# Properties

 Label 15.4.a.b Level $15$ Weight $4$ Character orbit 15.a Self dual yes Analytic conductor $0.885$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,4,Mod(1,15)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("15.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 15.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 9 q^{6} + 20 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10})$$ q + 3 * q^2 - 3 * q^3 + q^4 - 5 * q^5 - 9 * q^6 + 20 * q^7 - 21 * q^8 + 9 * q^9 $$q + 3 q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 9 q^{6} + 20 q^{7} - 21 q^{8} + 9 q^{9} - 15 q^{10} - 24 q^{11} - 3 q^{12} + 74 q^{13} + 60 q^{14} + 15 q^{15} - 71 q^{16} + 54 q^{17} + 27 q^{18} - 124 q^{19} - 5 q^{20} - 60 q^{21} - 72 q^{22} - 120 q^{23} + 63 q^{24} + 25 q^{25} + 222 q^{26} - 27 q^{27} + 20 q^{28} - 78 q^{29} + 45 q^{30} + 200 q^{31} - 45 q^{32} + 72 q^{33} + 162 q^{34} - 100 q^{35} + 9 q^{36} - 70 q^{37} - 372 q^{38} - 222 q^{39} + 105 q^{40} + 330 q^{41} - 180 q^{42} + 92 q^{43} - 24 q^{44} - 45 q^{45} - 360 q^{46} - 24 q^{47} + 213 q^{48} + 57 q^{49} + 75 q^{50} - 162 q^{51} + 74 q^{52} + 450 q^{53} - 81 q^{54} + 120 q^{55} - 420 q^{56} + 372 q^{57} - 234 q^{58} + 24 q^{59} + 15 q^{60} - 322 q^{61} + 600 q^{62} + 180 q^{63} + 433 q^{64} - 370 q^{65} + 216 q^{66} - 196 q^{67} + 54 q^{68} + 360 q^{69} - 300 q^{70} - 288 q^{71} - 189 q^{72} - 430 q^{73} - 210 q^{74} - 75 q^{75} - 124 q^{76} - 480 q^{77} - 666 q^{78} - 520 q^{79} + 355 q^{80} + 81 q^{81} + 990 q^{82} + 156 q^{83} - 60 q^{84} - 270 q^{85} + 276 q^{86} + 234 q^{87} + 504 q^{88} + 1026 q^{89} - 135 q^{90} + 1480 q^{91} - 120 q^{92} - 600 q^{93} - 72 q^{94} + 620 q^{95} + 135 q^{96} - 286 q^{97} + 171 q^{98} - 216 q^{99}+O(q^{100})$$ q + 3 * q^2 - 3 * q^3 + q^4 - 5 * q^5 - 9 * q^6 + 20 * q^7 - 21 * q^8 + 9 * q^9 - 15 * q^10 - 24 * q^11 - 3 * q^12 + 74 * q^13 + 60 * q^14 + 15 * q^15 - 71 * q^16 + 54 * q^17 + 27 * q^18 - 124 * q^19 - 5 * q^20 - 60 * q^21 - 72 * q^22 - 120 * q^23 + 63 * q^24 + 25 * q^25 + 222 * q^26 - 27 * q^27 + 20 * q^28 - 78 * q^29 + 45 * q^30 + 200 * q^31 - 45 * q^32 + 72 * q^33 + 162 * q^34 - 100 * q^35 + 9 * q^36 - 70 * q^37 - 372 * q^38 - 222 * q^39 + 105 * q^40 + 330 * q^41 - 180 * q^42 + 92 * q^43 - 24 * q^44 - 45 * q^45 - 360 * q^46 - 24 * q^47 + 213 * q^48 + 57 * q^49 + 75 * q^50 - 162 * q^51 + 74 * q^52 + 450 * q^53 - 81 * q^54 + 120 * q^55 - 420 * q^56 + 372 * q^57 - 234 * q^58 + 24 * q^59 + 15 * q^60 - 322 * q^61 + 600 * q^62 + 180 * q^63 + 433 * q^64 - 370 * q^65 + 216 * q^66 - 196 * q^67 + 54 * q^68 + 360 * q^69 - 300 * q^70 - 288 * q^71 - 189 * q^72 - 430 * q^73 - 210 * q^74 - 75 * q^75 - 124 * q^76 - 480 * q^77 - 666 * q^78 - 520 * q^79 + 355 * q^80 + 81 * q^81 + 990 * q^82 + 156 * q^83 - 60 * q^84 - 270 * q^85 + 276 * q^86 + 234 * q^87 + 504 * q^88 + 1026 * q^89 - 135 * q^90 + 1480 * q^91 - 120 * q^92 - 600 * q^93 - 72 * q^94 + 620 * q^95 + 135 * q^96 - 286 * q^97 + 171 * q^98 - 216 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
3.00000 −3.00000 1.00000 −5.00000 −9.00000 20.0000 −21.0000 9.00000 −15.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$

## 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 15.4.a.b 1
3.b odd 2 1 45.4.a.b 1
4.b odd 2 1 240.4.a.f 1
5.b even 2 1 75.4.a.a 1
5.c odd 4 2 75.4.b.a 2
7.b odd 2 1 735.4.a.i 1
8.b even 2 1 960.4.a.bi 1
8.d odd 2 1 960.4.a.l 1
9.c even 3 2 405.4.e.d 2
9.d odd 6 2 405.4.e.k 2
11.b odd 2 1 1815.4.a.a 1
12.b even 2 1 720.4.a.r 1
15.d odd 2 1 225.4.a.g 1
15.e even 4 2 225.4.b.d 2
20.d odd 2 1 1200.4.a.o 1
20.e even 4 2 1200.4.f.m 2
21.c even 2 1 2205.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 1.a even 1 1 trivial
45.4.a.b 1 3.b odd 2 1
75.4.a.a 1 5.b even 2 1
75.4.b.a 2 5.c odd 4 2
225.4.a.g 1 15.d odd 2 1
225.4.b.d 2 15.e even 4 2
240.4.a.f 1 4.b odd 2 1
405.4.e.d 2 9.c even 3 2
405.4.e.k 2 9.d odd 6 2
720.4.a.r 1 12.b even 2 1
735.4.a.i 1 7.b odd 2 1
960.4.a.l 1 8.d odd 2 1
960.4.a.bi 1 8.b even 2 1
1200.4.a.o 1 20.d odd 2 1
1200.4.f.m 2 20.e even 4 2
1815.4.a.a 1 11.b odd 2 1
2205.4.a.c 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(15))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T - 20$$
$11$ $$T + 24$$
$13$ $$T - 74$$
$17$ $$T - 54$$
$19$ $$T + 124$$
$23$ $$T + 120$$
$29$ $$T + 78$$
$31$ $$T - 200$$
$37$ $$T + 70$$
$41$ $$T - 330$$
$43$ $$T - 92$$
$47$ $$T + 24$$
$53$ $$T - 450$$
$59$ $$T - 24$$
$61$ $$T + 322$$
$67$ $$T + 196$$
$71$ $$T + 288$$
$73$ $$T + 430$$
$79$ $$T + 520$$
$83$ $$T - 156$$
$89$ $$T - 1026$$
$97$ $$T + 286$$