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Properties

Label 483.2.i.f
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $12$
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: \(12\)
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: \( x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 102623 \nu^{11} - 165571 \nu^{10} - 222384 \nu^{9} - 1691067 \nu^{8} - 1870676 \nu^{7} + \cdots - 1558388 ) / 32857894 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 387955 \nu^{11} - 1012381 \nu^{10} + 78220 \nu^{9} - 9799615 \nu^{8} + 1245446 \nu^{7} + \cdots - 9835940 ) / 65715788 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 389597 \nu^{11} - 492220 \nu^{10} + 2951205 \nu^{9} - 1391175 \nu^{8} + 14672007 \nu^{7} + \cdots + 62587576 ) / 32857894 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 782053 \nu^{11} - 392456 \nu^{10} + 5764204 \nu^{9} + 605046 \nu^{8} + 31455051 \nu^{7} + \cdots + 8030866 ) / 32857894 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2035745 \nu^{11} - 2108749 \nu^{10} - 12760488 \nu^{9} - 25032941 \nu^{8} - 80460962 \nu^{7} + \cdots - 38350448 ) / 65715788 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4015433 \nu^{11} + 5579539 \nu^{10} - 32908376 \nu^{9} + 23574707 \nu^{8} - 167438094 \nu^{7} + \cdots + 42936084 ) / 65715788 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3745062 \nu^{11} + 2035588 \nu^{10} - 28003566 \nu^{9} - 2006303 \nu^{8} - 150105166 \nu^{7} + \cdots - 39617942 ) / 32857894 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3881187 \nu^{11} + 6414399 \nu^{10} - 34645649 \nu^{9} + 33305993 \nu^{8} - 178419829 \nu^{7} + \cdots + 53681036 ) / 32857894 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12046299 \nu^{11} - 16738617 \nu^{10} + 98725128 \nu^{9} - 70724121 \nu^{8} + 502314282 \nu^{7} + \cdots + 68339112 ) / 65715788 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6206906 \nu^{11} + 4290544 \nu^{10} - 46524859 \nu^{9} + 1905036 \nu^{8} - 245693958 \nu^{7} + \cdots - 130991922 ) / 32857894 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 3\beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{10} - \beta_{9} - 12\beta_{7} + 5\beta_{4} - \beta_{3} - \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - 7\beta_{7} - 7\beta_{6} - 18\beta_{5} - 2\beta_{3} - 18\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{11} + 9\beta_{8} - 2\beta_{7} - 2\beta_{6} - 9\beta_{5} - 23\beta_{4} - 2\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{10} + 2\beta_{9} + 2\beta_{7} - 2\beta_{4} + 18\beta_{3} - 39\beta_{2} + 84\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 39 \beta_{11} + 105 \beta_{10} + 39 \beta_{9} - 59 \beta_{8} + 236 \beta_{7} + 22 \beta_{6} + 61 \beta_{5} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22\beta_{11} - 120\beta_{8} + 203\beta_{7} + 203\beta_{6} + 400\beta_{5} + 24\beta_{4} + 203\beta_{2} - 205 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -483\beta_{10} - 203\beta_{9} - 938\beta_{7} + 483\beta_{4} - 345\beta_{3} + 166\beta_{2} - 373\beta _1 + 483 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 166 \beta_{11} - 194 \beta_{10} - 166 \beta_{9} + 714 \beta_{8} - 1257 \beta_{7} - 1031 \beta_{6} + \cdots + 1257 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\) \(1\)

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
−1.02170 + 1.76964i
−0.682743 + 1.18255i
−0.144978 + 0.251110i
0.296764 − 0.514011i
0.916322 − 1.58712i
1.13633 − 1.96819i
−1.02170 − 1.76964i
−0.682743 − 1.18255i
−0.144978 − 0.251110i
0.296764 + 0.514011i
0.916322 + 1.58712i
1.13633 + 1.96819i
−1.02170 − 1.76964i 0.500000 − 0.866025i −1.08774 + 1.88403i −0.971745 − 1.68311i −2.04340 −1.16166 + 2.37709i 0.358585 −0.500000 − 0.866025i −1.98566 + 3.43927i
277.2 −0.682743 − 1.18255i 0.500000 − 0.866025i 0.0677246 − 0.117302i 0.663106 + 1.14853i −1.36549 2.35341 − 1.20891i −2.91593 −0.500000 − 0.866025i 0.905461 − 1.56830i
277.3 −0.144978 − 0.251110i 0.500000 − 0.866025i 0.957963 − 1.65924i −1.16311 − 2.01456i −0.289957 2.35341 − 1.20891i −1.13545 −0.500000 − 0.866025i −0.337250 + 0.584135i
277.4 0.296764 + 0.514011i 0.500000 − 0.866025i 0.823862 − 1.42697i 1.57560 + 2.72902i 0.593528 −1.69175 − 2.03420i 2.16503 −0.500000 − 0.866025i −0.935165 + 1.61975i
277.5 0.916322 + 1.58712i 0.500000 − 0.866025i −0.679293 + 1.17657i 0.471745 + 0.817086i 1.83264 −1.16166 + 2.37709i 1.17548 −0.500000 − 0.866025i −0.864540 + 1.49743i
277.6 1.13633 + 1.96819i 0.500000 − 0.866025i −1.58251 + 2.74099i −2.07560 − 3.59505i 2.27267 −1.69175 − 2.03420i −2.64772 −0.500000 − 0.866025i 4.71716 − 8.17036i
415.1 −1.02170 + 1.76964i 0.500000 + 0.866025i −1.08774 − 1.88403i −0.971745 + 1.68311i −2.04340 −1.16166 − 2.37709i 0.358585 −0.500000 + 0.866025i −1.98566 − 3.43927i
415.2 −0.682743 + 1.18255i 0.500000 + 0.866025i 0.0677246 + 0.117302i 0.663106 − 1.14853i −1.36549 2.35341 + 1.20891i −2.91593 −0.500000 + 0.866025i 0.905461 + 1.56830i
415.3 −0.144978 + 0.251110i 0.500000 + 0.866025i 0.957963 + 1.65924i −1.16311 + 2.01456i −0.289957 2.35341 + 1.20891i −1.13545 −0.500000 + 0.866025i −0.337250 − 0.584135i
415.4 0.296764 − 0.514011i 0.500000 + 0.866025i 0.823862 + 1.42697i 1.57560 − 2.72902i 0.593528 −1.69175 + 2.03420i 2.16503 −0.500000 + 0.866025i −0.935165 − 1.61975i
415.5 0.916322 − 1.58712i 0.500000 + 0.866025i −0.679293 − 1.17657i 0.471745 − 0.817086i 1.83264 −1.16166 − 2.37709i 1.17548 −0.500000 + 0.866025i −0.864540 − 1.49743i
415.6 1.13633 − 1.96819i 0.500000 + 0.866025i −1.58251 − 2.74099i −2.07560 + 3.59505i 2.27267 −1.69175 + 2.03420i −2.64772 −0.500000 + 0.866025i 4.71716 + 8.17036i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.6
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.f ✓ 12
7.c even 3 1 inner 483.2.i.f ✓ 12
7.c even 3 1 3381.2.a.bc 6
7.d odd 6 1 3381.2.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.f ✓ 12 1.a even 1 1 trivial
483.2.i.f ✓ 12 7.c even 3 1 inner
3381.2.a.bc 6 7.c even 3 1
3381.2.a.bd 6 7.d odd 6 1

Hecke kernels

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

\( T_{2}^{12} - T_{2}^{11} + 8 T_{2}^{10} - 3 T_{2}^{9} + 42 T_{2}^{8} - 15 T_{2}^{7} + 101 T_{2}^{6} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{12} + 3 T_{5}^{11} + 24 T_{5}^{10} + 25 T_{5}^{9} + 278 T_{5}^{8} + 283 T_{5}^{7} + 1647 T_{5}^{6} + \cdots + 5476 \) Copy content Toggle raw display
\( T_{11}^{12} - 14 T_{11}^{11} + 144 T_{11}^{10} - 776 T_{11}^{9} + 3360 T_{11}^{8} - 7584 T_{11}^{7} + \cdots + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 5476 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} + 2 T^{4} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} - 31 T^{4} + 52 T^{3} + \cdots - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 15 T^{11} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} + 6 T^{5} + \cdots - 4768)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 11 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$37$ \( T^{12} - 5 T^{11} + \cdots + 179776 \) Copy content Toggle raw display
$41$ \( (T^{6} - 18 T^{5} + \cdots + 52928)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 37 T^{5} + \cdots - 19796)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 3 T^{11} + \cdots + 22733824 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 5465349184 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 5805220864 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 9929723904 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1142372401 \) Copy content Toggle raw display
$71$ \( (T^{6} + 21 T^{5} + \cdots - 3208)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3064618881 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14531820304 \) Copy content Toggle raw display
$83$ \( (T^{6} - 12 T^{5} + \cdots - 2224)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 18 T^{11} + \cdots + 27290176 \) Copy content Toggle raw display
$97$ \( (T^{6} + 2 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
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