Properties

Label 13.3.d.a
Level 13
Weight 3
Character orbit 13.d
Analytic conductor 0.354
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 13.d (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.354224343668\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} \) \( + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{6} \) \( + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{7} \) \( + ( 9 - 9 \beta_{2} + 3 \beta_{3} ) q^{8} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{2} \) \( + ( -1 - \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{5} \) \( + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{6} \) \( + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{7} \) \( + ( 9 - 9 \beta_{2} + 3 \beta_{3} ) q^{8} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{9} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{10} \) \( + ( -1 + \beta_{2} - 4 \beta_{3} ) q^{11} \) \( + ( -\beta_{1} + 17 \beta_{2} - \beta_{3} ) q^{12} \) \( + ( 2 - 3 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} ) q^{13} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{14} \) \( + ( -7 - 5 \beta_{1} - 7 \beta_{2} ) q^{15} \) \( + ( -21 + 4 \beta_{1} - 4 \beta_{3} ) q^{16} \) \( + ( 6 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{17} \) \( + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{18} \) \( + 2 \beta_{1} q^{19} \) \( + ( 4 - 4 \beta_{2} - 5 \beta_{3} ) q^{20} \) \( + ( 8 - 8 \beta_{2} - 7 \beta_{3} ) q^{21} \) \( + ( 22 - 5 \beta_{1} + 5 \beta_{3} ) q^{22} \) \( + ( -3 \beta_{1} + 18 \beta_{2} - 3 \beta_{3} ) q^{23} \) \( + ( 6 - 6 \beta_{2} + 15 \beta_{3} ) q^{24} \) \( + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{25} \) \( + ( -22 + 7 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} ) q^{26} \) \( + ( -13 + 5 \beta_{1} - 5 \beta_{3} ) q^{27} \) \( + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{28} \) \( + ( 10 - 5 \beta_{1} + 5 \beta_{3} ) q^{29} \) \( + ( -2 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} ) q^{30} \) \( + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{31} \) \( + ( 5 - 17 \beta_{1} + 5 \beta_{2} ) q^{32} \) \( + ( -19 + 19 \beta_{2} + 2 \beta_{3} ) q^{33} \) \( + ( -27 + 27 \beta_{2} - 9 \beta_{3} ) q^{34} \) \( + ( -17 - 5 \beta_{1} + 5 \beta_{3} ) q^{35} \) \( + ( 2 \beta_{1} - 34 \beta_{2} + 2 \beta_{3} ) q^{36} \) \( + ( 10 - 10 \beta_{2} - 9 \beta_{3} ) q^{37} \) \( + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{38} \) \( + ( 23 + 11 \beta_{1} + 15 \beta_{2} + 10 \beta_{3} ) q^{39} \) \( + ( 21 + 3 \beta_{1} - 3 \beta_{3} ) q^{40} \) \( + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{41} \) \( + ( 19 + \beta_{1} - \beta_{3} ) q^{42} \) \( + ( -3 \beta_{1} - 21 \beta_{2} - 3 \beta_{3} ) q^{43} \) \( + ( -43 + 16 \beta_{1} - 43 \beta_{2} ) q^{44} \) \( + ( 14 + 10 \beta_{1} + 14 \beta_{2} ) q^{45} \) \( + ( 33 - 33 \beta_{2} + 24 \beta_{3} ) q^{46} \) \( + ( -1 + \beta_{2} + 23 \beta_{3} ) q^{47} \) \( + ( -19 + 17 \beta_{1} - 17 \beta_{3} ) q^{48} \) \( + ( -6 \beta_{1} + 26 \beta_{2} - 6 \beta_{3} ) q^{49} \) \( + ( -32 + 32 \beta_{2} - 20 \beta_{3} ) q^{50} \) \( + ( -9 \beta_{1} - 63 \beta_{2} - 9 \beta_{3} ) q^{51} \) \( + ( 20 - 30 \beta_{1} + 56 \beta_{2} + 7 \beta_{3} ) q^{52} \) \( + ( -20 - 5 \beta_{1} + 5 \beta_{3} ) q^{53} \) \( + ( 38 - 23 \beta_{1} + 38 \beta_{2} ) q^{54} \) \( + ( 16 + 7 \beta_{1} - 7 \beta_{3} ) q^{55} \) \( + 39 \beta_{2} q^{56} \) \( + ( -10 - 2 \beta_{1} - 10 \beta_{2} ) q^{57} \) \( + ( -35 + 20 \beta_{1} - 35 \beta_{2} ) q^{58} \) \( + ( 14 - 14 \beta_{2} - 28 \beta_{3} ) q^{59} \) \( + ( -29 + 29 \beta_{2} + 13 \beta_{3} ) q^{60} \) \( + ( -74 - 2 \beta_{1} + 2 \beta_{3} ) q^{61} \) \( + ( 16 \beta_{1} - 50 \beta_{2} + 16 \beta_{3} ) q^{62} \) \( + ( -16 + 16 \beta_{2} + 14 \beta_{3} ) q^{63} \) \( + ( 6 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} ) q^{64} \) \( + ( 14 - 8 \beta_{1} - 31 \beta_{2} - 12 \beta_{3} ) q^{65} \) \( + ( 28 - 17 \beta_{1} + 17 \beta_{3} ) q^{66} \) \( + ( -21 + 38 \beta_{1} - 21 \beta_{2} ) q^{67} \) \( + ( 111 - 12 \beta_{1} + 12 \beta_{3} ) q^{68} \) \( + ( -15 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} ) q^{69} \) \( + ( -8 - 7 \beta_{1} - 8 \beta_{2} ) q^{70} \) \( + ( 71 + 13 \beta_{1} + 71 \beta_{2} ) q^{71} \) \( + ( -12 + 12 \beta_{2} - 30 \beta_{3} ) q^{72} \) \( -20 \beta_{3} q^{73} \) \( + ( 25 + \beta_{1} - \beta_{3} ) q^{74} \) \( + ( 8 \beta_{1} - 28 \beta_{2} + 8 \beta_{3} ) q^{75} \) \( + ( 20 - 20 \beta_{2} + 6 \beta_{3} ) q^{76} \) \( + ( 11 \beta_{1} + 14 \beta_{2} + 11 \beta_{3} ) q^{77} \) \( + ( -58 + 22 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{78} \) \( + ( 16 - 11 \beta_{1} + 11 \beta_{3} ) q^{79} \) \( + ( -22 - 5 \beta_{1} - 22 \beta_{2} ) q^{80} \) \( + ( -55 - 10 \beta_{1} + 10 \beta_{3} ) q^{81} \) \( + ( 10 \beta_{1} - 26 \beta_{2} + 10 \beta_{3} ) q^{82} \) \( + ( -13 - 20 \beta_{1} - 13 \beta_{2} ) q^{83} \) \( + ( -46 - 11 \beta_{1} - 46 \beta_{2} ) q^{84} \) \( + ( -36 + 36 \beta_{2} + 27 \beta_{3} ) q^{85} \) \( + ( -6 + 6 \beta_{2} - 15 \beta_{3} ) q^{86} \) \( + ( 40 - 5 \beta_{1} + 5 \beta_{3} ) q^{87} \) \( + ( -39 \beta_{1} + 78 \beta_{2} - 39 \beta_{3} ) q^{88} \) \( + ( 50 - 50 \beta_{2} + 26 \beta_{3} ) q^{89} \) \( + ( 4 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} ) q^{90} \) \( + ( 39 + 13 \beta_{1} + 26 \beta_{2} - 13 \beta_{3} ) q^{91} \) \( + ( -114 + 45 \beta_{1} - 45 \beta_{3} ) q^{92} \) \( + ( 20 - 14 \beta_{1} + 20 \beta_{2} ) q^{93} \) \( + ( -113 + 22 \beta_{1} - 22 \beta_{3} ) q^{94} \) \( + ( 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} ) q^{95} \) \( + ( 80 + 7 \beta_{1} + 80 \beta_{2} ) q^{96} \) \( + ( -17 - 66 \beta_{1} - 17 \beta_{2} ) q^{97} \) \( + ( 56 - 56 \beta_{2} + 38 \beta_{3} ) q^{98} \) \( + ( 38 - 38 \beta_{2} - 4 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 28q^{15} \) \(\mathstrut -\mathstrut 84q^{16} \) \(\mathstrut +\mathstrut 32q^{18} \) \(\mathstrut +\mathstrut 16q^{20} \) \(\mathstrut +\mathstrut 32q^{21} \) \(\mathstrut +\mathstrut 88q^{22} \) \(\mathstrut +\mathstrut 24q^{24} \) \(\mathstrut -\mathstrut 88q^{26} \) \(\mathstrut -\mathstrut 52q^{27} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 40q^{29} \) \(\mathstrut +\mathstrut 40q^{31} \) \(\mathstrut +\mathstrut 20q^{32} \) \(\mathstrut -\mathstrut 76q^{33} \) \(\mathstrut -\mathstrut 108q^{34} \) \(\mathstrut -\mathstrut 68q^{35} \) \(\mathstrut +\mathstrut 40q^{37} \) \(\mathstrut +\mathstrut 92q^{39} \) \(\mathstrut +\mathstrut 84q^{40} \) \(\mathstrut +\mathstrut 32q^{41} \) \(\mathstrut +\mathstrut 76q^{42} \) \(\mathstrut -\mathstrut 172q^{44} \) \(\mathstrut +\mathstrut 56q^{45} \) \(\mathstrut +\mathstrut 132q^{46} \) \(\mathstrut -\mathstrut 4q^{47} \) \(\mathstrut -\mathstrut 76q^{48} \) \(\mathstrut -\mathstrut 128q^{50} \) \(\mathstrut +\mathstrut 80q^{52} \) \(\mathstrut -\mathstrut 80q^{53} \) \(\mathstrut +\mathstrut 152q^{54} \) \(\mathstrut +\mathstrut 64q^{55} \) \(\mathstrut -\mathstrut 40q^{57} \) \(\mathstrut -\mathstrut 140q^{58} \) \(\mathstrut +\mathstrut 56q^{59} \) \(\mathstrut -\mathstrut 116q^{60} \) \(\mathstrut -\mathstrut 296q^{61} \) \(\mathstrut -\mathstrut 64q^{63} \) \(\mathstrut +\mathstrut 56q^{65} \) \(\mathstrut +\mathstrut 112q^{66} \) \(\mathstrut -\mathstrut 84q^{67} \) \(\mathstrut +\mathstrut 444q^{68} \) \(\mathstrut -\mathstrut 32q^{70} \) \(\mathstrut +\mathstrut 284q^{71} \) \(\mathstrut -\mathstrut 48q^{72} \) \(\mathstrut +\mathstrut 100q^{74} \) \(\mathstrut +\mathstrut 80q^{76} \) \(\mathstrut -\mathstrut 232q^{78} \) \(\mathstrut +\mathstrut 64q^{79} \) \(\mathstrut -\mathstrut 88q^{80} \) \(\mathstrut -\mathstrut 220q^{81} \) \(\mathstrut -\mathstrut 52q^{83} \) \(\mathstrut -\mathstrut 184q^{84} \) \(\mathstrut -\mathstrut 144q^{85} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut +\mathstrut 160q^{87} \) \(\mathstrut +\mathstrut 200q^{89} \) \(\mathstrut +\mathstrut 156q^{91} \) \(\mathstrut -\mathstrut 456q^{92} \) \(\mathstrut +\mathstrut 80q^{93} \) \(\mathstrut -\mathstrut 452q^{94} \) \(\mathstrut +\mathstrut 320q^{96} \) \(\mathstrut -\mathstrut 68q^{97} \) \(\mathstrut +\mathstrut 224q^{98} \) \(\mathstrut +\mathstrut 152q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(5\) \(\beta_{3}\)

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
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
−2.58114 + 2.58114i 2.16228 9.32456i 0.418861 0.418861i −5.58114 + 5.58114i −1.41886 1.41886i 13.7434 + 13.7434i −4.32456 2.16228i
5.2 0.581139 0.581139i −4.16228 3.32456i 3.58114 3.58114i −2.41886 + 2.41886i −4.58114 4.58114i 4.25658 + 4.25658i 8.32456 4.16228i
8.1 −2.58114 2.58114i 2.16228 9.32456i 0.418861 + 0.418861i −5.58114 5.58114i −1.41886 + 1.41886i 13.7434 13.7434i −4.32456 2.16228i
8.2 0.581139 + 0.581139i −4.16228 3.32456i 3.58114 + 3.58114i −2.41886 2.41886i −4.58114 + 4.58114i 4.25658 4.25658i 8.32456 4.16228i
\(n\): 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.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(13, [\chi])\).