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Properties

Label 70.5.j.a
Level $70$
Weight $5$
Character orbit 70.j
Analytic conductor $7.236$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [70,5,Mod(31,70)]
 
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([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.j (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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{3} - 96 q^{4} + 76 q^{7} + 348 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{3} - 96 q^{4} + 76 q^{7} + 348 q^{9} + 84 q^{11} - 288 q^{12} - 96 q^{14} - 200 q^{15} - 768 q^{16} + 864 q^{17} - 512 q^{18} + 756 q^{19} - 200 q^{21} - 384 q^{22} + 1332 q^{23} + 768 q^{24} + 1500 q^{25} + 2880 q^{26} - 1312 q^{28} - 2112 q^{29} - 800 q^{30} - 1560 q^{31} - 384 q^{33} + 300 q^{35} - 5568 q^{36} - 808 q^{37} + 2880 q^{38} + 1968 q^{39} - 9152 q^{42} - 4008 q^{43} + 672 q^{44} + 600 q^{45} - 4736 q^{46} + 10296 q^{47} + 11696 q^{49} + 1592 q^{51} + 768 q^{52} + 3432 q^{53} - 4896 q^{54} - 768 q^{56} - 4320 q^{57} - 2048 q^{58} - 20016 q^{59} + 800 q^{60} - 10896 q^{61} + 11824 q^{63} + 12288 q^{64} + 6300 q^{65} + 22272 q^{66} + 8284 q^{67} - 6912 q^{68} + 2400 q^{70} + 5808 q^{71} - 4096 q^{72} - 6912 q^{73} - 384 q^{74} + 4500 q^{75} - 37248 q^{77} + 30976 q^{78} - 12152 q^{79} - 27404 q^{81} + 18432 q^{82} + 12992 q^{84} - 24000 q^{85} + 10080 q^{86} + 22212 q^{87} + 1536 q^{88} - 18216 q^{89} - 25408 q^{91} - 21312 q^{92} - 2776 q^{93} - 8736 q^{94} + 10200 q^{95} - 6144 q^{96} + 49920 q^{98} + 64968 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.41421 − 2.44949i −9.27207 − 5.35323i −4.00000 + 6.92820i −9.68246 + 5.59017i 30.2824i −2.45035 + 48.9387i 22.6274 16.8142 + 29.1230i 27.3861 + 15.8114i
31.2 −1.41421 − 2.44949i −1.47336 − 0.850642i −4.00000 + 6.92820i 9.68246 − 5.59017i 4.81196i −47.5891 + 11.6737i 22.6274 −39.0528 − 67.6415i −27.3861 − 15.8114i
31.3 −1.41421 − 2.44949i −0.950229 − 0.548615i −4.00000 + 6.92820i 9.68246 − 5.59017i 3.10343i 46.7485 + 14.6824i 22.6274 −39.8980 − 69.1054i −27.3861 − 15.8114i
31.4 −1.41421 − 2.44949i 4.31893 + 2.49353i −4.00000 + 6.92820i −9.68246 + 5.59017i − 14.1056i −26.3103 − 41.3373i 22.6274 −28.0646 − 48.6093i 27.3861 + 15.8114i
31.5 −1.41421 − 2.44949i 10.1550 + 5.86302i −4.00000 + 6.92820i −9.68246 + 5.59017i − 33.1663i 33.0212 + 36.2022i 22.6274 28.2500 + 48.9305i 27.3861 + 15.8114i
31.6 −1.41421 − 2.44949i 14.7070 + 8.49107i −4.00000 + 6.92820i 9.68246 − 5.59017i − 48.0327i 7.09468 − 48.4837i 22.6274 103.696 + 179.607i −27.3861 − 15.8114i
31.7 1.41421 + 2.44949i −12.1079 − 6.99053i −4.00000 + 6.92820i 9.68246 − 5.59017i − 39.5444i 46.3256 + 15.9668i −22.6274 57.2349 + 99.1338i 27.3861 + 15.8114i
31.8 1.41421 + 2.44949i −7.38100 − 4.26142i −4.00000 + 6.92820i −9.68246 + 5.59017i − 24.1063i 44.3801 − 20.7703i −22.6274 −4.18054 − 7.24091i −27.3861 − 15.8114i
31.9 1.41421 + 2.44949i −5.53636 − 3.19642i −4.00000 + 6.92820i 9.68246 − 5.59017i − 18.0817i −46.3868 − 15.7883i −22.6274 −20.0658 − 34.7551i 27.3861 + 15.8114i
31.10 1.41421 + 2.44949i 0.181996 + 0.105076i −4.00000 + 6.92820i −9.68246 + 5.59017i 0.594397i −17.7284 + 45.6804i −22.6274 −40.4779 − 70.1098i −27.3861 − 15.8114i
31.11 1.41421 + 2.44949i 10.4879 + 6.05522i −4.00000 + 6.92820i 9.68246 − 5.59017i 34.2535i 47.6639 + 11.3646i −22.6274 32.8313 + 56.8655i 27.3861 + 15.8114i
31.12 1.41421 + 2.44949i 14.8701 + 8.58525i −4.00000 + 6.92820i −9.68246 + 5.59017i 48.5655i −46.7691 + 14.6168i −22.6274 106.913 + 185.179i −27.3861 − 15.8114i
61.1 −1.41421 + 2.44949i −9.27207 + 5.35323i −4.00000 − 6.92820i −9.68246 − 5.59017i − 30.2824i −2.45035 − 48.9387i 22.6274 16.8142 − 29.1230i 27.3861 − 15.8114i
61.2 −1.41421 + 2.44949i −1.47336 + 0.850642i −4.00000 − 6.92820i 9.68246 + 5.59017i − 4.81196i −47.5891 − 11.6737i 22.6274 −39.0528 + 67.6415i −27.3861 + 15.8114i
61.3 −1.41421 + 2.44949i −0.950229 + 0.548615i −4.00000 − 6.92820i 9.68246 + 5.59017i − 3.10343i 46.7485 − 14.6824i 22.6274 −39.8980 + 69.1054i −27.3861 + 15.8114i
61.4 −1.41421 + 2.44949i 4.31893 − 2.49353i −4.00000 − 6.92820i −9.68246 − 5.59017i 14.1056i −26.3103 + 41.3373i 22.6274 −28.0646 + 48.6093i 27.3861 − 15.8114i
61.5 −1.41421 + 2.44949i 10.1550 − 5.86302i −4.00000 − 6.92820i −9.68246 − 5.59017i 33.1663i 33.0212 − 36.2022i 22.6274 28.2500 − 48.9305i 27.3861 − 15.8114i
61.6 −1.41421 + 2.44949i 14.7070 − 8.49107i −4.00000 − 6.92820i 9.68246 + 5.59017i 48.0327i 7.09468 + 48.4837i 22.6274 103.696 − 179.607i −27.3861 + 15.8114i
61.7 1.41421 − 2.44949i −12.1079 + 6.99053i −4.00000 − 6.92820i 9.68246 + 5.59017i 39.5444i 46.3256 − 15.9668i −22.6274 57.2349 − 99.1338i 27.3861 − 15.8114i
61.8 1.41421 − 2.44949i −7.38100 + 4.26142i −4.00000 − 6.92820i −9.68246 − 5.59017i 24.1063i 44.3801 + 20.7703i −22.6274 −4.18054 + 7.24091i −27.3861 + 15.8114i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d 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.j.a ✓ 24
5.b even 2 1 350.5.k.d 24
5.c odd 4 2 350.5.i.c 48
7.c even 3 1 490.5.b.c 24
7.d odd 6 1 inner 70.5.j.a ✓ 24
7.d odd 6 1 490.5.b.c 24
35.i odd 6 1 350.5.k.d 24
35.k even 12 2 350.5.i.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.j.a ✓ 24 1.a even 1 1 trivial
70.5.j.a ✓ 24 7.d odd 6 1 inner
350.5.i.c 48 5.c odd 4 2
350.5.i.c 48 35.k even 12 2
350.5.k.d 24 5.b even 2 1
350.5.k.d 24 35.i odd 6 1
490.5.b.c 24 7.c even 3 1
490.5.b.c 24 7.d odd 6 1

Hecke kernels

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