# Properties

 Label 48.6.k.a Level 48 Weight 6 Character orbit 48.k Analytic conductor 7.698 Analytic rank 0 Dimension 76 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 48.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.69842335102$$ Analytic rank: $$0$$ Dimension: $$76$$ Relative dimension: $$38$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$76q - 2q^{3} - 4q^{4} - 116q^{6} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$76q - 2q^{3} - 4q^{4} - 116q^{6} - 8q^{7} - 104q^{10} + 784q^{12} - 4q^{13} + 4560q^{16} - 4076q^{18} + 2356q^{19} - 488q^{21} + 2232q^{22} + 2568q^{24} - 3734q^{27} - 12392q^{28} + 628q^{30} - 4q^{33} + 26248q^{34} - 30748q^{36} - 4q^{37} - 44908q^{39} + 440q^{40} + 11256q^{42} + 652q^{43} - 6252q^{45} + 31320q^{46} - 9168q^{48} + 124844q^{49} + 8664q^{51} - 48448q^{52} - 19480q^{54} - 110056q^{55} + 76088q^{58} - 85408q^{60} + 48076q^{61} - 11080q^{64} + 91492q^{66} + 48924q^{67} + 484q^{69} + 86200q^{70} - 16376q^{72} - 69634q^{75} + 30920q^{76} + 13020q^{78} - 4q^{81} - 200768q^{82} + 111760q^{84} - 119904q^{85} + 282180q^{87} - 123616q^{88} - 17680q^{90} - 167288q^{91} + 181660q^{93} - 13200q^{94} - 61688q^{96} - 8q^{97} - 287860q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −5.65156 + 0.244697i −15.0380 4.10602i 31.8802 2.76584i −44.3040 + 44.3040i 85.9927 + 19.5257i −165.521 −179.496 + 23.4324i 209.281 + 123.492i 239.546 261.228i
11.2 −5.64437 + 0.375585i 0.900761 15.5624i 31.7179 4.23988i 4.84652 4.84652i 0.760775 + 88.1783i 213.483 −177.435 + 35.8442i −241.377 28.0360i −25.5353 + 29.1759i
11.3 −5.61292 0.703650i 15.5533 + 1.04571i 31.0098 + 7.89906i −58.6729 + 58.6729i −86.5639 16.8136i 13.6308 −168.497 66.1568i 240.813 + 32.5285i 370.612 288.041i
11.4 −5.46756 1.45114i 7.61435 + 13.6023i 27.7884 + 15.8684i 37.6198 37.6198i −21.8930 85.4207i 4.84822 −128.907 127.086i −127.043 + 207.145i −260.280 + 151.097i
11.5 −5.29446 + 1.99215i −11.7496 + 10.2444i 24.0627 21.0947i 20.1669 20.1669i 41.7994 77.6454i 131.682 −85.3753 + 159.622i 33.1052 240.734i −66.5975 + 146.948i
11.6 −5.17653 + 2.28112i 14.1218 6.60105i 21.5930 23.6166i 62.8699 62.8699i −58.0444 + 66.3841i −197.388 −57.9049 + 171.508i 155.852 186.438i −182.035 + 468.862i
11.7 −4.72725 3.10694i −15.0463 4.07524i 12.6938 + 29.3746i 75.4947 75.4947i 58.4663 + 66.0128i 8.00043 31.2584 178.300i 209.785 + 122.635i −591.440 + 122.325i
11.8 −4.50225 3.42488i 2.94311 15.3081i 8.54043 + 30.8393i −6.54819 + 6.54819i −65.6790 + 58.8411i −181.385 67.1696 168.096i −225.676 90.1068i 51.9083 7.05481i
11.9 −4.35628 + 3.60872i 4.78598 + 14.8356i 5.95430 31.4412i −31.7606 + 31.7606i −74.3865 47.3566i −71.3548 87.5237 + 158.454i −197.189 + 142.006i 23.7429 252.973i
11.10 −4.27922 3.69977i −7.94962 + 13.4091i 4.62345 + 31.6642i −37.0301 + 37.0301i 83.6287 27.9687i 35.4948 97.3655 152.604i −116.607 213.194i 295.462 21.4572i
11.11 −3.59855 + 4.36468i −9.65436 12.2390i −6.10082 31.4131i 13.2817 13.2817i 88.1611 + 1.90456i −46.0433 159.062 + 86.4135i −56.5866 + 236.320i 10.1754 + 105.765i
11.12 −3.29561 + 4.59771i 13.1034 8.44402i −10.2779 30.3045i −39.6034 + 39.6034i −4.36045 + 88.0738i 91.7189 173.203 + 52.6171i 100.397 221.290i −51.5676 312.602i
11.13 −2.92159 4.84400i 15.5263 1.39064i −14.9286 + 28.3043i 15.8062 15.8062i −52.0977 71.1465i 95.8790 180.722 10.3793i 239.132 43.1829i −122.744 30.3859i
11.14 −2.14294 5.23525i −9.18566 12.5946i −22.8156 + 22.4377i −45.3955 + 45.3955i −46.2514 + 75.0787i 149.114 166.359 + 71.3625i −74.2473 + 231.379i 334.936 + 140.376i
11.15 −1.61933 + 5.42012i 13.1651 + 8.34740i −26.7555 17.5540i 54.1063 54.1063i −66.5627 + 57.8395i 243.836 138.471 116.592i 103.642 + 219.789i 205.647 + 380.879i
11.16 −1.31086 + 5.50288i −10.2553 + 11.7400i −28.5633 14.4270i 55.5354 55.5354i −51.1607 71.8233i −177.545 116.833 138.268i −32.6571 240.796i 232.805 + 378.403i
11.17 −1.01378 + 5.56527i −15.2220 + 3.36009i −29.9445 11.2839i −66.9145 + 66.9145i −3.26809 88.1211i 97.7370 93.1548 155.210i 220.420 102.295i −304.561 440.234i
11.18 −1.01194 5.56561i 5.84525 + 14.4511i −29.9519 + 11.2641i 21.0919 21.0919i 74.5139 47.1560i −189.112 93.0014 + 155.302i −174.666 + 168.940i −138.733 96.0456i
11.19 −0.486958 5.63586i −15.0887 + 3.91558i −31.5257 + 5.48885i 7.73181 7.73181i 29.4152 + 83.1309i −59.0766 46.2861 + 175.002i 212.337 118.162i −47.3405 39.8103i
11.20 0.486958 + 5.63586i 3.91558 15.0887i −31.5257 + 5.48885i −7.73181 + 7.73181i 86.9443 + 14.7201i −59.0766 −46.2861 175.002i −212.337 118.162i −47.3405 39.8103i
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.38 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.k.a 76
3.b odd 2 1 inner 48.6.k.a 76
4.b odd 2 1 192.6.k.a 76
12.b even 2 1 192.6.k.a 76
16.e even 4 1 192.6.k.a 76
16.f odd 4 1 inner 48.6.k.a 76
48.i odd 4 1 192.6.k.a 76
48.k even 4 1 inner 48.6.k.a 76

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.k.a 76 1.a even 1 1 trivial
48.6.k.a 76 3.b odd 2 1 inner
48.6.k.a 76 16.f odd 4 1 inner
48.6.k.a 76 48.k even 4 1 inner
192.6.k.a 76 4.b odd 2 1
192.6.k.a 76 12.b even 2 1
192.6.k.a 76 16.e even 4 1
192.6.k.a 76 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database