Properties

Label 38.4.c.c
Level $38$
Weight $4$
Character orbit 38.c
Analytic conductor $2.242$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.24207258022\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{4} + 2) q^{2} + (2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{3} - 4 \beta_{4} q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{5} + (4 \beta_{4} - 2 \beta_1) q^{6} + (\beta_{2} + 9) q^{7} - 8 q^{8} + (2 \beta_{5} - 20 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{4} + 2) q^{2} + (2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{3} - 4 \beta_{4} q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{5} + (4 \beta_{4} - 2 \beta_1) q^{6} + (\beta_{2} + 9) q^{7} - 8 q^{8} + (2 \beta_{5} - 20 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{9} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{10} + (3 \beta_{3} - \beta_{2} + 2) q^{11} + (4 \beta_{2} + 8) q^{12} + (\beta_{5} + 44 \beta_{4} - \beta_{3} - 4 \beta_1) q^{13} + ( - 18 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 18) q^{14} + (3 \beta_{5} - 31 \beta_{4} - 3 \beta_{3} + 13 \beta_1) q^{15} + (16 \beta_{4} - 16) q^{16} + ( - 3 \beta_{5} + 18 \beta_{4} - 18) q^{17} + ( - 4 \beta_{3} - 8 \beta_{2} - 40) q^{18} + (3 \beta_{5} + 26 \beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 13 \beta_1 - 6) q^{19} + (4 \beta_{3} + 4 \beta_{2} + 4) q^{20} + ( - 2 \beta_{5} + 61 \beta_{4} - 11 \beta_{2} - 11 \beta_1 - 61) q^{21} + (6 \beta_{5} - 4 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{22} + (\beta_{5} + 17 \beta_{4} - \beta_{3} - 5 \beta_1) q^{23} + ( - 16 \beta_{4} + 8 \beta_{2} + 8 \beta_1 + 16) q^{24} + ( - 9 \beta_{5} - 113 \beta_{4} + 9 \beta_{3} + 10 \beta_1) q^{25} + ( - 2 \beta_{3} + 8 \beta_{2} + 88) q^{26} + (10 \beta_{3} + 21 \beta_{2} + 130) q^{27} + ( - 36 \beta_{4} + 4 \beta_1) q^{28} + (5 \beta_{5} - 35 \beta_{4} - 5 \beta_{3} - 25 \beta_1) q^{29} + ( - 6 \beta_{3} - 26 \beta_{2} - 62) q^{30} + ( - 6 \beta_{3} + 23 \beta_{2} - 11) q^{31} + 32 \beta_{4} q^{32} + ( - \beta_{5} - 81 \beta_{4} - 30 \beta_{2} - 30 \beta_1 + 81) q^{33} + ( - 6 \beta_{5} + 36 \beta_{4} + 6 \beta_{3}) q^{34} + ( - 10 \beta_{5} + 38 \beta_{4} - 20 \beta_{2} - 20 \beta_1 - 38) q^{35} + ( - 8 \beta_{5} + 80 \beta_{4} - 16 \beta_{2} - 16 \beta_1 - 80) q^{36} + (6 \beta_{3} + 5 \beta_{2} - 59) q^{37} + (2 \beta_{5} + 12 \beta_{4} - 6 \beta_{3} - 10 \beta_{2} + 16 \beta_1 + 40) q^{38} + ( - 7 \beta_{3} - 42 \beta_{2} - 274) q^{39} + (8 \beta_{5} - 8 \beta_{4} + 8 \beta_{2} + 8 \beta_1 + 8) q^{40} + ( - 4 \beta_{5} - 147 \beta_{4} - 30 \beta_{2} - 30 \beta_1 + 147) q^{41} + ( - 4 \beta_{5} + 122 \beta_{4} + 4 \beta_{3} - 22 \beta_1) q^{42} + (7 \beta_{5} + 8 \beta_{4} + 42 \beta_{2} + 42 \beta_1 - 8) q^{43} + (12 \beta_{5} - 8 \beta_{4} - 12 \beta_{3} - 4 \beta_1) q^{44} + (2 \beta_{3} + 60 \beta_{2} + 552) q^{45} + ( - 2 \beta_{3} + 10 \beta_{2} + 34) q^{46} + ( - 5 \beta_{5} - 85 \beta_{4} + 5 \beta_{3} + 19 \beta_1) q^{47} + ( - 32 \beta_{4} + 16 \beta_1) q^{48} + (2 \beta_{3} + 18 \beta_{2} - 219) q^{49} + (18 \beta_{3} - 20 \beta_{2} - 226) q^{50} + (3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 48 \beta_1) q^{51} + ( - 4 \beta_{5} - 176 \beta_{4} + 16 \beta_{2} + 16 \beta_1 + 176) q^{52} + (5 \beta_{5} - 5 \beta_{3} + 24 \beta_1) q^{53} + (20 \beta_{5} - 260 \beta_{4} + 42 \beta_{2} + 42 \beta_1 + 260) q^{54} + (32 \beta_{5} + 597 \beta_{4} + 15 \beta_{2} + 15 \beta_1 - 597) q^{55} + ( - 8 \beta_{2} - 72) q^{56} + ( - 17 \beta_{5} + 318 \beta_{4} + 29 \beta_{3} + 10 \beta_{2} + \cdots + 147) q^{57}+ \cdots + ( - 20 \beta_{5} - 1060 \beta_{4} + 20 \beta_{3} + 16 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 5 q^{3} - 12 q^{4} - q^{5} + 10 q^{6} + 52 q^{7} - 48 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 5 q^{3} - 12 q^{4} - q^{5} + 10 q^{6} + 52 q^{7} - 48 q^{8} - 54 q^{9} + 2 q^{10} + 8 q^{11} + 40 q^{12} + 129 q^{13} + 52 q^{14} - 77 q^{15} - 48 q^{16} - 51 q^{17} - 216 q^{18} + 40 q^{19} + 8 q^{20} - 170 q^{21} + 8 q^{22} + 47 q^{23} + 40 q^{24} - 338 q^{25} + 516 q^{26} + 718 q^{27} - 104 q^{28} - 125 q^{29} - 308 q^{30} - 100 q^{31} + 96 q^{32} + 274 q^{33} + 102 q^{34} - 84 q^{35} - 216 q^{36} - 376 q^{37} + 322 q^{38} - 1546 q^{39} + 8 q^{40} + 475 q^{41} + 340 q^{42} - 73 q^{43} - 16 q^{44} + 3188 q^{45} + 188 q^{46} - 241 q^{47} - 80 q^{48} - 1354 q^{49} - 1352 q^{50} + 69 q^{51} + 516 q^{52} + 29 q^{53} + 718 q^{54} - 1838 q^{55} - 416 q^{56} + 1755 q^{57} - 500 q^{58} - 1065 q^{59} - 308 q^{60} - 981 q^{61} - 100 q^{62} - 872 q^{63} + 384 q^{64} + 586 q^{65} - 548 q^{66} + 877 q^{67} + 408 q^{68} - 1526 q^{69} + 168 q^{70} + 2135 q^{71} + 432 q^{72} + 667 q^{73} - 376 q^{74} + 4584 q^{75} + 484 q^{76} - 492 q^{77} - 1546 q^{78} + 1671 q^{79} - 16 q^{80} - 1287 q^{81} - 950 q^{82} + 1176 q^{83} + 1360 q^{84} - 1929 q^{85} + 146 q^{86} - 6430 q^{87} - 64 q^{88} + 693 q^{89} + 3188 q^{90} + 1676 q^{91} + 188 q^{92} - 3138 q^{93} - 964 q^{94} + 4489 q^{95} - 320 q^{96} - 985 q^{97} - 1354 q^{98} - 3184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 64\nu^{4} + 4096\nu^{3} - 3984\nu^{2} + 945\nu - 60480 ) / 254031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{5} - 1344\nu^{4} + 1339\nu^{3} - 83664\nu^{2} + 19845\nu - 3641036 ) / 169354 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1344\nu^{5} - 1339\nu^{4} + 85696\nu^{3} + 64832\nu^{2} + 5334576\nu + 4800 ) / 1270155 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57792\nu^{5} - 57577\nu^{4} + 3684928\nu^{3} + 1517621\nu^{2} + 229386768\nu - 54410265 ) / 2540310 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 43\beta_{4} - 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 63\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 128\beta_{5} - 2737\beta_{4} - 128\beta_{3} - 48\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 224\beta_{5} - 3856\beta_{4} - 4017\beta_{2} - 4017\beta _1 + 3856 \) Copy content Toggle raw display

Character values

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

\(n\) \(21\)
\(\chi(n)\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−3.78825 + 6.56144i
0.118706 0.205606i
4.16954 7.22186i
−3.78825 6.56144i
0.118706 + 0.205606i
4.16954 + 7.22186i
1.00000 + 1.73205i −4.78825 8.29349i −2.00000 + 3.46410i −7.88908 13.6643i 9.57650 16.5870i 16.5765 −8.00000 −32.3546 + 56.0399i 15.7782 27.3286i
7.2 1.00000 + 1.73205i −0.881294 1.52645i −2.00000 + 3.46410i 10.3546 + 17.9347i 1.76259 3.05289i 8.76259 −8.00000 11.9466 20.6922i −20.7092 + 35.8694i
7.3 1.00000 + 1.73205i 3.16954 + 5.48981i −2.00000 + 3.46410i −2.96554 5.13646i −6.33908 + 10.9796i 0.660916 −8.00000 −6.59199 + 11.4177i 5.93108 10.2729i
11.1 1.00000 1.73205i −4.78825 + 8.29349i −2.00000 3.46410i −7.88908 + 13.6643i 9.57650 + 16.5870i 16.5765 −8.00000 −32.3546 56.0399i 15.7782 + 27.3286i
11.2 1.00000 1.73205i −0.881294 + 1.52645i −2.00000 3.46410i 10.3546 17.9347i 1.76259 + 3.05289i 8.76259 −8.00000 11.9466 + 20.6922i −20.7092 35.8694i
11.3 1.00000 1.73205i 3.16954 5.48981i −2.00000 3.46410i −2.96554 + 5.13646i −6.33908 10.9796i 0.660916 −8.00000 −6.59199 11.4177i 5.93108 + 10.2729i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.c.c 6
3.b odd 2 1 342.4.g.f 6
4.b odd 2 1 304.4.i.e 6
19.c even 3 1 inner 38.4.c.c 6
19.c even 3 1 722.4.a.j 3
19.d odd 6 1 722.4.a.k 3
57.h odd 6 1 342.4.g.f 6
76.g odd 6 1 304.4.i.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.c 6 1.a even 1 1 trivial
38.4.c.c 6 19.c even 3 1 inner
304.4.i.e 6 4.b odd 2 1
304.4.i.e 6 76.g odd 6 1
342.4.g.f 6 3.b odd 2 1
342.4.g.f 6 57.h odd 6 1
722.4.a.j 3 19.c even 3 1
722.4.a.k 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 5T_{3}^{5} + 80T_{3}^{4} - 61T_{3}^{3} + 3560T_{3}^{2} + 5885T_{3} + 11449 \) acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 5 T^{5} + 80 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 357 T^{4} + \cdots + 3755844 \) Copy content Toggle raw display
$7$ \( (T^{3} - 26 T^{2} + 162 T - 96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 3311 T + 49980)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 129 T^{5} + \cdots + 129322384 \) Copy content Toggle raw display
$17$ \( T^{6} + 51 T^{5} + \cdots + 11293737984 \) Copy content Toggle raw display
$19$ \( T^{6} - 40 T^{5} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{6} - 47 T^{5} + \cdots + 4555440036 \) Copy content Toggle raw display
$29$ \( T^{6} + 125 T^{5} + \cdots + 5937750562500 \) Copy content Toggle raw display
$31$ \( (T^{3} + 50 T^{2} - 52150 T + 3809848)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 188 T^{2} - 658 T - 88004)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 475 T^{5} + \cdots + 81183541856481 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 259289089231936 \) Copy content Toggle raw display
$47$ \( T^{6} + 241 T^{5} + \cdots + 25671752892900 \) Copy content Toggle raw display
$53$ \( T^{6} - 29 T^{5} + \cdots + 10857156800400 \) Copy content Toggle raw display
$59$ \( T^{6} + 1065 T^{5} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 356206057990084 \) Copy content Toggle raw display
$67$ \( T^{6} - 877 T^{5} + \cdots + 964239549849 \) Copy content Toggle raw display
$71$ \( T^{6} - 2135 T^{5} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} - 667 T^{5} + \cdots + 713681971209 \) Copy content Toggle raw display
$79$ \( T^{6} - 1671 T^{5} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} - 588 T^{2} - 848043 T - 162474984)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 693 T^{5} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{6} + 985 T^{5} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
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