# Properties

 Label 3675.2.a.bu Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ 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.

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

## 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
 −1.93185 −0.517638 0.517638 1.93185
−1.93185 1.00000 1.73205 0 −1.93185 0 0.517638 1.00000 0
1.2 −0.517638 1.00000 −1.73205 0 −0.517638 0 1.93185 1.00000 0
1.3 0.517638 1.00000 −1.73205 0 0.517638 0 −1.93185 1.00000 0
1.4 1.93185 1.00000 1.73205 0 1.93185 0 −0.517638 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.2.a.bu 4
5.b even 2 1 3675.2.a.bs 4
5.c odd 4 2 735.2.d.f 8
7.b odd 2 1 3675.2.a.bs 4
15.e even 4 2 2205.2.d.t 8
35.c odd 2 1 inner 3675.2.a.bu 4
35.f even 4 2 735.2.d.f 8
35.k even 12 2 735.2.q.c 8
35.k even 12 2 735.2.q.d 8
35.l odd 12 2 735.2.q.c 8
35.l odd 12 2 735.2.q.d 8
105.k odd 4 2 2205.2.d.t 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.d.f 8 5.c odd 4 2
735.2.d.f 8 35.f even 4 2
735.2.q.c 8 35.k even 12 2
735.2.q.c 8 35.l odd 12 2
735.2.q.d 8 35.k even 12 2
735.2.q.d 8 35.l odd 12 2
2205.2.d.t 8 15.e even 4 2
2205.2.d.t 8 105.k odd 4 2
3675.2.a.bs 4 5.b even 2 1
3675.2.a.bs 4 7.b odd 2 1
3675.2.a.bu 4 1.a even 1 1 trivial
3675.2.a.bu 4 35.c odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{2} + 1$$ $$T_{11}^{2} - 12$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -12 + T^{2} )^{2}$$
$13$ $$( 4 + T )^{4}$$
$17$ $$( 4 + T )^{4}$$
$19$ $$4 - 28 T^{2} + T^{4}$$
$23$ $$484 - 52 T^{2} + T^{4}$$
$29$ $$( -44 - 4 T + T^{2} )^{2}$$
$31$ $$( -54 + T^{2} )^{2}$$
$37$ $$64 - 112 T^{2} + T^{4}$$
$41$ $$( -72 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 6 + T )^{4}$$
$53$ $$( -54 + T^{2} )^{2}$$
$59$ $$64 - 112 T^{2} + T^{4}$$
$61$ $$36 - 84 T^{2} + T^{4}$$
$67$ $$7744 - 208 T^{2} + T^{4}$$
$71$ $$( -92 - 8 T + T^{2} )^{2}$$
$73$ $$( -32 + 8 T + T^{2} )^{2}$$
$79$ $$( 52 + 16 T + T^{2} )^{2}$$
$83$ $$( 6 + T )^{4}$$
$89$ $$4096 - 256 T^{2} + T^{4}$$
$97$ $$( 96 + 24 T + T^{2} )^{2}$$