# Properties

 Label 3675.2.a.bg 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + q^{3} + ( 2 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 6 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + q^{3} + ( 2 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 6 + 2 \beta ) q^{8} + q^{9} + ( -1 + \beta ) q^{11} + ( 2 + 2 \beta ) q^{12} + ( -4 + \beta ) q^{13} + ( 8 + 4 \beta ) q^{16} + ( -5 + \beta ) q^{17} + ( 1 + \beta ) q^{18} + ( 1 + 2 \beta ) q^{19} + 2 q^{22} + ( 3 + \beta ) q^{23} + ( 6 + 2 \beta ) q^{24} + ( -1 - 3 \beta ) q^{26} + q^{27} + ( 1 - 3 \beta ) q^{29} + ( 3 - 2 \beta ) q^{31} + ( 8 + 8 \beta ) q^{32} + ( -1 + \beta ) q^{33} + ( -2 - 4 \beta ) q^{34} + ( 2 + 2 \beta ) q^{36} + ( -2 + 3 \beta ) q^{37} + ( 7 + 3 \beta ) q^{38} + ( -4 + \beta ) q^{39} + ( 1 - \beta ) q^{41} + ( 2 - 3 \beta ) q^{43} + 4 q^{44} + ( 6 + 4 \beta ) q^{46} -2 q^{47} + ( 8 + 4 \beta ) q^{48} + ( -5 + \beta ) q^{51} + ( -2 - 6 \beta ) q^{52} + ( -2 - 6 \beta ) q^{53} + ( 1 + \beta ) q^{54} + ( 1 + 2 \beta ) q^{57} + ( -8 - 2 \beta ) q^{58} + ( 5 - 3 \beta ) q^{59} + 4 q^{61} + ( -3 + \beta ) q^{62} + ( 16 + 8 \beta ) q^{64} + 2 q^{66} + ( 6 + 5 \beta ) q^{67} + ( -4 - 8 \beta ) q^{68} + ( 3 + \beta ) q^{69} + ( 1 + 3 \beta ) q^{71} + ( 6 + 2 \beta ) q^{72} + ( -4 - 5 \beta ) q^{73} + ( 7 + \beta ) q^{74} + ( 14 + 6 \beta ) q^{76} + ( -1 - 3 \beta ) q^{78} + ( 3 - 6 \beta ) q^{79} + q^{81} -2 q^{82} + ( -3 - 7 \beta ) q^{83} + ( -7 - \beta ) q^{86} + ( 1 - 3 \beta ) q^{87} + 4 \beta q^{88} + ( 3 + 7 \beta ) q^{89} + ( 12 + 8 \beta ) q^{92} + ( 3 - 2 \beta ) q^{93} + ( -2 - 2 \beta ) q^{94} + ( 8 + 8 \beta ) q^{96} + ( -8 - 4 \beta ) q^{97} + ( -1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 4q^{4} + 2q^{6} + 12q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 4q^{4} + 2q^{6} + 12q^{8} + 2q^{9} - 2q^{11} + 4q^{12} - 8q^{13} + 16q^{16} - 10q^{17} + 2q^{18} + 2q^{19} + 4q^{22} + 6q^{23} + 12q^{24} - 2q^{26} + 2q^{27} + 2q^{29} + 6q^{31} + 16q^{32} - 2q^{33} - 4q^{34} + 4q^{36} - 4q^{37} + 14q^{38} - 8q^{39} + 2q^{41} + 4q^{43} + 8q^{44} + 12q^{46} - 4q^{47} + 16q^{48} - 10q^{51} - 4q^{52} - 4q^{53} + 2q^{54} + 2q^{57} - 16q^{58} + 10q^{59} + 8q^{61} - 6q^{62} + 32q^{64} + 4q^{66} + 12q^{67} - 8q^{68} + 6q^{69} + 2q^{71} + 12q^{72} - 8q^{73} + 14q^{74} + 28q^{76} - 2q^{78} + 6q^{79} + 2q^{81} - 4q^{82} - 6q^{83} - 14q^{86} + 2q^{87} + 6q^{89} + 24q^{92} + 6q^{93} - 4q^{94} + 16q^{96} - 16q^{97} - 2q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
−0.732051 1.00000 −1.46410 0 −0.732051 0 2.53590 1.00000 0
1.2 2.73205 1.00000 5.46410 0 2.73205 0 9.46410 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.bg 2
5.b even 2 1 735.2.a.g 2
7.b odd 2 1 3675.2.a.be 2
7.c even 3 2 525.2.i.f 4
15.d odd 2 1 2205.2.a.z 2
35.c odd 2 1 735.2.a.h 2
35.i odd 6 2 735.2.i.l 4
35.j even 6 2 105.2.i.d 4
35.l odd 12 2 525.2.r.a 4
35.l odd 12 2 525.2.r.f 4
105.g even 2 1 2205.2.a.ba 2
105.o odd 6 2 315.2.j.c 4
140.p odd 6 2 1680.2.bg.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 35.j even 6 2
315.2.j.c 4 105.o odd 6 2
525.2.i.f 4 7.c even 3 2
525.2.r.a 4 35.l odd 12 2
525.2.r.f 4 35.l odd 12 2
735.2.a.g 2 5.b even 2 1
735.2.a.h 2 35.c odd 2 1
735.2.i.l 4 35.i odd 6 2
1680.2.bg.o 4 140.p odd 6 2
2205.2.a.z 2 15.d odd 2 1
2205.2.a.ba 2 105.g even 2 1
3675.2.a.be 2 7.b odd 2 1
3675.2.a.bg 2 1.a even 1 1 trivial

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2 + 2 T + T^{2}$$
$13$ $$13 + 8 T + T^{2}$$
$17$ $$22 + 10 T + T^{2}$$
$19$ $$-11 - 2 T + T^{2}$$
$23$ $$6 - 6 T + T^{2}$$
$29$ $$-26 - 2 T + T^{2}$$
$31$ $$-3 - 6 T + T^{2}$$
$37$ $$-23 + 4 T + T^{2}$$
$41$ $$-2 - 2 T + T^{2}$$
$43$ $$-23 - 4 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$-104 + 4 T + T^{2}$$
$59$ $$-2 - 10 T + T^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$-39 - 12 T + T^{2}$$
$71$ $$-26 - 2 T + T^{2}$$
$73$ $$-59 + 8 T + T^{2}$$
$79$ $$-99 - 6 T + T^{2}$$
$83$ $$-138 + 6 T + T^{2}$$
$89$ $$-138 - 6 T + T^{2}$$
$97$ $$16 + 16 T + T^{2}$$