# Properties

 Label 4235.2.a.y Level $4235$ Weight $2$ Character orbit 4235.a Self dual yes Analytic conductor $33.817$ Analytic rank $1$ Dimension $5$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.173513.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{2} q^{3} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{4} - q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{6} + q^{7} + ( 2 + \beta_{3} + \beta_{4} ) q^{8} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + \beta_{2} q^{3} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{4} - q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{6} + q^{7} + ( 2 + \beta_{3} + \beta_{4} ) q^{8} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} -\beta_{4} q^{10} + ( 1 - 2 \beta_{3} - \beta_{4} ) q^{12} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + \beta_{4} q^{14} -\beta_{2} q^{15} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{16} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{17} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{18} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{20} + \beta_{2} q^{21} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 + \beta_{1} - \beta_{2} ) q^{24} + q^{25} + ( -5 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{26} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{27} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{28} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{29} + ( 1 + \beta_{2} - \beta_{3} ) q^{30} + ( -2 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{32} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{34} - q^{35} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{36} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{37} + ( -5 + \beta_{2} - 3 \beta_{3} - 5 \beta_{4} ) q^{38} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{39} + ( -2 - \beta_{3} - \beta_{4} ) q^{40} + ( 3 - 3 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -1 - \beta_{2} + \beta_{3} ) q^{42} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{43} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{45} + ( 6 - \beta_{1} + \beta_{2} + \beta_{3} + 7 \beta_{4} ) q^{46} + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{48} + q^{49} + \beta_{4} q^{50} + ( \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{4} ) q^{51} + ( -6 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{52} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{53} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{54} + ( 2 + \beta_{3} + \beta_{4} ) q^{56} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + \beta_{4} ) q^{57} + ( -10 - 5 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} ) q^{58} + ( -2 + \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} ) q^{59} + ( -1 + 2 \beta_{3} + \beta_{4} ) q^{60} + ( -6 + \beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{61} + ( -7 + \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} ) q^{62} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{63} + ( -6 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{64} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{65} + ( -1 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{67} + ( -3 + \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{68} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{69} -\beta_{4} q^{70} + ( -4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{71} + ( -8 - \beta_{2} - 2 \beta_{4} ) q^{72} + ( 1 - 5 \beta_{1} + 3 \beta_{3} ) q^{73} + ( 5 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{74} + \beta_{2} q^{75} + ( -15 + \beta_{1} - 7 \beta_{2} - 6 \beta_{4} ) q^{76} + ( 7 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{78} + ( -7 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{79} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{80} + ( 3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{82} + ( \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{83} + ( 1 - 2 \beta_{3} - \beta_{4} ) q^{84} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{85} + ( 9 - 5 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{86} + ( 6 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{87} + ( -5 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{89} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{90} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{91} + ( 16 - 4 \beta_{1} + 5 \beta_{2} + 9 \beta_{4} ) q^{92} + ( 5 - 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -8 + 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{94} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 2 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{96} + ( 3 + \beta_{1} + \beta_{2} + 7 \beta_{3} + 4 \beta_{4} ) q^{97} + \beta_{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10})$$ $$5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 3 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 2 \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 23 \beta_{1} + 14$$

## 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
 −0.812660 −1.52979 3.18986 0.859039 0.293545
−2.23053 2.39311 2.97525 −1.00000 −5.33790 1.00000 −2.17531 2.72699 2.23053
1.2 −1.65369 −1.14142 0.734678 −1.00000 1.88755 1.00000 2.09245 −1.69716 1.65369
1.3 −0.686507 0.347661 −1.52871 −1.00000 −0.238671 1.00000 2.42248 −2.87913 0.686507
1.4 0.164091 −3.27813 −1.97307 −1.00000 −0.537912 1.00000 −0.651947 7.74611 −0.164091
1.5 2.40663 −0.321225 3.79186 −1.00000 −0.773069 1.00000 4.31233 −2.89681 −2.40663
 $$n$$: 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
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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.y 5
11.b odd 2 1 4235.2.a.be yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.y 5 1.a even 1 1 trivial
4235.2.a.be yes 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4235))$$:

 $$T_{2}^{5} + 2 T_{2}^{4} - 5 T_{2}^{3} - 12 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{3}^{5} + 2 T_{3}^{4} - 7 T_{3}^{3} - 9 T_{3}^{2} + T_{3} + 1$$ $$T_{13}^{5} + 12 T_{13}^{4} + 12 T_{13}^{3} - 325 T_{13}^{2} - 1368 T_{13} - 1571$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T - 12 T^{2} - 5 T^{3} + 2 T^{4} + T^{5}$$
$3$ $$1 + T - 9 T^{2} - 7 T^{3} + 2 T^{4} + T^{5}$$
$5$ $$( 1 + T )^{5}$$
$7$ $$( -1 + T )^{5}$$
$11$ $$T^{5}$$
$13$ $$-1571 - 1368 T - 325 T^{2} + 12 T^{3} + 12 T^{4} + T^{5}$$
$17$ $$83 - 286 T - 35 T^{2} + 50 T^{3} + 14 T^{4} + T^{5}$$
$19$ $$2259 - 303 T - 446 T^{2} - 37 T^{3} + 9 T^{4} + T^{5}$$
$23$ $$-67 + 199 T - 207 T^{2} + 92 T^{3} - 17 T^{4} + T^{5}$$
$29$ $$3197 + 2257 T - 193 T^{2} - 104 T^{3} + 3 T^{4} + T^{5}$$
$31$ $$1359 + 1869 T + 55 T^{2} - 85 T^{3} - 2 T^{4} + T^{5}$$
$37$ $$1051 - 1211 T + 473 T^{2} - 59 T^{3} - 4 T^{4} + T^{5}$$
$41$ $$-4617 - 891 T + 999 T^{2} - 36 T^{3} - 15 T^{4} + T^{5}$$
$43$ $$10971 + 2391 T - 559 T^{2} - 119 T^{3} + 4 T^{4} + T^{5}$$
$47$ $$251 + 566 T - 260 T^{2} - 123 T^{3} + 2 T^{4} + T^{5}$$
$53$ $$-22913 + 4025 T + 739 T^{2} - 129 T^{3} - 6 T^{4} + T^{5}$$
$59$ $$68011 + 11074 T - 1511 T^{2} - 242 T^{3} + 6 T^{4} + T^{5}$$
$61$ $$1867 - 1336 T - 266 T^{2} + 87 T^{3} + 20 T^{4} + T^{5}$$
$67$ $$2309 + 721 T - 420 T^{2} - 133 T^{3} - 3 T^{4} + T^{5}$$
$71$ $$13 - 73 T + 127 T^{2} - 75 T^{3} + 6 T^{4} + T^{5}$$
$73$ $$-1097 - 2993 T - 2176 T^{2} - 173 T^{3} + 11 T^{4} + T^{5}$$
$79$ $$-20143 - 11379 T - 1708 T^{2} + 13 T^{3} + 19 T^{4} + T^{5}$$
$83$ $$279 - 1296 T - 707 T^{2} - 74 T^{3} + 8 T^{4} + T^{5}$$
$89$ $$-6047 + 3656 T + 571 T^{2} - 213 T^{3} - T^{4} + T^{5}$$
$97$ $$-271673 + 62192 T - 475 T^{2} - 463 T^{3} + 7 T^{4} + T^{5}$$