# Properties

 Label 315.2.i.b Level 315 Weight 2 Character orbit 315.i 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.i (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 2 + 2 \zeta_{6} ) q^{12} + \zeta_{6} q^{13} + ( 1 + \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} -3 q^{17} + 2 q^{19} + ( -2 + 2 \zeta_{6} ) q^{20} + ( -1 + 2 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -2 q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} - q^{35} + 6 q^{36} -4 q^{37} + ( 1 + \zeta_{6} ) q^{39} + ( 4 - 4 \zeta_{6} ) q^{43} + 3 q^{45} + ( -4 + 8 \zeta_{6} ) q^{48} -\zeta_{6} q^{49} + ( -6 + 3 \zeta_{6} ) q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} -6 q^{53} + ( 4 - 2 \zeta_{6} ) q^{57} -6 \zeta_{6} q^{59} + ( -2 + 4 \zeta_{6} ) q^{60} + ( -14 + 14 \zeta_{6} ) q^{61} + 3 \zeta_{6} q^{63} -8 q^{64} + ( -1 + \zeta_{6} ) q^{65} -14 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} + ( -6 - 6 \zeta_{6} ) q^{69} + 9 q^{71} + 11 q^{73} + ( -1 + 2 \zeta_{6} ) q^{75} + 4 \zeta_{6} q^{76} + ( -8 + 8 \zeta_{6} ) q^{79} -4 q^{80} -9 \zeta_{6} q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} + ( -4 + 2 \zeta_{6} ) q^{84} -3 \zeta_{6} q^{85} + ( 9 - 18 \zeta_{6} ) q^{87} -12 q^{89} - q^{91} + ( 12 - 12 \zeta_{6} ) q^{92} + ( 4 + 4 \zeta_{6} ) q^{93} + 2 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 2q^{4} + q^{5} - q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 2q^{4} + q^{5} - q^{7} + 3q^{9} + 6q^{12} + q^{13} + 3q^{15} - 4q^{16} - 6q^{17} + 4q^{19} - 2q^{20} - 6q^{23} - q^{25} - 4q^{28} + 9q^{29} + 4q^{31} - 2q^{35} + 12q^{36} - 8q^{37} + 3q^{39} + 4q^{43} + 6q^{45} - q^{49} - 9q^{51} - 2q^{52} - 12q^{53} + 6q^{57} - 6q^{59} - 14q^{61} + 3q^{63} - 16q^{64} - q^{65} - 14q^{67} - 6q^{68} - 18q^{69} + 18q^{71} + 22q^{73} + 4q^{76} - 8q^{79} - 8q^{80} - 9q^{81} - 9q^{83} - 6q^{84} - 3q^{85} - 24q^{89} - 2q^{91} + 12q^{92} + 12q^{93} + 2q^{95} - 2q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 1.00000 + 1.73205i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.50000 2.59808i 0
211.1 0 1.50000 + 0.866025i 1.00000 1.73205i 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.50000 + 2.59808i 0
 $$n$$: 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
9.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.i.b 2
3.b odd 2 1 945.2.i.a 2
9.c even 3 1 inner 315.2.i.b 2
9.c even 3 1 2835.2.a.c 1
9.d odd 6 1 945.2.i.a 2
9.d odd 6 1 2835.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.b 2 1.a even 1 1 trivial
315.2.i.b 2 9.c even 3 1 inner
945.2.i.a 2 3.b odd 2 1
945.2.i.a 2 9.d odd 6 1
2835.2.a.c 1 9.c even 3 1
2835.2.a.f 1 9.d odd 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

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + 4 T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 9 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 11 T + 73 T^{2} )^{2}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 12 T + 89 T^{2} )^{2}$$
$97$ $$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$