# Properties

 Label 9075.2.a.w Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−2.41421 1.00000 3.82843 0 −2.41421 −0.414214 −4.41421 1.00000 0
1.2 0.414214 1.00000 −1.82843 0 0.414214 2.41421 −1.58579 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.w 2
5.b even 2 1 9075.2.a.ca 2
11.b odd 2 1 825.2.a.f yes 2
33.d even 2 1 2475.2.a.l 2
55.d odd 2 1 825.2.a.d 2
55.e even 4 2 825.2.c.d 4
165.d even 2 1 2475.2.a.w 2
165.l odd 4 2 2475.2.c.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 55.d odd 2 1
825.2.a.f yes 2 11.b odd 2 1
825.2.c.d 4 55.e even 4 2
2475.2.a.l 2 33.d even 2 1
2475.2.a.w 2 165.d even 2 1
2475.2.c.o 4 165.l odd 4 2
9075.2.a.w 2 1.a even 1 1 trivial
9075.2.a.ca 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{7}^{2} - 2T_{7} - 1$$ T7^2 - 2*T7 - 1 $$T_{13}^{2} - 8$$ T13^2 - 8 $$T_{17}^{2} + 2T_{17} - 1$$ T17^2 + 2*T17 - 1 $$T_{19}^{2} + 10T_{19} + 23$$ T19^2 + 10*T19 + 23 $$T_{23} - 1$$ T23 - 1 $$T_{37}^{2} + 6T_{37} + 1$$ T37^2 + 6*T37 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T - 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} + 2T - 1$$
$19$ $$T^{2} + 10T + 23$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 8T + 8$$
$31$ $$T^{2} - 72$$
$37$ $$T^{2} + 6T + 1$$
$41$ $$T^{2} - 2T - 97$$
$43$ $$T^{2} - 12T + 4$$
$47$ $$T^{2} - 2T - 71$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$(T - 11)^{2}$$
$61$ $$T^{2} + 12T + 28$$
$67$ $$T^{2} + 12T + 4$$
$71$ $$T^{2} - 10T + 17$$
$73$ $$T^{2} - 12T + 28$$
$79$ $$T^{2} + 18T + 63$$
$83$ $$T^{2} + 8T - 56$$
$89$ $$T^{2} + 4T - 28$$
$97$ $$T^{2} - 6T + 1$$