# Properties

 Label 3675.2.a.bf 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 147) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + q^{3} + ( 1 + 2 \beta ) q^{4} + ( 1 + \beta ) q^{6} + ( 3 + \beta ) q^{8} + q^{9} -2 q^{11} + ( 1 + 2 \beta ) q^{12} + ( 4 + \beta ) q^{13} + 3 q^{16} + ( 2 + 3 \beta ) q^{17} + ( 1 + \beta ) q^{18} -2 \beta q^{19} + ( -2 - 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + ( 3 + \beta ) q^{24} + ( 6 + 5 \beta ) q^{26} + q^{27} + ( -4 + 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} -2 q^{33} + ( 8 + 5 \beta ) q^{34} + ( 1 + 2 \beta ) q^{36} + 4 q^{37} + ( -4 - 2 \beta ) q^{38} + ( 4 + \beta ) q^{39} + ( -2 + 3 \beta ) q^{41} + 4 \beta q^{43} + ( -2 - 4 \beta ) q^{44} + ( -6 - 2 \beta ) q^{46} + 2 \beta q^{47} + 3 q^{48} + ( 2 + 3 \beta ) q^{51} + ( 8 + 9 \beta ) q^{52} + 2 q^{53} + ( 1 + \beta ) q^{54} -2 \beta q^{57} -2 \beta q^{58} + ( 4 + 2 \beta ) q^{59} + ( -8 + 3 \beta ) q^{61} + ( 8 + 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( -2 - 2 \beta ) q^{66} -4 \beta q^{67} + ( 14 + 7 \beta ) q^{68} + ( 2 - 4 \beta ) q^{69} + ( -2 - 8 \beta ) q^{71} + ( 3 + \beta ) q^{72} + ( 4 - 7 \beta ) q^{73} + ( 4 + 4 \beta ) q^{74} + ( -8 - 2 \beta ) q^{76} + ( 6 + 5 \beta ) q^{78} + ( 8 - 4 \beta ) q^{79} + q^{81} + ( 4 + \beta ) q^{82} + ( -4 - 8 \beta ) q^{83} + ( 8 + 4 \beta ) q^{86} + ( -4 + 2 \beta ) q^{87} + ( -6 - 2 \beta ) q^{88} + ( 10 - 3 \beta ) q^{89} -14 q^{92} + ( 4 + 2 \beta ) q^{93} + ( 4 + 2 \beta ) q^{94} + ( -3 + \beta ) q^{96} + ( 4 + \beta ) q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{8} + 2q^{9} - 4q^{11} + 2q^{12} + 8q^{13} + 6q^{16} + 4q^{17} + 2q^{18} - 4q^{22} + 4q^{23} + 6q^{24} + 12q^{26} + 2q^{27} - 8q^{29} + 8q^{31} - 6q^{32} - 4q^{33} + 16q^{34} + 2q^{36} + 8q^{37} - 8q^{38} + 8q^{39} - 4q^{41} - 4q^{44} - 12q^{46} + 6q^{48} + 4q^{51} + 16q^{52} + 4q^{53} + 2q^{54} + 8q^{59} - 16q^{61} + 16q^{62} - 14q^{64} - 4q^{66} + 28q^{68} + 4q^{69} - 4q^{71} + 6q^{72} + 8q^{73} + 8q^{74} - 16q^{76} + 12q^{78} + 16q^{79} + 2q^{81} + 8q^{82} - 8q^{83} + 16q^{86} - 8q^{87} - 12q^{88} + 20q^{89} - 28q^{92} + 8q^{93} + 8q^{94} - 6q^{96} + 8q^{97} - 4q^{99} + O(q^{100})$$

## 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
−0.414214 1.00000 −1.82843 0 −0.414214 0 1.58579 1.00000 0
1.2 2.41421 1.00000 3.82843 0 2.41421 0 4.41421 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.bf 2
5.b even 2 1 147.2.a.d 2
7.b odd 2 1 3675.2.a.bd 2
15.d odd 2 1 441.2.a.j 2
20.d odd 2 1 2352.2.a.be 2
35.c odd 2 1 147.2.a.e yes 2
35.i odd 6 2 147.2.e.d 4
35.j even 6 2 147.2.e.e 4
40.e odd 2 1 9408.2.a.dq 2
40.f even 2 1 9408.2.a.ef 2
60.h even 2 1 7056.2.a.cv 2
105.g even 2 1 441.2.a.i 2
105.o odd 6 2 441.2.e.f 4
105.p even 6 2 441.2.e.g 4
140.c even 2 1 2352.2.a.bc 2
140.p odd 6 2 2352.2.q.bb 4
140.s even 6 2 2352.2.q.bd 4
280.c odd 2 1 9408.2.a.di 2
280.n even 2 1 9408.2.a.dt 2
420.o odd 2 1 7056.2.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 5.b even 2 1
147.2.a.e yes 2 35.c odd 2 1
147.2.e.d 4 35.i odd 6 2
147.2.e.e 4 35.j even 6 2
441.2.a.i 2 105.g even 2 1
441.2.a.j 2 15.d odd 2 1
441.2.e.f 4 105.o odd 6 2
441.2.e.g 4 105.p even 6 2
2352.2.a.bc 2 140.c even 2 1
2352.2.a.be 2 20.d odd 2 1
2352.2.q.bb 4 140.p odd 6 2
2352.2.q.bd 4 140.s even 6 2
3675.2.a.bd 2 7.b odd 2 1
3675.2.a.bf 2 1.a even 1 1 trivial
7056.2.a.cf 2 420.o odd 2 1
7056.2.a.cv 2 60.h even 2 1
9408.2.a.di 2 280.c odd 2 1
9408.2.a.dq 2 40.e odd 2 1
9408.2.a.dt 2 280.n even 2 1
9408.2.a.ef 2 40.f even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{11} + 2$$ $$T_{13}^{2} - 8 T_{13} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$14 - 8 T + T^{2}$$
$17$ $$-14 - 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-28 - 4 T + T^{2}$$
$29$ $$8 + 8 T + T^{2}$$
$31$ $$8 - 8 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$-14 + 4 T + T^{2}$$
$43$ $$-32 + T^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$8 - 8 T + T^{2}$$
$61$ $$46 + 16 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$-124 + 4 T + T^{2}$$
$73$ $$-82 - 8 T + T^{2}$$
$79$ $$32 - 16 T + T^{2}$$
$83$ $$-112 + 8 T + T^{2}$$
$89$ $$82 - 20 T + T^{2}$$
$97$ $$14 - 8 T + T^{2}$$