Properties

Label 5915.2.a.l
Level $5915$
Weight $2$
Character orbit 5915.a
Self dual yes
Analytic conductor $47.232$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.2315127956\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 2) q^{4} - q^{5} + 4 q^{6} + q^{7} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 2) q^{4} - q^{5} + 4 q^{6} + q^{7} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9} - \beta q^{10} + (\beta - 1) q^{11} + (2 \beta + 2) q^{12} + \beta q^{14} + ( - \beta + 1) q^{15} + 3 \beta q^{16} + (\beta - 3) q^{17} + (\beta - 4) q^{18} + (2 \beta + 2) q^{19} + ( - \beta - 2) q^{20} + (\beta - 1) q^{21} + 4 q^{22} + (2 \beta - 2) q^{23} + 4 \beta q^{24} + q^{25} + ( - \beta - 3) q^{27} + (\beta + 2) q^{28} + (3 \beta - 1) q^{29} - 4 q^{30} + (\beta + 4) q^{32} + ( - \beta + 5) q^{33} + ( - 2 \beta + 4) q^{34} - q^{35} - \beta q^{36} - 6 q^{37} + (4 \beta + 8) q^{38} + ( - \beta - 4) q^{40} - 2 \beta q^{41} + 4 q^{42} + ( - 2 \beta + 6) q^{43} + (2 \beta + 2) q^{44} + (\beta - 2) q^{45} + 8 q^{46} + (3 \beta + 1) q^{47} + 12 q^{48} + q^{49} + \beta q^{50} + ( - 3 \beta + 7) q^{51} - 2 \beta q^{53} + ( - 4 \beta - 4) q^{54} + ( - \beta + 1) q^{55} + (\beta + 4) q^{56} + (2 \beta + 6) q^{57} + (2 \beta + 12) q^{58} + 4 q^{59} + ( - 2 \beta - 2) q^{60} + 6 \beta q^{61} + ( - \beta + 2) q^{63} + ( - \beta + 4) q^{64} + (4 \beta - 4) q^{66} - 4 \beta q^{67} - 2 q^{68} + ( - 2 \beta + 10) q^{69} - \beta q^{70} - 8 q^{71} + ( - 3 \beta + 4) q^{72} + (4 \beta + 2) q^{73} - 6 \beta q^{74} + (\beta - 1) q^{75} + (8 \beta + 12) q^{76} + (\beta - 1) q^{77} + (\beta - 5) q^{79} - 3 \beta q^{80} - 7 q^{81} + ( - 2 \beta - 8) q^{82} - 4 q^{83} + (2 \beta + 2) q^{84} + ( - \beta + 3) q^{85} + (4 \beta - 8) q^{86} + ( - \beta + 13) q^{87} + 4 \beta q^{88} + (2 \beta - 4) q^{89} + ( - \beta + 4) q^{90} + (4 \beta + 4) q^{92} + (4 \beta + 12) q^{94} + ( - 2 \beta - 2) q^{95} + 4 \beta q^{96} + ( - 5 \beta + 7) q^{97} + \beta q^{98} + (2 \beta - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} + 5 q^{4} - 2 q^{5} + 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} + 5 q^{4} - 2 q^{5} + 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - q^{10} - q^{11} + 6 q^{12} + q^{14} + q^{15} + 3 q^{16} - 5 q^{17} - 7 q^{18} + 6 q^{19} - 5 q^{20} - q^{21} + 8 q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{25} - 7 q^{27} + 5 q^{28} + q^{29} - 8 q^{30} + 9 q^{32} + 9 q^{33} + 6 q^{34} - 2 q^{35} - q^{36} - 12 q^{37} + 20 q^{38} - 9 q^{40} - 2 q^{41} + 8 q^{42} + 10 q^{43} + 6 q^{44} - 3 q^{45} + 16 q^{46} + 5 q^{47} + 24 q^{48} + 2 q^{49} + q^{50} + 11 q^{51} - 2 q^{53} - 12 q^{54} + q^{55} + 9 q^{56} + 14 q^{57} + 26 q^{58} + 8 q^{59} - 6 q^{60} + 6 q^{61} + 3 q^{63} + 7 q^{64} - 4 q^{66} - 4 q^{67} - 4 q^{68} + 18 q^{69} - q^{70} - 16 q^{71} + 5 q^{72} + 8 q^{73} - 6 q^{74} - q^{75} + 32 q^{76} - q^{77} - 9 q^{79} - 3 q^{80} - 14 q^{81} - 18 q^{82} - 8 q^{83} + 6 q^{84} + 5 q^{85} - 12 q^{86} + 25 q^{87} + 4 q^{88} - 6 q^{89} + 7 q^{90} + 12 q^{92} + 28 q^{94} - 6 q^{95} + 4 q^{96} + 9 q^{97} + q^{98} - 10 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 −2.56155 0.438447 −1.00000 4.00000 1.00000 2.43845 3.56155 1.56155
1.2 2.56155 1.56155 4.56155 −1.00000 4.00000 1.00000 6.56155 −0.561553 −2.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 5915.2.a.l 2
13.b even 2 1 35.2.a.b 2
39.d odd 2 1 315.2.a.e 2
52.b odd 2 1 560.2.a.i 2
65.d even 2 1 175.2.a.f 2
65.h odd 4 2 175.2.b.b 4
91.b odd 2 1 245.2.a.d 2
91.r even 6 2 245.2.e.i 4
91.s odd 6 2 245.2.e.h 4
104.e even 2 1 2240.2.a.bh 2
104.h odd 2 1 2240.2.a.bd 2
143.d odd 2 1 4235.2.a.m 2
156.h even 2 1 5040.2.a.bt 2
195.e odd 2 1 1575.2.a.p 2
195.s even 4 2 1575.2.d.e 4
260.g odd 2 1 2800.2.a.bi 2
260.p even 4 2 2800.2.g.t 4
273.g even 2 1 2205.2.a.x 2
364.h even 2 1 3920.2.a.bs 2
455.h odd 2 1 1225.2.a.s 2
455.s even 4 2 1225.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 13.b even 2 1
175.2.a.f 2 65.d even 2 1
175.2.b.b 4 65.h odd 4 2
245.2.a.d 2 91.b odd 2 1
245.2.e.h 4 91.s odd 6 2
245.2.e.i 4 91.r even 6 2
315.2.a.e 2 39.d odd 2 1
560.2.a.i 2 52.b odd 2 1
1225.2.a.s 2 455.h odd 2 1
1225.2.b.f 4 455.s even 4 2
1575.2.a.p 2 195.e odd 2 1
1575.2.d.e 4 195.s even 4 2
2205.2.a.x 2 273.g even 2 1
2240.2.a.bd 2 104.h odd 2 1
2240.2.a.bh 2 104.e even 2 1
2800.2.a.bi 2 260.g odd 2 1
2800.2.g.t 4 260.p even 4 2
3920.2.a.bs 2 364.h even 2 1
4235.2.a.m 2 143.d odd 2 1
5040.2.a.bt 2 156.h even 2 1
5915.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T - 86 \) Copy content Toggle raw display
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