# Properties

 Label 15.3.c.a Level $15$ Weight $3$ Character orbit 15.c Analytic conductor $0.409$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.408720396540$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - \beta - 2) q^{3} - q^{4} - \beta q^{5} + ( - 2 \beta + 5) q^{6} - 6 q^{7} + 3 \beta q^{8} + (4 \beta - 1) q^{9} +O(q^{10})$$ q + b * q^2 + (-b - 2) * q^3 - q^4 - b * q^5 + (-2*b + 5) * q^6 - 6 * q^7 + 3*b * q^8 + (4*b - 1) * q^9 $$q + \beta q^{2} + ( - \beta - 2) q^{3} - q^{4} - \beta q^{5} + ( - 2 \beta + 5) q^{6} - 6 q^{7} + 3 \beta q^{8} + (4 \beta - 1) q^{9} + 5 q^{10} - 2 \beta q^{11} + (\beta + 2) q^{12} + 16 q^{13} - 6 \beta q^{14} + (2 \beta - 5) q^{15} - 19 q^{16} - 2 \beta q^{17} + ( - \beta - 20) q^{18} - 2 q^{19} + \beta q^{20} + (6 \beta + 12) q^{21} + 10 q^{22} - 6 \beta q^{23} + ( - 6 \beta + 15) q^{24} - 5 q^{25} + 16 \beta q^{26} + ( - 7 \beta + 22) q^{27} + 6 q^{28} + 14 \beta q^{29} + ( - 5 \beta - 10) q^{30} - 18 q^{31} - 7 \beta q^{32} + (4 \beta - 10) q^{33} + 10 q^{34} + 6 \beta q^{35} + ( - 4 \beta + 1) q^{36} - 16 q^{37} - 2 \beta q^{38} + ( - 16 \beta - 32) q^{39} + 15 q^{40} - 28 \beta q^{41} + (12 \beta - 30) q^{42} + 16 q^{43} + 2 \beta q^{44} + (\beta + 20) q^{45} + 30 q^{46} + 22 \beta q^{47} + (19 \beta + 38) q^{48} - 13 q^{49} - 5 \beta q^{50} + (4 \beta - 10) q^{51} - 16 q^{52} - 2 \beta q^{53} + (22 \beta + 35) q^{54} - 10 q^{55} - 18 \beta q^{56} + (2 \beta + 4) q^{57} - 70 q^{58} - 2 \beta q^{59} + ( - 2 \beta + 5) q^{60} + 82 q^{61} - 18 \beta q^{62} + ( - 24 \beta + 6) q^{63} - 41 q^{64} - 16 \beta q^{65} + ( - 10 \beta - 20) q^{66} + 24 q^{67} + 2 \beta q^{68} + (12 \beta - 30) q^{69} - 30 q^{70} + 56 \beta q^{71} + ( - 3 \beta - 60) q^{72} - 74 q^{73} - 16 \beta q^{74} + (5 \beta + 10) q^{75} + 2 q^{76} + 12 \beta q^{77} + ( - 32 \beta + 80) q^{78} + 138 q^{79} + 19 \beta q^{80} + ( - 8 \beta - 79) q^{81} + 140 q^{82} + 42 \beta q^{83} + ( - 6 \beta - 12) q^{84} - 10 q^{85} + 16 \beta q^{86} + ( - 28 \beta + 70) q^{87} + 30 q^{88} - 48 \beta q^{89} + (20 \beta - 5) q^{90} - 96 q^{91} + 6 \beta q^{92} + (18 \beta + 36) q^{93} - 110 q^{94} + 2 \beta q^{95} + (14 \beta - 35) q^{96} - 166 q^{97} - 13 \beta q^{98} + (2 \beta + 40) q^{99} +O(q^{100})$$ q + b * q^2 + (-b - 2) * q^3 - q^4 - b * q^5 + (-2*b + 5) * q^6 - 6 * q^7 + 3*b * q^8 + (4*b - 1) * q^9 + 5 * q^10 - 2*b * q^11 + (b + 2) * q^12 + 16 * q^13 - 6*b * q^14 + (2*b - 5) * q^15 - 19 * q^16 - 2*b * q^17 + (-b - 20) * q^18 - 2 * q^19 + b * q^20 + (6*b + 12) * q^21 + 10 * q^22 - 6*b * q^23 + (-6*b + 15) * q^24 - 5 * q^25 + 16*b * q^26 + (-7*b + 22) * q^27 + 6 * q^28 + 14*b * q^29 + (-5*b - 10) * q^30 - 18 * q^31 - 7*b * q^32 + (4*b - 10) * q^33 + 10 * q^34 + 6*b * q^35 + (-4*b + 1) * q^36 - 16 * q^37 - 2*b * q^38 + (-16*b - 32) * q^39 + 15 * q^40 - 28*b * q^41 + (12*b - 30) * q^42 + 16 * q^43 + 2*b * q^44 + (b + 20) * q^45 + 30 * q^46 + 22*b * q^47 + (19*b + 38) * q^48 - 13 * q^49 - 5*b * q^50 + (4*b - 10) * q^51 - 16 * q^52 - 2*b * q^53 + (22*b + 35) * q^54 - 10 * q^55 - 18*b * q^56 + (2*b + 4) * q^57 - 70 * q^58 - 2*b * q^59 + (-2*b + 5) * q^60 + 82 * q^61 - 18*b * q^62 + (-24*b + 6) * q^63 - 41 * q^64 - 16*b * q^65 + (-10*b - 20) * q^66 + 24 * q^67 + 2*b * q^68 + (12*b - 30) * q^69 - 30 * q^70 + 56*b * q^71 + (-3*b - 60) * q^72 - 74 * q^73 - 16*b * q^74 + (5*b + 10) * q^75 + 2 * q^76 + 12*b * q^77 + (-32*b + 80) * q^78 + 138 * q^79 + 19*b * q^80 + (-8*b - 79) * q^81 + 140 * q^82 + 42*b * q^83 + (-6*b - 12) * q^84 - 10 * q^85 + 16*b * q^86 + (-28*b + 70) * q^87 + 30 * q^88 - 48*b * q^89 + (20*b - 5) * q^90 - 96 * q^91 + 6*b * q^92 + (18*b + 36) * q^93 - 110 * q^94 + 2*b * q^95 + (14*b - 35) * q^96 - 166 * q^97 - 13*b * q^98 + (2*b + 40) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 2 q^{4} + 10 q^{6} - 12 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 2 * q^4 + 10 * q^6 - 12 * q^7 - 2 * q^9 $$2 q - 4 q^{3} - 2 q^{4} + 10 q^{6} - 12 q^{7} - 2 q^{9} + 10 q^{10} + 4 q^{12} + 32 q^{13} - 10 q^{15} - 38 q^{16} - 40 q^{18} - 4 q^{19} + 24 q^{21} + 20 q^{22} + 30 q^{24} - 10 q^{25} + 44 q^{27} + 12 q^{28} - 20 q^{30} - 36 q^{31} - 20 q^{33} + 20 q^{34} + 2 q^{36} - 32 q^{37} - 64 q^{39} + 30 q^{40} - 60 q^{42} + 32 q^{43} + 40 q^{45} + 60 q^{46} + 76 q^{48} - 26 q^{49} - 20 q^{51} - 32 q^{52} + 70 q^{54} - 20 q^{55} + 8 q^{57} - 140 q^{58} + 10 q^{60} + 164 q^{61} + 12 q^{63} - 82 q^{64} - 40 q^{66} + 48 q^{67} - 60 q^{69} - 60 q^{70} - 120 q^{72} - 148 q^{73} + 20 q^{75} + 4 q^{76} + 160 q^{78} + 276 q^{79} - 158 q^{81} + 280 q^{82} - 24 q^{84} - 20 q^{85} + 140 q^{87} + 60 q^{88} - 10 q^{90} - 192 q^{91} + 72 q^{93} - 220 q^{94} - 70 q^{96} - 332 q^{97} + 80 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 - 2 * q^4 + 10 * q^6 - 12 * q^7 - 2 * q^9 + 10 * q^10 + 4 * q^12 + 32 * q^13 - 10 * q^15 - 38 * q^16 - 40 * q^18 - 4 * q^19 + 24 * q^21 + 20 * q^22 + 30 * q^24 - 10 * q^25 + 44 * q^27 + 12 * q^28 - 20 * q^30 - 36 * q^31 - 20 * q^33 + 20 * q^34 + 2 * q^36 - 32 * q^37 - 64 * q^39 + 30 * q^40 - 60 * q^42 + 32 * q^43 + 40 * q^45 + 60 * q^46 + 76 * q^48 - 26 * q^49 - 20 * q^51 - 32 * q^52 + 70 * q^54 - 20 * q^55 + 8 * q^57 - 140 * q^58 + 10 * q^60 + 164 * q^61 + 12 * q^63 - 82 * q^64 - 40 * q^66 + 48 * q^67 - 60 * q^69 - 60 * q^70 - 120 * q^72 - 148 * q^73 + 20 * q^75 + 4 * q^76 + 160 * q^78 + 276 * q^79 - 158 * q^81 + 280 * q^82 - 24 * q^84 - 20 * q^85 + 140 * q^87 + 60 * q^88 - 10 * q^90 - 192 * q^91 + 72 * q^93 - 220 * q^94 - 70 * q^96 - 332 * q^97 + 80 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 − 2.23607i 2.23607i
2.23607i −2.00000 + 2.23607i −1.00000 2.23607i 5.00000 + 4.47214i −6.00000 6.70820i −1.00000 8.94427i 5.00000
11.2 2.23607i −2.00000 2.23607i −1.00000 2.23607i 5.00000 4.47214i −6.00000 6.70820i −1.00000 + 8.94427i 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
3.b odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5$$
$3$ $$T^{2} + 4T + 9$$
$5$ $$T^{2} + 5$$
$7$ $$(T + 6)^{2}$$
$11$ $$T^{2} + 20$$
$13$ $$(T - 16)^{2}$$
$17$ $$T^{2} + 20$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 180$$
$29$ $$T^{2} + 980$$
$31$ $$(T + 18)^{2}$$
$37$ $$(T + 16)^{2}$$
$41$ $$T^{2} + 3920$$
$43$ $$(T - 16)^{2}$$
$47$ $$T^{2} + 2420$$
$53$ $$T^{2} + 20$$
$59$ $$T^{2} + 20$$
$61$ $$(T - 82)^{2}$$
$67$ $$(T - 24)^{2}$$
$71$ $$T^{2} + 15680$$
$73$ $$(T + 74)^{2}$$
$79$ $$(T - 138)^{2}$$
$83$ $$T^{2} + 8820$$
$89$ $$T^{2} + 11520$$
$97$ $$(T + 166)^{2}$$