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Properties

Label 798.2.m.b
Level $798$
Weight $2$
Character orbit 798.m
Analytic conductor $6.372$
Analytic rank $0$
Dimension $28$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{3} - 28 q^{4} - 8 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{3} - 28 q^{4} - 8 q^{7} - 14 q^{9} + 4 q^{10} - 14 q^{12} - 2 q^{13} - 4 q^{14} + 6 q^{15} + 28 q^{16} - 6 q^{17} + 2 q^{21} - 12 q^{22} - 12 q^{23} - 28 q^{25} + 24 q^{26} - 28 q^{27} + 8 q^{28} + 18 q^{29} + 2 q^{30} - 4 q^{31} + 4 q^{34} - 4 q^{35} + 14 q^{36} + 24 q^{37} + 18 q^{38} + 2 q^{39} - 4 q^{40} - 4 q^{41} + 4 q^{42} - 14 q^{43} + 6 q^{45} + 12 q^{46} + 24 q^{47} + 14 q^{48} - 24 q^{49} - 6 q^{51} + 2 q^{52} - 18 q^{55} + 4 q^{56} + 12 q^{57} - 2 q^{58} + 22 q^{59} - 6 q^{60} - 6 q^{61} + 12 q^{62} + 10 q^{63} - 28 q^{64} + 24 q^{65} + 6 q^{68} - 24 q^{69} + 30 q^{70} - 6 q^{71} - 42 q^{73} - 6 q^{74} - 14 q^{75} + 40 q^{77} + 24 q^{78} - 14 q^{81} - 6 q^{82} - 2 q^{84} + 28 q^{85} - 54 q^{86} + 18 q^{87} + 12 q^{88} + 8 q^{89} - 2 q^{90} - 22 q^{91} + 12 q^{92} - 8 q^{93} + 18 q^{94} + 20 q^{95} + 40 q^{97} + 32 q^{98}+O(q^{100}) \) 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 − 3.25990i −0.866025 − 0.500000i −1.88485 − 1.85670i 1.00000i −0.500000 − 0.866025i −3.25990
145.2 − 1.00000i 0.500000 − 0.866025i −1.00000 − 2.71950i −0.866025 − 0.500000i −2.09808 + 1.61185i 1.00000i −0.500000 − 0.866025i −2.71950
145.3 − 1.00000i 0.500000 − 0.866025i −1.00000 − 0.212667i −0.866025 − 0.500000i 0.768966 + 2.53154i 1.00000i −0.500000 − 0.866025i −0.212667
145.4 − 1.00000i 0.500000 − 0.866025i −1.00000 0.553978i −0.866025 − 0.500000i 2.21615 − 1.44523i 1.00000i −0.500000 − 0.866025i 0.553978
145.5 − 1.00000i 0.500000 − 0.866025i −1.00000 2.36613i −0.866025 − 0.500000i 1.13558 + 2.38966i 1.00000i −0.500000 − 0.866025i 2.36613
145.6 − 1.00000i 0.500000 − 0.866025i −1.00000 2.96528i −0.866025 − 0.500000i −2.64115 − 0.155916i 1.00000i −0.500000 − 0.866025i 2.96528
145.7 − 1.00000i 0.500000 − 0.866025i −1.00000 3.03872i −0.866025 − 0.500000i −1.22866 − 2.34316i 1.00000i −0.500000 − 0.866025i 3.03872
145.8 1.00000i 0.500000 − 0.866025i −1.00000 − 3.46192i 0.866025 + 0.500000i 2.47235 + 0.942075i − 1.00000i −0.500000 − 0.866025i 3.46192
145.9 1.00000i 0.500000 − 0.866025i −1.00000 − 2.33851i 0.866025 + 0.500000i −0.397795 − 2.61568i − 1.00000i −0.500000 − 0.866025i 2.33851
145.10 1.00000i 0.500000 − 0.866025i −1.00000 − 0.758127i 0.866025 + 0.500000i −1.59981 + 2.10727i − 1.00000i −0.500000 − 0.866025i 0.758127
145.11 1.00000i 0.500000 − 0.866025i −1.00000 − 0.367048i 0.866025 + 0.500000i −1.05442 + 2.42656i − 1.00000i −0.500000 − 0.866025i 0.367048
145.12 1.00000i 0.500000 − 0.866025i −1.00000 1.15459i 0.866025 + 0.500000i 2.64324 + 0.115227i − 1.00000i −0.500000 − 0.866025i −1.15459
145.13 1.00000i 0.500000 − 0.866025i −1.00000 2.42722i 0.866025 + 0.500000i −0.919460 − 2.48085i − 1.00000i −0.500000 − 0.866025i −2.42722
145.14 1.00000i 0.500000 − 0.866025i −1.00000 4.07585i 0.866025 + 0.500000i −1.41205 + 2.23744i − 1.00000i −0.500000 − 0.866025i −4.07585
787.1 − 1.00000i 0.500000 + 0.866025i −1.00000 − 4.07585i 0.866025 − 0.500000i −1.41205 − 2.23744i 1.00000i −0.500000 + 0.866025i −4.07585
787.2 − 1.00000i 0.500000 + 0.866025i −1.00000 − 2.42722i 0.866025 − 0.500000i −0.919460 + 2.48085i 1.00000i −0.500000 + 0.866025i −2.42722
787.3 − 1.00000i 0.500000 + 0.866025i −1.00000 − 1.15459i 0.866025 − 0.500000i 2.64324 − 0.115227i 1.00000i −0.500000 + 0.866025i −1.15459
787.4 − 1.00000i 0.500000 + 0.866025i −1.00000 0.367048i 0.866025 − 0.500000i −1.05442 − 2.42656i 1.00000i −0.500000 + 0.866025i 0.367048
787.5 − 1.00000i 0.500000 + 0.866025i −1.00000 0.758127i 0.866025 − 0.500000i −1.59981 − 2.10727i 1.00000i −0.500000 + 0.866025i 0.758127
787.6 − 1.00000i 0.500000 + 0.866025i −1.00000 2.33851i 0.866025 − 0.500000i −0.397795 + 2.61568i 1.00000i −0.500000 + 0.866025i 2.33851
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.b ✓ 28
7.d odd 6 1 798.2.bc.b yes 28
19.d odd 6 1 798.2.bc.b yes 28
133.i even 6 1 inner 798.2.m.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.m.b ✓ 28 1.a even 1 1 trivial
798.2.m.b ✓ 28 133.i even 6 1 inner
798.2.bc.b yes 28 7.d odd 6 1
798.2.bc.b yes 28 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 84 T_{5}^{26} + 3104 T_{5}^{24} + 66476 T_{5}^{22} + 914134 T_{5}^{20} + 8440364 T_{5}^{18} + \cdots + 328329 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display