↑

Properties

Label 1815.2.c.k
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,2,Mod(364,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1815.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), 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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 2 q^{5} + 8 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 2 q^{5} + 8 q^{6} - 24 q^{9} + 6 q^{10} - 12 q^{14} + 48 q^{16} - 32 q^{19} - 2 q^{20} + 16 q^{21} - 24 q^{24} + 2 q^{25} + 32 q^{26} + 8 q^{30} - 12 q^{34} - 10 q^{35} + 24 q^{36} - 36 q^{39} - 34 q^{40} + 2 q^{45} + 56 q^{46} - 24 q^{49} - 46 q^{50} + 36 q^{51} - 8 q^{54} + 12 q^{56} - 40 q^{59} - 26 q^{60} + 40 q^{61} + 12 q^{64} - 10 q^{65} - 2 q^{70} + 64 q^{71} + 136 q^{74} + 20 q^{75} + 68 q^{76} - 64 q^{79} + 76 q^{80} + 24 q^{81} - 60 q^{84} - 72 q^{86} + 20 q^{89} - 6 q^{90} + 4 q^{94} - 64 q^{95} + 56 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1 − 2.72592i 1.00000i −5.43063 1.85250 − 1.25229i 2.72592 − 0.486331i 9.35163i −1.00000 −3.41365 − 5.04977i
364.2 − 2.59201i 1.00000i −4.71851 −0.0953833 + 2.23403i 2.59201 − 3.41835i 7.04639i −1.00000 5.79063 + 0.247234i
364.3 − 2.36377i − 1.00000i −3.58741 0.979185 + 2.01027i −2.36377 1.71423i 3.75227i −1.00000 4.75182 − 2.31457i
364.4 − 2.21395i − 1.00000i −2.90157 −2.14116 + 0.644543i −2.21395 1.59339i 1.99602i −1.00000 1.42699 + 4.74042i
364.5 − 2.10651i 1.00000i −2.43738 −0.823234 − 2.07901i 2.10651 3.31633i 0.921348i −1.00000 −4.37946 + 1.73415i
364.6 − 1.90743i 1.00000i −1.63829 −2.10308 + 0.759628i 1.90743 − 4.45734i − 0.689943i −1.00000 1.44894 + 4.01149i
364.7 − 1.27093i − 1.00000i 0.384732 1.01340 − 1.99324i −1.27093 − 0.483291i − 3.03083i −1.00000 −2.53328 − 1.28796i
364.8 − 0.803366i 1.00000i 1.35460 2.13623 − 0.660705i 0.803366 0.508505i − 2.69497i −1.00000 −0.530788 − 1.71617i
364.9 − 0.784131i − 1.00000i 1.38514 −1.83964 + 1.27111i −0.784131 1.51801i − 2.65439i −1.00000 0.996719 + 1.44252i
364.10 − 0.488299i 1.00000i 1.76156 2.10928 − 0.742259i 0.488299 − 5.10895i − 1.83677i −1.00000 −0.362444 − 1.02996i
364.11 − 0.298064i 1.00000i 1.91116 −1.68093 + 1.47460i 0.298064 2.32107i − 1.16578i −1.00000 0.439527 + 0.501026i
364.12 − 0.288813i − 1.00000i 1.91659 −0.407156 − 2.19869i −0.288813 − 3.66740i − 1.13116i −1.00000 −0.635009 + 0.117592i
364.13 0.288813i 1.00000i 1.91659 −0.407156 + 2.19869i −0.288813 3.66740i 1.13116i −1.00000 −0.635009 − 0.117592i
364.14 0.298064i − 1.00000i 1.91116 −1.68093 − 1.47460i 0.298064 − 2.32107i 1.16578i −1.00000 0.439527 − 0.501026i
364.15 0.488299i − 1.00000i 1.76156 2.10928 + 0.742259i 0.488299 5.10895i 1.83677i −1.00000 −0.362444 + 1.02996i
364.16 0.784131i 1.00000i 1.38514 −1.83964 − 1.27111i −0.784131 − 1.51801i 2.65439i −1.00000 0.996719 − 1.44252i
364.17 0.803366i − 1.00000i 1.35460 2.13623 + 0.660705i 0.803366 − 0.508505i 2.69497i −1.00000 −0.530788 + 1.71617i
364.18 1.27093i 1.00000i 0.384732 1.01340 + 1.99324i −1.27093 0.483291i 3.03083i −1.00000 −2.53328 + 1.28796i
364.19 1.90743i − 1.00000i −1.63829 −2.10308 − 0.759628i 1.90743 4.45734i 0.689943i −1.00000 1.44894 − 4.01149i
364.20 2.10651i − 1.00000i −2.43738 −0.823234 + 2.07901i 2.10651 − 3.31633i − 0.921348i −1.00000 −4.37946 − 1.73415i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.k 24
5.b even 2 1 inner 1815.2.c.k 24
5.c odd 4 1 9075.2.a.dx 12
5.c odd 4 1 9075.2.a.ea 12
11.b odd 2 1 1815.2.c.j 24
11.d odd 10 2 165.2.s.a ✓ 48
33.f even 10 2 495.2.ba.c 48
55.d odd 2 1 1815.2.c.j 24
55.e even 4 1 9075.2.a.dy 12
55.e even 4 1 9075.2.a.dz 12
55.h odd 10 2 165.2.s.a ✓ 48
55.l even 20 2 825.2.n.o 24
55.l even 20 2 825.2.n.p 24
165.r even 10 2 495.2.ba.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a ✓ 48 11.d odd 10 2
165.2.s.a ✓ 48 55.h odd 10 2
495.2.ba.c 48 33.f even 10 2
495.2.ba.c 48 165.r even 10 2
825.2.n.o 24 55.l even 20 2
825.2.n.p 24 55.l even 20 2
1815.2.c.j 24 11.b odd 2 1
1815.2.c.j 24 55.d odd 2 1
1815.2.c.k 24 1.a even 1 1 trivial
1815.2.c.k 24 5.b even 2 1 inner
9075.2.a.dx 12 5.c odd 4 1
9075.2.a.dy 12 55.e even 4 1
9075.2.a.dz 12 55.e even 4 1
9075.2.a.ea 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{24} + 36 T_{2}^{22} + 552 T_{2}^{20} + 4702 T_{2}^{18} + 24334 T_{2}^{16} + 78666 T_{2}^{14} + \cdots + 25 \) Copy content Toggle raw display
\( T_{19}^{12} + 16 T_{19}^{11} + 16 T_{19}^{10} - 942 T_{19}^{9} - 4470 T_{19}^{8} + 10358 T_{19}^{7} + \cdots + 734525 \) Copy content Toggle raw display