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Properties

Label 174.4.f.a
Level $174$
Weight $4$
Character orbit 174.f
Analytic conductor $10.266$
Analytic rank $0$
Dimension $60$
Inner twists $4$

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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] = [174,4,Mod(17,174)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(174, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) 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("174.17"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 174 = 2 \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 174.f (of order \(4\), degree \(2\), 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: \(10.2663323410\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 24 q^{10} + 88 q^{15} - 960 q^{16} + 48 q^{19} - 752 q^{21} + 64 q^{24} + 1500 q^{25} + 600 q^{27} + 312 q^{30} - 300 q^{31} - 224 q^{36} - 192 q^{37} - 172 q^{39} + 96 q^{40} + 888 q^{43} - 1532 q^{45}+ \cdots - 6000 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.41421 − 1.41421i −5.15122 + 0.681893i 4.00000i 7.07450 8.24926 + 6.32058i −8.70593 5.65685 − 5.65685i 26.0700 − 7.02516i −10.0049 − 10.0049i
17.2 −1.41421 − 1.41421i −4.66828 − 2.28191i 4.00000i −13.1057 3.37483 + 9.82906i −22.1141 5.65685 − 5.65685i 16.5857 + 21.3052i 18.5343 + 18.5343i
17.3 −1.41421 − 1.41421i −3.93655 + 3.39169i 4.00000i −6.62243 10.3637 + 0.770544i 32.6059 5.65685 − 5.65685i 3.99284 − 26.7031i 9.36553 + 9.36553i
17.4 −1.41421 − 1.41421i −3.68521 − 3.66323i 4.00000i −8.24792 0.0310808 + 10.3923i 32.9047 5.65685 − 5.65685i 0.161500 + 26.9995i 11.6643 + 11.6643i
17.5 −1.41421 − 1.41421i −2.99175 − 4.24846i 4.00000i 17.7535 −1.77727 + 10.2392i −4.74480 5.65685 − 5.65685i −9.09890 + 25.4207i −25.1073 − 25.1073i
17.6 −1.41421 − 1.41421i −2.48666 + 4.56251i 4.00000i −1.67768 9.96903 − 2.93570i −22.3099 5.65685 − 5.65685i −14.6331 − 22.6908i 2.37260 + 2.37260i
17.7 −1.41421 − 1.41421i −1.24130 + 5.04571i 4.00000i 21.0390 8.89117 − 5.38025i 17.3129 5.65685 − 5.65685i −23.9184 − 12.5265i −29.7537 − 29.7537i
17.8 −1.41421 − 1.41421i 0.228143 − 5.19114i 4.00000i 2.92456 −7.66403 + 7.01874i 3.24785 5.65685 − 5.65685i −26.8959 − 2.36865i −4.13595 − 4.13595i
17.9 −1.41421 − 1.41421i 0.762114 + 5.13996i 4.00000i −8.24082 6.19121 − 8.34679i 4.60548 5.65685 − 5.65685i −25.8384 + 7.83447i 11.6543 + 11.6543i
17.10 −1.41421 − 1.41421i 2.32730 − 4.64582i 4.00000i −19.3477 −9.86149 + 3.27888i −0.771607 5.65685 − 5.65685i −16.1673 − 21.6245i 27.3618 + 27.3618i
17.11 −1.41421 − 1.41421i 3.39930 + 3.92999i 4.00000i 1.42088 0.750511 − 10.3652i 6.13369 5.65685 − 5.65685i −3.88959 + 26.7184i −2.00943 − 2.00943i
17.12 −1.41421 − 1.41421i 3.52159 − 3.82078i 4.00000i 14.3966 −10.3837 + 0.423116i −25.6013 5.65685 − 5.65685i −2.19675 − 26.9105i −20.3599 − 20.3599i
17.13 −1.41421 − 1.41421i 5.06270 + 1.17009i 4.00000i 15.7560 −5.50498 − 8.81449i 15.8294 5.65685 − 5.65685i 24.2618 + 11.8476i −22.2824 − 22.2824i
17.14 −1.41421 − 1.41421i 5.08519 + 1.06812i 4.00000i −15.3181 −5.68099 − 8.70209i 5.90730 5.65685 − 5.65685i 24.7183 + 10.8631i 21.6631 + 21.6631i
17.15 −1.41421 − 1.41421i 5.18884 + 0.275608i 4.00000i −3.56208 −6.94836 − 7.72789i −34.2995 5.65685 − 5.65685i 26.8481 + 2.86017i 5.03755 + 5.03755i
17.16 1.41421 + 1.41421i −5.13996 − 0.762114i 4.00000i 8.24082 −6.19121 − 8.34679i 4.60548 −5.65685 + 5.65685i 25.8384 + 7.83447i 11.6543 + 11.6543i
17.17 1.41421 + 1.41421i −5.04571 + 1.24130i 4.00000i −21.0390 −8.89117 − 5.38025i 17.3129 −5.65685 + 5.65685i 23.9184 − 12.5265i −29.7537 − 29.7537i
17.18 1.41421 + 1.41421i −4.56251 + 2.48666i 4.00000i 1.67768 −9.96903 − 2.93570i −22.3099 −5.65685 + 5.65685i 14.6331 − 22.6908i 2.37260 + 2.37260i
17.19 1.41421 + 1.41421i −3.92999 − 3.39930i 4.00000i −1.42088 −0.750511 − 10.3652i 6.13369 −5.65685 + 5.65685i 3.88959 + 26.7184i −2.00943 − 2.00943i
17.20 1.41421 + 1.41421i −3.39169 + 3.93655i 4.00000i 6.62243 −10.3637 + 0.770544i 32.6059 −5.65685 + 5.65685i −3.99284 − 26.7031i 9.36553 + 9.36553i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.c odd 4 1 inner
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 174.4.f.a ✓ 60
3.b odd 2 1 inner 174.4.f.a ✓ 60
29.c odd 4 1 inner 174.4.f.a ✓ 60
87.f even 4 1 inner 174.4.f.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
174.4.f.a ✓ 60 1.a even 1 1 trivial
174.4.f.a ✓ 60 3.b odd 2 1 inner
174.4.f.a ✓ 60 29.c odd 4 1 inner
174.4.f.a ✓ 60 87.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(174, [\chi])\).