Properties

Label 484.6.a.d
Level $484$
Weight $6$
Character orbit 484.a
Self dual yes
Analytic conductor $77.626$
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] = [484,6,Mod(1,484)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(484, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("484.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 484.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: \(77.6257687895\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 44)
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 \(\beta = 4\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 3) q^{3} + (2 \beta - 11) q^{5} + (3 \beta - 134) q^{7} + ( - 6 \beta + 262) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 3) q^{3} + (2 \beta - 11) q^{5} + (3 \beta - 134) q^{7} + ( - 6 \beta + 262) q^{9} + (5 \beta - 616) q^{13} + ( - 17 \beta + 1025) q^{15} + ( - 47 \beta + 62) q^{17} + (31 \beta - 972) q^{19} + ( - 143 \beta + 1890) q^{21} + ( - 59 \beta + 1673) q^{23} + ( - 44 \beta - 1020) q^{25} + (37 \beta - 3033) q^{27} + (78 \beta + 3288) q^{29} + ( - 371 \beta + 1249) q^{31} + ( - 301 \beta + 4450) q^{35} + (94 \beta - 7337) q^{37} + ( - 631 \beta + 4328) q^{39} + (477 \beta + 3252) q^{41} + ( - 644 \beta - 5814) q^{43} + (590 \beta - 8834) q^{45} + ( - 522 \beta + 18408) q^{47} + ( - 804 \beta + 5613) q^{49} + (203 \beta - 23498) q^{51} + ( - 1150 \beta - 1646) q^{53} + ( - 1065 \beta + 18292) q^{57} + (585 \beta + 6063) q^{59} + (562 \beta - 36564) q^{61} + (1590 \beta - 44036) q^{63} + ( - 1287 \beta + 11736) q^{65} + (1733 \beta + 14667) q^{67} + (1850 \beta - 34283) q^{69} + ( - 951 \beta - 23061) q^{71} + ( - 799 \beta - 4120) q^{73} + ( - 888 \beta - 18764) q^{75} + (852 \beta + 7390) q^{79} + ( - 1686 \beta - 36215) q^{81} + ( - 802 \beta + 37282) q^{83} + (641 \beta - 47306) q^{85} + (3054 \beta + 28824) q^{87} + ( - 650 \beta - 16849) q^{89} + ( - 2518 \beta + 89984) q^{91} + (2362 \beta - 187763) q^{93} + ( - 2285 \beta + 41444) q^{95} + (3188 \beta + 63081) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 22 q^{5} - 268 q^{7} + 524 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 22 q^{5} - 268 q^{7} + 524 q^{9} - 1232 q^{13} + 2050 q^{15} + 124 q^{17} - 1944 q^{19} + 3780 q^{21} + 3346 q^{23} - 2040 q^{25} - 6066 q^{27} + 6576 q^{29} + 2498 q^{31} + 8900 q^{35} - 14674 q^{37} + 8656 q^{39} + 6504 q^{41} - 11628 q^{43} - 17668 q^{45} + 36816 q^{47} + 11226 q^{49} - 46996 q^{51} - 3292 q^{53} + 36584 q^{57} + 12126 q^{59} - 73128 q^{61} - 88072 q^{63} + 23472 q^{65} + 29334 q^{67} - 68566 q^{69} - 46122 q^{71} - 8240 q^{73} - 37528 q^{75} + 14780 q^{79} - 72430 q^{81} + 74564 q^{83} - 94612 q^{85} + 57648 q^{87} - 33698 q^{89} + 179968 q^{91} - 375526 q^{93} + 82888 q^{95} + 126162 q^{97}+O(q^{100}) \) 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
−5.56776
5.56776
0 −25.2711 0 −55.5421 0 −200.813 0 395.626 0
1.2 0 19.2711 0 33.5421 0 −67.1868 0 128.374 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 484.6.a.d 2
11.b odd 2 1 44.6.a.b 2
33.d even 2 1 396.6.a.f 2
44.c even 2 1 176.6.a.g 2
55.d odd 2 1 1100.6.a.b 2
55.e even 4 2 1100.6.b.c 4
88.b odd 2 1 704.6.a.n 2
88.g even 2 1 704.6.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.6.a.b 2 11.b odd 2 1
176.6.a.g 2 44.c even 2 1
396.6.a.f 2 33.d even 2 1
484.6.a.d 2 1.a even 1 1 trivial
704.6.a.m 2 88.g even 2 1
704.6.a.n 2 88.b odd 2 1
1100.6.a.b 2 55.d odd 2 1
1100.6.b.c 4 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(484))\):

\( T_{3}^{2} + 6T_{3} - 487 \) Copy content Toggle raw display
\( T_{7}^{2} + 268T_{7} + 13492 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T - 487 \) Copy content Toggle raw display
$5$ \( T^{2} + 22T - 1863 \) Copy content Toggle raw display
$7$ \( T^{2} + 268T + 13492 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1232 T + 367056 \) Copy content Toggle raw display
$17$ \( T^{2} - 124 T - 1091820 \) Copy content Toggle raw display
$19$ \( T^{2} + 1944 T + 468128 \) Copy content Toggle raw display
$23$ \( T^{2} - 3346 T + 1072353 \) Copy content Toggle raw display
$29$ \( T^{2} - 6576 T + 7793280 \) Copy content Toggle raw display
$31$ \( T^{2} - 2498 T - 66709935 \) Copy content Toggle raw display
$37$ \( T^{2} + 14674 T + 49448913 \) Copy content Toggle raw display
$41$ \( T^{2} - 6504 T - 102278880 \) Copy content Toggle raw display
$43$ \( T^{2} + 11628 T - 171906460 \) Copy content Toggle raw display
$47$ \( T^{2} - 36816 T + 203702400 \) Copy content Toggle raw display
$53$ \( T^{2} + 3292 T - 653250684 \) Copy content Toggle raw display
$59$ \( T^{2} - 12126 T - 132983631 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1180267472 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 1274510455 \) Copy content Toggle raw display
$71$ \( T^{2} + 46122 T + 83226825 \) Copy content Toggle raw display
$73$ \( T^{2} + 8240 T - 299672496 \) Copy content Toggle raw display
$79$ \( T^{2} - 14780 T - 305436284 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 1070918340 \) Copy content Toggle raw display
$89$ \( T^{2} + 33698 T + 74328801 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1061806063 \) Copy content Toggle raw display
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