LMFDB - Embedded newform 4004.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4004,2,Mod(1,4004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4004.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.246302029.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 15x^2 - 13*x - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73486$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73486 q^{3} +1.63787 q^{5} +1.00000 q^{7} +0.00974910 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.84148 q^{15} +4.66359 q^{17} +1.13950 q^{19} -1.73486 q^{21} -6.21421 q^{23} -2.31739 q^{25} +5.18768 q^{27} -5.84148 q^{29} +4.87694 q^{31} +1.73486 q^{33} +1.63787 q^{35} -8.62859 q^{37} +1.73486 q^{39} -3.59622 q^{41} +1.28549 q^{43} +0.0159677 q^{45} +7.07569 q^{47} +1.00000 q^{49} -8.09068 q^{51} +0.181146 q^{53} -1.63787 q^{55} -1.97687 q^{57} +0.206703 q^{59} -5.76708 q^{61} +0.00974910 q^{63} -1.63787 q^{65} -4.45947 q^{67} +10.7808 q^{69} +3.09421 q^{71} -6.97432 q^{73} +4.02035 q^{75} -1.00000 q^{77} +4.52507 q^{79} -9.02915 q^{81} -7.63478 q^{83} +7.63834 q^{85} +10.1342 q^{87} +0.284199 q^{89} -1.00000 q^{91} -8.46083 q^{93} +1.86634 q^{95} +0.615716 q^{97} -0.00974910 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

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
$3$	−1.73486	−1.00162	−0.500812	−	0.865556i	$-0.666965\pi$
$3$	−1.73486	−1.00162	−0.500812	+	0.865556i	$0.666965\pi$
$4$	0	0
$5$	1.63787	0.732477	0.366239	−	0.930521i	$-0.380645\pi$
$5$	1.63787	0.732477	0.366239	+	0.930521i	$0.380645\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.00974910	0.00324970
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.84148	−0.733666
$16$	0	0
$17$	4.66359	1.13109	0.565543	−	0.824719i	$-0.308667\pi$
$17$	4.66359	1.13109	0.565543	+	0.824719i	$0.308667\pi$
$18$	0	0
$19$	1.13950	0.261418	0.130709	−	0.991421i	$-0.458275\pi$
$19$	1.13950	0.261418	0.130709	+	0.991421i	$0.458275\pi$
$20$	0	0
$21$	−1.73486	−0.378578
$22$	0	0
$23$	−6.21421	−1.29575	−0.647876	−	0.761746i	$-0.724343\pi$
$23$	−6.21421	−1.29575	−0.647876	+	0.761746i	$0.724343\pi$
$24$	0	0
$25$	−2.31739	−0.463477
$26$	0	0
$27$	5.18768	0.998369
$28$	0	0
$29$	−5.84148	−1.08474	−0.542368	−	0.840141i	$-0.682472\pi$
$29$	−5.84148	−1.08474	−0.542368	+	0.840141i	$0.682472\pi$
$30$	0	0
$31$	4.87694	0.875925	0.437962	−	0.898993i	$-0.355700\pi$
$31$	4.87694	0.875925	0.437962	+	0.898993i	$0.355700\pi$
$32$	0	0
$33$	1.73486	0.302001
$34$	0	0
$35$	1.63787	0.276850
$36$	0	0
$37$	−8.62859	−1.41853	−0.709266	−	0.704941i	$-0.750973\pi$
$37$	−8.62859	−1.41853	−0.709266	+	0.704941i	$0.750973\pi$
$38$	0	0
$39$	1.73486	0.277800
$40$	0	0
$41$	−3.59622	−0.561635	−0.280817	−	0.959761i	$-0.590606\pi$
$41$	−3.59622	−0.561635	−0.280817	+	0.959761i	$0.590606\pi$
$42$	0	0
$43$	1.28549	0.196035	0.0980174	−	0.995185i	$-0.468750\pi$
$43$	1.28549	0.196035	0.0980174	+	0.995185i	$0.468750\pi$
$44$	0	0
$45$	0.0159677	0.00238033
$46$	0	0
$47$	7.07569	1.03210	0.516048	−	0.856560i	$-0.327403\pi$
$47$	7.07569	1.03210	0.516048	+	0.856560i	$0.327403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.09068	−1.13292
$52$	0	0
$53$	0.181146	0.0248823	0.0124411	−	0.999923i	$-0.496040\pi$
$53$	0.181146	0.0248823	0.0124411	+	0.999923i	$0.496040\pi$
$54$	0	0
$55$	−1.63787	−0.220850
$56$	0	0
$57$	−1.97687	−0.261843
$58$	0	0
$59$	0.206703	0.0269104	0.0134552	−	0.999909i	$-0.495717\pi$
$59$	0.206703	0.0269104	0.0134552	+	0.999909i	$0.495717\pi$
$60$	0	0
$61$	−5.76708	−0.738398	−0.369199	−	0.929350i	$-0.620368\pi$
$61$	−5.76708	−0.738398	−0.369199	+	0.929350i	$0.620368\pi$
$62$	0	0
$63$	0.00974910	0.00122827
$64$	0	0
$65$	−1.63787	−0.203153
$66$	0	0
$67$	−4.45947	−0.544811	−0.272405	−	0.962183i	$-0.587819\pi$
$67$	−4.45947	−0.544811	−0.272405	+	0.962183i	$0.587819\pi$
$68$	0	0
$69$	10.7808	1.29786
$70$	0	0
$71$	3.09421	0.367215	0.183608	−	0.983000i	$-0.441222\pi$
$71$	3.09421	0.367215	0.183608	+	0.983000i	$0.441222\pi$
$72$	0	0
$73$	−6.97432	−0.816282	−0.408141	−	0.912919i	$-0.633823\pi$
$73$	−6.97432	−0.816282	−0.408141	+	0.912919i	$0.633823\pi$
$74$	0	0
$75$	4.02035	0.464230
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.52507	0.509110	0.254555	−	0.967058i	$-0.418071\pi$
$79$	4.52507	0.509110	0.254555	+	0.967058i	$0.418071\pi$
$80$	0	0
$81$	−9.02915	−1.00324
$82$	0	0
$83$	−7.63478	−0.838026	−0.419013	−	0.907980i	$-0.637624\pi$
$83$	−7.63478	−0.838026	−0.419013	+	0.907980i	$0.637624\pi$
$84$	0	0
$85$	7.63834	0.828494
$86$	0	0
$87$	10.1342	1.08650
$88$	0	0
$89$	0.284199	0.0301250	0.0150625	−	0.999887i	$-0.495205\pi$
$89$	0.284199	0.0301250	0.0150625	+	0.999887i	$0.495205\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−8.46083	−0.877347
$94$	0	0
$95$	1.86634	0.191483
$96$	0	0
$97$	0.615716	0.0625165	0.0312583	−	0.999511i	$-0.490049\pi$
$97$	0.615716	0.0625165	0.0312583	+	0.999511i	$0.490049\pi$
$98$	0	0
$99$	−0.00974910	−0.000979821 0
$100$	0	0
$101$	−11.5796	−1.15221	−0.576106	−	0.817375i	$-0.695429\pi$
$101$	−11.5796	−1.15221	−0.576106	+	0.817375i	$0.695429\pi$
$102$	0	0
$103$	−3.86054	−0.380390	−0.190195	−	0.981746i	$-0.560912\pi$
$103$	−3.86054	−0.380390	−0.190195	+	0.981746i	$0.560912\pi$
$104$	0	0
$105$	−2.84148	−0.277300
$106$	0	0
$107$	14.4321	1.39520	0.697602	−	0.716485i	$-0.254250\pi$
$107$	14.4321	1.39520	0.697602	+	0.716485i	$0.254250\pi$
$108$	0	0
$109$	2.47269	0.236841	0.118420	−	0.992964i	$-0.462217\pi$
$109$	2.47269	0.236841	0.118420	+	0.992964i	$0.462217\pi$
$110$	0	0
$111$	14.9694	1.42083
$112$	0	0
$113$	−9.91686	−0.932900	−0.466450	−	0.884548i	$-0.654467\pi$
$113$	−9.91686	−0.932900	−0.466450	+	0.884548i	$0.654467\pi$
$114$	0	0
$115$	−10.1781	−0.949109
$116$	0	0
$117$	−0.00974910	−0.000901304 0
$118$	0	0
$119$	4.66359	0.427510
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.23895	0.562547
$124$	0	0
$125$	−11.9849	−1.07196
$126$	0	0
$127$	7.21679	0.640387	0.320193	−	0.947352i	$-0.396252\pi$
$127$	7.21679	0.640387	0.320193	+	0.947352i	$0.396252\pi$
$128$	0	0
$129$	−2.23014	−0.196353
$130$	0	0
$131$	2.52854	0.220920	0.110460	−	0.993881i	$-0.464768\pi$
$131$	2.52854	0.220920	0.110460	+	0.993881i	$0.464768\pi$
$132$	0	0
$133$	1.13950	0.0988068
$134$	0	0
$135$	8.49673	0.731282
$136$	0	0
$137$	−12.5679	−1.07375	−0.536875	−	0.843662i	$-0.680395\pi$
$137$	−12.5679	−1.07375	−0.536875	+	0.843662i	$0.680395\pi$
$138$	0	0
$139$	−6.68536	−0.567045	−0.283522	−	0.958966i	$-0.591503\pi$
$139$	−6.68536	−0.567045	−0.283522	+	0.958966i	$0.591503\pi$
$140$	0	0
$141$	−12.2754	−1.03377
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−9.56757	−0.794543
$146$	0	0
$147$	−1.73486	−0.143089
$148$	0	0
$149$	15.0016	1.22898	0.614491	−	0.788924i	$-0.289362\pi$
$149$	15.0016	1.22898	0.614491	+	0.788924i	$0.289362\pi$
$150$	0	0
$151$	−0.570661	−0.0464397	−0.0232199	−	0.999730i	$-0.507392\pi$
$151$	−0.570661	−0.0464397	−0.0232199	+	0.999730i	$0.507392\pi$
$152$	0	0
$153$	0.0454657	0.00367569
$154$	0	0
$155$	7.98779	0.641595
$156$	0	0
$157$	−4.89988	−0.391053	−0.195527	−	0.980698i	$-0.562642\pi$
$157$	−4.89988	−0.391053	−0.195527	+	0.980698i	$0.562642\pi$
$158$	0	0
$159$	−0.314263	−0.0249227
$160$	0	0
$161$	−6.21421	−0.489748
$162$	0	0
$163$	−7.41863	−0.581072	−0.290536	−	0.956864i	$-0.593834\pi$
$163$	−7.41863	−0.581072	−0.290536	+	0.956864i	$0.593834\pi$
$164$	0	0
$165$	2.84148	0.221209
$166$	0	0
$167$	−14.6073	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$167$	−14.6073	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.0111090	0.000849530 0
$172$	0	0
$173$	16.4903	1.25373	0.626865	−	0.779128i	$-0.284338\pi$
$173$	16.4903	1.25373	0.626865	+	0.779128i	$0.284338\pi$
$174$	0	0
$175$	−2.31739	−0.175178
$176$	0	0
$177$	−0.358601	−0.0269541
$178$	0	0
$179$	−15.7614	−1.17806	−0.589031	−	0.808110i	$-0.700490\pi$
$179$	−15.7614	−1.17806	−0.589031	+	0.808110i	$0.700490\pi$
$180$	0	0
$181$	−15.0789	−1.12081	−0.560405	−	0.828219i	$-0.689354\pi$
$181$	−15.0789	−1.12081	−0.560405	+	0.828219i	$0.689354\pi$
$182$	0	0
$183$	10.0051	0.739597
$184$	0	0
$185$	−14.1325	−1.03904
$186$	0	0
$187$	−4.66359	−0.341035
$188$	0	0
$189$	5.18768	0.377348
$190$	0	0
$191$	16.1683	1.16990	0.584950	−	0.811069i	$-0.301114\pi$
$191$	16.1683	1.16990	0.584950	+	0.811069i	$0.301114\pi$
$192$	0	0
$193$	7.68349	0.553070	0.276535	−	0.961004i	$-0.410814\pi$
$193$	7.68349	0.553070	0.276535	+	0.961004i	$0.410814\pi$
$194$	0	0
$195$	2.84148	0.203482
$196$	0	0
$197$	7.07865	0.504333	0.252167	−	0.967684i	$-0.418857\pi$
$197$	7.07865	0.504333	0.252167	+	0.967684i	$0.418857\pi$
$198$	0	0
$199$	−11.3781	−0.806570	−0.403285	−	0.915075i	$-0.632132\pi$
$199$	−11.3781	−0.806570	−0.403285	+	0.915075i	$0.632132\pi$
$200$	0	0
$201$	7.73656	0.545695
$202$	0	0
$203$	−5.84148	−0.409991
$204$	0	0
$205$	−5.89013	−0.411385
$206$	0	0
$207$	−0.0605829	−0.00421080
$208$	0	0
$209$	−1.13950	−0.0788205
$210$	0	0
$211$	−10.6730	−0.734761	−0.367381	−	0.930071i	$-0.619745\pi$
$211$	−10.6730	−0.734761	−0.367381	+	0.930071i	$0.619745\pi$
$212$	0	0
$213$	−5.36803	−0.367811
$214$	0	0
$215$	2.10546	0.143591
$216$	0	0
$217$	4.87694	0.331068
$218$	0	0
$219$	12.0995	0.817607
$220$	0	0
$221$	−4.66359	−0.313707
$222$	0	0
$223$	−17.0941	−1.14471	−0.572353	−	0.820008i	$-0.693969\pi$
$223$	−17.0941	−1.14471	−0.572353	+	0.820008i	$0.693969\pi$
$224$	0	0
$225$	−0.0225924	−0.00150616
$226$	0	0
$227$	−2.53908	−0.168525	−0.0842623	−	0.996444i	$-0.526853\pi$
$227$	−2.53908	−0.168525	−0.0842623	+	0.996444i	$0.526853\pi$
$228$	0	0
$229$	21.8923	1.44668	0.723342	−	0.690490i	$-0.242605\pi$
$229$	21.8923	1.44668	0.723342	+	0.690490i	$0.242605\pi$
$230$	0	0
$231$	1.73486	0.114146
$232$	0	0
$233$	−13.1460	−0.861225	−0.430612	−	0.902537i	$-0.641702\pi$
$233$	−13.1460	−0.861225	−0.430612	+	0.902537i	$0.641702\pi$
$234$	0	0
$235$	11.5891	0.755987
$236$	0	0
$237$	−7.85038	−0.509937
$238$	0	0
$239$	−16.3412	−1.05702	−0.528511	−	0.848926i	$-0.677250\pi$
$239$	−16.3412	−1.05702	−0.528511	+	0.848926i	$0.677250\pi$
$240$	0	0
$241$	−16.9504	−1.09187	−0.545934	−	0.837828i	$-0.683825\pi$
$241$	−16.9504	−1.09187	−0.545934	+	0.837828i	$0.683825\pi$
$242$	0	0
$243$	0.101315	0.00649937
$244$	0	0
$245$	1.63787	0.104640
$246$	0	0
$247$	−1.13950	−0.0725043
$248$	0	0
$249$	13.2453	0.839386
$250$	0	0
$251$	−8.85366	−0.558838	−0.279419	−	0.960169i	$-0.590142\pi$
$251$	−8.85366	−0.558838	−0.279419	+	0.960169i	$0.590142\pi$
$252$	0	0
$253$	6.21421	0.390684
$254$	0	0
$255$	−13.2515	−0.829839
$256$	0	0
$257$	−6.51481	−0.406383	−0.203191	−	0.979139i	$-0.565131\pi$
$257$	−6.51481	−0.406383	−0.203191	+	0.979139i	$0.565131\pi$
$258$	0	0
$259$	−8.62859	−0.536154
$260$	0	0
$261$	−0.0569491	−0.00352506
$262$	0	0
$263$	5.74465	0.354230	0.177115	−	0.984190i	$-0.443324\pi$
$263$	5.74465	0.354230	0.177115	+	0.984190i	$0.443324\pi$
$264$	0	0
$265$	0.296693	0.0182257
$266$	0	0
$267$	−0.493047	−0.0301740
$268$	0	0
$269$	−24.1717	−1.47378	−0.736888	−	0.676014i	$-0.763706\pi$
$269$	−24.1717	−1.47378	−0.736888	+	0.676014i	$0.763706\pi$
$270$	0	0
$271$	−3.93279	−0.238900	−0.119450	−	0.992840i	$-0.538113\pi$
$271$	−3.93279	−0.238900	−0.119450	+	0.992840i	$0.538113\pi$
$272$	0	0
$273$	1.73486	0.104999
$274$	0	0
$275$	2.31739	0.139744
$276$	0	0
$277$	−17.0368	−1.02364	−0.511820	−	0.859093i	$-0.671028\pi$
$277$	−17.0368	−1.02364	−0.511820	+	0.859093i	$0.671028\pi$
$278$	0	0
$279$	0.0475458	0.00284649
$280$	0	0
$281$	4.71275	0.281139	0.140569	−	0.990071i	$-0.455107\pi$
$281$	4.71275	0.281139	0.140569	+	0.990071i	$0.455107\pi$
$282$	0	0
$283$	−3.94578	−0.234552	−0.117276	−	0.993099i	$-0.537416\pi$
$283$	−3.94578	−0.234552	−0.117276	+	0.993099i	$0.537416\pi$
$284$	0	0
$285$	−3.23785	−0.191794
$286$	0	0
$287$	−3.59622	−0.212278
$288$	0	0
$289$	4.74903	0.279355
$290$	0	0
$291$	−1.06818	−0.0626180
$292$	0	0
$293$	−13.4035	−0.783042	−0.391521	−	0.920169i	$-0.628051\pi$
$293$	−13.4035	−0.783042	−0.391521	+	0.920169i	$0.628051\pi$
$294$	0	0
$295$	0.338552	0.0197113
$296$	0	0
$297$	−5.18768	−0.301019
$298$	0	0
$299$	6.21421	0.359377
$300$	0	0
$301$	1.28549	0.0740942
$302$	0	0
$303$	20.0890	1.15408
$304$	0	0
$305$	−9.44571	−0.540860
$306$	0	0
$307$	−16.4474	−0.938705	−0.469353	−	0.883011i	$-0.655513\pi$
$307$	−16.4474	−0.938705	−0.469353	+	0.883011i	$0.655513\pi$
$308$	0	0
$309$	6.69751	0.381008
$310$	0	0
$311$	7.81962	0.443410	0.221705	−	0.975114i	$-0.428838\pi$
$311$	7.81962	0.443410	0.221705	+	0.975114i	$0.428838\pi$
$312$	0	0
$313$	0.133838	0.00756499	0.00378250	−	0.999993i	$-0.498796\pi$
$313$	0.133838	0.00756499	0.00378250	+	0.999993i	$0.498796\pi$
$314$	0	0
$315$	0.0159677	0.000899680 0
$316$	0	0
$317$	−27.9340	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$317$	−27.9340	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$318$	0	0
$319$	5.84148	0.327060
$320$	0	0
$321$	−25.0377	−1.39747
$322$	0	0
$323$	5.31413	0.295686
$324$	0	0
$325$	2.31739	0.128546
$326$	0	0
$327$	−4.28978	−0.237225
$328$	0	0
$329$	7.07569	0.390096
$330$	0	0
$331$	−17.1098	−0.940441	−0.470221	−	0.882549i	$-0.655826\pi$
$331$	−17.1098	−0.940441	−0.470221	+	0.882549i	$0.655826\pi$
$332$	0	0
$333$	−0.0841209	−0.00460980
$334$	0	0
$335$	−7.30402	−0.399061
$336$	0	0
$337$	27.2128	1.48238	0.741189	−	0.671296i	$-0.234262\pi$
$337$	27.2128	1.48238	0.741189	+	0.671296i	$0.234262\pi$
$338$	0	0
$339$	17.2044	0.934414
$340$	0	0
$341$	−4.87694	−0.264101
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	17.6575	0.950649
$346$	0	0
$347$	−18.6097	−0.999022	−0.499511	−	0.866307i	$-0.666487\pi$
$347$	−18.6097	−0.999022	−0.499511	+	0.866307i	$0.666487\pi$
$348$	0	0
$349$	−28.7942	−1.54132	−0.770660	−	0.637247i	$-0.780073\pi$
$349$	−28.7942	−1.54132	−0.770660	+	0.637247i	$0.780073\pi$
$350$	0	0
$351$	−5.18768	−0.276898
$352$	0	0
$353$	−33.6656	−1.79184	−0.895919	−	0.444218i	$-0.853482\pi$
$353$	−33.6656	−1.79184	−0.895919	+	0.444218i	$0.853482\pi$
$354$	0	0
$355$	5.06791	0.268977
$356$	0	0
$357$	−8.09068	−0.428204
$358$	0	0
$359$	−5.30618	−0.280049	−0.140025	−	0.990148i	$-0.544718\pi$
$359$	−5.30618	−0.280049	−0.140025	+	0.990148i	$0.544718\pi$
$360$	0	0
$361$	−17.7016	−0.931661
$362$	0	0
$363$	−1.73486	−0.0910567
$364$	0	0
$365$	−11.4230	−0.597908
$366$	0	0
$367$	4.49510	0.234642	0.117321	−	0.993094i	$-0.462569\pi$
$367$	4.49510	0.234642	0.117321	+	0.993094i	$0.462569\pi$
$368$	0	0
$369$	−0.0350599	−0.00182514
$370$	0	0
$371$	0.181146	0.00940461
$372$	0	0
$373$	0.733587	0.0379837	0.0189918	−	0.999820i	$-0.493954\pi$
$373$	0.733587	0.0379837	0.0189918	+	0.999820i	$0.493954\pi$
$374$	0	0
$375$	20.7922	1.07370
$376$	0	0
$377$	5.84148	0.300851
$378$	0	0
$379$	−0.425767	−0.0218702	−0.0109351	−	0.999940i	$-0.503481\pi$
$379$	−0.425767	−0.0218702	−0.0109351	+	0.999940i	$0.503481\pi$
$380$	0	0
$381$	−12.5201	−0.641427
$382$	0	0
$383$	−6.69446	−0.342071	−0.171036	−	0.985265i	$-0.554711\pi$
$383$	−6.69446	−0.342071	−0.171036	+	0.985265i	$0.554711\pi$
$384$	0	0
$385$	−1.63787	−0.0834735
$386$	0	0
$387$	0.0125323	0.000637054 0
$388$	0	0
$389$	−17.7103	−0.897950	−0.448975	−	0.893544i	$-0.648211\pi$
$389$	−17.7103	−0.897950	−0.448975	+	0.893544i	$0.648211\pi$
$390$	0	0
$391$	−28.9805	−1.46561
$392$	0	0
$393$	−4.38667	−0.221278
$394$	0	0
$395$	7.41147	0.372911
$396$	0	0
$397$	25.4680	1.27820	0.639102	−	0.769122i	$-0.279306\pi$
$397$	25.4680	1.27820	0.639102	+	0.769122i	$0.279306\pi$
$398$	0	0
$399$	−1.97687	−0.0989672
$400$	0	0
$401$	−14.4696	−0.722577	−0.361289	−	0.932454i	$-0.617663\pi$
$401$	−14.4696	−0.722577	−0.361289	+	0.932454i	$0.617663\pi$
$402$	0	0
$403$	−4.87694	−0.242938
$404$	0	0
$405$	−14.7886	−0.734850
$406$	0	0
$407$	8.62859	0.427703
$408$	0	0
$409$	−3.37125	−0.166698	−0.0833488	−	0.996520i	$-0.526562\pi$
$409$	−3.37125	−0.166698	−0.0833488	+	0.996520i	$0.526562\pi$
$410$	0	0
$411$	21.8036	1.07549
$412$	0	0
$413$	0.206703	0.0101712
$414$	0	0
$415$	−12.5048	−0.613835
$416$	0	0
$417$	11.5982	0.567965
$418$	0	0
$419$	29.2787	1.43036	0.715179	−	0.698942i	$-0.246345\pi$
$419$	29.2787	1.43036	0.715179	+	0.698942i	$0.246345\pi$
$420$	0	0
$421$	−10.5789	−0.515585	−0.257793	−	0.966200i	$-0.582995\pi$
$421$	−10.5789	−0.515585	−0.257793	+	0.966200i	$0.582995\pi$
$422$	0	0
$423$	0.0689816	0.00335400
$424$	0	0
$425$	−10.8073	−0.524233
$426$	0	0
$427$	−5.76708	−0.279088
$428$	0	0
$429$	−1.73486	−0.0837600
$430$	0	0
$431$	5.79215	0.278998	0.139499	−	0.990222i	$-0.455451\pi$
$431$	5.79215	0.278998	0.139499	+	0.990222i	$0.455451\pi$
$432$	0	0
$433$	−4.12235	−0.198108	−0.0990538	−	0.995082i	$-0.531582\pi$
$433$	−4.12235	−0.198108	−0.0990538	+	0.995082i	$0.531582\pi$
$434$	0	0
$435$	16.5984	0.795833
$436$	0	0
$437$	−7.08106	−0.338733
$438$	0	0
$439$	−6.71845	−0.320654	−0.160327	−	0.987064i	$-0.551255\pi$
$439$	−6.71845	−0.320654	−0.160327	+	0.987064i	$0.551255\pi$
$440$	0	0
$441$	0.00974910	0.000464243 0
$442$	0	0
$443$	17.7031	0.841102	0.420551	−	0.907269i	$-0.361837\pi$
$443$	17.7031	0.841102	0.420551	+	0.907269i	$0.361837\pi$
$444$	0	0
$445$	0.465481	0.0220659
$446$	0	0
$447$	−26.0258	−1.23098
$448$	0	0
$449$	0.0333214	0.00157254	0.000786268	−	1.00000i	$-0.499750\pi$
$449$	0.0333214	0.00157254	0.000786268		1.00000i	$0.499750\pi$
$450$	0	0
$451$	3.59622	0.169339
$452$	0	0
$453$	0.990018	0.0465151
$454$	0	0
$455$	−1.63787	−0.0767845
$456$	0	0
$457$	2.79852	0.130909	0.0654546	−	0.997856i	$-0.479150\pi$
$457$	2.79852	0.130909	0.0654546	+	0.997856i	$0.479150\pi$
$458$	0	0
$459$	24.1932	1.12924
$460$	0	0
$461$	−22.7320	−1.05873	−0.529367	−	0.848393i	$-0.677571\pi$
$461$	−22.7320	−1.05873	−0.529367	+	0.848393i	$0.677571\pi$
$462$	0	0
$463$	2.41422	0.112198	0.0560992	−	0.998425i	$-0.482134\pi$
$463$	2.41422	0.112198	0.0560992	+	0.998425i	$0.482134\pi$
$464$	0	0
$465$	−13.8577	−0.642636
$466$	0	0
$467$	23.5247	1.08859	0.544296	−	0.838893i	$-0.316797\pi$
$467$	23.5247	1.08859	0.544296	+	0.838893i	$0.316797\pi$
$468$	0	0
$469$	−4.45947	−0.205919
$470$	0	0
$471$	8.50062	0.391688
$472$	0	0
$473$	−1.28549	−0.0591067
$474$	0	0
$475$	−2.64065	−0.121161
$476$	0	0
$477$	0.00176601	8.08598e−5 0
$478$	0	0
$479$	29.7473	1.35919	0.679594	−	0.733589i	$-0.262156\pi$
$479$	29.7473	1.35919	0.679594	+	0.733589i	$0.262156\pi$
$480$	0	0
$481$	8.62859	0.393430
$482$	0	0
$483$	10.7808	0.490543
$484$	0	0
$485$	1.00846	0.0457919
$486$	0	0
$487$	19.2523	0.872406	0.436203	−	0.899848i	$-0.356323\pi$
$487$	19.2523	0.872406	0.436203	+	0.899848i	$0.356323\pi$
$488$	0	0
$489$	12.8703	0.582016
$490$	0	0
$491$	−7.50585	−0.338734	−0.169367	−	0.985553i	$-0.554172\pi$
$491$	−7.50585	−0.338734	−0.169367	+	0.985553i	$0.554172\pi$
$492$	0	0
$493$	−27.2422	−1.22693
$494$	0	0
$495$	−0.0159677	−0.000717696 0
$496$	0	0
$497$	3.09421	0.138794
$498$	0	0
$499$	38.7993	1.73689	0.868447	−	0.495783i	$-0.165119\pi$
$499$	38.7993	1.73689	0.868447	+	0.495783i	$0.165119\pi$
$500$	0	0
$501$	25.3416	1.13218
$502$	0	0
$503$	26.2814	1.17183	0.585916	−	0.810372i	$-0.300735\pi$
$503$	26.2814	1.17183	0.585916	+	0.810372i	$0.300735\pi$
$504$	0	0
$505$	−18.9659	−0.843969
$506$	0	0
$507$	−1.73486	−0.0770480
$508$	0	0
$509$	35.7496	1.58457	0.792286	−	0.610150i	$-0.208891\pi$
$509$	35.7496	1.58457	0.792286	+	0.610150i	$0.208891\pi$
$510$	0	0
$511$	−6.97432	−0.308526
$512$	0	0
$513$	5.91133	0.260992
$514$	0	0
$515$	−6.32305	−0.278627
$516$	0	0
$517$	−7.07569	−0.311189
$518$	0	0
$519$	−28.6083	−1.25577
$520$	0	0
$521$	31.0441	1.36007	0.680034	−	0.733181i	$-0.261965\pi$
$521$	31.0441	1.36007	0.680034	+	0.733181i	$0.261965\pi$
$522$	0	0
$523$	24.1729	1.05701	0.528504	−	0.848931i	$-0.322753\pi$
$523$	24.1729	1.05701	0.528504	+	0.848931i	$0.322753\pi$
$524$	0	0
$525$	4.02035	0.175462
$526$	0	0
$527$	22.7440	0.990746
$528$	0	0
$529$	15.6164	0.678973
$530$	0	0
$531$	0.00201517	8.74508e−5 0
$532$	0	0
$533$	3.59622	0.155769
$534$	0	0
$535$	23.6379	1.02196
$536$	0	0
$537$	27.3439	1.17997
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−43.1903	−1.85690	−0.928448	−	0.371462i	$-0.878857\pi$
$541$	−43.1903	−1.85690	−0.928448	+	0.371462i	$0.878857\pi$
$542$	0	0
$543$	26.1599	1.12263
$544$	0	0
$545$	4.04994	0.173480
$546$	0	0
$547$	15.7546	0.673616	0.336808	−	0.941573i	$-0.390653\pi$
$547$	15.7546	0.673616	0.336808	+	0.941573i	$0.390653\pi$
$548$	0	0
$549$	−0.0562238	−0.00239957
$550$	0	0
$551$	−6.65634	−0.283569
$552$	0	0
$553$	4.52507	0.192426
$554$	0	0
$555$	24.5179	1.04073
$556$	0	0
$557$	−0.825920	−0.0349953	−0.0174977	−	0.999847i	$-0.505570\pi$
$557$	−0.825920	−0.0349953	−0.0174977	+	0.999847i	$0.505570\pi$
$558$	0	0
$559$	−1.28549	−0.0543703
$560$	0	0
$561$	8.09068	0.341589
$562$	0	0
$563$	−2.60467	−0.109774	−0.0548869	−	0.998493i	$-0.517480\pi$
$563$	−2.60467	−0.109774	−0.0548869	+	0.998493i	$0.517480\pi$
$564$	0	0
$565$	−16.2425	−0.683328
$566$	0	0
$567$	−9.02915	−0.379189
$568$	0	0
$569$	17.7400	0.743701	0.371850	−	0.928293i	$-0.378723\pi$
$569$	17.7400	0.743701	0.371850	+	0.928293i	$0.378723\pi$
$570$	0	0
$571$	−24.3546	−1.01921	−0.509604	−	0.860409i	$-0.670208\pi$
$571$	−24.3546	−1.01921	−0.509604	+	0.860409i	$0.670208\pi$
$572$	0	0
$573$	−28.0498	−1.17180
$574$	0	0
$575$	14.4007	0.600552
$576$	0	0
$577$	19.2930	0.803180	0.401590	−	0.915820i	$-0.368458\pi$
$577$	19.2930	0.803180	0.401590	+	0.915820i	$0.368458\pi$
$578$	0	0
$579$	−13.3298	−0.553968
$580$	0	0
$581$	−7.63478	−0.316744
$582$	0	0
$583$	−0.181146	−0.00750228
$584$	0	0
$585$	−0.0159677	−0.000660185 0
$586$	0	0
$587$	21.4812	0.886625	0.443312	−	0.896367i	$-0.353803\pi$
$587$	21.4812	0.886625	0.443312	+	0.896367i	$0.353803\pi$
$588$	0	0
$589$	5.55725	0.228983
$590$	0	0
$591$	−12.2805	−0.505152
$592$	0	0
$593$	7.21714	0.296373	0.148186	−	0.988959i	$-0.452656\pi$
$593$	7.21714	0.296373	0.148186	+	0.988959i	$0.452656\pi$
$594$	0	0
$595$	7.63834	0.313141
$596$	0	0
$597$	19.7394	0.807879
$598$	0	0
$599$	−21.7141	−0.887213	−0.443606	−	0.896222i	$-0.646301\pi$
$599$	−21.7141	−0.887213	−0.443606	+	0.896222i	$0.646301\pi$
$600$	0	0
$601$	29.2351	1.19252	0.596261	−	0.802790i	$-0.296652\pi$
$601$	29.2351	1.19252	0.596261	+	0.802790i	$0.296652\pi$
$602$	0	0
$603$	−0.0434758	−0.00177047
$604$	0	0
$605$	1.63787	0.0665888
$606$	0	0
$607$	13.0138	0.528212	0.264106	−	0.964494i	$-0.414923\pi$
$607$	13.0138	0.528212	0.264106	+	0.964494i	$0.414923\pi$
$608$	0	0
$609$	10.1342	0.410657
$610$	0	0
$611$	−7.07569	−0.286252
$612$	0	0
$613$	−3.02218	−0.122065	−0.0610323	−	0.998136i	$-0.519439\pi$
$613$	−3.02218	−0.122065	−0.0610323	+	0.998136i	$0.519439\pi$
$614$	0	0
$615$	10.2186	0.412053
$616$	0	0
$617$	8.56386	0.344768	0.172384	−	0.985030i	$-0.444853\pi$
$617$	8.56386	0.344768	0.172384	+	0.985030i	$0.444853\pi$
$618$	0	0
$619$	−18.6278	−0.748715	−0.374357	−	0.927284i	$-0.622137\pi$
$619$	−18.6278	−0.748715	−0.374357	+	0.927284i	$0.622137\pi$
$620$	0	0
$621$	−32.2373	−1.29364
$622$	0	0
$623$	0.284199	0.0113862
$624$	0	0
$625$	−8.04278	−0.321711
$626$	0	0
$627$	1.97687	0.0789485
$628$	0	0
$629$	−40.2402	−1.60448
$630$	0	0
$631$	−34.0341	−1.35488	−0.677438	−	0.735580i	$-0.736910\pi$
$631$	−34.0341	−1.35488	−0.677438	+	0.735580i	$0.736910\pi$
$632$	0	0
$633$	18.5162	0.735954
$634$	0	0
$635$	11.8202	0.469069
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0.0301658	0.00119334
$640$	0	0
$641$	36.1799	1.42902	0.714509	−	0.699626i	$-0.246650\pi$
$641$	36.1799	1.42902	0.714509	+	0.699626i	$0.246650\pi$
$642$	0	0
$643$	−19.2053	−0.757384	−0.378692	−	0.925523i	$-0.623626\pi$
$643$	−19.2053	−0.757384	−0.378692	+	0.925523i	$0.623626\pi$
$644$	0	0
$645$	−3.65268	−0.143824
$646$	0	0
$647$	11.9020	0.467914	0.233957	−	0.972247i	$-0.424832\pi$
$647$	11.9020	0.467914	0.233957	+	0.972247i	$0.424832\pi$
$648$	0	0
$649$	−0.206703	−0.00811380
$650$	0	0
$651$	−8.46083	−0.331606
$652$	0	0
$653$	−1.66683	−0.0652283	−0.0326141	−	0.999468i	$-0.510383\pi$
$653$	−1.66683	−0.0652283	−0.0326141	+	0.999468i	$0.510383\pi$
$654$	0	0
$655$	4.14142	0.161819
$656$	0	0
$657$	−0.0679933	−0.00265267
$658$	0	0
$659$	−4.02811	−0.156913	−0.0784564	−	0.996918i	$-0.524999\pi$
$659$	−4.02811	−0.156913	−0.0784564	+	0.996918i	$0.524999\pi$
$660$	0	0
$661$	−8.31714	−0.323499	−0.161750	−	0.986832i	$-0.551714\pi$
$661$	−8.31714	−0.323499	−0.161750	+	0.986832i	$0.551714\pi$
$662$	0	0
$663$	8.09068	0.314216
$664$	0	0
$665$	1.86634	0.0723737
$666$	0	0
$667$	36.3002	1.40555
$668$	0	0
$669$	29.6559	1.14656
$670$	0	0
$671$	5.76708	0.222635
$672$	0	0
$673$	−14.4505	−0.557027	−0.278513	−	0.960432i	$-0.589842\pi$
$673$	−14.4505	−0.557027	−0.278513	+	0.960432i	$0.589842\pi$
$674$	0	0
$675$	−12.0219	−0.462721
$676$	0	0
$677$	22.9742	0.882971	0.441485	−	0.897268i	$-0.354452\pi$
$677$	22.9742	0.882971	0.441485	+	0.897268i	$0.354452\pi$
$678$	0	0
$679$	0.615716	0.0236290
$680$	0	0
$681$	4.40495	0.168798
$682$	0	0
$683$	13.4897	0.516168	0.258084	−	0.966122i	$-0.416909\pi$
$683$	13.4897	0.516168	0.258084	+	0.966122i	$0.416909\pi$
$684$	0	0
$685$	−20.5846	−0.786497
$686$	0	0
$687$	−37.9801	−1.44903
$688$	0	0
$689$	−0.181146	−0.00690110
$690$	0	0
$691$	2.54231	0.0967141	0.0483571	−	0.998830i	$-0.484601\pi$
$691$	2.54231	0.0967141	0.0483571	+	0.998830i	$0.484601\pi$
$692$	0	0
$693$	−0.00974910	−0.000370337 0
$694$	0	0
$695$	−10.9497	−0.415347
$696$	0	0
$697$	−16.7713	−0.635257
$698$	0	0
$699$	22.8066	0.862623
$700$	0	0
$701$	9.10882	0.344035	0.172018	−	0.985094i	$-0.444971\pi$
$701$	9.10882	0.344035	0.172018	+	0.985094i	$0.444971\pi$
$702$	0	0
$703$	−9.83224	−0.370830
$704$	0	0
$705$	−20.1054	−0.757214
$706$	0	0
$707$	−11.5796	−0.435495
$708$	0	0
$709$	24.6986	0.927577	0.463788	−	0.885946i	$-0.346490\pi$
$709$	24.6986	0.927577	0.463788	+	0.885946i	$0.346490\pi$
$710$	0	0
$711$	0.0441153	0.00165445
$712$	0	0
$713$	−30.3063	−1.13498
$714$	0	0
$715$	1.63787	0.0612528
$716$	0	0
$717$	28.3497	1.05874
$718$	0	0
$719$	29.0789	1.08446	0.542230	−	0.840230i	$-0.317580\pi$
$719$	29.0789	1.08446	0.542230	+	0.840230i	$0.317580\pi$
$720$	0	0
$721$	−3.86054	−0.143774
$722$	0	0
$723$	29.4066	1.09364
$724$	0	0
$725$	13.5370	0.502750
$726$	0	0
$727$	23.9744	0.889160	0.444580	−	0.895739i	$-0.353353\pi$
$727$	23.9744	0.889160	0.444580	+	0.895739i	$0.353353\pi$
$728$	0	0
$729$	26.9117	0.996729
$730$	0	0
$731$	5.99497	0.221732
$732$	0	0
$733$	−20.4024	−0.753579	−0.376789	−	0.926299i	$-0.622972\pi$
$733$	−20.4024	−0.753579	−0.376789	+	0.926299i	$0.622972\pi$
$734$	0	0
$735$	−2.84148	−0.104809
$736$	0	0
$737$	4.45947	0.164267
$738$	0	0
$739$	−23.9158	−0.879755	−0.439878	−	0.898058i	$-0.644978\pi$
$739$	−23.9158	−0.879755	−0.439878	+	0.898058i	$0.644978\pi$
$740$	0	0
$741$	1.97687	0.0726221
$742$	0	0
$743$	39.8674	1.46259	0.731297	−	0.682059i	$-0.238915\pi$
$743$	39.8674	1.46259	0.731297	+	0.682059i	$0.238915\pi$
$744$	0	0
$745$	24.5707	0.900201
$746$	0	0
$747$	−0.0744322	−0.00272333
$748$	0	0
$749$	14.4321	0.527338
$750$	0	0
$751$	6.01398	0.219453	0.109727	−	0.993962i	$-0.465002\pi$
$751$	6.01398	0.219453	0.109727	+	0.993962i	$0.465002\pi$
$752$	0	0
$753$	15.3599	0.559746
$754$	0	0
$755$	−0.934667	−0.0340160
$756$	0	0
$757$	19.7106	0.716393	0.358196	−	0.933646i	$-0.383392\pi$
$757$	19.7106	0.716393	0.358196	+	0.933646i	$0.383392\pi$
$758$	0	0
$759$	−10.7808	−0.391318
$760$	0	0
$761$	54.9847	1.99319	0.996597	−	0.0824252i	$-0.0262665\pi$
$761$	54.9847	1.99319	0.996597	+	0.0824252i	$0.0262665\pi$
$762$	0	0
$763$	2.47269	0.0895174
$764$	0	0
$765$	0.0744669	0.00269236
$766$	0	0
$767$	−0.206703	−0.00746361
$768$	0	0
$769$	0.645787	0.0232877	0.0116438	−	0.999932i	$-0.496294\pi$
$769$	0.645787	0.0232877	0.0116438	+	0.999932i	$0.496294\pi$
$770$	0	0
$771$	11.3023	0.407043
$772$	0	0
$773$	39.1272	1.40731	0.703653	−	0.710543i	$-0.251551\pi$
$773$	39.1272	1.40731	0.703653	+	0.710543i	$0.251551\pi$
$774$	0	0
$775$	−11.3018	−0.405971
$776$	0	0
$777$	14.9694	0.537025
$778$	0	0
$779$	−4.09787	−0.146822
$780$	0	0
$781$	−3.09421	−0.110720
$782$	0	0
$783$	−30.3037	−1.08297
$784$	0	0
$785$	−8.02536	−0.286437
$786$	0	0
$787$	12.8801	0.459127	0.229564	−	0.973294i	$-0.426270\pi$
$787$	12.8801	0.459127	0.229564	+	0.973294i	$0.426270\pi$
$788$	0	0
$789$	−9.96617	−0.354805
$790$	0	0
$791$	−9.91686	−0.352603
$792$	0	0
$793$	5.76708	0.204795
$794$	0	0
$795$	−0.514721	−0.0182553
$796$	0	0
$797$	−4.59881	−0.162898	−0.0814491	−	0.996677i	$-0.525955\pi$
$797$	−4.59881	−0.162898	−0.0814491	+	0.996677i	$0.525955\pi$
$798$	0	0
$799$	32.9981	1.16739
$800$	0	0
$801$	0.00277068	9.78973e−5 0
$802$	0	0
$803$	6.97432	0.246118
$804$	0	0
$805$	−10.1781	−0.358729
$806$	0	0
$807$	41.9346	1.47617
$808$	0	0
$809$	29.8098	1.04806	0.524029	−	0.851700i	$-0.324428\pi$
$809$	29.8098	1.04806	0.524029	+	0.851700i	$0.324428\pi$
$810$	0	0
$811$	−29.8971	−1.04983	−0.524915	−	0.851155i	$-0.675903\pi$
$811$	−29.8971	−1.04983	−0.524915	+	0.851155i	$0.675903\pi$
$812$	0	0
$813$	6.82286	0.239288
$814$	0	0
$815$	−12.1507	−0.425622
$816$	0	0
$817$	1.46481	0.0512470
$818$	0	0
$819$	−0.00974910	−0.000340661 0
$820$	0	0
$821$	−29.0115	−1.01251	−0.506255	−	0.862384i	$-0.668970\pi$
$821$	−29.0115	−1.01251	−0.506255	+	0.862384i	$0.668970\pi$
$822$	0	0
$823$	31.8663	1.11079	0.555394	−	0.831587i	$-0.312567\pi$
$823$	31.8663	1.11079	0.555394	+	0.831587i	$0.312567\pi$
$824$	0	0
$825$	−4.02035	−0.139971
$826$	0	0
$827$	−47.6232	−1.65602	−0.828011	−	0.560712i	$-0.810527\pi$
$827$	−47.6232	−1.65602	−0.828011	+	0.560712i	$0.810527\pi$
$828$	0	0
$829$	−45.1325	−1.56752	−0.783758	−	0.621067i	$-0.786700\pi$
$829$	−45.1325	−1.56752	−0.783758	+	0.621067i	$0.786700\pi$
$830$	0	0
$831$	29.5564	1.02530
$832$	0	0
$833$	4.66359	0.161584
$834$	0	0
$835$	−23.9248	−0.827951
$836$	0	0
$837$	25.3000	0.874496
$838$	0	0
$839$	47.3419	1.63443	0.817213	−	0.576336i	$-0.195518\pi$
$839$	47.3419	1.63443	0.817213	+	0.576336i	$0.195518\pi$
$840$	0	0
$841$	5.12285	0.176650
$842$	0	0
$843$	−8.17597	−0.281595
$844$	0	0
$845$	1.63787	0.0563444
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	6.84539	0.234933
$850$	0	0
$851$	53.6199	1.83806
$852$	0	0
$853$	19.1241	0.654796	0.327398	−	0.944887i	$-0.393828\pi$
$853$	19.1241	0.654796	0.327398	+	0.944887i	$0.393828\pi$
$854$	0	0
$855$	0.0181952	0.000622261 0
$856$	0	0
$857$	12.5360	0.428223	0.214111	−	0.976809i	$-0.431314\pi$
$857$	12.5360	0.428223	0.214111	+	0.976809i	$0.431314\pi$
$858$	0	0
$859$	−11.3818	−0.388342	−0.194171	−	0.980968i	$-0.562202\pi$
$859$	−11.3818	−0.388342	−0.194171	+	0.980968i	$0.562202\pi$
$860$	0	0
$861$	6.23895	0.212623
$862$	0	0
$863$	−0.0466353	−0.00158748	−0.000793741	−	1.00000i	$-0.500253\pi$
$863$	−0.0466353	−0.00158748	−0.000793741		1.00000i	$0.500253\pi$
$864$	0	0
$865$	27.0089	0.918329
$866$	0	0
$867$	−8.23891	−0.279808
$868$	0	0
$869$	−4.52507	−0.153502
$870$	0	0
$871$	4.45947	0.151103
$872$	0	0
$873$	0.00600268	0.000203160 0
$874$	0	0
$875$	−11.9849	−0.405164
$876$	0	0
$877$	30.3083	1.02344	0.511719	−	0.859153i	$-0.329009\pi$
$877$	30.3083	1.02344	0.511719	+	0.859153i	$0.329009\pi$
$878$	0	0
$879$	23.2533	0.784313
$880$	0	0
$881$	21.9104	0.738179	0.369089	−	0.929394i	$-0.379670\pi$
$881$	21.9104	0.738179	0.369089	+	0.929394i	$0.379670\pi$
$882$	0	0
$883$	25.1127	0.845110	0.422555	−	0.906337i	$-0.361133\pi$
$883$	25.1127	0.845110	0.422555	+	0.906337i	$0.361133\pi$
$884$	0	0
$885$	−0.587341	−0.0197433
$886$	0	0
$887$	−5.10949	−0.171560	−0.0857799	−	0.996314i	$-0.527338\pi$
$887$	−5.10949	−0.171560	−0.0857799	+	0.996314i	$0.527338\pi$
$888$	0	0
$889$	7.21679	0.242043
$890$	0	0
$891$	9.02915	0.302488
$892$	0	0
$893$	8.06272	0.269809
$894$	0	0
$895$	−25.8151	−0.862903
$896$	0	0
$897$	−10.7808	−0.359960
$898$	0	0
$899$	−28.4886	−0.950146
$900$	0	0
$901$	0.844788	0.0281440
$902$	0	0
$903$	−2.23014	−0.0742145
$904$	0	0
$905$	−24.6973	−0.820967
$906$	0	0
$907$	7.19256	0.238825	0.119413	−	0.992845i	$-0.461899\pi$
$907$	7.19256	0.238825	0.119413	+	0.992845i	$0.461899\pi$
$908$	0	0
$909$	−0.112891	−0.00374434
$910$	0	0
$911$	30.6165	1.01437	0.507184	−	0.861838i	$-0.330686\pi$
$911$	30.6165	1.01437	0.507184	+	0.861838i	$0.330686\pi$
$912$	0	0
$913$	7.63478	0.252674
$914$	0	0
$915$	16.3870	0.541738
$916$	0	0
$917$	2.52854	0.0834997
$918$	0	0
$919$	−52.6554	−1.73694	−0.868470	−	0.495741i	$-0.834896\pi$
$919$	−52.6554	−1.73694	−0.868470	+	0.495741i	$0.834896\pi$
$920$	0	0
$921$	28.5341	0.940229
$922$	0	0
$923$	−3.09421	−0.101847
$924$	0	0
$925$	19.9958	0.657457
$926$	0	0
$927$	−0.0376368	−0.00123615
$928$	0	0
$929$	−17.9958	−0.590423	−0.295211	−	0.955432i	$-0.595390\pi$
$929$	−17.9958	−0.590423	−0.295211	+	0.955432i	$0.595390\pi$
$930$	0	0
$931$	1.13950	0.0373454
$932$	0	0
$933$	−13.5660	−0.444130
$934$	0	0
$935$	−7.63834	−0.249800
$936$	0	0
$937$	14.1337	0.461729	0.230865	−	0.972986i	$-0.425845\pi$
$937$	14.1337	0.461729	0.230865	+	0.972986i	$0.425845\pi$
$938$	0	0
$939$	−0.232191	−0.00757727
$940$	0	0
$941$	−4.85933	−0.158409	−0.0792047	−	0.996858i	$-0.525238\pi$
$941$	−4.85933	−0.158409	−0.0792047	+	0.996858i	$0.525238\pi$
$942$	0	0
$943$	22.3476	0.727740
$944$	0	0
$945$	8.49673	0.276399
$946$	0	0
$947$	−14.4556	−0.469745	−0.234873	−	0.972026i	$-0.575467\pi$
$947$	−14.4556	−0.469745	−0.234873	+	0.972026i	$0.575467\pi$
$948$	0	0
$949$	6.97432	0.226396
$950$	0	0
$951$	48.4616	1.57148
$952$	0	0
$953$	0.115449	0.00373975	0.00186987	−	0.999998i	$-0.499405\pi$
$953$	0.115449	0.00373975	0.00186987	+	0.999998i	$0.499405\pi$
$954$	0	0
$955$	26.4816	0.856925
$956$	0	0
$957$	−10.1342	−0.327591
$958$	0	0
$959$	−12.5679	−0.405839
$960$	0	0
$961$	−7.21543	−0.232756
$962$	0	0
$963$	0.140700	0.00453399
$964$	0	0
$965$	12.5846	0.405111
$966$	0	0
$967$	−6.83521	−0.219806	−0.109903	−	0.993942i	$-0.535054\pi$
$967$	−6.83521	−0.219806	−0.109903	+	0.993942i	$0.535054\pi$
$968$	0	0
$969$	−9.21929	−0.296166
$970$	0	0
$971$	18.9294	0.607474	0.303737	−	0.952756i	$-0.401766\pi$
$971$	18.9294	0.607474	0.303737	+	0.952756i	$0.401766\pi$
$972$	0	0
$973$	−6.68536	−0.214323
$974$	0	0
$975$	−4.02035	−0.128754
$976$	0	0
$977$	14.9280	0.477591	0.238795	−	0.971070i	$-0.423248\pi$
$977$	14.9280	0.477591	0.238795	+	0.971070i	$0.423248\pi$
$978$	0	0
$979$	−0.284199	−0.00908304
$980$	0	0
$981$	0.0241065	0.000769661 0
$982$	0	0
$983$	−26.3488	−0.840396	−0.420198	−	0.907433i	$-0.638039\pi$
$983$	−26.3488	−0.840396	−0.420198	+	0.907433i	$0.638039\pi$
$984$	0	0
$985$	11.5939	0.369412
$986$	0	0
$987$	−12.2754	−0.390729
$988$	0	0
$989$	−7.98828	−0.254012
$990$	0	0
$991$	34.7656	1.10437	0.552184	−	0.833723i	$-0.313795\pi$
$991$	34.7656	1.10437	0.552184	+	0.833723i	$0.313795\pi$
$992$	0	0
$993$	29.6832	0.941968
$994$	0	0
$995$	−18.6358	−0.590794
$996$	0	0
$997$	−45.0505	−1.42676	−0.713382	−	0.700776i	$-0.752837\pi$
$997$	−45.0505	−1.42676	−0.713382	+	0.700776i	$0.752837\pi$
$998$	0	0
$999$	−44.7623	−1.41622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.g.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.g.1.2	✓	6	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73486 q^{3} +1.63787 q^{5} +1.00000 q^{7} +0.00974910 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.84148 q^{15} +4.66359 q^{17} +1.13950 q^{19} -1.73486 q^{21} -6.21421 q^{23} -2.31739 q^{25} +5.18768 q^{27} -5.84148 q^{29} +4.87694 q^{31} +1.73486 q^{33} +1.63787 q^{35} -8.62859 q^{37} +1.73486 q^{39} -3.59622 q^{41} +1.28549 q^{43} +0.0159677 q^{45} +7.07569 q^{47} +1.00000 q^{49} -8.09068 q^{51} +0.181146 q^{53} -1.63787 q^{55} -1.97687 q^{57} +0.206703 q^{59} -5.76708 q^{61} +0.00974910 q^{63} -1.63787 q^{65} -4.45947 q^{67} +10.7808 q^{69} +3.09421 q^{71} -6.97432 q^{73} +4.02035 q^{75} -1.00000 q^{77} +4.52507 q^{79} -9.02915 q^{81} -7.63478 q^{83} +7.63834 q^{85} +10.1342 q^{87} +0.284199 q^{89} -1.00000 q^{91} -8.46083 q^{93} +1.86634 q^{95} +0.615716 q^{97} -0.00974910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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