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Properties

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

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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: \( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 4\nu^{5} + 25\nu^{4} - 2\nu^{3} - 40\nu^{2} + 15\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} + 4\nu^{6} - 25\nu^{5} + 2\nu^{4} + 40\nu^{3} - 15\nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 4\nu^{6} + 25\nu^{5} - 2\nu^{4} - 40\nu^{3} + 17\nu^{2} + 6\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + 6\nu^{7} - 4\nu^{6} - 31\nu^{5} + 44\nu^{4} + 34\nu^{3} - 69\nu^{2} + 16\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 8\nu^{7} + 14\nu^{6} + 33\nu^{5} - 106\nu^{4} + 2\nu^{3} + 169\nu^{2} - 84\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 22\nu^{6} + 17\nu^{5} + 106\nu^{4} - 94\nu^{3} - 147\nu^{2} + 128\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 5\nu^{7} + 28\nu^{5} - 23\nu^{4} - 36\nu^{3} + 40\nu^{2} - 8\nu + 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{5} + 2\beta_{4} + 3\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + \beta_{7} + \beta_{6} + 3\beta_{5} + 8\beta_{4} + 12\beta_{3} + 13\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} + 4\beta_{7} + 4\beta_{6} + 14\beta_{5} + 21\beta_{4} + 37\beta_{3} - 2\beta_{2} + 48\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41\beta_{8} + 18\beta_{7} + 20\beta_{6} + 45\beta_{5} + 69\beta_{4} + 125\beta_{3} - 4\beta_{2} + 135\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 143 \beta_{8} + 63 \beta_{7} + 71 \beta_{6} + 163 \beta_{5} + 204 \beta_{4} + 394 \beta_{3} - 22 \beta_{2} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 457 \beta_{8} + 226 \beta_{7} + 266 \beta_{6} + 528 \beta_{5} + 648 \beta_{4} + 1278 \beta_{3} + \cdots + 140 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.19178
2.21072
1.69494
1.14438
0.822927
−0.163909
−1.53422
−1.58860
−1.77801
−2.19178 1.00000 2.80388 −2.39677 −2.19178 1.35191 −1.76193 1.00000 5.25319
1.2 −1.21072 1.00000 −0.534161 1.79844 −1.21072 4.27693 3.06816 1.00000 −2.17740
1.3 −0.694939 1.00000 −1.51706 −2.01780 −0.694939 −2.39488 2.44414 1.00000 1.40225
1.4 −0.144378 1.00000 −1.97916 1.55424 −0.144378 −1.52752 0.574502 1.00000 −0.224397
1.5 0.177073 1.00000 −1.96865 −3.45143 0.177073 −3.79521 −0.702739 1.00000 −0.611154
1.6 1.16391 1.00000 −0.645316 3.11042 1.16391 1.74624 −3.07891 1.00000 3.62025
1.7 2.53422 1.00000 4.42225 1.08952 2.53422 1.73057 6.13851 1.00000 2.76107
1.8 2.58860 1.00000 4.70087 −1.14894 2.58860 2.76526 6.99148 1.00000 −2.97416
1.9 2.77801 1.00000 5.71733 −2.53766 2.77801 0.846706 10.3268 1.00000 −7.04965
\(n\): 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.r 9
3.b odd 2 1 7569.2.a.bj 9
29.b even 2 1 2523.2.a.o 9
29.d even 7 2 87.2.g.a ✓ 18
87.d odd 2 1 7569.2.a.bm 9
87.j odd 14 2 261.2.k.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.g.a ✓ 18 29.d even 7 2
261.2.k.c 18 87.j odd 14 2
2523.2.a.o 9 29.b even 2 1
2523.2.a.r 9 1.a even 1 1 trivial
7569.2.a.bj 9 3.b odd 2 1
7569.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2523))\):

\( T_{2}^{9} - 5T_{2}^{8} - 2T_{2}^{7} + 37T_{2}^{6} - 20T_{2}^{5} - 71T_{2}^{4} + 31T_{2}^{3} + 40T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{9} + 4T_{5}^{8} - 15T_{5}^{7} - 68T_{5}^{6} + 58T_{5}^{5} + 353T_{5}^{4} - 67T_{5}^{3} - 711T_{5}^{2} + 13T_{5} + 461 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 5 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots + 461 \) Copy content Toggle raw display
$7$ \( T^{9} - 5 T^{8} + \cdots + 568 \) Copy content Toggle raw display
$11$ \( T^{9} + T^{8} + \cdots + 1112 \) Copy content Toggle raw display
$13$ \( T^{9} - T^{8} + \cdots - 46697 \) Copy content Toggle raw display
$17$ \( T^{9} - 2 T^{8} + \cdots - 86143 \) Copy content Toggle raw display
$19$ \( T^{9} - 9 T^{8} + \cdots - 767768 \) Copy content Toggle raw display
$23$ \( T^{9} + 4 T^{8} + \cdots + 29672 \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( T^{9} - 8 T^{8} + \cdots + 13336 \) Copy content Toggle raw display
$37$ \( T^{9} - 27 T^{8} + \cdots - 137297 \) Copy content Toggle raw display
$41$ \( T^{9} - 12 T^{8} + \cdots - 3560291 \) Copy content Toggle raw display
$43$ \( T^{9} - 16 T^{8} + \cdots - 16472 \) Copy content Toggle raw display
$47$ \( T^{9} + 8 T^{8} + \cdots + 904 \) Copy content Toggle raw display
$53$ \( T^{9} + 8 T^{8} + \cdots + 2305981 \) Copy content Toggle raw display
$59$ \( T^{9} + 16 T^{8} + \cdots + 897224 \) Copy content Toggle raw display
$61$ \( T^{9} - 21 T^{8} + \cdots + 722639 \) Copy content Toggle raw display
$67$ \( T^{9} - 3 T^{8} + \cdots - 45704 \) Copy content Toggle raw display
$71$ \( T^{9} + 33 T^{8} + \cdots + 89667752 \) Copy content Toggle raw display
$73$ \( T^{9} - 3 T^{8} + \cdots + 247283 \) Copy content Toggle raw display
$79$ \( T^{9} - 3 T^{8} + \cdots + 93512 \) Copy content Toggle raw display
$83$ \( T^{9} - 13 T^{8} + \cdots + 803656 \) Copy content Toggle raw display
$89$ \( T^{9} - 6 T^{8} + \cdots + 46711771 \) Copy content Toggle raw display
$97$ \( T^{9} - 4 T^{8} + \cdots + 6388081 \) Copy content Toggle raw display
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