LMFDB - Embedded newform 936.2.w.g.811.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(307,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))

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

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

chi := DirichletCharacter("936.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.w (of order $4$, degree $2$, 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:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{26})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 169 $ x^4 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			811.1
Root			$-2.54951 - 2.54951i$ of defining polynomial
Character	$\chi$	$=$	936.811
Dual form			936.2.w.g.307.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} +(-2.00000 - 2.00000i) q^{8} -5.09902 q^{10} +(-1.00000 - 1.00000i) q^{11} +(2.54951 + 2.54951i) q^{13} +5.09902i q^{14} -4.00000 q^{16} +3.00000i q^{17} +(2.00000 - 2.00000i) q^{19} +(-5.09902 + 5.09902i) q^{20} -2.00000 q^{22} -5.09902 q^{23} +8.00000i q^{25} +5.09902 q^{26} +(5.09902 + 5.09902i) q^{28} +5.09902i q^{29} +(-5.09902 - 5.09902i) q^{31} +(-4.00000 + 4.00000i) q^{32} +(3.00000 + 3.00000i) q^{34} +13.0000 q^{35} +(-2.54951 + 2.54951i) q^{37} -4.00000i q^{38} +10.1980i q^{40} +(-6.00000 + 6.00000i) q^{41} -1.00000i q^{43} +(-2.00000 + 2.00000i) q^{44} +(-5.09902 + 5.09902i) q^{46} +(2.54951 - 2.54951i) q^{47} -6.00000i q^{49} +(8.00000 + 8.00000i) q^{50} +(5.09902 - 5.09902i) q^{52} -5.09902i q^{53} +5.09902i q^{55} +10.1980 q^{56} +(5.09902 + 5.09902i) q^{58} +(-8.00000 - 8.00000i) q^{59} -10.1980 q^{62} +8.00000i q^{64} -13.0000i q^{65} +(3.00000 - 3.00000i) q^{67} +6.00000 q^{68} +(13.0000 - 13.0000i) q^{70} +(-7.64853 - 7.64853i) q^{71} +(-6.00000 - 6.00000i) q^{73} +5.09902i q^{74} +(-4.00000 - 4.00000i) q^{76} +5.09902 q^{77} -5.09902i q^{79} +(10.1980 + 10.1980i) q^{80} +12.0000i q^{82} +(-5.00000 + 5.00000i) q^{83} +(7.64853 - 7.64853i) q^{85} +(-1.00000 - 1.00000i) q^{86} +4.00000i q^{88} +(2.00000 + 2.00000i) q^{89} -13.0000 q^{91} +10.1980i q^{92} -5.09902i q^{94} -10.1980 q^{95} +(-7.00000 + 7.00000i) q^{97} +(-6.00000 - 6.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 8 * q^8	$ 4 q + 4 q^{2} - 8 q^{8} - 4 q^{11} - 16 q^{16} + 8 q^{19} - 8 q^{22} - 16 q^{32} + 12 q^{34} + 52 q^{35} - 24 q^{41} - 8 q^{44} + 32 q^{50} - 32 q^{59} + 12 q^{67} + 24 q^{68} + 52 q^{70} - 24 q^{73} - 16 q^{76} - 20 q^{83} - 4 q^{86} + 8 q^{89} - 52 q^{91} - 28 q^{97} - 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^8 - 4 * q^11 - 16 * q^16 + 8 * q^19 - 8 * q^22 - 16 * q^32 + 12 * q^34 + 52 * q^35 - 24 * q^41 - 8 * q^44 + 32 * q^50 - 32 * q^59 + 12 * q^67 + 24 * q^68 + 52 * q^70 - 24 * q^73 - 16 * q^76 - 20 * q^83 - 4 * q^86 + 8 * q^89 - 52 * q^91 - 28 * q^97 - 24 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$

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 $

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	1.00000i
$5$	−2.54951	−	2.54951i	−1.14018	−	1.14018i	−0.988418	−	0.151758i	$-0.951507\pi$
$5$	−2.54951	−	2.54951i	−1.14018	−	1.14018i	−0.151758	−	0.988418i	$-0.548493\pi$
$6$		0			0
$7$	−2.54951	+	2.54951i	−0.963624	+	0.963624i	−0.999361	−	0.0357371i	$-0.988622\pi$
$7$	−2.54951	+	2.54951i	−0.963624	+	0.963624i	0.0357371	+	0.999361i	$0.488622\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$	−5.09902			−1.61245
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	0.540094	−	0.841605i	$-0.318389\pi$
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	−0.841605	+	0.540094i	$0.818389\pi$
$12$		0			0
$13$	2.54951	+	2.54951i	0.707107	+	0.707107i
$13$	2.54951	+	2.54951i	0.707107	+	0.707107i
$14$			5.09902i			1.36277i
$15$		0			0
$16$	−4.00000			−1.00000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	−0.439440	−	0.898272i	$-0.644823\pi$
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	0.898272	+	0.439440i	$0.144823\pi$
$20$	−5.09902	+	5.09902i	−1.14018	+	1.14018i
$21$		0			0
$22$	−2.00000			−0.426401
$23$	−5.09902			−1.06322			−0.531610	−	0.846990i	$-0.678413\pi$
$23$	−5.09902			−1.06322			−0.531610	+	0.846990i	$0.678413\pi$
$24$		0			0
$25$			8.00000i			1.60000i
$26$	5.09902			1.00000
$27$		0			0
$28$	5.09902	+	5.09902i	0.963624	+	0.963624i
$29$			5.09902i			0.946864i	0.880830	+	0.473432i	$0.156985\pi$
$29$			5.09902i			0.946864i	−0.880830	+	0.473432i	$0.843015\pi$
$30$		0			0
$31$	−5.09902	−	5.09902i	−0.915811	−	0.915811i	0.0809104	−	0.996721i	$-0.474217\pi$
$31$	−5.09902	−	5.09902i	−0.915811	−	0.915811i	−0.996721	+	0.0809104i	$0.974217\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	3.00000	+	3.00000i	0.514496	+	0.514496i
$35$	13.0000			2.19740
$36$		0			0
$37$	−2.54951	+	2.54951i	−0.419137	+	0.419137i	−0.884906	−	0.465769i	$-0.845778\pi$
$37$	−2.54951	+	2.54951i	−0.419137	+	0.419137i	0.465769	+	0.884906i	$0.345778\pi$
$38$		−	4.00000i		−	0.648886i
$39$		0			0
$40$			10.1980i			1.61245i
$41$	−6.00000	+	6.00000i	−0.937043	+	0.937043i	−0.998132	−	0.0610897i	$-0.980542\pi$
$41$	−6.00000	+	6.00000i	−0.937043	+	0.937043i	0.0610897	+	0.998132i	$0.480542\pi$
$42$		0			0
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−2.00000	+	2.00000i	−0.301511	+	0.301511i
$45$		0			0
$46$	−5.09902	+	5.09902i	−0.751809	+	0.751809i
$47$	2.54951	−	2.54951i	0.371884	−	0.371884i	−0.496279	−	0.868163i	$-0.665301\pi$
$47$	2.54951	−	2.54951i	0.371884	−	0.371884i	0.868163	+	0.496279i	$0.165301\pi$
$48$		0			0
$49$		−	6.00000i		−	0.857143i
$50$	8.00000	+	8.00000i	1.13137	+	1.13137i
$51$		0			0
$52$	5.09902	−	5.09902i	0.707107	−	0.707107i
$53$		−	5.09902i		−	0.700404i	−0.936674	−	0.350202i	$-0.886113\pi$
$53$		−	5.09902i		−	0.700404i	0.936674	−	0.350202i	$-0.113887\pi$
$54$		0			0
$55$			5.09902i			0.687552i
$56$	10.1980			1.36277
$57$		0			0
$58$	5.09902	+	5.09902i	0.669534	+	0.669534i
$59$	−8.00000	−	8.00000i	−1.04151	−	1.04151i	−0.999100	−	0.0424110i	$-0.986496\pi$
$59$	−8.00000	−	8.00000i	−1.04151	−	1.04151i	−0.0424110	−	0.999100i	$-0.513504\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−10.1980			−1.29515
$63$		0			0
$64$			8.00000i			1.00000i
$65$		−	13.0000i		−	1.61245i
$66$		0			0
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	−0.499694	−	0.866202i	$-0.666554\pi$
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	0.866202	+	0.499694i	$0.166554\pi$
$68$	6.00000			0.727607
$69$		0			0
$70$	13.0000	−	13.0000i	1.55380	−	1.55380i
$71$	−7.64853	−	7.64853i	−0.907713	−	0.907713i	0.0883739	−	0.996087i	$-0.471833\pi$
$71$	−7.64853	−	7.64853i	−0.907713	−	0.907713i	−0.996087	+	0.0883739i	$0.971833\pi$
$72$		0			0
$73$	−6.00000	−	6.00000i	−0.702247	−	0.702247i	0.262646	−	0.964892i	$-0.415405\pi$
$73$	−6.00000	−	6.00000i	−0.702247	−	0.702247i	−0.964892	+	0.262646i	$0.915405\pi$
$74$			5.09902i			0.592749i
$75$		0			0
$76$	−4.00000	−	4.00000i	−0.458831	−	0.458831i
$77$	5.09902			0.581087
$78$		0			0
$79$		−	5.09902i		−	0.573685i	−0.957978	−	0.286842i	$-0.907394\pi$
$79$		−	5.09902i		−	0.573685i	0.957978	−	0.286842i	$-0.0926056\pi$
$80$	10.1980	+	10.1980i	1.14018	+	1.14018i
$81$		0			0
$82$			12.0000i			1.32518i
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	−0.926100	−	0.377279i	$-0.876860\pi$
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	0.377279	+	0.926100i	$0.376860\pi$
$84$		0			0
$85$	7.64853	−	7.64853i	0.829599	−	0.829599i
$86$	−1.00000	−	1.00000i	−0.107833	−	0.107833i
$87$		0			0
$88$			4.00000i			0.426401i
$89$	2.00000	+	2.00000i	0.212000	+	0.212000i	0.805116	−	0.593117i	$-0.202103\pi$
$89$	2.00000	+	2.00000i	0.212000	+	0.212000i	−0.593117	+	0.805116i	$0.702103\pi$
$90$		0			0
$91$	−13.0000			−1.36277
$92$			10.1980i			1.06322i
$93$		0			0
$94$		−	5.09902i		−	0.525924i
$95$	−10.1980			−1.04630
$96$		0			0
$97$	−7.00000	+	7.00000i	−0.710742	+	0.710742i	−0.966691	−	0.255948i	$-0.917612\pi$
$97$	−7.00000	+	7.00000i	−0.710742	+	0.710742i	0.255948	+	0.966691i	$0.417612\pi$
$98$	−6.00000	−	6.00000i	−0.606092	−	0.606092i
$99$		0			0
$100$	16.0000			1.60000
$101$	10.1980			1.01474			0.507371	−	0.861727i	$-0.330617\pi$
$101$	10.1980			1.01474			0.507371	+	0.861727i	$0.330617\pi$
$102$		0			0
$103$	5.09902			0.502421			0.251211	−	0.967932i	$-0.419171\pi$
$103$	5.09902			0.502421			0.251211	+	0.967932i	$0.419171\pi$
$104$		−	10.1980i		−	1.00000i
$105$		0			0
$106$	−5.09902	−	5.09902i	−0.495261	−	0.495261i
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$	−2.54951	−	2.54951i	−0.244199	−	0.244199i	0.574386	−	0.818585i	$-0.305241\pi$
$109$	−2.54951	−	2.54951i	−0.244199	−	0.244199i	−0.818585	+	0.574386i	$0.805241\pi$
$110$	5.09902	+	5.09902i	0.486172	+	0.486172i
$111$		0			0
$112$	10.1980	−	10.1980i	0.963624	−	0.963624i
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$		0			0
$115$	13.0000	+	13.0000i	1.21226	+	1.21226i
$116$	10.1980			0.946864
$117$		0			0
$118$	−16.0000			−1.47292
$119$	−7.64853	−	7.64853i	−0.701140	−	0.701140i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$		0			0
$124$	−10.1980	+	10.1980i	−0.915811	+	0.915811i
$125$	7.64853	−	7.64853i	0.684105	−	0.684105i
$126$		0			0
$127$	−15.2971			−1.35739			−0.678697	−	0.734418i	$-0.737455\pi$
$127$	−15.2971			−1.35739			−0.678697	+	0.734418i	$0.737455\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$	−13.0000	−	13.0000i	−1.14018	−	1.14018i
$131$	−7.00000			−0.611593			−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000			−0.611593			−0.305796	+	0.952097i	$0.598923\pi$
$132$		0			0
$133$			10.1980i			0.884282i
$134$		−	6.00000i		−	0.518321i
$135$		0			0
$136$	6.00000	−	6.00000i	0.514496	−	0.514496i
$137$	5.00000	+	5.00000i	0.427179	+	0.427179i	0.887666	−	0.460487i	$-0.152325\pi$
$137$	5.00000	+	5.00000i	0.427179	+	0.427179i	−0.460487	+	0.887666i	$0.652325\pi$
$138$		0			0
$139$	−15.0000			−1.27228			−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000			−1.27228			−0.636142	+	0.771572i	$0.719471\pi$
$140$		−	26.0000i		−	2.19740i
$141$		0			0
$142$	−15.2971			−1.28370
$143$		−	5.09902i		−	0.426401i
$144$		0			0
$145$	13.0000	−	13.0000i	1.07959	−	1.07959i
$146$	−12.0000			−0.993127
$147$		0			0
$148$	5.09902	+	5.09902i	0.419137	+	0.419137i
$149$	−10.1980	−	10.1980i	−0.835456	−	0.835456i	0.152801	−	0.988257i	$-0.451171\pi$
$149$	−10.1980	−	10.1980i	−0.835456	−	0.835456i	−0.988257	+	0.152801i	$0.951171\pi$
$150$		0			0
$151$	7.64853	−	7.64853i	0.622428	−	0.622428i	−0.323723	−	0.946152i	$-0.604935\pi$
$151$	7.64853	−	7.64853i	0.622428	−	0.622428i	0.946152	+	0.323723i	$0.104935\pi$
$152$	−8.00000			−0.648886
$153$		0			0
$154$	5.09902	−	5.09902i	0.410891	−	0.410891i
$155$			26.0000i			2.08837i
$156$		0			0
$157$		−	10.1980i		−	0.813892i	−0.913452	−	0.406946i	$-0.866594\pi$
$157$		−	10.1980i		−	0.813892i	0.913452	−	0.406946i	$-0.133406\pi$
$158$	−5.09902	−	5.09902i	−0.405656	−	0.405656i
$159$		0			0
$160$	20.3961			1.61245
$161$	13.0000	−	13.0000i	1.02454	−	1.02454i
$162$		0			0
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	0.965465	−	0.260531i	$-0.0838976\pi$
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	−0.260531	+	0.965465i	$0.583898\pi$
$164$	12.0000	+	12.0000i	0.937043	+	0.937043i
$165$		0			0
$166$			10.0000i			0.776151i
$167$	15.2971	−	15.2971i	1.18372	−	1.18372i	0.204949	−	0.978773i	$-0.434297\pi$
$167$	15.2971	−	15.2971i	1.18372	−	1.18372i	0.978773	−	0.204949i	$-0.0657030\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		−	15.2971i		−	1.17323i
$171$		0			0
$172$	−2.00000			−0.152499
$173$	−5.09902			−0.387671			−0.193836	−	0.981034i	$-0.562093\pi$
$173$	−5.09902			−0.387671			−0.193836	+	0.981034i	$0.562093\pi$
$174$		0			0
$175$	−20.3961	−	20.3961i	−1.54180	−	1.54180i
$176$	4.00000	+	4.00000i	0.301511	+	0.301511i
$177$		0			0
$178$	4.00000			0.299813
$179$			9.00000i			0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$			9.00000i			0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$		0			0
$181$	15.2971			1.13702			0.568511	−	0.822676i	$-0.307520\pi$
$181$	15.2971			1.13702			0.568511	+	0.822676i	$0.307520\pi$
$182$	−13.0000	+	13.0000i	−0.963624	+	0.963624i
$183$		0			0
$184$	10.1980	+	10.1980i	0.751809	+	0.751809i
$185$	13.0000			0.955779
$186$		0			0
$187$	3.00000	−	3.00000i	0.219382	−	0.219382i
$188$	−5.09902	−	5.09902i	−0.371884	−	0.371884i
$189$		0			0
$190$	−10.1980	+	10.1980i	−0.739844	+	0.739844i
$191$			25.4951i			1.84476i	0.386283	+	0.922380i	$0.373759\pi$
$191$			25.4951i			1.84476i	−0.386283	+	0.922380i	$0.626241\pi$
$192$		0			0
$193$	9.00000	+	9.00000i	0.647834	+	0.647834i	0.952469	−	0.304635i	$-0.0985345\pi$
$193$	9.00000	+	9.00000i	0.647834	+	0.647834i	−0.304635	+	0.952469i	$0.598534\pi$
$194$			14.0000i			1.00514i
$195$		0			0
$196$	−12.0000			−0.857143
$197$	12.7475	+	12.7475i	0.908225	+	0.908225i	0.996129	−	0.0879037i	$-0.0280168\pi$
$197$	12.7475	+	12.7475i	0.908225	+	0.908225i	−0.0879037	+	0.996129i	$0.528017\pi$
$198$		0			0
$199$	−25.4951			−1.80730			−0.903650	−	0.428272i	$-0.859122\pi$
$199$	−25.4951			−1.80730			−0.903650	+	0.428272i	$0.859122\pi$
$200$	16.0000	−	16.0000i	1.13137	−	1.13137i
$201$		0			0
$202$	10.1980	−	10.1980i	0.717532	−	0.717532i
$203$	−13.0000	−	13.0000i	−0.912421	−	0.912421i
$204$		0			0
$205$	30.5941			2.13679
$206$	5.09902	−	5.09902i	0.355266	−	0.355266i
$207$		0			0
$208$	−10.1980	−	10.1980i	−0.707107	−	0.707107i
$209$	−4.00000			−0.276686
$210$		0			0
$211$	7.00000			0.481900			0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000			0.481900			0.240950	+	0.970538i	$0.422541\pi$
$212$	−10.1980			−0.700404
$213$		0			0
$214$	2.00000	−	2.00000i	0.136717	−	0.136717i
$215$	−2.54951	+	2.54951i	−0.173875	+	0.173875i
$216$		0			0
$217$	26.0000			1.76500
$218$	−5.09902			−0.345349
$219$		0			0
$220$	10.1980			0.687552
$221$	−7.64853	+	7.64853i	−0.514496	+	0.514496i
$222$		0			0
$223$	−7.64853	−	7.64853i	−0.512183	−	0.512183i	0.403012	−	0.915195i	$-0.367963\pi$
$223$	−7.64853	−	7.64853i	−0.512183	−	0.512183i	−0.915195	+	0.403012i	$0.867963\pi$
$224$		−	20.3961i		−	1.36277i
$225$		0			0
$226$	−4.00000	+	4.00000i	−0.266076	+	0.266076i
$227$	12.0000	−	12.0000i	0.796468	−	0.796468i	−0.186069	−	0.982537i	$-0.559575\pi$
$227$	12.0000	−	12.0000i	0.796468	−	0.796468i	0.982537	+	0.186069i	$0.0595747\pi$
$228$		0			0
$229$	2.54951	−	2.54951i	0.168476	−	0.168476i	−0.617833	−	0.786309i	$-0.711989\pi$
$229$	2.54951	−	2.54951i	0.168476	−	0.168476i	0.786309	+	0.617833i	$0.211989\pi$
$230$	26.0000			1.71439
$231$		0			0
$232$	10.1980	−	10.1980i	0.669534	−	0.669534i
$233$			11.0000i			0.720634i	0.932830	+	0.360317i	$0.117331\pi$
$233$			11.0000i			0.720634i	−0.932830	+	0.360317i	$0.882669\pi$
$234$		0			0
$235$	−13.0000			−0.848026
$236$	−16.0000	+	16.0000i	−1.04151	+	1.04151i
$237$		0			0
$238$	−15.2971			−0.991561
$239$	2.54951	+	2.54951i	0.164914	+	0.164914i	0.784740	−	0.619826i	$-0.212797\pi$
$239$	2.54951	+	2.54951i	0.164914	+	0.164914i	−0.619826	+	0.784740i	$0.712797\pi$
$240$		0			0
$241$	−14.0000	−	14.0000i	−0.901819	−	0.901819i	0.0937742	−	0.995593i	$-0.470107\pi$
$241$	−14.0000	−	14.0000i	−0.901819	−	0.901819i	−0.995593	+	0.0937742i	$0.970107\pi$
$242$	−9.00000	−	9.00000i	−0.578542	−	0.578542i
$243$		0			0
$244$		0			0
$245$	−15.2971	+	15.2971i	−0.977293	+	0.977293i
$246$		0			0
$247$	10.1980			0.648886
$248$			20.3961i			1.29515i
$249$		0			0
$250$		−	15.2971i		−	0.967471i
$251$		−	10.0000i		−	0.631194i	−0.948893	−	0.315597i	$-0.897795\pi$
$251$		−	10.0000i		−	0.631194i	0.948893	−	0.315597i	$-0.102205\pi$
$252$		0			0
$253$	5.09902	+	5.09902i	0.320573	+	0.320573i
$254$	−15.2971	+	15.2971i	−0.959823	+	0.959823i
$255$		0			0
$256$	16.0000			1.00000
$257$			3.00000i			0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$			3.00000i			0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$		0			0
$259$		−	13.0000i		−	0.807781i
$260$	−26.0000			−1.61245
$261$		0			0
$262$	−7.00000	+	7.00000i	−0.432461	+	0.432461i
$263$			20.3961i			1.25768i	0.777536	+	0.628838i	$0.216469\pi$
$263$			20.3961i			1.25768i	−0.777536	+	0.628838i	$0.783531\pi$
$264$		0			0
$265$	−13.0000	+	13.0000i	−0.798584	+	0.798584i
$266$	10.1980	+	10.1980i	0.625282	+	0.625282i
$267$		0			0
$268$	−6.00000	−	6.00000i	−0.366508	−	0.366508i
$269$		−	20.3961i		−	1.24357i	−0.783188	−	0.621785i	$-0.786408\pi$
$269$		−	20.3961i		−	1.24357i	0.783188	−	0.621785i	$-0.213592\pi$
$270$		0			0
$271$	7.64853	−	7.64853i	0.464615	−	0.464615i	−0.435550	−	0.900165i	$-0.643446\pi$
$271$	7.64853	−	7.64853i	0.464615	−	0.464615i	0.900165	+	0.435550i	$0.143446\pi$
$272$		−	12.0000i		−	0.727607i
$273$		0			0
$274$	10.0000			0.604122
$275$	8.00000	−	8.00000i	0.482418	−	0.482418i
$276$		0			0
$277$	10.1980			0.612741			0.306370	−	0.951912i	$-0.400885\pi$
$277$	10.1980			0.612741			0.306370	+	0.951912i	$0.400885\pi$
$278$	−15.0000	+	15.0000i	−0.899640	+	0.899640i
$279$		0			0
$280$	−26.0000	−	26.0000i	−1.55380	−	1.55380i
$281$	−16.0000	−	16.0000i	−0.954480	−	0.954480i	0.0445282	−	0.999008i	$-0.485822\pi$
$281$	−16.0000	−	16.0000i	−0.954480	−	0.954480i	−0.999008	+	0.0445282i	$0.985822\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−15.2971	+	15.2971i	−0.907713	+	0.907713i
$285$		0			0
$286$	−5.09902	−	5.09902i	−0.301511	−	0.301511i
$287$		−	30.5941i		−	1.80591i
$288$		0			0
$289$	8.00000			0.470588
$290$		−	26.0000i		−	1.52677i
$291$		0			0
$292$	−12.0000	+	12.0000i	−0.702247	+	0.702247i
$293$	12.7475	−	12.7475i	0.744720	−	0.744720i	−0.228763	−	0.973482i	$-0.573468\pi$
$293$	12.7475	−	12.7475i	0.744720	−	0.744720i	0.973482	+	0.228763i	$0.0734679\pi$
$294$		0			0
$295$			40.7922i			2.37501i
$296$	10.1980			0.592749
$297$		0			0
$298$	−20.3961			−1.18151
$299$	−13.0000	−	13.0000i	−0.751809	−	0.751809i
$300$		0			0
$301$	2.54951	+	2.54951i	0.146951	+	0.146951i
$302$		−	15.2971i		−	0.880247i
$303$		0			0
$304$	−8.00000	+	8.00000i	−0.458831	+	0.458831i
$305$		0			0
$306$		0			0
$307$	−10.0000	−	10.0000i	−0.570730	−	0.570730i	0.361602	−	0.932332i	$-0.382230\pi$
$307$	−10.0000	−	10.0000i	−0.570730	−	0.570730i	−0.932332	+	0.361602i	$0.882230\pi$
$308$		−	10.1980i		−	0.581087i
$309$		0			0
$310$	26.0000	+	26.0000i	1.47670	+	1.47670i
$311$	10.1980			0.578278			0.289139	−	0.957287i	$-0.406631\pi$
$311$	10.1980			0.578278			0.289139	+	0.957287i	$0.406631\pi$
$312$		0			0
$313$	−31.0000			−1.75222			−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000			−1.75222			−0.876112	+	0.482108i	$0.839871\pi$
$314$	−10.1980	−	10.1980i	−0.575509	−	0.575509i
$315$		0			0
$316$	−10.1980			−0.573685
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$	5.09902	−	5.09902i	0.285490	−	0.285490i
$320$	20.3961	−	20.3961i	1.14018	−	1.14018i
$321$		0			0
$322$		−	26.0000i		−	1.44892i
$323$	6.00000	+	6.00000i	0.333849	+	0.333849i
$324$		0			0
$325$	−20.3961	+	20.3961i	−1.13137	+	1.13137i
$326$	18.0000			0.996928
$327$		0			0
$328$	24.0000			1.32518
$329$			13.0000i			0.716713i
$330$		0			0
$331$	21.0000	−	21.0000i	1.15426	−	1.15426i	0.168576	−	0.985689i	$-0.446083\pi$
$331$	21.0000	−	21.0000i	1.15426	−	1.15426i	0.985689	−	0.168576i	$-0.0539168\pi$
$332$	10.0000	+	10.0000i	0.548821	+	0.548821i
$333$		0			0
$334$		−	30.5941i		−	1.67404i
$335$	−15.2971			−0.835768
$336$		0			0
$337$			17.0000i			0.926049i	0.886345	+	0.463025i	$0.153236\pi$
$337$			17.0000i			0.926049i	−0.886345	+	0.463025i	$0.846764\pi$
$338$	13.0000	+	13.0000i	0.707107	+	0.707107i
$339$		0			0
$340$	−15.2971	−	15.2971i	−0.829599	−	0.829599i
$341$			10.1980i			0.552255i
$342$		0			0
$343$	−2.54951	−	2.54951i	−0.137661	−	0.137661i
$344$	−2.00000	+	2.00000i	−0.107833	+	0.107833i
$345$		0			0
$346$	−5.09902	+	5.09902i	−0.274125	+	0.274125i
$347$	7.00000			0.375780			0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000			0.375780			0.187890	+	0.982190i	$0.439835\pi$
$348$		0			0
$349$	−22.9456	+	22.9456i	−1.22825	+	1.22825i	−0.263624	+	0.964626i	$0.584918\pi$
$349$	−22.9456	+	22.9456i	−1.22825	+	1.22825i	−0.964626	+	0.263624i	$0.915082\pi$
$350$	−40.7922			−2.18043
$351$		0			0
$352$	8.00000			0.426401
$353$	−5.00000	+	5.00000i	−0.266123	+	0.266123i	−0.827536	−	0.561413i	$-0.810258\pi$
$353$	−5.00000	+	5.00000i	−0.266123	+	0.266123i	0.561413	+	0.827536i	$0.310258\pi$
$354$		0			0
$355$			39.0000i			2.06991i
$356$	4.00000	−	4.00000i	0.212000	−	0.212000i
$357$		0			0
$358$	9.00000	+	9.00000i	0.475665	+	0.475665i
$359$	−15.2971	+	15.2971i	−0.807348	+	0.807348i	−0.984232	−	0.176884i	$-0.943398\pi$
$359$	−15.2971	+	15.2971i	−0.807348	+	0.807348i	0.176884	+	0.984232i	$0.443398\pi$
$360$		0			0
$361$			11.0000i			0.578947i
$362$	15.2971	−	15.2971i	0.803996	−	0.803996i
$363$		0			0
$364$			26.0000i			1.36277i
$365$			30.5941i			1.60137i
$366$		0			0
$367$			15.2971i			0.798500i	0.916842	+	0.399250i	$0.130729\pi$
$367$			15.2971i			0.798500i	−0.916842	+	0.399250i	$0.869271\pi$
$368$	20.3961			1.06322
$369$		0			0
$370$	13.0000	−	13.0000i	0.675838	−	0.675838i
$371$	13.0000	+	13.0000i	0.674926	+	0.674926i
$372$		0			0
$373$		−	20.3961i		−	1.05607i	−0.849223	−	0.528034i	$-0.822929\pi$
$373$		−	20.3961i		−	1.05607i	0.849223	−	0.528034i	$-0.177071\pi$
$374$		−	6.00000i		−	0.310253i
$375$		0			0
$376$	−10.1980			−0.525924
$377$	−13.0000	+	13.0000i	−0.669534	+	0.669534i
$378$		0			0
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.202057	+	0.979374i	$0.564763\pi$
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.979374	+	0.202057i	$0.935237\pi$
$380$			20.3961i			1.04630i
$381$		0			0
$382$	25.4951	+	25.4951i	1.30444	+	1.30444i
$383$	7.64853	+	7.64853i	0.390822	+	0.390822i	0.874980	−	0.484159i	$-0.160874\pi$
$383$	7.64853	+	7.64853i	0.390822	+	0.390822i	−0.484159	+	0.874980i	$0.660874\pi$
$384$		0			0
$385$	−13.0000	−	13.0000i	−0.662541	−	0.662541i
$386$	18.0000			0.916176
$387$		0			0
$388$	14.0000	+	14.0000i	0.710742	+	0.710742i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		−	15.2971i		−	0.773606i
$392$	−12.0000	+	12.0000i	−0.606092	+	0.606092i
$393$		0			0
$394$	25.4951			1.28442
$395$	−13.0000	+	13.0000i	−0.654101	+	0.654101i
$396$		0			0
$397$	10.1980	−	10.1980i	0.511825	−	0.511825i	−0.403260	−	0.915085i	$-0.632123\pi$
$397$	10.1980	−	10.1980i	0.511825	−	0.511825i	0.915085	+	0.403260i	$0.132123\pi$
$398$	−25.4951	+	25.4951i	−1.27795	+	1.27795i
$399$		0			0
$400$		−	32.0000i		−	1.60000i
$401$	−1.00000	−	1.00000i	−0.0499376	−	0.0499376i	0.681697	−	0.731635i	$-0.261242\pi$
$401$	−1.00000	−	1.00000i	−0.0499376	−	0.0499376i	−0.731635	+	0.681697i	$0.761242\pi$
$402$		0			0
$403$		−	26.0000i		−	1.29515i
$404$		−	20.3961i		−	1.01474i
$405$		0			0
$406$	−26.0000			−1.29036
$407$	5.09902			0.252749
$408$		0			0
$409$	27.0000	−	27.0000i	1.33506	−	1.33506i	0.434292	−	0.900772i	$-0.356999\pi$
$409$	27.0000	−	27.0000i	1.33506	−	1.33506i	0.900772	−	0.434292i	$-0.143001\pi$
$410$	30.5941	−	30.5941i	1.51094	−	1.51094i
$411$		0			0
$412$		−	10.1980i		−	0.502421i
$413$	40.7922			2.00725
$414$		0			0
$415$	25.4951			1.25151
$416$	−20.3961			−1.00000
$417$		0			0
$418$	−4.00000	+	4.00000i	−0.195646	+	0.195646i
$419$	−5.00000			−0.244266			−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000			−0.244266			−0.122133	+	0.992514i	$0.538973\pi$
$420$		0			0
$421$	7.64853	+	7.64853i	0.372767	+	0.372767i	0.868484	−	0.495717i	$-0.165095\pi$
$421$	7.64853	+	7.64853i	0.372767	+	0.372767i	−0.495717	+	0.868484i	$0.665095\pi$
$422$	7.00000	−	7.00000i	0.340755	−	0.340755i
$423$		0			0
$424$	−10.1980	+	10.1980i	−0.495261	+	0.495261i
$425$	−24.0000			−1.16417
$426$		0			0
$427$		0			0
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$			5.09902i			0.245897i
$431$	−7.64853	−	7.64853i	−0.368417	−	0.368417i	0.498483	−	0.866900i	$-0.333891\pi$
$431$	−7.64853	−	7.64853i	−0.368417	−	0.368417i	−0.866900	+	0.498483i	$0.833891\pi$
$432$		0			0
$433$			19.0000i			0.913082i	0.889702	+	0.456541i	$0.150912\pi$
$433$			19.0000i			0.913082i	−0.889702	+	0.456541i	$0.849088\pi$
$434$	26.0000	−	26.0000i	1.24804	−	1.24804i
$435$		0			0
$436$	−5.09902	+	5.09902i	−0.244199	+	0.244199i
$437$	−10.1980	+	10.1980i	−0.487838	+	0.487838i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	10.1980	−	10.1980i	0.486172	−	0.486172i
$441$		0			0
$442$			15.2971i			0.727607i
$443$	31.0000			1.47285			0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000			1.47285			0.736427	+	0.676517i	$0.236511\pi$
$444$		0			0
$445$		−	10.1980i		−	0.483433i
$446$	−15.2971			−0.724337
$447$		0			0
$448$	−20.3961	−	20.3961i	−0.963624	−	0.963624i
$449$	−3.00000	−	3.00000i	−0.141579	−	0.141579i	0.632765	−	0.774344i	$-0.281920\pi$
$449$	−3.00000	−	3.00000i	−0.141579	−	0.141579i	−0.774344	+	0.632765i	$0.781920\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$			8.00000i			0.376288i
$453$		0			0
$454$		−	24.0000i		−	1.12638i
$455$	33.1436	+	33.1436i	1.55380	+	1.55380i
$456$		0			0
$457$	−2.00000	+	2.00000i	−0.0935561	+	0.0935561i	−0.752336	−	0.658780i	$-0.771073\pi$
$457$	−2.00000	+	2.00000i	−0.0935561	+	0.0935561i	0.658780	+	0.752336i	$0.271073\pi$
$458$		−	5.09902i		−	0.238262i
$459$		0			0
$460$	26.0000	−	26.0000i	1.21226	−	1.21226i
$461$	17.8466	+	17.8466i	0.831198	+	0.831198i	0.987681	−	0.156483i	$-0.0500157\pi$
$461$	17.8466	+	17.8466i	0.831198	+	0.831198i	−0.156483	+	0.987681i	$0.550016\pi$
$462$		0			0
$463$	−25.4951	+	25.4951i	−1.18486	+	1.18486i	−0.206387	+	0.978470i	$0.566171\pi$
$463$	−25.4951	+	25.4951i	−1.18486	+	1.18486i	−0.978470	+	0.206387i	$0.933829\pi$
$464$		−	20.3961i		−	0.946864i
$465$		0			0
$466$	11.0000	+	11.0000i	0.509565	+	0.509565i
$467$		−	2.00000i		−	0.0925490i	−0.998929	−	0.0462745i	$-0.985265\pi$
$467$		−	2.00000i		−	0.0925490i	0.998929	−	0.0462745i	$-0.0147349\pi$
$468$		0			0
$469$			15.2971i			0.706353i
$470$	−13.0000	+	13.0000i	−0.599645	+	0.599645i
$471$		0			0
$472$			32.0000i			1.47292i
$473$	−1.00000	+	1.00000i	−0.0459800	+	0.0459800i
$474$		0			0
$475$	16.0000	+	16.0000i	0.734130	+	0.734130i
$476$	−15.2971	+	15.2971i	−0.701140	+	0.701140i
$477$		0			0
$478$	5.09902			0.233224
$479$	−2.54951	+	2.54951i	−0.116490	+	0.116490i	−0.762949	−	0.646459i	$-0.776249\pi$
$479$	−2.54951	+	2.54951i	−0.116490	+	0.116490i	0.646459	+	0.762949i	$0.276249\pi$
$480$		0			0
$481$	−13.0000			−0.592749
$482$	−28.0000			−1.27537
$483$		0			0
$484$	−18.0000			−0.818182
$485$	35.6931			1.62074
$486$		0			0
$487$	25.4951	+	25.4951i	1.15529	+	1.15529i	0.985476	+	0.169818i	$0.0543179\pi$
$487$	25.4951	+	25.4951i	1.15529	+	1.15529i	0.169818	+	0.985476i	$0.445682\pi$
$488$		0			0
$489$		0			0
$490$			30.5941i			1.38210i
$491$		−	5.00000i		−	0.225647i	−0.993615	−	0.112823i	$-0.964011\pi$
$491$		−	5.00000i		−	0.225647i	0.993615	−	0.112823i	$-0.0359894\pi$
$492$		0			0
$493$	−15.2971			−0.688945
$494$	10.1980	−	10.1980i	0.458831	−	0.458831i
$495$		0			0
$496$	20.3961	+	20.3961i	0.915811	+	0.915811i
$497$	39.0000			1.74939
$498$		0			0
$499$	2.00000	−	2.00000i	0.0895323	−	0.0895323i	−0.660922	−	0.750454i	$-0.729835\pi$
$499$	2.00000	−	2.00000i	0.0895323	−	0.0895323i	0.750454	+	0.660922i	$0.229835\pi$
$500$	−15.2971	−	15.2971i	−0.684105	−	0.684105i
$501$		0			0
$502$	−10.0000	−	10.0000i	−0.446322	−	0.446322i
$503$			20.3961i			0.909416i	0.890641	+	0.454708i	$0.150256\pi$
$503$			20.3961i			0.909416i	−0.890641	+	0.454708i	$0.849744\pi$
$504$		0			0
$505$	−26.0000	−	26.0000i	−1.15698	−	1.15698i
$506$	10.1980			0.453358
$507$		0			0
$508$			30.5941i			1.35739i
$509$	−10.1980	−	10.1980i	−0.452020	−	0.452020i	0.444004	−	0.896025i	$-0.353557\pi$
$509$	−10.1980	−	10.1980i	−0.452020	−	0.452020i	−0.896025	+	0.444004i	$0.853557\pi$
$510$		0			0
$511$	30.5941			1.35340
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	3.00000	+	3.00000i	0.132324	+	0.132324i
$515$	−13.0000	−	13.0000i	−0.572848	−	0.572848i
$516$		0			0
$517$	−5.09902			−0.224255
$518$	−13.0000	−	13.0000i	−0.571187	−	0.571187i
$519$		0			0
$520$	−26.0000	+	26.0000i	−1.14018	+	1.14018i
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	−6.00000			−0.262362			−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000			−0.262362			−0.131181	+	0.991358i	$0.541877\pi$
$524$			14.0000i			0.611593i
$525$		0			0
$526$	20.3961	+	20.3961i	0.889311	+	0.889311i
$527$	15.2971	−	15.2971i	0.666350	−	0.666350i
$528$		0			0
$529$	3.00000			0.130435
$530$			26.0000i			1.12937i
$531$		0			0
$532$	20.3961			0.884282
$533$	−30.5941			−1.32518
$534$		0			0
$535$	−5.09902	−	5.09902i	−0.220450	−	0.220450i
$536$	−12.0000			−0.518321
$537$		0			0
$538$	−20.3961	−	20.3961i	−0.879337	−	0.879337i
$539$	−6.00000	+	6.00000i	−0.258438	+	0.258438i
$540$		0			0
$541$	−17.8466	+	17.8466i	−0.767284	+	0.767284i	−0.977628	−	0.210344i	$-0.932542\pi$
$541$	−17.8466	+	17.8466i	−0.767284	+	0.767284i	0.210344	+	0.977628i	$0.432542\pi$
$542$		−	15.2971i		−	0.657065i
$543$		0			0
$544$	−12.0000	−	12.0000i	−0.514496	−	0.514496i
$545$			13.0000i			0.556859i
$546$		0			0
$547$	23.0000			0.983409			0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000			0.983409			0.491704	+	0.870762i	$0.336374\pi$
$548$	10.0000	−	10.0000i	0.427179	−	0.427179i
$549$		0			0
$550$		−	16.0000i		−	0.682242i
$551$	10.1980	+	10.1980i	0.434451	+	0.434451i
$552$		0			0
$553$	13.0000	+	13.0000i	0.552816	+	0.552816i
$554$	10.1980	−	10.1980i	0.433273	−	0.433273i
$555$		0			0
$556$			30.0000i			1.27228i
$557$	−22.9456	+	22.9456i	−0.972236	+	0.972236i	−0.999625	−	0.0273891i	$-0.991281\pi$
$557$	−22.9456	+	22.9456i	−0.972236	+	0.972236i	0.0273891	+	0.999625i	$0.491281\pi$
$558$		0			0
$559$	2.54951	−	2.54951i	0.107833	−	0.107833i
$560$	−52.0000			−2.19740
$561$		0			0
$562$	−32.0000			−1.34984
$563$		−	9.00000i		−	0.379305i	−0.981851	−	0.189652i	$-0.939264\pi$
$563$		−	9.00000i		−	0.379305i	0.981851	−	0.189652i	$-0.0607361\pi$
$564$		0			0
$565$	10.1980	+	10.1980i	0.429035	+	0.429035i
$566$	−16.0000	−	16.0000i	−0.672530	−	0.672530i
$567$		0			0
$568$			30.5941i			1.28370i
$569$		−	31.0000i		−	1.29959i	−0.760111	−	0.649794i	$-0.774855\pi$
$569$		−	31.0000i		−	1.29959i	0.760111	−	0.649794i	$-0.225145\pi$
$570$		0			0
$571$			15.0000i			0.627730i	0.949468	+	0.313865i	$0.101624\pi$
$571$			15.0000i			0.627730i	−0.949468	+	0.313865i	$0.898376\pi$
$572$	−10.1980			−0.426401
$573$		0			0
$574$	−30.5941	−	30.5941i	−1.27697	−	1.27697i
$575$		−	40.7922i		−	1.70115i
$576$		0			0
$577$	18.0000	−	18.0000i	0.749350	−	0.749350i	−0.225007	−	0.974357i	$-0.572241\pi$
$577$	18.0000	−	18.0000i	0.749350	−	0.749350i	0.974357	+	0.225007i	$0.0722406\pi$
$578$	8.00000	−	8.00000i	0.332756	−	0.332756i
$579$		0			0
$580$	−26.0000	−	26.0000i	−1.07959	−	1.07959i
$581$		−	25.4951i		−	1.05771i
$582$		0			0
$583$	−5.09902	+	5.09902i	−0.211180	+	0.211180i
$584$			24.0000i			0.993127i
$585$		0			0
$586$		−	25.4951i		−	1.05319i
$587$	7.00000	−	7.00000i	0.288921	−	0.288921i	−0.547733	−	0.836653i	$-0.684509\pi$
$587$	7.00000	−	7.00000i	0.288921	−	0.288921i	0.836653	+	0.547733i	$0.184509\pi$
$588$		0			0
$589$	−20.3961			−0.840406
$590$	40.7922	+	40.7922i	1.67939	+	1.67939i
$591$		0			0
$592$	10.1980	−	10.1980i	0.419137	−	0.419137i
$593$	−29.0000	−	29.0000i	−1.19089	−	1.19089i	−0.976819	−	0.214069i	$-0.931328\pi$
$593$	−29.0000	−	29.0000i	−1.19089	−	1.19089i	−0.214069	−	0.976819i	$-0.568672\pi$
$594$		0			0
$595$			39.0000i			1.59884i
$596$	−20.3961	+	20.3961i	−0.835456	+	0.835456i
$597$		0			0
$598$	−26.0000			−1.06322
$599$			30.5941i			1.25004i	0.780608	+	0.625021i	$0.214910\pi$
$599$			30.5941i			1.25004i	−0.780608	+	0.625021i	$0.785090\pi$
$600$		0			0
$601$	37.0000			1.50926			0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000			1.50926			0.754631	+	0.656150i	$0.227816\pi$
$602$	5.09902			0.207821
$603$		0			0
$604$	−15.2971	−	15.2971i	−0.622428	−	0.622428i
$605$	−22.9456	+	22.9456i	−0.932871	+	0.932871i
$606$		0			0
$607$		−	10.1980i		−	0.413926i	−0.978349	−	0.206963i	$-0.933642\pi$
$607$		−	10.1980i		−	0.413926i	0.978349	−	0.206963i	$-0.0663579\pi$
$608$			16.0000i			0.648886i
$609$		0			0
$610$		0			0
$611$	13.0000			0.525924
$612$		0			0
$613$	5.09902	+	5.09902i	0.205947	+	0.205947i	0.802542	−	0.596595i	$-0.203480\pi$
$613$	5.09902	+	5.09902i	0.205947	+	0.205947i	−0.596595	+	0.802542i	$0.703480\pi$
$614$	−20.0000			−0.807134
$615$		0			0
$616$	−10.1980	−	10.1980i	−0.410891	−	0.410891i
$617$	12.0000	−	12.0000i	0.483102	−	0.483102i	−0.423019	−	0.906121i	$-0.639030\pi$
$617$	12.0000	−	12.0000i	0.483102	−	0.483102i	0.906121	+	0.423019i	$0.139030\pi$
$618$		0			0
$619$	−22.0000	−	22.0000i	−0.884255	−	0.884255i	0.109709	−	0.993964i	$-0.465008\pi$
$619$	−22.0000	−	22.0000i	−0.884255	−	0.884255i	−0.993964	+	0.109709i	$0.965008\pi$
$620$	52.0000			2.08837
$621$		0			0
$622$	10.1980	−	10.1980i	0.408904	−	0.408904i
$623$	−10.1980			−0.408576
$624$		0			0
$625$	1.00000			0.0400000
$626$	−31.0000	+	31.0000i	−1.23901	+	1.23901i
$627$		0			0
$628$	−20.3961			−0.813892
$629$	−7.64853	−	7.64853i	−0.304967	−	0.304967i
$630$		0			0
$631$	−17.8466	+	17.8466i	−0.710461	+	0.710461i	−0.966632	−	0.256171i	$-0.917539\pi$
$631$	−17.8466	+	17.8466i	−0.710461	+	0.710461i	0.256171	+	0.966632i	$0.417539\pi$
$632$	−10.1980	+	10.1980i	−0.405656	+	0.405656i
$633$		0			0
$634$		0			0
$635$	39.0000	+	39.0000i	1.54767	+	1.54767i
$636$		0			0
$637$	15.2971	−	15.2971i	0.606092	−	0.606092i
$638$		−	10.1980i		−	0.403744i
$639$		0			0
$640$		−	40.7922i		−	1.61245i
$641$			40.0000i			1.57991i	0.613168	+	0.789953i	$0.289895\pi$
$641$			40.0000i			1.57991i	−0.613168	+	0.789953i	$0.710105\pi$
$642$		0			0
$643$	15.0000	−	15.0000i	0.591542	−	0.591542i	−0.346506	−	0.938048i	$-0.612632\pi$
$643$	15.0000	−	15.0000i	0.591542	−	0.591542i	0.938048	+	0.346506i	$0.112632\pi$
$644$	−26.0000	−	26.0000i	−1.02454	−	1.02454i
$645$		0			0
$646$	12.0000			0.472134
$647$	−10.1980			−0.400926			−0.200463	−	0.979701i	$-0.564245\pi$
$647$	−10.1980			−0.400926			−0.200463	+	0.979701i	$0.564245\pi$
$648$		0			0
$649$			16.0000i			0.628055i
$650$			40.7922i			1.60000i
$651$		0			0
$652$	18.0000	−	18.0000i	0.704934	−	0.704934i
$653$		−	30.5941i		−	1.19724i	−0.801033	−	0.598620i	$-0.795716\pi$
$653$		−	30.5941i		−	1.19724i	0.801033	−	0.598620i	$-0.204284\pi$
$654$		0			0
$655$	17.8466	+	17.8466i	0.697323	+	0.697323i
$656$	24.0000	−	24.0000i	0.937043	−	0.937043i
$657$		0			0
$658$	13.0000	+	13.0000i	0.506793	+	0.506793i
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−5.09902	+	5.09902i	−0.198329	+	0.198329i	−0.799283	−	0.600954i	$-0.794787\pi$
$661$	−5.09902	+	5.09902i	−0.198329	+	0.198329i	0.600954	+	0.799283i	$0.294787\pi$
$662$		−	42.0000i		−	1.63238i
$663$		0			0
$664$	20.0000			0.776151
$665$	26.0000	−	26.0000i	1.00824	−	1.00824i
$666$		0			0
$667$		−	26.0000i		−	1.00672i
$668$	−30.5941	−	30.5941i	−1.18372	−	1.18372i
$669$		0			0
$670$	−15.2971	+	15.2971i	−0.590977	+	0.590977i
$671$		0			0
$672$		0			0
$673$		−	21.0000i		−	0.809491i	−0.914429	−	0.404745i	$-0.867360\pi$
$673$		−	21.0000i		−	0.809491i	0.914429	−	0.404745i	$-0.132640\pi$
$674$	17.0000	+	17.0000i	0.654816	+	0.654816i
$675$		0			0
$676$	26.0000			1.00000
$677$		−	15.2971i		−	0.587914i	−0.955819	−	0.293957i	$-0.905028\pi$
$677$		−	15.2971i		−	0.587914i	0.955819	−	0.293957i	$-0.0949722\pi$
$678$		0			0
$679$		−	35.6931i		−	1.36978i
$680$	−30.5941			−1.17323
$681$		0			0
$682$	10.1980	+	10.1980i	0.390503	+	0.390503i
$683$	−29.0000	−	29.0000i	−1.10965	−	1.10965i	−0.993196	−	0.116459i	$-0.962846\pi$
$683$	−29.0000	−	29.0000i	−1.10965	−	1.10965i	−0.116459	−	0.993196i	$-0.537154\pi$
$684$		0			0
$685$		−	25.4951i		−	0.974118i
$686$	−5.09902			−0.194681
$687$		0			0
$688$			4.00000i			0.152499i
$689$	13.0000	−	13.0000i	0.495261	−	0.495261i
$690$		0			0
$691$	6.00000	−	6.00000i	0.228251	−	0.228251i	−0.583711	−	0.811962i	$-0.698400\pi$
$691$	6.00000	−	6.00000i	0.228251	−	0.228251i	0.811962	+	0.583711i	$0.198400\pi$
$692$			10.1980i			0.387671i
$693$		0			0
$694$	7.00000	−	7.00000i	0.265716	−	0.265716i
$695$	38.2426	+	38.2426i	1.45063	+	1.45063i
$696$		0			0
$697$	−18.0000	−	18.0000i	−0.681799	−	0.681799i
$698$			45.8912i			1.73701i
$699$		0			0
$700$	−40.7922	+	40.7922i	−1.54180	+	1.54180i
$701$	−15.2971			−0.577762			−0.288881	−	0.957365i	$-0.593283\pi$
$701$	−15.2971			−0.577762			−0.288881	+	0.957365i	$0.593283\pi$
$702$		0			0
$703$			10.1980i			0.384626i
$704$	8.00000	−	8.00000i	0.301511	−	0.301511i
$705$		0			0
$706$			10.0000i			0.376355i
$707$	−26.0000	+	26.0000i	−0.977831	+	0.977831i
$708$		0			0
$709$	−10.1980	+	10.1980i	−0.382995	+	0.382995i	−0.872180	−	0.489185i	$-0.837294\pi$
$709$	−10.1980	+	10.1980i	−0.382995	+	0.382995i	0.489185	+	0.872180i	$0.337294\pi$
$710$	39.0000	+	39.0000i	1.46364	+	1.46364i
$711$		0			0
$712$		−	8.00000i		−	0.299813i
$713$	26.0000	+	26.0000i	0.973708	+	0.973708i
$714$		0			0
$715$	−13.0000	+	13.0000i	−0.486172	+	0.486172i
$716$	18.0000			0.672692
$717$		0			0
$718$			30.5941i			1.14176i
$719$	−25.4951			−0.950807			−0.475403	−	0.879768i	$-0.657698\pi$
$719$	−25.4951			−0.950807			−0.475403	+	0.879768i	$0.657698\pi$
$720$		0			0
$721$	−13.0000	+	13.0000i	−0.484145	+	0.484145i
$722$	11.0000	+	11.0000i	0.409378	+	0.409378i
$723$		0			0
$724$		−	30.5941i		−	1.13702i
$725$	−40.7922			−1.51498
$726$		0			0
$727$	−15.2971			−0.567336			−0.283668	−	0.958922i	$-0.591551\pi$
$727$	−15.2971			−0.567336			−0.283668	+	0.958922i	$0.591551\pi$
$728$	26.0000	+	26.0000i	0.963624	+	0.963624i
$729$		0			0
$730$	30.5941	+	30.5941i	1.13234	+	1.13234i
$731$	3.00000			0.110959
$732$		0			0
$733$	−33.1436	−	33.1436i	−1.22419	−	1.22419i	−0.966129	−	0.258058i	$-0.916917\pi$
$733$	−33.1436	−	33.1436i	−1.22419	−	1.22419i	−0.258058	−	0.966129i	$-0.583083\pi$
$734$	15.2971	+	15.2971i	0.564625	+	0.564625i
$735$		0			0
$736$	20.3961	−	20.3961i	0.751809	−	0.751809i
$737$	−6.00000			−0.221013
$738$		0			0
$739$	−2.00000	−	2.00000i	−0.0735712	−	0.0735712i	0.669364	−	0.742935i	$-0.266567\pi$
$739$	−2.00000	−	2.00000i	−0.0735712	−	0.0735712i	−0.742935	+	0.669364i	$0.766567\pi$
$740$		−	26.0000i		−	0.955779i
$741$		0			0
$742$	26.0000			0.954490
$743$	7.64853	+	7.64853i	0.280597	+	0.280597i	0.833347	−	0.552750i	$-0.186422\pi$
$743$	7.64853	+	7.64853i	0.280597	+	0.280597i	−0.552750	+	0.833347i	$0.686422\pi$
$744$		0			0
$745$			52.0000i			1.90513i
$746$	−20.3961	−	20.3961i	−0.746753	−	0.746753i
$747$		0			0
$748$	−6.00000	−	6.00000i	−0.219382	−	0.219382i
$749$	−5.09902	+	5.09902i	−0.186314	+	0.186314i
$750$		0			0
$751$	−10.1980			−0.372132			−0.186066	−	0.982537i	$-0.559574\pi$
$751$	−10.1980			−0.372132			−0.186066	+	0.982537i	$0.559574\pi$
$752$	−10.1980	+	10.1980i	−0.371884	+	0.371884i
$753$		0			0
$754$			26.0000i			0.946864i
$755$	−39.0000			−1.41936
$756$		0			0
$757$		−	35.6931i		−	1.29729i	−0.761091	−	0.648645i	$-0.775336\pi$
$757$		−	35.6931i		−	1.29729i	0.761091	−	0.648645i	$-0.224664\pi$
$758$			46.0000i			1.67080i
$759$		0			0
$760$	20.3961	+	20.3961i	0.739844	+	0.739844i
$761$	4.00000	+	4.00000i	0.145000	+	0.145000i	0.775880	−	0.630880i	$-0.217306\pi$
$761$	4.00000	+	4.00000i	0.145000	+	0.145000i	−0.630880	+	0.775880i	$0.717306\pi$
$762$		0			0
$763$	13.0000			0.470632
$764$	50.9902			1.84476
$765$		0			0
$766$	15.2971			0.552705
$767$		−	40.7922i		−	1.47292i
$768$		0			0
$769$	−8.00000	+	8.00000i	−0.288487	+	0.288487i	−0.836482	−	0.547995i	$-0.815391\pi$
$769$	−8.00000	+	8.00000i	−0.288487	+	0.288487i	0.547995	+	0.836482i	$0.315391\pi$
$770$	−26.0000			−0.936975
$771$		0			0
$772$	18.0000	−	18.0000i	0.647834	−	0.647834i
$773$	7.64853	+	7.64853i	0.275098	+	0.275098i	0.831149	−	0.556050i	$-0.187684\pi$
$773$	7.64853	+	7.64853i	0.275098	+	0.275098i	−0.556050	+	0.831149i	$0.687684\pi$
$774$		0			0
$775$	40.7922	−	40.7922i	1.46530	−	1.46530i
$776$	28.0000			1.00514
$777$		0			0
$778$		0			0
$779$			24.0000i			0.859889i
$780$		0			0
$781$			15.2971i			0.547372i
$782$	−15.2971	−	15.2971i	−0.547022	−	0.547022i
$783$		0			0
$784$			24.0000i			0.857143i
$785$	−26.0000	+	26.0000i	−0.927980	+	0.927980i
$786$		0			0		</

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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} +(-2.00000 - 2.00000i) q^{8} -5.09902 q^{10} +(-1.00000 - 1.00000i) q^{11} +(2.54951 + 2.54951i) q^{13} +5.09902i q^{14} -4.00000 q^{16} +3.00000i q^{17} +(2.00000 - 2.00000i) q^{19} +(-5.09902 + 5.09902i) q^{20} -2.00000 q^{22} -5.09902 q^{23} +8.00000i q^{25} +5.09902 q^{26} +(5.09902 + 5.09902i) q^{28} +5.09902i q^{29} +(-5.09902 - 5.09902i) q^{31} +(-4.00000 + 4.00000i) q^{32} +(3.00000 + 3.00000i) q^{34} +13.0000 q^{35} +(-2.54951 + 2.54951i) q^{37} -4.00000i q^{38} +10.1980i q^{40} +(-6.00000 + 6.00000i) q^{41} -1.00000i q^{43} +(-2.00000 + 2.00000i) q^{44} +(-5.09902 + 5.09902i) q^{46} +(2.54951 - 2.54951i) q^{47} -6.00000i q^{49} +(8.00000 + 8.00000i) q^{50} +(5.09902 - 5.09902i) q^{52} -5.09902i q^{53} +5.09902i q^{55} +10.1980 q^{56} +(5.09902 + 5.09902i) q^{58} +(-8.00000 - 8.00000i) q^{59} -10.1980 q^{62} +8.00000i q^{64} -13.0000i q^{65} +(3.00000 - 3.00000i) q^{67} +6.00000 q^{68} +(13.0000 - 13.0000i) q^{70} +(-7.64853 - 7.64853i) q^{71} +(-6.00000 - 6.00000i) q^{73} +5.09902i q^{74} +(-4.00000 - 4.00000i) q^{76} +5.09902 q^{77} -5.09902i q^{79} +(10.1980 + 10.1980i) q^{80} +12.0000i q^{82} +(-5.00000 + 5.00000i) q^{83} +(7.64853 - 7.64853i) q^{85} +(-1.00000 - 1.00000i) q^{86} +4.00000i q^{88} +(2.00000 + 2.00000i) q^{89} -13.0000 q^{91} +10.1980i q^{92} -5.09902i q^{94} -10.1980 q^{95} +(-7.00000 + 7.00000i) q^{97} +(-6.00000 - 6.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 8 * q^8	\( 4 q + 4 q^{2} - 8 q^{8} - 4 q^{11} - 16 q^{16} + 8 q^{19} - 8 q^{22} - 16 q^{32} + 12 q^{34} + 52 q^{35} - 24 q^{41} - 8 q^{44} + 32 q^{50} - 32 q^{59} + 12 q^{67} + 24 q^{68} + 52 q^{70} - 24 q^{73} - 16 q^{76} - 20 q^{83} - 4 q^{86} + 8 q^{89} - 52 q^{91} - 28 q^{97} - 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^8 - 4 * q^11 - 16 * q^16 + 8 * q^19 - 8 * q^22 - 16 * q^32 + 12 * q^34 + 52 * q^35 - 24 * q^41 - 8 * q^44 + 32 * q^50 - 32 * q^59 + 12 * q^67 + 24 * q^68 + 52 * q^70 - 24 * q^73 - 16 * q^76 - 20 * q^83 - 4 * q^86 + 8 * q^89 - 52 * q^91 - 28 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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