Properties

Label 441.6.a.n
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
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] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, 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("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
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 \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + (10 \beta + 19) q^{5} + (24 \beta - 48) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + (10 \beta + 19) q^{5} + (24 \beta - 48) q^{8} + ( - 29 \beta - 389) q^{10} + ( - 23 \beta - 212) q^{11} + (28 \beta - 462) q^{13} + ( - 40 \beta - 1032) q^{16} + ( - 132 \beta + 1173) q^{17} + (277 \beta - 180) q^{19} + (98 \beta + 854) q^{20} + (235 \beta + 1063) q^{22} + (69 \beta + 6) q^{23} + (380 \beta + 936) q^{25} + (434 \beta - 574) q^{26} + (700 \beta + 3526) q^{29} + ( - 715 \beta + 1774) q^{31} + (304 \beta + 4048) q^{32} + ( - 1041 \beta + 3711) q^{34} + ( - 790 \beta + 5545) q^{37} + ( - 97 \beta - 10069) q^{38} + ( - 24 \beta + 7968) q^{40} + ( - 868 \beta - 1750) q^{41} + (1344 \beta - 6340) q^{43} + ( - 562 \beta - 2974) q^{44} + ( - 75 \beta - 2559) q^{46} + ( - 1635 \beta + 11478) q^{47} + ( - 1316 \beta - 14996) q^{50} + ( - 756 \beta - 700) q^{52} + (1818 \beta - 1521) q^{53} + ( - 2557 \beta - 12538) q^{55} + ( - 4226 \beta - 29426) q^{58} + ( - 531 \beta + 32904) q^{59} + ( - 4154 \beta - 21243) q^{61} + ( - 1059 \beta + 24681) q^{62} + ( - 3072 \beta + 17728) q^{64} + ( - 4088 \beta + 1582) q^{65} + (919 \beta + 21156) q^{67} + (1554 \beta - 2730) q^{68} + (2184 \beta + 1104) q^{71} + ( - 7372 \beta - 25253) q^{73} + ( - 4755 \beta + 23685) q^{74} + (1302 \beta + 19418) q^{76} + ( - 5193 \beta + 4502) q^{79} + ( - 11080 \beta - 34408) q^{80} + (2618 \beta + 33866) q^{82} + (4536 \beta + 52164) q^{83} + (9222 \beta - 26553) q^{85} + (4996 \beta - 43388) q^{86} + ( - 3984 \beta - 10248) q^{88} + ( - 9356 \beta + 13333) q^{89} + (426 \beta + 5142) q^{92} + ( - 9843 \beta + 49017) q^{94} + (3463 \beta + 99070) q^{95} + ( - 196 \beta + 104566) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100}) \) 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
3.54138
−2.54138
−7.08276 0 18.1655 79.8276 0 0 97.9863 0 −565.400
1.2 5.08276 0 −6.16553 −41.8276 0 0 −193.986 0 −212.600
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 441.6.a.n 2
3.b odd 2 1 49.6.a.d 2
7.b odd 2 1 441.6.a.m 2
7.c even 3 2 63.6.e.d 4
12.b even 2 1 784.6.a.ba 2
21.c even 2 1 49.6.a.e 2
21.g even 6 2 49.6.c.f 4
21.h odd 6 2 7.6.c.a 4
84.h odd 2 1 784.6.a.t 2
84.n even 6 2 112.6.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 21.h odd 6 2
49.6.a.d 2 3.b odd 2 1
49.6.a.e 2 21.c even 2 1
49.6.c.f 4 21.g even 6 2
63.6.e.d 4 7.c even 3 2
112.6.i.c 4 84.n even 6 2
441.6.a.m 2 7.b odd 2 1
441.6.a.n 2 1.a even 1 1 trivial
784.6.a.t 2 84.h odd 2 1
784.6.a.ba 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} + 2T_{2} - 36 \) Copy content Toggle raw display
\( T_{5}^{2} - 38T_{5} - 3339 \) Copy content Toggle raw display
\( T_{13}^{2} + 924T_{13} + 184436 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 36 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 38T - 3339 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 424T + 25371 \) Copy content Toggle raw display
$13$ \( T^{2} + 924T + 184436 \) Copy content Toggle raw display
$17$ \( T^{2} - 2346 T + 731241 \) Copy content Toggle raw display
$19$ \( T^{2} + 360 T - 2806573 \) Copy content Toggle raw display
$23$ \( T^{2} - 12T - 176121 \) Copy content Toggle raw display
$29$ \( T^{2} - 7052 T - 5697324 \) Copy content Toggle raw display
$31$ \( T^{2} - 3548 T - 15768249 \) Copy content Toggle raw display
$37$ \( T^{2} - 11090 T + 7655325 \) Copy content Toggle raw display
$41$ \( T^{2} + 3500 T - 24814188 \) Copy content Toggle raw display
$43$ \( T^{2} + 12680 T - 26638832 \) Copy content Toggle raw display
$47$ \( T^{2} - 22956 T + 32835159 \) Copy content Toggle raw display
$53$ \( T^{2} + 3042 T - 119976147 \) Copy content Toggle raw display
$59$ \( T^{2} - 65808 T + 1072240659 \) Copy content Toggle raw display
$61$ \( T^{2} + 42486 T - 187196443 \) Copy content Toggle raw display
$67$ \( T^{2} - 42312 T + 416327579 \) Copy content Toggle raw display
$71$ \( T^{2} - 2208 T - 175265856 \) Copy content Toggle raw display
$73$ \( T^{2} + 50506 T - 1373102199 \) Copy content Toggle raw display
$79$ \( T^{2} - 9004 T - 977520209 \) Copy content Toggle raw display
$83$ \( T^{2} - 104328 T + 1959796944 \) Copy content Toggle raw display
$89$ \( T^{2} - 26666 T - 3061016343 \) Copy content Toggle raw display
$97$ \( T^{2} - 209132 T + 10932626964 \) Copy content Toggle raw display
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