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 \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.4.c.a 2
3.b odd 2 1 117.4.g.c 2
4.b odd 2 1 208.4.i.b 2
13.b even 2 1 169.4.c.d 2
13.c even 3 1 inner 13.4.c.a 2
13.c even 3 1 169.4.a.d 1
13.d odd 4 2 169.4.e.c 4
13.e even 6 1 169.4.a.a 1
13.e even 6 1 169.4.c.d 2
13.f odd 12 2 169.4.b.c 2
13.f odd 12 2 169.4.e.c 4
39.h odd 6 1 1521.4.a.k 1
39.i odd 6 1 117.4.g.c 2
39.i odd 6 1 1521.4.a.b 1
52.j odd 6 1 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 1.a even 1 1 trivial
13.4.c.a 2 13.c even 3 1 inner
117.4.g.c 2 3.b odd 2 1
117.4.g.c 2 39.i odd 6 1
169.4.a.a 1 13.e even 6 1
169.4.a.d 1 13.c even 3 1
169.4.b.c 2 13.f odd 12 2
169.4.c.d 2 13.b even 2 1
169.4.c.d 2 13.e even 6 1
169.4.e.c 4 13.d odd 4 2
169.4.e.c 4 13.f odd 12 2
208.4.i.b 2 4.b odd 2 1
208.4.i.b 2 52.j odd 6 1
1521.4.a.b 1 39.i odd 6 1
1521.4.a.k 1 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T - 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$11$ \( T^{2} - 32T + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$19$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} + 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} + 197T + 38809 \) Copy content Toggle raw display
$31$ \( (T + 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 227T + 51529 \) Copy content Toggle raw display
$41$ \( T^{2} - 165T + 27225 \) Copy content Toggle raw display
$43$ \( T^{2} - 156T + 24336 \) Copy content Toggle raw display
$47$ \( (T + 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 864T + 746496 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} + 862T + 743044 \) Copy content Toggle raw display
$71$ \( T^{2} + 654T + 427716 \) Copy content Toggle raw display
$73$ \( (T - 215)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( (T - 628)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 266T + 70756 \) Copy content Toggle raw display
$97$ \( T^{2} + 238T + 56644 \) Copy content Toggle raw display
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