# Properties

 Label 25.6.b.a Level 25 Weight 6 Character orbit 25.b Analytic conductor 4.010 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.00959549532$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 4 i q^{3} + 28 q^{4} -8 q^{6} + 192 i q^{7} + 120 i q^{8} + 227 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} + 4 i q^{3} + 28 q^{4} -8 q^{6} + 192 i q^{7} + 120 i q^{8} + 227 q^{9} -148 q^{11} + 112 i q^{12} -286 i q^{13} -384 q^{14} + 656 q^{16} -1678 i q^{17} + 454 i q^{18} -1060 q^{19} -768 q^{21} -296 i q^{22} -2976 i q^{23} -480 q^{24} + 572 q^{26} + 1880 i q^{27} + 5376 i q^{28} + 3410 q^{29} -2448 q^{31} + 5152 i q^{32} -592 i q^{33} + 3356 q^{34} + 6356 q^{36} + 182 i q^{37} -2120 i q^{38} + 1144 q^{39} -9398 q^{41} -1536 i q^{42} + 1244 i q^{43} -4144 q^{44} + 5952 q^{46} -12088 i q^{47} + 2624 i q^{48} -20057 q^{49} + 6712 q^{51} -8008 i q^{52} -23846 i q^{53} -3760 q^{54} -23040 q^{56} -4240 i q^{57} + 6820 i q^{58} + 20020 q^{59} + 32302 q^{61} -4896 i q^{62} + 43584 i q^{63} + 10688 q^{64} + 1184 q^{66} + 60972 i q^{67} -46984 i q^{68} + 11904 q^{69} -32648 q^{71} + 27240 i q^{72} + 38774 i q^{73} -364 q^{74} -29680 q^{76} -28416 i q^{77} + 2288 i q^{78} + 33360 q^{79} + 47641 q^{81} -18796 i q^{82} -16716 i q^{83} -21504 q^{84} -2488 q^{86} + 13640 i q^{87} -17760 i q^{88} -101370 q^{89} + 54912 q^{91} -83328 i q^{92} -9792 i q^{93} + 24176 q^{94} -20608 q^{96} -119038 i q^{97} -40114 i q^{98} -33596 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 56q^{4} - 16q^{6} + 454q^{9} + O(q^{10})$$ $$2q + 56q^{4} - 16q^{6} + 454q^{9} - 296q^{11} - 768q^{14} + 1312q^{16} - 2120q^{19} - 1536q^{21} - 960q^{24} + 1144q^{26} + 6820q^{29} - 4896q^{31} + 6712q^{34} + 12712q^{36} + 2288q^{39} - 18796q^{41} - 8288q^{44} + 11904q^{46} - 40114q^{49} + 13424q^{51} - 7520q^{54} - 46080q^{56} + 40040q^{59} + 64604q^{61} + 21376q^{64} + 2368q^{66} + 23808q^{69} - 65296q^{71} - 728q^{74} - 59360q^{76} + 66720q^{79} + 95282q^{81} - 43008q^{84} - 4976q^{86} - 202740q^{89} + 109824q^{91} + 48352q^{94} - 41216q^{96} - 67192q^{99} + O(q^{100})$$

## 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
2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 0
24.2 2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 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 25.6.b.a 2
3.b odd 2 1 225.6.b.e 2
4.b odd 2 1 400.6.c.j 2
5.b even 2 1 inner 25.6.b.a 2
5.c odd 4 1 5.6.a.a 1
5.c odd 4 1 25.6.a.a 1
15.d odd 2 1 225.6.b.e 2
15.e even 4 1 45.6.a.b 1
15.e even 4 1 225.6.a.f 1
20.d odd 2 1 400.6.c.j 2
20.e even 4 1 80.6.a.e 1
20.e even 4 1 400.6.a.g 1
35.f even 4 1 245.6.a.b 1
40.i odd 4 1 320.6.a.j 1
40.k even 4 1 320.6.a.g 1
55.e even 4 1 605.6.a.a 1
60.l odd 4 1 720.6.a.a 1
65.h odd 4 1 845.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 5.c odd 4 1
25.6.a.a 1 5.c odd 4 1
25.6.b.a 2 1.a even 1 1 trivial
25.6.b.a 2 5.b even 2 1 inner
45.6.a.b 1 15.e even 4 1
80.6.a.e 1 20.e even 4 1
225.6.a.f 1 15.e even 4 1
225.6.b.e 2 3.b odd 2 1
225.6.b.e 2 15.d odd 2 1
245.6.a.b 1 35.f even 4 1
320.6.a.g 1 40.k even 4 1
320.6.a.j 1 40.i odd 4 1
400.6.a.g 1 20.e even 4 1
400.6.c.j 2 4.b odd 2 1
400.6.c.j 2 20.d odd 2 1
605.6.a.a 1 55.e even 4 1
720.6.a.a 1 60.l odd 4 1
845.6.a.b 1 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 60 T^{2} + 1024 T^{4}$$
$3$ $$1 - 470 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 3250 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 148 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 660790 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 24030 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 1060 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 4016110 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 3410 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 2448 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 138654790 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 9398 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 292469350 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 312570270 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 267759270 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 20020 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 32302 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 1017334570 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 32648 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 2642720110 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 33360 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7598656630 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 101370 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 3004635070 T^{2} + 73742412689492826049 T^{4}$$