# Properties

 Label 315.1.bg.a Level 315 Weight 1 Character orbit 315.bg Analytic conductor 0.157 Analytic rank 0 Dimension 2 Projective image $$D_{3}$$ CM discriminant -35 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 315.bg (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.157205478979$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.2835.1 Artin image $C_6\times S_3$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - q^{3} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + q^{9} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{12} + \zeta_{6}^{2} q^{13} + \zeta_{6}^{2} q^{15} -\zeta_{6} q^{16} + q^{17} + \zeta_{6} q^{20} -\zeta_{6} q^{21} -\zeta_{6} q^{25} - q^{27} - q^{28} -2 \zeta_{6} q^{29} -\zeta_{6} q^{33} + q^{35} + \zeta_{6}^{2} q^{36} -\zeta_{6}^{2} q^{39} - q^{44} -\zeta_{6}^{2} q^{45} -\zeta_{6} q^{47} + \zeta_{6} q^{48} + \zeta_{6}^{2} q^{49} - q^{51} -\zeta_{6} q^{52} + q^{55} -\zeta_{6} q^{60} + \zeta_{6} q^{63} + q^{64} + \zeta_{6} q^{65} + \zeta_{6}^{2} q^{68} - q^{71} + q^{73} + \zeta_{6} q^{75} + \zeta_{6}^{2} q^{77} + \zeta_{6} q^{79} - q^{80} + q^{81} -\zeta_{6} q^{83} + q^{84} -\zeta_{6}^{2} q^{85} + 2 \zeta_{6} q^{87} - q^{91} -\zeta_{6} q^{97} + \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - q^{4} + q^{5} + q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - q^{4} + q^{5} + q^{7} + 2q^{9} + q^{11} + q^{12} - q^{13} - q^{15} - q^{16} + 2q^{17} + q^{20} - q^{21} - q^{25} - 2q^{27} - 2q^{28} - 2q^{29} - q^{33} + 2q^{35} - q^{36} + q^{39} - 2q^{44} + q^{45} - q^{47} + q^{48} - q^{49} - 2q^{51} - q^{52} + 2q^{55} - q^{60} + q^{63} + 2q^{64} + q^{65} - q^{68} - 2q^{71} + 2q^{73} + q^{75} - q^{77} + q^{79} - 2q^{80} + 2q^{81} - q^{83} + 2q^{84} + q^{85} + 2q^{87} - 2q^{91} - q^{97} + q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i 0 1.00000 0
139.1 0 −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i 0 1.00000 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
9.c even 3 1 inner
315.bg odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.1.bg.a 2
3.b odd 2 1 945.1.bg.a 2
5.b even 2 1 315.1.bg.b yes 2
5.c odd 4 2 1575.1.y.a 4
7.b odd 2 1 315.1.bg.b yes 2
7.c even 3 1 2205.1.q.b 2
7.c even 3 1 2205.1.bn.b 2
7.d odd 6 1 2205.1.q.a 2
7.d odd 6 1 2205.1.bn.a 2
9.c even 3 1 inner 315.1.bg.a 2
9.c even 3 1 2835.1.e.a 1
9.d odd 6 1 945.1.bg.a 2
9.d odd 6 1 2835.1.e.c 1
15.d odd 2 1 945.1.bg.b 2
21.c even 2 1 945.1.bg.b 2
35.c odd 2 1 CM 315.1.bg.a 2
35.f even 4 2 1575.1.y.a 4
35.i odd 6 1 2205.1.q.b 2
35.i odd 6 1 2205.1.bn.b 2
35.j even 6 1 2205.1.q.a 2
35.j even 6 1 2205.1.bn.a 2
45.h odd 6 1 945.1.bg.b 2
45.h odd 6 1 2835.1.e.b 1
45.j even 6 1 315.1.bg.b yes 2
45.j even 6 1 2835.1.e.d 1
45.k odd 12 2 1575.1.y.a 4
63.g even 3 1 2205.1.q.b 2
63.h even 3 1 2205.1.bn.b 2
63.k odd 6 1 2205.1.q.a 2
63.l odd 6 1 315.1.bg.b yes 2
63.l odd 6 1 2835.1.e.d 1
63.o even 6 1 945.1.bg.b 2
63.o even 6 1 2835.1.e.b 1
63.t odd 6 1 2205.1.bn.a 2
105.g even 2 1 945.1.bg.a 2
315.q odd 6 1 2205.1.bn.b 2
315.r even 6 1 2205.1.bn.a 2
315.z even 6 1 945.1.bg.a 2
315.z even 6 1 2835.1.e.c 1
315.bg odd 6 1 inner 315.1.bg.a 2
315.bg odd 6 1 2835.1.e.a 1
315.bn odd 6 1 2205.1.q.b 2
315.bo even 6 1 2205.1.q.a 2
315.cb even 12 2 1575.1.y.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.1.bg.a 2 1.a even 1 1 trivial
315.1.bg.a 2 9.c even 3 1 inner
315.1.bg.a 2 35.c odd 2 1 CM
315.1.bg.a 2 315.bg odd 6 1 inner
315.1.bg.b yes 2 5.b even 2 1
315.1.bg.b yes 2 7.b odd 2 1
315.1.bg.b yes 2 45.j even 6 1
315.1.bg.b yes 2 63.l odd 6 1
945.1.bg.a 2 3.b odd 2 1
945.1.bg.a 2 9.d odd 6 1
945.1.bg.a 2 105.g even 2 1
945.1.bg.a 2 315.z even 6 1
945.1.bg.b 2 15.d odd 2 1
945.1.bg.b 2 21.c even 2 1
945.1.bg.b 2 45.h odd 6 1
945.1.bg.b 2 63.o even 6 1
1575.1.y.a 4 5.c odd 4 2
1575.1.y.a 4 35.f even 4 2
1575.1.y.a 4 45.k odd 12 2
1575.1.y.a 4 315.cb even 12 2
2205.1.q.a 2 7.d odd 6 1
2205.1.q.a 2 35.j even 6 1
2205.1.q.a 2 63.k odd 6 1
2205.1.q.a 2 315.bo even 6 1
2205.1.q.b 2 7.c even 3 1
2205.1.q.b 2 35.i odd 6 1
2205.1.q.b 2 63.g even 3 1
2205.1.q.b 2 315.bn odd 6 1
2205.1.bn.a 2 7.d odd 6 1
2205.1.bn.a 2 35.j even 6 1
2205.1.bn.a 2 63.t odd 6 1
2205.1.bn.a 2 315.r even 6 1
2205.1.bn.b 2 7.c even 3 1
2205.1.bn.b 2 35.i odd 6 1
2205.1.bn.b 2 63.h even 3 1
2205.1.bn.b 2 315.q odd 6 1
2835.1.e.a 1 9.c even 3 1
2835.1.e.a 1 315.bg odd 6 1
2835.1.e.b 1 45.h odd 6 1
2835.1.e.b 1 63.o even 6 1
2835.1.e.c 1 9.d odd 6 1
2835.1.e.c 1 315.z even 6 1
2835.1.e.d 1 45.j even 6 1
2835.1.e.d 1 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} + T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$13$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
$17$ $$( 1 - T + T^{2} )^{2}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$29$ $$( 1 + T + T^{2} )^{2}$$
$31$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$37$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$41$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$43$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$47$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
$53$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$59$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$61$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$67$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$71$ $$( 1 + T + T^{2} )^{2}$$
$73$ $$( 1 - T + T^{2} )^{2}$$
$79$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$83$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$
$89$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$97$ $$( 1 + T )^{2}( 1 - T + T^{2} )$$