Properties

Label 501.2.a.b
Level $501$
Weight $2$
Character orbit 501.a
Self dual yes
Analytic conductor $4.001$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [501,2,Mod(1,501)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(501, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("501.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 501 = 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 501.a (trivial)

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: yes
Analytic conductor: \(4.00050514127\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.36497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + q^{3} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} + q^{3} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{8} + q^{9} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{10} + (\beta_{4} + \beta_{2} - \beta_1 - 2) q^{11} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 1) q^{12} + (2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{14} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{15} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{16} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{17} + (\beta_{3} - 1) q^{18} + ( - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 3) q^{19} + ( - \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{20} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{21} + ( - \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{22} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{23} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{24} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{25} + ( - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 1) q^{26} + q^{27} + ( - 2 \beta_{3} - 3 \beta_{2} + 1) q^{28} + (\beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{29} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{30} + (3 \beta_{4} - \beta_{3} + 3 \beta_1 - 4) q^{31} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{32} + (\beta_{4} + \beta_{2} - \beta_1 - 2) q^{33} + (3 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 3) q^{34} + (4 \beta_{4} - \beta_{2} - \beta_1) q^{35} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 1) q^{36} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 1) q^{37} + (2 \beta_{4} - 8 \beta_{3} + 5 \beta_1 + 4) q^{38} + (2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 6) q^{40} + (\beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 - 4) q^{41} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{42} + ( - 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{43} + ( - 3 \beta_{4} + 9 \beta_{3} + 4 \beta_{2} - 7 \beta_1 + 1) q^{44} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{45} + ( - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{46} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{48} + (4 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 2) q^{50} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{51} + ( - 2 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 7 \beta_1 + 2) q^{52} + (3 \beta_{2} - 4 \beta_1 + 1) q^{53} + (\beta_{3} - 1) q^{54} + (\beta_{4} + \beta_{3} + \beta_{2} + 7 \beta_1 + 1) q^{55} + (5 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{56} + ( - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 3) q^{57} + (6 \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 1) q^{58} + (2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 7) q^{59} + ( - \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{60} + ( - 2 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 1) q^{61} + ( - \beta_{4} - 4 \beta_1 + 5) q^{62} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{63} + ( - 2 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{64} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_1 - 6) q^{65} + ( - \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{66} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 - 3) q^{67} + ( - 3 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 7 \beta_1 + 1) q^{68} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{69} + (\beta_{4} - 4 \beta_{2}) q^{70} + ( - 3 \beta_{4} + \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 5) q^{71} + ( - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 2) q^{72} + ( - \beta_{4} - 2 \beta_{3} - 1) q^{73} + (5 \beta_{4} - \beta_{3} + 2 \beta_1 + 3) q^{74} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{75} + ( - 2 \beta_{4} + 13 \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 9) q^{76} + ( - 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 9 \beta_1) q^{77} + ( - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 1) q^{78} + (3 \beta_{4} + \beta_{2} + 2 \beta_1 + 3) q^{79} + (\beta_{4} + 3 \beta_{3} + \beta_{2} - 5 \beta_1) q^{80} + q^{81} + (4 \beta_{2} - 4 \beta_1 + 11) q^{82} + (\beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{83} + ( - 2 \beta_{3} - 3 \beta_{2} + 1) q^{84} + (\beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 3 \beta_1 + 8) q^{85} + ( - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - \beta_1 - 6) q^{86} + (\beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{87} + (7 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 16 \beta_1 + 6) q^{88} + ( - 5 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 7) q^{89} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{90} + ( - 7 \beta_{4} - \beta_{2} - 6 \beta_1 - 6) q^{91} + ( - \beta_{4} + 6 \beta_{3} - \beta_{2} - 5 \beta_1 - 6) q^{92} + (3 \beta_{4} - \beta_{3} + 3 \beta_1 - 4) q^{93} + ( - 3 \beta_{3} + \beta_1 + 7) q^{94} + (3 \beta_{4} - \beta_{3} + \beta_1 + 9) q^{95} + (2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{96} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{97} + (6 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 8) q^{98} + (\beta_{4} + \beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{2} + 5 q^{3} + 4 q^{4} - 9 q^{5} - 4 q^{6} - 4 q^{7} - 12 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{2} + 5 q^{3} + 4 q^{4} - 9 q^{5} - 4 q^{6} - 4 q^{7} - 12 q^{8} + 5 q^{9} + 5 q^{10} - 15 q^{11} + 4 q^{12} - 3 q^{14} - 9 q^{15} + 10 q^{16} - 11 q^{17} - 4 q^{18} - 16 q^{19} - 9 q^{20} - 4 q^{21} + 7 q^{22} - 9 q^{23} - 12 q^{24} + 6 q^{25} + 7 q^{26} + 5 q^{27} + 9 q^{28} - q^{29} + 5 q^{30} - 18 q^{31} - 3 q^{32} - 15 q^{33} + 2 q^{34} - 4 q^{35} + 4 q^{36} + 7 q^{37} + 20 q^{38} + 32 q^{40} - 10 q^{41} - 3 q^{42} + 6 q^{43} - 5 q^{44} - 9 q^{45} + 13 q^{46} - 7 q^{47} + 10 q^{48} + 11 q^{49} + 3 q^{50} - 11 q^{51} - 6 q^{52} - 9 q^{53} - 4 q^{54} + 17 q^{55} - 15 q^{56} - 16 q^{57} - 19 q^{58} - 37 q^{59} - 9 q^{60} - 2 q^{61} + 18 q^{62} - 4 q^{63} + 14 q^{64} - 16 q^{65} + 7 q^{66} + 26 q^{68} - 9 q^{69} + 7 q^{70} + 13 q^{71} - 12 q^{72} - 6 q^{73} + 13 q^{74} + 6 q^{75} - 50 q^{76} - 2 q^{77} + 7 q^{78} + 14 q^{79} - 10 q^{80} + 5 q^{81} + 39 q^{82} - 5 q^{83} + 9 q^{84} + 29 q^{85} - 37 q^{86} - q^{87} + 48 q^{88} - 30 q^{89} + 5 q^{90} - 33 q^{91} - 31 q^{92} - 18 q^{93} + 34 q^{94} + 43 q^{95} - 3 q^{96} - 9 q^{97} + 36 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 7 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.33419
−1.55629
−0.506287
2.31801
0.410375
−2.65817 1.00000 5.06586 −1.19539 −2.65817 1.77240 −8.14956 1.00000 3.17754
1.2 −2.16509 1.00000 2.68763 −3.33576 −2.16509 −0.969164 −1.48879 1.00000 7.22224
1.3 −0.842403 1.00000 −1.29036 1.21255 −0.842403 −3.40756 2.77181 1.00000 −1.02146
1.4 0.559296 1.00000 −1.68719 −3.48658 0.559296 3.13190 −2.06223 1.00000 −1.95003
1.5 1.10637 1.00000 −0.775946 −2.19483 1.10637 −4.52759 −3.07122 1.00000 −2.42829
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(167\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 501.2.a.b 5
3.b odd 2 1 1503.2.a.d 5
4.b odd 2 1 8016.2.a.p 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.b 5 1.a even 1 1 trivial
1503.2.a.d 5 3.b odd 2 1
8016.2.a.p 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 4T_{2}^{4} + T_{2}^{3} - 8T_{2}^{2} - 2T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 4 T^{4} + T^{3} - 8 T^{2} - 2 T + 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 9 T^{4} + 25 T^{3} + 12 T^{2} + \cdots - 37 \) Copy content Toggle raw display
$7$ \( T^{5} + 4 T^{4} - 15 T^{3} - 49 T^{2} + \cdots + 83 \) Copy content Toggle raw display
$11$ \( T^{5} + 15 T^{4} + 69 T^{3} + \cdots - 761 \) Copy content Toggle raw display
$13$ \( T^{5} - 41 T^{3} - 32 T^{2} + \cdots + 687 \) Copy content Toggle raw display
$17$ \( T^{5} + 11 T^{4} - 10 T^{3} + \cdots + 2203 \) Copy content Toggle raw display
$19$ \( T^{5} + 16 T^{4} + 61 T^{3} + \cdots - 677 \) Copy content Toggle raw display
$23$ \( T^{5} + 9 T^{4} - T^{3} - 170 T^{2} + \cdots - 47 \) Copy content Toggle raw display
$29$ \( T^{5} + T^{4} - 89 T^{3} - 146 T^{2} + \cdots - 157 \) Copy content Toggle raw display
$31$ \( T^{5} + 18 T^{4} + 90 T^{3} + 79 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$37$ \( T^{5} - 7 T^{4} - 77 T^{3} + \cdots + 1809 \) Copy content Toggle raw display
$41$ \( T^{5} + 10 T^{4} - 120 T^{3} + \cdots - 87 \) Copy content Toggle raw display
$43$ \( T^{5} - 6 T^{4} - 126 T^{3} + \cdots - 127 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 57 T^{3} + \cdots + 1209 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} - 127 T^{3} + \cdots - 3061 \) Copy content Toggle raw display
$59$ \( T^{5} + 37 T^{4} + 469 T^{3} + \cdots - 617 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} - 231 T^{3} + \cdots + 69523 \) Copy content Toggle raw display
$67$ \( T^{5} - 219 T^{3} + 558 T^{2} + \cdots + 751 \) Copy content Toggle raw display
$71$ \( T^{5} - 13 T^{4} - 258 T^{3} + \cdots - 64423 \) Copy content Toggle raw display
$73$ \( T^{5} + 6 T^{4} - 15 T^{3} - 40 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$79$ \( T^{5} - 14 T^{4} + 44 T^{3} + \cdots + 923 \) Copy content Toggle raw display
$83$ \( T^{5} + 5 T^{4} - 169 T^{3} + \cdots - 10575 \) Copy content Toggle raw display
$89$ \( T^{5} + 30 T^{4} + 217 T^{3} + \cdots - 55203 \) Copy content Toggle raw display
$97$ \( T^{5} + 9 T^{4} - 48 T^{3} + \cdots + 3391 \) Copy content Toggle raw display
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