# Properties

 Label 16.15.f.a Level 16 Weight 15 Character orbit 16.f Analytic conductor 19.893 Analytic rank 0 Dimension 54 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 16.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.8926349043$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$27$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q - 2q^{2} - 2q^{3} + 15832q^{4} - 2q^{5} - 182528q^{6} - 4q^{7} + 2748484q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q - 2q^{2} - 2q^{3} + 15832q^{4} - 2q^{5} - 182528q^{6} - 4q^{7} + 2748484q^{8} + 16861596q^{10} - 10455538q^{11} - 87110828q^{12} - 2q^{13} + 66965260q^{14} - 1059602728q^{16} - 4q^{17} + 438192046q^{18} + 702532062q^{19} + 1755464396q^{20} - 9565940q^{21} - 4795498204q^{22} + 9748846012q^{23} + 6975250608q^{24} - 8671243576q^{26} - 17865634112q^{27} - 1297033864q^{28} - 13109213938q^{29} + 11736190852q^{30} - 33545826552q^{32} - 4q^{33} - 50990884972q^{34} + 125535560540q^{35} + 60739632140q^{36} + 115688878030q^{37} - 763264561040q^{38} + 207890167484q^{39} + 1298710786600q^{40} - 1441334806824q^{42} + 834144712302q^{43} - 1390640237388q^{44} + 12216597186q^{45} + 2618424802780q^{46} - 4460553354264q^{48} + 4069338437090q^{49} - 14467226554q^{50} - 109367594148q^{51} + 700335676036q^{52} - 1638253831762q^{53} - 6106962841312q^{54} + 7743608133372q^{55} + 892124789080q^{56} - 3111214006280q^{58} - 2886464056530q^{59} - 5948471456800q^{60} + 6325590293982q^{61} + 4387920975408q^{62} - 11796533376704q^{64} + 4831417084012q^{65} + 3599619796132q^{66} + 16035979669214q^{67} - 11417990783952q^{68} + 13777251815788q^{69} + 6360571783424q^{70} - 23491285382916q^{71} - 17358886385420q^{72} + 17391872660764q^{74} + 38233839801842q^{75} + 8965870021732q^{76} - 9967029629748q^{77} - 102277039258452q^{78} + 184151019521512q^{80} - 96590901476506q^{81} - 60118295271632q^{82} - 44676337597762q^{83} - 30278459356760q^{84} - 65266064531252q^{85} + 200817997017700q^{86} - 19584321079236q^{87} - 98329880319368q^{88} + 105728657951480q^{90} + 12075497546300q^{91} + 263822607742360q^{92} - 149536300814816q^{93} + 36585390208368q^{94} - 124834847268944q^{96} - 4q^{97} + 73723998799370q^{98} - 130415167656994q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −127.662 + 9.30177i 1995.33 1995.33i 16211.0 2374.96i −5949.31 + 5949.31i −236166. + 273287.i −842293. −2.04742e6 + 453982.i 3.17969e6i 704159. 814837.i
3.2 −123.321 34.2916i −39.1610 + 39.1610i 14032.2 + 8457.74i −4748.26 + 4748.26i 6172.27 3486.48i 903713. −1.44043e6 1.52420e6i 4.77990e6i 748385. 422735.i
3.3 −122.284 37.8242i −2920.51 + 2920.51i 13522.7 + 9250.58i −497.809 + 497.809i 467597. 246665.i −436996. −1.30371e6 1.64268e6i 1.22758e7i 79703.3 42044.8i
3.4 −118.943 + 47.2917i −1108.18 + 1108.18i 11911.0 11250.1i −55052.8 + 55052.8i 79403.1 184219.i 519580. −884698. + 1.90141e6i 2.32682e6i 3.94462e6 9.15171e6i
3.5 −108.596 + 67.7560i −1164.54 + 1164.54i 7202.24 14716.1i 98063.2 98063.2i 47560.0 205370.i −431458. 214968. + 2.08611e6i 2.07065e6i −4.00491e6 + 1.72937e7i
3.6 −99.8787 80.0515i 223.330 223.330i 3567.51 + 15990.9i 87742.2 87742.2i −40183.8 + 4428.01i −415149. 923776. 1.88273e6i 4.68322e6i −1.57875e7 + 1.73968e6i
3.7 −85.4809 + 95.2734i 2412.96 2412.96i −1770.02 16288.1i 56427.7 56427.7i 23628.8 + 436153.i 1.31174e6 1.70313e6 + 1.22369e6i 6.86182e6i 552565. + 1.01996e7i
3.8 −84.2589 96.3558i 2495.14 2495.14i −2184.86 + 16237.7i −43850.1 + 43850.1i −450659. 30183.2i 1.15759e6 1.74869e6 1.15764e6i 7.66847e6i 7.91997e6 + 530446.i
3.9 −79.3579 100.431i −417.666 + 417.666i −3788.66 + 15939.9i −101107. + 101107.i 75091.6 + 8801.42i −1.56037e6 1.90152e6 884462.i 4.43408e6i 1.81780e7 + 2.13062e6i
3.10 −72.6989 + 105.351i 692.853 692.853i −5813.74 15317.8i −62322.7 + 62322.7i 22623.2 + 123363.i −300008. 2.03640e6 + 501105.i 3.82288e6i −2.03498e6 1.10966e7i
3.11 −40.7689 + 121.334i −2416.37 + 2416.37i −13059.8 9893.30i −20189.0 + 20189.0i −194674. 391700.i −480275. 1.73283e6 1.18125e6i 6.89468e6i −1.62653e6 3.27270e6i
3.12 −36.7331 122.616i −1658.66 + 1658.66i −13685.4 + 9008.14i 7093.51 7093.51i 264306. + 142450.i 747019. 1.60725e6 + 1.34715e6i 719331.i −1.13034e6 609211.i
3.13 −11.9550 127.440i 1873.37 1873.37i −16098.2 + 3047.10i 56067.7 56067.7i −261139. 216347.i −548166. 580777. + 2.01513e6i 2.23606e6i −7.81558e6 6.47501e6i
3.14 −6.21757 + 127.849i 1390.71 1390.71i −16306.7 1589.82i 31054.7 31054.7i 169154. + 186448.i −1.17718e6 304644. 2.07491e6i 914816.i 3.77722e6 + 4.16339e6i
3.15 4.45002 + 127.923i −837.187 + 837.187i −16344.4 + 1138.52i 37060.2 37060.2i −110821. 103370.i 1.50447e6 −218375. 2.08575e6i 3.38120e6i 4.90575e6 + 4.57592e6i
3.16 42.3109 120.805i 859.991 859.991i −12803.6 10222.7i −55370.8 + 55370.8i −67504.0 140278.i 175518. −1.77668e6 + 1.11420e6i 3.30380e6i 4.34627e6 + 9.03184e6i
3.17 47.5205 + 118.852i 2009.03 2009.03i −11867.6 + 11295.8i −90795.6 + 90795.6i 334247. + 143307.i 656977. −1.90649e6 873704.i 3.28943e6i −1.51059e7 6.47658e6i
3.18 57.2769 114.470i −1878.56 + 1878.56i −9822.72 13113.0i 35992.2 35992.2i 107440. + 322636.i −784164. −2.06365e6 + 373337.i 2.27499e6i −2.05850e6 6.18154e6i
3.19 74.1318 + 104.348i −498.544 + 498.544i −5392.95 + 15471.0i 40758.2 40758.2i −88979.9 15064.0i −589264. −2.01415e6 + 584149.i 4.28588e6i 7.27451e6 + 1.23155e6i
3.20 83.7629 + 96.7872i −2050.98 + 2050.98i −2351.54 + 16214.4i −70340.6 + 70340.6i −370305. 26712.6i −283091. −1.76632e6 + 1.13056e6i 3.63010e6i −1.27000e7 916137.i
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.27 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 16.15.f.a 54
4.b odd 2 1 64.15.f.a 54
16.e even 4 1 64.15.f.a 54
16.f odd 4 1 inner 16.15.f.a 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.15.f.a 54 1.a even 1 1 trivial
16.15.f.a 54 16.f odd 4 1 inner
64.15.f.a 54 4.b odd 2 1
64.15.f.a 54 16.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{15}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database