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Properties

Label 460.2.e.a
Level $460$
Weight $2$
Character orbit 460.e
Analytic conductor $3.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [460,2,Mod(91,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.e (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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.7465802011608416256.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{8} - \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{6}+ \cdots + (\beta_{14} - \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{8} - \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{6}+ \cdots + (5 \beta_{15} - 2 \beta_{13} + \cdots - 9 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 14 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 14 q^{6} + 4 q^{9} - 30 q^{12} + 4 q^{13} + 4 q^{16} + 30 q^{18} + 2 q^{24} - 16 q^{25} - 54 q^{26} - 48 q^{29} + 34 q^{36} - 36 q^{41} - 40 q^{46} + 18 q^{48} + 68 q^{49} + 34 q^{52} - 40 q^{54} + 36 q^{58} + 6 q^{62} + 52 q^{64} + 40 q^{69} + 42 q^{70} - 78 q^{72} + 8 q^{73} + 72 q^{77} + 32 q^{78} + 40 q^{81} - 42 q^{82} + 12 q^{85} - 120 q^{93} + 20 q^{94} - 22 q^{96} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{15} + 5\nu^{13} + 3\nu^{11} + 4\nu^{9} + 172\nu^{7} - 96\nu^{5} - 112\nu^{3} - 256\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - \nu^{12} - 15\nu^{10} - 2\nu^{8} + 28\nu^{6} - 96\nu^{4} + 80\nu^{2} - 160 ) / 288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 7\nu^{13} + 9\nu^{11} - 16\nu^{9} + 44\nu^{7} + 144\nu^{5} - 80\nu^{3} + 640\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 15 \nu^{11} - 12 \nu^{10} - 27 \nu^{9} - 22 \nu^{8} + \cdots + 64 ) / 576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} + 11 \nu^{13} - 6 \nu^{12} - 15 \nu^{11} + 30 \nu^{10} + 22 \nu^{9} + \cdots + 768 ) / 1152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 15 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 24 \nu^{7} + \cdots - 64 ) / 576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 9\nu^{13} + 9\nu^{11} - 16\nu^{9} - 36\nu^{7} + 144\nu^{5} - 80\nu^{3} ) / 384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} - 4 \nu^{14} - 3 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 24 \nu^{10} - 54 \nu^{9} + \cdots + 128 ) / 1152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{14} + 9 \nu^{13} - 4 \nu^{12} - 9 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 36 \nu^{7} + \cdots - 64 ) / 576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{15} + 8 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} - 39 \nu^{11} + 36 \nu^{10} - 12 \nu^{9} + \cdots + 896 ) / 1152 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{15} - 6 \nu^{14} + 11 \nu^{13} + 30 \nu^{12} - 15 \nu^{11} - 6 \nu^{10} + 22 \nu^{9} + \cdots + 1152 ) / 1152 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 4\nu^{12} + 8\nu^{10} + 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} - 8\nu^{3} - 128\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + 4\nu^{12} + 5\nu^{9} + 20\nu^{6} - 96\nu^{4} - 8\nu^{3} + 96\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{15} + 8 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 39 \nu^{11} + 36 \nu^{10} + 12 \nu^{9} + \cdots + 896 ) / 1152 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{15} - 4\nu^{12} + 8\nu^{10} - 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} + 8\nu^{3} - 128\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} + \beta_{3} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{12} + \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} + 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} + \cdots + 2 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} + \beta_{12} + 5\beta_{11} + 5\beta_{7} + 3\beta_{6} + 5\beta_{5} - 3\beta_{4} + 6\beta_{2} + 5\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{14} + 5\beta_{10} - \beta_{9} - 10\beta_{8} + 7\beta_{7} - 3\beta_{6} + 6\beta_{4} + 5\beta_{3} + 26\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9 \beta_{15} - 25 \beta_{14} - \beta_{13} - 8 \beta_{12} - 18 \beta_{11} - 25 \beta_{10} + 9 \beta_{7} + \cdots + 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{15} + \beta_{12} - 21\beta_{9} - 9\beta_{8} - 13\beta_{7} + 6\beta_{6} - 6\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 49 \beta_{15} - 13 \beta_{14} + 43 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 13 \beta_{10} + 3 \beta_{7} + \cdots - 35 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22 \beta_{15} + 61 \beta_{14} - 22 \beta_{12} - 61 \beta_{10} + 7 \beta_{9} - 17 \beta_{7} + \cdots + 22 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 29 \beta_{15} - 29 \beta_{12} - \beta_{11} - \beta_{7} - 39 \beta_{6} - \beta_{5} + 39 \beta_{4} + \cdots - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 65 \beta_{14} - 65 \beta_{10} + 45 \beta_{9} + 130 \beta_{8} - 91 \beta_{7} - 25 \beta_{6} + \cdots - 50 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 43 \beta_{15} + 229 \beta_{14} + 109 \beta_{13} - 152 \beta_{12} - 86 \beta_{11} + 229 \beta_{10} + \cdots - 203 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -179\beta_{15} + 179\beta_{12} + 113\beta_{9} + 21\beta_{8} + 73\beta_{7} - 46\beta_{6} + 46\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.18353 − 0.774115i
−0.0786378 + 1.41203i
−0.0786378 − 1.41203i
1.18353 + 0.774115i
1.37379 − 0.335728i
0.977642 − 1.02187i
0.977642 + 1.02187i
1.37379 + 0.335728i
−0.977642 + 1.02187i
−1.37379 + 0.335728i
−1.37379 − 0.335728i
−0.977642 − 1.02187i
0.0786378 − 1.41203i
−1.18353 + 0.774115i
−1.18353 − 0.774115i
0.0786378 + 1.41203i
−1.35760 − 0.396143i 1.47175i 1.68614 + 1.07561i − 1.00000i 0.583024 − 1.99804i −3.53986 −1.86301 − 2.12819i 0.833952 −0.396143 + 1.35760i
91.2 −1.35760 − 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 − 1.99804i 3.53986 −1.86301 − 2.12819i 0.833952 0.396143 − 1.35760i
91.3 −1.35760 + 0.396143i − 1.47175i 1.68614 − 1.07561i − 1.00000i 0.583024 + 1.99804i 3.53986 −1.86301 + 2.12819i 0.833952 0.396143 + 1.35760i
91.4 −1.35760 + 0.396143i − 1.47175i 1.68614 − 1.07561i 1.00000i 0.583024 + 1.99804i −3.53986 −1.86301 + 2.12819i 0.833952 −0.396143 − 1.35760i
91.5 −0.637910 − 1.26217i 2.87247i −1.18614 + 1.61030i − 1.00000i 3.62554 − 1.83238i −2.68161 2.78912 + 0.469882i −5.25109 −1.26217 + 0.637910i
91.6 −0.637910 − 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 − 1.83238i 2.68161 2.78912 + 0.469882i −5.25109 1.26217 − 0.637910i
91.7 −0.637910 + 1.26217i − 2.87247i −1.18614 − 1.61030i − 1.00000i 3.62554 + 1.83238i 2.68161 2.78912 − 0.469882i −5.25109 1.26217 + 0.637910i
91.8 −0.637910 + 1.26217i − 2.87247i −1.18614 − 1.61030i 1.00000i 3.62554 + 1.83238i −2.68161 2.78912 − 0.469882i −5.25109 −1.26217 − 0.637910i
91.9 0.637910 − 1.26217i − 0.348132i −1.18614 − 1.61030i − 1.00000i −0.439402 − 0.222077i −4.89140 −2.78912 + 0.469882i 2.87880 −1.26217 − 0.637910i
91.10 0.637910 − 1.26217i − 0.348132i −1.18614 − 1.61030i 1.00000i −0.439402 − 0.222077i 4.89140 −2.78912 + 0.469882i 2.87880 1.26217 + 0.637910i
91.11 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i − 1.00000i −0.439402 + 0.222077i 4.89140 −2.78912 − 0.469882i 2.87880 1.26217 − 0.637910i
91.12 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i −4.89140 −2.78912 − 0.469882i 2.87880 −1.26217 + 0.637910i
91.13 1.35760 − 0.396143i − 0.679463i 1.68614 − 1.07561i − 1.00000i −0.269165 − 0.922437i 1.16300 1.86301 − 2.12819i 2.53833 −0.396143 − 1.35760i
91.14 1.35760 − 0.396143i − 0.679463i 1.68614 − 1.07561i 1.00000i −0.269165 − 0.922437i −1.16300 1.86301 − 2.12819i 2.53833 0.396143 + 1.35760i
91.15 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i − 1.00000i −0.269165 + 0.922437i −1.16300 1.86301 + 2.12819i 2.53833 0.396143 − 1.35760i
91.16 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i 1.16300 1.86301 + 2.12819i 2.53833 −0.396143 + 1.35760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.e.a ✓ 16
4.b odd 2 1 inner 460.2.e.a ✓ 16
23.b odd 2 1 inner 460.2.e.a ✓ 16
92.b even 2 1 inner 460.2.e.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.e.a ✓ 16 1.a even 1 1 trivial
460.2.e.a ✓ 16 4.b odd 2 1 inner
460.2.e.a ✓ 16 23.b odd 2 1 inner
460.2.e.a ✓ 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 11T_{3}^{6} + 24T_{3}^{4} + 11T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 11 T^{6} + 24 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} - 24 T^{2} + \cdots + 31)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 99 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{6} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots + 12)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 207 T^{6} + \cdots + 4124961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 108 T^{6} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 9 T^{3} + \cdots - 681)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 180 T^{6} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 362 T^{6} + \cdots + 163216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 180 T^{2} + 6912)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 76 T^{2} + 256)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 243 T^{6} + \cdots + 13089924)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 288 T^{6} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 215 T^{6} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + \cdots - 908)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 324 T^{6} + \cdots + 3779136)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 252 T^{2} + 5184)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 396 T^{6} + \cdots + 13483584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 531 T^{6} + \cdots + 29746116)^{2} \) Copy content Toggle raw display
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