# Properties

 Label 25.5.c.b 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{2} + 7 \beta_{3} q^{3} + 11 \beta_{2} q^{4} -63 q^{6} + 28 \beta_{1} q^{7} -15 \beta_{3} q^{8} -66 \beta_{2} q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{2} + 7 \beta_{3} q^{3} + 11 \beta_{2} q^{4} -63 q^{6} + 28 \beta_{1} q^{7} -15 \beta_{3} q^{8} -66 \beta_{2} q^{9} + 117 q^{11} -77 \beta_{1} q^{12} + 42 \beta_{3} q^{13} + 252 \beta_{2} q^{14} + 311 q^{16} -147 \beta_{1} q^{17} -198 \beta_{3} q^{18} -595 \beta_{2} q^{19} -588 q^{21} + 351 \beta_{1} q^{22} + 102 \beta_{3} q^{23} + 315 \beta_{2} q^{24} -378 q^{26} -105 \beta_{1} q^{27} + 308 \beta_{3} q^{28} + 1170 \beta_{2} q^{29} + 322 q^{31} + 693 \beta_{1} q^{32} + 819 \beta_{3} q^{33} -1323 \beta_{2} q^{34} + 726 q^{36} -472 \beta_{1} q^{37} -1785 \beta_{3} q^{38} -882 \beta_{2} q^{39} -63 q^{41} -1764 \beta_{1} q^{42} -1028 \beta_{3} q^{43} + 1287 \beta_{2} q^{44} -918 q^{46} + 378 \beta_{1} q^{47} + 2177 \beta_{3} q^{48} -49 \beta_{2} q^{49} + 3087 q^{51} -462 \beta_{1} q^{52} -1218 \beta_{3} q^{53} -945 \beta_{2} q^{54} + 1260 q^{56} + 4165 \beta_{1} q^{57} + 3510 \beta_{3} q^{58} + 1890 \beta_{2} q^{59} -5908 q^{61} + 966 \beta_{1} q^{62} -1848 \beta_{3} q^{63} + 1261 \beta_{2} q^{64} -7371 q^{66} -3897 \beta_{1} q^{67} -1617 \beta_{3} q^{68} -2142 \beta_{2} q^{69} + 2682 q^{71} -990 \beta_{1} q^{72} + 1477 \beta_{3} q^{73} -4248 \beta_{2} q^{74} + 6545 q^{76} + 3276 \beta_{1} q^{77} -2646 \beta_{3} q^{78} + 6520 \beta_{2} q^{79} + 7551 q^{81} -189 \beta_{1} q^{82} + 2037 \beta_{3} q^{83} -6468 \beta_{2} q^{84} + 9252 q^{86} -8190 \beta_{1} q^{87} -1755 \beta_{3} q^{88} + 5985 \beta_{2} q^{89} -3528 q^{91} -1122 \beta_{1} q^{92} + 2254 \beta_{3} q^{93} + 3402 \beta_{2} q^{94} -14553 q^{96} + 2828 \beta_{1} q^{97} -147 \beta_{3} q^{98} -7722 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 252q^{6} + O(q^{10})$$ $$4q - 252q^{6} + 468q^{11} + 1244q^{16} - 2352q^{21} - 1512q^{26} + 1288q^{31} + 2904q^{36} - 252q^{41} - 3672q^{46} + 12348q^{51} + 5040q^{56} - 23632q^{61} - 29484q^{66} + 10728q^{71} + 26180q^{76} + 30204q^{81} + 37008q^{86} - 14112q^{91} - 58212q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

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

## 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
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−3.67423 3.67423i 8.57321 8.57321i 11.0000i 0 −63.0000 −34.2929 34.2929i −18.3712 + 18.3712i 66.0000i 0
7.2 3.67423 + 3.67423i −8.57321 + 8.57321i 11.0000i 0 −63.0000 34.2929 + 34.2929i 18.3712 18.3712i 66.0000i 0
18.1 −3.67423 + 3.67423i 8.57321 + 8.57321i 11.0000i 0 −63.0000 −34.2929 + 34.2929i −18.3712 18.3712i 66.0000i 0
18.2 3.67423 3.67423i −8.57321 8.57321i 11.0000i 0 −63.0000 34.2929 34.2929i 18.3712 + 18.3712i 66.0000i 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.b 4
3.b odd 2 1 225.5.g.f 4
4.b odd 2 1 400.5.p.j 4
5.b even 2 1 inner 25.5.c.b 4
5.c odd 4 2 inner 25.5.c.b 4
15.d odd 2 1 225.5.g.f 4
15.e even 4 2 225.5.g.f 4
20.d odd 2 1 400.5.p.j 4
20.e even 4 2 400.5.p.j 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 729$$ acting on $$S_{5}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 487 T^{4} + 65536 T^{8}$$
$3$ $$1 - 12897 T^{4} + 43046721 T^{8}$$
$5$ 1
$7$ $$1 - 5527102 T^{4} + 33232930569601 T^{8}$$
$11$ $$( 1 - 117 T + 14641 T^{2} )^{4}$$
$13$ $$1 + 1054887458 T^{4} + 665416609183179841 T^{8}$$
$17$ $$1 - 3503608657 T^{4} + 48661191875666868481 T^{8}$$
$19$ $$( 1 + 93383 T^{2} + 16983563041 T^{4} )^{2}$$
$23$ $$1 + 122658570338 T^{4} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 45662 T^{2} + 500246412961 T^{4} )^{2}$$
$31$ $$( 1 - 322 T + 923521 T^{2} )^{4}$$
$37$ $$1 + 2461256293058 T^{4} +$$$$12\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 + 63 T + 2825761 T^{2} )^{4}$$
$43$ $$1 - 9927677992702 T^{4} +$$$$13\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 39439575780578 T^{4} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$1 + 3858356729378 T^{4} +$$$$38\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 20662622 T^{2} + 146830437604321 T^{4} )^{2}$$
$61$ $$( 1 + 5908 T + 13845841 T^{2} )^{4}$$
$67$ $$1 - 784493155081057 T^{4} +$$$$16\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 2682 T + 25411681 T^{2} )^{4}$$
$73$ $$1 + 912332767302863 T^{4} +$$$$65\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 - 35389762 T^{2} + 1517108809906561 T^{4} )^{2}$$
$83$ $$1 + 2296474800768143 T^{4} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 89664257 T^{2} + 3936588805702081 T^{4} )^{2}$$
$97$ $$1 + 7754275002202178 T^{4} +$$$$61\!\cdots\!21$$$$T^{8}$$