# Properties

 Label 25.5.c.c Level 25 Weight 5 Character orbit 25.c Analytic conductor 2.584 Analytic rank 0 Dimension 4 CM no Inner twists 4

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} -10 \beta_{2} q^{8} -39 \beta_{1} q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} -10 \beta_{2} q^{8} -39 \beta_{1} q^{9} -108 q^{11} -26 \beta_{3} q^{12} + 36 \beta_{2} q^{13} + 378 \beta_{1} q^{14} -4 q^{16} + 4 \beta_{3} q^{17} -39 \beta_{2} q^{18} -140 \beta_{1} q^{19} -378 q^{21} + 108 \beta_{3} q^{22} -79 \beta_{2} q^{23} -420 \beta_{1} q^{24} + 1512 q^{26} -120 \beta_{3} q^{27} + 234 \beta_{2} q^{28} -810 \beta_{1} q^{29} -728 q^{31} -156 \beta_{3} q^{32} -108 \beta_{2} q^{33} + 168 \beta_{1} q^{34} -1014 q^{36} + 144 \beta_{3} q^{37} -140 \beta_{2} q^{38} + 1512 \beta_{1} q^{39} + 1512 q^{41} + 378 \beta_{3} q^{42} -9 \beta_{2} q^{43} + 2808 \beta_{1} q^{44} -3318 q^{46} + 39 \beta_{3} q^{47} -4 \beta_{2} q^{48} -1001 \beta_{1} q^{49} -168 q^{51} -936 \beta_{3} q^{52} + 676 \beta_{2} q^{53} -5040 \beta_{1} q^{54} + 3780 q^{56} -140 \beta_{3} q^{57} -810 \beta_{2} q^{58} + 3780 \beta_{1} q^{59} + 4592 q^{61} + 728 \beta_{3} q^{62} + 351 \beta_{2} q^{63} -6616 \beta_{1} q^{64} -4536 q^{66} + 729 \beta_{3} q^{67} + 104 \beta_{2} q^{68} -3318 \beta_{1} q^{69} + 432 q^{71} + 390 \beta_{3} q^{72} -1404 \beta_{2} q^{73} + 6048 \beta_{1} q^{74} -3640 q^{76} -972 \beta_{3} q^{77} + 1512 \beta_{2} q^{78} + 8840 \beta_{1} q^{79} + 1881 q^{81} -1512 \beta_{3} q^{82} -169 \beta_{2} q^{83} + 9828 \beta_{1} q^{84} -378 q^{86} -810 \beta_{3} q^{87} + 1080 \beta_{2} q^{88} -13230 \beta_{1} q^{89} -13608 q^{91} + 2054 \beta_{3} q^{92} -728 \beta_{2} q^{93} + 1638 \beta_{1} q^{94} + 6552 q^{96} + 1764 \beta_{3} q^{97} -1001 \beta_{2} q^{98} + 4212 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 168q^{6} + O(q^{10})$$ $$4q + 168q^{6} - 432q^{11} - 16q^{16} - 1512q^{21} + 6048q^{26} - 2912q^{31} - 4056q^{36} + 6048q^{41} - 13272q^{46} - 672q^{51} + 15120q^{56} + 18368q^{61} - 18144q^{66} + 1728q^{71} - 14560q^{76} + 7524q^{81} - 1512q^{86} - 54432q^{91} + 26208q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 6 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu^{2} + 16 \nu + 55$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 10 \nu^{2} + 16 \nu - 55$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - 22$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} - 3 \beta_{2} + 16 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.79129i − 1.79129i − 2.79129i 1.79129i
−4.58258 4.58258i −4.58258 + 4.58258i 26.0000i 0 42.0000 41.2432 + 41.2432i 45.8258 45.8258i 39.0000i 0
7.2 4.58258 + 4.58258i 4.58258 4.58258i 26.0000i 0 42.0000 −41.2432 41.2432i −45.8258 + 45.8258i 39.0000i 0
18.1 −4.58258 + 4.58258i −4.58258 4.58258i 26.0000i 0 42.0000 41.2432 41.2432i 45.8258 + 45.8258i 39.0000i 0
18.2 4.58258 4.58258i 4.58258 + 4.58258i 26.0000i 0 42.0000 −41.2432 + 41.2432i −45.8258 45.8258i 39.0000i 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
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.5.c.c 4
3.b odd 2 1 225.5.g.j 4
4.b odd 2 1 400.5.p.l 4
5.b even 2 1 inner 25.5.c.c 4
5.c odd 4 2 inner 25.5.c.c 4
15.d odd 2 1 225.5.g.j 4
15.e even 4 2 225.5.g.j 4
20.d odd 2 1 400.5.p.l 4
20.e even 4 2 400.5.p.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.c.c 4 1.a even 1 1 trivial
25.5.c.c 4 5.b even 2 1 inner
25.5.c.c 4 5.c odd 4 2 inner
225.5.g.j 4 3.b odd 2 1
225.5.g.j 4 15.d odd 2 1
225.5.g.j 4 15.e even 4 2
400.5.p.l 4 4.b odd 2 1
400.5.p.l 4 20.d odd 2 1
400.5.p.l 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 1764$$ acting on $$S_{5}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 412 T^{4} + 65536 T^{8}$$
$3$ $$1 + 1278 T^{4} + 43046721 T^{8}$$
$5$ 1
$7$ $$1 - 9569602 T^{4} + 33232930569601 T^{8}$$
$11$ $$( 1 + 108 T + 14641 T^{2} )^{4}$$
$13$ $$1 - 1624225342 T^{4} + 665416609183179841 T^{8}$$
$17$ $$1 + 13727462018 T^{4} + 48661191875666868481 T^{8}$$
$19$ $$( 1 - 241042 T^{2} + 16983563041 T^{4} )^{2}$$
$23$ $$1 - 68080016962 T^{4} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 758462 T^{2} + 500246412961 T^{4} )^{2}$$
$31$ $$( 1 + 728 T + 923521 T^{2} )^{4}$$
$37$ $$1 + 1254529400258 T^{4} +$$$$12\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 - 1512 T + 2825761 T^{2} )^{4}$$
$43$ $$1 + 23329889084798 T^{4} +$$$$13\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 46379759106878 T^{4} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 112877432101822 T^{4} +$$$$38\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 9946322 T^{2} + 146830437604321 T^{4} )^{2}$$
$61$ $$( 1 - 4592 T + 13845841 T^{2} )^{4}$$
$67$ $$1 - 488793100954882 T^{4} +$$$$16\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 432 T + 25411681 T^{2} )^{4}$$
$73$ $$1 - 937201474520062 T^{4} +$$$$65\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 245438 T^{2} + 1517108809906561 T^{4} )^{2}$$
$83$ $$1 + 4278306619448318 T^{4} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 49548418 T^{2} + 3936588805702081 T^{4} )^{2}$$
$97$ $$1 - 13524937897425022 T^{4} +$$$$61\!\cdots\!21$$$$T^{8}$$