Properties

 Label 3025.2.a.v Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

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: $$4$$ Coefficient field: 4.4.2525.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 5$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 5 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{2} + \beta_1 q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{3} - \beta_{2} - 3) q^{7} + ( - \beta_{3} - \beta_1 - 2) q^{8} + (\beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + (-b3 - 1) * q^2 + b1 * q^3 + (b3 - b2 + b1 + 1) * q^4 + (-b3 - 2*b2 - b1 - 1) * q^6 + (b3 - b2 - 3) * q^7 + (-b3 - b1 - 2) * q^8 + (b2 + b1) * q^9 $$q + ( - \beta_{3} - 1) q^{2} + \beta_1 q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{3} - \beta_{2} - 3) q^{7} + ( - \beta_{3} - \beta_1 - 2) q^{8} + (\beta_{2} + \beta_1) q^{9} + (3 \beta_{2} + 2 \beta_1 + 4) q^{12} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 1) q^{14} + (\beta_{3} + 3 \beta_{2} + 3) q^{16} + (\beta_{3} + \beta_1 - 1) q^{17} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{18} + (\beta_{2} - \beta_1 + 4) q^{19} + (2 \beta_{2} - 3 \beta_1 + 1) q^{21} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{23} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 4) q^{24} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{26} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{27} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{28} + (3 \beta_{2} - \beta_1) q^{29} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{31} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{32} + ( - \beta_{2} - 2 \beta_1 - 2) q^{34} + (5 \beta_{2} + 3 \beta_1 + 3) q^{36} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{37} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{38} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{39} + ( - 3 \beta_{3} + \beta_1 - 3) q^{41} + (4 \beta_{3} + 4 \beta_{2} + \beta_1 + 2) q^{42} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{43} + ( - 2 \beta_{3} + 8 \beta_{2} + 7 \beta_1 + 3) q^{46} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{47} + (4 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{48} + ( - 5 \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{49} + (\beta_{3} + 3 \beta_{2} + 4) q^{51} + ( - 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 6) q^{52} + ( - \beta_{3} + \beta_1 + 2) q^{53} + (4 \beta_{2} - 3) q^{54} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{56} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{57} + (4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{58} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{59} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_{3} + 7 \beta_{2} + 3 \beta_1 - 1) q^{62} + ( - \beta_{3} - 2 \beta_1) q^{63} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 7) q^{64} + (3 \beta_{3} + 2 \beta_{2} + 2) q^{67} + (\beta_{3} + 5 \beta_{2} + \beta_1 + 6) q^{68} + ( - 4 \beta_{3} - 5 \beta_{2} - \beta_1 - 10) q^{69} + (3 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 6) q^{71} + ( - \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 4) q^{72} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 4) q^{73} + ( - 2 \beta_{3} + 7 \beta_{2} - 6) q^{74} + (4 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 1) q^{76} + (5 \beta_{2} + 2 \beta_1) q^{78} + (2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 2) q^{79} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{81} + (2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 8) q^{82} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 5) q^{83} + (\beta_{3} - 6 \beta_{2} - 3 \beta_1 - 13) q^{84} + (7 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{86} + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{87} + ( - 4 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 3) q^{89} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{91} + ( - 18 \beta_{2} - 7 \beta_1 - 10) q^{92} + ( - \beta_{3} - \beta_1 - 5) q^{93} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 - 4) q^{94} + (2 \beta_{2} - 3 \beta_1 - 4) q^{96} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - 7 \beta_{2} + 2 \beta_1 + 6) q^{98}+O(q^{100})$$ q + (-b3 - 1) * q^2 + b1 * q^3 + (b3 - b2 + b1 + 1) * q^4 + (-b3 - 2*b2 - b1 - 1) * q^6 + (b3 - b2 - 3) * q^7 + (-b3 - b1 - 2) * q^8 + (b2 + b1) * q^9 + (3*b2 + 2*b1 + 4) * q^12 + (-b3 + 2*b2 - 1) * q^13 + (2*b3 + 2*b2 + 1) * q^14 + (b3 + 3*b2 + 3) * q^16 + (b3 + b1 - 1) * q^17 + (-3*b2 - 2*b1 - 1) * q^18 + (b2 - b1 + 4) * q^19 + (2*b2 - 3*b1 + 1) * q^21 + (-b3 - 3*b2 - 3*b1 + 2) * q^23 + (-b3 - 3*b2 - 3*b1 - 4) * q^24 + (3*b3 - 3*b2 - b1 + 3) * q^26 + (b3 + b2 - 2*b1 + 3) * q^27 + (-b3 + 2*b2 - 4*b1 + 1) * q^28 + (3*b2 - b1) * q^29 + (b3 - 2*b2 - 2*b1 + 1) * q^31 + (2*b3 - 2*b2 - 2*b1 - 1) * q^32 + (-b2 - 2*b1 - 2) * q^34 + (5*b2 + 3*b1 + 3) * q^36 + (3*b3 - 2*b2 - b1 + 1) * q^37 + (-2*b3 + b2 - 3) * q^38 + (b3 - 2*b2 - b1 - 1) * q^39 + (-3*b3 + b1 - 3) * q^41 + (4*b3 + 4*b2 + b1 + 2) * q^42 + (b3 + 2*b2 - 2*b1 - 3) * q^43 + (-2*b3 + 8*b2 + 7*b1 + 3) * q^46 + (b3 - b2 + 3*b1 - 1) * q^47 + (4*b3 + 2*b2 + 3*b1 + 1) * q^48 + (-5*b3 + 4*b2 - b1 + 5) * q^49 + (b3 + 3*b2 + 4) * q^51 + (-3*b3 + 4*b2 + b1 - 6) * q^52 + (-b3 + b1 + 2) * q^53 + (4*b2 - 3) * q^54 + (b3 + b2 + 3*b1 + 3) * q^56 + (b3 - b2 + 3*b1 - 3) * q^57 + (4*b3 - b2 - 2*b1 + 1) * q^58 + (-3*b3 - 3*b2 + 2*b1 - 5) * q^59 + (2*b3 - b2 - 2*b1 + 2) * q^61 + (-b3 + 7*b2 + 3*b1 - 1) * q^62 + (-b3 - 2*b1) * q^63 + (-b3 + 2*b2 + 2*b1 - 7) * q^64 + (3*b3 + 2*b2 + 2) * q^67 + (b3 + 5*b2 + b1 + 6) * q^68 + (-4*b3 - 5*b2 - b1 - 10) * q^69 + (3*b3 - 8*b2 - 2*b1 - 6) * q^71 + (-b3 - 5*b2 - 4*b1 - 4) * q^72 + (-b3 + 6*b2 + 2*b1 + 4) * q^73 + (-2*b3 + 7*b2 - 6) * q^74 + (4*b3 - 5*b2 + 3*b1 - 1) * q^76 + (5*b2 + 2*b1) * q^78 + (2*b3 - 5*b2 + 3*b1 - 2) * q^79 + (2*b3 - 3*b2 - 2*b1 - 5) * q^81 + (2*b3 - 5*b2 + 2*b1 + 8) * q^82 + (-b3 - 3*b2 - b1 - 5) * q^83 + (b3 - 6*b2 - 3*b1 - 13) * q^84 + (7*b3 + 3*b2 - b1 + 3) * q^86 + (3*b3 - b2 - b1 - 3) * q^87 + (-4*b3 + 6*b2 - 2*b1 + 3) * q^89 + (-2*b2 + 2*b1 - 1) * q^91 + (-18*b2 - 7*b1 - 10) * q^92 + (-b3 - b1 - 5) * q^93 + (-3*b3 - 4*b2 - 3*b1 - 4) * q^94 + (2*b2 - 3*b1 - 4) * q^96 + (2*b3 + 2*b2 - 2*b1 + 1) * q^97 + (-7*b2 + 2*b1 + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 2 q^{3} + 7 q^{4} - q^{6} - 11 q^{7} - 9 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 + 2 * q^3 + 7 * q^4 - q^6 - 11 * q^7 - 9 * q^8 $$4 q - 3 q^{2} + 2 q^{3} + 7 q^{4} - q^{6} - 11 q^{7} - 9 q^{8} + 14 q^{12} - 7 q^{13} - 2 q^{14} + 5 q^{16} - 3 q^{17} - 2 q^{18} + 12 q^{19} - 6 q^{21} + 9 q^{23} - 15 q^{24} + 13 q^{26} + 5 q^{27} - 7 q^{28} - 8 q^{29} + 3 q^{31} - 6 q^{32} - 10 q^{34} + 8 q^{36} + 3 q^{37} - 12 q^{38} - 3 q^{39} - 7 q^{41} - 2 q^{42} - 21 q^{43} + 12 q^{46} + 3 q^{47} + 2 q^{48} + 15 q^{49} + 9 q^{51} - 27 q^{52} + 11 q^{53} - 20 q^{54} + 15 q^{56} - 5 q^{57} - 2 q^{58} - 7 q^{59} + 4 q^{61} - 11 q^{62} - 3 q^{63} - 27 q^{64} + q^{67} + 15 q^{68} - 28 q^{69} - 15 q^{71} - 13 q^{72} + 9 q^{73} - 36 q^{74} + 8 q^{76} - 6 q^{78} + 6 q^{79} - 20 q^{81} + 44 q^{82} - 15 q^{83} - 47 q^{84} - 3 q^{86} - 15 q^{87} + 4 q^{91} - 18 q^{92} - 21 q^{93} - 11 q^{94} - 26 q^{96} - 6 q^{97} + 42 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 + 2 * q^3 + 7 * q^4 - q^6 - 11 * q^7 - 9 * q^8 + 14 * q^12 - 7 * q^13 - 2 * q^14 + 5 * q^16 - 3 * q^17 - 2 * q^18 + 12 * q^19 - 6 * q^21 + 9 * q^23 - 15 * q^24 + 13 * q^26 + 5 * q^27 - 7 * q^28 - 8 * q^29 + 3 * q^31 - 6 * q^32 - 10 * q^34 + 8 * q^36 + 3 * q^37 - 12 * q^38 - 3 * q^39 - 7 * q^41 - 2 * q^42 - 21 * q^43 + 12 * q^46 + 3 * q^47 + 2 * q^48 + 15 * q^49 + 9 * q^51 - 27 * q^52 + 11 * q^53 - 20 * q^54 + 15 * q^56 - 5 * q^57 - 2 * q^58 - 7 * q^59 + 4 * q^61 - 11 * q^62 - 3 * q^63 - 27 * q^64 + q^67 + 15 * q^68 - 28 * q^69 - 15 * q^71 - 13 * q^72 + 9 * q^73 - 36 * q^74 + 8 * q^76 - 6 * q^78 + 6 * q^79 - 20 * q^81 + 44 * q^82 - 15 * q^83 - 47 * q^84 - 3 * q^86 - 15 * q^87 + 4 * q^91 - 18 * q^92 - 21 * q^93 - 11 * q^94 - 26 * q^96 - 6 * q^97 + 42 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 5x + 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu$$ v^3 - v^2 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 3$$ b3 + b2 + 4*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
 2.46673 −0.777484 −1.46673 1.77748
−2.52452 2.46673 4.37322 0 −6.22732 −2.09351 −5.99126 3.08477 0
1.2 −2.25800 −0.777484 3.09855 0 1.75556 −0.123970 −2.48051 −2.39552 0
1.3 −0.0935099 −1.46673 −1.99126 0 0.137154 −4.52452 0.373222 −0.848698 0
1.4 1.87603 1.77748 1.51949 0 3.33461 −4.25800 −0.901454 0.159450 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
$$5$$ $$1$$
$$11$$ $$1$$

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 3025.2.a.v 4
5.b even 2 1 605.2.a.l 4
11.b odd 2 1 3025.2.a.be 4
11.c even 5 2 275.2.h.b 8
15.d odd 2 1 5445.2.a.bg 4
20.d odd 2 1 9680.2.a.cs 4
55.d odd 2 1 605.2.a.i 4
55.h odd 10 2 605.2.g.g 8
55.h odd 10 2 605.2.g.n 8
55.j even 10 2 55.2.g.a 8
55.j even 10 2 605.2.g.j 8
55.k odd 20 4 275.2.z.b 16
165.d even 2 1 5445.2.a.bu 4
165.o odd 10 2 495.2.n.f 8
220.g even 2 1 9680.2.a.cv 4
220.n odd 10 2 880.2.bo.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a 8 55.j even 10 2
275.2.h.b 8 11.c even 5 2
275.2.z.b 16 55.k odd 20 4
495.2.n.f 8 165.o odd 10 2
605.2.a.i 4 55.d odd 2 1
605.2.a.l 4 5.b even 2 1
605.2.g.g 8 55.h odd 10 2
605.2.g.j 8 55.j even 10 2
605.2.g.n 8 55.h odd 10 2
880.2.bo.e 8 220.n odd 10 2
3025.2.a.v 4 1.a even 1 1 trivial
3025.2.a.be 4 11.b odd 2 1
5445.2.a.bg 4 15.d odd 2 1
5445.2.a.bu 4 165.d even 2 1
9680.2.a.cs 4 20.d odd 2 1
9680.2.a.cv 4 220.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{4} + 3T_{2}^{3} - 3T_{2}^{2} - 11T_{2} - 1$$ T2^4 + 3*T2^3 - 3*T2^2 - 11*T2 - 1 $$T_{3}^{4} - 2T_{3}^{3} - 4T_{3}^{2} + 5T_{3} + 5$$ T3^4 - 2*T3^3 - 4*T3^2 + 5*T3 + 5 $$T_{19}^{4} - 12T_{19}^{3} + 46T_{19}^{2} - 65T_{19} + 25$$ T19^4 - 12*T19^3 + 46*T19^2 - 65*T19 + 25

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} - 3 T^{2} - 11 T - 1$$
$3$ $$T^{4} - 2 T^{3} - 4 T^{2} + 5 T + 5$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11 T^{3} + 39 T^{2} + 45 T + 5$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 7 T^{3} + 7 T^{2} - 9 T - 11$$
$17$ $$T^{4} + 3 T^{3} - 8 T^{2} - 26 T - 11$$
$19$ $$T^{4} - 12 T^{3} + 46 T^{2} - 65 T + 25$$
$23$ $$T^{4} - 9 T^{3} - 54 T^{2} + \cdots - 1669$$
$29$ $$T^{4} + 8 T^{3} - 4 T^{2} - 95 T - 55$$
$31$ $$T^{4} - 3 T^{3} - 31 T^{2} + 3 T + 101$$
$37$ $$T^{4} - 3 T^{3} - 56 T^{2} + 28 T + 151$$
$41$ $$T^{4} + 7 T^{3} - 46 T^{2} - 382 T - 499$$
$43$ $$T^{4} + 21 T^{3} + 121 T^{2} + \cdots - 59$$
$47$ $$T^{4} - 3 T^{3} - 51 T^{2} + 133 T + 71$$
$53$ $$T^{4} - 11 T^{3} + 33 T^{2} - 27 T - 1$$
$59$ $$T^{4} + 7 T^{3} - 109 T^{2} + \cdots - 3025$$
$61$ $$T^{4} - 4 T^{3} - 41 T^{2} + 90 T + 55$$
$67$ $$T^{4} - T^{3} - 82 T^{2} + 238 T - 101$$
$71$ $$T^{4} + 15 T^{3} - 98 T^{2} + \cdots - 7799$$
$73$ $$T^{4} - 9 T^{3} - 74 T^{2} + 596 T - 389$$
$79$ $$T^{4} - 6 T^{3} - 96 T^{2} + \cdots - 2155$$
$83$ $$T^{4} + 15 T^{3} + 43 T^{2} - 25 T - 29$$
$89$ $$T^{4} - 150 T^{2} - 400 T + 725$$
$97$ $$T^{4} + 6 T^{3} - 56 T^{2} + 90 T - 25$$