Properties

 Label 38.10.a.e 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: $$x^{4} - 2 x^{3} - 8016 x^{2} - 155839 x + 2105804$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + ( 57 + \beta_{2} ) q^{3} + 256 q^{4} + ( 218 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( 912 + 16 \beta_{2} ) q^{6} + ( 672 + 20 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{7} + 4096 q^{8} + ( 12678 - 95 \beta_{1} + 53 \beta_{2} - 7 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 16 q^{2} + ( 57 + \beta_{2} ) q^{3} + 256 q^{4} + ( 218 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( 912 + 16 \beta_{2} ) q^{6} + ( 672 + 20 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{7} + 4096 q^{8} + ( 12678 - 95 \beta_{1} + 53 \beta_{2} - 7 \beta_{3} ) q^{9} + ( 3488 - 16 \beta_{1} + 48 \beta_{2} + 16 \beta_{3} ) q^{10} + ( 29748 + 75 \beta_{1} - 121 \beta_{2} - 23 \beta_{3} ) q^{11} + ( 14592 + 256 \beta_{2} ) q^{12} + ( -1568 + 275 \beta_{1} + 238 \beta_{2} - 48 \beta_{3} ) q^{13} + ( 10752 + 320 \beta_{1} + 144 \beta_{2} + 16 \beta_{3} ) q^{14} + ( 88972 - 489 \beta_{1} - 8 \beta_{2} + 49 \beta_{3} ) q^{15} + 65536 q^{16} + ( 168631 + 20 \beta_{1} - 2050 \beta_{2} + 164 \beta_{3} ) q^{17} + ( 202848 - 1520 \beta_{1} + 848 \beta_{2} - 112 \beta_{3} ) q^{18} -130321 q^{19} + ( 55808 - 256 \beta_{1} + 768 \beta_{2} + 256 \beta_{3} ) q^{20} + ( 394438 + 1230 \beta_{1} - 4492 \beta_{2} - 119 \beta_{3} ) q^{21} + ( 475968 + 1200 \beta_{1} - 1936 \beta_{2} - 368 \beta_{3} ) q^{22} + ( 724276 + 885 \beta_{1} - 7382 \beta_{2} - 286 \beta_{3} ) q^{23} + ( 233472 + 4096 \beta_{2} ) q^{24} + ( 67043 + 3659 \beta_{1} - 297 \beta_{2} + 1171 \beta_{3} ) q^{25} + ( -25088 + 4400 \beta_{1} + 3808 \beta_{2} - 768 \beta_{3} ) q^{26} + ( 709589 - 14725 \beta_{1} + 18149 \beta_{2} - 249 \beta_{3} ) q^{27} + ( 172032 + 5120 \beta_{1} + 2304 \beta_{2} + 256 \beta_{3} ) q^{28} + ( 2073202 + 550 \beta_{1} + 852 \beta_{2} - 2317 \beta_{3} ) q^{29} + ( 1423552 - 7824 \beta_{1} - 128 \beta_{2} + 784 \beta_{3} ) q^{30} + ( 870276 - 9775 \beta_{1} + 17818 \beta_{2} + 1575 \beta_{3} ) q^{31} + 1048576 q^{32} + ( -1318954 + 21855 \beta_{1} + 22986 \beta_{2} - 1075 \beta_{3} ) q^{33} + ( 2698096 + 320 \beta_{1} - 32800 \beta_{2} + 2624 \beta_{3} ) q^{34} + ( 1919394 - 6443 \beta_{1} - 18741 \beta_{2} + 5483 \beta_{3} ) q^{35} + ( 3245568 - 24320 \beta_{1} + 13568 \beta_{2} - 1792 \beta_{3} ) q^{36} + ( -261758 + 59215 \beta_{1} - 20754 \beta_{2} - 2407 \beta_{3} ) q^{37} -2085136 q^{38} + ( 8491292 + 11925 \beta_{1} - 45366 \beta_{2} - 6388 \beta_{3} ) q^{39} + ( 892928 - 4096 \beta_{1} + 12288 \beta_{2} + 4096 \beta_{3} ) q^{40} + ( 2148182 - 68935 \beta_{1} + 39302 \beta_{2} + 6677 \beta_{3} ) q^{41} + ( 6311008 + 19680 \beta_{1} - 71872 \beta_{2} - 1904 \beta_{3} ) q^{42} + ( 3472138 - 40335 \beta_{1} - 6087 \beta_{2} - 12765 \beta_{3} ) q^{43} + ( 7615488 + 19200 \beta_{1} - 30976 \beta_{2} - 5888 \beta_{3} ) q^{44} + ( -2179336 - 37513 \beta_{1} + 122429 \beta_{2} - 13557 \beta_{3} ) q^{45} + ( 11588416 + 14160 \beta_{1} - 118112 \beta_{2} - 4576 \beta_{3} ) q^{46} + ( -9150076 + 15365 \beta_{1} - 132675 \beta_{2} + 26833 \beta_{3} ) q^{47} + ( 3735552 + 65536 \beta_{2} ) q^{48} + ( -7756834 + 35580 \beta_{1} - 67764 \beta_{2} - 2112 \beta_{3} ) q^{49} + ( 1072688 + 58544 \beta_{1} - 4752 \beta_{2} + 18736 \beta_{3} ) q^{50} + ( -50917929 + 181350 \beta_{1} + 98679 \beta_{2} + 24726 \beta_{3} ) q^{51} + ( -401408 + 70400 \beta_{1} + 60928 \beta_{2} - 12288 \beta_{3} ) q^{52} + ( -28392182 - 211250 \beta_{1} + 200314 \beta_{2} - 5549 \beta_{3} ) q^{53} + ( 11353424 - 235600 \beta_{1} + 290384 \beta_{2} - 3984 \beta_{3} ) q^{54} + ( -44498004 - 113927 \beta_{1} + 74041 \beta_{2} + 17917 \beta_{3} ) q^{55} + ( 2752512 + 81920 \beta_{1} + 36864 \beta_{2} + 4096 \beta_{3} ) q^{56} + ( -7428297 - 130321 \beta_{2} ) q^{57} + ( 33171232 + 8800 \beta_{1} + 13632 \beta_{2} - 37072 \beta_{3} ) q^{58} + ( -99211973 + 255595 \beta_{1} - 37063 \beta_{2} - 23037 \beta_{3} ) q^{59} + ( 22776832 - 125184 \beta_{1} - 2048 \beta_{2} + 12544 \beta_{3} ) q^{60} + ( -74655920 + 201455 \beta_{1} - 215807 \beta_{2} - 40083 \beta_{3} ) q^{61} + ( 13924416 - 156400 \beta_{1} + 285088 \beta_{2} + 25200 \beta_{3} ) q^{62} + ( -114680548 + 178455 \beta_{1} + 751 \beta_{2} - 3235 \beta_{3} ) q^{63} + 16777216 q^{64} + ( -72911878 - 425824 \beta_{1} - 229918 \beta_{2} - 8146 \beta_{3} ) q^{65} + ( -21103264 + 349680 \beta_{1} + 367776 \beta_{2} - 17200 \beta_{3} ) q^{66} + ( -28773199 - 305650 \beta_{1} - 770537 \beta_{2} - 58278 \beta_{3} ) q^{67} + ( 43169536 + 5120 \beta_{1} - 524800 \beta_{2} + 41984 \beta_{3} ) q^{68} + ( -167542508 + 824925 \beta_{1} + 674842 \beta_{2} + 28060 \beta_{3} ) q^{69} + ( 30710304 - 103088 \beta_{1} - 299856 \beta_{2} + 87728 \beta_{3} ) q^{70} + ( 850912 - 669390 \beta_{1} - 627986 \beta_{2} + 94124 \beta_{3} ) q^{71} + ( 51929088 - 389120 \beta_{1} + 217088 \beta_{2} - 28672 \beta_{3} ) q^{72} + ( 33945965 - 709870 \beta_{1} - 207296 \beta_{2} - 55158 \beta_{3} ) q^{73} + ( -4188128 + 947440 \beta_{1} - 332064 \beta_{2} - 38512 \beta_{3} ) q^{74} + ( 6670777 + 315801 \beta_{1} - 1312583 \beta_{2} + 55069 \beta_{3} ) q^{75} -33362176 q^{76} + ( 30611478 + 918985 \beta_{1} + 947013 \beta_{2} - 99511 \beta_{3} ) q^{77} + ( 135860672 + 190800 \beta_{1} - 725856 \beta_{2} - 102208 \beta_{3} ) q^{78} + ( 17015018 - 62905 \beta_{1} + 402324 \beta_{2} + 39373 \beta_{3} ) q^{79} + ( 14286848 - 65536 \beta_{1} + 196608 \beta_{2} + 65536 \beta_{3} ) q^{80} + ( 247135793 - 1435640 \beta_{1} + 3150996 \beta_{2} + 83152 \beta_{3} ) q^{81} + ( 34370912 - 1102960 \beta_{1} + 628832 \beta_{2} + 106832 \beta_{3} ) q^{82} + ( 343756056 - 199770 \beta_{1} - 740496 \beta_{2} - 59700 \beta_{3} ) q^{83} + ( 100976128 + 314880 \beta_{1} - 1149952 \beta_{2} - 30464 \beta_{3} ) q^{84} + ( 159920954 + 1503517 \beta_{1} + 640819 \beta_{2} + 379483 \beta_{3} ) q^{85} + ( 55554208 - 645360 \beta_{1} - 97392 \beta_{2} - 204240 \beta_{3} ) q^{86} + ( 159148436 + 199125 \beta_{1} + 2979110 \beta_{2} - 157552 \beta_{3} ) q^{87} + ( 121847808 + 307200 \beta_{1} - 495616 \beta_{2} - 94208 \beta_{3} ) q^{88} + ( 389375140 + 1442325 \beta_{1} + 144650 \beta_{2} + 16075 \beta_{3} ) q^{89} + ( -34869376 - 600208 \beta_{1} + 1958864 \beta_{2} - 216912 \beta_{3} ) q^{90} + ( 355102557 + 871825 \beta_{1} - 1181751 \beta_{2} - 397111 \beta_{3} ) q^{91} + ( 185414656 + 226560 \beta_{1} - 1889792 \beta_{2} - 73216 \beta_{3} ) q^{92} + ( 510361898 - 2907810 \beta_{1} + 2380754 \beta_{2} + 34724 \beta_{3} ) q^{93} + ( -146401216 + 245840 \beta_{1} - 2122800 \beta_{2} + 429328 \beta_{3} ) q^{94} + ( -28409978 + 130321 \beta_{1} - 390963 \beta_{2} - 130321 \beta_{3} ) q^{95} + ( 59768832 + 1048576 \beta_{2} ) q^{96} + ( 244332850 + 247540 \beta_{1} + 756606 \beta_{2} - 183388 \beta_{3} ) q^{97} + ( -124109344 + 569280 \beta_{1} - 1084224 \beta_{2} - 33792 \beta_{3} ) q^{98} + ( 123874300 - 1175575 \beta_{1} - 3661725 \beta_{2} + 91877 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + 16384q^{8} + 50606q^{9} + O(q^{10})$$ $$4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + 16384q^{8} + 50606q^{9} + 13856q^{10} + 119234q^{11} + 57856q^{12} - 6748q^{13} + 42720q^{14} + 355904q^{15} + 262144q^{16} + 678624q^{17} + 809696q^{18} - 521284q^{19} + 221696q^{20} + 1586736q^{21} + 1907744q^{22} + 2911868q^{23} + 925696q^{24} + 268766q^{25} - 107968q^{26} + 2802058q^{27} + 683520q^{28} + 8291104q^{29} + 5694464q^{30} + 3445468q^{31} + 4194304q^{32} - 5321788q^{33} + 10857984q^{34} + 7715058q^{35} + 12955136q^{36} - 1005524q^{37} - 8340544q^{38} + 34055900q^{39} + 3547136q^{40} + 8514124q^{41} + 25387776q^{42} + 13900726q^{43} + 30523904q^{44} - 8962202q^{45} + 46589888q^{46} - 36334954q^{47} + 14811136q^{48} - 30891808q^{49} + 4300256q^{50} - 203869074q^{51} - 1727488q^{52} - 113969356q^{53} + 44832928q^{54} - 178140098q^{55} + 10936320q^{56} - 29452546q^{57} + 132657664q^{58} - 396773766q^{59} + 91111424q^{60} - 298192066q^{61} + 55127488q^{62} - 458723694q^{63} + 67108864q^{64} - 291187676q^{65} - 85148608q^{66} - 113551722q^{67} + 173727744q^{68} - 671519716q^{69} + 123440928q^{70} + 4659620q^{71} + 207282176q^{72} + 136198452q^{73} - 16088384q^{74} + 29308274q^{75} - 133448704q^{76} + 120551886q^{77} + 544894400q^{78} + 67255424q^{79} + 56754176q^{80} + 982241180q^{81} + 136225984q^{82} + 1376505216q^{83} + 406204416q^{84} + 638402178q^{85} + 222411616q^{86} + 630635524q^{87} + 488382464q^{88} + 1557211260q^{89} - 143395232q^{90} + 1422773730q^{91} + 745438208q^{92} + 2036686084q^{93} - 581359264q^{94} - 112857986q^{95} + 236978176q^{96} + 975818188q^{97} - 494268928q^{98} + 502820650q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 8016 x^{2} - 155839 x + 2105804$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} - 47 \nu^{2} - 12893 \nu - 62941$$$$)/693$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{2} + 57 \nu + 3970$$$$)/21$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{3} + 179 \nu^{2} + 22001 \nu - 470801$$$$)/693$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} + \beta_{1} + 14$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$19 \beta_{3} - 92 \beta_{2} + 19 \beta_{1} + 32026$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$3893 \beta_{3} + 9650 \beta_{2} + 8051 \beta_{1} + 1551688$$$$)/12$$

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
 −73.1023 97.8976 −31.9926 9.19740
16.0000 −206.847 256.000 −845.315 −3309.55 −6607.98 4096.00 23102.6 −13525.0
1.2 16.0000 55.3919 256.000 2263.93 886.270 5766.09 4096.00 −16614.7 36222.9
1.3 16.0000 110.471 256.000 −1298.23 1767.54 6626.40 4096.00 −7479.16 −20771.6
1.4 16.0000 266.984 256.000 745.613 4271.74 −3114.51 4096.00 51597.3 11929.8
 $$n$$: 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.e 4
3.b odd 2 1 342.10.a.i 4
4.b odd 2 1 304.10.a.d 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 226 T_{3}^{3} - 39131 T_{3}^{2} + 8791740 T_{3} - 337930956$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(38))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 + T )^{4}$$
$3$ $$-337930956 + 8791740 T - 39131 T^{2} - 226 T^{3} + T^{4}$$
$5$ $$1852446724000 + 315628300 T - 3665655 T^{2} - 866 T^{3} + T^{4}$$
$7$ $$786353549326443 + 116436017502 T - 61696860 T^{2} - 2670 T^{3} + T^{4}$$
$11$ $$308126839450989292 + 65933876389888 T + 2279476197 T^{2} - 119234 T^{3} + T^{4}$$
$13$ $$-7995567793303427024 - 1303658610803432 T - 21677117661 T^{2} + 6748 T^{3} + T^{4}$$
$17$ $$16\!\cdots\!09$$$$+ 102397547874735072 T - 173140916562 T^{2} - 678624 T^{3} + T^{4}$$
$19$ $$( 130321 + T )^{4}$$
$23$ $$-$$$$60\!\cdots\!88$$$$+ 2727384504786073504 T - 183984048477 T^{2} - 2911868 T^{3} + T^{4}$$
$29$ $$-$$$$13\!\cdots\!00$$$$+ 71525180313253109780 T + 6979347050031 T^{2} - 8291104 T^{3} + T^{4}$$
$31$ $$16\!\cdots\!44$$$$+ 31506328044595609280 T - 29678593269744 T^{2} - 3445468 T^{3} + T^{4}$$
$37$ $$88\!\cdots\!52$$$$-$$$$14\!\cdots\!28$$$$T - 457850038983408 T^{2} + 1005524 T^{3} + T^{4}$$
$41$ $$-$$$$18\!\cdots\!92$$$$+$$$$77\!\cdots\!12$$$$T - 733459745449164 T^{2} - 8514124 T^{3} + T^{4}$$
$43$ $$-$$$$70\!\cdots\!72$$$$+$$$$15\!\cdots\!88$$$$T - 747338384701767 T^{2} - 13900726 T^{3} + T^{4}$$
$47$ $$11\!\cdots\!00$$$$-$$$$99\!\cdots\!00$$$$T - 3143481498548271 T^{2} + 36334954 T^{3} + T^{4}$$
$53$ $$-$$$$13\!\cdots\!08$$$$-$$$$58\!\cdots\!32$$$$T - 2536079015140869 T^{2} + 113969356 T^{3} + T^{4}$$
$59$ $$-$$$$41\!\cdots\!00$$$$+$$$$15\!\cdots\!00$$$$T + 49363170632995341 T^{2} + 396773766 T^{3} + T^{4}$$
$61$ $$-$$$$32\!\cdots\!92$$$$-$$$$30\!\cdots\!68$$$$T + 21473548273899621 T^{2} + 298192066 T^{3} + T^{4}$$
$67$ $$-$$$$45\!\cdots\!48$$$$-$$$$10\!\cdots\!96$$$$T - 58688917279201671 T^{2} + 113551722 T^{3} + T^{4}$$
$71$ $$-$$$$13\!\cdots\!12$$$$+$$$$12\!\cdots\!48$$$$T - 116771198854642560 T^{2} - 4659620 T^{3} + T^{4}$$
$73$ $$14\!\cdots\!57$$$$+$$$$51\!\cdots\!00$$$$T - 76284324026492010 T^{2} - 136198452 T^{3} + T^{4}$$
$79$ $$32\!\cdots\!00$$$$+$$$$60\!\cdots\!60$$$$T - 12614020693462548 T^{2} - 67255424 T^{3} + T^{4}$$
$83$ $$94\!\cdots\!24$$$$-$$$$13\!\cdots\!80$$$$T + 658124224934927004 T^{2} - 1376505216 T^{3} + T^{4}$$
$89$ $$42\!\cdots\!00$$$$-$$$$31\!\cdots\!00$$$$T + 634805908820871600 T^{2} - 1557211260 T^{3} + T^{4}$$
$97$ $$-$$$$24\!\cdots\!84$$$$+$$$$20\!\cdots\!08$$$$T + 194682750107261220 T^{2} - 975818188 T^{3} + T^{4}$$