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Properties

Label 1520.2.j.f
Level $1520$
Weight $2$
Character orbit 1520.j
Analytic conductor $12.137$
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] = [1520,2,Mod(911,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.j (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: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 159x^{8} - 822x^{6} + 3096x^{4} - 5544x^{2} + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + \beta_{11} q^{7} + (\beta_{10} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + \beta_{11} q^{7} + (\beta_{10} + 2) q^{9} - \beta_{9} q^{13} + \beta_1 q^{15} + ( - \beta_{10} - 1) q^{17} + (\beta_{11} - \beta_{7} + \beta_{2}) q^{19} - \beta_{8} q^{21} + \beta_{11} q^{23} + q^{25} + ( - \beta_{7} + \beta_{4} + \cdots - 2 \beta_1) q^{27}+ \cdots + (\beta_{9} - \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} + 24 q^{9} - 12 q^{17} + 12 q^{25} - 24 q^{45} - 24 q^{49} + 12 q^{57} + 24 q^{61} + 60 q^{73} + 84 q^{81} + 12 q^{85} + 96 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 159x^{8} - 822x^{6} + 3096x^{4} - 5544x^{2} + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 187\nu^{11} - 467\nu^{9} - 353\nu^{7} + 122772\nu^{5} - 428628\nu^{3} + 1291080\nu ) / 477288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 233 \nu^{11} + 583 \nu^{10} + 3985 \nu^{9} - 7695 \nu^{8} - 42241 \nu^{7} + 81567 \nu^{6} + \cdots - 1889496 ) / 954576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 583\nu^{11} - 7695\nu^{9} + 81567\nu^{7} - 361248\nu^{5} + 1110960\nu^{3} - 934920\nu ) / 477288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 583 \nu^{11} + 583 \nu^{10} + 7695 \nu^{9} - 7695 \nu^{8} - 81567 \nu^{7} + 81567 \nu^{6} + \cdots - 1889496 ) / 954576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{10} - 265\nu^{8} + 2809\nu^{6} - 5160\nu^{4} + 9240\nu^{2} + 129336 ) / 22728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 285\nu^{10} - 6809\nu^{8} + 61569\nu^{6} - 342924\nu^{4} + 673536\nu^{2} - 1021104 ) / 238644 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 583\nu^{10} - 7695\nu^{8} + 81567\nu^{6} - 361248\nu^{4} + 1588248\nu^{2} - 1889496 ) / 477288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 583\nu^{11} - 7695\nu^{9} + 81567\nu^{7} - 361248\nu^{5} + 1349604\nu^{3} - 934920\nu ) / 238644 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -781\nu^{11} + 11309\nu^{9} - 109269\nu^{7} + 483936\nu^{5} - 1284144\nu^{3} + 1252440\nu ) / 318192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -35\nu^{10} + 371\nu^{8} - 3175\nu^{6} + 7224\nu^{4} - 12936\nu^{2} - 15156 ) / 11364 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 185\nu^{10} - 2908\nu^{8} + 29499\nu^{6} - 148983\nu^{4} + 449070\nu^{2} - 589554 ) / 59661 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} + 2\beta_{4} - \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + 5\beta_{7} + \beta_{6} - \beta_{5} + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{11} + \beta_{10} + 15\beta_{7} + 6\beta_{6} + 5\beta_{5} - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{9} + 5\beta_{8} - 6\beta_{7} - 21\beta_{4} - 16\beta_{3} + 33\beta_{2} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{10} + 42\beta_{5} - 219 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 30\beta_{9} - 30\beta_{8} - 69\beta_{7} - 159\beta_{4} + 129\beta_{3} + 297\beta_{2} - 90\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 525\beta_{11} + 159\beta_{10} - 750\beta_{7} - 501\beta_{6} + 342\beta_{5} - 1659 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 318\beta_{9} - 204\beta_{8} + 1047\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4077\beta_{11} - 1479\beta_{10} - 5778\beta_{7} - 4257\beta_{6} - 2778\beta_{5} + 13035 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2958\beta_{9} - 1500\beta_{8} + 5535\beta_{7} + 10035\beta_{4} + 8535\beta_{3} - 21105\beta_{2} + 8874\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
911.1
−1.84630 − 1.06596i
−1.84630 + 1.06596i
1.30003 − 0.750572i
1.30003 + 0.750572i
2.48014 + 1.43191i
2.48014 − 1.43191i
−2.48014 − 1.43191i
−2.48014 + 1.43191i
−1.30003 + 0.750572i
−1.30003 − 0.750572i
1.84630 + 1.06596i
1.84630 − 1.06596i
0 −3.28528 0 −1.00000 0 − 2.24200i 0 7.79306 0
911.2 0 −3.28528 0 −1.00000 0 2.24200i 0 7.79306 0
911.3 0 −1.85913 0 −1.00000 0 − 4.64003i 0 0.456363 0
911.4 0 −1.85913 0 −1.00000 0 4.64003i 0 0.456363 0
911.5 0 −0.866357 0 −1.00000 0 − 0.665985i 0 −2.24943 0
911.6 0 −0.866357 0 −1.00000 0 0.665985i 0 −2.24943 0
911.7 0 0.866357 0 −1.00000 0 − 0.665985i 0 −2.24943 0
911.8 0 0.866357 0 −1.00000 0 0.665985i 0 −2.24943 0
911.9 0 1.85913 0 −1.00000 0 − 4.64003i 0 0.456363 0
911.10 0 1.85913 0 −1.00000 0 4.64003i 0 0.456363 0
911.11 0 3.28528 0 −1.00000 0 − 2.24200i 0 7.79306 0
911.12 0 3.28528 0 −1.00000 0 2.24200i 0 7.79306 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.12
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 15T_{3}^{4} + 48T_{3}^{2} - 28 \) acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 15 T^{4} + \cdots - 28)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 27 T^{4} + \cdots + 48)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 45 T^{4} + \cdots + 756)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 3 T^{2} - 24 T + 12)^{4} \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{6} + 27 T^{4} + \cdots + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 129 T^{4} + \cdots + 1344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 144 T^{4} + \cdots - 28672)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 192 T^{4} + \cdots + 177744)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 240 T^{4} + \cdots + 344064)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 228 T^{4} + \cdots + 69312)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 132 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 45 T^{4} + \cdots + 756)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 219 T^{4} + \cdots - 64512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} + \cdots + 896)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 15 T^{4} + \cdots - 28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 192 T^{4} + \cdots - 64512)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 15 T^{2} + \cdots - 28)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 108 T^{4} + \cdots - 28672)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 216 T^{4} + \cdots + 27648)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 180 T^{4} + \cdots + 48384)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 192 T^{4} + \cdots + 177744)^{2} \) Copy content Toggle raw display
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