# Properties

 Label 39.4.k.a Level $39$ Weight $4$ Character orbit 39.k Analytic conductor $2.301$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.k (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9}+O(q^{10})$$ 48 * q - 2 * q^3 - 12 * q^4 - 50 * q^6 + 20 * q^7 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 2 q^{3} - 12 q^{4} - 50 q^{6} + 20 q^{7} - 2 q^{9} - 156 q^{10} - 80 q^{13} + 70 q^{15} + 260 q^{16} + 256 q^{18} + 260 q^{19} + 82 q^{21} + 212 q^{22} - 1194 q^{24} - 248 q^{27} - 756 q^{28} - 1062 q^{30} - 180 q^{31} + 10 q^{33} - 396 q^{34} + 3060 q^{36} + 1932 q^{37} + 538 q^{39} + 360 q^{40} + 968 q^{42} + 1416 q^{43} - 386 q^{45} - 144 q^{46} - 410 q^{48} - 3000 q^{49} - 4336 q^{52} + 1930 q^{54} - 1012 q^{55} - 1274 q^{57} + 908 q^{58} - 2860 q^{60} + 836 q^{61} - 5150 q^{63} + 1376 q^{66} - 136 q^{67} - 1674 q^{69} + 1808 q^{70} - 3900 q^{72} + 3572 q^{73} + 5796 q^{75} + 8400 q^{76} + 12292 q^{78} - 3760 q^{79} + 2494 q^{81} + 2544 q^{82} + 1084 q^{84} + 4980 q^{85} + 2318 q^{87} - 8436 q^{88} - 8908 q^{91} - 1214 q^{93} - 8464 q^{94} - 6968 q^{96} - 204 q^{97} - 13094 q^{99}+O(q^{100})$$ 48 * q - 2 * q^3 - 12 * q^4 - 50 * q^6 + 20 * q^7 - 2 * q^9 - 156 * q^10 - 80 * q^13 + 70 * q^15 + 260 * q^16 + 256 * q^18 + 260 * q^19 + 82 * q^21 + 212 * q^22 - 1194 * q^24 - 248 * q^27 - 756 * q^28 - 1062 * q^30 - 180 * q^31 + 10 * q^33 - 396 * q^34 + 3060 * q^36 + 1932 * q^37 + 538 * q^39 + 360 * q^40 + 968 * q^42 + 1416 * q^43 - 386 * q^45 - 144 * q^46 - 410 * q^48 - 3000 * q^49 - 4336 * q^52 + 1930 * q^54 - 1012 * q^55 - 1274 * q^57 + 908 * q^58 - 2860 * q^60 + 836 * q^61 - 5150 * q^63 + 1376 * q^66 - 136 * q^67 - 1674 * q^69 + 1808 * q^70 - 3900 * q^72 + 3572 * q^73 + 5796 * q^75 + 8400 * q^76 + 12292 * q^78 - 3760 * q^79 + 2494 * q^81 + 2544 * q^82 + 1084 * q^84 + 4980 * q^85 + 2318 * q^87 - 8436 * q^88 - 8908 * q^91 - 1214 * q^93 - 8464 * q^94 - 6968 * q^96 - 204 * q^97 - 13094 * q^99

## 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 −1.37323 + 5.12497i 0.625377 + 5.15838i −17.4513 10.0755i 5.80773 + 5.80773i −27.2953 3.87861i 20.4169 5.47069i 45.5877 45.5877i −26.2178 + 6.45187i −37.7398 + 21.7891i
2.2 −1.13863 + 4.24944i −3.90424 3.42883i −9.83302 5.67709i −8.66436 8.66436i 19.0161 12.6867i −1.76245 + 0.472246i 10.4342 10.4342i 3.48625 + 26.7740i 46.6841 26.9531i
2.3 −0.908381 + 3.39012i 2.68627 4.44791i −3.73959 2.15905i 12.1413 + 12.1413i 12.6388 + 13.1472i −3.81863 + 1.02320i −9.13753 + 9.13753i −12.5679 23.8966i −52.1894 + 30.1316i
2.4 −0.614396 + 2.29296i −3.60382 + 3.74332i 2.04803 + 1.18243i −2.27968 2.27968i −6.36910 10.5633i −25.9009 + 6.94012i −17.3981 + 17.3981i −1.02489 26.9805i 6.62784 3.82659i
2.5 −0.532104 + 1.98584i 4.60622 + 2.40472i 3.26777 + 1.88665i −4.41602 4.41602i −7.22639 + 7.86767i 3.85828 1.03382i −17.1153 + 17.1153i 15.4346 + 22.1534i 11.1193 6.41973i
2.6 −0.0560169 + 0.209058i −5.01878 1.34604i 6.88764 + 3.97658i 10.0436 + 10.0436i 0.562537 0.973816i 17.5010 4.68938i −2.44149 + 2.44149i 23.3764 + 13.5109i −2.66232 + 1.53709i
2.7 0.0560169 0.209058i 1.34369 5.01941i 6.88764 + 3.97658i −10.0436 10.0436i −0.974079 0.562081i 17.5010 4.68938i 2.44149 2.44149i −23.3890 13.4891i −2.66232 + 1.53709i
2.8 0.532104 1.98584i −0.220561 + 5.19147i 3.26777 + 1.88665i 4.41602 + 4.41602i 10.1921 + 3.20040i 3.85828 1.03382i 17.1153 17.1153i −26.9027 2.29007i 11.1193 6.41973i
2.9 0.614396 2.29296i 5.04372 1.24934i 2.04803 + 1.18243i 2.27968 + 2.27968i 0.234152 12.3326i −25.9009 + 6.94012i 17.3981 17.3981i 23.8783 12.6027i 6.62784 3.82659i
2.10 0.908381 3.39012i −5.19514 + 0.102423i −3.73959 2.15905i −12.1413 12.1413i −4.37194 + 17.7052i −3.81863 + 1.02320i 9.13753 9.13753i 26.9790 1.06421i −52.1894 + 30.1316i
2.11 1.13863 4.24944i −1.01733 5.09559i −9.83302 5.67709i 8.66436 + 8.66436i −22.8117 1.47892i −1.76245 + 0.472246i −10.4342 + 10.4342i −24.9301 + 10.3678i 46.6841 26.9531i
2.12 1.37323 5.12497i 4.15460 + 3.12078i −17.4513 10.0755i −5.80773 5.80773i 21.6992 17.0066i 20.4169 5.47069i −45.5877 + 45.5877i 7.52142 + 25.9312i −37.7398 + 21.7891i
11.1 −5.07899 1.36091i 4.03642 3.27221i 17.0159 + 9.82412i 0.833167 0.833167i −24.9541 + 11.1263i −6.38907 23.8443i −43.3091 43.3091i 5.58535 26.4160i −5.36552 + 3.09778i
11.2 −3.88819 1.04184i −4.68804 2.24104i 7.10439 + 4.10172i 0.168873 0.168873i 15.8932 + 13.5978i 5.26279 + 19.6410i −0.579067 0.579067i 16.9555 + 21.0122i −0.832548 + 0.480672i
11.3 −3.31082 0.887131i −2.02268 + 4.78631i 3.24631 + 1.87426i 9.09982 9.09982i 10.9428 14.0522i −7.29644 27.2307i 10.3043 + 10.3043i −18.8175 19.3624i −38.2006 + 22.0551i
11.4 −2.98884 0.800858i 3.86434 + 3.47374i 1.36361 + 0.787281i −6.51442 + 6.51442i −8.76795 13.4773i 4.63030 + 17.2805i 14.0588 + 14.0588i 2.86629 + 26.8474i 24.6877 14.2534i
11.5 −0.933566 0.250148i 0.0424052 5.19598i −6.11923 3.53294i −11.3224 + 11.3224i −1.33935 + 4.84018i −3.58928 13.3954i 10.2963 + 10.2963i −26.9964 0.440673i 13.4025 7.73791i
11.6 −0.385245 0.103226i 4.91747 1.67883i −6.79045 3.92047i 9.27643 9.27643i −2.06773 + 0.139148i 2.08748 + 7.79057i 4.46744 + 4.46744i 21.3631 16.5112i −4.53127 + 2.61613i
11.7 0.385245 + 0.103226i −3.91264 + 3.41924i −6.79045 3.92047i −9.27643 + 9.27643i −1.86028 + 0.913360i 2.08748 + 7.79057i −4.46744 4.46744i 3.61755 26.7566i −4.53127 + 2.61613i
11.8 0.933566 + 0.250148i −4.52105 2.56127i −6.11923 3.53294i 11.3224 11.3224i −3.58000 3.52204i −3.58928 13.3954i −10.2963 10.2963i 13.8798 + 23.1592i 13.4025 7.73791i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.12 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
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.k.a 48
3.b odd 2 1 inner 39.4.k.a 48
13.f odd 12 1 inner 39.4.k.a 48
39.k even 12 1 inner 39.4.k.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.k.a 48 1.a even 1 1 trivial
39.4.k.a 48 3.b odd 2 1 inner
39.4.k.a 48 13.f odd 12 1 inner
39.4.k.a 48 39.k even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(39, [\chi])$$.