Properties

Label 418.2.a.f
Level $418$
Weight $2$
Character orbit 418.a
Self dual yes
Analytic conductor $3.338$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.79129
−1.79129
1.00000 −2.79129 1.00000 −0.791288 −2.79129 −0.208712 1.00000 4.79129 −0.791288
1.2 1.00000 1.79129 1.00000 3.79129 1.79129 −4.79129 1.00000 0.208712 3.79129
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.a.f 2
3.b odd 2 1 3762.2.a.s 2
4.b odd 2 1 3344.2.a.l 2
11.b odd 2 1 4598.2.a.y 2
19.b odd 2 1 7942.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.f 2 1.a even 1 1 trivial
3344.2.a.l 2 4.b odd 2 1
3762.2.a.s 2 3.b odd 2 1
4598.2.a.y 2 11.b odd 2 1
7942.2.a.w 2 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 7 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 15 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T + 67 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 47 \) Copy content Toggle raw display
$47$ \( T^{2} - 84 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 27 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 60 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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