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Properties

Label 735.2.j.h
Level $735$
Weight $2$
Character orbit 735.j
Analytic conductor $5.869$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(197,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, 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("735.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} + 16 q^{10} - 16 q^{12} + 8 q^{13} - 16 q^{15} - 16 q^{16} - 20 q^{18} + 8 q^{22} - 16 q^{25} + 16 q^{27} + 20 q^{30} - 28 q^{33} + 16 q^{36} - 16 q^{37} - 64 q^{40} - 40 q^{43} - 20 q^{45}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.79963 + 1.79963i 1.66094 + 0.491204i − 4.47734i −1.87996 + 1.21069i −3.87306 + 2.10509i 0 4.45829 + 4.45829i 2.51744 + 1.63172i 1.20443 − 5.56202i
197.2 −1.54414 + 1.54414i 0.00622252 − 1.73204i − 2.76875i −0.252500 − 2.22177i 2.66491 + 2.68412i 0 1.18705 + 1.18705i −2.99992 − 0.0215553i 3.82062 + 3.04083i
197.3 −1.24414 + 1.24414i −1.66575 + 0.474620i − 1.09578i −1.67522 + 1.48109i 1.48194 − 2.66293i 0 −1.12498 − 1.12498i 2.54947 − 1.58120i 0.241524 − 3.92690i
197.4 −0.800553 + 0.800553i 1.34285 − 1.09397i 0.718229i 2.10480 + 0.754855i −0.199242 + 1.95080i 0 −2.17609 − 2.17609i 0.606476 − 2.93806i −2.28931 + 1.08070i
197.5 −0.347054 + 0.347054i −1.72305 + 0.176396i 1.75911i 1.16790 − 1.90683i 0.536770 − 0.659208i 0 −1.30461 − 1.30461i 2.93777 − 0.607876i 0.256447 + 1.06710i
197.6 −0.260263 + 0.260263i 1.52191 + 0.826909i 1.86453i −0.895238 − 2.04904i −0.611312 + 0.180884i 0 −1.00579 − 1.00579i 1.63244 + 2.51697i 0.766286 + 0.300291i
197.7 0.260263 − 0.260263i −0.826909 − 1.52191i 1.86453i 0.895238 + 2.04904i −0.611312 − 0.180884i 0 1.00579 + 1.00579i −1.63244 + 2.51697i 0.766286 + 0.300291i
197.8 0.347054 − 0.347054i −0.176396 + 1.72305i 1.75911i −1.16790 + 1.90683i 0.536770 + 0.659208i 0 1.30461 + 1.30461i −2.93777 − 0.607876i 0.256447 + 1.06710i
197.9 0.800553 − 0.800553i 1.09397 − 1.34285i 0.718229i −2.10480 − 0.754855i −0.199242 − 1.95080i 0 2.17609 + 2.17609i −0.606476 − 2.93806i −2.28931 + 1.08070i
197.10 1.24414 − 1.24414i −0.474620 + 1.66575i − 1.09578i 1.67522 − 1.48109i 1.48194 + 2.66293i 0 1.12498 + 1.12498i −2.54947 − 1.58120i 0.241524 − 3.92690i
197.11 1.54414 − 1.54414i 1.73204 − 0.00622252i − 2.76875i 0.252500 + 2.22177i 2.66491 − 2.68412i 0 −1.18705 − 1.18705i 2.99992 − 0.0215553i 3.82062 + 3.04083i
197.12 1.79963 − 1.79963i −0.491204 − 1.66094i − 4.47734i 1.87996 − 1.21069i −3.87306 − 2.10509i 0 −4.45829 − 4.45829i −2.51744 + 1.63172i 1.20443 − 5.56202i
638.1 −1.79963 − 1.79963i 1.66094 − 0.491204i 4.47734i −1.87996 − 1.21069i −3.87306 − 2.10509i 0 4.45829 − 4.45829i 2.51744 − 1.63172i 1.20443 + 5.56202i
638.2 −1.54414 − 1.54414i 0.00622252 + 1.73204i 2.76875i −0.252500 + 2.22177i 2.66491 − 2.68412i 0 1.18705 − 1.18705i −2.99992 + 0.0215553i 3.82062 − 3.04083i
638.3 −1.24414 − 1.24414i −1.66575 − 0.474620i 1.09578i −1.67522 − 1.48109i 1.48194 + 2.66293i 0 −1.12498 + 1.12498i 2.54947 + 1.58120i 0.241524 + 3.92690i
638.4 −0.800553 − 0.800553i 1.34285 + 1.09397i − 0.718229i 2.10480 − 0.754855i −0.199242 − 1.95080i 0 −2.17609 + 2.17609i 0.606476 + 2.93806i −2.28931 − 1.08070i
638.5 −0.347054 − 0.347054i −1.72305 − 0.176396i − 1.75911i 1.16790 + 1.90683i 0.536770 + 0.659208i 0 −1.30461 + 1.30461i 2.93777 + 0.607876i 0.256447 − 1.06710i
638.6 −0.260263 − 0.260263i 1.52191 − 0.826909i − 1.86453i −0.895238 + 2.04904i −0.611312 − 0.180884i 0 −1.00579 + 1.00579i 1.63244 − 2.51697i 0.766286 − 0.300291i
638.7 0.260263 + 0.260263i −0.826909 + 1.52191i − 1.86453i 0.895238 − 2.04904i −0.611312 + 0.180884i 0 1.00579 − 1.00579i −1.63244 − 2.51697i 0.766286 − 0.300291i
638.8 0.347054 + 0.347054i −0.176396 − 1.72305i − 1.75911i −1.16790 − 1.90683i 0.536770 − 0.659208i 0 1.30461 − 1.30461i −2.93777 + 0.607876i 0.256447 − 1.06710i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
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 735.2.j.h 24
3.b odd 2 1 inner 735.2.j.h 24
5.c odd 4 1 inner 735.2.j.h 24
7.b odd 2 1 105.2.j.a ✓ 24
7.c even 3 2 735.2.y.g 48
7.d odd 6 2 735.2.y.j 48
15.e even 4 1 inner 735.2.j.h 24
21.c even 2 1 105.2.j.a ✓ 24
21.g even 6 2 735.2.y.j 48
21.h odd 6 2 735.2.y.g 48
35.c odd 2 1 525.2.j.b 24
35.f even 4 1 105.2.j.a ✓ 24
35.f even 4 1 525.2.j.b 24
35.k even 12 2 735.2.y.j 48
35.l odd 12 2 735.2.y.g 48
105.g even 2 1 525.2.j.b 24
105.k odd 4 1 105.2.j.a ✓ 24
105.k odd 4 1 525.2.j.b 24
105.w odd 12 2 735.2.y.j 48
105.x even 12 2 735.2.y.g 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.j.a ✓ 24 7.b odd 2 1
105.2.j.a ✓ 24 21.c even 2 1
105.2.j.a ✓ 24 35.f even 4 1
105.2.j.a ✓ 24 105.k odd 4 1
525.2.j.b 24 35.c odd 2 1
525.2.j.b 24 35.f even 4 1
525.2.j.b 24 105.g even 2 1
525.2.j.b 24 105.k odd 4 1
735.2.j.h 24 1.a even 1 1 trivial
735.2.j.h 24 3.b odd 2 1 inner
735.2.j.h 24 5.c odd 4 1 inner
735.2.j.h 24 15.e even 4 1 inner
735.2.y.g 48 7.c even 3 2
735.2.y.g 48 21.h odd 6 2
735.2.y.g 48 35.l odd 12 2
735.2.y.g 48 105.x even 12 2
735.2.y.j 48 7.d odd 6 2
735.2.y.j 48 21.g even 6 2
735.2.y.j 48 35.k even 12 2
735.2.y.j 48 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{24} + 76T_{2}^{20} + 1702T_{2}^{16} + 11860T_{2}^{12} + 15921T_{2}^{8} + 1160T_{2}^{4} + 16 \) Copy content Toggle raw display
\( T_{13}^{12} - 4 T_{13}^{11} + 8 T_{13}^{10} + 124 T_{13}^{9} + 625 T_{13}^{8} - 160 T_{13}^{7} + \cdots + 33856 \) Copy content Toggle raw display
\( T_{17}^{24} + 1218T_{17}^{20} + 405105T_{17}^{16} + 26412784T_{17}^{12} + 52126816T_{17}^{8} + 244992T_{17}^{4} + 256 \) Copy content Toggle raw display