# Properties

 Label 315.2.d.a Level 315 Weight 2 Character orbit 315.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 2.00000 + 1.00000i 0 1.00000i 0 0 2.00000 4.00000i
64.2 2.00000i 0 −2.00000 2.00000 1.00000i 0 1.00000i 0 0 2.00000 + 4.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.d.a 2
3.b odd 2 1 35.2.b.a 2
4.b odd 2 1 5040.2.t.p 2
5.b even 2 1 inner 315.2.d.a 2
5.c odd 4 1 1575.2.a.a 1
5.c odd 4 1 1575.2.a.k 1
7.b odd 2 1 2205.2.d.b 2
12.b even 2 1 560.2.g.b 2
15.d odd 2 1 35.2.b.a 2
15.e even 4 1 175.2.a.a 1
15.e even 4 1 175.2.a.c 1
20.d odd 2 1 5040.2.t.p 2
21.c even 2 1 245.2.b.a 2
21.g even 6 2 245.2.j.d 4
21.h odd 6 2 245.2.j.e 4
24.f even 2 1 2240.2.g.g 2
24.h odd 2 1 2240.2.g.h 2
35.c odd 2 1 2205.2.d.b 2
60.h even 2 1 560.2.g.b 2
60.l odd 4 1 2800.2.a.l 1
60.l odd 4 1 2800.2.a.w 1
105.g even 2 1 245.2.b.a 2
105.k odd 4 1 1225.2.a.a 1
105.k odd 4 1 1225.2.a.i 1
105.o odd 6 2 245.2.j.e 4
105.p even 6 2 245.2.j.d 4
120.i odd 2 1 2240.2.g.h 2
120.m even 2 1 2240.2.g.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 3.b odd 2 1
35.2.b.a 2 15.d odd 2 1
175.2.a.a 1 15.e even 4 1
175.2.a.c 1 15.e even 4 1
245.2.b.a 2 21.c even 2 1
245.2.b.a 2 105.g even 2 1
245.2.j.d 4 21.g even 6 2
245.2.j.d 4 105.p even 6 2
245.2.j.e 4 21.h odd 6 2
245.2.j.e 4 105.o odd 6 2
315.2.d.a 2 1.a even 1 1 trivial
315.2.d.a 2 5.b even 2 1 inner
560.2.g.b 2 12.b even 2 1
560.2.g.b 2 60.h even 2 1
1225.2.a.a 1 105.k odd 4 1
1225.2.a.i 1 105.k odd 4 1
1575.2.a.a 1 5.c odd 4 1
1575.2.a.k 1 5.c odd 4 1
2205.2.d.b 2 7.b odd 2 1
2205.2.d.b 2 35.c odd 2 1
2240.2.g.g 2 24.f even 2 1
2240.2.g.g 2 120.m even 2 1
2240.2.g.h 2 24.h odd 2 1
2240.2.g.h 2 120.i odd 2 1
2800.2.a.l 1 60.l odd 4 1
2800.2.a.w 1 60.l odd 4 1
5040.2.t.p 2 4.b odd 2 1
5040.2.t.p 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{2} + 4$$ $$T_{11} - 3$$ $$T_{29} + 5$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$3$ 
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 + 15 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 - 10 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 5 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 - 85 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 10 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$
$79$ $$( 1 - 5 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$1 - 145 T^{2} + 9409 T^{4}$$