Properties

Label 1400.4.a.s
Level $1400$
Weight $4$
Character orbit 1400.a
Self dual yes
Analytic conductor $82.603$
Analytic rank $0$
Dimension $4$
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] = [1400,4,Mod(1,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1400.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: \(82.6026740080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 83x^{2} - 184x - 105 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + 7 q^{7} + (\beta_{3} - 3 \beta_1 + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + 7 q^{7} + (\beta_{3} - 3 \beta_1 + 16) q^{9} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 9) q^{11} + (\beta_{3} - 2 \beta_{2} - 6 \beta_1 + 23) q^{13} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 12) q^{17} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 - 22) q^{19} - 7 \beta_1 q^{21} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 15) q^{23} + (7 \beta_{3} - \beta_{2} - 12 \beta_1 + 112) q^{27} + ( - 7 \beta_{3} - 5 \beta_{2} + 12 \beta_1 - 25) q^{29} + ( - 3 \beta_{3} + 5 \beta_{2} - 25 \beta_1 + 114) q^{31} + (6 \beta_{3} + 5 \beta_{2} - 28 \beta_1 + 107) q^{33} + ( - 7 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 94) q^{37} + (8 \beta_{3} + 11 \beta_{2} - 47 \beta_1 + 231) q^{39} + (8 \beta_{3} - \beta_{2} - 26 \beta_1 - 121) q^{41} + ( - 12 \beta_{3} + 7 \beta_{2} + 12 \beta_1 - 58) q^{43} + ( - 12 \beta_{2} + 32 \beta_1 - 2) q^{47} + 49 q^{49} + (\beta_{3} - 7 \beta_{2} + 6 \beta_1 - 184) q^{51} + ( - 14 \beta_{3} - 5 \beta_{2} + 13 \beta_1 - 97) q^{53} + ( - 9 \beta_{3} + 7 \beta_{2} + 52 \beta_1 - 160) q^{57} + ( - 3 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 136) q^{59} + ( - 23 \beta_{3} + 2 \beta_{2} - 44 \beta_1 - 199) q^{61} + (7 \beta_{3} - 21 \beta_1 + 112) q^{63} + (5 \beta_{3} + 6 \beta_{2} + 60 \beta_1 - 46) q^{67} + (9 \beta_{3} + 10 \beta_{2} - 14 \beta_1 + 85) q^{69} + (4 \beta_{3} + 11 \beta_{2} - 10 \beta_1 - 162) q^{71} + (6 \beta_{3} + 13 \beta_{2} - 50 \beta_1 + 227) q^{73} + (7 \beta_{3} - 7 \beta_{2} - 21 \beta_1 + 63) q^{77} + ( - 14 \beta_{3} - 17 \beta_{2} - 36 \beta_1 + 26) q^{79} + (12 \beta_{3} - \beta_{2} - 161 \beta_1 - 40) q^{81} + ( - 6 \beta_{3} - 4 \beta_{2} - 41 \beta_1 + 480) q^{83} + ( - 45 \beta_{3} + 37 \beta_{2} + 179 \beta_1 - 422) q^{87} + ( - 2 \beta_{3} + 31 \beta_{2} + 26 \beta_1 + 239) q^{89} + (7 \beta_{3} - 14 \beta_{2} - 42 \beta_1 + 161) q^{91} + (18 \beta_{3} - 27 \beta_{2} - 167 \beta_1 + 1151) q^{93} + (\beta_{3} - 22 \beta_{2} - 10 \beta_1 + 255) q^{97} + (30 \beta_{3} - 9 \beta_{2} - 214 \beta_1 + 884) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 28 q^{7} + 65 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 28 q^{7} + 65 q^{9} + 36 q^{11} + 94 q^{13} - 53 q^{17} - 91 q^{19} + 7 q^{21} - 63 q^{23} + 445 q^{27} - 103 q^{29} + 492 q^{31} + 449 q^{33} + 387 q^{37} + 966 q^{39} - 475 q^{41} - 213 q^{43} - 52 q^{47} + 196 q^{49} - 751 q^{51} - 378 q^{53} - 667 q^{57} - 548 q^{59} - 704 q^{61} + 455 q^{63} - 248 q^{67} + 346 q^{69} - 635 q^{71} + 959 q^{73} + 252 q^{77} + 151 q^{79} - 24 q^{81} + 1969 q^{83} - 1740 q^{87} + 965 q^{89} + 658 q^{91} + 4708 q^{93} + 1006 q^{97} + 3681 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 83x^{2} - 184x - 105 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 2\nu^{2} - 81\nu - 97 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 85\nu + 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -10\nu^{3} + 22\nu^{2} + 804\nu + 885 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} + 23\beta _1 + 173 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} + 87\beta_{2} + 131\beta _1 + 815 ) / 4 \) 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
10.5788
−7.23450
−1.21600
−1.12826
0 −6.17273 0 0 0 7.00000 0 11.1026 0
1.2 0 −5.67920 0 0 0 7.00000 0 5.25336 0
1.3 0 3.25911 0 0 0 7.00000 0 −16.3782 0
1.4 0 9.59282 0 0 0 7.00000 0 65.0222 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 1400.4.a.s yes 4
5.b even 2 1 1400.4.a.p 4
5.c odd 4 2 1400.4.g.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.4.a.p 4 5.b even 2 1
1400.4.a.s yes 4 1.a even 1 1 trivial
1400.4.g.n 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1400))\):

\( T_{3}^{4} - T_{3}^{3} - 86T_{3}^{2} - 80T_{3} + 1096 \) Copy content Toggle raw display
\( T_{11}^{4} - 36T_{11}^{3} - 2078T_{11}^{2} + 38044T_{11} + 1178085 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 86 T^{2} - 80 T + 1096 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 36 T^{3} - 2078 T^{2} + \cdots + 1178085 \) Copy content Toggle raw display
$13$ \( T^{4} - 94 T^{3} - 4596 T^{2} + \cdots - 2441200 \) Copy content Toggle raw display
$17$ \( T^{4} + 53 T^{3} - 3432 T^{2} + \cdots - 2079168 \) Copy content Toggle raw display
$19$ \( T^{4} + 91 T^{3} - 1006 T^{2} + \cdots + 1139944 \) Copy content Toggle raw display
$23$ \( T^{4} + 63 T^{3} - 6905 T^{2} + \cdots + 15412512 \) Copy content Toggle raw display
$29$ \( T^{4} + 103 T^{3} + \cdots + 2900706682 \) Copy content Toggle raw display
$31$ \( T^{4} - 492 T^{3} + \cdots - 2059238496 \) Copy content Toggle raw display
$37$ \( T^{4} - 387 T^{3} + \cdots - 22873432 \) Copy content Toggle raw display
$41$ \( T^{4} + 475 T^{3} + \cdots - 4028990600 \) Copy content Toggle raw display
$43$ \( T^{4} + 213 T^{3} + \cdots + 4695739880 \) Copy content Toggle raw display
$47$ \( T^{4} + 52 T^{3} + \cdots - 3688005072 \) Copy content Toggle raw display
$53$ \( T^{4} + 378 T^{3} + \cdots + 6242755040 \) Copy content Toggle raw display
$59$ \( T^{4} + 548 T^{3} + \cdots - 56357856 \) Copy content Toggle raw display
$61$ \( T^{4} + 704 T^{3} + \cdots - 101039557936 \) Copy content Toggle raw display
$67$ \( T^{4} + 248 T^{3} + \cdots + 15750308065 \) Copy content Toggle raw display
$71$ \( T^{4} + 635 T^{3} + \cdots + 2484611010 \) Copy content Toggle raw display
$73$ \( T^{4} - 959 T^{3} + \cdots - 20029538840 \) Copy content Toggle raw display
$79$ \( T^{4} - 151 T^{3} + \cdots - 4819927306 \) Copy content Toggle raw display
$83$ \( T^{4} - 1969 T^{3} + \cdots + 1049518872 \) Copy content Toggle raw display
$89$ \( T^{4} - 965 T^{3} + \cdots - 12727152360 \) Copy content Toggle raw display
$97$ \( T^{4} - 1006 T^{3} + \cdots - 47666733552 \) Copy content Toggle raw display
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