LMFDB - Embedded newform 2700.3.g.o.701.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(701,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2700.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.g (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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.9292960.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 41x^{2} + 360 $ x^4 + 41*x^2 + 360
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.4
Root			$3.56902i$ of defining polynomial
Character	$\chi$	$=$	2700.701
Dual form			2700.3.g.o.701.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.26209 q^{7} +O(q^{10})$	$q+6.26209 q^{7} +1.03377i q^{11} -3.00000 q^{13} -15.7776i q^{17} -18.7863 q^{19} +34.6564i q^{23} -3.10132i q^{29} -17.7863 q^{31} -43.3104 q^{37} -46.2989i q^{41} -20.7379 q^{43} -78.8878i q^{47} -9.78626 q^{49} +44.2313i q^{53} +90.5302i q^{59} +8.78626 q^{61} +19.3104 q^{67} +56.6366i q^{71} -109.310 q^{73} +6.47359i q^{77} -39.0000 q^{79} +75.7865i q^{83} -90.5302i q^{89} -18.7863 q^{91} +8.88296 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7}+O(q^{10}) $ 4 * q - 6 * q^7	$ 4 q - 6 q^{7} - 12 q^{13} + 18 q^{19} + 22 q^{31} - 18 q^{37} - 114 q^{43} + 54 q^{49} - 58 q^{61} - 78 q^{67} - 282 q^{73} - 156 q^{79} + 18 q^{91} - 306 q^{97}+O(q^{100}) $ 4 * q - 6 * q^7 - 12 * q^13 + 18 * q^19 + 22 * q^31 - 18 * q^37 - 114 * q^43 + 54 * q^49 - 58 * q^61 - 78 * q^67 - 282 * q^73 - 156 * q^79 + 18 * q^91 - 306 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.26209	0.894584	0.447292	−	0.894388i	$-0.352388\pi$
$7$	6.26209	0.894584	0.447292	+	0.894388i	$0.352388\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.03377i	0.0939795i	0.998895	+	0.0469898i	$0.0149628\pi$
$11$	1.03377i	0.0939795i	−0.998895	+	0.0469898i	$0.985037\pi$
$12$	0	0
$13$	−3.00000	−0.230769	−0.115385	−	0.993321i	$-0.536810\pi$
$13$	−3.00000	−0.230769	−0.115385	+	0.993321i	$0.536810\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 15.7776i	− 0.928092i	−0.885811	−	0.464046i	$-0.846397\pi$
$17$	− 15.7776i	− 0.928092i	0.885811	−	0.464046i	$-0.153603\pi$
$18$	0	0
$19$	−18.7863	−0.988751	−0.494375	−	0.869249i	$-0.664603\pi$
$19$	−18.7863	−0.988751	−0.494375	+	0.869249i	$0.664603\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	34.6564i	1.50680i	0.657562	+	0.753401i	$0.271588\pi$
$23$	34.6564i	1.50680i	−0.657562	+	0.753401i	$0.728412\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.10132i	− 0.106942i	−0.998569	−	0.0534711i	$-0.982972\pi$
$29$	− 3.10132i	− 0.106942i	0.998569	−	0.0534711i	$-0.0170285\pi$
$30$	0	0
$31$	−17.7863	−0.573750	−0.286875	−	0.957968i	$-0.592617\pi$
$31$	−17.7863	−0.573750	−0.286875	+	0.957968i	$0.592617\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−43.3104	−1.17055	−0.585276	−	0.810834i	$-0.699014\pi$
$37$	−43.3104	−1.17055	−0.585276	+	0.810834i	$0.699014\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 46.2989i	− 1.12924i	−0.825351	−	0.564621i	$-0.809022\pi$
$41$	− 46.2989i	− 1.12924i	0.825351	−	0.564621i	$-0.190978\pi$
$42$	0	0
$43$	−20.7379	−0.482277	−0.241139	−	0.970491i	$-0.577521\pi$
$43$	−20.7379	−0.482277	−0.241139	+	0.970491i	$0.577521\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 78.8878i	− 1.67846i	−0.543774	−	0.839232i	$-0.683005\pi$
$47$	− 78.8878i	− 1.67846i	0.543774	−	0.839232i	$-0.316995\pi$
$48$	0	0
$49$	−9.78626	−0.199720
$50$	0	0
$51$	0	0
$52$	0	0
$53$	44.2313i	0.834554i	0.908779	+	0.417277i	$0.137015\pi$
$53$	44.2313i	0.834554i	−0.908779	+	0.417277i	$0.862985\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	90.5302i	1.53441i	0.641401	+	0.767205i	$0.278353\pi$
$59$	90.5302i	1.53441i	−0.641401	+	0.767205i	$0.721647\pi$
$60$	0	0
$61$	8.78626	0.144037	0.0720185	−	0.997403i	$-0.477056\pi$
$61$	8.78626	0.144037	0.0720185	+	0.997403i	$0.477056\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	19.3104	0.288215	0.144108	−	0.989562i	$-0.453969\pi$
$67$	19.3104	0.288215	0.144108	+	0.989562i	$0.453969\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	56.6366i	0.797699i	0.917016	+	0.398850i	$0.130590\pi$
$71$	56.6366i	0.797699i	−0.917016	+	0.398850i	$0.869410\pi$
$72$	0	0
$73$	−109.310	−1.49740	−0.748702	−	0.662907i	$-0.769322\pi$
$73$	−109.310	−1.49740	−0.748702	+	0.662907i	$0.769322\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.47359i	0.0840726i
$78$	0	0
$79$	−39.0000	−0.493671	−0.246835	−	0.969057i	$-0.579391\pi$
$79$	−39.0000	−0.493671	−0.246835	+	0.969057i	$0.579391\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	75.7865i	0.913090i	0.889700	+	0.456545i	$0.150913\pi$
$83$	75.7865i	0.913090i	−0.889700	+	0.456545i	$0.849087\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 90.5302i	− 1.01719i	−0.861005	−	0.508597i	$-0.830164\pi$
$89$	− 90.5302i	− 1.01719i	0.861005	−	0.508597i	$-0.169836\pi$
$90$	0	0
$91$	−18.7863	−0.206442
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.88296	0.0915769	0.0457885	−	0.998951i	$-0.485420\pi$
$97$	8.88296	0.0915769	0.0457885	+	0.998951i	$0.485420\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	43.1976i	0.427699i	0.976867	+	0.213849i	$0.0686002\pi$
$101$	43.1976i	0.427699i	−0.976867	+	0.213849i	$0.931400\pi$
$102$	0	0
$103$	−76.6896	−0.744559	−0.372279	−	0.928121i	$-0.621424\pi$
$103$	−76.6896	−0.744559	−0.372279	+	0.928121i	$0.621424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 38.0287i	− 0.355408i	−0.984084	−	0.177704i	$-0.943133\pi$
$107$	− 38.0287i	− 0.355408i	0.984084	−	0.177704i	$-0.0568670\pi$
$108$	0	0
$109$	146.718	1.34603	0.673016	−	0.739628i	$-0.264998\pi$
$109$	146.718	1.34603	0.673016	+	0.739628i	$0.264998\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 186.229i	− 1.64805i	−0.566555	−	0.824024i	$-0.691724\pi$
$113$	− 186.229i	− 1.64805i	0.566555	−	0.824024i	$-0.308276\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 98.8004i	− 0.830256i
$120$	0	0
$121$	119.931	0.991168
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−161.242	−1.26962	−0.634810	−	0.772668i	$-0.718922\pi$
$127$	−161.242	−1.26962	−0.634810	+	0.772668i	$0.718922\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 249.069i	− 1.90129i	−0.310285	−	0.950644i	$-0.600424\pi$
$131$	− 249.069i	− 1.90129i	0.310285	−	0.950644i	$-0.399576\pi$
$132$	0	0
$133$	−117.641	−0.884520
$134$	0	0
$135$	0	0
$136$	0	0
$137$	245.967i	1.79538i	0.440626	+	0.897691i	$0.354757\pi$
$137$	245.967i	1.79538i	−0.440626	+	0.897691i	$0.645243\pi$
$138$	0	0
$139$	169.931	1.22253	0.611264	−	0.791427i	$-0.290661\pi$
$139$	169.931	1.22253	0.611264	+	0.791427i	$0.290661\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3.10132i	− 0.0216876i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	162.674i	1.09177i	0.837861	+	0.545884i	$0.183806\pi$
$149$	162.674i	1.09177i	−0.837861	+	0.545884i	$0.816194\pi$
$150$	0	0
$151$	−110.145	−0.729437	−0.364719	−	0.931118i	$-0.618835\pi$
$151$	−110.145	−0.729437	−0.364719	+	0.931118i	$0.618835\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−129.262	−0.823325	−0.411663	−	0.911336i	$-0.635052\pi$
$157$	−129.262	−0.823325	−0.411663	+	0.911336i	$0.635052\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	217.022i	1.34796i
$162$	0	0
$163$	−245.621	−1.50688	−0.753438	−	0.657519i	$-0.771606\pi$
$163$	−245.621	−1.50688	−0.753438	+	0.657519i	$0.771606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	82.2600i	0.492575i	0.969197	+	0.246288i	$0.0792108\pi$
$167$	82.2600i	0.492575i	−0.969197	+	0.246288i	$0.920789\pi$
$168$	0	0
$169$	−160.000	−0.946746
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 220.886i	− 1.27680i	−0.769706	−	0.638398i	$-0.779597\pi$
$173$	− 220.886i	− 1.27680i	0.769706	−	0.638398i	$-0.220403\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 149.234i	− 0.833712i	−0.908973	−	0.416856i	$-0.863132\pi$
$179$	− 149.234i	− 0.833712i	0.908973	−	0.416856i	$-0.136868\pi$
$180$	0	0
$181$	137.931	0.762051	0.381026	−	0.924564i	$-0.375571\pi$
$181$	137.931	0.762051	0.381026	+	0.924564i	$0.375571\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.3104	0.0872216
$188$	0	0
$189$	0	0
$190$	0	0
$191$	201.736i	1.05621i	0.849179	+	0.528105i	$0.177097\pi$
$191$	201.736i	1.05621i	−0.849179	+	0.528105i	$0.822903\pi$
$192$	0	0
$193$	−368.883	−1.91131	−0.955655	−	0.294487i	$-0.904851\pi$
$193$	−368.883	−1.91131	−0.955655	+	0.294487i	$0.904851\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	170.181i	0.863862i	0.901907	+	0.431931i	$0.142168\pi$
$197$	170.181i	0.863862i	−0.901907	+	0.431931i	$0.857832\pi$
$198$	0	0
$199$	−136.863	−0.687752	−0.343876	−	0.939015i	$-0.611740\pi$
$199$	−136.863	−0.687752	−0.343876	+	0.939015i	$0.611740\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 19.4208i	− 0.0956688i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 19.4208i	− 0.0929223i
$210$	0	0
$211$	−39.9313	−0.189248	−0.0946240	−	0.995513i	$-0.530165\pi$
$211$	−39.9313	−0.189248	−0.0946240	+	0.995513i	$0.530165\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−111.379	−0.513268
$218$	0	0
$219$	0	0
$220$	0	0
$221$	47.3327i	0.214175i
$222$	0	0
$223$	−309.669	−1.38865	−0.694326	−	0.719661i	$-0.744297\pi$
$223$	−309.669	−1.38865	−0.694326	+	0.719661i	$0.744297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 107.342i	− 0.472870i	−0.971647	−	0.236435i	$-0.924021\pi$
$227$	− 107.342i	− 0.472870i	0.971647	−	0.236435i	$-0.0759791\pi$
$228$	0	0
$229$	88.8550	0.388013	0.194006	−	0.981000i	$-0.437852\pi$
$229$	88.8550	0.388013	0.194006	+	0.981000i	$0.437852\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 245.696i	− 1.05449i	−0.849713	−	0.527245i	$-0.823225\pi$
$233$	− 245.696i	− 1.05449i	0.849713	−	0.527245i	$-0.176775\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	237.697i	0.994549i	0.867593	+	0.497274i	$0.165666\pi$
$239$	237.697i	0.994549i	−0.867593	+	0.497274i	$0.834334\pi$
$240$	0	0
$241$	−161.000	−0.668050	−0.334025	−	0.942564i	$-0.608407\pi$
$241$	−161.000	−0.668050	−0.334025	+	0.942564i	$0.608407\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	56.3588	0.228173
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 91.5640i	− 0.364797i	−0.983225	−	0.182398i	$-0.941614\pi$
$251$	− 91.5640i	− 0.364797i	0.983225	−	0.182398i	$-0.0583861\pi$
$252$	0	0
$253$	−35.8269	−0.141608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	249.069i	0.969139i	0.874753	+	0.484569i	$0.161024\pi$
$257$	249.069i	0.969139i	−0.874753	+	0.484569i	$0.838976\pi$
$258$	0	0
$259$	−271.214	−1.04716
$260$	0	0
$261$	0	0
$262$	0	0
$263$	196.075i	0.745533i	0.927925	+	0.372767i	$0.121591\pi$
$263$	196.075i	0.745533i	−0.927925	+	0.372767i	$0.878409\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 166.809i	− 0.620106i	−0.950719	−	0.310053i	$-0.899653\pi$
$269$	− 166.809i	− 0.620106i	0.950719	−	0.310053i	$-0.100347\pi$
$270$	0	0
$271$	−89.9313	−0.331850	−0.165925	−	0.986138i	$-0.553061\pi$
$271$	−89.9313	−0.331850	−0.165925	+	0.986138i	$0.553061\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−420.814	−1.51919	−0.759593	−	0.650399i	$-0.774602\pi$
$277$	−420.814	−1.51919	−0.759593	+	0.650399i	$0.774602\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 308.807i	− 1.09896i	−0.835508	−	0.549478i	$-0.814827\pi$
$281$	− 308.807i	− 1.09896i	0.835508	−	0.549478i	$-0.185173\pi$
$282$	0	0
$283$	−59.1857	−0.209137	−0.104568	−	0.994518i	$-0.533346\pi$
$283$	−59.1857	−0.209137	−0.104568	+	0.994518i	$0.533346\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 289.928i	− 1.01020i
$288$	0	0
$289$	40.0687	0.138646
$290$	0	0
$291$	0	0
$292$	0	0
$293$	227.359i	0.775971i	0.921666	+	0.387985i	$0.126829\pi$
$293$	227.359i	0.775971i	−0.921666	+	0.387985i	$0.873171\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 103.969i	− 0.347723i
$300$	0	0
$301$	−129.863	−0.431437
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	301.221	0.981177	0.490589	−	0.871391i	$-0.336782\pi$
$307$	301.221	0.981177	0.490589	+	0.871391i	$0.336782\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 176.925i	− 0.568892i	−0.958692	−	0.284446i	$-0.908190\pi$
$311$	− 176.925i	− 0.568892i	0.958692	−	0.284446i	$-0.0918096\pi$
$312$	0	0
$313$	−349.310	−1.11601	−0.558004	−	0.829838i	$-0.688433\pi$
$313$	−349.310	−1.11601	−0.558004	+	0.829838i	$0.688433\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 371.917i	− 1.17324i	−0.809863	−	0.586620i	$-0.800458\pi$
$317$	− 371.917i	− 1.17324i	0.809863	−	0.586620i	$-0.199542\pi$
$318$	0	0
$319$	3.20607	0.0100504
$320$	0	0
$321$	0	0
$322$	0	0
$323$	296.401i	0.917651i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 494.002i	− 1.50153i
$330$	0	0
$331$	195.794	0.591522	0.295761	−	0.955262i	$-0.404427\pi$
$331$	195.794	0.591522	0.295761	+	0.955262i	$0.404427\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−471.173	−1.39814	−0.699070	−	0.715053i	$-0.746402\pi$
$337$	−471.173	−1.39814	−0.699070	+	0.715053i	$0.746402\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 18.3870i	− 0.0539208i
$342$	0	0
$343$	−368.125	−1.07325
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 353.038i	− 1.01740i	−0.860944	−	0.508700i	$-0.830126\pi$
$347$	− 353.038i	− 1.01740i	0.860944	−	0.508700i	$-0.169874\pi$
$348$	0	0
$349$	−617.931	−1.77058	−0.885288	−	0.465042i	$-0.846039\pi$
$349$	−617.931	−1.77058	−0.885288	+	0.465042i	$0.846039\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	52.9934i	0.150123i	0.997179	+	0.0750615i	$0.0239153\pi$
$353$	52.9934i	0.150123i	−0.997179	+	0.0750615i	$0.976085\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	618.426i	1.72264i	0.508067	+	0.861318i	$0.330360\pi$
$359$	618.426i	1.72264i	−0.508067	+	0.861318i	$0.669640\pi$
$360$	0	0
$361$	−8.07636	−0.0223722
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.1731	−0.0576923	−0.0288461	−	0.999584i	$-0.509183\pi$
$367$	−21.1731	−0.0576923	−0.0288461	+	0.999584i	$0.509183\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	276.981i	0.746578i
$372$	0	0
$373$	−578.298	−1.55040	−0.775198	−	0.631718i	$-0.782350\pi$
$373$	−578.298	−1.55040	−0.775198	+	0.631718i	$0.782350\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.30397i	0.0246790i
$378$	0	0
$379$	675.657	1.78273	0.891367	−	0.453281i	$-0.149747\pi$
$379$	675.657	1.78273	0.891367	+	0.453281i	$0.149747\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	167.892i	0.438361i	0.975684	+	0.219181i	$0.0703384\pi$
$383$	167.892i	0.438361i	−0.975684	+	0.219181i	$0.929662\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	676.909i	1.74013i	0.492940	+	0.870063i	$0.335922\pi$
$389$	676.909i	1.74013i	−0.492940	+	0.870063i	$0.664078\pi$
$390$	0	0
$391$	546.794	1.39845
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	63.8423	0.160812	0.0804059	−	0.996762i	$-0.474378\pi$
$397$	63.8423	0.160812	0.0804059	+	0.996762i	$0.474378\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	544.436i	1.35770i	0.734279	+	0.678848i	$0.237521\pi$
$401$	544.436i	1.35770i	−0.734279	+	0.678848i	$0.762479\pi$
$402$	0	0
$403$	53.3588	0.132404
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 44.7732i	− 0.110008i
$408$	0	0
$409$	746.725	1.82573	0.912867	−	0.408257i	$-0.133863\pi$
$409$	746.725	1.82573	0.912867	+	0.408257i	$0.133863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	566.908i	1.37266i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.0625i	0.0932279i	0.998913	+	0.0466139i	$0.0148431\pi$
$419$	39.0625i	0.0932279i	−0.998913	+	0.0466139i	$0.985157\pi$
$420$	0	0
$421$	−359.931	−0.854944	−0.427472	−	0.904029i	$-0.640596\pi$
$421$	−359.931	−0.854944	−0.427472	+	0.904029i	$0.640596\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	55.0203	0.128853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 431.163i	− 1.00038i	−0.865916	−	0.500189i	$-0.833264\pi$
$431$	− 431.163i	− 1.00038i	0.865916	−	0.500189i	$-0.166736\pi$
$432$	0	0
$433$	6.05602	0.0139862	0.00699309	−	0.999976i	$-0.497774\pi$
$433$	6.05602	0.0139862	0.00699309	+	0.999976i	$0.497774\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 651.065i	− 1.48985i
$438$	0	0
$439$	−493.282	−1.12365	−0.561825	−	0.827256i	$-0.689901\pi$
$439$	−493.282	−1.12365	−0.561825	+	0.827256i	$0.689901\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.8106i	0.0560058i	0.999608	+	0.0280029i	$0.00891477\pi$
$443$	24.8106i	0.0560058i	−0.999608	+	0.0280029i	$0.991085\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	131.881i	0.293722i	0.989157	+	0.146861i	$0.0469170\pi$
$449$	131.881i	0.293722i	−0.989157	+	0.146861i	$0.953083\pi$
$450$	0	0
$451$	47.8626	0.106126
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−303.379	−0.663849	−0.331925	−	0.943306i	$-0.607698\pi$
$457$	−303.379	−0.663849	−0.331925	+	0.943306i	$0.607698\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 650.252i	− 1.41053i	−0.708946	−	0.705263i	$-0.750829\pi$
$461$	− 650.252i	− 1.41053i	0.708946	−	0.705263i	$-0.249171\pi$
$462$	0	0
$463$	721.657	1.55865	0.779327	−	0.626618i	$-0.215561\pi$
$463$	721.657	1.55865	0.779327	+	0.626618i	$0.215561\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	341.445i	0.731147i	0.930783	+	0.365573i	$0.119127\pi$
$467$	341.445i	0.731147i	−0.930783	+	0.365573i	$0.880873\pi$
$468$	0	0
$469$	120.924	0.257833
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 21.4383i	− 0.0453242i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 639.102i	− 1.33424i	−0.744950	−	0.667121i	$-0.767527\pi$
$479$	− 639.102i	− 1.33424i	0.744950	−	0.667121i	$-0.232473\pi$
$480$	0	0
$481$	129.931	0.270127
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	411.931	0.845855	0.422927	−	0.906164i	$-0.361003\pi$
$487$	411.931	0.845855	0.422927	+	0.906164i	$0.361003\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	587.413i	1.19636i	0.801362	+	0.598180i	$0.204109\pi$
$491$	587.413i	1.19636i	−0.801362	+	0.598180i	$0.795891\pi$
$492$	0	0
$493$	−48.9313	−0.0992522
$494$	0	0
$495$	0	0
$496$	0	0
$497$	354.664i	0.713609i
$498$	0	0
$499$	774.580	1.55226	0.776132	−	0.630570i	$-0.217179\pi$
$499$	774.580	1.55226	0.776132	+	0.630570i	$0.217179\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 546.283i	− 1.08605i	−0.839717	−	0.543025i	$-0.817279\pi$
$503$	− 546.283i	− 1.08605i	0.839717	−	0.543025i	$-0.182721\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 696.772i	− 1.36890i	−0.729058	−	0.684452i	$-0.760042\pi$
$509$	− 696.772i	− 1.36890i	0.729058	−	0.684452i	$-0.239958\pi$
$510$	0	0
$511$	−684.512	−1.33955
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	81.5522	0.157741
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 558.909i	− 1.07276i	−0.843976	−	0.536381i	$-0.819791\pi$
$521$	− 558.909i	− 1.07276i	0.843976	−	0.536381i	$-0.180209\pi$
$522$	0	0
$523$	402.669	0.769922	0.384961	−	0.922933i	$-0.374215\pi$
$523$	402.669	0.769922	0.384961	+	0.922933i	$0.374215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	280.624i	0.532493i
$528$	0	0
$529$	−672.069	−1.27045
$530$	0	0
$531$	0	0
$532$	0	0
$533$	138.897i	0.260594i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 10.1168i	− 0.0187696i
$540$	0	0
$541$	636.649	1.17680	0.588400	−	0.808570i	$-0.299758\pi$
$541$	636.649	1.17680	0.588400	+	0.808570i	$0.299758\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	297.758	0.544348	0.272174	−	0.962248i	$-0.412257\pi$
$547$	297.758	0.544348	0.272174	+	0.962248i	$0.412257\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	58.2623i	0.105739i
$552$	0	0
$553$	−244.221	−0.441630
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 391.067i	− 0.702095i	−0.936358	−	0.351047i	$-0.885826\pi$
$557$	− 391.067i	− 0.702095i	0.936358	−	0.351047i	$-0.114174\pi$
$558$	0	0
$559$	62.2137	0.111295
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 410.216i	− 0.728626i	−0.931277	−	0.364313i	$-0.881304\pi$
$563$	− 410.216i	− 0.728626i	0.931277	−	0.364313i	$-0.118696\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	621.748i	1.09270i	0.837556	+	0.546352i	$0.183984\pi$
$569$	621.748i	1.09270i	−0.837556	+	0.546352i	$0.816016\pi$
$570$	0	0
$571$	396.649	0.694657	0.347328	−	0.937744i	$-0.387089\pi$
$571$	396.649	0.694657	0.347328	+	0.937744i	$0.387089\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−571.601	−0.990642	−0.495321	−	0.868710i	$-0.664950\pi$
$577$	−571.601	−0.990642	−0.495321	+	0.868710i	$0.664950\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	474.581i	0.816836i
$582$	0	0
$583$	−45.7252	−0.0784309
$584$	0	0
$585$	0	0
$586$	0	0
$587$	768.915i	1.30991i	0.755669	+	0.654953i	$0.227312\pi$
$587$	768.915i	1.30991i	−0.755669	+	0.654953i	$0.772688\pi$
$588$	0	0
$589$	334.137	0.567296
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 917.116i	− 1.54657i	−0.634059	−	0.773285i	$-0.718612\pi$
$593$	− 917.116i	− 1.54657i	0.634059	−	0.773285i	$-0.281388\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 883.814i	− 1.47548i	−0.675083	−	0.737741i	$-0.735892\pi$
$599$	− 883.814i	− 1.47548i	0.675083	−	0.737741i	$-0.264108\pi$
$600$	0	0
$601$	−675.649	−1.12421	−0.562104	−	0.827067i	$-0.690008\pi$
$601$	−675.649	−1.12421	−0.562104	+	0.827067i	$0.690008\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	687.173	1.13208	0.566040	−	0.824377i	$-0.308475\pi$
$607$	687.173	1.13208	0.566040	+	0.824377i	$0.308475\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	236.663i	0.387338i
$612$	0	0
$613$	193.896	0.316306	0.158153	−	0.987415i	$-0.449446\pi$
$613$	193.896	0.316306	0.158153	+	0.987415i	$0.449446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	557.604i	0.903735i	0.892085	+	0.451867i	$0.149242\pi$
$617$	557.604i	0.903735i	−0.892085	+	0.451867i	$0.850758\pi$
$618$	0	0
$619$	−1026.65	−1.65856	−0.829280	−	0.558833i	$-0.811249\pi$
$619$	−1026.65	−1.65856	−0.829280	+	0.558833i	$0.811249\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 566.908i	− 0.909965i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	683.333i	1.08638i
$630$	0	0
$631$	145.794	0.231052	0.115526	−	0.993304i	$-0.463145\pi$
$631$	145.794	0.231052	0.115526	+	0.993304i	$0.463145\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	29.3588	0.0460891
$638$	0	0
$639$	0	0
$640$	0	0
$641$	86.3951i	0.134782i	0.997727	+	0.0673909i	$0.0214675\pi$
$641$	86.3951i	0.134782i	−0.997727	+	0.0673909i	$0.978533\pi$
$642$	0	0
$643$	189.206	0.294255	0.147128	−	0.989118i	$-0.452997\pi$
$643$	189.206	0.294255	0.147128	+	0.989118i	$0.452997\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	801.554i	1.23888i	0.785045	+	0.619439i	$0.212640\pi$
$647$	801.554i	1.23888i	−0.785045	+	0.619439i	$0.787360\pi$
$648$	0	0
$649$	−93.5879	−0.144203
$650$	0	0
$651$	0	0
$652$	0	0
$653$	833.922i	1.27706i	0.769596	+	0.638531i	$0.220458\pi$
$653$	833.922i	1.27706i	−0.769596	+	0.638531i	$0.779542\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	152.557i	0.231497i	0.993279	+	0.115749i	$0.0369267\pi$
$659$	152.557i	0.231497i	−0.993279	+	0.115749i	$0.963073\pi$
$660$	0	0
$661$	−343.519	−0.519696	−0.259848	−	0.965650i	$-0.583672\pi$
$661$	−343.519	−0.519696	−0.259848	+	0.965650i	$0.583672\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	107.481	0.161141
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.08301i	0.0135365i
$672$	0	0
$673$	1067.38	1.58601	0.793003	−	0.609218i	$-0.208517\pi$
$673$	1067.38	1.58601	0.793003	+	0.609218i	$0.208517\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 125.137i	− 0.184840i	−0.995720	−	0.0924200i	$-0.970540\pi$
$677$	− 125.137i	− 0.184840i	0.995720	−	0.0924200i	$-0.0294602\pi$
$678$	0	0
$679$	55.6259	0.0819232
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 675.334i	− 0.988775i	−0.869241	−	0.494388i	$-0.835392\pi$
$683$	− 675.334i	− 0.988775i	0.869241	−	0.494388i	$-0.164608\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 132.694i	− 0.192589i
$690$	0	0
$691$	427.725	0.618995	0.309497	−	0.950900i	$-0.399839\pi$
$691$	427.725	0.618995	0.309497	+	0.950900i	$0.399839\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−730.483	−1.04804
$698$	0	0
$699$	0	0
$700$	0	0
$701$	410.708i	0.585889i	0.956129	+	0.292945i	$0.0946352\pi$
$701$	410.708i	0.585889i	−0.956129	+	0.292945i	$0.905365\pi$
$702$	0	0
$703$	813.641	1.15738
$704$	0	0
$705$	0	0
$706$	0	0
$707$	270.507i	0.382612i
$708$	0	0
$709$	−790.924	−1.11555	−0.557774	−	0.829993i	$-0.688344\pi$
$709$	−790.924	−1.11555	−0.557774	+	0.829993i	$0.688344\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 616.408i	− 0.864528i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	79.3797i	0.110403i	0.998475	+	0.0552014i	$0.0175801\pi$
$719$	79.3797i	0.110403i	−0.998475	+	0.0552014i	$0.982420\pi$
$720$	0	0
$721$	−480.237	−0.666070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1235.88	1.69997	0.849985	−	0.526807i	$-0.176611\pi$
$727$	1235.88	1.69997	0.849985	+	0.526807i	$0.176611\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	327.194i	0.447597i
$732$	0	0
$733$	1174.83	1.60277	0.801384	−	0.598150i	$-0.204097\pi$
$733$	1174.83	1.60277	0.801384	+	0.598150i	$0.204097\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.9626i	0.0270863i
$738$	0	0
$739$	425.863	0.576269	0.288134	−	0.957590i	$-0.406965\pi$
$739$	425.863	0.576269	0.288134	+	0.957590i	$0.406965\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 660.369i	− 0.888787i	−0.895832	−	0.444394i	$-0.853419\pi$
$743$	− 660.369i	− 0.888787i	0.895832	−	0.444394i	$-0.146581\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 238.139i	− 0.317943i
$750$	0	0
$751$	−76.7252	−0.102164	−0.0510821	−	0.998694i	$-0.516267\pi$
$751$	−76.7252	−0.102164	−0.0510821	+	0.998694i	$0.516267\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−530.532	−0.700835	−0.350417	−	0.936594i	$-0.613960\pi$
$757$	−530.532	−0.700835	−0.350417	+	0.936594i	$0.613960\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	733.325i	0.963633i	0.876272	+	0.481817i	$0.160023\pi$
$761$	733.325i	0.963633i	−0.876272	+	0.481817i	$0.839977\pi$
$762$	0	0
$763$	918.758	1.20414
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 271.591i	− 0.354095i
$768$	0	0
$769$	245.275	0.318953	0.159476	−	0.987202i	$-0.449019\pi$
$769$	245.275	0.318953	0.159476	+	0.987202i	$0.449019\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1211.23i	1.56692i	0.621443	+	0.783460i	$0.286547\pi$
$773$	1211.23i	1.56692i	−0.621443	+	0.783460i	$0.713453\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	869.783i	1.11654i
$780$	0	0
$781$	−58.5495	−0.0749674
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1203.06	1.52866	0.764330	−	0.644825i	$-0.223070\pi$
$787$	1203.06	1.52866	0.764330	+	0.644825i	$0.223070\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1166.18i	− 1.47432i
$792$	0	0
$793$	−26.3588	−0.0332393
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1014.34i	− 1.27270i	−0.771401	−	0.636349i	$-0.780444\pi$
$797$	− 1014.34i	− 1.27270i	0.771401	−	0.636349i	$-0.219556\pi$
$798$	0	0
$799$	−1244.66	−1.55777
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 113.002i	− 0.140725i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 951.230i	− 1.17581i	−0.808930	−	0.587905i	$-0.799953\pi$
$809$	− 951.230i	− 1.17581i	0.808930	−	0.587905i	$-0.200047\pi$
$810$	0	0
$811$	−404.580	−0.498866	−0.249433	−	0.968392i	$-0.580244\pi$
$811$	−404.580	−0.498866	−0.249433	+	0.968392i	$0.580244\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	389.588	0.476852
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1185.16i	− 1.44356i	−0.692122	−	0.721780i	$-0.743324\pi$
$821$	− 1185.16i	− 1.44356i	0.692122	−	0.721780i	$-0.256676\pi$
$822$	0	0
$823$	668.069	0.811748	0.405874	−	0.913929i	$-0.366967\pi$
$823$	668.069	0.811748	0.405874	+	0.913929i	$0.366967\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	427.199i	0.516564i	0.966070	+	0.258282i	$0.0831564\pi$
$827$	427.199i	0.516564i	−0.966070	+	0.258282i	$0.916844\pi$
$828$	0	0
$829$	1143.52	1.37940	0.689698	−	0.724097i	$-0.257743\pi$
$829$	1143.52	1.37940	0.689698	+	0.724097i	$0.257743\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	154.403i	0.185358i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 724.242i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$839$	− 724.242i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$840$	0	0
$841$	831.382	0.988563
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	751.020	0.886683
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1500.99i	− 1.76379i
$852$	0	0
$853$	488.654	0.572865	0.286433	−	0.958100i	$-0.407531\pi$
$853$	488.654	0.572865	0.286433	+	0.958100i	$0.407531\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 134.320i	− 0.156732i	−0.996925	−	0.0783662i	$-0.975030\pi$
$857$	− 134.320i	− 0.156732i	0.996925	−	0.0783662i	$-0.0249703\pi$
$858$	0	0
$859$	1321.38	1.53828	0.769140	−	0.639081i	$-0.220685\pi$
$859$	1321.38	1.53828	0.769140	+	0.639081i	$0.220685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	948.942i	1.09959i	0.835301	+	0.549793i	$0.185293\pi$
$863$	948.942i	1.09959i	−0.835301	+	0.549793i	$0.814707\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 40.3172i	− 0.0463949i
$870$	0	0
$871$	−57.9313	−0.0665113
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−54.3231	−0.0619420	−0.0309710	−	0.999520i	$-0.509860\pi$
$877$	−54.3231	−0.0619420	−0.0309710	+	0.999520i	$0.509860\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	762.713i	0.865735i	0.901458	+	0.432868i	$0.142498\pi$
$881$	762.713i	0.865735i	−0.901458	+	0.432868i	$0.857502\pi$
$882$	0	0
$883$	−95.0917	−0.107692	−0.0538458	−	0.998549i	$-0.517148\pi$
$883$	−95.0917	−0.107692	−0.0538458	+	0.998549i	$0.517148\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	672.774i	0.758483i	0.925298	+	0.379241i	$0.123815\pi$
$887$	672.774i	0.758483i	−0.925298	+	0.379241i	$0.876185\pi$
$888$	0	0
$889$	−1009.71	−1.13578
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1482.01i	1.65958i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	55.1610i	0.0613581i
$900$	0	0
$901$	697.863	0.774542
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	137.275	0.151350	0.0756752	−	0.997133i	$-0.475889\pi$
$907$	137.275	0.151350	0.0756752	+	0.997133i	$0.475889\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1708.55i	1.87547i	0.347351	+	0.937735i	$0.387081\pi$
$911$	1708.55i	1.87547i	−0.347351	+	0.937735i	$0.612919\pi$
$912$	0	0
$913$	−78.3461	−0.0858117
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1559.69i	− 1.70086i
$918$	0	0
$919$	−475.802	−0.517738	−0.258869	−	0.965912i	$-0.583350\pi$
$919$	−475.802	−0.517738	−0.258869	+	0.965912i	$0.583350\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 169.910i	− 0.184084i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	983.056i	1.05819i	0.848563	+	0.529094i	$0.177468\pi$
$929$	983.056i	1.05819i	−0.848563	+	0.529094i	$0.822532\pi$
$930$	0	0
$931$	183.847	0.197473
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−614.705	−0.656035	−0.328018	−	0.944672i	$-0.606381\pi$
$937$	−614.705	−0.656035	−0.328018	+	0.944672i	$0.606381\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1155.18i	− 1.22761i	−0.789456	−	0.613807i	$-0.789637\pi$
$941$	− 1155.18i	− 1.22761i	0.789456	−	0.613807i	$-0.210363\pi$
$942$	0	0
$943$	1604.55	1.70154
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1412.15i	− 1.49118i	−0.666402	−	0.745592i	$-0.732167\pi$
$947$	− 1412.15i	− 1.49118i	0.666402	−	0.745592i	$-0.267833\pi$
$948$	0	0
$949$	327.931	0.345555
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 243.408i	− 0.255412i	−0.991812	−	0.127706i	$-0.959239\pi$
$953$	− 243.408i	− 0.255412i	0.991812	−	0.127706i	$-0.0407614\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1540.27i	1.60612i
$960$	0	0
$961$	−644.649	−0.670810
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	881.448	0.911528	0.455764	−	0.890101i	$-0.349366\pi$
$967$	881.448	0.911528	0.455764	+	0.890101i	$0.349366\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1537.17i	1.58308i	0.611119	+	0.791538i	$0.290720\pi$
$971$	1537.17i	1.58308i	−0.611119	+	0.791538i	$0.709280\pi$
$972$	0	0
$973$	1064.12	1.09365
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 394.168i	− 0.403447i	−0.979442	−	0.201724i	$-0.935346\pi$
$977$	− 394.168i	− 0.403447i	0.979442	−	0.201724i	$-0.0646543\pi$
$978$	0	0
$979$	93.5879	0.0955954
$980$	0	0
$981$	0	0
$982$	0	0
$983$	728.869i	0.741474i	0.928738	+	0.370737i	$0.120895\pi$
$983$	728.869i	0.741474i	−0.928738	+	0.370737i	$0.879105\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 718.702i	− 0.726696i
$990$	0	0
$991$	−732.282	−0.738933	−0.369466	−	0.929244i	$-0.620460\pi$
$991$	−732.282	−0.738933	−0.369466	+	0.929244i	$0.620460\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−798.519	−0.800922	−0.400461	−	0.916314i	$-0.631150\pi$
$997$	−798.519	−0.800922	−0.400461	+	0.916314i	$0.631150\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.g.o.701.4		4
3.2	odd	2	inner	2700.3.g.o.701.3		4
5.2	odd	4		540.3.b.c.269.5	yes	8
5.3	odd	4		540.3.b.c.269.3	✓	8
5.4	even	2		2700.3.g.p.701.2		4
15.2	even	4		540.3.b.c.269.4	yes	8
15.8	even	4		540.3.b.c.269.6	yes	8
15.14	odd	2		2700.3.g.p.701.1		4
20.3	even	4		2160.3.c.n.1889.3		8
20.7	even	4		2160.3.c.n.1889.5		8
45.2	even	12		1620.3.t.e.1349.2		16
45.7	odd	12		1620.3.t.e.1349.7		16
45.13	odd	12		1620.3.t.e.269.2		16
45.22	odd	12		1620.3.t.e.269.1		16
45.23	even	12		1620.3.t.e.269.7		16
45.32	even	12		1620.3.t.e.269.8		16
45.38	even	12		1620.3.t.e.1349.1		16
45.43	odd	12		1620.3.t.e.1349.8		16
60.23	odd	4		2160.3.c.n.1889.6		8
60.47	odd	4		2160.3.c.n.1889.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.c.269.3	✓	8	5.3	odd	4
540.3.b.c.269.4	yes	8	15.2	even	4
540.3.b.c.269.5	yes	8	5.2	odd	4
540.3.b.c.269.6	yes	8	15.8	even	4
1620.3.t.e.269.1		16	45.22	odd	12
1620.3.t.e.269.2		16	45.13	odd	12
1620.3.t.e.269.7		16	45.23	even	12
1620.3.t.e.269.8		16	45.32	even	12
1620.3.t.e.1349.1		16	45.38	even	12
1620.3.t.e.1349.2		16	45.2	even	12
1620.3.t.e.1349.7		16	45.7	odd	12
1620.3.t.e.1349.8		16	45.43	odd	12
2160.3.c.n.1889.3		8	20.3	even	4
2160.3.c.n.1889.4		8	60.47	odd	4
2160.3.c.n.1889.5		8	20.7	even	4
2160.3.c.n.1889.6		8	60.23	odd	4
2700.3.g.o.701.3		4	3.2	odd	2	inner
2700.3.g.o.701.4		4	1.1	even	1	trivial
2700.3.g.p.701.1		4	15.14	odd	2
2700.3.g.p.701.2		4	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.26209 q^{7} +O(q^{10})\)	\(q+6.26209 q^{7} +1.03377i q^{11} -3.00000 q^{13} -15.7776i q^{17} -18.7863 q^{19} +34.6564i q^{23} -3.10132i q^{29} -17.7863 q^{31} -43.3104 q^{37} -46.2989i q^{41} -20.7379 q^{43} -78.8878i q^{47} -9.78626 q^{49} +44.2313i q^{53} +90.5302i q^{59} +8.78626 q^{61} +19.3104 q^{67} +56.6366i q^{71} -109.310 q^{73} +6.47359i q^{77} -39.0000 q^{79} +75.7865i q^{83} -90.5302i q^{89} -18.7863 q^{91} +8.88296 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7}+O(q^{10}) \) 4 * q - 6 * q^7	\( 4 q - 6 q^{7} - 12 q^{13} + 18 q^{19} + 22 q^{31} - 18 q^{37} - 114 q^{43} + 54 q^{49} - 58 q^{61} - 78 q^{67} - 282 q^{73} - 156 q^{79} + 18 q^{91} - 306 q^{97}+O(q^{100}) \) 4 * q - 6 * q^7 - 12 * q^13 + 18 * q^19 + 22 * q^31 - 18 * q^37 - 114 * q^43 + 54 * q^49 - 58 * q^61 - 78 * q^67 - 282 * q^73 - 156 * q^79 + 18 * q^91 - 306 * q^97

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