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Properties

Label 546.4.c.a
Level $546$
Weight $4$
Character orbit 546.c
Analytic conductor $32.215$
Analytic rank $0$
Dimension $10$
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] = [546,4,Mod(337,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.c (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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 650x^{8} + 157785x^{6} + 16980224x^{4} + 687898432x^{2} + 529552144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{5} q^{2} - 3 q^{3} - 4 q^{4} + \beta_1 q^{5} + 6 \beta_{5} q^{6} + 7 \beta_{5} q^{7} + 8 \beta_{5} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{5} q^{2} - 3 q^{3} - 4 q^{4} + \beta_1 q^{5} + 6 \beta_{5} q^{6} + 7 \beta_{5} q^{7} + 8 \beta_{5} q^{8} + 9 q^{9} - 2 \beta_{2} q^{10} + ( - \beta_{8} + \beta_{7} + \beta_{6}) q^{11} + 12 q^{12} + (\beta_{9} + \beta_{7} - 2 \beta_{5} + \cdots - 10) q^{13}+ \cdots + ( - 9 \beta_{8} + 9 \beta_{7} + 9 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 30 q^{3} - 40 q^{4} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 30 q^{3} - 40 q^{4} + 90 q^{9} + 120 q^{12} - 100 q^{13} + 140 q^{14} + 160 q^{16} + 142 q^{17} + 4 q^{22} + 136 q^{23} - 50 q^{25} - 44 q^{26} - 270 q^{27} - 80 q^{29} - 360 q^{36} - 288 q^{38} + 300 q^{39} - 420 q^{42} - 304 q^{43} - 480 q^{48} - 490 q^{49} - 426 q^{51} + 400 q^{52} - 1550 q^{53} - 902 q^{55} - 560 q^{56} - 678 q^{61} + 532 q^{62} - 640 q^{64} + 1394 q^{65} - 12 q^{66} - 568 q^{68} - 408 q^{69} + 388 q^{74} + 150 q^{75} - 14 q^{77} + 132 q^{78} - 4118 q^{79} + 810 q^{81} + 1064 q^{82} + 240 q^{87} - 16 q^{88} + 154 q^{91} - 544 q^{92} - 844 q^{94} - 4004 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 650x^{8} + 157785x^{6} + 16980224x^{4} + 687898432x^{2} + 529552144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -42\nu^{8} - 21547\nu^{6} - 3661885\nu^{4} - 207139168\nu^{2} - 162372672 ) / 2294716 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9291\nu^{8} - 4643573\nu^{6} - 772403346\nu^{4} - 43176007728\nu^{2} - 57383928120 ) / 417638312 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -15851\nu^{8} - 7899737\nu^{6} - 1304252014\nu^{4} - 71678858112\nu^{2} - 49182476864 ) / 178987848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1764\nu^{9} + 904974\nu^{7} + 154372849\nu^{5} + 8886290731\nu^{3} + 21781200544\nu ) / 13201501148 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24181621 \nu^{9} - 107063330 \nu^{8} - 12012845506 \nu^{7} - 51311340674 \nu^{6} + \cdots - 450001450931912 ) / 7208019626808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24181621 \nu^{9} + 107063330 \nu^{8} - 12012845506 \nu^{7} + 51311340674 \nu^{6} + \cdots + 450001450931912 ) / 7208019626808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3995375 \nu^{9} + 1971142139 \nu^{7} + 319784655112 \nu^{5} + 16896837870036 \nu^{3} - 27707836960552 \nu ) / 1029717089544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9359279 \nu^{9} - 4722952603 \nu^{7} - 785109554554 \nu^{5} - 42246685018454 \nu^{3} + 122962183292008 \nu ) / 2402673208936 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - 2\beta_{2} - 129 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{9} + 5\beta_{8} + 2\beta_{7} + 2\beta_{6} + 101\beta_{5} - 164\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -226\beta_{7} + 226\beta_{6} - 199\beta_{4} + 107\beta_{3} + 466\beta_{2} + 21213 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -1457\beta_{9} - 1793\beta_{8} - 818\beta_{7} - 818\beta_{6} - 31485\beta_{5} + 27556\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 47706\beta_{7} - 47706\beta_{6} + 39435\beta_{4} - 7855\beta_{3} - 103830\beta_{2} - 3546873 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 348161\beta_{9} + 462745\beta_{8} + 228646\beta_{7} + 228646\beta_{6} + 8193781\beta_{5} - 4710844\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -9701770\beta_{7} + 9701770\beta_{6} - 7812619\beta_{4} - 367401\beta_{3} + 22446910\beta_{2} + 602463953 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 76296409 \beta_{9} - 105676377 \beta_{8} - 55790406 \beta_{7} - 55790406 \beta_{6} + \cdots + 819083668 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 13.9431i
− 11.9342i
0.885951i
12.2791i
12.7123i
− 12.7123i
− 12.2791i
− 0.885951i
11.9342i
13.9431i
− 2.00000i −3.00000 −4.00000 − 13.9431i 6.00000i 7.00000i 8.00000i 9.00000 −27.8862
337.2 − 2.00000i −3.00000 −4.00000 − 11.9342i 6.00000i 7.00000i 8.00000i 9.00000 −23.8685
337.3 − 2.00000i −3.00000 −4.00000 0.885951i 6.00000i 7.00000i 8.00000i 9.00000 1.77190
337.4 − 2.00000i −3.00000 −4.00000 12.2791i 6.00000i 7.00000i 8.00000i 9.00000 24.5582
337.5 − 2.00000i −3.00000 −4.00000 12.7123i 6.00000i 7.00000i 8.00000i 9.00000 25.4246
337.6 2.00000i −3.00000 −4.00000 − 12.7123i − 6.00000i − 7.00000i − 8.00000i 9.00000 25.4246
337.7 2.00000i −3.00000 −4.00000 − 12.2791i − 6.00000i − 7.00000i − 8.00000i 9.00000 24.5582
337.8 2.00000i −3.00000 −4.00000 − 0.885951i − 6.00000i − 7.00000i − 8.00000i 9.00000 1.77190
337.9 2.00000i −3.00000 −4.00000 11.9342i − 6.00000i − 7.00000i − 8.00000i 9.00000 −23.8685
337.10 2.00000i −3.00000 −4.00000 13.9431i − 6.00000i − 7.00000i − 8.00000i 9.00000 −27.8862
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.c.a ✓ 10
13.b even 2 1 inner 546.4.c.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.c.a ✓ 10 1.a even 1 1 trivial
546.4.c.a ✓ 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 650T_{5}^{8} + 157785T_{5}^{6} + 16980224T_{5}^{4} + 687898432T_{5}^{2} + 529552144 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 529552144 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 38560391424 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( (T^{5} - 71 T^{4} + \cdots - 2055380096)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{5} - 68 T^{4} + \cdots - 2349009432)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 40 T^{4} + \cdots - 57520008)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{5} + 152 T^{4} + \cdots + 215782407680)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + 775 T^{4} + \cdots - 236673481392)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + 339 T^{4} + \cdots - 3783667744)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{5} + 2059 T^{4} + \cdots - 194045360832)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 97\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
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