LMFDB - Embedded newform 2456.1.h.a.613.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2456,1,Mod(613,2456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2456, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2456.613");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2456 = 2^{3} \cdot 307 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2456.h (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$1.22570367103$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{34})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 7x^{6} + 6x^{5} + 15x^{4} - 10x^{3} - 10x^{2} + 4x + 1 $ x^8 - x^7 - 7x^6 + 6x^5 + 15x^4 - 10x^3 - 10x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{17}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{17} - \cdots)$

Embedding invariants

Embedding label			613.2
Root			$-1.86494$ of defining polynomial
Character	$\chi$	$=$	2456.613

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.47802 q^{3} +1.00000 q^{4} +1.96595 q^{5} +1.47802 q^{6} -0.547326 q^{7} -1.00000 q^{8} +1.18454 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.47802 q^{3} +1.00000 q^{4} +1.96595 q^{5} +1.47802 q^{6} -0.547326 q^{7} -1.00000 q^{8} +1.18454 q^{9} -1.96595 q^{10} -1.47802 q^{12} -1.86494 q^{13} +0.547326 q^{14} -2.90570 q^{15} +1.00000 q^{16} +0.891477 q^{17} -1.18454 q^{18} +1.96595 q^{20} +0.808958 q^{21} +1.47802 q^{24} +2.86494 q^{25} +1.86494 q^{26} -0.272749 q^{27} -0.547326 q^{28} +1.20527 q^{29} +2.90570 q^{30} -1.00000 q^{32} -0.891477 q^{34} -1.07601 q^{35} +1.18454 q^{36} +2.75642 q^{39} -1.96595 q^{40} -1.20527 q^{41} -0.808958 q^{42} +1.70043 q^{43} +2.32874 q^{45} -1.47802 q^{48} -0.700434 q^{49} -2.86494 q^{50} -1.31762 q^{51} -1.86494 q^{52} +0.272749 q^{54} +0.547326 q^{56} -1.20527 q^{58} +1.70043 q^{59} -2.90570 q^{60} -0.184537 q^{61} -0.648328 q^{63} +1.00000 q^{64} -3.66638 q^{65} +0.547326 q^{67} +0.891477 q^{68} +1.07601 q^{70} +0.184537 q^{71} -1.18454 q^{72} -4.23444 q^{75} -2.75642 q^{78} +0.891477 q^{79} +1.96595 q^{80} -0.781409 q^{81} +1.20527 q^{82} +0.808958 q^{84} +1.75260 q^{85} -1.70043 q^{86} -1.78141 q^{87} -0.547326 q^{89} -2.32874 q^{90} +1.02073 q^{91} +1.47802 q^{96} -1.20527 q^{97} +0.700434 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + q^{5} - q^{6} - q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + q^5 - q^6 - q^7 - 8 * q^8 + 7 * q^9	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + q^{5} - q^{6} - q^{7} - 8 q^{8} + 7 q^{9} - q^{10} + q^{12} + q^{13} + q^{14} - 2 q^{15} + 8 q^{16} - q^{17} - 7 q^{18} + q^{20} + 2 q^{21} - q^{24} + 7 q^{25} - q^{26} + 2 q^{27} - q^{28} + q^{29} + 2 q^{30} - 8 q^{32} + q^{34} + 2 q^{35} + 7 q^{36} - 2 q^{39} - q^{40} - q^{41} - 2 q^{42} + q^{43} + 3 q^{45} + q^{48} + 7 q^{49} - 7 q^{50} + 2 q^{51} + q^{52} - 2 q^{54} + q^{56} - q^{58} + q^{59} - 2 q^{60} + q^{61} - 3 q^{63} + 8 q^{64} - 2 q^{65} + q^{67} - q^{68} - 2 q^{70} - q^{71} - 7 q^{72} + 3 q^{75} + 2 q^{78} - q^{79} + q^{80} + 6 q^{81} + q^{82} + 2 q^{84} + 2 q^{85} - q^{86} - 2 q^{87} - q^{89} - 3 q^{90} + 2 q^{91} - q^{96} - q^{97} - 7 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + q^5 - q^6 - q^7 - 8 * q^8 + 7 * q^9 - q^10 + q^12 + q^13 + q^14 - 2 * q^15 + 8 * q^16 - q^17 - 7 * q^18 + q^20 + 2 * q^21 - q^24 + 7 * q^25 - q^26 + 2 * q^27 - q^28 + q^29 + 2 * q^30 - 8 * q^32 + q^34 + 2 * q^35 + 7 * q^36 - 2 * q^39 - q^40 - q^41 - 2 * q^42 + q^43 + 3 * q^45 + q^48 + 7 * q^49 - 7 * q^50 + 2 * q^51 + q^52 - 2 * q^54 + q^56 - q^58 + q^59 - 2 * q^60 + q^61 - 3 * q^63 + 8 * q^64 - 2 * q^65 + q^67 - q^68 - 2 * q^70 - q^71 - 7 * q^72 + 3 * q^75 + 2 * q^78 - q^79 + q^80 + 6 * q^81 + q^82 + 2 * q^84 + 2 * q^85 - q^86 - 2 * q^87 - q^89 - 3 * q^90 + 2 * q^91 - q^96 - q^97 - 7 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2456\mathbb{Z}\right)^\times$.

$n$	$615$	$1229$	$1233$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.00000	−1.00000
$2$	−1.00000	−1.00000
$3$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$3$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$4$	1.00000	1.00000
$5$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$5$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$6$	1.47802	1.47802
$7$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$7$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$8$	−1.00000	−1.00000
$9$	1.18454	1.18454
$10$	−1.96595	−1.96595
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.47802	−1.47802
$13$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$13$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$14$	0.547326	0.547326
$15$	−2.90570	−2.90570
$16$	1.00000	1.00000
$17$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$17$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$18$	−1.18454	−1.18454
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.96595	1.96595
$21$	0.808958	0.808958
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.47802	1.47802
$25$	2.86494	2.86494
$26$	1.86494	1.86494
$27$	−0.272749	−0.272749
$28$	−0.547326	−0.547326
$29$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$29$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$30$	2.90570	2.90570
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	−0.891477	−0.891477
$35$	−1.07601	−1.07601
$36$	1.18454	1.18454
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	2.75642	2.75642
$40$	−1.96595	−1.96595
$41$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$41$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$42$	−0.808958	−0.808958
$43$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$43$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$44$	0	0
$45$	2.32874	2.32874
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.47802	−1.47802
$49$	−0.700434	−0.700434
$50$	−2.86494	−2.86494
$51$	−1.31762	−1.31762
$52$	−1.86494	−1.86494
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0.272749	0.272749
$55$	0	0
$56$	0.547326	0.547326
$57$	0	0
$58$	−1.20527	−1.20527
$59$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$59$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$60$	−2.90570	−2.90570
$61$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$61$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$62$	0	0
$63$	−0.648328	−0.648328
$64$	1.00000	1.00000
$65$	−3.66638	−3.66638
$66$	0	0
$67$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$67$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$68$	0.891477	0.891477
$69$	0	0
$70$	1.07601	1.07601
$71$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$71$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$72$	−1.18454	−1.18454
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−4.23444	−4.23444
$76$	0	0
$77$	0	0
$78$	−2.75642	−2.75642
$79$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$79$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$80$	1.96595	1.96595
$81$	−0.781409	−0.781409
$82$	1.20527	1.20527
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.808958	0.808958
$85$	1.75260	1.75260
$86$	−1.70043	−1.70043
$87$	−1.78141	−1.78141
$88$	0	0
$89$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$89$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$90$	−2.32874	−2.32874
$91$	1.02073	1.02073
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.47802	1.47802
$97$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$97$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$98$	0.700434	0.700434
$99$	0	0
$100$	2.86494	2.86494
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.31762	1.31762
$103$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$103$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$104$	1.86494	1.86494
$105$	1.59037	1.59037
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.272749	−0.272749
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−0.547326	−0.547326
$113$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$113$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$114$	0	0
$115$	0	0
$116$	1.20527	1.20527
$117$	−2.20910	−2.20910
$118$	−1.70043	−1.70043
$119$	−0.487928	−0.487928
$120$	2.90570	2.90570
$121$	1.00000	1.00000
$122$	0.184537	0.184537
$123$	1.78141	1.78141
$124$	0	0
$125$	3.66638	3.66638
$126$	0.648328	0.648328
$127$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$127$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$128$	−1.00000	−1.00000
$129$	−2.51327	−2.51327
$130$	3.66638	3.66638
$131$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$131$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$132$	0	0
$133$	0	0
$134$	−0.547326	−0.547326
$135$	−0.536209	−0.536209
$136$	−0.891477	−0.891477
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$139$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$140$	−1.07601	−1.07601
$141$	0	0
$142$	−0.184537	−0.184537
$143$	0	0
$144$	1.18454	1.18454
$145$	2.36949	2.36949
$146$	0	0
$147$	1.03525	1.03525
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	4.23444	4.23444
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	1.05599	1.05599
$154$	0	0
$155$	0	0
$156$	2.75642	2.75642
$157$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$157$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$158$	−0.891477	−0.891477
$159$	0	0
$160$	−1.96595	−1.96595
$161$	0	0
$162$	0.781409	0.781409
$163$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$163$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$164$	−1.20527	−1.20527
$165$	0	0
$166$	0	0
$167$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$167$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$168$	−0.808958	−0.808958
$169$	2.47802	2.47802
$170$	−1.75260	−1.75260
$171$	0	0
$172$	1.70043	1.70043
$173$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$173$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$174$	1.78141	1.78141
$175$	−1.56806	−1.56806
$176$	0	0
$177$	−2.51327	−2.51327
$178$	0.547326	0.547326
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	2.32874	2.32874
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.02073	−1.02073
$183$	0.272749	0.272749
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.149282	0.149282
$190$	0	0
$191$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$191$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$192$	−1.47802	−1.47802
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	1.20527	1.20527
$195$	5.41898	5.41898
$196$	−0.700434	−0.700434
$197$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$197$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$198$	0	0
$199$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$199$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$200$	−2.86494	−2.86494
$201$	−0.808958	−0.808958
$202$	0	0
$203$	−0.659675	−0.659675
$204$	−1.31762	−1.31762
$205$	−2.36949	−2.36949
$206$	−1.47802	−1.47802
$207$	0	0
$208$	−1.86494	−1.86494
$209$	0	0
$210$	−1.59037	−1.59037
$211$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$211$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$212$	0	0
$213$	−0.272749	−0.272749
$214$	0	0
$215$	3.34296	3.34296
$216$	0.272749	0.272749
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.66255	−1.66255
$222$	0	0
$223$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$223$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$224$	0.547326	0.547326
$225$	3.39363	3.39363
$226$	−1.47802	−1.47802
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−1.20527	−1.20527
$233$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$233$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$234$	2.20910	2.20910
$235$	0	0
$236$	1.70043	1.70043
$237$	−1.31762	−1.31762
$238$	0.487928	0.487928
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−2.90570	−2.90570
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.00000	−1.00000
$243$	1.42769	1.42769
$244$	−0.184537	−0.184537
$245$	−1.37702	−1.37702
$246$	−1.78141	−1.78141
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−3.66638	−3.66638
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.648328	−0.648328
$253$	0	0
$254$	−0.184537	−0.184537
$255$	−2.59037	−2.59037
$256$	1.00000	1.00000
$257$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$257$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$258$	2.51327	2.51327
$259$	0	0
$260$	−3.66638	−3.66638
$261$	1.42769	1.42769
$262$	1.86494	1.86494
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.808958	0.808958
$268$	0.547326	0.547326
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.536209	0.536209
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0.891477	0.891477
$273$	−1.50866	−1.50866
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.96595	−1.96595
$279$	0	0
$280$	1.07601	1.07601
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$283$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$284$	0.184537	0.184537
$285$	0	0
$286$	0	0
$287$	0.659675	0.659675
$288$	−1.18454	−1.18454
$289$	−0.205269	−0.205269
$290$	−2.36949	−2.36949
$291$	1.78141	1.78141
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.03525	−1.03525
$295$	3.34296	3.34296
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−4.23444	−4.23444
$301$	−0.930692	−0.930692
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.362789	−0.362789
$306$	−1.05599	−1.05599
$307$	−1.00000	−1.00000
$307$	−1.00000	−1.00000
$308$	0	0
$309$	−2.18454	−2.18454
$310$	0	0
$311$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$311$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$312$	−2.75642	−2.75642
$313$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$313$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$314$	−1.70043	−1.70043
$315$	−1.27458	−1.27458
$316$	0.891477	0.891477
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1.96595	1.96595
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.781409	−0.781409
$325$	−5.34296	−5.34296
$326$	0.891477	0.891477
$327$	0	0
$328$	1.20527	1.20527
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	1.70043	1.70043
$335$	1.07601	1.07601
$336$	0.808958	0.808958
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−2.47802	−2.47802
$339$	−2.18454	−2.18454
$340$	1.75260	1.75260
$341$	0	0
$342$	0	0
$343$	0.930692	0.930692
$344$	−1.70043	−1.70043
$345$	0	0
$346$	0.184537	0.184537
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.78141	−1.78141
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.56806	1.56806
$351$	0.508661	0.508661
$352$	0	0
$353$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$353$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$354$	2.51327	2.51327
$355$	0.362789	0.362789
$356$	−0.547326	−0.547326
$357$	0.721167	0.721167
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−2.32874	−2.32874
$361$	1.00000	1.00000
$362$	0	0
$363$	−1.47802	−1.47802
$364$	1.02073	1.02073
$365$	0	0
$366$	−0.272749	−0.272749
$367$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$367$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$368$	0	0
$369$	−1.42769	−1.42769
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−5.41898	−5.41898
$376$	0	0
$377$	−2.24776	−2.24776
$378$	−0.149282	−0.149282
$379$	−2.00000	−2.00000	−1.00000			$\pi$
$379$	−2.00000	−2.00000	−1.00000			$\pi$
$380$	0	0
$381$	−0.272749	−0.272749
$382$	1.96595	1.96595
$383$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$383$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$384$	1.47802	1.47802
$385$	0	0
$386$	0	0
$387$	2.01423	2.01423
$388$	−1.20527	−1.20527
$389$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$389$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$390$	−5.41898	−5.41898
$391$	0	0
$392$	0.700434	0.700434
$393$	2.75642	2.75642
$394$	0.891477	0.891477
$395$	1.75260	1.75260
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1.70043	1.70043
$399$	0	0
$400$	2.86494	2.86494
$401$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$401$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$402$	0.808958	0.808958
$403$	0	0
$404$	0	0
$405$	−1.53621	−1.53621
$406$	0.659675	0.659675
$407$	0	0
$408$	1.31762	1.31762
$409$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$409$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$410$	2.36949	2.36949
$411$	0	0
$412$	1.47802	1.47802
$413$	−0.930692	−0.930692
$414$	0	0
$415$	0	0
$416$	1.86494	1.86494
$417$	−2.90570	−2.90570
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	1.59037	1.59037
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.96595	−1.96595
$423$	0	0
$424$	0	0
$425$	2.55403	2.55403
$426$	0.272749	0.272749
$427$	0.101002	0.101002
$428$	0	0
$429$	0	0
$430$	−3.34296	−3.34296
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.272749	−0.272749
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−3.50216	−3.50216
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.829690	−0.829690
$442$	1.66255	1.66255
$443$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$443$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$444$	0	0
$445$	−1.07601	−1.07601
$446$	−1.86494	−1.86494
$447$	0	0
$448$	−0.547326	−0.547326
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−3.39363	−3.39363
$451$	0	0
$452$	1.47802	1.47802
$453$	0	0
$454$	0	0
$455$	2.00671	2.00671
$456$	0	0
$457$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$457$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$458$	0	0
$459$	−0.243149	−0.243149
$460$	0	0
$461$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$461$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$462$	0	0
$463$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$463$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$464$	1.20527	1.20527
$465$	0	0
$466$	−1.47802	−1.47802
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−2.20910	−2.20910
$469$	−0.299566	−0.299566
$470$	0	0
$471$	−2.51327	−2.51327
$472$	−1.70043	−1.70043
$473$	0	0
$474$	1.31762	1.31762
$475$	0	0
$476$	−0.487928	−0.487928
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	2.90570	2.90570
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.00000	1.00000
$485$	−2.36949	−2.36949
$486$	−1.42769	−1.42769
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0.184537	0.184537
$489$	1.31762	1.31762
$490$	1.37702	1.37702
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	1.78141	1.78141
$493$	1.07447	1.07447
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.101002	−0.101002
$498$	0	0
$499$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$499$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$500$	3.66638	3.66638
$501$	2.51327	2.51327
$502$	0	0
$503$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$503$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$504$	0.648328	0.648328
$505$	0	0
$506$	0	0
$507$	−3.66255	−3.66255
$508$	0.184537	0.184537
$509$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$509$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$510$	2.59037	2.59037
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	−1.86494	−1.86494
$515$	2.90570	2.90570
$516$	−2.51327	−2.51327
$517$	0	0
$518$	0	0
$519$	0.272749	0.272749
$520$	3.66638	3.66638
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.42769	−1.42769
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.86494	−1.86494
$525$	2.31762	2.31762
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	2.01423	2.01423
$532$	0	0
$533$	2.24776	2.24776
$534$	−0.808958	−0.808958
$535$	0	0
$536$	−0.547326	−0.547326
$537$	0	0
$538$	0	0
$539$	0	0
$540$	−0.536209	−0.536209
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−0.891477	−0.891477
$545$	0	0
$546$	1.50866	1.50866
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−0.218591	−0.218591
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−0.487928	−0.487928
$554$	0	0
$555$	0	0
$556$	1.96595	1.96595
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−3.17122	−3.17122
$560$	−1.07601	−1.07601
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	2.90570	2.90570
$566$	0.891477	0.891477
$567$	0.427686	0.427686
$568$	−0.184537	−0.184537
$569$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$569$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	2.90570	2.90570
$574$	−0.659675	−0.659675
$575$	0	0
$576$	1.18454	1.18454
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.205269	0.205269
$579$	0	0
$580$	2.36949	2.36949
$581$	0	0
$582$	−1.78141	−1.78141
$583$	0	0
$584$	0	0
$585$	−4.34296	−4.34296
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	1.03525	1.03525
$589$	0	0
$590$	−3.34296	−3.34296
$591$	1.31762	1.31762
$592$	0	0
$593$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$593$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$594$	0	0
$595$	−0.959241	−0.959241
$596$	0	0
$597$	2.51327	2.51327
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	4.23444	4.23444
$601$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$601$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$602$	0.930692	0.930692
$603$	0.648328	0.648328
$604$	0	0
$605$	1.96595	1.96595
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0.975012	0.975012
$610$	0.362789	0.362789
$611$	0	0
$612$	1.05599	1.05599
$613$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$613$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$614$	1.00000	1.00000
$615$	3.50216	3.50216
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	2.18454	2.18454
$619$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$619$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$620$	0	0
$621$	0	0
$622$	1.20527	1.20527
$623$	0.299566	0.299566
$624$	2.75642	2.75642
$625$	4.34296	4.34296
$626$	−0.184537	−0.184537
$627$	0	0
$628$	1.70043	1.70043
$629$	0	0
$630$	1.27458	1.27458
$631$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$631$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$632$	−0.891477	−0.891477
$633$	−2.90570	−2.90570
$634$	0	0
$635$	0.362789	0.362789
$636$	0	0
$637$	1.30627	1.30627
$638$	0	0
$639$	0.218591	0.218591
$640$	−1.96595	−1.96595
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$643$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$644$	0	0
$645$	−4.94096	−4.94096
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.781409	0.781409
$649$	0	0
$650$	5.34296	5.34296
$651$	0	0
$652$	−0.891477	−0.891477
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−3.66638	−3.66638
$656$	−1.20527	−1.20527
$657$	0	0
$658$	0	0
$659$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$659$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$660$	0	0
$661$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$661$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$662$	0	0
$663$	2.45729	2.45729
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.70043	−1.70043
$669$	−2.75642	−2.75642
$670$	−1.07601	−1.07601
$671$	0	0
$672$	−0.808958	−0.808958
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−0.781409	−0.781409
$676$	2.47802	2.47802
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	2.18454	2.18454
$679$	0.659675	0.659675
$680$	−1.75260	−1.75260
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	−0.930692	−0.930692
$687$	0	0
$688$	1.70043	1.70043
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.184537	−0.184537
$693$	0	0
$694$	0	0
$695$	3.86494	3.86494
$696$	1.78141	1.78141
$697$	−1.07447	−1.07447
$698$	0	0
$699$	−2.18454	−2.18454
$700$	−1.56806	−1.56806
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−0.508661	−0.508661
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1.96595	1.96595
$707$	0	0
$708$	−2.51327	−2.51327
$709$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$709$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$710$	−0.362789	−0.362789
$711$	1.05599	1.05599
$712$	0.547326	0.547326
$713$	0	0
$714$	−0.721167	−0.721167
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$719$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$720$	2.32874	2.32874
$721$	−0.808958	−0.808958
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0	0
$725$	3.45303	3.45303
$726$	1.47802	1.47802
$727$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$727$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$728$	−1.02073	−1.02073
$729$	−1.32874	−1.32874
$730$	0	0
$731$	1.51590	1.51590
$732$	0.272749	0.272749
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−0.891477	−0.891477
$735$	2.03525	2.03525
$736$	0	0
$737$	0	0
$738$	1.42769	1.42769
$739$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$739$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$743$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	5.41898	5.41898
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	2.24776	2.24776
$755$	0	0
$756$	0.149282	0.149282
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$758$	2.00000	2.00000
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0.272749	0.272749
$763$	0	0
$764$	−1.96595	−1.96595
$765$	2.07601	2.07601
$766$	0.547326	0.547326
$767$	−3.17122	−3.17122
$768$	−1.47802	−1.47802
$769$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$769$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$770$	0	0
$771$	−2.75642	−2.75642
$772$	0	0
$773$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$773$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$774$	−2.01423	−2.01423
$775$	0	0
$776$	1.20527	1.20527
$777$	0	0
$778$	1.86494	1.86494
$779$	0	0
$780$	5.41898	5.41898
$781$	0	0
$782$	0	0
$783$	−0.328735	−0.328735
$784$	−0.700434	−0.700434
$785$	3.34296	3.34296
$786$	−2.75642	−2.75642
$787$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$787$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$788$	−0.891477	−0.891477
$789$	0	0
$790$	−1.75260	−1.75260
$791$	−0.808958	−0.808958
$792$	0	0
$793$	0.344151	0.344151
$794$	0	0
$795$	0	0
$796$	−1.70043	−1.70043
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−2.86494	−2.86494
$801$	−0.648328	−0.648328
$802$	−1.86494	−1.86494
$803$	0	0
$804$	−0.808958	−0.808958
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.53621	1.53621
$811$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$811$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$812$	−0.659675	−0.659675
$813$	0	0
$814$	0	0
$815$	−1.75260	−1.75260
$816$	−1.31762	−1.31762
$817$	0	0
$818$	−0.184537	−0.184537
$819$	1.20910	1.20910
$820$	−2.36949	−2.36949
$821$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$821$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$822$	0	0
$823$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$823$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$824$	−1.47802	−1.47802
$825$	0	0
$826$	0.930692	0.930692
$827$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$827$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−1.86494	−1.86494
$833$	−0.624421	−0.624421
$834$	2.90570	2.90570
$835$	−3.34296	−3.34296
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.00000	2.00000	1.00000			$0$
$839$	2.00000	2.00000	1.00000			$0$
$840$	−1.59037	−1.59037
$841$	0.452674	0.452674
$842$	0	0
$843$	0	0
$844$	1.96595	1.96595
$845$	4.87165	4.87165
$846$	0	0
$847$	−0.547326	−0.547326
$848$	0	0
$849$	1.31762	1.31762
$850$	−2.55403	−2.55403
$851$	0	0
$852$	−0.272749	−0.272749
$853$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$853$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$854$	−0.101002	−0.101002
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$859$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$860$	3.34296	3.34296
$861$	−0.975012	−0.975012
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.272749	0.272749
$865$	−0.362789	−0.362789
$866$	0	0
$867$	0.303392	0.303392
$868$	0	0
$869$	0	0
$870$	3.50216	3.50216
$871$	−1.02073	−1.02073
$872$	0	0
$873$	−1.42769	−1.42769
$874$	0	0
$875$	−2.00671	−2.00671
$876$	0	0
$877$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$877$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.829690	0.829690
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−1.66255	−1.66255
$885$	−4.94096	−4.94096
$886$	−0.547326	−0.547326
$887$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$887$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$888$	0	0
$889$	−0.101002	−0.101002
$890$	1.07601	1.07601
$891$	0	0
$892$	1.86494	1.86494
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.547326	0.547326
$897$	0	0
$898$	0	0
$899$	0	0
$900$	3.39363	3.39363
$901$	0	0
$902$	0	0
$903$	1.37558	1.37558
$904$	−1.47802	−1.47802
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	−2.00671	−2.00671
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	1.70043	1.70043
$915$	0.536209	0.536209
$916$	0	0
$917$	1.02073	1.02073
$918$	0.243149	0.243149
$919$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$919$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$920$	0	0
$921$	1.47802	1.47802
$922$	0.891477	0.891477
$923$	−0.344151	−0.344151
$924$	0	0
$925$	0	0
$926$	1.70043	1.70043
$927$	1.75077	1.75077
$928$	−1.20527	−1.20527
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	1.47802	1.47802
$933$	1.78141	1.78141
$934$	0	0
$935$	0	0
$936$	2.20910	2.20910
$937$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$937$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$938$	0.299566	0.299566
$939$	−0.272749	−0.272749
$940$	0	0
$941$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$941$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$942$	2.51327	2.51327
$943$	0	0
$944$	1.70043	1.70043
$945$	0.293481	0.293481
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.31762	−1.31762
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0.487928	0.487928
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−3.86494	−3.86494
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−2.90570	−2.90570
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$967$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	2.36949	2.36949
$971$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$971$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$972$	1.42769	1.42769
$973$	−1.07601	−1.07601
$974$	0	0
$975$	7.89699	7.89699
$976$	−0.184537	−0.184537
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.31762	−1.31762
$979$	0	0
$980$	−1.37702	−1.37702
$981$	0	0
$982$	0	0
$983$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$983$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$984$	−1.78141	−1.78141
$985$	−1.75260	−1.75260
$986$	−1.07447	−1.07447
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$991$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$992$	0	0
$993$	0	0
$994$	0.101002	0.101002
$995$	−3.34296	−3.34296
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.86494	1.86494
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2456.1.h.a.613.2	✓	8
8.5	even	2		2456.1.h.b.613.7	yes	8
307.306	odd	2		2456.1.h.b.613.7	yes	8
2456.613	odd	2	CM	2456.1.h.a.613.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2456.1.h.a.613.2	✓	8	1.1	even	1	trivial
2456.1.h.a.613.2	✓	8	2456.613	odd	2	CM
2456.1.h.b.613.7	yes	8	8.5	even	2
2456.1.h.b.613.7	yes	8	307.306	odd	2

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:	\( N \)	\(=\)	\( 2456 = 2^{3} \cdot 307 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2456.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.47802 q^{3} +1.00000 q^{4} +1.96595 q^{5} +1.47802 q^{6} -0.547326 q^{7} -1.00000 q^{8} +1.18454 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.47802 q^{3} +1.00000 q^{4} +1.96595 q^{5} +1.47802 q^{6} -0.547326 q^{7} -1.00000 q^{8} +1.18454 q^{9} -1.96595 q^{10} -1.47802 q^{12} -1.86494 q^{13} +0.547326 q^{14} -2.90570 q^{15} +1.00000 q^{16} +0.891477 q^{17} -1.18454 q^{18} +1.96595 q^{20} +0.808958 q^{21} +1.47802 q^{24} +2.86494 q^{25} +1.86494 q^{26} -0.272749 q^{27} -0.547326 q^{28} +1.20527 q^{29} +2.90570 q^{30} -1.00000 q^{32} -0.891477 q^{34} -1.07601 q^{35} +1.18454 q^{36} +2.75642 q^{39} -1.96595 q^{40} -1.20527 q^{41} -0.808958 q^{42} +1.70043 q^{43} +2.32874 q^{45} -1.47802 q^{48} -0.700434 q^{49} -2.86494 q^{50} -1.31762 q^{51} -1.86494 q^{52} +0.272749 q^{54} +0.547326 q^{56} -1.20527 q^{58} +1.70043 q^{59} -2.90570 q^{60} -0.184537 q^{61} -0.648328 q^{63} +1.00000 q^{64} -3.66638 q^{65} +0.547326 q^{67} +0.891477 q^{68} +1.07601 q^{70} +0.184537 q^{71} -1.18454 q^{72} -4.23444 q^{75} -2.75642 q^{78} +0.891477 q^{79} +1.96595 q^{80} -0.781409 q^{81} +1.20527 q^{82} +0.808958 q^{84} +1.75260 q^{85} -1.70043 q^{86} -1.78141 q^{87} -0.547326 q^{89} -2.32874 q^{90} +1.02073 q^{91} +1.47802 q^{96} -1.20527 q^{97} +0.700434 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + q^{5} - q^{6} - q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + q^5 - q^6 - q^7 - 8 * q^8 + 7 * q^9	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + q^{5} - q^{6} - q^{7} - 8 q^{8} + 7 q^{9} - q^{10} + q^{12} + q^{13} + q^{14} - 2 q^{15} + 8 q^{16} - q^{17} - 7 q^{18} + q^{20} + 2 q^{21} - q^{24} + 7 q^{25} - q^{26} + 2 q^{27} - q^{28} + q^{29} + 2 q^{30} - 8 q^{32} + q^{34} + 2 q^{35} + 7 q^{36} - 2 q^{39} - q^{40} - q^{41} - 2 q^{42} + q^{43} + 3 q^{45} + q^{48} + 7 q^{49} - 7 q^{50} + 2 q^{51} + q^{52} - 2 q^{54} + q^{56} - q^{58} + q^{59} - 2 q^{60} + q^{61} - 3 q^{63} + 8 q^{64} - 2 q^{65} + q^{67} - q^{68} - 2 q^{70} - q^{71} - 7 q^{72} + 3 q^{75} + 2 q^{78} - q^{79} + q^{80} + 6 q^{81} + q^{82} + 2 q^{84} + 2 q^{85} - q^{86} - 2 q^{87} - q^{89} - 3 q^{90} + 2 q^{91} - q^{96} - q^{97} - 7 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + q^5 - q^6 - q^7 - 8 * q^8 + 7 * q^9 - q^10 + q^12 + q^13 + q^14 - 2 * q^15 + 8 * q^16 - q^17 - 7 * q^18 + q^20 + 2 * q^21 - q^24 + 7 * q^25 - q^26 + 2 * q^27 - q^28 + q^29 + 2 * q^30 - 8 * q^32 + q^34 + 2 * q^35 + 7 * q^36 - 2 * q^39 - q^40 - q^41 - 2 * q^42 + q^43 + 3 * q^45 + q^48 + 7 * q^49 - 7 * q^50 + 2 * q^51 + q^52 - 2 * q^54 + q^56 - q^58 + q^59 - 2 * q^60 + q^61 - 3 * q^63 + 8 * q^64 - 2 * q^65 + q^67 - q^68 - 2 * q^70 - q^71 - 7 * q^72 + 3 * q^75 + 2 * q^78 - q^79 + q^80 + 6 * q^81 + q^82 + 2 * q^84 + 2 * q^85 - q^86 - 2 * q^87 - q^89 - 3 * q^90 + 2 * q^91 - q^96 - q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more