# Properties

 Label 1.32.a.a Level $1$ Weight $32$ Character orbit 1.a Self dual yes Analytic conductor $6.088$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$32$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.08771328190$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 19980 - \beta ) q^{2} + ( 8681580 - 432 \beta ) q^{3} + ( 886267168 - 39960 \beta ) q^{4} + ( -9695609010 + 1418560 \beta ) q^{5} + ( 1311583748112 - 17312940 \beta ) q^{6} + ( 15128763788600 + 71928864 \beta ) q^{7} + ( 80077529352960 + 462815680 \beta ) q^{8} + ( -50633228151963 - 7500885120 \beta ) q^{9} +O(q^{10})$$ $$q +(19980 - \beta) q^{2} +(8681580 - 432 \beta) q^{3} +(886267168 - 39960 \beta) q^{4} +(-9695609010 + 1418560 \beta) q^{5} +(1311583748112 - 17312940 \beta) q^{6} +(15128763788600 + 71928864 \beta) q^{7} +(80077529352960 + 462815680 \beta) q^{8} +(-50633228151963 - 7500885120 \beta) q^{9} +(-3930986106140760 + 38038437810 \beta) q^{10} +(-3891176872559388 - 29000909200 \beta) q^{11} +(53173705477656960 - 729783353376 \beta) q^{12} +(37354476525130310 + 4661016429888 \beta) q^{13} +(112772481922620576 - 13691625085880 \beta) q^{14} +(-1698672911337290520 + 16503845217120 \beta) q^{15} +(-1522606456842450944 + 14982974507520 \beta) q^{16} +(8612303914493544690 - 45085348093056 \beta) q^{17} +(18749808114787989180 - 99234456545637 \beta) q^{18} +(-6185281664011082020 + 157805792764560 \beta) q^{19} +(-$$$$15\!\cdots\!80$$$$+ 1644659689877680 \beta) q^{20} +(49477478708035580832 - 5911169769550080 \beta) q^{21} +(-1341356516498345040 + 3311738706743388 \beta) q^{22} +($$$$94\!\cdots\!40$$$$+ 18557618179251808 \beta) q^{23} +($$$$16\!\cdots\!40$$$$- 30575521329304320 \beta) q^{24} +($$$$73\!\cdots\!75$$$$- 27507606234451200 \beta) q^{25} +(-$$$$11\!\cdots\!08$$$$+ 55772631744031930 \beta) q^{26} +($$$$27\!\cdots\!40$$$$+ 223588927516223520 \beta) q^{27} +($$$$58\!\cdots\!60$$$$- 540797210397718848 \beta) q^{28} +($$$$64\!\cdots\!70$$$$- 234192357384448960 \beta) q^{29} +(-$$$$77\!\cdots\!20$$$$+ 2028419738775348120 \beta) q^{30} +($$$$62\!\cdots\!32$$$$- 2412621171020361600 \beta) q^{31} +(-$$$$24\!\cdots\!20$$$$+ 828077182664699904 \beta) q^{32} +(-$$$$77\!\cdots\!40$$$$+ 1429214695653119616 \beta) q^{33} +($$$$29\!\cdots\!96$$$$- 9513109169392803570 \beta) q^{34} +($$$$12\!\cdots\!40$$$$+ 20763665018078951360 \beta) q^{35} +($$$$74\!\cdots\!16$$$$- 4624484415843298680 \beta) q^{36} +(-$$$$41\!\cdots\!30$$$$- 32224244113578511296 \beta) q^{37} +(-$$$$53\!\cdots\!60$$$$+ 9338241403446990820 \beta) q^{38} +(-$$$$49\!\cdots\!56$$$$+ 24327853158530769120 \beta) q^{39} +($$$$95\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$\beta) q^{40} +($$$$43\!\cdots\!42$$$$-$$$$18\!\cdots\!00$$$$\beta) q^{41} +($$$$16\!\cdots\!40$$$$-$$$$16\!\cdots\!32$$$$\beta) q^{42} +(-$$$$91\!\cdots\!00$$$$+$$$$34\!\cdots\!48$$$$\beta) q^{43} +(-$$$$39\!\cdots\!84$$$$+$$$$12\!\cdots\!80$$$$\beta) q^{44} +(-$$$$27\!\cdots\!70$$$$+ 899377225198297920 \beta) q^{45} +(-$$$$29\!\cdots\!28$$$$-$$$$57\!\cdots\!00$$$$\beta) q^{46} +($$$$47\!\cdots\!60$$$$-$$$$51\!\cdots\!16$$$$\beta) q^{47} +(-$$$$30\!\cdots\!60$$$$+$$$$78\!\cdots\!08$$$$\beta) q^{48} +($$$$84\!\cdots\!93$$$$+$$$$21\!\cdots\!00$$$$\beta) q^{49} +($$$$87\!\cdots\!00$$$$-$$$$12\!\cdots\!75$$$$\beta) q^{50} +($$$$12\!\cdots\!72$$$$-$$$$41\!\cdots\!60$$$$\beta) q^{51} +(-$$$$45\!\cdots\!00$$$$+$$$$26\!\cdots\!84$$$$\beta) q^{52} +($$$$97\!\cdots\!30$$$$+$$$$12\!\cdots\!68$$$$\beta) q^{53} +(-$$$$53\!\cdots\!20$$$$+$$$$17\!\cdots\!60$$$$\beta) q^{54} +(-$$$$70\!\cdots\!20$$$$-$$$$52\!\cdots\!80$$$$\beta) q^{55} +($$$$12\!\cdots\!20$$$$+$$$$12\!\cdots\!40$$$$\beta) q^{56} +(-$$$$23\!\cdots\!20$$$$+$$$$40\!\cdots\!40$$$$\beta) q^{57} +($$$$19\!\cdots\!60$$$$-$$$$68\!\cdots\!70$$$$\beta) q^{58} +(-$$$$99\!\cdots\!60$$$$+$$$$44\!\cdots\!80$$$$\beta) q^{59} +(-$$$$32\!\cdots\!60$$$$+$$$$82\!\cdots\!60$$$$\beta) q^{60} +(-$$$$60\!\cdots\!38$$$$-$$$$11\!\cdots\!00$$$$\beta) q^{61} +($$$$76\!\cdots\!60$$$$-$$$$11\!\cdots\!32$$$$\beta) q^{62} +(-$$$$21\!\cdots\!80$$$$-$$$$11\!\cdots\!32$$$$\beta) q^{63} +(-$$$$37\!\cdots\!52$$$$+$$$$22\!\cdots\!80$$$$\beta) q^{64} +($$$$17\!\cdots\!80$$$$+$$$$77\!\cdots\!20$$$$\beta) q^{65} +(-$$$$37\!\cdots\!56$$$$+$$$$29\!\cdots\!20$$$$\beta) q^{66} +(-$$$$48\!\cdots\!60$$$$+$$$$88\!\cdots\!44$$$$\beta) q^{67} +($$$$12\!\cdots\!80$$$$-$$$$38\!\cdots\!08$$$$\beta) q^{68} +(-$$$$12\!\cdots\!96$$$$-$$$$24\!\cdots\!40$$$$\beta) q^{69} +(-$$$$52\!\cdots\!60$$$$+$$$$29\!\cdots\!60$$$$\beta) q^{70} +($$$$27\!\cdots\!72$$$$+$$$$95\!\cdots\!00$$$$\beta) q^{71} +(-$$$$13\!\cdots\!80$$$$-$$$$62\!\cdots\!40$$$$\beta) q^{72} +($$$$31\!\cdots\!90$$$$-$$$$15\!\cdots\!92$$$$\beta) q^{73} +($$$$76\!\cdots\!36$$$$-$$$$22\!\cdots\!50$$$$\beta) q^{74} +($$$$37\!\cdots\!00$$$$-$$$$55\!\cdots\!00$$$$\beta) q^{75} +(-$$$$22\!\cdots\!60$$$$+$$$$38\!\cdots\!80$$$$\beta) q^{76} +(-$$$$64\!\cdots\!00$$$$-$$$$71\!\cdots\!32$$$$\beta) q^{77} +(-$$$$16\!\cdots\!00$$$$+$$$$54\!\cdots\!56$$$$\beta) q^{78} +(-$$$$59\!\cdots\!80$$$$-$$$$46\!\cdots\!60$$$$\beta) q^{79} +($$$$70\!\cdots\!40$$$$-$$$$23\!\cdots\!40$$$$\beta) q^{80} +(-$$$$19\!\cdots\!79$$$$+$$$$53\!\cdots\!60$$$$\beta) q^{81} +($$$$57\!\cdots\!60$$$$-$$$$80\!\cdots\!42$$$$\beta) q^{82} +(-$$$$13\!\cdots\!80$$$$+$$$$62\!\cdots\!28$$$$\beta) q^{83} +($$$$66\!\cdots\!76$$$$-$$$$72\!\cdots\!60$$$$\beta) q^{84} +(-$$$$25\!\cdots\!60$$$$+$$$$12\!\cdots\!60$$$$\beta) q^{85} +(-$$$$10\!\cdots\!68$$$$+$$$$16\!\cdots\!40$$$$\beta) q^{86} +($$$$82\!\cdots\!20$$$$-$$$$29\!\cdots\!40$$$$\beta) q^{87} +(-$$$$34\!\cdots\!80$$$$-$$$$41\!\cdots\!40$$$$\beta) q^{88} +(-$$$$10\!\cdots\!90$$$$-$$$$26\!\cdots\!80$$$$\beta) q^{89} +(-$$$$55\!\cdots\!20$$$$+$$$$27\!\cdots\!70$$$$\beta) q^{90} +($$$$14\!\cdots\!12$$$$+$$$$73\!\cdots\!40$$$$\beta) q^{91} +(-$$$$11\!\cdots\!60$$$$-$$$$21\!\cdots\!56$$$$\beta) q^{92} +($$$$32\!\cdots\!60$$$$-$$$$48\!\cdots\!24$$$$\beta) q^{93} +($$$$23\!\cdots\!56$$$$-$$$$57\!\cdots\!40$$$$\beta) q^{94} +($$$$64\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$\beta) q^{95} +(-$$$$30\!\cdots\!48$$$$+$$$$11\!\cdots\!60$$$$\beta) q^{96} +(-$$$$45\!\cdots\!90$$$$+$$$$16\!\cdots\!84$$$$\beta) q^{97} +(-$$$$40\!\cdots\!60$$$$-$$$$41\!\cdots\!93$$$$\beta) q^{98} +($$$$77\!\cdots\!44$$$$+$$$$30\!\cdots\!60$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 39960q^{2} + 17363160q^{3} + 1772534336q^{4} - 19391218020q^{5} + 2623167496224q^{6} + 30257527577200q^{7} + 160155058705920q^{8} - 101266456303926q^{9} + O(q^{10})$$ $$2q + 39960q^{2} + 17363160q^{3} + 1772534336q^{4} - 19391218020q^{5} + 2623167496224q^{6} + 30257527577200q^{7} + 160155058705920q^{8} - 101266456303926q^{9} - 7861972212281520q^{10} - 7782353745118776q^{11} + 106347410955313920q^{12} + 74708953050260620q^{13} + 225544963845241152q^{14} - 3397345822674581040q^{15} - 3045212913684901888q^{16} + 17224607828987089380q^{17} + 37499616229575978360q^{18} - 12370563328022164040q^{19} -$$$$31\!\cdots\!60$$$$q^{20} + 98954957416071161664q^{21} - 2682713032996690080q^{22} +$$$$18\!\cdots\!80$$$$q^{23} +$$$$33\!\cdots\!80$$$$q^{24} +$$$$14\!\cdots\!50$$$$q^{25} -$$$$23\!\cdots\!16$$$$q^{26} +$$$$54\!\cdots\!80$$$$q^{27} +$$$$11\!\cdots\!20$$$$q^{28} +$$$$12\!\cdots\!40$$$$q^{29} -$$$$15\!\cdots\!40$$$$q^{30} +$$$$12\!\cdots\!64$$$$q^{31} -$$$$48\!\cdots\!40$$$$q^{32} -$$$$15\!\cdots\!80$$$$q^{33} +$$$$58\!\cdots\!92$$$$q^{34} +$$$$24\!\cdots\!80$$$$q^{35} +$$$$14\!\cdots\!32$$$$q^{36} -$$$$83\!\cdots\!60$$$$q^{37} -$$$$10\!\cdots\!20$$$$q^{38} -$$$$99\!\cdots\!12$$$$q^{39} +$$$$19\!\cdots\!00$$$$q^{40} +$$$$87\!\cdots\!84$$$$q^{41} +$$$$33\!\cdots\!80$$$$q^{42} -$$$$18\!\cdots\!00$$$$q^{43} -$$$$79\!\cdots\!68$$$$q^{44} -$$$$55\!\cdots\!40$$$$q^{45} -$$$$59\!\cdots\!56$$$$q^{46} +$$$$95\!\cdots\!20$$$$q^{47} -$$$$60\!\cdots\!20$$$$q^{48} +$$$$16\!\cdots\!86$$$$q^{49} +$$$$17\!\cdots\!00$$$$q^{50} +$$$$25\!\cdots\!44$$$$q^{51} -$$$$91\!\cdots\!00$$$$q^{52} +$$$$19\!\cdots\!60$$$$q^{53} -$$$$10\!\cdots\!40$$$$q^{54} -$$$$14\!\cdots\!40$$$$q^{55} +$$$$25\!\cdots\!40$$$$q^{56} -$$$$46\!\cdots\!40$$$$q^{57} +$$$$38\!\cdots\!20$$$$q^{58} -$$$$19\!\cdots\!20$$$$q^{59} -$$$$64\!\cdots\!20$$$$q^{60} -$$$$12\!\cdots\!76$$$$q^{61} +$$$$15\!\cdots\!20$$$$q^{62} -$$$$43\!\cdots\!60$$$$q^{63} -$$$$74\!\cdots\!04$$$$q^{64} +$$$$34\!\cdots\!60$$$$q^{65} -$$$$75\!\cdots\!12$$$$q^{66} -$$$$96\!\cdots\!20$$$$q^{67} +$$$$24\!\cdots\!60$$$$q^{68} -$$$$25\!\cdots\!92$$$$q^{69} -$$$$10\!\cdots\!20$$$$q^{70} +$$$$55\!\cdots\!44$$$$q^{71} -$$$$26\!\cdots\!60$$$$q^{72} +$$$$62\!\cdots\!80$$$$q^{73} +$$$$15\!\cdots\!72$$$$q^{74} +$$$$75\!\cdots\!00$$$$q^{75} -$$$$44\!\cdots\!20$$$$q^{76} -$$$$12\!\cdots\!00$$$$q^{77} -$$$$32\!\cdots\!00$$$$q^{78} -$$$$11\!\cdots\!60$$$$q^{79} +$$$$14\!\cdots\!80$$$$q^{80} -$$$$39\!\cdots\!58$$$$q^{81} +$$$$11\!\cdots\!20$$$$q^{82} -$$$$26\!\cdots\!60$$$$q^{83} +$$$$13\!\cdots\!52$$$$q^{84} -$$$$50\!\cdots\!20$$$$q^{85} -$$$$21\!\cdots\!36$$$$q^{86} +$$$$16\!\cdots\!40$$$$q^{87} -$$$$69\!\cdots\!60$$$$q^{88} -$$$$21\!\cdots\!80$$$$q^{89} -$$$$11\!\cdots\!40$$$$q^{90} +$$$$28\!\cdots\!24$$$$q^{91} -$$$$22\!\cdots\!20$$$$q^{92} +$$$$65\!\cdots\!20$$$$q^{93} +$$$$46\!\cdots\!12$$$$q^{94} +$$$$12\!\cdots\!00$$$$q^{95} -$$$$60\!\cdots\!96$$$$q^{96} -$$$$90\!\cdots\!80$$$$q^{97} -$$$$80\!\cdots\!20$$$$q^{98} +$$$$15\!\cdots\!88$$$$q^{99} + O(q^{100})$$

## 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
−31347.9 −1.34921e7 −1.16479e9 6.31161e10 4.22947e11 1.88207e13 1.03833e14 −4.35638e14 −1.97855e15
1.2 71307.9 3.08552e7 2.93733e9 −8.25073e10 2.20022e12 1.14368e13 5.63222e13 3.34371e14 −5.88342e15
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 1.32.a.a 2
3.b odd 2 1 9.32.a.a 2
4.b odd 2 1 16.32.a.b 2
5.b even 2 1 25.32.a.a 2
5.c odd 4 2 25.32.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 1.a even 1 1 trivial
9.32.a.a 2 3.b odd 2 1
16.32.a.b 2 4.b odd 2 1
25.32.a.a 2 5.b even 2 1
25.32.b.a 4 5.c odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{32}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2235350016 - 39960 T + T^{2}$$
$3$ $$-416300505539184 - 17363160 T + T^{2}$$
$5$ $$-$$$$52\!\cdots\!00$$$$+ 19391218020 T + T^{2}$$
$7$ $$21\!\cdots\!64$$$$- 30257527577200 T + T^{2}$$
$11$ $$12\!\cdots\!44$$$$+ 7782353745118776 T + T^{2}$$
$13$ $$-$$$$55\!\cdots\!04$$$$- 74708953050260620 T + T^{2}$$
$17$ $$68\!\cdots\!24$$$$- 17224607828987089380 T + T^{2}$$
$19$ $$-$$$$27\!\cdots\!00$$$$+ 12370563328022164040 T + T^{2}$$
$23$ $$-$$$$73\!\cdots\!24$$$$-$$$$18\!\cdots\!80$$$$T + T^{2}$$
$29$ $$39\!\cdots\!00$$$$-$$$$12\!\cdots\!40$$$$T + T^{2}$$
$31$ $$-$$$$11\!\cdots\!76$$$$-$$$$12\!\cdots\!64$$$$T + T^{2}$$
$37$ $$-$$$$25\!\cdots\!56$$$$+$$$$83\!\cdots\!60$$$$T + T^{2}$$
$41$ $$-$$$$70\!\cdots\!36$$$$-$$$$87\!\cdots\!84$$$$T + T^{2}$$
$43$ $$-$$$$22\!\cdots\!64$$$$+$$$$18\!\cdots\!00$$$$T + T^{2}$$
$47$ $$15\!\cdots\!04$$$$-$$$$95\!\cdots\!20$$$$T + T^{2}$$
$53$ $$56\!\cdots\!16$$$$-$$$$19\!\cdots\!60$$$$T + T^{2}$$
$59$ $$-$$$$51\!\cdots\!00$$$$+$$$$19\!\cdots\!20$$$$T + T^{2}$$
$61$ $$36\!\cdots\!44$$$$+$$$$12\!\cdots\!76$$$$T + T^{2}$$
$67$ $$26\!\cdots\!24$$$$+$$$$96\!\cdots\!20$$$$T + T^{2}$$
$71$ $$-$$$$16\!\cdots\!16$$$$-$$$$55\!\cdots\!44$$$$T + T^{2}$$
$73$ $$90\!\cdots\!76$$$$-$$$$62\!\cdots\!80$$$$T + T^{2}$$
$79$ $$-$$$$53\!\cdots\!00$$$$+$$$$11\!\cdots\!60$$$$T + T^{2}$$
$83$ $$-$$$$86\!\cdots\!44$$$$+$$$$26\!\cdots\!60$$$$T + T^{2}$$
$89$ $$-$$$$62\!\cdots\!00$$$$+$$$$21\!\cdots\!80$$$$T + T^{2}$$
$97$ $$19\!\cdots\!04$$$$+$$$$90\!\cdots\!80$$$$T + T^{2}$$