Properties

Label 440.2.y.c
Level $440$
Weight $2$
Character orbit 440.y
Analytic conductor $3.513$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{3}) q^{3} - \beta_{6} q^{5} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{9} + 2 \beta_{7} + \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{3}) q^{3} - \beta_{6} q^{5} + (\beta_{11} - 2 \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + 2 \beta_{10} - 6 \beta_{8} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 3 q^{5} - q^{7} - 10 q^{9} + 4 q^{11} + 18 q^{13} + q^{15} + 3 q^{17} + 4 q^{19} - 28 q^{21} - 18 q^{23} - 3 q^{25} + 23 q^{27} + 15 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{35} + 6 q^{37} - 33 q^{39} + 2 q^{41} - 36 q^{43} - 10 q^{45} - 16 q^{47} - 16 q^{49} - 10 q^{51} + 19 q^{53} + 6 q^{55} + 62 q^{57} + 46 q^{59} + 18 q^{61} - 7 q^{63} + 2 q^{65} - 44 q^{67} - q^{69} - 6 q^{71} + 25 q^{73} + 4 q^{75} + 10 q^{77} + 19 q^{79} + 30 q^{81} - 3 q^{85} + 6 q^{87} - 50 q^{89} - 46 q^{91} - 37 q^{93} - 4 q^{95} - 31 q^{97} + 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3497434625 \nu^{11} - 2215518483724 \nu^{10} + 702407416002 \nu^{9} + \cdots - 708170152940520 ) / 249616873443718 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 469750424879 \nu^{11} - 1250785586077 \nu^{10} + 1635686014444 \nu^{9} + \cdots - 518561109011779 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2167683167019 \nu^{11} + 2869118757525 \nu^{10} - 9747395928812 \nu^{9} + \cdots + 19849150430431 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11524681849762 \nu^{11} + 12737270254139 \nu^{10} - 59052962251434 \nu^{9} + \cdots + 472887038077187 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19849150430431 \nu^{11} - 17681467263412 \nu^{10} + 96376633394630 \nu^{9} + \cdots - 5482407342022 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19849150430431 \nu^{11} + 17681467263412 \nu^{10} - 96376633394630 \nu^{9} + \cdots + 5482407342022 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30450870201976 \nu^{11} - 30852043974105 \nu^{10} + 152997188989378 \nu^{9} + \cdots - 818952899033937 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30972658508064 \nu^{11} + 29675899538053 \nu^{10} - 153987797348370 \nu^{9} + \cdots + 508820315621185 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 88405871699823 \nu^{11} - 83922796851166 \nu^{10} + 438655411068042 \nu^{9} + \cdots - 23\!\cdots\!76 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 89746069507409 \nu^{11} + 87864404405688 \nu^{10} - 443314313150206 \nu^{9} + \cdots + 14\!\cdots\!34 ) / 499233746887436 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12823391623530 \nu^{11} - 12647557096509 \nu^{10} + 63908603030210 \nu^{9} + \cdots - 340671582608193 ) / 45384886080676 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - 3\beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{9} - 6\beta_{8} - 2\beta_{7} + \beta_{6} - 5\beta_{5} + 6\beta_{4} - 6\beta_{3} - 5\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - 7\beta_{9} + 24\beta_{8} + 24\beta_{7} - 24\beta_{6} - 9\beta_{4} - 7\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{10} + 10\beta_{9} - 11\beta_{8} - 3\beta_{7} - 8\beta_{5} + 8\beta_{4} + 32\beta_{2} + 2\beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{11} - 48 \beta_{10} + 10 \beta_{9} + 62 \beta_{8} + 164 \beta_{6} + 70 \beta_{5} + \cdots - 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 83 \beta_{11} - 55 \beta_{10} - 110 \beta_{8} + 44 \beta_{7} - 99 \beta_{6} - 55 \beta_{4} + \cdots + 55 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 332 \beta_{11} - 83 \beta_{9} - 569 \beta_{8} - 1052 \beta_{7} + 458 \beta_{6} - 525 \beta_{5} + \cdots - 111 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 371 \beta_{11} + 371 \beta_{10} - 652 \beta_{9} + 3240 \beta_{8} + 1739 \beta_{7} - 2865 \beta_{6} + \cdots - 1739 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2324 \beta_{10} + 2324 \beta_{9} - 3253 \beta_{8} - 2797 \beta_{7} - 456 \beta_{5} + 456 \beta_{4} + \cdots + 11142 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2516 \beta_{11} - 4992 \beta_{10} + 2516 \beta_{9} + 4036 \beta_{8} + 21113 \beta_{6} + 11317 \beta_{5} + \cdots - 6641 \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{7}\)

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
1.85498 1.34772i
0.0307040 0.0223078i
−2.19470 + 1.59454i
1.85498 + 1.34772i
0.0307040 + 0.0223078i
−2.19470 1.59454i
0.830630 + 2.55642i
0.377272 + 1.16112i
−0.398885 1.22764i
0.830630 2.55642i
0.377272 1.16112i
−0.398885 + 1.22764i
0 −1.01756 + 3.13172i 0 0.809017 0.587785i 0 1.08622 + 3.34304i 0 −6.34518 4.61004i 0
81.2 0 −0.320745 + 0.987151i 0 0.809017 0.587785i 0 −0.804092 2.47474i 0 1.55546 + 1.13011i 0
81.3 0 0.529284 1.62897i 0 0.809017 0.587785i 0 1.14492 + 3.52372i 0 0.0536500 + 0.0389790i 0
201.1 0 −1.01756 3.13172i 0 0.809017 + 0.587785i 0 1.08622 3.34304i 0 −6.34518 + 4.61004i 0
201.2 0 −0.320745 0.987151i 0 0.809017 + 0.587785i 0 −0.804092 + 2.47474i 0 1.55546 1.13011i 0
201.3 0 0.529284 + 1.62897i 0 0.809017 + 0.587785i 0 1.14492 3.52372i 0 0.0536500 0.0389790i 0
361.1 0 −1.36560 + 0.992167i 0 −0.309017 0.951057i 0 0.156923 + 0.114011i 0 −0.0465816 + 0.143363i 0
361.2 0 −0.178694 + 0.129829i 0 −0.309017 0.951057i 0 1.68900 + 1.22713i 0 −0.911975 + 2.80677i 0
361.3 0 1.85331 1.34651i 0 −0.309017 0.951057i 0 −3.77298 2.74123i 0 0.694624 2.13783i 0
401.1 0 −1.36560 0.992167i 0 −0.309017 + 0.951057i 0 0.156923 0.114011i 0 −0.0465816 0.143363i 0
401.2 0 −0.178694 0.129829i 0 −0.309017 + 0.951057i 0 1.68900 1.22713i 0 −0.911975 2.80677i 0
401.3 0 1.85331 + 1.34651i 0 −0.309017 + 0.951057i 0 −3.77298 + 2.74123i 0 0.694624 + 2.13783i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.3
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.c 12
4.b odd 2 1 880.2.bo.i 12
11.c even 5 1 inner 440.2.y.c 12
11.c even 5 1 4840.2.a.bb 6
11.d odd 10 1 4840.2.a.ba 6
44.g even 10 1 9680.2.a.dd 6
44.h odd 10 1 880.2.bo.i 12
44.h odd 10 1 9680.2.a.dc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.c 12 1.a even 1 1 trivial
440.2.y.c 12 11.c even 5 1 inner
880.2.bo.i 12 4.b odd 2 1
880.2.bo.i 12 44.h odd 10 1
4840.2.a.ba 6 11.d odd 10 1
4840.2.a.bb 6 11.c even 5 1
9680.2.a.dc 6 44.h odd 10 1
9680.2.a.dd 6 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + T_{3}^{11} + 10 T_{3}^{10} - 6 T_{3}^{9} + 29 T_{3}^{8} + 54 T_{3}^{7} + 165 T_{3}^{6} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} - 18 T^{11} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 366025 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots + 95863681 \) Copy content Toggle raw display
$23$ \( (T^{6} + 9 T^{5} + \cdots + 5956)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} + \cdots + 1210000 \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 12931216 \) Copy content Toggle raw display
$37$ \( T^{12} - 6 T^{11} + \cdots + 3610000 \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} + \cdots + 14250625 \) Copy content Toggle raw display
$43$ \( (T^{6} + 18 T^{5} + \cdots + 3751)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 16 T^{11} + \cdots + 86192656 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 39921638416 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 138274081 \) Copy content Toggle raw display
$61$ \( T^{12} - 18 T^{11} + \cdots + 43824400 \) Copy content Toggle raw display
$67$ \( (T^{6} + 22 T^{5} + \cdots + 126475)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 6 T^{11} + \cdots + 384400 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 692275584961 \) Copy content Toggle raw display
$79$ \( T^{12} - 19 T^{11} + \cdots + 45212176 \) Copy content Toggle raw display
$83$ \( T^{12} + 180 T^{10} + \cdots + 1890625 \) Copy content Toggle raw display
$89$ \( (T^{6} + 25 T^{5} + \cdots - 22429)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27124443025 \) Copy content Toggle raw display
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