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Properties

Label 540.2.e.b
Level $540$
Weight $2$
Character orbit 540.e
Analytic conductor $4.312$
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] = [540,2,Mod(431,540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(540, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("540.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 540.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: \(4.31192170915\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.81094542259068665856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{7} + \beta_{6}) q^{7} + ( - \beta_{13} - \beta_{9} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{7} + \beta_{6}) q^{7} + ( - \beta_{13} - \beta_{9} + \beta_{3}) q^{8} + \beta_{6} q^{10} + (\beta_{15} + \beta_{14} + \cdots - \beta_{8}) q^{11}+ \cdots + (2 \beta_{13} - 2 \beta_{4} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} + 2 q^{10} - 16 q^{13} + 14 q^{16} + 4 q^{22} - 16 q^{25} - 28 q^{28} + 2 q^{34} + 16 q^{37} - 8 q^{40} + 22 q^{46} + 32 q^{49} + 44 q^{52} - 48 q^{58} + 32 q^{61} - 4 q^{64} - 24 q^{70} - 48 q^{73} + 18 q^{76} + 68 q^{82} - 16 q^{85} - 20 q^{88} - 12 q^{94} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} - 24\nu^{13} + 36\nu^{11} - 171\nu^{9} + 232\nu^{7} - 468\nu^{5} + 688\nu^{3} - 1328\nu ) / 1120 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} + 24\nu^{12} - 36\nu^{10} + 171\nu^{8} - 232\nu^{6} + 468\nu^{4} - 688\nu^{2} + 1328 ) / 560 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{15} + 9\nu^{13} - \nu^{11} + 16\nu^{9} - 12\nu^{7} + 48\nu^{5} - 48\nu^{3} + 128\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 9\nu^{11} + 6\nu^{9} - 12\nu^{7} + 8\nu^{5} - 48\nu^{3} - 32\nu ) / 320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{14} + 113\nu^{12} + 23\nu^{10} + 512\nu^{8} - 124\nu^{6} + 2816\nu^{4} + 1264\nu^{2} + 6976 ) / 2240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{14} - 39\nu^{12} - 169\nu^{10} - 116\nu^{8} - 708\nu^{6} - 848\nu^{4} - 912\nu^{2} - 2368 ) / 2240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} - \nu^{12} + 9\nu^{10} - 8\nu^{8} + 44\nu^{6} - 32\nu^{4} + 144\nu^{2} - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -37\nu^{15} + 153\nu^{13} - 177\nu^{11} + 972\nu^{9} - 1164\nu^{7} + 4016\nu^{5} - 1936\nu^{3} + 6016\nu ) / 4480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -37\nu^{15} + 153\nu^{13} - 177\nu^{11} + 972\nu^{9} - 1164\nu^{7} + 4016\nu^{5} - 1936\nu^{3} + 14976\nu ) / 4480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43\nu^{14} - 87\nu^{12} + 183\nu^{10} - 108\nu^{8} + 876\nu^{6} + 176\nu^{4} + 1584\nu^{2} + 576 ) / 2240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\nu^{14} - 27\nu^{12} + 163\nu^{10} - 188\nu^{8} + 436\nu^{6} - 544\nu^{4} + 1264\nu^{2} - 864 ) / 1120 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -3\nu^{14} + 2\nu^{12} - 10\nu^{10} + 23\nu^{8} - 52\nu^{6} + 60\nu^{4} - 160\nu^{2} + 176 ) / 112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\nu^{15} + 3\nu^{13} + 69\nu^{11} + 24\nu^{9} + 300\nu^{7} + 272\nu^{5} + 432\nu^{3} + 768\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -33\nu^{15} - 13\nu^{13} - 243\nu^{11} - 62\nu^{9} - 1076\nu^{7} + 184\nu^{5} - 3664\nu^{3} - 416\nu ) / 2240 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\nu^{15} - 47\nu^{13} + 263\nu^{11} - 418\nu^{9} + 956\nu^{7} - 1144\nu^{5} + 2864\nu^{3} - 3744\nu ) / 2240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{14} - 2\beta_{13} + \beta_{9} + \beta_{8} - \beta_{4} - 2\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{12} + \beta_{10} - 2\beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + 2\beta_{14} + 2\beta_{13} + 4\beta_{8} - \beta_{4} - 2\beta_{3} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{12} - 6\beta_{11} + \beta_{10} + \beta_{7} - 7\beta_{6} - 3\beta_{5} - \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{15} - \beta_{14} + 8\beta_{13} - 5\beta_{9} - \beta_{8} - 4\beta_{4} + 4\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2\beta_{12} - 2\beta_{11} + 4\beta_{10} + 11\beta_{7} + 6\beta_{6} - 5\beta_{5} + 14\beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -9\beta_{15} - 11\beta_{14} - 2\beta_{13} - 9\beta_{9} - \beta_{8} + 5\beta_{4} - 18\beta_{3} - 17\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 19\beta_{12} + 28\beta_{11} - 5\beta_{10} - 2\beta_{7} - 3\beta_{6} + 2\beta_{5} - \beta_{2} - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11\beta_{15} + 2\beta_{14} - 2\beta_{13} + 8\beta_{9} - 4\beta_{8} + 45\beta_{4} + 18\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -19\beta_{12} - 2\beta_{11} - 45\beta_{10} - 5\beta_{7} - 37\beta_{6} - \beta_{5} - 3\beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16\beta_{15} + 5\beta_{14} - 8\beta_{13} + \beta_{9} - 3\beta_{8} - 20\beta_{4} + 172\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -98\beta_{12} - 6\beta_{11} + 20\beta_{10} - 23\beta_{7} + 26\beta_{6} - 7\beta_{5} + 66\beta_{2} + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 85\beta_{15} + 23\beta_{14} - 6\beta_{13} - 3\beta_{9} - 11\beta_{8} - 97\beta_{4} - 86\beta_{3} - 155\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.929502 + 1.06584i
0.929502 − 1.06584i
1.12988 + 0.850516i
1.12988 − 0.850516i
−0.638735 − 1.26175i
−0.638735 + 1.26175i
1.30439 − 0.546424i
1.30439 + 0.546424i
−1.30439 − 0.546424i
−1.30439 + 0.546424i
0.638735 − 1.26175i
0.638735 + 1.26175i
−1.12988 + 0.850516i
−1.12988 − 0.850516i
−0.929502 + 1.06584i
−0.929502 − 1.06584i
−1.38780 − 0.272050i 0 1.85198 + 0.755103i − 1.00000i 0 3.36921i −2.36475 − 1.55176i 0 −0.272050 + 1.38780i
431.2 −1.38780 + 0.272050i 0 1.85198 − 0.755103i 1.00000i 0 − 3.36921i −2.36475 + 1.55176i 0 −0.272050 − 1.38780i
431.3 −1.30151 − 0.553244i 0 1.38784 + 1.44010i 1.00000i 0 0.620450i −1.00956 − 2.64212i 0 0.553244 − 1.30151i
431.4 −1.30151 + 0.553244i 0 1.38784 − 1.44010i − 1.00000i 0 − 0.620450i −1.00956 + 2.64212i 0 0.553244 + 1.30151i
431.5 −0.773342 − 1.18404i 0 −0.803884 + 1.83133i − 1.00000i 0 2.38519i 2.79004 − 0.464416i 0 −1.18404 + 0.773342i
431.6 −0.773342 + 1.18404i 0 −0.803884 − 1.83133i 1.00000i 0 − 2.38519i 2.79004 + 0.464416i 0 −1.18404 − 0.773342i
431.7 −0.178976 − 1.40284i 0 −1.93594 + 0.502151i 1.00000i 0 − 1.60447i 1.05092 + 2.62594i 0 1.40284 − 0.178976i
431.8 −0.178976 + 1.40284i 0 −1.93594 − 0.502151i − 1.00000i 0 1.60447i 1.05092 − 2.62594i 0 1.40284 + 0.178976i
431.9 0.178976 − 1.40284i 0 −1.93594 − 0.502151i 1.00000i 0 1.60447i −1.05092 + 2.62594i 0 1.40284 + 0.178976i
431.10 0.178976 + 1.40284i 0 −1.93594 + 0.502151i − 1.00000i 0 − 1.60447i −1.05092 − 2.62594i 0 1.40284 − 0.178976i
431.11 0.773342 − 1.18404i 0 −0.803884 − 1.83133i − 1.00000i 0 − 2.38519i −2.79004 − 0.464416i 0 −1.18404 − 0.773342i
431.12 0.773342 + 1.18404i 0 −0.803884 + 1.83133i 1.00000i 0 2.38519i −2.79004 + 0.464416i 0 −1.18404 + 0.773342i
431.13 1.30151 − 0.553244i 0 1.38784 − 1.44010i 1.00000i 0 − 0.620450i 1.00956 − 2.64212i 0 0.553244 + 1.30151i
431.14 1.30151 + 0.553244i 0 1.38784 + 1.44010i − 1.00000i 0 0.620450i 1.00956 + 2.64212i 0 0.553244 − 1.30151i
431.15 1.38780 − 0.272050i 0 1.85198 − 0.755103i − 1.00000i 0 − 3.36921i 2.36475 − 1.55176i 0 −0.272050 − 1.38780i
431.16 1.38780 + 0.272050i 0 1.85198 + 0.755103i 1.00000i 0 3.36921i 2.36475 + 1.55176i 0 −0.272050 + 1.38780i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.2.e.b ✓ 16
3.b odd 2 1 inner 540.2.e.b ✓ 16
4.b odd 2 1 inner 540.2.e.b ✓ 16
12.b even 2 1 inner 540.2.e.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.e.b ✓ 16 1.a even 1 1 trivial
540.2.e.b ✓ 16 3.b odd 2 1 inner
540.2.e.b ✓ 16 4.b odd 2 1 inner
540.2.e.b ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 20T_{7}^{6} + 116T_{7}^{4} + 208T_{7}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(540, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 20 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 52 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots - 56)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 64 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 64 T^{6} + \cdots + 47089)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 68 T^{6} + \cdots + 8281)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 156 T^{6} + \cdots + 817216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 200 T^{6} + \cdots + 5368489)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 2608)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 188 T^{6} + \cdots + 652864)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 196 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 248 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 216 T^{6} + \cdots + 10201)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 180 T^{6} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots + 757)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 368 T^{6} + \cdots + 1290496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 436 T^{6} + \cdots + 19998784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12 T^{3} - 46 T^{2} + \cdots - 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 344 T^{6} + \cdots + 121801)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 188 T^{6} + \cdots + 1079521)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 348 T^{6} + \cdots + 12559936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 12 T^{3} + \cdots + 32256)^{4} \) Copy content Toggle raw display
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