Properties

Label 13.4.a.b
Level 13
Weight 4
Character orbit 13.a
Self dual Yes
Analytic conductor 0.767
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.767024830075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + ( 4 - 3 \beta ) q^{3} \) \( + ( -4 + \beta ) q^{4} \) \( + ( -2 + \beta ) q^{5} \) \( + ( -12 + \beta ) q^{6} \) \( + ( -10 + 11 \beta ) q^{7} \) \( + ( 4 - 11 \beta ) q^{8} \) \( + ( 25 - 15 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + ( 4 - 3 \beta ) q^{3} \) \( + ( -4 + \beta ) q^{4} \) \( + ( -2 + \beta ) q^{5} \) \( + ( -12 + \beta ) q^{6} \) \( + ( -10 + 11 \beta ) q^{7} \) \( + ( 4 - 11 \beta ) q^{8} \) \( + ( 25 - 15 \beta ) q^{9} \) \( + ( 4 - \beta ) q^{10} \) \( + ( 34 + 12 \beta ) q^{11} \) \( + ( -28 + 13 \beta ) q^{12} \) \( -13 q^{13} \) \( + ( 44 + \beta ) q^{14} \) \( + ( -20 + 7 \beta ) q^{15} \) \( + ( -12 - 15 \beta ) q^{16} \) \( + ( 18 - 17 \beta ) q^{17} \) \( + ( -60 + 10 \beta ) q^{18} \) \( + ( -26 - 32 \beta ) q^{19} \) \( + ( 12 - 5 \beta ) q^{20} \) \( + ( -172 + 41 \beta ) q^{21} \) \( + ( 48 + 46 \beta ) q^{22} \) \( + ( 104 - 12 \beta ) q^{23} \) \( + ( 148 - 23 \beta ) q^{24} \) \( + ( -117 - 3 \beta ) q^{25} \) \( -13 \beta q^{26} \) \( + ( 172 - 9 \beta ) q^{27} \) \( + ( 84 - 43 \beta ) q^{28} \) \( + ( -70 + 96 \beta ) q^{29} \) \( + ( 28 - 13 \beta ) q^{30} \) \( + ( -26 - 34 \beta ) q^{31} \) \( + ( -92 + 61 \beta ) q^{32} \) \( + ( -8 - 90 \beta ) q^{33} \) \( + ( -68 + \beta ) q^{34} \) \( + ( 64 - 21 \beta ) q^{35} \) \( + ( -160 + 70 \beta ) q^{36} \) \( + ( 102 + 5 \beta ) q^{37} \) \( + ( -128 - 58 \beta ) q^{38} \) \( + ( -52 + 39 \beta ) q^{39} \) \( + ( -52 + 15 \beta ) q^{40} \) \( + ( -126 + 22 \beta ) q^{41} \) \( + ( 164 - 131 \beta ) q^{42} \) \( + ( 72 + 143 \beta ) q^{43} \) \( + ( -88 - 2 \beta ) q^{44} \) \( + ( -110 + 40 \beta ) q^{45} \) \( + ( -48 + 92 \beta ) q^{46} \) \( + ( 278 - 121 \beta ) q^{47} \) \( + ( 132 + 21 \beta ) q^{48} \) \( + ( 241 - 99 \beta ) q^{49} \) \( + ( -12 - 120 \beta ) q^{50} \) \( + ( 276 - 71 \beta ) q^{51} \) \( + ( 52 - 13 \beta ) q^{52} \) \( + ( -74 + 30 \beta ) q^{53} \) \( + ( -36 + 163 \beta ) q^{54} \) \( + ( -20 + 22 \beta ) q^{55} \) \( + ( -524 + 33 \beta ) q^{56} \) \( + ( 280 + 46 \beta ) q^{57} \) \( + ( 384 + 26 \beta ) q^{58} \) \( + ( -246 + 124 \beta ) q^{59} \) \( + ( 108 - 41 \beta ) q^{60} \) \( + ( -434 - 190 \beta ) q^{61} \) \( + ( -136 - 60 \beta ) q^{62} \) \( + ( -910 + 260 \beta ) q^{63} \) \( + ( 340 + 89 \beta ) q^{64} \) \( + ( 26 - 13 \beta ) q^{65} \) \( + ( -360 - 98 \beta ) q^{66} \) \( + ( 150 - 232 \beta ) q^{67} \) \( + ( -140 + 69 \beta ) q^{68} \) \( + ( 560 - 324 \beta ) q^{69} \) \( + ( -84 + 43 \beta ) q^{70} \) \( + ( 50 - 231 \beta ) q^{71} \) \( + ( 760 - 170 \beta ) q^{72} \) \( + ( 98 + 260 \beta ) q^{73} \) \( + ( 20 + 107 \beta ) q^{74} \) \( + ( -432 + 348 \beta ) q^{75} \) \( + ( -24 + 70 \beta ) q^{76} \) \( + ( 188 + 386 \beta ) q^{77} \) \( + ( 156 - 13 \beta ) q^{78} \) \( + ( -524 + 40 \beta ) q^{79} \) \( + ( -36 + 3 \beta ) q^{80} \) \( + ( 121 - 120 \beta ) q^{81} \) \( + ( 88 - 104 \beta ) q^{82} \) \( + ( 1070 - 182 \beta ) q^{83} \) \( + ( 852 - 295 \beta ) q^{84} \) \( + ( -104 + 35 \beta ) q^{85} \) \( + ( 572 + 215 \beta ) q^{86} \) \( + ( -1432 + 306 \beta ) q^{87} \) \( + ( -392 - 458 \beta ) q^{88} \) \( + ( -166 - 388 \beta ) q^{89} \) \( + ( 160 - 70 \beta ) q^{90} \) \( + ( 130 - 143 \beta ) q^{91} \) \( + ( -464 + 140 \beta ) q^{92} \) \( + ( 304 + 44 \beta ) q^{93} \) \( + ( -484 + 157 \beta ) q^{94} \) \( + ( -76 + 6 \beta ) q^{95} \) \( + ( -1100 + 337 \beta ) q^{96} \) \( + ( -718 + 508 \beta ) q^{97} \) \( + ( -396 + 142 \beta ) q^{98} \) \( + ( 130 - 390 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 23q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 35q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 7q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 23q^{6} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 35q^{9} \) \(\mathstrut +\mathstrut 7q^{10} \) \(\mathstrut +\mathstrut 80q^{11} \) \(\mathstrut -\mathstrut 43q^{12} \) \(\mathstrut -\mathstrut 26q^{13} \) \(\mathstrut +\mathstrut 89q^{14} \) \(\mathstrut -\mathstrut 33q^{15} \) \(\mathstrut -\mathstrut 39q^{16} \) \(\mathstrut +\mathstrut 19q^{17} \) \(\mathstrut -\mathstrut 110q^{18} \) \(\mathstrut -\mathstrut 84q^{19} \) \(\mathstrut +\mathstrut 19q^{20} \) \(\mathstrut -\mathstrut 303q^{21} \) \(\mathstrut +\mathstrut 142q^{22} \) \(\mathstrut +\mathstrut 196q^{23} \) \(\mathstrut +\mathstrut 273q^{24} \) \(\mathstrut -\mathstrut 237q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut +\mathstrut 335q^{27} \) \(\mathstrut +\mathstrut 125q^{28} \) \(\mathstrut -\mathstrut 44q^{29} \) \(\mathstrut +\mathstrut 43q^{30} \) \(\mathstrut -\mathstrut 86q^{31} \) \(\mathstrut -\mathstrut 123q^{32} \) \(\mathstrut -\mathstrut 106q^{33} \) \(\mathstrut -\mathstrut 135q^{34} \) \(\mathstrut +\mathstrut 107q^{35} \) \(\mathstrut -\mathstrut 250q^{36} \) \(\mathstrut +\mathstrut 209q^{37} \) \(\mathstrut -\mathstrut 314q^{38} \) \(\mathstrut -\mathstrut 65q^{39} \) \(\mathstrut -\mathstrut 89q^{40} \) \(\mathstrut -\mathstrut 230q^{41} \) \(\mathstrut +\mathstrut 197q^{42} \) \(\mathstrut +\mathstrut 287q^{43} \) \(\mathstrut -\mathstrut 178q^{44} \) \(\mathstrut -\mathstrut 180q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 435q^{47} \) \(\mathstrut +\mathstrut 285q^{48} \) \(\mathstrut +\mathstrut 383q^{49} \) \(\mathstrut -\mathstrut 144q^{50} \) \(\mathstrut +\mathstrut 481q^{51} \) \(\mathstrut +\mathstrut 91q^{52} \) \(\mathstrut -\mathstrut 118q^{53} \) \(\mathstrut +\mathstrut 91q^{54} \) \(\mathstrut -\mathstrut 18q^{55} \) \(\mathstrut -\mathstrut 1015q^{56} \) \(\mathstrut +\mathstrut 606q^{57} \) \(\mathstrut +\mathstrut 794q^{58} \) \(\mathstrut -\mathstrut 368q^{59} \) \(\mathstrut +\mathstrut 175q^{60} \) \(\mathstrut -\mathstrut 1058q^{61} \) \(\mathstrut -\mathstrut 332q^{62} \) \(\mathstrut -\mathstrut 1560q^{63} \) \(\mathstrut +\mathstrut 769q^{64} \) \(\mathstrut +\mathstrut 39q^{65} \) \(\mathstrut -\mathstrut 818q^{66} \) \(\mathstrut +\mathstrut 68q^{67} \) \(\mathstrut -\mathstrut 211q^{68} \) \(\mathstrut +\mathstrut 796q^{69} \) \(\mathstrut -\mathstrut 125q^{70} \) \(\mathstrut -\mathstrut 131q^{71} \) \(\mathstrut +\mathstrut 1350q^{72} \) \(\mathstrut +\mathstrut 456q^{73} \) \(\mathstrut +\mathstrut 147q^{74} \) \(\mathstrut -\mathstrut 516q^{75} \) \(\mathstrut +\mathstrut 22q^{76} \) \(\mathstrut +\mathstrut 762q^{77} \) \(\mathstrut +\mathstrut 299q^{78} \) \(\mathstrut -\mathstrut 1008q^{79} \) \(\mathstrut -\mathstrut 69q^{80} \) \(\mathstrut +\mathstrut 122q^{81} \) \(\mathstrut +\mathstrut 72q^{82} \) \(\mathstrut +\mathstrut 1958q^{83} \) \(\mathstrut +\mathstrut 1409q^{84} \) \(\mathstrut -\mathstrut 173q^{85} \) \(\mathstrut +\mathstrut 1359q^{86} \) \(\mathstrut -\mathstrut 2558q^{87} \) \(\mathstrut -\mathstrut 1242q^{88} \) \(\mathstrut -\mathstrut 720q^{89} \) \(\mathstrut +\mathstrut 250q^{90} \) \(\mathstrut +\mathstrut 117q^{91} \) \(\mathstrut -\mathstrut 788q^{92} \) \(\mathstrut +\mathstrut 652q^{93} \) \(\mathstrut -\mathstrut 811q^{94} \) \(\mathstrut -\mathstrut 146q^{95} \) \(\mathstrut -\mathstrut 1863q^{96} \) \(\mathstrut -\mathstrut 928q^{97} \) \(\mathstrut -\mathstrut 650q^{98} \) \(\mathstrut -\mathstrut 130q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
−1.56155 8.68466 −5.56155 −3.56155 −13.5616 −27.1771 21.1771 48.4233 5.56155
1.2 2.56155 −3.68466 −1.43845 0.561553 −9.43845 18.1771 −24.1771 −13.4233 1.43845
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut -\mathstrut 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\).