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Properties

Label 322.4.c.a
Level $322$
Weight $4$
Character orbit 322.c
Analytic conductor $18.999$
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] = [322,4,Mod(321,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.321");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.c (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: \(18.9986150218\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{2} + 96 q^{4} - 192 q^{8} - 248 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{2} + 96 q^{4} - 192 q^{8} - 248 q^{9} + 384 q^{16} + 496 q^{18} - 388 q^{23} + 1168 q^{25} + 256 q^{29} - 768 q^{32} + 652 q^{35} - 992 q^{36} - 8 q^{39} + 776 q^{46} + 1252 q^{49} - 2336 q^{50} - 512 q^{58} + 1536 q^{64} - 1304 q^{70} - 1824 q^{71} + 1984 q^{72} + 2596 q^{77} + 16 q^{78} + 5824 q^{81} + 5680 q^{85} - 1552 q^{92} - 3112 q^{93} + 2344 q^{95} - 2504 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 −2.00000 − 9.73722i 4.00000 −21.7256 19.4744i −16.2551 + 8.87534i −8.00000 −67.8134 43.4512
321.2 −2.00000 9.73722i 4.00000 −21.7256 − 19.4744i −16.2551 − 8.87534i −8.00000 −67.8134 43.4512
321.3 −2.00000 − 4.75737i 4.00000 18.5380 9.51474i −17.0849 + 7.14893i −8.00000 4.36741 −37.0761
321.4 −2.00000 4.75737i 4.00000 18.5380 − 9.51474i −17.0849 − 7.14893i −8.00000 4.36741 −37.0761
321.5 −2.00000 − 0.772667i 4.00000 10.8187 1.54533i −5.13301 − 17.7947i −8.00000 26.4030 −21.6374
321.6 −2.00000 0.772667i 4.00000 10.8187 − 1.54533i −5.13301 + 17.7947i −8.00000 26.4030 −21.6374
321.7 −2.00000 − 0.0141321i 4.00000 10.4228 0.0282642i 17.6918 + 5.47725i −8.00000 26.9998 −20.8455
321.8 −2.00000 0.0141321i 4.00000 10.4228 − 0.0282642i 17.6918 − 5.47725i −8.00000 26.9998 −20.8455
321.9 −2.00000 − 6.74897i 4.00000 −0.725649 13.4979i −5.05190 − 17.8179i −8.00000 −18.5486 1.45130
321.10 −2.00000 6.74897i 4.00000 −0.725649 − 13.4979i −5.05190 + 17.8179i −8.00000 −18.5486 1.45130
321.11 −2.00000 − 7.77227i 4.00000 0.366637 15.5445i −16.2638 − 8.85948i −8.00000 −33.4082 −0.733274
321.12 −2.00000 7.77227i 4.00000 0.366637 − 15.5445i −16.2638 + 8.85948i −8.00000 −33.4082 −0.733274
321.13 −2.00000 − 7.77227i 4.00000 −0.366637 15.5445i 16.2638 + 8.85948i −8.00000 −33.4082 0.733274
321.14 −2.00000 7.77227i 4.00000 −0.366637 − 15.5445i 16.2638 − 8.85948i −8.00000 −33.4082 0.733274
321.15 −2.00000 − 6.74897i 4.00000 0.725649 13.4979i 5.05190 + 17.8179i −8.00000 −18.5486 −1.45130
321.16 −2.00000 6.74897i 4.00000 0.725649 − 13.4979i 5.05190 − 17.8179i −8.00000 −18.5486 −1.45130
321.17 −2.00000 − 0.0141321i 4.00000 −10.4228 0.0282642i −17.6918 − 5.47725i −8.00000 26.9998 20.8455
321.18 −2.00000 0.0141321i 4.00000 −10.4228 − 0.0282642i −17.6918 + 5.47725i −8.00000 26.9998 20.8455
321.19 −2.00000 − 0.772667i 4.00000 −10.8187 1.54533i 5.13301 + 17.7947i −8.00000 26.4030 21.6374
321.20 −2.00000 0.772667i 4.00000 −10.8187 − 1.54533i 5.13301 − 17.7947i −8.00000 26.4030 21.6374
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.4.c.a ✓ 24
7.b odd 2 1 inner 322.4.c.a ✓ 24
23.b odd 2 1 inner 322.4.c.a ✓ 24
161.c even 2 1 inner 322.4.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.c.a ✓ 24 1.a even 1 1 trivial
322.4.c.a ✓ 24 7.b odd 2 1 inner
322.4.c.a ✓ 24 23.b odd 2 1 inner
322.4.c.a ✓ 24 161.c even 2 1 inner