# Properties

 Label 315.2.p.d Level 315 Weight 2 Character orbit 315.p Analytic conductor 2.515 Analytic rank 0 Dimension 8 CM no Inner twists 8

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.40960000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{4} + \beta_{5} q^{5} + ( -1 - \beta_{3} - \beta_{7} ) q^{7} -3 \beta_{6} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{4} + \beta_{5} q^{5} + ( -1 - \beta_{3} - \beta_{7} ) q^{7} -3 \beta_{6} q^{8} + \beta_{7} q^{10} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{11} + 2 \beta_{2} q^{13} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{14} + q^{16} + ( -\beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} + \beta_{7} ) q^{19} + \beta_{4} q^{20} + ( 3 + 3 \beta_{3} ) q^{22} -2 \beta_{6} q^{23} -5 q^{25} + 2 \beta_{5} q^{26} + ( -1 - \beta_{2} + \beta_{3} ) q^{28} + ( 4 \beta_{1} + 4 \beta_{6} ) q^{29} + ( -3 \beta_{2} - 3 \beta_{7} ) q^{31} -5 \beta_{1} q^{32} + ( -\beta_{2} + \beta_{7} ) q^{34} + ( 5 \beta_{1} + \beta_{4} - \beta_{5} ) q^{35} + ( 1 + \beta_{3} ) q^{37} + ( -\beta_{4} - \beta_{5} ) q^{38} + 3 \beta_{2} q^{40} -4 \beta_{5} q^{41} + ( -2 + 2 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{44} + 2 q^{46} + ( -4 \beta_{4} + 4 \beta_{5} ) q^{47} + ( -2 \beta_{2} - 3 \beta_{3} + 2 \beta_{7} ) q^{49} -5 \beta_{1} q^{50} -2 \beta_{7} q^{52} + 10 \beta_{6} q^{53} + ( 3 \beta_{2} + 3 \beta_{7} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} ) q^{56} + ( -4 + 4 \beta_{3} ) q^{58} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{62} -7 \beta_{3} q^{64} + 10 \beta_{6} q^{65} + ( 4 + 4 \beta_{3} ) q^{67} + ( \beta_{4} + \beta_{5} ) q^{68} + ( \beta_{2} + 5 \beta_{3} - \beta_{7} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{6} ) q^{71} + 2 \beta_{2} q^{73} + ( \beta_{1} + \beta_{6} ) q^{74} + ( \beta_{2} + \beta_{7} ) q^{76} + ( -6 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{77} -12 \beta_{3} q^{79} + \beta_{5} q^{80} -4 \beta_{7} q^{82} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -5 - 5 \beta_{3} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{86} + ( 9 - 9 \beta_{3} ) q^{88} -6 \beta_{4} q^{89} + ( 10 - 2 \beta_{2} - 2 \beta_{7} ) q^{91} -2 \beta_{1} q^{92} + ( -4 \beta_{2} + 4 \beta_{7} ) q^{94} + ( -5 \beta_{1} - 5 \beta_{6} ) q^{95} -2 \beta_{7} q^{97} + ( -2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} + 8q^{16} + 24q^{22} - 40q^{25} - 8q^{28} + 8q^{37} - 16q^{43} + 16q^{46} - 32q^{58} + 32q^{67} - 40q^{85} + 72q^{88} + 80q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} + 6 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} + 13 \nu^{3}$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} + 29 \nu^{3}$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 3 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{4} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{5} - 9 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} + 29 \beta_{6}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$ $$1$$

## 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
 −0.437016 + 0.437016i 1.14412 − 1.14412i 0.437016 − 0.437016i −1.14412 + 1.14412i 1.14412 + 1.14412i −0.437016 − 0.437016i −1.14412 − 1.14412i 0.437016 + 0.437016i
−0.707107 + 0.707107i 0 1.00000i 2.23607i 0 −2.58114 0.581139i −2.12132 2.12132i 0 1.58114 + 1.58114i
118.2 −0.707107 + 0.707107i 0 1.00000i 2.23607i 0 0.581139 + 2.58114i −2.12132 2.12132i 0 −1.58114 1.58114i
118.3 0.707107 0.707107i 0 1.00000i 2.23607i 0 0.581139 + 2.58114i 2.12132 + 2.12132i 0 −1.58114 1.58114i
118.4 0.707107 0.707107i 0 1.00000i 2.23607i 0 −2.58114 0.581139i 2.12132 + 2.12132i 0 1.58114 + 1.58114i
307.1 −0.707107 0.707107i 0 1.00000i 2.23607i 0 0.581139 2.58114i −2.12132 + 2.12132i 0 −1.58114 + 1.58114i
307.2 −0.707107 0.707107i 0 1.00000i 2.23607i 0 −2.58114 + 0.581139i −2.12132 + 2.12132i 0 1.58114 1.58114i
307.3 0.707107 + 0.707107i 0 1.00000i 2.23607i 0 −2.58114 + 0.581139i 2.12132 2.12132i 0 1.58114 1.58114i
307.4 0.707107 + 0.707107i 0 1.00000i 2.23607i 0 0.581139 2.58114i 2.12132 2.12132i 0 −1.58114 + 1.58114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 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.d 8
3.b odd 2 1 inner 315.2.p.d 8
5.c odd 4 1 inner 315.2.p.d 8
7.b odd 2 1 inner 315.2.p.d 8
15.e even 4 1 inner 315.2.p.d 8
21.c even 2 1 inner 315.2.p.d 8
35.f even 4 1 inner 315.2.p.d 8
105.k odd 4 1 inner 315.2.p.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.p.d 8 1.a even 1 1 trivial
315.2.p.d 8 3.b odd 2 1 inner
315.2.p.d 8 5.c odd 4 1 inner
315.2.p.d 8 7.b odd 2 1 inner
315.2.p.d 8 15.e even 4 1 inner
315.2.p.d 8 21.c even 2 1 inner
315.2.p.d 8 35.f even 4 1 inner
315.2.p.d 8 105.k odd 4 1 inner

## Hecke kernels

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

 $$T_{2}^{4} + 1$$ $$T_{17}^{4} + 100$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} + 16 T^{8} )^{2}$$
$3$ 
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 4 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 2 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 28 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 706 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 28 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 2 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 4 T + 8 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 62 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{8}$$
$61$ $$( 1 - 61 T^{2} )^{8}$$
$67$ $$( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 124 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 5218 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 14 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 2098 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 2 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 11458 T^{4} + 88529281 T^{8} )^{2}$$