# Properties

 Label 48.11.e.d Level $48$ Weight $11$ Character orbit 48.e Analytic conductor $30.497$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.4971481283$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{85})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 37 x^{2} + 38 x + 531$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{14}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 6) 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.

 $$f(q)$$ $$=$$ $$q + ( -21 - \beta_{1} ) q^{3} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 11278 - 43 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{7} + ( 39753 - 20 \beta_{1} - 62 \beta_{2} + 7 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -21 - \beta_{1} ) q^{3} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 11278 - 43 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{7} + ( 39753 - 20 \beta_{1} - 62 \beta_{2} + 7 \beta_{3} ) q^{9} + ( 21 \beta_{1} - 189 \beta_{2} - 35 \beta_{3} ) q^{11} + ( 68810 + 860 \beta_{1} - 100 \beta_{2} + 20 \beta_{3} ) q^{13} + ( 295200 + 743 \beta_{1} + 1877 \beta_{2} + 35 \beta_{3} ) q^{15} + ( 4276 \beta_{1} + 2860 \beta_{2} - 236 \beta_{3} ) q^{17} + ( 392182 + 4171 \beta_{1} - 485 \beta_{2} + 97 \beta_{3} ) q^{19} + ( 2407002 - 12952 \beta_{1} - 1674 \beta_{2} + 189 \beta_{3} ) q^{21} + ( 366 \beta_{1} - 4590 \beta_{2} - 826 \beta_{3} ) q^{23} + ( -8433095 - 18060 \beta_{1} + 2100 \beta_{2} - 420 \beta_{3} ) q^{25} + ( -8654877 - 39306 \beta_{1} - 5727 \beta_{2} + 1599 \beta_{3} ) q^{27} + ( -16812 \beta_{1} - 34494 \beta_{2} - 2947 \beta_{3} ) q^{29} + ( 5446462 - 130935 \beta_{1} + 15225 \beta_{2} - 3045 \beta_{3} ) q^{31} + ( 6493536 + 29456 \beta_{1} + 69146 \beta_{2} + 5033 \beta_{3} ) q^{33} + ( 129502 \beta_{1} + 71914 \beta_{2} - 9598 \beta_{3} ) q^{35} + ( -17753542 - 244412 \beta_{1} + 28420 \beta_{2} - 5684 \beta_{3} ) q^{37} + ( -54321810 - 35330 \beta_{1} + 33480 \beta_{2} - 3780 \beta_{3} ) q^{39} + ( 238792 \beta_{1} + 158308 \beta_{2} - 13414 \beta_{3} ) q^{41} + ( 117672166 - 291669 \beta_{1} + 33915 \beta_{2} - 6783 \beta_{3} ) q^{43} + ( 78079680 - 411492 \beta_{1} - 175386 \beta_{2} - 52581 \beta_{3} ) q^{45} + ( -246700 \beta_{1} - 159940 \beta_{2} + 14460 \beta_{3} ) q^{47} + ( -12514605 - 969908 \beta_{1} + 112780 \beta_{2} - 22556 \beta_{3} ) q^{49} + ( 177144192 + 78720 \beta_{1} + 346344 \beta_{2} - 98364 \beta_{3} ) q^{51} + ( 81716 \beta_{1} + 437750 \beta_{2} + 59339 \beta_{3} ) q^{53} + ( -675339840 - 1614564 \beta_{1} + 187740 \beta_{2} - 37548 \beta_{3} ) q^{55} + ( -264688302 - 229804 \beta_{1} + 162378 \beta_{2} - 18333 \beta_{3} ) q^{57} + ( 600237 \beta_{1} + 739875 \beta_{2} + 23273 \beta_{3} ) q^{59} + ( -296009686 + 844692 \beta_{1} - 98220 \beta_{2} + 19644 \beta_{3} ) q^{61} + ( 226251774 - 2903825 \beta_{1} - 924173 \beta_{2} + 130057 \beta_{3} ) q^{63} + ( -1412560 \beta_{1} - 2027020 \beta_{2} - 102410 \beta_{3} ) q^{65} + ( 74341462 + 2146431 \beta_{1} - 249585 \beta_{2} + 49917 \beta_{3} ) q^{67} + ( 149067072 + 698920 \beta_{1} + 1635604 \beta_{2} + 122926 \beta_{3} ) q^{69} + ( 478562 \beta_{1} - 803026 \beta_{2} - 213598 \beta_{3} ) q^{71} + ( 1633567250 - 1987632 \beta_{1} + 231120 \beta_{2} - 46224 \beta_{3} ) q^{73} + ( 1287507795 + 7730015 \beta_{1} - 703080 \beta_{2} + 79380 \beta_{3} ) q^{75} + ( 2259432 \beta_{1} + 1340556 \beta_{2} - 153146 \beta_{3} ) q^{77} + ( -49820642 + 6895609 \beta_{1} - 801815 \beta_{2} + 160363 \beta_{3} ) q^{79} + ( 364741137 + 6742260 \beta_{1} - 4229568 \beta_{2} + 397530 \beta_{3} ) q^{81} + ( -2981737 \beta_{1} - 651583 \beta_{2} + 388359 \beta_{3} ) q^{83} + ( -3220128000 + 23246832 \beta_{1} - 2703120 \beta_{2} + 540624 \beta_{3} ) q^{85} + ( -52567200 + 3019021 \beta_{1} + 6360919 \beta_{2} + 1018345 \beta_{3} ) q^{87} + ( 2016028 \beta_{1} + 5538400 \beta_{2} + 587062 \beta_{3} ) q^{89} + ( -2079308020 + 6740250 \beta_{1} - 783750 \beta_{2} + 156750 \beta_{3} ) q^{91} + ( 7936117098 - 10543792 \beta_{1} - 5097330 \beta_{2} + 575505 \beta_{3} ) q^{93} + ( -6617102 \beta_{1} - 9947954 \beta_{2} - 555142 \beta_{3} ) q^{95} + ( -9794088766 - 3429508 \beta_{1} + 398780 \beta_{2} - 79756 \beta_{3} ) q^{97} + ( 656728128 - 12335547 \beta_{1} - 11748765 \beta_{2} - 2000271 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + O(q^{10})$$ $$4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + 275240 q^{13} + 1180800 q^{15} + 1568728 q^{19} + 9628008 q^{21} - 33732380 q^{25} - 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} - 71014168 q^{37} - 217287240 q^{39} + 470688664 q^{43} + 312318720 q^{45} - 50058420 q^{49} + 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} - 1184038744 q^{61} + 905007096 q^{63} + 297365848 q^{67} + 596268288 q^{69} + 6534269000 q^{73} + 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} - 12880512000 q^{85} - 210268800 q^{87} - 8317232080 q^{91} + 31744468392 q^{93} - 39176355064 q^{97} + 2626912512 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 37 x^{2} + 38 x + 531$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{3} - 180 \nu^{2} + 1732 \nu + 2760$$$$)/31$$ $$\beta_{2}$$ $$=$$ $$($$$$260 \nu^{3} - 204 \nu^{2} - 5444 \nu - 840$$$$)/31$$ $$\beta_{3}$$ $$=$$ $$($$$$-256 \nu^{3} + 9312 \nu^{2} + 3712 \nu - 176016$$$$)/31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{3} + 7 \beta_{2} + 199 \beta_{1} + 5184$$$$)/10368$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 5616$$$$)/288$$ $$\nu^{3}$$ $$=$$ $$($$$$112 \beta_{3} + 1411 \beta_{2} + 4195 \beta_{1} + 300672$$$$)/10368$$

## 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
 5.10977 − 1.41421i 5.10977 + 1.41421i −4.10977 + 1.41421i −4.10977 − 1.41421i
0 −242.269 18.8335i 0 4818.41i 0 −670.530 0 58339.6 + 9125.53i 0
17.2 0 −242.269 + 18.8335i 0 4818.41i 0 −670.530 0 58339.6 9125.53i 0
17.3 0 200.269 137.627i 0 3630.47i 0 23226.5 0 21166.4 55125.0i 0
17.4 0 200.269 + 137.627i 0 3630.47i 0 23226.5 0 21166.4 + 55125.0i 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
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 48.11.e.d 4
3.b odd 2 1 inner 48.11.e.d 4
4.b odd 2 1 6.11.b.a 4
8.b even 2 1 192.11.e.h 4
8.d odd 2 1 192.11.e.g 4
12.b even 2 1 6.11.b.a 4
20.d odd 2 1 150.11.d.a 4
20.e even 4 2 150.11.b.a 8
24.f even 2 1 192.11.e.g 4
24.h odd 2 1 192.11.e.h 4
36.f odd 6 2 162.11.d.d 8
36.h even 6 2 162.11.d.d 8
60.h even 2 1 150.11.d.a 4
60.l odd 4 2 150.11.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 4.b odd 2 1
6.11.b.a 4 12.b even 2 1
48.11.e.d 4 1.a even 1 1 trivial
48.11.e.d 4 3.b odd 2 1 inner
150.11.b.a 8 20.e even 4 2
150.11.b.a 8 60.l odd 4 2
150.11.d.a 4 20.d odd 2 1
150.11.d.a 4 60.h even 2 1
162.11.d.d 8 36.f odd 6 2
162.11.d.d 8 36.h even 6 2
192.11.e.g 4 8.d odd 2 1
192.11.e.g 4 24.f even 2 1
192.11.e.h 4 8.b even 2 1
192.11.e.h 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 36397440 T_{5}^{2} +$$$$30\!\cdots\!00$$ acting on $$S_{11}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$3486784401 + 4960116 T - 75978 T^{2} + 84 T^{3} + T^{4}$$
$5$ $$306009247334400 + 36397440 T^{2} + T^{4}$$
$7$ $$( -15574076 - 22556 T + T^{2} )^{2}$$
$11$ $$21\!\cdots\!24$$$$+ 58313211264 T^{2} + T^{4}$$
$13$ $$( -52372127900 - 137620 T + T^{2} )^{2}$$
$17$ $$32\!\cdots\!84$$$$+ 7560967182336 T^{2} + T^{4}$$
$19$ $$( -1189491369116 - 784364 T + T^{2} )^{2}$$
$23$ $$61\!\cdots\!24$$$$+ 33055507478016 T^{2} + T^{4}$$
$29$ $$20\!\cdots\!00$$$$+ 907304099736960 T^{2} + T^{4}$$
$31$ $$( -1294078582786556 - 10892924 T + T^{2} )^{2}$$
$37$ $$( -4297319054834396 + 35507084 T + T^{2} )^{2}$$
$41$ $$28\!\cdots\!04$$$$+ 23561404561257984 T^{2} + T^{4}$$
$43$ $$( 7278142478596516 - 235344332 T + T^{2} )^{2}$$
$47$ $$26\!\cdots\!00$$$$+ 25124763600230400 T^{2} + T^{4}$$
$53$ $$37\!\cdots\!04$$$$+ 212636466457531776 T^{2} + T^{4}$$
$59$ $$20\!\cdots\!04$$$$+ 324064557407447424 T^{2} + T^{4}$$
$61$ $$( 32529703648081636 + 592019372 T + T^{2} )^{2}$$
$67$ $$( -350207761464045596 - 148682924 T + T^{2} )^{2}$$
$71$ $$49\!\cdots\!00$$$$+ 1847488216292328960 T^{2} + T^{4}$$
$73$ $$( 2363496913262627140 - 3267134500 T + T^{2} )^{2}$$
$79$ $$( -3668964988480567676 + 99641284 T + T^{2} )^{2}$$
$83$ $$57\!\cdots\!44$$$$+ 5977139602070968704 T^{2} + T^{4}$$
$89$ $$15\!\cdots\!84$$$$+ 27084125311735371264 T^{2} + T^{4}$$
$97$ $$( 95016028790224257796 + 19588177532 T + T^{2} )^{2}$$
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