# Properties

 Label 4025.2.a.l Level 4025 Weight 2 Character orbit 4025.a Self dual yes Analytic conductor 32.140 Analytic rank 1 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2777.1 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$

## 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
 −1.50848 −0.679643 0.825785 2.36234
−2.50848 −0.817356 4.29248 0 2.05032 1.00000 −5.75064 −2.33193 0
1.2 −1.67964 −3.26308 0.821201 0 5.48081 1.00000 1.97996 7.64767 0
1.3 −0.174215 0.596155 −1.96965 0 −0.103859 1.00000 0.691574 −2.64460 0
1.4 1.36234 −2.51572 −0.144030 0 −3.42727 1.00000 −2.92090 3.32886 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.l 4
5.b even 2 1 805.2.a.j 4
15.d odd 2 1 7245.2.a.bc 4
35.c odd 2 1 5635.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.j 4 5.b even 2 1
4025.2.a.l 4 1.a even 1 1 trivial
5635.2.a.w 4 35.c odd 2 1
7245.2.a.bc 4 15.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{4} + 3 T_{2}^{3} - T_{2}^{2} - 6 T_{2} - 1$$ $$T_{3}^{4} + 6 T_{3}^{3} + 9 T_{3}^{2} - T_{3} - 4$$ $$T_{11}^{4} - 7 T_{11}^{3} + 14 T_{11}^{2} - 5 T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 7 T^{2} + 12 T^{3} + 19 T^{4} + 24 T^{5} + 28 T^{6} + 24 T^{7} + 16 T^{8}$$
$3$ $$1 + 6 T + 21 T^{2} + 53 T^{3} + 104 T^{4} + 159 T^{5} + 189 T^{6} + 162 T^{7} + 81 T^{8}$$
$5$ 1
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 7 T + 58 T^{2} - 236 T^{3} + 1030 T^{4} - 2596 T^{5} + 7018 T^{6} - 9317 T^{7} + 14641 T^{8}$$
$13$ $$1 - 5 T + 52 T^{2} - 190 T^{3} + 1012 T^{4} - 2470 T^{5} + 8788 T^{6} - 10985 T^{7} + 28561 T^{8}$$
$17$ $$1 + 5 T + 56 T^{2} + 216 T^{3} + 1304 T^{4} + 3672 T^{5} + 16184 T^{6} + 24565 T^{7} + 83521 T^{8}$$
$19$ $$1 - 8 T + 73 T^{2} - 331 T^{3} + 1840 T^{4} - 6289 T^{5} + 26353 T^{6} - 54872 T^{7} + 130321 T^{8}$$
$23$ $$( 1 - T )^{4}$$
$29$ $$1 + 2 T + 80 T^{2} + 222 T^{3} + 2942 T^{4} + 6438 T^{5} + 67280 T^{6} + 48778 T^{7} + 707281 T^{8}$$
$31$ $$1 + 75 T^{2} + 117 T^{3} + 2696 T^{4} + 3627 T^{5} + 72075 T^{6} + 923521 T^{8}$$
$37$ $$1 + 13 T + 178 T^{2} + 1292 T^{3} + 9928 T^{4} + 47804 T^{5} + 243682 T^{6} + 658489 T^{7} + 1874161 T^{8}$$
$41$ $$1 - T + 121 T^{2} - 242 T^{3} + 6474 T^{4} - 9922 T^{5} + 203401 T^{6} - 68921 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 14 T + 161 T^{2} + 1421 T^{3} + 9720 T^{4} + 61103 T^{5} + 297689 T^{6} + 1113098 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 4 T + 109 T^{2} + 53 T^{3} + 5092 T^{4} + 2491 T^{5} + 240781 T^{6} + 415292 T^{7} + 4879681 T^{8}$$
$53$ $$1 - T + 202 T^{2} - 146 T^{3} + 15792 T^{4} - 7738 T^{5} + 567418 T^{6} - 148877 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 7 T + 162 T^{2} + 752 T^{3} + 12318 T^{4} + 44368 T^{5} + 563922 T^{6} + 1437653 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 7 T + 190 T^{2} + 984 T^{3} + 15464 T^{4} + 60024 T^{5} + 706990 T^{6} + 1588867 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 15 T + 182 T^{2} + 1634 T^{3} + 14382 T^{4} + 109478 T^{5} + 816998 T^{6} + 4511445 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 213 T^{2} - 199 T^{3} + 20156 T^{4} - 14129 T^{5} + 1073733 T^{6} + 25411681 T^{8}$$
$73$ $$1 + 3 T + 122 T^{2} + 960 T^{3} + 8920 T^{4} + 70080 T^{5} + 650138 T^{6} + 1167051 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 14 T + 319 T^{2} + 2683 T^{3} + 35728 T^{4} + 211957 T^{5} + 1990879 T^{6} + 6902546 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 3 T + 180 T^{2} + 1556 T^{3} + 14858 T^{4} + 129148 T^{5} + 1240020 T^{6} + 1715361 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 11 T + 232 T^{2} + 1102 T^{3} + 20340 T^{4} + 98078 T^{5} + 1837672 T^{6} + 7754659 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 9 T + 241 T^{2} + 2090 T^{3} + 31298 T^{4} + 202730 T^{5} + 2267569 T^{6} + 8214057 T^{7} + 88529281 T^{8}$$