Properties

Label 440.2.y.d
Level $440$
Weight $2$
Character orbit 440.y
Analytic conductor $3.513$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(81,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 141 x^{12} - 220 x^{11} + 1105 x^{10} - 1935 x^{9} + \cdots + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{12} - \beta_{10} + \beta_{4} - 1) q^{5} + (\beta_{12} - \beta_{10} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{14} - \beta_{13} - 2 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{12} - \beta_{10} + \beta_{4} - 1) q^{5} + (\beta_{12} - \beta_{10} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - 2 \beta_{15} + 4 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 4 q^{5} + 8 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{3} - 4 q^{5} + 8 q^{7} - 7 q^{9} - 7 q^{11} - 11 q^{13} - 3 q^{15} + 9 q^{17} - 2 q^{19} + 12 q^{21} + 20 q^{23} - 4 q^{25} - 9 q^{27} + q^{29} - 2 q^{31} - 32 q^{33} - 2 q^{35} - 16 q^{37} + 3 q^{39} - 11 q^{41} - 16 q^{43} + 38 q^{45} - 10 q^{47} + 4 q^{49} + 26 q^{51} - 9 q^{53} + 3 q^{55} - 50 q^{57} - 60 q^{59} + 30 q^{61} + 52 q^{63} + 24 q^{65} - 4 q^{67} + 41 q^{69} - 10 q^{71} + q^{73} + 2 q^{75} - 4 q^{77} + 19 q^{79} - 31 q^{81} + 64 q^{83} - 11 q^{85} + 30 q^{87} + 24 q^{89} - 9 q^{91} + 45 q^{93} - 2 q^{95} - 3 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 141 x^{12} - 220 x^{11} + 1105 x^{10} - 1935 x^{9} + \cdots + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\!\cdots\!45 \nu^{15} + \cdots + 62\!\cdots\!00 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 66\!\cdots\!11 \nu^{15} + \cdots + 89\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!47 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!36 \nu^{15} + \cdots + 20\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!75 \nu^{15} + \cdots - 31\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61\!\cdots\!01 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72\!\cdots\!27 \nu^{15} + \cdots + 87\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!59 \nu^{15} + \cdots + 98\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!49 \nu^{15} + \cdots - 60\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 58\!\cdots\!61 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!13 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!80 \nu^{15} + \cdots - 49\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 63\!\cdots\!95 \nu^{15} + \cdots + 87\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16\!\cdots\!94 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{13} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} + 4 \beta_{10} + \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{15} + 9 \beta_{14} - 41 \beta_{12} - 11 \beta_{11} + 3 \beta_{10} + \beta_{9} + 12 \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35 \beta_{15} - 20 \beta_{14} - 25 \beta_{13} + 14 \beta_{12} - 86 \beta_{11} - 14 \beta_{10} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 123 \beta_{15} + 81 \beta_{14} - 123 \beta_{13} - 114 \beta_{11} + 322 \beta_{10} - 19 \beta_{9} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 19 \beta_{15} + 128 \beta_{14} - 33 \beta_{13} - 126 \beta_{12} - 161 \beta_{11} - 161 \beta_{10} + \cdots - 35 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1005 \beta_{15} - 744 \beta_{14} + 483 \beta_{13} + 3436 \beta_{12} + 442 \beta_{11} - 3577 \beta_{10} + \cdots - 3577 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1322 \beta_{15} - 537 \beta_{14} - 1570 \beta_{13} + 4173 \beta_{12} + 5893 \beta_{11} - 1224 \beta_{10} + \cdots - 2173 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10075 \beta_{15} - 30 \beta_{14} + 10090 \beta_{13} + 3793 \beta_{12} + 16843 \beta_{11} + \cdots + 28771 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 25059 \beta_{15} - 6723 \beta_{14} + 31779 \beta_{13} + 38972 \beta_{11} - 32676 \beta_{10} + \cdots + 14851 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 54113 \beta_{15} - 64659 \beta_{14} - 11486 \beta_{13} - 42682 \beta_{12} + 53173 \beta_{11} + \cdots + 95855 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 278915 \beta_{15} + 135367 \beta_{14} + 121231 \beta_{13} - 526348 \beta_{12} - 23581 \beta_{11} + \cdots + 740761 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 608321 \beta_{15} + 617766 \beta_{14} - 18890 \beta_{13} - 3020414 \beta_{12} - 1524924 \beta_{11} + \cdots + 396864 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3297600 \beta_{15} - 931010 \beta_{14} - 2832095 \beta_{13} - 124649 \beta_{12} - 7807049 \beta_{11} + \cdots - 5590003 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.856564 2.63623i
0.507132 1.56079i
−0.220438 + 0.678438i
−0.952275 + 2.93080i
0.856564 + 2.63623i
0.507132 + 1.56079i
−0.220438 0.678438i
−0.952275 2.93080i
2.59852 1.88794i
1.52167 1.10556i
−0.460979 + 0.334921i
−2.35019 + 1.70752i
2.59852 + 1.88794i
1.52167 + 1.10556i
−0.460979 0.334921i
−2.35019 1.70752i
0 −0.856564 + 2.63623i 0 −0.809017 + 0.587785i 0 −1.40759 4.33212i 0 −3.78896 2.75284i 0
81.2 0 −0.507132 + 1.56079i 0 −0.809017 + 0.587785i 0 1.04337 + 3.21117i 0 0.248161 + 0.180299i 0
81.3 0 0.220438 0.678438i 0 −0.809017 + 0.587785i 0 0.116244 + 0.357761i 0 2.01537 + 1.46425i 0
81.4 0 0.952275 2.93080i 0 −0.809017 + 0.587785i 0 0.0119064 + 0.0366440i 0 −5.25571 3.81850i 0
201.1 0 −0.856564 2.63623i 0 −0.809017 0.587785i 0 −1.40759 + 4.33212i 0 −3.78896 + 2.75284i 0
201.2 0 −0.507132 1.56079i 0 −0.809017 0.587785i 0 1.04337 3.21117i 0 0.248161 0.180299i 0
201.3 0 0.220438 + 0.678438i 0 −0.809017 0.587785i 0 0.116244 0.357761i 0 2.01537 1.46425i 0
201.4 0 0.952275 + 2.93080i 0 −0.809017 0.587785i 0 0.0119064 0.0366440i 0 −5.25571 + 3.81850i 0
361.1 0 −2.59852 + 1.88794i 0 0.309017 + 0.951057i 0 3.97176 + 2.88565i 0 2.26096 6.95852i 0
361.2 0 −1.52167 + 1.10556i 0 0.309017 + 0.951057i 0 −2.97516 2.16158i 0 0.166167 0.511409i 0
361.3 0 0.460979 0.334921i 0 0.309017 + 0.951057i 0 1.47331 + 1.07042i 0 −0.826721 + 2.54439i 0
361.4 0 2.35019 1.70752i 0 0.309017 + 0.951057i 0 1.76616 + 1.28319i 0 1.68075 5.17281i 0
401.1 0 −2.59852 1.88794i 0 0.309017 0.951057i 0 3.97176 2.88565i 0 2.26096 + 6.95852i 0
401.2 0 −1.52167 1.10556i 0 0.309017 0.951057i 0 −2.97516 + 2.16158i 0 0.166167 + 0.511409i 0
401.3 0 0.460979 + 0.334921i 0 0.309017 0.951057i 0 1.47331 1.07042i 0 −0.826721 2.54439i 0
401.4 0 2.35019 + 1.70752i 0 0.309017 0.951057i 0 1.76616 1.28319i 0 1.68075 + 5.17281i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.y.d 16
4.b odd 2 1 880.2.bo.k 16
11.c even 5 1 inner 440.2.y.d 16
11.c even 5 1 4840.2.a.bg 8
11.d odd 10 1 4840.2.a.bh 8
44.g even 10 1 9680.2.a.de 8
44.h odd 10 1 880.2.bo.k 16
44.h odd 10 1 9680.2.a.df 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.d 16 1.a even 1 1 trivial
440.2.y.d 16 11.c even 5 1 inner
880.2.bo.k 16 4.b odd 2 1
880.2.bo.k 16 44.h odd 10 1
4840.2.a.bg 8 11.c even 5 1
4840.2.a.bh 8 11.d odd 10 1
9680.2.a.de 8 44.g even 10 1
9680.2.a.df 8 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 3 T_{3}^{15} + 14 T_{3}^{14} + 32 T_{3}^{13} + 141 T_{3}^{12} + 220 T_{3}^{11} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 10000 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 11 T^{15} + \cdots + 36772096 \) Copy content Toggle raw display
$17$ \( T^{16} - 9 T^{15} + \cdots + 9759376 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 5603271025 \) Copy content Toggle raw display
$23$ \( (T^{8} - 10 T^{7} + 8 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 4833030400 \) Copy content Toggle raw display
$31$ \( T^{16} + 2 T^{15} + \cdots + 6150400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1041256976400 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16066830025 \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{7} + \cdots - 76780)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 786279452176 \) Copy content Toggle raw display
$53$ \( T^{16} + 9 T^{15} + \cdots + 23232400 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 138039001 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 2548544045056 \) Copy content Toggle raw display
$67$ \( (T^{8} + 2 T^{7} + \cdots + 2034124)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 240100000000 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2853696400 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 153593389158400 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 61931296686736 \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots + 2898841)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
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