LMFDB - Embedded newform 729.3.b.a.728.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,3,Mod(728,729)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("729.728");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	729.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.8638112719$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			728.12
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.19

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16096i q^{2} +2.65217 q^{4} +5.38300i q^{5} -10.5344 q^{7} -7.72291i q^{8} +O(q^{10})$	\(q-1.16096i q^{2} +2.65217 q^{4} +5.38300i q^{5} -10.5344 q^{7} -7.72291i q^{8} +6.24946 q^{10} -11.7440i q^{11} +11.2329 q^{13} +12.2300i q^{14} +1.64265 q^{16} -3.15961i q^{17} -4.52102 q^{19} +14.2766i q^{20} -13.6343 q^{22} -19.9076i q^{23} -3.97672 q^{25} -13.0409i q^{26} -27.9389 q^{28} -25.3597i q^{29} +22.1215 q^{31} -32.7987i q^{32} -3.66818 q^{34} -56.7065i q^{35} +17.6561 q^{37} +5.24873i q^{38} +41.5725 q^{40} -12.4448i q^{41} +36.4484 q^{43} -31.1470i q^{44} -23.1120 q^{46} -12.3996i q^{47} +61.9729 q^{49} +4.61682i q^{50} +29.7914 q^{52} -84.6210i q^{53} +63.2178 q^{55} +81.3560i q^{56} -29.4417 q^{58} +68.2736i q^{59} +78.0720 q^{61} -25.6822i q^{62} -31.5075 q^{64} +60.4666i q^{65} -97.1238 q^{67} -8.37980i q^{68} -65.8341 q^{70} -121.664i q^{71} +91.1410 q^{73} -20.4981i q^{74} -11.9905 q^{76} +123.715i q^{77} -20.3879 q^{79} +8.84240i q^{80} -14.4479 q^{82} -31.6020i q^{83} +17.0082 q^{85} -42.3153i q^{86} -90.6977 q^{88} +69.0229i q^{89} -118.331 q^{91} -52.7983i q^{92} -14.3954 q^{94} -24.3367i q^{95} -79.8042 q^{97} -71.9482i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 48 q^{4}+O(q^{10}) $ 30 * q - 48 * q^4	$ 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67} - 150 q^{70} + 132 q^{73} - 24 q^{76} - 12 q^{82} + 96 q^{88} - 192 q^{91} - 24 q^{94}+O(q^{100}) $ 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.16096i	− 0.580481i	−0.956954	−	0.290241i	$-0.906265\pi$
$2$	− 1.16096i	− 0.580481i	0.956954	−	0.290241i	$-0.0937353\pi$
$3$	0	0
$3$	0	0
$4$	2.65217	0.663042
$5$	5.38300i	1.07660i	0.842753	+	0.538300i	$0.180933\pi$
$5$	5.38300i	1.07660i	−0.842753	+	0.538300i	$0.819067\pi$
$6$	0	0
$7$	−10.5344	−1.50491	−0.752455	−	0.658644i	$-0.771130\pi$
$7$	−10.5344	−1.50491	−0.752455	+	0.658644i	$0.771130\pi$
$8$	− 7.72291i	− 0.965364i
$9$	0	0
$10$	6.24946	0.624946
$11$	− 11.7440i	− 1.06763i	−0.845600	−	0.533817i	$-0.820757\pi$
$11$	− 11.7440i	− 1.06763i	0.845600	−	0.533817i	$-0.179243\pi$
$12$	0	0
$13$	11.2329	0.864067	0.432033	−	0.901858i	$-0.357796\pi$
$13$	11.2329	0.864067	0.432033	+	0.901858i	$0.357796\pi$
$14$	12.2300i	0.873572i
$15$	0	0
$16$	1.64265	0.102666
$17$	− 3.15961i	− 0.185859i	−0.995673	−	0.0929296i	$-0.970377\pi$
$17$	− 3.15961i	− 0.185859i	0.995673	−	0.0929296i	$-0.0296231\pi$
$18$	0	0
$19$	−4.52102	−0.237948	−0.118974	−	0.992897i	$-0.537961\pi$
$19$	−4.52102	−0.237948	−0.118974	+	0.992897i	$0.537961\pi$
$20$	14.2766i	0.713831i
$21$	0	0
$22$	−13.6343	−0.619741
$23$	− 19.9076i	− 0.865548i	−0.901502	−	0.432774i	$-0.857535\pi$
$23$	− 19.9076i	− 0.865548i	0.901502	−	0.432774i	$-0.142465\pi$
$24$	0	0
$25$	−3.97672	−0.159069
$26$	− 13.0409i	− 0.501575i
$27$	0	0
$28$	−27.9389	−0.997818
$29$	− 25.3597i	− 0.874472i	−0.899347	−	0.437236i	$-0.855957\pi$
$29$	− 25.3597i	− 0.874472i	0.899347	−	0.437236i	$-0.144043\pi$
$30$	0	0
$31$	22.1215	0.713595	0.356798	−	0.934182i	$-0.383869\pi$
$31$	22.1215	0.713595	0.356798	+	0.934182i	$0.383869\pi$
$32$	− 32.7987i	− 1.02496i
$33$	0	0
$34$	−3.66818	−0.107888
$35$	− 56.7065i	− 1.62019i
$36$	0	0
$37$	17.6561	0.477193	0.238596	−	0.971119i	$-0.423313\pi$
$37$	17.6561	0.477193	0.238596	+	0.971119i	$0.423313\pi$
$38$	5.24873i	0.138125i
$39$	0	0
$40$	41.5725	1.03931
$41$	− 12.4448i	− 0.303531i	−0.988417	−	0.151765i	$-0.951504\pi$
$41$	− 12.4448i	− 0.303531i	0.988417	−	0.151765i	$-0.0484958\pi$
$42$	0	0
$43$	36.4484	0.847638	0.423819	−	0.905747i	$-0.360689\pi$
$43$	36.4484	0.847638	0.423819	+	0.905747i	$0.360689\pi$
$44$	− 31.1470i	− 0.707886i
$45$	0	0
$46$	−23.1120	−0.502434
$47$	− 12.3996i	− 0.263821i	−0.991262	−	0.131910i	$-0.957889\pi$
$47$	− 12.3996i	− 0.263821i	0.991262	−	0.131910i	$-0.0421111\pi$
$48$	0	0
$49$	61.9729	1.26475
$50$	4.61682i	0.0923364i
$51$	0	0
$52$	29.7914	0.572912
$53$	− 84.6210i	− 1.59662i	−0.602245	−	0.798311i	$-0.705727\pi$
$53$	− 84.6210i	− 1.59662i	0.602245	−	0.798311i	$-0.294273\pi$
$54$	0	0
$55$	63.2178	1.14942
$56$	81.3560i	1.45279i
$57$	0	0
$58$	−29.4417	−0.507615
$59$	68.2736i	1.15718i	0.815619	+	0.578590i	$0.196397\pi$
$59$	68.2736i	1.15718i	−0.815619	+	0.578590i	$0.803603\pi$
$60$	0	0
$61$	78.0720	1.27987	0.639935	−	0.768429i	$-0.278961\pi$
$61$	78.0720	1.27987	0.639935	+	0.768429i	$0.278961\pi$
$62$	− 25.6822i	− 0.414229i
$63$	0	0
$64$	−31.5075	−0.492304
$65$	60.4666i	0.930255i
$66$	0	0
$67$	−97.1238	−1.44961	−0.724804	−	0.688955i	$-0.758070\pi$
$67$	−97.1238	−1.44961	−0.724804	+	0.688955i	$0.758070\pi$
$68$	− 8.37980i	− 0.123232i
$69$	0	0
$70$	−65.8341	−0.940488
$71$	− 121.664i	− 1.71358i	−0.515663	−	0.856791i	$-0.672455\pi$
$71$	− 121.664i	− 1.71358i	0.515663	−	0.856791i	$-0.327545\pi$
$72$	0	0
$73$	91.1410	1.24851	0.624254	−	0.781222i	$-0.285403\pi$
$73$	91.1410	1.24851	0.624254	+	0.781222i	$0.285403\pi$
$74$	− 20.4981i	− 0.277001i
$75$	0	0
$76$	−11.9905	−0.157770
$77$	123.715i	1.60669i
$78$	0	0
$79$	−20.3879	−0.258074	−0.129037	−	0.991640i	$-0.541189\pi$
$79$	−20.3879	−0.258074	−0.129037	+	0.991640i	$0.541189\pi$
$80$	8.84240i	0.110530i
$81$	0	0
$82$	−14.4479	−0.176194
$83$	− 31.6020i	− 0.380747i	−0.981712	−	0.190374i	$-0.939030\pi$
$83$	− 31.6020i	− 0.380747i	0.981712	−	0.190374i	$-0.0609700\pi$
$84$	0	0
$85$	17.0082	0.200096
$86$	− 42.3153i	− 0.492038i
$87$	0	0
$88$	−90.6977	−1.03066
$89$	69.0229i	0.775538i	0.921757	+	0.387769i	$0.126754\pi$
$89$	69.0229i	0.775538i	−0.921757	+	0.387769i	$0.873246\pi$
$90$	0	0
$91$	−118.331	−1.30034
$92$	− 52.7983i	− 0.573894i
$93$	0	0
$94$	−14.3954	−0.153143
$95$	− 24.3367i	− 0.256175i
$96$	0	0
$97$	−79.8042	−0.822724	−0.411362	−	0.911472i	$-0.634947\pi$
$97$	−79.8042	−0.822724	−0.411362	+	0.911472i	$0.634947\pi$
$98$	− 71.9482i	− 0.734165i
$99$	0	0
$100$	−10.5469	−0.105469
$101$	− 4.77669i	− 0.0472939i	−0.999720	−	0.0236470i	$-0.992472\pi$
$101$	− 4.77669i	− 0.0472939i	0.999720	−	0.0236470i	$-0.00752776\pi$
$102$	0	0
$103$	−72.2928	−0.701872	−0.350936	−	0.936400i	$-0.614136\pi$
$103$	−72.2928	−0.701872	−0.350936	+	0.936400i	$0.614136\pi$
$104$	− 86.7505i	− 0.834139i
$105$	0	0
$106$	−98.2418	−0.926809
$107$	− 158.133i	− 1.47788i	−0.673773	−	0.738938i	$-0.735328\pi$
$107$	− 158.133i	− 1.47788i	0.673773	−	0.738938i	$-0.264672\pi$
$108$	0	0
$109$	−18.5788	−0.170448	−0.0852240	−	0.996362i	$-0.527161\pi$
$109$	−18.5788	−0.170448	−0.0852240	+	0.996362i	$0.527161\pi$
$110$	− 73.3935i	− 0.667214i
$111$	0	0
$112$	−17.3043	−0.154503
$113$	18.0161i	0.159435i	0.996817	+	0.0797175i	$0.0254018\pi$
$113$	18.0161i	0.159435i	−0.996817	+	0.0797175i	$0.974598\pi$
$114$	0	0
$115$	107.163	0.931850
$116$	− 67.2581i	− 0.579811i
$117$	0	0
$118$	79.2631	0.671721
$119$	33.2844i	0.279701i
$120$	0	0
$121$	−16.9209	−0.139842
$122$	− 90.6387i	− 0.742940i
$123$	0	0
$124$	58.6698	0.473143
$125$	113.168i	0.905347i
$126$	0	0
$127$	−156.213	−1.23003	−0.615013	−	0.788517i	$-0.710849\pi$
$127$	−156.213	−1.23003	−0.615013	+	0.788517i	$0.710849\pi$
$128$	− 94.6159i	− 0.739187i
$129$	0	0
$130$	70.1994	0.539996
$131$	− 98.5190i	− 0.752054i	−0.926609	−	0.376027i	$-0.877290\pi$
$131$	− 98.5190i	− 0.752054i	0.926609	−	0.376027i	$-0.122710\pi$
$132$	0	0
$133$	47.6261	0.358091
$134$	112.757i	0.841471i
$135$	0	0
$136$	−24.4014	−0.179422
$137$	65.6564i	0.479244i	0.970866	+	0.239622i	$0.0770235\pi$
$137$	65.6564i	0.479244i	−0.970866	+	0.239622i	$0.922976\pi$
$138$	0	0
$139$	−125.301	−0.901443	−0.450722	−	0.892664i	$-0.648833\pi$
$139$	−125.301	−0.901443	−0.450722	+	0.892664i	$0.648833\pi$
$140$	− 150.395i	− 1.07425i
$141$	0	0
$142$	−141.248	−0.994702
$143$	− 131.919i	− 0.922507i
$144$	0	0
$145$	136.511	0.941457
$146$	− 105.811i	− 0.724735i
$147$	0	0
$148$	46.8270	0.316399
$149$	201.756i	1.35407i	0.735951	+	0.677035i	$0.236736\pi$
$149$	201.756i	1.35407i	−0.735951	+	0.677035i	$0.763264\pi$
$150$	0	0
$151$	128.141	0.848613	0.424306	−	0.905519i	$-0.360518\pi$
$151$	128.141	0.848613	0.424306	+	0.905519i	$0.360518\pi$
$152$	34.9154i	0.229707i
$153$	0	0
$154$	143.629	0.932655
$155$	119.080i	0.768257i
$156$	0	0
$157$	−138.946	−0.885008	−0.442504	−	0.896767i	$-0.645910\pi$
$157$	−138.946	−0.885008	−0.442504	+	0.896767i	$0.645910\pi$
$158$	23.6696i	0.149807i
$159$	0	0
$160$	176.556	1.10347
$161$	209.714i	1.30257i
$162$	0	0
$163$	39.8569	0.244521	0.122261	−	0.992498i	$-0.460986\pi$
$163$	39.8569	0.244521	0.122261	+	0.992498i	$0.460986\pi$
$164$	− 33.0056i	− 0.201254i
$165$	0	0
$166$	−36.6888	−0.221017
$167$	165.953i	0.993730i	0.867828	+	0.496865i	$0.165516\pi$
$167$	165.953i	0.993730i	−0.867828	+	0.496865i	$0.834484\pi$
$168$	0	0
$169$	−42.8226	−0.253388
$170$	− 19.7458i	− 0.116152i
$171$	0	0
$172$	96.6674	0.562019
$173$	− 10.8902i	− 0.0629489i	−0.999505	−	0.0314744i	$-0.989980\pi$
$173$	− 10.8902i	− 0.0629489i	0.999505	−	0.0314744i	$-0.0100203\pi$
$174$	0	0
$175$	41.8922	0.239384
$176$	− 19.2913i	− 0.109609i
$177$	0	0
$178$	80.1330	0.450185
$179$	282.579i	1.57865i	0.613974	+	0.789326i	$0.289570\pi$
$179$	282.579i	1.57865i	−0.613974	+	0.789326i	$0.710430\pi$
$180$	0	0
$181$	−108.799	−0.601100	−0.300550	−	0.953766i	$-0.597170\pi$
$181$	−108.799	−0.601100	−0.300550	+	0.953766i	$0.597170\pi$
$182$	137.378i	0.754824i
$183$	0	0
$184$	−153.745	−0.835569
$185$	95.0430i	0.513746i
$186$	0	0
$187$	−37.1063	−0.198429
$188$	− 32.8857i	− 0.174924i
$189$	0	0
$190$	−28.2539	−0.148705
$191$	− 108.629i	− 0.568740i	−0.958715	−	0.284370i	$-0.908216\pi$
$191$	− 108.629i	− 0.568740i	0.958715	−	0.284370i	$-0.0917844\pi$
$192$	0	0
$193$	200.108	1.03683	0.518415	−	0.855129i	$-0.326522\pi$
$193$	200.108	1.03683	0.518415	+	0.855129i	$0.326522\pi$
$194$	92.6496i	0.477576i
$195$	0	0
$196$	164.362	0.838584
$197$	292.710i	1.48584i	0.669383	+	0.742918i	$0.266559\pi$
$197$	292.710i	1.48584i	−0.669383	+	0.742918i	$0.733441\pi$
$198$	0	0
$199$	−82.3096	−0.413616	−0.206808	−	0.978382i	$-0.566308\pi$
$199$	−82.3096	−0.413616	−0.206808	+	0.978382i	$0.566308\pi$
$200$	30.7119i	0.153559i
$201$	0	0
$202$	−5.54555	−0.0274532
$203$	267.148i	1.31600i
$204$	0	0
$205$	66.9902	0.326781
$206$	83.9292i	0.407423i
$207$	0	0
$208$	18.4517	0.0887101
$209$	53.0947i	0.254042i
$210$	0	0
$211$	115.845	0.549029	0.274515	−	0.961583i	$-0.411483\pi$
$211$	115.845	0.549029	0.274515	+	0.961583i	$0.411483\pi$
$212$	− 224.429i	− 1.05863i
$213$	0	0
$214$	−183.586	−0.857879
$215$	196.202i	0.912568i
$216$	0	0
$217$	−233.036	−1.07390
$218$	21.5693i	0.0989418i
$219$	0	0
$220$	167.664	0.762110
$221$	− 35.4914i	− 0.160595i
$222$	0	0
$223$	−198.236	−0.888951	−0.444476	−	0.895791i	$-0.646610\pi$
$223$	−198.236	−0.888951	−0.444476	+	0.895791i	$0.646610\pi$
$224$	345.514i	1.54247i
$225$	0	0
$226$	20.9161	0.0925490
$227$	− 5.15186i	− 0.0226954i	−0.999936	−	0.0113477i	$-0.996388\pi$
$227$	− 5.15186i	− 0.0226954i	0.999936	−	0.0113477i	$-0.00361216\pi$
$228$	0	0
$229$	353.347	1.54300	0.771500	−	0.636229i	$-0.219507\pi$
$229$	353.347	1.54300	0.771500	+	0.636229i	$0.219507\pi$
$230$	− 124.412i	− 0.540921i
$231$	0	0
$232$	−195.851	−0.844184
$233$	− 140.561i	− 0.603265i	−0.953424	−	0.301632i	$-0.902469\pi$
$233$	− 140.561i	− 0.603265i	0.953424	−	0.301632i	$-0.0975315\pi$
$234$	0	0
$235$	66.7470	0.284030
$236$	181.073i	0.767258i
$237$	0	0
$238$	38.6420	0.162361
$239$	147.674i	0.617882i	0.951081	+	0.308941i	$0.0999746\pi$
$239$	147.674i	0.617882i	−0.951081	+	0.308941i	$0.900025\pi$
$240$	0	0
$241$	304.414	1.26313	0.631563	−	0.775324i	$-0.282414\pi$
$241$	304.414	1.26313	0.631563	+	0.775324i	$0.282414\pi$
$242$	19.6445i	0.0811755i
$243$	0	0
$244$	207.060	0.848606
$245$	333.600i	1.36163i
$246$	0	0
$247$	−50.7840	−0.205603
$248$	− 170.842i	− 0.688880i
$249$	0	0
$250$	131.384	0.525537
$251$	− 124.791i	− 0.497175i	−0.968609	−	0.248588i	$-0.920034\pi$
$251$	− 124.791i	− 0.497175i	0.968609	−	0.248588i	$-0.0799664\pi$
$252$	0	0
$253$	−233.794	−0.924088
$254$	181.358i	0.714007i
$255$	0	0
$256$	−235.875	−0.921388
$257$	− 457.031i	− 1.77833i	−0.457585	−	0.889166i	$-0.651285\pi$
$257$	− 457.031i	− 1.77833i	0.457585	−	0.889166i	$-0.348715\pi$
$258$	0	0
$259$	−185.996	−0.718132
$260$	160.367i	0.616798i
$261$	0	0
$262$	−114.377	−0.436553
$263$	− 467.947i	− 1.77927i	−0.456677	−	0.889633i	$-0.650960\pi$
$263$	− 467.947i	− 1.77927i	0.456677	−	0.889633i	$-0.349040\pi$
$264$	0	0
$265$	455.515	1.71892
$266$	− 55.2921i	− 0.207865i
$267$	0	0
$268$	−257.588	−0.961151
$269$	317.049i	1.17862i	0.807907	+	0.589310i	$0.200601\pi$
$269$	317.049i	1.17862i	−0.807907	+	0.589310i	$0.799399\pi$
$270$	0	0
$271$	115.698	0.426931	0.213465	−	0.976951i	$-0.431525\pi$
$271$	115.698	0.426931	0.213465	+	0.976951i	$0.431525\pi$
$272$	− 5.19013i	− 0.0190814i
$273$	0	0
$274$	76.2247	0.278192
$275$	46.7025i	0.169827i
$276$	0	0
$277$	−115.306	−0.416267	−0.208133	−	0.978100i	$-0.566739\pi$
$277$	−115.306	−0.416267	−0.208133	+	0.978100i	$0.566739\pi$
$278$	145.469i	0.523271i
$279$	0	0
$280$	−437.940	−1.56407
$281$	− 81.5920i	− 0.290363i	−0.989405	−	0.145181i	$-0.953623\pi$
$281$	− 81.5920i	− 0.290363i	0.989405	−	0.145181i	$-0.0463766\pi$
$282$	0	0
$283$	−417.139	−1.47399	−0.736995	−	0.675898i	$-0.763756\pi$
$283$	−417.139	−1.47399	−0.736995	+	0.675898i	$0.763756\pi$
$284$	− 322.674i	− 1.13618i
$285$	0	0
$286$	−153.152	−0.535498
$287$	131.098i	0.456786i
$288$	0	0
$289$	279.017	0.965456
$290$	− 158.484i	− 0.546498i
$291$	0	0
$292$	241.721	0.827812
$293$	345.833i	1.18032i	0.807287	+	0.590159i	$0.200935\pi$
$293$	345.833i	1.18032i	−0.807287	+	0.590159i	$0.799065\pi$
$294$	0	0
$295$	−367.517	−1.24582
$296$	− 136.357i	− 0.460665i
$297$	0	0
$298$	234.232	0.786012
$299$	− 223.620i	− 0.747892i
$300$	0	0
$301$	−383.961	−1.27562
$302$	− 148.766i	− 0.492604i
$303$	0	0
$304$	−7.42646	−0.0244291
$305$	420.262i	1.37791i
$306$	0	0
$307$	577.336	1.88057	0.940286	−	0.340384i	$-0.110557\pi$
$307$	577.336	1.88057	0.940286	+	0.340384i	$0.110557\pi$
$308$	328.114i	1.06530i
$309$	0	0
$310$	138.247	0.445959
$311$	449.103i	1.44406i	0.691861	+	0.722030i	$0.256791\pi$
$311$	449.103i	1.44406i	−0.691861	+	0.722030i	$0.743209\pi$
$312$	0	0
$313$	10.8376	0.0346248	0.0173124	−	0.999850i	$-0.494489\pi$
$313$	10.8376	0.0346248	0.0173124	+	0.999850i	$0.494489\pi$
$314$	161.311i	0.513730i
$315$	0	0
$316$	−54.0721	−0.171114
$317$	− 218.233i	− 0.688433i	−0.938890	−	0.344217i	$-0.888145\pi$
$317$	− 218.233i	− 0.688433i	0.938890	−	0.344217i	$-0.111855\pi$
$318$	0	0
$319$	−297.824	−0.933616
$320$	− 169.605i	− 0.530015i
$321$	0	0
$322$	243.470	0.756118
$323$	14.2846i	0.0442249i
$324$	0	0
$325$	−44.6700	−0.137446
$326$	− 46.2724i	− 0.141940i
$327$	0	0
$328$	−96.1099	−0.293018
$329$	130.622i	0.397026i
$330$	0	0
$331$	600.678	1.81474	0.907369	−	0.420335i	$-0.138087\pi$
$331$	600.678	1.81474	0.907369	+	0.420335i	$0.138087\pi$
$332$	− 83.8138i	− 0.252451i
$333$	0	0
$334$	192.665	0.576841
$335$	− 522.818i	− 1.56065i
$336$	0	0
$337$	449.269	1.33314	0.666571	−	0.745442i	$-0.267761\pi$
$337$	449.269	1.33314	0.666571	+	0.745442i	$0.267761\pi$
$338$	49.7154i	0.147087i
$339$	0	0
$340$	45.1085	0.132672
$341$	− 259.794i	− 0.761859i
$342$	0	0
$343$	−136.661	−0.398429
$344$	− 281.488i	− 0.818280i
$345$	0	0
$346$	−12.6431	−0.0365406
$347$	236.161i	0.680578i	0.940321	+	0.340289i	$0.110525\pi$
$347$	236.161i	0.680578i	−0.940321	+	0.340289i	$0.889475\pi$
$348$	0	0
$349$	239.784	0.687060	0.343530	−	0.939142i	$-0.388377\pi$
$349$	239.784	0.687060	0.343530	+	0.939142i	$0.388377\pi$
$350$	− 48.6353i	− 0.138958i
$351$	0	0
$352$	−385.187	−1.09428
$353$	− 25.5908i	− 0.0724951i	−0.999343	−	0.0362475i	$-0.988460\pi$
$353$	− 25.5908i	− 0.0724951i	0.999343	−	0.0362475i	$-0.0115405\pi$
$354$	0	0
$355$	654.920	1.84484
$356$	183.060i	0.514214i
$357$	0	0
$358$	328.063	0.916378
$359$	19.3755i	0.0539706i	0.999636	+	0.0269853i	$0.00859074\pi$
$359$	19.3755i	0.0539706i	−0.999636	+	0.0269853i	$0.991409\pi$
$360$	0	0
$361$	−340.560	−0.943381
$362$	126.312i	0.348927i
$363$	0	0
$364$	−313.834	−0.862181
$365$	490.612i	1.34414i
$366$	0	0
$367$	390.715	1.06462	0.532309	−	0.846550i	$-0.321324\pi$
$367$	390.715	1.06462	0.532309	+	0.846550i	$0.321324\pi$
$368$	− 32.7013i	− 0.0888622i
$369$	0	0
$370$	110.341	0.298220
$371$	891.428i	2.40277i
$372$	0	0
$373$	−178.823	−0.479418	−0.239709	−	0.970845i	$-0.577052\pi$
$373$	−178.823	−0.479418	−0.239709	+	0.970845i	$0.577052\pi$
$374$	43.0790i	0.115185i
$375$	0	0
$376$	−95.7609	−0.254683
$377$	− 284.862i	− 0.755603i
$378$	0	0
$379$	3.48118	0.00918518	0.00459259	−	0.999989i	$-0.498538\pi$
$379$	3.48118	0.00918518	0.00459259	+	0.999989i	$0.498538\pi$
$380$	− 64.5449i	− 0.169855i
$381$	0	0
$382$	−126.115	−0.330143
$383$	− 17.2265i	− 0.0449778i	−0.999747	−	0.0224889i	$-0.992841\pi$
$383$	− 17.2265i	− 0.0449778i	0.999747	−	0.0224889i	$-0.00715904\pi$
$384$	0	0
$385$	−665.960	−1.72977
$386$	− 232.318i	− 0.601860i
$387$	0	0
$388$	−211.654	−0.545500
$389$	316.773i	0.814326i	0.913356	+	0.407163i	$0.133482\pi$
$389$	316.773i	0.814326i	−0.913356	+	0.407163i	$0.866518\pi$
$390$	0	0
$391$	−62.9002	−0.160870
$392$	− 478.611i	− 1.22095i
$393$	0	0
$394$	339.825	0.862500
$395$	− 109.748i	− 0.277843i
$396$	0	0
$397$	268.082	0.675269	0.337634	−	0.941277i	$-0.390373\pi$
$397$	268.082	0.675269	0.337634	+	0.941277i	$0.390373\pi$
$398$	95.5584i	0.240096i
$399$	0	0
$400$	−6.53237	−0.0163309
$401$	− 97.4667i	− 0.243059i	−0.992588	−	0.121530i	$-0.961220\pi$
$401$	− 97.4667i	− 0.243059i	0.992588	−	0.121530i	$-0.0387799\pi$
$402$	0	0
$403$	248.487	0.616594
$404$	− 12.6686i	− 0.0313578i
$405$	0	0
$406$	310.149	0.763914
$407$	− 207.353i	− 0.509467i
$408$	0	0
$409$	−405.343	−0.991060	−0.495530	−	0.868591i	$-0.665026\pi$
$409$	−405.343	−0.991060	−0.495530	+	0.868591i	$0.665026\pi$
$410$	− 77.7731i	− 0.189691i
$411$	0	0
$412$	−191.732	−0.465370
$413$	− 719.219i	− 1.74145i
$414$	0	0
$415$	170.114	0.409913
$416$	− 368.424i	− 0.885634i
$417$	0	0
$418$	61.6410	0.147466
$419$	117.335i	0.280036i	0.990149	+	0.140018i	$0.0447160\pi$
$419$	117.335i	0.280036i	−0.990149	+	0.140018i	$0.955284\pi$
$420$	0	0
$421$	−521.334	−1.23832	−0.619161	−	0.785264i	$-0.712527\pi$
$421$	−521.334	−1.23832	−0.619161	+	0.785264i	$0.712527\pi$
$422$	− 134.492i	− 0.318701i
$423$	0	0
$424$	−653.521	−1.54132
$425$	12.5649i	0.0295644i
$426$	0	0
$427$	−822.439	−1.92609
$428$	− 419.394i	− 0.979893i
$429$	0	0
$430$	227.783	0.529729
$431$	263.580i	0.611555i	0.952103	+	0.305777i	$0.0989163\pi$
$431$	263.580i	0.611555i	−0.952103	+	0.305777i	$0.901084\pi$
$432$	0	0
$433$	−702.013	−1.62128	−0.810639	−	0.585547i	$-0.800880\pi$
$433$	−702.013	−1.62128	−0.810639	+	0.585547i	$0.800880\pi$
$434$	270.545i	0.623377i
$435$	0	0
$436$	−49.2741	−0.113014
$437$	90.0027i	0.205956i
$438$	0	0
$439$	−442.915	−1.00892	−0.504459	−	0.863436i	$-0.668308\pi$
$439$	−442.915	−1.00892	−0.504459	+	0.863436i	$0.668308\pi$
$440$	− 488.226i	− 1.10960i
$441$	0	0
$442$	−41.2042	−0.0932222
$443$	− 606.842i	− 1.36985i	−0.728615	−	0.684923i	$-0.759836\pi$
$443$	− 606.842i	− 1.36985i	0.728615	−	0.684923i	$-0.240164\pi$
$444$	0	0
$445$	−371.550	−0.834945
$446$	230.145i	0.516019i
$447$	0	0
$448$	331.911	0.740873
$449$	− 401.541i	− 0.894302i	−0.894459	−	0.447151i	$-0.852439\pi$
$449$	− 401.541i	− 0.894302i	0.894459	−	0.447151i	$-0.147561\pi$
$450$	0	0
$451$	−146.151	−0.324060
$452$	47.7818i	0.105712i
$453$	0	0
$454$	−5.98111	−0.0131743
$455$	− 636.977i	− 1.39995i
$456$	0	0
$457$	−147.113	−0.321909	−0.160955	−	0.986962i	$-0.551457\pi$
$457$	−147.113	−0.321909	−0.160955	+	0.986962i	$0.551457\pi$
$458$	− 410.223i	− 0.895683i
$459$	0	0
$460$	284.213	0.617855
$461$	785.412i	1.70371i	0.523774	+	0.851857i	$0.324524\pi$
$461$	785.412i	1.70371i	−0.523774	+	0.851857i	$0.675476\pi$
$462$	0	0
$463$	−358.453	−0.774196	−0.387098	−	0.922039i	$-0.626523\pi$
$463$	−358.453	−0.774196	−0.387098	+	0.922039i	$0.626523\pi$
$464$	− 41.6572i	− 0.0897783i
$465$	0	0
$466$	−163.186	−0.350184
$467$	− 101.929i	− 0.218263i	−0.994027	−	0.109131i	$-0.965193\pi$
$467$	− 101.929i	− 0.218263i	0.994027	−	0.109131i	$-0.0348069\pi$
$468$	0	0
$469$	1023.14	2.18153
$470$	− 77.4907i	− 0.164874i
$471$	0	0
$472$	527.271	1.11710
$473$	− 428.050i	− 0.904967i
$474$	0	0
$475$	17.9788	0.0378502
$476$	88.2759i	0.185453i
$477$	0	0
$478$	171.444	0.358669
$479$	126.370i	0.263820i	0.991262	+	0.131910i	$0.0421110\pi$
$479$	126.370i	0.263820i	−0.991262	+	0.131910i	$0.957889\pi$
$480$	0	0
$481$	198.329	0.412326
$482$	− 353.413i	− 0.733221i
$483$	0	0
$484$	−44.8769	−0.0927209
$485$	− 429.586i	− 0.885745i
$486$	0	0
$487$	454.010	0.932258	0.466129	−	0.884717i	$-0.345648\pi$
$487$	454.010	0.932258	0.466129	+	0.884717i	$0.345648\pi$
$488$	− 602.944i	− 1.23554i
$489$	0	0
$490$	387.297	0.790403
$491$	− 316.087i	− 0.643763i	−0.946780	−	0.321881i	$-0.895685\pi$
$491$	− 316.087i	− 0.643763i	0.946780	−	0.321881i	$-0.104315\pi$
$492$	0	0
$493$	−80.1266	−0.162529
$494$	58.9583i	0.119349i
$495$	0	0
$496$	36.3379	0.0732618
$497$	1281.66i	2.57879i
$498$	0	0
$499$	122.272	0.245034	0.122517	−	0.992466i	$-0.460903\pi$
$499$	122.272	0.245034	0.122517	+	0.992466i	$0.460903\pi$
$500$	300.141i	0.600283i
$501$	0	0
$502$	−144.878	−0.288601
$503$	451.536i	0.897685i	0.893611	+	0.448843i	$0.148164\pi$
$503$	451.536i	0.897685i	−0.893611	+	0.448843i	$0.851836\pi$
$504$	0	0
$505$	25.7129	0.0509167
$506$	271.426i	0.536416i
$507$	0	0
$508$	−414.304	−0.815559
$509$	499.710i	0.981748i	0.871231	+	0.490874i	$0.163322\pi$
$509$	499.710i	0.981748i	−0.871231	+	0.490874i	$0.836678\pi$
$510$	0	0
$511$	−960.113	−1.87889
$512$	− 104.621i	− 0.204338i
$513$	0	0
$514$	−530.596	−1.03229
$515$	− 389.152i	− 0.755635i
$516$	0	0
$517$	−145.620	−0.281664
$518$	215.935i	0.416862i
$519$	0	0
$520$	466.978	0.898035
$521$	− 687.777i	− 1.32011i	−0.751218	−	0.660054i	$-0.770533\pi$
$521$	− 687.777i	− 1.32011i	0.751218	−	0.660054i	$-0.229467\pi$
$522$	0	0
$523$	−520.436	−0.995098	−0.497549	−	0.867436i	$-0.665767\pi$
$523$	−520.436	−0.995098	−0.497549	+	0.867436i	$0.665767\pi$
$524$	− 261.289i	− 0.498643i
$525$	0	0
$526$	−543.269	−1.03283
$527$	− 69.8951i	− 0.132628i
$528$	0	0
$529$	132.687	0.250826
$530$	− 528.836i	− 0.997803i
$531$	0	0
$532$	126.312	0.237429
$533$	− 139.790i	− 0.262271i
$534$	0	0
$535$	851.229	1.59108
$536$	750.079i	1.39940i
$537$	0	0
$538$	368.082	0.684167
$539$	− 727.808i	− 1.35029i
$540$	0	0
$541$	803.120	1.48451	0.742255	−	0.670117i	$-0.233756\pi$
$541$	803.120	1.48451	0.742255	+	0.670117i	$0.233756\pi$
$542$	− 134.321i	− 0.247825i
$543$	0	0
$544$	−103.631	−0.190498
$545$	− 100.010i	− 0.183504i
$546$	0	0
$547$	−68.6886	−0.125573	−0.0627866	−	0.998027i	$-0.519999\pi$
$547$	−68.6886	−0.125573	−0.0627866	+	0.998027i	$0.519999\pi$
$548$	174.132i	0.317759i
$549$	0	0
$550$	54.2198	0.0985815
$551$	114.652i	0.208079i
$552$	0	0
$553$	214.773	0.388379
$554$	133.866i	0.241635i
$555$	0	0
$556$	−332.318	−0.597695
$557$	− 664.091i	− 1.19226i	−0.802886	−	0.596132i	$-0.796704\pi$
$557$	− 664.091i	− 1.19226i	0.802886	−	0.596132i	$-0.203296\pi$
$558$	0	0
$559$	409.421	0.732416
$560$	− 93.1491i	− 0.166338i
$561$	0	0
$562$	−94.7252	−0.168550
$563$	347.655i	0.617504i	0.951143	+	0.308752i	$0.0999114\pi$
$563$	347.655i	0.617504i	−0.951143	+	0.308752i	$0.900089\pi$
$564$	0	0
$565$	−96.9810	−0.171648
$566$	484.283i	0.855623i
$567$	0	0
$568$	−939.603	−1.65423
$569$	391.019i	0.687203i	0.939115	+	0.343602i	$0.111647\pi$
$569$	391.019i	0.687203i	−0.939115	+	0.343602i	$0.888353\pi$
$570$	0	0
$571$	855.725	1.49864	0.749321	−	0.662207i	$-0.230380\pi$
$571$	855.725	1.49864	0.749321	+	0.662207i	$0.230380\pi$
$572$	− 349.870i	− 0.611661i
$573$	0	0
$574$	152.200	0.265156
$575$	79.1670i	0.137682i
$576$	0	0
$577$	−627.491	−1.08751	−0.543753	−	0.839245i	$-0.682997\pi$
$577$	−627.491	−1.08751	−0.543753	+	0.839245i	$0.682997\pi$
$578$	− 323.928i	− 0.560429i
$579$	0	0
$580$	362.051	0.624225
$581$	332.907i	0.572990i
$582$	0	0
$583$	−993.786	−1.70461
$584$	− 703.874i	− 1.20526i
$585$	0	0
$586$	401.499	0.685152
$587$	− 744.557i	− 1.26841i	−0.773165	−	0.634205i	$-0.781327\pi$
$587$	− 744.557i	− 1.26841i	0.773165	−	0.634205i	$-0.218673\pi$
$588$	0	0
$589$	−100.012	−0.169799
$590$	426.673i	0.723175i
$591$	0	0
$592$	29.0029	0.0489913
$593$	− 625.722i	− 1.05518i	−0.849499	−	0.527591i	$-0.823096\pi$
$593$	− 625.722i	− 1.05518i	0.849499	−	0.527591i	$-0.176904\pi$
$594$	0	0
$595$	−179.170	−0.301126
$596$	535.092i	0.897805i
$597$	0	0
$598$	−259.614	−0.434137
$599$	573.042i	0.956664i	0.878179	+	0.478332i	$0.158758\pi$
$599$	573.042i	0.956664i	−0.878179	+	0.478332i	$0.841242\pi$
$600$	0	0
$601$	1131.48	1.88266	0.941330	−	0.337487i	$-0.109577\pi$
$601$	1131.48	1.88266	0.941330	+	0.337487i	$0.109577\pi$
$602$	445.765i	0.740473i
$603$	0	0
$604$	339.850	0.562666
$605$	− 91.0850i	− 0.150554i
$606$	0	0
$607$	−232.474	−0.382988	−0.191494	−	0.981494i	$-0.561333\pi$
$607$	−232.474	−0.382988	−0.191494	+	0.981494i	$0.561333\pi$
$608$	148.284i	0.243888i
$609$	0	0
$610$	487.908	0.799850
$611$	− 139.283i	− 0.227959i
$612$	0	0
$613$	−1067.78	−1.74189	−0.870945	−	0.491380i	$-0.836493\pi$
$613$	−1067.78	−1.74189	−0.870945	+	0.491380i	$0.836493\pi$
$614$	− 670.265i	− 1.09164i
$615$	0	0
$616$	955.443	1.55104
$617$	− 421.751i	− 0.683552i	−0.939782	−	0.341776i	$-0.888972\pi$
$617$	− 421.751i	− 0.683552i	0.939782	−	0.341776i	$-0.111028\pi$
$618$	0	0
$619$	600.839	0.970661	0.485331	−	0.874331i	$-0.338699\pi$
$619$	600.839	0.970661	0.485331	+	0.874331i	$0.338699\pi$
$620$	315.820i	0.509386i
$621$	0	0
$622$	521.392	0.838250
$623$	− 727.112i	− 1.16711i
$624$	0	0
$625$	−708.604	−1.13377
$626$	− 12.5820i	− 0.0200990i
$627$	0	0
$628$	−368.509	−0.586797
$629$	− 55.7864i	− 0.0886906i
$630$	0	0
$631$	−1109.24	−1.75791	−0.878956	−	0.476903i	$-0.841759\pi$
$631$	−1109.24	−1.75791	−0.878956	+	0.476903i	$0.841759\pi$
$632$	157.454i	0.249136i
$633$	0	0
$634$	−253.361	−0.399623
$635$	− 840.897i	− 1.32425i
$636$	0	0
$637$	696.133	1.09283
$638$	345.762i	0.541947i
$639$	0	0
$640$	509.318	0.795809
$641$	− 165.270i	− 0.257832i	−0.991656	−	0.128916i	$-0.958850\pi$
$641$	− 165.270i	− 0.257832i	0.991656	−	0.128916i	$-0.0411497\pi$
$642$	0	0
$643$	794.624	1.23581	0.617904	−	0.786254i	$-0.287982\pi$
$643$	794.624	1.23581	0.617904	+	0.786254i	$0.287982\pi$
$644$	556.197i	0.863659i
$645$	0	0
$646$	16.5839	0.0256717
$647$	− 222.504i	− 0.343900i	−0.985106	−	0.171950i	$-0.944993\pi$
$647$	− 222.504i	− 0.343900i	0.985106	−	0.171950i	$-0.0550068\pi$
$648$	0	0
$649$	801.803	1.23544
$650$	51.8602i	0.0797849i
$651$	0	0
$652$	105.707	0.162128
$653$	466.485i	0.714373i	0.934033	+	0.357186i	$0.116264\pi$
$653$	466.485i	0.714373i	−0.934033	+	0.357186i	$0.883736\pi$
$654$	0	0
$655$	530.328	0.809661
$656$	− 20.4424i	− 0.0311622i
$657$	0	0
$658$	151.647	0.230466
$659$	− 208.274i	− 0.316045i	−0.987436	−	0.158022i	$-0.949488\pi$
$659$	− 208.274i	− 0.316045i	0.987436	−	0.158022i	$-0.0505118\pi$
$660$	0	0
$661$	−633.782	−0.958823	−0.479411	−	0.877590i	$-0.659150\pi$
$661$	−633.782	−0.958823	−0.479411	+	0.877590i	$0.659150\pi$
$662$	− 697.365i	− 1.05342i
$663$	0	0
$664$	−244.060	−0.367560
$665$	256.371i	0.385521i
$666$	0	0
$667$	−504.851	−0.756898
$668$	440.135i	0.658884i
$669$	0	0
$670$	−606.972	−0.905928
$671$	− 916.876i	− 1.36643i
$672$	0	0
$673$	110.021	0.163478	0.0817390	−	0.996654i	$-0.473953\pi$
$673$	110.021	0.163478	0.0817390	+	0.996654i	$0.473953\pi$
$674$	− 521.584i	− 0.773864i
$675$	0	0
$676$	−113.573	−0.168007
$677$	143.039i	0.211283i	0.994404	+	0.105642i	$0.0336896\pi$
$677$	143.039i	0.211283i	−0.994404	+	0.105642i	$0.966310\pi$
$678$	0	0
$679$	840.686	1.23812
$680$	− 131.353i	− 0.193166i
$681$	0	0
$682$	−301.611	−0.442245
$683$	905.343i	1.32554i	0.748824	+	0.662769i	$0.230619\pi$
$683$	905.343i	1.32554i	−0.748824	+	0.662769i	$0.769381\pi$
$684$	0	0
$685$	−353.429	−0.515954
$686$	158.658i	0.231280i
$687$	0	0
$688$	59.8721	0.0870234
$689$	− 950.537i	− 1.37959i
$690$	0	0
$691$	532.010	0.769913	0.384957	−	0.922935i	$-0.374216\pi$
$691$	532.010	0.769913	0.384957	+	0.922935i	$0.374216\pi$
$692$	− 28.8825i	− 0.0417377i
$693$	0	0
$694$	274.173	0.395063
$695$	− 674.494i	− 0.970495i
$696$	0	0
$697$	−39.3205	−0.0564140
$698$	− 278.380i	− 0.398826i
$699$	0	0
$700$	111.105	0.158722
$701$	905.026i	1.29105i	0.763739	+	0.645525i	$0.223361\pi$
$701$	905.026i	1.29105i	−0.763739	+	0.645525i	$0.776639\pi$
$702$	0	0
$703$	−79.8237	−0.113547
$704$	370.023i	0.525601i
$705$	0	0
$706$	−29.7099	−0.0420820
$707$	50.3194i	0.0711731i
$708$	0	0
$709$	−363.866	−0.513211	−0.256605	−	0.966516i	$-0.582604\pi$
$709$	−363.866	−0.513211	−0.256605	+	0.966516i	$0.582604\pi$
$710$	− 760.337i	− 1.07090i
$711$	0	0
$712$	533.058	0.748677
$713$	− 440.385i	− 0.617651i
$714$	0	0
$715$	710.118	0.993172
$716$	749.446i	1.04671i
$717$	0	0
$718$	22.4942	0.0313289
$719$	449.557i	0.625253i	0.949876	+	0.312626i	$0.101209\pi$
$719$	449.557i	0.625253i	−0.949876	+	0.312626i	$0.898791\pi$
$720$	0	0
$721$	761.559	1.05625
$722$	395.378i	0.547615i
$723$	0	0
$724$	−288.553	−0.398555
$725$	100.848i	0.139101i
$726$	0	0
$727$	579.366	0.796927	0.398463	−	0.917184i	$-0.369544\pi$
$727$	579.366	0.796927	0.398463	+	0.917184i	$0.369544\pi$
$728$	913.862i	1.25530i
$729$	0	0
$730$	569.582	0.780250
$731$	− 115.163i	− 0.157541i
$732$	0	0
$733$	936.322	1.27738	0.638691	−	0.769463i	$-0.279476\pi$
$733$	936.322	1.27738	0.638691	+	0.769463i	$0.279476\pi$
$734$	− 453.605i	− 0.617991i
$735$	0	0
$736$	−652.944	−0.887152
$737$	1140.62i	1.54765i
$738$	0	0
$739$	527.835	0.714256	0.357128	−	0.934055i	$-0.383756\pi$
$739$	527.835	0.714256	0.357128	+	0.934055i	$0.383756\pi$
$740$	252.070i	0.340635i
$741$	0	0
$742$	1034.91	1.39476
$743$	− 36.8321i	− 0.0495721i	−0.999693	−	0.0247861i	$-0.992110\pi$
$743$	− 36.8321i	− 0.0495721i	0.999693	−	0.0247861i	$-0.00789045\pi$
$744$	0	0
$745$	−1086.06	−1.45779
$746$	207.607i	0.278293i
$747$	0	0
$748$	−98.4121	−0.131567
$749$	1665.83i	2.22407i
$750$	0	0
$751$	98.5014	0.131160	0.0655802	−	0.997847i	$-0.479110\pi$
$751$	98.5014	0.131160	0.0655802	+	0.997847i	$0.479110\pi$
$752$	− 20.3682i	− 0.0270854i
$753$	0	0
$754$	−330.714	−0.438613
$755$	689.781i	0.913617i
$756$	0	0
$757$	−1385.09	−1.82971	−0.914857	−	0.403779i	$-0.867697\pi$
$757$	−1385.09	−1.82971	−0.914857	+	0.403779i	$0.867697\pi$
$758$	− 4.04152i	− 0.00533182i
$759$	0	0
$760$	−187.950	−0.247303
$761$	− 1454.92i	− 1.91185i	−0.293608	−	0.955926i	$-0.594856\pi$
$761$	− 1454.92i	− 1.91185i	0.293608	−	0.955926i	$-0.405144\pi$
$762$	0	0
$763$	195.716	0.256509
$764$	− 288.103i	− 0.377098i
$765$	0	0
$766$	−19.9993	−0.0261088
$767$	766.908i	0.999880i
$768$	0	0
$769$	−100.066	−0.130125	−0.0650624	−	0.997881i	$-0.520725\pi$
$769$	−100.066	−0.130125	−0.0650624	+	0.997881i	$0.520725\pi$
$770$	773.154i	1.00410i
$771$	0	0
$772$	530.720	0.687461
$773$	632.693i	0.818491i	0.912424	+	0.409245i	$0.134208\pi$
$773$	632.693i	0.818491i	−0.912424	+	0.409245i	$0.865792\pi$
$774$	0	0
$775$	−87.9708	−0.113511
$776$	616.321i	0.794228i
$777$	0	0
$778$	367.761	0.472701
$779$	56.2630i	0.0722247i
$780$	0	0
$781$	−1428.82	−1.82948
$782$	73.0247i	0.0933820i
$783$	0	0
$784$	101.800	0.129847
$785$	− 747.948i	− 0.952800i
$786$	0	0
$787$	−815.667	−1.03643	−0.518213	−	0.855252i	$-0.673402\pi$
$787$	−815.667	−1.03643	−0.518213	+	0.855252i	$0.673402\pi$
$788$	776.315i	0.985171i
$789$	0	0
$790$	−127.413	−0.161283
$791$	− 189.789i	− 0.239935i
$792$	0	0
$793$	876.973	1.10589
$794$	− 311.233i	− 0.391981i
$795$	0	0
$796$	−218.299	−0.274245
$797$	1252.70i	1.57177i	0.618375	+	0.785883i	$0.287791\pi$
$797$	1252.70i	1.57177i	−0.618375	+	0.785883i	$0.712209\pi$
$798$	0	0
$799$	−39.1778	−0.0490335
$800$	130.431i	0.163039i
$801$	0	0
$802$	−113.155	−0.141091
$803$	− 1070.36i	− 1.33295i
$804$	0	0
$805$	−1128.89	−1.40235
$806$	− 288.485i	− 0.357921i
$807$	0	0
$808$	−36.8899	−0.0456559
$809$	− 1021.88i	− 1.26313i	−0.775321	−	0.631567i	$-0.782412\pi$
$809$	− 1021.88i	− 1.26313i	0.775321	−	0.631567i	$-0.217588\pi$
$810$	0	0
$811$	−365.788	−0.451034	−0.225517	−	0.974239i	$-0.572407\pi$
$811$	−365.788	−0.451034	−0.225517	+	0.974239i	$0.572407\pi$
$812$	708.522i	0.872564i
$813$	0	0
$814$	−240.729	−0.295736
$815$	214.550i	0.263252i
$816$	0	0
$817$	−164.784	−0.201694
$818$	470.589i	0.575292i
$819$	0	0
$820$	177.669	0.216670
$821$	− 9.69598i	− 0.0118100i	−0.999983	−	0.00590498i	$-0.998120\pi$
$821$	− 9.69598i	− 0.0118100i	0.999983	−	0.00590498i	$-0.00187962\pi$
$822$	0	0
$823$	0.893888	0.00108613	0.000543067	−	1.00000i	$-0.499827\pi$
$823$	0.893888	0.00108613	0.000543067		1.00000i	$0.499827\pi$
$824$	558.311i	0.677562i
$825$	0	0
$826$	−834.986	−1.01088
$827$	673.895i	0.814868i	0.913235	+	0.407434i	$0.133576\pi$
$827$	673.895i	0.814868i	−0.913235	+	0.407434i	$0.866424\pi$
$828$	0	0
$829$	28.9608	0.0349346	0.0174673	−	0.999847i	$-0.494440\pi$
$829$	28.9608	0.0349346	0.0174673	+	0.999847i	$0.494440\pi$
$830$	− 197.496i	− 0.237947i
$831$	0	0
$832$	−353.919	−0.425384
$833$	− 195.810i	− 0.235066i
$834$	0	0
$835$	−893.325	−1.06985
$836$	140.816i	0.168440i
$837$	0	0
$838$	136.221	0.162555
$839$	− 638.715i	− 0.761282i	−0.924723	−	0.380641i	$-0.875703\pi$
$839$	− 638.715i	− 0.761282i	0.924723	−	0.380641i	$-0.124297\pi$
$840$	0	0
$841$	197.886	0.235298
$842$	605.249i	0.718823i
$843$	0	0
$844$	307.241	0.364029
$845$	− 230.514i	− 0.272798i
$846$	0	0
$847$	178.251	0.210449
$848$	− 139.003i	− 0.163918i
$849$	0	0
$850$	14.5873	0.0171616
$851$	− 351.491i	− 0.413033i
$852$	0	0
$853$	777.081	0.910998	0.455499	−	0.890236i	$-0.349461\pi$
$853$	777.081	0.910998	0.455499	+	0.890236i	$0.349461\pi$
$854$	954.821i	1.11806i
$855$	0	0
$856$	−1221.25	−1.42669
$857$	1076.82i	1.25650i	0.778010	+	0.628252i	$0.216229\pi$
$857$	1076.82i	1.25650i	−0.778010	+	0.628252i	$0.783771\pi$
$858$	0	0
$859$	−80.9841	−0.0942772	−0.0471386	−	0.998888i	$-0.515010\pi$
$859$	−80.9841	−0.0942772	−0.0471386	+	0.998888i	$0.515010\pi$
$860$	520.361i	0.605070i
$861$	0	0
$862$	306.007	0.354996
$863$	1599.17i	1.85304i	0.376244	+	0.926520i	$0.377215\pi$
$863$	1599.17i	1.85304i	−0.376244	+	0.926520i	$0.622785\pi$
$864$	0	0
$865$	58.6217	0.0677708
$866$	815.011i	0.941121i
$867$	0	0
$868$	−618.049	−0.712038
$869$	239.435i	0.275529i
$870$	0	0
$871$	−1090.98	−1.25256
$872$	143.483i	0.164544i
$873$	0	0
$874$	104.490	0.119553
$875$	− 1192.16i	− 1.36247i
$876$	0	0
$877$	−378.583	−0.431679	−0.215840	−	0.976429i	$-0.569249\pi$
$877$	−378.583	−0.431679	−0.215840	+	0.976429i	$0.569249\pi$
$878$	514.208i	0.585658i
$879$	0	0
$880$	103.845	0.118006
$881$	1208.75i	1.37202i	0.727593	+	0.686009i	$0.240639\pi$
$881$	1208.75i	1.37202i	−0.727593	+	0.686009i	$0.759361\pi$
$882$	0	0
$883$	−596.289	−0.675299	−0.337649	−	0.941272i	$-0.609632\pi$
$883$	−596.289	−0.675299	−0.337649	+	0.941272i	$0.609632\pi$
$884$	− 94.1292i	− 0.106481i
$885$	0	0
$886$	−704.521	−0.795170
$887$	− 101.561i	− 0.114500i	−0.998360	−	0.0572499i	$-0.981767\pi$
$887$	− 101.561i	− 0.114500i	0.998360	−	0.0572499i	$-0.0182332\pi$
$888$	0	0
$889$	1645.61	1.85108
$890$	431.356i	0.484670i
$891$	0	0
$892$	−525.755	−0.589412
$893$	56.0587i	0.0627757i
$894$	0	0
$895$	−1521.12	−1.69958
$896$	996.718i	1.11241i
$897$	0	0
$898$	−466.174	−0.519125
$899$	− 560.993i	− 0.624019i
$900$	0	0
$901$	−267.369	−0.296747
$902$	169.676i	0.188111i
$903$	0	0
$904$	139.137	0.153913
$905$	− 585.666i	− 0.647145i
$906$	0	0
$907$	1725.63	1.90257	0.951285	−	0.308313i	$-0.0997644\pi$
$907$	1725.63	1.90257	0.951285	+	0.308313i	$0.0997644\pi$
$908$	− 13.6636i	− 0.0150480i
$909$	0	0
$910$	−739.506	−0.812644
$911$	312.709i	0.343259i	0.985162	+	0.171629i	$0.0549032\pi$
$911$	312.709i	0.343259i	−0.985162	+	0.171629i	$0.945097\pi$
$912$	0	0
$913$	−371.133	−0.406499
$914$	170.792i	0.186862i
$915$	0	0
$916$	937.135	1.02307
$917$	1037.84i	1.13177i
$918$	0	0
$919$	198.504	0.216000	0.108000	−	0.994151i	$-0.465555\pi$
$919$	198.504	0.216000	0.108000	+	0.994151i	$0.465555\pi$
$920$	− 827.609i	− 0.899574i
$921$	0	0
$922$	911.834	0.988974
$923$	− 1366.64i	− 1.48065i
$924$	0	0
$925$	−70.2135	−0.0759065
$926$	416.150i	0.449406i
$927$	0	0
$928$	−831.765	−0.896299
$929$	1329.75i	1.43138i	0.698420	+	0.715688i	$0.253887\pi$
$929$	1329.75i	1.43138i	−0.698420	+	0.715688i	$0.746113\pi$
$930$	0	0
$931$	−280.181	−0.300946
$932$	− 372.790i	− 0.399990i
$933$	0	0
$934$	−118.335	−0.126697
$935$	− 199.743i	− 0.213629i
$936$	0	0
$937$	1351.34	1.44220	0.721100	−	0.692831i	$-0.243637\pi$
$937$	1351.34	1.44220	0.721100	+	0.692831i	$0.243637\pi$
$938$	− 1187.82i	− 1.26634i
$939$	0	0
$940$	177.024	0.188323
$941$	− 777.265i	− 0.825999i	−0.910731	−	0.413000i	$-0.864481\pi$
$941$	− 777.265i	− 0.825999i	0.910731	−	0.413000i	$-0.135519\pi$
$942$	0	0
$943$	−247.746	−0.262721
$944$	112.150i	0.118803i
$945$	0	0
$946$	−496.949	−0.525316
$947$	274.291i	0.289642i	0.989458	+	0.144821i	$0.0462606\pi$
$947$	274.291i	0.289642i	−0.989458	+	0.144821i	$0.953739\pi$
$948$	0	0
$949$	1023.78	1.07879
$950$	− 20.8727i	− 0.0219713i
$951$	0	0
$952$	257.053	0.270014
$953$	991.095i	1.03997i	0.854174	+	0.519987i	$0.174063\pi$
$953$	991.095i	1.03997i	−0.854174	+	0.519987i	$0.825937\pi$
$954$	0	0
$955$	584.752	0.612306
$956$	391.655i	0.409681i
$957$	0	0
$958$	146.711	0.153143
$959$	− 691.649i	− 0.721219i
$960$	0	0
$961$	−471.641	−0.490782
$962$	− 230.253i	− 0.239348i
$963$	0	0
$964$	807.355	0.837506
$965$	1077.18i	1.11625i
$966$	0	0
$967$	312.452	0.323115	0.161557	−	0.986863i	$-0.448348\pi$
$967$	312.452	0.323115	0.161557	+	0.986863i	$0.448348\pi$
$968$	130.678i	0.134998i
$969$	0	0
$970$	−498.733	−0.514158
$971$	− 487.838i	− 0.502407i	−0.967934	−	0.251204i	$-0.919174\pi$
$971$	− 487.838i	− 0.502407i	0.967934	−	0.251204i	$-0.0808264\pi$
$972$	0	0
$973$	1319.96	1.35659
$974$	− 527.088i	− 0.541158i
$975$	0	0
$976$	128.245	0.131399
$977$	643.819i	0.658976i	0.944160	+	0.329488i	$0.106876\pi$
$977$	643.819i	0.658976i	−0.944160	+	0.329488i	$0.893124\pi$
$978$	0	0
$979$	810.603	0.827991
$980$	884.763i	0.902820i
$981$	0	0
$982$	−366.966	−0.373692
$983$	− 1539.82i	− 1.56645i	−0.621737	−	0.783226i	$-0.713573\pi$
$983$	− 1539.82i	− 1.56645i	0.621737	−	0.783226i	$-0.286427\pi$
$984$	0	0
$985$	−1575.66	−1.59965
$986$	93.0240i	0.0943448i
$987$	0	0
$988$	−134.688	−0.136324
$989$	− 725.601i	− 0.733672i
$990$	0	0
$991$	311.141	0.313967	0.156984	−	0.987601i	$-0.449823\pi$
$991$	311.141	0.313967	0.156984	+	0.987601i	$0.449823\pi$
$992$	− 725.555i	− 0.731407i
$993$	0	0
$994$	1487.96	1.49694
$995$	− 443.073i	− 0.445300i
$996$	0	0
$997$	−48.2298	−0.0483749	−0.0241875	−	0.999707i	$-0.507700\pi$
$997$	−48.2298	−0.0483749	−0.0241875	+	0.999707i	$0.507700\pi$
$998$	− 141.953i	− 0.142238i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.12		30
3.2	odd	2	inner	729.3.b.a.728.19		30
27.2	odd	18		243.3.f.b.53.4		30
27.4	even	9		243.3.f.a.107.4		30
27.5	odd	18		27.3.f.a.2.4	✓	30
27.7	even	9		243.3.f.d.134.2		30
27.11	odd	18		81.3.f.a.71.2		30
27.13	even	9		243.3.f.b.188.4		30
27.14	odd	18		243.3.f.c.188.2		30
27.16	even	9		27.3.f.a.14.4	yes	30
27.20	odd	18		243.3.f.a.134.4		30
27.22	even	9		81.3.f.a.8.2		30
27.23	odd	18		243.3.f.d.107.2		30
27.25	even	9		243.3.f.c.53.2		30
108.43	odd	18		432.3.bc.a.257.3		30
108.59	even	18		432.3.bc.a.353.3		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.4	✓	30	27.5	odd	18
27.3.f.a.14.4	yes	30	27.16	even	9
81.3.f.a.8.2		30	27.22	even	9
81.3.f.a.71.2		30	27.11	odd	18
243.3.f.a.107.4		30	27.4	even	9
243.3.f.a.134.4		30	27.20	odd	18
243.3.f.b.53.4		30	27.2	odd	18
243.3.f.b.188.4		30	27.13	even	9
243.3.f.c.53.2		30	27.25	even	9
243.3.f.c.188.2		30	27.14	odd	18
243.3.f.d.107.2		30	27.23	odd	18
243.3.f.d.134.2		30	27.7	even	9
432.3.bc.a.257.3		30	108.43	odd	18
432.3.bc.a.353.3		30	108.59	even	18
729.3.b.a.728.12		30	1.1	even	1	trivial
729.3.b.a.728.19		30	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.16096i q^{2} +2.65217 q^{4} +5.38300i q^{5} -10.5344 q^{7} -7.72291i q^{8} +O(q^{10})\)	\(q-1.16096i q^{2} +2.65217 q^{4} +5.38300i q^{5} -10.5344 q^{7} -7.72291i q^{8} +6.24946 q^{10} -11.7440i q^{11} +11.2329 q^{13} +12.2300i q^{14} +1.64265 q^{16} -3.15961i q^{17} -4.52102 q^{19} +14.2766i q^{20} -13.6343 q^{22} -19.9076i q^{23} -3.97672 q^{25} -13.0409i q^{26} -27.9389 q^{28} -25.3597i q^{29} +22.1215 q^{31} -32.7987i q^{32} -3.66818 q^{34} -56.7065i q^{35} +17.6561 q^{37} +5.24873i q^{38} +41.5725 q^{40} -12.4448i q^{41} +36.4484 q^{43} -31.1470i q^{44} -23.1120 q^{46} -12.3996i q^{47} +61.9729 q^{49} +4.61682i q^{50} +29.7914 q^{52} -84.6210i q^{53} +63.2178 q^{55} +81.3560i q^{56} -29.4417 q^{58} +68.2736i q^{59} +78.0720 q^{61} -25.6822i q^{62} -31.5075 q^{64} +60.4666i q^{65} -97.1238 q^{67} -8.37980i q^{68} -65.8341 q^{70} -121.664i q^{71} +91.1410 q^{73} -20.4981i q^{74} -11.9905 q^{76} +123.715i q^{77} -20.3879 q^{79} +8.84240i q^{80} -14.4479 q^{82} -31.6020i q^{83} +17.0082 q^{85} -42.3153i q^{86} -90.6977 q^{88} +69.0229i q^{89} -118.331 q^{91} -52.7983i q^{92} -14.3954 q^{94} -24.3367i q^{95} -79.8042 q^{97} -71.9482i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4}+O(q^{10}) \) 30 * q - 48 * q^4	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67} - 150 q^{70} + 132 q^{73} - 24 q^{76} - 12 q^{82} + 96 q^{88} - 192 q^{91} - 24 q^{94}+O(q^{100}) \) 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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