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Properties

Label 663.2.i.e
Level $663$
Weight $2$
Character orbit 663.i
Analytic conductor $5.294$
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] = [663,2,Mod(256,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, 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("663.256");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: 12.0.171567427640625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 8 x^{10} - 8 x^{9} + 30 x^{8} - 29 x^{7} + 64 x^{6} - 26 x^{5} + 48 x^{4} - 26 x^{3} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 8 x^{10} - 8 x^{9} + 30 x^{8} - 29 x^{7} + 64 x^{6} - 26 x^{5} + 48 x^{4} - 26 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 42328 \nu^{11} + 200434 \nu^{10} - 388204 \nu^{9} + 814736 \nu^{8} - 617214 \nu^{7} + \cdots - 1382783 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 98812 \nu^{11} + 671891 \nu^{10} - 220159 \nu^{9} + 4683512 \nu^{8} + 1314203 \nu^{7} + \cdots - 612080 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 690 \nu^{11} + 2001 \nu^{10} + 823 \nu^{9} + 17158 \nu^{8} + 9476 \nu^{7} + 63713 \nu^{6} + \cdots + 39602 ) / 20717 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 225796 \nu^{11} + 70589 \nu^{10} + 944453 \nu^{9} + 2239304 \nu^{8} + 3165845 \nu^{7} + \cdots + 3536269 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 560420 \nu^{11} - 195745 \nu^{10} - 2075998 \nu^{9} - 5676600 \nu^{8} - 7855265 \nu^{7} + \cdots - 8583809 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 825695 \nu^{11} - 1531356 \nu^{10} + 6108405 \nu^{9} - 5729968 \nu^{8} + 22872008 \nu^{7} + \cdots - 3750859 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 869515 \nu^{11} + 1010655 \nu^{10} - 5474008 \nu^{9} + 1650157 \nu^{8} - 20566517 \nu^{7} + \cdots + 98812 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1382783 \nu^{11} - 2807894 \nu^{10} + 11262698 \nu^{9} - 11450468 \nu^{8} + 42298226 \nu^{7} + \cdots - 7161513 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2153486 \nu^{11} + 4490440 \nu^{10} - 16956865 \nu^{9} + 17784137 \nu^{8} - 61550540 \nu^{7} + \cdots + 6648245 ) / 6442987 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2336966 \nu^{11} + 4662726 \nu^{10} - 18144006 \nu^{9} + 18007228 \nu^{8} - 67012146 \nu^{7} + \cdots + 9347613 ) / 6442987 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 4\beta_{10} - 3\beta_{9} + 4\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{4} - \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{9} + 5\beta_{8} + 6\beta_{7} + 5\beta_{3} - 11\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{6} - 8\beta_{5} - 16\beta_{4} + 16\beta_{3} - 7\beta_{2} + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8 \beta_{11} + 9 \beta_{10} + 7 \beta_{9} - 23 \beta_{8} - 30 \beta_{7} - 8 \beta_{6} - 30 \beta_{5} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -30\beta_{11} + 65\beta_{10} + 43\beta_{9} - 66\beta_{8} - 47\beta_{7} - 66\beta_{3} + 39\beta_{2} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 47\beta_{6} + 142\beta_{5} + 56\beta_{4} - 105\beta_{3} + 174\beta_{2} - 95 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 142 \beta_{11} - 269 \beta_{10} - 174 \beta_{9} + 279 \beta_{8} + 245 \beta_{7} + 142 \beta_{6} + \cdots - 443 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 245 \beta_{11} - 303 \beta_{10} - 200 \beta_{9} + 479 \beta_{8} + 656 \beta_{7} + 479 \beta_{3} + \cdots - 722 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
256.1
−0.903003 + 1.56405i
−0.464664 + 0.804822i
0.121928 − 0.211186i
0.341366 − 0.591263i
0.842736 − 1.45966i
1.06164 − 1.83881i
−0.903003 − 1.56405i
−0.464664 − 0.804822i
0.121928 + 0.211186i
0.341366 + 0.591263i
0.842736 + 1.45966i
1.06164 + 1.83881i
−0.903003 + 1.56405i −0.500000 + 0.866025i −0.630828 − 1.09263i 0.935674 −0.903003 − 1.56405i −1.28914 − 2.23285i −1.33345 −0.500000 − 0.866025i −0.844916 + 1.46344i
256.2 −0.464664 + 0.804822i −0.500000 + 0.866025i 0.568175 + 0.984107i 3.69408 −0.464664 − 0.804822i 1.98857 + 3.44431i −2.91470 −0.500000 − 0.866025i −1.71651 + 2.97308i
256.3 0.121928 − 0.211186i −0.500000 + 0.866025i 0.970267 + 1.68055i −1.48274 0.121928 + 0.211186i −2.19956 − 3.80975i 0.960924 −0.500000 − 0.866025i −0.180788 + 0.313134i
256.4 0.341366 − 0.591263i −0.500000 + 0.866025i 0.766938 + 1.32838i −1.08274 0.341366 + 0.591263i −0.665416 − 1.15253i 2.41269 −0.500000 − 0.866025i −0.369610 + 0.640183i
256.5 0.842736 − 1.45966i −0.500000 + 0.866025i −0.420408 − 0.728167i 2.02473 0.842736 + 1.45966i 0.638040 + 1.10512i 1.95377 −0.500000 − 0.866025i 1.70631 − 2.95542i
256.6 1.06164 − 1.83881i −0.500000 + 0.866025i −1.25414 − 2.17224i −0.0890049 1.06164 + 1.83881i −0.972496 − 1.68441i −1.07924 −0.500000 − 0.866025i −0.0944909 + 0.163663i
562.1 −0.903003 − 1.56405i −0.500000 − 0.866025i −0.630828 + 1.09263i 0.935674 −0.903003 + 1.56405i −1.28914 + 2.23285i −1.33345 −0.500000 + 0.866025i −0.844916 − 1.46344i
562.2 −0.464664 − 0.804822i −0.500000 − 0.866025i 0.568175 − 0.984107i 3.69408 −0.464664 + 0.804822i 1.98857 − 3.44431i −2.91470 −0.500000 + 0.866025i −1.71651 − 2.97308i
562.3 0.121928 + 0.211186i −0.500000 − 0.866025i 0.970267 − 1.68055i −1.48274 0.121928 − 0.211186i −2.19956 + 3.80975i 0.960924 −0.500000 + 0.866025i −0.180788 − 0.313134i
562.4 0.341366 + 0.591263i −0.500000 − 0.866025i 0.766938 − 1.32838i −1.08274 0.341366 − 0.591263i −0.665416 + 1.15253i 2.41269 −0.500000 + 0.866025i −0.369610 − 0.640183i
562.5 0.842736 + 1.45966i −0.500000 − 0.866025i −0.420408 + 0.728167i 2.02473 0.842736 − 1.45966i 0.638040 − 1.10512i 1.95377 −0.500000 + 0.866025i 1.70631 + 2.95542i
562.6 1.06164 + 1.83881i −0.500000 − 0.866025i −1.25414 + 2.17224i −0.0890049 1.06164 − 1.83881i −0.972496 + 1.68441i −1.07924 −0.500000 + 0.866025i −0.0944909 − 0.163663i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 256.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.i.e ✓ 12
13.c even 3 1 inner 663.2.i.e ✓ 12
13.c even 3 1 8619.2.a.ba 6
13.e even 6 1 8619.2.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.i.e ✓ 12 1.a even 1 1 trivial
663.2.i.e ✓ 12 13.c even 3 1 inner
8619.2.a.ba 6 13.c even 3 1
8619.2.a.bc 6 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):

\( T_{2}^{12} - 2 T_{2}^{11} + 8 T_{2}^{10} - 8 T_{2}^{9} + 30 T_{2}^{8} - 29 T_{2}^{7} + 64 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 4T_{5}^{5} - 3T_{5}^{4} + 15T_{5}^{3} + 4T_{5}^{2} - 11T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 4 T^{5} - 3 T^{4} + \cdots - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 22201 \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} - 13 T^{11} + \cdots + 73441 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + \cdots + 18395521 \) Copy content Toggle raw display
$29$ \( T^{12} - 10 T^{11} + \cdots + 71081761 \) Copy content Toggle raw display
$31$ \( (T^{6} + T^{5} + \cdots - 20039)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 10 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 227677921 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 466516801 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{5} - 207 T^{4} + \cdots - 7391)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 25 T^{5} + \cdots - 28261)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2893256521 \) Copy content Toggle raw display
$61$ \( T^{12} - 30 T^{11} + \cdots + 78765625 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 98056033321 \) Copy content Toggle raw display
$71$ \( T^{12} - 3 T^{11} + \cdots + 1125721 \) Copy content Toggle raw display
$73$ \( (T^{6} - 291 T^{4} + \cdots + 155611)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 26 T^{5} + \cdots - 34789)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 3 T^{5} + \cdots + 132439)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 432846883921 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 126990361 \) Copy content Toggle raw display
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