Properties

Label 33.8.a.a
Level $33$
Weight $8$
Character orbit 33.a
Self dual yes
Analytic conductor $10.309$
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] = [33,8,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(10.3087058410\)
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)
 
\(f(q)\) \(=\) \( q + 10 q^{2} + 27 q^{3} - 28 q^{4} - 410 q^{5} + 270 q^{6} - 1028 q^{7} - 1560 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{2} + 27 q^{3} - 28 q^{4} - 410 q^{5} + 270 q^{6} - 1028 q^{7} - 1560 q^{8} + 729 q^{9} - 4100 q^{10} - 1331 q^{11} - 756 q^{12} + 12958 q^{13} - 10280 q^{14} - 11070 q^{15} - 12016 q^{16} + 17062 q^{17} + 7290 q^{18} - 54168 q^{19} + 11480 q^{20} - 27756 q^{21} - 13310 q^{22} - 11488 q^{23} - 42120 q^{24} + 89975 q^{25} + 129580 q^{26} + 19683 q^{27} + 28784 q^{28} - 186654 q^{29} - 110700 q^{30} - 188672 q^{31} + 79520 q^{32} - 35937 q^{33} + 170620 q^{34} + 421480 q^{35} - 20412 q^{36} + 395886 q^{37} - 541680 q^{38} + 349866 q^{39} + 639600 q^{40} - 47546 q^{41} - 277560 q^{42} + 602088 q^{43} + 37268 q^{44} - 298890 q^{45} - 114880 q^{46} - 647200 q^{47} - 324432 q^{48} + 233241 q^{49} + 899750 q^{50} + 460674 q^{51} - 362824 q^{52} - 1312722 q^{53} + 196830 q^{54} + 545710 q^{55} + 1603680 q^{56} - 1462536 q^{57} - 1866540 q^{58} - 2681140 q^{59} + 309960 q^{60} + 551190 q^{61} - 1886720 q^{62} - 749412 q^{63} + 2333248 q^{64} - 5312780 q^{65} - 359370 q^{66} + 459260 q^{67} - 477736 q^{68} - 310176 q^{69} + 4214800 q^{70} - 18072 q^{71} - 1137240 q^{72} - 426062 q^{73} + 3958860 q^{74} + 2429325 q^{75} + 1516704 q^{76} + 1368268 q^{77} + 3498660 q^{78} + 297764 q^{79} + 4926560 q^{80} + 531441 q^{81} - 475460 q^{82} + 5684028 q^{83} + 777168 q^{84} - 6995420 q^{85} + 6020880 q^{86} - 5039658 q^{87} + 2076360 q^{88} - 6342966 q^{89} - 2988900 q^{90} - 13320824 q^{91} + 321664 q^{92} - 5094144 q^{93} - 6472000 q^{94} + 22208880 q^{95} + 2147040 q^{96} + 16651586 q^{97} + 2332410 q^{98} - 970299 q^{99}+O(q^{100}) \) 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
10.0000 27.0000 −28.0000 −410.000 270.000 −1028.00 −1560.00 729.000 −4100.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

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 33.8.a.a 1
3.b odd 2 1 99.8.a.a 1
4.b odd 2 1 528.8.a.a 1
11.b odd 2 1 363.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.a 1 1.a even 1 1 trivial
99.8.a.a 1 3.b odd 2 1
363.8.a.a 1 11.b odd 2 1
528.8.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 10 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T + 410 \) Copy content Toggle raw display
$7$ \( T + 1028 \) Copy content Toggle raw display
$11$ \( T + 1331 \) Copy content Toggle raw display
$13$ \( T - 12958 \) Copy content Toggle raw display
$17$ \( T - 17062 \) Copy content Toggle raw display
$19$ \( T + 54168 \) Copy content Toggle raw display
$23$ \( T + 11488 \) Copy content Toggle raw display
$29$ \( T + 186654 \) Copy content Toggle raw display
$31$ \( T + 188672 \) Copy content Toggle raw display
$37$ \( T - 395886 \) Copy content Toggle raw display
$41$ \( T + 47546 \) Copy content Toggle raw display
$43$ \( T - 602088 \) Copy content Toggle raw display
$47$ \( T + 647200 \) Copy content Toggle raw display
$53$ \( T + 1312722 \) Copy content Toggle raw display
$59$ \( T + 2681140 \) Copy content Toggle raw display
$61$ \( T - 551190 \) Copy content Toggle raw display
$67$ \( T - 459260 \) Copy content Toggle raw display
$71$ \( T + 18072 \) Copy content Toggle raw display
$73$ \( T + 426062 \) Copy content Toggle raw display
$79$ \( T - 297764 \) Copy content Toggle raw display
$83$ \( T - 5684028 \) Copy content Toggle raw display
$89$ \( T + 6342966 \) Copy content Toggle raw display
$97$ \( T - 16651586 \) Copy content Toggle raw display
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