Properties

Label 473.2.a.d
Level $473$
Weight $2$
Character orbit 473.a
Self dual yes
Analytic conductor $3.777$
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] = [473,2,Mod(1,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, 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("473.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.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: \(3.77692401561\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.173513.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{4} + 2\beta_{3} + \beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} + 9\beta_{3} + 2\beta_{2} + 23\beta _1 + 14 \) 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
−1.52979
3.18986
0.293545
−0.812660
0.859039
−2.74613 −0.876100 5.54124 3.27592 2.40589 −1.12390 −9.72471 −2.23245 −8.99610
1.2 −2.10899 2.87637 2.44784 −2.08087 −6.06623 −4.87637 −0.944480 5.27350 4.38854
1.3 −0.905706 −3.11308 −1.17970 −0.387840 2.81954 1.11308 2.87987 6.69128 0.351269
1.4 0.944788 0.417866 −1.10738 −1.13213 0.394795 −2.41787 −2.93581 −2.82539 −1.06962
1.5 1.81604 −0.305052 1.29800 −3.67508 −0.553987 −1.69495 −1.27487 −2.90694 −6.67408
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(43\) \(-1\)

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 473.2.a.d 5
3.b odd 2 1 4257.2.a.o 5
4.b odd 2 1 7568.2.a.bc 5
11.b odd 2 1 5203.2.a.i 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.a.d 5 1.a even 1 1 trivial
4257.2.a.o 5 3.b odd 2 1
5203.2.a.i 5 11.b odd 2 1
7568.2.a.bc 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(473))\):

\( T_{2}^{5} + 3T_{2}^{4} - 4T_{2}^{3} - 13T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 7T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 3 T^{4} - 4 T^{3} - 13 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} - 9 T^{3} - 7 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$5$ \( T^{5} + 4 T^{4} - 7 T^{3} - 41 T^{2} + \cdots - 11 \) Copy content Toggle raw display
$7$ \( T^{5} + 9 T^{4} + 23 T^{3} + 9 T^{2} + \cdots - 25 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 9 T^{4} - T^{3} - 132 T^{2} + \cdots + 193 \) Copy content Toggle raw display
$17$ \( T^{5} + 13 T^{4} + 29 T^{3} + \cdots + 279 \) Copy content Toggle raw display
$19$ \( T^{5} + 7 T^{4} - 35 T^{3} - 362 T^{2} + \cdots - 611 \) Copy content Toggle raw display
$23$ \( T^{5} - 8 T^{4} - 11 T^{3} + 138 T^{2} + \cdots - 361 \) Copy content Toggle raw display
$29$ \( T^{5} - 10 T^{4} - 15 T^{3} + \cdots - 1177 \) Copy content Toggle raw display
$31$ \( T^{5} + 22 T^{4} + 178 T^{3} + \cdots + 689 \) Copy content Toggle raw display
$37$ \( T^{5} + 13 T^{4} - 47 T^{3} + \cdots + 171 \) Copy content Toggle raw display
$41$ \( T^{5} - 10 T^{4} - 89 T^{3} + \cdots + 187 \) Copy content Toggle raw display
$43$ \( (T - 1)^{5} \) Copy content Toggle raw display
$47$ \( T^{5} + 37 T^{4} + 479 T^{3} + \cdots - 26021 \) Copy content Toggle raw display
$53$ \( T^{5} - 11 T^{4} - 81 T^{3} + \cdots + 2273 \) Copy content Toggle raw display
$59$ \( T^{5} + 13 T^{4} - 33 T^{3} + \cdots + 2239 \) Copy content Toggle raw display
$61$ \( T^{5} + 8 T^{4} - 5 T^{3} - 155 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$67$ \( T^{5} + 21 T^{4} + 6 T^{3} + \cdots - 22131 \) Copy content Toggle raw display
$71$ \( T^{5} + 3 T^{4} - 163 T^{3} + \cdots - 4675 \) Copy content Toggle raw display
$73$ \( T^{5} - 2 T^{4} - 209 T^{3} + \cdots - 15139 \) Copy content Toggle raw display
$79$ \( T^{5} + 2 T^{4} - 59 T^{3} + 112 T^{2} + \cdots - 521 \) Copy content Toggle raw display
$83$ \( T^{5} + 11 T^{4} - 9 T^{3} - 404 T^{2} + \cdots - 83 \) Copy content Toggle raw display
$89$ \( T^{5} - 14 T^{4} - 128 T^{3} + \cdots - 21589 \) Copy content Toggle raw display
$97$ \( T^{5} + 30 T^{4} + 211 T^{3} + \cdots - 26119 \) Copy content Toggle raw display
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