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Properties

Label 312.2.m.c
Level $312$
Weight $2$
Character orbit 312.m
Analytic conductor $2.491$
Analytic rank $0$
Dimension $24$
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] = [312,2,Mod(181,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.m (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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9} - 4 q^{10} + 4 q^{12} + 16 q^{14} + 28 q^{16} + 32 q^{17} - 44 q^{22} + 40 q^{25} + 4 q^{30} + 16 q^{38} - 8 q^{39} + 12 q^{40} - 32 q^{42} - 16 q^{48} - 72 q^{49} - 12 q^{52} + 16 q^{55} - 32 q^{56} - 88 q^{62} + 24 q^{64} + 28 q^{66} - 8 q^{68} - 24 q^{74} - 16 q^{78} - 64 q^{79} + 24 q^{81} + 68 q^{82} - 12 q^{88} + 4 q^{90} - 40 q^{92} + 44 q^{94} - 128 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −1.41087 − 0.0972570i 1.00000i 1.98108 + 0.274433i −0.859867 0.0972570 − 1.41087i 2.42446i −2.76835 − 0.579863i −1.00000 1.21316 + 0.0836282i
181.2 −1.41087 + 0.0972570i − 1.00000i 1.98108 − 0.274433i −0.859867 0.0972570 + 1.41087i − 2.42446i −2.76835 + 0.579863i −1.00000 1.21316 − 0.0836282i
181.3 −1.35838 − 0.393457i 1.00000i 1.69038 + 1.06893i 3.70799 0.393457 − 1.35838i − 4.20506i −1.87560 − 2.11710i −1.00000 −5.03685 − 1.45893i
181.4 −1.35838 + 0.393457i − 1.00000i 1.69038 − 1.06893i 3.70799 0.393457 + 1.35838i 4.20506i −1.87560 + 2.11710i −1.00000 −5.03685 + 1.45893i
181.5 −1.23796 − 0.683714i − 1.00000i 1.06507 + 1.69282i −3.44141 −0.683714 + 1.23796i 2.84073i −0.161106 − 2.82384i −1.00000 4.26031 + 2.35294i
181.6 −1.23796 + 0.683714i 1.00000i 1.06507 − 1.69282i −3.44141 −0.683714 − 1.23796i − 2.84073i −0.161106 + 2.82384i −1.00000 4.26031 − 2.35294i
181.7 −0.557063 − 1.29988i 1.00000i −1.37936 + 1.44823i 0.172184 1.29988 − 0.557063i − 1.91402i 2.65091 + 0.986248i −1.00000 −0.0959176 − 0.223819i
181.8 −0.557063 + 1.29988i − 1.00000i −1.37936 − 1.44823i 0.172184 1.29988 + 0.557063i 1.91402i 2.65091 − 0.986248i −1.00000 −0.0959176 + 0.223819i
181.9 −0.486010 − 1.32808i − 1.00000i −1.52759 + 1.29092i 3.48777 −1.32808 + 0.486010i − 0.644203i 2.45687 + 1.40136i −1.00000 −1.69509 − 4.63204i
181.10 −0.486010 + 1.32808i 1.00000i −1.52759 − 1.29092i 3.48777 −1.32808 − 0.486010i 0.644203i 2.45687 − 1.40136i −1.00000 −1.69509 + 4.63204i
181.11 −0.291905 − 1.38376i − 1.00000i −1.82958 + 0.807852i −1.21406 −1.38376 + 0.291905i 4.92861i 1.65194 + 2.29589i −1.00000 0.354391 + 1.67997i
181.12 −0.291905 + 1.38376i 1.00000i −1.82958 − 0.807852i −1.21406 −1.38376 − 0.291905i − 4.92861i 1.65194 − 2.29589i −1.00000 0.354391 − 1.67997i
181.13 0.291905 − 1.38376i 1.00000i −1.82958 − 0.807852i 1.21406 1.38376 + 0.291905i 4.92861i −1.65194 + 2.29589i −1.00000 0.354391 − 1.67997i
181.14 0.291905 + 1.38376i − 1.00000i −1.82958 + 0.807852i 1.21406 1.38376 − 0.291905i − 4.92861i −1.65194 − 2.29589i −1.00000 0.354391 + 1.67997i
181.15 0.486010 − 1.32808i 1.00000i −1.52759 − 1.29092i −3.48777 1.32808 + 0.486010i − 0.644203i −2.45687 + 1.40136i −1.00000 −1.69509 + 4.63204i
181.16 0.486010 + 1.32808i − 1.00000i −1.52759 + 1.29092i −3.48777 1.32808 − 0.486010i 0.644203i −2.45687 − 1.40136i −1.00000 −1.69509 − 4.63204i
181.17 0.557063 − 1.29988i − 1.00000i −1.37936 − 1.44823i −0.172184 −1.29988 − 0.557063i − 1.91402i −2.65091 + 0.986248i −1.00000 −0.0959176 + 0.223819i
181.18 0.557063 + 1.29988i 1.00000i −1.37936 + 1.44823i −0.172184 −1.29988 + 0.557063i 1.91402i −2.65091 − 0.986248i −1.00000 −0.0959176 − 0.223819i
181.19 1.23796 − 0.683714i 1.00000i 1.06507 − 1.69282i 3.44141 0.683714 + 1.23796i 2.84073i 0.161106 − 2.82384i −1.00000 4.26031 − 2.35294i
181.20 1.23796 + 0.683714i − 1.00000i 1.06507 + 1.69282i 3.44141 0.683714 − 1.23796i − 2.84073i 0.161106 + 2.82384i −1.00000 4.26031 + 2.35294i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.b even 2 1 inner
104.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.m.c ✓ 24
3.b odd 2 1 936.2.m.i 24
4.b odd 2 1 1248.2.m.c 24
8.b even 2 1 inner 312.2.m.c ✓ 24
8.d odd 2 1 1248.2.m.c 24
12.b even 2 1 3744.2.m.i 24
13.b even 2 1 inner 312.2.m.c ✓ 24
24.f even 2 1 3744.2.m.i 24
24.h odd 2 1 936.2.m.i 24
39.d odd 2 1 936.2.m.i 24
52.b odd 2 1 1248.2.m.c 24
104.e even 2 1 inner 312.2.m.c ✓ 24
104.h odd 2 1 1248.2.m.c 24
156.h even 2 1 3744.2.m.i 24
312.b odd 2 1 936.2.m.i 24
312.h even 2 1 3744.2.m.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.m.c ✓ 24 1.a even 1 1 trivial
312.2.m.c ✓ 24 8.b even 2 1 inner
312.2.m.c ✓ 24 13.b even 2 1 inner
312.2.m.c ✓ 24 104.e even 2 1 inner
936.2.m.i 24 3.b odd 2 1
936.2.m.i 24 24.h odd 2 1
936.2.m.i 24 39.d odd 2 1
936.2.m.i 24 312.b odd 2 1
1248.2.m.c 24 4.b odd 2 1
1248.2.m.c 24 8.d odd 2 1
1248.2.m.c 24 52.b odd 2 1
1248.2.m.c 24 104.h odd 2 1
3744.2.m.i 24 12.b even 2 1
3744.2.m.i 24 24.f even 2 1
3744.2.m.i 24 156.h even 2 1
3744.2.m.i 24 312.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 40T_{5}^{10} + 560T_{5}^{8} - 3088T_{5}^{6} + 4992T_{5}^{4} - 2304T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display