Properties

Label 38.4.c.a
Level $38$
Weight $4$
Character orbit 38.c
Analytic conductor $2.242$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + (5 \zeta_{6} - 5) q^{3} - 4 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - 10 \zeta_{6} q^{6} - 32 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + (5 \zeta_{6} - 5) q^{3} - 4 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - 10 \zeta_{6} q^{6} - 32 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} - 6 \zeta_{6} q^{10} + 4 q^{11} + 20 q^{12} + 69 \zeta_{6} q^{13} + ( - 64 \zeta_{6} + 64) q^{14} - 15 \zeta_{6} q^{15} + (16 \zeta_{6} - 16) q^{16} + (19 \zeta_{6} - 19) q^{17} - 4 q^{18} + (38 \zeta_{6} + 57) q^{19} + 12 q^{20} + ( - 160 \zeta_{6} + 160) q^{21} + (8 \zeta_{6} - 8) q^{22} - 67 \zeta_{6} q^{23} + (40 \zeta_{6} - 40) q^{24} + 116 \zeta_{6} q^{25} - 138 q^{26} - 145 q^{27} + 128 \zeta_{6} q^{28} - 51 \zeta_{6} q^{29} + 30 q^{30} - 132 q^{31} - 32 \zeta_{6} q^{32} + (20 \zeta_{6} - 20) q^{33} - 38 \zeta_{6} q^{34} + ( - 96 \zeta_{6} + 96) q^{35} + ( - 8 \zeta_{6} + 8) q^{36} - 14 q^{37} + (114 \zeta_{6} - 190) q^{38} - 345 q^{39} + (24 \zeta_{6} - 24) q^{40} + ( - 413 \zeta_{6} + 413) q^{41} + 320 \zeta_{6} q^{42} + (129 \zeta_{6} - 129) q^{43} - 16 \zeta_{6} q^{44} - 6 q^{45} + 134 q^{46} + 617 \zeta_{6} q^{47} - 80 \zeta_{6} q^{48} + 681 q^{49} - 232 q^{50} - 95 \zeta_{6} q^{51} + ( - 276 \zeta_{6} + 276) q^{52} - 383 \zeta_{6} q^{53} + ( - 290 \zeta_{6} + 290) q^{54} + (12 \zeta_{6} - 12) q^{55} - 256 q^{56} + (285 \zeta_{6} - 475) q^{57} + 102 q^{58} + ( - 599 \zeta_{6} + 599) q^{59} + (60 \zeta_{6} - 60) q^{60} + 217 \zeta_{6} q^{61} + ( - 264 \zeta_{6} + 264) q^{62} - 64 \zeta_{6} q^{63} + 64 q^{64} - 207 q^{65} - 40 \zeta_{6} q^{66} + 225 \zeta_{6} q^{67} + 76 q^{68} + 335 q^{69} + 192 \zeta_{6} q^{70} + (701 \zeta_{6} - 701) q^{71} + 16 \zeta_{6} q^{72} + (1015 \zeta_{6} - 1015) q^{73} + ( - 28 \zeta_{6} + 28) q^{74} - 580 q^{75} + ( - 380 \zeta_{6} + 152) q^{76} - 128 q^{77} + ( - 690 \zeta_{6} + 690) q^{78} + (349 \zeta_{6} - 349) q^{79} - 48 \zeta_{6} q^{80} + ( - 671 \zeta_{6} + 671) q^{81} + 826 \zeta_{6} q^{82} - 592 q^{83} - 640 q^{84} - 57 \zeta_{6} q^{85} - 258 \zeta_{6} q^{86} + 255 q^{87} + 32 q^{88} + 1349 \zeta_{6} q^{89} + ( - 12 \zeta_{6} + 12) q^{90} - 2208 \zeta_{6} q^{91} + (268 \zeta_{6} - 268) q^{92} + ( - 660 \zeta_{6} + 660) q^{93} - 1234 q^{94} + (171 \zeta_{6} - 285) q^{95} + 160 q^{96} + ( - 613 \zeta_{6} + 613) q^{97} + (1362 \zeta_{6} - 1362) q^{98} + 8 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 3 q^{5} - 10 q^{6} - 64 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 3 q^{5} - 10 q^{6} - 64 q^{7} + 16 q^{8} + 2 q^{9} - 6 q^{10} + 8 q^{11} + 40 q^{12} + 69 q^{13} + 64 q^{14} - 15 q^{15} - 16 q^{16} - 19 q^{17} - 8 q^{18} + 152 q^{19} + 24 q^{20} + 160 q^{21} - 8 q^{22} - 67 q^{23} - 40 q^{24} + 116 q^{25} - 276 q^{26} - 290 q^{27} + 128 q^{28} - 51 q^{29} + 60 q^{30} - 264 q^{31} - 32 q^{32} - 20 q^{33} - 38 q^{34} + 96 q^{35} + 8 q^{36} - 28 q^{37} - 266 q^{38} - 690 q^{39} - 24 q^{40} + 413 q^{41} + 320 q^{42} - 129 q^{43} - 16 q^{44} - 12 q^{45} + 268 q^{46} + 617 q^{47} - 80 q^{48} + 1362 q^{49} - 464 q^{50} - 95 q^{51} + 276 q^{52} - 383 q^{53} + 290 q^{54} - 12 q^{55} - 512 q^{56} - 665 q^{57} + 204 q^{58} + 599 q^{59} - 60 q^{60} + 217 q^{61} + 264 q^{62} - 64 q^{63} + 128 q^{64} - 414 q^{65} - 40 q^{66} + 225 q^{67} + 152 q^{68} + 670 q^{69} + 192 q^{70} - 701 q^{71} + 16 q^{72} - 1015 q^{73} + 28 q^{74} - 1160 q^{75} - 76 q^{76} - 256 q^{77} + 690 q^{78} - 349 q^{79} - 48 q^{80} + 671 q^{81} + 826 q^{82} - 1184 q^{83} - 1280 q^{84} - 57 q^{85} - 258 q^{86} + 510 q^{87} + 64 q^{88} + 1349 q^{89} + 12 q^{90} - 2208 q^{91} - 268 q^{92} + 660 q^{93} - 2468 q^{94} - 399 q^{95} + 320 q^{96} + 613 q^{97} - 1362 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i −2.50000 4.33013i −2.00000 + 3.46410i −1.50000 2.59808i −5.00000 + 8.66025i −32.0000 8.00000 1.00000 1.73205i −3.00000 + 5.19615i
11.1 −1.00000 + 1.73205i −2.50000 + 4.33013i −2.00000 3.46410i −1.50000 + 2.59808i −5.00000 8.66025i −32.0000 8.00000 1.00000 + 1.73205i −3.00000 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( (T + 32)^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$17$ \( T^{2} + 19T + 361 \) Copy content Toggle raw display
$19$ \( T^{2} - 152T + 6859 \) Copy content Toggle raw display
$23$ \( T^{2} + 67T + 4489 \) Copy content Toggle raw display
$29$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$31$ \( (T + 132)^{2} \) Copy content Toggle raw display
$37$ \( (T + 14)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 413T + 170569 \) Copy content Toggle raw display
$43$ \( T^{2} + 129T + 16641 \) Copy content Toggle raw display
$47$ \( T^{2} - 617T + 380689 \) Copy content Toggle raw display
$53$ \( T^{2} + 383T + 146689 \) Copy content Toggle raw display
$59$ \( T^{2} - 599T + 358801 \) Copy content Toggle raw display
$61$ \( T^{2} - 217T + 47089 \) Copy content Toggle raw display
$67$ \( T^{2} - 225T + 50625 \) Copy content Toggle raw display
$71$ \( T^{2} + 701T + 491401 \) Copy content Toggle raw display
$73$ \( T^{2} + 1015 T + 1030225 \) Copy content Toggle raw display
$79$ \( T^{2} + 349T + 121801 \) Copy content Toggle raw display
$83$ \( (T + 592)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1349 T + 1819801 \) Copy content Toggle raw display
$97$ \( T^{2} - 613T + 375769 \) Copy content Toggle raw display
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