# Properties

 Label 9075.2.a.df Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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.

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

## 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
 1.82709 −1.95630 −0.209057 1.33826
−1.95630 1.00000 1.82709 0 −1.95630 1.54732 0.338261 1.00000 0
1.2 −0.209057 1.00000 −1.95630 0 −0.209057 0.488830 0.827091 1.00000 0
1.3 1.33826 1.00000 −0.209057 0 1.33826 −3.78339 −2.95630 1.00000 0
1.4 1.82709 1.00000 1.33826 0 1.82709 1.74724 −1.20906 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 9075.2.a.df 4
5.b even 2 1 1815.2.a.q 4
11.b odd 2 1 9075.2.a.co 4
11.d odd 10 2 825.2.n.j 8
15.d odd 2 1 5445.2.a.bq 4
55.d odd 2 1 1815.2.a.u 4
55.h odd 10 2 165.2.m.c 8
55.l even 20 4 825.2.bx.e 16
165.d even 2 1 5445.2.a.bj 4
165.r even 10 2 495.2.n.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.c 8 55.h odd 10 2
495.2.n.c 8 165.r even 10 2
825.2.n.j 8 11.d odd 10 2
825.2.bx.e 16 55.l even 20 4
1815.2.a.q 4 5.b even 2 1
1815.2.a.u 4 55.d odd 2 1
5445.2.a.bj 4 165.d even 2 1
5445.2.a.bq 4 15.d odd 2 1
9075.2.a.co 4 11.b odd 2 1
9075.2.a.df 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} - T_{2}^{3} - 4 T_{2}^{2} + 4 T_{2} + 1$$ $$T_{7}^{4} - 10 T_{7}^{2} + 15 T_{7} - 5$$ $$T_{13}^{4} - 7 T_{13}^{3} - 16 T_{13}^{2} + 202 T_{13} - 359$$ $$T_{17}^{4} - 8 T_{17}^{3} - 6 T_{17}^{2} + 88 T_{17} - 59$$ $$T_{19}^{4} + 11 T_{19}^{3} + 6 T_{19}^{2} - 184 T_{19} - 239$$ $$T_{23}^{4} + 5 T_{23}^{3} - 25 T_{23}^{2} + 25 T_{23} - 5$$ $$T_{37}^{4} + 15 T_{37}^{3} + 65 T_{37}^{2} + 45 T_{37} - 155$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T - 4 T^{2} - T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$-5 + 15 T - 10 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-359 + 202 T - 16 T^{2} - 7 T^{3} + T^{4}$$
$17$ $$-59 + 88 T - 6 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$-239 - 184 T + 6 T^{2} + 11 T^{3} + T^{4}$$
$23$ $$-5 + 25 T - 25 T^{2} + 5 T^{3} + T^{4}$$
$29$ $$-419 + 38 T + 84 T^{2} + 17 T^{3} + T^{4}$$
$31$ $$-5 - 235 T - 45 T^{2} + 5 T^{3} + T^{4}$$
$37$ $$-155 + 45 T + 65 T^{2} + 15 T^{3} + T^{4}$$
$41$ $$-5 - 145 T - 10 T^{2} + 10 T^{3} + T^{4}$$
$43$ $$-569 - 541 T - 124 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$121 + 143 T - 16 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$25 - 25 T - 10 T^{2} + 10 T^{3} + T^{4}$$
$59$ $$1341 + 261 T - 69 T^{2} - 9 T^{3} + T^{4}$$
$61$ $$6571 + 2998 T + 504 T^{2} + 37 T^{3} + T^{4}$$
$67$ $$31 + 72 T - 46 T^{2} + 3 T^{3} + T^{4}$$
$71$ $$-389 + 83 T + 39 T^{2} - 13 T^{3} + T^{4}$$
$73$ $$-4205 + 1320 T - 40 T^{2} - 15 T^{3} + T^{4}$$
$79$ $$-305 + 10 T + 95 T^{2} + 20 T^{3} + T^{4}$$
$83$ $$-14669 - 5347 T - 261 T^{2} + 17 T^{3} + T^{4}$$
$89$ $$-31869 - 6066 T - 129 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$61 - 463 T + 284 T^{2} - 32 T^{3} + T^{4}$$