# Properties

 Label 25.8.b.c Level 25 Weight 8 Character orbit 25.b Analytic conductor 7.810 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.80962563710$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -\beta_{1} + 8 \beta_{3} ) q^{3} + ( -48 - 2 \beta_{2} ) q^{4} + ( -508 - 7 \beta_{2} ) q^{6} + ( -5 \beta_{1} + 56 \beta_{3} ) q^{7} + ( -72 \beta_{1} - 120 \beta_{3} ) q^{8} + ( -2777 + 16 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -\beta_{1} + 8 \beta_{3} ) q^{3} + ( -48 - 2 \beta_{2} ) q^{4} + ( -508 - 7 \beta_{2} ) q^{6} + ( -5 \beta_{1} + 56 \beta_{3} ) q^{7} + ( -72 \beta_{1} - 120 \beta_{3} ) q^{8} + ( -2777 + 16 \beta_{2} ) q^{9} + ( 2272 + 40 \beta_{2} ) q^{11} + ( -1168 \beta_{1} - 184 \beta_{3} ) q^{12} + ( -177 \beta_{1} + 608 \beta_{3} ) q^{13} + ( -3756 - 51 \beta_{2} ) q^{14} + ( 10176 - 64 \beta_{2} ) q^{16} + ( -1367 \beta_{1} - 1184 \beta_{3} ) q^{17} + ( -1561 \beta_{1} - 1177 \beta_{3} ) q^{18} + ( -19380 + 32 \beta_{2} ) q^{19} + ( -34548 + 96 \beta_{2} ) q^{21} + ( 5312 \beta_{1} + 6272 \beta_{3} ) q^{22} + ( 6207 \beta_{1} + 408 \beta_{3} ) q^{23} + ( 65760 + 456 \beta_{2} ) q^{24} + ( -28508 - 431 \beta_{2} ) q^{26} + ( 10318 \beta_{1} - 6320 \beta_{3} ) q^{27} + ( -8272 \beta_{1} - 1688 \beta_{3} ) q^{28} + ( 36130 - 1952 \beta_{2} ) q^{29} + ( 153412 - 280 \beta_{2} ) q^{31} + ( -3904 \beta_{1} - 11584 \beta_{3} ) q^{32} + ( 22048 \beta_{1} + 14176 \beta_{3} ) q^{33} + ( 226684 + 2551 \beta_{2} ) q^{34} + ( -109904 + 4786 \beta_{2} ) q^{36} + ( -6151 \beta_{1} + 25536 \beta_{3} ) q^{37} + ( -16948 \beta_{1} - 16180 \beta_{3} ) q^{38} + ( -387364 + 2024 \beta_{2} ) q^{39} + ( 132182 - 5680 \beta_{2} ) q^{41} + ( -27252 \beta_{1} - 24948 \beta_{3} ) q^{42} + ( -21165 \beta_{1} - 43192 \beta_{3} ) q^{43} + ( -717056 - 6464 \beta_{2} ) q^{44} + ( -651708 - 6615 \beta_{2} ) q^{46} + ( -5273 \beta_{1} + 45496 \beta_{3} ) q^{47} + ( -49088 \beta_{1} + 87808 \beta_{3} ) q^{48} + ( 582707 + 560 \beta_{2} ) q^{49} + ( 583172 + 9752 \beta_{2} ) q^{51} + ( -83920 \beta_{1} + 6216 \beta_{3} ) q^{52} + ( 119579 \beta_{1} + 53408 \beta_{3} ) q^{53} + ( -551480 - 3998 \beta_{2} ) q^{54} + ( 474720 + 3432 \beta_{2} ) q^{56} + ( 38836 \beta_{1} - 158240 \beta_{3} ) q^{57} + ( -112222 \beta_{1} - 159070 \beta_{3} ) q^{58} + ( 560060 + 22736 \beta_{2} ) q^{59} + ( 1128522 + 16000 \beta_{2} ) q^{61} + ( 132132 \beta_{1} + 125412 \beta_{3} ) q^{62} + ( 81981 \beta_{1} - 163512 \beta_{3} ) q^{63} + ( 2573312 + 7296 \beta_{2} ) q^{64} + ( -3282176 - 36224 \beta_{2} ) q^{66} + ( 225823 \beta_{1} - 79384 \beta_{3} ) q^{67} + ( 245584 \beta_{1} + 330232 \beta_{3} ) q^{68} + ( 372636 - 49248 \beta_{2} ) q^{69} + ( 310892 - 7000 \beta_{2} ) q^{71} + ( 54024 \beta_{1} + 218040 \beta_{3} ) q^{72} + ( -228453 \beta_{1} + 226208 \beta_{3} ) q^{73} + ( -1325636 - 19385 \beta_{2} ) q^{74} + ( 443840 + 37224 \beta_{2} ) q^{76} + ( 158880 \beta_{1} + 107232 \beta_{3} ) q^{77} + ( -233540 \beta_{1} - 184964 \beta_{3} ) q^{78} + ( -2166520 + 47248 \beta_{2} ) q^{79} + ( -1198939 - 53872 \beta_{2} ) q^{81} + ( -299498 \beta_{1} - 435818 \beta_{3} ) q^{82} + ( 489651 \beta_{1} - 490392 \beta_{3} ) q^{83} + ( 199104 + 64488 \beta_{2} ) q^{84} + ( 5399092 + 64357 \beta_{2} ) q^{86} + ( -1222946 \beta_{1} + 484240 \beta_{3} ) q^{87} + ( -528384 \beta_{1} - 560640 \beta_{3} ) q^{88} + ( -3012810 - 31776 \beta_{2} ) q^{89} + ( -2676148 + 12952 \beta_{2} ) q^{91} + ( -359952 \beta_{1} - 1260984 \beta_{3} ) q^{92} + ( -323652 \beta_{1} + 1255296 \beta_{3} ) q^{93} + ( -2930396 - 40223 \beta_{2} ) q^{94} + ( 6652672 + 19648 \beta_{2} ) q^{96} + ( 230477 \beta_{1} + 561696 \beta_{3} ) q^{97} + ( 625267 \beta_{1} + 638707 \beta_{3} ) q^{98} + ( -1445344 - 74728 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 192q^{4} - 2032q^{6} - 11108q^{9} + O(q^{10})$$ $$4q - 192q^{4} - 2032q^{6} - 11108q^{9} + 9088q^{11} - 15024q^{14} + 40704q^{16} - 77520q^{19} - 138192q^{21} + 263040q^{24} - 114032q^{26} + 144520q^{29} + 613648q^{31} + 906736q^{34} - 439616q^{36} - 1549456q^{39} + 528728q^{41} - 2868224q^{44} - 2606832q^{46} + 2330828q^{49} + 2332688q^{51} - 2205920q^{54} + 1898880q^{56} + 2240240q^{59} + 4514088q^{61} + 10293248q^{64} - 13128704q^{66} + 1490544q^{69} + 1243568q^{71} - 5302544q^{74} + 1775360q^{76} - 8666080q^{79} - 4795756q^{81} + 796416q^{84} + 21596368q^{86} - 12051240q^{89} - 10704592q^{91} - 11721584q^{94} + 26610688q^{96} - 5781376q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} - 8 \nu$$ $$\beta_{2}$$ $$=$$ $$-4 \nu^{3} + 56 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} - 18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 18$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 7 \beta_{1}$$$$)/10$$

## 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
 2.17945 − 0.500000i −2.17945 − 0.500000i −2.17945 + 0.500000i 2.17945 + 0.500000i
18.7178i 59.7424i −222.356 0 −1118.25 438.197i 1766.14i −1382.15 0
24.2 1.28220i 79.7424i 126.356 0 102.246 538.197i 326.136i −4171.85 0
24.3 1.28220i 79.7424i 126.356 0 102.246 538.197i 326.136i −4171.85 0
24.4 18.7178i 59.7424i −222.356 0 −1118.25 438.197i 1766.14i −1382.15 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 25.8.b.c 4
3.b odd 2 1 225.8.b.m 4
4.b odd 2 1 400.8.c.m 4
5.b even 2 1 inner 25.8.b.c 4
5.c odd 4 1 5.8.a.b 2
5.c odd 4 1 25.8.a.b 2
15.d odd 2 1 225.8.b.m 4
15.e even 4 1 45.8.a.h 2
15.e even 4 1 225.8.a.w 2
20.d odd 2 1 400.8.c.m 4
20.e even 4 1 80.8.a.g 2
20.e even 4 1 400.8.a.bb 2
35.f even 4 1 245.8.a.c 2
40.i odd 4 1 320.8.a.l 2
40.k even 4 1 320.8.a.u 2
55.e even 4 1 605.8.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.b 2 5.c odd 4 1
25.8.a.b 2 5.c odd 4 1
25.8.b.c 4 1.a even 1 1 trivial
25.8.b.c 4 5.b even 2 1 inner
45.8.a.h 2 15.e even 4 1
80.8.a.g 2 20.e even 4 1
225.8.a.w 2 15.e even 4 1
225.8.b.m 4 3.b odd 2 1
225.8.b.m 4 15.d odd 2 1
245.8.a.c 2 35.f even 4 1
320.8.a.l 2 40.i odd 4 1
320.8.a.u 2 40.k even 4 1
400.8.a.bb 2 20.e even 4 1
400.8.c.m 4 4.b odd 2 1
400.8.c.m 4 20.d odd 2 1
605.8.a.d 2 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 352 T_{2}^{2} + 576$$ acting on $$S_{8}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 160 T^{2} + 8768 T^{4} - 2621440 T^{6} + 268435456 T^{8}$$
$3$ $$1 + 1180 T^{2} + 7968438 T^{4} + 5643903420 T^{6} + 22876792454961 T^{8}$$
$5$ 1
$7$ $$1 - 2812500 T^{2} + 3331601848198 T^{4} - 1907502392387812500 T^{6} +$$$$45\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 4544 T + 31976326 T^{2} - 88549705024 T^{3} + 379749833583241 T^{4} )^{2}$$
$13$ $$1 - 188539340 T^{2} + 16409454868245078 T^{4} -$$$$74\!\cdots\!60$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$17$ $$1 - 1054534780 T^{2} + 535129713917980358 T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$19$ $$( 1 + 38760 T + 2155545478 T^{2} + 34646468603640 T^{3} + 799006685782884121 T^{4} )^{2}$$
$23$ $$1 - 5888629460 T^{2} + 31659696543257618118 T^{4} -$$$$68\!\cdots\!40$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8}$$
$29$ $$( 1 - 72260 T + 6846819118 T^{2} - 1246476062088340 T^{3} +$$$$29\!\cdots\!81$$$$T^{4} )^{2}$$
$31$ $$( 1 - 306824 T + 77964629966 T^{2} - 8441530311993464 T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$37$ $$1 - 273043279340 T^{2} +$$$$35\!\cdots\!78$$$$T^{4} -$$$$24\!\cdots\!60$$$$T^{6} +$$$$81\!\cdots\!21$$$$T^{8}$$
$41$ $$( 1 - 264364 T + 161786388886 T^{2} - 51486018860276684 T^{3} +$$$$37\!\cdots\!61$$$$T^{4} )^{2}$$
$43$ $$1 - 714119572100 T^{2} +$$$$24\!\cdots\!98$$$$T^{4} -$$$$52\!\cdots\!00$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 1706308901620 T^{2} +$$$$12\!\cdots\!38$$$$T^{4} -$$$$43\!\cdots\!80$$$$T^{6} +$$$$65\!\cdots\!61$$$$T^{8}$$
$53$ $$1 - 1405450112620 T^{2} +$$$$20\!\cdots\!38$$$$T^{4} -$$$$19\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!61$$$$T^{8}$$
$59$ $$( 1 - 1120120 T + 1362334883638 T^{2} - 2787588301175458280 T^{3} +$$$$61\!\cdots\!61$$$$T^{4} )^{2}$$
$61$ $$( 1 - 2257044 T + 5613447576526 T^{2} - 7093308861584181924 T^{3} +$$$$98\!\cdots\!41$$$$T^{4} )^{2}$$
$67$ $$1 - 13085764398180 T^{2} +$$$$10\!\cdots\!58$$$$T^{4} -$$$$48\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!41$$$$T^{8}$$
$71$ $$( 1 - 621784 T + 17914494152446 T^{2} - 5655200192564989544 T^{3} +$$$$82\!\cdots\!81$$$$T^{4} )^{2}$$
$73$ $$1 - 25973590426460 T^{2} +$$$$33\!\cdots\!18$$$$T^{4} -$$$$31\!\cdots\!40$$$$T^{6} +$$$$14\!\cdots\!81$$$$T^{8}$$
$79$ $$( 1 + 4333040 T + 26135588252318 T^{2} + 83211305793386393360 T^{3} +$$$$36\!\cdots\!81$$$$T^{4} )^{2}$$
$83$ $$1 - 24038967921380 T^{2} -$$$$13\!\cdots\!42$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$54\!\cdots\!41$$$$T^{8}$$
$89$ $$( 1 + 6025620 T + 89865866149558 T^{2} +$$$$26\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4} )^{2}$$
$97$ $$1 - 264612844151420 T^{2} +$$$$30\!\cdots\!38$$$$T^{4} -$$$$17\!\cdots\!80$$$$T^{6} +$$$$42\!\cdots\!61$$$$T^{8}$$