Properties

Label 8450.2.a.bt
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{2} q^{6} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{2} q^{6} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} + ( -2 + 3 \beta_{1} ) q^{11} -\beta_{2} q^{12} + ( -2 + \beta_{1} + \beta_{2} ) q^{14} + q^{16} + ( 1 + \beta_{1} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{18} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{19} + ( 2 - 3 \beta_{2} ) q^{21} + ( 2 - 3 \beta_{1} ) q^{22} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{23} + \beta_{2} q^{24} + ( 2 + 2 \beta_{2} ) q^{27} + ( 2 - \beta_{1} - \beta_{2} ) q^{28} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -1 - 3 \beta_{1} - 4 \beta_{2} ) q^{31} - q^{32} + ( 3 - 3 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -1 - \beta_{1} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{36} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{38} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{41} + ( -2 + 3 \beta_{2} ) q^{42} + 2 q^{43} + ( -2 + 3 \beta_{1} ) q^{44} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{46} + ( 2 + \beta_{1} + \beta_{2} ) q^{47} -\beta_{2} q^{48} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 1 - \beta_{1} - \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -2 - 2 \beta_{2} ) q^{54} + ( -2 + \beta_{1} + \beta_{2} ) q^{56} + ( 1 + \beta_{1} - 6 \beta_{2} ) q^{57} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 1 + 3 \beta_{1} + 4 \beta_{2} ) q^{62} + ( 3 - 2 \beta_{2} ) q^{63} + q^{64} + ( -3 + 3 \beta_{1} - 2 \beta_{2} ) q^{66} + ( 4 - 2 \beta_{2} ) q^{67} + ( 1 + \beta_{1} ) q^{68} + ( 6 - 4 \beta_{1} ) q^{69} + ( 6 + 2 \beta_{1} - 4 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{2} ) q^{72} + ( 3 - 3 \beta_{1} - 6 \beta_{2} ) q^{73} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{74} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{76} + ( -7 + 2 \beta_{1} - \beta_{2} ) q^{77} + ( -3 - 5 \beta_{1} - 4 \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{82} + ( 2 - 2 \beta_{1} + 5 \beta_{2} ) q^{83} + ( 2 - 3 \beta_{2} ) q^{84} -2 q^{86} + ( -8 + 4 \beta_{1} + 6 \beta_{2} ) q^{87} + ( 2 - 3 \beta_{1} ) q^{88} + ( -2 - \beta_{1} + \beta_{2} ) q^{89} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{92} + ( 9 - \beta_{1} - 3 \beta_{2} ) q^{93} + ( -2 - \beta_{1} - \beta_{2} ) q^{94} + \beta_{2} q^{96} + ( 9 - \beta_{1} + 2 \beta_{2} ) q^{97} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{98} + ( -3 - 4 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 5 q^{7} - 3 q^{8} - q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} + 5 q^{7} - 3 q^{8} - q^{9} - 3 q^{11} - 5 q^{14} + 3 q^{16} + 4 q^{17} + q^{18} + 13 q^{19} + 6 q^{21} + 3 q^{22} + 6 q^{27} + 5 q^{28} - 14 q^{29} - 6 q^{31} - 3 q^{32} + 6 q^{33} - 4 q^{34} - q^{36} + 5 q^{37} - 13 q^{38} - 2 q^{41} - 6 q^{42} + 6 q^{43} - 3 q^{44} + 7 q^{47} - 2 q^{49} + 2 q^{51} + 9 q^{53} - 6 q^{54} - 5 q^{56} + 4 q^{57} + 14 q^{58} - 8 q^{59} + 4 q^{61} + 6 q^{62} + 9 q^{63} + 3 q^{64} - 6 q^{66} + 12 q^{67} + 4 q^{68} + 14 q^{69} + 20 q^{71} + q^{72} + 6 q^{73} - 5 q^{74} + 13 q^{76} - 19 q^{77} - 14 q^{79} - 13 q^{81} + 2 q^{82} + 4 q^{83} + 6 q^{84} - 6 q^{86} - 20 q^{87} + 3 q^{88} - 7 q^{89} + 26 q^{93} - 7 q^{94} + 26 q^{97} + 2 q^{98} - 13 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
−1.48119
2.17009
0.311108
−1.00000 −1.67513 1.00000 0 1.67513 1.80606 −1.00000 −0.193937 0
1.2 −1.00000 −0.539189 1.00000 0 0.539189 −0.709275 −1.00000 −2.70928 0
1.3 −1.00000 2.21432 1.00000 0 −2.21432 3.90321 −1.00000 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 8450.2.a.bt 3
5.b even 2 1 8450.2.a.ca 3
5.c odd 4 2 1690.2.b.b 6
13.b even 2 1 8450.2.a.cb 3
13.e even 6 2 650.2.e.j 6
65.d even 2 1 8450.2.a.bu 3
65.f even 4 2 1690.2.c.c 6
65.h odd 4 2 1690.2.b.c 6
65.k even 4 2 1690.2.c.b 6
65.l even 6 2 650.2.e.k 6
65.r odd 12 4 130.2.n.a 12
195.bf even 12 4 1170.2.bp.h 12
260.bg even 12 4 1040.2.dh.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.n.a 12 65.r odd 12 4
650.2.e.j 6 13.e even 6 2
650.2.e.k 6 65.l even 6 2
1040.2.dh.b 12 260.bg even 12 4
1170.2.bp.h 12 195.bf even 12 4
1690.2.b.b 6 5.c odd 4 2
1690.2.b.c 6 65.h odd 4 2
1690.2.c.b 6 65.k even 4 2
1690.2.c.c 6 65.f even 4 2
8450.2.a.bt 3 1.a even 1 1 trivial
8450.2.a.bu 3 65.d even 2 1
8450.2.a.ca 3 5.b even 2 1
8450.2.a.cb 3 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{3} - 4 T_{3} - 2 \)
\( T_{7}^{3} - 5 T_{7}^{2} + 3 T_{7} + 5 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 27 T_{11} - 31 \)
\( T_{17}^{3} - 4 T_{17}^{2} + 2 T_{17} + 2 \)
\( T_{31}^{3} + 6 T_{31}^{2} - 58 T_{31} - 218 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( -2 - 4 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 5 + 3 T - 5 T^{2} + T^{3} \)
$11$ \( -31 - 27 T + 3 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 2 + 2 T - 4 T^{2} + T^{3} \)
$19$ \( -5 + 43 T - 13 T^{2} + T^{3} \)
$23$ \( 76 - 40 T + T^{3} \)
$29$ \( -152 + 28 T + 14 T^{2} + T^{3} \)
$31$ \( -218 - 58 T + 6 T^{2} + T^{3} \)
$37$ \( 107 - 29 T - 5 T^{2} + T^{3} \)
$41$ \( -20 - 44 T + 2 T^{2} + T^{3} \)
$43$ \( ( -2 + T )^{3} \)
$47$ \( -1 + 11 T - 7 T^{2} + T^{3} \)
$53$ \( 13 - 7 T - 9 T^{2} + T^{3} \)
$59$ \( -272 - 32 T + 8 T^{2} + T^{3} \)
$61$ \( 610 - 108 T - 4 T^{2} + T^{3} \)
$67$ \( -16 + 32 T - 12 T^{2} + T^{3} \)
$71$ \( 464 + 40 T - 20 T^{2} + T^{3} \)
$73$ \( -270 - 126 T - 6 T^{2} + T^{3} \)
$79$ \( -158 - 42 T + 14 T^{2} + T^{3} \)
$83$ \( 46 - 128 T - 4 T^{2} + T^{3} \)
$89$ \( -19 + 7 T + 7 T^{2} + T^{3} \)
$97$ \( -466 + 202 T - 26 T^{2} + T^{3} \)
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