# Properties

 Label 48.7.e.d Level 48 Weight 7 Character orbit 48.e Analytic conductor 11.043 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0425960138$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1173604352.2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{22}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{1} ) q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -27 + 3 \beta_{1} - \beta_{5} ) q^{7} + ( -13 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \beta_{1} ) q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -27 + 3 \beta_{1} - \beta_{5} ) q^{7} + ( -13 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( 9 - 15 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{11} + ( 10 + 48 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} ) q^{13} + ( -487 + \beta_{1} - 4 \beta_{2} + 10 \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{15} + ( 36 - 92 \beta_{1} + 16 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} ) q^{17} + ( 697 + 159 \beta_{1} - 21 \beta_{4} + 10 \beta_{5} ) q^{19} + ( -2514 + 21 \beta_{1} + 33 \beta_{2} - 42 \beta_{3} - 24 \beta_{4} - 12 \beta_{5} ) q^{21} + ( 54 - 274 \beta_{1} - 112 \beta_{2} - 64 \beta_{3} - 18 \beta_{4} ) q^{23} + ( 3521 + 120 \beta_{1} - 6 \beta_{4} - 22 \beta_{5} ) q^{25} + ( -6271 + 30 \beta_{1} + 4 \beta_{2} + 98 \beta_{3} - 19 \beta_{4} + 10 \beta_{5} ) q^{27} + ( 72 - 275 \beta_{1} - 59 \beta_{2} + 116 \beta_{3} - 24 \beta_{4} ) q^{29} + ( 12397 - 69 \beta_{1} + 18 \beta_{4} - 31 \beta_{5} ) q^{31} + ( -13838 - 61 \beta_{1} - 23 \beta_{2} - 145 \beta_{3} + 53 \beta_{4} + 55 \beta_{5} ) q^{33} + ( -90 + 634 \beta_{1} + 364 \beta_{2} - 130 \beta_{3} + 30 \beta_{4} ) q^{35} + ( 28554 - 96 \beta_{1} - 18 \beta_{4} + 86 \beta_{5} ) q^{37} + ( -33328 - 34 \beta_{1} + 204 \beta_{2} + 138 \beta_{3} + 102 \beta_{4} - 30 \beta_{5} ) q^{39} + ( -720 + 2090 \beta_{1} - 70 \beta_{2} + 112 \beta_{3} + 240 \beta_{4} ) q^{41} + ( 48977 - 1401 \beta_{1} + 159 \beta_{4} - 10 \beta_{5} ) q^{43} + ( -59356 + 207 \beta_{1} - 293 \beta_{2} - 10 \beta_{3} + 26 \beta_{4} - 98 \beta_{5} ) q^{45} + ( -540 + 1324 \beta_{1} - 296 \beta_{2} + 44 \beta_{3} + 180 \beta_{4} ) q^{47} + ( 87515 - 3672 \beta_{1} + 438 \beta_{4} - 90 \beta_{5} ) q^{49} + ( -75512 - 628 \beta_{1} - 1052 \beta_{2} - 286 \beta_{3} + 164 \beta_{4} - 20 \beta_{5} ) q^{51} + ( 864 - 1877 \beta_{1} + 715 \beta_{2} - 416 \beta_{3} - 288 \beta_{4} ) q^{53} + ( 124784 - 240 \beta_{1} - 54 \beta_{4} + 242 \beta_{5} ) q^{55} + ( -107944 - 763 \beta_{1} + 615 \beta_{2} + 609 \beta_{3} + 429 \beta_{4} - 69 \beta_{5} ) q^{57} + ( 81 - 1059 \beta_{1} - 816 \beta_{2} + 616 \beta_{3} - 27 \beta_{4} ) q^{59} + ( 96250 + 7296 \beta_{1} - 714 \beta_{4} - 290 \beta_{5} ) q^{61} + ( -168645 + 2673 \beta_{1} + 1632 \beta_{2} - 840 \beta_{3} - 408 \beta_{4} + 273 \beta_{5} ) q^{63} + ( -1080 + 2790 \beta_{1} - 450 \beta_{2} - 820 \beta_{3} + 360 \beta_{4} ) q^{65} + ( 93625 + 4575 \beta_{1} - 399 \beta_{4} - 328 \beta_{5} ) q^{67} + ( -163484 + 650 \beta_{1} + 790 \beta_{2} + 1022 \beta_{3} - 1570 \beta_{4} + 562 \beta_{5} ) q^{69} + ( 2106 - 4998 \beta_{1} + 1320 \beta_{2} + 1388 \beta_{3} - 702 \beta_{4} ) q^{71} + ( 188002 - 4176 \beta_{1} + 192 \beta_{4} + 816 \beta_{5} ) q^{73} + ( -85346 - 3689 \beta_{1} + 996 \beta_{2} - 870 \beta_{3} - 474 \beta_{4} - 318 \beta_{5} ) q^{75} + ( 4608 - 17050 \beta_{1} - 3226 \beta_{2} - 752 \beta_{3} - 1536 \beta_{4} ) q^{77} + ( 174269 + 4011 \beta_{1} - 270 \beta_{4} - 527 \beta_{5} ) q^{79} + ( -126379 + 4530 \beta_{1} - 3170 \beta_{2} + 338 \beta_{3} + 1112 \beta_{4} - 716 \beta_{5} ) q^{81} + ( 1827 - 3949 \beta_{1} + 1532 \beta_{2} - 282 \beta_{3} - 609 \beta_{4} ) q^{83} + ( 27776 + 15360 \beta_{1} - 1656 \beta_{4} - 152 \beta_{5} ) q^{85} + ( -209421 - 2445 \beta_{1} - 5028 \beta_{2} + 258 \beta_{3} + 240 \beta_{4} - 771 \beta_{5} ) q^{87} + ( -4140 + 16278 \beta_{1} + 3858 \beta_{2} - 126 \beta_{3} + 1380 \beta_{4} ) q^{89} + ( -138222 - 5154 \beta_{1} + 132 \beta_{4} + 1322 \beta_{5} ) q^{91} + ( 63338 - 12475 \beta_{1} + 213 \beta_{2} - 1464 \beta_{3} - 906 \beta_{4} - 210 \beta_{5} ) q^{93} + ( -3510 + 7282 \beta_{1} - 3248 \beta_{2} - 2480 \beta_{3} + 1170 \beta_{4} ) q^{95} + ( -125446 - 22920 \beta_{1} + 2934 \beta_{4} - 1162 \beta_{5} ) q^{97} + ( 470245 + 17433 \beta_{1} + 3368 \beta_{2} + 2812 \beta_{3} + 103 \beta_{4} + 1670 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 10q^{3} - 156q^{7} - 74q^{9} + O(q^{10})$$ $$6q + 10q^{3} - 156q^{7} - 74q^{9} + 156q^{13} - 2912q^{15} + 4500q^{19} - 15108q^{21} + 21366q^{25} - 37574q^{27} + 74244q^{31} - 83104q^{33} + 171132q^{37} - 200444q^{39} + 291060q^{43} - 355136q^{45} + 517746q^{49} - 452224q^{51} + 748224q^{55} - 650420q^{57} + 592092q^{61} - 1009788q^{63} + 570900q^{67} - 981184q^{69} + 1119660q^{73} - 521446q^{75} + 1053636q^{79} - 742874q^{81} + 197376q^{85} - 1251360q^{87} - 839640q^{91} + 354652q^{93} - 798516q^{97} + 2849600q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 14 x^{4} - 12 x^{3} + 112 x^{2} + 192 x + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 17 \nu^{4} - 40 \nu^{3} - 152 \nu^{2} + 138 \nu + 1143$$$$)/45$$ $$\beta_{2}$$ $$=$$ $$($$$$-53 \nu^{5} + 19 \nu^{4} + 1160 \nu^{3} + 896 \nu^{2} - 8994 \nu - 12069$$$$)/135$$ $$\beta_{3}$$ $$=$$ $$($$$$-128 \nu^{5} + 64 \nu^{4} + 1280 \nu^{3} + 896 \nu^{2} - 15744 \nu - 13824$$$$)/135$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{5} - 13 \nu^{4} - 200 \nu^{3} + 448 \nu^{2} + 1758 \nu - 792$$$$)/15$$ $$\beta_{5}$$ $$=$$ $$($$$$-23 \nu^{5} + 49 \nu^{4} + 200 \nu^{3} - 304 \nu^{2} - 54 \nu + 216$$$$)/15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$6 \beta_{5} + 2 \beta_{4} - 9 \beta_{3} - 36 \beta_{1} + 12$$$$)/1152$$ $$\nu^{2}$$ $$=$$ $$($$$$16 \beta_{4} + 3 \beta_{3} + 24 \beta_{2} + 24 \beta_{1} + 2688$$$$)/576$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} - 2 \beta_{4} - 18 \beta_{3} + 27 \beta_{2} - 18 \beta_{1} + 879$$$$)/144$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{5} + 59 \beta_{4} - 30 \beta_{3} + 192 \beta_{2} + 786 \beta_{1} - 2886$$$$)/288$$ $$\nu^{5}$$ $$=$$ $$($$$$-240 \beta_{5} - 32 \beta_{4} - 783 \beta_{3} + 1440 \beta_{2} + 2448 \beta_{1} - 11856$$$$)/576$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.624336 + 1.41421i −0.624336 − 1.41421i −2.80354 − 1.41421i −2.80354 + 1.41421i 3.42788 + 1.41421i 3.42788 − 1.41421i
0 −23.4408 13.3988i 0 100.147i 0 −56.7723 0 369.944 + 628.158i 0
17.2 0 −23.4408 + 13.3988i 0 100.147i 0 −56.7723 0 369.944 628.158i 0
17.3 0 6.43940 26.2209i 0 10.3581i 0 540.917 0 −646.068 337.693i 0
17.4 0 6.43940 + 26.2209i 0 10.3581i 0 540.917 0 −646.068 + 337.693i 0
17.5 0 22.0014 15.6505i 0 161.417i 0 −562.144 0 239.124 688.666i 0
17.6 0 22.0014 + 15.6505i 0 161.417i 0 −562.144 0 239.124 + 688.666i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.7.e.d 6
3.b odd 2 1 inner 48.7.e.d 6
4.b odd 2 1 24.7.e.a 6
8.b even 2 1 192.7.e.g 6
8.d odd 2 1 192.7.e.h 6
12.b even 2 1 24.7.e.a 6
24.f even 2 1 192.7.e.h 6
24.h odd 2 1 192.7.e.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.e.a 6 4.b odd 2 1
24.7.e.a 6 12.b even 2 1
48.7.e.d 6 1.a even 1 1 trivial
48.7.e.d 6 3.b odd 2 1 inner
192.7.e.g 6 8.b even 2 1
192.7.e.g 6 24.h odd 2 1
192.7.e.h 6 8.d odd 2 1
192.7.e.h 6 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 36192 T_{5}^{4} + 265190400 T_{5}^{2} + 28037120000$$ acting on $$S_{7}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 10 T + 87 T^{2} + 11988 T^{3} + 63423 T^{4} - 5314410 T^{5} + 387420489 T^{6}$$
$5$ $$1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} -$$$$34\!\cdots\!50$$$$T^{10} +$$$$14\!\cdots\!25$$$$T^{12}$$
$7$ $$( 1 + 78 T + 50079 T^{2} + 1090308 T^{3} + 5891744271 T^{4} + 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2}$$
$11$ $$1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} +$$$$40\!\cdots\!75$$$$T^{8} -$$$$32\!\cdots\!10$$$$T^{10} +$$$$30\!\cdots\!61$$$$T^{12}$$
$13$ $$( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} +$$$$11\!\cdots\!29$$$$T^{6} )^{2}$$
$17$ $$1 - 63219270 T^{2} + 2233832285753679 T^{4} -$$$$57\!\cdots\!32$$$$T^{6} +$$$$13\!\cdots\!19$$$$T^{8} -$$$$21\!\cdots\!70$$$$T^{10} +$$$$19\!\cdots\!81$$$$T^{12}$$
$19$ $$( 1 - 2250 T + 64193463 T^{2} - 323410900012 T^{3} + 3020038021275903 T^{4} - 4979958567898862250 T^{5} +$$$$10\!\cdots\!41$$$$T^{6} )^{2}$$
$23$ $$1 + 37480026 T^{2} + 40240367212774575 T^{4} +$$$$51\!\cdots\!12$$$$T^{6} +$$$$88\!\cdots\!75$$$$T^{8} +$$$$17\!\cdots\!66$$$$T^{10} +$$$$10\!\cdots\!61$$$$T^{12}$$
$29$ $$1 - 1660966710 T^{2} + 1523344801209102879 T^{4} -$$$$10\!\cdots\!08$$$$T^{6} +$$$$53\!\cdots\!39$$$$T^{8} -$$$$20\!\cdots\!10$$$$T^{10} +$$$$44\!\cdots\!21$$$$T^{12}$$
$31$ $$( 1 - 37122 T + 2847433071 T^{2} - 66134143254940 T^{3} + 2527107331913634351 T^{4} -$$$$29\!\cdots\!42$$$$T^{5} +$$$$69\!\cdots\!41$$$$T^{6} )^{2}$$
$37$ $$( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} -$$$$56\!\cdots\!46$$$$T^{5} +$$$$16\!\cdots\!29$$$$T^{6} )^{2}$$
$41$ $$1 - 7569104550 T^{2} + 1363608137405888943 T^{4} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$30\!\cdots\!83$$$$T^{8} -$$$$38\!\cdots\!50$$$$T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$
$43$ $$( 1 - 145530 T + 21837177927 T^{2} - 1677260136965132 T^{3} +$$$$13\!\cdots\!23$$$$T^{4} -$$$$58\!\cdots\!30$$$$T^{5} +$$$$25\!\cdots\!49$$$$T^{6} )^{2}$$
$47$ $$1 - 52339293510 T^{2} +$$$$12\!\cdots\!23$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$14\!\cdots\!43$$$$T^{8} -$$$$70\!\cdots\!10$$$$T^{10} +$$$$15\!\cdots\!21$$$$T^{12}$$
$53$ $$1 - 75107476374 T^{2} +$$$$29\!\cdots\!67$$$$T^{4} -$$$$77\!\cdots\!24$$$$T^{6} +$$$$14\!\cdots\!47$$$$T^{8} -$$$$18\!\cdots\!94$$$$T^{10} +$$$$11\!\cdots\!21$$$$T^{12}$$
$59$ $$1 - 180693503190 T^{2} +$$$$15\!\cdots\!31$$$$T^{4} -$$$$77\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!11$$$$T^{8} -$$$$57\!\cdots\!90$$$$T^{10} +$$$$56\!\cdots\!41$$$$T^{12}$$
$61$ $$( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} +$$$$21\!\cdots\!35$$$$T^{4} -$$$$78\!\cdots\!66$$$$T^{5} +$$$$13\!\cdots\!81$$$$T^{6} )^{2}$$
$67$ $$( 1 - 285450 T + 220735162839 T^{2} - 36790378201873708 T^{3} +$$$$19\!\cdots\!91$$$$T^{4} -$$$$23\!\cdots\!50$$$$T^{5} +$$$$74\!\cdots\!09$$$$T^{6} )^{2}$$
$71$ $$1 - 402983850726 T^{2} +$$$$77\!\cdots\!75$$$$T^{4} -$$$$10\!\cdots\!48$$$$T^{6} +$$$$12\!\cdots\!75$$$$T^{8} -$$$$10\!\cdots\!06$$$$T^{10} +$$$$44\!\cdots\!21$$$$T^{12}$$
$73$ $$( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} +$$$$49\!\cdots\!43$$$$T^{4} -$$$$12\!\cdots\!30$$$$T^{5} +$$$$34\!\cdots\!69$$$$T^{6} )^{2}$$
$79$ $$( 1 - 526818 T + 705484689807 T^{2} - 230000056692908636 T^{3} +$$$$17\!\cdots\!47$$$$T^{4} -$$$$31\!\cdots\!38$$$$T^{5} +$$$$14\!\cdots\!61$$$$T^{6} )^{2}$$
$83$ $$1 - 1779363154038 T^{2} +$$$$13\!\cdots\!67$$$$T^{4} -$$$$58\!\cdots\!36$$$$T^{6} +$$$$14\!\cdots\!87$$$$T^{8} -$$$$20\!\cdots\!98$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{12}$$
$89$ $$1 - 1464277161318 T^{2} +$$$$11\!\cdots\!39$$$$T^{4} -$$$$68\!\cdots\!36$$$$T^{6} +$$$$28\!\cdots\!19$$$$T^{8} -$$$$89\!\cdots\!38$$$$T^{10} +$$$$15\!\cdots\!61$$$$T^{12}$$
$97$ $$( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} +$$$$92\!\cdots\!03$$$$T^{4} +$$$$27\!\cdots\!78$$$$T^{5} +$$$$57\!\cdots\!89$$$$T^{6} )^{2}$$