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Properties

Label 663.2.z.c
Level $663$
Weight $2$
Character orbit 663.z
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(205,663)]
 
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, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.205");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.z (of order \(6\), 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 \(\Q(\zeta_{6})\)
Coefficient field: 12.0.5209545745815489.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 4x^{9} - 3x^{8} + 9x^{7} + 5x^{6} + 18x^{5} - 12x^{4} - 32x^{3} - 32x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 4x^{9} - 3x^{8} + 9x^{7} + 5x^{6} + 18x^{5} - 12x^{4} - 32x^{3} - 32x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 11 \nu^{7} + 3 \nu^{6} + 23 \nu^{5} + 28 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 2\nu^{10} + \nu^{9} - 4\nu^{8} - 11\nu^{7} - 8\nu^{6} + 4\nu^{5} + 25\nu^{4} + 26\nu^{3} - 16\nu - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - \nu^{9} - 4\nu^{8} - 7\nu^{7} + 6\nu^{6} + 14\nu^{5} + 23\nu^{4} + 6\nu^{3} - 28\nu^{2} - 16\nu - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} + 8 \nu^{9} - 19 \nu^{7} - 35 \nu^{6} - 27 \nu^{5} + 34 \nu^{4} + 88 \nu^{3} + \cdots - 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} - \nu^{9} + 5\nu^{7} + 2\nu^{6} + 8\nu^{5} - 7\nu^{4} - 14\nu^{3} - 6\nu^{2} - 20\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{11} + \nu^{10} + 2 \nu^{9} - 20 \nu^{8} - 47 \nu^{7} - 5 \nu^{6} + 23 \nu^{5} + 136 \nu^{4} + \cdots - 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{11} - \nu^{10} - 2 \nu^{9} + 5 \nu^{8} + 14 \nu^{7} + 7 \nu^{6} + 2 \nu^{5} - 32 \nu^{4} + \cdots + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{11} - 3 \nu^{10} - 4 \nu^{9} + 28 \nu^{8} + 67 \nu^{7} + 19 \nu^{6} - 9 \nu^{5} - 166 \nu^{4} + \cdots + 320 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + \nu^{9} - 2\nu^{8} - 7\nu^{7} - 2\nu^{6} - 4\nu^{5} + 17\nu^{4} + 20\nu^{3} + 4\nu^{2} + 8\nu - 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7 \nu^{11} - \nu^{10} + 36 \nu^{8} + 69 \nu^{7} + 9 \nu^{6} - 51 \nu^{5} - 238 \nu^{4} - 180 \nu^{3} + \cdots + 480 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{11} + 3 \nu^{9} + 18 \nu^{8} + 31 \nu^{7} - 14 \nu^{6} - 32 \nu^{5} - 89 \nu^{4} - 52 \nu^{3} + \cdots + 160 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{7} - \beta_{6} + 2\beta_{4} + \beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} + \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 3\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 4\beta_{3} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} - 2\beta_{7} - 4\beta_{6} + 3\beta_{5} + 5\beta_{4} + 4\beta_{3} - 6\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + 4\beta_{9} + 2\beta_{8} + 3\beta_{5} - 2\beta_{4} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5 \beta_{11} - 9 \beta_{10} - 3 \beta_{9} + 21 \beta_{8} - 16 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} + \cdots + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 6 \beta_{5} + 2 \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14 \beta_{11} + 9 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} - 26 \beta_{7} + 14 \beta_{6} + 24 \beta_{5} + \cdots + 2 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3 \beta_{11} - \beta_{10} + 8 \beta_{9} + 14 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( \beta_{11} - 27 \beta_{10} - 21 \beta_{9} + 45 \beta_{8} - 76 \beta_{7} - 26 \beta_{6} - 33 \beta_{5} + \cdots + 36 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
1.37664 − 0.323825i
−0.169576 − 1.40401i
−0.666967 + 1.24706i
−1.38675 − 0.277371i
1.39840 + 0.210906i
−0.0517491 + 1.41327i
1.37664 + 0.323825i
−0.169576 + 1.40401i
−0.666967 − 1.24706i
−1.38675 + 0.277371i
1.39840 − 0.210906i
−0.0517491 − 1.41327i
−2.10515 + 1.21541i 0.500000 + 0.866025i 1.95443 − 3.38517i 2.70824i −2.10515 − 1.21541i −1.78452 − 1.03029i 4.64008i −0.500000 + 0.866025i −3.29161 − 5.70123i
205.2 −1.60706 + 0.927836i 0.500000 + 0.866025i 0.721761 − 1.25013i 1.11030i −1.60706 − 0.927836i 1.47027 + 0.848862i − 1.03264i −0.500000 + 0.866025i −1.03017 − 1.78431i
205.3 −0.721286 + 0.416435i 0.500000 + 0.866025i −0.653165 + 1.13131i − 2.40228i −0.721286 − 0.416435i −0.0795345 − 0.0459193i − 2.75374i −0.500000 + 0.866025i 1.00039 + 1.73273i
205.4 0.852021 − 0.491915i 0.500000 + 0.866025i −0.516040 + 0.893808i − 2.12454i 0.852021 + 0.491915i 2.32033 + 1.33964i 2.98305i −0.500000 + 0.866025i −1.04509 − 1.81016i
205.5 1.38697 − 0.800767i 0.500000 + 0.866025i 0.282457 − 0.489230i 2.21119i 1.38697 + 0.800767i −2.28025 − 1.31650i 2.29834i −0.500000 + 0.866025i 1.77065 + 3.06686i
205.6 2.19450 − 1.26700i 0.500000 + 0.866025i 2.21056 − 3.82880i − 1.50290i 2.19450 + 1.26700i −1.14630 − 0.661817i − 6.13509i −0.500000 + 0.866025i −1.90417 − 3.29811i
511.1 −2.10515 − 1.21541i 0.500000 − 0.866025i 1.95443 + 3.38517i − 2.70824i −2.10515 + 1.21541i −1.78452 + 1.03029i − 4.64008i −0.500000 − 0.866025i −3.29161 + 5.70123i
511.2 −1.60706 − 0.927836i 0.500000 − 0.866025i 0.721761 + 1.25013i − 1.11030i −1.60706 + 0.927836i 1.47027 − 0.848862i 1.03264i −0.500000 − 0.866025i −1.03017 + 1.78431i
511.3 −0.721286 − 0.416435i 0.500000 − 0.866025i −0.653165 − 1.13131i 2.40228i −0.721286 + 0.416435i −0.0795345 + 0.0459193i 2.75374i −0.500000 − 0.866025i 1.00039 − 1.73273i
511.4 0.852021 + 0.491915i 0.500000 − 0.866025i −0.516040 − 0.893808i 2.12454i 0.852021 − 0.491915i 2.32033 − 1.33964i − 2.98305i −0.500000 − 0.866025i −1.04509 + 1.81016i
511.5 1.38697 + 0.800767i 0.500000 − 0.866025i 0.282457 + 0.489230i − 2.21119i 1.38697 − 0.800767i −2.28025 + 1.31650i − 2.29834i −0.500000 − 0.866025i 1.77065 − 3.06686i
511.6 2.19450 + 1.26700i 0.500000 − 0.866025i 2.21056 + 3.82880i 1.50290i 2.19450 − 1.26700i −1.14630 + 0.661817i 6.13509i −0.500000 − 0.866025i −1.90417 + 3.29811i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.z.c ✓ 12
13.e even 6 1 inner 663.2.z.c ✓ 12
13.f odd 12 2 8619.2.a.bm 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.c ✓ 12 1.a even 1 1 trivial
663.2.z.c ✓ 12 13.e even 6 1 inner
8619.2.a.bm 12 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 10T_{2}^{10} + 74T_{2}^{8} + 3T_{2}^{7} - 230T_{2}^{6} + 526T_{2}^{4} - 78T_{2}^{3} - 387T_{2}^{2} + 45T_{2} + 225 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 10 T^{10} + \cdots + 225 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 26 T^{10} + \cdots + 2601 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{12} - 15 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + \cdots + 335241 \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 6561 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 23409 \) Copy content Toggle raw display
$31$ \( T^{12} + 143 T^{10} + \cdots + 19158129 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots + 140625 \) Copy content Toggle raw display
$41$ \( T^{12} + 9 T^{11} + \cdots + 11309769 \) Copy content Toggle raw display
$43$ \( T^{12} + 4 T^{11} + \cdots + 762129 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1731142449 \) Copy content Toggle raw display
$53$ \( (T^{6} + 7 T^{5} + \cdots - 117)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 39 T^{11} + \cdots + 6095961 \) Copy content Toggle raw display
$61$ \( T^{12} - 6 T^{11} + \cdots + 94926049 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 324324081 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19424275641 \) Copy content Toggle raw display
$73$ \( T^{12} + 454 T^{10} + \cdots + 6985449 \) Copy content Toggle raw display
$79$ \( (T^{6} + 26 T^{5} + \cdots - 58469)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 177497903025 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 971755929 \) Copy content Toggle raw display
$97$ \( T^{12} + 3 T^{11} + \cdots + 7080921 \) Copy content Toggle raw display
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