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Properties

Label 58.4.d.b
Level $58$
Weight $4$
Character orbit 58.d
Analytic conductor $3.422$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,4,Mod(7,58)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("58.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.42211078033\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 5 q^{3} - 16 q^{4} + 16 q^{5} - 4 q^{6} + 71 q^{7} - 32 q^{8} - 27 q^{9} - 10 q^{10} + 110 q^{11} - 64 q^{12} - 107 q^{13} - 54 q^{14} + 417 q^{15} - 64 q^{16} - 128 q^{17} - 54 q^{18}+ \cdots + 10560 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.445042 − 1.94986i −7.53743 − 3.62983i −3.60388 + 1.73553i 2.19061 + 9.59767i −3.72318 + 16.3123i 3.58891 + 1.72833i 4.98792 + 6.25465i 26.8029 + 33.6097i 17.7392 − 8.54273i
7.2 −0.445042 − 1.94986i −0.764033 − 0.367939i −3.60388 + 1.73553i −1.81600 − 7.95642i −0.377401 + 1.65350i −17.0430 − 8.20747i 4.98792 + 6.25465i −16.3859 − 20.5472i −14.7057 + 7.08188i
7.3 −0.445042 − 1.94986i 3.55161 + 1.71037i −3.60388 + 1.73553i 4.17045 + 18.2719i 1.75435 − 7.68631i 13.7257 + 6.60997i 4.98792 + 6.25465i −7.14564 − 8.96035i 33.7716 − 16.2635i
7.4 −0.445042 − 1.94986i 7.05179 + 3.39596i −3.60388 + 1.73553i −3.69031 − 16.1683i 3.48329 − 15.2613i 22.0013 + 10.5953i 4.98792 + 6.25465i 21.3609 + 26.7857i −29.8835 + 14.3912i
23.1 1.24698 − 1.56366i −1.36256 + 5.96975i −0.890084 − 3.89971i −11.8479 + 14.8567i 7.63560 + 9.57474i −1.70791 + 7.48282i −7.20775 − 3.47107i −9.45521 − 4.55339i 8.45690 + 37.0521i
23.2 1.24698 − 1.56366i −0.504121 + 2.20870i −0.890084 − 3.89971i 7.88231 − 9.88411i 2.82503 + 3.54248i 2.98095 − 13.0604i −7.20775 − 3.47107i 19.7020 + 9.48796i −5.62633 − 24.6506i
23.3 1.24698 − 1.56366i 0.975670 − 4.27469i −0.890084 − 3.89971i −2.22476 + 2.78977i −5.46753 − 6.85607i 5.92598 − 25.9634i −7.20775 − 3.47107i 7.00512 + 3.37349i 1.58802 + 6.95756i
23.4 1.24698 − 1.56366i 1.83605 − 8.04426i −0.890084 − 3.89971i 3.15831 − 3.96040i −10.2890 − 12.9020i −7.28208 + 31.9049i −7.20775 − 3.47107i −37.0128 − 17.8244i −2.25438 − 9.87706i
25.1 −0.445042 + 1.94986i −7.53743 + 3.62983i −3.60388 − 1.73553i 2.19061 − 9.59767i −3.72318 − 16.3123i 3.58891 − 1.72833i 4.98792 − 6.25465i 26.8029 − 33.6097i 17.7392 + 8.54273i
25.2 −0.445042 + 1.94986i −0.764033 + 0.367939i −3.60388 − 1.73553i −1.81600 + 7.95642i −0.377401 − 1.65350i −17.0430 + 8.20747i 4.98792 − 6.25465i −16.3859 + 20.5472i −14.7057 − 7.08188i
25.3 −0.445042 + 1.94986i 3.55161 − 1.71037i −3.60388 − 1.73553i 4.17045 − 18.2719i 1.75435 + 7.68631i 13.7257 − 6.60997i 4.98792 − 6.25465i −7.14564 + 8.96035i 33.7716 + 16.2635i
25.4 −0.445042 + 1.94986i 7.05179 − 3.39596i −3.60388 − 1.73553i −3.69031 + 16.1683i 3.48329 + 15.2613i 22.0013 − 10.5953i 4.98792 − 6.25465i 21.3609 − 26.7857i −29.8835 − 14.3912i
45.1 −1.80194 − 0.867767i −6.26334 + 7.85398i 2.49396 + 3.12733i −7.20732 − 3.47086i 18.1016 − 8.71726i 14.4870 − 18.1662i −1.78017 − 7.79942i −16.4475 − 72.0612i 9.97524 + 12.5086i
45.2 −1.80194 − 0.867767i −0.791302 + 0.992261i 2.49396 + 3.12733i −2.81750 − 1.35683i 2.28693 − 1.10133i −12.3264 + 15.4568i −1.78017 − 7.79942i 5.64964 + 24.7527i 3.89953 + 4.88986i
45.3 −1.80194 − 0.867767i 0.477146 − 0.598322i 2.49396 + 3.12733i 1.70401 + 0.820609i −1.37899 + 0.664087i 16.1350 − 20.2326i −1.78017 − 7.79942i 5.87774 + 25.7521i −2.35842 − 2.95737i
45.4 −1.80194 − 0.867767i 5.83052 − 7.31124i 2.49396 + 3.12733i 18.4981 + 8.90819i −16.8507 + 8.11486i −4.98552 + 6.25165i −1.78017 − 7.79942i −13.4512 − 58.9335i −25.6021 − 32.1040i
49.1 −1.80194 + 0.867767i −6.26334 − 7.85398i 2.49396 − 3.12733i −7.20732 + 3.47086i 18.1016 + 8.71726i 14.4870 + 18.1662i −1.78017 + 7.79942i −16.4475 + 72.0612i 9.97524 − 12.5086i
49.2 −1.80194 + 0.867767i −0.791302 − 0.992261i 2.49396 − 3.12733i −2.81750 + 1.35683i 2.28693 + 1.10133i −12.3264 − 15.4568i −1.78017 + 7.79942i 5.64964 − 24.7527i 3.89953 − 4.88986i
49.3 −1.80194 + 0.867767i 0.477146 + 0.598322i 2.49396 − 3.12733i 1.70401 − 0.820609i −1.37899 − 0.664087i 16.1350 + 20.2326i −1.78017 + 7.79942i 5.87774 − 25.7521i −2.35842 + 2.95737i
49.4 −1.80194 + 0.867767i 5.83052 + 7.31124i 2.49396 − 3.12733i 18.4981 − 8.90819i −16.8507 − 8.11486i −4.98552 − 6.25165i −1.78017 + 7.79942i −13.4512 + 58.9335i −25.6021 + 32.1040i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.4.d.b ✓ 24
29.d even 7 1 inner 58.4.d.b ✓ 24
29.d even 7 1 1682.4.a.t 12
29.e even 14 1 1682.4.a.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.d.b ✓ 24 1.a even 1 1 trivial
58.4.d.b ✓ 24 29.d even 7 1 inner
1682.4.a.q 12 29.e even 14 1
1682.4.a.t 12 29.d even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 5 T_{3}^{23} + 80 T_{3}^{22} - 302 T_{3}^{21} + 9883 T_{3}^{20} - 60041 T_{3}^{19} + \cdots + 100462613794816 \) acting on \(S_{4}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display