LMFDB - Embedded newform 1280.2.c.m.769.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(769,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1280.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.c (of order $2$, degree $1$, 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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.4
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1280.769
Dual form			1280.2.c.m.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} +(1.00000 + 2.00000i) q^{5} -2.44949i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.44949i q^{3} +(1.00000 + 2.00000i) q^{5} -2.44949i q^{7} -3.00000 q^{9} +4.89898 q^{11} +(-4.89898 + 2.44949i) q^{15} +4.00000i q^{17} +4.89898 q^{19} +6.00000 q^{21} +2.44949i q^{23} +(-3.00000 + 4.00000i) q^{25} -8.00000 q^{29} +9.79796 q^{31} +12.0000i q^{33} +(4.89898 - 2.44949i) q^{35} +4.00000i q^{37} -8.00000 q^{41} +7.34847i q^{43} +(-3.00000 - 6.00000i) q^{45} -12.2474i q^{47} +1.00000 q^{49} -9.79796 q^{51} -8.00000i q^{53} +(4.89898 + 9.79796i) q^{55} +12.0000i q^{57} -4.89898 q^{59} +6.00000 q^{61} +7.34847i q^{63} -2.44949i q^{67} -6.00000 q^{69} -9.79796 q^{71} -4.00000i q^{73} +(-9.79796 - 7.34847i) q^{75} -12.0000i q^{77} -9.79796 q^{79} -9.00000 q^{81} -2.44949i q^{83} +(-8.00000 + 4.00000i) q^{85} -19.5959i q^{87} -2.00000 q^{89} +24.0000i q^{93} +(4.89898 + 9.79796i) q^{95} -4.00000i q^{97} -14.6969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 12 * q^9	$ 4 q + 4 q^{5} - 12 q^{9} + 24 q^{21} - 12 q^{25} - 32 q^{29} - 32 q^{41} - 12 q^{45} + 4 q^{49} + 24 q^{61} - 24 q^{69} - 36 q^{81} - 32 q^{85} - 8 q^{89}+O(q^{100}) $ 4 * q + 4 * q^5 - 12 * q^9 + 24 * q^21 - 12 * q^25 - 32 * q^29 - 32 * q^41 - 12 * q^45 + 4 * q^49 + 24 * q^61 - 24 * q^69 - 36 * q^81 - 32 * q^85 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.44949i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$3$			2.44949i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−4.89898	+	2.44949i	−1.26491	+	0.632456i
$16$		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
$19$	4.89898			1.12390			0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898			1.12390			0.561951	+	0.827170i	$0.310051\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$		0			0
$23$			2.44949i			0.510754i	0.966842	+	0.255377i	$0.0821996\pi$
$23$			2.44949i			0.510754i	−0.966842	+	0.255377i	$0.917800\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	9.79796			1.75977			0.879883	−	0.475191i	$-0.157621\pi$
$31$	9.79796			1.75977			0.879883	+	0.475191i	$0.157621\pi$
$32$		0			0
$33$			12.0000i			2.08893i
$34$		0			0
$35$	4.89898	−	2.44949i	0.828079	−	0.414039i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$			7.34847i			1.12063i	0.828279	+	0.560316i	$0.189320\pi$
$43$			7.34847i			1.12063i	−0.828279	+	0.560316i	$0.810680\pi$
$44$		0			0
$45$	−3.00000	−	6.00000i	−0.447214	−	0.894427i
$46$		0			0
$47$		−	12.2474i		−	1.78647i	−0.449586	−	0.893237i	$-0.648429\pi$
$47$		−	12.2474i		−	1.78647i	0.449586	−	0.893237i	$-0.351571\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−9.79796			−1.37199
$52$		0			0
$53$		−	8.00000i		−	1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$		−	8.00000i		−	1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$		0			0
$55$	4.89898	+	9.79796i	0.660578	+	1.32116i
$56$		0			0
$57$			12.0000i			1.58944i
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$			7.34847i			0.925820i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	2.44949i		−	0.299253i	−0.988743	−	0.149626i	$-0.952193\pi$
$67$		−	2.44949i		−	0.299253i	0.988743	−	0.149626i	$-0.0478071\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	−9.79796			−1.16280			−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796			−1.16280			−0.581402	+	0.813617i	$0.697496\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−9.79796	−	7.34847i	−1.13137	−	0.848528i
$76$		0			0
$77$		−	12.0000i		−	1.36753i
$78$		0			0
$79$	−9.79796			−1.10236			−0.551178	−	0.834388i	$-0.685822\pi$
$79$	−9.79796			−1.10236			−0.551178	+	0.834388i	$0.685822\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		−	2.44949i		−	0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$		−	2.44949i		−	0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$		0			0
$85$	−8.00000	+	4.00000i	−0.867722	+	0.433861i
$86$		0			0
$87$		−	19.5959i		−	2.10090i
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			24.0000i			2.48868i
$94$		0			0
$95$	4.89898	+	9.79796i	0.502625	+	1.00525i
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$		0			0
$99$	−14.6969			−1.47710
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$			2.44949i			0.241355i	0.992692	+	0.120678i	$0.0385068\pi$
$103$			2.44949i			0.241355i	−0.992692	+	0.120678i	$0.961493\pi$
$104$		0			0
$105$	6.00000	+	12.0000i	0.585540	+	1.17108i
$106$		0			0
$107$		−	7.34847i		−	0.710403i	−0.934790	−	0.355202i	$-0.884412\pi$
$107$		−	7.34847i		−	0.710403i	0.934790	−	0.355202i	$-0.115588\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	−9.79796			−0.929981
$112$		0			0
$113$			16.0000i			1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$			16.0000i			1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−4.89898	+	2.44949i	−0.456832	+	0.228416i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.79796			0.898177
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$		−	19.5959i		−	1.76690i
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$		−	7.34847i		−	0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$		−	7.34847i		−	0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$		0			0
$129$	−18.0000			−1.58481
$130$		0			0
$131$	4.89898			0.428026			0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898			0.428026			0.214013	+	0.976831i	$0.431347\pi$
$132$		0			0
$133$		−	12.0000i		−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	4.89898			0.415526			0.207763	−	0.978179i	$-0.433382\pi$
$139$	4.89898			0.415526			0.207763	+	0.978179i	$0.433382\pi$
$140$		0			0
$141$	30.0000			2.52646
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−8.00000	−	16.0000i	−0.664364	−	1.32873i
$146$		0			0
$147$			2.44949i			0.202031i
$148$		0			0
$149$	−2.00000			−0.163846			−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000			−0.163846			−0.0819232	+	0.996639i	$0.526106\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	12.0000i		−	0.970143i
$154$		0			0
$155$	9.79796	+	19.5959i	0.786991	+	1.57398i
$156$		0			0
$157$			20.0000i			1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$			20.0000i			1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$		0			0
$159$	19.5959			1.55406
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$			2.44949i			0.191859i	0.995388	+	0.0959294i	$0.0305823\pi$
$163$			2.44949i			0.191859i	−0.995388	+	0.0959294i	$0.969418\pi$
$164$		0			0
$165$	−24.0000	+	12.0000i	−1.86840	+	0.934199i
$166$		0			0
$167$			2.44949i			0.189547i	0.995499	+	0.0947736i	$0.0302127\pi$
$167$			2.44949i			0.189547i	−0.995499	+	0.0947736i	$0.969787\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−14.6969			−1.12390
$172$		0			0
$173$		−	4.00000i		−	0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$		−	4.00000i		−	0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$		0			0
$175$	9.79796	+	7.34847i	0.740656	+	0.555492i
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$	14.6969			1.09850			0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969			1.09850			0.549250	+	0.835658i	$0.314913\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$			14.6969i			1.08643i
$184$		0			0
$185$	−8.00000	+	4.00000i	−0.588172	+	0.294086i
$186$		0			0
$187$			19.5959i			1.43300i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.5959			−1.41791			−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959			−1.41791			−0.708955	+	0.705253i	$0.750833\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	−9.79796			−0.694559			−0.347279	−	0.937762i	$-0.612894\pi$
$199$	−9.79796			−0.694559			−0.347279	+	0.937762i	$0.612894\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$			19.5959i			1.37536i
$204$		0			0
$205$	−8.00000	−	16.0000i	−0.558744	−	1.11749i
$206$		0			0
$207$		−	7.34847i		−	0.510754i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	24.4949			1.68630			0.843149	−	0.537680i	$-0.180699\pi$
$211$	24.4949			1.68630			0.843149	+	0.537680i	$0.180699\pi$
$212$		0			0
$213$		−	24.0000i		−	1.64445i
$214$		0			0
$215$	−14.6969	+	7.34847i	−1.00232	+	0.501161i
$216$		0			0
$217$		−	24.0000i		−	1.62923i
$218$		0			0
$219$	9.79796			0.662085
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	7.34847i		−	0.492090i	−0.969258	−	0.246045i	$-0.920869\pi$
$223$		−	7.34847i		−	0.492090i	0.969258	−	0.246045i	$-0.0791311\pi$
$224$		0			0
$225$	9.00000	−	12.0000i	0.600000	−	0.800000i
$226$		0			0
$227$			2.44949i			0.162578i	0.996691	+	0.0812892i	$0.0259037\pi$
$227$			2.44949i			0.162578i	−0.996691	+	0.0812892i	$0.974096\pi$
$228$		0			0
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$		0			0
$231$	29.3939			1.93398
$232$		0			0
$233$		−	20.0000i		−	1.31024i	−0.755523	−	0.655122i	$-0.772617\pi$
$233$		−	20.0000i		−	1.31024i	0.755523	−	0.655122i	$-0.227383\pi$
$234$		0			0
$235$	24.4949	−	12.2474i	1.59787	−	0.798935i
$236$		0			0
$237$		−	24.0000i		−	1.55897i
$238$		0			0
$239$	9.79796			0.633777			0.316889	−	0.948463i	$-0.397362\pi$
$239$	9.79796			0.633777			0.316889	+	0.948463i	$0.397362\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		−	22.0454i		−	1.41421i
$244$		0			0
$245$	1.00000	+	2.00000i	0.0638877	+	0.127775i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	4.89898			0.309221			0.154610	−	0.987976i	$-0.450588\pi$
$251$	4.89898			0.309221			0.154610	+	0.987976i	$0.450588\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$	−9.79796	−	19.5959i	−0.613572	−	1.22714i
$256$		0			0
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$		0			0
$259$	9.79796			0.608816
$260$		0			0
$261$	24.0000			1.48556
$262$		0			0
$263$		−	22.0454i		−	1.35938i	−0.733500	−	0.679689i	$-0.762115\pi$
$263$		−	22.0454i		−	1.35938i	0.733500	−	0.679689i	$-0.237885\pi$
$264$		0			0
$265$	16.0000	−	8.00000i	0.982872	−	0.491436i
$266$		0			0
$267$		−	4.89898i		−	0.299813i
$268$		0			0
$269$	−26.0000			−1.58525			−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000			−1.58525			−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$	9.79796			0.595184			0.297592	−	0.954693i	$-0.403817\pi$
$271$	9.79796			0.595184			0.297592	+	0.954693i	$0.403817\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.6969	+	19.5959i	−0.886259	+	1.18168i
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$		0			0
$279$	−29.3939			−1.75977
$280$		0			0
$281$	−16.0000			−0.954480			−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000			−0.954480			−0.477240	+	0.878773i	$0.658363\pi$
$282$		0			0
$283$		−	26.9444i		−	1.60168i	−0.598880	−	0.800839i	$-0.704387\pi$
$283$		−	26.9444i		−	1.60168i	0.598880	−	0.800839i	$-0.295613\pi$
$284$		0			0
$285$	−24.0000	+	12.0000i	−1.42164	+	0.710819i
$286$		0			0
$287$			19.5959i			1.15671i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	9.79796			0.574367
$292$		0			0
$293$			20.0000i			1.16841i	0.811605	+	0.584206i	$0.198594\pi$
$293$			20.0000i			1.16841i	−0.811605	+	0.584206i	$0.801406\pi$
$294$		0			0
$295$	−4.89898	−	9.79796i	−0.285230	−	0.570459i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	18.0000			1.03750
$302$		0			0
$303$			19.5959i			1.12576i
$304$		0			0
$305$	6.00000	+	12.0000i	0.343559	+	0.687118i
$306$		0			0
$307$			17.1464i			0.978598i	0.872116	+	0.489299i	$0.162747\pi$
$307$			17.1464i			0.978598i	−0.872116	+	0.489299i	$0.837253\pi$
$308$		0			0
$309$	−6.00000			−0.341328
$310$		0			0
$311$	19.5959			1.11118			0.555591	−	0.831456i	$-0.312492\pi$
$311$	19.5959			1.11118			0.555591	+	0.831456i	$0.312492\pi$
$312$		0			0
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$		0			0
$315$	−14.6969	+	7.34847i	−0.828079	+	0.414039i
$316$		0			0
$317$		−	8.00000i		−	0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$		−	8.00000i		−	0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$		0			0
$319$	−39.1918			−2.19432
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$			19.5959i			1.09035i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			24.4949i			1.35457i
$328$		0			0
$329$	−30.0000			−1.65395
$330$		0			0
$331$	−4.89898			−0.269272			−0.134636	−	0.990895i	$-0.542987\pi$
$331$	−4.89898			−0.269272			−0.134636	+	0.990895i	$0.542987\pi$
$332$		0			0
$333$		−	12.0000i		−	0.657596i
$334$		0			0
$335$	4.89898	−	2.44949i	0.267660	−	0.133830i
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	−39.1918			−2.12861
$340$		0			0
$341$	48.0000			2.59935
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$	−6.00000	−	12.0000i	−0.323029	−	0.646058i
$346$		0			0
$347$		−	12.2474i		−	0.657477i	−0.944421	−	0.328739i	$-0.893376\pi$
$347$		−	12.2474i		−	0.657477i	0.944421	−	0.328739i	$-0.106624\pi$
$348$		0			0
$349$	−24.0000			−1.28469			−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000			−1.28469			−0.642345	+	0.766415i	$0.722038\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	−9.79796	−	19.5959i	−0.520022	−	1.04004i
$356$		0			0
$357$			24.0000i			1.27021i
$358$		0			0
$359$	19.5959			1.03423			0.517116	−	0.855915i	$-0.327005\pi$
$359$	19.5959			1.03423			0.517116	+	0.855915i	$0.327005\pi$
$360$		0			0
$361$	5.00000			0.263158
$362$		0			0
$363$			31.8434i			1.67134i
$364$		0			0
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		0			0
$367$			31.8434i			1.66221i	0.556115	+	0.831105i	$0.312291\pi$
$367$			31.8434i			1.66221i	−0.556115	+	0.831105i	$0.687709\pi$
$368$		0			0
$369$	24.0000			1.24939
$370$		0			0
$371$	−19.5959			−1.01737
$372$		0			0
$373$			20.0000i			1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$			20.0000i			1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$		0			0
$375$	4.89898	−	26.9444i	0.252982	−	1.39140i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−14.6969			−0.754931			−0.377466	−	0.926024i	$-0.623204\pi$
$379$	−14.6969			−0.754931			−0.377466	+	0.926024i	$0.623204\pi$
$380$		0			0
$381$	18.0000			0.922168
$382$		0			0
$383$		−	26.9444i		−	1.37679i	−0.725334	−	0.688397i	$-0.758315\pi$
$383$		−	26.9444i		−	1.37679i	0.725334	−	0.688397i	$-0.241685\pi$
$384$		0			0
$385$	24.0000	−	12.0000i	1.22315	−	0.611577i
$386$		0			0
$387$		−	22.0454i		−	1.12063i
$388$		0			0
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$		0			0
$391$	−9.79796			−0.495504
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$	−9.79796	−	19.5959i	−0.492989	−	0.985978i
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$	29.3939			1.47153
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$		0			0
$407$			19.5959i			0.971334i
$408$		0			0
$409$	24.0000			1.18672			0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000			1.18672			0.593362	+	0.804936i	$0.297800\pi$
$410$		0			0
$411$	−19.5959			−0.966595
$412$		0			0
$413$			12.0000i			0.590481i
$414$		0			0
$415$	4.89898	−	2.44949i	0.240481	−	0.120241i
$416$		0			0
$417$			12.0000i			0.587643i
$418$		0			0
$419$	14.6969			0.717992			0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969			0.717992			0.358996	+	0.933339i	$0.383119\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$			36.7423i			1.78647i
$424$		0			0
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$		−	14.6969i		−	0.711235i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	29.3939			1.41585			0.707927	−	0.706286i	$-0.249631\pi$
$431$	29.3939			1.41585			0.707927	+	0.706286i	$0.249631\pi$
$432$		0			0
$433$		−	36.0000i		−	1.73005i	−0.501729	−	0.865025i	$-0.667303\pi$
$433$		−	36.0000i		−	1.73005i	0.501729	−	0.865025i	$-0.332697\pi$
$434$		0			0
$435$	39.1918	−	19.5959i	1.87910	−	0.939552i
$436$		0			0
$437$			12.0000i			0.574038i
$438$		0			0
$439$	19.5959			0.935262			0.467631	−	0.883924i	$-0.345108\pi$
$439$	19.5959			0.935262			0.467631	+	0.883924i	$0.345108\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$		−	26.9444i		−	1.28017i	−0.768306	−	0.640083i	$-0.778900\pi$
$443$		−	26.9444i		−	1.28017i	0.768306	−	0.640083i	$-0.221100\pi$
$444$		0			0
$445$	−2.00000	−	4.00000i	−0.0948091	−	0.189618i
$446$		0			0
$447$		−	4.89898i		−	0.231714i
$448$		0			0
$449$	−32.0000			−1.51017			−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000			−1.51017			−0.755087	+	0.655625i	$0.772405\pi$
$450$		0			0
$451$	−39.1918			−1.84547
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.00000			0.372597			0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000			0.372597			0.186299	+	0.982493i	$0.440351\pi$
$462$		0			0
$463$		−	31.8434i		−	1.47989i	−0.672669	−	0.739943i	$-0.734852\pi$
$463$		−	31.8434i		−	1.47989i	0.672669	−	0.739943i	$-0.265148\pi$
$464$		0			0
$465$	−48.0000	+	24.0000i	−2.22595	+	1.11297i
$466$		0			0
$467$		−	36.7423i		−	1.70023i	−0.526595	−	0.850117i	$-0.676531\pi$
$467$		−	36.7423i		−	1.70023i	0.526595	−	0.850117i	$-0.323469\pi$
$468$		0			0
$469$	−6.00000			−0.277054
$470$		0			0
$471$	−48.9898			−2.25733
$472$		0			0
$473$			36.0000i			1.65528i
$474$		0			0
$475$	−14.6969	+	19.5959i	−0.674342	+	0.899122i
$476$		0			0
$477$			24.0000i			1.09888i
$478$		0			0
$479$	−19.5959			−0.895360			−0.447680	−	0.894194i	$-0.647750\pi$
$479$	−19.5959			−0.895360			−0.447680	+	0.894194i	$0.647750\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			14.6969i			0.668734i
$484$		0			0
$485$	8.00000	−	4.00000i	0.363261	−	0.181631i
$486$		0			0
$487$			17.1464i			0.776979i	0.921453	+	0.388489i	$0.127003\pi$
$487$			17.1464i			0.776979i	−0.921453	+	0.388489i	$0.872997\pi$
$488$		0			0
$489$	−6.00000			−0.271329
$490$		0			0
$491$	14.6969			0.663264			0.331632	−	0.943409i	$-0.392401\pi$
$491$	14.6969			0.663264			0.331632	+	0.943409i	$0.392401\pi$
$492$		0			0
$493$		−	32.0000i		−	1.44121i
$494$		0			0
$495$	−14.6969	−	29.3939i	−0.660578	−	1.32116i
$496$		0			0
$497$			24.0000i			1.07655i
$498$		0			0
$499$	−4.89898			−0.219308			−0.109654	−	0.993970i	$-0.534974\pi$
$499$	−4.89898			−0.219308			−0.109654	+	0.993970i	$0.534974\pi$
$500$		0			0
$501$	−6.00000			−0.268060
$502$		0			0
$503$		−	17.1464i		−	0.764521i	−0.924055	−	0.382261i	$-0.875146\pi$
$503$		−	17.1464i		−	0.764521i	0.924055	−	0.382261i	$-0.124854\pi$
$504$		0			0
$505$	8.00000	+	16.0000i	0.355995	+	0.711991i
$506$		0			0
$507$			31.8434i			1.41421i
$508$		0			0
$509$	−8.00000			−0.354594			−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000			−0.354594			−0.177297	+	0.984157i	$0.556735\pi$
$510$		0			0
$511$	−9.79796			−0.433436
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.89898	+	2.44949i	−0.215875	+	0.107937i
$516$		0			0
$517$		−	60.0000i		−	2.63880i
$518$		0			0
$519$	9.79796			0.430083
$520$		0			0
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$			31.8434i			1.39241i	0.717841	+	0.696207i	$0.245130\pi$
$523$			31.8434i			1.39241i	−0.717841	+	0.696207i	$0.754870\pi$
$524$		0			0
$525$	−18.0000	+	24.0000i	−0.785584	+	1.04745i
$526$		0			0
$527$			39.1918i			1.70722i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$	14.6969			0.637793
$532$		0			0
$533$		0			0
$534$		0			0
$535$	14.6969	−	7.34847i	0.635404	−	0.317702i
$536$		0			0
$537$			36.0000i			1.55351i
$538$		0			0
$539$	4.89898			0.211014
$540$		0			0
$541$	8.00000			0.343947			0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000			0.343947			0.171973	+	0.985102i	$0.444986\pi$
$542$		0			0
$543$			19.5959i			0.840941i
$544$		0			0
$545$	10.0000	+	20.0000i	0.428353	+	0.856706i
$546$		0			0
$547$			22.0454i			0.942594i	0.881975	+	0.471297i	$0.156214\pi$
$547$			22.0454i			0.942594i	−0.881975	+	0.471297i	$0.843786\pi$
$548$		0			0
$549$	−18.0000			−0.768221
$550$		0			0
$551$	−39.1918			−1.66963
$552$		0			0
$553$			24.0000i			1.02058i
$554$		0			0
$555$	−9.79796	−	19.5959i	−0.415900	−	0.831800i
$556$		0			0
$557$			4.00000i			0.169485i	0.996403	+	0.0847427i	$0.0270068\pi$
$557$			4.00000i			0.169485i	−0.996403	+	0.0847427i	$0.972993\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−48.0000			−2.02656
$562$		0			0
$563$			2.44949i			0.103234i	0.998667	+	0.0516168i	$0.0164375\pi$
$563$			2.44949i			0.103234i	−0.998667	+	0.0516168i	$0.983563\pi$
$564$		0			0
$565$	−32.0000	+	16.0000i	−1.34625	+	0.673125i
$566$		0			0
$567$			22.0454i			0.925820i
$568$		0			0
$569$	8.00000			0.335377			0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000			0.335377			0.167689	+	0.985840i	$0.446370\pi$
$570$		0			0
$571$	44.0908			1.84514			0.922572	−	0.385826i	$-0.126083\pi$
$571$	44.0908			1.84514			0.922572	+	0.385826i	$0.126083\pi$
$572$		0			0
$573$		−	48.0000i		−	2.00523i
$574$		0			0
$575$	−9.79796	−	7.34847i	−0.408603	−	0.306452i
$576$		0			0
$577$			24.0000i			0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$			24.0000i			0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$		0			0
$579$	29.3939			1.22157
$580$		0			0
$581$	−6.00000			−0.248922
$582$		0			0
$583$		−	39.1918i		−	1.62316i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12.2474i		−	0.505506i	−0.967531	−	0.252753i	$-0.918664\pi$
$587$		−	12.2474i		−	0.505506i	0.967531	−	0.252753i	$-0.0813361\pi$
$588$		0			0
$589$	48.0000			1.97781
$590$		0			0
$591$	−19.5959			−0.806068
$592$		0			0
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$		0			0
$595$	9.79796	+	19.5959i	0.401677	+	0.803354i
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$	9.79796			0.400334			0.200167	−	0.979762i	$-0.435852\pi$
$599$	9.79796			0.400334			0.200167	+	0.979762i	$0.435852\pi$
$600$		0			0
$601$	−16.0000			−0.652654			−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000			−0.652654			−0.326327	+	0.945257i	$0.605811\pi$
$602$		0			0
$603$			7.34847i			0.299253i
$604$		0			0
$605$	13.0000	+	26.0000i	0.528525	+	1.05705i
$606$		0			0
$607$			7.34847i			0.298265i	0.988817	+	0.149133i	$0.0476481\pi$
$607$			7.34847i			0.298265i	−0.988817	+	0.149133i	$0.952352\pi$
$608$		0			0
$609$	−48.0000			−1.94506
$610$		0			0
$611$		0			0
$612$		0			0
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$		0			0
$615$	39.1918	−	19.5959i	1.58037	−	0.790184i
$616$		0			0
$617$			28.0000i			1.12724i	0.826035	+	0.563619i	$0.190591\pi$
$617$			28.0000i			1.12724i	−0.826035	+	0.563619i	$0.809409\pi$
$618$		0			0
$619$	−24.4949			−0.984533			−0.492267	−	0.870445i	$-0.663831\pi$
$619$	−24.4949			−0.984533			−0.492267	+	0.870445i	$0.663831\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.89898i			0.196273i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			58.7878i			2.34776i
$628$		0			0
$629$	−16.0000			−0.637962
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$			60.0000i			2.38479i
$634$		0			0
$635$	14.6969	−	7.34847i	0.583230	−	0.291615i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	29.3939			1.16280
$640$		0			0
$641$	−16.0000			−0.631962			−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000			−0.631962			−0.315981	+	0.948766i	$0.602334\pi$
$642$		0			0
$643$		−	2.44949i		−	0.0965984i	−0.998833	−	0.0482992i	$-0.984620\pi$
$643$		−	2.44949i		−	0.0965984i	0.998833	−	0.0482992i	$-0.0153801\pi$
$644$		0			0
$645$	−18.0000	−	36.0000i	−0.708749	−	1.41750i
$646$		0			0
$647$		−	22.0454i		−	0.866694i	−0.901227	−	0.433347i	$-0.857332\pi$
$647$		−	22.0454i		−	0.866694i	0.901227	−	0.433347i	$-0.142668\pi$
$648$		0			0
$649$	−24.0000			−0.942082
$650$		0			0
$651$	58.7878			2.30407
$652$		0			0
$653$			16.0000i			0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$			16.0000i			0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$		0			0
$655$	4.89898	+	9.79796i	0.191419	+	0.382838i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$	14.6969			0.572511			0.286256	−	0.958153i	$-0.407589\pi$
$659$	14.6969			0.572511			0.286256	+	0.958153i	$0.407589\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.0000	−	12.0000i	0.930680	−	0.465340i
$666$		0			0
$667$		−	19.5959i		−	0.758757i
$668$		0			0
$669$	18.0000			0.695920
$670$		0			0
$671$	29.3939			1.13474
$672$		0			0
$673$		−	36.0000i		−	1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$		−	36.0000i		−	1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.0000i			1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$			32.0000i			1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$		0			0
$679$	−9.79796			−0.376011
$680$		0			0
$681$	−6.00000			−0.229920
$682$		0			0
$683$			7.34847i			0.281181i	0.990068	+	0.140591i	$0.0449002\pi$
$683$			7.34847i			0.281181i	−0.990068	+	0.140591i	$0.955100\pi$
$684$		0			0
$685$	−16.0000	+	8.00000i	−0.611329	+	0.305664i
$686$		0			0
$687$		−	19.5959i		−	0.747631i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−14.6969			−0.559098			−0.279549	−	0.960131i	$-0.590185\pi$
$691$	−14.6969			−0.559098			−0.279549	+	0.960131i	$0.590185\pi$
$692$		0			0
$693$			36.0000i			1.36753i
$694$		0			0
$695$	4.89898	+	9.79796i	0.185829	+	0.371658i
$696$		0			0
$697$		−	32.0000i		−	1.21209i
$698$		0			0
$699$	48.9898			1.85296
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$			19.5959i			0.739074i
$704$		0			0
$705$	30.0000	+	60.0000i	1.12987	+	2.25973i
$706$		0			0
$707$		−	19.5959i		−	0.736980i
$708$		0			0
$709$	40.0000			1.50223			0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000			1.50223			0.751116	+	0.660171i	$0.229516\pi$
$710$		0			0
$711$	29.3939			1.10236
$712$		0			0
$713$			24.0000i			0.898807i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			24.0000i			0.896296i
$718$		0			0
$719$	−48.9898			−1.82701			−0.913506	−	0.406826i	$-0.866635\pi$
$719$	−48.9898			−1.82701			−0.913506	+	0.406826i	$0.866635\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.0000	−	32.0000i	0.891338	−	1.18845i
$726$		0			0
$727$			36.7423i			1.36270i	0.731959	+	0.681349i	$0.238606\pi$
$727$			36.7423i			1.36270i	−0.731959	+	0.681349i	$0.761394\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−29.3939			−1.08717
$732$		0			0
$733$		−	12.0000i		−	0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$		−	12.0000i		−	0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$		0			0
$735$	−4.89898	+	2.44949i	−0.180702	+	0.0903508i
$736$		0			0
$737$		−	12.0000i		−	0.442026i
$738$		0			0
$739$	14.6969			0.540636			0.270318	−	0.962771i	$-0.412871\pi$
$739$	14.6969			0.540636			0.270318	+	0.962771i	$0.412871\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			17.1464i			0.629041i	0.949251	+	0.314521i	$0.101844\pi$
$743$			17.1464i			0.629041i	−0.949251	+	0.314521i	$0.898156\pi$
$744$		0			0
$745$	−2.00000	−	4.00000i	−0.0732743	−	0.146549i
$746$		0			0
$747$			7.34847i			0.268866i
$748$		0			0
$749$	−18.0000			−0.657706
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$		0			0
$759$	−29.3939			−1.06693
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$		−	24.4949i		−	0.886775i
$764$		0			0
$765$	24.0000	−	12.0000i	0.867722	−	0.433861i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$	19.5959			0.705730
$772$		0			0
$773$		−	8.00000i		−	0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$		−	8.00000i		−	0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$		0			0
$775$	−29.3939	+	39.1918i	−1.05586	+	1.40781i
$776$		0			0
$777$			24.0000i			0.860995i
$778$		0			0
$779$	−39.1918			−1.40419
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.0000	+	20.0000i	−1.42766	+	0.713831i
$786$		0			0
$787$		−	41.6413i		−	1.48435i	−0.670205	−	0.742176i	$-0.733794\pi$
$787$		−	41.6413i		−	1.48435i	0.670205	−	0.742176i	$-0.266206\pi$
$788$		0			0
$789$	54.0000			1.92245
$790$		0			0
$791$	39.1918			1.39350
$792$		0			0
$793$		0			0
$794$		0			0
$795$	19.5959	+	39.1918i	0.694996	+	1.38999i
$796$		0			0
$797$		−	8.00000i		−	0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$		−	8.00000i		−	0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$		0			0
$799$	48.9898			1.73313
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		−	19.5959i		−	0.691525i
$804$		0			0
$805$	6.00000	+	12.0000i	0.211472	+	0.422944i
$806$		0			0
$807$		−	63.6867i		−	2.24188i
$808$		0			0
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$	−53.8888			−1.89229			−0.946145	−	0.323742i	$-0.895059\pi$
$811$	−53.8888			−1.89229			−0.946145	+	0.323742i	$0.895059\pi$
$812$		0			0
$813$			24.0000i			0.841717i
$814$		0			0
$815$	−4.89898	+	2.44949i	−0.171604	+	0.0858019i
$816$		0			0
$817$			36.0000i			1.25948i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.0000			−0.488603			−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000			−0.488603			−0.244302	+	0.969699i	$0.578559\pi$
$822$		0			0
$823$			22.0454i			0.768455i	0.923239	+	0.384227i	$0.125532\pi$
$823$			22.0454i			0.768455i	−0.923239	+	0.384227i	$0.874468\pi$
$824$		0			0
$825$	−48.0000	−	36.0000i	−1.67115	−	1.25336i
$826$		0			0
$827$			46.5403i			1.61836i	0.587557	+	0.809182i	$0.300090\pi$
$827$			46.5403i			1.61836i	−0.587557	+	0.809182i	$0.699910\pi$
$828$		0			0
$829$	22.0000			0.764092			0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000			0.764092			0.382046	+	0.924143i	$0.375220\pi$
$830$		0			0
$831$	29.3939			1.01966
$832$		0			0
$833$			4.00000i			0.138592i
$834$		0			0
$835$	−4.89898	+	2.44949i	−0.169536	+	0.0847681i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−39.1918			−1.35305			−0.676526	−	0.736419i	$-0.736515\pi$
$839$	−39.1918			−1.35305			−0.676526	+	0.736419i	$0.736515\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		−	39.1918i		−	1.34984i
$844$		0			0
$845$	13.0000	+	26.0000i	0.447214	+	0.894427i
$846$		0			0
$847$		−	31.8434i		−	1.09415i
$848$		0			0
$849$	66.0000			2.26511
$850$		0			0
$851$	−9.79796			−0.335870
$852$		0			0
$853$			32.0000i			1.09566i	0.836590	+	0.547830i	$0.184546\pi$
$853$			32.0000i			1.09566i	−0.836590	+	0.547830i	$0.815454\pi$
$854$		0			0
$855$	−14.6969	−	29.3939i	−0.502625	−	1.00525i
$856$		0			0
$857$			32.0000i			1.09310i	0.837427	+	0.546550i	$0.184059\pi$
$857$			32.0000i			1.09310i	−0.837427	+	0.546550i	$0.815941\pi$
$858$		0			0
$859$	−24.4949			−0.835755			−0.417878	−	0.908503i	$-0.637226\pi$
$859$	−24.4949			−0.835755			−0.417878	+	0.908503i	$0.637226\pi$
$860$		0			0
$861$	−48.0000			−1.63584
$862$		0			0
$863$		−	51.4393i		−	1.75101i	−0.483206	−	0.875507i	$-0.660528\pi$
$863$		−	51.4393i		−	1.75101i	0.483206	−	0.875507i	$-0.339472\pi$
$864$		0			0
$865$	8.00000	−	4.00000i	0.272008	−	0.136004i
$866$		0			0
$867$			2.44949i			0.0831890i
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$		0			0
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$	−4.89898	+	26.9444i	−0.165616	+	0.910887i
$876$		0			0
$877$			28.0000i			0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$			28.0000i			0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$		0			0
$879$	−48.9898			−1.65238
$880$		0			0
$881$	−8.00000			−0.269527			−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000			−0.269527			−0.134763	+	0.990878i	$0.543027\pi$
$882$		0			0
$883$		−	36.7423i		−	1.23648i	−0.785990	−	0.618239i	$-0.787846\pi$
$883$		−	36.7423i		−	1.23648i	0.785990	−	0.618239i	$-0.212154\pi$
$884$		0			0
$885$	24.0000	−	12.0000i	0.806751	−	0.403376i
$886$		0			0
$887$		−	2.44949i		−	0.0822458i	−0.999154	−	0.0411229i	$-0.986906\pi$
$887$		−	2.44949i		−	0.0822458i	0.999154	−	0.0411229i	$-0.0130935\pi$
$888$		0			0
$889$	−18.0000			−0.603701
$890$		0			0
$891$	−44.0908			−1.47710
$892$		0			0
$893$		−	60.0000i		−	2.00782i
$894$		0			0
$895$	14.6969	+	29.3939i	0.491264	+	0.982529i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−78.3837			−2.61424
$900$		0			0
$901$	32.0000			1.06607
$902$		0			0
$903$			44.0908i			1.46725i
$904$		0			0
$905$	8.00000	+	16.0000i	0.265929	+	0.531858i
$906$		0			0
$907$			12.2474i			0.406670i	0.979109	+	0.203335i	$0.0651780\pi$
$907$			12.2474i			0.406670i	−0.979109	+	0.203335i	$0.934822\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$	−29.3939			−0.973863			−0.486931	−	0.873440i	$-0.661884\pi$
$911$	−29.3939			−0.973863			−0.486931	+	0.873440i	$0.661884\pi$
$912$		0			0
$913$		−	12.0000i		−	0.397142i
$914$		0			0
$915$	−29.3939	+	14.6969i	−0.971732	+	0.485866i
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	48.9898			1.61602			0.808012	−	0.589166i	$-0.200544\pi$
$919$	48.9898			1.61602			0.808012	+	0.589166i	$0.200544\pi$
$920$		0			0
$921$	−42.0000			−1.38395
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−16.0000	−	12.0000i	−0.526077	−	0.394558i
$926$		0			0
$927$		−	7.34847i		−	0.241355i
$928$		0			0
$929$	−16.0000			−0.524943			−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000			−0.524943			−0.262471	+	0.964940i	$0.584538\pi$
$930$		0			0
$931$	4.89898			0.160558
$932$		0			0
$933$			48.0000i			1.57145i
$934$		0			0
$935$	−39.1918	+	19.5959i	−1.28171	+	0.640855i
$936$		0			0
$937$		−	60.0000i		−	1.96011i	−0.198715	−	0.980057i	$-0.563677\pi$
$937$		−	60.0000i		−	1.96011i	0.198715	−	0.980057i	$-0.436323\pi$
$938$		0			0
$939$	58.7878			1.91847
$940$		0			0
$941$	8.00000			0.260793			0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000			0.260793			0.130396	+	0.991462i	$0.458375\pi$
$942$		0			0
$943$		−	19.5959i		−	0.638131i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			17.1464i			0.557184i	0.960410	+	0.278592i	$0.0898677\pi$
$947$			17.1464i			0.557184i	−0.960410	+	0.278592i	$0.910132\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	19.5959			0.635441
$952$		0			0
$953$		−	32.0000i		−	1.03658i	−0.855204	−	0.518291i	$-0.826568\pi$
$953$		−	32.0000i		−	1.03658i	0.855204	−	0.518291i	$-0.173432\pi$
$954$		0			0
$955$	−19.5959	−	39.1918i	−0.634109	−	1.26822i
$956$		0			0
$957$		−	96.0000i		−	3.10324i
$958$		0			0
$959$	19.5959			0.632785
$960$		0			0
$961$	65.0000			2.09677
$962$		0			0
$963$			22.0454i			0.710403i
$964$		0			0
$965$	24.0000	−	12.0000i	0.772587	−	0.386294i
$966$		0			0
$967$		−	17.1464i		−	0.551392i	−0.961245	−	0.275696i	$-0.911092\pi$
$967$		−	17.1464i		−	0.551392i	0.961245	−	0.275696i	$-0.0889083\pi$
$968$		0			0
$969$	−48.0000			−1.54198
$970$		0			0
$971$	−24.4949			−0.786079			−0.393039	−	0.919522i	$-0.628576\pi$
$971$	−24.4949			−0.786079			−0.393039	+	0.919522i	$0.628576\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	4.00000i		−	0.127971i	−0.997951	−	0.0639857i	$-0.979619\pi$
$977$		−	4.00000i		−	0.127971i	0.997951	−	0.0639857i	$-0.0203812\pi$
$978$		0			0
$979$	−9.79796			−0.313144
$980$		0			0
$981$	−30.0000			−0.957826
$982$		0			0
$983$		−	22.0454i		−	0.703139i	−0.936162	−	0.351570i	$-0.885648\pi$
$983$		−	22.0454i		−	0.703139i	0.936162	−	0.351570i	$-0.114352\pi$
$984$		0			0
$985$	−16.0000	+	8.00000i	−0.509802	+	0.254901i
$986$		0			0
$987$		−	73.4847i		−	2.33904i
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	−39.1918			−1.24497			−0.622485	−	0.782632i	$-0.713877\pi$
$991$	−39.1918			−1.24497			−0.622485	+	0.782632i	$0.713877\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$		0			0
$995$	−9.79796	−	19.5959i	−0.310616	−	0.621232i
$996$		0			0
$997$		−	56.0000i		−	1.77354i	−0.462213	−	0.886769i	$-0.652944\pi$
$997$		−	56.0000i		−	1.77354i	0.462213	−	0.886769i	$-0.347056\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.c.m.769.4		4
4.3	odd	2	inner	1280.2.c.m.769.2		4
5.2	odd	4		6400.2.a.bv.1.2		2
5.3	odd	4		6400.2.a.bx.1.1		2
5.4	even	2	inner	1280.2.c.m.769.1		4
8.3	odd	2		1280.2.c.e.769.3		4
8.5	even	2		1280.2.c.e.769.1		4
16.3	odd	4		640.2.f.g.449.3	yes	4
16.5	even	4		640.2.f.c.449.4	yes	4
16.11	odd	4		640.2.f.c.449.2	yes	4
16.13	even	4		640.2.f.g.449.1	yes	4
20.3	even	4		6400.2.a.bx.1.2		2
20.7	even	4		6400.2.a.bv.1.1		2
20.19	odd	2	inner	1280.2.c.m.769.3		4
40.3	even	4		6400.2.a.bw.1.1		2
40.13	odd	4		6400.2.a.bw.1.2		2
40.19	odd	2		1280.2.c.e.769.2		4
40.27	even	4		6400.2.a.bu.1.2		2
40.29	even	2		1280.2.c.e.769.4		4
40.37	odd	4		6400.2.a.bu.1.1		2
80.3	even	4		3200.2.d.l.1601.3		4
80.13	odd	4		3200.2.d.l.1601.2		4
80.19	odd	4		640.2.f.c.449.1	✓	4
80.27	even	4		3200.2.d.k.1601.4		4
80.29	even	4		640.2.f.c.449.3	yes	4
80.37	odd	4		3200.2.d.k.1601.1		4
80.43	even	4		3200.2.d.l.1601.1		4
80.53	odd	4		3200.2.d.l.1601.4		4
80.59	odd	4		640.2.f.g.449.4	yes	4
80.67	even	4		3200.2.d.k.1601.2		4
80.69	even	4		640.2.f.g.449.2	yes	4
80.77	odd	4		3200.2.d.k.1601.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.f.c.449.1	✓	4	80.19	odd	4
640.2.f.c.449.2	yes	4	16.11	odd	4
640.2.f.c.449.3	yes	4	80.29	even	4
640.2.f.c.449.4	yes	4	16.5	even	4
640.2.f.g.449.1	yes	4	16.13	even	4
640.2.f.g.449.2	yes	4	80.69	even	4
640.2.f.g.449.3	yes	4	16.3	odd	4
640.2.f.g.449.4	yes	4	80.59	odd	4
1280.2.c.e.769.1		4	8.5	even	2
1280.2.c.e.769.2		4	40.19	odd	2
1280.2.c.e.769.3		4	8.3	odd	2
1280.2.c.e.769.4		4	40.29	even	2
1280.2.c.m.769.1		4	5.4	even	2	inner
1280.2.c.m.769.2		4	4.3	odd	2	inner
1280.2.c.m.769.3		4	20.19	odd	2	inner
1280.2.c.m.769.4		4	1.1	even	1	trivial
3200.2.d.k.1601.1		4	80.37	odd	4
3200.2.d.k.1601.2		4	80.67	even	4
3200.2.d.k.1601.3		4	80.77	odd	4
3200.2.d.k.1601.4		4	80.27	even	4
3200.2.d.l.1601.1		4	80.43	even	4
3200.2.d.l.1601.2		4	80.13	odd	4
3200.2.d.l.1601.3		4	80.3	even	4
3200.2.d.l.1601.4		4	80.53	odd	4
6400.2.a.bu.1.1		2	40.37	odd	4
6400.2.a.bu.1.2		2	40.27	even	4
6400.2.a.bv.1.1		2	20.7	even	4
6400.2.a.bv.1.2		2	5.2	odd	4
6400.2.a.bw.1.1		2	40.3	even	4
6400.2.a.bw.1.2		2	40.13	odd	4
6400.2.a.bx.1.1		2	5.3	odd	4
6400.2.a.bx.1.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} +(1.00000 + 2.00000i) q^{5} -2.44949i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{3} +(1.00000 + 2.00000i) q^{5} -2.44949i q^{7} -3.00000 q^{9} +4.89898 q^{11} +(-4.89898 + 2.44949i) q^{15} +4.00000i q^{17} +4.89898 q^{19} +6.00000 q^{21} +2.44949i q^{23} +(-3.00000 + 4.00000i) q^{25} -8.00000 q^{29} +9.79796 q^{31} +12.0000i q^{33} +(4.89898 - 2.44949i) q^{35} +4.00000i q^{37} -8.00000 q^{41} +7.34847i q^{43} +(-3.00000 - 6.00000i) q^{45} -12.2474i q^{47} +1.00000 q^{49} -9.79796 q^{51} -8.00000i q^{53} +(4.89898 + 9.79796i) q^{55} +12.0000i q^{57} -4.89898 q^{59} +6.00000 q^{61} +7.34847i q^{63} -2.44949i q^{67} -6.00000 q^{69} -9.79796 q^{71} -4.00000i q^{73} +(-9.79796 - 7.34847i) q^{75} -12.0000i q^{77} -9.79796 q^{79} -9.00000 q^{81} -2.44949i q^{83} +(-8.00000 + 4.00000i) q^{85} -19.5959i q^{87} -2.00000 q^{89} +24.0000i q^{93} +(4.89898 + 9.79796i) q^{95} -4.00000i q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 12 * q^9	\( 4 q + 4 q^{5} - 12 q^{9} + 24 q^{21} - 12 q^{25} - 32 q^{29} - 32 q^{41} - 12 q^{45} + 4 q^{49} + 24 q^{61} - 24 q^{69} - 36 q^{81} - 32 q^{85} - 8 q^{89}+O(q^{100}) \) 4 * q + 4 * q^5 - 12 * q^9 + 24 * q^21 - 12 * q^25 - 32 * q^29 - 32 * q^41 - 12 * q^45 + 4 * q^49 + 24 * q^61 - 24 * q^69 - 36 * q^81 - 32 * q^85 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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