Properties

Label 418.2.f.d
Level $418$
Weight $2$
Character orbit 418.f
Analytic conductor $3.338$
Analytic rank $0$
Dimension $4$
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.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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - 2 \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{5} + 2 \zeta_{10} q^{6} + ( - 3 \zeta_{10} + 3) q^{7} + \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - 2 \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{5} + 2 \zeta_{10} q^{6} + ( - 3 \zeta_{10} + 3) q^{7} + \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{10} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{11} - 2 q^{12} + ( - 4 \zeta_{10}^{3} + 4) q^{13} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{14} + ( - 2 \zeta_{10} + 2) q^{15} - \zeta_{10} q^{16} + ( - \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{17} - \zeta_{10}^{3} q^{18} + \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{20} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2}) q^{21} + ( - 2 \zeta_{10}^{3} - \zeta_{10} - 2) q^{22} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3) q^{23} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{24} + ( - \zeta_{10}^{3} - 3 \zeta_{10}^{2} - \zeta_{10}) q^{25} + (4 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{26} - 4 \zeta_{10} q^{27} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{28} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{29} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{30} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{31} + q^{32} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{33} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 5) q^{34} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{35} + \zeta_{10}^{2} q^{36} - 4 \zeta_{10}^{3} q^{37} - \zeta_{10} q^{38} + ( - 8 \zeta_{10}^{2} - 8) q^{39} + (\zeta_{10} - 1) q^{40} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 2 \zeta_{10}) q^{41} + ( - 6 \zeta_{10}^{2} + 6 \zeta_{10}) q^{42} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 1) q^{43} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{44} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{45} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{46} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} + \zeta_{10}) q^{47} + 2 \zeta_{10}^{3} q^{48} + (9 \zeta_{10}^{2} - 11 \zeta_{10} + 9) q^{49} + (\zeta_{10}^{2} + 3 \zeta_{10} + 1) q^{50} + (10 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{51} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}) q^{52} + ( - 8 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 8) q^{53} + 4 q^{54} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{55} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{56} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{57} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 4 \zeta_{10}) q^{58} + (10 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{59} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{60} + ( - 9 \zeta_{10}^{2} + 4 \zeta_{10} - 9) q^{61} - 2 \zeta_{10}^{3} 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 q^{65} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4) q^{66} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 6) q^{67} + (5 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 5) q^{68} + ( - 6 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 6 \zeta_{10}) q^{69} + (3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{70} + (8 \zeta_{10}^{2} + 8) q^{71} - \zeta_{10} q^{72} + 14 \zeta_{10}^{3} q^{73} + 4 \zeta_{10}^{2} q^{74} + (8 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 8) q^{75} + q^{76} + (3 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 12 \zeta_{10}) q^{77} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 8) q^{78} + ( - 6 \zeta_{10}^{3} + 6) q^{79} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{80} + 11 \zeta_{10}^{3} q^{81} + (2 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{82} + ( - 11 \zeta_{10}^{2} + 7 \zeta_{10} - 11) q^{83} + (6 \zeta_{10} - 6) q^{84} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 4 \zeta_{10}) q^{85} + ( - \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 1) q^{86} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8) q^{87} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{88} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 8) q^{89} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{90} - 12 \zeta_{10}^{2} q^{91} + (3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{92} + 4 \zeta_{10} q^{93} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{94} + (\zeta_{10} - 1) q^{95} - 2 \zeta_{10}^{2} q^{96} + ( - 6 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 6) q^{97} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 2) q^{98} + ( - 2 \zeta_{10}^{3} - \zeta_{10} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + 2 q^{6} + 9 q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + 2 q^{6} + 9 q^{7} - q^{8} - q^{9} + 2 q^{10} - q^{11} - 8 q^{12} + 12 q^{13} + 9 q^{14} + 6 q^{15} - q^{16} - 7 q^{17} - q^{18} - q^{19} + 2 q^{20} + 12 q^{21} - 11 q^{22} - 18 q^{23} + 2 q^{24} + q^{25} - 8 q^{26} - 4 q^{27} - 6 q^{28} - 16 q^{29} + 6 q^{30} - 2 q^{31} + 4 q^{32} + 2 q^{33} + 18 q^{34} - 3 q^{35} - q^{36} - 4 q^{37} - q^{38} - 24 q^{39} - 3 q^{40} + 2 q^{41} + 12 q^{42} - 14 q^{43} + 4 q^{44} + 2 q^{45} - 3 q^{46} + 2 q^{48} + 16 q^{49} + 6 q^{50} + 4 q^{51} - 8 q^{52} + 12 q^{53} + 16 q^{54} + 7 q^{55} - 6 q^{56} + 2 q^{57} - 16 q^{58} - 2 q^{59} - 4 q^{60} - 23 q^{61} - 2 q^{62} + 9 q^{63} - q^{64} + 16 q^{65} - 18 q^{66} + 20 q^{67} - 7 q^{68} - 24 q^{69} + 12 q^{70} + 24 q^{71} - q^{72} + 14 q^{73} - 4 q^{74} - 12 q^{75} + 4 q^{76} + 24 q^{77} + 16 q^{78} + 18 q^{79} - 3 q^{80} + 11 q^{81} + 12 q^{82} - 26 q^{83} - 18 q^{84} - 11 q^{85} - 9 q^{86} - 48 q^{87} - q^{88} + 36 q^{89} + 2 q^{90} + 12 q^{91} + 12 q^{92} + 4 q^{93} - 5 q^{94} - 3 q^{95} + 2 q^{96} + 2 q^{97} + 26 q^{98} - 11 q^{99}+O(q^{100}) \) 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 1.61803 + 1.17557i −0.809017 + 0.587785i 0.500000 1.53884i −0.618034 + 1.90211i 3.92705 2.85317i −0.809017 0.587785i 0.309017 + 0.951057i 1.61803
191.1 −0.809017 0.587785i −0.618034 + 1.90211i 0.309017 + 0.951057i 0.500000 0.363271i 1.61803 1.17557i 0.572949 + 1.76336i 0.309017 0.951057i −0.809017 0.587785i −0.618034
229.1 0.309017 0.951057i 1.61803 1.17557i −0.809017 0.587785i 0.500000 + 1.53884i −0.618034 1.90211i 3.92705 + 2.85317i −0.809017 + 0.587785i 0.309017 0.951057i 1.61803
267.1 −0.809017 + 0.587785i −0.618034 1.90211i 0.309017 0.951057i 0.500000 + 0.363271i 1.61803 + 1.17557i 0.572949 1.76336i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.618034
\(n\): 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.d 4
11.c even 5 1 inner 418.2.f.d 4
11.c even 5 1 4598.2.a.bb 2
11.d odd 10 1 4598.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.d 4 1.a even 1 1 trivial
418.2.f.d 4 11.c even 5 1 inner
4598.2.a.t 2 11.d odd 10 1
4598.2.a.bb 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 4T_{3}^{2} - 8T_{3} + 16 \) 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
$3$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + 21 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361 \) 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
$29$ \( T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T - 19)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + 124 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$61$ \( T^{4} + 23 T^{3} + 304 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + 256 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + 456 T^{2} + \cdots + 22201 \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
show more
show less