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Properties

Label 1456.2.k.f
Level $1456$
Weight $2$
Character orbit 1456.k
Analytic conductor $11.626$
Analytic rank $0$
Dimension $12$
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] = [1456,2,Mod(337,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1456.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.k (of order \(2\), degree \(1\), not 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: \(11.6262185343\)
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: \( x^{12} + 21x^{10} + 152x^{8} + 456x^{6} + 532x^{4} + 192x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 728)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{3} + \beta_{2}) q^{9} + \beta_{7} q^{11} + ( - \beta_{10} - \beta_{6} - \beta_{4} + \cdots + 1) q^{13}+ \cdots + (\beta_{11} + \beta_{10} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 6 q^{9} + 6 q^{13} + 4 q^{17} + 20 q^{23} - 26 q^{25} - 26 q^{27} + 10 q^{29} - 2 q^{35} - 18 q^{39} + 6 q^{43} - 12 q^{49} + 12 q^{51} + 6 q^{53} + 40 q^{55} + 6 q^{61} + 38 q^{65} + 30 q^{69} + 58 q^{75} + 2 q^{77} - 40 q^{79} + 4 q^{81} - 8 q^{87} - 6 q^{91} - 54 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 152x^{8} + 456x^{6} + 532x^{4} + 192x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 7\nu^{8} + 82\nu^{6} + 708\nu^{4} + 1252\nu^{2} + 232 ) / 272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 7\nu^{8} + 82\nu^{6} + 708\nu^{4} + 1524\nu^{2} + 1048 ) / 272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{11} - 611\nu^{9} - 4422\nu^{7} - 13060\nu^{5} - 14012\nu^{3} - 3064\nu ) / 544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -21\nu^{11} - 419\nu^{9} - 2766\nu^{7} - 6892\nu^{5} - 4852\nu^{3} + 792\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{10} + 127\nu^{8} + 919\nu^{6} + 2654\nu^{4} + 2552\nu^{2} + 376 ) / 68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 21\nu^{9} - 152\nu^{7} - 456\nu^{5} - 532\nu^{3} - 184\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 64 \nu^{11} + 21 \nu^{10} - 1332 \nu^{9} + 419 \nu^{8} - 9508 \nu^{7} + 2766 \nu^{6} - 27856 \nu^{5} + \cdots + 2200 ) / 544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -15\nu^{10} - 309\nu^{8} - 2170\nu^{6} - 6176\nu^{4} - 6380\nu^{2} - 1416 ) / 136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 64 \nu^{11} + 21 \nu^{10} + 1332 \nu^{9} + 419 \nu^{8} + 9508 \nu^{7} + 2766 \nu^{6} + 27856 \nu^{5} + \cdots + 2200 ) / 544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -89\nu^{11} - 1847\nu^{9} - 13102\nu^{7} - 37764\nu^{5} - 39532\nu^{3} - 9272\nu ) / 544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{6} - 9\beta_{3} + 12\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{11} + 11\beta_{10} - 11\beta_{8} + 7\beta_{7} - 2\beta_{5} - 11\beta_{4} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{10} + 7\beta_{9} - 17\beta_{8} + 22\beta_{6} + 79\beta_{3} - 118\beta_{2} - 115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -167\beta_{11} - 101\beta_{10} + 101\beta_{8} - 45\beta_{7} + 34\beta_{5} + 123\beta_{4} - 418\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 201\beta_{10} - 35\beta_{9} + 201\beta_{8} - 202\beta_{6} - 699\beta_{3} + 1122\beta_{2} + 863 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1691\beta_{11} + 901\beta_{10} - 901\beta_{8} + 297\beta_{7} - 402\beta_{5} - 1327\beta_{4} + 3750\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2093\beta_{10} + 111\beta_{9} - 2093\beta_{8} + 1802\beta_{6} + 6251\beta_{3} - 10558\beta_{2} - 6959 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -16435\beta_{11} - 8053\beta_{10} + 8053\beta_{8} - 2065\beta_{7} + 4186\beta_{5} + 13655\beta_{4} - 34094\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 3.04217i
3.04217i
1.93919i
− 1.93919i
− 0.344051i
0.344051i
0.649715i
− 0.649715i
− 1.25097i
1.25097i
2.42473i
− 2.42473i
0 −3.04217 0 − 1.81349i 0 − 1.00000i 0 6.25481 0
337.2 0 −3.04217 0 1.81349i 0 1.00000i 0 6.25481 0
337.3 0 −1.93919 0 − 3.73905i 0 1.00000i 0 0.760458 0
337.4 0 −1.93919 0 3.73905i 0 − 1.00000i 0 0.760458 0
337.5 0 −0.344051 0 − 4.20126i 0 − 1.00000i 0 −2.88163 0
337.6 0 −0.344051 0 4.20126i 0 1.00000i 0 −2.88163 0
337.7 0 0.649715 0 − 1.23828i 0 − 1.00000i 0 −2.57787 0
337.8 0 0.649715 0 1.23828i 0 1.00000i 0 −2.57787 0
337.9 0 1.25097 0 − 2.55830i 0 1.00000i 0 −1.43508 0
337.10 0 1.25097 0 2.55830i 0 − 1.00000i 0 −1.43508 0
337.11 0 2.42473 0 − 0.0443237i 0 − 1.00000i 0 2.87931 0
337.12 0 2.42473 0 0.0443237i 0 1.00000i 0 2.87931 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.k.f 12
4.b odd 2 1 728.2.k.b ✓ 12
13.b even 2 1 inner 1456.2.k.f 12
52.b odd 2 1 728.2.k.b ✓ 12
52.f even 4 1 9464.2.a.bd 6
52.f even 4 1 9464.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.k.b ✓ 12 4.b odd 2 1
728.2.k.b ✓ 12 52.b odd 2 1
1456.2.k.f 12 1.a even 1 1 trivial
1456.2.k.f 12 13.b even 2 1 inner
9464.2.a.bd 6 52.f even 4 1
9464.2.a.be 6 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + T_{3}^{5} - 10T_{3}^{4} - 4T_{3}^{3} + 22T_{3}^{2} - 4T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} - 10 T^{4} + \cdots - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 43 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 45 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 2 T^{5} + \cdots - 304)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 139 T^{10} + \cdots + 5456896 \) Copy content Toggle raw display
$23$ \( (T^{6} - 10 T^{5} + \cdots + 3232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 5 T^{5} + \cdots + 5872)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12841422400 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 35244805696 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16294011904 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3 T^{5} + \cdots - 100480)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 200 T^{10} + \cdots + 1600 \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots - 24712)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 340 T^{10} + \cdots + 2310400 \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + \cdots - 84752)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 385 T^{10} + \cdots + 23658496 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 6338707456 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17571093136 \) Copy content Toggle raw display
$79$ \( (T^{6} + 20 T^{5} + \cdots + 97772)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2501200144 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 3324214336 \) Copy content Toggle raw display
$97$ \( T^{12} + 368 T^{10} + \cdots + 4963984 \) Copy content Toggle raw display
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