# Properties

 Label 315.2.b.a Level 315 Weight 2 Character orbit 315.b Analytic conductor 2.515 Analytic rank 0 Dimension 4 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0 −1.73205 −1.00000 0 1.00000 2.44949i 0.517638i 0 1.93185i
251.2 0.517638i 0 1.73205 −1.00000 0 1.00000 2.44949i 1.93185i 0 0.517638i
251.3 0.517638i 0 1.73205 −1.00000 0 1.00000 + 2.44949i 1.93185i 0 0.517638i
251.4 1.93185i 0 −1.73205 −1.00000 0 1.00000 + 2.44949i 0.517638i 0 1.93185i
 $$n$$: 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
21.c 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.b.a 4
3.b odd 2 1 315.2.b.b yes 4
4.b odd 2 1 5040.2.f.a 4
5.b even 2 1 1575.2.b.b 4
5.c odd 4 2 1575.2.g.c 8
7.b odd 2 1 315.2.b.b yes 4
12.b even 2 1 5040.2.f.c 4
15.d odd 2 1 1575.2.b.c 4
15.e even 4 2 1575.2.g.a 8
21.c even 2 1 inner 315.2.b.a 4
28.d even 2 1 5040.2.f.c 4
35.c odd 2 1 1575.2.b.c 4
35.f even 4 2 1575.2.g.a 8
84.h odd 2 1 5040.2.f.a 4
105.g even 2 1 1575.2.b.b 4
105.k odd 4 2 1575.2.g.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.b.a 4 1.a even 1 1 trivial
315.2.b.a 4 21.c even 2 1 inner
315.2.b.b yes 4 3.b odd 2 1
315.2.b.b yes 4 7.b odd 2 1
1575.2.b.b 4 5.b even 2 1
1575.2.b.b 4 105.g even 2 1
1575.2.b.c 4 15.d odd 2 1
1575.2.b.c 4 35.c odd 2 1
1575.2.g.a 8 15.e even 4 2
1575.2.g.a 8 35.f even 4 2
1575.2.g.c 8 5.c odd 4 2
1575.2.g.c 8 105.k odd 4 2
5040.2.f.a 4 4.b odd 2 1
5040.2.f.a 4 84.h odd 2 1
5040.2.f.c 4 12.b even 2 1
5040.2.f.c 4 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{47}^{2} - 12 T_{47} + 24$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8}$$
$3$ 
$5$ $$( 1 + T )^{4}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8}$$
$13$ $$1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8}$$
$17$ $$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$
$31$ $$1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8}$$