−	34.3791i		−	1.17712i	−0.808455	−	0.588558i	$-0.799696\pi$
$853$		−	34.3791i		−	1.17712i	0.808455	−	0.588558i	$-0.200304\pi$
$854$		0			0
$855$	−1.97241			−0.0674552
$856$			13.1044i			0.447899i
$857$	−16.5494			−0.565315			−0.282658	−	0.959221i	$-0.591216\pi$
$857$	−16.5494			−0.565315			−0.282658	+	0.959221i	$0.591216\pi$
$858$	−3.68133	+	6.98359i	−0.125679	+	0.238416i
$859$	−29.1372			−0.994149			−0.497074	−	0.867708i	$-0.665592\pi$
$859$	−29.1372			−0.994149			−0.497074	+	0.867708i	$0.665592\pi$
$860$		−	21.7141i		−	0.740446i
$861$		0			0
$862$	−5.65375			−0.192567
$863$			10.9836i			0.373886i	0.982371	+	0.186943i	$0.0598579\pi$
$863$			10.9836i			0.373886i	−0.982371	+	0.186943i	$0.940142\pi$
$864$			1.00000i			0.0340207i
$865$			25.3403i			0.861596i
$866$		−	19.7417i		−	0.670851i
$867$	−0.552195			−0.0187535
$868$		0			0
$869$			18.3463i			0.622354i
$870$	−4.53579			−0.153778
$871$	48.1760	+	25.3955i	1.63238	+	0.860493i
$872$	−7.04399			−0.238540
$873$			2.00000i			0.0676897i
$874$	−6.66492			−0.225444
$875$		0			0
$876$			3.81047i			0.128744i
$877$			0.637339i			0.0215214i	0.999942	+	0.0107607i	$0.00342530\pi$
$877$			0.637339i			0.0215214i	−0.999942	+	0.0107607i	$0.996575\pi$
$878$		−	19.7365i		−	0.666075i
$879$		−	21.7417i		−	0.733330i
$880$	−3.68133			−0.124098
$881$	27.4699			0.925484			0.462742	−	0.886493i	$-0.346866\pi$
$881$	27.4699			0.925484			0.462742	+	0.886493i	$0.346866\pi$
$882$		0			0
$883$	−18.6342			−0.627092			−0.313546	−	0.949573i	$-0.601517\pi$
$883$	−18.6342			−0.627092			−0.313546	+	0.949573i	$0.601517\pi$
$884$	7.04399	−	13.3627i	0.236915	−	0.449435i
$885$	9.01641			0.303083
$886$		−	22.4342i		−	0.753693i
$887$	31.3627			1.05306			0.526528	−	0.850158i	$-0.323494\pi$
$887$	31.3627			1.05306			0.526528	+	0.850158i	$0.323494\pi$
$888$	−3.68133			−0.123537
$889$		0			0
$890$			5.07158i			0.170000i
$891$			2.18953i			0.0733521i
$892$			0.258271i			0.00864754i
$893$	−2.79095			−0.0933956
$894$	23.4506			0.784307
$895$			15.3075i			0.511673i
$896$		0			0
$897$	9.55220	−	18.1208i	0.318939	−	0.605036i
$898$	−0.447805			−0.0149434
$899$		−	19.8625i		−	0.662452i
$900$	−2.17313			−0.0724376
$901$	12.6373			0.421011
$902$		−	20.4999i		−	0.682571i
$903$		0			0
$904$			7.39547i			0.245970i
$905$		−	15.6866i		−	0.521439i
$906$	10.0276			0.333144
$907$	−34.7253			−1.15304			−0.576518	−	0.817085i	$-0.695589\pi$
$907$	−34.7253			−1.15304			−0.576518	+	0.817085i	$0.695589\pi$
$908$			2.25827i			0.0749433i
$909$	−9.36266			−0.310540
$910$		0			0
$911$	−28.4723			−0.943329			−0.471664	−	0.881778i	$-0.656347\pi$
$911$	−28.4723			−0.943329			−0.471664	+	0.881778i	$0.656347\pi$
$912$		−	1.17313i		−	0.0388461i
$913$	6.60453			0.218578
$914$	−12.6373			−0.418006
$915$			1.17313i			0.0387824i
$916$			18.3791i			0.607262i
$917$		0			0
$918$		−	4.18953i		−	0.138275i
$919$	18.3463			0.605187			0.302594	−	0.953120i	$-0.402148\pi$
$919$	18.3463			0.605187			0.302594	+	0.953120i	$0.402148\pi$
$920$	9.55220			0.314927
$921$			27.4835i			0.905611i
$922$	38.5603			1.26991
$923$	16.0000	+	8.43424i	0.526646	+	0.277616i
$924$		0			0
$925$		−	8.00000i		−	0.263038i
$926$	40.1156			1.31828
$927$	−12.4395			−0.408566
$928$		−	2.69774i		−	0.0885576i
$929$			3.96719i			0.130159i	0.997880	+	0.0650796i	$0.0207301\pi$
$929$			3.96719i			0.130159i	−0.997880	+	0.0650796i	$0.979270\pi$
$930$			12.3791i			0.405926i
$931$		0			0
$932$	−17.7417			−0.581150
$933$	6.29108			0.205961
$934$		−	37.7282i		−	1.23450i
$935$	15.4231			0.504388
$936$	3.18953	+	1.68133i	0.104253	+	0.0549560i
$937$	40.6373			1.32756			0.663782	−	0.747926i	$-0.268950\pi$
$937$	40.6373			1.32756			0.663782	+	0.747926i	$0.268950\pi$
$938$		0			0
$939$	32.4342			1.05845
$940$	4.00000			0.130466
$941$		−	25.7417i		−	0.839156i	−0.907719	−	0.419578i	$-0.862178\pi$
$941$		−	25.7417i		−	0.839156i	0.907719	−	0.419578i	$-0.137822\pi$
$942$			18.0932i			0.589509i
$943$			53.1924i			1.73218i
$944$			5.36266i			0.174540i
$945$		0			0
$946$	28.2775			0.919381
$947$		−	41.2283i		−	1.33974i	−0.742479	−	0.669870i	$-0.766350\pi$
$947$		−	41.2283i		−	1.33974i	0.742479	−	0.669870i	$-0.233650\pi$
$948$	8.37907			0.272139
$949$	12.1536	+	6.40665i	0.394523	+	0.207969i
$950$	2.54935			0.0827120
$951$		−	29.7417i		−	0.964442i
$952$		0			0
$953$	−5.04922			−0.163560			−0.0817801	−	0.996650i	$-0.526061\pi$
$953$	−5.04922			−0.163560			−0.0817801	+	0.996650i	$0.526061\pi$
$954$			3.01641i			0.0976598i
$955$			10.8269i			0.350349i
$956$			17.0716i			0.552134i
$957$		−	5.90679i		−	0.190939i
$958$	−23.4559			−0.757825
$959$		0			0
$960$			1.68133i			0.0542647i
$961$	−23.2088			−0.748670
$962$	−6.18953	+	11.7417i	−0.199559	+	0.378569i
$963$	13.1044			0.422283
$964$			11.4506i			0.368800i
$965$	−5.21951			−0.168022
$966$		0			0
$967$			3.48869i			0.112189i	0.998425	+	0.0560943i	$0.0178647\pi$
$967$			3.48869i			0.112189i	−0.998425	+	0.0560943i	$0.982135\pi$
$968$			6.20594i			0.199466i
$969$			4.91486i			0.157888i
$970$			3.36266i			0.107969i
$971$	−37.3850			−1.19974			−0.599871	−	0.800097i	$-0.704782\pi$
$971$	−37.3850			−1.19974			−0.599871	+	0.800097i	$0.704782\pi$
$972$	1.00000			0.0320750
$973$		0			0
$974$	11.4835			0.367954
$975$	−3.65375	+	6.93126i	−0.117014	+	0.221978i
$976$	−0.697737			−0.0223340
$977$			6.92531i			0.221560i	0.993845	+	0.110780i	$0.0353350\pi$
$977$			6.92531i			0.221560i	−0.993845	+	0.110780i	$0.964665\pi$
$978$	−10.0328			−0.320814
$979$	−6.60453			−0.211082
$980$		0			0
$981$			7.04399i			0.224897i
$982$			17.4283i			0.556159i
$983$		−	25.6647i		−	0.818575i	−0.912405	−	0.409288i	$-0.865777\pi$
$983$		−	25.6647i		−	0.818575i	0.912405	−	0.409288i	$-0.134223\pi$
$984$	−9.36266			−0.298471
$985$	10.2911			0.327901
$986$			11.3023i			0.359937i
$987$		0			0
$988$	−3.74173	−	1.97241i	−0.119040	−	0.0627509i
$989$	−73.3736			−2.33314
$990$			3.68133i			0.117000i
$991$	1.96719			0.0624897			0.0312449	−	0.999512i	$-0.490053\pi$
$991$	1.96719			0.0624897			0.0312449	+	0.999512i	$0.490053\pi$
$992$	−7.36266			−0.233765
$993$		−	25.3955i		−	0.805901i
$994$		0			0
$995$			23.0580i			0.730988i
$996$		−	3.01641i		−	0.0955785i
$997$	15.6043			0.494192			0.247096	−	0.968991i	$-0.420524\pi$
$997$	15.6043			0.494192			0.247096	+	0.968991i	$0.420524\pi$
$998$	32.2416			1.02059
$999$			3.68133i			0.116472i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.c.i.883.3	✓	6
7.6	odd	2		3822.2.c.l.883.1	yes	6
13.12	even	2	inner	3822.2.c.i.883.4	yes	6
91.90	odd	2		3822.2.c.l.883.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.c.i.883.3	✓	6	1.1	even	1	trivial
3822.2.c.i.883.4	yes	6	13.12	even	2	inner
3822.2.c.l.883.1	yes	6	7.6	odd	2
3822.2.c.l.883.6	yes	6	91.90	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.68133i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +1.68133 q^{10} +2.18953i q^{11} +1.00000 q^{12} +(1.68133 - 3.18953i) q^{13} -1.68133i q^{15} +1.00000 q^{16} -4.18953 q^{17} -1.00000i q^{18} +1.17313i q^{19} -1.68133i q^{20} +2.18953 q^{22} -5.68133 q^{23} -1.00000i q^{24} +2.17313 q^{25} +(-3.18953 - 1.68133i) q^{26} -1.00000 q^{27} +2.69774 q^{29} -1.68133 q^{30} -7.36266i q^{31} -1.00000i q^{32} -2.18953i q^{33} +4.18953i q^{34} -1.00000 q^{36} -3.68133i q^{37} +1.17313 q^{38} +(-1.68133 + 3.18953i) q^{39} -1.68133 q^{40} -9.36266i q^{41} +12.9149 q^{43} -2.18953i q^{44} +1.68133i q^{45} +5.68133i q^{46} +2.37907i q^{47} -1.00000 q^{48} -2.17313i q^{50} +4.18953 q^{51} +(-1.68133 + 3.18953i) q^{52} -3.01641 q^{53} +1.00000i q^{54} -3.68133 q^{55} -1.17313i q^{57} -2.69774i q^{58} +5.36266i q^{59} +1.68133i q^{60} -0.697737 q^{61} -7.36266 q^{62} -1.00000 q^{64} +(5.36266 + 2.82687i) q^{65} -2.18953 q^{66} +15.1044i q^{67} +4.18953 q^{68} +5.68133 q^{69} +5.01641i q^{71} +1.00000i q^{72} +3.81047i q^{73} -3.68133 q^{74} -2.17313 q^{75} -1.17313i q^{76} +(3.18953 + 1.68133i) q^{78} +8.37907 q^{79} +1.68133i q^{80} +1.00000 q^{81} -9.36266 q^{82} -3.01641i q^{83} -7.04399i q^{85} -12.9149i q^{86} -2.69774 q^{87} -2.18953 q^{88} +3.01641i q^{89} +1.68133 q^{90} +5.68133 q^{92} +7.36266i q^{93} +2.37907 q^{94} -1.97241 q^{95} +1.00000i q^{96} +2.00000i q^{97} +2.18953i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 4 q^{10} + 6 q^{12} - 4 q^{13} + 6 q^{16} - 8 q^{17} - 4 q^{22} - 20 q^{23} + 2 q^{25} - 2 q^{26} - 6 q^{27} - 4 q^{29} + 4 q^{30} - 6 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40}+ \cdots - 48 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 4 * q^10 + 6 * q^12 - 4 * q^13 + 6 * q^16 - 8 * q^17 - 4 * q^22 - 20 * q^23 + 2 * q^25 - 2 * q^26 - 6 * q^27 - 4 * q^29 + 4 * q^30 - 6 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^40 + 4 * q^43 - 6 * q^48 + 8 * q^51 + 4 * q^52 - 12 * q^53 - 8 * q^55 + 16 * q^61 - 16 * q^62 - 6 * q^64 + 4 * q^65 + 4 * q^66 + 8 * q^68 + 20 * q^69 - 8 * q^74 - 2 * q^75 + 2 * q^78 + 16 * q^79 + 6 * q^81 - 28 * q^82 + 4 * q^87 + 4 * q^88 - 4 * q^90 + 20 * q^92 - 20 * q^94 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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