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Properties

Label 520.2.bc.c
Level $520$
Weight $2$
Character orbit 520.bc
Analytic conductor $4.152$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(363,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.363");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.bc (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(68\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 8 q^{3} + 4 q^{10} + 72 q^{16} + 32 q^{22} + 4 q^{26} + 40 q^{27} + 32 q^{30} - 80 q^{35} - 88 q^{36} - 12 q^{38} - 72 q^{40} - 80 q^{42} - 40 q^{43} - 68 q^{48} - 16 q^{51} - 12 q^{52} - 24 q^{56} + 76 q^{62} + 72 q^{66} + 64 q^{68} + 160 q^{75} - 4 q^{78} + 104 q^{81} + 36 q^{82} + 44 q^{88} - 184 q^{90} - 40 q^{91} + 36 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
363.1 −1.41369 + 0.0386564i −0.452140 − 0.452140i 1.99701 − 0.109296i 0.861981 + 2.06325i 0.656662 + 0.621706i 2.53690 − 2.53690i −2.81892 + 0.231707i − 2.59114i −1.29833 − 2.88346i
363.2 −1.41361 + 0.0413534i −0.832995 − 0.832995i 1.99658 − 0.116915i −2.20203 + 0.388683i 1.21198 + 1.14308i 0.00401385 − 0.00401385i −2.81755 + 0.247838i − 1.61224i 3.09673 − 0.640506i
363.3 −1.40622 + 0.150139i 1.89278 + 1.89278i 1.95492 − 0.422256i 2.11447 − 0.727346i −2.94585 − 2.37749i 1.55503 − 1.55503i −2.68565 + 0.887294i 4.16526i −2.86421 + 1.34027i
363.4 −1.40044 − 0.196879i −1.90481 − 1.90481i 1.92248 + 0.551436i −0.954713 − 2.02201i 2.29255 + 3.04259i 2.04797 − 2.04797i −2.58375 − 1.15075i 4.25657i 0.938928 + 3.01967i
363.5 −1.39130 − 0.253527i 1.48989 + 1.48989i 1.87145 + 0.705466i 0.116771 + 2.23302i −1.69516 − 2.45062i −1.18631 + 1.18631i −2.42490 − 1.45598i 1.43955i 0.403666 − 3.13641i
363.6 −1.37124 + 0.345966i 0.379781 + 0.379781i 1.76061 − 0.948808i 0.696554 − 2.12481i −0.652163 − 0.389381i −1.57858 + 1.57858i −2.08597 + 1.91016i − 2.71153i −0.220033 + 3.15461i
363.7 −1.36015 + 0.387292i −1.81010 − 1.81010i 1.70001 − 1.05355i 1.08934 − 1.95278i 3.16304 + 1.76097i −0.967797 + 0.967797i −1.90424 + 2.09138i 3.55293i −0.725369 + 3.07796i
363.8 −1.35004 + 0.421177i 1.48046 + 1.48046i 1.64522 − 1.13721i −1.83163 − 1.28262i −2.62222 − 1.37515i 2.62114 − 2.62114i −1.74215 + 2.22821i 1.38353i 3.01299 + 0.960149i
363.9 −1.34517 − 0.436475i −0.621562 − 0.621562i 1.61898 + 1.17427i 2.23577 + 0.0362530i 0.564812 + 1.10740i −3.35795 + 3.35795i −1.66527 − 2.28624i − 2.22732i −2.99168 − 1.02463i
363.10 −1.30653 + 0.541286i −1.27264 − 1.27264i 1.41402 − 1.41441i 0.865034 + 2.06197i 2.35160 + 0.973873i −1.18368 + 1.18368i −1.08185 + 2.61335i 0.239221i −2.24631 − 2.22578i
363.11 −1.30348 − 0.548590i 0.151178 + 0.151178i 1.39810 + 1.43015i −1.69114 − 1.46289i −0.114122 − 0.279991i −1.14900 + 1.14900i −1.03783 − 2.63114i − 2.95429i 1.40183 + 2.83458i
363.12 −1.25842 − 0.645271i −2.19956 − 2.19956i 1.16725 + 1.62405i 2.09373 + 0.785050i 1.34866 + 4.18730i 0.877661 − 0.877661i −0.420941 − 2.79693i 6.67617i −2.12822 − 2.33895i
363.13 −1.24821 − 0.664812i 0.278786 + 0.278786i 1.11605 + 1.65965i 0.395288 − 2.20085i −0.162643 − 0.533323i 2.50681 − 2.50681i −0.289712 − 2.81355i − 2.84456i −1.95655 + 2.48433i
363.14 −1.21528 + 0.723247i 0.630877 + 0.630877i 0.953828 − 1.75790i −1.73429 + 1.41147i −1.22297 − 0.310415i 0.474601 − 0.474601i 0.112224 + 2.82620i − 2.20399i 1.08681 − 2.96965i
363.15 −1.17268 − 0.790452i −1.25929 − 1.25929i 0.750372 + 1.85390i −1.75120 + 1.39044i 0.481339 + 2.47215i −1.10966 + 1.10966i 0.585469 − 2.76717i 0.171606i 3.15267 − 0.246303i
363.16 −1.08931 − 0.901889i 1.03432 + 1.03432i 0.373191 + 1.96487i 2.03762 + 0.920932i −0.193852 − 2.05953i 1.55134 − 1.55134i 1.36558 − 2.47693i − 0.860382i −1.38902 − 2.84089i
363.17 −1.02889 − 0.970247i 2.01502 + 2.01502i 0.117241 + 1.99656i 1.76410 − 1.37403i −0.118172 − 4.02831i −2.51611 + 2.51611i 1.81653 − 2.16800i 5.12062i −3.14822 − 0.297889i
363.18 −0.970247 − 1.02889i 2.01502 + 2.01502i −0.117241 + 1.99656i −1.76410 + 1.37403i 0.118172 − 4.02831i 2.51611 − 2.51611i 2.16800 − 1.81653i 5.12062i 3.12534 + 0.481926i
363.19 −0.901889 − 1.08931i 1.03432 + 1.03432i −0.373191 + 1.96487i −2.03762 − 0.920932i 0.193852 − 2.05953i −1.55134 + 1.55134i 2.47693 − 1.36558i − 0.860382i 0.834526 + 3.05017i
363.20 −0.790452 − 1.17268i −1.25929 − 1.25929i −0.750372 + 1.85390i 1.75120 − 1.39044i −0.481339 + 2.47215i 1.10966 − 1.10966i 2.76717 − 0.585469i 0.171606i −3.01478 − 0.954528i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 363.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
40.k even 4 1 inner
65.h odd 4 1 inner
104.h odd 2 1 inner
520.bc even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.bc.c ✓ 136
5.c odd 4 1 inner 520.2.bc.c ✓ 136
8.d odd 2 1 inner 520.2.bc.c ✓ 136
13.b even 2 1 inner 520.2.bc.c ✓ 136
40.k even 4 1 inner 520.2.bc.c ✓ 136
65.h odd 4 1 inner 520.2.bc.c ✓ 136
104.h odd 2 1 inner 520.2.bc.c ✓ 136
520.bc even 4 1 inner 520.2.bc.c ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bc.c ✓ 136 1.a even 1 1 trivial
520.2.bc.c ✓ 136 5.c odd 4 1 inner
520.2.bc.c ✓ 136 8.d odd 2 1 inner
520.2.bc.c ✓ 136 13.b even 2 1 inner
520.2.bc.c ✓ 136 40.k even 4 1 inner
520.2.bc.c ✓ 136 65.h odd 4 1 inner
520.2.bc.c ✓ 136 104.h odd 2 1 inner
520.2.bc.c ✓ 136 520.bc even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\):

\( T_{3}^{34} + 2 T_{3}^{33} + 2 T_{3}^{32} - 6 T_{3}^{31} + 173 T_{3}^{30} + 312 T_{3}^{29} + 296 T_{3}^{28} + \cdots + 8192 \) Copy content Toggle raw display
\( T_{7}^{68} + 1358 T_{7}^{64} + 732441 T_{7}^{60} + 210288220 T_{7}^{56} + 35741927056 T_{7}^{52} + \cdots + 1761205026816 \) Copy content Toggle raw display