Properties

Label 370.4.a.a
Level $370$
Weight $4$
Character orbit 370.a
Self dual yes
Analytic conductor $21.831$
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] = [370,4,Mod(1,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, 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("370.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 370.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: \(21.8307067021\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + 2 q^{7} + 8 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + 2 q^{7} + 8 q^{8} - 23 q^{9} + 10 q^{10} - 72 q^{11} - 8 q^{12} + 2 q^{13} + 4 q^{14} - 10 q^{15} + 16 q^{16} - 66 q^{17} - 46 q^{18} + 38 q^{19} + 20 q^{20} - 4 q^{21} - 144 q^{22} - 36 q^{23} - 16 q^{24} + 25 q^{25} + 4 q^{26} + 100 q^{27} + 8 q^{28} - 90 q^{29} - 20 q^{30} - 70 q^{31} + 32 q^{32} + 144 q^{33} - 132 q^{34} + 10 q^{35} - 92 q^{36} + 37 q^{37} + 76 q^{38} - 4 q^{39} + 40 q^{40} - 438 q^{41} - 8 q^{42} + 272 q^{43} - 288 q^{44} - 115 q^{45} - 72 q^{46} - 198 q^{47} - 32 q^{48} - 339 q^{49} + 50 q^{50} + 132 q^{51} + 8 q^{52} - 354 q^{53} + 200 q^{54} - 360 q^{55} + 16 q^{56} - 76 q^{57} - 180 q^{58} - 498 q^{59} - 40 q^{60} + 542 q^{61} - 140 q^{62} - 46 q^{63} + 64 q^{64} + 10 q^{65} + 288 q^{66} + 2 q^{67} - 264 q^{68} + 72 q^{69} + 20 q^{70} + 408 q^{71} - 184 q^{72} - 358 q^{73} + 74 q^{74} - 50 q^{75} + 152 q^{76} - 144 q^{77} - 8 q^{78} + 722 q^{79} + 80 q^{80} + 421 q^{81} - 876 q^{82} - 174 q^{83} - 16 q^{84} - 330 q^{85} + 544 q^{86} + 180 q^{87} - 576 q^{88} - 102 q^{89} - 230 q^{90} + 4 q^{91} - 144 q^{92} + 140 q^{93} - 396 q^{94} + 190 q^{95} - 64 q^{96} - 574 q^{97} - 678 q^{98} + 1656 q^{99}+O(q^{100}) \) 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 −2.00000 4.00000 5.00000 −4.00000 2.00000 8.00000 −23.0000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(37\) \( -1 \)

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 370.4.a.a 1
5.b even 2 1 1850.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.4.a.a 1 1.a even 1 1 trivial
1850.4.a.c 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(370))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 72 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 66 \) Copy content Toggle raw display
$19$ \( T - 38 \) Copy content Toggle raw display
$23$ \( T + 36 \) Copy content Toggle raw display
$29$ \( T + 90 \) Copy content Toggle raw display
$31$ \( T + 70 \) Copy content Toggle raw display
$37$ \( T - 37 \) Copy content Toggle raw display
$41$ \( T + 438 \) Copy content Toggle raw display
$43$ \( T - 272 \) Copy content Toggle raw display
$47$ \( T + 198 \) Copy content Toggle raw display
$53$ \( T + 354 \) Copy content Toggle raw display
$59$ \( T + 498 \) Copy content Toggle raw display
$61$ \( T - 542 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T - 408 \) Copy content Toggle raw display
$73$ \( T + 358 \) Copy content Toggle raw display
$79$ \( T - 722 \) Copy content Toggle raw display
$83$ \( T + 174 \) Copy content Toggle raw display
$89$ \( T + 102 \) Copy content Toggle raw display
$97$ \( T + 574 \) Copy content Toggle raw display
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