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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,3,Mod(27,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.27");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.03542500457\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + (2 \beta_{3} - 4 \beta_1 + 2) q^{7} + 2 \beta_{3} q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + (2 \beta_{3} - 4 \beta_1 + 2) q^{7} + 2 \beta_{3} q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{9} + ( - \beta_{3} + \beta_1) q^{10} + ( - 2 \beta_{3} + 4 \beta_1 - 10) q^{11} + ( - 2 \beta_{3} + 4 \beta_{2} - 2) q^{12} + ( - 6 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 10) q^{13} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{14} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 4 \beta_{3} - 7 \beta_{2} + 2 \beta_1 + 7) q^{17} + ( - 4 \beta_{3} - 8 \beta_{2} + 4) q^{18} + (15 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 3) q^{19} + 2 q^{20} + (6 \beta_{2} - 8 \beta_1 + 6) q^{21} + (4 \beta_{2} - 10 \beta_1 + 4) q^{22} + (9 \beta_{3} - 5 \beta_{2} + 9 \beta_1) q^{23} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{24} + 24 \beta_{2} q^{25} + (5 \beta_{3} - 10 \beta_1 + 12) q^{26} + (7 \beta_{3} - 18 \beta_{2} + 9) q^{27} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{28} + ( - 18 \beta_{3} - 11 \beta_{2} + 18 \beta_1 + 22) q^{29} + ( - \beta_{3} + 2 \beta_1 - 2) q^{30} + (36 \beta_{2} - 18) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + ( - 14 \beta_{2} + 16 \beta_1 - 14) q^{33} + ( - 7 \beta_{3} - 4 \beta_{2} + 7 \beta_1 + 8) q^{34} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{35} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 8) q^{36} + ( - 18 \beta_{3} + 4 \beta_{2} - 2) q^{37} + (2 \beta_{3} + 18 \beta_{2} + 3 \beta_1 - 30) q^{38} + ( - 11 \beta_{3} + 22 \beta_1 - 27) q^{39} + 2 \beta_1 q^{40} + (3 \beta_{2} - 42 \beta_1 + 3) q^{41} + (6 \beta_{3} - 16 \beta_{2} + 6 \beta_1) q^{42} + (46 \beta_{3} - 19 \beta_{2} - 23 \beta_1 + 19) q^{43} + (4 \beta_{3} - 20 \beta_{2} + 4 \beta_1) q^{44} + (2 \beta_{3} - 4 \beta_1 - 4) q^{45} + ( - 5 \beta_{3} + 36 \beta_{2} - 18) q^{46} + ( - 19 \beta_{3} - 35 \beta_{2} - 19 \beta_1) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 8) q^{48} + (8 \beta_{3} - 16 \beta_1 - 21) q^{49} + 24 \beta_{3} q^{50} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{51} + ( - 10 \beta_{2} + 12 \beta_1 - 10) q^{52} + ( - 48 \beta_{3} + 7 \beta_{2} + 48 \beta_1 - 14) q^{53} + ( - 18 \beta_{3} + 14 \beta_{2} + 9 \beta_1 - 14) q^{54} + ( - 4 \beta_{3} + 10 \beta_{2} + 2 \beta_1 - 10) q^{55} + (4 \beta_{3} - 16 \beta_{2} + 8) q^{56} + (22 \beta_{3} - 11 \beta_{2} - 24 \beta_1 + 31) q^{57} + ( - 11 \beta_{3} + 22 \beta_1 + 36) q^{58} + (7 \beta_{2} - 33 \beta_1 + 7) q^{59} + (2 \beta_{2} - 2 \beta_1 + 2) q^{60} + (16 \beta_{3} + 37 \beta_{2} + 16 \beta_1) q^{61} + (36 \beta_{3} - 18 \beta_1) q^{62} + (4 \beta_{3} + 16 \beta_{2} + 4 \beta_1) q^{63} - 8 q^{64} + ( - 6 \beta_{3} + 10 \beta_{2} - 5) q^{65} + ( - 14 \beta_{3} + 32 \beta_{2} - 14 \beta_1) q^{66} + ( - 63 \beta_{3} - 17 \beta_{2} + 63 \beta_1 + 34) q^{67} + ( - 4 \beta_{3} + 8 \beta_1 + 14) q^{68} + (32 \beta_{3} - 46 \beta_{2} + 23) q^{69} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 8) q^{70} + (51 \beta_{2} - 9 \beta_1 + 51) q^{71} + ( - 8 \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 16) q^{72} + (64 \beta_{3} - 49 \beta_{2} - 32 \beta_1 + 49) q^{73} + (4 \beta_{3} - 36 \beta_{2} - 2 \beta_1 + 36) q^{74} + ( - 24 \beta_{3} + 48 \beta_{2} - 24) q^{75} + (18 \beta_{3} + 10 \beta_{2} - 30 \beta_1 - 4) q^{76} + ( - 24 \beta_{3} + 48 \beta_1 - 44) q^{77} + (22 \beta_{2} - 27 \beta_1 + 22) q^{78} + ( - 21 \beta_{2} - 21 \beta_1 - 21) q^{79} + 4 \beta_{2} q^{80} + (68 \beta_{3} - 5 \beta_{2} - 34 \beta_1 + 5) q^{81} + (3 \beta_{3} - 84 \beta_{2} + 3 \beta_1) q^{82} + (6 \beta_{3} - 12 \beta_1 - 16) q^{83} + ( - 16 \beta_{3} + 24 \beta_{2} - 12) q^{84} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{85} + ( - 19 \beta_{3} + 46 \beta_{2} + 19 \beta_1 - 92) q^{86} + ( - 7 \beta_{3} + 14 \beta_1 - 3) q^{87} + ( - 20 \beta_{3} + 16 \beta_{2} - 8) q^{88} + (6 \beta_{3} + \beta_{2} - 6 \beta_1 - 2) q^{89} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{90} + ( - 42 \beta_{3} + 34 \beta_{2} + 42 \beta_1 - 68) q^{91} + (36 \beta_{3} - 10 \beta_{2} - 18 \beta_1 + 10) q^{92} + ( - 36 \beta_{3} + 54 \beta_{2} + 18 \beta_1 - 54) q^{93} + ( - 35 \beta_{3} - 76 \beta_{2} + 38) q^{94} + (6 \beta_{3} - 3 \beta_{2} + 9 \beta_1 + 5) q^{95} + (4 \beta_{3} - 8 \beta_1 + 8) q^{96} + ( - 81 \beta_{2} - 6 \beta_1 - 81) q^{97} + ( - 16 \beta_{2} - 21 \beta_1 - 16) q^{98} + (12 \beta_{3} + 16 \beta_{2} + 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{9} - 40 q^{11} - 30 q^{13} - 24 q^{14} + 6 q^{15} - 8 q^{16} + 14 q^{17} + 16 q^{19} + 8 q^{20} + 36 q^{21} + 24 q^{22} - 10 q^{23} + 8 q^{24} + 48 q^{25} + 48 q^{26} + 8 q^{28} + 66 q^{29} - 8 q^{30} - 84 q^{33} + 24 q^{34} + 4 q^{35} + 16 q^{36} - 84 q^{38} - 108 q^{39} + 18 q^{41} - 32 q^{42} + 38 q^{43} - 40 q^{44} - 16 q^{45} - 70 q^{47} - 24 q^{48} - 84 q^{49} + 18 q^{51} - 60 q^{52} - 42 q^{53} - 28 q^{54} - 20 q^{55} + 102 q^{57} + 144 q^{58} + 42 q^{59} + 12 q^{60} + 74 q^{61} + 32 q^{63} - 32 q^{64} + 64 q^{66} + 102 q^{67} + 56 q^{68} - 24 q^{70} + 306 q^{71} + 48 q^{72} + 98 q^{73} + 72 q^{74} + 4 q^{76} - 176 q^{77} + 132 q^{78} - 126 q^{79} + 8 q^{80} + 10 q^{81} - 168 q^{82} - 64 q^{83} - 14 q^{85} - 276 q^{86} - 12 q^{87} - 6 q^{89} - 24 q^{90} - 204 q^{91} + 20 q^{92} - 108 q^{93} + 14 q^{95} + 32 q^{96} - 486 q^{97} - 96 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + 13 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20 T + 76)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 30 T^{3} + 303 T^{2} + 90 T + 9 \) Copy content Toggle raw display
$17$ \( T^{4} - 14 T^{3} + 171 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} + 570 T^{2} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + 561 T^{2} + \cdots + 212521 \) Copy content Toggle raw display
$29$ \( T^{4} - 66 T^{3} + 1167 T^{2} + \cdots + 81225 \) Copy content Toggle raw display
$31$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 1320 T^{2} + 404496 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} - 3393 T^{2} + \cdots + 12257001 \) Copy content Toggle raw display
$43$ \( T^{4} - 38 T^{3} + 4257 T^{2} + \cdots + 7912969 \) Copy content Toggle raw display
$47$ \( T^{4} + 70 T^{3} + 5841 T^{2} + \cdots + 885481 \) Copy content Toggle raw display
$53$ \( T^{4} + 42 T^{3} - 3873 T^{2} + \cdots + 19900521 \) Copy content Toggle raw display
$59$ \( T^{4} - 42 T^{3} - 1443 T^{2} + \cdots + 4124961 \) Copy content Toggle raw display
$61$ \( T^{4} - 74 T^{3} + 5643 T^{2} + \cdots + 27889 \) Copy content Toggle raw display
$67$ \( T^{4} - 102 T^{3} + \cdots + 49999041 \) Copy content Toggle raw display
$71$ \( T^{4} - 306 T^{3} + \cdots + 58384881 \) Copy content Toggle raw display
$73$ \( T^{4} - 98 T^{3} + 13347 T^{2} + \cdots + 14010049 \) Copy content Toggle raw display
$79$ \( T^{4} + 126 T^{3} + 5733 T^{2} + \cdots + 194481 \) Copy content Toggle raw display
$83$ \( (T^{2} + 32 T + 40)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} - 57 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$97$ \( T^{4} + 486 T^{3} + \cdots + 384591321 \) Copy content Toggle raw display
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