# Properties

 Label 76.7.l.a Level $76$ Weight $7$ Character orbit 76.l Analytic conductor $17.484$ Analytic rank $0$ Dimension $348$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 76.l (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.4841103551$$ Analytic rank: $$0$$ Dimension: $$348$$ Relative dimension: $$58$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$348q - 6q^{2} + 192q^{4} - 12q^{5} - 492q^{6} - 3q^{8} + 1368q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$348q - 6q^{2} + 192q^{4} - 12q^{5} - 492q^{6} - 3q^{8} + 1368q^{9} + 2811q^{10} - 3q^{12} - 7572q^{13} + 8217q^{14} - 1392q^{16} - 12q^{17} - 12q^{18} + 33618q^{20} - 33546q^{21} - 198q^{22} - 7128q^{24} - 12q^{25} + 84453q^{26} + 13464q^{28} - 12q^{29} + 75510q^{30} + 9339q^{32} - 366q^{33} - 289248q^{34} - 419349q^{36} - 24q^{37} + 158040q^{38} + 264570q^{40} + 27360q^{41} + 206001q^{42} - 525639q^{44} - 6q^{45} - 269328q^{46} - 703179q^{48} + 2319360q^{49} + 264534q^{50} + 130809q^{52} - 182292q^{53} + 607083q^{54} - 705906q^{56} - 12q^{57} - 710004q^{58} + 1060416q^{60} - 325092q^{61} - 1967868q^{62} - 1611921q^{64} + 398994q^{65} - 2295684q^{66} + 1125642q^{68} - 1625190q^{69} + 3496059q^{70} + 4948626q^{72} - 2224812q^{73} + 2661159q^{74} - 968262q^{76} + 974412q^{77} - 340743q^{78} - 3996663q^{80} + 4277754q^{81} - 3214221q^{82} - 872109q^{84} - 3207900q^{85} + 3189360q^{86} + 827973q^{88} - 2016084q^{89} - 710406q^{90} - 2545116q^{92} + 4486254q^{93} - 14322630q^{94} + 2623530q^{96} - 1223112q^{97} + 10989009q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −7.99410 + 0.307308i −27.6808 + 4.88087i 63.8111 4.91330i −34.1068 + 28.6190i 219.783 47.5247i −83.9559 48.4720i −508.602 + 58.8871i 57.3665 20.8797i 263.858 239.264i
23.2 −7.91695 1.14977i 3.83980 0.677060i 61.3561 + 18.2054i −157.228 + 131.930i −31.1780 + 0.945358i 546.642 + 315.604i −464.821 214.676i −670.750 + 244.133i 1396.45 863.704i
23.3 −7.84835 + 1.55031i −22.9636 + 4.04910i 59.1931 24.3348i −26.2597 + 22.0345i 173.949 67.3795i −137.300 79.2704i −426.841 + 282.756i −174.105 + 63.3690i 171.935 213.645i
23.4 −7.76312 1.93236i 51.9029 9.15188i 56.5319 + 30.0023i −46.4533 + 38.9790i −420.613 29.2482i 69.3277 + 40.0264i −380.888 342.152i 1925.12 700.686i 435.944 212.834i
23.5 −7.72475 2.08045i 16.9114 2.98194i 55.3435 + 32.1418i 113.326 95.0915i −136.840 12.1485i −331.831 191.583i −360.645 363.427i −407.932 + 148.475i −1073.25 + 498.790i
23.6 −7.71608 + 2.11236i 26.4348 4.66117i 55.0759 32.5982i 6.31656 5.30022i −194.127 + 91.8057i −142.890 82.4974i −356.111 + 367.871i −7.96412 + 2.89870i −37.5431 + 54.2398i
23.7 −7.31367 + 3.24195i 23.5223 4.14762i 42.9795 47.4211i 104.114 87.3621i −158.588 + 106.593i 538.554 + 310.935i −160.601 + 486.160i −148.939 + 54.2093i −478.233 + 976.470i
23.8 −7.25479 3.37166i −16.9114 + 2.98194i 41.2638 + 48.9213i 113.326 95.0915i 132.743 + 35.3862i 331.831 + 191.583i −134.414 494.041i −407.932 + 148.475i −1142.77 + 307.773i
23.9 −7.25120 + 3.37936i −38.7503 + 6.83273i 41.1598 49.0089i 178.137 149.475i 257.896 180.497i −72.9496 42.1175i −132.840 + 494.467i 769.865 280.208i −786.579 + 1685.86i
23.10 −7.18899 3.50976i −51.9029 + 9.15188i 39.3632 + 50.4632i −46.4533 + 38.9790i 405.250 + 116.374i −69.3277 40.0264i −105.868 500.935i 1925.12 700.686i 470.759 117.180i
23.11 −6.80379 4.20814i −3.83980 + 0.677060i 28.5831 + 57.2626i −157.228 + 131.930i 28.9744 + 11.5518i −546.642 315.604i 46.4952 509.884i −670.750 + 244.133i 1624.92 235.986i
23.12 −6.50920 + 4.65084i 33.9040 5.97820i 20.7395 60.5465i −158.311 + 132.839i −192.885 + 196.595i −237.498 137.119i 146.594 + 490.565i 428.710 156.038i 412.668 1600.95i
23.13 −6.18379 + 5.07550i −9.73276 + 1.71615i 12.4786 62.7717i −96.0367 + 80.5844i 51.4751 60.0109i −51.8068 29.9106i 241.432 + 451.502i −593.255 + 215.927i 184.865 985.751i
23.14 −5.92630 5.37392i 27.6808 4.88087i 6.24202 + 63.6949i −34.1068 + 28.6190i −190.274 119.829i 83.9559 + 48.4720i 305.299 411.019i 57.3665 20.8797i 355.923 + 13.6824i
23.15 −5.75352 + 5.55851i −45.0914 + 7.95083i 2.20600 63.9620i −116.601 + 97.8402i 215.240 296.386i 418.318 + 241.516i 342.841 + 380.269i 1284.98 467.695i 127.023 1211.06i
23.16 −5.01566 6.23243i 22.9636 4.04910i −13.6863 + 62.5195i −26.2597 + 22.0345i −140.413 122.810i 137.300 + 79.2704i 458.294 228.277i −174.105 + 63.3690i 269.038 + 53.1441i
23.17 −4.61729 + 6.53304i 1.04918 0.184999i −21.3612 60.3299i 59.4109 49.8517i −3.63577 + 7.70853i −459.233 265.139i 492.769 + 139.007i −683.969 + 248.944i 51.3653 + 618.313i
23.18 −4.55307 6.57796i −26.4348 + 4.66117i −22.5392 + 59.8998i 6.31656 5.30022i 151.020 + 152.664i 142.890 + 82.4974i 496.641 124.466i −7.96412 + 2.89870i −63.6244 17.4178i
23.19 −4.53942 + 6.58739i 50.5024 8.90493i −22.7874 59.8058i 146.923 123.283i −170.591 + 373.102i −251.055 144.947i 497.406 + 121.374i 1786.16 650.108i 145.169 + 1527.48i
23.20 −4.45512 + 6.64469i −11.1633 + 1.96839i −24.3038 59.2058i 71.7659 60.2188i 36.6545 82.9459i 245.952 + 142.001i 501.680 + 102.278i −564.292 + 205.385i 80.4088 + 745.144i
See next 80 embeddings (of 348 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.58 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.e even 9 1 inner
76.l odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.l.a 348
4.b odd 2 1 inner 76.7.l.a 348
19.e even 9 1 inner 76.7.l.a 348
76.l odd 18 1 inner 76.7.l.a 348

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.l.a 348 1.a even 1 1 trivial
76.7.l.a 348 4.b odd 2 1 inner
76.7.l.a 348 19.e even 9 1 inner
76.7.l.a 348 76.l odd 18 1 inner

## Hecke kernels

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