Properties

Label 784.6.a.q
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
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] = [784,6,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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 \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{3} + ( - 5 \beta - 21) q^{5} + (14 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{3} + ( - 5 \beta - 21) q^{5} + (14 \beta - 1) q^{9} + (14 \beta + 358) q^{11} + (17 \beta + 357) q^{13} + (56 \beta + 1112) q^{15} + (54 \beta + 672) q^{17} + (93 \beta - 973) q^{19} + ( - 112 \beta + 896) q^{23} + (210 \beta + 2141) q^{25} + (146 \beta - 994) q^{27} + ( - 546 \beta - 600) q^{29} + ( - 446 \beta - 3402) q^{31} + ( - 456 \beta - 5208) q^{33} + ( - 154 \beta + 7320) q^{37} + ( - 476 \beta - 5780) q^{39} + (442 \beta - 3948) q^{41} + ( - 518 \beta - 262) q^{43} + ( - 289 \beta - 13489) q^{45} + ( - 746 \beta - 9198) q^{47} + ( - 1050 \beta - 15126) q^{51} + (1008 \beta + 22566) q^{53} + ( - 2084 \beta - 21028) q^{55} + (322 \beta - 11138) q^{57} + (365 \beta + 11291) q^{59} + ( - 1117 \beta + 26411) q^{61} + ( - 2142 \beta - 23902) q^{65} + ( - 224 \beta - 4924) q^{67} + ( - 112 \beta + 15344) q^{69} + ( - 4060 \beta + 420) q^{71} + (1204 \beta + 61026) q^{73} + ( - 3611 \beta - 55517) q^{75} + (1372 \beta - 15852) q^{79} + ( - 3430 \beta - 20977) q^{81} + (4185 \beta + 18487) q^{83} + ( - 4494 \beta - 66222) q^{85} + (4422 \beta + 109578) q^{87} + ( - 1512 \beta + 105294) q^{89} + (6524 \beta + 109892) q^{93} + (2912 \beta - 69312) q^{95} + ( - 8882 \beta + 22120) q^{97} + (4998 \beta + 37470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 42 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} - 42 q^{5} - 2 q^{9} + 716 q^{11} + 714 q^{13} + 2224 q^{15} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} - 10416 q^{33} + 14640 q^{37} - 11560 q^{39} - 7896 q^{41} - 524 q^{43} - 26978 q^{45} - 18396 q^{47} - 30252 q^{51} + 45132 q^{53} - 42056 q^{55} - 22276 q^{57} + 22582 q^{59} + 52822 q^{61} - 47804 q^{65} - 9848 q^{67} + 30688 q^{69} + 840 q^{71} + 122052 q^{73} - 111034 q^{75} - 31704 q^{79} - 41954 q^{81} + 36974 q^{83} - 132444 q^{85} + 219156 q^{87} + 210588 q^{89} + 219784 q^{93} - 138624 q^{95} + 44240 q^{97} + 74940 q^{99}+O(q^{100}) \) 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
7.44622
−6.44622
0 −20.8924 0 −90.4622 0 0 0 193.494 0
1.2 0 6.89244 0 48.4622 0 0 0 −195.494 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 784.6.a.q 2
4.b odd 2 1 392.6.a.e 2
7.b odd 2 1 112.6.a.j 2
21.c even 2 1 1008.6.a.bi 2
28.d even 2 1 56.6.a.d 2
28.f even 6 2 392.6.i.k 4
28.g odd 6 2 392.6.i.h 4
56.e even 2 1 448.6.a.x 2
56.h odd 2 1 448.6.a.r 2
84.h odd 2 1 504.6.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 28.d even 2 1
112.6.a.j 2 7.b odd 2 1
392.6.a.e 2 4.b odd 2 1
392.6.i.h 4 28.g odd 6 2
392.6.i.k 4 28.f even 6 2
448.6.a.r 2 56.h odd 2 1
448.6.a.x 2 56.e even 2 1
504.6.a.m 2 84.h odd 2 1
784.6.a.q 2 1.a even 1 1 trivial
1008.6.a.bi 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 14T_{3} - 144 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 14T - 144 \) Copy content Toggle raw display
$5$ \( T^{2} + 42T - 4384 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 716T + 90336 \) Copy content Toggle raw display
$13$ \( T^{2} - 714T + 71672 \) Copy content Toggle raw display
$17$ \( T^{2} - 1344 T - 111204 \) Copy content Toggle raw display
$19$ \( T^{2} + 1946 T - 722528 \) Copy content Toggle raw display
$23$ \( T^{2} - 1792 T - 1618176 \) Copy content Toggle raw display
$29$ \( T^{2} + 1200 T - 57176388 \) Copy content Toggle raw display
$31$ \( T^{2} + 6804 T - 26817184 \) Copy content Toggle raw display
$37$ \( T^{2} - 14640 T + 49005212 \) Copy content Toggle raw display
$41$ \( T^{2} + 7896 T - 22118548 \) Copy content Toggle raw display
$43$ \( T^{2} + 524 T - 51717888 \) Copy content Toggle raw display
$47$ \( T^{2} + 18396 T - 22804384 \) Copy content Toggle raw display
$53$ \( T^{2} - 45132 T + 313124004 \) Copy content Toggle raw display
$59$ \( T^{2} - 22582 T + 101774256 \) Copy content Toggle raw display
$61$ \( T^{2} - 52822 T + 456736944 \) Copy content Toggle raw display
$67$ \( T^{2} + 9848 T + 14561808 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 3181158400 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3444396788 \) Copy content Toggle raw display
$79$ \( T^{2} + 31704 T - 112014208 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3038476256 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10645600644 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 14736460932 \) Copy content Toggle raw display
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