Properties

Label 8450.2.a.bl
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 650)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + ( 3 - \beta ) q^{7} + q^{8} + \beta q^{9} +O(q^{10})\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + ( 3 - \beta ) q^{7} + q^{8} + \beta q^{9} + ( 3 - \beta ) q^{11} + \beta q^{12} + ( 3 - \beta ) q^{14} + q^{16} + ( 4 - 3 \beta ) q^{17} + \beta q^{18} + ( 5 - 2 \beta ) q^{19} + ( -3 + 2 \beta ) q^{21} + ( 3 - \beta ) q^{22} + ( 1 + 2 \beta ) q^{23} + \beta q^{24} + ( 3 - 2 \beta ) q^{27} + ( 3 - \beta ) q^{28} + ( 1 + \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + q^{32} + ( -3 + 2 \beta ) q^{33} + ( 4 - 3 \beta ) q^{34} + \beta q^{36} + ( -6 + \beta ) q^{37} + ( 5 - 2 \beta ) q^{38} - q^{41} + ( -3 + 2 \beta ) q^{42} + ( 3 - 5 \beta ) q^{43} + ( 3 - \beta ) q^{44} + ( 1 + 2 \beta ) q^{46} + ( 8 + \beta ) q^{47} + \beta q^{48} + ( 5 - 5 \beta ) q^{49} + ( -9 + \beta ) q^{51} + ( 3 + 4 \beta ) q^{53} + ( 3 - 2 \beta ) q^{54} + ( 3 - \beta ) q^{56} + ( -6 + 3 \beta ) q^{57} + ( 1 + \beta ) q^{58} + ( 1 + 4 \beta ) q^{59} + ( -5 + 4 \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + ( -3 + 2 \beta ) q^{63} + q^{64} + ( -3 + 2 \beta ) q^{66} + ( 1 + 2 \beta ) q^{67} + ( 4 - 3 \beta ) q^{68} + ( 6 + 3 \beta ) q^{69} + ( 6 + 2 \beta ) q^{71} + \beta q^{72} -8 q^{73} + ( -6 + \beta ) q^{74} + ( 5 - 2 \beta ) q^{76} + ( 12 - 5 \beta ) q^{77} + ( 7 + \beta ) q^{79} + ( -6 - 2 \beta ) q^{81} - q^{82} + ( 3 + 6 \beta ) q^{83} + ( -3 + 2 \beta ) q^{84} + ( 3 - 5 \beta ) q^{86} + ( 3 + 2 \beta ) q^{87} + ( 3 - \beta ) q^{88} + ( -7 + 3 \beta ) q^{89} + ( 1 + 2 \beta ) q^{92} + ( 6 + 6 \beta ) q^{93} + ( 8 + \beta ) q^{94} + \beta q^{96} + ( -14 + \beta ) q^{97} + ( 5 - 5 \beta ) q^{98} + ( -3 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 5 q^{7} + 2 q^{8} + q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 5 q^{7} + 2 q^{8} + q^{9} + 5 q^{11} + q^{12} + 5 q^{14} + 2 q^{16} + 5 q^{17} + q^{18} + 8 q^{19} - 4 q^{21} + 5 q^{22} + 4 q^{23} + q^{24} + 4 q^{27} + 5 q^{28} + 3 q^{29} + 10 q^{31} + 2 q^{32} - 4 q^{33} + 5 q^{34} + q^{36} - 11 q^{37} + 8 q^{38} - 2 q^{41} - 4 q^{42} + q^{43} + 5 q^{44} + 4 q^{46} + 17 q^{47} + q^{48} + 5 q^{49} - 17 q^{51} + 10 q^{53} + 4 q^{54} + 5 q^{56} - 9 q^{57} + 3 q^{58} + 6 q^{59} - 6 q^{61} + 10 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{66} + 4 q^{67} + 5 q^{68} + 15 q^{69} + 14 q^{71} + q^{72} - 16 q^{73} - 11 q^{74} + 8 q^{76} + 19 q^{77} + 15 q^{79} - 14 q^{81} - 2 q^{82} + 12 q^{83} - 4 q^{84} + q^{86} + 8 q^{87} + 5 q^{88} - 11 q^{89} + 4 q^{92} + 18 q^{93} + 17 q^{94} + q^{96} - 27 q^{97} + 5 q^{98} - 4 q^{99} + O(q^{100}) \)

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
−1.30278
2.30278
1.00000 −1.30278 1.00000 0 −1.30278 4.30278 1.00000 −1.30278 0
1.2 1.00000 2.30278 1.00000 0 2.30278 0.697224 1.00000 2.30278 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.bl 2
5.b even 2 1 8450.2.a.ba 2
13.b even 2 1 8450.2.a.bd 2
13.e even 6 2 650.2.e.g yes 4
65.d even 2 1 8450.2.a.bi 2
65.l even 6 2 650.2.e.e 4
65.r odd 12 4 650.2.o.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.e.e 4 65.l even 6 2
650.2.e.g yes 4 13.e even 6 2
650.2.o.h 8 65.r odd 12 4
8450.2.a.ba 2 5.b even 2 1
8450.2.a.bd 2 13.b even 2 1
8450.2.a.bi 2 65.d even 2 1
8450.2.a.bl 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{2} - T_{3} - 3 \)
\( T_{7}^{2} - 5 T_{7} + 3 \)
\( T_{11}^{2} - 5 T_{11} + 3 \)
\( T_{17}^{2} - 5 T_{17} - 23 \)
\( T_{31}^{2} - 10 T_{31} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -3 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 - 5 T + T^{2} \)
$11$ \( 3 - 5 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -23 - 5 T + T^{2} \)
$19$ \( 3 - 8 T + T^{2} \)
$23$ \( -9 - 4 T + T^{2} \)
$29$ \( -1 - 3 T + T^{2} \)
$31$ \( 12 - 10 T + T^{2} \)
$37$ \( 27 + 11 T + T^{2} \)
$41$ \( ( 1 + T )^{2} \)
$43$ \( -81 - T + T^{2} \)
$47$ \( 69 - 17 T + T^{2} \)
$53$ \( -27 - 10 T + T^{2} \)
$59$ \( -43 - 6 T + T^{2} \)
$61$ \( -43 + 6 T + T^{2} \)
$67$ \( -9 - 4 T + T^{2} \)
$71$ \( 36 - 14 T + T^{2} \)
$73$ \( ( 8 + T )^{2} \)
$79$ \( 53 - 15 T + T^{2} \)
$83$ \( -81 - 12 T + T^{2} \)
$89$ \( 1 + 11 T + T^{2} \)
$97$ \( 179 + 27 T + T^{2} \)
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