Properties

Label 38.4.c.b
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + ( - 5 \zeta_{6} + 5) q^{3} - 4 \zeta_{6} q^{4} + ( - 12 \zeta_{6} + 12) q^{5} + 10 \zeta_{6} q^{6} + 8 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + ( - 5 \zeta_{6} + 5) q^{3} - 4 \zeta_{6} q^{4} + ( - 12 \zeta_{6} + 12) q^{5} + 10 \zeta_{6} q^{6} + 8 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} + 24 \zeta_{6} q^{10} + 9 q^{11} - 20 q^{12} - 26 \zeta_{6} q^{13} + (16 \zeta_{6} - 16) q^{14} - 60 \zeta_{6} q^{15} + (16 \zeta_{6} - 16) q^{16} + (114 \zeta_{6} - 114) q^{17} - 4 q^{18} + ( - 57 \zeta_{6} - 38) q^{19} - 48 q^{20} + ( - 40 \zeta_{6} + 40) q^{21} + (18 \zeta_{6} - 18) q^{22} + 78 \zeta_{6} q^{23} + ( - 40 \zeta_{6} + 40) q^{24} - 19 \zeta_{6} q^{25} + 52 q^{26} + 145 q^{27} - 32 \zeta_{6} q^{28} + 204 \zeta_{6} q^{29} + 120 q^{30} + 98 q^{31} - 32 \zeta_{6} q^{32} + ( - 45 \zeta_{6} + 45) q^{33} - 228 \zeta_{6} q^{34} + ( - 96 \zeta_{6} + 96) q^{35} + ( - 8 \zeta_{6} + 8) q^{36} - 334 q^{37} + ( - 76 \zeta_{6} + 190) q^{38} - 130 q^{39} + ( - 96 \zeta_{6} + 96) q^{40} + (177 \zeta_{6} - 177) q^{41} + 80 \zeta_{6} q^{42} + ( - 316 \zeta_{6} + 316) q^{43} - 36 \zeta_{6} q^{44} + 24 q^{45} - 156 q^{46} + 492 \zeta_{6} q^{47} + 80 \zeta_{6} q^{48} - 279 q^{49} + 38 q^{50} + 570 \zeta_{6} q^{51} + (104 \zeta_{6} - 104) q^{52} - 678 \zeta_{6} q^{53} + (290 \zeta_{6} - 290) q^{54} + ( - 108 \zeta_{6} + 108) q^{55} + 64 q^{56} + (190 \zeta_{6} - 475) q^{57} - 408 q^{58} + ( - 579 \zeta_{6} + 579) q^{59} + (240 \zeta_{6} - 240) q^{60} + 352 \zeta_{6} q^{61} + (196 \zeta_{6} - 196) q^{62} + 16 \zeta_{6} q^{63} + 64 q^{64} - 312 q^{65} + 90 \zeta_{6} q^{66} - 755 \zeta_{6} q^{67} + 456 q^{68} + 390 q^{69} + 192 \zeta_{6} q^{70} + (6 \zeta_{6} - 6) q^{71} + 16 \zeta_{6} q^{72} + ( - 145 \zeta_{6} + 145) q^{73} + ( - 668 \zeta_{6} + 668) q^{74} - 95 q^{75} + (380 \zeta_{6} - 228) q^{76} + 72 q^{77} + ( - 260 \zeta_{6} + 260) q^{78} + ( - 316 \zeta_{6} + 316) q^{79} + 192 \zeta_{6} q^{80} + ( - 671 \zeta_{6} + 671) q^{81} - 354 \zeta_{6} q^{82} - 567 q^{83} - 160 q^{84} + 1368 \zeta_{6} q^{85} + 632 \zeta_{6} q^{86} + 1020 q^{87} + 72 q^{88} + 114 \zeta_{6} q^{89} + (48 \zeta_{6} - 48) q^{90} - 208 \zeta_{6} q^{91} + ( - 312 \zeta_{6} + 312) q^{92} + ( - 490 \zeta_{6} + 490) q^{93} - 984 q^{94} + (456 \zeta_{6} - 1140) q^{95} - 160 q^{96} + ( - 943 \zeta_{6} + 943) q^{97} + ( - 558 \zeta_{6} + 558) q^{98} + 18 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 12 q^{5} + 10 q^{6} + 16 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 12 q^{5} + 10 q^{6} + 16 q^{7} + 16 q^{8} + 2 q^{9} + 24 q^{10} + 18 q^{11} - 40 q^{12} - 26 q^{13} - 16 q^{14} - 60 q^{15} - 16 q^{16} - 114 q^{17} - 8 q^{18} - 133 q^{19} - 96 q^{20} + 40 q^{21} - 18 q^{22} + 78 q^{23} + 40 q^{24} - 19 q^{25} + 104 q^{26} + 290 q^{27} - 32 q^{28} + 204 q^{29} + 240 q^{30} + 196 q^{31} - 32 q^{32} + 45 q^{33} - 228 q^{34} + 96 q^{35} + 8 q^{36} - 668 q^{37} + 304 q^{38} - 260 q^{39} + 96 q^{40} - 177 q^{41} + 80 q^{42} + 316 q^{43} - 36 q^{44} + 48 q^{45} - 312 q^{46} + 492 q^{47} + 80 q^{48} - 558 q^{49} + 76 q^{50} + 570 q^{51} - 104 q^{52} - 678 q^{53} - 290 q^{54} + 108 q^{55} + 128 q^{56} - 760 q^{57} - 816 q^{58} + 579 q^{59} - 240 q^{60} + 352 q^{61} - 196 q^{62} + 16 q^{63} + 128 q^{64} - 624 q^{65} + 90 q^{66} - 755 q^{67} + 912 q^{68} + 780 q^{69} + 192 q^{70} - 6 q^{71} + 16 q^{72} + 145 q^{73} + 668 q^{74} - 190 q^{75} - 76 q^{76} + 144 q^{77} + 260 q^{78} + 316 q^{79} + 192 q^{80} + 671 q^{81} - 354 q^{82} - 1134 q^{83} - 320 q^{84} + 1368 q^{85} + 632 q^{86} + 2040 q^{87} + 144 q^{88} + 114 q^{89} - 48 q^{90} - 208 q^{91} + 312 q^{92} + 490 q^{93} - 1968 q^{94} - 1824 q^{95} - 320 q^{96} + 943 q^{97} + 558 q^{98} + 18 q^{99}+O(q^{100}) \) 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 6.00000 + 10.3923i 5.00000 8.66025i 8.00000 8.00000 1.00000 1.73205i 12.0000 20.7846i
11.1 −1.00000 + 1.73205i 2.50000 4.33013i −2.00000 3.46410i 6.00000 10.3923i 5.00000 + 8.66025i 8.00000 8.00000 1.00000 + 1.73205i 12.0000 + 20.7846i
\(n\): 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.b 2
3.b odd 2 1 342.4.g.c 2
4.b odd 2 1 304.4.i.a 2
19.c even 3 1 inner 38.4.c.b 2
19.c even 3 1 722.4.a.c 1
19.d odd 6 1 722.4.a.b 1
57.h odd 6 1 342.4.g.c 2
76.g odd 6 1 304.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.b 2 1.a even 1 1 trivial
38.4.c.b 2 19.c even 3 1 inner
304.4.i.a 2 4.b odd 2 1
304.4.i.a 2 76.g odd 6 1
342.4.g.c 2 3.b odd 2 1
342.4.g.c 2 57.h odd 6 1
722.4.a.b 1 19.d odd 6 1
722.4.a.c 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
$5$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$17$ \( T^{2} + 114T + 12996 \) Copy content Toggle raw display
$19$ \( T^{2} + 133T + 6859 \) Copy content Toggle raw display
$23$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} - 204T + 41616 \) Copy content Toggle raw display
$31$ \( (T - 98)^{2} \) Copy content Toggle raw display
$37$ \( (T + 334)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 177T + 31329 \) Copy content Toggle raw display
$43$ \( T^{2} - 316T + 99856 \) Copy content Toggle raw display
$47$ \( T^{2} - 492T + 242064 \) Copy content Toggle raw display
$53$ \( T^{2} + 678T + 459684 \) Copy content Toggle raw display
$59$ \( T^{2} - 579T + 335241 \) Copy content Toggle raw display
$61$ \( T^{2} - 352T + 123904 \) Copy content Toggle raw display
$67$ \( T^{2} + 755T + 570025 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 145T + 21025 \) Copy content Toggle raw display
$79$ \( T^{2} - 316T + 99856 \) Copy content Toggle raw display
$83$ \( (T + 567)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 114T + 12996 \) Copy content Toggle raw display
$97$ \( T^{2} - 943T + 889249 \) Copy content Toggle raw display
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