# Properties

 Label 25.4.b.b 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} -7 i q^{3} + 7 q^{4} + 7 q^{6} + 6 i q^{7} + 15 i q^{8} -22 q^{9} +O(q^{10})$$ $$q + i q^{2} -7 i q^{3} + 7 q^{4} + 7 q^{6} + 6 i q^{7} + 15 i q^{8} -22 q^{9} -43 q^{11} -49 i q^{12} + 28 i q^{13} -6 q^{14} + 41 q^{16} + 91 i q^{17} -22 i q^{18} + 35 q^{19} + 42 q^{21} -43 i q^{22} -162 i q^{23} + 105 q^{24} -28 q^{26} -35 i q^{27} + 42 i q^{28} -160 q^{29} + 42 q^{31} + 161 i q^{32} + 301 i q^{33} -91 q^{34} -154 q^{36} -314 i q^{37} + 35 i q^{38} + 196 q^{39} -203 q^{41} + 42 i q^{42} -92 i q^{43} -301 q^{44} + 162 q^{46} + 196 i q^{47} -287 i q^{48} + 307 q^{49} + 637 q^{51} + 196 i q^{52} -82 i q^{53} + 35 q^{54} -90 q^{56} -245 i q^{57} -160 i q^{58} + 280 q^{59} -518 q^{61} + 42 i q^{62} -132 i q^{63} + 167 q^{64} -301 q^{66} + 141 i q^{67} + 637 i q^{68} -1134 q^{69} + 412 q^{71} -330 i q^{72} + 763 i q^{73} + 314 q^{74} + 245 q^{76} -258 i q^{77} + 196 i q^{78} -510 q^{79} -839 q^{81} -203 i q^{82} -777 i q^{83} + 294 q^{84} + 92 q^{86} + 1120 i q^{87} -645 i q^{88} + 945 q^{89} -168 q^{91} -1134 i q^{92} -294 i q^{93} -196 q^{94} + 1127 q^{96} + 1246 i q^{97} + 307 i q^{98} + 946 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{4} + 14q^{6} - 44q^{9} + O(q^{10})$$ $$2q + 14q^{4} + 14q^{6} - 44q^{9} - 86q^{11} - 12q^{14} + 82q^{16} + 70q^{19} + 84q^{21} + 210q^{24} - 56q^{26} - 320q^{29} + 84q^{31} - 182q^{34} - 308q^{36} + 392q^{39} - 406q^{41} - 602q^{44} + 324q^{46} + 614q^{49} + 1274q^{51} + 70q^{54} - 180q^{56} + 560q^{59} - 1036q^{61} + 334q^{64} - 602q^{66} - 2268q^{69} + 824q^{71} + 628q^{74} + 490q^{76} - 1020q^{79} - 1678q^{81} + 588q^{84} + 184q^{86} + 1890q^{89} - 336q^{91} - 392q^{94} + 2254q^{96} + 1892q^{99} + O(q^{100})$$

## 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
1.00000i 7.00000i 7.00000 0 7.00000 6.00000i 15.0000i −22.0000 0
24.2 1.00000i 7.00000i 7.00000 0 7.00000 6.00000i 15.0000i −22.0000 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.b 2
3.b odd 2 1 225.4.b.f 2
4.b odd 2 1 400.4.c.e 2
5.b even 2 1 inner 25.4.b.b 2
5.c odd 4 1 25.4.a.a 1
5.c odd 4 1 25.4.a.b yes 1
15.d odd 2 1 225.4.b.f 2
15.e even 4 1 225.4.a.c 1
15.e even 4 1 225.4.a.e 1
20.d odd 2 1 400.4.c.e 2
20.e even 4 1 400.4.a.c 1
20.e even 4 1 400.4.a.s 1
35.f even 4 1 1225.4.a.h 1
35.f even 4 1 1225.4.a.i 1
40.i odd 4 1 1600.4.a.i 1
40.i odd 4 1 1600.4.a.bt 1
40.k even 4 1 1600.4.a.h 1
40.k even 4 1 1600.4.a.bs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 5.c odd 4 1
25.4.a.b yes 1 5.c odd 4 1
25.4.b.b 2 1.a even 1 1 trivial
25.4.b.b 2 5.b even 2 1 inner
225.4.a.c 1 15.e even 4 1
225.4.a.e 1 15.e even 4 1
225.4.b.f 2 3.b odd 2 1
225.4.b.f 2 15.d odd 2 1
400.4.a.c 1 20.e even 4 1
400.4.a.s 1 20.e even 4 1
400.4.c.e 2 4.b odd 2 1
400.4.c.e 2 20.d odd 2 1
1225.4.a.h 1 35.f even 4 1
1225.4.a.i 1 35.f even 4 1
1600.4.a.h 1 40.k even 4 1
1600.4.a.i 1 40.i odd 4 1
1600.4.a.bs 1 40.k even 4 1
1600.4.a.bt 1 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 15 T^{2} + 64 T^{4}$$
$3$ $$1 - 5 T^{2} + 729 T^{4}$$
$5$ 1
$7$ $$1 - 650 T^{2} + 117649 T^{4}$$
$11$ $$( 1 + 43 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 3610 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 1545 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 - 35 T + 6859 T^{2} )^{2}$$
$23$ $$1 + 1910 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 + 160 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 - 42 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 2710 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 203 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 150550 T^{2} + 6321363049 T^{4}$$
$47$ $$1 - 169230 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 291030 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 - 280 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 + 518 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 581645 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 412 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 195865 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 510 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 539845 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 - 945 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 272830 T^{2} + 832972004929 T^{4}$$