# Properties

 Label 3675.2.a.bo 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: 4.4.2624.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + q^{3} + \beta_{2} q^{4} + ( -1 + \beta_{1} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + q^{3} + \beta_{2} q^{4} + ( -1 + \beta_{1} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + \beta_{2} q^{12} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{16} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( -\beta_{1} - 3 \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{26} + q^{27} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{29} + ( -4 + 4 \beta_{1} - 3 \beta_{3} ) q^{31} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{32} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - \beta_{2} ) q^{34} + \beta_{2} q^{36} + ( -5 + 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( -8 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -1 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{43} + ( 4 - \beta_{2} - 3 \beta_{3} ) q^{44} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{46} + ( 2 + 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{48} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{51} + ( 4 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{52} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -\beta_{1} - 3 \beta_{2} ) q^{57} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{58} + ( 2 - 5 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -8 + 2 \beta_{3} ) q^{61} + ( 8 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{62} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{64} + ( -1 - \beta_{2} ) q^{66} + ( -7 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{67} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{68} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{71} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{72} + ( 4 - 7 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{73} + ( 7 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{74} + ( -8 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{76} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{78} + ( -5 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{79} + q^{81} + ( 8 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{82} + ( -2 - 6 \beta_{1} + 7 \beta_{3} ) q^{83} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{86} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{87} + ( -1 + 2 \beta_{3} ) q^{88} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{92} + ( -4 + 4 \beta_{1} - 3 \beta_{3} ) q^{93} + ( 4 + 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{94} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{96} + ( 8 - 4 \beta_{2} + \beta_{3} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 4q^{3} + 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 4q^{3} + 2q^{4} - 2q^{6} + 4q^{9} - 4q^{11} + 2q^{12} - 6q^{16} - 4q^{17} - 2q^{18} - 8q^{19} - 6q^{22} - 12q^{26} + 4q^{27} - 4q^{29} - 8q^{31} - 2q^{32} - 4q^{33} - 8q^{34} + 2q^{36} - 16q^{37} + 4q^{38} - 24q^{41} - 20q^{43} + 14q^{44} - 6q^{46} + 8q^{47} - 6q^{48} - 4q^{51} + 16q^{52} + 20q^{53} - 2q^{54} - 8q^{57} + 6q^{58} - 8q^{59} - 32q^{61} + 28q^{62} - 12q^{64} - 6q^{66} - 12q^{67} - 12q^{68} + 4q^{71} + 34q^{74} - 40q^{76} - 12q^{78} + 4q^{81} + 16q^{82} - 20q^{83} - 14q^{86} - 4q^{87} - 4q^{88} - 8q^{89} + 10q^{92} - 8q^{93} + 32q^{94} - 2q^{96} + 24q^{97} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 2 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 1$$

## 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
 −1.22833 −0.360409 0.814115 2.77462
−2.22833 1.00000 2.96545 0 −2.22833 0 −2.15133 1.00000 0
1.2 −1.36041 1.00000 −0.149286 0 −1.36041 0 2.92391 1.00000 0
1.3 −0.185885 1.00000 −1.96545 0 −0.185885 0 0.737118 1.00000 0
1.4 1.77462 1.00000 1.14929 0 1.77462 0 −1.50970 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.bo yes 4
5.b even 2 1 3675.2.a.by yes 4
7.b odd 2 1 3675.2.a.bm 4
35.c odd 2 1 3675.2.a.ca yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.2.a.bm 4 7.b odd 2 1
3675.2.a.bo yes 4 1.a even 1 1 trivial
3675.2.a.by yes 4 5.b even 2 1
3675.2.a.ca yes 4 35.c odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} - 3 T_{2}^{2} - 6 T_{2} - 1$$ $$T_{11}^{4} + 4 T_{11}^{3} - 8 T_{11}^{2} - 8 T_{11} + 7$$ $$T_{13}^{4} - 14 T_{13}^{2} - 16 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 6 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$7 - 8 T - 8 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$4 - 16 T - 14 T^{2} + T^{4}$$
$17$ $$-4 - 16 T - 10 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$196 - 224 T - 30 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$79 + 20 T - 32 T^{2} + T^{4}$$
$29$ $$463 - 64 T - 48 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$452 - 144 T - 36 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$-623 - 192 T + 46 T^{2} + 16 T^{3} + T^{4}$$
$41$ $$448 + 512 T + 184 T^{2} + 24 T^{3} + T^{4}$$
$43$ $$-9575 - 1980 T - 2 T^{2} + 20 T^{3} + T^{4}$$
$47$ $$-1028 + 904 T - 102 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$-784 + 80 T + 92 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$316 - 248 T - 78 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$( 56 + 16 T + T^{2} )^{2}$$
$67$ $$1489 - 404 T - 74 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$3383 + 504 T - 168 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$2884 + 160 T - 158 T^{2} + T^{4}$$
$79$ $$2009 - 202 T^{2} + T^{4}$$
$83$ $$-6692 - 1976 T - 40 T^{2} + 20 T^{3} + T^{4}$$
$89$ $$-6692 - 2504 T - 214 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$-3676 + 688 T + 108 T^{2} - 24 T^{3} + T^{4}$$