Properties

Label 6.10.a.a
Level $6$
Weight $10$
Character orbit 6.a
Self dual yes
Analytic conductor $3.090$
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] = [6,10,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(3.09021501698\)
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)
 
\(f(q)\) \(=\) \( q - 16 q^{2} + 81 q^{3} + 256 q^{4} + 2694 q^{5} - 1296 q^{6} - 3544 q^{7} - 4096 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + 81 q^{3} + 256 q^{4} + 2694 q^{5} - 1296 q^{6} - 3544 q^{7} - 4096 q^{8} + 6561 q^{9} - 43104 q^{10} + 29580 q^{11} + 20736 q^{12} - 44818 q^{13} + 56704 q^{14} + 218214 q^{15} + 65536 q^{16} - 101934 q^{17} - 104976 q^{18} - 895084 q^{19} + 689664 q^{20} - 287064 q^{21} - 473280 q^{22} - 1113000 q^{23} - 331776 q^{24} + 5304511 q^{25} + 717088 q^{26} + 531441 q^{27} - 907264 q^{28} - 2357346 q^{29} - 3491424 q^{30} + 175808 q^{31} - 1048576 q^{32} + 2395980 q^{33} + 1630944 q^{34} - 9547536 q^{35} + 1679616 q^{36} - 2919418 q^{37} + 14321344 q^{38} - 3630258 q^{39} - 11034624 q^{40} + 26218794 q^{41} + 4593024 q^{42} - 18762964 q^{43} + 7572480 q^{44} + 17675334 q^{45} + 17808000 q^{46} - 20966160 q^{47} + 5308416 q^{48} - 27793671 q^{49} - 84872176 q^{50} - 8256654 q^{51} - 11473408 q^{52} + 57251574 q^{53} - 8503056 q^{54} + 79688520 q^{55} + 14516224 q^{56} - 72501804 q^{57} + 37717536 q^{58} + 33587580 q^{59} + 55862784 q^{60} + 82260830 q^{61} - 2812928 q^{62} - 23252184 q^{63} + 16777216 q^{64} - 120739692 q^{65} - 38335680 q^{66} - 188455804 q^{67} - 26095104 q^{68} - 90153000 q^{69} + 152760576 q^{70} + 80924040 q^{71} - 26873856 q^{72} - 236140918 q^{73} + 46710688 q^{74} + 429665391 q^{75} - 229141504 q^{76} - 104831520 q^{77} + 58084128 q^{78} + 526909808 q^{79} + 176553984 q^{80} + 43046721 q^{81} - 419500704 q^{82} + 18346452 q^{83} - 73488384 q^{84} - 274610196 q^{85} + 300207424 q^{86} - 190945026 q^{87} - 121159680 q^{88} + 690643098 q^{89} - 282805344 q^{90} + 158834992 q^{91} - 284928000 q^{92} + 14240448 q^{93} + 335458560 q^{94} - 2411356296 q^{95} - 84934656 q^{96} - 438251038 q^{97} + 444698736 q^{98} + 194074380 q^{99}+O(q^{100}) \) 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
−16.0000 81.0000 256.000 2694.00 −1296.00 −3544.00 −4096.00 6561.00 −43104.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 6.10.a.a 1
3.b odd 2 1 18.10.a.c 1
4.b odd 2 1 48.10.a.d 1
5.b even 2 1 150.10.a.h 1
5.c odd 4 2 150.10.c.d 2
7.b odd 2 1 294.10.a.a 1
8.b even 2 1 192.10.a.a 1
8.d odd 2 1 192.10.a.h 1
9.c even 3 2 162.10.c.f 2
9.d odd 6 2 162.10.c.e 2
12.b even 2 1 144.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.10.a.a 1 1.a even 1 1 trivial
18.10.a.c 1 3.b odd 2 1
48.10.a.d 1 4.b odd 2 1
144.10.a.a 1 12.b even 2 1
150.10.a.h 1 5.b even 2 1
150.10.c.d 2 5.c odd 4 2
162.10.c.e 2 9.d odd 6 2
162.10.c.f 2 9.c even 3 2
192.10.a.a 1 8.b even 2 1
192.10.a.h 1 8.d odd 2 1
294.10.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 16 \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T - 2694 \) Copy content Toggle raw display
$7$ \( T + 3544 \) Copy content Toggle raw display
$11$ \( T - 29580 \) Copy content Toggle raw display
$13$ \( T + 44818 \) Copy content Toggle raw display
$17$ \( T + 101934 \) Copy content Toggle raw display
$19$ \( T + 895084 \) Copy content Toggle raw display
$23$ \( T + 1113000 \) Copy content Toggle raw display
$29$ \( T + 2357346 \) Copy content Toggle raw display
$31$ \( T - 175808 \) Copy content Toggle raw display
$37$ \( T + 2919418 \) Copy content Toggle raw display
$41$ \( T - 26218794 \) Copy content Toggle raw display
$43$ \( T + 18762964 \) Copy content Toggle raw display
$47$ \( T + 20966160 \) Copy content Toggle raw display
$53$ \( T - 57251574 \) Copy content Toggle raw display
$59$ \( T - 33587580 \) Copy content Toggle raw display
$61$ \( T - 82260830 \) Copy content Toggle raw display
$67$ \( T + 188455804 \) Copy content Toggle raw display
$71$ \( T - 80924040 \) Copy content Toggle raw display
$73$ \( T + 236140918 \) Copy content Toggle raw display
$79$ \( T - 526909808 \) Copy content Toggle raw display
$83$ \( T - 18346452 \) Copy content Toggle raw display
$89$ \( T - 690643098 \) Copy content Toggle raw display
$97$ \( T + 438251038 \) Copy content Toggle raw display
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