# Properties

 Label 322.2.c.a Level $322$ Weight $2$ Character orbit 322.c Analytic conductor $2.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} -\beta_{3} q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{1} q^{6} -\beta_{3} q^{7} - q^{8} + q^{9} + \beta_{2} q^{11} + \beta_{1} q^{12} + 3 \beta_{1} q^{13} + \beta_{3} q^{14} + q^{16} - q^{18} -2 \beta_{3} q^{19} -\beta_{2} q^{21} -\beta_{2} q^{22} + ( -3 - \beta_{2} ) q^{23} -\beta_{1} q^{24} -5 q^{25} -3 \beta_{1} q^{26} + 4 \beta_{1} q^{27} -\beta_{3} q^{28} + 6 q^{29} + 6 \beta_{1} q^{31} - q^{32} -2 \beta_{3} q^{33} + q^{36} + 3 \beta_{2} q^{37} + 2 \beta_{3} q^{38} -6 q^{39} -4 \beta_{1} q^{41} + \beta_{2} q^{42} -3 \beta_{2} q^{43} + \beta_{2} q^{44} + ( 3 + \beta_{2} ) q^{46} + 2 \beta_{1} q^{47} + \beta_{1} q^{48} + 7 q^{49} + 5 q^{50} + 3 \beta_{1} q^{52} + \beta_{2} q^{53} -4 \beta_{1} q^{54} + \beta_{3} q^{56} -2 \beta_{2} q^{57} -6 q^{58} -7 \beta_{1} q^{59} + 4 \beta_{3} q^{61} -6 \beta_{1} q^{62} -\beta_{3} q^{63} + q^{64} + 2 \beta_{3} q^{66} + 3 \beta_{2} q^{67} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{69} + 6 q^{71} - q^{72} -6 \beta_{1} q^{73} -3 \beta_{2} q^{74} -5 \beta_{1} q^{75} -2 \beta_{3} q^{76} -7 \beta_{1} q^{77} + 6 q^{78} -5 q^{81} + 4 \beta_{1} q^{82} + 6 \beta_{3} q^{83} -\beta_{2} q^{84} + 3 \beta_{2} q^{86} + 6 \beta_{1} q^{87} -\beta_{2} q^{88} + 6 \beta_{3} q^{89} -3 \beta_{2} q^{91} + ( -3 - \beta_{2} ) q^{92} -12 q^{93} -2 \beta_{1} q^{94} -\beta_{1} q^{96} -2 \beta_{3} q^{97} -7 q^{98} + \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} + 4q^{9} + 4q^{16} - 4q^{18} - 12q^{23} - 20q^{25} + 24q^{29} - 4q^{32} + 4q^{36} - 24q^{39} + 12q^{46} + 28q^{49} + 20q^{50} - 24q^{58} + 4q^{64} + 24q^{71} - 4q^{72} + 24q^{78} - 20q^{81} - 12q^{92} - 48q^{93} - 28q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 − 1.16372i 2.57794i 1.16372i − 2.57794i
−1.00000 1.41421i 1.00000 0 1.41421i −2.64575 −1.00000 1.00000 0
321.2 −1.00000 1.41421i 1.00000 0 1.41421i 2.64575 −1.00000 1.00000 0
321.3 −1.00000 1.41421i 1.00000 0 1.41421i −2.64575 −1.00000 1.00000 0
321.4 −1.00000 1.41421i 1.00000 0 1.41421i 2.64575 −1.00000 1.00000 0
 $$n$$: 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
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.c.a 4
3.b odd 2 1 2898.2.g.f 4
4.b odd 2 1 2576.2.f.d 4
7.b odd 2 1 inner 322.2.c.a 4
21.c even 2 1 2898.2.g.f 4
23.b odd 2 1 inner 322.2.c.a 4
28.d even 2 1 2576.2.f.d 4
69.c even 2 1 2898.2.g.f 4
92.b even 2 1 2576.2.f.d 4
161.c even 2 1 inner 322.2.c.a 4
483.c odd 2 1 2898.2.g.f 4
644.h odd 2 1 2576.2.f.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.a 4 1.a even 1 1 trivial
322.2.c.a 4 7.b odd 2 1 inner
322.2.c.a 4 23.b odd 2 1 inner
322.2.c.a 4 161.c even 2 1 inner
2576.2.f.d 4 4.b odd 2 1
2576.2.f.d 4 28.d even 2 1
2576.2.f.d 4 92.b even 2 1
2576.2.f.d 4 644.h odd 2 1
2898.2.g.f 4 3.b odd 2 1
2898.2.g.f 4 21.c even 2 1
2898.2.g.f 4 69.c even 2 1
2898.2.g.f 4 483.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$:

 $$T_{3}^{2} + 2$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$( 14 + T^{2} )^{2}$$
$13$ $$( 18 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -28 + T^{2} )^{2}$$
$23$ $$( 23 + 6 T + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 72 + T^{2} )^{2}$$
$37$ $$( 126 + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 126 + T^{2} )^{2}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( 14 + T^{2} )^{2}$$
$59$ $$( 98 + T^{2} )^{2}$$
$61$ $$( -112 + T^{2} )^{2}$$
$67$ $$( 126 + T^{2} )^{2}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$( 72 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( -252 + T^{2} )^{2}$$
$89$ $$( -252 + T^{2} )^{2}$$
$97$ $$( -28 + T^{2} )^{2}$$