Properties

Label 13.7.f.a
Level 13
Weight 7
Character orbit 13.f
Analytic conductor 2.991
Analytic rank 0
Dimension 24
CM No

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 13.f (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(2.99070308706\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 12q^{2} - 2q^{3} - 6q^{4} - 114q^{5} + 634q^{6} + 316q^{7} - 1398q^{8} - 1946q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 12q^{2} - 2q^{3} - 6q^{4} - 114q^{5} + 634q^{6} + 316q^{7} - 1398q^{8} - 1946q^{9} + 4674q^{10} + 228q^{11} - 1552q^{13} - 14616q^{14} + 7486q^{15} - 2138q^{16} + 20898q^{17} - 5630q^{18} - 14036q^{19} - 14634q^{20} - 314q^{21} + 37852q^{22} + 29634q^{23} + 47154q^{24} + 510q^{26} - 107060q^{27} - 40572q^{28} + 1026q^{29} - 247734q^{30} - 118116q^{31} + 180876q^{32} + 306046q^{33} + 101898q^{34} + 104862q^{35} + 105066q^{36} - 31470q^{37} + 28510q^{39} - 275328q^{40} - 357216q^{41} - 25636q^{42} - 480486q^{43} - 797712q^{44} + 464248q^{45} + 1012050q^{46} + 140028q^{47} + 676102q^{48} + 854562q^{49} + 756714q^{50} - 771704q^{52} - 1397076q^{53} - 1271048q^{54} - 205490q^{55} - 1461108q^{56} - 1483394q^{57} + 1133806q^{58} + 1568052q^{59} + 886808q^{60} + 768676q^{61} + 3741966q^{62} + 1336264q^{63} - 946968q^{65} - 5076928q^{66} - 1489892q^{67} - 569052q^{68} - 1475766q^{69} - 2626556q^{70} + 2136948q^{71} + 4787052q^{72} + 326326q^{73} + 2159718q^{74} + 3466008q^{75} + 1522158q^{76} - 1573544q^{78} - 1734344q^{79} - 5214300q^{80} - 773072q^{81} - 3540426q^{82} - 3642132q^{83} + 2952364q^{84} + 3509916q^{85} - 132324q^{86} + 531698q^{87} + 4478052q^{88} + 632754q^{89} - 2946788q^{91} - 1656060q^{92} - 295514q^{93} + 1552210q^{94} - 2753718q^{95} - 3979208q^{96} + 2258214q^{97} + 4984308q^{98} + 4336480q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −14.0703 3.77014i −17.8024 30.8347i 128.335 + 74.0942i −91.2022 + 91.2022i 134.235 + 500.973i 284.589 76.2554i −867.158 867.158i −269.354 + 466.535i 1627.09 939.401i
2.2 −9.37184 2.51118i 10.4091 + 18.0291i 26.0998 + 15.0687i 160.035 160.035i −52.2782 195.105i 162.036 43.4174i 232.320 + 232.320i 147.801 255.999i −1901.70 + 1097.95i
2.3 −4.77368 1.27910i 4.54257 + 7.86796i −34.2737 19.7879i −109.630 + 109.630i −11.6208 43.3696i −504.391 + 135.151i 361.954 + 361.954i 323.230 559.851i 663.567 383.110i
2.4 2.48669 + 0.666306i −16.9072 29.2841i −49.6860 28.6862i 34.0715 34.0715i −22.5307 84.0857i 190.342 51.0019i −220.944 220.944i −207.205 + 358.889i 107.427 62.0232i
2.5 5.89025 + 1.57829i 20.5701 + 35.6285i −23.2216 13.4070i −37.1940 + 37.1940i 64.9312 + 242.326i 588.562 157.705i −391.586 391.586i −481.762 + 834.436i −277.785 + 160.379i
2.6 12.5088 + 3.35172i −1.31220 2.27280i 89.8106 + 51.8522i 24.9454 24.9454i −8.79628 32.8282i −400.516 + 107.318i 363.576 + 363.576i 361.056 625.368i 395.648 228.427i
6.1 −2.92596 10.9198i 14.1081 24.4359i −55.2556 + 31.9018i −47.2751 + 47.2751i −308.115 82.5592i 27.7639 103.616i −1.56966 1.56966i −33.5751 58.1539i 654.560 + 377.911i
6.2 −2.74273 10.2360i −22.3986 + 38.7956i −41.8275 + 24.1491i −35.0169 + 35.0169i 458.545 + 122.867i −112.597 + 420.218i −117.658 117.658i −638.899 1106.61i 454.475 + 262.391i
6.3 −0.397723 1.48432i −0.503519 + 0.872121i 53.3806 30.8193i 84.5480 84.5480i 1.49477 + 0.400522i 41.0022 153.022i −136.519 136.519i 363.993 + 630.454i −159.123 91.8697i
6.4 1.76668 + 6.59335i −10.7542 + 18.6268i 15.0745 8.70327i −144.857 + 144.857i −141.812 37.9984i 49.5658 184.982i 392.922 + 392.922i 133.195 + 230.701i −1211.01 699.177i
6.5 1.99444 + 7.44337i 25.6304 44.3931i 3.99974 2.30925i −23.8066 + 23.8066i 381.553 + 102.237i −75.6362 + 282.278i 373.897 + 373.897i −949.333 1644.29i −224.682 129.720i
6.6 3.63541 + 13.5675i −6.58210 + 11.4005i −115.436 + 66.6469i 128.381 128.381i −178.606 47.8572i −92.7197 + 346.035i −688.234 688.234i 277.852 + 481.253i 2208.54 + 1275.10i
7.1 −14.0703 + 3.77014i −17.8024 + 30.8347i 128.335 74.0942i −91.2022 91.2022i 134.235 500.973i 284.589 + 76.2554i −867.158 + 867.158i −269.354 466.535i 1627.09 + 939.401i
7.2 −9.37184 + 2.51118i 10.4091 18.0291i 26.0998 15.0687i 160.035 + 160.035i −52.2782 + 195.105i 162.036 + 43.4174i 232.320 232.320i 147.801 + 255.999i −1901.70 1097.95i
7.3 −4.77368 + 1.27910i 4.54257 7.86796i −34.2737 + 19.7879i −109.630 109.630i −11.6208 + 43.3696i −504.391 135.151i 361.954 361.954i 323.230 + 559.851i 663.567 + 383.110i
7.4 2.48669 0.666306i −16.9072 + 29.2841i −49.6860 + 28.6862i 34.0715 + 34.0715i −22.5307 + 84.0857i 190.342 + 51.0019i −220.944 + 220.944i −207.205 358.889i 107.427 + 62.0232i
7.5 5.89025 1.57829i 20.5701 35.6285i −23.2216 + 13.4070i −37.1940 37.1940i 64.9312 242.326i 588.562 + 157.705i −391.586 + 391.586i −481.762 834.436i −277.785 160.379i
7.6 12.5088 3.35172i −1.31220 + 2.27280i 89.8106 51.8522i 24.9454 + 24.9454i −8.79628 + 32.8282i −400.516 107.318i 363.576 363.576i 361.056 + 625.368i 395.648 + 228.427i
11.1 −2.92596 + 10.9198i 14.1081 + 24.4359i −55.2556 31.9018i −47.2751 47.2751i −308.115 + 82.5592i 27.7639 + 103.616i −1.56966 + 1.56966i −33.5751 + 58.1539i 654.560 377.911i
11.2 −2.74273 + 10.2360i −22.3986 38.7956i −41.8275 24.1491i −35.0169 35.0169i 458.545 122.867i −112.597 420.218i −117.658 + 117.658i −638.899 + 1106.61i 454.475 262.391i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(13, [\chi])\).