LMFDB - Embedded newform 4016.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +O(q^{10})$	$q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +0.366288 q^{11} -3.69132 q^{13} +1.05396 q^{15} +5.91866 q^{17} -7.61514 q^{19} +0.891542 q^{21} +6.61559 q^{23} -4.89958 q^{25} -16.8337 q^{27} +5.48804 q^{29} +4.15696 q^{31} -1.21823 q^{33} +0.0849482 q^{35} +8.30100 q^{37} +12.2769 q^{39} +3.92223 q^{41} -6.01718 q^{43} -2.55464 q^{45} -1.79035 q^{47} -6.92814 q^{49} -19.6847 q^{51} -10.5522 q^{53} -0.116076 q^{55} +25.3270 q^{57} +3.99709 q^{59} +1.88567 q^{61} -2.16097 q^{63} +1.16977 q^{65} -1.28140 q^{67} -22.0026 q^{69} -8.15695 q^{71} -16.0154 q^{73} +16.2954 q^{75} -0.0981882 q^{77} -7.93927 q^{79} +31.8024 q^{81} +11.7328 q^{83} -1.87560 q^{85} -18.2525 q^{87} -1.39073 q^{89} +0.989507 q^{91} -13.8255 q^{93} +2.41321 q^{95} -9.53422 q^{97} +2.95281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

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	0
$2$	0	0
$3$	−3.32587	−1.92019	−0.960097	−	0.279668i	$-0.909776\pi$
$3$	−3.32587	−1.92019	−0.960097	+	0.279668i	$0.909776\pi$
$4$	0	0
$5$	−0.316897	−0.141721	−0.0708603	−	0.997486i	$-0.522574\pi$
$5$	−0.316897	−0.141721	−0.0708603	+	0.997486i	$0.522574\pi$
$6$	0	0
$7$	−0.268063	−0.101318	−0.0506591	−	0.998716i	$-0.516132\pi$
$7$	−0.268063	−0.101318	−0.0506591	+	0.998716i	$0.516132\pi$
$8$	0	0
$9$	8.06143	2.68714
$10$	0	0
$11$	0.366288	0.110440	0.0552200	−	0.998474i	$-0.482414\pi$
$11$	0.366288	0.110440	0.0552200	+	0.998474i	$0.482414\pi$
$12$	0	0
$13$	−3.69132	−1.02379	−0.511895	−	0.859048i	$-0.671056\pi$
$13$	−3.69132	−1.02379	−0.511895	+	0.859048i	$0.671056\pi$
$14$	0	0
$15$	1.05396	0.272131
$16$	0	0
$17$	5.91866	1.43549	0.717743	−	0.696308i	$-0.245175\pi$
$17$	5.91866	1.43549	0.717743	+	0.696308i	$0.245175\pi$
$18$	0	0
$19$	−7.61514	−1.74703	−0.873517	−	0.486794i	$-0.838166\pi$
$19$	−7.61514	−1.74703	−0.873517	+	0.486794i	$0.838166\pi$
$20$	0	0
$21$	0.891542	0.194551
$22$	0	0
$23$	6.61559	1.37945	0.689723	−	0.724073i	$-0.257732\pi$
$23$	6.61559	1.37945	0.689723	+	0.724073i	$0.257732\pi$
$24$	0	0
$25$	−4.89958	−0.979915
$26$	0	0
$27$	−16.8337	−3.23964
$28$	0	0
$29$	5.48804	1.01910	0.509552	−	0.860440i	$-0.329811\pi$
$29$	5.48804	1.01910	0.509552	+	0.860440i	$0.329811\pi$
$30$	0	0
$31$	4.15696	0.746613	0.373306	−	0.927708i	$-0.378224\pi$
$31$	4.15696	0.746613	0.373306	+	0.927708i	$0.378224\pi$
$32$	0	0
$33$	−1.21823	−0.212066
$34$	0	0
$35$	0.0849482	0.0143589
$36$	0	0
$37$	8.30100	1.36468	0.682338	−	0.731037i	$-0.260963\pi$
$37$	8.30100	1.36468	0.682338	+	0.731037i	$0.260963\pi$
$38$	0	0
$39$	12.2769	1.96587
$40$	0	0
$41$	3.92223	0.612550	0.306275	−	0.951943i	$-0.400917\pi$
$41$	3.92223	0.612550	0.306275	+	0.951943i	$0.400917\pi$
$42$	0	0
$43$	−6.01718	−0.917611	−0.458805	−	0.888537i	$-0.651722\pi$
$43$	−6.01718	−0.917611	−0.458805	+	0.888537i	$0.651722\pi$
$44$	0	0
$45$	−2.55464	−0.380823
$46$	0	0
$47$	−1.79035	−0.261149	−0.130574	−	0.991439i	$-0.541682\pi$
$47$	−1.79035	−0.261149	−0.130574	+	0.991439i	$0.541682\pi$
$48$	0	0
$49$	−6.92814	−0.989735
$50$	0	0
$51$	−19.6847	−2.75641
$52$	0	0
$53$	−10.5522	−1.44945	−0.724725	−	0.689039i	$-0.758033\pi$
$53$	−10.5522	−1.44945	−0.724725	+	0.689039i	$0.758033\pi$
$54$	0	0
$55$	−0.116076	−0.0156516
$56$	0	0
$57$	25.3270	3.35464
$58$	0	0
$59$	3.99709	0.520377	0.260188	−	0.965558i	$-0.416215\pi$
$59$	3.99709	0.520377	0.260188	+	0.965558i	$0.416215\pi$
$60$	0	0
$61$	1.88567	0.241436	0.120718	−	0.992687i	$-0.461480\pi$
$61$	1.88567	0.241436	0.120718	+	0.992687i	$0.461480\pi$
$62$	0	0
$63$	−2.16097	−0.272256
$64$	0	0
$65$	1.16977	0.145092
$66$	0	0
$67$	−1.28140	−0.156548	−0.0782739	−	0.996932i	$-0.524941\pi$
$67$	−1.28140	−0.156548	−0.0782739	+	0.996932i	$0.524941\pi$
$68$	0	0
$69$	−22.0026	−2.64880
$70$	0	0
$71$	−8.15695	−0.968052	−0.484026	−	0.875054i	$-0.660826\pi$
$71$	−8.15695	−0.968052	−0.484026	+	0.875054i	$0.660826\pi$
$72$	0	0
$73$	−16.0154	−1.87446	−0.937230	−	0.348712i	$-0.886619\pi$
$73$	−16.0154	−1.87446	−0.937230	+	0.348712i	$0.886619\pi$
$74$	0	0
$75$	16.2954	1.88163
$76$	0	0
$77$	−0.0981882	−0.0111896
$78$	0	0
$79$	−7.93927	−0.893237	−0.446619	−	0.894724i	$-0.647372\pi$
$79$	−7.93927	−0.893237	−0.446619	+	0.894724i	$0.647372\pi$
$80$	0	0
$81$	31.8024	3.53360
$82$	0	0
$83$	11.7328	1.28784	0.643919	−	0.765094i	$-0.277307\pi$
$83$	11.7328	1.28784	0.643919	+	0.765094i	$0.277307\pi$
$84$	0	0
$85$	−1.87560	−0.203438
$86$	0	0
$87$	−18.2525	−1.95688
$88$	0	0
$89$	−1.39073	−0.147417	−0.0737083	−	0.997280i	$-0.523483\pi$
$89$	−1.39073	−0.147417	−0.0737083	+	0.997280i	$0.523483\pi$
$90$	0	0
$91$	0.989507	0.103728
$92$	0	0
$93$	−13.8255	−1.43364
$94$	0	0
$95$	2.41321	0.247591
$96$	0	0
$97$	−9.53422	−0.968053	−0.484027	−	0.875053i	$-0.660826\pi$
$97$	−9.53422	−0.968053	−0.484027	+	0.875053i	$0.660826\pi$
$98$	0	0
$99$	2.95281	0.296768
$100$	0	0
$101$	−6.51231	−0.647999	−0.323999	−	0.946057i	$-0.605028\pi$
$101$	−6.51231	−0.647999	−0.323999	+	0.946057i	$0.605028\pi$
$102$	0	0
$103$	−7.24438	−0.713810	−0.356905	−	0.934141i	$-0.616168\pi$
$103$	−7.24438	−0.713810	−0.356905	+	0.934141i	$0.616168\pi$
$104$	0	0
$105$	−0.282527	−0.0275718
$106$	0	0
$107$	−9.89532	−0.956617	−0.478308	−	0.878192i	$-0.658750\pi$
$107$	−9.89532	−0.956617	−0.478308	+	0.878192i	$0.658750\pi$
$108$	0	0
$109$	14.9095	1.42807	0.714034	−	0.700111i	$-0.246866\pi$
$109$	14.9095	1.42807	0.714034	+	0.700111i	$0.246866\pi$
$110$	0	0
$111$	−27.6081	−2.62044
$112$	0	0
$113$	15.9835	1.50360	0.751798	−	0.659393i	$-0.229187\pi$
$113$	15.9835	1.50360	0.751798	+	0.659393i	$0.229187\pi$
$114$	0	0
$115$	−2.09646	−0.195496
$116$	0	0
$117$	−29.7574	−2.75107
$118$	0	0
$119$	−1.58657	−0.145441
$120$	0	0
$121$	−10.8658	−0.987803
$122$	0	0
$123$	−13.0448	−1.17621
$124$	0	0
$125$	3.13714	0.280595
$126$	0	0
$127$	9.21799	0.817964	0.408982	−	0.912542i	$-0.365884\pi$
$127$	9.21799	0.817964	0.408982	+	0.912542i	$0.365884\pi$
$128$	0	0
$129$	20.0124	1.76199
$130$	0	0
$131$	21.7817	1.90307	0.951536	−	0.307538i	$-0.0995050\pi$
$131$	21.7817	1.90307	0.951536	+	0.307538i	$0.0995050\pi$
$132$	0	0
$133$	2.04134	0.177006
$134$	0	0
$135$	5.33454	0.459124
$136$	0	0
$137$	7.50480	0.641178	0.320589	−	0.947218i	$-0.396119\pi$
$137$	7.50480	0.641178	0.320589	+	0.947218i	$0.396119\pi$
$138$	0	0
$139$	7.54416	0.639887	0.319944	−	0.947437i	$-0.396336\pi$
$139$	7.54416	0.639887	0.319944	+	0.947437i	$0.396336\pi$
$140$	0	0
$141$	5.95447	0.501457
$142$	0	0
$143$	−1.35209	−0.113067
$144$	0	0
$145$	−1.73914	−0.144428
$146$	0	0
$147$	23.0421	1.90048
$148$	0	0
$149$	−4.41372	−0.361586	−0.180793	−	0.983521i	$-0.557866\pi$
$149$	−4.41372	−0.361586	−0.180793	+	0.983521i	$0.557866\pi$
$150$	0	0
$151$	20.7013	1.68464	0.842322	−	0.538975i	$-0.181188\pi$
$151$	20.7013	1.68464	0.842322	+	0.538975i	$0.181188\pi$
$152$	0	0
$153$	47.7129	3.85736
$154$	0	0
$155$	−1.31733	−0.105810
$156$	0	0
$157$	−18.1328	−1.44715	−0.723576	−	0.690245i	$-0.757503\pi$
$157$	−18.1328	−1.44715	−0.723576	+	0.690245i	$0.757503\pi$
$158$	0	0
$159$	35.0951	2.78322
$160$	0	0
$161$	−1.77339	−0.139763
$162$	0	0
$163$	0.195775	0.0153343	0.00766715	−	0.999971i	$-0.497559\pi$
$163$	0.195775	0.0153343	0.00766715	+	0.999971i	$0.497559\pi$
$164$	0	0
$165$	0.386052	0.0300541
$166$	0	0
$167$	1.69068	0.130829	0.0654144	−	0.997858i	$-0.479163\pi$
$167$	1.69068	0.130829	0.0654144	+	0.997858i	$0.479163\pi$
$168$	0	0
$169$	0.625880	0.0481446
$170$	0	0
$171$	−61.3889	−4.69453
$172$	0	0
$173$	2.39382	0.181999	0.0909995	−	0.995851i	$-0.470994\pi$
$173$	2.39382	0.181999	0.0909995	+	0.995851i	$0.470994\pi$
$174$	0	0
$175$	1.31339	0.0992832
$176$	0	0
$177$	−13.2938	−0.999224
$178$	0	0
$179$	22.2730	1.66476	0.832380	−	0.554205i	$-0.186978\pi$
$179$	22.2730	1.66476	0.832380	+	0.554205i	$0.186978\pi$
$180$	0	0
$181$	18.2384	1.35565	0.677827	−	0.735222i	$-0.262922\pi$
$181$	18.2384	1.35565	0.677827	+	0.735222i	$0.262922\pi$
$182$	0	0
$183$	−6.27151	−0.463603
$184$	0	0
$185$	−2.63056	−0.193403
$186$	0	0
$187$	2.16793	0.158535
$188$	0	0
$189$	4.51248	0.328235
$190$	0	0
$191$	11.0429	0.799033	0.399517	−	0.916726i	$-0.369178\pi$
$191$	11.0429	0.799033	0.399517	+	0.916726i	$0.369178\pi$
$192$	0	0
$193$	−1.08590	−0.0781646	−0.0390823	−	0.999236i	$-0.512443\pi$
$193$	−1.08590	−0.0781646	−0.0390823	+	0.999236i	$0.512443\pi$
$194$	0	0
$195$	−3.89050	−0.278605
$196$	0	0
$197$	23.2852	1.65901	0.829503	−	0.558503i	$-0.188624\pi$
$197$	23.2852	1.65901	0.829503	+	0.558503i	$0.188624\pi$
$198$	0	0
$199$	17.8332	1.26416	0.632080	−	0.774903i	$-0.282201\pi$
$199$	17.8332	1.26416	0.632080	+	0.774903i	$0.282201\pi$
$200$	0	0
$201$	4.26177	0.300602
$202$	0	0
$203$	−1.47114	−0.103254
$204$	0	0
$205$	−1.24294	−0.0868109
$206$	0	0
$207$	53.3311	3.70677
$208$	0	0
$209$	−2.78934	−0.192942
$210$	0	0
$211$	26.6400	1.83397	0.916986	−	0.398920i	$-0.130615\pi$
$211$	26.6400	1.83397	0.916986	+	0.398920i	$0.130615\pi$
$212$	0	0
$213$	27.1290	1.85885
$214$	0	0
$215$	1.90682	0.130044
$216$	0	0
$217$	−1.11433	−0.0756454
$218$	0	0
$219$	53.2652	3.59933
$220$	0	0
$221$	−21.8477	−1.46963
$222$	0	0
$223$	−21.2507	−1.42305	−0.711525	−	0.702661i	$-0.751995\pi$
$223$	−21.2507	−1.42305	−0.711525	+	0.702661i	$0.751995\pi$
$224$	0	0
$225$	−39.4976	−2.63317
$226$	0	0
$227$	−9.08131	−0.602747	−0.301374	−	0.953506i	$-0.597445\pi$
$227$	−9.08131	−0.602747	−0.301374	+	0.953506i	$0.597445\pi$
$228$	0	0
$229$	−17.7753	−1.17463	−0.587313	−	0.809360i	$-0.699814\pi$
$229$	−17.7753	−1.17463	−0.587313	+	0.809360i	$0.699814\pi$
$230$	0	0
$231$	0.326561	0.0214862
$232$	0	0
$233$	−8.94497	−0.586004	−0.293002	−	0.956112i	$-0.594654\pi$
$233$	−8.94497	−0.586004	−0.293002	+	0.956112i	$0.594654\pi$
$234$	0	0
$235$	0.567355	0.0370102
$236$	0	0
$237$	26.4050	1.71519
$238$	0	0
$239$	−8.57703	−0.554802	−0.277401	−	0.960754i	$-0.589473\pi$
$239$	−8.57703	−0.554802	−0.277401	+	0.960754i	$0.589473\pi$
$240$	0	0
$241$	−0.371399	−0.0239239	−0.0119620	−	0.999928i	$-0.503808\pi$
$241$	−0.371399	−0.0239239	−0.0119620	+	0.999928i	$0.503808\pi$
$242$	0	0
$243$	−55.2696	−3.54555
$244$	0	0
$245$	2.19551	0.140266
$246$	0	0
$247$	28.1100	1.78859
$248$	0	0
$249$	−39.0217	−2.47290
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	2.42321	0.152346
$254$	0	0
$255$	6.23802	0.390640
$256$	0	0
$257$	19.0422	1.18782	0.593910	−	0.804531i	$-0.297583\pi$
$257$	19.0422	1.18782	0.593910	+	0.804531i	$0.297583\pi$
$258$	0	0
$259$	−2.22519	−0.138267
$260$	0	0
$261$	44.2414	2.73848
$262$	0	0
$263$	16.3403	1.00759	0.503795	−	0.863824i	$-0.331937\pi$
$263$	16.3403	1.00759	0.503795	+	0.863824i	$0.331937\pi$
$264$	0	0
$265$	3.34394	0.205417
$266$	0	0
$267$	4.62538	0.283068
$268$	0	0
$269$	22.5392	1.37424	0.687121	−	0.726543i	$-0.258874\pi$
$269$	22.5392	1.37424	0.687121	+	0.726543i	$0.258874\pi$
$270$	0	0
$271$	8.55125	0.519451	0.259726	−	0.965682i	$-0.416368\pi$
$271$	8.55125	0.519451	0.259726	+	0.965682i	$0.416368\pi$
$272$	0	0
$273$	−3.29097	−0.199179
$274$	0	0
$275$	−1.79466	−0.108222
$276$	0	0
$277$	−7.92242	−0.476012	−0.238006	−	0.971264i	$-0.576494\pi$
$277$	−7.92242	−0.476012	−0.238006	+	0.971264i	$0.576494\pi$
$278$	0	0
$279$	33.5111	2.00625
$280$	0	0
$281$	−5.74555	−0.342751	−0.171375	−	0.985206i	$-0.554821\pi$
$281$	−5.74555	−0.342751	−0.171375	+	0.985206i	$0.554821\pi$
$282$	0	0
$283$	−6.77595	−0.402789	−0.201394	−	0.979510i	$-0.564547\pi$
$283$	−6.77595	−0.402789	−0.201394	+	0.979510i	$0.564547\pi$
$284$	0	0
$285$	−8.02604	−0.475422
$286$	0	0
$287$	−1.05140	−0.0620624
$288$	0	0
$289$	18.0305	1.06062
$290$	0	0
$291$	31.7096	1.85885
$292$	0	0
$293$	−14.2153	−0.830464	−0.415232	−	0.909716i	$-0.636300\pi$
$293$	−14.2153	−0.830464	−0.415232	+	0.909716i	$0.636300\pi$
$294$	0	0
$295$	−1.26666	−0.0737481
$296$	0	0
$297$	−6.16597	−0.357786
$298$	0	0
$299$	−24.4203	−1.41226
$300$	0	0
$301$	1.61298	0.0929706
$302$	0	0
$303$	21.6591	1.24428
$304$	0	0
$305$	−0.597563	−0.0342164
$306$	0	0
$307$	23.7876	1.35763	0.678816	−	0.734308i	$-0.262493\pi$
$307$	23.7876	1.35763	0.678816	+	0.734308i	$0.262493\pi$
$308$	0	0
$309$	24.0939	1.37065
$310$	0	0
$311$	−9.00045	−0.510369	−0.255184	−	0.966892i	$-0.582136\pi$
$311$	−9.00045	−0.510369	−0.255184	+	0.966892i	$0.582136\pi$
$312$	0	0
$313$	22.8553	1.29186	0.645928	−	0.763398i	$-0.276471\pi$
$313$	22.8553	1.29186	0.645928	+	0.763398i	$0.276471\pi$
$314$	0	0
$315$	0.684804	0.0385843
$316$	0	0
$317$	24.6076	1.38210	0.691051	−	0.722806i	$-0.257148\pi$
$317$	24.6076	1.38210	0.691051	+	0.722806i	$0.257148\pi$
$318$	0	0
$319$	2.01020	0.112550
$320$	0	0
$321$	32.9106	1.83689
$322$	0	0
$323$	−45.0714	−2.50784
$324$	0	0
$325$	18.0859	1.00323
$326$	0	0
$327$	−49.5870	−2.74217
$328$	0	0
$329$	0.479925	0.0264591
$330$	0	0
$331$	17.7024	0.973013	0.486507	−	0.873677i	$-0.338271\pi$
$331$	17.7024	0.973013	0.486507	+	0.873677i	$0.338271\pi$
$332$	0	0
$333$	66.9179	3.66708
$334$	0	0
$335$	0.406071	0.0221860
$336$	0	0
$337$	32.2381	1.75612	0.878060	−	0.478551i	$-0.158838\pi$
$337$	32.2381	1.75612	0.878060	+	0.478551i	$0.158838\pi$
$338$	0	0
$339$	−53.1589	−2.88720
$340$	0	0
$341$	1.52265	0.0824559
$342$	0	0
$343$	3.73362	0.201596
$344$	0	0
$345$	6.97256	0.375390
$346$	0	0
$347$	7.82960	0.420315	0.210157	−	0.977668i	$-0.432602\pi$
$347$	7.82960	0.420315	0.210157	+	0.977668i	$0.432602\pi$
$348$	0	0
$349$	3.97080	0.212552	0.106276	−	0.994337i	$-0.466107\pi$
$349$	3.97080	0.212552	0.106276	+	0.994337i	$0.466107\pi$
$350$	0	0
$351$	62.1385	3.31671
$352$	0	0
$353$	0.605256	0.0322145	0.0161073	−	0.999870i	$-0.494873\pi$
$353$	0.605256	0.0322145	0.0161073	+	0.999870i	$0.494873\pi$
$354$	0	0
$355$	2.58491	0.137193
$356$	0	0
$357$	5.27674	0.279274
$358$	0	0
$359$	−20.2211	−1.06723	−0.533614	−	0.845728i	$-0.679167\pi$
$359$	−20.2211	−1.06723	−0.533614	+	0.845728i	$0.679167\pi$
$360$	0	0
$361$	38.9904	2.05213
$362$	0	0
$363$	36.1384	1.89677
$364$	0	0
$365$	5.07523	0.265650
$366$	0	0
$367$	−22.9860	−1.19986	−0.599931	−	0.800052i	$-0.704805\pi$
$367$	−22.9860	−1.19986	−0.599931	+	0.800052i	$0.704805\pi$
$368$	0	0
$369$	31.6188	1.64601
$370$	0	0
$371$	2.82864	0.146856
$372$	0	0
$373$	−32.6800	−1.69211	−0.846054	−	0.533098i	$-0.821028\pi$
$373$	−32.6800	−1.69211	−0.846054	+	0.533098i	$0.821028\pi$
$374$	0	0
$375$	−10.4337	−0.538796
$376$	0	0
$377$	−20.2581	−1.04335
$378$	0	0
$379$	−2.52675	−0.129790	−0.0648951	−	0.997892i	$-0.520671\pi$
$379$	−2.52675	−0.129790	−0.0648951	+	0.997892i	$0.520671\pi$
$380$	0	0
$381$	−30.6579	−1.57065
$382$	0	0
$383$	−26.2117	−1.33935	−0.669677	−	0.742652i	$-0.733568\pi$
$383$	−26.2117	−1.33935	−0.669677	+	0.742652i	$0.733568\pi$
$384$	0	0
$385$	0.0311155	0.00158579
$386$	0	0
$387$	−48.5070	−2.46575
$388$	0	0
$389$	−23.0102	−1.16667	−0.583333	−	0.812233i	$-0.698252\pi$
$389$	−23.0102	−1.16667	−0.583333	+	0.812233i	$0.698252\pi$
$390$	0	0
$391$	39.1554	1.98018
$392$	0	0
$393$	−72.4430	−3.65427
$394$	0	0
$395$	2.51593	0.126590
$396$	0	0
$397$	−1.25813	−0.0631439	−0.0315719	−	0.999501i	$-0.510051\pi$
$397$	−1.25813	−0.0631439	−0.0315719	+	0.999501i	$0.510051\pi$
$398$	0	0
$399$	−6.78922	−0.339886
$400$	0	0
$401$	4.13472	0.206478	0.103239	−	0.994657i	$-0.467079\pi$
$401$	4.13472	0.206478	0.103239	+	0.994657i	$0.467079\pi$
$402$	0	0
$403$	−15.3447	−0.764374
$404$	0	0
$405$	−10.0781	−0.500783
$406$	0	0
$407$	3.04056	0.150715
$408$	0	0
$409$	−12.8022	−0.633030	−0.316515	−	0.948588i	$-0.602513\pi$
$409$	−12.8022	−0.633030	−0.316515	+	0.948588i	$0.602513\pi$
$410$	0	0
$411$	−24.9600	−1.23119
$412$	0	0
$413$	−1.07147	−0.0527236
$414$	0	0
$415$	−3.71807	−0.182513
$416$	0	0
$417$	−25.0909	−1.22871
$418$	0	0
$419$	17.3329	0.846767	0.423384	−	0.905950i	$-0.360842\pi$
$419$	17.3329	0.846767	0.423384	+	0.905950i	$0.360842\pi$
$420$	0	0
$421$	−10.3663	−0.505222	−0.252611	−	0.967568i	$-0.581289\pi$
$421$	−10.3663	−0.505222	−0.252611	+	0.967568i	$0.581289\pi$
$422$	0	0
$423$	−14.4328	−0.701745
$424$	0	0
$425$	−28.9989	−1.40665
$426$	0	0
$427$	−0.505478	−0.0244618
$428$	0	0
$429$	4.49687	0.217111
$430$	0	0
$431$	3.16300	0.152357	0.0761783	−	0.997094i	$-0.475728\pi$
$431$	3.16300	0.152357	0.0761783	+	0.997094i	$0.475728\pi$
$432$	0	0
$433$	−18.9740	−0.911832	−0.455916	−	0.890023i	$-0.650688\pi$
$433$	−18.9740	−0.911832	−0.455916	+	0.890023i	$0.650688\pi$
$434$	0	0
$435$	5.78417	0.277330
$436$	0	0
$437$	−50.3787	−2.40994
$438$	0	0
$439$	29.5408	1.40991	0.704953	−	0.709254i	$-0.250968\pi$
$439$	29.5408	1.40991	0.704953	+	0.709254i	$0.250968\pi$
$440$	0	0
$441$	−55.8507	−2.65956
$442$	0	0
$443$	−12.3408	−0.586330	−0.293165	−	0.956062i	$-0.594709\pi$
$443$	−12.3408	−0.586330	−0.293165	+	0.956062i	$0.594709\pi$
$444$	0	0
$445$	0.440717	0.0208920
$446$	0	0
$447$	14.6795	0.694315
$448$	0	0
$449$	36.3313	1.71458	0.857289	−	0.514835i	$-0.172147\pi$
$449$	36.3313	1.71458	0.857289	+	0.514835i	$0.172147\pi$
$450$	0	0
$451$	1.43667	0.0676500
$452$	0	0
$453$	−68.8497	−3.23484
$454$	0	0
$455$	−0.313571	−0.0147005
$456$	0	0
$457$	12.9686	0.606646	0.303323	−	0.952888i	$-0.401904\pi$
$457$	12.9686	0.606646	0.303323	+	0.952888i	$0.401904\pi$
$458$	0	0
$459$	−99.6328	−4.65046
$460$	0	0
$461$	−6.19845	−0.288690	−0.144345	−	0.989527i	$-0.546108\pi$
$461$	−6.19845	−0.288690	−0.144345	+	0.989527i	$0.546108\pi$
$462$	0	0
$463$	35.9488	1.67068	0.835340	−	0.549733i	$-0.185271\pi$
$463$	35.9488	1.67068	0.835340	+	0.549733i	$0.185271\pi$
$464$	0	0
$465$	4.38127	0.203176
$466$	0	0
$467$	12.3324	0.570677	0.285339	−	0.958427i	$-0.407894\pi$
$467$	12.3324	0.570677	0.285339	+	0.958427i	$0.407894\pi$
$468$	0	0
$469$	0.343495	0.0158611
$470$	0	0
$471$	60.3072	2.77881
$472$	0	0
$473$	−2.20402	−0.101341
$474$	0	0
$475$	37.3110	1.71194
$476$	0	0
$477$	−85.0654	−3.89488
$478$	0	0
$479$	−20.4813	−0.935813	−0.467906	−	0.883778i	$-0.654992\pi$
$479$	−20.4813	−0.935813	−0.467906	+	0.883778i	$0.654992\pi$
$480$	0	0
$481$	−30.6417	−1.39714
$482$	0	0
$483$	5.89808	0.268372
$484$	0	0
$485$	3.02136	0.137193
$486$	0	0
$487$	13.4600	0.609930	0.304965	−	0.952364i	$-0.401355\pi$
$487$	13.4600	0.609930	0.304965	+	0.952364i	$0.401355\pi$
$488$	0	0
$489$	−0.651124	−0.0294448
$490$	0	0
$491$	−2.63215	−0.118787	−0.0593937	−	0.998235i	$-0.518917\pi$
$491$	−2.63215	−0.118787	−0.0593937	+	0.998235i	$0.518917\pi$
$492$	0	0
$493$	32.4818	1.46291
$494$	0	0
$495$	−0.935735	−0.0420581
$496$	0	0
$497$	2.18657	0.0980813
$498$	0	0
$499$	43.1846	1.93321	0.966604	−	0.256274i	$-0.0824950\pi$
$499$	43.1846	1.93321	0.966604	+	0.256274i	$0.0824950\pi$
$500$	0	0
$501$	−5.62299	−0.251217
$502$	0	0
$503$	4.19813	0.187185	0.0935926	−	0.995611i	$-0.470165\pi$
$503$	4.19813	0.187185	0.0935926	+	0.995611i	$0.470165\pi$
$504$	0	0
$505$	2.06373	0.0918348
$506$	0	0
$507$	−2.08160	−0.0924470
$508$	0	0
$509$	36.4290	1.61469	0.807344	−	0.590081i	$-0.200904\pi$
$509$	36.4290	1.61469	0.807344	+	0.590081i	$0.200904\pi$
$510$	0	0
$511$	4.29313	0.189917
$512$	0	0
$513$	128.191	5.65976
$514$	0	0
$515$	2.29572	0.101162
$516$	0	0
$517$	−0.655783	−0.0288413
$518$	0	0
$519$	−7.96155	−0.349473
$520$	0	0
$521$	−4.13667	−0.181231	−0.0906155	−	0.995886i	$-0.528883\pi$
$521$	−4.13667	−0.181231	−0.0906155	+	0.995886i	$0.528883\pi$
$522$	0	0
$523$	−9.67212	−0.422932	−0.211466	−	0.977385i	$-0.567824\pi$
$523$	−9.67212	−0.422932	−0.211466	+	0.977385i	$0.567824\pi$
$524$	0	0
$525$	−4.36818	−0.190643
$526$	0	0
$527$	24.6036	1.07175
$528$	0	0
$529$	20.7661	0.902873
$530$	0	0
$531$	32.2223	1.39833
$532$	0	0
$533$	−14.4782	−0.627122
$534$	0	0
$535$	3.13580	0.135572
$536$	0	0
$537$	−74.0771	−3.19666
$538$	0	0
$539$	−2.53770	−0.109306
$540$	0	0
$541$	15.6611	0.673324	0.336662	−	0.941626i	$-0.390702\pi$
$541$	15.6611	0.673324	0.336662	+	0.941626i	$0.390702\pi$
$542$	0	0
$543$	−60.6588	−2.60312
$544$	0	0
$545$	−4.72476	−0.202387
$546$	0	0
$547$	10.8854	0.465424	0.232712	−	0.972546i	$-0.425240\pi$
$547$	10.8854	0.465424	0.232712	+	0.972546i	$0.425240\pi$
$548$	0	0
$549$	15.2012	0.648772
$550$	0	0
$551$	−41.7922	−1.78041
$552$	0	0
$553$	2.12822	0.0905012
$554$	0	0
$555$	8.74891	0.371371
$556$	0	0
$557$	2.76645	0.117218	0.0586091	−	0.998281i	$-0.481333\pi$
$557$	2.76645	0.117218	0.0586091	+	0.998281i	$0.481333\pi$
$558$	0	0
$559$	22.2114	0.939440
$560$	0	0
$561$	−7.21027	−0.304418
$562$	0	0
$563$	6.67356	0.281257	0.140629	−	0.990062i	$-0.455088\pi$
$563$	6.67356	0.281257	0.140629	+	0.990062i	$0.455088\pi$
$564$	0	0
$565$	−5.06511	−0.213091
$566$	0	0
$567$	−8.52503	−0.358017
$568$	0	0
$569$	6.27226	0.262947	0.131473	−	0.991320i	$-0.458029\pi$
$569$	6.27226	0.262947	0.131473	+	0.991320i	$0.458029\pi$
$570$	0	0
$571$	33.7487	1.41234	0.706169	−	0.708043i	$-0.250422\pi$
$571$	33.7487	1.41234	0.706169	+	0.708043i	$0.250422\pi$
$572$	0	0
$573$	−36.7272	−1.53430
$574$	0	0
$575$	−32.4136	−1.35174
$576$	0	0
$577$	10.1858	0.424041	0.212021	−	0.977265i	$-0.431996\pi$
$577$	10.1858	0.424041	0.212021	+	0.977265i	$0.431996\pi$
$578$	0	0
$579$	3.61156	0.150091
$580$	0	0
$581$	−3.14511	−0.130481
$582$	0	0
$583$	−3.86513	−0.160077
$584$	0	0
$585$	9.43001	0.389883
$586$	0	0
$587$	−2.99433	−0.123589	−0.0617946	−	0.998089i	$-0.519682\pi$
$587$	−2.99433	−0.123589	−0.0617946	+	0.998089i	$0.519682\pi$
$588$	0	0
$589$	−31.6559	−1.30436
$590$	0	0
$591$	−77.4438	−3.18561
$592$	0	0
$593$	−0.634339	−0.0260492	−0.0130246	−	0.999915i	$-0.504146\pi$
$593$	−0.634339	−0.0260492	−0.0130246	+	0.999915i	$0.504146\pi$
$594$	0	0
$595$	0.502780	0.0206120
$596$	0	0
$597$	−59.3109	−2.42743
$598$	0	0
$599$	27.3229	1.11638	0.558192	−	0.829712i	$-0.311495\pi$
$599$	27.3229	1.11638	0.558192	+	0.829712i	$0.311495\pi$
$600$	0	0
$601$	37.4715	1.52849	0.764247	−	0.644924i	$-0.223111\pi$
$601$	37.4715	1.52849	0.764247	+	0.644924i	$0.223111\pi$
$602$	0	0
$603$	−10.3299	−0.420666
$604$	0	0
$605$	3.44335	0.139992
$606$	0	0
$607$	−11.8099	−0.479348	−0.239674	−	0.970853i	$-0.577041\pi$
$607$	−11.8099	−0.479348	−0.239674	+	0.970853i	$0.577041\pi$
$608$	0	0
$609$	4.89282	0.198267
$610$	0	0
$611$	6.60875	0.267362
$612$	0	0
$613$	−41.7932	−1.68801	−0.844005	−	0.536335i	$-0.819808\pi$
$613$	−41.7932	−1.68801	−0.844005	+	0.536335i	$0.819808\pi$
$614$	0	0
$615$	4.13387	0.166694
$616$	0	0
$617$	0.825343	0.0332270	0.0166135	−	0.999862i	$-0.494712\pi$
$617$	0.825343	0.0332270	0.0166135	+	0.999862i	$0.494712\pi$
$618$	0	0
$619$	−32.1673	−1.29291	−0.646456	−	0.762952i	$-0.723749\pi$
$619$	−32.1673	−1.29291	−0.646456	+	0.762952i	$0.723749\pi$
$620$	0	0
$621$	−111.365	−4.46891
$622$	0	0
$623$	0.372802	0.0149360
$624$	0	0
$625$	23.5037	0.940149
$626$	0	0
$627$	9.27697	0.370487
$628$	0	0
$629$	49.1308	1.95897
$630$	0	0
$631$	4.55806	0.181453	0.0907267	−	0.995876i	$-0.471081\pi$
$631$	4.55806	0.181453	0.0907267	+	0.995876i	$0.471081\pi$
$632$	0	0
$633$	−88.6012	−3.52158
$634$	0	0
$635$	−2.92115	−0.115922
$636$	0	0
$637$	25.5740	1.01328
$638$	0	0
$639$	−65.7567	−2.60129
$640$	0	0
$641$	−5.26084	−0.207791	−0.103895	−	0.994588i	$-0.533131\pi$
$641$	−5.26084	−0.207791	−0.103895	+	0.994588i	$0.533131\pi$
$642$	0	0
$643$	−7.88916	−0.311118	−0.155559	−	0.987827i	$-0.549718\pi$
$643$	−7.88916	−0.311118	−0.155559	+	0.987827i	$0.549718\pi$
$644$	0	0
$645$	−6.34185	−0.249710
$646$	0	0
$647$	13.2551	0.521113	0.260557	−	0.965459i	$-0.416094\pi$
$647$	13.2551	0.521113	0.260557	+	0.965459i	$0.416094\pi$
$648$	0	0
$649$	1.46409	0.0574704
$650$	0	0
$651$	3.70611	0.145254
$652$	0	0
$653$	1.98919	0.0778431	0.0389215	−	0.999242i	$-0.487608\pi$
$653$	1.98919	0.0778431	0.0389215	+	0.999242i	$0.487608\pi$
$654$	0	0
$655$	−6.90254	−0.269704
$656$	0	0
$657$	−129.107	−5.03694
$658$	0	0
$659$	27.2300	1.06073	0.530366	−	0.847769i	$-0.322055\pi$
$659$	27.2300	1.06073	0.530366	+	0.847769i	$0.322055\pi$
$660$	0	0
$661$	0.551019	0.0214322	0.0107161	−	0.999943i	$-0.496589\pi$
$661$	0.551019	0.0214322	0.0107161	+	0.999943i	$0.496589\pi$
$662$	0	0
$663$	72.6627	2.82198
$664$	0	0
$665$	−0.646893	−0.0250854
$666$	0	0
$667$	36.3066	1.40580
$668$	0	0
$669$	70.6770	2.73253
$670$	0	0
$671$	0.690699	0.0266641
$672$	0	0
$673$	11.0122	0.424488	0.212244	−	0.977217i	$-0.431923\pi$
$673$	11.0122	0.424488	0.212244	+	0.977217i	$0.431923\pi$
$674$	0	0
$675$	82.4779	3.17457
$676$	0	0
$677$	43.3014	1.66421	0.832105	−	0.554618i	$-0.187136\pi$
$677$	43.3014	1.66421	0.832105	+	0.554618i	$0.187136\pi$
$678$	0	0
$679$	2.55577	0.0980814
$680$	0	0
$681$	30.2033	1.15739
$682$	0	0
$683$	−45.7493	−1.75055	−0.875274	−	0.483627i	$-0.839319\pi$
$683$	−45.7493	−1.75055	−0.875274	+	0.483627i	$0.839319\pi$
$684$	0	0
$685$	−2.37825	−0.0908682
$686$	0	0
$687$	59.1184	2.25551
$688$	0	0
$689$	38.9514	1.48393
$690$	0	0
$691$	−12.2254	−0.465076	−0.232538	−	0.972587i	$-0.574703\pi$
$691$	−12.2254	−0.465076	−0.232538	+	0.972587i	$0.574703\pi$
$692$	0	0
$693$	−0.791537	−0.0300680
$694$	0	0
$695$	−2.39072	−0.0906852
$696$	0	0
$697$	23.2144	0.879307
$698$	0	0
$699$	29.7498	1.12524
$700$	0	0
$701$	−28.4854	−1.07588	−0.537939	−	0.842984i	$-0.680797\pi$
$701$	−28.4854	−1.07588	−0.537939	+	0.842984i	$0.680797\pi$
$702$	0	0
$703$	−63.2133	−2.38414
$704$	0	0
$705$	−1.88695	−0.0710667
$706$	0	0
$707$	1.74571	0.0656541
$708$	0	0
$709$	−11.5494	−0.433747	−0.216874	−	0.976200i	$-0.569586\pi$
$709$	−11.5494	−0.433747	−0.216874	+	0.976200i	$0.569586\pi$
$710$	0	0
$711$	−64.0018	−2.40026
$712$	0	0
$713$	27.5008	1.02991
$714$	0	0
$715$	0.428472	0.0160240
$716$	0	0
$717$	28.5261	1.06533
$718$	0	0
$719$	43.1550	1.60941	0.804704	−	0.593676i	$-0.202324\pi$
$719$	43.1550	1.60941	0.804704	+	0.593676i	$0.202324\pi$
$720$	0	0
$721$	1.94195	0.0723219
$722$	0	0
$723$	1.23523	0.0459385
$724$	0	0
$725$	−26.8891	−0.998635
$726$	0	0
$727$	18.8853	0.700416	0.350208	−	0.936672i	$-0.386111\pi$
$727$	18.8853	0.700416	0.350208	+	0.936672i	$0.386111\pi$
$728$	0	0
$729$	88.4125	3.27454
$730$	0	0
$731$	−35.6136	−1.31722
$732$	0	0
$733$	34.9088	1.28939	0.644693	−	0.764442i	$-0.276985\pi$
$733$	34.9088	1.28939	0.644693	+	0.764442i	$0.276985\pi$
$734$	0	0
$735$	−7.30197	−0.269337
$736$	0	0
$737$	−0.469361	−0.0172891
$738$	0	0
$739$	−5.98919	−0.220316	−0.110158	−	0.993914i	$-0.535136\pi$
$739$	−5.98919	−0.220316	−0.110158	+	0.993914i	$0.535136\pi$
$740$	0	0
$741$	−93.4902	−3.43445
$742$	0	0
$743$	1.68049	0.0616512	0.0308256	−	0.999525i	$-0.490186\pi$
$743$	1.68049	0.0616512	0.0308256	+	0.999525i	$0.490186\pi$
$744$	0	0
$745$	1.39869	0.0512442
$746$	0	0
$747$	94.5828	3.46060
$748$	0	0
$749$	2.65257	0.0969227
$750$	0	0
$751$	−19.3262	−0.705225	−0.352612	−	0.935770i	$-0.614707\pi$
$751$	−19.3262	−0.705225	−0.352612	+	0.935770i	$0.614707\pi$
$752$	0	0
$753$	−3.32587	−0.121202
$754$	0	0
$755$	−6.56016	−0.238749
$756$	0	0
$757$	−32.7816	−1.19147	−0.595734	−	0.803182i	$-0.703139\pi$
$757$	−32.7816	−1.19147	−0.595734	+	0.803182i	$0.703139\pi$
$758$	0	0
$759$	−8.05930	−0.292534
$760$	0	0
$761$	−5.10313	−0.184988	−0.0924942	−	0.995713i	$-0.529484\pi$
$761$	−5.10313	−0.184988	−0.0924942	+	0.995713i	$0.529484\pi$
$762$	0	0
$763$	−3.99667	−0.144689
$764$	0	0
$765$	−15.1201	−0.546667
$766$	0	0
$767$	−14.7546	−0.532756
$768$	0	0
$769$	−15.7298	−0.567232	−0.283616	−	0.958938i	$-0.591534\pi$
$769$	−15.7298	−0.567232	−0.283616	+	0.958938i	$0.591534\pi$
$770$	0	0
$771$	−63.3320	−2.28084
$772$	0	0
$773$	−15.2536	−0.548635	−0.274317	−	0.961639i	$-0.588452\pi$
$773$	−15.2536	−0.548635	−0.274317	+	0.961639i	$0.588452\pi$
$774$	0	0
$775$	−20.3674	−0.731617
$776$	0	0
$777$	7.40070	0.265498
$778$	0	0
$779$	−29.8684	−1.07015
$780$	0	0
$781$	−2.98779	−0.106912
$782$	0	0
$783$	−92.3839	−3.30153
$784$	0	0
$785$	5.74621	0.205091
$786$	0	0
$787$	−47.6696	−1.69924	−0.849619	−	0.527397i	$-0.823168\pi$
$787$	−47.6696	−1.69924	−0.849619	+	0.527397i	$0.823168\pi$
$788$	0	0
$789$	−54.3459	−1.93477
$790$	0	0
$791$	−4.28457	−0.152342
$792$	0	0
$793$	−6.96063	−0.247179
$794$	0	0
$795$	−11.1215	−0.394440
$796$	0	0
$797$	20.5952	0.729518	0.364759	−	0.931102i	$-0.381151\pi$
$797$	20.5952	0.729518	0.364759	+	0.931102i	$0.381151\pi$
$798$	0	0
$799$	−10.5965	−0.374876
$800$	0	0
$801$	−11.2112	−0.396130
$802$	0	0
$803$	−5.86625	−0.207015
$804$	0	0
$805$	0.561983	0.0198073
$806$	0	0
$807$	−74.9626	−2.63881
$808$	0	0
$809$	4.55199	0.160040	0.0800198	−	0.996793i	$-0.474502\pi$
$809$	4.55199	0.160040	0.0800198	+	0.996793i	$0.474502\pi$
$810$	0	0
$811$	5.19296	0.182349	0.0911747	−	0.995835i	$-0.470938\pi$
$811$	5.19296	0.182349	0.0911747	+	0.995835i	$0.470938\pi$
$812$	0	0
$813$	−28.4404	−0.997447
$814$	0	0
$815$	−0.0620406	−0.00217319
$816$	0	0
$817$	45.8216	1.60310
$818$	0	0
$819$	7.97684	0.278733
$820$	0	0
$821$	27.6078	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$821$	27.6078	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$822$	0	0
$823$	15.6633	0.545988	0.272994	−	0.962016i	$-0.411986\pi$
$823$	15.6633	0.545988	0.272994	+	0.962016i	$0.411986\pi$
$824$	0	0
$825$	5.96880	0.207807
$826$	0	0
$827$	−37.3658	−1.29933	−0.649667	−	0.760219i	$-0.725092\pi$
$827$	−37.3658	−1.29933	−0.649667	+	0.760219i	$0.725092\pi$
$828$	0	0
$829$	35.4986	1.23292	0.616459	−	0.787387i	$-0.288566\pi$
$829$	35.4986	1.23292	0.616459	+	0.787387i	$0.288566\pi$
$830$	0	0
$831$	26.3490	0.914036
$832$	0	0
$833$	−41.0053	−1.42075
$834$	0	0
$835$	−0.535771	−0.0185411
$836$	0	0
$837$	−69.9769	−2.41876
$838$	0	0
$839$	−13.2255	−0.456594	−0.228297	−	0.973592i	$-0.573316\pi$
$839$	−13.2255	−0.456594	−0.228297	+	0.973592i	$0.573316\pi$
$840$	0	0
$841$	1.11858	0.0385718
$842$	0	0
$843$	19.1090	0.658148
$844$	0	0
$845$	−0.198339	−0.00682308
$846$	0	0
$847$	2.91272	0.100082
$848$	0	0
$849$	22.5360	0.773432
$850$	0	0
$851$	54.9161	1.88250
$852$	0	0
$853$	−2.27716	−0.0779686	−0.0389843	−	0.999240i	$-0.512412\pi$
$853$	−2.27716	−0.0779686	−0.0389843	+	0.999240i	$0.512412\pi$
$854$	0	0
$855$	19.4540	0.665311
$856$	0	0
$857$	−20.4240	−0.697670	−0.348835	−	0.937184i	$-0.613423\pi$
$857$	−20.4240	−0.697670	−0.348835	+	0.937184i	$0.613423\pi$
$858$	0	0
$859$	0.815502	0.0278246	0.0139123	−	0.999903i	$-0.495571\pi$
$859$	0.815502	0.0278246	0.0139123	+	0.999903i	$0.495571\pi$
$860$	0	0
$861$	3.49684	0.119172
$862$	0	0
$863$	19.3454	0.658524	0.329262	−	0.944239i	$-0.393200\pi$
$863$	19.3454	0.658524	0.329262	+	0.944239i	$0.393200\pi$
$864$	0	0
$865$	−0.758595	−0.0257930
$866$	0	0
$867$	−59.9672	−2.03659
$868$	0	0
$869$	−2.90806	−0.0986491
$870$	0	0
$871$	4.73006	0.160272
$872$	0	0
$873$	−76.8594	−2.60130
$874$	0	0
$875$	−0.840951	−0.0284293
$876$	0	0
$877$	55.3799	1.87005	0.935023	−	0.354588i	$-0.115379\pi$
$877$	55.3799	1.87005	0.935023	+	0.354588i	$0.115379\pi$
$878$	0	0
$879$	47.2781	1.59465
$880$	0	0
$881$	11.2131	0.377779	0.188889	−	0.981998i	$-0.439511\pi$
$881$	11.2131	0.377779	0.188889	+	0.981998i	$0.439511\pi$
$882$	0	0
$883$	−21.0828	−0.709492	−0.354746	−	0.934963i	$-0.615433\pi$
$883$	−21.0828	−0.709492	−0.354746	+	0.934963i	$0.615433\pi$
$884$	0	0
$885$	4.21277	0.141611
$886$	0	0
$887$	−14.1584	−0.475391	−0.237696	−	0.971340i	$-0.576392\pi$
$887$	−14.1584	−0.475391	−0.237696	+	0.971340i	$0.576392\pi$
$888$	0	0
$889$	−2.47100	−0.0828746
$890$	0	0
$891$	11.6488	0.390250
$892$	0	0
$893$	13.6337	0.456236
$894$	0	0
$895$	−7.05824	−0.235931
$896$	0	0
$897$	81.2188	2.71182
$898$	0	0
$899$	22.8136	0.760875
$900$	0	0
$901$	−62.4546	−2.08066
$902$	0	0
$903$	−5.36457	−0.178522
$904$	0	0
$905$	−5.77971	−0.192124
$906$	0	0
$907$	15.9425	0.529363	0.264682	−	0.964336i	$-0.414733\pi$
$907$	15.9425	0.529363	0.264682	+	0.964336i	$0.414733\pi$
$908$	0	0
$909$	−52.4985	−1.74127
$910$	0	0
$911$	−3.68657	−0.122142	−0.0610708	−	0.998133i	$-0.519452\pi$
$911$	−3.68657	−0.122142	−0.0610708	+	0.998133i	$0.519452\pi$
$912$	0	0
$913$	4.29757	0.142229
$914$	0	0
$915$	1.98742	0.0657021
$916$	0	0
$917$	−5.83885	−0.192816
$918$	0	0
$919$	−31.4633	−1.03788	−0.518939	−	0.854811i	$-0.673673\pi$
$919$	−31.4633	−1.03788	−0.518939	+	0.854811i	$0.673673\pi$
$920$	0	0
$921$	−79.1146	−2.60692
$922$	0	0
$923$	30.1100	0.991082
$924$	0	0
$925$	−40.6714	−1.33727
$926$	0	0
$927$	−58.4000	−1.91811
$928$	0	0
$929$	15.5846	0.511315	0.255657	−	0.966767i	$-0.417708\pi$
$929$	15.5846	0.511315	0.255657	+	0.966767i	$0.417708\pi$
$930$	0	0
$931$	52.7588	1.72910
$932$	0	0
$933$	29.9344	0.980007
$934$	0	0
$935$	−0.687011	−0.0224677
$936$	0	0
$937$	−59.0343	−1.92857	−0.964283	−	0.264874i	$-0.914670\pi$
$937$	−59.0343	−1.92857	−0.964283	+	0.264874i	$0.914670\pi$
$938$	0	0
$939$	−76.0137	−2.48061
$940$	0	0
$941$	−26.6618	−0.869149	−0.434575	−	0.900636i	$-0.643101\pi$
$941$	−26.6618	−0.869149	−0.434575	+	0.900636i	$0.643101\pi$
$942$	0	0
$943$	25.9479	0.844980
$944$	0	0
$945$	−1.42999	−0.0465176
$946$	0	0
$947$	−31.3405	−1.01843	−0.509215	−	0.860640i	$-0.670064\pi$
$947$	−31.3405	−1.01843	−0.509215	+	0.860640i	$0.670064\pi$
$948$	0	0
$949$	59.1180	1.91905
$950$	0	0
$951$	−81.8419	−2.65391
$952$	0	0
$953$	47.3940	1.53524	0.767621	−	0.640904i	$-0.221440\pi$
$953$	47.3940	1.53524	0.767621	+	0.640904i	$0.221440\pi$
$954$	0	0
$955$	−3.49945	−0.113239
$956$	0	0
$957$	−6.68568	−0.216117
$958$	0	0
$959$	−2.01176	−0.0649630
$960$	0	0
$961$	−13.7197	−0.442570
$962$	0	0
$963$	−79.7704	−2.57057
$964$	0	0
$965$	0.344118	0.0110775
$966$	0	0
$967$	45.8553	1.47461	0.737303	−	0.675562i	$-0.236099\pi$
$967$	45.8553	1.47461	0.737303	+	0.675562i	$0.236099\pi$
$968$	0	0
$969$	149.902	4.81554
$970$	0	0
$971$	8.54398	0.274189	0.137095	−	0.990558i	$-0.456224\pi$
$971$	8.54398	0.274189	0.137095	+	0.990558i	$0.456224\pi$
$972$	0	0
$973$	−2.02231	−0.0648322
$974$	0	0
$975$	−60.1515	−1.92639
$976$	0	0
$977$	50.5432	1.61702	0.808510	−	0.588482i	$-0.200274\pi$
$977$	50.5432	1.61702	0.808510	+	0.588482i	$0.200274\pi$
$978$	0	0
$979$	−0.509406	−0.0162807
$980$	0	0
$981$	120.192	3.83742
$982$	0	0
$983$	40.1504	1.28060	0.640299	−	0.768125i	$-0.278810\pi$
$983$	40.1504	1.28060	0.640299	+	0.768125i	$0.278810\pi$
$984$	0	0
$985$	−7.37902	−0.235115
$986$	0	0
$987$	−1.59617	−0.0508067
$988$	0	0
$989$	−39.8072	−1.26580
$990$	0	0
$991$	−2.67733	−0.0850481	−0.0425241	−	0.999095i	$-0.513540\pi$
$991$	−2.67733	−0.0850481	−0.0425241	+	0.999095i	$0.513540\pi$
$992$	0	0
$993$	−58.8760	−1.86837
$994$	0	0
$995$	−5.65128	−0.179157
$996$	0	0
$997$	−38.1297	−1.20758	−0.603790	−	0.797143i	$-0.706343\pi$
$997$	−38.1297	−1.20758	−0.603790	+	0.797143i	$0.706343\pi$
$998$	0	0
$999$	−139.736	−4.42106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.1		23
4.3	odd	2		2008.2.a.d.1.23	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.23	✓	23	4.3	odd	2
4016.2.a.m.1.1		23	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +O(q^{10})\)	\(q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +0.366288 q^{11} -3.69132 q^{13} +1.05396 q^{15} +5.91866 q^{17} -7.61514 q^{19} +0.891542 q^{21} +6.61559 q^{23} -4.89958 q^{25} -16.8337 q^{27} +5.48804 q^{29} +4.15696 q^{31} -1.21823 q^{33} +0.0849482 q^{35} +8.30100 q^{37} +12.2769 q^{39} +3.92223 q^{41} -6.01718 q^{43} -2.55464 q^{45} -1.79035 q^{47} -6.92814 q^{49} -19.6847 q^{51} -10.5522 q^{53} -0.116076 q^{55} +25.3270 q^{57} +3.99709 q^{59} +1.88567 q^{61} -2.16097 q^{63} +1.16977 q^{65} -1.28140 q^{67} -22.0026 q^{69} -8.15695 q^{71} -16.0154 q^{73} +16.2954 q^{75} -0.0981882 q^{77} -7.93927 q^{79} +31.8024 q^{81} +11.7328 q^{83} -1.87560 q^{85} -18.2525 q^{87} -1.39073 q^{89} +0.989507 q^{91} -13.8255 q^{93} +2.41321 q^{95} -9.53422 q^{97} +2.95281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

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