# Properties

 Label 975.2.a.o Level $975$ Weight $2$ Character orbit 975.a Self dual yes Analytic conductor $7.785$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + \beta_1 q^{6} + \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + 3) * q^4 + b1 * q^6 + b2 * q^7 + (3*b1 - 2) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + \beta_1 q^{6} + \beta_{2} q^{7} + (3 \beta_1 - 2) q^{8} + q^{9} - \beta_{2} q^{11} + (\beta_{2} + 3) q^{12} - q^{13} + (2 \beta_1 - 2) q^{14} + (\beta_{2} - 2 \beta_1 + 9) q^{16} + ( - \beta_{2} - 2 \beta_1) q^{17} + \beta_1 q^{18} + (2 \beta_1 + 2) q^{19} + \beta_{2} q^{21} + ( - 2 \beta_1 + 2) q^{22} + ( - \beta_{2} - 2 \beta_1 + 2) q^{23} + (3 \beta_1 - 2) q^{24} - \beta_1 q^{26} + q^{27} + ( - 2 \beta_1 + 10) q^{28} + 6 q^{29} + ( - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{32} - \beta_{2} q^{33} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{34} + (\beta_{2} + 3) q^{36} + (\beta_{2} - 2 \beta_1 - 4) q^{37} + (2 \beta_{2} + 2 \beta_1 + 10) q^{38} - q^{39} + ( - \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_1 - 2) q^{42} + 4 \beta_1 q^{43} + (2 \beta_1 - 10) q^{44} + ( - 2 \beta_{2} - 8) q^{46} + ( - 2 \beta_1 + 6) q^{47} + (\beta_{2} - 2 \beta_1 + 9) q^{48} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} - 2 \beta_1) q^{51} + ( - \beta_{2} - 3) q^{52} + ( - \beta_{2} - 2 \beta_1 - 4) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{56} + (2 \beta_1 + 2) q^{57} + 6 \beta_1 q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (3 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{62} + \beta_{2} q^{63} + (3 \beta_{2} - 8 \beta_1 + 11) q^{64} + ( - 2 \beta_1 + 2) q^{66} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - 8 \beta_1 - 6) q^{68} + ( - \beta_{2} - 2 \beta_1 + 2) q^{69} + ( - \beta_{2} - 4) q^{71} + (3 \beta_1 - 2) q^{72} + (4 \beta_1 + 2) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 12) q^{74} + (2 \beta_{2} + 10 \beta_1 + 2) q^{76} + (3 \beta_{2} + 2 \beta_1 - 10) q^{77} - \beta_1 q^{78} + (\beta_{2} - 2 \beta_1 + 2) q^{79} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{82} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + ( - 2 \beta_1 + 10) q^{84} + (4 \beta_{2} + 20) q^{86} + 6 q^{87} + (2 \beta_{2} - 6 \beta_1 + 6) q^{88} + (\beta_{2} + 2 \beta_1 + 4) q^{89} - \beta_{2} q^{91} + (2 \beta_{2} - 8 \beta_1) q^{92} + ( - 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + 6 \beta_1 - 10) q^{94} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{96} + ( - \beta_{2} - 2 \beta_1 + 8) q^{97} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{98} - \beta_{2} q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + 3) * q^4 + b1 * q^6 + b2 * q^7 + (3*b1 - 2) * q^8 + q^9 - b2 * q^11 + (b2 + 3) * q^12 - q^13 + (2*b1 - 2) * q^14 + (b2 - 2*b1 + 9) * q^16 + (-b2 - 2*b1) * q^17 + b1 * q^18 + (2*b1 + 2) * q^19 + b2 * q^21 + (-2*b1 + 2) * q^22 + (-b2 - 2*b1 + 2) * q^23 + (3*b1 - 2) * q^24 - b1 * q^26 + q^27 + (-2*b1 + 10) * q^28 + 6 * q^29 + (-2*b1 + 2) * q^31 + (-2*b2 + 5*b1 - 8) * q^32 - b2 * q^33 + (-2*b2 - 2*b1 - 8) * q^34 + (b2 + 3) * q^36 + (b2 - 2*b1 - 4) * q^37 + (2*b2 + 2*b1 + 10) * q^38 - q^39 + (-b2 - 2*b1) * q^41 + (2*b1 - 2) * q^42 + 4*b1 * q^43 + (2*b1 - 10) * q^44 + (-2*b2 - 8) * q^46 + (-2*b1 + 6) * q^47 + (b2 - 2*b1 + 9) * q^48 + (-3*b2 - 2*b1 + 3) * q^49 + (-b2 - 2*b1) * q^51 + (-b2 - 3) * q^52 + (-b2 - 2*b1 - 4) * q^53 + b1 * q^54 + (-2*b2 + 6*b1 - 6) * q^56 + (2*b1 + 2) * q^57 + 6*b1 * q^58 + (2*b2 + 2*b1 - 2) * q^59 + (3*b2 + 2*b1 + 4) * q^61 + (-2*b2 + 2*b1 - 10) * q^62 + b2 * q^63 + (3*b2 - 8*b1 + 11) * q^64 + (-2*b1 + 2) * q^66 + (-2*b2 - 2*b1 - 2) * q^67 + (-8*b1 - 6) * q^68 + (-b2 - 2*b1 + 2) * q^69 + (-b2 - 4) * q^71 + (3*b1 - 2) * q^72 + (4*b1 + 2) * q^73 + (-2*b2 - 2*b1 - 12) * q^74 + (2*b2 + 10*b1 + 2) * q^76 + (3*b2 + 2*b1 - 10) * q^77 - b1 * q^78 + (b2 - 2*b1 + 2) * q^79 + q^81 + (-2*b2 - 2*b1 - 8) * q^82 + (2*b2 + 2*b1 - 2) * q^83 + (-2*b1 + 10) * q^84 + (4*b2 + 20) * q^86 + 6 * q^87 + (2*b2 - 6*b1 + 6) * q^88 + (b2 + 2*b1 + 4) * q^89 - b2 * q^91 + (2*b2 - 8*b1) * q^92 + (-2*b1 + 2) * q^93 + (-2*b2 + 6*b1 - 10) * q^94 + (-2*b2 + 5*b1 - 8) * q^96 + (-b2 - 2*b1 + 8) * q^97 + (-2*b2 - 3*b1 - 4) * q^98 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 8 * q^4 - q^7 - 6 * q^8 + 3 * q^9 $$3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9} + q^{11} + 8 q^{12} - 3 q^{13} - 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} - q^{21} + 6 q^{22} + 7 q^{23} - 6 q^{24} + 3 q^{27} + 30 q^{28} + 18 q^{29} + 6 q^{31} - 22 q^{32} + q^{33} - 22 q^{34} + 8 q^{36} - 13 q^{37} + 28 q^{38} - 3 q^{39} + q^{41} - 6 q^{42} - 30 q^{44} - 22 q^{46} + 18 q^{47} + 26 q^{48} + 12 q^{49} + q^{51} - 8 q^{52} - 11 q^{53} - 16 q^{56} + 6 q^{57} - 8 q^{59} + 9 q^{61} - 28 q^{62} - q^{63} + 30 q^{64} + 6 q^{66} - 4 q^{67} - 18 q^{68} + 7 q^{69} - 11 q^{71} - 6 q^{72} + 6 q^{73} - 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 3 q^{81} - 22 q^{82} - 8 q^{83} + 30 q^{84} + 56 q^{86} + 18 q^{87} + 16 q^{88} + 11 q^{89} + q^{91} - 2 q^{92} + 6 q^{93} - 28 q^{94} - 22 q^{96} + 25 q^{97} - 10 q^{98} + q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 8 * q^4 - q^7 - 6 * q^8 + 3 * q^9 + q^11 + 8 * q^12 - 3 * q^13 - 6 * q^14 + 26 * q^16 + q^17 + 6 * q^19 - q^21 + 6 * q^22 + 7 * q^23 - 6 * q^24 + 3 * q^27 + 30 * q^28 + 18 * q^29 + 6 * q^31 - 22 * q^32 + q^33 - 22 * q^34 + 8 * q^36 - 13 * q^37 + 28 * q^38 - 3 * q^39 + q^41 - 6 * q^42 - 30 * q^44 - 22 * q^46 + 18 * q^47 + 26 * q^48 + 12 * q^49 + q^51 - 8 * q^52 - 11 * q^53 - 16 * q^56 + 6 * q^57 - 8 * q^59 + 9 * q^61 - 28 * q^62 - q^63 + 30 * q^64 + 6 * q^66 - 4 * q^67 - 18 * q^68 + 7 * q^69 - 11 * q^71 - 6 * q^72 + 6 * q^73 - 34 * q^74 + 4 * q^76 - 33 * q^77 + 5 * q^79 + 3 * q^81 - 22 * q^82 - 8 * q^83 + 30 * q^84 + 56 * q^86 + 18 * q^87 + 16 * q^88 + 11 * q^89 + q^91 - 2 * q^92 + 6 * q^93 - 28 * q^94 - 22 * q^96 + 25 * q^97 - 10 * q^98 + q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 3$$ b1 + 3

## 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
 0.470683 −1.81361 2.34292
−2.77846 1.00000 5.71982 0 −2.77846 2.71982 −10.3354 1.00000 0
1.2 0.289169 1.00000 −1.91638 0 0.289169 −4.91638 −1.13249 1.00000 0
1.3 2.48929 1.00000 4.19656 0 2.48929 1.19656 5.46787 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 975.2.a.o 3
3.b odd 2 1 2925.2.a.bh 3
5.b even 2 1 195.2.a.e 3
5.c odd 4 2 975.2.c.i 6
15.d odd 2 1 585.2.a.n 3
15.e even 4 2 2925.2.c.w 6
20.d odd 2 1 3120.2.a.bj 3
35.c odd 2 1 9555.2.a.bq 3
60.h even 2 1 9360.2.a.dd 3
65.d even 2 1 2535.2.a.bc 3
195.e odd 2 1 7605.2.a.bx 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 5.b even 2 1
585.2.a.n 3 15.d odd 2 1
975.2.a.o 3 1.a even 1 1 trivial
975.2.c.i 6 5.c odd 4 2
2535.2.a.bc 3 65.d even 2 1
2925.2.a.bh 3 3.b odd 2 1
2925.2.c.w 6 15.e even 4 2
3120.2.a.bj 3 20.d odd 2 1
7605.2.a.bx 3 195.e odd 2 1
9360.2.a.dd 3 60.h even 2 1
9555.2.a.bq 3 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(975))$$:

 $$T_{2}^{3} - 7T_{2} + 2$$ T2^3 - 7*T2 + 2 $$T_{7}^{3} + T_{7}^{2} - 16T_{7} + 16$$ T7^3 + T7^2 - 16*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 7T + 2$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + T^{2} - 16 T + 16$$
$11$ $$T^{3} - T^{2} - 16 T - 16$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} - T^{2} - 32 T + 76$$
$19$ $$T^{3} - 6 T^{2} - 16 T + 64$$
$23$ $$T^{3} - 7 T^{2} - 16 T + 128$$
$29$ $$(T - 6)^{3}$$
$31$ $$T^{3} - 6 T^{2} - 16 T + 32$$
$37$ $$T^{3} + 13T^{2} - 316$$
$41$ $$T^{3} - T^{2} - 32 T + 76$$
$43$ $$T^{3} - 112T + 128$$
$47$ $$T^{3} - 18 T^{2} + 80 T - 64$$
$53$ $$T^{3} + 11 T^{2} + 8 T - 4$$
$59$ $$T^{3} + 8 T^{2} - 48 T - 128$$
$61$ $$T^{3} - 9 T^{2} - 112 T + 844$$
$67$ $$T^{3} + 4 T^{2} - 64 T - 128$$
$71$ $$T^{3} + 11 T^{2} + 24 T - 32$$
$73$ $$T^{3} - 6 T^{2} - 100 T + 344$$
$79$ $$T^{3} - 5 T^{2} - 48 T - 64$$
$83$ $$T^{3} + 8 T^{2} - 48 T - 128$$
$89$ $$T^{3} - 11 T^{2} + 8 T + 4$$
$97$ $$T^{3} - 25 T^{2} + 176 T - 244$$