LMFDB - Embedded newform 2008.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0812979 q^{3} +4.08000 q^{5} +4.55548 q^{7} -2.99339 q^{9} +O(q^{10})$	$q-0.0812979 q^{3} +4.08000 q^{5} +4.55548 q^{7} -2.99339 q^{9} +3.53247 q^{11} -1.88643 q^{13} -0.331695 q^{15} +7.25153 q^{17} -6.82679 q^{19} -0.370351 q^{21} -0.510976 q^{23} +11.6464 q^{25} +0.487250 q^{27} -0.891263 q^{29} +4.69490 q^{31} -0.287182 q^{33} +18.5864 q^{35} +5.85500 q^{37} +0.153363 q^{39} -10.5623 q^{41} -5.97649 q^{43} -12.2130 q^{45} -3.59740 q^{47} +13.7524 q^{49} -0.589534 q^{51} -7.13887 q^{53} +14.4125 q^{55} +0.555004 q^{57} -11.2148 q^{59} +5.92985 q^{61} -13.6363 q^{63} -7.69665 q^{65} -8.31848 q^{67} +0.0415413 q^{69} -8.26092 q^{71} -9.74189 q^{73} -0.946827 q^{75} +16.0921 q^{77} +10.0811 q^{79} +8.94056 q^{81} -8.97449 q^{83} +29.5862 q^{85} +0.0724579 q^{87} +3.15098 q^{89} -8.59362 q^{91} -0.381685 q^{93} -27.8533 q^{95} -5.31571 q^{97} -10.5741 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.0812979	−0.0469374	−0.0234687	−	0.999725i	$-0.507471\pi$
$3$	−0.0812979	−0.0469374	−0.0234687	+	0.999725i	$0.507471\pi$
$4$	0	0
$5$	4.08000	1.82463	0.912315	−	0.409489i	$-0.134293\pi$
$5$	4.08000	1.82463	0.912315	+	0.409489i	$0.134293\pi$
$6$	0	0
$7$	4.55548	1.72181	0.860905	−	0.508765i	$-0.169898\pi$
$7$	4.55548	1.72181	0.860905	+	0.508765i	$0.169898\pi$
$8$	0	0
$9$	−2.99339	−0.997797
$10$	0	0
$11$	3.53247	1.06508	0.532540	−	0.846405i	$-0.321238\pi$
$11$	3.53247	1.06508	0.532540	+	0.846405i	$0.321238\pi$
$12$	0	0
$13$	−1.88643	−0.523203	−0.261601	−	0.965176i	$-0.584251\pi$
$13$	−1.88643	−0.523203	−0.261601	+	0.965176i	$0.584251\pi$
$14$	0	0
$15$	−0.331695	−0.0856434
$16$	0	0
$17$	7.25153	1.75875	0.879377	−	0.476127i	$-0.157960\pi$
$17$	7.25153	1.75875	0.879377	+	0.476127i	$0.157960\pi$
$18$	0	0
$19$	−6.82679	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$19$	−6.82679	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$20$	0	0
$21$	−0.370351	−0.0808173
$22$	0	0
$23$	−0.510976	−0.106546	−0.0532729	−	0.998580i	$-0.516965\pi$
$23$	−0.510976	−0.106546	−0.0532729	+	0.998580i	$0.516965\pi$
$24$	0	0
$25$	11.6464	2.32928
$26$	0	0
$27$	0.487250	0.0937714
$28$	0	0
$29$	−0.891263	−0.165503	−0.0827517	−	0.996570i	$-0.526371\pi$
$29$	−0.891263	−0.165503	−0.0827517	+	0.996570i	$0.526371\pi$
$30$	0	0
$31$	4.69490	0.843229	0.421614	−	0.906775i	$-0.361464\pi$
$31$	4.69490	0.843229	0.421614	+	0.906775i	$0.361464\pi$
$32$	0	0
$33$	−0.287182	−0.0499920
$34$	0	0
$35$	18.5864	3.14167
$36$	0	0
$37$	5.85500	0.962557	0.481278	−	0.876568i	$-0.340173\pi$
$37$	5.85500	0.962557	0.481278	+	0.876568i	$0.340173\pi$
$38$	0	0
$39$	0.153363	0.0245578
$40$	0	0
$41$	−10.5623	−1.64955	−0.824775	−	0.565461i	$-0.808698\pi$
$41$	−10.5623	−1.64955	−0.824775	+	0.565461i	$0.808698\pi$
$42$	0	0
$43$	−5.97649	−0.911407	−0.455703	−	0.890132i	$-0.650612\pi$
$43$	−5.97649	−0.911407	−0.455703	+	0.890132i	$0.650612\pi$
$44$	0	0
$45$	−12.2130	−1.82061
$46$	0	0
$47$	−3.59740	−0.524734	−0.262367	−	0.964968i	$-0.584503\pi$
$47$	−3.59740	−0.524734	−0.262367	+	0.964968i	$0.584503\pi$
$48$	0	0
$49$	13.7524	1.96463
$50$	0	0
$51$	−0.589534	−0.0825513
$52$	0	0
$53$	−7.13887	−0.980600	−0.490300	−	0.871554i	$-0.663113\pi$
$53$	−7.13887	−0.980600	−0.490300	+	0.871554i	$0.663113\pi$
$54$	0	0
$55$	14.4125	1.94338
$56$	0	0
$57$	0.555004	0.0735121
$58$	0	0
$59$	−11.2148	−1.46004	−0.730021	−	0.683425i	$-0.760490\pi$
$59$	−11.2148	−1.46004	−0.730021	+	0.683425i	$0.760490\pi$
$60$	0	0
$61$	5.92985	0.759240	0.379620	−	0.925142i	$-0.376055\pi$
$61$	5.92985	0.759240	0.379620	+	0.925142i	$0.376055\pi$
$62$	0	0
$63$	−13.6363	−1.71802
$64$	0	0
$65$	−7.69665	−0.954652
$66$	0	0
$67$	−8.31848	−1.01626	−0.508132	−	0.861279i	$-0.669664\pi$
$67$	−8.31848	−1.01626	−0.508132	+	0.861279i	$0.669664\pi$
$68$	0	0
$69$	0.0415413	0.00500098
$70$	0	0
$71$	−8.26092	−0.980391	−0.490196	−	0.871613i	$-0.663075\pi$
$71$	−8.26092	−0.980391	−0.490196	+	0.871613i	$0.663075\pi$
$72$	0	0
$73$	−9.74189	−1.14020	−0.570101	−	0.821575i	$-0.693096\pi$
$73$	−9.74189	−1.14020	−0.570101	+	0.821575i	$0.693096\pi$
$74$	0	0
$75$	−0.946827	−0.109330
$76$	0	0
$77$	16.0921	1.83386
$78$	0	0
$79$	10.0811	1.13421	0.567107	−	0.823644i	$-0.308063\pi$
$79$	10.0811	1.13421	0.567107	+	0.823644i	$0.308063\pi$
$80$	0	0
$81$	8.94056	0.993396
$82$	0	0
$83$	−8.97449	−0.985078	−0.492539	−	0.870290i	$-0.663931\pi$
$83$	−8.97449	−0.985078	−0.492539	+	0.870290i	$0.663931\pi$
$84$	0	0
$85$	29.5862	3.20908
$86$	0	0
$87$	0.0724579	0.00776830
$88$	0	0
$89$	3.15098	0.334003	0.167002	−	0.985957i	$-0.446591\pi$
$89$	3.15098	0.334003	0.167002	+	0.985957i	$0.446591\pi$
$90$	0	0
$91$	−8.59362	−0.900856
$92$	0	0
$93$	−0.381685	−0.0395789
$94$	0	0
$95$	−27.8533	−2.85769
$96$	0	0
$97$	−5.31571	−0.539729	−0.269864	−	0.962898i	$-0.586979\pi$
$97$	−5.31571	−0.539729	−0.269864	+	0.962898i	$0.586979\pi$
$98$	0	0
$99$	−10.5741	−1.06273
$100$	0	0
$101$	8.10417	0.806395	0.403198	−	0.915113i	$-0.367899\pi$
$101$	8.10417	0.806395	0.403198	+	0.915113i	$0.367899\pi$
$102$	0	0
$103$	6.08265	0.599341	0.299670	−	0.954043i	$-0.403123\pi$
$103$	6.08265	0.599341	0.299670	+	0.954043i	$0.403123\pi$
$104$	0	0
$105$	−1.51103	−0.147462
$106$	0	0
$107$	4.24257	0.410145	0.205072	−	0.978747i	$-0.434257\pi$
$107$	4.24257	0.410145	0.205072	+	0.978747i	$0.434257\pi$
$108$	0	0
$109$	19.1124	1.83064	0.915320	−	0.402727i	$-0.131938\pi$
$109$	19.1124	1.83064	0.915320	+	0.402727i	$0.131938\pi$
$110$	0	0
$111$	−0.476000	−0.0451799
$112$	0	0
$113$	−0.811208	−0.0763121	−0.0381560	−	0.999272i	$-0.512148\pi$
$113$	−0.811208	−0.0763121	−0.0381560	+	0.999272i	$0.512148\pi$
$114$	0	0
$115$	−2.08478	−0.194407
$116$	0	0
$117$	5.64683	0.522050
$118$	0	0
$119$	33.0342	3.02824
$120$	0	0
$121$	1.47833	0.134393
$122$	0	0
$123$	0.858691	0.0774255
$124$	0	0
$125$	27.1172	2.42544
$126$	0	0
$127$	−20.0145	−1.77600	−0.887998	−	0.459848i	$-0.847904\pi$
$127$	−20.0145	−1.77600	−0.887998	+	0.459848i	$0.847904\pi$
$128$	0	0
$129$	0.485877	0.0427790
$130$	0	0
$131$	−15.7560	−1.37661	−0.688304	−	0.725422i	$-0.741645\pi$
$131$	−15.7560	−1.37661	−0.688304	+	0.725422i	$0.741645\pi$
$132$	0	0
$133$	−31.0993	−2.69665
$134$	0	0
$135$	1.98798	0.171098
$136$	0	0
$137$	19.6854	1.68183	0.840917	−	0.541165i	$-0.182016\pi$
$137$	19.6854	1.68183	0.840917	+	0.541165i	$0.182016\pi$
$138$	0	0
$139$	−8.46512	−0.718002	−0.359001	−	0.933337i	$-0.616883\pi$
$139$	−8.46512	−0.718002	−0.359001	+	0.933337i	$0.616883\pi$
$140$	0	0
$141$	0.292461	0.0246297
$142$	0	0
$143$	−6.66377	−0.557252
$144$	0	0
$145$	−3.63635	−0.301983
$146$	0	0
$147$	−1.11804	−0.0922146
$148$	0	0
$149$	3.07600	0.251996	0.125998	−	0.992031i	$-0.459787\pi$
$149$	3.07600	0.251996	0.125998	+	0.992031i	$0.459787\pi$
$150$	0	0
$151$	7.28405	0.592768	0.296384	−	0.955069i	$-0.404219\pi$
$151$	7.28405	0.592768	0.296384	+	0.955069i	$0.404219\pi$
$152$	0	0
$153$	−21.7066	−1.75488
$154$	0	0
$155$	19.1552	1.53858
$156$	0	0
$157$	−9.36041	−0.747042	−0.373521	−	0.927622i	$-0.621850\pi$
$157$	−9.36041	−0.747042	−0.373521	+	0.927622i	$0.621850\pi$
$158$	0	0
$159$	0.580376	0.0460268
$160$	0	0
$161$	−2.32774	−0.183452
$162$	0	0
$163$	0.991367	0.0776498	0.0388249	−	0.999246i	$-0.487639\pi$
$163$	0.991367	0.0776498	0.0388249	+	0.999246i	$0.487639\pi$
$164$	0	0
$165$	−1.17170	−0.0912170
$166$	0	0
$167$	−7.81810	−0.604982	−0.302491	−	0.953152i	$-0.597818\pi$
$167$	−7.81810	−0.604982	−0.302491	+	0.953152i	$0.597818\pi$
$168$	0	0
$169$	−9.44137	−0.726259
$170$	0	0
$171$	20.4353	1.56272
$172$	0	0
$173$	8.88338	0.675391	0.337695	−	0.941255i	$-0.390353\pi$
$173$	8.88338	0.675391	0.337695	+	0.941255i	$0.390353\pi$
$174$	0	0
$175$	53.0549	4.01057
$176$	0	0
$177$	0.911739	0.0685305
$178$	0	0
$179$	25.0766	1.87431	0.937156	−	0.348911i	$-0.113448\pi$
$179$	25.0766	1.87431	0.937156	+	0.348911i	$0.113448\pi$
$180$	0	0
$181$	13.1745	0.979254	0.489627	−	0.871932i	$-0.337133\pi$
$181$	13.1745	0.979254	0.489627	+	0.871932i	$0.337133\pi$
$182$	0	0
$183$	−0.482085	−0.0356367
$184$	0	0
$185$	23.8884	1.75631
$186$	0	0
$187$	25.6158	1.87321
$188$	0	0
$189$	2.21966	0.161456
$190$	0	0
$191$	−2.38371	−0.172479	−0.0862396	−	0.996274i	$-0.527485\pi$
$191$	−2.38371	−0.172479	−0.0862396	+	0.996274i	$0.527485\pi$
$192$	0	0
$193$	4.28291	0.308291	0.154145	−	0.988048i	$-0.450738\pi$
$193$	4.28291	0.308291	0.154145	+	0.988048i	$0.450738\pi$
$194$	0	0
$195$	0.625721	0.0448088
$196$	0	0
$197$	14.2113	1.01252	0.506258	−	0.862382i	$-0.331029\pi$
$197$	14.2113	1.01252	0.506258	+	0.862382i	$0.331029\pi$
$198$	0	0
$199$	25.4472	1.80390	0.901952	−	0.431837i	$-0.142134\pi$
$199$	25.4472	1.80390	0.901952	+	0.431837i	$0.142134\pi$
$200$	0	0
$201$	0.676275	0.0477008
$202$	0	0
$203$	−4.06013	−0.284966
$204$	0	0
$205$	−43.0940	−3.00982
$206$	0	0
$207$	1.52955	0.106311
$208$	0	0
$209$	−24.1154	−1.66810
$210$	0	0
$211$	−23.9641	−1.64976	−0.824878	−	0.565311i	$-0.808756\pi$
$211$	−23.9641	−1.64976	−0.824878	+	0.565311i	$0.808756\pi$
$212$	0	0
$213$	0.671596	0.0460170
$214$	0	0
$215$	−24.3841	−1.66298
$216$	0	0
$217$	21.3875	1.45188
$218$	0	0
$219$	0.791996	0.0535181
$220$	0	0
$221$	−13.6795	−0.920184
$222$	0	0
$223$	−11.8891	−0.796156	−0.398078	−	0.917352i	$-0.630323\pi$
$223$	−11.8891	−0.796156	−0.398078	+	0.917352i	$0.630323\pi$
$224$	0	0
$225$	−34.8622	−2.32414
$226$	0	0
$227$	0.0493739	0.00327706	0.00163853	−	0.999999i	$-0.499478\pi$
$227$	0.0493739	0.00327706	0.00163853	+	0.999999i	$0.499478\pi$
$228$	0	0
$229$	11.0320	0.729013	0.364507	−	0.931201i	$-0.381238\pi$
$229$	11.0320	0.729013	0.364507	+	0.931201i	$0.381238\pi$
$230$	0	0
$231$	−1.30825	−0.0860768
$232$	0	0
$233$	16.3230	1.06935	0.534677	−	0.845057i	$-0.320433\pi$
$233$	16.3230	1.06935	0.534677	+	0.845057i	$0.320433\pi$
$234$	0	0
$235$	−14.6774	−0.957446
$236$	0	0
$237$	−0.819574	−0.0532370
$238$	0	0
$239$	−5.21570	−0.337376	−0.168688	−	0.985670i	$-0.553953\pi$
$239$	−5.21570	−0.337376	−0.168688	+	0.985670i	$0.553953\pi$
$240$	0	0
$241$	−20.0438	−1.29114	−0.645568	−	0.763703i	$-0.723379\pi$
$241$	−20.0438	−1.29114	−0.645568	+	0.763703i	$0.723379\pi$
$242$	0	0
$243$	−2.18860	−0.140399
$244$	0	0
$245$	56.1098	3.58473
$246$	0	0
$247$	12.8783	0.819426
$248$	0	0
$249$	0.729608	0.0462370
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−1.80500	−0.113480
$254$	0	0
$255$	−2.40530	−0.150626
$256$	0	0
$257$	8.26409	0.515499	0.257750	−	0.966212i	$-0.417019\pi$
$257$	8.26409	0.515499	0.257750	+	0.966212i	$0.417019\pi$
$258$	0	0
$259$	26.6724	1.65734
$260$	0	0
$261$	2.66790	0.165139
$262$	0	0
$263$	6.14540	0.378942	0.189471	−	0.981886i	$-0.439323\pi$
$263$	6.14540	0.378942	0.189471	+	0.981886i	$0.439323\pi$
$264$	0	0
$265$	−29.1266	−1.78923
$266$	0	0
$267$	−0.256168	−0.0156772
$268$	0	0
$269$	−28.2396	−1.72180	−0.860901	−	0.508773i	$-0.830099\pi$
$269$	−28.2396	−1.72180	−0.860901	+	0.508773i	$0.830099\pi$
$270$	0	0
$271$	9.25843	0.562409	0.281205	−	0.959648i	$-0.409266\pi$
$271$	9.25843	0.562409	0.281205	+	0.959648i	$0.409266\pi$
$272$	0	0
$273$	0.698643	0.0422838
$274$	0	0
$275$	41.1405	2.48086
$276$	0	0
$277$	−18.8351	−1.13169	−0.565847	−	0.824510i	$-0.691451\pi$
$277$	−18.8351	−1.13169	−0.565847	+	0.824510i	$0.691451\pi$
$278$	0	0
$279$	−14.0537	−0.841371
$280$	0	0
$281$	1.15954	0.0691721	0.0345860	−	0.999402i	$-0.488989\pi$
$281$	1.15954	0.0691721	0.0345860	+	0.999402i	$0.488989\pi$
$282$	0	0
$283$	−24.2403	−1.44094	−0.720468	−	0.693488i	$-0.756073\pi$
$283$	−24.2403	−1.44094	−0.720468	+	0.693488i	$0.756073\pi$
$284$	0	0
$285$	2.26442	0.134132
$286$	0	0
$287$	−48.1162	−2.84021
$288$	0	0
$289$	35.5846	2.09321
$290$	0	0
$291$	0.432156	0.0253334
$292$	0	0
$293$	14.1841	0.828641	0.414321	−	0.910131i	$-0.364019\pi$
$293$	14.1841	0.828641	0.414321	+	0.910131i	$0.364019\pi$
$294$	0	0
$295$	−45.7563	−2.66404
$296$	0	0
$297$	1.72120	0.0998739
$298$	0	0
$299$	0.963922	0.0557450
$300$	0	0
$301$	−27.2258	−1.56927
$302$	0	0
$303$	−0.658852	−0.0378501
$304$	0	0
$305$	24.1938	1.38533
$306$	0	0
$307$	18.1114	1.03367	0.516836	−	0.856085i	$-0.327110\pi$
$307$	18.1114	1.03367	0.516836	+	0.856085i	$0.327110\pi$
$308$	0	0
$309$	−0.494507	−0.0281315
$310$	0	0
$311$	21.7357	1.23252	0.616261	−	0.787542i	$-0.288647\pi$
$311$	21.7357	1.23252	0.616261	+	0.787542i	$0.288647\pi$
$312$	0	0
$313$	−24.3780	−1.37793	−0.688963	−	0.724796i	$-0.741934\pi$
$313$	−24.3780	−1.37793	−0.688963	+	0.724796i	$0.741934\pi$
$314$	0	0
$315$	−55.6362	−3.13475
$316$	0	0
$317$	19.3696	1.08791	0.543953	−	0.839116i	$-0.316927\pi$
$317$	19.3696	1.08791	0.543953	+	0.839116i	$0.316927\pi$
$318$	0	0
$319$	−3.14836	−0.176274
$320$	0	0
$321$	−0.344912	−0.0192511
$322$	0	0
$323$	−49.5047	−2.75451
$324$	0	0
$325$	−21.9701	−1.21868
$326$	0	0
$327$	−1.55380	−0.0859255
$328$	0	0
$329$	−16.3879	−0.903493
$330$	0	0
$331$	5.55908	0.305555	0.152777	−	0.988261i	$-0.451178\pi$
$331$	5.55908	0.305555	0.152777	+	0.988261i	$0.451178\pi$
$332$	0	0
$333$	−17.5263	−0.960436
$334$	0	0
$335$	−33.9394	−1.85431
$336$	0	0
$337$	−30.5103	−1.66200	−0.831001	−	0.556271i	$-0.812232\pi$
$337$	−30.5103	−1.66200	−0.831001	+	0.556271i	$0.812232\pi$
$338$	0	0
$339$	0.0659496	0.00358189
$340$	0	0
$341$	16.5846	0.898105
$342$	0	0
$343$	30.7605	1.66091
$344$	0	0
$345$	0.169488	0.00912494
$346$	0	0
$347$	10.1759	0.546269	0.273134	−	0.961976i	$-0.411940\pi$
$347$	10.1759	0.546269	0.273134	+	0.961976i	$0.411940\pi$
$348$	0	0
$349$	−4.86972	−0.260670	−0.130335	−	0.991470i	$-0.541605\pi$
$349$	−4.86972	−0.260670	−0.130335	+	0.991470i	$0.541605\pi$
$350$	0	0
$351$	−0.919165	−0.0490614
$352$	0	0
$353$	−11.6941	−0.622415	−0.311208	−	0.950342i	$-0.600733\pi$
$353$	−11.6941	−0.622415	−0.311208	+	0.950342i	$0.600733\pi$
$354$	0	0
$355$	−33.7045	−1.78885
$356$	0	0
$357$	−2.68561	−0.142138
$358$	0	0
$359$	−6.69853	−0.353535	−0.176767	−	0.984253i	$-0.556564\pi$
$359$	−6.69853	−0.353535	−0.176767	+	0.984253i	$0.556564\pi$
$360$	0	0
$361$	27.6051	1.45290
$362$	0	0
$363$	−0.120185	−0.00630808
$364$	0	0
$365$	−39.7469	−2.08045
$366$	0	0
$367$	8.72221	0.455296	0.227648	−	0.973744i	$-0.426897\pi$
$367$	8.72221	0.455296	0.227648	+	0.973744i	$0.426897\pi$
$368$	0	0
$369$	31.6170	1.64592
$370$	0	0
$371$	−32.5210	−1.68841
$372$	0	0
$373$	−18.8557	−0.976309	−0.488154	−	0.872757i	$-0.662330\pi$
$373$	−18.8557	−0.976309	−0.488154	+	0.872757i	$0.662330\pi$
$374$	0	0
$375$	−2.20457	−0.113844
$376$	0	0
$377$	1.68131	0.0865918
$378$	0	0
$379$	−34.7023	−1.78254	−0.891268	−	0.453476i	$-0.850184\pi$
$379$	−34.7023	−1.78254	−0.891268	+	0.453476i	$0.850184\pi$
$380$	0	0
$381$	1.62713	0.0833606
$382$	0	0
$383$	12.9708	0.662774	0.331387	−	0.943495i	$-0.392483\pi$
$383$	12.9708	0.662774	0.331387	+	0.943495i	$0.392483\pi$
$384$	0	0
$385$	65.6557	3.34612
$386$	0	0
$387$	17.8900	0.909399
$388$	0	0
$389$	−16.5551	−0.839378	−0.419689	−	0.907668i	$-0.637861\pi$
$389$	−16.5551	−0.839378	−0.419689	+	0.907668i	$0.637861\pi$
$390$	0	0
$391$	−3.70535	−0.187388
$392$	0	0
$393$	1.28093	0.0646144
$394$	0	0
$395$	41.1309	2.06952
$396$	0	0
$397$	−37.4152	−1.87781	−0.938907	−	0.344172i	$-0.888160\pi$
$397$	−37.4152	−1.87781	−0.938907	+	0.344172i	$0.888160\pi$
$398$	0	0
$399$	2.52831	0.126574
$400$	0	0
$401$	−19.8151	−0.989518	−0.494759	−	0.869030i	$-0.664744\pi$
$401$	−19.8151	−0.989518	−0.494759	+	0.869030i	$0.664744\pi$
$402$	0	0
$403$	−8.85662	−0.441179
$404$	0	0
$405$	36.4775	1.81258
$406$	0	0
$407$	20.6826	1.02520
$408$	0	0
$409$	29.3248	1.45002	0.725010	−	0.688739i	$-0.241835\pi$
$409$	29.3248	1.45002	0.725010	+	0.688739i	$0.241835\pi$
$410$	0	0
$411$	−1.60038	−0.0789408
$412$	0	0
$413$	−51.0888	−2.51391
$414$	0	0
$415$	−36.6159	−1.79740
$416$	0	0
$417$	0.688197	0.0337012
$418$	0	0
$419$	−26.5237	−1.29577	−0.647884	−	0.761739i	$-0.724346\pi$
$419$	−26.5237	−1.29577	−0.647884	+	0.761739i	$0.724346\pi$
$420$	0	0
$421$	−8.62688	−0.420448	−0.210224	−	0.977653i	$-0.567419\pi$
$421$	−8.62688	−0.420448	−0.210224	+	0.977653i	$0.567419\pi$
$422$	0	0
$423$	10.7684	0.523578
$424$	0	0
$425$	84.4540	4.09662
$426$	0	0
$427$	27.0133	1.30727
$428$	0	0
$429$	0.541750	0.0261560
$430$	0	0
$431$	−7.33946	−0.353529	−0.176765	−	0.984253i	$-0.556563\pi$
$431$	−7.33946	−0.353529	−0.176765	+	0.984253i	$0.556563\pi$
$432$	0	0
$433$	37.8792	1.82036	0.910180	−	0.414213i	$-0.135943\pi$
$433$	37.8792	1.82036	0.910180	+	0.414213i	$0.135943\pi$
$434$	0	0
$435$	0.295628	0.0141743
$436$	0	0
$437$	3.48833	0.166869
$438$	0	0
$439$	11.5455	0.551035	0.275518	−	0.961296i	$-0.411151\pi$
$439$	11.5455	0.551035	0.275518	+	0.961296i	$0.411151\pi$
$440$	0	0
$441$	−41.1664	−1.96030
$442$	0	0
$443$	5.58800	0.265494	0.132747	−	0.991150i	$-0.457620\pi$
$443$	5.58800	0.265494	0.132747	+	0.991150i	$0.457620\pi$
$444$	0	0
$445$	12.8560	0.609432
$446$	0	0
$447$	−0.250072	−0.0118280
$448$	0	0
$449$	19.4504	0.917922	0.458961	−	0.888456i	$-0.348222\pi$
$449$	19.4504	0.917922	0.458961	+	0.888456i	$0.348222\pi$
$450$	0	0
$451$	−37.3109	−1.75690
$452$	0	0
$453$	−0.592178	−0.0278230
$454$	0	0
$455$	−35.0619	−1.64373
$456$	0	0
$457$	22.5002	1.05251	0.526257	−	0.850325i	$-0.323595\pi$
$457$	22.5002	1.05251	0.526257	+	0.850325i	$0.323595\pi$
$458$	0	0
$459$	3.53331	0.164921
$460$	0	0
$461$	−12.2605	−0.571027	−0.285514	−	0.958375i	$-0.592164\pi$
$461$	−12.2605	−0.571027	−0.285514	+	0.958375i	$0.592164\pi$
$462$	0	0
$463$	6.40785	0.297798	0.148899	−	0.988852i	$-0.452427\pi$
$463$	6.40785	0.297798	0.148899	+	0.988852i	$0.452427\pi$
$464$	0	0
$465$	−1.55728	−0.0722169
$466$	0	0
$467$	1.93134	0.0893718	0.0446859	−	0.999001i	$-0.485771\pi$
$467$	1.93134	0.0893718	0.0446859	+	0.999001i	$0.485771\pi$
$468$	0	0
$469$	−37.8947	−1.74981
$470$	0	0
$471$	0.760982	0.0350642
$472$	0	0
$473$	−21.1118	−0.970720
$474$	0	0
$475$	−79.5075	−3.64805
$476$	0	0
$477$	21.3694	0.978439
$478$	0	0
$479$	−32.7239	−1.49519	−0.747596	−	0.664154i	$-0.768792\pi$
$479$	−32.7239	−1.49519	−0.747596	+	0.664154i	$0.768792\pi$
$480$	0	0
$481$	−11.0451	−0.503612
$482$	0	0
$483$	0.189240	0.00861074
$484$	0	0
$485$	−21.6881	−0.984805
$486$	0	0
$487$	19.0302	0.862339	0.431170	−	0.902271i	$-0.358101\pi$
$487$	19.0302	0.862339	0.431170	+	0.902271i	$0.358101\pi$
$488$	0	0
$489$	−0.0805961	−0.00364468
$490$	0	0
$491$	35.1413	1.58591	0.792953	−	0.609283i	$-0.208543\pi$
$491$	35.1413	1.58591	0.792953	+	0.609283i	$0.208543\pi$
$492$	0	0
$493$	−6.46302	−0.291080
$494$	0	0
$495$	−43.1421	−1.93909
$496$	0	0
$497$	−37.6325	−1.68805
$498$	0	0
$499$	−11.3015	−0.505924	−0.252962	−	0.967476i	$-0.581405\pi$
$499$	−11.3015	−0.505924	−0.252962	+	0.967476i	$0.581405\pi$
$500$	0	0
$501$	0.635595	0.0283963
$502$	0	0
$503$	31.7607	1.41614	0.708069	−	0.706143i	$-0.249566\pi$
$503$	31.7607	1.41614	0.708069	+	0.706143i	$0.249566\pi$
$504$	0	0
$505$	33.0650	1.47137
$506$	0	0
$507$	0.767564	0.0340887
$508$	0	0
$509$	−29.8144	−1.32150	−0.660751	−	0.750605i	$-0.729762\pi$
$509$	−29.8144	−1.32150	−0.660751	+	0.750605i	$0.729762\pi$
$510$	0	0
$511$	−44.3790	−1.96321
$512$	0	0
$513$	−3.32636	−0.146862
$514$	0	0
$515$	24.8172	1.09358
$516$	0	0
$517$	−12.7077	−0.558883
$518$	0	0
$519$	−0.722200	−0.0317011
$520$	0	0
$521$	16.5762	0.726218	0.363109	−	0.931747i	$-0.381715\pi$
$521$	16.5762	0.726218	0.363109	+	0.931747i	$0.381715\pi$
$522$	0	0
$523$	−21.3611	−0.934056	−0.467028	−	0.884243i	$-0.654675\pi$
$523$	−21.3611	−0.934056	−0.467028	+	0.884243i	$0.654675\pi$
$524$	0	0
$525$	−4.31325	−0.188246
$526$	0	0
$527$	34.0452	1.48303
$528$	0	0
$529$	−22.7389	−0.988648
$530$	0	0
$531$	33.5702	1.45682
$532$	0	0
$533$	19.9250	0.863049
$534$	0	0
$535$	17.3097	0.748363
$536$	0	0
$537$	−2.03867	−0.0879753
$538$	0	0
$539$	48.5800	2.09249
$540$	0	0
$541$	−8.90934	−0.383042	−0.191521	−	0.981488i	$-0.561342\pi$
$541$	−8.90934	−0.383042	−0.191521	+	0.981488i	$0.561342\pi$
$542$	0	0
$543$	−1.07106	−0.0459636
$544$	0	0
$545$	77.9787	3.34024
$546$	0	0
$547$	−15.5948	−0.666785	−0.333392	−	0.942788i	$-0.608193\pi$
$547$	−15.5948	−0.666785	−0.333392	+	0.942788i	$0.608193\pi$
$548$	0	0
$549$	−17.7504	−0.757567
$550$	0	0
$551$	6.08447	0.259207
$552$	0	0
$553$	45.9243	1.95290
$554$	0	0
$555$	−1.94208	−0.0824366
$556$	0	0
$557$	−8.97903	−0.380454	−0.190227	−	0.981740i	$-0.560922\pi$
$557$	−8.97903	−0.380454	−0.190227	+	0.981740i	$0.560922\pi$
$558$	0	0
$559$	11.2743	0.476850
$560$	0	0
$561$	−2.08251	−0.0879236
$562$	0	0
$563$	−7.47683	−0.315111	−0.157555	−	0.987510i	$-0.550361\pi$
$563$	−7.47683	−0.315111	−0.157555	+	0.987510i	$0.550361\pi$
$564$	0	0
$565$	−3.30973	−0.139241
$566$	0	0
$567$	40.7286	1.71044
$568$	0	0
$569$	−22.0584	−0.924737	−0.462369	−	0.886688i	$-0.653000\pi$
$569$	−22.0584	−0.924737	−0.462369	+	0.886688i	$0.653000\pi$
$570$	0	0
$571$	34.0514	1.42500	0.712502	−	0.701670i	$-0.247562\pi$
$571$	34.0514	1.42500	0.712502	+	0.701670i	$0.247562\pi$
$572$	0	0
$573$	0.193791	0.00809572
$574$	0	0
$575$	−5.95102	−0.248175
$576$	0	0
$577$	−30.4513	−1.26771	−0.633853	−	0.773454i	$-0.718527\pi$
$577$	−30.4513	−1.26771	−0.633853	+	0.773454i	$0.718527\pi$
$578$	0	0
$579$	−0.348192	−0.0144704
$580$	0	0
$581$	−40.8831	−1.69612
$582$	0	0
$583$	−25.2178	−1.04442
$584$	0	0
$585$	23.0391	0.952548
$586$	0	0
$587$	−4.57228	−0.188718	−0.0943591	−	0.995538i	$-0.530080\pi$
$587$	−4.57228	−0.188718	−0.0943591	+	0.995538i	$0.530080\pi$
$588$	0	0
$589$	−32.0511	−1.32064
$590$	0	0
$591$	−1.15535	−0.0475248
$592$	0	0
$593$	33.4875	1.37517	0.687583	−	0.726105i	$-0.258672\pi$
$593$	33.4875	1.37517	0.687583	+	0.726105i	$0.258672\pi$
$594$	0	0
$595$	134.779	5.52542
$596$	0	0
$597$	−2.06880	−0.0846705
$598$	0	0
$599$	−4.13268	−0.168857	−0.0844283	−	0.996430i	$-0.526906\pi$
$599$	−4.13268	−0.168857	−0.0844283	+	0.996430i	$0.526906\pi$
$600$	0	0
$601$	18.1982	0.742321	0.371161	−	0.928569i	$-0.378960\pi$
$601$	18.1982	0.742321	0.371161	+	0.928569i	$0.378960\pi$
$602$	0	0
$603$	24.9005	1.01402
$604$	0	0
$605$	6.03157	0.245218
$606$	0	0
$607$	−35.6100	−1.44537	−0.722683	−	0.691180i	$-0.757091\pi$
$607$	−35.6100	−1.44537	−0.722683	+	0.691180i	$0.757091\pi$
$608$	0	0
$609$	0.330081	0.0133755
$610$	0	0
$611$	6.78625	0.274542
$612$	0	0
$613$	−4.36247	−0.176198	−0.0880992	−	0.996112i	$-0.528079\pi$
$613$	−4.36247	−0.176198	−0.0880992	+	0.996112i	$0.528079\pi$
$614$	0	0
$615$	3.50346	0.141273
$616$	0	0
$617$	−37.3669	−1.50433	−0.752167	−	0.658972i	$-0.770991\pi$
$617$	−37.3669	−1.50433	−0.752167	+	0.658972i	$0.770991\pi$
$618$	0	0
$619$	−15.3067	−0.615227	−0.307614	−	0.951511i	$-0.599530\pi$
$619$	−15.3067	−0.615227	−0.307614	+	0.951511i	$0.599530\pi$
$620$	0	0
$621$	−0.248973	−0.00999094
$622$	0	0
$623$	14.3542	0.575090
$624$	0	0
$625$	52.4063	2.09625
$626$	0	0
$627$	1.96053	0.0782962
$628$	0	0
$629$	42.4577	1.69290
$630$	0	0
$631$	3.68147	0.146557	0.0732786	−	0.997312i	$-0.476654\pi$
$631$	3.68147	0.146557	0.0732786	+	0.997312i	$0.476654\pi$
$632$	0	0
$633$	1.94823	0.0774352
$634$	0	0
$635$	−81.6589	−3.24054
$636$	0	0
$637$	−25.9430	−1.02790
$638$	0	0
$639$	24.7282	0.978231
$640$	0	0
$641$	−9.93361	−0.392354	−0.196177	−	0.980568i	$-0.562853\pi$
$641$	−9.93361	−0.392354	−0.196177	+	0.980568i	$0.562853\pi$
$642$	0	0
$643$	30.8635	1.21714	0.608569	−	0.793501i	$-0.291744\pi$
$643$	30.8635	1.21714	0.608569	+	0.793501i	$0.291744\pi$
$644$	0	0
$645$	1.98238	0.0780559
$646$	0	0
$647$	9.87949	0.388403	0.194201	−	0.980962i	$-0.437788\pi$
$647$	9.87949	0.388403	0.194201	+	0.980962i	$0.437788\pi$
$648$	0	0
$649$	−39.6159	−1.55506
$650$	0	0
$651$	−1.73876	−0.0681474
$652$	0	0
$653$	27.9126	1.09230	0.546152	−	0.837686i	$-0.316092\pi$
$653$	27.9126	1.09230	0.546152	+	0.837686i	$0.316092\pi$
$654$	0	0
$655$	−64.2845	−2.51180
$656$	0	0
$657$	29.1613	1.13769
$658$	0	0
$659$	−23.0433	−0.897641	−0.448820	−	0.893622i	$-0.648156\pi$
$659$	−23.0433	−0.897641	−0.448820	+	0.893622i	$0.648156\pi$
$660$	0	0
$661$	40.7362	1.58445	0.792227	−	0.610227i	$-0.208922\pi$
$661$	40.7362	1.58445	0.792227	+	0.610227i	$0.208922\pi$
$662$	0	0
$663$	1.11212	0.0431910
$664$	0	0
$665$	−126.885	−4.92040
$666$	0	0
$667$	0.455414	0.0176337
$668$	0	0
$669$	0.966563	0.0373695
$670$	0	0
$671$	20.9470	0.808651
$672$	0	0
$673$	−17.3600	−0.669177	−0.334589	−	0.942364i	$-0.608597\pi$
$673$	−17.3600	−0.669177	−0.334589	+	0.942364i	$0.608597\pi$
$674$	0	0
$675$	5.67470	0.218419
$676$	0	0
$677$	−42.2063	−1.62212	−0.811059	−	0.584964i	$-0.801109\pi$
$677$	−42.2063	−1.62212	−0.811059	+	0.584964i	$0.801109\pi$
$678$	0	0
$679$	−24.2156	−0.929310
$680$	0	0
$681$	−0.00401400	−0.000153817 0
$682$	0	0
$683$	5.97105	0.228476	0.114238	−	0.993453i	$-0.463557\pi$
$683$	5.97105	0.228476	0.114238	+	0.993453i	$0.463557\pi$
$684$	0	0
$685$	80.3162	3.06872
$686$	0	0
$687$	−0.896876	−0.0342180
$688$	0	0
$689$	13.4670	0.513052
$690$	0	0
$691$	18.8137	0.715706	0.357853	−	0.933778i	$-0.383509\pi$
$691$	18.8137	0.715706	0.357853	+	0.933778i	$0.383509\pi$
$692$	0	0
$693$	−48.1699	−1.82982
$694$	0	0
$695$	−34.5377	−1.31009
$696$	0	0
$697$	−76.5926	−2.90115
$698$	0	0
$699$	−1.32702	−0.0501927
$700$	0	0
$701$	−9.36923	−0.353871	−0.176936	−	0.984222i	$-0.556618\pi$
$701$	−9.36923	−0.353871	−0.176936	+	0.984222i	$0.556618\pi$
$702$	0	0
$703$	−39.9709	−1.50753
$704$	0	0
$705$	1.19324	0.0449400
$706$	0	0
$707$	36.9184	1.38846
$708$	0	0
$709$	−27.3166	−1.02590	−0.512948	−	0.858420i	$-0.671447\pi$
$709$	−27.3166	−1.02590	−0.512948	+	0.858420i	$0.671447\pi$
$710$	0	0
$711$	−30.1767	−1.13172
$712$	0	0
$713$	−2.39898	−0.0898424
$714$	0	0
$715$	−27.1882	−1.01678
$716$	0	0
$717$	0.424026	0.0158355
$718$	0	0
$719$	−23.3970	−0.872562	−0.436281	−	0.899810i	$-0.643705\pi$
$719$	−23.3970	−0.872562	−0.436281	+	0.899810i	$0.643705\pi$
$720$	0	0
$721$	27.7094	1.03195
$722$	0	0
$723$	1.62952	0.0606026
$724$	0	0
$725$	−10.3800	−0.385503
$726$	0	0
$727$	48.2868	1.79086	0.895429	−	0.445205i	$-0.146869\pi$
$727$	48.2868	1.79086	0.895429	+	0.445205i	$0.146869\pi$
$728$	0	0
$729$	−26.6437	−0.986806
$730$	0	0
$731$	−43.3387	−1.60294
$732$	0	0
$733$	24.0508	0.888337	0.444169	−	0.895943i	$-0.353499\pi$
$733$	24.0508	0.888337	0.444169	+	0.895943i	$0.353499\pi$
$734$	0	0
$735$	−4.56161	−0.168258
$736$	0	0
$737$	−29.3848	−1.08240
$738$	0	0
$739$	26.9486	0.991321	0.495660	−	0.868516i	$-0.334926\pi$
$739$	26.9486	0.991321	0.495660	+	0.868516i	$0.334926\pi$
$740$	0	0
$741$	−1.04698	−0.0384617
$742$	0	0
$743$	−5.23899	−0.192200	−0.0960999	−	0.995372i	$-0.530637\pi$
$743$	−5.23899	−0.192200	−0.0960999	+	0.995372i	$0.530637\pi$
$744$	0	0
$745$	12.5501	0.459799
$746$	0	0
$747$	26.8642	0.982908
$748$	0	0
$749$	19.3269	0.706191
$750$	0	0
$751$	−8.14855	−0.297345	−0.148672	−	0.988886i	$-0.547500\pi$
$751$	−8.14855	−0.297345	−0.148672	+	0.988886i	$0.547500\pi$
$752$	0	0
$753$	0.0812979	0.00296266
$754$	0	0
$755$	29.7189	1.08158
$756$	0	0
$757$	1.25379	0.0455698	0.0227849	−	0.999740i	$-0.492747\pi$
$757$	1.25379	0.0455698	0.0227849	+	0.999740i	$0.492747\pi$
$758$	0	0
$759$	0.146743	0.00532644
$760$	0	0
$761$	28.0354	1.01628	0.508141	−	0.861274i	$-0.330333\pi$
$761$	28.0354	1.01628	0.508141	+	0.861274i	$0.330333\pi$
$762$	0	0
$763$	87.0664	3.15202
$764$	0	0
$765$	−88.5631	−3.20201
$766$	0	0
$767$	21.1560	0.763898
$768$	0	0
$769$	33.2123	1.19767	0.598834	−	0.800873i	$-0.295631\pi$
$769$	33.2123	1.19767	0.598834	+	0.800873i	$0.295631\pi$
$770$	0	0
$771$	−0.671853	−0.0241962
$772$	0	0
$773$	−0.493202	−0.0177392	−0.00886962	−	0.999961i	$-0.502823\pi$
$773$	−0.493202	−0.0177392	−0.00886962	+	0.999961i	$0.502823\pi$
$774$	0	0
$775$	54.6786	1.96411
$776$	0	0
$777$	−2.16841	−0.0777912
$778$	0	0
$779$	72.1065	2.58348
$780$	0	0
$781$	−29.1814	−1.04419
$782$	0	0
$783$	−0.434268	−0.0155195
$784$	0	0
$785$	−38.1904	−1.36308
$786$	0	0
$787$	−22.7424	−0.810679	−0.405339	−	0.914166i	$-0.632847\pi$
$787$	−22.7424	−0.810679	−0.405339	+	0.914166i	$0.632847\pi$
$788$	0	0
$789$	−0.499608	−0.0177865
$790$	0	0
$791$	−3.69545	−0.131395
$792$	0	0
$793$	−11.1863	−0.397236
$794$	0	0
$795$	2.36793	0.0839819
$796$	0	0
$797$	−31.3453	−1.11031	−0.555154	−	0.831748i	$-0.687341\pi$
$797$	−31.3453	−1.11031	−0.555154	+	0.831748i	$0.687341\pi$
$798$	0	0
$799$	−26.0866	−0.922878
$800$	0	0
$801$	−9.43211	−0.333267
$802$	0	0
$803$	−34.4129	−1.21441
$804$	0	0
$805$	−9.49718	−0.334731
$806$	0	0
$807$	2.29582	0.0808168
$808$	0	0
$809$	20.6486	0.725967	0.362983	−	0.931796i	$-0.381758\pi$
$809$	20.6486	0.725967	0.362983	+	0.931796i	$0.381758\pi$
$810$	0	0
$811$	−6.41871	−0.225391	−0.112696	−	0.993630i	$-0.535949\pi$
$811$	−6.41871	−0.225391	−0.112696	+	0.993630i	$0.535949\pi$
$812$	0	0
$813$	−0.752691	−0.0263980
$814$	0	0
$815$	4.04477	0.141682
$816$	0	0
$817$	40.8003	1.42742
$818$	0	0
$819$	25.7241	0.898871
$820$	0	0
$821$	11.5844	0.404297	0.202149	−	0.979355i	$-0.435208\pi$
$821$	11.5844	0.404297	0.202149	+	0.979355i	$0.435208\pi$
$822$	0	0
$823$	−11.5089	−0.401173	−0.200587	−	0.979676i	$-0.564285\pi$
$823$	−11.5089	−0.401173	−0.200587	+	0.979676i	$0.564285\pi$
$824$	0	0
$825$	−3.34464	−0.116445
$826$	0	0
$827$	5.58063	0.194058	0.0970288	−	0.995282i	$-0.469066\pi$
$827$	5.58063	0.194058	0.0970288	+	0.995282i	$0.469066\pi$
$828$	0	0
$829$	17.3422	0.602320	0.301160	−	0.953574i	$-0.402626\pi$
$829$	17.3422	0.602320	0.301160	+	0.953574i	$0.402626\pi$
$830$	0	0
$831$	1.53126	0.0531188
$832$	0	0
$833$	99.7260	3.45530
$834$	0	0
$835$	−31.8978	−1.10387
$836$	0	0
$837$	2.28759	0.0790707
$838$	0	0
$839$	−14.0923	−0.486522	−0.243261	−	0.969961i	$-0.578217\pi$
$839$	−14.0923	−0.486522	−0.243261	+	0.969961i	$0.578217\pi$
$840$	0	0
$841$	−28.2056	−0.972609
$842$	0	0
$843$	−0.0942678	−0.00324676
$844$	0	0
$845$	−38.5208	−1.32515
$846$	0	0
$847$	6.73450	0.231400
$848$	0	0
$849$	1.97069	0.0676338
$850$	0	0
$851$	−2.99176	−0.102556
$852$	0	0
$853$	−23.0877	−0.790508	−0.395254	−	0.918572i	$-0.629343\pi$
$853$	−23.0877	−0.790508	−0.395254	+	0.918572i	$0.629343\pi$
$854$	0	0
$855$	83.3758	2.85139
$856$	0	0
$857$	25.4684	0.869983	0.434992	−	0.900434i	$-0.356751\pi$
$857$	25.4684	0.869983	0.434992	+	0.900434i	$0.356751\pi$
$858$	0	0
$859$	15.9645	0.544702	0.272351	−	0.962198i	$-0.412199\pi$
$859$	15.9645	0.544702	0.272351	+	0.962198i	$0.412199\pi$
$860$	0	0
$861$	3.91175	0.133312
$862$	0	0
$863$	34.8405	1.18598	0.592992	−	0.805208i	$-0.297946\pi$
$863$	34.8405	1.18598	0.592992	+	0.805208i	$0.297946\pi$
$864$	0	0
$865$	36.2442	1.23234
$866$	0	0
$867$	−2.89296	−0.0982499
$868$	0	0
$869$	35.6112	1.20803
$870$	0	0
$871$	15.6923	0.531712
$872$	0	0
$873$	15.9120	0.538540
$874$	0	0
$875$	123.532	4.17615
$876$	0	0
$877$	−37.9157	−1.28032	−0.640161	−	0.768241i	$-0.721132\pi$
$877$	−37.9157	−1.28032	−0.640161	+	0.768241i	$0.721132\pi$
$878$	0	0
$879$	−1.15313	−0.0388942
$880$	0	0
$881$	36.6026	1.23317	0.616586	−	0.787287i	$-0.288515\pi$
$881$	36.6026	1.23317	0.616586	+	0.787287i	$0.288515\pi$
$882$	0	0
$883$	−44.5586	−1.49952	−0.749758	−	0.661712i	$-0.769830\pi$
$883$	−44.5586	−1.49952	−0.749758	+	0.661712i	$0.769830\pi$
$884$	0	0
$885$	3.71989	0.125043
$886$	0	0
$887$	−7.66736	−0.257445	−0.128722	−	0.991681i	$-0.541088\pi$
$887$	−7.66736	−0.257445	−0.128722	+	0.991681i	$0.541088\pi$
$888$	0	0
$889$	−91.1755	−3.05793
$890$	0	0
$891$	31.5822	1.05804
$892$	0	0
$893$	24.5587	0.821825
$894$	0	0
$895$	102.312	3.41993
$896$	0	0
$897$	−0.0783648	−0.00261653
$898$	0	0
$899$	−4.18439	−0.139557
$900$	0	0
$901$	−51.7677	−1.72463
$902$	0	0
$903$	2.21340	0.0736574
$904$	0	0
$905$	53.7520	1.78678
$906$	0	0
$907$	−7.06462	−0.234577	−0.117288	−	0.993098i	$-0.537420\pi$
$907$	−7.06462	−0.234577	−0.117288	+	0.993098i	$0.537420\pi$
$908$	0	0
$909$	−24.2590	−0.804619
$910$	0	0
$911$	−7.80712	−0.258661	−0.129331	−	0.991602i	$-0.541283\pi$
$911$	−7.80712	−0.258661	−0.129331	+	0.991602i	$0.541283\pi$
$912$	0	0
$913$	−31.7021	−1.04919
$914$	0	0
$915$	−1.96691	−0.0650239
$916$	0	0
$917$	−71.7762	−2.37026
$918$	0	0
$919$	−50.5634	−1.66793	−0.833967	−	0.551815i	$-0.813936\pi$
$919$	−50.5634	−1.66793	−0.833967	+	0.551815i	$0.813936\pi$
$920$	0	0
$921$	−1.47242	−0.0485178
$922$	0	0
$923$	15.5837	0.512943
$924$	0	0
$925$	68.1896	2.24206
$926$	0	0
$927$	−18.2077	−0.598021
$928$	0	0
$929$	−22.5151	−0.738696	−0.369348	−	0.929291i	$-0.620419\pi$
$929$	−22.5151	−0.738696	−0.369348	+	0.929291i	$0.620419\pi$
$930$	0	0
$931$	−93.8849	−3.07695
$932$	0	0
$933$	−1.76707	−0.0578513
$934$	0	0
$935$	104.512	3.41792
$936$	0	0
$937$	34.7253	1.13443	0.567213	−	0.823571i	$-0.308022\pi$
$937$	34.7253	1.13443	0.567213	+	0.823571i	$0.308022\pi$
$938$	0	0
$939$	1.98188	0.0646763
$940$	0	0
$941$	−13.6640	−0.445432	−0.222716	−	0.974883i	$-0.571492\pi$
$941$	−13.6640	−0.445432	−0.222716	+	0.974883i	$0.571492\pi$
$942$	0	0
$943$	5.39706	0.175753
$944$	0	0
$945$	9.05621	0.294598
$946$	0	0
$947$	14.6968	0.477581	0.238790	−	0.971071i	$-0.423249\pi$
$947$	14.6968	0.477581	0.238790	+	0.971071i	$0.423249\pi$
$948$	0	0
$949$	18.3774	0.596557
$950$	0	0
$951$	−1.57471	−0.0510635
$952$	0	0
$953$	0.324327	0.0105060	0.00525299	−	0.999986i	$-0.498328\pi$
$953$	0.324327	0.0105060	0.00525299	+	0.999986i	$0.498328\pi$
$954$	0	0
$955$	−9.72553	−0.314711
$956$	0	0
$957$	0.255955	0.00827385
$958$	0	0
$959$	89.6763	2.89580
$960$	0	0
$961$	−8.95793	−0.288966
$962$	0	0
$963$	−12.6997	−0.409241
$964$	0	0
$965$	17.4743	0.562517
$966$	0	0
$967$	1.81111	0.0582413	0.0291206	−	0.999576i	$-0.490729\pi$
$967$	1.81111	0.0582413	0.0291206	+	0.999576i	$0.490729\pi$
$968$	0	0
$969$	4.02463	0.129290
$970$	0	0
$971$	−25.3189	−0.812521	−0.406260	−	0.913757i	$-0.633167\pi$
$971$	−25.3189	−0.812521	−0.406260	+	0.913757i	$0.633167\pi$
$972$	0	0
$973$	−38.5627	−1.23626
$974$	0	0
$975$	1.78613	0.0572018
$976$	0	0
$977$	33.0716	1.05805	0.529027	−	0.848605i	$-0.322557\pi$
$977$	33.0716	1.05805	0.529027	+	0.848605i	$0.322557\pi$
$978$	0	0
$979$	11.1307	0.355740
$980$	0	0
$981$	−57.2110	−1.82661
$982$	0	0
$983$	20.1354	0.642221	0.321110	−	0.947042i	$-0.395944\pi$
$983$	20.1354	0.642221	0.321110	+	0.947042i	$0.395944\pi$
$984$	0	0
$985$	57.9822	1.84747
$986$	0	0
$987$	1.33230	0.0424076
$988$	0	0
$989$	3.05384	0.0971065
$990$	0	0
$991$	−2.73796	−0.0869742	−0.0434871	−	0.999054i	$-0.513847\pi$
$991$	−2.73796	−0.0869742	−0.0434871	+	0.999054i	$0.513847\pi$
$992$	0	0
$993$	−0.451942	−0.0143419
$994$	0	0
$995$	103.824	3.29146
$996$	0	0
$997$	−31.8558	−1.00888	−0.504442	−	0.863446i	$-0.668302\pi$
$997$	−31.8558	−1.00888	−0.504442	+	0.863446i	$0.668302\pi$
$998$	0	0
$999$	2.85285	0.0902603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.12	✓	23
4.3	odd	2		4016.2.a.m.1.12		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.12	✓	23	1.1	even	1	trivial
4016.2.a.m.1.12		23	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0812979 q^{3} +4.08000 q^{5} +4.55548 q^{7} -2.99339 q^{9} +O(q^{10})\)	\(q-0.0812979 q^{3} +4.08000 q^{5} +4.55548 q^{7} -2.99339 q^{9} +3.53247 q^{11} -1.88643 q^{13} -0.331695 q^{15} +7.25153 q^{17} -6.82679 q^{19} -0.370351 q^{21} -0.510976 q^{23} +11.6464 q^{25} +0.487250 q^{27} -0.891263 q^{29} +4.69490 q^{31} -0.287182 q^{33} +18.5864 q^{35} +5.85500 q^{37} +0.153363 q^{39} -10.5623 q^{41} -5.97649 q^{43} -12.2130 q^{45} -3.59740 q^{47} +13.7524 q^{49} -0.589534 q^{51} -7.13887 q^{53} +14.4125 q^{55} +0.555004 q^{57} -11.2148 q^{59} +5.92985 q^{61} -13.6363 q^{63} -7.69665 q^{65} -8.31848 q^{67} +0.0415413 q^{69} -8.26092 q^{71} -9.74189 q^{73} -0.946827 q^{75} +16.0921 q^{77} +10.0811 q^{79} +8.94056 q^{81} -8.97449 q^{83} +29.5862 q^{85} +0.0724579 q^{87} +3.15098 q^{89} -8.59362 q^{91} -0.381685 q^{93} -27.8533 q^{95} -5.31571 q^{97} -10.5741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 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

Introduction

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