# Properties

 Label 153.10.a.c Level $153$ Weight $10$ Character orbit 153.a Self dual yes Analytic conductor $78.800$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 153.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.8004829331$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 7 - \beta_{1} ) q^{2} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( -309 + 26 \beta_{1} - \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{5} + ( -2625 - 44 \beta_{1} + 13 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} ) q^{7} + ( 8547 - 179 \beta_{1} - 28 \beta_{2} + 9 \beta_{3} - 67 \beta_{4} ) q^{8} +O(q^{10})$$ $$q + ( 7 - \beta_{1} ) q^{2} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( -309 + 26 \beta_{1} - \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{5} + ( -2625 - 44 \beta_{1} + 13 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} ) q^{7} + ( 8547 - 179 \beta_{1} - 28 \beta_{2} + 9 \beta_{3} - 67 \beta_{4} ) q^{8} + ( -18048 + 560 \beta_{1} - 104 \beta_{3} + 104 \beta_{4} ) q^{10} + ( 13582 + 138 \beta_{1} + 54 \beta_{2} - 48 \beta_{3} + 139 \beta_{4} ) q^{11} + ( -31799 - 150 \beta_{1} - 37 \beta_{2} + 198 \beta_{3} - 546 \beta_{4} ) q^{13} + ( 16212 + 2388 \beta_{1} + 368 \beta_{2} - 384 \beta_{3} + 256 \beta_{4} ) q^{14} + ( 73093 - 8421 \beta_{1} - 1244 \beta_{2} + 179 \beta_{3} - \beta_{4} ) q^{16} + 83521 q^{17} + ( -75406 + 4968 \beta_{1} + 234 \beta_{2} - 1832 \beta_{3} + 858 \beta_{4} ) q^{19} + ( -336512 + 28224 \beta_{1} + 1760 \beta_{2} - 2384 \beta_{3} - 976 \beta_{4} ) q^{20} + ( 23518 + 4366 \beta_{1} + 2600 \beta_{2} - 716 \beta_{3} + 2340 \beta_{4} ) q^{22} + ( -329681 + 6084 \beta_{1} - 167 \beta_{2} - 4900 \beta_{3} + 2325 \beta_{4} ) q^{23} + ( 642933 + 31460 \beta_{1} - 2518 \beta_{2} - 4932 \beta_{3} - 1648 \beta_{4} ) q^{25} + ( -134226 - 34018 \beta_{1} - 6048 \beta_{2} - 864 \beta_{3} - 4144 \beta_{4} ) q^{26} + ( 6524 + 88548 \beta_{1} + 5760 \beta_{2} - 10292 \beta_{3} + 9372 \beta_{4} ) q^{28} + ( -721211 - 24298 \beta_{1} + 8977 \beta_{2} - 2494 \beta_{3} - 5182 \beta_{4} ) q^{29} + ( -1493895 + 87056 \beta_{1} - 10641 \beta_{2} + 11180 \beta_{3} + 34533 \beta_{4} ) q^{31} + ( 1145387 - 92331 \beta_{1} - 16244 \beta_{2} + 23701 \beta_{3} - 17975 \beta_{4} ) q^{32} + ( 584647 - 83521 \beta_{1} ) q^{34} + ( 5284272 + 77964 \beta_{1} - 9164 \beta_{2} - 41232 \beta_{3} - 38410 \beta_{4} ) q^{35} + ( -6287685 + 22746 \beta_{1} + 15899 \beta_{2} - 5938 \beta_{3} - 12574 \beta_{4} ) q^{37} + ( -3848092 + 358532 \beta_{1} + 19808 \beta_{2} + 14752 \beta_{3} + 21536 \beta_{4} ) q^{38} + ( -10976352 + 468960 \beta_{1} + 43968 \beta_{2} + 28672 \beta_{3} + 59520 \beta_{4} ) q^{40} + ( 1421988 + 387124 \beta_{1} + 17594 \beta_{2} + 65172 \beta_{3} + 41192 \beta_{4} ) q^{41} + ( -11492294 + 231296 \beta_{1} - 64242 \beta_{2} + 13184 \beta_{3} + 123526 \beta_{4} ) q^{43} + ( -8739126 + 235702 \beta_{1} + 56336 \beta_{2} - 21246 \beta_{3} + 11738 \beta_{4} ) q^{44} + ( -6834352 + 1010368 \beta_{1} + 34192 \beta_{2} + 60872 \beta_{3} + 18728 \beta_{4} ) q^{46} + ( 3131634 + 723308 \beta_{1} - 2726 \beta_{2} - 102088 \beta_{3} + 22048 \beta_{4} ) q^{47} + ( -2354982 + 213486 \beta_{1} + 16821 \beta_{2} - 209650 \beta_{3} - 102102 \beta_{4} ) q^{49} + ( -17202051 - 68571 \beta_{1} - 53888 \beta_{2} + 86208 \beta_{3} + 19232 \beta_{4} ) q^{50} + ( 34854146 - 465666 \beta_{1} - 155904 \beta_{2} + 61890 \beta_{3} + 17850 \beta_{4} ) q^{52} + ( 16615966 - 152616 \beta_{1} - 258816 \beta_{2} + 164784 \beta_{3} - 55220 \beta_{4} ) q^{53} + ( 1354508 - 394556 \beta_{1} - 121312 \beta_{2} + 129312 \beta_{3} + 110278 \beta_{4} ) q^{55} + ( -64080268 + 1399244 \beta_{1} + 65968 \beta_{2} + 129724 \beta_{3} + 296940 \beta_{4} ) q^{56} + ( 12857952 + 327856 \beta_{1} + 183968 \beta_{2} - 92008 \beta_{3} + 88520 \beta_{4} ) q^{58} + ( 7188478 + 1081736 \beta_{1} + 101866 \beta_{2} - 28696 \beta_{3} - 838738 \beta_{4} ) q^{59} + ( -15322337 - 601494 \beta_{1} + 252183 \beta_{2} + 190702 \beta_{3} - 679618 \beta_{4} ) q^{61} + ( -66638204 + 4277044 \beta_{1} - 23840 \beta_{2} - 121104 \beta_{3} + 256624 \beta_{4} ) q^{62} + ( 26922421 - 1487477 \beta_{1} + 8468 \beta_{2} - 2789 \beta_{3} - 694297 \beta_{4} ) q^{64} + ( 7317776 + 2449816 \beta_{1} + 380232 \beta_{2} - 393496 \beta_{3} - 683460 \beta_{4} ) q^{65} + ( -61385498 + 1506284 \beta_{1} - 104058 \beta_{2} - 346368 \beta_{3} - 351296 \beta_{4} ) q^{67} + ( 14783217 - 1419857 \beta_{1} + 83521 \beta_{3} - 250563 \beta_{4} ) q^{68} + ( -23188592 - 4344416 \beta_{1} - 362288 \beta_{2} + 741056 \beta_{3} - 280640 \beta_{4} ) q^{70} + ( 96618793 - 3613408 \beta_{1} - 572997 \beta_{2} + 81388 \beta_{3} + 671613 \beta_{4} ) q^{71} + ( -58202062 - 413264 \beta_{1} - 551952 \beta_{2} + 652200 \beta_{3} - 1040600 \beta_{4} ) q^{73} + ( -54751612 + 6087540 \beta_{1} + 304736 \beta_{2} - 200648 \beta_{3} + 330696 \beta_{4} ) q^{74} + ( -208411364 + 5651492 \beta_{1} + 468864 \beta_{2} - 26692 \beta_{3} + 1230796 \beta_{4} ) q^{76} + ( 30391837 - 4369154 \beta_{1} + 318833 \beta_{2} + 241802 \beta_{3} - 84822 \beta_{4} ) q^{77} + ( -164909903 - 468556 \beta_{1} + 671111 \beta_{2} - 1041436 \beta_{3} + 399047 \beta_{4} ) q^{79} + ( -183928544 + 4263904 \beta_{1} + 515584 \beta_{2} - 560224 \beta_{3} + 3288352 \beta_{4} ) q^{80} + ( -221363594 + 545318 \beta_{1} + 491104 \beta_{2} - 1698608 \beta_{3} + 1925936 \beta_{4} ) q^{82} + ( -41276982 + 6790888 \beta_{1} - 96290 \beta_{2} - 987152 \beta_{3} - 1443942 \beta_{4} ) q^{83} + ( -25807989 + 2171546 \beta_{1} - 83521 \beta_{2} + 501126 \beta_{3} + 501126 \beta_{4} ) q^{85} + ( -242346664 + 22716488 \beta_{1} - 606336 \beta_{2} + 493432 \beta_{3} + 48088 \beta_{4} ) q^{86} + ( -208194130 + 12204402 \beta_{1} + 199752 \beta_{2} - 609190 \beta_{3} + 776354 \beta_{4} ) q^{88} + ( -66779497 - 20899250 \beta_{1} - 460683 \beta_{2} - 806402 \beta_{3} - 1206630 \beta_{4} ) q^{89} + ( 34422318 + 12123668 \beta_{1} - 679922 \beta_{2} - 1247592 \beta_{3} + 374208 \beta_{4} ) q^{91} + ( -505380000 + 10506368 \beta_{1} + 812448 \beta_{2} - 6688 \beta_{3} + 2827040 \beta_{4} ) q^{92} + ( -455043760 + 14752608 \beta_{1} + 519312 \beta_{2} + 710896 \beta_{3} + 2038064 \beta_{4} ) q^{94} + ( -290829908 - 24844352 \beta_{1} - 687740 \beta_{2} + 2235320 \beta_{3} + 886660 \beta_{4} ) q^{95} + ( 125658398 + 33525784 \beta_{1} + 236504 \beta_{2} - 1535904 \beta_{3} - 5543956 \beta_{4} ) q^{97} + ( -180869073 + 13890359 \beta_{1} + 425488 \beta_{2} + 2623040 \beta_{3} - 21392 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 33q^{2} + 853q^{4} - 1480q^{5} - 13202q^{7} + 42423q^{8} + O(q^{10})$$ $$5q + 33q^{2} + 853q^{4} - 1480q^{5} - 13202q^{7} + 42423q^{8} - 89328q^{10} + 68036q^{11} - 158862q^{13} + 84700q^{14} + 350225q^{16} + 417605q^{17} - 370992q^{19} - 1632640q^{20} + 122290q^{22} - 1645870q^{23} + 3270239q^{25} - 734846q^{26} + 183372q^{28} - 3668616q^{29} - 7262362q^{31} + 5605919q^{32} + 2756193q^{34} + 26503988q^{35} - 31420708q^{37} - 18513700q^{38} - 53930464q^{40} + 7996938q^{41} - 56908268q^{43} - 43323054q^{44} - 32063472q^{46} + 16903336q^{47} - 11784059q^{49} - 85921093q^{50} + 173619082q^{52} + 83362982q^{53} + 6363364q^{55} - 317409372q^{56} + 64577488q^{58} + 37946604q^{59} - 77685452q^{61} - 324855300q^{62} + 131623105q^{64} + 40321288q^{65} - 304503600q^{67} + 71243413q^{68} - 122787392q^{70} + 476602922q^{71} - 289980486q^{73} - 262289012q^{74} - 1031276084q^{76} + 143385648q^{77} - 828240610q^{79} - 912750944q^{80} - 1109615654q^{82} - 194681148q^{83} - 123611080q^{85} - 1164707144q^{86} - 1017979978q^{88} - 376848106q^{89} + 194543664q^{91} - 2506713088q^{92} - 2244811104q^{94} - 1498679864q^{95} + 692035246q^{97} - 871744055q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{4} + 207 \nu^{3} + 6301 \nu^{2} - 209023 \nu - 509736$$$$)/8096$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{4} + 345 \nu^{3} - 24845 \nu^{2} - 323145 \nu + 3450824$$$$)/16192$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{4} + 115 \nu^{3} - 13679 \nu^{2} - 123907 \nu + 4604568$$$$)/16192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-3 \beta_{4} + \beta_{3} - 3 \beta_{1} + 640$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{4} + 12 \beta_{3} + 28 \beta_{2} + 993 \beta_{1} - 1932$$ $$\nu^{4}$$ $$=$$ $$-3615 \beta_{4} + 1757 \beta_{3} - 460 \beta_{2} - 4475 \beta_{1} + 624596$$

## 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
 33.6330 18.8209 5.77274 −21.1654 −35.0613
−26.6330 0 197.318 1460.58 0 −446.232 8380.94 0 −38899.6
1.2 −11.8209 0 −372.266 −2390.67 0 −11355.8 10452.8 0 28259.9
1.3 1.22726 0 −510.494 1620.18 0 −1834.42 −1254.87 0 1988.39
1.4 28.1654 0 281.287 −762.851 0 5573.11 −6498.11 0 −21486.0
1.5 42.0613 0 1257.15 −1407.25 0 −5138.64 31342.2 0 −59190.7
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$-1$$

## 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 153.10.a.c 5
3.b odd 2 1 17.10.a.a 5
12.b even 2 1 272.10.a.f 5
51.c odd 2 1 289.10.a.a 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.a 5 3.b odd 2 1
153.10.a.c 5 1.a even 1 1 trivial
272.10.a.f 5 12.b even 2 1
289.10.a.a 5 51.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 33 T_{2}^{4} - 1162 T_{2}^{3} + 24920 T_{2}^{2} + 344192 T_{2} - 457728$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(153))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-457728 + 344192 T + 24920 T^{2} - 1162 T^{3} - 33 T^{4} + T^{5}$$
$3$ $$T^{5}$$
$5$ $$6073215799787520 + 6910711315456 T - 5931144800 T^{2} - 5422732 T^{3} + 1480 T^{4} + T^{5}$$
$7$ $$-266210160692281344 - 769174397227488 T - 392836554656 T^{2} - 7845586 T^{3} + 13202 T^{4} + T^{5}$$
$11$ $$-$$$$18\!\cdots\!36$$$$- 583150347803390760 T + 51654458131068 T^{2} + 161240150 T^{3} - 68036 T^{4} + T^{5}$$
$13$ $$46\!\cdots\!08$$$$+ 15605144350350783296 T - 1715294166968552 T^{2} - 8325650340 T^{3} + 158862 T^{4} + T^{5}$$
$17$ $$( -83521 + T )^{5}$$
$19$ $$-$$$$23\!\cdots\!16$$$$-$$$$17\!\cdots\!04$$$$T - 176324676443571104 T^{2} - 413150145000 T^{3} + 370992 T^{4} + T^{5}$$
$23$ $$-$$$$28\!\cdots\!00$$$$-$$$$13\!\cdots\!96$$$$T - 4397519995817681960 T^{2} - 2266461594322 T^{3} + 1645870 T^{4} + T^{5}$$
$29$ $$-$$$$14\!\cdots\!00$$$$-$$$$47\!\cdots\!52$$$$T - 43135444415107067232 T^{2} - 8671259026316 T^{3} + 3668616 T^{4} + T^{5}$$
$31$ $$-$$$$63\!\cdots\!72$$$$-$$$$15\!\cdots\!16$$$$T -$$$$61\!\cdots\!12$$$$T^{2} - 55088066425306 T^{3} + 7262362 T^{4} + T^{5}$$
$37$ $$12\!\cdots\!48$$$$+$$$$31\!\cdots\!60$$$$T +$$$$16\!\cdots\!40$$$$T^{2} + 352949672725220 T^{3} + 31420708 T^{4} + T^{5}$$
$41$ $$-$$$$84\!\cdots\!52$$$$+$$$$16\!\cdots\!32$$$$T +$$$$34\!\cdots\!32$$$$T^{2} - 944595954515528 T^{3} - 7996938 T^{4} + T^{5}$$
$43$ $$-$$$$35\!\cdots\!88$$$$-$$$$76\!\cdots\!72$$$$T -$$$$42\!\cdots\!64$$$$T^{2} + 104989881324440 T^{3} + 56908268 T^{4} + T^{5}$$
$47$ $$53\!\cdots\!68$$$$+$$$$13\!\cdots\!68$$$$T -$$$$13\!\cdots\!84$$$$T^{2} - 2082901585389616 T^{3} - 16903336 T^{4} + T^{5}$$
$53$ $$-$$$$46\!\cdots\!44$$$$-$$$$15\!\cdots\!36$$$$T +$$$$92\!\cdots\!16$$$$T^{2} - 9379424557396600 T^{3} - 83362982 T^{4} + T^{5}$$
$59$ $$-$$$$43\!\cdots\!28$$$$+$$$$40\!\cdots\!72$$$$T +$$$$23\!\cdots\!92$$$$T^{2} - 31008104213596424 T^{3} - 37946604 T^{4} + T^{5}$$
$61$ $$-$$$$81\!\cdots\!20$$$$+$$$$24\!\cdots\!92$$$$T -$$$$36\!\cdots\!32$$$$T^{2} - 33463527209865196 T^{3} + 77685452 T^{4} + T^{5}$$
$67$ $$-$$$$17\!\cdots\!00$$$$-$$$$58\!\cdots\!40$$$$T -$$$$55\!\cdots\!76$$$$T^{2} + 7020713481287504 T^{3} + 304503600 T^{4} + T^{5}$$
$71$ $$-$$$$27\!\cdots\!64$$$$-$$$$34\!\cdots\!28$$$$T +$$$$14\!\cdots\!84$$$$T^{2} + 9603673423875238 T^{3} - 476602922 T^{4} + T^{5}$$
$73$ $$39\!\cdots\!92$$$$+$$$$18\!\cdots\!28$$$$T -$$$$24\!\cdots\!16$$$$T^{2} - 108975991091496536 T^{3} + 289980486 T^{4} + T^{5}$$
$79$ $$88\!\cdots\!72$$$$-$$$$60\!\cdots\!44$$$$T -$$$$47\!\cdots\!40$$$$T^{2} + 104201235865665870 T^{3} + 828240610 T^{4} + T^{5}$$
$83$ $$48\!\cdots\!16$$$$-$$$$32\!\cdots\!40$$$$T -$$$$93\!\cdots\!08$$$$T^{2} - 285056400431908168 T^{3} + 194681148 T^{4} + T^{5}$$
$89$ $$-$$$$26\!\cdots\!96$$$$+$$$$21\!\cdots\!28$$$$T -$$$$27\!\cdots\!24$$$$T^{2} - 834395900612571220 T^{3} + 376848106 T^{4} + T^{5}$$
$97$ $$10\!\cdots\!28$$$$+$$$$30\!\cdots\!16$$$$T +$$$$49\!\cdots\!44$$$$T^{2} - 3215802378794508088 T^{3} - 692035246 T^{4} + T^{5}$$