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Properties

Label 474.2.n.a
Level $474$
Weight $2$
Character orbit 474.n
Analytic conductor $3.785$
Analytic rank $0$
Dimension $624$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [474,2,Mod(29,474)] 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("474.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(474, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([39, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 474 = 2 \cdot 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 474.n (of order \(78\), degree \(24\), 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: \(3.78490905581\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(26\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 26 q^{4} - 18 q^{7} + 4 q^{9} + 4 q^{10} + 8 q^{13} + 52 q^{15} + 26 q^{16} + 24 q^{18} - 8 q^{19} + 60 q^{21} - 22 q^{22} + 6 q^{25} - 78 q^{27} - 18 q^{28} + 6 q^{30} - 36 q^{31} - 12 q^{34} - 4 q^{36}+ \cdots + 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.979791 − 0.200026i −1.70043 − 0.329469i 0.919979 + 0.391967i −2.33325 − 2.24112i 1.60016 + 0.662939i 0.278083 + 3.44467i −0.822984 − 0.568065i 2.78290 + 1.12047i 1.83782 + 2.66254i
29.2 −0.979791 − 0.200026i −1.69450 − 0.358692i 0.919979 + 0.391967i 0.0174171 + 0.0167294i 1.58851 + 0.690387i 0.00445034 + 0.0551273i −0.822984 − 0.568065i 2.74268 + 1.21561i −0.0137188 − 0.0198752i
29.3 −0.979791 − 0.200026i −0.920406 + 1.46726i 0.919979 + 0.391967i 1.00056 + 0.961048i 1.19529 − 1.25350i −0.344091 − 4.26233i −0.822984 − 0.568065i −1.30571 − 2.70095i −0.788102 − 1.14176i
29.4 −0.979791 − 0.200026i −0.868321 + 1.49867i 0.919979 + 0.391967i −1.27225 − 1.22202i 1.15055 − 1.29470i 0.0750695 + 0.929903i −0.822984 − 0.568065i −1.49204 − 2.60266i 1.00211 + 1.45180i
29.5 −0.979791 − 0.200026i −0.816237 − 1.52766i 0.919979 + 0.391967i −2.44991 − 2.35317i 0.494170 + 1.66006i −0.372738 − 4.61718i −0.822984 − 0.568065i −1.66751 + 2.49387i 1.92970 + 2.79566i
29.6 −0.979791 − 0.200026i −0.732879 − 1.56936i 0.919979 + 0.391967i 1.50027 + 1.44103i 0.404156 + 1.68424i −0.0526760 − 0.652509i −0.822984 − 0.568065i −1.92578 + 2.30030i −1.18171 − 1.71200i
29.7 −0.979791 − 0.200026i 0.136983 − 1.72663i 0.919979 + 0.391967i −0.482194 − 0.463153i −0.479584 + 1.66433i 0.126796 + 1.57065i −0.822984 − 0.568065i −2.96247 − 0.473036i 0.379806 + 0.550244i
29.8 −0.979791 − 0.200026i 0.657102 + 1.60257i 0.919979 + 0.391967i −2.24206 − 2.15353i −0.323268 − 1.70162i −0.0793397 − 0.982798i −0.822984 − 0.568065i −2.13643 + 2.10610i 1.76599 + 2.55848i
29.9 −0.979791 − 0.200026i 0.805848 + 1.53317i 0.919979 + 0.391967i 2.16485 + 2.07936i −0.482889 − 1.66338i −0.199443 − 2.47055i −0.822984 − 0.568065i −1.70122 + 2.47100i −1.70517 − 2.47037i
29.10 −0.979791 − 0.200026i 0.807173 + 1.53247i 0.919979 + 0.391967i 1.05706 + 1.01532i −0.484327 − 1.66296i 0.320844 + 3.97437i −0.822984 − 0.568065i −1.69694 + 2.47394i −0.832608 − 1.20624i
29.11 −0.979791 − 0.200026i 1.00806 − 1.40848i 0.919979 + 0.391967i 3.00012 + 2.88166i −1.26942 + 1.17837i −0.290140 − 3.59403i −0.822984 − 0.568065i −0.967611 − 2.83967i −2.36309 − 3.42352i
29.12 −0.979791 − 0.200026i 1.58563 − 0.696969i 0.919979 + 0.391967i 0.868905 + 0.834595i −1.69300 + 0.365717i 0.291015 + 3.60486i −0.822984 − 0.568065i 2.02847 − 2.21028i −0.684405 − 0.991532i
29.13 −0.979791 − 0.200026i 1.73197 − 0.0168854i 0.919979 + 0.391967i −1.23081 − 1.18221i −1.70034 − 0.329894i −0.130069 − 1.61119i −0.822984 − 0.568065i 2.99943 − 0.0584900i 0.969465 + 1.40451i
29.14 0.979791 + 0.200026i −1.69956 + 0.333902i 0.919979 + 0.391967i 2.44991 + 2.35317i −1.73200 − 0.0128016i −0.372738 − 4.61718i 0.822984 + 0.568065i 2.77702 − 1.13497i 1.92970 + 2.79566i
29.15 0.979791 + 0.200026i −1.67914 + 0.424842i 0.919979 + 0.391967i −1.50027 − 1.44103i −1.73018 + 0.0803852i −0.0526760 − 0.652509i 0.822984 + 0.568065i 2.63902 − 1.42674i −1.18171 − 1.71200i
29.16 0.979791 + 0.200026i −1.34953 − 1.08572i 0.919979 + 0.391967i −0.0174171 − 0.0167294i −1.10508 − 1.33372i 0.00445034 + 0.0551273i 0.822984 + 0.568065i 0.642437 + 2.93041i −0.0137188 − 0.0198752i
29.17 0.979791 + 0.200026i −1.33063 − 1.10879i 0.919979 + 0.391967i 2.33325 + 2.24112i −1.08196 − 1.35254i 0.278083 + 3.44467i 0.822984 + 0.568065i 0.541178 + 2.95078i 1.83782 + 2.66254i
29.18 0.979791 + 0.200026i −1.25082 + 1.19810i 0.919979 + 0.391967i 0.482194 + 0.463153i −1.46519 + 0.923695i 0.126796 + 1.57065i 0.822984 + 0.568065i 0.129094 − 2.99722i 0.379806 + 0.550244i
29.19 0.979791 + 0.200026i −0.453467 + 1.67164i 0.919979 + 0.391967i −3.00012 − 2.88166i −0.778673 + 1.54715i −0.290140 − 3.59403i 0.822984 + 0.568065i −2.58874 − 1.51606i −2.36309 − 3.42352i
29.20 0.979791 + 0.200026i 0.462951 + 1.66903i 0.919979 + 0.391967i −0.868905 − 0.834595i 0.119745 + 1.72791i 0.291015 + 3.60486i 0.822984 + 0.568065i −2.57135 + 1.54536i −0.684405 − 0.991532i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
79.h odd 78 1 inner
237.n even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 474.2.n.a ✓ 624
3.b odd 2 1 inner 474.2.n.a ✓ 624
79.h odd 78 1 inner 474.2.n.a ✓ 624
237.n even 78 1 inner 474.2.n.a ✓ 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
474.2.n.a ✓ 624 1.a even 1 1 trivial
474.2.n.a ✓ 624 3.b odd 2 1 inner
474.2.n.a ✓ 624 79.h odd 78 1 inner
474.2.n.a ✓ 624 237.n even 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(474, [\chi])\).