# 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

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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.

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 $$T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( -4 + 4 T + T^{2} )^{2}$$
$17$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$( 36 - 6 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 17 - 10 T + T^{2} )^{2}$$
$43$ $$( 23 - 10 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$3969 + 378 T + 99 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$14161 + 2618 T + 365 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$( -56 - 8 T + T^{2} )^{2}$$
$73$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$79$ $$18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4}$$
$83$ $$( -161 + 2 T + T^{2} )^{2}$$
$89$ $$529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 4 - 12 T + T^{2} )^{2}$$