# Properties

 Label 25.34.a.a Level $25$ Weight $34$ Character orbit 25.a Self dual yes Analytic conductor $172.457$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$172.457072203$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 589050$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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 $$\beta = 72\sqrt{2356201}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 60840 - \beta ) q^{2} + ( -18959940 + 312 \beta ) q^{3} + ( 7326116992 - 121680 \beta ) q^{4} + ( -4964461096608 + 37942020 \beta ) q^{6} + ( 33576540033400 + 801594864 \beta ) q^{7} + ( 1409375292549120 - 6139193600 \beta ) q^{8} + ( -4010568477485427 - 11831002560 \beta ) q^{9} +O(q^{10})$$ $$q +(60840 - \beta) q^{2} +(-18959940 + 312 \beta) q^{3} +(7326116992 - 121680 \beta) q^{4} +(-4964461096608 + 37942020 \beta) q^{6} +(33576540033400 + 801594864 \beta) q^{7} +(1409375292549120 - 6139193600 \beta) q^{8} +(-4010568477485427 - 11831002560 \beta) q^{9} +(66935907720957132 + 1346611783000 \beta) q^{11} +(-602617716665233920 + 4592794000704 \beta) q^{12} +(1490805239129721970 - 3738595861728 \beta) q^{13} +(-7748320631234170176 + 15192491492360 \beta) q^{14} +(97802989555947175936 - 737660590018560 \beta) q^{16} +(39680574630587762670 - 2662090686805056 \beta) q^{17} +(-99492661364271659640 + 3290770281735027 \beta) q^{18} +(-$$$$68\!\cdots\!00$$$$- 4884703150768920 \beta) q^{19} +($$$$24\!\cdots\!12$$$$- 4722310035327360 \beta) q^{21} +(-$$$$12\!\cdots\!20$$$$+ 14991953156762868 \beta) q^{22} +(-$$$$13\!\cdots\!20$$$$+ 164629195362887632 \beta) q^{23} +(-$$$$50\!\cdots\!00$$$$+ 556123833579709440 \beta) q^{24} +($$$$13\!\cdots\!52$$$$- 1718261411357253490 \beta) q^{26} +($$$$13\!\cdots\!20$$$$- 2761409163063330000 \beta) q^{27} +(-$$$$94\!\cdots\!80$$$$+ 1786984362586217088 \beta) q^{28} +(-$$$$83\!\cdots\!50$$$$+ 6085145360184173920 \beta) q^{29} +(-$$$$31\!\cdots\!08$$$$- 32916497064471576000 \beta) q^{31} +($$$$28\!\cdots\!40$$$$- 89946988381051355136 \beta) q^{32} +($$$$38\!\cdots\!20$$$$- 4647675400034394816 \beta) q^{33} +($$$$34\!\cdots\!04$$$$-$$$$20\!\cdots\!10$$$$\beta) q^{34} +(-$$$$11\!\cdots\!84$$$$+$$$$40\!\cdots\!40$$$$\beta) q^{36} +($$$$52\!\cdots\!10$$$$+$$$$20\!\cdots\!04$$$$\beta) q^{37} +($$$$18\!\cdots\!80$$$$+$$$$38\!\cdots\!00$$$$\beta) q^{38} +(-$$$$42\!\cdots\!24$$$$+$$$$53\!\cdots\!60$$$$\beta) q^{39} +($$$$13\!\cdots\!22$$$$+$$$$32\!\cdots\!00$$$$\beta) q^{41} +($$$$20\!\cdots\!20$$$$-$$$$27\!\cdots\!12$$$$\beta) q^{42} +(-$$$$78\!\cdots\!00$$$$+$$$$88\!\cdots\!52$$$$\beta) q^{43} +(-$$$$15\!\cdots\!56$$$$+$$$$17\!\cdots\!40$$$$\beta) q^{44} +(-$$$$28\!\cdots\!88$$$$+$$$$23\!\cdots\!00$$$$\beta) q^{46} +(-$$$$27\!\cdots\!20$$$$+$$$$49\!\cdots\!84$$$$\beta) q^{47} +(-$$$$46\!\cdots\!20$$$$+$$$$44\!\cdots\!32$$$$\beta) q^{48} +($$$$12\!\cdots\!57$$$$+$$$$53\!\cdots\!00$$$$\beta) q^{49} +(-$$$$10\!\cdots\!48$$$$+$$$$62\!\cdots\!80$$$$\beta) q^{51} +($$$$16\!\cdots\!00$$$$-$$$$20\!\cdots\!76$$$$\beta) q^{52} +($$$$13\!\cdots\!10$$$$+$$$$20\!\cdots\!12$$$$\beta) q^{53} +($$$$42\!\cdots\!00$$$$-$$$$30\!\cdots\!20$$$$\beta) q^{54} +(-$$$$12\!\cdots\!00$$$$+$$$$92\!\cdots\!80$$$$\beta) q^{56} +(-$$$$57\!\cdots\!60$$$$-$$$$11\!\cdots\!00$$$$\beta) q^{57} +(-$$$$12\!\cdots\!80$$$$+$$$$12\!\cdots\!50$$$$\beta) q^{58} +(-$$$$15\!\cdots\!00$$$$-$$$$31\!\cdots\!60$$$$\beta) q^{59} +(-$$$$28\!\cdots\!18$$$$+$$$$63\!\cdots\!00$$$$\beta) q^{61} +($$$$21\!\cdots\!80$$$$+$$$$11\!\cdots\!08$$$$\beta) q^{62} +(-$$$$25\!\cdots\!60$$$$-$$$$36\!\cdots\!28$$$$\beta) q^{63} +($$$$43\!\cdots\!12$$$$-$$$$19\!\cdots\!60$$$$\beta) q^{64} +($$$$29\!\cdots\!44$$$$-$$$$41\!\cdots\!60$$$$\beta) q^{66} +(-$$$$79\!\cdots\!80$$$$+$$$$28\!\cdots\!44$$$$\beta) q^{67} +($$$$42\!\cdots\!60$$$$-$$$$24\!\cdots\!52$$$$\beta) q^{68} +($$$$87\!\cdots\!56$$$$-$$$$72\!\cdots\!20$$$$\beta) q^{69} +(-$$$$13\!\cdots\!88$$$$-$$$$18\!\cdots\!00$$$$\beta) q^{71} +(-$$$$47\!\cdots\!40$$$$+$$$$79\!\cdots\!00$$$$\beta) q^{72} +(-$$$$47\!\cdots\!70$$$$-$$$$59\!\cdots\!68$$$$\beta) q^{73} +($$$$72\!\cdots\!64$$$$-$$$$40\!\cdots\!50$$$$\beta) q^{74} +($$$$22\!\cdots\!00$$$$+$$$$46\!\cdots\!60$$$$\beta) q^{76} +($$$$15\!\cdots\!00$$$$+$$$$98\!\cdots\!48$$$$\beta) q^{77} +(-$$$$91\!\cdots\!00$$$$+$$$$75\!\cdots\!24$$$$\beta) q^{78} +(-$$$$42\!\cdots\!00$$$$+$$$$11\!\cdots\!20$$$$\beta) q^{79} +($$$$91\!\cdots\!21$$$$+$$$$16\!\cdots\!20$$$$\beta) q^{81} +(-$$$$31\!\cdots\!20$$$$+$$$$58\!\cdots\!78$$$$\beta) q^{82} +(-$$$$14\!\cdots\!60$$$$+$$$$24\!\cdots\!92$$$$\beta) q^{83} +($$$$24\!\cdots\!04$$$$-$$$$32\!\cdots\!80$$$$\beta) q^{84} +(-$$$$15\!\cdots\!68$$$$+$$$$13\!\cdots\!80$$$$\beta) q^{86} +($$$$39\!\cdots\!60$$$$-$$$$37\!\cdots\!00$$$$\beta) q^{87} +(-$$$$66\!\cdots\!60$$$$+$$$$14\!\cdots\!00$$$$\beta) q^{88} +($$$$66\!\cdots\!50$$$$+$$$$38\!\cdots\!60$$$$\beta) q^{89} +($$$$13\!\cdots\!72$$$$+$$$$10\!\cdots\!80$$$$\beta) q^{91} +(-$$$$34\!\cdots\!80$$$$+$$$$27\!\cdots\!44$$$$\beta) q^{92} +(-$$$$66\!\cdots\!80$$$$-$$$$34\!\cdots\!96$$$$\beta) q^{93} +(-$$$$22\!\cdots\!56$$$$+$$$$30\!\cdots\!80$$$$\beta) q^{94} +(-$$$$39\!\cdots\!88$$$$+$$$$25\!\cdots\!20$$$$\beta) q^{96} +($$$$18\!\cdots\!30$$$$+$$$$53\!\cdots\!84$$$$\beta) q^{97} +(-$$$$58\!\cdots\!20$$$$+$$$$20\!\cdots\!43$$$$\beta) q^{98} +(-$$$$46\!\cdots\!64$$$$-$$$$61\!\cdots\!20$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 121680q^{2} - 37919880q^{3} + 14652233984q^{4} - 9928922193216q^{6} + 67153080066800q^{7} + 2818750585098240q^{8} - 8021136954970854q^{9} + O(q^{10})$$ $$2q + 121680q^{2} - 37919880q^{3} + 14652233984q^{4} - 9928922193216q^{6} + 67153080066800q^{7} + 2818750585098240q^{8} - 8021136954970854q^{9} + 133871815441914264q^{11} - 1205235433330467840q^{12} + 2981610478259443940q^{13} - 15496641262468340352q^{14} +$$$$19\!\cdots\!72$$$$q^{16} + 79361149261175525340q^{17} -$$$$19\!\cdots\!80$$$$q^{18} -$$$$13\!\cdots\!00$$$$q^{19} +$$$$48\!\cdots\!24$$$$q^{21} -$$$$24\!\cdots\!40$$$$q^{22} -$$$$26\!\cdots\!40$$$$q^{23} -$$$$10\!\cdots\!00$$$$q^{24} +$$$$27\!\cdots\!04$$$$q^{26} +$$$$27\!\cdots\!40$$$$q^{27} -$$$$18\!\cdots\!60$$$$q^{28} -$$$$16\!\cdots\!00$$$$q^{29} -$$$$62\!\cdots\!16$$$$q^{31} +$$$$57\!\cdots\!80$$$$q^{32} +$$$$77\!\cdots\!40$$$$q^{33} +$$$$69\!\cdots\!08$$$$q^{34} -$$$$23\!\cdots\!68$$$$q^{36} +$$$$10\!\cdots\!20$$$$q^{37} +$$$$36\!\cdots\!60$$$$q^{38} -$$$$85\!\cdots\!48$$$$q^{39} +$$$$27\!\cdots\!44$$$$q^{41} +$$$$40\!\cdots\!40$$$$q^{42} -$$$$15\!\cdots\!00$$$$q^{43} -$$$$30\!\cdots\!12$$$$q^{44} -$$$$56\!\cdots\!76$$$$q^{46} -$$$$54\!\cdots\!40$$$$q^{47} -$$$$93\!\cdots\!40$$$$q^{48} +$$$$24\!\cdots\!14$$$$q^{49} -$$$$21\!\cdots\!96$$$$q^{51} +$$$$32\!\cdots\!00$$$$q^{52} +$$$$26\!\cdots\!20$$$$q^{53} +$$$$84\!\cdots\!00$$$$q^{54} -$$$$25\!\cdots\!00$$$$q^{56} -$$$$11\!\cdots\!20$$$$q^{57} -$$$$25\!\cdots\!60$$$$q^{58} -$$$$30\!\cdots\!00$$$$q^{59} -$$$$57\!\cdots\!36$$$$q^{61} +$$$$42\!\cdots\!60$$$$q^{62} -$$$$50\!\cdots\!20$$$$q^{63} +$$$$86\!\cdots\!24$$$$q^{64} +$$$$58\!\cdots\!88$$$$q^{66} -$$$$15\!\cdots\!60$$$$q^{67} +$$$$84\!\cdots\!20$$$$q^{68} +$$$$17\!\cdots\!12$$$$q^{69} -$$$$26\!\cdots\!76$$$$q^{71} -$$$$95\!\cdots\!80$$$$q^{72} -$$$$94\!\cdots\!40$$$$q^{73} +$$$$14\!\cdots\!28$$$$q^{74} +$$$$45\!\cdots\!00$$$$q^{76} +$$$$30\!\cdots\!00$$$$q^{77} -$$$$18\!\cdots\!00$$$$q^{78} -$$$$85\!\cdots\!00$$$$q^{79} +$$$$18\!\cdots\!42$$$$q^{81} -$$$$62\!\cdots\!40$$$$q^{82} -$$$$29\!\cdots\!20$$$$q^{83} +$$$$49\!\cdots\!08$$$$q^{84} -$$$$31\!\cdots\!36$$$$q^{86} +$$$$78\!\cdots\!20$$$$q^{87} -$$$$13\!\cdots\!20$$$$q^{88} +$$$$13\!\cdots\!00$$$$q^{89} +$$$$26\!\cdots\!44$$$$q^{91} -$$$$68\!\cdots\!60$$$$q^{92} -$$$$13\!\cdots\!60$$$$q^{93} -$$$$45\!\cdots\!12$$$$q^{94} -$$$$79\!\cdots\!76$$$$q^{96} +$$$$36\!\cdots\!60$$$$q^{97} -$$$$11\!\cdots\!40$$$$q^{98} -$$$$92\!\cdots\!28$$$$q^{99} + O(q^{100})$$

## 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
 767.996 −766.996
−49679.4 1.55221e7 −6.12189e9 0 −7.71130e11 1.22168e14 7.30875e14 −5.31812e15 0
1.2 171359. −5.34420e7 2.07741e10 0 −9.15779e12 −5.50153e13 2.08788e15 −2.70301e15 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 25.34.a.a 2
5.b even 2 1 1.34.a.a 2
5.c odd 4 2 25.34.b.a 4
15.d odd 2 1 9.34.a.b 2
20.d odd 2 1 16.34.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 5.b even 2 1
9.34.a.b 2 15.d odd 2 1
16.34.a.b 2 20.d odd 2 1
25.34.a.a 2 1.a even 1 1 trivial
25.34.b.a 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 121680 T_{2} - 8513040384$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8513040384 - 121680 T + T^{2}$$
$3$ $$-829533439462896 + 37919880 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-$$$$67\!\cdots\!64$$$$- 67153080066800 T + T^{2}$$
$11$ $$-$$$$17\!\cdots\!76$$$$- 133871815441914264 T + T^{2}$$
$13$ $$20\!\cdots\!44$$$$- 2981610478259443940 T + T^{2}$$
$17$ $$-$$$$84\!\cdots\!24$$$$- 79361149261175525340 T + T^{2}$$
$19$ $$17\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T + T^{2}$$
$23$ $$-$$$$15\!\cdots\!16$$$$+$$$$26\!\cdots\!40$$$$T + T^{2}$$
$29$ $$24\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$T + T^{2}$$
$31$ $$-$$$$35\!\cdots\!36$$$$+$$$$62\!\cdots\!16$$$$T + T^{2}$$
$37$ $$22\!\cdots\!56$$$$-$$$$10\!\cdots\!20$$$$T + T^{2}$$
$41$ $$-$$$$10\!\cdots\!16$$$$-$$$$27\!\cdots\!44$$$$T + T^{2}$$
$43$ $$-$$$$35\!\cdots\!36$$$$+$$$$15\!\cdots\!00$$$$T + T^{2}$$
$47$ $$70\!\cdots\!96$$$$+$$$$54\!\cdots\!40$$$$T + T^{2}$$
$53$ $$-$$$$33\!\cdots\!96$$$$-$$$$26\!\cdots\!20$$$$T + T^{2}$$
$59$ $$22\!\cdots\!00$$$$+$$$$30\!\cdots\!00$$$$T + T^{2}$$
$61$ $$-$$$$41\!\cdots\!76$$$$+$$$$57\!\cdots\!36$$$$T + T^{2}$$
$67$ $$54\!\cdots\!76$$$$+$$$$15\!\cdots\!60$$$$T + T^{2}$$
$71$ $$-$$$$26\!\cdots\!56$$$$+$$$$26\!\cdots\!76$$$$T + T^{2}$$
$73$ $$-$$$$42\!\cdots\!16$$$$+$$$$94\!\cdots\!40$$$$T + T^{2}$$
$79$ $$16\!\cdots\!00$$$$+$$$$85\!\cdots\!00$$$$T + T^{2}$$
$83$ $$-$$$$50\!\cdots\!76$$$$+$$$$29\!\cdots\!20$$$$T + T^{2}$$
$89$ $$43\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T + T^{2}$$
$97$ $$-$$$$31\!\cdots\!04$$$$-$$$$36\!\cdots\!60$$$$T + T^{2}$$
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