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Properties

Label 1170.2.bs.g
Level $1170$
Weight $2$
Character orbit 1170.bs
Analytic conductor $9.342$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(361,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.bs (of order \(6\), degree \(2\), 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: no
Analytic conductor: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{4} + (\beta_{6} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{6} + 4 \beta_{4} + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{10} + 6 q^{11} - 2 q^{13} - 4 q^{16} - 6 q^{17} - 6 q^{19} + 6 q^{22} - 12 q^{23} - 8 q^{25} - 30 q^{37} + 12 q^{38} + 8 q^{40} - 12 q^{41} + 4 q^{43} + 8 q^{49} - 10 q^{52} - 60 q^{53} + 6 q^{55} + 24 q^{59} + 26 q^{61} - 18 q^{62} - 8 q^{64} - 6 q^{65} + 24 q^{67} + 6 q^{68} - 6 q^{74} - 6 q^{76} + 60 q^{77} + 20 q^{79} - 18 q^{85} - 6 q^{88} + 24 q^{89} - 66 q^{91} - 24 q^{92} - 6 q^{95} - 6 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.27597 + 0.609843i
1.40994 − 0.109843i
1.20036 + 0.747754i
0.665665 − 1.24775i
−1.27597 − 0.609843i
1.40994 + 0.109843i
1.20036 − 0.747754i
0.665665 + 1.24775i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i − 1.00000i 0 −1.24653 − 0.719687i 1.00000i 0 0.500000 + 0.866025i
361.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i − 1.00000i 0 1.24653 + 0.719687i 1.00000i 0 0.500000 + 0.866025i
361.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.00000i 0 −3.45632 − 1.99551i − 1.00000i 0 0.500000 + 0.866025i
361.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.00000i 0 3.45632 + 1.99551i − 1.00000i 0 0.500000 + 0.866025i
901.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −1.24653 + 0.719687i − 1.00000i 0 0.500000 − 0.866025i
901.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 1.24653 − 0.719687i − 1.00000i 0 0.500000 − 0.866025i
901.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.00000i 0 −3.45632 + 1.99551i 1.00000i 0 0.500000 − 0.866025i
901.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.00000i 0 3.45632 − 1.99551i 1.00000i 0 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bs.g 8
3.b odd 2 1 130.2.l.b ✓ 8
12.b even 2 1 1040.2.da.d 8
13.e even 6 1 inner 1170.2.bs.g 8
15.d odd 2 1 650.2.m.c 8
15.e even 4 1 650.2.n.d 8
15.e even 4 1 650.2.n.e 8
39.d odd 2 1 1690.2.l.j 8
39.f even 4 1 1690.2.e.s 8
39.f even 4 1 1690.2.e.t 8
39.h odd 6 1 130.2.l.b ✓ 8
39.h odd 6 1 1690.2.d.k 8
39.i odd 6 1 1690.2.d.k 8
39.i odd 6 1 1690.2.l.j 8
39.k even 12 1 1690.2.a.t 4
39.k even 12 1 1690.2.a.u 4
39.k even 12 1 1690.2.e.s 8
39.k even 12 1 1690.2.e.t 8
156.r even 6 1 1040.2.da.d 8
195.y odd 6 1 650.2.m.c 8
195.bf even 12 1 650.2.n.d 8
195.bf even 12 1 650.2.n.e 8
195.bh even 12 1 8450.2.a.ci 4
195.bh even 12 1 8450.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b ✓ 8 3.b odd 2 1
130.2.l.b ✓ 8 39.h odd 6 1
650.2.m.c 8 15.d odd 2 1
650.2.m.c 8 195.y odd 6 1
650.2.n.d 8 15.e even 4 1
650.2.n.d 8 195.bf even 12 1
650.2.n.e 8 15.e even 4 1
650.2.n.e 8 195.bf even 12 1
1040.2.da.d 8 12.b even 2 1
1040.2.da.d 8 156.r even 6 1
1170.2.bs.g 8 1.a even 1 1 trivial
1170.2.bs.g 8 13.e even 6 1 inner
1690.2.a.t 4 39.k even 12 1
1690.2.a.u 4 39.k even 12 1
1690.2.d.k 8 39.h odd 6 1
1690.2.d.k 8 39.i odd 6 1
1690.2.e.s 8 39.f even 4 1
1690.2.e.s 8 39.k even 12 1
1690.2.e.t 8 39.f even 4 1
1690.2.e.t 8 39.k even 12 1
1690.2.l.j 8 39.d odd 2 1
1690.2.l.j 8 39.i odd 6 1
8450.2.a.ci 4 195.bh even 12 1
8450.2.a.cm 4 195.bh even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{8} - 18T_{7}^{6} + 291T_{7}^{4} - 594T_{7}^{2} + 1089 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} - 6T_{11}^{6} + 108T_{11}^{5} + 135T_{11}^{4} - 1620T_{11}^{3} + 2538T_{11}^{2} + 810T_{11} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{8} + 96 T^{6} + \cdots + 4460544 \) Copy content Toggle raw display
$31$ \( T^{8} + 144 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{8} + 30 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( (T^{4} + 30 T^{3} + \cdots - 579)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + \cdots + 394384 \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + \cdots + 9437184 \) Copy content Toggle raw display
$71$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 264 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots - 188)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 480 T^{6} + \cdots + 4511376 \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{7} + \cdots + 42849 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} + \cdots + 18558864 \) Copy content Toggle raw display
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