LMFDB - Embedded newform 416.2.u.a.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(47,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("416.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.u (of order $4$, 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:	$3.32177672409$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.1
Root			$-2.54951 - 2.54951i$ of defining polynomial
Character	$\chi$	$=$	416.239
Dual form			416.2.u.a.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} -2.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{3} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} -2.00000 q^{9} +(-1.00000 - 1.00000i) q^{11} +(-2.54951 - 2.54951i) q^{13} +(-2.54951 - 2.54951i) q^{15} -3.00000i q^{17} +(-2.00000 + 2.00000i) q^{19} +(-2.54951 + 2.54951i) q^{21} +5.09902 q^{23} +8.00000i q^{25} -5.00000 q^{27} +5.09902i q^{29} +(-5.09902 - 5.09902i) q^{31} +(-1.00000 - 1.00000i) q^{33} +13.0000 q^{35} +(2.54951 - 2.54951i) q^{37} +(-2.54951 - 2.54951i) q^{39} +(6.00000 - 6.00000i) q^{41} +1.00000i q^{43} +(5.09902 + 5.09902i) q^{45} +(-2.54951 + 2.54951i) q^{47} -6.00000i q^{49} -3.00000i q^{51} -5.09902i q^{53} +5.09902i q^{55} +(-2.00000 + 2.00000i) q^{57} +(-8.00000 - 8.00000i) q^{59} +(5.09902 - 5.09902i) q^{63} +13.0000i q^{65} +(-3.00000 + 3.00000i) q^{67} +5.09902 q^{69} +(7.64853 + 7.64853i) q^{71} +(-6.00000 - 6.00000i) q^{73} +8.00000i q^{75} +5.09902 q^{77} -5.09902i q^{79} +1.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} +(-7.64853 + 7.64853i) q^{85} +5.09902i q^{87} +(-2.00000 - 2.00000i) q^{89} +13.0000 q^{91} +(-5.09902 - 5.09902i) q^{93} +10.1980 q^{95} +(-7.00000 + 7.00000i) q^{97} +(2.00000 + 2.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 8 * q^9	$ 4 q + 4 q^{3} - 8 q^{9} - 4 q^{11} - 8 q^{19} - 20 q^{27} - 4 q^{33} + 52 q^{35} + 24 q^{41} - 8 q^{57} - 32 q^{59} - 12 q^{67} - 24 q^{73} + 4 q^{81} - 20 q^{83} - 8 q^{89} + 52 q^{91} - 28 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 8 * q^9 - 4 * q^11 - 8 * q^19 - 20 * q^27 - 4 * q^33 + 52 * q^35 + 24 * q^41 - 8 * q^57 - 32 * q^59 - 12 * q^67 - 24 * q^73 + 4 * q^81 - 20 * q^83 - 8 * q^89 + 52 * q^91 - 28 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$2$		0			0
$3$	1.00000			0.577350			0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000			0.577350			0.288675	+	0.957427i	$0.406785\pi$
$4$		0			0
$5$	−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	$0.488622\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	−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
$14$		0			0
$15$	−2.54951	−	2.54951i	−0.658281	−	0.658281i
$16$		0			0
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		0			0
$19$	−2.00000	+	2.00000i	−0.458831	+	0.458831i	−0.898272	−	0.439440i	$-0.855177\pi$
$19$	−2.00000	+	2.00000i	−0.458831	+	0.458831i	0.439440	+	0.898272i	$0.355177\pi$
$20$		0			0
$21$	−2.54951	+	2.54951i	−0.556349	+	0.556349i
$22$		0			0
$23$	5.09902			1.06322			0.531610	−	0.846990i	$-0.321587\pi$
$23$	5.09902			1.06322			0.531610	+	0.846990i	$0.321587\pi$
$24$		0			0
$25$			8.00000i			1.60000i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$			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	$0.974217\pi$
$32$		0			0
$33$	−1.00000	−	1.00000i	−0.174078	−	0.174078i
$34$		0			0
$35$	13.0000			2.19740
$36$		0			0
$37$	2.54951	−	2.54951i	0.419137	−	0.419137i	−0.465769	−	0.884906i	$-0.654222\pi$
$37$	2.54951	−	2.54951i	0.419137	−	0.419137i	0.884906	+	0.465769i	$0.154222\pi$
$38$		0			0
$39$	−2.54951	−	2.54951i	−0.408248	−	0.408248i
$40$		0			0
$41$	6.00000	−	6.00000i	0.937043	−	0.937043i	−0.0610897	−	0.998132i	$-0.519458\pi$
$41$	6.00000	−	6.00000i	0.937043	−	0.937043i	0.998132	+	0.0610897i	$0.0194576\pi$
$42$		0			0
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$		0			0
$45$	5.09902	+	5.09902i	0.760117	+	0.760117i
$46$		0			0
$47$	−2.54951	+	2.54951i	−0.371884	+	0.371884i	−0.868163	−	0.496279i	$-0.834699\pi$
$47$	−2.54951	+	2.54951i	−0.371884	+	0.371884i	0.496279	+	0.868163i	$0.334699\pi$
$48$		0			0
$49$		−	6.00000i		−	0.857143i
$50$		0			0
$51$		−	3.00000i		−	0.420084i
$52$		0			0
$53$		−	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
$56$		0			0
$57$	−2.00000	+	2.00000i	−0.264906	+	0.264906i
$58$		0			0
$59$	−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			$\pi$
$62$		0			0
$63$	5.09902	−	5.09902i	0.642416	−	0.642416i
$64$		0			0
$65$			13.0000i			1.61245i
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	−0.866202	−	0.499694i	$-0.833446\pi$
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	0.499694	+	0.866202i	$0.333446\pi$
$68$		0			0
$69$	5.09902			0.613850
$70$		0			0
$71$	7.64853	+	7.64853i	0.907713	+	0.907713i	0.996087	−	0.0883739i	$-0.0281670\pi$
$71$	7.64853	+	7.64853i	0.907713	+	0.907713i	−0.0883739	+	0.996087i	$0.528167\pi$
$72$		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	$0.915405\pi$
$74$		0			0
$75$			8.00000i			0.923760i
$76$		0			0
$77$	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	$-0.0926056\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−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
$86$		0			0
$87$			5.09902i			0.546672i
$88$		0			0
$89$	−2.00000	−	2.00000i	−0.212000	−	0.212000i	0.593117	−	0.805116i	$-0.297897\pi$
$89$	−2.00000	−	2.00000i	−0.212000	−	0.212000i	−0.805116	+	0.593117i	$0.797897\pi$
$90$		0			0
$91$	13.0000			1.36277
$92$		0			0
$93$	−5.09902	−	5.09902i	−0.528744	−	0.528744i
$94$		0			0
$95$	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	$0.417612\pi$
$98$		0			0
$99$	2.00000	+	2.00000i	0.201008	+	0.201008i
$100$		0			0
$101$	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	$0.419171\pi$
$104$		0			0
$105$	13.0000			1.26867
$106$		0			0
$107$	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	0.818585	−	0.574386i	$-0.194759\pi$
$109$	2.54951	+	2.54951i	0.244199	+	0.244199i	−0.574386	+	0.818585i	$0.694759\pi$
$110$		0			0
$111$	2.54951	−	2.54951i	0.241989	−	0.241989i
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−13.0000	−	13.0000i	−1.21226	−	1.21226i
$116$		0			0
$117$	5.09902	+	5.09902i	0.471405	+	0.471405i
$118$		0			0
$119$	7.64853	+	7.64853i	0.701140	+	0.701140i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$	6.00000	−	6.00000i	0.541002	−	0.541002i
$124$		0			0
$125$	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	$0.737455\pi$
$128$		0			0
$129$			1.00000i			0.0880451i
$130$		0			0
$131$	−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
$134$		0			0
$135$	12.7475	+	12.7475i	1.09713	+	1.09713i
$136$		0			0
$137$	−5.00000	−	5.00000i	−0.427179	−	0.427179i	0.460487	−	0.887666i	$-0.347675\pi$
$137$	−5.00000	−	5.00000i	−0.427179	−	0.427179i	−0.887666	+	0.460487i	$0.847675\pi$
$138$		0			0
$139$	15.0000			1.27228			0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000			1.27228			0.636142	+	0.771572i	$0.280529\pi$
$140$		0			0
$141$	−2.54951	+	2.54951i	−0.214707	+	0.214707i
$142$		0			0
$143$			5.09902i			0.426401i
$144$		0			0
$145$	13.0000	−	13.0000i	1.07959	−	1.07959i
$146$		0			0
$147$		−	6.00000i		−	0.494872i
$148$		0			0
$149$	−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	$0.104935\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$			26.0000i			2.08837i
$156$		0			0
$157$			10.1980i			0.813892i	0.913452	+	0.406946i	$0.133406\pi$
$157$			10.1980i			0.813892i	−0.913452	+	0.406946i	$0.866594\pi$
$158$		0			0
$159$		−	5.09902i		−	0.404379i
$160$		0			0
$161$	−13.0000	+	13.0000i	−1.02454	+	1.02454i
$162$		0			0
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	0.260531	−	0.965465i	$-0.416102\pi$
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	−0.965465	+	0.260531i	$0.916102\pi$
$164$		0			0
$165$			5.09902i			0.396958i
$166$		0			0
$167$	−15.2971	+	15.2971i	−1.18372	+	1.18372i	−0.204949	+	0.978773i	$0.565703\pi$
$167$	−15.2971	+	15.2971i	−1.18372	+	1.18372i	−0.978773	+	0.204949i	$0.934297\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	4.00000	−	4.00000i	0.305888	−	0.305888i
$172$		0			0
$173$	−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
$176$		0			0
$177$	−8.00000	−	8.00000i	−0.601317	−	0.601317i
$178$		0			0
$179$			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	$-0.692480\pi$
$181$	−15.2971			−1.13702			−0.568511	+	0.822676i	$0.692480\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.0000			−0.955779
$186$		0			0
$187$	−3.00000	+	3.00000i	−0.219382	+	0.219382i
$188$		0			0
$189$	12.7475	−	12.7475i	0.927248	−	0.927248i
$190$		0			0
$191$		−	25.4951i		−	1.84476i	−0.386283	−	0.922380i	$-0.626241\pi$
$191$		−	25.4951i		−	1.84476i	0.386283	−	0.922380i	$-0.373759\pi$
$192$		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	$0.598534\pi$
$194$		0			0
$195$			13.0000i			0.930949i
$196$		0			0
$197$	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	$0.859122\pi$
$200$		0			0
$201$	−3.00000	+	3.00000i	−0.211604	+	0.211604i
$202$		0			0
$203$	−13.0000	−	13.0000i	−0.912421	−	0.912421i
$204$		0			0
$205$	−30.5941			−2.13679
$206$		0			0
$207$	−10.1980			−0.708813
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−7.00000			−0.481900			−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000			−0.481900			−0.240950	+	0.970538i	$0.577459\pi$
$212$		0			0
$213$	7.64853	+	7.64853i	0.524069	+	0.524069i
$214$		0			0
$215$	2.54951	−	2.54951i	0.173875	−	0.173875i
$216$		0			0
$217$	26.0000			1.76500
$218$		0			0
$219$	−6.00000	−	6.00000i	−0.405442	−	0.405442i
$220$		0			0
$221$	−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	$0.867963\pi$
$224$		0			0
$225$		−	16.0000i		−	1.06667i
$226$		0			0
$227$	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	−0.786309	−	0.617833i	$-0.788011\pi$
$229$	−2.54951	+	2.54951i	−0.168476	+	0.168476i	0.617833	+	0.786309i	$0.288011\pi$
$230$		0			0
$231$	5.09902			0.335491
$232$		0			0
$233$		−	11.0000i		−	0.720634i	−0.932830	−	0.360317i	$-0.882669\pi$
$233$		−	11.0000i		−	0.720634i	0.932830	−	0.360317i	$-0.117331\pi$
$234$		0			0
$235$	13.0000			0.848026
$236$		0			0
$237$		−	5.09902i		−	0.331217i
$238$		0			0
$239$	−2.54951	−	2.54951i	−0.164914	−	0.164914i	0.619826	−	0.784740i	$-0.287203\pi$
$239$	−2.54951	−	2.54951i	−0.164914	−	0.164914i	−0.784740	+	0.619826i	$0.787203\pi$
$240$		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	$0.970107\pi$
$242$		0			0
$243$	16.0000			1.02640
$244$		0			0
$245$	−15.2971	+	15.2971i	−0.977293	+	0.977293i
$246$		0			0
$247$	10.1980			0.648886
$248$		0			0
$249$	−5.00000	+	5.00000i	−0.316862	+	0.316862i
$250$		0			0
$251$		−	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
$254$		0			0
$255$	−7.64853	+	7.64853i	−0.478969	+	0.478969i
$256$		0			0
$257$		−	3.00000i		−	0.187135i	−0.995613	−	0.0935674i	$-0.970173\pi$
$257$		−	3.00000i		−	0.187135i	0.995613	−	0.0935674i	$-0.0298271\pi$
$258$		0			0
$259$			13.0000i			0.807781i
$260$		0			0
$261$		−	10.1980i		−	0.631243i
$262$		0			0
$263$		−	20.3961i		−	1.25768i	−0.777536	−	0.628838i	$-0.783531\pi$
$263$		−	20.3961i		−	1.25768i	0.777536	−	0.628838i	$-0.216469\pi$
$264$		0			0
$265$	−13.0000	+	13.0000i	−0.798584	+	0.798584i
$266$		0			0
$267$	−2.00000	−	2.00000i	−0.122398	−	0.122398i
$268$		0			0
$269$		−	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	$0.143446\pi$
$272$		0			0
$273$	13.0000			0.786796
$274$		0			0
$275$	8.00000	−	8.00000i	0.482418	−	0.482418i
$276$		0			0
$277$	−10.1980			−0.612741			−0.306370	−	0.951912i	$-0.599115\pi$
$277$	−10.1980			−0.612741			−0.306370	+	0.951912i	$0.599115\pi$
$278$		0			0
$279$	10.1980	+	10.1980i	0.610541	+	0.610541i
$280$		0			0
$281$	16.0000	+	16.0000i	0.954480	+	0.954480i	0.999008	−	0.0445282i	$-0.0141784\pi$
$281$	16.0000	+	16.0000i	0.954480	+	0.954480i	−0.0445282	+	0.999008i	$0.514178\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$	10.1980			0.604080
$286$		0			0
$287$			30.5941i			1.80591i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$	−7.00000	+	7.00000i	−0.410347	+	0.410347i
$292$		0			0
$293$	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
$296$		0			0
$297$	5.00000	+	5.00000i	0.290129	+	0.290129i
$298$		0			0
$299$	−13.0000	−	13.0000i	−0.751809	−	0.751809i
$300$		0			0
$301$	−2.54951	−	2.54951i	−0.146951	−	0.146951i
$302$		0			0
$303$	10.1980			0.585862
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.0000	+	10.0000i	0.570730	+	0.570730i	0.932332	−	0.361602i	$-0.117770\pi$
$307$	10.0000	+	10.0000i	0.570730	+	0.570730i	−0.361602	+	0.932332i	$0.617770\pi$
$308$		0			0
$309$	5.09902			0.290073
$310$		0			0
$311$	−10.1980			−0.578278			−0.289139	−	0.957287i	$-0.593369\pi$
$311$	−10.1980			−0.578278			−0.289139	+	0.957287i	$0.593369\pi$
$312$		0			0
$313$	−31.0000			−1.75222			−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000			−1.75222			−0.876112	+	0.482108i	$0.839871\pi$
$314$		0			0
$315$	−26.0000			−1.46493
$316$		0			0
$317$		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
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$	6.00000	+	6.00000i	0.333849	+	0.333849i
$324$		0			0
$325$	20.3961	−	20.3961i	1.13137	−	1.13137i
$326$		0			0
$327$	2.54951	+	2.54951i	0.140988	+	0.140988i
$328$		0			0
$329$		−	13.0000i		−	0.716713i
$330$		0			0
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.168576	+	0.985689i	$0.553917\pi$
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.985689	+	0.168576i	$0.946083\pi$
$332$		0			0
$333$	−5.09902	+	5.09902i	−0.279425	+	0.279425i
$334$		0			0
$335$	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	$0.846764\pi$
$338$		0			0
$339$	4.00000			0.217250
$340$		0			0
$341$			10.1980i			0.552255i
$342$		0			0
$343$	−2.54951	−	2.54951i	−0.137661	−	0.137661i
$344$		0			0
$345$	−13.0000	−	13.0000i	−0.699896	−	0.699896i
$346$		0			0
$347$	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	$-0.415082\pi$
$349$	22.9456	−	22.9456i	1.22825	−	1.22825i	0.964626	−	0.263624i	$-0.0849177\pi$
$350$		0			0
$351$	12.7475	+	12.7475i	0.680414	+	0.680414i
$352$		0			0
$353$	5.00000	−	5.00000i	0.266123	−	0.266123i	−0.561413	−	0.827536i	$-0.689742\pi$
$353$	5.00000	−	5.00000i	0.266123	−	0.266123i	0.827536	+	0.561413i	$0.189742\pi$
$354$		0			0
$355$		−	39.0000i		−	2.06991i
$356$		0			0
$357$	7.64853	+	7.64853i	0.404803	+	0.404803i
$358$		0			0
$359$	15.2971	−	15.2971i	0.807348	−	0.807348i	−0.176884	−	0.984232i	$-0.556602\pi$
$359$	15.2971	−	15.2971i	0.807348	−	0.807348i	0.984232	+	0.176884i	$0.0566017\pi$
$360$		0			0
$361$			11.0000i			0.578947i
$362$		0			0
$363$		−	9.00000i		−	0.472377i
$364$		0			0
$365$			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	$0.869271\pi$
$368$		0			0
$369$	−12.0000	+	12.0000i	−0.624695	+	0.624695i
$370$		0			0
$371$	13.0000	+	13.0000i	0.674926	+	0.674926i
$372$		0			0
$373$			20.3961i			1.05607i	0.849223	+	0.528034i	$0.177071\pi$
$373$			20.3961i			1.05607i	−0.849223	+	0.528034i	$0.822929\pi$
$374$		0			0
$375$	7.64853	−	7.64853i	0.394968	−	0.394968i
$376$		0			0
$377$	13.0000	−	13.0000i	0.669534	−	0.669534i
$378$		0			0
$379$	23.0000	−	23.0000i	1.18143	−	1.18143i	0.202057	−	0.979374i	$-0.435237\pi$
$379$	23.0000	−	23.0000i	1.18143	−	1.18143i	0.979374	−	0.202057i	$-0.0647626\pi$
$380$		0			0
$381$	−15.2971			−0.783692
$382$		0			0
$383$	−7.64853	−	7.64853i	−0.390822	−	0.390822i	0.484159	−	0.874980i	$-0.339126\pi$
$383$	−7.64853	−	7.64853i	−0.390822	−	0.390822i	−0.874980	+	0.484159i	$0.839126\pi$
$384$		0			0
$385$	−13.0000	−	13.0000i	−0.662541	−	0.662541i
$386$		0			0
$387$		−	2.00000i		−	0.101666i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		−	15.2971i		−	0.773606i
$392$		0			0
$393$	−7.00000			−0.353103
$394$		0			0
$395$	−13.0000	+	13.0000i	−0.654101	+	0.654101i
$396$		0			0
$397$	−10.1980	+	10.1980i	−0.511825	+	0.511825i	−0.915085	−	0.403260i	$-0.867877\pi$
$397$	−10.1980	+	10.1980i	−0.511825	+	0.511825i	0.403260	+	0.915085i	$0.367877\pi$
$398$		0			0
$399$		−	10.1980i		−	0.510541i
$400$		0			0
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	0.731635	−	0.681697i	$-0.238758\pi$
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	−0.681697	+	0.731635i	$0.738758\pi$
$402$		0			0
$403$			26.0000i			1.29515i
$404$		0			0
$405$	−2.54951	−	2.54951i	−0.126686	−	0.126686i
$406$		0			0
$407$	−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	$-0.143001\pi$
$410$		0			0
$411$	−5.00000	−	5.00000i	−0.246632	−	0.246632i
$412$		0			0
$413$	40.7922			2.00725
$414$		0			0
$415$	25.4951			1.25151
$416$		0			0
$417$	15.0000			0.734553
$418$		0			0
$419$	−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	0.495717	−	0.868484i	$-0.334905\pi$
$421$	−7.64853	−	7.64853i	−0.372767	−	0.372767i	−0.868484	+	0.495717i	$0.834905\pi$
$422$		0			0
$423$	5.09902	−	5.09902i	0.247923	−	0.247923i
$424$		0			0
$425$	24.0000			1.16417
$426$		0			0
$427$		0			0
$428$		0			0
$429$			5.09902i			0.246183i
$430$		0			0
$431$	7.64853	+	7.64853i	0.368417	+	0.368417i	0.866900	−	0.498483i	$-0.166109\pi$
$431$	7.64853	+	7.64853i	0.368417	+	0.368417i	−0.498483	+	0.866900i	$0.666109\pi$
$432$		0			0
$433$			19.0000i			0.913082i	0.889702	+	0.456541i	$0.150912\pi$
$433$			19.0000i			0.913082i	−0.889702	+	0.456541i	$0.849088\pi$
$434$		0			0
$435$	13.0000	−	13.0000i	0.623302	−	0.623302i
$436$		0			0
$437$	−10.1980	+	10.1980i	−0.487838	+	0.487838i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$			12.0000i			0.571429i
$442$		0			0
$443$	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
$446$		0			0
$447$	−10.1980	−	10.1980i	−0.482351	−	0.482351i
$448$		0			0
$449$	3.00000	+	3.00000i	0.141579	+	0.141579i	0.774344	−	0.632765i	$-0.218080\pi$
$449$	3.00000	+	3.00000i	0.141579	+	0.141579i	−0.632765	+	0.774344i	$0.718080\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		0			0
$453$	7.64853	−	7.64853i	0.359359	−	0.359359i
$454$		0			0
$455$	−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	$0.271073\pi$
$458$		0			0
$459$			15.0000i			0.700140i
$460$		0			0
$461$	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	$0.933829\pi$
$464$		0			0
$465$			26.0000i			1.20572i
$466$		0			0
$467$		−	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
$470$		0			0
$471$			10.1980i			0.469901i
$472$		0			0
$473$	1.00000	−	1.00000i	0.0459800	−	0.0459800i
$474$		0			0
$475$	−16.0000	−	16.0000i	−0.734130	−	0.734130i
$476$		0			0
$477$			10.1980i			0.466936i
$478$		0			0
$479$	2.54951	−	2.54951i	0.116490	−	0.116490i	−0.646459	−	0.762949i	$-0.723751\pi$
$479$	2.54951	−	2.54951i	0.116490	−	0.116490i	0.762949	+	0.646459i	$0.223751\pi$
$480$		0			0
$481$	−13.0000			−0.592749
$482$		0			0
$483$	−13.0000	+	13.0000i	−0.591520	+	0.591520i
$484$		0			0
$485$	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
$489$	−9.00000	−	9.00000i	−0.406994	−	0.406994i
$490$		0			0
$491$		−	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
$494$		0			0
$495$		−	10.1980i		−	0.458368i
$496$		0			0
$497$	−39.0000			−1.74939
$498$		0			0
$499$	−2.00000	+	2.00000i	−0.0895323	+	0.0895323i	−0.750454	−	0.660922i	$-0.770165\pi$
$499$	−2.00000	+	2.00000i	−0.0895323	+	0.0895323i	0.660922	+	0.750454i	$0.270165\pi$
$500$		0			0
$501$	−15.2971	+	15.2971i	−0.683422	+	0.683422i
$502$		0			0
$503$		−	20.3961i		−	0.909416i	−0.890641	−	0.454708i	$-0.849744\pi$
$503$		−	20.3961i		−	0.909416i	0.890641	−	0.454708i	$-0.150256\pi$
$504$		0			0
$505$	−26.0000	−	26.0000i	−1.15698	−	1.15698i
$506$		0			0
$507$			13.0000i			0.577350i
$508$		0			0
$509$	−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
$512$		0			0
$513$	10.0000	−	10.0000i	0.441511	−	0.441511i
$514$		0			0
$515$	−13.0000	−	13.0000i	−0.572848	−	0.572848i
$516$		0			0
$517$	5.09902			0.224255
$518$		0			0
$519$	−5.09902			−0.223822
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	6.00000			0.262362			0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000			0.262362			0.131181	+	0.991358i	$0.458123\pi$
$524$		0			0
$525$	−20.3961	−	20.3961i	−0.890158	−	0.890158i
$526$		0			0
$527$	−15.2971	+	15.2971i	−0.666350	+	0.666350i
$528$		0			0
$529$	3.00000			0.130435
$530$		0			0
$531$	16.0000	+	16.0000i	0.694341	+	0.694341i
$532$		0			0
$533$	−30.5941			−1.32518
$534$		0			0
$535$	−5.09902	−	5.09902i	−0.220450	−	0.220450i
$536$		0			0
$537$			9.00000i			0.388379i
$538$		0			0
$539$	−6.00000	+	6.00000i	−0.258438	+	0.258438i
$540$		0			0
$541$	17.8466	−	17.8466i	0.767284	−	0.767284i	−0.210344	−	0.977628i	$-0.567458\pi$
$541$	17.8466	−	17.8466i	0.767284	−	0.767284i	0.977628	+	0.210344i	$0.0674583\pi$
$542$		0			0
$543$	−15.2971			−0.656460
$544$		0			0
$545$		−	13.0000i		−	0.556859i
$546$		0			0
$547$	−23.0000			−0.983409			−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000			−0.983409			−0.491704	+	0.870762i	$0.663626\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.1980	−	10.1980i	−0.434451	−	0.434451i
$552$		0			0
$553$	13.0000	+	13.0000i	0.552816	+	0.552816i
$554$		0			0
$555$	−13.0000			−0.551819
$556$		0			0
$557$	−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
$560$		0			0
$561$	−3.00000	+	3.00000i	−0.126660	+	0.126660i
$562$		0			0
$563$		−	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
$566$		0			0
$567$	−2.54951	+	2.54951i	−0.107069	+	0.107069i
$568$		0			0
$569$			31.0000i			1.29959i	0.760111	+	0.649794i	$0.225145\pi$
$569$			31.0000i			1.29959i	−0.760111	+	0.649794i	$0.774855\pi$
$570$		0			0
$571$		−	15.0000i		−	0.627730i	−0.949468	−	0.313865i	$-0.898376\pi$
$571$		−	15.0000i		−	0.627730i	0.949468	−	0.313865i	$-0.101624\pi$
$572$		0			0
$573$		−	25.4951i		−	1.06507i
$574$		0			0
$575$			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	$0.0722406\pi$
$578$		0			0
$579$	9.00000	+	9.00000i	0.374027	+	0.374027i
$580$		0			0
$581$		−	25.4951i		−	1.05771i
$582$		0			0
$583$	−5.09902	+	5.09902i	−0.211180	+	0.211180i
$584$		0			0
$585$		−	26.0000i		−	1.07497i
$586$		0			0
$587$	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
$590$		0			0
$591$	12.7475	+	12.7475i	0.524364	+	0.524364i
$592$		0			0
$593$	29.0000	+	29.0000i	1.19089	+	1.19089i	0.976819	+	0.214069i	$0.0686716\pi$
$593$	29.0000	+	29.0000i	1.19089	+	1.19089i	0.214069	+	0.976819i	$0.431328\pi$
$594$		0			0
$595$		−	39.0000i		−	1.59884i
$596$		0			0
$597$	−25.4951			−1.04344
$598$		0			0
$599$		−	30.5941i		−	1.25004i	−0.780608	−	0.625021i	$-0.785090\pi$
$599$		−	30.5941i		−	1.25004i	0.780608	−	0.625021i	$-0.214910\pi$
$600$		0			0
$601$	37.0000			1.50926			0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000			1.50926			0.754631	+	0.656150i	$0.227816\pi$
$602$		0			0
$603$	6.00000	−	6.00000i	0.244339	−	0.244339i
$604$		0			0
$605$	−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	$-0.0663579\pi$
$608$		0			0
$609$	−13.0000	−	13.0000i	−0.526787	−	0.526787i
$610$		0			0
$611$	13.0000			0.525924
$612$		0			0
$613$	−5.09902	−	5.09902i	−0.205947	−	0.205947i	0.596595	−	0.802542i	$-0.296520\pi$
$613$	−5.09902	−	5.09902i	−0.205947	−	0.205947i	−0.802542	+	0.596595i	$0.796520\pi$
$614$		0			0
$615$	−30.5941			−1.23367
$616$		0			0
$617$	−12.0000	+	12.0000i	−0.483102	+	0.483102i	−0.906121	−	0.423019i	$-0.860970\pi$
$617$	−12.0000	+	12.0000i	−0.483102	+	0.483102i	0.423019	+	0.906121i	$0.360970\pi$
$618$		0			0
$619$	22.0000	+	22.0000i	0.884255	+	0.884255i	0.993964	−	0.109709i	$-0.0349919\pi$
$619$	22.0000	+	22.0000i	0.884255	+	0.884255i	−0.109709	+	0.993964i	$0.534992\pi$
$620$		0			0
$621$	−25.4951			−1.02308
$622$		0			0
$623$	10.1980			0.408576
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	4.00000			0.159745
$628$		0			0
$629$	−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	$0.417539\pi$
$632$		0			0
$633$	−7.00000			−0.278225
$634$		0			0
$635$	39.0000	+	39.0000i	1.54767	+	1.54767i
$636$		0			0
$637$	−15.2971	+	15.2971i	−0.606092	+	0.606092i
$638$		0			0
$639$	−15.2971	−	15.2971i	−0.605142	−	0.605142i
$640$		0			0
$641$		−	40.0000i		−	1.57991i	−0.613168	−	0.789953i	$-0.710105\pi$
$641$		−	40.0000i		−	1.57991i	0.613168	−	0.789953i	$-0.289895\pi$
$642$		0			0
$643$	−15.0000	+	15.0000i	−0.591542	+	0.591542i	−0.938048	−	0.346506i	$-0.887368\pi$
$643$	−15.0000	+	15.0000i	−0.591542	+	0.591542i	0.346506	+	0.938048i	$0.387368\pi$
$644$		0			0
$645$	2.54951	−	2.54951i	0.100387	−	0.100387i
$646$		0			0
$647$	10.1980			0.400926			0.200463	−	0.979701i	$-0.435755\pi$
$647$	10.1980			0.400926			0.200463	+	0.979701i	$0.435755\pi$
$648$		0			0
$649$			16.0000i			0.628055i
$650$		0			0
$651$	26.0000			1.01902
$652$		0			0
$653$		−	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
$656$		0			0
$657$	12.0000	+	12.0000i	0.468165	+	0.468165i
$658$		0			0
$659$	−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	−0.600954	−	0.799283i	$-0.705213\pi$
$661$	5.09902	−	5.09902i	0.198329	−	0.198329i	0.799283	+	0.600954i	$0.205213\pi$
$662$		0			0
$663$	−7.64853	+	7.64853i	−0.297044	+	0.297044i
$664$		0			0
$665$	−26.0000	+	26.0000i	−1.00824	+	1.00824i
$666$		0			0
$667$			26.0000i			1.00672i
$668$		0			0
$669$	−7.64853	−	7.64853i	−0.295709	−	0.295709i
$670$		0			0
$671$		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	$-0.132640\pi$
$674$		0			0
$675$		−	40.0000i		−	1.53960i
$676$		0			0
$677$		−	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
$680$		0			0
$681$	12.0000	−	12.0000i	0.459841	−	0.459841i
$682$		0			0
$683$	−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
$686$		0			0
$687$	−2.54951	+	2.54951i	−0.0972699	+	0.0972699i
$688$		0			0
$689$	−13.0000	+	13.0000i	−0.495261	+	0.495261i
$690$		0			0
$691$	−6.00000	+	6.00000i	−0.228251	+	0.228251i	−0.811962	−	0.583711i	$-0.801600\pi$
$691$	−6.00000	+	6.00000i	−0.228251	+	0.228251i	0.583711	+	0.811962i	$0.301600\pi$
$692$		0			0
$693$	−10.1980			−0.387391
$694$		0			0
$695$	−38.2426	−	38.2426i	−1.45063	−	1.45063i
$696$		0			0
$697$	−18.0000	−	18.0000i	−0.681799	−	0.681799i
$698$		0			0
$699$		−	11.0000i		−	0.416058i
$700$		0			0
$701$	−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
$704$		0			0
$705$	13.0000			0.489608
$706$		0			0
$707$	−26.0000	+	26.0000i	−0.977831	+	0.977831i
$708$		0			0
$709$	10.1980	−	10.1980i	0.382995	−	0.382995i	−0.489185	−	0.872180i	$-0.662706\pi$
$709$	10.1980	−	10.1980i	0.382995	−	0.382995i	0.872180	+	0.489185i	$0.162706\pi$
$710$		0			0
$711$			10.1980i			0.382456i
$712$		0			0
$713$	−26.0000	−	26.0000i	−0.973708	−	0.973708i
$714$		0			0
$715$	13.0000	−	13.0000i	0.486172	−	0.486172i
$716$		0			0
$717$	−2.54951	−	2.54951i	−0.0952132	−	0.0952132i
$718$		0			0
$719$	25.4951			0.950807			0.475403	−	0.879768i	$-0.342302\pi$
$719$	25.4951			0.950807			0.475403	+	0.879768i	$0.342302\pi$
$720$		0			0
$721$	−13.0000	+	13.0000i	−0.484145	+	0.484145i
$722$		0			0
$723$	−14.0000	−	14.0000i	−0.520666	−	0.520666i
$724$		0			0
$725$	−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	$0.591551\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	3.00000			0.110959
$732$		0			0
$733$	33.1436	+	33.1436i	1.22419	+	1.22419i	0.966129	+	0.258058i	$0.0830827\pi$
$733$	33.1436	+	33.1436i	1.22419	+	1.22419i	0.258058	+	0.966129i	$0.416917\pi$
$734$		0			0
$735$	−15.2971	+	15.2971i	−0.564241	+	0.564241i
$736$		0			0
$737$	6.00000			0.221013
$738$		0			0
$739$	2.00000	+	2.00000i	0.0735712	+	0.0735712i	0.742935	−	0.669364i	$-0.233433\pi$
$739$	2.00000	+	2.00000i	0.0735712	+	0.0735712i	−0.669364	+	0.742935i	$0.733433\pi$
$740$		0			0
$741$	10.1980			0.374634
$742$		0			0
$743$	−7.64853	−	7.64853i	−0.280597	−	0.280597i	0.552750	−	0.833347i	$-0.313578\pi$
$743$	−7.64853	−	7.64853i	−0.280597	−	0.280597i	−0.833347	+	0.552750i	$0.813578\pi$
$744$		0			0
$745$			52.0000i			1.90513i
$746$		0			0
$747$	10.0000	−	10.0000i	0.365881	−	0.365881i
$748$		0			0
$749$	−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	$0.559574\pi$
$752$		0			0
$753$		−	10.0000i		−	0.364420i
$754$		0			0
$755$	−39.0000			−1.41936
$756$		0			0
$757$			35.6931i			1.29729i	0.761091	+	0.648645i	$0.224664\pi$
$757$			35.6931i			1.29729i	−0.761091	+	0.648645i	$0.775336\pi$
$758$		0			0
$759$	−5.09902	−	5.09902i	−0.185083	−	0.185083i
$760$		0			0
$761$	−4.00000	−	4.00000i	−0.145000	−	0.145000i	0.630880	−	0.775880i	$-0.282694\pi$
$761$	−4.00000	−	4.00000i	−0.145000	−	0.145000i	−0.775880	+	0.630880i	$0.782694\pi$
$762$		0			0
$763$	−13.0000			−0.470632
$764$		0			0
$765$	15.2971	−	15.2971i	0.553066	−	0.553066i
$766$		0			0
$767$			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	$0.315391\pi$
$770$		0			0
$771$		−	3.00000i		−	0.108042i
$772$		0			0
$773$	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
$776$		0			0
$777$			13.0000i			0.466372i
$778$		0			0
$779$			24.0000i			0.859889i
$780$		0			0
$781$		−	15.2971i		−	0.547372i
$782$		0			0
$783$		−	25.4951i		−	0.911120i
$784$		0			0
$785$	26.0000	−	26.0000i	0.927980	−	0.927980i
$786$		0			0
$787$	30.0000	+	30.0000i	1.06938	+	1.06938i	0.997406	+	0.0719783i	$0.0229312\pi$
$787$	30.0000	+	30.0000i	1.06938	+	1.06938i	0.0719783	+	0.997406i	$0.477069\pi$
$788$		0			0
$789$		−	20.3961i		−	0.726120i
$790$		0			0
$791$	−10.1980	+	10.1980i	−0.362601	+	0.362601i
$792$		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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.u (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(-2.54951 - 2.54951i) q^{5} +(-2.54951 + 2.54951i) q^{7} -2.00000 q^{9} +(-1.00000 - 1.00000i) q^{11} +(-2.54951 - 2.54951i) q^{13} +(-2.54951 - 2.54951i) q^{15} -3.00000i q^{17} +(-2.00000 + 2.00000i) q^{19} +(-2.54951 + 2.54951i) q^{21} +5.09902 q^{23} +8.00000i q^{25} -5.00000 q^{27} +5.09902i q^{29} +(-5.09902 - 5.09902i) q^{31} +(-1.00000 - 1.00000i) q^{33} +13.0000 q^{35} +(2.54951 - 2.54951i) q^{37} +(-2.54951 - 2.54951i) q^{39} +(6.00000 - 6.00000i) q^{41} +1.00000i q^{43} +(5.09902 + 5.09902i) q^{45} +(-2.54951 + 2.54951i) q^{47} -6.00000i q^{49} -3.00000i q^{51} -5.09902i q^{53} +5.09902i q^{55} +(-2.00000 + 2.00000i) q^{57} +(-8.00000 - 8.00000i) q^{59} +(5.09902 - 5.09902i) q^{63} +13.0000i q^{65} +(-3.00000 + 3.00000i) q^{67} +5.09902 q^{69} +(7.64853 + 7.64853i) q^{71} +(-6.00000 - 6.00000i) q^{73} +8.00000i q^{75} +5.09902 q^{77} -5.09902i q^{79} +1.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} +(-7.64853 + 7.64853i) q^{85} +5.09902i q^{87} +(-2.00000 - 2.00000i) q^{89} +13.0000 q^{91} +(-5.09902 - 5.09902i) q^{93} +10.1980 q^{95} +(-7.00000 + 7.00000i) q^{97} +(2.00000 + 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 8 * q^9	\( 4 q + 4 q^{3} - 8 q^{9} - 4 q^{11} - 8 q^{19} - 20 q^{27} - 4 q^{33} + 52 q^{35} + 24 q^{41} - 8 q^{57} - 32 q^{59} - 12 q^{67} - 24 q^{73} + 4 q^{81} - 20 q^{83} - 8 q^{89} + 52 q^{91} - 28 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 8 * q^9 - 4 * q^11 - 8 * q^19 - 20 * q^27 - 4 * q^33 + 52 * q^35 + 24 * q^41 - 8 * q^57 - 32 * q^59 - 12 * q^67 - 24 * q^73 + 4 * q^81 - 20 * q^83 - 8 * q^89 + 52 * q^91 - 28 * q^97 + 8 * q^99

Introduction

L-functions

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