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Properties

Label 930.2.be.b
Level $930$
Weight $2$
Character orbit 930.be
Analytic conductor $7.426$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,2,Mod(37,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.be (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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{7} - 4 q^{10} - 24 q^{14} - 8 q^{15} - 64 q^{16} - 4 q^{17} + 12 q^{20} - 24 q^{21} - 4 q^{22} - 32 q^{24} + 28 q^{25} + 8 q^{28} - 16 q^{29} + 8 q^{31} + 4 q^{33} + 24 q^{35} + 32 q^{36} + 16 q^{37} - 28 q^{38} - 8 q^{41} + 4 q^{42} - 40 q^{43} + 4 q^{44} - 12 q^{45} + 8 q^{47} + 60 q^{49} - 8 q^{50} - 24 q^{53} + 64 q^{54} + 44 q^{55} + 4 q^{57} - 52 q^{58} - 24 q^{59} + 20 q^{62} - 4 q^{63} + 44 q^{65} + 8 q^{66} - 44 q^{67} - 4 q^{68} + 12 q^{69} - 44 q^{70} + 8 q^{71} + 4 q^{73} - 12 q^{74} + 8 q^{75} - 8 q^{76} + 104 q^{77} - 56 q^{79} + 32 q^{81} - 16 q^{82} - 48 q^{83} - 32 q^{85} - 24 q^{86} - 32 q^{87} + 8 q^{88} + 176 q^{89} + 16 q^{93} + 64 q^{95} - 68 q^{97} + 32 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i −2.15172 + 0.608374i −0.866025 + 0.500000i 0.345988 − 1.29125i 0.707107 − 0.707107i −0.866025 − 0.500000i 1.95168 + 1.09131i
37.2 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i −1.90333 − 1.17360i −0.866025 + 0.500000i −0.365018 + 1.36227i 0.707107 − 0.707107i −0.866025 − 0.500000i 0.515994 + 2.17572i
37.3 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i −1.31828 − 1.80614i −0.866025 + 0.500000i 1.13468 − 4.23469i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.344964 + 2.20930i
37.4 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i −0.0950827 − 2.23405i −0.866025 + 0.500000i −0.582615 + 2.17435i 0.707107 − 0.707107i −0.866025 − 0.500000i −1.51248 + 1.64694i
37.5 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i 0.0920628 + 2.23417i −0.866025 + 0.500000i 0.111124 − 0.414720i 0.707107 − 0.707107i −0.866025 − 0.500000i 1.51470 − 1.64490i
37.6 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i 1.76107 + 1.37790i −0.866025 + 0.500000i 1.28122 − 4.78156i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.270942 − 2.21959i
37.7 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i 2.10363 − 0.758108i −0.866025 + 0.500000i −1.03988 + 3.88090i 0.707107 − 0.707107i −0.866025 − 0.500000i −2.02356 − 0.951430i
37.8 −0.707107 − 0.707107i 0.258819 − 0.965926i 1.00000i 2.11885 − 0.714484i −0.866025 + 0.500000i 0.511082 − 1.90738i 0.707107 − 0.707107i −0.866025 − 0.500000i −2.00347 − 0.993035i
37.9 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −2.08483 + 0.808396i −0.866025 + 0.500000i −0.0488226 + 0.182208i −0.707107 + 0.707107i −0.866025 − 0.500000i −2.04582 − 0.902572i
37.10 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −2.06875 − 0.848677i −0.866025 + 0.500000i 0.339513 − 1.26708i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.862725 − 2.06294i
37.11 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −0.604368 + 2.15284i −0.866025 + 0.500000i −0.350136 + 1.30673i −0.707107 + 0.707107i −0.866025 − 0.500000i −1.94964 + 1.09494i
37.12 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i −0.0918774 − 2.23418i −0.866025 + 0.500000i −0.264810 + 0.988286i −0.707107 + 0.707107i −0.866025 − 0.500000i 1.51484 − 1.64477i
37.13 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 0.260478 − 2.22084i −0.866025 + 0.500000i 0.587501 − 2.19258i −0.707107 + 0.707107i −0.866025 − 0.500000i 1.75456 − 1.38619i
37.14 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 1.24237 + 1.85917i −0.866025 + 0.500000i −0.725743 + 2.70851i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.436149 + 2.19312i
37.15 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 2.23583 − 0.0323803i −0.866025 + 0.500000i 0.933337 − 3.48326i −0.707107 + 0.707107i −0.866025 − 0.500000i 1.60387 + 1.55808i
37.16 0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 2.23599 − 0.0184061i −0.866025 + 0.500000i −0.867414 + 3.23723i −0.707107 + 0.707107i −0.866025 − 0.500000i 1.59410 + 1.56807i
223.1 −0.707107 + 0.707107i −0.965926 − 0.258819i − 1.00000i −2.21911 + 0.274898i 0.866025 − 0.500000i −4.03706 − 1.08173i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.37476 − 1.76353i
223.2 −0.707107 + 0.707107i −0.965926 − 0.258819i − 1.00000i −2.13285 − 0.671543i 0.866025 − 0.500000i 0.713991 + 0.191313i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.98300 − 1.03330i
223.3 −0.707107 + 0.707107i −0.965926 − 0.258819i − 1.00000i −1.41990 + 1.72739i 0.866025 − 0.500000i 4.37058 + 1.17109i 0.707107 + 0.707107i 0.866025 + 0.500000i −0.217429 − 2.22547i
223.4 −0.707107 + 0.707107i −0.965926 − 0.258819i − 1.00000i −0.0340238 − 2.23581i 0.866025 − 0.500000i −1.26918 − 0.340075i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.60501 + 1.55690i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.be.b yes 64
5.c odd 4 1 930.2.be.a ✓ 64
31.e odd 6 1 930.2.be.a ✓ 64
155.p even 12 1 inner 930.2.be.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.be.a ✓ 64 5.c odd 4 1
930.2.be.a ✓ 64 31.e odd 6 1
930.2.be.b yes 64 1.a even 1 1 trivial
930.2.be.b yes 64 155.p even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{64} - 4 T_{7}^{63} - 22 T_{7}^{62} + 132 T_{7}^{61} - 975 T_{7}^{60} + 824 T_{7}^{59} + \cdots + 86\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display