Properties

 Label 4025.2.a.m Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7537.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 4x + 3$$ x^4 - x^3 - 5*x^2 + 4*x + 3 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{6} - q^{7} - \beta_{3} q^{8} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b3 - 1) * q^3 + (b2 + 1) * q^4 + (b3 + b2 + b1) * q^6 - q^7 - b3 * q^8 + (b3 - b2 + b1 + 1) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} + \beta_{2} + \beta_1) q^{6} - q^{7} - \beta_{3} q^{8} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_1 + 3) q^{11} + ( - 2 \beta_{2} - \beta_1 - 1) q^{12} + (\beta_{3} + \beta_{2} + 1) q^{13} + \beta_1 q^{14} + (\beta_{3} - \beta_{2} - 2) q^{16} + ( - \beta_1 - 3) q^{17} + ( - 2 \beta_{2} - 3) q^{18} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{3} + 1) q^{21} + (\beta_{2} - 3 \beta_1 + 3) q^{22} + q^{23} + ( - \beta_{2} + \beta_1 + 3) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{26} + (2 \beta_{2} - \beta_1 - 1) q^{27} + ( - \beta_{2} - 1) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{29} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{31} + (2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{32} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{33} + (\beta_{2} + 3 \beta_1 + 3) q^{34} + (2 \beta_{2} + 3 \beta_1 - 2) q^{36} + (2 \beta_{2} + 3 \beta_1 + 1) q^{37} + ( - 3 \beta_1 - 3) q^{38} + ( - \beta_{2} - 2 \beta_1 - 4) q^{39} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{41} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{42} + ( - \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{43} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{44} - \beta_1 q^{46} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{47} + (\beta_{3} + 3 \beta_{2} - 1) q^{48} + q^{49} + (4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{51} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{52} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{53} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{54} + \beta_{3} q^{56} + ( - 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{57} + (2 \beta_{2} - 2 \beta_1) q^{58} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{59} + (5 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_1 + 3) q^{62} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{63} + ( - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 5) q^{64} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{66} + ( - \beta_{3} - \beta_{2} + 1) q^{67} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{68} + ( - \beta_{3} - 1) q^{69} + ( - 5 \beta_{3} + 3 \beta_{2} - 5 \beta_1) q^{71} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{72} + ( - 2 \beta_{3} - 5 \beta_{2} - 2) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 9) q^{74} + (2 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{76} + (\beta_1 - 3) q^{77} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 6) q^{78} + (\beta_{2} + 5 \beta_1 - 1) q^{79} + (\beta_{3} - 4 \beta_1 - 2) q^{81} + (3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 9) q^{82} + (3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 3) q^{83} + (2 \beta_{2} + \beta_1 + 1) q^{84} + ( - 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 6) q^{86} + (2 \beta_{3} - 6 \beta_{2} + 6) q^{87} + ( - 2 \beta_{3} + \beta_{2}) q^{88} + ( - \beta_1 + 9) q^{89} + ( - \beta_{3} - \beta_{2} - 1) q^{91} + (\beta_{2} + 1) q^{92} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 5) q^{93} + ( - 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{94} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1 - 6) q^{96} + (5 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{97} - \beta_1 q^{98} + (3 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b3 - 1) * q^3 + (b2 + 1) * q^4 + (b3 + b2 + b1) * q^6 - q^7 - b3 * q^8 + (b3 - b2 + b1 + 1) * q^9 + (-b1 + 3) * q^11 + (-2*b2 - b1 - 1) * q^12 + (b3 + b2 + 1) * q^13 + b1 * q^14 + (b3 - b2 - 2) * q^16 + (-b1 - 3) * q^17 + (-2*b2 - 3) * q^18 + (-b3 + b2 + b1 + 2) * q^19 + (b3 + 1) * q^21 + (b2 - 3*b1 + 3) * q^22 + q^23 + (-b2 + b1 + 3) * q^24 + (-2*b3 - b2 - 2*b1) * q^26 + (2*b2 - b1 - 1) * q^27 + (-b2 - 1) * q^28 + (-2*b3 + 2*b2) * q^29 + (b3 - b2 - b1 + 2) * q^31 + (2*b3 - b2 + 3*b1) * q^32 + (-2*b3 + b2 + b1 - 3) * q^33 + (b2 + 3*b1 + 3) * q^34 + (2*b2 + 3*b1 - 2) * q^36 + (2*b2 + 3*b1 + 1) * q^37 + (-3*b1 - 3) * q^38 + (-b2 - 2*b1 - 4) * q^39 + (-b3 - 2*b2 - 3*b1 + 6) * q^41 + (-b3 - b2 - b1) * q^42 + (-b3 + 4*b2 - 2*b1 + 1) * q^43 + (-b3 + 3*b2 - 2*b1 + 3) * q^44 - b1 * q^46 + (b3 + 3*b2 + b1) * q^47 + (b3 + 3*b2 - 1) * q^48 + q^49 + (4*b3 + b2 + b1 + 3) * q^51 + (b3 + 2*b2 + b1 + 4) * q^52 + (-2*b3 + b2 + 2*b1) * q^53 + (-2*b3 + b2 - b1 + 3) * q^54 + b3 * q^56 + (-2*b3 - 4*b2 - b1 + 1) * q^57 + (2*b2 - 2*b1) * q^58 + (-b3 - b2 + 2*b1 + 3) * q^59 + (5*b2 - 2*b1 + 2) * q^61 + (-b1 + 3) * q^62 + (-b3 + b2 - b1 - 1) * q^63 + (-3*b3 - 3*b2 + b1 - 5) * q^64 + (b3 + b2 + 2*b1 - 3) * q^66 + (-b3 - b2 + 1) * q^67 + (-b3 - 3*b2 - 2*b1 - 3) * q^68 + (-b3 - 1) * q^69 + (-5*b3 + 3*b2 - 5*b1) * q^71 + (-2*b3 + b2 - 3) * q^72 + (-2*b3 - 5*b2 - 2) * q^73 + (-2*b3 - 3*b2 - 3*b1 - 9) * q^74 + (2*b3 + b2 + b1 + 5) * q^76 + (b1 - 3) * q^77 + (b3 + 2*b2 + 5*b1 + 6) * q^78 + (b2 + 5*b1 - 1) * q^79 + (b3 - 4*b1 - 2) * q^81 + (3*b3 + 4*b2 - 4*b1 + 9) * q^82 + (3*b3 - 3*b2 + 4*b1 - 3) * q^83 + (2*b2 + b1 + 1) * q^84 + (-3*b3 + 3*b2 - 5*b1 + 6) * q^86 + (2*b3 - 6*b2 + 6) * q^87 + (-2*b3 + b2) * q^88 + (-b1 + 9) * q^89 + (-b3 - b2 - 1) * q^91 + (b2 + 1) * q^92 + (-2*b3 + 4*b2 + b1 - 5) * q^93 + (-4*b3 - 2*b2 - 3*b1 - 3) * q^94 + (-4*b3 + b2 - 4*b1 - 6) * q^96 + (5*b3 + 2*b2 - 3*b1 + 4) * q^97 - b1 * q^98 + (3*b3 - 5*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9 $$4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9} + 11 q^{11} - 3 q^{12} + 3 q^{13} + q^{14} - 7 q^{16} - 13 q^{17} - 10 q^{18} + 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{23} + 14 q^{24} - q^{26} - 7 q^{27} - 3 q^{28} - 2 q^{29} + 8 q^{31} + 4 q^{32} - 12 q^{33} + 14 q^{34} - 7 q^{36} + 5 q^{37} - 15 q^{38} - 17 q^{39} + 23 q^{41} - 2 q^{43} + 7 q^{44} - q^{46} - 2 q^{47} - 7 q^{48} + 4 q^{49} + 12 q^{51} + 15 q^{52} + q^{53} + 10 q^{54} + 7 q^{57} - 4 q^{58} + 15 q^{59} + q^{61} + 11 q^{62} - 6 q^{63} - 16 q^{64} - 11 q^{66} + 5 q^{67} - 11 q^{68} - 4 q^{69} - 8 q^{71} - 13 q^{72} - 3 q^{73} - 36 q^{74} + 20 q^{76} - 11 q^{77} + 27 q^{78} - 12 q^{81} + 28 q^{82} - 5 q^{83} + 3 q^{84} + 16 q^{86} + 30 q^{87} - q^{88} + 35 q^{89} - 3 q^{91} + 3 q^{92} - 23 q^{93} - 13 q^{94} - 29 q^{96} + 11 q^{97} - q^{98} + 8 q^{99}+O(q^{100})$$ 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9 + 11 * q^11 - 3 * q^12 + 3 * q^13 + q^14 - 7 * q^16 - 13 * q^17 - 10 * q^18 + 8 * q^19 + 4 * q^21 + 8 * q^22 + 4 * q^23 + 14 * q^24 - q^26 - 7 * q^27 - 3 * q^28 - 2 * q^29 + 8 * q^31 + 4 * q^32 - 12 * q^33 + 14 * q^34 - 7 * q^36 + 5 * q^37 - 15 * q^38 - 17 * q^39 + 23 * q^41 - 2 * q^43 + 7 * q^44 - q^46 - 2 * q^47 - 7 * q^48 + 4 * q^49 + 12 * q^51 + 15 * q^52 + q^53 + 10 * q^54 + 7 * q^57 - 4 * q^58 + 15 * q^59 + q^61 + 11 * q^62 - 6 * q^63 - 16 * q^64 - 11 * q^66 + 5 * q^67 - 11 * q^68 - 4 * q^69 - 8 * q^71 - 13 * q^72 - 3 * q^73 - 36 * q^74 + 20 * q^76 - 11 * q^77 + 27 * q^78 - 12 * q^81 + 28 * q^82 - 5 * q^83 + 3 * q^84 + 16 * q^86 + 30 * q^87 - q^88 + 35 * q^89 - 3 * q^91 + 3 * q^92 - 23 * q^93 - 13 * q^94 - 29 * q^96 + 11 * q^97 - q^98 + 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*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
 2.15976 1.37933 −0.491918 −2.04717
−2.15976 −2.43525 2.66454 0 5.25956 −1.00000 −1.43525 2.93047 0
1.2 −1.37933 1.89307 −0.0974383 0 −2.61117 −1.00000 2.89307 0.583705 0
1.3 0.491918 −2.84864 −1.75802 0 −1.40130 −1.00000 −1.84864 5.11474 0
1.4 2.04717 −0.609175 2.19091 0 −1.24709 −1.00000 0.390825 −2.62891 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.m 4
5.b even 2 1 805.2.a.i 4
15.d odd 2 1 7245.2.a.bd 4
35.c odd 2 1 5635.2.a.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.i 4 5.b even 2 1
4025.2.a.m 4 1.a even 1 1 trivial
5635.2.a.u 4 35.c odd 2 1
7245.2.a.bd 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{4} + T_{2}^{3} - 5T_{2}^{2} - 4T_{2} + 3$$ T2^4 + T2^3 - 5*T2^2 - 4*T2 + 3 $$T_{3}^{4} + 4T_{3}^{3} - T_{3}^{2} - 15T_{3} - 8$$ T3^4 + 4*T3^3 - T3^2 - 15*T3 - 8 $$T_{11}^{4} - 11T_{11}^{3} + 40T_{11}^{2} - 55T_{11} + 24$$ T11^4 - 11*T11^3 + 40*T11^2 - 55*T11 + 24

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 5 T^{2} - 4 T + 3$$
$3$ $$T^{4} + 4 T^{3} - T^{2} - 15 T - 8$$
$5$ $$T^{4}$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} - 11 T^{3} + 40 T^{2} - 55 T + 24$$
$13$ $$T^{4} - 3 T^{3} - 10 T^{2} + 23 T - 2$$
$17$ $$T^{4} + 13 T^{3} + 58 T^{2} + 101 T + 54$$
$19$ $$T^{4} - 8 T^{3} + 3 T^{2} + 81 T - 108$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} + 2 T^{3} - 52 T^{2} + 128 T - 48$$
$31$ $$T^{4} - 8 T^{3} + 3 T^{2} + 23 T + 8$$
$37$ $$T^{4} - 5 T^{3} - 72 T^{2} + 51 T + 526$$
$41$ $$T^{4} - 23 T^{3} + 113 T^{2} + \cdots - 4122$$
$43$ $$T^{4} + 2 T^{3} - 117 T^{2} + \cdots + 692$$
$47$ $$T^{4} + 2 T^{3} - 71 T^{2} - 209 T + 324$$
$53$ $$T^{4} - T^{3} - 62 T^{2} + 55 T + 366$$
$59$ $$T^{4} - 15 T^{3} + 50 T^{2} + \cdots - 228$$
$61$ $$T^{4} - T^{3} - 168 T^{2} + 15 T + 4882$$
$67$ $$T^{4} - 5 T^{3} - 4 T^{2} + 21 T - 4$$
$71$ $$T^{4} + 8 T^{3} - 299 T^{2} + \cdots + 13176$$
$73$ $$T^{4} + 3 T^{3} - 184 T^{2} + \cdots + 7102$$
$79$ $$T^{4} - 147 T^{2} - 121 T + 3436$$
$83$ $$T^{4} + 5 T^{3} - 170 T^{2} + \cdots + 2004$$
$89$ $$T^{4} - 35 T^{3} + 454 T^{2} + \cdots + 5466$$
$97$ $$T^{4} - 11 T^{3} - 211 T^{2} + \cdots - 13114$$