Properties

Label 13.4.e.a
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( -2 - 2 \zeta_{6} ) q^{2} \) \( + ( 7 - 7 \zeta_{6} ) q^{3} \) \( + 4 \zeta_{6} q^{4} \) \( + ( -8 + 16 \zeta_{6} ) q^{5} \) \( + ( -28 + 14 \zeta_{6} ) q^{6} \) \( + ( 26 - 13 \zeta_{6} ) q^{7} \) \( + ( -8 + 16 \zeta_{6} ) q^{8} \) \( -22 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 - 2 \zeta_{6} ) q^{2} \) \( + ( 7 - 7 \zeta_{6} ) q^{3} \) \( + 4 \zeta_{6} q^{4} \) \( + ( -8 + 16 \zeta_{6} ) q^{5} \) \( + ( -28 + 14 \zeta_{6} ) q^{6} \) \( + ( 26 - 13 \zeta_{6} ) q^{7} \) \( + ( -8 + 16 \zeta_{6} ) q^{8} \) \( -22 \zeta_{6} q^{9} \) \( + ( 48 - 48 \zeta_{6} ) q^{10} \) \( + ( -13 - 13 \zeta_{6} ) q^{11} \) \( + 28 q^{12} \) \( + ( -39 + 52 \zeta_{6} ) q^{13} \) \( -78 q^{14} \) \( + ( 56 + 56 \zeta_{6} ) q^{15} \) \( + ( 80 - 80 \zeta_{6} ) q^{16} \) \( -27 \zeta_{6} q^{17} \) \( + ( -44 + 88 \zeta_{6} ) q^{18} \) \( + ( -102 + 51 \zeta_{6} ) q^{19} \) \( + ( -64 + 32 \zeta_{6} ) q^{20} \) \( + ( 91 - 182 \zeta_{6} ) q^{21} \) \( + 78 \zeta_{6} q^{22} \) \( + ( -57 + 57 \zeta_{6} ) q^{23} \) \( + ( 56 + 56 \zeta_{6} ) q^{24} \) \( -67 q^{25} \) \( + ( 182 - 130 \zeta_{6} ) q^{26} \) \( + 35 q^{27} \) \( + ( 52 + 52 \zeta_{6} ) q^{28} \) \( + ( 69 - 69 \zeta_{6} ) q^{29} \) \( -336 \zeta_{6} q^{30} \) \( + ( 42 - 84 \zeta_{6} ) q^{31} \) \( + ( -192 + 96 \zeta_{6} ) q^{32} \) \( + ( -182 + 91 \zeta_{6} ) q^{33} \) \( + ( -54 + 108 \zeta_{6} ) q^{34} \) \( + 312 \zeta_{6} q^{35} \) \( + ( 88 - 88 \zeta_{6} ) q^{36} \) \( + ( -23 - 23 \zeta_{6} ) q^{37} \) \( + 306 q^{38} \) \( + ( 91 + 273 \zeta_{6} ) q^{39} \) \( -192 q^{40} \) \( + ( -227 - 227 \zeta_{6} ) q^{41} \) \( + ( -546 + 546 \zeta_{6} ) q^{42} \) \( + 85 \zeta_{6} q^{43} \) \( + ( 52 - 104 \zeta_{6} ) q^{44} \) \( + ( 352 - 176 \zeta_{6} ) q^{45} \) \( + ( 228 - 114 \zeta_{6} ) q^{46} \) \( + ( 198 - 396 \zeta_{6} ) q^{47} \) \( -560 \zeta_{6} q^{48} \) \( + ( 164 - 164 \zeta_{6} ) q^{49} \) \( + ( 134 + 134 \zeta_{6} ) q^{50} \) \( -189 q^{51} \) \( + ( -208 + 52 \zeta_{6} ) q^{52} \) \( + 426 q^{53} \) \( + ( -70 - 70 \zeta_{6} ) q^{54} \) \( + ( 312 - 312 \zeta_{6} ) q^{55} \) \( + 312 \zeta_{6} q^{56} \) \( + ( -357 + 714 \zeta_{6} ) q^{57} \) \( + ( -276 + 138 \zeta_{6} ) q^{58} \) \( + ( -22 + 11 \zeta_{6} ) q^{59} \) \( + ( -224 + 448 \zeta_{6} ) q^{60} \) \( + 17 \zeta_{6} q^{61} \) \( + ( -252 + 252 \zeta_{6} ) q^{62} \) \( + ( -286 - 286 \zeta_{6} ) q^{63} \) \( -64 q^{64} \) \( + ( -520 - 208 \zeta_{6} ) q^{65} \) \( + 546 q^{66} \) \( + ( 95 + 95 \zeta_{6} ) q^{67} \) \( + ( 108 - 108 \zeta_{6} ) q^{68} \) \( + 399 \zeta_{6} q^{69} \) \( + ( 624 - 1248 \zeta_{6} ) q^{70} \) \( + ( 674 - 337 \zeta_{6} ) q^{71} \) \( + ( 352 - 176 \zeta_{6} ) q^{72} \) \( + ( -580 + 1160 \zeta_{6} ) q^{73} \) \( + 138 \zeta_{6} q^{74} \) \( + ( -469 + 469 \zeta_{6} ) q^{75} \) \( + ( -204 - 204 \zeta_{6} ) q^{76} \) \( -507 q^{77} \) \( + ( 364 - 1274 \zeta_{6} ) q^{78} \) \( -1244 q^{79} \) \( + ( 640 + 640 \zeta_{6} ) q^{80} \) \( + ( 839 - 839 \zeta_{6} ) q^{81} \) \( + 1362 \zeta_{6} q^{82} \) \( + ( 246 - 492 \zeta_{6} ) q^{83} \) \( + ( 728 - 364 \zeta_{6} ) q^{84} \) \( + ( 432 - 216 \zeta_{6} ) q^{85} \) \( + ( 170 - 340 \zeta_{6} ) q^{86} \) \( -483 \zeta_{6} q^{87} \) \( + ( 312 - 312 \zeta_{6} ) q^{88} \) \( + ( 177 + 177 \zeta_{6} ) q^{89} \) \( -1056 q^{90} \) \( + ( -338 + 1183 \zeta_{6} ) q^{91} \) \( -228 q^{92} \) \( + ( -294 - 294 \zeta_{6} ) q^{93} \) \( + ( -1188 + 1188 \zeta_{6} ) q^{94} \) \( -1224 \zeta_{6} q^{95} \) \( + ( -672 + 1344 \zeta_{6} ) q^{96} \) \( + ( 1426 - 713 \zeta_{6} ) q^{97} \) \( + ( -656 + 328 \zeta_{6} ) q^{98} \) \( + ( -286 + 572 \zeta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 42q^{6} \) \(\mathstrut +\mathstrut 39q^{7} \) \(\mathstrut -\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 42q^{6} \) \(\mathstrut +\mathstrut 39q^{7} \) \(\mathstrut -\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut 48q^{10} \) \(\mathstrut -\mathstrut 39q^{11} \) \(\mathstrut +\mathstrut 56q^{12} \) \(\mathstrut -\mathstrut 26q^{13} \) \(\mathstrut -\mathstrut 156q^{14} \) \(\mathstrut +\mathstrut 168q^{15} \) \(\mathstrut +\mathstrut 80q^{16} \) \(\mathstrut -\mathstrut 27q^{17} \) \(\mathstrut -\mathstrut 153q^{19} \) \(\mathstrut -\mathstrut 96q^{20} \) \(\mathstrut +\mathstrut 78q^{22} \) \(\mathstrut -\mathstrut 57q^{23} \) \(\mathstrut +\mathstrut 168q^{24} \) \(\mathstrut -\mathstrut 134q^{25} \) \(\mathstrut +\mathstrut 234q^{26} \) \(\mathstrut +\mathstrut 70q^{27} \) \(\mathstrut +\mathstrut 156q^{28} \) \(\mathstrut +\mathstrut 69q^{29} \) \(\mathstrut -\mathstrut 336q^{30} \) \(\mathstrut -\mathstrut 288q^{32} \) \(\mathstrut -\mathstrut 273q^{33} \) \(\mathstrut +\mathstrut 312q^{35} \) \(\mathstrut +\mathstrut 88q^{36} \) \(\mathstrut -\mathstrut 69q^{37} \) \(\mathstrut +\mathstrut 612q^{38} \) \(\mathstrut +\mathstrut 455q^{39} \) \(\mathstrut -\mathstrut 384q^{40} \) \(\mathstrut -\mathstrut 681q^{41} \) \(\mathstrut -\mathstrut 546q^{42} \) \(\mathstrut +\mathstrut 85q^{43} \) \(\mathstrut +\mathstrut 528q^{45} \) \(\mathstrut +\mathstrut 342q^{46} \) \(\mathstrut -\mathstrut 560q^{48} \) \(\mathstrut +\mathstrut 164q^{49} \) \(\mathstrut +\mathstrut 402q^{50} \) \(\mathstrut -\mathstrut 378q^{51} \) \(\mathstrut -\mathstrut 364q^{52} \) \(\mathstrut +\mathstrut 852q^{53} \) \(\mathstrut -\mathstrut 210q^{54} \) \(\mathstrut +\mathstrut 312q^{55} \) \(\mathstrut +\mathstrut 312q^{56} \) \(\mathstrut -\mathstrut 414q^{58} \) \(\mathstrut -\mathstrut 33q^{59} \) \(\mathstrut +\mathstrut 17q^{61} \) \(\mathstrut -\mathstrut 252q^{62} \) \(\mathstrut -\mathstrut 858q^{63} \) \(\mathstrut -\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 1248q^{65} \) \(\mathstrut +\mathstrut 1092q^{66} \) \(\mathstrut +\mathstrut 285q^{67} \) \(\mathstrut +\mathstrut 108q^{68} \) \(\mathstrut +\mathstrut 399q^{69} \) \(\mathstrut +\mathstrut 1011q^{71} \) \(\mathstrut +\mathstrut 528q^{72} \) \(\mathstrut +\mathstrut 138q^{74} \) \(\mathstrut -\mathstrut 469q^{75} \) \(\mathstrut -\mathstrut 612q^{76} \) \(\mathstrut -\mathstrut 1014q^{77} \) \(\mathstrut -\mathstrut 546q^{78} \) \(\mathstrut -\mathstrut 2488q^{79} \) \(\mathstrut +\mathstrut 1920q^{80} \) \(\mathstrut +\mathstrut 839q^{81} \) \(\mathstrut +\mathstrut 1362q^{82} \) \(\mathstrut +\mathstrut 1092q^{84} \) \(\mathstrut +\mathstrut 648q^{85} \) \(\mathstrut -\mathstrut 483q^{87} \) \(\mathstrut +\mathstrut 312q^{88} \) \(\mathstrut +\mathstrut 531q^{89} \) \(\mathstrut -\mathstrut 2112q^{90} \) \(\mathstrut +\mathstrut 507q^{91} \) \(\mathstrut -\mathstrut 456q^{92} \) \(\mathstrut -\mathstrut 882q^{93} \) \(\mathstrut -\mathstrut 1188q^{94} \) \(\mathstrut -\mathstrut 1224q^{95} \) \(\mathstrut +\mathstrut 2139q^{97} \) \(\mathstrut -\mathstrut 984q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−3.00000 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i −21.0000 + 12.1244i 19.5000 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
10.1 −3.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i −21.0000 12.1244i 19.5000 + 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 6 T_{2} \) \(\mathstrut +\mathstrut 12 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).