# Properties

 Label 25.6.b.a Level $25$ Weight $6$ Character orbit 25.b Analytic conductor $4.010$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.00959549532$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 \beta q^{3} + 28 q^{4} - 8 q^{6} + 96 \beta q^{7} + 60 \beta q^{8} + 227 q^{9}+O(q^{10})$$ q + b * q^2 + 2*b * q^3 + 28 * q^4 - 8 * q^6 + 96*b * q^7 + 60*b * q^8 + 227 * q^9 $$q + \beta q^{2} + 2 \beta q^{3} + 28 q^{4} - 8 q^{6} + 96 \beta q^{7} + 60 \beta q^{8} + 227 q^{9} - 148 q^{11} + 56 \beta q^{12} - 143 \beta q^{13} - 384 q^{14} + 656 q^{16} - 839 \beta q^{17} + 227 \beta q^{18} - 1060 q^{19} - 768 q^{21} - 148 \beta q^{22} - 1488 \beta q^{23} - 480 q^{24} + 572 q^{26} + 940 \beta q^{27} + 2688 \beta q^{28} + 3410 q^{29} - 2448 q^{31} + 2576 \beta q^{32} - 296 \beta q^{33} + 3356 q^{34} + 6356 q^{36} + 91 \beta q^{37} - 1060 \beta q^{38} + 1144 q^{39} - 9398 q^{41} - 768 \beta q^{42} + 622 \beta q^{43} - 4144 q^{44} + 5952 q^{46} - 6044 \beta q^{47} + 1312 \beta q^{48} - 20057 q^{49} + 6712 q^{51} - 4004 \beta q^{52} - 11923 \beta q^{53} - 3760 q^{54} - 23040 q^{56} - 2120 \beta q^{57} + 3410 \beta q^{58} + 20020 q^{59} + 32302 q^{61} - 2448 \beta q^{62} + 21792 \beta q^{63} + 10688 q^{64} + 1184 q^{66} + 30486 \beta q^{67} - 23492 \beta q^{68} + 11904 q^{69} - 32648 q^{71} + 13620 \beta q^{72} + 19387 \beta q^{73} - 364 q^{74} - 29680 q^{76} - 14208 \beta q^{77} + 1144 \beta q^{78} + 33360 q^{79} + 47641 q^{81} - 9398 \beta q^{82} - 8358 \beta q^{83} - 21504 q^{84} - 2488 q^{86} + 6820 \beta q^{87} - 8880 \beta q^{88} - 101370 q^{89} + 54912 q^{91} - 41664 \beta q^{92} - 4896 \beta q^{93} + 24176 q^{94} - 20608 q^{96} - 59519 \beta q^{97} - 20057 \beta q^{98} - 33596 q^{99} +O(q^{100})$$ q + b * q^2 + 2*b * q^3 + 28 * q^4 - 8 * q^6 + 96*b * q^7 + 60*b * q^8 + 227 * q^9 - 148 * q^11 + 56*b * q^12 - 143*b * q^13 - 384 * q^14 + 656 * q^16 - 839*b * q^17 + 227*b * q^18 - 1060 * q^19 - 768 * q^21 - 148*b * q^22 - 1488*b * q^23 - 480 * q^24 + 572 * q^26 + 940*b * q^27 + 2688*b * q^28 + 3410 * q^29 - 2448 * q^31 + 2576*b * q^32 - 296*b * q^33 + 3356 * q^34 + 6356 * q^36 + 91*b * q^37 - 1060*b * q^38 + 1144 * q^39 - 9398 * q^41 - 768*b * q^42 + 622*b * q^43 - 4144 * q^44 + 5952 * q^46 - 6044*b * q^47 + 1312*b * q^48 - 20057 * q^49 + 6712 * q^51 - 4004*b * q^52 - 11923*b * q^53 - 3760 * q^54 - 23040 * q^56 - 2120*b * q^57 + 3410*b * q^58 + 20020 * q^59 + 32302 * q^61 - 2448*b * q^62 + 21792*b * q^63 + 10688 * q^64 + 1184 * q^66 + 30486*b * q^67 - 23492*b * q^68 + 11904 * q^69 - 32648 * q^71 + 13620*b * q^72 + 19387*b * q^73 - 364 * q^74 - 29680 * q^76 - 14208*b * q^77 + 1144*b * q^78 + 33360 * q^79 + 47641 * q^81 - 9398*b * q^82 - 8358*b * q^83 - 21504 * q^84 - 2488 * q^86 + 6820*b * q^87 - 8880*b * q^88 - 101370 * q^89 + 54912 * q^91 - 41664*b * q^92 - 4896*b * q^93 + 24176 * q^94 - 20608 * q^96 - 59519*b * q^97 - 20057*b * q^98 - 33596 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 56 q^{4} - 16 q^{6} + 454 q^{9}+O(q^{10})$$ 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9 $$2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99}+O(q^{100})$$ 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9 - 296 * q^11 - 768 * q^14 + 1312 * q^16 - 2120 * q^19 - 1536 * q^21 - 960 * q^24 + 1144 * q^26 + 6820 * q^29 - 4896 * q^31 + 6712 * q^34 + 12712 * q^36 + 2288 * q^39 - 18796 * q^41 - 8288 * q^44 + 11904 * q^46 - 40114 * q^49 + 13424 * q^51 - 7520 * q^54 - 46080 * q^56 + 40040 * q^59 + 64604 * q^61 + 21376 * q^64 + 2368 * q^66 + 23808 * q^69 - 65296 * q^71 - 728 * q^74 - 59360 * q^76 + 66720 * q^79 + 95282 * q^81 - 43008 * q^84 - 4976 * q^86 - 202740 * q^89 + 109824 * q^91 + 48352 * q^94 - 41216 * q^96 - 67192 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 1.00000i 1.00000i
2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 0
24.2 2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.6.b.a 2
3.b odd 2 1 225.6.b.e 2
4.b odd 2 1 400.6.c.j 2
5.b even 2 1 inner 25.6.b.a 2
5.c odd 4 1 5.6.a.a 1
5.c odd 4 1 25.6.a.a 1
15.d odd 2 1 225.6.b.e 2
15.e even 4 1 45.6.a.b 1
15.e even 4 1 225.6.a.f 1
20.d odd 2 1 400.6.c.j 2
20.e even 4 1 80.6.a.e 1
20.e even 4 1 400.6.a.g 1
35.f even 4 1 245.6.a.b 1
40.i odd 4 1 320.6.a.j 1
40.k even 4 1 320.6.a.g 1
55.e even 4 1 605.6.a.a 1
60.l odd 4 1 720.6.a.a 1
65.h odd 4 1 845.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 5.c odd 4 1
25.6.a.a 1 5.c odd 4 1
25.6.b.a 2 1.a even 1 1 trivial
25.6.b.a 2 5.b even 2 1 inner
45.6.a.b 1 15.e even 4 1
80.6.a.e 1 20.e even 4 1
225.6.a.f 1 15.e even 4 1
225.6.b.e 2 3.b odd 2 1
225.6.b.e 2 15.d odd 2 1
245.6.a.b 1 35.f even 4 1
320.6.a.g 1 40.k even 4 1
320.6.a.j 1 40.i odd 4 1
400.6.a.g 1 20.e even 4 1
400.6.c.j 2 4.b odd 2 1
400.6.c.j 2 20.d odd 2 1
605.6.a.a 1 55.e even 4 1
720.6.a.a 1 60.l odd 4 1
845.6.a.b 1 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36864$$
$11$ $$(T + 148)^{2}$$
$13$ $$T^{2} + 81796$$
$17$ $$T^{2} + 2815684$$
$19$ $$(T + 1060)^{2}$$
$23$ $$T^{2} + 8856576$$
$29$ $$(T - 3410)^{2}$$
$31$ $$(T + 2448)^{2}$$
$37$ $$T^{2} + 33124$$
$41$ $$(T + 9398)^{2}$$
$43$ $$T^{2} + 1547536$$
$47$ $$T^{2} + 146119744$$
$53$ $$T^{2} + 568631716$$
$59$ $$(T - 20020)^{2}$$
$61$ $$(T - 32302)^{2}$$
$67$ $$T^{2} + 3717584784$$
$71$ $$(T + 32648)^{2}$$
$73$ $$T^{2} + 1503423076$$
$79$ $$(T - 33360)^{2}$$
$83$ $$T^{2} + 279424656$$
$89$ $$(T + 101370)^{2}$$
$97$ $$T^{2} + 14170045444$$
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