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Properties

Label 626.4.a.d
Level $626$
Weight $4$
Character orbit 626.a
Self dual yes
Analytic conductor $36.935$
Analytic rank $0$
Dimension $25$
CM no
Inner twists $1$

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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] = [626,4,Mod(1,626)] 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("626.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(626, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 626 = 2 \cdot 313 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 626.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [25] 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: yes
Analytic conductor: \(36.9351956636\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 50 q^{2} + 16 q^{3} + 100 q^{4} + 40 q^{5} + 32 q^{6} + 51 q^{7} + 200 q^{8} + 351 q^{9} + 80 q^{10} + 178 q^{11} + 64 q^{12} + 142 q^{13} + 102 q^{14} + 151 q^{15} + 400 q^{16} + 339 q^{17} + 702 q^{18}+ \cdots + 2233 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 2.00000 −10.2867 4.00000 15.9859 −20.5735 24.3253 8.00000 78.8172 31.9718
1.2 2.00000 −9.46022 4.00000 −2.70387 −18.9204 15.8731 8.00000 62.4957 −5.40775
1.3 2.00000 −8.31749 4.00000 −11.7527 −16.6350 −28.8277 8.00000 42.1806 −23.5054
1.4 2.00000 −7.12760 4.00000 11.4194 −14.2552 −5.54798 8.00000 23.8027 22.8387
1.5 2.00000 −6.40677 4.00000 −8.10371 −12.8135 20.5177 8.00000 14.0467 −16.2074
1.6 2.00000 −4.67764 4.00000 −4.44587 −9.35528 −28.4359 8.00000 −5.11969 −8.89174
1.7 2.00000 −4.40018 4.00000 21.3896 −8.80037 −22.8106 8.00000 −7.63839 42.7793
1.8 2.00000 −3.14279 4.00000 −19.4719 −6.28559 −12.0721 8.00000 −17.1228 −38.9438
1.9 2.00000 −2.88411 4.00000 2.99385 −5.76821 26.2945 8.00000 −18.6819 5.98770
1.10 2.00000 −2.68968 4.00000 −3.51630 −5.37936 −33.0926 8.00000 −19.7656 −7.03261
1.11 2.00000 −2.41450 4.00000 13.1695 −4.82900 31.3107 8.00000 −21.1702 26.3390
1.12 2.00000 −1.12553 4.00000 −1.17535 −2.25106 7.91698 8.00000 −25.7332 −2.35069
1.13 2.00000 −0.609716 4.00000 −18.3472 −1.21943 −15.3017 8.00000 −26.6282 −36.6944
1.14 2.00000 2.29922 4.00000 17.7815 4.59844 −1.73132 8.00000 −21.7136 35.5629
1.15 2.00000 3.37128 4.00000 −16.2868 6.74257 22.5541 8.00000 −15.6344 −32.5737
1.16 2.00000 3.82424 4.00000 4.73313 7.64849 0.845675 8.00000 −12.3752 9.46626
1.17 2.00000 4.98227 4.00000 16.0637 9.96453 22.6960 8.00000 −2.17703 32.1273
1.18 2.00000 6.43902 4.00000 19.3778 12.8780 13.8483 8.00000 14.4610 38.7556
1.19 2.00000 6.75088 4.00000 6.78355 13.5018 −6.96009 8.00000 18.5744 13.5671
1.20 2.00000 7.44105 4.00000 1.97015 14.8821 28.8465 8.00000 28.3692 3.94030
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(313\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 626.4.a.d ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
626.4.a.d ✓ 25 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{25} - 16 T_{3}^{24} - 385 T_{3}^{23} + 7069 T_{3}^{22} + 58602 T_{3}^{21} - 1326517 T_{3}^{20} + \cdots + 80\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(626))\). Copy content Toggle raw display