Properties

Label 2299.4.a.b
Level $2299$
Weight $4$
Character orbit 2299.a
Self dual yes
Analytic conductor $135.645$
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] = [2299,4,Mod(1,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2299, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2299.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2299.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: \(135.645391103\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} - 5 q^{3} + q^{4} - 12 q^{5} - 15 q^{6} - 11 q^{7} - 21 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - 5 q^{3} + q^{4} - 12 q^{5} - 15 q^{6} - 11 q^{7} - 21 q^{8} - 2 q^{9} - 36 q^{10} - 5 q^{12} - 11 q^{13} - 33 q^{14} + 60 q^{15} - 71 q^{16} + 93 q^{17} - 6 q^{18} - 19 q^{19} - 12 q^{20} + 55 q^{21} + 183 q^{23} + 105 q^{24} + 19 q^{25} - 33 q^{26} + 145 q^{27} - 11 q^{28} + 249 q^{29} + 180 q^{30} + 56 q^{31} - 45 q^{32} + 279 q^{34} + 132 q^{35} - 2 q^{36} - 250 q^{37} - 57 q^{38} + 55 q^{39} + 252 q^{40} - 240 q^{41} + 165 q^{42} + 196 q^{43} + 24 q^{45} + 549 q^{46} - 168 q^{47} + 355 q^{48} - 222 q^{49} + 57 q^{50} - 465 q^{51} - 11 q^{52} + 435 q^{53} + 435 q^{54} + 231 q^{56} + 95 q^{57} + 747 q^{58} + 195 q^{59} + 60 q^{60} + 358 q^{61} + 168 q^{62} + 22 q^{63} + 433 q^{64} + 132 q^{65} - 961 q^{67} + 93 q^{68} - 915 q^{69} + 396 q^{70} - 246 q^{71} + 42 q^{72} - 353 q^{73} - 750 q^{74} - 95 q^{75} - 19 q^{76} + 165 q^{78} + 34 q^{79} + 852 q^{80} - 671 q^{81} - 720 q^{82} - 234 q^{83} + 55 q^{84} - 1116 q^{85} + 588 q^{86} - 1245 q^{87} - 168 q^{89} + 72 q^{90} + 121 q^{91} + 183 q^{92} - 280 q^{93} - 504 q^{94} + 228 q^{95} + 225 q^{96} + 758 q^{97} - 666 q^{98}+O(q^{100}) \) 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
3.00000 −5.00000 1.00000 −12.0000 −15.0000 −11.0000 −21.0000 −2.00000 −36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(19\) \(1\)

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 2299.4.a.b 1
11.b odd 2 1 19.4.a.a 1
33.d even 2 1 171.4.a.d 1
44.c even 2 1 304.4.a.b 1
55.d odd 2 1 475.4.a.e 1
55.e even 4 2 475.4.b.c 2
77.b even 2 1 931.4.a.a 1
88.b odd 2 1 1216.4.a.f 1
88.g even 2 1 1216.4.a.a 1
209.d even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 11.b odd 2 1
171.4.a.d 1 33.d even 2 1
304.4.a.b 1 44.c even 2 1
361.4.a.b 1 209.d even 2 1
475.4.a.e 1 55.d odd 2 1
475.4.b.c 2 55.e even 4 2
931.4.a.a 1 77.b even 2 1
1216.4.a.a 1 88.g even 2 1
1216.4.a.f 1 88.b odd 2 1
2299.4.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{5} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T + 5 \) Copy content Toggle raw display
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T + 11 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 11 \) Copy content Toggle raw display
$17$ \( T - 93 \) Copy content Toggle raw display
$19$ \( T + 19 \) Copy content Toggle raw display
$23$ \( T - 183 \) Copy content Toggle raw display
$29$ \( T - 249 \) Copy content Toggle raw display
$31$ \( T - 56 \) Copy content Toggle raw display
$37$ \( T + 250 \) Copy content Toggle raw display
$41$ \( T + 240 \) Copy content Toggle raw display
$43$ \( T - 196 \) Copy content Toggle raw display
$47$ \( T + 168 \) Copy content Toggle raw display
$53$ \( T - 435 \) Copy content Toggle raw display
$59$ \( T - 195 \) Copy content Toggle raw display
$61$ \( T - 358 \) Copy content Toggle raw display
$67$ \( T + 961 \) Copy content Toggle raw display
$71$ \( T + 246 \) Copy content Toggle raw display
$73$ \( T + 353 \) Copy content Toggle raw display
$79$ \( T - 34 \) Copy content Toggle raw display
$83$ \( T + 234 \) Copy content Toggle raw display
$89$ \( T + 168 \) Copy content Toggle raw display
$97$ \( T - 758 \) Copy content Toggle raw display
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