# Properties

 Label 38.10.a.d Level $38$ Weight $10$ Character orbit 38.a Self dual yes Analytic conductor $19.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 38.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5713617742$$ 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: $$2\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -16 q^{2} + ( 21 + \beta_{1} ) q^{3} + 256 q^{4} + ( -350 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{5} + ( -336 - 16 \beta_{1} ) q^{6} + ( 3075 + 27 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} ) q^{7} -4096 q^{8} + ( 4134 + 135 \beta_{1} - 18 \beta_{2} + 16 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -16 q^{2} + ( 21 + \beta_{1} ) q^{3} + 256 q^{4} + ( -350 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{5} + ( -336 - 16 \beta_{1} ) q^{6} + ( 3075 + 27 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} ) q^{7} -4096 q^{8} + ( 4134 + 135 \beta_{1} - 18 \beta_{2} + 16 \beta_{3} ) q^{9} + ( 5600 + 80 \beta_{1} + 16 \beta_{2} + 64 \beta_{3} ) q^{10} + ( -26056 - 3 \beta_{1} + 25 \beta_{2} ) q^{11} + ( 5376 + 256 \beta_{1} ) q^{12} + ( 30071 + 766 \beta_{1} - 33 \beta_{2} - 169 \beta_{3} ) q^{13} + ( -49200 - 432 \beta_{1} - 32 \beta_{2} + 144 \beta_{3} ) q^{14} + ( -147720 - 1380 \beta_{1} + 264 \beta_{2} - 54 \beta_{3} ) q^{15} + 65536 q^{16} + ( -103123 - 276 \beta_{1} - 587 \beta_{2} + 234 \beta_{3} ) q^{17} + ( -66144 - 2160 \beta_{1} + 288 \beta_{2} - 256 \beta_{3} ) q^{18} + 130321 q^{19} + ( -89600 - 1280 \beta_{1} - 256 \beta_{2} - 1024 \beta_{3} ) q^{20} + ( 609189 + 2976 \beta_{1} - 375 \beta_{2} + 125 \beta_{3} ) q^{21} + ( 416896 + 48 \beta_{1} - 400 \beta_{2} ) q^{22} + ( 752981 - 678 \beta_{1} + 1531 \beta_{2} + 93 \beta_{3} ) q^{23} + ( -86016 - 4096 \beta_{1} ) q^{24} + ( 2440945 + 6795 \beta_{1} + 1845 \beta_{2} + 1350 \beta_{3} ) q^{25} + ( -481136 - 12256 \beta_{1} + 528 \beta_{2} + 2704 \beta_{3} ) q^{26} + ( 3097563 + 12769 \beta_{1} - 1674 \beta_{2} + 3948 \beta_{3} ) q^{27} + ( 787200 + 6912 \beta_{1} + 512 \beta_{2} - 2304 \beta_{3} ) q^{28} + ( 1537183 - 1436 \beta_{1} - 1027 \beta_{2} - 3481 \beta_{3} ) q^{29} + ( 2363520 + 22080 \beta_{1} - 4224 \beta_{2} + 864 \beta_{3} ) q^{30} + ( 3192456 + 9482 \beta_{1} - 2050 \beta_{2} - 2150 \beta_{3} ) q^{31} -1048576 q^{32} + ( -815454 - 38998 \beta_{1} - 1596 \beta_{2} - 2198 \beta_{3} ) q^{33} + ( 1649968 + 4416 \beta_{1} + 9392 \beta_{2} - 3744 \beta_{3} ) q^{34} + ( 2453330 - 20875 \beta_{1} + 10051 \beta_{2} - 20156 \beta_{3} ) q^{35} + ( 1058304 + 34560 \beta_{1} - 4608 \beta_{2} + 4096 \beta_{3} ) q^{36} + ( 5128462 - 40262 \beta_{1} - 6770 \beta_{2} + 14570 \beta_{3} ) q^{37} -2085136 q^{38} + ( 17471739 + 93298 \beta_{1} - 7047 \beta_{2} + 12559 \beta_{3} ) q^{39} + ( 1433600 + 20480 \beta_{1} + 4096 \beta_{2} + 16384 \beta_{3} ) q^{40} + ( 2893200 - 11750 \beta_{1} - 19020 \beta_{2} - 28480 \beta_{3} ) q^{41} + ( -9747024 - 47616 \beta_{1} + 6000 \beta_{2} - 2000 \beta_{3} ) q^{42} + ( 1937238 - 59233 \beta_{1} + 24965 \beta_{2} + 25660 \beta_{3} ) q^{43} + ( -6670336 - 768 \beta_{1} + 6400 \beta_{2} ) q^{44} + ( -30988530 - 352695 \beta_{1} + 28557 \beta_{2} + 33138 \beta_{3} ) q^{45} + ( -12047696 + 10848 \beta_{1} - 24496 \beta_{2} - 1488 \beta_{3} ) q^{46} + ( -7894898 + 45347 \beta_{1} - 31111 \beta_{2} + 32602 \beta_{3} ) q^{47} + ( 1376256 + 65536 \beta_{1} ) q^{48} + ( 4720576 + 110482 \beta_{1} + 12239 \beta_{2} - 100318 \beta_{3} ) q^{49} + ( -39055120 - 108720 \beta_{1} - 29520 \beta_{2} - 21600 \beta_{3} ) q^{50} + ( -2126961 + 217655 \beta_{1} + 37392 \beta_{2} + 49576 \beta_{3} ) q^{51} + ( 7698176 + 196096 \beta_{1} - 8448 \beta_{2} - 43264 \beta_{3} ) q^{52} + ( 18108791 + 259234 \beta_{1} - 35357 \beta_{2} - 78901 \beta_{3} ) q^{53} + ( -49561008 - 204304 \beta_{1} + 26784 \beta_{2} - 63168 \beta_{3} ) q^{54} + ( 5350710 + 321855 \beta_{1} + 18501 \beta_{2} + 51834 \beta_{3} ) q^{55} + ( -12595200 - 110592 \beta_{1} - 8192 \beta_{2} + 36864 \beta_{3} ) q^{56} + ( 2736741 + 130321 \beta_{1} ) q^{57} + ( -24594928 + 22976 \beta_{1} + 16432 \beta_{2} + 55696 \beta_{3} ) q^{58} + ( -37327107 - 25343 \beta_{1} - 66072 \beta_{2} - 8236 \beta_{3} ) q^{59} + ( -37816320 - 353280 \beta_{1} + 67584 \beta_{2} - 13824 \beta_{3} ) q^{60} + ( 32233368 - 7739 \beta_{1} - 115309 \beta_{2} + 44408 \beta_{3} ) q^{61} + ( -51079296 - 151712 \beta_{1} + 32800 \beta_{2} + 34400 \beta_{3} ) q^{62} + ( 25788720 + 636137 \beta_{1} - 71559 \beta_{2} + 258888 \beta_{3} ) q^{63} + 16777216 q^{64} + ( 31222100 - 1113730 \beta_{1} + 309130 \beta_{2} - 112430 \beta_{3} ) q^{65} + ( 13047264 + 623968 \beta_{1} + 25536 \beta_{2} + 35168 \beta_{3} ) q^{66} + ( 33192661 - 873165 \beta_{1} + 156322 \beta_{2} + 19056 \beta_{3} ) q^{67} + ( -26399488 - 70656 \beta_{1} - 150272 \beta_{2} + 59904 \beta_{3} ) q^{68} + ( -11440611 - 73522 \beta_{1} - 91353 \beta_{2} - 141119 \beta_{3} ) q^{69} + ( -39253280 + 334000 \beta_{1} - 160816 \beta_{2} + 322496 \beta_{3} ) q^{70} + ( -11884254 + 50978 \beta_{1} + 8844 \beta_{2} - 407378 \beta_{3} ) q^{71} + ( -16932864 - 552960 \beta_{1} + 73728 \beta_{2} - 65536 \beta_{3} ) q^{72} + ( -9921749 - 1030550 \beta_{1} - 184019 \beta_{2} - 170182 \beta_{3} ) q^{73} + ( -82055392 + 644192 \beta_{1} + 108320 \beta_{2} - 233120 \beta_{3} ) q^{74} + ( 206079195 + 2611045 \beta_{1} - 280530 \beta_{2} - 29700 \beta_{3} ) q^{75} + 33362176 q^{76} + ( -41488292 - 940039 \beta_{1} - 63511 \beta_{2} + 44062 \beta_{3} ) q^{77} + ( -279547824 - 1492768 \beta_{1} + 112752 \beta_{2} - 200944 \beta_{3} ) q^{78} + ( -76710310 - 1674604 \beta_{1} + 293190 \beta_{2} - 123590 \beta_{3} ) q^{79} + ( -22937600 - 327680 \beta_{1} - 65536 \beta_{2} - 262144 \beta_{3} ) q^{80} + ( 326443161 + 3691188 \beta_{1} + 128340 \beta_{2} + 92560 \beta_{3} ) q^{81} + ( -46291200 + 188000 \beta_{1} + 304320 \beta_{2} + 455680 \beta_{3} ) q^{82} + ( -186639194 - 980280 \beta_{1} - 92806 \beta_{2} + 104262 \beta_{3} ) q^{83} + ( 155952384 + 761856 \beta_{1} - 96000 \beta_{2} + 32000 \beta_{3} ) q^{84} + ( -25807650 - 3902385 \beta_{1} + 26637 \beta_{2} + 1748748 \beta_{3} ) q^{85} + ( -30995808 + 947728 \beta_{1} - 399440 \beta_{2} - 410560 \beta_{3} ) q^{86} + ( -20486865 + 1052166 \beta_{1} + 187617 \beta_{2} + 13131 \beta_{3} ) q^{87} + ( 106725376 + 12288 \beta_{1} - 102400 \beta_{2} ) q^{88} + ( 72054508 - 224494 \beta_{1} + 201740 \beta_{2} + 1072810 \beta_{3} ) q^{89} + ( 495816480 + 5643120 \beta_{1} - 456912 \beta_{2} - 530208 \beta_{3} ) q^{90} + ( 788716179 + 2598565 \beta_{1} + 102178 \beta_{2} - 1018576 \beta_{3} ) q^{91} + ( 192763136 - 173568 \beta_{1} + 391936 \beta_{2} + 23808 \beta_{3} ) q^{92} + ( 288055008 + 4788454 \beta_{1} + 22674 \beta_{2} + 295762 \beta_{3} ) q^{93} + ( 126318368 - 725552 \beta_{1} + 497776 \beta_{2} - 521632 \beta_{3} ) q^{94} + ( -45612350 - 651605 \beta_{1} - 130321 \beta_{2} - 521284 \beta_{3} ) q^{95} + ( -22020096 - 1048576 \beta_{1} ) q^{96} + ( 197503898 - 3887418 \beta_{1} - 963354 \beta_{2} - 2541012 \beta_{3} ) q^{97} + ( -75529216 - 1767712 \beta_{1} - 195824 \beta_{2} + 1605088 \beta_{3} ) q^{98} + ( -420494730 - 4927511 \beta_{1} + 374571 \beta_{2} - 519682 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + O(q^{10})$$ $$4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + 9760585q^{25} - 1927776q^{26} + 12387978q^{27} + 3150592q^{28} + 6153240q^{29} + 9457440q^{30} + 12774024q^{31} - 4194304q^{32} - 3258022q^{33} + 6594224q^{34} + 9823425q^{35} + 4233728q^{36} + 20506048q^{37} - 8340544q^{38} + 69881444q^{39} + 5713920q^{40} + 11620300q^{41} - 38992096q^{42} + 7698327q^{43} - 26687744q^{44} - 124015815q^{45} - 48164800q^{46} - 31581083q^{47} + 5505024q^{48} + 18970383q^{49} - 156169360q^{50} - 8594812q^{51} + 30844416q^{52} + 72549422q^{53} - 198207648q^{54} + 21332505q^{55} - 50409472q^{56} + 10946964q^{57} - 98451840q^{58} - 149234120q^{59} - 151319040q^{60} + 129004373q^{61} - 204384384q^{62} + 102967551q^{63} + 67108864q^{64} + 124691700q^{65} + 52128352q^{66} + 132595266q^{67} - 105507584q^{68} - 45529972q^{69} - 157174800q^{70} - 47138482q^{71} - 67739648q^{72} - 39332795q^{73} - 328096768q^{74} + 824627010q^{75} + 133448704q^{76} - 165933719q^{77} - 1118103104q^{78} - 307010840q^{79} - 91422720q^{80} + 1305551744q^{81} - 185924800q^{82} - 746568232q^{83} + 623873536q^{84} - 105005985q^{85} - 123173232q^{86} - 82148208q^{87} + 427003904q^{88} + 286943482q^{89} + 1984253040q^{90} + 3155781114q^{91} + 770636800q^{92} + 1151901596q^{93} + 505297328q^{94} - 181797795q^{95} - 88080384q^{96} + 793519958q^{97} - 303526128q^{98} - 1681833809q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$9 \nu^{3} - 736 \nu^{2} - 252737 \nu - 5880008$$$$)/20632$$ $$\beta_{2}$$ $$=$$ $$($$$$-17 \nu^{3} + 244 \nu^{2} + 698613 \nu + 30780440$$$$)/20632$$ $$\beta_{3}$$ $$=$$ $$($$$$13 \nu^{3} - 490 \nu^{2} - 413779 \nu - 18350856$$$$)/10316$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$193 \beta_{3} + 85 \beta_{2} - 397 \beta_{1} + 103370$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$43865 \beta_{3} + 35033 \beta_{2} - 46793 \beta_{1} + 12429538$$$$)/6$$

## 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
 −124.888 219.264 −26.2676 −67.1081
−16.0000 −140.229 256.000 1263.95 2243.67 −3487.42 −4096.00 −18.7042 −20223.2
1.2 −16.0000 −66.6053 256.000 −2418.56 1065.68 −1533.95 −4096.00 −15246.7 38696.9
1.3 −16.0000 25.2570 256.000 2126.71 −404.112 11469.0 −4096.00 −19045.1 −34027.4
1.4 −16.0000 265.578 256.000 −2367.11 −4249.24 5859.40 −4096.00 50848.5 37873.7
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 84 T_{3}^{3} - 44107 T_{3}^{2} - 1329018 T_{3} + 62650008$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(38))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + T )^{4}$$
$3$ $$62650008 - 1329018 T - 44107 T^{2} - 84 T^{3} + T^{4}$$
$5$ $$15389064288000 - 6547343400 T - 7813530 T^{2} + 1395 T^{3} + T^{4}$$
$7$ $$359494671206216 + 244743771643 T - 14461281 T^{2} - 12307 T^{3} + T^{4}$$
$11$ $$115765319454159024 + 36343268701740 T + 3398840664 T^{2} + 104249 T^{3} + T^{4}$$
$13$ $$23\!\cdots\!00$$$$+ 2295095998230370 T - 30806569293 T^{2} - 120486 T^{3} + T^{4}$$
$17$ $$27\!\cdots\!62$$$$- 99501926037315363 T - 374372926323 T^{2} + 412139 T^{3} + T^{4}$$
$19$ $$( -130321 + T )^{4}$$
$23$ $$22\!\cdots\!08$$$$+ 2126564557696334952 T + 862408556595 T^{2} - 3010300 T^{3} + T^{4}$$
$29$ $$-$$$$34\!\cdots\!76$$$$+ 2981737641494670876 T + 8624111888511 T^{2} - 6153240 T^{3} + T^{4}$$
$31$ $$-$$$$14\!\cdots\!96$$$$- 63865994000171580800 T + 51883371775344 T^{2} - 12774024 T^{3} + T^{4}$$
$37$ $$-$$$$58\!\cdots\!72$$$$+$$$$22\!\cdots\!56$$$$T - 68206760068968 T^{2} - 20506048 T^{3} + T^{4}$$
$41$ $$-$$$$36\!\cdots\!00$$$$+$$$$67\!\cdots\!00$$$$T - 531906213139500 T^{2} - 11620300 T^{3} + T^{4}$$
$43$ $$-$$$$19\!\cdots\!12$$$$-$$$$93\!\cdots\!96$$$$T - 893995448146728 T^{2} - 7698327 T^{3} + T^{4}$$
$47$ $$52\!\cdots\!00$$$$-$$$$19\!\cdots\!60$$$$T - 1669221829649832 T^{2} + 31581083 T^{3} + T^{4}$$
$53$ $$-$$$$20\!\cdots\!64$$$$+$$$$31\!\cdots\!82$$$$T - 4017930015425685 T^{2} - 72549422 T^{3} + T^{4}$$
$59$ $$-$$$$27\!\cdots\!04$$$$-$$$$16\!\cdots\!70$$$$T + 3811888983551253 T^{2} + 149234120 T^{3} + T^{4}$$
$61$ $$43\!\cdots\!00$$$$+$$$$91\!\cdots\!60$$$$T - 10642541015571546 T^{2} - 129004373 T^{3} + T^{4}$$
$67$ $$-$$$$18\!\cdots\!00$$$$+$$$$84\!\cdots\!00$$$$T - 58349865773044743 T^{2} - 132595266 T^{3} + T^{4}$$
$71$ $$-$$$$32\!\cdots\!32$$$$-$$$$10\!\cdots\!44$$$$T - 74119300967107956 T^{2} + 47138482 T^{3} + T^{4}$$
$73$ $$35\!\cdots\!34$$$$-$$$$10\!\cdots\!87$$$$T - 87654713869556991 T^{2} + 39332795 T^{3} + T^{4}$$
$79$ $$29\!\cdots\!00$$$$+$$$$10\!\cdots\!40$$$$T - 229308708527002548 T^{2} + 307010840 T^{3} + T^{4}$$
$83$ $$-$$$$94\!\cdots\!68$$$$+$$$$21\!\cdots\!04$$$$T + 152802359319519228 T^{2} + 746568232 T^{3} + T^{4}$$
$89$ $$54\!\cdots\!84$$$$+$$$$99\!\cdots\!80$$$$T - 451932251304917424 T^{2} - 286943482 T^{3} + T^{4}$$
$97$ $$37\!\cdots\!00$$$$+$$$$15\!\cdots\!80$$$$T - 3736940299059182196 T^{2} - 793519958 T^{3} + T^{4}$$