Properties

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

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{3} \) \( -5 \beta_{1} q^{4} \) \( + ( -10 - 5 \beta_{3} ) q^{5} \) \( + ( 10 \beta_{1} + 14 \beta_{2} ) q^{6} \) \( + ( -\beta_{1} - 7 \beta_{2} ) q^{7} \) \( + ( -4 + 7 \beta_{3} ) q^{8} \) \( + ( -15 \beta_{1} - 10 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{2} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{3} \) \( -5 \beta_{1} q^{4} \) \( + ( -10 - 5 \beta_{3} ) q^{5} \) \( + ( 10 \beta_{1} + 14 \beta_{2} ) q^{6} \) \( + ( -\beta_{1} - 7 \beta_{2} ) q^{7} \) \( + ( -4 + 7 \beta_{3} ) q^{8} \) \( + ( -15 \beta_{1} - 10 \beta_{2} ) q^{9} \) \( + ( -5 \beta_{1} - 5 \beta_{3} ) q^{10} \) \( + ( -1 + 15 \beta_{1} + \beta_{2} + 15 \beta_{3} ) q^{11} \) \( + ( 60 - 20 \beta_{3} ) q^{12} \) \( + ( 49 + 5 \beta_{1} - 40 \beta_{2} - 2 \beta_{3} ) q^{13} \) \( + ( -18 + 10 \beta_{3} ) q^{14} \) \( + ( -50 - 10 \beta_{1} + 50 \beta_{2} - 10 \beta_{3} ) q^{15} \) \( + ( -36 - 15 \beta_{1} + 36 \beta_{2} - 15 \beta_{3} ) q^{16} \) \( + ( 16 \beta_{1} - 43 \beta_{2} ) q^{17} \) \( + ( -80 + 55 \beta_{3} ) q^{18} \) \( + ( -5 \beta_{1} + 73 \beta_{2} ) q^{19} \) \( + ( 25 \beta_{1} - 100 \beta_{2} ) q^{20} \) \( + ( 19 - 25 \beta_{3} ) q^{21} \) \( + ( 46 \beta_{1} + 62 \beta_{2} ) q^{22} \) \( + ( -89 - 33 \beta_{1} + 89 \beta_{2} - 33 \beta_{3} ) q^{23} \) \( + ( 88 - 40 \beta_{1} - 88 \beta_{2} - 40 \beta_{3} ) q^{24} \) \( + ( 75 + 75 \beta_{3} ) q^{25} \) \( + ( 46 - 55 \beta_{1} - 106 \beta_{2} - 30 \beta_{3} ) q^{26} \) \( + ( 163 - 9 \beta_{3} ) q^{27} \) \( + ( -20 + 40 \beta_{1} + 20 \beta_{2} + 40 \beta_{3} ) q^{28} \) \( + ( 13 + 60 \beta_{1} - 13 \beta_{2} + 60 \beta_{3} ) q^{29} \) \( + ( 20 \beta_{1} + 60 \beta_{2} ) q^{30} \) \( + ( -120 - 100 \beta_{3} ) q^{31} \) \( + ( -65 \beta_{1} - 20 \beta_{2} ) q^{32} \) \( + ( -63 \beta_{1} - 181 \beta_{2} ) q^{33} \) \( + ( -22 - 5 \beta_{3} ) q^{34} \) \( + ( -30 \beta_{1} + 50 \beta_{2} ) q^{35} \) \( + ( -300 + 125 \beta_{1} + 300 \beta_{2} + 125 \beta_{3} ) q^{36} \) \( + ( 73 - 44 \beta_{1} - 73 \beta_{2} - 44 \beta_{3} ) q^{37} \) \( + ( 126 - 58 \beta_{3} ) q^{38} \) \( + ( -93 + 155 \beta_{1} + 73 \beta_{2} + 55 \beta_{3} ) q^{39} \) \( + ( -100 - 15 \beta_{3} ) q^{40} \) \( + ( -259 + 20 \beta_{1} + 259 \beta_{2} + 20 \beta_{3} ) q^{41} \) \( + ( 138 - 94 \beta_{1} - 138 \beta_{2} - 94 \beta_{3} ) q^{42} \) \( + ( -97 \beta_{1} - 179 \beta_{2} ) q^{43} \) \( + ( 300 - 80 \beta_{3} ) q^{44} \) \( + ( 25 \beta_{1} - 200 \beta_{2} ) q^{45} \) \( + ( -10 \beta_{1} + 46 \beta_{2} ) q^{46} \) \( + ( 100 + 140 \beta_{3} ) q^{47} \) \( + ( -48 \beta_{1} + 144 \beta_{2} ) q^{48} \) \( + ( 290 + 15 \beta_{1} - 290 \beta_{2} + 15 \beta_{3} ) q^{49} \) \( + ( -150 + 150 \beta_{1} + 150 \beta_{2} + 150 \beta_{3} ) q^{50} \) \( + ( -149 - 65 \beta_{3} ) q^{51} \) \( + ( 100 - 80 \beta_{1} - 140 \beta_{2} + 175 \beta_{3} ) q^{52} \) \( + ( 190 - 165 \beta_{3} ) q^{53} \) \( + ( 362 - 190 \beta_{1} - 362 \beta_{2} - 190 \beta_{3} ) q^{54} \) \( + ( -290 - 70 \beta_{1} + 290 \beta_{2} - 70 \beta_{3} ) q^{55} \) \( + ( 60 \beta_{1} + 56 \beta_{2} ) q^{56} \) \( + ( -13 + 199 \beta_{3} ) q^{57} \) \( + ( 167 \beta_{1} + 214 \beta_{2} ) q^{58} \) \( + ( 55 \beta_{1} + 377 \beta_{2} ) q^{59} \) \( + ( -200 - 200 \beta_{3} ) q^{60} \) \( + ( 200 \beta_{1} - 351 \beta_{2} ) q^{61} \) \( + ( 160 - 180 \beta_{1} - 160 \beta_{2} - 180 \beta_{3} ) q^{62} \) \( + ( -130 + 130 \beta_{1} + 130 \beta_{2} + 130 \beta_{3} ) q^{63} \) \( + ( -588 + 95 \beta_{3} ) q^{64} \) \( + ( -450 - 225 \beta_{1} + 500 \beta_{2} - 235 \beta_{3} ) q^{65} \) \( + ( -614 + 370 \beta_{3} ) q^{66} \) \( + ( 283 + 91 \beta_{1} - 283 \beta_{2} + 91 \beta_{3} ) q^{67} \) \( + ( 320 + 135 \beta_{1} - 320 \beta_{2} + 135 \beta_{3} ) q^{68} \) \( + ( -135 \beta_{1} + 307 \beta_{2} ) q^{69} \) \( + ( -20 + 40 \beta_{3} ) q^{70} \) \( + ( -105 \beta_{1} - 11 \beta_{2} ) q^{71} \) \( + ( 235 \beta_{1} + 460 \beta_{2} ) q^{72} \) \( + ( 250 - 85 \beta_{3} ) q^{73} \) \( + ( -205 \beta_{1} - 322 \beta_{2} ) q^{74} \) \( + ( 825 - 75 \beta_{1} - 825 \beta_{2} - 75 \beta_{3} ) q^{75} \) \( + ( -100 - 340 \beta_{1} + 100 \beta_{2} - 340 \beta_{3} ) q^{76} \) \( + ( 67 - 121 \beta_{3} ) q^{77} \) \( + ( 360 + 258 \beta_{1} + 406 \beta_{2} - 280 \beta_{3} ) q^{78} \) \( + ( 140 + 40 \beta_{3} ) q^{79} \) \( + ( 660 + 255 \beta_{1} - 660 \beta_{2} + 255 \beta_{3} ) q^{80} \) \( + ( -1 + 120 \beta_{1} + \beta_{2} + 120 \beta_{3} ) q^{81} \) \( + ( 319 \beta_{1} + 598 \beta_{2} ) q^{82} \) \( + ( 180 + 100 \beta_{3} ) q^{83} \) \( + ( -220 \beta_{1} - 500 \beta_{2} ) q^{84} \) \( + ( -295 \beta_{1} + 750 \beta_{2} ) q^{85} \) \( + ( -746 + 470 \beta_{3} ) q^{86} \) \( + ( -201 \beta_{1} - 707 \beta_{2} ) q^{87} \) \( + ( 424 - 172 \beta_{1} - 424 \beta_{2} - 172 \beta_{3} ) q^{88} \) \( + ( -523 - 125 \beta_{1} + 523 \beta_{2} - 125 \beta_{3} ) q^{89} \) \( + ( -300 + 125 \beta_{3} ) q^{90} \) \( + ( -260 - 65 \beta_{1} - 91 \beta_{2} ) q^{91} \) \( + ( -660 - 280 \beta_{3} ) q^{92} \) \( + ( -1080 + 40 \beta_{1} + 1080 \beta_{2} + 40 \beta_{3} ) q^{93} \) \( + ( -360 + 320 \beta_{1} + 360 \beta_{2} + 320 \beta_{3} ) q^{94} \) \( + ( 390 \beta_{1} - 830 \beta_{2} ) q^{95} \) \( + ( 800 - 320 \beta_{3} ) q^{96} \) \( + ( 469 \beta_{1} - 27 \beta_{2} ) q^{97} \) \( + ( -245 \beta_{1} - 520 \beta_{2} ) q^{98} \) \( + ( 910 - 390 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 38q^{6} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut -\mathstrut 35q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 38q^{6} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut -\mathstrut 35q^{9} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 17q^{11} \) \(\mathstrut +\mathstrut 280q^{12} \) \(\mathstrut +\mathstrut 125q^{13} \) \(\mathstrut -\mathstrut 92q^{14} \) \(\mathstrut -\mathstrut 90q^{15} \) \(\mathstrut -\mathstrut 57q^{16} \) \(\mathstrut -\mathstrut 70q^{17} \) \(\mathstrut -\mathstrut 430q^{18} \) \(\mathstrut +\mathstrut 141q^{19} \) \(\mathstrut -\mathstrut 175q^{20} \) \(\mathstrut +\mathstrut 126q^{21} \) \(\mathstrut +\mathstrut 170q^{22} \) \(\mathstrut -\mathstrut 145q^{23} \) \(\mathstrut +\mathstrut 216q^{24} \) \(\mathstrut +\mathstrut 150q^{25} \) \(\mathstrut -\mathstrut 23q^{26} \) \(\mathstrut +\mathstrut 670q^{27} \) \(\mathstrut -\mathstrut 80q^{28} \) \(\mathstrut -\mathstrut 34q^{29} \) \(\mathstrut +\mathstrut 140q^{30} \) \(\mathstrut -\mathstrut 280q^{31} \) \(\mathstrut -\mathstrut 105q^{32} \) \(\mathstrut -\mathstrut 425q^{33} \) \(\mathstrut -\mathstrut 78q^{34} \) \(\mathstrut +\mathstrut 70q^{35} \) \(\mathstrut -\mathstrut 725q^{36} \) \(\mathstrut +\mathstrut 190q^{37} \) \(\mathstrut +\mathstrut 620q^{38} \) \(\mathstrut -\mathstrut 181q^{39} \) \(\mathstrut -\mathstrut 370q^{40} \) \(\mathstrut -\mathstrut 538q^{41} \) \(\mathstrut +\mathstrut 370q^{42} \) \(\mathstrut -\mathstrut 455q^{43} \) \(\mathstrut +\mathstrut 1360q^{44} \) \(\mathstrut -\mathstrut 375q^{45} \) \(\mathstrut +\mathstrut 82q^{46} \) \(\mathstrut +\mathstrut 120q^{47} \) \(\mathstrut +\mathstrut 240q^{48} \) \(\mathstrut +\mathstrut 565q^{49} \) \(\mathstrut -\mathstrut 450q^{50} \) \(\mathstrut -\mathstrut 466q^{51} \) \(\mathstrut -\mathstrut 310q^{52} \) \(\mathstrut +\mathstrut 1090q^{53} \) \(\mathstrut +\mathstrut 914q^{54} \) \(\mathstrut -\mathstrut 510q^{55} \) \(\mathstrut +\mathstrut 172q^{56} \) \(\mathstrut -\mathstrut 450q^{57} \) \(\mathstrut +\mathstrut 595q^{58} \) \(\mathstrut +\mathstrut 809q^{59} \) \(\mathstrut -\mathstrut 400q^{60} \) \(\mathstrut -\mathstrut 502q^{61} \) \(\mathstrut +\mathstrut 500q^{62} \) \(\mathstrut -\mathstrut 390q^{63} \) \(\mathstrut -\mathstrut 2542q^{64} \) \(\mathstrut -\mathstrut 555q^{65} \) \(\mathstrut -\mathstrut 3196q^{66} \) \(\mathstrut +\mathstrut 475q^{67} \) \(\mathstrut +\mathstrut 505q^{68} \) \(\mathstrut +\mathstrut 479q^{69} \) \(\mathstrut -\mathstrut 160q^{70} \) \(\mathstrut -\mathstrut 127q^{71} \) \(\mathstrut +\mathstrut 1155q^{72} \) \(\mathstrut +\mathstrut 1170q^{73} \) \(\mathstrut -\mathstrut 849q^{74} \) \(\mathstrut +\mathstrut 1725q^{75} \) \(\mathstrut +\mathstrut 140q^{76} \) \(\mathstrut +\mathstrut 510q^{77} \) \(\mathstrut +\mathstrut 3070q^{78} \) \(\mathstrut +\mathstrut 480q^{79} \) \(\mathstrut +\mathstrut 1065q^{80} \) \(\mathstrut -\mathstrut 122q^{81} \) \(\mathstrut +\mathstrut 1515q^{82} \) \(\mathstrut +\mathstrut 520q^{83} \) \(\mathstrut -\mathstrut 1220q^{84} \) \(\mathstrut +\mathstrut 1205q^{85} \) \(\mathstrut -\mathstrut 3924q^{86} \) \(\mathstrut -\mathstrut 1615q^{87} \) \(\mathstrut +\mathstrut 1020q^{88} \) \(\mathstrut -\mathstrut 921q^{89} \) \(\mathstrut -\mathstrut 1450q^{90} \) \(\mathstrut -\mathstrut 1287q^{91} \) \(\mathstrut -\mathstrut 2080q^{92} \) \(\mathstrut -\mathstrut 2200q^{93} \) \(\mathstrut -\mathstrut 1040q^{94} \) \(\mathstrut -\mathstrut 1270q^{95} \) \(\mathstrut +\mathstrut 3840q^{96} \) \(\mathstrut +\mathstrut 415q^{97} \) \(\mathstrut -\mathstrut 1285q^{98} \) \(\mathstrut +\mathstrut 4420q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(5\) \(x^{2}\mathstrut +\mathstrut \) \(4\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.780776 + 1.35234i
1.28078 2.21837i
−0.780776 1.35234i
1.28078 + 2.21837i
0.219224 + 0.379706i 1.84233 + 3.19101i 3.90388 6.76172i −17.8078 −0.807764 + 1.39909i −2.71922 + 4.70983i 6.93087 6.71165 11.6249i −3.90388 6.76172i
3.2 2.28078 + 3.95042i −4.34233 7.52113i −6.40388 + 11.0918i 2.80776 19.8078 34.3081i −4.78078 + 8.28055i −21.9309 −24.2116 + 41.9358i 6.40388 + 11.0918i
9.1 0.219224 0.379706i 1.84233 3.19101i 3.90388 + 6.76172i −17.8078 −0.807764 1.39909i −2.71922 4.70983i 6.93087 6.71165 + 11.6249i −3.90388 + 6.76172i
9.2 2.28078 3.95042i −4.34233 + 7.52113i −6.40388 11.0918i 2.80776 19.8078 + 34.3081i −4.78078 8.28055i −21.9309 −24.2116 41.9358i 6.40388 11.0918i
\(n\): 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.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 5 T_{2}^{3} \) \(\mathstrut +\mathstrut 23 T_{2}^{2} \) \(\mathstrut -\mathstrut 10 T_{2} \) \(\mathstrut +\mathstrut 4 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).