Properties

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

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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: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + 5 q^{4} + q^{5} + \beta q^{6} + (\beta - 1) q^{7} + 3 \beta q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + 5 q^{4} + q^{5} + \beta q^{6} + (\beta - 1) q^{7} + 3 \beta q^{8} + q^{9} + \beta q^{10} + ( - \beta + 3) q^{11} + 5 q^{12} + ( - \beta + 3) q^{13} + ( - \beta + 7) q^{14} + q^{15} + 11 q^{16} - 4 q^{17} + \beta q^{18} + 5 q^{20} + (\beta - 1) q^{21} + (3 \beta - 7) q^{22} + (2 \beta + 4) q^{23} + 3 \beta q^{24} + q^{25} + (3 \beta - 7) q^{26} + q^{27} + (5 \beta - 5) q^{28} + ( - 3 \beta - 1) q^{29} + \beta q^{30} - 6 q^{31} + 5 \beta q^{32} + ( - \beta + 3) q^{33} - 4 \beta q^{34} + (\beta - 1) q^{35} + 5 q^{36} + ( - \beta - 1) q^{37} + ( - \beta + 3) q^{39} + 3 \beta q^{40} + (\beta + 7) q^{41} + ( - \beta + 7) q^{42} + (\beta + 3) q^{43} + ( - 5 \beta + 15) q^{44} + q^{45} + (4 \beta + 14) q^{46} + ( - 2 \beta + 4) q^{47} + 11 q^{48} + ( - 2 \beta + 1) q^{49} + \beta q^{50} - 4 q^{51} + ( - 5 \beta + 15) q^{52} + ( - 2 \beta - 6) q^{53} + \beta q^{54} + ( - \beta + 3) q^{55} + ( - 3 \beta + 21) q^{56} + ( - \beta - 21) q^{58} + ( - 2 \beta - 6) q^{59} + 5 q^{60} + ( - 2 \beta - 6) q^{61} - 6 \beta q^{62} + (\beta - 1) q^{63} + 13 q^{64} + ( - \beta + 3) q^{65} + (3 \beta - 7) q^{66} + (4 \beta - 4) q^{67} - 20 q^{68} + (2 \beta + 4) q^{69} + ( - \beta + 7) q^{70} + ( - 2 \beta - 2) q^{71} + 3 \beta q^{72} + 10 q^{73} + ( - \beta - 7) q^{74} + q^{75} + (4 \beta - 10) q^{77} + (3 \beta - 7) q^{78} + ( - 4 \beta + 4) q^{79} + 11 q^{80} + q^{81} + (7 \beta + 7) q^{82} + 6 q^{83} + (5 \beta - 5) q^{84} - 4 q^{85} + (3 \beta + 7) q^{86} + ( - 3 \beta - 1) q^{87} + (9 \beta - 21) q^{88} + (3 \beta + 9) q^{89} + \beta q^{90} + (4 \beta - 10) q^{91} + (10 \beta + 20) q^{92} - 6 q^{93} + (4 \beta - 14) q^{94} + 5 \beta q^{96} + (3 \beta - 5) q^{97} + (\beta - 14) q^{98} + ( - \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 10 q^{12} + 6 q^{13} + 14 q^{14} + 2 q^{15} + 22 q^{16} - 8 q^{17} + 10 q^{20} - 2 q^{21} - 14 q^{22} + 8 q^{23} + 2 q^{25} - 14 q^{26} + 2 q^{27} - 10 q^{28} - 2 q^{29} - 12 q^{31} + 6 q^{33} - 2 q^{35} + 10 q^{36} - 2 q^{37} + 6 q^{39} + 14 q^{41} + 14 q^{42} + 6 q^{43} + 30 q^{44} + 2 q^{45} + 28 q^{46} + 8 q^{47} + 22 q^{48} + 2 q^{49} - 8 q^{51} + 30 q^{52} - 12 q^{53} + 6 q^{55} + 42 q^{56} - 42 q^{58} - 12 q^{59} + 10 q^{60} - 12 q^{61} - 2 q^{63} + 26 q^{64} + 6 q^{65} - 14 q^{66} - 8 q^{67} - 40 q^{68} + 8 q^{69} + 14 q^{70} - 4 q^{71} + 20 q^{73} - 14 q^{74} + 2 q^{75} - 20 q^{77} - 14 q^{78} + 8 q^{79} + 22 q^{80} + 2 q^{81} + 14 q^{82} + 12 q^{83} - 10 q^{84} - 8 q^{85} + 14 q^{86} - 2 q^{87} - 42 q^{88} + 18 q^{89} - 20 q^{91} + 40 q^{92} - 12 q^{93} - 28 q^{94} - 10 q^{97} - 28 q^{98} + 6 q^{99}+O(q^{100}) \) 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
−2.64575
2.64575
−2.64575 1.00000 5.00000 1.00000 −2.64575 −3.64575 −7.93725 1.00000 −2.64575
1.2 2.64575 1.00000 5.00000 1.00000 2.64575 1.64575 7.93725 1.00000 2.64575
\(n\): 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.s 2
19.b odd 2 1 285.2.a.d 2
57.d even 2 1 855.2.a.g 2
76.d even 2 1 4560.2.a.bo 2
95.d odd 2 1 1425.2.a.p 2
95.g even 4 2 1425.2.c.i 4
285.b even 2 1 4275.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 19.b odd 2 1
855.2.a.g 2 57.d even 2 1
1425.2.a.p 2 95.d odd 2 1
1425.2.c.i 4 95.g even 4 2
4275.2.a.u 2 285.b even 2 1
4560.2.a.bo 2 76.d even 2 1
5415.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{2} - 7 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 6 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 62 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 6 \) Copy content Toggle raw display
$41$ \( T^{2} - 14T + 42 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 96 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 24 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 96 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T - 38 \) Copy content Toggle raw display
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