# Properties

 Label 38.10.a.c Level $38$ Weight $10$ Character orbit 38.a Self dual yes Analytic conductor $19.571$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 4552 x + 85948$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -16 q^{2} + ( 1 + \beta_{1} ) q^{3} + 256 q^{4} + ( 162 + 2 \beta_{1} + \beta_{2} ) q^{5} + ( -16 - 16 \beta_{1} ) q^{6} + ( -4439 - 14 \beta_{1} - 10 \beta_{2} ) q^{7} -4096 q^{8} + ( 7632 - 72 \beta_{1} + 27 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -16 q^{2} + ( 1 + \beta_{1} ) q^{3} + 256 q^{4} + ( 162 + 2 \beta_{1} + \beta_{2} ) q^{5} + ( -16 - 16 \beta_{1} ) q^{6} + ( -4439 - 14 \beta_{1} - 10 \beta_{2} ) q^{7} -4096 q^{8} + ( 7632 - 72 \beta_{1} + 27 \beta_{2} ) q^{9} + ( -2592 - 32 \beta_{1} - 16 \beta_{2} ) q^{10} + ( 18656 - 386 \beta_{1} - 43 \beta_{2} ) q^{11} + ( 256 + 256 \beta_{1} ) q^{12} + ( 52727 - 204 \beta_{1} + 67 \beta_{2} ) q^{13} + ( 71024 + 224 \beta_{1} + 160 \beta_{2} ) q^{14} + ( 45220 + 319 \beta_{1} + 129 \beta_{2} ) q^{15} + 65536 q^{16} + ( -209697 + 858 \beta_{1} - 564 \beta_{2} ) q^{17} + ( -122112 + 1152 \beta_{1} - 432 \beta_{2} ) q^{18} -130321 q^{19} + ( 41472 + 512 \beta_{1} + 256 \beta_{2} ) q^{20} + ( -291135 - 6447 \beta_{1} - 1128 \beta_{2} ) q^{21} + ( -298496 + 6176 \beta_{1} + 688 \beta_{2} ) q^{22} + ( -308209 + 5110 \beta_{1} + 3289 \beta_{2} ) q^{23} + ( -4096 - 4096 \beta_{1} ) q^{24} + ( -1601839 + 2040 \beta_{1} + 425 \beta_{2} ) q^{25} + ( -843632 + 3264 \beta_{1} - 1072 \beta_{2} ) q^{26} + ( -2237049 + 1386 \beta_{1} + 81 \beta_{2} ) q^{27} + ( -1136384 - 3584 \beta_{1} - 2560 \beta_{2} ) q^{28} + ( -3279673 - 13893 \beta_{1} - 3628 \beta_{2} ) q^{29} + ( -723520 - 5104 \beta_{1} - 2064 \beta_{2} ) q^{30} + ( 454876 - 29685 \beta_{1} + 8839 \beta_{2} ) q^{31} -1048576 q^{32} + ( -10113038 + 33805 \beta_{1} - 13647 \beta_{2} ) q^{33} + ( 3355152 - 13728 \beta_{1} + 9024 \beta_{2} ) q^{34} + ( -3699190 - 24136 \beta_{1} - 6301 \beta_{2} ) q^{35} + ( 1953792 - 18432 \beta_{1} + 6912 \beta_{2} ) q^{36} + ( -763030 + 3757 \beta_{1} + 14761 \beta_{2} ) q^{37} + 2085136 q^{38} + ( -6160519 + 87920 \beta_{1} - 483 \beta_{2} ) q^{39} + ( -663552 - 8192 \beta_{1} - 4096 \beta_{2} ) q^{40} + ( -4299860 + 13925 \beta_{1} - 15333 \beta_{2} ) q^{41} + ( 4658160 + 103152 \beta_{1} + 18048 \beta_{2} ) q^{42} + ( -7459546 + 106246 \beta_{1} - 9441 \beta_{2} ) q^{43} + ( 4775936 - 98816 \beta_{1} - 11008 \beta_{2} ) q^{44} + ( 4335210 + 21654 \beta_{1} - 1395 \beta_{2} ) q^{45} + ( 4931344 - 81760 \beta_{1} - 52624 \beta_{2} ) q^{46} + ( 19632122 + 15016 \beta_{1} + 44215 \beta_{2} ) q^{47} + ( 65536 + 65536 \beta_{1} ) q^{48} + ( 7431658 + 242228 \beta_{1} + 84492 \beta_{2} ) q^{49} + ( 25629424 - 32640 \beta_{1} - 6800 \beta_{2} ) q^{50} + ( 28623195 - 443223 \beta_{1} - 19134 \beta_{2} ) q^{51} + ( 13498112 - 52224 \beta_{1} + 17152 \beta_{2} ) q^{52} + ( 2923543 - 299391 \beta_{1} - 106356 \beta_{2} ) q^{53} + ( 35792784 - 22176 \beta_{1} - 1296 \beta_{2} ) q^{54} + ( -24471934 - 131576 \beta_{1} - 31053 \beta_{2} ) q^{55} + ( 18182144 + 57344 \beta_{1} + 40960 \beta_{2} ) q^{56} + ( -130321 - 130321 \beta_{1} ) q^{57} + ( 52474768 + 222288 \beta_{1} + 58048 \beta_{2} ) q^{58} + ( 5475433 - 71558 \beta_{1} + 55609 \beta_{2} ) q^{59} + ( 11576320 + 81664 \beta_{1} + 33024 \beta_{2} ) q^{60} + ( -42281260 + 143100 \beta_{1} - 21479 \beta_{2} ) q^{61} + ( -7278016 + 474960 \beta_{1} - 141424 \beta_{2} ) q^{62} + ( -78216696 + 113274 \beta_{1} - 61839 \beta_{2} ) q^{63} + 16777216 q^{64} + ( 15089984 + 113280 \beta_{1} + 27084 \beta_{2} ) q^{65} + ( 161808608 - 540880 \beta_{1} + 218352 \beta_{2} ) q^{66} + ( -96025103 - 461027 \beta_{1} - 162254 \beta_{2} ) q^{67} + ( -53682432 + 219648 \beta_{1} - 144384 \beta_{2} ) q^{68} + ( 107790601 + 315328 \beta_{1} + 384645 \beta_{2} ) q^{69} + ( 59187040 + 386176 \beta_{1} + 100816 \beta_{2} ) q^{70} + ( 26040758 + 583150 \beta_{1} + 105562 \beta_{2} ) q^{71} + ( -31260672 + 294912 \beta_{1} - 110592 \beta_{2} ) q^{72} + ( -185980615 + 492830 \beta_{1} - 464234 \beta_{2} ) q^{73} + ( 12208480 - 60112 \beta_{1} - 236176 \beta_{2} ) q^{74} + ( 50051471 - 1621984 \beta_{1} + 86955 \beta_{2} ) q^{75} -33362176 q^{76} + ( 131337912 + 2609040 \beta_{1} + 350123 \beta_{2} ) q^{77} + ( 98568304 - 1406720 \beta_{1} + 7728 \beta_{2} ) q^{78} + ( -106740674 + 517415 \beta_{1} + 65849 \beta_{2} ) q^{79} + ( 10616832 + 131072 \beta_{1} + 65536 \beta_{2} ) q^{80} + ( -115375671 - 896508 \beta_{1} - 487944 \beta_{2} ) q^{81} + ( 68797760 - 222800 \beta_{1} + 245328 \beta_{2} ) q^{82} + ( 143497154 - 44200 \beta_{1} + 330414 \beta_{2} ) q^{83} + ( -74530560 - 1650432 \beta_{1} - 288768 \beta_{2} ) q^{84} + ( -127809414 - 757656 \beta_{1} - 103821 \beta_{2} ) q^{85} + ( 119352736 - 1699936 \beta_{1} + 151056 \beta_{2} ) q^{86} + ( -348033115 - 3364768 \beta_{1} - 647211 \beta_{2} ) q^{87} + ( -76414976 + 1581056 \beta_{1} + 176128 \beta_{2} ) q^{88} + ( 145896548 + 2108807 \beta_{1} + 726611 \beta_{2} ) q^{89} + ( -69363360 - 346464 \beta_{1} + 22320 \beta_{2} ) q^{90} + ( -336816729 - 28980 \beta_{1} - 463013 \beta_{2} ) q^{91} + ( -78901504 + 1308160 \beta_{1} + 841984 \beta_{2} ) q^{92} + ( -894950444 + 5300098 \beta_{1} - 138570 \beta_{2} ) q^{93} + ( -314113952 - 240256 \beta_{1} - 707440 \beta_{2} ) q^{94} + ( -21112002 - 260642 \beta_{1} - 130321 \beta_{2} ) q^{95} + ( -1048576 - 1048576 \beta_{1} ) q^{96} + ( -128317382 + 2792874 \beta_{1} + 1872892 \beta_{2} ) q^{97} + ( -118906528 - 3875648 \beta_{1} - 1351872 \beta_{2} ) q^{98} + ( 676632474 - 9118206 \beta_{1} + 735579 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} - 12288q^{8} + 22896q^{9} + O(q^{10})$$ $$3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} - 12288q^{8} + 22896q^{9} - 7776q^{10} + 55968q^{11} + 768q^{12} + 158181q^{13} + 213072q^{14} + 135660q^{15} + 196608q^{16} - 629091q^{17} - 366336q^{18} - 390963q^{19} + 124416q^{20} - 873405q^{21} - 895488q^{22} - 924627q^{23} - 12288q^{24} - 4805517q^{25} - 2530896q^{26} - 6711147q^{27} - 3409152q^{28} - 9839019q^{29} - 2170560q^{30} + 1364628q^{31} - 3145728q^{32} - 30339114q^{33} + 10065456q^{34} - 11097570q^{35} + 5861376q^{36} - 2289090q^{37} + 6255408q^{38} - 18481557q^{39} - 1990656q^{40} - 12899580q^{41} + 13974480q^{42} - 22378638q^{43} + 14327808q^{44} + 13005630q^{45} + 14794032q^{46} + 58896366q^{47} + 196608q^{48} + 22294974q^{49} + 76888272q^{50} + 85869585q^{51} + 40494336q^{52} + 8770629q^{53} + 107378352q^{54} - 73415802q^{55} + 54546432q^{56} - 390963q^{57} + 157424304q^{58} + 16426299q^{59} + 34728960q^{60} - 126843780q^{61} - 21834048q^{62} - 234650088q^{63} + 50331648q^{64} + 45269952q^{65} + 485425824q^{66} - 288075309q^{67} - 161047296q^{68} + 323371803q^{69} + 177561120q^{70} + 78122274q^{71} - 93782016q^{72} - 557941845q^{73} + 36625440q^{74} + 150154413q^{75} - 100086528q^{76} + 394013736q^{77} + 295704912q^{78} - 320222022q^{79} + 31850496q^{80} - 346127013q^{81} + 206393280q^{82} + 430491462q^{83} - 223591680q^{84} - 383428242q^{85} + 358058208q^{86} - 1044099345q^{87} - 229244928q^{88} + 437689644q^{89} - 208090080q^{90} - 1010450187q^{91} - 236704512q^{92} - 2684851332q^{93} - 942341856q^{94} - 63336006q^{95} - 3145728q^{96} - 384952146q^{97} - 356719584q^{98} + 2029897422q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4552 x + 85948$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + 24 \nu - 3043$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2} - 8 \beta_{1} + 3035$$

## 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
 −74.9878 20.7493 55.2385
−16.0000 −224.963 256.000 −29.7721 3599.41 −3877.06 −4096.00 30925.5 476.353
1.2 −16.0000 62.2478 256.000 −420.333 −995.965 1751.81 −4096.00 −15808.2 6725.32
1.3 −16.0000 165.716 256.000 936.105 −2651.45 −11191.8 −4096.00 7778.66 −14977.7
 $$n$$: 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.c 3
3.b odd 2 1 342.10.a.e 3
4.b odd 2 1 304.10.a.c 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 3 T_{3}^{2} - 40968 T_{3} + 2320596$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(38))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + T )^{3}$$
$3$ $$2320596 - 40968 T - 3 T^{2} + T^{3}$$
$5$ $$-11714584 - 408831 T - 486 T^{2} + T^{3}$$
$7$ $$-76013110089 + 16993347 T + 13317 T^{2} + T^{3}$$
$11$ $$153565439505290 - 5288499579 T - 55968 T^{2} + T^{3}$$
$13$ $$39757599966256 + 4532198760 T - 158181 T^{2} + T^{3}$$
$17$ $$-42219259563623643 - 33362060841 T + 629091 T^{2} + T^{3}$$
$19$ $$( 130321 + T )^{3}$$
$23$ $$-2803969222636749760 - 4424883980736 T + 924627 T^{2} + T^{3}$$
$29$ $$12251122398463289740 + 20791650832632 T + 9839019 T^{2} + T^{3}$$
$31$ $$-36228495064532263360 - 72790443364176 T - 1364628 T^{2} + T^{3}$$
$37$ $$15\!\cdots\!48$$$$- 80276054575188 T + 2289090 T^{2} + T^{3}$$
$41$ $$-$$$$74\!\cdots\!76$$$$- 48204763237836 T + 12899580 T^{2} + T^{3}$$
$43$ $$11\!\cdots\!48$$$$- 358321271024367 T + 22378638 T^{2} + T^{3}$$
$47$ $$12\!\cdots\!80$$$$+ 421048318071201 T - 58896366 T^{2} + T^{3}$$
$53$ $$23\!\cdots\!92$$$$- 7043447389420608 T - 8770629 T^{2} + T^{3}$$
$59$ $$25\!\cdots\!32$$$$- 1412593293705816 T - 16426299 T^{2} + T^{3}$$
$61$ $$41\!\cdots\!58$$$$+ 4260063288756141 T + 126843780 T^{2} + T^{3}$$
$67$ $$82\!\cdots\!08$$$$+ 11068901462426064 T + 288075309 T^{2} + T^{3}$$
$71$ $$-$$$$16\!\cdots\!60$$$$- 14377765914503364 T - 78122274 T^{2} + T^{3}$$
$73$ $$-$$$$22\!\cdots\!17$$$$+ 5114889505651731 T + 557941845 T^{2} + T^{3}$$
$79$ $$-$$$$19\!\cdots\!00$$$$+ 22537734732929904 T + 320222022 T^{2} + T^{3}$$
$83$ $$62\!\cdots\!44$$$$+ 19669089705706944 T - 430491462 T^{2} + T^{3}$$
$89$ $$-$$$$27\!\cdots\!80$$$$- 275557924910672976 T - 437689644 T^{2} + T^{3}$$
$97$ $$-$$$$40\!\cdots\!92$$$$- 1456799102471738880 T + 384952146 T^{2} + T^{3}$$