Properties

Label 418.2.e.i
Level $418$
Weight $2$
Character orbit 418.e
Analytic conductor $3.338$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.101617200.1
Defining polynomial: \( x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_1 q^{3} + (\beta_{4} - 1) q^{4} - 2 \beta_{4} q^{5} + ( - \beta_{2} - \beta_1) q^{6} + (\beta_{2} + 2) q^{7} - q^{8} + (\beta_{5} + 2 \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_1 q^{3} + (\beta_{4} - 1) q^{4} - 2 \beta_{4} q^{5} + ( - \beta_{2} - \beta_1) q^{6} + (\beta_{2} + 2) q^{7} - q^{8} + (\beta_{5} + 2 \beta_{4} - 2) q^{9} + ( - 2 \beta_{4} + 2) q^{10} + q^{11} - \beta_{2} q^{12} + (\beta_{5} + 2 \beta_{4} - 2) q^{13} + (2 \beta_{4} - \beta_1) q^{14} + (2 \beta_{2} + 2 \beta_1) q^{15} - \beta_{4} q^{16} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_1) q^{17} + (\beta_{3} - 2) q^{18} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{19} + 2 q^{20} + (\beta_{5} + 5 \beta_{4} - \beta_{3} - 2 \beta_1) q^{21} + \beta_{4} q^{22} + (4 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 4) q^{23} + \beta_1 q^{24} + ( - \beta_{4} + 1) q^{25} + (\beta_{3} - 2) q^{26} + ( - 2 \beta_{2} - 2) q^{27} + (2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{28} + ( - \beta_{4} + 1) q^{29} + 2 \beta_{2} q^{30} + (\beta_{3} + 2 \beta_{2} + 1) q^{31} + ( - \beta_{4} + 1) q^{32} - \beta_1 q^{33} + ( - \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 + 3) q^{34} + ( - 4 \beta_{4} + 2 \beta_1) q^{35} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{36} + ( - \beta_{3} - 3 \beta_{2} + 1) q^{37} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{38} + ( - 5 \beta_{2} - 2) q^{39} + 2 \beta_{4} q^{40} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_1) q^{41} + (\beta_{5} + 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 5) q^{42} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{43} + (\beta_{4} - 1) q^{44} + ( - 2 \beta_{3} + 4) q^{45} + (2 \beta_{2} - 4) q^{46} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 + 3) q^{47} + (\beta_{2} + \beta_1) q^{48} + ( - \beta_{3} + 4 \beta_{2} + 2) q^{49} + q^{50} + (\beta_{5} + 3 \beta_{4} + 6 \beta_{2} + 6 \beta_1 - 3) q^{51} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{52} + ( - 4 \beta_{2} - 4 \beta_1) q^{53} + ( - 2 \beta_{4} + 2 \beta_1) q^{54} - 2 \beta_{4} q^{55} + ( - \beta_{2} - 2) q^{56} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 + 5) q^{57} + q^{58} - 2 \beta_1 q^{60} + ( - \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 4) q^{61} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{62} + (2 \beta_{5} + 6 \beta_{4} - 5 \beta_{2} - 5 \beta_1 - 6) q^{63} + q^{64} + ( - 2 \beta_{3} + 4) q^{65} + ( - \beta_{2} - \beta_1) q^{66} + ( - 3 \beta_{2} - 3 \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} + 3) q^{68} + ( - 2 \beta_{3} - 4 \beta_{2} + 10) q^{69} + ( - 4 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 4) q^{70} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1) q^{71} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{72} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_1) q^{73} + (\beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_1) q^{74} + \beta_{2} q^{75} + ( - \beta_{5} - \beta_1 + 1) q^{76} + (\beta_{2} + 2) q^{77} + ( - 2 \beta_{4} + 5 \beta_1) q^{78} + ( - 2 \beta_{4} - 3 \beta_1) q^{79} + (2 \beta_{4} - 2) q^{80} + (\beta_{5} - 4 \beta_{4} - \beta_{3} + 2 \beta_1) q^{81} + ( - \beta_{5} - 3 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{82} + ( - \beta_{3} + 3 \beta_{2} + 2) q^{83} + (\beta_{3} - 2 \beta_{2} - 5) q^{84} + (2 \beta_{5} + 6 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (\beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{86} + \beta_{2} q^{87} - q^{88} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 1) q^{89} + (2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{90} + (2 \beta_{5} + 6 \beta_{4} - 5 \beta_{2} - 5 \beta_1 - 6) q^{91} + ( - 4 \beta_{4} - 2 \beta_1) q^{92} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{93} + ( - 2 \beta_{3} - \beta_{2} + 3) q^{94} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{95} + \beta_{2} q^{96} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{97} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_1) q^{98} + (\beta_{5} + 2 \beta_{4} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{7} - 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{5} + 12 q^{7} - 6 q^{8} - 7 q^{9} + 6 q^{10} + 6 q^{11} - 7 q^{13} + 6 q^{14} - 3 q^{16} - 10 q^{17} - 14 q^{18} - 5 q^{19} + 12 q^{20} + 16 q^{21} + 3 q^{22} - 12 q^{23} + 3 q^{25} - 14 q^{26} - 12 q^{27} - 6 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{32} + 10 q^{34} - 12 q^{35} - 7 q^{36} + 8 q^{37} + 2 q^{38} - 12 q^{39} + 6 q^{40} - 10 q^{41} - 16 q^{42} - 5 q^{43} - 3 q^{44} + 28 q^{45} - 24 q^{46} + 11 q^{47} + 14 q^{49} + 6 q^{50} - 10 q^{51} - 7 q^{52} - 6 q^{54} - 6 q^{55} - 12 q^{56} + 26 q^{57} + 6 q^{58} - 11 q^{61} + 2 q^{62} - 20 q^{63} + 6 q^{64} + 28 q^{65} + 20 q^{68} + 64 q^{69} + 12 q^{70} + 5 q^{71} + 7 q^{72} - 8 q^{73} + 4 q^{74} + 7 q^{76} + 12 q^{77} - 6 q^{78} - 6 q^{79} - 6 q^{80} - 11 q^{81} + 10 q^{82} + 14 q^{83} - 32 q^{84} - 20 q^{85} + 5 q^{86} - 6 q^{88} - q^{89} + 14 q^{90} - 20 q^{91} - 12 q^{92} + 26 q^{93} + 22 q^{94} - 4 q^{95} + 7 q^{97} + 7 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 8\nu^{2} - 2\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 8\nu^{3} - 2\nu^{2} + 64\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{5} - 40\nu^{3} + 26\nu^{2} - 320\nu + 80 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{5} - 40\beta_{4} + 8\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 26\beta_{4} - 64\beta_{2} - 64\beta _1 - 26 \) Copy content Toggle raw display

Character values

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

\(n\) \(287\) \(343\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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
1.34700 + 2.33307i
0.126000 + 0.218239i
−1.47300 2.55131i
1.34700 2.33307i
0.126000 0.218239i
−1.47300 + 2.55131i
0.500000 + 0.866025i −1.34700 2.33307i −0.500000 + 0.866025i −1.00000 1.73205i 1.34700 2.33307i −0.693995 −1.00000 −2.12880 + 3.68720i 1.00000 1.73205i
45.2 0.500000 + 0.866025i −0.126000 0.218239i −0.500000 + 0.866025i −1.00000 1.73205i 0.126000 0.218239i 1.74800 −1.00000 1.46825 2.54308i 1.00000 1.73205i
45.3 0.500000 + 0.866025i 1.47300 + 2.55131i −0.500000 + 0.866025i −1.00000 1.73205i −1.47300 + 2.55131i 4.94600 −1.00000 −2.83944 + 4.91806i 1.00000 1.73205i
353.1 0.500000 0.866025i −1.34700 + 2.33307i −0.500000 0.866025i −1.00000 + 1.73205i 1.34700 + 2.33307i −0.693995 −1.00000 −2.12880 3.68720i 1.00000 + 1.73205i
353.2 0.500000 0.866025i −0.126000 + 0.218239i −0.500000 0.866025i −1.00000 + 1.73205i 0.126000 + 0.218239i 1.74800 −1.00000 1.46825 + 2.54308i 1.00000 + 1.73205i
353.3 0.500000 0.866025i 1.47300 2.55131i −0.500000 0.866025i −1.00000 + 1.73205i −1.47300 2.55131i 4.94600 −1.00000 −2.83944 4.91806i 1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.3
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.i 6
19.c even 3 1 inner 418.2.e.i 6
19.c even 3 1 7942.2.a.bd 3
19.d odd 6 1 7942.2.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.i 6 1.a even 1 1 trivial
418.2.e.i 6 19.c even 3 1 inner
7942.2.a.bd 3 19.c even 3 1
7942.2.a.bj 3 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 \(S_{2}^{\mathrm{new}}(418, [\chi])\):

\( T_{3}^{6} + 8T_{3}^{4} + 4T_{3}^{3} + 64T_{3}^{2} + 16T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + 4 T^{3} + 64 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 6 T^{2} + 4 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 7 T^{5} + 54 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} + 90 T^{4} + \cdots + 2500 \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{5} - 16 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + 128 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2 T^{2} - 40 T + 116)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 4 T^{2} - 70 T - 150)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 10 T^{5} + 102 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{5} + 52 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$47$ \( T^{6} - 11 T^{5} + 162 T^{4} + \cdots + 251001 \) Copy content Toggle raw display
$53$ \( T^{6} + 128 T^{4} - 256 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 11 T^{5} + 146 T^{4} + \cdots + 140625 \) Copy content Toggle raw display
$67$ \( T^{6} + 72 T^{4} - 108 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$71$ \( T^{6} - 5 T^{5} + 52 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( T^{6} + 8 T^{5} + 78 T^{4} + \cdots + 10404 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + 96 T^{4} + \cdots + 36100 \) Copy content Toggle raw display
$83$ \( (T^{3} - 7 T^{2} - 95 T - 205)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + T^{5} + 94 T^{4} - 179 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} + 126 T^{4} + \cdots + 17161 \) Copy content Toggle raw display
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