# Properties

 Label 3675.2.a.bz 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: 4.4.11344.1 Defining polynomial: $$x^{4} - 2 x^{3} - 4 x^{2} + 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 4$$

## 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
 2.78165 1.28734 −0.552409 −1.51658
−1.78165 −1.00000 1.17429 0 1.78165 0 1.47113 1.00000 0
1.2 −0.287336 −1.00000 −1.91744 0 0.287336 0 1.12562 1.00000 0
1.3 1.55241 −1.00000 0.409975 0 −1.55241 0 −2.46837 1.00000 0
1.4 2.51658 −1.00000 4.33317 0 −2.51658 0 5.87162 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.bz 4
5.b even 2 1 3675.2.a.bp 4
5.c odd 4 2 735.2.d.d 8
7.b odd 2 1 3675.2.a.cb 4
7.c even 3 2 525.2.i.h 8
15.e even 4 2 2205.2.d.s 8
35.c odd 2 1 3675.2.a.bn 4
35.f even 4 2 735.2.d.e 8
35.j even 6 2 525.2.i.k 8
35.k even 12 4 735.2.q.g 16
35.l odd 12 4 105.2.q.a 16
105.k odd 4 2 2205.2.d.o 8
105.x even 12 4 315.2.bf.b 16
140.w even 12 4 1680.2.di.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 35.l odd 12 4
315.2.bf.b 16 105.x even 12 4
525.2.i.h 8 7.c even 3 2
525.2.i.k 8 35.j even 6 2
735.2.d.d 8 5.c odd 4 2
735.2.d.e 8 35.f even 4 2
735.2.q.g 16 35.k even 12 4
1680.2.di.d 16 140.w even 12 4
2205.2.d.o 8 105.k odd 4 2
2205.2.d.s 8 15.e even 4 2
3675.2.a.bn 4 35.c odd 2 1
3675.2.a.bp 4 5.b even 2 1
3675.2.a.bz 4 1.a even 1 1 trivial
3675.2.a.cb 4 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 6 T_{2} + 2$$ $$T_{11}^{4} - 18 T_{11}^{2} - 14 T_{11} + 30$$ $$T_{13}^{4} - 2 T_{13}^{3} - 28 T_{13}^{2} + 36 T_{13} + 127$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 6 T - 4 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$30 - 14 T - 18 T^{2} + T^{4}$$
$13$ $$127 + 36 T - 28 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$134 + 22 T - 28 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$9 + 40 T + 38 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$-1506 + 586 T - 32 T^{2} - 10 T^{3} + T^{4}$$
$29$ $$-22 - 190 T - 38 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$61 - 96 T - 6 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$907 - 722 T + 204 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$-10 + 146 T - 50 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-49 + 154 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$-120 - 64 T + 16 T^{2} + 10 T^{3} + T^{4}$$
$53$ $$96 - 208 T + 120 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$10 + 6 T - 6 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$500 - 700 T - 100 T^{2} + 8 T^{3} + T^{4}$$
$67$ $$-905 + 544 T - 72 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$-3202 - 1334 T - 90 T^{2} + 14 T^{3} + T^{4}$$
$73$ $$-1389 + 794 T - 64 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$7081 - 684 T - 154 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$362 - 238 T - 56 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$-534 + 986 T - 194 T^{2} - 8 T^{3} + T^{4}$$
$97$ $$-4 - 16 T - 8 T^{2} + 2 T^{3} + T^{4}$$