Properties

Label 112.5.l.b
Level $112$
Weight $5$
Character orbit 112.l
Analytic conductor $11.577$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(13,112)] 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("112.13"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120] 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 4 q^{2} - 72 q^{4} + 176 q^{8} - 512 q^{11} + 344 q^{14} - 8 q^{15} - 592 q^{16} - 2012 q^{18} - 164 q^{21} - 440 q^{22} + 960 q^{28} + 1600 q^{29} + 1112 q^{30} - 3064 q^{32} + 1340 q^{35} - 3680 q^{36}+ \cdots + 6152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −3.99986 + 0.0339225i −12.3740 + 12.3740i 15.9977 − 0.271370i 12.2236 + 12.2236i 49.0745 − 49.9141i −44.1493 − 21.2565i −63.9793 + 1.62812i − 225.233i −49.3072 − 48.4779i
13.2 −3.99986 + 0.0339225i 12.3740 − 12.3740i 15.9977 − 0.271370i −12.2236 − 12.2236i −49.0745 + 49.9141i 44.1493 − 21.2565i −63.9793 + 1.62812i − 225.233i 49.3072 + 48.4779i
13.3 −3.96960 − 0.492210i −3.82455 + 3.82455i 15.5155 + 3.90775i 30.0534 + 30.0534i 17.0644 − 13.2995i 16.3392 + 46.1956i −59.6667 − 23.1491i 51.7456i −104.507 − 134.092i
13.4 −3.96960 − 0.492210i 3.82455 − 3.82455i 15.5155 + 3.90775i −30.0534 − 30.0534i −17.0644 + 13.2995i −16.3392 + 46.1956i −59.6667 − 23.1491i 51.7456i 104.507 + 134.092i
13.5 −3.87338 + 0.998444i −5.86090 + 5.86090i 14.0062 − 7.73471i −10.4417 − 10.4417i 16.8497 − 28.5533i 45.5512 + 18.0579i −46.5288 + 43.9439i 12.2998i 50.8700 + 30.0192i
13.6 −3.87338 + 0.998444i 5.86090 − 5.86090i 14.0062 − 7.73471i 10.4417 + 10.4417i −16.8497 + 28.5533i −45.5512 + 18.0579i −46.5288 + 43.9439i 12.2998i −50.8700 − 30.0192i
13.7 −3.43667 + 2.04677i −4.98232 + 4.98232i 7.62147 − 14.0682i −23.0044 − 23.0044i 6.92495 − 27.3203i −30.2476 − 38.5497i 2.60177 + 63.9471i 31.3530i 126.143 + 31.9739i
13.8 −3.43667 + 2.04677i 4.98232 − 4.98232i 7.62147 − 14.0682i 23.0044 + 23.0044i −6.92495 + 27.3203i 30.2476 − 38.5497i 2.60177 + 63.9471i 31.3530i −126.143 − 31.9739i
13.9 −3.43409 − 2.05110i −5.64038 + 5.64038i 7.58595 + 14.0873i −13.1319 − 13.1319i 30.9386 − 7.80056i −42.9467 + 23.5920i 2.84378 − 63.9368i 17.3723i 18.1612 + 72.0309i
13.10 −3.43409 − 2.05110i 5.64038 − 5.64038i 7.58595 + 14.0873i 13.1319 + 13.1319i −30.9386 + 7.80056i 42.9467 + 23.5920i 2.84378 − 63.9368i 17.3723i −18.1612 − 72.0309i
13.11 −3.42143 − 2.07215i −9.70366 + 9.70366i 7.41241 + 14.1794i −17.2466 − 17.2466i 53.3079 − 13.0930i 48.3213 − 8.12732i 4.02080 − 63.8736i − 107.322i 23.2706 + 94.7456i
13.12 −3.42143 − 2.07215i 9.70366 − 9.70366i 7.41241 + 14.1794i 17.2466 + 17.2466i −53.3079 + 13.0930i −48.3213 − 8.12732i 4.02080 − 63.8736i − 107.322i −23.2706 − 94.7456i
13.13 −3.04191 + 2.59746i −5.92945 + 5.92945i 2.50645 − 15.8025i 13.1805 + 13.1805i 2.63538 − 33.4384i −27.6756 + 40.4359i 33.4218 + 54.5801i 10.6831i −74.3295 − 5.85814i
13.14 −3.04191 + 2.59746i 5.92945 − 5.92945i 2.50645 − 15.8025i −13.1805 − 13.1805i −2.63538 + 33.4384i 27.6756 + 40.4359i 33.4218 + 54.5801i 10.6831i 74.3295 + 5.85814i
13.15 −2.73351 − 2.92026i −5.22099 + 5.22099i −1.05581 + 15.9651i 28.9597 + 28.9597i 29.5183 + 0.974987i 4.78244 − 48.7661i 49.5083 − 40.5577i 26.4826i 5.40805 − 163.732i
13.16 −2.73351 − 2.92026i 5.22099 − 5.22099i −1.05581 + 15.9651i −28.9597 − 28.9597i −29.5183 − 0.974987i −4.78244 − 48.7661i 49.5083 − 40.5577i 26.4826i −5.40805 + 163.732i
13.17 −2.42135 + 3.18388i −10.4716 + 10.4716i −4.27416 − 15.4185i 11.9108 + 11.9108i −7.98492 − 58.6956i 45.6026 − 17.9277i 59.4400 + 23.7252i − 138.309i −66.7625 + 9.08233i
13.18 −2.42135 + 3.18388i 10.4716 − 10.4716i −4.27416 − 15.4185i −11.9108 − 11.9108i 7.98492 + 58.6956i −45.6026 − 17.9277i 59.4400 + 23.7252i − 138.309i 66.7625 − 9.08233i
13.19 −2.24080 − 3.31343i −5.49536 + 5.49536i −5.95763 + 14.8495i 2.11435 + 2.11435i 30.5225 + 5.89449i −31.3946 + 37.6216i 62.5525 − 13.5345i 20.6021i 2.26792 − 11.7436i
13.20 −2.24080 − 3.31343i 5.49536 − 5.49536i −5.95763 + 14.8495i −2.11435 − 2.11435i −30.5225 − 5.89449i 31.3946 + 37.6216i 62.5525 − 13.5345i 20.6021i −2.26792 + 11.7436i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.60
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.l.b ✓ 120
7.b odd 2 1 inner 112.5.l.b ✓ 120
16.e even 4 1 inner 112.5.l.b ✓ 120
112.l odd 4 1 inner 112.5.l.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.l.b ✓ 120 1.a even 1 1 trivial
112.5.l.b ✓ 120 7.b odd 2 1 inner
112.5.l.b ✓ 120 16.e even 4 1 inner
112.5.l.b ✓ 120 112.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 613820 T_{3}^{116} + 172101311152 T_{3}^{112} + \cdots + 47\!\cdots\!00 \) acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display