Properties

Label 49.6.a.f
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
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] = [49,6,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 5) q^{2} + ( - 6 \beta + 6) q^{3} + ( - 9 \beta + 7) q^{4} + (10 \beta + 4) q^{5} + ( - 30 \beta + 114) q^{6} + ( - 11 \beta + 1) q^{8} + ( - 36 \beta + 297) q^{9} + (36 \beta - 120) q^{10}+ \cdots + (27468 \beta - 22104) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9} - 204 q^{10} + 396 q^{11} + 1554 q^{12} + 350 q^{13} - 1656 q^{15} + 113 q^{16} - 1800 q^{17} + 3537 q^{18} + 3266 q^{19} - 2520 q^{20}+ \cdots - 16740 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0.725083 −19.6495 −31.4743 46.7492 −14.2475 0 −46.0241 143.103 33.8970
1.2 8.27492 25.6495 36.4743 −28.7492 212.248 0 37.0241 414.897 −237.897
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.6.a.f 2
3.b odd 2 1 441.6.a.l 2
4.b odd 2 1 784.6.a.v 2
7.b odd 2 1 7.6.a.b 2
7.c even 3 2 49.6.c.d 4
7.d odd 6 2 49.6.c.e 4
21.c even 2 1 63.6.a.f 2
28.d even 2 1 112.6.a.h 2
35.c odd 2 1 175.6.a.c 2
35.f even 4 2 175.6.b.c 4
56.e even 2 1 448.6.a.u 2
56.h odd 2 1 448.6.a.w 2
77.b even 2 1 847.6.a.c 2
84.h odd 2 1 1008.6.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 7.b odd 2 1
49.6.a.f 2 1.a even 1 1 trivial
49.6.c.d 4 7.c even 3 2
49.6.c.e 4 7.d odd 6 2
63.6.a.f 2 21.c even 2 1
112.6.a.h 2 28.d even 2 1
175.6.a.c 2 35.c odd 2 1
175.6.b.c 4 35.f even 4 2
441.6.a.l 2 3.b odd 2 1
448.6.a.u 2 56.e even 2 1
448.6.a.w 2 56.h odd 2 1
784.6.a.v 2 4.b odd 2 1
847.6.a.c 2 77.b even 2 1
1008.6.a.bq 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} - 9T_{2} + 6 \) Copy content Toggle raw display
\( T_{3}^{2} - 6T_{3} - 504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 504 \) Copy content Toggle raw display
$5$ \( T^{2} - 18T - 1344 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 396T - 179904 \) Copy content Toggle raw display
$13$ \( T^{2} - 350T - 195608 \) Copy content Toggle raw display
$17$ \( T^{2} + 1800 T + 727692 \) Copy content Toggle raw display
$19$ \( T^{2} - 3266 T + 2662072 \) Copy content Toggle raw display
$23$ \( T^{2} - 2088 T - 3507456 \) Copy content Toggle raw display
$29$ \( T^{2} - 6696 T + 10304172 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 4155200 \) Copy content Toggle raw display
$37$ \( T^{2} - 6232 T + 5554156 \) Copy content Toggle raw display
$41$ \( T^{2} - 6048 T - 7848036 \) Copy content Toggle raw display
$43$ \( T^{2} + 3020 T - 324400352 \) Copy content Toggle raw display
$47$ \( T^{2} + 11700 T - 165954432 \) Copy content Toggle raw display
$53$ \( T^{2} - 9468 T + 21794244 \) Copy content Toggle raw display
$59$ \( T^{2} - 43938 T + 422751336 \) Copy content Toggle raw display
$61$ \( T^{2} - 64754 T + 719128816 \) Copy content Toggle raw display
$67$ \( T^{2} - 24784 T + 99708976 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2121099264 \) Copy content Toggle raw display
$73$ \( T^{2} + 17452 T - 317520812 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2508546944 \) Copy content Toggle raw display
$83$ \( T^{2} + 117558 T - 79919784 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5252421468 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1000631156 \) Copy content Toggle raw display
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