# Properties

 Label 342.10.a.l Level $342$ Weight $10$ Character orbit 342.a Self dual yes Analytic conductor $176.142$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$176.142255968$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 256 q^{4} + ( 350 + 3 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} + ( 3075 + 6 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{7} + 4096 q^{8} +O(q^{10})$$ $$q + 16 q^{2} + 256 q^{4} + ( 350 + 3 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} + ( 3075 + 6 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{7} + 4096 q^{8} + ( 5600 + 48 \beta_{1} + 16 \beta_{2} - 64 \beta_{3} ) q^{10} + ( 26056 + \beta_{1} - 25 \beta_{2} ) q^{11} + ( 30071 + 199 \beta_{1} - 33 \beta_{2} + 169 \beta_{3} ) q^{13} + ( 49200 + 96 \beta_{1} + 32 \beta_{2} + 144 \beta_{3} ) q^{14} + 65536 q^{16} + ( 103123 + 14 \beta_{1} + 587 \beta_{2} + 234 \beta_{3} ) q^{17} + 130321 q^{19} + ( 89600 + 768 \beta_{1} + 256 \beta_{2} - 1024 \beta_{3} ) q^{20} + ( 416896 + 16 \beta_{1} - 400 \beta_{2} ) q^{22} + ( -752981 + 195 \beta_{1} - 1531 \beta_{2} + 93 \beta_{3} ) q^{23} + ( 2440945 + 2715 \beta_{1} + 1845 \beta_{2} - 1350 \beta_{3} ) q^{25} + ( 481136 + 3184 \beta_{1} - 528 \beta_{2} + 2704 \beta_{3} ) q^{26} + ( 787200 + 1536 \beta_{1} + 512 \beta_{2} + 2304 \beta_{3} ) q^{28} + ( -1537183 + 1639 \beta_{1} + 1027 \beta_{2} - 3481 \beta_{3} ) q^{29} + ( 3192456 + 2444 \beta_{1} - 2050 \beta_{2} + 2150 \beta_{3} ) q^{31} + 1048576 q^{32} + ( 1649968 + 224 \beta_{1} + 9392 \beta_{2} + 3744 \beta_{3} ) q^{34} + ( -2453330 + 13677 \beta_{1} - 10051 \beta_{2} - 20156 \beta_{3} ) q^{35} + ( 5128462 - 8564 \beta_{1} - 6770 \beta_{2} - 14570 \beta_{3} ) q^{37} + 2085136 q^{38} + ( 1433600 + 12288 \beta_{1} + 4096 \beta_{2} - 16384 \beta_{3} ) q^{40} + ( -2893200 + 13410 \beta_{1} + 19020 \beta_{2} - 28480 \beta_{3} ) q^{41} + ( 1937238 - 11191 \beta_{1} + 24965 \beta_{2} - 25660 \beta_{3} ) q^{43} + ( 6670336 + 256 \beta_{1} - 6400 \beta_{2} ) q^{44} + ( -12047696 + 3120 \beta_{1} - 24496 \beta_{2} + 1488 \beta_{3} ) q^{46} + ( 7894898 - 25983 \beta_{1} + 31111 \beta_{2} + 32602 \beta_{3} ) q^{47} + ( 4720576 + 3388 \beta_{1} + 12239 \beta_{2} + 100318 \beta_{3} ) q^{49} + ( 39055120 + 43440 \beta_{1} + 29520 \beta_{2} - 21600 \beta_{3} ) q^{50} + ( 7698176 + 50944 \beta_{1} - 8448 \beta_{2} + 43264 \beta_{3} ) q^{52} + ( -18108791 - 60111 \beta_{1} + 35357 \beta_{2} - 78901 \beta_{3} ) q^{53} + ( 5350710 + 124563 \beta_{1} + 18501 \beta_{2} - 51834 \beta_{3} ) q^{55} + ( 12595200 + 24576 \beta_{1} + 8192 \beta_{2} + 36864 \beta_{3} ) q^{56} + ( -24594928 + 26224 \beta_{1} + 16432 \beta_{2} - 55696 \beta_{3} ) q^{58} + ( 37327107 + 11193 \beta_{1} + 66072 \beta_{2} - 8236 \beta_{3} ) q^{59} + ( 32233368 + 12223 \beta_{1} - 115309 \beta_{2} - 44408 \beta_{3} ) q^{61} + ( 51079296 + 39104 \beta_{1} - 32800 \beta_{2} + 34400 \beta_{3} ) q^{62} + 16777216 q^{64} + ( -31222100 + 408720 \beta_{1} - 309130 \beta_{2} - 112430 \beta_{3} ) q^{65} + ( 33192661 - 284703 \beta_{1} + 156322 \beta_{2} - 19056 \beta_{3} ) q^{67} + ( 26399488 + 3584 \beta_{1} + 150272 \beta_{2} + 59904 \beta_{3} ) q^{68} + ( -39253280 + 218832 \beta_{1} - 160816 \beta_{2} - 322496 \beta_{3} ) q^{70} + ( 11884254 + 118800 \beta_{1} - 8844 \beta_{2} - 407378 \beta_{3} ) q^{71} + ( -9921749 - 400244 \beta_{1} - 184019 \beta_{2} + 170182 \beta_{3} ) q^{73} + ( 82055392 - 137024 \beta_{1} - 108320 \beta_{2} - 233120 \beta_{3} ) q^{74} + 33362176 q^{76} + ( 41488292 + 298659 \beta_{1} + 63511 \beta_{2} + 44062 \beta_{3} ) q^{77} + ( -76710310 - 599398 \beta_{1} + 293190 \beta_{2} + 123590 \beta_{3} ) q^{79} + ( 22937600 + 196608 \beta_{1} + 65536 \beta_{2} - 262144 \beta_{3} ) q^{80} + ( -46291200 + 214560 \beta_{1} + 304320 \beta_{2} - 455680 \beta_{3} ) q^{82} + ( 186639194 + 292006 \beta_{1} + 92806 \beta_{2} + 104262 \beta_{3} ) q^{83} + ( -25807650 - 717879 \beta_{1} + 26637 \beta_{2} - 1748748 \beta_{3} ) q^{85} + ( 30995808 - 179056 \beta_{1} + 399440 \beta_{2} - 410560 \beta_{3} ) q^{86} + ( 106725376 + 4096 \beta_{1} - 102400 \beta_{2} ) q^{88} + ( -72054508 - 282772 \beta_{1} - 201740 \beta_{2} + 1072810 \beta_{3} ) q^{89} + ( 788716179 + 526663 \beta_{1} + 102178 \beta_{2} + 1018576 \beta_{3} ) q^{91} + ( -192763136 + 49920 \beta_{1} - 391936 \beta_{2} + 23808 \beta_{3} ) q^{92} + ( 126318368 - 415728 \beta_{1} + 497776 \beta_{2} + 521632 \beta_{3} ) q^{94} + ( 45612350 + 390963 \beta_{1} + 130321 \beta_{2} - 521284 \beta_{3} ) q^{95} + ( 197503898 - 2142810 \beta_{1} - 963354 \beta_{2} + 2541012 \beta_{3} ) q^{97} + ( 75529216 + 54208 \beta_{1} + 195824 \beta_{2} + 1605088 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 64q^{2} + 1024q^{4} + 1395q^{5} + 12307q^{7} + 16384q^{8} + O(q^{10})$$ $$4q + 64q^{2} + 1024q^{4} + 1395q^{5} + 12307q^{7} + 16384q^{8} + 22320q^{10} + 104249q^{11} + 120486q^{13} + 196912q^{14} + 262144q^{16} + 412139q^{17} + 521284q^{19} + 357120q^{20} + 1667984q^{22} - 3010300q^{23} + 9760585q^{25} + 1927776q^{26} + 3150592q^{28} - 6153240q^{29} + 12774024q^{31} + 4194304q^{32} + 6594224q^{34} - 9823425q^{35} + 20506048q^{37} + 8340544q^{38} + 5713920q^{40} - 11620300q^{41} + 7698327q^{43} + 26687744q^{44} - 48164800q^{46} + 31581083q^{47} + 18970383q^{49} + 156169360q^{50} + 30844416q^{52} - 72549422q^{53} + 21332505q^{55} + 50409472q^{56} - 98451840q^{58} + 149234120q^{59} + 129004373q^{61} + 204384384q^{62} + 67108864q^{64} - 124691700q^{65} + 132595266q^{67} + 105507584q^{68} - 157174800q^{70} + 47138482q^{71} - 39332795q^{73} + 328096768q^{74} + 133448704q^{76} + 165933719q^{77} - 307010840q^{79} + 91422720q^{80} - 185924800q^{82} + 746568232q^{83} - 105005985q^{85} + 123173232q^{86} + 427003904q^{88} - 286943482q^{89} + 3155781114q^{91} - 770636800q^{92} + 505297328q^{94} + 181797795q^{95} + 793519958q^{97} + 303526128q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$27 \nu^{3} - 2208 \nu^{2} - 758211 \nu - 17640024$$$$)/20632$$ $$\beta_{2}$$ $$=$$ $$($$$$-17 \nu^{3} + 244 \nu^{2} + 698613 \nu + 30780440$$$$)/20632$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{3} + 244 \nu^{2} + 574821 \nu + 30821704$$$$)/20632$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-193 \beta_{3} + 85 \beta_{2} - 68 \beta_{1} + 103370$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-43865 \beta_{3} + 35033 \beta_{2} - 976 \beta_{1} + 12429538$$$$)/6$$

## 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
 −26.2676 −124.888 −67.1081 219.264
16.0000 0 256.000 −2126.71 0 11469.0 4096.00 0 −34027.4
1.2 16.0000 0 256.000 −1263.95 0 −3487.42 4096.00 0 −20223.2
1.3 16.0000 0 256.000 2367.11 0 5859.40 4096.00 0 37873.7
1.4 16.0000 0 256.000 2418.56 0 −1533.95 4096.00 0 38696.9
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 342.10.a.l 4
3.b odd 2 1 38.10.a.d 4
12.b even 2 1 304.10.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.d 4 3.b odd 2 1
304.10.a.e 4 12.b even 2 1
342.10.a.l 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 1395 T_{5}^{3} - 7813530 T_{5}^{2} + 6547343400 T_{5} +$$$$15\!\cdots\!00$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(342))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$15389064288000 + 6547343400 T - 7813530 T^{2} - 1395 T^{3} + T^{4}$$
$7$ $$359494671206216 + 244743771643 T - 14461281 T^{2} - 12307 T^{3} + T^{4}$$
$11$ $$115765319454159024 - 36343268701740 T + 3398840664 T^{2} - 104249 T^{3} + T^{4}$$
$13$ $$23\!\cdots\!00$$$$+ 2295095998230370 T - 30806569293 T^{2} - 120486 T^{3} + T^{4}$$
$17$ $$27\!\cdots\!62$$$$+ 99501926037315363 T - 374372926323 T^{2} - 412139 T^{3} + T^{4}$$
$19$ $$( -130321 + T )^{4}$$
$23$ $$22\!\cdots\!08$$$$- 2126564557696334952 T + 862408556595 T^{2} + 3010300 T^{3} + T^{4}$$
$29$ $$-$$$$34\!\cdots\!76$$$$- 2981737641494670876 T + 8624111888511 T^{2} + 6153240 T^{3} + T^{4}$$
$31$ $$-$$$$14\!\cdots\!96$$$$- 63865994000171580800 T + 51883371775344 T^{2} - 12774024 T^{3} + T^{4}$$
$37$ $$-$$$$58\!\cdots\!72$$$$+$$$$22\!\cdots\!56$$$$T - 68206760068968 T^{2} - 20506048 T^{3} + T^{4}$$
$41$ $$-$$$$36\!\cdots\!00$$$$-$$$$67\!\cdots\!00$$$$T - 531906213139500 T^{2} + 11620300 T^{3} + T^{4}$$
$43$ $$-$$$$19\!\cdots\!12$$$$-$$$$93\!\cdots\!96$$$$T - 893995448146728 T^{2} - 7698327 T^{3} + T^{4}$$
$47$ $$52\!\cdots\!00$$$$+$$$$19\!\cdots\!60$$$$T - 1669221829649832 T^{2} - 31581083 T^{3} + T^{4}$$
$53$ $$-$$$$20\!\cdots\!64$$$$-$$$$31\!\cdots\!82$$$$T - 4017930015425685 T^{2} + 72549422 T^{3} + T^{4}$$
$59$ $$-$$$$27\!\cdots\!04$$$$+$$$$16\!\cdots\!70$$$$T + 3811888983551253 T^{2} - 149234120 T^{3} + T^{4}$$
$61$ $$43\!\cdots\!00$$$$+$$$$91\!\cdots\!60$$$$T - 10642541015571546 T^{2} - 129004373 T^{3} + T^{4}$$
$67$ $$-$$$$18\!\cdots\!00$$$$+$$$$84\!\cdots\!00$$$$T - 58349865773044743 T^{2} - 132595266 T^{3} + T^{4}$$
$71$ $$-$$$$32\!\cdots\!32$$$$+$$$$10\!\cdots\!44$$$$T - 74119300967107956 T^{2} - 47138482 T^{3} + T^{4}$$
$73$ $$35\!\cdots\!34$$$$-$$$$10\!\cdots\!87$$$$T - 87654713869556991 T^{2} + 39332795 T^{3} + T^{4}$$
$79$ $$29\!\cdots\!00$$$$+$$$$10\!\cdots\!40$$$$T - 229308708527002548 T^{2} + 307010840 T^{3} + T^{4}$$
$83$ $$-$$$$94\!\cdots\!68$$$$-$$$$21\!\cdots\!04$$$$T + 152802359319519228 T^{2} - 746568232 T^{3} + T^{4}$$
$89$ $$54\!\cdots\!84$$$$-$$$$99\!\cdots\!80$$$$T - 451932251304917424 T^{2} + 286943482 T^{3} + T^{4}$$
$97$ $$37\!\cdots\!00$$$$+$$$$15\!\cdots\!80$$$$T - 3736940299059182196 T^{2} - 793519958 T^{3} + T^{4}$$