# Properties

 Label 25.9.c.a Level $25$ Weight $9$ Character orbit 25.c Analytic conductor $10.184$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1844652515$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{141})$$ Defining polynomial: $$x^{4} + 71 x^{2} + 1225$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 282 q^{6} -231 \beta_{3} q^{7} + 230 \beta_{2} q^{8} -6279 \beta_{1} q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 282 q^{6} -231 \beta_{3} q^{7} + 230 \beta_{2} q^{8} -6279 \beta_{1} q^{9} + 12132 q^{11} -26 \beta_{3} q^{12} -204 \beta_{2} q^{13} -65142 \beta_{1} q^{14} + 71516 q^{16} + 6484 \beta_{3} q^{17} -6279 \beta_{2} q^{18} -168380 \beta_{1} q^{19} + 65142 q^{21} -12132 \beta_{3} q^{22} + 18641 \beta_{2} q^{23} + 64860 \beta_{1} q^{24} -57528 q^{26} -12840 \beta_{3} q^{27} -6006 \beta_{2} q^{28} + 666630 \beta_{1} q^{29} -1042808 q^{31} -12636 \beta_{3} q^{32} + 12132 \beta_{2} q^{33} + 1828488 \beta_{1} q^{34} -163254 q^{36} + 174864 \beta_{3} q^{37} -168380 \beta_{2} q^{38} -57528 \beta_{1} q^{39} -1321128 q^{41} -65142 \beta_{3} q^{42} + 234231 \beta_{2} q^{43} -315432 \beta_{1} q^{44} + 5256762 q^{46} -305721 \beta_{3} q^{47} + 71516 \beta_{2} q^{48} -9283001 \beta_{1} q^{49} -1828488 q^{51} + 5304 \beta_{3} q^{52} -264524 \beta_{2} q^{53} -3620880 \beta_{1} q^{54} + 14982660 q^{56} -168380 \beta_{3} q^{57} + 666630 \beta_{2} q^{58} -6498540 \beta_{1} q^{59} -14393968 q^{61} + 1042808 \beta_{3} q^{62} -1450449 \beta_{2} q^{63} + 14744744 \beta_{1} q^{64} + 3421224 q^{66} + 964809 \beta_{3} q^{67} + 168584 \beta_{2} q^{68} + 5256762 \beta_{1} q^{69} -23065488 q^{71} -1444170 \beta_{3} q^{72} + 1468116 \beta_{2} q^{73} + 49311648 \beta_{1} q^{74} -4377880 q^{76} -2802492 \beta_{3} q^{77} -57528 \beta_{2} q^{78} + 2760680 \beta_{1} q^{79} -37575639 q^{81} + 1321128 \beta_{3} q^{82} -970489 \beta_{2} q^{83} -1693692 \beta_{1} q^{84} + 66053142 q^{86} + 666630 \beta_{3} q^{87} + 2790360 \beta_{2} q^{88} -26130510 \beta_{1} q^{89} -13288968 q^{91} -484666 \beta_{3} q^{92} -1042808 \beta_{2} q^{93} -86213322 \beta_{1} q^{94} + 3563352 q^{96} + 6815604 \beta_{3} q^{97} -9283001 \beta_{2} q^{98} -76176828 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 1128q^{6} + O(q^{10})$$ $$4q + 1128q^{6} + 48528q^{11} + 286064q^{16} + 260568q^{21} - 230112q^{26} - 4171232q^{31} - 653016q^{36} - 5284512q^{41} + 21027048q^{46} - 7313952q^{51} + 59930640q^{56} - 57575872q^{61} + 13684896q^{66} - 92261952q^{71} - 17511520q^{76} - 150302556q^{81} + 264212568q^{86} - 53155872q^{91} + 14253408q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 71 x^{2} + 1225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 36 \nu$$$$)/35$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 70 \nu^{2} + 106 \nu + 2485$$$$)/35$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 70 \nu^{2} + 106 \nu - 2485$$$$)/35$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - 142$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$-9 \beta_{3} - 9 \beta_{2} + 53 \beta_{1}$$

## 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
 6.43717i − 5.43717i − 6.43717i 5.43717i
−11.8743 11.8743i −11.8743 + 11.8743i 26.0000i 0 282.000 −2742.97 2742.97i −2731.10 + 2731.10i 6279.00i 0
7.2 11.8743 + 11.8743i 11.8743 11.8743i 26.0000i 0 282.000 2742.97 + 2742.97i 2731.10 2731.10i 6279.00i 0
18.1 −11.8743 + 11.8743i −11.8743 11.8743i 26.0000i 0 282.000 −2742.97 + 2742.97i −2731.10 2731.10i 6279.00i 0
18.2 11.8743 11.8743i 11.8743 + 11.8743i 26.0000i 0 282.000 2742.97 2742.97i 2731.10 + 2731.10i 6279.00i 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 25.9.c.a 4
5.b even 2 1 inner 25.9.c.a 4
5.c odd 4 2 inner 25.9.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.9.c.a 4 1.a even 1 1 trivial
25.9.c.a 4 5.b even 2 1 inner
25.9.c.a 4 5.c odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 79524$$ acting on $$S_{9}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$79524 + T^{4}$$
$3$ $$79524 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$226436345031204 + T^{4}$$
$11$ $$( -12132 + T )^{4}$$
$13$ $$137726936146944 + T^{4}$$
$17$ $$14\!\cdots\!64$$$$+ T^{4}$$
$19$ $$( 28351824400 + T^{2} )^{2}$$
$23$ $$96\!\cdots\!64$$$$+ T^{4}$$
$29$ $$( 444395556900 + T^{2} )^{2}$$
$31$ $$( 1042808 + T )^{4}$$
$37$ $$74\!\cdots\!84$$$$+ T^{4}$$
$41$ $$( 1321128 + T )^{4}$$
$43$ $$23\!\cdots\!04$$$$+ T^{4}$$
$47$ $$69\!\cdots\!44$$$$+ T^{4}$$
$53$ $$38\!\cdots\!24$$$$+ T^{4}$$
$59$ $$( 42231022131600 + T^{2} )^{2}$$
$61$ $$( 14393968 + T )^{4}$$
$67$ $$68\!\cdots\!64$$$$+ T^{4}$$
$71$ $$( 23065488 + T )^{4}$$
$73$ $$36\!\cdots\!64$$$$+ T^{4}$$
$79$ $$( 7621354062400 + T^{2} )^{2}$$
$83$ $$70\!\cdots\!84$$$$+ T^{4}$$
$89$ $$( 682803552860100 + T^{2} )^{2}$$
$97$ $$17\!\cdots\!44$$$$+ T^{4}$$