↑

Properties

Label 315.3.ca.c
Level $315$
Weight $3$
Character orbit 315.ca
Analytic conductor $8.583$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,3,Mod(37,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.ca (of order \(12\), degree \(4\), 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: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{7} - 16 q^{10} + 176 q^{16} - 96 q^{22} + 68 q^{25} + 220 q^{28} - 88 q^{31} - 100 q^{37} - 124 q^{40} - 328 q^{43} - 240 q^{46} + 172 q^{52} - 440 q^{55} - 328 q^{58} - 216 q^{61} + 304 q^{67} - 808 q^{70} - 316 q^{73} + 1120 q^{76} + 556 q^{82} + 1000 q^{85} - 420 q^{88} - 680 q^{91} - 800 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −3.79810 + 1.01770i 0 9.92576 − 5.73064i 3.96530 + 3.04571i 0 5.72104 + 4.03357i −20.7454 + 20.7454i 0 −18.1602 − 7.53245i
37.2 −3.08266 + 0.825995i 0 5.35640 − 3.09252i −4.73821 + 1.59668i 0 −0.960348 − 6.93381i −4.93088 + 4.93088i 0 13.2874 − 8.83575i
37.3 −3.02701 + 0.811086i 0 5.04084 − 2.91033i −1.67751 − 4.71020i 0 −1.04427 + 6.92167i −4.03446 + 4.03446i 0 8.89821 + 12.8972i
37.4 −2.31746 + 0.620960i 0 1.52090 − 0.878095i 4.98838 + 0.340652i 0 −6.88026 − 1.28919i 3.80661 − 3.80661i 0 −11.7719 + 2.30814i
37.5 −1.91414 + 0.512891i 0 −0.0632444 + 0.0365141i 3.49070 − 3.57981i 0 5.81222 − 3.90104i 5.70731 − 5.70731i 0 −4.84562 + 8.64258i
37.6 −1.31073 + 0.351210i 0 −1.86943 + 1.07931i 0.516380 + 4.97326i 0 6.99901 − 0.117775i 5.90935 − 5.90935i 0 −2.42350 − 6.33727i
37.7 −0.916706 + 0.245631i 0 −2.68409 + 1.54966i −4.98577 + 0.376974i 0 −0.575993 + 6.97626i 4.76418 − 4.76418i 0 4.47789 − 1.57023i
37.8 −0.317144 + 0.0849784i 0 −3.37074 + 1.94610i −1.03356 − 4.89201i 0 −5.70537 − 4.05571i 1.83229 − 1.83229i 0 0.743502 + 1.46364i
37.9 0.317144 − 0.0849784i 0 −3.37074 + 1.94610i 1.03356 + 4.89201i 0 −5.70537 − 4.05571i −1.83229 + 1.83229i 0 0.743502 + 1.46364i
37.10 0.916706 − 0.245631i 0 −2.68409 + 1.54966i 4.98577 − 0.376974i 0 −0.575993 + 6.97626i −4.76418 + 4.76418i 0 4.47789 − 1.57023i
37.11 1.31073 − 0.351210i 0 −1.86943 + 1.07931i −0.516380 − 4.97326i 0 6.99901 − 0.117775i −5.90935 + 5.90935i 0 −2.42350 − 6.33727i
37.12 1.91414 − 0.512891i 0 −0.0632444 + 0.0365141i −3.49070 + 3.57981i 0 5.81222 − 3.90104i −5.70731 + 5.70731i 0 −4.84562 + 8.64258i
37.13 2.31746 − 0.620960i 0 1.52090 − 0.878095i −4.98838 − 0.340652i 0 −6.88026 − 1.28919i −3.80661 + 3.80661i 0 −11.7719 + 2.30814i
37.14 3.02701 − 0.811086i 0 5.04084 − 2.91033i 1.67751 + 4.71020i 0 −1.04427 + 6.92167i 4.03446 − 4.03446i 0 8.89821 + 12.8972i
37.15 3.08266 − 0.825995i 0 5.35640 − 3.09252i 4.73821 − 1.59668i 0 −0.960348 − 6.93381i 4.93088 − 4.93088i 0 13.2874 − 8.83575i
37.16 3.79810 − 1.01770i 0 9.92576 − 5.73064i −3.96530 − 3.04571i 0 5.72104 + 4.03357i 20.7454 − 20.7454i 0 −18.1602 − 7.53245i
163.1 −1.01770 − 3.79810i 0 −9.92576 + 5.73064i 4.62032 + 1.91120i 0 4.03357 − 5.72104i 20.7454 + 20.7454i 0 2.55683 − 19.4935i
163.2 −0.825995 − 3.08266i 0 −5.35640 + 3.09252i −0.986340 − 4.90175i 0 −6.93381 + 0.960348i 4.93088 + 4.93088i 0 −14.2957 + 7.08937i
163.3 −0.811086 − 3.02701i 0 −5.04084 + 2.91033i −4.91791 + 0.902335i 0 6.92167 + 1.04427i 4.03446 + 4.03446i 0 6.72022 + 14.1547i
163.4 −0.620960 − 2.31746i 0 −1.52090 + 0.878095i 2.78920 + 4.14974i 0 −1.28919 + 6.88026i −3.80661 − 3.80661i 0 7.88485 − 9.04068i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
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
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.ca.c ✓ 64
3.b odd 2 1 inner 315.3.ca.c ✓ 64
5.c odd 4 1 inner 315.3.ca.c ✓ 64
7.c even 3 1 inner 315.3.ca.c ✓ 64
15.e even 4 1 inner 315.3.ca.c ✓ 64
21.h odd 6 1 inner 315.3.ca.c ✓ 64
35.l odd 12 1 inner 315.3.ca.c ✓ 64
105.x even 12 1 inner 315.3.ca.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.3.ca.c ✓ 64 1.a even 1 1 trivial
315.3.ca.c ✓ 64 3.b odd 2 1 inner
315.3.ca.c ✓ 64 5.c odd 4 1 inner
315.3.ca.c ✓ 64 7.c even 3 1 inner
315.3.ca.c ✓ 64 15.e even 4 1 inner
315.3.ca.c ✓ 64 21.h odd 6 1 inner
315.3.ca.c ✓ 64 35.l odd 12 1 inner
315.3.ca.c ✓ 64 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 492 T_{2}^{60} + 160310 T_{2}^{56} - 28697896 T_{2}^{52} + 3679776583 T_{2}^{48} + \cdots + 15\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display