# Properties

 Label 25.32.a.a Level $25$ Weight $32$ Character orbit 25.a Self dual yes Analytic conductor $152.193$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$32$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$152.192832048$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 4573872$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{18295489}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -19980 - \beta ) q^{2} + ( -8681580 - 432 \beta ) q^{3} + ( 886267168 + 39960 \beta ) q^{4} + ( 1311583748112 + 17312940 \beta ) q^{6} + ( -15128763788600 + 71928864 \beta ) q^{7} + ( -80077529352960 + 462815680 \beta ) q^{8} + ( -50633228151963 + 7500885120 \beta ) q^{9} +O(q^{10})$$ $$q +(-19980 - \beta) q^{2} +(-8681580 - 432 \beta) q^{3} +(886267168 + 39960 \beta) q^{4} +(1311583748112 + 17312940 \beta) q^{6} +(-15128763788600 + 71928864 \beta) q^{7} +(-80077529352960 + 462815680 \beta) q^{8} +(-50633228151963 + 7500885120 \beta) q^{9} +(-3891176872559388 + 29000909200 \beta) q^{11} +(-53173705477656960 - 729783353376 \beta) q^{12} +(-37354476525130310 + 4661016429888 \beta) q^{13} +(112772481922620576 + 13691625085880 \beta) q^{14} +(-1522606456842450944 - 14982974507520 \beta) q^{16} +(-8612303914493544690 - 45085348093056 \beta) q^{17} +(-18749808114787989180 - 99234456545637 \beta) q^{18} +(-6185281664011082020 - 157805792764560 \beta) q^{19} +(49477478708035580832 + 5911169769550080 \beta) q^{21} +(1341356516498345040 + 3311738706743388 \beta) q^{22} +(-$$$$94\!\cdots\!40$$$$+ 18557618179251808 \beta) q^{23} +($$$$16\!\cdots\!40$$$$+ 30575521329304320 \beta) q^{24} +(-$$$$11\!\cdots\!08$$$$- 55772631744031930 \beta) q^{26} +(-$$$$27\!\cdots\!40$$$$+ 223588927516223520 \beta) q^{27} +(-$$$$58\!\cdots\!60$$$$- 540797210397718848 \beta) q^{28} +($$$$64\!\cdots\!70$$$$+ 234192357384448960 \beta) q^{29} +($$$$62\!\cdots\!32$$$$+ 2412621171020361600 \beta) q^{31} +($$$$24\!\cdots\!20$$$$+ 828077182664699904 \beta) q^{32} +($$$$77\!\cdots\!40$$$$+ 1429214695653119616 \beta) q^{33} +($$$$29\!\cdots\!96$$$$+ 9513109169392803570 \beta) q^{34} +($$$$74\!\cdots\!16$$$$+ 4624484415843298680 \beta) q^{36} +($$$$41\!\cdots\!30$$$$- 32224244113578511296 \beta) q^{37} +($$$$53\!\cdots\!60$$$$+ 9338241403446990820 \beta) q^{38} +(-$$$$49\!\cdots\!56$$$$- 24327853158530769120 \beta) q^{39} +($$$$43\!\cdots\!42$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{41} +(-$$$$16\!\cdots\!40$$$$-$$$$16\!\cdots\!32$$$$\beta) q^{42} +($$$$91\!\cdots\!00$$$$+$$$$34\!\cdots\!48$$$$\beta) q^{43} +(-$$$$39\!\cdots\!84$$$$-$$$$12\!\cdots\!80$$$$\beta) q^{44} +(-$$$$29\!\cdots\!28$$$$+$$$$57\!\cdots\!00$$$$\beta) q^{46} +(-$$$$47\!\cdots\!60$$$$-$$$$51\!\cdots\!16$$$$\beta) q^{47} +($$$$30\!\cdots\!60$$$$+$$$$78\!\cdots\!08$$$$\beta) q^{48} +($$$$84\!\cdots\!93$$$$-$$$$21\!\cdots\!00$$$$\beta) q^{49} +($$$$12\!\cdots\!72$$$$+$$$$41\!\cdots\!60$$$$\beta) q^{51} +($$$$45\!\cdots\!00$$$$+$$$$26\!\cdots\!84$$$$\beta) q^{52} +(-$$$$97\!\cdots\!30$$$$+$$$$12\!\cdots\!68$$$$\beta) q^{53} +(-$$$$53\!\cdots\!20$$$$-$$$$17\!\cdots\!60$$$$\beta) q^{54} +($$$$12\!\cdots\!20$$$$-$$$$12\!\cdots\!40$$$$\beta) q^{56} +($$$$23\!\cdots\!20$$$$+$$$$40\!\cdots\!40$$$$\beta) q^{57} +(-$$$$19\!\cdots\!60$$$$-$$$$68\!\cdots\!70$$$$\beta) q^{58} +(-$$$$99\!\cdots\!60$$$$-$$$$44\!\cdots\!80$$$$\beta) q^{59} +(-$$$$60\!\cdots\!38$$$$+$$$$11\!\cdots\!00$$$$\beta) q^{61} +(-$$$$76\!\cdots\!60$$$$-$$$$11\!\cdots\!32$$$$\beta) q^{62} +($$$$21\!\cdots\!80$$$$-$$$$11\!\cdots\!32$$$$\beta) q^{63} +(-$$$$37\!\cdots\!52$$$$-$$$$22\!\cdots\!80$$$$\beta) q^{64} +(-$$$$37\!\cdots\!56$$$$-$$$$29\!\cdots\!20$$$$\beta) q^{66} +($$$$48\!\cdots\!60$$$$+$$$$88\!\cdots\!44$$$$\beta) q^{67} +(-$$$$12\!\cdots\!80$$$$-$$$$38\!\cdots\!08$$$$\beta) q^{68} +(-$$$$12\!\cdots\!96$$$$+$$$$24\!\cdots\!40$$$$\beta) q^{69} +($$$$27\!\cdots\!72$$$$-$$$$95\!\cdots\!00$$$$\beta) q^{71} +($$$$13\!\cdots\!80$$$$-$$$$62\!\cdots\!40$$$$\beta) q^{72} +(-$$$$31\!\cdots\!90$$$$-$$$$15\!\cdots\!92$$$$\beta) q^{73} +($$$$76\!\cdots\!36$$$$+$$$$22\!\cdots\!50$$$$\beta) q^{74} +(-$$$$22\!\cdots\!60$$$$-$$$$38\!\cdots\!80$$$$\beta) q^{76} +($$$$64\!\cdots\!00$$$$-$$$$71\!\cdots\!32$$$$\beta) q^{77} +($$$$16\!\cdots\!00$$$$+$$$$54\!\cdots\!56$$$$\beta) q^{78} +(-$$$$59\!\cdots\!80$$$$+$$$$46\!\cdots\!60$$$$\beta) q^{79} +(-$$$$19\!\cdots\!79$$$$-$$$$53\!\cdots\!60$$$$\beta) q^{81} +(-$$$$57\!\cdots\!60$$$$-$$$$80\!\cdots\!42$$$$\beta) q^{82} +($$$$13\!\cdots\!80$$$$+$$$$62\!\cdots\!28$$$$\beta) q^{83} +($$$$66\!\cdots\!76$$$$+$$$$72\!\cdots\!60$$$$\beta) q^{84} +(-$$$$10\!\cdots\!68$$$$-$$$$16\!\cdots\!40$$$$\beta) q^{86} +(-$$$$82\!\cdots\!20$$$$-$$$$29\!\cdots\!40$$$$\beta) q^{87} +($$$$34\!\cdots\!80$$$$-$$$$41\!\cdots\!40$$$$\beta) q^{88} +(-$$$$10\!\cdots\!90$$$$+$$$$26\!\cdots\!80$$$$\beta) q^{89} +($$$$14\!\cdots\!12$$$$-$$$$73\!\cdots\!40$$$$\beta) q^{91} +($$$$11\!\cdots\!60$$$$-$$$$21\!\cdots\!56$$$$\beta) q^{92} +(-$$$$32\!\cdots\!60$$$$-$$$$48\!\cdots\!24$$$$\beta) q^{93} +($$$$23\!\cdots\!56$$$$+$$$$57\!\cdots\!40$$$$\beta) q^{94} +(-$$$$30\!\cdots\!48$$$$-$$$$11\!\cdots\!60$$$$\beta) q^{96} +($$$$45\!\cdots\!90$$$$+$$$$16\!\cdots\!84$$$$\beta) q^{97} +($$$$40\!\cdots\!60$$$$-$$$$41\!\cdots\!93$$$$\beta) q^{98} +($$$$77\!\cdots\!44$$$$-$$$$30\!\cdots\!60$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 39960q^{2} - 17363160q^{3} + 1772534336q^{4} + 2623167496224q^{6} - 30257527577200q^{7} - 160155058705920q^{8} - 101266456303926q^{9} + O(q^{10})$$ $$2q - 39960q^{2} - 17363160q^{3} + 1772534336q^{4} + 2623167496224q^{6} - 30257527577200q^{7} - 160155058705920q^{8} - 101266456303926q^{9} - 7782353745118776q^{11} - 106347410955313920q^{12} - 74708953050260620q^{13} + 225544963845241152q^{14} - 3045212913684901888q^{16} - 17224607828987089380q^{17} - 37499616229575978360q^{18} - 12370563328022164040q^{19} + 98954957416071161664q^{21} + 2682713032996690080q^{22} -$$$$18\!\cdots\!80$$$$q^{23} +$$$$33\!\cdots\!80$$$$q^{24} -$$$$23\!\cdots\!16$$$$q^{26} -$$$$54\!\cdots\!80$$$$q^{27} -$$$$11\!\cdots\!20$$$$q^{28} +$$$$12\!\cdots\!40$$$$q^{29} +$$$$12\!\cdots\!64$$$$q^{31} +$$$$48\!\cdots\!40$$$$q^{32} +$$$$15\!\cdots\!80$$$$q^{33} +$$$$58\!\cdots\!92$$$$q^{34} +$$$$14\!\cdots\!32$$$$q^{36} +$$$$83\!\cdots\!60$$$$q^{37} +$$$$10\!\cdots\!20$$$$q^{38} -$$$$99\!\cdots\!12$$$$q^{39} +$$$$87\!\cdots\!84$$$$q^{41} -$$$$33\!\cdots\!80$$$$q^{42} +$$$$18\!\cdots\!00$$$$q^{43} -$$$$79\!\cdots\!68$$$$q^{44} -$$$$59\!\cdots\!56$$$$q^{46} -$$$$95\!\cdots\!20$$$$q^{47} +$$$$60\!\cdots\!20$$$$q^{48} +$$$$16\!\cdots\!86$$$$q^{49} +$$$$25\!\cdots\!44$$$$q^{51} +$$$$91\!\cdots\!00$$$$q^{52} -$$$$19\!\cdots\!60$$$$q^{53} -$$$$10\!\cdots\!40$$$$q^{54} +$$$$25\!\cdots\!40$$$$q^{56} +$$$$46\!\cdots\!40$$$$q^{57} -$$$$38\!\cdots\!20$$$$q^{58} -$$$$19\!\cdots\!20$$$$q^{59} -$$$$12\!\cdots\!76$$$$q^{61} -$$$$15\!\cdots\!20$$$$q^{62} +$$$$43\!\cdots\!60$$$$q^{63} -$$$$74\!\cdots\!04$$$$q^{64} -$$$$75\!\cdots\!12$$$$q^{66} +$$$$96\!\cdots\!20$$$$q^{67} -$$$$24\!\cdots\!60$$$$q^{68} -$$$$25\!\cdots\!92$$$$q^{69} +$$$$55\!\cdots\!44$$$$q^{71} +$$$$26\!\cdots\!60$$$$q^{72} -$$$$62\!\cdots\!80$$$$q^{73} +$$$$15\!\cdots\!72$$$$q^{74} -$$$$44\!\cdots\!20$$$$q^{76} +$$$$12\!\cdots\!00$$$$q^{77} +$$$$32\!\cdots\!00$$$$q^{78} -$$$$11\!\cdots\!60$$$$q^{79} -$$$$39\!\cdots\!58$$$$q^{81} -$$$$11\!\cdots\!20$$$$q^{82} +$$$$26\!\cdots\!60$$$$q^{83} +$$$$13\!\cdots\!52$$$$q^{84} -$$$$21\!\cdots\!36$$$$q^{86} -$$$$16\!\cdots\!40$$$$q^{87} +$$$$69\!\cdots\!60$$$$q^{88} -$$$$21\!\cdots\!80$$$$q^{89} +$$$$28\!\cdots\!24$$$$q^{91} +$$$$22\!\cdots\!20$$$$q^{92} -$$$$65\!\cdots\!20$$$$q^{93} +$$$$46\!\cdots\!12$$$$q^{94} -$$$$60\!\cdots\!96$$$$q^{96} +$$$$90\!\cdots\!80$$$$q^{97} +$$$$80\!\cdots\!20$$$$q^{98} +$$$$15\!\cdots\!88$$$$q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2139.16 −2138.16
−71307.9 −3.08552e7 2.93733e9 0 2.20022e12 −1.14368e13 −5.63222e13 3.34371e14 0
1.2 31347.9 1.34921e7 −1.16479e9 0 4.22947e11 −1.88207e13 −1.03833e14 −4.35638e14 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.32.a.a 2
5.b even 2 1 1.32.a.a 2
5.c odd 4 2 25.32.b.a 4
15.d odd 2 1 9.32.a.a 2
20.d odd 2 1 16.32.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 5.b even 2 1
9.32.a.a 2 15.d odd 2 1
16.32.a.b 2 20.d odd 2 1
25.32.a.a 2 1.a even 1 1 trivial
25.32.b.a 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 39960 T_{2} - 2235350016$$ acting on $$S_{32}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2235350016 + 39960 T + T^{2}$$
$3$ $$-416300505539184 + 17363160 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$21\!\cdots\!64$$$$+ 30257527577200 T + T^{2}$$
$11$ $$12\!\cdots\!44$$$$+ 7782353745118776 T + T^{2}$$
$13$ $$-$$$$55\!\cdots\!04$$$$+ 74708953050260620 T + T^{2}$$
$17$ $$68\!\cdots\!24$$$$+ 17224607828987089380 T + T^{2}$$
$19$ $$-$$$$27\!\cdots\!00$$$$+ 12370563328022164040 T + T^{2}$$
$23$ $$-$$$$73\!\cdots\!24$$$$+$$$$18\!\cdots\!80$$$$T + T^{2}$$
$29$ $$39\!\cdots\!00$$$$-$$$$12\!\cdots\!40$$$$T + T^{2}$$
$31$ $$-$$$$11\!\cdots\!76$$$$-$$$$12\!\cdots\!64$$$$T + T^{2}$$
$37$ $$-$$$$25\!\cdots\!56$$$$-$$$$83\!\cdots\!60$$$$T + T^{2}$$
$41$ $$-$$$$70\!\cdots\!36$$$$-$$$$87\!\cdots\!84$$$$T + T^{2}$$
$43$ $$-$$$$22\!\cdots\!64$$$$-$$$$18\!\cdots\!00$$$$T + T^{2}$$
$47$ $$15\!\cdots\!04$$$$+$$$$95\!\cdots\!20$$$$T + T^{2}$$
$53$ $$56\!\cdots\!16$$$$+$$$$19\!\cdots\!60$$$$T + T^{2}$$
$59$ $$-$$$$51\!\cdots\!00$$$$+$$$$19\!\cdots\!20$$$$T + T^{2}$$
$61$ $$36\!\cdots\!44$$$$+$$$$12\!\cdots\!76$$$$T + T^{2}$$
$67$ $$26\!\cdots\!24$$$$-$$$$96\!\cdots\!20$$$$T + T^{2}$$
$71$ $$-$$$$16\!\cdots\!16$$$$-$$$$55\!\cdots\!44$$$$T + T^{2}$$
$73$ $$90\!\cdots\!76$$$$+$$$$62\!\cdots\!80$$$$T + T^{2}$$
$79$ $$-$$$$53\!\cdots\!00$$$$+$$$$11\!\cdots\!60$$$$T + T^{2}$$
$83$ $$-$$$$86\!\cdots\!44$$$$-$$$$26\!\cdots\!60$$$$T + T^{2}$$
$89$ $$-$$$$62\!\cdots\!00$$$$+$$$$21\!\cdots\!80$$$$T + T^{2}$$
$97$ $$19\!\cdots\!04$$$$-$$$$90\!\cdots\!80$$$$T + T^{2}$$