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Properties

Label 403.2.k.c
Level 403
Weight 2
Character orbit 403.k
Analytic conductor 3.218
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.k (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{2} + ( 1 - \zeta_{10}^{3} ) q^{3} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{4} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{6} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{7} + ( 4 - 4 \zeta_{10} + \zeta_{10}^{3} ) q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{2} + ( 1 - \zeta_{10}^{3} ) q^{3} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{4} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{6} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{7} + ( 4 - 4 \zeta_{10} + \zeta_{10}^{3} ) q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{3} ) q^{9} + \zeta_{10} q^{10} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} -3 \zeta_{10} q^{12} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{13} + ( 6 - 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{14} + ( 3 - \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{15} + ( -5 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{16} -\zeta_{10}^{3} q^{17} + ( -2 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{18} + ( -2 - 4 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{19} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{20} -3 \zeta_{10} q^{21} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{22} + ( 4 - 4 \zeta_{10} ) q^{23} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{25} + ( 1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{26} + ( 4 \zeta_{10} - \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{27} + ( -9 + 18 \zeta_{10} - 18 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{28} + ( 3 + \zeta_{10} + 3 \zeta_{10}^{2} ) q^{29} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{30} + ( 3 - 2 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{31} + ( -6 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{32} + ( 3 + 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{33} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{34} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{35} + ( -3 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{36} + ( -7 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{37} + ( 2 \zeta_{10} - 6 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{38} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{39} + ( 3 - 3 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{40} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{41} + ( 3 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{42} + ( 1 - 5 \zeta_{10} + \zeta_{10}^{2} ) q^{43} + ( 6 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{44} + \zeta_{10}^{3} q^{45} + ( -4 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{46} + ( 7 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{47} + ( -5 + 5 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{48} + ( -2 + 11 \zeta_{10} - 11 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{49} + ( 6 - 9 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{50} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{51} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{52} + ( -11 + 11 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{53} + ( -5 + 5 \zeta_{10} - \zeta_{10}^{3} ) q^{54} + ( 5 \zeta_{10} + 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{55} + ( -12 + 15 \zeta_{10}^{2} - 15 \zeta_{10}^{3} ) q^{56} + ( -8 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{57} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{58} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{59} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{60} + 11 q^{61} + ( 3 + 2 \zeta_{10} - 7 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{62} + ( -3 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{63} + ( 6 - 5 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{64} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{65} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{66} + ( -5 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{67} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68} -4 \zeta_{10}^{2} q^{69} + ( 3 - 3 \zeta_{10} ) q^{70} + ( -2 + 2 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{71} + ( 9 - 14 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{72} + ( 2 \zeta_{10} - 7 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{73} + ( 5 - 11 \zeta_{10} + 5 \zeta_{10}^{2} ) q^{74} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{75} + ( -12 + 12 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{76} + ( 6 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{77} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{78} + ( 2 - 2 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{79} + ( -7 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{80} + ( 6 - 2 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{81} + ( -\zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{82} + ( -6 + 4 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{83} + ( -9 + 9 \zeta_{10} ) q^{84} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{85} + ( 6 \zeta_{10} - 11 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{86} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{87} + ( 5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{88} + ( -2 \zeta_{10} + 7 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{89} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{90} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{91} + ( -12 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{92} + ( 7 + 4 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{93} + ( 10 - 13 \zeta_{10}^{2} + 13 \zeta_{10}^{3} ) q^{94} + ( -8 - 6 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{95} + ( -3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{96} + 12 \zeta_{10}^{2} q^{97} + ( -11 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{98} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 3q^{3} - 9q^{4} + 6q^{5} - 2q^{6} - 9q^{7} + 13q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - q^{2} + 3q^{3} - 9q^{4} + 6q^{5} - 2q^{6} - 9q^{7} + 13q^{8} + 4q^{9} + q^{10} + 3q^{11} - 3q^{12} + q^{13} + 21q^{14} + 7q^{15} + q^{16} - q^{17} + 4q^{18} - 10q^{19} - 6q^{20} - 3q^{21} - 2q^{22} + 12q^{23} + 6q^{24} - 6q^{25} + 6q^{26} + 9q^{27} + 9q^{28} + 10q^{29} + 2q^{30} + 4q^{31} - 30q^{32} + 11q^{33} - q^{34} - 6q^{35} - 24q^{36} - 26q^{37} + 10q^{38} + 2q^{39} + 7q^{40} + 5q^{41} + 12q^{42} - 2q^{43} + 12q^{44} + q^{45} + 12q^{46} + q^{47} - 13q^{48} + 16q^{49} + 9q^{50} - 2q^{51} - 6q^{52} - 28q^{53} - 16q^{54} + 7q^{55} - 78q^{56} - 20q^{57} + 5q^{58} - 12q^{60} + 44q^{61} + 29q^{62} - 24q^{63} + 13q^{64} + 4q^{65} + q^{66} - 18q^{67} + 6q^{68} + 4q^{69} + 9q^{70} + 2q^{71} + 13q^{72} + 11q^{73} + 4q^{74} + 3q^{75} - 30q^{76} + 12q^{77} + 2q^{78} + 9q^{79} - 11q^{80} + 16q^{81} - 5q^{82} - 14q^{83} - 27q^{84} + q^{85} + 23q^{86} + 20q^{87} + 16q^{88} - 11q^{89} + q^{90} - 6q^{91} - 72q^{92} + 23q^{93} + 66q^{94} - 30q^{95} - 15q^{96} - 12q^{97} - 84q^{98} - 2q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1
−0.309017 − 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 − 0.587785i
−0.809017 − 2.48990i 0.190983 − 0.587785i −3.92705 + 2.85317i 0.381966 −1.61803 −3.92705 + 2.85317i 6.04508 + 4.39201i 2.11803 + 1.53884i −0.309017 − 0.951057i
157.1 0.309017 + 0.224514i 1.30902 − 0.951057i −0.572949 − 1.76336i 2.61803 0.618034 −0.572949 − 1.76336i 0.454915 − 1.40008i −0.118034 + 0.363271i 0.809017 + 0.587785i
287.1 −0.809017 + 2.48990i 0.190983 + 0.587785i −3.92705 − 2.85317i 0.381966 −1.61803 −3.92705 − 2.85317i 6.04508 − 4.39201i 2.11803 − 1.53884i −0.309017 + 0.951057i
326.1 0.309017 − 0.224514i 1.30902 + 0.951057i −0.572949 + 1.76336i 2.61803 0.618034 −0.572949 + 1.76336i 0.454915 + 1.40008i −0.118034 − 0.363271i 0.809017 − 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 6 T_{2}^{2} - 4 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).