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Properties

Label 603.2.g.h
Level $603$
Weight $2$
Character orbit 603.g
Analytic conductor $4.815$
Analytic rank $0$
Dimension $12$
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] = [603,2,Mod(37,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("603.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.g (of order \(3\), degree \(2\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 11x^{10} + 89x^{8} + 326x^{6} + 881x^{4} + 416x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{9} + 2 \beta_{6} + \beta_{3}) q^{4} + \beta_{10} q^{5} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{7} + ( - \beta_{10} - \beta_{5} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{9} + 2 \beta_{6} + \beta_{3}) q^{4} + \beta_{10} q^{5} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{7} + ( - \beta_{10} - \beta_{5} + \beta_1) q^{8} + (\beta_{9} + \beta_{6} - \beta_{2} + 1) q^{10} - 2 \beta_{5} q^{11} + (\beta_{9} - 2 \beta_{6} - 2) q^{13} + (\beta_{11} + 2 \beta_{10} + \cdots - \beta_1) q^{14}+ \cdots + (\beta_{11} + 5 \beta_{10} + \cdots - 5 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{4} + 6 q^{7} + 4 q^{10} - 14 q^{13} - 6 q^{16} - 10 q^{19} - 88 q^{22} + 16 q^{25} + 20 q^{28} - 26 q^{34} - 38 q^{37} - 84 q^{40} + 16 q^{43} + 2 q^{46} - 24 q^{49} - 20 q^{52} - 8 q^{55} + 12 q^{58} + 18 q^{61} - 64 q^{64} + 44 q^{67} + 148 q^{70} + 24 q^{73} + 80 q^{76} + 42 q^{79} + 56 q^{82} + 42 q^{85} + 52 q^{88} - 8 q^{91} - 40 q^{94} + 62 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 11x^{10} + 89x^{8} + 326x^{6} + 881x^{4} + 416x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 298\nu^{10} - 4301\nu^{8} - 34799\nu^{6} - 253566\nu^{4} - 344471\nu^{2} - 162656 ) / 319943 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -121\nu^{10} - 979\nu^{8} - 7921\nu^{6} - 9691\nu^{4} - 4576\nu^{2} + 258308 ) / 73833 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -121\nu^{11} - 979\nu^{9} - 7921\nu^{7} - 9691\nu^{5} - 4576\nu^{3} + 332141\nu ) / 73833 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2848\nu^{11} + 29755\nu^{9} + 240745\nu^{7} + 825475\nu^{5} + 2383105\nu^{3} + 1125280\nu ) / 959829 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2848\nu^{10} + 29755\nu^{8} + 240745\nu^{6} + 825475\nu^{4} + 2383105\nu^{2} + 165451 ) / 959829 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -374\nu^{10} - 3026\nu^{8} - 17771\nu^{6} - 29954\nu^{4} - 14144\nu^{2} + 19804 ) / 73833 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 616\nu^{11} + 4984\nu^{9} + 33613\nu^{7} + 49336\nu^{5} + 23296\nu^{3} - 684086\nu ) / 73833 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3273\nu^{10} - 35431\nu^{8} - 286669\nu^{6} - 1058639\nu^{4} - 2837701\nu^{2} - 1339936 ) / 319943 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -14240\nu^{11} - 148775\nu^{9} - 1203725\nu^{7} - 4127375\nu^{5} - 10955696\nu^{3} - 827255\nu ) / 959829 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1880\nu^{11} + 21923\nu^{9} + 177377\nu^{7} + 674114\nu^{5} + 1755833\nu^{3} + 829088\nu ) / 73833 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 4\beta_{6} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 5\beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} - 21\beta_{6} + \beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - 8\beta_{10} - 27\beta_{5} + 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} - 34\beta_{3} + 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{8} - 56\beta_{4} + 150\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 195\beta_{9} - 89\beta_{7} + 656\beta_{6} + 195\beta_{3} - 89\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 89\beta_{11} + 373\beta_{10} + 89\beta_{8} + 851\beta_{5} - 851\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1135\beta_{9} - 3777\beta_{6} + 640\beta_{2} - 3777 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -640\beta_{11} - 2415\beta_{10} - 4912\beta_{5} + 2415\beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(470\)
\(\chi(n)\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.23628 + 2.14130i
1.04923 + 1.81733i
0.347450 + 0.601802i
−0.347450 − 0.601802i
−1.04923 − 1.81733i
−1.23628 − 2.14130i
1.23628 − 2.14130i
1.04923 − 1.81733i
0.347450 − 0.601802i
−0.347450 + 0.601802i
−1.04923 + 1.81733i
−1.23628 + 2.14130i
−1.23628 − 2.14130i 0 −2.05677 + 3.56243i −2.75329 0 2.34706 − 4.06522i 5.22584 0 3.40383 + 5.89560i
37.2 −1.04923 − 1.81733i 0 −1.20179 + 2.08156i 1.25158 0 −1.51499 + 2.62404i 0.846891 0 −1.31320 − 2.27453i
37.3 −0.347450 − 0.601802i 0 0.758557 − 1.31386i 3.13894 0 0.667930 − 1.15689i −2.44404 0 −1.09063 − 1.88902i
37.4 0.347450 + 0.601802i 0 0.758557 − 1.31386i −3.13894 0 0.667930 − 1.15689i 2.44404 0 −1.09063 − 1.88902i
37.5 1.04923 + 1.81733i 0 −1.20179 + 2.08156i −1.25158 0 −1.51499 + 2.62404i −0.846891 0 −1.31320 − 2.27453i
37.6 1.23628 + 2.14130i 0 −2.05677 + 3.56243i 2.75329 0 2.34706 − 4.06522i −5.22584 0 3.40383 + 5.89560i
163.1 −1.23628 + 2.14130i 0 −2.05677 − 3.56243i −2.75329 0 2.34706 + 4.06522i 5.22584 0 3.40383 − 5.89560i
163.2 −1.04923 + 1.81733i 0 −1.20179 − 2.08156i 1.25158 0 −1.51499 − 2.62404i 0.846891 0 −1.31320 + 2.27453i
163.3 −0.347450 + 0.601802i 0 0.758557 + 1.31386i 3.13894 0 0.667930 + 1.15689i −2.44404 0 −1.09063 + 1.88902i
163.4 0.347450 − 0.601802i 0 0.758557 + 1.31386i −3.13894 0 0.667930 + 1.15689i 2.44404 0 −1.09063 + 1.88902i
163.5 1.04923 − 1.81733i 0 −1.20179 − 2.08156i −1.25158 0 −1.51499 − 2.62404i −0.846891 0 −1.31320 + 2.27453i
163.6 1.23628 − 2.14130i 0 −2.05677 − 3.56243i 2.75329 0 2.34706 + 4.06522i −5.22584 0 3.40383 − 5.89560i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.c even 3 1 inner
201.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.g.h ✓ 12
3.b odd 2 1 inner 603.2.g.h ✓ 12
67.c even 3 1 inner 603.2.g.h ✓ 12
201.g odd 6 1 inner 603.2.g.h ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.g.h ✓ 12 1.a even 1 1 trivial
603.2.g.h ✓ 12 3.b odd 2 1 inner
603.2.g.h ✓ 12 67.c even 3 1 inner
603.2.g.h ✓ 12 201.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 11T_{2}^{10} + 89T_{2}^{8} + 326T_{2}^{6} + 881T_{2}^{4} + 416T_{2}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 11 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 19 T^{4} + \cdots - 117)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 3 T^{5} + \cdots + 361)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 44 T^{10} + \cdots + 692224 \) Copy content Toggle raw display
$13$ \( (T^{6} + 7 T^{5} + 41 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 34 T^{10} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} + 35 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 390971529 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 577777369 \) Copy content Toggle raw display
$31$ \( (T^{6} + 60 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 19 T^{5} + \cdots + 80089)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1143318969 \) Copy content Toggle raw display
$43$ \( (T^{3} - 4 T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + 125 T^{10} + \cdots + 66015625 \) Copy content Toggle raw display
$53$ \( (T^{6} - 106 T^{4} + \cdots - 33813)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 156 T^{4} + \cdots - 1053)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} + \cdots + 169)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 22 T^{5} + \cdots + 300763)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 390971529 \) Copy content Toggle raw display
$73$ \( (T^{6} - 12 T^{5} + \cdots + 1852321)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 21 T^{5} + \cdots + 113569)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 26 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$89$ \( (T^{6} - 164 T^{4} + \cdots - 89557)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 31 T^{5} + \cdots + 400689)^{2} \) Copy content Toggle raw display
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