Properties

Label 14.8.a.b
Level $14$
Weight $8$
Character orbit 14.a
Self dual yes
Analytic conductor $4.373$
Analytic rank $1$
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] = [14,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(4.37339035678\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 8 q^{2} - 66 q^{3} + 64 q^{4} - 400 q^{5} - 528 q^{6} - 343 q^{7} + 512 q^{8} + 2169 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} - 66 q^{3} + 64 q^{4} - 400 q^{5} - 528 q^{6} - 343 q^{7} + 512 q^{8} + 2169 q^{9} - 3200 q^{10} + 40 q^{11} - 4224 q^{12} - 4452 q^{13} - 2744 q^{14} + 26400 q^{15} + 4096 q^{16} + 36502 q^{17} + 17352 q^{18} - 46222 q^{19} - 25600 q^{20} + 22638 q^{21} + 320 q^{22} - 105200 q^{23} - 33792 q^{24} + 81875 q^{25} - 35616 q^{26} + 1188 q^{27} - 21952 q^{28} - 126334 q^{29} + 211200 q^{30} - 170964 q^{31} + 32768 q^{32} - 2640 q^{33} + 292016 q^{34} + 137200 q^{35} + 138816 q^{36} + 20954 q^{37} - 369776 q^{38} + 293832 q^{39} - 204800 q^{40} + 318486 q^{41} + 181104 q^{42} + 77744 q^{43} + 2560 q^{44} - 867600 q^{45} - 841600 q^{46} + 703716 q^{47} - 270336 q^{48} + 117649 q^{49} + 655000 q^{50} - 2409132 q^{51} - 284928 q^{52} + 1603278 q^{53} + 9504 q^{54} - 16000 q^{55} - 175616 q^{56} + 3050652 q^{57} - 1010672 q^{58} - 1171894 q^{59} + 1689600 q^{60} - 2068872 q^{61} - 1367712 q^{62} - 743967 q^{63} + 262144 q^{64} + 1780800 q^{65} - 21120 q^{66} - 994268 q^{67} + 2336128 q^{68} + 6943200 q^{69} + 1097600 q^{70} + 33280 q^{71} + 1110528 q^{72} - 2971454 q^{73} + 167632 q^{74} - 5403750 q^{75} - 2958208 q^{76} - 13720 q^{77} + 2350656 q^{78} - 2376168 q^{79} - 1638400 q^{80} - 4822011 q^{81} + 2547888 q^{82} - 2122358 q^{83} + 1448832 q^{84} - 14600800 q^{85} + 621952 q^{86} + 8338044 q^{87} + 20480 q^{88} + 6920346 q^{89} - 6940800 q^{90} + 1527036 q^{91} - 6732800 q^{92} + 11283624 q^{93} + 5629728 q^{94} + 18488800 q^{95} - 2162688 q^{96} + 4952710 q^{97} + 941192 q^{98} + 86760 q^{99}+O(q^{100}) \) 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
8.00000 −66.0000 64.0000 −400.000 −528.000 −343.000 512.000 2169.00 −3200.00
\(n\): 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 14.8.a.b 1
3.b odd 2 1 126.8.a.c 1
4.b odd 2 1 112.8.a.d 1
5.b even 2 1 350.8.a.d 1
5.c odd 4 2 350.8.c.b 2
7.b odd 2 1 98.8.a.c 1
7.c even 3 2 98.8.c.b 2
7.d odd 6 2 98.8.c.a 2
8.b even 2 1 448.8.a.i 1
8.d odd 2 1 448.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.8.a.b 1 1.a even 1 1 trivial
98.8.a.c 1 7.b odd 2 1
98.8.c.a 2 7.d odd 6 2
98.8.c.b 2 7.c even 3 2
112.8.a.d 1 4.b odd 2 1
126.8.a.c 1 3.b odd 2 1
350.8.a.d 1 5.b even 2 1
350.8.c.b 2 5.c odd 4 2
448.8.a.b 1 8.d odd 2 1
448.8.a.i 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 66 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T + 66 \) Copy content Toggle raw display
$5$ \( T + 400 \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T - 40 \) Copy content Toggle raw display
$13$ \( T + 4452 \) Copy content Toggle raw display
$17$ \( T - 36502 \) Copy content Toggle raw display
$19$ \( T + 46222 \) Copy content Toggle raw display
$23$ \( T + 105200 \) Copy content Toggle raw display
$29$ \( T + 126334 \) Copy content Toggle raw display
$31$ \( T + 170964 \) Copy content Toggle raw display
$37$ \( T - 20954 \) Copy content Toggle raw display
$41$ \( T - 318486 \) Copy content Toggle raw display
$43$ \( T - 77744 \) Copy content Toggle raw display
$47$ \( T - 703716 \) Copy content Toggle raw display
$53$ \( T - 1603278 \) Copy content Toggle raw display
$59$ \( T + 1171894 \) Copy content Toggle raw display
$61$ \( T + 2068872 \) Copy content Toggle raw display
$67$ \( T + 994268 \) Copy content Toggle raw display
$71$ \( T - 33280 \) Copy content Toggle raw display
$73$ \( T + 2971454 \) Copy content Toggle raw display
$79$ \( T + 2376168 \) Copy content Toggle raw display
$83$ \( T + 2122358 \) Copy content Toggle raw display
$89$ \( T - 6920346 \) Copy content Toggle raw display
$97$ \( T - 4952710 \) Copy content Toggle raw display
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