Properties

Label 5415.2.a.n
Level $5415$
Weight $2$
Character orbit 5415.a
Self dual yes
Analytic conductor $43.239$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 1 - \beta ) q^{6} + \beta q^{7} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 1 - \beta ) q^{6} + \beta q^{7} + ( -3 + \beta ) q^{8} + q^{9} + ( 1 - \beta ) q^{10} + ( 2 + 3 \beta ) q^{11} + ( -1 + 2 \beta ) q^{12} + ( 2 - \beta ) q^{13} + ( 2 - \beta ) q^{14} + q^{15} + 3 q^{16} + ( 4 + 2 \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( -1 + 2 \beta ) q^{20} -\beta q^{21} + ( 4 - \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + ( 3 - \beta ) q^{24} + q^{25} + ( -4 + 3 \beta ) q^{26} - q^{27} + ( -4 + \beta ) q^{28} + \beta q^{29} + ( -1 + \beta ) q^{30} + ( 6 + 2 \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( -2 - 3 \beta ) q^{33} + 2 \beta q^{34} -\beta q^{35} + ( 1 - 2 \beta ) q^{36} + ( 2 - \beta ) q^{37} + ( -2 + \beta ) q^{39} + ( 3 - \beta ) q^{40} + ( -4 - 3 \beta ) q^{41} + ( -2 + \beta ) q^{42} + ( 8 - 3 \beta ) q^{43} + ( -10 - \beta ) q^{44} - q^{45} + ( -10 + 6 \beta ) q^{46} + ( -2 + 4 \beta ) q^{47} -3 q^{48} -5 q^{49} + ( -1 + \beta ) q^{50} + ( -4 - 2 \beta ) q^{51} + ( 6 - 5 \beta ) q^{52} -8 q^{53} + ( 1 - \beta ) q^{54} + ( -2 - 3 \beta ) q^{55} + ( 2 - 3 \beta ) q^{56} + ( 2 - \beta ) q^{58} + ( -4 + 6 \beta ) q^{59} + ( 1 - 2 \beta ) q^{60} + ( -4 - 8 \beta ) q^{61} + ( -2 + 4 \beta ) q^{62} + \beta q^{63} + ( -7 + 2 \beta ) q^{64} + ( -2 + \beta ) q^{65} + ( -4 + \beta ) q^{66} + ( 4 - 4 \beta ) q^{67} + ( -4 - 6 \beta ) q^{68} + ( -2 + 4 \beta ) q^{69} + ( -2 + \beta ) q^{70} + ( 8 - 2 \beta ) q^{71} + ( -3 + \beta ) q^{72} + ( -2 + 4 \beta ) q^{73} + ( -4 + 3 \beta ) q^{74} - q^{75} + ( 6 + 2 \beta ) q^{77} + ( 4 - 3 \beta ) q^{78} -3 q^{80} + q^{81} + ( -2 - \beta ) q^{82} + ( 10 - 2 \beta ) q^{83} + ( 4 - \beta ) q^{84} + ( -4 - 2 \beta ) q^{85} + ( -14 + 11 \beta ) q^{86} -\beta q^{87} -7 \beta q^{88} + ( 4 + 7 \beta ) q^{89} + ( 1 - \beta ) q^{90} + ( -2 + 2 \beta ) q^{91} + ( 18 - 8 \beta ) q^{92} + ( -6 - 2 \beta ) q^{93} + ( 10 - 6 \beta ) q^{94} + ( -3 - \beta ) q^{96} + ( 14 + 3 \beta ) q^{97} + ( 5 - 5 \beta ) q^{98} + ( 2 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + 2q^{10} + 4q^{11} - 2q^{12} + 4q^{13} + 4q^{14} + 2q^{15} + 6q^{16} + 8q^{17} - 2q^{18} - 2q^{20} + 8q^{22} + 4q^{23} + 6q^{24} + 2q^{25} - 8q^{26} - 2q^{27} - 8q^{28} - 2q^{30} + 12q^{31} + 6q^{32} - 4q^{33} + 2q^{36} + 4q^{37} - 4q^{39} + 6q^{40} - 8q^{41} - 4q^{42} + 16q^{43} - 20q^{44} - 2q^{45} - 20q^{46} - 4q^{47} - 6q^{48} - 10q^{49} - 2q^{50} - 8q^{51} + 12q^{52} - 16q^{53} + 2q^{54} - 4q^{55} + 4q^{56} + 4q^{58} - 8q^{59} + 2q^{60} - 8q^{61} - 4q^{62} - 14q^{64} - 4q^{65} - 8q^{66} + 8q^{67} - 8q^{68} - 4q^{69} - 4q^{70} + 16q^{71} - 6q^{72} - 4q^{73} - 8q^{74} - 2q^{75} + 12q^{77} + 8q^{78} - 6q^{80} + 2q^{81} - 4q^{82} + 20q^{83} + 8q^{84} - 8q^{85} - 28q^{86} + 8q^{89} + 2q^{90} - 4q^{91} + 36q^{92} - 12q^{93} + 20q^{94} - 6q^{96} + 28q^{97} + 10q^{98} + 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 −1.00000 3.82843 −1.00000 2.41421 −1.41421 −4.41421 1.00000 2.41421
1.2 0.414214 −1.00000 −1.82843 −1.00000 −0.414214 1.41421 −1.58579 1.00000 −0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(19\) \(-1\)

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 5415.2.a.n 2
19.b odd 2 1 285.2.a.g 2
57.d even 2 1 855.2.a.d 2
76.d even 2 1 4560.2.a.bf 2
95.d odd 2 1 1425.2.a.k 2
95.g even 4 2 1425.2.c.l 4
285.b even 2 1 4275.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 19.b odd 2 1
855.2.a.d 2 57.d even 2 1
1425.2.a.k 2 95.d odd 2 1
1425.2.c.l 4 95.g even 4 2
4275.2.a.y 2 285.b even 2 1
4560.2.a.bf 2 76.d even 2 1
5415.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{7}^{2} - 2 \)
\( T_{11}^{2} - 4 T_{11} - 14 \)
\( T_{13}^{2} - 4 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -14 - 4 T + T^{2} \)
$13$ \( 2 - 4 T + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -28 - 4 T + T^{2} \)
$29$ \( -2 + T^{2} \)
$31$ \( 28 - 12 T + T^{2} \)
$37$ \( 2 - 4 T + T^{2} \)
$41$ \( -2 + 8 T + T^{2} \)
$43$ \( 46 - 16 T + T^{2} \)
$47$ \( -28 + 4 T + T^{2} \)
$53$ \( ( 8 + T )^{2} \)
$59$ \( -56 + 8 T + T^{2} \)
$61$ \( -112 + 8 T + T^{2} \)
$67$ \( -16 - 8 T + T^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 92 - 20 T + T^{2} \)
$89$ \( -82 - 8 T + T^{2} \)
$97$ \( 178 - 28 T + T^{2} \)
show more
show less