# Properties

 Label 15.3.d.b Level $15$ Weight $3$ Character orbit 15.d Self dual yes Analytic conductor $0.409$ Analytic rank $0$ Dimension $1$ CM discriminant -15 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 15.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.408720396540$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} - 3 q^{4} + 5 q^{5} - 3 q^{6} - 7 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 - 3 * q^4 + 5 * q^5 - 3 * q^6 - 7 * q^8 + 9 * q^9 $$q + q^{2} - 3 q^{3} - 3 q^{4} + 5 q^{5} - 3 q^{6} - 7 q^{8} + 9 q^{9} + 5 q^{10} + 9 q^{12} - 15 q^{15} + 5 q^{16} - 14 q^{17} + 9 q^{18} - 22 q^{19} - 15 q^{20} + 34 q^{23} + 21 q^{24} + 25 q^{25} - 27 q^{27} - 15 q^{30} + 2 q^{31} + 33 q^{32} - 14 q^{34} - 27 q^{36} - 22 q^{38} - 35 q^{40} + 45 q^{45} + 34 q^{46} - 14 q^{47} - 15 q^{48} + 49 q^{49} + 25 q^{50} + 42 q^{51} - 86 q^{53} - 27 q^{54} + 66 q^{57} + 45 q^{60} - 118 q^{61} + 2 q^{62} + 13 q^{64} + 42 q^{68} - 102 q^{69} - 63 q^{72} - 75 q^{75} + 66 q^{76} + 98 q^{79} + 25 q^{80} + 81 q^{81} + 154 q^{83} - 70 q^{85} + 45 q^{90} - 102 q^{92} - 6 q^{93} - 14 q^{94} - 110 q^{95} - 99 q^{96} + 49 q^{98}+O(q^{100})$$ q + q^2 - 3 * q^3 - 3 * q^4 + 5 * q^5 - 3 * q^6 - 7 * q^8 + 9 * q^9 + 5 * q^10 + 9 * q^12 - 15 * q^15 + 5 * q^16 - 14 * q^17 + 9 * q^18 - 22 * q^19 - 15 * q^20 + 34 * q^23 + 21 * q^24 + 25 * q^25 - 27 * q^27 - 15 * q^30 + 2 * q^31 + 33 * q^32 - 14 * q^34 - 27 * q^36 - 22 * q^38 - 35 * q^40 + 45 * q^45 + 34 * q^46 - 14 * q^47 - 15 * q^48 + 49 * q^49 + 25 * q^50 + 42 * q^51 - 86 * q^53 - 27 * q^54 + 66 * q^57 + 45 * q^60 - 118 * q^61 + 2 * q^62 + 13 * q^64 + 42 * q^68 - 102 * q^69 - 63 * q^72 - 75 * q^75 + 66 * q^76 + 98 * q^79 + 25 * q^80 + 81 * q^81 + 154 * q^83 - 70 * q^85 + 45 * q^90 - 102 * q^92 - 6 * q^93 - 14 * q^94 - 110 * q^95 - 99 * q^96 + 49 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 0
1.00000 −3.00000 −3.00000 5.00000 −3.00000 0 −7.00000 9.00000 5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.3.d.b yes 1
3.b odd 2 1 15.3.d.a 1
4.b odd 2 1 240.3.c.b 1
5.b even 2 1 15.3.d.a 1
5.c odd 4 2 75.3.c.d 2
8.b even 2 1 960.3.c.c 1
8.d odd 2 1 960.3.c.a 1
9.c even 3 2 405.3.h.a 2
9.d odd 6 2 405.3.h.b 2
12.b even 2 1 240.3.c.a 1
15.d odd 2 1 CM 15.3.d.b yes 1
15.e even 4 2 75.3.c.d 2
20.d odd 2 1 240.3.c.a 1
20.e even 4 2 1200.3.l.l 2
24.f even 2 1 960.3.c.d 1
24.h odd 2 1 960.3.c.b 1
40.e odd 2 1 960.3.c.d 1
40.f even 2 1 960.3.c.b 1
45.h odd 6 2 405.3.h.a 2
45.j even 6 2 405.3.h.b 2
60.h even 2 1 240.3.c.b 1
60.l odd 4 2 1200.3.l.l 2
120.i odd 2 1 960.3.c.c 1
120.m even 2 1 960.3.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 3.b odd 2 1
15.3.d.a 1 5.b even 2 1
15.3.d.b yes 1 1.a even 1 1 trivial
15.3.d.b yes 1 15.d odd 2 1 CM
75.3.c.d 2 5.c odd 4 2
75.3.c.d 2 15.e even 4 2
240.3.c.a 1 12.b even 2 1
240.3.c.a 1 20.d odd 2 1
240.3.c.b 1 4.b odd 2 1
240.3.c.b 1 60.h even 2 1
405.3.h.a 2 9.c even 3 2
405.3.h.a 2 45.h odd 6 2
405.3.h.b 2 9.d odd 6 2
405.3.h.b 2 45.j even 6 2
960.3.c.a 1 8.d odd 2 1
960.3.c.a 1 120.m even 2 1
960.3.c.b 1 24.h odd 2 1
960.3.c.b 1 40.f even 2 1
960.3.c.c 1 8.b even 2 1
960.3.c.c 1 120.i odd 2 1
960.3.c.d 1 24.f even 2 1
960.3.c.d 1 40.e odd 2 1
1200.3.l.l 2 20.e even 4 2
1200.3.l.l 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{3}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 14$$
$19$ $$T + 22$$
$23$ $$T - 34$$
$29$ $$T$$
$31$ $$T - 2$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T + 14$$
$53$ $$T + 86$$
$59$ $$T$$
$61$ $$T + 118$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T - 98$$
$83$ $$T - 154$$
$89$ $$T$$
$97$ $$T$$
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