Properties

Label 3630.2.a.bt
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.43025.1
Defining polynomial: \(x^{4} - x^{3} - 21 x^{2} + 10 x + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 21 x^{2} + 10 x + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 11 \nu \)\()/10\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 10 \beta_{2} + 12 \beta_{1} + 11\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.48636
−2.45511
2.86832
4.07314
1.00000 1.00000 1.00000 1.00000 1.00000 −3.48636 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 −2.45511 1.00000 1.00000 1.00000
1.3 1.00000 1.00000 1.00000 1.00000 1.00000 2.86832 1.00000 1.00000 1.00000
1.4 1.00000 1.00000 1.00000 1.00000 1.00000 4.07314 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 3630.2.a.bt 4
11.b odd 2 1 3630.2.a.br 4
11.c even 5 2 330.2.m.e 8
33.h odd 10 2 990.2.n.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.e 8 11.c even 5 2
990.2.n.k 8 33.h odd 10 2
3630.2.a.br 4 11.b odd 2 1
3630.2.a.bt 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{4} - T_{7}^{3} - 21 T_{7}^{2} + 10 T_{7} + 100 \)
\( T_{13}^{4} - 4 T_{13}^{3} - 27 T_{13}^{2} + 112 T_{13} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 100 + 10 T - 21 T^{2} - T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -11 + 112 T - 27 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 16 - 100 T - 23 T^{2} + 5 T^{3} + T^{4} \)
$19$ \( -220 + 190 T - 19 T^{2} - 7 T^{3} + T^{4} \)
$23$ \( -605 + 330 T - 29 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( 100 + 10 T - 21 T^{2} - T^{3} + T^{4} \)
$31$ \( -220 + 190 T - 21 T^{2} - 9 T^{3} + T^{4} \)
$37$ \( ( 19 - 11 T + T^{2} )^{2} \)
$41$ \( 2596 + 158 T - 101 T^{2} - 3 T^{3} + T^{4} \)
$43$ \( 404 + 242 T - 107 T^{2} + T^{3} + T^{4} \)
$47$ \( 2651 + 152 T - 151 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( 596 + 240 T - 87 T^{2} - 5 T^{3} + T^{4} \)
$59$ \( -55 + 760 T - 31 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 704 - 432 T - 12 T^{2} + 14 T^{3} + T^{4} \)
$67$ \( -26476 + 3662 T + 83 T^{2} - 29 T^{3} + T^{4} \)
$71$ \( 1600 - 160 T - 116 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( ( 6 + T )^{4} \)
$79$ \( 5620 - 140 T - 209 T^{2} - 3 T^{3} + T^{4} \)
$83$ \( ( 16 + 12 T + T^{2} )^{2} \)
$89$ \( 500 + 350 T - 145 T^{2} - 5 T^{3} + T^{4} \)
$97$ \( -5104 + 520 T + 192 T^{2} - 30 T^{3} + T^{4} \)
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