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Properties

Label 1.250.a.a
Level $1$
Weight $250$
Character orbit 1.a
Self dual yes
Analytic conductor $392.628$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,250,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 250, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 250);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 250 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

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: yes
Analytic conductor: \(392.628030028\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + \cdots - 15\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{762}\cdot 3^{297}\cdot 5^{100}\cdot 7^{47}\cdot 11^{20}\cdot 13^{13}\cdot 17^{6}\cdot 19^{7}\cdot 23^{5}\cdot 31^{7}\cdot 41^{3}\cdot 83^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 57\!\cdots\!40) q^{2}+ \cdots + (\beta_{8} - 196027 \beta_{7} + \cdots + 19\!\cdots\!53) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 57\!\cdots\!40) q^{2}+ \cdots + ( - 43\!\cdots\!04 \beta_{19} + \cdots - 97\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 11\!\cdots\!00 q^{2}+ \cdots + 38\!\cdots\!60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 11\!\cdots\!00 q^{2}+ \cdots - 19\!\cdots\!80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + \cdots - 15\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 144\nu - 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\!\cdots\!17 \nu^{19} + \cdots - 15\!\cdots\!40 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!39 \nu^{19} + \cdots - 56\!\cdots\!60 ) / 44\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51\!\cdots\!47 \nu^{19} + \cdots - 49\!\cdots\!76 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!73 \nu^{19} + \cdots - 18\!\cdots\!16 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!77 \nu^{19} + \cdots + 74\!\cdots\!16 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!71 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!93 \nu^{19} + \cdots + 80\!\cdots\!44 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56\!\cdots\!71 \nu^{19} + \cdots - 28\!\cdots\!32 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!93 \nu^{19} + \cdots - 20\!\cdots\!44 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!03 \nu^{19} + \cdots - 87\!\cdots\!24 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!53 \nu^{19} + \cdots + 37\!\cdots\!24 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 36\!\cdots\!97 \nu^{19} + \cdots - 81\!\cdots\!16 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 84\!\cdots\!63 \nu^{19} + \cdots + 20\!\cdots\!04 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!61 \nu^{19} + \cdots - 41\!\cdots\!88 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{19} + \cdots - 28\!\cdots\!68 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 91\!\cdots\!89 \nu^{19} + \cdots + 20\!\cdots\!12 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 19\!\cdots\!49 \nu^{19} + \cdots + 89\!\cdots\!08 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 56\!\cdots\!49 \nu^{19} + \cdots + 20\!\cdots\!92 ) / 61\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 36 ) / 144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 40646296539264 \beta_{2} + \cdots + 13\!\cdots\!00 ) / 20736 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 634320 \beta_{6} + 20891974110149 \beta_{5} + \cdots - 31\!\cdots\!12 ) / 2985984 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + 337 \beta_{14} - 649312 \beta_{13} + 66811291676 \beta_{12} + 346012944588063 \beta_{11} + \cdots + 28\!\cdots\!16 ) / 429981696 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29\!\cdots\!68 \beta_{19} + \cdots - 63\!\cdots\!44 ) / 3869835264 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16\!\cdots\!12 \beta_{19} + \cdots + 27\!\cdots\!60 ) / 34828517376 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 34\!\cdots\!04 \beta_{19} + \cdots - 70\!\cdots\!72 ) / 313456656384 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12\!\cdots\!84 \beta_{19} + \cdots + 99\!\cdots\!32 ) / 940369969152 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 51\!\cdots\!20 \beta_{19} + \cdots - 24\!\cdots\!68 ) / 8463329722368 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20\!\cdots\!08 \beta_{19} + \cdots + 11\!\cdots\!00 ) / 76169967501312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 93\!\cdots\!68 \beta_{19} + \cdots - 26\!\cdots\!24 ) / 685529707511808 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10\!\cdots\!88 \beta_{19} + \cdots + 43\!\cdots\!64 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!96 \beta_{19} + \cdots - 99\!\cdots\!36 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 15\!\cdots\!20 \beta_{19} + \cdots + 58\!\cdots\!20 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 27\!\cdots\!96 \beta_{19} + \cdots - 14\!\cdots\!48 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 24\!\cdots\!44 \beta_{19} + \cdots + 87\!\cdots\!96 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 19\!\cdots\!16 \beta_{19} + \cdots - 68\!\cdots\!32 ) / 61\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 13\!\cdots\!48 \beta_{19} + \cdots + 43\!\cdots\!40 ) / 61\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 11\!\cdots\!20 \beta_{19} + \cdots - 31\!\cdots\!28 ) / 18\!\cdots\!16 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.97763e35
−3.62545e35
−3.19211e35
−3.05855e35
−2.63821e35
−2.34462e35
−1.58599e35
−1.12406e35
−8.81799e34
2.10990e34
4.29625e34
1.00479e35
1.23581e35
1.58570e35
2.18195e35
2.56070e35
2.77773e35
2.91339e35
3.56538e35
3.96237e35
−5.78520e37 −3.53206e58 2.44222e75 −1.12402e86 2.04337e96 2.85959e105 −8.89530e112 −6.23137e118 6.50268e123
1.2 −5.27805e37 3.69691e59 1.88115e75 1.05681e87 −1.95125e97 −2.65777e105 −5.15417e112 7.31103e118 −5.57790e124
1.3 −4.65405e37 −3.92180e59 1.26139e75 −2.30413e86 1.82522e97 −1.18348e105 −1.66039e112 9.02441e118 1.07235e124
1.4 −4.46171e37 6.40726e58 1.08606e75 −7.41004e86 −2.85873e96 −1.36061e105 −8.09509e111 −5.94559e118 3.30614e124
1.5 −3.85642e37 4.02028e59 5.82570e74 −9.12671e86 −1.55039e97 2.78909e105 1.24198e112 9.80649e118 3.51964e124
1.6 −3.43366e37 −1.20165e59 2.74376e74 1.80813e87 4.12605e96 7.31436e104 2.16406e112 −4.91216e118 −6.20852e124
1.7 −2.34123e37 −8.80131e57 −3.56488e74 −1.79351e87 2.06059e95 −6.76452e104 2.95256e112 −6.34838e118 4.19902e124
1.8 −1.67605e37 2.87540e59 −6.23710e74 6.28468e86 −4.81932e96 −2.02660e104 2.56157e112 1.91180e118 −1.05335e124
1.9 −1.32719e37 −3.41570e59 −7.28482e74 −5.08422e86 4.53329e96 2.01818e105 2.16745e112 5.31088e118 6.74774e123
1.10 2.46424e36 −3.52304e59 −8.98553e74 6.86099e86 −8.68161e95 −3.05923e105 −4.44347e111 6.05567e118 1.69071e123
1.11 5.61259e36 3.47447e57 −8.73124e74 2.38550e86 1.95008e94 −8.68917e104 −9.97779e111 −6.35492e118 1.33889e123
1.12 1.38949e37 4.84372e59 −7.11556e74 −1.74161e87 6.73032e96 −2.65178e105 −2.24568e112 1.71055e119 −2.41995e124
1.13 1.72216e37 2.96594e59 −6.08043e74 6.01542e86 5.10781e96 2.31664e105 −2.60505e112 2.44065e118 1.03595e124
1.14 2.22601e37 −1.45096e58 −4.09112e74 −1.44350e87 −3.22986e95 1.56544e105 −2.92440e112 −6.33507e118 −3.21325e124
1.15 3.08461e37 −2.87660e59 4.68537e73 1.52371e87 −8.87317e96 1.68139e105 −2.64589e112 1.91869e118 4.70005e124
1.16 3.63001e37 −4.64644e59 4.13070e74 −1.60246e87 −1.68666e97 −5.50906e104 −1.78435e112 1.52333e119 −5.81696e124
1.17 3.94254e37 3.13461e59 6.49733e74 1.98914e87 1.23583e97 −1.96514e105 −1.00493e112 3.46966e118 7.84224e124
1.18 4.13788e37 −2.49281e58 8.07578e74 −3.93975e86 −1.03149e96 −1.76341e105 −4.01572e111 −6.29398e118 −1.63022e124
1.19 5.07675e37 3.05232e59 1.67271e75 −5.91236e86 1.54958e97 1.18944e105 3.89939e112 2.96051e118 −3.00156e124
1.20 5.64840e37 −2.26177e59 2.28582e75 4.00884e86 −1.27754e97 4.22618e104 7.80155e112 −1.24052e118 2.26436e124
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.250.a.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.250.a.a ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{250}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 16\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots - 25\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots - 50\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 30\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 88\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 34\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 24\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
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Additional information

This newform has the largest known coefficient ring index among those with $Nk^2$ $\le 4000$.