# Properties

 Label 3525.2.a.v Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.2379008.1 Defining polynomial: $$x^{5} - 10 x^{3} + 23 x - 4$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 10 x^{3} + 23 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu + 4$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 10 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 7 \beta_{2} + 22$$

## 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
 −2.35575 −2.15072 0.176288 1.70768 2.62251
−2.35575 −1.00000 3.54957 0 2.35575 0.844227 −3.65041 1.00000 0
1.2 −2.15072 −1.00000 2.62562 0 2.15072 −2.17958 −1.34553 1.00000 0
1.3 0.176288 −1.00000 −1.96892 0 −0.176288 −5.09296 −0.699672 1.00000 0
1.4 1.70768 −1.00000 0.916167 0 −1.70768 0.474685 −1.85084 1.00000 0
1.5 2.62251 −1.00000 4.87757 0 −2.62251 −4.04637 7.54645 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.v 5
5.b even 2 1 705.2.a.l 5
15.d odd 2 1 2115.2.a.q 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.l 5 5.b even 2 1
2115.2.a.q 5 15.d odd 2 1
3525.2.a.v 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{5} - 10 T_{2}^{3} + 23 T_{2} - 4$$ $$T_{7}^{5} + 10 T_{7}^{4} + 26 T_{7}^{3} - 4 T_{7}^{2} - 43 T_{7} + 18$$ $$T_{11}^{5} + 2 T_{11}^{4} - 20 T_{11}^{3} - 8 T_{11}^{2} + 20 T_{11} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + 23 T - 10 T^{3} + T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$T^{5}$$
$7$ $$18 - 43 T - 4 T^{2} + 26 T^{3} + 10 T^{4} + T^{5}$$
$11$ $$8 + 20 T - 8 T^{2} - 20 T^{3} + 2 T^{4} + T^{5}$$
$13$ $$557 + 39 T - 134 T^{2} - 14 T^{3} + 7 T^{4} + T^{5}$$
$17$ $$6 - 17 T - 20 T^{2} + 30 T^{3} - 10 T^{4} + T^{5}$$
$19$ $$-342 + 259 T + 40 T^{2} - 38 T^{3} - 2 T^{4} + T^{5}$$
$23$ $$-81 + 301 T - 26 T^{2} - 38 T^{3} + 3 T^{4} + T^{5}$$
$29$ $$-10 + 81 T - 26 T^{3} + 2 T^{4} + T^{5}$$
$31$ $$632 + 1868 T - 384 T^{2} - 92 T^{3} + 6 T^{4} + T^{5}$$
$37$ $$7552 + 1460 T - 544 T^{2} - 76 T^{3} + 8 T^{4} + T^{5}$$
$41$ $$-17076 + 8233 T + 196 T^{2} - 182 T^{3} + T^{5}$$
$43$ $$320 + 704 T - 368 T^{2} - 16 T^{3} + 13 T^{4} + T^{5}$$
$47$ $$( 1 + T )^{5}$$
$53$ $$-366 + 367 T + 928 T^{2} - 126 T^{3} - 10 T^{4} + T^{5}$$
$59$ $$-9351 + 6103 T - 686 T^{2} - 118 T^{3} + 11 T^{4} + T^{5}$$
$61$ $$-10055 + 509 T + 786 T^{2} - 62 T^{3} - 11 T^{4} + T^{5}$$
$67$ $$61168 - 7620 T - 2984 T^{2} - 12 T^{3} + 24 T^{4} + T^{5}$$
$71$ $$-2903 - 3305 T - 1166 T^{2} - 102 T^{3} + 11 T^{4} + T^{5}$$
$73$ $$-33380 - 23788 T - 4108 T^{2} - 136 T^{3} + 17 T^{4} + T^{5}$$
$79$ $$-684 + 692 T - 92 T^{2} - 44 T^{3} + 5 T^{4} + T^{5}$$
$83$ $$8192 + 1520 T - 672 T^{2} - 152 T^{3} + T^{5}$$
$89$ $$20 + 132 T - 92 T^{2} - 40 T^{3} + 3 T^{4} + T^{5}$$
$97$ $$-160 + 368 T + 176 T^{2} - 88 T^{3} - 14 T^{4} + T^{5}$$