↑

Properties

Label 798.2.m.a
Level $798$
Weight $2$
Character orbit 798.m
Analytic conductor $6.372$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,2,Mod(145,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.m (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{3} - 28 q^{4} + 4 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{3} - 28 q^{4} + 4 q^{7} - 14 q^{9} - 4 q^{10} + 14 q^{12} + 2 q^{13} + 4 q^{14} - 6 q^{15} + 28 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{21} - 12 q^{22} - 28 q^{25} + 28 q^{27} - 4 q^{28} + 18 q^{29} + 2 q^{30} + 4 q^{31} + 8 q^{34} + 32 q^{35} + 14 q^{36} + 12 q^{37} + 18 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{42} + 10 q^{43} + 6 q^{45} - 24 q^{46} - 12 q^{47} - 14 q^{48} + 6 q^{51} - 2 q^{52} + 30 q^{55} - 4 q^{56} - 12 q^{57} - 2 q^{58} + 2 q^{59} + 6 q^{60} - 6 q^{61} - 12 q^{62} - 2 q^{63} - 28 q^{64} - 96 q^{65} + 6 q^{68} - 30 q^{70} - 42 q^{71} - 18 q^{73} - 6 q^{74} + 14 q^{75} - 12 q^{76} + 16 q^{77} - 14 q^{81} + 6 q^{82} + 2 q^{84} - 8 q^{85} + 30 q^{86} - 18 q^{87} + 12 q^{88} - 8 q^{89} + 2 q^{90} + 46 q^{91} - 8 q^{93} + 18 q^{94} + 92 q^{95} + 8 q^{97} - 8 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 − 1.00000i −0.500000 + 0.866025i −1.00000 − 4.15255i 0.866025 + 0.500000i 2.46654 − 0.957163i 1.00000i −0.500000 − 0.866025i −4.15255
145.2 − 1.00000i −0.500000 + 0.866025i −1.00000 − 1.53962i 0.866025 + 0.500000i −2.47775 − 0.927776i 1.00000i −0.500000 − 0.866025i −1.53962
145.3 − 1.00000i −0.500000 + 0.866025i −1.00000 − 1.46815i 0.866025 + 0.500000i −0.0182682 + 2.64569i 1.00000i −0.500000 − 0.866025i −1.46815
145.4 − 1.00000i −0.500000 + 0.866025i −1.00000 − 0.865783i 0.866025 + 0.500000i −1.47106 + 2.19908i 1.00000i −0.500000 − 0.866025i −0.865783
145.5 − 1.00000i −0.500000 + 0.866025i −1.00000 2.40297i 0.866025 + 0.500000i 0.924173 − 2.47909i 1.00000i −0.500000 − 0.866025i 2.40297
145.6 − 1.00000i −0.500000 + 0.866025i −1.00000 2.70816i 0.866025 + 0.500000i 2.44311 + 1.01548i 1.00000i −0.500000 − 0.866025i 2.70816
145.7 − 1.00000i −0.500000 + 0.866025i −1.00000 3.64701i 0.866025 + 0.500000i −2.59880 − 0.496218i 1.00000i −0.500000 − 0.866025i 3.64701
145.8 1.00000i −0.500000 + 0.866025i −1.00000 − 2.86414i −0.866025 − 0.500000i −0.00184546 + 2.64575i − 1.00000i −0.500000 − 0.866025i 2.86414
145.9 1.00000i −0.500000 + 0.866025i −1.00000 − 1.47305i −0.866025 − 0.500000i −1.15317 − 2.38122i − 1.00000i −0.500000 − 0.866025i 1.47305
145.10 1.00000i −0.500000 + 0.866025i −1.00000 − 0.953104i −0.866025 − 0.500000i 2.56076 + 0.665210i − 1.00000i −0.500000 − 0.866025i 0.953104
145.11 1.00000i −0.500000 + 0.866025i −1.00000 0.525465i −0.866025 − 0.500000i −2.53656 − 0.752229i − 1.00000i −0.500000 − 0.866025i −0.525465
145.12 1.00000i −0.500000 + 0.866025i −1.00000 1.41174i −0.866025 − 0.500000i 1.61371 − 2.09665i − 1.00000i −0.500000 − 0.866025i −1.41174
145.13 1.00000i −0.500000 + 0.866025i −1.00000 1.81645i −0.866025 − 0.500000i 0.235464 + 2.63525i − 1.00000i −0.500000 − 0.866025i −1.81645
145.14 1.00000i −0.500000 + 0.866025i −1.00000 4.26868i −0.866025 − 0.500000i 2.01369 − 1.71612i − 1.00000i −0.500000 − 0.866025i −4.26868
787.1 − 1.00000i −0.500000 − 0.866025i −1.00000 − 4.26868i −0.866025 + 0.500000i 2.01369 + 1.71612i 1.00000i −0.500000 + 0.866025i −4.26868
787.2 − 1.00000i −0.500000 − 0.866025i −1.00000 − 1.81645i −0.866025 + 0.500000i 0.235464 − 2.63525i 1.00000i −0.500000 + 0.866025i −1.81645
787.3 − 1.00000i −0.500000 − 0.866025i −1.00000 − 1.41174i −0.866025 + 0.500000i 1.61371 + 2.09665i 1.00000i −0.500000 + 0.866025i −1.41174
787.4 − 1.00000i −0.500000 − 0.866025i −1.00000 − 0.525465i −0.866025 + 0.500000i −2.53656 + 0.752229i 1.00000i −0.500000 + 0.866025i −0.525465
787.5 − 1.00000i −0.500000 − 0.866025i −1.00000 0.953104i −0.866025 + 0.500000i 2.56076 − 0.665210i 1.00000i −0.500000 + 0.866025i 0.953104
787.6 − 1.00000i −0.500000 − 0.866025i −1.00000 1.47305i −0.866025 + 0.500000i −1.15317 + 2.38122i 1.00000i −0.500000 + 0.866025i 1.47305
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.m.a ✓ 28
7.d odd 6 1 798.2.bc.a yes 28
19.d odd 6 1 798.2.bc.a yes 28
133.i even 6 1 inner 798.2.m.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.m.a ✓ 28 1.a even 1 1 trivial
798.2.m.a ✓ 28 133.i even 6 1 inner
798.2.bc.a yes 28 7.d odd 6 1
798.2.bc.a yes 28 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 84 T_{5}^{26} + 3032 T_{5}^{24} + 62036 T_{5}^{22} + 800854 T_{5}^{20} + 6882500 T_{5}^{18} + \cdots + 19900521 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display