# Properties

 Label 315.2.a.d Level 315 Weight 2 Character orbit 315.a Self dual yes Analytic conductor 2.515 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} + q^{5} + q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} + q^{5} + q^{7} + ( 1 - 2 \beta ) q^{8} + ( 1 - 2 \beta ) q^{10} + ( -4 + 4 \beta ) q^{11} + ( 2 - 4 \beta ) q^{13} + ( 1 - 2 \beta ) q^{14} - q^{16} + 2 q^{17} + 4 \beta q^{19} + 3 q^{20} + ( -12 + 4 \beta ) q^{22} -4 q^{23} + q^{25} + 10 q^{26} + 3 q^{28} + 2 q^{29} + ( 4 + 4 \beta ) q^{31} + ( -3 + 6 \beta ) q^{32} + ( 2 - 4 \beta ) q^{34} + q^{35} + ( -2 + 8 \beta ) q^{37} + ( -8 - 4 \beta ) q^{38} + ( 1 - 2 \beta ) q^{40} + 2 q^{41} + ( 4 - 8 \beta ) q^{43} + ( -12 + 12 \beta ) q^{44} + ( -4 + 8 \beta ) q^{46} + ( -8 + 8 \beta ) q^{47} + q^{49} + ( 1 - 2 \beta ) q^{50} + ( 6 - 12 \beta ) q^{52} + ( 6 + 4 \beta ) q^{53} + ( -4 + 4 \beta ) q^{55} + ( 1 - 2 \beta ) q^{56} + ( 2 - 4 \beta ) q^{58} + ( 4 - 8 \beta ) q^{59} -2 q^{61} + ( -4 - 12 \beta ) q^{62} -13 q^{64} + ( 2 - 4 \beta ) q^{65} -4 q^{67} + 6 q^{68} + ( 1 - 2 \beta ) q^{70} + ( -8 - 4 \beta ) q^{71} + ( -10 + 4 \beta ) q^{73} + ( -18 - 4 \beta ) q^{74} + 12 \beta q^{76} + ( -4 + 4 \beta ) q^{77} + ( 8 - 8 \beta ) q^{79} - q^{80} + ( 2 - 4 \beta ) q^{82} + ( 12 - 8 \beta ) q^{83} + 2 q^{85} + 20 q^{86} + ( -12 + 4 \beta ) q^{88} + 2 q^{89} + ( 2 - 4 \beta ) q^{91} -12 q^{92} + ( -24 + 8 \beta ) q^{94} + 4 \beta q^{95} + ( 6 - 4 \beta ) q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{4} + 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q + 6q^{4} + 2q^{5} + 2q^{7} - 4q^{11} - 2q^{16} + 4q^{17} + 4q^{19} + 6q^{20} - 20q^{22} - 8q^{23} + 2q^{25} + 20q^{26} + 6q^{28} + 4q^{29} + 12q^{31} + 2q^{35} + 4q^{37} - 20q^{38} + 4q^{41} - 12q^{44} - 8q^{47} + 2q^{49} + 16q^{53} - 4q^{55} - 4q^{61} - 20q^{62} - 26q^{64} - 8q^{67} + 12q^{68} - 20q^{71} - 16q^{73} - 40q^{74} + 12q^{76} - 4q^{77} + 8q^{79} - 2q^{80} + 16q^{83} + 4q^{85} + 40q^{86} - 20q^{88} + 4q^{89} - 24q^{92} - 40q^{94} + 4q^{95} + 8q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−2.23607 0 3.00000 1.00000 0 1.00000 −2.23607 0 −2.23607
1.2 2.23607 0 3.00000 1.00000 0 1.00000 2.23607 0 2.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.a.d 2
3.b odd 2 1 105.2.a.b 2
4.b odd 2 1 5040.2.a.bw 2
5.b even 2 1 1575.2.a.r 2
5.c odd 4 2 1575.2.d.d 4
7.b odd 2 1 2205.2.a.w 2
12.b even 2 1 1680.2.a.v 2
15.d odd 2 1 525.2.a.g 2
15.e even 4 2 525.2.d.c 4
21.c even 2 1 735.2.a.k 2
21.g even 6 2 735.2.i.i 4
21.h odd 6 2 735.2.i.k 4
24.f even 2 1 6720.2.a.cs 2
24.h odd 2 1 6720.2.a.cx 2
60.h even 2 1 8400.2.a.cx 2
105.g even 2 1 3675.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 3.b odd 2 1
315.2.a.d 2 1.a even 1 1 trivial
525.2.a.g 2 15.d odd 2 1
525.2.d.c 4 15.e even 4 2
735.2.a.k 2 21.c even 2 1
735.2.i.i 4 21.g even 6 2
735.2.i.k 4 21.h odd 6 2
1575.2.a.r 2 5.b even 2 1
1575.2.d.d 4 5.c odd 4 2
1680.2.a.v 2 12.b even 2 1
2205.2.a.w 2 7.b odd 2 1
3675.2.a.y 2 105.g even 2 1
5040.2.a.bw 2 4.b odd 2 1
6720.2.a.cs 2 24.f even 2 1
6720.2.a.cx 2 24.h odd 2 1
8400.2.a.cx 2 60.h even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ 
$5$ $$( 1 - T )^{2}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 + 6 T^{2} + 1849 T^{4}$$
$47$ $$1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 16 T + 150 T^{2} - 848 T^{3} + 2809 T^{4}$$
$59$ $$1 + 38 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4}$$
$73$ $$1 + 16 T + 190 T^{2} + 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 16 T + 150 T^{2} - 1328 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 - 8 T + 190 T^{2} - 776 T^{3} + 9409 T^{4}$$