Properties

Label 414.2.e.a
Level $414$
Weight $2$
Character orbit 414.e
Analytic conductor $3.306$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.30580664368\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 2) q^{6} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 2) q^{6} - q^{8} + 3 \zeta_{6} q^{9} - 4 q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} - 4 \zeta_{6} q^{13} + ( - 8 \zeta_{6} + 4) q^{15} + (\zeta_{6} - 1) q^{16} + 7 q^{17} + 3 q^{18} - 5 q^{19} + (4 \zeta_{6} - 4) q^{20} - 3 \zeta_{6} q^{22} - \zeta_{6} q^{23} + ( - \zeta_{6} - 1) q^{24} + (11 \zeta_{6} - 11) q^{25} - 4 q^{26} + (6 \zeta_{6} - 3) q^{27} + ( - 4 \zeta_{6} + 4) q^{29} + ( - 4 \zeta_{6} - 4) q^{30} + 2 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 6) q^{33} + ( - 7 \zeta_{6} + 7) q^{34} + ( - 3 \zeta_{6} + 3) q^{36} + 8 q^{37} + (5 \zeta_{6} - 5) q^{38} + ( - 8 \zeta_{6} + 4) q^{39} + 4 \zeta_{6} q^{40} + 7 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 q^{44} + ( - 12 \zeta_{6} + 12) q^{45} - q^{46} + (10 \zeta_{6} - 10) q^{47} + (\zeta_{6} - 2) q^{48} + 7 \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + (7 \zeta_{6} + 7) q^{51} + (4 \zeta_{6} - 4) q^{52} + 10 q^{53} + (3 \zeta_{6} + 3) q^{54} - 12 q^{55} + ( - 5 \zeta_{6} - 5) q^{57} - 4 \zeta_{6} q^{58} - 13 \zeta_{6} q^{59} + (4 \zeta_{6} - 8) q^{60} + (2 \zeta_{6} - 2) q^{61} + 2 q^{62} + q^{64} + (16 \zeta_{6} - 16) q^{65} + ( - 6 \zeta_{6} + 3) q^{66} + 7 \zeta_{6} q^{67} - 7 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 1) q^{69} - 3 \zeta_{6} q^{72} - 9 q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + (11 \zeta_{6} - 22) q^{75} + 5 \zeta_{6} q^{76} + ( - 4 \zeta_{6} - 4) q^{78} + (2 \zeta_{6} - 2) q^{79} + 4 q^{80} + (9 \zeta_{6} - 9) q^{81} + 7 q^{82} + (4 \zeta_{6} - 4) q^{83} - 28 \zeta_{6} q^{85} - \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 8) q^{87} + (3 \zeta_{6} - 3) q^{88} + 18 q^{89} - 12 \zeta_{6} q^{90} + (\zeta_{6} - 1) q^{92} + (4 \zeta_{6} - 2) q^{93} + 10 \zeta_{6} q^{94} + 20 \zeta_{6} q^{95} + (2 \zeta_{6} - 1) q^{96} + (7 \zeta_{6} - 7) q^{97} + 7 q^{98} + 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 3 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 3 q^{6} - 2 q^{8} + 3 q^{9} - 8 q^{10} + 3 q^{11} - 4 q^{13} - q^{16} + 14 q^{17} + 6 q^{18} - 10 q^{19} - 4 q^{20} - 3 q^{22} - q^{23} - 3 q^{24} - 11 q^{25} - 8 q^{26} + 4 q^{29} - 12 q^{30} + 2 q^{31} + q^{32} + 9 q^{33} + 7 q^{34} + 3 q^{36} + 16 q^{37} - 5 q^{38} + 4 q^{40} + 7 q^{41} + q^{43} - 6 q^{44} + 12 q^{45} - 2 q^{46} - 10 q^{47} - 3 q^{48} + 7 q^{49} + 11 q^{50} + 21 q^{51} - 4 q^{52} + 20 q^{53} + 9 q^{54} - 24 q^{55} - 15 q^{57} - 4 q^{58} - 13 q^{59} - 12 q^{60} - 2 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} + 7 q^{67} - 7 q^{68} - 3 q^{72} - 18 q^{73} + 8 q^{74} - 33 q^{75} + 5 q^{76} - 12 q^{78} - 2 q^{79} + 8 q^{80} - 9 q^{81} + 14 q^{82} - 4 q^{83} - 28 q^{85} - q^{86} + 12 q^{87} - 3 q^{88} + 36 q^{89} - 12 q^{90} - q^{92} + 10 q^{94} + 20 q^{95} - 7 q^{97} + 14 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i −2.00000 + 3.46410i 1.50000 + 0.866025i 0 −1.00000 1.50000 2.59808i −4.00000
277.1 0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i −2.00000 3.46410i 1.50000 0.866025i 0 −1.00000 1.50000 + 2.59808i −4.00000
\(n\): e.g. 2-40 or 990-1000
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 414.2.e.a 2
3.b odd 2 1 1242.2.e.a 2
9.c even 3 1 inner 414.2.e.a 2
9.c even 3 1 3726.2.a.d 1
9.d odd 6 1 1242.2.e.a 2
9.d odd 6 1 3726.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.a 2 1.a even 1 1 trivial
414.2.e.a 2 9.c even 3 1 inner
1242.2.e.a 2 3.b odd 2 1
1242.2.e.a 2 9.d odd 6 1
3726.2.a.d 1 9.c even 3 1
3726.2.a.e 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( (T + 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
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