# Properties

 Label 3025.2.a.bk Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $8$ 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: $$8$$ Coefficient field: 8.8.1480160000.1 Defining polynomial: $$x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11$$ x^8 - 9*x^6 + 27*x^4 - 31*x^2 + 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{6} - \beta_{4} q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b7 - b6 - b1) * q^3 + b2 * q^4 + (-b5 - 2*b3 - b2 - 2) * q^6 - b4 * q^7 + (b6 + b4 - b1) * q^8 + (-b5 + b3 + 2*b2) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{6} - \beta_{4} q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{9} + ( - 2 \beta_{6} - \beta_{4} - \beta_1) q^{12} + (2 \beta_{6} + \beta_1) q^{13} + (2 \beta_{5} - 2 \beta_{2} + 1) q^{14} + (\beta_{3} - \beta_{2} - 2) q^{16} + (3 \beta_{7} + \beta_{6} + 2 \beta_1) q^{17} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{18} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{5} + \beta_{3} + 1) q^{21} + ( - \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{23} + (2 \beta_{3} - \beta_{2} + 1) q^{24} + (4 \beta_{5} + 2 \beta_{3} + \beta_{2} + 4) q^{26} + (2 \beta_{7} - \beta_{4}) q^{27} - \beta_1 q^{28} + ( - \beta_{5} - \beta_{3} - 4) q^{29} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 2) q^{31} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{4} - \beta_1) q^{32} + ( - \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 2) q^{34} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 3) q^{36} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{37} + (\beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{38} + (\beta_{5} - 5 \beta_{2} - 2) q^{39} + (5 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - 2) q^{41} + (\beta_{7} + 3 \beta_{6} + \beta_1) q^{42} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_1) q^{43} + ( - 6 \beta_{5} - \beta_{3} + 2 \beta_{2} - 7) q^{46} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} + \beta_1) q^{47} + (2 \beta_{7} + 5 \beta_{6} + \beta_{4} + 2 \beta_1) q^{48} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 4) q^{49} + (2 \beta_{5} - 3 \beta_{3} - \beta_{2} - 7) q^{51} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} + 3 \beta_1) q^{52} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - 3 \beta_1) q^{53} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{54} + ( - 4 \beta_{5} + 3 \beta_{2} - 4) q^{56} + (\beta_{7} + 5 \beta_{6} + \beta_{4} + \beta_1) q^{57} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_1) q^{58} + (2 \beta_{5} - 3 \beta_{3} - 5 \beta_{2}) q^{59} + ( - 5 \beta_{5} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{61} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_1) q^{62} + (\beta_{7} + \beta_{4} - 2 \beta_1) q^{63} + (\beta_{5} - 3 \beta_{3} - 5 \beta_{2} + 2) q^{64} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{4} - 4 \beta_1) q^{67} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{68} + ( - 3 \beta_{5} + \beta_{3} + 4 \beta_{2} + 2) q^{69} + ( - 3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 9) q^{71} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{72} + (\beta_{7} - 5 \beta_{4} + 2 \beta_1) q^{73} + (3 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 2) q^{74} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - 3) q^{76} + ( - 4 \beta_{6} - 5 \beta_{4} - 7 \beta_1) q^{78} + ( - 5 \beta_{5} - \beta_{3} + 4 \beta_{2} - 6) q^{79} + (7 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 1) q^{81} + ( - 2 \beta_{7} - 3 \beta_{4} - 5 \beta_1) q^{82} + (\beta_{7} - 3 \beta_{6} - \beta_{4} - \beta_1) q^{83} + (\beta_{5} + 2 \beta_{3} + \beta_{2} + 2) q^{84} + ( - 6 \beta_{5} + 3 \beta_{2} + 2) q^{86} + (3 \beta_{7} + 3 \beta_{6} + \beta_{4} + 5 \beta_1) q^{87} + ( - 2 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 5) q^{89} + ( - 4 \beta_{3} - 2 \beta_{2} + 1) q^{91} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{4} - \beta_1) q^{92} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{93} + ( - 7 \beta_{5} - 2 \beta_{3} + 7 \beta_{2} - 1) q^{94} + (6 \beta_{5} + 3 \beta_{3} + 6 \beta_{2} + 4) q^{96} + (3 \beta_{7} + \beta_{4} - \beta_1) q^{97} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b7 - b6 - b1) * q^3 + b2 * q^4 + (-b5 - 2*b3 - b2 - 2) * q^6 - b4 * q^7 + (b6 + b4 - b1) * q^8 + (-b5 + b3 + 2*b2) * q^9 + (-2*b6 - b4 - b1) * q^12 + (2*b6 + b1) * q^13 + (2*b5 - 2*b2 + 1) * q^14 + (b3 - b2 - 2) * q^16 + (3*b7 + b6 + 2*b1) * q^17 + (b7 + 2*b6 + 2*b4 + 2*b1) * q^18 + (b5 + b3 - 2*b2) * q^19 + (2*b5 + b3 + 1) * q^21 + (-b6 + 2*b4 - 2*b1) * q^23 + (2*b3 - b2 + 1) * q^24 + (4*b5 + 2*b3 + b2 + 4) * q^26 + (2*b7 - b4) * q^27 - b1 * q^28 + (-b5 - b3 - 4) * q^29 + (-b5 + b3 - b2 + 2) * q^31 + (b7 - 2*b6 - 3*b4 - b1) * q^32 + (-b5 + 4*b3 + 2*b2 + 2) * q^34 + (b5 + b3 + 2*b2 + 3) * q^36 + (-b7 + 3*b6 + 2*b4) * q^37 + (b7 - 2*b4 - 2*b1) * q^38 + (b5 - 5*b2 - 2) * q^39 + (5*b5 - 2*b3 - 3*b2 - 2) * q^41 + (b7 + 3*b6 + b1) * q^42 + (2*b7 - 2*b6 + 3*b1) * q^43 + (-6*b5 - b3 + 2*b2 - 7) * q^46 + (-b7 - b6 + 3*b4 + b1) * q^47 + (2*b7 + 5*b6 + b4 + 2*b1) * q^48 + (b5 - 2*b3 - b2 - 4) * q^49 + (2*b5 - 3*b3 - b2 - 7) * q^51 + (2*b7 + 3*b6 + b4 + 3*b1) * q^52 + (2*b7 + 3*b6 + b4 - 3*b1) * q^53 + (2*b3 - 2*b2 - 1) * q^54 + (-4*b5 + 3*b2 - 4) * q^56 + (b7 + 5*b6 + b4 + b1) * q^57 + (-b7 - 2*b6 - 4*b1) * q^58 + (2*b5 - 3*b3 - 5*b2) * q^59 + (-5*b5 + 4*b3 + 4*b2 - 4) * q^61 + (b7 - b6 - b4 + b1) * q^62 + (b7 + b4 - 2*b1) * q^63 + (b5 - 3*b3 - 5*b2 + 2) * q^64 + (-3*b7 - 2*b6 - b4 - 4*b1) * q^67 + (-2*b7 + 3*b6 + 2*b4) * q^68 + (-3*b5 + b3 + 4*b2 + 2) * q^69 + (-3*b5 - 3*b3 + 3*b2 - 9) * q^71 + (-b7 - 2*b4 + b1) * q^72 + (b7 - 5*b4 + 2*b1) * q^73 + (3*b5 + 2*b3 + 4*b2 + 2) * q^74 + (b5 - b3 - 2*b2 - 3) * q^76 + (-4*b6 - 5*b4 - 7*b1) * q^78 + (-5*b5 - b3 + 4*b2 - 6) * q^79 + (7*b5 - 2*b3 - 4*b2 - 1) * q^81 + (-2*b7 - 3*b4 - 5*b1) * q^82 + (b7 - 3*b6 - b4 - b1) * q^83 + (b5 + 2*b3 + b2 + 2) * q^84 + (-6*b5 + 3*b2 + 2) * q^86 + (3*b7 + 3*b6 + b4 + 5*b1) * q^87 + (-2*b5 - 6*b3 - 2*b2 - 5) * q^89 + (-4*b3 - 2*b2 + 1) * q^91 + (-b7 - 3*b6 - 2*b4 - b1) * q^92 + (-3*b7 + b6 + 2*b4 - 2*b1) * q^93 + (-7*b5 - 2*b3 + 7*b2 - 1) * q^94 + (6*b5 + 3*b3 + 6*b2 + 4) * q^96 + (3*b7 + b4 - b1) * q^97 + (-2*b7 - 2*b6 - b4 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10})$$ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 $$8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100})$$ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 - 12 * q^19 - 4 * q^21 - 2 * q^24 + 10 * q^26 - 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 - 34 * q^41 - 24 * q^46 - 30 * q^49 - 54 * q^51 - 20 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 + 4 * q^74 - 28 * q^76 - 16 * q^79 - 36 * q^81 + 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 + 42 * q^94 + 8 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 4$$ v^4 - 5*v^2 + 4 $$\beta_{4}$$ $$=$$ $$-\nu^{7} + 7\nu^{5} - 13\nu^{3} + 5\nu$$ -v^7 + 7*v^5 - 13*v^3 + 5*v $$\beta_{5}$$ $$=$$ $$\nu^{6} - 7\nu^{4} + 14\nu^{2} - 8$$ v^6 - 7*v^4 + 14*v^2 - 8 $$\beta_{6}$$ $$=$$ $$\nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu$$ v^7 - 7*v^5 + 14*v^3 - 8*v $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 8\nu^{5} - 19\nu^{3} + 12\nu$$ -v^7 + 8*v^5 - 19*v^3 + 12*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} + 3\beta_1$$ b6 + b4 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5\beta_{2} + 6$$ b3 + 5*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 6\beta_{6} + 5\beta_{4} + 11\beta_1$$ b7 + 6*b6 + 5*b4 + 11*b1 $$\nu^{6}$$ $$=$$ $$\beta_{5} + 7\beta_{3} + 21\beta_{2} + 22$$ b5 + 7*b3 + 21*b2 + 22 $$\nu^{7}$$ $$=$$ $$7\beta_{7} + 29\beta_{6} + 21\beta_{4} + 43\beta_1$$ 7*b7 + 29*b6 + 21*b4 + 43*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
 −2.02368 −1.65458 −1.23399 −0.802699 0.802699 1.23399 1.65458 2.02368
−2.02368 2.62059 2.09529 0 −5.30325 0.965823 −0.192845 3.86752 0
1.2 −1.65458 −1.97479 0.737640 0 3.26745 2.24307 2.08868 0.899788 0
1.3 −1.23399 −0.363982 −0.477260 0 0.449152 −2.58558 3.05692 −2.86752 0
1.4 −0.802699 1.76074 −1.35567 0 −1.41335 −0.592103 2.69360 0.100212 0
1.5 0.802699 −1.76074 −1.35567 0 −1.41335 0.592103 −2.69360 0.100212 0
1.6 1.23399 0.363982 −0.477260 0 0.449152 2.58558 −3.05692 −2.86752 0
1.7 1.65458 1.97479 0.737640 0 3.26745 −2.24307 −2.08868 0.899788 0
1.8 2.02368 −2.62059 2.09529 0 −5.30325 −0.965823 0.192845 3.86752 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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
5.b even 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.bk 8
5.b even 2 1 inner 3025.2.a.bk 8
5.c odd 4 2 605.2.b.f 8
11.b odd 2 1 3025.2.a.bl 8
11.d odd 10 2 275.2.h.d 16
55.d odd 2 1 3025.2.a.bl 8
55.e even 4 2 605.2.b.g 8
55.h odd 10 2 275.2.h.d 16
55.k odd 20 4 605.2.j.d 16
55.k odd 20 4 605.2.j.g 16
55.l even 20 4 55.2.j.a 16
55.l even 20 4 605.2.j.h 16
165.u odd 20 4 495.2.ba.a 16
220.w odd 20 4 880.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 55.l even 20 4
275.2.h.d 16 11.d odd 10 2
275.2.h.d 16 55.h odd 10 2
495.2.ba.a 16 165.u odd 20 4
605.2.b.f 8 5.c odd 4 2
605.2.b.g 8 55.e even 4 2
605.2.j.d 16 55.k odd 20 4
605.2.j.g 16 55.k odd 20 4
605.2.j.h 16 55.l even 20 4
880.2.cd.c 16 220.w odd 20 4
3025.2.a.bk 8 1.a even 1 1 trivial
3025.2.a.bk 8 5.b even 2 1 inner
3025.2.a.bl 8 11.b odd 2 1
3025.2.a.bl 8 55.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{8} - 9T_{2}^{6} + 27T_{2}^{4} - 31T_{2}^{2} + 11$$ T2^8 - 9*T2^6 + 27*T2^4 - 31*T2^2 + 11 $$T_{3}^{8} - 14T_{3}^{6} + 62T_{3}^{4} - 91T_{3}^{2} + 11$$ T3^8 - 14*T3^6 + 62*T3^4 - 91*T3^2 + 11 $$T_{19}^{4} + 6T_{19}^{3} - 4T_{19}^{2} - 39T_{19} + 11$$ T19^4 + 6*T19^3 - 4*T19^2 - 39*T19 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 9 T^{6} + 27 T^{4} - 31 T^{2} + \cdots + 11$$
$3$ $$T^{8} - 14 T^{6} + 62 T^{4} - 91 T^{2} + \cdots + 11$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 13 T^{6} + 49 T^{4} - 47 T^{2} + \cdots + 11$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 45 T^{6} + 675 T^{4} + \cdots + 6875$$
$17$ $$T^{8} - 81 T^{6} + 1842 T^{4} + \cdots + 40931$$
$19$ $$(T^{4} + 6 T^{3} - 4 T^{2} - 39 T + 11)^{2}$$
$23$ $$T^{8} - 99 T^{6} + 2232 T^{4} + \cdots + 26411$$
$29$ $$(T^{4} + 12 T^{3} + 48 T^{2} + 67 T + 11)^{2}$$
$31$ $$(T^{4} - 7 T^{3} + 5 T^{2} + T - 1)^{2}$$
$37$ $$T^{8} - 101 T^{6} + 2822 T^{4} + \cdots + 3971$$
$41$ $$(T^{4} + 17 T^{3} + 48 T^{2} - 438 T - 1969)^{2}$$
$43$ $$T^{8} - 173 T^{6} + 8319 T^{4} + \cdots + 212531$$
$47$ $$T^{8} - 191 T^{6} + 7351 T^{4} + \cdots + 244211$$
$53$ $$T^{8} - 249 T^{6} + 15237 T^{4} + \cdots + 489731$$
$59$ $$(T^{4} + 3 T^{3} - 75 T^{2} - 29 T - 1)^{2}$$
$61$ $$(T^{4} + 10 T^{3} - 61 T^{2} - 10 T + 209)^{2}$$
$67$ $$T^{8} - 149 T^{6} + 5736 T^{4} + \cdots + 18491$$
$71$ $$(T^{4} + 21 T^{3} + 90 T^{2} - 432 T - 2511)^{2}$$
$73$ $$T^{8} - 297 T^{6} + 25504 T^{4} + \cdots + 1279091$$
$79$ $$(T^{4} + 8 T^{3} - 52 T^{2} - 377 T - 319)^{2}$$
$83$ $$T^{8} - 111 T^{6} + 3931 T^{4} + \cdots + 212531$$
$89$ $$(T^{4} + 6 T^{3} - 128 T^{2} - 486 T + 1871)^{2}$$
$97$ $$T^{8} - 162 T^{6} + 5588 T^{4} + \cdots + 161051$$