# Properties

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

# Related objects

## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} + q^{5} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 + q^5 - q^7 - 2 * q^9 $$q + q^{3} - 2 q^{4} + q^{5} - q^{7} - 2 q^{9} + 3 q^{11} - 2 q^{12} + q^{15} + 4 q^{16} + 3 q^{17} - 2 q^{19} - 2 q^{20} - q^{21} - 6 q^{23} + q^{25} - 5 q^{27} + 2 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 4 q^{36} - 2 q^{37} + 12 q^{41} - 10 q^{43} - 6 q^{44} - 2 q^{45} - 9 q^{47} + 4 q^{48} + q^{49} + 3 q^{51} + 12 q^{53} + 3 q^{55} - 2 q^{57} - 2 q^{60} + 8 q^{61} + 2 q^{63} - 8 q^{64} + 4 q^{67} - 6 q^{68} - 6 q^{69} - 2 q^{73} + q^{75} + 4 q^{76} - 3 q^{77} - q^{79} + 4 q^{80} + q^{81} - 12 q^{83} + 2 q^{84} + 3 q^{85} + 3 q^{87} + 12 q^{89} + 12 q^{92} + 4 q^{93} - 2 q^{95} + q^{97} - 6 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^4 + q^5 - q^7 - 2 * q^9 + 3 * q^11 - 2 * q^12 + q^15 + 4 * q^16 + 3 * q^17 - 2 * q^19 - 2 * q^20 - q^21 - 6 * q^23 + q^25 - 5 * q^27 + 2 * q^28 + 3 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 4 * q^36 - 2 * q^37 + 12 * q^41 - 10 * q^43 - 6 * q^44 - 2 * q^45 - 9 * q^47 + 4 * q^48 + q^49 + 3 * q^51 + 12 * q^53 + 3 * q^55 - 2 * q^57 - 2 * q^60 + 8 * q^61 + 2 * q^63 - 8 * q^64 + 4 * q^67 - 6 * q^68 - 6 * q^69 - 2 * q^73 + q^75 + 4 * q^76 - 3 * q^77 - q^79 + 4 * q^80 + q^81 - 12 * q^83 + 2 * q^84 + 3 * q^85 + 3 * q^87 + 12 * q^89 + 12 * q^92 + 4 * q^93 - 2 * q^95 + q^97 - 6 * q^99

## 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
 0
0 1.00000 −2.00000 1.00000 0 −1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 1
13.b even 2 1 35.2.a.a 1
39.d odd 2 1 315.2.a.b 1
52.b odd 2 1 560.2.a.b 1
65.d even 2 1 175.2.a.b 1
65.h odd 4 2 175.2.b.a 2
91.b odd 2 1 245.2.a.c 1
91.r even 6 2 245.2.e.a 2
91.s odd 6 2 245.2.e.b 2
104.e even 2 1 2240.2.a.k 1
104.h odd 2 1 2240.2.a.u 1
143.d odd 2 1 4235.2.a.c 1
156.h even 2 1 5040.2.a.v 1
195.e odd 2 1 1575.2.a.f 1
195.s even 4 2 1575.2.d.c 2
260.g odd 2 1 2800.2.a.z 1
260.p even 4 2 2800.2.g.l 2
273.g even 2 1 2205.2.a.e 1
364.h even 2 1 3920.2.a.ba 1
455.h odd 2 1 1225.2.a.e 1
455.s even 4 2 1225.2.b.d 2

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

 $$T_{2}$$ T2 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T + 2$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T - 12$$
$43$ $$T + 10$$
$47$ $$T + 9$$
$53$ $$T - 12$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 1$$
$83$ $$T + 12$$
$89$ $$T - 12$$
$97$ $$T - 1$$