Properties

Label 666.4.a.j
Level $666$
Weight $4$
Character orbit 666.a
Self dual yes
Analytic conductor $39.295$
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] = [666,4,Mod(1,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("666.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 666.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: \(39.2952720638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
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 \(\beta = \sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (3 \beta + 7) q^{5} + (4 \beta + 2) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (3 \beta + 7) q^{5} + (4 \beta + 2) q^{7} + 8 q^{8} + (6 \beta + 14) q^{10} + ( - 10 \beta + 22) q^{11} + (6 \beta - 16) q^{13} + (8 \beta + 4) q^{14} + 16 q^{16} + (13 \beta + 57) q^{17} + ( - 14 \beta - 6) q^{19} + (12 \beta + 28) q^{20} + ( - 20 \beta + 44) q^{22} + (9 \beta + 83) q^{23} + (42 \beta + 23) q^{25} + (12 \beta - 32) q^{26} + (16 \beta + 8) q^{28} + ( - 23 \beta + 57) q^{29} + ( - 64 \beta + 16) q^{31} + 32 q^{32} + (26 \beta + 114) q^{34} + (34 \beta + 146) q^{35} - 37 q^{37} + ( - 28 \beta - 12) q^{38} + (24 \beta + 56) q^{40} + ( - 4 \beta + 134) q^{41} + ( - 68 \beta - 240) q^{43} + ( - 40 \beta + 88) q^{44} + (18 \beta + 166) q^{46} + ( - 88 \beta + 284) q^{47} + (16 \beta - 163) q^{49} + (84 \beta + 46) q^{50} + (24 \beta - 64) q^{52} + ( - 46 \beta + 220) q^{53} + ( - 4 \beta - 176) q^{55} + (32 \beta + 16) q^{56} + ( - 46 \beta + 114) q^{58} + ( - 65 \beta + 373) q^{59} + (28 \beta + 110) q^{61} + ( - 128 \beta + 32) q^{62} + 64 q^{64} + ( - 6 \beta + 86) q^{65} + ( - 148 \beta + 150) q^{67} + (52 \beta + 228) q^{68} + (68 \beta + 292) q^{70} + 148 \beta q^{71} + ( - 118 \beta + 510) q^{73} - 74 q^{74} + ( - 56 \beta - 24) q^{76} + (68 \beta - 396) q^{77} + (78 \beta + 390) q^{79} + (48 \beta + 112) q^{80} + ( - 8 \beta + 268) q^{82} + (34 \beta - 410) q^{83} + (262 \beta + 828) q^{85} + ( - 136 \beta - 480) q^{86} + ( - 80 \beta + 176) q^{88} + ( - 127 \beta + 45) q^{89} + ( - 52 \beta + 232) q^{91} + (36 \beta + 332) q^{92} + ( - 176 \beta + 568) q^{94} + ( - 116 \beta - 504) q^{95} + (530 \beta + 64) q^{97} + (32 \beta - 326) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 14 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 14 q^{5} + 4 q^{7} + 16 q^{8} + 28 q^{10} + 44 q^{11} - 32 q^{13} + 8 q^{14} + 32 q^{16} + 114 q^{17} - 12 q^{19} + 56 q^{20} + 88 q^{22} + 166 q^{23} + 46 q^{25} - 64 q^{26} + 16 q^{28} + 114 q^{29} + 32 q^{31} + 64 q^{32} + 228 q^{34} + 292 q^{35} - 74 q^{37} - 24 q^{38} + 112 q^{40} + 268 q^{41} - 480 q^{43} + 176 q^{44} + 332 q^{46} + 568 q^{47} - 326 q^{49} + 92 q^{50} - 128 q^{52} + 440 q^{53} - 352 q^{55} + 32 q^{56} + 228 q^{58} + 746 q^{59} + 220 q^{61} + 64 q^{62} + 128 q^{64} + 172 q^{65} + 300 q^{67} + 456 q^{68} + 584 q^{70} + 1020 q^{73} - 148 q^{74} - 48 q^{76} - 792 q^{77} + 780 q^{79} + 224 q^{80} + 536 q^{82} - 820 q^{83} + 1656 q^{85} - 960 q^{86} + 352 q^{88} + 90 q^{89} + 464 q^{91} + 664 q^{92} + 1136 q^{94} - 1008 q^{95} + 128 q^{97} - 652 q^{98}+O(q^{100}) \) 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
−3.31662
3.31662
2.00000 0 4.00000 −2.94987 0 −11.2665 8.00000 0 −5.89975
1.2 2.00000 0 4.00000 16.9499 0 15.2665 8.00000 0 33.8997
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(37\) \(1\)

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 666.4.a.j 2
3.b odd 2 1 222.4.a.d 2
12.b even 2 1 1776.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.4.a.d 2 3.b odd 2 1
666.4.a.j 2 1.a even 1 1 trivial
1776.4.a.h 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 14T_{5} - 50 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(666))\). 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
$5$ \( T^{2} - 14T - 50 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 172 \) Copy content Toggle raw display
$11$ \( T^{2} - 44T - 616 \) Copy content Toggle raw display
$13$ \( T^{2} + 32T - 140 \) Copy content Toggle raw display
$17$ \( T^{2} - 114T + 1390 \) Copy content Toggle raw display
$19$ \( T^{2} + 12T - 2120 \) Copy content Toggle raw display
$23$ \( T^{2} - 166T + 5998 \) Copy content Toggle raw display
$29$ \( T^{2} - 114T - 2570 \) Copy content Toggle raw display
$31$ \( T^{2} - 32T - 44800 \) Copy content Toggle raw display
$37$ \( (T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 268T + 17780 \) Copy content Toggle raw display
$43$ \( T^{2} + 480T + 6736 \) Copy content Toggle raw display
$47$ \( T^{2} - 568T - 4528 \) Copy content Toggle raw display
$53$ \( T^{2} - 440T + 25124 \) Copy content Toggle raw display
$59$ \( T^{2} - 746T + 92654 \) Copy content Toggle raw display
$61$ \( T^{2} - 220T + 3476 \) Copy content Toggle raw display
$67$ \( T^{2} - 300T - 218444 \) Copy content Toggle raw display
$71$ \( T^{2} - 240944 \) Copy content Toggle raw display
$73$ \( T^{2} - 1020 T + 106936 \) Copy content Toggle raw display
$79$ \( T^{2} - 780T + 85176 \) Copy content Toggle raw display
$83$ \( T^{2} + 820T + 155384 \) Copy content Toggle raw display
$89$ \( T^{2} - 90T - 175394 \) Copy content Toggle raw display
$97$ \( T^{2} - 128 T - 3085804 \) Copy content Toggle raw display
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