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Properties

Label 840.2.bj.b
Level $840$
Weight $2$
Character orbit 840.bj
Analytic conductor $6.707$
Analytic rank $0$
Dimension $184$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(13,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bj (of order \(4\), degree \(2\), 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(92\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 184 q + 8 q^{2} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q + 8 q^{2} + 4 q^{7} + 8 q^{8} - 16 q^{15} + 64 q^{16} + 8 q^{18} + 16 q^{23} - 32 q^{25} - 4 q^{28} + 24 q^{30} - 32 q^{32} - 16 q^{36} - 20 q^{42} - 80 q^{46} + 80 q^{50} + 56 q^{58} - 56 q^{60} + 4 q^{63} + 32 q^{65} - 44 q^{70} + 32 q^{71} - 8 q^{72} + 8 q^{78} - 184 q^{81} - 16 q^{86} - 64 q^{88} + 104 q^{92} - 96 q^{95} + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.40929 + 0.117921i −0.707107 − 0.707107i 1.97219 − 0.332370i 0.361094 + 2.20672i 1.07990 + 0.913135i 2.47159 + 0.944062i −2.74019 + 0.700969i 1.00000i −0.769104 − 3.06732i
13.2 −1.40929 + 0.117921i 0.707107 + 0.707107i 1.97219 − 0.332370i −0.361094 − 2.20672i −1.07990 − 0.913135i −0.944062 − 2.47159i −2.74019 + 0.700969i 1.00000i 0.769104 + 3.06732i
13.3 −1.40309 − 0.177021i −0.707107 − 0.707107i 1.93733 + 0.496754i 1.27167 − 1.83926i 0.866962 + 1.11731i 2.63949 − 0.181931i −2.63031 − 1.03994i 1.00000i −2.10985 + 2.35553i
13.4 −1.40309 − 0.177021i 0.707107 + 0.707107i 1.93733 + 0.496754i −1.27167 + 1.83926i −0.866962 − 1.11731i 0.181931 − 2.63949i −2.63031 − 1.03994i 1.00000i 2.10985 − 2.35553i
13.5 −1.39560 − 0.228667i −0.707107 − 0.707107i 1.89542 + 0.638257i −1.53990 − 1.62133i 0.825150 + 1.14853i 0.369731 + 2.61979i −2.49931 − 1.32417i 1.00000i 1.77834 + 2.61486i
13.6 −1.39560 − 0.228667i 0.707107 + 0.707107i 1.89542 + 0.638257i 1.53990 + 1.62133i −0.825150 − 1.14853i −2.61979 − 0.369731i −2.49931 − 1.32417i 1.00000i −1.77834 − 2.61486i
13.7 −1.38448 − 0.288447i −0.707107 − 0.707107i 1.83360 + 0.798701i −1.97588 + 1.04685i 0.775016 + 1.18294i 0.379700 − 2.61836i −2.30820 − 1.63468i 1.00000i 3.03754 − 0.879417i
13.8 −1.38448 − 0.288447i 0.707107 + 0.707107i 1.83360 + 0.798701i 1.97588 − 1.04685i −0.775016 − 1.18294i 2.61836 − 0.379700i −2.30820 − 1.63468i 1.00000i −3.03754 + 0.879417i
13.9 −1.36723 + 0.361491i −0.707107 − 0.707107i 1.73865 − 0.988485i −0.0291073 + 2.23588i 1.22239 + 0.711166i −2.56683 + 0.641375i −2.01981 + 1.97999i 1.00000i −0.768454 − 3.06749i
13.10 −1.36723 + 0.361491i 0.707107 + 0.707107i 1.73865 − 0.988485i 0.0291073 − 2.23588i −1.22239 − 0.711166i −0.641375 + 2.56683i −2.01981 + 1.97999i 1.00000i 0.768454 + 3.06749i
13.11 −1.30929 + 0.534572i −0.707107 − 0.707107i 1.42846 − 1.39982i 1.34687 − 1.78492i 1.30381 + 0.547806i −2.37457 + 1.16680i −1.12197 + 2.59638i 1.00000i −0.809265 + 3.05697i
13.12 −1.30929 + 0.534572i 0.707107 + 0.707107i 1.42846 − 1.39982i −1.34687 + 1.78492i −1.30381 − 0.547806i −1.16680 + 2.37457i −1.12197 + 2.59638i 1.00000i 0.809265 − 3.05697i
13.13 −1.29398 + 0.570620i −0.707107 − 0.707107i 1.34879 − 1.47675i 2.22180 + 0.252237i 1.31847 + 0.511495i 0.496782 − 2.59869i −0.902645 + 2.68053i 1.00000i −3.01890 + 0.941410i
13.14 −1.29398 + 0.570620i 0.707107 + 0.707107i 1.34879 − 1.47675i −2.22180 − 0.252237i −1.31847 − 0.511495i 2.59869 − 0.496782i −0.902645 + 2.68053i 1.00000i 3.01890 − 0.941410i
13.15 −1.27271 − 0.616616i −0.707107 − 0.707107i 1.23957 + 1.56954i 2.03587 + 0.924781i 0.463927 + 1.33595i −1.65217 − 2.06648i −0.609805 − 2.76191i 1.00000i −2.02084 − 2.43233i
13.16 −1.27271 − 0.616616i 0.707107 + 0.707107i 1.23957 + 1.56954i −2.03587 − 0.924781i −0.463927 − 1.33595i 2.06648 + 1.65217i −0.609805 − 2.76191i 1.00000i 2.02084 + 2.43233i
13.17 −1.18486 + 0.772078i −0.707107 − 0.707107i 0.807792 − 1.82961i −0.563926 − 2.16379i 1.38376 + 0.291882i 2.63180 + 0.271325i 0.455480 + 2.79151i 1.00000i 2.33879 + 2.12840i
13.18 −1.18486 + 0.772078i 0.707107 + 0.707107i 0.807792 − 1.82961i 0.563926 + 2.16379i −1.38376 − 0.291882i −0.271325 − 2.63180i 0.455480 + 2.79151i 1.00000i −2.33879 − 2.12840i
13.19 −1.08726 − 0.904365i −0.707107 − 0.707107i 0.364249 + 1.96655i −0.998967 − 2.00052i 0.129323 + 1.40829i −2.62852 + 0.301481i 1.38245 − 2.46756i 1.00000i −0.723063 + 3.07850i
13.20 −1.08726 − 0.904365i 0.707107 + 0.707107i 0.364249 + 1.96655i 0.998967 + 2.00052i −0.129323 − 1.40829i −0.301481 + 2.62852i 1.38245 − 2.46756i 1.00000i 0.723063 − 3.07850i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
56.h odd 2 1 inner
280.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.bj.b ✓ 184
5.c odd 4 1 inner 840.2.bj.b ✓ 184
7.b odd 2 1 inner 840.2.bj.b ✓ 184
8.b even 2 1 inner 840.2.bj.b ✓ 184
35.f even 4 1 inner 840.2.bj.b ✓ 184
40.i odd 4 1 inner 840.2.bj.b ✓ 184
56.h odd 2 1 inner 840.2.bj.b ✓ 184
280.s even 4 1 inner 840.2.bj.b ✓ 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bj.b ✓ 184 1.a even 1 1 trivial
840.2.bj.b ✓ 184 5.c odd 4 1 inner
840.2.bj.b ✓ 184 7.b odd 2 1 inner
840.2.bj.b ✓ 184 8.b even 2 1 inner
840.2.bj.b ✓ 184 35.f even 4 1 inner
840.2.bj.b ✓ 184 40.i odd 4 1 inner
840.2.bj.b ✓ 184 56.h odd 2 1 inner
840.2.bj.b ✓ 184 280.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{46} + 262 T_{11}^{44} + 31388 T_{11}^{42} + 2283176 T_{11}^{40} + 112958976 T_{11}^{38} + \cdots + 12\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display