# Properties

 Label 25.4.b.a Level $25$ Weight $4$ Character orbit 25.b Analytic conductor $1.475$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47504775014$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 2 i q^{3} -8 q^{4} -8 q^{6} -6 i q^{7} + 23 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 2 i q^{3} -8 q^{4} -8 q^{6} -6 i q^{7} + 23 q^{9} + 32 q^{11} -16 i q^{12} -38 i q^{13} + 24 q^{14} -64 q^{16} -26 i q^{17} + 92 i q^{18} -100 q^{19} + 12 q^{21} + 128 i q^{22} -78 i q^{23} + 152 q^{26} + 100 i q^{27} + 48 i q^{28} + 50 q^{29} -108 q^{31} -256 i q^{32} + 64 i q^{33} + 104 q^{34} -184 q^{36} -266 i q^{37} -400 i q^{38} + 76 q^{39} + 22 q^{41} + 48 i q^{42} + 442 i q^{43} -256 q^{44} + 312 q^{46} + 514 i q^{47} -128 i q^{48} + 307 q^{49} + 52 q^{51} + 304 i q^{52} + 2 i q^{53} -400 q^{54} -200 i q^{57} + 200 i q^{58} -500 q^{59} -518 q^{61} -432 i q^{62} -138 i q^{63} + 512 q^{64} -256 q^{66} -126 i q^{67} + 208 i q^{68} + 156 q^{69} + 412 q^{71} -878 i q^{73} + 1064 q^{74} + 800 q^{76} -192 i q^{77} + 304 i q^{78} -600 q^{79} + 421 q^{81} + 88 i q^{82} + 282 i q^{83} -96 q^{84} -1768 q^{86} + 100 i q^{87} + 150 q^{89} -228 q^{91} + 624 i q^{92} -216 i q^{93} -2056 q^{94} + 512 q^{96} -386 i q^{97} + 1228 i q^{98} + 736 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{4} - 16q^{6} + 46q^{9} + O(q^{10})$$ $$2q - 16q^{4} - 16q^{6} + 46q^{9} + 64q^{11} + 48q^{14} - 128q^{16} - 200q^{19} + 24q^{21} + 304q^{26} + 100q^{29} - 216q^{31} + 208q^{34} - 368q^{36} + 152q^{39} + 44q^{41} - 512q^{44} + 624q^{46} + 614q^{49} + 104q^{51} - 800q^{54} - 1000q^{59} - 1036q^{61} + 1024q^{64} - 512q^{66} + 312q^{69} + 824q^{71} + 2128q^{74} + 1600q^{76} - 1200q^{79} + 842q^{81} - 192q^{84} - 3536q^{86} + 300q^{89} - 456q^{91} - 4112q^{94} + 1024q^{96} + 1472q^{99} + O(q^{100})$$

## 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
4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 0
24.2 4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 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 25.4.b.a 2
3.b odd 2 1 225.4.b.c 2
4.b odd 2 1 400.4.c.k 2
5.b even 2 1 inner 25.4.b.a 2
5.c odd 4 1 5.4.a.a 1
5.c odd 4 1 25.4.a.c 1
15.d odd 2 1 225.4.b.c 2
15.e even 4 1 45.4.a.d 1
15.e even 4 1 225.4.a.b 1
20.d odd 2 1 400.4.c.k 2
20.e even 4 1 80.4.a.d 1
20.e even 4 1 400.4.a.m 1
35.f even 4 1 245.4.a.a 1
35.f even 4 1 1225.4.a.k 1
35.k even 12 2 245.4.e.g 2
35.l odd 12 2 245.4.e.f 2
40.i odd 4 1 320.4.a.g 1
40.i odd 4 1 1600.4.a.bi 1
40.k even 4 1 320.4.a.h 1
40.k even 4 1 1600.4.a.s 1
45.k odd 12 2 405.4.e.l 2
45.l even 12 2 405.4.e.c 2
55.e even 4 1 605.4.a.d 1
60.l odd 4 1 720.4.a.u 1
65.h odd 4 1 845.4.a.b 1
80.i odd 4 1 1280.4.d.e 2
80.j even 4 1 1280.4.d.l 2
80.s even 4 1 1280.4.d.l 2
80.t odd 4 1 1280.4.d.e 2
85.g odd 4 1 1445.4.a.a 1
95.g even 4 1 1805.4.a.h 1
105.k odd 4 1 2205.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 5.c odd 4 1
25.4.a.c 1 5.c odd 4 1
25.4.b.a 2 1.a even 1 1 trivial
25.4.b.a 2 5.b even 2 1 inner
45.4.a.d 1 15.e even 4 1
80.4.a.d 1 20.e even 4 1
225.4.a.b 1 15.e even 4 1
225.4.b.c 2 3.b odd 2 1
225.4.b.c 2 15.d odd 2 1
245.4.a.a 1 35.f even 4 1
245.4.e.f 2 35.l odd 12 2
245.4.e.g 2 35.k even 12 2
320.4.a.g 1 40.i odd 4 1
320.4.a.h 1 40.k even 4 1
400.4.a.m 1 20.e even 4 1
400.4.c.k 2 4.b odd 2 1
400.4.c.k 2 20.d odd 2 1
405.4.e.c 2 45.l even 12 2
405.4.e.l 2 45.k odd 12 2
605.4.a.d 1 55.e even 4 1
720.4.a.u 1 60.l odd 4 1
845.4.a.b 1 65.h odd 4 1
1225.4.a.k 1 35.f even 4 1
1280.4.d.e 2 80.i odd 4 1
1280.4.d.e 2 80.t odd 4 1
1280.4.d.l 2 80.j even 4 1
1280.4.d.l 2 80.s even 4 1
1445.4.a.a 1 85.g odd 4 1
1600.4.a.s 1 40.k even 4 1
1600.4.a.bi 1 40.i odd 4 1
1805.4.a.h 1 95.g even 4 1
2205.4.a.q 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 4 T + 8 T^{2} )( 1 + 4 T + 8 T^{2} )$$
$3$ $$1 - 50 T^{2} + 729 T^{4}$$
$5$ 1
$7$ $$1 - 650 T^{2} + 117649 T^{4}$$
$11$ $$( 1 - 32 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 2950 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 9150 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 + 100 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 18250 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 - 50 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 108 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 30550 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 - 22 T + 68921 T^{2} )^{2}$$
$43$ $$1 + 36350 T^{2} + 6321363049 T^{4}$$
$47$ $$1 + 56550 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 297750 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 + 500 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 + 518 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 585650 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 412 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 7150 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 600 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1064050 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 - 150 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1676350 T^{2} + 832972004929 T^{4}$$