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Properties

Label 312.4.q.e
Level $312$
Weight $4$
Character orbit 312.q
Analytic conductor $18.409$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [312,4,Mod(217,312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312.217"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.4085959218\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} + 161 x^{10} - 1480 x^{9} + 25918 x^{8} - 119864 x^{7} + 547597 x^{6} - 230908 x^{5} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 3 \beta_{4} q^{3} + (\beta_1 + 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7} + (9 \beta_{4} - 9) q^{9} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{11} + (\beta_{11} + \beta_{10} + \beta_{8} + \cdots + 7) q^{13}+ \cdots + (9 \beta_{3} - 9 \beta_1 + 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{3} + 8 q^{5} - 10 q^{7} - 54 q^{9} - 60 q^{11} + 86 q^{13} - 12 q^{15} - 56 q^{17} - 48 q^{19} + 60 q^{21} + 80 q^{23} + 508 q^{25} + 324 q^{27} - 164 q^{29} + 668 q^{31} - 180 q^{33} + 552 q^{35}+ \cdots + 1080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 161 x^{10} - 1480 x^{9} + 25918 x^{8} - 119864 x^{7} + 547597 x^{6} - 230908 x^{5} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!57 \nu^{11} + \cdots - 63\!\cdots\!22 ) / 42\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!05 \nu^{11} + \cdots - 31\!\cdots\!34 ) / 21\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 73\!\cdots\!81 \nu^{11} + \cdots - 43\!\cdots\!06 ) / 42\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37\!\cdots\!96 \nu^{11} + \cdots + 84\!\cdots\!29 ) / 70\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!91 \nu^{11} + \cdots + 14\!\cdots\!24 ) / 86\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!07 \nu^{11} + \cdots + 12\!\cdots\!68 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!22 \nu^{11} + \cdots - 16\!\cdots\!07 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 98\!\cdots\!91 \nu^{11} + \cdots + 63\!\cdots\!51 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!96 \nu^{11} + \cdots + 24\!\cdots\!01 ) / 69\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!94 \nu^{11} + \cdots + 35\!\cdots\!09 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!43 \nu^{11} + \cdots - 47\!\cdots\!93 ) / 34\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} - \beta _1 + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} - 5 \beta_{7} + \beta_{6} + 13 \beta_{5} + \cdots - 215 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{11} - 18 \beta_{10} + 127 \beta_{9} - 163 \beta_{8} - 36 \beta_{6} - 18 \beta_{5} + \cdots + 2932 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 129 \beta_{11} - 169 \beta_{10} - 175 \beta_{9} + 279 \beta_{7} + 46 \beta_{6} - 579 \beta_{5} + \cdots + 129 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 26305 \beta_{11} + 31945 \beta_{10} + 4344 \beta_{9} + 30649 \beta_{8} - 22745 \beta_{7} + 4344 \beta_{6} + \cdots - 805881 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 36281 \beta_{11} - 36281 \beta_{10} + 129701 \beta_{9} - 202263 \beta_{8} - 72562 \beta_{6} + \cdots + 6631942 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4297815 \beta_{11} - 5441129 \beta_{10} - 5268017 \beta_{9} + 5306343 \beta_{7} + 970202 \beta_{6} + \cdots + 4297815 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9100462 \beta_{11} + 11043674 \beta_{10} + 1860424 \beta_{9} + 10960886 \beta_{8} - 10726354 \beta_{7} + \cdots - 346410119 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 209623312 \beta_{11} - 209623312 \beta_{10} + 883123505 \beta_{9} - 1302370129 \beta_{8} + \cdots + 38241655712 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6244562425 \beta_{11} - 7916908202 \beta_{10} - 7800013562 \beta_{9} + 8862513037 \beta_{7} + \cdots + 6244562425 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 229557218705 \beta_{11} + 279034214779 \beta_{10} + 44708673410 \beta_{9} + 274265892115 \beta_{8} + \cdots - 8339698790755 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
2.71781 − 4.70739i
0.652533 − 1.13022i
−0.206972 + 0.358486i
−7.27366 + 12.5983i
4.43122 − 7.67510i
−0.320934 + 0.555874i
2.71781 + 4.70739i
0.652533 + 1.13022i
−0.206972 − 0.358486i
−7.27366 − 12.5983i
4.43122 + 7.67510i
−0.320934 − 0.555874i
0 −1.50000 + 2.59808i 0 −20.9870 0 −14.2108 − 24.6139i 0 −4.50000 − 7.79423i 0
217.2 0 −1.50000 + 2.59808i 0 −8.79544 0 6.16853 + 10.6842i 0 −4.50000 − 7.79423i 0
217.3 0 −1.50000 + 2.59808i 0 −3.80742 0 1.42598 + 2.46987i 0 −4.50000 − 7.79423i 0
217.4 0 −1.50000 + 2.59808i 0 11.9116 0 −12.1124 − 20.9793i 0 −4.50000 − 7.79423i 0
217.5 0 −1.50000 + 2.59808i 0 12.5989 0 −4.48285 − 7.76453i 0 −4.50000 − 7.79423i 0
217.6 0 −1.50000 + 2.59808i 0 13.0795 0 18.2116 + 31.5434i 0 −4.50000 − 7.79423i 0
289.1 0 −1.50000 − 2.59808i 0 −20.9870 0 −14.2108 + 24.6139i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 − 2.59808i 0 −8.79544 0 6.16853 − 10.6842i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 − 2.59808i 0 −3.80742 0 1.42598 − 2.46987i 0 −4.50000 + 7.79423i 0
289.4 0 −1.50000 − 2.59808i 0 11.9116 0 −12.1124 + 20.9793i 0 −4.50000 + 7.79423i 0
289.5 0 −1.50000 − 2.59808i 0 12.5989 0 −4.48285 + 7.76453i 0 −4.50000 + 7.79423i 0
289.6 0 −1.50000 − 2.59808i 0 13.0795 0 18.2116 − 31.5434i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.4.q.e ✓ 12
4.b odd 2 1 624.4.q.n 12
13.c even 3 1 inner 312.4.q.e ✓ 12
52.j odd 6 1 624.4.q.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.q.e ✓ 12 1.a even 1 1 trivial
312.4.q.e ✓ 12 13.c even 3 1 inner
624.4.q.n 12 4.b odd 2 1
624.4.q.n 12 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 4T_{5}^{5} - 494T_{5}^{4} + 3348T_{5}^{3} + 47897T_{5}^{2} - 254120T_{5} - 1379524 \) acting on \(S_{4}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 4 T^{5} + \cdots - 1379524)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 62582781821184 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 13672166336768)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 42091344197568)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 1081428091200)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 96\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 785538442110969)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 29\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 865905241260800)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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