# Properties

 Label 25.8.a.d Level 25 Weight 8 Character orbit 25.a Self dual yes Analytic conductor 7.810 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.80962563710$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{29})$$ Defining polynomial: $$x^{2} - x - 7$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + 3 \beta q^{3} -12 q^{4} -348 q^{6} + 39 \beta q^{7} + 140 \beta q^{8} -1143 q^{9} +O(q^{10})$$ $$q -\beta q^{2} + 3 \beta q^{3} -12 q^{4} -348 q^{6} + 39 \beta q^{7} + 140 \beta q^{8} -1143 q^{9} -6828 q^{11} -36 \beta q^{12} -942 \beta q^{13} -4524 q^{14} -14704 q^{16} -1456 \beta q^{17} + 1143 \beta q^{18} -6860 q^{19} + 13572 q^{21} + 6828 \beta q^{22} + 2713 \beta q^{23} + 48720 q^{24} + 109272 q^{26} -9990 \beta q^{27} -468 \beta q^{28} -25590 q^{29} + 82112 q^{31} -3216 \beta q^{32} -20484 \beta q^{33} + 168896 q^{34} + 13716 q^{36} + 20754 \beta q^{37} + 6860 \beta q^{38} -327816 q^{39} -533118 q^{41} -13572 \beta q^{42} + 65823 \beta q^{43} + 81936 q^{44} -314708 q^{46} -541 \beta q^{47} -44112 \beta q^{48} -647107 q^{49} -506688 q^{51} + 11304 \beta q^{52} -54722 \beta q^{53} + 1158840 q^{54} + 633360 q^{56} -20580 \beta q^{57} + 25590 \beta q^{58} -1438980 q^{59} + 1381022 q^{61} -82112 \beta q^{62} -44577 \beta q^{63} + 2255168 q^{64} + 2376144 q^{66} + 252069 \beta q^{67} + 17472 \beta q^{68} + 944124 q^{69} -481608 q^{71} -160020 \beta q^{72} + 137988 \beta q^{73} -2407464 q^{74} + 82320 q^{76} -266292 \beta q^{77} + 327816 \beta q^{78} + 1059760 q^{79} -976779 q^{81} + 533118 \beta q^{82} -241757 \beta q^{83} -162864 q^{84} -7635468 q^{86} -76770 \beta q^{87} -955920 \beta q^{88} -5644170 q^{89} -4261608 q^{91} -32556 \beta q^{92} + 246336 \beta q^{93} + 62756 q^{94} -1119168 q^{96} -1115016 \beta q^{97} + 647107 \beta q^{98} + 7804404 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 24q^{4} - 696q^{6} - 2286q^{9} + O(q^{10})$$ $$2q - 24q^{4} - 696q^{6} - 2286q^{9} - 13656q^{11} - 9048q^{14} - 29408q^{16} - 13720q^{19} + 27144q^{21} + 97440q^{24} + 218544q^{26} - 51180q^{29} + 164224q^{31} + 337792q^{34} + 27432q^{36} - 655632q^{39} - 1066236q^{41} + 163872q^{44} - 629416q^{46} - 1294214q^{49} - 1013376q^{51} + 2317680q^{54} + 1266720q^{56} - 2877960q^{59} + 2762044q^{61} + 4510336q^{64} + 4752288q^{66} + 1888248q^{69} - 963216q^{71} - 4814928q^{74} + 164640q^{76} + 2119520q^{79} - 1953558q^{81} - 325728q^{84} - 15270936q^{86} - 11288340q^{89} - 8523216q^{91} + 125512q^{94} - 2238336q^{96} + 15608808q^{99} + O(q^{100})$$

## 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
 3.19258 −2.19258
−10.7703 32.3110 −12.0000 0 −348.000 420.043 1507.85 −1143.00 0
1.2 10.7703 −32.3110 −12.0000 0 −348.000 −420.043 −1507.85 −1143.00 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
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.8.a.d 2
3.b odd 2 1 225.8.a.n 2
4.b odd 2 1 400.8.a.y 2
5.b even 2 1 inner 25.8.a.d 2
5.c odd 4 2 5.8.b.a 2
15.d odd 2 1 225.8.a.n 2
15.e even 4 2 45.8.b.a 2
20.d odd 2 1 400.8.a.y 2
20.e even 4 2 80.8.c.a 2
40.i odd 4 2 320.8.c.d 2
40.k even 4 2 320.8.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 5.c odd 4 2
25.8.a.d 2 1.a even 1 1 trivial
25.8.a.d 2 5.b even 2 1 inner
45.8.b.a 2 15.e even 4 2
80.8.c.a 2 20.e even 4 2
225.8.a.n 2 3.b odd 2 1
225.8.a.n 2 15.d odd 2 1
320.8.c.c 2 40.k even 4 2
320.8.c.d 2 40.i odd 4 2
400.8.a.y 2 4.b odd 2 1
400.8.a.y 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 116$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 140 T^{2} + 16384 T^{4}$$
$3$ $$1 + 3330 T^{2} + 4782969 T^{4}$$
$5$ 1
$7$ $$1 + 1470650 T^{2} + 678223072849 T^{4}$$
$11$ $$( 1 + 6828 T + 19487171 T^{2} )^{2}$$
$13$ $$1 + 22562810 T^{2} + 3937376385699289 T^{4}$$
$17$ $$1 + 574764770 T^{2} + 168377826559400929 T^{4}$$
$19$ $$( 1 + 6860 T + 893871739 T^{2} )^{2}$$
$23$ $$1 + 5955848090 T^{2} + 11592836324538749809 T^{4}$$
$29$ $$( 1 + 25590 T + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 - 82112 T + 27512614111 T^{2} )^{2}$$
$37$ $$1 + 139899246410 T^{2} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$( 1 + 533118 T + 194754273881 T^{2} )^{2}$$
$43$ $$1 + 41047812050 T^{2} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 1013212289930 T^{2} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 + 2002060594730 T^{2} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$( 1 + 1438980 T + 2488651484819 T^{2} )^{2}$$
$61$ $$( 1 - 1381022 T + 3142742836021 T^{2} )^{2}$$
$67$ $$1 + 4750924642370 T^{2} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$( 1 + 481608 T + 9095120158391 T^{2} )^{2}$$
$73$ $$1 + 19886077213490 T^{2} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$( 1 - 1059760 T + 19203908986159 T^{2} )^{2}$$
$83$ $$1 + 47492314121570 T^{2} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$( 1 + 5644170 T + 44231334895529 T^{2} )^{2}$$
$97$ $$1 + 17378330046530 T^{2} +$$$$65\!\cdots\!69$$$$T^{4}$$