# Properties

 Label 315.2.bb.a Level 315 Weight 2 Character orbit 315.bb Analytic conductor 2.515 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{4} q^{4} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 \beta_{4} q^{4} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} -2 \beta_{2} q^{11} + \beta_{3} q^{13} + ( -4 + 4 \beta_{4} ) q^{16} + ( -2 \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{4} ) q^{19} + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{20} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{23} + ( -2 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{25} + 2 \beta_{3} q^{28} + \beta_{7} q^{29} + ( 6 - 3 \beta_{4} ) q^{31} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{35} + ( -\beta_{3} + \beta_{6} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{41} + ( \beta_{3} + 2 \beta_{6} ) q^{43} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{44} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{47} -7 \beta_{4} q^{49} -2 \beta_{6} q^{52} + ( -4 \beta_{1} + 2 \beta_{2} - 8 \beta_{5} + 4 \beta_{7} ) q^{53} + ( 2 - 2 \beta_{3} - 4 \beta_{4} ) q^{55} + ( 3 \beta_{2} + 6 \beta_{7} ) q^{59} + ( -6 - 6 \beta_{4} ) q^{61} -8 q^{64} + ( -\beta_{1} + 4 \beta_{2} + \beta_{5} + 3 \beta_{7} ) q^{65} + ( -6 \beta_{3} - 3 \beta_{6} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{68} -8 \beta_{7} q^{71} -5 \beta_{6} q^{73} + ( -2 + 4 \beta_{4} ) q^{76} + ( 4 \beta_{5} - 2 \beta_{7} ) q^{77} + ( 7 - 7 \beta_{4} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{7} ) q^{80} + ( -8 \beta_{5} + 4 \beta_{7} ) q^{83} + ( -7 + \beta_{3} + 2 \beta_{6} ) q^{85} + ( 7 \beta_{2} - 7 \beta_{7} ) q^{89} + ( 7 - 7 \beta_{4} ) q^{91} + ( 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} - 2 \beta_{7} ) q^{92} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{95} -2 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + O(q^{10})$$ $$8q + 8q^{4} - 16q^{16} + 12q^{19} + 8q^{25} + 36q^{31} - 28q^{49} - 72q^{61} - 64q^{64} + 28q^{79} - 56q^{85} + 28q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 203 \nu$$$$)/165$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 148$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$-8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{7} + 55 \nu^{5} - 440 \nu^{3} + 81 \nu$$$$)/495$$ $$\beta_{6}$$ $$=$$ $$($$$$-23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656$$$$)/495$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 55 \nu^{5} + 341 \nu^{3} - 81 \nu$$$$)/297$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 4 \beta_{4} + \beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 5 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{6} - 23 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$-24 \beta_{7} - 31 \beta_{5} - 24 \beta_{2} - 31 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-55 \beta_{3} - 148$$ $$\nu^{7}$$ $$=$$ $$-165 \beta_{2} - 203 \beta_{1}$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −1.00781 + 0.581861i −2.23256 + 1.28897i 2.23256 − 1.28897i 1.00781 − 0.581861i −1.00781 − 0.581861i −2.23256 − 1.28897i 2.23256 + 1.28897i 1.00781 + 0.581861i
0 0 1.00000 + 1.73205i −2.23256 + 0.125246i 0 −1.32288 + 2.29129i 0 0 0
89.2 0 0 1.00000 + 1.73205i −1.00781 1.99607i 0 1.32288 2.29129i 0 0 0
89.3 0 0 1.00000 + 1.73205i 1.00781 + 1.99607i 0 1.32288 2.29129i 0 0 0
89.4 0 0 1.00000 + 1.73205i 2.23256 0.125246i 0 −1.32288 + 2.29129i 0 0 0
269.1 0 0 1.00000 1.73205i −2.23256 0.125246i 0 −1.32288 2.29129i 0 0 0
269.2 0 0 1.00000 1.73205i −1.00781 + 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.3 0 0 1.00000 1.73205i 1.00781 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.4 0 0 1.00000 1.73205i 2.23256 + 0.125246i 0 −1.32288 2.29129i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p 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.bb.a 8
3.b odd 2 1 inner 315.2.bb.a 8
5.b even 2 1 inner 315.2.bb.a 8
5.c odd 4 2 1575.2.bk.d 8
7.c even 3 1 2205.2.g.a 8
7.d odd 6 1 inner 315.2.bb.a 8
7.d odd 6 1 2205.2.g.a 8
15.d odd 2 1 inner 315.2.bb.a 8
15.e even 4 2 1575.2.bk.d 8
21.g even 6 1 inner 315.2.bb.a 8
21.g even 6 1 2205.2.g.a 8
21.h odd 6 1 2205.2.g.a 8
35.i odd 6 1 inner 315.2.bb.a 8
35.i odd 6 1 2205.2.g.a 8
35.j even 6 1 2205.2.g.a 8
35.k even 12 2 1575.2.bk.d 8
105.o odd 6 1 2205.2.g.a 8
105.p even 6 1 inner 315.2.bb.a 8
105.p even 6 1 2205.2.g.a 8
105.w odd 12 2 1575.2.bk.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.a 8 1.a even 1 1 trivial
315.2.bb.a 8 3.b odd 2 1 inner
315.2.bb.a 8 5.b even 2 1 inner
315.2.bb.a 8 7.d odd 6 1 inner
315.2.bb.a 8 15.d odd 2 1 inner
315.2.bb.a 8 21.g even 6 1 inner
315.2.bb.a 8 35.i odd 6 1 inner
315.2.bb.a 8 105.p even 6 1 inner
1575.2.bk.d 8 5.c odd 4 2
1575.2.bk.d 8 15.e even 4 2
1575.2.bk.d 8 35.k even 12 2
1575.2.bk.d 8 105.w odd 12 2
2205.2.g.a 8 7.c even 3 1
2205.2.g.a 8 7.d odd 6 1
2205.2.g.a 8 21.g even 6 1
2205.2.g.a 8 21.h odd 6 1
2205.2.g.a 8 35.i odd 6 1
2205.2.g.a 8 35.j even 6 1
2205.2.g.a 8 105.o odd 6 1
2205.2.g.a 8 105.p even 6 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$
$3$ 1
$5$ $$1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8}$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 19 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 56 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 53 T^{2} + 1440 T^{4} + 72557 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 58 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 65 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 80 T^{2} + 4191 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 62 T^{2} + 1035 T^{4} + 174158 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 55 T^{2} - 1464 T^{4} - 246895 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 14 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 58 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 116 T^{2} + 5535 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 166 T^{2} + 9409 T^{4} )^{4}$$