# Properties

 Label 4235.2.a.bc 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: 5.5.288385.1 Defining polynomial: $$x^{5} - 7 x^{3} - 2 x^{2} + 10 x + 3$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 7 x^{3} - 2 x^{2} + 10 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 12$$

## 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
 2.43438 1.35681 −0.300703 −1.58597 −1.90452
−2.43438 2.36534 3.92619 1.00000 −5.75812 −1.00000 −4.68908 2.59481 −2.43438
1.2 −1.35681 −0.242974 −0.159077 1.00000 0.329669 −1.00000 2.92945 −2.94096 −1.35681
1.3 0.300703 2.68115 −1.90958 1.00000 0.806228 −1.00000 −1.17562 4.18855 0.300703
1.4 1.58597 −3.01851 0.515286 1.00000 −4.78725 −1.00000 −2.35471 6.11138 1.58597
1.5 1.90452 0.214998 1.62718 1.00000 0.409467 −1.00000 −0.710046 −2.95378 1.90452
 $$n$$: 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.bc 5
11.b odd 2 1 4235.2.a.bd yes 5

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

 $$T_{2}^{5} - 7 T_{2}^{3} + 2 T_{2}^{2} + 10 T_{2} - 3$$ $$T_{3}^{5} - 2 T_{3}^{4} - 9 T_{3}^{3} + 19 T_{3}^{2} + T_{3} - 1$$ $$T_{13}^{5} + 6 T_{13}^{4} - 8 T_{13}^{3} - 119 T_{13}^{2} - 260 T_{13} - 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + 10 T + 2 T^{2} - 7 T^{3} + T^{5}$$
$3$ $$-1 + T + 19 T^{2} - 9 T^{3} - 2 T^{4} + T^{5}$$
$5$ $$( -1 + T )^{5}$$
$7$ $$( 1 + T )^{5}$$
$11$ $$T^{5}$$
$13$ $$-169 - 260 T - 119 T^{2} - 8 T^{3} + 6 T^{4} + T^{5}$$
$17$ $$-75 - 170 T - 123 T^{2} - 28 T^{3} + 2 T^{4} + T^{5}$$
$19$ $$73 + 9 T - 56 T^{2} - T^{3} + 7 T^{4} + T^{5}$$
$23$ $$81 - 339 T + 223 T^{2} - 34 T^{3} - 5 T^{4} + T^{5}$$
$29$ $$81 - 243 T - 349 T^{2} - 16 T^{3} + 11 T^{4} + T^{5}$$
$31$ $$-327 + 1313 T - 181 T^{2} - 89 T^{3} + 2 T^{4} + T^{5}$$
$37$ $$-9801 + 2651 T + 529 T^{2} - 139 T^{3} - 2 T^{4} + T^{5}$$
$41$ $$-2475 - 1935 T - 409 T^{2} + 16 T^{3} + 13 T^{4} + T^{5}$$
$43$ $$-965 - 5325 T - 1761 T^{2} - 123 T^{3} + 10 T^{4} + T^{5}$$
$47$ $$12609 + 3198 T - 442 T^{2} - 127 T^{3} + 2 T^{4} + T^{5}$$
$53$ $$-309 - 413 T - 67 T^{2} + 45 T^{3} + 14 T^{4} + T^{5}$$
$59$ $$81 + 640 T + 563 T^{2} - 100 T^{3} - 6 T^{4} + T^{5}$$
$61$ $$393 + 778 T + 510 T^{2} + 149 T^{3} + 20 T^{4} + T^{5}$$
$67$ $$43677 + 1039 T - 2386 T^{2} - 219 T^{3} + 9 T^{4} + T^{5}$$
$71$ $$3753 + 1191 T - 421 T^{2} - 71 T^{3} + 10 T^{4} + T^{5}$$
$73$ $$-103 - 1067 T - 386 T^{2} + 13 T^{3} + 15 T^{4} + T^{5}$$
$79$ $$41845 - 4755 T - 2612 T^{2} - 23 T^{3} + 23 T^{4} + T^{5}$$
$83$ $$-26499 + 10186 T - 265 T^{2} - 198 T^{3} + 8 T^{4} + T^{5}$$
$89$ $$1587 - 3380 T + 1589 T^{2} - 195 T^{3} - 7 T^{4} + T^{5}$$
$97$ $$1297 - 1506 T + 209 T^{2} + 101 T^{3} - 21 T^{4} + T^{5}$$