Properties

Label 266.2.f.a
Level $266$
Weight $2$
Character orbit 266.f
Analytic conductor $2.124$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} - \zeta_{6} q^{10} - 2 q^{11} + 5 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} + (\zeta_{6} - 1) q^{16} - 3 q^{18} + (5 \zeta_{6} - 3) q^{19} + q^{20} + ( - 2 \zeta_{6} + 2) q^{22} - \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} - 5 q^{26} - \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + 4 q^{31} - \zeta_{6} q^{32} + (\zeta_{6} - 1) q^{35} + ( - 3 \zeta_{6} + 3) q^{36} - 4 q^{37} + ( - 3 \zeta_{6} - 2) q^{38} + (\zeta_{6} - 1) q^{40} + ( - 2 \zeta_{6} + 2) q^{41} + ( - 8 \zeta_{6} + 8) q^{43} + 2 \zeta_{6} q^{44} - 3 q^{45} + q^{46} + q^{49} - 4 q^{50} + ( - 5 \zeta_{6} + 5) q^{52} + 2 \zeta_{6} q^{53} + ( - 2 \zeta_{6} + 2) q^{55} + q^{56} + 6 q^{58} + ( - 7 \zeta_{6} + 7) q^{59} + 7 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + 3 \zeta_{6} q^{63} + q^{64} - 5 q^{65} - 12 \zeta_{6} q^{67} - \zeta_{6} q^{70} + ( - 15 \zeta_{6} + 15) q^{71} + 3 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + ( - 2 \zeta_{6} + 5) q^{76} - 2 q^{77} + ( - 4 \zeta_{6} + 4) q^{79} - \zeta_{6} q^{80} + (9 \zeta_{6} - 9) q^{81} + 2 \zeta_{6} q^{82} - 7 q^{83} + 8 \zeta_{6} q^{86} - 2 q^{88} + ( - 3 \zeta_{6} + 3) q^{90} + 5 \zeta_{6} q^{91} + (\zeta_{6} - 1) q^{92} + ( - 3 \zeta_{6} - 2) q^{95} + (12 \zeta_{6} - 12) q^{97} + (\zeta_{6} - 1) q^{98} - 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} + 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{5} + 2 q^{7} + 2 q^{8} + 3 q^{9} - q^{10} - 4 q^{11} + 5 q^{13} - q^{14} - q^{16} - 6 q^{18} - q^{19} + 2 q^{20} + 2 q^{22} - q^{23} + 4 q^{25} - 10 q^{26} - q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - q^{35} + 3 q^{36} - 8 q^{37} - 7 q^{38} - q^{40} + 2 q^{41} + 8 q^{43} + 2 q^{44} - 6 q^{45} + 2 q^{46} + 2 q^{49} - 8 q^{50} + 5 q^{52} + 2 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 7 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{63} + 2 q^{64} - 10 q^{65} - 12 q^{67} - q^{70} + 15 q^{71} + 3 q^{72} + 14 q^{73} + 4 q^{74} + 8 q^{76} - 4 q^{77} + 4 q^{79} - q^{80} - 9 q^{81} + 2 q^{82} - 14 q^{83} + 8 q^{86} - 4 q^{88} + 3 q^{90} + 5 q^{91} - q^{92} - 7 q^{95} - 12 q^{97} - q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000 1.00000 1.50000 2.59808i −0.500000 + 0.866025i
239.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000 1.00000 1.50000 + 2.59808i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 266.2.f.a 2
3.b odd 2 1 2394.2.o.i 2
19.c even 3 1 inner 266.2.f.a 2
19.c even 3 1 5054.2.a.b 1
19.d odd 6 1 5054.2.a.a 1
57.h odd 6 1 2394.2.o.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.a 2 1.a even 1 1 trivial
266.2.f.a 2 19.c even 3 1 inner
2394.2.o.i 2 3.b odd 2 1
2394.2.o.i 2 57.h odd 6 1
5054.2.a.a 1 19.d odd 6 1
5054.2.a.b 1 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 7)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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