# Properties

 Label 25.8.b.a Level 25 Weight 8 Character orbit 25.b Analytic conductor 7.810 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.80962563710$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 14 i q^{2} -48 i q^{3} -68 q^{4} + 672 q^{6} + 1644 i q^{7} + 840 i q^{8} -117 q^{9} +O(q^{10})$$ $$q + 14 i q^{2} -48 i q^{3} -68 q^{4} + 672 q^{6} + 1644 i q^{7} + 840 i q^{8} -117 q^{9} + 172 q^{11} + 3264 i q^{12} + 3862 i q^{13} -23016 q^{14} -20464 q^{16} + 12254 i q^{17} -1638 i q^{18} + 25940 q^{19} + 78912 q^{21} + 2408 i q^{22} + 12972 i q^{23} + 40320 q^{24} -54068 q^{26} -99360 i q^{27} -111792 i q^{28} + 81610 q^{29} -156888 q^{31} -178976 i q^{32} -8256 i q^{33} -171556 q^{34} + 7956 q^{36} -110126 i q^{37} + 363160 i q^{38} + 185376 q^{39} + 467882 q^{41} + 1104768 i q^{42} -499208 i q^{43} -11696 q^{44} -181608 q^{46} + 396884 i q^{47} + 982272 i q^{48} -1879193 q^{49} + 588192 q^{51} -262616 i q^{52} -1280498 i q^{53} + 1391040 q^{54} -1380960 q^{56} -1245120 i q^{57} + 1142540 i q^{58} + 1337420 q^{59} -923978 q^{61} -2196432 i q^{62} -192348 i q^{63} -113728 q^{64} + 115584 q^{66} + 797304 i q^{67} -833272 i q^{68} + 622656 q^{69} + 5103392 q^{71} -98280 i q^{72} -4267478 i q^{73} + 1541764 q^{74} -1763920 q^{76} + 282768 i q^{77} + 2595264 i q^{78} + 960 q^{79} -5025159 q^{81} + 6550348 i q^{82} + 6140832 i q^{83} -5366016 q^{84} + 6988912 q^{86} -3917280 i q^{87} + 144480 i q^{88} -2010570 q^{89} -6349128 q^{91} -882096 i q^{92} + 7530624 i q^{93} -5556376 q^{94} -8590848 q^{96} + 4881934 i q^{97} -26308702 i q^{98} -20124 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 136q^{4} + 1344q^{6} - 234q^{9} + O(q^{10})$$ $$2q - 136q^{4} + 1344q^{6} - 234q^{9} + 344q^{11} - 46032q^{14} - 40928q^{16} + 51880q^{19} + 157824q^{21} + 80640q^{24} - 108136q^{26} + 163220q^{29} - 313776q^{31} - 343112q^{34} + 15912q^{36} + 370752q^{39} + 935764q^{41} - 23392q^{44} - 363216q^{46} - 3758386q^{49} + 1176384q^{51} + 2782080q^{54} - 2761920q^{56} + 2674840q^{59} - 1847956q^{61} - 227456q^{64} + 231168q^{66} + 1245312q^{69} + 10206784q^{71} + 3083528q^{74} - 3527840q^{76} + 1920q^{79} - 10050318q^{81} - 10732032q^{84} + 13977824q^{86} - 4021140q^{89} - 12698256q^{91} - 11112752q^{94} - 17181696q^{96} - 40248q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
14.0000i 48.0000i −68.0000 0 672.000 1644.00i 840.000i −117.000 0
24.2 14.0000i 48.0000i −68.0000 0 672.000 1644.00i 840.000i −117.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.a 2
3.b odd 2 1 225.8.b.b 2
4.b odd 2 1 400.8.c.e 2
5.b even 2 1 inner 25.8.b.a 2
5.c odd 4 1 5.8.a.a 1
5.c odd 4 1 25.8.a.a 1
15.d odd 2 1 225.8.b.b 2
15.e even 4 1 45.8.a.f 1
15.e even 4 1 225.8.a.b 1
20.d odd 2 1 400.8.c.e 2
20.e even 4 1 80.8.a.d 1
20.e even 4 1 400.8.a.e 1
35.f even 4 1 245.8.a.a 1
40.i odd 4 1 320.8.a.h 1
40.k even 4 1 320.8.a.a 1
55.e even 4 1 605.8.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.a 1 5.c odd 4 1
25.8.a.a 1 5.c odd 4 1
25.8.b.a 2 1.a even 1 1 trivial
25.8.b.a 2 5.b even 2 1 inner
45.8.a.f 1 15.e even 4 1
80.8.a.d 1 20.e even 4 1
225.8.a.b 1 15.e even 4 1
225.8.b.b 2 3.b odd 2 1
225.8.b.b 2 15.d odd 2 1
245.8.a.a 1 35.f even 4 1
320.8.a.a 1 40.k even 4 1
320.8.a.h 1 40.i odd 4 1
400.8.a.e 1 20.e even 4 1
400.8.c.e 2 4.b odd 2 1
400.8.c.e 2 20.d odd 2 1
605.8.a.c 1 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 196$$ acting on $$S_{8}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 60 T^{2} + 16384 T^{4}$$
$3$ $$1 - 2070 T^{2} + 4782969 T^{4}$$
$5$ 1
$7$ $$1 + 1055650 T^{2} + 678223072849 T^{4}$$
$11$ $$( 1 - 172 T + 19487171 T^{2} )^{2}$$
$13$ $$1 - 110581990 T^{2} + 3937376385699289 T^{4}$$
$17$ $$1 - 670516830 T^{2} + 168377826559400929 T^{4}$$
$19$ $$( 1 - 25940 T + 893871739 T^{2} )^{2}$$
$23$ $$1 - 6641378110 T^{2} + 11592836324538749809 T^{4}$$
$29$ $$( 1 - 81610 T + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 + 156888 T + 27512614111 T^{2} )^{2}$$
$37$ $$1 - 177736018390 T^{2} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$( 1 - 467882 T + 194754273881 T^{2} )^{2}$$
$43$ $$1 - 294428594950 T^{2} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 855729331470 T^{2} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 - 709747151670 T^{2} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$( 1 - 1337420 T + 2488651484819 T^{2} )^{2}$$
$61$ $$( 1 + 923978 T + 3142742836021 T^{2} )^{2}$$
$67$ $$1 - 11485729542230 T^{2} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$( 1 - 5103392 T + 9095120158391 T^{2} )^{2}$$
$73$ $$1 - 3883428557710 T^{2} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$( 1 - 960 T + 19203908986159 T^{2} )^{2}$$
$83$ $$1 - 16562284327030 T^{2} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$( 1 + 2010570 T + 44231334895529 T^{2} )^{2}$$
$97$ $$1 - 137763289375870 T^{2} +$$$$65\!\cdots\!69$$$$T^{4}$$