# Properties

 Label 3675.2.a.bk Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} + \beta_{3} ) q^{2} - q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} ) q^{6} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} + \beta_{3} ) q^{2} - q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} ) q^{6} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} + ( 2 + 2 \beta_{2} ) q^{11} + ( -2 - \beta_{1} + \beta_{2} ) q^{12} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{16} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} ) q^{18} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{19} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{24} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{26} - q^{27} + ( 2 - 4 \beta_{1} ) q^{29} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{31} + ( -7 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{32} + ( -2 - 2 \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{34} + ( 2 + \beta_{1} - \beta_{2} ) q^{36} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{38} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{44} + ( 3 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{46} + ( -2 - 6 \beta_{3} ) q^{47} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{48} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 8 - 2 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{53} + ( 1 + \beta_{1} - \beta_{3} ) q^{54} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{57} + ( 6 + 2 \beta_{1} + 6 \beta_{3} ) q^{58} + ( 4 + 4 \beta_{1} - 6 \beta_{3} ) q^{59} + ( -8 + \beta_{3} ) q^{61} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{62} + ( 6 - \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{64} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{66} + ( 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{67} + ( -6 + 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{68} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{72} + ( -4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -8 + 2 \beta_{2} - 6 \beta_{3} ) q^{74} + ( 2 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{78} + ( -2 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{82} + ( -10 - 2 \beta_{3} ) q^{83} + ( -8 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{86} + ( -2 + 4 \beta_{1} ) q^{87} + ( 10 - 6 \beta_{2} - 10 \beta_{3} ) q^{88} + ( -2 - 2 \beta_{3} ) q^{89} + ( 2 - 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{92} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{93} + ( -4 + 8 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 7 - \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{96} + ( 4 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{97} + ( 2 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} + 8q^{4} + 4q^{6} - 12q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} + 8q^{4} + 4q^{6} - 12q^{8} + 4q^{9} + 8q^{11} - 8q^{12} + 12q^{16} - 8q^{17} - 4q^{18} - 8q^{19} + 12q^{24} - 4q^{27} + 8q^{29} - 8q^{31} - 28q^{32} - 8q^{33} - 8q^{34} + 8q^{36} - 8q^{37} - 4q^{38} + 8q^{43} - 16q^{44} + 12q^{46} - 8q^{47} - 12q^{48} + 8q^{51} + 32q^{52} - 8q^{53} + 4q^{54} + 8q^{57} + 24q^{58} + 16q^{59} - 32q^{61} + 20q^{62} + 24q^{64} - 24q^{68} - 8q^{71} - 12q^{72} - 32q^{74} + 8q^{76} + 4q^{81} + 8q^{82} - 40q^{83} - 32q^{86} - 8q^{87} + 40q^{88} - 8q^{89} + 8q^{92} + 8q^{93} - 16q^{94} + 28q^{96} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 3$$

## 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
 0.334904 2.68554 −1.74912 −1.27133
−2.74912 −1.00000 5.55765 0 2.74912 0 −9.78039 1.00000 0
1.2 −2.27133 −1.00000 3.15894 0 2.27133 0 −2.63234 1.00000 0
1.3 −0.665096 −1.00000 −1.55765 0 0.665096 0 2.36618 1.00000 0
1.4 1.68554 −1.00000 0.841058 0 −1.68554 0 −1.95345 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.bk 4
5.b even 2 1 735.2.a.o yes 4
7.b odd 2 1 3675.2.a.bl 4
15.d odd 2 1 2205.2.a.bg 4
35.c odd 2 1 735.2.a.n 4
35.i odd 6 2 735.2.i.n 8
35.j even 6 2 735.2.i.m 8
105.g even 2 1 2205.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.a.n 4 35.c odd 2 1
735.2.a.o yes 4 5.b even 2 1
735.2.i.m 8 35.j even 6 2
735.2.i.n 8 35.i odd 6 2
2205.2.a.bf 4 105.g even 2 1
2205.2.a.bg 4 15.d odd 2 1
3675.2.a.bk 4 1.a even 1 1 trivial
3675.2.a.bl 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{4} + 4 T_{2}^{3} - 12 T_{2} - 7$$ $$T_{11}^{4} - 8 T_{11}^{3} - 8 T_{11}^{2} + 160 T_{11} - 224$$ $$T_{13}^{4} - 32 T_{13}^{2} + 64 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 - 12 T + 4 T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$-224 + 160 T - 8 T^{2} - 8 T^{3} + T^{4}$$
$13$ $$16 + 64 T - 32 T^{2} + T^{4}$$
$17$ $$128 - 128 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$-284 - 176 T - 12 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$196 + 64 T - 44 T^{2} + T^{4}$$
$29$ $$-368 + 608 T - 72 T^{2} - 8 T^{3} + T^{4}$$
$31$ $$-28 - 48 T - 12 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$784 - 288 T - 56 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$16 + 128 T - 56 T^{2} + T^{4}$$
$43$ $$64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$( -68 + 4 T + T^{2} )^{2}$$
$53$ $$-28 - 80 T - 4 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$-4544 + 1408 T - 48 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$( 62 + 16 T + T^{2} )^{2}$$
$67$ $$3136 + 512 T - 176 T^{2} + T^{4}$$
$71$ $$32 - 96 T - 72 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$3088 + 128 T - 128 T^{2} + T^{4}$$
$79$ $$2576 - 384 T - 184 T^{2} + T^{4}$$
$83$ $$( 92 + 20 T + T^{2} )^{2}$$
$89$ $$( -4 + 4 T + T^{2} )^{2}$$
$97$ $$-2032 + 1280 T - 192 T^{2} + T^{4}$$