# Properties

 Label 3520.2.a.bn Level $3520$ Weight $2$ Character orbit 3520.a Self dual yes Analytic conductor $28.107$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1073415115$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} -2 q^{7} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + q^{5} -2 q^{7} + 5 q^{9} - q^{11} + ( 4 - \beta ) q^{13} + \beta q^{15} + ( 4 + \beta ) q^{17} -2 \beta q^{21} -\beta q^{23} + q^{25} + 2 \beta q^{27} + ( -2 + 2 \beta ) q^{29} -\beta q^{33} -2 q^{35} + ( 2 + 2 \beta ) q^{37} + ( -8 + 4 \beta ) q^{39} + 6 q^{41} + 6 q^{43} + 5 q^{45} + \beta q^{47} -3 q^{49} + ( 8 + 4 \beta ) q^{51} + ( -6 - 2 \beta ) q^{53} - q^{55} + ( 4 - 2 \beta ) q^{59} + ( -2 + 4 \beta ) q^{61} -10 q^{63} + ( 4 - \beta ) q^{65} + ( -4 - 3 \beta ) q^{67} -8 q^{69} + 4 \beta q^{71} + ( -4 + \beta ) q^{73} + \beta q^{75} + 2 q^{77} + 4 q^{79} + q^{81} + 6 q^{83} + ( 4 + \beta ) q^{85} + ( 16 - 2 \beta ) q^{87} + ( -2 - 4 \beta ) q^{89} + ( -8 + 2 \beta ) q^{91} + ( -2 + 2 \beta ) q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 4 q^{7} + 10 q^{9} + O(q^{10})$$ $$2 q + 2 q^{5} - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 8 q^{17} + 2 q^{25} - 4 q^{29} - 4 q^{35} + 4 q^{37} - 16 q^{39} + 12 q^{41} + 12 q^{43} + 10 q^{45} - 6 q^{49} + 16 q^{51} - 12 q^{53} - 2 q^{55} + 8 q^{59} - 4 q^{61} - 20 q^{63} + 8 q^{65} - 8 q^{67} - 16 q^{69} - 8 q^{73} + 4 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} + 8 q^{85} + 32 q^{87} - 4 q^{89} - 16 q^{91} - 4 q^{97} - 10 q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −2.82843 0 1.00000 0 −2.00000 0 5.00000 0
1.2 0 2.82843 0 1.00000 0 −2.00000 0 5.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.2.a.bn 2
4.b odd 2 1 3520.2.a.bo 2
8.b even 2 1 55.2.a.b 2
8.d odd 2 1 880.2.a.m 2
24.f even 2 1 7920.2.a.ch 2
24.h odd 2 1 495.2.a.b 2
40.e odd 2 1 4400.2.a.bn 2
40.f even 2 1 275.2.a.c 2
40.i odd 4 2 275.2.b.d 4
40.k even 4 2 4400.2.b.q 4
56.h odd 2 1 2695.2.a.f 2
88.b odd 2 1 605.2.a.d 2
88.g even 2 1 9680.2.a.bn 2
88.o even 10 4 605.2.g.f 8
88.p odd 10 4 605.2.g.l 8
104.e even 2 1 9295.2.a.g 2
120.i odd 2 1 2475.2.a.x 2
120.w even 4 2 2475.2.c.l 4
264.m even 2 1 5445.2.a.y 2
440.o odd 2 1 3025.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 8.b even 2 1
275.2.a.c 2 40.f even 2 1
275.2.b.d 4 40.i odd 4 2
495.2.a.b 2 24.h odd 2 1
605.2.a.d 2 88.b odd 2 1
605.2.g.f 8 88.o even 10 4
605.2.g.l 8 88.p odd 10 4
880.2.a.m 2 8.d odd 2 1
2475.2.a.x 2 120.i odd 2 1
2475.2.c.l 4 120.w even 4 2
2695.2.a.f 2 56.h odd 2 1
3025.2.a.o 2 440.o odd 2 1
3520.2.a.bn 2 1.a even 1 1 trivial
3520.2.a.bo 2 4.b odd 2 1
4400.2.a.bn 2 40.e odd 2 1
4400.2.b.q 4 40.k even 4 2
5445.2.a.y 2 264.m even 2 1
7920.2.a.ch 2 24.f even 2 1
9295.2.a.g 2 104.e even 2 1
9680.2.a.bn 2 88.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3520))$$:

 $$T_{3}^{2} - 8$$ $$T_{7} + 2$$ $$T_{13}^{2} - 8 T_{13} + 8$$ $$T_{17}^{2} - 8 T_{17} + 8$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$8 - 8 T + T^{2}$$
$17$ $$8 - 8 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-28 + 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$4 + 12 T + T^{2}$$
$59$ $$-16 - 8 T + T^{2}$$
$61$ $$-124 + 4 T + T^{2}$$
$67$ $$-56 + 8 T + T^{2}$$
$71$ $$-128 + T^{2}$$
$73$ $$8 + 8 T + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$-124 + 4 T + T^{2}$$
$97$ $$-28 + 4 T + T^{2}$$