# Properties

 Label 315.2.d.e Level $315$ Weight $2$ Character orbit 315.d Analytic conductor $2.515$ Analytic rank $0$ Dimension $6$ 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{3} + \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{3} + \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{8} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{10} -2 q^{11} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{13} + \beta_{2} q^{14} + ( 3 - 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{17} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{19} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{20} + 2 \beta_{1} q^{22} + ( -\beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{23} + ( -1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( -4 - 4 \beta_{2} + \beta_{3} + \beta_{5} ) q^{26} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{28} + ( 2 + 2 \beta_{3} + 2 \beta_{5} ) q^{29} + ( 2 + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{31} + ( -\beta_{1} - 2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{32} + ( 4 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{34} + ( \beta_{2} - \beta_{5} ) q^{35} -4 \beta_{1} q^{37} + ( \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{38} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{40} + ( 4 \beta_{2} - \beta_{3} - \beta_{5} ) q^{41} + ( 4 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} ) q^{43} + ( 2 - 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 + \beta_{3} + \beta_{5} ) q^{46} + ( 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{47} - q^{49} + ( 2 + \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{50} + ( 2 \beta_{1} - \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 2 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{56} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} ) q^{59} + ( -2 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{61} + ( 3 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} ) q^{62} + ( -1 + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} ) q^{64} + ( -6 - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{65} + ( -2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{70} -2 q^{71} + ( 2 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{73} + ( -12 + 4 \beta_{3} + 4 \beta_{5} ) q^{74} + ( -2 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{76} -2 \beta_{4} q^{77} + ( -4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{79} + ( -4 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{80} + ( -2 \beta_{1} + 3 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} ) q^{82} + ( -4 \beta_{1} - 4 \beta_{4} ) q^{83} + ( 6 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{85} + ( 4 + 8 \beta_{2} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{88} + ( 4 - \beta_{3} - \beta_{5} ) q^{89} + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{92} + ( 4 - 2 \beta_{3} - 2 \beta_{5} ) q^{94} + ( -2 - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{95} + ( 2 \beta_{1} + 3 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{5} + O(q^{10})$$ $$6 q - 10 q^{4} - 2 q^{5} - 12 q^{10} - 12 q^{11} - 2 q^{14} + 26 q^{16} - 12 q^{19} + 30 q^{20} - 2 q^{25} - 20 q^{26} + 4 q^{29} + 4 q^{31} + 24 q^{34} + 4 q^{40} - 4 q^{41} + 20 q^{44} - 16 q^{46} - 6 q^{49} + 16 q^{50} + 4 q^{55} + 18 q^{56} + 32 q^{59} - 12 q^{61} - 26 q^{64} - 32 q^{65} + 10 q^{70} - 12 q^{71} - 88 q^{74} + 4 q^{76} - 24 q^{79} - 46 q^{80} + 32 q^{85} + 8 q^{86} + 28 q^{89} + 12 q^{91} + 32 q^{94} - 4 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + \nu^{4} + 11 \nu^{3} - 26 \nu^{2} + 6 \nu - 1$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{5} + 9 \nu^{4} - 16 \nu^{3} - 4 \nu^{2} + 8 \nu - 9$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{5} - 2 \nu^{4} + \nu^{3} + 6 \nu^{2} + 80 \nu + 2$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-16 \nu^{5} + 36 \nu^{4} - 41 \nu^{3} - 16 \nu^{2} - 60 \nu + 56$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 8 \beta_{1} - 9$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.854638 + 0.854638i 1.45161 + 1.45161i 0.403032 − 0.403032i 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 − 0.854638i
2.70928i 0 −5.34017 −2.17009 0.539189i 0 1.00000i 9.04945i 0 −1.46081 + 5.87936i
64.2 1.90321i 0 −1.62222 −0.311108 2.21432i 0 1.00000i 0.719004i 0 −4.21432 + 0.592104i
64.3 0.193937i 0 1.96239 1.48119 1.67513i 0 1.00000i 0.768452i 0 −0.324869 0.287258i
64.4 0.193937i 0 1.96239 1.48119 + 1.67513i 0 1.00000i 0.768452i 0 −0.324869 + 0.287258i
64.5 1.90321i 0 −1.62222 −0.311108 + 2.21432i 0 1.00000i 0.719004i 0 −4.21432 0.592104i
64.6 2.70928i 0 −5.34017 −2.17009 + 0.539189i 0 1.00000i 9.04945i 0 −1.46081 5.87936i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.6 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.b 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.d.e 6
3.b odd 2 1 105.2.d.b 6
4.b odd 2 1 5040.2.t.v 6
5.b even 2 1 inner 315.2.d.e 6
5.c odd 4 1 1575.2.a.w 3
5.c odd 4 1 1575.2.a.x 3
7.b odd 2 1 2205.2.d.l 6
12.b even 2 1 1680.2.t.k 6
15.d odd 2 1 105.2.d.b 6
15.e even 4 1 525.2.a.j 3
15.e even 4 1 525.2.a.k 3
20.d odd 2 1 5040.2.t.v 6
21.c even 2 1 735.2.d.b 6
21.g even 6 2 735.2.q.f 12
21.h odd 6 2 735.2.q.e 12
35.c odd 2 1 2205.2.d.l 6
60.h even 2 1 1680.2.t.k 6
60.l odd 4 1 8400.2.a.dg 3
60.l odd 4 1 8400.2.a.dj 3
105.g even 2 1 735.2.d.b 6
105.k odd 4 1 3675.2.a.bi 3
105.k odd 4 1 3675.2.a.bj 3
105.o odd 6 2 735.2.q.e 12
105.p even 6 2 735.2.q.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 3.b odd 2 1
105.2.d.b 6 15.d odd 2 1
315.2.d.e 6 1.a even 1 1 trivial
315.2.d.e 6 5.b even 2 1 inner
525.2.a.j 3 15.e even 4 1
525.2.a.k 3 15.e even 4 1
735.2.d.b 6 21.c even 2 1
735.2.d.b 6 105.g even 2 1
735.2.q.e 12 21.h odd 6 2
735.2.q.e 12 105.o odd 6 2
735.2.q.f 12 21.g even 6 2
735.2.q.f 12 105.p even 6 2
1575.2.a.w 3 5.c odd 4 1
1575.2.a.x 3 5.c odd 4 1
1680.2.t.k 6 12.b even 2 1
1680.2.t.k 6 60.h even 2 1
2205.2.d.l 6 7.b odd 2 1
2205.2.d.l 6 35.c odd 2 1
3675.2.a.bi 3 105.k odd 4 1
3675.2.a.bj 3 105.k odd 4 1
5040.2.t.v 6 4.b odd 2 1
5040.2.t.v 6 20.d odd 2 1
8400.2.a.dg 3 60.l odd 4 1
8400.2.a.dj 3 60.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 11 T_{2}^{4} + 27 T_{2}^{2} + 1$$ $$T_{11} + 2$$ $$T_{29}^{3} - 2 T_{29}^{2} - 52 T_{29} + 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 27 T^{2} + 11 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$125 + 50 T + 15 T^{2} + 12 T^{3} + 3 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( 2 + T )^{6}$$
$13$ $$64 + 112 T^{2} + 44 T^{4} + T^{6}$$
$17$ $$256 + 256 T^{2} + 32 T^{4} + T^{6}$$
$19$ $$( -40 - 4 T + 6 T^{2} + T^{3} )^{2}$$
$23$ $$256 + 192 T^{2} + 32 T^{4} + T^{6}$$
$29$ $$( 40 - 52 T - 2 T^{2} + T^{3} )^{2}$$
$31$ $$( 184 - 52 T - 2 T^{2} + T^{3} )^{2}$$
$37$ $$4096 + 6912 T^{2} + 176 T^{4} + T^{6}$$
$41$ $$( -200 - 60 T + 2 T^{2} + T^{3} )^{2}$$
$43$ $$692224 + 27392 T^{2} + 304 T^{4} + T^{6}$$
$47$ $$16384 + 3072 T^{2} + 128 T^{4} + T^{6}$$
$53$ $$87616 + 8432 T^{2} + 172 T^{4} + T^{6}$$
$59$ $$( 1280 - 64 T - 16 T^{2} + T^{3} )^{2}$$
$61$ $$( -248 - 52 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$16384 + 3072 T^{2} + 128 T^{4} + T^{6}$$
$71$ $$( 2 + T )^{6}$$
$73$ $$10816 + 4720 T^{2} + 140 T^{4} + T^{6}$$
$79$ $$( -320 - 16 T + 12 T^{2} + T^{3} )^{2}$$
$83$ $$65536 + 8192 T^{2} + 192 T^{4} + T^{6}$$
$89$ $$( -40 + 52 T - 14 T^{2} + T^{3} )^{2}$$
$97$ $$3474496 + 83312 T^{2} + 556 T^{4} + T^{6}$$