Properties

Label 2523.2.a.q
Level $2523$
Weight $2$
Character orbit 2523.a
Self dual yes
Analytic conductor $20.146$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2523,2,Mod(1,2523)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2523, 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("2523.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2523.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: \(20.1462564300\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 33\nu^{4} - 44\nu^{3} - 37\nu^{2} + 33\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 9\nu^{6} - 28\nu^{5} - 25\nu^{4} + 79\nu^{3} + 20\nu^{2} - 61\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{8} - 8\nu^{7} - 26\nu^{6} + 69\nu^{5} + 67\nu^{4} - 178\nu^{3} - 41\nu^{2} + 125\nu - 15 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{8} + 16\nu^{7} + 42\nu^{6} - 143\nu^{5} - 109\nu^{4} + 382\nu^{3} + 83\nu^{2} - 267\nu + 21 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -9\nu^{8} + 26\nu^{7} + 80\nu^{6} - 233\nu^{5} - 221\nu^{4} + 624\nu^{3} + 177\nu^{2} - 447\nu + 35 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - \beta_{7} + \beta_{6} - 4\beta_{5} + \beta_{4} + 8\beta_{3} + 9\beta_{2} + 19\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} - 10\beta_{7} + 4\beta_{6} - 14\beta_{5} + 9\beta_{4} + 12\beta_{3} + 36\beta_{2} + 21\beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{8} - 13\beta_{7} + 15\beta_{6} - 48\beta_{5} + 14\beta_{4} + 57\beta_{3} + 69\beta_{2} + 102\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105 \beta_{8} - 76 \beta_{7} + 53 \beta_{6} - 135 \beta_{5} + 70 \beta_{4} + 109 \beta_{3} + 228 \beta_{2} + \cdots + 240 \) 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
−2.07337
−1.83470
−1.79470
−1.15439
0.0831558
0.915678
1.98078
2.19659
2.68096
−2.07337 1.00000 2.29887 1.89543 −2.07337 −4.71271 −0.619659 1.00000 −3.92993
1.2 −1.83470 1.00000 1.36614 −2.25851 −1.83470 −0.744275 1.16295 1.00000 4.14369
1.3 −1.79470 1.00000 1.22096 1.34531 −1.79470 −1.11399 1.39815 1.00000 −2.41444
1.4 −1.15439 1.00000 −0.667390 −3.44957 −1.15439 4.51754 3.07920 1.00000 3.98215
1.5 0.0831558 1.00000 −1.99309 1.93038 0.0831558 0.343925 −0.332048 1.00000 0.160522
1.6 0.915678 1.00000 −1.16153 0.285007 0.915678 5.03918 −2.89495 1.00000 0.260975
1.7 1.98078 1.00000 1.92348 1.21902 1.98078 0.393238 −0.151564 1.00000 2.41460
1.8 2.19659 1.00000 2.82501 −4.07769 2.19659 3.64733 1.81221 1.00000 −8.95700
1.9 2.68096 1.00000 5.18756 3.11062 2.68096 −2.37023 8.54571 1.00000 8.33945
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(29\) \( +1 \)

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 2523.2.a.q 9
3.b odd 2 1 7569.2.a.bk 9
29.b even 2 1 2523.2.a.p 9
29.e even 14 2 87.2.g.b 18
87.d odd 2 1 7569.2.a.bl 9
87.h odd 14 2 261.2.k.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.g.b 18 29.e even 14 2
261.2.k.b 18 87.h odd 14 2
2523.2.a.p 9 29.b even 2 1
2523.2.a.q 9 1.a even 1 1 trivial
7569.2.a.bk 9 3.b odd 2 1
7569.2.a.bl 9 87.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2523))\):

\( T_{2}^{9} - T_{2}^{8} - 14T_{2}^{7} + 9T_{2}^{6} + 70T_{2}^{5} - 23T_{2}^{4} - 141T_{2}^{3} + 14T_{2}^{2} + 84T_{2} - 7 \) Copy content Toggle raw display
\( T_{5}^{9} - 27T_{5}^{7} + 24T_{5}^{6} + 222T_{5}^{5} - 383T_{5}^{4} - 383T_{5}^{3} + 1293T_{5}^{2} - 923T_{5} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - T^{8} - 14 T^{7} + \cdots - 7 \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 27 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$7$ \( T^{9} - 5 T^{8} + \cdots - 104 \) Copy content Toggle raw display
$11$ \( T^{9} - 3 T^{8} + \cdots + 7496 \) Copy content Toggle raw display
$13$ \( T^{9} - 5 T^{8} + \cdots - 841 \) Copy content Toggle raw display
$17$ \( T^{9} - 16 T^{8} + \cdots + 271727 \) Copy content Toggle raw display
$19$ \( T^{9} + T^{8} - 27 T^{7} + \cdots - 8 \) Copy content Toggle raw display
$23$ \( T^{9} + 10 T^{8} + \cdots + 13448 \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( T^{9} + 4 T^{8} + \cdots - 465592 \) Copy content Toggle raw display
$37$ \( T^{9} - 25 T^{8} + \cdots + 111257 \) Copy content Toggle raw display
$41$ \( T^{9} - 34 T^{8} + \cdots - 583681 \) Copy content Toggle raw display
$43$ \( T^{9} - 12 T^{8} + \cdots + 7736 \) Copy content Toggle raw display
$47$ \( T^{9} - 8 T^{8} + \cdots + 7004024 \) Copy content Toggle raw display
$53$ \( T^{9} + 32 T^{8} + \cdots - 1284907 \) Copy content Toggle raw display
$59$ \( T^{9} - 10 T^{8} + \cdots - 56161784 \) Copy content Toggle raw display
$61$ \( T^{9} - 51 T^{8} + \cdots - 642691 \) Copy content Toggle raw display
$67$ \( T^{9} - 7 T^{8} + \cdots + 51076472 \) Copy content Toggle raw display
$71$ \( T^{9} - 7 T^{8} + \cdots + 661256 \) Copy content Toggle raw display
$73$ \( T^{9} - 17 T^{8} + \cdots + 1284473 \) Copy content Toggle raw display
$79$ \( T^{9} - 13 T^{8} + \cdots - 31304 \) Copy content Toggle raw display
$83$ \( T^{9} + 31 T^{8} + \cdots - 4705112 \) Copy content Toggle raw display
$89$ \( T^{9} + 32 T^{8} + \cdots - 237040103 \) Copy content Toggle raw display
$97$ \( T^{9} - 16 T^{8} + \cdots - 18124457 \) Copy content Toggle raw display
show more
show less