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Properties

Label 4805.2.a.t
Level $4805$
Weight $2$
Character orbit 4805.a
Self dual yes
Analytic conductor $38.368$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4805,2,Mod(1,4805)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4805.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4805 = 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4805.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-1,8,13,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.3681181712\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 18 x^{10} + 14 x^{9} + 119 x^{8} - 63 x^{7} - 352 x^{6} + 86 x^{5} + 453 x^{4} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + 1) q^{4} + q^{5} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots - 1) q^{6} + \beta_{11} q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+ \cdots + ( - 2 \beta_{10} + 3 \beta_{9} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 8 q^{3} + 13 q^{4} + 12 q^{5} - 4 q^{7} - 9 q^{8} + 20 q^{9} - q^{10} + 5 q^{11} + 44 q^{12} + 6 q^{13} + 4 q^{14} + 8 q^{15} + 15 q^{16} + 24 q^{17} - q^{19} + 13 q^{20} + 8 q^{21} + 12 q^{23}+ \cdots + 83 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 18 x^{10} + 14 x^{9} + 119 x^{8} - 63 x^{7} - 352 x^{6} + 86 x^{5} + 453 x^{4} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} - 23 \nu^{9} - 22 \nu^{8} + 163 \nu^{7} + 85 \nu^{6} - 438 \nu^{5} - 172 \nu^{4} + \cdots - 30 ) / 33 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8 \nu^{11} - 17 \nu^{10} - 129 \nu^{9} + 253 \nu^{8} + 721 \nu^{7} - 1278 \nu^{6} - 1535 \nu^{5} + \cdots + 321 ) / 99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10 \nu^{11} + 13 \nu^{10} + 186 \nu^{9} - 209 \nu^{8} - 1256 \nu^{7} + 1119 \nu^{6} + 3775 \nu^{5} + \cdots + 564 ) / 99 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{11} + 15 \nu^{10} + 152 \nu^{9} - 231 \nu^{8} - 917 \nu^{7} + 1226 \nu^{6} + 2369 \nu^{5} + \cdots + 6 ) / 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37 \nu^{11} - 58 \nu^{10} - 642 \nu^{9} + 902 \nu^{8} + 4007 \nu^{7} - 4797 \nu^{6} - 10882 \nu^{5} + \cdots - 879 ) / 99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28 \nu^{11} - 43 \nu^{10} - 479 \nu^{9} + 638 \nu^{8} + 2980 \nu^{7} - 3230 \nu^{6} - 8172 \nu^{5} + \cdots - 642 ) / 33 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 109 \nu^{11} + 145 \nu^{10} + 1935 \nu^{9} - 2222 \nu^{8} - 12476 \nu^{7} + 11613 \nu^{6} + \cdots + 3039 ) / 99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 40 \nu^{11} + 52 \nu^{10} + 700 \nu^{9} - 770 \nu^{8} - 4452 \nu^{7} + 3871 \nu^{6} + 12449 \nu^{5} + \cdots + 1035 ) / 33 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 181 \nu^{11} + 265 \nu^{10} + 3129 \nu^{9} - 3971 \nu^{8} - 19625 \nu^{7} + 20343 \nu^{6} + \cdots + 3516 ) / 99 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + 5\beta_{7} + 7\beta_{6} + 6\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{6} + 3\beta_{5} + \beta_{4} + \beta_{3} + 9\beta_{2} + 29\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{11} + 11 \beta_{10} - \beta_{9} + 3 \beta_{8} + 25 \beta_{7} + 47 \beta_{6} + 37 \beta_{5} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{11} + 16 \beta_{10} - 13 \beta_{9} + 15 \beta_{8} - 4 \beta_{7} + 27 \beta_{6} + 37 \beta_{5} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 66 \beta_{11} + 97 \beta_{10} - 18 \beta_{9} + 43 \beta_{8} + 125 \beta_{7} + 314 \beta_{6} + \cdots + 506 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 32 \beta_{11} + 172 \beta_{10} - 122 \beta_{9} + 158 \beta_{8} - 56 \beta_{7} + 269 \beta_{6} + \cdots + 427 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 459 \beta_{11} + 789 \beta_{10} - 208 \beta_{9} + 439 \beta_{8} + 613 \beta_{7} + 2112 \beta_{6} + \cdots + 3098 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 351 \beta_{11} + 1579 \beta_{10} - 1020 \beta_{9} + 1440 \beta_{8} - 552 \beta_{7} + 2380 \beta_{6} + \cdots + 3083 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
2.72565
2.37598
2.10921
1.43452
1.09915
−0.184289
−0.407227
−0.493843
−1.12671
−1.75065
−2.38103
−2.40075
−2.72565 2.96835 5.42917 1.00000 −8.09068 2.01322 −9.34672 5.81108 −2.72565
1.2 −2.37598 2.85169 3.64527 1.00000 −6.77556 −2.48646 −3.90913 5.13216 −2.37598
1.3 −2.10921 −1.84546 2.44876 1.00000 3.89245 −3.13208 −0.946528 0.405706 −2.10921
1.4 −1.43452 −0.641052 0.0578430 1.00000 0.919601 0.229344 2.78606 −2.58905 −1.43452
1.5 −1.09915 1.32639 −0.791879 1.00000 −1.45790 0.437824 3.06868 −1.24069 −1.09915
1.6 0.184289 −2.06829 −1.96604 1.00000 −0.381163 −0.923413 −0.730897 1.27782 0.184289
1.7 0.407227 0.0975796 −1.83417 1.00000 0.0397370 −4.74250 −1.56137 −2.99048 0.407227
1.8 0.493843 2.22037 −1.75612 1.00000 1.09651 4.52236 −1.85493 1.93003 0.493843
1.9 1.12671 −2.94549 −0.730516 1.00000 −3.31873 −0.268137 −3.07651 5.67592 1.12671
1.10 1.75065 0.545091 1.06477 1.00000 0.954263 4.52073 −1.63725 −2.70288 1.75065
1.11 2.38103 3.07392 3.66931 1.00000 7.31909 −2.68021 3.97469 6.44896 2.38103
1.12 2.40075 2.41690 3.76358 1.00000 5.80237 −1.49067 4.23392 2.84142 2.40075
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(31\) \( +1 \)

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 4805.2.a.t 12
31.b odd 2 1 4805.2.a.s 12
31.d even 5 2 155.2.h.a ✓ 24
155.n even 10 2 775.2.k.e 24
155.s odd 20 4 775.2.bf.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.h.a ✓ 24 31.d even 5 2
775.2.k.e 24 155.n even 10 2
775.2.bf.d 48 155.s odd 20 4
4805.2.a.s 12 31.b odd 2 1
4805.2.a.t 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4805))\):

\( T_{2}^{12} + T_{2}^{11} - 18 T_{2}^{10} - 14 T_{2}^{9} + 119 T_{2}^{8} + 63 T_{2}^{7} - 352 T_{2}^{6} + \cdots - 9 \) Copy content Toggle raw display
\( T_{3}^{12} - 8 T_{3}^{11} + 4 T_{3}^{10} + 117 T_{3}^{9} - 260 T_{3}^{8} - 417 T_{3}^{7} + 1659 T_{3}^{6} + \cdots + 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{11} + \cdots + 71 \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots - 151 \) Copy content Toggle raw display
$11$ \( T^{12} - 5 T^{11} + \cdots - 37809 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots - 2888819 \) Copy content Toggle raw display
$17$ \( T^{12} - 24 T^{11} + \cdots + 106839 \) Copy content Toggle raw display
$19$ \( T^{12} + T^{11} + \cdots - 135845 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots - 92781 \) Copy content Toggle raw display
$29$ \( T^{12} + T^{11} + \cdots + 4469355 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} - 18 T^{11} + \cdots - 4090896 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 120637449 \) Copy content Toggle raw display
$43$ \( T^{12} - 25 T^{11} + \cdots - 239809 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 142318179 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 531316089 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 1563683445 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 83834704 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 3617387264 \) Copy content Toggle raw display
$71$ \( T^{12} - 11 T^{11} + \cdots - 9534429 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 193440439 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 510895495 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 2159109009 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 104989455 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 167663099 \) Copy content Toggle raw display
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