Properties

Label 76.7.g.a
Level $76$
Weight $7$
Character orbit 76.g
Analytic conductor $17.484$
Analytic rank $0$
Dimension $116$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116q - q^{2} - 67q^{4} - 2q^{5} + 33q^{6} + 962q^{8} + 12660q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 116q - q^{2} - 67q^{4} - 2q^{5} + 33q^{6} + 962q^{8} + 12660q^{9} - 812q^{10} - 1462q^{12} + 2518q^{13} - 2742q^{14} + 461q^{16} + 6838q^{17} - 13948q^{18} - 24396q^{20} + 11176q^{21} + 22293q^{22} + 33905q^{24} - 156252q^{25} - 78080q^{26} - 34362q^{28} - 2q^{29} + 208344q^{30} - 10941q^{32} + 110316q^{33} + 64648q^{34} + 78574q^{36} + 55432q^{37} + 221994q^{38} + 242590q^{40} - 9126q^{41} + 187550q^{42} + 138081q^{44} + 59576q^{45} - 686052q^{46} - 141617q^{48} - 1306636q^{49} + 167498q^{50} - 35414q^{52} + 60758q^{53} - 348601q^{54} + 543732q^{56} - 466566q^{57} + 1208536q^{58} - 319142q^{60} - 839474q^{61} + 628548q^{62} - 470026q^{64} + 151836q^{65} + 1471119q^{66} + 341484q^{68} - 2169836q^{69} - 68364q^{70} + 1236042q^{72} + 207538q^{73} + 336144q^{74} - 97437q^{76} + 1977632q^{77} - 2740948q^{78} + 1250050q^{80} - 3334682q^{81} - 2244893q^{82} - 2404612q^{84} + 788426q^{85} + 910490q^{86} + 5123058q^{88} + 360166q^{89} + 2446616q^{90} + 2518692q^{92} + 2079592q^{93} - 2221200q^{94} - 5927450q^{96} + 3567778q^{97} + 5270703q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −7.98715 + 0.453260i 2.84653 1.64344i 63.5891 7.24051i −102.375 177.320i −21.9907 + 14.4167i 574.394i −504.614 + 86.6535i −359.098 + 621.976i 898.060 + 1369.87i
7.2 −7.98630 + 0.467925i −26.4172 + 15.2520i 63.5621 7.47398i −21.3081 36.9067i 203.839 134.168i 91.3832i −504.129 + 89.4318i 100.747 174.500i 187.442 + 284.777i
7.3 −7.94531 + 0.933840i 13.1287 7.57984i 62.2559 14.8393i 69.1678 + 119.802i −97.2330 + 72.4842i 258.851i −480.785 + 176.040i −249.592 + 432.306i −661.436 887.274i
7.4 −7.83923 1.59575i 36.3885 21.0089i 58.9072 + 25.0189i 30.8964 + 53.5142i −318.783 + 106.627i 234.954i −421.863 290.130i 518.249 897.634i −156.809 468.813i
7.5 −7.69093 2.20219i 30.2193 17.4471i 54.3007 + 33.8737i −100.842 174.663i −270.837 + 67.6361i 567.863i −343.027 380.101i 244.306 423.150i 390.926 + 1565.39i
7.6 −7.63062 + 2.40284i −38.8799 + 22.4473i 52.4527 36.6703i 102.646 + 177.787i 242.740 264.709i 16.9433i −312.134 + 405.852i 643.264 1114.17i −1210.44 1109.99i
7.7 −7.48364 2.82757i −21.0608 + 12.1595i 48.0097 + 42.3210i 39.6150 + 68.6152i 191.993 31.4462i 411.817i −239.622 452.466i −68.7942 + 119.155i −102.450 625.506i
7.8 −7.07398 + 3.73615i −0.173689 + 0.100280i 36.0824 52.8588i −61.9026 107.218i 0.854016 1.35830i 472.399i −57.7582 + 508.732i −364.480 + 631.298i 838.481 + 527.184i
7.9 −7.00051 3.87205i −41.5950 + 24.0149i 34.0144 + 54.2127i −83.5383 144.693i 384.173 7.05848i 406.670i −28.2037 511.223i 788.930 1366.47i 24.5537 + 1336.39i
7.10 −6.97428 + 3.91912i 22.3528 12.9054i 33.2810 54.6660i 17.9822 + 31.1462i −105.317 + 177.609i 75.3694i −17.8687 + 511.688i −31.4026 + 54.3909i −247.479 146.747i
7.11 −6.73237 4.32147i 0.162821 0.0940049i 26.6497 + 58.1876i −28.3957 49.1827i −1.50241 0.0707515i 19.6656i 72.0399 506.907i −364.482 + 631.302i −21.3716 + 453.828i
7.12 −6.65177 4.44454i −5.80264 + 3.35015i 24.4921 + 59.1281i 84.8928 + 147.039i 53.4877 + 3.50559i 460.277i 99.8813 502.163i −342.053 + 592.453i 88.8316 1355.38i
7.13 −6.29518 + 4.93667i 44.1231 25.4745i 15.2586 62.1545i −59.7633 103.513i −152.004 + 378.187i 116.972i 210.781 + 466.600i 933.397 1616.69i 887.230 + 356.601i
7.14 −5.97766 + 5.31672i −4.45729 + 2.57342i 7.46487 63.5632i 64.1001 + 111.025i 12.9620 39.0812i 508.357i 293.325 + 419.648i −351.255 + 608.392i −973.457 322.865i
7.15 −5.68771 + 5.62583i −33.2985 + 19.2249i 0.700063 63.9962i −72.8107 126.112i 81.2362 296.678i 173.537i 356.050 + 367.930i 374.695 648.990i 1123.61 + 307.667i
7.16 −4.96370 6.27389i 28.5054 16.4576i −14.7234 + 62.2834i 89.7900 + 155.521i −244.746 97.1494i 293.448i 463.842 216.783i 177.207 306.931i 530.031 1335.29i
7.17 −4.82765 6.37917i 18.7719 10.8380i −17.3875 + 61.5928i −60.5457 104.868i −159.762 67.4273i 211.698i 476.852 186.431i −129.576 + 224.432i −376.678 + 892.498i
7.18 −4.43934 6.65524i −40.1020 + 23.1529i −24.5845 + 59.0898i 19.2234 + 33.2960i 332.115 + 164.105i 622.494i 502.396 98.7039i 707.614 1225.62i 136.253 275.749i
7.19 −4.27334 + 6.76303i −22.3261 + 12.8900i −27.4772 57.8014i 38.8259 + 67.2485i 8.23162 206.075i 608.804i 508.332 + 61.1760i −32.1974 + 55.7675i −620.720 24.7945i
7.20 −3.54394 7.17220i 40.1020 23.1529i −38.8810 + 50.8357i 19.2234 + 33.2960i −308.176 205.567i 622.494i 502.396 + 98.7039i 707.614 1225.62i 170.679 255.873i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.g.a 116
4.b odd 2 1 inner 76.7.g.a 116
19.c even 3 1 inner 76.7.g.a 116
76.g odd 6 1 inner 76.7.g.a 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.g.a 116 1.a even 1 1 trivial
76.7.g.a 116 4.b odd 2 1 inner
76.7.g.a 116 19.c even 3 1 inner
76.7.g.a 116 76.g odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(76, [\chi])\).