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Properties

Label 70.5.l.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{2} - 8 q^{3} - 10 q^{5} + 64 q^{6} - 34 q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{2} - 8 q^{3} - 10 q^{5} + 64 q^{6} - 34 q^{7} - 512 q^{8} - 4 q^{10} - 228 q^{11} - 64 q^{12} + 96 q^{13} - 676 q^{15} + 1024 q^{16} - 30 q^{17} - 376 q^{18} - 128 q^{20} - 784 q^{21} + 912 q^{22} - 612 q^{23} - 1726 q^{25} - 192 q^{26} + 808 q^{27} + 416 q^{28} + 1012 q^{30} + 2092 q^{31} + 2048 q^{32} + 1010 q^{33} - 2532 q^{35} + 3008 q^{36} - 1186 q^{37} - 1600 q^{38} + 288 q^{40} + 352 q^{41} - 3248 q^{42} + 4 q^{43} + 5214 q^{45} - 2448 q^{46} - 3886 q^{47} - 1024 q^{48} + 7848 q^{50} + 5860 q^{51} - 384 q^{52} + 5898 q^{53} - 8012 q^{55} + 1088 q^{56} + 6396 q^{57} - 6368 q^{58} + 1344 q^{60} + 11832 q^{61} - 8368 q^{62} + 13296 q^{63} + 358 q^{65} + 4040 q^{66} + 12544 q^{67} + 240 q^{68} - 10548 q^{70} - 42248 q^{71} - 3008 q^{72} - 10690 q^{73} - 9952 q^{75} + 12800 q^{76} - 6902 q^{77} - 11920 q^{78} + 640 q^{80} - 53208 q^{81} - 352 q^{82} + 2428 q^{83} + 46792 q^{85} - 8 q^{86} - 16710 q^{87} + 3648 q^{88} - 6736 q^{90} - 7104 q^{91} + 9792 q^{92} - 23062 q^{93} + 26414 q^{95} + 2048 q^{96} + 1600 q^{97} + 30144 q^{98}+O(q^{100}) \) 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 −3.40292 + 12.6999i −6.92820 + 4.00000i −13.9969 + 20.7144i −37.1878 48.1641 + 9.01224i −16.0000 − 16.0000i −79.5586 − 45.9332i −66.8392 − 23.0763i
23.2 0.732051 + 2.73205i −2.58908 + 9.66257i −6.92820 + 4.00000i −14.4126 − 20.4273i −28.2940 −39.9480 − 28.3753i −16.0000 − 16.0000i −16.5139 − 9.53428i 45.2577 − 54.3299i
23.3 0.732051 + 2.73205i −2.24155 + 8.36560i −6.92820 + 4.00000i 21.8112 + 12.2176i −24.4962 −37.4924 + 31.5487i −16.0000 − 16.0000i 5.18941 + 2.99611i −17.4123 + 68.5333i
23.4 0.732051 + 2.73205i 0.0918937 − 0.342952i −6.92820 + 4.00000i 11.3103 − 22.2952i 1.00423 47.9780 + 9.95525i −16.0000 − 16.0000i 70.0389 + 40.4370i 69.1914 + 14.5790i
23.5 0.732051 + 2.73205i 1.08199 − 4.03806i −6.92820 + 4.00000i 22.8378 + 10.1702i 11.8243 9.65706 − 48.0390i −16.0000 − 16.0000i 55.0129 + 31.7617i −11.0670 + 69.8392i
23.6 0.732051 + 2.73205i 2.32594 − 8.68051i −6.92820 + 4.00000i −23.9376 + 7.21050i 25.4183 13.1745 − 47.1957i −16.0000 − 16.0000i 0.206744 + 0.119364i −37.2230 − 60.1203i
23.7 0.732051 + 2.73205i 2.55066 − 9.51919i −6.92820 + 4.00000i −12.0472 + 21.9058i 27.8741 −25.0179 + 42.1320i −16.0000 − 16.0000i −13.9610 − 8.06041i −68.6670 − 16.8775i
23.8 0.732051 + 2.73205i 3.64717 − 13.6114i −6.92820 + 4.00000i 9.39916 − 23.1658i 39.8570 −44.9340 + 19.5432i −16.0000 − 16.0000i −101.821 − 58.7863i 70.1709 + 8.72041i
37.1 −2.73205 + 0.732051i −13.6114 − 3.64717i 6.92820 − 4.00000i 15.3626 − 19.7228i 39.8570 −19.5432 − 44.9340i −16.0000 + 16.0000i 101.821 + 58.7863i −27.5333 + 65.1300i
37.2 −2.73205 + 0.732051i −9.51919 − 2.55066i 6.92820 − 4.00000i −12.9474 + 21.3861i 27.8741 −42.1320 − 25.0179i −16.0000 + 16.0000i 13.9610 + 8.06041i 19.7172 − 67.9061i
37.3 −2.73205 + 0.732051i −8.68051 − 2.32594i 6.92820 − 4.00000i 5.72432 + 24.3358i 25.4183 47.1957 + 13.1745i −16.0000 + 16.0000i −0.206744 − 0.119364i −33.4542 − 62.2962i
37.4 −2.73205 + 0.732051i −4.03806 − 1.08199i 6.92820 − 4.00000i −20.2266 − 14.6931i 11.8243 48.0390 + 9.65706i −16.0000 + 16.0000i −55.0129 − 31.7617i 66.0161 + 25.3353i
37.5 −2.73205 + 0.732051i −0.342952 − 0.0918937i 6.92820 − 4.00000i 13.6531 − 20.9426i 1.00423 −9.95525 + 47.9780i −16.0000 + 16.0000i −70.0389 − 40.4370i −21.9699 + 67.2110i
37.6 −2.73205 + 0.732051i 8.36560 + 2.24155i 6.92820 − 4.00000i −21.4864 − 12.7803i −24.4962 −31.5487 − 37.4924i −16.0000 + 16.0000i −5.18941 − 2.99611i 68.0577 + 19.1872i
37.7 −2.73205 + 0.732051i 9.66257 + 2.58908i 6.92820 − 4.00000i 24.8969 + 2.26805i −28.2940 28.3753 − 39.9480i −16.0000 + 16.0000i 16.5139 + 9.53428i −69.6799 + 12.0294i
37.8 −2.73205 + 0.732051i 12.6999 + 3.40292i 6.92820 − 4.00000i −10.9407 + 22.4789i −37.1878 −9.01224 + 48.1641i −16.0000 + 16.0000i 79.5586 + 45.9332i 13.4349 − 69.4226i
53.1 −2.73205 − 0.732051i −13.6114 + 3.64717i 6.92820 + 4.00000i 15.3626 + 19.7228i 39.8570 −19.5432 + 44.9340i −16.0000 − 16.0000i 101.821 − 58.7863i −27.5333 − 65.1300i
53.2 −2.73205 − 0.732051i −9.51919 + 2.55066i 6.92820 + 4.00000i −12.9474 − 21.3861i 27.8741 −42.1320 + 25.0179i −16.0000 − 16.0000i 13.9610 − 8.06041i 19.7172 + 67.9061i
53.3 −2.73205 − 0.732051i −8.68051 + 2.32594i 6.92820 + 4.00000i 5.72432 − 24.3358i 25.4183 47.1957 − 13.1745i −16.0000 − 16.0000i −0.206744 + 0.119364i −33.4542 + 62.2962i
53.4 −2.73205 − 0.732051i −4.03806 + 1.08199i 6.92820 + 4.00000i −20.2266 + 14.6931i 11.8243 48.0390 − 9.65706i −16.0000 − 16.0000i −55.0129 + 31.7617i 66.0161 − 25.3353i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 8 T_{3}^{31} + 32 T_{3}^{30} + 680 T_{3}^{29} - 45679 T_{3}^{28} - 432704 T_{3}^{27} + \cdots + 19\!\cdots\!00 \) acting on \(S_{5}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display