Properties

Label 1587.2.a.r
Level $1587$
Weight $2$
Character orbit 1587.a
Self dual yes
Analytic conductor $12.672$
Analytic rank $0$
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] = [1587,2,Mod(1,1587)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1587, 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("1587.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1587 = 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1587.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: \(12.6722588008\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 69)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) 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
0.284630
1.30972
1.91899
−1.68251
−0.830830
−2.51334 −1.00000 4.31686 4.05529 2.51334 1.37703 −5.82306 1.00000 −10.1923
1.2 0.236479 −1.00000 −1.94408 −1.00714 −0.236479 2.05529 −0.932691 1.00000 −0.238168
1.3 0.478891 −1.00000 −1.77066 3.37703 −0.478891 −4.53843 −1.80574 1.00000 1.61723
1.4 1.59435 −1.00000 0.541956 −2.53843 −1.59435 1.11325 −2.32463 1.00000 −4.04715
1.5 2.20362 −1.00000 2.85592 3.11325 −2.20362 −3.00714 1.88612 1.00000 6.86040
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(-1\)

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 1587.2.a.r 5
3.b odd 2 1 4761.2.a.bm 5
23.b odd 2 1 1587.2.a.q 5
23.d odd 22 2 69.2.e.b 10
69.c even 2 1 4761.2.a.bp 5
69.g even 22 2 207.2.i.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.b 10 23.d odd 22 2
207.2.i.a 10 69.g even 22 2
1587.2.a.q 5 23.b odd 2 1
1587.2.a.r 5 1.a even 1 1 trivial
4761.2.a.bm 5 3.b odd 2 1
4761.2.a.bp 5 69.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\):

\( T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 13T_{2}^{2} - 7T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{5} - 7T_{5}^{4} + 2T_{5}^{3} + 61T_{5}^{2} - 57T_{5} - 109 \) Copy content Toggle raw display
\( T_{7}^{5} + 3T_{7}^{4} - 14T_{7}^{3} - 15T_{7}^{2} + 67T_{7} - 43 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 5 T^{3} + 13 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 7 T^{4} + 2 T^{3} + 61 T^{2} + \cdots - 109 \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} - 14 T^{3} - 15 T^{2} + \cdots - 43 \) Copy content Toggle raw display
$11$ \( T^{5} - 2 T^{4} - 27 T^{3} + 13 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{5} + 7 T^{4} + 13 T^{3} + 5 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{5} - 16 T^{4} + 98 T^{3} + \cdots - 199 \) Copy content Toggle raw display
$19$ \( T^{5} + T^{4} - 37 T^{3} - 58 T^{2} + \cdots + 331 \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 18 T^{4} + 101 T^{3} + \cdots + 331 \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} - 67 T^{3} + 309 T^{2} + \cdots + 109 \) Copy content Toggle raw display
$37$ \( T^{5} - 26 T^{4} + 255 T^{3} + \cdots - 2047 \) Copy content Toggle raw display
$41$ \( T^{5} - 9 T^{4} - 93 T^{3} + \cdots - 14279 \) Copy content Toggle raw display
$43$ \( T^{5} - 22 T^{4} + 143 T^{3} + \cdots + 1199 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} - 148 T^{3} + \cdots - 13133 \) Copy content Toggle raw display
$53$ \( T^{5} - 35 T^{4} + 413 T^{3} + \cdots + 7481 \) Copy content Toggle raw display
$59$ \( T^{5} - 6 T^{4} - 199 T^{3} + \cdots + 21781 \) Copy content Toggle raw display
$61$ \( T^{5} + 7 T^{4} - 163 T^{3} + \cdots + 769 \) Copy content Toggle raw display
$67$ \( T^{5} - 5 T^{4} - 133 T^{3} + \cdots + 2507 \) Copy content Toggle raw display
$71$ \( T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 4663 \) Copy content Toggle raw display
$73$ \( T^{5} - 4 T^{4} - 196 T^{3} + \cdots + 241 \) Copy content Toggle raw display
$79$ \( T^{5} - 13 T^{4} - 115 T^{3} + \cdots + 5897 \) Copy content Toggle raw display
$83$ \( T^{5} - 24 T^{4} + 83 T^{3} + \cdots + 11903 \) Copy content Toggle raw display
$89$ \( T^{5} + 7 T^{4} - 97 T^{3} - 127 T^{2} + \cdots + 263 \) Copy content Toggle raw display
$97$ \( T^{5} - 6 T^{4} - 56 T^{3} + 197 T^{2} + \cdots + 199 \) Copy content Toggle raw display
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