# Properties

 Label 3675.2.a.s Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 735) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} + ( 1 - \beta ) q^{6} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} + ( 1 - \beta ) q^{6} + ( -3 + \beta ) q^{8} + q^{9} -2 \beta q^{11} + ( -1 + 2 \beta ) q^{12} + ( 2 - 2 \beta ) q^{13} + 3 q^{16} + ( 2 - 4 \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( 2 + 2 \beta ) q^{19} + ( -4 + 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + ( 3 - \beta ) q^{24} + ( -6 + 4 \beta ) q^{26} - q^{27} + 6 q^{29} + ( -2 - 6 \beta ) q^{31} + ( 3 + \beta ) q^{32} + 2 \beta q^{33} + ( -10 + 6 \beta ) q^{34} + ( 1 - 2 \beta ) q^{36} + ( -2 - 4 \beta ) q^{37} + 2 q^{38} + ( -2 + 2 \beta ) q^{39} + ( 6 - 4 \beta ) q^{41} -8 q^{43} + ( 8 - 2 \beta ) q^{44} + ( -10 + 6 \beta ) q^{46} + 4 \beta q^{47} -3 q^{48} + ( -2 + 4 \beta ) q^{51} + ( 10 - 6 \beta ) q^{52} -6 \beta q^{53} + ( 1 - \beta ) q^{54} + ( -2 - 2 \beta ) q^{57} + ( -6 + 6 \beta ) q^{58} + ( 8 + 4 \beta ) q^{59} + ( -10 + 4 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 4 - 2 \beta ) q^{66} + ( 18 - 8 \beta ) q^{68} + ( -2 + 4 \beta ) q^{69} + 2 \beta q^{71} + ( -3 + \beta ) q^{72} + ( 14 + 2 \beta ) q^{73} + ( -6 + 2 \beta ) q^{74} + ( -6 - 2 \beta ) q^{76} + ( 6 - 4 \beta ) q^{78} -8 q^{79} + q^{81} + ( -14 + 10 \beta ) q^{82} + ( -4 - 4 \beta ) q^{83} + ( 8 - 8 \beta ) q^{86} -6 q^{87} + ( -4 + 6 \beta ) q^{88} + ( 6 + 8 \beta ) q^{89} + ( 18 - 8 \beta ) q^{92} + ( 2 + 6 \beta ) q^{93} + ( 8 - 4 \beta ) q^{94} + ( -3 - \beta ) q^{96} + ( 2 + 6 \beta ) q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{8} + 2q^{9} - 2q^{12} + 4q^{13} + 6q^{16} + 4q^{17} - 2q^{18} + 4q^{19} - 8q^{22} + 4q^{23} + 6q^{24} - 12q^{26} - 2q^{27} + 12q^{29} - 4q^{31} + 6q^{32} - 20q^{34} + 2q^{36} - 4q^{37} + 4q^{38} - 4q^{39} + 12q^{41} - 16q^{43} + 16q^{44} - 20q^{46} - 6q^{48} - 4q^{51} + 20q^{52} + 2q^{54} - 4q^{57} - 12q^{58} + 16q^{59} - 20q^{62} - 14q^{64} + 8q^{66} + 36q^{68} - 4q^{69} - 6q^{72} + 28q^{73} - 12q^{74} - 12q^{76} + 12q^{78} - 16q^{79} + 2q^{81} - 28q^{82} - 8q^{83} + 16q^{86} - 12q^{87} - 8q^{88} + 12q^{89} + 36q^{92} + 4q^{93} + 16q^{94} - 6q^{96} + 4q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.41421 −1.00000 3.82843 0 2.41421 0 −4.41421 1.00000 0
1.2 0.414214 −1.00000 −1.82843 0 −0.414214 0 −1.58579 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 3675.2.a.s 2
5.b even 2 1 735.2.a.m yes 2
7.b odd 2 1 3675.2.a.t 2
15.d odd 2 1 2205.2.a.o 2
35.c odd 2 1 735.2.a.l 2
35.i odd 6 2 735.2.i.h 4
35.j even 6 2 735.2.i.g 4
105.g even 2 1 2205.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.a.l 2 35.c odd 2 1
735.2.a.m yes 2 5.b even 2 1
735.2.i.g 4 35.j even 6 2
735.2.i.h 4 35.i odd 6 2
2205.2.a.o 2 15.d odd 2 1
2205.2.a.r 2 105.g even 2 1
3675.2.a.s 2 1.a even 1 1 trivial
3675.2.a.t 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{11}^{2} - 8$$ $$T_{13}^{2} - 4 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$-28 - 4 T + T^{2}$$
$19$ $$-4 - 4 T + T^{2}$$
$23$ $$-28 - 4 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$-68 + 4 T + T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$4 - 12 T + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-32 + T^{2}$$
$53$ $$-72 + T^{2}$$
$59$ $$32 - 16 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$-8 + T^{2}$$
$73$ $$188 - 28 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-16 + 8 T + T^{2}$$
$89$ $$-92 - 12 T + T^{2}$$
$97$ $$-68 - 4 T + T^{2}$$