↑

Properties

Label 325.2.x.c
Level $325$
Weight $2$
Character orbit 325.x
Analytic conductor $2.595$
Analytic rank $0$
Dimension $40$
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] = [325,2,Mod(7,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 11])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{4} - 12 q^{6} - 24 q^{9} + 8 q^{11} - 32 q^{16} + 24 q^{19} + 32 q^{21} - 56 q^{24} + 76 q^{26} + 36 q^{29} + 8 q^{31} - 44 q^{34} - 60 q^{36} - 44 q^{39} - 52 q^{41} + 80 q^{44} - 60 q^{46}+ \cdots - 208 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.20489 − 1.27299i −0.473014 − 1.76531i 2.24102 + 3.88157i 0 −1.20429 + 4.49446i 1.77897 + 3.08127i − 6.31926i −0.294506 + 0.170033i 0
7.2 −1.89212 − 1.09242i 0.360067 + 1.34379i 1.38675 + 2.40193i 0 0.786686 − 2.93595i −1.29136 − 2.23670i − 1.68998i 0.921960 − 0.532294i 0
7.3 −1.52518 − 0.880562i −0.845033 − 3.15371i 0.550780 + 0.953979i 0 −1.48821 + 5.55407i 0.694909 + 1.20362i 1.58226i −6.63370 + 3.82997i 0
7.4 −1.10480 − 0.637857i 0.381652 + 1.42434i −0.186277 − 0.322641i 0 0.486878 − 1.81705i −0.0287274 − 0.0497573i 3.02670i 0.714982 − 0.412795i 0
7.5 −0.107612 − 0.0621297i −0.652423 − 2.43488i −0.992280 − 1.71868i 0 −0.0810697 + 0.302556i −1.31142 − 2.27145i 0.495119i −2.90489 + 1.67714i 0
7.6 0.107612 + 0.0621297i 0.652423 + 2.43488i −0.992280 − 1.71868i 0 −0.0810697 + 0.302556i 1.31142 + 2.27145i − 0.495119i −2.90489 + 1.67714i 0
7.7 1.10480 + 0.637857i −0.381652 − 1.42434i −0.186277 − 0.322641i 0 0.486878 − 1.81705i 0.0287274 + 0.0497573i − 3.02670i 0.714982 − 0.412795i 0
7.8 1.52518 + 0.880562i 0.845033 + 3.15371i 0.550780 + 0.953979i 0 −1.48821 + 5.55407i −0.694909 − 1.20362i − 1.58226i −6.63370 + 3.82997i 0
7.9 1.89212 + 1.09242i −0.360067 − 1.34379i 1.38675 + 2.40193i 0 0.786686 − 2.93595i 1.29136 + 2.23670i 1.68998i 0.921960 − 0.532294i 0
7.10 2.20489 + 1.27299i 0.473014 + 1.76531i 2.24102 + 3.88157i 0 −1.20429 + 4.49446i −1.77897 − 3.08127i 6.31926i −0.294506 + 0.170033i 0
93.1 −2.20489 + 1.27299i −0.473014 + 1.76531i 2.24102 − 3.88157i 0 −1.20429 − 4.49446i 1.77897 − 3.08127i 6.31926i −0.294506 − 0.170033i 0
93.2 −1.89212 + 1.09242i 0.360067 − 1.34379i 1.38675 − 2.40193i 0 0.786686 + 2.93595i −1.29136 + 2.23670i 1.68998i 0.921960 + 0.532294i 0
93.3 −1.52518 + 0.880562i −0.845033 + 3.15371i 0.550780 − 0.953979i 0 −1.48821 − 5.55407i 0.694909 − 1.20362i − 1.58226i −6.63370 − 3.82997i 0
93.4 −1.10480 + 0.637857i 0.381652 − 1.42434i −0.186277 + 0.322641i 0 0.486878 + 1.81705i −0.0287274 + 0.0497573i − 3.02670i 0.714982 + 0.412795i 0
93.5 −0.107612 + 0.0621297i −0.652423 + 2.43488i −0.992280 + 1.71868i 0 −0.0810697 − 0.302556i −1.31142 + 2.27145i − 0.495119i −2.90489 − 1.67714i 0
93.6 0.107612 − 0.0621297i 0.652423 − 2.43488i −0.992280 + 1.71868i 0 −0.0810697 − 0.302556i 1.31142 − 2.27145i 0.495119i −2.90489 − 1.67714i 0
93.7 1.10480 − 0.637857i −0.381652 + 1.42434i −0.186277 + 0.322641i 0 0.486878 + 1.81705i 0.0287274 − 0.0497573i 3.02670i 0.714982 + 0.412795i 0
93.8 1.52518 − 0.880562i 0.845033 − 3.15371i 0.550780 − 0.953979i 0 −1.48821 − 5.55407i −0.694909 + 1.20362i 1.58226i −6.63370 − 3.82997i 0
93.9 1.89212 − 1.09242i −0.360067 + 1.34379i 1.38675 − 2.40193i 0 0.786686 + 2.93595i 1.29136 − 2.23670i − 1.68998i 0.921960 + 0.532294i 0
93.10 2.20489 − 1.27299i 0.473014 − 1.76531i 2.24102 − 3.88157i 0 −1.20429 − 4.49446i −1.77897 + 3.08127i − 6.31926i −0.294506 − 0.170033i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.x.c yes 40
5.b even 2 1 inner 325.2.x.c yes 40
5.c odd 4 2 325.2.s.c ✓ 40
13.f odd 12 1 325.2.s.c ✓ 40
65.o even 12 1 inner 325.2.x.c yes 40
65.s odd 12 1 325.2.s.c ✓ 40
65.t even 12 1 inner 325.2.x.c yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.s.c ✓ 40 5.c odd 4 2
325.2.s.c ✓ 40 13.f odd 12 1
325.2.s.c ✓ 40 65.s odd 12 1
325.2.x.c yes 40 1.a even 1 1 trivial
325.2.x.c yes 40 5.b even 2 1 inner
325.2.x.c yes 40 65.o even 12 1 inner
325.2.x.c yes 40 65.t even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 32 T_{2}^{38} + 596 T_{2}^{36} - 7464 T_{2}^{34} + 69980 T_{2}^{32} - 505888 T_{2}^{30} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display