LMFDB - Embedded newform 4840.2.a.bb.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-6,0,7,0,5,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.25903625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 $ x^6 - 3x^5 - 7x^4 + 17x^3 + 16x^2 - 20*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.68797$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68797 q^{3} -1.00000 q^{5} -0.193967 q^{7} -0.150741 q^{9} -0.495133 q^{13} -1.68797 q^{15} -4.98130 q^{17} +7.26172 q^{19} -0.327411 q^{21} -2.41321 q^{23} +1.00000 q^{25} -5.31837 q^{27} +2.10646 q^{29} -6.56591 q^{31} +0.193967 q^{35} +7.28844 q^{37} -0.835771 q^{39} +1.10864 q^{41} -5.46058 q^{43} +0.150741 q^{45} -5.50315 q^{47} -6.96238 q^{49} -8.40832 q^{51} -6.73973 q^{53} +12.2576 q^{57} -7.73492 q^{59} +13.1238 q^{61} +0.0292388 q^{63} +0.495133 q^{65} +4.49499 q^{67} -4.07343 q^{69} +0.379781 q^{71} -14.6213 q^{73} +1.68797 q^{75} +1.65379 q^{79} -8.52505 q^{81} -8.68029 q^{83} +4.98130 q^{85} +3.55564 q^{87} +1.83811 q^{89} +0.0960393 q^{91} -11.0831 q^{93} -7.26172 q^{95} -12.8755 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 + 7 * q^7 + 5 * q^9 - q^13 + 3 * q^15 - 6 * q^17 + 7 * q^19 - 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 - 10 * q^29 + q^31 - 7 * q^35 + 3 * q^37 + q^39 + 6 * q^41 - 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 - 15 * q^51 - 23 * q^53 - 9 * q^57 - 2 * q^59 - 6 * q^61 + 49 * q^63 + q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 - 10 * q^73 - 3 * q^75 + 22 * q^79 + 10 * q^81 - 10 * q^83 + 6 * q^85 + 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 - 7 * q^95 - 33 * q^97

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
$3$	1.68797	0.974553	0.487276	−	0.873248i	$-0.337990\pi$
$3$	1.68797	0.974553	0.487276	+	0.873248i	$0.337990\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.193967	−0.0733126	−0.0366563	−	0.999328i	$-0.511671\pi$
$7$	−0.193967	−0.0733126	−0.0366563	+	0.999328i	$0.511671\pi$
$8$	0	0
$9$	−0.150741	−0.0502470
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.495133	−0.137325	−0.0686625	−	0.997640i	$-0.521873\pi$
$13$	−0.495133	−0.137325	−0.0686625	+	0.997640i	$0.521873\pi$
$14$	0	0
$15$	−1.68797	−0.435833
$16$	0	0
$17$	−4.98130	−1.20814	−0.604072	−	0.796930i	$-0.706456\pi$
$17$	−4.98130	−1.20814	−0.604072	+	0.796930i	$0.706456\pi$
$18$	0	0
$19$	7.26172	1.66595	0.832977	−	0.553308i	$-0.186635\pi$
$19$	7.26172	1.66595	0.832977	+	0.553308i	$0.186635\pi$
$20$	0	0
$21$	−0.327411	−0.0714470
$22$	0	0
$23$	−2.41321	−0.503189	−0.251594	−	0.967833i	$-0.580955\pi$
$23$	−2.41321	−0.503189	−0.251594	+	0.967833i	$0.580955\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.31837	−1.02352
$28$	0	0
$29$	2.10646	0.391159	0.195579	−	0.980688i	$-0.437341\pi$
$29$	2.10646	0.391159	0.195579	+	0.980688i	$0.437341\pi$
$30$	0	0
$31$	−6.56591	−1.17927	−0.589636	−	0.807669i	$-0.700729\pi$
$31$	−6.56591	−1.17927	−0.589636	+	0.807669i	$0.700729\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.193967	0.0327864
$36$	0	0
$37$	7.28844	1.19821	0.599106	−	0.800670i	$-0.295523\pi$
$37$	7.28844	1.19821	0.599106	+	0.800670i	$0.295523\pi$
$38$	0	0
$39$	−0.835771	−0.133831
$40$	0	0
$41$	1.10864	0.173140	0.0865702	−	0.996246i	$-0.472409\pi$
$41$	1.10864	0.173140	0.0865702	+	0.996246i	$0.472409\pi$
$42$	0	0
$43$	−5.46058	−0.832731	−0.416365	−	0.909197i	$-0.636696\pi$
$43$	−5.46058	−0.832731	−0.416365	+	0.909197i	$0.636696\pi$
$44$	0	0
$45$	0.150741	0.0224712
$46$	0	0
$47$	−5.50315	−0.802717	−0.401359	−	0.915921i	$-0.631462\pi$
$47$	−5.50315	−0.802717	−0.401359	+	0.915921i	$0.631462\pi$
$48$	0	0
$49$	−6.96238	−0.994625
$50$	0	0
$51$	−8.40832	−1.17740
$52$	0	0
$53$	−6.73973	−0.925773	−0.462887	−	0.886417i	$-0.653186\pi$
$53$	−6.73973	−0.925773	−0.462887	+	0.886417i	$0.653186\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.2576	1.62356
$58$	0	0
$59$	−7.73492	−1.00700	−0.503500	−	0.863995i	$-0.667955\pi$
$59$	−7.73492	−1.00700	−0.503500	+	0.863995i	$0.667955\pi$
$60$	0	0
$61$	13.1238	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$61$	13.1238	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$62$	0	0
$63$	0.0292388	0.00368374
$64$	0	0
$65$	0.495133	0.0614136
$66$	0	0
$67$	4.49499	0.549151	0.274575	−	0.961566i	$-0.411463\pi$
$67$	4.49499	0.549151	0.274575	+	0.961566i	$0.411463\pi$
$68$	0	0
$69$	−4.07343	−0.490384
$70$	0	0
$71$	0.379781	0.0450717	0.0225359	−	0.999746i	$-0.492826\pi$
$71$	0.379781	0.0450717	0.0225359	+	0.999746i	$0.492826\pi$
$72$	0	0
$73$	−14.6213	−1.71129	−0.855647	−	0.517560i	$-0.826841\pi$
$73$	−14.6213	−1.71129	−0.855647	+	0.517560i	$0.826841\pi$
$74$	0	0
$75$	1.68797	0.194911
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.65379	0.186066	0.0930329	−	0.995663i	$-0.470344\pi$
$79$	1.65379	0.186066	0.0930329	+	0.995663i	$0.470344\pi$
$80$	0	0
$81$	−8.52505	−0.947228
$82$	0	0
$83$	−8.68029	−0.952785	−0.476393	−	0.879233i	$-0.658056\pi$
$83$	−8.68029	−0.952785	−0.476393	+	0.879233i	$0.658056\pi$
$84$	0	0
$85$	4.98130	0.540298
$86$	0	0
$87$	3.55564	0.381205
$88$	0	0
$89$	1.83811	0.194840	0.0974198	−	0.995243i	$-0.468941\pi$
$89$	1.83811	0.194840	0.0974198	+	0.995243i	$0.468941\pi$
$90$	0	0
$91$	0.0960393	0.0100677
$92$	0	0
$93$	−11.0831	−1.14926
$94$	0	0
$95$	−7.26172	−0.745037
$96$	0	0
$97$	−12.8755	−1.30730	−0.653652	−	0.756795i	$-0.726764\pi$
$97$	−12.8755	−1.30730	−0.653652	+	0.756795i	$0.726764\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.998793	−0.0993836	−0.0496918	−	0.998765i	$-0.515824\pi$
$101$	−0.998793	−0.0993836	−0.0496918	+	0.998765i	$0.515824\pi$
$102$	0	0
$103$	−9.19951	−0.906454	−0.453227	−	0.891395i	$-0.649727\pi$
$103$	−9.19951	−0.906454	−0.453227	+	0.891395i	$0.649727\pi$
$104$	0	0
$105$	0.327411	0.0319521
$106$	0	0
$107$	2.84972	0.275493	0.137747	−	0.990468i	$-0.456014\pi$
$107$	2.84972	0.275493	0.137747	+	0.990468i	$0.456014\pi$
$108$	0	0
$109$	1.23155	0.117962	0.0589808	−	0.998259i	$-0.481215\pi$
$109$	1.23155	0.117962	0.0589808	+	0.998259i	$0.481215\pi$
$110$	0	0
$111$	12.3027	1.16772
$112$	0	0
$113$	−9.61697	−0.904689	−0.452344	−	0.891843i	$-0.649412\pi$
$113$	−9.61697	−0.904689	−0.452344	+	0.891843i	$0.649412\pi$
$114$	0	0
$115$	2.41321	0.225033
$116$	0	0
$117$	0.0746368	0.00690018
$118$	0	0
$119$	0.966208	0.0885722
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1.87136	0.168735
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.0637	−1.51416	−0.757081	−	0.653321i	$-0.773375\pi$
$127$	−17.0637	−1.51416	−0.757081	+	0.653321i	$0.773375\pi$
$128$	0	0
$129$	−9.21732	−0.811540
$130$	0	0
$131$	−10.8561	−0.948506	−0.474253	−	0.880389i	$-0.657282\pi$
$131$	−10.8561	−0.948506	−0.474253	+	0.880389i	$0.657282\pi$
$132$	0	0
$133$	−1.40853	−0.122135
$134$	0	0
$135$	5.31837	0.457733
$136$	0	0
$137$	−9.60844	−0.820905	−0.410452	−	0.911882i	$-0.634629\pi$
$137$	−9.60844	−0.820905	−0.410452	+	0.911882i	$0.634629\pi$
$138$	0	0
$139$	15.5588	1.31968	0.659839	−	0.751407i	$-0.270625\pi$
$139$	15.5588	1.31968	0.659839	+	0.751407i	$0.270625\pi$
$140$	0	0
$141$	−9.28918	−0.782290
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.10646	−0.174932
$146$	0	0
$147$	−11.7523	−0.969315
$148$	0	0
$149$	−6.06239	−0.496651	−0.248325	−	0.968677i	$-0.579880\pi$
$149$	−6.06239	−0.496651	−0.248325	+	0.968677i	$0.579880\pi$
$150$	0	0
$151$	17.4419	1.41940	0.709700	−	0.704504i	$-0.248830\pi$
$151$	17.4419	1.41940	0.709700	+	0.704504i	$0.248830\pi$
$152$	0	0
$153$	0.750887	0.0607057
$154$	0	0
$155$	6.56591	0.527387
$156$	0	0
$157$	19.0907	1.52361	0.761804	−	0.647808i	$-0.224314\pi$
$157$	19.0907	1.52361	0.761804	+	0.647808i	$0.224314\pi$
$158$	0	0
$159$	−11.3765	−0.902215
$160$	0	0
$161$	0.468082	0.0368901
$162$	0	0
$163$	−0.100999	−0.00791089	−0.00395544	−	0.999992i	$-0.501259\pi$
$163$	−0.100999	−0.00791089	−0.00395544	+	0.999992i	$0.501259\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.2432	0.870023	0.435011	−	0.900425i	$-0.356744\pi$
$167$	11.2432	0.870023	0.435011	+	0.900425i	$0.356744\pi$
$168$	0	0
$169$	−12.7548	−0.981142
$170$	0	0
$171$	−1.09464	−0.0837092
$172$	0	0
$173$	8.11033	0.616617	0.308308	−	0.951286i	$-0.400237\pi$
$173$	8.11033	0.616617	0.308308	+	0.951286i	$0.400237\pi$
$174$	0	0
$175$	−0.193967	−0.0146625
$176$	0	0
$177$	−13.0563	−0.981375
$178$	0	0
$179$	−21.2782	−1.59041	−0.795203	−	0.606343i	$-0.792636\pi$
$179$	−21.2782	−1.59041	−0.795203	+	0.606343i	$0.792636\pi$
$180$	0	0
$181$	8.62159	0.640838	0.320419	−	0.947276i	$-0.396176\pi$
$181$	8.62159	0.640838	0.320419	+	0.947276i	$0.396176\pi$
$182$	0	0
$183$	22.1527	1.63758
$184$	0	0
$185$	−7.28844	−0.535857
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.03159	0.0750370
$190$	0	0
$191$	−20.1441	−1.45758	−0.728789	−	0.684739i	$-0.759916\pi$
$191$	−20.1441	−1.45758	−0.728789	+	0.684739i	$0.759916\pi$
$192$	0	0
$193$	9.71381	0.699216	0.349608	−	0.936896i	$-0.386315\pi$
$193$	9.71381	0.699216	0.349608	+	0.936896i	$0.386315\pi$
$194$	0	0
$195$	0.835771	0.0598508
$196$	0	0
$197$	−14.4934	−1.03261	−0.516304	−	0.856405i	$-0.672693\pi$
$197$	−14.4934	−1.03261	−0.516304	+	0.856405i	$0.672693\pi$
$198$	0	0
$199$	−20.8055	−1.47486	−0.737431	−	0.675423i	$-0.763961\pi$
$199$	−20.8055	−1.47486	−0.737431	+	0.675423i	$0.763961\pi$
$200$	0	0
$201$	7.58744	0.535176
$202$	0	0
$203$	−0.408583	−0.0286769
$204$	0	0
$205$	−1.10864	−0.0774308
$206$	0	0
$207$	0.363770	0.0252837
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.57224	0.590138	0.295069	−	0.955476i	$-0.404657\pi$
$211$	8.57224	0.590138	0.295069	+	0.955476i	$0.404657\pi$
$212$	0	0
$213$	0.641061	0.0439248
$214$	0	0
$215$	5.46058	0.372409
$216$	0	0
$217$	1.27357	0.0864555
$218$	0	0
$219$	−24.6804	−1.66775
$220$	0	0
$221$	2.46641	0.165908
$222$	0	0
$223$	0.347927	0.0232989	0.0116495	−	0.999932i	$-0.496292\pi$
$223$	0.347927	0.0232989	0.0116495	+	0.999932i	$0.496292\pi$
$224$	0	0
$225$	−0.150741	−0.0100494
$226$	0	0
$227$	−15.1340	−1.00448	−0.502240	−	0.864728i	$-0.667491\pi$
$227$	−15.1340	−1.00448	−0.502240	+	0.864728i	$0.667491\pi$
$228$	0	0
$229$	−26.7980	−1.77086	−0.885430	−	0.464773i	$-0.846136\pi$
$229$	−26.7980	−1.77086	−0.885430	+	0.464773i	$0.846136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.842261	0.0551783	0.0275892	−	0.999619i	$-0.491217\pi$
$233$	0.842261	0.0551783	0.0275892	+	0.999619i	$0.491217\pi$
$234$	0	0
$235$	5.50315	0.358986
$236$	0	0
$237$	2.79155	0.181331
$238$	0	0
$239$	5.82655	0.376888	0.188444	−	0.982084i	$-0.439656\pi$
$239$	5.82655	0.376888	0.188444	+	0.982084i	$0.439656\pi$
$240$	0	0
$241$	−26.5420	−1.70972	−0.854860	−	0.518860i	$-0.826357\pi$
$241$	−26.5420	−1.70972	−0.854860	+	0.518860i	$0.826357\pi$
$242$	0	0
$243$	1.56504	0.100397
$244$	0	0
$245$	6.96238	0.444810
$246$	0	0
$247$	−3.59552	−0.228777
$248$	0	0
$249$	−14.6521	−0.928539
$250$	0	0
$251$	21.3841	1.34975	0.674875	−	0.737932i	$-0.264198\pi$
$251$	21.3841	1.34975	0.674875	+	0.737932i	$0.264198\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.40832	0.526549
$256$	0	0
$257$	21.0242	1.31145	0.655727	−	0.754998i	$-0.272362\pi$
$257$	21.0242	1.31145	0.655727	+	0.754998i	$0.272362\pi$
$258$	0	0
$259$	−1.41372	−0.0878440
$260$	0	0
$261$	−0.317529	−0.0196546
$262$	0	0
$263$	−1.64662	−0.101535	−0.0507675	−	0.998710i	$-0.516167\pi$
$263$	−1.64662	−0.101535	−0.0507675	+	0.998710i	$0.516167\pi$
$264$	0	0
$265$	6.73973	0.414018
$266$	0	0
$267$	3.10269	0.189881
$268$	0	0
$269$	16.1548	0.984974	0.492487	−	0.870320i	$-0.336088\pi$
$269$	16.1548	0.984974	0.492487	+	0.870320i	$0.336088\pi$
$270$	0	0
$271$	4.90093	0.297710	0.148855	−	0.988859i	$-0.452441\pi$
$271$	4.90093	0.297710	0.148855	+	0.988859i	$0.452441\pi$
$272$	0	0
$273$	0.162112	0.00981146
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.5435	0.873836	0.436918	−	0.899501i	$-0.356070\pi$
$277$	14.5435	0.873836	0.436918	+	0.899501i	$0.356070\pi$
$278$	0	0
$279$	0.989753	0.0592550
$280$	0	0
$281$	−3.46414	−0.206653	−0.103327	−	0.994647i	$-0.532949\pi$
$281$	−3.46414	−0.206653	−0.103327	+	0.994647i	$0.532949\pi$
$282$	0	0
$283$	1.45530	0.0865083	0.0432542	−	0.999064i	$-0.486227\pi$
$283$	1.45530	0.0865083	0.0432542	+	0.999064i	$0.486227\pi$
$284$	0	0
$285$	−12.2576	−0.726078
$286$	0	0
$287$	−0.215039	−0.0126934
$288$	0	0
$289$	7.81339	0.459611
$290$	0	0
$291$	−21.7334	−1.27404
$292$	0	0
$293$	31.8597	1.86126	0.930631	−	0.365959i	$-0.119259\pi$
$293$	31.8597	1.86126	0.930631	+	0.365959i	$0.119259\pi$
$294$	0	0
$295$	7.73492	0.450344
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.19486	0.0691004
$300$	0	0
$301$	1.05917	0.0610497
$302$	0	0
$303$	−1.68594	−0.0968545
$304$	0	0
$305$	−13.1238	−0.751469
$306$	0	0
$307$	16.1632	0.922480	0.461240	−	0.887275i	$-0.347405\pi$
$307$	16.1632	0.922480	0.461240	+	0.887275i	$0.347405\pi$
$308$	0	0
$309$	−15.5285	−0.883387
$310$	0	0
$311$	22.5292	1.27751	0.638757	−	0.769408i	$-0.279449\pi$
$311$	22.5292	1.27751	0.638757	+	0.769408i	$0.279449\pi$
$312$	0	0
$313$	17.5467	0.991798	0.495899	−	0.868380i	$-0.334839\pi$
$313$	17.5467	0.991798	0.495899	+	0.868380i	$0.334839\pi$
$314$	0	0
$315$	−0.0292388	−0.00164742
$316$	0	0
$317$	−28.1627	−1.58178	−0.790888	−	0.611961i	$-0.790381\pi$
$317$	−28.1627	−1.58178	−0.790888	+	0.611961i	$0.790381\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.81026	0.268483
$322$	0	0
$323$	−36.1728	−2.01271
$324$	0	0
$325$	−0.495133	−0.0274650
$326$	0	0
$327$	2.07883	0.114960
$328$	0	0
$329$	1.06743	0.0588493
$330$	0	0
$331$	0.970270	0.0533309	0.0266654	−	0.999644i	$-0.491511\pi$
$331$	0.970270	0.0533309	0.0266654	+	0.999644i	$0.491511\pi$
$332$	0	0
$333$	−1.09867	−0.0602066
$334$	0	0
$335$	−4.49499	−0.245588
$336$	0	0
$337$	5.55386	0.302538	0.151269	−	0.988493i	$-0.451664\pi$
$337$	5.55386	0.302538	0.151269	+	0.988493i	$0.451664\pi$
$338$	0	0
$339$	−16.2332	−0.881667
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.70824	0.146231
$344$	0	0
$345$	4.07343	0.219306
$346$	0	0
$347$	−16.6620	−0.894465	−0.447233	−	0.894418i	$-0.647590\pi$
$347$	−16.6620	−0.894465	−0.447233	+	0.894418i	$0.647590\pi$
$348$	0	0
$349$	−33.1078	−1.77222	−0.886110	−	0.463475i	$-0.846603\pi$
$349$	−33.1078	−1.77222	−0.886110	+	0.463475i	$0.846603\pi$
$350$	0	0
$351$	2.63330	0.140555
$352$	0	0
$353$	−6.70224	−0.356724	−0.178362	−	0.983965i	$-0.557080\pi$
$353$	−6.70224	−0.356724	−0.178362	+	0.983965i	$0.557080\pi$
$354$	0	0
$355$	−0.379781	−0.0201567
$356$	0	0
$357$	1.63094	0.0863182
$358$	0	0
$359$	24.0596	1.26982	0.634908	−	0.772588i	$-0.281038\pi$
$359$	24.0596	1.26982	0.634908	+	0.772588i	$0.281038\pi$
$360$	0	0
$361$	33.7326	1.77540
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.6213	0.765314
$366$	0	0
$367$	−8.30379	−0.433454	−0.216727	−	0.976232i	$-0.569538\pi$
$367$	−8.30379	−0.433454	−0.216727	+	0.976232i	$0.569538\pi$
$368$	0	0
$369$	−0.167118	−0.00869980
$370$	0	0
$371$	1.30729	0.0678709
$372$	0	0
$373$	−16.0339	−0.830202	−0.415101	−	0.909775i	$-0.636254\pi$
$373$	−16.0339	−0.830202	−0.415101	+	0.909775i	$0.636254\pi$
$374$	0	0
$375$	−1.68797	−0.0871666
$376$	0	0
$377$	−1.04297	−0.0537159
$378$	0	0
$379$	26.2598	1.34888	0.674439	−	0.738331i	$-0.264386\pi$
$379$	26.2598	1.34888	0.674439	+	0.738331i	$0.264386\pi$
$380$	0	0
$381$	−28.8032	−1.47563
$382$	0	0
$383$	−13.6392	−0.696929	−0.348465	−	0.937322i	$-0.613297\pi$
$383$	−13.6392	−0.696929	−0.348465	+	0.937322i	$0.613297\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.823134	0.0418423
$388$	0	0
$389$	−31.6588	−1.60517	−0.802583	−	0.596541i	$-0.796541\pi$
$389$	−31.6588	−1.60517	−0.802583	+	0.596541i	$0.796541\pi$
$390$	0	0
$391$	12.0209	0.607924
$392$	0	0
$393$	−18.3249	−0.924369
$394$	0	0
$395$	−1.65379	−0.0832111
$396$	0	0
$397$	33.2898	1.67077	0.835383	−	0.549668i	$-0.185246\pi$
$397$	33.2898	1.67077	0.835383	+	0.549668i	$0.185246\pi$
$398$	0	0
$399$	−2.37757	−0.119027
$400$	0	0
$401$	−23.1007	−1.15359	−0.576797	−	0.816887i	$-0.695698\pi$
$401$	−23.1007	−1.15359	−0.576797	+	0.816887i	$0.695698\pi$
$402$	0	0
$403$	3.25100	0.161944
$404$	0	0
$405$	8.52505	0.423613
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−30.8815	−1.52699	−0.763496	−	0.645813i	$-0.776519\pi$
$409$	−30.8815	−1.52699	−0.763496	+	0.645813i	$0.776519\pi$
$410$	0	0
$411$	−16.2188	−0.800015
$412$	0	0
$413$	1.50032	0.0738258
$414$	0	0
$415$	8.68029	0.426098
$416$	0	0
$417$	26.2628	1.28610
$418$	0	0
$419$	11.9155	0.582109	0.291054	−	0.956707i	$-0.405994\pi$
$419$	11.9155	0.582109	0.291054	+	0.956707i	$0.405994\pi$
$420$	0	0
$421$	−4.56495	−0.222482	−0.111241	−	0.993793i	$-0.535483\pi$
$421$	−4.56495	−0.222482	−0.111241	+	0.993793i	$0.535483\pi$
$422$	0	0
$423$	0.829551	0.0403342
$424$	0	0
$425$	−4.98130	−0.241629
$426$	0	0
$427$	−2.54559	−0.123190
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.58188	−0.365206	−0.182603	−	0.983187i	$-0.558452\pi$
$431$	−7.58188	−0.365206	−0.182603	+	0.983187i	$0.558452\pi$
$432$	0	0
$433$	−10.5815	−0.508514	−0.254257	−	0.967137i	$-0.581831\pi$
$433$	−10.5815	−0.508514	−0.254257	+	0.967137i	$0.581831\pi$
$434$	0	0
$435$	−3.55564	−0.170480
$436$	0	0
$437$	−17.5240	−0.838289
$438$	0	0
$439$	7.04399	0.336191	0.168096	−	0.985771i	$-0.446238\pi$
$439$	7.04399	0.336191	0.168096	+	0.985771i	$0.446238\pi$
$440$	0	0
$441$	1.04952	0.0499770
$442$	0	0
$443$	1.16448	0.0553259	0.0276630	−	0.999617i	$-0.491193\pi$
$443$	1.16448	0.0553259	0.0276630	+	0.999617i	$0.491193\pi$
$444$	0	0
$445$	−1.83811	−0.0871349
$446$	0	0
$447$	−10.2332	−0.484012
$448$	0	0
$449$	12.3659	0.583583	0.291791	−	0.956482i	$-0.405749\pi$
$449$	12.3659	0.583583	0.291791	+	0.956482i	$0.405749\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	29.4415	1.38328
$454$	0	0
$455$	−0.0960393	−0.00450239
$456$	0	0
$457$	−36.0429	−1.68602	−0.843009	−	0.537900i	$-0.819218\pi$
$457$	−36.0429	−1.68602	−0.843009	+	0.537900i	$0.819218\pi$
$458$	0	0
$459$	26.4924	1.23656
$460$	0	0
$461$	−13.6603	−0.636222	−0.318111	−	0.948053i	$-0.603049\pi$
$461$	−13.6603	−0.636222	−0.318111	+	0.948053i	$0.603049\pi$
$462$	0	0
$463$	29.3515	1.36408	0.682041	−	0.731314i	$-0.261093\pi$
$463$	29.3515	1.36408	0.682041	+	0.731314i	$0.261093\pi$
$464$	0	0
$465$	11.0831	0.513966
$466$	0	0
$467$	24.1479	1.11743	0.558717	−	0.829358i	$-0.311294\pi$
$467$	24.1479	1.11743	0.558717	+	0.829358i	$0.311294\pi$
$468$	0	0
$469$	−0.871880	−0.0402597
$470$	0	0
$471$	32.2247	1.48484
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.26172	0.333191
$476$	0	0
$477$	1.01595	0.0465174
$478$	0	0
$479$	40.2629	1.83966	0.919829	−	0.392319i	$-0.128327\pi$
$479$	40.2629	1.83966	0.919829	+	0.392319i	$0.128327\pi$
$480$	0	0
$481$	−3.60874	−0.164545
$482$	0	0
$483$	0.790111	0.0359513
$484$	0	0
$485$	12.8755	0.584644
$486$	0	0
$487$	42.1964	1.91210	0.956051	−	0.293201i	$-0.0947207\pi$
$487$	42.1964	1.91210	0.956051	+	0.293201i	$0.0947207\pi$
$488$	0	0
$489$	−0.170485	−0.00770958
$490$	0	0
$491$	36.2831	1.63743	0.818716	−	0.574198i	$-0.194686\pi$
$491$	36.2831	1.63743	0.818716	+	0.574198i	$0.194686\pi$
$492$	0	0
$493$	−10.4929	−0.472576
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.0736649	−0.00330432
$498$	0	0
$499$	15.6747	0.701698	0.350849	−	0.936432i	$-0.385893\pi$
$499$	15.6747	0.701698	0.350849	+	0.936432i	$0.385893\pi$
$500$	0	0
$501$	18.9782	0.847883
$502$	0	0
$503$	25.6886	1.14540	0.572698	−	0.819766i	$-0.305897\pi$
$503$	25.6886	1.14540	0.572698	+	0.819766i	$0.305897\pi$
$504$	0	0
$505$	0.998793	0.0444457
$506$	0	0
$507$	−21.5299	−0.956174
$508$	0	0
$509$	−21.8032	−0.966408	−0.483204	−	0.875508i	$-0.660527\pi$
$509$	−21.8032	−0.966408	−0.483204	+	0.875508i	$0.660527\pi$
$510$	0	0
$511$	2.83605	0.125459
$512$	0	0
$513$	−38.6205	−1.70514
$514$	0	0
$515$	9.19951	0.405379
$516$	0	0
$517$	0	0
$518$	0	0
$519$	13.6900	0.600925
$520$	0	0
$521$	39.8455	1.74566	0.872832	−	0.488020i	$-0.162281\pi$
$521$	39.8455	1.74566	0.872832	+	0.488020i	$0.162281\pi$
$522$	0	0
$523$	−11.4599	−0.501106	−0.250553	−	0.968103i	$-0.580612\pi$
$523$	−11.4599	−0.501106	−0.250553	+	0.968103i	$0.580612\pi$
$524$	0	0
$525$	−0.327411	−0.0142894
$526$	0	0
$527$	32.7068	1.42473
$528$	0	0
$529$	−17.1764	−0.746801
$530$	0	0
$531$	1.16597	0.0505988
$532$	0	0
$533$	−0.548924	−0.0237765
$534$	0	0
$535$	−2.84972	−0.123204
$536$	0	0
$537$	−35.9170	−1.54993
$538$	0	0
$539$	0	0
$540$	0	0
$541$	43.5728	1.87334	0.936670	−	0.350212i	$-0.113890\pi$
$541$	43.5728	1.87334	0.936670	+	0.350212i	$0.113890\pi$
$542$	0	0
$543$	14.5530	0.624530
$544$	0	0
$545$	−1.23155	−0.0527540
$546$	0	0
$547$	5.10019	0.218068	0.109034	−	0.994038i	$-0.465224\pi$
$547$	5.10019	0.218068	0.109034	+	0.994038i	$0.465224\pi$
$548$	0	0
$549$	−1.97830	−0.0844319
$550$	0	0
$551$	15.2965	0.651652
$552$	0	0
$553$	−0.320780	−0.0136410
$554$	0	0
$555$	−12.3027	−0.522220
$556$	0	0
$557$	−4.53850	−0.192302	−0.0961512	−	0.995367i	$-0.530653\pi$
$557$	−4.53850	−0.192302	−0.0961512	+	0.995367i	$0.530653\pi$
$558$	0	0
$559$	2.70371	0.114355
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.4510	−1.24121	−0.620605	−	0.784123i	$-0.713113\pi$
$563$	−29.4510	−1.24121	−0.620605	+	0.784123i	$0.713113\pi$
$564$	0	0
$565$	9.61697	0.404589
$566$	0	0
$567$	1.65358	0.0694438
$568$	0	0
$569$	10.6590	0.446849	0.223424	−	0.974721i	$-0.428276\pi$
$569$	10.6590	0.446849	0.223424	+	0.974721i	$0.428276\pi$
$570$	0	0
$571$	−35.1373	−1.47045	−0.735224	−	0.677824i	$-0.762923\pi$
$571$	−35.1373	−1.47045	−0.735224	+	0.677824i	$0.762923\pi$
$572$	0	0
$573$	−34.0028	−1.42049
$574$	0	0
$575$	−2.41321	−0.100638
$576$	0	0
$577$	34.7058	1.44482	0.722410	−	0.691465i	$-0.243034\pi$
$577$	34.7058	1.44482	0.722410	+	0.691465i	$0.243034\pi$
$578$	0	0
$579$	16.3967	0.681422
$580$	0	0
$581$	1.68369	0.0698512
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.0746368	−0.00308585
$586$	0	0
$587$	28.4171	1.17290	0.586451	−	0.809985i	$-0.300525\pi$
$587$	28.4171	1.17290	0.586451	+	0.809985i	$0.300525\pi$
$588$	0	0
$589$	−47.6798	−1.96461
$590$	0	0
$591$	−24.4644	−1.00633
$592$	0	0
$593$	−19.6141	−0.805456	−0.402728	−	0.915320i	$-0.631938\pi$
$593$	−19.6141	−0.805456	−0.402728	+	0.915320i	$0.631938\pi$
$594$	0	0
$595$	−0.966208	−0.0396107
$596$	0	0
$597$	−35.1191	−1.43733
$598$	0	0
$599$	−16.5958	−0.678087	−0.339044	−	0.940771i	$-0.610103\pi$
$599$	−16.5958	−0.678087	−0.339044	+	0.940771i	$0.610103\pi$
$600$	0	0
$601$	−10.8128	−0.441065	−0.220532	−	0.975380i	$-0.570779\pi$
$601$	−10.8128	−0.441065	−0.220532	+	0.975380i	$0.570779\pi$
$602$	0	0
$603$	−0.677580	−0.0275932
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.5733	−0.550922	−0.275461	−	0.961312i	$-0.588831\pi$
$607$	−13.5733	−0.550922	−0.275461	+	0.961312i	$0.588831\pi$
$608$	0	0
$609$	−0.689677	−0.0279471
$610$	0	0
$611$	2.72479	0.110233
$612$	0	0
$613$	−12.5331	−0.506208	−0.253104	−	0.967439i	$-0.581451\pi$
$613$	−12.5331	−0.506208	−0.253104	+	0.967439i	$0.581451\pi$
$614$	0	0
$615$	−1.87136	−0.0754604
$616$	0	0
$617$	−46.5587	−1.87438	−0.937191	−	0.348816i	$-0.886584\pi$
$617$	−46.5587	−1.87438	−0.937191	+	0.348816i	$0.886584\pi$
$618$	0	0
$619$	−21.1339	−0.849444	−0.424722	−	0.905324i	$-0.639628\pi$
$619$	−21.1339	−0.849444	−0.424722	+	0.905324i	$0.639628\pi$
$620$	0	0
$621$	12.8343	0.515024
$622$	0	0
$623$	−0.356533	−0.0142842
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−36.3059	−1.44761
$630$	0	0
$631$	11.6752	0.464783	0.232391	−	0.972622i	$-0.425345\pi$
$631$	11.6752	0.464783	0.232391	+	0.972622i	$0.425345\pi$
$632$	0	0
$633$	14.4697	0.575120
$634$	0	0
$635$	17.0637	0.677154
$636$	0	0
$637$	3.44730	0.136587
$638$	0	0
$639$	−0.0572486	−0.00226472
$640$	0	0
$641$	8.55924	0.338070	0.169035	−	0.985610i	$-0.445935\pi$
$641$	8.55924	0.338070	0.169035	+	0.985610i	$0.445935\pi$
$642$	0	0
$643$	−35.0691	−1.38299	−0.691496	−	0.722380i	$-0.743048\pi$
$643$	−35.0691	−1.38299	−0.691496	+	0.722380i	$0.743048\pi$
$644$	0	0
$645$	9.21732	0.362932
$646$	0	0
$647$	−25.1028	−0.986892	−0.493446	−	0.869776i	$-0.664263\pi$
$647$	−25.1028	−0.986892	−0.493446	+	0.869776i	$0.664263\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.14975	0.0842555
$652$	0	0
$653$	47.4918	1.85850	0.929249	−	0.369455i	$-0.120456\pi$
$653$	47.4918	1.85850	0.929249	+	0.369455i	$0.120456\pi$
$654$	0	0
$655$	10.8561	0.424185
$656$	0	0
$657$	2.20403	0.0859875
$658$	0	0
$659$	41.5603	1.61896	0.809479	−	0.587148i	$-0.199749\pi$
$659$	41.5603	1.61896	0.809479	+	0.587148i	$0.199749\pi$
$660$	0	0
$661$	38.3584	1.49197	0.745985	−	0.665963i	$-0.231979\pi$
$661$	38.3584	1.49197	0.745985	+	0.665963i	$0.231979\pi$
$662$	0	0
$663$	4.16323	0.161687
$664$	0	0
$665$	1.40853	0.0546206
$666$	0	0
$667$	−5.08331	−0.196827
$668$	0	0
$669$	0.587292	0.0227060
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29.0562	−1.12003	−0.560017	−	0.828481i	$-0.689205\pi$
$673$	−29.0562	−1.12003	−0.560017	+	0.828481i	$0.689205\pi$
$674$	0	0
$675$	−5.31837	−0.204704
$676$	0	0
$677$	39.0099	1.49927	0.749635	−	0.661851i	$-0.230229\pi$
$677$	39.0099	1.49927	0.749635	+	0.661851i	$0.230229\pi$
$678$	0	0
$679$	2.49741	0.0958419
$680$	0	0
$681$	−25.5458	−0.978919
$682$	0	0
$683$	−27.9569	−1.06974	−0.534871	−	0.844934i	$-0.679640\pi$
$683$	−27.9569	−1.06974	−0.534871	+	0.844934i	$0.679640\pi$
$684$	0	0
$685$	9.60844	0.367120
$686$	0	0
$687$	−45.2343	−1.72580
$688$	0	0
$689$	3.33706	0.127132
$690$	0	0
$691$	29.5794	1.12526	0.562628	−	0.826710i	$-0.309790\pi$
$691$	29.5794	1.12526	0.562628	+	0.826710i	$0.309790\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.5588	−0.590178
$696$	0	0
$697$	−5.52247	−0.209179
$698$	0	0
$699$	1.42171	0.0537742
$700$	0	0
$701$	12.6712	0.478584	0.239292	−	0.970948i	$-0.423085\pi$
$701$	12.6712	0.478584	0.239292	+	0.970948i	$0.423085\pi$
$702$	0	0
$703$	52.9266	1.99617
$704$	0	0
$705$	9.28918	0.349851
$706$	0	0
$707$	0.193733	0.00728607
$708$	0	0
$709$	−15.6326	−0.587093	−0.293547	−	0.955945i	$-0.594836\pi$
$709$	−15.6326	−0.587093	−0.293547	+	0.955945i	$0.594836\pi$
$710$	0	0
$711$	−0.249294	−0.00934925
$712$	0	0
$713$	15.8449	0.593396
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.83507	0.367297
$718$	0	0
$719$	−42.3705	−1.58015	−0.790077	−	0.613008i	$-0.789959\pi$
$719$	−42.3705	−1.58015	−0.790077	+	0.613008i	$0.789959\pi$
$720$	0	0
$721$	1.78440	0.0664545
$722$	0	0
$723$	−44.8022	−1.66621
$724$	0	0
$725$	2.10646	0.0782318
$726$	0	0
$727$	−22.6871	−0.841417	−0.420709	−	0.907196i	$-0.638219\pi$
$727$	−22.6871	−0.841417	−0.420709	+	0.907196i	$0.638219\pi$
$728$	0	0
$729$	28.2169	1.04507
$730$	0	0
$731$	27.2008	1.00606
$732$	0	0
$733$	−12.5379	−0.463098	−0.231549	−	0.972823i	$-0.574379\pi$
$733$	−12.5379	−0.463098	−0.231549	+	0.972823i	$0.574379\pi$
$734$	0	0
$735$	11.7523	0.433491
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14.5517	0.535293	0.267647	−	0.963517i	$-0.413754\pi$
$739$	14.5517	0.535293	0.267647	+	0.963517i	$0.413754\pi$
$740$	0	0
$741$	−6.06914	−0.222955
$742$	0	0
$743$	3.32326	0.121919	0.0609593	−	0.998140i	$-0.480584\pi$
$743$	3.32326	0.121919	0.0609593	+	0.998140i	$0.480584\pi$
$744$	0	0
$745$	6.06239	0.222109
$746$	0	0
$747$	1.30848	0.0478746
$748$	0	0
$749$	−0.552752	−0.0201971
$750$	0	0
$751$	22.3432	0.815315	0.407658	−	0.913135i	$-0.366346\pi$
$751$	22.3432	0.815315	0.407658	+	0.913135i	$0.366346\pi$
$752$	0	0
$753$	36.0957	1.31540
$754$	0	0
$755$	−17.4419	−0.634775
$756$	0	0
$757$	−24.5701	−0.893016	−0.446508	−	0.894780i	$-0.647333\pi$
$757$	−24.5701	−0.893016	−0.446508	+	0.894780i	$0.647333\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−44.4186	−1.61017	−0.805086	−	0.593158i	$-0.797881\pi$
$761$	−44.4186	−1.61017	−0.805086	+	0.593158i	$0.797881\pi$
$762$	0	0
$763$	−0.238881	−0.00864807
$764$	0	0
$765$	−0.750887	−0.0271484
$766$	0	0
$767$	3.82981	0.138286
$768$	0	0
$769$	−7.95061	−0.286706	−0.143353	−	0.989672i	$-0.545788\pi$
$769$	−7.95061	−0.286706	−0.143353	+	0.989672i	$0.545788\pi$
$770$	0	0
$771$	35.4883	1.27808
$772$	0	0
$773$	−37.0953	−1.33422	−0.667112	−	0.744958i	$-0.732470\pi$
$773$	−37.0953	−1.33422	−0.667112	+	0.744958i	$0.732470\pi$
$774$	0	0
$775$	−6.56591	−0.235855
$776$	0	0
$777$	−2.38632	−0.0856086
$778$	0	0
$779$	8.05064	0.288444
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.2029	−0.400359
$784$	0	0
$785$	−19.0907	−0.681378
$786$	0	0
$787$	−39.7143	−1.41566	−0.707831	−	0.706382i	$-0.750326\pi$
$787$	−39.7143	−1.41566	−0.707831	+	0.706382i	$0.750326\pi$
$788$	0	0
$789$	−2.77945	−0.0989512
$790$	0	0
$791$	1.86537	0.0663251
$792$	0	0
$793$	−6.49804	−0.230752
$794$	0	0
$795$	11.3765	0.403483
$796$	0	0
$797$	19.0325	0.674165	0.337083	−	0.941475i	$-0.390560\pi$
$797$	19.0325	0.674165	0.337083	+	0.941475i	$0.390560\pi$
$798$	0	0
$799$	27.4129	0.969798
$800$	0	0
$801$	−0.277079	−0.00979011
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−0.468082	−0.0164977
$806$	0	0
$807$	27.2688	0.959909
$808$	0	0
$809$	−28.7883	−1.01214	−0.506072	−	0.862491i	$-0.668903\pi$
$809$	−28.7883	−1.01214	−0.506072	+	0.862491i	$0.668903\pi$
$810$	0	0
$811$	29.9629	1.05214	0.526071	−	0.850441i	$-0.323665\pi$
$811$	29.9629	1.05214	0.526071	+	0.850441i	$0.323665\pi$
$812$	0	0
$813$	8.27264	0.290134
$814$	0	0
$815$	0.100999	0.00353786
$816$	0	0
$817$	−39.6532	−1.38729
$818$	0	0
$819$	−0.0144771	−0.000505870 0
$820$	0	0
$821$	−14.5080	−0.506332	−0.253166	−	0.967423i	$-0.581472\pi$
$821$	−14.5080	−0.506332	−0.253166	+	0.967423i	$0.581472\pi$
$822$	0	0
$823$	5.87388	0.204751	0.102375	−	0.994746i	$-0.467356\pi$
$823$	5.87388	0.204751	0.102375	+	0.994746i	$0.467356\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.9979	−0.904034	−0.452017	−	0.892009i	$-0.649295\pi$
$827$	−25.9979	−0.904034	−0.452017	+	0.892009i	$0.649295\pi$
$828$	0	0
$829$	38.0995	1.32325	0.661626	−	0.749834i	$-0.269867\pi$
$829$	38.0995	1.32325	0.661626	+	0.749834i	$0.269867\pi$
$830$	0	0
$831$	24.5491	0.851599
$832$	0	0
$833$	34.6817	1.20165
$834$	0	0
$835$	−11.2432	−0.389086
$836$	0	0
$837$	34.9200	1.20701
$838$	0	0
$839$	−51.1449	−1.76572	−0.882858	−	0.469640i	$-0.844384\pi$
$839$	−51.1449	−1.76572	−0.882858	+	0.469640i	$0.844384\pi$
$840$	0	0
$841$	−24.5628	−0.846995
$842$	0	0
$843$	−5.84737	−0.201394
$844$	0	0
$845$	12.7548	0.438780
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.45650	0.0843069
$850$	0	0
$851$	−17.5885	−0.602926
$852$	0	0
$853$	−16.2720	−0.557142	−0.278571	−	0.960416i	$-0.589861\pi$
$853$	−16.2720	−0.557142	−0.278571	+	0.960416i	$0.589861\pi$
$854$	0	0
$855$	1.09464	0.0374359
$856$	0	0
$857$	14.0851	0.481139	0.240569	−	0.970632i	$-0.422666\pi$
$857$	14.0851	0.481139	0.240569	+	0.970632i	$0.422666\pi$
$858$	0	0
$859$	15.4255	0.526312	0.263156	−	0.964753i	$-0.415237\pi$
$859$	15.4255	0.526312	0.263156	+	0.964753i	$0.415237\pi$
$860$	0	0
$861$	−0.362981	−0.0123704
$862$	0	0
$863$	−0.0688844	−0.00234485	−0.00117243	−	0.999999i	$-0.500373\pi$
$863$	−0.0688844	−0.00234485	−0.00117243	+	0.999999i	$0.500373\pi$
$864$	0	0
$865$	−8.11033	−0.275759
$866$	0	0
$867$	13.1888	0.447915
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.22562	−0.0754122
$872$	0	0
$873$	1.94086	0.0656882
$874$	0	0
$875$	0.193967	0.00655728
$876$	0	0
$877$	−36.8625	−1.24476	−0.622379	−	0.782716i	$-0.713834\pi$
$877$	−36.8625	−1.24476	−0.622379	+	0.782716i	$0.713834\pi$
$878$	0	0
$879$	53.7783	1.81390
$880$	0	0
$881$	10.5610	0.355810	0.177905	−	0.984048i	$-0.443068\pi$
$881$	10.5610	0.355810	0.177905	+	0.984048i	$0.443068\pi$
$882$	0	0
$883$	−23.2630	−0.782862	−0.391431	−	0.920207i	$-0.628020\pi$
$883$	−23.2630	−0.782862	−0.391431	+	0.920207i	$0.628020\pi$
$884$	0	0
$885$	13.0563	0.438884
$886$	0	0
$887$	39.4597	1.32493	0.662464	−	0.749094i	$-0.269511\pi$
$887$	39.4597	1.32493	0.662464	+	0.749094i	$0.269511\pi$
$888$	0	0
$889$	3.30980	0.111007
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−39.9624	−1.33729
$894$	0	0
$895$	21.2782	0.711251
$896$	0	0
$897$	2.01689	0.0673420
$898$	0	0
$899$	−13.8308	−0.461283
$900$	0	0
$901$	33.5727	1.11847
$902$	0	0
$903$	1.78786	0.0594961
$904$	0	0
$905$	−8.62159	−0.286591
$906$	0	0
$907$	−25.3952	−0.843233	−0.421616	−	0.906774i	$-0.638537\pi$
$907$	−25.3952	−0.843233	−0.421616	+	0.906774i	$0.638537\pi$
$908$	0	0
$909$	0.150559	0.00499373
$910$	0	0
$911$	0.482847	0.0159974	0.00799871	−	0.999968i	$-0.497454\pi$
$911$	0.482847	0.0159974	0.00799871	+	0.999968i	$0.497454\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−22.1527	−0.732346
$916$	0	0
$917$	2.10573	0.0695374
$918$	0	0
$919$	30.2756	0.998699	0.499349	−	0.866401i	$-0.333572\pi$
$919$	30.2756	0.998699	0.499349	+	0.866401i	$0.333572\pi$
$920$	0	0
$921$	27.2830	0.899006
$922$	0	0
$923$	−0.188042	−0.00618948
$924$	0	0
$925$	7.28844	0.239642
$926$	0	0
$927$	1.38674	0.0455466
$928$	0	0
$929$	9.58302	0.314409	0.157204	−	0.987566i	$-0.449752\pi$
$929$	9.58302	0.314409	0.157204	+	0.987566i	$0.449752\pi$
$930$	0	0
$931$	−50.5588	−1.65700
$932$	0	0
$933$	38.0288	1.24501
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.7376	0.742805	0.371403	−	0.928472i	$-0.378877\pi$
$937$	22.7376	0.742805	0.371403	+	0.928472i	$0.378877\pi$
$938$	0	0
$939$	29.6184	0.966560
$940$	0	0
$941$	59.4480	1.93795	0.968975	−	0.247160i	$-0.0794974\pi$
$941$	59.4480	1.93795	0.968975	+	0.247160i	$0.0794974\pi$
$942$	0	0
$943$	−2.67538	−0.0871223
$944$	0	0
$945$	−1.03159	−0.0335576
$946$	0	0
$947$	−21.0694	−0.684662	−0.342331	−	0.939579i	$-0.611216\pi$
$947$	−21.0694	−0.684662	−0.342331	+	0.939579i	$0.611216\pi$
$948$	0	0
$949$	7.23948	0.235004
$950$	0	0
$951$	−47.5380	−1.54152
$952$	0	0
$953$	−6.39354	−0.207107	−0.103554	−	0.994624i	$-0.533021\pi$
$953$	−6.39354	−0.207107	−0.103554	+	0.994624i	$0.533021\pi$
$954$	0	0
$955$	20.1441	0.651848
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.86372	0.0601827
$960$	0	0
$961$	12.1112	0.390684
$962$	0	0
$963$	−0.429570	−0.0138427
$964$	0	0
$965$	−9.71381	−0.312699
$966$	0	0
$967$	48.1138	1.54724	0.773618	−	0.633652i	$-0.218445\pi$
$967$	48.1138	1.54724	0.773618	+	0.633652i	$0.218445\pi$
$968$	0	0
$969$	−61.0589	−1.96149
$970$	0	0
$971$	−28.6249	−0.918616	−0.459308	−	0.888277i	$-0.651903\pi$
$971$	−28.6249	−0.918616	−0.459308	+	0.888277i	$0.651903\pi$
$972$	0	0
$973$	−3.01789	−0.0967490
$974$	0	0
$975$	−0.835771	−0.0267661
$976$	0	0
$977$	−3.30958	−0.105883	−0.0529414	−	0.998598i	$-0.516860\pi$
$977$	−3.30958	−0.105883	−0.0529414	+	0.998598i	$0.516860\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−0.185646	−0.00592722
$982$	0	0
$983$	11.4847	0.366305	0.183152	−	0.983085i	$-0.441370\pi$
$983$	11.4847	0.366305	0.183152	+	0.983085i	$0.441370\pi$
$984$	0	0
$985$	14.4934	0.461797
$986$	0	0
$987$	1.80179	0.0573517
$988$	0	0
$989$	13.1775	0.419021
$990$	0	0
$991$	28.8261	0.915693	0.457846	−	0.889031i	$-0.348621\pi$
$991$	28.8261	0.915693	0.457846	+	0.889031i	$0.348621\pi$
$992$	0	0
$993$	1.63779	0.0519737
$994$	0	0
$995$	20.8055	0.659578
$996$	0	0
$997$	25.2674	0.800226	0.400113	−	0.916466i	$-0.368971\pi$
$997$	25.2674	0.800226	0.400113	+	0.916466i	$0.368971\pi$
$998$	0	0
$999$	−38.7626	−1.22640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bb.1.5		6
4.3	odd	2		9680.2.a.dc.1.2		6
11.3	even	5		440.2.y.c.361.1	✓	12
11.4	even	5		440.2.y.c.401.1	yes	12
11.10	odd	2		4840.2.a.ba.1.5		6
44.3	odd	10		880.2.bo.i.801.3		12
44.15	odd	10		880.2.bo.i.401.3		12
44.43	even	2		9680.2.a.dd.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.c.361.1	✓	12	11.3	even	5
440.2.y.c.401.1	yes	12	11.4	even	5
880.2.bo.i.401.3		12	44.15	odd	10
880.2.bo.i.801.3		12	44.3	odd	10
4840.2.a.ba.1.5		6	11.10	odd	2
4840.2.a.bb.1.5		6	1.1	even	1	trivial
9680.2.a.dc.1.2		6	4.3	odd	2
9680.2.a.dd.1.2		6	44.43	even	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68797 q^{3} -1.00000 q^{5} -0.193967 q^{7} -0.150741 q^{9} -0.495133 q^{13} -1.68797 q^{15} -4.98130 q^{17} +7.26172 q^{19} -0.327411 q^{21} -2.41321 q^{23} +1.00000 q^{25} -5.31837 q^{27} +2.10646 q^{29} -6.56591 q^{31} +0.193967 q^{35} +7.28844 q^{37} -0.835771 q^{39} +1.10864 q^{41} -5.46058 q^{43} +0.150741 q^{45} -5.50315 q^{47} -6.96238 q^{49} -8.40832 q^{51} -6.73973 q^{53} +12.2576 q^{57} -7.73492 q^{59} +13.1238 q^{61} +0.0292388 q^{63} +0.495133 q^{65} +4.49499 q^{67} -4.07343 q^{69} +0.379781 q^{71} -14.6213 q^{73} +1.68797 q^{75} +1.65379 q^{79} -8.52505 q^{81} -8.68029 q^{83} +4.98130 q^{85} +3.55564 q^{87} +1.83811 q^{89} +0.0960393 q^{91} -11.0831 q^{93} -7.26172 q^{95} -12.8755 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 + 7 * q^7 + 5 * q^9 - q^13 + 3 * q^15 - 6 * q^17 + 7 * q^19 - 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 - 10 * q^29 + q^31 - 7 * q^35 + 3 * q^37 + q^39 + 6 * q^41 - 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 - 15 * q^51 - 23 * q^53 - 9 * q^57 - 2 * q^59 - 6 * q^61 + 49 * q^63 + q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 - 10 * q^73 - 3 * q^75 + 22 * q^79 + 10 * q^81 - 10 * q^83 + 6 * q^85 + 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 - 7 * q^95 - 33 * q^97

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