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Properties

Label 930.2.be.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{7} - 4 q^{10} + 24 q^{14} + 8 q^{15} - 64 q^{16} + 4 q^{17} - 12 q^{20} - 24 q^{21} - 8 q^{22} + 32 q^{24} - 20 q^{25} - 4 q^{28} + 16 q^{29} + 8 q^{31} + 4 q^{33} + 24 q^{35} + 32 q^{36} + 56 q^{37} - 4 q^{38} - 8 q^{41} + 8 q^{42} - 56 q^{43} - 4 q^{44} + 12 q^{45} + 8 q^{47} - 60 q^{49} - 8 q^{50} + 24 q^{53} - 64 q^{54} + 16 q^{55} - 28 q^{57} + 52 q^{58} + 24 q^{59} - 4 q^{62} - 4 q^{63} + 100 q^{65} + 8 q^{66} + 76 q^{67} + 4 q^{68} - 12 q^{69} - 44 q^{70} + 8 q^{71} - 52 q^{73} + 12 q^{74} - 8 q^{75} - 8 q^{76} - 104 q^{77} + 56 q^{79} + 32 q^{81} + 32 q^{82} + 24 q^{83} + 32 q^{85} - 24 q^{86} - 20 q^{87} + 4 q^{88} - 176 q^{89} + 8 q^{93} + 64 q^{95} - 68 q^{97} - 64 q^{98} - 4 q^{99}+O(q^{100}) \) 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 −2.04325 − 0.908360i 0.866025 − 0.500000i −0.239418 + 0.893522i 0.707107 − 0.707107i −0.866025 − 0.500000i 0.802490 + 2.08711i
37.2 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i −1.98218 + 1.03487i 0.866025 − 0.500000i 0.211801 − 0.790453i 0.707107 − 0.707107i −0.866025 − 0.500000i 2.13338 + 0.669853i
37.3 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i −1.08971 − 1.95257i 0.866025 − 0.500000i 0.294543 − 1.09925i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.610134 + 2.15122i
37.4 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i −0.786015 + 2.09337i 0.866025 − 0.500000i −1.17109 + 4.37058i 0.707107 − 0.707107i −0.866025 − 0.500000i 2.03603 − 0.924436i
37.5 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i 0.303557 − 2.21537i 0.866025 − 0.500000i −1.35687 + 5.06392i 0.707107 − 0.707107i −0.866025 − 0.500000i −1.78115 + 1.35185i
37.6 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i 0.871485 + 2.05925i 0.866025 − 0.500000i 1.08173 − 4.03706i 0.707107 − 0.707107i −0.866025 − 0.500000i 0.839878 − 2.07234i
37.7 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i 1.64800 + 1.51133i 0.866025 − 0.500000i −0.191313 + 0.713991i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.0966398 − 2.23398i
37.8 −0.707107 − 0.707107i −0.258819 + 0.965926i 1.00000i 1.95328 − 1.08844i 0.866025 − 0.500000i 0.340075 − 1.26918i 0.707107 − 0.707107i −0.866025 − 0.500000i −2.15082 − 0.611534i
37.9 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i −2.17687 + 0.511134i 0.866025 − 0.500000i 0.572203 − 2.13549i −0.707107 + 0.707107i −0.866025 − 0.500000i −1.90070 − 1.17785i
37.10 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i −1.66175 + 1.49620i 0.866025 − 0.500000i 1.33377 − 4.97771i −0.707107 + 0.707107i −0.866025 − 0.500000i −2.23300 − 0.117061i
37.11 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i −1.62734 − 1.53354i 0.866025 − 0.500000i −0.664038 + 2.47822i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.0663294 − 2.23508i
37.12 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i 0.247692 + 2.22231i 0.866025 − 0.500000i −1.08066 + 4.03306i −0.707107 + 0.707107i −0.866025 − 0.500000i −1.39626 + 1.74655i
37.13 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i 0.795067 + 2.08994i 0.866025 − 0.500000i −0.0433914 + 0.161939i −0.707107 + 0.707107i −0.866025 − 0.500000i −0.915617 + 2.04001i
37.14 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i 1.01126 − 1.99433i 0.866025 − 0.500000i −0.754227 + 2.81481i −0.707107 + 0.707107i −0.866025 − 0.500000i 2.12527 − 0.695132i
37.15 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i 1.20229 − 1.88534i 0.866025 − 0.500000i 0.614903 − 2.29485i −0.707107 + 0.707107i −0.866025 − 0.500000i 2.18328 − 0.482987i
37.16 0.707107 + 0.707107i 0.258819 − 0.965926i 1.00000i 1.60244 + 1.55955i 0.866025 − 0.500000i 0.784034 − 2.92605i −0.707107 + 0.707107i −0.866025 − 0.500000i 0.0303243 + 2.23586i
223.1 −0.707107 + 0.707107i 0.965926 + 0.258819i − 1.00000i −2.07384 − 0.836183i −0.866025 + 0.500000i −4.78156 − 1.28122i 0.707107 + 0.707107i 0.866025 + 0.500000i 2.05769 − 0.875153i
223.2 −0.707107 + 0.707107i 0.965926 + 0.258819i − 1.00000i −1.98088 + 1.03736i −0.866025 + 0.500000i −0.414720 − 0.111124i 0.707107 + 0.707107i 0.866025 + 0.500000i 0.667172 − 2.13422i
223.3 −0.707107 + 0.707107i 0.965926 + 0.258819i − 1.00000i −0.440662 − 2.19222i −0.866025 + 0.500000i −1.90738 − 0.511082i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.86173 + 1.23854i
223.4 −0.707107 + 0.707107i 0.965926 + 0.258819i − 1.00000i −0.395276 − 2.20085i −0.866025 + 0.500000i 3.88090 + 1.03988i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.83574 + 1.27674i
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.a ✓ 64
5.c odd 4 1 930.2.be.b yes 64
31.e odd 6 1 930.2.be.b yes 64
155.p even 12 1 inner 930.2.be.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.be.a ✓ 64 1.a even 1 1 trivial
930.2.be.a ✓ 64 155.p even 12 1 inner
930.2.be.b yes 64 5.c odd 4 1
930.2.be.b yes 64 31.e odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{64} + 8 T_{7}^{63} + 62 T_{7}^{62} + 348 T_{7}^{61} + 657 T_{7}^{60} + 200 T_{7}^{59} + \cdots + 86\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display