Properties

Label 70.5.h.a
Level $70$
Weight $5$
Character orbit 70.h
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(19,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.h (of order \(6\), degree \(2\), 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: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 128 q^{4} + 54 q^{5} - 516 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 128 q^{4} + 54 q^{5} - 516 q^{9} - 96 q^{10} - 144 q^{11} + 96 q^{14} - 76 q^{15} - 1024 q^{16} - 816 q^{19} + 1084 q^{21} + 768 q^{24} + 294 q^{25} - 576 q^{26} + 912 q^{29} + 608 q^{30} + 6468 q^{31} + 1398 q^{35} - 8256 q^{36} - 2424 q^{39} - 768 q^{40} + 1152 q^{44} - 8256 q^{45} + 1664 q^{46} - 7712 q^{49} + 15744 q^{50} + 2708 q^{51} + 7776 q^{54} - 768 q^{56} + 3924 q^{59} - 304 q^{60} + 41556 q^{61} - 16384 q^{64} + 936 q^{65} - 15360 q^{66} + 15104 q^{70} - 53808 q^{71} - 6336 q^{74} - 29058 q^{75} - 24076 q^{79} - 3456 q^{80} + 17680 q^{81} - 19232 q^{84} + 8324 q^{85} + 6624 q^{86} - 41436 q^{89} + 12928 q^{91} + 62880 q^{94} + 42990 q^{95} + 6144 q^{96} + 101048 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.44949 1.41421i −6.32716 10.9590i 4.00000 + 6.92820i −18.7989 16.4803i 35.7918i −21.6321 43.9665i 22.6274i −39.5658 + 68.5300i 22.7412 + 66.9540i
19.2 −2.44949 1.41421i −6.27030 10.8605i 4.00000 + 6.92820i 10.1380 + 22.8521i 35.4702i −32.1106 + 37.0123i 22.6274i −38.1334 + 66.0490i 7.48477 70.3134i
19.3 −2.44949 1.41421i −4.27603 7.40629i 4.00000 + 6.92820i 22.3515 11.1986i 24.1889i 48.9684 1.76024i 22.6274i 3.93122 6.80907i −70.5871 4.17884i
19.4 −2.44949 1.41421i 0.694226 + 1.20243i 4.00000 + 6.92820i −9.07375 + 23.2952i 3.92713i −24.8459 42.2337i 22.6274i 39.5361 68.4785i 55.1705 44.2292i
19.5 −2.44949 1.41421i 2.92307 + 5.06291i 4.00000 + 6.92820i −16.0876 19.1361i 16.5354i −17.3754 + 45.8159i 22.6274i 23.4113 40.5496i 12.3439 + 69.6249i
19.6 −2.44949 1.41421i 3.98422 + 6.90087i 4.00000 + 6.92820i 24.9999 + 0.0650489i 22.5381i −33.9906 + 35.2936i 22.6274i 8.75204 15.1590i −61.1450 35.5146i
19.7 −2.44949 1.41421i 5.75172 + 9.96228i 4.00000 + 6.92820i 1.98309 24.9212i 32.5367i 17.7931 45.6553i 22.6274i −25.6646 + 44.4524i −40.1015 + 58.2398i
19.8 −2.44949 1.41421i 8.41923 + 14.5825i 4.00000 + 6.92820i 2.88670 + 24.8328i 47.6264i 48.4962 + 7.00864i 22.6274i −101.267 + 175.399i 28.0479 64.9101i
19.9 2.44949 + 1.41421i −8.41923 14.5825i 4.00000 + 6.92820i 22.9492 9.91643i 47.6264i −48.4962 7.00864i 22.6274i −101.267 + 175.399i 70.2377 + 8.16483i
19.10 2.44949 + 1.41421i −5.75172 9.96228i 4.00000 + 6.92820i −20.5909 + 14.1780i 32.5367i −17.7931 + 45.6553i 22.6274i −25.6646 + 44.4524i −70.4879 + 5.60903i
19.11 2.44949 + 1.41421i −3.98422 6.90087i 4.00000 + 6.92820i 12.5563 + 21.6180i 22.5381i 33.9906 35.2936i 22.6274i 8.75204 15.1590i 0.183986 + 70.7104i
19.12 2.44949 + 1.41421i −2.92307 5.06291i 4.00000 + 6.92820i −24.6161 4.36422i 16.5354i 17.3754 45.8159i 22.6274i 23.4113 40.5496i −54.1250 45.5026i
19.13 2.44949 + 1.41421i −0.694226 1.20243i 4.00000 + 6.92820i 15.6374 19.5057i 3.92713i 24.8459 + 42.2337i 22.6274i 39.5361 68.4785i 65.8888 25.6645i
19.14 2.44949 + 1.41421i 4.27603 + 7.40629i 4.00000 + 6.92820i 1.47744 + 24.9563i 24.1889i −48.9684 + 1.76024i 22.6274i 3.93122 6.80907i −31.6746 + 63.2196i
19.15 2.44949 + 1.41421i 6.27030 + 10.8605i 4.00000 + 6.92820i 24.8596 2.64627i 35.4702i 32.1106 37.0123i 22.6274i −38.1334 + 66.0490i 64.6356 + 28.6747i
19.16 2.44949 + 1.41421i 6.32716 + 10.9590i 4.00000 + 6.92820i −23.6718 8.04022i 35.7918i 21.6321 + 43.9665i 22.6274i −39.5658 + 68.5300i −46.6133 53.1714i
59.1 −2.44949 + 1.41421i −6.32716 + 10.9590i 4.00000 6.92820i −18.7989 + 16.4803i 35.7918i −21.6321 + 43.9665i 22.6274i −39.5658 68.5300i 22.7412 66.9540i
59.2 −2.44949 + 1.41421i −6.27030 + 10.8605i 4.00000 6.92820i 10.1380 22.8521i 35.4702i −32.1106 37.0123i 22.6274i −38.1334 66.0490i 7.48477 + 70.3134i
59.3 −2.44949 + 1.41421i −4.27603 + 7.40629i 4.00000 6.92820i 22.3515 + 11.1986i 24.1889i 48.9684 + 1.76024i 22.6274i 3.93122 + 6.80907i −70.5871 + 4.17884i
59.4 −2.44949 + 1.41421i 0.694226 1.20243i 4.00000 6.92820i −9.07375 23.2952i 3.92713i −24.8459 + 42.2337i 22.6274i 39.5361 + 68.4785i 55.1705 + 44.2292i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.5.h.a 32
5.b even 2 1 inner 70.5.h.a 32
5.c odd 4 2 350.5.k.e 32
7.c even 3 1 490.5.d.a 32
7.d odd 6 1 inner 70.5.h.a 32
7.d odd 6 1 490.5.d.a 32
35.i odd 6 1 inner 70.5.h.a 32
35.i odd 6 1 490.5.d.a 32
35.j even 6 1 490.5.d.a 32
35.k even 12 2 350.5.k.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.h.a 32 1.a even 1 1 trivial
70.5.h.a 32 5.b even 2 1 inner
70.5.h.a 32 7.d odd 6 1 inner
70.5.h.a 32 35.i odd 6 1 inner
350.5.k.e 32 5.c odd 4 2
350.5.k.e 32 35.k even 12 2
490.5.d.a 32 7.c even 3 1
490.5.d.a 32 7.d odd 6 1
490.5.d.a 32 35.i odd 6 1
490.5.d.a 32 35.j even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(70, [\chi])\).