LMFDB - Embedded newform 2804.1.d.a.2803.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2804,1,Mod(2803,2804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2804, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2804.2803");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2804 = 2^{2} \cdot 701 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2804.d (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:	$1.39937829542$
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			2803.1
Root			$1.96595$ of defining polynomial
Character	$\chi$	$=$	2804.2803

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.86494 q^{3} +1.00000 q^{4} -1.20527 q^{5} +1.86494 q^{6} -1.00000 q^{8} +2.47802 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.86494 q^{3} +1.00000 q^{4} -1.20527 q^{5} +1.86494 q^{6} -1.00000 q^{8} +2.47802 q^{9} +1.20527 q^{10} +1.96595 q^{11} -1.86494 q^{12} +0.891477 q^{13} +2.24776 q^{15} +1.00000 q^{16} +0.184537 q^{17} -2.47802 q^{18} -1.20527 q^{20} -1.96595 q^{22} -0.184537 q^{23} +1.86494 q^{24} +0.452674 q^{25} -0.891477 q^{26} -2.75642 q^{27} -1.70043 q^{29} -2.24776 q^{30} -1.00000 q^{32} -3.66638 q^{33} -0.184537 q^{34} +2.47802 q^{36} -1.66255 q^{39} +1.20527 q^{40} -0.547326 q^{41} +1.96595 q^{44} -2.98668 q^{45} +0.184537 q^{46} +1.70043 q^{47} -1.86494 q^{48} +1.00000 q^{49} -0.452674 q^{50} -0.344151 q^{51} +0.891477 q^{52} +2.75642 q^{54} -2.36949 q^{55} +1.70043 q^{58} -1.47802 q^{59} +2.24776 q^{60} +1.00000 q^{64} -1.07447 q^{65} +3.66638 q^{66} +1.96595 q^{67} +0.184537 q^{68} +0.344151 q^{69} +1.20527 q^{71} -2.47802 q^{72} -0.844212 q^{75} +1.66255 q^{78} +0.547326 q^{79} -1.20527 q^{80} +2.66255 q^{81} +0.547326 q^{82} -0.222416 q^{85} +3.17122 q^{87} -1.96595 q^{88} +1.47802 q^{89} +2.98668 q^{90} -0.184537 q^{92} -1.70043 q^{94} +1.86494 q^{96} -1.70043 q^{97} -1.00000 q^{98} +4.87165 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} - q^{5} - q^{6} - 8 q^{8} + 7 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 - q^5 - q^6 - 8 * q^8 + 7 * q^9	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} - q^{5} - q^{6} - 8 q^{8} + 7 q^{9} + q^{10} + q^{11} + q^{12} - q^{13} + 2 q^{15} + 8 q^{16} - q^{17} - 7 q^{18} - q^{20} - q^{22} + q^{23} - q^{24} + 7 q^{25} + q^{26} + 2 q^{27} - q^{29} - 2 q^{30} - 8 q^{32} - 2 q^{33} + q^{34} + 7 q^{36} + 2 q^{39} + q^{40} - q^{41} + q^{44} - 3 q^{45} - q^{46} + q^{47} + q^{48} + 8 q^{49} - 7 q^{50} + 2 q^{51} - q^{52} - 2 q^{54} + 2 q^{55} + q^{58} + q^{59} + 2 q^{60} + 8 q^{64} - 2 q^{65} + 2 q^{66} + q^{67} - q^{68} - 2 q^{69} + q^{71} - 7 q^{72} + 3 q^{75} - 2 q^{78} + q^{79} - q^{80} + 6 q^{81} + q^{82} - 2 q^{85} + 2 q^{87} - q^{88} - q^{89} + 3 q^{90} + q^{92} - q^{94} - q^{96} - q^{97} - 8 q^{98} + 3 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 - q^5 - q^6 - 8 * q^8 + 7 * q^9 + q^10 + q^11 + q^12 - q^13 + 2 * q^15 + 8 * q^16 - q^17 - 7 * q^18 - q^20 - q^22 + q^23 - q^24 + 7 * q^25 + q^26 + 2 * q^27 - q^29 - 2 * q^30 - 8 * q^32 - 2 * q^33 + q^34 + 7 * q^36 + 2 * q^39 + q^40 - q^41 + q^44 - 3 * q^45 - q^46 + q^47 + q^48 + 8 * q^49 - 7 * q^50 + 2 * q^51 - q^52 - 2 * q^54 + 2 * q^55 + q^58 + q^59 + 2 * q^60 + 8 * q^64 - 2 * q^65 + 2 * q^66 + q^67 - q^68 - 2 * q^69 + q^71 - 7 * q^72 + 3 * q^75 - 2 * q^78 + q^79 - q^80 + 6 * q^81 + q^82 - 2 * q^85 + 2 * q^87 - q^88 - q^89 + 3 * q^90 + q^92 - q^94 - q^96 - q^97 - 8 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2804\mathbb{Z}\right)^\times$.

$n$	$1403$	$2105$
$\chi(n)$	$-1$	$-1$

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
$3$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$3$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$4$	1.00000	1.00000
$5$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$5$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$6$	1.86494	1.86494
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.00000	−1.00000
$9$	2.47802	2.47802
$10$	1.20527	1.20527
$11$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$11$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$12$	−1.86494	−1.86494
$13$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$13$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$14$	0	0
$15$	2.24776	2.24776
$16$	1.00000	1.00000
$17$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$17$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$18$	−2.47802	−2.47802
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.20527	−1.20527
$21$	0	0
$22$	−1.96595	−1.96595
$23$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$23$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$24$	1.86494	1.86494
$25$	0.452674	0.452674
$26$	−0.891477	−0.891477
$27$	−2.75642	−2.75642
$28$	0	0
$29$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$29$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$30$	−2.24776	−2.24776
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	−3.66638	−3.66638
$34$	−0.184537	−0.184537
$35$	0	0
$36$	2.47802	2.47802
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−1.66255	−1.66255
$40$	1.20527	1.20527
$41$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$41$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.96595	1.96595
$45$	−2.98668	−2.98668
$46$	0.184537	0.184537
$47$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$47$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$48$	−1.86494	−1.86494
$49$	1.00000	1.00000
$50$	−0.452674	−0.452674
$51$	−0.344151	−0.344151
$52$	0.891477	0.891477
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	2.75642	2.75642
$55$	−2.36949	−2.36949
$56$	0	0
$57$	0	0
$58$	1.70043	1.70043
$59$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$59$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$60$	2.24776	2.24776
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	−1.07447	−1.07447
$66$	3.66638	3.66638
$67$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$67$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$68$	0.184537	0.184537
$69$	0.344151	0.344151
$70$	0	0
$71$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$71$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$72$	−2.47802	−2.47802
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−0.844212	−0.844212
$76$	0	0
$77$	0	0
$78$	1.66255	1.66255
$79$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$79$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$80$	−1.20527	−1.20527
$81$	2.66255	2.66255
$82$	0.547326	0.547326
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−0.222416	−0.222416
$86$	0	0
$87$	3.17122	3.17122
$88$	−1.96595	−1.96595
$89$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$89$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$90$	2.98668	2.98668
$91$	0	0
$92$	−0.184537	−0.184537
$93$	0	0
$94$	−1.70043	−1.70043
$95$	0	0
$96$	1.86494	1.86494
$97$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$97$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$98$	−1.00000	−1.00000
$99$	4.87165	4.87165
$100$	0.452674	0.452674
$101$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$101$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$102$	0.344151	0.344151
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.891477	−0.891477
$105$	0	0
$106$	0	0
$107$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$107$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$108$	−2.75642	−2.75642
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	2.36949	2.36949
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0.222416	0.222416
$116$	−1.70043	−1.70043
$117$	2.20910	2.20910
$118$	1.47802	1.47802
$119$	0	0
$120$	−2.24776	−2.24776
$121$	2.86494	2.86494
$122$	0	0
$123$	1.02073	1.02073
$124$	0	0
$125$	0.659675	0.659675
$126$	0	0
$127$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$127$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	1.07447	1.07447
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−3.66638	−3.66638
$133$	0	0
$134$	−1.96595	−1.96595
$135$	3.32223	3.32223
$136$	−0.184537	−0.184537
$137$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$137$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$138$	−0.344151	−0.344151
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−3.17122	−3.17122
$142$	−1.20527	−1.20527
$143$	1.75260	1.75260
$144$	2.47802	2.47802
$145$	2.04948	2.04948
$146$	0	0
$147$	−1.86494	−1.86494
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.844212	0.844212
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0.457285	0.457285
$154$	0	0
$155$	0	0
$156$	−1.66255	−1.66255
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−0.547326	−0.547326
$159$	0	0
$160$	1.20527	1.20527
$161$	0	0
$162$	−2.66255	−2.66255
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−0.547326	−0.547326
$165$	4.41898	4.41898
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.205269	−0.205269
$170$	0.222416	0.222416
$171$	0	0
$172$	0	0
$173$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$173$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$174$	−3.17122	−3.17122
$175$	0	0
$176$	1.96595	1.96595
$177$	2.75642	2.75642
$178$	−1.47802	−1.47802
$179$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$179$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$180$	−2.98668	−2.98668
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0.184537	0.184537
$185$	0	0
$186$	0	0
$187$	0.362789	0.362789
$188$	1.70043	1.70043
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.86494	−1.86494
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	1.70043	1.70043
$195$	2.00383	2.00383
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−4.87165	−4.87165
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.452674	−0.452674
$201$	−3.66638	−3.66638
$202$	1.96595	1.96595
$203$	0	0
$204$	−0.344151	−0.344151
$205$	0.659675	0.659675
$206$	0	0
$207$	−0.457285	−0.457285
$208$	0.891477	0.891477
$209$	0	0
$210$	0	0
$211$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$211$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$212$	0	0
$213$	−2.24776	−2.24776
$214$	1.47802	1.47802
$215$	0	0
$216$	2.75642	2.75642
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−2.36949	−2.36949
$221$	0.164510	0.164510
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	1.12173	1.12173
$226$	0	0
$227$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$227$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$228$	0	0
$229$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$229$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$230$	−0.222416	−0.222416
$231$	0	0
$232$	1.70043	1.70043
$233$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$233$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$234$	−2.20910	−2.20910
$235$	−2.04948	−2.04948
$236$	−1.47802	−1.47802
$237$	−1.02073	−1.02073
$238$	0	0
$239$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$239$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$240$	2.24776	2.24776
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−2.86494	−2.86494
$243$	−2.20910	−2.20910
$244$	0	0
$245$	−1.20527	−1.20527
$246$	−1.02073	−1.02073
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−0.659675	−0.659675
$251$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$251$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$252$	0	0
$253$	−0.362789	−0.362789
$254$	0.891477	0.891477
$255$	0.414794	0.414794
$256$	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	$0.117647\pi$
$258$	0	0
$259$	0	0
$260$	−1.07447	−1.07447
$261$	−4.21371	−4.21371
$262$	0	0
$263$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$263$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$264$	3.66638	3.66638
$265$	0	0
$266$	0	0
$267$	−2.75642	−2.75642
$268$	1.96595	1.96595
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−3.32223	−3.32223
$271$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$271$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$272$	0.184537	0.184537
$273$	0	0
$274$	−1.86494	−1.86494
$275$	0.889933	0.889933
$276$	0.344151	0.344151
$277$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$277$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	3.17122	3.17122
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.20527	1.20527
$285$	0	0
$286$	−1.75260	−1.75260
$287$	0	0
$288$	−2.47802	−2.47802
$289$	−0.965946	−0.965946
$290$	−2.04948	−2.04948
$291$	3.17122	3.17122
$292$	0	0
$293$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$293$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$294$	1.86494	1.86494
$295$	1.78141	1.78141
$296$	0	0
$297$	−5.41898	−5.41898
$298$	0	0
$299$	−0.164510	−0.164510
$300$	−0.844212	−0.844212
$301$	0	0
$302$	0	0
$303$	3.66638	3.66638
$304$	0	0
$305$	0	0
$306$	−0.457285	−0.457285
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	1.66255	1.66255
$313$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$313$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$314$	0	0
$315$	0	0
$316$	0.547326	0.547326
$317$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$317$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$318$	0	0
$319$	−3.34296	−3.34296
$320$	−1.20527	−1.20527
$321$	2.75642	2.75642
$322$	0	0
$323$	0	0
$324$	2.66255	2.66255
$325$	0.403548	0.403548
$326$	0	0
$327$	0	0
$328$	0.547326	0.547326
$329$	0	0
$330$	−4.41898	−4.41898
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.36949	−2.36949
$336$	0	0
$337$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$337$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$338$	0.205269	0.205269
$339$	0	0
$340$	−0.222416	−0.222416
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.414794	−0.414794
$346$	0.547326	0.547326
$347$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$347$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$348$	3.17122	3.17122
$349$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$349$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$350$	0	0
$351$	−2.45729	−2.45729
$352$	−1.96595	−1.96595
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−2.75642	−2.75642
$355$	−1.45267	−1.45267
$356$	1.47802	1.47802
$357$	0	0
$358$	−1.20527	−1.20527
$359$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$359$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$360$	2.98668	2.98668
$361$	1.00000	1.00000
$362$	0	0
$363$	−5.34296	−5.34296
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$367$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$368$	−0.184537	−0.184537
$369$	−1.35628	−1.35628
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−0.362789	−0.362789
$375$	−1.23026	−1.23026
$376$	−1.70043	−1.70043
$377$	−1.51590	−1.51590
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.66255	1.66255
$382$	0	0
$383$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$383$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$384$	1.86494	1.86494
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−1.70043	−1.70043
$389$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$389$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$390$	−2.00383	−2.00383
$391$	−0.0340538	−0.0340538
$392$	−1.00000	−1.00000
$393$	0	0
$394$	0	0
$395$	−0.659675	−0.659675
$396$	4.87165	4.87165
$397$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$397$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$398$	0	0
$399$	0	0
$400$	0.452674	0.452674
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	3.66638	3.66638
$403$	0	0
$404$	−1.96595	−1.96595
$405$	−3.20910	−3.20910
$406$	0	0
$407$	0	0
$408$	0.344151	0.344151
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−0.659675	−0.659675
$411$	−3.47802	−3.47802
$412$	0	0
$413$	0	0
$414$	0.457285	0.457285
$415$	0	0
$416$	−0.891477	−0.891477
$417$	0	0
$418$	0	0
$419$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$419$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.70043	−1.70043
$423$	4.21371	4.21371
$424$	0	0
$425$	0.0835350	0.0835350
$426$	2.24776	2.24776
$427$	0	0
$428$	−1.47802	−1.47802
$429$	−3.26849	−3.26849
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−2.75642	−2.75642
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−3.82217	−3.82217
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$439$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$440$	2.36949	2.36949
$441$	2.47802	2.47802
$442$	−0.164510	−0.164510
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−1.78141	−1.78141
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$449$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$450$	−1.12173	−1.12173
$451$	−1.07601	−1.07601
$452$	0	0
$453$	0	0
$454$	−0.547326	−0.547326
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−0.184537	−0.184537
$459$	−0.508661	−0.508661
$460$	0.222416	0.222416
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−2.00000	−2.00000	−1.00000			$\pi$
$463$	−2.00000	−2.00000	−1.00000			$\pi$
$464$	−1.70043	−1.70043
$465$	0	0
$466$	−0.891477	−0.891477
$467$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$467$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$468$	2.20910	2.20910
$469$	0	0
$470$	2.04948	2.04948
$471$	0	0
$472$	1.47802	1.47802
$473$	0	0
$474$	1.02073	1.02073
$475$	0	0
$476$	0	0
$477$	0	0
$478$	1.86494	1.86494
$479$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$479$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$480$	−2.24776	−2.24776
$481$	0	0
$482$	0	0
$483$	0	0
$484$	2.86494	2.86494
$485$	2.04948	2.04948
$486$	2.20910	2.20910
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	1.20527	1.20527
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	1.02073	1.02073
$493$	−0.313793	−0.313793
$494$	0	0
$495$	−5.87165	−5.87165
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$499$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$500$	0.659675	0.659675
$501$	0	0
$502$	−0.547326	−0.547326
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	2.36949	2.36949
$506$	0.362789	0.362789
$507$	0.382816	0.382816
$508$	−0.891477	−0.891477
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.414794	−0.414794
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	−1.86494	−1.86494
$515$	0	0
$516$	0	0
$517$	3.34296	3.34296
$518$	0	0
$519$	1.02073	1.02073
$520$	1.07447	1.07447
$521$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$521$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$522$	4.21371	4.21371
$523$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$523$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$524$	0	0
$525$	0	0
$526$	0.891477	0.891477
$527$	0	0
$528$	−3.66638	−3.66638
$529$	−0.965946	−0.965946
$530$	0	0
$531$	−3.66255	−3.66255
$532$	0	0
$533$	−0.487928	−0.487928
$534$	2.75642	2.75642
$535$	1.78141	1.78141
$536$	−1.96595	−1.96595
$537$	−2.24776	−2.24776
$538$	0	0
$539$	1.96595	1.96595
$540$	3.32223	3.32223
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.70043	−1.70043
$543$	0	0
$544$	−0.184537	−0.184537
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	1.86494	1.86494
$549$	0	0
$550$	−0.889933	−0.889933
$551$	0	0
$552$	−0.344151	−0.344151
$553$	0	0
$554$	−1.86494	−1.86494
$555$	0	0
$556$	0	0
$557$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$557$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.676582	−0.676582
$562$	0	0
$563$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$563$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$564$	−3.17122	−3.17122
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−1.20527	−1.20527
$569$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$569$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$570$	0	0
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$571$	−2.00000	−2.00000	−1.00000			$\pi$
$572$	1.75260	1.75260
$573$	0	0
$574$	0	0
$575$	−0.0835350	−0.0835350
$576$	2.47802	2.47802
$577$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$577$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$578$	0.965946	0.965946
$579$	0	0
$580$	2.04948	2.04948
$581$	0	0
$582$	−3.17122	−3.17122
$583$	0	0
$584$	0	0
$585$	−2.66255	−2.66255
$586$	−1.47802	−1.47802
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−1.86494	−1.86494
$589$	0	0
$590$	−1.78141	−1.78141
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	5.41898	5.41898
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0.164510	0.164510
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.844212	0.844212
$601$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$601$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$602$	0	0
$603$	4.87165	4.87165
$604$	0	0
$605$	−3.45303	−3.45303
$606$	−3.66638	−3.66638
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.51590	1.51590
$612$	0.457285	0.457285
$613$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$613$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$614$	0	0
$615$	−1.23026	−1.23026
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$619$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$620$	0	0
$621$	0.508661	0.508661
$622$	0	0
$623$	0	0
$624$	−1.66255	−1.66255
$625$	−1.24776	−1.24776
$626$	−1.47802	−1.47802
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$631$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$632$	−0.547326	−0.547326
$633$	−3.17122	−3.17122
$634$	−1.47802	−1.47802
$635$	1.07447	1.07447
$636$	0	0
$637$	0.891477	0.891477
$638$	3.34296	3.34296
$639$	2.98668	2.98668
$640$	1.20527	1.20527
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−2.75642	−2.75642
$643$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$643$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.66255	−2.66255
$649$	−2.90570	−2.90570
$650$	−0.403548	−0.403548
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−0.547326	−0.547326
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	4.41898	4.41898
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	−0.306802	−0.306802
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.313793	0.313793
$668$	0	0
$669$	0	0
$670$	2.36949	2.36949
$671$	0	0
$672$	0	0
$673$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$673$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$674$	0.547326	0.547326
$675$	−1.24776	−1.24776
$676$	−0.205269	−0.205269
$677$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$677$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$678$	0	0
$679$	0	0
$680$	0.222416	0.222416
$681$	−1.02073	−1.02073
$682$	0	0
$683$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$683$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$684$	0	0
$685$	−2.24776	−2.24776
$686$	0	0
$687$	−0.344151	−0.344151
$688$	0	0
$689$	0	0
$690$	0.414794	0.414794
$691$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$691$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$692$	−0.547326	−0.547326
$693$	0	0
$694$	−1.70043	−1.70043
$695$	0	0
$696$	−3.17122	−3.17122
$697$	−0.101002	−0.101002
$698$	1.96595	1.96595
$699$	−1.66255	−1.66255
$700$	0	0
$701$	1.00000	1.00000
$701$	1.00000	1.00000
$702$	2.45729	2.45729
$703$	0	0
$704$	1.96595	1.96595
$705$	3.82217	3.82217
$706$	0	0
$707$	0	0
$708$	2.75642	2.75642
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	1.45267	1.45267
$711$	1.35628	1.35628
$712$	−1.47802	−1.47802
$713$	0	0
$714$	0	0
$715$	−2.11235	−2.11235
$716$	1.20527	1.20527
$717$	3.47802	3.47802
$718$	0.891477	0.891477
$719$	1.96595	1.96595	0.982973	−	0.183750i	$-0.0588235\pi$
$719$	1.96595	1.96595	0.982973	+	0.183750i	$0.0588235\pi$
$720$	−2.98668	−2.98668
$721$	0	0
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0	0
$725$	−0.769742	−0.769742
$726$	5.34296	5.34296
$727$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$727$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$728$	0	0
$729$	1.45729	1.45729
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.86494	1.86494
$735$	2.24776	2.24776
$736$	0.184537	0.184537
$737$	3.86494	3.86494
$738$	1.35628	1.35628
$739$	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
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0.362789	0.362789
$749$	0	0
$750$	1.23026	1.23026
$751$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$751$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$752$	1.70043	1.70043
$753$	−1.02073	−1.02073
$754$	1.51590	1.51590
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0.676582	0.676582
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−1.66255	−1.66255
$763$	0	0
$764$	0	0
$765$	−0.551152	−0.551152
$766$	0.184537	0.184537
$767$	−1.31762	−1.31762
$768$	−1.86494	−1.86494
$769$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$769$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$770$	0	0
$771$	−3.47802	−3.47802
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	1.70043	1.70043
$777$	0	0
$778$	−0.891477	−0.891477
$779$	0	0
$780$	2.00383	2.00383
$781$	2.36949	2.36949
$782$	0.0340538	0.0340538
$783$	4.68711	4.68711
$784$	1.00000	1.00000
$785$	0	0
$786$	0	0
$787$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$787$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$788$	0	0
$789$	1.66255	1.66255
$790$	0.659675	0.659675
$791$	0	0
$792$	−4.87165	−4.87165
$793$	0	0
$794$	−1.86494	−1.86494
$795$	0	0
$796$	0	0
$797$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$797$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$798$	0	0
$799$	0.313793	0.313793
$800$	−0.452674	−0.452674
$801$	3.66255	3.66255
$802$	0	0
$803$	0	0
$804$	−3.66638	−3.66638
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1.96595	1.96595
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	3.20910	3.20910
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−3.17122	−3.17122
$814$	0	0
$815$	0	0
$816$	−0.344151	−0.344151
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0.659675	0.659675
$821$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$821$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$822$	3.47802	3.47802
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−1.65968	−1.65968
$826$	0	0
$827$	1.20527	1.20527	0.602635	−	0.798017i	$-0.294118\pi$
$827$	1.20527	1.20527	0.602635	+	0.798017i	$0.294118\pi$
$828$	−0.457285	−0.457285
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−3.47802	−3.47802
$832$	0.891477	0.891477
$833$	0.184537	0.184537
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−0.547326	−0.547326
$839$	1.70043	1.70043	0.850217	−	0.526432i	$-0.176471\pi$
$839$	1.70043	1.70043	0.850217	+	0.526432i	$0.176471\pi$
$840$	0	0
$841$	1.89148	1.89148
$842$	0	0
$843$	0	0
$844$	1.70043	1.70043
$845$	0.247405	0.247405
$846$	−4.21371	−4.21371
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−0.0835350	−0.0835350
$851$	0	0
$852$	−2.24776	−2.24776
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	1.47802	1.47802
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	3.26849	3.26849
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.891477	−0.891477	−0.445738	−	0.895163i	$-0.647059\pi$
$863$	−0.891477	−0.891477	−0.445738	+	0.895163i	$0.647059\pi$
$864$	2.75642	2.75642
$865$	0.659675	0.659675
$866$	0	0
$867$	1.80144	1.80144
$868$	0	0
$869$	1.07601	1.07601
$870$	3.82217	3.82217
$871$	1.75260	1.75260
$872$	0	0
$873$	−4.21371	−4.21371
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.96595	−1.96595
$879$	−2.75642	−2.75642
$880$	−2.36949	−2.36949
$881$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$881$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$882$	−2.47802	−2.47802
$883$	0.547326	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$883$	0.547326	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$884$	0.164510	0.164510
$885$	−3.32223	−3.32223
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	1.78141	1.78141
$891$	5.23444	5.23444
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−1.45267	−1.45267
$896$	0	0
$897$	0.306802	0.306802
$898$	0.547326	0.547326
$899$	0	0
$900$	1.12173	1.12173
$901$	0	0
$902$	1.07601	1.07601
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$907$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$908$	0.547326	0.547326
$909$	−4.87165	−4.87165
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0.184537	0.184537
$917$	0	0
$918$	0.508661	0.508661
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−0.222416	−0.222416
$921$	0	0
$922$	0	0
$923$	1.07447	1.07447
$924$	0	0
$925$	0	0
$926$	2.00000	2.00000
$927$	0	0
$928$	1.70043	1.70043
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0.891477	0.891477
$933$	0	0
$934$	−1.20527	−1.20527
$935$	−0.437259	−0.437259
$936$	−2.20910	−2.20910
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−2.75642	−2.75642
$940$	−2.04948	−2.04948
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0.101002	0.101002
$944$	−1.47802	−1.47802
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.02073	−1.02073
$949$	0	0
$950$	0	0
$951$	−2.75642	−2.75642
$952$	0	0
$953$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$953$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$954$	0	0
$955$	0	0
$956$	−1.86494	−1.86494
$957$	6.23444	6.23444
$958$	1.47802	1.47802
$959$	0	0
$960$	2.24776	2.24776
$961$	1.00000	1.00000
$962$	0	0
$963$	−3.66255	−3.66255
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.184537	−0.184537	−0.0922684	−	0.995734i	$-0.529412\pi$
$967$	−0.184537	−0.184537	−0.0922684	+	0.995734i	$0.529412\pi$
$968$	−2.86494	−2.86494
$969$	0	0
$970$	−2.04948	−2.04948
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.20910	−2.20910
$973$	0	0
$974$	0	0
$975$	−0.752595	−0.752595
$976$	0	0
$977$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$977$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$978$	0	0
$979$	2.90570	2.90570
$980$	−1.20527	−1.20527
$981$	0	0
$982$	0	0
$983$	−1.47802	−1.47802	−0.739009	−	0.673696i	$-0.764706\pi$
$983$	−1.47802	−1.47802	−0.739009	+	0.673696i	$0.764706\pi$
$984$	−1.02073	−1.02073
$985$	0	0
$986$	0.313793	0.313793
$987$	0	0
$988$	0	0
$989$	0	0
$990$	5.87165	5.87165
$991$	−1.86494	−1.86494	−0.932472	−	0.361242i	$-0.882353\pi$
$991$	−1.86494	−1.86494	−0.932472	+	0.361242i	$0.882353\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$997$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$998$	1.47802	1.47802
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2804.1.d.a.2803.1	✓	8
4.3	odd	2		2804.1.d.b.2803.8	yes	8
701.700	even	2		2804.1.d.b.2803.8	yes	8
2804.2803	odd	2	CM	2804.1.d.a.2803.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2804.1.d.a.2803.1	✓	8	1.1	even	1	trivial
2804.1.d.a.2803.1	✓	8	2804.2803	odd	2	CM
2804.1.d.b.2803.8	yes	8	4.3	odd	2
2804.1.d.b.2803.8	yes	8	701.700	even	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:	\( N \)	\(=\)	\( 2804 = 2^{2} \cdot 701 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2804.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.86494 q^{3} +1.00000 q^{4} -1.20527 q^{5} +1.86494 q^{6} -1.00000 q^{8} +2.47802 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.86494 q^{3} +1.00000 q^{4} -1.20527 q^{5} +1.86494 q^{6} -1.00000 q^{8} +2.47802 q^{9} +1.20527 q^{10} +1.96595 q^{11} -1.86494 q^{12} +0.891477 q^{13} +2.24776 q^{15} +1.00000 q^{16} +0.184537 q^{17} -2.47802 q^{18} -1.20527 q^{20} -1.96595 q^{22} -0.184537 q^{23} +1.86494 q^{24} +0.452674 q^{25} -0.891477 q^{26} -2.75642 q^{27} -1.70043 q^{29} -2.24776 q^{30} -1.00000 q^{32} -3.66638 q^{33} -0.184537 q^{34} +2.47802 q^{36} -1.66255 q^{39} +1.20527 q^{40} -0.547326 q^{41} +1.96595 q^{44} -2.98668 q^{45} +0.184537 q^{46} +1.70043 q^{47} -1.86494 q^{48} +1.00000 q^{49} -0.452674 q^{50} -0.344151 q^{51} +0.891477 q^{52} +2.75642 q^{54} -2.36949 q^{55} +1.70043 q^{58} -1.47802 q^{59} +2.24776 q^{60} +1.00000 q^{64} -1.07447 q^{65} +3.66638 q^{66} +1.96595 q^{67} +0.184537 q^{68} +0.344151 q^{69} +1.20527 q^{71} -2.47802 q^{72} -0.844212 q^{75} +1.66255 q^{78} +0.547326 q^{79} -1.20527 q^{80} +2.66255 q^{81} +0.547326 q^{82} -0.222416 q^{85} +3.17122 q^{87} -1.96595 q^{88} +1.47802 q^{89} +2.98668 q^{90} -0.184537 q^{92} -1.70043 q^{94} +1.86494 q^{96} -1.70043 q^{97} -1.00000 q^{98} +4.87165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} - q^{5} - q^{6} - 8 q^{8} + 7 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 - q^5 - q^6 - 8 * q^8 + 7 * q^9	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} - q^{5} - q^{6} - 8 q^{8} + 7 q^{9} + q^{10} + q^{11} + q^{12} - q^{13} + 2 q^{15} + 8 q^{16} - q^{17} - 7 q^{18} - q^{20} - q^{22} + q^{23} - q^{24} + 7 q^{25} + q^{26} + 2 q^{27} - q^{29} - 2 q^{30} - 8 q^{32} - 2 q^{33} + q^{34} + 7 q^{36} + 2 q^{39} + q^{40} - q^{41} + q^{44} - 3 q^{45} - q^{46} + q^{47} + q^{48} + 8 q^{49} - 7 q^{50} + 2 q^{51} - q^{52} - 2 q^{54} + 2 q^{55} + q^{58} + q^{59} + 2 q^{60} + 8 q^{64} - 2 q^{65} + 2 q^{66} + q^{67} - q^{68} - 2 q^{69} + q^{71} - 7 q^{72} + 3 q^{75} - 2 q^{78} + q^{79} - q^{80} + 6 q^{81} + q^{82} - 2 q^{85} + 2 q^{87} - q^{88} - q^{89} + 3 q^{90} + q^{92} - q^{94} - q^{96} - q^{97} - 8 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 - q^5 - q^6 - 8 * q^8 + 7 * q^9 + q^10 + q^11 + q^12 - q^13 + 2 * q^15 + 8 * q^16 - q^17 - 7 * q^18 - q^20 - q^22 + q^23 - q^24 + 7 * q^25 + q^26 + 2 * q^27 - q^29 - 2 * q^30 - 8 * q^32 - 2 * q^33 + q^34 + 7 * q^36 + 2 * q^39 + q^40 - q^41 + q^44 - 3 * q^45 - q^46 + q^47 + q^48 + 8 * q^49 - 7 * q^50 + 2 * q^51 - q^52 - 2 * q^54 + 2 * q^55 + q^58 + q^59 + 2 * q^60 + 8 * q^64 - 2 * q^65 + 2 * q^66 + q^67 - q^68 - 2 * q^69 + q^71 - 7 * q^72 + 3 * q^75 - 2 * q^78 + q^79 - q^80 + 6 * q^81 + q^82 - 2 * q^85 + 2 * q^87 - q^88 - q^89 + 3 * q^90 + q^92 - q^94 - q^96 - q^97 - 8 * q^98 + 3 * q^99

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