# Properties

 Label 318.2.c.a Level $318$ Weight $2$ Character orbit 318.c Analytic conductor $2.539$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$318 = 2 \cdot 3 \cdot 53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 318.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.53924278428$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} -4 i q^{5} - q^{6} + q^{7} -i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} - q^{4} -4 i q^{5} - q^{6} + q^{7} -i q^{8} - q^{9} + 4 q^{10} + 5 q^{11} -i q^{12} + 2 q^{13} + i q^{14} + 4 q^{15} + q^{16} -i q^{18} -i q^{19} + 4 i q^{20} + i q^{21} + 5 i q^{22} -3 i q^{23} + q^{24} -11 q^{25} + 2 i q^{26} -i q^{27} - q^{28} + 5 q^{29} + 4 i q^{30} + 6 i q^{31} + i q^{32} + 5 i q^{33} -4 i q^{35} + q^{36} -10 q^{37} + q^{38} + 2 i q^{39} -4 q^{40} -3 i q^{41} - q^{42} + 6 q^{43} -5 q^{44} + 4 i q^{45} + 3 q^{46} + 6 q^{47} + i q^{48} -6 q^{49} -11 i q^{50} -2 q^{52} + ( 7 - 2 i ) q^{53} + q^{54} -20 i q^{55} -i q^{56} + q^{57} + 5 i q^{58} -12 q^{59} -4 q^{60} + 13 i q^{61} -6 q^{62} - q^{63} - q^{64} -8 i q^{65} -5 q^{66} -13 i q^{67} + 3 q^{69} + 4 q^{70} -7 i q^{71} + i q^{72} + 4 i q^{73} -10 i q^{74} -11 i q^{75} + i q^{76} + 5 q^{77} -2 q^{78} + 16 i q^{79} -4 i q^{80} + q^{81} + 3 q^{82} + 12 i q^{83} -i q^{84} + 6 i q^{86} + 5 i q^{87} -5 i q^{88} -14 q^{89} -4 q^{90} + 2 q^{91} + 3 i q^{92} -6 q^{93} + 6 i q^{94} -4 q^{95} - q^{96} + 5 q^{97} -6 i q^{98} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} + 2q^{7} - 2q^{9} + 8q^{10} + 10q^{11} + 4q^{13} + 8q^{15} + 2q^{16} + 2q^{24} - 22q^{25} - 2q^{28} + 10q^{29} + 2q^{36} - 20q^{37} + 2q^{38} - 8q^{40} - 2q^{42} + 12q^{43} - 10q^{44} + 6q^{46} + 12q^{47} - 12q^{49} - 4q^{52} + 14q^{53} + 2q^{54} + 2q^{57} - 24q^{59} - 8q^{60} - 12q^{62} - 2q^{63} - 2q^{64} - 10q^{66} + 6q^{69} + 8q^{70} + 10q^{77} - 4q^{78} + 2q^{81} + 6q^{82} - 28q^{89} - 8q^{90} + 4q^{91} - 12q^{93} - 8q^{95} - 2q^{96} + 10q^{97} - 10q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/318\mathbb{Z}\right)^\times$$.

 $$n$$ $$55$$ $$107$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 4.00000i −1.00000 1.00000 1.00000i −1.00000 4.00000
211.2 1.00000i 1.00000i −1.00000 4.00000i −1.00000 1.00000 1.00000i −1.00000 4.00000
 $$n$$: 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
53.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 318.2.c.a 2
3.b odd 2 1 954.2.c.d 2
4.b odd 2 1 2544.2.i.a 2
53.b even 2 1 inner 318.2.c.a 2
159.d odd 2 1 954.2.c.d 2
212.d odd 2 1 2544.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
318.2.c.a 2 1.a even 1 1 trivial
318.2.c.a 2 53.b even 2 1 inner
954.2.c.d 2 3.b odd 2 1
954.2.c.d 2 159.d odd 2 1
2544.2.i.a 2 4.b odd 2 1
2544.2.i.a 2 212.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(318, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$1 + T^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$9 + T^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$53 - 14 T + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$169 + T^{2}$$
$67$ $$169 + T^{2}$$
$71$ $$49 + T^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$256 + T^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$( -5 + T )^{2}$$