LMFDB - Embedded newform 4016.2.a.c.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.737640$ of defining polynomial
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)

$f(q)$	$=$	$q-1.19353 q^{3} -0.837853 q^{5} +2.93117 q^{7} -1.57549 q^{9} +O(q^{10})$	$q-1.19353 q^{3} -0.837853 q^{5} +2.93117 q^{7} -1.57549 q^{9} +2.35567 q^{11} -2.42451 q^{13} +1.00000 q^{15} -0.637428 q^{17} -2.02448 q^{19} -3.49843 q^{21} +0.293740 q^{23} -4.29800 q^{25} +5.46097 q^{27} +1.68079 q^{29} +7.67390 q^{31} -2.81156 q^{33} -2.45589 q^{35} -5.17413 q^{37} +2.89371 q^{39} +1.88863 q^{41} -1.58799 q^{43} +1.32003 q^{45} -7.90488 q^{47} +1.59174 q^{49} +0.760787 q^{51} +6.20978 q^{53} -1.97371 q^{55} +2.41627 q^{57} -10.4772 q^{59} -5.88039 q^{61} -4.61803 q^{63} +2.03138 q^{65} -10.2794 q^{67} -0.350586 q^{69} -5.47347 q^{71} +3.71510 q^{73} +5.12978 q^{75} +6.90488 q^{77} +12.5272 q^{79} -1.79134 q^{81} +5.72384 q^{83} +0.534071 q^{85} -2.00607 q^{87} -4.97056 q^{89} -7.10664 q^{91} -9.15900 q^{93} +1.69622 q^{95} +12.5705 q^{97} -3.71135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9	$ 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9} + 3 q^{11} - 12 q^{13} + 4 q^{15} + q^{17} + 9 q^{19} - 7 q^{21} - 4 q^{23} - 7 q^{25} - q^{27} - 12 q^{29} + 2 q^{31} - 5 q^{35} - 13 q^{37} - 13 q^{39} + q^{41} + 5 q^{43} + 11 q^{45} - 12 q^{47} - 9 q^{49} - 2 q^{51} + 5 q^{53} + 3 q^{55} + 16 q^{57} - 6 q^{59} - 21 q^{61} - 14 q^{63} + q^{65} - 17 q^{67} - 13 q^{69} + 10 q^{71} - 2 q^{73} - 13 q^{75} + 8 q^{77} + 21 q^{79} - 8 q^{81} + q^{83} - 17 q^{85} - 31 q^{87} + 5 q^{89} + 2 q^{91} - 23 q^{93} - 12 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9 + 3 * q^11 - 12 * q^13 + 4 * q^15 + q^17 + 9 * q^19 - 7 * q^21 - 4 * q^23 - 7 * q^25 - q^27 - 12 * q^29 + 2 * q^31 - 5 * q^35 - 13 * q^37 - 13 * q^39 + q^41 + 5 * q^43 + 11 * q^45 - 12 * q^47 - 9 * q^49 - 2 * q^51 + 5 * q^53 + 3 * q^55 + 16 * q^57 - 6 * q^59 - 21 * q^61 - 14 * q^63 + q^65 - 17 * q^67 - 13 * q^69 + 10 * q^71 - 2 * q^73 - 13 * q^75 + 8 * q^77 + 21 * q^79 - 8 * q^81 + q^83 - 17 * q^85 - 31 * q^87 + 5 * q^89 + 2 * q^91 - 23 * q^93 - 12 * q^95 + 6 * q^97 - 2 * 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
$3$	−1.19353	−0.689083	−0.344542	−	0.938771i	$-0.611966\pi$
$3$	−1.19353	−0.689083	−0.344542	+	0.938771i	$0.611966\pi$
$4$	0	0
$5$	−0.837853	−0.374699	−0.187350	−	0.982293i	$-0.559990\pi$
$5$	−0.837853	−0.374699	−0.187350	+	0.982293i	$0.559990\pi$
$6$	0	0
$7$	2.93117	1.10788	0.553939	−	0.832558i	$-0.313124\pi$
$7$	2.93117	1.10788	0.553939	+	0.832558i	$0.313124\pi$
$8$	0	0
$9$	−1.57549	−0.525164
$10$	0	0
$11$	2.35567	0.710263	0.355131	−	0.934816i	$-0.384436\pi$
$11$	2.35567	0.710263	0.355131	+	0.934816i	$0.384436\pi$
$12$	0	0
$13$	−2.42451	−0.672437	−0.336219	−	0.941784i	$-0.609148\pi$
$13$	−2.42451	−0.672437	−0.336219	+	0.941784i	$0.609148\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.637428	−0.154599	−0.0772995	−	0.997008i	$-0.524630\pi$
$17$	−0.637428	−0.154599	−0.0772995	+	0.997008i	$0.524630\pi$
$18$	0	0
$19$	−2.02448	−0.464448	−0.232224	−	0.972662i	$-0.574600\pi$
$19$	−2.02448	−0.464448	−0.232224	+	0.972662i	$0.574600\pi$
$20$	0	0
$21$	−3.49843	−0.763420
$22$	0	0
$23$	0.293740	0.0612489	0.0306245	−	0.999531i	$-0.490250\pi$
$23$	0.293740	0.0612489	0.0306245	+	0.999531i	$0.490250\pi$
$24$	0	0
$25$	−4.29800	−0.859601
$26$	0	0
$27$	5.46097	1.05097
$28$	0	0
$29$	1.68079	0.312116	0.156058	−	0.987748i	$-0.450121\pi$
$29$	1.68079	0.312116	0.156058	+	0.987748i	$0.450121\pi$
$30$	0	0
$31$	7.67390	1.37827	0.689136	−	0.724632i	$-0.257990\pi$
$31$	7.67390	1.37827	0.689136	+	0.724632i	$0.257990\pi$
$32$	0	0
$33$	−2.81156	−0.489430
$34$	0	0
$35$	−2.45589	−0.415121
$36$	0	0
$37$	−5.17413	−0.850622	−0.425311	−	0.905047i	$-0.639835\pi$
$37$	−5.17413	−0.850622	−0.425311	+	0.905047i	$0.639835\pi$
$38$	0	0
$39$	2.89371	0.463365
$40$	0	0
$41$	1.88863	0.294954	0.147477	−	0.989065i	$-0.452885\pi$
$41$	1.88863	0.294954	0.147477	+	0.989065i	$0.452885\pi$
$42$	0	0
$43$	−1.58799	−0.242166	−0.121083	−	0.992642i	$-0.538637\pi$
$43$	−1.58799	−0.242166	−0.121083	+	0.992642i	$0.538637\pi$
$44$	0	0
$45$	1.32003	0.196779
$46$	0	0
$47$	−7.90488	−1.15304	−0.576522	−	0.817081i	$-0.695591\pi$
$47$	−7.90488	−1.15304	−0.576522	+	0.817081i	$0.695591\pi$
$48$	0	0
$49$	1.59174	0.227392
$50$	0	0
$51$	0.760787	0.106532
$52$	0	0
$53$	6.20978	0.852978	0.426489	−	0.904493i	$-0.359750\pi$
$53$	6.20978	0.852978	0.426489	+	0.904493i	$0.359750\pi$
$54$	0	0
$55$	−1.97371	−0.266135
$56$	0	0
$57$	2.41627	0.320043
$58$	0	0
$59$	−10.4772	−1.36402	−0.682009	−	0.731344i	$-0.738894\pi$
$59$	−10.4772	−1.36402	−0.682009	+	0.731344i	$0.738894\pi$
$60$	0	0
$61$	−5.88039	−0.752907	−0.376454	−	0.926435i	$-0.622857\pi$
$61$	−5.88039	−0.752907	−0.376454	+	0.926435i	$0.622857\pi$
$62$	0	0
$63$	−4.61803	−0.581818
$64$	0	0
$65$	2.03138	0.251962
$66$	0	0
$67$	−10.2794	−1.25583	−0.627916	−	0.778281i	$-0.716092\pi$
$67$	−10.2794	−1.25583	−0.627916	+	0.778281i	$0.716092\pi$
$68$	0	0
$69$	−0.350586	−0.0422056
$70$	0	0
$71$	−5.47347	−0.649581	−0.324791	−	0.945786i	$-0.605294\pi$
$71$	−5.47347	−0.649581	−0.324791	+	0.945786i	$0.605294\pi$
$72$	0	0
$73$	3.71510	0.434820	0.217410	−	0.976080i	$-0.430239\pi$
$73$	3.71510	0.434820	0.217410	+	0.976080i	$0.430239\pi$
$74$	0	0
$75$	5.12978	0.592336
$76$	0	0
$77$	6.90488	0.786884
$78$	0	0
$79$	12.5272	1.40942	0.704709	−	0.709497i	$-0.251078\pi$
$79$	12.5272	1.40942	0.704709	+	0.709497i	$0.251078\pi$
$80$	0	0
$81$	−1.79134	−0.199038
$82$	0	0
$83$	5.72384	0.628274	0.314137	−	0.949378i	$-0.398285\pi$
$83$	5.72384	0.628274	0.314137	+	0.949378i	$0.398285\pi$
$84$	0	0
$85$	0.534071	0.0579281
$86$	0	0
$87$	−2.00607	−0.215074
$88$	0	0
$89$	−4.97056	−0.526879	−0.263439	−	0.964676i	$-0.584857\pi$
$89$	−4.97056	−0.526879	−0.263439	+	0.964676i	$0.584857\pi$
$90$	0	0
$91$	−7.10664	−0.744978
$92$	0	0
$93$	−9.15900	−0.949744
$94$	0	0
$95$	1.69622	0.174028
$96$	0	0
$97$	12.5705	1.27634	0.638172	−	0.769893i	$-0.279691\pi$
$97$	12.5705	1.27634	0.638172	+	0.769893i	$0.279691\pi$
$98$	0	0
$99$	−3.71135	−0.373005
$100$	0	0
$101$	−19.2290	−1.91336	−0.956679	−	0.291145i	$-0.905964\pi$
$101$	−19.2290	−1.91336	−0.956679	+	0.291145i	$0.905964\pi$
$102$	0	0
$103$	8.55429	0.842879	0.421440	−	0.906856i	$-0.361525\pi$
$103$	8.55429	0.842879	0.421440	+	0.906856i	$0.361525\pi$
$104$	0	0
$105$	2.93117	0.286053
$106$	0	0
$107$	2.39020	0.231069	0.115535	−	0.993303i	$-0.463142\pi$
$107$	2.39020	0.231069	0.115535	+	0.993303i	$0.463142\pi$
$108$	0	0
$109$	1.66385	0.159368	0.0796841	−	0.996820i	$-0.474609\pi$
$109$	1.66385	0.159368	0.0796841	+	0.996820i	$0.474609\pi$
$110$	0	0
$111$	6.17547	0.586150
$112$	0	0
$113$	9.82915	0.924648	0.462324	−	0.886711i	$-0.347016\pi$
$113$	9.82915	0.924648	0.462324	+	0.886711i	$0.347016\pi$
$114$	0	0
$115$	−0.246111	−0.0229499
$116$	0	0
$117$	3.81979	0.353140
$118$	0	0
$119$	−1.86841	−0.171277
$120$	0	0
$121$	−5.45080	−0.495527
$122$	0	0
$123$	−2.25413	−0.203248
$124$	0	0
$125$	7.79036	0.696791
$126$	0	0
$127$	−1.36369	−0.121008	−0.0605040	−	0.998168i	$-0.519271\pi$
$127$	−1.36369	−0.121008	−0.0605040	+	0.998168i	$0.519271\pi$
$128$	0	0
$129$	1.89531	0.166873
$130$	0	0
$131$	15.0866	1.31812	0.659059	−	0.752091i	$-0.270955\pi$
$131$	15.0866	1.31812	0.659059	+	0.752091i	$0.270955\pi$
$132$	0	0
$133$	−5.93410	−0.514551
$134$	0	0
$135$	−4.57549	−0.393796
$136$	0	0
$137$	−15.7101	−1.34221	−0.671104	−	0.741363i	$-0.734180\pi$
$137$	−15.7101	−1.34221	−0.671104	+	0.741363i	$0.734180\pi$
$138$	0	0
$139$	−7.71644	−0.654500	−0.327250	−	0.944938i	$-0.606122\pi$
$139$	−7.71644	−0.654500	−0.327250	+	0.944938i	$0.606122\pi$
$140$	0	0
$141$	9.43468	0.794544
$142$	0	0
$143$	−5.71135	−0.477607
$144$	0	0
$145$	−1.40826	−0.116949
$146$	0	0
$147$	−1.89979	−0.156692
$148$	0	0
$149$	−14.3026	−1.17172	−0.585858	−	0.810414i	$-0.699242\pi$
$149$	−14.3026	−1.17172	−0.585858	+	0.810414i	$0.699242\pi$
$150$	0	0
$151$	14.2385	1.15871	0.579357	−	0.815074i	$-0.303304\pi$
$151$	14.2385	1.15871	0.579357	+	0.815074i	$0.303304\pi$
$152$	0	0
$153$	1.00426	0.0811899
$154$	0	0
$155$	−6.42960	−0.516438
$156$	0	0
$157$	−8.70610	−0.694822	−0.347411	−	0.937713i	$-0.612939\pi$
$157$	−8.70610	−0.694822	−0.347411	+	0.937713i	$0.612939\pi$
$158$	0	0
$159$	−7.41154	−0.587773
$160$	0	0
$161$	0.861000	0.0678563
$162$	0	0
$163$	−22.1493	−1.73487	−0.867434	−	0.497552i	$-0.834232\pi$
$163$	−22.1493	−1.73487	−0.867434	+	0.497552i	$0.834232\pi$
$164$	0	0
$165$	2.35567	0.183389
$166$	0	0
$167$	−2.00741	−0.155338	−0.0776689	−	0.996979i	$-0.524748\pi$
$167$	−2.00741	−0.155338	−0.0776689	+	0.996979i	$0.524748\pi$
$168$	0	0
$169$	−7.12177	−0.547828
$170$	0	0
$171$	3.18956	0.243912
$172$	0	0
$173$	5.98013	0.454661	0.227330	−	0.973818i	$-0.427000\pi$
$173$	5.98013	0.454661	0.227330	+	0.973818i	$0.427000\pi$
$174$	0	0
$175$	−12.5982	−0.952332
$176$	0	0
$177$	12.5049	0.939922
$178$	0	0
$179$	−8.61114	−0.643627	−0.321813	−	0.946803i	$-0.604292\pi$
$179$	−8.61114	−0.643627	−0.321813	+	0.946803i	$0.604292\pi$
$180$	0	0
$181$	−16.0734	−1.19473	−0.597365	−	0.801970i	$-0.703786\pi$
$181$	−16.0734	−1.19473	−0.597365	+	0.801970i	$0.703786\pi$
$182$	0	0
$183$	7.01841	0.518816
$184$	0	0
$185$	4.33516	0.318727
$186$	0	0
$187$	−1.50157	−0.109806
$188$	0	0
$189$	16.0070	1.16434
$190$	0	0
$191$	4.74670	0.343459	0.171730	−	0.985144i	$-0.445064\pi$
$191$	4.74670	0.343459	0.171730	+	0.985144i	$0.445064\pi$
$192$	0	0
$193$	−14.5794	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$193$	−14.5794	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$194$	0	0
$195$	−2.42451	−0.173623
$196$	0	0
$197$	−12.5748	−0.895918	−0.447959	−	0.894054i	$-0.647849\pi$
$197$	−12.5748	−0.895918	−0.447959	+	0.894054i	$0.647849\pi$
$198$	0	0
$199$	−19.5166	−1.38350	−0.691748	−	0.722139i	$-0.743159\pi$
$199$	−19.5166	−1.38350	−0.691748	+	0.722139i	$0.743159\pi$
$200$	0	0
$201$	12.2688	0.865373
$202$	0	0
$203$	4.92669	0.345786
$204$	0	0
$205$	−1.58239	−0.110519
$206$	0	0
$207$	−0.462785	−0.0321658
$208$	0	0
$209$	−4.76902	−0.329880
$210$	0	0
$211$	−17.0246	−1.17202	−0.586011	−	0.810303i	$-0.699303\pi$
$211$	−17.0246	−1.17202	−0.586011	+	0.810303i	$0.699303\pi$
$212$	0	0
$213$	6.53274	0.447616
$214$	0	0
$215$	1.33050	0.0907394
$216$	0	0
$217$	22.4935	1.52696
$218$	0	0
$219$	−4.43407	−0.299627
$220$	0	0
$221$	1.54545	0.103958
$222$	0	0
$223$	−15.3334	−1.02680	−0.513399	−	0.858150i	$-0.671614\pi$
$223$	−15.3334	−1.02680	−0.513399	+	0.858150i	$0.671614\pi$
$224$	0	0
$225$	6.77147	0.451432
$226$	0	0
$227$	−11.8895	−0.789131	−0.394565	−	0.918868i	$-0.629105\pi$
$227$	−11.8895	−0.789131	−0.394565	+	0.918868i	$0.629105\pi$
$228$	0	0
$229$	−21.0317	−1.38982	−0.694908	−	0.719099i	$-0.744555\pi$
$229$	−21.0317	−1.38982	−0.694908	+	0.719099i	$0.744555\pi$
$230$	0	0
$231$	−8.24116	−0.542228
$232$	0	0
$233$	−23.0079	−1.50730	−0.753649	−	0.657278i	$-0.771708\pi$
$233$	−23.0079	−1.50730	−0.753649	+	0.657278i	$0.771708\pi$
$234$	0	0
$235$	6.62312	0.432045
$236$	0	0
$237$	−14.9515	−0.971206
$238$	0	0
$239$	15.4360	0.998473	0.499237	−	0.866466i	$-0.333614\pi$
$239$	15.4360	0.998473	0.499237	+	0.866466i	$0.333614\pi$
$240$	0	0
$241$	−12.5523	−0.808562	−0.404281	−	0.914635i	$-0.632478\pi$
$241$	−12.5523	−0.808562	−0.404281	+	0.914635i	$0.632478\pi$
$242$	0	0
$243$	−14.2449	−0.913811
$244$	0	0
$245$	−1.33365	−0.0852035
$246$	0	0
$247$	4.90837	0.312312
$248$	0	0
$249$	−6.83156	−0.432933
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.691955	0.0435028
$254$	0	0
$255$	−0.637428	−0.0399173
$256$	0	0
$257$	−1.47701	−0.0921332	−0.0460666	−	0.998938i	$-0.514669\pi$
$257$	−1.47701	−0.0921332	−0.0460666	+	0.998938i	$0.514669\pi$
$258$	0	0
$259$	−15.1663	−0.942385
$260$	0	0
$261$	−2.64808	−0.163912
$262$	0	0
$263$	−10.9120	−0.672862	−0.336431	−	0.941708i	$-0.609220\pi$
$263$	−10.9120	−0.672862	−0.336431	+	0.941708i	$0.609220\pi$
$264$	0	0
$265$	−5.20288	−0.319610
$266$	0	0
$267$	5.93250	0.363063
$268$	0	0
$269$	−20.7010	−1.26216	−0.631082	−	0.775717i	$-0.717389\pi$
$269$	−20.7010	−1.26216	−0.631082	+	0.775717i	$0.717389\pi$
$270$	0	0
$271$	19.8976	1.20869	0.604347	−	0.796722i	$-0.293434\pi$
$271$	19.8976	1.20869	0.604347	+	0.796722i	$0.293434\pi$
$272$	0	0
$273$	8.48196	0.513352
$274$	0	0
$275$	−10.1247	−0.610542
$276$	0	0
$277$	−8.49947	−0.510684	−0.255342	−	0.966851i	$-0.582188\pi$
$277$	−8.49947	−0.510684	−0.255342	+	0.966851i	$0.582188\pi$
$278$	0	0
$279$	−12.0902	−0.723820
$280$	0	0
$281$	30.1332	1.79760	0.898798	−	0.438363i	$-0.144442\pi$
$281$	30.1332	1.79760	0.898798	+	0.438363i	$0.144442\pi$
$282$	0	0
$283$	6.06763	0.360683	0.180342	−	0.983604i	$-0.442280\pi$
$283$	6.06763	0.360683	0.180342	+	0.983604i	$0.442280\pi$
$284$	0	0
$285$	−2.02448	−0.119920
$286$	0	0
$287$	5.53588	0.326773
$288$	0	0
$289$	−16.5937	−0.976099
$290$	0	0
$291$	−15.0033	−0.879508
$292$	0	0
$293$	5.89190	0.344209	0.172104	−	0.985079i	$-0.444943\pi$
$293$	5.89190	0.344209	0.172104	+	0.985079i	$0.444943\pi$
$294$	0	0
$295$	8.77837	0.511097
$296$	0	0
$297$	12.8643	0.746461
$298$	0	0
$299$	−0.712174	−0.0411861
$300$	0	0
$301$	−4.65466	−0.268290
$302$	0	0
$303$	22.9503	1.31846
$304$	0	0
$305$	4.92690	0.282114
$306$	0	0
$307$	27.0271	1.54252	0.771258	−	0.636522i	$-0.219628\pi$
$307$	27.0271	1.54252	0.771258	+	0.636522i	$0.219628\pi$
$308$	0	0
$309$	−10.2098	−0.580814
$310$	0	0
$311$	1.58217	0.0897169	0.0448584	−	0.998993i	$-0.485716\pi$
$311$	1.58217	0.0897169	0.0448584	+	0.998993i	$0.485716\pi$
$312$	0	0
$313$	−5.56148	−0.314353	−0.157177	−	0.987570i	$-0.550239\pi$
$313$	−5.56148	−0.314353	−0.157177	+	0.987570i	$0.550239\pi$
$314$	0	0
$315$	3.86923	0.218007
$316$	0	0
$317$	17.6482	0.991224	0.495612	−	0.868544i	$-0.334944\pi$
$317$	17.6482	0.991224	0.495612	+	0.868544i	$0.334944\pi$
$318$	0	0
$319$	3.95940	0.221684
$320$	0	0
$321$	−2.85277	−0.159226
$322$	0	0
$323$	1.29046	0.0718032
$324$	0	0
$325$	10.4205	0.578027
$326$	0	0
$327$	−1.98585	−0.109818
$328$	0	0
$329$	−23.1705	−1.27743
$330$	0	0
$331$	−5.74204	−0.315611	−0.157805	−	0.987470i	$-0.550442\pi$
$331$	−5.74204	−0.315611	−0.157805	+	0.987470i	$0.550442\pi$
$332$	0	0
$333$	8.15181	0.446717
$334$	0	0
$335$	8.61265	0.470559
$336$	0	0
$337$	12.1812	0.663550	0.331775	−	0.943358i	$-0.392352\pi$
$337$	12.1812	0.663550	0.331775	+	0.943358i	$0.392352\pi$
$338$	0	0
$339$	−11.7314	−0.637160
$340$	0	0
$341$	18.0772	0.978935
$342$	0	0
$343$	−15.8525	−0.855955
$344$	0	0
$345$	0.293740	0.0158144
$346$	0	0
$347$	−16.2643	−0.873114	−0.436557	−	0.899677i	$-0.643802\pi$
$347$	−16.2643	−0.873114	−0.436557	+	0.899677i	$0.643802\pi$
$348$	0	0
$349$	−11.2964	−0.604680	−0.302340	−	0.953200i	$-0.597768\pi$
$349$	−11.2964	−0.604680	−0.302340	+	0.953200i	$0.597768\pi$
$350$	0	0
$351$	−13.2402	−0.706708
$352$	0	0
$353$	11.8627	0.631389	0.315695	−	0.948861i	$-0.397762\pi$
$353$	11.8627	0.631389	0.315695	+	0.948861i	$0.397762\pi$
$354$	0	0
$355$	4.58596	0.243398
$356$	0	0
$357$	2.23000	0.118024
$358$	0	0
$359$	−9.91324	−0.523201	−0.261600	−	0.965176i	$-0.584250\pi$
$359$	−9.91324	−0.523201	−0.261600	+	0.965176i	$0.584250\pi$
$360$	0	0
$361$	−14.9015	−0.784288
$362$	0	0
$363$	6.50568	0.341459
$364$	0	0
$365$	−3.11271	−0.162927
$366$	0	0
$367$	−34.6451	−1.80846	−0.904230	−	0.427045i	$-0.859555\pi$
$367$	−34.6451	−1.80846	−0.904230	+	0.427045i	$0.859555\pi$
$368$	0	0
$369$	−2.97552	−0.154899
$370$	0	0
$371$	18.2019	0.944995
$372$	0	0
$373$	−9.12520	−0.472485	−0.236243	−	0.971694i	$-0.575916\pi$
$373$	−9.12520	−0.472485	−0.236243	+	0.971694i	$0.575916\pi$
$374$	0	0
$375$	−9.29800	−0.480147
$376$	0	0
$377$	−4.07510	−0.209878
$378$	0	0
$379$	−17.2046	−0.883741	−0.441871	−	0.897079i	$-0.645685\pi$
$379$	−17.2046	−0.883741	−0.441871	+	0.897079i	$0.645685\pi$
$380$	0	0
$381$	1.62760	0.0833846
$382$	0	0
$383$	7.95596	0.406531	0.203265	−	0.979124i	$-0.434845\pi$
$383$	7.95596	0.406531	0.203265	+	0.979124i	$0.434845\pi$
$384$	0	0
$385$	−5.78527	−0.294845
$386$	0	0
$387$	2.50187	0.127177
$388$	0	0
$389$	−23.5671	−1.19490	−0.597450	−	0.801906i	$-0.703819\pi$
$389$	−23.5671	−1.19490	−0.597450	+	0.801906i	$0.703819\pi$
$390$	0	0
$391$	−0.187238	−0.00946902
$392$	0	0
$393$	−18.0062	−0.908293
$394$	0	0
$395$	−10.4959	−0.528107
$396$	0	0
$397$	−5.25711	−0.263847	−0.131923	−	0.991260i	$-0.542115\pi$
$397$	−5.25711	−0.263847	−0.131923	+	0.991260i	$0.542115\pi$
$398$	0	0
$399$	7.08250	0.354569
$400$	0	0
$401$	15.2877	0.763430	0.381715	−	0.924280i	$-0.375334\pi$
$401$	15.2877	0.763430	0.381715	+	0.924280i	$0.375334\pi$
$402$	0	0
$403$	−18.6054	−0.926802
$404$	0	0
$405$	1.50088	0.0745794
$406$	0	0
$407$	−12.1886	−0.604165
$408$	0	0
$409$	25.8643	1.27891	0.639454	−	0.768829i	$-0.279160\pi$
$409$	25.8643	1.27891	0.639454	+	0.768829i	$0.279160\pi$
$410$	0	0
$411$	18.7505	0.924893
$412$	0	0
$413$	−30.7105	−1.51116
$414$	0	0
$415$	−4.79574	−0.235414
$416$	0	0
$417$	9.20978	0.451005
$418$	0	0
$419$	3.81795	0.186519	0.0932595	−	0.995642i	$-0.470271\pi$
$419$	3.81795	0.186519	0.0932595	+	0.995642i	$0.470271\pi$
$420$	0	0
$421$	−12.2724	−0.598120	−0.299060	−	0.954234i	$-0.596673\pi$
$421$	−12.2724	−0.598120	−0.299060	+	0.954234i	$0.596673\pi$
$422$	0	0
$423$	12.4541	0.605538
$424$	0	0
$425$	2.73967	0.132893
$426$	0	0
$427$	−17.2364	−0.834129
$428$	0	0
$429$	6.81665	0.329111
$430$	0	0
$431$	−14.0257	−0.675594	−0.337797	−	0.941219i	$-0.609682\pi$
$431$	−14.0257	−0.675594	−0.337797	+	0.941219i	$0.609682\pi$
$432$	0	0
$433$	0.485400	0.0233268	0.0116634	−	0.999932i	$-0.496287\pi$
$433$	0.485400	0.0233268	0.0116634	+	0.999932i	$0.496287\pi$
$434$	0	0
$435$	1.68079	0.0805879
$436$	0	0
$437$	−0.594670	−0.0284469
$438$	0	0
$439$	18.3646	0.876497	0.438248	−	0.898854i	$-0.355599\pi$
$439$	18.3646	0.876497	0.438248	+	0.898854i	$0.355599\pi$
$440$	0	0
$441$	−2.50778	−0.119418
$442$	0	0
$443$	36.7617	1.74660	0.873301	−	0.487181i	$-0.161975\pi$
$443$	36.7617	1.74660	0.873301	+	0.487181i	$0.161975\pi$
$444$	0	0
$445$	4.16460	0.197421
$446$	0	0
$447$	17.0706	0.807410
$448$	0	0
$449$	25.8808	1.22139	0.610696	−	0.791865i	$-0.290890\pi$
$449$	25.8808	1.22139	0.610696	+	0.791865i	$0.290890\pi$
$450$	0	0
$451$	4.44899	0.209495
$452$	0	0
$453$	−16.9941	−0.798451
$454$	0	0
$455$	5.95431	0.279143
$456$	0	0
$457$	−1.93311	−0.0904271	−0.0452136	−	0.998977i	$-0.514397\pi$
$457$	−1.93311	−0.0904271	−0.0452136	+	0.998977i	$0.514397\pi$
$458$	0	0
$459$	−3.48098	−0.162478
$460$	0	0
$461$	15.5188	0.722782	0.361391	−	0.932414i	$-0.382302\pi$
$461$	15.5188	0.722782	0.361391	+	0.932414i	$0.382302\pi$
$462$	0	0
$463$	−1.99379	−0.0926594	−0.0463297	−	0.998926i	$-0.514752\pi$
$463$	−1.99379	−0.0926594	−0.0463297	+	0.998926i	$0.514752\pi$
$464$	0	0
$465$	7.67390	0.355868
$466$	0	0
$467$	38.4888	1.78105	0.890525	−	0.454933i	$-0.150337\pi$
$467$	38.4888	1.78105	0.890525	+	0.454933i	$0.150337\pi$
$468$	0	0
$469$	−30.1307	−1.39131
$470$	0	0
$471$	10.3910	0.478790
$472$	0	0
$473$	−3.74078	−0.172001
$474$	0	0
$475$	8.70123	0.399240
$476$	0	0
$477$	−9.78346	−0.447954
$478$	0	0
$479$	26.5064	1.21111	0.605555	−	0.795804i	$-0.292951\pi$
$479$	26.5064	1.21111	0.605555	+	0.795804i	$0.292951\pi$
$480$	0	0
$481$	12.5447	0.571990
$482$	0	0
$483$	−1.02763	−0.0467586
$484$	0	0
$485$	−10.5323	−0.478245
$486$	0	0
$487$	−15.0469	−0.681838	−0.340919	−	0.940093i	$-0.610738\pi$
$487$	−15.0469	−0.681838	−0.340919	+	0.940093i	$0.610738\pi$
$488$	0	0
$489$	26.4358	1.19547
$490$	0	0
$491$	17.5627	0.792595	0.396298	−	0.918122i	$-0.370295\pi$
$491$	17.5627	0.792595	0.396298	+	0.918122i	$0.370295\pi$
$492$	0	0
$493$	−1.07138	−0.0482527
$494$	0	0
$495$	3.10956	0.139764
$496$	0	0
$497$	−16.0437	−0.719656
$498$	0	0
$499$	−15.3593	−0.687577	−0.343788	−	0.939047i	$-0.611710\pi$
$499$	−15.3593	−0.687577	−0.343788	+	0.939047i	$0.611710\pi$
$500$	0	0
$501$	2.39590	0.107041
$502$	0	0
$503$	−9.44219	−0.421006	−0.210503	−	0.977593i	$-0.567510\pi$
$503$	−9.44219	−0.421006	−0.210503	+	0.977593i	$0.567510\pi$
$504$	0	0
$505$	16.1111	0.716934
$506$	0	0
$507$	8.50002	0.377499
$508$	0	0
$509$	20.4522	0.906529	0.453265	−	0.891376i	$-0.350259\pi$
$509$	20.4522	0.906529	0.453265	+	0.891376i	$0.350259\pi$
$510$	0	0
$511$	10.8896	0.481727
$512$	0	0
$513$	−11.0556	−0.488119
$514$	0	0
$515$	−7.16724	−0.315826
$516$	0	0
$517$	−18.6213	−0.818964
$518$	0	0
$519$	−7.13745	−0.313299
$520$	0	0
$521$	37.6156	1.64797	0.823985	−	0.566611i	$-0.191746\pi$
$521$	37.6156	1.64797	0.823985	+	0.566611i	$0.191746\pi$
$522$	0	0
$523$	−40.2945	−1.76196	−0.880978	−	0.473157i	$-0.843114\pi$
$523$	−40.2945	−1.76196	−0.880978	+	0.473157i	$0.843114\pi$
$524$	0	0
$525$	15.0363	0.656236
$526$	0	0
$527$	−4.89155	−0.213079
$528$	0	0
$529$	−22.9137	−0.996249
$530$	0	0
$531$	16.5068	0.716334
$532$	0	0
$533$	−4.57899	−0.198338
$534$	0	0
$535$	−2.00263	−0.0865815
$536$	0	0
$537$	10.2776	0.443512
$538$	0	0
$539$	3.74963	0.161508
$540$	0	0
$541$	27.2369	1.17100	0.585502	−	0.810671i	$-0.300897\pi$
$541$	27.2369	1.17100	0.585502	+	0.810671i	$0.300897\pi$
$542$	0	0
$543$	19.1841	0.823268
$544$	0	0
$545$	−1.39406	−0.0597151
$546$	0	0
$547$	23.5769	1.00807	0.504037	−	0.863682i	$-0.331847\pi$
$547$	23.5769	1.00807	0.504037	+	0.863682i	$0.331847\pi$
$548$	0	0
$549$	9.26452	0.395400
$550$	0	0
$551$	−3.40274	−0.144961
$552$	0	0
$553$	36.7192	1.56146
$554$	0	0
$555$	−5.17413	−0.219630
$556$	0	0
$557$	−32.8729	−1.39287	−0.696434	−	0.717621i	$-0.745231\pi$
$557$	−32.8729	−1.39287	−0.696434	+	0.717621i	$0.745231\pi$
$558$	0	0
$559$	3.85009	0.162841
$560$	0	0
$561$	1.79217	0.0756654
$562$	0	0
$563$	−15.1097	−0.636798	−0.318399	−	0.947957i	$-0.603145\pi$
$563$	−15.1097	−0.636798	−0.318399	+	0.947957i	$0.603145\pi$
$564$	0	0
$565$	−8.23538	−0.346465
$566$	0	0
$567$	−5.25072	−0.220510
$568$	0	0
$569$	22.3526	0.937068	0.468534	−	0.883445i	$-0.344782\pi$
$569$	22.3526	0.937068	0.468534	+	0.883445i	$0.344782\pi$
$570$	0	0
$571$	39.4575	1.65125	0.825623	−	0.564223i	$-0.190824\pi$
$571$	39.4575	1.65125	0.825623	+	0.564223i	$0.190824\pi$
$572$	0	0
$573$	−5.66531	−0.236672
$574$	0	0
$575$	−1.26249	−0.0526496
$576$	0	0
$577$	−9.99310	−0.416018	−0.208009	−	0.978127i	$-0.566698\pi$
$577$	−9.99310	−0.416018	−0.208009	+	0.978127i	$0.566698\pi$
$578$	0	0
$579$	17.4009	0.723156
$580$	0	0
$581$	16.7775	0.696050
$582$	0	0
$583$	14.6282	0.605839
$584$	0	0
$585$	−3.20042	−0.132321
$586$	0	0
$587$	12.4509	0.513902	0.256951	−	0.966424i	$-0.417282\pi$
$587$	12.4509	0.513902	0.256951	+	0.966424i	$0.417282\pi$
$588$	0	0
$589$	−15.5357	−0.640136
$590$	0	0
$591$	15.0084	0.617362
$592$	0	0
$593$	26.6859	1.09586	0.547929	−	0.836525i	$-0.315416\pi$
$593$	26.6859	1.09586	0.547929	+	0.836525i	$0.315416\pi$
$594$	0	0
$595$	1.56545	0.0641772
$596$	0	0
$597$	23.2936	0.953344
$598$	0	0
$599$	−32.6261	−1.33307	−0.666534	−	0.745475i	$-0.732223\pi$
$599$	−32.6261	−1.33307	−0.666534	+	0.745475i	$0.732223\pi$
$600$	0	0
$601$	−23.3508	−0.952500	−0.476250	−	0.879310i	$-0.658004\pi$
$601$	−23.3508	−0.952500	−0.476250	+	0.879310i	$0.658004\pi$
$602$	0	0
$603$	16.1952	0.659519
$604$	0	0
$605$	4.56697	0.185674
$606$	0	0
$607$	−30.6518	−1.24412	−0.622059	−	0.782971i	$-0.713704\pi$
$607$	−30.6518	−1.24412	−0.622059	+	0.782971i	$0.713704\pi$
$608$	0	0
$609$	−5.88014	−0.238275
$610$	0	0
$611$	19.1654	0.775350
$612$	0	0
$613$	33.8146	1.36576	0.682879	−	0.730531i	$-0.260727\pi$
$613$	33.8146	1.36576	0.682879	+	0.730531i	$0.260727\pi$
$614$	0	0
$615$	1.88863	0.0761568
$616$	0	0
$617$	1.87203	0.0753650	0.0376825	−	0.999290i	$-0.488002\pi$
$617$	1.87203	0.0753650	0.0376825	+	0.999290i	$0.488002\pi$
$618$	0	0
$619$	−24.6466	−0.990630	−0.495315	−	0.868713i	$-0.664947\pi$
$619$	−24.6466	−0.990630	−0.495315	+	0.868713i	$0.664947\pi$
$620$	0	0
$621$	1.60410	0.0643705
$622$	0	0
$623$	−14.5696	−0.583717
$624$	0	0
$625$	14.9628	0.598514
$626$	0	0
$627$	5.69195	0.227315
$628$	0	0
$629$	3.29814	0.131505
$630$	0	0
$631$	6.18905	0.246382	0.123191	−	0.992383i	$-0.460687\pi$
$631$	6.18905	0.246382	0.123191	+	0.992383i	$0.460687\pi$
$632$	0	0
$633$	20.3193	0.807621
$634$	0	0
$635$	1.14257	0.0453416
$636$	0	0
$637$	−3.85919	−0.152907
$638$	0	0
$639$	8.62342	0.341137
$640$	0	0
$641$	31.3785	1.23938	0.619688	−	0.784848i	$-0.287259\pi$
$641$	31.3785	1.23938	0.619688	+	0.784848i	$0.287259\pi$
$642$	0	0
$643$	32.5618	1.28411	0.642056	−	0.766658i	$-0.278082\pi$
$643$	32.5618	1.28411	0.642056	+	0.766658i	$0.278082\pi$
$644$	0	0
$645$	−1.58799	−0.0625270
$646$	0	0
$647$	34.8763	1.37113	0.685564	−	0.728013i	$-0.259556\pi$
$647$	34.8763	1.37113	0.685564	+	0.728013i	$0.259556\pi$
$648$	0	0
$649$	−24.6809	−0.968811
$650$	0	0
$651$	−26.8466	−1.05220
$652$	0	0
$653$	34.7905	1.36146	0.680729	−	0.732535i	$-0.261663\pi$
$653$	34.7905	1.36146	0.680729	+	0.732535i	$0.261663\pi$
$654$	0	0
$655$	−12.6403	−0.493898
$656$	0	0
$657$	−5.85312	−0.228352
$658$	0	0
$659$	31.7180	1.23556	0.617780	−	0.786351i	$-0.288032\pi$
$659$	31.7180	1.23556	0.617780	+	0.786351i	$0.288032\pi$
$660$	0	0
$661$	−47.1908	−1.83551	−0.917756	−	0.397146i	$-0.870001\pi$
$661$	−47.1908	−1.83551	−0.917756	+	0.397146i	$0.870001\pi$
$662$	0	0
$663$	−1.84453	−0.0716358
$664$	0	0
$665$	4.97190	0.192802
$666$	0	0
$667$	0.493716	0.0191167
$668$	0	0
$669$	18.3008	0.707549
$670$	0	0
$671$	−13.8523	−0.534762
$672$	0	0
$673$	−33.1722	−1.27870	−0.639348	−	0.768918i	$-0.720796\pi$
$673$	−33.1722	−1.27870	−0.639348	+	0.768918i	$0.720796\pi$
$674$	0	0
$675$	−23.4713	−0.903410
$676$	0	0
$677$	−4.55940	−0.175232	−0.0876161	−	0.996154i	$-0.527925\pi$
$677$	−4.55940	−0.175232	−0.0876161	+	0.996154i	$0.527925\pi$
$678$	0	0
$679$	36.8464	1.41403
$680$	0	0
$681$	14.1904	0.543777
$682$	0	0
$683$	39.9242	1.52766	0.763829	−	0.645419i	$-0.223317\pi$
$683$	39.9242	1.52766	0.763829	+	0.645419i	$0.223317\pi$
$684$	0	0
$685$	13.1628	0.502924
$686$	0	0
$687$	25.1019	0.957699
$688$	0	0
$689$	−15.0556	−0.573574
$690$	0	0
$691$	−47.0803	−1.79102	−0.895509	−	0.445044i	$-0.853188\pi$
$691$	−47.0803	−1.79102	−0.895509	+	0.445044i	$0.853188\pi$
$692$	0	0
$693$	−10.8786	−0.413243
$694$	0	0
$695$	6.46524	0.245240
$696$	0	0
$697$	−1.20386	−0.0455996
$698$	0	0
$699$	27.4605	1.03865
$700$	0	0
$701$	−6.59110	−0.248942	−0.124471	−	0.992223i	$-0.539723\pi$
$701$	−6.59110	−0.248942	−0.124471	+	0.992223i	$0.539723\pi$
$702$	0	0
$703$	10.4749	0.395070
$704$	0	0
$705$	−7.90488	−0.297715
$706$	0	0
$707$	−56.3634	−2.11977
$708$	0	0
$709$	−23.7935	−0.893584	−0.446792	−	0.894638i	$-0.647434\pi$
$709$	−23.7935	−0.893584	−0.446792	+	0.894638i	$0.647434\pi$
$710$	0	0
$711$	−19.7365	−0.740176
$712$	0	0
$713$	2.25413	0.0844177
$714$	0	0
$715$	4.78527	0.178959
$716$	0	0
$717$	−18.4233	−0.688031
$718$	0	0
$719$	−9.08282	−0.338732	−0.169366	−	0.985553i	$-0.554172\pi$
$719$	−9.08282	−0.338732	−0.169366	+	0.985553i	$0.554172\pi$
$720$	0	0
$721$	25.0741	0.933807
$722$	0	0
$723$	14.9815	0.557167
$724$	0	0
$725$	−7.22406	−0.268295
$726$	0	0
$727$	−5.95708	−0.220936	−0.110468	−	0.993880i	$-0.535235\pi$
$727$	−5.95708	−0.220936	−0.110468	+	0.993880i	$0.535235\pi$
$728$	0	0
$729$	22.3757	0.828730
$730$	0	0
$731$	1.01223	0.0374386
$732$	0	0
$733$	−1.41300	−0.0521902	−0.0260951	−	0.999659i	$-0.508307\pi$
$733$	−1.41300	−0.0521902	−0.0260951	+	0.999659i	$0.508307\pi$
$734$	0	0
$735$	1.59174	0.0587123
$736$	0	0
$737$	−24.2150	−0.891971
$738$	0	0
$739$	−3.79225	−0.139500	−0.0697501	−	0.997564i	$-0.522220\pi$
$739$	−3.79225	−0.139500	−0.0697501	+	0.997564i	$0.522220\pi$
$740$	0	0
$741$	−5.85827	−0.215209
$742$	0	0
$743$	−43.8071	−1.60713	−0.803563	−	0.595220i	$-0.797065\pi$
$743$	−43.8071	−1.60713	−0.803563	+	0.595220i	$0.797065\pi$
$744$	0	0
$745$	11.9835	0.439041
$746$	0	0
$747$	−9.01788	−0.329947
$748$	0	0
$749$	7.00607	0.255996
$750$	0	0
$751$	48.4559	1.76818	0.884091	−	0.467315i	$-0.154778\pi$
$751$	48.4559	1.76818	0.884091	+	0.467315i	$0.154778\pi$
$752$	0	0
$753$	−1.19353	−0.0434945
$754$	0	0
$755$	−11.9298	−0.434169
$756$	0	0
$757$	−23.4104	−0.850864	−0.425432	−	0.904990i	$-0.639878\pi$
$757$	−23.4104	−0.850864	−0.425432	+	0.904990i	$0.639878\pi$
$758$	0	0
$759$	−0.825867	−0.0299771
$760$	0	0
$761$	39.1366	1.41870	0.709351	−	0.704855i	$-0.248988\pi$
$761$	39.1366	1.41870	0.709351	+	0.704855i	$0.248988\pi$
$762$	0	0
$763$	4.87703	0.176560
$764$	0	0
$765$	−0.841425	−0.0304218
$766$	0	0
$767$	25.4021	0.917217
$768$	0	0
$769$	−0.469466	−0.0169294	−0.00846469	−	0.999964i	$-0.502694\pi$
$769$	−0.469466	−0.0169294	−0.00846469	+	0.999964i	$0.502694\pi$
$770$	0	0
$771$	1.76285	0.0634874
$772$	0	0
$773$	2.01580	0.0725033	0.0362516	−	0.999343i	$-0.488458\pi$
$773$	2.01580	0.0725033	0.0362516	+	0.999343i	$0.488458\pi$
$774$	0	0
$775$	−32.9824	−1.18476
$776$	0	0
$777$	18.1013	0.649382
$778$	0	0
$779$	−3.82349	−0.136991
$780$	0	0
$781$	−12.8937	−0.461373
$782$	0	0
$783$	9.17877	0.328023
$784$	0	0
$785$	7.29443	0.260349
$786$	0	0
$787$	23.9902	0.855157	0.427579	−	0.903978i	$-0.359367\pi$
$787$	23.9902	0.855157	0.427579	+	0.903978i	$0.359367\pi$
$788$	0	0
$789$	13.0238	0.463658
$790$	0	0
$791$	28.8109	1.02440
$792$	0	0
$793$	14.2571	0.506283
$794$	0	0
$795$	6.20978	0.220238
$796$	0	0
$797$	−33.4710	−1.18560	−0.592801	−	0.805349i	$-0.701978\pi$
$797$	−33.4710	−1.18560	−0.592801	+	0.805349i	$0.701978\pi$
$798$	0	0
$799$	5.03879	0.178259
$800$	0	0
$801$	7.83109	0.276698
$802$	0	0
$803$	8.75157	0.308836
$804$	0	0
$805$	−0.721391	−0.0254257
$806$	0	0
$807$	24.7072	0.869735
$808$	0	0
$809$	−13.2213	−0.464834	−0.232417	−	0.972616i	$-0.574663\pi$
$809$	−13.2213	−0.464834	−0.232417	+	0.972616i	$0.574663\pi$
$810$	0	0
$811$	32.7878	1.15134	0.575668	−	0.817683i	$-0.304742\pi$
$811$	32.7878	1.15134	0.575668	+	0.817683i	$0.304742\pi$
$812$	0	0
$813$	−23.7483	−0.832890
$814$	0	0
$815$	18.5579	0.650054
$816$	0	0
$817$	3.21486	0.112474
$818$	0	0
$819$	11.1965	0.391236
$820$	0	0
$821$	−46.9790	−1.63958	−0.819789	−	0.572666i	$-0.805909\pi$
$821$	−46.9790	−1.63958	−0.819789	+	0.572666i	$0.805909\pi$
$822$	0	0
$823$	5.63351	0.196372	0.0981860	−	0.995168i	$-0.468696\pi$
$823$	5.63351	0.196372	0.0981860	+	0.995168i	$0.468696\pi$
$824$	0	0
$825$	12.0841	0.420714
$826$	0	0
$827$	44.3501	1.54221	0.771103	−	0.636711i	$-0.219706\pi$
$827$	44.3501	1.54221	0.771103	+	0.636711i	$0.219706\pi$
$828$	0	0
$829$	38.1840	1.32618	0.663092	−	0.748538i	$-0.269244\pi$
$829$	38.1840	1.32618	0.663092	+	0.748538i	$0.269244\pi$
$830$	0	0
$831$	10.1443	0.351903
$832$	0	0
$833$	−1.01462	−0.0351545
$834$	0	0
$835$	1.68191	0.0582050
$836$	0	0
$837$	41.9070	1.44852
$838$	0	0
$839$	−30.0216	−1.03646	−0.518231	−	0.855241i	$-0.673409\pi$
$839$	−30.0216	−1.03646	−0.518231	+	0.855241i	$0.673409\pi$
$840$	0	0
$841$	−26.1749	−0.902584
$842$	0	0
$843$	−35.9648	−1.23869
$844$	0	0
$845$	5.96699	0.205271
$846$	0	0
$847$	−15.9772	−0.548983
$848$	0	0
$849$	−7.24188	−0.248541
$850$	0	0
$851$	−1.51985	−0.0520997
$852$	0	0
$853$	30.4841	1.04376	0.521878	−	0.853020i	$-0.325232\pi$
$853$	30.4841	1.04376	0.521878	+	0.853020i	$0.325232\pi$
$854$	0	0
$855$	−2.67238	−0.0913935
$856$	0	0
$857$	−16.4612	−0.562305	−0.281153	−	0.959663i	$-0.590717\pi$
$857$	−16.4612	−0.562305	−0.281153	+	0.959663i	$0.590717\pi$
$858$	0	0
$859$	−16.7542	−0.571644	−0.285822	−	0.958283i	$-0.592267\pi$
$859$	−16.7542	−0.571644	−0.285822	+	0.958283i	$0.592267\pi$
$860$	0	0
$861$	−6.60722	−0.225174
$862$	0	0
$863$	−31.2406	−1.06344	−0.531721	−	0.846920i	$-0.678454\pi$
$863$	−31.2406	−1.06344	−0.531721	+	0.846920i	$0.678454\pi$
$864$	0	0
$865$	−5.01047	−0.170361
$866$	0	0
$867$	19.8050	0.672614
$868$	0	0
$869$	29.5099	1.00106
$870$	0	0
$871$	24.9226	0.844469
$872$	0	0
$873$	−19.8048	−0.670291
$874$	0	0
$875$	22.8348	0.771959
$876$	0	0
$877$	−11.4109	−0.385320	−0.192660	−	0.981266i	$-0.561711\pi$
$877$	−11.4109	−0.385320	−0.192660	+	0.981266i	$0.561711\pi$
$878$	0	0
$879$	−7.03215	−0.237188
$880$	0	0
$881$	34.1884	1.15184	0.575918	−	0.817507i	$-0.304645\pi$
$881$	34.1884	1.15184	0.575918	+	0.817507i	$0.304645\pi$
$882$	0	0
$883$	−12.1126	−0.407621	−0.203810	−	0.979010i	$-0.565333\pi$
$883$	−12.1126	−0.407621	−0.203810	+	0.979010i	$0.565333\pi$
$884$	0	0
$885$	−10.4772	−0.352188
$886$	0	0
$887$	29.2401	0.981786	0.490893	−	0.871220i	$-0.336671\pi$
$887$	29.2401	0.981786	0.490893	+	0.871220i	$0.336671\pi$
$888$	0	0
$889$	−3.99721	−0.134062
$890$	0	0
$891$	−4.21982	−0.141369
$892$	0	0
$893$	16.0033	0.535529
$894$	0	0
$895$	7.21486	0.241166
$896$	0	0
$897$	0.849999	0.0283806
$898$	0	0
$899$	12.8982	0.430180
$900$	0	0
$901$	−3.95828	−0.131870
$902$	0	0
$903$	5.55546	0.184874
$904$	0	0
$905$	13.4672	0.447664
$906$	0	0
$907$	50.2089	1.66716	0.833580	−	0.552399i	$-0.186287\pi$
$907$	50.2089	1.66716	0.833580	+	0.552399i	$0.186287\pi$
$908$	0	0
$909$	30.2952	1.00483
$910$	0	0
$911$	−37.4099	−1.23944	−0.619722	−	0.784821i	$-0.712755\pi$
$911$	−37.4099	−1.23944	−0.619722	+	0.784821i	$0.712755\pi$
$912$	0	0
$913$	13.4835	0.446239
$914$	0	0
$915$	−5.88039	−0.194400
$916$	0	0
$917$	44.2212	1.46031
$918$	0	0
$919$	−50.3419	−1.66062	−0.830312	−	0.557298i	$-0.811838\pi$
$919$	−50.3419	−1.66062	−0.830312	+	0.557298i	$0.811838\pi$
$920$	0	0
$921$	−32.2575	−1.06292
$922$	0	0
$923$	13.2705	0.436803
$924$	0	0
$925$	22.2384	0.731195
$926$	0	0
$927$	−13.4772	−0.442650
$928$	0	0
$929$	−23.5473	−0.772561	−0.386280	−	0.922381i	$-0.626240\pi$
$929$	−23.5473	−0.772561	−0.386280	+	0.922381i	$0.626240\pi$
$930$	0	0
$931$	−3.22245	−0.105612
$932$	0	0
$933$	−1.88837	−0.0618224
$934$	0	0
$935$	1.25810	0.0411442
$936$	0	0
$937$	38.6166	1.26155	0.630774	−	0.775966i	$-0.282737\pi$
$937$	38.6166	1.26155	0.630774	+	0.775966i	$0.282737\pi$
$938$	0	0
$939$	6.63778	0.216616
$940$	0	0
$941$	52.0595	1.69709	0.848545	−	0.529123i	$-0.177479\pi$
$941$	52.0595	1.69709	0.848545	+	0.529123i	$0.177479\pi$
$942$	0	0
$943$	0.554764	0.0180656
$944$	0	0
$945$	−13.4115	−0.436277
$946$	0	0
$947$	−41.8401	−1.35962	−0.679810	−	0.733388i	$-0.737938\pi$
$947$	−41.8401	−1.35962	−0.679810	+	0.733388i	$0.737938\pi$
$948$	0	0
$949$	−9.00729	−0.292389
$950$	0	0
$951$	−21.0637	−0.683036
$952$	0	0
$953$	−33.2493	−1.07705	−0.538525	−	0.842609i	$-0.681018\pi$
$953$	−33.2493	−1.07705	−0.538525	+	0.842609i	$0.681018\pi$
$954$	0	0
$955$	−3.97703	−0.128694
$956$	0	0
$957$	−4.72565	−0.152759
$958$	0	0
$959$	−46.0491	−1.48700
$960$	0	0
$961$	27.8887	0.899635
$962$	0	0
$963$	−3.76574	−0.121349
$964$	0	0
$965$	12.2154	0.393227
$966$	0	0
$967$	50.3290	1.61847	0.809236	−	0.587484i	$-0.199881\pi$
$967$	50.3290	1.61847	0.809236	+	0.587484i	$0.199881\pi$
$968$	0	0
$969$	−1.54020	−0.0494784
$970$	0	0
$971$	−3.56225	−0.114318	−0.0571591	−	0.998365i	$-0.518204\pi$
$971$	−3.56225	−0.114318	−0.0571591	+	0.998365i	$0.518204\pi$
$972$	0	0
$973$	−22.6182	−0.725105
$974$	0	0
$975$	−12.4372	−0.398309
$976$	0	0
$977$	33.6163	1.07548	0.537740	−	0.843111i	$-0.319278\pi$
$977$	33.6163	1.07548	0.537740	+	0.843111i	$0.319278\pi$
$978$	0	0
$979$	−11.7090	−0.374222
$980$	0	0
$981$	−2.62139	−0.0836945
$982$	0	0
$983$	−33.4765	−1.06773	−0.533867	−	0.845568i	$-0.679262\pi$
$983$	−33.4765	−1.06773	−0.533867	+	0.845568i	$0.679262\pi$
$984$	0	0
$985$	10.5358	0.335700
$986$	0	0
$987$	27.6546	0.880257
$988$	0	0
$989$	−0.466455	−0.0148324
$990$	0	0
$991$	−30.6863	−0.974782	−0.487391	−	0.873184i	$-0.662051\pi$
$991$	−30.6863	−0.974782	−0.487391	+	0.873184i	$0.662051\pi$
$992$	0	0
$993$	6.85328	0.217482
$994$	0	0
$995$	16.3521	0.518395
$996$	0	0
$997$	22.1638	0.701936	0.350968	−	0.936388i	$-0.385853\pi$
$997$	22.1638	0.701936	0.350968	+	0.936388i	$0.385853\pi$
$998$	0	0
$999$	−28.2558	−0.893974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.c.1.1		4
4.3	odd	2		251.2.a.a.1.2	✓	4
12.11	even	2		2259.2.a.f.1.4		4
20.19	odd	2		6275.2.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.a.1.2	✓	4	4.3	odd	2
2259.2.a.f.1.4		4	12.11	even	2
4016.2.a.c.1.1		4	1.1	even	1	trivial
6275.2.a.c.1.3		4	20.19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19353 q^{3} -0.837853 q^{5} +2.93117 q^{7} -1.57549 q^{9} +O(q^{10})\)	\(q-1.19353 q^{3} -0.837853 q^{5} +2.93117 q^{7} -1.57549 q^{9} +2.35567 q^{11} -2.42451 q^{13} +1.00000 q^{15} -0.637428 q^{17} -2.02448 q^{19} -3.49843 q^{21} +0.293740 q^{23} -4.29800 q^{25} +5.46097 q^{27} +1.68079 q^{29} +7.67390 q^{31} -2.81156 q^{33} -2.45589 q^{35} -5.17413 q^{37} +2.89371 q^{39} +1.88863 q^{41} -1.58799 q^{43} +1.32003 q^{45} -7.90488 q^{47} +1.59174 q^{49} +0.760787 q^{51} +6.20978 q^{53} -1.97371 q^{55} +2.41627 q^{57} -10.4772 q^{59} -5.88039 q^{61} -4.61803 q^{63} +2.03138 q^{65} -10.2794 q^{67} -0.350586 q^{69} -5.47347 q^{71} +3.71510 q^{73} +5.12978 q^{75} +6.90488 q^{77} +12.5272 q^{79} -1.79134 q^{81} +5.72384 q^{83} +0.534071 q^{85} -2.00607 q^{87} -4.97056 q^{89} -7.10664 q^{91} -9.15900 q^{93} +1.69622 q^{95} +12.5705 q^{97} -3.71135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9	\( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9} + 3 q^{11} - 12 q^{13} + 4 q^{15} + q^{17} + 9 q^{19} - 7 q^{21} - 4 q^{23} - 7 q^{25} - q^{27} - 12 q^{29} + 2 q^{31} - 5 q^{35} - 13 q^{37} - 13 q^{39} + q^{41} + 5 q^{43} + 11 q^{45} - 12 q^{47} - 9 q^{49} - 2 q^{51} + 5 q^{53} + 3 q^{55} + 16 q^{57} - 6 q^{59} - 21 q^{61} - 14 q^{63} + q^{65} - 17 q^{67} - 13 q^{69} + 10 q^{71} - 2 q^{73} - 13 q^{75} + 8 q^{77} + 21 q^{79} - 8 q^{81} + q^{83} - 17 q^{85} - 31 q^{87} + 5 q^{89} + 2 q^{91} - 23 q^{93} - 12 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - 4 * q^9 + 3 * q^11 - 12 * q^13 + 4 * q^15 + q^17 + 9 * q^19 - 7 * q^21 - 4 * q^23 - 7 * q^25 - q^27 - 12 * q^29 + 2 * q^31 - 5 * q^35 - 13 * q^37 - 13 * q^39 + q^41 + 5 * q^43 + 11 * q^45 - 12 * q^47 - 9 * q^49 - 2 * q^51 + 5 * q^53 + 3 * q^55 + 16 * q^57 - 6 * q^59 - 21 * q^61 - 14 * q^63 + q^65 - 17 * q^67 - 13 * q^69 + 10 * q^71 - 2 * q^73 - 13 * q^75 + 8 * q^77 + 21 * q^79 - 8 * q^81 + q^83 - 17 * q^85 - 31 * q^87 + 5 * q^89 + 2 * q^91 - 23 * q^93 - 12 * q^95 + 6 * q^97 - 2 * q^99

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