Properties

Label 38.3.d.a
Level 38
Weight 3
Character orbit 38.d
Analytic conductor 1.035
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.03542500457\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{3} ) q^{10} + ( -10 + 4 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{12} + ( -10 + 6 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{13} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( 7 + 2 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 4 - 8 \beta_{2} - 4 \beta_{3} ) q^{18} + ( 3 - 6 \beta_{1} + 2 \beta_{2} + 15 \beta_{3} ) q^{19} + 2 q^{20} + ( 6 - 8 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 4 - 10 \beta_{1} + 4 \beta_{2} ) q^{22} + ( 9 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} ) q^{23} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{24} + 24 \beta_{2} q^{25} + ( 12 - 10 \beta_{1} + 5 \beta_{3} ) q^{26} + ( 9 - 18 \beta_{2} + 7 \beta_{3} ) q^{27} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{28} + ( 22 + 18 \beta_{1} - 11 \beta_{2} - 18 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{30} + ( -18 + 36 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -14 + 16 \beta_{1} - 14 \beta_{2} ) q^{33} + ( 8 + 7 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{34} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{35} + ( 8 + 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{36} + ( -2 + 4 \beta_{2} - 18 \beta_{3} ) q^{37} + ( -30 + 3 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -27 + 22 \beta_{1} - 11 \beta_{3} ) q^{39} + 2 \beta_{1} q^{40} + ( 3 - 42 \beta_{1} + 3 \beta_{2} ) q^{41} + ( 6 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} ) q^{42} + ( 19 - 23 \beta_{1} - 19 \beta_{2} + 46 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -4 - 4 \beta_{1} + 2 \beta_{3} ) q^{45} + ( -18 + 36 \beta_{2} - 5 \beta_{3} ) q^{46} + ( -19 \beta_{1} - 35 \beta_{2} - 19 \beta_{3} ) q^{47} + ( -8 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -21 - 16 \beta_{1} + 8 \beta_{3} ) q^{49} + 24 \beta_{3} q^{50} + ( 6 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{51} + ( -10 + 12 \beta_{1} - 10 \beta_{2} ) q^{52} + ( -14 + 48 \beta_{1} + 7 \beta_{2} - 48 \beta_{3} ) q^{53} + ( -14 + 9 \beta_{1} + 14 \beta_{2} - 18 \beta_{3} ) q^{54} + ( -10 + 2 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 8 - 16 \beta_{2} + 4 \beta_{3} ) q^{56} + ( 31 - 24 \beta_{1} - 11 \beta_{2} + 22 \beta_{3} ) q^{57} + ( 36 + 22 \beta_{1} - 11 \beta_{3} ) q^{58} + ( 7 - 33 \beta_{1} + 7 \beta_{2} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{60} + ( 16 \beta_{1} + 37 \beta_{2} + 16 \beta_{3} ) q^{61} + ( -18 \beta_{1} + 36 \beta_{3} ) q^{62} + ( 4 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} ) q^{63} -8 q^{64} + ( -5 + 10 \beta_{2} - 6 \beta_{3} ) q^{65} + ( -14 \beta_{1} + 32 \beta_{2} - 14 \beta_{3} ) q^{66} + ( 34 + 63 \beta_{1} - 17 \beta_{2} - 63 \beta_{3} ) q^{67} + ( 14 + 8 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 23 - 46 \beta_{2} + 32 \beta_{3} ) q^{69} + ( -8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{70} + ( 51 - 9 \beta_{1} + 51 \beta_{2} ) q^{71} + ( 16 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{72} + ( 49 - 32 \beta_{1} - 49 \beta_{2} + 64 \beta_{3} ) q^{73} + ( 36 - 2 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -24 + 48 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -4 - 30 \beta_{1} + 10 \beta_{2} + 18 \beta_{3} ) q^{76} + ( -44 + 48 \beta_{1} - 24 \beta_{3} ) q^{77} + ( 22 - 27 \beta_{1} + 22 \beta_{2} ) q^{78} + ( -21 - 21 \beta_{1} - 21 \beta_{2} ) q^{79} + 4 \beta_{2} q^{80} + ( 5 - 34 \beta_{1} - 5 \beta_{2} + 68 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 84 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -16 - 12 \beta_{1} + 6 \beta_{3} ) q^{83} + ( -12 + 24 \beta_{2} - 16 \beta_{3} ) q^{84} + ( -2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -92 + 19 \beta_{1} + 46 \beta_{2} - 19 \beta_{3} ) q^{86} + ( -3 + 14 \beta_{1} - 7 \beta_{3} ) q^{87} + ( -8 + 16 \beta_{2} - 20 \beta_{3} ) q^{88} + ( -2 - 6 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{89} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{90} + ( -68 + 42 \beta_{1} + 34 \beta_{2} - 42 \beta_{3} ) q^{91} + ( 10 - 18 \beta_{1} - 10 \beta_{2} + 36 \beta_{3} ) q^{92} + ( -54 + 18 \beta_{1} + 54 \beta_{2} - 36 \beta_{3} ) q^{93} + ( 38 - 76 \beta_{2} - 35 \beta_{3} ) q^{94} + ( 5 + 9 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 8 - 8 \beta_{1} + 4 \beta_{3} ) q^{96} + ( -81 - 6 \beta_{1} - 81 \beta_{2} ) q^{97} + ( -16 - 21 \beta_{1} - 16 \beta_{2} ) q^{98} + ( 12 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} + 4q^{4} + 2q^{5} - 4q^{6} + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 4q + 6q^{3} + 4q^{4} + 2q^{5} - 4q^{6} + 8q^{7} - 8q^{9} - 40q^{11} - 30q^{13} - 24q^{14} + 6q^{15} - 8q^{16} + 14q^{17} + 16q^{19} + 8q^{20} + 36q^{21} + 24q^{22} - 10q^{23} + 8q^{24} + 48q^{25} + 48q^{26} + 8q^{28} + 66q^{29} - 8q^{30} - 84q^{33} + 24q^{34} + 4q^{35} + 16q^{36} - 84q^{38} - 108q^{39} + 18q^{41} - 32q^{42} + 38q^{43} - 40q^{44} - 16q^{45} - 70q^{47} - 24q^{48} - 84q^{49} + 18q^{51} - 60q^{52} - 42q^{53} - 28q^{54} - 20q^{55} + 102q^{57} + 144q^{58} + 42q^{59} + 12q^{60} + 74q^{61} + 32q^{63} - 32q^{64} + 64q^{66} + 102q^{67} + 56q^{68} - 24q^{70} + 306q^{71} + 48q^{72} + 98q^{73} + 72q^{74} + 4q^{76} - 176q^{77} + 132q^{78} - 126q^{79} + 8q^{80} + 10q^{81} - 168q^{82} - 64q^{83} - 14q^{85} - 276q^{86} - 12q^{87} - 6q^{89} - 24q^{90} - 204q^{91} + 20q^{92} - 108q^{93} + 14q^{95} + 32q^{96} - 486q^{97} - 96q^{98} + 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(21\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 2.72474 + 1.57313i 1.00000 + 1.73205i 0.500000 0.866025i −2.22474 3.85337i 6.89898 2.82843i 0.449490 + 0.778539i −1.22474 + 0.707107i
27.2 1.22474 + 0.707107i 0.275255 + 0.158919i 1.00000 + 1.73205i 0.500000 0.866025i 0.224745 + 0.389270i −2.89898 2.82843i −4.44949 7.70674i 1.22474 0.707107i
31.1 −1.22474 + 0.707107i 2.72474 1.57313i 1.00000 1.73205i 0.500000 + 0.866025i −2.22474 + 3.85337i 6.89898 2.82843i 0.449490 0.778539i −1.22474 0.707107i
31.2 1.22474 0.707107i 0.275255 0.158919i 1.00000 1.73205i 0.500000 + 0.866025i 0.224745 0.389270i −2.89898 2.82843i −4.44949 + 7.70674i 1.22474 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.3.d.a 4
3.b odd 2 1 342.3.m.a 4
4.b odd 2 1 304.3.r.a 4
19.c even 3 1 722.3.b.b 4
19.d odd 6 1 inner 38.3.d.a 4
19.d odd 6 1 722.3.b.b 4
57.f even 6 1 342.3.m.a 4
76.f even 6 1 304.3.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.d.a 4 1.a even 1 1 trivial
38.3.d.a 4 19.d odd 6 1 inner
304.3.r.a 4 4.b odd 2 1
304.3.r.a 4 76.f even 6 1
342.3.m.a 4 3.b odd 2 1
342.3.m.a 4 57.f even 6 1
722.3.b.b 4 19.c even 3 1
722.3.b.b 4 19.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( 1 - 6 T + 31 T^{2} - 114 T^{3} + 388 T^{4} - 1026 T^{5} + 2511 T^{6} - 4374 T^{7} + 6561 T^{8} \)
$5$ \( ( 1 - T - 24 T^{2} - 25 T^{3} + 625 T^{4} )^{2} \)
$7$ \( ( 1 - 4 T + 78 T^{2} - 196 T^{3} + 2401 T^{4} )^{2} \)
$11$ \( ( 1 + 20 T + 318 T^{2} + 2420 T^{3} + 14641 T^{4} )^{2} \)
$13$ \( 1 + 30 T + 641 T^{2} + 10230 T^{3} + 138420 T^{4} + 1728870 T^{5} + 18307601 T^{6} + 144804270 T^{7} + 815730721 T^{8} \)
$17$ \( 1 - 14 T - 407 T^{2} - 350 T^{3} + 223444 T^{4} - 101150 T^{5} - 33993047 T^{6} - 337925966 T^{7} + 6975757441 T^{8} \)
$19$ \( 1 - 16 T + 570 T^{2} - 5776 T^{3} + 130321 T^{4} \)
$23$ \( 1 + 10 T - 497 T^{2} - 4610 T^{3} + 23668 T^{4} - 2438690 T^{5} - 139080977 T^{6} + 1480358890 T^{7} + 78310985281 T^{8} \)
$29$ \( 1 - 66 T + 2849 T^{2} - 92202 T^{3} + 2465460 T^{4} - 77541882 T^{5} + 2015043569 T^{6} - 39258339186 T^{7} + 500246412961 T^{8} \)
$31$ \( ( 1 - 950 T^{2} + 923521 T^{4} )^{2} \)
$37$ \( 1 - 4156 T^{2} + 8035302 T^{4} - 7789013116 T^{6} + 3512479453921 T^{8} \)
$41$ \( 1 - 18 T - 31 T^{2} + 2502 T^{3} - 2624892 T^{4} + 4205862 T^{5} - 87598591 T^{6} - 85501876338 T^{7} + 7984925229121 T^{8} \)
$43$ \( 1 - 38 T + 559 T^{2} + 106894 T^{3} - 5305532 T^{4} + 197647006 T^{5} + 1911109759 T^{6} - 240211795862 T^{7} + 11688200277601 T^{8} \)
$47$ \( 1 + 70 T + 1423 T^{2} - 65870 T^{3} - 3614252 T^{4} - 145506830 T^{5} + 6943786063 T^{6} + 754545073030 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 + 42 T + 1745 T^{2} + 48594 T^{3} - 4900140 T^{4} + 136500546 T^{5} + 13768889345 T^{6} + 930903167418 T^{7} + 62259690411361 T^{8} \)
$59$ \( 1 - 42 T + 5519 T^{2} - 207102 T^{3} + 14244228 T^{4} - 720922062 T^{5} + 66875715359 T^{6} - 1771582412922 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 - 74 T - 1799 T^{2} + 12358 T^{3} + 18703588 T^{4} + 45984118 T^{5} - 24908667959 T^{6} - 3812507702714 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 - 102 T + 5375 T^{2} - 194514 T^{3} - 946620 T^{4} - 873173346 T^{5} + 108312275375 T^{6} - 9226754981238 T^{7} + 406067677556641 T^{8} \)
$71$ \( 1 - 306 T + 48935 T^{2} - 5423238 T^{3} + 446032740 T^{4} - 27338542758 T^{5} + 1243520609735 T^{6} - 39198686879826 T^{7} + 645753531245761 T^{8} \)
$73$ \( 1 - 98 T + 2689 T^{2} + 366814 T^{3} - 31760732 T^{4} + 1954751806 T^{5} + 76362870049 T^{6} - 14830754176322 T^{7} + 806460091894081 T^{8} \)
$79$ \( 1 + 126 T + 18215 T^{2} + 1628298 T^{3} + 161081220 T^{4} + 10162207818 T^{5} + 709475725415 T^{6} + 30629019395646 T^{7} + 1517108809906561 T^{8} \)
$83$ \( ( 1 + 32 T + 13818 T^{2} + 220448 T^{3} + 47458321 T^{4} )^{2} \)
$89$ \( 1 + 6 T + 15785 T^{2} + 94638 T^{3} + 186140340 T^{4} + 749627598 T^{5} + 990386274185 T^{6} + 2981887745766 T^{7} + 3936588805702081 T^{8} \)
$97$ \( 1 + 486 T + 117161 T^{2} + 18676494 T^{3} + 2129048148 T^{4} + 175727132046 T^{5} + 10372179091241 T^{6} + 404824394395494 T^{7} + 7837433594376961 T^{8} \)
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