# Properties

 Label 25.34.b.a Level $25$ Weight $34$ Character orbit 25.b Analytic conductor $172.457$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.457072203$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Defining polynomial: $$x^{4} + 1178101 x^{2} + 346979902500$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 6084 \beta_{1} + \beta_{2} ) q^{2} + ( 1895994 \beta_{1} + 312 \beta_{2} ) q^{3} + ( -7326116992 + 12168 \beta_{3} ) q^{4} + ( -4964461096608 + 3794202 \beta_{3} ) q^{6} + ( 3357654003340 \beta_{1} - 801594864 \beta_{2} ) q^{7} + ( -140937529254912 \beta_{1} - 6139193600 \beta_{2} ) q^{8} + ( 4010568477485427 + 1183100256 \beta_{3} ) q^{9} +O(q^{10})$$ $$q +(6084 \beta_{1} + \beta_{2}) q^{2} +(1895994 \beta_{1} + 312 \beta_{2}) q^{3} +(-7326116992 + 12168 \beta_{3}) q^{4} +(-4964461096608 + 3794202 \beta_{3}) q^{6} +(3357654003340 \beta_{1} - 801594864 \beta_{2}) q^{7} +(-140937529254912 \beta_{1} - 6139193600 \beta_{2}) q^{8} +(4010568477485427 + 1183100256 \beta_{3}) q^{9} +(66935907720957132 + 134661178300 \beta_{3}) q^{11} +(-60261771666523392 \beta_{1} - 4592794000704 \beta_{2}) q^{12} +(-149080523912972197 \beta_{1} - 3738595861728 \beta_{2}) q^{13} +(7748320631234170176 - 1519249149236 \beta_{3}) q^{14} +(97802989555947175936 - 73766059001856 \beta_{3}) q^{16} +(3968057463058776267 \beta_{1} + 2662090686805056 \beta_{2}) q^{17} +(9949266136427165964 \beta_{1} + 3290770281735027 \beta_{2}) q^{18} +($$$$68\!\cdots\!00$$$$+ 488470315076892 \beta_{3}) q^{19} +($$$$24\!\cdots\!12$$$$- 472231003532736 \beta_{3}) q^{21} +(-$$$$12\!\cdots\!12$$$$\beta_{1} - 14991953156762868 \beta_{2}) q^{22} +($$$$13\!\cdots\!72$$$$\beta_{1} + 164629195362887632 \beta_{2}) q^{23} +($$$$50\!\cdots\!00$$$$- 55612383357970944 \beta_{3}) q^{24} +($$$$13\!\cdots\!52$$$$- 171826141135725349 \beta_{3}) q^{26} +($$$$13\!\cdots\!52$$$$\beta_{1} + 2761409163063330000 \beta_{2}) q^{27} +($$$$94\!\cdots\!88$$$$\beta_{1} + 1786984362586217088 \beta_{2}) q^{28} +($$$$83\!\cdots\!50$$$$- 608514536018417392 \beta_{3}) q^{29} +(-$$$$31\!\cdots\!08$$$$- 3291649706447157600 \beta_{3}) q^{31} +($$$$28\!\cdots\!24$$$$\beta_{1} + 89946988381051355136 \beta_{2}) q^{32} +(-$$$$38\!\cdots\!92$$$$\beta_{1} - 4647675400034394816 \beta_{2}) q^{33} +(-$$$$34\!\cdots\!04$$$$+ 20164217201580736971 \beta_{3}) q^{34} +(-$$$$11\!\cdots\!84$$$$+ 40133066345321525784 \beta_{3}) q^{36} +($$$$52\!\cdots\!81$$$$\beta_{1} -$$$$20\!\cdots\!04$$$$\beta_{2}) q^{37} +(-$$$$18\!\cdots\!28$$$$\beta_{1} +$$$$38\!\cdots\!00$$$$\beta_{2}) q^{38} +($$$$42\!\cdots\!24$$$$- 53601478783108443096 \beta_{3}) q^{39} +($$$$13\!\cdots\!22$$$$+$$$$32\!\cdots\!00$$$$\beta_{3}) q^{41} +($$$$20\!\cdots\!32$$$$\beta_{1} +$$$$27\!\cdots\!12$$$$\beta_{2}) q^{42} +($$$$78\!\cdots\!90$$$$\beta_{1} +$$$$88\!\cdots\!52$$$$\beta_{2}) q^{43} +($$$$15\!\cdots\!56$$$$-$$$$17\!\cdots\!24$$$$\beta_{3}) q^{44} +(-$$$$28\!\cdots\!88$$$$+$$$$23\!\cdots\!60$$$$\beta_{3}) q^{46} +(-$$$$27\!\cdots\!32$$$$\beta_{1} -$$$$49\!\cdots\!84$$$$\beta_{2}) q^{47} +($$$$46\!\cdots\!32$$$$\beta_{1} +$$$$44\!\cdots\!32$$$$\beta_{2}) q^{48} +(-$$$$12\!\cdots\!57$$$$-$$$$53\!\cdots\!20$$$$\beta_{3}) q^{49} +(-$$$$10\!\cdots\!48$$$$+$$$$62\!\cdots\!68$$$$\beta_{3}) q^{51} +($$$$16\!\cdots\!60$$$$\beta_{1} +$$$$20\!\cdots\!76$$$$\beta_{2}) q^{52} +(-$$$$13\!\cdots\!61$$$$\beta_{1} +$$$$20\!\cdots\!12$$$$\beta_{2}) q^{53} +(-$$$$42\!\cdots\!00$$$$+$$$$30\!\cdots\!52$$$$\beta_{3}) q^{54} +(-$$$$12\!\cdots\!00$$$$+$$$$92\!\cdots\!68$$$$\beta_{3}) q^{56} +(-$$$$57\!\cdots\!36$$$$\beta_{1} +$$$$11\!\cdots\!00$$$$\beta_{2}) q^{57} +($$$$12\!\cdots\!28$$$$\beta_{1} +$$$$12\!\cdots\!50$$$$\beta_{2}) q^{58} +($$$$15\!\cdots\!00$$$$+$$$$31\!\cdots\!36$$$$\beta_{3}) q^{59} +(-$$$$28\!\cdots\!18$$$$+$$$$63\!\cdots\!00$$$$\beta_{3}) q^{61} +($$$$21\!\cdots\!28$$$$\beta_{1} -$$$$11\!\cdots\!08$$$$\beta_{2}) q^{62} +($$$$25\!\cdots\!36$$$$\beta_{1} -$$$$36\!\cdots\!28$$$$\beta_{2}) q^{63} +(-$$$$43\!\cdots\!12$$$$+$$$$19\!\cdots\!96$$$$\beta_{3}) q^{64} +($$$$29\!\cdots\!44$$$$-$$$$41\!\cdots\!36$$$$\beta_{3}) q^{66} +(-$$$$79\!\cdots\!98$$$$\beta_{1} -$$$$28\!\cdots\!44$$$$\beta_{2}) q^{67} +(-$$$$42\!\cdots\!36$$$$\beta_{1} -$$$$24\!\cdots\!52$$$$\beta_{2}) q^{68} +(-$$$$87\!\cdots\!56$$$$+$$$$72\!\cdots\!72$$$$\beta_{3}) q^{69} +(-$$$$13\!\cdots\!88$$$$-$$$$18\!\cdots\!00$$$$\beta_{3}) q^{71} +(-$$$$47\!\cdots\!24$$$$\beta_{1} -$$$$79\!\cdots\!00$$$$\beta_{2}) q^{72} +($$$$47\!\cdots\!17$$$$\beta_{1} -$$$$59\!\cdots\!68$$$$\beta_{2}) q^{73} +(-$$$$72\!\cdots\!64$$$$+$$$$40\!\cdots\!45$$$$\beta_{3}) q^{74} +($$$$22\!\cdots\!00$$$$+$$$$46\!\cdots\!36$$$$\beta_{3}) q^{76} +($$$$15\!\cdots\!80$$$$\beta_{1} -$$$$98\!\cdots\!48$$$$\beta_{2}) q^{77} +($$$$91\!\cdots\!80$$$$\beta_{1} +$$$$75\!\cdots\!24$$$$\beta_{2}) q^{78} +($$$$42\!\cdots\!00$$$$-$$$$11\!\cdots\!12$$$$\beta_{3}) q^{79} +($$$$91\!\cdots\!21$$$$+$$$$16\!\cdots\!12$$$$\beta_{3}) q^{81} +(-$$$$31\!\cdots\!52$$$$\beta_{1} -$$$$58\!\cdots\!78$$$$\beta_{2}) q^{82} +($$$$14\!\cdots\!46$$$$\beta_{1} +$$$$24\!\cdots\!92$$$$\beta_{2}) q^{83} +(-$$$$24\!\cdots\!04$$$$+$$$$32\!\cdots\!28$$$$\beta_{3}) q^{84} +(-$$$$15\!\cdots\!68$$$$+$$$$13\!\cdots\!58$$$$\beta_{3}) q^{86} +($$$$39\!\cdots\!36$$$$\beta_{1} +$$$$37\!\cdots\!00$$$$\beta_{2}) q^{87} +($$$$66\!\cdots\!16$$$$\beta_{1} +$$$$14\!\cdots\!00$$$$\beta_{2}) q^{88} +(-$$$$66\!\cdots\!50$$$$-$$$$38\!\cdots\!16$$$$\beta_{3}) q^{89} +($$$$13\!\cdots\!72$$$$+$$$$10\!\cdots\!88$$$$\beta_{3}) q^{91} +(-$$$$34\!\cdots\!08$$$$\beta_{1} -$$$$27\!\cdots\!44$$$$\beta_{2}) q^{92} +($$$$66\!\cdots\!48$$$$\beta_{1} -$$$$34\!\cdots\!96$$$$\beta_{2}) q^{93} +($$$$22\!\cdots\!56$$$$-$$$$30\!\cdots\!88$$$$\beta_{3}) q^{94} +(-$$$$39\!\cdots\!88$$$$+$$$$25\!\cdots\!72$$$$\beta_{3}) q^{96} +($$$$18\!\cdots\!03$$$$\beta_{1} -$$$$53\!\cdots\!84$$$$\beta_{2}) q^{97} +($$$$58\!\cdots\!92$$$$\beta_{1} +$$$$20\!\cdots\!43$$$$\beta_{2}) q^{98} +($$$$46\!\cdots\!64$$$$+$$$$61\!\cdots\!92$$$$\beta_{3}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 29304467968q^{4} - 19857844386432q^{6} + 16042273909941708q^{9} + O(q^{10})$$ $$4q - 29304467968q^{4} - 19857844386432q^{6} + 16042273909941708q^{9} + 267743630883828528q^{11} + 30993282524936680704q^{14} +$$$$39\!\cdots\!44$$$$q^{16} +$$$$27\!\cdots\!00$$$$q^{19} +$$$$96\!\cdots\!48$$$$q^{21} +$$$$20\!\cdots\!00$$$$q^{24} +$$$$54\!\cdots\!08$$$$q^{26} +$$$$33\!\cdots\!00$$$$q^{29} -$$$$12\!\cdots\!32$$$$q^{31} -$$$$13\!\cdots\!16$$$$q^{34} -$$$$47\!\cdots\!36$$$$q^{36} +$$$$17\!\cdots\!96$$$$q^{39} +$$$$55\!\cdots\!88$$$$q^{41} +$$$$60\!\cdots\!24$$$$q^{44} -$$$$11\!\cdots\!52$$$$q^{46} -$$$$49\!\cdots\!28$$$$q^{49} -$$$$43\!\cdots\!92$$$$q^{51} -$$$$16\!\cdots\!00$$$$q^{54} -$$$$51\!\cdots\!00$$$$q^{56} +$$$$61\!\cdots\!00$$$$q^{59} -$$$$11\!\cdots\!72$$$$q^{61} -$$$$17\!\cdots\!48$$$$q^{64} +$$$$11\!\cdots\!76$$$$q^{66} -$$$$35\!\cdots\!24$$$$q^{69} -$$$$53\!\cdots\!52$$$$q^{71} -$$$$29\!\cdots\!56$$$$q^{74} +$$$$91\!\cdots\!00$$$$q^{76} +$$$$17\!\cdots\!00$$$$q^{79} +$$$$36\!\cdots\!84$$$$q^{81} -$$$$98\!\cdots\!16$$$$q^{84} -$$$$62\!\cdots\!72$$$$q^{86} -$$$$26\!\cdots\!00$$$$q^{89} +$$$$53\!\cdots\!88$$$$q^{91} +$$$$90\!\cdots\!24$$$$q^{94} -$$$$15\!\cdots\!52$$$$q^{96} +$$$$18\!\cdots\!56$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 1178101 x^{2} + 346979902500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 589051 \nu$$$$)/58905$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 7068604 \nu$$$$)/32725$$ $$\beta_{3}$$ $$=$$ $$1440 \nu^{2} + 848232720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$5 \beta_{2} + 36 \beta_{1}$$$$)/720$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 848232720$$$$)/1440$$ $$\nu^{3}$$ $$=$$ $$($$$$-2945255 \beta_{2} - 63617436 \beta_{1}$$$$)/720$$

## 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
 − 767.996i − 766.996i 766.996i 767.996i
171359.i 5.34420e7i −2.07741e10 0 −9.15779e12 5.50153e13i 2.08788e15i 2.70301e15 0
24.2 49679.4i 1.55221e7i 6.12189e9 0 −7.71130e11 1.22168e14i 7.30875e14i 5.31812e15 0
24.3 49679.4i 1.55221e7i 6.12189e9 0 −7.71130e11 1.22168e14i 7.30875e14i 5.31812e15 0
24.4 171359.i 5.34420e7i −2.07741e10 0 −9.15779e12 5.50153e13i 2.08788e15i 2.70301e15 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 25.34.b.a 4
5.b even 2 1 inner 25.34.b.a 4
5.c odd 4 1 1.34.a.a 2
5.c odd 4 1 25.34.a.a 2
15.e even 4 1 9.34.a.b 2
20.e even 4 1 16.34.a.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 31832103168 T_{2}^{2} +$$$$72\!\cdots\!56$$ acting on $$S_{34}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$72471856579614867456 + 31832103168 T^{2} + T^{4}$$
$3$ $$68\!\cdots\!16$$$$+ 3096984178140192 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$45\!\cdots\!96$$$$+$$$$17\!\cdots\!28$$$$T^{2} + T^{4}$$
$11$ $$( -$$$$17\!\cdots\!76$$$$- 133871815441914264 T + T^{2} )^{2}$$
$13$ $$42\!\cdots\!36$$$$+$$$$47\!\cdots\!12$$$$T^{2} + T^{4}$$
$17$ $$72\!\cdots\!76$$$$+$$$$17\!\cdots\!48$$$$T^{2} + T^{4}$$
$19$ $$($$$$17\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T + T^{2} )^{2}$$
$23$ $$25\!\cdots\!56$$$$+$$$$10\!\cdots\!32$$$$T^{2} + T^{4}$$
$29$ $$($$$$24\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$T + T^{2} )^{2}$$
$31$ $$( -$$$$35\!\cdots\!36$$$$+$$$$62\!\cdots\!16$$$$T + T^{2} )^{2}$$
$37$ $$50\!\cdots\!36$$$$+$$$$64\!\cdots\!88$$$$T^{2} + T^{4}$$
$41$ $$( -$$$$10\!\cdots\!16$$$$-$$$$27\!\cdots\!44$$$$T + T^{2} )^{2}$$
$43$ $$12\!\cdots\!96$$$$+$$$$31\!\cdots\!72$$$$T^{2} + T^{4}$$
$47$ $$49\!\cdots\!16$$$$+$$$$15\!\cdots\!08$$$$T^{2} + T^{4}$$
$53$ $$10\!\cdots\!16$$$$+$$$$13\!\cdots\!92$$$$T^{2} + T^{4}$$
$59$ $$($$$$22\!\cdots\!00$$$$-$$$$30\!\cdots\!00$$$$T + T^{2} )^{2}$$
$61$ $$( -$$$$41\!\cdots\!76$$$$+$$$$57\!\cdots\!36$$$$T + T^{2} )^{2}$$
$67$ $$29\!\cdots\!76$$$$+$$$$14\!\cdots\!48$$$$T^{2} + T^{4}$$
$71$ $$( -$$$$26\!\cdots\!56$$$$+$$$$26\!\cdots\!76$$$$T + T^{2} )^{2}$$
$73$ $$18\!\cdots\!56$$$$+$$$$86\!\cdots\!32$$$$T^{2} + T^{4}$$
$79$ $$($$$$16\!\cdots\!00$$$$-$$$$85\!\cdots\!00$$$$T + T^{2} )^{2}$$
$83$ $$25\!\cdots\!76$$$$+$$$$18\!\cdots\!52$$$$T^{2} + T^{4}$$
$89$ $$($$$$43\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T + T^{2} )^{2}$$
$97$ $$97\!\cdots\!16$$$$+$$$$75\!\cdots\!08$$$$T^{2} + T^{4}$$