# Properties

 Label 3675.2.a.z 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 105) 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.

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

## 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
−1.41421 1.00000 0 0 −1.41421 0 2.82843 1.00000 0
1.2 1.41421 1.00000 0 0 1.41421 0 −2.82843 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.z 2
5.b even 2 1 735.2.a.i 2
7.b odd 2 1 3675.2.a.x 2
7.c even 3 2 525.2.i.g 4
15.d odd 2 1 2205.2.a.u 2
35.c odd 2 1 735.2.a.j 2
35.i odd 6 2 735.2.i.j 4
35.j even 6 2 105.2.i.c 4
35.l odd 12 4 525.2.r.g 8
105.g even 2 1 2205.2.a.s 2
105.o odd 6 2 315.2.j.d 4
140.p odd 6 2 1680.2.bg.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.c 4 35.j even 6 2
315.2.j.d 4 105.o odd 6 2
525.2.i.g 4 7.c even 3 2
525.2.r.g 8 35.l odd 12 4
735.2.a.i 2 5.b even 2 1
735.2.a.j 2 35.c odd 2 1
735.2.i.j 4 35.i odd 6 2
1680.2.bg.p 4 140.p odd 6 2
2205.2.a.s 2 105.g even 2 1
2205.2.a.u 2 15.d odd 2 1
3675.2.a.x 2 7.b odd 2 1
3675.2.a.z 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} - 2$$ $$T_{11}^{2} - 4 T_{11} + 2$$ $$T_{13}^{2} - 6 T_{13} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$2 - 4 T + T^{2}$$
$13$ $$7 - 6 T + T^{2}$$
$17$ $$-14 - 4 T + T^{2}$$
$19$ $$-31 + 2 T + T^{2}$$
$23$ $$-14 - 4 T + T^{2}$$
$29$ $$-2 - 8 T + T^{2}$$
$31$ $$1 + 6 T + T^{2}$$
$37$ $$47 - 14 T + T^{2}$$
$41$ $$-14 + 4 T + T^{2}$$
$43$ $$79 - 18 T + T^{2}$$
$47$ $$-124 - 4 T + T^{2}$$
$53$ $$8 + 8 T + T^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-56 - 8 T + T^{2}$$
$67$ $$-161 - 2 T + T^{2}$$
$71$ $$2 + 4 T + T^{2}$$
$73$ $$-25 - 10 T + T^{2}$$
$79$ $$-31 - 2 T + T^{2}$$
$83$ $$14 + 8 T + T^{2}$$
$89$ $$46 + 16 T + T^{2}$$
$97$ $$56 - 16 T + T^{2}$$