Properties

Label 418.2.e.g
Level $418$
Weight $2$
Character orbit 418.e
Analytic conductor $3.338$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{3} q^{7} - q^{8} + (3 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{3} q^{7} - q^{8} + (3 \beta_1 - 3) q^{9} - q^{11} + \beta_{3} q^{12} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{13} + \beta_{2} q^{14} - \beta_1 q^{16} + ( - \beta_{2} + 2 \beta_1) q^{17} - 3 q^{18} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} - 6 \beta_1 q^{21} - \beta_1 q^{22} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{23} + \beta_{2} q^{24} + ( - 5 \beta_1 + 5) q^{25} + (2 \beta_{3} + 1) q^{26} + ( - \beta_{3} + \beta_{2}) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{29} + 2 q^{31} + ( - \beta_1 + 1) q^{32} + \beta_{2} q^{33} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{34} - 3 \beta_1 q^{36} + ( - \beta_{3} + 6) q^{37} + (\beta_{3} - 2 \beta_{2} + 1) q^{38} + ( - \beta_{3} - 12) q^{39} + (\beta_{2} + 10 \beta_1) q^{41} + ( - 6 \beta_1 + 6) q^{42} + (\beta_{2} - 7 \beta_1) q^{43} + ( - \beta_1 + 1) q^{44} + (2 \beta_{3} + 4) q^{46} + (\beta_{3} - \beta_{2} - 9 \beta_1 + 9) q^{47} + ( - \beta_{3} + \beta_{2}) q^{48} - q^{49} + 5 q^{50} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{51} + (2 \beta_{2} + \beta_1) q^{52} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{53} - \beta_{3} q^{56} + ( - \beta_{3} + 12 \beta_1 - 6) q^{57} + ( - 2 \beta_{3} - 3) q^{58} + (2 \beta_{2} + 2 \beta_1) q^{59} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{61} + 2 \beta_1 q^{62} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{63} + q^{64} + ( - \beta_{3} + \beta_{2}) q^{66} + ( - \beta_{3} + \beta_{2} + 12 \beta_1 - 12) q^{67} + (\beta_{3} - 2) q^{68} + ( - 4 \beta_{3} - 12) q^{69} + (\beta_{2} - 5 \beta_1) q^{71} + ( - 3 \beta_1 + 3) q^{72} + ( - \beta_{2} + 12 \beta_1) q^{73} + ( - \beta_{2} + 6 \beta_1) q^{74} - 5 \beta_{3} q^{75} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{76} - \beta_{3} q^{77} + ( - \beta_{2} - 12 \beta_1) q^{78} + (\beta_{2} - 2 \beta_1) q^{79} + 9 \beta_1 q^{81} + ( - \beta_{3} + \beta_{2} + 10 \beta_1 - 10) q^{82} + (\beta_{3} - 3) q^{83} + 6 q^{84} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 + 7) q^{86} + (3 \beta_{3} + 12) q^{87} + q^{88} + ( - 9 \beta_1 + 9) q^{89} + (\beta_{3} - \beta_{2} - 12 \beta_1 + 12) q^{91} + (2 \beta_{2} + 4 \beta_1) q^{92} - 2 \beta_{2} q^{93} + (\beta_{3} + 9) q^{94} - \beta_{3} q^{96} + (2 \beta_{2} + \beta_1) q^{97} - \beta_1 q^{98} + ( - 3 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 6 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{16} + 4 q^{17} - 12 q^{18} + 2 q^{19} - 12 q^{21} - 2 q^{22} + 8 q^{23} + 10 q^{25} + 4 q^{26} - 6 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} - 6 q^{36} + 24 q^{37} + 4 q^{38} - 48 q^{39} + 20 q^{41} + 12 q^{42} - 14 q^{43} + 2 q^{44} + 16 q^{46} + 18 q^{47} - 4 q^{49} + 20 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} - 12 q^{58} + 4 q^{59} - 6 q^{61} + 4 q^{62} + 4 q^{64} - 24 q^{67} - 8 q^{68} - 48 q^{69} - 10 q^{71} + 6 q^{72} + 24 q^{73} + 12 q^{74} + 2 q^{76} - 24 q^{78} - 4 q^{79} + 18 q^{81} - 20 q^{82} - 12 q^{83} + 24 q^{84} + 14 q^{86} + 48 q^{87} + 4 q^{88} + 18 q^{89} + 24 q^{91} + 8 q^{92} + 36 q^{94} + 2 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0.500000 + 0.866025i −1.22474 2.12132i −0.500000 + 0.866025i 0 1.22474 2.12132i 2.44949 −1.00000 −1.50000 + 2.59808i 0
45.2 0.500000 + 0.866025i 1.22474 + 2.12132i −0.500000 + 0.866025i 0 −1.22474 + 2.12132i −2.44949 −1.00000 −1.50000 + 2.59808i 0
353.1 0.500000 0.866025i −1.22474 + 2.12132i −0.500000 0.866025i 0 1.22474 + 2.12132i 2.44949 −1.00000 −1.50000 2.59808i 0
353.2 0.500000 0.866025i 1.22474 2.12132i −0.500000 0.866025i 0 −1.22474 2.12132i −2.44949 −1.00000 −1.50000 2.59808i 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.g 4
19.c even 3 1 inner 418.2.e.g 4
19.c even 3 1 7942.2.a.v 2
19.d odd 6 1 7942.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.g 4 1.a even 1 1 trivial
418.2.e.g 4 19.c even 3 1 inner
7942.2.a.v 2 19.c even 3 1
7942.2.a.y 2 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 \(S_{2}^{\mathrm{new}}(418, [\chi])\):

\( T_{3}^{4} + 6T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 15 T^{2} - 38 T + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 72 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 20 T^{3} + 306 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 153 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} - 18 T^{3} + 249 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + 108 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$67$ \( T^{4} + 24 T^{3} + 438 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 438 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 18 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T + 3)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529 \) Copy content Toggle raw display
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