↑

Properties

Label 7.73.d.a
Level $7$
Weight $73$
Character orbit 7.d
Analytic conductor $229.811$
Analytic rank $0$
Dimension $94$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7,73,Mod(3,7)] 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("7.3"); S:= CuspForms(chi, 73); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 73, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 73 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\), degree \(2\), 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: \(229.811218987\)
Analytic rank: \(0\)
Dimension: \(94\)
Relative dimension: \(47\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 94 q - 40225339396 q^{2} - 3 q^{3} - 11\!\cdots\!24 q^{4} + 10\!\cdots\!25 q^{5} + 37\!\cdots\!18 q^{7} + 31\!\cdots\!28 q^{8} + 33\!\cdots\!54 q^{9} + 75\!\cdots\!96 q^{10} - 97\!\cdots\!53 q^{11} + 26\!\cdots\!60 q^{12}+ \cdots - 33\!\cdots\!20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −6.73181e10 − 1.16598e11i 9.92736e16 + 5.73157e16i −6.70228e21 + 1.16087e22i −5.62149e24 + 3.24557e24i − 1.54335e28i 2.39748e30 − 1.13304e30i 1.16894e33 −4.69403e33 − 8.13030e33i 7.56857e35 + 4.36972e35i
3.2 −6.45949e10 − 1.11882e11i −2.06821e17 − 1.19408e17i −5.98382e21 + 1.03643e22i −1.83518e25 + 1.05954e25i 3.08527e28i −2.49376e30 + 9.01579e29i 9.36017e32 1.72525e34 + 2.98822e34i 2.37087e36 + 1.36882e36i
3.3 −6.33615e10 − 1.09745e11i −2.08729e17 − 1.20510e17i −5.66819e21 + 9.81759e21i 2.40344e25 − 1.38763e25i 3.05427e28i 2.56584e30 − 6.69453e29i 8.38147e32 1.77810e34 + 3.07976e34i −3.04572e36 − 1.75845e36i
3.4 −5.95630e10 − 1.03166e11i 3.67605e16 + 2.12237e16i −4.73432e21 + 8.20008e21i 7.35030e24 − 4.24370e24i − 5.05659e27i −2.36696e30 − 1.19548e30i 5.65404e32 −1.03633e34 − 1.79498e34i −8.75612e35 − 5.05535e35i
3.5 −5.69777e10 − 9.86883e10i −5.98994e16 − 3.45830e16i −4.13174e21 + 7.15638e21i 4.41304e23 − 2.54787e23i 7.88183e27i 8.31430e29 + 2.51802e30i 4.03528e32 −8.87224e33 − 1.53672e34i −5.02890e34 − 2.90344e34i
3.6 −5.68332e10 − 9.84380e10i 2.13954e17 + 1.23526e17i −4.09884e21 + 7.09941e21i 1.94955e25 − 1.12557e25i − 2.80816e28i −1.19895e30 + 2.36521e30i 3.95028e32 1.92532e34 + 3.33476e34i −2.21598e36 − 1.27940e36i
3.7 −5.40647e10 − 9.36427e10i 1.43064e17 + 8.25981e16i −3.48479e21 + 6.03583e21i −2.26065e25 + 1.30519e25i − 1.78625e28i −7.15525e29 + 2.55337e30i 2.42990e32 2.38069e33 + 4.12348e33i 2.44443e36 + 1.41129e36i
3.8 −5.19047e10 − 8.99016e10i −1.35838e17 − 7.84263e16i −3.02702e21 + 5.24295e21i −6.66349e24 + 3.84717e24i 1.62828e28i 4.70173e29 − 2.60972e30i 1.38240e32 1.03717e33 + 1.79643e33i 6.91733e35 + 3.99372e35i
3.9 −4.91517e10 − 8.51332e10i 2.34372e17 + 1.35315e17i −2.47059e21 + 4.27918e21i 6.29003e23 − 3.63155e23i − 2.66038e28i 1.95924e30 − 1.78691e30i 2.15097e31 2.53560e34 + 4.39179e34i −6.18331e34 − 3.56993e34i
3.10 −4.26066e10 − 7.37968e10i −9.31537e16 − 5.37823e16i −1.26946e21 + 2.19877e21i −1.89205e25 + 1.09237e25i 9.16592e27i 2.46490e30 − 9.77727e29i −1.86058e32 −5.47913e33 − 9.49012e33i 1.61227e36 + 9.30847e35i
3.11 −4.21026e10 − 7.29239e10i −1.63894e17 − 9.46244e16i −1.18408e21 + 2.05088e21i 1.06660e25 − 6.15803e24i 1.59357e28i −2.14341e30 + 1.56125e30i −1.98237e32 6.64336e33 + 1.15066e34i −8.98134e35 − 5.18538e35i
3.12 −4.12816e10 − 7.15018e10i 8.55669e15 + 4.94020e15i −1.04715e21 + 1.81372e21i 9.17502e24 − 5.29720e24i − 8.15758e26i 2.04067e30 + 1.69333e30i −2.16981e32 −1.12154e34 − 1.94256e34i −7.57518e35 − 4.37353e35i
3.13 −4.10314e10 − 7.10684e10i 1.65207e17 + 9.53821e16i −1.00596e21 + 1.74238e21i −1.33118e25 + 7.68557e24i − 1.56546e28i −1.69176e30 − 2.04197e30i −2.22426e32 6.93130e33 + 1.20054e34i 1.09240e36 + 6.30698e35i
3.14 −3.40245e10 − 5.89322e10i 6.33477e16 + 3.65738e16i 4.58501e19 − 7.94147e19i 2.35800e25 − 1.36139e25i − 4.97763e27i 6.36491e29 − 2.57421e30i −3.27592e32 −8.58891e33 − 1.48764e34i −1.60459e36 − 9.26412e35i
3.15 −2.79302e10 − 4.83765e10i −8.49540e15 − 4.90482e15i 8.00990e20 − 1.38736e21i −1.05772e25 + 6.10674e24i 5.47971e26i −2.51894e30 − 8.28638e29i −3.53281e32 −1.12161e34 − 1.94268e34i 5.90846e35 + 3.41125e35i
3.16 −2.76945e10 − 4.79682e10i −2.49335e17 − 1.43954e17i 8.27215e20 − 1.43278e21i −9.81207e24 + 5.66500e24i 1.59469e28i 1.42891e30 + 2.23381e30i −3.53204e32 3.01812e34 + 5.22754e34i 5.43481e35 + 3.13779e35i
3.17 −2.62943e10 − 4.55431e10i −1.95303e17 − 1.12758e17i 9.78400e20 − 1.69464e21i 8.15167e24 − 4.70637e24i 1.18596e28i −6.80614e29 − 2.56290e30i −3.51248e32 1.41645e34 + 2.45337e34i −4.28685e35 − 2.47502e35i
3.18 −2.50499e10 − 4.33877e10i 1.30422e17 + 7.52990e16i 1.10619e21 − 1.91597e21i −9.42167e23 + 5.43961e23i − 7.54493e27i 2.59301e30 + 5.54945e29i −3.47429e32 7.56699e31 + 1.31064e32i 4.72024e34 + 2.72523e34i
3.19 −1.83735e10 − 3.18238e10i 1.60301e17 + 9.25500e16i 1.68601e21 − 2.92026e21i 3.72525e24 − 2.15078e24i − 6.80186e27i −1.55243e30 + 2.14980e30i −2.97444e32 5.86680e33 + 1.01616e34i −1.36892e35 − 7.90345e34i
3.20 −1.31988e10 − 2.28611e10i −6.77581e16 − 3.91202e16i 2.01276e21 − 3.48621e21i −1.91507e25 + 1.10566e25i 2.06536e27i −5.93295e29 + 2.58451e30i −2.30924e32 −8.20343e33 − 1.42087e34i 5.05533e35 + 2.91870e35i
See all 94 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.47
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.73.d.a ✓ 94
7.d odd 6 1 inner 7.73.d.a ✓ 94
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.73.d.a ✓ 94 1.a even 1 1 trivial
7.73.d.a ✓ 94 7.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{73}^{\mathrm{new}}(7, [\chi])\).