# Properties

 Label 315.2.d.d 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + ( 2 + i ) q^{5} -i q^{7} + 3 i q^{8} +O(q^{10})$$ $$q + i q^{2} + q^{4} + ( 2 + i ) q^{5} -i q^{7} + 3 i q^{8} + ( -1 + 2 i ) q^{10} -4 i q^{13} + q^{14} - q^{16} + 2 i q^{17} + ( 2 + i ) q^{20} + ( 3 + 4 i ) q^{25} + 4 q^{26} -i q^{28} -8 q^{29} -4 q^{31} + 5 i q^{32} -2 q^{34} + ( 1 - 2 i ) q^{35} + 8 i q^{37} + ( -3 + 6 i ) q^{40} + 4 q^{41} -8 i q^{43} -12 i q^{47} - q^{49} + ( -4 + 3 i ) q^{50} -4 i q^{52} -6 i q^{53} + 3 q^{56} -8 i q^{58} -8 q^{59} + 10 q^{61} -4 i q^{62} -7 q^{64} + ( 4 - 8 i ) q^{65} + 8 i q^{67} + 2 i q^{68} + ( 2 + i ) q^{70} -16 q^{71} -12 i q^{73} -8 q^{74} + 8 q^{79} + ( -2 - i ) q^{80} + 4 i q^{82} -16 i q^{83} + ( -2 + 4 i ) q^{85} + 8 q^{86} + 12 q^{89} -4 q^{91} + 12 q^{94} + 4 i q^{97} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 4q^{5} + O(q^{10})$$ $$2q + 2q^{4} + 4q^{5} - 2q^{10} + 2q^{14} - 2q^{16} + 4q^{20} + 6q^{25} + 8q^{26} - 16q^{29} - 8q^{31} - 4q^{34} + 2q^{35} - 6q^{40} + 8q^{41} - 2q^{49} - 8q^{50} + 6q^{56} - 16q^{59} + 20q^{61} - 14q^{64} + 8q^{65} + 4q^{70} - 32q^{71} - 16q^{74} + 16q^{79} - 4q^{80} - 4q^{85} + 16q^{86} + 24q^{89} - 8q^{91} + 24q^{94} + O(q^{100})$$

## 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
1.00000i 0 1.00000 2.00000 1.00000i 0 1.00000i 3.00000i 0 −1.00000 2.00000i
64.2 1.00000i 0 1.00000 2.00000 + 1.00000i 0 1.00000i 3.00000i 0 −1.00000 + 2.00000i
 $$n$$: 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.d yes 2
3.b odd 2 1 315.2.d.b 2
4.b odd 2 1 5040.2.t.r 2
5.b even 2 1 inner 315.2.d.d yes 2
5.c odd 4 1 1575.2.a.d 1
5.c odd 4 1 1575.2.a.g 1
7.b odd 2 1 2205.2.d.c 2
12.b even 2 1 5040.2.t.c 2
15.d odd 2 1 315.2.d.b 2
15.e even 4 1 1575.2.a.b 1
15.e even 4 1 1575.2.a.j 1
20.d odd 2 1 5040.2.t.r 2
21.c even 2 1 2205.2.d.g 2
35.c odd 2 1 2205.2.d.c 2
60.h even 2 1 5040.2.t.c 2
105.g even 2 1 2205.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.d.b 2 3.b odd 2 1
315.2.d.b 2 15.d odd 2 1
315.2.d.d yes 2 1.a even 1 1 trivial
315.2.d.d yes 2 5.b even 2 1 inner
1575.2.a.b 1 15.e even 4 1
1575.2.a.d 1 5.c odd 4 1
1575.2.a.g 1 5.c odd 4 1
1575.2.a.j 1 15.e even 4 1
2205.2.d.c 2 7.b odd 2 1
2205.2.d.c 2 35.c odd 2 1
2205.2.d.g 2 21.c even 2 1
2205.2.d.g 2 105.g even 2 1
5040.2.t.c 2 12.b even 2 1
5040.2.t.c 2 60.h even 2 1
5040.2.t.r 2 4.b odd 2 1
5040.2.t.r 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 $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{11}$$ $$T_{29} + 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 4 T^{4}$$
$3$ 
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 4 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 + 50 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 70 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 16 T + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 90 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 12 T + 89 T^{2} )^{2}$$
$97$ $$1 - 178 T^{2} + 9409 T^{4}$$