↑

Properties

Label 930.2.o.d
Level $930$
Weight $2$
Character orbit 930.o
Analytic conductor $7.426$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,2,Mod(161,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.o (of order \(6\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{4} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{4} - 8 q^{7} + 8 q^{9} - 18 q^{10} + 36 q^{16} - 4 q^{18} - 4 q^{19} + 72 q^{21} - 18 q^{22} + 18 q^{25} + 8 q^{28} + 52 q^{31} - 36 q^{33} + 18 q^{34} - 8 q^{36} - 54 q^{37} - 8 q^{39} + 18 q^{40} + 12 q^{42} - 18 q^{43} + 4 q^{45} - 62 q^{49} - 10 q^{51} - 18 q^{55} + 12 q^{57} + 88 q^{63} - 36 q^{64} - 4 q^{66} + 10 q^{67} + 46 q^{69} + 16 q^{70} + 4 q^{72} + 24 q^{73} + 4 q^{76} + 92 q^{78} + 42 q^{79} + 4 q^{81} - 12 q^{82} - 72 q^{84} - 16 q^{87} + 18 q^{88} + 8 q^{90} + 48 q^{93} - 92 q^{94} - 80 q^{97} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 − 1.00000i −1.71477 + 0.244025i −1.00000 −0.866025 − 0.500000i 0.244025 + 1.71477i 0.0693993 + 0.120203i 1.00000i 2.88090 − 0.836897i −0.500000 + 0.866025i
161.2 − 1.00000i −1.57795 + 0.714201i −1.00000 −0.866025 − 0.500000i 0.714201 + 1.57795i −2.21698 − 3.83992i 1.00000i 1.97984 − 2.25394i −0.500000 + 0.866025i
161.3 − 1.00000i −1.06943 − 1.36247i −1.00000 −0.866025 − 0.500000i −1.36247 + 1.06943i −2.35643 − 4.08145i 1.00000i −0.712629 + 2.91413i −0.500000 + 0.866025i
161.4 − 1.00000i −0.763610 + 1.55464i −1.00000 −0.866025 − 0.500000i 1.55464 + 0.763610i 0.262744 + 0.455085i 1.00000i −1.83380 − 2.37427i −0.500000 + 0.866025i
161.5 − 1.00000i 0.475358 + 1.66554i −1.00000 −0.866025 − 0.500000i 1.66554 − 0.475358i −1.64767 − 2.85386i 1.00000i −2.54807 + 1.58346i −0.500000 + 0.866025i
161.6 − 1.00000i 0.652600 − 1.60440i −1.00000 −0.866025 − 0.500000i −1.60440 − 0.652600i 2.20176 + 3.81356i 1.00000i −2.14823 − 2.09407i −0.500000 + 0.866025i
161.7 − 1.00000i 0.761277 − 1.55578i −1.00000 −0.866025 − 0.500000i −1.55578 − 0.761277i −0.777017 − 1.34583i 1.00000i −1.84092 − 2.36876i −0.500000 + 0.866025i
161.8 − 1.00000i 1.55555 + 0.761750i −1.00000 −0.866025 − 0.500000i 0.761750 − 1.55555i 0.295093 + 0.511116i 1.00000i 1.83947 + 2.36988i −0.500000 + 0.866025i
161.9 − 1.00000i 1.68098 − 0.417505i −1.00000 −0.866025 − 0.500000i −0.417505 − 1.68098i 2.16910 + 3.75699i 1.00000i 2.65138 − 1.40363i −0.500000 + 0.866025i
161.10 1.00000i −1.72816 − 0.116014i −1.00000 0.866025 + 0.500000i 0.116014 − 1.72816i 0.262744 + 0.455085i − 1.00000i 2.97308 + 0.400981i −0.500000 + 0.866025i
161.11 1.00000i −1.40749 + 1.00944i −1.00000 0.866025 + 0.500000i −1.00944 − 1.40749i −2.21698 − 3.83992i − 1.00000i 0.962053 − 2.84156i −0.500000 + 0.866025i
161.12 1.00000i −1.20472 − 1.24444i −1.00000 0.866025 + 0.500000i 1.24444 − 1.20472i −1.64767 − 2.85386i − 1.00000i −0.0972822 + 2.99842i −0.500000 + 0.866025i
161.13 1.00000i −1.06872 + 1.36303i −1.00000 0.866025 + 0.500000i −1.36303 − 1.06872i 0.0693993 + 0.120203i − 1.00000i −0.715677 − 2.91338i −0.500000 + 0.866025i
161.14 1.00000i 0.118080 − 1.72802i −1.00000 0.866025 + 0.500000i 1.72802 + 0.118080i 0.295093 + 0.511116i − 1.00000i −2.97211 − 0.408090i −0.500000 + 0.866025i
161.15 1.00000i 0.645214 + 1.60739i −1.00000 0.866025 + 0.500000i −1.60739 + 0.645214i −2.35643 − 4.08145i − 1.00000i −2.16740 + 2.07422i −0.500000 + 0.866025i
161.16 1.00000i 1.20206 − 1.24702i −1.00000 0.866025 + 0.500000i 1.24702 + 1.20206i 2.16910 + 3.75699i − 1.00000i −0.110108 − 2.99798i −0.500000 + 0.866025i
161.17 1.00000i 1.71575 + 0.237034i −1.00000 0.866025 + 0.500000i −0.237034 + 1.71575i 2.20176 + 3.81356i − 1.00000i 2.88763 + 0.813386i −0.500000 + 0.866025i
161.18 1.00000i 1.72799 + 0.118606i −1.00000 0.866025 + 0.500000i −0.118606 + 1.72799i −0.777017 − 1.34583i − 1.00000i 2.97187 + 0.409900i −0.500000 + 0.866025i
491.1 − 1.00000i −1.72816 + 0.116014i −1.00000 0.866025 − 0.500000i 0.116014 + 1.72816i 0.262744 − 0.455085i 1.00000i 2.97308 − 0.400981i −0.500000 − 0.866025i
491.2 − 1.00000i −1.40749 − 1.00944i −1.00000 0.866025 − 0.500000i −1.00944 + 1.40749i −2.21698 + 3.83992i 1.00000i 0.962053 + 2.84156i −0.500000 − 0.866025i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.o.d ✓ 36
3.b odd 2 1 inner 930.2.o.d ✓ 36
31.e odd 6 1 inner 930.2.o.d ✓ 36
93.g even 6 1 inner 930.2.o.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.o.d ✓ 36 1.a even 1 1 trivial
930.2.o.d ✓ 36 3.b odd 2 1 inner
930.2.o.d ✓ 36 31.e odd 6 1 inner
930.2.o.d ✓ 36 93.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{18} + 4 T_{7}^{17} + 55 T_{7}^{16} + 156 T_{7}^{15} + 1754 T_{7}^{14} + 4564 T_{7}^{13} + \cdots + 7744 \) Copy content Toggle raw display
\( T_{11}^{36} + 143 T_{11}^{34} + 12407 T_{11}^{32} + 701046 T_{11}^{30} + 29350491 T_{11}^{28} + \cdots + 3170478249 \) Copy content Toggle raw display