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Properties

Label 312.4.q.c
Level $312$
Weight $4$
Character orbit 312.q
Analytic conductor $18.409$
Analytic rank $0$
Dimension $8$
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 = [8] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 58x^{6} + 143x^{5} + 2768x^{4} + 3327x^{3} + 26815x^{2} - 18834x + 191844 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{2} + 3) q^{3} + (\beta_{7} - \beta_{5}) q^{5} + (\beta_{7} + \beta_{6} - 3 \beta_{2}) q^{7} - 9 \beta_{2} q^{9} + ( - 2 \beta_{5} + 6 \beta_{2} + \cdots - 6) q^{11} + (3 \beta_{7} + \beta_{5} + 4 \beta_{4} + \cdots + 6) q^{13}+ \cdots + ( - 18 \beta_{7} + 18 \beta_{5} + \cdots + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} - 2 q^{5} - 13 q^{7} - 36 q^{9} - 24 q^{11} - 12 q^{13} - 3 q^{15} + 73 q^{17} + 54 q^{19} - 78 q^{21} - 220 q^{23} - 330 q^{25} - 216 q^{27} + 5 q^{29} - 102 q^{31} + 72 q^{33} + 180 q^{35}+ \cdots + 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 58x^{6} + 143x^{5} + 2768x^{4} + 3327x^{3} + 26815x^{2} - 18834x + 191844 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7946981 \nu^{7} - 64729045 \nu^{6} - 336918776 \nu^{5} - 4847995681 \nu^{4} + \cdots - 70917424026 ) / 1721029659042 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2926380 \nu^{7} - 40088254 \nu^{6} + 149450740 \nu^{5} + 312917300 \nu^{4} + \cdots + 1428208043658 ) / 51080788967 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 137405071 \nu^{7} - 312741289 \nu^{6} + 7086236020 \nu^{5} + 10694473739 \nu^{4} + \cdots - 3120078669954 ) / 1721029659042 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 388781363 \nu^{7} - 543356075 \nu^{6} - 2828202760 \nu^{5} - 168619413446 \nu^{4} + \cdots - 15042170538180 ) / 3728897594591 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 551576327 \nu^{7} - 7163100757 \nu^{6} - 11142170132 \nu^{5} - 472865326893 \nu^{4} + \cdots - 13027686585402 ) / 3728897594591 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 626481832 \nu^{7} - 3556653871 \nu^{6} + 49021466725 \nu^{5} - 60057498121 \nu^{4} + \cdots - 20693002060158 ) / 3728897594591 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + 3\beta_{4} - 117\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -26\beta_{7} + 26\beta_{5} + 75\beta_{4} + 13\beta_{3} - 75\beta_{2} - 75\beta _1 - 270 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -46\beta_{6} + 35\beta_{5} + 46\beta_{3} + 1254\beta_{2} - 83\beta _1 - 1254 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 854\beta_{7} - 527\beta_{6} - 1930\beta_{4} + 14584\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9930\beta_{7} - 9930\beta_{5} - 24695\beta_{4} - 10787\beta_{3} + 24695\beta_{2} + 24695\beta _1 + 272898 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73083\beta_{6} - 101918\beta_{5} - 73083\beta_{3} - 1812882\beta_{2} + 226141\beta _1 + 1812882 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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
−2.63834 + 4.56975i
1.24593 − 2.15801i
−2.09075 + 3.62128i
3.98316 − 6.89904i
−2.63834 − 4.56975i
1.24593 + 2.15801i
−2.09075 − 3.62128i
3.98316 + 6.89904i
0 1.50000 − 2.59808i 0 −11.0767 0 −15.5695 − 26.9672i 0 −4.50000 − 7.79423i 0
217.2 0 1.50000 − 2.59808i 0 −7.54200 0 9.56729 + 16.5710i 0 −4.50000 − 7.79423i 0
217.3 0 1.50000 − 2.59808i 0 8.47975 0 8.72839 + 15.1180i 0 −4.50000 − 7.79423i 0
217.4 0 1.50000 − 2.59808i 0 9.13891 0 −9.22613 − 15.9801i 0 −4.50000 − 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 −11.0767 0 −15.5695 + 26.9672i 0 −4.50000 + 7.79423i 0
289.2 0 1.50000 + 2.59808i 0 −7.54200 0 9.56729 − 16.5710i 0 −4.50000 + 7.79423i 0
289.3 0 1.50000 + 2.59808i 0 8.47975 0 8.72839 − 15.1180i 0 −4.50000 + 7.79423i 0
289.4 0 1.50000 + 2.59808i 0 9.13891 0 −9.22613 + 15.9801i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.4
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.c ✓ 8
4.b odd 2 1 624.4.q.j 8
13.c even 3 1 inner 312.4.q.c ✓ 8
52.j odd 6 1 624.4.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.q.c ✓ 8 1.a even 1 1 trivial
312.4.q.c ✓ 8 13.c even 3 1 inner
624.4.q.j 8 4.b odd 2 1
624.4.q.j 8 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + T_{5}^{3} - 167T_{5}^{2} - 29T_{5} + 6474 \) acting on \(S_{4}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} - 167 T^{2} + \cdots + 6474)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 36836357184 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 2083739904 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3461564438784 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 361417208472576 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 135898192221444 \) Copy content Toggle raw display
$31$ \( (T^{4} + 51 T^{3} + \cdots + 119066944)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{4} + 276 T^{3} + \cdots + 2604993072)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 657 T^{3} + \cdots + 11631180912)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + 662 T^{3} + \cdots - 769668021)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 243 T^{3} + \cdots + 83653014720)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1162 T^{3} + \cdots - 93299454144)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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