↑

Properties

Label 80.14.n.c
Level $80$
Weight $14$
Character orbit 80.n
Analytic conductor $85.785$
Analytic rank $0$
Dimension $52$
Inner twists $4$

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] = [80,14,Mod(47,80)] 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("80.47"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(85.7847431615\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 62716 q^{5} - 43227596 q^{13} - 10158316 q^{17} + 2202830248 q^{21} - 1097115132 q^{25} + 27081370264 q^{33} - 55596330460 q^{37} + 58262341064 q^{41} + 125937904744 q^{45} + 108209257012 q^{53}+ \cdots + 15136403213972 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1565.59 − 1565.59i 0 34115.5 − 7539.06i 0 76910.7 − 76910.7i 0 3.30784e6i 0
47.2 0 −1545.65 − 1545.65i 0 15378.5 − 31372.1i 0 −358370. + 358370.i 0 3.18374e6i 0
47.3 0 −1463.66 − 1463.66i 0 −31990.7 + 14046.3i 0 102008. − 102008.i 0 2.69029e6i 0
47.4 0 −1421.82 − 1421.82i 0 1532.63 + 34904.9i 0 18782.2 − 18782.2i 0 2.44884e6i 0
47.5 0 −1062.98 − 1062.98i 0 18015.3 + 29935.8i 0 −176547. + 176547.i 0 665513.i 0
47.6 0 −962.037 − 962.037i 0 −29562.4 − 18621.7i 0 113638. − 113638.i 0 256707.i 0
47.7 0 −901.283 − 901.283i 0 34926.1 − 931.724i 0 337272. − 337272.i 0 30300.2i 0
47.8 0 −835.990 − 835.990i 0 −19673.6 − 28873.1i 0 250393. − 250393.i 0 − 196565.i 0
47.9 0 −800.198 − 800.198i 0 −3918.86 − 34718.1i 0 −313530. + 313530.i 0 − 313689.i 0
47.10 0 −440.767 − 440.767i 0 −22264.1 + 26926.1i 0 −348507. + 348507.i 0 − 1.20577e6i 0
47.11 0 −290.854 − 290.854i 0 −29513.4 + 18699.3i 0 −217272. + 217272.i 0 − 1.42513e6i 0
47.12 0 −106.498 − 106.498i 0 25101.0 − 24303.2i 0 −68827.3 + 68827.3i 0 − 1.57164e6i 0
47.13 0 −96.4564 − 96.4564i 0 23532.9 + 25824.5i 0 230129. − 230129.i 0 − 1.57572e6i 0
47.14 0 96.4564 + 96.4564i 0 23532.9 + 25824.5i 0 −230129. + 230129.i 0 − 1.57572e6i 0
47.15 0 106.498 + 106.498i 0 25101.0 − 24303.2i 0 68827.3 − 68827.3i 0 − 1.57164e6i 0
47.16 0 290.854 + 290.854i 0 −29513.4 + 18699.3i 0 217272. − 217272.i 0 − 1.42513e6i 0
47.17 0 440.767 + 440.767i 0 −22264.1 + 26926.1i 0 348507. − 348507.i 0 − 1.20577e6i 0
47.18 0 800.198 + 800.198i 0 −3918.86 − 34718.1i 0 313530. − 313530.i 0 − 313689.i 0
47.19 0 835.990 + 835.990i 0 −19673.6 − 28873.1i 0 −250393. + 250393.i 0 − 196565.i 0
47.20 0 901.283 + 901.283i 0 34926.1 − 931.724i 0 −337272. + 337272.i 0 30300.2i 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.14.n.c ✓ 52
4.b odd 2 1 inner 80.14.n.c ✓ 52
5.c odd 4 1 inner 80.14.n.c ✓ 52
20.e even 4 1 inner 80.14.n.c ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.14.n.c ✓ 52 1.a even 1 1 trivial
80.14.n.c ✓ 52 4.b odd 2 1 inner
80.14.n.c ✓ 52 5.c odd 4 1 inner
80.14.n.c ✓ 52 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{52} + 96512937596228 T_{3}^{48} + \cdots + 18\!\cdots\!56 \) acting on \(S_{14}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display