# Properties

 Label 15.4.b.a Level $15$ Weight $4$ Character orbit 15.b Analytic conductor $0.885$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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, 1]))

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

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

chi := DirichletCharacter("15.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 15.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 5) q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{5} + ( - \beta_{3} + 5) q^{6} + ( - 4 \beta_{2} + 6 \beta_1) q^{7} + (8 \beta_{2} + \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q - b1 * q^2 + b2 * q^3 + (b3 - 5) * q^4 + (-b3 - 2*b2 + b1 + 2) * q^5 + (-b3 + 5) * q^6 + (-4*b2 + 6*b1) * q^7 + (8*b2 + b1) * q^8 - 9 * q^9 $$q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 5) q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{5} + ( - \beta_{3} + 5) q^{6} + ( - 4 \beta_{2} + 6 \beta_1) q^{7} + (8 \beta_{2} + \beta_1) q^{8} - 9 q^{9} + (\beta_{3} - 8 \beta_{2} - 6 \beta_1 + 3) q^{10} + (2 \beta_{3} - 22) q^{11} - 9 \beta_1 q^{12} + (16 \beta_{2} - 6 \beta_1) q^{13} + ( - 2 \beta_{3} + 58) q^{14} + (\beta_{3} - 3 \beta_{2} + 9 \beta_1 + 13) q^{15} + ( - \beta_{3} + 13) q^{16} + ( - 12 \beta_{2} - 10 \beta_1) q^{17} + 9 \beta_1 q^{18} + ( - 8 \beta_{3} - 24) q^{19} + (6 \beta_{3} - 8 \beta_{2} + \cdots - 102) q^{20}+ \cdots + ( - 18 \beta_{3} + 198) q^{99}+O(q^{100})$$ q - b1 * q^2 + b2 * q^3 + (b3 - 5) * q^4 + (-b3 - 2*b2 + b1 + 2) * q^5 + (-b3 + 5) * q^6 + (-4*b2 + 6*b1) * q^7 + (8*b2 + b1) * q^8 - 9 * q^9 + (b3 - 8*b2 - 6*b1 + 3) * q^10 + (2*b3 - 22) * q^11 - 9*b1 * q^12 + (16*b2 - 6*b1) * q^13 + (-2*b3 + 58) * q^14 + (b3 - 3*b2 + 9*b1 + 13) * q^15 + (-b3 + 13) * q^16 + (-12*b2 - 10*b1) * q^17 + 9*b1 * q^18 + (-8*b3 - 24) * q^19 + (6*b3 - 8*b2 + 9*b1 - 102) * q^20 + (6*b3 + 6) * q^21 + (16*b2 + 30*b1) * q^22 + (-12*b2 + 8*b1) * q^23 + (b3 - 77) * q^24 + (-6*b3 + 28*b2 - 24*b1 + 67) * q^25 + (-10*b3 + 2) * q^26 - 9*b2 * q^27 + (-48*b2 - 18*b1) * q^28 + (14*b3 + 152) * q^29 + (-6*b3 + 8*b2 - 9*b1 + 102) * q^30 + (4*b3 + 24) * q^31 + (56*b2 - 9*b1) * q^32 + (-12*b2 - 18*b1) * q^33 + (22*b3 - 190) * q^34 + (-10*b3 + 60*b2 - 70) * q^35 + (-9*b3 + 45) * q^36 + (-24*b2 - 54*b1) * q^37 + (-64*b2 - 8*b1) * q^38 + (-6*b3 - 114) * q^39 + (7*b3 - 16*b2 + 78*b1 + 101) * q^40 + (4*b3 - 206) * q^41 + (48*b2 + 18*b1) * q^42 + (-28*b2 + 96*b1) * q^43 + (-30*b3 + 294) * q^44 + (9*b3 + 18*b2 - 9*b1 - 18) * q^45 + (4*b3 + 44) * q^46 + (-76*b2 + 92*b1) * q^47 + (8*b2 + 9*b1) * q^48 + (-12*b3 - 29) * q^49 + (-4*b3 - 48*b2 - 91*b1 - 172) * q^50 + (-10*b3 + 158) * q^51 + (48*b2 - 90*b1) * q^52 + (108*b2 - 82*b1) * q^53 + (9*b3 - 45) * q^54 + (24*b3 + 8*b2 + 6*b1 - 228) * q^55 + (50*b3 - 10) * q^56 + (-64*b2 + 72*b1) * q^57 + (112*b2 - 96*b1) * q^58 + (-2*b3 + 94) * q^59 + (9*b3 - 72*b2 - 54*b1 + 27) * q^60 + (-32*b3 + 186) * q^61 + (32*b2 - 8*b1) * q^62 + (36*b2 - 54*b1) * q^63 + (-55*b3 + 267) * q^64 + (22*b3 - 96*b2 + 108*b1 + 226) * q^65 + (30*b3 - 294) * q^66 + (-92*b2 - 60*b1) * q^67 + (80*b2 + 198*b1) * q^68 + (8*b3 + 68) * q^69 + (-60*b3 - 80*b2 + 30*b1 + 300) * q^70 + (-60*b3 + 12) * q^71 + (-72*b2 - 9*b1) * q^72 + (168*b2 + 108*b1) * q^73 + (78*b3 - 822) * q^74 + (-24*b3 + 37*b2 + 54*b1 - 132) * q^75 + (8*b3 - 616) * q^76 + (-48*b2 - 108*b1) * q^77 + (-48*b2 + 90*b1) * q^78 + (100*b3 - 240) * q^79 + (-14*b3 - 8*b2 - b1 + 118) * q^80 + 81 * q^81 + (32*b2 + 222*b1) * q^82 + (-60*b2 - 208*b1) * q^83 + (-18*b3 + 522) * q^84 + (-2*b3 - 44*b2 - 168*b1 - 126) * q^85 + (-68*b3 + 1108) * q^86 + (222*b2 - 126*b1) * q^87 + (-112*b2 - 174*b1) * q^88 + (-48*b3 - 534) * q^89 + (-9*b3 + 72*b2 + 54*b1 - 27) * q^90 + (84*b3 + 444) * q^91 + (-64*b2 + 36*b1) * q^92 + (44*b2 - 36*b1) * q^93 + (-16*b3 + 816) * q^94 + (16*b3 + 192*b2 - 136*b1 + 688) * q^95 + (-9*b3 - 459) * q^96 + (8*b2 + 240*b1) * q^97 + (-96*b2 - 19*b1) * q^98 + (-18*b3 + 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9 $$4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9} + 14 q^{10} - 84 q^{11} + 228 q^{14} + 54 q^{15} + 50 q^{16} - 112 q^{19} - 396 q^{20} + 36 q^{21} - 306 q^{24} + 256 q^{25} - 12 q^{26} + 636 q^{29} + 396 q^{30} + 104 q^{31} - 716 q^{34} - 300 q^{35} + 162 q^{36} - 468 q^{39} + 418 q^{40} - 816 q^{41} + 1116 q^{44} - 54 q^{45} + 184 q^{46} - 140 q^{49} - 696 q^{50} + 612 q^{51} - 162 q^{54} - 864 q^{55} + 60 q^{56} + 372 q^{59} + 126 q^{60} + 680 q^{61} + 958 q^{64} + 948 q^{65} - 1116 q^{66} + 288 q^{69} + 1080 q^{70} - 72 q^{71} - 3132 q^{74} - 576 q^{75} - 2448 q^{76} - 760 q^{79} + 444 q^{80} + 324 q^{81} + 2052 q^{84} - 508 q^{85} + 4296 q^{86} - 2232 q^{89} - 126 q^{90} + 1944 q^{91} + 3232 q^{94} + 2784 q^{95} - 1854 q^{96} + 756 q^{99}+O(q^{100})$$ 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9 + 14 * q^10 - 84 * q^11 + 228 * q^14 + 54 * q^15 + 50 * q^16 - 112 * q^19 - 396 * q^20 + 36 * q^21 - 306 * q^24 + 256 * q^25 - 12 * q^26 + 636 * q^29 + 396 * q^30 + 104 * q^31 - 716 * q^34 - 300 * q^35 + 162 * q^36 - 468 * q^39 + 418 * q^40 - 816 * q^41 + 1116 * q^44 - 54 * q^45 + 184 * q^46 - 140 * q^49 - 696 * q^50 + 612 * q^51 - 162 * q^54 - 864 * q^55 + 60 * q^56 + 372 * q^59 + 126 * q^60 + 680 * q^61 + 958 * q^64 + 948 * q^65 - 1116 * q^66 + 288 * q^69 + 1080 * q^70 - 72 * q^71 - 3132 * q^74 - 576 * q^75 - 2448 * q^76 - 760 * q^79 + 444 * q^80 + 324 * q^81 + 2052 * q^84 - 508 * q^85 + 4296 * q^86 - 2232 * q^89 - 126 * q^90 + 1944 * q^91 + 3232 * q^94 + 2784 * q^95 - 1854 * q^96 + 756 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu ) / 10$$ (v^3 + v) / 10 $$\beta_{2}$$ $$=$$ $$( 3\nu^{3} + 33\nu ) / 10$$ (3*v^3 + 33*v) / 10 $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 32$$ 3*v^2 + 32
 $$\nu$$ $$=$$ $$( \beta_{2} - 3\beta_1 ) / 3$$ (b2 - 3*b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 32 ) / 3$$ (b3 - 32) / 3 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 33\beta_1 ) / 3$$ (-b2 + 33*b1) / 3

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 3.00000i −14.1047 11.1047 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 52.2094i
4.2 1.70156i 3.00000i 5.10469 −8.10469 + 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 + 13.7906i
4.3 1.70156i 3.00000i 5.10469 −8.10469 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 13.7906i
4.4 4.70156i 3.00000i −14.1047 11.1047 + 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 + 52.2094i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.4.b.a 4
3.b odd 2 1 45.4.b.b 4
4.b odd 2 1 240.4.f.f 4
5.b even 2 1 inner 15.4.b.a 4
5.c odd 4 1 75.4.a.c 2
5.c odd 4 1 75.4.a.f 2
8.b even 2 1 960.4.f.q 4
8.d odd 2 1 960.4.f.p 4
12.b even 2 1 720.4.f.j 4
15.d odd 2 1 45.4.b.b 4
15.e even 4 1 225.4.a.i 2
15.e even 4 1 225.4.a.o 2
20.d odd 2 1 240.4.f.f 4
20.e even 4 1 1200.4.a.bn 2
20.e even 4 1 1200.4.a.bt 2
40.e odd 2 1 960.4.f.p 4
40.f even 2 1 960.4.f.q 4
60.h even 2 1 720.4.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 1.a even 1 1 trivial
15.4.b.a 4 5.b even 2 1 inner
45.4.b.b 4 3.b odd 2 1
45.4.b.b 4 15.d odd 2 1
75.4.a.c 2 5.c odd 4 1
75.4.a.f 2 5.c odd 4 1
225.4.a.i 2 15.e even 4 1
225.4.a.o 2 15.e even 4 1
240.4.f.f 4 4.b odd 2 1
240.4.f.f 4 20.d odd 2 1
720.4.f.j 4 12.b even 2 1
720.4.f.j 4 60.h even 2 1
960.4.f.p 4 8.d odd 2 1
960.4.f.p 4 40.e odd 2 1
960.4.f.q 4 8.b even 2 1
960.4.f.q 4 40.f even 2 1
1200.4.a.bn 2 20.e even 4 1
1200.4.a.bt 2 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25T^{2} + 64$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4} - 6 T^{3} + \cdots + 15625$$
$7$ $$T^{4} + 756 T^{2} + 129600$$
$11$ $$(T^{2} + 42 T + 72)^{2}$$
$13$ $$T^{4} + 3780 T^{2} + 1327104$$
$17$ $$T^{4} + 7252 T^{2} + 2483776$$
$19$ $$(T^{2} + 56 T - 5120)^{2}$$
$23$ $$T^{4} + 2464 T^{2} + 6400$$
$29$ $$(T^{2} - 318 T + 7200)^{2}$$
$31$ $$(T^{2} - 52 T - 800)^{2}$$
$37$ $$T^{4} + 106596 T^{2} + 41990400$$
$41$ $$(T^{2} + 408 T + 40140)^{2}$$
$43$ $$T^{4} + \cdots + 8256266496$$
$47$ $$T^{4} + \cdots + 6186766336$$
$53$ $$T^{4} + 218644 T^{2} + 813390400$$
$59$ $$(T^{2} - 186 T + 8280)^{2}$$
$61$ $$(T^{2} - 340 T - 65564)^{2}$$
$67$ $$T^{4} + \cdots + 9419867136$$
$71$ $$(T^{2} + 36 T - 331776)^{2}$$
$73$ $$T^{4} + \cdots + 104976000000$$
$79$ $$(T^{2} + 380 T - 886400)^{2}$$
$83$ $$T^{4} + \cdots + 40558737664$$
$89$ $$(T^{2} + 1116 T + 98820)^{2}$$
$97$ $$T^{4} + \cdots + 196199387136$$