# Properties

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

# 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: 4.0.2312.1 Defining polynomial: $$x^{4} - x^{3} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} + 2 q^{5} -\beta_{2} q^{6} + ( 1 - \beta_{1} ) q^{7} + q^{8} + ( -2 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} + 2 q^{5} -\beta_{2} q^{6} + ( 1 - \beta_{1} ) q^{7} + q^{8} + ( -2 + \beta_{1} - \beta_{3} ) q^{9} + 2 q^{10} + \beta_{2} q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} ) q^{14} -2 \beta_{2} q^{15} + q^{16} + ( -3 + \beta_{1} - \beta_{3} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{18} -4 q^{19} + 2 q^{20} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{21} + \beta_{2} q^{22} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{23} -\beta_{2} q^{24} - q^{25} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{27} + ( 1 - \beta_{1} ) q^{28} -2 q^{29} -2 \beta_{2} q^{30} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{31} + q^{32} + ( 5 - \beta_{1} + \beta_{3} ) q^{33} + ( -3 + \beta_{1} - \beta_{3} ) q^{34} + ( 2 - 2 \beta_{1} ) q^{35} + ( -2 + \beta_{1} - \beta_{3} ) q^{36} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} -4 q^{38} + ( 1 - \beta_{1} + \beta_{3} ) q^{39} + 2 q^{40} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{42} + \beta_{2} q^{43} + \beta_{2} q^{44} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{45} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} -\beta_{2} q^{48} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{49} - q^{50} + ( -2 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{52} + ( 1 - \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{54} + 2 \beta_{2} q^{55} + ( 1 - \beta_{1} ) q^{56} + 4 \beta_{2} q^{57} -2 q^{58} + 3 \beta_{2} q^{59} -2 \beta_{2} q^{60} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{61} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{62} + ( -8 - 2 \beta_{2} - 3 \beta_{3} ) q^{63} + q^{64} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 5 - \beta_{1} + \beta_{3} ) q^{66} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( -3 + \beta_{1} - \beta_{3} ) q^{68} + ( 5 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} ) q^{70} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{3} ) q^{72} -4 \beta_{2} q^{73} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + \beta_{2} q^{75} -4 q^{76} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{77} + ( 1 - \beta_{1} + \beta_{3} ) q^{78} + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{79} + 2 q^{80} + ( 6 - \beta_{1} + \beta_{3} ) q^{81} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{82} -4 q^{83} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{84} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{85} + \beta_{2} q^{86} + 2 \beta_{2} q^{87} + \beta_{2} q^{88} -2 q^{89} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{90} + ( 4 + \beta_{2} - 2 \beta_{3} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( -8 + 4 \beta_{1} - 4 \beta_{3} ) q^{93} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{94} -8 q^{95} -\beta_{2} q^{96} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{98} + ( 2 - 2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} + 8q^{5} + 2q^{7} + 4q^{8} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} + 8q^{5} + 2q^{7} + 4q^{8} - 8q^{9} + 8q^{10} + 2q^{14} + 4q^{16} - 12q^{17} - 8q^{18} - 16q^{19} + 8q^{20} + 8q^{21} + 2q^{23} - 4q^{25} + 2q^{28} - 8q^{29} + 4q^{32} + 20q^{33} - 12q^{34} + 4q^{35} - 8q^{36} - 16q^{38} + 4q^{39} + 8q^{40} + 8q^{42} - 16q^{45} + 2q^{46} + 8q^{49} - 4q^{50} + 2q^{56} - 8q^{58} - 16q^{61} - 38q^{63} + 4q^{64} + 20q^{66} - 12q^{68} + 16q^{69} + 4q^{70} + 12q^{71} - 8q^{72} - 16q^{76} - 8q^{77} + 4q^{78} + 8q^{80} + 24q^{81} - 16q^{83} + 8q^{84} - 24q^{85} - 8q^{89} - 16q^{90} + 12q^{91} + 2q^{92} - 32q^{93} - 32q^{95} + 16q^{97} + 8q^{98} + O(q^{100})$$

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

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

## 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
 −0.780776 + 1.17915i 1.28078 − 0.599676i 1.28078 + 0.599676i −0.780776 − 1.17915i
1.00000 3.02045i 1.00000 2.00000 3.02045i 2.56155 + 0.662153i 1.00000 −6.12311 2.00000
321.2 1.00000 0.936426i 1.00000 2.00000 0.936426i −1.56155 + 2.13578i 1.00000 2.12311 2.00000
321.3 1.00000 0.936426i 1.00000 2.00000 0.936426i −1.56155 2.13578i 1.00000 2.12311 2.00000
321.4 1.00000 3.02045i 1.00000 2.00000 3.02045i 2.56155 0.662153i 1.00000 −6.12311 2.00000
 $$n$$: 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
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.d yes 4
3.b odd 2 1 2898.2.g.a 4
4.b odd 2 1 2576.2.f.e 4
7.b odd 2 1 322.2.c.c 4
21.c even 2 1 2898.2.g.b 4
23.b odd 2 1 322.2.c.c 4
28.d even 2 1 2576.2.f.b 4
69.c even 2 1 2898.2.g.b 4
92.b even 2 1 2576.2.f.b 4
161.c even 2 1 inner 322.2.c.d yes 4
483.c odd 2 1 2898.2.g.a 4
644.h odd 2 1 2576.2.f.e 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 10 T_{3}^{2} + 8$$ $$T_{5} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$8 + 10 T^{2} + T^{4}$$
$5$ $$( -2 + T )^{4}$$
$7$ $$49 - 14 T - 2 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$8 + 10 T^{2} + T^{4}$$
$13$ $$32 + 14 T^{2} + T^{4}$$
$17$ $$( -8 + 6 T + T^{2} )^{2}$$
$19$ $$( 4 + T )^{4}$$
$23$ $$529 - 46 T + 30 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$5408 + 148 T^{2} + T^{4}$$
$37$ $$32 + 14 T^{2} + T^{4}$$
$41$ $$512 + 56 T^{2} + T^{4}$$
$43$ $$8 + 10 T^{2} + T^{4}$$
$47$ $$2048 + 92 T^{2} + T^{4}$$
$53$ $$32 + 190 T^{2} + T^{4}$$
$59$ $$648 + 90 T^{2} + T^{4}$$
$61$ $$( -52 + 8 T + T^{2} )^{2}$$
$67$ $$8 + 58 T^{2} + T^{4}$$
$71$ $$( -144 - 6 T + T^{2} )^{2}$$
$73$ $$2048 + 160 T^{2} + T^{4}$$
$79$ $$2592 + 180 T^{2} + T^{4}$$
$83$ $$( 4 + T )^{4}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( -52 - 8 T + T^{2} )^{2}$$