Properties

Label 230.4.a.a
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
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] = [230,4,Mod(1,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, 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("230.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(13.5704393013\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 5 q^{5} + 10 q^{6} + 12 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 5 q^{5} + 10 q^{6} + 12 q^{7} - 8 q^{8} - 2 q^{9} + 10 q^{10} + 22 q^{11} - 20 q^{12} + 19 q^{13} - 24 q^{14} + 25 q^{15} + 16 q^{16} + 96 q^{17} + 4 q^{18} - 98 q^{19} - 20 q^{20} - 60 q^{21} - 44 q^{22} + 23 q^{23} + 40 q^{24} + 25 q^{25} - 38 q^{26} + 145 q^{27} + 48 q^{28} - 227 q^{29} - 50 q^{30} - 285 q^{31} - 32 q^{32} - 110 q^{33} - 192 q^{34} - 60 q^{35} - 8 q^{36} - 398 q^{37} + 196 q^{38} - 95 q^{39} + 40 q^{40} + 271 q^{41} + 120 q^{42} - 100 q^{43} + 88 q^{44} + 10 q^{45} - 46 q^{46} - 285 q^{47} - 80 q^{48} - 199 q^{49} - 50 q^{50} - 480 q^{51} + 76 q^{52} + 18 q^{53} - 290 q^{54} - 110 q^{55} - 96 q^{56} + 490 q^{57} + 454 q^{58} - 352 q^{59} + 100 q^{60} - 478 q^{61} + 570 q^{62} - 24 q^{63} + 64 q^{64} - 95 q^{65} + 220 q^{66} + 330 q^{67} + 384 q^{68} - 115 q^{69} + 120 q^{70} + 835 q^{71} + 16 q^{72} - 1127 q^{73} + 796 q^{74} - 125 q^{75} - 392 q^{76} + 264 q^{77} + 190 q^{78} + 322 q^{79} - 80 q^{80} - 671 q^{81} - 542 q^{82} + 572 q^{83} - 240 q^{84} - 480 q^{85} + 200 q^{86} + 1135 q^{87} - 176 q^{88} - 504 q^{89} - 20 q^{90} + 228 q^{91} + 92 q^{92} + 1425 q^{93} + 570 q^{94} + 490 q^{95} + 160 q^{96} + 1712 q^{97} + 398 q^{98} - 44 q^{99}+O(q^{100}) \) 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
−2.00000 −5.00000 4.00000 −5.00000 10.0000 12.0000 −8.00000 −2.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 230.4.a.a 1
3.b odd 2 1 2070.4.a.o 1
4.b odd 2 1 1840.4.a.g 1
5.b even 2 1 1150.4.a.i 1
5.c odd 4 2 1150.4.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.a 1 1.a even 1 1 trivial
1150.4.a.i 1 5.b even 2 1
1150.4.b.h 2 5.c odd 4 2
1840.4.a.g 1 4.b odd 2 1
2070.4.a.o 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 5 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 12 \) Copy content Toggle raw display
$11$ \( T - 22 \) Copy content Toggle raw display
$13$ \( T - 19 \) Copy content Toggle raw display
$17$ \( T - 96 \) Copy content Toggle raw display
$19$ \( T + 98 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T + 227 \) Copy content Toggle raw display
$31$ \( T + 285 \) Copy content Toggle raw display
$37$ \( T + 398 \) Copy content Toggle raw display
$41$ \( T - 271 \) Copy content Toggle raw display
$43$ \( T + 100 \) Copy content Toggle raw display
$47$ \( T + 285 \) Copy content Toggle raw display
$53$ \( T - 18 \) Copy content Toggle raw display
$59$ \( T + 352 \) Copy content Toggle raw display
$61$ \( T + 478 \) Copy content Toggle raw display
$67$ \( T - 330 \) Copy content Toggle raw display
$71$ \( T - 835 \) Copy content Toggle raw display
$73$ \( T + 1127 \) Copy content Toggle raw display
$79$ \( T - 322 \) Copy content Toggle raw display
$83$ \( T - 572 \) Copy content Toggle raw display
$89$ \( T + 504 \) Copy content Toggle raw display
$97$ \( T - 1712 \) Copy content Toggle raw display
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