Properties

Label 666.2.bj.c
Level $666$
Weight $2$
Character orbit 666.bj
Analytic conductor $5.318$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.bj (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} + \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - 2 \zeta_{36}^{11} + \zeta_{36}^{10} + \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36} q^{2} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} + \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - 2 \zeta_{36}^{11} + \zeta_{36}^{10} + \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8} + ( - \zeta_{36}^{10} + \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{6} + \zeta_{36}^{4} + \zeta_{36}^{3} - \zeta_{36}^{2}) q^{10} + (\zeta_{36}^{10} + 4 \zeta_{36}^{9} + \zeta_{36}^{8} - \zeta_{36}^{6} - 2 \zeta_{36}^{3} - \zeta_{36}^{2} + 1) q^{11} + (2 \zeta_{36}^{11} + \zeta_{36}^{10} + 2 \zeta_{36}^{8} + \zeta_{36}^{6} - \zeta_{36}^{4} - \zeta_{36}^{3} - \zeta_{36}^{2} + \zeta_{36}) q^{13} + (\zeta_{36}^{11} - \zeta_{36}^{6} - \zeta_{36} + 2) q^{14} + \zeta_{36}^{4} q^{16} + ( - 2 \zeta_{36}^{11} - \zeta_{36}^{10} - 2 \zeta_{36}^{7} - \zeta_{36}^{4} - 2 \zeta_{36}^{3}) q^{17} + ( - 2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} - \zeta_{36}^{8} - 2 \zeta_{36}^{7} - \zeta_{36}^{6} + 2 \zeta_{36}^{5} - \zeta_{36}^{4} + \cdots - 1) q^{19}+ \cdots + (2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} + 2 \zeta_{36}^{6} + 2 \zeta_{36}^{4} - \zeta_{36}^{3} + \zeta_{36} - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 18 q^{14} - 18 q^{19} - 18 q^{25} - 12 q^{26} - 6 q^{28} - 18 q^{29} + 12 q^{34} - 18 q^{35} + 30 q^{37} + 24 q^{38} + 12 q^{40} - 24 q^{41} - 6 q^{44} + 30 q^{46} - 6 q^{47} + 12 q^{49} + 36 q^{50} - 12 q^{52} + 12 q^{53} - 18 q^{55} + 6 q^{58} - 36 q^{61} + 6 q^{64} - 36 q^{65} - 30 q^{67} - 12 q^{70} - 12 q^{71} + 48 q^{74} - 18 q^{76} - 12 q^{77} + 6 q^{79} + 48 q^{83} + 18 q^{85} + 36 q^{86} - 36 q^{88} + 18 q^{89} - 6 q^{91} - 18 q^{92} + 36 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\zeta_{36}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.342020 0.939693i
0.342020 + 0.939693i
−0.984808 + 0.173648i
0.984808 0.173648i
−0.642788 0.766044i
0.642788 + 0.766044i
−0.984808 0.173648i
0.984808 + 0.173648i
−0.642788 + 0.766044i
0.642788 0.766044i
−0.342020 + 0.939693i
0.342020 0.939693i
−0.342020 0.939693i 0 −0.766044 + 0.642788i −0.839712 + 0.148064i 0 0.240460 + 1.36372i 0.866025 + 0.500000i 0 0.426333 + 0.738430i
289.2 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −2.57176 + 0.453471i 0 −0.361075 2.04776i −0.866025 0.500000i 0 −1.30572 2.26157i
361.1 −0.984808 + 0.173648i 0 0.939693 0.342020i 0.247315 0.294739i 0 −2.50048 2.09815i −0.866025 + 0.500000i 0 −0.192377 + 0.333207i
361.2 0.984808 0.173648i 0 0.939693 0.342020i 1.97937 2.35892i 0 0.153180 + 0.128533i 0.866025 0.500000i 0 1.53967 2.66679i
469.1 −0.642788 0.766044i 0 −0.173648 + 0.984808i 1.45842 + 4.00698i 0 −3.39364 + 1.23518i 0.866025 0.500000i 0 2.13207 3.69285i
469.2 0.642788 + 0.766044i 0 −0.173648 + 0.984808i −0.273629 0.751790i 0 −0.138449 + 0.0503913i −0.866025 + 0.500000i 0 0.400019 0.692853i
559.1 −0.984808 0.173648i 0 0.939693 + 0.342020i 0.247315 + 0.294739i 0 −2.50048 + 2.09815i −0.866025 0.500000i 0 −0.192377 0.333207i
559.2 0.984808 + 0.173648i 0 0.939693 + 0.342020i 1.97937 + 2.35892i 0 0.153180 0.128533i 0.866025 + 0.500000i 0 1.53967 + 2.66679i
595.1 −0.642788 + 0.766044i 0 −0.173648 0.984808i 1.45842 4.00698i 0 −3.39364 1.23518i 0.866025 + 0.500000i 0 2.13207 + 3.69285i
595.2 0.642788 0.766044i 0 −0.173648 0.984808i −0.273629 + 0.751790i 0 −0.138449 0.0503913i −0.866025 0.500000i 0 0.400019 + 0.692853i
613.1 −0.342020 + 0.939693i 0 −0.766044 0.642788i −0.839712 0.148064i 0 0.240460 1.36372i 0.866025 0.500000i 0 0.426333 0.738430i
613.2 0.342020 0.939693i 0 −0.766044 0.642788i −2.57176 0.453471i 0 −0.361075 + 2.04776i −0.866025 + 0.500000i 0 −1.30572 + 2.26157i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.bj.c 12
3.b odd 2 1 74.2.h.a 12
12.b even 2 1 592.2.bq.b 12
37.h even 18 1 inner 666.2.bj.c 12
111.n odd 18 1 74.2.h.a 12
111.q even 36 1 2738.2.a.r 6
111.q even 36 1 2738.2.a.s 6
444.ba even 18 1 592.2.bq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 3.b odd 2 1
74.2.h.a 12 111.n odd 18 1
592.2.bq.b 12 12.b even 2 1
592.2.bq.b 12 444.ba even 18 1
666.2.bj.c 12 1.a even 1 1 trivial
666.2.bj.c 12 37.h even 18 1 inner
2738.2.a.r 6 111.q even 36 1
2738.2.a.s 6 111.q even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{10} + 72 T_{5}^{9} + 36 T_{5}^{8} + 162 T_{5}^{7} + 1476 T_{5}^{6} + 2268 T_{5}^{5} + 1701 T_{5}^{4} + 810 T_{5}^{3} + 81 T_{5}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} + 72 T^{9} + 36 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + 66 T^{10} + 218 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + 63 T^{10} + \cdots + 408321 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + 30 T^{10} + \cdots + 288369 \) Copy content Toggle raw display
$17$ \( T^{12} + 36 T^{10} - 234 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 10439361 \) Copy content Toggle raw display
$23$ \( T^{12} - 63 T^{10} + 3303 T^{8} + \cdots + 431649 \) Copy content Toggle raw display
$29$ \( T^{12} + 18 T^{11} + 126 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( T^{12} + 210 T^{10} + \cdots + 317445489 \) Copy content Toggle raw display
$37$ \( T^{12} - 30 T^{11} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} + 24 T^{11} + 216 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( T^{12} + 156 T^{10} + 8910 T^{8} + \cdots + 2277081 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots + 2027430729 \) Copy content Toggle raw display
$53$ \( T^{12} - 12 T^{11} + \cdots + 45041148441 \) Copy content Toggle raw display
$59$ \( T^{12} - 1404 T^{9} + \cdots + 17477104401 \) Copy content Toggle raw display
$61$ \( T^{12} + 36 T^{11} + 756 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$67$ \( T^{12} + 30 T^{11} + \cdots + 59697637561 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots + 4131885551616 \) Copy content Toggle raw display
$73$ \( (T^{6} - 366 T^{4} + 322 T^{3} + \cdots + 94609)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + 12 T^{10} + \cdots + 47961 \) Copy content Toggle raw display
$83$ \( T^{12} - 48 T^{11} + 1008 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$89$ \( T^{12} - 18 T^{11} + \cdots + 687331089 \) Copy content Toggle raw display
$97$ \( T^{12} - 36 T^{11} + \cdots + 27455164416 \) Copy content Toggle raw display
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