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Properties

Label 14.14.c.a
Level $14$
Weight $14$
Character orbit 14.c
Analytic conductor $15.012$
Analytic rank $0$
Dimension $8$
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] = [14,14,Mod(9,14)] 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("14.9"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-256] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.0123300533\)
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: \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - 64 \beta_1 - 64) q^{2} + (\beta_{4} + 45 \beta_1) q^{3} + 4096 \beta_1 q^{4} + (\beta_{7} + 5 \beta_{4} + \cdots - 16102) q^{5} + ( - 64 \beta_{2} + 2880) q^{6} + (7 \beta_{7} - 3 \beta_{6} + \cdots + 11651) q^{7}+ \cdots + ( - 30229148 \beta_{6} + \cdots - 3231197379261) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 256 q^{2} - 182 q^{3} - 16384 q^{4} - 64400 q^{5} + 23296 q^{6} - 113736 q^{7} + 2097152 q^{8} - 3983752 q^{9} - 4121600 q^{10} - 1008790 q^{11} - 745472 q^{12} + 53807264 q^{13} + 23506560 q^{14}+ \cdots - 25837834576328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27\!\cdots\!26 \nu^{7} + \cdots - 92\!\cdots\!05 ) / 90\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!70 \nu^{7} + \cdots - 42\!\cdots\!05 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!30 \nu^{7} + \cdots - 84\!\cdots\!45 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99\!\cdots\!39 \nu^{7} + \cdots - 33\!\cdots\!05 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!37 \nu^{7} + \cdots - 14\!\cdots\!70 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!67 \nu^{7} + \cdots + 21\!\cdots\!15 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{7} + \cdots - 35\!\cdots\!95 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 2\beta_{5} - 9\beta_{4} - 4\beta_{3} + 8\beta_{2} + 171\beta _1 + 174 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -87\beta_{7} + 386\beta_{6} + 193\beta_{5} - 5751\beta_{4} + 193\beta_{3} + 193\beta_{2} + 29070783\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 813235\beta_{6} - 813235\beta_{5} + 2971559\beta_{3} - 16649389\beta_{2} - 56301080754 ) / 336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 26739030 \beta_{7} - 47369614 \beta_{6} - 94739228 \beta_{5} + 1122208434 \beta_{4} + \cdots - 5071555289829 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 859127137203 \beta_{7} - 1474522292642 \beta_{6} - 737261146321 \beta_{5} + \cdots - 64\!\cdots\!31 \beta_1 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 176934179918233 \beta_{6} + 176934179918233 \beta_{5} - 502838085390899 \beta_{3} + \cdots + 17\!\cdots\!76 ) / 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69\!\cdots\!09 \beta_{7} + \cdots + 63\!\cdots\!34 ) / 336 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−3.53342 + 6.12007i
−258.759 + 448.183i
481.229 − 833.513i
−217.937 + 377.477i
−3.53342 − 6.12007i
−258.759 − 448.183i
481.229 + 833.513i
−217.937 − 377.477i
−32.0000 + 55.4256i −855.669 − 1482.06i −2048.00 − 3547.24i 12545.8 − 21729.9i 109526. 48813.9 − 307419.i 262144. −667176. + 1.15558e6i 802928. + 1.39071e6i
9.2 −32.0000 + 55.4256i −679.656 − 1177.20i −2048.00 − 3547.24i −29011.3 + 50249.0i 86996.0 −173759. + 258257.i 262144. −126703. + 219456.i −1.85672e6 − 3.21593e6i
9.3 −32.0000 + 55.4256i 301.970 + 523.027i −2048.00 − 3547.24i 2521.96 − 4368.16i −38652.2 302038. + 75247.5i 262144. 614790. − 1.06485e6i 161406. + 279563.i
9.4 −32.0000 + 55.4256i 1142.35 + 1978.62i −2048.00 − 3547.24i −18256.5 + 31621.1i −146221. −233961. − 205308.i 262144. −1.81279e6 + 3.13984e6i −1.16841e6 − 2.02375e6i
11.1 −32.0000 − 55.4256i −855.669 + 1482.06i −2048.00 + 3547.24i 12545.8 + 21729.9i 109526. 48813.9 + 307419.i 262144. −667176. − 1.15558e6i 802928. − 1.39071e6i
11.2 −32.0000 − 55.4256i −679.656 + 1177.20i −2048.00 + 3547.24i −29011.3 − 50249.0i 86996.0 −173759. − 258257.i 262144. −126703. − 219456.i −1.85672e6 + 3.21593e6i
11.3 −32.0000 − 55.4256i 301.970 − 523.027i −2048.00 + 3547.24i 2521.96 + 4368.16i −38652.2 302038. − 75247.5i 262144. 614790. + 1.06485e6i 161406. − 279563.i
11.4 −32.0000 − 55.4256i 1142.35 − 1978.62i −2048.00 + 3547.24i −18256.5 − 31621.1i −146221. −233961. + 205308.i 262144. −1.81279e6 − 3.13984e6i −1.16841e6 + 2.02375e6i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.14.c.a ✓ 8
3.b odd 2 1 126.14.g.d 8
7.b odd 2 1 98.14.c.n 8
7.c even 3 1 inner 14.14.c.a ✓ 8
7.c even 3 1 98.14.a.l 4
7.d odd 6 1 98.14.a.k 4
7.d odd 6 1 98.14.c.n 8
21.h odd 6 1 126.14.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.a ✓ 8 1.a even 1 1 trivial
14.14.c.a ✓ 8 7.c even 3 1 inner
98.14.a.k 4 7.d odd 6 1
98.14.a.l 4 7.c even 3 1
98.14.c.n 8 7.b odd 2 1
98.14.c.n 8 7.d odd 6 1
126.14.g.d 8 3.b odd 2 1
126.14.g.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 182 T_{3}^{7} + 5197084 T_{3}^{6} + 4025617932 T_{3}^{5} + 23908528667637 T_{3}^{4} + \cdots + 10\!\cdots\!25 \) acting on \(S_{14}^{\mathrm{new}}(14, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64 T + 4096)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 71\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 88\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 94\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 47\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 79\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
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