LMFDB - Embedded newform 2015.1.h.b.2014.5

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			2014.5
Root			$-1.13613$ 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)

$f(q)$	$=$	$q+1.13613 q^{2} -1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} -2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +O(q^{10})$	\(q+1.13613 q^{2} -1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} -2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +1.13613 q^{10} +0.241073 q^{11} -0.564681 q^{12} +1.00000 q^{13} -1.70081 q^{14} -1.94188 q^{15} -1.20623 q^{16} -0.709210 q^{17} +3.14811 q^{18} +0.290790 q^{20} +2.90704 q^{21} +0.273891 q^{22} +1.77091 q^{23} +1.56468 q^{24} +1.00000 q^{25} +1.13613 q^{26} -3.43891 q^{27} -0.435319 q^{28} -2.20623 q^{30} +1.00000 q^{31} -0.564681 q^{32} -0.468136 q^{33} -0.805754 q^{34} -1.49702 q^{35} +0.805754 q^{36} -1.94188 q^{39} -0.805754 q^{40} +3.30278 q^{42} +1.13613 q^{43} +0.0701018 q^{44} +2.77091 q^{45} +2.01199 q^{46} +1.77091 q^{47} +2.34236 q^{48} +1.24107 q^{49} +1.13613 q^{50} +1.37720 q^{51} +0.290790 q^{52} +0.241073 q^{53} -3.90704 q^{54} +0.241073 q^{55} +1.20623 q^{56} -0.564681 q^{60} +1.13613 q^{62} -4.14811 q^{63} +0.564681 q^{64} +1.00000 q^{65} -0.531864 q^{66} +0.241073 q^{67} -0.206231 q^{68} -3.43891 q^{69} -1.70081 q^{70} -2.23267 q^{72} -1.94188 q^{75} -0.360892 q^{77} -2.20623 q^{78} -1.20623 q^{80} +3.90704 q^{81} +0.845339 q^{84} -0.709210 q^{85} +1.29079 q^{86} -0.194246 q^{88} -1.49702 q^{89} +3.14811 q^{90} -1.49702 q^{91} +0.514964 q^{92} -1.94188 q^{93} +2.01199 q^{94} +1.09654 q^{96} -1.94188 q^{97} +1.41002 q^{98} +0.667993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$2$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$3$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$3$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$4$	0.290790	0.290790
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	−2.20623	−2.20623
$7$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$7$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$8$	−0.805754	−0.805754
$9$	2.77091	2.77091
$10$	1.13613	1.13613
$11$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$11$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$12$	−0.564681	−0.564681
$13$	1.00000	1.00000
$13$	1.00000	1.00000
$14$	−1.70081	−1.70081
$15$	−1.94188	−1.94188
$16$	−1.20623	−1.20623
$17$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$17$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$18$	3.14811	3.14811
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.290790	0.290790
$21$	2.90704	2.90704
$22$	0.273891	0.273891
$23$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$23$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$24$	1.56468	1.56468
$25$	1.00000	1.00000
$26$	1.13613	1.13613
$27$	−3.43891	−3.43891
$28$	−0.435319	−0.435319
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−2.20623	−2.20623
$31$	1.00000	1.00000
$31$	1.00000	1.00000
$32$	−0.564681	−0.564681
$33$	−0.468136	−0.468136
$34$	−0.805754	−0.805754
$35$	−1.49702	−1.49702
$36$	0.805754	0.805754
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−1.94188	−1.94188
$40$	−0.805754	−0.805754
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	3.30278	3.30278
$43$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$43$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$44$	0.0701018	0.0701018
$45$	2.77091	2.77091
$46$	2.01199	2.01199
$47$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$47$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$48$	2.34236	2.34236
$49$	1.24107	1.24107
$50$	1.13613	1.13613
$51$	1.37720	1.37720
$52$	0.290790	0.290790
$53$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$53$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$54$	−3.90704	−3.90704
$55$	0.241073	0.241073
$56$	1.20623	1.20623
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.564681	−0.564681
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.13613	1.13613
$63$	−4.14811	−4.14811
$64$	0.564681	0.564681
$65$	1.00000	1.00000
$66$	−0.531864	−0.531864
$67$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$67$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$68$	−0.206231	−0.206231
$69$	−3.43891	−3.43891
$70$	−1.70081	−1.70081
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.23267	−2.23267
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.94188	−1.94188
$76$	0	0
$77$	−0.360892	−0.360892
$78$	−2.20623	−2.20623
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.20623	−1.20623
$81$	3.90704	3.90704
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.845339	0.845339
$85$	−0.709210	−0.709210
$86$	1.29079	1.29079
$87$	0	0
$88$	−0.194246	−0.194246
$89$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$89$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$90$	3.14811	3.14811
$91$	−1.49702	−1.49702
$92$	0.514964	0.514964
$93$	−1.94188	−1.94188
$94$	2.01199	2.01199
$95$	0	0
$96$	1.09654	1.09654
$97$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$97$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$98$	1.41002	1.41002
$99$	0.667993	0.667993
$100$	0.290790	0.290790
$101$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$101$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$102$	1.56468	1.56468
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.805754	−0.805754
$105$	2.90704	2.90704
$106$	0.273891	0.273891
$107$	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			$\pi$
$110$	0.273891	0.273891
$111$	0	0
$112$	1.80575	1.80575
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	1.77091	1.77091
$116$	0	0
$117$	2.77091	2.77091
$118$	0	0
$119$	1.06170	1.06170
$120$	1.56468	1.56468
$121$	−0.941884	−0.941884
$122$	0	0
$123$	0	0
$124$	0.290790	0.290790
$125$	1.00000	1.00000
$126$	−4.71280	−4.71280
$127$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$127$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$128$	1.20623	1.20623
$129$	−2.20623	−2.20623
$130$	1.13613	1.13613
$131$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$131$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$132$	−0.136129	−0.136129
$133$	0	0
$134$	0.273891	0.273891
$135$	−3.43891	−3.43891
$136$	0.571449	0.571449
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−3.90704	−3.90704
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−0.435319	−0.435319
$141$	−3.43891	−3.43891
$142$	0	0
$143$	0.241073	0.241073
$144$	−3.34236	−3.34236
$145$	0	0
$146$	0	0
$147$	−2.41002	−2.41002
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.20623	−2.20623
$151$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$151$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$152$	0	0
$153$	−1.96516	−1.96516
$154$	−0.410020	−0.410020
$155$	1.00000	1.00000
$156$	−0.564681	−0.564681
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−0.468136	−0.468136
$160$	−0.564681	−0.564681
$161$	−2.65109	−2.65109
$162$	4.43891	4.43891
$163$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$163$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$164$	0	0
$165$	−0.468136	−0.468136
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−2.34236	−2.34236
$169$	1.00000	1.00000
$170$	−0.805754	−0.805754
$171$	0	0
$172$	0.330375	0.330375
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.49702	−1.49702
$176$	−0.290790	−0.290790
$177$	0	0
$178$	−1.70081	−1.70081
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0.805754	0.805754
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.70081	−1.70081
$183$	0	0
$184$	−1.42692	−1.42692
$185$	0	0
$186$	−2.20623	−2.20623
$187$	−0.170972	−0.170972
$188$	0.514964	0.514964
$189$	5.14811	5.14811
$190$	0	0
$191$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$191$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$192$	−1.09654	−1.09654
$193$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$193$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$194$	−2.20623	−2.20623
$195$	−1.94188	−1.94188
$196$	0.360892	0.360892
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.758927	0.758927
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.805754	−0.805754
$201$	−0.468136	−0.468136
$202$	1.29079	1.29079
$203$	0	0
$204$	0.400477	0.400477
$205$	0	0
$206$	0	0
$207$	4.90704	4.90704
$208$	−1.20623	−1.20623
$209$	0	0
$210$	3.30278	3.30278
$211$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$211$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$212$	0.0701018	0.0701018
$213$	0	0
$214$	0	0
$215$	1.13613	1.13613
$216$	2.77091	2.77091
$217$	−1.49702	−1.49702
$218$	0	0
$219$	0	0
$220$	0.0701018	0.0701018
$221$	−0.709210	−0.709210
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.845339	0.845339
$225$	2.77091	2.77091
$226$	0	0
$227$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$227$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$228$	0	0
$229$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$229$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$230$	2.01199	2.01199
$231$	0.700810	0.700810
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	3.14811	3.14811
$235$	1.77091	1.77091
$236$	0	0
$237$	0	0
$238$	1.20623	1.20623
$239$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$239$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$240$	2.34236	2.34236
$241$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$241$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$242$	−1.07010	−1.07010
$243$	−4.14811	−4.14811
$244$	0	0
$245$	1.24107	1.24107
$246$	0	0
$247$	0	0
$248$	−0.805754	−0.805754
$249$	0	0
$250$	1.13613	1.13613
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.20623	−1.20623
$253$	0.426920	0.426920
$254$	−0.805754	−0.805754
$255$	1.37720	1.37720
$256$	0.805754	0.805754
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−2.50656	−2.50656
$259$	0	0
$260$	0.290790	0.290790
$261$	0	0
$262$	−0.805754	−0.805754
$263$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$263$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$264$	0.377203	0.377203
$265$	0.241073	0.241073
$266$	0	0
$267$	2.90704	2.90704
$268$	0.0701018	0.0701018
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−3.90704	−3.90704
$271$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$271$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$272$	0.855471	0.855471
$273$	2.90704	2.90704
$274$	0	0
$275$	0.241073	0.241073
$276$	−1.00000	−1.00000
$277$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$277$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$278$	0	0
$279$	2.77091	2.77091
$280$	1.20623	1.20623
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−3.90704	−3.90704
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0.273891	0.273891
$287$	0	0
$288$	−1.56468	−1.56468
$289$	−0.497021	−0.497021
$290$	0	0
$291$	3.77091	3.77091
$292$	0	0
$293$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$293$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$294$	−2.73809	−2.73809
$295$	0	0
$296$	0	0
$297$	−0.829028	−0.829028
$298$	0	0
$299$	1.77091	1.77091
$300$	−0.564681	−0.564681
$301$	−1.70081	−1.70081
$302$	1.29079	1.29079
$303$	−2.20623	−2.20623
$304$	0	0
$305$	0	0
$306$	−2.23267	−2.23267
$307$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$307$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$308$	−0.104944	−0.104944
$309$	0	0
$310$	1.13613	1.13613
$311$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$311$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$312$	1.56468	1.56468
$313$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$313$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$314$	0	0
$315$	−4.14811	−4.14811
$316$	0	0
$317$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$317$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$318$	−0.531864	−0.531864
$319$	0	0
$320$	0.564681	0.564681
$321$	0	0
$322$	−3.01199	−3.01199
$323$	0	0
$324$	1.13613	1.13613
$325$	1.00000	1.00000
$326$	−0.805754	−0.805754
$327$	0	0
$328$	0	0
$329$	−2.65109	−2.65109
$330$	−0.531864	−0.531864
$331$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$331$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.241073	0.241073
$336$	−3.50656	−3.50656
$337$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$337$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$338$	1.13613	1.13613
$339$	0	0
$340$	−0.206231	−0.206231
$341$	0.241073	0.241073
$342$	0	0
$343$	−0.360892	−0.360892
$344$	−0.915441	−0.915441
$345$	−3.43891	−3.43891
$346$	0	0
$347$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$347$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.70081	−1.70081
$351$	−3.43891	−3.43891
$352$	−0.136129	−0.136129
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−0.435319	−0.435319
$357$	−2.06170	−2.06170
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−2.23267	−2.23267
$361$	1.00000	1.00000
$362$	0	0
$363$	1.82903	1.82903
$364$	−0.435319	−0.435319
$365$	0	0
$366$	0	0
$367$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$367$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$368$	−2.13613	−2.13613
$369$	0	0
$370$	0	0
$371$	−0.360892	−0.360892
$372$	−0.564681	−0.564681
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−0.194246	−0.194246
$375$	−1.94188	−1.94188
$376$	−1.42692	−1.42692
$377$	0	0
$378$	5.84893	5.84893
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.37720	1.37720
$382$	1.29079	1.29079
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−2.34236	−2.34236
$385$	−0.360892	−0.360892
$386$	2.01199	2.01199
$387$	3.14811	3.14811
$388$	−0.564681	−0.564681
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−2.20623	−2.20623
$391$	−1.25595	−1.25595
$392$	−1.00000	−1.00000
$393$	1.37720	1.37720
$394$	0	0
$395$	0	0
$396$	0.194246	0.194246
$397$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$397$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$398$	0	0
$399$	0	0
$400$	−1.20623	−1.20623
$401$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$401$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$402$	−0.531864	−0.531864
$403$	1.00000	1.00000
$404$	0.330375	0.330375
$405$	3.90704	3.90704
$406$	0	0
$407$	0	0
$408$	−1.10969	−1.10969
$409$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$409$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	5.57503	5.57503
$415$	0	0
$416$	−0.564681	−0.564681
$417$	0	0
$418$	0	0
$419$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$419$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$420$	0.845339	0.845339
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−2.20623	−2.20623
$423$	4.90704	4.90704
$424$	−0.194246	−0.194246
$425$	−0.709210	−0.709210
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.468136	−0.468136
$430$	1.29079	1.29079
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	4.14811	4.14811
$433$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$433$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$434$	−1.70081	−1.70081
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$439$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$440$	−0.194246	−0.194246
$441$	3.43891	3.43891
$442$	−0.805754	−0.805754
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−1.49702	−1.49702
$446$	0	0
$447$	0	0
$448$	−0.845339	−0.845339
$449$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$449$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$450$	3.14811	3.14811
$451$	0	0
$452$	0	0
$453$	−2.20623	−2.20623
$454$	−0.805754	−0.805754
$455$	−1.49702	−1.49702
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.70081	−1.70081
$459$	2.43891	2.43891
$460$	0.514964	0.514964
$461$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$461$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$462$	0.796211	0.796211
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	−1.94188	−1.94188
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0.805754	0.805754
$469$	−0.360892	−0.360892
$470$	2.01199	2.01199
$471$	0	0
$472$	0	0
$473$	0.273891	0.273891
$474$	0	0
$475$	0	0
$476$	0.308733	0.308733
$477$	0.667993	0.667993
$478$	−1.70081	−1.70081
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	1.09654	1.09654
$481$	0	0
$482$	−2.20623	−2.20623
$483$	5.14811	5.14811
$484$	−0.273891	−0.273891
$485$	−1.94188	−1.94188
$486$	−4.71280	−4.71280
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.37720	1.37720
$490$	1.41002	1.41002
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.667993	0.667993
$496$	−1.20623	−1.20623
$497$	0	0
$498$	0	0
$499$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$499$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$500$	0.290790	0.290790
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	3.34236	3.34236
$505$	1.13613	1.13613
$506$	0.485036	0.485036
$507$	−1.94188	−1.94188
$508$	−0.206231	−0.206231
$509$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$509$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$510$	1.56468	1.56468
$511$	0	0
$512$	−0.290790	−0.290790
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−0.641550	−0.641550
$517$	0.426920	0.426920
$518$	0	0
$519$	0	0
$520$	−0.805754	−0.805754
$521$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$521$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$522$	0	0
$523$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$523$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$524$	−0.206231	−0.206231
$525$	2.90704	2.90704
$526$	1.29079	1.29079
$527$	−0.709210	−0.709210
$528$	0.564681	0.564681
$529$	2.13613	2.13613
$530$	0.273891	0.273891
$531$	0	0
$532$	0	0
$533$	0	0
$534$	3.30278	3.30278
$535$	0	0
$536$	−0.194246	−0.194246
$537$	0	0
$538$	0	0
$539$	0.299190	0.299190
$540$	−1.00000	−1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−0.805754	−0.805754
$543$	0	0
$544$	0.400477	0.400477
$545$	0	0
$546$	3.30278	3.30278
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0.273891	0.273891
$551$	0	0
$552$	2.77091	2.77091
$553$	0	0
$554$	2.01199	2.01199
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	3.14811	3.14811
$559$	1.13613	1.13613
$560$	1.80575	1.80575
$561$	0.332007	0.332007
$562$	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
$567$	−5.84893	−5.84893
$568$	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			$\pi$
$572$	0.0701018	0.0701018
$573$	−2.20623	−2.20623
$574$	0	0
$575$	1.77091	1.77091
$576$	1.56468	1.56468
$577$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$577$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$578$	−0.564681	−0.564681
$579$	−3.43891	−3.43891
$580$	0	0
$581$	0	0
$582$	4.28424	4.28424
$583$	0.0581164	0.0581164
$584$	0	0
$585$	2.77091	2.77091
$586$	−1.70081	−1.70081
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.700810	−0.700810
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$593$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$594$	−0.941884	−0.941884
$595$	1.06170	1.06170
$596$	0	0
$597$	0	0
$598$	2.01199	2.01199
$599$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$599$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$600$	1.56468	1.56468
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.93234	−1.93234
$603$	0.667993	0.667993
$604$	0.330375	0.330375
$605$	−0.941884	−0.941884
$606$	−2.50656	−2.50656
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.77091	1.77091
$612$	−0.571449	−0.571449
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.29079	1.29079
$615$	0	0
$616$	0.290790	0.290790
$617$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$617$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$618$	0	0
$619$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$619$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$620$	0.290790	0.290790
$621$	−6.09000	−6.09000
$622$	2.01199	2.01199
$623$	2.24107	2.24107
$624$	2.34236	2.34236
$625$	1.00000	1.00000
$626$	2.01199	2.01199
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−4.71280	−4.71280
$631$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$631$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$632$	0	0
$633$	3.77091	3.77091
$634$	−0.805754	−0.805754
$635$	−0.709210	−0.709210
$636$	−0.136129	−0.136129
$637$	1.24107	1.24107
$638$	0	0
$639$	0	0
$640$	1.20623	1.20623
$641$	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			$\pi$
$644$	−0.770912	−0.770912
$645$	−2.20623	−2.20623
$646$	0	0
$647$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$647$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$648$	−3.14811	−3.14811
$649$	0	0
$650$	1.13613	1.13613
$651$	2.90704	2.90704
$652$	−0.206231	−0.206231
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−0.709210	−0.709210
$656$	0	0
$657$	0	0
$658$	−3.01199	−3.01199
$659$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$659$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$660$	−0.136129	−0.136129
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.805754	−0.805754
$663$	1.37720	1.37720
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0.273891	0.273891
$671$	0	0
$672$	−1.64155	−1.64155
$673$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$673$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$674$	0.273891	0.273891
$675$	−3.43891	−3.43891
$676$	0.290790	0.290790
$677$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$677$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$678$	0	0
$679$	2.90704	2.90704
$680$	0.571449	0.571449
$681$	1.37720	1.37720
$682$	0.273891	0.273891
$683$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$683$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$684$	0	0
$685$	0	0
$686$	−0.410020	−0.410020
$687$	2.90704	2.90704
$688$	−1.37043	−1.37043
$689$	0.241073	0.241073
$690$	−3.90704	−3.90704
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−1.00000	−1.00000
$694$	0.273891	0.273891
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−0.435319	−0.435319
$701$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$701$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$702$	−3.90704	−3.90704
$703$	0	0
$704$	0.136129	0.136129
$705$	−3.43891	−3.43891
$706$	0	0
$707$	−1.70081	−1.70081
$708$	0	0
$709$	2.00000	2.00000	1.00000			$0$
$709$	2.00000	2.00000	1.00000			$0$
$710$	0	0
$711$	0	0
$712$	1.20623	1.20623
$713$	1.77091	1.77091
$714$	−2.34236	−2.34236
$715$	0.241073	0.241073
$716$	0	0
$717$	2.90704	2.90704
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−3.34236	−3.34236
$721$	0	0
$722$	1.13613	1.13613
$723$	3.77091	3.77091
$724$	0	0
$725$	0	0
$726$	2.07801	2.07801
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1.20623	1.20623
$729$	4.14811	4.14811
$730$	0	0
$731$	−0.805754	−0.805754
$732$	0	0
$733$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$733$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$734$	−1.70081	−1.70081
$735$	−2.41002	−2.41002
$736$	−1.00000	−1.00000
$737$	0.0581164	0.0581164
$738$	0	0
$739$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$739$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$740$	0	0
$741$	0	0
$742$	−0.410020	−0.410020
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.56468	1.56468
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−0.0497169	−0.0497169
$749$	0	0
$750$	−2.20623	−2.20623
$751$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$751$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$752$	−2.13613	−2.13613
$753$	0	0
$754$	0	0
$755$	1.13613	1.13613
$756$	1.49702	1.49702
$757$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$757$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$758$	0	0
$759$	−0.829028	−0.829028
$760$	0	0
$761$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$761$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$762$	1.56468	1.56468
$763$	0	0
$764$	0.330375	0.330375
$765$	−1.96516	−1.96516
$766$	0	0
$767$	0	0
$768$	−1.56468	−1.56468
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	−0.410020	−0.410020
$771$	0	0
$772$	0.514964	0.514964
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	3.57667	3.57667
$775$	1.00000	1.00000
$776$	1.56468	1.56468
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−0.564681	−0.564681
$781$	0	0
$782$	−1.42692	−1.42692
$783$	0	0
$784$	−1.49702	−1.49702
$785$	0	0
$786$	1.56468	1.56468
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−2.20623	−2.20623
$790$	0	0
$791$	0	0
$792$	−0.538238	−0.538238
$793$	0	0
$794$	0.273891	0.273891
$795$	−0.468136	−0.468136
$796$	0	0
$797$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$797$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$798$	0	0
$799$	−1.25595	−1.25595
$800$	−0.564681	−0.564681
$801$	−4.14811	−4.14811
$802$	2.01199	2.01199
$803$	0	0
$804$	−0.136129	−0.136129
$805$	−2.65109	−2.65109
$806$	1.13613	1.13613
$807$	0	0
$808$	−0.915441	−0.915441
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	4.43891	4.43891
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	1.37720	1.37720
$814$	0	0
$815$	−0.709210	−0.709210
$816$	−1.66123	−1.66123
$817$	0	0
$818$	2.01199	2.01199
$819$	−4.14811	−4.14811
$820$	0	0
$821$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$821$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$822$	0	0
$823$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$823$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$824$	0	0
$825$	−0.468136	−0.468136
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.42692	1.42692
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−3.43891	−3.43891
$832$	0.564681	0.564681
$833$	−0.880181	−0.880181
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.43891	−3.43891
$838$	−2.20623	−2.20623
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−2.34236	−2.34236
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	−0.564681	−0.564681
$845$	1.00000	1.00000
$846$	5.57503	5.57503
$847$	1.41002	1.41002
$848$	−0.290790	−0.290790
$849$	0	0
$850$	−0.805754	−0.805754
$851$	0	0
$852$	0	0
$853$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$853$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−0.531864	−0.531864
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0.330375	0.330375
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.94188	1.94188
$865$	0	0
$866$	1.29079	1.29079
$867$	0.965158	0.965158
$868$	−0.435319	−0.435319
$869$	0	0
$870$	0	0
$871$	0.241073	0.241073
$872$	0	0
$873$	−5.38079	−5.38079
$874$	0	0
$875$	−1.49702	−1.49702
$876$	0	0
$877$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$877$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$878$	2.01199	2.01199
$879$	2.90704	2.90704
$880$	−0.290790	−0.290790
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3.90704	3.90704
$883$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$883$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$884$	−0.206231	−0.206231
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	1.06170	1.06170
$890$	−1.70081	−1.70081
$891$	0.941884	0.941884
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.80575	−1.80575
$897$	−3.43891	−3.43891
$898$	1.29079	1.29079
$899$	0	0
$900$	0.805754	0.805754
$901$	−0.170972	−0.170972
$902$	0	0
$903$	3.30278	3.30278
$904$	0	0
$905$	0	0
$906$	−2.50656	−2.50656
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.206231	−0.206231
$909$	3.14811	3.14811
$910$	−1.70081	−1.70081
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−0.435319	−0.435319
$917$	1.06170	1.06170
$918$	2.77091	2.77091
$919$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$919$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$920$	−1.42692	−1.42692
$921$	−2.20623	−2.20623
$922$	−2.20623	−2.20623
$923$	0	0
$924$	0.203789	0.203789
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$929$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$930$	−2.20623	−2.20623
$931$	0	0
$932$	0	0
$933$	−3.43891	−3.43891
$934$	0	0
$935$	−0.170972	−0.170972
$936$	−2.23267	−2.23267
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−0.410020	−0.410020
$939$	−3.43891	−3.43891
$940$	0.514964	0.514964
$941$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$941$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.14811	5.14811
$946$	0.311175	0.311175
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.37720	1.37720
$952$	−0.855471	−0.855471
$953$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$953$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$954$	0.758927	0.758927
$955$	1.13613	1.13613
$956$	−0.435319	−0.435319
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−1.09654	−1.09654
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−0.564681	−0.564681
$965$	1.77091	1.77091
$966$	5.84893	5.84893
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.758927	0.758927
$969$	0	0
$970$	−2.20623	−2.20623
$971$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$971$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$972$	−1.20623	−1.20623
$973$	0	0
$974$	0	0
$975$	−1.94188	−1.94188
$976$	0	0
$977$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$977$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$978$	1.56468	1.56468
$979$	−0.360892	−0.360892
$980$	0.360892	0.360892
$981$	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
$987$	5.14811	5.14811
$988$	0	0
$989$	2.01199	2.01199
$990$	0.758927	0.758927
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−0.564681	−0.564681
$993$	1.37720	1.37720
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.29079	1.29079
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.1.h.b.2014.5	✓	6
5.4	even	2		2015.1.h.e.2014.2	yes	6
13.12	even	2		2015.1.h.d.2014.2	yes	6
31.30	odd	2		2015.1.h.c.2014.5	yes	6
65.64	even	2		2015.1.h.c.2014.5	yes	6
155.154	odd	2		2015.1.h.d.2014.2	yes	6
403.402	odd	2		2015.1.h.e.2014.2	yes	6
2015.2014	odd	2	CM	2015.1.h.b.2014.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.1.h.b.2014.5	✓	6	1.1	even	1	trivial
2015.1.h.b.2014.5	✓	6	2015.2014	odd	2	CM
2015.1.h.c.2014.5	yes	6	31.30	odd	2
2015.1.h.c.2014.5	yes	6	65.64	even	2
2015.1.h.d.2014.2	yes	6	13.12	even	2
2015.1.h.d.2014.2	yes	6	155.154	odd	2
2015.1.h.e.2014.2	yes	6	5.4	even	2
2015.1.h.e.2014.2	yes	6	403.402	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:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2015.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.13613 q^{2} -1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} -2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +O(q^{10})\)	\(q+1.13613 q^{2} -1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} -2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +1.13613 q^{10} +0.241073 q^{11} -0.564681 q^{12} +1.00000 q^{13} -1.70081 q^{14} -1.94188 q^{15} -1.20623 q^{16} -0.709210 q^{17} +3.14811 q^{18} +0.290790 q^{20} +2.90704 q^{21} +0.273891 q^{22} +1.77091 q^{23} +1.56468 q^{24} +1.00000 q^{25} +1.13613 q^{26} -3.43891 q^{27} -0.435319 q^{28} -2.20623 q^{30} +1.00000 q^{31} -0.564681 q^{32} -0.468136 q^{33} -0.805754 q^{34} -1.49702 q^{35} +0.805754 q^{36} -1.94188 q^{39} -0.805754 q^{40} +3.30278 q^{42} +1.13613 q^{43} +0.0701018 q^{44} +2.77091 q^{45} +2.01199 q^{46} +1.77091 q^{47} +2.34236 q^{48} +1.24107 q^{49} +1.13613 q^{50} +1.37720 q^{51} +0.290790 q^{52} +0.241073 q^{53} -3.90704 q^{54} +0.241073 q^{55} +1.20623 q^{56} -0.564681 q^{60} +1.13613 q^{62} -4.14811 q^{63} +0.564681 q^{64} +1.00000 q^{65} -0.531864 q^{66} +0.241073 q^{67} -0.206231 q^{68} -3.43891 q^{69} -1.70081 q^{70} -2.23267 q^{72} -1.94188 q^{75} -0.360892 q^{77} -2.20623 q^{78} -1.20623 q^{80} +3.90704 q^{81} +0.845339 q^{84} -0.709210 q^{85} +1.29079 q^{86} -0.194246 q^{88} -1.49702 q^{89} +3.14811 q^{90} -1.49702 q^{91} +0.514964 q^{92} -1.94188 q^{93} +2.01199 q^{94} +1.09654 q^{96} -1.94188 q^{97} +1.41002 q^{98} +0.667993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - q^{3} + 5 q^{4} + 6 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) 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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