# Properties

 Label 4235.2.a.v Level $4235$ Weight $2$ Character orbit 4235.a Self dual yes Analytic conductor $33.817$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4235 = 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4235.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.8166452560$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15952.1 Defining polynomial: $$x^{4} - 6 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + q^{5} + ( 1 - \beta_{1} ) q^{6} - q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + q^{5} + ( 1 - \beta_{1} ) q^{6} - q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + \beta_{1} q^{10} + ( -3 + \beta_{2} - \beta_{3} ) q^{12} + ( 5 - \beta_{2} + \beta_{3} ) q^{13} -\beta_{1} q^{14} -\beta_{2} q^{15} + ( 3 + 2 \beta_{1} ) q^{16} + ( -\beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{18} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{20} + \beta_{2} q^{21} + ( -1 + \beta_{1} + \beta_{2} ) q^{23} + ( -2 - \beta_{1} + \beta_{3} ) q^{24} + q^{25} + ( 5 \beta_{1} - \beta_{3} ) q^{26} + ( 3 + \beta_{3} ) q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{28} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 1 - \beta_{1} ) q^{30} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} + ( 4 + \beta_{1} + 2 \beta_{3} ) q^{32} + ( 3 - 3 \beta_{1} + 2 \beta_{3} ) q^{34} - q^{35} + ( -7 - 2 \beta_{1} ) q^{36} + ( -3 + 5 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -3 + 4 \beta_{1} - \beta_{3} ) q^{38} + ( 4 - \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{39} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{40} + ( 4 - 3 \beta_{1} - \beta_{3} ) q^{41} + ( -1 + \beta_{1} ) q^{42} + ( 3 - 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{45} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + ( 1 + 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{48} + q^{49} + \beta_{1} q^{50} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{51} + ( 6 + 4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 9 + \beta_{1} - \beta_{2} ) q^{53} + ( -1 + 4 \beta_{1} - \beta_{3} ) q^{54} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{56} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 8 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -4 + \beta_{1} - 3 \beta_{3} ) q^{59} + ( -3 + \beta_{2} - \beta_{3} ) q^{60} + ( -1 - 3 \beta_{1} + 2 \beta_{3} ) q^{61} + ( -6 - \beta_{2} - 2 \beta_{3} ) q^{62} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{63} + ( -5 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{64} + ( 5 - \beta_{2} + \beta_{3} ) q^{65} + ( 5 - 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -11 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{68} + ( -3 + 2 \beta_{2} + \beta_{3} ) q^{69} -\beta_{1} q^{70} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -4 - 5 \beta_{1} - 2 \beta_{3} ) q^{72} + ( 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{73} + ( 12 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{74} -\beta_{2} q^{75} + ( 11 + 3 \beta_{3} ) q^{76} + ( 5 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{78} + ( -2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + ( 3 + 2 \beta_{1} ) q^{80} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{81} + ( -8 - 3 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 1 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{83} + ( 3 - \beta_{2} + \beta_{3} ) q^{84} + ( -\beta_{2} - 2 \beta_{3} ) q^{85} + ( -6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{86} + ( 10 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 1 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{90} + ( -5 + \beta_{2} - \beta_{3} ) q^{91} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{92} + ( -9 + 3 \beta_{1} + 5 \beta_{2} ) q^{93} + ( 14 + 3 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{94} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{95} + ( 1 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{96} + ( -7 - 2 \beta_{2} + \beta_{3} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} + O(q^{10})$$ $$4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} - 8 q^{12} + 16 q^{13} - 2 q^{15} + 12 q^{16} + 2 q^{17} - 6 q^{18} + 6 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{23} - 10 q^{24} + 4 q^{25} + 2 q^{26} + 10 q^{27} - 4 q^{28} + 12 q^{29} + 4 q^{30} - 6 q^{31} + 12 q^{32} + 8 q^{34} - 4 q^{35} - 28 q^{36} - 6 q^{37} - 10 q^{38} + 12 q^{39} + 6 q^{40} + 18 q^{41} - 4 q^{42} + 6 q^{43} + 4 q^{45} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{51} + 30 q^{52} + 34 q^{53} - 2 q^{54} - 6 q^{56} - 28 q^{57} + 32 q^{58} - 10 q^{59} - 8 q^{60} - 8 q^{61} - 22 q^{62} - 4 q^{63} - 16 q^{64} + 16 q^{65} + 14 q^{67} - 44 q^{68} - 10 q^{69} - 12 q^{72} + 2 q^{73} + 48 q^{74} - 2 q^{75} + 38 q^{76} + 14 q^{78} - 8 q^{79} + 12 q^{80} - 12 q^{81} - 34 q^{82} - 10 q^{83} + 8 q^{84} + 2 q^{85} - 24 q^{86} + 32 q^{87} + 10 q^{89} - 6 q^{90} - 16 q^{91} + 10 q^{92} - 26 q^{93} + 54 q^{94} + 6 q^{95} + 8 q^{96} - 34 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 6 \nu - 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 6 \beta_{1} + 1$$

## 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
 −2.21161 −0.637875 0.275377 2.57411
−2.21161 −1.45216 2.89124 1.00000 3.21161 −1.00000 −1.97107 −0.891236 −2.21161
1.2 −0.637875 −2.56771 −1.59312 1.00000 1.63787 −1.00000 2.29196 3.59312 −0.637875
1.3 0.275377 2.63138 −1.92417 1.00000 0.724623 −1.00000 −1.08063 3.92417 0.275377
1.4 2.57411 −0.611516 4.62605 1.00000 −1.57411 −1.00000 6.75974 −2.62605 2.57411
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 4235.2.a.v yes 4
11.b odd 2 1 4235.2.a.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.u 4 11.b odd 2 1
4235.2.a.v yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4235))$$:

 $$T_{2}^{4} - 6 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{3}^{4} + 2 T_{3}^{3} - 6 T_{3}^{2} - 14 T_{3} - 6$$ $$T_{13}^{4} - 16 T_{13}^{3} + 70 T_{13}^{2} + 14 T_{13} - 442$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T - 6 T^{2} + T^{4}$$
$3$ $$-6 - 14 T - 6 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$T^{4}$$
$13$ $$-442 + 14 T + 70 T^{2} - 16 T^{3} + T^{4}$$
$17$ $$246 + 98 T - 50 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$-136 + 128 T - 20 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$12 - 8 T - 8 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$624 + 304 T - 16 T^{2} - 12 T^{3} + T^{4}$$
$31$ $$-102 - 166 T - 36 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$1636 - 880 T - 144 T^{2} + 6 T^{3} + T^{4}$$
$41$ $$-1298 + 322 T + 60 T^{2} - 18 T^{3} + T^{4}$$
$43$ $$972 + 216 T - 72 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$-262 + 702 T - 130 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$4036 - 2160 T + 416 T^{2} - 34 T^{3} + T^{4}$$
$59$ $$-702 - 574 T - 96 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$-2066 - 1026 T - 96 T^{2} + 8 T^{3} + T^{4}$$
$67$ $$316 + 400 T - 12 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$1872 + 160 T - 120 T^{2} + T^{4}$$
$73$ $$326 - 302 T - 122 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$-788 - 372 T - 28 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$-1304 - 2208 T - 256 T^{2} + 10 T^{3} + T^{4}$$
$89$ $$5448 + 824 T - 136 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$2168 + 1648 T + 380 T^{2} + 34 T^{3} + T^{4}$$