LMFDB - Newform orbit 465.2.bm.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [465,2,Mod(44,465)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("465.44"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([15, 15, 11])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.bm (of order $30$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,-3,3,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.71304369399$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{15}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{15}^{6} - \zeta_{15}^{5} + \cdots - 1) q^{2} + (2 \zeta_{15}^{7} + \zeta_{15}^{2}) q^{3} + (\zeta_{15}^{7} + \zeta_{15}^{5}) q^{4} + (\zeta_{15}^{5} - 2 \zeta_{15}^{4} + \cdots + 1) q^{5}+ \cdots + (7 \zeta_{15}^{3} + 7 \zeta_{15}^{2}) q^{98}+O(q^{100}) $ q + (-z^6 - z^5 - z - 1) * q^2 + (2z^7 + z^2) q^3 + (z^7 + z^5) * q^4 + (z^5 - 2z^4 - 2z + 1) * q^5 + (-2z^7 + z^5 - z^4 + 2z^3 + z^2 - z + 1) * q^6 + (z^3 - z^2 - z + 1) * q^8 - 3z^4 q^9 + (4z^7 - z^6 - z^5 + 2z^4 - 2z^3 + 4z^2 + 2z - 4) q^10 + (-2z^7 + z^6 - 2z^4 + z^3 - 3z^2 + 1) q^12 + (-3z^7 + 2z^6 + 4z^5 - 4z^4 + 2z^3 - z^2 + 4) q^15 + (3z^7 + 3z^6 - 3z^5 + z^4 + z^3 + z^2 + 4z - 2) * q^16 + (-5z^7 + 3z^6 + z^4 + z^3 - 3z^2 - 2z + 1) * q^17 + (3z^7 - 3z^6 + 3z^4 - 3z^3 + 3z^2 - 6) q^18 + (6z^7 - z^6 - 3z^5 + 2z^4 - 4z^3 + 2z^2 + 4z - 3) * q^19 + (-4z^7 + 2z^6 + 2z^5 - 2z^4 + 4z^3 - 3z^2 + 3) * q^20 + (6z^7 + 4z^6 + 2z^4 - 3z^3 + 4z^2 + 2z + 1) * q^23 + (-2z^7 + 2z^6 + z^5 - 3z^4 + 3z^3 - z^2 - 2z + 2) q^24 + 5z^5 q^25 + (3z^6 + 6z) * q^27 + (2z^7 - 6z^6 - 3z^5 + z^4 - 5z^3 + 2z^2 + z - 5) q^30 + (2z^7 - 2z^6 + 2z^5 + 2z^4 + 3z^3 - 2z) * q^31 + (-7z^7 + 2z^5 - 2z^4 + 5z^3 - 3z^2 - 2z + 4) * q^32 + (5z^7 + 3z^6 - 3z^5 - 3z^3 + z^2 + 6z) q^34 + (-3z^7 + 6z^6 + 3z^3 - 3z^2 + 3z + 3) q^36 + (-2z^7 - 3z^6 + z^5 + 4z^4 + 4z^3 + z^2 - 3z - 2) q^38 + (-2z^7 + z^6 + 2z^5 - 3z^4 + 2z^3 + z^2 - 2z) q^40 + (3z^7 + 3z^6 + 3z^4 - 3z^3 - 3z^2 + 6z - 3) * q^45 + (-6z^7 - 4z^6 + 5z^3 + 5z^2 + 1) * q^46 + (5z^7 - 5z^6 + 3z^5 + 5z^4 - 6z^3 + 4z^2 + 2z - 1) q^47 + (3z^7 - 6z^5 - 6z^3 + 3z^2 + 3z - 6) q^48 + 7z^7 q^49 + (5z + 5) q^50 + (-6z^7 + 6z^5 - 9z + 6) q^51 + (7z^7 - 8z^6 - 4z^5 + 2z^4 + 8z^2 - 3z - 4) * q^53 + (-6z^7 - 6z^6 - 3z^2 - 3z) * q^54 + (4z^7 - 5z^5 - z^4 - z^3 + z^2 + 5z + 1) q^57 + (4z^6 - 2z^5 + 3z^4 - 3z^2 + 2z - 4) q^60 + (4z^7 - 12z^6 - 7z^5 + 3z^4 - 11z^3 + 4z^2 + 3z - 11) q^61 + (-6z^7 + 7z^6 + 3z^5 - 8z^4 + 7z^3 - 3z^2 - z + 12) * q^62 + (z^6 - 4z^5 - z^4 - z^2 - 4z + 1) * q^64 + (2z^7 - 9z^6 + 4z^5 + z^4 - 2z^3 + 8z^2 - 8z) * q^68 + (4z^7 - 4z^6 + 3z^5 - 8z^4 - 8z^3 + 3z^2 - 4z + 4) q^69 + (-3z^7 + 3z^6 + 3z^5 - 3z^4) * q^72 + (-5z^7 - 10z^2) * q^75 + (-5z^7 + 5z^6 + 3z^5 - 5z^4 + 2z^3 - 5z^2 + 8) * q^76 + (-7z^7 - 6z^5 + 4z^4 - 2z^3 - 11z^2 - 7z - 2) * q^79 + (-7z^7 + 7z^6 + z^5 - 4z^4 - z^3 - 3z^2 + 4z + 7) q^80 + (9z^7 - 9z^5 + 9z^4 - 9z^3 + 9z - 9) q^81 + (-6z^7 + 3z^6 - 5z^4 + 11z^3 - 8z + 3) q^83 + (-z^7 - 7z^6 + 6z^4 - 3z^3 + 7z^2 - 8z - 3) q^85 + (3z^7 - 3z^6 - 3z^5 + 6z^4 + 3z^3 + 3z^2 + 6z) q^90 + (-6z^7 - 4z^6 + 9z^5 - 11z^4 + z^3 - 6z^2 - 11z + 8) * q^92 + (-6z^7 - z^5 - 6z^4 + 6z^3 - 6z^2 - 6z - 2) q^93 + (-3z^7 - 2z^6 - z^5 + 3z^4 + 4z^3 - 5z^2 - z + 6) q^94 + (10z^6 + 3z^5 - 4z^4 + 4z^3 - 4z^2 + 3z + 10) * q^95 + (3z^7 + z^6 - z^5 + 7z^4 + z^3 + z^2 - 7) * q^96 + (7z^3 + 7z^2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{6} + 4 q^{8} - 3 q^{9} - 10 q^{10} - 3 q^{12} - 3 q^{16} - 9 q^{17} - 27 q^{18} + 12 q^{19} - 5 q^{20} + 20 q^{23} - 6 q^{24} - 20 q^{25} - 8 q^{31} + 24 q^{34}+ \cdots - 7 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^6 + 4 * q^8 - 3 * q^9 - 10 * q^10 - 3 * q^12 - 3 * q^16 - 9 * q^17 - 27 * q^18 + 12 * q^19 - 5 * q^20 + 20 * q^23 - 6 * q^24 - 20 * q^25 - 8 * q^31 + 24 * q^34 + 3 * q^36 - 22 * q^38 - 20 * q^40 - 15 * q^45 + 5 * q^46 + 18 * q^47 - 3 * q^48 + 7 * q^49 + 45 * q^50 + 9 * q^51 + 14 * q^53 + 39 * q^57 - 30 * q^60 + 38 * q^62 + 16 * q^64 + 9 * q^68 + 39 * q^69 - 24 * q^72 - 15 * q^75 + 23 * q^76 - 9 * q^79 + 30 * q^80 + 9 * q^81 - 23 * q^83 + 30 * q^90 - 48 * q^93 + 42 * q^94 + 35 * q^95 - 45 * q^96 - 7 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/465\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$406$
$\chi(n)$	$-1$	$-1$	$-\zeta_{15}^{4}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

44.1

−0.978148	+	0.207912i
−0.978148	−	0.207912i
0.913545	+	0.406737i
−0.104528	−	0.994522i
−0.104528	+	0.994522i
0.669131	−	0.743145i
0.669131	+	0.743145i
0.913545	−	0.406737i

0.169131

−

0.122881i

0.704489

1.58231i

−0.604528

1.86055i

1.11803

1.93649i

0.313585

0.181049i

0.255585

0.786610i

−2.00739

2.22943i

0.427051

0.190135i

74.1

0.169131

0.122881i

0.704489

−

1.58231i

−0.604528

−

1.86055i

1.11803

−

1.93649i

0.313585

−

0.181049i

0.255585

−

0.786610i

−2.00739

−

2.22943i

0.427051

−

0.190135i

104.1

−0.604528

−

1.86055i

−1.28716

1.15897i

−1.47815

1.07394i

−1.11803

−

1.93649i

2.93444

1.69420i

−0.273659

−

0.198825i

0.313585

−

2.98357i

−2.92705

3.25082i

179.1

0.413545

1.27276i

0.360114

1.69420i

0.169131

−

0.122881i

−1.11803

1.93649i

−2.00739

1.15897i

2.39169

1.73767i

−2.74064

1.22021i

−2.92705

−

0.622164i

239.1

0.413545

−

1.27276i

0.360114

−

1.69420i

0.169131

0.122881i

−1.11803

−

1.93649i

−2.00739

−

1.15897i

2.39169

−

1.73767i

−2.74064

−

1.22021i

−2.92705

0.622164i

269.1

−1.47815

−

1.07394i

1.72256

−

0.181049i

0.413545

1.27276i

1.11803

1.93649i

−2.74064

−

1.58231i

−0.373619

1.14988i

2.93444

−

0.623735i

0.427051

−

4.06312i

344.1

−1.47815

1.07394i

1.72256

0.181049i

0.413545

−

1.27276i

1.11803

−

1.93649i

−2.74064

1.58231i

−0.373619

−

1.14988i

2.93444

0.623735i

0.427051

4.06312i

389.1

−0.604528

1.86055i

−1.28716

−

1.15897i

−1.47815

−

1.07394i

−1.11803

1.93649i

2.93444

−

1.69420i

−0.273659

0.198825i

0.313585

2.98357i

−2.92705

−

3.25082i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
31.h	odd	30	1	inner
465.bm	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	465.2.bm.a	✓	8
3.b	odd	2	1		465.2.bm.c	yes	8
5.b	even	2	1		465.2.bm.c	yes	8
15.d	odd	2	1	CM	465.2.bm.a	✓	8
31.h	odd	30	1	inner	465.2.bm.a	✓	8
93.p	even	30	1		465.2.bm.c	yes	8
155.v	odd	30	1		465.2.bm.c	yes	8
465.bm	even	30	1	inner	465.2.bm.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
465.2.bm.a	✓	8	1.a	even	1	1	trivial
465.2.bm.a	✓	8	15.d	odd	2	1	CM
465.2.bm.a	✓	8	31.h	odd	30	1	inner
465.2.bm.a	✓	8	465.bm	even	30	1	inner
465.2.bm.c	yes	8	3.b	odd	2	1
465.2.bm.c	yes	8	5.b	even	2	1
465.2.bm.c	yes	8	93.p	even	30	1
465.2.bm.c	yes	8	155.v	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 3T_{2}^{7} + 8T_{2}^{6} + 11T_{2}^{5} + 15T_{2}^{4} + 11T_{2}^{3} + 18T_{2}^{2} - 7T_{2} + 1 $ T2^8 + 3*T2^7 + 8*T2^6 + 11*T2^5 + 15*T2^4 + 11*T2^3 + 18*T2^2 - 7*T2 + 1 acting on $S_{2}^{\mathrm{new}}(465, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + 8T^6 + 11T^5 + 15T^4 + 11T^3 + 18T^2 - 7*T + 1
$3$	$ T^{8} - 3 T^{7} + \cdots + 81 $ T^8 - 3T^7 + 6T^6 - 9T^5 + 9T^4 - 27T^3 + 54T^2 - 81*T + 81
$5$	$ (T^{4} + 5 T^{2} + 25)^{2} $ (T^4 + 5*T^2 + 25)^2
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ T^{8} + 9 T^{7} + \cdots + 81 $ T^8 + 9T^7 + 84T^6 + 513T^5 + 3429T^4 + 3591T^3 + 1836T^2 + 567*T + 81
$19$	$ T^{8} - 12 T^{7} + \cdots + 841 $ T^8 - 12T^7 + 20T^6 + 248T^5 - 261T^4 - 478T^3 + 3315T^2 - 493*T + 841
$23$	$ T^{8} - 20 T^{7} + \cdots + 1985281 $ T^8 - 20T^7 + 103T^6 - 240T^5 + 8854T^4 - 77620T^3 + 93492T^2 + 338160*T + 1985281
$29$	$ T^{8} $ T^8
$31$	$ T^{8} + 8 T^{7} + \cdots + 923521 $ T^8 + 8T^7 + 33T^6 + 16T^5 - 895T^4 + 496T^3 + 31713T^2 + 238328*T + 923521
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ T^{8} - 18 T^{7} + \cdots + 259081 $ T^8 - 18T^7 + 263T^6 - 1991T^5 + 9690T^4 - 31211T^3 + 136383T^2 - 278423*T + 259081
$53$	$ T^{8} - 14 T^{7} + \cdots + 841 $ T^8 - 14T^7 + 314T^6 - 3878T^5 + 42309T^4 - 247386T^3 + 1041351T^2 - 58667*T + 841
$59$	$ T^{8} $ T^8
$61$	$ T^{8} + 607 T^{6} + \cdots + 1042441 $ T^8 + 607T^6 + 119894T^4 + 7643528*T^2 + 1042441
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} $ T^8
$79$	$ T^{8} + \cdots + 1458552481 $ T^8 + 9T^7 + 264T^6 - 2212T^5 + 7234T^4 - 769539T^3 + 1171396T^2 - 97692578*T + 1458552481
$83$	$ T^{8} + 23 T^{7} + \cdots + 73441 $ T^8 + 23T^7 + 446T^6 + 4724T^5 - 681T^4 - 105228T^3 + 699474T^2 - 192139*T + 73441
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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Level:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.bm (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{15}^{6} - \zeta_{15}^{5} + \cdots - 1) q^{2} + (2 \zeta_{15}^{7} + \zeta_{15}^{2}) q^{3} + (\zeta_{15}^{7} + \zeta_{15}^{5}) q^{4} + (\zeta_{15}^{5} - 2 \zeta_{15}^{4} + \cdots + 1) q^{5}+ \cdots + (7 \zeta_{15}^{3} + 7 \zeta_{15}^{2}) q^{98}+O(q^{100}) \) q + (-z^6 - z^5 - z - 1) * q^2 + (2z^7 + z^2) q^3 + (z^7 + z^5) * q^4 + (z^5 - 2z^4 - 2z + 1) * q^5 + (-2z^7 + z^5 - z^4 + 2z^3 + z^2 - z + 1) * q^6 + (z^3 - z^2 - z + 1) * q^8 - 3z^4 q^9 + (4z^7 - z^6 - z^5 + 2z^4 - 2z^3 + 4z^2 + 2z - 4) q^10 + (-2z^7 + z^6 - 2z^4 + z^3 - 3z^2 + 1) q^12 + (-3z^7 + 2z^6 + 4z^5 - 4z^4 + 2z^3 - z^2 + 4) q^15 + (3z^7 + 3z^6 - 3z^5 + z^4 + z^3 + z^2 + 4z - 2) * q^16 + (-5z^7 + 3z^6 + z^4 + z^3 - 3z^2 - 2z + 1) * q^17 + (3z^7 - 3z^6 + 3z^4 - 3z^3 + 3z^2 - 6) q^18 + (6z^7 - z^6 - 3z^5 + 2z^4 - 4z^3 + 2z^2 + 4z - 3) * q^19 + (-4z^7 + 2z^6 + 2z^5 - 2z^4 + 4z^3 - 3z^2 + 3) * q^20 + (6z^7 + 4z^6 + 2z^4 - 3z^3 + 4z^2 + 2z + 1) * q^23 + (-2z^7 + 2z^6 + z^5 - 3z^4 + 3z^3 - z^2 - 2z + 2) q^24 + 5z^5 q^25 + (3z^6 + 6z) * q^27 + (2z^7 - 6z^6 - 3z^5 + z^4 - 5z^3 + 2z^2 + z - 5) q^30 + (2z^7 - 2z^6 + 2z^5 + 2z^4 + 3z^3 - 2z) * q^31 + (-7z^7 + 2z^5 - 2z^4 + 5z^3 - 3z^2 - 2z + 4) * q^32 + (5z^7 + 3z^6 - 3z^5 - 3z^3 + z^2 + 6z) q^34 + (-3z^7 + 6z^6 + 3z^3 - 3z^2 + 3z + 3) q^36 + (-2z^7 - 3z^6 + z^5 + 4z^4 + 4z^3 + z^2 - 3z - 2) q^38 + (-2z^7 + z^6 + 2z^5 - 3z^4 + 2z^3 + z^2 - 2z) q^40 + (3z^7 + 3z^6 + 3z^4 - 3z^3 - 3z^2 + 6z - 3) * q^45 + (-6z^7 - 4z^6 + 5z^3 + 5z^2 + 1) * q^46 + (5z^7 - 5z^6 + 3z^5 + 5z^4 - 6z^3 + 4z^2 + 2z - 1) q^47 + (3z^7 - 6z^5 - 6z^3 + 3z^2 + 3z - 6) q^48 + 7z^7 q^49 + (5z + 5) q^50 + (-6z^7 + 6z^5 - 9z + 6) q^51 + (7z^7 - 8z^6 - 4z^5 + 2z^4 + 8z^2 - 3z - 4) * q^53 + (-6z^7 - 6z^6 - 3z^2 - 3z) * q^54 + (4z^7 - 5z^5 - z^4 - z^3 + z^2 + 5z + 1) q^57 + (4z^6 - 2z^5 + 3z^4 - 3z^2 + 2z - 4) q^60 + (4z^7 - 12z^6 - 7z^5 + 3z^4 - 11z^3 + 4z^2 + 3z - 11) q^61 + (-6z^7 + 7z^6 + 3z^5 - 8z^4 + 7z^3 - 3z^2 - z + 12) * q^62 + (z^6 - 4z^5 - z^4 - z^2 - 4z + 1) * q^64 + (2z^7 - 9z^6 + 4z^5 + z^4 - 2z^3 + 8z^2 - 8z) * q^68 + (4z^7 - 4z^6 + 3z^5 - 8z^4 - 8z^3 + 3z^2 - 4z + 4) q^69 + (-3z^7 + 3z^6 + 3z^5 - 3z^4) * q^72 + (-5z^7 - 10z^2) * q^75 + (-5z^7 + 5z^6 + 3z^5 - 5z^4 + 2z^3 - 5z^2 + 8) * q^76 + (-7z^7 - 6z^5 + 4z^4 - 2z^3 - 11z^2 - 7z - 2) * q^79 + (-7z^7 + 7z^6 + z^5 - 4z^4 - z^3 - 3z^2 + 4z + 7) q^80 + (9z^7 - 9z^5 + 9z^4 - 9z^3 + 9z - 9) q^81 + (-6z^7 + 3z^6 - 5z^4 + 11z^3 - 8z + 3) q^83 + (-z^7 - 7z^6 + 6z^4 - 3z^3 + 7z^2 - 8z - 3) q^85 + (3z^7 - 3z^6 - 3z^5 + 6z^4 + 3z^3 + 3z^2 + 6z) q^90 + (-6z^7 - 4z^6 + 9z^5 - 11z^4 + z^3 - 6z^2 - 11z + 8) * q^92 + (-6z^7 - z^5 - 6z^4 + 6z^3 - 6z^2 - 6z - 2) q^93 + (-3z^7 - 2z^6 - z^5 + 3z^4 + 4z^3 - 5z^2 - z + 6) q^94 + (10z^6 + 3z^5 - 4z^4 + 4z^3 - 4z^2 + 3z + 10) * q^95 + (3z^7 + z^6 - z^5 + 7z^4 + z^3 + z^2 - 7) * q^96 + (7z^3 + 7z^2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{6} + 4 q^{8} - 3 q^{9} - 10 q^{10} - 3 q^{12} - 3 q^{16} - 9 q^{17} - 27 q^{18} + 12 q^{19} - 5 q^{20} + 20 q^{23} - 6 q^{24} - 20 q^{25} - 8 q^{31} + 24 q^{34}+ \cdots - 7 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^6 + 4 * q^8 - 3 * q^9 - 10 * q^10 - 3 * q^12 - 3 * q^16 - 9 * q^17 - 27 * q^18 + 12 * q^19 - 5 * q^20 + 20 * q^23 - 6 * q^24 - 20 * q^25 - 8 * q^31 + 24 * q^34 + 3 * q^36 - 22 * q^38 - 20 * q^40 - 15 * q^45 + 5 * q^46 + 18 * q^47 - 3 * q^48 + 7 * q^49 + 45 * q^50 + 9 * q^51 + 14 * q^53 + 39 * q^57 - 30 * q^60 + 38 * q^62 + 16 * q^64 + 9 * q^68 + 39 * q^69 - 24 * q^72 - 15 * q^75 + 23 * q^76 - 9 * q^79 + 30 * q^80 + 9 * q^81 - 23 * q^83 + 30 * q^90 - 48 * q^93 + 42 * q^94 + 35 * q^95 - 45 * q^96 - 7 * q^98

Introduction

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Newspace parameters

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Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	465.2.bm.a

Level	$465$
Weight	$2$
Character orbit	465.bm
Analytic conductor	$3.713$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-15
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.71304369399\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

\(n\)	\(187\)	\(311\)	\(406\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{15}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 44.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + 8T^6 + 11T^5 + 15T^4 + 11T^3 + 18T^2 - 7*T + 1
$3$	\( T^{8} - 3 T^{7} + \cdots + 81 \) T^8 - 3T^7 + 6T^6 - 9T^5 + 9T^4 - 27T^3 + 54T^2 - 81*T + 81
$5$	\( (T^{4} + 5 T^{2} + 25)^{2} \) (T^4 + 5*T^2 + 25)^2
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} \) T^8
$17$	\( T^{8} + 9 T^{7} + \cdots + 81 \) T^8 + 9T^7 + 84T^6 + 513T^5 + 3429T^4 + 3591T^3 + 1836T^2 + 567*T + 81
$19$	\( T^{8} - 12 T^{7} + \cdots + 841 \) T^8 - 12T^7 + 20T^6 + 248T^5 - 261T^4 - 478T^3 + 3315T^2 - 493*T + 841
$23$	\( T^{8} - 20 T^{7} + \cdots + 1985281 \) T^8 - 20T^7 + 103T^6 - 240T^5 + 8854T^4 - 77620T^3 + 93492T^2 + 338160*T + 1985281
$29$	\( T^{8} \) T^8
$31$	\( T^{8} + 8 T^{7} + \cdots + 923521 \) T^8 + 8T^7 + 33T^6 + 16T^5 - 895T^4 + 496T^3 + 31713T^2 + 238328*T + 923521
$37$	\( T^{8} \) T^8
$41$	\( T^{8} \) T^8
$43$	\( T^{8} \) T^8
$47$	\( T^{8} - 18 T^{7} + \cdots + 259081 \) T^8 - 18T^7 + 263T^6 - 1991T^5 + 9690T^4 - 31211T^3 + 136383T^2 - 278423*T + 259081
$53$	\( T^{8} - 14 T^{7} + \cdots + 841 \) T^8 - 14T^7 + 314T^6 - 3878T^5 + 42309T^4 - 247386T^3 + 1041351T^2 - 58667*T + 841
$59$	\( T^{8} \) T^8
$61$	\( T^{8} + 607 T^{6} + \cdots + 1042441 \) T^8 + 607T^6 + 119894T^4 + 7643528*T^2 + 1042441
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( T^{8} \) T^8
$79$	\( T^{8} + \cdots + 1458552481 \) T^8 + 9T^7 + 264T^6 - 2212T^5 + 7234T^4 - 769539T^3 + 1171396T^2 - 97692578*T + 1458552481
$83$	\( T^{8} + 23 T^{7} + \cdots + 73441 \) T^8 + 23T^7 + 446T^6 + 4724T^5 - 681T^4 - 105228T^3 + 699474T^2 - 192139*T + 73441
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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