Properties

Label 418.2.f.c
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + ( - \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{5} + \zeta_{10} q^{6} + (\zeta_{10} - 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + ( - \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{5} + \zeta_{10} q^{6} + (\zeta_{10} - 1) q^{7} + \zeta_{10}^{2} q^{8} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{10} + ( - 2 \zeta_{10}^{3} - \zeta_{10} - 2) q^{11} - q^{12} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{13} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{14} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{15} - \zeta_{10} q^{16} + (2 \zeta_{10}^{2} + \zeta_{10} + 2) q^{17} + 2 \zeta_{10}^{3} q^{18} - \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{20} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{21} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{22} - q^{23} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10}) q^{25} + (3 \zeta_{10} - 3) q^{26} - 5 \zeta_{10} q^{27} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{28} + ( - 4 \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{29} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{30} + ( - \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{31} + q^{32} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{33} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 3) q^{34} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{35} - 2 \zeta_{10}^{2} q^{36} + ( - 11 \zeta_{10}^{3} - \zeta_{10} + 1) q^{37} + \zeta_{10} q^{38} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{39} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{40} + (5 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 5 \zeta_{10}) q^{41} + (\zeta_{10}^{2} - \zeta_{10}) q^{42} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 2) q^{43} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{44} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{45} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{46} + (6 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 6 \zeta_{10}) q^{47} + \zeta_{10}^{3} q^{48} + (\zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{49} + (3 \zeta_{10}^{2} + 3) q^{50} + ( - 3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{51} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{52} + (11 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 11) q^{53} + 5 q^{54} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 7 \zeta_{10} + 4) q^{55} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{56} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{57} + (8 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8 \zeta_{10}) q^{58} + (\zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{59} + (\zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{60} + (9 \zeta_{10}^{2} - 9 \zeta_{10} + 9) q^{61} + (\zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{62} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 3) q^{65} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{66} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 4) q^{67} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 3) q^{68} + \zeta_{10}^{2} q^{69} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{70} + ( - 2 \zeta_{10}^{2} - 11 \zeta_{10} - 2) q^{71} + 2 \zeta_{10} q^{72} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{73} + (\zeta_{10}^{3} + 10 \zeta_{10}^{2} + \zeta_{10}) q^{74} + (3 \zeta_{10}^{3} - 3) q^{75} - q^{76} + (\zeta_{10}^{2} - 3 \zeta_{10} + 4) q^{77} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{78} + (15 \zeta_{10}^{3} - 15 \zeta_{10}^{2} + 15 \zeta_{10} - 15) q^{79} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{80} - \zeta_{10}^{3} q^{81} + ( - 5 \zeta_{10}^{2} + 3 \zeta_{10} - 5) q^{82} + (11 \zeta_{10}^{2} - 2 \zeta_{10} + 11) q^{83} + ( - \zeta_{10} + 1) q^{84} + (\zeta_{10}^{3} + \zeta_{10}) q^{85} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{86} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 4) q^{87} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{88} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 1) q^{89} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{90} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 3 \zeta_{10}) q^{91} + \zeta_{10}^{3} q^{92} + ( - 6 \zeta_{10}^{2} + 5 \zeta_{10} - 6) q^{93} + ( - 6 \zeta_{10}^{2} + 5 \zeta_{10} - 6) q^{94} + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{95} - \zeta_{10}^{2} q^{96} + ( - 7 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} + 7) q^{97} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 6) q^{98} + (4 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 4 \zeta_{10} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3 q^{7} - q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3 q^{7} - q^{8} + 2 q^{9} - 6 q^{10} - 11 q^{11} - 4 q^{12} + 6 q^{13} - 3 q^{14} - 4 q^{15} - q^{16} + 7 q^{17} + 2 q^{18} + q^{19} - q^{20} - 2 q^{21} + 4 q^{22} - 4 q^{23} + q^{24} - 6 q^{25} - 9 q^{26} - 5 q^{27} + 2 q^{28} + 20 q^{29} - 4 q^{30} + 13 q^{31} + 4 q^{32} - 9 q^{33} - 8 q^{34} - 3 q^{35} + 2 q^{36} - 8 q^{37} + q^{38} - 6 q^{39} + 4 q^{40} + 13 q^{41} - 2 q^{42} - 4 q^{43} - q^{44} + 12 q^{45} + q^{46} + 17 q^{47} + q^{48} + 8 q^{49} + 9 q^{50} + 3 q^{51} - 9 q^{52} - 19 q^{53} + 20 q^{54} + 4 q^{55} + 2 q^{56} - q^{57} + 20 q^{58} + 10 q^{59} + q^{60} + 18 q^{61} - 17 q^{62} + 6 q^{63} - q^{64} - 24 q^{65} + q^{66} - 18 q^{67} + 7 q^{68} - q^{69} + 7 q^{70} - 17 q^{71} + 2 q^{72} + q^{73} - 8 q^{74} - 9 q^{75} - 4 q^{76} + 12 q^{77} - 6 q^{78} - 15 q^{79} + 4 q^{80} - q^{81} - 12 q^{82} + 31 q^{83} + 3 q^{84} + 2 q^{85} - 14 q^{86} + 9 q^{88} + 20 q^{89} + 2 q^{90} - 12 q^{91} + q^{92} - 13 q^{93} - 13 q^{94} - 4 q^{95} + q^{96} + 27 q^{97} - 22 q^{98} - 8 q^{99}+O(q^{100}) \) 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 0.809017 + 0.587785i −0.809017 + 0.587785i −0.809017 + 2.48990i −0.309017 + 0.951057i −1.30902 + 0.951057i −0.809017 0.587785i −0.618034 1.90211i −2.61803
191.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.309017 0.224514i 0.809017 0.587785i −0.190983 0.587785i 0.309017 0.951057i 1.61803 + 1.17557i −0.381966
229.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.809017 2.48990i −0.309017 0.951057i −1.30902 0.951057i −0.809017 + 0.587785i −0.618034 + 1.90211i −2.61803
267.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.309017 + 0.224514i 0.809017 + 0.587785i −0.190983 + 0.587785i 0.309017 + 0.951057i 1.61803 1.17557i −0.381966
\(n\): 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.c 4
11.c even 5 1 inner 418.2.f.c 4
11.c even 5 1 4598.2.a.bd 2
11.d odd 10 1 4598.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.c 4 1.a even 1 1 trivial
418.2.f.c 4 11.c even 5 1 inner
4598.2.a.u 2 11.d odd 10 1
4598.2.a.bd 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1 \) 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
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + 6 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( T^{4} - 13 T^{3} + 139 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 114 T^{2} + \cdots + 11881 \) Copy content Toggle raw display
$41$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 17 T^{3} + 139 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$53$ \( T^{4} + 19 T^{3} + 141 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 40 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$61$ \( T^{4} - 18 T^{3} + 324 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$67$ \( (T^{2} + 9 T + 19)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 17 T^{3} + 159 T^{2} + \cdots + 19321 \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$79$ \( T^{4} + 15 T^{3} + 225 T^{2} + \cdots + 50625 \) Copy content Toggle raw display
$83$ \( T^{4} - 31 T^{3} + 466 T^{2} + \cdots + 19321 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T - 155)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + 379 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
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