LMFDB - Embedded newform 8004.2.a.e.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8004,2,Mod(1,8004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8004.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-9,0,-3,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.25739$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.650765 q^{5} -1.33403 q^{7} +1.00000 q^{9} +5.13354 q^{11} +1.17838 q^{13} +0.650765 q^{15} -0.132625 q^{17} -1.33048 q^{19} +1.33403 q^{21} -1.00000 q^{23} -4.57650 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.92836 q^{31} -5.13354 q^{33} +0.868143 q^{35} -2.05624 q^{37} -1.17838 q^{39} -10.6659 q^{41} +5.90191 q^{43} -0.650765 q^{45} -6.07807 q^{47} -5.22035 q^{49} +0.132625 q^{51} +13.1354 q^{53} -3.34073 q^{55} +1.33048 q^{57} -5.88852 q^{59} +9.63773 q^{61} -1.33403 q^{63} -0.766847 q^{65} -5.90527 q^{67} +1.00000 q^{69} -7.38396 q^{71} +4.38565 q^{73} +4.57650 q^{75} -6.84832 q^{77} -7.87595 q^{79} +1.00000 q^{81} +17.7631 q^{83} +0.0863074 q^{85} -1.00000 q^{87} +9.08883 q^{89} -1.57200 q^{91} +1.92836 q^{93} +0.865829 q^{95} -16.7482 q^{97} +5.13354 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.650765	−0.291031	−0.145515	−	0.989356i	$-0.546484\pi$
$5$	−0.650765	−0.291031	−0.145515	+	0.989356i	$0.546484\pi$
$6$	0	0
$7$	−1.33403	−0.504218	−0.252109	−	0.967699i	$-0.581124\pi$
$7$	−1.33403	−0.504218	−0.252109	+	0.967699i	$0.581124\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.13354	1.54782	0.773911	−	0.633294i	$-0.218298\pi$
$11$	5.13354	1.54782	0.773911	+	0.633294i	$0.218298\pi$
$12$	0	0
$13$	1.17838	0.326823	0.163412	−	0.986558i	$-0.447750\pi$
$13$	1.17838	0.326823	0.163412	+	0.986558i	$0.447750\pi$
$14$	0	0
$15$	0.650765	0.168027
$16$	0	0
$17$	−0.132625	−0.0321662	−0.0160831	−	0.999871i	$-0.505120\pi$
$17$	−0.132625	−0.0321662	−0.0160831	+	0.999871i	$0.505120\pi$
$18$	0	0
$19$	−1.33048	−0.305233	−0.152616	−	0.988286i	$-0.548770\pi$
$19$	−1.33048	−0.305233	−0.152616	+	0.988286i	$0.548770\pi$
$20$	0	0
$21$	1.33403	0.291110
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.57650	−0.915301
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.92836	−0.346343	−0.173172	−	0.984892i	$-0.555402\pi$
$31$	−1.92836	−0.346343	−0.173172	+	0.984892i	$0.555402\pi$
$32$	0	0
$33$	−5.13354	−0.893635
$34$	0	0
$35$	0.868143	0.146743
$36$	0	0
$37$	−2.05624	−0.338043	−0.169022	−	0.985612i	$-0.554061\pi$
$37$	−2.05624	−0.338043	−0.169022	+	0.985612i	$0.554061\pi$
$38$	0	0
$39$	−1.17838	−0.188691
$40$	0	0
$41$	−10.6659	−1.66574	−0.832868	−	0.553472i	$-0.813303\pi$
$41$	−10.6659	−1.66574	−0.832868	+	0.553472i	$0.813303\pi$
$42$	0	0
$43$	5.90191	0.900033	0.450016	−	0.893020i	$-0.351418\pi$
$43$	5.90191	0.900033	0.450016	+	0.893020i	$0.351418\pi$
$44$	0	0
$45$	−0.650765	−0.0970103
$46$	0	0
$47$	−6.07807	−0.886578	−0.443289	−	0.896379i	$-0.646188\pi$
$47$	−6.07807	−0.886578	−0.443289	+	0.896379i	$0.646188\pi$
$48$	0	0
$49$	−5.22035	−0.745765
$50$	0	0
$51$	0.132625	0.0185712
$52$	0	0
$53$	13.1354	1.80428	0.902141	−	0.431442i	$-0.141995\pi$
$53$	13.1354	1.80428	0.902141	+	0.431442i	$0.141995\pi$
$54$	0	0
$55$	−3.34073	−0.450464
$56$	0	0
$57$	1.33048	0.176226
$58$	0	0
$59$	−5.88852	−0.766620	−0.383310	−	0.923620i	$-0.625216\pi$
$59$	−5.88852	−0.766620	−0.383310	+	0.923620i	$0.625216\pi$
$60$	0	0
$61$	9.63773	1.23399	0.616993	−	0.786969i	$-0.288351\pi$
$61$	9.63773	1.23399	0.616993	+	0.786969i	$0.288351\pi$
$62$	0	0
$63$	−1.33403	−0.168073
$64$	0	0
$65$	−0.766847	−0.0951156
$66$	0	0
$67$	−5.90527	−0.721444	−0.360722	−	0.932673i	$-0.617470\pi$
$67$	−5.90527	−0.721444	−0.360722	+	0.932673i	$0.617470\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.38396	−0.876314	−0.438157	−	0.898898i	$-0.644369\pi$
$71$	−7.38396	−0.876314	−0.438157	+	0.898898i	$0.644369\pi$
$72$	0	0
$73$	4.38565	0.513302	0.256651	−	0.966504i	$-0.417381\pi$
$73$	4.38565	0.513302	0.256651	+	0.966504i	$0.417381\pi$
$74$	0	0
$75$	4.57650	0.528449
$76$	0	0
$77$	−6.84832	−0.780439
$78$	0	0
$79$	−7.87595	−0.886114	−0.443057	−	0.896493i	$-0.646106\pi$
$79$	−7.87595	−0.886114	−0.443057	+	0.896493i	$0.646106\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.7631	1.94975	0.974877	−	0.222744i	$-0.0715014\pi$
$83$	17.7631	1.94975	0.974877	+	0.222744i	$0.0715014\pi$
$84$	0	0
$85$	0.0863074	0.00936136
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	9.08883	0.963414	0.481707	−	0.876332i	$-0.340017\pi$
$89$	9.08883	0.963414	0.481707	+	0.876332i	$0.340017\pi$
$90$	0	0
$91$	−1.57200	−0.164790
$92$	0	0
$93$	1.92836	0.199961
$94$	0	0
$95$	0.865829	0.0888321
$96$	0	0
$97$	−16.7482	−1.70053	−0.850263	−	0.526358i	$-0.823557\pi$
$97$	−16.7482	−1.70053	−0.850263	+	0.526358i	$0.823557\pi$
$98$	0	0
$99$	5.13354	0.515941
$100$	0	0
$101$	9.82078	0.977204	0.488602	−	0.872507i	$-0.337507\pi$
$101$	9.82078	0.977204	0.488602	+	0.872507i	$0.337507\pi$
$102$	0	0
$103$	7.80111	0.768667	0.384333	−	0.923194i	$-0.374431\pi$
$103$	7.80111	0.768667	0.384333	+	0.923194i	$0.374431\pi$
$104$	0	0
$105$	−0.868143	−0.0847221
$106$	0	0
$107$	−3.93159	−0.380081	−0.190041	−	0.981776i	$-0.560862\pi$
$107$	−3.93159	−0.380081	−0.190041	+	0.981776i	$0.560862\pi$
$108$	0	0
$109$	−10.5756	−1.01296	−0.506480	−	0.862251i	$-0.669054\pi$
$109$	−10.5756	−1.01296	−0.506480	+	0.862251i	$0.669054\pi$
$110$	0	0
$111$	2.05624	0.195169
$112$	0	0
$113$	−12.8588	−1.20966	−0.604829	−	0.796355i	$-0.706759\pi$
$113$	−12.8588	−1.20966	−0.604829	+	0.796355i	$0.706759\pi$
$114$	0	0
$115$	0.650765	0.0606841
$116$	0	0
$117$	1.17838	0.108941
$118$	0	0
$119$	0.176926	0.0162188
$120$	0	0
$121$	15.3533	1.39575
$122$	0	0
$123$	10.6659	0.961713
$124$	0	0
$125$	6.23205	0.557412
$126$	0	0
$127$	1.91677	0.170086	0.0850428	−	0.996377i	$-0.472897\pi$
$127$	1.91677	0.170086	0.0850428	+	0.996377i	$0.472897\pi$
$128$	0	0
$129$	−5.90191	−0.519634
$130$	0	0
$131$	1.66409	0.145392	0.0726962	−	0.997354i	$-0.476840\pi$
$131$	1.66409	0.145392	0.0726962	+	0.997354i	$0.476840\pi$
$132$	0	0
$133$	1.77490	0.153904
$134$	0	0
$135$	0.650765	0.0560089
$136$	0	0
$137$	20.0439	1.71247	0.856233	−	0.516589i	$-0.172799\pi$
$137$	20.0439	1.71247	0.856233	+	0.516589i	$0.172799\pi$
$138$	0	0
$139$	10.0030	0.848445	0.424223	−	0.905558i	$-0.360547\pi$
$139$	10.0030	0.848445	0.424223	+	0.905558i	$0.360547\pi$
$140$	0	0
$141$	6.07807	0.511866
$142$	0	0
$143$	6.04925	0.505864
$144$	0	0
$145$	−0.650765	−0.0540431
$146$	0	0
$147$	5.22035	0.430567
$148$	0	0
$149$	4.34210	0.355719	0.177859	−	0.984056i	$-0.443083\pi$
$149$	4.34210	0.355719	0.177859	+	0.984056i	$0.443083\pi$
$150$	0	0
$151$	−5.85205	−0.476233	−0.238117	−	0.971237i	$-0.576530\pi$
$151$	−5.85205	−0.476233	−0.238117	+	0.971237i	$0.576530\pi$
$152$	0	0
$153$	−0.132625	−0.0107221
$154$	0	0
$155$	1.25491	0.100797
$156$	0	0
$157$	−0.295720	−0.0236010	−0.0118005	−	0.999930i	$-0.503756\pi$
$157$	−0.295720	−0.0236010	−0.0118005	+	0.999930i	$0.503756\pi$
$158$	0	0
$159$	−13.1354	−1.04170
$160$	0	0
$161$	1.33403	0.105137
$162$	0	0
$163$	−2.66668	−0.208871	−0.104435	−	0.994532i	$-0.533304\pi$
$163$	−2.66668	−0.208871	−0.104435	+	0.994532i	$0.533304\pi$
$164$	0	0
$165$	3.34073	0.260076
$166$	0	0
$167$	−19.6691	−1.52204	−0.761020	−	0.648728i	$-0.775301\pi$
$167$	−19.6691	−1.52204	−0.761020	+	0.648728i	$0.775301\pi$
$168$	0	0
$169$	−11.6114	−0.893187
$170$	0	0
$171$	−1.33048	−0.101744
$172$	0	0
$173$	−12.4733	−0.948325	−0.474163	−	0.880437i	$-0.657249\pi$
$173$	−12.4733	−0.948325	−0.474163	+	0.880437i	$0.657249\pi$
$174$	0	0
$175$	6.10521	0.461511
$176$	0	0
$177$	5.88852	0.442608
$178$	0	0
$179$	−4.84197	−0.361906	−0.180953	−	0.983492i	$-0.557918\pi$
$179$	−4.84197	−0.361906	−0.180953	+	0.983492i	$0.557918\pi$
$180$	0	0
$181$	−4.99870	−0.371551	−0.185775	−	0.982592i	$-0.559480\pi$
$181$	−4.99870	−0.371551	−0.185775	+	0.982592i	$0.559480\pi$
$182$	0	0
$183$	−9.63773	−0.712442
$184$	0	0
$185$	1.33813	0.0983811
$186$	0	0
$187$	−0.680834	−0.0497875
$188$	0	0
$189$	1.33403	0.0970367
$190$	0	0
$191$	−7.01032	−0.507249	−0.253624	−	0.967303i	$-0.581623\pi$
$191$	−7.01032	−0.507249	−0.253624	+	0.967303i	$0.581623\pi$
$192$	0	0
$193$	−13.6972	−0.985947	−0.492973	−	0.870044i	$-0.664090\pi$
$193$	−13.6972	−0.985947	−0.492973	+	0.870044i	$0.664090\pi$
$194$	0	0
$195$	0.766847	0.0549150
$196$	0	0
$197$	−8.40065	−0.598521	−0.299261	−	0.954171i	$-0.596740\pi$
$197$	−8.40065	−0.598521	−0.299261	+	0.954171i	$0.596740\pi$
$198$	0	0
$199$	16.5621	1.17406	0.587029	−	0.809566i	$-0.300298\pi$
$199$	16.5621	1.17406	0.587029	+	0.809566i	$0.300298\pi$
$200$	0	0
$201$	5.90527	0.416526
$202$	0	0
$203$	−1.33403	−0.0936309
$204$	0	0
$205$	6.94100	0.484781
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−6.83007	−0.472446
$210$	0	0
$211$	8.47612	0.583520	0.291760	−	0.956492i	$-0.405759\pi$
$211$	8.47612	0.583520	0.291760	+	0.956492i	$0.405759\pi$
$212$	0	0
$213$	7.38396	0.505940
$214$	0	0
$215$	−3.84076	−0.261937
$216$	0	0
$217$	2.57249	0.174632
$218$	0	0
$219$	−4.38565	−0.296355
$220$	0	0
$221$	−0.156282	−0.0105127
$222$	0	0
$223$	14.2177	0.952085	0.476042	−	0.879422i	$-0.342071\pi$
$223$	14.2177	0.952085	0.476042	+	0.879422i	$0.342071\pi$
$224$	0	0
$225$	−4.57650	−0.305100
$226$	0	0
$227$	5.21362	0.346040	0.173020	−	0.984918i	$-0.444647\pi$
$227$	5.21362	0.346040	0.173020	+	0.984918i	$0.444647\pi$
$228$	0	0
$229$	−19.0310	−1.25761	−0.628803	−	0.777565i	$-0.716455\pi$
$229$	−19.0310	−1.25761	−0.628803	+	0.777565i	$0.716455\pi$
$230$	0	0
$231$	6.84832	0.450587
$232$	0	0
$233$	−26.7954	−1.75543	−0.877713	−	0.479186i	$-0.840932\pi$
$233$	−26.7954	−1.75543	−0.877713	+	0.479186i	$0.840932\pi$
$234$	0	0
$235$	3.95540	0.258022
$236$	0	0
$237$	7.87595	0.511598
$238$	0	0
$239$	2.26026	0.146204	0.0731020	−	0.997324i	$-0.476710\pi$
$239$	2.26026	0.146204	0.0731020	+	0.997324i	$0.476710\pi$
$240$	0	0
$241$	6.92524	0.446094	0.223047	−	0.974808i	$-0.428400\pi$
$241$	6.92524	0.446094	0.223047	+	0.974808i	$0.428400\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.39722	0.217041
$246$	0	0
$247$	−1.56781	−0.0997571
$248$	0	0
$249$	−17.7631	−1.12569
$250$	0	0
$251$	15.1859	0.958527	0.479264	−	0.877671i	$-0.340904\pi$
$251$	15.1859	0.958527	0.479264	+	0.877671i	$0.340904\pi$
$252$	0	0
$253$	−5.13354	−0.322743
$254$	0	0
$255$	−0.0863074	−0.00540478
$256$	0	0
$257$	−15.2246	−0.949688	−0.474844	−	0.880070i	$-0.657495\pi$
$257$	−15.2246	−0.949688	−0.474844	+	0.880070i	$0.657495\pi$
$258$	0	0
$259$	2.74309	0.170447
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−5.21935	−0.321839	−0.160920	−	0.986968i	$-0.551446\pi$
$263$	−5.21935	−0.321839	−0.160920	+	0.986968i	$0.551446\pi$
$264$	0	0
$265$	−8.54804	−0.525102
$266$	0	0
$267$	−9.08883	−0.556227
$268$	0	0
$269$	−29.4706	−1.79685	−0.898426	−	0.439125i	$-0.855289\pi$
$269$	−29.4706	−1.79685	−0.898426	+	0.439125i	$0.855289\pi$
$270$	0	0
$271$	1.04180	0.0632850	0.0316425	−	0.999499i	$-0.489926\pi$
$271$	1.04180	0.0632850	0.0316425	+	0.999499i	$0.489926\pi$
$272$	0	0
$273$	1.57200	0.0951415
$274$	0	0
$275$	−23.4937	−1.41672
$276$	0	0
$277$	−10.2229	−0.614233	−0.307117	−	0.951672i	$-0.599364\pi$
$277$	−10.2229	−0.614233	−0.307117	+	0.951672i	$0.599364\pi$
$278$	0	0
$279$	−1.92836	−0.115448
$280$	0	0
$281$	−13.3230	−0.794783	−0.397392	−	0.917649i	$-0.630085\pi$
$281$	−13.3230	−0.794783	−0.397392	+	0.917649i	$0.630085\pi$
$282$	0	0
$283$	−18.7477	−1.11443	−0.557217	−	0.830367i	$-0.688131\pi$
$283$	−18.7477	−1.11443	−0.557217	+	0.830367i	$0.688131\pi$
$284$	0	0
$285$	−0.865829	−0.0512873
$286$	0	0
$287$	14.2287	0.839893
$288$	0	0
$289$	−16.9824	−0.998965
$290$	0	0
$291$	16.7482	0.981799
$292$	0	0
$293$	−6.11769	−0.357399	−0.178700	−	0.983904i	$-0.557189\pi$
$293$	−6.11769	−0.357399	−0.178700	+	0.983904i	$0.557189\pi$
$294$	0	0
$295$	3.83204	0.223110
$296$	0	0
$297$	−5.13354	−0.297878
$298$	0	0
$299$	−1.17838	−0.0681473
$300$	0	0
$301$	−7.87335	−0.453812
$302$	0	0
$303$	−9.82078	−0.564189
$304$	0	0
$305$	−6.27190	−0.359128
$306$	0	0
$307$	9.66557	0.551643	0.275821	−	0.961209i	$-0.411050\pi$
$307$	9.66557	0.551643	0.275821	+	0.961209i	$0.411050\pi$
$308$	0	0
$309$	−7.80111	−0.443790
$310$	0	0
$311$	−22.9305	−1.30027	−0.650134	−	0.759820i	$-0.725287\pi$
$311$	−22.9305	−1.30027	−0.650134	+	0.759820i	$0.725287\pi$
$312$	0	0
$313$	28.5054	1.61122	0.805610	−	0.592446i	$-0.201837\pi$
$313$	28.5054	1.61122	0.805610	+	0.592446i	$0.201837\pi$
$314$	0	0
$315$	0.868143	0.0489143
$316$	0	0
$317$	−23.3183	−1.30968	−0.654842	−	0.755766i	$-0.727265\pi$
$317$	−23.3183	−1.30968	−0.654842	+	0.755766i	$0.727265\pi$
$318$	0	0
$319$	5.13354	0.287423
$320$	0	0
$321$	3.93159	0.219440
$322$	0	0
$323$	0.176454	0.00981817
$324$	0	0
$325$	−5.39285	−0.299141
$326$	0	0
$327$	10.5756	0.584833
$328$	0	0
$329$	8.10836	0.447028
$330$	0	0
$331$	13.7752	0.757153	0.378577	−	0.925570i	$-0.376414\pi$
$331$	13.7752	0.757153	0.378577	+	0.925570i	$0.376414\pi$
$332$	0	0
$333$	−2.05624	−0.112681
$334$	0	0
$335$	3.84294	0.209962
$336$	0	0
$337$	−12.1266	−0.660580	−0.330290	−	0.943879i	$-0.607147\pi$
$337$	−12.1266	−0.660580	−0.330290	+	0.943879i	$0.607147\pi$
$338$	0	0
$339$	12.8588	0.698397
$340$	0	0
$341$	−9.89931	−0.536077
$342$	0	0
$343$	16.3024	0.880245
$344$	0	0
$345$	−0.650765	−0.0350360
$346$	0	0
$347$	−27.6195	−1.48269	−0.741346	−	0.671123i	$-0.765812\pi$
$347$	−27.6195	−1.48269	−0.741346	+	0.671123i	$0.765812\pi$
$348$	0	0
$349$	−20.5169	−1.09825	−0.549123	−	0.835742i	$-0.685038\pi$
$349$	−20.5169	−1.09825	−0.549123	+	0.835742i	$0.685038\pi$
$350$	0	0
$351$	−1.17838	−0.0628971
$352$	0	0
$353$	32.1730	1.71240	0.856198	−	0.516648i	$-0.172821\pi$
$353$	32.1730	1.71240	0.856198	+	0.516648i	$0.172821\pi$
$354$	0	0
$355$	4.80522	0.255035
$356$	0	0
$357$	−0.176926	−0.00936390
$358$	0	0
$359$	5.54317	0.292557	0.146279	−	0.989243i	$-0.453270\pi$
$359$	5.54317	0.292557	0.146279	+	0.989243i	$0.453270\pi$
$360$	0	0
$361$	−17.2298	−0.906833
$362$	0	0
$363$	−15.3533	−0.805838
$364$	0	0
$365$	−2.85403	−0.149387
$366$	0	0
$367$	−20.1473	−1.05168	−0.525840	−	0.850584i	$-0.676249\pi$
$367$	−20.1473	−1.05168	−0.525840	+	0.850584i	$0.676249\pi$
$368$	0	0
$369$	−10.6659	−0.555245
$370$	0	0
$371$	−17.5230	−0.909751
$372$	0	0
$373$	1.00565	0.0520704	0.0260352	−	0.999661i	$-0.491712\pi$
$373$	1.00565	0.0520704	0.0260352	+	0.999661i	$0.491712\pi$
$374$	0	0
$375$	−6.23205	−0.321822
$376$	0	0
$377$	1.17838	0.0606895
$378$	0	0
$379$	5.99274	0.307827	0.153913	−	0.988084i	$-0.450812\pi$
$379$	5.99274	0.307827	0.153913	+	0.988084i	$0.450812\pi$
$380$	0	0
$381$	−1.91677	−0.0981989
$382$	0	0
$383$	−11.6728	−0.596453	−0.298226	−	0.954495i	$-0.596395\pi$
$383$	−11.6728	−0.596453	−0.298226	+	0.954495i	$0.596395\pi$
$384$	0	0
$385$	4.45665	0.227132
$386$	0	0
$387$	5.90191	0.300011
$388$	0	0
$389$	14.8000	0.750387	0.375194	−	0.926946i	$-0.377576\pi$
$389$	14.8000	0.750387	0.375194	+	0.926946i	$0.377576\pi$
$390$	0	0
$391$	0.132625	0.00670711
$392$	0	0
$393$	−1.66409	−0.0839424
$394$	0	0
$395$	5.12539	0.257887
$396$	0	0
$397$	21.7493	1.09157	0.545783	−	0.837927i	$-0.316232\pi$
$397$	21.7493	1.09157	0.545783	+	0.837927i	$0.316232\pi$
$398$	0	0
$399$	−1.77490	−0.0888563
$400$	0	0
$401$	−30.6226	−1.52922	−0.764610	−	0.644494i	$-0.777068\pi$
$401$	−30.6226	−1.52922	−0.764610	+	0.644494i	$0.777068\pi$
$402$	0	0
$403$	−2.27233	−0.113193
$404$	0	0
$405$	−0.650765	−0.0323368
$406$	0	0
$407$	−10.5558	−0.523231
$408$	0	0
$409$	−3.73481	−0.184674	−0.0923372	−	0.995728i	$-0.529434\pi$
$409$	−3.73481	−0.184674	−0.0923372	+	0.995728i	$0.529434\pi$
$410$	0	0
$411$	−20.0439	−0.988693
$412$	0	0
$413$	7.85549	0.386543
$414$	0	0
$415$	−11.5596	−0.567439
$416$	0	0
$417$	−10.0030	−0.489850
$418$	0	0
$419$	19.6995	0.962383	0.481192	−	0.876615i	$-0.340204\pi$
$419$	19.6995	0.962383	0.481192	+	0.876615i	$0.340204\pi$
$420$	0	0
$421$	−11.9228	−0.581080	−0.290540	−	0.956863i	$-0.593835\pi$
$421$	−11.9228	−0.581080	−0.290540	+	0.956863i	$0.593835\pi$
$422$	0	0
$423$	−6.07807	−0.295526
$424$	0	0
$425$	0.606957	0.0294417
$426$	0	0
$427$	−12.8571	−0.622197
$428$	0	0
$429$	−6.04925	−0.292061
$430$	0	0
$431$	−25.0991	−1.20898	−0.604489	−	0.796613i	$-0.706623\pi$
$431$	−25.0991	−1.20898	−0.604489	+	0.796613i	$0.706623\pi$
$432$	0	0
$433$	1.59603	0.0767003	0.0383502	−	0.999264i	$-0.487790\pi$
$433$	1.59603	0.0767003	0.0383502	+	0.999264i	$0.487790\pi$
$434$	0	0
$435$	0.650765	0.0312018
$436$	0	0
$437$	1.33048	0.0636454
$438$	0	0
$439$	9.17612	0.437953	0.218976	−	0.975730i	$-0.429728\pi$
$439$	9.17612	0.437953	0.218976	+	0.975730i	$0.429728\pi$
$440$	0	0
$441$	−5.22035	−0.248588
$442$	0	0
$443$	−40.2215	−1.91098	−0.955490	−	0.295025i	$-0.904672\pi$
$443$	−40.2215	−1.91098	−0.955490	+	0.295025i	$0.904672\pi$
$444$	0	0
$445$	−5.91469	−0.280383
$446$	0	0
$447$	−4.34210	−0.205374
$448$	0	0
$449$	−37.2563	−1.75823	−0.879117	−	0.476605i	$-0.841867\pi$
$449$	−37.2563	−1.75823	−0.879117	+	0.476605i	$0.841867\pi$
$450$	0	0
$451$	−54.7539	−2.57826
$452$	0	0
$453$	5.85205	0.274953
$454$	0	0
$455$	1.02300	0.0479590
$456$	0	0
$457$	−6.82461	−0.319242	−0.159621	−	0.987178i	$-0.551027\pi$
$457$	−6.82461	−0.319242	−0.159621	+	0.987178i	$0.551027\pi$
$458$	0	0
$459$	0.132625	0.00619039
$460$	0	0
$461$	−12.4719	−0.580873	−0.290436	−	0.956894i	$-0.593800\pi$
$461$	−12.4719	−0.580873	−0.290436	+	0.956894i	$0.593800\pi$
$462$	0	0
$463$	−7.66529	−0.356236	−0.178118	−	0.984009i	$-0.557001\pi$
$463$	−7.66529	−0.356236	−0.178118	+	0.984009i	$0.557001\pi$
$464$	0	0
$465$	−1.25491	−0.0581949
$466$	0	0
$467$	−0.864313	−0.0399956	−0.0199978	−	0.999800i	$-0.506366\pi$
$467$	−0.864313	−0.0399956	−0.0199978	+	0.999800i	$0.506366\pi$
$468$	0	0
$469$	7.87783	0.363765
$470$	0	0
$471$	0.295720	0.0136260
$472$	0	0
$473$	30.2977	1.39309
$474$	0	0
$475$	6.08894	0.279380
$476$	0	0
$477$	13.1354	0.601427
$478$	0	0
$479$	−1.22116	−0.0557960	−0.0278980	−	0.999611i	$-0.508881\pi$
$479$	−1.22116	−0.0557960	−0.0278980	+	0.999611i	$0.508881\pi$
$480$	0	0
$481$	−2.42302	−0.110480
$482$	0	0
$483$	−1.33403	−0.0607007
$484$	0	0
$485$	10.8992	0.494906
$486$	0	0
$487$	30.8011	1.39573	0.697865	−	0.716229i	$-0.254134\pi$
$487$	30.8011	1.39573	0.697865	+	0.716229i	$0.254134\pi$
$488$	0	0
$489$	2.66668	0.120592
$490$	0	0
$491$	−17.1501	−0.773973	−0.386986	−	0.922085i	$-0.626484\pi$
$491$	−17.1501	−0.773973	−0.386986	+	0.922085i	$0.626484\pi$
$492$	0	0
$493$	−0.132625	−0.00597311
$494$	0	0
$495$	−3.34073	−0.150155
$496$	0	0
$497$	9.85045	0.441853
$498$	0	0
$499$	9.43136	0.422206	0.211103	−	0.977464i	$-0.432294\pi$
$499$	9.43136	0.422206	0.211103	+	0.977464i	$0.432294\pi$
$500$	0	0
$501$	19.6691	0.878750
$502$	0	0
$503$	28.4809	1.26990	0.634950	−	0.772553i	$-0.281021\pi$
$503$	28.4809	1.26990	0.634950	+	0.772553i	$0.281021\pi$
$504$	0	0
$505$	−6.39102	−0.284397
$506$	0	0
$507$	11.6114	0.515682
$508$	0	0
$509$	37.5818	1.66578	0.832892	−	0.553436i	$-0.186684\pi$
$509$	37.5818	1.66578	0.832892	+	0.553436i	$0.186684\pi$
$510$	0	0
$511$	−5.85061	−0.258816
$512$	0	0
$513$	1.33048	0.0587420
$514$	0	0
$515$	−5.07669	−0.223706
$516$	0	0
$517$	−31.2021	−1.37226
$518$	0	0
$519$	12.4733	0.547516
$520$	0	0
$521$	−34.7827	−1.52386	−0.761929	−	0.647660i	$-0.775748\pi$
$521$	−34.7827	−1.52386	−0.761929	+	0.647660i	$0.775748\pi$
$522$	0	0
$523$	14.9186	0.652345	0.326172	−	0.945310i	$-0.394241\pi$
$523$	14.9186	0.652345	0.326172	+	0.945310i	$0.394241\pi$
$524$	0	0
$525$	−6.10521	−0.266453
$526$	0	0
$527$	0.255748	0.0111405
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.88852	−0.255540
$532$	0	0
$533$	−12.5685	−0.544401
$534$	0	0
$535$	2.55854	0.110615
$536$	0	0
$537$	4.84197	0.208946
$538$	0	0
$539$	−26.7989	−1.15431
$540$	0	0
$541$	−6.89030	−0.296237	−0.148119	−	0.988970i	$-0.547322\pi$
$541$	−6.89030	−0.296237	−0.148119	+	0.988970i	$0.547322\pi$
$542$	0	0
$543$	4.99870	0.214515
$544$	0	0
$545$	6.88224	0.294803
$546$	0	0
$547$	19.7438	0.844185	0.422092	−	0.906553i	$-0.361296\pi$
$547$	19.7438	0.844185	0.422092	+	0.906553i	$0.361296\pi$
$548$	0	0
$549$	9.63773	0.411328
$550$	0	0
$551$	−1.33048	−0.0566803
$552$	0	0
$553$	10.5068	0.446794
$554$	0	0
$555$	−1.33813	−0.0568004
$556$	0	0
$557$	2.04602	0.0866928	0.0433464	−	0.999060i	$-0.486198\pi$
$557$	2.04602	0.0866928	0.0433464	+	0.999060i	$0.486198\pi$
$558$	0	0
$559$	6.95467	0.294151
$560$	0	0
$561$	0.680834	0.0287448
$562$	0	0
$563$	−43.4542	−1.83138	−0.915688	−	0.401890i	$-0.868353\pi$
$563$	−43.4542	−1.83138	−0.915688	+	0.401890i	$0.868353\pi$
$564$	0	0
$565$	8.36809	0.352048
$566$	0	0
$567$	−1.33403	−0.0560242
$568$	0	0
$569$	−2.78371	−0.116699	−0.0583496	−	0.998296i	$-0.518584\pi$
$569$	−2.78371	−0.116699	−0.0583496	+	0.998296i	$0.518584\pi$
$570$	0	0
$571$	17.4200	0.729004	0.364502	−	0.931203i	$-0.381239\pi$
$571$	17.4200	0.729004	0.364502	+	0.931203i	$0.381239\pi$
$572$	0	0
$573$	7.01032	0.292860
$574$	0	0
$575$	4.57650	0.190853
$576$	0	0
$577$	25.1597	1.04741	0.523706	−	0.851899i	$-0.324549\pi$
$577$	25.1597	1.04741	0.523706	+	0.851899i	$0.324549\pi$
$578$	0	0
$579$	13.6972	0.569237
$580$	0	0
$581$	−23.6966	−0.983100
$582$	0	0
$583$	67.4310	2.79271
$584$	0	0
$585$	−0.766847	−0.0317052
$586$	0	0
$587$	−11.2993	−0.466373	−0.233187	−	0.972432i	$-0.574915\pi$
$587$	−11.2993	−0.466373	−0.233187	+	0.972432i	$0.574915\pi$
$588$	0	0
$589$	2.56564	0.105715
$590$	0	0
$591$	8.40065	0.345556
$592$	0	0
$593$	21.8453	0.897077	0.448539	−	0.893763i	$-0.351945\pi$
$593$	21.8453	0.897077	0.448539	+	0.893763i	$0.351945\pi$
$594$	0	0
$595$	−0.115137	−0.00472016
$596$	0	0
$597$	−16.5621	−0.677842
$598$	0	0
$599$	−13.5125	−0.552104	−0.276052	−	0.961143i	$-0.589026\pi$
$599$	−13.5125	−0.552104	−0.276052	+	0.961143i	$0.589026\pi$
$600$	0	0
$601$	−34.8960	−1.42344	−0.711719	−	0.702464i	$-0.752083\pi$
$601$	−34.8960	−1.42344	−0.711719	+	0.702464i	$0.752083\pi$
$602$	0	0
$603$	−5.90527	−0.240481
$604$	0	0
$605$	−9.99137	−0.406207
$606$	0	0
$607$	24.2382	0.983798	0.491899	−	0.870652i	$-0.336303\pi$
$607$	24.2382	0.983798	0.491899	+	0.870652i	$0.336303\pi$
$608$	0	0
$609$	1.33403	0.0540578
$610$	0	0
$611$	−7.16226	−0.289754
$612$	0	0
$613$	−31.7548	−1.28256	−0.641282	−	0.767305i	$-0.721597\pi$
$613$	−31.7548	−1.28256	−0.641282	+	0.767305i	$0.721597\pi$
$614$	0	0
$615$	−6.94100	−0.279888
$616$	0	0
$617$	10.6331	0.428071	0.214036	−	0.976826i	$-0.431339\pi$
$617$	10.6331	0.428071	0.214036	+	0.976826i	$0.431339\pi$
$618$	0	0
$619$	30.3614	1.22033	0.610163	−	0.792276i	$-0.291104\pi$
$619$	30.3614	1.22033	0.610163	+	0.792276i	$0.291104\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−12.1248	−0.485770
$624$	0	0
$625$	18.8269	0.753077
$626$	0	0
$627$	6.83007	0.272767
$628$	0	0
$629$	0.272708	0.0108736
$630$	0	0
$631$	22.5240	0.896667	0.448334	−	0.893866i	$-0.352018\pi$
$631$	22.5240	0.896667	0.448334	+	0.893866i	$0.352018\pi$
$632$	0	0
$633$	−8.47612	−0.336895
$634$	0	0
$635$	−1.24736	−0.0495001
$636$	0	0
$637$	−6.15154	−0.243733
$638$	0	0
$639$	−7.38396	−0.292105
$640$	0	0
$641$	−15.2156	−0.600981	−0.300491	−	0.953785i	$-0.597150\pi$
$641$	−15.2156	−0.600981	−0.300491	+	0.953785i	$0.597150\pi$
$642$	0	0
$643$	−24.7646	−0.976621	−0.488311	−	0.872670i	$-0.662387\pi$
$643$	−24.7646	−0.976621	−0.488311	+	0.872670i	$0.662387\pi$
$644$	0	0
$645$	3.84076	0.151230
$646$	0	0
$647$	29.8978	1.17541	0.587703	−	0.809077i	$-0.300032\pi$
$647$	29.8978	1.17541	0.587703	+	0.809077i	$0.300032\pi$
$648$	0	0
$649$	−30.2290	−1.18659
$650$	0	0
$651$	−2.57249	−0.100824
$652$	0	0
$653$	−15.5766	−0.609558	−0.304779	−	0.952423i	$-0.598583\pi$
$653$	−15.5766	−0.609558	−0.304779	+	0.952423i	$0.598583\pi$
$654$	0	0
$655$	−1.08293	−0.0423137
$656$	0	0
$657$	4.38565	0.171101
$658$	0	0
$659$	−28.9820	−1.12898	−0.564490	−	0.825440i	$-0.690927\pi$
$659$	−28.9820	−1.12898	−0.564490	+	0.825440i	$0.690927\pi$
$660$	0	0
$661$	22.7430	0.884601	0.442301	−	0.896867i	$-0.354162\pi$
$661$	22.7430	0.884601	0.442301	+	0.896867i	$0.354162\pi$
$662$	0	0
$663$	0.156282	0.00606948
$664$	0	0
$665$	−1.15505	−0.0447907
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−14.2177	−0.549686
$670$	0	0
$671$	49.4757	1.90999
$672$	0	0
$673$	−20.2767	−0.781611	−0.390805	−	0.920473i	$-0.627803\pi$
$673$	−20.2767	−0.781611	−0.390805	+	0.920473i	$0.627803\pi$
$674$	0	0
$675$	4.57650	0.176150
$676$	0	0
$677$	−2.62787	−0.100997	−0.0504987	−	0.998724i	$-0.516081\pi$
$677$	−2.62787	−0.100997	−0.0504987	+	0.998724i	$0.516081\pi$
$678$	0	0
$679$	22.3427	0.857435
$680$	0	0
$681$	−5.21362	−0.199786
$682$	0	0
$683$	21.8387	0.835633	0.417817	−	0.908531i	$-0.362795\pi$
$683$	21.8387	0.835633	0.417817	+	0.908531i	$0.362795\pi$
$684$	0	0
$685$	−13.0439	−0.498381
$686$	0	0
$687$	19.0310	0.726079
$688$	0	0
$689$	15.4784	0.589681
$690$	0	0
$691$	−34.7850	−1.32328	−0.661642	−	0.749820i	$-0.730140\pi$
$691$	−34.7850	−1.32328	−0.661642	+	0.749820i	$0.730140\pi$
$692$	0	0
$693$	−6.84832	−0.260146
$694$	0	0
$695$	−6.50961	−0.246924
$696$	0	0
$697$	1.41456	0.0535804
$698$	0	0
$699$	26.7954	1.01350
$700$	0	0
$701$	−8.72029	−0.329361	−0.164680	−	0.986347i	$-0.552659\pi$
$701$	−8.72029	−0.329361	−0.164680	+	0.986347i	$0.552659\pi$
$702$	0	0
$703$	2.73578	0.103182
$704$	0	0
$705$	−3.95540	−0.148969
$706$	0	0
$707$	−13.1013	−0.492723
$708$	0	0
$709$	−2.64918	−0.0994922	−0.0497461	−	0.998762i	$-0.515841\pi$
$709$	−2.64918	−0.0994922	−0.0497461	+	0.998762i	$0.515841\pi$
$710$	0	0
$711$	−7.87595	−0.295371
$712$	0	0
$713$	1.92836	0.0722175
$714$	0	0
$715$	−3.93664	−0.147222
$716$	0	0
$717$	−2.26026	−0.0844109
$718$	0	0
$719$	14.3115	0.533728	0.266864	−	0.963734i	$-0.414013\pi$
$719$	14.3115	0.533728	0.266864	+	0.963734i	$0.414013\pi$
$720$	0	0
$721$	−10.4070	−0.387575
$722$	0	0
$723$	−6.92524	−0.257553
$724$	0	0
$725$	−4.57650	−0.169967
$726$	0	0
$727$	19.5436	0.724831	0.362415	−	0.932017i	$-0.381952\pi$
$727$	19.5436	0.724831	0.362415	+	0.932017i	$0.381952\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.782738	−0.0289506
$732$	0	0
$733$	48.5435	1.79299	0.896497	−	0.443050i	$-0.146104\pi$
$733$	48.5435	1.79299	0.896497	+	0.443050i	$0.146104\pi$
$734$	0	0
$735$	−3.39722	−0.125308
$736$	0	0
$737$	−30.3150	−1.11667
$738$	0	0
$739$	−8.43523	−0.310295	−0.155148	−	0.987891i	$-0.549585\pi$
$739$	−8.43523	−0.310295	−0.155148	+	0.987891i	$0.549585\pi$
$740$	0	0
$741$	1.56781	0.0575948
$742$	0	0
$743$	−52.5713	−1.92865	−0.964326	−	0.264717i	$-0.914721\pi$
$743$	−52.5713	−1.92865	−0.964326	+	0.264717i	$0.914721\pi$
$744$	0	0
$745$	−2.82569	−0.103525
$746$	0	0
$747$	17.7631	0.649918
$748$	0	0
$749$	5.24488	0.191644
$750$	0	0
$751$	44.3346	1.61779	0.808897	−	0.587951i	$-0.200065\pi$
$751$	44.3346	1.61779	0.808897	+	0.587951i	$0.200065\pi$
$752$	0	0
$753$	−15.1859	−0.553406
$754$	0	0
$755$	3.80831	0.138599
$756$	0	0
$757$	−31.5742	−1.14759	−0.573793	−	0.819001i	$-0.694528\pi$
$757$	−31.5742	−1.14759	−0.573793	+	0.819001i	$0.694528\pi$
$758$	0	0
$759$	5.13354	0.186336
$760$	0	0
$761$	−17.7051	−0.641811	−0.320905	−	0.947111i	$-0.603987\pi$
$761$	−17.7051	−0.641811	−0.320905	+	0.947111i	$0.603987\pi$
$762$	0	0
$763$	14.1082	0.510753
$764$	0	0
$765$	0.0863074	0.00312045
$766$	0	0
$767$	−6.93890	−0.250549
$768$	0	0
$769$	−16.5687	−0.597481	−0.298741	−	0.954334i	$-0.596567\pi$
$769$	−16.5687	−0.597481	−0.298741	+	0.954334i	$0.596567\pi$
$770$	0	0
$771$	15.2246	0.548302
$772$	0	0
$773$	−10.9271	−0.393021	−0.196511	−	0.980502i	$-0.562961\pi$
$773$	−10.9271	−0.393021	−0.196511	+	0.980502i	$0.562961\pi$
$774$	0	0
$775$	8.82514	0.317008
$776$	0	0
$777$	−2.74309	−0.0984079
$778$	0	0
$779$	14.1908	0.508437
$780$	0	0
$781$	−37.9059	−1.35638
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	0.192444	0.00686862
$786$	0	0
$787$	22.0867	0.787307	0.393654	−	0.919259i	$-0.371211\pi$
$787$	22.0867	0.787307	0.393654	+	0.919259i	$0.371211\pi$
$788$	0	0
$789$	5.21935	0.185814
$790$	0	0
$791$	17.1541	0.609931
$792$	0	0
$793$	11.3569	0.403295
$794$	0	0
$795$	8.54804	0.303168
$796$	0	0
$797$	41.7258	1.47800	0.739001	−	0.673704i	$-0.235298\pi$
$797$	41.7258	1.47800	0.739001	+	0.673704i	$0.235298\pi$
$798$	0	0
$799$	0.806102	0.0285178
$800$	0	0
$801$	9.08883	0.321138
$802$	0	0
$803$	22.5139	0.794500
$804$	0	0
$805$	−0.868143	−0.0305980
$806$	0	0
$807$	29.4706	1.03741
$808$	0	0
$809$	7.73463	0.271935	0.135967	−	0.990713i	$-0.456586\pi$
$809$	7.73463	0.271935	0.135967	+	0.990713i	$0.456586\pi$
$810$	0	0
$811$	−38.0623	−1.33655	−0.668274	−	0.743915i	$-0.732967\pi$
$811$	−38.0623	−1.33655	−0.668274	+	0.743915i	$0.732967\pi$
$812$	0	0
$813$	−1.04180	−0.0365376
$814$	0	0
$815$	1.73538	0.0607878
$816$	0	0
$817$	−7.85236	−0.274719
$818$	0	0
$819$	−1.57200	−0.0549300
$820$	0	0
$821$	−27.6350	−0.964469	−0.482234	−	0.876042i	$-0.660175\pi$
$821$	−27.6350	−0.964469	−0.482234	+	0.876042i	$0.660175\pi$
$822$	0	0
$823$	14.1245	0.492349	0.246174	−	0.969226i	$-0.420826\pi$
$823$	14.1245	0.492349	0.246174	+	0.969226i	$0.420826\pi$
$824$	0	0
$825$	23.4937	0.817945
$826$	0	0
$827$	18.3713	0.638834	0.319417	−	0.947614i	$-0.396513\pi$
$827$	18.3713	0.638834	0.319417	+	0.947614i	$0.396513\pi$
$828$	0	0
$829$	−30.7412	−1.06769	−0.533844	−	0.845583i	$-0.679253\pi$
$829$	−30.7412	−1.06769	−0.533844	+	0.845583i	$0.679253\pi$
$830$	0	0
$831$	10.2229	0.354628
$832$	0	0
$833$	0.692347	0.0239884
$834$	0	0
$835$	12.8000	0.442961
$836$	0	0
$837$	1.92836	0.0666538
$838$	0	0
$839$	−29.4426	−1.01647	−0.508235	−	0.861218i	$-0.669702\pi$
$839$	−29.4426	−1.01647	−0.508235	+	0.861218i	$0.669702\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	13.3230	0.458868
$844$	0	0
$845$	7.55631	0.259945
$846$	0	0
$847$	−20.4818	−0.703763
$848$	0	0
$849$	18.7477	0.643418
$850$	0	0
$851$	2.05624	0.0704869
$852$	0	0
$853$	−6.03614	−0.206673	−0.103337	−	0.994646i	$-0.532952\pi$
$853$	−6.03614	−0.206673	−0.103337	+	0.994646i	$0.532952\pi$
$854$	0	0
$855$	0.865829	0.0296107
$856$	0	0
$857$	−40.2469	−1.37481	−0.687404	−	0.726276i	$-0.741250\pi$
$857$	−40.2469	−1.37481	−0.687404	+	0.726276i	$0.741250\pi$
$858$	0	0
$859$	−12.2606	−0.418326	−0.209163	−	0.977881i	$-0.567074\pi$
$859$	−12.2606	−0.418326	−0.209163	+	0.977881i	$0.567074\pi$
$860$	0	0
$861$	−14.2287	−0.484913
$862$	0	0
$863$	48.5646	1.65316	0.826579	−	0.562820i	$-0.190284\pi$
$863$	48.5646	1.65316	0.826579	+	0.562820i	$0.190284\pi$
$864$	0	0
$865$	8.11717	0.275992
$866$	0	0
$867$	16.9824	0.576753
$868$	0	0
$869$	−40.4316	−1.37155
$870$	0	0
$871$	−6.95864	−0.235784
$872$	0	0
$873$	−16.7482	−0.566842
$874$	0	0
$875$	−8.31377	−0.281057
$876$	0	0
$877$	49.9111	1.68538	0.842688	−	0.538402i	$-0.180972\pi$
$877$	49.9111	1.68538	0.842688	+	0.538402i	$0.180972\pi$
$878$	0	0
$879$	6.11769	0.206345
$880$	0	0
$881$	−34.6328	−1.16681	−0.583405	−	0.812181i	$-0.698280\pi$
$881$	−34.6328	−1.16681	−0.583405	+	0.812181i	$0.698280\pi$
$882$	0	0
$883$	56.3365	1.89587	0.947936	−	0.318460i	$-0.103166\pi$
$883$	56.3365	1.89587	0.947936	+	0.318460i	$0.103166\pi$
$884$	0	0
$885$	−3.83204	−0.128813
$886$	0	0
$887$	−15.9989	−0.537192	−0.268596	−	0.963253i	$-0.586560\pi$
$887$	−15.9989	−0.537192	−0.268596	+	0.963253i	$0.586560\pi$
$888$	0	0
$889$	−2.55703	−0.0857601
$890$	0	0
$891$	5.13354	0.171980
$892$	0	0
$893$	8.08674	0.270613
$894$	0	0
$895$	3.15099	0.105326
$896$	0	0
$897$	1.17838	0.0393449
$898$	0	0
$899$	−1.92836	−0.0643143
$900$	0	0
$901$	−1.74207	−0.0580369
$902$	0	0
$903$	7.87335	0.262009
$904$	0	0
$905$	3.25298	0.108133
$906$	0	0
$907$	−38.8639	−1.29046	−0.645228	−	0.763990i	$-0.723238\pi$
$907$	−38.8639	−1.29046	−0.645228	+	0.763990i	$0.723238\pi$
$908$	0	0
$909$	9.82078	0.325735
$910$	0	0
$911$	−36.2327	−1.20044	−0.600222	−	0.799834i	$-0.704921\pi$
$911$	−36.2327	−1.20044	−0.600222	+	0.799834i	$0.704921\pi$
$912$	0	0
$913$	91.1877	3.01787
$914$	0	0
$915$	6.27190	0.207343
$916$	0	0
$917$	−2.21996	−0.0733094
$918$	0	0
$919$	−46.8721	−1.54617	−0.773083	−	0.634304i	$-0.781287\pi$
$919$	−46.8721	−1.54617	−0.773083	+	0.634304i	$0.781287\pi$
$920$	0	0
$921$	−9.66557	−0.318491
$922$	0	0
$923$	−8.70109	−0.286400
$924$	0	0
$925$	9.41038	0.309412
$926$	0	0
$927$	7.80111	0.256222
$928$	0	0
$929$	27.8959	0.915233	0.457617	−	0.889150i	$-0.348703\pi$
$929$	27.8959	0.915233	0.457617	+	0.889150i	$0.348703\pi$
$930$	0	0
$931$	6.94556	0.227632
$932$	0	0
$933$	22.9305	0.750710
$934$	0	0
$935$	0.443063	0.0144897
$936$	0	0
$937$	−43.6319	−1.42539	−0.712697	−	0.701472i	$-0.752526\pi$
$937$	−43.6319	−1.42539	−0.712697	+	0.701472i	$0.752526\pi$
$938$	0	0
$939$	−28.5054	−0.930239
$940$	0	0
$941$	−41.5445	−1.35431	−0.677155	−	0.735840i	$-0.736787\pi$
$941$	−41.5445	−1.35431	−0.677155	+	0.735840i	$0.736787\pi$
$942$	0	0
$943$	10.6659	0.347330
$944$	0	0
$945$	−0.868143	−0.0282407
$946$	0	0
$947$	52.4514	1.70444	0.852220	−	0.523184i	$-0.175256\pi$
$947$	52.4514	1.70444	0.852220	+	0.523184i	$0.175256\pi$
$948$	0	0
$949$	5.16795	0.167759
$950$	0	0
$951$	23.3183	0.756146
$952$	0	0
$953$	−6.34579	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$953$	−6.34579	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$954$	0	0
$955$	4.56207	0.147625
$956$	0	0
$957$	−5.13354	−0.165944
$958$	0	0
$959$	−26.7393	−0.863456
$960$	0	0
$961$	−27.2814	−0.880046
$962$	0	0
$963$	−3.93159	−0.126694
$964$	0	0
$965$	8.91366	0.286941
$966$	0	0
$967$	−0.221567	−0.00712511	−0.00356255	−	0.999994i	$-0.501134\pi$
$967$	−0.221567	−0.00712511	−0.00356255	+	0.999994i	$0.501134\pi$
$968$	0	0
$969$	−0.176454	−0.00566852
$970$	0	0
$971$	−19.4630	−0.624598	−0.312299	−	0.949984i	$-0.601099\pi$
$971$	−19.4630	−0.624598	−0.312299	+	0.949984i	$0.601099\pi$
$972$	0	0
$973$	−13.3444	−0.427801
$974$	0	0
$975$	5.39285	0.172709
$976$	0	0
$977$	−17.3891	−0.556328	−0.278164	−	0.960534i	$-0.589726\pi$
$977$	−17.3891	−0.556328	−0.278164	+	0.960534i	$0.589726\pi$
$978$	0	0
$979$	46.6579	1.49119
$980$	0	0
$981$	−10.5756	−0.337654
$982$	0	0
$983$	34.9182	1.11372	0.556859	−	0.830607i	$-0.312007\pi$
$983$	34.9182	1.11372	0.556859	+	0.830607i	$0.312007\pi$
$984$	0	0
$985$	5.46685	0.174188
$986$	0	0
$987$	−8.10836	−0.258092
$988$	0	0
$989$	−5.90191	−0.187670
$990$	0	0
$991$	39.5383	1.25598	0.627988	−	0.778223i	$-0.283879\pi$
$991$	39.5383	1.25598	0.627988	+	0.778223i	$0.283879\pi$
$992$	0	0
$993$	−13.7752	−0.437143
$994$	0	0
$995$	−10.7780	−0.341687
$996$	0	0
$997$	29.9035	0.947053	0.473526	−	0.880780i	$-0.342981\pi$
$997$	29.9035	0.947053	0.473526	+	0.880780i	$0.342981\pi$
$998$	0	0
$999$	2.05624	0.0650565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.e.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.4	✓	9	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.650765 q^{5} -1.33403 q^{7} +1.00000 q^{9} +5.13354 q^{11} +1.17838 q^{13} +0.650765 q^{15} -0.132625 q^{17} -1.33048 q^{19} +1.33403 q^{21} -1.00000 q^{23} -4.57650 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.92836 q^{31} -5.13354 q^{33} +0.868143 q^{35} -2.05624 q^{37} -1.17838 q^{39} -10.6659 q^{41} +5.90191 q^{43} -0.650765 q^{45} -6.07807 q^{47} -5.22035 q^{49} +0.132625 q^{51} +13.1354 q^{53} -3.34073 q^{55} +1.33048 q^{57} -5.88852 q^{59} +9.63773 q^{61} -1.33403 q^{63} -0.766847 q^{65} -5.90527 q^{67} +1.00000 q^{69} -7.38396 q^{71} +4.38565 q^{73} +4.57650 q^{75} -6.84832 q^{77} -7.87595 q^{79} +1.00000 q^{81} +17.7631 q^{83} +0.0863074 q^{85} -1.00000 q^{87} +9.08883 q^{89} -1.57200 q^{91} +1.92836 q^{93} +0.865829 q^{95} -16.7482 q^{97} +5.13354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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