Properties

Label 8450.2.a.bf
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 −0.732051 1.00000 0 0.732051 3.00000 −1.00000 −2.46410 0
1.2 −1.00000 2.73205 1.00000 0 −2.73205 3.00000 −1.00000 4.46410 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.bf 2
5.b even 2 1 1690.2.a.m 2
13.b even 2 1 8450.2.a.bm 2
13.f odd 12 2 650.2.m.a 4
65.d even 2 1 1690.2.a.j 2
65.g odd 4 2 1690.2.d.f 4
65.l even 6 2 1690.2.e.n 4
65.n even 6 2 1690.2.e.l 4
65.o even 12 2 650.2.n.b 4
65.s odd 12 2 130.2.l.a 4
65.s odd 12 2 1690.2.l.g 4
65.t even 12 2 650.2.n.a 4
195.bh even 12 2 1170.2.bs.c 4
260.bc even 12 2 1040.2.da.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 65.s odd 12 2
650.2.m.a 4 13.f odd 12 2
650.2.n.a 4 65.t even 12 2
650.2.n.b 4 65.o even 12 2
1040.2.da.a 4 260.bc even 12 2
1170.2.bs.c 4 195.bh even 12 2
1690.2.a.j 2 65.d even 2 1
1690.2.a.m 2 5.b even 2 1
1690.2.d.f 4 65.g odd 4 2
1690.2.e.l 4 65.n even 6 2
1690.2.e.n 4 65.l even 6 2
1690.2.l.g 4 65.s odd 12 2
8450.2.a.bf 2 1.a even 1 1 trivial
8450.2.a.bm 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{7} - 3 \)
\( T_{11} - 3 \)
\( T_{17}^{2} - 6 T_{17} - 18 \)
\( T_{31}^{2} - 6 T_{31} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( -18 - 6 T + T^{2} \)
$19$ \( -3 - 6 T + T^{2} \)
$23$ \( 24 - 12 T + T^{2} \)
$29$ \( 24 + 12 T + T^{2} \)
$31$ \( 6 - 6 T + T^{2} \)
$37$ \( 9 - 12 T + T^{2} \)
$41$ \( -108 + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( -3 - 6 T + T^{2} \)
$59$ \( -108 + T^{2} \)
$61$ \( -26 - 2 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( -66 - 6 T + T^{2} \)
$79$ \( -26 - 2 T + T^{2} \)
$83$ \( -18 - 6 T + T^{2} \)
$89$ \( 117 - 24 T + T^{2} \)
$97$ \( -138 - 6 T + T^{2} \)
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