Properties

Label 1386.4.a.v
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{113}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - \beta - 7) q^{5} + 7 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + ( - \beta - 7) q^{5} + 7 q^{7} + 8 q^{8} + ( - 2 \beta - 14) q^{10} + 11 q^{11} + ( - 2 \beta - 52) q^{13} + 14 q^{14} + 16 q^{16} + (5 \beta + 11) q^{17} + ( - 11 \beta + 1) q^{19} + ( - 4 \beta - 28) q^{20} + 22 q^{22} + ( - 12 \beta + 20) q^{23} + (14 \beta + 37) q^{25} + ( - 4 \beta - 104) q^{26} + 28 q^{28} + (14 \beta + 48) q^{29} + ( - 9 \beta + 119) q^{31} + 32 q^{32} + (10 \beta + 22) q^{34} + ( - 7 \beta - 49) q^{35} + ( - 10 \beta + 196) q^{37} + ( - 22 \beta + 2) q^{38} + ( - 8 \beta - 56) q^{40} + (25 \beta + 55) q^{41} + (20 \beta - 144) q^{43} + 44 q^{44} + ( - 24 \beta + 40) q^{46} + ( - 3 \beta + 413) q^{47} + 49 q^{49} + (28 \beta + 74) q^{50} + ( - 8 \beta - 208) q^{52} + ( - 22 \beta + 236) q^{53} + ( - 11 \beta - 77) q^{55} + 56 q^{56} + (28 \beta + 96) q^{58} + ( - 70 \beta - 50) q^{59} + (38 \beta - 252) q^{61} + ( - 18 \beta + 238) q^{62} + 64 q^{64} + (66 \beta + 590) q^{65} + ( - 84 \beta + 160) q^{67} + (20 \beta + 44) q^{68} + ( - 14 \beta - 98) q^{70} + (38 \beta + 454) q^{71} + ( - 7 \beta - 869) q^{73} + ( - 20 \beta + 392) q^{74} + ( - 44 \beta + 4) q^{76} + 77 q^{77} + (60 \beta + 492) q^{79} + ( - 16 \beta - 112) q^{80} + (50 \beta + 110) q^{82} + ( - 11 \beta + 121) q^{83} + ( - 46 \beta - 642) q^{85} + (40 \beta - 288) q^{86} + 88 q^{88} + (12 \beta + 1442) q^{89} + ( - 14 \beta - 364) q^{91} + ( - 48 \beta + 80) q^{92} + ( - 6 \beta + 826) q^{94} + (76 \beta + 1236) q^{95} + (88 \beta + 314) q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{5} + 14 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{5} + 14 q^{7} + 16 q^{8} - 28 q^{10} + 22 q^{11} - 104 q^{13} + 28 q^{14} + 32 q^{16} + 22 q^{17} + 2 q^{19} - 56 q^{20} + 44 q^{22} + 40 q^{23} + 74 q^{25} - 208 q^{26} + 56 q^{28} + 96 q^{29} + 238 q^{31} + 64 q^{32} + 44 q^{34} - 98 q^{35} + 392 q^{37} + 4 q^{38} - 112 q^{40} + 110 q^{41} - 288 q^{43} + 88 q^{44} + 80 q^{46} + 826 q^{47} + 98 q^{49} + 148 q^{50} - 416 q^{52} + 472 q^{53} - 154 q^{55} + 112 q^{56} + 192 q^{58} - 100 q^{59} - 504 q^{61} + 476 q^{62} + 128 q^{64} + 1180 q^{65} + 320 q^{67} + 88 q^{68} - 196 q^{70} + 908 q^{71} - 1738 q^{73} + 784 q^{74} + 8 q^{76} + 154 q^{77} + 984 q^{79} - 224 q^{80} + 220 q^{82} + 242 q^{83} - 1284 q^{85} - 576 q^{86} + 176 q^{88} + 2884 q^{89} - 728 q^{91} + 160 q^{92} + 1652 q^{94} + 2472 q^{95} + 628 q^{97} + 196 q^{98}+O(q^{100}) \) 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
5.81507
−4.81507
2.00000 0 4.00000 −17.6301 0 7.00000 8.00000 0 −35.2603
1.2 2.00000 0 4.00000 3.63015 0 7.00000 8.00000 0 7.26029
\(n\): 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.v 2
3.b odd 2 1 462.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.l 2 3.b odd 2 1
1386.4.a.v 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{2} + 14T_{5} - 64 \) Copy content Toggle raw display
\( T_{13}^{2} + 104T_{13} + 2252 \) 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
$5$ \( T^{2} + 14T - 64 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 104T + 2252 \) Copy content Toggle raw display
$17$ \( T^{2} - 22T - 2704 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 13672 \) Copy content Toggle raw display
$23$ \( T^{2} - 40T - 15872 \) Copy content Toggle raw display
$29$ \( T^{2} - 96T - 19844 \) Copy content Toggle raw display
$31$ \( T^{2} - 238T + 5008 \) Copy content Toggle raw display
$37$ \( T^{2} - 392T + 27116 \) Copy content Toggle raw display
$41$ \( T^{2} - 110T - 67600 \) Copy content Toggle raw display
$43$ \( T^{2} + 288T - 24464 \) Copy content Toggle raw display
$47$ \( T^{2} - 826T + 169552 \) Copy content Toggle raw display
$53$ \( T^{2} - 472T + 1004 \) Copy content Toggle raw display
$59$ \( T^{2} + 100T - 551200 \) Copy content Toggle raw display
$61$ \( T^{2} + 504T - 99668 \) Copy content Toggle raw display
$67$ \( T^{2} - 320T - 771728 \) Copy content Toggle raw display
$71$ \( T^{2} - 908T + 42944 \) Copy content Toggle raw display
$73$ \( T^{2} + 1738 T + 749624 \) Copy content Toggle raw display
$79$ \( T^{2} - 984T - 164736 \) Copy content Toggle raw display
$83$ \( T^{2} - 242T + 968 \) Copy content Toggle raw display
$89$ \( T^{2} - 2884 T + 2063092 \) Copy content Toggle raw display
$97$ \( T^{2} - 628T - 776476 \) Copy content Toggle raw display
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