LMFDB - Embedded newform 1089.3.c.f.604.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.f.604.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +1.00000 q^{4} +4.89898 q^{5} +7.07107i q^{7} +8.66025i q^{8} +O(q^{10})$	$q+1.73205i q^{2} +1.00000 q^{4} +4.89898 q^{5} +7.07107i q^{7} +8.66025i q^{8} +8.48528i q^{10} -7.07107i q^{13} -12.2474 q^{14} -11.0000 q^{16} +27.7128i q^{17} +15.5563i q^{19} +4.89898 q^{20} -24.4949 q^{23} -1.00000 q^{25} +12.2474 q^{26} +7.07107i q^{28} -34.6410i q^{29} +12.0000 q^{31} +15.5885i q^{32} -48.0000 q^{34} +34.6410i q^{35} -60.0000 q^{37} -26.9444 q^{38} +42.4264i q^{40} +34.6410i q^{41} -49.4975i q^{43} -42.4264i q^{46} +83.2827 q^{47} -1.00000 q^{49} -1.73205i q^{50} -7.07107i q^{52} +83.2827 q^{53} -61.2372 q^{56} +60.0000 q^{58} +48.9898 q^{59} +26.8701i q^{61} +20.7846i q^{62} -71.0000 q^{64} -34.6410i q^{65} -120.000 q^{67} +27.7128i q^{68} -60.0000 q^{70} -24.4949 q^{71} +120.208i q^{73} -103.923i q^{74} +15.5563i q^{76} -128.693i q^{79} -53.8888 q^{80} -60.0000 q^{82} +76.2102i q^{83} +135.765i q^{85} +85.7321 q^{86} +97.9796 q^{89} +50.0000 q^{91} -24.4949 q^{92} +144.250i q^{94} +76.2102i q^{95} +70.0000 q^{97} -1.73205i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} - 44 q^{16} - 4 q^{25} + 48 q^{31} - 192 q^{34} - 240 q^{37} - 4 q^{49} + 240 q^{58} - 284 q^{64} - 480 q^{67} - 240 q^{70} - 240 q^{82} + 200 q^{91} + 280 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 44 * q^16 - 4 * q^25 + 48 * q^31 - 192 * q^34 - 240 * q^37 - 4 * q^49 + 240 * q^58 - 284 * q^64 - 480 * q^67 - 240 * q^70 - 240 * q^82 + 200 * q^91 + 280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.73205i	0.866025i	0.901388	+	0.433013i	$0.142549\pi$
$2$	1.73205i	0.866025i	−0.901388	+	0.433013i	$0.857451\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.250000
$5$	4.89898	0.979796	0.489898	−	0.871780i	$-0.337034\pi$
$5$	4.89898	0.979796	0.489898	+	0.871780i	$0.337034\pi$
$6$	0	0
$7$	7.07107i	1.01015i	0.863075	+	0.505076i	$0.168536\pi$
$7$	7.07107i	1.01015i	−0.863075	+	0.505076i	$0.831464\pi$
$8$	8.66025i	1.08253i
$9$	0	0
$10$	8.48528i	0.848528i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 7.07107i	− 0.543928i	−0.962307	−	0.271964i	$-0.912327\pi$
$13$	− 7.07107i	− 0.543928i	0.962307	−	0.271964i	$-0.0876732\pi$
$14$	−12.2474	−0.874818
$15$	0	0
$16$	−11.0000	−0.687500
$17$	27.7128i	1.63017i	0.579345	+	0.815083i	$0.303309\pi$
$17$	27.7128i	1.63017i	−0.579345	+	0.815083i	$0.696691\pi$
$18$	0	0
$19$	15.5563i	0.818755i	0.912365	+	0.409378i	$0.134254\pi$
$19$	15.5563i	0.818755i	−0.912365	+	0.409378i	$0.865746\pi$
$20$	4.89898	0.244949
$21$	0	0
$22$	0	0
$23$	−24.4949	−1.06500	−0.532498	−	0.846431i	$-0.678747\pi$
$23$	−24.4949	−1.06500	−0.532498	+	0.846431i	$0.678747\pi$
$24$	0	0
$25$	−1.00000	−0.0400000
$26$	12.2474	0.471056
$27$	0	0
$28$	7.07107i	0.252538i
$29$	− 34.6410i	− 1.19452i	−0.802049	−	0.597259i	$-0.796256\pi$
$29$	− 34.6410i	− 1.19452i	0.802049	−	0.597259i	$-0.203744\pi$
$30$	0	0
$31$	12.0000	0.387097	0.193548	−	0.981091i	$-0.438000\pi$
$31$	12.0000	0.387097	0.193548	+	0.981091i	$0.438000\pi$
$32$	15.5885i	0.487139i
$33$	0	0
$34$	−48.0000	−1.41176
$35$	34.6410i	0.989743i
$36$	0	0
$37$	−60.0000	−1.62162	−0.810811	−	0.585308i	$-0.800973\pi$
$37$	−60.0000	−1.62162	−0.810811	+	0.585308i	$0.800973\pi$
$38$	−26.9444	−0.709063
$39$	0	0
$40$	42.4264i	1.06066i
$41$	34.6410i	0.844903i	0.906386	+	0.422451i	$0.138830\pi$
$41$	34.6410i	0.844903i	−0.906386	+	0.422451i	$0.861170\pi$
$42$	0	0
$43$	− 49.4975i	− 1.15110i	−0.817765	−	0.575552i	$-0.804787\pi$
$43$	− 49.4975i	− 1.15110i	0.817765	−	0.575552i	$-0.195213\pi$
$44$	0	0
$45$	0	0
$46$	− 42.4264i	− 0.922313i
$47$	83.2827	1.77197	0.885986	−	0.463713i	$-0.153483\pi$
$47$	83.2827	1.77197	0.885986	+	0.463713i	$0.153483\pi$
$48$	0	0
$49$	−1.00000	−0.0204082
$50$	− 1.73205i	− 0.0346410i
$51$	0	0
$52$	− 7.07107i	− 0.135982i
$53$	83.2827	1.57137	0.785685	−	0.618626i	$-0.212310\pi$
$53$	83.2827	1.57137	0.785685	+	0.618626i	$0.212310\pi$
$54$	0	0
$55$	0	0
$56$	−61.2372	−1.09352
$57$	0	0
$58$	60.0000	1.03448
$59$	48.9898	0.830336	0.415168	−	0.909745i	$-0.363723\pi$
$59$	48.9898	0.830336	0.415168	+	0.909745i	$0.363723\pi$
$60$	0	0
$61$	26.8701i	0.440493i	0.975444	+	0.220246i	$0.0706861\pi$
$61$	26.8701i	0.440493i	−0.975444	+	0.220246i	$0.929314\pi$
$62$	20.7846i	0.335236i
$63$	0	0
$64$	−71.0000	−1.10938
$65$	− 34.6410i	− 0.532939i
$66$	0	0
$67$	−120.000	−1.79104	−0.895522	−	0.445016i	$-0.853198\pi$
$67$	−120.000	−1.79104	−0.895522	+	0.445016i	$0.853198\pi$
$68$	27.7128i	0.407541i
$69$	0	0
$70$	−60.0000	−0.857143
$71$	−24.4949	−0.344999	−0.172499	−	0.985010i	$-0.555184\pi$
$71$	−24.4949	−0.344999	−0.172499	+	0.985010i	$0.555184\pi$
$72$	0	0
$73$	120.208i	1.64669i	0.567543	+	0.823344i	$0.307894\pi$
$73$	120.208i	1.64669i	−0.567543	+	0.823344i	$0.692106\pi$
$74$	− 103.923i	− 1.40437i
$75$	0	0
$76$	15.5563i	0.204689i
$77$	0	0
$78$	0	0
$79$	− 128.693i	− 1.62903i	−0.580142	−	0.814515i	$-0.697003\pi$
$79$	− 128.693i	− 1.62903i	0.580142	−	0.814515i	$-0.302997\pi$
$80$	−53.8888	−0.673610
$81$	0	0
$82$	−60.0000	−0.731707
$83$	76.2102i	0.918196i	0.888386	+	0.459098i	$0.151827\pi$
$83$	76.2102i	0.918196i	−0.888386	+	0.459098i	$0.848173\pi$
$84$	0	0
$85$	135.765i	1.59723i
$86$	85.7321	0.996885
$87$	0	0
$88$	0	0
$89$	97.9796	1.10089	0.550447	−	0.834870i	$-0.314457\pi$
$89$	97.9796	1.10089	0.550447	+	0.834870i	$0.314457\pi$
$90$	0	0
$91$	50.0000	0.549451
$92$	−24.4949	−0.266249
$93$	0	0
$94$	144.250i	1.53457i
$95$	76.2102i	0.802213i
$96$	0	0
$97$	70.0000	0.721649	0.360825	−	0.932634i	$-0.382495\pi$
$97$	70.0000	0.721649	0.360825	+	0.932634i	$0.382495\pi$
$98$	− 1.73205i	− 0.0176740i
$99$	0	0
$100$	−1.00000	−0.0100000
$101$	69.2820i	0.685961i	0.939343	+	0.342980i	$0.111436\pi$
$101$	69.2820i	0.685961i	−0.939343	+	0.342980i	$0.888564\pi$
$102$	0	0
$103$	−50.0000	−0.485437	−0.242718	−	0.970097i	$-0.578039\pi$
$103$	−50.0000	−0.485437	−0.242718	+	0.970097i	$0.578039\pi$
$104$	61.2372	0.588820
$105$	0	0
$106$	144.250i	1.36085i
$107$	96.9948i	0.906494i	0.891385	+	0.453247i	$0.149734\pi$
$107$	96.9948i	0.906494i	−0.891385	+	0.453247i	$0.850266\pi$
$108$	0	0
$109$	− 43.8406i	− 0.402208i	−0.979570	−	0.201104i	$-0.935547\pi$
$109$	− 43.8406i	− 0.402208i	0.979570	−	0.201104i	$-0.0644528\pi$
$110$	0	0
$111$	0	0
$112$	− 77.7817i	− 0.694480i
$113$	48.9898	0.433538	0.216769	−	0.976223i	$-0.430448\pi$
$113$	48.9898	0.433538	0.216769	+	0.976223i	$0.430448\pi$
$114$	0	0
$115$	−120.000	−1.04348
$116$	− 34.6410i	− 0.298629i
$117$	0	0
$118$	84.8528i	0.719092i
$119$	−195.959	−1.64672
$120$	0	0
$121$	0	0
$122$	−46.5403	−0.381478
$123$	0	0
$124$	12.0000	0.0967742
$125$	−127.373	−1.01899
$126$	0	0
$127$	− 91.9239i	− 0.723810i	−0.932215	−	0.361905i	$-0.882126\pi$
$127$	− 91.9239i	− 0.723810i	0.932215	−	0.361905i	$-0.117874\pi$
$128$	− 60.6218i	− 0.473608i
$129$	0	0
$130$	60.0000	0.461538
$131$	34.6410i	0.264435i	0.991221	+	0.132218i	$0.0422098\pi$
$131$	34.6410i	0.264435i	−0.991221	+	0.132218i	$0.957790\pi$
$132$	0	0
$133$	−110.000	−0.827068
$134$	− 207.846i	− 1.55109i
$135$	0	0
$136$	−240.000	−1.76471
$137$	88.1816	0.643662	0.321831	−	0.946797i	$-0.395702\pi$
$137$	88.1816	0.643662	0.321831	+	0.946797i	$0.395702\pi$
$138$	0	0
$139$	69.2965i	0.498536i	0.968435	+	0.249268i	$0.0801900\pi$
$139$	69.2965i	0.498536i	−0.968435	+	0.249268i	$0.919810\pi$
$140$	34.6410i	0.247436i
$141$	0	0
$142$	− 42.4264i	− 0.298778i
$143$	0	0
$144$	0	0
$145$	− 169.706i	− 1.17038i
$146$	−208.207	−1.42607
$147$	0	0
$148$	−60.0000	−0.405405
$149$	− 242.487i	− 1.62743i	−0.581264	−	0.813715i	$-0.697442\pi$
$149$	− 242.487i	− 1.62743i	0.581264	−	0.813715i	$-0.302558\pi$
$150$	0	0
$151$	26.8701i	0.177947i	0.996034	+	0.0889737i	$0.0283587\pi$
$151$	26.8701i	0.177947i	−0.996034	+	0.0889737i	$0.971641\pi$
$152$	−134.722	−0.886329
$153$	0	0
$154$	0	0
$155$	58.7878	0.379276
$156$	0	0
$157$	60.0000	0.382166	0.191083	−	0.981574i	$-0.438800\pi$
$157$	60.0000	0.382166	0.191083	+	0.981574i	$0.438800\pi$
$158$	222.904	1.41078
$159$	0	0
$160$	76.3675i	0.477297i
$161$	− 173.205i	− 1.07581i
$162$	0	0
$163$	−190.000	−1.16564	−0.582822	−	0.812600i	$-0.698052\pi$
$163$	−190.000	−1.16564	−0.582822	+	0.812600i	$0.698052\pi$
$164$	34.6410i	0.211226i
$165$	0	0
$166$	−132.000	−0.795181
$167$	− 180.133i	− 1.07864i	−0.842100	−	0.539321i	$-0.818681\pi$
$167$	− 180.133i	− 1.07864i	0.842100	−	0.539321i	$-0.181319\pi$
$168$	0	0
$169$	119.000	0.704142
$170$	−235.151	−1.38324
$171$	0	0
$172$	− 49.4975i	− 0.287776i
$173$	− 76.2102i	− 0.440522i	−0.975441	−	0.220261i	$-0.929309\pi$
$173$	− 76.2102i	− 0.440522i	0.975441	−	0.220261i	$-0.0706908\pi$
$174$	0	0
$175$	− 7.07107i	− 0.0404061i
$176$	0	0
$177$	0	0
$178$	169.706i	0.953402i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	310.000	1.71271	0.856354	−	0.516390i	$-0.172724\pi$
$181$	310.000	1.71271	0.856354	+	0.516390i	$0.172724\pi$
$182$	86.6025i	0.475838i
$183$	0	0
$184$	− 212.132i	− 1.15289i
$185$	−293.939	−1.58886
$186$	0	0
$187$	0	0
$188$	83.2827	0.442993
$189$	0	0
$190$	−132.000	−0.694737
$191$	−367.423	−1.92368	−0.961842	−	0.273607i	$-0.911783\pi$
$191$	−367.423	−1.92368	−0.961842	+	0.273607i	$0.911783\pi$
$192$	0	0
$193$	− 35.3553i	− 0.183188i	−0.995796	−	0.0915941i	$-0.970804\pi$
$193$	− 35.3553i	− 0.183188i	0.995796	−	0.0915941i	$-0.0291962\pi$
$194$	121.244i	0.624967i
$195$	0	0
$196$	−1.00000	−0.00510204
$197$	41.5692i	0.211011i	0.994419	+	0.105506i	$0.0336461\pi$
$197$	41.5692i	0.211011i	−0.994419	+	0.105506i	$0.966354\pi$
$198$	0	0
$199$	−180.000	−0.904523	−0.452261	−	0.891885i	$-0.649383\pi$
$199$	−180.000	−0.904523	−0.452261	+	0.891885i	$0.649383\pi$
$200$	− 8.66025i	− 0.0433013i
$201$	0	0
$202$	−120.000	−0.594059
$203$	244.949	1.20665
$204$	0	0
$205$	169.706i	0.827832i
$206$	− 86.6025i	− 0.420401i
$207$	0	0
$208$	77.7817i	0.373951i
$209$	0	0
$210$	0	0
$211$	100.409i	0.475873i	0.971281	+	0.237936i	$0.0764710\pi$
$211$	100.409i	0.475873i	−0.971281	+	0.237936i	$0.923529\pi$
$212$	83.2827	0.392843
$213$	0	0
$214$	−168.000	−0.785047
$215$	− 242.487i	− 1.12785i
$216$	0	0
$217$	84.8528i	0.391027i
$218$	75.9342	0.348322
$219$	0	0
$220$	0	0
$221$	195.959	0.886693
$222$	0	0
$223$	190.000	0.852018	0.426009	−	0.904719i	$-0.359919\pi$
$223$	190.000	0.852018	0.426009	+	0.904719i	$0.359919\pi$
$224$	−110.227	−0.492085
$225$	0	0
$226$	84.8528i	0.375455i
$227$	374.123i	1.64812i	0.566503	+	0.824059i	$0.308296\pi$
$227$	374.123i	1.64812i	−0.566503	+	0.824059i	$0.691704\pi$
$228$	0	0
$229$	180.000	0.786026	0.393013	−	0.919533i	$-0.371433\pi$
$229$	180.000	0.786026	0.393013	+	0.919533i	$0.371433\pi$
$230$	− 207.846i	− 0.903679i
$231$	0	0
$232$	300.000	1.29310
$233$	214.774i	0.921778i	0.887458	+	0.460889i	$0.152469\pi$
$233$	214.774i	0.921778i	−0.887458	+	0.460889i	$0.847531\pi$
$234$	0	0
$235$	408.000	1.73617
$236$	48.9898	0.207584
$237$	0	0
$238$	− 339.411i	− 1.42610i
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	− 1.41421i	− 0.00586811i	−0.999996	−	0.00293405i	$-0.999066\pi$
$241$	− 1.41421i	− 0.00586811i	0.999996	−	0.00293405i	$-0.000933939\pi$
$242$	0	0
$243$	0	0
$244$	26.8701i	0.110123i
$245$	−4.89898	−0.0199958
$246$	0	0
$247$	110.000	0.445344
$248$	103.923i	0.419045i
$249$	0	0
$250$	− 220.617i	− 0.882469i
$251$	440.908	1.75661	0.878303	−	0.478104i	$-0.158676\pi$
$251$	440.908	1.75661	0.878303	+	0.478104i	$0.158676\pi$
$252$	0	0
$253$	0	0
$254$	159.217	0.626838
$255$	0	0
$256$	−179.000	−0.699219
$257$	293.939	1.14373	0.571865	−	0.820348i	$-0.306220\pi$
$257$	293.939	1.14373	0.571865	+	0.820348i	$0.306220\pi$
$258$	0	0
$259$	− 424.264i	− 1.63809i
$260$	− 34.6410i	− 0.133235i
$261$	0	0
$262$	−60.0000	−0.229008
$263$	131.636i	0.500517i	0.968179	+	0.250258i	$0.0805155\pi$
$263$	131.636i	0.500517i	−0.968179	+	0.250258i	$0.919484\pi$
$264$	0	0
$265$	408.000	1.53962
$266$	− 190.526i	− 0.716262i
$267$	0	0
$268$	−120.000	−0.447761
$269$	−24.4949	−0.0910591	−0.0455295	−	0.998963i	$-0.514498\pi$
$269$	−24.4949	−0.0910591	−0.0455295	+	0.998963i	$0.514498\pi$
$270$	0	0
$271$	− 298.399i	− 1.10110i	−0.834801	−	0.550552i	$-0.814417\pi$
$271$	− 298.399i	− 1.10110i	0.834801	−	0.550552i	$-0.185583\pi$
$272$	− 304.841i	− 1.12074i
$273$	0	0
$274$	152.735i	0.557427i
$275$	0	0
$276$	0	0
$277$	162.635i	0.587128i	0.955939	+	0.293564i	$0.0948414\pi$
$277$	162.635i	0.587128i	−0.955939	+	0.293564i	$0.905159\pi$
$278$	−120.025	−0.431745
$279$	0	0
$280$	−300.000	−1.07143
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	304.056i	1.07440i	0.843454	+	0.537201i	$0.180518\pi$
$283$	304.056i	1.07440i	−0.843454	+	0.537201i	$0.819482\pi$
$284$	−24.4949	−0.0862496
$285$	0	0
$286$	0	0
$287$	−244.949	−0.853481
$288$	0	0
$289$	−479.000	−1.65744
$290$	293.939	1.01358
$291$	0	0
$292$	120.208i	0.411672i
$293$	110.851i	0.378332i	0.981945	+	0.189166i	$0.0605784\pi$
$293$	110.851i	0.378332i	−0.981945	+	0.189166i	$0.939422\pi$
$294$	0	0
$295$	240.000	0.813559
$296$	− 519.615i	− 1.75546i
$297$	0	0
$298$	420.000	1.40940
$299$	173.205i	0.579281i
$300$	0	0
$301$	350.000	1.16279
$302$	−46.5403	−0.154107
$303$	0	0
$304$	− 171.120i	− 0.562894i
$305$	131.636i	0.431593i
$306$	0	0
$307$	− 35.3553i	− 0.115164i	−0.998341	−	0.0575820i	$-0.981661\pi$
$307$	− 35.3553i	− 0.115164i	0.998341	−	0.0575820i	$-0.0183391\pi$
$308$	0	0
$309$	0	0
$310$	101.823i	0.328463i
$311$	269.444	0.866379	0.433190	−	0.901303i	$-0.357388\pi$
$311$	269.444	0.866379	0.433190	+	0.901303i	$0.357388\pi$
$312$	0	0
$313$	480.000	1.53355	0.766773	−	0.641918i	$-0.221861\pi$
$313$	480.000	1.53355	0.766773	+	0.641918i	$0.221861\pi$
$314$	103.923i	0.330965i
$315$	0	0
$316$	− 128.693i	− 0.407258i
$317$	122.474	0.386355	0.193177	−	0.981164i	$-0.438121\pi$
$317$	122.474	0.386355	0.193177	+	0.981164i	$0.438121\pi$
$318$	0	0
$319$	0	0
$320$	−347.828	−1.08696
$321$	0	0
$322$	300.000	0.931677
$323$	−431.110	−1.33471
$324$	0	0
$325$	7.07107i	0.0217571i
$326$	− 329.090i	− 1.00948i
$327$	0	0
$328$	−300.000	−0.914634
$329$	588.897i	1.78996i
$330$	0	0
$331$	290.000	0.876133	0.438066	−	0.898943i	$-0.355663\pi$
$331$	290.000	0.876133	0.438066	+	0.898943i	$0.355663\pi$
$332$	76.2102i	0.229549i
$333$	0	0
$334$	312.000	0.934132
$335$	−587.878	−1.75486
$336$	0	0
$337$	− 289.914i	− 0.860278i	−0.902763	−	0.430139i	$-0.858464\pi$
$337$	− 289.914i	− 0.860278i	0.902763	−	0.430139i	$-0.141536\pi$
$338$	206.114i	0.609805i
$339$	0	0
$340$	135.765i	0.399307i
$341$	0	0
$342$	0	0
$343$	339.411i	0.989537i
$344$	428.661	1.24611
$345$	0	0
$346$	132.000	0.381503
$347$	339.482i	0.978334i	0.872190	+	0.489167i	$0.162699\pi$
$347$	339.482i	0.978334i	−0.872190	+	0.489167i	$0.837301\pi$
$348$	0	0
$349$	− 482.247i	− 1.38180i	−0.722952	−	0.690898i	$-0.757215\pi$
$349$	− 482.247i	− 1.38180i	0.722952	−	0.690898i	$-0.242785\pi$
$350$	12.2474	0.0349927
$351$	0	0
$352$	0	0
$353$	−205.757	−0.582881	−0.291441	−	0.956589i	$-0.594135\pi$
$353$	−205.757	−0.582881	−0.291441	+	0.956589i	$0.594135\pi$
$354$	0	0
$355$	−120.000	−0.338028
$356$	97.9796	0.275224
$357$	0	0
$358$	0	0
$359$	− 658.179i	− 1.83337i	−0.399612	−	0.916684i	$-0.630855\pi$
$359$	− 658.179i	− 1.83337i	0.399612	−	0.916684i	$-0.369145\pi$
$360$	0	0
$361$	119.000	0.329640
$362$	536.936i	1.48325i
$363$	0	0
$364$	50.0000	0.137363
$365$	588.897i	1.61342i
$366$	0	0
$367$	530.000	1.44414	0.722071	−	0.691819i	$-0.243190\pi$
$367$	530.000	1.44414	0.722071	+	0.691819i	$0.243190\pi$
$368$	269.444	0.732184
$369$	0	0
$370$	− 509.117i	− 1.37599i
$371$	588.897i	1.58732i
$372$	0	0
$373$	− 176.777i	− 0.473932i	−0.971518	−	0.236966i	$-0.923847\pi$
$373$	− 176.777i	− 0.473932i	0.971518	−	0.236966i	$-0.0761530\pi$
$374$	0	0
$375$	0	0
$376$	721.249i	1.91822i
$377$	−244.949	−0.649732
$378$	0	0
$379$	242.000	0.638522	0.319261	−	0.947667i	$-0.396565\pi$
$379$	242.000	0.638522	0.319261	+	0.947667i	$0.396565\pi$
$380$	76.2102i	0.200553i
$381$	0	0
$382$	− 636.396i	− 1.66596i
$383$	−63.6867	−0.166284	−0.0831419	−	0.996538i	$-0.526495\pi$
$383$	−63.6867	−0.166284	−0.0831419	+	0.996538i	$0.526495\pi$
$384$	0	0
$385$	0	0
$386$	61.2372	0.158646
$387$	0	0
$388$	70.0000	0.180412
$389$	612.372	1.57422	0.787111	−	0.616811i	$-0.211576\pi$
$389$	612.372	1.57422	0.787111	+	0.616811i	$0.211576\pi$
$390$	0	0
$391$	− 678.823i	− 1.73612i
$392$	− 8.66025i	− 0.0220925i
$393$	0	0
$394$	−72.0000	−0.182741
$395$	− 630.466i	− 1.59612i
$396$	0	0
$397$	−310.000	−0.780856	−0.390428	−	0.920633i	$-0.627673\pi$
$397$	−310.000	−0.780856	−0.390428	+	0.920633i	$0.627673\pi$
$398$	− 311.769i	− 0.783340i
$399$	0	0
$400$	11.0000	0.0275000
$401$	293.939	0.733014	0.366507	−	0.930415i	$-0.380554\pi$
$401$	293.939	0.733014	0.366507	+	0.930415i	$0.380554\pi$
$402$	0	0
$403$	− 84.8528i	− 0.210553i
$404$	69.2820i	0.171490i
$405$	0	0
$406$	424.264i	1.04499i
$407$	0	0
$408$	0	0
$409$	− 253.144i	− 0.618935i	−0.950910	−	0.309467i	$-0.899849\pi$
$409$	− 253.144i	− 0.618935i	0.950910	−	0.309467i	$-0.100151\pi$
$410$	−293.939	−0.716924
$411$	0	0
$412$	−50.0000	−0.121359
$413$	346.410i	0.838766i
$414$	0	0
$415$	373.352i	0.899644i
$416$	110.227	0.264969
$417$	0	0
$418$	0	0
$419$	−685.857	−1.63689	−0.818445	−	0.574585i	$-0.805164\pi$
$419$	−685.857	−1.63689	−0.818445	+	0.574585i	$0.805164\pi$
$420$	0	0
$421$	108.000	0.256532	0.128266	−	0.991740i	$-0.459059\pi$
$421$	108.000	0.256532	0.128266	+	0.991740i	$0.459059\pi$
$422$	−173.914	−0.412118
$423$	0	0
$424$	721.249i	1.70106i
$425$	− 27.7128i	− 0.0652066i
$426$	0	0
$427$	−190.000	−0.444965
$428$	96.9948i	0.226623i
$429$	0	0
$430$	420.000	0.976744
$431$	− 103.923i	− 0.241121i	−0.992706	−	0.120560i	$-0.961531\pi$
$431$	− 103.923i	− 0.241121i	0.992706	−	0.120560i	$-0.0384691\pi$
$432$	0	0
$433$	50.0000	0.115473	0.0577367	−	0.998332i	$-0.481612\pi$
$433$	50.0000	0.115473	0.0577367	+	0.998332i	$0.481612\pi$
$434$	−146.969	−0.338639
$435$	0	0
$436$	− 43.8406i	− 0.100552i
$437$	− 381.051i	− 0.871971i
$438$	0	0
$439$	− 651.952i	− 1.48509i	−0.669799	−	0.742543i	$-0.733620\pi$
$439$	− 651.952i	− 1.48509i	0.669799	−	0.742543i	$-0.266380\pi$
$440$	0	0
$441$	0	0
$442$	339.411i	0.767899i
$443$	−235.151	−0.530815	−0.265407	−	0.964136i	$-0.585506\pi$
$443$	−235.151	−0.530815	−0.265407	+	0.964136i	$0.585506\pi$
$444$	0	0
$445$	480.000	1.07865
$446$	329.090i	0.737869i
$447$	0	0
$448$	− 502.046i	− 1.12064i
$449$	489.898	1.09109	0.545543	−	0.838083i	$-0.316323\pi$
$449$	489.898	1.09109	0.545543	+	0.838083i	$0.316323\pi$
$450$	0	0
$451$	0	0
$452$	48.9898	0.108385
$453$	0	0
$454$	−648.000	−1.42731
$455$	244.949	0.538349
$456$	0	0
$457$	− 799.031i	− 1.74843i	−0.485543	−	0.874213i	$-0.661378\pi$
$457$	− 799.031i	− 1.74843i	0.485543	−	0.874213i	$-0.338622\pi$
$458$	311.769i	0.680719i
$459$	0	0
$460$	−120.000	−0.260870
$461$	− 277.128i	− 0.601146i	−0.953759	−	0.300573i	$-0.902822\pi$
$461$	− 277.128i	− 0.601146i	0.953759	−	0.300573i	$-0.0971778\pi$
$462$	0	0
$463$	430.000	0.928726	0.464363	−	0.885645i	$-0.346283\pi$
$463$	430.000	0.928726	0.464363	+	0.885645i	$0.346283\pi$
$464$	381.051i	0.821231i
$465$	0	0
$466$	−372.000	−0.798283
$467$	293.939	0.629419	0.314710	−	0.949188i	$-0.398093\pi$
$467$	293.939	0.629419	0.314710	+	0.949188i	$0.398093\pi$
$468$	0	0
$469$	− 848.528i	− 1.80923i
$470$	706.677i	1.50357i
$471$	0	0
$472$	424.264i	0.898865i
$473$	0	0
$474$	0	0
$475$	− 15.5563i	− 0.0327502i
$476$	−195.959	−0.411679
$477$	0	0
$478$	0	0
$479$	103.923i	0.216958i	0.994099	+	0.108479i	$0.0345981\pi$
$479$	103.923i	0.216958i	−0.994099	+	0.108479i	$0.965402\pi$
$480$	0	0
$481$	424.264i	0.882046i
$482$	2.44949	0.00508193
$483$	0	0
$484$	0	0
$485$	342.929	0.707069
$486$	0	0
$487$	−540.000	−1.10883	−0.554415	−	0.832240i	$-0.687058\pi$
$487$	−540.000	−1.10883	−0.554415	+	0.832240i	$0.687058\pi$
$488$	−232.702	−0.476847
$489$	0	0
$490$	− 8.48528i	− 0.0173169i
$491$	554.256i	1.12883i	0.825491	+	0.564416i	$0.190898\pi$
$491$	554.256i	1.12883i	−0.825491	+	0.564416i	$0.809102\pi$
$492$	0	0
$493$	960.000	1.94726
$494$	190.526i	0.385679i
$495$	0	0
$496$	−132.000	−0.266129
$497$	− 173.205i	− 0.348501i
$498$	0	0
$499$	240.000	0.480962	0.240481	−	0.970654i	$-0.422695\pi$
$499$	240.000	0.480962	0.240481	+	0.970654i	$0.422695\pi$
$500$	−127.373	−0.254747
$501$	0	0
$502$	763.675i	1.52127i
$503$	− 374.123i	− 0.743783i	−0.928276	−	0.371892i	$-0.878709\pi$
$503$	− 374.123i	− 0.743783i	0.928276	−	0.371892i	$-0.121291\pi$
$504$	0	0
$505$	339.411i	0.672101i
$506$	0	0
$507$	0	0
$508$	− 91.9239i	− 0.180953i
$509$	612.372	1.20309	0.601545	−	0.798839i	$-0.294552\pi$
$509$	612.372	1.20309	0.601545	+	0.798839i	$0.294552\pi$
$510$	0	0
$511$	−850.000	−1.66341
$512$	− 552.524i	− 1.07915i
$513$	0	0
$514$	509.117i	0.990500i
$515$	−244.949	−0.475629
$516$	0	0
$517$	0	0
$518$	734.847	1.41862
$519$	0	0
$520$	300.000	0.576923
$521$	−538.888	−1.03433	−0.517167	−	0.855885i	$-0.673013\pi$
$521$	−538.888	−1.03433	−0.517167	+	0.855885i	$0.673013\pi$
$522$	0	0
$523$	219.203i	0.419126i	0.977795	+	0.209563i	$0.0672042\pi$
$523$	219.203i	0.419126i	−0.977795	+	0.209563i	$0.932796\pi$
$524$	34.6410i	0.0661088i
$525$	0	0
$526$	−228.000	−0.433460
$527$	332.554i	0.631032i
$528$	0	0
$529$	71.0000	0.134216
$530$	706.677i	1.33335i
$531$	0	0
$532$	−110.000	−0.206767
$533$	244.949	0.459567
$534$	0	0
$535$	475.176i	0.888179i
$536$	− 1039.23i	− 1.93886i
$537$	0	0
$538$	− 42.4264i	− 0.0788595i
$539$	0	0
$540$	0	0
$541$	− 281.428i	− 0.520201i	−0.965582	−	0.260100i	$-0.916244\pi$
$541$	− 281.428i	− 0.520201i	0.965582	−	0.260100i	$-0.0837556\pi$
$542$	516.842	0.953584
$543$	0	0
$544$	−432.000	−0.794118
$545$	− 214.774i	− 0.394081i
$546$	0	0
$547$	− 120.208i	− 0.219759i	−0.993945	−	0.109879i	$-0.964954\pi$
$547$	− 120.208i	− 0.219759i	0.993945	−	0.109879i	$-0.0350465\pi$
$548$	88.1816	0.160915
$549$	0	0
$550$	0	0
$551$	538.888	0.978018
$552$	0	0
$553$	910.000	1.64557
$554$	−281.691	−0.508468
$555$	0	0
$556$	69.2965i	0.124634i
$557$	− 512.687i	− 0.920444i	−0.887804	−	0.460222i	$-0.847770\pi$
$557$	− 512.687i	− 0.920444i	0.887804	−	0.460222i	$-0.152230\pi$
$558$	0	0
$559$	−350.000	−0.626118
$560$	− 381.051i	− 0.680449i
$561$	0	0
$562$	0	0
$563$	− 859.097i	− 1.52593i	−0.646441	−	0.762964i	$-0.723743\pi$
$563$	− 859.097i	− 1.52593i	0.646441	−	0.762964i	$-0.276257\pi$
$564$	0	0
$565$	240.000	0.424779
$566$	−526.640	−0.930460
$567$	0	0
$568$	− 212.132i	− 0.373472i
$569$	− 415.692i	− 0.730566i	−0.930896	−	0.365283i	$-0.880972\pi$
$569$	− 415.692i	− 0.730566i	0.930896	−	0.365283i	$-0.119028\pi$
$570$	0	0
$571$	100.409i	0.175848i	0.996127	+	0.0879240i	$0.0280233\pi$
$571$	100.409i	0.175848i	−0.996127	+	0.0879240i	$0.971977\pi$
$572$	0	0
$573$	0	0
$574$	− 424.264i	− 0.739136i
$575$	24.4949	0.0425998
$576$	0	0
$577$	480.000	0.831889	0.415945	−	0.909390i	$-0.363451\pi$
$577$	480.000	0.831889	0.415945	+	0.909390i	$0.363451\pi$
$578$	− 829.652i	− 1.43538i
$579$	0	0
$580$	− 169.706i	− 0.292596i
$581$	−538.888	−0.927518
$582$	0	0
$583$	0	0
$584$	−1041.03	−1.78259
$585$	0	0
$586$	−192.000	−0.327645
$587$	636.867	1.08495	0.542476	−	0.840071i	$-0.317487\pi$
$587$	636.867	1.08495	0.542476	+	0.840071i	$0.317487\pi$
$588$	0	0
$589$	186.676i	0.316938i
$590$	415.692i	0.704563i
$591$	0	0
$592$	660.000	1.11486
$593$	− 651.251i	− 1.09823i	−0.835746	−	0.549116i	$-0.814965\pi$
$593$	− 651.251i	− 1.09823i	0.835746	−	0.549116i	$-0.185035\pi$
$594$	0	0
$595$	−960.000	−1.61345
$596$	− 242.487i	− 0.406858i
$597$	0	0
$598$	−300.000	−0.501672
$599$	−955.301	−1.59483	−0.797413	−	0.603434i	$-0.793799\pi$
$599$	−955.301	−1.59483	−0.797413	+	0.603434i	$0.793799\pi$
$600$	0	0
$601$	832.972i	1.38598i	0.720949	+	0.692988i	$0.243706\pi$
$601$	832.972i	1.38598i	−0.720949	+	0.692988i	$0.756294\pi$
$602$	606.218i	1.00701i
$603$	0	0
$604$	26.8701i	0.0444869i
$605$	0	0
$606$	0	0
$607$	261.630i	0.431021i	0.976502	+	0.215510i	$0.0691415\pi$
$607$	261.630i	0.431021i	−0.976502	+	0.215510i	$0.930859\pi$
$608$	−242.499	−0.398848
$609$	0	0
$610$	−228.000	−0.373770
$611$	− 588.897i	− 0.963825i
$612$	0	0
$613$	841.457i	1.37269i	0.727278	+	0.686343i	$0.240785\pi$
$613$	841.457i	1.37269i	−0.727278	+	0.686343i	$0.759215\pi$
$614$	61.2372	0.0997349
$615$	0	0
$616$	0	0
$617$	587.878	0.952800	0.476400	−	0.879229i	$-0.341941\pi$
$617$	587.878	0.952800	0.476400	+	0.879229i	$0.341941\pi$
$618$	0	0
$619$	170.000	0.274637	0.137318	−	0.990527i	$-0.456152\pi$
$619$	170.000	0.274637	0.137318	+	0.990527i	$0.456152\pi$
$620$	58.7878	0.0948190
$621$	0	0
$622$	466.690i	0.750306i
$623$	692.820i	1.11207i
$624$	0	0
$625$	−599.000	−0.958400
$626$	831.384i	1.32809i
$627$	0	0
$628$	60.0000	0.0955414
$629$	− 1662.77i	− 2.64351i
$630$	0	0
$631$	−300.000	−0.475436	−0.237718	−	0.971334i	$-0.576399\pi$
$631$	−300.000	−0.475436	−0.237718	+	0.971334i	$0.576399\pi$
$632$	1114.52	1.76348
$633$	0	0
$634$	212.132i	0.334593i
$635$	− 450.333i	− 0.709186i
$636$	0	0
$637$	7.07107i	0.0111006i
$638$	0	0
$639$	0	0
$640$	− 296.985i	− 0.464039i
$641$	−832.827	−1.29926	−0.649631	−	0.760250i	$-0.725076\pi$
$641$	−832.827	−1.29926	−0.649631	+	0.760250i	$0.725076\pi$
$642$	0	0
$643$	310.000	0.482115	0.241058	−	0.970511i	$-0.422506\pi$
$643$	310.000	0.482115	0.241058	+	0.970511i	$0.422506\pi$
$644$	− 173.205i	− 0.268952i
$645$	0	0
$646$	− 746.705i	− 1.15589i
$647$	−328.232	−0.507313	−0.253657	−	0.967294i	$-0.581633\pi$
$647$	−328.232	−0.507313	−0.253657	+	0.967294i	$0.581633\pi$
$648$	0	0
$649$	0	0
$650$	−12.2474	−0.0188422
$651$	0	0
$652$	−190.000	−0.291411
$653$	−171.464	−0.262579	−0.131290	−	0.991344i	$-0.541912\pi$
$653$	−171.464	−0.262579	−0.131290	+	0.991344i	$0.541912\pi$
$654$	0	0
$655$	169.706i	0.259093i
$656$	− 381.051i	− 0.580871i
$657$	0	0
$658$	−1020.00	−1.55015
$659$	− 207.846i	− 0.315396i	−0.987487	−	0.157698i	$-0.949593\pi$
$659$	− 207.846i	− 0.315396i	0.987487	−	0.157698i	$-0.0504073\pi$
$660$	0	0
$661$	−1020.00	−1.54312	−0.771558	−	0.636159i	$-0.780522\pi$
$661$	−1020.00	−1.54312	−0.771558	+	0.636159i	$0.780522\pi$
$662$	502.295i	0.758753i
$663$	0	0
$664$	−660.000	−0.993976
$665$	−538.888	−0.810358
$666$	0	0
$667$	848.528i	1.27216i
$668$	− 180.133i	− 0.269661i
$669$	0	0
$670$	− 1018.23i	− 1.51975i
$671$	0	0
$672$	0	0
$673$	− 1308.15i	− 1.94376i	−0.235486	−	0.971878i	$-0.575668\pi$
$673$	− 1308.15i	− 1.94376i	0.235486	−	0.971878i	$-0.424332\pi$
$674$	502.145	0.745023
$675$	0	0
$676$	119.000	0.176036
$677$	− 110.851i	− 0.163739i	−0.996643	−	0.0818695i	$-0.973911\pi$
$677$	− 110.851i	− 0.163739i	0.996643	−	0.0818695i	$-0.0260891\pi$
$678$	0	0
$679$	494.975i	0.728976i
$680$	−1175.76	−1.72905
$681$	0	0
$682$	0	0
$683$	1038.58	1.52062	0.760310	−	0.649560i	$-0.225047\pi$
$683$	1038.58	1.52062	0.760310	+	0.649560i	$0.225047\pi$
$684$	0	0
$685$	432.000	0.630657
$686$	−587.878	−0.856964
$687$	0	0
$688$	544.472i	0.791384i
$689$	− 588.897i	− 0.854713i
$690$	0	0
$691$	218.000	0.315485	0.157742	−	0.987480i	$-0.449578\pi$
$691$	218.000	0.315485	0.157742	+	0.987480i	$0.449578\pi$
$692$	− 76.2102i	− 0.110130i
$693$	0	0
$694$	−588.000	−0.847262
$695$	339.482i	0.488463i
$696$	0	0
$697$	−960.000	−1.37733
$698$	835.276	1.19667
$699$	0	0
$700$	− 7.07107i	− 0.0101015i
$701$	554.256i	0.790665i	0.918538	+	0.395333i	$0.129371\pi$
$701$	554.256i	0.790665i	−0.918538	+	0.395333i	$0.870629\pi$
$702$	0	0
$703$	− 933.381i	− 1.32771i
$704$	0	0
$705$	0	0
$706$	− 356.382i	− 0.504790i
$707$	−489.898	−0.692925
$708$	0	0
$709$	−132.000	−0.186178	−0.0930889	−	0.995658i	$-0.529674\pi$
$709$	−132.000	−0.186178	−0.0930889	+	0.995658i	$0.529674\pi$
$710$	− 207.846i	− 0.292741i
$711$	0	0
$712$	848.528i	1.19175i
$713$	−293.939	−0.412256
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	1140.00	1.58774
$719$	−171.464	−0.238476	−0.119238	−	0.992866i	$-0.538045\pi$
$719$	−171.464	−0.238476	−0.119238	+	0.992866i	$0.538045\pi$
$720$	0	0
$721$	− 353.553i	− 0.490365i
$722$	206.114i	0.285477i
$723$	0	0
$724$	310.000	0.428177
$725$	34.6410i	0.0477807i
$726$	0	0
$727$	−300.000	−0.412655	−0.206327	−	0.978483i	$-0.566151\pi$
$727$	−300.000	−0.412655	−0.206327	+	0.978483i	$0.566151\pi$
$728$	433.013i	0.594798i
$729$	0	0
$730$	−1020.00	−1.39726
$731$	1371.71	1.87649
$732$	0	0
$733$	− 91.9239i	− 0.125408i	−0.998032	−	0.0627039i	$-0.980028\pi$
$733$	− 91.9239i	− 0.125408i	0.998032	−	0.0627039i	$-0.0199724\pi$
$734$	917.987i	1.25066i
$735$	0	0
$736$	− 381.838i	− 0.518801i
$737$	0	0
$738$	0	0
$739$	− 1.41421i	− 0.00191369i	−1.00000		0.000956843i	$-0.999695\pi$
$739$	− 1.41421i	− 0.00191369i	1.00000		0.000956843i	$-0.000304573\pi$
$740$	−293.939	−0.397215
$741$	0	0
$742$	−1020.00	−1.37466
$743$	1392.57i	1.87425i	0.348992	+	0.937126i	$0.386524\pi$
$743$	1392.57i	1.87425i	−0.348992	+	0.937126i	$0.613476\pi$
$744$	0	0
$745$	− 1187.94i	− 1.59455i
$746$	306.186	0.410437
$747$	0	0
$748$	0	0
$749$	−685.857	−0.915697
$750$	0	0
$751$	−540.000	−0.719041	−0.359521	−	0.933137i	$-0.617060\pi$
$751$	−540.000	−0.719041	−0.359521	+	0.933137i	$0.617060\pi$
$752$	−916.109	−1.21823
$753$	0	0
$754$	− 424.264i	− 0.562684i
$755$	131.636i	0.174352i
$756$	0	0
$757$	60.0000	0.0792602	0.0396301	−	0.999214i	$-0.487382\pi$
$757$	60.0000	0.0792602	0.0396301	+	0.999214i	$0.487382\pi$
$758$	419.156i	0.552977i
$759$	0	0
$760$	−660.000	−0.868421
$761$	− 69.2820i	− 0.0910408i	−0.998963	−	0.0455204i	$-0.985505\pi$
$761$	− 69.2820i	− 0.0910408i	0.998963	−	0.0455204i	$-0.0144946\pi$
$762$	0	0
$763$	310.000	0.406291
$764$	−367.423	−0.480921
$765$	0	0
$766$	− 110.309i	− 0.144006i
$767$	− 346.410i	− 0.451643i
$768$	0	0
$769$	779.232i	1.01331i	0.862150	+	0.506653i	$0.169117\pi$
$769$	779.232i	1.01331i	−0.862150	+	0.506653i	$0.830883\pi$
$770$	0	0
$771$	0	0
$772$	− 35.3553i	− 0.0457971i
$773$	−1151.26	−1.48934	−0.744670	−	0.667433i	$-0.767393\pi$
$773$	−1151.26	−1.48934	−0.744670	+	0.667433i	$0.767393\pi$
$774$	0	0
$775$	−12.0000	−0.0154839
$776$	606.218i	0.781208i
$777$	0	0
$778$	1060.66i	1.36332i
$779$	−538.888	−0.691769
$780$	0	0
$781$	0	0
$782$	1175.76	1.50352
$783$	0	0
$784$	11.0000	0.0140306
$785$	293.939	0.374444
$786$	0	0
$787$	898.026i	1.14107i	0.821272	+	0.570537i	$0.193265\pi$
$787$	898.026i	1.14107i	−0.821272	+	0.570537i	$0.806735\pi$
$788$	41.5692i	0.0527528i
$789$	0	0
$790$	1092.00	1.38228
$791$	346.410i	0.437940i
$792$	0	0
$793$	190.000	0.239596
$794$	− 536.936i	− 0.676241i
$795$	0	0
$796$	−180.000	−0.226131
$797$	−710.352	−0.891282	−0.445641	−	0.895212i	$-0.647024\pi$
$797$	−710.352	−0.891282	−0.445641	+	0.895212i	$0.647024\pi$
$798$	0	0
$799$	2308.00i	2.88861i
$800$	− 15.5885i	− 0.0194856i
$801$	0	0
$802$	509.117i	0.634809i
$803$	0	0
$804$	0	0
$805$	− 848.528i	− 1.05407i
$806$	146.969	0.182344
$807$	0	0
$808$	−600.000	−0.742574
$809$	− 381.051i	− 0.471015i	−0.971873	−	0.235508i	$-0.924325\pi$
$809$	− 381.051i	− 0.471015i	0.971873	−	0.235508i	$-0.0756752\pi$
$810$	0	0
$811$	1104.50i	1.36190i	0.732330	+	0.680950i	$0.238433\pi$
$811$	1104.50i	1.36190i	−0.732330	+	0.680950i	$0.761567\pi$
$812$	244.949	0.301661
$813$	0	0
$814$	0	0
$815$	−930.806	−1.14209
$816$	0	0
$817$	770.000	0.942472
$818$	438.459	0.536013
$819$	0	0
$820$	169.706i	0.206958i
$821$	554.256i	0.675099i	0.941308	+	0.337549i	$0.109598\pi$
$821$	554.256i	0.675099i	−0.941308	+	0.337549i	$0.890402\pi$
$822$	0	0
$823$	−530.000	−0.643985	−0.321993	−	0.946742i	$-0.604353\pi$
$823$	−530.000	−0.643985	−0.321993	+	0.946742i	$0.604353\pi$
$824$	− 433.013i	− 0.525501i
$825$	0	0
$826$	−600.000	−0.726392
$827$	− 131.636i	− 0.159173i	−0.996828	−	0.0795864i	$-0.974640\pi$
$827$	− 131.636i	− 0.159173i	0.996828	−	0.0795864i	$-0.0253599\pi$
$828$	0	0
$829$	−410.000	−0.494572	−0.247286	−	0.968943i	$-0.579539\pi$
$829$	−410.000	−0.494572	−0.247286	+	0.968943i	$0.579539\pi$
$830$	−646.665	−0.779115
$831$	0	0
$832$	502.046i	0.603420i
$833$	− 27.7128i	− 0.0332687i
$834$	0	0
$835$	− 882.469i	− 1.05685i
$836$	0	0
$837$	0	0
$838$	− 1187.94i	− 1.41759i
$839$	220.454	0.262758	0.131379	−	0.991332i	$-0.458059\pi$
$839$	220.454	0.262758	0.131379	+	0.991332i	$0.458059\pi$
$840$	0	0
$841$	−359.000	−0.426873
$842$	187.061i	0.222163i
$843$	0	0
$844$	100.409i	0.118968i
$845$	582.979	0.689915
$846$	0	0
$847$	0	0
$848$	−916.109	−1.08032
$849$	0	0
$850$	48.0000	0.0564706
$851$	1469.69	1.72702
$852$	0	0
$853$	1110.16i	1.30147i	0.759303	+	0.650737i	$0.225540\pi$
$853$	1110.16i	1.30147i	−0.759303	+	0.650737i	$0.774460\pi$
$854$	− 329.090i	− 0.385351i
$855$	0	0
$856$	−840.000	−0.981308
$857$	− 1170.87i	− 1.36624i	−0.730307	−	0.683119i	$-0.760623\pi$
$857$	− 1170.87i	− 1.36624i	0.730307	−	0.683119i	$-0.239377\pi$
$858$	0	0
$859$	−50.0000	−0.0582072	−0.0291036	−	0.999576i	$-0.509265\pi$
$859$	−50.0000	−0.0582072	−0.0291036	+	0.999576i	$0.509265\pi$
$860$	− 242.487i	− 0.281962i
$861$	0	0
$862$	180.000	0.208817
$863$	455.605	0.527932	0.263966	−	0.964532i	$-0.414969\pi$
$863$	455.605	0.527932	0.263966	+	0.964532i	$0.414969\pi$
$864$	0	0
$865$	− 373.352i	− 0.431621i
$866$	86.6025i	0.100003i
$867$	0	0
$868$	84.8528i	0.0977567i
$869$	0	0
$870$	0	0
$871$	848.528i	0.974200i
$872$	379.671	0.435402
$873$	0	0
$874$	660.000	0.755149
$875$	− 900.666i	− 1.02933i
$876$	0	0
$877$	261.630i	0.298323i	0.988813	+	0.149162i	$0.0476575\pi$
$877$	261.630i	0.298323i	−0.988813	+	0.149162i	$0.952343\pi$
$878$	1129.21	1.28612
$879$	0	0
$880$	0	0
$881$	−1077.78	−1.22335	−0.611677	−	0.791107i	$-0.709505\pi$
$881$	−1077.78	−1.22335	−0.611677	+	0.791107i	$0.709505\pi$
$882$	0	0
$883$	−1200.00	−1.35900	−0.679502	−	0.733674i	$-0.737804\pi$
$883$	−1200.00	−1.35900	−0.679502	+	0.733674i	$0.737804\pi$
$884$	195.959	0.221673
$885$	0	0
$886$	− 407.294i	− 0.459699i
$887$	− 1309.43i	− 1.47625i	−0.674666	−	0.738123i	$-0.735712\pi$
$887$	− 1309.43i	− 1.47625i	0.674666	−	0.738123i	$-0.264288\pi$
$888$	0	0
$889$	650.000	0.731159
$890$	831.384i	0.934140i
$891$	0	0
$892$	190.000	0.213004
$893$	1295.57i	1.45081i
$894$	0	0
$895$	0	0
$896$	428.661	0.478416
$897$	0	0
$898$	848.528i	0.944909i
$899$	− 415.692i	− 0.462394i
$900$	0	0
$901$	2308.00i	2.56159i
$902$	0	0
$903$	0	0
$904$	424.264i	0.469319i
$905$	1518.68	1.67810
$906$	0	0
$907$	−650.000	−0.716648	−0.358324	−	0.933597i	$-0.616652\pi$
$907$	−650.000	−0.716648	−0.358324	+	0.933597i	$0.616652\pi$
$908$	374.123i	0.412030i
$909$	0	0
$910$	424.264i	0.466224i
$911$	857.321	0.941077	0.470539	−	0.882379i	$-0.344060\pi$
$911$	857.321	0.941077	0.470539	+	0.882379i	$0.344060\pi$
$912$	0	0
$913$	0	0
$914$	1383.96	1.51418
$915$	0	0
$916$	180.000	0.196507
$917$	−244.949	−0.267120
$918$	0	0
$919$	− 57.9828i	− 0.0630933i	−0.999502	−	0.0315467i	$-0.989957\pi$
$919$	− 57.9828i	− 0.0630933i	0.999502	−	0.0315467i	$-0.0100433\pi$
$920$	− 1039.23i	− 1.12960i
$921$	0	0
$922$	480.000	0.520607
$923$	173.205i	0.187654i
$924$	0	0
$925$	60.0000	0.0648649
$926$	744.782i	0.804300i
$927$	0	0
$928$	540.000	0.581897
$929$	−1224.74	−1.31835	−0.659174	−	0.751991i	$-0.729094\pi$
$929$	−1224.74	−1.31835	−0.659174	+	0.751991i	$0.729094\pi$
$930$	0	0
$931$	− 15.5563i	− 0.0167093i
$932$	214.774i	0.230445i
$933$	0	0
$934$	509.117i	0.545093i
$935$	0	0
$936$	0	0
$937$	− 1661.70i	− 1.77343i	−0.462320	−	0.886713i	$-0.652983\pi$
$937$	− 1661.70i	− 1.77343i	0.462320	−	0.886713i	$-0.347017\pi$
$938$	1469.69	1.56684
$939$	0	0
$940$	408.000	0.434043
$941$	415.692i	0.441756i	0.975301	+	0.220878i	$0.0708922\pi$
$941$	415.692i	0.441756i	−0.975301	+	0.220878i	$0.929108\pi$
$942$	0	0
$943$	− 848.528i	− 0.899818i
$944$	−538.888	−0.570856
$945$	0	0
$946$	0	0
$947$	−774.039	−0.817359	−0.408679	−	0.912678i	$-0.634011\pi$
$947$	−774.039	−0.817359	−0.408679	+	0.912678i	$0.634011\pi$
$948$	0	0
$949$	850.000	0.895680
$950$	26.9444	0.0283625
$951$	0	0
$952$	− 1697.06i	− 1.78262i
$953$	1517.28i	1.59211i	0.605227	+	0.796053i	$0.293082\pi$
$953$	1517.28i	1.59211i	−0.605227	+	0.796053i	$0.706918\pi$
$954$	0	0
$955$	−1800.00	−1.88482
$956$	0	0
$957$	0	0
$958$	−180.000	−0.187891
$959$	623.538i	0.650196i
$960$	0	0
$961$	−817.000	−0.850156
$962$	−734.847	−0.763874
$963$	0	0
$964$	− 1.41421i	− 0.00146703i
$965$	− 173.205i	− 0.179487i
$966$	0	0
$967$	− 1449.57i	− 1.49904i	−0.661983	−	0.749519i	$-0.730285\pi$
$967$	− 1449.57i	− 1.49904i	0.661983	−	0.749519i	$-0.269715\pi$
$968$	0	0
$969$	0	0
$970$	593.970i	0.612340i
$971$	342.929	0.353171	0.176585	−	0.984285i	$-0.443495\pi$
$971$	342.929	0.353171	0.176585	+	0.984285i	$0.443495\pi$
$972$	0	0
$973$	−490.000	−0.503597
$974$	− 935.307i	− 0.960275i
$975$	0	0
$976$	− 295.571i	− 0.302839i
$977$	1822.42	1.86532	0.932661	−	0.360753i	$-0.117480\pi$
$977$	1822.42	1.86532	0.932661	+	0.360753i	$0.117480\pi$
$978$	0	0
$979$	0	0
$980$	−4.89898	−0.00499896
$981$	0	0
$982$	−960.000	−0.977597
$983$	−671.160	−0.682767	−0.341384	−	0.939924i	$-0.610896\pi$
$983$	−671.160	−0.682767	−0.341384	+	0.939924i	$0.610896\pi$
$984$	0	0
$985$	203.647i	0.206748i
$986$	1662.77i	1.68638i
$987$	0	0
$988$	110.000	0.111336
$989$	1212.44i	1.22592i
$990$	0	0
$991$	−468.000	−0.472250	−0.236125	−	0.971723i	$-0.575878\pi$
$991$	−468.000	−0.472250	−0.236125	+	0.971723i	$0.575878\pi$
$992$	187.061i	0.188570i
$993$	0	0
$994$	300.000	0.301811
$995$	−881.816	−0.886248
$996$	0	0
$997$	− 1025.30i	− 1.02839i	−0.857673	−	0.514195i	$-0.828091\pi$
$997$	− 1025.30i	− 1.02839i	0.857673	−	0.514195i	$-0.171909\pi$
$998$	415.692i	0.416525i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.f.604.4	yes	4
3.2	odd	2	inner	1089.3.c.f.604.1	✓	4
11.10	odd	2	inner	1089.3.c.f.604.2	yes	4
33.32	even	2	inner	1089.3.c.f.604.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.c.f.604.1	✓	4	3.2	odd	2	inner
1089.3.c.f.604.2	yes	4	11.10	odd	2	inner
1089.3.c.f.604.3	yes	4	33.32	even	2	inner
1089.3.c.f.604.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000 q^{4} +4.89898 q^{5} +7.07107i q^{7} +8.66025i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} +1.00000 q^{4} +4.89898 q^{5} +7.07107i q^{7} +8.66025i q^{8} +8.48528i q^{10} -7.07107i q^{13} -12.2474 q^{14} -11.0000 q^{16} +27.7128i q^{17} +15.5563i q^{19} +4.89898 q^{20} -24.4949 q^{23} -1.00000 q^{25} +12.2474 q^{26} +7.07107i q^{28} -34.6410i q^{29} +12.0000 q^{31} +15.5885i q^{32} -48.0000 q^{34} +34.6410i q^{35} -60.0000 q^{37} -26.9444 q^{38} +42.4264i q^{40} +34.6410i q^{41} -49.4975i q^{43} -42.4264i q^{46} +83.2827 q^{47} -1.00000 q^{49} -1.73205i q^{50} -7.07107i q^{52} +83.2827 q^{53} -61.2372 q^{56} +60.0000 q^{58} +48.9898 q^{59} +26.8701i q^{61} +20.7846i q^{62} -71.0000 q^{64} -34.6410i q^{65} -120.000 q^{67} +27.7128i q^{68} -60.0000 q^{70} -24.4949 q^{71} +120.208i q^{73} -103.923i q^{74} +15.5563i q^{76} -128.693i q^{79} -53.8888 q^{80} -60.0000 q^{82} +76.2102i q^{83} +135.765i q^{85} +85.7321 q^{86} +97.9796 q^{89} +50.0000 q^{91} -24.4949 q^{92} +144.250i q^{94} +76.2102i q^{95} +70.0000 q^{97} -1.73205i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} - 44 q^{16} - 4 q^{25} + 48 q^{31} - 192 q^{34} - 240 q^{37} - 4 q^{49} + 240 q^{58} - 284 q^{64} - 480 q^{67} - 240 q^{70} - 240 q^{82} + 200 q^{91} + 280 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 44 * q^16 - 4 * q^25 + 48 * q^31 - 192 * q^34 - 240 * q^37 - 4 * q^49 + 240 * q^58 - 284 * q^64 - 480 * q^67 - 240 * q^70 - 240 * q^82 + 200 * q^91 + 280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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