LMFDB - Embedded newform 2015.1.h.d.2014.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2015,1,Mod(2014,2015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2015.2014");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2015 = 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2015.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:	$1.00561600046$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{26})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1 $ x^6 - x^5 - 5x^4 + 4x^3 + 6x^2 - 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{13}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{13} - \cdots)$

Embedding invariants

Embedding label			2014.1
Root			$-1.77091$ of defining polynomial
Character	$\chi$	$=$	2015.2014

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77091 q^{2} +0.241073 q^{3} +2.13613 q^{4} -1.00000 q^{5} -0.426920 q^{6} +0.709210 q^{7} -2.01199 q^{8} -0.941884 q^{9} +O(q^{10})$	\(q-1.77091 q^{2} +0.241073 q^{3} +2.13613 q^{4} -1.00000 q^{5} -0.426920 q^{6} +0.709210 q^{7} -2.01199 q^{8} -0.941884 q^{9} +1.77091 q^{10} +1.49702 q^{11} +0.514964 q^{12} +1.00000 q^{13} -1.25595 q^{14} -0.241073 q^{15} +1.42692 q^{16} +1.13613 q^{17} +1.66799 q^{18} -2.13613 q^{20} +0.170972 q^{21} -2.65109 q^{22} -1.94188 q^{23} -0.485036 q^{24} +1.00000 q^{25} -1.77091 q^{26} -0.468136 q^{27} +1.51496 q^{28} +0.426920 q^{30} -1.00000 q^{31} -0.514964 q^{32} +0.360892 q^{33} -2.01199 q^{34} -0.709210 q^{35} -2.01199 q^{36} +0.241073 q^{39} +2.01199 q^{40} -0.302776 q^{42} +1.77091 q^{43} +3.19783 q^{44} +0.941884 q^{45} +3.43891 q^{46} +1.94188 q^{47} +0.343992 q^{48} -0.497021 q^{49} -1.77091 q^{50} +0.273891 q^{51} +2.13613 q^{52} -1.49702 q^{53} +0.829028 q^{54} -1.49702 q^{55} -1.42692 q^{56} -0.514964 q^{60} +1.77091 q^{62} -0.667993 q^{63} -0.514964 q^{64} -1.00000 q^{65} -0.639108 q^{66} +1.49702 q^{67} +2.42692 q^{68} -0.468136 q^{69} +1.25595 q^{70} +1.89506 q^{72} +0.241073 q^{75} +1.06170 q^{77} -0.426920 q^{78} -1.42692 q^{80} +0.829028 q^{81} +0.365217 q^{84} -1.13613 q^{85} -3.13613 q^{86} -3.01199 q^{88} +0.709210 q^{89} -1.66799 q^{90} +0.709210 q^{91} -4.14811 q^{92} -0.241073 q^{93} -3.43891 q^{94} -0.124144 q^{96} -0.241073 q^{97} +0.880181 q^{98} -1.41002 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - q^{3} + 5 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) $ 6 * q + q^2 - q^3 + 5 * q^4 - 6 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 5 * q^9	$ 6 q + q^{2} - q^{3} + 5 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 5 q^{9} - q^{10} + q^{11} - 3 q^{12} + 6 q^{13} - 2 q^{14} + q^{15} + 4 q^{16} - q^{17} + 3 q^{18} - 5 q^{20} + 2 q^{21} - 2 q^{22} - q^{23} - 9 q^{24} + 6 q^{25} + q^{26} - 2 q^{27} + 3 q^{28} - 2 q^{30} - 6 q^{31} + 3 q^{32} + 2 q^{33} + 2 q^{34} - q^{35} + 2 q^{36} - q^{39} - 2 q^{40} + 9 q^{42} - q^{43} + 3 q^{44} - 5 q^{45} + 2 q^{46} + q^{47} - 5 q^{48} + 5 q^{49} + q^{50} - 2 q^{51} + 5 q^{52} - q^{53} + 4 q^{54} - q^{55} - 4 q^{56} + 3 q^{60} - q^{62} + 3 q^{63} + 3 q^{64} - 6 q^{65} - 4 q^{66} + q^{67} + 10 q^{68} - 2 q^{69} + 2 q^{70} + 6 q^{72} - q^{75} - 2 q^{77} + 2 q^{78} - 4 q^{80} + 4 q^{81} + 6 q^{84} + q^{85} - 11 q^{86} - 4 q^{88} + q^{89} - 3 q^{90} + q^{91} - 3 q^{92} + q^{93} - 2 q^{94} - 7 q^{96} + q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100}) $ 6 * q + q^2 - q^3 + 5 * q^4 - 6 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 5 * q^9 - q^10 + q^11 - 3 * q^12 + 6 * q^13 - 2 * q^14 + q^15 + 4 * q^16 - q^17 + 3 * q^18 - 5 * q^20 + 2 * q^21 - 2 * q^22 - q^23 - 9 * q^24 + 6 * q^25 + q^26 - 2 * q^27 + 3 * q^28 - 2 * q^30 - 6 * q^31 + 3 * q^32 + 2 * q^33 + 2 * q^34 - q^35 + 2 * q^36 - q^39 - 2 * q^40 + 9 * q^42 - q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + q^47 - 5 * q^48 + 5 * q^49 + q^50 - 2 * q^51 + 5 * q^52 - q^53 + 4 * q^54 - q^55 - 4 * q^56 + 3 * q^60 - q^62 + 3 * q^63 + 3 * q^64 - 6 * q^65 - 4 * q^66 + q^67 + 10 * q^68 - 2 * q^69 + 2 * q^70 + 6 * q^72 - q^75 - 2 * q^77 + 2 * q^78 - 4 * q^80 + 4 * q^81 + 6 * q^84 + q^85 - 11 * q^86 - 4 * q^88 + q^89 - 3 * q^90 + q^91 - 3 * q^92 + q^93 - 2 * q^94 - 7 * q^96 + q^97 + 3 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times$.

$n$	$716$	$807$	$1861$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$2$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$3$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$3$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$4$	2.13613	2.13613
$5$	−1.00000	−1.00000
$5$	−1.00000	−1.00000
$6$	−0.426920	−0.426920
$7$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$7$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$8$	−2.01199	−2.01199
$9$	−0.941884	−0.941884
$10$	1.77091	1.77091
$11$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$11$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$12$	0.514964	0.514964
$13$	1.00000	1.00000
$13$	1.00000	1.00000
$14$	−1.25595	−1.25595
$15$	−0.241073	−0.241073
$16$	1.42692	1.42692
$17$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$17$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$18$	1.66799	1.66799
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−2.13613	−2.13613
$21$	0.170972	0.170972
$22$	−2.65109	−2.65109
$23$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$23$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$24$	−0.485036	−0.485036
$25$	1.00000	1.00000
$26$	−1.77091	−1.77091
$27$	−0.468136	−0.468136
$28$	1.51496	1.51496
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0.426920	0.426920
$31$	−1.00000	−1.00000
$31$	−1.00000	−1.00000
$32$	−0.514964	−0.514964
$33$	0.360892	0.360892
$34$	−2.01199	−2.01199
$35$	−0.709210	−0.709210
$36$	−2.01199	−2.01199
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0.241073	0.241073
$40$	2.01199	2.01199
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.302776	−0.302776
$43$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$43$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$44$	3.19783	3.19783
$45$	0.941884	0.941884
$46$	3.43891	3.43891
$47$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$47$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$48$	0.343992	0.343992
$49$	−0.497021	−0.497021
$50$	−1.77091	−1.77091
$51$	0.273891	0.273891
$52$	2.13613	2.13613
$53$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$53$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$54$	0.829028	0.829028
$55$	−1.49702	−1.49702
$56$	−1.42692	−1.42692
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.514964	−0.514964
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.77091	1.77091
$63$	−0.667993	−0.667993
$64$	−0.514964	−0.514964
$65$	−1.00000	−1.00000
$66$	−0.639108	−0.639108
$67$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$67$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$68$	2.42692	2.42692
$69$	−0.468136	−0.468136
$70$	1.25595	1.25595
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.89506	1.89506
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0.241073	0.241073
$76$	0	0
$77$	1.06170	1.06170
$78$	−0.426920	−0.426920
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.42692	−1.42692
$81$	0.829028	0.829028
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.365217	0.365217
$85$	−1.13613	−1.13613
$86$	−3.13613	−3.13613
$87$	0	0
$88$	−3.01199	−3.01199
$89$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$89$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$90$	−1.66799	−1.66799
$91$	0.709210	0.709210
$92$	−4.14811	−4.14811
$93$	−0.241073	−0.241073
$94$	−3.43891	−3.43891
$95$	0	0
$96$	−0.124144	−0.124144
$97$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$97$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$98$	0.880181	0.880181
$99$	−1.41002	−1.41002
$100$	2.13613	2.13613
$101$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$101$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$102$	−0.485036	−0.485036
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−2.01199	−2.01199
$105$	−0.170972	−0.170972
$106$	2.65109	2.65109
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.00000	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	2.65109	2.65109
$111$	0	0
$112$	1.01199	1.01199
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	1.94188	1.94188
$116$	0	0
$117$	−0.941884	−0.941884
$118$	0	0
$119$	0.805754	0.805754
$120$	0.485036	0.485036
$121$	1.24107	1.24107
$122$	0	0
$123$	0	0
$124$	−2.13613	−2.13613
$125$	−1.00000	−1.00000
$126$	1.18296	1.18296
$127$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$127$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$128$	1.42692	1.42692
$129$	0.426920	0.426920
$130$	1.77091	1.77091
$131$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$131$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$132$	0.770912	0.770912
$133$	0	0
$134$	−2.65109	−2.65109
$135$	0.468136	0.468136
$136$	−2.28588	−2.28588
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.829028	0.829028
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.51496	−1.51496
$141$	0.468136	0.468136
$142$	0	0
$143$	1.49702	1.49702
$144$	−1.34399	−1.34399
$145$	0	0
$146$	0	0
$147$	−0.119819	−0.119819
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.426920	−0.426920
$151$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$151$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$152$	0	0
$153$	−1.07010	−1.07010
$154$	−1.88018	−1.88018
$155$	1.00000	1.00000
$156$	0.514964	0.514964
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−0.360892	−0.360892
$160$	0.514964	0.514964
$161$	−1.37720	−1.37720
$162$	−1.46814	−1.46814
$163$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$163$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$164$	0	0
$165$	−0.360892	−0.360892
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.343992	−0.343992
$169$	1.00000	1.00000
$170$	2.01199	2.01199
$171$	0	0
$172$	3.78290	3.78290
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0.709210	0.709210
$176$	2.13613	2.13613
$177$	0	0
$178$	−1.25595	−1.25595
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	2.01199	2.01199
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.25595	−1.25595
$183$	0	0
$184$	3.90704	3.90704
$185$	0	0
$186$	0.426920	0.426920
$187$	1.70081	1.70081
$188$	4.14811	4.14811
$189$	−0.332007	−0.332007
$190$	0	0
$191$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$191$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$192$	−0.124144	−0.124144
$193$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$193$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$194$	0.426920	0.426920
$195$	−0.241073	−0.241073
$196$	−1.06170	−1.06170
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	2.49702	2.49702
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−2.01199	−2.01199
$201$	0.360892	0.360892
$202$	−3.13613	−3.13613
$203$	0	0
$204$	0.585066	0.585066
$205$	0	0
$206$	0	0
$207$	1.82903	1.82903
$208$	1.42692	1.42692
$209$	0	0
$210$	0.302776	0.302776
$211$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$211$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$212$	−3.19783	−3.19783
$213$	0	0
$214$	0	0
$215$	−1.77091	−1.77091
$216$	0.941884	0.941884
$217$	−0.709210	−0.709210
$218$	0	0
$219$	0	0
$220$	−3.19783	−3.19783
$221$	1.13613	1.13613
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.365217	−0.365217
$225$	−0.941884	−0.941884
$226$	0	0
$227$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$227$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$228$	0	0
$229$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$229$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$230$	−3.43891	−3.43891
$231$	0.255948	0.255948
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.66799	1.66799
$235$	−1.94188	−1.94188
$236$	0	0
$237$	0	0
$238$	−1.42692	−1.42692
$239$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$239$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$240$	−0.343992	−0.343992
$241$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$241$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$242$	−2.19783	−2.19783
$243$	0.667993	0.667993
$244$	0	0
$245$	0.497021	0.497021
$246$	0	0
$247$	0	0
$248$	2.01199	2.01199
$249$	0	0
$250$	1.77091	1.77091
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.42692	−1.42692
$253$	−2.90704	−2.90704
$254$	−2.01199	−2.01199
$255$	−0.273891	−0.273891
$256$	−2.01199	−2.01199
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−0.756037	−0.756037
$259$	0	0
$260$	−2.13613	−2.13613
$261$	0	0
$262$	−2.01199	−2.01199
$263$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$263$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$264$	−0.726109	−0.726109
$265$	1.49702	1.49702
$266$	0	0
$267$	0.170972	0.170972
$268$	3.19783	3.19783
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.829028	−0.829028
$271$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$271$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$272$	1.62117	1.62117
$273$	0.170972	0.170972
$274$	0	0
$275$	1.49702	1.49702
$276$	−1.00000	−1.00000
$277$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$277$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$278$	0	0
$279$	0.941884	0.941884
$280$	1.42692	1.42692
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−0.829028	−0.829028
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−2.65109	−2.65109
$287$	0	0
$288$	0.485036	0.485036
$289$	0.290790	0.290790
$290$	0	0
$291$	−0.0581164	−0.0581164
$292$	0	0
$293$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$293$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$294$	0.212188	0.212188
$295$	0	0
$296$	0	0
$297$	−0.700810	−0.700810
$298$	0	0
$299$	−1.94188	−1.94188
$300$	0.514964	0.514964
$301$	1.25595	1.25595
$302$	3.13613	3.13613
$303$	0.426920	0.426920
$304$	0	0
$305$	0	0
$306$	1.89506	1.89506
$307$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$307$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$308$	2.26793	2.26793
$309$	0	0
$310$	−1.77091	−1.77091
$311$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$311$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$312$	−0.485036	−0.485036
$313$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$313$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$314$	0	0
$315$	0.667993	0.667993
$316$	0	0
$317$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$317$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$318$	0.639108	0.639108
$319$	0	0
$320$	0.514964	0.514964
$321$	0	0
$322$	2.43891	2.43891
$323$	0	0
$324$	1.77091	1.77091
$325$	1.00000	1.00000
$326$	2.01199	2.01199
$327$	0	0
$328$	0	0
$329$	1.37720	1.37720
$330$	0.639108	0.639108
$331$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$331$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.49702	−1.49702
$336$	0.243963	0.243963
$337$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$337$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$338$	−1.77091	−1.77091
$339$	0	0
$340$	−2.42692	−2.42692
$341$	−1.49702	−1.49702
$342$	0	0
$343$	−1.06170	−1.06170
$344$	−3.56305	−3.56305
$345$	0.468136	0.468136
$346$	0	0
$347$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$347$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.25595	−1.25595
$351$	−0.468136	−0.468136
$352$	−0.770912	−0.770912
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	1.51496	1.51496
$357$	0.194246	0.194246
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.89506	−1.89506
$361$	1.00000	1.00000
$362$	0	0
$363$	0.299190	0.299190
$364$	1.51496	1.51496
$365$	0	0
$366$	0	0
$367$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$367$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$368$	−2.77091	−2.77091
$369$	0	0
$370$	0	0
$371$	−1.06170	−1.06170
$372$	−0.514964	−0.514964
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−3.01199	−3.01199
$375$	−0.241073	−0.241073
$376$	−3.90704	−3.90704
$377$	0	0
$378$	0.587955	0.587955
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0.273891	0.273891
$382$	−3.13613	−3.13613
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.343992	0.343992
$385$	−1.06170	−1.06170
$386$	−3.43891	−3.43891
$387$	−1.66799	−1.66799
$388$	−0.514964	−0.514964
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0.426920	0.426920
$391$	−2.20623	−2.20623
$392$	1.00000	1.00000
$393$	0.273891	0.273891
$394$	0	0
$395$	0	0
$396$	−3.01199	−3.01199
$397$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$397$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$398$	0	0
$399$	0	0
$400$	1.42692	1.42692
$401$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$401$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$402$	−0.639108	−0.639108
$403$	−1.00000	−1.00000
$404$	3.78290	3.78290
$405$	−0.829028	−0.829028
$406$	0	0
$407$	0	0
$408$	−0.551064	−0.551064
$409$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$409$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−3.23905	−3.23905
$415$	0	0
$416$	−0.514964	−0.514964
$417$	0	0
$418$	0	0
$419$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$419$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$420$	−0.365217	−0.365217
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−0.426920	−0.426920
$423$	−1.82903	−1.82903
$424$	3.01199	3.01199
$425$	1.13613	1.13613
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.360892	0.360892
$430$	3.13613	3.13613
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.667993	−0.667993
$433$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$433$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$434$	1.25595	1.25595
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$439$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$440$	3.01199	3.01199
$441$	0.468136	0.468136
$442$	−2.01199	−2.01199
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−0.709210	−0.709210
$446$	0	0
$447$	0	0
$448$	−0.365217	−0.365217
$449$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$449$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$450$	1.66799	1.66799
$451$	0	0
$452$	0	0
$453$	−0.426920	−0.426920
$454$	2.01199	2.01199
$455$	−0.709210	−0.709210
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.25595	−1.25595
$459$	−0.531864	−0.531864
$460$	4.14811	4.14811
$461$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$461$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$462$	−0.453262	−0.453262
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0.241073	0.241073
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−2.01199	−2.01199
$469$	1.06170	1.06170
$470$	3.43891	3.43891
$471$	0	0
$472$	0	0
$473$	2.65109	2.65109
$474$	0	0
$475$	0	0
$476$	1.72120	1.72120
$477$	1.41002	1.41002
$478$	−1.25595	−1.25595
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0.124144	0.124144
$481$	0	0
$482$	0.426920	0.426920
$483$	−0.332007	−0.332007
$484$	2.65109	2.65109
$485$	0.241073	0.241073
$486$	−1.18296	−1.18296
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−0.273891	−0.273891
$490$	−0.880181	−0.880181
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.41002	1.41002
$496$	−1.42692	−1.42692
$497$	0	0
$498$	0	0
$499$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$499$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$500$	−2.13613	−2.13613
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.34399	1.34399
$505$	−1.77091	−1.77091
$506$	5.14811	5.14811
$507$	0.241073	0.241073
$508$	2.42692	2.42692
$509$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$509$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$510$	0.485036	0.485036
$511$	0	0
$512$	2.13613	2.13613
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0.911956	0.911956
$517$	2.90704	2.90704
$518$	0	0
$519$	0	0
$520$	2.01199	2.01199
$521$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$521$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$522$	0	0
$523$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$523$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$524$	2.42692	2.42692
$525$	0.170972	0.170972
$526$	−3.13613	−3.13613
$527$	−1.13613	−1.13613
$528$	0.514964	0.514964
$529$	2.77091	2.77091
$530$	−2.65109	−2.65109
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−0.302776	−0.302776
$535$	0	0
$536$	−3.01199	−3.01199
$537$	0	0
$538$	0	0
$539$	−0.744052	−0.744052
$540$	1.00000	1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	2.01199	2.01199
$543$	0	0
$544$	−0.585066	−0.585066
$545$	0	0
$546$	−0.302776	−0.302776
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2.65109	−2.65109
$551$	0	0
$552$	0.941884	0.941884
$553$	0	0
$554$	3.43891	3.43891
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−1.66799	−1.66799
$559$	1.77091	1.77091
$560$	−1.01199	−1.01199
$561$	0.410020	0.410020
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.00000	1.00000
$565$	0	0
$566$	0	0
$567$	0.587955	0.587955
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	3.19783	3.19783
$573$	0.426920	0.426920
$574$	0	0
$575$	−1.94188	−1.94188
$576$	0.485036	0.485036
$577$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$577$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$578$	−0.514964	−0.514964
$579$	0.468136	0.468136
$580$	0	0
$581$	0	0
$582$	0.102919	0.102919
$583$	−2.24107	−2.24107
$584$	0	0
$585$	0.941884	0.941884
$586$	−1.25595	−1.25595
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.255948	−0.255948
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$593$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$594$	1.24107	1.24107
$595$	−0.805754	−0.805754
$596$	0	0
$597$	0	0
$598$	3.43891	3.43891
$599$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$599$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$600$	−0.485036	−0.485036
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−2.22417	−2.22417
$603$	−1.41002	−1.41002
$604$	−3.78290	−3.78290
$605$	−1.24107	−1.24107
$606$	−0.756037	−0.756037
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.94188	1.94188
$612$	−2.28588	−2.28588
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	3.13613	3.13613
$615$	0	0
$616$	−2.13613	−2.13613
$617$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$617$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$618$	0	0
$619$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$619$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$620$	2.13613	2.13613
$621$	0.909066	0.909066
$622$	3.43891	3.43891
$623$	0.502979	0.502979
$624$	0.343992	0.343992
$625$	1.00000	1.00000
$626$	3.43891	3.43891
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−1.18296	−1.18296
$631$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$631$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$632$	0	0
$633$	0.0581164	0.0581164
$634$	2.01199	2.01199
$635$	−1.13613	−1.13613
$636$	−0.770912	−0.770912
$637$	−0.497021	−0.497021
$638$	0	0
$639$	0	0
$640$	−1.42692	−1.42692
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−2.94188	−2.94188
$645$	−0.426920	−0.426920
$646$	0	0
$647$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$647$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$648$	−1.66799	−1.66799
$649$	0	0
$650$	−1.77091	−1.77091
$651$	−0.170972	−0.170972
$652$	−2.42692	−2.42692
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−1.13613	−1.13613
$656$	0	0
$657$	0	0
$658$	−2.43891	−2.43891
$659$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$659$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$660$	−0.770912	−0.770912
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	2.01199	2.01199
$663$	0.273891	0.273891
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	2.65109	2.65109
$671$	0	0
$672$	−0.0880442	−0.0880442
$673$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$673$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$674$	2.65109	2.65109
$675$	−0.468136	−0.468136
$676$	2.13613	2.13613
$677$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$677$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$678$	0	0
$679$	−0.170972	−0.170972
$680$	2.28588	2.28588
$681$	−0.273891	−0.273891
$682$	2.65109	2.65109
$683$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$683$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$684$	0	0
$685$	0	0
$686$	1.88018	1.88018
$687$	0.170972	0.170972
$688$	2.52695	2.52695
$689$	−1.49702	−1.49702
$690$	−0.829028	−0.829028
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−1.00000	−1.00000
$694$	2.65109	2.65109
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1.51496	1.51496
$701$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$701$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$702$	0.829028	0.829028
$703$	0	0
$704$	−0.770912	−0.770912
$705$	−0.468136	−0.468136
$706$	0	0
$707$	1.25595	1.25595
$708$	0	0
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−1.42692	−1.42692
$713$	1.94188	1.94188
$714$	−0.343992	−0.343992
$715$	−1.49702	−1.49702
$716$	0	0
$717$	0.170972	0.170972
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1.34399	1.34399
$721$	0	0
$722$	−1.77091	−1.77091
$723$	−0.0581164	−0.0581164
$724$	0	0
$725$	0	0
$726$	−0.529839	−0.529839
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−1.42692	−1.42692
$729$	−0.667993	−0.667993
$730$	0	0
$731$	2.01199	2.01199
$732$	0	0
$733$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$733$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$734$	1.25595	1.25595
$735$	0.119819	0.119819
$736$	1.00000	1.00000
$737$	2.24107	2.24107
$738$	0	0
$739$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$739$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$740$	0	0
$741$	0	0
$742$	1.88018	1.88018
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0.485036	0.485036
$745$	0	0
$746$	0	0
$747$	0	0
$748$	3.63315	3.63315
$749$	0	0
$750$	0.426920	0.426920
$751$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$751$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$752$	2.77091	2.77091
$753$	0	0
$754$	0	0
$755$	1.77091	1.77091
$756$	−0.709210	−0.709210
$757$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$757$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$758$	0	0
$759$	−0.700810	−0.700810
$760$	0	0
$761$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$761$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$762$	−0.485036	−0.485036
$763$	0	0
$764$	3.78290	3.78290
$765$	1.07010	1.07010
$766$	0	0
$767$	0	0
$768$	−0.485036	−0.485036
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	1.88018	1.88018
$771$	0	0
$772$	4.14811	4.14811
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.95387	2.95387
$775$	−1.00000	−1.00000
$776$	0.485036	0.485036
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−0.514964	−0.514964
$781$	0	0
$782$	3.90704	3.90704
$783$	0	0
$784$	−0.709210	−0.709210
$785$	0	0
$786$	−0.485036	−0.485036
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0.426920	0.426920
$790$	0	0
$791$	0	0
$792$	2.83694	2.83694
$793$	0	0
$794$	−2.65109	−2.65109
$795$	0.360892	0.360892
$796$	0	0
$797$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$797$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$798$	0	0
$799$	2.20623	2.20623
$800$	−0.514964	−0.514964
$801$	−0.667993	−0.667993
$802$	−3.43891	−3.43891
$803$	0	0
$804$	0.770912	0.770912
$805$	1.37720	1.37720
$806$	1.77091	1.77091
$807$	0	0
$808$	−3.56305	−3.56305
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.46814	1.46814
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−0.273891	−0.273891
$814$	0	0
$815$	1.13613	1.13613
$816$	0.390820	0.390820
$817$	0	0
$818$	−3.43891	−3.43891
$819$	−0.667993	−0.667993
$820$	0	0
$821$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$821$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$822$	0	0
$823$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$823$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$824$	0	0
$825$	0.360892	0.360892
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	3.90704	3.90704
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−0.468136	−0.468136
$832$	−0.514964	−0.514964
$833$	−0.564681	−0.564681
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.468136	0.468136
$838$	−0.426920	−0.426920
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0.343992	0.343992
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0.514964	0.514964
$845$	−1.00000	−1.00000
$846$	3.23905	3.23905
$847$	0.880181	0.880181
$848$	−2.13613	−2.13613
$849$	0	0
$850$	−2.01199	−2.01199
$851$	0	0
$852$	0	0
$853$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$853$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−0.639108	−0.639108
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−3.78290	−3.78290
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.241073	0.241073
$865$	0	0
$866$	−3.13613	−3.13613
$867$	0.0701018	0.0701018
$868$	−1.51496	−1.51496
$869$	0	0
$870$	0	0
$871$	1.49702	1.49702
$872$	0	0
$873$	0.227063	0.227063
$874$	0	0
$875$	−0.709210	−0.709210
$876$	0	0
$877$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$877$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$878$	3.43891	3.43891
$879$	0.170972	0.170972
$880$	−2.13613	−2.13613
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.829028	−0.829028
$883$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$883$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$884$	2.42692	2.42692
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0.805754	0.805754
$890$	1.25595	1.25595
$891$	1.24107	1.24107
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.01199	1.01199
$897$	−0.468136	−0.468136
$898$	3.13613	3.13613
$899$	0	0
$900$	−2.01199	−2.01199
$901$	−1.70081	−1.70081
$902$	0	0
$903$	0.302776	0.302776
$904$	0	0
$905$	0	0
$906$	0.756037	0.756037
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−2.42692	−2.42692
$909$	−1.66799	−1.66799
$910$	1.25595	1.25595
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1.51496	1.51496
$917$	0.805754	0.805754
$918$	0.941884	0.941884
$919$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$919$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$920$	−3.90704	−3.90704
$921$	−0.426920	−0.426920
$922$	0.426920	0.426920
$923$	0	0
$924$	0.546738	0.546738
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$929$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$930$	−0.426920	−0.426920
$931$	0	0
$932$	0	0
$933$	−0.468136	−0.468136
$934$	0	0
$935$	−1.70081	−1.70081
$936$	1.89506	1.89506
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−1.88018	−1.88018
$939$	−0.468136	−0.468136
$940$	−4.14811	−4.14811
$941$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$941$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0.332007	0.332007
$946$	−4.69485	−4.69485
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−0.273891	−0.273891
$952$	−1.62117	−1.62117
$953$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$953$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$954$	−2.49702	−2.49702
$955$	−1.77091	−1.77091
$956$	1.51496	1.51496
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0.124144	0.124144
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−0.514964	−0.514964
$965$	−1.94188	−1.94188
$966$	0.587955	0.587955
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.49702	−2.49702
$969$	0	0
$970$	−0.426920	−0.426920
$971$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$971$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$972$	1.42692	1.42692
$973$	0	0
$974$	0	0
$975$	0.241073	0.241073
$976$	0	0
$977$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$977$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$978$	0.485036	0.485036
$979$	1.06170	1.06170
$980$	1.06170	1.06170
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.332007	0.332007
$988$	0	0
$989$	−3.43891	−3.43891
$990$	−2.49702	−2.49702
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0.514964	0.514964
$993$	−0.273891	−0.273891
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	3.13613	3.13613
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.1.h.d.2014.1	yes	6
5.4	even	2		2015.1.h.c.2014.6	yes	6
13.12	even	2		2015.1.h.b.2014.6	✓	6
31.30	odd	2		2015.1.h.e.2014.1	yes	6
65.64	even	2		2015.1.h.e.2014.1	yes	6
155.154	odd	2		2015.1.h.b.2014.6	✓	6
403.402	odd	2		2015.1.h.c.2014.6	yes	6
2015.2014	odd	2	CM	2015.1.h.d.2014.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.1.h.b.2014.6	✓	6	13.12	even	2
2015.1.h.b.2014.6	✓	6	155.154	odd	2
2015.1.h.c.2014.6	yes	6	5.4	even	2
2015.1.h.c.2014.6	yes	6	403.402	odd	2
2015.1.h.d.2014.1	yes	6	1.1	even	1	trivial
2015.1.h.d.2014.1	yes	6	2015.2014	odd	2	CM
2015.1.h.e.2014.1	yes	6	31.30	odd	2
2015.1.h.e.2014.1	yes	6	65.64	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:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2015.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.77091 q^{2} +0.241073 q^{3} +2.13613 q^{4} -1.00000 q^{5} -0.426920 q^{6} +0.709210 q^{7} -2.01199 q^{8} -0.941884 q^{9} +O(q^{10})\)	\(q-1.77091 q^{2} +0.241073 q^{3} +2.13613 q^{4} -1.00000 q^{5} -0.426920 q^{6} +0.709210 q^{7} -2.01199 q^{8} -0.941884 q^{9} +1.77091 q^{10} +1.49702 q^{11} +0.514964 q^{12} +1.00000 q^{13} -1.25595 q^{14} -0.241073 q^{15} +1.42692 q^{16} +1.13613 q^{17} +1.66799 q^{18} -2.13613 q^{20} +0.170972 q^{21} -2.65109 q^{22} -1.94188 q^{23} -0.485036 q^{24} +1.00000 q^{25} -1.77091 q^{26} -0.468136 q^{27} +1.51496 q^{28} +0.426920 q^{30} -1.00000 q^{31} -0.514964 q^{32} +0.360892 q^{33} -2.01199 q^{34} -0.709210 q^{35} -2.01199 q^{36} +0.241073 q^{39} +2.01199 q^{40} -0.302776 q^{42} +1.77091 q^{43} +3.19783 q^{44} +0.941884 q^{45} +3.43891 q^{46} +1.94188 q^{47} +0.343992 q^{48} -0.497021 q^{49} -1.77091 q^{50} +0.273891 q^{51} +2.13613 q^{52} -1.49702 q^{53} +0.829028 q^{54} -1.49702 q^{55} -1.42692 q^{56} -0.514964 q^{60} +1.77091 q^{62} -0.667993 q^{63} -0.514964 q^{64} -1.00000 q^{65} -0.639108 q^{66} +1.49702 q^{67} +2.42692 q^{68} -0.468136 q^{69} +1.25595 q^{70} +1.89506 q^{72} +0.241073 q^{75} +1.06170 q^{77} -0.426920 q^{78} -1.42692 q^{80} +0.829028 q^{81} +0.365217 q^{84} -1.13613 q^{85} -3.13613 q^{86} -3.01199 q^{88} +0.709210 q^{89} -1.66799 q^{90} +0.709210 q^{91} -4.14811 q^{92} -0.241073 q^{93} -3.43891 q^{94} -0.124144 q^{96} -0.241073 q^{97} +0.880181 q^{98} -1.41002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - q^{3} + 5 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) 6 * q + q^2 - q^3 + 5 * q^4 - 6 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 5 * q^9	\( 6 q + q^{2} - q^{3} + 5 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 5 q^{9} - q^{10} + q^{11} - 3 q^{12} + 6 q^{13} - 2 q^{14} + q^{15} + 4 q^{16} - q^{17} + 3 q^{18} - 5 q^{20} + 2 q^{21} - 2 q^{22} - q^{23} - 9 q^{24} + 6 q^{25} + q^{26} - 2 q^{27} + 3 q^{28} - 2 q^{30} - 6 q^{31} + 3 q^{32} + 2 q^{33} + 2 q^{34} - q^{35} + 2 q^{36} - q^{39} - 2 q^{40} + 9 q^{42} - q^{43} + 3 q^{44} - 5 q^{45} + 2 q^{46} + q^{47} - 5 q^{48} + 5 q^{49} + q^{50} - 2 q^{51} + 5 q^{52} - q^{53} + 4 q^{54} - q^{55} - 4 q^{56} + 3 q^{60} - q^{62} + 3 q^{63} + 3 q^{64} - 6 q^{65} - 4 q^{66} + q^{67} + 10 q^{68} - 2 q^{69} + 2 q^{70} + 6 q^{72} - q^{75} - 2 q^{77} + 2 q^{78} - 4 q^{80} + 4 q^{81} + 6 q^{84} + q^{85} - 11 q^{86} - 4 q^{88} + q^{89} - 3 q^{90} + q^{91} - 3 q^{92} + q^{93} - 2 q^{94} - 7 q^{96} + q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100}) \) 6 * q + q^2 - q^3 + 5 * q^4 - 6 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 5 * q^9 - q^10 + q^11 - 3 * q^12 + 6 * q^13 - 2 * q^14 + q^15 + 4 * q^16 - q^17 + 3 * q^18 - 5 * q^20 + 2 * q^21 - 2 * q^22 - q^23 - 9 * q^24 + 6 * q^25 + q^26 - 2 * q^27 + 3 * q^28 - 2 * q^30 - 6 * q^31 + 3 * q^32 + 2 * q^33 + 2 * q^34 - q^35 + 2 * q^36 - q^39 - 2 * q^40 + 9 * q^42 - q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + q^47 - 5 * q^48 + 5 * q^49 + q^50 - 2 * q^51 + 5 * q^52 - q^53 + 4 * q^54 - q^55 - 4 * q^56 + 3 * q^60 - q^62 + 3 * q^63 + 3 * q^64 - 6 * q^65 - 4 * q^66 + q^67 + 10 * q^68 - 2 * q^69 + 2 * q^70 + 6 * q^72 - q^75 - 2 * q^77 + 2 * q^78 - 4 * q^80 + 4 * q^81 + 6 * q^84 + q^85 - 11 * q^86 - 4 * q^88 + q^89 - 3 * q^90 + q^91 - 3 * q^92 + q^93 - 2 * q^94 - 7 * q^96 + q^97 + 3 * q^98 + 3 * q^99

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