LMFDB - Embedded newform 416.2.k.a.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(31,416)]

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, 0, 3]))

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

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

chi := DirichletCharacter("416.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.k (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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			255.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	416.255
Dual form			416.2.k.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{7} -1.00000 q^{9} +(-3.00000 - 3.00000i) q^{11} +(-3.00000 + 2.00000i) q^{13} +(-2.00000 + 2.00000i) q^{15} +4.00000i q^{17} +(3.00000 - 3.00000i) q^{19} +(-2.00000 + 2.00000i) q^{21} -3.00000i q^{25} -4.00000i q^{27} -6.00000 q^{29} +(-3.00000 + 3.00000i) q^{31} +(-6.00000 + 6.00000i) q^{33} +2.00000i q^{35} +(3.00000 - 3.00000i) q^{37} +(4.00000 + 6.00000i) q^{39} +(1.00000 + 1.00000i) q^{41} +4.00000 q^{43} +(1.00000 + 1.00000i) q^{45} +(-5.00000 - 5.00000i) q^{47} -5.00000i q^{49} +8.00000 q^{51} +6.00000 q^{53} +6.00000i q^{55} +(-6.00000 - 6.00000i) q^{57} +(-7.00000 - 7.00000i) q^{59} +14.0000 q^{61} +(1.00000 + 1.00000i) q^{63} +(5.00000 + 1.00000i) q^{65} +(-5.00000 + 5.00000i) q^{67} +(5.00000 - 5.00000i) q^{71} +(9.00000 - 9.00000i) q^{73} -6.00000 q^{75} +6.00000i q^{77} +6.00000i q^{79} -11.0000 q^{81} +(7.00000 - 7.00000i) q^{83} +(4.00000 - 4.00000i) q^{85} +12.0000i q^{87} +(5.00000 - 5.00000i) q^{89} +(5.00000 + 1.00000i) q^{91} +(6.00000 + 6.00000i) q^{93} -6.00000 q^{95} +(13.0000 + 13.0000i) q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} - 4 q^{15} + 6 q^{19} - 4 q^{21} - 12 q^{29} - 6 q^{31} - 12 q^{33} + 6 q^{37} + 8 q^{39} + 2 q^{41} + 8 q^{43} + 2 q^{45} - 10 q^{47} + 16 q^{51} + 12 q^{53} - 12 q^{57} - 14 q^{59} + 28 q^{61} + 2 q^{63} + 10 q^{65} - 10 q^{67} + 10 q^{71} + 18 q^{73} - 12 q^{75} - 22 q^{81} + 14 q^{83} + 8 q^{85} + 10 q^{89} + 10 q^{91} + 12 q^{93} - 12 q^{95} + 26 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 - 2 * q^9 - 6 * q^11 - 6 * q^13 - 4 * q^15 + 6 * q^19 - 4 * q^21 - 12 * q^29 - 6 * q^31 - 12 * q^33 + 6 * q^37 + 8 * q^39 + 2 * q^41 + 8 * q^43 + 2 * q^45 - 10 * q^47 + 16 * q^51 + 12 * q^53 - 12 * q^57 - 14 * q^59 + 28 * q^61 + 2 * q^63 + 10 * q^65 - 10 * q^67 + 10 * q^71 + 18 * q^73 - 12 * q^75 - 22 * q^81 + 14 * q^83 + 8 * q^85 + 10 * q^89 + 10 * q^91 + 12 * q^93 - 12 * q^95 + 26 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$		0			0
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	0.447214	−	0.894427i	$-0.352416\pi$
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	0.492403	−	0.870367i	$-0.336119\pi$
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	−0.870367	+	0.492403i	$0.836119\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	0.0912903	−	0.995824i	$-0.470901\pi$
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	−0.995824	+	0.0912903i	$0.970901\pi$
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$	−2.00000	+	2.00000i	−0.516398	+	0.516398i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	−0.273597	−	0.961844i	$-0.588214\pi$
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	0.961844	+	0.273597i	$0.0882135\pi$
$20$		0			0
$21$	−2.00000	+	2.00000i	−0.436436	+	0.436436i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−3.00000	+	3.00000i	−0.538816	+	0.538816i	−0.923181	−	0.384365i	$-0.874420\pi$
$31$	−3.00000	+	3.00000i	−0.538816	+	0.538816i	0.384365	+	0.923181i	$0.374420\pi$
$32$		0			0
$33$	−6.00000	+	6.00000i	−1.04447	+	1.04447i
$34$		0			0
$35$			2.00000i			0.338062i
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$	4.00000	+	6.00000i	0.640513	+	0.960769i
$40$		0			0
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	0.780869	−	0.624695i	$-0.214777\pi$
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	1.00000	+	1.00000i	0.149071	+	0.149071i
$46$		0			0
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	0.241160	−	0.970485i	$-0.422472\pi$
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	−0.970485	+	0.241160i	$0.922472\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$	8.00000			1.12022
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$			6.00000i			0.809040i
$56$		0			0
$57$	−6.00000	−	6.00000i	−0.794719	−	0.794719i
$58$		0			0
$59$	−7.00000	−	7.00000i	−0.911322	−	0.911322i	0.0850540	−	0.996376i	$-0.472894\pi$
$59$	−7.00000	−	7.00000i	−0.911322	−	0.911322i	−0.996376	+	0.0850540i	$0.972894\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	1.00000	+	1.00000i	0.125988	+	0.125988i
$64$		0			0
$65$	5.00000	+	1.00000i	0.620174	+	0.124035i
$66$		0			0
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	−0.943167	−	0.332320i	$-0.892169\pi$
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	0.332320	+	0.943167i	$0.392169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.00000	−	5.00000i	0.593391	−	0.593391i	−0.345155	−	0.938546i	$-0.612174\pi$
$71$	5.00000	−	5.00000i	0.593391	−	0.593391i	0.938546	+	0.345155i	$0.112174\pi$
$72$		0			0
$73$	9.00000	−	9.00000i	1.05337	−	1.05337i	0.0548772	−	0.998493i	$-0.482523\pi$
$73$	9.00000	−	9.00000i	1.05337	−	1.05337i	0.998493	−	0.0548772i	$-0.0174767\pi$
$74$		0			0
$75$	−6.00000			−0.692820
$76$		0			0
$77$			6.00000i			0.683763i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	7.00000	−	7.00000i	0.768350	−	0.768350i	−0.209466	−	0.977816i	$-0.567173\pi$
$83$	7.00000	−	7.00000i	0.768350	−	0.768350i	0.977816	+	0.209466i	$0.0671726\pi$
$84$		0			0
$85$	4.00000	−	4.00000i	0.433861	−	0.433861i
$86$		0			0
$87$			12.0000i			1.28654i
$88$		0			0
$89$	5.00000	−	5.00000i	0.529999	−	0.529999i	−0.390573	−	0.920572i	$-0.627723\pi$
$89$	5.00000	−	5.00000i	0.529999	−	0.529999i	0.920572	+	0.390573i	$0.127723\pi$
$90$		0			0
$91$	5.00000	+	1.00000i	0.524142	+	0.104828i
$92$		0			0
$93$	6.00000	+	6.00000i	0.622171	+	0.622171i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.913812	+	0.406138i	$0.133125\pi$
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$		0			0
$101$		−	16.0000i		−	1.59206i	−0.605257	−	0.796030i	$-0.706930\pi$
$101$		−	16.0000i		−	1.59206i	0.605257	−	0.796030i	$-0.293070\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$	4.00000			0.390360
$106$		0			0
$107$			14.0000i			1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$			14.0000i			1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$		0			0
$109$	−9.00000	−	9.00000i	−0.862044	−	0.862044i	0.129532	−	0.991575i	$-0.458653\pi$
$109$	−9.00000	−	9.00000i	−0.862044	−	0.862044i	−0.991575	+	0.129532i	$0.958653\pi$
$110$		0			0
$111$	−6.00000	−	6.00000i	−0.569495	−	0.569495i
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$		0			0
$119$	4.00000	−	4.00000i	0.366679	−	0.366679i
$120$		0			0
$121$			7.00000i			0.636364i
$122$		0			0
$123$	2.00000	−	2.00000i	0.180334	−	0.180334i
$124$		0			0
$125$	−8.00000	+	8.00000i	−0.715542	+	0.715542i
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$		−	8.00000i		−	0.704361i
$130$		0			0
$131$			14.0000i			1.22319i	0.791173	+	0.611593i	$0.209471\pi$
$131$			14.0000i			1.22319i	−0.791173	+	0.611593i	$0.790529\pi$
$132$		0			0
$133$	−6.00000			−0.520266
$134$		0			0
$135$	−4.00000	+	4.00000i	−0.344265	+	0.344265i
$136$		0			0
$137$	−3.00000	+	3.00000i	−0.256307	+	0.256307i	−0.823550	−	0.567243i	$-0.808010\pi$
$137$	−3.00000	+	3.00000i	−0.256307	+	0.256307i	0.567243	+	0.823550i	$0.308010\pi$
$138$		0			0
$139$			22.0000i			1.86602i	0.359856	+	0.933008i	$0.382826\pi$
$139$			22.0000i			1.86602i	−0.359856	+	0.933008i	$0.617174\pi$
$140$		0			0
$141$	−10.0000	+	10.0000i	−0.842152	+	0.842152i
$142$		0			0
$143$	15.0000	+	3.00000i	1.25436	+	0.250873i
$144$		0			0
$145$	6.00000	+	6.00000i	0.498273	+	0.498273i
$146$		0			0
$147$	−10.0000			−0.824786
$148$		0			0
$149$	−13.0000	−	13.0000i	−1.06500	−	1.06500i	−0.997735	−	0.0672664i	$-0.978572\pi$
$149$	−13.0000	−	13.0000i	−1.06500	−	1.06500i	−0.0672664	−	0.997735i	$-0.521428\pi$
$150$		0			0
$151$	−9.00000	−	9.00000i	−0.732410	−	0.732410i	0.238687	−	0.971097i	$-0.423283\pi$
$151$	−9.00000	−	9.00000i	−0.732410	−	0.732410i	−0.971097	+	0.238687i	$0.923283\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	6.00000			0.481932
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$		−	12.0000i		−	0.951662i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.00000	−	3.00000i	−0.234978	−	0.234978i	0.579789	−	0.814767i	$-0.303135\pi$
$163$	−3.00000	−	3.00000i	−0.234978	−	0.234978i	−0.814767	+	0.579789i	$0.803135\pi$
$164$		0			0
$165$	12.0000			0.934199
$166$		0			0
$167$	−13.0000	−	13.0000i	−1.00597	−	1.00597i	−0.999982	−	0.00598813i	$-0.998094\pi$
$167$	−13.0000	−	13.0000i	−1.00597	−	1.00597i	−0.00598813	−	0.999982i	$-0.501906\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$	−3.00000	+	3.00000i	−0.229416	+	0.229416i
$172$		0			0
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$		0			0
$175$	−3.00000	+	3.00000i	−0.226779	+	0.226779i
$176$		0			0
$177$	−14.0000	+	14.0000i	−1.05230	+	1.05230i
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$		−	4.00000i		−	0.297318i	−0.988889	−	0.148659i	$-0.952504\pi$
$181$		−	4.00000i		−	0.297318i	0.988889	−	0.148659i	$-0.0474956\pi$
$182$		0			0
$183$		−	28.0000i		−	2.06982i
$184$		0			0
$185$	−6.00000			−0.441129
$186$		0			0
$187$	12.0000	−	12.0000i	0.877527	−	0.877527i
$188$		0			0
$189$	−4.00000	+	4.00000i	−0.290957	+	0.290957i
$190$		0			0
$191$		−	10.0000i		−	0.723575i	−0.932261	−	0.361787i	$-0.882167\pi$
$191$		−	10.0000i		−	0.723575i	0.932261	−	0.361787i	$-0.117833\pi$
$192$		0			0
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	0.863779	+	0.503871i	$0.168091\pi$
$194$		0			0
$195$	2.00000	−	10.0000i	0.143223	−	0.716115i
$196$		0			0
$197$	7.00000	+	7.00000i	0.498729	+	0.498729i	0.911042	−	0.412313i	$-0.135279\pi$
$197$	7.00000	+	7.00000i	0.498729	+	0.498729i	−0.412313	+	0.911042i	$0.635279\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	10.0000	+	10.0000i	0.705346	+	0.705346i
$202$		0			0
$203$	6.00000	+	6.00000i	0.421117	+	0.421117i
$204$		0			0
$205$		−	2.00000i		−	0.139686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−18.0000			−1.24509
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$		0			0
$213$	−10.0000	−	10.0000i	−0.685189	−	0.685189i
$214$		0			0
$215$	−4.00000	−	4.00000i	−0.272798	−	0.272798i
$216$		0			0
$217$	6.00000			0.407307
$218$		0			0
$219$	−18.0000	−	18.0000i	−1.21633	−	1.21633i
$220$		0			0
$221$	−8.00000	−	12.0000i	−0.538138	−	0.807207i
$222$		0			0
$223$	−7.00000	+	7.00000i	−0.468755	+	0.468755i	−0.901511	−	0.432756i	$-0.857541\pi$
$223$	−7.00000	+	7.00000i	−0.468755	+	0.468755i	0.432756	+	0.901511i	$0.357541\pi$
$224$		0			0
$225$			3.00000i			0.200000i
$226$		0			0
$227$	−13.0000	+	13.0000i	−0.862840	+	0.862840i	−0.991667	−	0.128827i	$-0.958879\pi$
$227$	−13.0000	+	13.0000i	−0.862840	+	0.862840i	0.128827	+	0.991667i	$0.458879\pi$
$228$		0			0
$229$	3.00000	−	3.00000i	0.198246	−	0.198246i	−0.601002	−	0.799248i	$-0.705232\pi$
$229$	3.00000	−	3.00000i	0.198246	−	0.198246i	0.799248	+	0.601002i	$0.205232\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$			10.0000i			0.652328i
$236$		0			0
$237$	12.0000			0.779484
$238$		0			0
$239$	17.0000	−	17.0000i	1.09964	−	1.09964i	0.105186	−	0.994453i	$-0.466456\pi$
$239$	17.0000	−	17.0000i	1.09964	−	1.09964i	0.994453	−	0.105186i	$-0.0335438\pi$
$240$		0			0
$241$	−15.0000	+	15.0000i	−0.966235	+	0.966235i	−0.999448	−	0.0332133i	$-0.989426\pi$
$241$	−15.0000	+	15.0000i	−0.966235	+	0.966235i	0.0332133	+	0.999448i	$0.489426\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	−5.00000	+	5.00000i	−0.319438	+	0.319438i
$246$		0			0
$247$	−3.00000	+	15.0000i	−0.190885	+	0.954427i
$248$		0			0
$249$	−14.0000	−	14.0000i	−0.887214	−	0.887214i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−8.00000	−	8.00000i	−0.500979	−	0.500979i
$256$		0			0
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$	−6.00000			−0.372822
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	18.0000i		−	1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$		−	18.0000i		−	1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$		0			0
$265$	−6.00000	−	6.00000i	−0.368577	−	0.368577i
$266$		0			0
$267$	−10.0000	−	10.0000i	−0.611990	−	0.611990i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	19.0000	+	19.0000i	1.15417	+	1.15417i	0.985709	+	0.168459i	$0.0538791\pi$
$271$	19.0000	+	19.0000i	1.15417	+	1.15417i	0.168459	+	0.985709i	$0.446121\pi$
$272$		0			0
$273$	2.00000	−	10.0000i	0.121046	−	0.605228i
$274$		0			0
$275$	−9.00000	+	9.00000i	−0.542720	+	0.542720i
$276$		0			0
$277$			16.0000i			0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$			16.0000i			0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$		0			0
$279$	3.00000	−	3.00000i	0.179605	−	0.179605i
$280$		0			0
$281$	1.00000	−	1.00000i	0.0596550	−	0.0596550i	−0.676650	−	0.736305i	$-0.736569\pi$
$281$	1.00000	−	1.00000i	0.0596550	−	0.0596550i	0.736305	+	0.676650i	$0.236569\pi$
$282$		0			0
$283$	20.0000			1.18888			0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000			1.18888			0.594438	+	0.804141i	$0.297374\pi$
$284$		0			0
$285$			12.0000i			0.710819i
$286$		0			0
$287$		−	2.00000i		−	0.118056i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	26.0000	−	26.0000i	1.52415	−	1.52415i
$292$		0			0
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	−0.919313	−	0.393527i	$-0.871255\pi$
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	0.393527	+	0.919313i	$0.371255\pi$
$294$		0			0
$295$			14.0000i			0.815112i
$296$		0			0
$297$	−12.0000	+	12.0000i	−0.696311	+	0.696311i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−4.00000	−	4.00000i	−0.230556	−	0.230556i
$302$		0			0
$303$	−32.0000			−1.83835
$304$		0			0
$305$	−14.0000	−	14.0000i	−0.801638	−	0.801638i
$306$		0			0
$307$	5.00000	+	5.00000i	0.285365	+	0.285365i	0.835244	−	0.549879i	$-0.185326\pi$
$307$	5.00000	+	5.00000i	0.285365	+	0.285365i	−0.549879	+	0.835244i	$0.685326\pi$
$308$		0			0
$309$		−	16.0000i		−	0.910208i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		−	2.00000i		−	0.112687i
$316$		0			0
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	0.786318	−	0.617822i	$-0.211985\pi$
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$	18.0000	+	18.0000i	1.00781	+	1.00781i
$320$		0			0
$321$	28.0000			1.56281
$322$		0			0
$323$	12.0000	+	12.0000i	0.667698	+	0.667698i
$324$		0			0
$325$	6.00000	+	9.00000i	0.332820	+	0.499230i
$326$		0			0
$327$	−18.0000	+	18.0000i	−0.995402	+	0.995402i
$328$		0			0
$329$			10.0000i			0.551318i
$330$		0			0
$331$	−17.0000	+	17.0000i	−0.934405	+	0.934405i	−0.997977	−	0.0635727i	$-0.979751\pi$
$331$	−17.0000	+	17.0000i	−0.934405	+	0.934405i	0.0635727	+	0.997977i	$0.479751\pi$
$332$		0			0
$333$	−3.00000	+	3.00000i	−0.164399	+	0.164399i
$334$		0			0
$335$	10.0000			0.546358
$336$		0			0
$337$		−	20.0000i		−	1.08947i	−0.838608	−	0.544735i	$-0.816630\pi$
$337$		−	20.0000i		−	1.08947i	0.838608	−	0.544735i	$-0.183370\pi$
$338$		0			0
$339$			4.00000i			0.217250i
$340$		0			0
$341$	18.0000			0.974755
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.00000i			0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$			6.00000i			0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$	3.00000	−	3.00000i	0.160586	−	0.160586i	−0.622240	−	0.782826i	$-0.713777\pi$
$349$	3.00000	−	3.00000i	0.160586	−	0.160586i	0.782826	+	0.622240i	$0.213777\pi$
$350$		0			0
$351$	8.00000	+	12.0000i	0.427008	+	0.640513i
$352$		0			0
$353$	17.0000	+	17.0000i	0.904819	+	0.904819i	0.995848	−	0.0910295i	$-0.0290158\pi$
$353$	17.0000	+	17.0000i	0.904819	+	0.904819i	−0.0910295	+	0.995848i	$0.529016\pi$
$354$		0			0
$355$	−10.0000			−0.530745
$356$		0			0
$357$	−8.00000	−	8.00000i	−0.423405	−	0.423405i
$358$		0			0
$359$	−1.00000	−	1.00000i	−0.0527780	−	0.0527780i	0.680225	−	0.733003i	$-0.261882\pi$
$359$	−1.00000	−	1.00000i	−0.0527780	−	0.0527780i	−0.733003	+	0.680225i	$0.761882\pi$
$360$		0			0
$361$			1.00000i			0.0526316i
$362$		0			0
$363$	14.0000			0.734809
$364$		0			0
$365$	−18.0000			−0.942163
$366$		0			0
$367$		−	10.0000i		−	0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$		−	10.0000i		−	0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$		0			0
$369$	−1.00000	−	1.00000i	−0.0520579	−	0.0520579i
$370$		0			0
$371$	−6.00000	−	6.00000i	−0.311504	−	0.311504i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	16.0000	+	16.0000i	0.826236	+	0.826236i
$376$		0			0
$377$	18.0000	−	12.0000i	0.927047	−	0.618031i
$378$		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
$381$		−	32.0000i		−	1.63941i
$382$		0			0
$383$	9.00000	−	9.00000i	0.459879	−	0.459879i	−0.438737	−	0.898616i	$-0.644574\pi$
$383$	9.00000	−	9.00000i	0.459879	−	0.459879i	0.898616	+	0.438737i	$0.144574\pi$
$384$		0			0
$385$	6.00000	−	6.00000i	0.305788	−	0.305788i
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	28.0000			1.41241
$394$		0			0
$395$	6.00000	−	6.00000i	0.301893	−	0.301893i
$396$		0			0
$397$	7.00000	−	7.00000i	0.351320	−	0.351320i	−0.509281	−	0.860601i	$-0.670088\pi$
$397$	7.00000	−	7.00000i	0.351320	−	0.351320i	0.860601	+	0.509281i	$0.170088\pi$
$398$		0			0
$399$			12.0000i			0.600751i
$400$		0			0
$401$	−7.00000	+	7.00000i	−0.349563	+	0.349563i	−0.859947	−	0.510384i	$-0.829503\pi$
$401$	−7.00000	+	7.00000i	−0.349563	+	0.349563i	0.510384	+	0.859947i	$0.329503\pi$
$402$		0			0
$403$	3.00000	−	15.0000i	0.149441	−	0.747203i
$404$		0			0
$405$	11.0000	+	11.0000i	0.546594	+	0.546594i
$406$		0			0
$407$	−18.0000			−0.892227
$408$		0			0
$409$	−3.00000	−	3.00000i	−0.148340	−	0.148340i	0.629036	−	0.777376i	$-0.283450\pi$
$409$	−3.00000	−	3.00000i	−0.148340	−	0.148340i	−0.777376	+	0.629036i	$0.783450\pi$
$410$		0			0
$411$	6.00000	+	6.00000i	0.295958	+	0.295958i
$412$		0			0
$413$			14.0000i			0.688895i
$414$		0			0
$415$	−14.0000			−0.687233
$416$		0			0
$417$	44.0000			2.15469
$418$		0			0
$419$		−	2.00000i		−	0.0977064i	−0.998806	−	0.0488532i	$-0.984443\pi$
$419$		−	2.00000i		−	0.0977064i	0.998806	−	0.0488532i	$-0.0155566\pi$
$420$		0			0
$421$	3.00000	+	3.00000i	0.146211	+	0.146211i	0.776423	−	0.630212i	$-0.217032\pi$
$421$	3.00000	+	3.00000i	0.146211	+	0.146211i	−0.630212	+	0.776423i	$0.717032\pi$
$422$		0			0
$423$	5.00000	+	5.00000i	0.243108	+	0.243108i
$424$		0			0
$425$	12.0000			0.582086
$426$		0			0
$427$	−14.0000	−	14.0000i	−0.677507	−	0.677507i
$428$		0			0
$429$	6.00000	−	30.0000i	0.289683	−	1.44841i
$430$		0			0
$431$	−11.0000	+	11.0000i	−0.529851	+	0.529851i	−0.920528	−	0.390677i	$-0.872241\pi$
$431$	−11.0000	+	11.0000i	−0.529851	+	0.529851i	0.390677	+	0.920528i	$0.372241\pi$
$432$		0			0
$433$		−	24.0000i		−	1.15337i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$		−	24.0000i		−	1.15337i	0.816968	−	0.576683i	$-0.195653\pi$
$434$		0			0
$435$	12.0000	−	12.0000i	0.575356	−	0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	40.0000			1.90910			0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000			1.90910			0.954548	+	0.298057i	$0.0963387\pi$
$440$		0			0
$441$			5.00000i			0.238095i
$442$		0			0
$443$			30.0000i			1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$			30.0000i			1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$		0			0
$445$	−10.0000			−0.474045
$446$		0			0
$447$	−26.0000	+	26.0000i	−1.22976	+	1.22976i
$448$		0			0
$449$	1.00000	−	1.00000i	0.0471929	−	0.0471929i	−0.683117	−	0.730309i	$-0.739376\pi$
$449$	1.00000	−	1.00000i	0.0471929	−	0.0471929i	0.730309	+	0.683117i	$0.239376\pi$
$450$		0			0
$451$		−	6.00000i		−	0.282529i
$452$		0			0
$453$	−18.0000	+	18.0000i	−0.845714	+	0.845714i
$454$		0			0
$455$	−4.00000	−	6.00000i	−0.187523	−	0.281284i
$456$		0			0
$457$	13.0000	+	13.0000i	0.608114	+	0.608114i	0.942453	−	0.334339i	$-0.108513\pi$
$457$	13.0000	+	13.0000i	0.608114	+	0.608114i	−0.334339	+	0.942453i	$0.608513\pi$
$458$		0			0
$459$	16.0000			0.746816
$460$		0			0
$461$	11.0000	+	11.0000i	0.512321	+	0.512321i	0.915237	−	0.402916i	$-0.132003\pi$
$461$	11.0000	+	11.0000i	0.512321	+	0.512321i	−0.402916	+	0.915237i	$0.632003\pi$
$462$		0			0
$463$	−5.00000	−	5.00000i	−0.232370	−	0.232370i	0.581311	−	0.813681i	$-0.302540\pi$
$463$	−5.00000	−	5.00000i	−0.232370	−	0.232370i	−0.813681	+	0.581311i	$0.802540\pi$
$464$		0			0
$465$		−	12.0000i		−	0.556487i
$466$		0			0
$467$	−4.00000			−0.185098			−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000			−0.185098			−0.0925490	+	0.995708i	$0.529501\pi$
$468$		0			0
$469$	10.0000			0.461757
$470$		0			0
$471$		−	4.00000i		−	0.184310i
$472$		0			0
$473$	−12.0000	−	12.0000i	−0.551761	−	0.551761i
$474$		0			0
$475$	−9.00000	−	9.00000i	−0.412948	−	0.412948i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	23.0000	+	23.0000i	1.05090	+	1.05090i	0.998633	+	0.0522635i	$0.0166436\pi$
$479$	23.0000	+	23.0000i	1.05090	+	1.05090i	0.0522635	+	0.998633i	$0.483356\pi$
$480$		0			0
$481$	−3.00000	+	15.0000i	−0.136788	+	0.683941i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	26.0000i		−	1.18060i
$486$		0			0
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.0431614	+	0.999068i	$0.513743\pi$
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.999068	+	0.0431614i	$0.986257\pi$
$488$		0			0
$489$	−6.00000	+	6.00000i	−0.271329	+	0.271329i
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$		−	24.0000i		−	1.08091i
$494$		0			0
$495$		−	6.00000i		−	0.269680i
$496$		0			0
$497$	−10.0000			−0.448561
$498$		0			0
$499$	27.0000	−	27.0000i	1.20869	−	1.20869i	0.237233	−	0.971453i	$-0.423759\pi$
$499$	27.0000	−	27.0000i	1.20869	−	1.20869i	0.971453	−	0.237233i	$-0.0762406\pi$
$500$		0			0
$501$	−26.0000	+	26.0000i	−1.16159	+	1.16159i
$502$		0			0
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$	−16.0000	+	16.0000i	−0.711991	+	0.711991i
$506$		0			0
$507$	−24.0000	−	10.0000i	−1.06588	−	0.444116i
$508$		0			0
$509$	11.0000	+	11.0000i	0.487566	+	0.487566i	0.907537	−	0.419971i	$-0.137960\pi$
$509$	11.0000	+	11.0000i	0.487566	+	0.487566i	−0.419971	+	0.907537i	$0.637960\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		0			0
$513$	−12.0000	−	12.0000i	−0.529813	−	0.529813i
$514$		0			0
$515$	−8.00000	−	8.00000i	−0.352522	−	0.352522i
$516$		0			0
$517$			30.0000i			1.31940i
$518$		0			0
$519$	48.0000			2.10697
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$		−	34.0000i		−	1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$		−	34.0000i		−	1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$		0			0
$525$	6.00000	+	6.00000i	0.261861	+	0.261861i
$526$		0			0
$527$	−12.0000	−	12.0000i	−0.522728	−	0.522728i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	7.00000	+	7.00000i	0.303774	+	0.303774i
$532$		0			0
$533$	−5.00000	−	1.00000i	−0.216574	−	0.0433148i
$534$		0			0
$535$	14.0000	−	14.0000i	0.605273	−	0.605273i
$536$		0			0
$537$			40.0000i			1.72613i
$538$		0			0
$539$	−15.0000	+	15.0000i	−0.646096	+	0.646096i
$540$		0			0
$541$	−1.00000	+	1.00000i	−0.0429934	+	0.0429934i	−0.728277	−	0.685283i	$-0.759678\pi$
$541$	−1.00000	+	1.00000i	−0.0429934	+	0.0429934i	0.685283	+	0.728277i	$0.259678\pi$
$542$		0			0
$543$	−8.00000			−0.343313
$544$		0			0
$545$			18.0000i			0.771035i
$546$		0			0
$547$			30.0000i			1.28271i	0.767245	+	0.641354i	$0.221627\pi$
$547$			30.0000i			1.28271i	−0.767245	+	0.641354i	$0.778373\pi$
$548$		0			0
$549$	−14.0000			−0.597505
$550$		0			0
$551$	−18.0000	+	18.0000i	−0.766826	+	0.766826i
$552$		0			0
$553$	6.00000	−	6.00000i	0.255146	−	0.255146i
$554$		0			0
$555$			12.0000i			0.509372i
$556$		0			0
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	−0.313889	−	0.949460i	$-0.601632\pi$
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	0.949460	+	0.313889i	$0.101632\pi$
$558$		0			0
$559$	−12.0000	+	8.00000i	−0.507546	+	0.338364i
$560$		0			0
$561$	−24.0000	−	24.0000i	−1.01328	−	1.01328i
$562$		0			0
$563$	20.0000			0.842900			0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000			0.842900			0.421450	+	0.906852i	$0.361521\pi$
$564$		0			0
$565$	2.00000	+	2.00000i	0.0841406	+	0.0841406i
$566$		0			0
$567$	11.0000	+	11.0000i	0.461957	+	0.461957i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$	−20.0000			−0.835512
$574$		0			0
$575$		0			0
$576$		0			0
$577$	5.00000	+	5.00000i	0.208153	+	0.208153i	0.803482	−	0.595329i	$-0.202978\pi$
$577$	5.00000	+	5.00000i	0.208153	+	0.208153i	−0.595329	+	0.803482i	$0.702978\pi$
$578$		0			0
$579$	−10.0000	−	10.0000i	−0.415586	−	0.415586i
$580$		0			0
$581$	−14.0000			−0.580818
$582$		0			0
$583$	−18.0000	−	18.0000i	−0.745484	−	0.745484i
$584$		0			0
$585$	−5.00000	−	1.00000i	−0.206725	−	0.0413449i
$586$		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
$589$			18.0000i			0.741677i
$590$		0			0
$591$	14.0000	−	14.0000i	0.575883	−	0.575883i
$592$		0			0
$593$	−3.00000	+	3.00000i	−0.123195	+	0.123195i	−0.766016	−	0.642821i	$-0.777764\pi$
$593$	−3.00000	+	3.00000i	−0.123195	+	0.123195i	0.642821	+	0.766016i	$0.277764\pi$
$594$		0			0
$595$	−8.00000			−0.327968
$596$		0			0
$597$			16.0000i			0.654836i
$598$		0			0
$599$		−	26.0000i		−	1.06233i	−0.847268	−	0.531166i	$-0.821754\pi$
$599$		−	26.0000i		−	1.06233i	0.847268	−	0.531166i	$-0.178246\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	5.00000	−	5.00000i	0.203616	−	0.203616i
$604$		0			0
$605$	7.00000	−	7.00000i	0.284590	−	0.284590i
$606$		0			0
$607$		−	42.0000i		−	1.70473i	−0.522949	−	0.852364i	$-0.675168\pi$
$607$		−	42.0000i		−	1.70473i	0.522949	−	0.852364i	$-0.324832\pi$
$608$		0			0
$609$	12.0000	−	12.0000i	0.486265	−	0.486265i
$610$		0			0
$611$	25.0000	+	5.00000i	1.01139	+	0.202278i
$612$		0			0
$613$	−9.00000	−	9.00000i	−0.363507	−	0.363507i	0.501596	−	0.865102i	$-0.332747\pi$
$613$	−9.00000	−	9.00000i	−0.363507	−	0.363507i	−0.865102	+	0.501596i	$0.832747\pi$
$614$		0			0
$615$	−4.00000			−0.161296
$616$		0			0
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	0.0714958	−	0.997441i	$-0.477223\pi$
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	−0.997441	+	0.0714958i	$0.977223\pi$
$618$		0			0
$619$	−3.00000	−	3.00000i	−0.120580	−	0.120580i	0.644242	−	0.764822i	$-0.277173\pi$
$619$	−3.00000	−	3.00000i	−0.120580	−	0.120580i	−0.764822	+	0.644242i	$0.777173\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.0000			−0.400642
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			36.0000i			1.43770i
$628$		0			0
$629$	12.0000	+	12.0000i	0.478471	+	0.478471i
$630$		0			0
$631$	−1.00000	−	1.00000i	−0.0398094	−	0.0398094i	0.686922	−	0.726731i	$-0.258961\pi$
$631$	−1.00000	−	1.00000i	−0.0398094	−	0.0398094i	−0.726731	+	0.686922i	$0.758961\pi$
$632$		0			0
$633$	−20.0000			−0.794929
$634$		0			0
$635$	−16.0000	−	16.0000i	−0.634941	−	0.634941i
$636$		0			0
$637$	10.0000	+	15.0000i	0.396214	+	0.594322i
$638$		0			0
$639$	−5.00000	+	5.00000i	−0.197797	+	0.197797i
$640$		0			0
$641$		−	36.0000i		−	1.42191i	−0.703235	−	0.710957i	$-0.748262\pi$
$641$		−	36.0000i		−	1.42191i	0.703235	−	0.710957i	$-0.251738\pi$
$642$		0			0
$643$	7.00000	−	7.00000i	0.276053	−	0.276053i	−0.555478	−	0.831531i	$-0.687465\pi$
$643$	7.00000	−	7.00000i	0.276053	−	0.276053i	0.831531	+	0.555478i	$0.187465\pi$
$644$		0			0
$645$	−8.00000	+	8.00000i	−0.315000	+	0.315000i
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$			42.0000i			1.64864i
$650$		0			0
$651$		−	12.0000i		−	0.470317i
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	14.0000	−	14.0000i	0.547025	−	0.547025i
$656$		0			0
$657$	−9.00000	+	9.00000i	−0.351123	+	0.351123i
$658$		0			0
$659$		−	34.0000i		−	1.32445i	−0.749304	−	0.662226i	$-0.769612\pi$
$659$		−	34.0000i		−	1.32445i	0.749304	−	0.662226i	$-0.230388\pi$
$660$		0			0
$661$	19.0000	−	19.0000i	0.739014	−	0.739014i	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	19.0000	−	19.0000i	0.739014	−	0.739014i	0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$	−24.0000	+	16.0000i	−0.932083	+	0.621389i
$664$		0			0
$665$	6.00000	+	6.00000i	0.232670	+	0.232670i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	14.0000	+	14.0000i	0.541271	+	0.541271i
$670$		0			0
$671$	−42.0000	−	42.0000i	−1.62139	−	1.62139i
$672$		0			0
$673$		−	44.0000i		−	1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$		−	44.0000i		−	1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$		0			0
$675$	−12.0000			−0.461880
$676$		0			0
$677$	22.0000			0.845529			0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000			0.845529			0.422764	+	0.906240i	$0.361060\pi$
$678$		0			0
$679$		−	26.0000i		−	0.997788i
$680$		0			0
$681$	26.0000	+	26.0000i	0.996322	+	0.996322i
$682$		0			0
$683$	5.00000	+	5.00000i	0.191320	+	0.191320i	0.796266	−	0.604946i	$-0.206805\pi$
$683$	5.00000	+	5.00000i	0.191320	+	0.191320i	−0.604946	+	0.796266i	$0.706805\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$	−6.00000	−	6.00000i	−0.228914	−	0.228914i
$688$		0			0
$689$	−18.0000	+	12.0000i	−0.685745	+	0.457164i
$690$		0			0
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	−0.982919	−	0.184041i	$-0.941082\pi$
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	0.184041	+	0.982919i	$0.441082\pi$
$692$		0			0
$693$		−	6.00000i		−	0.227921i
$694$		0			0
$695$	22.0000	−	22.0000i	0.834508	−	0.834508i
$696$		0			0
$697$	−4.00000	+	4.00000i	−0.151511	+	0.151511i
$698$		0			0
$699$	−32.0000			−1.21035
$700$		0			0
$701$		−	28.0000i		−	1.05755i	−0.848763	−	0.528773i	$-0.822652\pi$
$701$		−	28.0000i		−	1.05755i	0.848763	−	0.528773i	$-0.177348\pi$
$702$		0			0
$703$		−	18.0000i		−	0.678883i
$704$		0			0
$705$	20.0000			0.753244
$706$		0			0
$707$	−16.0000	+	16.0000i	−0.601742	+	0.601742i
$708$		0			0
$709$	−33.0000	+	33.0000i	−1.23934	+	1.23934i	−0.279070	+	0.960271i	$0.590026\pi$
$709$	−33.0000	+	33.0000i	−1.23934	+	1.23934i	−0.960271	+	0.279070i	$0.909974\pi$
$710$		0			0
$711$		−	6.00000i		−	0.225018i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−12.0000	−	18.0000i	−0.448775	−	0.673162i
$716$		0			0
$717$	−34.0000	−	34.0000i	−1.26975	−	1.26975i
$718$		0			0
$719$	40.0000			1.49175			0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000			1.49175			0.745874	+	0.666087i	$0.232032\pi$
$720$		0			0
$721$	−8.00000	−	8.00000i	−0.297936	−	0.297936i
$722$		0			0
$723$	30.0000	+	30.0000i	1.11571	+	1.11571i
$724$		0			0
$725$			18.0000i			0.668503i
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$			16.0000i			0.591781i
$732$		0			0
$733$	−17.0000	−	17.0000i	−0.627909	−	0.627909i	0.319632	−	0.947542i	$-0.396441\pi$
$733$	−17.0000	−	17.0000i	−0.627909	−	0.627909i	−0.947542	+	0.319632i	$0.896441\pi$
$734$		0			0
$735$	10.0000	+	10.0000i	0.368856	+	0.368856i
$736$		0			0
$737$	30.0000			1.10506
$738$		0			0
$739$	−7.00000	−	7.00000i	−0.257499	−	0.257499i	0.566537	−	0.824036i	$-0.308283\pi$
$739$	−7.00000	−	7.00000i	−0.257499	−	0.257499i	−0.824036	+	0.566537i	$0.808283\pi$
$740$		0			0
$741$	30.0000	+	6.00000i	1.10208	+	0.220416i
$742$		0			0
$743$	13.0000	−	13.0000i	0.476924	−	0.476924i	−0.427223	−	0.904146i	$-0.640508\pi$
$743$	13.0000	−	13.0000i	0.476924	−	0.476924i	0.904146	+	0.427223i	$0.140508\pi$
$744$		0			0
$745$			26.0000i			0.952566i
$746$		0			0
$747$	−7.00000	+	7.00000i	−0.256117	+	0.256117i
$748$		0			0
$749$	14.0000	−	14.0000i	0.511549	−	0.511549i
$750$		0			0
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		−	8.00000i		−	0.291536i
$754$		0			0
$755$			18.0000i			0.655087i
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	29.0000	−	29.0000i	1.05125	−	1.05125i	0.0526354	−	0.998614i	$-0.483238\pi$
$761$	29.0000	−	29.0000i	1.05125	−	1.05125i	0.998614	−	0.0526354i	$-0.0167621\pi$
$762$		0			0
$763$			18.0000i			0.651644i
$764$		0			0
$765$	−4.00000	+	4.00000i	−0.144620	+	0.144620i
$766$		0			0
$767$	35.0000	+	7.00000i	1.26378	+	0.252755i
$768$		0			0
$769$	17.0000	+	17.0000i	0.613036	+	0.613036i	0.943736	−	0.330700i	$-0.107285\pi$
$769$	17.0000	+	17.0000i	0.613036	+	0.613036i	−0.330700	+	0.943736i	$0.607285\pi$
$770$		0			0
$771$	48.0000			1.72868
$772$		0			0
$773$	7.00000	+	7.00000i	0.251773	+	0.251773i	0.821697	−	0.569925i	$-0.193028\pi$
$773$	7.00000	+	7.00000i	0.251773	+	0.251773i	−0.569925	+	0.821697i	$0.693028\pi$
$774$		0			0
$775$	9.00000	+	9.00000i	0.323290	+	0.323290i
$776$		0			0
$777$			12.0000i			0.430498i
$778$		0			0
$779$	6.00000			0.214972
$780$		0			0
$781$	−30.0000			−1.07348
$782$		0			0
$783$			24.0000i			0.857690i
$784$		0			0
$785$	−2.00000	−	2.00000i	−0.0713831	−	0.0713831i
$786$		0			0
$787$	−3.00000	−	3.00000i	−0.106938	−	0.106938i	0.651613	−	0.758552i	$-0.274093\pi$
$787$	−3.00000	−	3.00000i	−0.106938	−	0.106938i	−0.758552	+	0.651613i	$0.774093\pi$
$788$		0			0
$789$	−36.0000			−1.28163
$790$		0			0
$791$	2.00000	+	2.00000i	0.0711118	+	0.0711118i

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.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.00000 - 1.00000i) q^{7} -1.00000 q^{9} +(-3.00000 - 3.00000i) q^{11} +(-3.00000 + 2.00000i) q^{13} +(-2.00000 + 2.00000i) q^{15} +4.00000i q^{17} +(3.00000 - 3.00000i) q^{19} +(-2.00000 + 2.00000i) q^{21} -3.00000i q^{25} -4.00000i q^{27} -6.00000 q^{29} +(-3.00000 + 3.00000i) q^{31} +(-6.00000 + 6.00000i) q^{33} +2.00000i q^{35} +(3.00000 - 3.00000i) q^{37} +(4.00000 + 6.00000i) q^{39} +(1.00000 + 1.00000i) q^{41} +4.00000 q^{43} +(1.00000 + 1.00000i) q^{45} +(-5.00000 - 5.00000i) q^{47} -5.00000i q^{49} +8.00000 q^{51} +6.00000 q^{53} +6.00000i q^{55} +(-6.00000 - 6.00000i) q^{57} +(-7.00000 - 7.00000i) q^{59} +14.0000 q^{61} +(1.00000 + 1.00000i) q^{63} +(5.00000 + 1.00000i) q^{65} +(-5.00000 + 5.00000i) q^{67} +(5.00000 - 5.00000i) q^{71} +(9.00000 - 9.00000i) q^{73} -6.00000 q^{75} +6.00000i q^{77} +6.00000i q^{79} -11.0000 q^{81} +(7.00000 - 7.00000i) q^{83} +(4.00000 - 4.00000i) q^{85} +12.0000i q^{87} +(5.00000 - 5.00000i) q^{89} +(5.00000 + 1.00000i) q^{91} +(6.00000 + 6.00000i) q^{93} -6.00000 q^{95} +(13.0000 + 13.0000i) q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} - 4 q^{15} + 6 q^{19} - 4 q^{21} - 12 q^{29} - 6 q^{31} - 12 q^{33} + 6 q^{37} + 8 q^{39} + 2 q^{41} + 8 q^{43} + 2 q^{45} - 10 q^{47} + 16 q^{51} + 12 q^{53} - 12 q^{57} - 14 q^{59} + 28 q^{61} + 2 q^{63} + 10 q^{65} - 10 q^{67} + 10 q^{71} + 18 q^{73} - 12 q^{75} - 22 q^{81} + 14 q^{83} + 8 q^{85} + 10 q^{89} + 10 q^{91} + 12 q^{93} - 12 q^{95} + 26 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 - 2 * q^9 - 6 * q^11 - 6 * q^13 - 4 * q^15 + 6 * q^19 - 4 * q^21 - 12 * q^29 - 6 * q^31 - 12 * q^33 + 6 * q^37 + 8 * q^39 + 2 * q^41 + 8 * q^43 + 2 * q^45 - 10 * q^47 + 16 * q^51 + 12 * q^53 - 12 * q^57 - 14 * q^59 + 28 * q^61 + 2 * q^63 + 10 * q^65 - 10 * q^67 + 10 * q^71 + 18 * q^73 - 12 * q^75 - 22 * q^81 + 14 * q^83 + 8 * q^85 + 10 * q^89 + 10 * q^91 + 12 * q^93 - 12 * q^95 + 26 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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