Properties

Label 13.4.e.b
Level 13
Weight 4
Character orbit 13.e
Analytic conductor 0.767
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.e (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \zeta_{6} ) q^{2} \) \( + ( -2 + 2 \zeta_{6} ) q^{3} \) \( -5 \zeta_{6} q^{4} \) \( + ( 1 - 2 \zeta_{6} ) q^{5} \) \( + ( -4 + 2 \zeta_{6} ) q^{6} \) \( + ( -16 + 8 \zeta_{6} ) q^{7} \) \( + ( 13 - 26 \zeta_{6} ) q^{8} \) \( + 23 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \zeta_{6} ) q^{2} \) \( + ( -2 + 2 \zeta_{6} ) q^{3} \) \( -5 \zeta_{6} q^{4} \) \( + ( 1 - 2 \zeta_{6} ) q^{5} \) \( + ( -4 + 2 \zeta_{6} ) q^{6} \) \( + ( -16 + 8 \zeta_{6} ) q^{7} \) \( + ( 13 - 26 \zeta_{6} ) q^{8} \) \( + 23 \zeta_{6} q^{9} \) \( + ( 3 - 3 \zeta_{6} ) q^{10} \) \( + ( 8 + 8 \zeta_{6} ) q^{11} \) \( + 10 q^{12} \) \( + ( 39 + 13 \zeta_{6} ) q^{13} \) \( -24 q^{14} \) \( + ( 2 + 2 \zeta_{6} ) q^{15} \) \( + ( -1 + \zeta_{6} ) q^{16} \) \( -117 \zeta_{6} q^{17} \) \( + ( -23 + 46 \zeta_{6} ) q^{18} \) \( + ( -132 + 66 \zeta_{6} ) q^{19} \) \( + ( -10 + 5 \zeta_{6} ) q^{20} \) \( + ( 16 - 32 \zeta_{6} ) q^{21} \) \( + 24 \zeta_{6} q^{22} \) \( + ( 78 - 78 \zeta_{6} ) q^{23} \) \( + ( 26 + 26 \zeta_{6} ) q^{24} \) \( + 122 q^{25} \) \( + ( 26 + 65 \zeta_{6} ) q^{26} \) \( -100 q^{27} \) \( + ( 40 + 40 \zeta_{6} ) q^{28} \) \( + ( 141 - 141 \zeta_{6} ) q^{29} \) \( + 6 \zeta_{6} q^{30} \) \( + ( -90 + 180 \zeta_{6} ) q^{31} \) \( + ( -210 + 105 \zeta_{6} ) q^{32} \) \( + ( -32 + 16 \zeta_{6} ) q^{33} \) \( + ( 117 - 234 \zeta_{6} ) q^{34} \) \( + 24 \zeta_{6} q^{35} \) \( + ( 115 - 115 \zeta_{6} ) q^{36} \) \( + ( -83 - 83 \zeta_{6} ) q^{37} \) \( -198 q^{38} \) \( + ( -104 + 78 \zeta_{6} ) q^{39} \) \( -39 q^{40} \) \( + ( 157 + 157 \zeta_{6} ) q^{41} \) \( + ( 48 - 48 \zeta_{6} ) q^{42} \) \( -104 \zeta_{6} q^{43} \) \( + ( 40 - 80 \zeta_{6} ) q^{44} \) \( + ( 46 - 23 \zeta_{6} ) q^{45} \) \( + ( 156 - 78 \zeta_{6} ) q^{46} \) \( + ( -174 + 348 \zeta_{6} ) q^{47} \) \( -2 \zeta_{6} q^{48} \) \( + ( -151 + 151 \zeta_{6} ) q^{49} \) \( + ( 122 + 122 \zeta_{6} ) q^{50} \) \( + 234 q^{51} \) \( + ( 65 - 260 \zeta_{6} ) q^{52} \) \( + 93 q^{53} \) \( + ( -100 - 100 \zeta_{6} ) q^{54} \) \( + ( 24 - 24 \zeta_{6} ) q^{55} \) \( + 312 \zeta_{6} q^{56} \) \( + ( 132 - 264 \zeta_{6} ) q^{57} \) \( + ( 282 - 141 \zeta_{6} ) q^{58} \) \( + ( -328 + 164 \zeta_{6} ) q^{59} \) \( + ( 10 - 20 \zeta_{6} ) q^{60} \) \( -145 \zeta_{6} q^{61} \) \( + ( -270 + 270 \zeta_{6} ) q^{62} \) \( + ( -184 - 184 \zeta_{6} ) q^{63} \) \( -307 q^{64} \) \( + ( 65 - 91 \zeta_{6} ) q^{65} \) \( -48 q^{66} \) \( + ( -454 - 454 \zeta_{6} ) q^{67} \) \( + ( -585 + 585 \zeta_{6} ) q^{68} \) \( + 156 \zeta_{6} q^{69} \) \( + ( -24 + 48 \zeta_{6} ) q^{70} \) \( + ( 1220 - 610 \zeta_{6} ) q^{71} \) \( + ( 598 - 299 \zeta_{6} ) q^{72} \) \( + ( -265 + 530 \zeta_{6} ) q^{73} \) \( -249 \zeta_{6} q^{74} \) \( + ( -244 + 244 \zeta_{6} ) q^{75} \) \( + ( 330 + 330 \zeta_{6} ) q^{76} \) \( -192 q^{77} \) \( + ( -182 + 52 \zeta_{6} ) q^{78} \) \( + 1276 q^{79} \) \( + ( 1 + \zeta_{6} ) q^{80} \) \( + ( -421 + 421 \zeta_{6} ) q^{81} \) \( + 471 \zeta_{6} q^{82} \) \( + ( 456 - 912 \zeta_{6} ) q^{83} \) \( + ( -160 + 80 \zeta_{6} ) q^{84} \) \( + ( -234 + 117 \zeta_{6} ) q^{85} \) \( + ( 104 - 208 \zeta_{6} ) q^{86} \) \( + 282 \zeta_{6} q^{87} \) \( + ( 312 - 312 \zeta_{6} ) q^{88} \) \( + ( -564 - 564 \zeta_{6} ) q^{89} \) \( + 69 q^{90} \) \( + ( -728 + 208 \zeta_{6} ) q^{91} \) \( -390 q^{92} \) \( + ( -180 - 180 \zeta_{6} ) q^{93} \) \( + ( -522 + 522 \zeta_{6} ) q^{94} \) \( + 198 \zeta_{6} q^{95} \) \( + ( 210 - 420 \zeta_{6} ) q^{96} \) \( + ( 232 - 116 \zeta_{6} ) q^{97} \) \( + ( -302 + 151 \zeta_{6} ) q^{98} \) \( + ( -184 + 368 \zeta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut +\mathstrut 23q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut +\mathstrut 23q^{9} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 24q^{11} \) \(\mathstrut +\mathstrut 20q^{12} \) \(\mathstrut +\mathstrut 91q^{13} \) \(\mathstrut -\mathstrut 48q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 117q^{17} \) \(\mathstrut -\mathstrut 198q^{19} \) \(\mathstrut -\mathstrut 15q^{20} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut +\mathstrut 78q^{23} \) \(\mathstrut +\mathstrut 78q^{24} \) \(\mathstrut +\mathstrut 244q^{25} \) \(\mathstrut +\mathstrut 117q^{26} \) \(\mathstrut -\mathstrut 200q^{27} \) \(\mathstrut +\mathstrut 120q^{28} \) \(\mathstrut +\mathstrut 141q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 315q^{32} \) \(\mathstrut -\mathstrut 48q^{33} \) \(\mathstrut +\mathstrut 24q^{35} \) \(\mathstrut +\mathstrut 115q^{36} \) \(\mathstrut -\mathstrut 249q^{37} \) \(\mathstrut -\mathstrut 396q^{38} \) \(\mathstrut -\mathstrut 130q^{39} \) \(\mathstrut -\mathstrut 78q^{40} \) \(\mathstrut +\mathstrut 471q^{41} \) \(\mathstrut +\mathstrut 48q^{42} \) \(\mathstrut -\mathstrut 104q^{43} \) \(\mathstrut +\mathstrut 69q^{45} \) \(\mathstrut +\mathstrut 234q^{46} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 151q^{49} \) \(\mathstrut +\mathstrut 366q^{50} \) \(\mathstrut +\mathstrut 468q^{51} \) \(\mathstrut -\mathstrut 130q^{52} \) \(\mathstrut +\mathstrut 186q^{53} \) \(\mathstrut -\mathstrut 300q^{54} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut +\mathstrut 312q^{56} \) \(\mathstrut +\mathstrut 423q^{58} \) \(\mathstrut -\mathstrut 492q^{59} \) \(\mathstrut -\mathstrut 145q^{61} \) \(\mathstrut -\mathstrut 270q^{62} \) \(\mathstrut -\mathstrut 552q^{63} \) \(\mathstrut -\mathstrut 614q^{64} \) \(\mathstrut +\mathstrut 39q^{65} \) \(\mathstrut -\mathstrut 96q^{66} \) \(\mathstrut -\mathstrut 1362q^{67} \) \(\mathstrut -\mathstrut 585q^{68} \) \(\mathstrut +\mathstrut 156q^{69} \) \(\mathstrut +\mathstrut 1830q^{71} \) \(\mathstrut +\mathstrut 897q^{72} \) \(\mathstrut -\mathstrut 249q^{74} \) \(\mathstrut -\mathstrut 244q^{75} \) \(\mathstrut +\mathstrut 990q^{76} \) \(\mathstrut -\mathstrut 384q^{77} \) \(\mathstrut -\mathstrut 312q^{78} \) \(\mathstrut +\mathstrut 2552q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut -\mathstrut 421q^{81} \) \(\mathstrut +\mathstrut 471q^{82} \) \(\mathstrut -\mathstrut 240q^{84} \) \(\mathstrut -\mathstrut 351q^{85} \) \(\mathstrut +\mathstrut 282q^{87} \) \(\mathstrut +\mathstrut 312q^{88} \) \(\mathstrut -\mathstrut 1692q^{89} \) \(\mathstrut +\mathstrut 138q^{90} \) \(\mathstrut -\mathstrut 1248q^{91} \) \(\mathstrut -\mathstrut 780q^{92} \) \(\mathstrut -\mathstrut 540q^{93} \) \(\mathstrut -\mathstrut 522q^{94} \) \(\mathstrut +\mathstrut 198q^{95} \) \(\mathstrut +\mathstrut 348q^{97} \) \(\mathstrut -\mathstrut 453q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i −1.00000 + 1.73205i −2.50000 4.33013i 1.73205i −3.00000 + 1.73205i −12.0000 + 6.92820i 22.5167i 11.5000 + 19.9186i 1.50000 2.59808i
10.1 1.50000 0.866025i −1.00000 1.73205i −2.50000 + 4.33013i 1.73205i −3.00000 1.73205i −12.0000 6.92820i 22.5167i 11.5000 19.9186i 1.50000 + 2.59808i
\(n\): 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
13.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 3 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).