Properties

Label 2.18.a.a
Level $2$
Weight $18$
Character orbit 2.a
Self dual yes
Analytic conductor $3.664$
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] = [2,18,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(3.66444174689\)
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)
 
\(f(q)\) \(=\) \( q + 256 q^{2} + 6084 q^{3} + 65536 q^{4} + 1255110 q^{5} + 1557504 q^{6} - 22465912 q^{7} + 16777216 q^{8} - 92125107 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{2} + 6084 q^{3} + 65536 q^{4} + 1255110 q^{5} + 1557504 q^{6} - 22465912 q^{7} + 16777216 q^{8} - 92125107 q^{9} + 321308160 q^{10} + 172399692 q^{11} + 398721024 q^{12} - 2180149426 q^{13} - 5751273472 q^{14} + 7636089240 q^{15} + 4294967296 q^{16} + 30163933458 q^{17} - 23584027392 q^{18} - 76275766060 q^{19} + 82254888960 q^{20} - 136682608608 q^{21} + 44134321152 q^{22} + 130466597784 q^{23} + 102072582144 q^{24} + 812361658975 q^{25} - 558118253056 q^{26} - 1346177902680 q^{27} - 1472326008832 q^{28} + 803134463070 q^{29} + 1954838845440 q^{30} + 2045336056352 q^{31} + 1099511627776 q^{32} + 1048879726128 q^{33} + 7721966965248 q^{34} - 28197190810320 q^{35} - 6037511012352 q^{36} + 33855367078118 q^{37} - 19526596111360 q^{38} - 13264029107784 q^{39} + 21057251573760 q^{40} + 53206442755242 q^{41} - 34990747803648 q^{42} + 26590357792364 q^{43} + 11298386214912 q^{44} - 115627143046770 q^{45} + 33399449032704 q^{46} - 232565394320592 q^{47} + 26130581028864 q^{48} + 272086688004537 q^{49} + 207964584697600 q^{50} + 183517371158472 q^{51} - 142878272782336 q^{52} - 163277861935626 q^{53} - 344621543086080 q^{54} + 216380577426120 q^{55} - 376915458260992 q^{56} - 464061760709040 q^{57} + 205602422545920 q^{58} + 697820734313340 q^{59} + 500438744432640 q^{60} - 898968337037698 q^{61} + 523606030426112 q^{62} + 20\!\cdots\!84 q^{63}+ \cdots - 15\!\cdots\!44 q^{99}+O(q^{100}) \) 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
256.000 6084.00 65536.0 1.25511e6 1.55750e6 −2.24659e7 1.67772e7 −9.21251e7 3.21308e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2.18.a.a 1
3.b odd 2 1 18.18.a.a 1
4.b odd 2 1 16.18.a.a 1
5.b even 2 1 50.18.a.a 1
5.c odd 4 2 50.18.b.b 2
7.b odd 2 1 98.18.a.a 1
8.b even 2 1 64.18.a.a 1
8.d odd 2 1 64.18.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.18.a.a 1 1.a even 1 1 trivial
16.18.a.a 1 4.b odd 2 1
18.18.a.a 1 3.b odd 2 1
50.18.a.a 1 5.b even 2 1
50.18.b.b 2 5.c odd 4 2
64.18.a.a 1 8.b even 2 1
64.18.a.e 1 8.d odd 2 1
98.18.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T - 6084 \) Copy content Toggle raw display
$5$ \( T - 1255110 \) Copy content Toggle raw display
$7$ \( T + 22465912 \) Copy content Toggle raw display
$11$ \( T - 172399692 \) Copy content Toggle raw display
$13$ \( T + 2180149426 \) Copy content Toggle raw display
$17$ \( T - 30163933458 \) Copy content Toggle raw display
$19$ \( T + 76275766060 \) Copy content Toggle raw display
$23$ \( T - 130466597784 \) Copy content Toggle raw display
$29$ \( T - 803134463070 \) Copy content Toggle raw display
$31$ \( T - 2045336056352 \) Copy content Toggle raw display
$37$ \( T - 33855367078118 \) Copy content Toggle raw display
$41$ \( T - 53206442755242 \) Copy content Toggle raw display
$43$ \( T - 26590357792364 \) Copy content Toggle raw display
$47$ \( T + 232565394320592 \) Copy content Toggle raw display
$53$ \( T + 163277861935626 \) Copy content Toggle raw display
$59$ \( T - 697820734313340 \) Copy content Toggle raw display
$61$ \( T + 898968337037698 \) Copy content Toggle raw display
$67$ \( T + 2667002109080572 \) Copy content Toggle raw display
$71$ \( T - 3910637666678472 \) Copy content Toggle raw display
$73$ \( T - 5855931724867274 \) Copy content Toggle raw display
$79$ \( T + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T + 13\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T + 30\!\cdots\!10 \) Copy content Toggle raw display
$97$ \( T - 57\!\cdots\!78 \) Copy content Toggle raw display
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