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Properties

Label 325.2.s.c
Level $325$
Weight $2$
Character orbit 325.s
Analytic conductor $2.595$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,2,Mod(32,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.s (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{4} - 12 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{4} - 12 q^{6} + 24 q^{9} + 8 q^{11} - 32 q^{16} - 24 q^{19} + 32 q^{21} + 56 q^{24} + 76 q^{26} - 36 q^{29} + 8 q^{31} + 44 q^{34} - 60 q^{36} + 44 q^{39} - 52 q^{41} - 80 q^{44} - 60 q^{46} + 28 q^{49} - 68 q^{54} + 72 q^{56} - 72 q^{59} + 144 q^{64} + 24 q^{66} - 80 q^{69} - 52 q^{71} + 168 q^{74} + 8 q^{76} - 20 q^{81} - 248 q^{84} - 168 q^{86} + 60 q^{89} - 100 q^{91} - 156 q^{94} - 36 q^{96} + 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −1.39065 + 2.40867i 0.196249 + 0.732412i −2.86780 − 4.96717i 0 −2.03705 − 0.545827i 1.71325 − 0.989146i 10.3898 2.10016 − 1.21253i 0
32.2 −1.14665 + 1.98606i 0.483980 + 1.80624i −1.62962 − 2.82258i 0 −4.14225 − 1.10991i −2.78443 + 1.60759i 2.88781 −0.430177 + 0.248363i 0
32.3 −0.616283 + 1.06743i −0.608074 − 2.26936i 0.240390 + 0.416368i 0 2.79714 + 0.749491i 2.09393 − 1.20893i −3.05773 −2.18217 + 1.25988i 0
32.4 −0.457039 + 0.791616i 0.0872808 + 0.325736i 0.582230 + 1.00845i 0 −0.297749 − 0.0797815i 3.61133 − 2.08500i −2.89257 2.49959 − 1.44314i 0
32.5 −0.403239 + 0.698431i −0.724266 − 2.70300i 0.674796 + 1.16878i 0 2.17991 + 0.584105i −3.71184 + 2.14303i −2.70137 −4.18355 + 2.41538i 0
32.6 0.403239 − 0.698431i 0.724266 + 2.70300i 0.674796 + 1.16878i 0 2.17991 + 0.584105i 3.71184 − 2.14303i 2.70137 −4.18355 + 2.41538i 0
32.7 0.457039 − 0.791616i −0.0872808 − 0.325736i 0.582230 + 1.00845i 0 −0.297749 − 0.0797815i −3.61133 + 2.08500i 2.89257 2.49959 − 1.44314i 0
32.8 0.616283 − 1.06743i 0.608074 + 2.26936i 0.240390 + 0.416368i 0 2.79714 + 0.749491i −2.09393 + 1.20893i 3.05773 −2.18217 + 1.25988i 0
32.9 1.14665 − 1.98606i −0.483980 − 1.80624i −1.62962 − 2.82258i 0 −4.14225 − 1.10991i 2.78443 − 1.60759i −2.88781 −0.430177 + 0.248363i 0
32.10 1.39065 − 2.40867i −0.196249 − 0.732412i −2.86780 − 4.96717i 0 −2.03705 − 0.545827i −1.71325 + 0.989146i −10.3898 2.10016 − 1.21253i 0
132.1 −1.27299 − 2.20489i 1.76531 + 0.473014i −2.24102 + 3.88157i 0 −1.20429 − 4.49446i 3.08127 + 1.77897i 6.31926 0.294506 + 0.170033i 0
132.2 −1.09242 − 1.89212i −1.34379 − 0.360067i −1.38675 + 2.40193i 0 0.786686 + 2.93595i −2.23670 − 1.29136i 1.68998 −0.921960 − 0.532294i 0
132.3 −0.880562 − 1.52518i 3.15371 + 0.845033i −0.550780 + 0.953979i 0 −1.48821 − 5.55407i 1.20362 + 0.694909i −1.58226 6.63370 + 3.82997i 0
132.4 −0.637857 − 1.10480i −1.42434 − 0.381652i 0.186277 − 0.322641i 0 0.486878 + 1.81705i −0.0497573 − 0.0287274i −3.02670 −0.714982 − 0.412795i 0
132.5 −0.0621297 − 0.107612i 2.43488 + 0.652423i 0.992280 − 1.71868i 0 −0.0810697 − 0.302556i −2.27145 − 1.31142i −0.495119 2.90489 + 1.67714i 0
132.6 0.0621297 + 0.107612i −2.43488 − 0.652423i 0.992280 − 1.71868i 0 −0.0810697 − 0.302556i 2.27145 + 1.31142i 0.495119 2.90489 + 1.67714i 0
132.7 0.637857 + 1.10480i 1.42434 + 0.381652i 0.186277 − 0.322641i 0 0.486878 + 1.81705i 0.0497573 + 0.0287274i 3.02670 −0.714982 − 0.412795i 0
132.8 0.880562 + 1.52518i −3.15371 − 0.845033i −0.550780 + 0.953979i 0 −1.48821 − 5.55407i −1.20362 − 0.694909i 1.58226 6.63370 + 3.82997i 0
132.9 1.09242 + 1.89212i 1.34379 + 0.360067i −1.38675 + 2.40193i 0 0.786686 + 2.93595i 2.23670 + 1.29136i −1.68998 −0.921960 − 0.532294i 0
132.10 1.27299 + 2.20489i −1.76531 − 0.473014i −2.24102 + 3.88157i 0 −1.20429 − 4.49446i −3.08127 − 1.77897i −6.31926 0.294506 + 0.170033i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.10
Significant digits:
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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.s.c ✓ 40
5.b even 2 1 inner 325.2.s.c ✓ 40
5.c odd 4 2 325.2.x.c yes 40
13.f odd 12 1 325.2.x.c yes 40
65.o even 12 1 inner 325.2.s.c ✓ 40
65.s odd 12 1 325.2.x.c yes 40
65.t even 12 1 inner 325.2.s.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.s.c ✓ 40 1.a even 1 1 trivial
325.2.s.c ✓ 40 5.b even 2 1 inner
325.2.s.c ✓ 40 65.o even 12 1 inner
325.2.s.c ✓ 40 65.t even 12 1 inner
325.2.x.c yes 40 5.c odd 4 2
325.2.x.c yes 40 13.f odd 12 1
325.2.x.c yes 40 65.s odd 12 1

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