Properties

Label 418.2.f.b
Level $418$
Weight $2$
Character orbit 418.f
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.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{3} q^{4} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{5} + ( - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{3} q^{4} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{5} + ( - \zeta_{10} + 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{9} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2) q^{10} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{11} + (4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{13} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{14} - \zeta_{10} q^{16} + ( - 5 \zeta_{10}^{2} + 4 \zeta_{10} - 5) q^{17} + 3 \zeta_{10}^{3} q^{18} + \zeta_{10}^{2} q^{19} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{20} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{22} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3) q^{23} + (3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 3 \zeta_{10}) q^{25} + ( - 4 \zeta_{10} + 4) q^{26} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{28} + ( - 2 \zeta_{10} + 2) q^{29} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{31} + q^{32} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 1) q^{34} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 3) q^{35} - 3 \zeta_{10}^{2} q^{36} + (8 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{37} - \zeta_{10} q^{38} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{40} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{41} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{43} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{44} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 6) q^{45} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{46} + ( - \zeta_{10}^{3} - 10 \zeta_{10}^{2} - \zeta_{10}) q^{47} + (\zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{49} + ( - 3 \zeta_{10}^{2} - 5 \zeta_{10} - 3) q^{50} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{52} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{53} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 5 \zeta_{10} + 5) q^{55} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{56} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{58} + 10 \zeta_{10}^{3} q^{59} + ( - 5 \zeta_{10}^{2} - 2 \zeta_{10} - 5) q^{61} + (2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{62} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 12) q^{65} - 8 q^{67} + (\zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{68} + (3 \zeta_{10}^{3} - \zeta_{10} + 1) q^{70} + ( - 4 \zeta_{10}^{2} - 2 \zeta_{10} - 4) q^{71} + 3 \zeta_{10} q^{72} + (6 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{73} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 2 \zeta_{10}) q^{74} + q^{76} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{77} + ( - 12 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 12) q^{79} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{80} - 9 \zeta_{10}^{3} q^{81} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{82} + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{83} + (2 \zeta_{10}^{3} - 19 \zeta_{10}^{2} + 2 \zeta_{10}) q^{85} + ( - \zeta_{10}^{3} + 1) q^{86} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 3 \zeta_{10} + 1) q^{88} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} - 8) q^{89} + (6 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} - 6) q^{90} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 4 \zeta_{10}) q^{91} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{92} + (\zeta_{10}^{2} + 10 \zeta_{10} + 1) q^{94} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{95} + ( - 10 \zeta_{10}^{3} + 14 \zeta_{10}^{2} - 14 \zeta_{10} + 10) q^{97} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 6) q^{98} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 3 \zeta_{10}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{4} + 8 q^{5} + 3 q^{7} - q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - q^{4} + 8 q^{5} + 3 q^{7} - q^{8} + 3 q^{9} - 2 q^{10} - q^{11} - 8 q^{13} + 3 q^{14} - q^{16} - 11 q^{17} + 3 q^{18} - q^{19} + 8 q^{20} - q^{22} + 18 q^{23} + q^{25} + 12 q^{26} - 2 q^{28} + 6 q^{29} + 14 q^{31} + 4 q^{32} - 6 q^{34} + q^{35} + 3 q^{36} + 2 q^{37} - q^{38} - 7 q^{40} + 4 q^{41} - 2 q^{43} + 4 q^{44} + 6 q^{45} + 3 q^{46} + 8 q^{47} + 8 q^{49} - 14 q^{50} + 12 q^{52} - 2 q^{53} + 13 q^{55} - 2 q^{56} + 6 q^{58} + 10 q^{59} - 17 q^{61} - 16 q^{62} - 9 q^{63} - q^{64} - 56 q^{65} - 32 q^{67} - 11 q^{68} + 6 q^{70} - 14 q^{71} + 3 q^{72} - 6 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} + 12 q^{79} - 7 q^{80} - 9 q^{81} - 16 q^{82} - 8 q^{83} + 23 q^{85} + 3 q^{86} + 9 q^{88} - 12 q^{89} - 24 q^{90} - 16 q^{91} - 12 q^{92} + 13 q^{94} - 7 q^{95} + 2 q^{97} - 22 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(287\) \(343\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−0.309017 + 0.951057i
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.881966 2.71441i 0 1.30902 0.951057i −0.809017 0.587785i −0.927051 2.85317i 2.85410
191.1 −0.809017 0.587785i 0 0.309017 + 0.951057i 3.11803 2.26538i 0 0.190983 + 0.587785i 0.309017 0.951057i 2.42705 + 1.76336i −3.85410
229.1 0.309017 0.951057i 0 −0.809017 0.587785i 0.881966 + 2.71441i 0 1.30902 + 0.951057i −0.809017 + 0.587785i −0.927051 + 2.85317i 2.85410
267.1 −0.809017 + 0.587785i 0 0.309017 0.951057i 3.11803 + 2.26538i 0 0.190983 0.587785i 0.309017 + 0.951057i 2.42705 1.76336i −3.85410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.b 4
11.c even 5 1 inner 418.2.f.b 4
11.c even 5 1 4598.2.a.bh 2
11.d odd 10 1 4598.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.b 4 1.a even 1 1 trivial
418.2.f.b 4 11.c even 5 1 inner
4598.2.a.z 2 11.d odd 10 1
4598.2.a.bh 2 11.c even 5 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} - 9 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + 64 T^{2} + 192 T + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 11 T^{3} + 96 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( (T^{2} - 9 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 96 T^{2} + 256 T + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + 114 T^{2} + \cdots + 11881 \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$61$ \( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$67$ \( (T + 8)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} + 76 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} + 76 T^{2} + 56 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + 144 T^{2} + \cdots + 20736 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 116)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + 204 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
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