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Properties

Label 798.2.bt.a
Level $798$
Weight $2$
Character orbit 798.bt
Analytic conductor $6.372$
Analytic rank $0$
Dimension $324$
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] = [798,2,Mod(5,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 15, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bt (of order \(18\), degree \(6\), 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: \(324\)
Relative dimension: \(54\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 324 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 324 q - 6 q^{13} - 6 q^{15} + 12 q^{18} + 42 q^{19} - 12 q^{22} + 12 q^{25} - 18 q^{28} + 36 q^{34} - 12 q^{37} + 18 q^{39} + 18 q^{43} - 54 q^{45} - 24 q^{46} - 96 q^{49} - 6 q^{52} + 84 q^{57} - 48 q^{60} + 42 q^{61} + 36 q^{63} + 162 q^{64} - 72 q^{66} + 18 q^{67} + 48 q^{70} - 12 q^{73} + 24 q^{78} + 84 q^{79} + 120 q^{81} - 18 q^{84} - 36 q^{85} - 36 q^{91} - 84 q^{93} + 144 q^{94} + 78 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.984808 − 0.173648i −1.73146 − 0.0450853i 0.939693 + 0.342020i 3.54542 − 1.29043i 1.69733 + 0.345066i 2.59862 + 0.497155i −0.866025 − 0.500000i 2.99593 + 0.156127i −3.71564 + 0.655168i
5.2 −0.984808 − 0.173648i −1.72928 + 0.0979859i 0.939693 + 0.342020i −1.62077 + 0.589912i 1.72002 + 0.203788i 2.41768 − 1.07462i −0.866025 − 0.500000i 2.98080 − 0.338890i 1.69858 − 0.299506i
5.3 −0.984808 − 0.173648i −1.69753 + 0.344103i 0.939693 + 0.342020i 1.39224 − 0.506735i 1.73149 − 0.0441030i −1.88305 − 1.85853i −0.866025 − 0.500000i 2.76319 − 1.16825i −1.45909 + 0.257276i
5.4 −0.984808 − 0.173648i −1.66089 − 0.491354i 0.939693 + 0.342020i −2.78797 + 1.01474i 1.55034 + 0.772300i −1.68017 + 2.04378i −0.866025 − 0.500000i 2.51714 + 1.63217i 2.92183 − 0.515197i
5.5 −0.984808 − 0.173648i −1.48409 + 0.893007i 0.939693 + 0.342020i −1.32965 + 0.483955i 1.61662 − 0.621730i −1.30520 + 2.30140i −0.866025 − 0.500000i 1.40508 − 2.65062i 1.39349 − 0.245710i
5.6 −0.984808 − 0.173648i −1.20477 − 1.24440i 0.939693 + 0.342020i −1.87610 + 0.682846i 0.970378 + 1.43470i 0.534943 − 2.59111i −0.866025 − 0.500000i −0.0970619 + 2.99843i 1.96618 − 0.346690i
5.7 −0.984808 − 0.173648i −1.14284 + 1.30151i 0.939693 + 0.342020i 3.55285 − 1.29313i 1.35148 − 1.08328i −1.88071 + 1.86089i −0.866025 − 0.500000i −0.387831 − 2.97483i −3.72342 + 0.656539i
5.8 −0.984808 − 0.173648i −1.10342 + 1.33509i 0.939693 + 0.342020i 0.801875 − 0.291859i 1.31849 − 1.12320i 2.37737 + 1.16109i −0.866025 − 0.500000i −0.564924 − 2.94633i −0.840374 + 0.148181i
5.9 −0.984808 − 0.173648i −1.10320 − 1.33527i 0.939693 + 0.342020i 1.68957 − 0.614955i 0.854578 + 1.50655i −2.57971 − 0.587435i −0.866025 − 0.500000i −0.565879 + 2.94615i −1.77069 + 0.312221i
5.10 −0.984808 − 0.173648i −0.828058 − 1.52129i 0.939693 + 0.342020i 1.69718 − 0.617724i 0.551309 + 1.64197i −0.218323 + 2.63673i −0.866025 − 0.500000i −1.62864 + 2.51943i −1.77867 + 0.313627i
5.11 −0.984808 − 0.173648i −0.689937 + 1.58871i 0.939693 + 0.342020i −1.04392 + 0.379955i 0.955332 − 1.44476i 0.0152613 − 2.64571i −0.866025 − 0.500000i −2.04797 − 2.19221i 1.09404 − 0.192908i
5.12 −0.984808 − 0.173648i −0.240402 − 1.71529i 0.939693 + 0.342020i −0.721795 + 0.262712i −0.0611068 + 1.73097i 2.53985 + 0.741044i −0.866025 − 0.500000i −2.88441 + 0.824716i 0.756448 − 0.133382i
5.13 −0.984808 − 0.173648i 0.0108782 + 1.73202i 0.939693 + 0.342020i −0.566128 + 0.206054i 0.290049 − 1.70759i −2.24043 + 1.40730i −0.866025 − 0.500000i −2.99976 + 0.0376826i 0.593308 − 0.104616i
5.14 −0.984808 − 0.173648i 0.205411 − 1.71983i 0.939693 + 0.342020i 2.87411 − 1.04609i −0.500935 + 1.65803i 1.01877 − 2.44174i −0.866025 − 0.500000i −2.91561 − 0.706542i −3.01210 + 0.531114i
5.15 −0.984808 − 0.173648i 0.365205 − 1.69311i 0.939693 + 0.342020i −3.01095 + 1.09590i −0.653663 + 1.60397i −0.886115 − 2.49295i −0.866025 − 0.500000i −2.73325 − 1.23667i 3.15551 − 0.556401i
5.16 −0.984808 − 0.173648i 0.426084 + 1.67882i 0.939693 + 0.342020i −0.693921 + 0.252567i −0.128086 − 1.72731i 1.73023 − 2.00157i −0.866025 − 0.500000i −2.63690 + 1.43064i 0.727237 − 0.128232i
5.17 −0.984808 − 0.173648i 0.477676 + 1.66488i 0.939693 + 0.342020i −3.94363 + 1.43536i −0.181315 − 1.72253i 1.27510 + 2.31821i −0.866025 − 0.500000i −2.54365 + 1.59055i 4.13297 − 0.728753i
5.18 −0.984808 − 0.173648i 0.553250 + 1.64131i 0.939693 + 0.342020i 2.95464 − 1.07540i −0.259834 − 1.71245i −2.39060 − 1.13360i −0.866025 − 0.500000i −2.38783 + 1.81612i −3.09649 + 0.545995i
5.19 −0.984808 − 0.173648i 0.624782 − 1.61544i 0.939693 + 0.342020i −2.78451 + 1.01348i −0.895808 + 1.48241i 1.38450 + 2.25459i −0.866025 − 0.500000i −2.21930 − 2.01860i 2.91820 − 0.514557i
5.20 −0.984808 − 0.173648i 0.910976 − 1.47313i 0.939693 + 0.342020i 3.55282 − 1.29312i −1.15294 + 1.29256i 0.0205122 + 2.64567i −0.866025 − 0.500000i −1.34025 − 2.68398i −3.72340 + 0.656535i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
133.z odd 18 1 inner
399.cb even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bt.a ✓ 324
3.b odd 2 1 inner 798.2.bt.a ✓ 324
7.d odd 6 1 798.2.cb.a yes 324
19.e even 9 1 798.2.cb.a yes 324
21.g even 6 1 798.2.cb.a yes 324
57.l odd 18 1 798.2.cb.a yes 324
133.z odd 18 1 inner 798.2.bt.a ✓ 324
399.cb even 18 1 inner 798.2.bt.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bt.a ✓ 324 1.a even 1 1 trivial
798.2.bt.a ✓ 324 3.b odd 2 1 inner
798.2.bt.a ✓ 324 133.z odd 18 1 inner
798.2.bt.a ✓ 324 399.cb even 18 1 inner
798.2.cb.a yes 324 7.d odd 6 1
798.2.cb.a yes 324 19.e even 9 1
798.2.cb.a yes 324 21.g even 6 1
798.2.cb.a yes 324 57.l odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(798, [\chi])\).