# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} + ( -3 + 3 \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} + ( -3 + 3 \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} -3 q^{9} + ( -1 - \zeta_{6} ) q^{10} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 2 - \zeta_{6} ) q^{12} + ( -4 - 4 \zeta_{6} ) q^{13} + ( 4 + \zeta_{6} ) q^{14} + ( 1 - 2 \zeta_{6} ) q^{15} + ( 5 - 5 \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 3 + 3 \zeta_{6} ) q^{18} + \zeta_{6} q^{20} + ( -1 + 5 \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{22} + ( 3 - 6 \zeta_{6} ) q^{23} + 3 q^{24} + q^{25} + 12 \zeta_{6} q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( -3 + 3 \zeta_{6} ) q^{30} + ( -4 + 2 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} + 6 q^{33} + ( 12 - 6 \zeta_{6} ) q^{34} + ( -3 + \zeta_{6} ) q^{35} -3 \zeta_{6} q^{36} + 2 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{39} + ( -1 + 2 \zeta_{6} ) q^{40} + ( -6 + 6 \zeta_{6} ) q^{41} + ( 6 - 9 \zeta_{6} ) q^{42} -\zeta_{6} q^{43} + ( -4 + 2 \zeta_{6} ) q^{44} -3 q^{45} + ( -9 + 9 \zeta_{6} ) q^{46} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -5 - 5 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -1 - \zeta_{6} ) q^{50} + ( 6 + 6 \zeta_{6} ) q^{51} + ( 4 - 8 \zeta_{6} ) q^{52} + ( -2 - 2 \zeta_{6} ) q^{53} + ( 9 - 9 \zeta_{6} ) q^{54} + ( -2 + 4 \zeta_{6} ) q^{55} + ( 1 - 5 \zeta_{6} ) q^{56} + ( 2 - \zeta_{6} ) q^{60} + ( -7 - 7 \zeta_{6} ) q^{61} + 6 q^{62} + ( 9 - 3 \zeta_{6} ) q^{63} - q^{64} + ( -4 - 4 \zeta_{6} ) q^{65} + ( -6 - 6 \zeta_{6} ) q^{66} -5 \zeta_{6} q^{67} -6 q^{68} -9 q^{69} + ( 4 + \zeta_{6} ) q^{70} + ( -4 + 8 \zeta_{6} ) q^{71} + ( 3 - 6 \zeta_{6} ) q^{72} + ( -6 - 6 \zeta_{6} ) q^{73} + ( 2 - 4 \zeta_{6} ) q^{74} + ( 1 - 2 \zeta_{6} ) q^{75} + ( 2 - 10 \zeta_{6} ) q^{77} + ( 24 - 12 \zeta_{6} ) q^{78} + ( 10 - 10 \zeta_{6} ) q^{79} + ( 5 - 5 \zeta_{6} ) q^{80} + 9 q^{81} + ( 12 - 6 \zeta_{6} ) q^{82} + 12 \zeta_{6} q^{83} + ( -5 + 4 \zeta_{6} ) q^{84} + ( -6 + 6 \zeta_{6} ) q^{85} + ( -1 + 2 \zeta_{6} ) q^{86} -6 q^{88} -15 \zeta_{6} q^{89} + ( 3 + 3 \zeta_{6} ) q^{90} + ( 16 + 4 \zeta_{6} ) q^{91} + ( 6 - 3 \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{93} + ( -18 + 9 \zeta_{6} ) q^{94} + 9 \zeta_{6} q^{96} + ( -8 + 4 \zeta_{6} ) q^{97} + ( -13 + 2 \zeta_{6} ) q^{98} + ( 6 - 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} + q^{4} + 2q^{5} - 3q^{6} - 5q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 3q^{2} + q^{4} + 2q^{5} - 3q^{6} - 5q^{7} - 6q^{9} - 3q^{10} + 3q^{12} - 12q^{13} + 9q^{14} + 5q^{16} - 6q^{17} + 9q^{18} + q^{20} + 3q^{21} + 6q^{22} + 6q^{24} + 2q^{25} + 12q^{26} - 4q^{28} - 3q^{30} - 6q^{31} - 9q^{32} + 12q^{33} + 18q^{34} - 5q^{35} - 3q^{36} + 2q^{37} - 12q^{39} - 6q^{41} + 3q^{42} - q^{43} - 6q^{44} - 6q^{45} - 9q^{46} + 9q^{47} - 15q^{48} + 11q^{49} - 3q^{50} + 18q^{51} - 6q^{53} + 9q^{54} - 3q^{56} + 3q^{60} - 21q^{61} + 12q^{62} + 15q^{63} - 2q^{64} - 12q^{65} - 18q^{66} - 5q^{67} - 12q^{68} - 18q^{69} + 9q^{70} - 18q^{73} - 6q^{77} + 36q^{78} + 10q^{79} + 5q^{80} + 18q^{81} + 18q^{82} + 12q^{83} - 6q^{84} - 6q^{85} - 12q^{88} - 15q^{89} + 9q^{90} + 36q^{91} + 9q^{92} + 6q^{93} - 27q^{94} + 9q^{96} - 12q^{97} - 24q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$ $$1 - \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
236.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 1.73205i 0.500000 0.866025i 1.00000 −1.50000 2.59808i −2.50000 0.866025i 1.73205i −3.00000 −1.50000 + 0.866025i
311.1 −1.50000 0.866025i 1.73205i 0.500000 + 0.866025i 1.00000 −1.50000 + 2.59808i −2.50000 + 0.866025i 1.73205i −3.00000 −1.50000 0.866025i
 $$n$$: 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
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.be.a yes 2
3.b odd 2 1 945.2.be.a 2
7.d odd 6 1 315.2.t.a 2
9.c even 3 1 945.2.t.a 2
9.d odd 6 1 315.2.t.a 2
21.g even 6 1 945.2.t.a 2
63.k odd 6 1 945.2.be.a 2
63.s even 6 1 inner 315.2.be.a yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.a 2 7.d odd 6 1
315.2.t.a 2 9.d odd 6 1
315.2.be.a yes 2 1.a even 1 1 trivial
315.2.be.a yes 2 63.s even 6 1 inner
945.2.t.a 2 9.c even 3 1
945.2.t.a 2 21.g even 6 1
945.2.be.a 2 3.b odd 2 1
945.2.be.a 2 63.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T^{2}$$
$5$ $$( 1 - T )^{2}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 19 T^{2} + 361 T^{4}$$
$23$ $$1 - 19 T^{2} + 529 T^{4}$$
$29$ $$1 + 29 T^{2} + 841 T^{4}$$
$31$ $$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$
$41$ $$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$
$47$ $$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$
$53$ $$1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$1 - 94 T^{2} + 5041 T^{4}$$
$73$ $$1 + 18 T + 181 T^{2} + 1314 T^{3} + 5329 T^{4}$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 + 12 T + 145 T^{2} + 1164 T^{3} + 9409 T^{4}$$