Properties

Label 13.5.d.a
Level 13
Weight 5
Character orbit 13.d
Analytic conductor 1.344
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 13.d (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.34380952009\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.53039932416.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{4} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 5 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( 10 + 10 \beta_{2} - \beta_{3} - 5 \beta_{4} - 5 \beta_{5} ) q^{7} + ( -12 - 12 \beta_{2} - 11 \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} + ( -16 + 9 \beta_{1} + 9 \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{4} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 5 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( 10 + 10 \beta_{2} - \beta_{3} - 5 \beta_{4} - 5 \beta_{5} ) q^{7} + ( -12 - 12 \beta_{2} - 11 \beta_{3} - \beta_{4} - \beta_{5} ) q^{8} + ( -16 + 9 \beta_{1} + 9 \beta_{3} + \beta_{4} ) q^{9} + ( 6 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} ) q^{10} + ( -15 - 15 \beta_{2} + 22 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} ) q^{11} + ( -6 \beta_{1} + 55 \beta_{2} + 6 \beta_{3} + 5 \beta_{5} ) q^{12} + ( 13 - 26 \beta_{1} - 39 \beta_{2} - 13 \beta_{3} - 13 \beta_{5} ) q^{13} + ( -33 - 21 \beta_{1} - 21 \beta_{3} + 19 \beta_{4} ) q^{14} + ( 62 + 15 \beta_{1} - 62 \beta_{2} + \beta_{4} - \beta_{5} ) q^{15} + ( 193 - 17 \beta_{1} - 17 \beta_{3} - 23 \beta_{4} ) q^{16} + ( -57 \beta_{1} + 51 \beta_{2} + 57 \beta_{3} + 9 \beta_{5} ) q^{17} + ( -148 + 36 \beta_{1} + 148 \beta_{2} + 7 \beta_{4} - 7 \beta_{5} ) q^{18} + ( 94 + 90 \beta_{1} - 94 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} ) q^{19} + ( 69 + 69 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{20} + ( -325 - 325 \beta_{2} - 87 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{21} + ( -304 + 51 \beta_{1} + 51 \beta_{3} - 6 \beta_{4} ) q^{22} + ( 13 \beta_{1} + 216 \beta_{2} - 13 \beta_{3} - 10 \beta_{5} ) q^{23} + ( 3 + 3 \beta_{2} + 15 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} ) q^{24} + ( 7 \beta_{1} + 482 \beta_{2} - 7 \beta_{3} - \beta_{5} ) q^{25} + ( 156 - 52 \beta_{1} - 507 \beta_{2} + 26 \beta_{3} + 13 \beta_{4} + 52 \beta_{5} ) q^{26} + ( 71 - 18 \beta_{1} - 18 \beta_{3} - 59 \beta_{4} ) q^{27} + ( 292 + 45 \beta_{1} - 292 \beta_{2} + 21 \beta_{4} - 21 \beta_{5} ) q^{28} + ( 708 - 57 \beta_{1} - 57 \beta_{3} + 82 \beta_{4} ) q^{29} + ( -45 \beta_{1} + 245 \beta_{2} + 45 \beta_{3} - 11 \beta_{5} ) q^{30} + ( -548 - 58 \beta_{1} + 548 \beta_{2} - 32 \beta_{4} + 32 \beta_{5} ) q^{31} + ( 366 - 97 \beta_{1} - 366 \beta_{2} + 45 \beta_{4} - 45 \beta_{5} ) q^{32} + ( 353 + 353 \beta_{2} + 60 \beta_{3} + 43 \beta_{4} + 43 \beta_{5} ) q^{33} + ( -924 - 924 \beta_{2} + 81 \beta_{3} + 39 \beta_{4} + 39 \beta_{5} ) q^{34} + ( -657 - 64 \beta_{1} - 64 \beta_{3} - 3 \beta_{4} ) q^{35} + ( 54 \beta_{1} + 286 \beta_{2} - 54 \beta_{3} + 8 \beta_{5} ) q^{36} + ( -323 - 323 \beta_{2} + 33 \beta_{3} + 57 \beta_{4} + 57 \beta_{5} ) q^{37} + ( -16 \beta_{1} + 1590 \beta_{2} + 16 \beta_{3} - 114 \beta_{5} ) q^{38} + ( 182 + 195 \beta_{1} - 858 \beta_{2} - 78 \beta_{3} - 65 \beta_{4} - 39 \beta_{5} ) q^{39} + ( -123 + 38 \beta_{1} + 38 \beta_{3} - 61 \beta_{4} ) q^{40} + ( 876 - 140 \beta_{1} - 876 \beta_{2} - 113 \beta_{4} + 113 \beta_{5} ) q^{41} + ( 1459 + 234 \beta_{1} + 234 \beta_{3} - 79 \beta_{4} ) q^{42} + ( 370 \beta_{1} + 537 \beta_{2} - 370 \beta_{3} - 169 \beta_{5} ) q^{43} + ( -657 + 42 \beta_{1} + 657 \beta_{2} - 49 \beta_{4} + 49 \beta_{5} ) q^{44} + ( 170 - 108 \beta_{1} - 170 \beta_{2} + 13 \beta_{4} - 13 \beta_{5} ) q^{45} + ( 171 + 171 \beta_{2} - 262 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} ) q^{46} + ( -1230 - 1230 \beta_{2} + 361 \beta_{3} - 107 \beta_{4} - 107 \beta_{5} ) q^{47} + ( -1495 - 156 \beta_{1} - 156 \beta_{3} + 79 \beta_{4} ) q^{48} + ( -409 \beta_{1} + 916 \beta_{2} + 409 \beta_{3} - 11 \beta_{5} ) q^{49} + ( 114 + 114 \beta_{2} - 498 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} ) q^{50} + ( 90 \beta_{1} - 45 \beta_{2} - 90 \beta_{3} + 297 \beta_{5} ) q^{51} + ( 507 + 52 \beta_{1} - 481 \beta_{2} + 273 \beta_{3} + 104 \beta_{4} - 26 \beta_{5} ) q^{52} + ( 6 + 363 \beta_{1} + 363 \beta_{3} + 310 \beta_{4} ) q^{53} + ( 11 - 225 \beta_{1} - 11 \beta_{2} + 100 \beta_{4} - 100 \beta_{5} ) q^{54} + ( 1070 - 35 \beta_{1} - 35 \beta_{3} + 64 \beta_{4} ) q^{55} + ( 131 \beta_{1} + 27 \beta_{2} - 131 \beta_{3} + 343 \beta_{5} ) q^{56} + ( -928 - 378 \beta_{1} + 928 \beta_{2} + 262 \beta_{4} - 262 \beta_{5} ) q^{57} + ( 1379 - 658 \beta_{1} - 1379 \beta_{2} - 221 \beta_{4} + 221 \beta_{5} ) q^{58} + ( 714 + 714 \beta_{2} + 284 \beta_{3} - 240 \beta_{4} - 240 \beta_{5} ) q^{59} + ( 172 + 172 \beta_{2} + 63 \beta_{3} + 83 \beta_{4} + 83 \beta_{5} ) q^{60} + ( 1250 - 318 \beta_{1} - 318 \beta_{3} - 120 \beta_{4} ) q^{61} + ( 426 \beta_{1} - 666 \beta_{2} - 426 \beta_{3} - 70 \beta_{5} ) q^{62} + ( -178 - 178 \beta_{2} + 306 \beta_{3} - 98 \beta_{4} - 98 \beta_{5} ) q^{63} + ( -101 \beta_{1} + 989 \beta_{2} + 101 \beta_{3} - 91 \beta_{5} ) q^{64} + ( -780 + 247 \beta_{1} - 429 \beta_{2} - 325 \beta_{3} - 91 \beta_{4} ) q^{65} + ( -590 - 207 \beta_{1} - 207 \beta_{3} - 112 \beta_{4} ) q^{66} + ( -209 + 500 \beta_{1} + 209 \beta_{2} + 163 \beta_{4} - 163 \beta_{5} ) q^{67} + ( -1803 + 171 \beta_{1} + 171 \beta_{3} - 219 \beta_{4} ) q^{68} + ( 51 \beta_{1} - 726 \beta_{2} - 51 \beta_{3} + 144 \beta_{5} ) q^{69} + ( 1073 + 523 \beta_{1} - 1073 \beta_{2} - 58 \beta_{4} + 58 \beta_{5} ) q^{70} + ( -2730 + 813 \beta_{1} + 2730 \beta_{2} + 41 \beta_{4} - 41 \beta_{5} ) q^{71} + ( -1410 - 1410 \beta_{2} + 198 \beta_{3} + 42 \beta_{4} + 42 \beta_{5} ) q^{72} + ( 910 + 910 \beta_{2} + 162 \beta_{3} + 45 \beta_{4} + 45 \beta_{5} ) q^{73} + ( 9 + 470 \beta_{1} + 470 \beta_{3} - 195 \beta_{4} ) q^{74} + ( -12 \beta_{1} - 476 \beta_{2} + 12 \beta_{3} + 452 \beta_{5} ) q^{75} + ( 662 + 662 \beta_{2} - 346 \beta_{3} + 148 \beta_{4} + 148 \beta_{5} ) q^{76} + ( 139 \beta_{1} - 3984 \beta_{2} - 139 \beta_{3} - 366 \beta_{5} ) q^{77} + ( 806 - 195 \beta_{1} + 3445 \beta_{2} + 507 \beta_{3} + 130 \beta_{4} - 247 \beta_{5} ) q^{78} + ( 770 - 465 \beta_{1} - 465 \beta_{3} - 72 \beta_{4} ) q^{79} + ( -2055 + 29 \beta_{1} + 2055 \beta_{2} + 96 \beta_{4} - 96 \beta_{5} ) q^{80} + ( -2371 - 1206 \beta_{1} - 1206 \beta_{3} - 200 \beta_{4} ) q^{81} + ( -1242 \beta_{1} - 1250 \beta_{2} + 1242 \beta_{3} - 312 \beta_{5} ) q^{82} + ( 3549 - 514 \beta_{1} - 3549 \beta_{2} - 501 \beta_{4} + 501 \beta_{5} ) q^{83} + ( 827 + 243 \beta_{1} - 827 \beta_{2} + 424 \beta_{4} - 424 \beta_{5} ) q^{84} + ( 777 + 777 \beta_{2} - 489 \beta_{3} + 240 \beta_{4} + 240 \beta_{5} ) q^{85} + ( 5445 + 5445 \beta_{2} - 1615 \beta_{3} - 32 \beta_{4} - 32 \beta_{5} ) q^{86} + ( 5110 + 909 \beta_{1} + 909 \beta_{3} + 644 \beta_{4} ) q^{87} + ( -215 \beta_{1} - 3660 \beta_{2} + 215 \beta_{3} - 334 \beta_{5} ) q^{88} + ( -702 - 702 \beta_{2} + 1464 \beta_{3} + 149 \beta_{4} + 149 \beta_{5} ) q^{89} + ( -252 \beta_{1} - 1966 \beta_{2} + 252 \beta_{3} + 160 \beta_{5} ) q^{90} + ( -4433 - 1456 \beta_{1} + 3406 \beta_{2} - 832 \beta_{3} + 143 \beta_{4} + 468 \beta_{5} ) q^{91} + ( 1068 - 211 \beta_{1} - 211 \beta_{3} - 130 \beta_{4} ) q^{92} + ( -1210 - 402 \beta_{1} + 1210 \beta_{2} - 728 \beta_{4} + 728 \beta_{5} ) q^{93} + ( -7207 + 1377 \beta_{1} + 1377 \beta_{3} + 789 \beta_{4} ) q^{94} + ( -542 \beta_{1} + 642 \beta_{2} + 542 \beta_{3} - 434 \beta_{5} ) q^{95} + ( 2999 + 1101 \beta_{1} - 2999 \beta_{2} + 262 \beta_{4} - 262 \beta_{5} ) q^{96} + ( -4415 + 1870 \beta_{1} + 4415 \beta_{2} - 202 \beta_{4} + 202 \beta_{5} ) q^{97} + ( -7008 - 7008 \beta_{2} - 120 \beta_{3} + 431 \beta_{4} + 431 \beta_{5} ) q^{98} + ( 3314 + 3314 \beta_{2} - 1188 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 4q^{3} - 14q^{5} + 32q^{6} + 48q^{7} - 96q^{8} - 58q^{9} + O(q^{10}) \) \( 6q - 2q^{2} - 4q^{3} - 14q^{5} + 32q^{6} + 48q^{7} - 96q^{8} - 58q^{9} - 32q^{11} - 244q^{14} + 404q^{15} + 1044q^{16} - 802q^{18} + 732q^{19} + 428q^{20} - 2128q^{21} - 1632q^{22} - 24q^{24} + 910q^{26} + 236q^{27} + 1884q^{28} + 4184q^{29} - 3468q^{31} + 2092q^{32} + 2324q^{33} - 5304q^{34} - 4204q^{35} - 1758q^{37} + 1196q^{39} - 708q^{40} + 4750q^{41} + 9532q^{42} - 3956q^{44} + 830q^{45} + 516q^{46} - 6872q^{47} - 9436q^{48} - 322q^{50} + 3900q^{52} + 2108q^{53} - 184q^{54} + 6408q^{55} - 5800q^{57} + 6516q^{58} + 4372q^{59} + 1324q^{60} + 5988q^{61} - 652q^{63} - 5018q^{65} - 4592q^{66} + 72q^{67} - 10572q^{68} + 7368q^{70} - 14672q^{71} - 7980q^{72} + 5874q^{73} + 1544q^{74} + 3576q^{76} + 5720q^{78} + 2616q^{79} - 12080q^{80} - 19450q^{81} + 19264q^{83} + 6296q^{84} + 4164q^{85} + 29376q^{86} + 35584q^{87} - 986q^{89} - 30888q^{91} + 5304q^{92} - 9520q^{93} - 36156q^{94} + 20720q^{96} - 23154q^{97} - 41426q^{98} + 17492q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} - 12 x^{3} + 529 x^{2} - 1334 x + 1682\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -581 \nu^{5} + 437 \nu^{4} + 114 \nu^{3} - 11153 \nu^{2} - 302999 \nu + 392573 \)\()/424531\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{5} + 44 \nu^{4} - 625 \nu^{3} + 150 \nu^{2} - 13189 \nu + 33698 \)\()/14639\)
\(\beta_{4}\)\(=\)\((\)\( 31 \nu^{5} + 531 \nu^{4} + 775 \nu^{3} - 186 \nu^{2} - 1798 \nu + 173115 \)\()/14639\)
\(\beta_{5}\)\(=\)\((\)\( -9152 \nu^{5} + 6153 \nu^{4} + 20063 \nu^{3} + 230580 \nu^{2} - 4343971 \nu + 5696499 \)\()/424531\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{3} - 17 \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - 21 \beta_{3} + 12 \beta_{2} + 12\)
\(\nu^{4}\)\(=\)\(25 \beta_{4} + 31 \beta_{3} + 31 \beta_{1} - 367\)
\(\nu^{5}\)\(=\)\(-19 \beta_{5} + 19 \beta_{4} - 402 \beta_{2} - 479 \beta_{1} + 402\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
3.18200 3.18200i
1.30633 1.30633i
−3.48832 + 3.48832i
3.18200 + 3.18200i
1.30633 + 1.30633i
−3.48832 3.48832i
−3.18200 + 3.18200i −10.6142 4.25023i −9.43223 + 9.43223i 33.7744 33.7744i 54.8891 + 54.8891i −37.3878 37.3878i 31.6617 60.0267i
5.2 −1.30633 + 1.30633i 9.97438 12.5870i 9.28070 9.28070i −13.0298 + 13.0298i −46.1782 46.1782i −37.3440 37.3440i 18.4882 24.2472i
5.3 3.48832 3.48832i −1.36015 8.33680i −6.84848 + 6.84848i −4.74466 + 4.74466i 15.2891 + 15.2891i 26.7317 + 26.7317i −79.1500 47.7794i
8.1 −3.18200 3.18200i −10.6142 4.25023i −9.43223 9.43223i 33.7744 + 33.7744i 54.8891 54.8891i −37.3878 + 37.3878i 31.6617 60.0267i
8.2 −1.30633 1.30633i 9.97438 12.5870i 9.28070 + 9.28070i −13.0298 13.0298i −46.1782 + 46.1782i −37.3440 + 37.3440i 18.4882 24.2472i
8.3 3.48832 + 3.48832i −1.36015 8.33680i −6.84848 6.84848i −4.74466 4.74466i 15.2891 15.2891i 26.7317 26.7317i −79.1500 47.7794i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(13, [\chi])\).