# Properties

 Label 3675.2.a.bv Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4400.1 Defining polynomial: $$x^{4} - 7 x^{2} + 11$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + q^{3} + ( 2 - \beta_{3} ) q^{4} -\beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + q^{3} + ( 2 - \beta_{3} ) q^{4} -\beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{8} + q^{9} + 2 \beta_{3} q^{11} + ( 2 - \beta_{3} ) q^{12} + ( 3 - \beta_{3} ) q^{13} + ( 1 - 2 \beta_{3} ) q^{16} + ( 3 + \beta_{3} ) q^{17} -\beta_{2} q^{18} + ( \beta_{1} - 2 \beta_{2} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{22} + \beta_{1} q^{23} + ( \beta_{1} - \beta_{2} ) q^{24} + ( \beta_{1} - 4 \beta_{2} ) q^{26} + q^{27} + 2 q^{29} + ( \beta_{1} + 2 \beta_{2} ) q^{31} -\beta_{2} q^{32} + 2 \beta_{3} q^{33} + ( -\beta_{1} - 2 \beta_{2} ) q^{34} + ( 2 - \beta_{3} ) q^{36} -2 \beta_{1} q^{37} + ( 9 - 5 \beta_{3} ) q^{38} + ( 3 - \beta_{3} ) q^{39} + ( -\beta_{1} - 4 \beta_{2} ) q^{41} + 4 \beta_{2} q^{43} + ( -10 + 4 \beta_{3} ) q^{44} + ( 1 - 3 \beta_{3} ) q^{46} + ( 4 + 4 \beta_{3} ) q^{47} + ( 1 - 2 \beta_{3} ) q^{48} + ( 3 + \beta_{3} ) q^{51} + ( 11 - 5 \beta_{3} ) q^{52} + ( -\beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( \beta_{1} - 2 \beta_{2} ) q^{57} -2 \beta_{2} q^{58} + 4 \beta_{2} q^{59} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{61} + ( -7 - \beta_{3} ) q^{62} + ( 2 + 3 \beta_{3} ) q^{64} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{66} + 2 \beta_{1} q^{67} + ( 1 - \beta_{3} ) q^{68} + \beta_{1} q^{69} + ( -4 - 2 \beta_{3} ) q^{71} + ( \beta_{1} - \beta_{2} ) q^{72} + ( 5 + \beta_{3} ) q^{73} + ( -2 + 6 \beta_{3} ) q^{74} + ( 3 \beta_{1} - 10 \beta_{2} ) q^{76} + ( \beta_{1} - 4 \beta_{2} ) q^{78} + ( 4 - 4 \beta_{3} ) q^{79} + q^{81} + ( 15 - \beta_{3} ) q^{82} + ( 2 - 2 \beta_{3} ) q^{83} + ( -16 + 4 \beta_{3} ) q^{86} + 2 q^{87} + 10 \beta_{2} q^{88} -3 \beta_{1} q^{89} + ( \beta_{1} - 4 \beta_{2} ) q^{92} + ( \beta_{1} + 2 \beta_{2} ) q^{93} -4 \beta_{1} q^{94} -\beta_{2} q^{96} + ( -1 - 5 \beta_{3} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 8q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 8q^{4} + 4q^{9} + 8q^{12} + 12q^{13} + 4q^{16} + 12q^{17} + 4q^{27} + 8q^{29} + 8q^{36} + 36q^{38} + 12q^{39} - 40q^{44} + 4q^{46} + 16q^{47} + 4q^{48} + 12q^{51} + 44q^{52} - 28q^{62} + 8q^{64} + 4q^{68} - 16q^{71} + 20q^{73} - 8q^{74} + 16q^{79} + 4q^{81} + 60q^{82} + 8q^{83} - 64q^{86} + 8q^{87} - 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2 \beta_{1}$$

## 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
 −1.54336 2.14896 −2.14896 1.54336
−2.49721 1.00000 4.23607 0 −2.49721 0 −5.58394 1.00000 0
1.2 −1.32813 1.00000 −0.236068 0 −1.32813 0 2.96979 1.00000 0
1.3 1.32813 1.00000 −0.236068 0 1.32813 0 −2.96979 1.00000 0
1.4 2.49721 1.00000 4.23607 0 2.49721 0 5.58394 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.bv 4
5.b even 2 1 3675.2.a.bt 4
5.c odd 4 2 735.2.d.c 8
7.b odd 2 1 3675.2.a.bt 4
15.e even 4 2 2205.2.d.m 8
35.c odd 2 1 inner 3675.2.a.bv 4
35.f even 4 2 735.2.d.c 8
35.k even 12 4 735.2.q.h 16
35.l odd 12 4 735.2.q.h 16
105.k odd 4 2 2205.2.d.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.d.c 8 5.c odd 4 2
735.2.d.c 8 35.f even 4 2
735.2.q.h 16 35.k even 12 4
735.2.q.h 16 35.l odd 12 4
2205.2.d.m 8 15.e even 4 2
2205.2.d.m 8 105.k odd 4 2
3675.2.a.bt 4 5.b even 2 1
3675.2.a.bt 4 7.b odd 2 1
3675.2.a.bv 4 1.a even 1 1 trivial
3675.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{4} - 8 T_{2}^{2} + 11$$ $$T_{11}^{2} - 20$$ $$T_{13}^{2} - 6 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 - 8 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -20 + T^{2} )^{2}$$
$13$ $$( 4 - 6 T + T^{2} )^{2}$$
$17$ $$( 4 - 6 T + T^{2} )^{2}$$
$19$ $$176 - 68 T^{2} + T^{4}$$
$23$ $$176 - 28 T^{2} + T^{4}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$176 - 52 T^{2} + T^{4}$$
$37$ $$2816 - 112 T^{2} + T^{4}$$
$41$ $$4400 - 140 T^{2} + T^{4}$$
$43$ $$2816 - 128 T^{2} + T^{4}$$
$47$ $$( -64 - 8 T + T^{2} )^{2}$$
$53$ $$176 - 52 T^{2} + T^{4}$$
$59$ $$2816 - 128 T^{2} + T^{4}$$
$61$ $$2816 - 208 T^{2} + T^{4}$$
$67$ $$2816 - 112 T^{2} + T^{4}$$
$71$ $$( -4 + 8 T + T^{2} )^{2}$$
$73$ $$( 20 - 10 T + T^{2} )^{2}$$
$79$ $$( -64 - 8 T + T^{2} )^{2}$$
$83$ $$( -16 - 4 T + T^{2} )^{2}$$
$89$ $$14256 - 252 T^{2} + T^{4}$$
$97$ $$( -124 + 2 T + T^{2} )^{2}$$