# Properties

 Label 315.2.p.a Level 315 Weight 2 Character orbit 315.p Analytic conductor 2.515 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( -2 + \beta_{2} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( -2 + \beta_{2} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( -2 \beta_{1} + \beta_{3} ) q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} + 2 \beta_{1} q^{13} + ( -5 + \beta_{1} + \beta_{3} ) q^{14} + q^{16} + ( 5 - 5 \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{3} ) q^{19} + ( -3 - 6 \beta_{2} ) q^{20} + ( -5 - 5 \beta_{2} ) q^{22} + 2 \beta_{3} q^{23} + ( 3 - 4 \beta_{2} ) q^{25} + 10 \beta_{2} q^{26} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{28} + ( -\beta_{1} - \beta_{3} ) q^{31} + 3 \beta_{1} q^{32} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{34} + ( -3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{35} + ( -3 - 3 \beta_{2} ) q^{37} + ( 5 + 5 \beta_{2} ) q^{38} + ( -\beta_{1} - 2 \beta_{3} ) q^{40} + ( 6 - 6 \beta_{2} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{44} -10 q^{46} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{50} + 6 \beta_{3} q^{52} -2 \beta_{3} q^{53} + ( \beta_{1} - 3 \beta_{3} ) q^{55} + ( -\beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{56} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( 5 - 5 \beta_{2} ) q^{62} + 13 \beta_{2} q^{64} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 8 + 8 \beta_{2} ) q^{67} + ( 15 + 15 \beta_{2} ) q^{68} + ( 10 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{71} -6 \beta_{1} q^{73} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{74} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{76} + ( 5 - 2 \beta_{1} - 5 \beta_{2} ) q^{77} + 4 \beta_{2} q^{79} + ( -2 + \beta_{2} ) q^{80} + ( -10 - 10 \beta_{2} ) q^{83} + ( -5 + 15 \beta_{2} ) q^{85} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{86} + ( 5 - 5 \beta_{2} ) q^{88} + 10 q^{89} + ( -10 + 2 \beta_{1} + 2 \beta_{3} ) q^{91} -6 \beta_{1} q^{92} + ( -\beta_{1} + 3 \beta_{3} ) q^{95} -2 \beta_{3} q^{97} + ( -10 - 10 \beta_{2} - 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} + 4q^{7} + O(q^{10})$$ $$4q - 8q^{5} + 4q^{7} - 20q^{14} + 4q^{16} + 20q^{17} - 12q^{20} - 20q^{22} + 12q^{25} - 12q^{28} - 12q^{35} - 12q^{37} + 20q^{38} + 24q^{43} - 40q^{46} + 20q^{62} + 32q^{67} + 60q^{68} + 40q^{70} + 20q^{77} - 8q^{80} - 40q^{83} - 20q^{85} + 20q^{88} + 40q^{89} - 40q^{91} - 40q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
−1.58114 + 1.58114i 0 3.00000i −2.00000 1.00000i 0 2.58114 + 0.581139i 1.58114 + 1.58114i 0 4.74342 1.58114i
118.2 1.58114 1.58114i 0 3.00000i −2.00000 1.00000i 0 −0.581139 2.58114i −1.58114 1.58114i 0 −4.74342 + 1.58114i
307.1 −1.58114 1.58114i 0 3.00000i −2.00000 + 1.00000i 0 2.58114 0.581139i 1.58114 1.58114i 0 4.74342 + 1.58114i
307.2 1.58114 + 1.58114i 0 3.00000i −2.00000 + 1.00000i 0 −0.581139 + 2.58114i −1.58114 + 1.58114i 0 −4.74342 1.58114i
 $$n$$: 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
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.a 4
3.b odd 2 1 315.2.p.b yes 4
5.c odd 4 1 315.2.p.b yes 4
7.b odd 2 1 315.2.p.b yes 4
15.e even 4 1 inner 315.2.p.a 4
21.c even 2 1 inner 315.2.p.a 4
35.f even 4 1 inner 315.2.p.a 4
105.k odd 4 1 315.2.p.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.p.a 4 1.a even 1 1 trivial
315.2.p.a 4 15.e even 4 1 inner
315.2.p.a 4 21.c even 2 1 inner
315.2.p.a 4 35.f even 4 1 inner
315.2.p.b yes 4 3.b odd 2 1
315.2.p.b yes 4 5.c odd 4 1
315.2.p.b yes 4 7.b odd 2 1
315.2.p.b yes 4 105.k odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 25$$ $$T_{17}^{2} - 10 T_{17} + 50$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{4} + 16 T^{8}$$
$3$ 
$5$ $$( 1 + 4 T + 5 T^{2} )^{2}$$
$7$ $$1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4}$$
$11$ $$( 1 + 12 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 12 T + 72 T^{2} - 276 T^{3} + 529 T^{4} )( 1 + 12 T + 72 T^{2} + 276 T^{3} + 529 T^{4} )$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 2209 T^{4} )^{2}$$
$53$ $$1 + 1778 T^{4} + 7890481 T^{8}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 + 38 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 16 T + 128 T^{2} - 1072 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 52 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 - 9502 T^{4} + 28398241 T^{8}$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{4}$$
$97$ $$1 + 11458 T^{4} + 88529281 T^{8}$$