Properties

Label 1386.4.a.y
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,4,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

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: yes
Analytic conductor: \(81.7766472680\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - \beta - 1) q^{5} + 7 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + ( - \beta - 1) q^{5} + 7 q^{7} + 8 q^{8} + ( - 2 \beta - 2) q^{10} - 11 q^{11} + (3 \beta - 1) q^{13} + 14 q^{14} + 16 q^{16} + ( - 2 \beta - 56) q^{17} + (13 \beta - 45) q^{19} + ( - 4 \beta - 4) q^{20} - 22 q^{22} + (14 \beta - 22) q^{23} + (3 \beta - 80) q^{25} + (6 \beta - 2) q^{26} + 28 q^{28} + ( - 27 \beta - 39) q^{29} + ( - 32 \beta - 24) q^{31} + 32 q^{32} + ( - 4 \beta - 112) q^{34} + ( - 7 \beta - 7) q^{35} + (25 \beta - 87) q^{37} + (26 \beta - 90) q^{38} + ( - 8 \beta - 8) q^{40} + (70 \beta - 8) q^{41} + (34 \beta - 102) q^{43} - 44 q^{44} + (28 \beta - 44) q^{46} + (31 \beta - 131) q^{47} + 49 q^{49} + (6 \beta - 160) q^{50} + (12 \beta - 4) q^{52} + ( - 32 \beta - 326) q^{53} + (11 \beta + 11) q^{55} + 56 q^{56} + ( - 54 \beta - 78) q^{58} + ( - 97 \beta - 175) q^{59} + ( - 12 \beta + 626) q^{61} + ( - 64 \beta - 48) q^{62} + 64 q^{64} + ( - 5 \beta - 131) q^{65} + ( - 49 \beta + 145) q^{67} + ( - 8 \beta - 224) q^{68} + ( - 14 \beta - 14) q^{70} - 528 q^{71} + ( - 95 \beta + 141) q^{73} + (50 \beta - 174) q^{74} + (52 \beta - 180) q^{76} - 77 q^{77} + ( - 32 \beta + 48) q^{79} + ( - 16 \beta - 16) q^{80} + (140 \beta - 16) q^{82} + ( - 36 \beta - 624) q^{83} + (60 \beta + 144) q^{85} + (68 \beta - 204) q^{86} - 88 q^{88} + ( - 56 \beta - 938) q^{89} + (21 \beta - 7) q^{91} + (56 \beta - 88) q^{92} + (62 \beta - 262) q^{94} + (19 \beta - 527) q^{95} + (132 \beta + 446) q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 3 q^{5} + 14 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 3 q^{5} + 14 q^{7} + 16 q^{8} - 6 q^{10} - 22 q^{11} + q^{13} + 28 q^{14} + 32 q^{16} - 114 q^{17} - 77 q^{19} - 12 q^{20} - 44 q^{22} - 30 q^{23} - 157 q^{25} + 2 q^{26} + 56 q^{28} - 105 q^{29} - 80 q^{31} + 64 q^{32} - 228 q^{34} - 21 q^{35} - 149 q^{37} - 154 q^{38} - 24 q^{40} + 54 q^{41} - 170 q^{43} - 88 q^{44} - 60 q^{46} - 231 q^{47} + 98 q^{49} - 314 q^{50} + 4 q^{52} - 684 q^{53} + 33 q^{55} + 112 q^{56} - 210 q^{58} - 447 q^{59} + 1240 q^{61} - 160 q^{62} + 128 q^{64} - 267 q^{65} + 241 q^{67} - 456 q^{68} - 42 q^{70} - 1056 q^{71} + 187 q^{73} - 298 q^{74} - 308 q^{76} - 154 q^{77} + 64 q^{79} - 48 q^{80} + 108 q^{82} - 1284 q^{83} + 348 q^{85} - 340 q^{86} - 176 q^{88} - 1932 q^{89} + 7 q^{91} - 120 q^{92} - 462 q^{94} - 1035 q^{95} + 1024 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.15207
−6.15207
2.00000 0 4.00000 −8.15207 0 7.00000 8.00000 0 −16.3041
1.2 2.00000 0 4.00000 5.15207 0 7.00000 8.00000 0 10.3041
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.4.a.y 2
3.b odd 2 1 462.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.j 2 3.b odd 2 1
1386.4.a.y 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{2} + 3T_{5} - 42 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 398 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 42 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 398 \) Copy content Toggle raw display
$17$ \( T^{2} + 114T + 3072 \) Copy content Toggle raw display
$19$ \( T^{2} + 77T - 5996 \) Copy content Toggle raw display
$23$ \( T^{2} + 30T - 8448 \) Copy content Toggle raw display
$29$ \( T^{2} + 105T - 29502 \) Copy content Toggle raw display
$31$ \( T^{2} + 80T - 43712 \) Copy content Toggle raw display
$37$ \( T^{2} + 149T - 22106 \) Copy content Toggle raw display
$41$ \( T^{2} - 54T - 216096 \) Copy content Toggle raw display
$43$ \( T^{2} + 170T - 43928 \) Copy content Toggle raw display
$47$ \( T^{2} + 231T - 29184 \) Copy content Toggle raw display
$53$ \( T^{2} + 684T + 71652 \) Copy content Toggle raw display
$59$ \( T^{2} + 447T - 366396 \) Copy content Toggle raw display
$61$ \( T^{2} - 1240 T + 378028 \) Copy content Toggle raw display
$67$ \( T^{2} - 241T - 91724 \) Copy content Toggle raw display
$71$ \( (T + 528)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 187T - 390614 \) Copy content Toggle raw display
$79$ \( T^{2} - 64T - 44288 \) Copy content Toggle raw display
$83$ \( T^{2} + 1284 T + 354816 \) Copy content Toggle raw display
$89$ \( T^{2} + 1932 T + 794388 \) Copy content Toggle raw display
$97$ \( T^{2} - 1024 T - 508868 \) Copy content Toggle raw display
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