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Properties

Label 312.6.q.a
Level $312$
Weight $6$
Character orbit 312.q
Analytic conductor $50.040$
Analytic rank $0$
Dimension $16$
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] = [312,6,Mod(217,312)] 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("312.217"); S:= CuspForms(chi, 6); N := Newforms(S);
 
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, 6, names="a")
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 312.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.0397517816\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 27696 x^{14} - 39172 x^{13} + 293413850 x^{12} + 1900944112 x^{11} + \cdots + 23\!\cdots\!33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 9) q^{3} + ( - \beta_{2} - 12) q^{5} + ( - \beta_{10} + 17 \beta_1) q^{7} - 81 \beta_1 q^{9} + (\beta_{10} + \beta_{6} - \beta_{4} + \cdots + 77) q^{11} + (\beta_{11} - \beta_{8} - \beta_{5} + \cdots - 59) q^{13}+ \cdots + (81 \beta_{12} + 81 \beta_{4} + \cdots - 6237) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 72 q^{3} - 188 q^{5} + 138 q^{7} - 648 q^{9} + 612 q^{11} - 228 q^{13} + 846 q^{15} + 1450 q^{17} + 1064 q^{19} - 2484 q^{21} + 1768 q^{23} + 7628 q^{25} + 11664 q^{27} - 3338 q^{29} - 3724 q^{31}+ \cdots - 99144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 27696 x^{14} - 39172 x^{13} + 293413850 x^{12} + 1900944112 x^{11} + \cdots + 23\!\cdots\!33 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26\!\cdots\!30 \nu^{15} + \cdots + 12\!\cdots\!86 ) / 97\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!30 \nu^{15} + \cdots - 12\!\cdots\!86 ) / 97\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 70\!\cdots\!27 \nu^{15} + \cdots - 19\!\cdots\!54 ) / 32\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82\!\cdots\!33 \nu^{15} + \cdots + 10\!\cdots\!54 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!81 \nu^{15} + \cdots - 57\!\cdots\!27 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!81 \nu^{15} + \cdots - 12\!\cdots\!37 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!16 \nu^{15} + \cdots - 77\!\cdots\!53 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58\!\cdots\!47 \nu^{15} + \cdots - 50\!\cdots\!38 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 60\!\cdots\!28 \nu^{15} + \cdots + 28\!\cdots\!35 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66\!\cdots\!92 \nu^{15} + \cdots + 42\!\cdots\!95 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 78\!\cdots\!83 \nu^{15} + \cdots - 18\!\cdots\!50 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!89 \nu^{15} + \cdots + 51\!\cdots\!74 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 88\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!43 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!57 \nu^{15} + \cdots + 57\!\cdots\!74 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 87\!\cdots\!41 \nu^{15} + \cdots - 35\!\cdots\!85 ) / 53\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + 4\beta_{11} + 3\beta_{8} - 14\beta_{4} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 3459 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9 \beta_{15} + 238 \beta_{14} - 81 \beta_{13} - 166 \beta_{12} + 144 \beta_{11} - 42 \beta_{10} + \cdots + 24366 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 626 \beta_{15} + 8469 \beta_{14} - 1636 \beta_{13} + 12017 \beta_{12} + 49008 \beta_{11} + \cdots + 22582912 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 58465 \beta_{15} + 2625166 \beta_{14} - 1033883 \beta_{13} - 1586790 \beta_{12} + 1887904 \beta_{11} + \cdots + 291574642 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9469002 \beta_{15} + 79913516 \beta_{14} - 24192080 \beta_{13} + 107150236 \beta_{12} + \cdots + 179019415541 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 376546436 \beta_{15} + 24519625498 \beta_{14} - 10255696008 \beta_{13} - 13010471102 \beta_{12} + \cdots + 3327556013116 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 113507624358 \beta_{15} + 796523693927 \beta_{14} - 283097127204 \beta_{13} + 887274209379 \beta_{12} + \cdots + 15\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2441100493539 \beta_{15} + 221250299063526 \beta_{14} - 94740169901797 \beta_{13} + \cdots + 36\!\cdots\!18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12\!\cdots\!66 \beta_{15} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13\!\cdots\!47 \beta_{15} + \cdots + 38\!\cdots\!50 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13\!\cdots\!64 \beta_{15} + \cdots + 11\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 41\!\cdots\!68 \beta_{15} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 13\!\cdots\!36 \beta_{15} + \cdots + 10\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 52\!\cdots\!63 \beta_{15} + \cdots + 40\!\cdots\!46 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
96.7832 + 0.866025i
57.7710 + 0.866025i
45.6563 + 0.866025i
−3.19246 + 0.866025i
−8.72707 + 0.866025i
−42.4540 + 0.866025i
−53.3737 + 0.866025i
−90.4632 + 0.866025i
96.7832 − 0.866025i
57.7710 − 0.866025i
45.6563 − 0.866025i
−3.19246 − 0.866025i
−8.72707 − 0.866025i
−42.4540 − 0.866025i
−53.3737 − 0.866025i
−90.4632 − 0.866025i
0 −4.50000 + 7.79423i 0 −108.283 0 111.879 + 193.781i 0 −40.5000 − 70.1481i 0
217.2 0 −4.50000 + 7.79423i 0 −69.2710 0 −86.5651 − 149.935i 0 −40.5000 − 70.1481i 0
217.3 0 −4.50000 + 7.79423i 0 −57.1563 0 28.2598 + 48.9474i 0 −40.5000 − 70.1481i 0
217.4 0 −4.50000 + 7.79423i 0 −8.30754 0 −49.9909 − 86.5868i 0 −40.5000 − 70.1481i 0
217.5 0 −4.50000 + 7.79423i 0 −2.77293 0 −21.0025 − 36.3773i 0 −40.5000 − 70.1481i 0
217.6 0 −4.50000 + 7.79423i 0 30.9540 0 88.3260 + 152.985i 0 −40.5000 − 70.1481i 0
217.7 0 −4.50000 + 7.79423i 0 41.8737 0 −62.6700 − 108.548i 0 −40.5000 − 70.1481i 0
217.8 0 −4.50000 + 7.79423i 0 78.9632 0 60.7633 + 105.245i 0 −40.5000 − 70.1481i 0
289.1 0 −4.50000 − 7.79423i 0 −108.283 0 111.879 − 193.781i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 − 7.79423i 0 −69.2710 0 −86.5651 + 149.935i 0 −40.5000 + 70.1481i 0
289.3 0 −4.50000 − 7.79423i 0 −57.1563 0 28.2598 − 48.9474i 0 −40.5000 + 70.1481i 0
289.4 0 −4.50000 − 7.79423i 0 −8.30754 0 −49.9909 + 86.5868i 0 −40.5000 + 70.1481i 0
289.5 0 −4.50000 − 7.79423i 0 −2.77293 0 −21.0025 + 36.3773i 0 −40.5000 + 70.1481i 0
289.6 0 −4.50000 − 7.79423i 0 30.9540 0 88.3260 − 152.985i 0 −40.5000 + 70.1481i 0
289.7 0 −4.50000 − 7.79423i 0 41.8737 0 −62.6700 + 108.548i 0 −40.5000 + 70.1481i 0
289.8 0 −4.50000 − 7.79423i 0 78.9632 0 60.7633 − 105.245i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.8
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 312.6.q.a ✓ 16
13.c even 3 1 inner 312.6.q.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.6.q.a ✓ 16 1.a even 1 1 trivial
312.6.q.a ✓ 16 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 94 T_{5}^{7} - 9989 T_{5}^{6} - 817852 T_{5}^{5} + 27280051 T_{5}^{4} + 1589521822 T_{5}^{3} + \cdots - 1010810592324 \) acting on \(S_{6}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 1010810592324)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 24\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 24\!\cdots\!43)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
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