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Properties

Label 666.2.e.c
Level $666$
Weight $2$
Character orbit 666.e
Analytic conductor $5.318$
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] = [666,2,Mod(223,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("666.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.e (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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: 12.0.298306404859401.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 12 x^{10} - 22 x^{9} + 37 x^{8} - 53 x^{7} + 71 x^{6} - 106 x^{5} + 148 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 12 x^{10} - 22 x^{9} + 37 x^{8} - 53 x^{7} + 71 x^{6} - 106 x^{5} + 148 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} - 19 \nu^{10} + 36 \nu^{9} - 66 \nu^{8} + 115 \nu^{7} - 155 \nu^{6} + 185 \nu^{5} + \cdots - 352 ) / 112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{11} - 17 \nu^{10} + 44 \nu^{9} - 62 \nu^{8} + 97 \nu^{7} - 177 \nu^{6} + 195 \nu^{5} + \cdots - 480 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13 \nu^{11} - 9 \nu^{10} + 62 \nu^{9} - 74 \nu^{8} + 179 \nu^{7} - 321 \nu^{6} + 207 \nu^{5} + \cdots - 1440 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{11} - 19 \nu^{10} + 43 \nu^{9} - 66 \nu^{8} + 108 \nu^{7} - 155 \nu^{6} + 213 \nu^{5} + \cdots - 352 ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} + 7 \nu^{10} - 8 \nu^{9} + 14 \nu^{8} - 15 \nu^{7} + 15 \nu^{6} - 41 \nu^{5} + 50 \nu^{4} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29 \nu^{11} - 93 \nu^{10} + 160 \nu^{9} - 326 \nu^{8} + 473 \nu^{7} - 573 \nu^{6} + 879 \nu^{5} + \cdots - 768 ) / 224 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{11} - 66 \nu^{10} + 121 \nu^{9} - 216 \nu^{8} + 349 \nu^{7} - 450 \nu^{6} + 650 \nu^{5} + \cdots - 816 ) / 112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41 \nu^{11} + 145 \nu^{10} - 260 \nu^{9} + 458 \nu^{8} - 773 \nu^{7} + 953 \nu^{6} - 1319 \nu^{5} + \cdots + 1696 ) / 224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34 \nu^{11} - 131 \nu^{10} + 253 \nu^{9} - 444 \nu^{8} + 724 \nu^{7} - 911 \nu^{6} + 1291 \nu^{5} + \cdots - 1584 ) / 112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43 \nu^{11} - 149 \nu^{10} + 272 \nu^{9} - 480 \nu^{8} + 795 \nu^{7} - 993 \nu^{6} + 1425 \nu^{5} + \cdots - 1776 ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19 \nu^{11} - 67 \nu^{10} + 124 \nu^{9} - 222 \nu^{8} + 351 \nu^{7} - 443 \nu^{6} + 637 \nu^{5} + \cdots - 704 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{5} + 3\beta_{3} - 5\beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 5 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{11} - \beta_{10} + 6 \beta_{9} + \beta_{8} + 6 \beta_{7} + \beta_{6} + 5 \beta_{4} + \cdots - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 10 \beta_{11} - \beta_{10} + 17 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + 6 \beta_{6} - 5 \beta_{5} + \cdots - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} - 5 \beta_{10} + 3 \beta_{9} - 7 \beta_{8} + 6 \beta_{7} - 10 \beta_{6} + 6 \beta_{5} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12 \beta_{11} - \beta_{10} + 16 \beta_{9} - 11 \beta_{8} + 7 \beta_{7} - 13 \beta_{6} - \beta_{5} + \cdots + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{11} - 19 \beta_{10} + 41 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} - 9 \beta_{6} - 23 \beta_{5} + \cdots - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16 \beta_{11} + \beta_{10} - 2 \beta_{9} + 32 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} + 23 \beta_{5} + \cdots + 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( \beta_{11} + 18 \beta_{10} - 22 \beta_{9} - 6 \beta_{8} + 5 \beta_{7} + 17 \beta_{6} + 4 \beta_{5} + \cdots + 35 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(-1 + \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
−1.08615 − 0.905697i
1.40854 − 0.126504i
0.0638770 − 1.41277i
−0.323702 + 1.37667i
1.15997 + 0.808993i
1.27746 − 0.606716i
−1.08615 + 0.905697i
1.40854 + 0.126504i
0.0638770 + 1.41277i
−0.323702 − 1.37667i
1.15997 − 0.808993i
1.27746 + 0.606716i
−0.500000 + 0.866025i −1.65693 + 0.504552i −0.500000 − 0.866025i 0.829504 + 1.43674i 0.391512 − 1.68722i 1.82743 − 3.16520i 1.00000 2.49086 − 1.67202i −1.65901
223.2 −0.500000 + 0.866025i −0.748126 − 1.56215i −0.500000 − 0.866025i 1.84284 + 3.19190i 1.72692 + 0.133178i −0.0947164 + 0.164054i 1.00000 −1.88061 + 2.33737i −3.68569
223.3 −0.500000 + 0.866025i −0.212828 + 1.71893i −0.500000 − 0.866025i −0.478729 − 0.829183i −1.38222 − 1.04378i 1.69156 − 2.92986i 1.00000 −2.90941 − 0.731670i 0.957457
223.4 −0.500000 + 0.866025i 0.924181 − 1.46489i −0.500000 − 0.866025i 0.606199 + 1.04997i 0.806538 + 1.53281i −0.530379 + 0.918644i 1.00000 −1.29178 − 2.70764i −1.21240
223.5 −0.500000 + 0.866025i 1.05750 + 1.37175i −0.500000 − 0.866025i 0.723090 + 1.25243i −1.71672 + 0.229952i −0.780593 + 1.35203i 1.00000 −0.763374 + 2.90125i −1.44618
223.6 −0.500000 + 0.866025i 1.63620 − 0.568189i −0.500000 − 0.866025i −1.02291 − 1.77173i −0.326035 + 1.70109i 0.386703 − 0.669789i 1.00000 2.35432 − 1.85935i 2.04581
445.1 −0.500000 − 0.866025i −1.65693 − 0.504552i −0.500000 + 0.866025i 0.829504 − 1.43674i 0.391512 + 1.68722i 1.82743 + 3.16520i 1.00000 2.49086 + 1.67202i −1.65901
445.2 −0.500000 − 0.866025i −0.748126 + 1.56215i −0.500000 + 0.866025i 1.84284 − 3.19190i 1.72692 − 0.133178i −0.0947164 − 0.164054i 1.00000 −1.88061 − 2.33737i −3.68569
445.3 −0.500000 − 0.866025i −0.212828 − 1.71893i −0.500000 + 0.866025i −0.478729 + 0.829183i −1.38222 + 1.04378i 1.69156 + 2.92986i 1.00000 −2.90941 + 0.731670i 0.957457
445.4 −0.500000 − 0.866025i 0.924181 + 1.46489i −0.500000 + 0.866025i 0.606199 − 1.04997i 0.806538 − 1.53281i −0.530379 − 0.918644i 1.00000 −1.29178 + 2.70764i −1.21240
445.5 −0.500000 − 0.866025i 1.05750 − 1.37175i −0.500000 + 0.866025i 0.723090 − 1.25243i −1.71672 − 0.229952i −0.780593 − 1.35203i 1.00000 −0.763374 − 2.90125i −1.44618
445.6 −0.500000 − 0.866025i 1.63620 + 0.568189i −0.500000 + 0.866025i −1.02291 + 1.77173i −0.326035 − 1.70109i 0.386703 + 0.669789i 1.00000 2.35432 + 1.85935i 2.04581
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.e.c ✓ 12
3.b odd 2 1 1998.2.e.c 12
9.c even 3 1 inner 666.2.e.c ✓ 12
9.c even 3 1 5994.2.a.t 6
9.d odd 6 1 1998.2.e.c 12
9.d odd 6 1 5994.2.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.e.c ✓ 12 1.a even 1 1 trivial
666.2.e.c ✓ 12 9.c even 3 1 inner
1998.2.e.c 12 3.b odd 2 1
1998.2.e.c 12 9.d odd 6 1
5994.2.a.s 6 9.d odd 6 1
5994.2.a.t 6 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 5 T_{5}^{11} + 25 T_{5}^{10} - 50 T_{5}^{9} + 148 T_{5}^{8} - 248 T_{5}^{7} + 577 T_{5}^{6} + \cdots + 441 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 5 T^{11} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{12} - 5 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{12} - 5 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} + \cdots - 1809)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + \cdots - 353)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 51 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 1791574929 \) Copy content Toggle raw display
$31$ \( T^{12} - 19 T^{11} + \cdots + 24649 \) Copy content Toggle raw display
$37$ \( (T - 1)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 8498442969 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 207043321 \) Copy content Toggle raw display
$47$ \( T^{12} - 2 T^{11} + \cdots + 859329 \) Copy content Toggle raw display
$53$ \( (T^{6} + 12 T^{5} + \cdots + 13599)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 121154049 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 24014771089 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10308137841 \) Copy content Toggle raw display
$71$ \( (T^{6} + 3 T^{5} + \cdots + 1593)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 22 T^{5} + \cdots + 53824)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 3296711889 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 9516197601 \) Copy content Toggle raw display
$89$ \( (T^{6} + 5 T^{5} + \cdots + 282537)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2752786089 \) Copy content Toggle raw display
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