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

Newform invariants

Self dual: no
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} - 5 q^{3} - 5 q^{4} - 30 q^{5} + 38 q^{6} - 15 q^{7} - 30 q^{8} - 35 q^{9} + O(q^{10}) \) \( 4 q + 5 q^{2} - 5 q^{3} - 5 q^{4} - 30 q^{5} + 38 q^{6} - 15 q^{7} - 30 q^{8} - 35 q^{9} + 5 q^{10} - 17 q^{11} + 280 q^{12} + 125 q^{13} - 92 q^{14} - 90 q^{15} - 57 q^{16} - 70 q^{17} - 430 q^{18} + 141 q^{19} - 175 q^{20} + 126 q^{21} + 170 q^{22} - 145 q^{23} + 216 q^{24} + 150 q^{25} - 23 q^{26} + 670 q^{27} - 80 q^{28} - 34 q^{29} + 140 q^{30} - 280 q^{31} - 105 q^{32} - 425 q^{33} - 78 q^{34} + 70 q^{35} - 725 q^{36} + 190 q^{37} + 620 q^{38} - 181 q^{39} - 370 q^{40} - 538 q^{41} + 370 q^{42} - 455 q^{43} + 1360 q^{44} - 375 q^{45} + 82 q^{46} + 120 q^{47} + 240 q^{48} + 565 q^{49} - 450 q^{50} - 466 q^{51} - 310 q^{52} + 1090 q^{53} + 914 q^{54} - 510 q^{55} + 172 q^{56} - 450 q^{57} + 595 q^{58} + 809 q^{59} - 400 q^{60} - 502 q^{61} + 500 q^{62} - 390 q^{63} - 2542 q^{64} - 555 q^{65} - 3196 q^{66} + 475 q^{67} + 505 q^{68} + 479 q^{69} - 160 q^{70} - 127 q^{71} + 1155 q^{72} + 1170 q^{73} - 849 q^{74} + 1725 q^{75} + 140 q^{76} + 510 q^{77} + 3070 q^{78} + 480 q^{79} + 1065 q^{80} - 122 q^{81} + 1515 q^{82} + 520 q^{83} - 1220 q^{84} + 1205 q^{85} - 3924 q^{86} - 1615 q^{87} + 1020 q^{88} - 921 q^{89} - 1450 q^{90} - 1287 q^{91} - 2080 q^{92} - 2200 q^{93} - 1040 q^{94} - 1270 q^{95} + 3840 q^{96} + 415 q^{97} - 1285 q^{98} + 4420 q^{99} + O(q^{100}) \)

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

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

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 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.b 4
3.b odd 2 1 117.4.g.d 4
4.b odd 2 1 208.4.i.e 4
13.b even 2 1 169.4.c.f 4
13.c even 3 1 inner 13.4.c.b 4
13.c even 3 1 169.4.a.f 2
13.d odd 4 2 169.4.e.g 8
13.e even 6 1 169.4.a.j 2
13.e even 6 1 169.4.c.f 4
13.f odd 12 2 169.4.b.e 4
13.f odd 12 2 169.4.e.g 8
39.h odd 6 1 1521.4.a.l 2
39.i odd 6 1 117.4.g.d 4
39.i odd 6 1 1521.4.a.t 2
52.j odd 6 1 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 1.a even 1 1 trivial
13.4.c.b 4 13.c even 3 1 inner
117.4.g.d 4 3.b odd 2 1
117.4.g.d 4 39.i odd 6 1
169.4.a.f 2 13.c even 3 1
169.4.a.j 2 13.e even 6 1
169.4.b.e 4 13.f odd 12 2
169.4.c.f 4 13.b even 2 1
169.4.c.f 4 13.e even 6 1
169.4.e.g 8 13.d odd 4 2
169.4.e.g 8 13.f odd 12 2
208.4.i.e 4 4.b odd 2 1
208.4.i.e 4 52.j odd 6 1
1521.4.a.l 2 39.h odd 6 1
1521.4.a.t 2 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} \)
$3$ \( 1024 - 160 T + 57 T^{2} + 5 T^{3} + T^{4} \)
$5$ \( ( -50 + 15 T + T^{2} )^{2} \)
$7$ \( 2704 + 780 T + 173 T^{2} + 15 T^{3} + T^{4} \)
$11$ \( 781456 - 15028 T + 1173 T^{2} + 17 T^{3} + T^{4} \)
$13$ \( 4826809 - 274625 T + 7956 T^{2} - 125 T^{3} + T^{4} \)
$17$ \( 18769 + 9590 T + 4763 T^{2} + 70 T^{3} + T^{4} \)
$19$ \( 23658496 - 685824 T + 15017 T^{2} - 141 T^{3} + T^{4} \)
$23$ \( 394384 + 91060 T + 20397 T^{2} + 145 T^{3} + T^{4} \)
$29$ \( 225330121 - 510374 T + 16167 T^{2} + 34 T^{3} + T^{4} \)
$31$ \( ( -37600 + 140 T + T^{2} )^{2} \)
$37$ \( 635209 - 151430 T + 35303 T^{2} - 190 T^{3} + T^{4} \)
$41$ \( 4992976921 + 38015618 T + 218783 T^{2} + 538 T^{3} + T^{4} \)
$43$ \( 138485824 + 5354440 T + 195257 T^{2} + 455 T^{3} + T^{4} \)
$47$ \( ( -82400 - 60 T + T^{2} )^{2} \)
$53$ \( ( -41450 - 545 T + T^{2} )^{2} \)
$59$ \( 22729783696 - 121968076 T + 503717 T^{2} - 809 T^{3} + T^{4} \)
$61$ \( 11448786001 - 53713498 T + 359003 T^{2} + 502 T^{3} + T^{4} \)
$67$ \( 449948944 - 10075700 T + 204413 T^{2} - 475 T^{3} + T^{4} \)
$71$ \( 1833894976 - 5438648 T + 58953 T^{2} + 127 T^{3} + T^{4} \)
$73$ \( ( 54850 - 585 T + T^{2} )^{2} \)
$79$ \( ( 7600 - 240 T + T^{2} )^{2} \)
$83$ \( ( -25600 - 260 T + T^{2} )^{2} \)
$89$ \( 21215087716 + 134147334 T + 702587 T^{2} + 921 T^{3} + T^{4} \)
$97$ \( 795268001284 + 370087870 T + 1064003 T^{2} - 415 T^{3} + T^{4} \)
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