# Properties

 Label 320.6.a.u Level 320 Weight 6 Character orbit 320.a Self dual yes Analytic conductor 51.323 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.3228223402$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{85})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{85}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + 25 q^{5} -3 \beta q^{7} + 97 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + 25 q^{5} -3 \beta q^{7} + 97 q^{9} + 26 \beta q^{11} -506 q^{13} -25 \beta q^{15} -1838 q^{17} + 112 \beta q^{19} + 1020 q^{21} + 103 \beta q^{23} + 625 q^{25} + 146 \beta q^{27} + 4530 q^{29} + 198 \beta q^{31} -8840 q^{33} -75 \beta q^{35} -338 q^{37} + 506 \beta q^{39} -6330 q^{41} -985 \beta q^{43} + 2425 q^{45} -219 \beta q^{47} -13747 q^{49} + 1838 \beta q^{51} + 15486 q^{53} + 650 \beta q^{55} -38080 q^{57} -396 \beta q^{59} + 16750 q^{61} -291 \beta q^{63} -12650 q^{65} + 741 \beta q^{67} -35020 q^{69} -2346 \beta q^{71} -20806 q^{73} -625 \beta q^{75} -26520 q^{77} -3788 \beta q^{79} -73211 q^{81} -5765 \beta q^{83} -45950 q^{85} -4530 \beta q^{87} -18310 q^{89} + 1518 \beta q^{91} -67320 q^{93} + 2800 \beta q^{95} + 49978 q^{97} + 2522 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 50q^{5} + 194q^{9} + O(q^{10})$$ $$2q + 50q^{5} + 194q^{9} - 1012q^{13} - 3676q^{17} + 2040q^{21} + 1250q^{25} + 9060q^{29} - 17680q^{33} - 676q^{37} - 12660q^{41} + 4850q^{45} - 27494q^{49} + 30972q^{53} - 76160q^{57} + 33500q^{61} - 25300q^{65} - 70040q^{69} - 41612q^{73} - 53040q^{77} - 146422q^{81} - 91900q^{85} - 36620q^{89} - 134640q^{93} + 99956q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 5.10977 −4.10977
0 −18.4391 0 25.0000 0 −55.3173 0 97.0000 0
1.2 0 18.4391 0 25.0000 0 55.3173 0 97.0000 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.a.u 2
4.b odd 2 1 inner 320.6.a.u 2
8.b even 2 1 160.6.a.c 2
8.d odd 2 1 160.6.a.c 2
40.e odd 2 1 800.6.a.j 2
40.f even 2 1 800.6.a.j 2
40.i odd 4 2 800.6.c.e 4
40.k even 4 2 800.6.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.c 2 8.b even 2 1
160.6.a.c 2 8.d odd 2 1
320.6.a.u 2 1.a even 1 1 trivial
320.6.a.u 2 4.b odd 2 1 inner
800.6.a.j 2 40.e odd 2 1
800.6.a.j 2 40.f even 2 1
800.6.c.e 4 40.i odd 4 2
800.6.c.e 4 40.k even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 340$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 146 T^{2} + 59049 T^{4}$$
$5$ $$( 1 - 25 T )^{2}$$
$7$ $$1 + 30554 T^{2} + 282475249 T^{4}$$
$11$ $$1 + 92262 T^{2} + 25937424601 T^{4}$$
$13$ $$( 1 + 506 T + 371293 T^{2} )^{2}$$
$17$ $$( 1 + 1838 T + 1419857 T^{2} )^{2}$$
$19$ $$1 + 687238 T^{2} + 6131066257801 T^{4}$$
$23$ $$1 + 9265626 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 4530 T + 20511149 T^{2} )^{2}$$
$31$ $$1 + 43928942 T^{2} + 819628286980801 T^{4}$$
$37$ $$( 1 + 338 T + 69343957 T^{2} )^{2}$$
$41$ $$( 1 + 6330 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 35859614 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 442383274 T^{2} + 52599132235830049 T^{4}$$
$53$ $$( 1 - 15486 T + 418195493 T^{2} )^{2}$$
$59$ $$1 + 1376531158 T^{2} + 511116753300641401 T^{4}$$
$61$ $$( 1 - 16750 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 2513562674 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$1 + 1737195262 T^{2} + 3255243551009881201 T^{4}$$
$73$ $$( 1 + 20806 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 + 1275471838 T^{2} + 9468276082626847201 T^{4}$$
$83$ $$1 - 3421895214 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 18310 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 - 49978 T + 8587340257 T^{2} )^{2}$$
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