LMFDB - Embedded newform 4008.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.71099$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.71099 q^{5} -0.775439 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.71099 q^{5} -0.775439 q^{7} +1.00000 q^{9} +3.70739 q^{11} -6.16847 q^{13} -2.71099 q^{15} +6.36353 q^{17} +0.922209 q^{19} -0.775439 q^{21} -4.47047 q^{23} +2.34945 q^{25} +1.00000 q^{27} -5.18433 q^{29} +1.35539 q^{31} +3.70739 q^{33} +2.10220 q^{35} +11.5076 q^{37} -6.16847 q^{39} -8.90644 q^{41} +5.52092 q^{43} -2.71099 q^{45} +8.07634 q^{47} -6.39869 q^{49} +6.36353 q^{51} -2.23790 q^{53} -10.0507 q^{55} +0.922209 q^{57} +12.3284 q^{59} -5.08112 q^{61} -0.775439 q^{63} +16.7226 q^{65} +0.628889 q^{67} -4.47047 q^{69} +3.07457 q^{71} +3.83167 q^{73} +2.34945 q^{75} -2.87486 q^{77} +15.8763 q^{79} +1.00000 q^{81} -4.76483 q^{83} -17.2514 q^{85} -5.18433 q^{87} +0.390848 q^{89} +4.78327 q^{91} +1.35539 q^{93} -2.50010 q^{95} +15.5125 q^{97} +3.70739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.71099	−1.21239	−0.606195	−	0.795316i	$-0.707305\pi$
$5$	−2.71099	−1.21239	−0.606195	+	0.795316i	$0.707305\pi$
$6$	0	0
$7$	−0.775439	−0.293088	−0.146544	−	0.989204i	$-0.546815\pi$
$7$	−0.775439	−0.293088	−0.146544	+	0.989204i	$0.546815\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.70739	1.11782	0.558911	−	0.829228i	$-0.311219\pi$
$11$	3.70739	1.11782	0.558911	+	0.829228i	$0.311219\pi$
$12$	0	0
$13$	−6.16847	−1.71083	−0.855413	−	0.517947i	$-0.826697\pi$
$13$	−6.16847	−1.71083	−0.855413	+	0.517947i	$0.826697\pi$
$14$	0	0
$15$	−2.71099	−0.699974
$16$	0	0
$17$	6.36353	1.54338	0.771691	−	0.635998i	$-0.219411\pi$
$17$	6.36353	1.54338	0.771691	+	0.635998i	$0.219411\pi$
$18$	0	0
$19$	0.922209	0.211569	0.105785	−	0.994389i	$-0.466265\pi$
$19$	0.922209	0.211569	0.105785	+	0.994389i	$0.466265\pi$
$20$	0	0
$21$	−0.775439	−0.169215
$22$	0	0
$23$	−4.47047	−0.932157	−0.466079	−	0.884743i	$-0.654334\pi$
$23$	−4.47047	−0.932157	−0.466079	+	0.884743i	$0.654334\pi$
$24$	0	0
$25$	2.34945	0.469889
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.18433	−0.962705	−0.481353	−	0.876527i	$-0.659854\pi$
$29$	−5.18433	−0.962705	−0.481353	+	0.876527i	$0.659854\pi$
$30$	0	0
$31$	1.35539	0.243436	0.121718	−	0.992565i	$-0.461160\pi$
$31$	1.35539	0.243436	0.121718	+	0.992565i	$0.461160\pi$
$32$	0	0
$33$	3.70739	0.645375
$34$	0	0
$35$	2.10220	0.355337
$36$	0	0
$37$	11.5076	1.89183	0.945915	−	0.324414i	$-0.105167\pi$
$37$	11.5076	1.89183	0.945915	+	0.324414i	$0.105167\pi$
$38$	0	0
$39$	−6.16847	−0.987746
$40$	0	0
$41$	−8.90644	−1.39095	−0.695476	−	0.718549i	$-0.744806\pi$
$41$	−8.90644	−1.39095	−0.695476	+	0.718549i	$0.744806\pi$
$42$	0	0
$43$	5.52092	0.841933	0.420966	−	0.907076i	$-0.361691\pi$
$43$	5.52092	0.841933	0.420966	+	0.907076i	$0.361691\pi$
$44$	0	0
$45$	−2.71099	−0.404130
$46$	0	0
$47$	8.07634	1.17806	0.589028	−	0.808113i	$-0.299511\pi$
$47$	8.07634	1.17806	0.589028	+	0.808113i	$0.299511\pi$
$48$	0	0
$49$	−6.39869	−0.914099
$50$	0	0
$51$	6.36353	0.891072
$52$	0	0
$53$	−2.23790	−0.307399	−0.153700	−	0.988118i	$-0.549119\pi$
$53$	−2.23790	−0.307399	−0.153700	+	0.988118i	$0.549119\pi$
$54$	0	0
$55$	−10.0507	−1.35524
$56$	0	0
$57$	0.922209	0.122150
$58$	0	0
$59$	12.3284	1.60502	0.802509	−	0.596640i	$-0.203498\pi$
$59$	12.3284	1.60502	0.802509	+	0.596640i	$0.203498\pi$
$60$	0	0
$61$	−5.08112	−0.650571	−0.325285	−	0.945616i	$-0.605460\pi$
$61$	−5.08112	−0.650571	−0.325285	+	0.945616i	$0.605460\pi$
$62$	0	0
$63$	−0.775439	−0.0976961
$64$	0	0
$65$	16.7226	2.07419
$66$	0	0
$67$	0.628889	0.0768311	0.0384155	−	0.999262i	$-0.487769\pi$
$67$	0.628889	0.0768311	0.0384155	+	0.999262i	$0.487769\pi$
$68$	0	0
$69$	−4.47047	−0.538181
$70$	0	0
$71$	3.07457	0.364884	0.182442	−	0.983217i	$-0.441600\pi$
$71$	3.07457	0.364884	0.182442	+	0.983217i	$0.441600\pi$
$72$	0	0
$73$	3.83167	0.448463	0.224232	−	0.974536i	$-0.428013\pi$
$73$	3.83167	0.448463	0.224232	+	0.974536i	$0.428013\pi$
$74$	0	0
$75$	2.34945	0.271291
$76$	0	0
$77$	−2.87486	−0.327620
$78$	0	0
$79$	15.8763	1.78622	0.893111	−	0.449837i	$-0.148518\pi$
$79$	15.8763	1.78622	0.893111	+	0.449837i	$0.148518\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.76483	−0.523009	−0.261504	−	0.965202i	$-0.584219\pi$
$83$	−4.76483	−0.523009	−0.261504	+	0.965202i	$0.584219\pi$
$84$	0	0
$85$	−17.2514	−1.87118
$86$	0	0
$87$	−5.18433	−0.555818
$88$	0	0
$89$	0.390848	0.0414298	0.0207149	−	0.999785i	$-0.493406\pi$
$89$	0.390848	0.0414298	0.0207149	+	0.999785i	$0.493406\pi$
$90$	0	0
$91$	4.78327	0.501423
$92$	0	0
$93$	1.35539	0.140548
$94$	0	0
$95$	−2.50010	−0.256504
$96$	0	0
$97$	15.5125	1.57506	0.787529	−	0.616277i	$-0.211360\pi$
$97$	15.5125	1.57506	0.787529	+	0.616277i	$0.211360\pi$
$98$	0	0
$99$	3.70739	0.372607
$100$	0	0
$101$	14.2133	1.41428	0.707140	−	0.707074i	$-0.249985\pi$
$101$	14.2133	1.41428	0.707140	+	0.707074i	$0.249985\pi$
$102$	0	0
$103$	−11.6148	−1.14444	−0.572219	−	0.820101i	$-0.693917\pi$
$103$	−11.6148	−1.14444	−0.572219	+	0.820101i	$0.693917\pi$
$104$	0	0
$105$	2.10220	0.205154
$106$	0	0
$107$	13.1267	1.26901	0.634503	−	0.772920i	$-0.281205\pi$
$107$	13.1267	1.26901	0.634503	+	0.772920i	$0.281205\pi$
$108$	0	0
$109$	3.37165	0.322945	0.161473	−	0.986877i	$-0.448376\pi$
$109$	3.37165	0.322945	0.161473	+	0.986877i	$0.448376\pi$
$110$	0	0
$111$	11.5076	1.09225
$112$	0	0
$113$	−12.9419	−1.21747	−0.608735	−	0.793374i	$-0.708323\pi$
$113$	−12.9419	−1.21747	−0.608735	+	0.793374i	$0.708323\pi$
$114$	0	0
$115$	12.1194	1.13014
$116$	0	0
$117$	−6.16847	−0.570275
$118$	0	0
$119$	−4.93452	−0.452347
$120$	0	0
$121$	2.74477	0.249525
$122$	0	0
$123$	−8.90644	−0.803066
$124$	0	0
$125$	7.18562	0.642701
$126$	0	0
$127$	−5.24079	−0.465045	−0.232522	−	0.972591i	$-0.574698\pi$
$127$	−5.24079	−0.465045	−0.232522	+	0.972591i	$0.574698\pi$
$128$	0	0
$129$	5.52092	0.486090
$130$	0	0
$131$	4.29290	0.375072	0.187536	−	0.982258i	$-0.439950\pi$
$131$	4.29290	0.375072	0.187536	+	0.982258i	$0.439950\pi$
$132$	0	0
$133$	−0.715117	−0.0620085
$134$	0	0
$135$	−2.71099	−0.233325
$136$	0	0
$137$	−2.17341	−0.185687	−0.0928433	−	0.995681i	$-0.529596\pi$
$137$	−2.17341	−0.185687	−0.0928433	+	0.995681i	$0.529596\pi$
$138$	0	0
$139$	18.0691	1.53260	0.766300	−	0.642483i	$-0.222096\pi$
$139$	18.0691	1.53260	0.766300	+	0.642483i	$0.222096\pi$
$140$	0	0
$141$	8.07634	0.680150
$142$	0	0
$143$	−22.8689	−1.91240
$144$	0	0
$145$	14.0546	1.16717
$146$	0	0
$147$	−6.39869	−0.527755
$148$	0	0
$149$	−0.330735	−0.0270949	−0.0135474	−	0.999908i	$-0.504312\pi$
$149$	−0.330735	−0.0270949	−0.0135474	+	0.999908i	$0.504312\pi$
$150$	0	0
$151$	−3.69073	−0.300347	−0.150174	−	0.988660i	$-0.547983\pi$
$151$	−3.69073	−0.300347	−0.150174	+	0.988660i	$0.547983\pi$
$152$	0	0
$153$	6.36353	0.514461
$154$	0	0
$155$	−3.67445	−0.295139
$156$	0	0
$157$	15.7517	1.25712	0.628560	−	0.777761i	$-0.283645\pi$
$157$	15.7517	1.25712	0.628560	+	0.777761i	$0.283645\pi$
$158$	0	0
$159$	−2.23790	−0.177477
$160$	0	0
$161$	3.46657	0.273204
$162$	0	0
$163$	15.5320	1.21656	0.608280	−	0.793722i	$-0.291860\pi$
$163$	15.5320	1.21656	0.608280	+	0.793722i	$0.291860\pi$
$164$	0	0
$165$	−10.0507	−0.782446
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	25.0500	1.92692
$170$	0	0
$171$	0.922209	0.0705231
$172$	0	0
$173$	16.8363	1.28004	0.640020	−	0.768358i	$-0.278926\pi$
$173$	16.8363	1.28004	0.640020	+	0.768358i	$0.278926\pi$
$174$	0	0
$175$	−1.82185	−0.137719
$176$	0	0
$177$	12.3284	0.926658
$178$	0	0
$179$	−14.7443	−1.10204	−0.551020	−	0.834492i	$-0.685761\pi$
$179$	−14.7443	−1.10204	−0.551020	+	0.834492i	$0.685761\pi$
$180$	0	0
$181$	11.7286	0.871783	0.435891	−	0.899999i	$-0.356433\pi$
$181$	11.7286	0.871783	0.435891	+	0.899999i	$0.356433\pi$
$182$	0	0
$183$	−5.08112	−0.375607
$184$	0	0
$185$	−31.1968	−2.29364
$186$	0	0
$187$	23.5921	1.72523
$188$	0	0
$189$	−0.775439	−0.0564049
$190$	0	0
$191$	1.61629	0.116951	0.0584753	−	0.998289i	$-0.481376\pi$
$191$	1.61629	0.116951	0.0584753	+	0.998289i	$0.481376\pi$
$192$	0	0
$193$	18.7212	1.34758	0.673792	−	0.738921i	$-0.264664\pi$
$193$	18.7212	1.34758	0.673792	+	0.738921i	$0.264664\pi$
$194$	0	0
$195$	16.7226	1.19753
$196$	0	0
$197$	−12.1950	−0.868856	−0.434428	−	0.900707i	$-0.643049\pi$
$197$	−12.1950	−0.868856	−0.434428	+	0.900707i	$0.643049\pi$
$198$	0	0
$199$	−1.48491	−0.105262	−0.0526312	−	0.998614i	$-0.516761\pi$
$199$	−1.48491	−0.105262	−0.0526312	+	0.998614i	$0.516761\pi$
$200$	0	0
$201$	0.628889	0.0443584
$202$	0	0
$203$	4.02013	0.282158
$204$	0	0
$205$	24.1452	1.68638
$206$	0	0
$207$	−4.47047	−0.310719
$208$	0	0
$209$	3.41899	0.236497
$210$	0	0
$211$	14.9344	1.02813	0.514063	−	0.857752i	$-0.328140\pi$
$211$	14.9344	1.02813	0.514063	+	0.857752i	$0.328140\pi$
$212$	0	0
$213$	3.07457	0.210666
$214$	0	0
$215$	−14.9671	−1.02075
$216$	0	0
$217$	−1.05102	−0.0713482
$218$	0	0
$219$	3.83167	0.258920
$220$	0	0
$221$	−39.2532	−2.64046
$222$	0	0
$223$	−10.3483	−0.692975	−0.346487	−	0.938055i	$-0.612626\pi$
$223$	−10.3483	−0.692975	−0.346487	+	0.938055i	$0.612626\pi$
$224$	0	0
$225$	2.34945	0.156630
$226$	0	0
$227$	−7.31500	−0.485513	−0.242757	−	0.970087i	$-0.578052\pi$
$227$	−7.31500	−0.485513	−0.242757	+	0.970087i	$0.578052\pi$
$228$	0	0
$229$	17.3272	1.14502	0.572508	−	0.819899i	$-0.305970\pi$
$229$	17.3272	1.14502	0.572508	+	0.819899i	$0.305970\pi$
$230$	0	0
$231$	−2.87486	−0.189152
$232$	0	0
$233$	−16.4242	−1.07599	−0.537994	−	0.842949i	$-0.680818\pi$
$233$	−16.4242	−1.07599	−0.537994	+	0.842949i	$0.680818\pi$
$234$	0	0
$235$	−21.8948	−1.42826
$236$	0	0
$237$	15.8763	1.03128
$238$	0	0
$239$	−4.16870	−0.269651	−0.134825	−	0.990869i	$-0.543047\pi$
$239$	−4.16870	−0.269651	−0.134825	+	0.990869i	$0.543047\pi$
$240$	0	0
$241$	10.1175	0.651725	0.325862	−	0.945417i	$-0.394345\pi$
$241$	10.1175	0.651725	0.325862	+	0.945417i	$0.394345\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	17.3468	1.10824
$246$	0	0
$247$	−5.68862	−0.361958
$248$	0	0
$249$	−4.76483	−0.301959
$250$	0	0
$251$	11.9898	0.756790	0.378395	−	0.925644i	$-0.376476\pi$
$251$	11.9898	0.756790	0.378395	+	0.925644i	$0.376476\pi$
$252$	0	0
$253$	−16.5738	−1.04199
$254$	0	0
$255$	−17.2514	−1.08033
$256$	0	0
$257$	12.0253	0.750117	0.375059	−	0.927001i	$-0.377623\pi$
$257$	12.0253	0.750117	0.375059	+	0.927001i	$0.377623\pi$
$258$	0	0
$259$	−8.92340	−0.554473
$260$	0	0
$261$	−5.18433	−0.320902
$262$	0	0
$263$	−0.941771	−0.0580721	−0.0290360	−	0.999578i	$-0.509244\pi$
$263$	−0.941771	−0.0580721	−0.0290360	+	0.999578i	$0.509244\pi$
$264$	0	0
$265$	6.06691	0.372688
$266$	0	0
$267$	0.390848	0.0239195
$268$	0	0
$269$	3.56306	0.217243	0.108622	−	0.994083i	$-0.465356\pi$
$269$	3.56306	0.217243	0.108622	+	0.994083i	$0.465356\pi$
$270$	0	0
$271$	21.3869	1.29916	0.649582	−	0.760292i	$-0.274944\pi$
$271$	21.3869	1.29916	0.649582	+	0.760292i	$0.274944\pi$
$272$	0	0
$273$	4.78327	0.289497
$274$	0	0
$275$	8.71032	0.525252
$276$	0	0
$277$	28.5165	1.71339	0.856695	−	0.515823i	$-0.172514\pi$
$277$	28.5165	1.71339	0.856695	+	0.515823i	$0.172514\pi$
$278$	0	0
$279$	1.35539	0.0811453
$280$	0	0
$281$	−26.7452	−1.59549	−0.797744	−	0.602997i	$-0.793973\pi$
$281$	−26.7452	−1.59549	−0.797744	+	0.602997i	$0.793973\pi$
$282$	0	0
$283$	−2.76889	−0.164594	−0.0822968	−	0.996608i	$-0.526226\pi$
$283$	−2.76889	−0.164594	−0.0822968	+	0.996608i	$0.526226\pi$
$284$	0	0
$285$	−2.50010	−0.148093
$286$	0	0
$287$	6.90640	0.407672
$288$	0	0
$289$	23.4945	1.38203
$290$	0	0
$291$	15.5125	0.909360
$292$	0	0
$293$	−17.3417	−1.01311	−0.506555	−	0.862208i	$-0.669081\pi$
$293$	−17.3417	−1.01311	−0.506555	+	0.862208i	$0.669081\pi$
$294$	0	0
$295$	−33.4221	−1.94591
$296$	0	0
$297$	3.70739	0.215125
$298$	0	0
$299$	27.5759	1.59476
$300$	0	0
$301$	−4.28114	−0.246761
$302$	0	0
$303$	14.2133	0.816535
$304$	0	0
$305$	13.7748	0.788745
$306$	0	0
$307$	−17.3038	−0.987578	−0.493789	−	0.869582i	$-0.664388\pi$
$307$	−17.3038	−0.987578	−0.493789	+	0.869582i	$0.664388\pi$
$308$	0	0
$309$	−11.6148	−0.660742
$310$	0	0
$311$	23.3959	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$311$	23.3959	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$312$	0	0
$313$	0.191470	0.0108225	0.00541127	−	0.999985i	$-0.498278\pi$
$313$	0.191470	0.0108225	0.00541127	+	0.999985i	$0.498278\pi$
$314$	0	0
$315$	2.10220	0.118446
$316$	0	0
$317$	−3.25612	−0.182882	−0.0914410	−	0.995810i	$-0.529147\pi$
$317$	−3.25612	−0.182882	−0.0914410	+	0.995810i	$0.529147\pi$
$318$	0	0
$319$	−19.2203	−1.07613
$320$	0	0
$321$	13.1267	0.732661
$322$	0	0
$323$	5.86850	0.326532
$324$	0	0
$325$	−14.4925	−0.803898
$326$	0	0
$327$	3.37165	0.186452
$328$	0	0
$329$	−6.26271	−0.345274
$330$	0	0
$331$	−30.7585	−1.69064	−0.845320	−	0.534260i	$-0.820590\pi$
$331$	−30.7585	−1.69064	−0.845320	+	0.534260i	$0.820590\pi$
$332$	0	0
$333$	11.5076	0.630610
$334$	0	0
$335$	−1.70491	−0.0931492
$336$	0	0
$337$	2.60466	0.141885	0.0709425	−	0.997480i	$-0.477399\pi$
$337$	2.60466	0.141885	0.0709425	+	0.997480i	$0.477399\pi$
$338$	0	0
$339$	−12.9419	−0.702906
$340$	0	0
$341$	5.02498	0.272118
$342$	0	0
$343$	10.3899	0.561000
$344$	0	0
$345$	12.1194	0.652485
$346$	0	0
$347$	−20.1851	−1.08359	−0.541796	−	0.840510i	$-0.682255\pi$
$347$	−20.1851	−1.08359	−0.541796	+	0.840510i	$0.682255\pi$
$348$	0	0
$349$	13.7950	0.738431	0.369216	−	0.929344i	$-0.379626\pi$
$349$	13.7950	0.738431	0.369216	+	0.929344i	$0.379626\pi$
$350$	0	0
$351$	−6.16847	−0.329249
$352$	0	0
$353$	−6.65454	−0.354185	−0.177093	−	0.984194i	$-0.556669\pi$
$353$	−6.65454	−0.354185	−0.177093	+	0.984194i	$0.556669\pi$
$354$	0	0
$355$	−8.33510	−0.442381
$356$	0	0
$357$	−4.93452	−0.261163
$358$	0	0
$359$	18.5216	0.977535	0.488768	−	0.872414i	$-0.337447\pi$
$359$	18.5216	0.977535	0.488768	+	0.872414i	$0.337447\pi$
$360$	0	0
$361$	−18.1495	−0.955238
$362$	0	0
$363$	2.74477	0.144063
$364$	0	0
$365$	−10.3876	−0.543712
$366$	0	0
$367$	−17.6276	−0.920152	−0.460076	−	0.887880i	$-0.652178\pi$
$367$	−17.6276	−0.920152	−0.460076	+	0.887880i	$0.652178\pi$
$368$	0	0
$369$	−8.90644	−0.463651
$370$	0	0
$371$	1.73535	0.0900951
$372$	0	0
$373$	4.37087	0.226315	0.113158	−	0.993577i	$-0.463904\pi$
$373$	4.37087	0.226315	0.113158	+	0.993577i	$0.463904\pi$
$374$	0	0
$375$	7.18562	0.371064
$376$	0	0
$377$	31.9794	1.64702
$378$	0	0
$379$	−31.2802	−1.60676	−0.803379	−	0.595468i	$-0.796967\pi$
$379$	−31.2802	−1.60676	−0.803379	+	0.595468i	$0.796967\pi$
$380$	0	0
$381$	−5.24079	−0.268494
$382$	0	0
$383$	−6.10351	−0.311875	−0.155937	−	0.987767i	$-0.549840\pi$
$383$	−6.10351	−0.311875	−0.155937	+	0.987767i	$0.549840\pi$
$384$	0	0
$385$	7.79370	0.397204
$386$	0	0
$387$	5.52092	0.280644
$388$	0	0
$389$	−7.97649	−0.404424	−0.202212	−	0.979342i	$-0.564813\pi$
$389$	−7.97649	−0.404424	−0.202212	+	0.979342i	$0.564813\pi$
$390$	0	0
$391$	−28.4479	−1.43867
$392$	0	0
$393$	4.29290	0.216548
$394$	0	0
$395$	−43.0404	−2.16560
$396$	0	0
$397$	24.2101	1.21507	0.607535	−	0.794293i	$-0.292158\pi$
$397$	24.2101	1.21507	0.607535	+	0.794293i	$0.292158\pi$
$398$	0	0
$399$	−0.715117	−0.0358006
$400$	0	0
$401$	−17.9420	−0.895982	−0.447991	−	0.894038i	$-0.647860\pi$
$401$	−17.9420	−0.895982	−0.447991	+	0.894038i	$0.647860\pi$
$402$	0	0
$403$	−8.36071	−0.416476
$404$	0	0
$405$	−2.71099	−0.134710
$406$	0	0
$407$	42.6630	2.11473
$408$	0	0
$409$	−10.9929	−0.543562	−0.271781	−	0.962359i	$-0.587613\pi$
$409$	−10.9929	−0.543562	−0.271781	+	0.962359i	$0.587613\pi$
$410$	0	0
$411$	−2.17341	−0.107206
$412$	0	0
$413$	−9.55990	−0.470412
$414$	0	0
$415$	12.9174	0.634090
$416$	0	0
$417$	18.0691	0.884847
$418$	0	0
$419$	−20.5571	−1.00428	−0.502139	−	0.864787i	$-0.667453\pi$
$419$	−20.5571	−1.00428	−0.502139	+	0.864787i	$0.667453\pi$
$420$	0	0
$421$	15.2776	0.744583	0.372291	−	0.928116i	$-0.378572\pi$
$421$	15.2776	0.744583	0.372291	+	0.928116i	$0.378572\pi$
$422$	0	0
$423$	8.07634	0.392685
$424$	0	0
$425$	14.9508	0.725218
$426$	0	0
$427$	3.94010	0.190675
$428$	0	0
$429$	−22.8689	−1.10412
$430$	0	0
$431$	−24.6247	−1.18613	−0.593066	−	0.805154i	$-0.702083\pi$
$431$	−24.6247	−1.18613	−0.593066	+	0.805154i	$0.702083\pi$
$432$	0	0
$433$	−30.2365	−1.45307	−0.726537	−	0.687127i	$-0.758872\pi$
$433$	−30.2365	−1.45307	−0.726537	+	0.687127i	$0.758872\pi$
$434$	0	0
$435$	14.0546	0.673868
$436$	0	0
$437$	−4.12271	−0.197216
$438$	0	0
$439$	5.07328	0.242134	0.121067	−	0.992644i	$-0.461368\pi$
$439$	5.07328	0.242134	0.121067	+	0.992644i	$0.461368\pi$
$440$	0	0
$441$	−6.39869	−0.304700
$442$	0	0
$443$	−11.3097	−0.537339	−0.268670	−	0.963232i	$-0.586584\pi$
$443$	−11.3097	−0.537339	−0.268670	+	0.963232i	$0.586584\pi$
$444$	0	0
$445$	−1.05958	−0.0502291
$446$	0	0
$447$	−0.330735	−0.0156432
$448$	0	0
$449$	−23.0503	−1.08781	−0.543906	−	0.839146i	$-0.683055\pi$
$449$	−23.0503	−1.08781	−0.543906	+	0.839146i	$0.683055\pi$
$450$	0	0
$451$	−33.0197	−1.55484
$452$	0	0
$453$	−3.69073	−0.173406
$454$	0	0
$455$	−12.9674	−0.607920
$456$	0	0
$457$	38.1775	1.78587	0.892935	−	0.450185i	$-0.148642\pi$
$457$	38.1775	1.78587	0.892935	+	0.450185i	$0.148642\pi$
$458$	0	0
$459$	6.36353	0.297024
$460$	0	0
$461$	−3.19692	−0.148895	−0.0744477	−	0.997225i	$-0.523719\pi$
$461$	−3.19692	−0.148895	−0.0744477	+	0.997225i	$0.523719\pi$
$462$	0	0
$463$	29.8591	1.38767	0.693834	−	0.720135i	$-0.255920\pi$
$463$	29.8591	1.38767	0.693834	+	0.720135i	$0.255920\pi$
$464$	0	0
$465$	−3.67445	−0.170399
$466$	0	0
$467$	−34.0326	−1.57484	−0.787421	−	0.616415i	$-0.788584\pi$
$467$	−34.0326	−1.57484	−0.787421	+	0.616415i	$0.788584\pi$
$468$	0	0
$469$	−0.487665	−0.0225183
$470$	0	0
$471$	15.7517	0.725798
$472$	0	0
$473$	20.4682	0.941131
$474$	0	0
$475$	2.16668	0.0994141
$476$	0	0
$477$	−2.23790	−0.102466
$478$	0	0
$479$	−30.2683	−1.38300	−0.691498	−	0.722378i	$-0.743049\pi$
$479$	−30.2683	−1.38300	−0.691498	+	0.722378i	$0.743049\pi$
$480$	0	0
$481$	−70.9840	−3.23659
$482$	0	0
$483$	3.46657	0.157735
$484$	0	0
$485$	−42.0542	−1.90958
$486$	0	0
$487$	−18.1702	−0.823369	−0.411684	−	0.911327i	$-0.635059\pi$
$487$	−18.1702	−0.823369	−0.411684	+	0.911327i	$0.635059\pi$
$488$	0	0
$489$	15.5320	0.702381
$490$	0	0
$491$	−32.7025	−1.47584	−0.737922	−	0.674886i	$-0.764193\pi$
$491$	−32.7025	−1.47584	−0.737922	+	0.674886i	$0.764193\pi$
$492$	0	0
$493$	−32.9906	−1.48582
$494$	0	0
$495$	−10.0507	−0.451745
$496$	0	0
$497$	−2.38414	−0.106943
$498$	0	0
$499$	−3.70087	−0.165674	−0.0828368	−	0.996563i	$-0.526398\pi$
$499$	−3.70087	−0.165674	−0.0828368	+	0.996563i	$0.526398\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	17.4615	0.778569	0.389285	−	0.921118i	$-0.372722\pi$
$503$	17.4615	0.778569	0.389285	+	0.921118i	$0.372722\pi$
$504$	0	0
$505$	−38.5322	−1.71466
$506$	0	0
$507$	25.0500	1.11251
$508$	0	0
$509$	12.8296	0.568662	0.284331	−	0.958726i	$-0.408229\pi$
$509$	12.8296	0.568662	0.284331	+	0.958726i	$0.408229\pi$
$510$	0	0
$511$	−2.97123	−0.131439
$512$	0	0
$513$	0.922209	0.0407165
$514$	0	0
$515$	31.4875	1.38751
$516$	0	0
$517$	29.9422	1.31686
$518$	0	0
$519$	16.8363	0.739032
$520$	0	0
$521$	14.7753	0.647316	0.323658	−	0.946174i	$-0.395087\pi$
$521$	14.7753	0.647316	0.323658	+	0.946174i	$0.395087\pi$
$522$	0	0
$523$	18.6247	0.814402	0.407201	−	0.913339i	$-0.366505\pi$
$523$	18.6247	0.814402	0.407201	+	0.913339i	$0.366505\pi$
$524$	0	0
$525$	−1.82185	−0.0795121
$526$	0	0
$527$	8.62508	0.375715
$528$	0	0
$529$	−3.01492	−0.131083
$530$	0	0
$531$	12.3284	0.535006
$532$	0	0
$533$	54.9391	2.37968
$534$	0	0
$535$	−35.5863	−1.53853
$536$	0	0
$537$	−14.7443	−0.636263
$538$	0	0
$539$	−23.7225	−1.02180
$540$	0	0
$541$	12.5963	0.541558	0.270779	−	0.962641i	$-0.412719\pi$
$541$	12.5963	0.541558	0.270779	+	0.962641i	$0.412719\pi$
$542$	0	0
$543$	11.7286	0.503324
$544$	0	0
$545$	−9.14048	−0.391535
$546$	0	0
$547$	11.0198	0.471172	0.235586	−	0.971853i	$-0.424299\pi$
$547$	11.0198	0.471172	0.235586	+	0.971853i	$0.424299\pi$
$548$	0	0
$549$	−5.08112	−0.216857
$550$	0	0
$551$	−4.78103	−0.203679
$552$	0	0
$553$	−12.3111	−0.523521
$554$	0	0
$555$	−31.1968	−1.32423
$556$	0	0
$557$	25.8382	1.09480	0.547401	−	0.836871i	$-0.315617\pi$
$557$	25.8382	1.09480	0.547401	+	0.836871i	$0.315617\pi$
$558$	0	0
$559$	−34.0556	−1.44040
$560$	0	0
$561$	23.5921	0.996059
$562$	0	0
$563$	34.8221	1.46757	0.733787	−	0.679379i	$-0.237751\pi$
$563$	34.8221	1.46757	0.733787	+	0.679379i	$0.237751\pi$
$564$	0	0
$565$	35.0853	1.47605
$566$	0	0
$567$	−0.775439	−0.0325654
$568$	0	0
$569$	−2.62357	−0.109986	−0.0549929	−	0.998487i	$-0.517514\pi$
$569$	−2.62357	−0.109986	−0.0549929	+	0.998487i	$0.517514\pi$
$570$	0	0
$571$	9.52229	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$571$	9.52229	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$572$	0	0
$573$	1.61629	0.0675215
$574$	0	0
$575$	−10.5031	−0.438010
$576$	0	0
$577$	31.2255	1.29994	0.649968	−	0.759962i	$-0.274782\pi$
$577$	31.2255	1.29994	0.649968	+	0.759962i	$0.274782\pi$
$578$	0	0
$579$	18.7212	0.778028
$580$	0	0
$581$	3.69484	0.153288
$582$	0	0
$583$	−8.29678	−0.343617
$584$	0	0
$585$	16.7226	0.691396
$586$	0	0
$587$	6.02985	0.248878	0.124439	−	0.992227i	$-0.460287\pi$
$587$	6.02985	0.248878	0.124439	+	0.992227i	$0.460287\pi$
$588$	0	0
$589$	1.24996	0.0515036
$590$	0	0
$591$	−12.1950	−0.501634
$592$	0	0
$593$	27.7573	1.13986	0.569929	−	0.821694i	$-0.306971\pi$
$593$	27.7573	1.13986	0.569929	+	0.821694i	$0.306971\pi$
$594$	0	0
$595$	13.3774	0.548421
$596$	0	0
$597$	−1.48491	−0.0607733
$598$	0	0
$599$	−35.4573	−1.44874	−0.724372	−	0.689409i	$-0.757870\pi$
$599$	−35.4573	−1.44874	−0.724372	+	0.689409i	$0.757870\pi$
$600$	0	0
$601$	31.3416	1.27845	0.639225	−	0.769020i	$-0.279255\pi$
$601$	31.3416	1.27845	0.639225	+	0.769020i	$0.279255\pi$
$602$	0	0
$603$	0.628889	0.0256104
$604$	0	0
$605$	−7.44104	−0.302521
$606$	0	0
$607$	−31.9080	−1.29510	−0.647552	−	0.762021i	$-0.724207\pi$
$607$	−31.9080	−1.29510	−0.647552	+	0.762021i	$0.724207\pi$
$608$	0	0
$609$	4.02013	0.162904
$610$	0	0
$611$	−49.8187	−2.01545
$612$	0	0
$613$	3.53486	0.142772	0.0713859	−	0.997449i	$-0.477258\pi$
$613$	3.53486	0.142772	0.0713859	+	0.997449i	$0.477258\pi$
$614$	0	0
$615$	24.1452	0.973630
$616$	0	0
$617$	−6.53527	−0.263100	−0.131550	−	0.991310i	$-0.541995\pi$
$617$	−6.53527	−0.263100	−0.131550	+	0.991310i	$0.541995\pi$
$618$	0	0
$619$	−27.3866	−1.10076	−0.550381	−	0.834914i	$-0.685517\pi$
$619$	−27.3866	−1.10076	−0.550381	+	0.834914i	$0.685517\pi$
$620$	0	0
$621$	−4.47047	−0.179394
$622$	0	0
$623$	−0.303079	−0.0121426
$624$	0	0
$625$	−31.2273	−1.24909
$626$	0	0
$627$	3.41899	0.136541
$628$	0	0
$629$	73.2286	2.91982
$630$	0	0
$631$	−12.8036	−0.509704	−0.254852	−	0.966980i	$-0.582027\pi$
$631$	−12.8036	−0.509704	−0.254852	+	0.966980i	$0.582027\pi$
$632$	0	0
$633$	14.9344	0.593589
$634$	0	0
$635$	14.2077	0.563815
$636$	0	0
$637$	39.4702	1.56386
$638$	0	0
$639$	3.07457	0.121628
$640$	0	0
$641$	−2.32208	−0.0917165	−0.0458583	−	0.998948i	$-0.514602\pi$
$641$	−2.32208	−0.0917165	−0.0458583	+	0.998948i	$0.514602\pi$
$642$	0	0
$643$	1.60321	0.0632245	0.0316122	−	0.999500i	$-0.489936\pi$
$643$	1.60321	0.0632245	0.0316122	+	0.999500i	$0.489936\pi$
$644$	0	0
$645$	−14.9671	−0.589331
$646$	0	0
$647$	43.0337	1.69183	0.845915	−	0.533317i	$-0.179055\pi$
$647$	43.0337	1.69183	0.845915	+	0.533317i	$0.179055\pi$
$648$	0	0
$649$	45.7062	1.79412
$650$	0	0
$651$	−1.05102	−0.0411929
$652$	0	0
$653$	18.3118	0.716596	0.358298	−	0.933607i	$-0.383357\pi$
$653$	18.3118	0.716596	0.358298	+	0.933607i	$0.383357\pi$
$654$	0	0
$655$	−11.6380	−0.454734
$656$	0	0
$657$	3.83167	0.149488
$658$	0	0
$659$	4.81736	0.187658	0.0938289	−	0.995588i	$-0.470089\pi$
$659$	4.81736	0.187658	0.0938289	+	0.995588i	$0.470089\pi$
$660$	0	0
$661$	42.4346	1.65051	0.825257	−	0.564757i	$-0.191030\pi$
$661$	42.4346	1.65051	0.825257	+	0.564757i	$0.191030\pi$
$662$	0	0
$663$	−39.2532	−1.52447
$664$	0	0
$665$	1.93867	0.0751785
$666$	0	0
$667$	23.1764	0.897392
$668$	0	0
$669$	−10.3483	−0.400089
$670$	0	0
$671$	−18.8377	−0.727222
$672$	0	0
$673$	23.4211	0.902815	0.451408	−	0.892318i	$-0.350922\pi$
$673$	23.4211	0.902815	0.451408	+	0.892318i	$0.350922\pi$
$674$	0	0
$675$	2.34945	0.0904302
$676$	0	0
$677$	31.8529	1.22421	0.612104	−	0.790777i	$-0.290323\pi$
$677$	31.8529	1.22421	0.612104	+	0.790777i	$0.290323\pi$
$678$	0	0
$679$	−12.0290	−0.461631
$680$	0	0
$681$	−7.31500	−0.280311
$682$	0	0
$683$	35.9939	1.37727	0.688635	−	0.725108i	$-0.258210\pi$
$683$	35.9939	1.37727	0.688635	+	0.725108i	$0.258210\pi$
$684$	0	0
$685$	5.89207	0.225124
$686$	0	0
$687$	17.3272	0.661076
$688$	0	0
$689$	13.8044	0.525906
$690$	0	0
$691$	−27.9417	−1.06295	−0.531476	−	0.847073i	$-0.678362\pi$
$691$	−27.9417	−1.06295	−0.531476	+	0.847073i	$0.678362\pi$
$692$	0	0
$693$	−2.87486	−0.109207
$694$	0	0
$695$	−48.9850	−1.85811
$696$	0	0
$697$	−56.6763	−2.14677
$698$	0	0
$699$	−16.4242	−0.621222
$700$	0	0
$701$	−11.3135	−0.427306	−0.213653	−	0.976910i	$-0.568536\pi$
$701$	−11.3135	−0.427306	−0.213653	+	0.976910i	$0.568536\pi$
$702$	0	0
$703$	10.6124	0.400253
$704$	0	0
$705$	−21.8948	−0.824607
$706$	0	0
$707$	−11.0216	−0.414509
$708$	0	0
$709$	−33.8151	−1.26995	−0.634976	−	0.772532i	$-0.718990\pi$
$709$	−33.8151	−1.26995	−0.634976	+	0.772532i	$0.718990\pi$
$710$	0	0
$711$	15.8763	0.595407
$712$	0	0
$713$	−6.05924	−0.226920
$714$	0	0
$715$	61.9974	2.31857
$716$	0	0
$717$	−4.16870	−0.155683
$718$	0	0
$719$	−38.3845	−1.43150	−0.715749	−	0.698357i	$-0.753915\pi$
$719$	−38.3845	−1.43150	−0.715749	+	0.698357i	$0.753915\pi$
$720$	0	0
$721$	9.00655	0.335421
$722$	0	0
$723$	10.1175	0.376273
$724$	0	0
$725$	−12.1803	−0.452365
$726$	0	0
$727$	−12.6807	−0.470301	−0.235150	−	0.971959i	$-0.575558\pi$
$727$	−12.6807	−0.470301	−0.235150	+	0.971959i	$0.575558\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.1325	1.29942
$732$	0	0
$733$	4.51823	0.166885	0.0834423	−	0.996513i	$-0.473409\pi$
$733$	4.51823	0.166885	0.0834423	+	0.996513i	$0.473409\pi$
$734$	0	0
$735$	17.3468	0.639845
$736$	0	0
$737$	2.33154	0.0858834
$738$	0	0
$739$	49.4318	1.81838	0.909189	−	0.416383i	$-0.136703\pi$
$739$	49.4318	1.81838	0.909189	+	0.416383i	$0.136703\pi$
$740$	0	0
$741$	−5.68862	−0.208977
$742$	0	0
$743$	−11.0292	−0.404623	−0.202312	−	0.979321i	$-0.564845\pi$
$743$	−11.0292	−0.404623	−0.202312	+	0.979321i	$0.564845\pi$
$744$	0	0
$745$	0.896619	0.0328496
$746$	0	0
$747$	−4.76483	−0.174336
$748$	0	0
$749$	−10.1790	−0.371931
$750$	0	0
$751$	50.2471	1.83354	0.916771	−	0.399412i	$-0.130786\pi$
$751$	50.2471	1.83354	0.916771	+	0.399412i	$0.130786\pi$
$752$	0	0
$753$	11.9898	0.436933
$754$	0	0
$755$	10.0055	0.364138
$756$	0	0
$757$	−40.6067	−1.47588	−0.737938	−	0.674869i	$-0.764200\pi$
$757$	−40.6067	−1.47588	−0.737938	+	0.674869i	$0.764200\pi$
$758$	0	0
$759$	−16.5738	−0.601590
$760$	0	0
$761$	−29.3817	−1.06509	−0.532543	−	0.846403i	$-0.678764\pi$
$761$	−29.3817	−1.06509	−0.532543	+	0.846403i	$0.678764\pi$
$762$	0	0
$763$	−2.61450	−0.0946514
$764$	0	0
$765$	−17.2514	−0.623727
$766$	0	0
$767$	−76.0473	−2.74591
$768$	0	0
$769$	29.1206	1.05012	0.525058	−	0.851067i	$-0.324044\pi$
$769$	29.1206	1.05012	0.525058	+	0.851067i	$0.324044\pi$
$770$	0	0
$771$	12.0253	0.433081
$772$	0	0
$773$	−0.665280	−0.0239285	−0.0119642	−	0.999928i	$-0.503808\pi$
$773$	−0.665280	−0.0239285	−0.0119642	+	0.999928i	$0.503808\pi$
$774$	0	0
$775$	3.18442	0.114388
$776$	0	0
$777$	−8.92340	−0.320125
$778$	0	0
$779$	−8.21360	−0.294283
$780$	0	0
$781$	11.3986	0.407875
$782$	0	0
$783$	−5.18433	−0.185273
$784$	0	0
$785$	−42.7025	−1.52412
$786$	0	0
$787$	−32.0421	−1.14218	−0.571089	−	0.820888i	$-0.693479\pi$
$787$	−32.0421	−1.14218	−0.571089	+	0.820888i	$0.693479\pi$
$788$	0	0
$789$	−0.941771	−0.0335279
$790$	0	0
$791$	10.0356	0.356826
$792$	0	0
$793$	31.3427	1.11301
$794$	0	0
$795$	6.06691	0.215171
$796$	0	0
$797$	−41.1598	−1.45795	−0.728977	−	0.684538i	$-0.760004\pi$
$797$	−41.1598	−1.45795	−0.728977	+	0.684538i	$0.760004\pi$
$798$	0	0
$799$	51.3940	1.81819
$800$	0	0
$801$	0.390848	0.0138099
$802$	0	0
$803$	14.2055	0.501302
$804$	0	0
$805$	−9.39783	−0.331230
$806$	0	0
$807$	3.56306	0.125426
$808$	0	0
$809$	31.4775	1.10669	0.553345	−	0.832952i	$-0.313351\pi$
$809$	31.4775	1.10669	0.553345	+	0.832952i	$0.313351\pi$
$810$	0	0
$811$	−15.1659	−0.532546	−0.266273	−	0.963898i	$-0.585792\pi$
$811$	−15.1659	−0.532546	−0.266273	+	0.963898i	$0.585792\pi$
$812$	0	0
$813$	21.3869	0.750072
$814$	0	0
$815$	−42.1070	−1.47495
$816$	0	0
$817$	5.09145	0.178127
$818$	0	0
$819$	4.78327	0.167141
$820$	0	0
$821$	−50.3086	−1.75578	−0.877890	−	0.478862i	$-0.841050\pi$
$821$	−50.3086	−1.75578	−0.877890	+	0.478862i	$0.841050\pi$
$822$	0	0
$823$	−31.2887	−1.09066	−0.545328	−	0.838222i	$-0.683595\pi$
$823$	−31.2887	−1.09066	−0.545328	+	0.838222i	$0.683595\pi$
$824$	0	0
$825$	8.71032	0.303254
$826$	0	0
$827$	21.5291	0.748639	0.374320	−	0.927300i	$-0.377876\pi$
$827$	21.5291	0.748639	0.374320	+	0.927300i	$0.377876\pi$
$828$	0	0
$829$	−30.7206	−1.06697	−0.533486	−	0.845809i	$-0.679118\pi$
$829$	−30.7206	−1.06697	−0.533486	+	0.845809i	$0.679118\pi$
$830$	0	0
$831$	28.5165	0.989227
$832$	0	0
$833$	−40.7183	−1.41080
$834$	0	0
$835$	2.71099	0.0938175
$836$	0	0
$837$	1.35539	0.0468493
$838$	0	0
$839$	6.94057	0.239615	0.119808	−	0.992797i	$-0.461772\pi$
$839$	6.94057	0.239615	0.119808	+	0.992797i	$0.461772\pi$
$840$	0	0
$841$	−2.12277	−0.0731989
$842$	0	0
$843$	−26.7452	−0.921155
$844$	0	0
$845$	−67.9103	−2.33618
$846$	0	0
$847$	−2.12840	−0.0731328
$848$	0	0
$849$	−2.76889	−0.0950281
$850$	0	0
$851$	−51.4442	−1.76348
$852$	0	0
$853$	−5.88185	−0.201391	−0.100695	−	0.994917i	$-0.532107\pi$
$853$	−5.88185	−0.201391	−0.100695	+	0.994917i	$0.532107\pi$
$854$	0	0
$855$	−2.50010	−0.0855015
$856$	0	0
$857$	−24.9088	−0.850869	−0.425435	−	0.904989i	$-0.639879\pi$
$857$	−24.9088	−0.850869	−0.425435	+	0.904989i	$0.639879\pi$
$858$	0	0
$859$	9.86029	0.336429	0.168214	−	0.985750i	$-0.446200\pi$
$859$	9.86029	0.336429	0.168214	+	0.985750i	$0.446200\pi$
$860$	0	0
$861$	6.90640	0.235369
$862$	0	0
$863$	37.4522	1.27489	0.637444	−	0.770497i	$-0.279992\pi$
$863$	37.4522	1.27489	0.637444	+	0.770497i	$0.279992\pi$
$864$	0	0
$865$	−45.6430	−1.55191
$866$	0	0
$867$	23.4945	0.797914
$868$	0	0
$869$	58.8597	1.99668
$870$	0	0
$871$	−3.87928	−0.131445
$872$	0	0
$873$	15.5125	0.525019
$874$	0	0
$875$	−5.57200	−0.188368
$876$	0	0
$877$	−33.7918	−1.14107	−0.570534	−	0.821274i	$-0.693264\pi$
$877$	−33.7918	−1.14107	−0.570534	+	0.821274i	$0.693264\pi$
$878$	0	0
$879$	−17.3417	−0.584920
$880$	0	0
$881$	50.8128	1.71193	0.855963	−	0.517038i	$-0.172965\pi$
$881$	50.8128	1.71193	0.855963	+	0.517038i	$0.172965\pi$
$882$	0	0
$883$	46.4246	1.56231	0.781156	−	0.624336i	$-0.214630\pi$
$883$	46.4246	1.56231	0.781156	+	0.624336i	$0.214630\pi$
$884$	0	0
$885$	−33.4221	−1.12347
$886$	0	0
$887$	7.20134	0.241797	0.120899	−	0.992665i	$-0.461422\pi$
$887$	7.20134	0.241797	0.120899	+	0.992665i	$0.461422\pi$
$888$	0	0
$889$	4.06391	0.136299
$890$	0	0
$891$	3.70739	0.124202
$892$	0	0
$893$	7.44807	0.249240
$894$	0	0
$895$	39.9716	1.33610
$896$	0	0
$897$	27.5759	0.920734
$898$	0	0
$899$	−7.02680	−0.234357
$900$	0	0
$901$	−14.2409	−0.474434
$902$	0	0
$903$	−4.28114	−0.142467
$904$	0	0
$905$	−31.7962	−1.05694
$906$	0	0
$907$	−35.0271	−1.16305	−0.581527	−	0.813527i	$-0.697545\pi$
$907$	−35.0271	−1.16305	−0.581527	+	0.813527i	$0.697545\pi$
$908$	0	0
$909$	14.2133	0.471427
$910$	0	0
$911$	−30.7886	−1.02007	−0.510036	−	0.860153i	$-0.670368\pi$
$911$	−30.7886	−1.02007	−0.510036	+	0.860153i	$0.670368\pi$
$912$	0	0
$913$	−17.6651	−0.584630
$914$	0	0
$915$	13.7748	0.455382
$916$	0	0
$917$	−3.32888	−0.109929
$918$	0	0
$919$	23.8564	0.786949	0.393474	−	0.919336i	$-0.371273\pi$
$919$	23.8564	0.786949	0.393474	+	0.919336i	$0.371273\pi$
$920$	0	0
$921$	−17.3038	−0.570178
$922$	0	0
$923$	−18.9654	−0.624252
$924$	0	0
$925$	27.0364	0.888951
$926$	0	0
$927$	−11.6148	−0.381479
$928$	0	0
$929$	−41.8969	−1.37459	−0.687297	−	0.726377i	$-0.741203\pi$
$929$	−41.8969	−1.37459	−0.687297	+	0.726377i	$0.741203\pi$
$930$	0	0
$931$	−5.90094	−0.193395
$932$	0	0
$933$	23.3959	0.765947
$934$	0	0
$935$	−63.9579	−2.09165
$936$	0	0
$937$	−38.6906	−1.26397	−0.631983	−	0.774983i	$-0.717759\pi$
$937$	−38.6906	−1.26397	−0.631983	+	0.774983i	$0.717759\pi$
$938$	0	0
$939$	0.191470	0.00624839
$940$	0	0
$941$	−49.1394	−1.60190	−0.800949	−	0.598732i	$-0.795671\pi$
$941$	−49.1394	−1.60190	−0.800949	+	0.598732i	$0.795671\pi$
$942$	0	0
$943$	39.8159	1.29659
$944$	0	0
$945$	2.10220	0.0683847
$946$	0	0
$947$	60.1778	1.95552	0.977758	−	0.209736i	$-0.0672604\pi$
$947$	60.1778	1.95552	0.977758	+	0.209736i	$0.0672604\pi$
$948$	0	0
$949$	−23.6355	−0.767242
$950$	0	0
$951$	−3.25612	−0.105587
$952$	0	0
$953$	6.03764	0.195578	0.0977892	−	0.995207i	$-0.468823\pi$
$953$	6.03764	0.195578	0.0977892	+	0.995207i	$0.468823\pi$
$954$	0	0
$955$	−4.38174	−0.141790
$956$	0	0
$957$	−19.2203	−0.621305
$958$	0	0
$959$	1.68534	0.0544225
$960$	0	0
$961$	−29.1629	−0.940739
$962$	0	0
$963$	13.1267	0.423002
$964$	0	0
$965$	−50.7530	−1.63380
$966$	0	0
$967$	−17.4074	−0.559785	−0.279892	−	0.960031i	$-0.590299\pi$
$967$	−17.4074	−0.559785	−0.279892	+	0.960031i	$0.590299\pi$
$968$	0	0
$969$	5.86850	0.188523
$970$	0	0
$971$	−18.2201	−0.584711	−0.292356	−	0.956310i	$-0.594439\pi$
$971$	−18.2201	−0.584711	−0.292356	+	0.956310i	$0.594439\pi$
$972$	0	0
$973$	−14.0115	−0.449187
$974$	0	0
$975$	−14.4925	−0.464131
$976$	0	0
$977$	−40.7926	−1.30507	−0.652535	−	0.757758i	$-0.726295\pi$
$977$	−40.7926	−1.30507	−0.652535	+	0.757758i	$0.726295\pi$
$978$	0	0
$979$	1.44903	0.0463111
$980$	0	0
$981$	3.37165	0.107648
$982$	0	0
$983$	56.5978	1.80519	0.902594	−	0.430492i	$-0.141660\pi$
$983$	56.5978	1.80519	0.902594	+	0.430492i	$0.141660\pi$
$984$	0	0
$985$	33.0604	1.05339
$986$	0	0
$987$	−6.26271	−0.199344
$988$	0	0
$989$	−24.6811	−0.784814
$990$	0	0
$991$	−15.0500	−0.478081	−0.239040	−	0.971010i	$-0.576833\pi$
$991$	−15.0500	−0.478081	−0.239040	+	0.971010i	$0.576833\pi$
$992$	0	0
$993$	−30.7585	−0.976092
$994$	0	0
$995$	4.02557	0.127619
$996$	0	0
$997$	−23.2414	−0.736063	−0.368031	−	0.929813i	$-0.619968\pi$
$997$	−23.2414	−0.736063	−0.368031	+	0.929813i	$0.619968\pi$
$998$	0	0
$999$	11.5076	0.364083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.m.1.4	✓	13
4.3	odd	2		8016.2.a.bg.1.4		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.4	✓	13	1.1	even	1	trivial
8016.2.a.bg.1.4		13	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.71099 q^{5} -0.775439 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.71099 q^{5} -0.775439 q^{7} +1.00000 q^{9} +3.70739 q^{11} -6.16847 q^{13} -2.71099 q^{15} +6.36353 q^{17} +0.922209 q^{19} -0.775439 q^{21} -4.47047 q^{23} +2.34945 q^{25} +1.00000 q^{27} -5.18433 q^{29} +1.35539 q^{31} +3.70739 q^{33} +2.10220 q^{35} +11.5076 q^{37} -6.16847 q^{39} -8.90644 q^{41} +5.52092 q^{43} -2.71099 q^{45} +8.07634 q^{47} -6.39869 q^{49} +6.36353 q^{51} -2.23790 q^{53} -10.0507 q^{55} +0.922209 q^{57} +12.3284 q^{59} -5.08112 q^{61} -0.775439 q^{63} +16.7226 q^{65} +0.628889 q^{67} -4.47047 q^{69} +3.07457 q^{71} +3.83167 q^{73} +2.34945 q^{75} -2.87486 q^{77} +15.8763 q^{79} +1.00000 q^{81} -4.76483 q^{83} -17.2514 q^{85} -5.18433 q^{87} +0.390848 q^{89} +4.78327 q^{91} +1.35539 q^{93} -2.50010 q^{95} +15.5125 q^{97} +3.70739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

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