LMFDB - Embedded newform 6664.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.30091192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 10x^{3} + 18x^{2} - 23x + 2 $ x^6 - x^5 - 9x^4 + 10x^3 + 18x^2 - 23x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36506$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36506 q^{3} +1.89992 q^{5} +2.59349 q^{9} +3.76870 q^{11} +0.189847 q^{13} -4.49341 q^{15} +1.00000 q^{17} +0.880612 q^{19} -3.97038 q^{23} -1.39032 q^{25} +0.961410 q^{27} -6.30040 q^{29} +5.88017 q^{31} -8.91320 q^{33} -10.8935 q^{37} -0.448999 q^{39} -3.28277 q^{41} -0.477269 q^{43} +4.92742 q^{45} -2.44464 q^{47} -2.36506 q^{51} -7.16378 q^{53} +7.16022 q^{55} -2.08270 q^{57} +4.53281 q^{59} -11.3227 q^{61} +0.360693 q^{65} +8.76314 q^{67} +9.39016 q^{69} -6.72185 q^{71} -0.695543 q^{73} +3.28818 q^{75} -3.98071 q^{79} -10.0543 q^{81} +7.77110 q^{83} +1.89992 q^{85} +14.9008 q^{87} -0.237007 q^{89} -13.9069 q^{93} +1.67309 q^{95} +6.81746 q^{97} +9.77411 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} + q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} - 5 q^{23} + q^{25} + 8 q^{27} - 15 q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{37} - 2 q^{39} - 4 q^{41} + 24 q^{43} - 2 q^{45} + q^{47}+ \cdots + 2 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 + q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 - 5 * q^23 + q^25 + 8 * q^27 - 15 * q^29 - 8 * q^31 - 11 * q^33 - 16 * q^37 - 2 * q^39 - 4 * q^41 + 24 * q^43 - 2 * q^45 + q^47 - q^51 - 32 * q^53 + 25 * q^55 - 19 * q^57 + 6 * q^59 - 10 * q^61 - 26 * q^65 + 27 * q^67 + 22 * q^69 - 12 * q^71 - 15 * q^73 - 24 * q^75 - 21 * q^79 - 30 * q^81 + 20 * q^83 - q^85 + 9 * q^87 + 33 * q^89 - 29 * q^93 - 15 * q^95 + 9 * q^97 + 2 * q^99

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
$3$	−2.36506	−1.36547	−0.682733	−	0.730668i	$-0.739209\pi$
$3$	−2.36506	−1.36547	−0.682733	+	0.730668i	$0.739209\pi$
$4$	0	0
$5$	1.89992	0.849668	0.424834	−	0.905271i	$-0.360332\pi$
$5$	1.89992	0.849668	0.424834	+	0.905271i	$0.360332\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.59349	0.864498
$10$	0	0
$11$	3.76870	1.13631	0.568154	−	0.822923i	$-0.307658\pi$
$11$	3.76870	1.13631	0.568154	+	0.822923i	$0.307658\pi$
$12$	0	0
$13$	0.189847	0.0526541	0.0263270	−	0.999653i	$-0.491619\pi$
$13$	0.189847	0.0526541	0.0263270	+	0.999653i	$0.491619\pi$
$14$	0	0
$15$	−4.49341	−1.16019
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.880612	0.202026	0.101013	−	0.994885i	$-0.467792\pi$
$19$	0.880612	0.202026	0.101013	+	0.994885i	$0.467792\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.97038	−0.827881	−0.413940	−	0.910304i	$-0.635848\pi$
$23$	−3.97038	−0.827881	−0.413940	+	0.910304i	$0.635848\pi$
$24$	0	0
$25$	−1.39032	−0.278063
$26$	0	0
$27$	0.961410	0.185023
$28$	0	0
$29$	−6.30040	−1.16996	−0.584978	−	0.811049i	$-0.698897\pi$
$29$	−6.30040	−1.16996	−0.584978	+	0.811049i	$0.698897\pi$
$30$	0	0
$31$	5.88017	1.05611	0.528055	−	0.849210i	$-0.322922\pi$
$31$	5.88017	1.05611	0.528055	+	0.849210i	$0.322922\pi$
$32$	0	0
$33$	−8.91320	−1.55159
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.8935	−1.79088	−0.895440	−	0.445182i	$-0.853139\pi$
$37$	−10.8935	−1.79088	−0.895440	+	0.445182i	$0.853139\pi$
$38$	0	0
$39$	−0.448999	−0.0718973
$40$	0	0
$41$	−3.28277	−0.512683	−0.256341	−	0.966586i	$-0.582517\pi$
$41$	−3.28277	−0.512683	−0.256341	+	0.966586i	$0.582517\pi$
$42$	0	0
$43$	−0.477269	−0.0727828	−0.0363914	−	0.999338i	$-0.511586\pi$
$43$	−0.477269	−0.0727828	−0.0363914	+	0.999338i	$0.511586\pi$
$44$	0	0
$45$	4.92742	0.734537
$46$	0	0
$47$	−2.44464	−0.356587	−0.178293	−	0.983977i	$-0.557058\pi$
$47$	−2.44464	−0.356587	−0.178293	+	0.983977i	$0.557058\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.36506	−0.331174
$52$	0	0
$53$	−7.16378	−0.984021	−0.492011	−	0.870589i	$-0.663738\pi$
$53$	−7.16378	−0.984021	−0.492011	+	0.870589i	$0.663738\pi$
$54$	0	0
$55$	7.16022	0.965484
$56$	0	0
$57$	−2.08270	−0.275860
$58$	0	0
$59$	4.53281	0.590121	0.295061	−	0.955479i	$-0.404660\pi$
$59$	4.53281	0.590121	0.295061	+	0.955479i	$0.404660\pi$
$60$	0	0
$61$	−11.3227	−1.44973	−0.724863	−	0.688894i	$-0.758097\pi$
$61$	−11.3227	−1.44973	−0.724863	+	0.688894i	$0.758097\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.360693	0.0447385
$66$	0	0
$67$	8.76314	1.07059	0.535294	−	0.844666i	$-0.320201\pi$
$67$	8.76314	1.07059	0.535294	+	0.844666i	$0.320201\pi$
$68$	0	0
$69$	9.39016	1.13044
$70$	0	0
$71$	−6.72185	−0.797737	−0.398868	−	0.917008i	$-0.630597\pi$
$71$	−6.72185	−0.797737	−0.398868	+	0.917008i	$0.630597\pi$
$72$	0	0
$73$	−0.695543	−0.0814072	−0.0407036	−	0.999171i	$-0.512960\pi$
$73$	−0.695543	−0.0814072	−0.0407036	+	0.999171i	$0.512960\pi$
$74$	0	0
$75$	3.28818	0.379686
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.98071	−0.447865	−0.223933	−	0.974605i	$-0.571890\pi$
$79$	−3.98071	−0.447865	−0.223933	+	0.974605i	$0.571890\pi$
$80$	0	0
$81$	−10.0543	−1.11714
$82$	0	0
$83$	7.77110	0.852989	0.426495	−	0.904490i	$-0.359748\pi$
$83$	7.77110	0.852989	0.426495	+	0.904490i	$0.359748\pi$
$84$	0	0
$85$	1.89992	0.206075
$86$	0	0
$87$	14.9008	1.59753
$88$	0	0
$89$	−0.237007	−0.0251227	−0.0125613	−	0.999921i	$-0.503999\pi$
$89$	−0.237007	−0.0251227	−0.0125613	+	0.999921i	$0.503999\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.9069	−1.44208
$94$	0	0
$95$	1.67309	0.171655
$96$	0	0
$97$	6.81746	0.692208	0.346104	−	0.938196i	$-0.387504\pi$
$97$	6.81746	0.692208	0.346104	+	0.938196i	$0.387504\pi$
$98$	0	0
$99$	9.77411	0.982335
$100$	0	0
$101$	−16.9513	−1.68672	−0.843360	−	0.537349i	$-0.819426\pi$
$101$	−16.9513	−1.68672	−0.843360	+	0.537349i	$0.819426\pi$
$102$	0	0
$103$	−2.47025	−0.243401	−0.121701	−	0.992567i	$-0.538835\pi$
$103$	−2.47025	−0.243401	−0.121701	+	0.992567i	$0.538835\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.48790	−0.240514	−0.120257	−	0.992743i	$-0.538372\pi$
$107$	−2.48790	−0.240514	−0.120257	+	0.992743i	$0.538372\pi$
$108$	0	0
$109$	7.81194	0.748248	0.374124	−	0.927379i	$-0.377943\pi$
$109$	7.81194	0.748248	0.374124	+	0.927379i	$0.377943\pi$
$110$	0	0
$111$	25.7637	2.44539
$112$	0	0
$113$	7.15811	0.673378	0.336689	−	0.941616i	$-0.390693\pi$
$113$	7.15811	0.673378	0.336689	+	0.941616i	$0.390693\pi$
$114$	0	0
$115$	−7.54338	−0.703424
$116$	0	0
$117$	0.492367	0.0455193
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.20313	0.291194
$122$	0	0
$123$	7.76394	0.700051
$124$	0	0
$125$	−12.1411	−1.08593
$126$	0	0
$127$	5.32958	0.472924	0.236462	−	0.971641i	$-0.424012\pi$
$127$	5.32958	0.472924	0.236462	+	0.971641i	$0.424012\pi$
$128$	0	0
$129$	1.12877	0.0993825
$130$	0	0
$131$	5.96791	0.521419	0.260710	−	0.965417i	$-0.416044\pi$
$131$	5.96791	0.521419	0.260710	+	0.965417i	$0.416044\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.82660	0.157209
$136$	0	0
$137$	17.0184	1.45398	0.726989	−	0.686649i	$-0.240919\pi$
$137$	17.0184	1.45398	0.726989	+	0.686649i	$0.240919\pi$
$138$	0	0
$139$	−8.57639	−0.727440	−0.363720	−	0.931508i	$-0.618494\pi$
$139$	−8.57639	−0.727440	−0.363720	+	0.931508i	$0.618494\pi$
$140$	0	0
$141$	5.78170	0.486907
$142$	0	0
$143$	0.715477	0.0598312
$144$	0	0
$145$	−11.9702	−0.994074
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.0212	−1.72213	−0.861063	−	0.508497i	$-0.830201\pi$
$149$	−21.0212	−1.72213	−0.861063	+	0.508497i	$0.830201\pi$
$150$	0	0
$151$	−23.5151	−1.91363	−0.956817	−	0.290691i	$-0.906115\pi$
$151$	−23.5151	−1.91363	−0.956817	+	0.290691i	$0.906115\pi$
$152$	0	0
$153$	2.59349	0.209672
$154$	0	0
$155$	11.1718	0.897343
$156$	0	0
$157$	18.8236	1.50229	0.751143	−	0.660140i	$-0.229503\pi$
$157$	18.8236	1.50229	0.751143	+	0.660140i	$0.229503\pi$
$158$	0	0
$159$	16.9428	1.34365
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.8182	−1.31730	−0.658652	−	0.752448i	$-0.728873\pi$
$163$	−16.8182	−1.31730	−0.658652	+	0.752448i	$0.728873\pi$
$164$	0	0
$165$	−16.9343	−1.31834
$166$	0	0
$167$	10.9092	0.844182	0.422091	−	0.906554i	$-0.361296\pi$
$167$	10.9092	0.844182	0.422091	+	0.906554i	$0.361296\pi$
$168$	0	0
$169$	−12.9640	−0.997228
$170$	0	0
$171$	2.28386	0.174651
$172$	0	0
$173$	11.3416	0.862289	0.431145	−	0.902283i	$-0.358110\pi$
$173$	11.3416	0.862289	0.431145	+	0.902283i	$0.358110\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.7203	−0.805790
$178$	0	0
$179$	9.01985	0.674175	0.337088	−	0.941473i	$-0.390558\pi$
$179$	9.01985	0.674175	0.337088	+	0.941473i	$0.390558\pi$
$180$	0	0
$181$	−3.13099	−0.232725	−0.116363	−	0.993207i	$-0.537123\pi$
$181$	−3.13099	−0.232725	−0.116363	+	0.993207i	$0.537123\pi$
$182$	0	0
$183$	26.7789	1.97955
$184$	0	0
$185$	−20.6967	−1.52165
$186$	0	0
$187$	3.76870	0.275595
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.1352	−1.45693	−0.728466	−	0.685082i	$-0.759766\pi$
$191$	−20.1352	−1.45693	−0.728466	+	0.685082i	$0.759766\pi$
$192$	0	0
$193$	21.1031	1.51903	0.759517	−	0.650487i	$-0.225435\pi$
$193$	21.1031	1.51903	0.759517	+	0.650487i	$0.225435\pi$
$194$	0	0
$195$	−0.853060	−0.0610889
$196$	0	0
$197$	16.8747	1.20227	0.601136	−	0.799147i	$-0.294715\pi$
$197$	16.8747	1.20227	0.601136	+	0.799147i	$0.294715\pi$
$198$	0	0
$199$	−9.42684	−0.668251	−0.334125	−	0.942529i	$-0.608441\pi$
$199$	−9.42684	−0.668251	−0.334125	+	0.942529i	$0.608441\pi$
$200$	0	0
$201$	−20.7253	−1.46185
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.23699	−0.435611
$206$	0	0
$207$	−10.2971	−0.715701
$208$	0	0
$209$	3.31877	0.229564
$210$	0	0
$211$	−0.0770744	−0.00530602	−0.00265301	−	0.999996i	$-0.500844\pi$
$211$	−0.0770744	−0.00530602	−0.00265301	+	0.999996i	$0.500844\pi$
$212$	0	0
$213$	15.8976	1.08928
$214$	0	0
$215$	−0.906771	−0.0618413
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.64500	0.111159
$220$	0	0
$221$	0.189847	0.0127705
$222$	0	0
$223$	−10.0086	−0.670226	−0.335113	−	0.942178i	$-0.608774\pi$
$223$	−10.0086	−0.670226	−0.335113	+	0.942178i	$0.608774\pi$
$224$	0	0
$225$	−3.60578	−0.240385
$226$	0	0
$227$	−10.6886	−0.709428	−0.354714	−	0.934975i	$-0.615422\pi$
$227$	−10.6886	−0.709428	−0.354714	+	0.934975i	$0.615422\pi$
$228$	0	0
$229$	15.7536	1.04103	0.520513	−	0.853854i	$-0.325741\pi$
$229$	15.7536	1.04103	0.520513	+	0.853854i	$0.325741\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.5422	1.21474	0.607369	−	0.794420i	$-0.292225\pi$
$233$	18.5422	1.21474	0.607369	+	0.794420i	$0.292225\pi$
$234$	0	0
$235$	−4.64460	−0.302980
$236$	0	0
$237$	9.41461	0.611545
$238$	0	0
$239$	−13.8718	−0.897295	−0.448647	−	0.893709i	$-0.648094\pi$
$239$	−13.8718	−0.897295	−0.448647	+	0.893709i	$0.648094\pi$
$240$	0	0
$241$	−7.38211	−0.475523	−0.237762	−	0.971324i	$-0.576414\pi$
$241$	−7.38211	−0.475523	−0.237762	+	0.971324i	$0.576414\pi$
$242$	0	0
$243$	20.8947	1.34040
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.167182	0.0106375
$248$	0	0
$249$	−18.3791	−1.16473
$250$	0	0
$251$	25.1206	1.58560	0.792799	−	0.609484i	$-0.208623\pi$
$251$	25.1206	1.58560	0.792799	+	0.609484i	$0.208623\pi$
$252$	0	0
$253$	−14.9632	−0.940727
$254$	0	0
$255$	−4.49341	−0.281388
$256$	0	0
$257$	25.1909	1.57137	0.785684	−	0.618628i	$-0.212311\pi$
$257$	25.1909	1.57137	0.785684	+	0.618628i	$0.212311\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−16.3401	−1.01142
$262$	0	0
$263$	−5.26914	−0.324909	−0.162455	−	0.986716i	$-0.551941\pi$
$263$	−5.26914	−0.324909	−0.162455	+	0.986716i	$0.551941\pi$
$264$	0	0
$265$	−13.6106	−0.836092
$266$	0	0
$267$	0.560534	0.0343041
$268$	0	0
$269$	−13.8188	−0.842545	−0.421272	−	0.906934i	$-0.638416\pi$
$269$	−13.8188	−0.842545	−0.421272	+	0.906934i	$0.638416\pi$
$270$	0	0
$271$	−29.7525	−1.80734	−0.903669	−	0.428231i	$-0.859137\pi$
$271$	−29.7525	−1.80734	−0.903669	+	0.428231i	$0.859137\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.23969	−0.315965
$276$	0	0
$277$	−14.5268	−0.872830	−0.436415	−	0.899745i	$-0.643752\pi$
$277$	−14.5268	−0.872830	−0.436415	+	0.899745i	$0.643752\pi$
$278$	0	0
$279$	15.2502	0.913005
$280$	0	0
$281$	−17.3065	−1.03242	−0.516209	−	0.856462i	$-0.672657\pi$
$281$	−17.3065	−1.03242	−0.516209	+	0.856462i	$0.672657\pi$
$282$	0	0
$283$	23.1015	1.37324	0.686621	−	0.727016i	$-0.259093\pi$
$283$	23.1015	1.37324	0.686621	+	0.727016i	$0.259093\pi$
$284$	0	0
$285$	−3.95695	−0.234390
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−16.1237	−0.945187
$292$	0	0
$293$	−31.4518	−1.83743	−0.918717	−	0.394917i	$-0.870773\pi$
$293$	−31.4518	−1.83743	−0.918717	+	0.394917i	$0.870773\pi$
$294$	0	0
$295$	8.61195	0.501407
$296$	0	0
$297$	3.62327	0.210243
$298$	0	0
$299$	−0.753764	−0.0435913
$300$	0	0
$301$	0	0
$302$	0	0
$303$	40.0909	2.30316
$304$	0	0
$305$	−21.5122	−1.23179
$306$	0	0
$307$	14.9725	0.854527	0.427263	−	0.904127i	$-0.359478\pi$
$307$	14.9725	0.854527	0.427263	+	0.904127i	$0.359478\pi$
$308$	0	0
$309$	5.84228	0.332356
$310$	0	0
$311$	−0.412653	−0.0233994	−0.0116997	−	0.999932i	$-0.503724\pi$
$311$	−0.412653	−0.0233994	−0.0116997	+	0.999932i	$0.503724\pi$
$312$	0	0
$313$	−4.32928	−0.244705	−0.122353	−	0.992487i	$-0.539044\pi$
$313$	−4.32928	−0.244705	−0.122353	+	0.992487i	$0.539044\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.9791	−1.68380	−0.841898	−	0.539637i	$-0.818562\pi$
$317$	−29.9791	−1.68380	−0.841898	+	0.539637i	$0.818562\pi$
$318$	0	0
$319$	−23.7443	−1.32943
$320$	0	0
$321$	5.88402	0.328414
$322$	0	0
$323$	0.880612	0.0489986
$324$	0	0
$325$	−0.263947	−0.0146412
$326$	0	0
$327$	−18.4757	−1.02171
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.7406	1.30490	0.652450	−	0.757832i	$-0.273741\pi$
$331$	23.7406	1.30490	0.652450	+	0.757832i	$0.273741\pi$
$332$	0	0
$333$	−28.2522	−1.54821
$334$	0	0
$335$	16.6492	0.909645
$336$	0	0
$337$	−11.1783	−0.608921	−0.304460	−	0.952525i	$-0.598476\pi$
$337$	−11.1783	−0.608921	−0.304460	+	0.952525i	$0.598476\pi$
$338$	0	0
$339$	−16.9293	−0.919475
$340$	0	0
$341$	22.1606	1.20006
$342$	0	0
$343$	0	0
$344$	0	0
$345$	17.8405	0.960502
$346$	0	0
$347$	−12.3139	−0.661047	−0.330523	−	0.943798i	$-0.607225\pi$
$347$	−12.3139	−0.661047	−0.330523	+	0.943798i	$0.607225\pi$
$348$	0	0
$349$	20.2052	1.08156	0.540780	−	0.841164i	$-0.318129\pi$
$349$	20.2052	1.08156	0.540780	+	0.841164i	$0.318129\pi$
$350$	0	0
$351$	0.182521	0.00974223
$352$	0	0
$353$	−0.256048	−0.0136280	−0.00681402	−	0.999977i	$-0.502169\pi$
$353$	−0.256048	−0.0136280	−0.00681402	+	0.999977i	$0.502169\pi$
$354$	0	0
$355$	−12.7709	−0.677812
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.3851	−0.811993	−0.405997	−	0.913875i	$-0.633076\pi$
$359$	−15.3851	−0.811993	−0.405997	+	0.913875i	$0.633076\pi$
$360$	0	0
$361$	−18.2245	−0.959185
$362$	0	0
$363$	−7.57558	−0.397615
$364$	0	0
$365$	−1.32147	−0.0691691
$366$	0	0
$367$	2.43345	0.127025	0.0635125	−	0.997981i	$-0.479770\pi$
$367$	2.43345	0.127025	0.0635125	+	0.997981i	$0.479770\pi$
$368$	0	0
$369$	−8.51385	−0.443213
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.3919	−1.05585	−0.527926	−	0.849290i	$-0.677030\pi$
$373$	−20.3919	−1.05585	−0.527926	+	0.849290i	$0.677030\pi$
$374$	0	0
$375$	28.7143	1.48280
$376$	0	0
$377$	−1.19611	−0.0616029
$378$	0	0
$379$	−27.1309	−1.39362	−0.696810	−	0.717256i	$-0.745398\pi$
$379$	−27.1309	−1.39362	−0.696810	+	0.717256i	$0.745398\pi$
$380$	0	0
$381$	−12.6048	−0.645762
$382$	0	0
$383$	18.6749	0.954244	0.477122	−	0.878837i	$-0.341680\pi$
$383$	18.6749	0.954244	0.477122	+	0.878837i	$0.341680\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.23779	−0.0629206
$388$	0	0
$389$	3.15646	0.160039	0.0800195	−	0.996793i	$-0.474502\pi$
$389$	3.15646	0.160039	0.0800195	+	0.996793i	$0.474502\pi$
$390$	0	0
$391$	−3.97038	−0.200791
$392$	0	0
$393$	−14.1145	−0.711980
$394$	0	0
$395$	−7.56302	−0.380537
$396$	0	0
$397$	−28.6610	−1.43845	−0.719226	−	0.694776i	$-0.755503\pi$
$397$	−28.6610	−1.43845	−0.719226	+	0.694776i	$0.755503\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.3670	−0.617579	−0.308790	−	0.951130i	$-0.599924\pi$
$401$	−12.3670	−0.617579	−0.308790	+	0.951130i	$0.599924\pi$
$402$	0	0
$403$	1.11633	0.0556085
$404$	0	0
$405$	−19.1023	−0.949200
$406$	0	0
$407$	−41.0544	−2.03499
$408$	0	0
$409$	−27.6207	−1.36576	−0.682878	−	0.730533i	$-0.739272\pi$
$409$	−27.6207	−1.36576	−0.682878	+	0.730533i	$0.739272\pi$
$410$	0	0
$411$	−40.2494	−1.98536
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14.7644	0.724758
$416$	0	0
$417$	20.2837	0.993295
$418$	0	0
$419$	21.1268	1.03211	0.516055	−	0.856555i	$-0.327400\pi$
$419$	21.1268	1.03211	0.516055	+	0.856555i	$0.327400\pi$
$420$	0	0
$421$	−4.03124	−0.196471	−0.0982353	−	0.995163i	$-0.531320\pi$
$421$	−4.03124	−0.196471	−0.0982353	+	0.995163i	$0.531320\pi$
$422$	0	0
$423$	−6.34015	−0.308268
$424$	0	0
$425$	−1.39032	−0.0674403
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.69214	−0.0816975
$430$	0	0
$431$	11.1630	0.537700	0.268850	−	0.963182i	$-0.413356\pi$
$431$	11.1630	0.537700	0.268850	+	0.963182i	$0.413356\pi$
$432$	0	0
$433$	−30.3717	−1.45957	−0.729785	−	0.683676i	$-0.760380\pi$
$433$	−30.3717	−1.45957	−0.729785	+	0.683676i	$0.760380\pi$
$434$	0	0
$435$	28.3103	1.35737
$436$	0	0
$437$	−3.49636	−0.167254
$438$	0	0
$439$	−17.4776	−0.834160	−0.417080	−	0.908870i	$-0.636946\pi$
$439$	−17.4776	−0.834160	−0.417080	+	0.908870i	$0.636946\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.04197	−0.429597	−0.214798	−	0.976658i	$-0.568909\pi$
$443$	−9.04197	−0.429597	−0.214798	+	0.976658i	$0.568909\pi$
$444$	0	0
$445$	−0.450293	−0.0213459
$446$	0	0
$447$	49.7164	2.35151
$448$	0	0
$449$	−20.6592	−0.974966	−0.487483	−	0.873132i	$-0.662085\pi$
$449$	−20.6592	−0.974966	−0.487483	+	0.873132i	$0.662085\pi$
$450$	0	0
$451$	−12.3718	−0.582565
$452$	0	0
$453$	55.6146	2.61300
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.5009	−1.00577	−0.502885	−	0.864353i	$-0.667728\pi$
$457$	−21.5009	−1.00577	−0.502885	+	0.864353i	$0.667728\pi$
$458$	0	0
$459$	0.961410	0.0448748
$460$	0	0
$461$	−15.7330	−0.732758	−0.366379	−	0.930466i	$-0.619403\pi$
$461$	−15.7330	−0.732758	−0.366379	+	0.930466i	$0.619403\pi$
$462$	0	0
$463$	41.7714	1.94128	0.970640	−	0.240537i	$-0.0773236\pi$
$463$	41.7714	1.94128	0.970640	+	0.240537i	$0.0773236\pi$
$464$	0	0
$465$	−26.4220	−1.22529
$466$	0	0
$467$	−35.8993	−1.66122	−0.830611	−	0.556854i	$-0.812008\pi$
$467$	−35.8993	−1.66122	−0.830611	+	0.556854i	$0.812008\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−44.5188	−2.05132
$472$	0	0
$473$	−1.79868	−0.0827036
$474$	0	0
$475$	−1.22433	−0.0561762
$476$	0	0
$477$	−18.5792	−0.850684
$478$	0	0
$479$	−7.27453	−0.332382	−0.166191	−	0.986094i	$-0.553147\pi$
$479$	−7.27453	−0.332382	−0.166191	+	0.986094i	$0.553147\pi$
$480$	0	0
$481$	−2.06810	−0.0942971
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.9526	0.588148
$486$	0	0
$487$	−11.7260	−0.531354	−0.265677	−	0.964062i	$-0.585595\pi$
$487$	−11.7260	−0.531354	−0.265677	+	0.964062i	$0.585595\pi$
$488$	0	0
$489$	39.7760	1.79873
$490$	0	0
$491$	0.539527	0.0243485	0.0121742	−	0.999926i	$-0.496125\pi$
$491$	0.539527	0.0243485	0.0121742	+	0.999926i	$0.496125\pi$
$492$	0	0
$493$	−6.30040	−0.283756
$494$	0	0
$495$	18.5700	0.834659
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.7194	0.882763	0.441382	−	0.897319i	$-0.354489\pi$
$499$	19.7194	0.882763	0.441382	+	0.897319i	$0.354489\pi$
$500$	0	0
$501$	−25.8010	−1.15270
$502$	0	0
$503$	−23.4213	−1.04430	−0.522152	−	0.852852i	$-0.674871\pi$
$503$	−23.4213	−1.04430	−0.522152	+	0.852852i	$0.674871\pi$
$504$	0	0
$505$	−32.2061	−1.43315
$506$	0	0
$507$	30.6605	1.36168
$508$	0	0
$509$	20.5876	0.912531	0.456266	−	0.889844i	$-0.349187\pi$
$509$	20.5876	0.912531	0.456266	+	0.889844i	$0.349187\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.846629	0.0373796
$514$	0	0
$515$	−4.69327	−0.206810
$516$	0	0
$517$	−9.21311	−0.405192
$518$	0	0
$519$	−26.8236	−1.17743
$520$	0	0
$521$	−1.40857	−0.0617107	−0.0308553	−	0.999524i	$-0.509823\pi$
$521$	−1.40857	−0.0617107	−0.0308553	+	0.999524i	$0.509823\pi$
$522$	0	0
$523$	−25.8772	−1.13153	−0.565766	−	0.824566i	$-0.691419\pi$
$523$	−25.8772	−1.13153	−0.565766	+	0.824566i	$0.691419\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.88017	0.256144
$528$	0	0
$529$	−7.23612	−0.314614
$530$	0	0
$531$	11.7558	0.510158
$532$	0	0
$533$	−0.623224	−0.0269948
$534$	0	0
$535$	−4.72680	−0.204357
$536$	0	0
$537$	−21.3325	−0.920564
$538$	0	0
$539$	0	0
$540$	0	0
$541$	27.0596	1.16338	0.581691	−	0.813410i	$-0.302391\pi$
$541$	27.0596	1.16338	0.581691	+	0.813410i	$0.302391\pi$
$542$	0	0
$543$	7.40498	0.317778
$544$	0	0
$545$	14.8420	0.635763
$546$	0	0
$547$	0.235008	0.0100482	0.00502410	−	0.999987i	$-0.498401\pi$
$547$	0.235008	0.0100482	0.00502410	+	0.999987i	$0.498401\pi$
$548$	0	0
$549$	−29.3654	−1.25328
$550$	0	0
$551$	−5.54821	−0.236362
$552$	0	0
$553$	0	0
$554$	0	0
$555$	48.9490	2.07777
$556$	0	0
$557$	28.3966	1.20320	0.601600	−	0.798797i	$-0.294530\pi$
$557$	28.3966	1.20320	0.601600	+	0.798797i	$0.294530\pi$
$558$	0	0
$559$	−0.0906080	−0.00383231
$560$	0	0
$561$	−8.91320	−0.376316
$562$	0	0
$563$	−13.3285	−0.561730	−0.280865	−	0.959747i	$-0.590621\pi$
$563$	−13.3285	−0.561730	−0.280865	+	0.959747i	$0.590621\pi$
$564$	0	0
$565$	13.5998	0.572148
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.8400	1.87979	0.939896	−	0.341461i	$-0.110922\pi$
$569$	44.8400	1.87979	0.939896	+	0.341461i	$0.110922\pi$
$570$	0	0
$571$	17.9036	0.749242	0.374621	−	0.927178i	$-0.377773\pi$
$571$	17.9036	0.749242	0.374621	+	0.927178i	$0.377773\pi$
$572$	0	0
$573$	47.6209	1.98939
$574$	0	0
$575$	5.52008	0.230203
$576$	0	0
$577$	−24.9986	−1.04071	−0.520353	−	0.853951i	$-0.674200\pi$
$577$	−24.9986	−1.04071	−0.520353	+	0.853951i	$0.674200\pi$
$578$	0	0
$579$	−49.9100	−2.07419
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−26.9982	−1.11815
$584$	0	0
$585$	0.935456	0.0386763
$586$	0	0
$587$	−40.3825	−1.66676	−0.833382	−	0.552697i	$-0.813599\pi$
$587$	−40.3825	−1.66676	−0.833382	+	0.552697i	$0.813599\pi$
$588$	0	0
$589$	5.17815	0.213362
$590$	0	0
$591$	−39.9096	−1.64166
$592$	0	0
$593$	5.22209	0.214446	0.107223	−	0.994235i	$-0.465804\pi$
$593$	5.22209	0.214446	0.107223	+	0.994235i	$0.465804\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.2950	0.912474
$598$	0	0
$599$	1.09789	0.0448588	0.0224294	−	0.999748i	$-0.492860\pi$
$599$	1.09789	0.0448588	0.0224294	+	0.999748i	$0.492860\pi$
$600$	0	0
$601$	−16.3752	−0.667958	−0.333979	−	0.942581i	$-0.608391\pi$
$601$	−16.3752	−0.667958	−0.333979	+	0.942581i	$0.608391\pi$
$602$	0	0
$603$	22.7272	0.925522
$604$	0	0
$605$	6.08568	0.247418
$606$	0	0
$607$	−42.1987	−1.71279	−0.856396	−	0.516319i	$-0.827302\pi$
$607$	−42.1987	−1.71279	−0.856396	+	0.516319i	$0.827302\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.464106	−0.0187757
$612$	0	0
$613$	30.9122	1.24853	0.624265	−	0.781213i	$-0.285399\pi$
$613$	30.9122	1.24853	0.624265	+	0.781213i	$0.285399\pi$
$614$	0	0
$615$	14.7508	0.594811
$616$	0	0
$617$	−11.2653	−0.453526	−0.226763	−	0.973950i	$-0.572814\pi$
$617$	−11.2653	−0.453526	−0.226763	+	0.973950i	$0.572814\pi$
$618$	0	0
$619$	31.0025	1.24609	0.623047	−	0.782184i	$-0.285894\pi$
$619$	31.0025	1.24609	0.623047	+	0.782184i	$0.285894\pi$
$620$	0	0
$621$	−3.81716	−0.153177
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−16.1154	−0.644617
$626$	0	0
$627$	−7.84907	−0.313462
$628$	0	0
$629$	−10.8935	−0.434352
$630$	0	0
$631$	25.0599	0.997621	0.498810	−	0.866711i	$-0.333770\pi$
$631$	25.0599	0.997621	0.498810	+	0.866711i	$0.333770\pi$
$632$	0	0
$633$	0.182285	0.00724520
$634$	0	0
$635$	10.1258	0.401829
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17.4331	−0.689642
$640$	0	0
$641$	−33.2570	−1.31357	−0.656787	−	0.754077i	$-0.728085\pi$
$641$	−33.2570	−1.31357	−0.656787	+	0.754077i	$0.728085\pi$
$642$	0	0
$643$	−21.0140	−0.828710	−0.414355	−	0.910115i	$-0.635993\pi$
$643$	−21.0140	−0.828710	−0.414355	+	0.910115i	$0.635993\pi$
$644$	0	0
$645$	2.14456	0.0844421
$646$	0	0
$647$	21.2791	0.836567	0.418284	−	0.908316i	$-0.362632\pi$
$647$	21.2791	0.836567	0.418284	+	0.908316i	$0.362632\pi$
$648$	0	0
$649$	17.0828	0.670559
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.61889	0.0633520	0.0316760	−	0.999498i	$-0.489916\pi$
$653$	1.61889	0.0633520	0.0316760	+	0.999498i	$0.489916\pi$
$654$	0	0
$655$	11.3385	0.443033
$656$	0	0
$657$	−1.80389	−0.0703764
$658$	0	0
$659$	9.90958	0.386022	0.193011	−	0.981197i	$-0.438175\pi$
$659$	9.90958	0.386022	0.193011	+	0.981197i	$0.438175\pi$
$660$	0	0
$661$	12.1768	0.473624	0.236812	−	0.971555i	$-0.423897\pi$
$661$	12.1768	0.473624	0.236812	+	0.971555i	$0.423897\pi$
$662$	0	0
$663$	−0.448999	−0.0174377
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.0150	0.968583
$668$	0	0
$669$	23.6709	0.915171
$670$	0	0
$671$	−42.6720	−1.64733
$672$	0	0
$673$	4.19259	0.161612	0.0808062	−	0.996730i	$-0.474251\pi$
$673$	4.19259	0.161612	0.0808062	+	0.996730i	$0.474251\pi$
$674$	0	0
$675$	−1.33666	−0.0514483
$676$	0	0
$677$	16.2205	0.623406	0.311703	−	0.950180i	$-0.399101\pi$
$677$	16.2205	0.623406	0.311703	+	0.950180i	$0.399101\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	25.2792	0.968701
$682$	0	0
$683$	18.1844	0.695806	0.347903	−	0.937530i	$-0.386894\pi$
$683$	18.1844	0.695806	0.347903	+	0.937530i	$0.386894\pi$
$684$	0	0
$685$	32.3335	1.23540
$686$	0	0
$687$	−37.2581	−1.42149
$688$	0	0
$689$	−1.36002	−0.0518127
$690$	0	0
$691$	−34.5376	−1.31387	−0.656936	−	0.753946i	$-0.728148\pi$
$691$	−34.5376	−1.31387	−0.656936	+	0.753946i	$0.728148\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.2944	−0.618083
$696$	0	0
$697$	−3.28277	−0.124344
$698$	0	0
$699$	−43.8533	−1.65868
$700$	0	0
$701$	−13.4147	−0.506665	−0.253333	−	0.967379i	$-0.581527\pi$
$701$	−13.4147	−0.506665	−0.253333	+	0.967379i	$0.581527\pi$
$702$	0	0
$703$	−9.59295	−0.361805
$704$	0	0
$705$	10.9847	0.413710
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.9084	−0.484786	−0.242393	−	0.970178i	$-0.577932\pi$
$709$	−12.9084	−0.484786	−0.242393	+	0.970178i	$0.577932\pi$
$710$	0	0
$711$	−10.3240	−0.387179
$712$	0	0
$713$	−23.3465	−0.874333
$714$	0	0
$715$	1.35935	0.0508367
$716$	0	0
$717$	32.8077	1.22523
$718$	0	0
$719$	15.9545	0.595004	0.297502	−	0.954721i	$-0.403847\pi$
$719$	15.9545	0.595004	0.297502	+	0.954721i	$0.403847\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	17.4591	0.649311
$724$	0	0
$725$	8.75956	0.325322
$726$	0	0
$727$	−11.5506	−0.428389	−0.214195	−	0.976791i	$-0.568713\pi$
$727$	−11.5506	−0.428389	−0.214195	+	0.976791i	$0.568713\pi$
$728$	0	0
$729$	−19.2543	−0.713123
$730$	0	0
$731$	−0.477269	−0.0176524
$732$	0	0
$733$	−7.36061	−0.271870	−0.135935	−	0.990718i	$-0.543404\pi$
$733$	−7.36061	−0.271870	−0.135935	+	0.990718i	$0.543404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.0257	1.21652
$738$	0	0
$739$	41.9796	1.54424	0.772122	−	0.635474i	$-0.219195\pi$
$739$	41.9796	1.54424	0.772122	+	0.635474i	$0.219195\pi$
$740$	0	0
$741$	−0.395394	−0.0145252
$742$	0	0
$743$	33.2804	1.22094	0.610469	−	0.792040i	$-0.290981\pi$
$743$	33.2804	1.22094	0.610469	+	0.792040i	$0.290981\pi$
$744$	0	0
$745$	−39.9386	−1.46324
$746$	0	0
$747$	20.1543	0.737407
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.5957	−0.751548	−0.375774	−	0.926711i	$-0.622623\pi$
$751$	−20.5957	−0.751548	−0.375774	+	0.926711i	$0.622623\pi$
$752$	0	0
$753$	−59.4116	−2.16508
$754$	0	0
$755$	−44.6768	−1.62595
$756$	0	0
$757$	38.5859	1.40243	0.701214	−	0.712950i	$-0.252642\pi$
$757$	38.5859	1.40243	0.701214	+	0.712950i	$0.252642\pi$
$758$	0	0
$759$	35.3888	1.28453
$760$	0	0
$761$	−8.83878	−0.320406	−0.160203	−	0.987084i	$-0.551215\pi$
$761$	−8.83878	−0.320406	−0.160203	+	0.987084i	$0.551215\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.92742	0.178151
$766$	0	0
$767$	0.860539	0.0310723
$768$	0	0
$769$	7.92516	0.285789	0.142894	−	0.989738i	$-0.454359\pi$
$769$	7.92516	0.285789	0.142894	+	0.989738i	$0.454359\pi$
$770$	0	0
$771$	−59.5780	−2.14565
$772$	0	0
$773$	49.7380	1.78895	0.894475	−	0.447117i	$-0.147549\pi$
$773$	49.7380	1.78895	0.894475	+	0.447117i	$0.147549\pi$
$774$	0	0
$775$	−8.17530	−0.293666
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.89085	−0.103575
$780$	0	0
$781$	−25.3327	−0.906474
$782$	0	0
$783$	−6.05727	−0.216469
$784$	0	0
$785$	35.7632	1.27644
$786$	0	0
$787$	−12.5655	−0.447913	−0.223956	−	0.974599i	$-0.571897\pi$
$787$	−12.5655	−0.447913	−0.223956	+	0.974599i	$0.571897\pi$
$788$	0	0
$789$	12.4618	0.443653
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.14958	−0.0763339
$794$	0	0
$795$	32.1898	1.14165
$796$	0	0
$797$	2.30988	0.0818200	0.0409100	−	0.999163i	$-0.486974\pi$
$797$	2.30988	0.0818200	0.0409100	+	0.999163i	$0.486974\pi$
$798$	0	0
$799$	−2.44464	−0.0864850
$800$	0	0
$801$	−0.614676	−0.0217185
$802$	0	0
$803$	−2.62130	−0.0925036
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.6822	1.15047
$808$	0	0
$809$	−49.7710	−1.74986	−0.874928	−	0.484254i	$-0.839091\pi$
$809$	−49.7710	−1.74986	−0.874928	+	0.484254i	$0.839091\pi$
$810$	0	0
$811$	−28.5230	−1.00158	−0.500789	−	0.865569i	$-0.666957\pi$
$811$	−28.5230	−1.00158	−0.500789	+	0.865569i	$0.666957\pi$
$812$	0	0
$813$	70.3665	2.46786
$814$	0	0
$815$	−31.9532	−1.11927
$816$	0	0
$817$	−0.420289	−0.0147040
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.1298	−1.26094	−0.630469	−	0.776215i	$-0.717137\pi$
$821$	−36.1298	−1.26094	−0.630469	+	0.776215i	$0.717137\pi$
$822$	0	0
$823$	−50.6288	−1.76481	−0.882404	−	0.470493i	$-0.844076\pi$
$823$	−50.6288	−1.76481	−0.882404	+	0.470493i	$0.844076\pi$
$824$	0	0
$825$	12.3922	0.431440
$826$	0	0
$827$	−12.6809	−0.440957	−0.220479	−	0.975392i	$-0.570762\pi$
$827$	−12.6809	−0.440957	−0.220479	+	0.975392i	$0.570762\pi$
$828$	0	0
$829$	23.6020	0.819733	0.409866	−	0.912146i	$-0.365575\pi$
$829$	23.6020	0.819733	0.409866	+	0.912146i	$0.365575\pi$
$830$	0	0
$831$	34.3567	1.19182
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.7266	0.717275
$836$	0	0
$837$	5.65325	0.195405
$838$	0	0
$839$	40.8581	1.41058	0.705290	−	0.708919i	$-0.250817\pi$
$839$	40.8581	1.41058	0.705290	+	0.708919i	$0.250817\pi$
$840$	0	0
$841$	10.6951	0.368795
$842$	0	0
$843$	40.9308	1.40973
$844$	0	0
$845$	−24.6304	−0.847313
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−54.6363	−1.87511
$850$	0	0
$851$	43.2513	1.48263
$852$	0	0
$853$	25.7967	0.883261	0.441630	−	0.897197i	$-0.354400\pi$
$853$	25.7967	0.883261	0.441630	+	0.897197i	$0.354400\pi$
$854$	0	0
$855$	4.33915	0.148396
$856$	0	0
$857$	−41.4448	−1.41573	−0.707864	−	0.706349i	$-0.750341\pi$
$857$	−41.4448	−1.41573	−0.707864	+	0.706349i	$0.750341\pi$
$858$	0	0
$859$	46.6359	1.59120	0.795598	−	0.605825i	$-0.207157\pi$
$859$	46.6359	1.59120	0.795598	+	0.605825i	$0.207157\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.02837	−0.171168	−0.0855839	−	0.996331i	$-0.527276\pi$
$863$	−5.02837	−0.171168	−0.0855839	+	0.996331i	$0.527276\pi$
$864$	0	0
$865$	21.5482	0.732660
$866$	0	0
$867$	−2.36506	−0.0803215
$868$	0	0
$869$	−15.0021	−0.508912
$870$	0	0
$871$	1.66366	0.0563708
$872$	0	0
$873$	17.6810	0.598413
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.3965	−1.26279	−0.631396	−	0.775461i	$-0.717518\pi$
$877$	−37.3965	−1.26279	−0.631396	+	0.775461i	$0.717518\pi$
$878$	0	0
$879$	74.3853	2.50895
$880$	0	0
$881$	−52.7508	−1.77722	−0.888609	−	0.458665i	$-0.848328\pi$
$881$	−52.7508	−1.77722	−0.888609	+	0.458665i	$0.848328\pi$
$882$	0	0
$883$	45.9674	1.54692	0.773462	−	0.633842i	$-0.218523\pi$
$883$	45.9674	1.54692	0.773462	+	0.633842i	$0.218523\pi$
$884$	0	0
$885$	−20.3678	−0.684655
$886$	0	0
$887$	−5.46886	−0.183626	−0.0918131	−	0.995776i	$-0.529266\pi$
$887$	−5.46886	−0.183626	−0.0918131	+	0.995776i	$0.529266\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−37.8916	−1.26942
$892$	0	0
$893$	−2.15278	−0.0720399
$894$	0	0
$895$	17.1370	0.572826
$896$	0	0
$897$	1.78269	0.0595224
$898$	0	0
$899$	−37.0474	−1.23560
$900$	0	0
$901$	−7.16378	−0.238660
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.94863	−0.197739
$906$	0	0
$907$	50.7671	1.68569	0.842846	−	0.538154i	$-0.180878\pi$
$907$	50.7671	1.68569	0.842846	+	0.538154i	$0.180878\pi$
$908$	0	0
$909$	−43.9632	−1.45817
$910$	0	0
$911$	15.2983	0.506856	0.253428	−	0.967354i	$-0.418442\pi$
$911$	15.2983	0.506856	0.253428	+	0.967354i	$0.418442\pi$
$912$	0	0
$913$	29.2870	0.969258
$914$	0	0
$915$	50.8776	1.68196
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32.7857	−1.08150	−0.540750	−	0.841183i	$-0.681860\pi$
$919$	−32.7857	−1.08150	−0.540750	+	0.841183i	$0.681860\pi$
$920$	0	0
$921$	−35.4109	−1.16683
$922$	0	0
$923$	−1.27612	−0.0420041
$924$	0	0
$925$	15.1454	0.497978
$926$	0	0
$927$	−6.40658	−0.210420
$928$	0	0
$929$	10.3173	0.338499	0.169249	−	0.985573i	$-0.445866\pi$
$929$	10.3173	0.338499	0.169249	+	0.985573i	$0.445866\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0.975949	0.0319511
$934$	0	0
$935$	7.16022	0.234164
$936$	0	0
$937$	38.7047	1.26443	0.632214	−	0.774794i	$-0.282146\pi$
$937$	38.7047	1.26443	0.632214	+	0.774794i	$0.282146\pi$
$938$	0	0
$939$	10.2390	0.334137
$940$	0	0
$941$	43.3796	1.41413	0.707067	−	0.707146i	$-0.250018\pi$
$941$	43.3796	1.41413	0.707067	+	0.707146i	$0.250018\pi$
$942$	0	0
$943$	13.0338	0.424440
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.2852	−1.73154	−0.865768	−	0.500445i	$-0.833170\pi$
$947$	−53.2852	−1.73154	−0.865768	+	0.500445i	$0.833170\pi$
$948$	0	0
$949$	−0.132047	−0.00428642
$950$	0	0
$951$	70.9023	2.29917
$952$	0	0
$953$	−54.3851	−1.76170	−0.880852	−	0.473391i	$-0.843030\pi$
$953$	−54.3851	−1.76170	−0.880852	+	0.473391i	$0.843030\pi$
$954$	0	0
$955$	−38.2552	−1.23791
$956$	0	0
$957$	56.1567	1.81529
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.57639	0.115367
$962$	0	0
$963$	−6.45235	−0.207924
$964$	0	0
$965$	40.0941	1.29068
$966$	0	0
$967$	38.2418	1.22977	0.614887	−	0.788615i	$-0.289202\pi$
$967$	38.2418	1.22977	0.614887	+	0.788615i	$0.289202\pi$
$968$	0	0
$969$	−2.08270	−0.0669059
$970$	0	0
$971$	5.15029	0.165281	0.0826404	−	0.996579i	$-0.473665\pi$
$971$	5.15029	0.165281	0.0826404	+	0.996579i	$0.473665\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.624251	0.0199920
$976$	0	0
$977$	19.4591	0.622552	0.311276	−	0.950319i	$-0.399244\pi$
$977$	19.4591	0.622552	0.311276	+	0.950319i	$0.399244\pi$
$978$	0	0
$979$	−0.893208	−0.0285471
$980$	0	0
$981$	20.2602	0.646859
$982$	0	0
$983$	−18.3002	−0.583684	−0.291842	−	0.956467i	$-0.594268\pi$
$983$	−18.3002	−0.583684	−0.291842	+	0.956467i	$0.594268\pi$
$984$	0	0
$985$	32.0605	1.02153
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.89494	0.0602555
$990$	0	0
$991$	44.6995	1.41993	0.709963	−	0.704239i	$-0.248712\pi$
$991$	44.6995	1.41993	0.709963	+	0.704239i	$0.248712\pi$
$992$	0	0
$993$	−56.1478	−1.78180
$994$	0	0
$995$	−17.9102	−0.567792
$996$	0	0
$997$	14.9063	0.472088	0.236044	−	0.971742i	$-0.424149\pi$
$997$	14.9063	0.472088	0.236044	+	0.971742i	$0.424149\pi$
$998$	0	0
$999$	−10.4731	−0.331355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.r.1.1		6
7.3	odd	6		952.2.q.c.681.1	yes	12
7.5	odd	6		952.2.q.c.137.1	✓	12
7.6	odd	2		6664.2.a.s.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.c.137.1	✓	12	7.5	odd	6
952.2.q.c.681.1	yes	12	7.3	odd	6
6664.2.a.r.1.1		6	1.1	even	1	trivial
6664.2.a.s.1.6		6	7.6	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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36506 q^{3} +1.89992 q^{5} +2.59349 q^{9} +3.76870 q^{11} +0.189847 q^{13} -4.49341 q^{15} +1.00000 q^{17} +0.880612 q^{19} -3.97038 q^{23} -1.39032 q^{25} +0.961410 q^{27} -6.30040 q^{29} +5.88017 q^{31} -8.91320 q^{33} -10.8935 q^{37} -0.448999 q^{39} -3.28277 q^{41} -0.477269 q^{43} +4.92742 q^{45} -2.44464 q^{47} -2.36506 q^{51} -7.16378 q^{53} +7.16022 q^{55} -2.08270 q^{57} +4.53281 q^{59} -11.3227 q^{61} +0.360693 q^{65} +8.76314 q^{67} +9.39016 q^{69} -6.72185 q^{71} -0.695543 q^{73} +3.28818 q^{75} -3.98071 q^{79} -10.0543 q^{81} +7.77110 q^{83} +1.89992 q^{85} +14.9008 q^{87} -0.237007 q^{89} -13.9069 q^{93} +1.67309 q^{95} +6.81746 q^{97} +9.77411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} + q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} - 5 q^{23} + q^{25} + 8 q^{27} - 15 q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{37} - 2 q^{39} - 4 q^{41} + 24 q^{43} - 2 q^{45} + q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 + q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 - 5 * q^23 + q^25 + 8 * q^27 - 15 * q^29 - 8 * q^31 - 11 * q^33 - 16 * q^37 - 2 * q^39 - 4 * q^41 + 24 * q^43 - 2 * q^45 + q^47 - q^51 - 32 * q^53 + 25 * q^55 - 19 * q^57 + 6 * q^59 - 10 * q^61 - 26 * q^65 + 27 * q^67 + 22 * q^69 - 12 * q^71 - 15 * q^73 - 24 * q^75 - 21 * q^79 - 30 * q^81 + 20 * q^83 - q^85 + 9 * q^87 + 33 * q^89 - 29 * q^93 - 15 * q^95 + 9 * q^97 + 2 * q^99

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