# Properties

 Label 3025.2.a.bf 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: $$\Q(\zeta_{36})^+$$ Defining polynomial: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{36} + \zeta_{36}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 4$$ v^4 - 5*v^2 + 4 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 4\nu$$ v^5 - 5*v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 6$$ b4 + 5*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5\beta_{3} + 11\beta_1$$ b5 + 5*b3 + 11*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
 −1.96962 −1.28558 −0.684040 0.684040 1.28558 1.96962
−1.96962 −0.652704 1.87939 0 1.28558 −0.0825054 0.237565 −2.57398 0
1.2 −1.28558 0.532089 −0.347296 0 −0.684040 −4.62327 3.01763 −2.71688 0
1.3 −0.684040 −2.87939 −1.53209 0 1.96962 4.54077 2.41609 5.29086 0
1.4 0.684040 −2.87939 −1.53209 0 −1.96962 −4.54077 −2.41609 5.29086 0
1.5 1.28558 0.532089 −0.347296 0 0.684040 4.62327 −3.01763 −2.71688 0
1.6 1.96962 −0.652704 1.87939 0 −1.28558 0.0825054 −0.237565 −2.57398 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.bf 6
5.b even 2 1 3025.2.a.bh yes 6
11.b odd 2 1 inner 3025.2.a.bf 6
55.d odd 2 1 3025.2.a.bh yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.2.a.bf 6 1.a even 1 1 trivial
3025.2.a.bf 6 11.b odd 2 1 inner
3025.2.a.bh yes 6 5.b even 2 1
3025.2.a.bh yes 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{6} - 6T_{2}^{4} + 9T_{2}^{2} - 3$$ T2^6 - 6*T2^4 + 9*T2^2 - 3 $$T_{3}^{3} + 3T_{3}^{2} - 1$$ T3^3 + 3*T3^2 - 1 $$T_{19}^{6} - 87T_{19}^{4} + 1242T_{19}^{2} - 1083$$ T19^6 - 87*T19^4 + 1242*T19^2 - 1083

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 6 T^{4} + 9 T^{2} - 3$$
$3$ $$(T^{3} + 3 T^{2} - 1)^{2}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 42 T^{4} + 441 T^{2} - 3$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 27 T^{4} + 162 T^{2} + \cdots - 243$$
$17$ $$T^{6} - 42 T^{4} + 441 T^{2} - 3$$
$19$ $$T^{6} - 87 T^{4} + 1242 T^{2} + \cdots - 1083$$
$23$ $$(T^{3} + 12 T^{2} + 27 T - 3)^{2}$$
$29$ $$T^{6} - 135 T^{4} + 4914 T^{2} + \cdots - 36963$$
$31$ $$(T^{3} - 12 T^{2} + 12 T + 152)^{2}$$
$37$ $$(T^{3} - 93 T + 289)^{2}$$
$41$ $$T^{6} - 177 T^{4} + 6939 T^{2} + \cdots - 15987$$
$43$ $$T^{6} - 105 T^{4} + 1899 T^{2} + \cdots - 8427$$
$47$ $$(T^{3} + 12 T^{2} + 45 T + 51)^{2}$$
$53$ $$(T^{3} + 3 T^{2} - 54 T - 219)^{2}$$
$59$ $$(T^{3} - 3 T^{2} - 90 T + 381)^{2}$$
$61$ $$T^{6} - 15 T^{4} + 54 T^{2} - 3$$
$67$ $$(T^{3} + 6 T^{2} - 9 T - 17)^{2}$$
$71$ $$(T^{3} + 12 T^{2} - 45 T - 597)^{2}$$
$73$ $$T^{6} - 267 T^{4} + 17226 T^{2} + \cdots - 220323$$
$79$ $$T^{6} - 330 T^{4} + 23373 T^{2} + \cdots - 282747$$
$83$ $$T^{6} - 411 T^{4} + 44946 T^{2} + \cdots - 604803$$
$89$ $$(T^{3} + 3 T^{2} - 72 T + 51)^{2}$$
$97$ $$(T^{3} + 6 T^{2} - 159 T + 323)^{2}$$