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Properties

Label 14.50.c.a
Level $14$
Weight $50$
Character orbit 14.c
Analytic conductor $212.893$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [14,50,Mod(9,14)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 50, names="a")
 
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, 50); N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-268435456] 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: \(212.892687139\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 268435456 q^{2} - 339592472912 q^{3} - 45\!\cdots\!96 q^{4} - 15\!\cdots\!08 q^{5} + 11\!\cdots\!84 q^{6} - 10\!\cdots\!92 q^{7} + 15\!\cdots\!72 q^{8} - 11\!\cdots\!64 q^{9} - 26\!\cdots\!28 q^{10}+ \cdots + 41\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −8.38861e6 + 1.45295e7i −4.27353e11 − 7.40198e11i −1.40737e14 − 2.43764e14i 1.32602e17 − 2.29673e17i 1.43396e19 7.23922e19 + 5.01680e20i 4.72237e21 −2.45612e23 + 4.25412e23i 2.22469e24 + 3.85327e24i
9.2 −8.38861e6 + 1.45295e7i −3.97856e11 − 6.89106e11i −1.40737e14 − 2.43764e14i −6.17949e16 + 1.07032e17i 1.33498e19 5.06844e20 − 5.70075e18i 4.72237e21 −1.96928e23 + 3.41090e23i −1.03675e24 − 1.79570e24i
9.3 −8.38861e6 + 1.45295e7i −3.69650e11 − 6.40252e11i −1.40737e14 − 2.43764e14i −6.61451e16 + 1.14567e17i 1.24034e19 −4.91791e20 − 1.22740e20i 4.72237e21 −1.53632e23 + 2.66099e23i −1.10973e24 − 1.92211e24i
9.4 −8.38861e6 + 1.45295e7i −3.35575e11 − 5.81233e11i −1.40737e14 − 2.43764e14i 5.24436e16 − 9.08349e16i 1.12600e19 −2.84063e20 − 4.19800e20i 4.72237e21 −1.05572e23 + 1.82856e23i 8.79857e23 + 1.52396e24i
9.5 −8.38861e6 + 1.45295e7i −2.66782e11 − 4.62081e11i −1.40737e14 − 2.43764e14i 1.37804e15 − 2.38684e15i 8.95173e18 −2.64114e19 + 5.06188e20i 4.72237e21 −2.26961e22 + 3.93108e22i 2.31198e22 + 4.00446e22i
9.6 −8.38861e6 + 1.45295e7i −1.33415e11 − 2.31081e11i −1.40737e14 − 2.43764e14i 6.73220e16 − 1.16605e17i 4.47666e18 2.30631e20 − 4.51368e20i 4.72237e21 8.40506e22 − 1.45580e23i 1.12948e24 + 1.95631e24i
9.7 −8.38861e6 + 1.45295e7i −3.47729e10 − 6.02285e10i −1.40737e14 − 2.43764e14i −1.02602e17 + 1.77713e17i 1.16679e18 −1.64679e20 − 4.79379e20i 4.72237e21 1.17231e23 − 2.03051e23i −1.72138e24 − 2.98152e24i
9.8 −8.38861e6 + 1.45295e7i −1.33330e10 − 2.30934e10i −1.40737e14 − 2.43764e14i −9.67735e16 + 1.67617e17i 4.47380e17 4.52496e20 − 2.28409e20i 4.72237e21 1.19294e23 − 2.06623e23i −1.62359e24 − 2.81214e24i
9.9 −8.38861e6 + 1.45295e7i 1.34514e10 + 2.32986e10i −1.40737e14 − 2.43764e14i 2.86502e16 − 4.96236e16i −4.51355e17 −4.48250e20 + 2.36634e20i 4.72237e21 1.19288e23 − 2.06612e23i 4.80671e23 + 8.32547e23i
9.10 −8.38861e6 + 1.45295e7i 5.10941e10 + 8.84976e10i −1.40737e14 − 2.43764e14i −6.36376e16 + 1.10224e17i −1.71443e18 −7.33050e19 + 5.01548e20i 4.72237e21 1.14428e23 − 1.98196e23i −1.06766e24 − 1.84924e24i
9.11 −8.38861e6 + 1.45295e7i 1.18607e11 + 2.05434e11i −1.40737e14 − 2.43764e14i 1.24153e17 − 2.15040e17i −3.97981e18 −5.01750e20 − 7.19092e19i 4.72237e21 9.15142e22 − 1.58507e23i 2.08295e24 + 3.60777e24i
9.12 −8.38861e6 + 1.45295e7i 2.32940e11 + 4.03463e11i −1.40737e14 − 2.43764e14i 6.54039e16 − 1.13283e17i −7.81616e18 3.98026e20 + 3.13846e20i 4.72237e21 1.11279e22 − 1.92740e22i 1.09730e24 + 1.90057e24i
9.13 −8.38861e6 + 1.45295e7i 2.75646e11 + 4.77432e11i −1.40737e14 − 2.43764e14i −6.53558e16 + 1.13200e17i −9.24913e18 −5.01005e20 + 7.69273e19i 4.72237e21 −3.23113e22 + 5.59647e22i −1.09649e24 − 1.89917e24i
9.14 −8.38861e6 + 1.45295e7i 3.24593e11 + 5.62211e11i −1.40737e14 − 2.43764e14i 2.22157e16 − 3.84787e16i −1.08915e19 3.26450e20 − 3.87755e20i 4.72237e21 −9.10710e22 + 1.57740e23i 3.72717e23 + 6.45565e23i
9.15 −8.38861e6 + 1.45295e7i 3.73043e11 + 6.46130e11i −1.40737e14 − 2.43764e14i −1.13076e16 + 1.95854e16i −1.25173e19 −2.30279e20 − 4.51547e20i 4.72237e21 −1.58673e23 + 2.74830e23i −1.89711e23 − 3.28589e23i
9.16 −8.38861e6 + 1.45295e7i 4.19567e11 + 7.26711e11i −1.40737e14 − 2.43764e14i −1.05072e17 + 1.81991e17i −1.40783e19 2.32831e20 + 4.50237e20i 4.72237e21 −2.32423e23 + 4.02568e23i −1.76282e24 − 3.05329e24i
11.1 −8.38861e6 − 1.45295e7i −4.27353e11 + 7.40198e11i −1.40737e14 + 2.43764e14i 1.32602e17 + 2.29673e17i 1.43396e19 7.23922e19 − 5.01680e20i 4.72237e21 −2.45612e23 − 4.25412e23i 2.22469e24 − 3.85327e24i
11.2 −8.38861e6 − 1.45295e7i −3.97856e11 + 6.89106e11i −1.40737e14 + 2.43764e14i −6.17949e16 − 1.07032e17i 1.33498e19 5.06844e20 + 5.70075e18i 4.72237e21 −1.96928e23 − 3.41090e23i −1.03675e24 + 1.79570e24i
11.3 −8.38861e6 − 1.45295e7i −3.69650e11 + 6.40252e11i −1.40737e14 + 2.43764e14i −6.61451e16 − 1.14567e17i 1.24034e19 −4.91791e20 + 1.22740e20i 4.72237e21 −1.53632e23 − 2.66099e23i −1.10973e24 + 1.92211e24i
11.4 −8.38861e6 − 1.45295e7i −3.35575e11 + 5.81233e11i −1.40737e14 + 2.43764e14i 5.24436e16 + 9.08349e16i 1.12600e19 −2.84063e20 + 4.19800e20i 4.72237e21 −1.05572e23 − 1.82856e23i 8.79857e23 − 1.52396e24i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.16
Significant digits:
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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 14.50.c.a ✓ 32
7.c even 3 1 inner 14.50.c.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.50.c.a ✓ 32 1.a even 1 1 trivial
14.50.c.a ✓ 32 7.c even 3 1 inner