Properties

Label 418.2.b.a
Level $418$
Weight $2$
Character orbit 418.b
Analytic conductor $3.338$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-10}) \)
Defining polynomial: \( x^{2} + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2 q^{5} + \beta q^{7} - q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - 2 q^{5} + \beta q^{7} - q^{8} + 3 q^{9} + 2 q^{10} + ( - \beta + 1) q^{11} - 6 q^{13} - \beta q^{14} + q^{16} + 2 \beta q^{17} - 3 q^{18} + (\beta - 3) q^{19} - 2 q^{20} + (\beta - 1) q^{22} - 4 q^{23} - q^{25} + 6 q^{26} + \beta q^{28} + 6 q^{29} + 3 \beta q^{31} - q^{32} - 2 \beta q^{34} - 2 \beta q^{35} + 3 q^{36} + 3 \beta q^{37} + ( - \beta + 3) q^{38} + 2 q^{40} + ( - \beta + 1) q^{44} - 6 q^{45} + 4 q^{46} - 8 q^{47} - 3 q^{49} + q^{50} - 6 q^{52} - 3 \beta q^{53} + (2 \beta - 2) q^{55} - \beta q^{56} - 6 q^{58} - \beta q^{61} - 3 \beta q^{62} + 3 \beta q^{63} + q^{64} + 12 q^{65} + 2 \beta q^{68} + 2 \beta q^{70} + 3 \beta q^{71} - 3 q^{72} - 3 \beta q^{74} + (\beta - 3) q^{76} + (\beta + 10) q^{77} - 2 q^{80} + 9 q^{81} - 2 \beta q^{83} - 4 \beta q^{85} + (\beta - 1) q^{88} + 6 q^{90} - 6 \beta q^{91} - 4 q^{92} + 8 q^{94} + ( - 2 \beta + 6) q^{95} - 6 \beta q^{97} + 3 q^{98} + ( - 3 \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{11} - 12 q^{13} + 2 q^{16} - 6 q^{18} - 6 q^{19} - 4 q^{20} - 2 q^{22} - 8 q^{23} - 2 q^{25} + 12 q^{26} + 12 q^{29} - 2 q^{32} + 6 q^{36} + 6 q^{38} + 4 q^{40} + 2 q^{44} - 12 q^{45} + 8 q^{46} - 16 q^{47} - 6 q^{49} + 2 q^{50} - 12 q^{52} - 4 q^{55} - 12 q^{58} + 2 q^{64} + 24 q^{65} - 6 q^{72} - 6 q^{76} + 20 q^{77} - 4 q^{80} + 18 q^{81} - 2 q^{88} + 12 q^{90} - 8 q^{92} + 16 q^{94} + 12 q^{95} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(287\) \(343\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
3.16228i
3.16228i
−1.00000 0 1.00000 −2.00000 0 3.16228i −1.00000 3.00000 2.00000
417.2 −1.00000 0 1.00000 −2.00000 0 3.16228i −1.00000 3.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.b.a 2
3.b odd 2 1 3762.2.g.f 2
11.b odd 2 1 418.2.b.b yes 2
19.b odd 2 1 418.2.b.b yes 2
33.d even 2 1 3762.2.g.c 2
57.d even 2 1 3762.2.g.c 2
209.d even 2 1 inner 418.2.b.a 2
627.b odd 2 1 3762.2.g.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.b.a 2 1.a even 1 1 trivial
418.2.b.a 2 209.d even 2 1 inner
418.2.b.b yes 2 11.b odd 2 1
418.2.b.b yes 2 19.b odd 2 1
3762.2.g.c 2 33.d even 2 1
3762.2.g.c 2 57.d even 2 1
3762.2.g.f 2 3.b odd 2 1
3762.2.g.f 2 627.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 11 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 40 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 19 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 90 \) Copy content Toggle raw display
$37$ \( T^{2} + 90 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 90 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 10 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 90 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 40 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 360 \) Copy content Toggle raw display
show more
show less