Properties

Label 38.8.a.b
Level 38
Weight 8
Character orbit 38.a
Self dual yes
Analytic conductor 11.871
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2737}) \)
Defining polynomial: \(x^{2} - x - 684\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{2737})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} + ( -30 - \beta ) q^{3} + 64 q^{4} + ( 95 - 15 \beta ) q^{5} + ( 240 + 8 \beta ) q^{6} + ( -1293 - 6 \beta ) q^{7} -512 q^{8} + ( -603 + 61 \beta ) q^{9} +O(q^{10})\) \( q -8 q^{2} + ( -30 - \beta ) q^{3} + 64 q^{4} + ( 95 - 15 \beta ) q^{5} + ( 240 + 8 \beta ) q^{6} + ( -1293 - 6 \beta ) q^{7} -512 q^{8} + ( -603 + 61 \beta ) q^{9} + ( -760 + 120 \beta ) q^{10} + ( 659 - 273 \beta ) q^{11} + ( -1920 - 64 \beta ) q^{12} + ( 7280 + 87 \beta ) q^{13} + ( 10344 + 48 \beta ) q^{14} + ( 7410 + 370 \beta ) q^{15} + 4096 q^{16} + ( 14853 - 90 \beta ) q^{17} + ( 4824 - 488 \beta ) q^{18} -6859 q^{19} + ( 6080 - 960 \beta ) q^{20} + ( 42894 + 1479 \beta ) q^{21} + ( -5272 + 2184 \beta ) q^{22} + ( 35684 - 1383 \beta ) q^{23} + ( 15360 + 512 \beta ) q^{24} + ( 84800 - 2625 \beta ) q^{25} + ( -58240 - 696 \beta ) q^{26} + ( 41976 + 899 \beta ) q^{27} + ( -82752 - 384 \beta ) q^{28} + ( 72346 - 5871 \beta ) q^{29} + ( -59280 - 2960 \beta ) q^{30} + ( -102140 + 4884 \beta ) q^{31} -32768 q^{32} + ( 166962 + 7804 \beta ) q^{33} + ( -118824 + 720 \beta ) q^{34} + ( -61275 + 18915 \beta ) q^{35} + ( -38592 + 3904 \beta ) q^{36} + ( -26678 - 14484 \beta ) q^{37} + 54872 q^{38} + ( -277908 - 9977 \beta ) q^{39} + ( -48640 + 7680 \beta ) q^{40} + ( -264518 - 10314 \beta ) q^{41} + ( -343152 - 11832 \beta ) q^{42} + ( 300991 + 657 \beta ) q^{43} + ( 42176 - 17472 \beta ) q^{44} + ( -683145 + 13925 \beta ) q^{45} + ( -285472 + 11064 \beta ) q^{46} + ( -502075 - 27825 \beta ) q^{47} + ( -122880 - 4096 \beta ) q^{48} + ( 872930 + 15552 \beta ) q^{49} + ( -678400 + 21000 \beta ) q^{50} + ( -384030 - 12063 \beta ) q^{51} + ( 465920 + 5568 \beta ) q^{52} + ( 1050056 + 38151 \beta ) q^{53} + ( -335808 - 7192 \beta ) q^{54} + ( 2863585 - 31725 \beta ) q^{55} + ( 662016 + 3072 \beta ) q^{56} + ( 205770 + 6859 \beta ) q^{57} + ( -578768 + 46968 \beta ) q^{58} + ( -1966314 - 3741 \beta ) q^{59} + ( 474240 + 23680 \beta ) q^{60} + ( -553597 + 79539 \beta ) q^{61} + ( 817120 - 39072 \beta ) q^{62} + ( 529335 - 75621 \beta ) q^{63} + 262144 q^{64} + ( -201020 - 102240 \beta ) q^{65} + ( -1335696 - 62432 \beta ) q^{66} + ( 351408 + 62133 \beta ) q^{67} + ( 950592 - 5760 \beta ) q^{68} + ( -124548 + 7189 \beta ) q^{69} + ( 490200 - 151320 \beta ) q^{70} + ( -1852738 + 133392 \beta ) q^{71} + ( 308736 - 31232 \beta ) q^{72} + ( 4535793 - 2064 \beta ) q^{73} + ( 213424 + 115872 \beta ) q^{74} + ( -748500 - 3425 \beta ) q^{75} -438976 q^{76} + ( 268305 + 350673 \beta ) q^{77} + ( 2223264 + 79816 \beta ) q^{78} + ( -1237816 - 277782 \beta ) q^{79} + ( 389120 - 61440 \beta ) q^{80} + ( -555435 - 203252 \beta ) q^{81} + ( 2116144 + 82512 \beta ) q^{82} + ( -3811596 - 19854 \beta ) q^{83} + ( 2745216 + 94656 \beta ) q^{84} + ( 2334435 - 229995 \beta ) q^{85} + ( -2407928 - 5256 \beta ) q^{86} + ( 1845384 + 109655 \beta ) q^{87} + ( -337408 + 139776 \beta ) q^{88} + ( 677700 + 38220 \beta ) q^{89} + ( 5465160 - 111400 \beta ) q^{90} + ( -9770088 - 156693 \beta ) q^{91} + ( 2283776 - 88512 \beta ) q^{92} + ( -276456 - 49264 \beta ) q^{93} + ( 4016600 + 222600 \beta ) q^{94} + ( -651605 + 102885 \beta ) q^{95} + ( 983040 + 32768 \beta ) q^{96} + ( -3609776 + 298086 \beta ) q^{97} + ( -6983440 - 124416 \beta ) q^{98} + ( -11788029 + 188165 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{2} - 61q^{3} + 128q^{4} + 175q^{5} + 488q^{6} - 2592q^{7} - 1024q^{8} - 1145q^{9} + O(q^{10}) \) \( 2q - 16q^{2} - 61q^{3} + 128q^{4} + 175q^{5} + 488q^{6} - 2592q^{7} - 1024q^{8} - 1145q^{9} - 1400q^{10} + 1045q^{11} - 3904q^{12} + 14647q^{13} + 20736q^{14} + 15190q^{15} + 8192q^{16} + 29616q^{17} + 9160q^{18} - 13718q^{19} + 11200q^{20} + 87267q^{21} - 8360q^{22} + 69985q^{23} + 31232q^{24} + 166975q^{25} - 117176q^{26} + 84851q^{27} - 165888q^{28} + 138821q^{29} - 121520q^{30} - 199396q^{31} - 65536q^{32} + 341728q^{33} - 236928q^{34} - 103635q^{35} - 73280q^{36} - 67840q^{37} + 109744q^{38} - 565793q^{39} - 89600q^{40} - 539350q^{41} - 698136q^{42} + 602639q^{43} + 66880q^{44} - 1352365q^{45} - 559880q^{46} - 1031975q^{47} - 249856q^{48} + 1761412q^{49} - 1335800q^{50} - 780123q^{51} + 937408q^{52} + 2138263q^{53} - 678808q^{54} + 5695445q^{55} + 1327104q^{56} + 418399q^{57} - 1110568q^{58} - 3936369q^{59} + 972160q^{60} - 1027655q^{61} + 1595168q^{62} + 983049q^{63} + 524288q^{64} - 504280q^{65} - 2733824q^{66} + 764949q^{67} + 1895424q^{68} - 241907q^{69} + 829080q^{70} - 3572084q^{71} + 586240q^{72} + 9069522q^{73} + 542720q^{74} - 1500425q^{75} - 877952q^{76} + 887283q^{77} + 4526344q^{78} - 2753414q^{79} + 716800q^{80} - 1314122q^{81} + 4314800q^{82} - 7643046q^{83} + 5585088q^{84} + 4438875q^{85} - 4821112q^{86} + 3800423q^{87} - 535040q^{88} + 1393620q^{89} + 10818920q^{90} - 19696869q^{91} + 4479040q^{92} - 602176q^{93} + 8255800q^{94} - 1200325q^{95} + 1998848q^{96} - 6921466q^{97} - 14091296q^{98} - 23387893q^{99} + O(q^{100}) \)

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
26.6582
−25.6582
−8.00000 −56.6582 64.0000 −304.873 453.265 −1452.95 −512.000 1023.15 2438.98
1.2 −8.00000 −4.34183 64.0000 479.873 34.7346 −1139.05 −512.000 −2168.15 −3838.98
\(n\): 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.b 2
3.b odd 2 1 342.8.a.h 2
4.b odd 2 1 304.8.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.b 2 1.a even 1 1 trivial
304.8.a.c 2 4.b odd 2 1
342.8.a.h 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 61 T_{3} + 246 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T )^{2} \)
$3$ \( 1 + 61 T + 4620 T^{2} + 133407 T^{3} + 4782969 T^{4} \)
$5$ \( 1 - 175 T + 9950 T^{2} - 13671875 T^{3} + 6103515625 T^{4} \)
$7$ \( 1 + 2592 T + 3302069 T^{2} + 2134623456 T^{3} + 678223072849 T^{4} \)
$11$ \( 1 - 1045 T - 11749120 T^{2} - 20364093695 T^{3} + 379749833583241 T^{4} \)
$13$ \( 1 - 14647 T + 173951598 T^{2} - 919077528499 T^{3} + 3937376385699289 T^{4} \)
$17$ \( 1 - 29616 T + 1034411785 T^{2} - 12152590139568 T^{3} + 168377826559400929 T^{4} \)
$19$ \( ( 1 + 6859 T )^{2} \)
$23$ \( 1 - 69985 T + 6725368502 T^{2} - 238286708908295 T^{3} + 11592836324538749809 T^{4} \)
$29$ \( 1 - 138821 T + 15732402524 T^{2} - 2394645079091689 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + 199396 T + 48643192158 T^{2} + 5485905203276956 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 67840 T + 47468074998 T^{2} + 6440178544702720 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + 539350 T + 389443599074 T^{2} + 105040717617717350 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 602639 T + 634135307466 T^{2} - 163808495978911373 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 1031975 T + 749722035926 T^{2} + 522822394739804425 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - 2138263 T + 2496539389382 T^{2} - 2511841366001283131 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 + 3936369 T + 8841477061504 T^{2} + 9796250556645482211 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 1027655 T + 2220629234304 T^{2} + 3229655389151160755 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - 764949 T + 9626156199098 T^{2} - 4636135281780223527 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + 3572084 T + 9205034831954 T^{2} + 32488533195865956844 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - 9069522 T + 42655939394627 T^{2} - \)\(10\!\cdots\!34\)\( T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + 2753414 T - 12495532808130 T^{2} + 52876311857215996826 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + 7643046 T + 68606421453310 T^{2} + \)\(20\!\cdots\!42\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 - 1393620 T + 87948683189458 T^{2} - 61641672937107124980 T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + 6921466 T + 112774027874802 T^{2} + \)\(55\!\cdots\!58\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \)
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