# Properties

 Label 9075.2.a.v 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 165) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 1) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + q^3 + (-2*b + 1) * q^4 + (b - 1) * q^6 + (-2*b - 2) * q^7 + (b - 3) * q^8 + q^9 $$q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} + (\beta - 1) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9} + ( - 2 \beta + 1) q^{12} + 4 \beta q^{13} - 2 q^{14} + 3 q^{16} + ( - 2 \beta - 4) q^{17} + (\beta - 1) q^{18} + ( - 2 \beta + 4) q^{19} + ( - 2 \beta - 2) q^{21} + 4 q^{23} + (\beta - 3) q^{24} + ( - 4 \beta + 8) q^{26} + q^{27} + (2 \beta + 6) q^{28} + ( - 2 \beta + 2) q^{29} + (\beta + 3) q^{32} - 2 \beta q^{34} + ( - 2 \beta + 1) q^{36} + (4 \beta - 6) q^{37} + (6 \beta - 8) q^{38} + 4 \beta q^{39} + (2 \beta - 2) q^{41} - 2 q^{42} + (2 \beta - 6) q^{43} + (4 \beta - 4) q^{46} + 4 q^{47} + 3 q^{48} + (8 \beta + 5) q^{49} + ( - 2 \beta - 4) q^{51} + (4 \beta - 16) q^{52} + (8 \beta + 2) q^{53} + (\beta - 1) q^{54} + (4 \beta + 2) q^{56} + ( - 2 \beta + 4) q^{57} + (4 \beta - 6) q^{58} - 4 q^{59} + ( - 4 \beta + 6) q^{61} + ( - 2 \beta - 2) q^{63} + (2 \beta - 7) q^{64} - 4 \beta q^{67} + (6 \beta + 4) q^{68} + 4 q^{69} + (4 \beta + 8) q^{71} + (\beta - 3) q^{72} - 8 \beta q^{73} + ( - 10 \beta + 14) q^{74} + ( - 10 \beta + 12) q^{76} + ( - 4 \beta + 8) q^{78} + 6 \beta q^{79} + q^{81} + ( - 4 \beta + 6) q^{82} - 10 q^{83} + (2 \beta + 6) q^{84} + ( - 8 \beta + 10) q^{86} + ( - 2 \beta + 2) q^{87} + ( - 4 \beta - 2) q^{89} + ( - 8 \beta - 16) q^{91} + ( - 8 \beta + 4) q^{92} + (4 \beta - 4) q^{94} + (\beta + 3) q^{96} + (4 \beta - 6) q^{97} + ( - 3 \beta + 11) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + q^3 + (-2*b + 1) * q^4 + (b - 1) * q^6 + (-2*b - 2) * q^7 + (b - 3) * q^8 + q^9 + (-2*b + 1) * q^12 + 4*b * q^13 - 2 * q^14 + 3 * q^16 + (-2*b - 4) * q^17 + (b - 1) * q^18 + (-2*b + 4) * q^19 + (-2*b - 2) * q^21 + 4 * q^23 + (b - 3) * q^24 + (-4*b + 8) * q^26 + q^27 + (2*b + 6) * q^28 + (-2*b + 2) * q^29 + (b + 3) * q^32 - 2*b * q^34 + (-2*b + 1) * q^36 + (4*b - 6) * q^37 + (6*b - 8) * q^38 + 4*b * q^39 + (2*b - 2) * q^41 - 2 * q^42 + (2*b - 6) * q^43 + (4*b - 4) * q^46 + 4 * q^47 + 3 * q^48 + (8*b + 5) * q^49 + (-2*b - 4) * q^51 + (4*b - 16) * q^52 + (8*b + 2) * q^53 + (b - 1) * q^54 + (4*b + 2) * q^56 + (-2*b + 4) * q^57 + (4*b - 6) * q^58 - 4 * q^59 + (-4*b + 6) * q^61 + (-2*b - 2) * q^63 + (2*b - 7) * q^64 - 4*b * q^67 + (6*b + 4) * q^68 + 4 * q^69 + (4*b + 8) * q^71 + (b - 3) * q^72 - 8*b * q^73 + (-10*b + 14) * q^74 + (-10*b + 12) * q^76 + (-4*b + 8) * q^78 + 6*b * q^79 + q^81 + (-4*b + 6) * q^82 - 10 * q^83 + (2*b + 6) * q^84 + (-8*b + 10) * q^86 + (-2*b + 2) * q^87 + (-4*b - 2) * q^89 + (-8*b - 16) * q^91 + (-8*b + 4) * q^92 + (4*b - 4) * q^94 + (b + 3) * q^96 + (4*b - 6) * q^97 + (-3*b + 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} + 8 q^{19} - 4 q^{21} + 8 q^{23} - 6 q^{24} + 16 q^{26} + 2 q^{27} + 12 q^{28} + 4 q^{29} + 6 q^{32} + 2 q^{36} - 12 q^{37} - 16 q^{38} - 4 q^{41} - 4 q^{42} - 12 q^{43} - 8 q^{46} + 8 q^{47} + 6 q^{48} + 10 q^{49} - 8 q^{51} - 32 q^{52} + 4 q^{53} - 2 q^{54} + 4 q^{56} + 8 q^{57} - 12 q^{58} - 8 q^{59} + 12 q^{61} - 4 q^{63} - 14 q^{64} + 8 q^{68} + 8 q^{69} + 16 q^{71} - 6 q^{72} + 28 q^{74} + 24 q^{76} + 16 q^{78} + 2 q^{81} + 12 q^{82} - 20 q^{83} + 12 q^{84} + 20 q^{86} + 4 q^{87} - 4 q^{89} - 32 q^{91} + 8 q^{92} - 8 q^{94} + 6 q^{96} - 12 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^18 + 8 * q^19 - 4 * q^21 + 8 * q^23 - 6 * q^24 + 16 * q^26 + 2 * q^27 + 12 * q^28 + 4 * q^29 + 6 * q^32 + 2 * q^36 - 12 * q^37 - 16 * q^38 - 4 * q^41 - 4 * q^42 - 12 * q^43 - 8 * q^46 + 8 * q^47 + 6 * q^48 + 10 * q^49 - 8 * q^51 - 32 * q^52 + 4 * q^53 - 2 * q^54 + 4 * q^56 + 8 * q^57 - 12 * q^58 - 8 * q^59 + 12 * q^61 - 4 * q^63 - 14 * q^64 + 8 * q^68 + 8 * q^69 + 16 * q^71 - 6 * q^72 + 28 * q^74 + 24 * q^76 + 16 * q^78 + 2 * q^81 + 12 * q^82 - 20 * q^83 + 12 * q^84 + 20 * q^86 + 4 * q^87 - 4 * q^89 - 32 * q^91 + 8 * q^92 - 8 * q^94 + 6 * q^96 - 12 * q^97 + 22 * 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 0.828427 −4.41421 1.00000 0
1.2 0.414214 1.00000 −1.82843 0 0.414214 −4.82843 −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.v 2
5.b even 2 1 1815.2.a.k 2
11.b odd 2 1 825.2.a.g 2
15.d odd 2 1 5445.2.a.m 2
33.d even 2 1 2475.2.a.m 2
55.d odd 2 1 165.2.a.a 2
55.e even 4 2 825.2.c.e 4
165.d even 2 1 495.2.a.d 2
165.l odd 4 2 2475.2.c.m 4
220.g even 2 1 2640.2.a.bb 2
385.h even 2 1 8085.2.a.ba 2
660.g odd 2 1 7920.2.a.cg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 55.d odd 2 1
495.2.a.d 2 165.d even 2 1
825.2.a.g 2 11.b odd 2 1
825.2.c.e 4 55.e even 4 2
1815.2.a.k 2 5.b even 2 1
2475.2.a.m 2 33.d even 2 1
2475.2.c.m 4 165.l odd 4 2
2640.2.a.bb 2 220.g even 2 1
5445.2.a.m 2 15.d odd 2 1
7920.2.a.cg 2 660.g odd 2 1
8085.2.a.ba 2 385.h even 2 1
9075.2.a.v 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{7}^{2} + 4T_{7} - 4$$ T7^2 + 4*T7 - 4 $$T_{13}^{2} - 32$$ T13^2 - 32 $$T_{17}^{2} + 8T_{17} + 8$$ T17^2 + 8*T17 + 8 $$T_{19}^{2} - 8T_{19} + 8$$ T19^2 - 8*T19 + 8 $$T_{23} - 4$$ T23 - 4 $$T_{37}^{2} + 12T_{37} + 4$$ T37^2 + 12*T37 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} + 8T + 8$$
$19$ $$T^{2} - 8T + 8$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 4T - 4$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 12T + 4$$
$41$ $$T^{2} + 4T - 4$$
$43$ $$T^{2} + 12T + 28$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} - 4T - 124$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} - 12T + 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} - 16T + 32$$
$73$ $$T^{2} - 128$$
$79$ $$T^{2} - 72$$
$83$ $$(T + 10)^{2}$$
$89$ $$T^{2} + 4T - 28$$
$97$ $$T^{2} + 12T + 4$$