LMFDB - Embedded newform 2015.1.h.c.2014.3

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			2014.3
Root			$0.709210$ 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)

$f(q)$	$=$	$q-0.709210 q^{2} -1.77091 q^{3} -0.497021 q^{4} +1.00000 q^{5} +1.25595 q^{6} +0.241073 q^{7} +1.06170 q^{8} +2.13613 q^{9} +O(q^{10})$	\(q-0.709210 q^{2} -1.77091 q^{3} -0.497021 q^{4} +1.00000 q^{5} +1.25595 q^{6} +0.241073 q^{7} +1.06170 q^{8} +2.13613 q^{9} -0.709210 q^{10} +1.94188 q^{11} +0.880181 q^{12} -1.00000 q^{13} -0.170972 q^{14} -1.77091 q^{15} -0.255948 q^{16} +1.49702 q^{17} -1.51496 q^{18} -0.497021 q^{20} -0.426920 q^{21} -1.37720 q^{22} -1.13613 q^{23} -1.88018 q^{24} +1.00000 q^{25} +0.709210 q^{26} -2.01199 q^{27} -0.119819 q^{28} +1.25595 q^{30} -1.00000 q^{31} -0.880181 q^{32} -3.43891 q^{33} -1.06170 q^{34} +0.241073 q^{35} -1.06170 q^{36} +1.77091 q^{39} +1.06170 q^{40} +0.302776 q^{42} +0.709210 q^{43} -0.965158 q^{44} +2.13613 q^{45} +0.805754 q^{46} +1.13613 q^{47} +0.453262 q^{48} -0.941884 q^{49} -0.709210 q^{50} -2.65109 q^{51} +0.497021 q^{52} +1.94188 q^{53} +1.42692 q^{54} +1.94188 q^{55} +0.255948 q^{56} +0.880181 q^{60} +0.709210 q^{62} +0.514964 q^{63} +0.880181 q^{64} -1.00000 q^{65} +2.43891 q^{66} -1.94188 q^{67} -0.744052 q^{68} +2.01199 q^{69} -0.170972 q^{70} +2.26793 q^{72} -1.77091 q^{75} +0.468136 q^{77} -1.25595 q^{78} -0.255948 q^{80} +1.42692 q^{81} +0.212188 q^{84} +1.49702 q^{85} -0.502979 q^{86} +2.06170 q^{88} -0.241073 q^{89} -1.51496 q^{90} -0.241073 q^{91} +0.564681 q^{92} +1.77091 q^{93} -0.805754 q^{94} +1.55872 q^{96} +1.77091 q^{97} +0.667993 q^{98} +4.14811 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$2$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$3$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$3$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$4$	−0.497021	−0.497021
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	1.25595	1.25595
$7$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$7$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$8$	1.06170	1.06170
$9$	2.13613	2.13613
$10$	−0.709210	−0.709210
$11$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$11$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$12$	0.880181	0.880181
$13$	−1.00000	−1.00000
$13$	−1.00000	−1.00000
$14$	−0.170972	−0.170972
$15$	−1.77091	−1.77091
$16$	−0.255948	−0.255948
$17$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$17$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$18$	−1.51496	−1.51496
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−0.497021	−0.497021
$21$	−0.426920	−0.426920
$22$	−1.37720	−1.37720
$23$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$23$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$24$	−1.88018	−1.88018
$25$	1.00000	1.00000
$26$	0.709210	0.709210
$27$	−2.01199	−2.01199
$28$	−0.119819	−0.119819
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	1.25595	1.25595
$31$	−1.00000	−1.00000
$31$	−1.00000	−1.00000
$32$	−0.880181	−0.880181
$33$	−3.43891	−3.43891
$34$	−1.06170	−1.06170
$35$	0.241073	0.241073
$36$	−1.06170	−1.06170
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	1.77091	1.77091
$40$	1.06170	1.06170
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0.302776	0.302776
$43$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$43$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$44$	−0.965158	−0.965158
$45$	2.13613	2.13613
$46$	0.805754	0.805754
$47$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$47$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$48$	0.453262	0.453262
$49$	−0.941884	−0.941884
$50$	−0.709210	−0.709210
$51$	−2.65109	−2.65109
$52$	0.497021	0.497021
$53$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$53$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$54$	1.42692	1.42692
$55$	1.94188	1.94188
$56$	0.255948	0.255948
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0.880181	0.880181
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0.709210	0.709210
$63$	0.514964	0.514964
$64$	0.880181	0.880181
$65$	−1.00000	−1.00000
$66$	2.43891	2.43891
$67$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$67$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$68$	−0.744052	−0.744052
$69$	2.01199	2.01199
$70$	−0.170972	−0.170972
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	2.26793	2.26793
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.77091	−1.77091
$76$	0	0
$77$	0.468136	0.468136
$78$	−1.25595	−1.25595
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.255948	−0.255948
$81$	1.42692	1.42692
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.212188	0.212188
$85$	1.49702	1.49702
$86$	−0.502979	−0.502979
$87$	0	0
$88$	2.06170	2.06170
$89$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$89$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$90$	−1.51496	−1.51496
$91$	−0.241073	−0.241073
$92$	0.564681	0.564681
$93$	1.77091	1.77091
$94$	−0.805754	−0.805754
$95$	0	0
$96$	1.55872	1.55872
$97$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$97$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$98$	0.667993	0.667993
$99$	4.14811	4.14811
$100$	−0.497021	−0.497021
$101$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$101$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$102$	1.88018	1.88018
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−1.06170	−1.06170
$105$	−0.426920	−0.426920
$106$	−1.37720	−1.37720
$107$	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			$\pi$
$110$	−1.37720	−1.37720
$111$	0	0
$112$	−0.0617023	−0.0617023
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.13613	−1.13613
$116$	0	0
$117$	−2.13613	−2.13613
$118$	0	0
$119$	0.360892	0.360892
$120$	−1.88018	−1.88018
$121$	2.77091	2.77091
$122$	0	0
$123$	0	0
$124$	0.497021	0.497021
$125$	1.00000	1.00000
$126$	−0.365217	−0.365217
$127$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$127$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$128$	0.255948	0.255948
$129$	−1.25595	−1.25595
$130$	0.709210	0.709210
$131$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$131$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$132$	1.70921	1.70921
$133$	0	0
$134$	1.37720	1.37720
$135$	−2.01199	−2.01199
$136$	1.58939	1.58939
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.42692	−1.42692
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−0.119819	−0.119819
$141$	−2.01199	−2.01199
$142$	0	0
$143$	−1.94188	−1.94188
$144$	−0.546738	−0.546738
$145$	0	0
$146$	0	0
$147$	1.66799	1.66799
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.25595	1.25595
$151$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$151$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$152$	0	0
$153$	3.19783	3.19783
$154$	−0.332007	−0.332007
$155$	−1.00000	−1.00000
$156$	−0.880181	−0.880181
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−3.43891	−3.43891
$160$	−0.880181	−0.880181
$161$	−0.273891	−0.273891
$162$	−1.01199	−1.01199
$163$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$163$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$164$	0	0
$165$	−3.43891	−3.43891
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.453262	−0.453262
$169$	1.00000	1.00000
$170$	−1.06170	−1.06170
$171$	0	0
$172$	−0.352493	−0.352493
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0.241073	0.241073
$176$	−0.497021	−0.497021
$177$	0	0
$178$	0.170972	0.170972
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.06170	−1.06170
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0.170972	0.170972
$183$	0	0
$184$	−1.20623	−1.20623
$185$	0	0
$186$	−1.25595	−1.25595
$187$	2.90704	2.90704
$188$	−0.564681	−0.564681
$189$	−0.485036	−0.485036
$190$	0	0
$191$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$191$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$192$	−1.55872	−1.55872
$193$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$193$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$194$	−1.25595	−1.25595
$195$	1.77091	1.77091
$196$	0.468136	0.468136
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−2.94188	−2.94188
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.06170	1.06170
$201$	3.43891	3.43891
$202$	0.502979	0.502979
$203$	0	0
$204$	1.31765	1.31765
$205$	0	0
$206$	0	0
$207$	−2.42692	−2.42692
$208$	0.255948	0.255948
$209$	0	0
$210$	0.302776	0.302776
$211$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$211$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$212$	−0.965158	−0.965158
$213$	0	0
$214$	0	0
$215$	0.709210	0.709210
$216$	−2.13613	−2.13613
$217$	−0.241073	−0.241073
$218$	0	0
$219$	0	0
$220$	−0.965158	−0.965158
$221$	−1.49702	−1.49702
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.212188	−0.212188
$225$	2.13613	2.13613
$226$	0	0
$227$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$227$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$228$	0	0
$229$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$229$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$230$	0.805754	0.805754
$231$	−0.829028	−0.829028
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.51496	1.51496
$235$	1.13613	1.13613
$236$	0	0
$237$	0	0
$238$	−0.255948	−0.255948
$239$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$239$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$240$	0.453262	0.453262
$241$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$241$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$242$	−1.96516	−1.96516
$243$	−0.514964	−0.514964
$244$	0	0
$245$	−0.941884	−0.941884
$246$	0	0
$247$	0	0
$248$	−1.06170	−1.06170
$249$	0	0
$250$	−0.709210	−0.709210
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.255948	−0.255948
$253$	−2.20623	−2.20623
$254$	−1.06170	−1.06170
$255$	−2.65109	−2.65109
$256$	−1.06170	−1.06170
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.890731	0.890731
$259$	0	0
$260$	0.497021	0.497021
$261$	0	0
$262$	1.06170	1.06170
$263$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$263$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$264$	−3.65109	−3.65109
$265$	1.94188	1.94188
$266$	0	0
$267$	0.426920	0.426920
$268$	0.965158	0.965158
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.42692	1.42692
$271$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$271$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$272$	−0.383160	−0.383160
$273$	0.426920	0.426920
$274$	0	0
$275$	1.94188	1.94188
$276$	−1.00000	−1.00000
$277$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$277$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$278$	0	0
$279$	−2.13613	−2.13613
$280$	0.255948	0.255948
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.42692	1.42692
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	1.37720	1.37720
$287$	0	0
$288$	−1.88018	−1.88018
$289$	1.24107	1.24107
$290$	0	0
$291$	−3.13613	−3.13613
$292$	0	0
$293$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$293$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$294$	−1.18296	−1.18296
$295$	0	0
$296$	0	0
$297$	−3.90704	−3.90704
$298$	0	0
$299$	1.13613	1.13613
$300$	0.880181	0.880181
$301$	0.170972	0.170972
$302$	−0.502979	−0.502979
$303$	1.25595	1.25595
$304$	0	0
$305$	0	0
$306$	−2.26793	−2.26793
$307$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$307$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$308$	−0.232674	−0.232674
$309$	0	0
$310$	0.709210	0.709210
$311$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$311$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$312$	1.88018	1.88018
$313$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$313$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$314$	0	0
$315$	0.514964	0.514964
$316$	0	0
$317$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$317$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$318$	2.43891	2.43891
$319$	0	0
$320$	0.880181	0.880181
$321$	0	0
$322$	0.194246	0.194246
$323$	0	0
$324$	−0.709210	−0.709210
$325$	−1.00000	−1.00000
$326$	1.06170	1.06170
$327$	0	0
$328$	0	0
$329$	0.273891	0.273891
$330$	2.43891	2.43891
$331$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$331$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.94188	−1.94188
$336$	0.109269	0.109269
$337$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$337$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$338$	−0.709210	−0.709210
$339$	0	0
$340$	−0.744052	−0.744052
$341$	−1.94188	−1.94188
$342$	0	0
$343$	−0.468136	−0.468136
$344$	0.752970	0.752970
$345$	2.01199	2.01199
$346$	0	0
$347$	1.94188	1.94188	0.970942	−	0.239316i	$-0.0769231\pi$
$347$	1.94188	1.94188	0.970942	+	0.239316i	$0.0769231\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−0.170972	−0.170972
$351$	2.01199	2.01199
$352$	−1.70921	−1.70921
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0.119819	0.119819
$357$	−0.639108	−0.639108
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	2.26793	2.26793
$361$	1.00000	1.00000
$362$	0	0
$363$	−4.90704	−4.90704
$364$	0.119819	0.119819
$365$	0	0
$366$	0	0
$367$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$367$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$368$	0.290790	0.290790
$369$	0	0
$370$	0	0
$371$	0.468136	0.468136
$372$	−0.880181	−0.880181
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−2.06170	−2.06170
$375$	−1.77091	−1.77091
$376$	1.20623	1.20623
$377$	0	0
$378$	0.343992	0.343992
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−2.65109	−2.65109
$382$	0.502979	0.502979
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.453262	−0.453262
$385$	0.468136	0.468136
$386$	−0.805754	−0.805754
$387$	1.51496	1.51496
$388$	−0.880181	−0.880181
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−1.25595	−1.25595
$391$	−1.70081	−1.70081
$392$	−1.00000	−1.00000
$393$	2.65109	2.65109
$394$	0	0
$395$	0	0
$396$	−2.06170	−2.06170
$397$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$397$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$398$	0	0
$399$	0	0
$400$	−0.255948	−0.255948
$401$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$401$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$402$	−2.43891	−2.43891
$403$	1.00000	1.00000
$404$	0.352493	0.352493
$405$	1.42692	1.42692
$406$	0	0
$407$	0	0
$408$	−2.81467	−2.81467
$409$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$409$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.72120	1.72120
$415$	0	0
$416$	0.880181	0.880181
$417$	0	0
$418$	0	0
$419$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$419$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$420$	0.212188	0.212188
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.25595	−1.25595
$423$	2.42692	2.42692
$424$	2.06170	2.06170
$425$	1.49702	1.49702
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.43891	3.43891
$430$	−0.502979	−0.502979
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.514964	0.514964
$433$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$433$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$434$	0.170972	0.170972
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$439$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$440$	2.06170	2.06170
$441$	−2.01199	−2.01199
$442$	1.06170	1.06170
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−0.241073	−0.241073
$446$	0	0
$447$	0	0
$448$	0.212188	0.212188
$449$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$449$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$450$	−1.51496	−1.51496
$451$	0	0
$452$	0	0
$453$	−1.25595	−1.25595
$454$	1.06170	1.06170
$455$	−0.241073	−0.241073
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.170972	0.170972
$459$	−3.01199	−3.01199
$460$	0.564681	0.564681
$461$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$461$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$462$	0.587955	0.587955
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	1.77091	1.77091
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.06170	1.06170
$469$	−0.468136	−0.468136
$470$	−0.805754	−0.805754
$471$	0	0
$472$	0	0
$473$	1.37720	1.37720
$474$	0	0
$475$	0	0
$476$	−0.179371	−0.179371
$477$	4.14811	4.14811
$478$	0.170972	0.170972
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	1.55872	1.55872
$481$	0	0
$482$	1.25595	1.25595
$483$	0.485036	0.485036
$484$	−1.37720	−1.37720
$485$	1.77091	1.77091
$486$	0.365217	0.365217
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	2.65109	2.65109
$490$	0.667993	0.667993
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.14811	4.14811
$496$	0.255948	0.255948
$497$	0	0
$498$	0	0
$499$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$499$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$500$	−0.497021	−0.497021
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0.546738	0.546738
$505$	−0.709210	−0.709210
$506$	1.56468	1.56468
$507$	−1.77091	−1.77091
$508$	−0.744052	−0.744052
$509$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$509$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$510$	1.88018	1.88018
$511$	0	0
$512$	0.497021	0.497021
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0.624233	0.624233
$517$	2.20623	2.20623
$518$	0	0
$519$	0	0
$520$	−1.06170	−1.06170
$521$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$521$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$522$	0	0
$523$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$523$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$524$	0.744052	0.744052
$525$	−0.426920	−0.426920
$526$	−0.502979	−0.502979
$527$	−1.49702	−1.49702
$528$	0.880181	0.880181
$529$	0.290790	0.290790
$530$	−1.37720	−1.37720
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−0.302776	−0.302776
$535$	0	0
$536$	−2.06170	−2.06170
$537$	0	0
$538$	0	0
$539$	−1.82903	−1.82903
$540$	1.00000	1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.06170	−1.06170
$543$	0	0
$544$	−1.31765	−1.31765
$545$	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
$550$	−1.37720	−1.37720
$551$	0	0
$552$	2.13613	2.13613
$553$	0	0
$554$	0.805754	0.805754
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.51496	1.51496
$559$	−0.709210	−0.709210
$560$	−0.0617023	−0.0617023
$561$	−5.14811	−5.14811
$562$	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
$567$	0.343992	0.343992
$568$	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			$\pi$
$572$	0.965158	0.965158
$573$	1.25595	1.25595
$574$	0	0
$575$	−1.13613	−1.13613
$576$	1.88018	1.88018
$577$	−0.709210	−0.709210	−0.354605	−	0.935016i	$-0.615385\pi$
$577$	−0.709210	−0.709210	−0.354605	+	0.935016i	$0.615385\pi$
$578$	−0.880181	−0.880181
$579$	−2.01199	−2.01199
$580$	0	0
$581$	0	0
$582$	2.22417	2.22417
$583$	3.77091	3.77091
$584$	0	0
$585$	−2.13613	−2.13613
$586$	−0.170972	−0.170972
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.829028	−0.829028
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$593$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$594$	2.77091	2.77091
$595$	0.360892	0.360892
$596$	0	0
$597$	0	0
$598$	−0.805754	−0.805754
$599$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$599$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$600$	−1.88018	−1.88018
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.121255	−0.121255
$603$	−4.14811	−4.14811
$604$	−0.352493	−0.352493
$605$	2.77091	2.77091
$606$	−0.890731	−0.890731
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.13613	−1.13613
$612$	−1.58939	−1.58939
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0.502979	0.502979
$615$	0	0
$616$	0.497021	0.497021
$617$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$617$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$618$	0	0
$619$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$619$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$620$	0.497021	0.497021
$621$	2.28588	2.28588
$622$	−0.805754	−0.805754
$623$	−0.0581164	−0.0581164
$624$	−0.453262	−0.453262
$625$	1.00000	1.00000
$626$	0.805754	0.805754
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−0.365217	−0.365217
$631$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$631$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$632$	0	0
$633$	−3.13613	−3.13613
$634$	1.06170	1.06170
$635$	1.49702	1.49702
$636$	1.70921	1.70921
$637$	0.941884	0.941884
$638$	0	0
$639$	0	0
$640$	0.255948	0.255948
$641$	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			$\pi$
$644$	0.136129	0.136129
$645$	−1.25595	−1.25595
$646$	0	0
$647$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$647$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$648$	1.51496	1.51496
$649$	0	0
$650$	0.709210	0.709210
$651$	0.426920	0.426920
$652$	0.744052	0.744052
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−1.49702	−1.49702
$656$	0	0
$657$	0	0
$658$	−0.194246	−0.194246
$659$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$659$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$660$	1.70921	1.70921
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.06170	−1.06170
$663$	2.65109	2.65109
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	1.37720	1.37720
$671$	0	0
$672$	0.375767	0.375767
$673$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$673$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$674$	−1.37720	−1.37720
$675$	−2.01199	−2.01199
$676$	−0.497021	−0.497021
$677$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$677$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$678$	0	0
$679$	0.426920	0.426920
$680$	1.58939	1.58939
$681$	2.65109	2.65109
$682$	1.37720	1.37720
$683$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$683$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$684$	0	0
$685$	0	0
$686$	0.332007	0.332007
$687$	0.426920	0.426920
$688$	−0.181521	−0.181521
$689$	−1.94188	−1.94188
$690$	−1.42692	−1.42692
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	1.00000	1.00000
$694$	−1.37720	−1.37720
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−0.119819	−0.119819
$701$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$701$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$702$	−1.42692	−1.42692
$703$	0	0
$704$	1.70921	1.70921
$705$	−2.01199	−2.01199
$706$	0	0
$707$	−0.170972	−0.170972
$708$	0	0
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−0.255948	−0.255948
$713$	1.13613	1.13613
$714$	0.453262	0.453262
$715$	−1.94188	−1.94188
$716$	0	0
$717$	0.426920	0.426920
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.546738	−0.546738
$721$	0	0
$722$	−0.709210	−0.709210
$723$	3.13613	3.13613
$724$	0	0
$725$	0	0
$726$	3.48012	3.48012
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−0.255948	−0.255948
$729$	−0.514964	−0.514964
$730$	0	0
$731$	1.06170	1.06170
$732$	0	0
$733$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$733$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$734$	0.170972	0.170972
$735$	1.66799	1.66799
$736$	1.00000	1.00000
$737$	−3.77091	−3.77091
$738$	0	0
$739$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$739$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$740$	0	0
$741$	0	0
$742$	−0.332007	−0.332007
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.88018	1.88018
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−1.44486	−1.44486
$749$	0	0
$750$	1.25595	1.25595
$751$	0.241073	0.241073	0.120537	−	0.992709i	$-0.461538\pi$
$751$	0.241073	0.241073	0.120537	+	0.992709i	$0.461538\pi$
$752$	−0.290790	−0.290790
$753$	0	0
$754$	0	0
$755$	0.709210	0.709210
$756$	0.241073	0.241073
$757$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$757$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$758$	0	0
$759$	3.90704	3.90704
$760$	0	0
$761$	0.709210	0.709210	0.354605	−	0.935016i	$-0.384615\pi$
$761$	0.709210	0.709210	0.354605	+	0.935016i	$0.384615\pi$
$762$	1.88018	1.88018
$763$	0	0
$764$	0.352493	0.352493
$765$	3.19783	3.19783
$766$	0	0
$767$	0	0
$768$	1.88018	1.88018
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	−0.332007	−0.332007
$771$	0	0
$772$	−0.564681	−0.564681
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.07443	−1.07443
$775$	−1.00000	−1.00000
$776$	1.88018	1.88018
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−0.880181	−0.880181
$781$	0	0
$782$	1.20623	1.20623
$783$	0	0
$784$	0.241073	0.241073
$785$	0	0
$786$	−1.88018	−1.88018
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−1.25595	−1.25595
$790$	0	0
$791$	0	0
$792$	4.40406	4.40406
$793$	0	0
$794$	1.37720	1.37720
$795$	−3.43891	−3.43891
$796$	0	0
$797$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$797$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$798$	0	0
$799$	1.70081	1.70081
$800$	−0.880181	−0.880181
$801$	−0.514964	−0.514964
$802$	0.805754	0.805754
$803$	0	0
$804$	−1.70921	−1.70921
$805$	−0.273891	−0.273891
$806$	−0.709210	−0.709210
$807$	0	0
$808$	−0.752970	−0.752970
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−1.01199	−1.01199
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−2.65109	−2.65109
$814$	0	0
$815$	−1.49702	−1.49702
$816$	0.678542	0.678542
$817$	0	0
$818$	0.805754	0.805754
$819$	−0.514964	−0.514964
$820$	0	0
$821$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$821$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$822$	0	0
$823$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$823$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$824$	0	0
$825$	−3.43891	−3.43891
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.20623	1.20623
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	2.01199	2.01199
$832$	−0.880181	−0.880181
$833$	−1.41002	−1.41002
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.01199	2.01199
$838$	−1.25595	−1.25595
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−0.453262	−0.453262
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	−0.880181	−0.880181
$845$	1.00000	1.00000
$846$	−1.72120	−1.72120
$847$	0.667993	0.667993
$848$	−0.497021	−0.497021
$849$	0	0
$850$	−1.06170	−1.06170
$851$	0	0
$852$	0	0
$853$	1.77091	1.77091	0.885456	−	0.464723i	$-0.153846\pi$
$853$	1.77091	1.77091	0.885456	+	0.464723i	$0.153846\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−2.43891	−2.43891
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−0.352493	−0.352493
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.77091	1.77091
$865$	0	0
$866$	−0.502979	−0.502979
$867$	−2.19783	−2.19783
$868$	0.119819	0.119819
$869$	0	0
$870$	0	0
$871$	1.94188	1.94188
$872$	0	0
$873$	3.78290	3.78290
$874$	0	0
$875$	0.241073	0.241073
$876$	0	0
$877$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$877$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$878$	−0.805754	−0.805754
$879$	−0.426920	−0.426920
$880$	−0.497021	−0.497021
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.42692	1.42692
$883$	1.49702	1.49702	0.748511	−	0.663123i	$-0.230769\pi$
$883$	1.49702	1.49702	0.748511	+	0.663123i	$0.230769\pi$
$884$	0.744052	0.744052
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0.360892	0.360892
$890$	0.170972	0.170972
$891$	2.77091	2.77091
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.0617023	0.0617023
$897$	−2.01199	−2.01199
$898$	−0.502979	−0.502979
$899$	0	0
$900$	−1.06170	−1.06170
$901$	2.90704	2.90704
$902$	0	0
$903$	−0.302776	−0.302776
$904$	0	0
$905$	0	0
$906$	0.890731	0.890731
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0.744052	0.744052
$909$	−1.51496	−1.51496
$910$	0.170972	0.170972
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0.119819	0.119819
$917$	−0.360892	−0.360892
$918$	2.13613	2.13613
$919$	−1.94188	−1.94188	−0.970942	−	0.239316i	$-0.923077\pi$
$919$	−1.94188	−1.94188	−0.970942	+	0.239316i	$0.923077\pi$
$920$	−1.20623	−1.20623
$921$	1.25595	1.25595
$922$	1.25595	1.25595
$923$	0	0
$924$	0.412045	0.412045
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.13613	−1.13613	−0.568065	−	0.822984i	$-0.692308\pi$
$929$	−1.13613	−1.13613	−0.568065	+	0.822984i	$0.692308\pi$
$930$	−1.25595	−1.25595
$931$	0	0
$932$	0	0
$933$	−2.01199	−2.01199
$934$	0	0
$935$	2.90704	2.90704
$936$	−2.26793	−2.26793
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0.332007	0.332007
$939$	2.01199	2.01199
$940$	−0.564681	−0.564681
$941$	−1.77091	−1.77091	−0.885456	−	0.464723i	$-0.846154\pi$
$941$	−1.77091	−1.77091	−0.885456	+	0.464723i	$0.846154\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.485036	−0.485036
$946$	−0.976726	−0.976726
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	2.65109	2.65109
$952$	0.383160	0.383160
$953$	−0.241073	−0.241073	−0.120537	−	0.992709i	$-0.538462\pi$
$953$	−0.241073	−0.241073	−0.120537	+	0.992709i	$0.538462\pi$
$954$	−2.94188	−2.94188
$955$	−0.709210	−0.709210
$956$	0.119819	0.119819
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−1.55872	−1.55872
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.880181	0.880181
$965$	1.13613	1.13613
$966$	−0.343992	−0.343992
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	2.94188	2.94188
$969$	0	0
$970$	−1.25595	−1.25595
$971$	1.13613	1.13613	0.568065	−	0.822984i	$-0.307692\pi$
$971$	1.13613	1.13613	0.568065	+	0.822984i	$0.307692\pi$
$972$	0.255948	0.255948
$973$	0	0
$974$	0	0
$975$	1.77091	1.77091
$976$	0	0
$977$	−1.49702	−1.49702	−0.748511	−	0.663123i	$-0.769231\pi$
$977$	−1.49702	−1.49702	−0.748511	+	0.663123i	$0.769231\pi$
$978$	−1.88018	−1.88018
$979$	−0.468136	−0.468136
$980$	0.468136	0.468136
$981$	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
$987$	−0.485036	−0.485036
$988$	0	0
$989$	−0.805754	−0.805754
$990$	−2.94188	−2.94188
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0.880181	0.880181
$993$	−2.65109	−2.65109
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−0.502979	−0.502979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.1.h.c.2014.3	yes	6
5.4	even	2		2015.1.h.d.2014.4	yes	6
13.12	even	2		2015.1.h.e.2014.4	yes	6
31.30	odd	2		2015.1.h.b.2014.3	✓	6
65.64	even	2		2015.1.h.b.2014.3	✓	6
155.154	odd	2		2015.1.h.e.2014.4	yes	6
403.402	odd	2		2015.1.h.d.2014.4	yes	6
2015.2014	odd	2	CM	2015.1.h.c.2014.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.1.h.b.2014.3	✓	6	31.30	odd	2
2015.1.h.b.2014.3	✓	6	65.64	even	2
2015.1.h.c.2014.3	yes	6	1.1	even	1	trivial
2015.1.h.c.2014.3	yes	6	2015.2014	odd	2	CM
2015.1.h.d.2014.4	yes	6	5.4	even	2
2015.1.h.d.2014.4	yes	6	403.402	odd	2
2015.1.h.e.2014.4	yes	6	13.12	even	2
2015.1.h.e.2014.4	yes	6	155.154	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)

\(f(q)\)	\(=\)	\(q-0.709210 q^{2} -1.77091 q^{3} -0.497021 q^{4} +1.00000 q^{5} +1.25595 q^{6} +0.241073 q^{7} +1.06170 q^{8} +2.13613 q^{9} +O(q^{10})\)	\(q-0.709210 q^{2} -1.77091 q^{3} -0.497021 q^{4} +1.00000 q^{5} +1.25595 q^{6} +0.241073 q^{7} +1.06170 q^{8} +2.13613 q^{9} -0.709210 q^{10} +1.94188 q^{11} +0.880181 q^{12} -1.00000 q^{13} -0.170972 q^{14} -1.77091 q^{15} -0.255948 q^{16} +1.49702 q^{17} -1.51496 q^{18} -0.497021 q^{20} -0.426920 q^{21} -1.37720 q^{22} -1.13613 q^{23} -1.88018 q^{24} +1.00000 q^{25} +0.709210 q^{26} -2.01199 q^{27} -0.119819 q^{28} +1.25595 q^{30} -1.00000 q^{31} -0.880181 q^{32} -3.43891 q^{33} -1.06170 q^{34} +0.241073 q^{35} -1.06170 q^{36} +1.77091 q^{39} +1.06170 q^{40} +0.302776 q^{42} +0.709210 q^{43} -0.965158 q^{44} +2.13613 q^{45} +0.805754 q^{46} +1.13613 q^{47} +0.453262 q^{48} -0.941884 q^{49} -0.709210 q^{50} -2.65109 q^{51} +0.497021 q^{52} +1.94188 q^{53} +1.42692 q^{54} +1.94188 q^{55} +0.255948 q^{56} +0.880181 q^{60} +0.709210 q^{62} +0.514964 q^{63} +0.880181 q^{64} -1.00000 q^{65} +2.43891 q^{66} -1.94188 q^{67} -0.744052 q^{68} +2.01199 q^{69} -0.170972 q^{70} +2.26793 q^{72} -1.77091 q^{75} +0.468136 q^{77} -1.25595 q^{78} -0.255948 q^{80} +1.42692 q^{81} +0.212188 q^{84} +1.49702 q^{85} -0.502979 q^{86} +2.06170 q^{88} -0.241073 q^{89} -1.51496 q^{90} -0.241073 q^{91} +0.564681 q^{92} +1.77091 q^{93} -0.805754 q^{94} +1.55872 q^{96} +1.77091 q^{97} +0.667993 q^{98} +4.14811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + q^{3} + 5 q^{4} + 6 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) 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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