↑

Properties

Label 155.2.q.b
Level $155$
Weight $2$
Character orbit 155.q
Analytic conductor $1.238$
Analytic rank $0$
Dimension $40$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [155,2,Mod(41,155)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(155, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("155.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{2} + 3 q^{3} - 2 q^{4} - 20 q^{5} - 12 q^{6} - 2 q^{7} - 31 q^{8} + 6 q^{9} - q^{10} + 16 q^{11} - q^{12} + 2 q^{13} - 10 q^{14} - 6 q^{15} - 16 q^{16} + 4 q^{17} + 37 q^{18} - 9 q^{19} + q^{20}+ \cdots + 17 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −1.68173 − 1.22185i 0.111961 + 1.06524i 0.717276 + 2.20755i −0.500000 − 0.866025i 1.11327 − 1.92825i 0.859323 + 0.182655i 0.206298 − 0.634920i 1.81225 − 0.385205i −0.217287 + 2.06735i
41.2 −0.674820 − 0.490285i −0.345257 − 3.28490i −0.403032 − 1.24040i −0.500000 − 0.866025i −1.37755 + 2.38599i 4.81427 + 1.02330i −0.851695 + 2.62125i −7.73691 + 1.64453i −0.0871896 + 0.829554i
41.3 0.302062 + 0.219461i −0.0950600 − 0.904435i −0.574956 − 1.76953i −0.500000 − 0.866025i 0.169774 − 0.294057i −3.29862 − 0.701142i 0.445425 − 1.37088i 2.12548 − 0.451784i 0.0390276 − 0.371323i
41.4 1.59955 + 1.16214i 0.217822 + 2.07244i 0.589956 + 1.81570i −0.500000 − 0.866025i −2.06005 + 3.56811i −2.00916 − 0.427061i 0.0555144 − 0.170856i −1.31311 + 0.279109i 0.206669 − 1.96632i
41.5 2.03762 + 1.48042i −0.203052 − 1.93191i 1.34222 + 4.13093i −0.500000 − 0.866025i 2.44628 − 4.23709i 0.816826 + 0.173622i −1.82396 + 5.61358i −0.756592 + 0.160819i 0.263269 − 2.50484i
51.1 −0.658531 − 2.02675i −1.58243 − 1.75746i −2.05602 + 1.49378i −0.500000 − 0.866025i −2.51986 + 4.36452i −0.00518954 − 0.0493751i 0.933366 + 0.678130i −0.271016 + 2.57854i −1.42595 + 1.58368i
51.2 −0.437416 − 1.34623i 2.08671 + 2.31753i −0.00296283 + 0.00215262i −0.500000 − 0.866025i 2.20716 − 3.82291i −0.0992859 − 0.944642i −2.28615 − 1.66098i −0.702986 + 6.68846i −0.947160 + 1.05193i
51.3 −0.113347 − 0.348848i −0.527922 − 0.586317i 1.50919 − 1.09649i −0.500000 − 0.866025i −0.144697 + 0.250622i 0.101473 + 0.965447i −1.14706 − 0.833391i 0.248520 − 2.36451i −0.245437 + 0.272586i
51.4 0.335014 + 1.03107i 1.33393 + 1.48148i 0.667170 − 0.484727i −0.500000 − 0.866025i −1.08062 + 1.87168i −0.0611663 − 0.581958i 2.47745 + 1.79997i −0.101824 + 0.968793i 0.725423 − 0.805664i
51.5 0.809678 + 2.49193i 0.697102 + 0.774210i −3.93612 + 2.85976i −0.500000 − 0.866025i −1.36485 + 2.36399i 0.0288113 + 0.274121i −6.07379 − 4.41287i 0.200135 − 1.90416i 1.75324 − 1.94717i
71.1 −0.533804 + 1.64288i −1.18744 − 0.252397i −0.796073 − 0.578381i −0.500000 − 0.866025i 1.04852 − 1.81608i −4.30971 + 1.91880i −1.41988 + 1.03160i −1.39434 − 0.620799i 1.68968 − 0.359152i
71.2 −0.0791076 + 0.243468i −3.22924 − 0.686396i 1.56502 + 1.13705i −0.500000 − 0.866025i 0.422573 − 0.731918i 2.97452 − 1.32434i −0.814853 + 0.592025i 7.21620 + 3.21286i 0.250404 − 0.0532249i
71.3 −0.0597895 + 0.184013i 1.06671 + 0.226736i 1.58775 + 1.15357i −0.500000 − 0.866025i −0.105500 + 0.182732i 1.34512 − 0.598885i −0.620264 + 0.450648i −1.65418 − 0.736487i 0.189255 − 0.0402273i
71.4 0.494524 − 1.52199i 2.08260 + 0.442670i −0.453859 − 0.329748i −0.500000 − 0.866025i 1.70363 − 2.95078i −3.40184 + 1.51460i 1.86304 − 1.35358i 1.40063 + 0.623601i −1.56534 + 0.332724i
71.5 0.742779 − 2.28604i −1.66708 − 0.354348i −3.05622 − 2.22048i −0.500000 − 0.866025i −2.04833 + 3.54780i 0.691197 − 0.307741i −3.45695 + 2.51162i −0.0870493 − 0.0387568i −2.35116 + 0.499754i
76.1 −0.658531 + 2.02675i −1.58243 + 1.75746i −2.05602 − 1.49378i −0.500000 + 0.866025i −2.51986 − 4.36452i −0.00518954 + 0.0493751i 0.933366 − 0.678130i −0.271016 − 2.57854i −1.42595 − 1.58368i
76.2 −0.437416 + 1.34623i 2.08671 − 2.31753i −0.00296283 − 0.00215262i −0.500000 + 0.866025i 2.20716 + 3.82291i −0.0992859 + 0.944642i −2.28615 + 1.66098i −0.702986 − 6.68846i −0.947160 − 1.05193i
76.3 −0.113347 + 0.348848i −0.527922 + 0.586317i 1.50919 + 1.09649i −0.500000 + 0.866025i −0.144697 − 0.250622i 0.101473 − 0.965447i −1.14706 + 0.833391i 0.248520 + 2.36451i −0.245437 − 0.272586i
76.4 0.335014 − 1.03107i 1.33393 − 1.48148i 0.667170 + 0.484727i −0.500000 + 0.866025i −1.08062 − 1.87168i −0.0611663 + 0.581958i 2.47745 − 1.79997i −0.101824 − 0.968793i 0.725423 + 0.805664i
76.5 0.809678 − 2.49193i 0.697102 − 0.774210i −3.93612 − 2.85976i −0.500000 + 0.866025i −1.36485 − 2.36399i 0.0288113 − 0.274121i −6.07379 + 4.41287i 0.200135 + 1.90416i 1.75324 + 1.94717i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.q.b ✓ 40
5.b even 2 1 775.2.bl.b 40
5.c odd 4 2 775.2.ck.b 80
31.g even 15 1 inner 155.2.q.b ✓ 40
31.g even 15 1 4805.2.a.ba 20
31.h odd 30 1 4805.2.a.z 20
155.u even 30 1 775.2.bl.b 40
155.w odd 60 2 775.2.ck.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.b ✓ 40 1.a even 1 1 trivial
155.2.q.b ✓ 40 31.g even 15 1 inner
775.2.bl.b 40 5.b even 2 1
775.2.bl.b 40 155.u even 30 1
775.2.ck.b 80 5.c odd 4 2
775.2.ck.b 80 155.w odd 60 2
4805.2.a.z 20 31.h odd 30 1
4805.2.a.ba 20 31.g even 15 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 2 T_{2}^{39} + 13 T_{2}^{38} - 9 T_{2}^{37} + 104 T_{2}^{36} - 35 T_{2}^{35} + 934 T_{2}^{34} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display