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Properties

Label 35.8.b.a
Level $35$
Weight $8$
Character orbit 35.b
Analytic conductor $10.933$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,8,Mod(29,35)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35.29"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9334758919\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 1588 q^{4} - 304 q^{5} + 108 q^{6} - 22696 q^{9} - 4456 q^{10} + 13602 q^{11} - 10976 q^{14} + 6998 q^{15} + 109428 q^{16} - 32256 q^{19} - 112708 q^{20} + 8918 q^{21} + 482664 q^{24} + 189458 q^{25}+ \cdots + 32815284 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 − 21.9392i − 73.1454i −353.327 262.944 + 94.7915i −1604.75 − 343.000i 4943.48i −3163.26 2079.65 − 5768.77i
29.2 − 19.7733i − 10.5143i −262.984 −235.288 − 150.880i −207.903 343.000i 2669.08i 2076.45 −2983.40 + 4652.42i
29.3 − 18.4977i 33.5740i −214.164 −137.823 + 243.166i 621.041 − 343.000i 1593.83i 1059.79 4498.02 + 2549.40i
29.4 − 18.2265i 80.5319i −204.206 103.178 − 259.768i 1467.82 − 343.000i 1388.97i −4298.39 −4734.67 − 1880.57i
29.5 − 15.7239i 22.0881i −119.241 278.937 + 17.8583i 347.312 343.000i − 137.728i 1699.11 280.802 − 4385.98i
29.6 − 11.9881i − 90.2514i −15.7137 −264.459 + 90.4793i −1081.94 343.000i − 1346.10i −5958.31 1084.67 + 3170.35i
29.7 − 11.6247i − 45.0853i −7.13462 50.5805 − 274.894i −524.105 − 343.000i − 1405.03i 154.316 −3195.57 − 587.985i
29.8 − 9.10894i 81.2795i 45.0272 −169.354 + 222.361i 740.370 343.000i − 1576.09i −4419.35 2025.47 + 1542.64i
29.9 − 5.48728i 13.4870i 97.8898 −275.350 − 48.0359i 74.0067 − 343.000i − 1239.52i 2005.10 −263.587 + 1510.92i
29.10 − 3.94501i 46.1650i 112.437 −25.1332 − 278.376i 182.121 343.000i − 948.527i 55.7959 −1098.20 + 99.1508i
29.11 − 0.763848i 52.4047i 127.417 259.766 + 103.182i 40.0293 − 343.000i − 195.100i −559.253 78.8151 − 198.422i
29.12 0.763848i − 52.4047i 127.417 259.766 − 103.182i 40.0293 343.000i 195.100i −559.253 78.8151 + 198.422i
29.13 3.94501i − 46.1650i 112.437 −25.1332 + 278.376i 182.121 − 343.000i 948.527i 55.7959 −1098.20 − 99.1508i
29.14 5.48728i − 13.4870i 97.8898 −275.350 + 48.0359i 74.0067 343.000i 1239.52i 2005.10 −263.587 − 1510.92i
29.15 9.10894i − 81.2795i 45.0272 −169.354 − 222.361i 740.370 − 343.000i 1576.09i −4419.35 2025.47 − 1542.64i
29.16 11.6247i 45.0853i −7.13462 50.5805 + 274.894i −524.105 343.000i 1405.03i 154.316 −3195.57 + 587.985i
29.17 11.9881i 90.2514i −15.7137 −264.459 − 90.4793i −1081.94 − 343.000i 1346.10i −5958.31 1084.67 − 3170.35i
29.18 15.7239i − 22.0881i −119.241 278.937 − 17.8583i 347.312 − 343.000i 137.728i 1699.11 280.802 + 4385.98i
29.19 18.2265i − 80.5319i −204.206 103.178 + 259.768i 1467.82 343.000i − 1388.97i −4298.39 −4734.67 + 1880.57i
29.20 18.4977i − 33.5740i −214.164 −137.823 − 243.166i 621.041 343.000i − 1593.83i 1059.79 4498.02 − 2549.40i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.8.b.a ✓ 22
5.b even 2 1 inner 35.8.b.a ✓ 22
5.c odd 4 1 175.8.a.k 11
5.c odd 4 1 175.8.a.l 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.b.a ✓ 22 1.a even 1 1 trivial
35.8.b.a ✓ 22 5.b even 2 1 inner
175.8.a.k 11 5.c odd 4 1
175.8.a.l 11 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(35, [\chi])\).