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Properties

Label 930.2.j.g
Level $930$
Weight $2$
Character orbit 930.j
Analytic conductor $7.426$
Analytic rank $0$
Dimension $40$
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] = [930,2,Mod(497,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.j (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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 8 q^{7} + 8 q^{10} + 4 q^{12} - 20 q^{13} - 44 q^{15} - 40 q^{16} + 16 q^{18} - 32 q^{21} - 4 q^{22} + 8 q^{25} + 8 q^{27} - 8 q^{28} - 4 q^{30} + 40 q^{31} + 48 q^{33} + 64 q^{37} - 4 q^{40} - 20 q^{42} - 48 q^{43} - 4 q^{45} - 80 q^{46} + 4 q^{48} + 48 q^{51} + 20 q^{52} + 164 q^{55} + 40 q^{57} + 56 q^{58} + 24 q^{60} - 128 q^{61} + 56 q^{63} + 8 q^{66} - 4 q^{67} - 56 q^{70} - 16 q^{72} - 48 q^{73} - 48 q^{75} - 40 q^{76} - 36 q^{78} - 24 q^{81} - 16 q^{82} - 56 q^{85} - 12 q^{87} - 4 q^{88} - 28 q^{90} - 32 q^{91} - 4 q^{93} - 52 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1 −0.707107 + 0.707107i −1.65310 + 0.516967i − 1.00000i −0.633448 − 2.14447i 0.803369 − 1.53447i −2.44874 − 2.44874i 0.707107 + 0.707107i 2.46549 − 1.70920i 1.96428 + 1.06845i
497.2 −0.707107 + 0.707107i −1.47409 − 0.909430i − 1.00000i 2.03224 − 0.932730i 1.68540 − 0.399274i −2.61297 − 2.61297i 0.707107 + 0.707107i 1.34588 + 2.68116i −0.777474 + 2.09655i
497.3 −0.707107 + 0.707107i −0.919406 + 1.46789i − 1.00000i 2.22068 + 0.261841i −0.387835 − 1.68807i 3.32500 + 3.32500i 0.707107 + 0.707107i −1.30939 − 2.69917i −1.75541 + 1.38511i
497.4 −0.707107 + 0.707107i −0.731811 + 1.56986i − 1.00000i −1.45516 + 1.69779i −0.592589 − 1.62753i 1.76311 + 1.76311i 0.707107 + 0.707107i −1.92891 − 2.29768i −0.171568 − 2.22948i
497.5 −0.707107 + 0.707107i −0.673165 − 1.59589i − 1.00000i −1.55932 − 1.60266i 1.60446 + 0.652462i −0.700251 − 0.700251i 0.707107 + 0.707107i −2.09370 + 2.14859i 2.23586 + 0.0306496i
497.6 −0.707107 + 0.707107i 0.0444250 − 1.73148i − 1.00000i −0.142194 − 2.23154i 1.19293 + 1.25576i 2.34383 + 2.34383i 0.707107 + 0.707107i −2.99605 − 0.153842i 1.67849 + 1.47739i
497.7 −0.707107 + 0.707107i 0.765006 + 1.55395i − 1.00000i 1.73301 + 1.41304i −1.63975 − 0.557869i 1.07773 + 1.07773i 0.707107 + 0.707107i −1.82953 + 2.37756i −2.22459 + 0.226250i
497.8 −0.707107 + 0.707107i 0.853874 + 1.50695i − 1.00000i −2.20708 − 0.358875i −1.66935 − 0.461795i −1.72132 − 1.72132i 0.707107 + 0.707107i −1.54180 + 2.57349i 1.81441 − 1.30688i
497.9 −0.707107 + 0.707107i 1.05623 − 1.37273i − 1.00000i −0.0887404 + 2.23431i 0.223801 + 1.71753i −1.91575 − 1.91575i 0.707107 + 0.707107i −0.768771 − 2.89983i −1.51714 − 1.64264i
497.10 −0.707107 + 0.707107i 1.73204 − 0.00608872i − 1.00000i −2.02131 + 0.956187i −1.22043 + 1.22904i −1.11064 − 1.11064i 0.707107 + 0.707107i 2.99993 − 0.0210918i 0.753158 − 2.10541i
497.11 0.707107 − 0.707107i −1.56986 + 0.731811i − 1.00000i 1.45516 − 1.69779i −0.592589 + 1.62753i 1.76311 + 1.76311i −0.707107 − 0.707107i 1.92891 − 2.29768i −0.171568 − 2.22948i
497.12 0.707107 − 0.707107i −1.55395 − 0.765006i − 1.00000i −1.73301 − 1.41304i −1.63975 + 0.557869i 1.07773 + 1.07773i −0.707107 − 0.707107i 1.82953 + 2.37756i −2.22459 + 0.226250i
497.13 0.707107 − 0.707107i −1.50695 − 0.853874i − 1.00000i 2.20708 + 0.358875i −1.66935 + 0.461795i −1.72132 − 1.72132i −0.707107 − 0.707107i 1.54180 + 2.57349i 1.81441 − 1.30688i
497.14 0.707107 − 0.707107i −1.46789 + 0.919406i − 1.00000i −2.22068 − 0.261841i −0.387835 + 1.68807i 3.32500 + 3.32500i −0.707107 − 0.707107i 1.30939 − 2.69917i −1.75541 + 1.38511i
497.15 0.707107 − 0.707107i −0.516967 + 1.65310i − 1.00000i 0.633448 + 2.14447i 0.803369 + 1.53447i −2.44874 − 2.44874i −0.707107 − 0.707107i −2.46549 − 1.70920i 1.96428 + 1.06845i
497.16 0.707107 − 0.707107i 0.00608872 − 1.73204i − 1.00000i 2.02131 − 0.956187i −1.22043 − 1.22904i −1.11064 − 1.11064i −0.707107 − 0.707107i −2.99993 − 0.0210918i 0.753158 − 2.10541i
497.17 0.707107 − 0.707107i 0.909430 + 1.47409i − 1.00000i −2.03224 + 0.932730i 1.68540 + 0.399274i −2.61297 − 2.61297i −0.707107 − 0.707107i −1.34588 + 2.68116i −0.777474 + 2.09655i
497.18 0.707107 − 0.707107i 1.37273 − 1.05623i − 1.00000i 0.0887404 − 2.23431i 0.223801 − 1.71753i −1.91575 − 1.91575i −0.707107 − 0.707107i 0.768771 − 2.89983i −1.51714 − 1.64264i
497.19 0.707107 − 0.707107i 1.59589 + 0.673165i − 1.00000i 1.55932 + 1.60266i 1.60446 − 0.652462i −0.700251 − 0.700251i −0.707107 − 0.707107i 2.09370 + 2.14859i 2.23586 + 0.0306496i
497.20 0.707107 − 0.707107i 1.73148 − 0.0444250i − 1.00000i 0.142194 + 2.23154i 1.19293 − 1.25576i 2.34383 + 2.34383i −0.707107 − 0.707107i 2.99605 − 0.153842i 1.67849 + 1.47739i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 497.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.g ✓ 40
3.b odd 2 1 inner 930.2.j.g ✓ 40
5.c odd 4 1 inner 930.2.j.g ✓ 40
15.e even 4 1 inner 930.2.j.g ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.g ✓ 40 1.a even 1 1 trivial
930.2.j.g ✓ 40 3.b odd 2 1 inner
930.2.j.g ✓ 40 5.c odd 4 1 inner
930.2.j.g ✓ 40 15.e even 4 1 inner

Hecke kernels

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

\( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} + 24 T_{7}^{17} + 604 T_{7}^{16} + 2664 T_{7}^{15} + \cdots + 60466176 \) Copy content Toggle raw display
\( T_{11}^{20} + 184 T_{11}^{18} + 14417 T_{11}^{16} + 628792 T_{11}^{14} + 16758208 T_{11}^{12} + \cdots + 11906737924 \) Copy content Toggle raw display
\( T_{17}^{40} + 12210 T_{17}^{36} + 54185201 T_{17}^{32} + 105212553808 T_{17}^{28} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display