# Properties

 Label 4025.2.a.k Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 805) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + ( - 2 \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 2 \beta_1 - 1) q^{6} + q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + ( - 3 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (-2*b2 + b1 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + 2*b1 - 1) * q^6 + q^7 + (2*b2 - b1 + 2) * q^8 + (-3*b2 + 2*b1) * q^9 $$q + ( - \beta_1 + 1) q^{2} + ( - 2 \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 2 \beta_1 - 1) q^{6} + q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + ( - 3 \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_1 - 2) q^{11} + (2 \beta_{2} + \beta_1 - 2) q^{12} + (3 \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_1 + 1) q^{14} + ( - \beta_{2} + \beta_1) q^{16} + (2 \beta_{2} - 3 \beta_1) q^{17} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{18} + (\beta_{2} - 2 \beta_1 - 1) q^{19} + ( - 2 \beta_{2} + \beta_1 - 1) q^{21} + (\beta_{2} + \beta_1) q^{22} + q^{23} + (\beta_{2} - \beta_1 - 4) q^{24} + ( - \beta_{2} + 2 \beta_1 - 6) q^{26} + ( - 2 \beta_{2} + \beta_1 + 6) q^{27} + (\beta_{2} - 2 \beta_1 + 1) q^{28} + (2 \beta_{2} + 2 \beta_1 - 4) q^{29} + (\beta_{2} - 1) q^{31} + ( - 5 \beta_{2} + 3 \beta_1 - 5) q^{32} + (5 \beta_{2} - \beta_1 + 2) q^{33} + (3 \beta_{2} - 3 \beta_1 + 4) q^{34} + (4 \beta_{2} - \beta_1 - 3) q^{36} - 3 \beta_1 q^{37} + (2 \beta_{2} - \beta_1 + 2) q^{38} + (7 \beta_{2} - 8 \beta_1 - 2) q^{39} + ( - 4 \beta_{2} + 6 \beta_1 - 3) q^{41} + ( - \beta_{2} + 2 \beta_1 - 1) q^{42} + (\beta_1 - 1) q^{43} + ( - \beta_{2} + 3 \beta_1 + 1) q^{44} + ( - \beta_1 + 1) q^{46} + (5 \beta_{2} - 4 \beta_1 - 1) q^{47} + ( - 3 \beta_{2} + \beta_1 + 1) q^{48} + q^{49} + (7 \beta_{2} - \beta_1 - 2) q^{51} + ( - 8 \beta_{2} + 6 \beta_1 - 7) q^{52} + ( - 5 \beta_{2} + 10 \beta_1 - 2) q^{53} + ( - \beta_{2} - 5 \beta_1 + 6) q^{54} + (2 \beta_{2} - \beta_1 + 2) q^{56} + (6 \beta_{2} - \beta_1) q^{57} + ( - 2 \beta_{2} + 6 \beta_1 - 10) q^{58} + (3 \beta_{2} - 3 \beta_1 - 5) q^{59} + ( - 5 \beta_{2} + 4 \beta_1 - 6) q^{61} + (\beta_1 - 2) q^{62} + ( - 3 \beta_{2} + 2 \beta_1) q^{63} + ( - \beta_{2} + 6 \beta_1 - 6) q^{64} + (\beta_{2} - 3 \beta_1 - 1) q^{66} + (5 \beta_{2} + \beta_1 - 3) q^{67} + ( - \beta_{2} - \beta_1 + 7) q^{68} + ( - 2 \beta_{2} + \beta_1 - 1) q^{69} + (\beta_{2} + 6 \beta_1 - 7) q^{71} + (5 \beta_{2} - 2 \beta_1 - 3) q^{72} + ( - 5 \beta_{2} + 4 \beta_1) q^{73} + (3 \beta_{2} - 3 \beta_1 + 6) q^{74} + ( - \beta_{2} + \beta_1 + 4) q^{76} + ( - \beta_1 - 2) q^{77} + (8 \beta_{2} - 6 \beta_1 + 7) q^{78} + (3 \beta_{2} + 3 \beta_1 - 6) q^{79} + ( - 8 \beta_{2} + 3 \beta_1 - 4) q^{81} + ( - 6 \beta_{2} + 9 \beta_1 - 11) q^{82} + (7 \beta_{2} - 3 \beta_1 + 13) q^{83} + (2 \beta_{2} + \beta_1 - 2) q^{84} + ( - \beta_{2} + 2 \beta_1 - 3) q^{86} + (10 \beta_{2} - 10 \beta_1 + 2) q^{87} + ( - 5 \beta_{2} - 4) q^{88} + (4 \beta_{2} - \beta_1 + 4) q^{89} + (3 \beta_{2} + \beta_1 - 1) q^{91} + (\beta_{2} - 2 \beta_1 + 1) q^{92} + (4 \beta_{2} - 3 \beta_1) q^{93} + (4 \beta_{2} - 3 \beta_1 + 2) q^{94} + ( - 3 \beta_{2} + 2 \beta_1 + 10) q^{96} + ( - 8 \beta_{2} - 3) q^{97} + ( - \beta_1 + 1) q^{98} + (7 \beta_{2} - 4 \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (-2*b2 + b1 - 1) * q^3 + (b2 - 2*b1 + 1) * q^4 + (-b2 + 2*b1 - 1) * q^6 + q^7 + (2*b2 - b1 + 2) * q^8 + (-3*b2 + 2*b1) * q^9 + (-b1 - 2) * q^11 + (2*b2 + b1 - 2) * q^12 + (3*b2 + b1 - 1) * q^13 + (-b1 + 1) * q^14 + (-b2 + b1) * q^16 + (2*b2 - 3*b1) * q^17 + (-2*b2 + 2*b1 - 1) * q^18 + (b2 - 2*b1 - 1) * q^19 + (-2*b2 + b1 - 1) * q^21 + (b2 + b1) * q^22 + q^23 + (b2 - b1 - 4) * q^24 + (-b2 + 2*b1 - 6) * q^26 + (-2*b2 + b1 + 6) * q^27 + (b2 - 2*b1 + 1) * q^28 + (2*b2 + 2*b1 - 4) * q^29 + (b2 - 1) * q^31 + (-5*b2 + 3*b1 - 5) * q^32 + (5*b2 - b1 + 2) * q^33 + (3*b2 - 3*b1 + 4) * q^34 + (4*b2 - b1 - 3) * q^36 - 3*b1 * q^37 + (2*b2 - b1 + 2) * q^38 + (7*b2 - 8*b1 - 2) * q^39 + (-4*b2 + 6*b1 - 3) * q^41 + (-b2 + 2*b1 - 1) * q^42 + (b1 - 1) * q^43 + (-b2 + 3*b1 + 1) * q^44 + (-b1 + 1) * q^46 + (5*b2 - 4*b1 - 1) * q^47 + (-3*b2 + b1 + 1) * q^48 + q^49 + (7*b2 - b1 - 2) * q^51 + (-8*b2 + 6*b1 - 7) * q^52 + (-5*b2 + 10*b1 - 2) * q^53 + (-b2 - 5*b1 + 6) * q^54 + (2*b2 - b1 + 2) * q^56 + (6*b2 - b1) * q^57 + (-2*b2 + 6*b1 - 10) * q^58 + (3*b2 - 3*b1 - 5) * q^59 + (-5*b2 + 4*b1 - 6) * q^61 + (b1 - 2) * q^62 + (-3*b2 + 2*b1) * q^63 + (-b2 + 6*b1 - 6) * q^64 + (b2 - 3*b1 - 1) * q^66 + (5*b2 + b1 - 3) * q^67 + (-b2 - b1 + 7) * q^68 + (-2*b2 + b1 - 1) * q^69 + (b2 + 6*b1 - 7) * q^71 + (5*b2 - 2*b1 - 3) * q^72 + (-5*b2 + 4*b1) * q^73 + (3*b2 - 3*b1 + 6) * q^74 + (-b2 + b1 + 4) * q^76 + (-b1 - 2) * q^77 + (8*b2 - 6*b1 + 7) * q^78 + (3*b2 + 3*b1 - 6) * q^79 + (-8*b2 + 3*b1 - 4) * q^81 + (-6*b2 + 9*b1 - 11) * q^82 + (7*b2 - 3*b1 + 13) * q^83 + (2*b2 + b1 - 2) * q^84 + (-b2 + 2*b1 - 3) * q^86 + (10*b2 - 10*b1 + 2) * q^87 + (-5*b2 - 4) * q^88 + (4*b2 - b1 + 4) * q^89 + (3*b2 + b1 - 1) * q^91 + (b2 - 2*b1 + 1) * q^92 + (4*b2 - 3*b1) * q^93 + (4*b2 - 3*b1 + 2) * q^94 + (-3*b2 + 2*b1 + 10) * q^96 + (-8*b2 - 3) * q^97 + (-b1 + 1) * q^98 + (7*b2 - 4*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 $$3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9} - 7 q^{11} - 7 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{16} - 5 q^{17} + q^{18} - 6 q^{19} + 3 q^{23} - 14 q^{24} - 15 q^{26} + 21 q^{27} - 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 14 q^{36} - 3 q^{37} + 3 q^{38} - 21 q^{39} + q^{41} - 2 q^{43} + 7 q^{44} + 2 q^{46} - 12 q^{47} + 7 q^{48} + 3 q^{49} - 14 q^{51} - 7 q^{52} + 9 q^{53} + 14 q^{54} + 3 q^{56} - 7 q^{57} - 22 q^{58} - 21 q^{59} - 9 q^{61} - 5 q^{62} + 5 q^{63} - 11 q^{64} - 7 q^{66} - 13 q^{67} + 21 q^{68} - 16 q^{71} - 16 q^{72} + 9 q^{73} + 12 q^{74} + 14 q^{76} - 7 q^{77} + 7 q^{78} - 18 q^{79} - q^{81} - 18 q^{82} + 29 q^{83} - 7 q^{84} - 6 q^{86} - 14 q^{87} - 7 q^{88} + 7 q^{89} - 5 q^{91} - 7 q^{93} - q^{94} + 35 q^{96} - q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 - 7 * q^11 - 7 * q^12 - 5 * q^13 + 2 * q^14 + 2 * q^16 - 5 * q^17 + q^18 - 6 * q^19 + 3 * q^23 - 14 * q^24 - 15 * q^26 + 21 * q^27 - 12 * q^29 - 4 * q^31 - 7 * q^32 + 6 * q^34 - 14 * q^36 - 3 * q^37 + 3 * q^38 - 21 * q^39 + q^41 - 2 * q^43 + 7 * q^44 + 2 * q^46 - 12 * q^47 + 7 * q^48 + 3 * q^49 - 14 * q^51 - 7 * q^52 + 9 * q^53 + 14 * q^54 + 3 * q^56 - 7 * q^57 - 22 * q^58 - 21 * q^59 - 9 * q^61 - 5 * q^62 + 5 * q^63 - 11 * q^64 - 7 * q^66 - 13 * q^67 + 21 * q^68 - 16 * q^71 - 16 * q^72 + 9 * q^73 + 12 * q^74 + 14 * q^76 - 7 * q^77 + 7 * q^78 - 18 * q^79 - q^81 - 18 * q^82 + 29 * q^83 - 7 * q^84 - 6 * q^86 - 14 * q^87 - 7 * q^88 + 7 * q^89 - 5 * q^91 - 7 * q^93 - q^94 + 35 * q^96 - q^97 + 2 * q^98 - 14 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.80194 0.445042 −1.24698
−0.801938 −1.69202 −1.35690 0 1.35690 1.00000 2.69202 −0.137063 0
1.2 0.554958 3.04892 −1.69202 0 1.69202 1.00000 −2.04892 6.29590 0
1.3 2.24698 −1.35690 3.04892 0 −3.04892 1.00000 2.35690 −1.15883 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
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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 4025.2.a.k 3
5.b even 2 1 805.2.a.f 3
15.d odd 2 1 7245.2.a.ba 3
35.c odd 2 1 5635.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.f 3 5.b even 2 1
4025.2.a.k 3 1.a even 1 1 trivial
5635.2.a.r 3 35.c odd 2 1
7245.2.a.ba 3 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{3} - 2T_{2}^{2} - T_{2} + 1$$ T2^3 - 2*T2^2 - T2 + 1 $$T_{3}^{3} - 7T_{3} - 7$$ T3^3 - 7*T3 - 7 $$T_{11}^{3} + 7T_{11}^{2} + 14T_{11} + 7$$ T11^3 + 7*T11^2 + 14*T11 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2T^{2} - T + 1$$
$3$ $$T^{3} - 7T - 7$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3} + 7 T^{2} + 14 T + 7$$
$13$ $$T^{3} + 5 T^{2} - 22 T - 97$$
$17$ $$T^{3} + 5 T^{2} - 8 T - 41$$
$19$ $$T^{3} + 6 T^{2} + 5 T - 13$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} + 12 T^{2} + 20 T - 104$$
$31$ $$T^{3} + 4 T^{2} + 3 T - 1$$
$37$ $$T^{3} + 3 T^{2} - 18 T - 27$$
$41$ $$T^{3} - T^{2} - 65 T + 169$$
$43$ $$T^{3} + 2T^{2} - T - 1$$
$47$ $$T^{3} + 12 T^{2} - T - 41$$
$53$ $$T^{3} - 9 T^{2} - 148 T + 1373$$
$59$ $$T^{3} + 21 T^{2} + 126 T + 203$$
$61$ $$T^{3} + 9 T^{2} - 22 T - 211$$
$67$ $$T^{3} + 13 T^{2} - 16 T - 377$$
$71$ $$T^{3} + 16 T^{2} - 15 T - 463$$
$73$ $$T^{3} - 9 T^{2} - 22 T + 29$$
$79$ $$T^{3} + 18 T^{2} + 45 T - 351$$
$83$ $$T^{3} - 29 T^{2} + 194 T + 211$$
$89$ $$T^{3} - 7 T^{2} - 14 T + 91$$
$97$ $$T^{3} + T^{2} - 149 T + 83$$