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Properties

Label 945.2.r.b
Level $945$
Weight $2$
Character orbit 945.r
Analytic conductor $7.546$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(424,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.r (of order \(6\), degree \(2\), not 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 88 q^{4} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 88 q^{4} - 3 q^{5} + 6 q^{10} - 12 q^{11} - 8 q^{14} + 72 q^{16} + 8 q^{19} + 10 q^{20} - 5 q^{25} + 40 q^{26} + 10 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{35} + 4 q^{40} + 30 q^{41} + 4 q^{44} + 4 q^{46} + 8 q^{49} - 42 q^{50} - 54 q^{55} + 18 q^{56} + 84 q^{59} - 44 q^{61} - 28 q^{64} + 16 q^{65} + q^{70} + 4 q^{71} - 54 q^{74} + 24 q^{76} - 48 q^{79} + 9 q^{80} + q^{85} - 46 q^{86} - 46 q^{89} - 44 q^{91} + 16 q^{94} - 50 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1 − 2.71825i 0 −5.38890 2.07547 + 0.832131i 0 −2.57523 − 0.606776i 9.21190i 0 2.26194 − 5.64164i
424.2 − 2.61859i 0 −4.85701 1.02882 − 1.98533i 0 2.61471 − 0.404081i 7.48135i 0 −5.19877 − 2.69405i
424.3 − 2.61386i 0 −4.83226 −2.17617 + 0.514103i 0 1.37276 + 2.26175i 7.40313i 0 1.34379 + 5.68819i
424.4 − 2.42945i 0 −3.90223 −1.12953 + 1.92981i 0 0.958372 − 2.46607i 4.62136i 0 4.68837 + 2.74414i
424.5 − 2.37032i 0 −3.61843 −2.03667 − 0.923024i 0 0.987053 − 2.45474i 3.83621i 0 −2.18786 + 4.82757i
424.6 − 2.19537i 0 −2.81965 0.0446370 + 2.23562i 0 0.555178 + 2.58685i 1.79943i 0 4.90802 − 0.0979947i
424.7 − 2.12646i 0 −2.52185 2.23351 + 0.106887i 0 −2.47990 + 0.921996i 1.10970i 0 0.227292 − 4.74948i
424.8 − 2.02504i 0 −2.10077 −2.21568 − 0.301270i 0 −2.64524 − 0.0518041i 0.204070i 0 −0.610083 + 4.48683i
424.9 − 1.80957i 0 −1.27455 1.92051 + 1.14527i 0 0.174487 + 2.63999i − 1.31275i 0 2.07246 − 3.47530i
424.10 − 1.80560i 0 −1.26019 1.93209 − 1.12562i 0 2.01307 − 1.71685i − 1.33580i 0 −2.03243 − 3.48858i
424.11 − 1.64358i 0 −0.701356 0.289890 − 2.21720i 0 −1.64618 − 2.07125i − 2.13443i 0 −3.64414 − 0.476457i
424.12 − 1.62167i 0 −0.629802 −0.321745 − 2.21280i 0 −1.53778 + 2.15296i − 2.22200i 0 −3.58842 + 0.521762i
424.13 − 1.36154i 0 0.146205 −1.24734 + 1.85584i 0 −1.93746 − 1.80173i − 2.92215i 0 2.52681 + 1.69830i
424.14 − 1.08755i 0 0.817229 0.940709 + 2.02856i 0 −1.34619 − 2.27767i − 3.06389i 0 2.20617 − 1.02307i
424.15 − 1.07565i 0 0.842975 −0.440420 + 2.19227i 0 2.63194 − 0.269937i − 3.05805i 0 2.35811 + 0.473738i
424.16 − 0.872874i 0 1.23809 −2.16932 + 0.542260i 0 2.41635 + 1.07761i − 2.82644i 0 0.473324 + 1.89354i
424.17 − 0.724192i 0 1.47555 2.21636 + 0.296230i 0 2.56533 − 0.647375i − 2.51696i 0 0.214528 − 1.60507i
424.18 − 0.672284i 0 1.54803 −2.23061 + 0.156193i 0 −1.87376 + 1.86790i − 2.38529i 0 0.105006 + 1.49960i
424.19 − 0.286902i 0 1.91769 0.815278 − 2.08214i 0 −0.0701001 + 2.64482i − 1.12399i 0 −0.597370 − 0.233905i
424.20 − 0.257756i 0 1.93356 1.95987 − 1.07653i 0 −2.46150 − 0.970066i − 1.01390i 0 −0.277481 − 0.505168i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.h even 3 1 inner
315.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.r.b 84
3.b odd 2 1 315.2.r.b ✓ 84
5.b even 2 1 inner 945.2.r.b 84
7.c even 3 1 945.2.bo.b 84
9.c even 3 1 945.2.bo.b 84
9.d odd 6 1 315.2.bo.b yes 84
15.d odd 2 1 315.2.r.b ✓ 84
21.h odd 6 1 315.2.bo.b yes 84
35.j even 6 1 945.2.bo.b 84
45.h odd 6 1 315.2.bo.b yes 84
45.j even 6 1 945.2.bo.b 84
63.h even 3 1 inner 945.2.r.b 84
63.j odd 6 1 315.2.r.b ✓ 84
105.o odd 6 1 315.2.bo.b yes 84
315.r even 6 1 inner 945.2.r.b 84
315.br odd 6 1 315.2.r.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.b ✓ 84 3.b odd 2 1
315.2.r.b ✓ 84 15.d odd 2 1
315.2.r.b ✓ 84 63.j odd 6 1
315.2.r.b ✓ 84 315.br odd 6 1
315.2.bo.b yes 84 9.d odd 6 1
315.2.bo.b yes 84 21.h odd 6 1
315.2.bo.b yes 84 45.h odd 6 1
315.2.bo.b yes 84 105.o odd 6 1
945.2.r.b 84 1.a even 1 1 trivial
945.2.r.b 84 5.b even 2 1 inner
945.2.r.b 84 63.h even 3 1 inner
945.2.r.b 84 315.r even 6 1 inner
945.2.bo.b 84 7.c even 3 1
945.2.bo.b 84 9.c even 3 1
945.2.bo.b 84 35.j even 6 1
945.2.bo.b 84 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 64 T_{2}^{40} + 1889 T_{2}^{38} + 34121 T_{2}^{36} + 422073 T_{2}^{34} + 3791037 T_{2}^{32} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display