# Properties

 Label 3025.2.a.bg Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.27433728.1 Defining polynomial: $$x^{6} - 9x^{4} + 15x^{2} - 3$$ x^6 - 9*x^4 + 15*x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 605) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{4} - 2 \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 - 1) * q^3 + (b2 + 1) * q^4 + (b4 - 2*b1) * q^6 - b1 * q^7 + (b5 + b4 + b1) * q^8 + (-2*b3 - b2 + 2) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{4} - 2 \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{5} + \beta_{4} + \beta_1) q^{8} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + ( - \beta_{3} - \beta_{2} - 3) q^{12} + ( - \beta_{5} + \beta_{4} - 2 \beta_1) q^{13} + ( - \beta_{2} - 3) q^{14} + (2 \beta_{3} + 2 \beta_{2} + 3) q^{16} + ( - 2 \beta_{4} + 2 \beta_1) q^{17} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{18} + (\beta_{5} - \beta_{4}) q^{19} + ( - \beta_{4} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{23} + ( - \beta_{5} - 4 \beta_{4}) q^{24} + ( - 3 \beta_{2} - 6) q^{26} + (3 \beta_{3} + 3 \beta_{2} - 5) q^{27} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{28} + ( - 2 \beta_{5} + 2 \beta_1) q^{29} + (2 \beta_{3} - \beta_{2}) q^{31} + (2 \beta_{4} + 3 \beta_1) q^{32} + ( - 2 \beta_{3} + 4) q^{34} + ( - \beta_{2} - 2) q^{36} - 2 \beta_{3} q^{37} + \beta_{2} q^{38} + ( - \beta_{5} + \beta_{4} + 4 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{41} + ( - \beta_{3} + \beta_{2} + 5) q^{42} + (2 \beta_{5} - \beta_1) q^{43} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_1) q^{46} + ( - \beta_{3} - 7) q^{47} + ( - 3 \beta_{3} - 4 \beta_{2} + 1) q^{48} + (\beta_{2} - 4) q^{49} + (2 \beta_{5} + 4 \beta_{4} - 6 \beta_1) q^{51} + ( - \beta_{5} - 5 \beta_{4} - 8 \beta_1) q^{52} + ( - 4 \beta_{3} - 3 \beta_{2} - 4) q^{53} + (3 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{54} + ( - 2 \beta_{3} - 4 \beta_{2} - 5) q^{56} + (\beta_{5} - 3 \beta_{4}) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{58} - \beta_{2} q^{59} - \beta_{5} q^{61} + ( - \beta_{5} + \beta_{4} - 4 \beta_1) q^{62} + (\beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{63} + ( - 2 \beta_{3} + \beta_{2} + 5) q^{64} + ( - \beta_{3} - 5) q^{67} + (2 \beta_{4} + 2 \beta_1) q^{68} + (2 \beta_{3} + 3 \beta_{2} - 4) q^{69} + \beta_{2} q^{71} + (\beta_{5} + 5 \beta_{4} - 8 \beta_1) q^{72} + (4 \beta_{4} - 4 \beta_1) q^{73} + ( - 2 \beta_{4} + 2 \beta_1) q^{74} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_1) q^{76} + (3 \beta_{2} + 12) q^{78} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{79} + ( - 8 \beta_{3} - 3 \beta_{2} + 5) q^{81} + ( - \beta_{3} - 2 \beta_{2} - 7) q^{82} + ( - \beta_{5} - 3 \beta_{4} + 6 \beta_1) q^{83} + (\beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{84} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{86} + (10 \beta_{4} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 5) q^{89} + (3 \beta_{2} + 6) q^{91} + ( - 5 \beta_{2} - 6) q^{92} + ( - \beta_{2} + 10) q^{93} + ( - \beta_{4} - 6 \beta_1) q^{94} + ( - 2 \beta_{5} + \beta_{4} - 4 \beta_1) q^{96} + (4 \beta_{3} + \beta_{2} - 4) q^{97} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 1) * q^3 + (b2 + 1) * q^4 + (b4 - 2*b1) * q^6 - b1 * q^7 + (b5 + b4 + b1) * q^8 + (-2*b3 - b2 + 2) * q^9 + (-b3 - b2 - 3) * q^12 + (-b5 + b4 - 2*b1) * q^13 + (-b2 - 3) * q^14 + (2*b3 + 2*b2 + 3) * q^16 + (-2*b4 + 2*b1) * q^17 + (-b5 - 3*b4 + 2*b1) * q^18 + (b5 - b4) * q^19 + (-b4 + 2*b1) * q^21 + (-2*b3 - b2 - 2) * q^23 + (-b5 - 4*b4) * q^24 + (-3*b2 - 6) * q^26 + (3*b3 + 3*b2 - 5) * q^27 + (-b5 - b4 - 3*b1) * q^28 + (-2*b5 + 2*b1) * q^29 + (2*b3 - b2) * q^31 + (2*b4 + 3*b1) * q^32 + (-2*b3 + 4) * q^34 + (-b2 - 2) * q^36 - 2*b3 * q^37 + b2 * q^38 + (-b5 + b4 + 4*b1) * q^39 + (b5 - 2*b4 - 2*b1) * q^41 + (-b3 + b2 + 5) * q^42 + (2*b5 - b1) * q^43 + (-b5 - 3*b4 - 2*b1) * q^46 + (-b3 - 7) * q^47 + (-3*b3 - 4*b2 + 1) * q^48 + (b2 - 4) * q^49 + (2*b5 + 4*b4 - 6*b1) * q^51 + (-b5 - 5*b4 - 8*b1) * q^52 + (-4*b3 - 3*b2 - 4) * q^53 + (3*b5 + 6*b4 - 2*b1) * q^54 + (-2*b3 - 4*b2 - 5) * q^56 + (b5 - 3*b4) * q^57 + (-2*b3 - 2*b2 + 4) * q^58 - b2 * q^59 - b5 * q^61 + (-b5 + b4 - 4*b1) * q^62 + (b5 + 3*b4 - 2*b1) * q^63 + (-2*b3 + b2 + 5) * q^64 + (-b3 - 5) * q^67 + (2*b4 + 2*b1) * q^68 + (2*b3 + 3*b2 - 4) * q^69 + b2 * q^71 + (b5 + 5*b4 - 8*b1) * q^72 + (4*b4 - 4*b1) * q^73 + (-2*b4 + 2*b1) * q^74 + (-b5 + 3*b4 + 2*b1) * q^76 + (3*b2 + 12) * q^78 + (2*b5 + 2*b4 + 2*b1) * q^79 + (-8*b3 - 3*b2 + 5) * q^81 + (-b3 - 2*b2 - 7) * q^82 + (-b5 - 3*b4 + 6*b1) * q^83 + (b5 + 2*b4 + 4*b1) * q^84 + (2*b3 + 3*b2 - 1) * q^86 + (10*b4 - 6*b1) * q^87 + (-2*b3 + 2*b2 - 5) * q^89 + (3*b2 + 6) * q^91 + (-5*b2 - 6) * q^92 + (-b2 + 10) * q^93 + (-b4 - 6*b1) * q^94 + (-2*b5 + b4 - 4*b1) * q^96 + (4*b3 + b2 - 4) * q^97 + (b5 + b4 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9 $$6 q - 6 q^{3} + 6 q^{4} + 12 q^{9} - 18 q^{12} - 18 q^{14} + 18 q^{16} - 12 q^{23} - 36 q^{26} - 30 q^{27} + 24 q^{34} - 12 q^{36} + 30 q^{42} - 42 q^{47} + 6 q^{48} - 24 q^{49} - 24 q^{53} - 30 q^{56} + 24 q^{58} + 30 q^{64} - 30 q^{67} - 24 q^{69} + 72 q^{78} + 30 q^{81} - 42 q^{82} - 6 q^{86} - 30 q^{89} + 36 q^{91} - 36 q^{92} + 60 q^{93} - 24 q^{97}+O(q^{100})$$ 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9 - 18 * q^12 - 18 * q^14 + 18 * q^16 - 12 * q^23 - 36 * q^26 - 30 * q^27 + 24 * q^34 - 12 * q^36 + 30 * q^42 - 42 * q^47 + 6 * q^48 - 24 * q^49 - 24 * q^53 - 30 * q^56 + 24 * q^58 + 30 * q^64 - 30 * q^67 - 24 * q^69 + 72 * q^78 + 30 * q^81 - 42 * q^82 - 6 * q^86 - 30 * q^89 + 36 * q^91 - 36 * q^92 + 60 * q^93 - 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9x^{4} + 15x^{2} - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 8\nu^{2} + 7 ) / 2$$ (v^4 - 8*v^2 + 7) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 8\nu^{3} + 9\nu ) / 2$$ (v^5 - 8*v^3 + 9*v) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 10\nu^{3} - 19\nu ) / 2$$ (-v^5 + 10*v^3 - 19*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 5\beta_1$$ b5 + b4 + 5*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{3} + 8\beta_{2} + 17$$ 2*b3 + 8*b2 + 17 $$\nu^{5}$$ $$=$$ $$8\beta_{5} + 10\beta_{4} + 31\beta_1$$ 8*b5 + 10*b4 + 31*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
 −2.62383 −1.37268 −0.480901 0.480901 1.37268 2.62383
−2.62383 −1.33988 4.88448 0 3.51561 2.62383 −7.56839 −1.20473 0
1.2 −1.37268 −3.26180 −0.115749 0 4.47741 1.37268 2.90425 7.63935 0
1.3 −0.480901 1.60168 −1.76873 0 −0.770249 0.480901 1.81239 −0.434624 0
1.4 0.480901 1.60168 −1.76873 0 0.770249 −0.480901 −1.81239 −0.434624 0
1.5 1.37268 −3.26180 −0.115749 0 −4.47741 −1.37268 −2.90425 7.63935 0
1.6 2.62383 −1.33988 4.88448 0 −3.51561 −2.62383 7.56839 −1.20473 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.bg 6
5.b even 2 1 605.2.a.m 6
11.b odd 2 1 inner 3025.2.a.bg 6
15.d odd 2 1 5445.2.a.bx 6
20.d odd 2 1 9680.2.a.cw 6
55.d odd 2 1 605.2.a.m 6
55.h odd 10 4 605.2.g.q 24
55.j even 10 4 605.2.g.q 24
165.d even 2 1 5445.2.a.bx 6
220.g even 2 1 9680.2.a.cw 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 5.b even 2 1
605.2.a.m 6 55.d odd 2 1
605.2.g.q 24 55.h odd 10 4
605.2.g.q 24 55.j even 10 4
3025.2.a.bg 6 1.a even 1 1 trivial
3025.2.a.bg 6 11.b odd 2 1 inner
5445.2.a.bx 6 15.d odd 2 1
5445.2.a.bx 6 165.d even 2 1
9680.2.a.cw 6 20.d odd 2 1
9680.2.a.cw 6 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{6} - 9T_{2}^{4} + 15T_{2}^{2} - 3$$ T2^6 - 9*T2^4 + 15*T2^2 - 3 $$T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 7$$ T3^3 + 3*T3^2 - 3*T3 - 7 $$T_{19}^{6} - 36T_{19}^{4} + 96T_{19}^{2} - 48$$ T19^6 - 36*T19^4 + 96*T19^2 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 9 T^{4} + 15 T^{2} - 3$$
$3$ $$(T^{3} + 3 T^{2} - 3 T - 7)^{2}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 9 T^{4} + 15 T^{2} - 3$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 72 T^{4} + 1296 T^{2} + \cdots - 3888$$
$17$ $$T^{6} - 48 T^{4} + 384 T^{2} + \cdots - 768$$
$19$ $$T^{6} - 36 T^{4} + 96 T^{2} - 48$$
$23$ $$(T^{3} + 6 T^{2} - 12 T - 84)^{2}$$
$29$ $$T^{6} - 144 T^{4} + 5184 T^{2} + \cdots - 6912$$
$31$ $$(T^{3} - 48 T - 124)^{2}$$
$37$ $$(T^{3} - 24 T + 16)^{2}$$
$41$ $$T^{6} - 105 T^{4} + 2499 T^{2} + \cdots - 7203$$
$43$ $$T^{6} - 129 T^{4} + 4695 T^{2} + \cdots - 43923$$
$47$ $$(T^{3} + 21 T^{2} + 141 T + 303)^{2}$$
$53$ $$(T^{3} + 12 T^{2} - 84 T - 732)^{2}$$
$59$ $$(T^{3} - 12 T + 12)^{2}$$
$61$ $$T^{6} - 33 T^{4} + 339 T^{2} + \cdots - 1083$$
$67$ $$(T^{3} + 15 T^{2} + 69 T + 97)^{2}$$
$71$ $$(T^{3} - 12 T - 12)^{2}$$
$73$ $$T^{6} - 192 T^{4} + 6144 T^{2} + \cdots - 49152$$
$79$ $$T^{6} - 276 T^{4} + 11184 T^{2} + \cdots - 101568$$
$83$ $$T^{6} - 312 T^{4} + 14640 T^{2} + \cdots - 40368$$
$89$ $$(T^{3} + 15 T^{2} - 21 T - 147)^{2}$$
$97$ $$(T^{3} + 12 T^{2} - 36 T - 76)^{2}$$