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Properties

Label 182.2.bb.a
Level $182$
Weight $2$
Character orbit 182.bb
Analytic conductor $1.453$
Analytic rank $0$
Dimension $32$
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] = [182,2,Mod(5,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("182.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.bb (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: \(1.45327731679\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} + 8 q^{9} + 12 q^{11} - 4 q^{14} + 32 q^{15} + 16 q^{16} + 16 q^{18} - 36 q^{19} - 8 q^{22} - 4 q^{28} - 56 q^{29} - 12 q^{31} - 84 q^{33} - 56 q^{35} + 24 q^{37} + 8 q^{39} - 48 q^{42} + 12 q^{44} - 12 q^{46} - 36 q^{47} - 48 q^{50} + 12 q^{52} + 20 q^{53} - 36 q^{54} + 16 q^{57} - 8 q^{58} + 16 q^{60} + 12 q^{61} - 60 q^{63} + 20 q^{65} - 8 q^{67} + 12 q^{68} + 8 q^{70} + 40 q^{71} - 16 q^{72} - 12 q^{73} + 40 q^{74} + 32 q^{78} + 56 q^{79} + 32 q^{81} + 128 q^{85} + 4 q^{86} + 48 q^{87} + 108 q^{89} + 8 q^{91} - 16 q^{92} + 4 q^{93} + 132 q^{94} - 8 q^{98} + 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.258819 − 0.965926i −1.62621 + 0.938890i −0.866025 + 0.500000i 1.06962 − 0.286602i 1.32779 + 1.32779i −1.79012 + 1.94820i 0.707107 + 0.707107i 0.263029 − 0.455579i −0.553674 − 0.958991i
5.2 −0.258819 − 0.965926i −0.516378 + 0.298131i −0.866025 + 0.500000i −4.11903 + 1.10369i 0.421621 + 0.421621i 2.62588 − 0.323637i 0.707107 + 0.707107i −1.32224 + 2.29018i 2.13217 + 3.69303i
5.3 −0.258819 − 0.965926i 0.346434 − 0.200014i −0.866025 + 0.500000i 2.51241 − 0.673198i −0.282862 − 0.282862i 2.13341 + 1.56478i 0.707107 + 0.707107i −1.41999 + 2.45949i −1.30052 − 2.25256i
5.4 −0.258819 − 0.965926i 1.79615 − 1.03701i −0.866025 + 0.500000i 0.537010 − 0.143891i −1.46655 − 1.46655i −1.00325 − 2.44816i 0.707107 + 0.707107i 0.650769 − 1.12716i −0.277977 − 0.481470i
5.5 0.258819 + 0.965926i −2.68453 + 1.54991i −0.866025 + 0.500000i −1.33913 + 0.358820i −2.19191 − 2.19191i 0.0841064 − 2.64441i −0.707107 − 0.707107i 3.30447 − 5.72351i −0.693187 − 1.20063i
5.6 0.258819 + 0.965926i −0.954449 + 0.551051i −0.866025 + 0.500000i −0.383866 + 0.102857i −0.779304 − 0.779304i −0.0182534 + 2.64569i −0.707107 − 0.707107i −0.892685 + 1.54618i −0.198704 − 0.344165i
5.7 0.258819 + 0.965926i 1.77549 − 1.02508i −0.866025 + 0.500000i −0.789614 + 0.211577i 1.44968 + 1.44968i 2.55319 + 0.693694i −0.707107 − 0.707107i 0.601582 − 1.04197i −0.408734 − 0.707949i
5.8 0.258819 + 0.965926i 1.86349 − 1.07589i −0.866025 + 0.500000i 2.51261 − 0.673253i 1.52153 + 1.52153i −2.58497 + 0.563850i −0.707107 − 0.707107i 0.815058 − 1.41172i 1.30062 + 2.25275i
31.1 −0.965926 − 0.258819i −2.68453 − 1.54991i 0.866025 + 0.500000i −0.358820 + 1.33913i 2.19191 + 2.19191i −2.64441 + 0.0841064i −0.707107 − 0.707107i 3.30447 + 5.72351i 0.693187 − 1.20063i
31.2 −0.965926 − 0.258819i −0.954449 − 0.551051i 0.866025 + 0.500000i −0.102857 + 0.383866i 0.779304 + 0.779304i 2.64569 − 0.0182534i −0.707107 − 0.707107i −0.892685 − 1.54618i 0.198704 − 0.344165i
31.3 −0.965926 − 0.258819i 1.77549 + 1.02508i 0.866025 + 0.500000i −0.211577 + 0.789614i −1.44968 − 1.44968i 0.693694 + 2.55319i −0.707107 − 0.707107i 0.601582 + 1.04197i 0.408734 − 0.707949i
31.4 −0.965926 − 0.258819i 1.86349 + 1.07589i 0.866025 + 0.500000i 0.673253 − 2.51261i −1.52153 − 1.52153i 0.563850 − 2.58497i −0.707107 − 0.707107i 0.815058 + 1.41172i −1.30062 + 2.25275i
31.5 0.965926 + 0.258819i −1.62621 − 0.938890i 0.866025 + 0.500000i 0.286602 − 1.06962i −1.32779 − 1.32779i 1.94820 − 1.79012i 0.707107 + 0.707107i 0.263029 + 0.455579i 0.553674 − 0.958991i
31.6 0.965926 + 0.258819i −0.516378 − 0.298131i 0.866025 + 0.500000i −1.10369 + 4.11903i −0.421621 − 0.421621i −0.323637 + 2.62588i 0.707107 + 0.707107i −1.32224 − 2.29018i −2.13217 + 3.69303i
31.7 0.965926 + 0.258819i 0.346434 + 0.200014i 0.866025 + 0.500000i 0.673198 − 2.51241i 0.282862 + 0.282862i 1.56478 + 2.13341i 0.707107 + 0.707107i −1.41999 − 2.45949i 1.30052 − 2.25256i
31.8 0.965926 + 0.258819i 1.79615 + 1.03701i 0.866025 + 0.500000i 0.143891 − 0.537010i 1.46655 + 1.46655i −2.44816 − 1.00325i 0.707107 + 0.707107i 0.650769 + 1.12716i 0.277977 − 0.481470i
47.1 −0.965926 + 0.258819i −2.68453 + 1.54991i 0.866025 − 0.500000i −0.358820 − 1.33913i 2.19191 − 2.19191i −2.64441 − 0.0841064i −0.707107 + 0.707107i 3.30447 − 5.72351i 0.693187 + 1.20063i
47.2 −0.965926 + 0.258819i −0.954449 + 0.551051i 0.866025 − 0.500000i −0.102857 − 0.383866i 0.779304 − 0.779304i 2.64569 + 0.0182534i −0.707107 + 0.707107i −0.892685 + 1.54618i 0.198704 + 0.344165i
47.3 −0.965926 + 0.258819i 1.77549 − 1.02508i 0.866025 − 0.500000i −0.211577 − 0.789614i −1.44968 + 1.44968i 0.693694 − 2.55319i −0.707107 + 0.707107i 0.601582 − 1.04197i 0.408734 + 0.707949i
47.4 −0.965926 + 0.258819i 1.86349 − 1.07589i 0.866025 − 0.500000i 0.673253 + 2.51261i −1.52153 + 1.52153i 0.563850 + 2.58497i −0.707107 + 0.707107i 0.815058 − 1.41172i −1.30062 − 2.25275i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.bb.a ✓ 32
7.c even 3 1 1274.2.i.d 32
7.d odd 6 1 inner 182.2.bb.a ✓ 32
7.d odd 6 1 1274.2.i.d 32
13.d odd 4 1 inner 182.2.bb.a ✓ 32
91.z odd 12 1 1274.2.i.d 32
91.bb even 12 1 inner 182.2.bb.a ✓ 32
91.bb even 12 1 1274.2.i.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.bb.a ✓ 32 1.a even 1 1 trivial
182.2.bb.a ✓ 32 7.d odd 6 1 inner
182.2.bb.a ✓ 32 13.d odd 4 1 inner
182.2.bb.a ✓ 32 91.bb even 12 1 inner
1274.2.i.d 32 7.c even 3 1
1274.2.i.d 32 7.d odd 6 1
1274.2.i.d 32 91.z odd 12 1
1274.2.i.d 32 91.bb even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(182, [\chi])\).