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Properties

Label 832.3.d.e
Level $832$
Weight $3$
Character orbit 832.d
Analytic conductor $22.670$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [832,3,Mod(703,832)] 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("832.703"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 832.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-28,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.6703579948\)
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: \( x^{12} + 18 x^{10} - 6 x^{9} + 94 x^{8} + 414 x^{7} - 212 x^{6} + 2938 x^{5} - 4875 x^{4} + \cdots + 211588 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 416)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_1 q^{7} + ( - \beta_{10} - \beta_{7} - 2) q^{9} + (\beta_{8} - \beta_{6} + \beta_{2}) q^{11} + \beta_{7} q^{13} + (\beta_{8} + \beta_{3} + \beta_{2} + \beta_1) q^{15}+ \cdots + (\beta_{8} - 2 \beta_{6} + \cdots + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{9} - 8 q^{17} + 32 q^{21} - 20 q^{25} - 144 q^{29} + 96 q^{33} + 32 q^{37} - 8 q^{41} + 128 q^{45} - 52 q^{49} + 312 q^{53} + 96 q^{57} - 280 q^{61} - 192 q^{69} - 120 q^{73} + 120 q^{77} + 28 q^{81}+ \cdots - 312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18 x^{10} - 6 x^{9} + 94 x^{8} + 414 x^{7} - 212 x^{6} + 2938 x^{5} - 4875 x^{4} + \cdots + 211588 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1224582794581 \nu^{11} + 11339878454885 \nu^{10} + 20515693177679 \nu^{9} + \cdots + 42\!\cdots\!48 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 731300329528 \nu^{11} - 3755727487639 \nu^{10} - 26404122012070 \nu^{9} + \cdots - 12\!\cdots\!04 ) / 92\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11596480 \nu^{11} - 51285863 \nu^{10} + 27097048 \nu^{9} - 1076684159 \nu^{8} + \cdots - 1693141344676 ) / 662862341106 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 191735111471 \nu^{11} - 796303488257 \nu^{10} + 5575215647907 \nu^{9} + \cdots - 16\!\cdots\!84 ) / 87\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 135157062238 \nu^{11} + 423178637845 \nu^{10} + 2080591410328 \nu^{9} + \cdots + 21\!\cdots\!56 ) / 43\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2890181265623 \nu^{11} + 3013087225031 \nu^{10} + 70031048066951 \nu^{9} + \cdots - 31\!\cdots\!56 ) / 92\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1998557163 \nu^{11} + 4783319697 \nu^{10} + 39506592163 \nu^{9} + 59677007399 \nu^{8} + \cdots - 30784058136364 ) / 57437341631696 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11659039704161 \nu^{11} - 8469873873527 \nu^{10} - 164036428212461 \nu^{9} + \cdots + 74\!\cdots\!36 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3079426328672 \nu^{11} - 4576144935181 \nu^{10} + 45768376998122 \nu^{9} + \cdots - 63\!\cdots\!60 ) / 46\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 757057124933 \nu^{11} - 1973689973889 \nu^{10} - 13188322910905 \nu^{9} + \cdots + 59\!\cdots\!32 ) / 87\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1212595697257 \nu^{11} - 971062708191 \nu^{10} - 22774282758857 \nu^{9} + \cdots + 10\!\cdots\!88 ) / 87\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{10} - 2\beta_{8} + 3\beta_{7} - \beta_{5} - \beta_{4} - 2\beta_{3} - 2\beta_{2} - 2\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} - 3 \beta_{10} - 6 \beta_{9} - 4 \beta_{8} - 41 \beta_{7} + 9 \beta_{6} + 10 \beta_{5} + \cdots + 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{11} + 8 \beta_{10} + 11 \beta_{9} + 10 \beta_{8} - 44 \beta_{7} - 15 \beta_{6} + 48 \beta_{5} + \cdots + 92 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13 \beta_{11} + 7 \beta_{10} + 16 \beta_{9} + 56 \beta_{8} + 339 \beta_{7} - 39 \beta_{6} - 244 \beta_{5} + \cdots - 1781 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11 \beta_{11} - 9 \beta_{10} - 236 \beta_{9} - 160 \beta_{8} + 322 \beta_{7} + 365 \beta_{6} + \cdots - 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 749 \beta_{11} - 821 \beta_{10} - 20 \beta_{9} + 620 \beta_{8} - 4393 \beta_{7} + 929 \beta_{6} + \cdots + 28255 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1232 \beta_{11} - 134 \beta_{10} + 3393 \beta_{9} + 542 \beta_{8} + 3676 \beta_{7} - 7047 \beta_{6} + \cdots - 5260 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8759 \beta_{11} + 6987 \beta_{10} - 7068 \beta_{9} - 8456 \beta_{8} + 76235 \beta_{7} + 7937 \beta_{6} + \cdots - 319229 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7157 \beta_{11} - 5159 \beta_{10} - 35042 \beta_{9} - 11272 \beta_{8} - 114002 \beta_{7} + \cdots + 301150 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 62021 \beta_{11} - 4461 \beta_{10} + 242104 \beta_{9} + 131580 \beta_{8} - 1002245 \beta_{7} + \cdots + 3120643 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
−2.73805 − 0.585101i
2.02483 − 2.50336i
1.80373 − 0.842321i
−0.986456 + 3.01736i
−0.817269 − 3.85968i
0.713221 + 1.91826i
0.713221 − 1.91826i
−0.817269 + 3.85968i
−0.986456 − 3.01736i
1.80373 + 0.842321i
2.02483 + 2.50336i
−2.73805 + 0.585101i
0 − 5.47610i 0 1.17020 0 3.19201i 0 −20.9877 0
703.2 0 − 4.04966i 0 −5.00673 0 11.3887i 0 −7.39976 0
703.3 0 − 3.60745i 0 −1.68464 0 − 9.07224i 0 −4.01370 0
703.4 0 − 1.97291i 0 −6.03472 0 2.42336i 0 5.10762 0
703.5 0 − 1.63454i 0 7.71936 0 − 8.89004i 0 6.32828 0
703.6 0 − 1.42644i 0 3.83652 0 − 3.59118i 0 6.96526 0
703.7 0 1.42644i 0 3.83652 0 3.59118i 0 6.96526 0
703.8 0 1.63454i 0 7.71936 0 8.89004i 0 6.32828 0
703.9 0 1.97291i 0 −6.03472 0 − 2.42336i 0 5.10762 0
703.10 0 3.60745i 0 −1.68464 0 9.07224i 0 −4.01370 0
703.11 0 4.04966i 0 −5.00673 0 − 11.3887i 0 −7.39976 0
703.12 0 5.47610i 0 1.17020 0 − 3.19201i 0 −20.9877 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.12
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 832.3.d.e 12
4.b odd 2 1 inner 832.3.d.e 12
8.b even 2 1 416.3.d.b ✓ 12
8.d odd 2 1 416.3.d.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.3.d.b ✓ 12 8.b even 2 1
416.3.d.b ✓ 12 8.d odd 2 1
832.3.d.e 12 1.a even 1 1 trivial
832.3.d.e 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 68T_{3}^{10} + 1630T_{3}^{8} + 17252T_{3}^{6} + 82313T_{3}^{4} + 175216T_{3}^{2} + 135424 \) acting on \(S_{3}^{\mathrm{new}}(832, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 68 T^{10} + \cdots + 135424 \) Copy content Toggle raw display
$5$ \( (T^{6} - 70 T^{4} + \cdots - 1764)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 651066256 \) Copy content Toggle raw display
$11$ \( T^{12} + 636 T^{10} + \cdots + 1183744 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{6} \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 17690404)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 337179971584 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47126797090816 \) Copy content Toggle raw display
$29$ \( (T^{6} + 72 T^{5} + \cdots + 885493696)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} + \cdots - 21217876)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots - 538750976)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} - 156 T^{5} + \cdots + 40817664)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 78\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} + 140 T^{5} + \cdots - 273870848)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + 60 T^{5} + \cdots + 21029516352)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 98\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{6} - 100 T^{5} + \cdots - 7718819264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 156 T^{5} + \cdots - 85837216448)^{2} \) Copy content Toggle raw display
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