LMFDB - Embedded newform 4840.2.a.bh.1.7

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 = [8,0,1,0,8,0,6,0,19,0,0,0,-12] 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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
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			1.7
Root			$3.08163$ 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)

$f(q)$	$=$	$q+3.08163 q^{3} +1.00000 q^{5} -0.0385298 q^{7} +6.49642 q^{9} -3.45332 q^{13} +3.08163 q^{15} -0.897632 q^{17} -1.20724 q^{19} -0.118734 q^{21} +7.60421 q^{23} +1.00000 q^{25} +10.7747 q^{27} +7.56188 q^{29} -5.95574 q^{31} -0.0385298 q^{35} +10.7125 q^{37} -10.6419 q^{39} -3.08927 q^{41} +9.68676 q^{43} +6.49642 q^{45} -1.30605 q^{47} -6.99852 q^{49} -2.76617 q^{51} +2.03218 q^{53} -3.72025 q^{57} -0.185620 q^{59} +9.77911 q^{61} -0.250306 q^{63} -3.45332 q^{65} +8.10586 q^{67} +23.4333 q^{69} -7.18945 q^{71} -12.7078 q^{73} +3.08163 q^{75} +12.6594 q^{79} +13.7142 q^{81} +10.7423 q^{83} -0.897632 q^{85} +23.3029 q^{87} -2.78161 q^{89} +0.133056 q^{91} -18.3534 q^{93} -1.20724 q^{95} +17.7130 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) $ 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 - 12 * q^13 + q^15 - 2 * q^17 + 6 * q^19 - 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 - 8 * q^29 + 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 + 3 * q^41 + 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 - 25 * q^57 + 25 * q^59 - 10 * q^61 + 64 * q^63 - 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 - 38 * q^73 + q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 - 2 * q^85 - 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * 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
$3$	3.08163	1.77918	0.889589	−	0.456762i	$-0.150991\pi$
$3$	3.08163	1.77918	0.889589	+	0.456762i	$0.150991\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0385298	−0.0145629	−0.00728144	−	0.999973i	$-0.502318\pi$
$7$	−0.0385298	−0.0145629	−0.00728144	+	0.999973i	$0.502318\pi$
$8$	0	0
$9$	6.49642	2.16547
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.45332	−0.957780	−0.478890	−	0.877875i	$-0.658961\pi$
$13$	−3.45332	−0.957780	−0.478890	+	0.877875i	$0.658961\pi$
$14$	0	0
$15$	3.08163	0.795672
$16$	0	0
$17$	−0.897632	−0.217708	−0.108854	−	0.994058i	$-0.534718\pi$
$17$	−0.897632	−0.217708	−0.108854	+	0.994058i	$0.534718\pi$
$18$	0	0
$19$	−1.20724	−0.276959	−0.138479	−	0.990365i	$-0.544221\pi$
$19$	−1.20724	−0.276959	−0.138479	+	0.990365i	$0.544221\pi$
$20$	0	0
$21$	−0.118734	−0.0259100
$22$	0	0
$23$	7.60421	1.58559	0.792794	−	0.609490i	$-0.208626\pi$
$23$	7.60421	1.58559	0.792794	+	0.609490i	$0.208626\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.7747	2.07358
$28$	0	0
$29$	7.56188	1.40421	0.702103	−	0.712076i	$-0.252245\pi$
$29$	7.56188	1.40421	0.702103	+	0.712076i	$0.252245\pi$
$30$	0	0
$31$	−5.95574	−1.06968	−0.534841	−	0.844953i	$-0.679628\pi$
$31$	−5.95574	−1.06968	−0.534841	+	0.844953i	$0.679628\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.0385298	−0.00651272
$36$	0	0
$37$	10.7125	1.76112	0.880560	−	0.473935i	$-0.157167\pi$
$37$	10.7125	1.76112	0.880560	+	0.473935i	$0.157167\pi$
$38$	0	0
$39$	−10.6419	−1.70406
$40$	0	0
$41$	−3.08927	−0.482463	−0.241231	−	0.970468i	$-0.577551\pi$
$41$	−3.08927	−0.482463	−0.241231	+	0.970468i	$0.577551\pi$
$42$	0	0
$43$	9.68676	1.47722	0.738609	−	0.674135i	$-0.235483\pi$
$43$	9.68676	1.47722	0.738609	+	0.674135i	$0.235483\pi$
$44$	0	0
$45$	6.49642	0.968429
$46$	0	0
$47$	−1.30605	−0.190507	−0.0952537	−	0.995453i	$-0.530366\pi$
$47$	−1.30605	−0.190507	−0.0952537	+	0.995453i	$0.530366\pi$
$48$	0	0
$49$	−6.99852	−0.999788
$50$	0	0
$51$	−2.76617	−0.387341
$52$	0	0
$53$	2.03218	0.279142	0.139571	−	0.990212i	$-0.455428\pi$
$53$	2.03218	0.279142	0.139571	+	0.990212i	$0.455428\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.72025	−0.492759
$58$	0	0
$59$	−0.185620	−0.0241657	−0.0120828	−	0.999927i	$-0.503846\pi$
$59$	−0.185620	−0.0241657	−0.0120828	+	0.999927i	$0.503846\pi$
$60$	0	0
$61$	9.77911	1.25209	0.626043	−	0.779788i	$-0.284673\pi$
$61$	9.77911	1.25209	0.626043	+	0.779788i	$0.284673\pi$
$62$	0	0
$63$	−0.250306	−0.0315355
$64$	0	0
$65$	−3.45332	−0.428332
$66$	0	0
$67$	8.10586	0.990288	0.495144	−	0.868811i	$-0.335115\pi$
$67$	8.10586	0.990288	0.495144	+	0.868811i	$0.335115\pi$
$68$	0	0
$69$	23.4333	2.82104
$70$	0	0
$71$	−7.18945	−0.853231	−0.426616	−	0.904433i	$-0.640294\pi$
$71$	−7.18945	−0.853231	−0.426616	+	0.904433i	$0.640294\pi$
$72$	0	0
$73$	−12.7078	−1.48733	−0.743665	−	0.668553i	$-0.766914\pi$
$73$	−12.7078	−1.48733	−0.743665	+	0.668553i	$0.766914\pi$
$74$	0	0
$75$	3.08163	0.355836
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6594	1.42430	0.712148	−	0.702029i	$-0.247722\pi$
$79$	12.6594	1.42430	0.712148	+	0.702029i	$0.247722\pi$
$80$	0	0
$81$	13.7142	1.52380
$82$	0	0
$83$	10.7423	1.17912	0.589562	−	0.807723i	$-0.299300\pi$
$83$	10.7423	1.17912	0.589562	+	0.807723i	$0.299300\pi$
$84$	0	0
$85$	−0.897632	−0.0973619
$86$	0	0
$87$	23.3029	2.49833
$88$	0	0
$89$	−2.78161	−0.294850	−0.147425	−	0.989073i	$-0.547099\pi$
$89$	−2.78161	−0.294850	−0.147425	+	0.989073i	$0.547099\pi$
$90$	0	0
$91$	0.133056	0.0139480
$92$	0	0
$93$	−18.3534	−1.90315
$94$	0	0
$95$	−1.20724	−0.123860
$96$	0	0
$97$	17.7130	1.79848	0.899242	−	0.437452i	$-0.144119\pi$
$97$	17.7130	1.79848	0.899242	+	0.437452i	$0.144119\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.57715	0.256436	0.128218	−	0.991746i	$-0.459074\pi$
$101$	2.57715	0.256436	0.128218	+	0.991746i	$0.459074\pi$
$102$	0	0
$103$	3.70321	0.364888	0.182444	−	0.983216i	$-0.441599\pi$
$103$	3.70321	0.364888	0.182444	+	0.983216i	$0.441599\pi$
$104$	0	0
$105$	−0.118734	−0.0115873
$106$	0	0
$107$	−19.2082	−1.85692	−0.928462	−	0.371427i	$-0.878869\pi$
$107$	−19.2082	−1.85692	−0.928462	+	0.371427i	$0.878869\pi$
$108$	0	0
$109$	6.51583	0.624104	0.312052	−	0.950065i	$-0.398984\pi$
$109$	6.51583	0.624104	0.312052	+	0.950065i	$0.398984\pi$
$110$	0	0
$111$	33.0118	3.13334
$112$	0	0
$113$	−12.1185	−1.14001	−0.570006	−	0.821641i	$-0.693059\pi$
$113$	−12.1185	−1.14001	−0.570006	+	0.821641i	$0.693059\pi$
$114$	0	0
$115$	7.60421	0.709096
$116$	0	0
$117$	−22.4342	−2.07405
$118$	0	0
$119$	0.0345856	0.00317045
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−9.51997	−0.858387
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.88142	0.255685	0.127842	−	0.991795i	$-0.459195\pi$
$127$	2.88142	0.255685	0.127842	+	0.991795i	$0.459195\pi$
$128$	0	0
$129$	29.8510	2.62823
$130$	0	0
$131$	−3.67705	−0.321266	−0.160633	−	0.987014i	$-0.551353\pi$
$131$	−3.67705	−0.321266	−0.160633	+	0.987014i	$0.551353\pi$
$132$	0	0
$133$	0.0465145	0.00403332
$134$	0	0
$135$	10.7747	0.927335
$136$	0	0
$137$	−6.40503	−0.547218	−0.273609	−	0.961841i	$-0.588217\pi$
$137$	−6.40503	−0.547218	−0.273609	+	0.961841i	$0.588217\pi$
$138$	0	0
$139$	−9.78896	−0.830288	−0.415144	−	0.909756i	$-0.636269\pi$
$139$	−9.78896	−0.830288	−0.415144	+	0.909756i	$0.636269\pi$
$140$	0	0
$141$	−4.02477	−0.338947
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.56188	0.627980
$146$	0	0
$147$	−21.5668	−1.77880
$148$	0	0
$149$	−19.5271	−1.59972	−0.799862	−	0.600183i	$-0.795094\pi$
$149$	−19.5271	−1.59972	−0.799862	+	0.600183i	$0.795094\pi$
$150$	0	0
$151$	4.51138	0.367131	0.183565	−	0.983007i	$-0.441236\pi$
$151$	4.51138	0.367131	0.183565	+	0.983007i	$0.441236\pi$
$152$	0	0
$153$	−5.83140	−0.471440
$154$	0	0
$155$	−5.95574	−0.478376
$156$	0	0
$157$	4.54155	0.362455	0.181228	−	0.983441i	$-0.441993\pi$
$157$	4.54155	0.362455	0.181228	+	0.983441i	$0.441993\pi$
$158$	0	0
$159$	6.26243	0.496643
$160$	0	0
$161$	−0.292988	−0.0230907
$162$	0	0
$163$	−24.6645	−1.93187	−0.965936	−	0.258780i	$-0.916680\pi$
$163$	−24.6645	−1.93187	−0.965936	+	0.258780i	$0.916680\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.6827	−1.29094	−0.645472	−	0.763784i	$-0.723339\pi$
$167$	−16.6827	−1.29094	−0.645472	+	0.763784i	$0.723339\pi$
$168$	0	0
$169$	−1.07455	−0.0826581
$170$	0	0
$171$	−7.84271	−0.599747
$172$	0	0
$173$	13.2199	1.00509	0.502544	−	0.864552i	$-0.332398\pi$
$173$	13.2199	1.00509	0.502544	+	0.864552i	$0.332398\pi$
$174$	0	0
$175$	−0.0385298	−0.00291258
$176$	0	0
$177$	−0.572012	−0.0429951
$178$	0	0
$179$	20.6661	1.54465	0.772327	−	0.635225i	$-0.219092\pi$
$179$	20.6661	1.54465	0.772327	+	0.635225i	$0.219092\pi$
$180$	0	0
$181$	−3.76425	−0.279795	−0.139897	−	0.990166i	$-0.544677\pi$
$181$	−3.76425	−0.279795	−0.139897	+	0.990166i	$0.544677\pi$
$182$	0	0
$183$	30.1356	2.22768
$184$	0	0
$185$	10.7125	0.787597
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.415145	−0.0301974
$190$	0	0
$191$	−9.47011	−0.685233	−0.342616	−	0.939475i	$-0.611313\pi$
$191$	−9.47011	−0.685233	−0.342616	+	0.939475i	$0.611313\pi$
$192$	0	0
$193$	−16.3716	−1.17845	−0.589225	−	0.807969i	$-0.700567\pi$
$193$	−16.3716	−1.17845	−0.589225	+	0.807969i	$0.700567\pi$
$194$	0	0
$195$	−10.6419	−0.762079
$196$	0	0
$197$	15.1615	1.08021	0.540107	−	0.841596i	$-0.318384\pi$
$197$	15.1615	1.08021	0.540107	+	0.841596i	$0.318384\pi$
$198$	0	0
$199$	2.94883	0.209037	0.104519	−	0.994523i	$-0.466670\pi$
$199$	2.94883	0.209037	0.104519	+	0.994523i	$0.466670\pi$
$200$	0	0
$201$	24.9792	1.76190
$202$	0	0
$203$	−0.291357	−0.0204493
$204$	0	0
$205$	−3.08927	−0.215764
$206$	0	0
$207$	49.4001	3.43355
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−17.4402	−1.20064	−0.600318	−	0.799762i	$-0.704959\pi$
$211$	−17.4402	−1.20064	−0.600318	+	0.799762i	$0.704959\pi$
$212$	0	0
$213$	−22.1552	−1.51805
$214$	0	0
$215$	9.68676	0.660632
$216$	0	0
$217$	0.229473	0.0155777
$218$	0	0
$219$	−39.1605	−2.64622
$220$	0	0
$221$	3.09981	0.208516
$222$	0	0
$223$	−22.8815	−1.53226	−0.766130	−	0.642686i	$-0.777820\pi$
$223$	−22.8815	−1.53226	−0.766130	+	0.642686i	$0.777820\pi$
$224$	0	0
$225$	6.49642	0.433095
$226$	0	0
$227$	−19.8808	−1.31954	−0.659769	−	0.751468i	$-0.729346\pi$
$227$	−19.8808	−1.31954	−0.659769	+	0.751468i	$0.729346\pi$
$228$	0	0
$229$	−15.3967	−1.01744	−0.508721	−	0.860932i	$-0.669881\pi$
$229$	−15.3967	−1.01744	−0.508721	+	0.860932i	$0.669881\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.5192	−0.951185	−0.475592	−	0.879666i	$-0.657766\pi$
$233$	−14.5192	−0.951185	−0.475592	+	0.879666i	$0.657766\pi$
$234$	0	0
$235$	−1.30605	−0.0851975
$236$	0	0
$237$	39.0116	2.53408
$238$	0	0
$239$	−5.26252	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$239$	−5.26252	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$240$	0	0
$241$	−10.2035	−0.657265	−0.328633	−	0.944458i	$-0.606588\pi$
$241$	−10.2035	−0.657265	−0.328633	+	0.944458i	$0.606588\pi$
$242$	0	0
$243$	9.93811	0.637530
$244$	0	0
$245$	−6.99852	−0.447119
$246$	0	0
$247$	4.16897	0.265266
$248$	0	0
$249$	33.1038	2.09787
$250$	0	0
$251$	8.52003	0.537780	0.268890	−	0.963171i	$-0.413343\pi$
$251$	8.52003	0.537780	0.268890	+	0.963171i	$0.413343\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.76617	−0.173224
$256$	0	0
$257$	11.5515	0.720565	0.360282	−	0.932843i	$-0.382680\pi$
$257$	11.5515	0.720565	0.360282	+	0.932843i	$0.382680\pi$
$258$	0	0
$259$	−0.412749	−0.0256470
$260$	0	0
$261$	49.1251	3.04077
$262$	0	0
$263$	2.91262	0.179600	0.0898000	−	0.995960i	$-0.471377\pi$
$263$	2.91262	0.179600	0.0898000	+	0.995960i	$0.471377\pi$
$264$	0	0
$265$	2.03218	0.124836
$266$	0	0
$267$	−8.57188	−0.524591
$268$	0	0
$269$	−0.306960	−0.0187157	−0.00935784	−	0.999956i	$-0.502979\pi$
$269$	−0.306960	−0.0187157	−0.00935784	+	0.999956i	$0.502979\pi$
$270$	0	0
$271$	16.4190	0.997382	0.498691	−	0.866780i	$-0.333814\pi$
$271$	16.4190	0.997382	0.498691	+	0.866780i	$0.333814\pi$
$272$	0	0
$273$	0.410028	0.0248160
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.11565	0.187201	0.0936007	−	0.995610i	$-0.470162\pi$
$277$	3.11565	0.187201	0.0936007	+	0.995610i	$0.470162\pi$
$278$	0	0
$279$	−38.6910	−2.31637
$280$	0	0
$281$	25.6494	1.53012	0.765058	−	0.643961i	$-0.222710\pi$
$281$	25.6494	1.53012	0.765058	+	0.643961i	$0.222710\pi$
$282$	0	0
$283$	20.3137	1.20752	0.603761	−	0.797165i	$-0.293668\pi$
$283$	20.3137	1.20752	0.603761	+	0.797165i	$0.293668\pi$
$284$	0	0
$285$	−3.72025	−0.220368
$286$	0	0
$287$	0.119029	0.00702605
$288$	0	0
$289$	−16.1943	−0.952603
$290$	0	0
$291$	54.5849	3.19982
$292$	0	0
$293$	−30.1012	−1.75853	−0.879266	−	0.476331i	$-0.841966\pi$
$293$	−30.1012	−1.75853	−0.879266	+	0.476331i	$0.841966\pi$
$294$	0	0
$295$	−0.185620	−0.0108072
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−26.2598	−1.51864
$300$	0	0
$301$	−0.373229	−0.0215125
$302$	0	0
$303$	7.94180	0.456244
$304$	0	0
$305$	9.77911	0.559950
$306$	0	0
$307$	5.73211	0.327149	0.163574	−	0.986531i	$-0.447698\pi$
$307$	5.73211	0.327149	0.163574	+	0.986531i	$0.447698\pi$
$308$	0	0
$309$	11.4119	0.649200
$310$	0	0
$311$	−26.2777	−1.49007	−0.745035	−	0.667025i	$-0.767567\pi$
$311$	−26.2777	−1.49007	−0.745035	+	0.667025i	$0.767567\pi$
$312$	0	0
$313$	10.1095	0.571421	0.285711	−	0.958316i	$-0.407770\pi$
$313$	10.1095	0.571421	0.285711	+	0.958316i	$0.407770\pi$
$314$	0	0
$315$	−0.250306	−0.0141031
$316$	0	0
$317$	3.56813	0.200406	0.100203	−	0.994967i	$-0.468051\pi$
$317$	3.56813	0.200406	0.100203	+	0.994967i	$0.468051\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−59.1924	−3.30380
$322$	0	0
$323$	1.08365	0.0602961
$324$	0	0
$325$	−3.45332	−0.191556
$326$	0	0
$327$	20.0794	1.11039
$328$	0	0
$329$	0.0503219	0.00277434
$330$	0	0
$331$	8.06105	0.443075	0.221538	−	0.975152i	$-0.428892\pi$
$331$	8.06105	0.443075	0.221538	+	0.975152i	$0.428892\pi$
$332$	0	0
$333$	69.5927	3.81366
$334$	0	0
$335$	8.10586	0.442870
$336$	0	0
$337$	−27.3424	−1.48944	−0.744719	−	0.667379i	$-0.767416\pi$
$337$	−27.3424	−1.48944	−0.744719	+	0.667379i	$0.767416\pi$
$338$	0	0
$339$	−37.3447	−2.02828
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.539360	0.0291227
$344$	0	0
$345$	23.4333	1.26161
$346$	0	0
$347$	30.8345	1.65528	0.827642	−	0.561256i	$-0.189682\pi$
$347$	30.8345	1.65528	0.827642	+	0.561256i	$0.189682\pi$
$348$	0	0
$349$	23.7919	1.27355	0.636776	−	0.771049i	$-0.280268\pi$
$349$	23.7919	1.27355	0.636776	+	0.771049i	$0.280268\pi$
$350$	0	0
$351$	−37.2084	−1.98604
$352$	0	0
$353$	2.87310	0.152920	0.0764598	−	0.997073i	$-0.475638\pi$
$353$	2.87310	0.152920	0.0764598	+	0.997073i	$0.475638\pi$
$354$	0	0
$355$	−7.18945	−0.381577
$356$	0	0
$357$	0.106580	0.00564080
$358$	0	0
$359$	1.76912	0.0933708	0.0466854	−	0.998910i	$-0.485134\pi$
$359$	1.76912	0.0933708	0.0466854	+	0.998910i	$0.485134\pi$
$360$	0	0
$361$	−17.5426	−0.923294
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.7078	−0.665154
$366$	0	0
$367$	5.71808	0.298481	0.149241	−	0.988801i	$-0.452317\pi$
$367$	5.71808	0.298481	0.149241	+	0.988801i	$0.452317\pi$
$368$	0	0
$369$	−20.0692	−1.04476
$370$	0	0
$371$	−0.0782996	−0.00406511
$372$	0	0
$373$	−18.1456	−0.939543	−0.469772	−	0.882788i	$-0.655664\pi$
$373$	−18.1456	−0.939543	−0.469772	+	0.882788i	$0.655664\pi$
$374$	0	0
$375$	3.08163	0.159134
$376$	0	0
$377$	−26.1136	−1.34492
$378$	0	0
$379$	−31.5044	−1.61827	−0.809136	−	0.587622i	$-0.800064\pi$
$379$	−31.5044	−1.61827	−0.809136	+	0.587622i	$0.800064\pi$
$380$	0	0
$381$	8.87946	0.454908
$382$	0	0
$383$	2.88227	0.147277	0.0736386	−	0.997285i	$-0.476539\pi$
$383$	2.88227	0.147277	0.0736386	+	0.997285i	$0.476539\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	62.9293	3.19887
$388$	0	0
$389$	−34.2856	−1.73835	−0.869174	−	0.494506i	$-0.835349\pi$
$389$	−34.2856	−1.73835	−0.869174	+	0.494506i	$0.835349\pi$
$390$	0	0
$391$	−6.82578	−0.345195
$392$	0	0
$393$	−11.3313	−0.571589
$394$	0	0
$395$	12.6594	0.636965
$396$	0	0
$397$	−14.4405	−0.724746	−0.362373	−	0.932033i	$-0.618033\pi$
$397$	−14.4405	−0.724746	−0.362373	+	0.932033i	$0.618033\pi$
$398$	0	0
$399$	0.143340	0.00717599
$400$	0	0
$401$	4.44966	0.222206	0.111103	−	0.993809i	$-0.464562\pi$
$401$	4.44966	0.222206	0.111103	+	0.993809i	$0.464562\pi$
$402$	0	0
$403$	20.5671	1.02452
$404$	0	0
$405$	13.7142	0.681465
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−30.4633	−1.50631	−0.753157	−	0.657841i	$-0.771470\pi$
$409$	−30.4633	−1.50631	−0.753157	+	0.657841i	$0.771470\pi$
$410$	0	0
$411$	−19.7379	−0.973599
$412$	0	0
$413$	0.00715190	0.000351922 0
$414$	0	0
$415$	10.7423	0.527320
$416$	0	0
$417$	−30.1659	−1.47723
$418$	0	0
$419$	−14.5620	−0.711402	−0.355701	−	0.934600i	$-0.615758\pi$
$419$	−14.5620	−0.711402	−0.355701	+	0.934600i	$0.615758\pi$
$420$	0	0
$421$	0.651674	0.0317607	0.0158803	−	0.999874i	$-0.494945\pi$
$421$	0.651674	0.0317607	0.0158803	+	0.999874i	$0.494945\pi$
$422$	0	0
$423$	−8.48467	−0.412539
$424$	0	0
$425$	−0.897632	−0.0435416
$426$	0	0
$427$	−0.376787	−0.0182340
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.9668	1.63612	0.818062	−	0.575131i	$-0.195049\pi$
$431$	33.9668	1.63612	0.818062	+	0.575131i	$0.195049\pi$
$432$	0	0
$433$	−31.8266	−1.52949	−0.764743	−	0.644335i	$-0.777134\pi$
$433$	−31.8266	−1.52949	−0.764743	+	0.644335i	$0.777134\pi$
$434$	0	0
$435$	23.3029	1.11729
$436$	0	0
$437$	−9.18007	−0.439142
$438$	0	0
$439$	1.74428	0.0832501	0.0416251	−	0.999133i	$-0.486747\pi$
$439$	1.74428	0.0832501	0.0416251	+	0.999133i	$0.486747\pi$
$440$	0	0
$441$	−45.4653	−2.16501
$442$	0	0
$443$	0.920539	0.0437361	0.0218681	−	0.999761i	$-0.493039\pi$
$443$	0.920539	0.0437361	0.0218681	+	0.999761i	$0.493039\pi$
$444$	0	0
$445$	−2.78161	−0.131861
$446$	0	0
$447$	−60.1753	−2.84619
$448$	0	0
$449$	5.00340	0.236125	0.118062	−	0.993006i	$-0.462332\pi$
$449$	5.00340	0.236125	0.118062	+	0.993006i	$0.462332\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.9024	0.653191
$454$	0	0
$455$	0.133056	0.00623775
$456$	0	0
$457$	10.9781	0.513532	0.256766	−	0.966474i	$-0.417343\pi$
$457$	10.9781	0.513532	0.256766	+	0.966474i	$0.417343\pi$
$458$	0	0
$459$	−9.67168	−0.451436
$460$	0	0
$461$	8.89371	0.414221	0.207111	−	0.978318i	$-0.433594\pi$
$461$	8.89371	0.414221	0.207111	+	0.978318i	$0.433594\pi$
$462$	0	0
$463$	−8.95007	−0.415945	−0.207972	−	0.978135i	$-0.566686\pi$
$463$	−8.95007	−0.415945	−0.207972	+	0.978135i	$0.566686\pi$
$464$	0	0
$465$	−18.3534	−0.851117
$466$	0	0
$467$	12.1931	0.564229	0.282114	−	0.959381i	$-0.408964\pi$
$467$	12.1931	0.564229	0.282114	+	0.959381i	$0.408964\pi$
$468$	0	0
$469$	−0.312317	−0.0144215
$470$	0	0
$471$	13.9954	0.644873
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.20724	−0.0553918
$476$	0	0
$477$	13.2019	0.604474
$478$	0	0
$479$	42.6446	1.94848	0.974241	−	0.225508i	$-0.0724042\pi$
$479$	42.6446	1.94848	0.974241	+	0.225508i	$0.0724042\pi$
$480$	0	0
$481$	−36.9936	−1.68676
$482$	0	0
$483$	−0.902881	−0.0410825
$484$	0	0
$485$	17.7130	0.804306
$486$	0	0
$487$	11.0636	0.501342	0.250671	−	0.968072i	$-0.419349\pi$
$487$	11.0636	0.501342	0.250671	+	0.968072i	$0.419349\pi$
$488$	0	0
$489$	−76.0068	−3.43714
$490$	0	0
$491$	19.6823	0.888248	0.444124	−	0.895965i	$-0.353515\pi$
$491$	19.6823	0.888248	0.444124	+	0.895965i	$0.353515\pi$
$492$	0	0
$493$	−6.78778	−0.305706
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.277008	0.0124255
$498$	0	0
$499$	−11.2051	−0.501609	−0.250804	−	0.968038i	$-0.580695\pi$
$499$	−11.2051	−0.501609	−0.250804	+	0.968038i	$0.580695\pi$
$500$	0	0
$501$	−51.4098	−2.29682
$502$	0	0
$503$	25.3693	1.13116	0.565580	−	0.824693i	$-0.308652\pi$
$503$	25.3693	1.13116	0.565580	+	0.824693i	$0.308652\pi$
$504$	0	0
$505$	2.57715	0.114681
$506$	0	0
$507$	−3.31138	−0.147063
$508$	0	0
$509$	−37.3796	−1.65682	−0.828410	−	0.560122i	$-0.810754\pi$
$509$	−37.3796	−1.65682	−0.828410	+	0.560122i	$0.810754\pi$
$510$	0	0
$511$	0.489627	0.0216598
$512$	0	0
$513$	−13.0076	−0.574297
$514$	0	0
$515$	3.70321	0.163183
$516$	0	0
$517$	0	0
$518$	0	0
$519$	40.7387	1.78823
$520$	0	0
$521$	26.5085	1.16136	0.580678	−	0.814133i	$-0.302787\pi$
$521$	26.5085	1.16136	0.580678	+	0.814133i	$0.302787\pi$
$522$	0	0
$523$	−18.3371	−0.801827	−0.400914	−	0.916116i	$-0.631307\pi$
$523$	−18.3371	−0.801827	−0.400914	+	0.916116i	$0.631307\pi$
$524$	0	0
$525$	−0.118734	−0.00518199
$526$	0	0
$527$	5.34606	0.232878
$528$	0	0
$529$	34.8240	1.51409
$530$	0	0
$531$	−1.20587	−0.0523302
$532$	0	0
$533$	10.6682	0.462093
$534$	0	0
$535$	−19.2082	−0.830442
$536$	0	0
$537$	63.6851	2.74822
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.70731	−0.245376	−0.122688	−	0.992445i	$-0.539152\pi$
$541$	−5.70731	−0.245376	−0.122688	+	0.992445i	$0.539152\pi$
$542$	0	0
$543$	−11.6000	−0.497805
$544$	0	0
$545$	6.51583	0.279108
$546$	0	0
$547$	−13.6622	−0.584152	−0.292076	−	0.956395i	$-0.594346\pi$
$547$	−13.6622	−0.584152	−0.292076	+	0.956395i	$0.594346\pi$
$548$	0	0
$549$	63.5292	2.71136
$550$	0	0
$551$	−9.12897	−0.388907
$552$	0	0
$553$	−0.487765	−0.0207419
$554$	0	0
$555$	33.0118	1.40127
$556$	0	0
$557$	−14.0848	−0.596790	−0.298395	−	0.954442i	$-0.596451\pi$
$557$	−14.0848	−0.596790	−0.298395	+	0.954442i	$0.596451\pi$
$558$	0	0
$559$	−33.4515	−1.41485
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.32334	0.350787	0.175394	−	0.984498i	$-0.443880\pi$
$563$	8.32334	0.350787	0.175394	+	0.984498i	$0.443880\pi$
$564$	0	0
$565$	−12.1185	−0.509829
$566$	0	0
$567$	−0.528406	−0.0221909
$568$	0	0
$569$	−17.1243	−0.717888	−0.358944	−	0.933359i	$-0.616863\pi$
$569$	−17.1243	−0.717888	−0.358944	+	0.933359i	$0.616863\pi$
$570$	0	0
$571$	33.6964	1.41015	0.705076	−	0.709132i	$-0.250913\pi$
$571$	33.6964	1.41015	0.705076	+	0.709132i	$0.250913\pi$
$572$	0	0
$573$	−29.1833	−1.21915
$574$	0	0
$575$	7.60421	0.317117
$576$	0	0
$577$	−30.4030	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$577$	−30.4030	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$578$	0	0
$579$	−50.4510	−2.09667
$580$	0	0
$581$	−0.413899	−0.0171714
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−22.4342	−0.927542
$586$	0	0
$587$	−9.38173	−0.387226	−0.193613	−	0.981078i	$-0.562021\pi$
$587$	−9.38173	−0.387226	−0.193613	+	0.981078i	$0.562021\pi$
$588$	0	0
$589$	7.18998	0.296258
$590$	0	0
$591$	46.7222	1.92189
$592$	0	0
$593$	−30.1444	−1.23788	−0.618941	−	0.785437i	$-0.712438\pi$
$593$	−30.1444	−1.23788	−0.618941	+	0.785437i	$0.712438\pi$
$594$	0	0
$595$	0.0345856	0.00141787
$596$	0	0
$597$	9.08720	0.371914
$598$	0	0
$599$	10.2032	0.416891	0.208446	−	0.978034i	$-0.433160\pi$
$599$	10.2032	0.416891	0.208446	+	0.978034i	$0.433160\pi$
$600$	0	0
$601$	17.5371	0.715353	0.357676	−	0.933846i	$-0.383569\pi$
$601$	17.5371	0.715353	0.357676	+	0.933846i	$0.383569\pi$
$602$	0	0
$603$	52.6591	2.14444
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.32563	0.175572	0.0877859	−	0.996139i	$-0.472021\pi$
$607$	4.32563	0.175572	0.0877859	+	0.996139i	$0.472021\pi$
$608$	0	0
$609$	−0.897854	−0.0363829
$610$	0	0
$611$	4.51022	0.182464
$612$	0	0
$613$	−12.5463	−0.506741	−0.253370	−	0.967369i	$-0.581539\pi$
$613$	−12.5463	−0.506741	−0.253370	+	0.967369i	$0.581539\pi$
$614$	0	0
$615$	−9.51997	−0.383882
$616$	0	0
$617$	17.1256	0.689452	0.344726	−	0.938703i	$-0.387972\pi$
$617$	17.1256	0.689452	0.344726	+	0.938703i	$0.387972\pi$
$618$	0	0
$619$	−7.29199	−0.293090	−0.146545	−	0.989204i	$-0.546815\pi$
$619$	−7.29199	−0.293090	−0.146545	+	0.989204i	$0.546815\pi$
$620$	0	0
$621$	81.9328	3.28785
$622$	0	0
$623$	0.107175	0.00429387
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.61586	−0.383409
$630$	0	0
$631$	27.5808	1.09797	0.548987	−	0.835831i	$-0.315014\pi$
$631$	27.5808	1.09797	0.548987	+	0.835831i	$0.315014\pi$
$632$	0	0
$633$	−53.7443	−2.13614
$634$	0	0
$635$	2.88142	0.114346
$636$	0	0
$637$	24.1681	0.957577
$638$	0	0
$639$	−46.7057	−1.84765
$640$	0	0
$641$	24.6956	0.975417	0.487708	−	0.873007i	$-0.337833\pi$
$641$	24.6956	0.975417	0.487708	+	0.873007i	$0.337833\pi$
$642$	0	0
$643$	−25.7228	−1.01441	−0.507205	−	0.861825i	$-0.669321\pi$
$643$	−25.7228	−1.01441	−0.507205	+	0.861825i	$0.669321\pi$
$644$	0	0
$645$	29.8510	1.17538
$646$	0	0
$647$	9.16022	0.360125	0.180063	−	0.983655i	$-0.442370\pi$
$647$	9.16022	0.360125	0.180063	+	0.983655i	$0.442370\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.707151	0.0277154
$652$	0	0
$653$	19.9381	0.780238	0.390119	−	0.920764i	$-0.372434\pi$
$653$	19.9381	0.780238	0.390119	+	0.920764i	$0.372434\pi$
$654$	0	0
$655$	−3.67705	−0.143674
$656$	0	0
$657$	−82.5549	−3.22077
$658$	0	0
$659$	21.8117	0.849662	0.424831	−	0.905273i	$-0.360333\pi$
$659$	21.8117	0.849662	0.424831	+	0.905273i	$0.360333\pi$
$660$	0	0
$661$	32.2443	1.25416	0.627079	−	0.778955i	$-0.284250\pi$
$661$	32.2443	1.25416	0.627079	+	0.778955i	$0.284250\pi$
$662$	0	0
$663$	9.55247	0.370987
$664$	0	0
$665$	0.0465145	0.00180376
$666$	0	0
$667$	57.5021	2.22649
$668$	0	0
$669$	−70.5123	−2.72616
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.74689	0.0673378	0.0336689	−	0.999433i	$-0.489281\pi$
$673$	1.74689	0.0673378	0.0336689	+	0.999433i	$0.489281\pi$
$674$	0	0
$675$	10.7747	0.414717
$676$	0	0
$677$	−8.13336	−0.312590	−0.156295	−	0.987710i	$-0.549955\pi$
$677$	−8.13336	−0.312590	−0.156295	+	0.987710i	$0.549955\pi$
$678$	0	0
$679$	−0.682478	−0.0261911
$680$	0	0
$681$	−61.2653	−2.34769
$682$	0	0
$683$	24.4004	0.933655	0.466827	−	0.884348i	$-0.345397\pi$
$683$	24.4004	0.933655	0.466827	+	0.884348i	$0.345397\pi$
$684$	0	0
$685$	−6.40503	−0.244724
$686$	0	0
$687$	−47.4468	−1.81021
$688$	0	0
$689$	−7.01779	−0.267356
$690$	0	0
$691$	18.0728	0.687523	0.343762	−	0.939057i	$-0.388299\pi$
$691$	18.0728	0.687523	0.343762	+	0.939057i	$0.388299\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.78896	−0.371316
$696$	0	0
$697$	2.77303	0.105036
$698$	0	0
$699$	−44.7428	−1.69233
$700$	0	0
$701$	−16.2777	−0.614799	−0.307399	−	0.951581i	$-0.599459\pi$
$701$	−16.2777	−0.614799	−0.307399	+	0.951581i	$0.599459\pi$
$702$	0	0
$703$	−12.9325	−0.487758
$704$	0	0
$705$	−4.02477	−0.151582
$706$	0	0
$707$	−0.0992968	−0.00373444
$708$	0	0
$709$	−33.7908	−1.26904	−0.634521	−	0.772906i	$-0.718802\pi$
$709$	−33.7908	−1.26904	−0.634521	+	0.772906i	$0.718802\pi$
$710$	0	0
$711$	82.2409	3.08428
$712$	0	0
$713$	−45.2887	−1.69607
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.2171	−0.605639
$718$	0	0
$719$	23.5131	0.876890	0.438445	−	0.898758i	$-0.355529\pi$
$719$	23.5131	0.876890	0.438445	+	0.898758i	$0.355529\pi$
$720$	0	0
$721$	−0.142684	−0.00531382
$722$	0	0
$723$	−31.4434	−1.16939
$724$	0	0
$725$	7.56188	0.280841
$726$	0	0
$727$	2.84884	0.105658	0.0528288	−	0.998604i	$-0.483176\pi$
$727$	2.84884	0.105658	0.0528288	+	0.998604i	$0.483176\pi$
$728$	0	0
$729$	−10.5171	−0.389523
$730$	0	0
$731$	−8.69515	−0.321602
$732$	0	0
$733$	25.2162	0.931381	0.465691	−	0.884948i	$-0.345806\pi$
$733$	25.2162	0.931381	0.465691	+	0.884948i	$0.345806\pi$
$734$	0	0
$735$	−21.5668	−0.795504
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−36.3847	−1.33843	−0.669216	−	0.743068i	$-0.733370\pi$
$739$	−36.3847	−1.33843	−0.669216	+	0.743068i	$0.733370\pi$
$740$	0	0
$741$	12.8472	0.471955
$742$	0	0
$743$	24.6526	0.904414	0.452207	−	0.891913i	$-0.350637\pi$
$743$	24.6526	0.904414	0.452207	+	0.891913i	$0.350637\pi$
$744$	0	0
$745$	−19.5271	−0.715419
$746$	0	0
$747$	69.7867	2.55336
$748$	0	0
$749$	0.740087	0.0270422
$750$	0	0
$751$	39.4928	1.44111	0.720557	−	0.693396i	$-0.243886\pi$
$751$	39.4928	1.44111	0.720557	+	0.693396i	$0.243886\pi$
$752$	0	0
$753$	26.2556	0.956806
$754$	0	0
$755$	4.51138	0.164186
$756$	0	0
$757$	−9.31713	−0.338637	−0.169318	−	0.985561i	$-0.554157\pi$
$757$	−9.31713	−0.338637	−0.169318	+	0.985561i	$0.554157\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.425026	0.0154072	0.00770359	−	0.999970i	$-0.497548\pi$
$761$	0.425026	0.0154072	0.00770359	+	0.999970i	$0.497548\pi$
$762$	0	0
$763$	−0.251054	−0.00908875
$764$	0	0
$765$	−5.83140	−0.210835
$766$	0	0
$767$	0.641007	0.0231454
$768$	0	0
$769$	−5.37492	−0.193825	−0.0969123	−	0.995293i	$-0.530897\pi$
$769$	−5.37492	−0.193825	−0.0969123	+	0.995293i	$0.530897\pi$
$770$	0	0
$771$	35.5975	1.28201
$772$	0	0
$773$	3.38581	0.121779	0.0608896	−	0.998145i	$-0.480606\pi$
$773$	3.38581	0.121779	0.0608896	+	0.998145i	$0.480606\pi$
$774$	0	0
$775$	−5.95574	−0.213936
$776$	0	0
$777$	−1.27194	−0.0456305
$778$	0	0
$779$	3.72948	0.133622
$780$	0	0
$781$	0	0
$782$	0	0
$783$	81.4767	2.91174
$784$	0	0
$785$	4.54155	0.162095
$786$	0	0
$787$	36.5717	1.30364	0.651820	−	0.758373i	$-0.274006\pi$
$787$	36.5717	1.30364	0.651820	+	0.758373i	$0.274006\pi$
$788$	0	0
$789$	8.97561	0.319540
$790$	0	0
$791$	0.466923	0.0166019
$792$	0	0
$793$	−33.7704	−1.19922
$794$	0	0
$795$	6.26243	0.222106
$796$	0	0
$797$	26.4768	0.937855	0.468928	−	0.883237i	$-0.344641\pi$
$797$	26.4768	0.937855	0.468928	+	0.883237i	$0.344641\pi$
$798$	0	0
$799$	1.17236	0.0414750
$800$	0	0
$801$	−18.0705	−0.638490
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−0.292988	−0.0103265
$806$	0	0
$807$	−0.945936	−0.0332985
$808$	0	0
$809$	−11.4341	−0.402003	−0.201001	−	0.979591i	$-0.564420\pi$
$809$	−11.4341	−0.402003	−0.201001	+	0.979591i	$0.564420\pi$
$810$	0	0
$811$	−4.43533	−0.155746	−0.0778728	−	0.996963i	$-0.524813\pi$
$811$	−4.43533	−0.155746	−0.0778728	+	0.996963i	$0.524813\pi$
$812$	0	0
$813$	50.5972	1.77452
$814$	0	0
$815$	−24.6645	−0.863960
$816$	0	0
$817$	−11.6942	−0.409128
$818$	0	0
$819$	0.864386	0.0302041
$820$	0	0
$821$	−48.2156	−1.68273	−0.841367	−	0.540463i	$-0.818249\pi$
$821$	−48.2156	−1.68273	−0.841367	+	0.540463i	$0.818249\pi$
$822$	0	0
$823$	42.1249	1.46838	0.734191	−	0.678943i	$-0.237562\pi$
$823$	42.1249	1.46838	0.734191	+	0.678943i	$0.237562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.0987	1.35959	0.679797	−	0.733401i	$-0.262068\pi$
$827$	39.0987	1.35959	0.679797	+	0.733401i	$0.262068\pi$
$828$	0	0
$829$	28.0646	0.974725	0.487363	−	0.873200i	$-0.337959\pi$
$829$	28.0646	0.974725	0.487363	+	0.873200i	$0.337959\pi$
$830$	0	0
$831$	9.60127	0.333064
$832$	0	0
$833$	6.28209	0.217662
$834$	0	0
$835$	−16.6827	−0.577328
$836$	0	0
$837$	−64.1710	−2.21808
$838$	0	0
$839$	−34.3628	−1.18634	−0.593168	−	0.805079i	$-0.702123\pi$
$839$	−34.3628	−1.18634	−0.593168	+	0.805079i	$0.702123\pi$
$840$	0	0
$841$	28.1820	0.971792
$842$	0	0
$843$	79.0419	2.72235
$844$	0	0
$845$	−1.07455	−0.0369658
$846$	0	0
$847$	0	0
$848$	0	0
$849$	62.5992	2.14840
$850$	0	0
$851$	81.4599	2.79241
$852$	0	0
$853$	16.3946	0.561339	0.280669	−	0.959804i	$-0.409444\pi$
$853$	16.3946	0.561339	0.280669	+	0.959804i	$0.409444\pi$
$854$	0	0
$855$	−7.84271	−0.268215
$856$	0	0
$857$	6.89777	0.235623	0.117812	−	0.993036i	$-0.462412\pi$
$857$	6.89777	0.235623	0.117812	+	0.993036i	$0.462412\pi$
$858$	0	0
$859$	−1.17514	−0.0400954	−0.0200477	−	0.999799i	$-0.506382\pi$
$859$	−1.17514	−0.0400954	−0.0200477	+	0.999799i	$0.506382\pi$
$860$	0	0
$861$	0.366802	0.0125006
$862$	0	0
$863$	−9.66842	−0.329117	−0.164558	−	0.986367i	$-0.552620\pi$
$863$	−9.66842	−0.329117	−0.164558	+	0.986367i	$0.552620\pi$
$864$	0	0
$865$	13.2199	0.449489
$866$	0	0
$867$	−49.9046	−1.69485
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−27.9922	−0.948478
$872$	0	0
$873$	115.071	3.89457
$874$	0	0
$875$	−0.0385298	−0.00130254
$876$	0	0
$877$	−6.09017	−0.205650	−0.102825	−	0.994699i	$-0.532788\pi$
$877$	−6.09017	−0.205650	−0.102825	+	0.994699i	$0.532788\pi$
$878$	0	0
$879$	−92.7607	−3.12874
$880$	0	0
$881$	55.4058	1.86667	0.933335	−	0.359008i	$-0.116885\pi$
$881$	55.4058	1.86667	0.933335	+	0.359008i	$0.116885\pi$
$882$	0	0
$883$	−40.0880	−1.34907	−0.674534	−	0.738244i	$-0.735655\pi$
$883$	−40.0880	−1.34907	−0.674534	+	0.738244i	$0.735655\pi$
$884$	0	0
$885$	−0.572012	−0.0192280
$886$	0	0
$887$	−35.5736	−1.19445	−0.597223	−	0.802075i	$-0.703729\pi$
$887$	−35.5736	−1.19445	−0.597223	+	0.802075i	$0.703729\pi$
$888$	0	0
$889$	−0.111020	−0.00372350
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.57671	0.0527627
$894$	0	0
$895$	20.6661	0.690791
$896$	0	0
$897$	−80.9229	−2.70194
$898$	0	0
$899$	−45.0366	−1.50205
$900$	0	0
$901$	−1.82415	−0.0607714
$902$	0	0
$903$	−1.15015	−0.0382746
$904$	0	0
$905$	−3.76425	−0.125128
$906$	0	0
$907$	12.2245	0.405907	0.202954	−	0.979188i	$-0.434946\pi$
$907$	12.2245	0.405907	0.202954	+	0.979188i	$0.434946\pi$
$908$	0	0
$909$	16.7422	0.555304
$910$	0	0
$911$	−17.1201	−0.567214	−0.283607	−	0.958941i	$-0.591531\pi$
$911$	−17.1201	−0.567214	−0.283607	+	0.958941i	$0.591531\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	30.1356	0.996251
$916$	0	0
$917$	0.141676	0.00467855
$918$	0	0
$919$	30.8276	1.01691	0.508454	−	0.861089i	$-0.330217\pi$
$919$	30.8276	1.01691	0.508454	+	0.861089i	$0.330217\pi$
$920$	0	0
$921$	17.6642	0.582056
$922$	0	0
$923$	24.8275	0.817208
$924$	0	0
$925$	10.7125	0.352224
$926$	0	0
$927$	24.0576	0.790155
$928$	0	0
$929$	7.29211	0.239246	0.119623	−	0.992819i	$-0.461831\pi$
$929$	7.29211	0.239246	0.119623	+	0.992819i	$0.461831\pi$
$930$	0	0
$931$	8.44886	0.276900
$932$	0	0
$933$	−80.9780	−2.65110
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.4143	−0.862917	−0.431458	−	0.902133i	$-0.642001\pi$
$937$	−26.4143	−0.862917	−0.431458	+	0.902133i	$0.642001\pi$
$938$	0	0
$939$	31.1536	1.01666
$940$	0	0
$941$	−54.6703	−1.78220	−0.891100	−	0.453807i	$-0.850065\pi$
$941$	−54.6703	−1.78220	−0.891100	+	0.453807i	$0.850065\pi$
$942$	0	0
$943$	−23.4915	−0.764987
$944$	0	0
$945$	−0.415145	−0.0135047
$946$	0	0
$947$	6.86882	0.223207	0.111603	−	0.993753i	$-0.464401\pi$
$947$	6.86882	0.223207	0.111603	+	0.993753i	$0.464401\pi$
$948$	0	0
$949$	43.8840	1.42453
$950$	0	0
$951$	10.9956	0.356558
$952$	0	0
$953$	−43.5319	−1.41014	−0.705068	−	0.709140i	$-0.749083\pi$
$953$	−43.5319	−1.41014	−0.705068	+	0.709140i	$0.749083\pi$
$954$	0	0
$955$	−9.47011	−0.306445
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.246784	0.00796908
$960$	0	0
$961$	4.47081	0.144220
$962$	0	0
$963$	−124.784	−4.02112
$964$	0	0
$965$	−16.3716	−0.527019
$966$	0	0
$967$	31.8547	1.02438	0.512188	−	0.858873i	$-0.328835\pi$
$967$	31.8547	1.02438	0.512188	+	0.858873i	$0.328835\pi$
$968$	0	0
$969$	3.33942	0.107277
$970$	0	0
$971$	−19.2035	−0.616271	−0.308136	−	0.951342i	$-0.599705\pi$
$971$	−19.2035	−0.616271	−0.308136	+	0.951342i	$0.599705\pi$
$972$	0	0
$973$	0.377166	0.0120914
$974$	0	0
$975$	−10.6419	−0.340812
$976$	0	0
$977$	11.6680	0.373292	0.186646	−	0.982427i	$-0.440238\pi$
$977$	11.6680	0.373292	0.186646	+	0.982427i	$0.440238\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	42.3296	1.35148
$982$	0	0
$983$	−20.7424	−0.661581	−0.330790	−	0.943704i	$-0.607315\pi$
$983$	−20.7424	−0.661581	−0.330790	+	0.943704i	$0.607315\pi$
$984$	0	0
$985$	15.1615	0.483087
$986$	0	0
$987$	0.155073	0.00493604
$988$	0	0
$989$	73.6602	2.34226
$990$	0	0
$991$	−31.5161	−1.00114	−0.500572	−	0.865695i	$-0.666877\pi$
$991$	−31.5161	−1.00114	−0.500572	+	0.865695i	$0.666877\pi$
$992$	0	0
$993$	24.8411	0.788310
$994$	0	0
$995$	2.94883	0.0934843
$996$	0	0
$997$	−23.2775	−0.737206	−0.368603	−	0.929587i	$-0.620164\pi$
$997$	−23.2775	−0.737206	−0.368603	+	0.929587i	$0.620164\pi$
$998$	0	0
$999$	115.423	3.65183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bh.1.7		8
4.3	odd	2		9680.2.a.de.1.2		8
11.2	odd	10		440.2.y.d.81.4	✓	16
11.6	odd	10		440.2.y.d.201.4	yes	16
11.10	odd	2		4840.2.a.bg.1.7		8
44.35	even	10		880.2.bo.k.81.1		16
44.39	even	10		880.2.bo.k.641.1		16
44.43	even	2		9680.2.a.df.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.81.4	✓	16	11.2	odd	10
440.2.y.d.201.4	yes	16	11.6	odd	10
880.2.bo.k.81.1		16	44.35	even	10
880.2.bo.k.641.1		16	44.39	even	10
4840.2.a.bg.1.7		8	11.10	odd	2
4840.2.a.bh.1.7		8	1.1	even	1	trivial
9680.2.a.de.1.2		8	4.3	odd	2
9680.2.a.df.1.2		8	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)

\(f(q)\)	\(=\)	\(q+3.08163 q^{3} +1.00000 q^{5} -0.0385298 q^{7} +6.49642 q^{9} -3.45332 q^{13} +3.08163 q^{15} -0.897632 q^{17} -1.20724 q^{19} -0.118734 q^{21} +7.60421 q^{23} +1.00000 q^{25} +10.7747 q^{27} +7.56188 q^{29} -5.95574 q^{31} -0.0385298 q^{35} +10.7125 q^{37} -10.6419 q^{39} -3.08927 q^{41} +9.68676 q^{43} +6.49642 q^{45} -1.30605 q^{47} -6.99852 q^{49} -2.76617 q^{51} +2.03218 q^{53} -3.72025 q^{57} -0.185620 q^{59} +9.77911 q^{61} -0.250306 q^{63} -3.45332 q^{65} +8.10586 q^{67} +23.4333 q^{69} -7.18945 q^{71} -12.7078 q^{73} +3.08163 q^{75} +12.6594 q^{79} +13.7142 q^{81} +10.7423 q^{83} -0.897632 q^{85} +23.3029 q^{87} -2.78161 q^{89} +0.133056 q^{91} -18.3534 q^{93} -1.20724 q^{95} +17.7130 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 8 * q + q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 - 12 * q^13 + q^15 - 2 * q^17 + 6 * q^19 - 6 * q^21 + 10 * q^23 + 8 * q^25 + 13 * q^27 - 8 * q^29 + 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 + 3 * q^41 + 8 * q^43 + 19 * q^45 + 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 - 25 * q^57 + 25 * q^59 - 10 * q^61 + 64 * q^63 - 12 * q^65 - 2 * q^67 + 18 * q^69 + 25 * q^71 - 38 * q^73 + q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 - 2 * q^85 - 15 * q^87 + 12 * q^89 + 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * q^97

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