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Properties

Label 483.2.i.g
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $16$
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] = [483,2,Mod(277,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.i (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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 16 x^{14} - 7 x^{13} + 161 x^{12} - 50 x^{11} + 929 x^{10} - 47 x^{9} + 3741 x^{8} + \cdots + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{9} - 2 \beta_{4} + \beta_{3} - 2) q^{4} + ( - \beta_{14} + \beta_{12} + \beta_{4}) q^{5} + \beta_{5} q^{6} + \beta_{11} q^{7} + ( - \beta_{15} + \beta_{14} - 2 \beta_{8} + \cdots + 2) q^{8}+ \cdots + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{9} - 2 \beta_{4} + \beta_{3} - 2) q^{4} + ( - \beta_{14} + \beta_{12} + \beta_{4}) q^{5} + \beta_{5} q^{6} + \beta_{11} q^{7} + ( - \beta_{15} + \beta_{14} - 2 \beta_{8} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{11} - \beta_{10} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 8 q^{3} - 15 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} + 18 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - 8 q^{3} - 15 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} + 18 q^{8} - 8 q^{9} + 3 q^{10} - 10 q^{11} - 15 q^{12} - 12 q^{13} - 17 q^{14} + 10 q^{15} - 13 q^{16} - 21 q^{17} - q^{18} - 5 q^{19} - 2 q^{20} + 4 q^{21} + 36 q^{22} + 8 q^{23} - 9 q^{24} - 27 q^{25} - 3 q^{26} + 16 q^{27} + 6 q^{28} + 4 q^{29} + 3 q^{30} + 13 q^{31} - 29 q^{32} - 10 q^{33} - 38 q^{34} + 27 q^{35} + 30 q^{36} - 13 q^{37} + 6 q^{38} + 6 q^{39} - 7 q^{40} + 32 q^{41} + 25 q^{42} + 30 q^{43} - 24 q^{44} - 5 q^{45} + q^{46} + q^{47} + 26 q^{48} + 16 q^{49} + 32 q^{50} - 21 q^{51} + 19 q^{52} - 3 q^{53} - q^{54} - 20 q^{55} + 33 q^{56} + 10 q^{57} + 40 q^{58} - 26 q^{59} + q^{60} + 14 q^{61} - 28 q^{62} - 2 q^{63} + 98 q^{64} + 3 q^{65} - 18 q^{66} - 38 q^{67} - 43 q^{68} - 16 q^{69} + 40 q^{70} + 18 q^{71} - 9 q^{72} + 6 q^{73} - 32 q^{74} - 27 q^{75} - 28 q^{76} + 56 q^{77} + 6 q^{78} - 23 q^{79} + 17 q^{80} - 8 q^{81} - 20 q^{82} + 60 q^{83} + 3 q^{84} - 74 q^{85} + 28 q^{86} - 2 q^{87} - 86 q^{88} - 12 q^{89} - 6 q^{90} + 26 q^{91} - 30 q^{92} + 13 q^{93} + 45 q^{94} - 16 q^{95} - 29 q^{96} + 28 q^{97} + 26 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 16 x^{14} - 7 x^{13} + 161 x^{12} - 50 x^{11} + 929 x^{10} - 47 x^{9} + 3741 x^{8} + \cdots + 6400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\!\cdots\!31 \nu^{15} + \cdots - 21\!\cdots\!20 ) / 81\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 86\!\cdots\!96 \nu^{15} + \cdots + 32\!\cdots\!88 ) / 10\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73\!\cdots\!61 \nu^{15} + \cdots + 24\!\cdots\!40 ) / 81\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!48 \nu^{15} + \cdots - 58\!\cdots\!80 ) / 10\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\!\cdots\!17 \nu^{15} + \cdots - 53\!\cdots\!00 ) / 81\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!17 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 81\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 68\!\cdots\!50 \nu^{15} + \cdots - 37\!\cdots\!18 ) / 20\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73\!\cdots\!61 \nu^{15} + \cdots - 24\!\cdots\!40 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!11 \nu^{15} + \cdots + 16\!\cdots\!80 ) / 40\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42\!\cdots\!67 \nu^{15} + \cdots - 33\!\cdots\!40 ) / 81\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53\!\cdots\!37 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 81\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 68\!\cdots\!26 \nu^{15} + \cdots + 49\!\cdots\!50 ) / 10\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!36 \nu^{15} + \cdots + 76\!\cdots\!62 ) / 20\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!91 \nu^{15} + \cdots + 62\!\cdots\!80 ) / 81\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2\beta_{8} - 5\beta_{5} + \beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{9} + \beta_{6} - 22\beta_{4} + 8\beta_{3} - \beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{15} + 8 \beta_{14} - 8 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} + \cdots - 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{15} + 12\beta_{13} + 12\beta_{5} - 57\beta_{3} + 2\beta_{2} + 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{15} + 55 \beta_{12} + 81 \beta_{11} - 83 \beta_{10} + 81 \beta_{9} - 134 \beta_{7} - 4 \beta_{6} + \cdots + 164 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 38 \beta_{15} - 2 \beta_{14} - 109 \beta_{13} + 2 \beta_{12} + 38 \beta_{11} - 36 \beta_{10} + \cdots + 107 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 620 \beta_{15} - 368 \beta_{14} - 72 \beta_{13} + 32 \beta_{11} + 32 \beta_{10} + 948 \beta_{8} + \cdots - 1264 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 40 \beta_{15} - 36 \beta_{12} - 432 \beta_{11} + 472 \beta_{10} - 2849 \beta_{9} + 120 \beta_{7} + \cdots - 6288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5009 \beta_{15} + 2457 \beta_{14} + 868 \beta_{13} - 2457 \beta_{12} - 5009 \beta_{11} + \cdots - 8113 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4368 \beta_{15} + 428 \beta_{14} + 7213 \beta_{13} - 516 \beta_{11} - 516 \beta_{10} - 1548 \beta_{8} + \cdots + 43790 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3248 \beta_{15} + 16472 \beta_{12} + 34838 \beta_{11} - 38086 \beta_{10} + 35882 \beta_{9} + \cdots + 71260 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 45794 \beta_{15} - 4292 \beta_{14} - 56452 \beta_{13} + 4292 \beta_{12} + 45794 \beta_{11} + \cdots + 52496 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 259981 \beta_{15} - 111063 \beta_{14} - 81808 \beta_{13} + 28110 \beta_{11} + 28110 \beta_{10} + \cdots - 532272 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.37292 − 2.37797i
1.25839 − 2.17960i
0.736956 − 1.27644i
0.595971 − 1.03225i
−0.409272 + 0.708880i
−0.838903 + 1.45302i
−0.939958 + 1.62805i
−1.27611 + 2.21029i
1.37292 + 2.37797i
1.25839 + 2.17960i
0.736956 + 1.27644i
0.595971 + 1.03225i
−0.409272 − 0.708880i
−0.838903 − 1.45302i
−0.939958 − 1.62805i
−1.27611 − 2.21029i
−1.37292 − 2.37797i −0.500000 + 0.866025i −2.76984 + 4.79751i 0.957403 + 1.65827i 2.74585 1.61005 − 2.09946i 9.71944 −0.500000 − 0.866025i 2.62888 − 4.55336i
277.2 −1.25839 − 2.17960i −0.500000 + 0.866025i −2.16710 + 3.75353i −1.20917 − 2.09434i 2.51679 −0.750699 + 2.53702i 5.87470 −0.500000 − 0.866025i −3.04322 + 5.27101i
277.3 −0.736956 − 1.27644i −0.500000 + 0.866025i −0.0862077 + 0.149316i −1.81170 − 3.13796i 1.47391 −0.366412 − 2.62026i −2.69370 −0.500000 − 0.866025i −2.67029 + 4.62508i
277.4 −0.595971 − 1.03225i −0.500000 + 0.866025i 0.289637 − 0.501666i 1.89410 + 3.28068i 1.19194 2.62328 + 0.344130i −3.07435 −0.500000 − 0.866025i 2.25766 − 3.91038i
277.5 0.409272 + 0.708880i −0.500000 + 0.866025i 0.664993 − 1.15180i −1.86020 − 3.22197i −0.818544 −2.34029 + 1.23412i 2.72574 −0.500000 − 0.866025i 1.52266 − 2.63732i
277.6 0.838903 + 1.45302i −0.500000 + 0.866025i −0.407518 + 0.705841i −1.47412 − 2.55325i −1.67781 2.59341 − 0.523657i 1.98814 −0.500000 − 0.866025i 2.47329 − 4.28386i
277.7 0.939958 + 1.62805i −0.500000 + 0.866025i −0.767041 + 1.32855i 1.32777 + 2.29977i −1.87992 −1.98456 − 1.74973i 0.875886 −0.500000 − 0.866025i −2.49610 + 4.32337i
277.8 1.27611 + 2.21029i −0.500000 + 0.866025i −2.25692 + 3.90910i −0.324080 − 0.561322i −2.55222 −2.38478 + 1.14579i −6.41586 −0.500000 − 0.866025i 0.827123 − 1.43262i
415.1 −1.37292 + 2.37797i −0.500000 − 0.866025i −2.76984 − 4.79751i 0.957403 − 1.65827i 2.74585 1.61005 + 2.09946i 9.71944 −0.500000 + 0.866025i 2.62888 + 4.55336i
415.2 −1.25839 + 2.17960i −0.500000 − 0.866025i −2.16710 − 3.75353i −1.20917 + 2.09434i 2.51679 −0.750699 − 2.53702i 5.87470 −0.500000 + 0.866025i −3.04322 − 5.27101i
415.3 −0.736956 + 1.27644i −0.500000 − 0.866025i −0.0862077 − 0.149316i −1.81170 + 3.13796i 1.47391 −0.366412 + 2.62026i −2.69370 −0.500000 + 0.866025i −2.67029 − 4.62508i
415.4 −0.595971 + 1.03225i −0.500000 − 0.866025i 0.289637 + 0.501666i 1.89410 − 3.28068i 1.19194 2.62328 − 0.344130i −3.07435 −0.500000 + 0.866025i 2.25766 + 3.91038i
415.5 0.409272 − 0.708880i −0.500000 − 0.866025i 0.664993 + 1.15180i −1.86020 + 3.22197i −0.818544 −2.34029 − 1.23412i 2.72574 −0.500000 + 0.866025i 1.52266 + 2.63732i
415.6 0.838903 − 1.45302i −0.500000 − 0.866025i −0.407518 − 0.705841i −1.47412 + 2.55325i −1.67781 2.59341 + 0.523657i 1.98814 −0.500000 + 0.866025i 2.47329 + 4.28386i
415.7 0.939958 − 1.62805i −0.500000 − 0.866025i −0.767041 − 1.32855i 1.32777 − 2.29977i −1.87992 −1.98456 + 1.74973i 0.875886 −0.500000 + 0.866025i −2.49610 − 4.32337i
415.8 1.27611 − 2.21029i −0.500000 − 0.866025i −2.25692 − 3.90910i −0.324080 + 0.561322i −2.55222 −2.38478 − 1.14579i −6.41586 −0.500000 + 0.866025i 0.827123 + 1.43262i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.g ✓ 16
7.c even 3 1 inner 483.2.i.g ✓ 16
7.c even 3 1 3381.2.a.bf 8
7.d odd 6 1 3381.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.g ✓ 16 1.a even 1 1 trivial
483.2.i.g ✓ 16 7.c even 3 1 inner
3381.2.a.be 8 7.d odd 6 1
3381.2.a.bf 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{16} + T_{2}^{15} + 16 T_{2}^{14} + 7 T_{2}^{13} + 161 T_{2}^{12} + 50 T_{2}^{11} + 929 T_{2}^{10} + \cdots + 6400 \) Copy content Toggle raw display
\( T_{5}^{16} + 5 T_{5}^{15} + 46 T_{5}^{14} + 149 T_{5}^{13} + 995 T_{5}^{12} + 2790 T_{5}^{11} + \cdots + 1440000 \) Copy content Toggle raw display
\( T_{11}^{16} + 10 T_{11}^{15} + 104 T_{11}^{14} + 472 T_{11}^{13} + 2880 T_{11}^{12} + 8968 T_{11}^{11} + \cdots + 19219456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 6400 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + \cdots + 1440000 \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 10 T^{15} + \cdots + 19219456 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 2127)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 4513421124 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 9822395664 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} - 2 T^{7} + \cdots + 34688)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 118722816 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 391090176 \) Copy content Toggle raw display
$41$ \( (T^{8} - 16 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 15 T^{7} + \cdots - 937796)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1666159476804 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 886371984 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1182003840000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 21489214464 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5849089043121 \) Copy content Toggle raw display
$71$ \( (T^{8} - 9 T^{7} + \cdots + 44639898)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 91705403241 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 219581777797264 \) Copy content Toggle raw display
$83$ \( (T^{8} - 30 T^{7} + \cdots + 9331200)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{8} - 14 T^{7} + \cdots + 1304736)^{2} \) Copy content Toggle raw display
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