# Properties

 Label 625.2.b.c Level $625$ Weight $2$ Character orbit 625.b Analytic conductor $4.991$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( \beta_{5} + \beta_{6} ) q^{3} + ( -1 - \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( \beta_{5} + \beta_{6} ) q^{3} + ( -1 - \beta_{3} ) q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{2} ) q^{9} + 2 q^{11} + ( -\beta_{4} + \beta_{6} + \beta_{7} ) q^{12} + \beta_{5} q^{13} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{14} + ( -1 + \beta_{2} - \beta_{3} ) q^{16} + ( -\beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{17} + ( \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{18} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{19} + ( \beta_{2} - \beta_{3} ) q^{21} -2 \beta_{4} q^{22} + ( \beta_{5} - \beta_{6} ) q^{23} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{24} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + ( 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{27} + ( \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{28} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{29} + ( 2 - \beta_{2} - 2 \beta_{3} ) q^{31} + ( \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{32} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{33} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{36} + ( \beta_{4} + \beta_{5} + 4 \beta_{7} ) q^{37} + ( \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{5} - \beta_{6} ) q^{42} + ( \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{43} + ( -2 - 2 \beta_{3} ) q^{44} + ( 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{46} + ( 3 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} + ( \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{48} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{49} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -2 \beta_{5} + \beta_{6} ) q^{52} + ( -3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{53} + ( 4 + \beta_{3} ) q^{54} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{56} + ( 3 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{57} + ( 2 \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{58} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{61} + ( \beta_{4} - \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{62} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{63} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{64} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{66} + ( 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{67} + ( -\beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{68} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{69} + ( 8 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{71} + ( 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{72} + ( -\beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{74} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{76} + ( -2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{78} + ( -2 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{81} + ( -3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{82} + ( -4 \beta_{4} - \beta_{5} - 5 \beta_{6} - 4 \beta_{7} ) q^{83} + ( 4 \beta_{1} - \beta_{3} ) q^{84} + ( -4 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{86} + ( 3 \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{87} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{88} + ( -1 - 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{92} + ( -4 \beta_{4} + 5 \beta_{5} + 7 \beta_{6} ) q^{93} + ( 8 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{94} + ( -8 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{96} + ( -4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{97} + ( -2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{98} + ( -2 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{4} + 6q^{6} - 4q^{9} + O(q^{10})$$ $$8q - 6q^{4} + 6q^{6} - 4q^{9} + 16q^{11} - 12q^{14} - 2q^{16} - 10q^{19} + 6q^{21} - 20q^{24} + 6q^{26} - 20q^{29} + 16q^{31} - 2q^{34} - 12q^{36} - 18q^{39} + 26q^{41} - 12q^{44} + 6q^{46} + 14q^{49} - 4q^{51} + 30q^{54} + 10q^{56} - 30q^{59} + 6q^{61} + 44q^{64} + 12q^{66} - 8q^{69} + 46q^{71} - 12q^{74} - 20q^{76} - 10q^{79} - 32q^{81} + 18q^{84} - 14q^{86} - 30q^{89} - 14q^{91} + 68q^{94} - 54q^{96} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-14 \nu^{7} + 62 \nu^{6} + 49 \nu^{5} + 28 \nu^{4} - 194 \nu^{3} - 180 \nu^{2} + 10 \nu + 2465$$$$)/1355$$ $$\beta_{2}$$ $$=$$ $$($$$$-45 \nu^{7} + 238 \nu^{6} - 249 \nu^{5} + 632 \nu^{4} - 701 \nu^{3} - 927 \nu^{2} + 1000 \nu + 2600$$$$)/1355$$ $$\beta_{3}$$ $$=$$ $$($$$$78 \nu^{7} - 268 \nu^{6} + 269 \nu^{5} - 427 \nu^{4} + 926 \nu^{3} + 306 \nu^{2} - 2185 \nu - 2700$$$$)/1355$$ $$\beta_{4}$$ $$=$$ $$($$$$163 \nu^{7} - 296 \nu^{6} + 378 \nu^{5} - 1139 \nu^{4} + 1407 \nu^{3} + 973 \nu^{2} + 1045 \nu - 3245$$$$)/1355$$ $$\beta_{5}$$ $$=$$ $$($$$$227 \nu^{7} - 502 \nu^{6} + 696 \nu^{5} - 1538 \nu^{4} + 2139 \nu^{3} + 1099 \nu^{2} + 1580 \nu - 4835$$$$)/1355$$ $$\beta_{6}$$ $$=$$ $$($$$$826 \nu^{7} - 1490 \nu^{6} + 1445 \nu^{5} - 3820 \nu^{4} + 4400 \nu^{3} + 10078 \nu^{2} + 2120 \nu - 18065$$$$)/1355$$ $$\beta_{7}$$ $$=$$ $$($$$$1442 \nu^{7} - 2592 \nu^{6} + 2541 \nu^{5} - 6678 \nu^{4} + 6974 \nu^{3} + 16914 \nu^{2} + 4390 \nu - 30320$$$$)/1355$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 5 \beta_{6} + \beta_{5} + \beta_{2} - \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} - 9 \beta_{4} + \beta_{3} + 4 \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 6 \beta_{4} + 10 \beta_{1} - 14$$ $$\nu^{6}$$ $$=$$ $$($$$$-8 \beta_{7} + 14 \beta_{6} - 9 \beta_{5} + 16 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} + 31 \beta_{1} - 52$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{7} - 11 \beta_{6} - 15 \beta_{5} + 23 \beta_{4} + 23 \beta_{3} + 38 \beta_{2} + 12 \beta_{1} - 15$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
624.1
 −0.357358 − 1.86824i −0.983224 + 0.644389i 1.66637 − 0.917186i 1.17421 + 0.0566033i 1.17421 − 0.0566033i 1.66637 + 0.917186i −0.983224 − 0.644389i −0.357358 + 1.86824i
2.30927i 0.474903i −3.33275 0 1.09668 3.03582i 3.07768i 2.77447 0
624.2 2.08529i 2.19849i −2.34841 0 4.58448 0.992398i 0.726543i −1.83337 0
624.3 1.13370i 2.60278i 0.714715 0 −2.95078 0.407162i 3.07768i −3.77447 0
624.4 0.183172i 1.47195i 1.96645 0 0.269620 3.26086i 0.726543i 0.833366 0
624.5 0.183172i 1.47195i 1.96645 0 0.269620 3.26086i 0.726543i 0.833366 0
624.6 1.13370i 2.60278i 0.714715 0 −2.95078 0.407162i 3.07768i −3.77447 0
624.7 2.08529i 2.19849i −2.34841 0 4.58448 0.992398i 0.726543i −1.83337 0
624.8 2.30927i 0.474903i −3.33275 0 1.09668 3.03582i 3.07768i 2.77447 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 624.8 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 625.2.b.c 8
5.b even 2 1 inner 625.2.b.c 8
5.c odd 4 2 625.2.a.f 8
15.e even 4 2 5625.2.a.x 8
20.e even 4 2 10000.2.a.bj 8
25.d even 5 1 25.2.e.a 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 25.2.e.a 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 4 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.d even 5 1
25.2.e.a 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 5.c odd 4 2
625.2.b.c 8 1.a even 1 1 trivial
625.2.b.c 8 5.b even 2 1 inner
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 15.e even 4 2
10000.2.a.bj 8 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{8} + 11 T_{2}^{6} + 36 T_{2}^{4} + 31 T_{2}^{2} + 1$$ $$T_{3}^{8} + 14 T_{3}^{6} + 61 T_{3}^{4} + 84 T_{3}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 31 T^{2} + 36 T^{4} + 11 T^{6} + T^{8}$$
$3$ $$16 + 84 T^{2} + 61 T^{4} + 14 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8}$$
$11$ $$( -2 + T )^{8}$$
$13$ $$1 + 14 T^{2} + 31 T^{4} + 14 T^{6} + T^{8}$$
$17$ $$1936 + 1636 T^{2} + 441 T^{4} + 41 T^{6} + T^{8}$$
$19$ $$( -20 - 30 T - 5 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$23$ $$256 + 544 T^{2} + 301 T^{4} + 34 T^{6} + T^{8}$$
$29$ $$( -695 - 290 T - 5 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$31$ $$( -44 + 328 T - 41 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$37$ $$116281 + 39331 T^{2} + 3556 T^{4} + 111 T^{6} + T^{8}$$
$41$ $$( 116 + 148 T + 19 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$43$ $$246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8}$$
$47$ $$65536 + 47616 T^{2} + 4661 T^{4} + 141 T^{6} + T^{8}$$
$53$ $$8755681 + 722619 T^{2} + 20356 T^{4} + 239 T^{6} + T^{8}$$
$59$ $$( -2020 - 630 T + 5 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$61$ $$( 341 - 237 T - 146 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$67$ $$246016 + 84736 T^{2} + 7776 T^{4} + 176 T^{6} + T^{8}$$
$71$ $$( -4924 + 798 T + 99 T^{2} - 23 T^{3} + T^{4} )^{2}$$
$73$ $$1 + 19 T^{2} + 76 T^{4} + 79 T^{6} + T^{8}$$
$79$ $$( 5780 - 195 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$83$ $$99856 + 138164 T^{2} + 35061 T^{4} + 374 T^{6} + T^{8}$$
$89$ $$( 1180 - 530 T - 35 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$97$ $$301334881 + 12038986 T^{2} + 146971 T^{4} + 666 T^{6} + T^{8}$$