Properties

Label 38.8.a.c
Level 38
Weight 8
Character orbit 38.a
Self dual yes
Analytic conductor 11.871
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8706309684\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17953}) \)
Defining polynomial: \(x^{2} - x - 4488\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17953})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} + ( -5 - \beta ) q^{3} + 64 q^{4} + ( -36 + 3 \beta ) q^{5} + ( 40 + 8 \beta ) q^{6} + ( -181 + 14 \beta ) q^{7} -512 q^{8} + ( 2326 + 11 \beta ) q^{9} +O(q^{10})\) \( q -8 q^{2} + ( -5 - \beta ) q^{3} + 64 q^{4} + ( -36 + 3 \beta ) q^{5} + ( 40 + 8 \beta ) q^{6} + ( -181 + 14 \beta ) q^{7} -512 q^{8} + ( 2326 + 11 \beta ) q^{9} + ( 288 - 24 \beta ) q^{10} + ( 1362 - \beta ) q^{11} + ( -320 - 64 \beta ) q^{12} + ( -7045 - 23 \beta ) q^{13} + ( 1448 - 112 \beta ) q^{14} + ( -13284 + 18 \beta ) q^{15} + 4096 q^{16} + ( -19341 + 122 \beta ) q^{17} + ( -18608 - 88 \beta ) q^{18} + 6859 q^{19} + ( -2304 + 192 \beta ) q^{20} + ( -61927 + 97 \beta ) q^{21} + ( -10896 + 8 \beta ) q^{22} + ( -37137 + 377 \beta ) q^{23} + ( 2560 + 512 \beta ) q^{24} + ( -36437 - 207 \beta ) q^{25} + ( 56360 + 184 \beta ) q^{26} + ( -50063 - 205 \beta ) q^{27} + ( -11584 + 896 \beta ) q^{28} + ( 80871 - 1929 \beta ) q^{29} + ( 106272 - 144 \beta ) q^{30} + ( -129736 + 4 \beta ) q^{31} -32768 q^{32} + ( -2322 - 1356 \beta ) q^{33} + ( 154728 - 976 \beta ) q^{34} + ( 195012 - 1005 \beta ) q^{35} + ( 148864 + 704 \beta ) q^{36} + ( -262378 - 3412 \beta ) q^{37} -54872 q^{38} + ( 138449 + 7183 \beta ) q^{39} + ( 18432 - 1536 \beta ) q^{40} + ( 503280 - 910 \beta ) q^{41} + ( 495416 - 776 \beta ) q^{42} + ( 139412 + 7393 \beta ) q^{43} + ( 87168 - 64 \beta ) q^{44} + ( 64368 + 6615 \beta ) q^{45} + ( 297096 - 3016 \beta ) q^{46} + ( -699120 + 731 \beta ) q^{47} + ( -20480 - 4096 \beta ) q^{48} + ( 88866 - 4872 \beta ) q^{49} + ( 291496 + 1656 \beta ) q^{50} + ( -450831 + 18609 \beta ) q^{51} + ( -450880 - 1472 \beta ) q^{52} + ( -682701 - 20567 \beta ) q^{53} + ( 400504 + 1640 \beta ) q^{54} + ( -62496 + 4119 \beta ) q^{55} + ( 92672 - 7168 \beta ) q^{56} + ( -34295 - 6859 \beta ) q^{57} + ( -646968 + 15432 \beta ) q^{58} + ( 1361619 - 22285 \beta ) q^{59} + ( -850176 + 1152 \beta ) q^{60} + ( -1977358 - 21231 \beta ) q^{61} + ( 1037888 - 32 \beta ) q^{62} + ( 270146 + 30727 \beta ) q^{63} + 262144 q^{64} + ( -56052 - 20376 \beta ) q^{65} + ( 18576 + 10848 \beta ) q^{66} + ( -40657 - 53243 \beta ) q^{67} + ( -1237824 + 7808 \beta ) q^{68} + ( -1506291 + 34875 \beta ) q^{69} + ( -1560096 + 8040 \beta ) q^{70} + ( 2087898 + 26944 \beta ) q^{71} + ( -1190912 - 5632 \beta ) q^{72} + ( 434897 + 30704 \beta ) q^{73} + ( 2099024 + 27296 \beta ) q^{74} + ( 1111201 + 37679 \beta ) q^{75} + 438976 q^{76} + ( -309354 + 19235 \beta ) q^{77} + ( -1107592 - 57464 \beta ) q^{78} + ( -3440194 - 13342 \beta ) q^{79} + ( -147456 + 12288 \beta ) q^{80} + ( -3916607 + 27236 \beta ) q^{81} + ( -4026240 + 7280 \beta ) q^{82} + ( -1220838 - 23654 \beta ) q^{83} + ( -3963328 + 6208 \beta ) q^{84} + ( 2338884 - 62049 \beta ) q^{85} + ( -1115296 - 59144 \beta ) q^{86} + ( 8252997 - 69297 \beta ) q^{87} + ( -697344 + 512 \beta ) q^{88} + ( 8690568 + 50588 \beta ) q^{89} + ( -514944 - 52920 \beta ) q^{90} + ( -169991 - 94789 \beta ) q^{91} + ( -2376768 + 24128 \beta ) q^{92} + ( 630728 + 129712 \beta ) q^{93} + ( 5592960 - 5848 \beta ) q^{94} + ( -246924 + 20577 \beta ) q^{95} + ( 163840 + 32768 \beta ) q^{96} + ( 3253238 - 154542 \beta ) q^{97} + ( -710928 + 38976 \beta ) q^{98} + ( 3118644 + 12645 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{2} - 11q^{3} + 128q^{4} - 69q^{5} + 88q^{6} - 348q^{7} - 1024q^{8} + 4663q^{9} + O(q^{10}) \) \( 2q - 16q^{2} - 11q^{3} + 128q^{4} - 69q^{5} + 88q^{6} - 348q^{7} - 1024q^{8} + 4663q^{9} + 552q^{10} + 2723q^{11} - 704q^{12} - 14113q^{13} + 2784q^{14} - 26550q^{15} + 8192q^{16} - 38560q^{17} - 37304q^{18} + 13718q^{19} - 4416q^{20} - 123757q^{21} - 21784q^{22} - 73897q^{23} + 5632q^{24} - 73081q^{25} + 112904q^{26} - 100331q^{27} - 22272q^{28} + 159813q^{29} + 212400q^{30} - 259468q^{31} - 65536q^{32} - 6000q^{33} + 308480q^{34} + 389019q^{35} + 298432q^{36} - 528168q^{37} - 109744q^{38} + 284081q^{39} + 35328q^{40} + 1005650q^{41} + 990056q^{42} + 286217q^{43} + 174272q^{44} + 135351q^{45} + 591176q^{46} - 1397509q^{47} - 45056q^{48} + 172860q^{49} + 584648q^{50} - 883053q^{51} - 903232q^{52} - 1385969q^{53} + 802648q^{54} - 120873q^{55} + 178176q^{56} - 75449q^{57} - 1278504q^{58} + 2700953q^{59} - 1699200q^{60} - 3975947q^{61} + 2075744q^{62} + 571019q^{63} + 524288q^{64} - 132480q^{65} + 48000q^{66} - 134557q^{67} - 2467840q^{68} - 2977707q^{69} - 3112152q^{70} + 4202740q^{71} - 2387456q^{72} + 900498q^{73} + 4225344q^{74} + 2260081q^{75} + 877952q^{76} - 599473q^{77} - 2272648q^{78} - 6893730q^{79} - 282624q^{80} - 7805978q^{81} - 8045200q^{82} - 2465330q^{83} - 7920448q^{84} + 4615719q^{85} - 2289736q^{86} + 16436697q^{87} - 1394176q^{88} + 17431724q^{89} - 1082808q^{90} - 434771q^{91} - 4729408q^{92} + 1391168q^{93} + 11180072q^{94} - 473271q^{95} + 360448q^{96} + 6351934q^{97} - 1382880q^{98} + 6249933q^{99} + O(q^{100}) \)

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
67.4944
−66.4944
−8.00000 −72.4944 64.0000 166.483 579.955 763.922 −512.000 3068.44 −1331.87
1.2 −8.00000 61.4944 64.0000 −235.483 −491.955 −1111.92 −512.000 1594.56 1883.87
\(n\): 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 38.8.a.c 2
3.b odd 2 1 342.8.a.i 2
4.b odd 2 1 304.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.c 2 1.a even 1 1 trivial
304.8.a.b 2 4.b odd 2 1
342.8.a.i 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 11 T_{3} - 4458 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T )^{2} \)
$3$ \( 1 + 11 T - 84 T^{2} + 24057 T^{3} + 4782969 T^{4} \)
$5$ \( 1 + 69 T + 117046 T^{2} + 5390625 T^{3} + 6103515625 T^{4} \)
$7$ \( 1 + 348 T + 797665 T^{2} + 286592964 T^{3} + 678223072849 T^{4} \)
$11$ \( 1 - 2723 T + 40823536 T^{2} - 53063566633 T^{3} + 379749833583241 T^{4} \)
$13$ \( 1 + 14113 T + 172916942 T^{2} + 885569820421 T^{3} + 3937376385699289 T^{4} \)
$17$ \( 1 + 38560 T + 1125592633 T^{2} + 15822659230880 T^{3} + 168377826559400929 T^{4} \)
$19$ \( ( 1 - 6859 T )^{2} \)
$23$ \( 1 + 73897 T + 7536932062 T^{2} + 251606386056959 T^{3} + 11592836324538749809 T^{4} \)
$29$ \( 1 - 159813 T + 24183839092 T^{2} - 2756754482570217 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + 259468 T + 71856067166 T^{2} + 7138642958152948 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 528168 T + 207353055814 T^{2} + 50139979681582344 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - 1005650 T + 638624808562 T^{2} - 195854635528427650 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 286217 T + 318805457762 T^{2} - 77799107415212219 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 1397509 T + 1499105746438 T^{2} + 708010370455126667 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 1385969 T + 931113363910 T^{2} + 1628113223768747053 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 2700953 T + 4572129533584 T^{2} - 6721730693876332507 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 3975947 T + 8214421559736 T^{2} + 12495378950649186887 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 134557 T - 597418012966 T^{2} + 815511171477446911 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - 4202740 T + 19347620336530 T^{2} - 38224425294476191340 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - 900498 T + 18066288071683 T^{2} - 9948160271649810306 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + 6893730 T + 49489747567870 T^{2} + \)\(13\!\cdots\!70\)\( T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + 2465330 T + 53280336522142 T^{2} + 66899320586257131910 T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 - 17431724 T + 152942834308594 T^{2} - \)\(77\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - 6351934 T + 64489429353042 T^{2} - \)\(51\!\cdots\!42\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \)
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