# Properties

 Label 3630.2.a.br Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9856959337$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.43025.1 Defining polynomial: $$x^{4} - x^{3} - 21 x^{2} + 10 x + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 330) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} -\beta_{1} q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} -\beta_{1} q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + ( -2 - \beta_{2} - \beta_{3} ) q^{13} + \beta_{1} q^{14} + q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} - q^{18} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{19} + q^{20} -\beta_{1} q^{21} + ( 3 + 3 \beta_{2} - \beta_{3} ) q^{23} - q^{24} + q^{25} + ( 2 + \beta_{2} + \beta_{3} ) q^{26} + q^{27} -\beta_{1} q^{28} + ( -1 + \beta_{1} - \beta_{2} ) q^{29} - q^{30} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{31} - q^{32} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} -\beta_{1} q^{35} + q^{36} + ( 4 - 3 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{38} + ( -2 - \beta_{2} - \beta_{3} ) q^{39} - q^{40} + ( \beta_{1} + 2 \beta_{3} ) q^{41} + \beta_{1} q^{42} + ( 3 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{43} + q^{45} + ( -3 - 3 \beta_{2} + \beta_{3} ) q^{46} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + q^{48} + ( 4 + \beta_{1} + \beta_{3} ) q^{49} - q^{50} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -2 - \beta_{2} - \beta_{3} ) q^{52} + ( -2 + \beta_{1} - 6 \beta_{2} ) q^{53} - q^{54} + \beta_{1} q^{56} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} ) q^{58} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{59} + q^{60} + ( 4 - 2 \beta_{1} ) q^{61} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{62} -\beta_{1} q^{63} + q^{64} + ( -2 - \beta_{2} - \beta_{3} ) q^{65} + ( 11 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} + ( 3 + 3 \beta_{2} - \beta_{3} ) q^{69} + \beta_{1} q^{70} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{71} - q^{72} + 6 q^{73} + ( -4 + 3 \beta_{2} ) q^{74} + q^{75} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{76} + ( 2 + \beta_{2} + \beta_{3} ) q^{78} + ( -5 + \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{79} + q^{80} + q^{81} + ( -\beta_{1} - 2 \beta_{3} ) q^{82} + ( 8 + 4 \beta_{2} ) q^{83} -\beta_{1} q^{84} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -3 - \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{86} + ( -1 + \beta_{1} - \beta_{2} ) q^{87} + ( 4 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{89} - q^{90} + ( 1 + 2 \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{91} + ( 3 + 3 \beta_{2} - \beta_{3} ) q^{92} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{94} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95} - q^{96} + ( 4 + 2 \beta_{1} - 6 \beta_{2} ) q^{97} + ( -4 - \beta_{1} - \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - q^{7} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - q^{7} - 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} - 4q^{13} + q^{14} + 4q^{15} + 4q^{16} + 5q^{17} - 4q^{18} - 7q^{19} + 4q^{20} - q^{21} + 8q^{23} - 4q^{24} + 4q^{25} + 4q^{26} + 4q^{27} - q^{28} - q^{29} - 4q^{30} + 9q^{31} - 4q^{32} - 5q^{34} - q^{35} + 4q^{36} + 22q^{37} + 7q^{38} - 4q^{39} - 4q^{40} - 3q^{41} + q^{42} + q^{43} + 4q^{45} - 8q^{46} + 2q^{47} + 4q^{48} + 15q^{49} - 4q^{50} + 5q^{51} - 4q^{52} + 5q^{53} - 4q^{54} + q^{56} - 7q^{57} + q^{58} + 16q^{59} + 4q^{60} + 14q^{61} - 9q^{62} - q^{63} + 4q^{64} - 4q^{65} + 29q^{67} + 5q^{68} + 8q^{69} + q^{70} - 6q^{71} - 4q^{72} + 24q^{73} - 22q^{74} + 4q^{75} - 7q^{76} + 4q^{78} - 3q^{79} + 4q^{80} + 4q^{81} + 3q^{82} + 24q^{83} - q^{84} + 5q^{85} - q^{86} - q^{87} + 5q^{89} - 4q^{90} - 16q^{91} + 8q^{92} + 9q^{93} - 2q^{94} - 7q^{95} - 4q^{96} + 30q^{97} - 15q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 21 x^{2} + 10 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 11 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 11$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 10 \beta_{2} + 12 \beta_{1} + 11$$

## 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
 4.07314 2.86832 −2.45511 −3.48636
−1.00000 1.00000 1.00000 1.00000 −1.00000 −4.07314 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −2.86832 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 2.45511 −1.00000 1.00000 −1.00000
1.4 −1.00000 1.00000 1.00000 1.00000 −1.00000 3.48636 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 3630.2.a.br 4
11.b odd 2 1 3630.2.a.bt 4
11.d odd 10 2 330.2.m.e 8
33.f even 10 2 990.2.n.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.e 8 11.d odd 10 2
990.2.n.k 8 33.f even 10 2
3630.2.a.br 4 1.a even 1 1 trivial
3630.2.a.bt 4 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7}^{4} + T_{7}^{3} - 21 T_{7}^{2} - 10 T_{7} + 100$$ $$T_{13}^{4} + 4 T_{13}^{3} - 27 T_{13}^{2} - 112 T_{13} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$100 - 10 T - 21 T^{2} + T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-11 - 112 T - 27 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$16 + 100 T - 23 T^{2} - 5 T^{3} + T^{4}$$
$19$ $$-220 - 190 T - 19 T^{2} + 7 T^{3} + T^{4}$$
$23$ $$-605 + 330 T - 29 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$100 - 10 T - 21 T^{2} + T^{3} + T^{4}$$
$31$ $$-220 + 190 T - 21 T^{2} - 9 T^{3} + T^{4}$$
$37$ $$( 19 - 11 T + T^{2} )^{2}$$
$41$ $$2596 - 158 T - 101 T^{2} + 3 T^{3} + T^{4}$$
$43$ $$404 - 242 T - 107 T^{2} - T^{3} + T^{4}$$
$47$ $$2651 + 152 T - 151 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$596 + 240 T - 87 T^{2} - 5 T^{3} + T^{4}$$
$59$ $$-55 + 760 T - 31 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$704 + 432 T - 12 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$-26476 + 3662 T + 83 T^{2} - 29 T^{3} + T^{4}$$
$71$ $$1600 - 160 T - 116 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$5620 + 140 T - 209 T^{2} + 3 T^{3} + T^{4}$$
$83$ $$( 16 - 12 T + T^{2} )^{2}$$
$89$ $$500 + 350 T - 145 T^{2} - 5 T^{3} + T^{4}$$
$97$ $$-5104 + 520 T + 192 T^{2} - 30 T^{3} + T^{4}$$