↑

Properties

Label 1520.2.bq.q
Level $1520$
Weight $2$
Character orbit 1520.bq
Analytic conductor $12.137$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(31,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1520.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.bq (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 8x^{10} + 53x^{8} + 86x^{6} + 113x^{4} + 11x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{8} + \cdots + 2 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + (\beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - 3 \beta_{11} + 5 \beta_{10} + \cdots - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} - 4 q^{9} + 18 q^{13} + 18 q^{17} + 24 q^{21} - 6 q^{25} + 24 q^{29} - 18 q^{33} + 12 q^{41} + 8 q^{45} - 28 q^{49} - 18 q^{53} - 62 q^{57} + 26 q^{61} + 16 q^{73} + 56 q^{77} - 66 q^{81} + 18 q^{85} - 36 q^{89} - 20 q^{93} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8x^{10} + 53x^{8} + 86x^{6} + 113x^{4} + 11x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -61\nu^{11} - 650\nu^{9} - 4798\nu^{7} - 15860\nu^{5} - 33277\nu^{3} - 31874\nu ) / 5901 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -14\nu^{10} - 163\nu^{8} - 1115\nu^{6} - 3640\nu^{4} - 4445\nu^{2} - 3280 ) / 843 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -162\nu^{10} - 1565\nu^{8} - 10614\nu^{6} - 26384\nu^{4} - 31203\nu^{2} - 5840 ) / 5901 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 424\nu^{11} + 2809\nu^{9} + 17872\nu^{7} + 5989\nu^{5} + 583\nu^{3} - 60311\nu ) / 5901 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -583\nu^{10} - 4600\nu^{8} - 30475\nu^{6} - 47329\nu^{4} - 64975\nu^{2} - 424 ) / 5901 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -583\nu^{11} - 4600\nu^{9} - 30475\nu^{7} - 47329\nu^{5} - 64975\nu^{3} - 424\nu ) / 5901 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -620\nu^{10} - 5091\nu^{8} - 33482\nu^{6} - 56949\nu^{4} - 62813\nu^{2} - 3312 ) / 5901 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1847\nu^{10} - 14941\nu^{8} - 99230\nu^{6} - 167467\nu^{4} - 220139\nu^{2} - 24232 ) / 5901 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2952\nu^{11} + 23491\nu^{9} + 155382\nu^{7} + 246265\nu^{5} + 316812\nu^{3} - 893\nu ) / 5901 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6325\nu^{11} + 50017\nu^{9} + 330625\nu^{7} + 513475\nu^{5} + 667396\nu^{3} + 4600\nu ) / 5901 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 3\beta_{6} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - 6\beta_{7} - \beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} + \beta_{8} + 16\beta_{6} + 7\beta_{4} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{11} - 8\beta_{10} + 37\beta_{7} + \beta_{5} - 8\beta_{2} - 37\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{8} - 53\beta_{4} + 37\beta_{3} + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -53\beta_{11} + 106\beta_{10} + 45\beta_{5} - 53\beta_{2} + 231\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 231\beta_{9} - 106\beta_{8} - 595\beta_{6} + 53\beta_{4} - 231\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53\beta_{11} - 337\beta_{10} - 1447\beta_{7} - 337\beta_{5} + 674\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1447\beta_{9} + 337\beta_{8} + 3720\beta_{6} + 1784\beta_{4} - 3720 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1784\beta_{11} - 2121\beta_{10} + 9072\beta_{7} + 337\beta_{5} - 2121\beta_{2} - 9072\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.156347 + 0.270800i
0.638510 + 1.10593i
1.25214 + 2.16877i
−1.25214 − 2.16877i
−0.638510 − 1.10593i
−0.156347 − 0.270800i
0.156347 − 0.270800i
0.638510 − 1.10593i
1.25214 − 2.16877i
−1.25214 + 2.16877i
−0.638510 + 1.10593i
−0.156347 + 0.270800i
0 −1.59901 − 2.76957i 0 −0.500000 − 0.866025i 0 2.37033i 0 −3.61367 + 6.25906i 0
31.2 0 −0.391537 − 0.678161i 0 −0.500000 − 0.866025i 0 1.46162i 0 1.19340 − 2.06703i 0
31.3 0 −0.199658 − 0.345817i 0 −0.500000 − 0.866025i 0 − 4.49947i 0 1.42027 − 2.45999i 0
31.4 0 0.199658 + 0.345817i 0 −0.500000 − 0.866025i 0 4.49947i 0 1.42027 − 2.45999i 0
31.5 0 0.391537 + 0.678161i 0 −0.500000 − 0.866025i 0 − 1.46162i 0 1.19340 − 2.06703i 0
31.6 0 1.59901 + 2.76957i 0 −0.500000 − 0.866025i 0 − 2.37033i 0 −3.61367 + 6.25906i 0
1471.1 0 −1.59901 + 2.76957i 0 −0.500000 + 0.866025i 0 − 2.37033i 0 −3.61367 − 6.25906i 0
1471.2 0 −0.391537 + 0.678161i 0 −0.500000 + 0.866025i 0 − 1.46162i 0 1.19340 + 2.06703i 0
1471.3 0 −0.199658 + 0.345817i 0 −0.500000 + 0.866025i 0 4.49947i 0 1.42027 + 2.45999i 0
1471.4 0 0.199658 − 0.345817i 0 −0.500000 + 0.866025i 0 − 4.49947i 0 1.42027 + 2.45999i 0
1471.5 0 0.391537 − 0.678161i 0 −0.500000 + 0.866025i 0 1.46162i 0 1.19340 + 2.06703i 0
1471.6 0 1.59901 − 2.76957i 0 −0.500000 + 0.866025i 0 2.37033i 0 −3.61367 − 6.25906i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.bq.q ✓ 12
4.b odd 2 1 inner 1520.2.bq.q ✓ 12
19.d odd 6 1 inner 1520.2.bq.q ✓ 12
76.f even 6 1 inner 1520.2.bq.q ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.bq.q ✓ 12 1.a even 1 1 trivial
1520.2.bq.q ✓ 12 4.b odd 2 1 inner
1520.2.bq.q ✓ 12 19.d odd 6 1 inner
1520.2.bq.q ✓ 12 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{12} + 11T_{3}^{10} + 113T_{3}^{8} + 86T_{3}^{6} + 53T_{3}^{4} + 8T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 28T_{7}^{4} + 169T_{7}^{2} + 243 \) Copy content Toggle raw display
\( T_{11}^{6} + 37T_{11}^{4} + 40T_{11}^{2} + 3 \) Copy content Toggle raw display
\( T_{13}^{6} - 9T_{13}^{5} + 8T_{13}^{4} + 171T_{13}^{3} - 80T_{13}^{2} - 2793T_{13} + 7203 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 11 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 28 T^{4} + \cdots + 243)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 37 T^{4} + 40 T^{2} + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 9 T^{5} + \cdots + 7203)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 2 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 22 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( (T^{6} - 12 T^{5} + \cdots + 11907)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 133 T^{4} + \cdots - 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 182 T^{4} + \cdots + 128547)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 6 T^{5} + \cdots + 243)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 105445026729 \) Copy content Toggle raw display
$53$ \( (T^{6} + 9 T^{5} + 29 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 32015587041 \) Copy content Toggle raw display
$61$ \( (T^{6} - 13 T^{5} + \cdots + 1560001)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 796594176 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 25344958401 \) Copy content Toggle raw display
$73$ \( (T^{6} - 8 T^{5} + \cdots + 194481)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 767544201216 \) Copy content Toggle raw display
$83$ \( (T^{6} + 303 T^{4} + \cdots + 615627)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 12)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} - 21 T^{5} + 167 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
show more
show less