# Properties

 Label 3525.2.a.t Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 705) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} + \beta_{3} ) q^{2} + q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} ) q^{6} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} + \beta_{3} ) q^{2} + q^{3} + ( 2 + \beta_{1} - \beta_{2} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} ) q^{6} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{11} + ( 2 + \beta_{1} - \beta_{2} ) q^{12} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{13} + ( 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{14} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{16} + ( -1 + \beta_{1} - 3 \beta_{3} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} ) q^{18} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{19} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{22} + ( -4 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{23} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{24} + ( -\beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{26} + q^{27} + ( -5 + 4 \beta_{2} + \beta_{3} ) q^{28} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 2 \beta_{2} - 5 \beta_{3} ) q^{31} + ( -7 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{32} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{33} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 2 + \beta_{1} - \beta_{2} ) q^{36} + ( 1 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 - 2 \beta_{2} - 5 \beta_{3} ) q^{38} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{39} + ( -2 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{42} + ( -6 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -5 - \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{44} + ( 8 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{46} + q^{47} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{48} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{49} + ( -1 + \beta_{1} - 3 \beta_{3} ) q^{51} + ( 7 - \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -1 - \beta_{1} + \beta_{3} ) q^{54} + ( 10 - 7 \beta_{2} - \beta_{3} ) q^{56} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( 1 + \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{58} + ( -3 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -7 + 4 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -3 + 5 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{62} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( 6 - \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{64} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{66} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{67} + ( 1 + \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -4 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{69} + ( 4 + \beta_{2} + 4 \beta_{3} ) q^{71} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{72} + ( 6 + 2 \beta_{2} - \beta_{3} ) q^{73} + ( -14 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{74} + ( -2 + 2 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{76} + ( -1 + 7 \beta_{1} - \beta_{2} - 10 \beta_{3} ) q^{77} + ( -\beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{78} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{79} + q^{81} + ( 1 + 7 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{82} + ( -6 + 2 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -5 + 4 \beta_{2} + \beta_{3} ) q^{84} + ( -2 + 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{86} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 10 + 4 \beta_{1} - 8 \beta_{2} - 11 \beta_{3} ) q^{88} + ( -2 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -10 + 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{91} + ( -7 - 6 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{92} + ( 2 \beta_{2} - 5 \beta_{3} ) q^{93} + ( -1 - \beta_{1} + \beta_{3} ) q^{94} + ( -7 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{96} + ( 2 \beta_{2} + 6 \beta_{3} ) q^{97} + ( 8 - 4 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} ) q^{98} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{3} + 8q^{4} - 4q^{6} - 8q^{7} - 12q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{3} + 8q^{4} - 4q^{6} - 8q^{7} - 12q^{8} + 4q^{9} + 4q^{11} + 8q^{12} + 4q^{13} + 12q^{16} - 4q^{17} - 4q^{18} - 8q^{19} - 8q^{21} + 16q^{22} - 16q^{23} - 12q^{24} + 4q^{27} - 20q^{28} + 4q^{29} - 28q^{32} + 4q^{33} - 16q^{34} + 8q^{36} + 4q^{37} - 4q^{38} + 4q^{39} - 8q^{41} - 24q^{43} - 20q^{44} + 32q^{46} + 4q^{47} + 12q^{48} + 8q^{49} - 4q^{51} + 28q^{52} - 12q^{53} - 4q^{54} + 40q^{56} - 8q^{57} + 4q^{58} - 28q^{61} - 12q^{62} - 8q^{63} + 24q^{64} + 16q^{66} + 8q^{67} + 4q^{68} - 16q^{69} + 16q^{71} - 12q^{72} + 24q^{73} - 56q^{74} - 8q^{76} - 4q^{77} - 4q^{79} + 4q^{81} + 4q^{82} - 24q^{83} - 20q^{84} - 8q^{86} + 4q^{87} + 40q^{88} - 8q^{89} - 40q^{91} - 28q^{92} - 4q^{94} - 28q^{96} + 32q^{98} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - 4 x + 2$$:

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

## 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
 0.334904 2.68554 −1.74912 −1.27133
−2.74912 1.00000 5.55765 0 −2.74912 −3.47363 −9.78039 1.00000 0
1.2 −2.27133 1.00000 3.15894 0 −2.27133 0.797933 −2.63234 1.00000 0
1.3 −0.665096 1.00000 −1.55765 0 −0.665096 −0.526374 2.36618 1.00000 0
1.4 1.68554 1.00000 0.841058 0 1.68554 −4.79793 −1.95345 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$47$$ $$-1$$

## 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 3525.2.a.t 4
5.b even 2 1 705.2.a.k 4
15.d odd 2 1 2115.2.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.k 4 5.b even 2 1
2115.2.a.o 4 15.d odd 2 1
3525.2.a.t 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{4} + 4 T_{2}^{3} - 12 T_{2} - 7$$ $$T_{7}^{4} + 8 T_{7}^{3} + 14 T_{7}^{2} - 8 T_{7} - 7$$ $$T_{11}^{4} - 4 T_{11}^{3} - 24 T_{11}^{2} + 120 T_{11} - 124$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 - 12 T + 4 T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$-7 - 8 T + 14 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$-124 + 120 T - 24 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$169 + 52 T - 24 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$89 - 36 T - 24 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$-383 - 192 T - 8 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$-287 + 16 T + 70 T^{2} + 16 T^{3} + T^{4}$$
$29$ $$-167 + 236 T - 50 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$196 - 256 T - 92 T^{2} + T^{4}$$
$37$ $$4772 + 280 T - 136 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$41 - 408 T - 54 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$-2944 - 256 T + 144 T^{2} + 24 T^{3} + T^{4}$$
$47$ $$( -1 + T )^{4}$$
$53$ $$-1607 - 892 T - 56 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$-599 - 504 T - 112 T^{2} + T^{4}$$
$61$ $$-431 + 252 T + 214 T^{2} + 28 T^{3} + T^{4}$$
$67$ $$-92 + 208 T - 20 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$-79 + 520 T + 8 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$356 - 528 T + 188 T^{2} - 24 T^{3} + T^{4}$$
$79$ $$-28 - 40 T - 8 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$-1264 + 96 T + 160 T^{2} + 24 T^{3} + T^{4}$$
$89$ $$-3452 - 2544 T - 236 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$6928 + 448 T - 224 T^{2} + T^{4}$$