# Properties

 Label 315.2.bf.a Level 315 Weight 2 Character orbit 315.bf Analytic conductor 2.515 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} -\zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} -\zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} + ( \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} -2 \zeta_{12}^{3} q^{13} + ( -1 - 2 \zeta_{12}^{2} ) q^{14} + ( 1 - \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( 6 - 6 \zeta_{12}^{2} ) q^{19} + ( -1 + 2 \zeta_{12}^{3} ) q^{20} + 3 \zeta_{12} q^{23} + ( -4 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( 2 - 2 \zeta_{12}^{2} ) q^{26} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + 7 q^{29} -2 \zeta_{12}^{2} q^{31} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{32} -2 q^{34} + ( 2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{35} + 8 \zeta_{12} q^{37} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{38} + ( -6 - 3 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{40} -5 q^{41} -7 \zeta_{12}^{3} q^{43} + 3 \zeta_{12}^{2} q^{46} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( -4 + 3 \zeta_{12}^{3} ) q^{50} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -6 + 9 \zeta_{12}^{2} ) q^{56} + 7 \zeta_{12} q^{58} -10 \zeta_{12}^{2} q^{59} + ( -7 + 7 \zeta_{12}^{2} ) q^{61} -2 \zeta_{12}^{3} q^{62} -7 q^{64} + ( -4 - 2 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{65} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{67} + 2 \zeta_{12} q^{68} + ( -3 + 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} + 2 q^{71} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + 8 \zeta_{12}^{2} q^{74} -6 q^{76} + ( -2 + 2 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{80} -5 \zeta_{12} q^{82} -11 \zeta_{12}^{3} q^{83} + ( 4 + 2 \zeta_{12}^{3} ) q^{85} + ( 7 - 7 \zeta_{12}^{2} ) q^{86} + ( -9 + 9 \zeta_{12}^{2} ) q^{89} + ( -4 + 6 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12}^{3} q^{92} + ( -12 \zeta_{12} - 6 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{95} -16 \zeta_{12}^{3} q^{97} + ( 5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + 2q^{5} + O(q^{10})$$ $$4q - 2q^{4} + 2q^{5} - 4q^{10} - 8q^{14} + 2q^{16} + 12q^{19} - 4q^{20} + 6q^{25} + 4q^{26} + 28q^{29} - 4q^{31} - 8q^{34} + 16q^{35} - 12q^{40} - 20q^{41} + 6q^{46} + 26q^{49} - 16q^{50} - 6q^{56} - 20q^{59} - 14q^{61} - 28q^{64} - 8q^{65} - 10q^{70} + 8q^{71} + 16q^{74} - 24q^{76} - 4q^{79} - 2q^{80} + 16q^{85} + 14q^{86} - 18q^{89} - 4q^{91} - 12q^{95} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 −0.500000 + 0.866025i 2.23205 0.133975i 0 2.59808 0.500000i 3.00000i 0 −1.86603 + 1.23205i
109.2 0.866025 0.500000i 0 −0.500000 + 0.866025i −1.23205 + 1.86603i 0 −2.59808 + 0.500000i 3.00000i 0 −0.133975 + 2.23205i
289.1 −0.866025 0.500000i 0 −0.500000 0.866025i 2.23205 + 0.133975i 0 2.59808 + 0.500000i 3.00000i 0 −1.86603 1.23205i
289.2 0.866025 + 0.500000i 0 −0.500000 0.866025i −1.23205 1.86603i 0 −2.59808 0.500000i 3.00000i 0 −0.133975 2.23205i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bf.a 4
3.b odd 2 1 35.2.j.a 4
5.b even 2 1 inner 315.2.bf.a 4
7.c even 3 1 inner 315.2.bf.a 4
7.c even 3 1 2205.2.d.d 2
7.d odd 6 1 2205.2.d.e 2
12.b even 2 1 560.2.bw.b 4
15.d odd 2 1 35.2.j.a 4
15.e even 4 1 175.2.e.a 2
15.e even 4 1 175.2.e.b 2
21.c even 2 1 245.2.j.c 4
21.g even 6 1 245.2.b.b 2
21.g even 6 1 245.2.j.c 4
21.h odd 6 1 35.2.j.a 4
21.h odd 6 1 245.2.b.c 2
35.i odd 6 1 2205.2.d.e 2
35.j even 6 1 inner 315.2.bf.a 4
35.j even 6 1 2205.2.d.d 2
60.h even 2 1 560.2.bw.b 4
84.n even 6 1 560.2.bw.b 4
105.g even 2 1 245.2.j.c 4
105.o odd 6 1 35.2.j.a 4
105.o odd 6 1 245.2.b.c 2
105.p even 6 1 245.2.b.b 2
105.p even 6 1 245.2.j.c 4
105.w odd 12 1 1225.2.a.b 1
105.w odd 12 1 1225.2.a.g 1
105.x even 12 1 175.2.e.a 2
105.x even 12 1 175.2.e.b 2
105.x even 12 1 1225.2.a.d 1
105.x even 12 1 1225.2.a.f 1
420.ba even 6 1 560.2.bw.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 3.b odd 2 1
35.2.j.a 4 15.d odd 2 1
35.2.j.a 4 21.h odd 6 1
35.2.j.a 4 105.o odd 6 1
175.2.e.a 2 15.e even 4 1
175.2.e.a 2 105.x even 12 1
175.2.e.b 2 15.e even 4 1
175.2.e.b 2 105.x even 12 1
245.2.b.b 2 21.g even 6 1
245.2.b.b 2 105.p even 6 1
245.2.b.c 2 21.h odd 6 1
245.2.b.c 2 105.o odd 6 1
245.2.j.c 4 21.c even 2 1
245.2.j.c 4 21.g even 6 1
245.2.j.c 4 105.g even 2 1
245.2.j.c 4 105.p even 6 1
315.2.bf.a 4 1.a even 1 1 trivial
315.2.bf.a 4 5.b even 2 1 inner
315.2.bf.a 4 7.c even 3 1 inner
315.2.bf.a 4 35.j even 6 1 inner
560.2.bw.b 4 12.b even 2 1
560.2.bw.b 4 60.h even 2 1
560.2.bw.b 4 84.n even 6 1
560.2.bw.b 4 420.ba even 6 1
1225.2.a.b 1 105.w odd 12 1
1225.2.a.d 1 105.x even 12 1
1225.2.a.f 1 105.x even 12 1
1225.2.a.g 1 105.w odd 12 1
2205.2.d.d 2 7.c even 3 1
2205.2.d.d 2 35.j even 6 1
2205.2.d.e 2 7.d odd 6 1
2205.2.d.e 2 35.i odd 6 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$
$3$ 
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 7 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 5 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 37 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} )$$
$79$ $$( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 45 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 62 T^{2} + 9409 T^{4} )^{2}$$