# Properties

 Label 3675.2.a.bj Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} + 2 q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{12} + ( 2 - \beta_{1} + \beta_{2} ) q^{13} + ( 3 + 4 \beta_{1} ) q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -2 - \beta_{1} + \beta_{2} ) q^{19} + 2 \beta_{1} q^{22} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{24} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{26} + q^{27} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{31} + ( 8 + 3 \beta_{1} + 2 \beta_{2} ) q^{32} + 2 q^{33} + ( 4 - \beta_{1} + \beta_{2} ) q^{34} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} -4 \beta_{1} q^{37} + ( -4 - \beta_{1} - \beta_{2} ) q^{38} + ( 2 - \beta_{1} + \beta_{2} ) q^{39} + ( -3 \beta_{1} + \beta_{2} ) q^{41} + 4 \beta_{2} q^{43} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -2 - \beta_{1} - \beta_{2} ) q^{46} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + 4 \beta_{1} ) q^{48} + ( \beta_{1} - \beta_{2} ) q^{51} + ( 6 - \beta_{1} + \beta_{2} ) q^{52} + ( 6 - 3 \beta_{1} - \beta_{2} ) q^{53} + \beta_{1} q^{54} + ( -2 - \beta_{1} + \beta_{2} ) q^{57} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 - 4 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 8 + 3 \beta_{1} + 3 \beta_{2} ) q^{62} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{64} + 2 \beta_{1} q^{66} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{68} + ( 2 - \beta_{1} - \beta_{2} ) q^{69} + 2 q^{71} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{72} + ( 6 + \beta_{1} - \beta_{2} ) q^{73} + ( -12 - 4 \beta_{1} - 4 \beta_{2} ) q^{74} + ( 2 - 5 \beta_{1} - 3 \beta_{2} ) q^{76} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{78} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( -10 - \beta_{1} - 3 \beta_{2} ) q^{82} + ( -4 + 4 \beta_{1} ) q^{83} + ( -4 + 8 \beta_{1} ) q^{86} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{88} + ( -4 - \beta_{1} - \beta_{2} ) q^{89} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{92} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{93} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 8 + 3 \beta_{1} + 2 \beta_{2} ) q^{96} + ( 10 - 5 \beta_{1} - 3 \beta_{2} ) q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + q^{2} + 3 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} + 13 q^{16} + q^{18} - 6 q^{19} + 2 q^{22} + 4 q^{23} + 9 q^{24} - 10 q^{26} + 3 q^{27} + 2 q^{29} - 2 q^{31} + 29 q^{32} + 6 q^{33} + 12 q^{34} + 5 q^{36} - 4 q^{37} - 14 q^{38} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 10 q^{44} - 8 q^{46} - 8 q^{47} + 13 q^{48} + 18 q^{52} + 14 q^{53} + q^{54} - 6 q^{57} - 18 q^{58} - 16 q^{59} + 6 q^{61} + 30 q^{62} + 13 q^{64} + 2 q^{66} + 8 q^{67} - 8 q^{68} + 4 q^{69} + 6 q^{71} + 9 q^{72} + 18 q^{73} - 44 q^{74} - 2 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 34 q^{82} - 8 q^{83} - 4 q^{86} + 2 q^{87} + 18 q^{88} - 14 q^{89} - 20 q^{92} - 2 q^{93} + 16 q^{94} + 29 q^{96} + 22 q^{97} + 6 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$

## 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
 0.311108 −1.48119 2.17009
−1.90321 1.00000 1.62222 0 −1.90321 0 0.719004 1.00000 0
1.2 0.193937 1.00000 −1.96239 0 0.193937 0 −0.768452 1.00000 0
1.3 2.70928 1.00000 5.34017 0 2.70928 0 9.04945 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.bj 3
5.b even 2 1 3675.2.a.bi 3
5.c odd 4 2 735.2.d.b 6
7.b odd 2 1 525.2.a.k 3
15.e even 4 2 2205.2.d.l 6
21.c even 2 1 1575.2.a.w 3
28.d even 2 1 8400.2.a.dj 3
35.c odd 2 1 525.2.a.j 3
35.f even 4 2 105.2.d.b 6
35.k even 12 4 735.2.q.e 12
35.l odd 12 4 735.2.q.f 12
105.g even 2 1 1575.2.a.x 3
105.k odd 4 2 315.2.d.e 6
140.c even 2 1 8400.2.a.dg 3
140.j odd 4 2 1680.2.t.k 6
420.w even 4 2 5040.2.t.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 35.f even 4 2
315.2.d.e 6 105.k odd 4 2
525.2.a.j 3 35.c odd 2 1
525.2.a.k 3 7.b odd 2 1
735.2.d.b 6 5.c odd 4 2
735.2.q.e 12 35.k even 12 4
735.2.q.f 12 35.l odd 12 4
1575.2.a.w 3 21.c even 2 1
1575.2.a.x 3 105.g even 2 1
1680.2.t.k 6 140.j odd 4 2
2205.2.d.l 6 15.e even 4 2
3675.2.a.bi 3 5.b even 2 1
3675.2.a.bj 3 1.a even 1 1 trivial
5040.2.t.v 6 420.w even 4 2
8400.2.a.dg 3 140.c even 2 1
8400.2.a.dj 3 28.d even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 5 T_{2} + 1$$ $$T_{11} - 2$$ $$T_{13}^{3} - 6 T_{13}^{2} - 4 T_{13} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T - T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$( -2 + T )^{3}$$
$13$ $$8 - 4 T - 6 T^{2} + T^{3}$$
$17$ $$16 - 16 T + T^{3}$$
$19$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$16 - 8 T - 4 T^{2} + T^{3}$$
$29$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$31$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$37$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$41$ $$-200 - 60 T + 2 T^{2} + T^{3}$$
$43$ $$832 - 144 T - 4 T^{2} + T^{3}$$
$47$ $$-128 - 32 T + 8 T^{2} + T^{3}$$
$53$ $$296 + 12 T - 14 T^{2} + T^{3}$$
$59$ $$-1280 - 64 T + 16 T^{2} + T^{3}$$
$61$ $$248 - 52 T - 6 T^{2} + T^{3}$$
$67$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$71$ $$( -2 + T )^{3}$$
$73$ $$-104 + 92 T - 18 T^{2} + T^{3}$$
$79$ $$320 - 16 T - 12 T^{2} + T^{3}$$
$83$ $$-256 - 64 T + 8 T^{2} + T^{3}$$
$89$ $$40 + 52 T + 14 T^{2} + T^{3}$$
$97$ $$1864 - 36 T - 22 T^{2} + T^{3}$$