# Properties

 Label 9075.2.a.cl 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: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 - 2) q^{2} - q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{4} + ( - \beta_{2} - \beta_1 + 2) q^{6} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{8} + q^{9}+O(q^{10})$$ q + (b2 + b1 - 2) * q^2 - q^3 + (b3 - 2*b2 - b1 + 3) * q^4 + (-b2 - b1 + 2) * q^6 + (b3 + b2 - 2*b1) * q^7 + (-3*b3 + 3*b2 + b1 - 4) * q^8 + q^9 $$q + (\beta_{2} + \beta_1 - 2) q^{2} - q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{4} + ( - \beta_{2} - \beta_1 + 2) q^{6} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{8} + q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{12} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{13} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{14} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 5) q^{16} - 5 q^{17} + (\beta_{2} + \beta_1 - 2) q^{18} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{19} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{24} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{26} - q^{27} + (9 \beta_{3} - 5 \beta_1 - 4) q^{28} + (4 \beta_{2} + \beta_1 - 1) q^{29} + ( - 2 \beta_{3} + 3 \beta_1) q^{31} + ( - 9 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 7) q^{32} + ( - 5 \beta_{2} - 5 \beta_1 + 10) q^{34} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{36} + (5 \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - 9 \beta_{3} - 3 \beta_{2} + 7 \beta_1 + 4) q^{38} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + ( - 5 \beta_{3} + \beta_{2} + 4 \beta_1 + 6) q^{41} + (4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{42} + ( - 6 \beta_{3} + \beta_{2} + 3) q^{43} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{46} + ( - 3 \beta_{3} - \beta_{2} + 4 \beta_1 + 6) q^{47} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 5) q^{48} + ( - 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{49} + 5 q^{51} + ( - \beta_1 + 2) q^{52} + (5 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 1) q^{53} + ( - \beta_{2} - \beta_1 + 2) q^{54} + ( - 15 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 6) q^{56} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{57} + (\beta_{3} - 4 \beta_{2} + 3 \beta_1 + 6) q^{58} + (6 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{59} + ( - 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{61} + (7 \beta_{3} + \beta_{2} - 4 \beta_1) q^{62} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{63} + (16 \beta_{3} - 2 \beta_{2} - 11 \beta_1 + 8) q^{64} + ( - \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 5) q^{67} + ( - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 - 15) q^{68} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{69} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{71} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{72} + ( - 4 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 2) q^{73} + ( - 11 \beta_{3} + 5 \beta_{2} + 9 \beta_1 - 1) q^{74} + (19 \beta_{3} + \beta_{2} - 11 \beta_1 - 9) q^{76} + (\beta_{3} - 2 \beta_{2} - 1) q^{78} + ( - 7 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{79} + q^{81} + (14 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 11) q^{82} + (8 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 10) q^{83} + ( - 9 \beta_{3} + 5 \beta_1 + 4) q^{84} + (12 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 5) q^{86} + ( - 4 \beta_{2} - \beta_1 + 1) q^{87} + (3 \beta_{3} + 2 \beta_1 + 2) q^{89} + (4 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 6) q^{91} + (3 \beta_{3} - 2 \beta_1 - 1) q^{92} + (2 \beta_{3} - 3 \beta_1) q^{93} + (10 \beta_{3} + 8 \beta_{2} - \beta_1 - 13) q^{94} + (9 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 7) q^{96} + (3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 1) q^{97} + (6 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 7) q^{98}+O(q^{100})$$ q + (b2 + b1 - 2) * q^2 - q^3 + (b3 - 2*b2 - b1 + 3) * q^4 + (-b2 - b1 + 2) * q^6 + (b3 + b2 - 2*b1) * q^7 + (-3*b3 + 3*b2 + b1 - 4) * q^8 + q^9 + (-b3 + 2*b2 + b1 - 3) * q^12 + (b3 - b2 + b1 - 1) * q^13 + (-4*b3 - 2*b2 + 3*b1 + 1) * q^14 + (5*b3 - 5*b2 - 5*b1 + 5) * q^16 - 5 * q^17 + (b2 + b1 - 2) * q^18 + (3*b3 + 2*b2 - 3*b1 - 1) * q^19 + (-b3 - b2 + 2*b1) * q^21 + (b3 + b2 + b1) * q^23 + (3*b3 - 3*b2 - b1 + 4) * q^24 + (-b3 + 2*b2 + 1) * q^26 - q^27 + (9*b3 - 5*b1 - 4) * q^28 + (4*b2 + b1 - 1) * q^29 + (-2*b3 + 3*b1) * q^31 + (-9*b3 + 4*b2 + 8*b1 - 7) * q^32 + (-5*b2 - 5*b1 + 10) * q^34 + (b3 - 2*b2 - b1 + 3) * q^36 + (5*b3 - b2 - b1) * q^37 + (-9*b3 - 3*b2 + 7*b1 + 4) * q^38 + (-b3 + b2 - b1 + 1) * q^39 + (-5*b3 + b2 + 4*b1 + 6) * q^41 + (4*b3 + 2*b2 - 3*b1 - 1) * q^42 + (-6*b3 + b2 + 3) * q^43 + (-b3 + b2 + 3*b1 + 1) * q^46 + (-3*b3 - b2 + 4*b1 + 6) * q^47 + (-5*b3 + 5*b2 + 5*b1 - 5) * q^48 + (-4*b3 + b2 - 2*b1 + 4) * q^49 + 5 * q^51 + (-b1 + 2) * q^52 + (5*b3 - 3*b2 - 6*b1 - 1) * q^53 + (-b2 - b1 + 2) * q^54 + (-15*b3 + 4*b2 + 8*b1 + 6) * q^56 + (-3*b3 - 2*b2 + 3*b1 + 1) * q^57 + (b3 - 4*b2 + 3*b1 + 6) * q^58 + (6*b3 + b2 + b1 - 5) * q^59 + (-2*b3 - 6*b2 - 3*b1 + 3) * q^61 + (7*b3 + b2 - 4*b1) * q^62 + (b3 + b2 - 2*b1) * q^63 + (16*b3 - 2*b2 - 11*b1 + 8) * q^64 + (-b3 - 5*b2 + 5*b1 + 5) * q^67 + (-5*b3 + 10*b2 + 5*b1 - 15) * q^68 + (-b3 - b2 - b1) * q^69 + (-4*b3 - 2*b2 + 3*b1 - 4) * q^71 + (-3*b3 + 3*b2 + b1 - 4) * q^72 + (-4*b3 - 5*b2 + 3*b1 - 2) * q^73 + (-11*b3 + 5*b2 + 9*b1 - 1) * q^74 + (19*b3 + b2 - 11*b1 - 9) * q^76 + (b3 - 2*b2 - 1) * q^78 + (-7*b3 + 2*b2 + 2*b1 + 2) * q^79 + q^81 + (14*b3 + 4*b2 - 3*b1 - 11) * q^82 + (8*b3 - 2*b2 - 5*b1 - 10) * q^83 + (-9*b3 + 5*b1 + 4) * q^84 + (12*b3 - 4*b2 - 8*b1 - 5) * q^86 + (-4*b2 - b1 + 1) * q^87 + (3*b3 + 2*b1 + 2) * q^89 + (4*b3 - 5*b2 + 2*b1 - 6) * q^91 + (3*b3 - 2*b1 - 1) * q^92 + (2*b3 - 3*b1) * q^93 + (10*b3 + 8*b2 - b1 - 13) * q^94 + (9*b3 - 4*b2 - 8*b1 + 7) * q^96 + (3*b3 - 3*b2 - 4*b1 + 1) * q^97 + (6*b3 - 3*b2 - 3*b1 - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 $$4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98}+O(q^{100})$$ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 - 9 * q^12 - 3 * q^13 - 5 * q^14 + 15 * q^16 - 20 * q^17 - 5 * q^18 + 3 * q^19 - 2 * q^21 + 5 * q^23 + 15 * q^24 + 6 * q^26 - 4 * q^27 - 3 * q^28 + 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 + 9 * q^36 + 7 * q^37 - q^38 + 3 * q^39 + 20 * q^41 + 5 * q^42 + 2 * q^43 + 7 * q^46 + 20 * q^47 - 15 * q^48 + 8 * q^49 + 20 * q^51 + 7 * q^52 - 6 * q^53 + 5 * q^54 + 10 * q^56 - 3 * q^57 + 21 * q^58 - 5 * q^59 - 7 * q^61 + 12 * q^62 + 2 * q^63 + 49 * q^64 + 13 * q^67 - 45 * q^68 - 5 * q^69 - 25 * q^71 - 15 * q^72 - 23 * q^73 - 7 * q^74 - 7 * q^76 - 6 * q^78 + 4 * q^81 - 11 * q^82 - 33 * q^83 + 3 * q^84 - 12 * q^86 - 5 * q^87 + 16 * q^89 - 24 * q^91 + q^93 - 17 * q^94 + 30 * q^96 - 25 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 1$$ v^2 - v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 2\nu + 1$$ v^3 - v^2 - 2*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 1$$ b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3\beta_1$$ b3 + b2 + 3*b1

## 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
 −0.477260 0.737640 −1.35567 2.09529
−2.77222 −1.00000 5.68522 0 2.77222 2.27759 −10.2163 1.00000 0
1.2 −2.45589 −1.00000 4.03138 0 2.45589 −3.28684 −4.98884 1.00000 0
1.3 −1.16215 −1.00000 −0.649414 0 1.16215 4.28684 3.07901 1.00000 0
1.4 1.39026 −1.00000 −0.0671858 0 −1.39026 −1.27759 −2.87392 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.cl 4
5.b even 2 1 1815.2.a.x 4
11.b odd 2 1 9075.2.a.dj 4
11.c even 5 2 825.2.n.k 8
15.d odd 2 1 5445.2.a.be 4
55.d odd 2 1 1815.2.a.o 4
55.j even 10 2 165.2.m.a 8
55.k odd 20 4 825.2.bx.h 16
165.d even 2 1 5445.2.a.bv 4
165.o odd 10 2 495.2.n.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 55.j even 10 2
495.2.n.d 8 165.o odd 10 2
825.2.n.k 8 11.c even 5 2
825.2.bx.h 16 55.k odd 20 4
1815.2.a.o 4 55.d odd 2 1
1815.2.a.x 4 5.b even 2 1
5445.2.a.be 4 15.d odd 2 1
5445.2.a.bv 4 165.d even 2 1
9075.2.a.cl 4 1.a even 1 1 trivial
9075.2.a.dj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} + 5T_{2}^{3} + 4T_{2}^{2} - 10T_{2} - 11$$ T2^4 + 5*T2^3 + 4*T2^2 - 10*T2 - 11 $$T_{7}^{4} - 2T_{7}^{3} - 16T_{7}^{2} + 17T_{7} + 41$$ T7^4 - 2*T7^3 - 16*T7^2 + 17*T7 + 41 $$T_{13}^{4} + 3T_{13}^{3} - 10T_{13}^{2} + 6T_{13} - 1$$ T13^4 + 3*T13^3 - 10*T13^2 + 6*T13 - 1 $$T_{17} + 5$$ T17 + 5 $$T_{19}^{4} - 3T_{19}^{3} - 50T_{19}^{2} + 204T_{19} - 31$$ T19^4 - 3*T19^3 - 50*T19^2 + 204*T19 - 31 $$T_{23}^{4} - 5T_{23}^{3} - T_{23}^{2} + 5T_{23} - 1$$ T23^4 - 5*T23^3 - T23^2 + 5*T23 - 1 $$T_{37}^{4} - 7T_{37}^{3} - 37T_{37}^{2} + 133T_{37} + 431$$ T37^4 - 7*T37^3 - 37*T37^2 + 133*T37 + 431

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + 4 T^{2} - 10 T - 11$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 2 T^{3} - 16 T^{2} + 17 T + 41$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 3 T^{3} - 10 T^{2} + 6 T - 1$$
$17$ $$(T + 5)^{4}$$
$19$ $$T^{4} - 3 T^{3} - 50 T^{2} + 204 T - 31$$
$23$ $$T^{4} - 5 T^{3} - T^{2} + 5 T - 1$$
$29$ $$T^{4} - 5 T^{3} - 44 T^{2} + 140 T + 539$$
$31$ $$T^{4} + T^{3} - 25 T^{2} - 7 T + 139$$
$37$ $$T^{4} - 7 T^{3} - 37 T^{2} + 133 T + 431$$
$41$ $$T^{4} - 20 T^{3} + 86 T^{2} + \cdots - 2071$$
$43$ $$T^{4} - 2 T^{3} - 92 T^{2} + \cdots + 1861$$
$47$ $$T^{4} - 20 T^{3} + 94 T^{2} + 25 T - 11$$
$53$ $$T^{4} + 6 T^{3} - 100 T^{2} + \cdots - 1271$$
$59$ $$T^{4} + 5 T^{3} - 101 T^{2} + \cdots + 2299$$
$61$ $$T^{4} + 7 T^{3} - 136 T^{2} + \cdots + 1891$$
$67$ $$T^{4} - 13 T^{3} - 136 T^{2} + \cdots - 3379$$
$71$ $$T^{4} + 25 T^{3} + 171 T^{2} + \cdots - 2351$$
$73$ $$T^{4} + 23 T^{3} + 48 T^{2} + \cdots - 9199$$
$79$ $$T^{4} - 109 T^{2} + 210 T + 1199$$
$83$ $$T^{4} + 33 T^{3} + 265 T^{2} + \cdots - 12221$$
$89$ $$T^{4} - 16 T^{3} + 45 T^{2} + \cdots - 271$$
$97$ $$T^{4} - 60 T^{2} - 125 T - 25$$