Properties

Label 38.2.c.a
Level $38$
Weight $2$
Character orbit 38.c
Analytic conductor $0.303$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - 4 q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - 4 q^{7} - q^{8} + 2 \zeta_{6} q^{9} + 3 q^{11} + q^{12} - 2 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{14} + (\zeta_{6} - 1) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + 2 q^{18} + ( - 3 \zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 4) q^{21} + ( - 3 \zeta_{6} + 3) q^{22} + 6 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + 5 \zeta_{6} q^{25} - 2 q^{26} - 5 q^{27} + 4 \zeta_{6} q^{28} + 2 q^{31} + \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{33} - 6 \zeta_{6} q^{34} + ( - 2 \zeta_{6} + 2) q^{36} - 10 q^{37} + (2 \zeta_{6} - 5) q^{38} + 2 q^{39} + (9 \zeta_{6} - 9) q^{41} - 4 \zeta_{6} q^{42} + ( - 4 \zeta_{6} + 4) q^{43} - 3 \zeta_{6} q^{44} + 6 q^{46} - \zeta_{6} q^{48} + 9 q^{49} + 5 q^{50} + 6 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} - 6 \zeta_{6} q^{53} + (5 \zeta_{6} - 5) q^{54} + 4 q^{56} + ( - 2 \zeta_{6} + 5) q^{57} + ( - 9 \zeta_{6} + 9) q^{59} + 4 \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{62} - 8 \zeta_{6} q^{63} + q^{64} + 3 \zeta_{6} q^{66} + 7 \zeta_{6} q^{67} - 6 q^{68} - 6 q^{69} + ( - 6 \zeta_{6} + 6) q^{71} - 2 \zeta_{6} q^{72} + ( - \zeta_{6} + 1) q^{73} + (10 \zeta_{6} - 10) q^{74} - 5 q^{75} + (5 \zeta_{6} - 3) q^{76} - 12 q^{77} + ( - 2 \zeta_{6} + 2) q^{78} + ( - 4 \zeta_{6} + 4) q^{79} + (\zeta_{6} - 1) q^{81} + 9 \zeta_{6} q^{82} + 3 q^{83} - 4 q^{84} - 4 \zeta_{6} q^{86} - 3 q^{88} - 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( - 6 \zeta_{6} + 6) q^{92} + (2 \zeta_{6} - 2) q^{93} - q^{96} + (17 \zeta_{6} - 17) q^{97} + ( - 9 \zeta_{6} + 9) q^{98} + 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 8 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 8 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{14} - q^{16} + 6 q^{17} + 4 q^{18} - 7 q^{19} + 4 q^{21} + 3 q^{22} + 6 q^{23} + q^{24} + 5 q^{25} - 4 q^{26} - 10 q^{27} + 4 q^{28} + 4 q^{31} + q^{32} - 3 q^{33} - 6 q^{34} + 2 q^{36} - 20 q^{37} - 8 q^{38} + 4 q^{39} - 9 q^{41} - 4 q^{42} + 4 q^{43} - 3 q^{44} + 12 q^{46} - q^{48} + 18 q^{49} + 10 q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} - 5 q^{54} + 8 q^{56} + 8 q^{57} + 9 q^{59} + 4 q^{61} + 2 q^{62} - 8 q^{63} + 2 q^{64} + 3 q^{66} + 7 q^{67} - 12 q^{68} - 12 q^{69} + 6 q^{71} - 2 q^{72} + q^{73} - 10 q^{74} - 10 q^{75} - q^{76} - 24 q^{77} + 2 q^{78} + 4 q^{79} - q^{81} + 9 q^{82} + 6 q^{83} - 8 q^{84} - 4 q^{86} - 6 q^{88} - 6 q^{89} + 8 q^{91} + 6 q^{92} - 2 q^{93} - 2 q^{96} - 17 q^{97} + 9 q^{98} + 6 q^{99}+O(q^{100}) \) 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
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −4.00000 −1.00000 1.00000 1.73205i 0
11.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −4.00000 −1.00000 1.00000 + 1.73205i 0
\(n\): 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 38.2.c.a 2
3.b odd 2 1 342.2.g.b 2
4.b odd 2 1 304.2.i.c 2
5.b even 2 1 950.2.e.d 2
5.c odd 4 2 950.2.j.e 4
8.b even 2 1 1216.2.i.h 2
8.d odd 2 1 1216.2.i.d 2
12.b even 2 1 2736.2.s.m 2
19.b odd 2 1 722.2.c.b 2
19.c even 3 1 inner 38.2.c.a 2
19.c even 3 1 722.2.a.c 1
19.d odd 6 1 722.2.a.d 1
19.d odd 6 1 722.2.c.b 2
19.e even 9 6 722.2.e.j 6
19.f odd 18 6 722.2.e.i 6
57.f even 6 1 6498.2.a.e 1
57.h odd 6 1 342.2.g.b 2
57.h odd 6 1 6498.2.a.s 1
76.f even 6 1 5776.2.a.n 1
76.g odd 6 1 304.2.i.c 2
76.g odd 6 1 5776.2.a.g 1
95.i even 6 1 950.2.e.d 2
95.m odd 12 2 950.2.j.e 4
152.k odd 6 1 1216.2.i.d 2
152.p even 6 1 1216.2.i.h 2
228.m even 6 1 2736.2.s.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 1.a even 1 1 trivial
38.2.c.a 2 19.c even 3 1 inner
304.2.i.c 2 4.b odd 2 1
304.2.i.c 2 76.g odd 6 1
342.2.g.b 2 3.b odd 2 1
342.2.g.b 2 57.h odd 6 1
722.2.a.c 1 19.c even 3 1
722.2.a.d 1 19.d odd 6 1
722.2.c.b 2 19.b odd 2 1
722.2.c.b 2 19.d odd 6 1
722.2.e.i 6 19.f odd 18 6
722.2.e.j 6 19.e even 9 6
950.2.e.d 2 5.b even 2 1
950.2.e.d 2 95.i even 6 1
950.2.j.e 4 5.c odd 4 2
950.2.j.e 4 95.m odd 12 2
1216.2.i.d 2 8.d odd 2 1
1216.2.i.d 2 152.k odd 6 1
1216.2.i.h 2 8.b even 2 1
1216.2.i.h 2 152.p even 6 1
2736.2.s.m 2 12.b even 2 1
2736.2.s.m 2 228.m even 6 1
5776.2.a.g 1 76.g odd 6 1
5776.2.a.n 1 76.f even 6 1
6498.2.a.e 1 57.f even 6 1
6498.2.a.s 1 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T - 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
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