Properties

Label 38.5.b.a
Level $38$
Weight $5$
Character orbit 38.b
Analytic conductor $3.928$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 38.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92805859719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 450 x^{6} + 68229 x^{4} + 4001228 x^{2} + 77475204\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} -8 q^{4} + ( 2 - \beta_{6} ) q^{5} + ( -4 - \beta_{4} ) q^{6} + ( -20 + \beta_{5} ) q^{7} -8 \beta_{2} q^{8} + ( -33 - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} -8 q^{4} + ( 2 - \beta_{6} ) q^{5} + ( -4 - \beta_{4} ) q^{6} + ( -20 + \beta_{5} ) q^{7} -8 \beta_{2} q^{8} + ( -33 - \beta_{4} + \beta_{5} + \beta_{6} ) q^{9} + ( 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{10} + ( -2 - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{11} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{12} + ( 7 \beta_{1} + 6 \beta_{2} + \beta_{3} + \beta_{7} ) q^{13} + ( -\beta_{1} - 21 \beta_{2} + \beta_{7} ) q^{14} + ( 12 \beta_{1} + 50 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{15} + 64 q^{16} + ( 64 + 7 \beta_{4} + \beta_{5} ) q^{17} + ( -12 \beta_{1} - 40 \beta_{2} + \beta_{3} + \beta_{7} ) q^{18} + ( -1 + 14 \beta_{1} + 57 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{19} + ( -16 + 8 \beta_{6} ) q^{20} + ( -47 \beta_{1} - 46 \beta_{2} + 5 \beta_{3} - \beta_{7} ) q^{21} + ( -5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{7} ) q^{22} + ( -48 + 14 \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{23} + ( 32 + 8 \beta_{4} ) q^{24} + ( 431 + 12 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{25} + ( -28 - 11 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} ) q^{26} + ( 3 \beta_{1} - 143 \beta_{2} + 8 \beta_{3} + 2 \beta_{7} ) q^{27} + ( 160 - 8 \beta_{5} ) q^{28} + ( 15 \beta_{1} + 24 \beta_{2} - 9 \beta_{3} + 3 \beta_{7} ) q^{29} + ( -336 - 4 \beta_{4} + 16 \beta_{5} + 16 \beta_{6} ) q^{30} + ( -30 \beta_{1} + 222 \beta_{2} ) q^{31} + 64 \beta_{2} q^{32} + ( 66 \beta_{1} - 40 \beta_{2} - 14 \beta_{3} - 2 \beta_{7} ) q^{33} + ( 55 \beta_{1} + 91 \beta_{2} + \beta_{7} ) q^{34} + ( 128 - 58 \beta_{4} + 2 \beta_{5} + 9 \beta_{6} ) q^{35} + ( 264 + 8 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} ) q^{36} + ( -76 \beta_{1} - 246 \beta_{2} + 8 \beta_{3} + 2 \beta_{7} ) q^{37} + ( -404 - 32 \beta_{1} - 18 \beta_{2} + 3 \beta_{3} - 19 \beta_{4} + 8 \beta_{5} - 16 \beta_{6} - \beta_{7} ) q^{38} + ( -834 + 6 \beta_{4} + 3 \beta_{5} - 45 \beta_{6} ) q^{39} + ( -24 \beta_{1} - 32 \beta_{2} + 8 \beta_{3} ) q^{40} + ( 42 \beta_{1} - 234 \beta_{2} - 12 \beta_{3} ) q^{41} + ( 164 + 33 \beta_{4} + 8 \beta_{5} - 40 \beta_{6} ) q^{42} + ( -1078 + 48 \beta_{4} + 15 \beta_{6} ) q^{43} + ( 16 + 16 \beta_{4} + 16 \beta_{5} + 24 \beta_{6} ) q^{44} + ( -1278 - 74 \beta_{4} + 20 \beta_{5} + 35 \beta_{6} ) q^{45} + ( 92 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} - \beta_{7} ) q^{46} + ( 398 - 24 \beta_{4} + 18 \beta_{5} - 31 \beta_{6} ) q^{47} + ( 64 \beta_{1} + 64 \beta_{2} ) q^{48} + ( 1137 - 57 \beta_{4} - 39 \beta_{5} - 24 \beta_{6} ) q^{49} + ( 107 \beta_{1} + 487 \beta_{2} - 3 \beta_{3} - 2 \beta_{7} ) q^{50} + ( 93 \beta_{1} + 857 \beta_{2} - 2 \beta_{3} - 8 \beta_{7} ) q^{51} + ( -56 \beta_{1} - 48 \beta_{2} - 8 \beta_{3} - 8 \beta_{7} ) q^{52} + ( -183 \beta_{1} - 114 \beta_{2} + 3 \beta_{3} + 15 \beta_{7} ) q^{53} + ( 1116 - 29 \beta_{4} - 16 \beta_{5} - 64 \beta_{6} ) q^{54} + ( 2160 + 144 \beta_{4} + 22 \beta_{5} + 45 \beta_{6} ) q^{55} + ( 8 \beta_{1} + 168 \beta_{2} - 8 \beta_{7} ) q^{56} + ( -1758 - 28 \beta_{1} - 470 \beta_{2} + 4 \beta_{3} - 69 \beta_{4} - 21 \beta_{5} + 27 \beta_{6} + 10 \beta_{7} ) q^{57} + ( -108 + 9 \beta_{4} - 24 \beta_{5} + 72 \beta_{6} ) q^{58} + ( 159 \beta_{1} - 459 \beta_{2} - 6 \beta_{3} + 6 \beta_{7} ) q^{59} + ( -96 \beta_{1} - 400 \beta_{2} + 16 \beta_{3} + 16 \beta_{7} ) q^{60} + ( 178 + 150 \beta_{4} - 20 \beta_{5} + 75 \beta_{6} ) q^{61} + ( -1896 + 30 \beta_{4} ) q^{62} + ( 3714 + 34 \beta_{4} - 40 \beta_{5} - 73 \beta_{6} ) q^{63} -512 q^{64} + ( -408 \beta_{1} - 606 \beta_{2} - 30 \beta_{3} - 18 \beta_{7} ) q^{65} + ( 648 - 22 \beta_{4} + 16 \beta_{5} + 112 \beta_{6} ) q^{66} + ( -275 \beta_{1} - 1017 \beta_{2} + 22 \beta_{3} - 26 \beta_{7} ) q^{67} + ( -512 - 56 \beta_{4} - 8 \beta_{5} ) q^{68} + ( 21 \beta_{1} + 1280 \beta_{2} - 5 \beta_{3} + \beta_{7} ) q^{69} + ( -493 \beta_{1} - 124 \beta_{2} + 9 \beta_{3} + 2 \beta_{7} ) q^{70} + ( 66 \beta_{1} - 912 \beta_{2} + 66 \beta_{3} + 18 \beta_{7} ) q^{71} + ( 96 \beta_{1} + 320 \beta_{2} - 8 \beta_{3} - 8 \beta_{7} ) q^{72} + ( 2972 - 69 \beta_{4} + 39 \beta_{5} + 150 \beta_{6} ) q^{73} + ( 1624 + 50 \beta_{4} - 16 \beta_{5} - 64 \beta_{6} ) q^{74} + ( 611 \beta_{1} + 2031 \beta_{2} - 28 \beta_{3} - 16 \beta_{7} ) q^{75} + ( 8 - 112 \beta_{1} - 456 \beta_{2} - 16 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} - 24 \beta_{6} + 8 \beta_{7} ) q^{76} + ( -5564 + 6 \beta_{4} + 60 \beta_{5} - 5 \beta_{6} ) q^{77} + ( 180 \beta_{1} - 723 \beta_{2} - 45 \beta_{3} + 3 \beta_{7} ) q^{78} + ( 218 \beta_{1} + 852 \beta_{2} + 50 \beta_{3} + 2 \beta_{7} ) q^{79} + ( 128 - 64 \beta_{6} ) q^{80} + ( -2571 + 86 \beta_{4} - 2 \beta_{5} - 98 \beta_{6} ) q^{81} + ( 2088 - 6 \beta_{4} + 96 \beta_{6} ) q^{82} + ( -1362 - 64 \beta_{4} - 70 \beta_{5} - 158 \beta_{6} ) q^{83} + ( 376 \beta_{1} + 368 \beta_{2} - 40 \beta_{3} + 8 \beta_{7} ) q^{84} + ( 2592 - 30 \beta_{4} - 110 \beta_{5} - 159 \beta_{6} ) q^{85} + ( 339 \beta_{1} - 916 \beta_{2} + 15 \beta_{3} ) q^{86} + ( -1746 + 12 \beta_{4} + 159 \beta_{5} + 15 \beta_{6} ) q^{87} + ( 40 \beta_{1} + 16 \beta_{2} + 24 \beta_{3} + 16 \beta_{7} ) q^{88} + ( 480 \beta_{1} + 2346 \beta_{2} - 6 \beta_{3} + 18 \beta_{7} ) q^{89} + ( -717 \beta_{1} - 1664 \beta_{2} + 35 \beta_{3} + 20 \beta_{7} ) q^{90} + ( -389 \beta_{1} + 2595 \beta_{2} + 22 \beta_{3} - 32 \beta_{7} ) q^{91} + ( 384 - 112 \beta_{4} + 8 \beta_{5} - 56 \beta_{6} ) q^{92} + ( 2412 - 222 \beta_{4} - 30 \beta_{5} - 30 \beta_{6} ) q^{93} + ( -117 \beta_{1} + 346 \beta_{2} - 31 \beta_{3} + 18 \beta_{7} ) q^{94} + ( -4310 + 582 \beta_{1} - 1050 \beta_{2} - 66 \beta_{3} + 10 \beta_{4} + 52 \beta_{5} + 51 \beta_{6} - 36 \beta_{7} ) q^{95} + ( -256 - 64 \beta_{4} ) q^{96} + ( 194 \beta_{1} - 2946 \beta_{2} - 64 \beta_{3} - 4 \beta_{7} ) q^{97} + ( -345 \beta_{1} + 996 \beta_{2} - 24 \beta_{3} - 39 \beta_{7} ) q^{98} + ( -7062 - 146 \beta_{4} + 68 \beta_{5} + 83 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 64q^{4} + 18q^{5} - 32q^{6} - 162q^{7} - 268q^{9} + O(q^{10}) \) \( 8q - 64q^{4} + 18q^{5} - 32q^{6} - 162q^{7} - 268q^{9} - 6q^{11} + 512q^{16} + 510q^{17} - 12q^{19} - 144q^{20} - 396q^{23} + 256q^{24} + 3458q^{25} - 192q^{26} + 1296q^{28} - 2752q^{30} + 1002q^{35} + 2144q^{36} - 3216q^{38} - 6588q^{39} + 1376q^{42} - 8654q^{43} + 48q^{44} - 10334q^{45} + 3210q^{47} + 9222q^{49} + 9088q^{54} + 17146q^{55} - 14076q^{57} - 960q^{58} + 1314q^{61} - 15168q^{62} + 29938q^{63} - 4096q^{64} + 4928q^{66} - 4080q^{68} + 23398q^{73} + 13152q^{74} + 96q^{76} - 44622q^{77} + 1152q^{80} - 20368q^{81} + 16512q^{82} - 10440q^{83} + 21274q^{85} - 14316q^{87} + 3168q^{92} + 19416q^{93} - 34686q^{95} - 2048q^{96} - 56798q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 450 x^{6} + 68229 x^{4} + 4001228 x^{2} + 77475204\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 450 \nu^{5} - 59427 \nu^{3} - 770894 \nu \)\()/1249884\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 450 \nu^{5} + 59427 \nu^{3} + 2020778 \nu \)\()/624942\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2484 \nu^{5} + 1017351 \nu^{3} + 84215350 \nu \)\()/2499768\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 225 \nu^{2} + 8802 \)\()/71\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 352 \nu^{4} - 33685 \nu^{2} - 733602 \)\()/3408\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 352 \nu^{4} + 37093 \nu^{2} + 1115298 \)\()/3408\)
\(\beta_{7}\)\(=\)\((\)\( 151 \nu^{7} + 55236 \nu^{5} + 5973951 \nu^{3} + 176424854 \nu \)\()/833256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - 112\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 13 \beta_{3} - 257 \beta_{2} - 294 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(-225 \beta_{6} - 225 \beta_{5} + 71 \beta_{4} + 16398\)
\(\nu^{5}\)\(=\)\((\)\(-367 \beta_{7} - 3067 \beta_{3} + 65353 \beta_{2} + 49114 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(45515 \beta_{6} + 42107 \beta_{5} - 24992 \beta_{4} - 2732978\)
\(\nu^{7}\)\(=\)\((\)\(105723 \beta_{7} + 607599 \beta_{3} - 14907005 \beta_{2} - 8671318 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
13.9305i
6.38941i
8.07810i
12.2418i
12.2418i
8.07810i
6.38941i
13.9305i
2.82843i 15.3447i −8.00000 41.6240 −43.4013 −62.4342 22.6274i −154.460 117.730i
37.2 2.82843i 7.80363i −8.00000 −33.0971 −22.0720 16.0783 22.6274i 20.1034 93.6127i
37.3 2.82843i 6.66389i −8.00000 26.7027 18.8483 51.4469 22.6274i 36.5926 75.5266i
37.4 2.82843i 10.8276i −8.00000 −26.2296 30.6250 −86.0910 22.6274i −36.2364 74.1886i
37.5 2.82843i 10.8276i −8.00000 −26.2296 30.6250 −86.0910 22.6274i −36.2364 74.1886i
37.6 2.82843i 6.66389i −8.00000 26.7027 18.8483 51.4469 22.6274i 36.5926 75.5266i
37.7 2.82843i 7.80363i −8.00000 −33.0971 −22.0720 16.0783 22.6274i 20.1034 93.6127i
37.8 2.82843i 15.3447i −8.00000 41.6240 −43.4013 −62.4342 22.6274i −154.460 117.730i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.5.b.a 8
3.b odd 2 1 342.5.d.a 8
4.b odd 2 1 304.5.e.e 8
19.b odd 2 1 inner 38.5.b.a 8
57.d even 2 1 342.5.d.a 8
76.d even 2 1 304.5.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.b.a 8 1.a even 1 1 trivial
38.5.b.a 8 19.b odd 2 1 inner
304.5.e.e 8 4.b odd 2 1
304.5.e.e 8 76.d even 2 1
342.5.d.a 8 3.b odd 2 1
342.5.d.a 8 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T^{2} )^{4} \)
$3$ \( 74649600 + 3860640 T^{2} + 67449 T^{4} + 458 T^{6} + T^{8} \)
$5$ \( ( 964896 + 6624 T - 2074 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$7$ \( ( 4446118 - 240093 T - 3827 T^{2} + 81 T^{3} + T^{4} )^{2} \)
$11$ \( ( 76301016 + 2109492 T - 37690 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$13$ \( 57445091172827136 + 34129514947104 T^{2} + 5282600121 T^{4} + 140586 T^{6} + T^{8} \)
$17$ \( ( 403423998 + 14985243 T - 67555 T^{2} - 255 T^{3} + T^{4} )^{2} \)
$19$ \( \)\(28\!\cdots\!81\)\( + 26559779028793932 T + 5155530396725960 T^{2} - 2634460387644 T^{3} + 46020337038 T^{4} - 20215164 T^{5} + 303560 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( ( 18350234964 - 14380020 T - 326059 T^{2} + 198 T^{3} + T^{4} )^{2} \)
$29$ \( \)\(26\!\cdots\!00\)\( + 1941717474110086560 T^{2} + 4545080369001 T^{4} + 3774042 T^{6} + T^{8} \)
$31$ \( \)\(13\!\cdots\!16\)\( + 290480962963064832 T^{2} + 1470564161424 T^{4} + 2202408 T^{6} + T^{8} \)
$37$ \( \)\(61\!\cdots\!36\)\( + 68770887827876352 T^{2} + 1219007010960 T^{4} + 6003528 T^{6} + T^{8} \)
$41$ \( \)\(14\!\cdots\!04\)\( + 5375602779897139200 T^{2} + 14417398812816 T^{4} + 7187688 T^{6} + T^{8} \)
$43$ \( ( 407751532960 - 2817187520 T + 3365214 T^{2} + 4327 T^{3} + T^{4} )^{2} \)
$47$ \( ( -98774187816 - 1482029532 T - 3671746 T^{2} - 1605 T^{3} + T^{4} )^{2} \)
$53$ \( \)\(82\!\cdots\!84\)\( + \)\(33\!\cdots\!60\)\( T^{2} + 356018538506121 T^{4} + 36847098 T^{6} + T^{8} \)
$59$ \( \)\(20\!\cdots\!00\)\( + 17355507167396547360 T^{2} + 42239279002761 T^{4} + 25952346 T^{6} + T^{8} \)
$61$ \( ( 195962902247296 + 64039717152 T - 41732066 T^{2} - 657 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(13\!\cdots\!96\)\( + \)\(17\!\cdots\!60\)\( T^{2} + 7964319437470377 T^{4} + 149621370 T^{6} + T^{8} \)
$71$ \( \)\(22\!\cdots\!56\)\( + \)\(26\!\cdots\!40\)\( T^{2} + 10801833589824912 T^{4} + 175804200 T^{6} + T^{8} \)
$73$ \( ( -1571012619241250 + 629910296275 T - 25007871 T^{2} - 11699 T^{3} + T^{4} )^{2} \)
$79$ \( \)\(50\!\cdots\!76\)\( + \)\(58\!\cdots\!16\)\( T^{2} + 1855389240525456 T^{4} + 113259624 T^{6} + T^{8} \)
$83$ \( ( 57645106800768 - 4083728832 T - 57758860 T^{2} + 5220 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(70\!\cdots\!96\)\( + \)\(80\!\cdots\!60\)\( T^{2} + 25689634455831552 T^{4} + 282427200 T^{6} + T^{8} \)
$97$ \( \)\(11\!\cdots\!56\)\( + \)\(10\!\cdots\!88\)\( T^{2} + 51581376150900624 T^{4} + 436011432 T^{6} + T^{8} \)
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