Properties

Label 440.2.g.b
Level $440$
Weight $2$
Character orbit 440.g
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(221,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.g (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.192526503153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} - x^{10} + 2x^{9} + 6x^{8} - 20x^{6} + 24x^{4} + 16x^{3} - 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + (\beta_{3} + \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{8} q^{5} + ( - \beta_{8} - \beta_{7}) q^{6} + (\beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{10} - \beta_{9} + \beta_{6} + \beta_{3} + 1) q^{8} + (\beta_{5} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + (\beta_{3} + \beta_1) q^{3} - \beta_{5} q^{4} - \beta_{8} q^{5} + ( - \beta_{8} - \beta_{7}) q^{6} + (\beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{10} - \beta_{9} + \beta_{6} + \beta_{3} + 1) q^{8} + (\beta_{5} + \beta_{2} - 1) q^{9} + \beta_{3} q^{10} - \beta_{8} q^{11} + (\beta_{11} - \beta_{8} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{12} + (\beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8}) q^{13} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{14} + ( - \beta_{11} - \beta_{9}) q^{15} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{16} + (\beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{17} + ( - 2 \beta_{11} + \beta_{10} - \beta_{6} - \beta_{3} - 1) q^{18} + ( - \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{19} + (\beta_{8} - \beta_{7}) q^{20} + (2 \beta_{10} - \beta_{7} - \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{21} + \beta_{3} q^{22} + ( - 3 \beta_{11} - 3 \beta_{9} - \beta_{5} - 2 \beta_{4} - \beta_{2} + 2) q^{23} + (2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{24} - q^{25} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3) q^{26} + ( - \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{27} + (\beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 1) q^{28} + ( - \beta_{11} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{29} + (\beta_{5} - 2) q^{30} + ( - \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{31} + ( - \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots + 1) q^{32}+ \cdots + ( - \beta_{8} + \beta_{7} + \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{4} - 4 q^{6} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{4} - 4 q^{6} + 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{12} + 6 q^{14} - 6 q^{16} - 16 q^{17} - 6 q^{18} - 4 q^{20} - 2 q^{22} + 12 q^{23} + 8 q^{24} - 12 q^{25} + 32 q^{26} - 22 q^{28} - 18 q^{30} + 12 q^{31} + 10 q^{32} + 18 q^{34} - 36 q^{36} - 16 q^{38} - 8 q^{39} + 8 q^{40} + 64 q^{41} + 28 q^{42} - 4 q^{44} - 28 q^{46} + 4 q^{47} + 44 q^{48} + 32 q^{49} - 24 q^{52} - 24 q^{54} - 12 q^{55} + 14 q^{56} + 36 q^{57} - 6 q^{60} - 10 q^{62} + 40 q^{63} - 6 q^{64} - 20 q^{65} - 18 q^{66} - 10 q^{68} - 12 q^{70} - 52 q^{71} - 16 q^{72} - 28 q^{73} + 12 q^{74} - 44 q^{76} - 44 q^{78} - 8 q^{79} + 16 q^{80} - 36 q^{81} + 36 q^{82} - 20 q^{84} - 12 q^{86} + 12 q^{87} + 8 q^{88} + 44 q^{89} - 10 q^{90} + 16 q^{92} + 12 q^{95} + 24 q^{96} - 24 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} - x^{10} + 2x^{9} + 6x^{8} - 20x^{6} + 24x^{4} + 16x^{3} - 16x^{2} - 64x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 2\nu^{10} - \nu^{9} + 2\nu^{8} + 6\nu^{7} - 20\nu^{5} + 24\nu^{3} + 16\nu^{2} - 16\nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{11} + 7 \nu^{9} + 8 \nu^{8} - 14 \nu^{7} - 28 \nu^{6} + 20 \nu^{5} + 56 \nu^{4} + 24 \nu^{3} - 64 \nu^{2} - 80 \nu + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + \nu^{10} + 3\nu^{9} - \nu^{8} - 8\nu^{7} - 6\nu^{6} + 20\nu^{5} + 20\nu^{4} - 24\nu^{3} - 40\nu^{2} + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} + \nu^{9} + 12 \nu^{8} + 18 \nu^{7} - 12 \nu^{6} - 60 \nu^{5} - 8 \nu^{4} + 88 \nu^{3} + 96 \nu^{2} - 48 \nu - 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{10} + 2 \nu^{9} - 7 \nu^{8} - 14 \nu^{7} + 6 \nu^{6} + 36 \nu^{5} + 12 \nu^{4} - 48 \nu^{3} - 56 \nu^{2} + 16 \nu + 96 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} + 4 \nu^{8} - 30 \nu^{7} - 44 \nu^{6} + 52 \nu^{5} + 88 \nu^{4} - 8 \nu^{3} - 128 \nu^{2} - 112 \nu + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - \nu^{10} - 7 \nu^{9} - 7 \nu^{8} + 14 \nu^{7} + 30 \nu^{6} - 16 \nu^{5} - 52 \nu^{4} - 16 \nu^{3} + 56 \nu^{2} + 80 \nu - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{11} + 3 \nu^{10} + 9 \nu^{9} + 5 \nu^{8} - 20 \nu^{7} - 34 \nu^{6} + 32 \nu^{5} + 68 \nu^{4} + 8 \nu^{3} - 88 \nu^{2} - 96 \nu + 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 22 \nu^{7} - 24 \nu^{6} + 44 \nu^{5} + 56 \nu^{4} - 24 \nu^{3} - 96 \nu^{2} - 64 \nu + 160 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} + \beta_{9} - 4\beta_{8} + 2\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + 3 \beta_{10} + 6 \beta_{9} - \beta_{8} - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - 8 \beta_{3} - 3 \beta_{2} - 3 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9 \beta_{11} + 5 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 5 \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6 \beta_{11} + 8 \beta_{10} + \beta_{9} - 16 \beta_{8} + 2 \beta_{7} + 7 \beta_{6} + \beta_{5} - 5 \beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 3 \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 19 \beta_{11} + 15 \beta_{10} - 6 \beta_{9} - \beta_{8} + \beta_{7} - 11 \beta_{6} - 10 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta _1 + 15 \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−0.856842 1.12509i
−0.856842 + 1.12509i
1.12994 + 0.850428i
1.12994 0.850428i
−1.37658 0.324093i
−1.37658 + 0.324093i
1.40059 0.195848i
1.40059 + 0.195848i
1.18373 0.773803i
1.18373 + 0.773803i
−0.480846 + 1.32996i
−0.480846 1.32996i
−1.12509 0.856842i 2.25017i 0.531643 + 1.92804i 1.00000i −1.92804 + 2.53164i −4.09451 1.05388 2.62475i −2.06329 0.856842 1.12509i
221.2 −1.12509 + 0.856842i 2.25017i 0.531643 1.92804i 1.00000i −1.92804 2.53164i −4.09451 1.05388 + 2.62475i −2.06329 0.856842 + 1.12509i
221.3 −0.850428 1.12994i 1.70086i −0.553545 + 1.92187i 1.00000i 1.92187 1.44646i 1.51507 2.64236 1.00894i 0.107089 −1.12994 + 0.850428i
221.4 −0.850428 + 1.12994i 1.70086i −0.553545 1.92187i 1.00000i 1.92187 + 1.44646i 1.51507 2.64236 + 1.00894i 0.107089 −1.12994 0.850428i
221.5 −0.324093 1.37658i 0.648186i −1.78993 + 0.892278i 1.00000i −0.892278 + 0.210073i 1.46292 1.80839 + 2.17479i 2.57985 1.37658 0.324093i
221.6 −0.324093 + 1.37658i 0.648186i −1.78993 0.892278i 1.00000i −0.892278 0.210073i 1.46292 1.80839 2.17479i 2.57985 1.37658 + 0.324093i
221.7 0.195848 1.40059i 0.391695i −1.92329 0.548603i 1.00000i −0.548603 0.0767126i 5.05421 −1.14504 + 2.58629i 2.84657 −1.40059 0.195848i
221.8 0.195848 + 1.40059i 0.391695i −1.92329 + 0.548603i 1.00000i −0.548603 + 0.0767126i 5.05421 −1.14504 2.58629i 2.84657 −1.40059 + 0.195848i
221.9 0.773803 1.18373i 1.54761i −0.802457 1.83196i 1.00000i −1.83196 1.19754i −3.13415 −2.78949 0.467677i 0.604914 −1.18373 0.773803i
221.10 0.773803 + 1.18373i 1.54761i −0.802457 + 1.83196i 1.00000i −1.83196 + 1.19754i −3.13415 −2.78949 + 0.467677i 0.604914 −1.18373 + 0.773803i
221.11 1.32996 0.480846i 2.65991i 1.53757 1.27901i 1.00000i 1.27901 + 3.53757i 1.19647 1.42990 2.44037i −4.07515 0.480846 + 1.32996i
221.12 1.32996 + 0.480846i 2.65991i 1.53757 + 1.27901i 1.00000i 1.27901 3.53757i 1.19647 1.42990 + 2.44037i −4.07515 0.480846 1.32996i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.g.b 12
4.b odd 2 1 1760.2.g.b 12
8.b even 2 1 inner 440.2.g.b 12
8.d odd 2 1 1760.2.g.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.g.b 12 1.a even 1 1 trivial
440.2.g.b 12 8.b even 2 1 inner
1760.2.g.b 12 4.b odd 2 1
1760.2.g.b 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 18T_{3}^{10} + 117T_{3}^{8} + 336T_{3}^{6} + 412T_{3}^{4} + 160T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{10} - 2 T^{9} + 6 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 18 T^{10} + 117 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} - 27 T^{4} + 44 T^{3} + \cdots + 172)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 80 T^{10} + 2460 T^{8} + \cdots + 565504 \) Copy content Toggle raw display
$17$ \( (T^{6} + 8 T^{5} - 5 T^{4} - 110 T^{3} + \cdots + 124)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 90 T^{10} + 1753 T^{8} + \cdots + 102400 \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} - 58 T^{4} + 228 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 122 T^{10} + 4281 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} - 95 T^{4} + 416 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 138 T^{10} + \cdots + 24127744 \) Copy content Toggle raw display
$41$ \( (T^{6} - 32 T^{5} + 332 T^{4} - 1008 T^{3} + \cdots - 6016)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 240 T^{10} + \cdots + 266864896 \) Copy content Toggle raw display
$47$ \( (T^{6} - 2 T^{5} - 194 T^{4} + 740 T^{3} + \cdots + 67552)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 410 T^{10} + \cdots + 9231366400 \) Copy content Toggle raw display
$59$ \( T^{12} + 600 T^{10} + \cdots + 365628227584 \) Copy content Toggle raw display
$61$ \( T^{12} + 418 T^{10} + \cdots + 491065600 \) Copy content Toggle raw display
$67$ \( T^{12} + 568 T^{10} + \cdots + 255437246464 \) Copy content Toggle raw display
$71$ \( (T^{6} + 26 T^{5} + 77 T^{4} + \cdots + 181904)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 14 T^{5} - 94 T^{4} - 1268 T^{3} + \cdots - 18064)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 4 T^{5} - 120 T^{4} - 240 T^{3} + \cdots - 5888)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 416 T^{10} + \cdots + 10653542656 \) Copy content Toggle raw display
$89$ \( (T^{6} - 22 T^{5} - 75 T^{4} + 3712 T^{3} + \cdots - 14576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 12 T^{5} - 132 T^{4} + \cdots - 33728)^{2} \) Copy content Toggle raw display
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