LMFDB - Embedded newform 6561.2.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6561.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6561, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.a (trivial)

Newform invariants

comment:select newform

sage:traces = [72,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.3898487662$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6561.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.66924 q^{2} +5.12486 q^{4} +1.81561 q^{5} -2.58238 q^{7} -8.34102 q^{8} -4.84630 q^{10} -3.66030 q^{11} +0.378476 q^{13} +6.89301 q^{14} +12.0145 q^{16} -2.27635 q^{17} +1.40631 q^{19} +9.30474 q^{20} +9.77023 q^{22} +3.54005 q^{23} -1.70357 q^{25} -1.01025 q^{26} -13.2344 q^{28} -6.60809 q^{29} +9.77392 q^{31} -15.3876 q^{32} +6.07613 q^{34} -4.68860 q^{35} -0.605354 q^{37} -3.75377 q^{38} -15.1440 q^{40} -3.34990 q^{41} +2.66812 q^{43} -18.7585 q^{44} -9.44925 q^{46} +0.150740 q^{47} -0.331292 q^{49} +4.54724 q^{50} +1.93964 q^{52} +13.4514 q^{53} -6.64567 q^{55} +21.5397 q^{56} +17.6386 q^{58} -4.24844 q^{59} +7.45019 q^{61} -26.0890 q^{62} +17.0442 q^{64} +0.687165 q^{65} +10.0915 q^{67} -11.6660 q^{68} +12.5150 q^{70} -1.75794 q^{71} +5.36123 q^{73} +1.61584 q^{74} +7.20712 q^{76} +9.45230 q^{77} +4.24959 q^{79} +21.8136 q^{80} +8.94171 q^{82} +2.60462 q^{83} -4.13296 q^{85} -7.12186 q^{86} +30.5306 q^{88} -3.19024 q^{89} -0.977371 q^{91} +18.1423 q^{92} -0.402362 q^{94} +2.55330 q^{95} +15.3717 q^{97} +0.884298 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

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 $

$2$	−2.66924	−1.88744	−0.943720	−	0.330745i	$-0.892700\pi$
$2$	−2.66924	−1.88744	−0.943720	+	0.330745i	$0.892700\pi$
$3$	0	0
$3$	0	0
$4$	5.12486	2.56243
$5$	1.81561	0.811965	0.405982	−	0.913881i	$-0.366929\pi$
$5$	1.81561	0.811965	0.405982	+	0.913881i	$0.366929\pi$
$6$	0	0
$7$	−2.58238	−0.976050	−0.488025	−	0.872830i	$-0.662282\pi$
$7$	−2.58238	−0.976050	−0.488025	+	0.872830i	$0.662282\pi$
$8$	−8.34102	−2.94900
$9$	0	0
$10$	−4.84630	−1.53253
$11$	−3.66030	−1.10362	−0.551811	−	0.833969i	$-0.686063\pi$
$11$	−3.66030	−1.10362	−0.551811	+	0.833969i	$0.686063\pi$
$12$	0	0
$13$	0.378476	0.104970	0.0524852	−	0.998622i	$-0.483286\pi$
$13$	0.378476	0.104970	0.0524852	+	0.998622i	$0.483286\pi$
$14$	6.89301	1.84224
$15$	0	0
$16$	12.0145	3.00362
$17$	−2.27635	−0.552096	−0.276048	−	0.961144i	$-0.589025\pi$
$17$	−2.27635	−0.552096	−0.276048	+	0.961144i	$0.589025\pi$
$18$	0	0
$19$	1.40631	0.322629	0.161314	−	0.986903i	$-0.448427\pi$
$19$	1.40631	0.322629	0.161314	+	0.986903i	$0.448427\pi$
$20$	9.30474	2.08060
$21$	0	0
$22$	9.77023	2.08302
$23$	3.54005	0.738151	0.369075	−	0.929399i	$-0.379674\pi$
$23$	3.54005	0.738151	0.369075	+	0.929399i	$0.379674\pi$
$24$	0	0
$25$	−1.70357	−0.340713
$26$	−1.01025	−0.198125
$27$	0	0
$28$	−13.2344	−2.50106
$29$	−6.60809	−1.22709	−0.613546	−	0.789659i	$-0.710258\pi$
$29$	−6.60809	−1.22709	−0.613546	+	0.789659i	$0.710258\pi$
$30$	0	0
$31$	9.77392	1.75545	0.877724	−	0.479166i	$-0.159061\pi$
$31$	9.77392	1.75545	0.877724	+	0.479166i	$0.159061\pi$
$32$	−15.3876	−2.72016
$33$	0	0
$34$	6.07613	1.04205
$35$	−4.68860	−0.792518
$36$	0	0
$37$	−0.605354	−0.0995196	−0.0497598	−	0.998761i	$-0.515846\pi$
$37$	−0.605354	−0.0995196	−0.0497598	+	0.998761i	$0.515846\pi$
$38$	−3.75377	−0.608942
$39$	0	0
$40$	−15.1440	−2.39448
$41$	−3.34990	−0.523167	−0.261583	−	0.965181i	$-0.584245\pi$
$41$	−3.34990	−0.523167	−0.261583	+	0.965181i	$0.584245\pi$
$42$	0	0
$43$	2.66812	0.406884	0.203442	−	0.979087i	$-0.434787\pi$
$43$	2.66812	0.406884	0.203442	+	0.979087i	$0.434787\pi$
$44$	−18.7585	−2.82796
$45$	0	0
$46$	−9.44925	−1.39322
$47$	0.150740	0.0219877	0.0109938	−	0.999940i	$-0.496500\pi$
$47$	0.150740	0.0219877	0.0109938	+	0.999940i	$0.496500\pi$
$48$	0	0
$49$	−0.331292	−0.0473274
$50$	4.54724	0.643076
$51$	0	0
$52$	1.93964	0.268980
$53$	13.4514	1.84770	0.923849	−	0.382756i	$-0.125025\pi$
$53$	13.4514	1.84770	0.923849	+	0.382756i	$0.125025\pi$
$54$	0	0
$55$	−6.64567	−0.896102
$56$	21.5397	2.87837
$57$	0	0
$58$	17.6386	2.31606
$59$	−4.24844	−0.553100	−0.276550	−	0.961000i	$-0.589191\pi$
$59$	−4.24844	−0.553100	−0.276550	+	0.961000i	$0.589191\pi$
$60$	0	0
$61$	7.45019	0.953899	0.476949	−	0.878931i	$-0.341742\pi$
$61$	7.45019	0.953899	0.476949	+	0.878931i	$0.341742\pi$
$62$	−26.0890	−3.31330
$63$	0	0
$64$	17.0442	2.13052
$65$	0.687165	0.0852323
$66$	0	0
$67$	10.0915	1.23287	0.616435	−	0.787406i	$-0.288576\pi$
$67$	10.0915	1.23287	0.616435	+	0.787406i	$0.288576\pi$
$68$	−11.6660	−1.41471
$69$	0	0
$70$	12.5150	1.49583
$71$	−1.75794	−0.208629	−0.104315	−	0.994544i	$-0.533265\pi$
$71$	−1.75794	−0.208629	−0.104315	+	0.994544i	$0.533265\pi$
$72$	0	0
$73$	5.36123	0.627484	0.313742	−	0.949508i	$-0.398417\pi$
$73$	5.36123	0.627484	0.313742	+	0.949508i	$0.398417\pi$
$74$	1.61584	0.187837
$75$	0	0
$76$	7.20712	0.826714
$77$	9.45230	1.07719
$78$	0	0
$79$	4.24959	0.478116	0.239058	−	0.971005i	$-0.423161\pi$
$79$	4.24959	0.478116	0.239058	+	0.971005i	$0.423161\pi$
$80$	21.8136	2.43884
$81$	0	0
$82$	8.94171	0.987446
$83$	2.60462	0.285894	0.142947	−	0.989730i	$-0.454342\pi$
$83$	2.60462	0.285894	0.142947	+	0.989730i	$0.454342\pi$
$84$	0	0
$85$	−4.13296	−0.448282
$86$	−7.12186	−0.767970
$87$	0	0
$88$	30.5306	3.25458
$89$	−3.19024	−0.338164	−0.169082	−	0.985602i	$-0.554080\pi$
$89$	−3.19024	−0.338164	−0.169082	+	0.985602i	$0.554080\pi$
$90$	0	0
$91$	−0.977371	−0.102456
$92$	18.1423	1.89146
$93$	0	0
$94$	−0.402362	−0.0415005
$95$	2.55330	0.261963
$96$	0	0
$97$	15.3717	1.56076	0.780382	−	0.625303i	$-0.215025\pi$
$97$	15.3717	1.56076	0.780382	+	0.625303i	$0.215025\pi$
$98$	0.884298	0.0893276
$99$	0	0
$100$	−8.73055	−0.873055
$101$	−13.0448	−1.29801	−0.649004	−	0.760785i	$-0.724814\pi$
$101$	−13.0448	−1.29801	−0.649004	+	0.760785i	$0.724814\pi$
$102$	0	0
$103$	−13.7208	−1.35195	−0.675973	−	0.736927i	$-0.736276\pi$
$103$	−13.7208	−1.35195	−0.675973	+	0.736927i	$0.736276\pi$
$104$	−3.15688	−0.309558
$105$	0	0
$106$	−35.9052	−3.48742
$107$	2.30536	0.222868	0.111434	−	0.993772i	$-0.464456\pi$
$107$	2.30536	0.222868	0.111434	+	0.993772i	$0.464456\pi$
$108$	0	0
$109$	9.24936	0.885928	0.442964	−	0.896539i	$-0.353927\pi$
$109$	9.24936	0.885928	0.442964	+	0.896539i	$0.353927\pi$
$110$	17.7389	1.69134
$111$	0	0
$112$	−31.0260	−2.93169
$113$	9.49882	0.893574	0.446787	−	0.894640i	$-0.352568\pi$
$113$	9.49882	0.893574	0.446787	+	0.894640i	$0.352568\pi$
$114$	0	0
$115$	6.42734	0.599352
$116$	−33.8656	−3.14434
$117$	0	0
$118$	11.3401	1.04394
$119$	5.87841	0.538873
$120$	0	0
$121$	2.39780	0.217982
$122$	−19.8864	−1.80043
$123$	0	0
$124$	50.0900	4.49822
$125$	−12.1711	−1.08861
$126$	0	0
$127$	12.6498	1.12249	0.561243	−	0.827651i	$-0.310323\pi$
$127$	12.6498	1.12249	0.561243	+	0.827651i	$0.310323\pi$
$128$	−14.7200	−1.30107
$129$	0	0
$130$	−1.83421	−0.160871
$131$	−14.8788	−1.29997	−0.649985	−	0.759947i	$-0.725225\pi$
$131$	−14.8788	−1.29997	−0.649985	+	0.759947i	$0.725225\pi$
$132$	0	0
$133$	−3.63162	−0.314901
$134$	−26.9366	−2.32697
$135$	0	0
$136$	18.9871	1.62813
$137$	−16.2475	−1.38812	−0.694061	−	0.719916i	$-0.744180\pi$
$137$	−16.2475	−1.38812	−0.694061	+	0.719916i	$0.744180\pi$
$138$	0	0
$139$	−16.5075	−1.40015	−0.700073	−	0.714071i	$-0.746849\pi$
$139$	−16.5075	−1.40015	−0.700073	+	0.714071i	$0.746849\pi$
$140$	−24.0284	−2.03077
$141$	0	0
$142$	4.69237	0.393775
$143$	−1.38534	−0.115848
$144$	0	0
$145$	−11.9977	−0.996356
$146$	−14.3104	−1.18434
$147$	0	0
$148$	−3.10236	−0.255012
$149$	−3.40747	−0.279151	−0.139576	−	0.990211i	$-0.544574\pi$
$149$	−3.40747	−0.279151	−0.139576	+	0.990211i	$0.544574\pi$
$150$	0	0
$151$	−8.64100	−0.703195	−0.351597	−	0.936151i	$-0.614361\pi$
$151$	−8.64100	−0.703195	−0.351597	+	0.936151i	$0.614361\pi$
$152$	−11.7300	−0.951431
$153$	0	0
$154$	−25.2305	−2.03313
$155$	17.7456	1.42536
$156$	0	0
$157$	3.88458	0.310023	0.155012	−	0.987913i	$-0.450458\pi$
$157$	3.88458	0.310023	0.155012	+	0.987913i	$0.450458\pi$
$158$	−11.3432	−0.902416
$159$	0	0
$160$	−27.9378	−2.20868
$161$	−9.14176	−0.720472
$162$	0	0
$163$	−23.8379	−1.86713	−0.933564	−	0.358412i	$-0.883318\pi$
$163$	−23.8379	−1.86713	−0.933564	+	0.358412i	$0.883318\pi$
$164$	−17.1678	−1.34058
$165$	0	0
$166$	−6.95237	−0.539609
$167$	12.7883	0.989587	0.494793	−	0.869011i	$-0.335244\pi$
$167$	12.7883	0.989587	0.494793	+	0.869011i	$0.335244\pi$
$168$	0	0
$169$	−12.8568	−0.988981
$170$	11.0319	0.846106
$171$	0	0
$172$	13.6737	1.04261
$173$	−5.09857	−0.387637	−0.193818	−	0.981037i	$-0.562087\pi$
$173$	−5.09857	−0.387637	−0.193818	+	0.981037i	$0.562087\pi$
$174$	0	0
$175$	4.39927	0.332553
$176$	−43.9767	−3.31487
$177$	0	0
$178$	8.51552	0.638265
$179$	−1.17952	−0.0881611	−0.0440806	−	0.999028i	$-0.514036\pi$
$179$	−1.17952	−0.0881611	−0.0440806	+	0.999028i	$0.514036\pi$
$180$	0	0
$181$	11.4538	0.851356	0.425678	−	0.904875i	$-0.360036\pi$
$181$	11.4538	0.851356	0.425678	+	0.904875i	$0.360036\pi$
$182$	2.60884	0.193380
$183$	0	0
$184$	−29.5276	−2.17680
$185$	−1.09909	−0.0808064
$186$	0	0
$187$	8.33212	0.609305
$188$	0.772522	0.0563419
$189$	0	0
$190$	−6.81538	−0.494440
$191$	−15.3834	−1.11310	−0.556552	−	0.830813i	$-0.687876\pi$
$191$	−15.3834	−1.11310	−0.556552	+	0.830813i	$0.687876\pi$
$192$	0	0
$193$	−17.4472	−1.25588	−0.627939	−	0.778263i	$-0.716101\pi$
$193$	−17.4472	−1.25588	−0.627939	+	0.778263i	$0.716101\pi$
$194$	−41.0309	−2.94585
$195$	0	0
$196$	−1.69782	−0.121273
$197$	9.15442	0.652226	0.326113	−	0.945331i	$-0.394261\pi$
$197$	9.15442	0.652226	0.326113	+	0.945331i	$0.394261\pi$
$198$	0	0
$199$	−23.1814	−1.64329	−0.821644	−	0.570001i	$-0.806943\pi$
$199$	−23.1814	−1.64329	−0.821644	+	0.570001i	$0.806943\pi$
$200$	14.2095	1.00476
$201$	0	0
$202$	34.8198	2.44991
$203$	17.0646	1.19770
$204$	0	0
$205$	−6.08211	−0.424793
$206$	36.6240	2.55172
$207$	0	0
$208$	4.54720	0.315292
$209$	−5.14750	−0.356060
$210$	0	0
$211$	−25.8278	−1.77806	−0.889029	−	0.457851i	$-0.848619\pi$
$211$	−25.8278	−1.77806	−0.889029	+	0.457851i	$0.848619\pi$
$212$	68.9368	4.73460
$213$	0	0
$214$	−6.15357	−0.420649
$215$	4.84426	0.330376
$216$	0	0
$217$	−25.2400	−1.71340
$218$	−24.6888	−1.67214
$219$	0	0
$220$	−34.0582	−2.29620
$221$	−0.861544	−0.0579537
$222$	0	0
$223$	4.07909	0.273156	0.136578	−	0.990629i	$-0.456390\pi$
$223$	4.07909	0.273156	0.136578	+	0.990629i	$0.456390\pi$
$224$	39.7366	2.65502
$225$	0	0
$226$	−25.3547	−1.68657
$227$	−21.2872	−1.41288	−0.706439	−	0.707774i	$-0.749700\pi$
$227$	−21.2872	−1.41288	−0.706439	+	0.707774i	$0.749700\pi$
$228$	0	0
$229$	12.1209	0.800974	0.400487	−	0.916302i	$-0.368841\pi$
$229$	12.1209	0.800974	0.400487	+	0.916302i	$0.368841\pi$
$230$	−17.1561	−1.13124
$231$	0	0
$232$	55.1183	3.61869
$233$	24.1097	1.57948	0.789741	−	0.613441i	$-0.210215\pi$
$233$	24.1097	1.57948	0.789741	+	0.613441i	$0.210215\pi$
$234$	0	0
$235$	0.273685	0.0178532
$236$	−21.7727	−1.41728
$237$	0	0
$238$	−15.6909	−1.01709
$239$	−23.7385	−1.53552	−0.767759	−	0.640738i	$-0.778628\pi$
$239$	−23.7385	−1.53552	−0.767759	+	0.640738i	$0.778628\pi$
$240$	0	0
$241$	−12.6749	−0.816465	−0.408232	−	0.912878i	$-0.633855\pi$
$241$	−12.6749	−0.816465	−0.408232	+	0.912878i	$0.633855\pi$
$242$	−6.40031	−0.411428
$243$	0	0
$244$	38.1812	2.44430
$245$	−0.601496	−0.0384281
$246$	0	0
$247$	0.532253	0.0338665
$248$	−81.5245	−5.17681
$249$	0	0
$250$	32.4875	2.05469
$251$	13.5656	0.856250	0.428125	−	0.903719i	$-0.359174\pi$
$251$	13.5656	0.856250	0.428125	+	0.903719i	$0.359174\pi$
$252$	0	0
$253$	−12.9576	−0.814639
$254$	−33.7654	−2.11863
$255$	0	0
$256$	5.20281	0.325175
$257$	2.79905	0.174600	0.0872999	−	0.996182i	$-0.472176\pi$
$257$	2.79905	0.174600	0.0872999	+	0.996182i	$0.472176\pi$
$258$	0	0
$259$	1.56326	0.0971360
$260$	3.52163	0.218402
$261$	0	0
$262$	39.7152	2.45362
$263$	−1.48546	−0.0915975	−0.0457987	−	0.998951i	$-0.514583\pi$
$263$	−1.48546	−0.0915975	−0.0457987	+	0.998951i	$0.514583\pi$
$264$	0	0
$265$	24.4226	1.50027
$266$	9.69368	0.594358
$267$	0	0
$268$	51.7175	3.15915
$269$	9.12375	0.556285	0.278142	−	0.960540i	$-0.410281\pi$
$269$	9.12375	0.556285	0.278142	+	0.960540i	$0.410281\pi$
$270$	0	0
$271$	5.89686	0.358209	0.179105	−	0.983830i	$-0.442680\pi$
$271$	5.89686	0.358209	0.179105	+	0.983830i	$0.442680\pi$
$272$	−27.3492	−1.65829
$273$	0	0
$274$	43.3687	2.62000
$275$	6.23557	0.376019
$276$	0	0
$277$	−4.94979	−0.297404	−0.148702	−	0.988882i	$-0.547510\pi$
$277$	−4.94979	−0.297404	−0.148702	+	0.988882i	$0.547510\pi$
$278$	44.0625	2.64269
$279$	0	0
$280$	39.1077	2.33713
$281$	13.6087	0.811824	0.405912	−	0.913912i	$-0.366954\pi$
$281$	13.6087	0.811824	0.405912	+	0.913912i	$0.366954\pi$
$282$	0	0
$283$	−11.8258	−0.702970	−0.351485	−	0.936193i	$-0.614323\pi$
$283$	−11.8258	−0.702970	−0.351485	+	0.936193i	$0.614323\pi$
$284$	−9.00920	−0.534598
$285$	0	0
$286$	3.69780	0.218656
$287$	8.65073	0.510637
$288$	0	0
$289$	−11.8182	−0.695190
$290$	32.0248	1.88056
$291$	0	0
$292$	27.4756	1.60789
$293$	−16.2791	−0.951033	−0.475517	−	0.879707i	$-0.657739\pi$
$293$	−16.2791	−0.951033	−0.475517	+	0.879707i	$0.657739\pi$
$294$	0	0
$295$	−7.71350	−0.449097
$296$	5.04927	0.293483
$297$	0	0
$298$	9.09538	0.526881
$299$	1.33982	0.0774840
$300$	0	0
$301$	−6.89011	−0.397139
$302$	23.0649	1.32724
$303$	0	0
$304$	16.8960	0.969055
$305$	13.5266	0.774532
$306$	0	0
$307$	−24.9492	−1.42392	−0.711962	−	0.702218i	$-0.752193\pi$
$307$	−24.9492	−1.42392	−0.711962	+	0.702218i	$0.752193\pi$
$308$	48.4418	2.76023
$309$	0	0
$310$	−47.3674	−2.69029
$311$	8.29393	0.470306	0.235153	−	0.971958i	$-0.424441\pi$
$311$	8.29393	0.470306	0.235153	+	0.971958i	$0.424441\pi$
$312$	0	0
$313$	5.53438	0.312822	0.156411	−	0.987692i	$-0.450008\pi$
$313$	5.53438	0.312822	0.156411	+	0.987692i	$0.450008\pi$
$314$	−10.3689	−0.585151
$315$	0	0
$316$	21.7786	1.22514
$317$	−22.0932	−1.24088	−0.620438	−	0.784256i	$-0.713045\pi$
$317$	−22.0932	−1.24088	−0.620438	+	0.784256i	$0.713045\pi$
$318$	0	0
$319$	24.1876	1.35425
$320$	30.9456	1.72991
$321$	0	0
$322$	24.4016	1.35985
$323$	−3.20124	−0.178122
$324$	0	0
$325$	−0.644760	−0.0357648
$326$	63.6291	3.52409
$327$	0	0
$328$	27.9416	1.54282
$329$	−0.389269	−0.0214611
$330$	0	0
$331$	2.86313	0.157372	0.0786859	−	0.996899i	$-0.474928\pi$
$331$	2.86313	0.157372	0.0786859	+	0.996899i	$0.474928\pi$
$332$	13.3483	0.732585
$333$	0	0
$334$	−34.1350	−1.86779
$335$	18.3222	1.00105
$336$	0	0
$337$	−16.1655	−0.880591	−0.440295	−	0.897853i	$-0.645126\pi$
$337$	−16.1655	−0.880591	−0.440295	+	0.897853i	$0.645126\pi$
$338$	34.3178	1.86664
$339$	0	0
$340$	−21.1808	−1.14869
$341$	−35.7755	−1.93735
$342$	0	0
$343$	18.9322	1.02224
$344$	−22.2548	−1.19990
$345$	0	0
$346$	13.6093	0.731642
$347$	14.7401	0.791289	0.395645	−	0.918404i	$-0.370521\pi$
$347$	14.7401	0.791289	0.395645	+	0.918404i	$0.370521\pi$
$348$	0	0
$349$	−2.47371	−0.132415	−0.0662074	−	0.997806i	$-0.521090\pi$
$349$	−2.47371	−0.132415	−0.0662074	+	0.997806i	$0.521090\pi$
$350$	−11.7427	−0.627674
$351$	0	0
$352$	56.3231	3.00203
$353$	15.6647	0.833750	0.416875	−	0.908964i	$-0.363125\pi$
$353$	15.6647	0.833750	0.416875	+	0.908964i	$0.363125\pi$
$354$	0	0
$355$	−3.19173	−0.169399
$356$	−16.3495	−0.866523
$357$	0	0
$358$	3.14841	0.166399
$359$	−0.235713	−0.0124405	−0.00622023	−	0.999981i	$-0.501980\pi$
$359$	−0.235713	−0.0124405	−0.00622023	+	0.999981i	$0.501980\pi$
$360$	0	0
$361$	−17.0223	−0.895911
$362$	−30.5731	−1.60688
$363$	0	0
$364$	−5.00889	−0.262537
$365$	9.73389	0.509495
$366$	0	0
$367$	−9.63450	−0.502917	−0.251458	−	0.967868i	$-0.580910\pi$
$367$	−9.63450	−0.502917	−0.251458	+	0.967868i	$0.580910\pi$
$368$	42.5319	2.21713
$369$	0	0
$370$	2.93373	0.152517
$371$	−34.7368	−1.80345
$372$	0	0
$373$	−0.0602067	−0.00311738	−0.00155869	−	0.999999i	$-0.500496\pi$
$373$	−0.0602067	−0.00311738	−0.00155869	+	0.999999i	$0.500496\pi$
$374$	−22.2405	−1.15003
$375$	0	0
$376$	−1.25733	−0.0648416
$377$	−2.50101	−0.128808
$378$	0	0
$379$	2.47228	0.126992	0.0634961	−	0.997982i	$-0.479775\pi$
$379$	2.47228	0.126992	0.0634961	+	0.997982i	$0.479775\pi$
$380$	13.0853	0.671262
$381$	0	0
$382$	41.0620	2.10092
$383$	−27.5899	−1.40978	−0.704889	−	0.709318i	$-0.749003\pi$
$383$	−27.5899	−1.40978	−0.704889	+	0.709318i	$0.749003\pi$
$384$	0	0
$385$	17.1617	0.874640
$386$	46.5709	2.37039
$387$	0	0
$388$	78.7781	3.99935
$389$	10.8071	0.547944	0.273972	−	0.961738i	$-0.411662\pi$
$389$	10.8071	0.547944	0.273972	+	0.961738i	$0.411662\pi$
$390$	0	0
$391$	−8.05838	−0.407530
$392$	2.76331	0.139568
$393$	0	0
$394$	−24.4354	−1.23104
$395$	7.71559	0.388214
$396$	0	0
$397$	−19.1601	−0.961619	−0.480810	−	0.876825i	$-0.659657\pi$
$397$	−19.1601	−0.961619	−0.480810	+	0.876825i	$0.659657\pi$
$398$	61.8769	3.10161
$399$	0	0
$400$	−20.4675	−1.02337
$401$	31.5068	1.57338	0.786688	−	0.617351i	$-0.211794\pi$
$401$	31.5068	1.57338	0.786688	+	0.617351i	$0.211794\pi$
$402$	0	0
$403$	3.69920	0.184270
$404$	−66.8529	−3.32606
$405$	0	0
$406$	−45.5497	−2.26059
$407$	2.21578	0.109832
$408$	0	0
$409$	−0.915697	−0.0452783	−0.0226392	−	0.999744i	$-0.507207\pi$
$409$	−0.915697	−0.0452783	−0.0226392	+	0.999744i	$0.507207\pi$
$410$	16.2346	0.801771
$411$	0	0
$412$	−70.3170	−3.46427
$413$	10.9711	0.539853
$414$	0	0
$415$	4.72897	0.232136
$416$	−5.82383	−0.285537
$417$	0	0
$418$	13.7399	0.672042
$419$	−12.1995	−0.595985	−0.297992	−	0.954568i	$-0.596317\pi$
$419$	−12.1995	−0.595985	−0.297992	+	0.954568i	$0.596317\pi$
$420$	0	0
$421$	31.3354	1.52719	0.763597	−	0.645694i	$-0.223432\pi$
$421$	31.3354	1.52719	0.763597	+	0.645694i	$0.223432\pi$
$422$	68.9407	3.35598
$423$	0	0
$424$	−112.199	−5.44886
$425$	3.87791	0.188106
$426$	0	0
$427$	−19.2392	−0.931052
$428$	11.8147	0.571083
$429$	0	0
$430$	−12.9305	−0.623564
$431$	−13.8065	−0.665035	−0.332518	−	0.943097i	$-0.607898\pi$
$431$	−13.8065	−0.665035	−0.332518	+	0.943097i	$0.607898\pi$
$432$	0	0
$433$	−24.2825	−1.16694	−0.583472	−	0.812133i	$-0.698306\pi$
$433$	−24.2825	−1.16694	−0.583472	+	0.812133i	$0.698306\pi$
$434$	67.3718	3.23395
$435$	0	0
$436$	47.4017	2.27013
$437$	4.97839	0.238148
$438$	0	0
$439$	12.7095	0.606591	0.303295	−	0.952897i	$-0.401913\pi$
$439$	12.7095	0.606591	0.303295	+	0.952897i	$0.401913\pi$
$440$	55.4317	2.64260
$441$	0	0
$442$	2.29967	0.109384
$443$	25.5174	1.21237	0.606185	−	0.795324i	$-0.292699\pi$
$443$	25.5174	1.21237	0.606185	+	0.795324i	$0.292699\pi$
$444$	0	0
$445$	−5.79222	−0.274578
$446$	−10.8881	−0.515566
$447$	0	0
$448$	−44.0147	−2.07950
$449$	3.20974	0.151477	0.0757386	−	0.997128i	$-0.475869\pi$
$449$	3.20974	0.151477	0.0757386	+	0.997128i	$0.475869\pi$
$450$	0	0
$451$	12.2616	0.577378
$452$	48.6801	2.28972
$453$	0	0
$454$	56.8206	2.66672
$455$	−1.77452	−0.0831909
$456$	0	0
$457$	−2.13314	−0.0997838	−0.0498919	−	0.998755i	$-0.515888\pi$
$457$	−2.13314	−0.0997838	−0.0498919	+	0.998755i	$0.515888\pi$
$458$	−32.3538	−1.51179
$459$	0	0
$460$	32.9392	1.53580
$461$	22.9372	1.06829	0.534146	−	0.845392i	$-0.320633\pi$
$461$	22.9372	1.06829	0.534146	+	0.845392i	$0.320633\pi$
$462$	0	0
$463$	28.9514	1.34548	0.672742	−	0.739877i	$-0.265116\pi$
$463$	28.9514	1.34548	0.672742	+	0.739877i	$0.265116\pi$
$464$	−79.3929	−3.68572
$465$	0	0
$466$	−64.3548	−2.98118
$467$	2.08928	0.0966806	0.0483403	−	0.998831i	$-0.484607\pi$
$467$	2.08928	0.0966806	0.0483403	+	0.998831i	$0.484607\pi$
$468$	0	0
$469$	−26.0601	−1.20334
$470$	−0.730531	−0.0336969
$471$	0	0
$472$	35.4363	1.63109
$473$	−9.76612	−0.449047
$474$	0	0
$475$	−2.39574	−0.109924
$476$	30.1260	1.38082
$477$	0	0
$478$	63.3640	2.89820
$479$	−21.7385	−0.993256	−0.496628	−	0.867964i	$-0.665429\pi$
$479$	−21.7385	−0.993256	−0.496628	+	0.867964i	$0.665429\pi$
$480$	0	0
$481$	−0.229112	−0.0104466
$482$	33.8325	1.54103
$483$	0	0
$484$	12.2884	0.558563
$485$	27.9091	1.26729
$486$	0	0
$487$	−3.65843	−0.165779	−0.0828896	−	0.996559i	$-0.526415\pi$
$487$	−3.65843	−0.165779	−0.0828896	+	0.996559i	$0.526415\pi$
$488$	−62.1422	−2.81304
$489$	0	0
$490$	1.60554	0.0725308
$491$	−17.3672	−0.783769	−0.391884	−	0.920014i	$-0.628177\pi$
$491$	−17.3672	−0.783769	−0.391884	+	0.920014i	$0.628177\pi$
$492$	0	0
$493$	15.0423	0.677472
$494$	−1.42071	−0.0639209
$495$	0	0
$496$	117.429	5.27271
$497$	4.53968	0.203632
$498$	0	0
$499$	−16.9180	−0.757355	−0.378678	−	0.925529i	$-0.623621\pi$
$499$	−16.9180	−0.757355	−0.378678	+	0.925529i	$0.623621\pi$
$500$	−62.3750	−2.78949
$501$	0	0
$502$	−36.2098	−1.61612
$503$	28.8154	1.28482	0.642408	−	0.766363i	$-0.277935\pi$
$503$	28.8154	1.28482	0.642408	+	0.766363i	$0.277935\pi$
$504$	0	0
$505$	−23.6843	−1.05394
$506$	34.5871	1.53758
$507$	0	0
$508$	64.8284	2.87630
$509$	−11.9779	−0.530912	−0.265456	−	0.964123i	$-0.585522\pi$
$509$	−11.9779	−0.530912	−0.265456	+	0.964123i	$0.585522\pi$
$510$	0	0
$511$	−13.8448	−0.612456
$512$	15.5524	0.687325
$513$	0	0
$514$	−7.47134	−0.329547
$515$	−24.9115	−1.09773
$516$	0	0
$517$	−0.551754	−0.0242661
$518$	−4.17271	−0.183338
$519$	0	0
$520$	−5.73166	−0.251350
$521$	3.58360	0.157000	0.0785001	−	0.996914i	$-0.474987\pi$
$521$	3.58360	0.157000	0.0785001	+	0.996914i	$0.474987\pi$
$522$	0	0
$523$	−26.9569	−1.17874	−0.589372	−	0.807861i	$-0.700625\pi$
$523$	−26.9569	−1.17874	−0.589372	+	0.807861i	$0.700625\pi$
$524$	−76.2520	−3.33108
$525$	0	0
$526$	3.96506	0.172885
$527$	−22.2489	−0.969176
$528$	0	0
$529$	−10.4681	−0.455134
$530$	−65.1898	−2.83166
$531$	0	0
$532$	−18.6116	−0.806913
$533$	−1.26786	−0.0549171
$534$	0	0
$535$	4.18563	0.180961
$536$	−84.1733	−3.63573
$537$	0	0
$538$	−24.3535	−1.04995
$539$	1.21263	0.0522315
$540$	0	0
$541$	−16.6701	−0.716702	−0.358351	−	0.933587i	$-0.616661\pi$
$541$	−16.6701	−0.716702	−0.358351	+	0.933587i	$0.616661\pi$
$542$	−15.7402	−0.676098
$543$	0	0
$544$	35.0275	1.50179
$545$	16.7932	0.719342
$546$	0	0
$547$	−38.7221	−1.65564	−0.827819	−	0.560995i	$-0.810419\pi$
$547$	−38.7221	−1.65564	−0.827819	+	0.560995i	$0.810419\pi$
$548$	−83.2665	−3.55697
$549$	0	0
$550$	−16.6443	−0.709713
$551$	−9.29300	−0.395895
$552$	0	0
$553$	−10.9741	−0.466665
$554$	13.2122	0.561333
$555$	0	0
$556$	−84.5986	−3.58778
$557$	19.7122	0.835231	0.417616	−	0.908624i	$-0.362866\pi$
$557$	19.7122	0.835231	0.417616	+	0.908624i	$0.362866\pi$
$558$	0	0
$559$	1.00982	0.0427108
$560$	−56.3311	−2.38042
$561$	0	0
$562$	−36.3248	−1.53227
$563$	0.734504	0.0309557	0.0154778	−	0.999880i	$-0.495073\pi$
$563$	0.734504	0.0309557	0.0154778	+	0.999880i	$0.495073\pi$
$564$	0	0
$565$	17.2461	0.725550
$566$	31.5659	1.32681
$567$	0	0
$568$	14.6630	0.615246
$569$	−20.5750	−0.862547	−0.431273	−	0.902221i	$-0.641936\pi$
$569$	−20.5750	−0.862547	−0.431273	+	0.902221i	$0.641936\pi$
$570$	0	0
$571$	2.34611	0.0981818	0.0490909	−	0.998794i	$-0.484368\pi$
$571$	2.34611	0.0981818	0.0490909	+	0.998794i	$0.484368\pi$
$572$	−7.09966	−0.296852
$573$	0	0
$574$	−23.0909	−0.963796
$575$	−6.03071	−0.251498
$576$	0	0
$577$	−32.4208	−1.34969	−0.674847	−	0.737958i	$-0.735790\pi$
$577$	−32.4208	−1.34969	−0.674847	+	0.737958i	$0.735790\pi$
$578$	31.5457	1.31213
$579$	0	0
$580$	−61.4866	−2.55309
$581$	−6.72614	−0.279047
$582$	0	0
$583$	−49.2363	−2.03916
$584$	−44.7181	−1.85045
$585$	0	0
$586$	43.4528	1.79502
$587$	−18.5558	−0.765880	−0.382940	−	0.923773i	$-0.625088\pi$
$587$	−18.5558	−0.765880	−0.382940	+	0.923773i	$0.625088\pi$
$588$	0	0
$589$	13.7451	0.566358
$590$	20.5892	0.847645
$591$	0	0
$592$	−7.27302	−0.298919
$593$	−40.8678	−1.67824	−0.839120	−	0.543946i	$-0.816930\pi$
$593$	−40.8678	−1.67824	−0.839120	+	0.543946i	$0.816930\pi$
$594$	0	0
$595$	10.6729	0.437546
$596$	−17.4628	−0.715305
$597$	0	0
$598$	−3.57632	−0.146246
$599$	9.13507	0.373249	0.186624	−	0.982431i	$-0.440245\pi$
$599$	9.13507	0.373249	0.186624	+	0.982431i	$0.440245\pi$
$600$	0	0
$601$	−37.0943	−1.51311	−0.756554	−	0.653932i	$-0.773118\pi$
$601$	−37.0943	−1.51311	−0.756554	+	0.653932i	$0.773118\pi$
$602$	18.3914	0.749577
$603$	0	0
$604$	−44.2839	−1.80189
$605$	4.35346	0.176994
$606$	0	0
$607$	−33.4475	−1.35759	−0.678796	−	0.734326i	$-0.737498\pi$
$607$	−33.4475	−1.35759	−0.678796	+	0.734326i	$0.737498\pi$
$608$	−21.6396	−0.877603
$609$	0	0
$610$	−36.1059	−1.46188
$611$	0.0570515	0.00230806
$612$	0	0
$613$	46.0232	1.85886	0.929429	−	0.369001i	$-0.120300\pi$
$613$	46.0232	1.85886	0.929429	+	0.369001i	$0.120300\pi$
$614$	66.5954	2.68757
$615$	0	0
$616$	−78.8419	−3.17663
$617$	42.5107	1.71142	0.855709	−	0.517457i	$-0.173121\pi$
$617$	42.5107	1.71142	0.855709	+	0.517457i	$0.173121\pi$
$618$	0	0
$619$	42.0677	1.69084	0.845422	−	0.534099i	$-0.179349\pi$
$619$	42.0677	1.69084	0.845422	+	0.534099i	$0.179349\pi$
$620$	90.9439	3.65239
$621$	0	0
$622$	−22.1385	−0.887674
$623$	8.23842	0.330065
$624$	0	0
$625$	−13.5800	−0.543201
$626$	−14.7726	−0.590432
$627$	0	0
$628$	19.9079	0.794414
$629$	1.37800	0.0549443
$630$	0	0
$631$	1.33597	0.0531840	0.0265920	−	0.999646i	$-0.491535\pi$
$631$	1.33597	0.0531840	0.0265920	+	0.999646i	$0.491535\pi$
$632$	−35.4459	−1.40996
$633$	0	0
$634$	58.9720	2.34208
$635$	22.9671	0.911420
$636$	0	0
$637$	−0.125386	−0.00496797
$638$	−64.5626	−2.55606
$639$	0	0
$640$	−26.7257	−1.05643
$641$	−9.62855	−0.380305	−0.190152	−	0.981755i	$-0.560898\pi$
$641$	−9.62855	−0.380305	−0.190152	+	0.981755i	$0.560898\pi$
$642$	0	0
$643$	−18.3614	−0.724101	−0.362051	−	0.932158i	$-0.617923\pi$
$643$	−18.3614	−0.724101	−0.362051	+	0.932158i	$0.617923\pi$
$644$	−46.8503	−1.84616
$645$	0	0
$646$	8.54490	0.336194
$647$	−24.2705	−0.954170	−0.477085	−	0.878857i	$-0.658307\pi$
$647$	−24.2705	−0.954170	−0.477085	+	0.878857i	$0.658307\pi$
$648$	0	0
$649$	15.5506	0.610413
$650$	1.72102	0.0675040
$651$	0	0
$652$	−122.166	−4.78439
$653$	42.5587	1.66545	0.832724	−	0.553688i	$-0.186780\pi$
$653$	42.5587	1.66545	0.832724	+	0.553688i	$0.186780\pi$
$654$	0	0
$655$	−27.0141	−1.05553
$656$	−40.2474	−1.57140
$657$	0	0
$658$	1.03905	0.0405065
$659$	1.88588	0.0734636	0.0367318	−	0.999325i	$-0.488305\pi$
$659$	1.88588	0.0734636	0.0367318	+	0.999325i	$0.488305\pi$
$660$	0	0
$661$	42.0954	1.63732	0.818661	−	0.574277i	$-0.194717\pi$
$661$	42.0954	1.63732	0.818661	+	0.574277i	$0.194717\pi$
$662$	−7.64239	−0.297030
$663$	0	0
$664$	−21.7252	−0.843102
$665$	−6.59360	−0.255689
$666$	0	0
$667$	−23.3930	−0.905779
$668$	65.5382	2.53575
$669$	0	0
$670$	−48.9064	−1.88942
$671$	−27.2699	−1.05274
$672$	0	0
$673$	10.7690	0.415114	0.207557	−	0.978223i	$-0.433449\pi$
$673$	10.7690	0.415114	0.207557	+	0.978223i	$0.433449\pi$
$674$	43.1497	1.66206
$675$	0	0
$676$	−65.8891	−2.53420
$677$	−10.4892	−0.403131	−0.201565	−	0.979475i	$-0.564603\pi$
$677$	−10.4892	−0.403131	−0.201565	+	0.979475i	$0.564603\pi$
$678$	0	0
$679$	−39.6957	−1.52338
$680$	34.4731	1.32198
$681$	0	0
$682$	95.4935	3.65664
$683$	−24.1810	−0.925259	−0.462629	−	0.886552i	$-0.653094\pi$
$683$	−24.1810	−0.925259	−0.462629	+	0.886552i	$0.653094\pi$
$684$	0	0
$685$	−29.4992	−1.12711
$686$	−50.5347	−1.92942
$687$	0	0
$688$	32.0561	1.22213
$689$	5.09106	0.193954
$690$	0	0
$691$	−8.56348	−0.325770	−0.162885	−	0.986645i	$-0.552080\pi$
$691$	−8.56348	−0.325770	−0.162885	+	0.986645i	$0.552080\pi$
$692$	−26.1295	−0.993293
$693$	0	0
$694$	−39.3449	−1.49351
$695$	−29.9711	−1.13687
$696$	0	0
$697$	7.62555	0.288838
$698$	6.60295	0.249925
$699$	0	0
$700$	22.5456	0.852145
$701$	2.42039	0.0914168	0.0457084	−	0.998955i	$-0.485445\pi$
$701$	2.42039	0.0914168	0.0457084	+	0.998955i	$0.485445\pi$
$702$	0	0
$703$	−0.851312	−0.0321079
$704$	−62.3869	−2.35129
$705$	0	0
$706$	−41.8130	−1.57365
$707$	33.6867	1.26692
$708$	0	0
$709$	49.7759	1.86937	0.934686	−	0.355475i	$-0.115681\pi$
$709$	49.7759	1.86937	0.934686	+	0.355475i	$0.115681\pi$
$710$	8.51950	0.319731
$711$	0	0
$712$	26.6098	0.997246
$713$	34.6001	1.29579
$714$	0	0
$715$	−2.51523	−0.0940643
$716$	−6.04486	−0.225907
$717$	0	0
$718$	0.629175	0.0234806
$719$	−52.3613	−1.95275	−0.976373	−	0.216093i	$-0.930669\pi$
$719$	−52.3613	−1.95275	−0.976373	+	0.216093i	$0.930669\pi$
$720$	0	0
$721$	35.4323	1.31957
$722$	45.4367	1.69098
$723$	0	0
$724$	58.6993	2.18154
$725$	11.2573	0.418087
$726$	0	0
$727$	−35.2007	−1.30552	−0.652760	−	0.757564i	$-0.726389\pi$
$727$	−35.2007	−1.30552	−0.652760	+	0.757564i	$0.726389\pi$
$728$	8.15228	0.302143
$729$	0	0
$730$	−25.9821	−0.961642
$731$	−6.07357	−0.224639
$732$	0	0
$733$	23.7927	0.878802	0.439401	−	0.898291i	$-0.355191\pi$
$733$	23.7927	0.878802	0.439401	+	0.898291i	$0.355191\pi$
$734$	25.7168	0.949225
$735$	0	0
$736$	−54.4727	−2.00789
$737$	−36.9379	−1.36062
$738$	0	0
$739$	−43.5839	−1.60326	−0.801631	−	0.597820i	$-0.796034\pi$
$739$	−43.5839	−1.60326	−0.801631	+	0.597820i	$0.796034\pi$
$740$	−5.63266	−0.207061
$741$	0	0
$742$	92.7210	3.40390
$743$	10.1694	0.373079	0.186539	−	0.982447i	$-0.440273\pi$
$743$	10.1694	0.373079	0.186539	+	0.982447i	$0.440273\pi$
$744$	0	0
$745$	−6.18663	−0.226661
$746$	0.160706	0.00588388
$747$	0	0
$748$	42.7010	1.56130
$749$	−5.95332	−0.217530
$750$	0	0
$751$	−16.7213	−0.610170	−0.305085	−	0.952325i	$-0.598685\pi$
$751$	−16.7213	−0.610170	−0.305085	+	0.952325i	$0.598685\pi$
$752$	1.81107	0.0660427
$753$	0	0
$754$	6.67580	0.243118
$755$	−15.6887	−0.570969
$756$	0	0
$757$	−40.6578	−1.47773	−0.738867	−	0.673852i	$-0.764639\pi$
$757$	−40.6578	−1.47773	−0.738867	+	0.673852i	$0.764639\pi$
$758$	−6.59911	−0.239690
$759$	0	0
$760$	−21.2971	−0.772528
$761$	13.6161	0.493583	0.246791	−	0.969069i	$-0.420624\pi$
$761$	13.6161	0.493583	0.246791	+	0.969069i	$0.420624\pi$
$762$	0	0
$763$	−23.8854	−0.864709
$764$	−78.8378	−2.85225
$765$	0	0
$766$	73.6441	2.66087
$767$	−1.60793	−0.0580591
$768$	0	0
$769$	44.9223	1.61994	0.809970	−	0.586471i	$-0.199483\pi$
$769$	44.9223	1.61994	0.809970	+	0.586471i	$0.199483\pi$
$770$	−45.8087	−1.65083
$771$	0	0
$772$	−89.4146	−3.21810
$773$	23.2895	0.837664	0.418832	−	0.908064i	$-0.362440\pi$
$773$	23.2895	0.837664	0.418832	+	0.908064i	$0.362440\pi$
$774$	0	0
$775$	−16.6505	−0.598105
$776$	−128.216	−4.60269
$777$	0	0
$778$	−28.8469	−1.03421
$779$	−4.71099	−0.168789
$780$	0	0
$781$	6.43459	0.230248
$782$	21.5098	0.769188
$783$	0	0
$784$	−3.98030	−0.142154
$785$	7.05288	0.251728
$786$	0	0
$787$	2.98243	0.106312	0.0531560	−	0.998586i	$-0.483072\pi$
$787$	2.98243	0.106312	0.0531560	+	0.998586i	$0.483072\pi$
$788$	46.9152	1.67128
$789$	0	0
$790$	−20.5948	−0.732730
$791$	−24.5296	−0.872172
$792$	0	0
$793$	2.81972	0.100131
$794$	51.1430	1.81500
$795$	0	0
$796$	−118.802	−4.21081
$797$	−23.9545	−0.848511	−0.424256	−	0.905542i	$-0.639464\pi$
$797$	−23.9545	−0.848511	−0.424256	+	0.905542i	$0.639464\pi$
$798$	0	0
$799$	−0.343137	−0.0121393
$800$	26.2138	0.926797
$801$	0	0
$802$	−84.0994	−2.96965
$803$	−19.6237	−0.692506
$804$	0	0
$805$	−16.5979	−0.584997
$806$	−9.87407	−0.347799
$807$	0	0
$808$	108.807	3.82782
$809$	36.1049	1.26938	0.634691	−	0.772766i	$-0.281128\pi$
$809$	36.1049	1.26938	0.634691	+	0.772766i	$0.281128\pi$
$810$	0	0
$811$	−41.1368	−1.44451	−0.722255	−	0.691627i	$-0.756894\pi$
$811$	−41.1368	−1.44451	−0.722255	+	0.691627i	$0.756894\pi$
$812$	87.4539	3.06903
$813$	0	0
$814$	−5.91445	−0.207301
$815$	−43.2803	−1.51604
$816$	0	0
$817$	3.75219	0.131273
$818$	2.44422	0.0854601
$819$	0	0
$820$	−31.1700	−1.08850
$821$	−41.7681	−1.45772	−0.728859	−	0.684664i	$-0.759949\pi$
$821$	−41.7681	−1.45772	−0.728859	+	0.684664i	$0.759949\pi$
$822$	0	0
$823$	18.6180	0.648982	0.324491	−	0.945889i	$-0.394807\pi$
$823$	18.6180	0.648982	0.324491	+	0.945889i	$0.394807\pi$
$824$	114.445	3.98688
$825$	0	0
$826$	−29.2845	−1.01894
$827$	−4.92478	−0.171251	−0.0856256	−	0.996327i	$-0.527289\pi$
$827$	−4.92478	−0.171251	−0.0856256	+	0.996327i	$0.527289\pi$
$828$	0	0
$829$	−9.14266	−0.317538	−0.158769	−	0.987316i	$-0.550752\pi$
$829$	−9.14266	−0.317538	−0.158769	+	0.987316i	$0.550752\pi$
$830$	−12.6228	−0.438143
$831$	0	0
$832$	6.45083	0.223642
$833$	0.754135	0.0261292
$834$	0	0
$835$	23.2185	0.803510
$836$	−26.3802	−0.912379
$837$	0	0
$838$	32.5634	1.12489
$839$	36.2122	1.25018	0.625092	−	0.780551i	$-0.285061\pi$
$839$	36.2122	1.25018	0.625092	+	0.780551i	$0.285061\pi$
$840$	0	0
$841$	14.6669	0.505755
$842$	−83.6418	−2.88249
$843$	0	0
$844$	−132.364	−4.55615
$845$	−23.3428	−0.803018
$846$	0	0
$847$	−6.19204	−0.212761
$848$	161.612	5.54979
$849$	0	0
$850$	−10.3511	−0.355040
$851$	−2.14298	−0.0734604
$852$	0	0
$853$	1.85396	0.0634786	0.0317393	−	0.999496i	$-0.489895\pi$
$853$	1.85396	0.0634786	0.0317393	+	0.999496i	$0.489895\pi$
$854$	51.3542	1.75731
$855$	0	0
$856$	−19.2291	−0.657235
$857$	1.01274	0.0345947	0.0172973	−	0.999850i	$-0.494494\pi$
$857$	1.01274	0.0345947	0.0172973	+	0.999850i	$0.494494\pi$
$858$	0	0
$859$	5.87727	0.200530	0.100265	−	0.994961i	$-0.468031\pi$
$859$	5.87727	0.200530	0.100265	+	0.994961i	$0.468031\pi$
$860$	24.8262	0.846565
$861$	0	0
$862$	36.8529	1.25521
$863$	−27.1699	−0.924875	−0.462438	−	0.886652i	$-0.653025\pi$
$863$	−27.1699	−0.924875	−0.462438	+	0.886652i	$0.653025\pi$
$864$	0	0
$865$	−9.25700	−0.314747
$866$	64.8160	2.20254
$867$	0	0
$868$	−129.352	−4.39048
$869$	−15.5548	−0.527660
$870$	0	0
$871$	3.81939	0.129415
$872$	−77.1491	−2.61260
$873$	0	0
$874$	−13.2885	−0.449491
$875$	31.4303	1.06254
$876$	0	0
$877$	6.05949	0.204615	0.102307	−	0.994753i	$-0.467377\pi$
$877$	6.05949	0.204615	0.102307	+	0.994753i	$0.467377\pi$
$878$	−33.9247	−1.14490
$879$	0	0
$880$	−79.8444	−2.69155
$881$	−27.6566	−0.931776	−0.465888	−	0.884844i	$-0.654265\pi$
$881$	−27.6566	−0.931776	−0.465888	+	0.884844i	$0.654265\pi$
$882$	0	0
$883$	46.1501	1.55307	0.776537	−	0.630072i	$-0.216975\pi$
$883$	46.1501	1.55307	0.776537	+	0.630072i	$0.216975\pi$
$884$	−4.41530	−0.148503
$885$	0	0
$886$	−68.1123	−2.28828
$887$	−10.5727	−0.354995	−0.177498	−	0.984121i	$-0.556800\pi$
$887$	−10.5727	−0.354995	−0.177498	+	0.984121i	$0.556800\pi$
$888$	0	0
$889$	−32.6666	−1.09560
$890$	15.4608	0.518249
$891$	0	0
$892$	20.9048	0.699944
$893$	0.211986	0.00709386
$894$	0	0
$895$	−2.14154	−0.0715837
$896$	38.0126	1.26991
$897$	0	0
$898$	−8.56759	−0.285904
$899$	−64.5870	−2.15410
$900$	0	0
$901$	−30.6202	−1.02011
$902$	−32.7293	−1.08977
$903$	0	0
$904$	−79.2298	−2.63515
$905$	20.7957	0.691271
$906$	0	0
$907$	−6.61129	−0.219524	−0.109762	−	0.993958i	$-0.535009\pi$
$907$	−6.61129	−0.219524	−0.109762	+	0.993958i	$0.535009\pi$
$908$	−109.094	−3.62040
$909$	0	0
$910$	4.73664	0.157018
$911$	−9.17704	−0.304049	−0.152024	−	0.988377i	$-0.548579\pi$
$911$	−9.17704	−0.304049	−0.152024	+	0.988377i	$0.548579\pi$
$912$	0	0
$913$	−9.53370	−0.315519
$914$	5.69386	0.188336
$915$	0	0
$916$	62.1182	2.05244
$917$	38.4229	1.26883
$918$	0	0
$919$	−14.9497	−0.493146	−0.246573	−	0.969124i	$-0.579305\pi$
$919$	−14.9497	−0.493146	−0.246573	+	0.969124i	$0.579305\pi$
$920$	−53.6106	−1.76749
$921$	0	0
$922$	−61.2250	−2.01634
$923$	−0.665339	−0.0218999
$924$	0	0
$925$	1.03126	0.0339077
$926$	−77.2782	−2.53952
$927$	0	0
$928$	101.683	3.33789
$929$	−23.1734	−0.760293	−0.380147	−	0.924926i	$-0.624126\pi$
$929$	−23.1734	−0.760293	−0.380147	+	0.924926i	$0.624126\pi$
$930$	0	0
$931$	−0.465897	−0.0152692
$932$	123.559	4.04731
$933$	0	0
$934$	−5.57681	−0.182479
$935$	15.1279	0.494734
$936$	0	0
$937$	24.7831	0.809628	0.404814	−	0.914399i	$-0.367336\pi$
$937$	24.7831	0.809628	0.404814	+	0.914399i	$0.367336\pi$
$938$	69.5607	2.27124
$939$	0	0
$940$	1.40260	0.0457477
$941$	41.9888	1.36879	0.684397	−	0.729109i	$-0.260065\pi$
$941$	41.9888	1.36879	0.684397	+	0.729109i	$0.260065\pi$
$942$	0	0
$943$	−11.8588	−0.386176
$944$	−51.0428	−1.66130
$945$	0	0
$946$	26.0681	0.847549
$947$	−5.40036	−0.175488	−0.0877440	−	0.996143i	$-0.527966\pi$
$947$	−5.40036	−0.175488	−0.0877440	+	0.996143i	$0.527966\pi$
$948$	0	0
$949$	2.02910	0.0658673
$950$	6.39480	0.207475
$951$	0	0
$952$	−49.0319	−1.58913
$953$	26.7994	0.868119	0.434059	−	0.900884i	$-0.357081\pi$
$953$	26.7994	0.868119	0.434059	+	0.900884i	$0.357081\pi$
$954$	0	0
$955$	−27.9302	−0.903801
$956$	−121.657	−3.93466
$957$	0	0
$958$	58.0253	1.87471
$959$	41.9574	1.35488
$960$	0	0
$961$	64.5296	2.08160
$962$	0.611556	0.0197174
$963$	0	0
$964$	−64.9573	−2.09214
$965$	−31.6773	−1.01973
$966$	0	0
$967$	25.1527	0.808855	0.404428	−	0.914570i	$-0.367471\pi$
$967$	25.1527	0.808855	0.404428	+	0.914570i	$0.367471\pi$
$968$	−20.0001	−0.642828
$969$	0	0
$970$	−74.4961	−2.39193
$971$	−42.8042	−1.37365	−0.686826	−	0.726822i	$-0.740997\pi$
$971$	−42.8042	−1.37365	−0.686826	+	0.726822i	$0.740997\pi$
$972$	0	0
$973$	42.6287	1.36661
$974$	9.76524	0.312899
$975$	0	0
$976$	89.5102	2.86515
$977$	−26.6227	−0.851736	−0.425868	−	0.904785i	$-0.640031\pi$
$977$	−26.6227	−0.851736	−0.425868	+	0.904785i	$0.640031\pi$
$978$	0	0
$979$	11.6772	0.373206
$980$	−3.08258	−0.0984695
$981$	0	0
$982$	46.3572	1.47932
$983$	59.3588	1.89325	0.946625	−	0.322336i	$-0.104468\pi$
$983$	59.3588	1.89325	0.946625	+	0.322336i	$0.104468\pi$
$984$	0	0
$985$	16.6208	0.529584
$986$	−40.1516	−1.27869
$987$	0	0
$988$	2.72773	0.0867805
$989$	9.44526	0.300342
$990$	0	0
$991$	−18.1773	−0.577420	−0.288710	−	0.957417i	$-0.593226\pi$
$991$	−18.1773	−0.577420	−0.288710	+	0.957417i	$0.593226\pi$
$992$	−150.397	−4.77511
$993$	0	0
$994$	−12.1175	−0.384344
$995$	−42.0884	−1.33429
$996$	0	0
$997$	9.69823	0.307146	0.153573	−	0.988137i	$-0.450922\pi$
$997$	9.69823	0.307146	0.153573	+	0.988137i	$0.450922\pi$
$998$	45.1584	1.42946
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.3		72
3.2	odd	2		6561.2.a.d.1.70		72
81.5	odd	54		729.2.g.b.460.8		144
81.11	odd	54		243.2.g.a.91.1		144
81.16	even	27		729.2.g.c.271.1		144
81.22	even	27		81.2.g.a.79.8	yes	144
81.32	odd	54		729.2.g.a.703.1		144
81.38	odd	54		729.2.g.a.28.1		144
81.43	even	27		729.2.g.d.28.8		144
81.49	even	27		729.2.g.d.703.8		144
81.59	odd	54		243.2.g.a.235.1		144
81.65	odd	54		729.2.g.b.271.8		144
81.70	even	27		81.2.g.a.40.8	✓	144
81.76	even	27		729.2.g.c.460.1		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.40.8	✓	144	81.70	even	27
81.2.g.a.79.8	yes	144	81.22	even	27
243.2.g.a.91.1		144	81.11	odd	54
243.2.g.a.235.1		144	81.59	odd	54
729.2.g.a.28.1		144	81.38	odd	54
729.2.g.a.703.1		144	81.32	odd	54
729.2.g.b.271.8		144	81.65	odd	54
729.2.g.b.460.8		144	81.5	odd	54
729.2.g.c.271.1		144	81.16	even	27
729.2.g.c.460.1		144	81.76	even	27
729.2.g.d.28.8		144	81.43	even	27
729.2.g.d.703.8		144	81.49	even	27
6561.2.a.c.1.3		72	1.1	even	1	trivial
6561.2.a.d.1.70		72	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66924 q^{2} +5.12486 q^{4} +1.81561 q^{5} -2.58238 q^{7} -8.34102 q^{8} -4.84630 q^{10} -3.66030 q^{11} +0.378476 q^{13} +6.89301 q^{14} +12.0145 q^{16} -2.27635 q^{17} +1.40631 q^{19} +9.30474 q^{20} +9.77023 q^{22} +3.54005 q^{23} -1.70357 q^{25} -1.01025 q^{26} -13.2344 q^{28} -6.60809 q^{29} +9.77392 q^{31} -15.3876 q^{32} +6.07613 q^{34} -4.68860 q^{35} -0.605354 q^{37} -3.75377 q^{38} -15.1440 q^{40} -3.34990 q^{41} +2.66812 q^{43} -18.7585 q^{44} -9.44925 q^{46} +0.150740 q^{47} -0.331292 q^{49} +4.54724 q^{50} +1.93964 q^{52} +13.4514 q^{53} -6.64567 q^{55} +21.5397 q^{56} +17.6386 q^{58} -4.24844 q^{59} +7.45019 q^{61} -26.0890 q^{62} +17.0442 q^{64} +0.687165 q^{65} +10.0915 q^{67} -11.6660 q^{68} +12.5150 q^{70} -1.75794 q^{71} +5.36123 q^{73} +1.61584 q^{74} +7.20712 q^{76} +9.45230 q^{77} +4.24959 q^{79} +21.8136 q^{80} +8.94171 q^{82} +2.60462 q^{83} -4.13296 q^{85} -7.12186 q^{86} +30.5306 q^{88} -3.19024 q^{89} -0.977371 q^{91} +18.1423 q^{92} -0.402362 q^{94} +2.55330 q^{95} +15.3717 q^{97} +0.884298 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

Introduction

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