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Properties

Label 520.2.ca.a
Level $520$
Weight $2$
Character orbit 520.ca
Analytic conductor $4.152$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(101,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.ca (of order \(6\), 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: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{4} - 56 q^{5} - 5 q^{6} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{4} - 56 q^{5} - 5 q^{6} + 28 q^{9} + 8 q^{11} + 6 q^{12} - 4 q^{14} - 10 q^{16} - 18 q^{18} + 16 q^{19} + 6 q^{20} - 6 q^{22} + 12 q^{23} + 2 q^{24} + 56 q^{25} + 11 q^{26} + 6 q^{28} + 5 q^{30} + 16 q^{34} - 21 q^{36} - 4 q^{37} - 24 q^{39} + 29 q^{42} - 24 q^{44} - 28 q^{45} - 11 q^{46} + 3 q^{48} + 20 q^{49} + 18 q^{52} - 49 q^{54} - 8 q^{55} + 61 q^{56} - 47 q^{58} + 16 q^{59} - 6 q^{60} - 2 q^{62} - 30 q^{64} + 14 q^{66} + 36 q^{67} + 33 q^{68} + 4 q^{70} - 51 q^{72} - 2 q^{74} - 48 q^{76} - 35 q^{78} + 10 q^{80} - 28 q^{81} - 21 q^{82} - 40 q^{83} - 61 q^{84} + 28 q^{86} - 36 q^{87} + 41 q^{88} + 18 q^{90} - 16 q^{91} - 18 q^{92} - 41 q^{94} - 16 q^{95} + 48 q^{96} + 24 q^{97} + 28 q^{98} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.39780 + 0.214856i −1.78301 + 1.02942i 1.90767 − 0.600650i −1.00000 2.27111 − 1.82202i 1.47046 + 0.848970i −2.53749 + 1.24946i 0.619423 − 1.07287i 1.39780 − 0.214856i
101.2 −1.38390 − 0.291221i 1.44998 − 0.837149i 1.83038 + 0.806043i −1.00000 −2.25043 + 0.736268i 1.28278 + 0.740615i −2.29833 − 1.64853i −0.0983634 + 0.170370i 1.38390 + 0.291221i
101.3 −1.37890 − 0.314043i 1.00502 − 0.580249i 1.80275 + 0.866072i −1.00000 −1.56805 + 0.484488i −2.74313 − 1.58374i −2.21384 − 1.76037i −0.826622 + 1.43175i 1.37890 + 0.314043i
101.4 −1.24568 − 0.669541i −2.59349 + 1.49735i 1.10343 + 1.66806i −1.00000 4.23319 − 0.128773i −0.668489 − 0.385952i −0.257683 − 2.81666i 2.98412 − 5.16865i 1.24568 + 0.669541i
101.5 −1.13099 + 0.849042i −0.176551 + 0.101932i 0.558256 − 1.92051i −1.00000 0.113132 − 0.265182i 0.820502 + 0.473717i 0.999213 + 2.64605i −1.47922 + 2.56208i 1.13099 − 0.849042i
101.6 −1.09478 + 0.895236i −0.815452 + 0.470802i 0.397104 − 1.96018i −1.00000 0.471265 − 1.24545i −2.82007 − 1.62817i 1.32008 + 2.50148i −1.05669 + 1.83024i 1.09478 − 0.895236i
101.7 −1.00973 − 0.990170i 2.90546 − 1.67747i 0.0391258 + 1.99962i −1.00000 −4.59472 − 1.18310i 0.132211 + 0.0763322i 1.94045 − 2.05782i 4.12781 − 7.14957i 1.00973 + 0.990170i
101.8 −0.936579 − 1.05963i 0.700368 − 0.404358i −0.245639 + 1.98486i −1.00000 −1.08442 − 0.363419i 3.38125 + 1.95217i 2.33328 − 1.59869i −1.17299 + 2.03168i 0.936579 + 1.05963i
101.9 −0.819242 + 1.15275i 2.50839 − 1.44822i −0.657684 − 1.88877i −1.00000 −0.385538 + 4.07801i −2.08112 − 1.20154i 2.71609 + 0.789211i 2.69469 − 4.66734i 0.819242 − 1.15275i
101.10 −0.683296 + 1.23819i −2.80828 + 1.62136i −1.06621 − 1.69210i −1.00000 −0.0886625 − 4.58505i 3.76770 + 2.17528i 2.82367 − 0.163970i 3.75763 − 6.50841i 0.683296 − 1.23819i
101.11 −0.567872 − 1.29519i −0.654624 + 0.377948i −1.35504 + 1.47101i −1.00000 0.861257 + 0.633238i 0.518819 + 0.299540i 2.67472 + 0.919699i −1.21431 + 2.10325i 0.567872 + 1.29519i
101.12 −0.316120 + 1.37843i 0.855456 − 0.493898i −1.80014 − 0.871498i −1.00000 0.410377 + 1.33532i −1.13840 − 0.657256i 1.77036 − 2.20586i −1.01213 + 1.75306i 0.316120 − 1.37843i
101.13 −0.137476 + 1.40752i −1.00895 + 0.582519i −1.96220 − 0.386999i −1.00000 −0.681198 − 1.50020i 0.786953 + 0.454347i 0.814463 − 2.70863i −0.821344 + 1.42261i 0.137476 − 1.40752i
101.14 0.0259673 − 1.41398i 1.48693 − 0.858477i −1.99865 − 0.0734343i −1.00000 −1.17525 − 2.12477i −3.27445 − 1.89050i −0.155734 + 2.82414i −0.0260346 + 0.0450933i −0.0259673 + 1.41398i
101.15 0.0710713 − 1.41243i −2.41987 + 1.39711i −1.98990 − 0.200766i −1.00000 1.80134 + 3.51719i −3.43922 − 1.98564i −0.424992 + 2.79632i 2.40385 − 4.16359i −0.0710713 + 1.41243i
101.16 0.164552 + 1.40461i 1.79930 − 1.03883i −1.94585 + 0.462263i −1.00000 1.75522 + 2.35637i 3.65440 + 2.10987i −0.969491 − 2.65708i 0.658316 − 1.14024i −0.164552 − 1.40461i
101.17 0.237097 − 1.39420i −0.158158 + 0.0913126i −1.88757 − 0.661121i −1.00000 0.0898089 + 0.242153i 2.61251 + 1.50833i −1.36927 + 2.47489i −1.48332 + 2.56919i −0.237097 + 1.39420i
101.18 0.491578 + 1.32603i −2.26293 + 1.30650i −1.51670 + 1.30369i −1.00000 −2.84486 − 2.35846i −3.24721 − 1.87478i −2.47431 − 1.37032i 1.91389 − 3.31496i −0.491578 − 1.32603i
101.19 0.738148 − 1.20629i 2.77854 − 1.60419i −0.910276 − 1.78084i −1.00000 0.115852 − 4.53585i 1.83925 + 1.06189i −2.82013 − 0.216468i 3.64684 − 6.31652i −0.738148 + 1.20629i
101.20 0.795042 + 1.16958i 0.127180 − 0.0734274i −0.735818 + 1.85972i −1.00000 0.186992 + 0.0903689i −2.93738 − 1.69590i −2.76009 + 0.617962i −1.48922 + 2.57940i −0.795042 − 1.16958i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.ca.a ✓ 56
8.b even 2 1 520.2.ca.b yes 56
13.e even 6 1 520.2.ca.b yes 56
104.s even 6 1 inner 520.2.ca.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.ca.a ✓ 56 1.a even 1 1 trivial
520.2.ca.a ✓ 56 104.s even 6 1 inner
520.2.ca.b yes 56 8.b even 2 1
520.2.ca.b yes 56 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 56 T_{3}^{54} + 1764 T_{3}^{52} + 36 T_{3}^{51} - 38040 T_{3}^{50} - 1500 T_{3}^{49} + \cdots + 61465600 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display