LMFDB - Embedded newform 1040.2.f.a.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(129,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1040.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.f (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:	$8.30444181021$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.a.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +2.00000i q^{11} +(3.00000 + 2.00000i) q^{13} +(4.00000 - 2.00000i) q^{15} -6.00000i q^{19} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +4.00000i q^{27} +6.00000 q^{29} +6.00000i q^{31} -4.00000 q^{33} +6.00000 q^{37} +(-4.00000 + 6.00000i) q^{39} +8.00000i q^{41} -6.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +8.00000 q^{47} -7.00000 q^{49} +12.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +12.0000 q^{57} +2.00000i q^{59} +6.00000 q^{61} +(1.00000 - 8.00000i) q^{65} -12.0000 q^{67} -12.0000 q^{69} -2.00000i q^{71} -6.00000 q^{73} +(-8.00000 - 6.00000i) q^{75} -11.0000 q^{81} +4.00000 q^{83} +12.0000i q^{87} +8.00000i q^{89} -12.0000 q^{93} +(-12.0000 + 6.00000i) q^{95} -6.00000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{9} + 6 q^{13} + 8 q^{15} - 6 q^{25} + 12 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 2 q^{45} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 24 q^{57} + 12 q^{61} + 2 q^{65} - 24 q^{67} - 24 q^{69} - 12 q^{73} - 16 q^{75} - 22 q^{81} + 8 q^{83} - 24 q^{93} - 24 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 6 * q^13 + 8 * q^15 - 6 * q^25 + 12 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 2 * q^45 + 16 * q^47 - 14 * q^49 + 8 * q^55 + 24 * q^57 + 12 * q^61 + 2 * q^65 - 24 * q^67 - 24 * q^69 - 12 * q^73 - 16 * q^75 - 22 * q^81 + 8 * q^83 - 24 * q^93 - 24 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$	4.00000	−	2.00000i	1.03280	−	0.516398i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			6.00000i			1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$			6.00000i			1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$		0			0
$33$	−4.00000			−0.696311
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	−4.00000	+	6.00000i	−0.640513	+	0.960769i
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$	4.00000	−	2.00000i	0.539360	−	0.269680i
$56$		0			0
$57$	12.0000			1.58944
$58$		0			0
$59$			2.00000i			0.260378i	0.991489	+	0.130189i	$0.0415584\pi$
$59$			2.00000i			0.260378i	−0.991489	+	0.130189i	$0.958442\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$	1.00000	−	8.00000i	0.124035	−	0.992278i
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$		−	2.00000i		−	0.237356i	−0.992933	−	0.118678i	$-0.962134\pi$
$71$		−	2.00000i		−	0.237356i	0.992933	−	0.118678i	$-0.0378657\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	−8.00000	−	6.00000i	−0.923760	−	0.692820i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			12.0000i			1.28654i
$88$		0			0
$89$			8.00000i			0.847998i	0.905663	+	0.423999i	$0.139374\pi$
$89$			8.00000i			0.847998i	−0.905663	+	0.423999i	$0.860626\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−12.0000			−1.24434
$94$		0			0
$95$	−12.0000	+	6.00000i	−1.23117	+	0.615587i
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$		−	2.00000i		−	0.201008i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$		0			0
$111$			12.0000i			1.13899i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	12.0000	−	6.00000i	1.11901	−	0.559503i
$116$		0			0
$117$	−3.00000	−	2.00000i	−0.277350	−	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	−16.0000			−1.44267
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	12.0000			1.05654
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	8.00000	−	4.00000i	0.688530	−	0.344265i
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			16.0000i			1.34744i
$142$		0			0
$143$	−4.00000	+	6.00000i	−0.334497	+	0.501745i
$144$		0			0
$145$	−6.00000	−	12.0000i	−0.498273	−	0.996546i
$146$		0			0
$147$		−	14.0000i		−	1.15470i
$148$		0			0
$149$		−	20.0000i		−	1.63846i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$		−	20.0000i		−	1.63846i	0.573462	−	0.819232i	$-0.305600\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.0000	−	6.00000i	0.963863	−	0.481932i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		0			0
$159$	−24.0000			−1.90332
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$		0			0
$165$	4.00000	+	8.00000i	0.311400	+	0.622799i
$166$		0			0
$167$	16.0000			1.23812			0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000			1.23812			0.619059	+	0.785345i	$0.287514\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$			6.00000i			0.458831i
$172$		0			0
$173$		−	12.0000i		−	0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$		−	12.0000i		−	0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.00000			−0.300658
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$			12.0000i			0.887066i
$184$		0			0
$185$	−6.00000	−	12.0000i	−0.441129	−	0.882258i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$		0			0
$195$	16.0000	+	2.00000i	1.14578	+	0.143223i
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$		−	24.0000i		−	1.69283i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	16.0000	−	8.00000i	1.11749	−	0.558744i
$206$		0			0
$207$		−	6.00000i		−	0.417029i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$	4.00000			0.274075
$214$		0			0
$215$	−12.0000	+	6.00000i	−0.818393	+	0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$	4.00000			0.265489			0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000			0.265489			0.132745	+	0.991150i	$0.457621\pi$
$228$		0			0
$229$			12.0000i			0.792982i	0.918039	+	0.396491i	$0.129772\pi$
$229$			12.0000i			0.792982i	−0.918039	+	0.396491i	$0.870228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.0000i			1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$			24.0000i			1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$		0			0
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.0000i		−	0.646846i	−0.946254	−	0.323423i	$-0.895166\pi$
$239$		−	10.0000i		−	0.646846i	0.946254	−	0.323423i	$-0.104834\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$	7.00000	+	14.0000i	0.447214	+	0.894427i
$246$		0			0
$247$	12.0000	−	18.0000i	0.763542	−	1.14531i
$248$		0			0
$249$			8.00000i			0.506979i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$	24.0000	−	12.0000i	1.47431	−	0.737154i
$266$		0			0
$267$	−16.0000			−0.979184
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$			6.00000i			0.364474i	0.983255	+	0.182237i	$0.0583338\pi$
$271$			6.00000i			0.364474i	−0.983255	+	0.182237i	$0.941666\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000	−	6.00000i	−0.482418	−	0.361814i
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		0			0
$279$		−	6.00000i		−	0.359211i
$280$		0			0
$281$		−	8.00000i		−	0.477240i	−0.971113	−	0.238620i	$-0.923305\pi$
$281$		−	8.00000i		−	0.477240i	0.971113	−	0.238620i	$-0.0766950\pi$
$282$		0			0
$283$		−	22.0000i		−	1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$		−	22.0000i		−	1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$		0			0
$285$	−12.0000	−	24.0000i	−0.710819	−	1.42164i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$	4.00000	−	2.00000i	0.232889	−	0.116445i
$296$		0			0
$297$	−8.00000			−0.464207
$298$		0			0
$299$	−12.0000	+	18.0000i	−0.693978	+	1.04097i
$300$		0			0
$301$		0			0
$302$		0			0
$303$			12.0000i			0.689382i
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	−12.0000			−0.682656
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.00000			−0.112331			−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000			−0.112331			−0.0561656	+	0.998421i	$0.517887\pi$
$318$		0			0
$319$			12.0000i			0.671871i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−17.0000	+	6.00000i	−0.942990	+	0.332820i
$326$		0			0
$327$	24.0000			1.32720
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	30.0000i		−	1.64895i	−0.565899	−	0.824475i	$-0.691471\pi$
$331$		−	30.0000i		−	1.64895i	0.565899	−	0.824475i	$-0.308529\pi$
$332$		0			0
$333$	−6.00000			−0.328798
$334$		0			0
$335$	12.0000	+	24.0000i	0.655630	+	1.31126i
$336$		0			0
$337$		−	32.0000i		−	1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$		−	32.0000i		−	1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$		0			0
$344$		0			0
$345$	12.0000	+	24.0000i	0.646058	+	1.29212i
$346$		0			0
$347$		−	6.00000i		−	0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$		−	6.00000i		−	0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$		0			0
$349$		−	12.0000i		−	0.642345i	−0.947021	−	0.321173i	$-0.895923\pi$
$349$		−	12.0000i		−	0.642345i	0.947021	−	0.321173i	$-0.104077\pi$
$350$		0			0
$351$	−8.00000	+	12.0000i	−0.427008	+	0.640513i
$352$		0			0
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	−4.00000	+	2.00000i	−0.212298	+	0.106149i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2.00000i		−	0.105556i	−0.998606	−	0.0527780i	$-0.983192\pi$
$359$		−	2.00000i		−	0.105556i	0.998606	−	0.0527780i	$-0.0168076\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$			14.0000i			0.734809i
$364$		0			0
$365$	6.00000	+	12.0000i	0.314054	+	0.628109i
$366$		0			0
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$		0			0
$369$		−	8.00000i		−	0.416463i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$	−4.00000	+	22.0000i	−0.206559	+	1.13608i
$376$		0			0
$377$	18.0000	+	12.0000i	0.927047	+	0.618031i
$378$		0			0
$379$			18.0000i			0.924598i	0.886724	+	0.462299i	$0.152975\pi$
$379$			18.0000i			0.924598i	−0.886724	+	0.462299i	$0.847025\pi$
$380$		0			0
$381$	4.00000			0.204926
$382$		0			0
$383$	−8.00000			−0.408781			−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000			−0.408781			−0.204390	+	0.978889i	$0.565521\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			6.00000i			0.304997i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			24.0000i			1.21064i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	16.0000i		−	0.799002i	−0.916733	−	0.399501i	$-0.869183\pi$
$401$		−	16.0000i		−	0.799002i	0.916733	−	0.399501i	$-0.130817\pi$
$402$		0			0
$403$	−12.0000	+	18.0000i	−0.597763	+	0.896644i
$404$		0			0
$405$	11.0000	+	22.0000i	0.546594	+	1.09319i
$406$		0			0
$407$			12.0000i			0.594818i
$408$		0			0
$409$			24.0000i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$409$			24.0000i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$410$		0			0
$411$			4.00000i			0.197305i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000	−	8.00000i	−0.196352	−	0.392705i
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$		−	36.0000i		−	1.75453i	−0.480004	−	0.877266i	$-0.659365\pi$
$421$		−	36.0000i		−	1.75453i	0.480004	−	0.877266i	$-0.340635\pi$
$422$		0			0
$423$	−8.00000			−0.388973
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−12.0000	−	8.00000i	−0.579365	−	0.386244i
$430$		0			0
$431$		−	10.0000i		−	0.481683i	−0.970564	−	0.240842i	$-0.922577\pi$
$431$		−	10.0000i		−	0.481683i	0.970564	−	0.240842i	$-0.0774234\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$		0			0
$435$	24.0000	−	12.0000i	1.15071	−	0.575356i
$436$		0			0
$437$	36.0000			1.72211
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$		0			0
$445$	16.0000	−	8.00000i	0.758473	−	0.379236i
$446$		0			0
$447$	40.0000			1.89194
$448$		0			0
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	36.0000			1.69143
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−30.0000			−1.40334			−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000			−1.40334			−0.701670	+	0.712502i	$0.747562\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.00000i			0.186299i	0.995652	+	0.0931493i	$0.0296934\pi$
$461$			4.00000i			0.186299i	−0.995652	+	0.0931493i	$0.970307\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$	12.0000	+	24.0000i	0.556487	+	1.11297i
$466$		0			0
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	24.0000			1.10586
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	24.0000	+	18.0000i	1.10120	+	0.825897i
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$			22.0000i			1.00521i	0.864517	+	0.502603i	$0.167624\pi$
$479$			22.0000i			1.00521i	−0.864517	+	0.502603i	$0.832376\pi$
$480$		0			0
$481$	18.0000	+	12.0000i	0.820729	+	0.547153i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000	+	12.0000i	0.272446	+	0.544892i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		−	24.0000i		−	1.08532i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−4.00000	+	2.00000i	−0.179787	+	0.0898933i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$			32.0000i			1.42965i
$502$		0			0
$503$			6.00000i			0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$			6.00000i			0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$		0			0
$505$	−6.00000	−	12.0000i	−0.266996	−	0.533993i
$506$		0			0
$507$	−24.0000	+	10.0000i	−1.06588	+	0.444116i
$508$		0			0
$509$			20.0000i			0.886484i	0.896402	+	0.443242i	$0.146172\pi$
$509$			20.0000i			0.886484i	−0.896402	+	0.443242i	$0.853828\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	24.0000			1.05963
$514$		0			0
$515$	12.0000	−	6.00000i	0.528783	−	0.264392i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$	24.0000			1.05348
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$			42.0000i			1.83653i	0.395964	+	0.918266i	$0.370410\pi$
$523$			42.0000i			1.83653i	−0.395964	+	0.918266i	$0.629590\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		−	2.00000i		−	0.0867926i
$532$		0			0
$533$	−16.0000	+	24.0000i	−0.693037	+	1.03956i
$534$		0			0
$535$	−12.0000	+	6.00000i	−0.518805	+	0.259403i
$536$		0			0
$537$			24.0000i			1.03568i
$538$		0			0
$539$		−	14.0000i		−	0.603023i
$540$		0			0
$541$		−	12.0000i		−	0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$		−	12.0000i		−	0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$		0			0
$543$		−	4.00000i		−	0.171656i
$544$		0			0
$545$	−24.0000	+	12.0000i	−1.02805	+	0.514024i
$546$		0			0
$547$			18.0000i			0.769624i	0.922995	+	0.384812i	$0.125734\pi$
$547$			18.0000i			0.769624i	−0.922995	+	0.384812i	$0.874266\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$		−	36.0000i		−	1.53365i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	24.0000	−	12.0000i	1.01874	−	0.509372i
$556$		0			0
$557$	14.0000			0.593199			0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000			0.593199			0.296600	+	0.955002i	$0.404147\pi$
$558$		0			0
$559$	12.0000	−	18.0000i	0.507546	−	0.761319i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	30.0000i		−	1.26435i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$		−	30.0000i		−	1.26435i	0.774826	−	0.632175i	$-0.217837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		−	12.0000i		−	0.498703i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−24.0000			−0.993978
$584$		0			0
$585$	−1.00000	+	8.00000i	−0.0413449	+	0.330759i
$586$		0			0
$587$	−20.0000			−0.825488			−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000			−0.825488			−0.412744	+	0.910847i	$0.635430\pi$
$588$		0			0
$589$	36.0000			1.48335
$590$		0			0
$591$		−	4.00000i		−	0.164538i
$592$		0			0
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			48.0000i			1.96451i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	−7.00000	−	14.0000i	−0.284590	−	0.569181i
$606$		0			0
$607$		−	18.0000i		−	0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$		−	18.0000i		−	0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000	+	16.0000i	0.970936	+	0.647291i
$612$		0			0
$613$	30.0000			1.21169			0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000			1.21169			0.605844	+	0.795583i	$0.292835\pi$
$614$		0			0
$615$	16.0000	+	32.0000i	0.645182	+	1.29036i
$616$		0			0
$617$	34.0000			1.36879			0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000			1.36879			0.684394	+	0.729112i	$0.260067\pi$
$618$		0			0
$619$			18.0000i			0.723481i	0.932279	+	0.361741i	$0.117817\pi$
$619$			18.0000i			0.723481i	−0.932279	+	0.361741i	$0.882183\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			24.0000i			0.958468i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$		0			0
$633$			24.0000i			0.953914i
$634$		0			0
$635$	−4.00000	+	2.00000i	−0.158735	+	0.0793676i
$636$		0			0
$637$	−21.0000	−	14.0000i	−0.832050	−	0.554700i
$638$		0			0
$639$			2.00000i			0.0791188i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	36.0000			1.41970			0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000			1.41970			0.709851	+	0.704352i	$0.248762\pi$
$644$		0			0
$645$	−12.0000	−	24.0000i	−0.472500	−	0.944999i
$646$		0			0
$647$			6.00000i			0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$			6.00000i			0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$		0			0
$652$		0			0
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$		0			0
$657$	6.00000			0.234082
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$			12.0000i			0.466746i	0.972387	+	0.233373i	$0.0749763\pi$
$661$			12.0000i			0.466746i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			36.0000i			1.39393i
$668$		0			0
$669$		−	48.0000i		−	1.85579i
$670$		0			0
$671$			12.0000i			0.463255i
$672$		0			0
$673$		−	48.0000i		−	1.85026i	−0.379646	−	0.925132i	$-0.623954\pi$
$673$		−	48.0000i		−	1.85026i	0.379646	−	0.925132i	$-0.376046\pi$
$674$		0			0
$675$	−16.0000	−	12.0000i	−0.615840	−	0.461880i
$676$		0			0
$677$		−	36.0000i		−	1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$		−	36.0000i		−	1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$		0			0
$685$	−2.00000	−	4.00000i	−0.0764161	−	0.152832i
$686$		0			0
$687$	−24.0000			−0.915657
$688$		0			0
$689$	−24.0000	+	36.0000i	−0.914327	+	1.37149i
$690$		0			0
$691$			42.0000i			1.59776i	0.601494	+	0.798878i	$0.294573\pi$
$691$			42.0000i			1.59776i	−0.601494	+	0.798878i	$0.705427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.00000	−	8.00000i	−0.151729	−	0.303457i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−48.0000			−1.81553
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		−	36.0000i		−	1.35777i
$704$		0			0
$705$	32.0000	−	16.0000i	1.20519	−	0.602595i
$706$		0			0
$707$		0			0
$708$		0			0
$709$			12.0000i			0.450669i	0.974281	+	0.225335i	$0.0723476\pi$
$709$			12.0000i			0.450669i	−0.974281	+	0.225335i	$0.927652\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−36.0000			−1.34821
$714$		0			0
$715$	16.0000	+	2.00000i	0.598366	+	0.0747958i
$716$		0			0
$717$	20.0000			0.746914
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	+	24.0000i	−0.668503	+	0.891338i
$726$		0			0
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−42.0000			−1.55131			−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000			−1.55131			−0.775653	+	0.631160i	$0.782579\pi$
$734$		0			0
$735$	−28.0000	+	14.0000i	−1.03280	+	0.516398i
$736$		0			0
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$		−	6.00000i		−	0.220714i	−0.993892	−	0.110357i	$-0.964801\pi$
$739$		−	6.00000i		−	0.220714i	0.993892	−	0.110357i	$-0.0351994\pi$
$740$		0			0
$741$	36.0000	+	24.0000i	1.32249	+	0.881662i
$742$		0			0
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$		0			0
$745$	−40.0000	+	20.0000i	−1.46549	+	0.732743i
$746$		0			0
$747$	−4.00000			−0.146352
$748$		0			0
$749$		0			0
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$		−	24.0000i		−	0.874609i
$754$		0			0
$755$	−36.0000	+	18.0000i	−1.31017	+	0.655087i
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		0			0
$759$		−	24.0000i		−	0.871145i
$760$		0			0
$761$			40.0000i			1.45000i	0.688749	+	0.724999i	$0.258160\pi$
$761$			40.0000i			1.45000i	−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	6.00000i	−0.144432	+	0.216647i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.0000			1.36677			0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000			1.36677			0.683383	+	0.730061i	$0.260508\pi$
$774$		0			0
$775$	−24.0000	−	18.0000i	−0.862105	−	0.646579i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	48.0000			1.71978
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$			24.0000i			0.857690i
$784$		0			0
$785$	−24.0000	+	12.0000i	−0.856597	+	0.428298i
$786$		0			0
$787$	−12.0000			−0.427754			−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000			−0.427754			−0.213877	+	0.976861i	$0.568609\pi$
$788$		0			0
$789$	−12.0000			−0.427211
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18.0000	+	12.0000i	0.639199	+	0.426132i
$794$		0			0
$795$	24.0000	+	48.0000i	0.851192	+	1.70238i
$796$		0			0
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	8.00000i		−	0.282666i
$802$		0			0
$803$		−	12.0000i		−	0.423471i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	36.0000i		−	1.26726i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	30.0000i		−	1.05344i	−0.850038	−	0.526721i	$-0.823421\pi$
$811$		−	30.0000i		−	1.05344i	0.850038	−	0.526721i	$-0.176579\pi$
$812$		0			0
$813$	−12.0000			−0.420858
$814$		0			0
$815$	12.0000	+	24.0000i	0.420342	+	0.840683i
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.0000i		−	0.698005i	−0.937122	−	0.349002i	$-0.886521\pi$
$821$		−	20.0000i		−	0.698005i	0.937122	−	0.349002i	$-0.113479\pi$
$822$		0			0
$823$		−	42.0000i		−	1.46403i	−0.681290	−	0.732014i	$-0.738581\pi$
$823$		−	42.0000i		−	1.46403i	0.681290	−	0.732014i	$-0.261419\pi$
$824$		0			0
$825$	12.0000	−	16.0000i	0.417786	−	0.557048i
$826$		0			0
$827$	−4.00000			−0.139094			−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000			−0.139094			−0.0695468	+	0.997579i	$0.522155\pi$
$828$		0			0
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$		0			0
$831$	−24.0000			−0.832551
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−16.0000	−	32.0000i	−0.553703	−	1.10741i
$836$		0			0
$837$	−24.0000			−0.829561
$838$		0			0
$839$			46.0000i			1.58810i	0.607855	+	0.794048i	$0.292030\pi$
$839$			46.0000i			1.58810i	−0.607855	+	0.794048i	$0.707970\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	16.0000			0.551069
$844$		0			0
$845$	19.0000	−	22.0000i	0.653620	−	0.756823i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	44.0000			1.51008
$850$		0			0
$851$			36.0000i			1.23406i
$852$		0			0
$853$	54.0000			1.84892			0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000			1.84892			0.924462	+	0.381273i	$0.124514\pi$
$854$		0			0
$855$	12.0000	−	6.00000i	0.410391	−	0.205196i
$856$		0			0
$857$		−	24.0000i		−	0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$		−	24.0000i		−	0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$		0			0
$859$	36.0000			1.22830			0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000			1.22830			0.614152	+	0.789188i	$0.289498\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.00000			0.272323			0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000			0.272323			0.136162	+	0.990687i	$0.456523\pi$
$864$		0			0
$865$	−24.0000	+	12.0000i	−0.816024	+	0.408012i
$866$		0			0
$867$			34.0000i			1.15470i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−36.0000	−	24.0000i	−1.21981	−	0.813209i
$872$		0			0
$873$	6.00000			0.203069
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.00000			0.202606			0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000			0.202606			0.101303	+	0.994856i	$0.467699\pi$
$878$		0			0
$879$		−	52.0000i		−	1.75392i
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$			2.00000i			0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$			2.00000i			0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$		0			0
$885$	4.00000	+	8.00000i	0.134459	+	0.268917i
$886$		0			0
$887$		−	42.0000i		−	1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$		−	42.0000i		−	1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	22.0000i		−	0.737028i
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$	−12.0000	−	24.0000i	−0.401116	−	0.802232i
$896$		0			0
$897$	−36.0000	−	24.0000i	−1.20201	−	0.801337i
$898$		0			0
$899$			36.0000i			1.20067i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.00000	+	4.00000i	0.0664822	+	0.132964i
$906$		0			0
$907$			10.0000i			0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$			10.0000i			0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$			8.00000i			0.264761i
$914$		0			0
$915$	24.0000	−	12.0000i	0.793416	−	0.396708i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$		−	24.0000i		−	0.790827i
$922$		0			0
$923$	4.00000	−	6.00000i	0.131662	−	0.197492i
$924$		0			0
$925$	−18.0000	+	24.0000i	−0.591836	+	0.789115i
$926$		0			0
$927$		−	6.00000i		−	0.197066i
$928$		0			0
$929$		−	16.0000i		−	0.524943i	−0.964940	−	0.262471i	$-0.915462\pi$
$929$		−	16.0000i		−	0.524943i	0.964940	−	0.262471i	$-0.0845376\pi$
$930$		0			0
$931$			42.0000i			1.37649i
$932$		0			0
$933$		−	48.0000i		−	1.57145i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			56.0000i			1.82944i	0.404088	+	0.914720i	$0.367589\pi$
$937$			56.0000i			1.82944i	−0.404088	+	0.914720i	$0.632411\pi$
$938$		0			0
$939$	16.0000			0.522140
$940$		0			0
$941$		−	28.0000i		−	0.912774i	−0.889781	−	0.456387i	$-0.849143\pi$
$941$		−	28.0000i		−	0.912774i	0.889781	−	0.456387i	$-0.150857\pi$
$942$		0			0
$943$	−48.0000			−1.56310
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.0000			−0.909878			−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000			−0.909878			−0.454939	+	0.890523i	$0.650339\pi$
$948$		0			0
$949$	−18.0000	−	12.0000i	−0.584305	−	0.389536i
$950$		0			0
$951$		−	4.00000i		−	0.129709i
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−24.0000			−0.775810
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$			6.00000i			0.193347i
$964$		0			0
$965$	6.00000	+	12.0000i	0.193147	+	0.386294i
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−12.0000	−	34.0000i	−0.384308	−	1.08887i
$976$		0			0
$977$	34.0000			1.08776			0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000			1.08776			0.543878	+	0.839164i	$0.316955\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$			12.0000i			0.383131i
$982$		0			0
$983$	−16.0000			−0.510321			−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000			−0.510321			−0.255160	+	0.966899i	$0.582128\pi$
$984$		0			0
$985$	2.00000	+	4.00000i	0.0637253	+	0.127451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.0000			1.14473
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	60.0000			1.90404
$994$		0			0
$995$	−24.0000	−	48.0000i	−0.760851	−	1.52170i
$996$		0			0
$997$			60.0000i			1.90022i	0.311916	+	0.950110i	$0.399029\pi$
$997$			60.0000i			1.90022i	−0.311916	+	0.950110i	$0.600971\pi$
$998$		0			0
$999$			24.0000i			0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.a.129.2		2
4.3	odd	2		65.2.d.a.64.1	✓	2
5.4	even	2		1040.2.f.b.129.1		2
12.11	even	2		585.2.h.c.64.2		2
13.12	even	2		1040.2.f.b.129.2		2
20.3	even	4		325.2.c.e.51.2		2
20.7	even	4		325.2.c.b.51.1		2
20.19	odd	2		65.2.d.b.64.2	yes	2
52.3	odd	6		845.2.l.b.654.2		4
52.7	even	12		845.2.n.b.484.1		4
52.11	even	12		845.2.n.b.529.2		4
52.15	even	12		845.2.n.a.529.1		4
52.19	even	12		845.2.n.a.484.2		4
52.23	odd	6		845.2.l.a.654.2		4
52.31	even	4		845.2.b.a.339.2		2
52.35	odd	6		845.2.l.b.699.1		4
52.43	odd	6		845.2.l.a.699.1		4
52.47	even	4		845.2.b.b.339.1		2
52.51	odd	2		65.2.d.b.64.1	yes	2
60.59	even	2		585.2.h.b.64.2		2
65.64	even	2	inner	1040.2.f.a.129.1		2
156.155	even	2		585.2.h.b.64.1		2
260.19	even	12		845.2.n.a.484.1		4
260.47	odd	4		4225.2.a.k.1.1		1
260.59	even	12		845.2.n.b.484.2		4
260.83	odd	4		4225.2.a.m.1.1		1
260.99	even	4		845.2.b.b.339.2		2
260.103	even	4		325.2.c.e.51.1		2
260.119	even	12		845.2.n.a.529.2		4
260.139	odd	6		845.2.l.a.699.2		4
260.159	odd	6		845.2.l.a.654.1		4
260.179	odd	6		845.2.l.b.654.1		4
260.187	odd	4		4225.2.a.e.1.1		1
260.199	odd	6		845.2.l.b.699.2		4
260.203	odd	4		4225.2.a.h.1.1		1
260.207	even	4		325.2.c.b.51.2		2
260.219	even	12		845.2.n.b.529.1		4
260.239	even	4		845.2.b.a.339.1		2
260.259	odd	2		65.2.d.a.64.2	yes	2
780.779	even	2		585.2.h.c.64.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.d.a.64.1	✓	2	4.3	odd	2
65.2.d.a.64.2	yes	2	260.259	odd	2
65.2.d.b.64.1	yes	2	52.51	odd	2
65.2.d.b.64.2	yes	2	20.19	odd	2
325.2.c.b.51.1		2	20.7	even	4
325.2.c.b.51.2		2	260.207	even	4
325.2.c.e.51.1		2	260.103	even	4
325.2.c.e.51.2		2	20.3	even	4
585.2.h.b.64.1		2	156.155	even	2
585.2.h.b.64.2		2	60.59	even	2
585.2.h.c.64.1		2	780.779	even	2
585.2.h.c.64.2		2	12.11	even	2
845.2.b.a.339.1		2	260.239	even	4
845.2.b.a.339.2		2	52.31	even	4
845.2.b.b.339.1		2	52.47	even	4
845.2.b.b.339.2		2	260.99	even	4
845.2.l.a.654.1		4	260.159	odd	6
845.2.l.a.654.2		4	52.23	odd	6
845.2.l.a.699.1		4	52.43	odd	6
845.2.l.a.699.2		4	260.139	odd	6
845.2.l.b.654.1		4	260.179	odd	6
845.2.l.b.654.2		4	52.3	odd	6
845.2.l.b.699.1		4	52.35	odd	6
845.2.l.b.699.2		4	260.199	odd	6
845.2.n.a.484.1		4	260.19	even	12
845.2.n.a.484.2		4	52.19	even	12
845.2.n.a.529.1		4	52.15	even	12
845.2.n.a.529.2		4	260.119	even	12
845.2.n.b.484.1		4	52.7	even	12
845.2.n.b.484.2		4	260.59	even	12
845.2.n.b.529.1		4	260.219	even	12
845.2.n.b.529.2		4	52.11	even	12
1040.2.f.a.129.1		2	65.64	even	2	inner
1040.2.f.a.129.2		2	1.1	even	1	trivial
1040.2.f.b.129.1		2	5.4	even	2
1040.2.f.b.129.2		2	13.12	even	2
4225.2.a.e.1.1		1	260.187	odd	4
4225.2.a.h.1.1		1	260.203	odd	4
4225.2.a.k.1.1		1	260.47	odd	4
4225.2.a.m.1.1		1	260.83	odd	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +2.00000i q^{11} +(3.00000 + 2.00000i) q^{13} +(4.00000 - 2.00000i) q^{15} -6.00000i q^{19} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +4.00000i q^{27} +6.00000 q^{29} +6.00000i q^{31} -4.00000 q^{33} +6.00000 q^{37} +(-4.00000 + 6.00000i) q^{39} +8.00000i q^{41} -6.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +8.00000 q^{47} -7.00000 q^{49} +12.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +12.0000 q^{57} +2.00000i q^{59} +6.00000 q^{61} +(1.00000 - 8.00000i) q^{65} -12.0000 q^{67} -12.0000 q^{69} -2.00000i q^{71} -6.00000 q^{73} +(-8.00000 - 6.00000i) q^{75} -11.0000 q^{81} +4.00000 q^{83} +12.0000i q^{87} +8.00000i q^{89} -12.0000 q^{93} +(-12.0000 + 6.00000i) q^{95} -6.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} + 6 q^{13} + 8 q^{15} - 6 q^{25} + 12 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 2 q^{45} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 24 q^{57} + 12 q^{61} + 2 q^{65} - 24 q^{67} - 24 q^{69} - 12 q^{73} - 16 q^{75} - 22 q^{81} + 8 q^{83} - 24 q^{93} - 24 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 6 * q^13 + 8 * q^15 - 6 * q^25 + 12 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 2 * q^45 + 16 * q^47 - 14 * q^49 + 8 * q^55 + 24 * q^57 + 12 * q^61 + 2 * q^65 - 24 * q^67 - 24 * q^69 - 12 * q^73 - 16 * q^75 - 22 * q^81 + 8 * q^83 - 24 * q^93 - 24 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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