Properties

Label 84.6.a.c
Level $84$
Weight $6$
Character orbit 84.a
Self dual yes
Analytic conductor $13.472$
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] = [84,6,Mod(1,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("84.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 84.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: \(13.4722408643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5569}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{5569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + ( - \beta - 3) q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + ( - \beta - 3) q^{5} - 49 q^{7} + 81 q^{9} + (7 \beta - 45) q^{11} + ( - 6 \beta + 384) q^{13} + (9 \beta + 27) q^{15} + (\beta + 963) q^{17} + ( - 24 \beta + 1124) q^{19} + 441 q^{21} + (\beta + 3177) q^{23} + (6 \beta + 2453) q^{25} - 729 q^{27} + ( - 40 \beta + 5286) q^{29} + (72 \beta - 1656) q^{31} + ( - 63 \beta + 405) q^{33} + (49 \beta + 147) q^{35} + (150 \beta + 1052) q^{37} + (54 \beta - 3456) q^{39} + ( - 33 \beta + 633) q^{41} + ( - 240 \beta - 2884) q^{43} + ( - 81 \beta - 243) q^{45} + ( - 162 \beta + 7806) q^{47} + 2401 q^{49} + ( - 9 \beta - 8667) q^{51} + (234 \beta + 8256) q^{53} + (24 \beta - 38848) q^{55} + (216 \beta - 10116) q^{57} + (50 \beta - 6570) q^{59} + (264 \beta - 2898) q^{61} - 3969 q^{63} + ( - 366 \beta + 32262) q^{65} + ( - 462 \beta - 28058) q^{67} + ( - 9 \beta - 28593) q^{69} + (895 \beta + 5511) q^{71} + ( - 174 \beta - 42692) q^{73} + ( - 54 \beta - 22077) q^{75} + ( - 343 \beta + 2205) q^{77} + (78 \beta - 9810) q^{79} + 6561 q^{81} + ( - 320 \beta - 22212) q^{83} + ( - 966 \beta - 8458) q^{85} + (360 \beta - 47574) q^{87} + (399 \beta + 105609) q^{89} + (294 \beta - 18816) q^{91} + ( - 648 \beta + 14904) q^{93} + ( - 1052 \beta + 130284) q^{95} + (1614 \beta + 22432) q^{97} + (567 \beta - 3645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33} + 294 q^{35} + 2104 q^{37} - 6912 q^{39} + 1266 q^{41} - 5768 q^{43} - 486 q^{45} + 15612 q^{47} + 4802 q^{49} - 17334 q^{51} + 16512 q^{53} - 77696 q^{55} - 20232 q^{57} - 13140 q^{59} - 5796 q^{61} - 7938 q^{63} + 64524 q^{65} - 56116 q^{67} - 57186 q^{69} + 11022 q^{71} - 85384 q^{73} - 44154 q^{75} + 4410 q^{77} - 19620 q^{79} + 13122 q^{81} - 44424 q^{83} - 16916 q^{85} - 95148 q^{87} + 211218 q^{89} - 37632 q^{91} + 29808 q^{93} + 260568 q^{95} + 44864 q^{97} - 7290 q^{99}+O(q^{100}) \) 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
37.8129
−36.8129
0 −9.00000 0 −77.6257 0 −49.0000 0 81.0000 0
1.2 0 −9.00000 0 71.6257 0 −49.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 84.6.a.c 2
3.b odd 2 1 252.6.a.h 2
4.b odd 2 1 336.6.a.x 2
7.b odd 2 1 588.6.a.k 2
7.c even 3 2 588.6.i.l 4
7.d odd 6 2 588.6.i.i 4
12.b even 2 1 1008.6.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.c 2 1.a even 1 1 trivial
252.6.a.h 2 3.b odd 2 1
336.6.a.x 2 4.b odd 2 1
588.6.a.k 2 7.b odd 2 1
588.6.i.i 4 7.d odd 6 2
588.6.i.l 4 7.c even 3 2
1008.6.a.bo 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} - 5560 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(84))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T - 5560 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 90T - 270856 \) Copy content Toggle raw display
$13$ \( T^{2} - 768T - 53028 \) Copy content Toggle raw display
$17$ \( T^{2} - 1926 T + 921800 \) Copy content Toggle raw display
$19$ \( T^{2} - 2248 T - 1944368 \) Copy content Toggle raw display
$23$ \( T^{2} - 6354 T + 10087760 \) Copy content Toggle raw display
$29$ \( T^{2} - 10572 T + 19031396 \) Copy content Toggle raw display
$31$ \( T^{2} + 3312 T - 26127360 \) Copy content Toggle raw display
$37$ \( T^{2} - 2104 T - 124195796 \) Copy content Toggle raw display
$41$ \( T^{2} - 1266 T - 5663952 \) Copy content Toggle raw display
$43$ \( T^{2} + 5768 T - 312456944 \) Copy content Toggle raw display
$47$ \( T^{2} - 15612 T - 85219200 \) Copy content Toggle raw display
$53$ \( T^{2} - 16512 T - 236774628 \) Copy content Toggle raw display
$59$ \( T^{2} + 13140 T + 29242400 \) Copy content Toggle raw display
$61$ \( T^{2} + 5796 T - 379738620 \) Copy content Toggle raw display
$67$ \( T^{2} + 56116 T - 401418272 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 4430537104 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1653999820 \) Copy content Toggle raw display
$79$ \( T^{2} + 19620 T + 62354304 \) Copy content Toggle raw display
$83$ \( T^{2} + 44424 T - 76892656 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10266670512 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 14004028100 \) Copy content Toggle raw display
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