LMFDB - Embedded newform 8001.2.a.x.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.775659 q^{2} -1.39835 q^{4} +0.833166 q^{5} +1.00000 q^{7} -2.63596 q^{8} +O(q^{10})$	\(q+0.775659 q^{2} -1.39835 q^{4} +0.833166 q^{5} +1.00000 q^{7} -2.63596 q^{8} +0.646253 q^{10} +0.821214 q^{11} -3.40878 q^{13} +0.775659 q^{14} +0.752101 q^{16} -2.50310 q^{17} +1.40296 q^{19} -1.16506 q^{20} +0.636982 q^{22} +0.947115 q^{23} -4.30583 q^{25} -2.64405 q^{26} -1.39835 q^{28} +1.78577 q^{29} +8.51944 q^{31} +5.85530 q^{32} -1.94155 q^{34} +0.833166 q^{35} -2.89168 q^{37} +1.08822 q^{38} -2.19620 q^{40} +2.30169 q^{41} +5.58041 q^{43} -1.14835 q^{44} +0.734638 q^{46} +2.14718 q^{47} +1.00000 q^{49} -3.33986 q^{50} +4.76668 q^{52} -0.992903 q^{53} +0.684208 q^{55} -2.63596 q^{56} +1.38515 q^{58} -13.0882 q^{59} -7.19849 q^{61} +6.60817 q^{62} +3.03751 q^{64} -2.84008 q^{65} -5.77977 q^{67} +3.50022 q^{68} +0.646253 q^{70} -0.675593 q^{71} -10.0038 q^{73} -2.24296 q^{74} -1.96183 q^{76} +0.821214 q^{77} +1.57957 q^{79} +0.626625 q^{80} +1.78533 q^{82} +3.31531 q^{83} -2.08550 q^{85} +4.32849 q^{86} -2.16469 q^{88} -1.39169 q^{89} -3.40878 q^{91} -1.32440 q^{92} +1.66548 q^{94} +1.16890 q^{95} -4.55016 q^{97} +0.775659 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) $ 22 * q + 10 * q^4 + 22 * q^7	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0.775659	0.548473	0.274237	−	0.961662i	$-0.411575\pi$
$2$	0.775659	0.548473	0.274237	+	0.961662i	$0.411575\pi$
$3$	0	0
$3$	0	0
$4$	−1.39835	−0.699177
$5$	0.833166	0.372603	0.186302	−	0.982493i	$-0.440350\pi$
$5$	0.833166	0.372603	0.186302	+	0.982493i	$0.440350\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.63596	−0.931953
$9$	0	0
$10$	0.646253	0.204363
$11$	0.821214	0.247605	0.123803	−	0.992307i	$-0.460491\pi$
$11$	0.821214	0.247605	0.123803	+	0.992307i	$0.460491\pi$
$12$	0	0
$13$	−3.40878	−0.945424	−0.472712	−	0.881217i	$-0.656725\pi$
$13$	−3.40878	−0.945424	−0.472712	+	0.881217i	$0.656725\pi$
$14$	0.775659	0.207303
$15$	0	0
$16$	0.752101	0.188025
$17$	−2.50310	−0.607091	−0.303546	−	0.952817i	$-0.598170\pi$
$17$	−2.50310	−0.607091	−0.303546	+	0.952817i	$0.598170\pi$
$18$	0	0
$19$	1.40296	0.321861	0.160931	−	0.986966i	$-0.448550\pi$
$19$	1.40296	0.321861	0.160931	+	0.986966i	$0.448550\pi$
$20$	−1.16506	−0.260516
$21$	0	0
$22$	0.636982	0.135805
$23$	0.947115	0.197487	0.0987436	−	0.995113i	$-0.468518\pi$
$23$	0.947115	0.197487	0.0987436	+	0.995113i	$0.468518\pi$
$24$	0	0
$25$	−4.30583	−0.861167
$26$	−2.64405	−0.518540
$27$	0	0
$28$	−1.39835	−0.264264
$29$	1.78577	0.331610	0.165805	−	0.986159i	$-0.446978\pi$
$29$	1.78577	0.331610	0.165805	+	0.986159i	$0.446978\pi$
$30$	0	0
$31$	8.51944	1.53014	0.765068	−	0.643949i	$-0.222705\pi$
$31$	8.51944	1.53014	0.765068	+	0.643949i	$0.222705\pi$
$32$	5.85530	1.03508
$33$	0	0
$34$	−1.94155	−0.332973
$35$	0.833166	0.140831
$36$	0	0
$37$	−2.89168	−0.475390	−0.237695	−	0.971340i	$-0.576392\pi$
$37$	−2.89168	−0.475390	−0.237695	+	0.971340i	$0.576392\pi$
$38$	1.08822	0.176532
$39$	0	0
$40$	−2.19620	−0.347249
$41$	2.30169	0.359464	0.179732	−	0.983716i	$-0.442477\pi$
$41$	2.30169	0.359464	0.179732	+	0.983716i	$0.442477\pi$
$42$	0	0
$43$	5.58041	0.851004	0.425502	−	0.904957i	$-0.360098\pi$
$43$	5.58041	0.851004	0.425502	+	0.904957i	$0.360098\pi$
$44$	−1.14835	−0.173120
$45$	0	0
$46$	0.734638	0.108316
$47$	2.14718	0.313198	0.156599	−	0.987662i	$-0.449947\pi$
$47$	2.14718	0.313198	0.156599	+	0.987662i	$0.449947\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.33986	−0.472327
$51$	0	0
$52$	4.76668	0.661019
$53$	−0.992903	−0.136386	−0.0681929	−	0.997672i	$-0.521723\pi$
$53$	−0.992903	−0.136386	−0.0681929	+	0.997672i	$0.521723\pi$
$54$	0	0
$55$	0.684208	0.0922586
$56$	−2.63596	−0.352245
$57$	0	0
$58$	1.38515	0.181879
$59$	−13.0882	−1.70394	−0.851971	−	0.523589i	$-0.824593\pi$
$59$	−13.0882	−1.70394	−0.851971	+	0.523589i	$0.824593\pi$
$60$	0	0
$61$	−7.19849	−0.921672	−0.460836	−	0.887485i	$-0.652450\pi$
$61$	−7.19849	−0.921672	−0.460836	+	0.887485i	$0.652450\pi$
$62$	6.60817	0.839239
$63$	0	0
$64$	3.03751	0.379689
$65$	−2.84008	−0.352268
$66$	0	0
$67$	−5.77977	−0.706111	−0.353056	−	0.935602i	$-0.614857\pi$
$67$	−5.77977	−0.706111	−0.353056	+	0.935602i	$0.614857\pi$
$68$	3.50022	0.424464
$69$	0	0
$70$	0.646253	0.0772420
$71$	−0.675593	−0.0801781	−0.0400891	−	0.999196i	$-0.512764\pi$
$71$	−0.675593	−0.0801781	−0.0400891	+	0.999196i	$0.512764\pi$
$72$	0	0
$73$	−10.0038	−1.17086	−0.585429	−	0.810724i	$-0.699074\pi$
$73$	−10.0038	−1.17086	−0.585429	+	0.810724i	$0.699074\pi$
$74$	−2.24296	−0.260739
$75$	0	0
$76$	−1.96183	−0.225038
$77$	0.821214	0.0935860
$78$	0	0
$79$	1.57957	0.177716	0.0888578	−	0.996044i	$-0.471678\pi$
$79$	1.57957	0.177716	0.0888578	+	0.996044i	$0.471678\pi$
$80$	0.626625	0.0700589
$81$	0	0
$82$	1.78533	0.197156
$83$	3.31531	0.363902	0.181951	−	0.983308i	$-0.441759\pi$
$83$	3.31531	0.363902	0.181951	+	0.983308i	$0.441759\pi$
$84$	0	0
$85$	−2.08550	−0.226204
$86$	4.32849	0.466753
$87$	0	0
$88$	−2.16469	−0.230757
$89$	−1.39169	−0.147519	−0.0737595	−	0.997276i	$-0.523500\pi$
$89$	−1.39169	−0.147519	−0.0737595	+	0.997276i	$0.523500\pi$
$90$	0	0
$91$	−3.40878	−0.357337
$92$	−1.32440	−0.138078
$93$	0	0
$94$	1.66548	0.171781
$95$	1.16890	0.119927
$96$	0	0
$97$	−4.55016	−0.461999	−0.230999	−	0.972954i	$-0.574200\pi$
$97$	−4.55016	−0.461999	−0.230999	+	0.972954i	$0.574200\pi$
$98$	0.775659	0.0783533
$99$	0	0
$100$	6.02108	0.602108
$101$	8.82048	0.877670	0.438835	−	0.898568i	$-0.355391\pi$
$101$	8.82048	0.877670	0.438835	+	0.898568i	$0.355391\pi$
$102$	0	0
$103$	5.65003	0.556714	0.278357	−	0.960478i	$-0.410210\pi$
$103$	5.65003	0.556714	0.278357	+	0.960478i	$0.410210\pi$
$104$	8.98540	0.881091
$105$	0	0
$106$	−0.770154	−0.0748039
$107$	−20.2463	−1.95729	−0.978643	−	0.205567i	$-0.934096\pi$
$107$	−20.2463	−1.95729	−0.978643	+	0.205567i	$0.934096\pi$
$108$	0	0
$109$	−0.369111	−0.0353545	−0.0176772	−	0.999844i	$-0.505627\pi$
$109$	−0.369111	−0.0353545	−0.0176772	+	0.999844i	$0.505627\pi$
$110$	0.530712	0.0506014
$111$	0	0
$112$	0.752101	0.0710669
$113$	−10.8983	−1.02523	−0.512613	−	0.858620i	$-0.671323\pi$
$113$	−10.8983	−1.02523	−0.512613	+	0.858620i	$0.671323\pi$
$114$	0	0
$115$	0.789105	0.0735844
$116$	−2.49714	−0.231854
$117$	0	0
$118$	−10.1520	−0.934567
$119$	−2.50310	−0.229459
$120$	0	0
$121$	−10.3256	−0.938692
$122$	−5.58357	−0.505513
$123$	0	0
$124$	−11.9132	−1.06984
$125$	−7.75331	−0.693477
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.35453	−0.826831
$129$	0	0
$130$	−2.20293	−0.193210
$131$	−6.34339	−0.554224	−0.277112	−	0.960838i	$-0.589377\pi$
$131$	−6.34339	−0.554224	−0.277112	+	0.960838i	$0.589377\pi$
$132$	0	0
$133$	1.40296	0.121652
$134$	−4.48313	−0.387283
$135$	0	0
$136$	6.59808	0.565781
$137$	−16.5664	−1.41536	−0.707682	−	0.706531i	$-0.750259\pi$
$137$	−16.5664	−1.41536	−0.707682	+	0.706531i	$0.750259\pi$
$138$	0	0
$139$	−14.6965	−1.24654	−0.623271	−	0.782006i	$-0.714197\pi$
$139$	−14.6965	−1.24654	−0.623271	+	0.782006i	$0.714197\pi$
$140$	−1.16506	−0.0984657
$141$	0	0
$142$	−0.524029	−0.0439756
$143$	−2.79933	−0.234092
$144$	0	0
$145$	1.48785	0.123559
$146$	−7.75955	−0.642185
$147$	0	0
$148$	4.04359	0.332381
$149$	12.6949	1.04001	0.520003	−	0.854164i	$-0.325931\pi$
$149$	12.6949	1.04001	0.520003	+	0.854164i	$0.325931\pi$
$150$	0	0
$151$	−0.318150	−0.0258906	−0.0129453	−	0.999916i	$-0.504121\pi$
$151$	−0.318150	−0.0258906	−0.0129453	+	0.999916i	$0.504121\pi$
$152$	−3.69815	−0.299960
$153$	0	0
$154$	0.636982	0.0513294
$155$	7.09811	0.570134
$156$	0	0
$157$	8.24209	0.657790	0.328895	−	0.944366i	$-0.393324\pi$
$157$	8.24209	0.657790	0.328895	+	0.944366i	$0.393324\pi$
$158$	1.22521	0.0974723
$159$	0	0
$160$	4.87844	0.385674
$161$	0.947115	0.0746431
$162$	0	0
$163$	12.7768	1.00076	0.500379	−	0.865806i	$-0.333194\pi$
$163$	12.7768	1.00076	0.500379	+	0.865806i	$0.333194\pi$
$164$	−3.21858	−0.251329
$165$	0	0
$166$	2.57155	0.199591
$167$	19.9015	1.54002	0.770010	−	0.638031i	$-0.220251\pi$
$167$	19.9015	1.54002	0.770010	+	0.638031i	$0.220251\pi$
$168$	0	0
$169$	−1.38025	−0.106173
$170$	−1.61764	−0.124067
$171$	0	0
$172$	−7.80338	−0.595002
$173$	6.38620	0.485534	0.242767	−	0.970085i	$-0.421945\pi$
$173$	6.38620	0.485534	0.242767	+	0.970085i	$0.421945\pi$
$174$	0	0
$175$	−4.30583	−0.325490
$176$	0.617636	0.0465561
$177$	0	0
$178$	−1.07948	−0.0809102
$179$	−13.5632	−1.01376	−0.506879	−	0.862017i	$-0.669201\pi$
$179$	−13.5632	−1.01376	−0.506879	+	0.862017i	$0.669201\pi$
$180$	0	0
$181$	−7.27667	−0.540871	−0.270435	−	0.962738i	$-0.587168\pi$
$181$	−7.27667	−0.540871	−0.270435	+	0.962738i	$0.587168\pi$
$182$	−2.64405	−0.195990
$183$	0	0
$184$	−2.49656	−0.184049
$185$	−2.40925	−0.177132
$186$	0	0
$187$	−2.05558	−0.150319
$188$	−3.00251	−0.218981
$189$	0	0
$190$	0.906667	0.0657765
$191$	−2.22153	−0.160744	−0.0803721	−	0.996765i	$-0.525611\pi$
$191$	−2.22153	−0.160744	−0.0803721	+	0.996765i	$0.525611\pi$
$192$	0	0
$193$	−15.4286	−1.11057	−0.555287	−	0.831659i	$-0.687392\pi$
$193$	−15.4286	−1.11057	−0.555287	+	0.831659i	$0.687392\pi$
$194$	−3.52937	−0.253394
$195$	0	0
$196$	−1.39835	−0.0998824
$197$	12.8028	0.912162	0.456081	−	0.889938i	$-0.349253\pi$
$197$	12.8028	0.912162	0.456081	+	0.889938i	$0.349253\pi$
$198$	0	0
$199$	3.89117	0.275838	0.137919	−	0.990444i	$-0.455959\pi$
$199$	3.89117	0.275838	0.137919	+	0.990444i	$0.455959\pi$
$200$	11.3500	0.802567
$201$	0	0
$202$	6.84168	0.481379
$203$	1.78577	0.125337
$204$	0	0
$205$	1.91769	0.133937
$206$	4.38249	0.305343
$207$	0	0
$208$	−2.56374	−0.177764
$209$	1.15213	0.0796945
$210$	0	0
$211$	23.1279	1.59219	0.796095	−	0.605172i	$-0.206896\pi$
$211$	23.1279	1.59219	0.796095	+	0.605172i	$0.206896\pi$
$212$	1.38843	0.0953577
$213$	0	0
$214$	−15.7042	−1.07352
$215$	4.64941	0.317087
$216$	0	0
$217$	8.51944	0.578337
$218$	−0.286304	−0.0193910
$219$	0	0
$220$	−0.956765	−0.0645051
$221$	8.53251	0.573959
$222$	0	0
$223$	6.28503	0.420877	0.210438	−	0.977607i	$-0.432511\pi$
$223$	6.28503	0.420877	0.210438	+	0.977607i	$0.432511\pi$
$224$	5.85530	0.391224
$225$	0	0
$226$	−8.45336	−0.562309
$227$	7.52578	0.499503	0.249752	−	0.968310i	$-0.419651\pi$
$227$	7.52578	0.499503	0.249752	+	0.968310i	$0.419651\pi$
$228$	0	0
$229$	−27.0451	−1.78719	−0.893596	−	0.448872i	$-0.851826\pi$
$229$	−27.0451	−1.78719	−0.893596	+	0.448872i	$0.851826\pi$
$230$	0.612076	0.0403591
$231$	0	0
$232$	−4.70723	−0.309045
$233$	−20.8606	−1.36663	−0.683313	−	0.730126i	$-0.739461\pi$
$233$	−20.8606	−1.36663	−0.683313	+	0.730126i	$0.739461\pi$
$234$	0	0
$235$	1.78896	0.116699
$236$	18.3020	1.19136
$237$	0	0
$238$	−1.94155	−0.125852
$239$	20.7716	1.34360	0.671800	−	0.740732i	$-0.265521\pi$
$239$	20.7716	1.34360	0.671800	+	0.740732i	$0.265521\pi$
$240$	0	0
$241$	−26.7249	−1.72150	−0.860750	−	0.509028i	$-0.830005\pi$
$241$	−26.7249	−1.72150	−0.860750	+	0.509028i	$0.830005\pi$
$242$	−8.00915	−0.514847
$243$	0	0
$244$	10.0660	0.644412
$245$	0.833166	0.0532291
$246$	0	0
$247$	−4.78238	−0.304295
$248$	−22.4569	−1.42602
$249$	0	0
$250$	−6.01392	−0.380354
$251$	−24.2026	−1.52765	−0.763827	−	0.645420i	$-0.776682\pi$
$251$	−24.2026	−1.52765	−0.763827	+	0.645420i	$0.776682\pi$
$252$	0	0
$253$	0.777784	0.0488989
$254$	0.775659	0.0486691
$255$	0	0
$256$	−13.3309	−0.833184
$257$	−29.2037	−1.82168	−0.910838	−	0.412765i	$-0.864563\pi$
$257$	−29.2037	−1.82168	−0.910838	+	0.412765i	$0.864563\pi$
$258$	0	0
$259$	−2.89168	−0.179680
$260$	3.97143	0.246298
$261$	0	0
$262$	−4.92030	−0.303977
$263$	8.46412	0.521920	0.260960	−	0.965350i	$-0.415961\pi$
$263$	8.46412	0.521920	0.260960	+	0.965350i	$0.415961\pi$
$264$	0	0
$265$	−0.827253	−0.0508178
$266$	1.08822	0.0667229
$267$	0	0
$268$	8.08217	0.493697
$269$	−25.1306	−1.53224	−0.766119	−	0.642699i	$-0.777815\pi$
$269$	−25.1306	−1.53224	−0.766119	+	0.642699i	$0.777815\pi$
$270$	0	0
$271$	0.485582	0.0294970	0.0147485	−	0.999891i	$-0.495305\pi$
$271$	0.485582	0.0294970	0.0147485	+	0.999891i	$0.495305\pi$
$272$	−1.88259	−0.114149
$273$	0	0
$274$	−12.8499	−0.776290
$275$	−3.53601	−0.213229
$276$	0	0
$277$	−12.5162	−0.752027	−0.376013	−	0.926614i	$-0.622705\pi$
$277$	−12.5162	−0.752027	−0.376013	+	0.926614i	$0.622705\pi$
$278$	−11.3995	−0.683695
$279$	0	0
$280$	−2.19620	−0.131248
$281$	4.50129	0.268524	0.134262	−	0.990946i	$-0.457134\pi$
$281$	4.50129	0.268524	0.134262	+	0.990946i	$0.457134\pi$
$282$	0	0
$283$	−4.68047	−0.278225	−0.139112	−	0.990277i	$-0.544425\pi$
$283$	−4.68047	−0.278225	−0.139112	+	0.990277i	$0.544425\pi$
$284$	0.944718	0.0560587
$285$	0	0
$286$	−2.17133	−0.128393
$287$	2.30169	0.135865
$288$	0	0
$289$	−10.7345	−0.631440
$290$	1.15406	0.0677688
$291$	0	0
$292$	13.9889	0.818637
$293$	21.7063	1.26809	0.634047	−	0.773295i	$-0.281393\pi$
$293$	21.7063	1.26809	0.634047	+	0.773295i	$0.281393\pi$
$294$	0	0
$295$	−10.9047	−0.634894
$296$	7.62236	0.443041
$297$	0	0
$298$	9.84690	0.570416
$299$	−3.22850	−0.186709
$300$	0	0
$301$	5.58041	0.321649
$302$	−0.246775	−0.0142003
$303$	0	0
$304$	1.05517	0.0605180
$305$	−5.99754	−0.343418
$306$	0	0
$307$	7.76773	0.443328	0.221664	−	0.975123i	$-0.428851\pi$
$307$	7.76773	0.443328	0.221664	+	0.975123i	$0.428851\pi$
$308$	−1.14835	−0.0654332
$309$	0	0
$310$	5.50571	0.312703
$311$	3.35408	0.190192	0.0950961	−	0.995468i	$-0.469684\pi$
$311$	3.35408	0.190192	0.0950961	+	0.995468i	$0.469684\pi$
$312$	0	0
$313$	4.56356	0.257948	0.128974	−	0.991648i	$-0.458832\pi$
$313$	4.56356	0.257948	0.128974	+	0.991648i	$0.458832\pi$
$314$	6.39305	0.360780
$315$	0	0
$316$	−2.20880	−0.124255
$317$	0.418733	0.0235184	0.0117592	−	0.999931i	$-0.496257\pi$
$317$	0.418733	0.0235184	0.0117592	+	0.999931i	$0.496257\pi$
$318$	0	0
$319$	1.46650	0.0821083
$320$	2.53075	0.141473
$321$	0	0
$322$	0.734638	0.0409398
$323$	−3.51175	−0.195399
$324$	0	0
$325$	14.6776	0.814168
$326$	9.91046	0.548889
$327$	0	0
$328$	−6.06717	−0.335003
$329$	2.14718	0.118378
$330$	0	0
$331$	13.6681	0.751266	0.375633	−	0.926768i	$-0.377425\pi$
$331$	13.6681	0.751266	0.375633	+	0.926768i	$0.377425\pi$
$332$	−4.63597	−0.254432
$333$	0	0
$334$	15.4367	0.844660
$335$	−4.81551	−0.263099
$336$	0	0
$337$	−13.1006	−0.713637	−0.356819	−	0.934174i	$-0.616139\pi$
$337$	−13.1006	−0.713637	−0.356819	+	0.934174i	$0.616139\pi$
$338$	−1.07060	−0.0582329
$339$	0	0
$340$	2.91627	0.158157
$341$	6.99628	0.378870
$342$	0	0
$343$	1.00000	0.0539949
$344$	−14.7097	−0.793096
$345$	0	0
$346$	4.95351	0.266302
$347$	−3.79071	−0.203496	−0.101748	−	0.994810i	$-0.532444\pi$
$347$	−3.79071	−0.203496	−0.101748	+	0.994810i	$0.532444\pi$
$348$	0	0
$349$	−18.5522	−0.993075	−0.496537	−	0.868015i	$-0.665395\pi$
$349$	−18.5522	−0.993075	−0.496537	+	0.868015i	$0.665395\pi$
$350$	−3.33986	−0.178523
$351$	0	0
$352$	4.80845	0.256291
$353$	3.95098	0.210289	0.105145	−	0.994457i	$-0.466469\pi$
$353$	3.95098	0.210289	0.105145	+	0.994457i	$0.466469\pi$
$354$	0	0
$355$	−0.562881	−0.0298746
$356$	1.94608	0.103142
$357$	0	0
$358$	−10.5204	−0.556019
$359$	−5.14938	−0.271774	−0.135887	−	0.990724i	$-0.543388\pi$
$359$	−5.14938	−0.271774	−0.135887	+	0.990724i	$0.543388\pi$
$360$	0	0
$361$	−17.0317	−0.896405
$362$	−5.64421	−0.296653
$363$	0	0
$364$	4.76668	0.249842
$365$	−8.33485	−0.436266
$366$	0	0
$367$	−31.8116	−1.66055	−0.830276	−	0.557353i	$-0.811817\pi$
$367$	−31.8116	−1.66055	−0.830276	+	0.557353i	$0.811817\pi$
$368$	0.712326	0.0371326
$369$	0	0
$370$	−1.86876	−0.0971521
$371$	−0.992903	−0.0515490
$372$	0	0
$373$	−17.4919	−0.905695	−0.452848	−	0.891588i	$-0.649592\pi$
$373$	−17.4919	−0.905695	−0.452848	+	0.891588i	$0.649592\pi$
$374$	−1.59443	−0.0824460
$375$	0	0
$376$	−5.65988	−0.291886
$377$	−6.08730	−0.313512
$378$	0	0
$379$	−23.3687	−1.20037	−0.600185	−	0.799861i	$-0.704906\pi$
$379$	−23.3687	−1.20037	−0.600185	+	0.799861i	$0.704906\pi$
$380$	−1.63453	−0.0838499
$381$	0	0
$382$	−1.72315	−0.0881639
$383$	12.7842	0.653242	0.326621	−	0.945155i	$-0.394090\pi$
$383$	12.7842	0.653242	0.326621	+	0.945155i	$0.394090\pi$
$384$	0	0
$385$	0.684208	0.0348705
$386$	−11.9673	−0.609121
$387$	0	0
$388$	6.36273	0.323019
$389$	−31.1873	−1.58126	−0.790630	−	0.612294i	$-0.790247\pi$
$389$	−31.1873	−1.58126	−0.790630	+	0.612294i	$0.790247\pi$
$390$	0	0
$391$	−2.37073	−0.119893
$392$	−2.63596	−0.133136
$393$	0	0
$394$	9.93060	0.500296
$395$	1.31605	0.0662174
$396$	0	0
$397$	36.1511	1.81437	0.907186	−	0.420729i	$-0.138226\pi$
$397$	36.1511	1.81437	0.907186	+	0.420729i	$0.138226\pi$
$398$	3.01822	0.151290
$399$	0	0
$400$	−3.23842	−0.161921
$401$	22.0119	1.09922	0.549611	−	0.835421i	$-0.314776\pi$
$401$	22.0119	1.09922	0.549611	+	0.835421i	$0.314776\pi$
$402$	0	0
$403$	−29.0409	−1.44663
$404$	−12.3341	−0.613647
$405$	0	0
$406$	1.38515	0.0687438
$407$	−2.37469	−0.117709
$408$	0	0
$409$	−23.7967	−1.17667	−0.588335	−	0.808617i	$-0.700216\pi$
$409$	−23.7967	−1.17667	−0.588335	+	0.808617i	$0.700216\pi$
$410$	1.48747	0.0734611
$411$	0	0
$412$	−7.90074	−0.389242
$413$	−13.0882	−0.644029
$414$	0	0
$415$	2.76220	0.135591
$416$	−19.9594	−0.978590
$417$	0	0
$418$	0.893660	0.0437103
$419$	19.9178	0.973048	0.486524	−	0.873667i	$-0.338265\pi$
$419$	19.9178	0.973048	0.486524	+	0.873667i	$0.338265\pi$
$420$	0	0
$421$	−7.26730	−0.354187	−0.177093	−	0.984194i	$-0.556669\pi$
$421$	−7.26730	−0.354187	−0.177093	+	0.984194i	$0.556669\pi$
$422$	17.9393	0.873274
$423$	0	0
$424$	2.61725	0.127105
$425$	10.7779	0.522807
$426$	0	0
$427$	−7.19849	−0.348359
$428$	28.3115	1.36849
$429$	0	0
$430$	3.60635	0.173914
$431$	11.2632	0.542529	0.271264	−	0.962505i	$-0.412558\pi$
$431$	11.2632	0.542529	0.271264	+	0.962505i	$0.412558\pi$
$432$	0	0
$433$	−37.4552	−1.79998	−0.899992	−	0.435907i	$-0.856428\pi$
$433$	−37.4552	−1.79998	−0.899992	+	0.435907i	$0.856428\pi$
$434$	6.60817	0.317203
$435$	0	0
$436$	0.516148	0.0247190
$437$	1.32876	0.0635634
$438$	0	0
$439$	−7.93341	−0.378641	−0.189321	−	0.981915i	$-0.560629\pi$
$439$	−7.93341	−0.378641	−0.189321	+	0.981915i	$0.560629\pi$
$440$	−1.80355	−0.0859807
$441$	0	0
$442$	6.61832	0.314801
$443$	8.55610	0.406513	0.203256	−	0.979126i	$-0.434848\pi$
$443$	8.55610	0.406513	0.203256	+	0.979126i	$0.434848\pi$
$444$	0	0
$445$	−1.15951	−0.0549661
$446$	4.87504	0.230840
$447$	0	0
$448$	3.03751	0.143509
$449$	38.8738	1.83457	0.917285	−	0.398232i	$-0.130376\pi$
$449$	38.8738	1.83457	0.917285	+	0.398232i	$0.130376\pi$
$450$	0	0
$451$	1.89018	0.0890051
$452$	15.2397	0.716815
$453$	0	0
$454$	5.83743	0.273964
$455$	−2.84008	−0.133145
$456$	0	0
$457$	−4.65559	−0.217779	−0.108890	−	0.994054i	$-0.534730\pi$
$457$	−4.65559	−0.217779	−0.108890	+	0.994054i	$0.534730\pi$
$458$	−20.9778	−0.980227
$459$	0	0
$460$	−1.10345	−0.0514485
$461$	0.338860	0.0157823	0.00789114	−	0.999969i	$-0.497488\pi$
$461$	0.338860	0.0157823	0.00789114	+	0.999969i	$0.497488\pi$
$462$	0	0
$463$	−34.1206	−1.58572	−0.792858	−	0.609406i	$-0.791408\pi$
$463$	−34.1206	−1.58572	−0.792858	+	0.609406i	$0.791408\pi$
$464$	1.34308	0.0623510
$465$	0	0
$466$	−16.1807	−0.749558
$467$	15.2172	0.704166	0.352083	−	0.935969i	$-0.385473\pi$
$467$	15.2172	0.704166	0.352083	+	0.935969i	$0.385473\pi$
$468$	0	0
$469$	−5.77977	−0.266885
$470$	1.38762	0.0640061
$471$	0	0
$472$	34.5001	1.58799
$473$	4.58271	0.210713
$474$	0	0
$475$	−6.04091	−0.277176
$476$	3.50022	0.160432
$477$	0	0
$478$	16.1116	0.736929
$479$	−41.7641	−1.90825	−0.954126	−	0.299405i	$-0.903212\pi$
$479$	−41.7641	−1.90825	−0.954126	+	0.299405i	$0.903212\pi$
$480$	0	0
$481$	9.85710	0.449445
$482$	−20.7294	−0.944197
$483$	0	0
$484$	14.4389	0.656312
$485$	−3.79104	−0.172142
$486$	0	0
$487$	12.0202	0.544689	0.272344	−	0.962200i	$-0.412201\pi$
$487$	12.0202	0.544689	0.272344	+	0.962200i	$0.412201\pi$
$488$	18.9749	0.858956
$489$	0	0
$490$	0.646253	0.0291947
$491$	20.4192	0.921506	0.460753	−	0.887528i	$-0.347579\pi$
$491$	20.4192	0.921506	0.460753	+	0.887528i	$0.347579\pi$
$492$	0	0
$493$	−4.46997	−0.201317
$494$	−3.70949	−0.166898
$495$	0	0
$496$	6.40748	0.287704
$497$	−0.675593	−0.0303045
$498$	0	0
$499$	−23.7661	−1.06392	−0.531958	−	0.846771i	$-0.678544\pi$
$499$	−23.7661	−1.06392	−0.531958	+	0.846771i	$0.678544\pi$
$500$	10.8419	0.484863
$501$	0	0
$502$	−18.7730	−0.837878
$503$	17.3725	0.774604	0.387302	−	0.921953i	$-0.373407\pi$
$503$	17.3725	0.774604	0.387302	+	0.921953i	$0.373407\pi$
$504$	0	0
$505$	7.34892	0.327023
$506$	0.603295	0.0268197
$507$	0	0
$508$	−1.39835	−0.0620419
$509$	−3.06618	−0.135906	−0.0679531	−	0.997689i	$-0.521647\pi$
$509$	−3.06618	−0.135906	−0.0679531	+	0.997689i	$0.521647\pi$
$510$	0	0
$511$	−10.0038	−0.442543
$512$	8.36880	0.369852
$513$	0	0
$514$	−22.6521	−0.999140
$515$	4.70742	0.207433
$516$	0	0
$517$	1.76329	0.0775495
$518$	−2.24296	−0.0985499
$519$	0	0
$520$	7.48634	0.328298
$521$	−26.9322	−1.17992	−0.589961	−	0.807431i	$-0.700857\pi$
$521$	−26.9322	−1.17992	−0.589961	+	0.807431i	$0.700857\pi$
$522$	0	0
$523$	7.30944	0.319619	0.159810	−	0.987148i	$-0.448912\pi$
$523$	7.30944	0.319619	0.159810	+	0.987148i	$0.448912\pi$
$524$	8.87030	0.387501
$525$	0	0
$526$	6.56527	0.286259
$527$	−21.3250	−0.928932
$528$	0	0
$529$	−22.1030	−0.960999
$530$	−0.641666	−0.0278722
$531$	0	0
$532$	−1.96183	−0.0850563
$533$	−7.84595	−0.339846
$534$	0	0
$535$	−16.8686	−0.729291
$536$	15.2353	0.658063
$537$	0	0
$538$	−19.4927	−0.840392
$539$	0.821214	0.0353722
$540$	0	0
$541$	−30.5215	−1.31222	−0.656111	−	0.754664i	$-0.727800\pi$
$541$	−30.5215	−1.31222	−0.656111	+	0.754664i	$0.727800\pi$
$542$	0.376646	0.0161783
$543$	0	0
$544$	−14.6564	−0.628388
$545$	−0.307531	−0.0131732
$546$	0	0
$547$	−38.3344	−1.63906	−0.819531	−	0.573035i	$-0.805766\pi$
$547$	−38.3344	−1.63906	−0.819531	+	0.573035i	$0.805766\pi$
$548$	23.1657	0.989590
$549$	0	0
$550$	−2.74274	−0.116951
$551$	2.50537	0.106732
$552$	0	0
$553$	1.57957	0.0671702
$554$	−9.70831	−0.412467
$555$	0	0
$556$	20.5509	0.871553
$557$	0.188070	0.00796880	0.00398440	−	0.999992i	$-0.498732\pi$
$557$	0.188070	0.00796880	0.00398440	+	0.999992i	$0.498732\pi$
$558$	0	0
$559$	−19.0224	−0.804560
$560$	0.626625	0.0264798
$561$	0	0
$562$	3.49146	0.147278
$563$	20.4436	0.861594	0.430797	−	0.902449i	$-0.358232\pi$
$563$	20.4436	0.861594	0.430797	+	0.902449i	$0.358232\pi$
$564$	0	0
$565$	−9.08010	−0.382003
$566$	−3.63045	−0.152599
$567$	0	0
$568$	1.78084	0.0747223
$569$	27.1530	1.13831	0.569157	−	0.822229i	$-0.307270\pi$
$569$	27.1530	1.13831	0.569157	+	0.822229i	$0.307270\pi$
$570$	0	0
$571$	6.65084	0.278329	0.139165	−	0.990269i	$-0.455558\pi$
$571$	6.65084	0.278329	0.139165	+	0.990269i	$0.455558\pi$
$572$	3.91446	0.163672
$573$	0	0
$574$	1.78533	0.0745181
$575$	−4.07812	−0.170069
$576$	0	0
$577$	36.7405	1.52953	0.764764	−	0.644310i	$-0.222856\pi$
$577$	36.7405	1.52953	0.764764	+	0.644310i	$0.222856\pi$
$578$	−8.32629	−0.346328
$579$	0	0
$580$	−2.08054	−0.0863895
$581$	3.31531	0.137542
$582$	0	0
$583$	−0.815386	−0.0337698
$584$	26.3697	1.09119
$585$	0	0
$586$	16.8366	0.695515
$587$	12.4810	0.515145	0.257572	−	0.966259i	$-0.417077\pi$
$587$	12.4810	0.515145	0.257572	+	0.966259i	$0.417077\pi$
$588$	0	0
$589$	11.9524	0.492491
$590$	−8.45830	−0.348223
$591$	0	0
$592$	−2.17484	−0.0893853
$593$	44.7690	1.83844	0.919221	−	0.393742i	$-0.128820\pi$
$593$	44.7690	1.83844	0.919221	+	0.393742i	$0.128820\pi$
$594$	0	0
$595$	−2.08550	−0.0854972
$596$	−17.7519	−0.727148
$597$	0	0
$598$	−2.50422	−0.102405
$599$	−39.1831	−1.60098	−0.800490	−	0.599346i	$-0.795427\pi$
$599$	−39.1831	−1.60098	−0.800490	+	0.599346i	$0.795427\pi$
$600$	0	0
$601$	−9.78843	−0.399279	−0.199639	−	0.979869i	$-0.563977\pi$
$601$	−9.78843	−0.399279	−0.199639	+	0.979869i	$0.563977\pi$
$602$	4.32849	0.176416
$603$	0	0
$604$	0.444886	0.0181021
$605$	−8.60295	−0.349760
$606$	0	0
$607$	−11.8224	−0.479857	−0.239929	−	0.970791i	$-0.577124\pi$
$607$	−11.8224	−0.479857	−0.239929	+	0.970791i	$0.577124\pi$
$608$	8.21475	0.333152
$609$	0	0
$610$	−4.65204	−0.188356
$611$	−7.31924	−0.296105
$612$	0	0
$613$	20.7346	0.837461	0.418730	−	0.908111i	$-0.362475\pi$
$613$	20.7346	0.837461	0.418730	+	0.908111i	$0.362475\pi$
$614$	6.02511	0.243154
$615$	0	0
$616$	−2.16469	−0.0872178
$617$	−29.3288	−1.18073	−0.590367	−	0.807135i	$-0.701017\pi$
$617$	−29.3288	−1.18073	−0.590367	+	0.807135i	$0.701017\pi$
$618$	0	0
$619$	15.0032	0.603029	0.301515	−	0.953462i	$-0.402508\pi$
$619$	15.0032	0.603029	0.301515	+	0.953462i	$0.402508\pi$
$620$	−9.92567	−0.398624
$621$	0	0
$622$	2.60162	0.104315
$623$	−1.39169	−0.0557569
$624$	0	0
$625$	15.0694	0.602775
$626$	3.53977	0.141477
$627$	0	0
$628$	−11.5254	−0.459912
$629$	7.23817	0.288605
$630$	0	0
$631$	11.3359	0.451273	0.225637	−	0.974212i	$-0.427554\pi$
$631$	11.3359	0.451273	0.225637	+	0.974212i	$0.427554\pi$
$632$	−4.16369	−0.165623
$633$	0	0
$634$	0.324794	0.0128992
$635$	0.833166	0.0330632
$636$	0	0
$637$	−3.40878	−0.135061
$638$	1.13750	0.0450342
$639$	0	0
$640$	−7.79388	−0.308080
$641$	12.5109	0.494149	0.247074	−	0.968997i	$-0.420531\pi$
$641$	12.5109	0.494149	0.247074	+	0.968997i	$0.420531\pi$
$642$	0	0
$643$	−15.8249	−0.624073	−0.312037	−	0.950070i	$-0.601011\pi$
$643$	−15.8249	−0.624073	−0.312037	+	0.950070i	$0.601011\pi$
$644$	−1.32440	−0.0521888
$645$	0	0
$646$	−2.72392	−0.107171
$647$	17.2085	0.676534	0.338267	−	0.941050i	$-0.390159\pi$
$647$	17.2085	0.676534	0.338267	+	0.941050i	$0.390159\pi$
$648$	0	0
$649$	−10.7482	−0.421905
$650$	11.3848	0.446550
$651$	0	0
$652$	−17.8665	−0.699707
$653$	16.6436	0.651315	0.325657	−	0.945488i	$-0.394414\pi$
$653$	16.6436	0.651315	0.325657	+	0.945488i	$0.394414\pi$
$654$	0	0
$655$	−5.28510	−0.206506
$656$	1.73110	0.0675883
$657$	0	0
$658$	1.66548	0.0649270
$659$	35.9180	1.39917	0.699583	−	0.714551i	$-0.253369\pi$
$659$	35.9180	1.39917	0.699583	+	0.714551i	$0.253369\pi$
$660$	0	0
$661$	28.3781	1.10378	0.551890	−	0.833917i	$-0.313907\pi$
$661$	28.3781	1.10378	0.551890	+	0.833917i	$0.313907\pi$
$662$	10.6018	0.412049
$663$	0	0
$664$	−8.73903	−0.339140
$665$	1.16890	0.0453280
$666$	0	0
$667$	1.69133	0.0654887
$668$	−27.8293	−1.07675
$669$	0	0
$670$	−3.73519	−0.144303
$671$	−5.91150	−0.228211
$672$	0	0
$673$	−45.0138	−1.73515	−0.867577	−	0.497303i	$-0.834324\pi$
$673$	−45.0138	−1.73515	−0.867577	+	0.497303i	$0.834324\pi$
$674$	−10.1616	−0.391411
$675$	0	0
$676$	1.93007	0.0742335
$677$	−24.6030	−0.945570	−0.472785	−	0.881178i	$-0.656751\pi$
$677$	−24.6030	−0.945570	−0.472785	+	0.881178i	$0.656751\pi$
$678$	0	0
$679$	−4.55016	−0.174619
$680$	5.49730	0.210812
$681$	0	0
$682$	5.42673	0.207800
$683$	17.5156	0.670217	0.335109	−	0.942179i	$-0.391227\pi$
$683$	17.5156	0.670217	0.335109	+	0.942179i	$0.391227\pi$
$684$	0	0
$685$	−13.8026	−0.527369
$686$	0.775659	0.0296148
$687$	0	0
$688$	4.19703	0.160010
$689$	3.38458	0.128942
$690$	0	0
$691$	16.1962	0.616132	0.308066	−	0.951365i	$-0.400318\pi$
$691$	16.1962	0.616132	0.308066	+	0.951365i	$0.400318\pi$
$692$	−8.93017	−0.339474
$693$	0	0
$694$	−2.94029	−0.111612
$695$	−12.2446	−0.464466
$696$	0	0
$697$	−5.76136	−0.218227
$698$	−14.3901	−0.544675
$699$	0	0
$700$	6.02108	0.227575
$701$	40.1314	1.51574	0.757871	−	0.652405i	$-0.226240\pi$
$701$	40.1314	1.51574	0.757871	+	0.652405i	$0.226240\pi$
$702$	0	0
$703$	−4.05691	−0.153009
$704$	2.49444	0.0940129
$705$	0	0
$706$	3.06461	0.115338
$707$	8.82048	0.331728
$708$	0	0
$709$	−30.6274	−1.15024	−0.575118	−	0.818070i	$-0.695044\pi$
$709$	−30.6274	−1.15024	−0.575118	+	0.818070i	$0.695044\pi$
$710$	−0.436604	−0.0163854
$711$	0	0
$712$	3.66844	0.137481
$713$	8.06889	0.302182
$714$	0	0
$715$	−2.33231	−0.0872235
$716$	18.9661	0.708796
$717$	0	0
$718$	−3.99416	−0.149061
$719$	−31.6921	−1.18192	−0.590958	−	0.806702i	$-0.701250\pi$
$719$	−31.6921	−1.18192	−0.590958	+	0.806702i	$0.701250\pi$
$720$	0	0
$721$	5.65003	0.210418
$722$	−13.2108	−0.491655
$723$	0	0
$724$	10.1754	0.378164
$725$	−7.68924	−0.285571
$726$	0	0
$727$	−24.4231	−0.905801	−0.452901	−	0.891561i	$-0.649611\pi$
$727$	−24.4231	−0.905801	−0.452901	+	0.891561i	$0.649611\pi$
$728$	8.98540	0.333021
$729$	0	0
$730$	−6.46499	−0.239280
$731$	−13.9683	−0.516637
$732$	0	0
$733$	11.9558	0.441599	0.220800	−	0.975319i	$-0.429133\pi$
$733$	11.9558	0.441599	0.220800	+	0.975319i	$0.429133\pi$
$734$	−24.6749	−0.910769
$735$	0	0
$736$	5.54564	0.204415
$737$	−4.74643	−0.174837
$738$	0	0
$739$	−15.0507	−0.553648	−0.276824	−	0.960921i	$-0.589282\pi$
$739$	−15.0507	−0.553648	−0.276824	+	0.960921i	$0.589282\pi$
$740$	3.36899	0.123846
$741$	0	0
$742$	−0.770154	−0.0282732
$743$	53.7505	1.97191	0.985957	−	0.166999i	$-0.0534075\pi$
$743$	53.7505	1.97191	0.985957	+	0.166999i	$0.0534075\pi$
$744$	0	0
$745$	10.5770	0.387510
$746$	−13.5677	−0.496750
$747$	0	0
$748$	2.87443	0.105100
$749$	−20.2463	−0.739785
$750$	0	0
$751$	9.76059	0.356169	0.178085	−	0.984015i	$-0.443010\pi$
$751$	9.76059	0.356169	0.178085	+	0.984015i	$0.443010\pi$
$752$	1.61489	0.0588891
$753$	0	0
$754$	−4.72167	−0.171953
$755$	−0.265072	−0.00964694
$756$	0	0
$757$	32.3529	1.17589	0.587944	−	0.808902i	$-0.299938\pi$
$757$	32.3529	1.17589	0.587944	+	0.808902i	$0.299938\pi$
$758$	−18.1261	−0.658371
$759$	0	0
$760$	−3.08117	−0.111766
$761$	−29.5981	−1.07293	−0.536465	−	0.843923i	$-0.680241\pi$
$761$	−29.5981	−1.07293	−0.536465	+	0.843923i	$0.680241\pi$
$762$	0	0
$763$	−0.369111	−0.0133627
$764$	3.10648	0.112389
$765$	0	0
$766$	9.91617	0.358286
$767$	44.6148	1.61095
$768$	0	0
$769$	−23.0353	−0.830676	−0.415338	−	0.909667i	$-0.636337\pi$
$769$	−23.0353	−0.830676	−0.415338	+	0.909667i	$0.636337\pi$
$770$	0.530712	0.0191255
$771$	0	0
$772$	21.5746	0.776488
$773$	22.4412	0.807154	0.403577	−	0.914946i	$-0.367767\pi$
$773$	22.4412	0.807154	0.403577	+	0.914946i	$0.367767\pi$
$774$	0	0
$775$	−36.6833	−1.31770
$776$	11.9940	0.430561
$777$	0	0
$778$	−24.1907	−0.867279
$779$	3.22918	0.115697
$780$	0	0
$781$	−0.554806	−0.0198525
$782$	−1.83887	−0.0657580
$783$	0	0
$784$	0.752101	0.0268608
$785$	6.86703	0.245095
$786$	0	0
$787$	34.2193	1.21979	0.609894	−	0.792483i	$-0.291212\pi$
$787$	34.2193	1.21979	0.609894	+	0.792483i	$0.291212\pi$
$788$	−17.9028	−0.637762
$789$	0	0
$790$	1.02080	0.0363185
$791$	−10.8983	−0.387499
$792$	0	0
$793$	24.5380	0.871371
$794$	28.0409	0.995135
$795$	0	0
$796$	−5.44123	−0.192859
$797$	37.5468	1.32998	0.664988	−	0.746854i	$-0.268437\pi$
$797$	37.5468	1.32998	0.664988	+	0.746854i	$0.268437\pi$
$798$	0	0
$799$	−5.37460	−0.190140
$800$	−25.2119	−0.891377
$801$	0	0
$802$	17.0737	0.602894
$803$	−8.21527	−0.289911
$804$	0	0
$805$	0.789105	0.0278123
$806$	−22.5258	−0.793437
$807$	0	0
$808$	−23.2504	−0.817948
$809$	18.2474	0.641544	0.320772	−	0.947156i	$-0.396058\pi$
$809$	18.2474	0.641544	0.320772	+	0.947156i	$0.396058\pi$
$810$	0	0
$811$	−27.7675	−0.975048	−0.487524	−	0.873110i	$-0.662100\pi$
$811$	−27.7675	−0.975048	−0.487524	+	0.873110i	$0.662100\pi$
$812$	−2.49714	−0.0876325
$813$	0	0
$814$	−1.84195	−0.0645602
$815$	10.6452	0.372886
$816$	0	0
$817$	7.82909	0.273905
$818$	−18.4581	−0.645372
$819$	0	0
$820$	−2.68161	−0.0936459
$821$	−0.530588	−0.0185176	−0.00925882	−	0.999957i	$-0.502947\pi$
$821$	−0.530588	−0.0185176	−0.00925882	+	0.999957i	$0.502947\pi$
$822$	0	0
$823$	38.3917	1.33825	0.669125	−	0.743150i	$-0.266669\pi$
$823$	38.3917	1.33825	0.669125	+	0.743150i	$0.266669\pi$
$824$	−14.8933	−0.518831
$825$	0	0
$826$	−10.1520	−0.353233
$827$	−19.9999	−0.695464	−0.347732	−	0.937594i	$-0.613048\pi$
$827$	−19.9999	−0.695464	−0.347732	+	0.937594i	$0.613048\pi$
$828$	0	0
$829$	8.81614	0.306197	0.153099	−	0.988211i	$-0.451075\pi$
$829$	8.81614	0.306197	0.153099	+	0.988211i	$0.451075\pi$
$830$	2.14253	0.0743682
$831$	0	0
$832$	−10.3542	−0.358967
$833$	−2.50310	−0.0867273
$834$	0	0
$835$	16.5812	0.573817
$836$	−1.61109	−0.0557206
$837$	0	0
$838$	15.4494	0.533691
$839$	−22.1358	−0.764211	−0.382106	−	0.924119i	$-0.624801\pi$
$839$	−22.1358	−0.764211	−0.382106	+	0.924119i	$0.624801\pi$
$840$	0	0
$841$	−25.8110	−0.890035
$842$	−5.63695	−0.194262
$843$	0	0
$844$	−32.3410	−1.11322
$845$	−1.14997	−0.0395603
$846$	0	0
$847$	−10.3256	−0.354792
$848$	−0.746763	−0.0256440
$849$	0	0
$850$	8.36000	0.286746
$851$	−2.73876	−0.0938833
$852$	0	0
$853$	−8.56457	−0.293245	−0.146623	−	0.989192i	$-0.546840\pi$
$853$	−8.56457	−0.293245	−0.146623	+	0.989192i	$0.546840\pi$
$854$	−5.58357	−0.191066
$855$	0	0
$856$	53.3685	1.82410
$857$	−6.57137	−0.224474	−0.112237	−	0.993681i	$-0.535802\pi$
$857$	−6.57137	−0.224474	−0.112237	+	0.993681i	$0.535802\pi$
$858$	0	0
$859$	14.7406	0.502944	0.251472	−	0.967865i	$-0.419085\pi$
$859$	14.7406	0.502944	0.251472	+	0.967865i	$0.419085\pi$
$860$	−6.50152	−0.221700
$861$	0	0
$862$	8.73639	0.297563
$863$	20.0421	0.682242	0.341121	−	0.940019i	$-0.389193\pi$
$863$	20.0421	0.682242	0.341121	+	0.940019i	$0.389193\pi$
$864$	0	0
$865$	5.32077	0.180912
$866$	−29.0525	−0.987243
$867$	0	0
$868$	−11.9132	−0.404360
$869$	1.29717	0.0440033
$870$	0	0
$871$	19.7019	0.667575
$872$	0.972964	0.0329487
$873$	0	0
$874$	1.03067	0.0348628
$875$	−7.75331	−0.262110
$876$	0	0
$877$	−31.0175	−1.04739	−0.523694	−	0.851907i	$-0.675446\pi$
$877$	−31.0175	−1.04739	−0.523694	+	0.851907i	$0.675446\pi$
$878$	−6.15362	−0.207675
$879$	0	0
$880$	0.514594	0.0173469
$881$	−4.04652	−0.136331	−0.0681654	−	0.997674i	$-0.521715\pi$
$881$	−4.04652	−0.136331	−0.0681654	+	0.997674i	$0.521715\pi$
$882$	0	0
$883$	55.7648	1.87663	0.938317	−	0.345777i	$-0.112385\pi$
$883$	55.7648	1.87663	0.938317	+	0.345777i	$0.112385\pi$
$884$	−11.9315	−0.401299
$885$	0	0
$886$	6.63661	0.222961
$887$	56.9316	1.91158	0.955788	−	0.294058i	$-0.0950058\pi$
$887$	56.9316	1.91158	0.955788	+	0.294058i	$0.0950058\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.899384	−0.0301474
$891$	0	0
$892$	−8.78870	−0.294267
$893$	3.01240	0.100806
$894$	0	0
$895$	−11.3004	−0.377730
$896$	−9.35453	−0.312513
$897$	0	0
$898$	30.1528	1.00621
$899$	15.2138	0.507408
$900$	0	0
$901$	2.48534	0.0827986
$902$	1.46613	0.0488169
$903$	0	0
$904$	28.7275	0.955463
$905$	−6.06268	−0.201530
$906$	0	0
$907$	−15.6800	−0.520645	−0.260323	−	0.965522i	$-0.583829\pi$
$907$	−15.6800	−0.520645	−0.260323	+	0.965522i	$0.583829\pi$
$908$	−10.5237	−0.349241
$909$	0	0
$910$	−2.20293	−0.0730264
$911$	−23.8935	−0.791627	−0.395814	−	0.918331i	$-0.629537\pi$
$911$	−23.8935	−0.791627	−0.395814	+	0.918331i	$0.629537\pi$
$912$	0	0
$913$	2.72258	0.0901042
$914$	−3.61115	−0.119446
$915$	0	0
$916$	37.8186	1.24956
$917$	−6.34339	−0.209477
$918$	0	0
$919$	−18.0890	−0.596702	−0.298351	−	0.954456i	$-0.596437\pi$
$919$	−18.0890	−0.596702	−0.298351	+	0.954456i	$0.596437\pi$
$920$	−2.08005	−0.0685772
$921$	0	0
$922$	0.262840	0.00865616
$923$	2.30294	0.0758024
$924$	0	0
$925$	12.4511	0.409390
$926$	−26.4659	−0.869723
$927$	0	0
$928$	10.4562	0.343243
$929$	10.9992	0.360873	0.180437	−	0.983587i	$-0.442249\pi$
$929$	10.9992	0.360873	0.180437	+	0.983587i	$0.442249\pi$
$930$	0	0
$931$	1.40296	0.0459802
$932$	29.1706	0.955513
$933$	0	0
$934$	11.8033	0.386216
$935$	−1.71264	−0.0560094
$936$	0	0
$937$	51.9182	1.69609	0.848046	−	0.529923i	$-0.177779\pi$
$937$	51.9182	1.69609	0.848046	+	0.529923i	$0.177779\pi$
$938$	−4.48313	−0.146379
$939$	0	0
$940$	−2.50159	−0.0815930
$941$	−13.1964	−0.430192	−0.215096	−	0.976593i	$-0.569006\pi$
$941$	−13.1964	−0.430192	−0.215096	+	0.976593i	$0.569006\pi$
$942$	0	0
$943$	2.17997	0.0709895
$944$	−9.84367	−0.320384
$945$	0	0
$946$	3.55462	0.115571
$947$	−39.0756	−1.26979	−0.634893	−	0.772600i	$-0.718956\pi$
$947$	−39.0756	−1.26979	−0.634893	+	0.772600i	$0.718956\pi$
$948$	0	0
$949$	34.1008	1.10696
$950$	−4.68568	−0.152024
$951$	0	0
$952$	6.59808	0.213845
$953$	4.31917	0.139912	0.0699559	−	0.997550i	$-0.477714\pi$
$953$	4.31917	0.139912	0.0699559	+	0.997550i	$0.477714\pi$
$954$	0	0
$955$	−1.85090	−0.0598938
$956$	−29.0460	−0.939415
$957$	0	0
$958$	−32.3947	−1.04663
$959$	−16.5664	−0.534957
$960$	0	0
$961$	41.5808	1.34132
$962$	7.64574	0.246509
$963$	0	0
$964$	37.3708	1.20363
$965$	−12.8546	−0.413804
$966$	0	0
$967$	53.6032	1.72376	0.861881	−	0.507110i	$-0.169286\pi$
$967$	53.6032	1.72376	0.861881	+	0.507110i	$0.169286\pi$
$968$	27.2179	0.874817
$969$	0	0
$970$	−2.94055	−0.0944155
$971$	5.46228	0.175293	0.0876465	−	0.996152i	$-0.472065\pi$
$971$	5.46228	0.175293	0.0876465	+	0.996152i	$0.472065\pi$
$972$	0	0
$973$	−14.6965	−0.471149
$974$	9.32359	0.298747
$975$	0	0
$976$	−5.41399	−0.173298
$977$	32.4956	1.03963	0.519813	−	0.854280i	$-0.326001\pi$
$977$	32.4956	1.03963	0.519813	+	0.854280i	$0.326001\pi$
$978$	0	0
$979$	−1.14288	−0.0365265
$980$	−1.16506	−0.0372165
$981$	0	0
$982$	15.8383	0.505422
$983$	3.17977	0.101419	0.0507095	−	0.998713i	$-0.483852\pi$
$983$	3.17977	0.101419	0.0507095	+	0.998713i	$0.483852\pi$
$984$	0	0
$985$	10.6669	0.339875
$986$	−3.46717	−0.110417
$987$	0	0
$988$	6.68745	0.212756
$989$	5.28529	0.168062
$990$	0	0
$991$	−38.3965	−1.21971	−0.609853	−	0.792515i	$-0.708771\pi$
$991$	−38.3965	−1.21971	−0.609853	+	0.792515i	$0.708771\pi$
$992$	49.8838	1.58381
$993$	0	0
$994$	−0.524029	−0.0166212
$995$	3.24199	0.102778
$996$	0	0
$997$	24.5944	0.778912	0.389456	−	0.921045i	$-0.372663\pi$
$997$	24.5944	0.778912	0.389456	+	0.921045i	$0.372663\pi$
$998$	−18.4344	−0.583530
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.16	yes	22
3.2	odd	2	inner	8001.2.a.x.1.7	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.7	✓	22	3.2	odd	2	inner
8001.2.a.x.1.16	yes	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.775659 q^{2} -1.39835 q^{4} +0.833166 q^{5} +1.00000 q^{7} -2.63596 q^{8} +O(q^{10})\)	\(q+0.775659 q^{2} -1.39835 q^{4} +0.833166 q^{5} +1.00000 q^{7} -2.63596 q^{8} +0.646253 q^{10} +0.821214 q^{11} -3.40878 q^{13} +0.775659 q^{14} +0.752101 q^{16} -2.50310 q^{17} +1.40296 q^{19} -1.16506 q^{20} +0.636982 q^{22} +0.947115 q^{23} -4.30583 q^{25} -2.64405 q^{26} -1.39835 q^{28} +1.78577 q^{29} +8.51944 q^{31} +5.85530 q^{32} -1.94155 q^{34} +0.833166 q^{35} -2.89168 q^{37} +1.08822 q^{38} -2.19620 q^{40} +2.30169 q^{41} +5.58041 q^{43} -1.14835 q^{44} +0.734638 q^{46} +2.14718 q^{47} +1.00000 q^{49} -3.33986 q^{50} +4.76668 q^{52} -0.992903 q^{53} +0.684208 q^{55} -2.63596 q^{56} +1.38515 q^{58} -13.0882 q^{59} -7.19849 q^{61} +6.60817 q^{62} +3.03751 q^{64} -2.84008 q^{65} -5.77977 q^{67} +3.50022 q^{68} +0.646253 q^{70} -0.675593 q^{71} -10.0038 q^{73} -2.24296 q^{74} -1.96183 q^{76} +0.821214 q^{77} +1.57957 q^{79} +0.626625 q^{80} +1.78533 q^{82} +3.31531 q^{83} -2.08550 q^{85} +4.32849 q^{86} -2.16469 q^{88} -1.39169 q^{89} -3.40878 q^{91} -1.32440 q^{92} +1.66548 q^{94} +1.16890 q^{95} -4.55016 q^{97} +0.775659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 22 * q + 10 * q^4 + 22 * q^7	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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