Properties

Label 418.2.e.f
Level $418$
Weight $2$
Character orbit 418.e
Analytic conductor $3.338$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{7} - q^{8} - \zeta_{6} q^{9} - 3 \zeta_{6} q^{10} + q^{11} - 2 q^{12} + 4 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} - 6 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} - q^{18} + (3 \zeta_{6} - 5) q^{19} - 3 q^{20} + (2 \zeta_{6} - 2) q^{21} + ( - \zeta_{6} + 1) q^{22} + (2 \zeta_{6} - 2) q^{24} - 4 \zeta_{6} q^{25} + 4 q^{26} + 4 q^{27} + \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} - 6 q^{30} - 4 q^{31} + \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + (3 \zeta_{6} - 3) q^{35} + (\zeta_{6} - 1) q^{36} - q^{37} + (5 \zeta_{6} - 2) q^{38} + 8 q^{39} + (3 \zeta_{6} - 3) q^{40} + ( - 6 \zeta_{6} + 6) q^{41} + 2 \zeta_{6} q^{42} + ( - 4 \zeta_{6} + 4) q^{43} - \zeta_{6} q^{44} - 3 q^{45} - 6 \zeta_{6} q^{47} + 2 \zeta_{6} q^{48} - 6 q^{49} - 4 q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 3 \zeta_{6} q^{53} + ( - 4 \zeta_{6} + 4) q^{54} + ( - 3 \zeta_{6} + 3) q^{55} + q^{56} + (10 \zeta_{6} - 4) q^{57} + 6 q^{58} + (6 \zeta_{6} - 6) q^{60} + 4 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + \zeta_{6} q^{63} + q^{64} + 12 q^{65} - 2 \zeta_{6} q^{66} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{70} + ( - 6 \zeta_{6} + 6) q^{71} + \zeta_{6} q^{72} + (2 \zeta_{6} - 2) q^{73} + (\zeta_{6} - 1) q^{74} - 8 q^{75} + (2 \zeta_{6} + 3) q^{76} - q^{77} + ( - 8 \zeta_{6} + 8) q^{78} + (11 \zeta_{6} - 11) q^{79} + 3 \zeta_{6} q^{80} + ( - 11 \zeta_{6} + 11) q^{81} - 6 \zeta_{6} q^{82} - 3 q^{83} + 2 q^{84} - 4 \zeta_{6} q^{86} + 12 q^{87} - q^{88} + 18 \zeta_{6} q^{89} + (3 \zeta_{6} - 3) q^{90} - 4 \zeta_{6} q^{91} + (8 \zeta_{6} - 8) q^{93} - 6 q^{94} + (15 \zeta_{6} - 6) q^{95} + 2 q^{96} + ( - 19 \zeta_{6} + 19) q^{97} + (6 \zeta_{6} - 6) q^{98} - \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} + 4 q^{13} - q^{14} - 6 q^{15} - q^{16} - 2 q^{18} - 7 q^{19} - 6 q^{20} - 2 q^{21} + q^{22} - 2 q^{24} - 4 q^{25} + 8 q^{26} + 8 q^{27} + q^{28} + 6 q^{29} - 12 q^{30} - 8 q^{31} + q^{32} + 2 q^{33} - 3 q^{35} - q^{36} - 2 q^{37} + q^{38} + 16 q^{39} - 3 q^{40} + 6 q^{41} + 2 q^{42} + 4 q^{43} - q^{44} - 6 q^{45} - 6 q^{47} + 2 q^{48} - 12 q^{49} - 8 q^{50} + 4 q^{52} - 3 q^{53} + 4 q^{54} + 3 q^{55} + 2 q^{56} + 2 q^{57} + 12 q^{58} - 6 q^{60} + 4 q^{61} - 4 q^{62} + q^{63} + 2 q^{64} + 24 q^{65} - 2 q^{66} + 4 q^{67} + 3 q^{70} + 6 q^{71} + q^{72} - 2 q^{73} - q^{74} - 16 q^{75} + 8 q^{76} - 2 q^{77} + 8 q^{78} - 11 q^{79} + 3 q^{80} + 11 q^{81} - 6 q^{82} - 6 q^{83} + 4 q^{84} - 4 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} - 3 q^{90} - 4 q^{91} - 8 q^{93} - 12 q^{94} + 3 q^{95} + 4 q^{96} + 19 q^{97} - 6 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i 1.50000 + 2.59808i −1.00000 + 1.73205i −1.00000 −1.00000 −0.500000 + 0.866025i −1.50000 + 2.59808i
353.1 0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i 1.50000 2.59808i −1.00000 1.73205i −1.00000 −1.00000 −0.500000 0.866025i −1.50000 2.59808i
\(n\): 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 418.2.e.f 2
19.c even 3 1 inner 418.2.e.f 2
19.c even 3 1 7942.2.a.b 1
19.d odd 6 1 7942.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.f 2 1.a even 1 1 trivial
418.2.e.f 2 19.c even 3 1 inner
7942.2.a.b 1 19.c even 3 1
7942.2.a.r 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
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