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Properties

Label 520.2.ca.b
Level $520$
Weight $2$
Character orbit 520.ca
Analytic conductor $4.152$
Analytic rank $0$
Dimension $56$
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{4} + 56 q^{5} - 13 q^{6} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{4} + 56 q^{5} - 13 q^{6} + 28 q^{9} - 8 q^{11} + 6 q^{12} - 4 q^{14} + 14 q^{16} + 18 q^{18} - 16 q^{19} + 6 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{24} + 56 q^{25} - 37 q^{26} - 12 q^{28} - 13 q^{30} - 30 q^{32} - 16 q^{34} + 15 q^{36} + 4 q^{37} - 24 q^{39} - 61 q^{42} + 24 q^{44} + 28 q^{45} - 19 q^{46} - 51 q^{48} + 20 q^{49} - 64 q^{52} - 5 q^{54} - 8 q^{55} - 23 q^{56} - q^{58} - 16 q^{59} + 6 q^{60} + 10 q^{62} - 30 q^{64} + 14 q^{66} - 36 q^{67} - 51 q^{68} - 4 q^{70} - 81 q^{72} + 70 q^{74} - 60 q^{76} + 143 q^{78} + 14 q^{80} - 28 q^{81} + 21 q^{82} + 40 q^{83} + 31 q^{84} - 28 q^{86} - 36 q^{87} - 19 q^{88} + 18 q^{90} + 16 q^{91} - 18 q^{92} + 43 q^{94} - 16 q^{95} - 48 q^{96} + 24 q^{97} + 56 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −1.41395 − 0.0273420i 2.80828 − 1.62136i 1.99850 + 0.0773203i 1.00000 −4.01510 + 2.21574i 3.76770 + 2.17528i −2.82367 − 0.163970i 3.75763 − 6.50841i −1.41395 − 0.0273420i
101.2 −1.40794 + 0.133107i −2.50839 + 1.44822i 1.96456 − 0.374813i 1.00000 3.33889 − 2.37289i −2.08112 − 1.20154i −2.71609 + 0.789211i 2.69469 − 4.66734i −1.40794 + 0.133107i
101.3 −1.35182 − 0.415447i −0.855456 + 0.493898i 1.65481 + 1.12321i 1.00000 1.36161 − 0.312262i −1.13840 − 0.657256i −1.77036 − 2.20586i −1.01213 + 1.75306i −1.35182 − 0.415447i
101.4 −1.32269 + 0.500493i 0.815452 − 0.470802i 1.49901 − 1.32399i 1.00000 −0.842957 + 1.03085i −2.82007 − 1.62817i −1.32008 + 2.50148i −1.05669 + 1.83024i −1.32269 + 0.500493i
101.5 −1.30078 + 0.554941i 0.176551 − 0.101932i 1.38408 − 1.44372i 1.00000 −0.173088 + 0.230566i 0.820502 + 0.473717i −0.999213 + 2.64605i −1.47922 + 2.56208i −1.30078 + 0.554941i
101.6 −1.28768 − 0.584700i 1.00895 − 0.582519i 1.31625 + 1.50582i 1.00000 −1.63981 + 0.160165i 0.786953 + 0.454347i −0.814463 − 2.70863i −0.821344 + 1.42261i −1.28768 − 0.584700i
101.7 −1.13415 − 0.844810i −1.79930 + 1.03883i 0.572591 + 1.91628i 1.00000 2.91828 + 0.341882i 3.65440 + 2.10987i 0.969491 − 2.65708i 0.658316 − 1.14024i −1.13415 − 0.844810i
101.8 −0.902585 − 1.08873i 2.26293 − 1.30650i −0.370681 + 1.96535i 1.00000 −3.46492 − 1.28450i −3.24721 − 1.87478i 2.47431 − 1.37032i 1.91389 − 3.31496i −0.902585 − 1.08873i
101.9 −0.884969 + 1.10310i 1.78301 − 1.02942i −0.433658 − 1.95242i 1.00000 −0.442356 + 2.87785i 1.47046 + 0.848970i 2.53749 + 1.24946i 0.619423 − 1.07287i −0.884969 + 1.10310i
101.10 −0.615362 − 1.27331i −0.127180 + 0.0734274i −1.24266 + 1.56710i 1.00000 0.171758 + 0.116756i −2.93738 − 1.69590i 2.76009 + 0.617962i −1.48922 + 2.57940i −0.615362 − 1.27331i
101.11 −0.439747 + 1.34411i −1.44998 + 0.837149i −1.61324 − 1.18213i 1.00000 −0.487590 − 2.31707i 1.28278 + 0.740615i 2.29833 − 1.64853i −0.0983634 + 0.170370i −0.439747 + 1.34411i
101.12 −0.417482 + 1.35119i −1.00502 + 0.580249i −1.65142 − 1.12819i 1.00000 −0.364447 − 1.60022i −2.74313 − 1.58374i 2.21384 − 1.76037i −0.826622 + 1.43175i −0.417482 + 1.35119i
101.13 −0.378853 − 1.36252i 1.76273 − 1.01771i −1.71294 + 1.03239i 1.00000 −2.05447 − 2.01620i 2.98263 + 1.72202i 2.05561 + 1.94280i 0.571474 − 0.989823i −0.378853 − 1.36252i
101.14 −0.0643054 − 1.41275i −0.756863 + 0.436975i −1.99173 + 0.181695i 1.00000 0.666007 + 1.04116i −0.0742471 − 0.0428666i 0.384769 + 2.80213i −1.11811 + 1.93662i −0.0643054 − 1.41275i
101.15 −0.0430002 + 1.41356i 2.59349 − 1.49735i −1.99630 − 0.121567i 1.00000 2.00507 + 3.73044i −0.668489 − 0.385952i 0.257683 − 2.81666i 2.98412 − 5.16865i −0.0430002 + 1.41356i
101.16 0.352646 + 1.36954i −2.90546 + 1.67747i −1.75128 + 0.965925i 1.00000 −3.32196 − 3.38760i 0.132211 + 0.0763322i −1.94045 − 2.05782i 4.12781 − 7.14957i 0.352646 + 1.36954i
101.17 0.386680 − 1.36032i −2.17544 + 1.25599i −1.70096 − 1.05202i 1.00000 0.867355 + 3.44497i 0.0859851 + 0.0496435i −2.08881 + 1.90705i 1.65504 − 2.86661i 0.386680 − 1.36032i
101.18 0.449378 + 1.34092i −0.700368 + 0.404358i −1.59612 + 1.20516i 1.00000 −0.856940 − 0.757426i 3.38125 + 1.95217i −2.33328 − 1.59869i −1.17299 + 2.03168i 0.449378 + 1.34092i
101.19 0.724783 − 1.21437i 1.50547 − 0.869185i −0.949379 − 1.76031i 1.00000 0.0356307 − 2.45817i 0.987404 + 0.570078i −2.82575 − 0.122945i 0.0109656 − 0.0189930i 0.724783 − 1.21437i
101.20 0.837733 + 1.13939i 0.654624 − 0.377948i −0.596406 + 1.90900i 1.00000 0.979029 + 0.429251i 0.518819 + 0.299540i −2.67472 + 0.919699i −1.21431 + 2.10325i 0.837733 + 1.13939i
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.b yes 56
8.b even 2 1 520.2.ca.a ✓ 56
13.e even 6 1 520.2.ca.a ✓ 56
104.s even 6 1 inner 520.2.ca.b yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.ca.a ✓ 56 8.b even 2 1
520.2.ca.a ✓ 56 13.e even 6 1
520.2.ca.b yes 56 1.a even 1 1 trivial
520.2.ca.b yes 56 104.s even 6 1 inner

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