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Properties

Label 104.2.u.a
Level $104$
Weight $2$
Character orbit 104.u
Analytic conductor $0.830$
Analytic rank $0$
Dimension $48$
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] = [104,2,Mod(11,104)] 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("104.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.u (of order \(12\), degree \(4\), 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: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 4 q^{3} - 6 q^{4} - 6 q^{6} - 10 q^{8} - 20 q^{9} - 6 q^{10} - 8 q^{11} + 8 q^{14} - 10 q^{16} - 12 q^{17} - 6 q^{18} - 8 q^{19} + 10 q^{20} - 20 q^{22} + 46 q^{24} - 2 q^{26} + 8 q^{27}+ \cdots + 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.35755 + 0.396300i −1.57612 − 2.72991i 1.68589 − 1.07599i −2.13831 + 2.13831i 3.22152 + 3.08138i −1.43298 + 0.383965i −1.86227 + 2.12884i −3.46828 + 6.00723i 2.05545 − 3.75028i
11.2 −1.28449 − 0.591686i −0.223033 − 0.386304i 1.29981 + 1.52003i 0.612771 − 0.612771i 0.0579118 + 0.628168i 2.00697 − 0.537767i −0.770215 − 2.72154i 1.40051 − 2.42576i −1.14967 + 0.424528i
11.3 −1.22577 + 0.705331i 0.0597334 + 0.103461i 1.00502 − 1.72914i 2.08890 − 2.08890i −0.146194 − 0.0846877i −1.83497 + 0.491678i −0.0122979 + 2.82840i 1.49286 − 2.58572i −1.08714 + 4.03388i
11.4 −0.899259 + 1.09148i 1.05427 + 1.82605i −0.382666 − 1.96305i −2.29020 + 2.29020i −2.94117 − 0.491377i 1.21025 − 0.324286i 2.48675 + 1.34762i −0.722984 + 1.25224i −0.440228 − 4.55919i
11.5 −0.376293 − 1.36323i 1.11333 + 1.92835i −1.71681 + 1.02595i 0.0693382 − 0.0693382i 2.20985 − 2.24336i 3.36551 − 0.901786i 2.04463 + 1.95435i −0.979029 + 1.69573i −0.120616 − 0.0684326i
11.6 −0.195127 − 1.40069i −0.928193 − 1.60768i −1.92385 + 0.546624i 1.51817 − 1.51817i −2.07074 + 1.61381i −2.97768 + 0.797866i 1.14104 + 2.58805i −0.223086 + 0.386396i −2.42271 − 1.83024i
11.7 0.233040 + 1.39488i 1.05427 + 1.82605i −1.89138 + 0.650127i 2.29020 − 2.29020i −2.30144 + 1.89613i −1.21025 + 0.324286i −1.34762 − 2.48675i −0.722984 + 1.25224i 3.72826 + 2.66085i
11.8 0.708881 + 1.22372i 0.0597334 + 0.103461i −0.994975 + 1.73494i −2.08890 + 2.08890i −0.0842636 + 0.146439i 1.83497 − 0.491678i −2.82840 + 0.0122979i 1.49286 − 2.58572i −4.03702 − 1.07545i
11.9 0.869329 − 1.11547i −0.928193 − 1.60768i −0.488535 − 1.93942i −1.51817 + 1.51817i −2.60022 − 0.362231i 2.97768 − 0.797866i −2.58805 − 1.14104i −0.223086 + 0.386396i 0.373680 + 3.01325i
11.10 0.977524 + 1.02198i −1.57612 − 2.72991i −0.0888919 + 1.99802i 2.13831 − 2.13831i 1.24923 − 4.27932i 1.43298 − 0.383965i −2.12884 + 1.86227i −3.46828 + 6.00723i 4.27556 + 0.0950627i
11.11 1.00750 − 0.992447i 1.11333 + 1.92835i 0.0300962 − 1.99977i −0.0693382 + 0.0693382i 3.03547 + 0.837881i −3.36551 + 0.901786i −1.95435 − 2.04463i −0.979029 + 1.69573i −0.00104344 + 0.138673i
11.12 1.40824 + 0.129828i −0.223033 − 0.386304i 1.96629 + 0.365659i −0.612771 + 0.612771i −0.263931 − 0.572966i −2.00697 + 0.537767i 2.72154 + 0.770215i 1.40051 − 2.42576i −0.942485 + 0.783375i
19.1 −1.35755 − 0.396300i −1.57612 + 2.72991i 1.68589 + 1.07599i −2.13831 − 2.13831i 3.22152 − 3.08138i −1.43298 − 0.383965i −1.86227 − 2.12884i −3.46828 − 6.00723i 2.05545 + 3.75028i
19.2 −1.28449 + 0.591686i −0.223033 + 0.386304i 1.29981 − 1.52003i 0.612771 + 0.612771i 0.0579118 − 0.628168i 2.00697 + 0.537767i −0.770215 + 2.72154i 1.40051 + 2.42576i −1.14967 − 0.424528i
19.3 −1.22577 − 0.705331i 0.0597334 − 0.103461i 1.00502 + 1.72914i 2.08890 + 2.08890i −0.146194 + 0.0846877i −1.83497 − 0.491678i −0.0122979 − 2.82840i 1.49286 + 2.58572i −1.08714 − 4.03388i
19.4 −0.899259 − 1.09148i 1.05427 − 1.82605i −0.382666 + 1.96305i −2.29020 − 2.29020i −2.94117 + 0.491377i 1.21025 + 0.324286i 2.48675 − 1.34762i −0.722984 − 1.25224i −0.440228 + 4.55919i
19.5 −0.376293 + 1.36323i 1.11333 − 1.92835i −1.71681 − 1.02595i 0.0693382 + 0.0693382i 2.20985 + 2.24336i 3.36551 + 0.901786i 2.04463 − 1.95435i −0.979029 − 1.69573i −0.120616 + 0.0684326i
19.6 −0.195127 + 1.40069i −0.928193 + 1.60768i −1.92385 − 0.546624i 1.51817 + 1.51817i −2.07074 − 1.61381i −2.97768 − 0.797866i 1.14104 − 2.58805i −0.223086 − 0.386396i −2.42271 + 1.83024i
19.7 0.233040 − 1.39488i 1.05427 − 1.82605i −1.89138 − 0.650127i 2.29020 + 2.29020i −2.30144 − 1.89613i −1.21025 − 0.324286i −1.34762 + 2.48675i −0.722984 − 1.25224i 3.72826 − 2.66085i
19.8 0.708881 − 1.22372i 0.0597334 − 0.103461i −0.994975 − 1.73494i −2.08890 − 2.08890i −0.0842636 − 0.146439i 1.83497 + 0.491678i −2.82840 − 0.0122979i 1.49286 + 2.58572i −4.03702 + 1.07545i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.f odd 12 1 inner
104.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.2.u.a ✓ 48
3.b odd 2 1 936.2.ed.d 48
4.b odd 2 1 416.2.bk.a 48
8.b even 2 1 416.2.bk.a 48
8.d odd 2 1 inner 104.2.u.a ✓ 48
13.f odd 12 1 inner 104.2.u.a ✓ 48
24.f even 2 1 936.2.ed.d 48
39.k even 12 1 936.2.ed.d 48
52.l even 12 1 416.2.bk.a 48
104.u even 12 1 inner 104.2.u.a ✓ 48
104.x odd 12 1 416.2.bk.a 48
312.bq odd 12 1 936.2.ed.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.u.a ✓ 48 1.a even 1 1 trivial
104.2.u.a ✓ 48 8.d odd 2 1 inner
104.2.u.a ✓ 48 13.f odd 12 1 inner
104.2.u.a ✓ 48 104.u even 12 1 inner
416.2.bk.a 48 4.b odd 2 1
416.2.bk.a 48 8.b even 2 1
416.2.bk.a 48 52.l even 12 1
416.2.bk.a 48 104.x odd 12 1
936.2.ed.d 48 3.b odd 2 1
936.2.ed.d 48 24.f even 2 1
936.2.ed.d 48 39.k even 12 1
936.2.ed.d 48 312.bq odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(104, [\chi])\).