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Properties

Label 5635.2.a.bk
Level $5635$
Weight $2$
Character orbit 5635.a
Self dual yes
Analytic conductor $44.996$
Analytic rank $1$
Dimension $16$
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] = [5635,2,Mod(1,5635)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5635, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5635.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5635 = 5 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5635.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: \(44.9957015390\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 19 x^{14} + 36 x^{13} + 142 x^{12} - 248 x^{11} - 536 x^{10} + 820 x^{9} + 1097 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - 19 x^{14} + 36 x^{13} + 142 x^{12} - 248 x^{11} - 536 x^{10} + 820 x^{9} + 1097 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 348 \nu^{15} + 891 \nu^{14} + 5869 \nu^{13} - 14598 \nu^{12} - 37406 \nu^{11} + 87696 \nu^{10} + \cdots - 29328 ) / 8078 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 478 \nu^{15} + 4079 \nu^{14} + 7249 \nu^{13} - 76458 \nu^{12} - 38334 \nu^{11} + 548590 \nu^{10} + \cdots - 6022 ) / 8078 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 695 \nu^{15} - 2441 \nu^{14} - 11106 \nu^{13} + 43012 \nu^{12} + 62750 \nu^{11} - 288232 \nu^{10} + \cdots - 1874 ) / 8078 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 678 \nu^{15} + 1527 \nu^{14} + 12479 \nu^{13} - 27466 \nu^{12} - 87919 \nu^{11} + 187012 \nu^{10} + \cdots + 939 ) / 4039 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1551 \nu^{15} - 3797 \nu^{14} - 27028 \nu^{13} + 66942 \nu^{12} + 177230 \nu^{11} - 447398 \nu^{10} + \cdots - 486 ) / 8078 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1705 \nu^{15} + 753 \nu^{14} - 35498 \nu^{13} - 16988 \nu^{12} + 289930 \nu^{11} + 143878 \nu^{10} + \cdots - 25112 ) / 8078 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1965 \nu^{15} + 6494 \nu^{14} - 42297 \nu^{13} - 127530 \nu^{12} + 348332 \nu^{11} + 958860 \nu^{10} + \cdots - 47490 ) / 8078 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 370 \nu^{15} + 241 \nu^{14} + 7079 \nu^{13} - 3762 \nu^{12} - 52352 \nu^{11} + 21692 \nu^{10} + \cdots - 422 ) / 1154 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3175 \nu^{15} + 123 \nu^{14} - 59802 \nu^{13} - 10338 \nu^{12} + 427186 \nu^{11} + 128870 \nu^{10} + \cdots + 23286 ) / 8078 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 499 \nu^{15} + 49 \nu^{14} + 9558 \nu^{13} + 188 \nu^{12} - 70068 \nu^{11} - 9270 \nu^{10} + \cdots + 1480 ) / 1154 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4140 \nu^{15} + 6143 \nu^{14} + 75949 \nu^{13} - 107092 \nu^{12} - 531632 \nu^{11} + 712082 \nu^{10} + \cdots + 1516 ) / 8078 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 9843 \nu^{15} - 8175 \nu^{14} - 186510 \nu^{13} + 131490 \nu^{12} + 1361004 \nu^{11} - 784056 \nu^{10} + \cdots + 11978 ) / 8078 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} + 8\beta_{3} + \beta_{2} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{14} - 10 \beta_{13} - \beta_{12} - 11 \beta_{11} - 9 \beta_{10} - \beta_{9} - 9 \beta_{8} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} + 3 \beta_{14} - 13 \beta_{13} - 11 \beta_{12} - 11 \beta_{11} - 2 \beta_{10} - 11 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{15} + 77 \beta_{14} - 77 \beta_{13} - 14 \beta_{12} - 88 \beta_{11} - 64 \beta_{10} + \cdots + 401 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14 \beta_{15} + 45 \beta_{14} - 120 \beta_{13} - 90 \beta_{12} - 93 \beta_{11} - 29 \beta_{10} + \cdots + 51 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32 \beta_{15} + 545 \beta_{14} - 546 \beta_{13} - 137 \beta_{12} - 630 \beta_{11} - 424 \beta_{10} + \cdots + 2277 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 137 \beta_{15} + 459 \beta_{14} - 968 \beta_{13} - 666 \beta_{12} - 719 \beta_{11} - 289 \beta_{10} + \cdots + 571 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 336 \beta_{15} + 3723 \beta_{14} - 3747 \beta_{13} - 1154 \beta_{12} - 4303 \beta_{11} - 2741 \beta_{10} + \cdots + 13245 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1157 \beta_{15} + 3987 \beta_{14} - 7297 \beta_{13} - 4716 \beta_{12} - 5332 \beta_{11} - 2470 \beta_{10} + \cdots + 5321 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2949 \beta_{15} + 25009 \beta_{14} - 25350 \beta_{13} - 8978 \beta_{12} - 28762 \beta_{11} + \cdots + 78515 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 9036 \beta_{15} + 31806 \beta_{14} - 52919 \beta_{13} - 32658 \beta_{12} - 38624 \beta_{11} + \cdots + 44788 \) 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
2.58566
2.28572
1.91695
1.89217
1.80159
0.945293
0.702602
0.197422
−0.156103
−0.483494
−0.683219
−1.03543
−1.57995
−1.71435
−2.29584
−2.37902
−2.58566 0.477770 4.68561 1.00000 −1.23535 0 −6.94407 −2.77174 −2.58566
1.2 −2.28572 2.49219 3.22452 1.00000 −5.69645 0 −2.79890 3.21102 −2.28572
1.3 −1.91695 −2.45705 1.67470 1.00000 4.71004 0 0.623576 3.03708 −1.91695
1.4 −1.89217 −1.79093 1.58032 1.00000 3.38875 0 0.794111 0.207440 −1.89217
1.5 −1.80159 0.533115 1.24574 1.00000 −0.960456 0 1.35887 −2.71579 −1.80159
1.6 −0.945293 2.63255 −1.10642 1.00000 −2.48853 0 2.93648 3.93031 −0.945293
1.7 −0.702602 −1.41751 −1.50635 1.00000 0.995947 0 2.46357 −0.990659 −0.702602
1.8 −0.197422 −0.223910 −1.96102 1.00000 0.0442048 0 0.781993 −2.94986 −0.197422
1.9 0.156103 1.73443 −1.97563 1.00000 0.270749 0 −0.620606 0.00825121 0.156103
1.10 0.483494 2.57707 −1.76623 1.00000 1.24600 0 −1.82095 3.64131 0.483494
1.11 0.683219 −2.74899 −1.53321 1.00000 −1.87817 0 −2.41396 4.55696 0.683219
1.12 1.03543 −0.678266 −0.927878 1.00000 −0.702299 0 −3.03162 −2.53996 1.03543
1.13 1.57995 −2.88034 0.496228 1.00000 −4.55078 0 −2.37588 5.29636 1.57995
1.14 1.71435 1.00416 0.939008 1.00000 1.72148 0 −1.81892 −1.99166 1.71435
1.15 2.29584 −1.42849 3.27089 1.00000 −3.27959 0 2.91775 −0.959416 2.29584
1.16 2.37902 0.174203 3.65974 1.00000 0.414433 0 3.94855 −2.96965 2.37902
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 5635.2.a.bk ✓ 16
7.b odd 2 1 5635.2.a.bl yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5635.2.a.bk ✓ 16 1.a even 1 1 trivial
5635.2.a.bl yes 16 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):

\( T_{2}^{16} + 2 T_{2}^{15} - 19 T_{2}^{14} - 36 T_{2}^{13} + 142 T_{2}^{12} + 248 T_{2}^{11} - 536 T_{2}^{10} + \cdots + 4 \) Copy content Toggle raw display
\( T_{3}^{16} + 2 T_{3}^{15} - 25 T_{3}^{14} - 48 T_{3}^{13} + 240 T_{3}^{12} + 440 T_{3}^{11} - 1104 T_{3}^{10} + \cdots + 14 \) Copy content Toggle raw display
\( T_{11}^{16} - 2 T_{11}^{15} - 49 T_{11}^{14} + 44 T_{11}^{13} + 933 T_{11}^{12} + 30 T_{11}^{11} + \cdots + 658 \) Copy content Toggle raw display
\( T_{17}^{16} + 22 T_{17}^{15} + 149 T_{17}^{14} - 80 T_{17}^{13} - 5255 T_{17}^{12} - 17230 T_{17}^{11} + \cdots - 3742864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 14 \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} + \cdots + 658 \) Copy content Toggle raw display
$13$ \( T^{16} + 18 T^{15} + \cdots + 6967996 \) Copy content Toggle raw display
$17$ \( T^{16} + 22 T^{15} + \cdots - 3742864 \) Copy content Toggle raw display
$19$ \( T^{16} + 24 T^{15} + \cdots - 752864 \) Copy content Toggle raw display
$23$ \( (T - 1)^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 14 T^{15} + \cdots + 9834496 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 508748576 \) Copy content Toggle raw display
$37$ \( T^{16} - 8 T^{15} + \cdots - 38839312 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 19400060252 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 6606000352 \) Copy content Toggle raw display
$47$ \( T^{16} + 18 T^{15} + \cdots - 50847986 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3865848248 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 245968959712 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 11865610312 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 1659780110864 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 52936996864 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 162090160192 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 40938085502 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 130527575264 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1151863496 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 46505408610752 \) Copy content Toggle raw display
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