# Properties

 Label 25.10.b.a Level 25 Weight 10 Character orbit 25.b Analytic conductor 12.876 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8758959041$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{2} -114 i q^{3} + 448 q^{4} + 912 q^{6} -4242 i q^{7} + 7680 i q^{8} + 6687 q^{9} +O(q^{10})$$ $$q + 8 i q^{2} -114 i q^{3} + 448 q^{4} + 912 q^{6} -4242 i q^{7} + 7680 i q^{8} + 6687 q^{9} -46208 q^{11} -51072 i q^{12} -115934 i q^{13} + 33936 q^{14} + 167936 q^{16} -494842 i q^{17} + 53496 i q^{18} + 1008740 q^{19} -483588 q^{21} -369664 i q^{22} -532554 i q^{23} + 875520 q^{24} + 927472 q^{26} -3006180 i q^{27} -1900416 i q^{28} -4196390 q^{29} -3365028 q^{31} + 5275648 i q^{32} + 5267712 i q^{33} + 3958736 q^{34} + 2995776 q^{36} + 14931358 i q^{37} + 8069920 i q^{38} -13216476 q^{39} + 11056262 q^{41} -3868704 i q^{42} -6396794 i q^{43} -20701184 q^{44} + 4260432 q^{46} + 35559158 i q^{47} -19144704 i q^{48} + 22359043 q^{49} -56411988 q^{51} -51938432 i q^{52} + 39738586 i q^{53} + 24049440 q^{54} + 32578560 q^{56} -114996360 i q^{57} -33571120 i q^{58} + 85185620 q^{59} + 45748642 q^{61} -26920224 i q^{62} -28366254 i q^{63} + 43778048 q^{64} -42141696 q^{66} + 45286158 i q^{67} -221689216 i q^{68} -60711156 q^{69} -189967468 q^{71} + 51356160 i q^{72} + 412170946 i q^{73} -119450864 q^{74} + 451915520 q^{76} + 196014336 i q^{77} -105731808 i q^{78} -95040840 q^{79} -211084299 q^{81} + 88450096 i q^{82} + 261706326 i q^{83} -216647424 q^{84} + 51174352 q^{86} + 478388460 i q^{87} -354877440 i q^{88} + 19938630 q^{89} -491792028 q^{91} -238584192 i q^{92} + 383613192 i q^{93} -284473264 q^{94} + 601423872 q^{96} + 19503358 i q^{97} + 178872344 i q^{98} -308992896 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 896q^{4} + 1824q^{6} + 13374q^{9} + O(q^{10})$$ $$2q + 896q^{4} + 1824q^{6} + 13374q^{9} - 92416q^{11} + 67872q^{14} + 335872q^{16} + 2017480q^{19} - 967176q^{21} + 1751040q^{24} + 1854944q^{26} - 8392780q^{29} - 6730056q^{31} + 7917472q^{34} + 5991552q^{36} - 26432952q^{39} + 22112524q^{41} - 41402368q^{44} + 8520864q^{46} + 44718086q^{49} - 112823976q^{51} + 48098880q^{54} + 65157120q^{56} + 170371240q^{59} + 91497284q^{61} + 87556096q^{64} - 84283392q^{66} - 121422312q^{69} - 379934936q^{71} - 238901728q^{74} + 903831040q^{76} - 190081680q^{79} - 422168598q^{81} - 433294848q^{84} + 102348704q^{86} + 39877260q^{89} - 983584056q^{91} - 568946528q^{94} + 1202847744q^{96} - 617985792q^{99} + O(q^{100})$$

## Character values

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

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

## 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
8.00000i 114.000i 448.000 0 912.000 4242.00i 7680.00i 6687.00 0
24.2 8.00000i 114.000i 448.000 0 912.000 4242.00i 7680.00i 6687.00 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 25.10.b.a 2
3.b odd 2 1 225.10.b.d 2
4.b odd 2 1 400.10.c.e 2
5.b even 2 1 inner 25.10.b.a 2
5.c odd 4 1 5.10.a.a 1
5.c odd 4 1 25.10.a.a 1
15.d odd 2 1 225.10.b.d 2
15.e even 4 1 45.10.a.c 1
15.e even 4 1 225.10.a.b 1
20.d odd 2 1 400.10.c.e 2
20.e even 4 1 80.10.a.d 1
20.e even 4 1 400.10.a.c 1
35.f even 4 1 245.10.a.a 1
40.i odd 4 1 320.10.a.h 1
40.k even 4 1 320.10.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.a 1 5.c odd 4 1
25.10.a.a 1 5.c odd 4 1
25.10.b.a 2 1.a even 1 1 trivial
25.10.b.a 2 5.b even 2 1 inner
45.10.a.c 1 15.e even 4 1
80.10.a.d 1 20.e even 4 1
225.10.a.b 1 15.e even 4 1
225.10.b.d 2 3.b odd 2 1
225.10.b.d 2 15.d odd 2 1
245.10.a.a 1 35.f even 4 1
320.10.a.c 1 40.k even 4 1
320.10.a.h 1 40.i odd 4 1
400.10.a.c 1 20.e even 4 1
400.10.c.e 2 4.b odd 2 1
400.10.c.e 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 64$$ acting on $$S_{10}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 960 T^{2} + 262144 T^{4}$$
$3$ $$1 - 26370 T^{2} + 387420489 T^{4}$$
$5$ 1
$7$ $$1 - 62712650 T^{2} + 1628413597910449 T^{4}$$
$11$ $$( 1 + 46208 T + 2357947691 T^{2} )^{2}$$
$13$ $$1 - 7768306390 T^{2} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 7692851970 T^{2} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$( 1 - 1008740 T + 322687697779 T^{2} )^{2}$$
$23$ $$1 - 3318691560010 T^{2} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$( 1 + 4196390 T + 14507145975869 T^{2} )^{2}$$
$31$ $$( 1 + 3365028 T + 26439622160671 T^{2} )^{2}$$
$37$ $$1 - 36978027865990 T^{2} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$( 1 - 11056262 T + 327381934393961 T^{2} )^{2}$$
$43$ $$1 - 964266250395250 T^{2} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 973807228536570 T^{2} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 5020371966324870 T^{2} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$( 1 - 85185620 T + 8662995818654939 T^{2} )^{2}$$
$61$ $$( 1 - 45748642 T + 11694146092834141 T^{2} )^{2}$$
$67$ $$1 - 52362232686188930 T^{2} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$( 1 + 189967468 T + 45848500718449031 T^{2} )^{2}$$
$73$ $$1 + 52141715309999090 T^{2} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$( 1 + 95040840 T + 119851595982618319 T^{2} )^{2}$$
$83$ $$1 - 305390309466662530 T^{2} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$( 1 - 19938630 T + 350356403707485209 T^{2} )^{2}$$
$97$ $$1 - 1520081736335854270 T^{2} +$$$$57\!\cdots\!89$$$$T^{4}$$