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Properties

Label 502.2.e.a
Level $502$
Weight $2$
Character orbit 502.e
Analytic conductor $4.008$
Analytic rank $0$
Dimension $200$
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] = [502,2,Mod(5,502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(502, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("502.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 502 = 2 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 502.e (of order \(25\), degree \(20\), 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: \(4.00849018147\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(10\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q + 50 q^{2} - 5 q^{3} - 50 q^{4} + 5 q^{5} + 5 q^{6} + 15 q^{7} + 50 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 200 q + 50 q^{2} - 5 q^{3} - 50 q^{4} + 5 q^{5} + 5 q^{6} + 15 q^{7} + 50 q^{8} - 5 q^{9} - 5 q^{10} - 10 q^{11} - 5 q^{12} + 10 q^{14} - 5 q^{15} - 50 q^{16} - 20 q^{18} + 10 q^{19} - 20 q^{20} + 10 q^{22} - 60 q^{23} - 20 q^{24} - 65 q^{25} - 35 q^{27} + 15 q^{28} + 30 q^{29} + 5 q^{30} + 35 q^{31} - 200 q^{32} + 20 q^{33} - 15 q^{35} - 5 q^{36} - 5 q^{37} - 10 q^{38} + 45 q^{39} - 5 q^{40} + 25 q^{41} - 25 q^{43} + 40 q^{44} - 35 q^{45} - 15 q^{46} - 5 q^{48} - 5 q^{49} + 15 q^{50} + 55 q^{51} - 20 q^{53} + 35 q^{54} + 10 q^{55} + 10 q^{56} + 30 q^{57} - 30 q^{58} - 25 q^{59} + 20 q^{60} - 15 q^{61} + 65 q^{62} - 70 q^{63} - 50 q^{64} - 15 q^{65} - 20 q^{66} - 45 q^{67} - 60 q^{69} - 35 q^{70} + 15 q^{71} + 5 q^{72} - 15 q^{73} + 5 q^{74} + 20 q^{75} + 10 q^{76} - 5 q^{77} - 70 q^{78} - 15 q^{79} + 5 q^{80} + 115 q^{81} + 25 q^{82} + 75 q^{83} - 60 q^{85} - 100 q^{86} - 85 q^{87} + 10 q^{88} - 95 q^{89} + 10 q^{90} + 235 q^{91} + 15 q^{92} - 65 q^{93} - 10 q^{95} + 5 q^{96} - 45 q^{97} + 80 q^{98} + 115 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.809017 + 0.587785i −1.56534 − 2.46658i 0.309017 + 0.951057i −0.465896 + 1.43388i 0.183433 − 2.91558i 3.75478 − 0.474339i −0.309017 + 0.951057i −2.35638 + 5.00756i −1.21973 + 0.886187i
5.2 0.809017 + 0.587785i −1.46916 − 2.31503i 0.309017 + 0.951057i 1.06181 − 3.26793i 0.172163 − 2.73645i 2.02372 − 0.255655i −0.309017 + 0.951057i −1.92358 + 4.08782i 2.77987 − 2.01969i
5.3 0.809017 + 0.587785i −1.19628 − 1.88504i 0.309017 + 0.951057i −0.612715 + 1.88574i 0.140185 − 2.22818i −1.53427 + 0.193824i −0.309017 + 0.951057i −0.844940 + 1.79559i −1.60411 + 1.16545i
5.4 0.809017 + 0.587785i −0.309833 − 0.488219i 0.309017 + 0.951057i 0.972920 − 2.99434i 0.0363076 − 0.577093i −0.243910 + 0.0308130i −0.309017 + 0.951057i 1.13498 − 2.41195i 2.54714 − 1.85060i
5.5 0.809017 + 0.587785i −0.0309743 − 0.0488077i 0.309017 + 0.951057i 0.300001 − 0.923308i 0.00362970 − 0.0576925i 0.707861 − 0.0894237i −0.309017 + 0.951057i 1.27592 − 2.71146i 0.785412 − 0.570635i
5.6 0.809017 + 0.587785i 0.227724 + 0.358835i 0.309017 + 0.951057i −0.313629 + 0.965249i −0.0266857 + 0.424156i 4.47269 − 0.565032i −0.309017 + 0.951057i 1.20043 − 2.55105i −0.821090 + 0.596557i
5.7 0.809017 + 0.587785i 0.558644 + 0.880282i 0.309017 + 0.951057i −0.807656 + 2.48571i −0.0654644 + 1.04053i −3.56309 + 0.450123i −0.309017 + 0.951057i 0.814524 − 1.73095i −2.11447 + 1.53625i
5.8 0.809017 + 0.587785i 1.39812 + 2.20309i 0.309017 + 0.951057i −1.17018 + 3.60143i −0.163838 + 2.60413i 1.90429 − 0.240568i −0.309017 + 0.951057i −1.62152 + 3.44591i −3.06356 + 2.22581i
5.9 0.809017 + 0.587785i 1.43325 + 2.25844i 0.309017 + 0.951057i 0.0405156 − 0.124694i −0.167955 + 2.66956i −2.33008 + 0.294357i −0.309017 + 0.951057i −1.76901 + 3.75935i 0.106071 − 0.0770653i
5.10 0.809017 + 0.587785i 1.45600 + 2.29429i 0.309017 + 0.951057i 0.724612 − 2.23013i −0.170621 + 2.71194i 1.52756 − 0.192976i −0.309017 + 0.951057i −1.86650 + 3.96652i 1.89706 − 1.37829i
25.1 −0.309017 − 0.951057i −1.36337 + 2.89730i −0.809017 + 0.587785i −0.824930 − 0.599346i 3.17680 + 0.401323i −1.90581 + 0.489329i 0.809017 + 0.587785i −4.62332 − 5.58864i −0.315095 + 0.969763i
25.2 −0.309017 − 0.951057i −1.03050 + 2.18992i −0.809017 + 0.587785i 1.41019 + 1.02456i 2.40118 + 0.303339i 3.89180 − 0.999244i 0.809017 + 0.587785i −1.82155 − 2.20187i 0.538644 − 1.65778i
25.3 −0.309017 − 0.951057i −0.824046 + 1.75119i −0.809017 + 0.587785i 3.16302 + 2.29807i 1.92012 + 0.242568i −2.89284 + 0.742754i 0.809017 + 0.587785i −0.475333 − 0.574579i 1.20816 − 3.71835i
25.4 −0.309017 − 0.951057i −0.384989 + 0.818143i −0.809017 + 0.587785i 0.435768 + 0.316604i 0.897068 + 0.113326i −3.23576 + 0.830803i 0.809017 + 0.587785i 1.39113 + 1.68159i 0.166449 − 0.512276i
25.5 −0.309017 − 0.951057i −0.257351 + 0.546899i −0.809017 + 0.587785i −1.39944 − 1.01675i 0.599658 + 0.0757544i 0.997462 − 0.256105i 0.809017 + 0.587785i 1.67940 + 2.03005i −0.534539 + 1.64514i
25.6 −0.309017 − 0.951057i 0.113209 − 0.240581i −0.809017 + 0.587785i −1.01525 − 0.737619i −0.263789 − 0.0333244i −1.80255 + 0.462816i 0.809017 + 0.587785i 1.86721 + 2.25707i −0.387789 + 1.19349i
25.7 −0.309017 − 0.951057i 0.502241 − 1.06732i −0.809017 + 0.587785i 0.407345 + 0.295954i −1.17028 − 0.147841i 3.53970 − 0.908840i 0.809017 + 0.587785i 1.02535 + 1.23944i 0.155592 − 0.478863i
25.8 −0.309017 − 0.951057i 0.718887 − 1.52771i −0.809017 + 0.587785i 2.81894 + 2.04808i −1.67509 − 0.211613i −1.74703 + 0.448561i 0.809017 + 0.587785i 0.0951669 + 0.115037i 1.07674 − 3.31387i
25.9 −0.309017 − 0.951057i 1.02422 − 2.17657i −0.809017 + 0.587785i −2.81527 − 2.04541i −2.38654 − 0.301491i −2.18971 + 0.562221i 0.809017 + 0.587785i −1.77617 − 2.14702i −1.07534 + 3.30954i
25.10 −0.309017 − 0.951057i 1.18125 − 2.51028i −0.809017 + 0.587785i 0.233913 + 0.169948i −2.75245 − 0.347715i 2.35153 − 0.603770i 0.809017 + 0.587785i −2.99390 − 3.61900i 0.0893469 − 0.274982i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
251.e even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 502.2.e.a ✓ 200
251.e even 25 1 inner 502.2.e.a ✓ 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.e.a ✓ 200 1.a even 1 1 trivial
502.2.e.a ✓ 200 251.e even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{200} + 5 T_{3}^{199} + 15 T_{3}^{198} + 45 T_{3}^{197} + 90 T_{3}^{196} + 131 T_{3}^{195} + \cdots + 12\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(502, [\chi])\). Copy content Toggle raw display