Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(381,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("760.381"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 2 q^{4} - 6 q^{6} + 4 q^{7} + 8 q^{8} - 12 q^{9} + 4 q^{12} + 4 q^{14} - 6 q^{16} - 24 q^{17} - 14 q^{18} - 4 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{24} - 28 q^{25} - 6 q^{26} + 10 q^{28}+ \cdots - 82 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
381.1 −1.40331 − 0.175237i − 0.0798397i 1.93858 + 0.491825i 1.00000i −0.0139909 + 0.112040i 3.91506 −2.63426 − 1.02990i 2.99363 0.175237 − 1.40331i
381.2 −1.40331 + 0.175237i 0.0798397i 1.93858 − 0.491825i − 1.00000i −0.0139909 − 0.112040i 3.91506 −2.63426 + 1.02990i 2.99363 0.175237 + 1.40331i
381.3 −1.18545 − 0.771171i 1.74129i 0.810590 + 1.82837i 1.00000i 1.34283 − 2.06421i −4.26759 0.449072 − 2.79255i −0.0320903 0.771171 − 1.18545i
381.4 −1.18545 + 0.771171i − 1.74129i 0.810590 − 1.82837i − 1.00000i 1.34283 + 2.06421i −4.26759 0.449072 + 2.79255i −0.0320903 0.771171 + 1.18545i
381.5 −1.17171 − 0.791889i − 0.823852i 0.745825 + 1.85573i − 1.00000i −0.652399 + 0.965318i −0.523009 0.595641 − 2.76500i 2.32127 −0.791889 + 1.17171i
381.6 −1.17171 + 0.791889i 0.823852i 0.745825 − 1.85573i 1.00000i −0.652399 − 0.965318i −0.523009 0.595641 + 2.76500i 2.32127 −0.791889 − 1.17171i
381.7 −0.743283 − 1.20313i − 1.92849i −0.895061 + 1.78854i 1.00000i −2.32023 + 1.43341i 0.210397 2.81713 − 0.252512i −0.719076 1.20313 − 0.743283i
381.8 −0.743283 + 1.20313i 1.92849i −0.895061 − 1.78854i − 1.00000i −2.32023 − 1.43341i 0.210397 2.81713 + 0.252512i −0.719076 1.20313 + 0.743283i
381.9 −0.560741 − 1.29829i − 2.48816i −1.37114 + 1.45602i − 1.00000i −3.23036 + 1.39521i 3.58757 2.65919 + 0.963693i −3.19093 −1.29829 + 0.560741i
381.10 −0.560741 + 1.29829i 2.48816i −1.37114 − 1.45602i 1.00000i −3.23036 − 1.39521i 3.58757 2.65919 − 0.963693i −3.19093 −1.29829 − 0.560741i
381.11 −0.453222 − 1.33962i 0.782983i −1.58918 + 1.21429i − 1.00000i 1.04890 − 0.354865i −3.28528 2.34694 + 1.57856i 2.38694 −1.33962 + 0.453222i
381.12 −0.453222 + 1.33962i − 0.782983i −1.58918 − 1.21429i 1.00000i 1.04890 + 0.354865i −3.28528 2.34694 − 1.57856i 2.38694 −1.33962 − 0.453222i
381.13 −0.193159 − 1.40096i 2.50151i −1.92538 + 0.541217i 1.00000i 3.50452 − 0.483191i 0.698520 1.13013 + 2.59284i −3.25757 1.40096 − 0.193159i
381.14 −0.193159 + 1.40096i − 2.50151i −1.92538 − 0.541217i − 1.00000i 3.50452 + 0.483191i 0.698520 1.13013 − 2.59284i −3.25757 1.40096 + 0.193159i
381.15 0.172291 − 1.40368i − 3.19922i −1.94063 − 0.483683i 1.00000i −4.49068 − 0.551197i −0.255739 −1.01329 + 2.64069i −7.23503 1.40368 + 0.172291i
381.16 0.172291 + 1.40368i 3.19922i −1.94063 + 0.483683i − 1.00000i −4.49068 + 0.551197i −0.255739 −1.01329 − 2.64069i −7.23503 1.40368 − 0.172291i
381.17 0.530413 − 1.31098i 1.71180i −1.43732 − 1.39072i − 1.00000i 2.24413 + 0.907961i −2.02692 −2.58558 + 1.14664i 0.0697451 −1.31098 − 0.530413i
381.18 0.530413 + 1.31098i − 1.71180i −1.43732 + 1.39072i 1.00000i 2.24413 − 0.907961i −2.02692 −2.58558 − 1.14664i 0.0697451 −1.31098 + 0.530413i
381.19 0.950066 − 1.04756i 0.485865i −0.194751 − 1.99050i 1.00000i 0.508971 + 0.461603i 3.31744 −2.27018 − 1.68709i 2.76394 1.04756 + 0.950066i
381.20 0.950066 + 1.04756i − 0.485865i −0.194751 + 1.99050i − 1.00000i 0.508971 − 0.461603i 3.31744 −2.27018 + 1.68709i 2.76394 1.04756 − 0.950066i
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 381.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.f.a ✓ 28
4.b odd 2 1 3040.2.f.a 28
8.b even 2 1 inner 760.2.f.a ✓ 28
8.d odd 2 1 3040.2.f.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.f.a ✓ 28 1.a even 1 1 trivial
760.2.f.a ✓ 28 8.b even 2 1 inner
3040.2.f.a 28 4.b odd 2 1
3040.2.f.a 28 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 48 T_{3}^{26} + 1006 T_{3}^{24} + 12140 T_{3}^{22} + 93605 T_{3}^{20} + 483500 T_{3}^{18} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display