Properties

Label 418.2.e.d
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + 2 \zeta_{6} q^{6} + 2 q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + 2 \zeta_{6} q^{6} + 2 q^{7} + q^{8} - \zeta_{6} q^{9} + q^{11} - 2 q^{12} + \zeta_{6} q^{13} + (2 \zeta_{6} - 2) q^{14} + (\zeta_{6} - 1) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + q^{18} + ( - 3 \zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 4) q^{21} + (\zeta_{6} - 1) q^{22} + ( - 2 \zeta_{6} + 2) q^{24} + 5 \zeta_{6} q^{25} - q^{26} + 4 q^{27} - 2 \zeta_{6} q^{28} - 9 \zeta_{6} q^{29} + 8 q^{31} - \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + 6 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{36} - 10 q^{37} + ( - 2 \zeta_{6} + 5) q^{38} + 2 q^{39} + ( - 6 \zeta_{6} + 6) q^{41} + 4 \zeta_{6} q^{42} + (5 \zeta_{6} - 5) q^{43} - \zeta_{6} q^{44} + 3 \zeta_{6} q^{47} + 2 \zeta_{6} q^{48} - 3 q^{49} - 5 q^{50} - 12 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + 6 \zeta_{6} q^{53} + (4 \zeta_{6} - 4) q^{54} + 2 q^{56} + (4 \zeta_{6} - 10) q^{57} + 9 q^{58} + (6 \zeta_{6} - 6) q^{59} + 13 \zeta_{6} q^{61} + (8 \zeta_{6} - 8) q^{62} - 2 \zeta_{6} q^{63} + q^{64} + 2 \zeta_{6} q^{66} + 10 \zeta_{6} q^{67} - 6 q^{68} + (9 \zeta_{6} - 9) q^{71} - \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 10 \zeta_{6} + 10) q^{74} + 10 q^{75} + (5 \zeta_{6} - 3) q^{76} + 2 q^{77} + (2 \zeta_{6} - 2) q^{78} + (8 \zeta_{6} - 8) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 \zeta_{6} q^{82} - 9 q^{83} - 4 q^{84} - 5 \zeta_{6} q^{86} - 18 q^{87} + q^{88} - 15 \zeta_{6} q^{89} + 2 \zeta_{6} q^{91} + ( - 16 \zeta_{6} + 16) q^{93} - 3 q^{94} - 2 q^{96} + ( - \zeta_{6} + 1) q^{97} + ( - 3 \zeta_{6} + 3) q^{98} - \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9} + 2 q^{11} - 4 q^{12} + q^{13} - 2 q^{14} - q^{16} + 6 q^{17} + 2 q^{18} - 7 q^{19} + 4 q^{21} - q^{22} + 2 q^{24} + 5 q^{25} - 2 q^{26} + 8 q^{27} - 2 q^{28} - 9 q^{29} + 16 q^{31} - q^{32} + 2 q^{33} + 6 q^{34} - q^{36} - 20 q^{37} + 8 q^{38} + 4 q^{39} + 6 q^{41} + 4 q^{42} - 5 q^{43} - q^{44} + 3 q^{47} + 2 q^{48} - 6 q^{49} - 10 q^{50} - 12 q^{51} + q^{52} + 6 q^{53} - 4 q^{54} + 4 q^{56} - 16 q^{57} + 18 q^{58} - 6 q^{59} + 13 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{66} + 10 q^{67} - 12 q^{68} - 9 q^{71} - q^{72} + 4 q^{73} + 10 q^{74} + 20 q^{75} - q^{76} + 4 q^{77} - 2 q^{78} - 8 q^{79} + 11 q^{81} + 6 q^{82} - 18 q^{83} - 8 q^{84} - 5 q^{86} - 36 q^{87} + 2 q^{88} - 15 q^{89} + 2 q^{91} + 16 q^{93} - 6 q^{94} - 4 q^{96} + q^{97} + 3 q^{98} - q^{99}+O(q^{100}) \) 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 0 1.00000 1.73205i 2.00000 1.00000 −0.500000 + 0.866025i 0
353.1 −0.500000 + 0.866025i 1.00000 1.73205i −0.500000 0.866025i 0 1.00000 + 1.73205i 2.00000 1.00000 −0.500000 0.866025i 0
\(n\): 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.d 2
19.c even 3 1 inner 418.2.e.d 2
19.c even 3 1 7942.2.a.l 1
19.d odd 6 1 7942.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.d 2 1.a even 1 1 trivial
418.2.e.d 2 19.c even 3 1 inner
7942.2.a.j 1 19.d odd 6 1
7942.2.a.l 1 19.c even 3 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
\( T_{5} \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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