LMFDB - Embedded newform 4024.2.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.502871 q^{3} -1.85170 q^{5} +1.22131 q^{7} -2.74712 q^{9} +O(q^{10})$	$q-0.502871 q^{3} -1.85170 q^{5} +1.22131 q^{7} -2.74712 q^{9} +4.21482 q^{11} -6.01706 q^{13} +0.931168 q^{15} -6.53168 q^{17} -3.84005 q^{19} -0.614162 q^{21} +3.97402 q^{23} -1.57120 q^{25} +2.89006 q^{27} +1.39246 q^{29} +4.43535 q^{31} -2.11951 q^{33} -2.26150 q^{35} +2.19690 q^{37} +3.02581 q^{39} +3.88073 q^{41} +9.46434 q^{43} +5.08685 q^{45} -6.20672 q^{47} -5.50840 q^{49} +3.28460 q^{51} -13.9641 q^{53} -7.80459 q^{55} +1.93105 q^{57} +3.95939 q^{59} +11.2661 q^{61} -3.35509 q^{63} +11.1418 q^{65} +2.46928 q^{67} -1.99842 q^{69} +13.6474 q^{71} +5.80859 q^{73} +0.790110 q^{75} +5.14760 q^{77} +1.22002 q^{79} +6.78803 q^{81} -0.451577 q^{83} +12.0947 q^{85} -0.700227 q^{87} +4.12204 q^{89} -7.34870 q^{91} -2.23041 q^{93} +7.11063 q^{95} +1.44281 q^{97} -11.5786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * 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
$3$	−0.502871	−0.290333	−0.145166	−	0.989407i	$-0.546372\pi$
$3$	−0.502871	−0.290333	−0.145166	+	0.989407i	$0.546372\pi$
$4$	0	0
$5$	−1.85170	−0.828107	−0.414053	−	0.910253i	$-0.635887\pi$
$5$	−1.85170	−0.828107	−0.414053	+	0.910253i	$0.635887\pi$
$6$	0	0
$7$	1.22131	0.461612	0.230806	−	0.973000i	$-0.425864\pi$
$7$	1.22131	0.461612	0.230806	+	0.973000i	$0.425864\pi$
$8$	0	0
$9$	−2.74712	−0.915707
$10$	0	0
$11$	4.21482	1.27082	0.635408	−	0.772177i	$-0.280832\pi$
$11$	4.21482	1.27082	0.635408	+	0.772177i	$0.280832\pi$
$12$	0	0
$13$	−6.01706	−1.66883	−0.834416	−	0.551134i	$-0.814195\pi$
$13$	−6.01706	−1.66883	−0.834416	+	0.551134i	$0.814195\pi$
$14$	0	0
$15$	0.931168	0.240427
$16$	0	0
$17$	−6.53168	−1.58417	−0.792083	−	0.610414i	$-0.791003\pi$
$17$	−6.53168	−1.58417	−0.792083	+	0.610414i	$0.791003\pi$
$18$	0	0
$19$	−3.84005	−0.880968	−0.440484	−	0.897761i	$-0.645193\pi$
$19$	−3.84005	−0.880968	−0.440484	+	0.897761i	$0.645193\pi$
$20$	0	0
$21$	−0.614162	−0.134021
$22$	0	0
$23$	3.97402	0.828641	0.414321	−	0.910131i	$-0.364019\pi$
$23$	3.97402	0.828641	0.414321	+	0.910131i	$0.364019\pi$
$24$	0	0
$25$	−1.57120	−0.314239
$26$	0	0
$27$	2.89006	0.556193
$28$	0	0
$29$	1.39246	0.258573	0.129286	−	0.991607i	$-0.458731\pi$
$29$	1.39246	0.258573	0.129286	+	0.991607i	$0.458731\pi$
$30$	0	0
$31$	4.43535	0.796612	0.398306	−	0.917253i	$-0.369598\pi$
$31$	4.43535	0.796612	0.398306	+	0.917253i	$0.369598\pi$
$32$	0	0
$33$	−2.11951	−0.368959
$34$	0	0
$35$	−2.26150	−0.382264
$36$	0	0
$37$	2.19690	0.361169	0.180584	−	0.983560i	$-0.442201\pi$
$37$	2.19690	0.361169	0.180584	+	0.983560i	$0.442201\pi$
$38$	0	0
$39$	3.02581	0.484517
$40$	0	0
$41$	3.88073	0.606068	0.303034	−	0.952980i	$-0.402000\pi$
$41$	3.88073	0.606068	0.303034	+	0.952980i	$0.402000\pi$
$42$	0	0
$43$	9.46434	1.44330	0.721649	−	0.692259i	$-0.243384\pi$
$43$	9.46434	1.44330	0.721649	+	0.692259i	$0.243384\pi$
$44$	0	0
$45$	5.08685	0.758303
$46$	0	0
$47$	−6.20672	−0.905343	−0.452671	−	0.891677i	$-0.649529\pi$
$47$	−6.20672	−0.905343	−0.452671	+	0.891677i	$0.649529\pi$
$48$	0	0
$49$	−5.50840	−0.786914
$50$	0	0
$51$	3.28460	0.459936
$52$	0	0
$53$	−13.9641	−1.91812	−0.959058	−	0.283210i	$-0.908601\pi$
$53$	−13.9641	−1.91812	−0.959058	+	0.283210i	$0.908601\pi$
$54$	0	0
$55$	−7.80459	−1.05237
$56$	0	0
$57$	1.93105	0.255774
$58$	0	0
$59$	3.95939	0.515468	0.257734	−	0.966216i	$-0.417024\pi$
$59$	3.95939	0.515468	0.257734	+	0.966216i	$0.417024\pi$
$60$	0	0
$61$	11.2661	1.44248	0.721239	−	0.692686i	$-0.243573\pi$
$61$	11.2661	1.44248	0.721239	+	0.692686i	$0.243573\pi$
$62$	0	0
$63$	−3.35509	−0.422701
$64$	0	0
$65$	11.1418	1.38197
$66$	0	0
$67$	2.46928	0.301671	0.150836	−	0.988559i	$-0.451804\pi$
$67$	2.46928	0.301671	0.150836	+	0.988559i	$0.451804\pi$
$68$	0	0
$69$	−1.99842	−0.240582
$70$	0	0
$71$	13.6474	1.61964	0.809822	−	0.586676i	$-0.199564\pi$
$71$	13.6474	1.61964	0.809822	+	0.586676i	$0.199564\pi$
$72$	0	0
$73$	5.80859	0.679844	0.339922	−	0.940454i	$-0.389599\pi$
$73$	5.80859	0.679844	0.339922	+	0.940454i	$0.389599\pi$
$74$	0	0
$75$	0.790110	0.0912340
$76$	0	0
$77$	5.14760	0.586624
$78$	0	0
$79$	1.22002	0.137262	0.0686312	−	0.997642i	$-0.478137\pi$
$79$	1.22002	0.137262	0.0686312	+	0.997642i	$0.478137\pi$
$80$	0	0
$81$	6.78803	0.754226
$82$	0	0
$83$	−0.451577	−0.0495670	−0.0247835	−	0.999693i	$-0.507890\pi$
$83$	−0.451577	−0.0495670	−0.0247835	+	0.999693i	$0.507890\pi$
$84$	0	0
$85$	12.0947	1.31186
$86$	0	0
$87$	−0.700227	−0.0750722
$88$	0	0
$89$	4.12204	0.436935	0.218468	−	0.975844i	$-0.429894\pi$
$89$	4.12204	0.436935	0.218468	+	0.975844i	$0.429894\pi$
$90$	0	0
$91$	−7.34870	−0.770353
$92$	0	0
$93$	−2.23041	−0.231283
$94$	0	0
$95$	7.11063	0.729535
$96$	0	0
$97$	1.44281	0.146495	0.0732477	−	0.997314i	$-0.476664\pi$
$97$	1.44281	0.146495	0.0732477	+	0.997314i	$0.476664\pi$
$98$	0	0
$99$	−11.5786	−1.16369
$100$	0	0
$101$	−12.7500	−1.26868	−0.634338	−	0.773056i	$-0.718727\pi$
$101$	−12.7500	−1.26868	−0.634338	+	0.773056i	$0.718727\pi$
$102$	0	0
$103$	−12.7269	−1.25402	−0.627011	−	0.779011i	$-0.715722\pi$
$103$	−12.7269	−1.25402	−0.627011	+	0.779011i	$0.715722\pi$
$104$	0	0
$105$	1.13725	0.110984
$106$	0	0
$107$	1.70081	0.164424	0.0822118	−	0.996615i	$-0.473802\pi$
$107$	1.70081	0.164424	0.0822118	+	0.996615i	$0.473802\pi$
$108$	0	0
$109$	−9.16832	−0.878165	−0.439083	−	0.898447i	$-0.644696\pi$
$109$	−9.16832	−0.878165	−0.439083	+	0.898447i	$0.644696\pi$
$110$	0	0
$111$	−1.10476	−0.104859
$112$	0	0
$113$	−16.4252	−1.54515	−0.772577	−	0.634921i	$-0.781033\pi$
$113$	−16.4252	−1.54515	−0.772577	+	0.634921i	$0.781033\pi$
$114$	0	0
$115$	−7.35871	−0.686203
$116$	0	0
$117$	16.5296	1.52816
$118$	0	0
$119$	−7.97721	−0.731270
$120$	0	0
$121$	6.76468	0.614971
$122$	0	0
$123$	−1.95151	−0.175962
$124$	0	0
$125$	12.1679	1.08833
$126$	0	0
$127$	8.67901	0.770137	0.385069	−	0.922888i	$-0.374178\pi$
$127$	8.67901	0.770137	0.385069	+	0.922888i	$0.374178\pi$
$128$	0	0
$129$	−4.75935	−0.419037
$130$	0	0
$131$	−4.27041	−0.373108	−0.186554	−	0.982445i	$-0.559732\pi$
$131$	−4.27041	−0.373108	−0.186554	+	0.982445i	$0.559732\pi$
$132$	0	0
$133$	−4.68989	−0.406665
$134$	0	0
$135$	−5.35154	−0.460587
$136$	0	0
$137$	8.10572	0.692519	0.346259	−	0.938139i	$-0.387452\pi$
$137$	8.10572	0.692519	0.346259	+	0.938139i	$0.387452\pi$
$138$	0	0
$139$	15.7478	1.33571	0.667855	−	0.744292i	$-0.267213\pi$
$139$	15.7478	1.33571	0.667855	+	0.744292i	$0.267213\pi$
$140$	0	0
$141$	3.12118	0.262851
$142$	0	0
$143$	−25.3608	−2.12078
$144$	0	0
$145$	−2.57842	−0.214126
$146$	0	0
$147$	2.77002	0.228467
$148$	0	0
$149$	11.5303	0.944597	0.472299	−	0.881439i	$-0.343424\pi$
$149$	11.5303	0.944597	0.472299	+	0.881439i	$0.343424\pi$
$150$	0	0
$151$	−11.1568	−0.907927	−0.453964	−	0.891020i	$-0.649990\pi$
$151$	−11.1568	−0.907927	−0.453964	+	0.891020i	$0.649990\pi$
$152$	0	0
$153$	17.9433	1.45063
$154$	0	0
$155$	−8.21295	−0.659680
$156$	0	0
$157$	19.8327	1.58282	0.791410	−	0.611286i	$-0.209347\pi$
$157$	19.8327	1.58282	0.791410	+	0.611286i	$0.209347\pi$
$158$	0	0
$159$	7.02214	0.556892
$160$	0	0
$161$	4.85352	0.382511
$162$	0	0
$163$	11.9357	0.934877	0.467439	−	0.884025i	$-0.345177\pi$
$163$	11.9357	0.934877	0.467439	+	0.884025i	$0.345177\pi$
$164$	0	0
$165$	3.92470	0.305538
$166$	0	0
$167$	1.16238	0.0899480	0.0449740	−	0.998988i	$-0.485680\pi$
$167$	1.16238	0.0899480	0.0449740	+	0.998988i	$0.485680\pi$
$168$	0	0
$169$	23.2050	1.78500
$170$	0	0
$171$	10.5491	0.806708
$172$	0	0
$173$	11.8853	0.903621	0.451810	−	0.892114i	$-0.350778\pi$
$173$	11.8853	0.903621	0.451810	+	0.892114i	$0.350778\pi$
$174$	0	0
$175$	−1.91892	−0.145057
$176$	0	0
$177$	−1.99106	−0.149657
$178$	0	0
$179$	17.0215	1.27225	0.636125	−	0.771586i	$-0.280536\pi$
$179$	17.0215	1.27225	0.636125	+	0.771586i	$0.280536\pi$
$180$	0	0
$181$	5.41457	0.402462	0.201231	−	0.979544i	$-0.435506\pi$
$181$	5.41457	0.402462	0.201231	+	0.979544i	$0.435506\pi$
$182$	0	0
$183$	−5.66541	−0.418799
$184$	0	0
$185$	−4.06801	−0.299086
$186$	0	0
$187$	−27.5298	−2.01318
$188$	0	0
$189$	3.52966	0.256745
$190$	0	0
$191$	12.0658	0.873049	0.436524	−	0.899692i	$-0.356209\pi$
$191$	12.0658	0.873049	0.436524	+	0.899692i	$0.356209\pi$
$192$	0	0
$193$	−12.9406	−0.931486	−0.465743	−	0.884920i	$-0.654213\pi$
$193$	−12.9406	−0.931486	−0.465743	+	0.884920i	$0.654213\pi$
$194$	0	0
$195$	−5.60290	−0.401232
$196$	0	0
$197$	17.9387	1.27808	0.639039	−	0.769174i	$-0.279332\pi$
$197$	17.9387	1.27808	0.639039	+	0.769174i	$0.279332\pi$
$198$	0	0
$199$	17.9805	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$199$	17.9805	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$200$	0	0
$201$	−1.24173	−0.0875850
$202$	0	0
$203$	1.70062	0.119360
$204$	0	0
$205$	−7.18596	−0.501889
$206$	0	0
$207$	−10.9171	−0.758792
$208$	0	0
$209$	−16.1851	−1.11955
$210$	0	0
$211$	8.37520	0.576573	0.288286	−	0.957544i	$-0.406914\pi$
$211$	8.37520	0.576573	0.288286	+	0.957544i	$0.406914\pi$
$212$	0	0
$213$	−6.86287	−0.470236
$214$	0	0
$215$	−17.5252	−1.19521
$216$	0	0
$217$	5.41694	0.367726
$218$	0	0
$219$	−2.92097	−0.197381
$220$	0	0
$221$	39.3015	2.64371
$222$	0	0
$223$	−5.67169	−0.379805	−0.189902	−	0.981803i	$-0.560817\pi$
$223$	−5.67169	−0.379805	−0.189902	+	0.981803i	$0.560817\pi$
$224$	0	0
$225$	4.31627	0.287751
$226$	0	0
$227$	14.6346	0.971331	0.485665	−	0.874145i	$-0.338577\pi$
$227$	14.6346	0.971331	0.485665	+	0.874145i	$0.338577\pi$
$228$	0	0
$229$	0.780709	0.0515907	0.0257953	−	0.999667i	$-0.491788\pi$
$229$	0.780709	0.0515907	0.0257953	+	0.999667i	$0.491788\pi$
$230$	0	0
$231$	−2.58858	−0.170316
$232$	0	0
$233$	3.62716	0.237623	0.118811	−	0.992917i	$-0.462092\pi$
$233$	3.62716	0.237623	0.118811	+	0.992917i	$0.462092\pi$
$234$	0	0
$235$	11.4930	0.749720
$236$	0	0
$237$	−0.613511	−0.0398518
$238$	0	0
$239$	9.16918	0.593105	0.296552	−	0.955017i	$-0.404163\pi$
$239$	9.16918	0.593105	0.296552	+	0.955017i	$0.404163\pi$
$240$	0	0
$241$	−5.08353	−0.327459	−0.163730	−	0.986505i	$-0.552352\pi$
$241$	−5.08353	−0.327459	−0.163730	+	0.986505i	$0.552352\pi$
$242$	0	0
$243$	−12.0837	−0.775169
$244$	0	0
$245$	10.1999	0.651649
$246$	0	0
$247$	23.1058	1.47019
$248$	0	0
$249$	0.227085	0.0143909
$250$	0	0
$251$	−5.53339	−0.349264	−0.174632	−	0.984634i	$-0.555874\pi$
$251$	−5.53339	−0.349264	−0.174632	+	0.984634i	$0.555874\pi$
$252$	0	0
$253$	16.7498	1.05305
$254$	0	0
$255$	−6.08210	−0.380876
$256$	0	0
$257$	5.48145	0.341924	0.170962	−	0.985278i	$-0.445313\pi$
$257$	5.48145	0.341924	0.170962	+	0.985278i	$0.445313\pi$
$258$	0	0
$259$	2.68310	0.166720
$260$	0	0
$261$	−3.82525	−0.236777
$262$	0	0
$263$	7.23059	0.445857	0.222929	−	0.974835i	$-0.428438\pi$
$263$	7.23059	0.445857	0.222929	+	0.974835i	$0.428438\pi$
$264$	0	0
$265$	25.8574	1.58840
$266$	0	0
$267$	−2.07286	−0.126857
$268$	0	0
$269$	1.76103	0.107372	0.0536861	−	0.998558i	$-0.482903\pi$
$269$	1.76103	0.107372	0.0536861	+	0.998558i	$0.482903\pi$
$270$	0	0
$271$	27.2906	1.65779	0.828894	−	0.559406i	$-0.188971\pi$
$271$	27.2906	1.65779	0.828894	+	0.559406i	$0.188971\pi$
$272$	0	0
$273$	3.69545	0.223659
$274$	0	0
$275$	−6.62231	−0.399340
$276$	0	0
$277$	0.494771	0.0297279	0.0148639	−	0.999890i	$-0.495268\pi$
$277$	0.494771	0.0297279	0.0148639	+	0.999890i	$0.495268\pi$
$278$	0	0
$279$	−12.1844	−0.729463
$280$	0	0
$281$	−31.5571	−1.88254	−0.941268	−	0.337660i	$-0.890365\pi$
$281$	−31.5571	−1.88254	−0.941268	+	0.337660i	$0.890365\pi$
$282$	0	0
$283$	−10.7781	−0.640691	−0.320345	−	0.947301i	$-0.603799\pi$
$283$	−10.7781	−0.640691	−0.320345	+	0.947301i	$0.603799\pi$
$284$	0	0
$285$	−3.57573	−0.211808
$286$	0	0
$287$	4.73958	0.279769
$288$	0	0
$289$	25.6629	1.50958
$290$	0	0
$291$	−0.725549	−0.0425325
$292$	0	0
$293$	6.57891	0.384344	0.192172	−	0.981361i	$-0.438447\pi$
$293$	6.57891	0.384344	0.192172	+	0.981361i	$0.438447\pi$
$294$	0	0
$295$	−7.33160	−0.426863
$296$	0	0
$297$	12.1811	0.706818
$298$	0	0
$299$	−23.9119	−1.38286
$300$	0	0
$301$	11.5589	0.666244
$302$	0	0
$303$	6.41162	0.368338
$304$	0	0
$305$	−20.8615	−1.19453
$306$	0	0
$307$	−31.7686	−1.81313	−0.906565	−	0.422066i	$-0.861305\pi$
$307$	−31.7686	−1.81313	−0.906565	+	0.422066i	$0.861305\pi$
$308$	0	0
$309$	6.40001	0.364084
$310$	0	0
$311$	33.4947	1.89931	0.949656	−	0.313296i	$-0.101433\pi$
$311$	33.4947	1.89931	0.949656	+	0.313296i	$0.101433\pi$
$312$	0	0
$313$	−19.5558	−1.10536	−0.552678	−	0.833395i	$-0.686394\pi$
$313$	−19.5558	−1.10536	−0.552678	+	0.833395i	$0.686394\pi$
$314$	0	0
$315$	6.21262	0.350042
$316$	0	0
$317$	10.0205	0.562807	0.281403	−	0.959590i	$-0.409200\pi$
$317$	10.0205	0.562807	0.281403	+	0.959590i	$0.409200\pi$
$318$	0	0
$319$	5.86895	0.328598
$320$	0	0
$321$	−0.855289	−0.0477376
$322$	0	0
$323$	25.0820	1.39560
$324$	0	0
$325$	9.45399	0.524413
$326$	0	0
$327$	4.61048	0.254960
$328$	0	0
$329$	−7.58033	−0.417917
$330$	0	0
$331$	−4.67434	−0.256925	−0.128462	−	0.991714i	$-0.541004\pi$
$331$	−4.67434	−0.256925	−0.128462	+	0.991714i	$0.541004\pi$
$332$	0	0
$333$	−6.03516	−0.330725
$334$	0	0
$335$	−4.57238	−0.249816
$336$	0	0
$337$	−21.1097	−1.14992	−0.574959	−	0.818182i	$-0.694982\pi$
$337$	−21.1097	−1.14992	−0.574959	+	0.818182i	$0.694982\pi$
$338$	0	0
$339$	8.25977	0.448609
$340$	0	0
$341$	18.6942	1.01235
$342$	0	0
$343$	−15.2766	−0.824861
$344$	0	0
$345$	3.70048	0.199227
$346$	0	0
$347$	17.4328	0.935841	0.467920	−	0.883771i	$-0.345003\pi$
$347$	17.4328	0.935841	0.467920	+	0.883771i	$0.345003\pi$
$348$	0	0
$349$	−29.8948	−1.60023	−0.800116	−	0.599845i	$-0.795229\pi$
$349$	−29.8948	−1.60023	−0.800116	+	0.599845i	$0.795229\pi$
$350$	0	0
$351$	−17.3897	−0.928193
$352$	0	0
$353$	8.40961	0.447599	0.223799	−	0.974635i	$-0.428154\pi$
$353$	8.40961	0.447599	0.223799	+	0.974635i	$0.428154\pi$
$354$	0	0
$355$	−25.2709	−1.34124
$356$	0	0
$357$	4.01151	0.212312
$358$	0	0
$359$	33.0122	1.74232	0.871158	−	0.491002i	$-0.163369\pi$
$359$	33.0122	1.74232	0.871158	+	0.491002i	$0.163369\pi$
$360$	0	0
$361$	−4.25403	−0.223896
$362$	0	0
$363$	−3.40177	−0.178546
$364$	0	0
$365$	−10.7558	−0.562983
$366$	0	0
$367$	−31.9314	−1.66681	−0.833404	−	0.552665i	$-0.813611\pi$
$367$	−31.9314	−1.66681	−0.833404	+	0.552665i	$0.813611\pi$
$368$	0	0
$369$	−10.6608	−0.554981
$370$	0	0
$371$	−17.0545	−0.885425
$372$	0	0
$373$	10.5612	0.546839	0.273419	−	0.961895i	$-0.411845\pi$
$373$	10.5612	0.546839	0.273419	+	0.961895i	$0.411845\pi$
$374$	0	0
$375$	−6.11889	−0.315978
$376$	0	0
$377$	−8.37850	−0.431515
$378$	0	0
$379$	−12.4432	−0.639165	−0.319583	−	0.947558i	$-0.603543\pi$
$379$	−12.4432	−0.639165	−0.319583	+	0.947558i	$0.603543\pi$
$380$	0	0
$381$	−4.36442	−0.223596
$382$	0	0
$383$	5.91094	0.302035	0.151018	−	0.988531i	$-0.451745\pi$
$383$	5.91094	0.302035	0.151018	+	0.988531i	$0.451745\pi$
$384$	0	0
$385$	−9.53183	−0.485787
$386$	0	0
$387$	−25.9997	−1.32164
$388$	0	0
$389$	−14.5597	−0.738206	−0.369103	−	0.929388i	$-0.620335\pi$
$389$	−14.5597	−0.738206	−0.369103	+	0.929388i	$0.620335\pi$
$390$	0	0
$391$	−25.9571	−1.31270
$392$	0	0
$393$	2.14747	0.108325
$394$	0	0
$395$	−2.25911	−0.113668
$396$	0	0
$397$	7.36575	0.369677	0.184838	−	0.982769i	$-0.440824\pi$
$397$	7.36575	0.369677	0.184838	+	0.982769i	$0.440824\pi$
$398$	0	0
$399$	2.35841	0.118068
$400$	0	0
$401$	18.5359	0.925638	0.462819	−	0.886453i	$-0.346838\pi$
$401$	18.5359	0.925638	0.462819	+	0.886453i	$0.346838\pi$
$402$	0	0
$403$	−26.6878	−1.32941
$404$	0	0
$405$	−12.5694	−0.624579
$406$	0	0
$407$	9.25954	0.458979
$408$	0	0
$409$	−24.3341	−1.20324	−0.601622	−	0.798781i	$-0.705479\pi$
$409$	−24.3341	−1.20324	−0.601622	+	0.798781i	$0.705479\pi$
$410$	0	0
$411$	−4.07614	−0.201061
$412$	0	0
$413$	4.83564	0.237946
$414$	0	0
$415$	0.836186	0.0410468
$416$	0	0
$417$	−7.91911	−0.387801
$418$	0	0
$419$	22.9419	1.12079	0.560393	−	0.828227i	$-0.310650\pi$
$419$	22.9419	1.12079	0.560393	+	0.828227i	$0.310650\pi$
$420$	0	0
$421$	−3.32299	−0.161953	−0.0809763	−	0.996716i	$-0.525804\pi$
$421$	−3.32299	−0.161953	−0.0809763	+	0.996716i	$0.525804\pi$
$422$	0	0
$423$	17.0506	0.829028
$424$	0	0
$425$	10.2626	0.497807
$426$	0	0
$427$	13.7594	0.665865
$428$	0	0
$429$	12.7532	0.615732
$430$	0	0
$431$	20.9210	1.00773	0.503864	−	0.863783i	$-0.331911\pi$
$431$	20.9210	1.00773	0.503864	+	0.863783i	$0.331911\pi$
$432$	0	0
$433$	−0.542592	−0.0260753	−0.0130377	−	0.999915i	$-0.504150\pi$
$433$	−0.542592	−0.0260753	−0.0130377	+	0.999915i	$0.504150\pi$
$434$	0	0
$435$	1.29661	0.0621678
$436$	0	0
$437$	−15.2604	−0.730006
$438$	0	0
$439$	1.16784	0.0557380	0.0278690	−	0.999612i	$-0.491128\pi$
$439$	1.16784	0.0557380	0.0278690	+	0.999612i	$0.491128\pi$
$440$	0	0
$441$	15.1322	0.720583
$442$	0	0
$443$	28.8412	1.37029	0.685144	−	0.728408i	$-0.259739\pi$
$443$	28.8412	1.37029	0.685144	+	0.728408i	$0.259739\pi$
$444$	0	0
$445$	−7.63279	−0.361829
$446$	0	0
$447$	−5.79825	−0.274248
$448$	0	0
$449$	−19.7006	−0.929727	−0.464864	−	0.885382i	$-0.653897\pi$
$449$	−19.7006	−0.929727	−0.464864	+	0.885382i	$0.653897\pi$
$450$	0	0
$451$	16.3566	0.770201
$452$	0	0
$453$	5.61044	0.263601
$454$	0	0
$455$	13.6076	0.637935
$456$	0	0
$457$	10.6170	0.496641	0.248321	−	0.968678i	$-0.420121\pi$
$457$	10.6170	0.496641	0.248321	+	0.968678i	$0.420121\pi$
$458$	0	0
$459$	−18.8770	−0.881102
$460$	0	0
$461$	20.1370	0.937875	0.468937	−	0.883231i	$-0.344637\pi$
$461$	20.1370	0.937875	0.468937	+	0.883231i	$0.344637\pi$
$462$	0	0
$463$	−30.3526	−1.41061	−0.705303	−	0.708906i	$-0.749189\pi$
$463$	−30.3526	−1.41061	−0.705303	+	0.708906i	$0.749189\pi$
$464$	0	0
$465$	4.13006	0.191527
$466$	0	0
$467$	27.8800	1.29013	0.645067	−	0.764126i	$-0.276829\pi$
$467$	27.8800	1.29013	0.645067	+	0.764126i	$0.276829\pi$
$468$	0	0
$469$	3.01576	0.139255
$470$	0	0
$471$	−9.97328	−0.459545
$472$	0	0
$473$	39.8905	1.83417
$474$	0	0
$475$	6.03347	0.276835
$476$	0	0
$477$	38.3610	1.75643
$478$	0	0
$479$	−41.1827	−1.88169	−0.940843	−	0.338843i	$-0.889965\pi$
$479$	−41.1827	−1.88169	−0.940843	+	0.338843i	$0.889965\pi$
$480$	0	0
$481$	−13.2189	−0.602730
$482$	0	0
$483$	−2.44069	−0.111055
$484$	0	0
$485$	−2.67166	−0.121314
$486$	0	0
$487$	−2.39414	−0.108489	−0.0542443	−	0.998528i	$-0.517275\pi$
$487$	−2.39414	−0.108489	−0.0542443	+	0.998528i	$0.517275\pi$
$488$	0	0
$489$	−6.00213	−0.271426
$490$	0	0
$491$	7.50302	0.338606	0.169303	−	0.985564i	$-0.445848\pi$
$491$	7.50302	0.338606	0.169303	+	0.985564i	$0.445848\pi$
$492$	0	0
$493$	−9.09509	−0.409622
$494$	0	0
$495$	21.4401	0.963663
$496$	0	0
$497$	16.6677	0.747647
$498$	0	0
$499$	29.3451	1.31367	0.656833	−	0.754036i	$-0.271896\pi$
$499$	29.3451	1.31367	0.656833	+	0.754036i	$0.271896\pi$
$500$	0	0
$501$	−0.584530	−0.0261149
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	23.6093	1.05060
$506$	0	0
$507$	−11.6692	−0.518245
$508$	0	0
$509$	−7.26012	−0.321799	−0.160900	−	0.986971i	$-0.551440\pi$
$509$	−7.26012	−0.321799	−0.160900	+	0.986971i	$0.551440\pi$
$510$	0	0
$511$	7.09409	0.313824
$512$	0	0
$513$	−11.0980	−0.489988
$514$	0	0
$515$	23.5665	1.03846
$516$	0	0
$517$	−26.1602	−1.15052
$518$	0	0
$519$	−5.97676	−0.262351
$520$	0	0
$521$	−44.5673	−1.95253	−0.976263	−	0.216587i	$-0.930508\pi$
$521$	−44.5673	−1.95253	−0.976263	+	0.216587i	$0.930508\pi$
$522$	0	0
$523$	−21.1505	−0.924849	−0.462424	−	0.886659i	$-0.653020\pi$
$523$	−21.1505	−0.924849	−0.462424	+	0.886659i	$0.653020\pi$
$524$	0	0
$525$	0.964970	0.0421147
$526$	0	0
$527$	−28.9703	−1.26197
$528$	0	0
$529$	−7.20714	−0.313354
$530$	0	0
$531$	−10.8769	−0.472018
$532$	0	0
$533$	−23.3506	−1.01143
$534$	0	0
$535$	−3.14940	−0.136160
$536$	0	0
$537$	−8.55965	−0.369376
$538$	0	0
$539$	−23.2169	−1.00002
$540$	0	0
$541$	45.9194	1.97423	0.987114	−	0.160019i	$-0.0511557\pi$
$541$	45.9194	1.97423	0.987114	+	0.160019i	$0.0511557\pi$
$542$	0	0
$543$	−2.72283	−0.116848
$544$	0	0
$545$	16.9770	0.727215
$546$	0	0
$547$	34.5102	1.47555	0.737774	−	0.675048i	$-0.235877\pi$
$547$	34.5102	1.47555	0.737774	+	0.675048i	$0.235877\pi$
$548$	0	0
$549$	−30.9494	−1.32089
$550$	0	0
$551$	−5.34710	−0.227794
$552$	0	0
$553$	1.49002	0.0633620
$554$	0	0
$555$	2.04569	0.0868346
$556$	0	0
$557$	20.6084	0.873207	0.436603	−	0.899654i	$-0.356181\pi$
$557$	20.6084	0.873207	0.436603	+	0.899654i	$0.356181\pi$
$558$	0	0
$559$	−56.9476	−2.40862
$560$	0	0
$561$	13.8440	0.584493
$562$	0	0
$563$	2.94528	0.124129	0.0620644	−	0.998072i	$-0.480232\pi$
$563$	2.94528	0.124129	0.0620644	+	0.998072i	$0.480232\pi$
$564$	0	0
$565$	30.4146	1.27955
$566$	0	0
$567$	8.29030	0.348160
$568$	0	0
$569$	5.10275	0.213918	0.106959	−	0.994263i	$-0.465889\pi$
$569$	5.10275	0.213918	0.106959	+	0.994263i	$0.465889\pi$
$570$	0	0
$571$	4.51107	0.188782	0.0943912	−	0.995535i	$-0.469910\pi$
$571$	4.51107	0.188782	0.0943912	+	0.995535i	$0.469910\pi$
$572$	0	0
$573$	−6.06753	−0.253475
$574$	0	0
$575$	−6.24397	−0.260392
$576$	0	0
$577$	−41.3780	−1.72259	−0.861294	−	0.508107i	$-0.830346\pi$
$577$	−41.3780	−1.72259	−0.861294	+	0.508107i	$0.830346\pi$
$578$	0	0
$579$	6.50747	0.270441
$580$	0	0
$581$	−0.551516	−0.0228807
$582$	0	0
$583$	−58.8561	−2.43757
$584$	0	0
$585$	−30.6079	−1.26548
$586$	0	0
$587$	1.89400	0.0781739	0.0390869	−	0.999236i	$-0.487555\pi$
$587$	1.89400	0.0781739	0.0390869	+	0.999236i	$0.487555\pi$
$588$	0	0
$589$	−17.0320	−0.701789
$590$	0	0
$591$	−9.02085	−0.371068
$592$	0	0
$593$	−11.8856	−0.488082	−0.244041	−	0.969765i	$-0.578473\pi$
$593$	−11.8856	−0.488082	−0.244041	+	0.969765i	$0.578473\pi$
$594$	0	0
$595$	14.7714	0.605570
$596$	0	0
$597$	−9.04187	−0.370059
$598$	0	0
$599$	19.4594	0.795090	0.397545	−	0.917583i	$-0.369862\pi$
$599$	19.4594	0.795090	0.397545	+	0.917583i	$0.369862\pi$
$600$	0	0
$601$	13.3289	0.543697	0.271849	−	0.962340i	$-0.412365\pi$
$601$	13.3289	0.543697	0.271849	+	0.962340i	$0.412365\pi$
$602$	0	0
$603$	−6.78342	−0.276242
$604$	0	0
$605$	−12.5262	−0.509262
$606$	0	0
$607$	−29.8607	−1.21201	−0.606005	−	0.795461i	$-0.707229\pi$
$607$	−29.8607	−1.21201	−0.606005	+	0.795461i	$0.707229\pi$
$608$	0	0
$609$	−0.855194	−0.0346542
$610$	0	0
$611$	37.3462	1.51087
$612$	0	0
$613$	30.6023	1.23601	0.618007	−	0.786173i	$-0.287940\pi$
$613$	30.6023	1.23601	0.618007	+	0.786173i	$0.287940\pi$
$614$	0	0
$615$	3.61361	0.145715
$616$	0	0
$617$	−2.47971	−0.0998292	−0.0499146	−	0.998753i	$-0.515895\pi$
$617$	−2.47971	−0.0998292	−0.0499146	+	0.998753i	$0.515895\pi$
$618$	0	0
$619$	−42.9839	−1.72767	−0.863835	−	0.503775i	$-0.831944\pi$
$619$	−42.9839	−1.72767	−0.863835	+	0.503775i	$0.831944\pi$
$620$	0	0
$621$	11.4852	0.460884
$622$	0	0
$623$	5.03429	0.201695
$624$	0	0
$625$	−14.6754	−0.587014
$626$	0	0
$627$	8.13902	0.325041
$628$	0	0
$629$	−14.3495	−0.572151
$630$	0	0
$631$	−12.6885	−0.505121	−0.252561	−	0.967581i	$-0.581273\pi$
$631$	−12.6885	−0.505121	−0.252561	+	0.967581i	$0.581273\pi$
$632$	0	0
$633$	−4.21165	−0.167398
$634$	0	0
$635$	−16.0709	−0.637756
$636$	0	0
$637$	33.1444	1.31323
$638$	0	0
$639$	−37.4909	−1.48312
$640$	0	0
$641$	−20.0042	−0.790117	−0.395059	−	0.918656i	$-0.629276\pi$
$641$	−20.0042	−0.790117	−0.395059	+	0.918656i	$0.629276\pi$
$642$	0	0
$643$	−10.6939	−0.421727	−0.210864	−	0.977516i	$-0.567628\pi$
$643$	−10.6939	−0.421727	−0.210864	+	0.977516i	$0.567628\pi$
$644$	0	0
$645$	8.81290	0.347007
$646$	0	0
$647$	12.4438	0.489215	0.244608	−	0.969622i	$-0.421341\pi$
$647$	12.4438	0.489215	0.244608	+	0.969622i	$0.421341\pi$
$648$	0	0
$649$	16.6881	0.655065
$650$	0	0
$651$	−2.72402	−0.106763
$652$	0	0
$653$	12.7924	0.500607	0.250304	−	0.968167i	$-0.419470\pi$
$653$	12.7924	0.500607	0.250304	+	0.968167i	$0.419470\pi$
$654$	0	0
$655$	7.90754	0.308973
$656$	0	0
$657$	−15.9569	−0.622537
$658$	0	0
$659$	−16.4856	−0.642188	−0.321094	−	0.947047i	$-0.604050\pi$
$659$	−16.4856	−0.642188	−0.321094	+	0.947047i	$0.604050\pi$
$660$	0	0
$661$	6.14716	0.239097	0.119548	−	0.992828i	$-0.461855\pi$
$661$	6.14716	0.239097	0.119548	+	0.992828i	$0.461855\pi$
$662$	0	0
$663$	−19.7636	−0.767556
$664$	0	0
$665$	8.68429	0.336762
$666$	0	0
$667$	5.53366	0.214264
$668$	0	0
$669$	2.85213	0.110270
$670$	0	0
$671$	47.4846	1.83312
$672$	0	0
$673$	−2.44712	−0.0943295	−0.0471648	−	0.998887i	$-0.515019\pi$
$673$	−2.44712	−0.0943295	−0.0471648	+	0.998887i	$0.515019\pi$
$674$	0	0
$675$	−4.54086	−0.174778
$676$	0	0
$677$	42.1453	1.61978	0.809888	−	0.586584i	$-0.199528\pi$
$677$	42.1453	1.61978	0.809888	+	0.586584i	$0.199528\pi$
$678$	0	0
$679$	1.76212	0.0676241
$680$	0	0
$681$	−7.35931	−0.282009
$682$	0	0
$683$	−6.19027	−0.236864	−0.118432	−	0.992962i	$-0.537787\pi$
$683$	−6.19027	−0.236864	−0.118432	+	0.992962i	$0.537787\pi$
$684$	0	0
$685$	−15.0094	−0.573479
$686$	0	0
$687$	−0.392596	−0.0149785
$688$	0	0
$689$	84.0228	3.20102
$690$	0	0
$691$	−2.03146	−0.0772804	−0.0386402	−	0.999253i	$-0.512303\pi$
$691$	−2.03146	−0.0772804	−0.0386402	+	0.999253i	$0.512303\pi$
$692$	0	0
$693$	−14.1411	−0.537175
$694$	0	0
$695$	−29.1602	−1.10611
$696$	0	0
$697$	−25.3477	−0.960113
$698$	0	0
$699$	−1.82399	−0.0689897
$700$	0	0
$701$	−6.49497	−0.245311	−0.122656	−	0.992449i	$-0.539141\pi$
$701$	−6.49497	−0.245311	−0.122656	+	0.992449i	$0.539141\pi$
$702$	0	0
$703$	−8.43621	−0.318178
$704$	0	0
$705$	−5.77950	−0.217668
$706$	0	0
$707$	−15.5717	−0.585636
$708$	0	0
$709$	−9.90116	−0.371846	−0.185923	−	0.982564i	$-0.559527\pi$
$709$	−9.90116	−0.371846	−0.185923	+	0.982564i	$0.559527\pi$
$710$	0	0
$711$	−3.35153	−0.125692
$712$	0	0
$713$	17.6262	0.660106
$714$	0	0
$715$	46.9607	1.75623
$716$	0	0
$717$	−4.61092	−0.172198
$718$	0	0
$719$	48.7629	1.81855	0.909275	−	0.416196i	$-0.136637\pi$
$719$	48.7629	1.81855	0.909275	+	0.416196i	$0.136637\pi$
$720$	0	0
$721$	−15.5435	−0.578871
$722$	0	0
$723$	2.55636	0.0950721
$724$	0	0
$725$	−2.18782	−0.0812537
$726$	0	0
$727$	−17.0112	−0.630909	−0.315455	−	0.948941i	$-0.602157\pi$
$727$	−17.0112	−0.630909	−0.315455	+	0.948941i	$0.602157\pi$
$728$	0	0
$729$	−14.2876	−0.529169
$730$	0	0
$731$	−61.8181	−2.28642
$732$	0	0
$733$	−48.5302	−1.79250	−0.896251	−	0.443547i	$-0.853720\pi$
$733$	−48.5302	−1.79250	−0.896251	+	0.443547i	$0.853720\pi$
$734$	0	0
$735$	−5.12925	−0.189195
$736$	0	0
$737$	10.4076	0.383368
$738$	0	0
$739$	−8.57201	−0.315327	−0.157663	−	0.987493i	$-0.550396\pi$
$739$	−8.57201	−0.315327	−0.157663	+	0.987493i	$0.550396\pi$
$740$	0	0
$741$	−11.6193	−0.426844
$742$	0	0
$743$	−15.9505	−0.585166	−0.292583	−	0.956240i	$-0.594515\pi$
$743$	−15.9505	−0.585166	−0.292583	+	0.956240i	$0.594515\pi$
$744$	0	0
$745$	−21.3506	−0.782227
$746$	0	0
$747$	1.24054	0.0453888
$748$	0	0
$749$	2.07722	0.0758999
$750$	0	0
$751$	48.1045	1.75536	0.877678	−	0.479250i	$-0.159091\pi$
$751$	48.1045	1.75536	0.877678	+	0.479250i	$0.159091\pi$
$752$	0	0
$753$	2.78258	0.101403
$754$	0	0
$755$	20.6591	0.751861
$756$	0	0
$757$	27.9865	1.01719	0.508593	−	0.861007i	$-0.330166\pi$
$757$	27.9865	1.01719	0.508593	+	0.861007i	$0.330166\pi$
$758$	0	0
$759$	−8.42299	−0.305735
$760$	0	0
$761$	53.1084	1.92518	0.962590	−	0.270963i	$-0.0873420\pi$
$761$	53.1084	1.92518	0.962590	+	0.270963i	$0.0873420\pi$
$762$	0	0
$763$	−11.1974	−0.405372
$764$	0	0
$765$	−33.2257	−1.20128
$766$	0	0
$767$	−23.8239	−0.860230
$768$	0	0
$769$	−1.67056	−0.0602418	−0.0301209	−	0.999546i	$-0.509589\pi$
$769$	−1.67056	−0.0602418	−0.0301209	+	0.999546i	$0.509589\pi$
$770$	0	0
$771$	−2.75646	−0.0992717
$772$	0	0
$773$	−23.7799	−0.855305	−0.427652	−	0.903943i	$-0.640659\pi$
$773$	−23.7799	−0.855305	−0.427652	+	0.903943i	$0.640659\pi$
$774$	0	0
$775$	−6.96881	−0.250327
$776$	0	0
$777$	−1.34925	−0.0484042
$778$	0	0
$779$	−14.9022	−0.533927
$780$	0	0
$781$	57.5211	2.05827
$782$	0	0
$783$	4.02429	0.143816
$784$	0	0
$785$	−36.7242	−1.31074
$786$	0	0
$787$	12.0846	0.430768	0.215384	−	0.976529i	$-0.430900\pi$
$787$	12.0846	0.430768	0.215384	+	0.976529i	$0.430900\pi$
$788$	0	0
$789$	−3.63606	−0.129447
$790$	0	0
$791$	−20.0603	−0.713262
$792$	0	0
$793$	−67.7889	−2.40725
$794$	0	0
$795$	−13.0029	−0.461166
$796$	0	0
$797$	−7.78088	−0.275613	−0.137806	−	0.990459i	$-0.544005\pi$
$797$	−7.78088	−0.275613	−0.137806	+	0.990459i	$0.544005\pi$
$798$	0	0
$799$	40.5403	1.43421
$800$	0	0
$801$	−11.3237	−0.400105
$802$	0	0
$803$	24.4821	0.863955
$804$	0	0
$805$	−8.98727	−0.316760
$806$	0	0
$807$	−0.885574	−0.0311737
$808$	0	0
$809$	−2.15102	−0.0756260	−0.0378130	−	0.999285i	$-0.512039\pi$
$809$	−2.15102	−0.0756260	−0.0378130	+	0.999285i	$0.512039\pi$
$810$	0	0
$811$	47.8624	1.68068	0.840339	−	0.542062i	$-0.182356\pi$
$811$	47.8624	1.68068	0.840339	+	0.542062i	$0.182356\pi$
$812$	0	0
$813$	−13.7237	−0.481310
$814$	0	0
$815$	−22.1014	−0.774178
$816$	0	0
$817$	−36.3435	−1.27150
$818$	0	0
$819$	20.1878	0.705418
$820$	0	0
$821$	−54.2698	−1.89403	−0.947014	−	0.321193i	$-0.895916\pi$
$821$	−54.2698	−1.89403	−0.947014	+	0.321193i	$0.895916\pi$
$822$	0	0
$823$	−11.8639	−0.413550	−0.206775	−	0.978389i	$-0.566297\pi$
$823$	−11.8639	−0.413550	−0.206775	+	0.978389i	$0.566297\pi$
$824$	0	0
$825$	3.33017	0.115942
$826$	0	0
$827$	9.86474	0.343031	0.171515	−	0.985181i	$-0.445134\pi$
$827$	9.86474	0.343031	0.171515	+	0.985181i	$0.445134\pi$
$828$	0	0
$829$	1.84772	0.0641739	0.0320870	−	0.999485i	$-0.489785\pi$
$829$	1.84772	0.0641739	0.0320870	+	0.999485i	$0.489785\pi$
$830$	0	0
$831$	−0.248806	−0.00863099
$832$	0	0
$833$	35.9791	1.24660
$834$	0	0
$835$	−2.15239	−0.0744865
$836$	0	0
$837$	12.8184	0.443070
$838$	0	0
$839$	6.86442	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$839$	6.86442	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$840$	0	0
$841$	−27.0611	−0.933140
$842$	0	0
$843$	15.8691	0.546562
$844$	0	0
$845$	−42.9688	−1.47817
$846$	0	0
$847$	8.26178	0.283878
$848$	0	0
$849$	5.41999	0.186014
$850$	0	0
$851$	8.73054	0.299279
$852$	0	0
$853$	−1.98820	−0.0680746	−0.0340373	−	0.999421i	$-0.510836\pi$
$853$	−1.98820	−0.0680746	−0.0340373	+	0.999421i	$0.510836\pi$
$854$	0	0
$855$	−19.5338	−0.668040
$856$	0	0
$857$	−23.6812	−0.808936	−0.404468	−	0.914552i	$-0.632543\pi$
$857$	−23.6812	−0.808936	−0.404468	+	0.914552i	$0.632543\pi$
$858$	0	0
$859$	1.65466	0.0564563	0.0282282	−	0.999602i	$-0.491014\pi$
$859$	1.65466	0.0564563	0.0282282	+	0.999602i	$0.491014\pi$
$860$	0	0
$861$	−2.38340	−0.0812260
$862$	0	0
$863$	3.33561	0.113545	0.0567727	−	0.998387i	$-0.481919\pi$
$863$	3.33561	0.113545	0.0567727	+	0.998387i	$0.481919\pi$
$864$	0	0
$865$	−22.0080	−0.748294
$866$	0	0
$867$	−12.9051	−0.438281
$868$	0	0
$869$	5.14214	0.174435
$870$	0	0
$871$	−14.8578	−0.503439
$872$	0	0
$873$	−3.96358	−0.134147
$874$	0	0
$875$	14.8608	0.502386
$876$	0	0
$877$	−13.3800	−0.451809	−0.225905	−	0.974149i	$-0.572534\pi$
$877$	−13.3800	−0.451809	−0.225905	+	0.974149i	$0.572534\pi$
$878$	0	0
$879$	−3.30835	−0.111588
$880$	0	0
$881$	−37.7165	−1.27070	−0.635351	−	0.772224i	$-0.719144\pi$
$881$	−37.7165	−1.27070	−0.635351	+	0.772224i	$0.719144\pi$
$882$	0	0
$883$	−46.3426	−1.55955	−0.779776	−	0.626059i	$-0.784667\pi$
$883$	−46.3426	−1.55955	−0.779776	+	0.626059i	$0.784667\pi$
$884$	0	0
$885$	3.68685	0.123932
$886$	0	0
$887$	−17.6527	−0.592721	−0.296360	−	0.955076i	$-0.595773\pi$
$887$	−17.6527	−0.592721	−0.296360	+	0.955076i	$0.595773\pi$
$888$	0	0
$889$	10.5998	0.355505
$890$	0	0
$891$	28.6103	0.958481
$892$	0	0
$893$	23.8341	0.797577
$894$	0	0
$895$	−31.5188	−1.05356
$896$	0	0
$897$	12.0246	0.401491
$898$	0	0
$899$	6.17603	0.205982
$900$	0	0
$901$	91.2090	3.03861
$902$	0	0
$903$	−5.81264	−0.193433
$904$	0	0
$905$	−10.0262	−0.333281
$906$	0	0
$907$	36.1702	1.20101	0.600506	−	0.799620i	$-0.294966\pi$
$907$	36.1702	1.20101	0.600506	+	0.799620i	$0.294966\pi$
$908$	0	0
$909$	35.0259	1.16173
$910$	0	0
$911$	37.4723	1.24151	0.620756	−	0.784004i	$-0.286826\pi$
$911$	37.4723	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$912$	0	0
$913$	−1.90331	−0.0629905
$914$	0	0
$915$	10.4906	0.346810
$916$	0	0
$917$	−5.21550	−0.172231
$918$	0	0
$919$	18.1712	0.599413	0.299707	−	0.954031i	$-0.403111\pi$
$919$	18.1712	0.599413	0.299707	+	0.954031i	$0.403111\pi$
$920$	0	0
$921$	15.9755	0.526411
$922$	0	0
$923$	−82.1170	−2.70292
$924$	0	0
$925$	−3.45177	−0.113493
$926$	0	0
$927$	34.9624	1.14832
$928$	0	0
$929$	0.291725	0.00957120	0.00478560	−	0.999989i	$-0.498477\pi$
$929$	0.291725	0.00957120	0.00478560	+	0.999989i	$0.498477\pi$
$930$	0	0
$931$	21.1525	0.693246
$932$	0	0
$933$	−16.8435	−0.551433
$934$	0	0
$935$	50.9771	1.66713
$936$	0	0
$937$	−23.8841	−0.780259	−0.390129	−	0.920760i	$-0.627570\pi$
$937$	−23.8841	−0.780259	−0.390129	+	0.920760i	$0.627570\pi$
$938$	0	0
$939$	9.83403	0.320921
$940$	0	0
$941$	8.95666	0.291979	0.145989	−	0.989286i	$-0.453363\pi$
$941$	8.95666	0.291979	0.145989	+	0.989286i	$0.453363\pi$
$942$	0	0
$943$	15.4221	0.502213
$944$	0	0
$945$	−6.53589	−0.212612
$946$	0	0
$947$	−18.5212	−0.601860	−0.300930	−	0.953646i	$-0.597297\pi$
$947$	−18.5212	−0.601860	−0.300930	+	0.953646i	$0.597297\pi$
$948$	0	0
$949$	−34.9506	−1.13455
$950$	0	0
$951$	−5.03902	−0.163401
$952$	0	0
$953$	32.3162	1.04682	0.523412	−	0.852080i	$-0.324659\pi$
$953$	32.3162	1.04682	0.523412	+	0.852080i	$0.324659\pi$
$954$	0	0
$955$	−22.3422	−0.722978
$956$	0	0
$957$	−2.95133	−0.0954029
$958$	0	0
$959$	9.89961	0.319675
$960$	0	0
$961$	−11.3277	−0.365409
$962$	0	0
$963$	−4.67233	−0.150564
$964$	0	0
$965$	23.9622	0.771370
$966$	0	0
$967$	−39.5321	−1.27127	−0.635633	−	0.771991i	$-0.719261\pi$
$967$	−39.5321	−1.27127	−0.635633	+	0.771991i	$0.719261\pi$
$968$	0	0
$969$	−12.6130	−0.405188
$970$	0	0
$971$	−33.5506	−1.07669	−0.538344	−	0.842725i	$-0.680950\pi$
$971$	−33.5506	−1.07669	−0.538344	+	0.842725i	$0.680950\pi$
$972$	0	0
$973$	19.2329	0.616580
$974$	0	0
$975$	−4.75414	−0.152254
$976$	0	0
$977$	36.9643	1.18259	0.591297	−	0.806454i	$-0.298616\pi$
$977$	36.9643	1.18259	0.591297	+	0.806454i	$0.298616\pi$
$978$	0	0
$979$	17.3736	0.555264
$980$	0	0
$981$	25.1865	0.804142
$982$	0	0
$983$	−0.678201	−0.0216313	−0.0108156	−	0.999942i	$-0.503443\pi$
$983$	−0.678201	−0.0216313	−0.0108156	+	0.999942i	$0.503443\pi$
$984$	0	0
$985$	−33.2171	−1.05839
$986$	0	0
$987$	3.81193	0.121335
$988$	0	0
$989$	37.6115	1.19598
$990$	0	0
$991$	0.821791	0.0261051	0.0130525	−	0.999915i	$-0.495845\pi$
$991$	0.821791	0.0261051	0.0130525	+	0.999915i	$0.495845\pi$
$992$	0	0
$993$	2.35059	0.0745937
$994$	0	0
$995$	−33.2945	−1.05551
$996$	0	0
$997$	−41.4500	−1.31273	−0.656367	−	0.754442i	$-0.727908\pi$
$997$	−41.4500	−1.31273	−0.656367	+	0.754442i	$0.727908\pi$
$998$	0	0
$999$	6.34919	0.200879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.14	✓	33
4.3	odd	2		8048.2.a.y.1.20		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.14	✓	33	1.1	even	1	trivial
8048.2.a.y.1.20		33	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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.502871 q^{3} -1.85170 q^{5} +1.22131 q^{7} -2.74712 q^{9} +O(q^{10})\)	\(q-0.502871 q^{3} -1.85170 q^{5} +1.22131 q^{7} -2.74712 q^{9} +4.21482 q^{11} -6.01706 q^{13} +0.931168 q^{15} -6.53168 q^{17} -3.84005 q^{19} -0.614162 q^{21} +3.97402 q^{23} -1.57120 q^{25} +2.89006 q^{27} +1.39246 q^{29} +4.43535 q^{31} -2.11951 q^{33} -2.26150 q^{35} +2.19690 q^{37} +3.02581 q^{39} +3.88073 q^{41} +9.46434 q^{43} +5.08685 q^{45} -6.20672 q^{47} -5.50840 q^{49} +3.28460 q^{51} -13.9641 q^{53} -7.80459 q^{55} +1.93105 q^{57} +3.95939 q^{59} +11.2661 q^{61} -3.35509 q^{63} +11.1418 q^{65} +2.46928 q^{67} -1.99842 q^{69} +13.6474 q^{71} +5.80859 q^{73} +0.790110 q^{75} +5.14760 q^{77} +1.22002 q^{79} +6.78803 q^{81} -0.451577 q^{83} +12.0947 q^{85} -0.700227 q^{87} +4.12204 q^{89} -7.34870 q^{91} -2.23041 q^{93} +7.11063 q^{95} +1.44281 q^{97} -11.5786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

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