# Properties

 Label 315.2.j.e 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (b2 + 1) * q^5 + (-b3 - b2 + b1) * q^7 + (b3 - 3) * q^8 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + (\beta_{3} + \beta_{2} + \beta_1) q^{10} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{3} - 2) q^{13} + (4 \beta_{2} + 2 \beta_1 + 3) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} - 2 \beta_1 q^{19} + (2 \beta_{3} - 1) q^{20} - 2 q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{23} + \beta_{2} q^{25} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{26} + (2 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 3) q^{28} + q^{29} - 6 \beta_{2} q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + ( - 4 \beta_{3} + 6) q^{34} + (\beta_{3} + 2 \beta_1 + 1) q^{35} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{38} + ( - 3 \beta_{2} - \beta_1 - 3) q^{40} + ( - 2 \beta_{3} + 5) q^{41} + (\beta_{3} + 5) q^{43} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{44} + \beta_{2} q^{46} + (2 \beta_{2} + 2) q^{47} + ( - 4 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 5) q^{49} + (\beta_{3} - 1) q^{50} + ( - 6 \beta_{3} - 10 \beta_{2} - 6 \beta_1) q^{52} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{53} + (2 \beta_{3} + 2) q^{55} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{56} + (\beta_{2} + \beta_1 + 1) q^{58} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{59} + (3 \beta_{2} - 6 \beta_1 + 3) q^{61} + ( - 6 \beta_{3} + 6) q^{62} + (2 \beta_{3} - 7) q^{64} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{65} + ( - \beta_{3} + 11 \beta_{2} - \beta_1) q^{67} + (10 \beta_{2} + 6 \beta_1 + 10) q^{68} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{70} + (6 \beta_{3} + 4) q^{71} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{73} + ( - 2 \beta_{3} + 8) q^{76} + ( - 4 \beta_{3} + 2 \beta_{2} - 6) q^{77} + ( - 12 \beta_{2} - 2 \beta_1 - 12) q^{79} - 3 \beta_{2} q^{80} + (9 \beta_{2} + 7 \beta_1 + 9) q^{82} + (9 \beta_{3} - 1) q^{83} + ( - 2 \beta_{3} + 2) q^{85} + (3 \beta_{2} + 4 \beta_1 + 3) q^{86} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{88} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{89} + (4 \beta_{3} - 2 \beta_{2} - 8) q^{91} + ( - \beta_{3} - 3) q^{92} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{94} + ( - 2 \beta_{3} - 2 \beta_1) q^{95} + ( - 4 \beta_{3} + 6) q^{97} + (3 \beta_{3} + 9 \beta_{2} + 7 \beta_1 + 8) q^{98}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (2*b3 + b2 + 2*b1) * q^4 + (b2 + 1) * q^5 + (-b3 - b2 + b1) * q^7 + (b3 - 3) * q^8 + (b3 + b2 + b1) * q^10 + (2*b3 - 2*b2 + 2*b1) * q^11 + (2*b3 - 2) * q^13 + (4*b2 + 2*b1 + 3) * q^14 + (-3*b2 - 3) * q^16 + (-2*b3 - 2*b2 - 2*b1) * q^17 - 2*b1 * q^19 + (2*b3 - 1) * q^20 - 2 * q^22 + (-b2 + b1 - 1) * q^23 + b2 * q^25 + (-6*b2 - 4*b1 - 6) * q^26 + (2*b3 + 5*b2 + 3*b1 - 3) * q^28 + q^29 - 6*b2 * q^31 + (-b3 + 3*b2 - b1) * q^32 + (-4*b3 + 6) * q^34 + (b3 + 2*b1 + 1) * q^35 + (-2*b3 - 4*b2 - 2*b1) * q^38 + (-3*b2 - b1 - 3) * q^40 + (-2*b3 + 5) * q^41 + (b3 + 5) * q^43 + (-6*b2 + 2*b1 - 6) * q^44 + b2 * q^46 + (2*b2 + 2) * q^47 + (-4*b3 + 5*b2 - 2*b1 + 5) * q^49 + (b3 - 1) * q^50 + (-6*b3 - 10*b2 - 6*b1) * q^52 + (-2*b3 + 4*b2 - 2*b1) * q^53 + (2*b3 + 2) * q^55 + (4*b3 + b2 - 2*b1 - 4) * q^56 + (b2 + b1 + 1) * q^58 + (-6*b3 + 4*b2 - 6*b1) * q^59 + (3*b2 - 6*b1 + 3) * q^61 + (-6*b3 + 6) * q^62 + (2*b3 - 7) * q^64 + (-2*b2 - 2*b1 - 2) * q^65 + (-b3 + 11*b2 - b1) * q^67 + (10*b2 + 6*b1 + 10) * q^68 + (2*b3 + 3*b2 + 2*b1 - 1) * q^70 + (6*b3 + 4) * q^71 + (-2*b3 + 2*b2 - 2*b1) * q^73 + (-2*b3 + 8) * q^76 + (-4*b3 + 2*b2 - 6) * q^77 + (-12*b2 - 2*b1 - 12) * q^79 - 3*b2 * q^80 + (9*b2 + 7*b1 + 9) * q^82 + (9*b3 - 1) * q^83 + (-2*b3 + 2) * q^85 + (3*b2 + 4*b1 + 3) * q^86 + (-4*b3 + 2*b2 - 4*b1) * q^88 + (-3*b2 - 4*b1 - 3) * q^89 + (4*b3 - 2*b2 - 8) * q^91 + (-b3 - 3) * q^92 + (2*b3 + 2*b2 + 2*b1) * q^94 + (-2*b3 - 2*b1) * q^95 + (-4*b3 + 6) * q^97 + (3*b3 + 9*b2 + 7*b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} - 6 q^{16} + 4 q^{17} - 4 q^{20} - 8 q^{22} - 2 q^{23} - 2 q^{25} - 12 q^{26} - 22 q^{28} + 4 q^{29} + 12 q^{31} - 6 q^{32} + 24 q^{34} + 4 q^{35} + 8 q^{38} - 6 q^{40} + 20 q^{41} + 20 q^{43} - 12 q^{44} - 2 q^{46} + 4 q^{47} + 10 q^{49} - 4 q^{50} + 20 q^{52} - 8 q^{53} + 8 q^{55} - 18 q^{56} + 2 q^{58} - 8 q^{59} + 6 q^{61} + 24 q^{62} - 28 q^{64} - 4 q^{65} - 22 q^{67} + 20 q^{68} - 10 q^{70} + 16 q^{71} - 4 q^{73} + 32 q^{76} - 28 q^{77} - 24 q^{79} + 6 q^{80} + 18 q^{82} - 4 q^{83} + 8 q^{85} + 6 q^{86} - 4 q^{88} - 6 q^{89} - 28 q^{91} - 12 q^{92} - 4 q^{94} + 24 q^{97} + 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 - 2 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 - 6 * q^16 + 4 * q^17 - 4 * q^20 - 8 * q^22 - 2 * q^23 - 2 * q^25 - 12 * q^26 - 22 * q^28 + 4 * q^29 + 12 * q^31 - 6 * q^32 + 24 * q^34 + 4 * q^35 + 8 * q^38 - 6 * q^40 + 20 * q^41 + 20 * q^43 - 12 * q^44 - 2 * q^46 + 4 * q^47 + 10 * q^49 - 4 * q^50 + 20 * q^52 - 8 * q^53 + 8 * q^55 - 18 * q^56 + 2 * q^58 - 8 * q^59 + 6 * q^61 + 24 * q^62 - 28 * q^64 - 4 * q^65 - 22 * q^67 + 20 * q^68 - 10 * q^70 + 16 * q^71 - 4 * q^73 + 32 * q^76 - 28 * q^77 - 24 * q^79 + 6 * q^80 + 18 * q^82 - 4 * q^83 + 8 * q^85 + 6 * q^86 - 4 * q^88 - 6 * q^89 - 28 * q^91 - 12 * q^92 - 4 * q^94 + 24 * q^97 + 14 * q^98

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

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

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i 0 0.914214 1.58346i 0.500000 + 0.866025i 0 −1.62132 2.09077i −1.58579 0 0.207107 0.358719i
46.2 1.20711 + 2.09077i 0 −1.91421 + 3.31552i 0.500000 + 0.866025i 0 2.62132 + 0.358719i −4.41421 0 −1.20711 + 2.09077i
226.1 −0.207107 + 0.358719i 0 0.914214 + 1.58346i 0.500000 0.866025i 0 −1.62132 + 2.09077i −1.58579 0 0.207107 + 0.358719i
226.2 1.20711 2.09077i 0 −1.91421 3.31552i 0.500000 0.866025i 0 2.62132 0.358719i −4.41421 0 −1.20711 2.09077i
 $$n$$: 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.e 4
3.b odd 2 1 35.2.e.a 4
7.c even 3 1 inner 315.2.j.e 4
7.c even 3 1 2205.2.a.n 2
7.d odd 6 1 2205.2.a.q 2
12.b even 2 1 560.2.q.k 4
15.d odd 2 1 175.2.e.c 4
15.e even 4 2 175.2.k.a 8
21.c even 2 1 245.2.e.e 4
21.g even 6 1 245.2.a.g 2
21.g even 6 1 245.2.e.e 4
21.h odd 6 1 35.2.e.a 4
21.h odd 6 1 245.2.a.h 2
84.j odd 6 1 3920.2.a.bv 2
84.n even 6 1 560.2.q.k 4
84.n even 6 1 3920.2.a.bq 2
105.o odd 6 1 175.2.e.c 4
105.o odd 6 1 1225.2.a.k 2
105.p even 6 1 1225.2.a.m 2
105.w odd 12 2 1225.2.b.h 4
105.x even 12 2 175.2.k.a 8
105.x even 12 2 1225.2.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 3.b odd 2 1
35.2.e.a 4 21.h odd 6 1
175.2.e.c 4 15.d odd 2 1
175.2.e.c 4 105.o odd 6 1
175.2.k.a 8 15.e even 4 2
175.2.k.a 8 105.x even 12 2
245.2.a.g 2 21.g even 6 1
245.2.a.h 2 21.h odd 6 1
245.2.e.e 4 21.c even 2 1
245.2.e.e 4 21.g even 6 1
315.2.j.e 4 1.a even 1 1 trivial
315.2.j.e 4 7.c even 3 1 inner
560.2.q.k 4 12.b even 2 1
560.2.q.k 4 84.n even 6 1
1225.2.a.k 2 105.o odd 6 1
1225.2.a.m 2 105.p even 6 1
1225.2.b.g 4 105.x even 12 2
1225.2.b.h 4 105.w odd 12 2
2205.2.a.n 2 7.c even 3 1
2205.2.a.q 2 7.d odd 6 1
3920.2.a.bq 2 84.n even 6 1
3920.2.a.bv 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49$$
$11$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$13$ $$(T^{2} + 4 T - 4)^{2}$$
$17$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} - 6 T + 36)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - 10 T + 17)^{2}$$
$43$ $$(T^{2} - 10 T + 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$59$ $$T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$61$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$67$ $$T^{4} + 22 T^{3} + 365 T^{2} + \cdots + 14161$$
$71$ $$(T^{2} - 8 T - 56)^{2}$$
$73$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$79$ $$T^{4} + 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$83$ $$(T^{2} + 2 T - 161)^{2}$$
$89$ $$T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529$$
$97$ $$(T^{2} - 12 T + 4)^{2}$$