Properties

Label 4.31.b.a
Level $4$
Weight $31$
Character orbit 4.b
Analytic conductor $22.806$
Analytic rank $0$
Dimension $14$
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] = [4,31,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 31, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 4.b (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: \(22.8057047611\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} + 7040347091761 x^{12} - 42242082550475 x^{11} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{182}\cdot 3^{19}\cdot 5^{5} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1748) q^{2} + ( - \beta_{2} + 30 \beta_1 + 9) q^{3} + ( - \beta_{3} + 14 \beta_{2} - 1845 \beta_1 - 58628180) q^{4} + (\beta_{4} - 2 \beta_{3} - 11 \beta_{2} - 84189 \beta_1 + 1061747275) q^{5} + ( - \beta_{5} - \beta_{4} - 15 \beta_{3} + 4394 \beta_{2} + \cdots - 32433446090) q^{6}+ \cdots + (\beta_{9} - \beta_{8} - 26 \beta_{6} - 37 \beta_{5} + \cdots - 51584237429141) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1748) q^{2} + ( - \beta_{2} + 30 \beta_1 + 9) q^{3} + ( - \beta_{3} + 14 \beta_{2} - 1845 \beta_1 - 58628180) q^{4} + (\beta_{4} - 2 \beta_{3} - 11 \beta_{2} - 84189 \beta_1 + 1061747275) q^{5} + ( - \beta_{5} - \beta_{4} - 15 \beta_{3} + 4394 \beta_{2} + \cdots - 32433446090) q^{6}+ \cdots + ( - 361733216611760 \beta_{13} + \cdots - 32\!\cdots\!37) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 24476 q^{2} - 820787056 q^{4} + 14864798540 q^{5} - 454068012192 q^{6} - 31343321296064 q^{8} - 722181861644898 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 24476 q^{2} - 820787056 q^{4} + 14864798540 q^{5} - 454068012192 q^{6} - 31343321296064 q^{8} - 722181861644898 q^{9} - 12\!\cdots\!60 q^{10}+ \cdots + 47\!\cdots\!36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} + 7040347091761 x^{12} - 42242082550475 x^{11} + \cdots + 48\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28\!\cdots\!75 \nu^{13} + \cdots - 75\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 28\!\cdots\!75 \nu^{13} + \cdots - 75\!\cdots\!60 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!11 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!97 \nu^{13} + \cdots + 77\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 49\!\cdots\!51 \nu^{13} + \cdots - 12\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!13 \nu^{13} + \cdots - 97\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!37 \nu^{13} + \cdots + 24\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 90\!\cdots\!79 \nu^{13} + \cdots + 57\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32\!\cdots\!37 \nu^{13} + \cdots + 37\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!27 \nu^{13} + \cdots + 16\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!65 \nu^{13} + \cdots - 16\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{13} + \cdots + 15\!\cdots\!00 ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 43\!\cdots\!29 \nu^{13} + \cdots - 30\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 30\beta _1 - 1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - 26 \beta_{6} - 37 \beta_{5} - 854 \beta_{4} + 19227 \beta_{3} + 84319 \beta_{2} + 634535194 \beta _1 - 257475369523870 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1481411 \beta_{13} + 514624 \beta_{12} + 173503 \beta_{11} - 7009552 \beta_{10} + 688151 \beta_{9} + 30883214 \beta_{8} + 28660282 \beta_{7} + 81675800 \beta_{6} + \cdots - 468002778268567 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1832211023272 \beta_{13} - 1785748076572 \beta_{12} + 1832200560008 \beta_{11} - 6450780653063 \beta_{10} - 177033784140878 \beta_{9} + \cdots + 31\!\cdots\!62 ) / 16384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17\!\cdots\!95 \beta_{13} + \cdots - 10\!\cdots\!93 ) / 131072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 31\!\cdots\!12 \beta_{13} + \cdots - 48\!\cdots\!30 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16\!\cdots\!92 \beta_{13} + \cdots + 17\!\cdots\!44 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 66\!\cdots\!88 \beta_{13} + \cdots + 12\!\cdots\!48 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23\!\cdots\!80 \beta_{13} + \cdots - 27\!\cdots\!20 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 13\!\cdots\!36 \beta_{13} + \cdots - 33\!\cdots\!82 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!68 \beta_{13} + \cdots + 13\!\cdots\!80 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 24\!\cdots\!44 \beta_{13} + \cdots + 92\!\cdots\!76 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11\!\cdots\!35 \beta_{13} + \cdots - 73\!\cdots\!57 ) / 131072 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 287382.i
0.500000 + 287382.i
0.500000 + 1.69534e6i
0.500000 1.69534e6i
0.500000 872651.i
0.500000 + 872651.i
0.500000 + 868866.i
0.500000 868866.i
0.500000 + 174763.i
0.500000 174763.i
0.500000 1.39431e6i
0.500000 + 1.39431e6i
0.500000 + 769735.i
0.500000 769735.i
−31548.7 8855.74i 4.59811e6i 9.16894e8 + 5.58773e8i −1.46052e10 4.07197e10 1.45064e11i 9.18461e12i −2.39784e13 2.57483e13i 1.84749e14 4.60774e14 + 1.29340e14i
3.2 −31548.7 + 8855.74i 4.59811e6i 9.16894e8 5.58773e8i −1.46052e10 4.07197e10 + 1.45064e11i 9.18461e12i −2.39784e13 + 2.57483e13i 1.84749e14 4.60774e14 1.29340e14i
3.3 −26662.2 19049.1i 2.71254e7i 3.48003e8 + 1.01578e9i 3.98373e10 −5.16715e11 + 7.23222e11i 1.87458e12i 1.00713e13 3.37122e13i −5.29895e14 −1.06215e15 7.58866e14i
3.4 −26662.2 + 19049.1i 2.71254e7i 3.48003e8 1.01578e9i 3.98373e10 −5.16715e11 7.23222e11i 1.87458e12i 1.00713e13 + 3.37122e13i −5.29895e14 −1.06215e15 + 7.58866e14i
3.5 −16991.5 28018.4i 1.39624e7i −4.96318e8 + 9.52150e8i 8.38288e9 3.91204e11 2.37243e11i 4.67445e12i 3.51109e13 2.27243e12i 1.09422e13 −1.42438e14 2.34875e14i
3.6 −16991.5 + 28018.4i 1.39624e7i −4.96318e8 9.52150e8i 8.38288e9 3.91204e11 + 2.37243e11i 4.67445e12i 3.51109e13 + 2.27243e12i 1.09422e13 −1.42438e14 + 2.34875e14i
3.7 −713.932 32760.2i 1.39019e7i −1.07272e9 + 4.67771e7i −5.42656e10 −4.55428e11 + 9.92498e9i 5.92817e11i 2.29828e12 + 3.51092e13i 1.26296e13 3.87420e13 + 1.77775e15i
3.8 −713.932 + 32760.2i 1.39019e7i −1.07272e9 4.67771e7i −5.42656e10 −4.55428e11 9.92498e9i 5.92817e11i 2.29828e12 3.51092e13i 1.26296e13 3.87420e13 1.77775e15i
3.9 10610.7 31002.5i 2.79620e6i −8.48568e8 6.57916e8i 4.95390e10 −8.66893e10 2.96697e10i 4.58323e12i −2.94010e13 + 1.93268e13i 1.98072e14 5.25643e14 1.53583e15i
3.10 10610.7 + 31002.5i 2.79620e6i −8.48568e8 + 6.57916e8i 4.95390e10 −8.66893e10 + 2.96697e10i 4.58323e12i −2.94010e13 1.93268e13i 1.98072e14 5.25643e14 + 1.53583e15i
3.11 22243.3 24061.9i 2.23090e7i −8.42119e7 1.07043e9i −2.29401e10 5.36798e11 + 4.96227e11i 1.19922e12i −2.76299e13 2.17837e13i −2.91801e14 −5.10263e14 + 5.51982e14i
3.12 22243.3 + 24061.9i 2.23090e7i −8.42119e7 + 1.07043e9i −2.29401e10 5.36798e11 4.96227e11i 1.19922e12i −2.76299e13 + 2.17837e13i −2.91801e14 −5.10263e14 5.51982e14i
3.13 30824.3 11117.8i 1.23158e7i 8.26531e8 6.85396e8i 1.48412e9 −1.36924e11 3.79624e11i 2.60240e12i 1.78571e13 3.03161e13i 5.42133e13 4.57469e13 1.65001e13i
3.14 30824.3 + 11117.8i 1.23158e7i 8.26531e8 + 6.85396e8i 1.48412e9 −1.36924e11 + 3.79624e11i 2.60240e12i 1.78571e13 + 3.03161e13i 5.42133e13 4.57469e13 + 1.65001e13i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.31.b.a 14
3.b odd 2 1 36.31.d.c 14
4.b odd 2 1 inner 4.31.b.a 14
12.b even 2 1 36.31.d.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.31.b.a 14 1.a even 1 1 trivial
4.31.b.a 14 4.b odd 2 1 inner
36.31.d.c 14 3.b odd 2 1
36.31.d.c 14 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{31}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 24476 T^{13} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{7} - 7432399270 T^{6} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 41\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
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