# 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

## 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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

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.

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

## 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.

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 $$T_{2}^{4} + 2 T_{2}^{3} + 6 T_{2}^{2} - 4 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$( 13 - 8 T + T^{2} )^{2}$$
$17$ $$484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4}$$
$19$ $$121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$( -26 + 2 T + T^{2} )^{2}$$
$31$ $$9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$( -2 + 2 T + T^{2} )^{2}$$
$43$ $$( -23 + 4 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$10816 + 416 T + 120 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$4 + 20 T + 102 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$( 16 + 4 T + T^{2} )^{2}$$
$67$ $$1521 + 468 T + 183 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$( -26 + 2 T + T^{2} )^{2}$$
$73$ $$3481 - 472 T + 123 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$9801 - 594 T + 135 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$( -138 + 6 T + T^{2} )^{2}$$
$89$ $$19044 + 828 T + 174 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$( 16 - 16 T + T^{2} )^{2}$$