# Properties

 Label 3675.2.a.bi Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ 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: $$1$$ 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} - 3q^{3} + 5q^{4} + q^{6} - 9q^{8} + 3q^{9} + O(q^{10})$$ $$3q - q^{2} - 3q^{3} + 5q^{4} + q^{6} - 9q^{8} + 3q^{9} + 6q^{11} - 5q^{12} - 6q^{13} + 13q^{16} - q^{18} - 6q^{19} - 2q^{22} - 4q^{23} + 9q^{24} - 10q^{26} - 3q^{27} + 2q^{29} - 2q^{31} - 29q^{32} - 6q^{33} + 12q^{34} + 5q^{36} + 4q^{37} + 14q^{38} + 6q^{39} - 2q^{41} - 4q^{43} + 10q^{44} - 8q^{46} + 8q^{47} - 13q^{48} - 18q^{52} - 14q^{53} + q^{54} + 6q^{57} + 18q^{58} - 16q^{59} + 6q^{61} - 30q^{62} + 13q^{64} + 2q^{66} - 8q^{67} + 8q^{68} + 4q^{69} + 6q^{71} - 9q^{72} - 18q^{73} - 44q^{74} - 2q^{76} + 10q^{78} + 12q^{79} + 3q^{81} + 34q^{82} + 8q^{83} - 4q^{86} - 2q^{87} - 18q^{88} - 14q^{89} + 20q^{92} + 2q^{93} + 16q^{94} + 29q^{96} - 22q^{97} + 6q^{99} + O(q^{100})$$

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

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