# Properties

 Label 25.5.c.a Level $25$ Weight $5$ Character orbit 25.c Analytic conductor $2.584$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.58424907710$$ 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: $$1$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + ( 6 - 6 i ) q^{3} -14 i q^{4} + 12 q^{6} + ( 26 + 26 i ) q^{7} + ( 30 - 30 i ) q^{8} + 9 i q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + ( 6 - 6 i ) q^{3} -14 i q^{4} + 12 q^{6} + ( 26 + 26 i ) q^{7} + ( 30 - 30 i ) q^{8} + 9 i q^{9} -8 q^{11} + ( -84 - 84 i ) q^{12} + ( -139 + 139 i ) q^{13} + 52 i q^{14} -164 q^{16} + ( 1 + i ) q^{17} + ( -9 + 9 i ) q^{18} + 180 i q^{19} + 312 q^{21} + ( -8 - 8 i ) q^{22} + ( 166 - 166 i ) q^{23} -360 i q^{24} -278 q^{26} + ( 540 + 540 i ) q^{27} + ( 364 - 364 i ) q^{28} -480 i q^{29} + 572 q^{31} + ( -644 - 644 i ) q^{32} + ( -48 + 48 i ) q^{33} + 2 i q^{34} + 126 q^{36} + ( 251 + 251 i ) q^{37} + ( -180 + 180 i ) q^{38} + 1668 i q^{39} -1688 q^{41} + ( 312 + 312 i ) q^{42} + ( -1474 + 1474 i ) q^{43} + 112 i q^{44} + 332 q^{46} + ( -2474 - 2474 i ) q^{47} + ( -984 + 984 i ) q^{48} -1049 i q^{49} + 12 q^{51} + ( 1946 + 1946 i ) q^{52} + ( 3331 - 3331 i ) q^{53} + 1080 i q^{54} + 1560 q^{56} + ( 1080 + 1080 i ) q^{57} + ( 480 - 480 i ) q^{58} -3660 i q^{59} + 1592 q^{61} + ( 572 + 572 i ) q^{62} + ( -234 + 234 i ) q^{63} + 1336 i q^{64} -96 q^{66} + ( -874 - 874 i ) q^{67} + ( 14 - 14 i ) q^{68} -1992 i q^{69} -6068 q^{71} + ( 270 + 270 i ) q^{72} + ( 791 - 791 i ) q^{73} + 502 i q^{74} + 2520 q^{76} + ( -208 - 208 i ) q^{77} + ( -1668 + 1668 i ) q^{78} + 9120 i q^{79} + 5751 q^{81} + ( -1688 - 1688 i ) q^{82} + ( -5654 + 5654 i ) q^{83} -4368 i q^{84} -2948 q^{86} + ( -2880 - 2880 i ) q^{87} + ( -240 + 240 i ) q^{88} + 2160 i q^{89} -7228 q^{91} + ( -2324 - 2324 i ) q^{92} + ( 3432 - 3432 i ) q^{93} -4948 i q^{94} -7728 q^{96} + ( 6551 + 6551 i ) q^{97} + ( 1049 - 1049 i ) q^{98} -72 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 12q^{3} + 24q^{6} + 52q^{7} + 60q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 12q^{3} + 24q^{6} + 52q^{7} + 60q^{8} - 16q^{11} - 168q^{12} - 278q^{13} - 328q^{16} + 2q^{17} - 18q^{18} + 624q^{21} - 16q^{22} + 332q^{23} - 556q^{26} + 1080q^{27} + 728q^{28} + 1144q^{31} - 1288q^{32} - 96q^{33} + 252q^{36} + 502q^{37} - 360q^{38} - 3376q^{41} + 624q^{42} - 2948q^{43} + 664q^{46} - 4948q^{47} - 1968q^{48} + 24q^{51} + 3892q^{52} + 6662q^{53} + 3120q^{56} + 2160q^{57} + 960q^{58} + 3184q^{61} + 1144q^{62} - 468q^{63} - 192q^{66} - 1748q^{67} + 28q^{68} - 12136q^{71} + 540q^{72} + 1582q^{73} + 5040q^{76} - 416q^{77} - 3336q^{78} + 11502q^{81} - 3376q^{82} - 11308q^{83} - 5896q^{86} - 5760q^{87} - 480q^{88} - 14456q^{91} - 4648q^{92} + 6864q^{93} - 15456q^{96} + 13102q^{97} + 2098q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.00000i − 1.00000i
1.00000 + 1.00000i 6.00000 6.00000i 14.0000i 0 12.0000 26.0000 + 26.0000i 30.0000 30.0000i 9.00000i 0
18.1 1.00000 1.00000i 6.00000 + 6.00000i 14.0000i 0 12.0000 26.0000 26.0000i 30.0000 + 30.0000i 9.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.5.c.a 2
3.b odd 2 1 225.5.g.b 2
4.b odd 2 1 400.5.p.a 2
5.b even 2 1 5.5.c.a 2
5.c odd 4 1 5.5.c.a 2
5.c odd 4 1 inner 25.5.c.a 2
15.d odd 2 1 45.5.g.b 2
15.e even 4 1 45.5.g.b 2
15.e even 4 1 225.5.g.b 2
20.d odd 2 1 80.5.p.d 2
20.e even 4 1 80.5.p.d 2
20.e even 4 1 400.5.p.a 2
40.e odd 2 1 320.5.p.c 2
40.f even 2 1 320.5.p.h 2
40.i odd 4 1 320.5.p.h 2
40.k even 4 1 320.5.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.5.c.a 2 5.b even 2 1
5.5.c.a 2 5.c odd 4 1
25.5.c.a 2 1.a even 1 1 trivial
25.5.c.a 2 5.c odd 4 1 inner
45.5.g.b 2 15.d odd 2 1
45.5.g.b 2 15.e even 4 1
80.5.p.d 2 20.d odd 2 1
80.5.p.d 2 20.e even 4 1
225.5.g.b 2 3.b odd 2 1
225.5.g.b 2 15.e even 4 1
320.5.p.c 2 40.e odd 2 1
320.5.p.c 2 40.k even 4 1
320.5.p.h 2 40.f even 2 1
320.5.p.h 2 40.i odd 4 1
400.5.p.a 2 4.b odd 2 1
400.5.p.a 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} + 2$$ acting on $$S_{5}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 32 T^{3} + 256 T^{4}$$
$3$ $$1 - 12 T + 72 T^{2} - 972 T^{3} + 6561 T^{4}$$
$5$ 1
$7$ $$1 - 52 T + 1352 T^{2} - 124852 T^{3} + 5764801 T^{4}$$
$11$ $$( 1 + 8 T + 14641 T^{2} )^{2}$$
$13$ $$1 + 278 T + 38642 T^{2} + 7939958 T^{3} + 815730721 T^{4}$$
$17$ $$1 - 2 T + 2 T^{2} - 167042 T^{3} + 6975757441 T^{4}$$
$19$ $$1 - 228242 T^{2} + 16983563041 T^{4}$$
$23$ $$1 - 332 T + 55112 T^{2} - 92907212 T^{3} + 78310985281 T^{4}$$
$29$ $$1 - 1184162 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 - 572 T + 923521 T^{2} )^{2}$$
$37$ $$1 - 502 T + 126002 T^{2} - 940828822 T^{3} + 3512479453921 T^{4}$$
$41$ $$( 1 + 1688 T + 2825761 T^{2} )^{2}$$
$43$ $$1 + 2948 T + 4345352 T^{2} + 10078625348 T^{3} + 11688200277601 T^{4}$$
$47$ $$1 + 4948 T + 12241352 T^{2} + 24144661588 T^{3} + 23811286661761 T^{4}$$
$53$ $$1 - 6662 T + 22191122 T^{2} - 52566384422 T^{3} + 62259690411361 T^{4}$$
$59$ $$1 - 10839122 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 1592 T + 13845841 T^{2} )^{2}$$
$67$ $$1 + 1748 T + 1527752 T^{2} + 35224159508 T^{3} + 406067677556641 T^{4}$$
$71$ $$( 1 + 6068 T + 25411681 T^{2} )^{2}$$
$73$ $$1 - 1582 T + 1251362 T^{2} - 44926017262 T^{3} + 806460091894081 T^{4}$$
$79$ $$1 + 5274238 T^{2} + 1517108809906561 T^{4}$$
$83$ $$1 + 11308 T + 63935432 T^{2} + 536658693868 T^{3} + 2252292232139041 T^{4}$$
$89$ $$1 - 120818882 T^{2} + 3936588805702081 T^{4}$$
$97$ $$1 - 13102 T + 85831202 T^{2} - 1159910639662 T^{3} + 7837433594376961 T^{4}$$