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Properties

Label 17.44.d.a
Level $17$
Weight $44$
Character orbit 17.d
Analytic conductor $199.088$
Analytic rank $0$
Dimension $252$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [17,44,Mod(2,17)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 44, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17.2"); S:= CuspForms(chi, 44); N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(199.087672088\)
Analytic rank: \(0\)
Dimension: \(252\)
Relative dimension: \(63\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 252 q - 4 q^{2} - 4 q^{3} + 18\!\cdots\!80 q^{5} + 22\!\cdots\!40 q^{6} - 4 q^{7} + 35184372088828 q^{8} + 63\!\cdots\!24 q^{9} - 91\!\cdots\!16 q^{10} + 31\!\cdots\!84 q^{11} + 16\!\cdots\!88 q^{12} - 10\!\cdots\!24 q^{14}+ \cdots + 44\!\cdots\!04 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −4.09676e6 − 4.09676e6i −3.15385e10 + 1.30637e10i 2.47708e13i −1.80315e14 − 4.35320e14i 1.82724e17 + 7.56869e16i −1.01629e18 + 2.45353e18i 6.54443e19 − 6.54443e19i 5.91905e20 − 5.91905e20i −1.04469e21 + 2.52211e21i
2.2 −3.97442e6 − 3.97442e6i −7.19330e9 + 2.97956e9i 2.27960e13i −1.19474e14 − 2.88437e14i 4.04313e16 + 1.67472e16i 4.77964e17 − 1.15391e18i 5.56415e19 − 5.56415e19i −1.89247e20 + 1.89247e20i −6.71528e20 + 1.62121e21i
2.3 −3.96922e6 − 3.96922e6i 9.09897e9 − 3.76891e9i 2.27134e13i 6.73776e14 + 1.62664e15i −5.10755e16 − 2.11562e16i −1.81557e16 + 4.38318e16i 5.52408e19 − 5.52408e19i −1.63526e20 + 1.63526e20i 3.78213e21 − 9.13086e21i
2.4 −3.80683e6 − 3.80683e6i 2.44978e10 − 1.01473e10i 2.01878e13i −5.36168e13 − 1.29442e14i −1.31888e17 − 5.46298e16i −6.31645e17 + 1.52492e18i 4.33664e19 − 4.33664e19i 2.65060e20 − 2.65060e20i −2.88655e20 + 6.96875e20i
2.5 −3.60575e6 − 3.60575e6i 1.23830e10 − 5.12919e9i 1.72067e13i −5.94068e14 − 1.43421e15i −6.31444e16 − 2.61553e16i −2.26819e17 + 5.47588e17i 3.03267e19 − 3.03267e19i −1.05084e20 + 1.05084e20i −3.02933e21 + 7.31345e21i
2.6 −3.48662e6 − 3.48662e6i 2.29447e10 − 9.50399e9i 1.55169e13i 2.88501e14 + 6.96503e14i −1.13136e17 − 4.68624e16i 9.32651e17 − 2.25162e18i 2.34328e19 − 2.34328e19i 2.04019e20 − 2.04019e20i 1.42255e21 − 3.43433e21i
2.7 −3.45514e6 − 3.45514e6i −6.95589e9 + 2.88122e9i 1.50799e13i −2.51273e14 − 6.06627e14i 3.39886e16 + 1.40785e16i −1.03940e17 + 2.50933e17i 2.17115e19 − 2.17115e19i −1.92030e20 + 1.92030e20i −1.22780e21 + 2.96417e21i
2.8 −3.37788e6 − 3.37788e6i −2.88895e10 + 1.19664e10i 1.40240e13i 4.65586e14 + 1.12402e15i 1.38006e17 + 5.71640e16i 1.00699e18 − 2.43109e18i 1.76593e19 − 1.76593e19i 4.59294e20 − 4.59294e20i 2.22412e21 − 5.36951e21i
2.9 −3.30953e6 − 3.30953e6i −1.25934e10 + 5.21635e9i 1.31099e13i 4.61615e14 + 1.11444e15i 5.89418e16 + 2.44145e16i −7.15817e17 + 1.72813e18i 1.42765e19 − 1.42765e19i −1.00730e20 + 1.00730e20i 2.16054e21 − 5.21599e21i
2.10 −3.18305e6 − 3.18305e6i −2.05920e10 + 8.52950e9i 1.14675e13i −7.09250e14 − 1.71228e15i 9.26951e16 + 3.83956e16i 6.03290e17 − 1.45647e18i 8.50312e18 − 8.50312e18i 1.19166e20 − 1.19166e20i −3.19269e21 + 7.70785e21i
2.11 −3.17742e6 − 3.17742e6i 2.68747e10 − 1.11319e10i 1.13958e13i −5.29509e14 − 1.27835e15i −1.20763e17 − 5.00216e16i 8.49333e17 − 2.05047e18i 8.26049e18 − 8.26049e18i 3.66220e20 − 3.66220e20i −2.37937e21 + 5.74432e21i
2.12 −3.13678e6 − 3.13678e6i −1.51005e10 + 6.25485e9i 1.08827e13i 5.61563e14 + 1.35573e15i 6.69872e16 + 2.77470e16i −1.25921e17 + 3.04000e17i 6.54523e18 − 6.54523e18i −4.32097e19 + 4.32097e19i 2.49114e21 − 6.01414e21i
2.13 −2.73867e6 − 2.73867e6i 3.46866e9 − 1.43676e9i 6.20458e12i −1.03368e14 − 2.49552e14i −1.34344e16 − 5.56469e15i −9.66419e17 + 2.33314e18i −7.09731e18 + 7.09731e18i −2.22145e20 + 2.22145e20i −4.00352e20 + 9.66534e20i
2.14 −2.58405e6 − 2.58405e6i 2.39234e10 − 9.90941e9i 4.55856e12i 5.45301e14 + 1.31647e15i −8.74259e16 − 3.62130e16i −4.35755e17 + 1.05201e18i −1.09500e19 + 1.09500e19i 2.42022e20 − 2.42022e20i 1.99275e21 − 4.81092e21i
2.15 −2.55044e6 − 2.55044e6i −2.41593e10 + 1.00071e10i 4.21338e12i −1.28853e14 − 3.11079e14i 8.71394e16 + 3.60943e16i −2.61539e16 + 6.31412e16i −1.16879e19 + 1.16879e19i 2.51418e20 − 2.51418e20i −4.64756e20 + 1.12202e21i
2.16 −2.53351e6 − 2.53351e6i 6.01529e9 − 2.49162e9i 4.04122e12i 8.06913e13 + 1.94806e14i −2.15523e16 − 8.92726e15i 3.73543e17 − 9.01812e17i −1.20465e19 + 1.20465e19i −2.02137e20 + 2.02137e20i 2.89110e20 − 6.97974e20i
2.17 −2.19852e6 − 2.19852e6i −2.04695e10 + 8.47875e9i 8.70890e11i −5.58529e14 − 1.34841e15i 6.36433e16 + 2.63619e16i −6.66859e17 + 1.60994e18i −1.74237e19 + 1.74237e19i 1.14999e20 − 1.14999e20i −1.73657e21 + 4.19244e21i
2.18 −2.10968e6 − 2.10968e6i 6.34475e9 − 2.62808e9i 1.05407e11i 3.30196e14 + 7.97165e14i −1.89298e16 − 7.84098e15i 6.00667e17 − 1.45014e18i −1.83346e19 + 1.83346e19i −1.98764e20 + 1.98764e20i 9.85153e20 − 2.37837e21i
2.19 −2.03028e6 − 2.03028e6i 3.13117e10 − 1.29697e10i − 5.52058e11i −1.81415e14 − 4.37975e14i −8.99034e16 − 3.72392e16i −2.07922e17 + 5.01968e17i −1.89793e19 + 1.89793e19i 5.80094e20 − 5.80094e20i −5.20887e20 + 1.25753e21i
2.20 −1.99866e6 − 1.99866e6i 1.06333e10 − 4.40446e9i − 8.06823e11i −5.67672e14 − 1.37048e15i −3.00554e16 − 1.24493e16i −6.22846e17 + 1.50368e18i −1.91929e19 + 1.91929e19i −1.38445e20 + 1.38445e20i −1.60454e21 + 3.87371e21i
See next 80 embeddings (of 252 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.63
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.44.d.a ✓ 252
17.d even 8 1 inner 17.44.d.a ✓ 252
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.44.d.a ✓ 252 1.a even 1 1 trivial
17.44.d.a ✓ 252 17.d even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{44}^{\mathrm{new}}(17, [\chi])\).