Properties

Label 603.2.g.f
Level $603$
Weight $2$
Character orbit 603.g
Analytic conductor $4.815$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.3665654523963.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 8x^{8} + 21x^{6} - 5x^{5} + 26x^{4} + 4x^{3} + 13x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 201)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_1) q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{8} - \beta_{5} + \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_1) q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{8} - \beta_{5} + \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 1) q^{8} + (\beta_{9} - \beta_{7} + 3 \beta_{5} - \beta_{3} - \beta_1 - 3) q^{10} + (\beta_{9} - \beta_{8} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{11} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6}) q^{13} + (\beta_{6} + 1) q^{14} + ( - \beta_{8} - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} - 5) q^{16} + ( - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 1) q^{17} + ( - \beta_{9} + \beta_{7} + 2 \beta_{5} - 3 \beta_{3} - 3 \beta_1 - 2) q^{19} + (2 \beta_{9} + \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \beta_1) q^{20} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2) q^{22} + ( - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 3) q^{23} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{25} + (2 \beta_{9} + 2 \beta_{8} + 5 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{26} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 2) q^{28} + (\beta_{9} + 2 \beta_{8} + 4 \beta_{5} - 2 \beta_{4} + \beta_{2} - 3 \beta_1) q^{29} + ( - \beta_{9} - 3 \beta_{8} - 4 \beta_{5} - 2 \beta_{4} - \beta_{2}) q^{31} + (5 \beta_{9} + \beta_{5} + 5 \beta_{2}) q^{32} + (4 \beta_{9} + 3 \beta_{8} + \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{34} + ( - \beta_{8} - \beta_{4} + \beta_1) q^{35} + ( - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} - 2 \beta_{3} + \cdots - 6) q^{37}+ \cdots + ( - 6 \beta_{9} - 3 \beta_{8} - 4 \beta_{5} - 6 \beta_{2} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 6 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{4} - 6 q^{5} - q^{7} - 12 q^{10} - 6 q^{11} - q^{13} + 6 q^{14} - 24 q^{16} - 8 q^{17} - 5 q^{19} + 8 q^{20} + 22 q^{22} + 7 q^{23} - 8 q^{25} + 17 q^{26} + 3 q^{28} + 12 q^{29} - 12 q^{31} + 5 q^{32} + 5 q^{35} - 17 q^{37} - 30 q^{38} + 34 q^{40} - 13 q^{41} - 4 q^{43} - 43 q^{44} - 26 q^{46} + 25 q^{47} + 16 q^{49} + 25 q^{50} + 64 q^{52} + 12 q^{53} - 14 q^{55} + 11 q^{56} + 4 q^{58} + 12 q^{59} + 9 q^{61} - 46 q^{62} + 64 q^{64} + 14 q^{65} + 2 q^{67} + 98 q^{68} + 2 q^{70} - 29 q^{71} + 12 q^{73} - 15 q^{74} - 6 q^{76} - 4 q^{77} - q^{79} + 13 q^{80} - 2 q^{82} + 6 q^{83} + 9 q^{85} + 21 q^{86} + 18 q^{88} + 4 q^{89} - 40 q^{91} + 30 q^{94} + 14 q^{95} + 11 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 8x^{8} + 21x^{6} - 5x^{5} + 26x^{4} + 4x^{3} + 13x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1127 \nu^{9} + 7711 \nu^{8} - 25937 \nu^{7} + 48871 \nu^{6} - 53276 \nu^{5} + 58183 \nu^{4} - 144911 \nu^{3} + 39957 \nu^{2} - 9814 \nu - 47489 ) / 44248 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1899 \nu^{9} - 6083 \nu^{8} + 20461 \nu^{7} - 17723 \nu^{6} + 42028 \nu^{5} - 45899 \nu^{4} + 89131 \nu^{3} - 31521 \nu^{2} + 7742 \nu - 12403 ) / 44248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3473 \nu^{9} + 2993 \nu^{8} - 22135 \nu^{7} - 25035 \nu^{6} - 95096 \nu^{5} - 47811 \nu^{4} - 116057 \nu^{3} - 66953 \nu^{2} - 90294 \nu - 6827 ) / 22124 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12403 \nu^{9} - 22907 \nu^{8} + 93141 \nu^{7} + 20461 \nu^{6} + 242740 \nu^{5} - 19987 \nu^{4} + 276579 \nu^{3} + 138743 \nu^{2} + 129718 \nu + 14781 ) / 44248 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14781 \nu^{9} + 41965 \nu^{8} - 141155 \nu^{7} + 93141 \nu^{6} - 289940 \nu^{5} + 316645 \nu^{4} - 404293 \nu^{3} + 217455 \nu^{2} - 53410 \nu + 129813 ) / 44248 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3911 \nu^{9} - 7629 \nu^{8} + 31695 \nu^{7} - 1369 \nu^{6} + 89918 \nu^{5} - 22367 \nu^{4} + 104757 \nu^{3} + 1699 \nu^{2} + 52952 \nu - 12251 ) / 11062 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2133 \nu^{9} - 2953 \nu^{8} + 14961 \nu^{7} + 9085 \nu^{6} + 49566 \nu^{5} + 14424 \nu^{4} + 58697 \nu^{3} + 31963 \nu^{2} + 49012 \nu + 3317 ) / 5531 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9205 \nu^{9} + 14061 \nu^{8} - 61375 \nu^{7} - 41255 \nu^{6} - 165532 \nu^{5} - 38715 \nu^{4} - 146785 \nu^{3} - 130277 \nu^{2} - 56110 \nu + 12005 ) / 22124 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} + 2\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + 2\beta_{4} + 7\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 7\beta_{8} - 2\beta_{5} + 8\beta_{4} + 2\beta_{2} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{9} + 18\beta_{8} - 20\beta_{7} - 18\beta_{6} - \beta_{5} - 53\beta_{3} - 53\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -61\beta_{7} - 53\beta_{6} - 61\beta_{4} - 145\beta_{3} - 20\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -61\beta_{9} - 145\beta_{8} + 11\beta_{5} - 165\beta_{4} - 61\beta_{2} + 411\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -165\beta_{9} - 411\beta_{8} + 472\beta_{7} + 411\beta_{6} + 40\beta_{5} + 1143\beta_{3} + 1143\beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1308\beta_{7} + 1143\beta_{6} + 1308\beta_{4} + 3209\beta_{3} + 472\beta_{2} - 95 \) Copy content Toggle raw display

Character values

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

\(n\) \(136\) \(470\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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
−0.393795 + 0.682072i
0.133354 0.230976i
1.39901 2.42316i
−0.725148 + 1.25599i
0.586579 1.01598i
−0.393795 0.682072i
0.133354 + 0.230976i
1.39901 + 2.42316i
−0.725148 1.25599i
0.586579 + 1.01598i
−1.31682 2.28079i 0 −2.46802 + 4.27473i 0.846046 0 0.241054 0.417518i 7.73244 0 −1.11409 1.92966i
37.2 −0.538290 0.932346i 0 0.420487 0.728305i 0.343289 0 −1.74135 + 3.01611i −3.05854 0 −0.184789 0.320064i
37.3 0.0732181 + 0.126817i 0 0.989278 1.71348i 1.65158 0 1.22031 2.11364i 0.582605 0 0.120926 + 0.209450i
37.4 0.453162 + 0.784899i 0 0.589289 1.02068i −3.35662 0 −0.380391 + 0.658856i 2.88082 0 −1.52109 2.63461i
37.5 1.32873 + 2.30142i 0 −2.53104 + 4.38389i −2.48430 0 0.160379 0.277784i −8.13733 0 −3.30096 5.71743i
163.1 −1.31682 + 2.28079i 0 −2.46802 4.27473i 0.846046 0 0.241054 + 0.417518i 7.73244 0 −1.11409 + 1.92966i
163.2 −0.538290 + 0.932346i 0 0.420487 + 0.728305i 0.343289 0 −1.74135 3.01611i −3.05854 0 −0.184789 + 0.320064i
163.3 0.0732181 0.126817i 0 0.989278 + 1.71348i 1.65158 0 1.22031 + 2.11364i 0.582605 0 0.120926 0.209450i
163.4 0.453162 0.784899i 0 0.589289 + 1.02068i −3.35662 0 −0.380391 0.658856i 2.88082 0 −1.52109 + 2.63461i
163.5 1.32873 2.30142i 0 −2.53104 4.38389i −2.48430 0 0.160379 + 0.277784i −8.13733 0 −3.30096 + 5.71743i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.g.f 10
3.b odd 2 1 201.2.e.c 10
67.c even 3 1 inner 603.2.g.f 10
201.g odd 6 1 201.2.e.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.e.c 10 3.b odd 2 1
201.2.e.c 10 201.g odd 6 1
603.2.g.f 10 1.a even 1 1 trivial
603.2.g.f 10 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8T_{2}^{8} + 57T_{2}^{6} - T_{2}^{5} + 56T_{2}^{4} - 16T_{2}^{3} + 49T_{2}^{2} - 7T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{8} + 57 T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} + 3 T^{4} - 6 T^{3} - 11 T^{2} + 16 T - 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + 10 T^{8} - 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 6 T^{9} + 53 T^{8} + 230 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{10} + T^{9} + 26 T^{8} + 35 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} + 8 T^{9} + 80 T^{8} + \cdots + 208849 \) Copy content Toggle raw display
$19$ \( T^{10} + 5 T^{9} + 104 T^{8} + \cdots + 37613689 \) Copy content Toggle raw display
$23$ \( T^{10} - 7 T^{9} + 87 T^{8} + \cdots + 7198489 \) Copy content Toggle raw display
$29$ \( T^{10} - 12 T^{9} + 194 T^{8} + \cdots + 3690241 \) Copy content Toggle raw display
$31$ \( T^{10} + 12 T^{9} + 181 T^{8} + \cdots + 2166784 \) Copy content Toggle raw display
$37$ \( T^{10} + 17 T^{9} + 264 T^{8} + \cdots + 47761921 \) Copy content Toggle raw display
$41$ \( T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 26122321 \) Copy content Toggle raw display
$43$ \( (T^{5} + 2 T^{4} - 79 T^{3} - 131 T^{2} + \cdots + 2032)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 25 T^{9} + \cdots + 560884489 \) Copy content Toggle raw display
$53$ \( (T^{5} - 6 T^{4} - 109 T^{3} + 391 T^{2} + \cdots - 1964)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 6 T^{4} - 125 T^{3} + 707 T^{2} + \cdots - 20672)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} - 9 T^{9} + 162 T^{8} + \cdots + 3916441 \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + 29 T^{9} + 527 T^{8} + \cdots + 9591409 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + 122 T^{8} + \cdots + 100489 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + 232 T^{8} + \cdots + 7678441 \) Copy content Toggle raw display
$83$ \( T^{10} - 6 T^{9} + 82 T^{8} + \cdots + 375769 \) Copy content Toggle raw display
$89$ \( (T^{5} - 2 T^{4} - 407 T^{3} + \cdots - 130084)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 11 T^{9} + 108 T^{8} + \cdots + 3481 \) Copy content Toggle raw display
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