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Properties

Label 70.5.l.b
Level $70$
Weight $5$
Character orbit 70.l
Analytic conductor $7.236$
Analytic rank $0$
Dimension $32$
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] = [70,5,Mod(23,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.23");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.l (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: \(7.23589741587\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} + 8 q^{3} - 26 q^{5} + 64 q^{6} + 14 q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} + 8 q^{3} - 26 q^{5} + 64 q^{6} + 14 q^{7} + 512 q^{8} + 68 q^{10} - 228 q^{11} + 64 q^{12} - 96 q^{13} + 1084 q^{15} + 1024 q^{16} - 30 q^{17} + 1016 q^{18} + 128 q^{20} - 1904 q^{21} - 912 q^{22} - 708 q^{23} - 478 q^{25} - 192 q^{26} - 1528 q^{27} + 1184 q^{28} + 2252 q^{30} + 2092 q^{31} - 2048 q^{32} + 1010 q^{33} - 5076 q^{35} + 8128 q^{36} - 514 q^{37} - 320 q^{38} - 288 q^{40} - 12448 q^{41} - 272 q^{42} - 8924 q^{43} + 3614 q^{45} + 2832 q^{46} + 226 q^{47} + 1024 q^{48} + 4440 q^{50} + 3140 q^{51} + 384 q^{52} - 3318 q^{53} - 17516 q^{55} + 448 q^{56} - 12996 q^{57} - 7712 q^{58} - 4672 q^{60} + 312 q^{61} + 8368 q^{62} + 23264 q^{63} + 10502 q^{65} - 4040 q^{66} - 2384 q^{67} + 240 q^{68} - 19788 q^{70} + 43832 q^{71} + 8128 q^{72} + 2270 q^{73} + 9696 q^{75} - 2560 q^{76} + 14282 q^{77} - 36720 q^{78} + 1664 q^{80} + 44392 q^{81} - 12448 q^{82} + 65372 q^{83} + 15400 q^{85} - 17848 q^{86} - 63830 q^{87} - 3648 q^{88} - 36784 q^{90} + 22656 q^{91} + 11328 q^{92} + 13642 q^{93} - 1442 q^{95} + 2048 q^{96} + 78400 q^{97} + 35136 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.732051 − 2.73205i −3.93762 + 14.6954i −6.92820 + 4.00000i 1.06831 + 24.9772i 43.0311 −18.5109 − 45.3690i 16.0000 + 16.0000i −130.302 − 75.2298i 67.4568 − 21.2032i
23.2 −0.732051 − 2.73205i −3.51712 + 13.1261i −6.92820 + 4.00000i −8.10772 − 23.6488i 38.4358 23.2447 + 43.1356i 16.0000 + 16.0000i −89.7757 − 51.8320i −58.6744 + 39.4628i
23.3 −0.732051 − 2.73205i −1.91483 + 7.14624i −6.92820 + 4.00000i 19.1814 − 16.0335i 20.9256 −48.8315 + 4.06041i 16.0000 + 16.0000i 22.7459 + 13.1323i −57.8462 − 40.6672i
23.4 −0.732051 − 2.73205i −0.992115 + 3.70262i −6.92820 + 4.00000i −24.9759 − 1.09776i 10.8420 43.5404 − 22.4774i 16.0000 + 16.0000i 57.4229 + 33.1531i 15.2845 + 69.0390i
23.5 −0.732051 − 2.73205i 0.894853 − 3.33964i −6.92820 + 4.00000i 24.9720 + 1.18339i −9.77914 7.04779 − 48.4905i 16.0000 + 16.0000i 59.7956 + 34.5230i −15.0477 − 69.0910i
23.6 −0.732051 − 2.73205i 1.13153 − 4.22294i −6.92820 + 4.00000i 5.57172 + 24.3712i −12.3656 9.38735 + 48.0924i 16.0000 + 16.0000i 53.5952 + 30.9432i 62.5046 − 33.0632i
23.7 −0.732051 − 2.73205i 2.38468 − 8.89973i −6.92820 + 4.00000i −13.4658 − 21.0635i −26.0602 −42.3455 + 24.6548i 16.0000 + 16.0000i −3.37054 − 1.94598i −47.6889 + 52.2089i
23.8 −0.732051 − 2.73205i 4.48652 − 16.7439i −6.92820 + 4.00000i −14.2080 + 20.5702i −49.0296 −10.7356 − 47.8095i 16.0000 + 16.0000i −190.082 − 109.744i 66.5997 + 23.7587i
37.1 2.73205 − 0.732051i −16.7439 − 4.48652i 6.92820 − 4.00000i −10.7103 + 22.5896i −49.0296 47.8095 − 10.7356i 16.0000 − 16.0000i 190.082 + 109.744i −12.7242 + 69.5564i
37.2 2.73205 − 0.732051i −8.89973 − 2.38468i 6.92820 − 4.00000i 24.9744 + 1.13000i −26.0602 −24.6548 − 42.3455i 16.0000 − 16.0000i 3.37054 + 1.94598i 69.0587 − 15.1953i
37.3 2.73205 − 0.732051i −4.22294 − 1.13153i 6.92820 − 4.00000i −23.8919 + 7.36036i −12.3656 −48.0924 + 9.38735i 16.0000 − 16.0000i −53.5952 − 30.9432i −59.8859 + 37.5990i
37.4 2.73205 − 0.732051i −3.33964 − 0.894853i 6.92820 − 4.00000i −13.5108 − 21.0347i −9.77914 48.4905 + 7.04779i 16.0000 − 16.0000i −59.7956 − 34.5230i −52.3107 − 47.5772i
37.5 2.73205 − 0.732051i 3.70262 + 0.992115i 6.92820 − 4.00000i 13.4386 + 21.0809i 10.8420 22.4774 + 43.5404i 16.0000 − 16.0000i −57.4229 − 33.1531i 52.1473 + 47.7563i
37.6 2.73205 − 0.732051i 7.14624 + 1.91483i 6.92820 − 4.00000i 4.29476 − 24.6283i 20.9256 −4.06041 − 48.8315i 16.0000 − 16.0000i −22.7459 − 13.1323i −6.29569 − 70.4299i
37.7 2.73205 − 0.732051i 13.1261 + 3.51712i 6.92820 − 4.00000i 24.5343 − 4.80290i 38.4358 −43.1356 + 23.2447i 16.0000 − 16.0000i 89.7757 + 51.8320i 63.5130 − 31.0821i
37.8 2.73205 − 0.732051i 14.6954 + 3.93762i 6.92820 − 4.00000i −22.1650 + 11.5634i 43.0311 45.3690 − 18.5109i 16.0000 − 16.0000i 130.302 + 75.2298i −52.0910 + 47.8177i
53.1 2.73205 + 0.732051i −16.7439 + 4.48652i 6.92820 + 4.00000i −10.7103 − 22.5896i −49.0296 47.8095 + 10.7356i 16.0000 + 16.0000i 190.082 − 109.744i −12.7242 − 69.5564i
53.2 2.73205 + 0.732051i −8.89973 + 2.38468i 6.92820 + 4.00000i 24.9744 − 1.13000i −26.0602 −24.6548 + 42.3455i 16.0000 + 16.0000i 3.37054 − 1.94598i 69.0587 + 15.1953i
53.3 2.73205 + 0.732051i −4.22294 + 1.13153i 6.92820 + 4.00000i −23.8919 − 7.36036i −12.3656 −48.0924 − 9.38735i 16.0000 + 16.0000i −53.5952 + 30.9432i −59.8859 − 37.5990i
53.4 2.73205 + 0.732051i −3.33964 + 0.894853i 6.92820 + 4.00000i −13.5108 + 21.0347i −9.77914 48.4905 − 7.04779i 16.0000 + 16.0000i −59.7956 + 34.5230i −52.3107 + 47.5772i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.5.l.b ✓ 32
5.c odd 4 1 inner 70.5.l.b ✓ 32
7.c even 3 1 inner 70.5.l.b ✓ 32
35.l odd 12 1 inner 70.5.l.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.l.b ✓ 32 1.a even 1 1 trivial
70.5.l.b ✓ 32 5.c odd 4 1 inner
70.5.l.b ✓ 32 7.c even 3 1 inner
70.5.l.b ✓ 32 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 8 T_{3}^{31} + 32 T_{3}^{30} - 440 T_{3}^{29} - 91439 T_{3}^{28} + 649984 T_{3}^{27} + \cdots + 40\!\cdots\!00 \) acting on \(S_{5}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display