Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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.

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

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.bm 2
5.b even 2 1 1690.2.a.j 2
13.b even 2 1 8450.2.a.bf 2
13.f odd 12 2 650.2.m.a 4
65.d even 2 1 1690.2.a.m 2
65.g odd 4 2 1690.2.d.f 4
65.l even 6 2 1690.2.e.l 4
65.n even 6 2 1690.2.e.n 4
65.o even 12 2 650.2.n.a 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.b 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.o even 12 2
650.2.n.b 4 65.t 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 5.b even 2 1
1690.2.a.m 2 65.d even 2 1
1690.2.d.f 4 65.g odd 4 2
1690.2.e.l 4 65.l even 6 2
1690.2.e.n 4 65.n even 6 2
1690.2.l.g 4 65.s odd 12 2
8450.2.a.bf 2 13.b even 2 1
8450.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 18 \) Copy content Toggle raw display
\( T_{31}^{2} + 6T_{31} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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