# Properties

 Label 315.2.j.a Level 315 Weight 2 Character orbit 315.j Analytic conductor 2.515 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{10} + ( -6 + 6 \zeta_{6} ) q^{11} -3 q^{13} + ( 6 - 2 \zeta_{6} ) q^{14} + 4 \zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} -2 q^{20} + 12 q^{22} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} + ( -2 - 4 \zeta_{6} ) q^{28} + 8 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( 8 - 8 \zeta_{6} ) q^{32} + 8 q^{34} + ( -3 + \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + 6 q^{41} + q^{43} -12 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} + 2 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 2 q^{50} + ( 6 - 6 \zeta_{6} ) q^{52} + ( 4 - 4 \zeta_{6} ) q^{53} -6 q^{55} -16 \zeta_{6} q^{58} + ( -8 + 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + 2 q^{62} -8 q^{64} -3 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} -8 \zeta_{6} q^{68} + ( 2 + 4 \zeta_{6} ) q^{70} -6 q^{71} + ( -1 + \zeta_{6} ) q^{73} + ( -14 + 14 \zeta_{6} ) q^{74} + 2 q^{76} + ( -6 - 12 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} -12 \zeta_{6} q^{82} -2 q^{83} -4 q^{85} -2 \zeta_{6} q^{86} -12 \zeta_{6} q^{89} + ( 6 - 9 \zeta_{6} ) q^{91} + 8 q^{92} + ( 4 - 4 \zeta_{6} ) q^{94} + ( 1 - \zeta_{6} ) q^{95} -6 q^{97} + ( -6 + 16 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{4} + q^{5} - q^{7} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{4} + q^{5} - q^{7} + 2q^{10} - 6q^{11} - 6q^{13} + 10q^{14} + 4q^{16} - 4q^{17} - q^{19} - 4q^{20} + 24q^{22} - 4q^{23} - q^{25} + 6q^{26} - 8q^{28} + 16q^{29} - q^{31} + 8q^{32} + 16q^{34} - 5q^{35} - 7q^{37} - 2q^{38} + 12q^{41} + 2q^{43} - 12q^{44} - 8q^{46} + 2q^{47} - 13q^{49} + 4q^{50} + 6q^{52} + 4q^{53} - 12q^{55} - 16q^{58} - 8q^{59} + 14q^{61} + 4q^{62} - 16q^{64} - 3q^{65} - 7q^{67} - 8q^{68} + 8q^{70} - 12q^{71} - q^{73} - 14q^{74} + 4q^{76} - 24q^{77} + q^{79} - 4q^{80} - 12q^{82} - 4q^{83} - 8q^{85} - 2q^{86} - 12q^{89} + 3q^{91} + 16q^{92} + 4q^{94} + q^{95} - 12q^{97} + 4q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −1.00000 + 1.73205i 0.500000 + 0.866025i 0 −0.500000 + 2.59808i 0 0 1.00000 1.73205i
226.1 −1.00000 + 1.73205i 0 −1.00000 1.73205i 0.500000 0.866025i 0 −0.500000 2.59808i 0 0 1.00000 + 1.73205i
 $$n$$: 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.a 2
3.b odd 2 1 105.2.i.b 2
7.c even 3 1 inner 315.2.j.a 2
7.c even 3 1 2205.2.a.k 1
7.d odd 6 1 2205.2.a.m 1
12.b even 2 1 1680.2.bg.l 2
15.d odd 2 1 525.2.i.a 2
15.e even 4 2 525.2.r.d 4
21.c even 2 1 735.2.i.f 2
21.g even 6 1 735.2.a.a 1
21.g even 6 1 735.2.i.f 2
21.h odd 6 1 105.2.i.b 2
21.h odd 6 1 735.2.a.b 1
84.n even 6 1 1680.2.bg.l 2
105.o odd 6 1 525.2.i.a 2
105.o odd 6 1 3675.2.a.o 1
105.p even 6 1 3675.2.a.p 1
105.x even 12 2 525.2.r.d 4

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ 
$5$ $$1 - T + T^{2}$$
$7$ $$1 + T + 7 T^{2}$$
$11$ $$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 3 T + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{2}$$
$31$ $$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - T + 43 T^{2} )^{2}$$
$47$ $$1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4}$$
$53$ $$1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4}$$
$59$ $$1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$
$79$ $$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 2 T + 83 T^{2} )^{2}$$
$89$ $$1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{2}$$