# Properties

 Label 3675.2.a.bq Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.88404.1 Defining polynomial: $$x^{4} - x^{3} - 7 x^{2} + 5 x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} + ( -2 + \beta_{1} + \beta_{3} ) q^{17} -\beta_{1} q^{18} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( 4 - 3 \beta_{1} - \beta_{3} ) q^{22} + 2 \beta_{2} q^{23} + ( 2 \beta_{1} + \beta_{3} ) q^{24} + ( 4 + 3 \beta_{1} ) q^{26} - q^{27} + 4 q^{29} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -4 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -2 + \beta_{1} - \beta_{3} ) q^{33} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -3 + 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 8 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{38} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( 8 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{44} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{46} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{48} + ( 2 - \beta_{1} - \beta_{3} ) q^{51} + ( -8 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{52} + ( 2 - 3 \beta_{1} + \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( 2 \beta_{1} - \beta_{2} ) q^{57} -4 \beta_{1} q^{58} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{59} + 5 q^{61} + ( 12 + 4 \beta_{2} ) q^{62} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{64} + ( -4 + 3 \beta_{1} + \beta_{3} ) q^{66} + \beta_{2} q^{67} + ( 7 \beta_{1} + \beta_{3} ) q^{68} -2 \beta_{2} q^{69} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -2 \beta_{1} - \beta_{3} ) q^{72} + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( -8 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( 8 - 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{76} + ( -4 - 3 \beta_{1} ) q^{78} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{79} + q^{81} + ( -4 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 4 - 2 \beta_{1} - 2 \beta_{3} ) q^{86} -4 q^{87} + ( -4 - 9 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{88} + ( 2 + \beta_{1} + \beta_{3} ) q^{89} + ( 16 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{93} + ( -4 - 5 \beta_{1} - \beta_{3} ) q^{94} + ( 4 + 3 \beta_{1} + 2 \beta_{2} ) q^{96} + ( -3 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 2 - \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} - 4q^{3} + 7q^{4} + q^{6} - 3q^{8} + 4q^{9} + O(q^{10})$$ $$4q - q^{2} - 4q^{3} + 7q^{4} + q^{6} - 3q^{8} + 4q^{9} + 8q^{11} - 7q^{12} - 7q^{13} + 17q^{16} - 6q^{17} - q^{18} - 3q^{19} + 12q^{22} - 2q^{23} + 3q^{24} + 19q^{26} - 4q^{27} + 16q^{29} - 9q^{31} - 17q^{32} - 8q^{33} - 14q^{34} + 7q^{36} - 8q^{37} + 27q^{38} + 7q^{39} - 4q^{41} - 5q^{43} + 26q^{44} - 6q^{46} + 6q^{47} - 17q^{48} + 6q^{51} - 35q^{52} + 6q^{53} + q^{54} + 3q^{57} - 4q^{58} - 10q^{59} + 20q^{61} + 44q^{62} + 21q^{64} - 12q^{66} - q^{67} + 8q^{68} + 2q^{69} + 22q^{71} - 3q^{72} + 4q^{73} - 21q^{74} + 23q^{76} - 19q^{78} + 8q^{79} + 4q^{81} - 8q^{82} + 2q^{83} + 12q^{86} - 16q^{87} - 28q^{88} + 10q^{89} + 66q^{92} + 9q^{93} - 22q^{94} + 17q^{96} - 12q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 7 x^{2} + 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$

## 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
 2.70285 1.22868 −0.494173 −2.43736
−2.70285 −1.00000 5.30542 0 2.70285 0 −8.93406 1.00000 0
1.2 −1.22868 −1.00000 −0.490347 0 1.22868 0 3.05984 1.00000 0
1.3 0.494173 −1.00000 −1.75579 0 −0.494173 0 −1.85601 1.00000 0
1.4 2.43736 −1.00000 3.94072 0 −2.43736 0 4.73024 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.bq 4
5.b even 2 1 3675.2.a.bx 4
7.b odd 2 1 3675.2.a.br 4
7.d odd 6 2 525.2.i.j yes 8
35.c odd 2 1 3675.2.a.bw 4
35.i odd 6 2 525.2.i.i 8
35.k even 12 4 525.2.r.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i 8 35.i odd 6 2
525.2.i.j yes 8 7.d odd 6 2
525.2.r.h 16 35.k even 12 4
3675.2.a.bq 4 1.a even 1 1 trivial
3675.2.a.br 4 7.b odd 2 1
3675.2.a.bw 4 35.c odd 2 1
3675.2.a.bx 4 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + T_{2}^{3} - 7 T_{2}^{2} - 5 T_{2} + 4$$ $$T_{11}^{4} - 8 T_{11}^{3} - 8 T_{11}^{2} + 180 T_{11} - 328$$ $$T_{13}^{4} + 7 T_{13}^{3} - 19 T_{13}^{2} - 179 T_{13} - 194$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 5 T - 7 T^{2} + T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$-328 + 180 T - 8 T^{2} - 8 T^{3} + T^{4}$$
$13$ $$-194 - 179 T - 19 T^{2} + 7 T^{3} + T^{4}$$
$17$ $$40 - 108 T - 20 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$-196 - 175 T - 37 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$960 - 72 T - 68 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$-768 - 576 T - 56 T^{2} + 9 T^{3} + T^{4}$$
$37$ $$773 - 144 T - 62 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$-200 - 236 T - 60 T^{2} + 4 T^{3} + T^{4}$$
$43$ $$160 + 32 T - 52 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$-856 + 500 T - 64 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$1112 + 292 T - 76 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$-8 - 44 T - 40 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$( -5 + T )^{4}$$
$67$ $$60 - 9 T - 17 T^{2} + T^{3} + T^{4}$$
$71$ $$-12736 + 2880 T - 32 T^{2} - 22 T^{3} + T^{4}$$
$73$ $$-1567 + 924 T - 122 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$153 + 132 T - 70 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$-312 - 396 T - 112 T^{2} - 2 T^{3} + T^{4}$$
$89$ $$24 + 84 T + 4 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$-79 - 196 T - 58 T^{2} + 12 T^{3} + T^{4}$$