Properties

 Label 315.2.j.c Level $315$ Weight $2$ Character orbit 315.j Analytic conductor $2.515$ 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] = [315,2,Mod(46,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("315.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 315.j (of order $$3$$, degree $$2$$, 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: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{4} + \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{8}+O(q^{10})$$ q + (z^3 - z^2 + z) * q^2 + (-4*z^3 + 2*z^2 + 2*z - 2) * q^4 + z^2 * q^5 + (3*z^3 - z) * q^7 + (2*z^3 - 4*z + 6) * q^8 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{4} + \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{8} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 1) q^{10} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 1) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 4) q^{13} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 5) q^{14} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4 \zeta_{12}) q^{16} + ( - 2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{17} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12}) q^{19} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{20} - 2 q^{22} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{23} + (\zeta_{12}^{2} - 1) q^{25} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12}) q^{26} + ( - 2 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{28} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 1) q^{29} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{31} + ( - 16 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{32} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 2) q^{34} + (2 \zeta_{12}^{3} - 3 \zeta_{12}) q^{35} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12}) q^{37} + ( - 6 \zeta_{12}^{3} + 7 \zeta_{12}^{2} + 3 \zeta_{12} - 7) q^{38} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{40} + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{41} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 2) q^{43} + 4 \zeta_{12}^{2} q^{44} + ( - 8 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 4 \zeta_{12} - 6) q^{46} + 2 \zeta_{12}^{2} q^{47} + ( - 5 \zeta_{12}^{2} - 3) q^{49} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{50} + ( - 12 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{52} + (12 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12} - 1) q^{55} + (18 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 6 \zeta_{12} + 8) q^{56} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 2 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 3 \zeta_{12} + 5) q^{59} - 4 \zeta_{12}^{2} q^{61} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{62} + (8 \zeta_{12}^{3} - 16 \zeta_{12} + 16) q^{64} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12}) q^{65} + (10 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 5 \zeta_{12} + 6) q^{67} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12}) q^{68} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{70} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 1) q^{71} + ( - 10 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{73} + (2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - \zeta_{12} + 7) q^{74} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 14) q^{76} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{77} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12}) q^{79} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 4 \zeta_{12} + 8) q^{80} - 2 \zeta_{12}^{2} q^{82} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} - 3) q^{83} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 5) q^{85} + (\zeta_{12}^{3} - 7 \zeta_{12}^{2} + \zeta_{12}) q^{86} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{88} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12}) q^{89} + (12 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{91} + (8 \zeta_{12}^{3} - 16 \zeta_{12} + 12) q^{92} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{94} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{95} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 8) q^{97} + ( - 13 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 2 \zeta_{12} - 5) q^{98} +O(q^{100})$$ q + (z^3 - z^2 + z) * q^2 + (-4*z^3 + 2*z^2 + 2*z - 2) * q^4 + z^2 * q^5 + (3*z^3 - z) * q^7 + (2*z^3 - 4*z + 6) * q^8 + (2*z^3 - z^2 - z + 1) * q^10 + (2*z^3 + z^2 - z - 1) * q^11 + (-z^3 + 2*z + 4) * q^13 + (-2*z^3 + z^2 + 3*z - 5) * q^14 + (4*z^3 - 8*z^2 + 4*z) * q^16 + (-2*z^3 - 5*z^2 + z + 5) * q^17 + (2*z^3 - z^2 + 2*z) * q^19 + (-2*z^3 + 4*z - 2) * q^20 - 2 * q^22 + (z^3 - 3*z^2 + z) * q^23 + (z^2 - 1) * q^25 + (3*z^3 - z^2 + 3*z) * q^26 + (-2*z^3 + 8*z^2 - 4*z + 2) * q^28 + (3*z^3 - 6*z - 1) * q^29 + (4*z^3 + 3*z^2 - 2*z - 3) * q^31 + (-16*z^3 + 8*z^2 + 8*z - 8) * q^32 + (-4*z^3 + 8*z - 2) * q^34 + (2*z^3 - 3*z) * q^35 + (-3*z^3 - 2*z^2 - 3*z) * q^37 + (-6*z^3 + 7*z^2 + 3*z - 7) * q^38 + (-2*z^3 + 6*z^2 - 2*z) * q^40 + (z^3 - 2*z - 1) * q^41 + (3*z^3 - 6*z - 2) * q^43 + 4*z^2 * q^44 + (-8*z^3 + 6*z^2 + 4*z - 6) * q^46 + 2*z^2 * q^47 + (-5*z^2 - 3) * q^49 + (z^3 - 2*z + 1) * q^50 + (-12*z^3 + 2*z^2 + 6*z - 2) * q^52 + (12*z^3 - 2*z^2 - 6*z + 2) * q^53 + (z^3 - 2*z - 1) * q^55 + (18*z^3 - 10*z^2 - 6*z + 8) * q^56 + (2*z^3 - 8*z^2 + 2*z) * q^58 + (-6*z^3 - 5*z^2 + 3*z + 5) * q^59 - 4*z^2 * q^61 + (z^3 - 2*z - 3) * q^62 + (8*z^3 - 16*z + 16) * q^64 + (z^3 + 4*z^2 + z) * q^65 + (10*z^3 - 6*z^2 - 5*z + 6) * q^67 + (-8*z^3 + 4*z^2 - 8*z) * q^68 + (z^3 - 4*z^2 + 2*z - 1) * q^70 + (-3*z^3 + 6*z - 1) * q^71 + (-10*z^3 + 4*z^2 + 5*z - 4) * q^73 + (2*z^3 - 7*z^2 - z + 7) * q^74 + (6*z^3 - 12*z + 14) * q^76 + (-z^3 - 4*z^2 - 2*z - 1) * q^77 + (-6*z^3 - 3*z^2 - 6*z) * q^79 + (8*z^3 - 8*z^2 - 4*z + 8) * q^80 - 2*z^2 * q^82 + (-7*z^3 + 14*z - 3) * q^83 + (-z^3 + 2*z + 5) * q^85 + (z^3 - 7*z^2 + z) * q^86 + (8*z^3 - 4*z) * q^88 + (-7*z^3 + 3*z^2 - 7*z) * q^89 + (12*z^3 + 5*z^2 - 4*z - 4) * q^91 + (8*z^3 - 16*z + 12) * q^92 + (4*z^3 - 2*z^2 - 2*z + 2) * q^94 + (4*z^3 - z^2 - 2*z + 1) * q^95 + (4*z^3 - 8*z + 8) * q^97 + (-13*z^3 + 8*z^2 + 2*z - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 24 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 4 * q^4 + 2 * q^5 + 24 * q^8 $$4 q - 2 q^{2} - 4 q^{4} + 2 q^{5} + 24 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{13} - 18 q^{14} - 16 q^{16} + 10 q^{17} - 2 q^{19} - 8 q^{20} - 8 q^{22} - 6 q^{23} - 2 q^{25} - 2 q^{26} + 24 q^{28} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 8 q^{34} - 4 q^{37} - 14 q^{38} + 12 q^{40} - 4 q^{41} - 8 q^{43} + 8 q^{44} - 12 q^{46} + 4 q^{47} - 22 q^{49} + 4 q^{50} - 4 q^{52} + 4 q^{53} - 4 q^{55} + 12 q^{56} - 16 q^{58} + 10 q^{59} - 8 q^{61} - 12 q^{62} + 64 q^{64} + 8 q^{65} + 12 q^{67} + 8 q^{68} - 12 q^{70} - 4 q^{71} - 8 q^{73} + 14 q^{74} + 56 q^{76} - 12 q^{77} - 6 q^{79} + 16 q^{80} - 4 q^{82} - 12 q^{83} + 20 q^{85} - 14 q^{86} + 6 q^{89} - 6 q^{91} + 48 q^{92} + 4 q^{94} + 2 q^{95} + 32 q^{97} - 4 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 4 * q^4 + 2 * q^5 + 24 * q^8 + 2 * q^10 - 2 * q^11 + 16 * q^13 - 18 * q^14 - 16 * q^16 + 10 * q^17 - 2 * q^19 - 8 * q^20 - 8 * q^22 - 6 * q^23 - 2 * q^25 - 2 * q^26 + 24 * q^28 - 4 * q^29 - 6 * q^31 - 16 * q^32 - 8 * q^34 - 4 * q^37 - 14 * q^38 + 12 * q^40 - 4 * q^41 - 8 * q^43 + 8 * q^44 - 12 * q^46 + 4 * q^47 - 22 * q^49 + 4 * q^50 - 4 * q^52 + 4 * q^53 - 4 * q^55 + 12 * q^56 - 16 * q^58 + 10 * q^59 - 8 * q^61 - 12 * q^62 + 64 * q^64 + 8 * q^65 + 12 * q^67 + 8 * q^68 - 12 * q^70 - 4 * q^71 - 8 * q^73 + 14 * q^74 + 56 * q^76 - 12 * q^77 - 6 * q^79 + 16 * q^80 - 4 * q^82 - 12 * q^83 + 20 * q^85 - 14 * q^86 + 6 * q^89 - 6 * q^91 + 48 * q^92 + 4 * q^94 + 2 * q^95 + 32 * q^97 - 4 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}^{2}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.36603 2.36603i 0 −2.73205 + 4.73205i 0.500000 + 0.866025i 0 0.866025 2.50000i 9.46410 0 1.36603 2.36603i
46.2 0.366025 + 0.633975i 0 0.732051 1.26795i 0.500000 + 0.866025i 0 −0.866025 + 2.50000i 2.53590 0 −0.366025 + 0.633975i
226.1 −1.36603 + 2.36603i 0 −2.73205 4.73205i 0.500000 0.866025i 0 0.866025 + 2.50000i 9.46410 0 1.36603 + 2.36603i
226.2 0.366025 0.633975i 0 0.732051 + 1.26795i 0.500000 0.866025i 0 −0.866025 2.50000i 2.53590 0 −0.366025 0.633975i
 $$n$$: 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
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.c 4
3.b odd 2 1 105.2.i.d 4
7.c even 3 1 inner 315.2.j.c 4
7.c even 3 1 2205.2.a.z 2
7.d odd 6 1 2205.2.a.ba 2
12.b even 2 1 1680.2.bg.o 4
15.d odd 2 1 525.2.i.f 4
15.e even 4 1 525.2.r.a 4
15.e even 4 1 525.2.r.f 4
21.c even 2 1 735.2.i.l 4
21.g even 6 1 735.2.a.h 2
21.g even 6 1 735.2.i.l 4
21.h odd 6 1 105.2.i.d 4
21.h odd 6 1 735.2.a.g 2
84.n even 6 1 1680.2.bg.o 4
105.o odd 6 1 525.2.i.f 4
105.o odd 6 1 3675.2.a.bg 2
105.p even 6 1 3675.2.a.be 2
105.x even 12 1 525.2.r.a 4
105.x even 12 1 525.2.r.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 3.b odd 2 1
105.2.i.d 4 21.h odd 6 1
315.2.j.c 4 1.a even 1 1 trivial
315.2.j.c 4 7.c even 3 1 inner
525.2.i.f 4 15.d odd 2 1
525.2.i.f 4 105.o odd 6 1
525.2.r.a 4 15.e even 4 1
525.2.r.a 4 105.x even 12 1
525.2.r.f 4 15.e even 4 1
525.2.r.f 4 105.x even 12 1
735.2.a.g 2 21.h odd 6 1
735.2.a.h 2 21.g even 6 1
735.2.i.l 4 21.c even 2 1
735.2.i.l 4 21.g even 6 1
1680.2.bg.o 4 12.b even 2 1
1680.2.bg.o 4 84.n even 6 1
2205.2.a.z 2 7.c even 3 1
2205.2.a.ba 2 7.d odd 6 1
3675.2.a.be 2 105.p even 6 1
3675.2.a.bg 2 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$13$ $$(T^{2} - 8 T + 13)^{2}$$
$17$ $$T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484$$
$19$ $$T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} + 2 T - 26)^{2}$$
$31$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$37$ $$T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529$$
$41$ $$(T^{2} + 2 T - 2)^{2}$$
$43$ $$(T^{2} + 4 T - 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} - 4 T^{3} + 120 T^{2} + \cdots + 10816$$
$59$ $$T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$71$ $$(T^{2} + 2 T - 26)^{2}$$
$73$ $$T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481$$
$79$ $$T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801$$
$83$ $$(T^{2} + 6 T - 138)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$97$ $$(T^{2} - 16 T + 16)^{2}$$