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Properties

Label 440.2.y.b
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: \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 1) q^{5} + (\beta_{9} - \beta_{8} + \cdots - \beta_{6}) q^{7}+ \cdots + (\beta_{11} + \beta_{10} - \beta_{9} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 1) q^{5} + (\beta_{9} - \beta_{8} + \cdots - \beta_{6}) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} + 3 q^{19} + 4 q^{21} + 36 q^{23} - 3 q^{25} + 8 q^{27} + 13 q^{29} + 2 q^{31} - 19 q^{33} - 2 q^{35} - 22 q^{37} - q^{39} + 7 q^{41} + 6 q^{43} - 20 q^{45} - 2 q^{47} - 19 q^{49} - 33 q^{51} + 3 q^{53} + 4 q^{55} - 25 q^{57} + 19 q^{59} + 22 q^{61} - 2 q^{63} + 12 q^{65} + 22 q^{67} + 21 q^{69} + 44 q^{71} + 17 q^{73} - q^{75} - 38 q^{77} - 43 q^{79} - 85 q^{81} - 15 q^{83} + 18 q^{85} + 50 q^{87} + 2 q^{89} + 59 q^{91} + 5 q^{93} - 3 q^{95} - 20 q^{97} - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 183142278447 \nu^{11} - 2277678635906 \nu^{10} + 1497463471915 \nu^{9} + \cdots - 16\!\cdots\!00 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 147497735591 \nu^{11} - 5588599527022 \nu^{10} + 8587750298190 \nu^{9} + \cdots - 66\!\cdots\!00 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1378720916147 \nu^{11} - 1842154717364 \nu^{10} + 18603057405925 \nu^{9} + \cdots - 924275028852200 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1378720916147 \nu^{11} - 1842154717364 \nu^{10} + 18603057405925 \nu^{9} + \cdots - 924275028852200 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1424041493464 \nu^{11} + 3116845670683 \nu^{10} - 20053332036635 \nu^{9} + \cdots + 850171561945900 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4121259436790 \nu^{11} + 8425661152027 \nu^{10} - 59541212915944 \nu^{9} + \cdots + 48\!\cdots\!80 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8501715619459 \nu^{11} + 11307265265062 \nu^{10} - 115058351609153 \nu^{9} + \cdots - 31294691684920 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8983553542643 \nu^{11} - 18514673951448 \nu^{10} + 135296709681109 \nu^{9} + \cdots - 96\!\cdots\!20 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18735151134651 \nu^{11} + 22105897203158 \nu^{10} - 244513689144357 \nu^{9} + \cdots - 101022986909080 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1941584272431 \nu^{11} - 2386060570628 \nu^{10} + 28042643669239 \nu^{9} + \cdots + 56036595164120 ) / 521205462484916 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5268988587323 \nu^{11} - 10133555287451 \nu^{10} + 80319172840620 \nu^{9} + \cdots - 94\!\cdots\!00 ) / 13\!\cdots\!90 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 4\beta_{8} - 4\beta_{7} + 6\beta_{6} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + 7\beta_{4} - 2\beta_{3} + \beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} + 9\beta_{9} - 22\beta_{8} + 9\beta_{7} - 46\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 9\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{11} - 13\beta_{10} + 6\beta_{7} - 6\beta_{6} - 25\beta_{5} - 25\beta_{4} - 23\beta_{2} + 30\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 38 \beta_{10} - 78 \beta_{9} + 210 \beta_{8} + 92 \beta_{7} + 210 \beta_{6} + 16 \beta_{5} + \cdots + 16 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 137 \beta_{11} + 94 \beta_{10} - 94 \beta_{9} + 52 \beta_{8} - 146 \beta_{7} + 158 \beta_{6} + \cdots - 158 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 385 \beta_{11} + 689 \beta_{10} + 385 \beta_{9} - 1336 \beta_{8} - 1523 \beta_{7} + 295 \beta_{4} + \cdots - 1908 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 889 \beta_{11} + 1369 \beta_{9} - 1390 \beta_{8} + 1369 \beta_{7} - 2170 \beta_{6} - 3933 \beta_{5} + \cdots + 1390 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6191 \beta_{11} - 6191 \beta_{10} + 17038 \beta_{7} - 17038 \beta_{6} - 2279 \beta_{5} - 2279 \beta_{4} + \cdots + 28184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8470 \beta_{10} - 13392 \beta_{9} + 16194 \beta_{8} - 3794 \beta_{7} + 16194 \beta_{6} + \cdots + 20700 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−0.866917 − 2.66810i
0.421568 + 1.29745i
0.945349 + 2.90948i
−0.866917 + 2.66810i
0.421568 − 1.29745i
0.945349 − 2.90948i
1.65720 + 1.20403i
0.359793 + 0.261405i
−1.51700 − 1.10216i
1.65720 − 1.20403i
0.359793 − 0.261405i
−1.51700 + 1.10216i
0 −0.535784 + 1.64897i 0 0.809017 − 0.587785i 0 −0.386966 − 1.19096i 0 −0.00499969 − 0.00363249i 0
81.2 0 0.260543 − 0.801870i 0 0.809017 − 0.587785i 0 1.46997 + 4.52411i 0 1.85194 + 1.34551i 0
81.3 0 0.584258 − 1.79816i 0 0.809017 − 0.587785i 0 −0.846938 − 2.60661i 0 −0.464972 − 0.337822i 0
201.1 0 −0.535784 − 1.64897i 0 0.809017 + 0.587785i 0 −0.386966 + 1.19096i 0 −0.00499969 + 0.00363249i 0
201.2 0 0.260543 + 0.801870i 0 0.809017 + 0.587785i 0 1.46997 − 4.52411i 0 1.85194 − 1.34551i 0
201.3 0 0.584258 + 1.79816i 0 0.809017 + 0.587785i 0 −0.846938 + 2.60661i 0 −0.464972 + 0.337822i 0
361.1 0 −2.68141 + 1.94816i 0 −0.309017 − 0.951057i 0 −0.150443 − 0.109303i 0 2.46759 − 7.59446i 0
361.2 0 −0.582157 + 0.422962i 0 −0.309017 − 0.951057i 0 −3.38507 − 2.45940i 0 −0.767041 + 2.36071i 0
361.3 0 2.45455 − 1.78334i 0 −0.309017 − 0.951057i 0 −0.700550 − 0.508979i 0 1.91748 − 5.90141i 0
401.1 0 −2.68141 − 1.94816i 0 −0.309017 + 0.951057i 0 −0.150443 + 0.109303i 0 2.46759 + 7.59446i 0
401.2 0 −0.582157 − 0.422962i 0 −0.309017 + 0.951057i 0 −3.38507 + 2.45940i 0 −0.767041 − 2.36071i 0
401.3 0 2.45455 + 1.78334i 0 −0.309017 + 0.951057i 0 −0.700550 + 0.508979i 0 1.91748 + 5.90141i 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.b ✓ 12
4.b odd 2 1 880.2.bo.j 12
11.c even 5 1 inner 440.2.y.b ✓ 12
11.c even 5 1 4840.2.a.bf 6
11.d odd 10 1 4840.2.a.be 6
44.g even 10 1 9680.2.a.cy 6
44.h odd 10 1 880.2.bo.j 12
44.h odd 10 1 9680.2.a.cx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.b ✓ 12 1.a even 1 1 trivial
440.2.y.b ✓ 12 11.c even 5 1 inner
880.2.bo.j 12 4.b odd 2 1
880.2.bo.j 12 44.h odd 10 1
4840.2.a.be 6 11.d odd 10 1
4840.2.a.bf 6 11.c even 5 1
9680.2.a.cx 6 44.h odd 10 1
9680.2.a.cy 6 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + T_{3}^{11} - T_{3}^{9} + 79 T_{3}^{8} + 19 T_{3}^{7} + 500 T_{3}^{6} + 311 T_{3}^{5} + \cdots + 400 \) 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
$3$ \( T^{12} + T^{11} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 7 T^{11} + \cdots + 22801 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{11} + \cdots + 234256 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 3025 \) Copy content Toggle raw display
$23$ \( (T^{6} - 18 T^{5} + \cdots + 9505)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 13 T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots + 8880400 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 219662041 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1401828481 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3 T^{5} + \cdots + 75284)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + \cdots + 990025 \) Copy content Toggle raw display
$53$ \( T^{12} - 3 T^{11} + \cdots + 50625 \) Copy content Toggle raw display
$59$ \( T^{12} - 19 T^{11} + \cdots + 101761 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 4935905536 \) Copy content Toggle raw display
$67$ \( (T^{6} - 11 T^{5} + \cdots - 19900)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 4157154576 \) Copy content Toggle raw display
$73$ \( T^{12} - 17 T^{11} + \cdots + 10523536 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 220938496 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 115379784976 \) Copy content Toggle raw display
$89$ \( (T^{6} - T^{5} + \cdots + 179771)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1613062564096 \) Copy content Toggle raw display
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