# Properties

 Label 4025.2.a.q Level 4025 Weight 2 Character orbit 4025.a Self dual yes Analytic conductor 32.140 Analytic rank 1 Dimension 5 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: $$5$$ Coefficient field: 5.5.255877.1 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{6} - q^{7} + ( -1 + \beta_{3} ) q^{8} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{6} - q^{7} + ( -1 + \beta_{3} ) q^{8} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -2 - 2 \beta_{2} + \beta_{4} ) q^{11} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{12} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{13} -\beta_{1} q^{14} + ( -2 - \beta_{2} + \beta_{4} ) q^{16} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} ) q^{17} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{18} + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{19} + ( -1 - \beta_{4} ) q^{21} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{22} - q^{23} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{24} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{26} + ( 3 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -3 + 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{29} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{32} + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{34} + ( 6 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{36} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{37} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{38} + ( -3 - \beta_{1} - 3 \beta_{2} + 3 \beta_{4} ) q^{39} + ( -3 + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{42} + ( -2 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{43} + ( -6 - \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{44} -\beta_{1} q^{46} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{47} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{48} + q^{49} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( -3 - 3 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{52} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{53} + ( -10 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} ) q^{54} + ( 1 - \beta_{3} ) q^{56} + ( -6 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{57} + ( 8 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{58} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{59} + ( -4 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{61} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{62} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + ( -3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} ) q^{64} + ( 8 - 7 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{66} + ( \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{67} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{68} + ( -1 - \beta_{4} ) q^{69} + ( -5 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{71} + ( -6 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{72} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{73} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{74} + ( -2 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{76} + ( 2 + 2 \beta_{2} - \beta_{4} ) q^{77} + ( -6 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} ) q^{78} + ( 6 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{79} + ( 1 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{81} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{82} + ( 6 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{83} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{84} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{86} + ( -11 + 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{87} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{88} + ( -2 - 4 \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{89} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( -7 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{93} + ( 6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{94} + ( 7 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{96} + ( -6 + 5 \beta_{3} - \beta_{4} ) q^{97} + \beta_{1} q^{98} + ( -8 + 5 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + q^{2} + 4q^{3} + 3q^{4} - 5q^{6} - 5q^{7} - 3q^{8} + 5q^{9} + O(q^{10})$$ $$5q + q^{2} + 4q^{3} + 3q^{4} - 5q^{6} - 5q^{7} - 3q^{8} + 5q^{9} - 7q^{11} + 10q^{12} + 5q^{13} - q^{14} - 9q^{16} + 7q^{17} - 11q^{18} - 10q^{19} - 4q^{21} - 4q^{22} - 5q^{23} - 4q^{24} - 2q^{26} + 7q^{27} - 3q^{28} - 14q^{29} - 10q^{31} - 4q^{33} - 10q^{34} + 20q^{36} + 3q^{37} + 15q^{38} - 13q^{39} - 15q^{41} + 5q^{42} - 8q^{43} - 27q^{44} - q^{46} + 10q^{47} + 2q^{48} + 5q^{49} - 18q^{51} - 18q^{52} + 9q^{53} - 39q^{54} + 3q^{56} - 23q^{57} + 31q^{58} - 19q^{59} - 21q^{61} + 10q^{62} - 5q^{63} - 7q^{64} + 25q^{66} - 5q^{67} + 15q^{68} - 4q^{69} - 16q^{71} - 26q^{72} - q^{73} - 16q^{74} + 7q^{77} - 28q^{78} + 20q^{79} - 3q^{81} - 11q^{82} + 31q^{83} - 10q^{84} + 10q^{86} - 38q^{87} + 3q^{88} - 21q^{89} - 5q^{91} - 3q^{92} - 23q^{93} + 28q^{94} + 24q^{96} - 19q^{97} + q^{98} - 26q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 6 x^{3} + 6 x^{2} + 6 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2} + 12$$

## 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
 −2.19548 −1.06459 0.698160 1.45275 2.10917
−2.19548 3.13309 2.82015 0 −6.87865 −1.00000 −1.80062 6.81626 0
1.2 −1.06459 −0.382286 −0.866643 0 0.406979 −1.00000 3.05181 −2.85386 0
1.3 0.698160 1.80045 −1.51257 0 1.25700 −1.00000 −2.45234 0.241606 0
1.4 1.45275 −2.09827 0.110473 0 −3.04825 −1.00000 −2.74500 1.40273 0
1.5 2.10917 1.54702 2.44859 0 3.26292 −1.00000 0.946160 −0.606735 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.q 5
5.b even 2 1 805.2.a.l 5
15.d odd 2 1 7245.2.a.bh 5
35.c odd 2 1 5635.2.a.y 5

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

 $$T_{2}^{5} - T_{2}^{4} - 6 T_{2}^{3} + 6 T_{2}^{2} + 6 T_{2} - 5$$ $$T_{3}^{5} - 4 T_{3}^{4} - 2 T_{3}^{3} + 19 T_{3}^{2} - 11 T_{3} - 7$$ $$T_{11}^{5} + 7 T_{11}^{4} - 4 T_{11}^{3} - 107 T_{11}^{2} - 156 T_{11} + 68$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 4 T^{2} - 2 T^{3} + 10 T^{4} - 5 T^{5} + 20 T^{6} - 8 T^{7} + 32 T^{8} - 16 T^{9} + 32 T^{10}$$
$3$ $$1 - 4 T + 13 T^{2} - 29 T^{3} + 61 T^{4} - 109 T^{5} + 183 T^{6} - 261 T^{7} + 351 T^{8} - 324 T^{9} + 243 T^{10}$$
$5$ 
$7$ $$( 1 + T )^{5}$$
$11$ $$1 + 7 T + 51 T^{2} + 201 T^{3} + 922 T^{4} + 2796 T^{5} + 10142 T^{6} + 24321 T^{7} + 67881 T^{8} + 102487 T^{9} + 161051 T^{10}$$
$13$ $$1 - 5 T + 44 T^{2} - 134 T^{3} + 749 T^{4} - 1789 T^{5} + 9737 T^{6} - 22646 T^{7} + 96668 T^{8} - 142805 T^{9} + 371293 T^{10}$$
$17$ $$1 - 7 T + 47 T^{2} - 133 T^{3} + 566 T^{4} - 1192 T^{5} + 9622 T^{6} - 38437 T^{7} + 230911 T^{8} - 584647 T^{9} + 1419857 T^{10}$$
$19$ $$1 + 10 T + 116 T^{2} + 713 T^{3} + 4699 T^{4} + 19998 T^{5} + 89281 T^{6} + 257393 T^{7} + 795644 T^{8} + 1303210 T^{9} + 2476099 T^{10}$$
$23$ $$( 1 + T )^{5}$$
$29$ $$1 + 14 T + 162 T^{2} + 1292 T^{3} + 9157 T^{4} + 50996 T^{5} + 265553 T^{6} + 1086572 T^{7} + 3951018 T^{8} + 9901934 T^{9} + 20511149 T^{10}$$
$31$ $$1 + 10 T + 169 T^{2} + 1209 T^{3} + 10855 T^{4} + 55721 T^{5} + 336505 T^{6} + 1161849 T^{7} + 5034679 T^{8} + 9235210 T^{9} + 28629151 T^{10}$$
$37$ $$1 - 3 T + 127 T^{2} - 247 T^{3} + 7314 T^{4} - 10492 T^{5} + 270618 T^{6} - 338143 T^{7} + 6432931 T^{8} - 5622483 T^{9} + 69343957 T^{10}$$
$41$ $$1 + 15 T + 215 T^{2} + 2090 T^{3} + 18157 T^{4} + 121401 T^{5} + 744437 T^{6} + 3513290 T^{7} + 14818015 T^{8} + 42386415 T^{9} + 115856201 T^{10}$$
$43$ $$1 + 8 T + 200 T^{2} + 1169 T^{3} + 16157 T^{4} + 70810 T^{5} + 694751 T^{6} + 2161481 T^{7} + 15901400 T^{8} + 27350408 T^{9} + 147008443 T^{10}$$
$47$ $$1 - 10 T + 219 T^{2} - 1665 T^{3} + 19993 T^{4} - 112659 T^{5} + 939671 T^{6} - 3677985 T^{7} + 22737237 T^{8} - 48796810 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 9 T + 135 T^{2} - 103 T^{3} + 846 T^{4} + 46032 T^{5} + 44838 T^{6} - 289327 T^{7} + 20098395 T^{8} - 71014329 T^{9} + 418195493 T^{10}$$
$59$ $$1 + 19 T + 369 T^{2} + 4255 T^{3} + 47176 T^{4} + 369412 T^{5} + 2783384 T^{6} + 14811655 T^{7} + 75784851 T^{8} + 230229859 T^{9} + 714924299 T^{10}$$
$61$ $$1 + 21 T + 405 T^{2} + 4851 T^{3} + 53432 T^{4} + 434312 T^{5} + 3259352 T^{6} + 18050571 T^{7} + 91927305 T^{8} + 290762661 T^{9} + 844596301 T^{10}$$
$67$ $$1 + 5 T + 191 T^{2} + 1087 T^{3} + 21248 T^{4} + 94260 T^{5} + 1423616 T^{6} + 4879543 T^{7} + 57445733 T^{8} + 100755605 T^{9} + 1350125107 T^{10}$$
$71$ $$1 + 16 T + 367 T^{2} + 4221 T^{3} + 52733 T^{4} + 439265 T^{5} + 3744043 T^{6} + 21278061 T^{7} + 131353337 T^{8} + 406586896 T^{9} + 1804229351 T^{10}$$
$73$ $$1 + T + 150 T^{2} + 302 T^{3} + 16641 T^{4} + 10797 T^{5} + 1214793 T^{6} + 1609358 T^{7} + 58352550 T^{8} + 28398241 T^{9} + 2073071593 T^{10}$$
$79$ $$1 - 20 T + 426 T^{2} - 5881 T^{3} + 69513 T^{4} - 679554 T^{5} + 5491527 T^{6} - 36703321 T^{7} + 210034614 T^{8} - 779001620 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - 31 T + 617 T^{2} - 7921 T^{3} + 85250 T^{4} - 777204 T^{5} + 7075750 T^{6} - 54567769 T^{7} + 352792579 T^{8} - 1471207951 T^{9} + 3939040643 T^{10}$$
$89$ $$1 + 21 T + 417 T^{2} + 4765 T^{3} + 54724 T^{4} + 484844 T^{5} + 4870436 T^{6} + 37743565 T^{7} + 293972073 T^{8} + 1317587061 T^{9} + 5584059449 T^{10}$$
$97$ $$1 + 19 T + 298 T^{2} + 2293 T^{3} + 25987 T^{4} + 186112 T^{5} + 2520739 T^{6} + 21574837 T^{7} + 271976554 T^{8} + 1682056339 T^{9} + 8587340257 T^{10}$$