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Properties

Label 174.5.b.a
Level $174$
Weight $5$
Character orbit 174.b
Analytic conductor $17.986$
Analytic rank $0$
Dimension $36$
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] = [174,5,Mod(59,174)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(174, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("174.59"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 174 = 2 \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 174.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: \(17.9863735766\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 12 q^{3} - 288 q^{4} - 32 q^{6} - 184 q^{7} + 84 q^{9} - 96 q^{12} + 320 q^{13} - 724 q^{15} + 2304 q^{16} + 576 q^{18} + 912 q^{19} + 832 q^{21} - 704 q^{22} + 256 q^{24} - 5764 q^{25} - 1800 q^{27}+ \cdots - 18012 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 − 2.82843i −8.99123 + 0.397256i −8.00000 27.6771i 1.12361 + 25.4310i 18.1599 22.6274i 80.6844 − 7.14364i 78.2828
59.2 − 2.82843i −8.91167 + 1.25783i −8.00000 − 47.2893i 3.55767 + 25.2060i 64.8652 22.6274i 77.8358 − 22.4186i −133.754
59.3 − 2.82843i −8.61787 − 2.59467i −8.00000 − 11.6973i −7.33884 + 24.3750i −38.4486 22.6274i 67.5353 + 44.7211i −33.0849
59.4 − 2.82843i −7.19116 + 5.41177i −8.00000 − 21.6550i 15.3068 + 20.3397i −96.6764 22.6274i 22.4254 − 77.8338i −61.2495
59.5 − 2.82843i −5.68562 − 6.97666i −8.00000 − 9.60105i −19.7330 + 16.0814i 82.1570 22.6274i −16.3475 + 79.3332i −27.1559
59.6 − 2.82843i −4.68873 − 7.68217i −8.00000 31.2712i −21.7285 + 13.2617i −39.9280 22.6274i −37.0315 + 72.0393i 88.4483
59.7 − 2.82843i −4.47543 + 7.80836i −8.00000 39.9714i 22.0854 + 12.6584i −82.8769 22.6274i −40.9411 − 69.8916i 113.056
59.8 − 2.82843i −1.44324 − 8.88353i −8.00000 − 16.8061i −25.1264 + 4.08210i 24.6279 22.6274i −76.8341 + 25.6421i −47.5347
59.9 − 2.82843i 1.26693 − 8.91038i −8.00000 − 33.9779i −25.2024 − 3.58342i −68.2531 22.6274i −77.7898 − 22.5776i −96.1040
59.10 − 2.82843i 2.39326 + 8.67596i −8.00000 − 4.41077i 24.5393 − 6.76917i −13.1103 22.6274i −69.5446 + 41.5277i −12.4755
59.11 − 2.82843i 2.85378 + 8.53557i −8.00000 39.9975i 24.1422 − 8.07172i 64.5344 22.6274i −64.7118 + 48.7173i 113.130
59.12 − 2.82843i 4.64330 + 7.70972i −8.00000 − 37.7316i 21.8064 − 13.1332i −29.4140 22.6274i −37.8795 + 71.5971i −106.721
59.13 − 2.82843i 4.88614 − 7.55815i −8.00000 18.8279i −21.3777 − 13.8201i 57.7115 22.6274i −33.2513 − 73.8603i 53.2534
59.14 − 2.82843i 5.65245 − 7.00356i −8.00000 26.6999i −19.8090 − 15.9875i −43.5937 22.6274i −17.0996 − 79.1745i 75.5186
59.15 − 2.82843i 8.33744 + 3.38925i −8.00000 28.5478i 9.58625 − 23.5819i −63.2135 22.6274i 58.0260 + 56.5154i 80.7453
59.16 − 2.82843i 8.46271 − 3.06308i −8.00000 − 37.9596i −8.66371 − 23.9362i 17.2775 22.6274i 62.2350 − 51.8440i −107.366
59.17 − 2.82843i 8.57661 + 2.72797i −8.00000 0.912561i 7.71587 − 24.2583i −19.2855 22.6274i 66.1163 + 46.7935i 2.58111
59.18 − 2.82843i 8.93232 + 1.10166i −8.00000 7.22309i 3.11596 − 25.2644i 73.4665 22.6274i 78.5727 + 19.6807i 20.4300
59.19 2.82843i −8.99123 − 0.397256i −8.00000 − 27.6771i 1.12361 − 25.4310i 18.1599 − 22.6274i 80.6844 + 7.14364i 78.2828
59.20 2.82843i −8.91167 − 1.25783i −8.00000 47.2893i 3.55767 − 25.2060i 64.8652 − 22.6274i 77.8358 + 22.4186i −133.754
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.36
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 174.5.b.a ✓ 36
3.b odd 2 1 inner 174.5.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
174.5.b.a ✓ 36 1.a even 1 1 trivial
174.5.b.a ✓ 36 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(174, [\chi])\).