Properties

Label 403.2.k.b
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 \(5\), degree \(4\), minimal)

Newform invariants

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

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

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
0.309017 + 0.951057i 0.618034 1.90211i 0.809017 0.587785i 1.23607 2.00000 −0.309017 + 0.224514i 2.42705 + 1.76336i −0.809017 0.587785i 0.381966 + 1.17557i
157.1 −0.809017 0.587785i −1.61803 + 1.17557i −0.309017 0.951057i −3.23607 2.00000 0.809017 + 2.48990i −0.927051 + 2.85317i 0.309017 0.951057i 2.61803 + 1.90211i
287.1 0.309017 0.951057i 0.618034 + 1.90211i 0.809017 + 0.587785i 1.23607 2.00000 −0.309017 0.224514i 2.42705 1.76336i −0.809017 + 0.587785i 0.381966 1.17557i
326.1 −0.809017 + 0.587785i −1.61803 1.17557i −0.309017 + 0.951057i −3.23607 2.00000 0.809017 2.48990i −0.927051 2.85317i 0.309017 + 0.951057i 2.61803 1.90211i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.k.b 4
31.d even 5 1 inner 403.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.b 4 1.a even 1 1 trivial
403.2.k.b 4 31.d even 5 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \)
$3$ \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 119 T^{5} - 49 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4} )( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} ) \)
$13$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$17$ \( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 901 T^{5} - 289 T^{6} + 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \)
$23$ \( 1 - 2 T + T^{2} - 106 T^{3} + 729 T^{4} - 2438 T^{5} + 529 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 10 T + 71 T^{2} + 520 T^{3} + 3641 T^{4} + 15080 T^{5} + 59711 T^{6} + 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 70 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 18 T + 101 T^{2} - 174 T^{3} + 49 T^{4} - 7482 T^{5} + 186749 T^{6} - 1431126 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 34686 T^{5} + 130331 T^{6} - 1661168 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 10812 T^{5} - 137641 T^{6} - 297754 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 15 T + 41 T^{2} - 45 T^{3} + 1156 T^{4} - 2655 T^{5} + 142721 T^{6} + 3080685 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + 23 T + 253 T^{2} + 1403 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 11 T + 163 T^{2} - 737 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 2 T - 67 T^{2} - 74 T^{3} + 5255 T^{4} - 5254 T^{5} - 337747 T^{6} - 715822 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 62050 T^{5} + 335727 T^{6} + 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 24 T + 297 T^{2} - 3382 T^{3} + 35205 T^{4} - 267178 T^{5} + 1853577 T^{6} - 11832936 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - T - 67 T^{2} + 515 T^{3} + 5516 T^{4} + 42745 T^{5} - 461563 T^{6} - 571787 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 4 T - 73 T^{2} + 648 T^{3} + 3905 T^{4} + 57672 T^{5} - 578233 T^{6} - 2819876 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 28 T + 287 T^{2} - 1970 T^{3} + 17821 T^{4} - 191090 T^{5} + 2700383 T^{6} - 25554844 T^{7} + 88529281 T^{8} \)
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