Properties

Label 14.6.a.a
Level $14$
Weight $6$
Character orbit 14.a
Self dual yes
Analytic conductor $2.245$
Analytic rank $0$
Dimension $1$
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] = [14,6,Mod(1,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, 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("14.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 14.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: \(2.24537347738\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 10 q^{3} + 16 q^{4} + 84 q^{5} - 40 q^{6} + 49 q^{7} - 64 q^{8} - 143 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 10 q^{3} + 16 q^{4} + 84 q^{5} - 40 q^{6} + 49 q^{7} - 64 q^{8} - 143 q^{9} - 336 q^{10} - 336 q^{11} + 160 q^{12} + 584 q^{13} - 196 q^{14} + 840 q^{15} + 256 q^{16} - 1458 q^{17} + 572 q^{18} + 470 q^{19} + 1344 q^{20} + 490 q^{21} + 1344 q^{22} - 4200 q^{23} - 640 q^{24} + 3931 q^{25} - 2336 q^{26} - 3860 q^{27} + 784 q^{28} + 4866 q^{29} - 3360 q^{30} - 7372 q^{31} - 1024 q^{32} - 3360 q^{33} + 5832 q^{34} + 4116 q^{35} - 2288 q^{36} + 14330 q^{37} - 1880 q^{38} + 5840 q^{39} - 5376 q^{40} + 6222 q^{41} - 1960 q^{42} + 3704 q^{43} - 5376 q^{44} - 12012 q^{45} + 16800 q^{46} - 1812 q^{47} + 2560 q^{48} + 2401 q^{49} - 15724 q^{50} - 14580 q^{51} + 9344 q^{52} - 37242 q^{53} + 15440 q^{54} - 28224 q^{55} - 3136 q^{56} + 4700 q^{57} - 19464 q^{58} + 34302 q^{59} + 13440 q^{60} + 24476 q^{61} + 29488 q^{62} - 7007 q^{63} + 4096 q^{64} + 49056 q^{65} + 13440 q^{66} - 17452 q^{67} - 23328 q^{68} - 42000 q^{69} - 16464 q^{70} + 28224 q^{71} + 9152 q^{72} + 3602 q^{73} - 57320 q^{74} + 39310 q^{75} + 7520 q^{76} - 16464 q^{77} - 23360 q^{78} + 42872 q^{79} + 21504 q^{80} - 3851 q^{81} - 24888 q^{82} - 35202 q^{83} + 7840 q^{84} - 122472 q^{85} - 14816 q^{86} + 48660 q^{87} + 21504 q^{88} + 26730 q^{89} + 48048 q^{90} + 28616 q^{91} - 67200 q^{92} - 73720 q^{93} + 7248 q^{94} + 39480 q^{95} - 10240 q^{96} - 16978 q^{97} - 9604 q^{98} + 48048 q^{99}+O(q^{100}) \) 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
0
−4.00000 10.0000 16.0000 84.0000 −40.0000 49.0000 −64.0000 −143.000 −336.000
\(n\): 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 14.6.a.a 1
3.b odd 2 1 126.6.a.f 1
4.b odd 2 1 112.6.a.c 1
5.b even 2 1 350.6.a.i 1
5.c odd 4 2 350.6.c.d 2
7.b odd 2 1 98.6.a.a 1
7.c even 3 2 98.6.c.c 2
7.d odd 6 2 98.6.c.d 2
8.b even 2 1 448.6.a.e 1
8.d odd 2 1 448.6.a.l 1
12.b even 2 1 1008.6.a.b 1
21.c even 2 1 882.6.a.x 1
28.d even 2 1 784.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 1.a even 1 1 trivial
98.6.a.a 1 7.b odd 2 1
98.6.c.c 2 7.c even 3 2
98.6.c.d 2 7.d odd 6 2
112.6.a.c 1 4.b odd 2 1
126.6.a.f 1 3.b odd 2 1
350.6.a.i 1 5.b even 2 1
350.6.c.d 2 5.c odd 4 2
448.6.a.e 1 8.b even 2 1
448.6.a.l 1 8.d odd 2 1
784.6.a.i 1 28.d even 2 1
882.6.a.x 1 21.c even 2 1
1008.6.a.b 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 10 \) Copy content Toggle raw display
$5$ \( T - 84 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 336 \) Copy content Toggle raw display
$13$ \( T - 584 \) Copy content Toggle raw display
$17$ \( T + 1458 \) Copy content Toggle raw display
$19$ \( T - 470 \) Copy content Toggle raw display
$23$ \( T + 4200 \) Copy content Toggle raw display
$29$ \( T - 4866 \) Copy content Toggle raw display
$31$ \( T + 7372 \) Copy content Toggle raw display
$37$ \( T - 14330 \) Copy content Toggle raw display
$41$ \( T - 6222 \) Copy content Toggle raw display
$43$ \( T - 3704 \) Copy content Toggle raw display
$47$ \( T + 1812 \) Copy content Toggle raw display
$53$ \( T + 37242 \) Copy content Toggle raw display
$59$ \( T - 34302 \) Copy content Toggle raw display
$61$ \( T - 24476 \) Copy content Toggle raw display
$67$ \( T + 17452 \) Copy content Toggle raw display
$71$ \( T - 28224 \) Copy content Toggle raw display
$73$ \( T - 3602 \) Copy content Toggle raw display
$79$ \( T - 42872 \) Copy content Toggle raw display
$83$ \( T + 35202 \) Copy content Toggle raw display
$89$ \( T - 26730 \) Copy content Toggle raw display
$97$ \( T + 16978 \) Copy content Toggle raw display
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