Properties

Label 1950.4.a.a
Level $1950$
Weight $4$
Character orbit 1950.a
Self dual yes
Analytic conductor $115.054$
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] = [1950,4,Mod(1,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1950.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1950.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: \(115.053724511\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{6} - 4 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{6} - 4 q^{7} - 8 q^{8} + 9 q^{9} + 2 q^{11} - 12 q^{12} + 13 q^{13} + 8 q^{14} + 16 q^{16} + 6 q^{17} - 18 q^{18} - 36 q^{19} + 12 q^{21} - 4 q^{22} + 20 q^{23} + 24 q^{24} - 26 q^{26} - 27 q^{27} - 16 q^{28} - 14 q^{29} - 152 q^{31} - 32 q^{32} - 6 q^{33} - 12 q^{34} + 36 q^{36} + 258 q^{37} + 72 q^{38} - 39 q^{39} + 84 q^{41} - 24 q^{42} + 188 q^{43} + 8 q^{44} - 40 q^{46} - 254 q^{47} - 48 q^{48} - 327 q^{49} - 18 q^{51} + 52 q^{52} - 366 q^{53} + 54 q^{54} + 32 q^{56} + 108 q^{57} + 28 q^{58} + 550 q^{59} - 14 q^{61} + 304 q^{62} - 36 q^{63} + 64 q^{64} + 12 q^{66} - 448 q^{67} + 24 q^{68} - 60 q^{69} + 926 q^{71} - 72 q^{72} - 254 q^{73} - 516 q^{74} - 144 q^{76} - 8 q^{77} + 78 q^{78} + 1328 q^{79} + 81 q^{81} - 168 q^{82} - 186 q^{83} + 48 q^{84} - 376 q^{86} + 42 q^{87} - 16 q^{88} - 336 q^{89} - 52 q^{91} + 80 q^{92} + 456 q^{93} + 508 q^{94} + 96 q^{96} - 614 q^{97} + 654 q^{98} + 18 q^{99}+O(q^{100}) \) 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 −3.00000 4.00000 0 6.00000 −4.00000 −8.00000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

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 1950.4.a.a 1
5.b even 2 1 78.4.a.f 1
15.d odd 2 1 234.4.a.c 1
20.d odd 2 1 624.4.a.c 1
40.e odd 2 1 2496.4.a.l 1
40.f even 2 1 2496.4.a.c 1
60.h even 2 1 1872.4.a.f 1
65.d even 2 1 1014.4.a.e 1
65.g odd 4 2 1014.4.b.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.f 1 5.b even 2 1
234.4.a.c 1 15.d odd 2 1
624.4.a.c 1 20.d odd 2 1
1014.4.a.e 1 65.d even 2 1
1014.4.b.g 2 65.g odd 4 2
1872.4.a.f 1 60.h even 2 1
1950.4.a.a 1 1.a even 1 1 trivial
2496.4.a.c 1 40.f even 2 1
2496.4.a.l 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1950))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 36 \) Copy content Toggle raw display
$23$ \( T - 20 \) Copy content Toggle raw display
$29$ \( T + 14 \) Copy content Toggle raw display
$31$ \( T + 152 \) Copy content Toggle raw display
$37$ \( T - 258 \) Copy content Toggle raw display
$41$ \( T - 84 \) Copy content Toggle raw display
$43$ \( T - 188 \) Copy content Toggle raw display
$47$ \( T + 254 \) Copy content Toggle raw display
$53$ \( T + 366 \) Copy content Toggle raw display
$59$ \( T - 550 \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T + 448 \) Copy content Toggle raw display
$71$ \( T - 926 \) Copy content Toggle raw display
$73$ \( T + 254 \) Copy content Toggle raw display
$79$ \( T - 1328 \) Copy content Toggle raw display
$83$ \( T + 186 \) Copy content Toggle raw display
$89$ \( T + 336 \) Copy content Toggle raw display
$97$ \( T + 614 \) Copy content Toggle raw display
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