↑

Properties

Label 60.7.f.a
Level $60$
Weight $7$
Character orbit 60.f
Analytic conductor $13.803$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,7,Mod(19,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.f (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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 66 q^{4} + 44 q^{5} - 162 q^{6} + 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 66 q^{4} + 44 q^{5} - 162 q^{6} + 8748 q^{9} - 1422 q^{10} + 15484 q^{14} + 1266 q^{16} - 2188 q^{20} - 9234 q^{24} + 13812 q^{25} - 32052 q^{26} - 36920 q^{29} - 24300 q^{30} - 156204 q^{34} - 16038 q^{36} + 7674 q^{40} + 341848 q^{41} + 74892 q^{44} + 10692 q^{45} + 478080 q^{46} + 482556 q^{49} - 191448 q^{50} - 39366 q^{54} - 926132 q^{56} + 126846 q^{60} - 455976 q^{61} - 618 q^{64} - 624192 q^{65} - 127332 q^{66} - 541728 q^{69} - 1871304 q^{70} - 612324 q^{74} - 1566456 q^{76} + 2360972 q^{80} + 2125764 q^{81} + 858276 q^{84} + 2280864 q^{85} + 5157592 q^{86} - 1806104 q^{89} - 345546 q^{90} - 3121992 q^{94} - 4040118 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −7.93273 − 1.03526i 15.5885 61.8565 + 16.4248i −80.2400 + 95.8465i −123.659 − 16.1381i −143.606 −473.687 − 194.331i 243.000 735.748 − 677.255i
19.2 −7.93273 + 1.03526i 15.5885 61.8565 − 16.4248i −80.2400 − 95.8465i −123.659 + 16.1381i −143.606 −473.687 + 194.331i 243.000 735.748 + 677.255i
19.3 −7.65640 − 2.31937i −15.5885 53.2410 + 35.5161i 87.3911 + 89.3745i 119.352 + 36.1554i −36.3122 −325.260 − 395.411i 243.000 −461.809 − 886.980i
19.4 −7.65640 + 2.31937i −15.5885 53.2410 − 35.5161i 87.3911 − 89.3745i 119.352 − 36.1554i −36.3122 −325.260 + 395.411i 243.000 −461.809 + 886.980i
19.5 −7.34967 − 3.15947i 15.5885 44.0354 + 46.4422i 111.509 − 56.4858i −114.570 − 49.2513i 231.647 −176.913 − 480.464i 243.000 −998.024 + 62.8414i
19.6 −7.34967 + 3.15947i 15.5885 44.0354 − 46.4422i 111.509 + 56.4858i −114.570 + 49.2513i 231.647 −176.913 + 480.464i 243.000 −998.024 − 62.8414i
19.7 −6.43144 − 4.75779i −15.5885 18.7270 + 61.1989i −121.123 + 30.8887i 100.256 + 74.1665i −549.446 170.730 − 482.696i 243.000 925.960 + 377.621i
19.8 −6.43144 + 4.75779i −15.5885 18.7270 − 61.1989i −121.123 − 30.8887i 100.256 − 74.1665i −549.446 170.730 + 482.696i 243.000 925.960 − 377.621i
19.9 −5.26743 − 6.02115i −15.5885 −8.50840 + 63.4319i −13.2381 − 124.297i 82.1111 + 93.8604i 380.004 426.750 − 282.893i 243.000 −678.680 + 734.434i
19.10 −5.26743 + 6.02115i −15.5885 −8.50840 − 63.4319i −13.2381 + 124.297i 82.1111 − 93.8604i 380.004 426.750 + 282.893i 243.000 −678.680 − 734.434i
19.11 −4.30858 − 6.74063i 15.5885 −26.8722 + 58.0851i −108.897 − 61.3721i −67.1642 − 105.076i −118.127 507.312 − 69.1287i 243.000 55.5036 + 998.458i
19.12 −4.30858 + 6.74063i 15.5885 −26.8722 − 58.0851i −108.897 + 61.3721i −67.1642 + 105.076i −118.127 507.312 + 69.1287i 243.000 55.5036 − 998.458i
19.13 −3.70638 − 7.08962i 15.5885 −36.5255 + 52.5537i 108.680 + 61.7555i −57.7768 − 110.516i −629.059 507.963 + 64.1675i 243.000 35.0151 − 999.387i
19.14 −3.70638 + 7.08962i 15.5885 −36.5255 − 52.5537i 108.680 − 61.7555i −57.7768 + 110.516i −629.059 507.963 − 64.1675i 243.000 35.0151 + 999.387i
19.15 −1.63894 − 7.83032i −15.5885 −58.6278 + 25.6668i −44.2074 + 116.922i 25.5485 + 122.063i 42.6864 297.066 + 417.008i 243.000 987.988 + 154.531i
19.16 −1.63894 + 7.83032i −15.5885 −58.6278 − 25.6668i −44.2074 − 116.922i 25.5485 − 122.063i 42.6864 297.066 − 417.008i 243.000 987.988 − 154.531i
19.17 −0.294917 − 7.99456i 15.5885 −63.8260 + 4.71546i 71.1255 − 102.792i −4.59730 − 124.623i 496.078 56.5214 + 508.871i 243.000 −842.752 − 538.302i
19.18 −0.294917 + 7.99456i 15.5885 −63.8260 − 4.71546i 71.1255 + 102.792i −4.59730 + 124.623i 496.078 56.5214 − 508.871i 243.000 −842.752 + 538.302i
19.19 0.294917 − 7.99456i −15.5885 −63.8260 − 4.71546i 71.1255 − 102.792i −4.59730 + 124.623i −496.078 −56.5214 + 508.871i 243.000 −800.800 − 598.932i
19.20 0.294917 + 7.99456i −15.5885 −63.8260 + 4.71546i 71.1255 + 102.792i −4.59730 − 124.623i −496.078 −56.5214 − 508.871i 243.000 −800.800 + 598.932i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.f.a ✓ 36
3.b odd 2 1 180.7.f.g 36
4.b odd 2 1 inner 60.7.f.a ✓ 36
5.b even 2 1 inner 60.7.f.a ✓ 36
12.b even 2 1 180.7.f.g 36
15.d odd 2 1 180.7.f.g 36
20.d odd 2 1 inner 60.7.f.a ✓ 36
60.h even 2 1 180.7.f.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.f.a ✓ 36 1.a even 1 1 trivial
60.7.f.a ✓ 36 4.b odd 2 1 inner
60.7.f.a ✓ 36 5.b even 2 1 inner
60.7.f.a ✓ 36 20.d odd 2 1 inner
180.7.f.g 36 3.b odd 2 1
180.7.f.g 36 12.b even 2 1
180.7.f.g 36 15.d odd 2 1
180.7.f.g 36 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(60, [\chi])\).