# Properties

 Label 48.7.l.a Level 48 Weight 7 Character orbit 48.l Analytic conductor 11.043 Analytic rank 0 Dimension 48 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 48.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0425960138$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 180q^{4} - 1932q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 180q^{4} - 1932q^{8} + 744q^{10} - 2720q^{11} - 1944q^{12} - 15388q^{14} + 3744q^{16} + 4860q^{18} - 3936q^{19} - 14000q^{20} - 41184q^{22} + 26240q^{23} + 18468q^{24} - 5300q^{26} - 96120q^{28} + 66400q^{29} - 33048q^{30} - 52960q^{32} + 122208q^{34} + 162336q^{35} + 32076q^{36} - 7200q^{37} + 16968q^{38} - 309072q^{40} - 231660q^{42} + 340704q^{43} - 193192q^{44} - 450264q^{46} + 299376q^{48} + 806736q^{49} + 537764q^{50} + 80352q^{51} + 1126224q^{52} + 443680q^{53} + 78732q^{54} + 232704q^{55} - 420448q^{56} - 1295664q^{58} - 886144q^{59} - 627912q^{60} - 326496q^{61} - 719652q^{62} - 192024q^{64} - 372832q^{65} + 775656q^{66} - 962112q^{67} + 3197632q^{68} + 541728q^{69} + 642816q^{70} + 534016q^{71} + 82620q^{72} - 4894836q^{74} - 1073088q^{75} - 3162552q^{76} - 932960q^{77} - 337284q^{78} + 4668072q^{80} - 2834352q^{81} + 5077560q^{82} - 2497760q^{83} + 1312200q^{84} - 372000q^{85} + 4142928q^{86} - 2794272q^{88} - 507384q^{90} + 775008q^{91} - 9470992q^{92} - 5050728q^{94} + 1879200q^{96} + 12708584q^{98} - 660960q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −7.91026 + 1.19493i 11.0227 11.0227i 61.1443 18.9044i 156.277 156.277i −74.0211 + 100.364i 23.3623 −461.078 + 222.601i 243.000i −1049.45 + 1422.93i
19.2 −7.86116 1.48398i −11.0227 + 11.0227i 59.5956 + 23.3316i 52.2146 52.2146i 103.009 70.2938i −282.592 −433.867 271.852i 243.000i −487.953 + 332.982i
19.3 −7.47563 + 2.84868i −11.0227 + 11.0227i 47.7700 42.5914i −55.7840 + 55.7840i 51.0014 113.802i 496.753 −235.781 + 454.479i 243.000i 258.109 575.931i
19.4 −7.29112 + 3.29235i 11.0227 11.0227i 42.3208 48.0099i −73.3870 + 73.3870i −44.0772 + 116.658i 291.401 −150.501 + 489.381i 243.000i 293.457 776.689i
19.5 −7.21738 3.45101i 11.0227 11.0227i 40.1811 + 49.8145i −26.8357 + 26.8357i −117.594 + 41.5155i −81.4485 −118.092 498.195i 243.000i 286.294 101.073i
19.6 −5.74110 5.57133i −11.0227 + 11.0227i 1.92050 + 63.9712i −160.484 + 160.484i 124.694 1.87131i −53.5181 345.379 377.965i 243.000i 1815.46 27.2451i
19.7 −4.50221 + 6.61288i 11.0227 11.0227i −23.4602 59.5451i −9.26689 + 9.26689i 23.2652 + 122.518i −320.373 499.387 + 112.945i 243.000i −19.5593 103.002i
19.8 −3.75499 7.06400i −11.0227 + 11.0227i −35.8001 + 53.0505i 107.837 107.837i 119.254 + 36.4742i 5.36994 509.177 + 53.6875i 243.000i −1166.68 356.832i
19.9 −3.30175 + 7.28687i −11.0227 + 11.0227i −42.1969 48.1188i −98.7574 + 98.7574i −43.9268 116.715i −218.381 489.959 148.607i 243.000i −393.560 1045.70i
19.10 −2.85723 7.47236i 11.0227 11.0227i −47.6725 + 42.7006i −99.4317 + 99.4317i −113.860 50.8712i 407.926 455.285 + 234.221i 243.000i 1027.09 + 458.891i
19.11 −0.382617 + 7.99085i −11.0227 + 11.0227i −63.7072 6.11486i 122.335 122.335i −83.8632 92.2982i 392.292 73.2383 506.735i 243.000i 930.754 + 1024.37i
19.12 0.181149 + 7.99795i 11.0227 11.0227i −63.9344 + 2.89763i −124.587 + 124.587i 90.1558 + 86.1623i 200.172 −34.7567 510.819i 243.000i −1019.01 973.874i
19.13 1.02416 7.93417i 11.0227 11.0227i −61.9022 16.2518i 145.472 145.472i −76.1670 98.7451i 535.877 −192.342 + 474.498i 243.000i −1005.21 1303.19i
19.14 1.27618 + 7.89755i 11.0227 11.0227i −60.7427 + 20.1574i 132.791 132.791i 101.119 + 72.9854i −560.492 −236.713 453.995i 243.000i 1218.19 + 879.260i
19.15 1.61327 7.83565i −11.0227 + 11.0227i −58.7947 25.2820i −15.5658 + 15.5658i 68.5874 + 104.153i 10.2991 −292.952 + 419.908i 243.000i 96.8564 + 147.080i
19.16 3.36544 7.25767i 11.0227 11.0227i −41.3476 48.8505i −26.6395 + 26.6395i −42.9029 117.095i −403.840 −493.694 + 135.684i 243.000i 103.687 + 282.994i
19.17 4.87132 + 6.34588i −11.0227 + 11.0227i −16.5404 + 61.8257i −95.7535 + 95.7535i −123.644 16.2536i 338.697 −472.912 + 196.210i 243.000i −1074.09 141.194i
19.18 5.00967 + 6.23724i −11.0227 + 11.0227i −13.8064 + 62.4931i 45.7781 45.7781i −123.971 13.5312i −565.544 −458.950 + 226.956i 243.000i 514.863 + 56.1959i
19.19 5.29238 + 5.99922i 11.0227 11.0227i −7.98136 + 63.5004i 47.0845 47.0845i 124.464 + 7.79129i 400.703 −423.193 + 288.186i 243.000i 531.659 + 33.2812i
19.20 5.65268 5.66103i −11.0227 + 11.0227i −0.0944246 63.9999i 29.4894 29.4894i 0.0919959 + 124.708i 461.206 −362.839 361.237i 243.000i −0.246120 333.634i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.7.l.a 48
4.b odd 2 1 192.7.l.a 48
8.b even 2 1 384.7.l.b 48
8.d odd 2 1 384.7.l.a 48
16.e even 4 1 192.7.l.a 48
16.e even 4 1 384.7.l.a 48
16.f odd 4 1 inner 48.7.l.a 48
16.f odd 4 1 384.7.l.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.l.a 48 1.a even 1 1 trivial
48.7.l.a 48 16.f odd 4 1 inner
192.7.l.a 48 4.b odd 2 1
192.7.l.a 48 16.e even 4 1
384.7.l.a 48 8.d odd 2 1
384.7.l.a 48 16.e even 4 1
384.7.l.b 48 8.b even 2 1
384.7.l.b 48 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database