Properties

Label 1840.4.a.n
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $5$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{3} - 5 q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{3} - 5 q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 16) q^{9} + (\beta_{3} - 2 \beta_{2} - 6 \beta_1 - 7) q^{11} + (4 \beta_{4} - 3 \beta_{3} + \beta_{2} + \cdots + 26) q^{13}+ \cdots + ( - 153 \beta_{4} + 208 \beta_{3} + \cdots - 436) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35} - 38 q^{37} - 335 q^{39} - 12 q^{41} + 566 q^{43} - 385 q^{45} - 919 q^{47} - 738 q^{49} + 993 q^{51} + 1156 q^{53} + 115 q^{55} + 114 q^{57} - 1324 q^{59} - 1673 q^{61} - 270 q^{63} - 660 q^{65} - 558 q^{67} - 92 q^{69} + 108 q^{71} + 1173 q^{73} - 100 q^{75} + 2608 q^{77} - 656 q^{79} - 319 q^{81} + 82 q^{83} - 115 q^{85} - 2389 q^{87} + 570 q^{89} + 1589 q^{91} + 911 q^{93} - 805 q^{95} + 633 q^{97} - 2021 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 7\nu^{3} + 9\nu^{2} - 77\nu - 2 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + \nu^{3} - 9\nu^{2} - 43\nu - 78 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - \nu^{3} + 25\nu^{2} + 43\nu - 98 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{4} + 5\nu^{3} + 59\nu^{2} - 55\nu - 182 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + 2\beta_{3} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{4} + 30\beta_{3} + 8\beta_{2} + 23\beta _1 + 55 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{4} + 23\beta_{3} + 23\beta_{2} + 5\beta _1 + 174 \) 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
−2.49214
3.41740
4.60878
−3.93900
−0.595043
0 −9.02447 0 −5.00000 0 −4.33445 0 54.4411 0
1.2 0 −7.84147 0 −5.00000 0 8.97260 0 34.4886 0
1.3 0 1.89520 0 −5.00000 0 −11.4426 0 −23.4082 0
1.4 0 3.85751 0 −5.00000 0 23.5932 0 −12.1196 0
1.5 0 7.11323 0 −5.00000 0 −13.7888 0 23.5981 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 1840.4.a.n 5
4.b odd 2 1 115.4.a.e 5
12.b even 2 1 1035.4.a.k 5
20.d odd 2 1 575.4.a.j 5
20.e even 4 2 575.4.b.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 4.b odd 2 1
575.4.a.j 5 20.d odd 2 1
575.4.b.i 10 20.e even 4 2
1035.4.a.k 5 12.b even 2 1
1840.4.a.n 5 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{5} + 4T_{3}^{4} - 98T_{3}^{3} - 149T_{3}^{2} + 2536T_{3} - 3680 \) Copy content Toggle raw display
\( T_{7}^{5} - 3T_{7}^{4} - 484T_{7}^{3} - 1757T_{7}^{2} + 34281T_{7} + 144774 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 4 T^{4} + \cdots - 3680 \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 3 T^{4} + \cdots + 144774 \) Copy content Toggle raw display
$11$ \( T^{5} + 23 T^{4} + \cdots + 74136848 \) Copy content Toggle raw display
$13$ \( T^{5} - 132 T^{4} + \cdots + 1550116 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 1039045340 \) Copy content Toggle raw display
$19$ \( T^{5} - 161 T^{4} + \cdots + 801280 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 6149898500 \) Copy content Toggle raw display
$31$ \( T^{5} + 32 T^{4} + \cdots + 438072447 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 1590700778176 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 114116030755 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 504784881664 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 117787714816 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 5720332226904 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 24279649927232 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 34095834816896 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 5644442112 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 15638892903635 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 100895881632176 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 90481602379776 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 18307318870176 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 115104799418880 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 480989167569272 \) Copy content Toggle raw display
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