LMFDB - Embedded newform 4016.2.a.m.1.12

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			1.12
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)

$f(q)$	$=$	$q+0.0812979 q^{3} +4.08000 q^{5} -4.55548 q^{7} -2.99339 q^{9} +O(q^{10})$	$q+0.0812979 q^{3} +4.08000 q^{5} -4.55548 q^{7} -2.99339 q^{9} -3.53247 q^{11} -1.88643 q^{13} +0.331695 q^{15} +7.25153 q^{17} +6.82679 q^{19} -0.370351 q^{21} +0.510976 q^{23} +11.6464 q^{25} -0.487250 q^{27} -0.891263 q^{29} -4.69490 q^{31} -0.287182 q^{33} -18.5864 q^{35} +5.85500 q^{37} -0.153363 q^{39} -10.5623 q^{41} +5.97649 q^{43} -12.2130 q^{45} +3.59740 q^{47} +13.7524 q^{49} +0.589534 q^{51} -7.13887 q^{53} -14.4125 q^{55} +0.555004 q^{57} +11.2148 q^{59} +5.92985 q^{61} +13.6363 q^{63} -7.69665 q^{65} +8.31848 q^{67} +0.0415413 q^{69} +8.26092 q^{71} -9.74189 q^{73} +0.946827 q^{75} +16.0921 q^{77} -10.0811 q^{79} +8.94056 q^{81} +8.97449 q^{83} +29.5862 q^{85} -0.0724579 q^{87} +3.15098 q^{89} +8.59362 q^{91} -0.381685 q^{93} +27.8533 q^{95} -5.31571 q^{97} +10.5741 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.492529\pi$
$3$	0.0812979	0.0469374	0.0234687	+	0.999725i	$0.492529\pi$
$4$	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	$-0.830102\pi$
$7$	−4.55548	−1.72181	−0.860905	+	0.508765i	$0.830102\pi$
$8$	0	0
$9$	−2.99339	−0.997797
$10$	0	0
$11$	−3.53247	−1.06508	−0.532540	−	0.846405i	$-0.678762\pi$
$11$	−3.53247	−1.06508	−0.532540	+	0.846405i	$0.678762\pi$
$12$	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	$-0.213644\pi$
$19$	6.82679	1.56617	0.783087	+	0.621912i	$0.213644\pi$
$20$	0	0
$21$	−0.370351	−0.0808173
$22$	0	0
$23$	0.510976	0.106546	0.0532729	−	0.998580i	$-0.483035\pi$
$23$	0.510976	0.106546	0.0532729	+	0.998580i	$0.483035\pi$
$24$	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	$-0.638536\pi$
$31$	−4.69490	−0.843229	−0.421614	+	0.906775i	$0.638536\pi$
$32$	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	$-0.349388\pi$
$43$	5.97649	0.911407	0.455703	+	0.890132i	$0.349388\pi$
$44$	0	0
$45$	−12.2130	−1.82061
$46$	0	0
$47$	3.59740	0.524734	0.262367	−	0.964968i	$-0.415497\pi$
$47$	3.59740	0.524734	0.262367	+	0.964968i	$0.415497\pi$
$48$	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	$-0.239510\pi$
$59$	11.2148	1.46004	0.730021	+	0.683425i	$0.239510\pi$
$60$	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	$-0.330336\pi$
$67$	8.31848	1.01626	0.508132	+	0.861279i	$0.330336\pi$
$68$	0	0
$69$	0.0415413	0.00500098
$70$	0	0
$71$	8.26092	0.980391	0.490196	−	0.871613i	$-0.336925\pi$
$71$	8.26092	0.980391	0.490196	+	0.871613i	$0.336925\pi$
$72$	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	$-0.691937\pi$
$79$	−10.0811	−1.13421	−0.567107	+	0.823644i	$0.691937\pi$
$80$	0	0
$81$	8.94056	0.993396
$82$	0	0
$83$	8.97449	0.985078	0.492539	−	0.870290i	$-0.336069\pi$
$83$	8.97449	0.985078	0.492539	+	0.870290i	$0.336069\pi$
$84$	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	$-0.596877\pi$
$103$	−6.08265	−0.599341	−0.299670	+	0.954043i	$0.596877\pi$
$104$	0	0
$105$	−1.51103	−0.147462
$106$	0	0
$107$	−4.24257	−0.410145	−0.205072	−	0.978747i	$-0.565743\pi$
$107$	−4.24257	−0.410145	−0.205072	+	0.978747i	$0.565743\pi$
$108$	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	$-0.152096\pi$
$127$	20.0145	1.77600	0.887998	+	0.459848i	$0.152096\pi$
$128$	0	0
$129$	0.485877	0.0427790
$130$	0	0
$131$	15.7560	1.37661	0.688304	−	0.725422i	$-0.258355\pi$
$131$	15.7560	1.37661	0.688304	+	0.725422i	$0.258355\pi$
$132$	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	$-0.383117\pi$
$139$	8.46512	0.718002	0.359001	+	0.933337i	$0.383117\pi$
$140$	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	$-0.595781\pi$
$151$	−7.28405	−0.592768	−0.296384	+	0.955069i	$0.595781\pi$
$152$	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	$-0.512361\pi$
$163$	−0.991367	−0.0776498	−0.0388249	+	0.999246i	$0.512361\pi$
$164$	0	0
$165$	−1.17170	−0.0912170
$166$	0	0
$167$	7.81810	0.604982	0.302491	−	0.953152i	$-0.402182\pi$
$167$	7.81810	0.604982	0.302491	+	0.953152i	$0.402182\pi$
$168$	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	$-0.886552\pi$
$179$	−25.0766	−1.87431	−0.937156	+	0.348911i	$0.886552\pi$
$180$	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	$-0.472515\pi$
$191$	2.38371	0.172479	0.0862396	+	0.996274i	$0.472515\pi$
$192$	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	$-0.857866\pi$
$199$	−25.4472	−1.80390	−0.901952	+	0.431837i	$0.857866\pi$
$200$	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	$-0.191244\pi$
$211$	23.9641	1.64976	0.824878	+	0.565311i	$0.191244\pi$
$212$	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	$-0.369677\pi$
$223$	11.8891	0.796156	0.398078	+	0.917352i	$0.369677\pi$
$224$	0	0
$225$	−34.8622	−2.32414
$226$	0	0
$227$	−0.0493739	−0.00327706	−0.00163853	−	0.999999i	$-0.500522\pi$
$227$	−0.0493739	−0.00327706	−0.00163853	+	0.999999i	$0.500522\pi$
$228$	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	$-0.446047\pi$
$239$	5.21570	0.337376	0.168688	+	0.985670i	$0.446047\pi$
$240$	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	$-0.560677\pi$
$263$	−6.14540	−0.378942	−0.189471	+	0.981886i	$0.560677\pi$
$264$	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	$-0.590734\pi$
$271$	−9.25843	−0.562409	−0.281205	+	0.959648i	$0.590734\pi$
$272$	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	$-0.243927\pi$
$283$	24.2403	1.44094	0.720468	+	0.693488i	$0.243927\pi$
$284$	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	$-0.672890\pi$
$307$	−18.1114	−1.03367	−0.516836	+	0.856085i	$0.672890\pi$
$308$	0	0
$309$	−0.494507	−0.0281315
$310$	0	0
$311$	−21.7357	−1.23252	−0.616261	−	0.787542i	$-0.711353\pi$
$311$	−21.7357	−1.23252	−0.616261	+	0.787542i	$0.711353\pi$
$312$	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	$-0.548822\pi$
$331$	−5.55908	−0.305555	−0.152777	+	0.988261i	$0.548822\pi$
$332$	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	$-0.588060\pi$
$347$	−10.1759	−0.546269	−0.273134	+	0.961976i	$0.588060\pi$
$348$	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	$-0.443436\pi$
$359$	6.69853	0.353535	0.176767	+	0.984253i	$0.443436\pi$
$360$	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	$-0.573103\pi$
$367$	−8.72221	−0.455296	−0.227648	+	0.973744i	$0.573103\pi$
$368$	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	$-0.149816\pi$
$379$	34.7023	1.78254	0.891268	+	0.453476i	$0.149816\pi$
$380$	0	0
$381$	1.62713	0.0833606
$382$	0	0
$383$	−12.9708	−0.662774	−0.331387	−	0.943495i	$-0.607517\pi$
$383$	−12.9708	−0.662774	−0.331387	+	0.943495i	$0.607517\pi$
$384$	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	$-0.275654\pi$
$419$	26.5237	1.29577	0.647884	+	0.761739i	$0.275654\pi$
$420$	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	$-0.443437\pi$
$431$	7.33946	0.353529	0.176765	+	0.984253i	$0.443437\pi$
$432$	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	$-0.588849\pi$
$439$	−11.5455	−0.551035	−0.275518	+	0.961296i	$0.588849\pi$
$440$	0	0
$441$	−41.1664	−1.96030
$442$	0	0
$443$	−5.58800	−0.265494	−0.132747	−	0.991150i	$-0.542380\pi$
$443$	−5.58800	−0.265494	−0.132747	+	0.991150i	$0.542380\pi$
$444$	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	$-0.547573\pi$
$463$	−6.40785	−0.297798	−0.148899	+	0.988852i	$0.547573\pi$
$464$	0	0
$465$	−1.55728	−0.0722169
$466$	0	0
$467$	−1.93134	−0.0893718	−0.0446859	−	0.999001i	$-0.514229\pi$
$467$	−1.93134	−0.0893718	−0.0446859	+	0.999001i	$0.514229\pi$
$468$	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	$-0.231208\pi$
$479$	32.7239	1.49519	0.747596	+	0.664154i	$0.231208\pi$
$480$	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	$-0.641899\pi$
$487$	−19.0302	−0.862339	−0.431170	+	0.902271i	$0.641899\pi$
$488$	0	0
$489$	−0.0805961	−0.00364468
$490$	0	0
$491$	−35.1413	−1.58591	−0.792953	−	0.609283i	$-0.791457\pi$
$491$	−35.1413	−1.58591	−0.792953	+	0.609283i	$0.791457\pi$
$492$	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	$-0.418595\pi$
$499$	11.3015	0.505924	0.252962	+	0.967476i	$0.418595\pi$
$500$	0	0
$501$	0.635595	0.0283963
$502$	0	0
$503$	−31.7607	−1.41614	−0.708069	−	0.706143i	$-0.750434\pi$
$503$	−31.7607	−1.41614	−0.708069	+	0.706143i	$0.750434\pi$
$504$	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	$-0.345325\pi$
$523$	21.3611	0.934056	0.467028	+	0.884243i	$0.345325\pi$
$524$	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	$-0.391807\pi$
$547$	15.5948	0.666785	0.333392	+	0.942788i	$0.391807\pi$
$548$	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	$-0.449639\pi$
$563$	7.47683	0.315111	0.157555	+	0.987510i	$0.449639\pi$
$564$	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	$-0.752438\pi$
$571$	−34.0514	−1.42500	−0.712502	+	0.701670i	$0.752438\pi$
$572$	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	$-0.469920\pi$
$587$	4.57228	0.188718	0.0943591	+	0.995538i	$0.469920\pi$
$588$	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	$-0.473094\pi$
$599$	4.13268	0.168857	0.0844283	+	0.996430i	$0.473094\pi$
$600$	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	$-0.242909\pi$
$607$	35.6100	1.44537	0.722683	+	0.691180i	$0.242909\pi$
$608$	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	$-0.400470\pi$
$619$	15.3067	0.615227	0.307614	+	0.951511i	$0.400470\pi$
$620$	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	$-0.523346\pi$
$631$	−3.68147	−0.146557	−0.0732786	+	0.997312i	$0.523346\pi$
$632$	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	$-0.708256\pi$
$643$	−30.8635	−1.21714	−0.608569	+	0.793501i	$0.708256\pi$
$644$	0	0
$645$	1.98238	0.0780559
$646$	0	0
$647$	−9.87949	−0.388403	−0.194201	−	0.980962i	$-0.562212\pi$
$647$	−9.87949	−0.388403	−0.194201	+	0.980962i	$0.562212\pi$
$648$	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	$-0.351844\pi$
$659$	23.0433	0.897641	0.448820	+	0.893622i	$0.351844\pi$
$660$	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	$-0.536443\pi$
$683$	−5.97105	−0.228476	−0.114238	+	0.993453i	$0.536443\pi$
$684$	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	$-0.616491\pi$
$691$	−18.8137	−0.715706	−0.357853	+	0.933778i	$0.616491\pi$
$692$	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	$-0.356295\pi$
$719$	23.3970	0.872562	0.436281	+	0.899810i	$0.356295\pi$
$720$	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	$-0.853131\pi$
$727$	−48.2868	−1.79086	−0.895429	+	0.445205i	$0.853131\pi$
$728$	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	$-0.665074\pi$
$739$	−26.9486	−0.991321	−0.495660	+	0.868516i	$0.665074\pi$
$740$	0	0
$741$	−1.04698	−0.0384617
$742$	0	0
$743$	5.23899	0.192200	0.0960999	−	0.995372i	$-0.469363\pi$
$743$	5.23899	0.192200	0.0960999	+	0.995372i	$0.469363\pi$
$744$	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	$-0.452500\pi$
$751$	8.14855	0.297345	0.148672	+	0.988886i	$0.452500\pi$
$752$	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	$-0.367153\pi$
$787$	22.7424	0.810679	0.405339	+	0.914166i	$0.367153\pi$
$788$	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	$-0.464051\pi$
$811$	6.41871	0.225391	0.112696	+	0.993630i	$0.464051\pi$
$812$	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	$-0.435715\pi$
$823$	11.5089	0.401173	0.200587	+	0.979676i	$0.435715\pi$
$824$	0	0
$825$	−3.34464	−0.116445
$826$	0	0
$827$	−5.58063	−0.194058	−0.0970288	−	0.995282i	$-0.530934\pi$
$827$	−5.58063	−0.194058	−0.0970288	+	0.995282i	$0.530934\pi$
$828$	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	$-0.421783\pi$
$839$	14.0923	0.486522	0.243261	+	0.969961i	$0.421783\pi$
$840$	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	$-0.587801\pi$
$859$	−15.9645	−0.544702	−0.272351	+	0.962198i	$0.587801\pi$
$860$	0	0
$861$	3.91175	0.133312
$862$	0	0
$863$	−34.8405	−1.18598	−0.592992	−	0.805208i	$-0.702054\pi$
$863$	−34.8405	−1.18598	−0.592992	+	0.805208i	$0.702054\pi$
$864$	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	$-0.230170\pi$
$883$	44.5586	1.49952	0.749758	+	0.661712i	$0.230170\pi$
$884$	0	0
$885$	3.71989	0.125043
$886$	0	0
$887$	7.66736	0.257445	0.128722	−	0.991681i	$-0.458912\pi$
$887$	7.66736	0.257445	0.128722	+	0.991681i	$0.458912\pi$
$888$	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	$-0.462580\pi$
$907$	7.06462	0.234577	0.117288	+	0.993098i	$0.462580\pi$
$908$	0	0
$909$	−24.2590	−0.804619
$910$	0	0
$911$	7.80712	0.258661	0.129331	−	0.991602i	$-0.458717\pi$
$911$	7.80712	0.258661	0.129331	+	0.991602i	$0.458717\pi$
$912$	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	$-0.186064\pi$
$919$	50.5634	1.66793	0.833967	+	0.551815i	$0.186064\pi$
$920$	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	$-0.576751\pi$
$947$	−14.6968	−0.477581	−0.238790	+	0.971071i	$0.576751\pi$
$948$	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	$-0.509271\pi$
$967$	−1.81111	−0.0582413	−0.0291206	+	0.999576i	$0.509271\pi$
$968$	0	0
$969$	4.02463	0.129290
$970$	0	0
$971$	25.3189	0.812521	0.406260	−	0.913757i	$-0.366833\pi$
$971$	25.3189	0.812521	0.406260	+	0.913757i	$0.366833\pi$
$972$	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	$-0.604056\pi$
$983$	−20.1354	−0.642221	−0.321110	+	0.947042i	$0.604056\pi$
$984$	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	$-0.486153\pi$
$991$	2.73796	0.0869742	0.0434871	+	0.999054i	$0.486153\pi$
$992$	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	4016.2.a.m.1.12		23
4.3	odd	2		2008.2.a.d.1.12	✓	23

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

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

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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