# Properties

 Label 25.38.b.a Level 25 Weight 38 Character orbit 25.b Analytic conductor 216.785 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$38$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$216.785095312$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Defining polynomial: $$x^{4} + 31868761 x^{2} + 253904465984400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}\cdot 5^{4}$$ Twist minimal: no (minimal twist has level 1) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - 1944 \beta_{2} ) q^{2} + ( 72 \beta_{1} - 139914 \beta_{2} ) q^{3} + ( -18860134912 + 3888 \beta_{3} ) q^{4} + ( -11253271923648 + 279882 \beta_{3} ) q^{6} + ( 9650004336 \beta_{1} - 34484439534860 \beta_{2} ) q^{7} + ( 99683138560 \beta_{1} + 340440430436352 \beta_{2} ) q^{8} + ( 449473689200884707 + 20147616 \beta_{3} ) q^{9} +O(q^{10})$$ $$q +(\beta_{1} - 1944 \beta_{2}) q^{2} +(72 \beta_{1} - 139914 \beta_{2}) q^{3} +(-18860134912 + 3888 \beta_{3}) q^{4} +(-11253271923648 + 279882 \beta_{3}) q^{6} +(9650004336 \beta_{1} - 34484439534860 \beta_{2}) q^{7} +(99683138560 \beta_{1} + 340440430436352 \beta_{2}) q^{8} +(449473689200884707 + 20147616 \beta_{3}) q^{9} +(-13367018177424269028 - 643726129420 \beta_{3}) q^{11} +(-2717893793664 \beta_{1} + 43747747983700992 \beta_{2}) q^{12} +(-724443400725408 \beta_{1} - 5305817416539331783 \beta_{2}) q^{13} +(-$$$$15\!\cdots\!24$$$$+ 53244047964044 \beta_{3}) q^{14} +(-$$$$15\!\cdots\!04$$$$+ 387706242023424 \beta_{3}) q^{16} +(24633489938571456 \beta_{1} -$$$$89\!\cdots\!27$$$$\beta_{2}) q^{17} +(449375771787124707 \beta_{1} -$$$$87\!\cdots\!64$$$$\beta_{2}) q^{18} +(-$$$$18\!\cdots\!60$$$$- 25859523645743148 \beta_{3}) q^{19} +(-$$$$11\!\cdots\!28$$$$+ 3833050353177024 \beta_{3}) q^{21} +(-10238509188443069028 \beta_{1} -$$$$68\!\cdots\!48$$$$\beta_{2}) q^{22} +(-34024622863285905488 \beta_{1} +$$$$26\!\cdots\!28$$$$\beta_{2}) q^{23} +(-$$$$93\!\cdots\!80$$$$- 10564644342933504 \beta_{3}) q^{24} +($$$$80\!\cdots\!72$$$$+ 3897499445529138631 \beta_{3}) q^{26} +(64775499512752949040 \beta_{1} -$$$$12\!\cdots\!12$$$$\beta_{2}) q^{27} +(-$$$$51\!\cdots\!32$$$$\beta_{1} +$$$$61\!\cdots\!32$$$$\beta_{2}) q^{28} +($$$$63\!\cdots\!10$$$$+ 46166393183925012208 \beta_{3}) q^{29} +($$$$13\!\cdots\!12$$$$+$$$$12\!\cdots\!40$$$$\beta_{3}) q^{31} +(-$$$$37\!\cdots\!84$$$$\beta_{1} +$$$$13\!\cdots\!36$$$$\beta_{2}) q^{32} +(-$$$$73\!\cdots\!16$$$$\beta_{1} -$$$$49\!\cdots\!68$$$$\beta_{2}) q^{33} +(-$$$$79\!\cdots\!04$$$$+$$$$94\!\cdots\!91$$$$\beta_{3}) q^{34} +(-$$$$84\!\cdots\!84$$$$+$$$$17\!\cdots\!24$$$$\beta_{3}) q^{36} +(-$$$$25\!\cdots\!04$$$$\beta_{1} -$$$$68\!\cdots\!81$$$$\beta_{2}) q^{37} +(-$$$$60\!\cdots\!60$$$$\beta_{1} -$$$$34\!\cdots\!92$$$$\beta_{2}) q^{38} +($$$$58\!\cdots\!84$$$$+$$$$28\!\cdots\!64$$$$\beta_{3}) q^{39} +(-$$$$63\!\cdots\!18$$$$+$$$$11\!\cdots\!40$$$$\beta_{3}) q^{41} +(-$$$$13\!\cdots\!28$$$$\beta_{1} +$$$$78\!\cdots\!48$$$$\beta_{2}) q^{42} +($$$$42\!\cdots\!52$$$$\beta_{1} +$$$$25\!\cdots\!50$$$$\beta_{2}) q^{43} +(-$$$$66\!\cdots\!64$$$$-$$$$39\!\cdots\!24$$$$\beta_{3}) q^{44} +($$$$62\!\cdots\!92$$$$-$$$$32\!\cdots\!00$$$$\beta_{3}) q^{46} +($$$$38\!\cdots\!16$$$$\beta_{1} +$$$$42\!\cdots\!32$$$$\beta_{2}) q^{47} +(-$$$$12\!\cdots\!88$$$$\beta_{1} +$$$$62\!\cdots\!08$$$$\beta_{2}) q^{48} +($$$$19\!\cdots\!43$$$$+$$$$66\!\cdots\!20$$$$\beta_{3}) q^{49} +(-$$$$57\!\cdots\!88$$$$+$$$$67\!\cdots\!28$$$$\beta_{3}) q^{51} +(-$$$$37\!\cdots\!04$$$$\beta_{1} -$$$$31\!\cdots\!40$$$$\beta_{2}) q^{52} +($$$$11\!\cdots\!72$$$$\beta_{1} -$$$$15\!\cdots\!39$$$$\beta_{2}) q^{53} +(-$$$$10\!\cdots\!60$$$$+$$$$25\!\cdots\!72$$$$\beta_{3}) q^{54} +(-$$$$11\!\cdots\!40$$$$+$$$$15\!\cdots\!28$$$$\beta_{3}) q^{56} +(-$$$$43\!\cdots\!20$$$$\beta_{1} -$$$$24\!\cdots\!64$$$$\beta_{2}) q^{57} +($$$$41\!\cdots\!10$$$$\beta_{1} +$$$$55\!\cdots\!32$$$$\beta_{2}) q^{58} +($$$$11\!\cdots\!20$$$$+$$$$96\!\cdots\!16$$$$\beta_{3}) q^{59} +($$$$50\!\cdots\!22$$$$+$$$$57\!\cdots\!00$$$$\beta_{3}) q^{61} +(-$$$$49\!\cdots\!88$$$$\beta_{1} +$$$$18\!\cdots\!32$$$$\beta_{2}) q^{62} +($$$$43\!\cdots\!52$$$$\beta_{1} -$$$$15\!\cdots\!36$$$$\beta_{2}) q^{63} +(-$$$$93\!\cdots\!32$$$$-$$$$88\!\cdots\!04$$$$\beta_{3}) q^{64} +($$$$84\!\cdots\!44$$$$+$$$$35\!\cdots\!64$$$$\beta_{3}) q^{66} +($$$$78\!\cdots\!56$$$$\beta_{1} -$$$$10\!\cdots\!62$$$$\beta_{2}) q^{67} +(-$$$$91\!\cdots\!72$$$$\beta_{1} +$$$$30\!\cdots\!76$$$$\beta_{2}) q^{68} +($$$$45\!\cdots\!24$$$$-$$$$23\!\cdots\!48$$$$\beta_{3}) q^{69} +(-$$$$37\!\cdots\!08$$$$+$$$$34\!\cdots\!00$$$$\beta_{3}) q^{71} +($$$$44\!\cdots\!20$$$$\beta_{1} +$$$$15\!\cdots\!04$$$$\beta_{2}) q^{72} +(-$$$$11\!\cdots\!88$$$$\beta_{1} -$$$$19\!\cdots\!77$$$$\beta_{2}) q^{73} +($$$$33\!\cdots\!36$$$$+$$$$19\!\cdots\!05$$$$\beta_{3}) q^{74} +(-$$$$33\!\cdots\!80$$$$-$$$$23\!\cdots\!04$$$$\beta_{3}) q^{76} +(-$$$$73\!\cdots\!08$$$$\beta_{1} -$$$$45\!\cdots\!00$$$$\beta_{2}) q^{77} +($$$$44\!\cdots\!84$$$$\beta_{1} +$$$$29\!\cdots\!80$$$$\beta_{2}) q^{78} +(-$$$$13\!\cdots\!40$$$$+$$$$81\!\cdots\!48$$$$\beta_{3}) q^{79} +($$$$20\!\cdots\!21$$$$+$$$$27\!\cdots\!32$$$$\beta_{3}) q^{81} +(-$$$$68\!\cdots\!18$$$$\beta_{1} +$$$$29\!\cdots\!52$$$$\beta_{2}) q^{82} +($$$$23\!\cdots\!32$$$$\beta_{1} +$$$$47\!\cdots\!34$$$$\beta_{2}) q^{83} +($$$$76\!\cdots\!36$$$$-$$$$51\!\cdots\!52$$$$\beta_{3}) q^{84} +(-$$$$50\!\cdots\!68$$$$-$$$$17\!\cdots\!62$$$$\beta_{3}) q^{86} +($$$$29\!\cdots\!20$$$$\beta_{1} +$$$$39\!\cdots\!44$$$$\beta_{2}) q^{87} +(-$$$$18\!\cdots\!80$$$$\beta_{1} -$$$$13\!\cdots\!56$$$$\beta_{2}) q^{88} +($$$$66\!\cdots\!30$$$$+$$$$46\!\cdots\!04$$$$\beta_{3}) q^{89} +($$$$56\!\cdots\!92$$$$+$$$$26\!\cdots\!08$$$$\beta_{3}) q^{91} +($$$$31\!\cdots\!56$$$$\beta_{1} -$$$$24\!\cdots\!32$$$$\beta_{2}) q^{92} +(-$$$$35\!\cdots\!36$$$$\beta_{1} +$$$$13\!\cdots\!52$$$$\beta_{2}) q^{93} +(-$$$$35\!\cdots\!44$$$$-$$$$35\!\cdots\!28$$$$\beta_{3}) q^{94} +($$$$86\!\cdots\!32$$$$-$$$$10\!\cdots\!68$$$$\beta_{3}) q^{96} +(-$$$$10\!\cdots\!84$$$$\beta_{1} +$$$$60\!\cdots\!77$$$$\beta_{2}) q^{97} +(-$$$$13\!\cdots\!57$$$$\beta_{1} +$$$$94\!\cdots\!88$$$$\beta_{2}) q^{98} +(-$$$$60\!\cdots\!96$$$$-$$$$28\!\cdots\!88$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 75440539648q^{4} - 45013087694592q^{6} + 1797894756803538828q^{9} + O(q^{10})$$ $$4q - 75440539648q^{4} - 45013087694592q^{6} + 1797894756803538828q^{9} - 53468072709697076112q^{11} -$$$$63\!\cdots\!96$$$$q^{14} -$$$$62\!\cdots\!16$$$$q^{16} -$$$$74\!\cdots\!40$$$$q^{19} -$$$$45\!\cdots\!12$$$$q^{21} -$$$$37\!\cdots\!20$$$$q^{24} +$$$$32\!\cdots\!88$$$$q^{26} +$$$$25\!\cdots\!40$$$$q^{29} +$$$$52\!\cdots\!48$$$$q^{31} -$$$$31\!\cdots\!16$$$$q^{34} -$$$$33\!\cdots\!36$$$$q^{36} +$$$$23\!\cdots\!36$$$$q^{39} -$$$$25\!\cdots\!72$$$$q^{41} -$$$$26\!\cdots\!56$$$$q^{44} +$$$$25\!\cdots\!68$$$$q^{46} +$$$$76\!\cdots\!72$$$$q^{49} -$$$$22\!\cdots\!52$$$$q^{51} -$$$$40\!\cdots\!40$$$$q^{54} -$$$$44\!\cdots\!60$$$$q^{56} +$$$$47\!\cdots\!80$$$$q^{59} +$$$$20\!\cdots\!88$$$$q^{61} -$$$$37\!\cdots\!28$$$$q^{64} +$$$$33\!\cdots\!76$$$$q^{66} +$$$$18\!\cdots\!96$$$$q^{69} -$$$$14\!\cdots\!32$$$$q^{71} +$$$$13\!\cdots\!44$$$$q^{74} -$$$$13\!\cdots\!20$$$$q^{76} -$$$$54\!\cdots\!60$$$$q^{79} +$$$$80\!\cdots\!84$$$$q^{81} +$$$$30\!\cdots\!44$$$$q^{84} -$$$$20\!\cdots\!72$$$$q^{86} +$$$$26\!\cdots\!20$$$$q^{89} +$$$$22\!\cdots\!68$$$$q^{91} -$$$$14\!\cdots\!76$$$$q^{94} +$$$$34\!\cdots\!28$$$$q^{96} -$$$$24\!\cdots\!84$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 31868761 x^{2} + 253904465984400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{3} + 191212564 \nu$$$$)/1327865$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{3} + 79671905 \nu$$$$)/1593438$$ $$\beta_{3}$$ $$=$$ $$4800 \nu^{2} + 76485026400$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-24 \beta_{2} + 25 \beta_{1}$$$$)/2400$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 76485026400$$$$)/4800$$ $$\nu^{3}$$ $$=$$ $$($$$$1147275384 \beta_{2} - 398359525 \beta_{1}$$$$)/2400$$

## 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
 − 3992.29i − 3991.29i 3991.29i 3992.29i
480412.i 3.45869e7i −9.33565e10 0 −1.66160e13 5.42222e15i 2.11777e16i 4.49088e17 0
24.2 286012.i 2.05955e7i 5.56362e10 0 −5.89057e12 1.97377e15i 5.52218e16i 4.49860e17 0
24.3 286012.i 2.05955e7i 5.56362e10 0 −5.89057e12 1.97377e15i 5.52218e16i 4.49860e17 0
24.4 480412.i 3.45869e7i −9.33565e10 0 −1.66160e13 5.42222e15i 2.11777e16i 4.49088e17 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 25.38.b.a 4
5.b even 2 1 inner 25.38.b.a 4
5.c odd 4 1 1.38.a.a 2
5.c odd 4 1 25.38.a.a 2
15.e even 4 1 9.38.a.a 2
20.e even 4 1 16.38.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 5.c odd 4 1
9.38.a.a 2 15.e even 4 1
16.38.a.b 2 20.e even 4 1
25.38.a.a 2 5.c odd 4 1
25.38.b.a 4 1.a even 1 1 trivial
25.38.b.a 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 312598176768 T_{2}^{2} +$$$$18\!\cdots\!56$$ acting on $$S_{38}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 237157637120 T^{2} +$$$$46\!\cdots\!68$$$$T^{4} -$$$$44\!\cdots\!80$$$$T^{6} +$$$$35\!\cdots\!56$$$$T^{8}$$
$3$ $$1 - 1799515190183764140 T^{2} +$$$$12\!\cdots\!38$$$$T^{4} -$$$$36\!\cdots\!60$$$$T^{6} +$$$$41\!\cdots\!61$$$$T^{8}$$
$5$ 1
$7$ $$1 -$$$$40\!\cdots\!00$$$$T^{2} +$$$$94\!\cdots\!98$$$$T^{4} -$$$$14\!\cdots\!00$$$$T^{6} +$$$$11\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 26734036354848538056 T +$$$$70\!\cdots\!26$$$$T^{2} +$$$$90\!\cdots\!76$$$$T^{3} +$$$$11\!\cdots\!41$$$$T^{4} )^{2}$$
$13$ $$1 -$$$$36\!\cdots\!80$$$$T^{2} +$$$$65\!\cdots\!78$$$$T^{4} -$$$$98\!\cdots\!20$$$$T^{6} +$$$$73\!\cdots\!21$$$$T^{8}$$
$17$ $$1 -$$$$92\!\cdots\!60$$$$T^{2} +$$$$43\!\cdots\!58$$$$T^{4} -$$$$10\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$19$ $$( 1 +$$$$37\!\cdots\!20$$$$T +$$$$20\!\cdots\!78$$$$T^{2} +$$$$76\!\cdots\!80$$$$T^{3} +$$$$42\!\cdots\!21$$$$T^{4} )^{2}$$
$23$ $$1 -$$$$28\!\cdots\!20$$$$T^{2} +$$$$20\!\cdots\!18$$$$T^{4} -$$$$16\!\cdots\!80$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$29$ $$( 1 -$$$$12\!\cdots\!20$$$$T +$$$$21\!\cdots\!18$$$$T^{2} -$$$$16\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!81$$$$T^{4} )^{2}$$
$31$ $$( 1 -$$$$26\!\cdots\!24$$$$T +$$$$24\!\cdots\!66$$$$T^{2} -$$$$39\!\cdots\!64$$$$T^{3} +$$$$22\!\cdots\!21$$$$T^{4} )^{2}$$
$37$ $$1 -$$$$21\!\cdots\!80$$$$T^{2} +$$$$29\!\cdots\!78$$$$T^{4} -$$$$23\!\cdots\!20$$$$T^{6} +$$$$12\!\cdots\!21$$$$T^{8}$$
$41$ $$( 1 +$$$$12\!\cdots\!36$$$$T +$$$$12\!\cdots\!86$$$$T^{2} +$$$$59\!\cdots\!16$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4} )^{2}$$
$43$ $$1 -$$$$22\!\cdots\!00$$$$T^{2} -$$$$13\!\cdots\!02$$$$T^{4} -$$$$17\!\cdots\!00$$$$T^{6} +$$$$56\!\cdots\!01$$$$T^{8}$$
$47$ $$1 -$$$$28\!\cdots\!40$$$$T^{2} +$$$$30\!\cdots\!38$$$$T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$29\!\cdots\!61$$$$T^{8}$$
$53$ $$1 -$$$$84\!\cdots\!40$$$$T^{2} +$$$$46\!\cdots\!38$$$$T^{4} -$$$$33\!\cdots\!60$$$$T^{6} +$$$$15\!\cdots\!61$$$$T^{8}$$
$59$ $$( 1 -$$$$23\!\cdots\!40$$$$T +$$$$64\!\cdots\!38$$$$T^{2} -$$$$79\!\cdots\!60$$$$T^{3} +$$$$11\!\cdots\!61$$$$T^{4} )^{2}$$
$61$ $$( 1 -$$$$10\!\cdots\!44$$$$T +$$$$10\!\cdots\!26$$$$T^{2} -$$$$11\!\cdots\!24$$$$T^{3} +$$$$13\!\cdots\!41$$$$T^{4} )^{2}$$
$67$ $$1 -$$$$69\!\cdots\!60$$$$T^{2} +$$$$28\!\cdots\!58$$$$T^{4} -$$$$93\!\cdots\!40$$$$T^{6} +$$$$18\!\cdots\!41$$$$T^{8}$$
$71$ $$( 1 +$$$$74\!\cdots\!16$$$$T +$$$$58\!\cdots\!46$$$$T^{2} +$$$$23\!\cdots\!56$$$$T^{3} +$$$$98\!\cdots\!81$$$$T^{4} )^{2}$$
$73$ $$1 -$$$$33\!\cdots\!20$$$$T^{2} +$$$$42\!\cdots\!18$$$$T^{4} -$$$$25\!\cdots\!80$$$$T^{6} +$$$$59\!\cdots\!81$$$$T^{8}$$
$79$ $$( 1 +$$$$27\!\cdots\!80$$$$T +$$$$50\!\cdots\!18$$$$T^{2} +$$$$44\!\cdots\!20$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4} )^{2}$$
$83$ $$1 -$$$$27\!\cdots\!60$$$$T^{2} +$$$$38\!\cdots\!58$$$$T^{4} -$$$$28\!\cdots\!40$$$$T^{6} +$$$$10\!\cdots\!41$$$$T^{8}$$
$89$ $$( 1 -$$$$13\!\cdots\!60$$$$T +$$$$23\!\cdots\!58$$$$T^{2} -$$$$17\!\cdots\!40$$$$T^{3} +$$$$17\!\cdots\!41$$$$T^{4} )^{2}$$
$97$ $$1 -$$$$78\!\cdots\!40$$$$T^{2} +$$$$30\!\cdots\!38$$$$T^{4} -$$$$82\!\cdots\!60$$$$T^{6} +$$$$11\!\cdots\!61$$$$T^{8}$$