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Properties

Label 760.2.v.a
Level $760$
Weight $2$
Character orbit 760.v
Analytic conductor $6.069$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(113,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.v (of order \(4\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} + ( - 2 \beta_{5} - 1) q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (2 \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} + ( - 2 \beta_{5} - 1) q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (2 \beta_{5} + \beta_{4}) q^{9} + ( - \beta_{11} - \beta_{6}) q^{11} + ( - \beta_{10} - \beta_1) q^{13} + ( - \beta_{10} - 2 \beta_{2}) q^{15} + ( - \beta_{11} - \beta_{6} + \beta_{4} + 1) q^{17} + (\beta_{9} - \beta_{5} - \beta_{4}) q^{19} + ( - \beta_{10} + \beta_{9} + \cdots + 3 \beta_1) q^{21}+ \cdots + (2 \beta_{9} - 2 \beta_{8} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 4 q^{7} - 8 q^{11} + 4 q^{17} + 4 q^{23} - 36 q^{25} - 4 q^{35} - 4 q^{43} + 56 q^{45} - 4 q^{47} + 8 q^{55} + 16 q^{57} - 64 q^{61} - 28 q^{63} + 52 q^{73} + 40 q^{77} - 12 q^{81} + 4 q^{83} + 4 q^{85} - 80 q^{87} - 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} - 2\nu^{9} - 9\nu^{7} + 6\nu^{5} - 32\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} + 2\nu^{7} - \nu^{5} - 6\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 3\nu^{7} + 24\nu^{5} - 28\nu^{3} + 112\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 14\nu^{8} - 9\nu^{6} + 22\nu^{4} + 64\nu^{2} - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 34\nu^{4} - 64\nu^{2} + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 2\nu^{9} + 4\nu^{7} - 3\nu^{6} - 18\nu^{5} + 8\nu^{4} + 20\nu^{3} - 12\nu^{2} - 32\nu + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{11} + 14\nu^{9} - 29\nu^{7} + 54\nu^{5} - 48\nu^{3} + 32\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{10} + 8 \nu^{9} + 18 \nu^{8} - 16 \nu^{7} - 43 \nu^{6} + 72 \nu^{5} + 106 \nu^{4} + \cdots + 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} - 2 \nu^{9} - 18 \nu^{8} + 11 \nu^{7} + 43 \nu^{6} - 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} + 5\nu^{9} - 15\nu^{7} + 29\nu^{5} - 46\nu^{3} + 40\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{10} + 2 \nu^{9} + 8 \nu^{8} - 4 \nu^{7} - 23 \nu^{6} + 18 \nu^{5} + 48 \nu^{4} - 20 \nu^{3} + \cdots + 64 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + \beta_{3} - \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{3} - 3\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 11\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{10} + 5\beta_{9} + 5\beta_{8} + 6\beta_{7} + 3\beta_{3} + 5\beta_{2} + 7\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{11} - 9 \beta_{9} + \beta_{8} - 9 \beta_{7} - 9 \beta_{6} - 19 \beta_{5} + 3 \beta_{4} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{10} + 3\beta_{9} + 3\beta_{8} + 10\beta_{7} + 3\beta_{3} + 15\beta_{2} - 11\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11 \beta_{11} - 7 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} - 9 \beta_{6} + 11 \beta_{5} + 15 \beta_{4} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4\beta_{10} + 3\beta_{9} + 3\beta_{8} + 2\beta_{7} - 11\beta_{3} - 13\beta_{2} - 39\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15 \beta_{11} + 25 \beta_{9} + 31 \beta_{8} + 25 \beta_{7} + 41 \beta_{6} + 19 \beta_{5} + 13 \beta_{4} + \cdots - 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22\beta_{10} + 29\beta_{9} + 29\beta_{8} - 58\beta_{7} - 19\beta_{3} - 47\beta_{2} - 5\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.892524 − 1.09700i
−1.16947 − 0.795191i
−1.35489 − 0.405301i
1.35489 + 0.405301i
1.16947 + 0.795191i
−0.892524 + 1.09700i
0.892524 + 1.09700i
−1.16947 + 0.795191i
−1.35489 + 0.405301i
1.35489 − 0.405301i
1.16947 − 0.795191i
−0.892524 − 1.09700i
0 −2.19399 + 2.19399i 0 −1.00000 + 2.00000i 0 2.10278 − 2.10278i 0 − 6.62721i 0
113.2 0 −1.59038 + 1.59038i 0 −1.00000 + 2.00000i 0 −3.24914 + 3.24914i 0 − 2.05863i 0
113.3 0 −0.810603 + 0.810603i 0 −1.00000 + 2.00000i 0 0.146365 − 0.146365i 0 1.68585i 0
113.4 0 0.810603 − 0.810603i 0 −1.00000 + 2.00000i 0 0.146365 − 0.146365i 0 1.68585i 0
113.5 0 1.59038 − 1.59038i 0 −1.00000 + 2.00000i 0 −3.24914 + 3.24914i 0 − 2.05863i 0
113.6 0 2.19399 − 2.19399i 0 −1.00000 + 2.00000i 0 2.10278 − 2.10278i 0 − 6.62721i 0
417.1 0 −2.19399 − 2.19399i 0 −1.00000 − 2.00000i 0 2.10278 + 2.10278i 0 6.62721i 0
417.2 0 −1.59038 − 1.59038i 0 −1.00000 − 2.00000i 0 −3.24914 − 3.24914i 0 2.05863i 0
417.3 0 −0.810603 − 0.810603i 0 −1.00000 − 2.00000i 0 0.146365 + 0.146365i 0 − 1.68585i 0
417.4 0 0.810603 + 0.810603i 0 −1.00000 − 2.00000i 0 0.146365 + 0.146365i 0 − 1.68585i 0
417.5 0 1.59038 + 1.59038i 0 −1.00000 − 2.00000i 0 −3.24914 − 3.24914i 0 2.05863i 0
417.6 0 2.19399 + 2.19399i 0 −1.00000 − 2.00000i 0 2.10278 + 2.10278i 0 6.62721i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.v.a ✓ 12
5.c odd 4 1 inner 760.2.v.a ✓ 12
19.b odd 2 1 inner 760.2.v.a ✓ 12
95.g even 4 1 inner 760.2.v.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.v.a ✓ 12 1.a even 1 1 trivial
760.2.v.a ✓ 12 5.c odd 4 1 inner
760.2.v.a ✓ 12 19.b odd 2 1 inner
760.2.v.a ✓ 12 95.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 120T_{3}^{8} + 2576T_{3}^{4} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 120 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 16 T - 16)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + 776 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 42 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 160 T^{4} + \cdots - 147968)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 104 T^{4} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 236421376 \) Copy content Toggle raw display
$41$ \( (T^{6} + 136 T^{4} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 7269949696 \) Copy content Toggle raw display
$59$ \( (T^{6} - 256 T^{4} + \cdots - 524288)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 16 T^{2} + \cdots - 736)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3584872677376 \) Copy content Toggle raw display
$71$ \( (T^{6} + 72 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 26 T^{5} + \cdots + 4021448)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 136 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 2 T^{5} + \cdots + 42632)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 296 T^{4} + \cdots - 476288)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 24584 T^{8} + \cdots + 3748096 \) Copy content Toggle raw display
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