Properties

Label 38.6.a.d
Level 38
Weight 6
Character orbit 38.a
Self dual yes
Analytic conductor 6.095
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 454 x + 3760\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( 4 + \beta_{1} ) q^{3} + 16 q^{4} + ( 27 - \beta_{1} + \beta_{2} ) q^{5} + ( 16 + 4 \beta_{1} ) q^{6} + ( 79 - 4 \beta_{1} - 5 \beta_{2} ) q^{7} + 64 q^{8} + ( 79 - 3 \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + 4 q^{2} + ( 4 + \beta_{1} ) q^{3} + 16 q^{4} + ( 27 - \beta_{1} + \beta_{2} ) q^{5} + ( 16 + 4 \beta_{1} ) q^{6} + ( 79 - 4 \beta_{1} - 5 \beta_{2} ) q^{7} + 64 q^{8} + ( 79 - 3 \beta_{1} + 2 \beta_{2} ) q^{9} + ( 108 - 4 \beta_{1} + 4 \beta_{2} ) q^{10} + ( 121 - 17 \beta_{1} + 17 \beta_{2} ) q^{11} + ( 64 + 16 \beta_{1} ) q^{12} + ( 178 - 5 \beta_{1} - 28 \beta_{2} ) q^{13} + ( 316 - 16 \beta_{1} - 20 \beta_{2} ) q^{14} + ( -242 + 42 \beta_{1} + 14 \beta_{2} ) q^{15} + 256 q^{16} + ( -429 + 60 \beta_{1} + 21 \beta_{2} ) q^{17} + ( 316 - 12 \beta_{1} + 8 \beta_{2} ) q^{18} -361 q^{19} + ( 432 - 16 \beta_{1} + 16 \beta_{2} ) q^{20} + ( -688 + 67 \beta_{1} - 88 \beta_{2} ) q^{21} + ( 484 - 68 \beta_{1} + 68 \beta_{2} ) q^{22} + ( -318 - 157 \beta_{1} + 34 \beta_{2} ) q^{23} + ( 256 + 64 \beta_{1} ) q^{24} + ( -1284 - 55 \beta_{1} + 25 \beta_{2} ) q^{25} + ( 712 - 20 \beta_{1} - 112 \beta_{2} ) q^{26} + ( -1662 - 127 \beta_{1} + 26 \beta_{2} ) q^{27} + ( 1264 - 64 \beta_{1} - 80 \beta_{2} ) q^{28} + ( -2844 + 121 \beta_{1} + 62 \beta_{2} ) q^{29} + ( -968 + 168 \beta_{1} + 56 \beta_{2} ) q^{30} + ( -2388 + 28 \beta_{1} - 196 \beta_{2} ) q^{31} + 1024 q^{32} + ( -5466 + 376 \beta_{1} + 238 \beta_{2} ) q^{33} + ( -1716 + 240 \beta_{1} + 84 \beta_{2} ) q^{34} + ( -277 - 353 \beta_{1} - \beta_{2} ) q^{35} + ( 1264 - 48 \beta_{1} + 32 \beta_{2} ) q^{36} + ( -510 - 236 \beta_{1} + 116 \beta_{2} ) q^{37} -1444 q^{38} + ( 414 - 11 \beta_{1} - 458 \beta_{2} ) q^{39} + ( 1728 - 64 \beta_{1} + 64 \beta_{2} ) q^{40} + ( 3478 - 66 \beta_{1} - 228 \beta_{2} ) q^{41} + ( -2752 + 268 \beta_{1} - 352 \beta_{2} ) q^{42} + ( 913 + 549 \beta_{1} + 489 \beta_{2} ) q^{43} + ( 1936 - 272 \beta_{1} + 272 \beta_{2} ) q^{44} + ( 4707 - 181 \beta_{1} + 65 \beta_{2} ) q^{45} + ( -1272 - 628 \beta_{1} + 136 \beta_{2} ) q^{46} + ( 11055 + 71 \beta_{1} - 5 \beta_{2} ) q^{47} + ( 1024 + 256 \beta_{1} ) q^{48} + ( 10520 + 162 \beta_{1} - 453 \beta_{2} ) q^{49} + ( -5136 - 220 \beta_{1} + 100 \beta_{2} ) q^{50} + ( 15720 - 681 \beta_{1} + 456 \beta_{2} ) q^{51} + ( 2848 - 80 \beta_{1} - 448 \beta_{2} ) q^{52} + ( 10106 + 867 \beta_{1} - 156 \beta_{2} ) q^{53} + ( -6648 - 508 \beta_{1} + 104 \beta_{2} ) q^{54} + ( 22171 - 597 \beta_{1} + 87 \beta_{2} ) q^{55} + ( 5056 - 256 \beta_{1} - 320 \beta_{2} ) q^{56} + ( -1444 - 361 \beta_{1} ) q^{57} + ( -11376 + 484 \beta_{1} + 248 \beta_{2} ) q^{58} + ( 5920 + 2405 \beta_{1} + 244 \beta_{2} ) q^{59} + ( -3872 + 672 \beta_{1} + 224 \beta_{2} ) q^{60} + ( 5779 - 623 \beta_{1} + 401 \beta_{2} ) q^{61} + ( -9552 + 112 \beta_{1} - 784 \beta_{2} ) q^{62} + ( 2425 - 889 \beta_{1} - 59 \beta_{2} ) q^{63} + 4096 q^{64} + ( -14780 - 912 \beta_{1} - 96 \beta_{2} ) q^{65} + ( -21864 + 1504 \beta_{1} + 952 \beta_{2} ) q^{66} + ( -610 + 2095 \beta_{1} - 1054 \beta_{2} ) q^{67} + ( -6864 + 960 \beta_{1} + 336 \beta_{2} ) q^{68} + ( -50810 + 1053 \beta_{1} + 230 \beta_{2} ) q^{69} + ( -1108 - 1412 \beta_{1} - 4 \beta_{2} ) q^{70} + ( 7326 - 1996 \beta_{1} - 818 \beta_{2} ) q^{71} + ( 5056 - 192 \beta_{1} + 128 \beta_{2} ) q^{72} + ( -24541 - 1898 \beta_{1} - 739 \beta_{2} ) q^{73} + ( -2040 - 944 \beta_{1} + 464 \beta_{2} ) q^{74} + ( -23066 - 699 \beta_{1} + 290 \beta_{2} ) q^{75} -5776 q^{76} + ( -31411 - 4649 \beta_{1} + 1673 \beta_{2} ) q^{77} + ( 1656 - 44 \beta_{1} - 1832 \beta_{2} ) q^{78} + ( 22212 + 758 \beta_{1} + 964 \beta_{2} ) q^{79} + ( 6912 - 256 \beta_{1} + 256 \beta_{2} ) q^{80} + ( -65851 + 164 \beta_{1} - 324 \beta_{2} ) q^{81} + ( 13912 - 264 \beta_{1} - 912 \beta_{2} ) q^{82} + ( 600 + 4278 \beta_{1} + 684 \beta_{2} ) q^{83} + ( -11008 + 1072 \beta_{1} - 1408 \beta_{2} ) q^{84} + ( -16581 + 3567 \beta_{1} + 339 \beta_{2} ) q^{85} + ( 3652 + 2196 \beta_{1} + 1956 \beta_{2} ) q^{86} + ( 22922 - 3195 \beta_{1} + 1234 \beta_{2} ) q^{87} + ( 7744 - 1088 \beta_{1} + 1088 \beta_{2} ) q^{88} + ( -29512 + 4384 \beta_{1} - 1354 \beta_{2} ) q^{89} + ( 18828 - 724 \beta_{1} + 260 \beta_{2} ) q^{90} + ( 114674 + 3409 \beta_{1} - 2398 \beta_{2} ) q^{91} + ( -5088 - 2512 \beta_{1} + 544 \beta_{2} ) q^{92} + ( 7640 - 4152 \beta_{1} - 3080 \beta_{2} ) q^{93} + ( 44220 + 284 \beta_{1} - 20 \beta_{2} ) q^{94} + ( -9747 + 361 \beta_{1} - 361 \beta_{2} ) q^{95} + ( 4096 + 1024 \beta_{1} ) q^{96} + ( 33168 + 2218 \beta_{1} + 3302 \beta_{2} ) q^{97} + ( 42080 + 648 \beta_{1} - 1812 \beta_{2} ) q^{98} + ( 53317 - 2063 \beta_{1} + 429 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 12q^{2} + 13q^{3} + 48q^{4} + 81q^{5} + 52q^{6} + 228q^{7} + 192q^{8} + 236q^{9} + O(q^{10}) \) \( 3q + 12q^{2} + 13q^{3} + 48q^{4} + 81q^{5} + 52q^{6} + 228q^{7} + 192q^{8} + 236q^{9} + 324q^{10} + 363q^{11} + 208q^{12} + 501q^{13} + 912q^{14} - 670q^{15} + 768q^{16} - 1206q^{17} + 944q^{18} - 1083q^{19} + 1296q^{20} - 2085q^{21} + 1452q^{22} - 1077q^{23} + 832q^{24} - 3882q^{25} + 2004q^{26} - 5087q^{27} + 3648q^{28} - 8349q^{29} - 2680q^{30} - 7332q^{31} + 3072q^{32} - 15784q^{33} - 4824q^{34} - 1185q^{35} + 3776q^{36} - 1650q^{37} - 4332q^{38} + 773q^{39} + 5184q^{40} + 10140q^{41} - 8340q^{42} + 3777q^{43} + 5808q^{44} + 14005q^{45} - 4308q^{46} + 33231q^{47} + 3328q^{48} + 31269q^{49} - 15528q^{50} + 46935q^{51} + 8016q^{52} + 31029q^{53} - 20348q^{54} + 66003q^{55} + 14592q^{56} - 4693q^{57} - 33396q^{58} + 20409q^{59} - 10720q^{60} + 17115q^{61} - 29328q^{62} + 6327q^{63} + 12288q^{64} - 45348q^{65} - 63136q^{66} - 789q^{67} - 19296q^{68} - 151147q^{69} - 4740q^{70} + 19164q^{71} + 15104q^{72} - 76260q^{73} - 6600q^{74} - 69607q^{75} - 17328q^{76} - 97209q^{77} + 3092q^{78} + 68358q^{79} + 20736q^{80} - 197713q^{81} + 40560q^{82} + 6762q^{83} - 33360q^{84} - 45837q^{85} + 15108q^{86} + 66805q^{87} + 23232q^{88} - 85506q^{89} + 56020q^{90} + 345033q^{91} - 17232q^{92} + 15688q^{93} + 132924q^{94} - 29241q^{95} + 13312q^{96} + 105024q^{97} + 125076q^{98} + 158317q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 454 x + 3760\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 11 \nu - 306 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} - 11 \beta_{1} + 306\)

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
−24.1916
10.7990
14.3926
4.00000 −20.1916 16.0000 57.7548 −80.7665 142.950 64.0000 164.701 231.019
1.2 4.00000 14.7990 16.0000 −19.0956 59.1959 212.287 64.0000 −23.9901 −76.3823
1.3 4.00000 18.3926 16.0000 42.3408 73.5705 −127.237 64.0000 95.2889 169.363
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 38.6.a.d 3
3.b odd 2 1 342.6.a.l 3
4.b odd 2 1 304.6.a.h 3
5.b even 2 1 950.6.a.f 3
19.b odd 2 1 722.6.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.d 3 1.a even 1 1 trivial
304.6.a.h 3 4.b odd 2 1
342.6.a.l 3 3.b odd 2 1
722.6.a.d 3 19.b odd 2 1
950.6.a.f 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 13 T_{3}^{2} - 398 T_{3} + 5496 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T )^{3} \)
$3$ \( 1 - 13 T + 331 T^{2} - 822 T^{3} + 80433 T^{4} - 767637 T^{5} + 14348907 T^{6} \)
$5$ \( 1 - 81 T + 9909 T^{2} - 459554 T^{3} + 30965625 T^{4} - 791015625 T^{5} + 30517578125 T^{6} \)
$7$ \( 1 - 228 T + 35568 T^{2} - 3802776 T^{3} + 597791376 T^{4} - 64404356772 T^{5} + 4747561509943 T^{6} \)
$11$ \( 1 - 363 T + 49359 T^{2} + 45957094 T^{3} + 7949316309 T^{4} - 9415285130163 T^{5} + 4177248169415651 T^{6} \)
$13$ \( 1 - 501 T + 350229 T^{2} - 278952890 T^{3} + 130037576097 T^{4} - 69067104416349 T^{5} + 51185893014090757 T^{6} \)
$17$ \( 1 + 1206 T + 2771250 T^{2} + 2460853566 T^{3} + 3934778711250 T^{4} + 2431288643941494 T^{5} + 2862423051509815793 T^{6} \)
$19$ \( ( 1 + 361 T )^{3} \)
$23$ \( 1 + 1077 T + 6641373 T^{2} - 4780608698 T^{3} + 42746154618939 T^{4} + 44616352577099973 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \)
$29$ \( 1 + 8349 T + 74783679 T^{2} + 327686176502 T^{3} + 1533899182737171 T^{4} + 3512484690823378149 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + 7332 T + 61101597 T^{2} + 255516763064 T^{3} + 1749286846854147 T^{4} + 6009514600143232932 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + 1650 T + 165941883 T^{2} + 209325534188 T^{3} + 11507066799251031 T^{4} + 7934164214489450850 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \)
$41$ \( 1 - 10140 T + 325081167 T^{2} - 2185083057016 T^{3} + 37662669025266567 T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 - 3777 T + 79236057 T^{2} - 3338955229894 T^{3} + 11648369369029251 T^{4} - 81626568697274608473 T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - 33231 T + 1053777477 T^{2} - 16574372980994 T^{3} + 241678602839007339 T^{4} - \)\(17\!\cdots\!19\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$53$ \( 1 - 31029 T + 1192371741 T^{2} - 20622372968282 T^{3} + 498644488066763313 T^{4} - \)\(54\!\cdots\!21\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \)
$59$ \( 1 - 20409 T - 343016529 T^{2} + 27176512433970 T^{3} - 245230851540738171 T^{4} - \)\(10\!\cdots\!09\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - 17115 T + 2252343099 T^{2} - 25812537224162 T^{3} + 1902320649998276799 T^{4} - \)\(12\!\cdots\!15\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 + 789 T + 601630785 T^{2} - 4061691437586 T^{3} + 812276827972618995 T^{4} + \)\(14\!\cdots\!61\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \)
$71$ \( 1 - 19164 T + 3181554489 T^{2} - 35510810109608 T^{3} + 5740253990859606639 T^{4} - \)\(62\!\cdots\!64\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 + 76260 T + 6085343520 T^{2} + 293061463314558 T^{3} + 12615352784958627360 T^{4} + \)\(32\!\cdots\!40\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \)
$79$ \( 1 - 68358 T + 9600289101 T^{2} - 418395905475764 T^{3} + 29540631010482007299 T^{4} - \)\(64\!\cdots\!58\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - 6762 T + 3337936953 T^{2} + 130668532187172 T^{3} + 13148269321638580779 T^{4} - \)\(10\!\cdots\!38\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \)
$89$ \( 1 + 85506 T + 7798835091 T^{2} + 891678014164068 T^{3} + 43549158781091324859 T^{4} + \)\(26\!\cdots\!06\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - 105024 T + 16181794191 T^{2} - 1792277878659808 T^{3} + \)\(13\!\cdots\!87\)\( T^{4} - \)\(77\!\cdots\!76\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \)
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