Properties

Label 13.4.c.a
Level 13
Weight 4
Character orbit 13.c
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.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -4 + 4 \zeta_{6} ) q^{2} \) \( + ( -2 + 2 \zeta_{6} ) q^{3} \) \( -8 \zeta_{6} q^{4} \) \( + 17 q^{5} \) \( -8 \zeta_{6} q^{6} \) \( -20 \zeta_{6} q^{7} \) \( + 23 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -4 + 4 \zeta_{6} ) q^{2} \) \( + ( -2 + 2 \zeta_{6} ) q^{3} \) \( -8 \zeta_{6} q^{4} \) \( + 17 q^{5} \) \( -8 \zeta_{6} q^{6} \) \( -20 \zeta_{6} q^{7} \) \( + 23 \zeta_{6} q^{9} \) \( + ( -68 + 68 \zeta_{6} ) q^{10} \) \( + ( 32 - 32 \zeta_{6} ) q^{11} \) \( + 16 q^{12} \) \( + ( -39 - 13 \zeta_{6} ) q^{13} \) \( + 80 q^{14} \) \( + ( -34 + 34 \zeta_{6} ) q^{15} \) \( + ( 64 - 64 \zeta_{6} ) q^{16} \) \( + 13 \zeta_{6} q^{17} \) \( -92 q^{18} \) \( -30 \zeta_{6} q^{19} \) \( -136 \zeta_{6} q^{20} \) \( + 40 q^{21} \) \( + 128 \zeta_{6} q^{22} \) \( + ( -78 + 78 \zeta_{6} ) q^{23} \) \( + 164 q^{25} \) \( + ( 208 - 156 \zeta_{6} ) q^{26} \) \( -100 q^{27} \) \( + ( -160 + 160 \zeta_{6} ) q^{28} \) \( + ( -197 + 197 \zeta_{6} ) q^{29} \) \( -136 \zeta_{6} q^{30} \) \( -74 q^{31} \) \( + 256 \zeta_{6} q^{32} \) \( + 64 \zeta_{6} q^{33} \) \( -52 q^{34} \) \( -340 \zeta_{6} q^{35} \) \( + ( 184 - 184 \zeta_{6} ) q^{36} \) \( + ( 227 - 227 \zeta_{6} ) q^{37} \) \( + 120 q^{38} \) \( + ( 104 - 78 \zeta_{6} ) q^{39} \) \( + ( 165 - 165 \zeta_{6} ) q^{41} \) \( + ( -160 + 160 \zeta_{6} ) q^{42} \) \( + 156 \zeta_{6} q^{43} \) \( -256 q^{44} \) \( + 391 \zeta_{6} q^{45} \) \( -312 \zeta_{6} q^{46} \) \( -162 q^{47} \) \( + 128 \zeta_{6} q^{48} \) \( + ( -57 + 57 \zeta_{6} ) q^{49} \) \( + ( -656 + 656 \zeta_{6} ) q^{50} \) \( -26 q^{51} \) \( + ( -104 + 416 \zeta_{6} ) q^{52} \) \( + 93 q^{53} \) \( + ( 400 - 400 \zeta_{6} ) q^{54} \) \( + ( 544 - 544 \zeta_{6} ) q^{55} \) \( + 60 q^{57} \) \( -788 \zeta_{6} q^{58} \) \( + 864 \zeta_{6} q^{59} \) \( + 272 q^{60} \) \( -145 \zeta_{6} q^{61} \) \( + ( 296 - 296 \zeta_{6} ) q^{62} \) \( + ( 460 - 460 \zeta_{6} ) q^{63} \) \( -512 q^{64} \) \( + ( -663 - 221 \zeta_{6} ) q^{65} \) \( -256 q^{66} \) \( + ( -862 + 862 \zeta_{6} ) q^{67} \) \( + ( 104 - 104 \zeta_{6} ) q^{68} \) \( -156 \zeta_{6} q^{69} \) \( + 1360 q^{70} \) \( -654 \zeta_{6} q^{71} \) \( + 215 q^{73} \) \( + 908 \zeta_{6} q^{74} \) \( + ( -328 + 328 \zeta_{6} ) q^{75} \) \( + ( -240 + 240 \zeta_{6} ) q^{76} \) \( -640 q^{77} \) \( + ( -104 + 416 \zeta_{6} ) q^{78} \) \( -76 q^{79} \) \( + ( 1088 - 1088 \zeta_{6} ) q^{80} \) \( + ( -421 + 421 \zeta_{6} ) q^{81} \) \( + 660 \zeta_{6} q^{82} \) \( + 628 q^{83} \) \( -320 \zeta_{6} q^{84} \) \( + 221 \zeta_{6} q^{85} \) \( -624 q^{86} \) \( -394 \zeta_{6} q^{87} \) \( + ( 266 - 266 \zeta_{6} ) q^{89} \) \( -1564 q^{90} \) \( + ( -260 + 1040 \zeta_{6} ) q^{91} \) \( + 624 q^{92} \) \( + ( 148 - 148 \zeta_{6} ) q^{93} \) \( + ( 648 - 648 \zeta_{6} ) q^{94} \) \( -510 \zeta_{6} q^{95} \) \( -512 q^{96} \) \( -238 \zeta_{6} q^{97} \) \( -228 \zeta_{6} q^{98} \) \( + 736 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 34q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 23q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 34q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 23q^{9} \) \(\mathstrut -\mathstrut 68q^{10} \) \(\mathstrut +\mathstrut 32q^{11} \) \(\mathstrut +\mathstrut 32q^{12} \) \(\mathstrut -\mathstrut 91q^{13} \) \(\mathstrut +\mathstrut 160q^{14} \) \(\mathstrut -\mathstrut 34q^{15} \) \(\mathstrut +\mathstrut 64q^{16} \) \(\mathstrut +\mathstrut 13q^{17} \) \(\mathstrut -\mathstrut 184q^{18} \) \(\mathstrut -\mathstrut 30q^{19} \) \(\mathstrut -\mathstrut 136q^{20} \) \(\mathstrut +\mathstrut 80q^{21} \) \(\mathstrut +\mathstrut 128q^{22} \) \(\mathstrut -\mathstrut 78q^{23} \) \(\mathstrut +\mathstrut 328q^{25} \) \(\mathstrut +\mathstrut 260q^{26} \) \(\mathstrut -\mathstrut 200q^{27} \) \(\mathstrut -\mathstrut 160q^{28} \) \(\mathstrut -\mathstrut 197q^{29} \) \(\mathstrut -\mathstrut 136q^{30} \) \(\mathstrut -\mathstrut 148q^{31} \) \(\mathstrut +\mathstrut 256q^{32} \) \(\mathstrut +\mathstrut 64q^{33} \) \(\mathstrut -\mathstrut 104q^{34} \) \(\mathstrut -\mathstrut 340q^{35} \) \(\mathstrut +\mathstrut 184q^{36} \) \(\mathstrut +\mathstrut 227q^{37} \) \(\mathstrut +\mathstrut 240q^{38} \) \(\mathstrut +\mathstrut 130q^{39} \) \(\mathstrut +\mathstrut 165q^{41} \) \(\mathstrut -\mathstrut 160q^{42} \) \(\mathstrut +\mathstrut 156q^{43} \) \(\mathstrut -\mathstrut 512q^{44} \) \(\mathstrut +\mathstrut 391q^{45} \) \(\mathstrut -\mathstrut 312q^{46} \) \(\mathstrut -\mathstrut 324q^{47} \) \(\mathstrut +\mathstrut 128q^{48} \) \(\mathstrut -\mathstrut 57q^{49} \) \(\mathstrut -\mathstrut 656q^{50} \) \(\mathstrut -\mathstrut 52q^{51} \) \(\mathstrut +\mathstrut 208q^{52} \) \(\mathstrut +\mathstrut 186q^{53} \) \(\mathstrut +\mathstrut 400q^{54} \) \(\mathstrut +\mathstrut 544q^{55} \) \(\mathstrut +\mathstrut 120q^{57} \) \(\mathstrut -\mathstrut 788q^{58} \) \(\mathstrut +\mathstrut 864q^{59} \) \(\mathstrut +\mathstrut 544q^{60} \) \(\mathstrut -\mathstrut 145q^{61} \) \(\mathstrut +\mathstrut 296q^{62} \) \(\mathstrut +\mathstrut 460q^{63} \) \(\mathstrut -\mathstrut 1024q^{64} \) \(\mathstrut -\mathstrut 1547q^{65} \) \(\mathstrut -\mathstrut 512q^{66} \) \(\mathstrut -\mathstrut 862q^{67} \) \(\mathstrut +\mathstrut 104q^{68} \) \(\mathstrut -\mathstrut 156q^{69} \) \(\mathstrut +\mathstrut 2720q^{70} \) \(\mathstrut -\mathstrut 654q^{71} \) \(\mathstrut +\mathstrut 430q^{73} \) \(\mathstrut +\mathstrut 908q^{74} \) \(\mathstrut -\mathstrut 328q^{75} \) \(\mathstrut -\mathstrut 240q^{76} \) \(\mathstrut -\mathstrut 1280q^{77} \) \(\mathstrut +\mathstrut 208q^{78} \) \(\mathstrut -\mathstrut 152q^{79} \) \(\mathstrut +\mathstrut 1088q^{80} \) \(\mathstrut -\mathstrut 421q^{81} \) \(\mathstrut +\mathstrut 660q^{82} \) \(\mathstrut +\mathstrut 1256q^{83} \) \(\mathstrut -\mathstrut 320q^{84} \) \(\mathstrut +\mathstrut 221q^{85} \) \(\mathstrut -\mathstrut 1248q^{86} \) \(\mathstrut -\mathstrut 394q^{87} \) \(\mathstrut +\mathstrut 266q^{89} \) \(\mathstrut -\mathstrut 3128q^{90} \) \(\mathstrut +\mathstrut 520q^{91} \) \(\mathstrut +\mathstrut 1248q^{92} \) \(\mathstrut +\mathstrut 148q^{93} \) \(\mathstrut +\mathstrut 648q^{94} \) \(\mathstrut -\mathstrut 510q^{95} \) \(\mathstrut -\mathstrut 1024q^{96} \) \(\mathstrut -\mathstrut 238q^{97} \) \(\mathstrut -\mathstrut 228q^{98} \) \(\mathstrut +\mathstrut 1472q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 0.866025i
0.500000 + 0.866025i
−2.00000 3.46410i −1.00000 1.73205i −4.00000 + 6.92820i 17.0000 −4.00000 + 6.92820i −10.0000 + 17.3205i 0 11.5000 19.9186i −34.0000 58.8897i
9.1 −2.00000 + 3.46410i −1.00000 + 1.73205i −4.00000 6.92820i 17.0000 −4.00000 6.92820i −10.0000 17.3205i 0 11.5000 + 19.9186i −34.0000 + 58.8897i
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 4 T_{2} \) \(\mathstrut +\mathstrut 16 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).