Properties

Label 501.2.a.e
Level $501$
Weight $2$
Character orbit 501.a
Self dual yes
Analytic conductor $4.001$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 38\nu^{5} + 2\nu^{4} - 147\nu^{3} - 11\nu^{2} + 168\nu + 15 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 15\nu^{5} + 3\nu^{4} - 70\nu^{3} - 20\nu^{2} + 98\nu + 19 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} - 16\nu^{5} + 64\nu^{4} + 21\nu^{3} - 135\nu^{2} + 21\nu + 25 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} + 24\nu^{5} - 61\nu^{4} - 28\nu^{3} + 122\nu^{2} - 42\nu - 27 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 18\nu^{4} - 25\nu^{3} + 37\nu^{2} + 8\nu - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 7\nu^{6} - 39\nu^{5} + 58\nu^{4} + 98\nu^{3} - 95\nu^{2} - 56\nu - 13 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} + 7\beta_{5} + \beta_{4} + 8\beta_{3} - 2\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 9\beta_{6} + 8\beta_{4} + 2\beta_{3} - 9\beta_{2} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{7} + \beta_{6} + 45\beta_{5} + 8\beta_{4} + 57\beta_{3} - 21\beta_{2} - 8\beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 65\beta_{6} + \beta_{5} + 53\beta_{4} + 27\beta_{3} - 71\beta_{2} + 165\beta _1 + 4 \) 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
−2.45154
−1.71120
−0.459587
−0.0452510
1.23856
1.60389
2.18779
2.63734
−2.45154 1.00000 4.01004 1.48532 −2.45154 −5.10210 −4.92768 1.00000 −3.64131
1.2 −1.71120 1.00000 0.928220 2.45529 −1.71120 2.43610 1.83403 1.00000 −4.20151
1.3 −0.459587 1.00000 −1.78878 −2.63865 −0.459587 −0.604857 1.74127 1.00000 1.21269
1.4 −0.0452510 1.00000 −1.99795 1.52778 −0.0452510 0.428515 0.180911 1.00000 −0.0691336
1.5 1.23856 1.00000 −0.465971 3.06821 1.23856 −2.07562 −3.05425 1.00000 3.80015
1.6 1.60389 1.00000 0.572476 −0.122001 1.60389 3.47013 −2.28960 1.00000 −0.195676
1.7 2.18779 1.00000 2.78642 2.52133 2.18779 −0.307143 1.72052 1.00000 5.51613
1.8 2.63734 1.00000 4.95555 −1.29727 2.63734 −2.24503 7.79478 1.00000 −3.42135
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.e 8
3.b odd 2 1 1503.2.a.e 8
4.b odd 2 1 8016.2.a.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.e 8 1.a even 1 1 trivial
1503.2.a.e 8 3.b odd 2 1
8016.2.a.x 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 3T_{2}^{7} - 8T_{2}^{6} + 28T_{2}^{5} + 9T_{2}^{4} - 64T_{2}^{3} + 17T_{2}^{2} + 23T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} - 8 T^{6} + 28 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + 7 T^{6} + 50 T^{5} + \cdots - 18 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} - 19 T^{6} - 65 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 13 T^{7} + 41 T^{6} + \cdots - 1596 \) Copy content Toggle raw display
$13$ \( T^{8} - 37 T^{6} - 52 T^{5} + 320 T^{4} + \cdots + 56 \) Copy content Toggle raw display
$17$ \( T^{8} - 11 T^{7} - 20 T^{6} + \cdots + 7822 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + 5 T^{6} + \cdots + 4384 \) Copy content Toggle raw display
$23$ \( T^{8} - 7 T^{7} - 19 T^{6} + 114 T^{5} + \cdots + 384 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} - 103 T^{6} + 8 T^{5} + \cdots + 88 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} - 206 T^{6} + \cdots - 1017696 \) Copy content Toggle raw display
$37$ \( T^{8} + 9 T^{7} - 107 T^{6} + \cdots - 42872 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} - 188 T^{6} + \cdots - 859446 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} - 38 T^{6} - 23 T^{5} + \cdots + 114 \) Copy content Toggle raw display
$47$ \( T^{8} - 17 T^{7} - 33 T^{6} + \cdots - 68732 \) Copy content Toggle raw display
$53$ \( T^{8} - 9 T^{7} - 53 T^{6} + \cdots + 20326 \) Copy content Toggle raw display
$59$ \( T^{8} - 29 T^{7} + 269 T^{6} + \cdots - 21056 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} - 99 T^{6} + \cdots + 51244 \) Copy content Toggle raw display
$67$ \( T^{8} - 177 T^{6} - 320 T^{5} + \cdots - 1226366 \) Copy content Toggle raw display
$71$ \( T^{8} - 13 T^{7} - 34 T^{6} + \cdots + 32816 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} - 169 T^{6} + \cdots - 9386184 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} - 464 T^{6} + \cdots + 39988958 \) Copy content Toggle raw display
$83$ \( T^{8} - 33 T^{7} + 257 T^{6} + \cdots - 2562224 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} - 199 T^{6} + \cdots - 216312 \) Copy content Toggle raw display
$97$ \( T^{8} + 31 T^{7} + 50 T^{6} + \cdots - 24584 \) Copy content Toggle raw display
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