# Properties

 Label 315.2.p.c 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

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

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
1.00000 1.00000i 0 0 −1.58114 + 1.58114i 0 2.58114 + 0.581139i 2.00000 + 2.00000i 0 3.16228i
118.2 1.00000 1.00000i 0 0 1.58114 1.58114i 0 −0.581139 2.58114i 2.00000 + 2.00000i 0 3.16228i
307.1 1.00000 + 1.00000i 0 0 −1.58114 1.58114i 0 2.58114 0.581139i 2.00000 2.00000i 0 3.16228i
307.2 1.00000 + 1.00000i 0 0 1.58114 + 1.58114i 0 −0.581139 + 2.58114i 2.00000 2.00000i 0 3.16228i
 $$n$$: 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.c odd 4 1 inner
7.b odd 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.c 4
3.b odd 2 1 35.2.f.a 4
5.c odd 4 1 inner 315.2.p.c 4
7.b odd 2 1 inner 315.2.p.c 4
12.b even 2 1 560.2.bj.a 4
15.d odd 2 1 175.2.f.c 4
15.e even 4 1 35.2.f.a 4
15.e even 4 1 175.2.f.c 4
21.c even 2 1 35.2.f.a 4
21.g even 6 2 245.2.l.c 8
21.h odd 6 2 245.2.l.c 8
35.f even 4 1 inner 315.2.p.c 4
60.l odd 4 1 560.2.bj.a 4
84.h odd 2 1 560.2.bj.a 4
105.g even 2 1 175.2.f.c 4
105.k odd 4 1 35.2.f.a 4
105.k odd 4 1 175.2.f.c 4
105.w odd 12 2 245.2.l.c 8
105.x even 12 2 245.2.l.c 8
420.w even 4 1 560.2.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a 4 3.b odd 2 1
35.2.f.a 4 15.e even 4 1
35.2.f.a 4 21.c even 2 1
35.2.f.a 4 105.k odd 4 1
175.2.f.c 4 15.d odd 2 1
175.2.f.c 4 15.e even 4 1
175.2.f.c 4 105.g even 2 1
175.2.f.c 4 105.k odd 4 1
245.2.l.c 8 21.g even 6 2
245.2.l.c 8 21.h odd 6 2
245.2.l.c 8 105.w odd 12 2
245.2.l.c 8 105.x even 12 2
315.2.p.c 4 1.a even 1 1 trivial
315.2.p.c 4 5.c odd 4 1 inner
315.2.p.c 4 7.b odd 2 1 inner
315.2.p.c 4 35.f even 4 1 inner
560.2.bj.a 4 12.b even 2 1
560.2.bj.a 4 60.l odd 4 1
560.2.bj.a 4 84.h odd 2 1
560.2.bj.a 4 420.w even 4 1

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$1 + 25 T^{4}$$
$7$ $$1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4}$$
$11$ $$( 1 - T + 11 T^{2} )^{4}$$
$13$ $$1 + 103 T^{4} + 28561 T^{8}$$
$17$ $$1 + 263 T^{4} + 83521 T^{8}$$
$19$ $$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 8 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 2017 T^{4} + 4879681 T^{8}$$
$53$ $$( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 82 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 2 T + 2 T^{2} + 134 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 7538 T^{4} + 47458321 T^{8}$$
$89$ $$( 1 + 138 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$1 + 16903 T^{4} + 88529281 T^{8}$$