# Properties

 Label 25.2.d.a Level $25$ Weight $2$ Character orbit 25.d Analytic conductor $0.200$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.199626005053$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{3} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{5} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{3} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{5} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10} q^{9} + ( 2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( 2 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11} + ( 1 + \zeta_{10}^{2} ) q^{12} + ( 3 + 3 \zeta_{10}^{2} ) q^{13} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{14} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{15} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{16} + ( 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{17} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( 3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{20} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{21} + ( -4 + 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{22} + ( -7 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{23} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{24} + 5 \zeta_{10}^{2} q^{25} -3 q^{26} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{27} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{28} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{3} ) q^{29} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{30} -3 \zeta_{10}^{2} q^{31} + ( 5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{32} + ( 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{33} + ( 4 - 6 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{34} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{35} + ( -2 + 2 \zeta_{10}^{3} ) q^{36} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{37} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{38} + ( 3 - 3 \zeta_{10}^{3} ) q^{39} + 5 \zeta_{10}^{3} q^{40} + ( 2 + 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{41} -\zeta_{10}^{2} q^{42} + ( 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{43} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{45} + ( 5 - 5 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{46} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{47} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{48} + ( -5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{49} + ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{50} + ( -2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{51} + ( -6 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{52} + ( 4 - 4 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{53} + ( -5 + 5 \zeta_{10} ) q^{54} + ( 2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{55} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56} + ( -4 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{57} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{58} + ( -3 + 9 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{59} + ( -\zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{60} + ( 1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{61} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{62} + ( -2 - 2 \zeta_{10}^{2} ) q^{63} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -3 \zeta_{10} - 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{65} + ( 4 - 6 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{66} + ( 2 \zeta_{10} + 6 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{68} + ( 2 \zeta_{10} + 5 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{69} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{70} + ( 1 - \zeta_{10} + 5 \zeta_{10}^{3} ) q^{71} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{72} + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{73} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} + 5 q^{75} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{76} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{77} + 3 \zeta_{10}^{3} q^{78} + ( 5 - 5 \zeta_{10} ) q^{79} + ( -3 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{80} + \zeta_{10}^{2} q^{81} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{82} + ( 2 \zeta_{10} - 5 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{83} + ( -1 - \zeta_{10} - \zeta_{10}^{2} ) q^{84} + ( -6 + 6 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{85} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{86} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{87} + ( -6 + 8 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{88} + ( -4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{89} + ( -2 + 6 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{90} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{91} + ( -7 \zeta_{10} - 2 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{92} -3 q^{93} -\zeta_{10}^{2} q^{94} + ( -5 + 5 \zeta_{10} + 10 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} - 5 \zeta_{10}^{3} ) q^{96} + ( 3 - 3 \zeta_{10} - \zeta_{10}^{3} ) q^{97} + ( -1 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{98} + ( 4 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - q^{3} - 2q^{4} - 5q^{5} + 3q^{6} - 2q^{7} - 5q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - q^{3} - 2q^{4} - 5q^{5} + 3q^{6} - 2q^{7} - 5q^{8} + 2q^{9} + 10q^{10} - 2q^{11} + 3q^{12} + 9q^{13} + q^{14} - 5q^{15} - 6q^{16} + 8q^{17} + 4q^{18} - 5q^{19} + 5q^{20} - 2q^{21} - 14q^{22} - 11q^{23} - 5q^{25} - 12q^{26} + 5q^{27} + q^{28} + 5q^{29} - 5q^{30} + 3q^{31} + 18q^{32} + 8q^{33} + 6q^{34} + 5q^{35} - 6q^{36} - 7q^{37} + 5q^{38} + 9q^{39} + 5q^{40} + 8q^{41} + q^{42} - 6q^{43} + 6q^{44} - 10q^{45} + 13q^{46} - 2q^{47} - 6q^{48} - 22q^{49} - 10q^{50} - 12q^{51} - 12q^{52} + 9q^{53} - 15q^{54} + 20q^{55} - 10q^{57} - 10q^{58} + 13q^{61} + 6q^{62} - 6q^{63} + 3q^{64} + 6q^{66} - 2q^{67} - 4q^{68} - q^{69} + 8q^{71} + 10q^{72} + 9q^{73} + 6q^{74} + 20q^{75} + 20q^{76} - 4q^{77} + 3q^{78} + 15q^{79} - 15q^{80} - q^{81} - 4q^{82} + 9q^{83} - 4q^{84} - 20q^{85} + 3q^{86} - 10q^{88} - 20q^{89} - 12q^{91} - 12q^{92} - 12q^{93} + q^{94} - 5q^{95} - 2q^{96} + 8q^{97} + 11q^{98} + 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
−0.500000 1.53884i −0.809017 + 0.587785i −0.500000 + 0.363271i −0.690983 + 2.12663i 1.30902 + 0.951057i 0.618034 −1.80902 1.31433i −0.618034 + 1.90211i 3.61803
11.1 −0.500000 + 0.363271i 0.309017 0.951057i −0.500000 + 1.53884i −1.80902 1.31433i 0.190983 + 0.587785i −1.61803 −0.690983 2.12663i 1.61803 + 1.17557i 1.38197
16.1 −0.500000 0.363271i 0.309017 + 0.951057i −0.500000 1.53884i −1.80902 + 1.31433i 0.190983 0.587785i −1.61803 −0.690983 + 2.12663i 1.61803 1.17557i 1.38197
21.1 −0.500000 + 1.53884i −0.809017 0.587785i −0.500000 0.363271i −0.690983 2.12663i 1.30902 0.951057i 0.618034 −1.80902 + 1.31433i −0.618034 1.90211i 3.61803
 $$n$$: 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.2.d.a 4
3.b odd 2 1 225.2.h.b 4
4.b odd 2 1 400.2.u.b 4
5.b even 2 1 125.2.d.a 4
5.c odd 4 2 125.2.e.a 8
25.d even 5 1 inner 25.2.d.a 4
25.d even 5 1 625.2.a.b 2
25.d even 5 2 625.2.d.h 4
25.e even 10 1 125.2.d.a 4
25.e even 10 1 625.2.a.c 2
25.e even 10 2 625.2.d.b 4
25.f odd 20 2 125.2.e.a 8
25.f odd 20 2 625.2.b.a 4
25.f odd 20 4 625.2.e.c 8
75.h odd 10 1 5625.2.a.d 2
75.j odd 10 1 225.2.h.b 4
75.j odd 10 1 5625.2.a.f 2
100.h odd 10 1 10000.2.a.l 2
100.j odd 10 1 400.2.u.b 4
100.j odd 10 1 10000.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 1.a even 1 1 trivial
25.2.d.a 4 25.d even 5 1 inner
125.2.d.a 4 5.b even 2 1
125.2.d.a 4 25.e even 10 1
125.2.e.a 8 5.c odd 4 2
125.2.e.a 8 25.f odd 20 2
225.2.h.b 4 3.b odd 2 1
225.2.h.b 4 75.j odd 10 1
400.2.u.b 4 4.b odd 2 1
400.2.u.b 4 100.j odd 10 1
625.2.a.b 2 25.d even 5 1
625.2.a.c 2 25.e even 10 1
625.2.b.a 4 25.f odd 20 2
625.2.d.b 4 25.e even 10 2
625.2.d.h 4 25.d even 5 2
625.2.e.c 8 25.f odd 20 4
5625.2.a.d 2 75.h odd 10 1
5625.2.a.f 2 75.j odd 10 1
10000.2.a.c 2 100.j odd 10 1
10000.2.a.l 2 100.h odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ $$1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 15 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8}$$
$5$ $$1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T + 13 T^{2} + 7 T^{3} + 49 T^{4} )^{2}$$
$11$ $$1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 195 T^{5} + 3887 T^{6} - 19773 T^{7} + 28561 T^{8}$$
$17$ $$1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 1870 T^{5} + 2023 T^{6} - 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8}$$
$23$ $$1 + 11 T + 28 T^{2} - 245 T^{3} - 2259 T^{4} - 5635 T^{5} + 14812 T^{6} + 133837 T^{7} + 279841 T^{8}$$
$29$ $$1 - 5 T - 19 T^{2} + 145 T^{3} - 4 T^{4} + 4205 T^{5} - 15979 T^{6} - 121945 T^{7} + 707281 T^{8}$$
$31$ $$1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 4929 T^{5} - 21142 T^{6} - 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 5365 T^{5} - 24642 T^{6} + 354571 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 10414 T^{5} - 28577 T^{6} - 551368 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 2 T - 43 T^{2} + 50 T^{3} + 2351 T^{4} + 2350 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 9 T + 8 T^{2} - 315 T^{3} + 5131 T^{4} - 16695 T^{5} + 22472 T^{6} - 1339893 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 31 T^{2} + 210 T^{3} + 2851 T^{4} + 12390 T^{5} + 107911 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 57401 T^{5} + 290238 T^{6} - 2950753 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 27470 T^{5} - 13467 T^{6} + 601526 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 49274 T^{5} - 186517 T^{6} - 2863288 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 42705 T^{5} + 42632 T^{6} - 3501153 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 11455 T^{5} + 131061 T^{6} - 7395585 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 9 T - 52 T^{2} + 675 T^{3} + 121 T^{4} + 56025 T^{5} - 358228 T^{6} - 5146083 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 142400 T^{5} + 1196071 T^{6} + 14099380 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 1940 T^{5} - 592767 T^{6} - 7301384 T^{7} + 88529281 T^{8}$$