LMFDB - Embedded newform 4024.2.a.e.1.6

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:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.34298 q^{3} -1.32410 q^{5} +0.439798 q^{7} +2.48955 q^{9} +O(q^{10})$	$q-2.34298 q^{3} -1.32410 q^{5} +0.439798 q^{7} +2.48955 q^{9} +1.15199 q^{11} -2.14274 q^{13} +3.10233 q^{15} +2.70230 q^{17} -1.78473 q^{19} -1.03044 q^{21} -2.60579 q^{23} -3.24677 q^{25} +1.19598 q^{27} +5.83751 q^{29} -1.27372 q^{31} -2.69908 q^{33} -0.582335 q^{35} -0.680971 q^{37} +5.02040 q^{39} +2.61489 q^{41} +2.38073 q^{43} -3.29640 q^{45} -5.77071 q^{47} -6.80658 q^{49} -6.33142 q^{51} +9.90638 q^{53} -1.52534 q^{55} +4.18158 q^{57} +5.85057 q^{59} +13.8393 q^{61} +1.09490 q^{63} +2.83719 q^{65} +6.48169 q^{67} +6.10530 q^{69} -1.07311 q^{71} +12.2928 q^{73} +7.60711 q^{75} +0.506641 q^{77} -16.6546 q^{79} -10.2708 q^{81} -10.2356 q^{83} -3.57810 q^{85} -13.6772 q^{87} -3.55197 q^{89} -0.942374 q^{91} +2.98430 q^{93} +2.36315 q^{95} +2.15081 q^{97} +2.86792 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−2.34298	−1.35272	−0.676360	−	0.736572i	$-0.736443\pi$
$3$	−2.34298	−1.35272	−0.676360	+	0.736572i	$0.736443\pi$
$4$	0	0
$5$	−1.32410	−0.592153	−0.296077	−	0.955164i	$-0.595678\pi$
$5$	−1.32410	−0.592153	−0.296077	+	0.955164i	$0.595678\pi$
$6$	0	0
$7$	0.439798	0.166228	0.0831140	−	0.996540i	$-0.473513\pi$
$7$	0.439798	0.166228	0.0831140	+	0.996540i	$0.473513\pi$
$8$	0	0
$9$	2.48955	0.829849
$10$	0	0
$11$	1.15199	0.347337	0.173668	−	0.984804i	$-0.444438\pi$
$11$	1.15199	0.347337	0.173668	+	0.984804i	$0.444438\pi$
$12$	0	0
$13$	−2.14274	−0.594289	−0.297145	−	0.954832i	$-0.596034\pi$
$13$	−2.14274	−0.594289	−0.297145	+	0.954832i	$0.596034\pi$
$14$	0	0
$15$	3.10233	0.801017
$16$	0	0
$17$	2.70230	0.655403	0.327702	−	0.944781i	$-0.393726\pi$
$17$	2.70230	0.655403	0.327702	+	0.944781i	$0.393726\pi$
$18$	0	0
$19$	−1.78473	−0.409445	−0.204722	−	0.978820i	$-0.565629\pi$
$19$	−1.78473	−0.409445	−0.204722	+	0.978820i	$0.565629\pi$
$20$	0	0
$21$	−1.03044	−0.224860
$22$	0	0
$23$	−2.60579	−0.543344	−0.271672	−	0.962390i	$-0.587577\pi$
$23$	−2.60579	−0.543344	−0.271672	+	0.962390i	$0.587577\pi$
$24$	0	0
$25$	−3.24677	−0.649354
$26$	0	0
$27$	1.19598	0.230167
$28$	0	0
$29$	5.83751	1.08400	0.541999	−	0.840379i	$-0.317668\pi$
$29$	5.83751	1.08400	0.541999	+	0.840379i	$0.317668\pi$
$30$	0	0
$31$	−1.27372	−0.228767	−0.114383	−	0.993437i	$-0.536489\pi$
$31$	−1.27372	−0.228767	−0.114383	+	0.993437i	$0.536489\pi$
$32$	0	0
$33$	−2.69908	−0.469849
$34$	0	0
$35$	−0.582335	−0.0984325
$36$	0	0
$37$	−0.680971	−0.111951	−0.0559755	−	0.998432i	$-0.517827\pi$
$37$	−0.680971	−0.111951	−0.0559755	+	0.998432i	$0.517827\pi$
$38$	0	0
$39$	5.02040	0.803907
$40$	0	0
$41$	2.61489	0.408378	0.204189	−	0.978932i	$-0.434544\pi$
$41$	2.61489	0.408378	0.204189	+	0.978932i	$0.434544\pi$
$42$	0	0
$43$	2.38073	0.363058	0.181529	−	0.983386i	$-0.441895\pi$
$43$	2.38073	0.363058	0.181529	+	0.983386i	$0.441895\pi$
$44$	0	0
$45$	−3.29640	−0.491398
$46$	0	0
$47$	−5.77071	−0.841745	−0.420873	−	0.907120i	$-0.638276\pi$
$47$	−5.77071	−0.841745	−0.420873	+	0.907120i	$0.638276\pi$
$48$	0	0
$49$	−6.80658	−0.972368
$50$	0	0
$51$	−6.33142	−0.886576
$52$	0	0
$53$	9.90638	1.36075	0.680373	−	0.732866i	$-0.261818\pi$
$53$	9.90638	1.36075	0.680373	+	0.732866i	$0.261818\pi$
$54$	0	0
$55$	−1.52534	−0.205677
$56$	0	0
$57$	4.18158	0.553864
$58$	0	0
$59$	5.85057	0.761679	0.380840	−	0.924641i	$-0.375635\pi$
$59$	5.85057	0.761679	0.380840	+	0.924641i	$0.375635\pi$
$60$	0	0
$61$	13.8393	1.77194	0.885968	−	0.463746i	$-0.153495\pi$
$61$	13.8393	1.77194	0.885968	+	0.463746i	$0.153495\pi$
$62$	0	0
$63$	1.09490	0.137944
$64$	0	0
$65$	2.83719	0.351911
$66$	0	0
$67$	6.48169	0.791864	0.395932	−	0.918280i	$-0.370422\pi$
$67$	6.48169	0.791864	0.395932	+	0.918280i	$0.370422\pi$
$68$	0	0
$69$	6.10530	0.734992
$70$	0	0
$71$	−1.07311	−0.127354	−0.0636771	−	0.997971i	$-0.520283\pi$
$71$	−1.07311	−0.127354	−0.0636771	+	0.997971i	$0.520283\pi$
$72$	0	0
$73$	12.2928	1.43877	0.719384	−	0.694613i	$-0.244424\pi$
$73$	12.2928	1.43877	0.719384	+	0.694613i	$0.244424\pi$
$74$	0	0
$75$	7.60711	0.878394
$76$	0	0
$77$	0.506641	0.0577371
$78$	0	0
$79$	−16.6546	−1.87379	−0.936893	−	0.349617i	$-0.886312\pi$
$79$	−16.6546	−1.87379	−0.936893	+	0.349617i	$0.886312\pi$
$80$	0	0
$81$	−10.2708	−1.14120
$82$	0	0
$83$	−10.2356	−1.12350	−0.561751	−	0.827306i	$-0.689872\pi$
$83$	−10.2356	−1.12350	−0.561751	+	0.827306i	$0.689872\pi$
$84$	0	0
$85$	−3.57810	−0.388099
$86$	0	0
$87$	−13.6772	−1.46635
$88$	0	0
$89$	−3.55197	−0.376508	−0.188254	−	0.982120i	$-0.560283\pi$
$89$	−3.55197	−0.376508	−0.188254	+	0.982120i	$0.560283\pi$
$90$	0	0
$91$	−0.942374	−0.0987876
$92$	0	0
$93$	2.98430	0.309457
$94$	0	0
$95$	2.36315	0.242454
$96$	0	0
$97$	2.15081	0.218381	0.109191	−	0.994021i	$-0.465174\pi$
$97$	2.15081	0.218381	0.109191	+	0.994021i	$0.465174\pi$
$98$	0	0
$99$	2.86792	0.288237
$100$	0	0
$101$	0.867595	0.0863289	0.0431645	−	0.999068i	$-0.486256\pi$
$101$	0.867595	0.0863289	0.0431645	+	0.999068i	$0.486256\pi$
$102$	0	0
$103$	5.85049	0.576466	0.288233	−	0.957560i	$-0.406932\pi$
$103$	5.85049	0.576466	0.288233	+	0.957560i	$0.406932\pi$
$104$	0	0
$105$	1.36440	0.133152
$106$	0	0
$107$	4.98347	0.481771	0.240885	−	0.970554i	$-0.422562\pi$
$107$	4.98347	0.481771	0.240885	+	0.970554i	$0.422562\pi$
$108$	0	0
$109$	−1.13105	−0.108335	−0.0541675	−	0.998532i	$-0.517250\pi$
$109$	−1.13105	−0.108335	−0.0541675	+	0.998532i	$0.517250\pi$
$110$	0	0
$111$	1.59550	0.151438
$112$	0	0
$113$	−6.06921	−0.570943	−0.285471	−	0.958387i	$-0.592150\pi$
$113$	−6.06921	−0.570943	−0.285471	+	0.958387i	$0.592150\pi$
$114$	0	0
$115$	3.45031	0.321743
$116$	0	0
$117$	−5.33445	−0.493170
$118$	0	0
$119$	1.18846	0.108946
$120$	0	0
$121$	−9.67293	−0.879357
$122$	0	0
$123$	−6.12663	−0.552420
$124$	0	0
$125$	10.9195	0.976671
$126$	0	0
$127$	−3.61352	−0.320648	−0.160324	−	0.987064i	$-0.551254\pi$
$127$	−3.61352	−0.320648	−0.160324	+	0.987064i	$0.551254\pi$
$128$	0	0
$129$	−5.57800	−0.491116
$130$	0	0
$131$	−6.98332	−0.610135	−0.305068	−	0.952331i	$-0.598679\pi$
$131$	−6.98332	−0.610135	−0.305068	+	0.952331i	$0.598679\pi$
$132$	0	0
$133$	−0.784920	−0.0680612
$134$	0	0
$135$	−1.58359	−0.136294
$136$	0	0
$137$	2.85893	0.244255	0.122127	−	0.992514i	$-0.461028\pi$
$137$	2.85893	0.244255	0.122127	+	0.992514i	$0.461028\pi$
$138$	0	0
$139$	−13.7945	−1.17003	−0.585016	−	0.811022i	$-0.698912\pi$
$139$	−13.7945	−1.17003	−0.585016	+	0.811022i	$0.698912\pi$
$140$	0	0
$141$	13.5207	1.13864
$142$	0	0
$143$	−2.46841	−0.206419
$144$	0	0
$145$	−7.72942	−0.641894
$146$	0	0
$147$	15.9477	1.31534
$148$	0	0
$149$	15.1802	1.24361	0.621804	−	0.783173i	$-0.286400\pi$
$149$	15.1802	1.24361	0.621804	+	0.783173i	$0.286400\pi$
$150$	0	0
$151$	−10.2141	−0.831210	−0.415605	−	0.909545i	$-0.636430\pi$
$151$	−10.2141	−0.831210	−0.415605	+	0.909545i	$0.636430\pi$
$152$	0	0
$153$	6.72749	0.543886
$154$	0	0
$155$	1.68653	0.135465
$156$	0	0
$157$	21.6932	1.73131	0.865655	−	0.500642i	$-0.166903\pi$
$157$	21.6932	1.73131	0.865655	+	0.500642i	$0.166903\pi$
$158$	0	0
$159$	−23.2104	−1.84071
$160$	0	0
$161$	−1.14602	−0.0903191
$162$	0	0
$163$	−11.6082	−0.909223	−0.454612	−	0.890690i	$-0.650222\pi$
$163$	−11.6082	−0.909223	−0.454612	+	0.890690i	$0.650222\pi$
$164$	0	0
$165$	3.57384	0.278223
$166$	0	0
$167$	−13.9621	−1.08042	−0.540208	−	0.841531i	$-0.681655\pi$
$167$	−13.9621	−1.08042	−0.540208	+	0.841531i	$0.681655\pi$
$168$	0	0
$169$	−8.40866	−0.646820
$170$	0	0
$171$	−4.44316	−0.339777
$172$	0	0
$173$	−8.84542	−0.672505	−0.336252	−	0.941772i	$-0.609160\pi$
$173$	−8.84542	−0.672505	−0.336252	+	0.941772i	$0.609160\pi$
$174$	0	0
$175$	−1.42792	−0.107941
$176$	0	0
$177$	−13.7078	−1.03034
$178$	0	0
$179$	−20.5436	−1.53550	−0.767749	−	0.640750i	$-0.778623\pi$
$179$	−20.5436	−1.53550	−0.767749	+	0.640750i	$0.778623\pi$
$180$	0	0
$181$	−23.0697	−1.71476	−0.857378	−	0.514687i	$-0.827908\pi$
$181$	−23.0697	−1.71476	−0.857378	+	0.514687i	$0.827908\pi$
$182$	0	0
$183$	−32.4251	−2.39693
$184$	0	0
$185$	0.901671	0.0662922
$186$	0	0
$187$	3.11301	0.227646
$188$	0	0
$189$	0.525990	0.0382602
$190$	0	0
$191$	−20.8453	−1.50831	−0.754156	−	0.656696i	$-0.771954\pi$
$191$	−20.8453	−1.50831	−0.754156	+	0.656696i	$0.771954\pi$
$192$	0	0
$193$	−4.54755	−0.327339	−0.163670	−	0.986515i	$-0.552333\pi$
$193$	−4.54755	−0.327339	−0.163670	+	0.986515i	$0.552333\pi$
$194$	0	0
$195$	−6.64748	−0.476036
$196$	0	0
$197$	−8.80529	−0.627351	−0.313675	−	0.949530i	$-0.601560\pi$
$197$	−8.80529	−0.627351	−0.313675	+	0.949530i	$0.601560\pi$
$198$	0	0
$199$	20.3591	1.44322	0.721608	−	0.692302i	$-0.243403\pi$
$199$	20.3591	1.44322	0.721608	+	0.692302i	$0.243403\pi$
$200$	0	0
$201$	−15.1865	−1.07117
$202$	0	0
$203$	2.56733	0.180191
$204$	0	0
$205$	−3.46237	−0.241822
$206$	0	0
$207$	−6.48723	−0.450894
$208$	0	0
$209$	−2.05598	−0.142215
$210$	0	0
$211$	5.02993	0.346275	0.173137	−	0.984898i	$-0.444610\pi$
$211$	5.02993	0.346275	0.173137	+	0.984898i	$0.444610\pi$
$212$	0	0
$213$	2.51426	0.172275
$214$	0	0
$215$	−3.15232	−0.214986
$216$	0	0
$217$	−0.560179	−0.0380274
$218$	0	0
$219$	−28.8018	−1.94625
$220$	0	0
$221$	−5.79032	−0.389499
$222$	0	0
$223$	16.7399	1.12099	0.560494	−	0.828158i	$-0.310611\pi$
$223$	16.7399	1.12099	0.560494	+	0.828158i	$0.310611\pi$
$224$	0	0
$225$	−8.08299	−0.538866
$226$	0	0
$227$	−3.67014	−0.243595	−0.121798	−	0.992555i	$-0.538866\pi$
$227$	−3.67014	−0.243595	−0.121798	+	0.992555i	$0.538866\pi$
$228$	0	0
$229$	23.6405	1.56221	0.781104	−	0.624401i	$-0.214657\pi$
$229$	23.6405	1.56221	0.781104	+	0.624401i	$0.214657\pi$
$230$	0	0
$231$	−1.18705	−0.0781021
$232$	0	0
$233$	−13.6307	−0.892979	−0.446490	−	0.894789i	$-0.647326\pi$
$233$	−13.6307	−0.892979	−0.446490	+	0.894789i	$0.647326\pi$
$234$	0	0
$235$	7.64098	0.498442
$236$	0	0
$237$	39.0213	2.53471
$238$	0	0
$239$	6.18760	0.400242	0.200121	−	0.979771i	$-0.435866\pi$
$239$	6.18760	0.400242	0.200121	+	0.979771i	$0.435866\pi$
$240$	0	0
$241$	0.316186	0.0203673	0.0101837	−	0.999948i	$-0.496758\pi$
$241$	0.316186	0.0203673	0.0101837	+	0.999948i	$0.496758\pi$
$242$	0	0
$243$	20.4763	1.31356
$244$	0	0
$245$	9.01256	0.575791
$246$	0	0
$247$	3.82421	0.243329
$248$	0	0
$249$	23.9818	1.51978
$250$	0	0
$251$	−24.1829	−1.52641	−0.763205	−	0.646156i	$-0.776376\pi$
$251$	−24.1829	−1.52641	−0.763205	+	0.646156i	$0.776376\pi$
$252$	0	0
$253$	−3.00183	−0.188724
$254$	0	0
$255$	8.38341	0.524989
$256$	0	0
$257$	4.20310	0.262182	0.131091	−	0.991370i	$-0.458152\pi$
$257$	4.20310	0.262182	0.131091	+	0.991370i	$0.458152\pi$
$258$	0	0
$259$	−0.299490	−0.0186094
$260$	0	0
$261$	14.5328	0.899555
$262$	0	0
$263$	−27.2114	−1.67793	−0.838963	−	0.544188i	$-0.816838\pi$
$263$	−27.2114	−1.67793	−0.838963	+	0.544188i	$0.816838\pi$
$264$	0	0
$265$	−13.1170	−0.805771
$266$	0	0
$267$	8.32218	0.509309
$268$	0	0
$269$	−21.5556	−1.31427	−0.657134	−	0.753774i	$-0.728231\pi$
$269$	−21.5556	−1.31427	−0.657134	+	0.753774i	$0.728231\pi$
$270$	0	0
$271$	−23.8933	−1.45141	−0.725706	−	0.688005i	$-0.758487\pi$
$271$	−23.8933	−1.45141	−0.725706	+	0.688005i	$0.758487\pi$
$272$	0	0
$273$	2.20796	0.133632
$274$	0	0
$275$	−3.74024	−0.225545
$276$	0	0
$277$	−9.08269	−0.545726	−0.272863	−	0.962053i	$-0.587971\pi$
$277$	−9.08269	−0.545726	−0.272863	+	0.962053i	$0.587971\pi$
$278$	0	0
$279$	−3.17098	−0.189842
$280$	0	0
$281$	−18.4384	−1.09994	−0.549971	−	0.835183i	$-0.685361\pi$
$281$	−18.4384	−1.09994	−0.549971	+	0.835183i	$0.685361\pi$
$282$	0	0
$283$	−12.3479	−0.734005	−0.367003	−	0.930220i	$-0.619616\pi$
$283$	−12.3479	−0.734005	−0.367003	+	0.930220i	$0.619616\pi$
$284$	0	0
$285$	−5.53681	−0.327972
$286$	0	0
$287$	1.15002	0.0678838
$288$	0	0
$289$	−9.69759	−0.570447
$290$	0	0
$291$	−5.03929	−0.295409
$292$	0	0
$293$	−28.3276	−1.65492	−0.827458	−	0.561528i	$-0.810214\pi$
$293$	−28.3276	−1.65492	−0.827458	+	0.561528i	$0.810214\pi$
$294$	0	0
$295$	−7.74671	−0.451031
$296$	0	0
$297$	1.37775	0.0799454
$298$	0	0
$299$	5.58353	0.322904
$300$	0	0
$301$	1.04704	0.0603505
$302$	0	0
$303$	−2.03276	−0.116779
$304$	0	0
$305$	−18.3245	−1.04926
$306$	0	0
$307$	−11.8306	−0.675210	−0.337605	−	0.941288i	$-0.609617\pi$
$307$	−11.8306	−0.675210	−0.337605	+	0.941288i	$0.609617\pi$
$308$	0	0
$309$	−13.7076	−0.779796
$310$	0	0
$311$	7.50455	0.425544	0.212772	−	0.977102i	$-0.431751\pi$
$311$	7.50455	0.425544	0.212772	+	0.977102i	$0.431751\pi$
$312$	0	0
$313$	25.5133	1.44210	0.721050	−	0.692884i	$-0.243660\pi$
$313$	25.5133	1.44210	0.721050	+	0.692884i	$0.243660\pi$
$314$	0	0
$315$	−1.44975	−0.0816841
$316$	0	0
$317$	1.54130	0.0865682	0.0432841	−	0.999063i	$-0.486218\pi$
$317$	1.54130	0.0865682	0.0432841	+	0.999063i	$0.486218\pi$
$318$	0	0
$319$	6.72473	0.376513
$320$	0	0
$321$	−11.6762	−0.651700
$322$	0	0
$323$	−4.82286	−0.268351
$324$	0	0
$325$	6.95699	0.385904
$326$	0	0
$327$	2.65003	0.146547
$328$	0	0
$329$	−2.53795	−0.139922
$330$	0	0
$331$	−16.5162	−0.907811	−0.453905	−	0.891050i	$-0.649970\pi$
$331$	−16.5162	−0.907811	−0.453905	+	0.891050i	$0.649970\pi$
$332$	0	0
$333$	−1.69531	−0.0929024
$334$	0	0
$335$	−8.58238	−0.468905
$336$	0	0
$337$	19.2247	1.04724	0.523619	−	0.851953i	$-0.324582\pi$
$337$	19.2247	1.04724	0.523619	+	0.851953i	$0.324582\pi$
$338$	0	0
$339$	14.2200	0.772325
$340$	0	0
$341$	−1.46731	−0.0794591
$342$	0	0
$343$	−6.07211	−0.327863
$344$	0	0
$345$	−8.08401	−0.435228
$346$	0	0
$347$	14.3478	0.770230	0.385115	−	0.922869i	$-0.374162\pi$
$347$	14.3478	0.770230	0.385115	+	0.922869i	$0.374162\pi$
$348$	0	0
$349$	33.0799	1.77073	0.885364	−	0.464899i	$-0.153910\pi$
$349$	33.0799	1.77073	0.885364	+	0.464899i	$0.153910\pi$
$350$	0	0
$351$	−2.56268	−0.136786
$352$	0	0
$353$	−8.26383	−0.439839	−0.219920	−	0.975518i	$-0.570580\pi$
$353$	−8.26383	−0.439839	−0.219920	+	0.975518i	$0.570580\pi$
$354$	0	0
$355$	1.42089	0.0754133
$356$	0	0
$357$	−2.78455	−0.147374
$358$	0	0
$359$	8.13647	0.429426	0.214713	−	0.976677i	$-0.431118\pi$
$359$	8.13647	0.429426	0.214713	+	0.976677i	$0.431118\pi$
$360$	0	0
$361$	−15.8147	−0.832355
$362$	0	0
$363$	22.6635	1.18952
$364$	0	0
$365$	−16.2769	−0.851971
$366$	0	0
$367$	−24.1130	−1.25869	−0.629343	−	0.777127i	$-0.716676\pi$
$367$	−24.1130	−1.25869	−0.629343	+	0.777127i	$0.716676\pi$
$368$	0	0
$369$	6.50990	0.338892
$370$	0	0
$371$	4.35681	0.226194
$372$	0	0
$373$	−1.12754	−0.0583819	−0.0291909	−	0.999574i	$-0.509293\pi$
$373$	−1.12754	−0.0583819	−0.0291909	+	0.999574i	$0.509293\pi$
$374$	0	0
$375$	−25.5842	−1.32116
$376$	0	0
$377$	−12.5083	−0.644209
$378$	0	0
$379$	−0.862115	−0.0442839	−0.0221419	−	0.999755i	$-0.507049\pi$
$379$	−0.862115	−0.0442839	−0.0221419	+	0.999755i	$0.507049\pi$
$380$	0	0
$381$	8.46639	0.433746
$382$	0	0
$383$	3.40953	0.174219	0.0871094	−	0.996199i	$-0.472237\pi$
$383$	3.40953	0.174219	0.0871094	+	0.996199i	$0.472237\pi$
$384$	0	0
$385$	−0.670842	−0.0341892
$386$	0	0
$387$	5.92694	0.301284
$388$	0	0
$389$	−7.10129	−0.360050	−0.180025	−	0.983662i	$-0.557618\pi$
$389$	−7.10129	−0.360050	−0.180025	+	0.983662i	$0.557618\pi$
$390$	0	0
$391$	−7.04161	−0.356110
$392$	0	0
$393$	16.3618	0.825341
$394$	0	0
$395$	22.0522	1.10957
$396$	0	0
$397$	−0.193447	−0.00970883	−0.00485442	−	0.999988i	$-0.501545\pi$
$397$	−0.193447	−0.00970883	−0.00485442	+	0.999988i	$0.501545\pi$
$398$	0	0
$399$	1.83905	0.0920677
$400$	0	0
$401$	20.8310	1.04025	0.520124	−	0.854090i	$-0.325886\pi$
$401$	20.8310	1.04025	0.520124	+	0.854090i	$0.325886\pi$
$402$	0	0
$403$	2.72925	0.135954
$404$	0	0
$405$	13.5995	0.675765
$406$	0	0
$407$	−0.784470	−0.0388847
$408$	0	0
$409$	9.64723	0.477025	0.238512	−	0.971139i	$-0.423340\pi$
$409$	9.64723	0.477025	0.238512	+	0.971139i	$0.423340\pi$
$410$	0	0
$411$	−6.69841	−0.330408
$412$	0	0
$413$	2.57307	0.126612
$414$	0	0
$415$	13.5529	0.665286
$416$	0	0
$417$	32.3202	1.58273
$418$	0	0
$419$	−9.97715	−0.487416	−0.243708	−	0.969849i	$-0.578364\pi$
$419$	−9.97715	−0.487416	−0.243708	+	0.969849i	$0.578364\pi$
$420$	0	0
$421$	−35.4611	−1.72827	−0.864135	−	0.503260i	$-0.832134\pi$
$421$	−35.4611	−1.72827	−0.864135	+	0.503260i	$0.832134\pi$
$422$	0	0
$423$	−14.3665	−0.698521
$424$	0	0
$425$	−8.77374	−0.425589
$426$	0	0
$427$	6.08648	0.294546
$428$	0	0
$429$	5.78343	0.279226
$430$	0	0
$431$	10.9210	0.526047	0.263024	−	0.964789i	$-0.415280\pi$
$431$	10.9210	0.526047	0.263024	+	0.964789i	$0.415280\pi$
$432$	0	0
$433$	16.3195	0.784265	0.392132	−	0.919909i	$-0.371737\pi$
$433$	16.3195	0.784265	0.392132	+	0.919909i	$0.371737\pi$
$434$	0	0
$435$	18.1099	0.868302
$436$	0	0
$437$	4.65062	0.222469
$438$	0	0
$439$	−24.5668	−1.17251	−0.586254	−	0.810128i	$-0.699398\pi$
$439$	−24.5668	−1.17251	−0.586254	+	0.810128i	$0.699398\pi$
$440$	0	0
$441$	−16.9453	−0.806919
$442$	0	0
$443$	−24.1030	−1.14517	−0.572583	−	0.819847i	$-0.694059\pi$
$443$	−24.1030	−1.14517	−0.572583	+	0.819847i	$0.694059\pi$
$444$	0	0
$445$	4.70314	0.222950
$446$	0	0
$447$	−35.5668	−1.68225
$448$	0	0
$449$	−17.4763	−0.824755	−0.412378	−	0.911013i	$-0.635302\pi$
$449$	−17.4763	−0.824755	−0.412378	+	0.911013i	$0.635302\pi$
$450$	0	0
$451$	3.01232	0.141845
$452$	0	0
$453$	23.9314	1.12439
$454$	0	0
$455$	1.24779	0.0584974
$456$	0	0
$457$	0.0764875	0.00357793	0.00178897	−	0.999998i	$-0.499431\pi$
$457$	0.0764875	0.00357793	0.00178897	+	0.999998i	$0.499431\pi$
$458$	0	0
$459$	3.23189	0.150852
$460$	0	0
$461$	5.25610	0.244801	0.122400	−	0.992481i	$-0.460941\pi$
$461$	5.25610	0.244801	0.122400	+	0.992481i	$0.460941\pi$
$462$	0	0
$463$	15.4720	0.719044	0.359522	−	0.933137i	$-0.382940\pi$
$463$	15.4720	0.719044	0.359522	+	0.933137i	$0.382940\pi$
$464$	0	0
$465$	−3.95149	−0.183246
$466$	0	0
$467$	13.4123	0.620648	0.310324	−	0.950631i	$-0.399562\pi$
$467$	13.4123	0.620648	0.310324	+	0.950631i	$0.399562\pi$
$468$	0	0
$469$	2.85064	0.131630
$470$	0	0
$471$	−50.8268	−2.34197
$472$	0	0
$473$	2.74257	0.126104
$474$	0	0
$475$	5.79460	0.265875
$476$	0	0
$477$	24.6624	1.12921
$478$	0	0
$479$	22.8209	1.04271	0.521356	−	0.853339i	$-0.325426\pi$
$479$	22.8209	1.04271	0.521356	+	0.853339i	$0.325426\pi$
$480$	0	0
$481$	1.45915	0.0665313
$482$	0	0
$483$	2.68510	0.122176
$484$	0	0
$485$	−2.84787	−0.129315
$486$	0	0
$487$	26.1863	1.18661	0.593307	−	0.804976i	$-0.297822\pi$
$487$	26.1863	1.18661	0.593307	+	0.804976i	$0.297822\pi$
$488$	0	0
$489$	27.1977	1.22992
$490$	0	0
$491$	−3.67100	−0.165670	−0.0828350	−	0.996563i	$-0.526397\pi$
$491$	−3.67100	−0.165670	−0.0828350	+	0.996563i	$0.526397\pi$
$492$	0	0
$493$	15.7747	0.710456
$494$	0	0
$495$	−3.79740	−0.170681
$496$	0	0
$497$	−0.471950	−0.0211698
$498$	0	0
$499$	28.6220	1.28130	0.640648	−	0.767834i	$-0.278666\pi$
$499$	28.6220	1.28130	0.640648	+	0.767834i	$0.278666\pi$
$500$	0	0
$501$	32.7128	1.46150
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−1.14878	−0.0511200
$506$	0	0
$507$	19.7013	0.874966
$508$	0	0
$509$	−15.8949	−0.704527	−0.352264	−	0.935901i	$-0.614588\pi$
$509$	−15.8949	−0.704527	−0.352264	+	0.935901i	$0.614588\pi$
$510$	0	0
$511$	5.40636	0.239163
$512$	0	0
$513$	−2.13450	−0.0942405
$514$	0	0
$515$	−7.74660	−0.341356
$516$	0	0
$517$	−6.64778	−0.292369
$518$	0	0
$519$	20.7246	0.909710
$520$	0	0
$521$	−32.7939	−1.43673	−0.718363	−	0.695669i	$-0.755108\pi$
$521$	−32.7939	−1.43673	−0.718363	+	0.695669i	$0.755108\pi$
$522$	0	0
$523$	4.46109	0.195070	0.0975349	−	0.995232i	$-0.468904\pi$
$523$	4.46109	0.195070	0.0975349	+	0.995232i	$0.468904\pi$
$524$	0	0
$525$	3.34559	0.146014
$526$	0	0
$527$	−3.44197	−0.149934
$528$	0	0
$529$	−16.2099	−0.704777
$530$	0	0
$531$	14.5653	0.632079
$532$	0	0
$533$	−5.60304	−0.242694
$534$	0	0
$535$	−6.59859	−0.285282
$536$	0	0
$537$	48.1332	2.07710
$538$	0	0
$539$	−7.84108	−0.337739
$540$	0	0
$541$	3.61782	0.155542	0.0777712	−	0.996971i	$-0.475220\pi$
$541$	3.61782	0.155542	0.0777712	+	0.996971i	$0.475220\pi$
$542$	0	0
$543$	54.0518	2.31958
$544$	0	0
$545$	1.49762	0.0641509
$546$	0	0
$547$	−16.9964	−0.726713	−0.363357	−	0.931650i	$-0.618369\pi$
$547$	−16.9964	−0.726713	−0.363357	+	0.931650i	$0.618369\pi$
$548$	0	0
$549$	34.4535	1.47044
$550$	0	0
$551$	−10.4184	−0.443837
$552$	0	0
$553$	−7.32465	−0.311476
$554$	0	0
$555$	−2.11260	−0.0896747
$556$	0	0
$557$	−24.6624	−1.04498	−0.522490	−	0.852645i	$-0.674997\pi$
$557$	−24.6624	−1.04498	−0.522490	+	0.852645i	$0.674997\pi$
$558$	0	0
$559$	−5.10129	−0.215762
$560$	0	0
$561$	−7.29371	−0.307941
$562$	0	0
$563$	−25.7543	−1.08541	−0.542707	−	0.839922i	$-0.682601\pi$
$563$	−25.7543	−1.08541	−0.542707	+	0.839922i	$0.682601\pi$
$564$	0	0
$565$	8.03621	0.338086
$566$	0	0
$567$	−4.51708	−0.189699
$568$	0	0
$569$	−13.8887	−0.582246	−0.291123	−	0.956686i	$-0.594029\pi$
$569$	−13.8887	−0.582246	−0.291123	+	0.956686i	$0.594029\pi$
$570$	0	0
$571$	−3.37988	−0.141444	−0.0707218	−	0.997496i	$-0.522530\pi$
$571$	−3.37988	−0.141444	−0.0707218	+	0.997496i	$0.522530\pi$
$572$	0	0
$573$	48.8400	2.04032
$574$	0	0
$575$	8.46040	0.352823
$576$	0	0
$577$	6.88935	0.286807	0.143404	−	0.989664i	$-0.454195\pi$
$577$	6.88935	0.286807	0.143404	+	0.989664i	$0.454195\pi$
$578$	0	0
$579$	10.6548	0.442798
$580$	0	0
$581$	−4.50159	−0.186758
$582$	0	0
$583$	11.4120	0.472637
$584$	0	0
$585$	7.06333	0.292033
$586$	0	0
$587$	15.0844	0.622598	0.311299	−	0.950312i	$-0.399236\pi$
$587$	15.0844	0.622598	0.311299	+	0.950312i	$0.399236\pi$
$588$	0	0
$589$	2.27324	0.0936673
$590$	0	0
$591$	20.6306	0.848629
$592$	0	0
$593$	−42.4931	−1.74498	−0.872490	−	0.488631i	$-0.837496\pi$
$593$	−42.4931	−1.74498	−0.872490	+	0.488631i	$0.837496\pi$
$594$	0	0
$595$	−1.57364	−0.0645130
$596$	0	0
$597$	−47.7009	−1.95227
$598$	0	0
$599$	27.6633	1.13029	0.565145	−	0.824992i	$-0.308820\pi$
$599$	27.6633	1.13029	0.565145	+	0.824992i	$0.308820\pi$
$600$	0	0
$601$	43.1790	1.76131	0.880654	−	0.473759i	$-0.157103\pi$
$601$	43.1790	1.76131	0.880654	+	0.473759i	$0.157103\pi$
$602$	0	0
$603$	16.1365	0.657128
$604$	0	0
$605$	12.8079	0.520714
$606$	0	0
$607$	21.4395	0.870204	0.435102	−	0.900381i	$-0.356712\pi$
$607$	21.4395	0.870204	0.435102	+	0.900381i	$0.356712\pi$
$608$	0	0
$609$	−6.01519	−0.243748
$610$	0	0
$611$	12.3651	0.500240
$612$	0	0
$613$	8.55194	0.345410	0.172705	−	0.984974i	$-0.444749\pi$
$613$	8.55194	0.345410	0.172705	+	0.984974i	$0.444749\pi$
$614$	0	0
$615$	8.11225	0.327117
$616$	0	0
$617$	−4.60056	−0.185212	−0.0926058	−	0.995703i	$-0.529520\pi$
$617$	−4.60056	−0.185212	−0.0926058	+	0.995703i	$0.529520\pi$
$618$	0	0
$619$	21.8111	0.876661	0.438331	−	0.898814i	$-0.355570\pi$
$619$	21.8111	0.876661	0.438331	+	0.898814i	$0.355570\pi$
$620$	0	0
$621$	−3.11647	−0.125060
$622$	0	0
$623$	−1.56215	−0.0625861
$624$	0	0
$625$	1.77538	0.0710152
$626$	0	0
$627$	4.81712	0.192377
$628$	0	0
$629$	−1.84019	−0.0733730
$630$	0	0
$631$	−45.4773	−1.81042	−0.905211	−	0.424963i	$-0.860287\pi$
$631$	−45.4773	−1.81042	−0.905211	+	0.424963i	$0.860287\pi$
$632$	0	0
$633$	−11.7850	−0.468412
$634$	0	0
$635$	4.78464	0.189873
$636$	0	0
$637$	14.5847	0.577868
$638$	0	0
$639$	−2.67155	−0.105685
$640$	0	0
$641$	38.7681	1.53125	0.765623	−	0.643289i	$-0.222431\pi$
$641$	38.7681	1.53125	0.765623	+	0.643289i	$0.222431\pi$
$642$	0	0
$643$	−34.7667	−1.37107	−0.685533	−	0.728042i	$-0.740431\pi$
$643$	−34.7667	−1.37107	−0.685533	+	0.728042i	$0.740431\pi$
$644$	0	0
$645$	7.38581	0.290816
$646$	0	0
$647$	40.4560	1.59049	0.795245	−	0.606288i	$-0.207342\pi$
$647$	40.4560	1.59049	0.795245	+	0.606288i	$0.207342\pi$
$648$	0	0
$649$	6.73977	0.264559
$650$	0	0
$651$	1.31249	0.0514404
$652$	0	0
$653$	5.60455	0.219323	0.109661	−	0.993969i	$-0.465023\pi$
$653$	5.60455	0.219323	0.109661	+	0.993969i	$0.465023\pi$
$654$	0	0
$655$	9.24658	0.361294
$656$	0	0
$657$	30.6036	1.19396
$658$	0	0
$659$	−4.74426	−0.184810	−0.0924051	−	0.995721i	$-0.529455\pi$
$659$	−4.74426	−0.184810	−0.0924051	+	0.995721i	$0.529455\pi$
$660$	0	0
$661$	−16.2235	−0.631021	−0.315511	−	0.948922i	$-0.602176\pi$
$661$	−16.2235	−0.631021	−0.315511	+	0.948922i	$0.602176\pi$
$662$	0	0
$663$	13.5666	0.526883
$664$	0	0
$665$	1.03931	0.0403027
$666$	0	0
$667$	−15.2113	−0.588985
$668$	0	0
$669$	−39.2213	−1.51638
$670$	0	0
$671$	15.9426	0.615459
$672$	0	0
$673$	37.7380	1.45469	0.727347	−	0.686270i	$-0.240753\pi$
$673$	37.7380	1.45469	0.727347	+	0.686270i	$0.240753\pi$
$674$	0	0
$675$	−3.88308	−0.149460
$676$	0	0
$677$	−0.907362	−0.0348728	−0.0174364	−	0.999848i	$-0.505550\pi$
$677$	−0.907362	−0.0348728	−0.0174364	+	0.999848i	$0.505550\pi$
$678$	0	0
$679$	0.945921	0.0363011
$680$	0	0
$681$	8.59905	0.329516
$682$	0	0
$683$	−38.9401	−1.49000	−0.745001	−	0.667063i	$-0.767551\pi$
$683$	−38.9401	−1.49000	−0.745001	+	0.667063i	$0.767551\pi$
$684$	0	0
$685$	−3.78550	−0.144636
$686$	0	0
$687$	−55.3892	−2.11323
$688$	0	0
$689$	−21.2268	−0.808677
$690$	0	0
$691$	43.2396	1.64491	0.822457	−	0.568828i	$-0.192603\pi$
$691$	43.2396	1.64491	0.822457	+	0.568828i	$0.192603\pi$
$692$	0	0
$693$	1.26131	0.0479131
$694$	0	0
$695$	18.2652	0.692839
$696$	0	0
$697$	7.06621	0.267652
$698$	0	0
$699$	31.9365	1.20795
$700$	0	0
$701$	−9.33652	−0.352635	−0.176318	−	0.984333i	$-0.556419\pi$
$701$	−9.33652	−0.352635	−0.176318	+	0.984333i	$0.556419\pi$
$702$	0	0
$703$	1.21535	0.0458377
$704$	0	0
$705$	−17.9026	−0.674252
$706$	0	0
$707$	0.381567	0.0143503
$708$	0	0
$709$	−17.7160	−0.665337	−0.332669	−	0.943044i	$-0.607949\pi$
$709$	−17.7160	−0.665337	−0.332669	+	0.943044i	$0.607949\pi$
$710$	0	0
$711$	−41.4623	−1.55496
$712$	0	0
$713$	3.31904	0.124299
$714$	0	0
$715$	3.26841	0.122232
$716$	0	0
$717$	−14.4974	−0.541415
$718$	0	0
$719$	−2.51383	−0.0937500	−0.0468750	−	0.998901i	$-0.514926\pi$
$719$	−2.51383	−0.0937500	−0.0468750	+	0.998901i	$0.514926\pi$
$720$	0	0
$721$	2.57303	0.0958248
$722$	0	0
$723$	−0.740817	−0.0275513
$724$	0	0
$725$	−18.9531	−0.703899
$726$	0	0
$727$	34.3698	1.27471	0.637353	−	0.770572i	$-0.280029\pi$
$727$	34.3698	1.27471	0.637353	+	0.770572i	$0.280029\pi$
$728$	0	0
$729$	−17.1632	−0.635673
$730$	0	0
$731$	6.43344	0.237950
$732$	0	0
$733$	40.4039	1.49235	0.746176	−	0.665749i	$-0.231888\pi$
$733$	40.4039	1.49235	0.746176	+	0.665749i	$0.231888\pi$
$734$	0	0
$735$	−21.1162	−0.778884
$736$	0	0
$737$	7.46682	0.275044
$738$	0	0
$739$	13.3906	0.492580	0.246290	−	0.969196i	$-0.420788\pi$
$739$	13.3906	0.492580	0.246290	+	0.969196i	$0.420788\pi$
$740$	0	0
$741$	−8.96004	−0.329155
$742$	0	0
$743$	−15.9962	−0.586844	−0.293422	−	0.955983i	$-0.594794\pi$
$743$	−15.9962	−0.586844	−0.293422	+	0.955983i	$0.594794\pi$
$744$	0	0
$745$	−20.1000	−0.736407
$746$	0	0
$747$	−25.4820	−0.932337
$748$	0	0
$749$	2.19172	0.0800838
$750$	0	0
$751$	−47.9494	−1.74970	−0.874850	−	0.484394i	$-0.839040\pi$
$751$	−47.9494	−1.74970	−0.874850	+	0.484394i	$0.839040\pi$
$752$	0	0
$753$	56.6600	2.06481
$754$	0	0
$755$	13.5244	0.492204
$756$	0	0
$757$	−43.8506	−1.59378	−0.796889	−	0.604125i	$-0.793523\pi$
$757$	−43.8506	−1.59378	−0.796889	+	0.604125i	$0.793523\pi$
$758$	0	0
$759$	7.03323	0.255290
$760$	0	0
$761$	−11.3296	−0.410699	−0.205350	−	0.978689i	$-0.565833\pi$
$761$	−11.3296	−0.410699	−0.205350	+	0.978689i	$0.565833\pi$
$762$	0	0
$763$	−0.497434	−0.0180083
$764$	0	0
$765$	−8.90784	−0.322064
$766$	0	0
$767$	−12.5363	−0.452658
$768$	0	0
$769$	−24.4766	−0.882648	−0.441324	−	0.897348i	$-0.645491\pi$
$769$	−24.4766	−0.882648	−0.441324	+	0.897348i	$0.645491\pi$
$770$	0	0
$771$	−9.84777	−0.354659
$772$	0	0
$773$	−25.1509	−0.904616	−0.452308	−	0.891862i	$-0.649399\pi$
$773$	−25.1509	−0.904616	−0.452308	+	0.891862i	$0.649399\pi$
$774$	0	0
$775$	4.13547	0.148551
$776$	0	0
$777$	0.701698	0.0251733
$778$	0	0
$779$	−4.66687	−0.167208
$780$	0	0
$781$	−1.23620	−0.0442348
$782$	0	0
$783$	6.98155	0.249500
$784$	0	0
$785$	−28.7239	−1.02520
$786$	0	0
$787$	27.2263	0.970513	0.485256	−	0.874372i	$-0.338726\pi$
$787$	27.2263	0.970513	0.485256	+	0.874372i	$0.338726\pi$
$788$	0	0
$789$	63.7557	2.26976
$790$	0	0
$791$	−2.66923	−0.0949067
$792$	0	0
$793$	−29.6540	−1.05304
$794$	0	0
$795$	30.7328	1.08998
$796$	0	0
$797$	−10.9055	−0.386293	−0.193147	−	0.981170i	$-0.561869\pi$
$797$	−10.9055	−0.386293	−0.193147	+	0.981170i	$0.561869\pi$
$798$	0	0
$799$	−15.5942	−0.551682
$800$	0	0
$801$	−8.84278	−0.312444
$802$	0	0
$803$	14.1612	0.499737
$804$	0	0
$805$	1.51744	0.0534828
$806$	0	0
$807$	50.5043	1.77784
$808$	0	0
$809$	8.58814	0.301943	0.150971	−	0.988538i	$-0.451760\pi$
$809$	8.58814	0.301943	0.150971	+	0.988538i	$0.451760\pi$
$810$	0	0
$811$	27.7965	0.976066	0.488033	−	0.872825i	$-0.337715\pi$
$811$	27.7965	0.976066	0.488033	+	0.872825i	$0.337715\pi$
$812$	0	0
$813$	55.9814	1.96335
$814$	0	0
$815$	15.3703	0.538400
$816$	0	0
$817$	−4.24896	−0.148652
$818$	0	0
$819$	−2.34608	−0.0819788
$820$	0	0
$821$	46.7466	1.63147	0.815733	−	0.578428i	$-0.196334\pi$
$821$	46.7466	1.63147	0.815733	+	0.578428i	$0.196334\pi$
$822$	0	0
$823$	9.23480	0.321905	0.160952	−	0.986962i	$-0.448543\pi$
$823$	9.23480	0.321905	0.160952	+	0.986962i	$0.448543\pi$
$824$	0	0
$825$	8.76329	0.305099
$826$	0	0
$827$	9.39433	0.326673	0.163336	−	0.986570i	$-0.447774\pi$
$827$	9.39433	0.326673	0.163336	+	0.986570i	$0.447774\pi$
$828$	0	0
$829$	16.3092	0.566442	0.283221	−	0.959055i	$-0.408597\pi$
$829$	16.3092	0.566442	0.283221	+	0.959055i	$0.408597\pi$
$830$	0	0
$831$	21.2805	0.738214
$832$	0	0
$833$	−18.3934	−0.637293
$834$	0	0
$835$	18.4871	0.639772
$836$	0	0
$837$	−1.52334	−0.0526545
$838$	0	0
$839$	−43.1276	−1.48893	−0.744464	−	0.667662i	$-0.767295\pi$
$839$	−43.1276	−1.48893	−0.744464	+	0.667662i	$0.767295\pi$
$840$	0	0
$841$	5.07654	0.175053
$842$	0	0
$843$	43.2008	1.48791
$844$	0	0
$845$	11.1339	0.383017
$846$	0	0
$847$	−4.25414	−0.146174
$848$	0	0
$849$	28.9308	0.992903
$850$	0	0
$851$	1.77447	0.0608280
$852$	0	0
$853$	13.7616	0.471190	0.235595	−	0.971851i	$-0.424296\pi$
$853$	13.7616	0.471190	0.235595	+	0.971851i	$0.424296\pi$
$854$	0	0
$855$	5.88317	0.201200
$856$	0	0
$857$	−5.05164	−0.172561	−0.0862804	−	0.996271i	$-0.527498\pi$
$857$	−5.05164	−0.172561	−0.0862804	+	0.996271i	$0.527498\pi$
$858$	0	0
$859$	5.92455	0.202143	0.101072	−	0.994879i	$-0.467773\pi$
$859$	5.92455	0.202143	0.101072	+	0.994879i	$0.467773\pi$
$860$	0	0
$861$	−2.69448	−0.0918277
$862$	0	0
$863$	26.1946	0.891676	0.445838	−	0.895114i	$-0.352906\pi$
$863$	26.1946	0.891676	0.445838	+	0.895114i	$0.352906\pi$
$864$	0	0
$865$	11.7122	0.398226
$866$	0	0
$867$	22.7212	0.771654
$868$	0	0
$869$	−19.1858	−0.650835
$870$	0	0
$871$	−13.8886	−0.470597
$872$	0	0
$873$	5.35453	0.181224
$874$	0	0
$875$	4.80238	0.162350
$876$	0	0
$877$	8.67566	0.292956	0.146478	−	0.989214i	$-0.453206\pi$
$877$	8.67566	0.292956	0.146478	+	0.989214i	$0.453206\pi$
$878$	0	0
$879$	66.3709	2.23864
$880$	0	0
$881$	−56.3216	−1.89752	−0.948761	−	0.315994i	$-0.897662\pi$
$881$	−56.3216	−1.89752	−0.948761	+	0.315994i	$0.897662\pi$
$882$	0	0
$883$	12.0941	0.406998	0.203499	−	0.979075i	$-0.434769\pi$
$883$	12.0941	0.406998	0.203499	+	0.979075i	$0.434769\pi$
$884$	0	0
$885$	18.1504	0.610118
$886$	0	0
$887$	20.3663	0.683834	0.341917	−	0.939730i	$-0.388924\pi$
$887$	20.3663	0.683834	0.341917	+	0.939730i	$0.388924\pi$
$888$	0	0
$889$	−1.58922	−0.0533007
$890$	0	0
$891$	−11.8318	−0.396381
$892$	0	0
$893$	10.2992	0.344648
$894$	0	0
$895$	27.2017	0.909251
$896$	0	0
$897$	−13.0821	−0.436798
$898$	0	0
$899$	−7.43535	−0.247983
$900$	0	0
$901$	26.7700	0.891837
$902$	0	0
$903$	−2.45320	−0.0816372
$904$	0	0
$905$	30.5465	1.01540
$906$	0	0
$907$	−27.8279	−0.924010	−0.462005	−	0.886877i	$-0.652870\pi$
$907$	−27.8279	−0.924010	−0.462005	+	0.886877i	$0.652870\pi$
$908$	0	0
$909$	2.15992	0.0716400
$910$	0	0
$911$	9.28747	0.307708	0.153854	−	0.988094i	$-0.450831\pi$
$911$	9.28747	0.307708	0.153854	+	0.988094i	$0.450831\pi$
$912$	0	0
$913$	−11.7913	−0.390234
$914$	0	0
$915$	42.9339	1.41935
$916$	0	0
$917$	−3.07125	−0.101422
$918$	0	0
$919$	23.7295	0.782765	0.391383	−	0.920228i	$-0.371997\pi$
$919$	23.7295	0.782765	0.391383	+	0.920228i	$0.371997\pi$
$920$	0	0
$921$	27.7189	0.913369
$922$	0	0
$923$	2.29939	0.0756853
$924$	0	0
$925$	2.21096	0.0726959
$926$	0	0
$927$	14.5651	0.478379
$928$	0	0
$929$	1.66705	0.0546940	0.0273470	−	0.999626i	$-0.491294\pi$
$929$	1.66705	0.0546940	0.0273470	+	0.999626i	$0.491294\pi$
$930$	0	0
$931$	12.1479	0.398131
$932$	0	0
$933$	−17.5830	−0.575642
$934$	0	0
$935$	−4.12192	−0.134801
$936$	0	0
$937$	7.70547	0.251727	0.125863	−	0.992048i	$-0.459830\pi$
$937$	7.70547	0.251727	0.125863	+	0.992048i	$0.459830\pi$
$938$	0	0
$939$	−59.7772	−1.95075
$940$	0	0
$941$	6.39827	0.208578	0.104289	−	0.994547i	$-0.466743\pi$
$941$	6.39827	0.208578	0.104289	+	0.994547i	$0.466743\pi$
$942$	0	0
$943$	−6.81385	−0.221890
$944$	0	0
$945$	−0.696461	−0.0226559
$946$	0	0
$947$	24.3619	0.791655	0.395828	−	0.918325i	$-0.370458\pi$
$947$	24.3619	0.791655	0.395828	+	0.918325i	$0.370458\pi$
$948$	0	0
$949$	−26.3404	−0.855044
$950$	0	0
$951$	−3.61124	−0.117103
$952$	0	0
$953$	−21.5093	−0.696756	−0.348378	−	0.937354i	$-0.613267\pi$
$953$	−21.5093	−0.696756	−0.348378	+	0.937354i	$0.613267\pi$
$954$	0	0
$955$	27.6011	0.893152
$956$	0	0
$957$	−15.7559	−0.509316
$958$	0	0
$959$	1.25735	0.0406020
$960$	0	0
$961$	−29.3776	−0.947666
$962$	0	0
$963$	12.4066	0.399797
$964$	0	0
$965$	6.02138	0.193835
$966$	0	0
$967$	19.3302	0.621616	0.310808	−	0.950473i	$-0.399400\pi$
$967$	19.3302	0.621616	0.310808	+	0.950473i	$0.399400\pi$
$968$	0	0
$969$	11.2999	0.363004
$970$	0	0
$971$	8.55744	0.274621	0.137311	−	0.990528i	$-0.456154\pi$
$971$	8.55744	0.274621	0.137311	+	0.990528i	$0.456154\pi$
$972$	0	0
$973$	−6.06679	−0.194492
$974$	0	0
$975$	−16.3001	−0.522020
$976$	0	0
$977$	5.83351	0.186630	0.0933152	−	0.995637i	$-0.470254\pi$
$977$	5.83351	0.186630	0.0933152	+	0.995637i	$0.470254\pi$
$978$	0	0
$979$	−4.09182	−0.130775
$980$	0	0
$981$	−2.81580	−0.0899016
$982$	0	0
$983$	2.35005	0.0749550	0.0374775	−	0.999297i	$-0.488068\pi$
$983$	2.35005	0.0749550	0.0374775	+	0.999297i	$0.488068\pi$
$984$	0	0
$985$	11.6590	0.371488
$986$	0	0
$987$	5.94636	0.189275
$988$	0	0
$989$	−6.20368	−0.197266
$990$	0	0
$991$	−17.3067	−0.549765	−0.274882	−	0.961478i	$-0.588639\pi$
$991$	−17.3067	−0.549765	−0.274882	+	0.961478i	$0.588639\pi$
$992$	0	0
$993$	38.6970	1.22801
$994$	0	0
$995$	−26.9574	−0.854606
$996$	0	0
$997$	−2.67507	−0.0847202	−0.0423601	−	0.999102i	$-0.513488\pi$
$997$	−2.67507	−0.0847202	−0.0423601	+	0.999102i	$0.513488\pi$
$998$	0	0
$999$	−0.814429	−0.0257674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.6	✓	29
4.3	odd	2		8048.2.a.w.1.24		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.6	✓	29	1.1	even	1	trivial
8048.2.a.w.1.24		29	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)

\(f(q)\)	\(=\)	\(q-2.34298 q^{3} -1.32410 q^{5} +0.439798 q^{7} +2.48955 q^{9} +O(q^{10})\)	\(q-2.34298 q^{3} -1.32410 q^{5} +0.439798 q^{7} +2.48955 q^{9} +1.15199 q^{11} -2.14274 q^{13} +3.10233 q^{15} +2.70230 q^{17} -1.78473 q^{19} -1.03044 q^{21} -2.60579 q^{23} -3.24677 q^{25} +1.19598 q^{27} +5.83751 q^{29} -1.27372 q^{31} -2.69908 q^{33} -0.582335 q^{35} -0.680971 q^{37} +5.02040 q^{39} +2.61489 q^{41} +2.38073 q^{43} -3.29640 q^{45} -5.77071 q^{47} -6.80658 q^{49} -6.33142 q^{51} +9.90638 q^{53} -1.52534 q^{55} +4.18158 q^{57} +5.85057 q^{59} +13.8393 q^{61} +1.09490 q^{63} +2.83719 q^{65} +6.48169 q^{67} +6.10530 q^{69} -1.07311 q^{71} +12.2928 q^{73} +7.60711 q^{75} +0.506641 q^{77} -16.6546 q^{79} -10.2708 q^{81} -10.2356 q^{83} -3.57810 q^{85} -13.6772 q^{87} -3.55197 q^{89} -0.942374 q^{91} +2.98430 q^{93} +2.36315 q^{95} +2.15081 q^{97} +2.86792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

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