LMFDB - Embedded newform 475.2.j.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(49,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("475.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.j (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	475.49
Dual form			475.2.j.a.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 + 1.00000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +4.00000i q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(1.73205 + 1.00000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +4.00000i q^{7} +(0.500000 + 0.866025i) q^{9} +3.00000 q^{11} -4.00000i q^{12} +(1.73205 - 1.00000i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(5.19615 + 3.00000i) q^{17} +(3.50000 + 2.59808i) q^{19} +(-4.00000 + 6.92820i) q^{21} -4.00000i q^{27} +(6.92820 - 4.00000i) q^{28} +(-1.50000 - 2.59808i) q^{29} -7.00000 q^{31} +(5.19615 + 3.00000i) q^{33} +(1.00000 - 1.73205i) q^{36} -8.00000i q^{37} +4.00000 q^{39} +(3.00000 - 5.19615i) q^{41} +(3.46410 + 2.00000i) q^{43} +(-3.00000 - 5.19615i) q^{44} +(-5.19615 + 3.00000i) q^{47} +(-6.92820 + 4.00000i) q^{48} -9.00000 q^{49} +(6.00000 + 10.3923i) q^{51} +(-3.46410 - 2.00000i) q^{52} +(-5.19615 + 3.00000i) q^{53} +(3.46410 + 8.00000i) q^{57} +(-7.50000 + 12.9904i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-3.46410 + 2.00000i) q^{63} +8.00000 q^{64} +(-1.73205 + 1.00000i) q^{67} -12.0000i q^{68} +(1.50000 - 2.59808i) q^{71} +(-6.92820 - 4.00000i) q^{73} +(1.00000 - 8.66025i) q^{76} +12.0000i q^{77} +(2.50000 - 4.33013i) q^{79} +(5.50000 - 9.52628i) q^{81} +12.0000i q^{83} +16.0000 q^{84} -6.00000i q^{87} +(-7.50000 - 12.9904i) q^{89} +(4.00000 + 6.92820i) q^{91} +(-12.1244 - 7.00000i) q^{93} +(6.92820 + 4.00000i) q^{97} +(1.50000 + 2.59808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 2 * q^9	$ 4 q - 4 q^{4} + 2 q^{9} + 12 q^{11} - 8 q^{16} + 14 q^{19} - 16 q^{21} - 6 q^{29} - 28 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 12 q^{44} - 36 q^{49} + 24 q^{51} - 30 q^{59} - 10 q^{61} + 32 q^{64} + 6 q^{71} + 4 q^{76} + 10 q^{79} + 22 q^{81} + 64 q^{84} - 30 q^{89} + 16 q^{91} + 6 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^9 + 12 * q^11 - 8 * q^16 + 14 * q^19 - 16 * q^21 - 6 * q^29 - 28 * q^31 + 4 * q^36 + 16 * q^39 + 12 * q^41 - 12 * q^44 - 36 * q^49 + 24 * q^51 - 30 * q^59 - 10 * q^61 + 32 * q^64 + 6 * q^71 + 4 * q^76 + 10 * q^79 + 22 * q^81 + 64 * q^84 - 30 * q^89 + 16 * q^91 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	1.73205	+	1.00000i	1.00000	+	0.577350i	0.908248	−	0.418432i	$-0.137420\pi$
$3$	1.73205	+	1.00000i	1.00000	+	0.577350i	0.0917517	+	0.995782i	$0.470753\pi$
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	0.500000	+	0.866025i	0.166667	+	0.288675i
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		−	4.00000i		−	1.15470i
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	0.720577	+	0.693375i	$0.243877\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	5.19615	+	3.00000i	1.26025	+	0.727607i	0.973123	−	0.230285i	$-0.0739659\pi$
$17$	5.19615	+	3.00000i	1.26025	+	0.727607i	0.287129	+	0.957892i	$0.407299\pi$
$18$		0			0
$19$	3.50000	+	2.59808i	0.802955	+	0.596040i
$19$	3.50000	+	2.59808i	0.802955	+	0.596040i
$20$		0			0
$21$	−4.00000	+	6.92820i	−0.872872	+	1.51186i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$	6.92820	−	4.00000i	1.30931	−	0.755929i
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$	5.19615	+	3.00000i	0.904534	+	0.522233i
$34$		0			0
$35$		0			0
$36$	1.00000	−	1.73205i	0.166667	−	0.288675i
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		0			0
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	0.740312	−	0.672264i	$-0.234678\pi$
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	−0.212041	+	0.977261i	$0.568011\pi$
$44$	−3.00000	−	5.19615i	−0.452267	−	0.783349i
$45$		0			0
$46$		0			0
$47$	−5.19615	+	3.00000i	−0.757937	+	0.437595i	−0.828554	−	0.559908i	$-0.810836\pi$
$47$	−5.19615	+	3.00000i	−0.757937	+	0.437595i	0.0706177	+	0.997503i	$0.477503\pi$
$48$	−6.92820	+	4.00000i	−1.00000	+	0.577350i
$49$	−9.00000			−1.28571
$50$		0			0
$51$	6.00000	+	10.3923i	0.840168	+	1.45521i
$52$	−3.46410	−	2.00000i	−0.480384	−	0.277350i
$53$	−5.19615	+	3.00000i	−0.713746	+	0.412082i	−0.812447	−	0.583036i	$-0.801865\pi$
$53$	−5.19615	+	3.00000i	−0.713746	+	0.412082i	0.0987002	+	0.995117i	$0.468532\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.46410	+	8.00000i	0.458831	+	1.05963i
$58$		0			0
$59$	−7.50000	+	12.9904i	−0.976417	+	1.69120i	−0.301239	+	0.953549i	$0.597400\pi$
$59$	−7.50000	+	12.9904i	−0.976417	+	1.69120i	−0.675178	+	0.737655i	$0.735933\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$	−3.46410	+	2.00000i	−0.436436	+	0.251976i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−1.73205	+	1.00000i	−0.211604	+	0.122169i	−0.602056	−	0.798454i	$-0.705652\pi$
$67$	−1.73205	+	1.00000i	−0.211604	+	0.122169i	0.390453	+	0.920623i	$0.372318\pi$
$68$		−	12.0000i		−	1.45521i
$69$		0			0
$70$		0			0
$71$	1.50000	−	2.59808i	0.178017	−	0.308335i	−0.763184	−	0.646181i	$-0.776365\pi$
$71$	1.50000	−	2.59808i	0.178017	−	0.308335i	0.941201	+	0.337846i	$0.109698\pi$
$72$		0			0
$73$	−6.92820	−	4.00000i	−0.810885	−	0.468165i	0.0363782	−	0.999338i	$-0.488418\pi$
$73$	−6.92820	−	4.00000i	−0.810885	−	0.468165i	−0.847263	+	0.531174i	$0.821751\pi$
$74$		0			0
$75$		0			0
$76$	1.00000	−	8.66025i	0.114708	−	0.993399i
$77$			12.0000i			1.36753i
$78$		0			0
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	−0.690426	−	0.723403i	$-0.742577\pi$
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	0.971698	+	0.236225i	$0.0759104\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	16.0000			1.74574
$85$		0			0
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	−7.50000	−	12.9904i	−0.794998	−	1.37698i	−0.922840	−	0.385183i	$-0.874138\pi$
$89$	−7.50000	−	12.9904i	−0.794998	−	1.37698i	0.127842	−	0.991795i	$-0.459195\pi$
$90$		0			0
$91$	4.00000	+	6.92820i	0.419314	+	0.726273i
$92$		0			0
$93$	−12.1244	−	7.00000i	−1.25724	−	0.725866i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.92820	+	4.00000i	0.703452	+	0.406138i	0.808632	−	0.588315i	$-0.200208\pi$
$97$	6.92820	+	4.00000i	0.703452	+	0.406138i	−0.105180	+	0.994453i	$0.533542\pi$
$98$		0			0
$99$	1.50000	+	2.59808i	0.150756	+	0.261116i
$100$		0			0
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−6.92820	+	4.00000i	−0.666667	+	0.384900i
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	−0.472708	−	0.881219i	$-0.656723\pi$
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	0.999512	−	0.0312328i	$-0.00994332\pi$
$110$		0			0
$111$	8.00000	−	13.8564i	0.759326	−	1.31519i
$112$	−13.8564	−	8.00000i	−1.30931	−	0.755929i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$	−3.00000	+	5.19615i	−0.278543	+	0.482451i
$117$	1.73205	+	1.00000i	0.160128	+	0.0924500i
$118$		0			0
$119$	−12.0000	+	20.7846i	−1.10004	+	1.90532i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	10.3923	−	6.00000i	0.937043	−	0.541002i
$124$	7.00000	+	12.1244i	0.628619	+	1.08880i
$125$		0			0
$126$		0			0
$127$	−1.73205	+	1.00000i	−0.153695	+	0.0887357i	−0.574875	−	0.818241i	$-0.694949\pi$
$127$	−1.73205	+	1.00000i	−0.153695	+	0.0887357i	0.421180	+	0.906977i	$0.361616\pi$
$128$		0			0
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		−	12.0000i		−	1.04447i
$133$	−10.3923	+	14.0000i	−0.901127	+	1.21395i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.0146279	−	0.999893i	$-0.495344\pi$
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.873247	+	0.487278i	$0.162010\pi$
$138$		0			0
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$	−12.0000			−1.01058
$142$		0			0
$143$	5.19615	−	3.00000i	0.434524	−	0.250873i
$144$	−4.00000			−0.333333
$145$		0			0
$146$		0			0
$147$	−15.5885	−	9.00000i	−1.28571	−	0.742307i
$148$	−13.8564	+	8.00000i	−1.13899	+	0.657596i
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	−0.798019	−	0.602632i	$-0.794119\pi$
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	0.920904	+	0.389789i	$0.127452\pi$
$150$		0			0
$151$	17.0000			1.38344			0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000			1.38344			0.691720	+	0.722166i	$0.256853\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$	−4.00000	−	6.92820i	−0.320256	−	0.554700i
$157$	1.73205	+	1.00000i	0.138233	+	0.0798087i	0.567521	−	0.823359i	$-0.307902\pi$
$157$	1.73205	+	1.00000i	0.138233	+	0.0798087i	−0.429289	+	0.903167i	$0.641236\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−4.50000	+	7.79423i	−0.346154	+	0.599556i
$170$		0			0
$171$	−0.500000	+	4.33013i	−0.0382360	+	0.331133i
$172$		−	8.00000i		−	0.609994i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	−6.00000	+	10.3923i	−0.452267	+	0.783349i
$177$	−25.9808	+	15.0000i	−1.95283	+	1.12747i
$178$		0			0
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	0.826465	−	0.562988i	$-0.190348\pi$
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	−0.900794	+	0.434246i	$0.857015\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	15.5885	+	9.00000i	1.13994	+	0.658145i
$188$	10.3923	+	6.00000i	0.757937	+	0.437595i
$189$	16.0000			1.16383
$190$		0			0
$191$	−15.0000			−1.08536			−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000			−1.08536			−0.542681	+	0.839939i	$0.682591\pi$
$192$	13.8564	+	8.00000i	1.00000	+	0.577350i
$193$	13.8564	+	8.00000i	0.997406	+	0.575853i	0.907480	−	0.420096i	$-0.138004\pi$
$193$	13.8564	+	8.00000i	0.997406	+	0.575853i	0.0899262	+	0.995948i	$0.471337\pi$
$194$		0			0
$195$		0			0
$196$	9.00000	+	15.5885i	0.642857	+	1.11346i
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−9.50000	−	16.4545i	−0.673437	−	1.16643i	−0.976923	−	0.213591i	$-0.931484\pi$
$199$	−9.50000	−	16.4545i	−0.673437	−	1.16643i	0.303486	−	0.952836i	$-0.401849\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$	10.3923	−	6.00000i	0.729397	−	0.421117i
$204$	12.0000	−	20.7846i	0.840168	−	1.45521i
$205$		0			0
$206$		0			0
$207$		0			0
$208$			8.00000i			0.554700i
$209$	10.5000	+	7.79423i	0.726300	+	0.539138i
$210$		0			0
$211$	3.50000	−	6.06218i	0.240950	−	0.417338i	−0.720035	−	0.693938i	$-0.755874\pi$
$211$	3.50000	−	6.06218i	0.240950	−	0.417338i	0.960985	+	0.276600i	$0.0892077\pi$
$212$	10.3923	+	6.00000i	0.713746	+	0.412082i
$213$	5.19615	−	3.00000i	0.356034	−	0.205557i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	28.0000i		−	1.90076i
$218$		0			0
$219$	−8.00000	−	13.8564i	−0.540590	−	0.936329i
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$	3.46410	+	2.00000i	0.231973	+	0.133930i	0.611482	−	0.791258i	$-0.290574\pi$
$223$	3.46410	+	2.00000i	0.231973	+	0.133930i	−0.379509	+	0.925188i	$0.623907\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			24.0000i			1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$			24.0000i			1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	10.3923	−	14.0000i	0.688247	−	0.927173i
$229$	7.00000			0.462573			0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000			0.462573			0.231287	+	0.972886i	$0.425707\pi$
$230$		0			0
$231$	−12.0000	+	20.7846i	−0.789542	+	1.36753i
$232$		0			0
$233$	−5.19615	−	3.00000i	−0.340411	−	0.196537i	0.320043	−	0.947403i	$-0.396303\pi$
$233$	−5.19615	−	3.00000i	−0.340411	−	0.196537i	−0.660454	+	0.750867i	$0.729636\pi$
$234$		0			0
$235$		0			0
$236$	30.0000			1.95283
$237$	8.66025	−	5.00000i	0.562544	−	0.324785i
$238$		0			0
$239$	3.00000			0.194054			0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000			0.194054			0.0970269	+	0.995282i	$0.469067\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$		0			0
$243$	8.66025	−	5.00000i	0.555556	−	0.320750i
$244$	−5.00000	+	8.66025i	−0.320092	+	0.554416i
$245$		0			0
$246$		0			0
$247$	8.66025	+	1.00000i	0.551039	+	0.0636285i
$248$		0			0
$249$	−12.0000	+	20.7846i	−0.760469	+	1.31717i
$250$		0			0
$251$	7.50000	+	12.9904i	0.473396	+	0.819946i	0.999536	−	0.0304521i	$-0.00969471\pi$
$251$	7.50000	+	12.9904i	0.473396	+	0.819946i	−0.526140	+	0.850398i	$0.676361\pi$
$252$	6.92820	+	4.00000i	0.436436	+	0.251976i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	15.5885	−	9.00000i	0.972381	−	0.561405i	0.0724199	−	0.997374i	$-0.476928\pi$
$257$	15.5885	−	9.00000i	0.972381	−	0.561405i	0.899961	+	0.435970i	$0.143595\pi$
$258$		0			0
$259$	32.0000			1.98838
$260$		0			0
$261$	1.50000	−	2.59808i	0.0928477	−	0.160817i
$262$		0			0
$263$	15.5885	+	9.00000i	0.961225	+	0.554964i	0.896550	−	0.442943i	$-0.146065\pi$
$263$	15.5885	+	9.00000i	0.961225	+	0.554964i	0.0646755	+	0.997906i	$0.479399\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	30.0000i		−	1.83597i
$268$	3.46410	+	2.00000i	0.211604	+	0.122169i
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	0.345192	+	0.938532i	$0.387814\pi$
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	−0.985389	+	0.170321i	$0.945520\pi$
$270$		0			0
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	−0.983312	−	0.181928i	$-0.941766\pi$
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	0.649211	+	0.760609i	$0.275099\pi$
$272$	−20.7846	+	12.0000i	−1.26025	+	0.727607i
$273$			16.0000i			0.968364i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$	−3.50000	−	6.06218i	−0.209540	−	0.362933i
$280$		0			0
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	0.762561	−	0.646916i	$-0.223942\pi$
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	−0.941526	+	0.336939i	$0.890608\pi$
$282$		0			0
$283$	−12.1244	−	7.00000i	−0.720718	−	0.416107i	0.0942988	−	0.995544i	$-0.469939\pi$
$283$	−12.1244	−	7.00000i	−0.720718	−	0.416107i	−0.815017	+	0.579437i	$0.803272\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$		0			0
$287$	20.7846	+	12.0000i	1.22688	+	0.708338i
$288$		0			0
$289$	9.50000	+	16.4545i	0.558824	+	0.967911i
$290$		0			0
$291$	8.00000	+	13.8564i	0.468968	+	0.812277i
$292$			16.0000i			0.936329i
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	12.0000i		−	0.696311i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000	+	13.8564i	−0.461112	+	0.798670i
$302$		0			0
$303$		−	30.0000i		−	1.72345i
$304$	−16.0000	+	6.92820i	−0.917663	+	0.397360i
$305$		0			0
$306$		0			0
$307$	−29.4449	−	17.0000i	−1.68051	−	0.970241i	−0.961324	−	0.275421i	$-0.911183\pi$
$307$	−29.4449	−	17.0000i	−1.68051	−	0.970241i	−0.719183	−	0.694820i	$-0.755484\pi$
$308$	20.7846	−	12.0000i	1.18431	−	0.683763i
$309$	16.0000	−	27.7128i	0.910208	−	1.57653i
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	−8.66025	+	5.00000i	−0.489506	+	0.282617i	−0.724370	−	0.689412i	$-0.757869\pi$
$313$	−8.66025	+	5.00000i	−0.489506	+	0.282617i	0.234863	+	0.972028i	$0.424536\pi$
$314$		0			0
$315$		0			0
$316$	−10.0000			−0.562544
$317$	−20.7846	+	12.0000i	−1.16738	+	0.673987i	−0.953062	−	0.302777i	$-0.902086\pi$
$317$	−20.7846	+	12.0000i	−1.16738	+	0.673987i	−0.214318	+	0.976764i	$0.568753\pi$
$318$		0			0
$319$	−4.50000	−	7.79423i	−0.251952	−	0.436393i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.3923	+	24.0000i	0.578243	+	1.33540i
$324$	−22.0000			−1.22222
$325$		0			0
$326$		0			0
$327$	19.0526	−	11.0000i	1.05361	−	0.608301i
$328$		0			0
$329$	−12.0000	−	20.7846i	−0.661581	−	1.14589i
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$	20.7846	−	12.0000i	1.14070	−	0.658586i
$333$	6.92820	−	4.00000i	0.379663	−	0.219199i
$334$		0			0
$335$		0			0
$336$	−16.0000	−	27.7128i	−0.872872	−	1.51186i
$337$	−13.8564	−	8.00000i	−0.754807	−	0.435788i	0.0726214	−	0.997360i	$-0.476864\pi$
$337$	−13.8564	−	8.00000i	−0.754807	−	0.435788i	−0.827428	+	0.561572i	$0.810197\pi$
$338$		0			0
$339$	−6.00000	+	10.3923i	−0.325875	+	0.564433i
$340$		0			0
$341$	−21.0000			−1.13721
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.3923	+	6.00000i	0.557888	+	0.322097i	0.752297	−	0.658824i	$-0.228946\pi$
$347$	10.3923	+	6.00000i	0.557888	+	0.322097i	−0.194409	+	0.980921i	$0.562279\pi$
$348$	−10.3923	+	6.00000i	−0.557086	+	0.321634i
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−4.00000	−	6.92820i	−0.213504	−	0.369800i
$352$		0			0
$353$			12.0000i			0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$			12.0000i			0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$		0			0
$355$		0			0
$356$	−15.0000	+	25.9808i	−0.794998	+	1.37698i
$357$	−41.5692	+	24.0000i	−2.20008	+	1.27021i
$358$		0			0
$359$	12.0000	−	20.7846i	0.633336	−	1.09697i	−0.353529	−	0.935423i	$-0.615019\pi$
$359$	12.0000	−	20.7846i	0.633336	−	1.09697i	0.986865	−	0.161546i	$-0.0516481\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$	−3.46410	−	2.00000i	−0.181818	−	0.104973i
$364$	8.00000	−	13.8564i	0.419314	−	0.726273i
$365$		0			0
$366$		0			0
$367$	3.46410	−	2.00000i	0.180825	−	0.104399i	−0.406855	−	0.913493i	$-0.633375\pi$
$367$	3.46410	−	2.00000i	0.180825	−	0.104399i	0.587680	+	0.809093i	$0.300041\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−12.0000	−	20.7846i	−0.623009	−	1.07908i
$372$			28.0000i			1.45173i
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.19615	−	3.00000i	−0.267615	−	0.154508i
$378$		0			0
$379$	37.0000			1.90056			0.950281	−	0.311393i	$-0.100796\pi$
$379$	37.0000			1.90056			0.950281	+	0.311393i	$0.100796\pi$
$380$		0			0
$381$	−4.00000			−0.204926
$382$		0			0
$383$	−25.9808	−	15.0000i	−1.32755	−	0.766464i	−0.342634	−	0.939469i	$-0.611319\pi$
$383$	−25.9808	−	15.0000i	−1.32755	−	0.766464i	−0.984921	+	0.173005i	$0.944652\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			4.00000i			0.203331i
$388$		−	16.0000i		−	0.812277i
$389$	7.50000	+	12.9904i	0.380265	+	0.658638i	0.991100	−	0.133120i	$-0.0424994\pi$
$389$	7.50000	+	12.9904i	0.380265	+	0.658638i	−0.610835	+	0.791758i	$0.709166\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−20.7846	+	12.0000i	−1.04844	+	0.605320i
$394$		0			0
$395$		0			0
$396$	3.00000	−	5.19615i	0.150756	−	0.261116i
$397$	6.92820	+	4.00000i	0.347717	+	0.200754i	0.663679	−	0.748017i	$-0.268994\pi$
$397$	6.92820	+	4.00000i	0.347717	+	0.200754i	−0.315963	+	0.948772i	$0.602327\pi$
$398$		0			0
$399$	−32.0000	+	13.8564i	−1.60200	+	0.693688i
$400$		0			0
$401$	16.5000	−	28.5788i	0.823971	−	1.42716i	−0.0787327	−	0.996896i	$-0.525087\pi$
$401$	16.5000	−	28.5788i	0.823971	−	1.42716i	0.902703	−	0.430263i	$-0.141579\pi$
$402$		0			0
$403$	−12.1244	+	7.00000i	−0.603957	+	0.348695i
$404$	−15.0000	+	25.9808i	−0.746278	+	1.29259i
$405$		0			0
$406$		0			0
$407$		−	24.0000i		−	1.18964i
$408$		0			0
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	0.996701	+	0.0811615i	$0.0258630\pi$
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	−0.428063	+	0.903749i	$0.640804\pi$
$410$		0			0
$411$	24.0000			1.18383
$412$	−27.7128	+	16.0000i	−1.36531	+	0.788263i
$413$	−51.9615	−	30.0000i	−2.55686	−	1.47620i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	32.0000i		−	1.56705i
$418$		0			0
$419$	3.00000			0.146560			0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000			0.146560			0.0732798	+	0.997311i	$0.476653\pi$
$420$		0			0
$421$	9.50000	−	16.4545i	0.463002	−	0.801942i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	9.50000	−	16.4545i	0.463002	−	0.801942i	0.999109	+	0.0422075i	$0.0134391\pi$
$422$		0			0
$423$	−5.19615	−	3.00000i	−0.252646	−	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.3205	−	10.0000i	0.838198	−	0.483934i
$428$		0			0
$429$	12.0000			0.579365
$430$		0			0
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	0.988169	−	0.153370i	$-0.0490126\pi$
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	−0.626907	+	0.779094i	$0.715679\pi$
$432$	13.8564	+	8.00000i	0.666667	+	0.384900i
$433$	6.92820	−	4.00000i	0.332948	−	0.192228i	−0.324201	−	0.945988i	$-0.605095\pi$
$433$	6.92820	−	4.00000i	0.332948	−	0.192228i	0.657149	+	0.753760i	$0.271762\pi$
$434$		0			0
$435$		0			0
$436$	−22.0000			−1.05361
$437$		0			0
$438$		0			0
$439$	−6.50000	+	11.2583i	−0.310228	+	0.537331i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−6.50000	+	11.2583i	−0.310228	+	0.537331i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−4.50000	−	7.79423i	−0.214286	−	0.371154i
$442$		0			0
$443$	−10.3923	+	6.00000i	−0.493753	+	0.285069i	−0.726130	−	0.687557i	$-0.758683\pi$
$443$	−10.3923	+	6.00000i	−0.493753	+	0.285069i	0.232377	+	0.972626i	$0.425350\pi$
$444$	−32.0000			−1.51865
$445$		0			0
$446$		0			0
$447$	5.19615	−	3.00000i	0.245770	−	0.141895i
$448$			32.0000i			1.51186i
$449$	9.00000			0.424736			0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000			0.424736			0.212368	+	0.977190i	$0.431882\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$	10.3923	−	6.00000i	0.488813	−	0.282216i
$453$	29.4449	+	17.0000i	1.38344	+	0.798730i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$	12.0000	−	20.7846i	0.560112	−	0.970143i
$460$		0			0
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	−0.741998	−	0.670402i	$-0.766122\pi$
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	0.951584	+	0.307388i	$0.0994551\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	12.0000			0.557086
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		−	4.00000i		−	0.184900i
$469$	−4.00000	−	6.92820i	−0.184703	−	0.319915i
$470$		0			0
$471$	2.00000	+	3.46410i	0.0921551	+	0.159617i
$472$		0			0
$473$	10.3923	+	6.00000i	0.477839	+	0.275880i
$474$		0			0
$475$		0			0
$476$	48.0000			2.20008
$477$	−5.19615	−	3.00000i	−0.237915	−	0.137361i
$478$		0			0
$479$	7.50000	+	12.9904i	0.342684	+	0.593546i	0.984930	−	0.172953i	$-0.0553307\pi$
$479$	7.50000	+	12.9904i	0.342684	+	0.593546i	−0.642246	+	0.766498i	$0.721997\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$		0			0
$483$		0			0
$484$	2.00000	+	3.46410i	0.0909091	+	0.157459i
$485$		0			0
$486$		0			0
$487$		−	2.00000i		−	0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$		−	2.00000i		−	0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$		0			0
$489$	10.0000	−	17.3205i	0.452216	−	0.783260i
$490$		0			0
$491$	7.50000	−	12.9904i	0.338470	−	0.586248i	−0.645675	−	0.763612i	$-0.723424\pi$
$491$	7.50000	−	12.9904i	0.338470	−	0.586248i	0.984145	+	0.177365i	$0.0567572\pi$
$492$	−20.7846	−	12.0000i	−0.937043	−	0.541002i
$493$		−	18.0000i		−	0.810679i
$494$		0			0
$495$		0			0
$496$	14.0000	−	24.2487i	0.628619	−	1.08880i
$497$	10.3923	+	6.00000i	0.466159	+	0.269137i
$498$		0			0
$499$	10.0000	−	17.3205i	0.447661	−	0.775372i	−0.550572	−	0.834788i	$-0.685590\pi$
$499$	10.0000	−	17.3205i	0.447661	−	0.775372i	0.998233	+	0.0594153i	$0.0189236\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−36.3731	+	21.0000i	−1.62179	+	0.936344i	−0.635355	+	0.772220i	$0.719146\pi$
$503$	−36.3731	+	21.0000i	−1.62179	+	0.936344i	−0.986440	+	0.164124i	$0.947520\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.5885	+	9.00000i	−0.692308	+	0.399704i
$508$	3.46410	+	2.00000i	0.153695	+	0.0887357i
$509$	−9.00000	−	15.5885i	−0.398918	−	0.690946i	0.594675	−	0.803966i	$-0.297281\pi$
$509$	−9.00000	−	15.5885i	−0.398918	−	0.690946i	−0.993593	+	0.113020i	$0.963948\pi$
$510$		0			0
$511$	16.0000	−	27.7128i	0.707798	−	1.22594i
$512$		0			0
$513$	10.3923	−	14.0000i	0.458831	−	0.618115i
$514$		0			0
$515$		0			0
$516$	8.00000	−	13.8564i	0.352180	−	0.609994i
$517$	−15.5885	+	9.00000i	−0.685580	+	0.395820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−45.0000			−1.97149			−0.985743	−	0.168259i	$-0.946186\pi$
$521$	−45.0000			−1.97149			−0.985743	+	0.168259i	$0.946186\pi$
$522$		0			0
$523$	22.5167	−	13.0000i	0.984585	−	0.568450i	0.0809336	−	0.996719i	$-0.474210\pi$
$523$	22.5167	−	13.0000i	0.984585	−	0.568450i	0.903651	+	0.428269i	$0.140876\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$		0			0
$527$	−36.3731	−	21.0000i	−1.58444	−	0.914774i
$528$	−20.7846	+	12.0000i	−0.904534	+	0.522233i
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$	−15.0000			−0.650945
$532$	34.6410	+	4.00000i	1.50188	+	0.173422i
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	15.5885	+	9.00000i	0.672692	+	0.388379i
$538$		0			0
$539$	−27.0000			−1.16297
$540$		0			0
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	0.922599	+	0.385759i	$0.126061\pi$
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	−0.127222	+	0.991874i	$0.540606\pi$
$542$		0			0
$543$		−	4.00000i		−	0.171656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.46410	−	2.00000i	0.148114	−	0.0855138i	−0.424111	−	0.905610i	$-0.639413\pi$
$547$	3.46410	−	2.00000i	0.148114	−	0.0855138i	0.572226	+	0.820096i	$0.306080\pi$
$548$	−20.7846	−	12.0000i	−0.887875	−	0.512615i
$549$	2.50000	−	4.33013i	0.106697	−	0.184805i
$550$		0			0
$551$	1.50000	−	12.9904i	0.0639021	−	0.553409i
$552$		0			0
$553$	17.3205	+	10.0000i	0.736543	+	0.425243i
$554$		0			0
$555$		0			0
$556$	−16.0000	+	27.7128i	−0.678551	+	1.17529i
$557$	−10.3923	+	6.00000i	−0.440336	+	0.254228i	−0.703740	−	0.710457i	$-0.748488\pi$
$557$	−10.3923	+	6.00000i	−0.440336	+	0.254228i	0.263404	+	0.964686i	$0.415155\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	18.0000	+	31.1769i	0.759961	+	1.31629i
$562$		0			0
$563$			6.00000i			0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$			6.00000i			0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	12.0000	+	20.7846i	0.505291	+	0.875190i
$565$		0			0
$566$		0			0
$567$	38.1051	+	22.0000i	1.60026	+	0.923913i
$568$		0			0
$569$	−39.0000			−1.63497			−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000			−1.63497			−0.817483	+	0.575953i	$0.804631\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$	−10.3923	−	6.00000i	−0.434524	−	0.250873i
$573$	−25.9808	−	15.0000i	−1.08536	−	0.626634i
$574$		0			0
$575$		0			0
$576$	4.00000	+	6.92820i	0.166667	+	0.288675i
$577$			16.0000i			0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		0			0
$579$	16.0000	+	27.7128i	0.664937	+	1.15171i
$580$		0			0
$581$	−48.0000			−1.99138
$582$		0			0
$583$	−15.5885	+	9.00000i	−0.645608	+	0.372742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−31.1769	−	18.0000i	−1.28681	−	0.742940i	−0.308725	−	0.951151i	$-0.599902\pi$
$587$	−31.1769	−	18.0000i	−1.28681	−	0.742940i	−0.978084	+	0.208212i	$0.933236\pi$
$588$			36.0000i			1.48461i
$589$	−24.5000	−	18.1865i	−1.00950	−	0.749363i
$590$		0			0
$591$	−18.0000	+	31.1769i	−0.740421	+	1.28245i
$592$	27.7128	+	16.0000i	1.13899	+	0.657596i
$593$	10.3923	−	6.00000i	0.426761	−	0.246390i	−0.271205	−	0.962522i	$-0.587422\pi$
$593$	10.3923	−	6.00000i	0.426761	−	0.246390i	0.697966	+	0.716131i	$0.254089\pi$
$594$		0			0
$595$		0			0
$596$	−6.00000			−0.245770
$597$		−	38.0000i		−	1.55524i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−31.0000			−1.26452			−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000			−1.26452			−0.632258	+	0.774758i	$0.717872\pi$
$602$		0			0
$603$	−1.73205	−	1.00000i	−0.0705346	−	0.0407231i
$604$	−17.0000	−	29.4449i	−0.691720	−	1.19809i
$605$		0			0
$606$		0			0
$607$		−	2.00000i		−	0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$		−	2.00000i		−	0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$	−6.00000	+	10.3923i	−0.242734	+	0.420428i
$612$	10.3923	−	6.00000i	0.420084	−	0.242536i
$613$	−1.73205	−	1.00000i	−0.0699569	−	0.0403896i	0.464614	−	0.885514i	$-0.346193\pi$
$613$	−1.73205	−	1.00000i	−0.0699569	−	0.0403896i	−0.534570	+	0.845124i	$0.679527\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.972125	−	0.234464i	$-0.924666\pi$
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.283011	+	0.959117i	$0.591333\pi$
$618$		0			0
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	51.9615	−	30.0000i	2.08179	−	1.20192i
$624$	−8.00000	+	13.8564i	−0.320256	+	0.554700i
$625$		0			0
$626$		0			0
$627$	10.3923	+	24.0000i	0.415029	+	0.958468i
$628$		−	4.00000i		−	0.159617i
$629$	24.0000	−	41.5692i	0.956943	−	1.65747i
$630$		0			0
$631$	9.50000	+	16.4545i	0.378189	+	0.655043i	0.990799	−	0.135343i	$-0.0432136\pi$
$631$	9.50000	+	16.4545i	0.378189	+	0.655043i	−0.612610	+	0.790386i	$0.709880\pi$
$632$		0			0
$633$	12.1244	−	7.00000i	0.481900	−	0.278225i
$634$		0			0
$635$		0			0
$636$	12.0000	+	20.7846i	0.475831	+	0.824163i
$637$	−15.5885	+	9.00000i	−0.617637	+	0.356593i
$638$		0			0
$639$	3.00000			0.118678
$640$		0			0
$641$	−4.50000	+	7.79423i	−0.177739	+	0.307854i	−0.941106	−	0.338112i	$-0.890212\pi$
$641$	−4.50000	+	7.79423i	−0.177739	+	0.307854i	0.763367	+	0.645966i	$0.223545\pi$
$642$		0			0
$643$	29.4449	+	17.0000i	1.16119	+	0.670415i	0.951589	−	0.307372i	$-0.0994496\pi$
$643$	29.4449	+	17.0000i	1.16119	+	0.670415i	0.209603	+	0.977787i	$0.432783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$	−22.5000	+	38.9711i	−0.883202	+	1.52975i
$650$		0			0
$651$	28.0000	−	48.4974i	1.09741	−	1.90076i
$652$	−17.3205	+	10.0000i	−0.678323	+	0.391630i
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$		0			0
$656$	12.0000	+	20.7846i	0.468521	+	0.811503i
$657$		−	8.00000i		−	0.312110i
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	−2.50000	−	4.33013i	−0.0972387	−	0.168422i	0.813302	−	0.581842i	$-0.197668\pi$
$661$	−2.50000	−	4.33013i	−0.0972387	−	0.168422i	−0.910541	+	0.413419i	$0.864334\pi$
$662$		0			0
$663$	20.7846	+	12.0000i	0.807207	+	0.466041i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.00000	+	6.92820i	0.154649	+	0.267860i
$670$		0			0
$671$	−7.50000	−	12.9904i	−0.289534	−	0.501488i
$672$		0			0
$673$		−	4.00000i		−	0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$		−	4.00000i		−	0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$		0			0
$675$		0			0
$676$	18.0000			0.692308
$677$		−	48.0000i		−	1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$		−	48.0000i		−	1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$		0			0
$679$	−16.0000	+	27.7128i	−0.614024	+	1.06352i
$680$		0			0
$681$	−24.0000	+	41.5692i	−0.919682	+	1.59294i
$682$		0			0
$683$		−	18.0000i		−	0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$		−	18.0000i		−	0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	8.00000	−	3.46410i	0.305888	−	0.132453i
$685$		0			0
$686$		0			0
$687$	12.1244	+	7.00000i	0.462573	+	0.267067i
$688$	−13.8564	+	8.00000i	−0.528271	+	0.304997i
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	−37.0000			−1.40755			−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000			−1.40755			−0.703773	+	0.710425i	$0.748503\pi$
$692$		0			0
$693$	−10.3923	+	6.00000i	−0.394771	+	0.227921i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	31.1769	−	18.0000i	1.18091	−	0.681799i
$698$		0			0
$699$	−6.00000	−	10.3923i	−0.226941	−	0.393073i
$700$		0			0
$701$	−3.00000	+	5.19615i	−0.113308	+	0.196256i	−0.917102	−	0.398652i	$-0.869478\pi$
$701$	−3.00000	+	5.19615i	−0.113308	+	0.196256i	0.803794	+	0.594908i	$0.202811\pi$
$702$		0			0
$703$	20.7846	−	28.0000i	0.783906	−	1.05604i
$704$	24.0000			0.904534
$705$		0			0
$706$		0			0
$707$	51.9615	−	30.0000i	1.95421	−	1.12827i
$708$	51.9615	+	30.0000i	1.95283	+	1.12747i
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	0.413114	−	0.910679i	$-0.364441\pi$
$710$		0			0
$711$	5.00000			0.187515
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−9.00000	−	15.5885i	−0.336346	−	0.582568i
$717$	5.19615	+	3.00000i	0.194054	+	0.112037i
$718$		0			0
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	−0.691608	−	0.722273i	$-0.743097\pi$
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	0.971311	+	0.237814i	$0.0764307\pi$
$720$		0			0
$721$	64.0000			2.38348
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$	−2.00000	+	3.46410i	−0.0743294	+	0.128742i
$725$		0			0
$726$		0			0
$727$	12.1244	+	7.00000i	0.449667	+	0.259616i	0.707690	−	0.706523i	$-0.249737\pi$
$727$	12.1244	+	7.00000i	0.449667	+	0.259616i	−0.258022	+	0.966139i	$0.583071\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	12.0000	+	20.7846i	0.443836	+	0.768747i
$732$	−17.3205	+	10.0000i	−0.640184	+	0.369611i
$733$			32.0000i			1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$733$			32.0000i			1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.19615	+	3.00000i	−0.191403	+	0.110506i
$738$		0			0
$739$	17.5000	−	30.3109i	0.643748	−	1.11500i	−0.340841	−	0.940121i	$-0.610712\pi$
$739$	17.5000	−	30.3109i	0.643748	−	1.11500i	0.984589	−	0.174883i	$-0.0559548\pi$
$740$		0			0
$741$	14.0000	+	10.3923i	0.514303	+	0.381771i
$742$		0			0
$743$	20.7846	+	12.0000i	0.762513	+	0.440237i	0.830197	−	0.557470i	$-0.188228\pi$
$743$	20.7846	+	12.0000i	0.762513	+	0.440237i	−0.0676840	+	0.997707i	$0.521561\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.3923	+	6.00000i	−0.380235	+	0.219529i
$748$		−	36.0000i		−	1.31629i
$749$		0			0
$750$		0			0
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	0.922792	−	0.385299i	$-0.125902\pi$
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	−0.795075	+	0.606511i	$0.792568\pi$
$752$		−	24.0000i		−	0.875190i
$753$			30.0000i			1.09326i
$754$		0			0
$755$		0			0
$756$	−16.0000	−	27.7128i	−0.581914	−	1.00791i
$757$	−8.66025	−	5.00000i	−0.314762	−	0.181728i	0.334293	−	0.942469i	$-0.391502\pi$
$757$	−8.66025	−	5.00000i	−0.314762	−	0.181728i	−0.649056	+	0.760741i	$0.724836\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	38.1051	+	22.0000i	1.37950	+	0.796453i
$764$	15.0000	+	25.9808i	0.542681	+	0.939951i
$765$		0			0
$766$		0			0
$767$			30.0000i			1.08324i
$768$		−	32.0000i		−	1.15470i
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	0.999645	+	0.0266282i	$0.00847701\pi$
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	−0.476762	+	0.879032i	$0.658190\pi$
$770$		0			0
$771$	36.0000			1.29651
$772$		−	32.0000i		−	1.15171i
$773$	25.9808	−	15.0000i	0.934463	−	0.539513i	0.0462427	−	0.998930i	$-0.485275\pi$
$773$	25.9808	−	15.0000i	0.934463	−	0.539513i	0.888220	+	0.459418i	$0.151942\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	55.4256	+	32.0000i	1.98838	+	1.14799i
$778$		0			0
$779$	24.0000	−	10.3923i	0.859889	−	0.372343i
$780$		0			0
$781$	4.50000	−	7.79423i	0.161023	−	0.278899i
$782$		0			0
$783$	−10.3923	+	6.00000i	−0.371391	+	0.214423i
$784$	18.0000	−	31.1769i	0.642857	−	1.11346i
$785$		0			0
$786$		0			0
$787$			34.0000i			1.21197i	0.795476	+	0.605985i	$0.207221\pi$
$787$			34.0000i			1.21197i	−0.795476	+	0.605985i	$0.792779\pi$
$788$	31.1769	−	18.0000i	1.11063	−	0.641223i
$789$	18.0000	+	31.1769i	0.640817	+	1.10993

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 + 1.00000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +4.00000i q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(1.73205 + 1.00000i) q^{3} +(-1.00000 - 1.73205i) q^{4} +4.00000i q^{7} +(0.500000 + 0.866025i) q^{9} +3.00000 q^{11} -4.00000i q^{12} +(1.73205 - 1.00000i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(5.19615 + 3.00000i) q^{17} +(3.50000 + 2.59808i) q^{19} +(-4.00000 + 6.92820i) q^{21} -4.00000i q^{27} +(6.92820 - 4.00000i) q^{28} +(-1.50000 - 2.59808i) q^{29} -7.00000 q^{31} +(5.19615 + 3.00000i) q^{33} +(1.00000 - 1.73205i) q^{36} -8.00000i q^{37} +4.00000 q^{39} +(3.00000 - 5.19615i) q^{41} +(3.46410 + 2.00000i) q^{43} +(-3.00000 - 5.19615i) q^{44} +(-5.19615 + 3.00000i) q^{47} +(-6.92820 + 4.00000i) q^{48} -9.00000 q^{49} +(6.00000 + 10.3923i) q^{51} +(-3.46410 - 2.00000i) q^{52} +(-5.19615 + 3.00000i) q^{53} +(3.46410 + 8.00000i) q^{57} +(-7.50000 + 12.9904i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-3.46410 + 2.00000i) q^{63} +8.00000 q^{64} +(-1.73205 + 1.00000i) q^{67} -12.0000i q^{68} +(1.50000 - 2.59808i) q^{71} +(-6.92820 - 4.00000i) q^{73} +(1.00000 - 8.66025i) q^{76} +12.0000i q^{77} +(2.50000 - 4.33013i) q^{79} +(5.50000 - 9.52628i) q^{81} +12.0000i q^{83} +16.0000 q^{84} -6.00000i q^{87} +(-7.50000 - 12.9904i) q^{89} +(4.00000 + 6.92820i) q^{91} +(-12.1244 - 7.00000i) q^{93} +(6.92820 + 4.00000i) q^{97} +(1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 2 * q^9	\( 4 q - 4 q^{4} + 2 q^{9} + 12 q^{11} - 8 q^{16} + 14 q^{19} - 16 q^{21} - 6 q^{29} - 28 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 12 q^{44} - 36 q^{49} + 24 q^{51} - 30 q^{59} - 10 q^{61} + 32 q^{64} + 6 q^{71} + 4 q^{76} + 10 q^{79} + 22 q^{81} + 64 q^{84} - 30 q^{89} + 16 q^{91} + 6 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^9 + 12 * q^11 - 8 * q^16 + 14 * q^19 - 16 * q^21 - 6 * q^29 - 28 * q^31 + 4 * q^36 + 16 * q^39 + 12 * q^41 - 12 * q^44 - 36 * q^49 + 24 * q^51 - 30 * q^59 - 10 * q^61 + 32 * q^64 + 6 * q^71 + 4 * q^76 + 10 * q^79 + 22 * q^81 + 64 * q^84 - 30 * q^89 + 16 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more