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Properties

Label 385.2.h.c
Level $385$
Weight $2$
Character orbit 385.h
Analytic conductor $3.074$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,2,Mod(384,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.384");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.h (of order \(2\), degree \(1\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{4} + 40 q^{9} - 8 q^{11} - 16 q^{14} - 28 q^{15} - 56 q^{16} - 20 q^{25} - 8 q^{36} + 96 q^{44} - 24 q^{49} + 24 q^{56} - 8 q^{60} - 56 q^{64} - 12 q^{70} + 56 q^{71} - 144 q^{86} + 80 q^{91} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
384.1 −2.25989 −1.54268 3.10710 1.88196 − 1.20757i 3.48629 −1.04725 + 2.42966i −2.50191 −0.620134 −4.25302 + 2.72898i
384.2 −2.25989 −1.54268 3.10710 1.88196 + 1.20757i 3.48629 −1.04725 − 2.42966i −2.50191 −0.620134 −4.25302 − 2.72898i
384.3 −2.25989 1.54268 3.10710 −1.88196 − 1.20757i −3.48629 −1.04725 + 2.42966i −2.50191 −0.620134 4.25302 + 2.72898i
384.4 −2.25989 1.54268 3.10710 −1.88196 + 1.20757i −3.48629 −1.04725 − 2.42966i −2.50191 −0.620134 4.25302 − 2.72898i
384.5 −1.94620 −1.98235 1.78771 −1.36734 − 1.76929i 3.85806 2.59047 − 0.538005i 0.413159 0.929724 2.66112 + 3.44340i
384.6 −1.94620 −1.98235 1.78771 −1.36734 + 1.76929i 3.85806 2.59047 + 0.538005i 0.413159 0.929724 2.66112 − 3.44340i
384.7 −1.94620 1.98235 1.78771 1.36734 − 1.76929i −3.85806 2.59047 − 0.538005i 0.413159 0.929724 −2.66112 + 3.44340i
384.8 −1.94620 1.98235 1.78771 1.36734 + 1.76929i −3.85806 2.59047 + 0.538005i 0.413159 0.929724 −2.66112 − 3.44340i
384.9 −1.21402 −3.12201 −0.526158 0.513816 − 2.17623i 3.79018 −1.54786 − 2.14573i 3.06680 6.74697 −0.623783 + 2.64199i
384.10 −1.21402 −3.12201 −0.526158 0.513816 + 2.17623i 3.79018 −1.54786 + 2.14573i 3.06680 6.74697 −0.623783 − 2.64199i
384.11 −1.21402 3.12201 −0.526158 −0.513816 − 2.17623i −3.79018 −1.54786 − 2.14573i 3.06680 6.74697 0.623783 + 2.64199i
384.12 −1.21402 3.12201 −0.526158 −0.513816 + 2.17623i −3.79018 −1.54786 + 2.14573i 3.06680 6.74697 0.623783 − 2.64199i
384.13 −0.794576 −0.971308 −1.36865 1.75345 − 1.38759i 0.771778 1.51554 + 2.16867i 2.67665 −2.05656 −1.39325 + 1.10254i
384.14 −0.794576 −0.971308 −1.36865 1.75345 + 1.38759i 0.771778 1.51554 − 2.16867i 2.67665 −2.05656 −1.39325 − 1.10254i
384.15 −0.794576 0.971308 −1.36865 −1.75345 − 1.38759i −0.771778 1.51554 + 2.16867i 2.67665 −2.05656 1.39325 + 1.10254i
384.16 −0.794576 0.971308 −1.36865 −1.75345 + 1.38759i −0.771778 1.51554 − 2.16867i 2.67665 −2.05656 1.39325 − 1.10254i
384.17 0.794576 −0.971308 −1.36865 1.75345 − 1.38759i −0.771778 −1.51554 − 2.16867i −2.67665 −2.05656 1.39325 − 1.10254i
384.18 0.794576 −0.971308 −1.36865 1.75345 + 1.38759i −0.771778 −1.51554 + 2.16867i −2.67665 −2.05656 1.39325 + 1.10254i
384.19 0.794576 0.971308 −1.36865 −1.75345 − 1.38759i 0.771778 −1.51554 − 2.16867i −2.67665 −2.05656 −1.39325 − 1.10254i
384.20 0.794576 0.971308 −1.36865 −1.75345 + 1.38759i 0.771778 −1.51554 + 2.16867i −2.67665 −2.05656 −1.39325 + 1.10254i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 384.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
11.b odd 2 1 inner
35.c odd 2 1 inner
55.d odd 2 1 inner
77.b even 2 1 inner
385.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.h.c ✓ 32
5.b even 2 1 inner 385.2.h.c ✓ 32
7.b odd 2 1 inner 385.2.h.c ✓ 32
11.b odd 2 1 inner 385.2.h.c ✓ 32
35.c odd 2 1 inner 385.2.h.c ✓ 32
55.d odd 2 1 inner 385.2.h.c ✓ 32
77.b even 2 1 inner 385.2.h.c ✓ 32
385.h even 2 1 inner 385.2.h.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.h.c ✓ 32 1.a even 1 1 trivial
385.2.h.c ✓ 32 5.b even 2 1 inner
385.2.h.c ✓ 32 7.b odd 2 1 inner
385.2.h.c ✓ 32 11.b odd 2 1 inner
385.2.h.c ✓ 32 35.c odd 2 1 inner
385.2.h.c ✓ 32 55.d odd 2 1 inner
385.2.h.c ✓ 32 77.b even 2 1 inner
385.2.h.c ✓ 32 385.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 11T_{2}^{6} + 39T_{2}^{4} - 49T_{2}^{2} + 18 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display