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Properties

Label 14.12.c.b
Level $14$
Weight $12$
Character orbit 14.c
Analytic conductor $10.757$
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] = [14,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128] 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: \(10.7568045278\)
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: \( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{3} \)
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.

\(f(q)\) \(=\) \( q + 32 \beta_{2} q^{2} + (67 \beta_{2} + \beta_1 - 67) q^{3} + (1024 \beta_{2} - 1024) q^{4} + ( - \beta_{5} + 952 \beta_{2}) q^{5} + ( - 32 \beta_{3} + 32 \beta_1 - 2144) q^{6} + ( - \beta_{6} - \beta_{5} + \cdots + 12827) q^{7}+ \cdots + (706864 \beta_{7} + 883580 \beta_{6} + \cdots - 68138460309) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} - 266 q^{3} - 4096 q^{4} + 3808 q^{5} - 17024 q^{6} + 110328 q^{7} - 262144 q^{8} - 503848 q^{9} - 121856 q^{10} + 920150 q^{11} - 272384 q^{12} + 997472 q^{13} + 1396800 q^{14} - 500444 q^{15}+ \cdots - 545487662552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35\!\cdots\!25 \nu^{7} + \cdots - 26\!\cdots\!44 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 170605955991169 \nu^{7} + \cdots - 87\!\cdots\!00 ) / 21\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76\!\cdots\!75 \nu^{7} + \cdots + 72\!\cdots\!68 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!37 \nu^{7} + \cdots - 17\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43\!\cdots\!45 \nu^{7} + \cdots + 27\!\cdots\!48 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{7} + \cdots + 15\!\cdots\!40 ) / 77\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} + \beta_{6} + 25\beta_{5} - \beta_{4} + 57\beta_{3} - 298656\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1756 \beta_{7} + 2195 \beta_{6} - 3301 \beta_{5} - 3740 \beta_{4} + 504127 \beta_{3} + 878 \beta_{2} + \cdots - 16931881 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 299125 \beta_{7} + 1196500 \beta_{6} + 299125 \beta_{5} + 7762535 \beta_{4} + 75274567413 \beta_{2} + \cdots - 75273969163 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 178225175 \beta_{7} - 35645035 \beta_{6} + 207672605 \beta_{5} + 35645035 \beta_{4} - 34326018287 \beta_{3} + \cdots + 71290070 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 169793717028 \beta_{7} - 212242146285 \beta_{6} - 1059317406417 \beta_{5} - 1016868977160 \beta_{4} + \cdots + 10\!\cdots\!91 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10735791588669 \beta_{7} - 42943166354676 \beta_{6} - 10735791588669 \beta_{5} + 23495063646081 \beta_{4} + \cdots + 81\!\cdots\!23 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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
−178.705 − 309.527i
−62.4133 − 108.103i
51.3434 + 88.9294i
190.275 + 329.566i
−178.705 + 309.527i
−62.4133 + 108.103i
51.3434 − 88.9294i
190.275 − 329.566i
16.0000 − 27.7128i −390.910 − 677.077i −512.000 − 886.810i 4992.28 − 8646.88i −25018.3 24330.9 − 37220.1i −32768.0 −217048. + 375939.i −159753. − 276700.i
9.2 16.0000 − 27.7128i −158.327 − 274.230i −512.000 − 886.810i −6009.47 + 10408.7i −10132.9 34168.3 + 28457.9i −32768.0 38438.9 − 66578.1i 192303. + 333079.i
9.3 16.0000 − 27.7128i 69.1868 + 119.835i −512.000 − 886.810i 274.515 − 475.474i 4427.96 −25454.3 − 36461.0i −32768.0 78999.9 − 136832.i −8784.49 − 15215.2i
9.4 16.0000 − 27.7128i 347.050 + 601.109i −512.000 − 886.810i 2646.67 − 4584.17i 22211.2 22119.1 + 38575.5i −32768.0 −152314. + 263816.i −84693.5 − 146693.i
11.1 16.0000 + 27.7128i −390.910 + 677.077i −512.000 + 886.810i 4992.28 + 8646.88i −25018.3 24330.9 + 37220.1i −32768.0 −217048. − 375939.i −159753. + 276700.i
11.2 16.0000 + 27.7128i −158.327 + 274.230i −512.000 + 886.810i −6009.47 − 10408.7i −10132.9 34168.3 − 28457.9i −32768.0 38438.9 + 66578.1i 192303. − 333079.i
11.3 16.0000 + 27.7128i 69.1868 − 119.835i −512.000 + 886.810i 274.515 + 475.474i 4427.96 −25454.3 + 36461.0i −32768.0 78999.9 + 136832.i −8784.49 + 15215.2i
11.4 16.0000 + 27.7128i 347.050 − 601.109i −512.000 + 886.810i 2646.67 + 4584.17i 22211.2 22119.1 − 38575.5i −32768.0 −152314. − 263816.i −84693.5 + 146693.i
\(n\): e.g. 2-40 or 990-1000
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 14.12.c.b ✓ 8
3.b odd 2 1 126.12.g.c 8
4.b odd 2 1 112.12.i.b 8
7.b odd 2 1 98.12.c.n 8
7.c even 3 1 inner 14.12.c.b ✓ 8
7.c even 3 1 98.12.a.i 4
7.d odd 6 1 98.12.a.h 4
7.d odd 6 1 98.12.c.n 8
21.h odd 6 1 126.12.g.c 8
28.g odd 6 1 112.12.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.c.b ✓ 8 1.a even 1 1 trivial
14.12.c.b ✓ 8 7.c even 3 1 inner
98.12.a.h 4 7.d odd 6 1
98.12.a.i 4 7.c even 3 1
98.12.c.n 8 7.b odd 2 1
98.12.c.n 8 7.d odd 6 1
112.12.i.b 8 4.b odd 2 1
112.12.i.b 8 28.g odd 6 1
126.12.g.c 8 3.b odd 2 1
126.12.g.c 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 266 T_{3}^{7} + 641596 T_{3}^{6} + 49334964 T_{3}^{5} + 328837501557 T_{3}^{4} + \cdots + 56\!\cdots\!25 \) acting on \(S_{12}^{\mathrm{new}}(14, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32 T + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 40\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 42\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 23\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
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