LMFDB - Embedded newform 2456.1.h.a.613.8

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			613.8
Root			$-0.184537$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.96595 q^{3} +1.00000 q^{4} -1.47802 q^{5} -1.96595 q^{6} +0.891477 q^{7} -1.00000 q^{8} +2.86494 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.96595 q^{3} +1.00000 q^{4} -1.47802 q^{5} -1.96595 q^{6} +0.891477 q^{7} -1.00000 q^{8} +2.86494 q^{9} +1.47802 q^{10} +1.96595 q^{12} -0.184537 q^{13} -0.891477 q^{14} -2.90570 q^{15} +1.00000 q^{16} -0.547326 q^{17} -2.86494 q^{18} -1.47802 q^{20} +1.75260 q^{21} -1.96595 q^{24} +1.18454 q^{25} +0.184537 q^{26} +3.66638 q^{27} +0.891477 q^{28} +1.70043 q^{29} +2.90570 q^{30} -1.00000 q^{32} +0.547326 q^{34} -1.31762 q^{35} +2.86494 q^{36} -0.362789 q^{39} +1.47802 q^{40} -1.70043 q^{41} -1.75260 q^{42} +1.20527 q^{43} -4.23444 q^{45} +1.96595 q^{48} -0.205269 q^{49} -1.18454 q^{50} -1.07601 q^{51} -0.184537 q^{52} -3.66638 q^{54} -0.891477 q^{56} -1.70043 q^{58} +1.20527 q^{59} -2.90570 q^{60} -1.86494 q^{61} +2.55403 q^{63} +1.00000 q^{64} +0.272749 q^{65} -0.891477 q^{67} -0.547326 q^{68} +1.31762 q^{70} +1.86494 q^{71} -2.86494 q^{72} +2.32874 q^{75} +0.362789 q^{78} -0.547326 q^{79} -1.47802 q^{80} +4.34296 q^{81} +1.70043 q^{82} +1.75260 q^{84} +0.808958 q^{85} -1.20527 q^{86} +3.34296 q^{87} +0.891477 q^{89} +4.23444 q^{90} -0.164510 q^{91} -1.96595 q^{96} -1.70043 q^{97} +0.205269 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$3$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$4$	1.00000	1.00000
$5$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$5$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$6$	−1.96595	−1.96595
$7$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$7$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$8$	−1.00000	−1.00000
$9$	2.86494	2.86494
$10$	1.47802	1.47802
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.96595	1.96595
$13$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$13$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$14$	−0.891477	−0.891477
$15$	−2.90570	−2.90570
$16$	1.00000	1.00000
$17$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$17$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$18$	−2.86494	−2.86494
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.47802	−1.47802
$21$	1.75260	1.75260
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.96595	−1.96595
$25$	1.18454	1.18454
$26$	0.184537	0.184537
$27$	3.66638	3.66638
$28$	0.891477	0.891477
$29$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$29$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$30$	2.90570	2.90570
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	0.547326	0.547326
$35$	−1.31762	−1.31762
$36$	2.86494	2.86494
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−0.362789	−0.362789
$40$	1.47802	1.47802
$41$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$41$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$42$	−1.75260	−1.75260
$43$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$43$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$44$	0	0
$45$	−4.23444	−4.23444
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.96595	1.96595
$49$	−0.205269	−0.205269
$50$	−1.18454	−1.18454
$51$	−1.07601	−1.07601
$52$	−0.184537	−0.184537
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−3.66638	−3.66638
$55$	0	0
$56$	−0.891477	−0.891477
$57$	0	0
$58$	−1.70043	−1.70043
$59$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$59$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$60$	−2.90570	−2.90570
$61$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$61$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$62$	0	0
$63$	2.55403	2.55403
$64$	1.00000	1.00000
$65$	0.272749	0.272749
$66$	0	0
$67$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$67$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$68$	−0.547326	−0.547326
$69$	0	0
$70$	1.31762	1.31762
$71$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$71$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$72$	−2.86494	−2.86494
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	2.32874	2.32874
$76$	0	0
$77$	0	0
$78$	0.362789	0.362789
$79$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$79$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$80$	−1.47802	−1.47802
$81$	4.34296	4.34296
$82$	1.70043	1.70043
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	1.75260	1.75260
$85$	0.808958	0.808958
$86$	−1.20527	−1.20527
$87$	3.34296	3.34296
$88$	0	0
$89$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$89$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$90$	4.23444	4.23444
$91$	−0.164510	−0.164510
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.96595	−1.96595
$97$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$97$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$98$	0.205269	0.205269
$99$	0	0
$100$	1.18454	1.18454
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.07601	1.07601
$103$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$103$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$104$	0.184537	0.184537
$105$	−2.59037	−2.59037
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	3.66638	3.66638
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0.891477	0.891477
$113$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$113$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$114$	0	0
$115$	0	0
$116$	1.70043	1.70043
$117$	−0.528687	−0.528687
$118$	−1.20527	−1.20527
$119$	−0.487928	−0.487928
$120$	2.90570	2.90570
$121$	1.00000	1.00000
$122$	1.86494	1.86494
$123$	−3.34296	−3.34296
$124$	0	0
$125$	−0.272749	−0.272749
$126$	−2.55403	−2.55403
$127$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$127$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$128$	−1.00000	−1.00000
$129$	2.36949	2.36949
$130$	−0.272749	−0.272749
$131$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$131$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$132$	0	0
$133$	0	0
$134$	0.891477	0.891477
$135$	−5.41898	−5.41898
$136$	0.547326	0.547326
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$139$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$140$	−1.31762	−1.31762
$141$	0	0
$142$	−1.86494	−1.86494
$143$	0	0
$144$	2.86494	2.86494
$145$	−2.51327	−2.51327
$146$	0	0
$147$	−0.403548	−0.403548
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.32874	−2.32874
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−1.56806	−1.56806
$154$	0	0
$155$	0	0
$156$	−0.362789	−0.362789
$157$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$157$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$158$	0.547326	0.547326
$159$	0	0
$160$	1.47802	1.47802
$161$	0	0
$162$	−4.34296	−4.34296
$163$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$163$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$164$	−1.70043	−1.70043
$165$	0	0
$166$	0	0
$167$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$167$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$168$	−1.75260	−1.75260
$169$	−0.965946	−0.965946
$170$	−0.808958	−0.808958
$171$	0	0
$172$	1.20527	1.20527
$173$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$173$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$174$	−3.34296	−3.34296
$175$	1.05599	1.05599
$176$	0	0
$177$	2.36949	2.36949
$178$	−0.891477	−0.891477
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−4.23444	−4.23444
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0.164510	0.164510
$183$	−3.66638	−3.66638
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.26849	3.26849
$190$	0	0
$191$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$191$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$192$	1.96595	1.96595
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	1.70043	1.70043
$195$	0.536209	0.536209
$196$	−0.205269	−0.205269
$197$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$197$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$198$	0	0
$199$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$199$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$200$	−1.18454	−1.18454
$201$	−1.75260	−1.75260
$202$	0	0
$203$	1.51590	1.51590
$204$	−1.07601	−1.07601
$205$	2.51327	2.51327
$206$	1.96595	1.96595
$207$	0	0
$208$	−0.184537	−0.184537
$209$	0	0
$210$	2.59037	2.59037
$211$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$211$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$212$	0	0
$213$	3.66638	3.66638
$214$	0	0
$215$	−1.78141	−1.78141
$216$	−3.66638	−3.66638
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.101002	0.101002
$222$	0	0
$223$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$223$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$224$	−0.891477	−0.891477
$225$	3.39363	3.39363
$226$	1.96595	1.96595
$227$	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
$232$	−1.70043	−1.70043
$233$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$233$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$234$	0.528687	0.528687
$235$	0	0
$236$	1.20527	1.20527
$237$	−1.07601	−1.07601
$238$	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
$243$	4.87165	4.87165
$244$	−1.86494	−1.86494
$245$	0.303392	0.303392
$246$	3.34296	3.34296
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0.272749	0.272749
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	2.55403	2.55403
$253$	0	0
$254$	−1.86494	−1.86494
$255$	1.59037	1.59037
$256$	1.00000	1.00000
$257$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$257$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$258$	−2.36949	−2.36949
$259$	0	0
$260$	0.272749	0.272749
$261$	4.87165	4.87165
$262$	0.184537	0.184537
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.75260	1.75260
$268$	−0.891477	−0.891477
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	5.41898	5.41898
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−0.547326	−0.547326
$273$	−0.323418	−0.323418
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.47802	1.47802
$279$	0	0
$280$	1.31762	1.31762
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$283$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$284$	1.86494	1.86494
$285$	0	0
$286$	0	0
$287$	−1.51590	−1.51590
$288$	−2.86494	−2.86494
$289$	−0.700434	−0.700434
$290$	2.51327	2.51327
$291$	−3.34296	−3.34296
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.403548	0.403548
$295$	−1.78141	−1.78141
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	2.32874	2.32874
$301$	1.07447	1.07447
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.75642	2.75642
$306$	1.56806	1.56806
$307$	−1.00000	−1.00000
$307$	−1.00000	−1.00000
$308$	0	0
$309$	−3.86494	−3.86494
$310$	0	0
$311$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$311$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$312$	0.362789	0.362789
$313$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$313$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$314$	−1.20527	−1.20527
$315$	−3.77490	−3.77490
$316$	−0.547326	−0.547326
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−1.47802	−1.47802
$321$	0	0
$322$	0	0
$323$	0	0
$324$	4.34296	4.34296
$325$	−0.218591	−0.218591
$326$	−0.547326	−0.547326
$327$	0	0
$328$	1.70043	1.70043
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	1.20527	1.20527
$335$	1.31762	1.31762
$336$	1.75260	1.75260
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.965946	0.965946
$339$	−3.86494	−3.86494
$340$	0.808958	0.808958
$341$	0	0
$342$	0	0
$343$	−1.07447	−1.07447
$344$	−1.20527	−1.20527
$345$	0	0
$346$	1.86494	1.86494
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	3.34296	3.34296
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.05599	−1.05599
$351$	−0.676582	−0.676582
$352$	0	0
$353$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$353$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$354$	−2.36949	−2.36949
$355$	−2.75642	−2.75642
$356$	0.891477	0.891477
$357$	−0.959241	−0.959241
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	4.23444	4.23444
$361$	1.00000	1.00000
$362$	0	0
$363$	1.96595	1.96595
$364$	−0.164510	−0.164510
$365$	0	0
$366$	3.66638	3.66638
$367$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$367$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$368$	0	0
$369$	−4.87165	−4.87165
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.536209	−0.536209
$376$	0	0
$377$	−0.313793	−0.313793
$378$	−3.26849	−3.26849
$379$	−2.00000	−2.00000	−1.00000			$\pi$
$379$	−2.00000	−2.00000	−1.00000			$\pi$
$380$	0	0
$381$	3.66638	3.66638
$382$	−1.47802	−1.47802
$383$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$383$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$384$	−1.96595	−1.96595
$385$	0	0
$386$	0	0
$387$	3.45303	3.45303
$388$	−1.70043	−1.70043
$389$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$389$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$390$	−0.536209	−0.536209
$391$	0	0
$392$	0.205269	0.205269
$393$	−0.362789	−0.362789
$394$	−0.547326	−0.547326
$395$	0.808958	0.808958
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1.20527	1.20527
$399$	0	0
$400$	1.18454	1.18454
$401$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$401$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$402$	1.75260	1.75260
$403$	0	0
$404$	0	0
$405$	−6.41898	−6.41898
$406$	−1.51590	−1.51590
$407$	0	0
$408$	1.07601	1.07601
$409$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$409$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$410$	−2.51327	−2.51327
$411$	0	0
$412$	−1.96595	−1.96595
$413$	1.07447	1.07447
$414$	0	0
$415$	0	0
$416$	0.184537	0.184537
$417$	−2.90570	−2.90570
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	−2.59037	−2.59037
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.47802	1.47802
$423$	0	0
$424$	0	0
$425$	−0.648328	−0.648328
$426$	−3.66638	−3.66638
$427$	−1.66255	−1.66255
$428$	0	0
$429$	0	0
$430$	1.78141	1.78141
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	3.66638	3.66638
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−4.94096	−4.94096
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.588085	−0.588085
$442$	−0.101002	−0.101002
$443$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$443$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$444$	0	0
$445$	−1.31762	−1.31762
$446$	−0.184537	−0.184537
$447$	0	0
$448$	0.891477	0.891477
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−3.39363	−3.39363
$451$	0	0
$452$	−1.96595	−1.96595
$453$	0	0
$454$	0	0
$455$	0.243149	0.243149
$456$	0	0
$457$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$457$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$458$	0	0
$459$	−2.00671	−2.00671
$460$	0	0
$461$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$461$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$462$	0	0
$463$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$463$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$464$	1.70043	1.70043
$465$	0	0
$466$	1.96595	1.96595
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−0.528687	−0.528687
$469$	−0.794731	−0.794731
$470$	0	0
$471$	2.36949	2.36949
$472$	−1.20527	−1.20527
$473$	0	0
$474$	1.07601	1.07601
$475$	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
$485$	2.51327	2.51327
$486$	−4.87165	−4.87165
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.86494	1.86494
$489$	1.07601	1.07601
$490$	−0.303392	−0.303392
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−3.34296	−3.34296
$493$	−0.930692	−0.930692
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.66255	1.66255
$498$	0	0
$499$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$499$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$500$	−0.272749	−0.272749
$501$	−2.36949	−2.36949
$502$	0	0
$503$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$503$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$504$	−2.55403	−2.55403
$505$	0	0
$506$	0	0
$507$	−1.89900	−1.89900
$508$	1.86494	1.86494
$509$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$509$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$510$	−1.59037	−1.59037
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	−0.184537	−0.184537
$515$	2.90570	2.90570
$516$	2.36949	2.36949
$517$	0	0
$518$	0	0
$519$	−3.66638	−3.66638
$520$	−0.272749	−0.272749
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−4.87165	−4.87165
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.184537	−0.184537
$525$	2.07601	2.07601
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	3.45303	3.45303
$532$	0	0
$533$	0.313793	0.313793
$534$	−1.75260	−1.75260
$535$	0	0
$536$	0.891477	0.891477
$537$	0	0
$538$	0	0
$539$	0	0
$540$	−5.41898	−5.41898
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0.547326	0.547326
$545$	0	0
$546$	0.323418	0.323418
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−5.34296	−5.34296
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−0.487928	−0.487928
$554$	0	0
$555$	0	0
$556$	−1.47802	−1.47802
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−0.222416	−0.222416
$560$	−1.31762	−1.31762
$561$	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
$566$	−0.547326	−0.547326
$567$	3.87165	3.87165
$568$	−1.86494	−1.86494
$569$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$569$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	2.90570	2.90570
$574$	1.51590	1.51590
$575$	0	0
$576$	2.86494	2.86494
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.700434	0.700434
$579$	0	0
$580$	−2.51327	−2.51327
$581$	0	0
$582$	3.34296	3.34296
$583$	0	0
$584$	0	0
$585$	0.781409	0.781409
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.403548	−0.403548
$589$	0	0
$590$	1.78141	1.78141
$591$	1.07601	1.07601
$592$	0	0
$593$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$593$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$594$	0	0
$595$	0.721167	0.721167
$596$	0	0
$597$	−2.36949	−2.36949
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−2.32874	−2.32874
$601$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$601$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$602$	−1.07447	−1.07447
$603$	−2.55403	−2.55403
$604$	0	0
$605$	−1.47802	−1.47802
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	2.98017	2.98017
$610$	−2.75642	−2.75642
$611$	0	0
$612$	−1.56806	−1.56806
$613$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$613$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$614$	1.00000	1.00000
$615$	4.94096	4.94096
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	3.86494	3.86494
$619$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$619$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$620$	0	0
$621$	0	0
$622$	1.70043	1.70043
$623$	0.794731	0.794731
$624$	−0.362789	−0.362789
$625$	−0.781409	−0.781409
$626$	−1.86494	−1.86494
$627$	0	0
$628$	1.20527	1.20527
$629$	0	0
$630$	3.77490	3.77490
$631$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$631$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$632$	0.547326	0.547326
$633$	−2.90570	−2.90570
$634$	0	0
$635$	−2.75642	−2.75642
$636$	0	0
$637$	0.0378797	0.0378797
$638$	0	0
$639$	5.34296	5.34296
$640$	1.47802	1.47802
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$643$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$644$	0	0
$645$	−3.50216	−3.50216
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−4.34296	−4.34296
$649$	0	0
$650$	0.218591	0.218591
$651$	0	0
$652$	0.547326	0.547326
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0.272749	0.272749
$656$	−1.70043	−1.70043
$657$	0	0
$658$	0	0
$659$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$659$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$660$	0	0
$661$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$661$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$662$	0	0
$663$	0.198564	0.198564
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.20527	−1.20527
$669$	0.362789	0.362789
$670$	−1.31762	−1.31762
$671$	0	0
$672$	−1.75260	−1.75260
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	4.34296	4.34296
$676$	−0.965946	−0.965946
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	3.86494	3.86494
$679$	−1.51590	−1.51590
$680$	−0.808958	−0.808958
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	1.07447	1.07447
$687$	0	0
$688$	1.20527	1.20527
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−1.86494	−1.86494
$693$	0	0
$694$	0	0
$695$	2.18454	2.18454
$696$	−3.34296	−3.34296
$697$	0.930692	0.930692
$698$	0	0
$699$	−3.86494	−3.86494
$700$	1.05599	1.05599
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.676582	0.676582
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.47802	−1.47802
$707$	0	0
$708$	2.36949	2.36949
$709$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$709$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$710$	2.75642	2.75642
$711$	−1.56806	−1.56806
$712$	−0.891477	−0.891477
$713$	0	0
$714$	0.959241	0.959241
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$719$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$720$	−4.23444	−4.23444
$721$	−1.75260	−1.75260
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0	0
$725$	2.01423	2.01423
$726$	−1.96595	−1.96595
$727$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$727$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$728$	0.164510	0.164510
$729$	5.23444	5.23444
$730$	0	0
$731$	−0.659675	−0.659675
$732$	−3.66638	−3.66638
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0.547326	0.547326
$735$	0.596452	0.596452
$736$	0	0
$737$	0	0
$738$	4.87165	4.87165
$739$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$739$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$743$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0.536209	0.536209
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0.313793	0.313793
$755$	0	0
$756$	3.26849	3.26849
$757$	−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			$\pi$
$762$	−3.66638	−3.66638
$763$	0	0
$764$	1.47802	1.47802
$765$	2.31762	2.31762
$766$	−0.891477	−0.891477
$767$	−0.222416	−0.222416
$768$	1.96595	1.96595
$769$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$769$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$770$	0	0
$771$	0.362789	0.362789
$772$	0	0
$773$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$773$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$774$	−3.45303	−3.45303
$775$	0	0
$776$	1.70043	1.70043
$777$	0	0
$778$	0.184537	0.184537
$779$	0	0
$780$	0.536209	0.536209
$781$	0	0
$782$	0	0
$783$	6.23444	6.23444
$784$	−0.205269	−0.205269
$785$	−1.78141	−1.78141
$786$	0.362789	0.362789
$787$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$787$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$788$	0.547326	0.547326
$789$	0	0
$790$	−0.808958	−0.808958
$791$	−1.75260	−1.75260
$792$	0	0
$793$	0.344151	0.344151
$794$	0	0
$795$	0	0
$796$	−1.20527	−1.20527
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.18454	−1.18454
$801$	2.55403	2.55403
$802$	−0.184537	−0.184537
$803$	0	0
$804$	−1.75260	−1.75260
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	6.41898	6.41898
$811$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$811$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$812$	1.51590	1.51590
$813$	0	0
$814$	0	0
$815$	−0.808958	−0.808958
$816$	−1.07601	−1.07601
$817$	0	0
$818$	−1.86494	−1.86494
$819$	−0.471313	−0.471313
$820$	2.51327	2.51327
$821$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$821$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$822$	0	0
$823$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$823$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$824$	1.96595	1.96595
$825$	0	0
$826$	−1.07447	−1.07447
$827$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$827$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−0.184537	−0.184537
$833$	0.112349	0.112349
$834$	2.90570	2.90570
$835$	1.78141	1.78141
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.00000	2.00000	1.00000			$0$
$839$	2.00000	2.00000	1.00000			$0$
$840$	2.59037	2.59037
$841$	1.89148	1.89148
$842$	0	0
$843$	0	0
$844$	−1.47802	−1.47802
$845$	1.42769	1.42769
$846$	0	0
$847$	0.891477	0.891477
$848$	0	0
$849$	1.07601	1.07601
$850$	0.648328	0.648328
$851$	0	0
$852$	3.66638	3.66638
$853$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$853$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$854$	1.66255	1.66255
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$859$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$860$	−1.78141	−1.78141
$861$	−2.98017	−2.98017
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−3.66638	−3.66638
$865$	2.75642	2.75642
$866$	0	0
$867$	−1.37702	−1.37702
$868$	0	0
$869$	0	0
$870$	4.94096	4.94096
$871$	0.164510	0.164510
$872$	0	0
$873$	−4.87165	−4.87165
$874$	0	0
$875$	−0.243149	−0.243149
$876$	0	0
$877$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$877$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.588085	0.588085
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0.101002	0.101002
$885$	−3.50216	−3.50216
$886$	0.891477	0.891477
$887$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$887$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$888$	0	0
$889$	1.66255	1.66255
$890$	1.31762	1.31762
$891$	0	0
$892$	0.184537	0.184537
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−0.891477	−0.891477
$897$	0	0
$898$	0	0
$899$	0	0
$900$	3.39363	3.39363
$901$	0	0
$902$	0	0
$903$	2.11235	2.11235
$904$	1.96595	1.96595
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	−0.243149	−0.243149
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	1.20527	1.20527
$915$	5.41898	5.41898
$916$	0	0
$917$	−0.164510	−0.164510
$918$	2.00671	2.00671
$919$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$919$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$920$	0	0
$921$	−1.96595	−1.96595
$922$	−0.547326	−0.547326
$923$	−0.344151	−0.344151
$924$	0	0
$925$	0	0
$926$	1.20527	1.20527
$927$	−5.63233	−5.63233
$928$	−1.70043	−1.70043
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−1.96595	−1.96595
$933$	−3.34296	−3.34296
$934$	0	0
$935$	0	0
$936$	0.528687	0.528687
$937$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$937$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$938$	0.794731	0.794731
$939$	3.66638	3.66638
$940$	0	0
$941$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$941$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$942$	−2.36949	−2.36949
$943$	0	0
$944$	1.20527	1.20527
$945$	−4.83089	−4.83089
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.07601	−1.07601
$949$	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
$955$	−2.18454	−2.18454
$956$	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
$967$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$967$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	−2.51327	−2.51327
$971$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$971$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$972$	4.87165	4.87165
$973$	−1.31762	−1.31762
$974$	0	0
$975$	−0.429737	−0.429737
$976$	−1.86494	−1.86494
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.07601	−1.07601
$979$	0	0
$980$	0.303392	0.303392
$981$	0	0
$982$	0	0
$983$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$983$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$984$	3.34296	3.34296
$985$	−0.808958	−0.808958
$986$	0.930692	0.930692
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$991$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$992$	0	0
$993$	0	0
$994$	−1.66255	−1.66255
$995$	1.78141	1.78141
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.184537	0.184537
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2456.1.h.a.613.8	✓	8
8.5	even	2		2456.1.h.b.613.1	yes	8
307.306	odd	2		2456.1.h.b.613.1	yes	8
2456.613	odd	2	CM	2456.1.h.a.613.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2456.1.h.a.613.8	✓	8	1.1	even	1	trivial
2456.1.h.a.613.8	✓	8	2456.613	odd	2	CM
2456.1.h.b.613.1	yes	8	8.5	even	2
2456.1.h.b.613.1	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.96595 q^{3} +1.00000 q^{4} -1.47802 q^{5} -1.96595 q^{6} +0.891477 q^{7} -1.00000 q^{8} +2.86494 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.96595 q^{3} +1.00000 q^{4} -1.47802 q^{5} -1.96595 q^{6} +0.891477 q^{7} -1.00000 q^{8} +2.86494 q^{9} +1.47802 q^{10} +1.96595 q^{12} -0.184537 q^{13} -0.891477 q^{14} -2.90570 q^{15} +1.00000 q^{16} -0.547326 q^{17} -2.86494 q^{18} -1.47802 q^{20} +1.75260 q^{21} -1.96595 q^{24} +1.18454 q^{25} +0.184537 q^{26} +3.66638 q^{27} +0.891477 q^{28} +1.70043 q^{29} +2.90570 q^{30} -1.00000 q^{32} +0.547326 q^{34} -1.31762 q^{35} +2.86494 q^{36} -0.362789 q^{39} +1.47802 q^{40} -1.70043 q^{41} -1.75260 q^{42} +1.20527 q^{43} -4.23444 q^{45} +1.96595 q^{48} -0.205269 q^{49} -1.18454 q^{50} -1.07601 q^{51} -0.184537 q^{52} -3.66638 q^{54} -0.891477 q^{56} -1.70043 q^{58} +1.20527 q^{59} -2.90570 q^{60} -1.86494 q^{61} +2.55403 q^{63} +1.00000 q^{64} +0.272749 q^{65} -0.891477 q^{67} -0.547326 q^{68} +1.31762 q^{70} +1.86494 q^{71} -2.86494 q^{72} +2.32874 q^{75} +0.362789 q^{78} -0.547326 q^{79} -1.47802 q^{80} +4.34296 q^{81} +1.70043 q^{82} +1.75260 q^{84} +0.808958 q^{85} -1.20527 q^{86} +3.34296 q^{87} +0.891477 q^{89} +4.23444 q^{90} -0.164510 q^{91} -1.96595 q^{96} -1.70043 q^{97} +0.205269 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + q^{5} - q^{6} - q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) \) 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

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