LMFDB - Embedded newform 8004.2.a.f.1.7

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)

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);

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")

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,-1,0,-5] 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} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.09926$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.09926 q^{5} -1.62400 q^{7} +1.00000 q^{9} -4.16402 q^{11} +3.95343 q^{13} +1.09926 q^{15} -2.36393 q^{17} +4.37074 q^{19} -1.62400 q^{21} -1.00000 q^{23} -3.79163 q^{25} +1.00000 q^{27} -1.00000 q^{29} -8.09478 q^{31} -4.16402 q^{33} -1.78520 q^{35} +3.52968 q^{37} +3.95343 q^{39} -3.81475 q^{41} +1.25079 q^{43} +1.09926 q^{45} +6.41448 q^{47} -4.36263 q^{49} -2.36393 q^{51} -3.55412 q^{53} -4.57734 q^{55} +4.37074 q^{57} +5.96232 q^{59} -12.3425 q^{61} -1.62400 q^{63} +4.34585 q^{65} -12.8312 q^{67} -1.00000 q^{69} +8.30445 q^{71} -2.26071 q^{73} -3.79163 q^{75} +6.76236 q^{77} -14.7456 q^{79} +1.00000 q^{81} +9.49232 q^{83} -2.59857 q^{85} -1.00000 q^{87} -4.52312 q^{89} -6.42036 q^{91} -8.09478 q^{93} +4.80459 q^{95} -2.79546 q^{97} -4.16402 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * 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
$5$	1.09926	0.491604	0.245802	−	0.969320i	$-0.420949\pi$
$5$	1.09926	0.491604	0.245802	+	0.969320i	$0.420949\pi$
$6$	0	0
$7$	−1.62400	−0.613814	−0.306907	−	0.951740i	$-0.599294\pi$
$7$	−1.62400	−0.613814	−0.306907	+	0.951740i	$0.599294\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.16402	−1.25550	−0.627750	−	0.778415i	$-0.716024\pi$
$11$	−4.16402	−1.25550	−0.627750	+	0.778415i	$0.716024\pi$
$12$	0	0
$13$	3.95343	1.09648	0.548242	−	0.836320i	$-0.315297\pi$
$13$	3.95343	1.09648	0.548242	+	0.836320i	$0.315297\pi$
$14$	0	0
$15$	1.09926	0.283828
$16$	0	0
$17$	−2.36393	−0.573337	−0.286668	−	0.958030i	$-0.592548\pi$
$17$	−2.36393	−0.573337	−0.286668	+	0.958030i	$0.592548\pi$
$18$	0	0
$19$	4.37074	1.00272	0.501359	−	0.865240i	$-0.332834\pi$
$19$	4.37074	1.00272	0.501359	+	0.865240i	$0.332834\pi$
$20$	0	0
$21$	−1.62400	−0.354386
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.79163	−0.758325
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.09478	−1.45387	−0.726933	−	0.686708i	$-0.759055\pi$
$31$	−8.09478	−1.45387	−0.726933	+	0.686708i	$0.759055\pi$
$32$	0	0
$33$	−4.16402	−0.724863
$34$	0	0
$35$	−1.78520	−0.301754
$36$	0	0
$37$	3.52968	0.580276	0.290138	−	0.956985i	$-0.406299\pi$
$37$	3.52968	0.580276	0.290138	+	0.956985i	$0.406299\pi$
$38$	0	0
$39$	3.95343	0.633055
$40$	0	0
$41$	−3.81475	−0.595764	−0.297882	−	0.954603i	$-0.596280\pi$
$41$	−3.81475	−0.595764	−0.297882	+	0.954603i	$0.596280\pi$
$42$	0	0
$43$	1.25079	0.190744	0.0953721	−	0.995442i	$-0.469596\pi$
$43$	1.25079	0.190744	0.0953721	+	0.995442i	$0.469596\pi$
$44$	0	0
$45$	1.09926	0.163868
$46$	0	0
$47$	6.41448	0.935649	0.467824	−	0.883821i	$-0.345038\pi$
$47$	6.41448	0.935649	0.467824	+	0.883821i	$0.345038\pi$
$48$	0	0
$49$	−4.36263	−0.623233
$50$	0	0
$51$	−2.36393	−0.331016
$52$	0	0
$53$	−3.55412	−0.488196	−0.244098	−	0.969751i	$-0.578492\pi$
$53$	−3.55412	−0.488196	−0.244098	+	0.969751i	$0.578492\pi$
$54$	0	0
$55$	−4.57734	−0.617209
$56$	0	0
$57$	4.37074	0.578919
$58$	0	0
$59$	5.96232	0.776228	0.388114	−	0.921611i	$-0.373127\pi$
$59$	5.96232	0.776228	0.388114	+	0.921611i	$0.373127\pi$
$60$	0	0
$61$	−12.3425	−1.58030	−0.790149	−	0.612915i	$-0.789997\pi$
$61$	−12.3425	−1.58030	−0.790149	+	0.612915i	$0.789997\pi$
$62$	0	0
$63$	−1.62400	−0.204605
$64$	0	0
$65$	4.34585	0.539036
$66$	0	0
$67$	−12.8312	−1.56758	−0.783791	−	0.621024i	$-0.786717\pi$
$67$	−12.8312	−1.56758	−0.783791	+	0.621024i	$0.786717\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	8.30445	0.985557	0.492778	−	0.870155i	$-0.335981\pi$
$71$	8.30445	0.985557	0.492778	+	0.870155i	$0.335981\pi$
$72$	0	0
$73$	−2.26071	−0.264596	−0.132298	−	0.991210i	$-0.542236\pi$
$73$	−2.26071	−0.264596	−0.132298	+	0.991210i	$0.542236\pi$
$74$	0	0
$75$	−3.79163	−0.437819
$76$	0	0
$77$	6.76236	0.770643
$78$	0	0
$79$	−14.7456	−1.65900	−0.829502	−	0.558504i	$-0.811376\pi$
$79$	−14.7456	−1.65900	−0.829502	+	0.558504i	$0.811376\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.49232	1.04192	0.520959	−	0.853582i	$-0.325575\pi$
$83$	9.49232	1.04192	0.520959	+	0.853582i	$0.325575\pi$
$84$	0	0
$85$	−2.59857	−0.281855
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−4.52312	−0.479450	−0.239725	−	0.970841i	$-0.577057\pi$
$89$	−4.52312	−0.479450	−0.239725	+	0.970841i	$0.577057\pi$
$90$	0	0
$91$	−6.42036	−0.673036
$92$	0	0
$93$	−8.09478	−0.839390
$94$	0	0
$95$	4.80459	0.492940
$96$	0	0
$97$	−2.79546	−0.283836	−0.141918	−	0.989878i	$-0.545327\pi$
$97$	−2.79546	−0.283836	−0.141918	+	0.989878i	$0.545327\pi$
$98$	0	0
$99$	−4.16402	−0.418500
$100$	0	0
$101$	−2.71767	−0.270418	−0.135209	−	0.990817i	$-0.543171\pi$
$101$	−2.71767	−0.270418	−0.135209	+	0.990817i	$0.543171\pi$
$102$	0	0
$103$	0.577486	0.0569014	0.0284507	−	0.999595i	$-0.490943\pi$
$103$	0.577486	0.0569014	0.0284507	+	0.999595i	$0.490943\pi$
$104$	0	0
$105$	−1.78520	−0.174217
$106$	0	0
$107$	7.55434	0.730306	0.365153	−	0.930948i	$-0.381017\pi$
$107$	7.55434	0.730306	0.365153	+	0.930948i	$0.381017\pi$
$108$	0	0
$109$	−5.65806	−0.541944	−0.270972	−	0.962587i	$-0.587345\pi$
$109$	−5.65806	−0.541944	−0.270972	+	0.962587i	$0.587345\pi$
$110$	0	0
$111$	3.52968	0.335022
$112$	0	0
$113$	−10.3321	−0.971959	−0.485979	−	0.873970i	$-0.661537\pi$
$113$	−10.3321	−0.971959	−0.485979	+	0.873970i	$0.661537\pi$
$114$	0	0
$115$	−1.09926	−0.102507
$116$	0	0
$117$	3.95343	0.365494
$118$	0	0
$119$	3.83902	0.351922
$120$	0	0
$121$	6.33907	0.576279
$122$	0	0
$123$	−3.81475	−0.343964
$124$	0	0
$125$	−9.66429	−0.864400
$126$	0	0
$127$	5.87590	0.521402	0.260701	−	0.965420i	$-0.416046\pi$
$127$	5.87590	0.521402	0.260701	+	0.965420i	$0.416046\pi$
$128$	0	0
$129$	1.25079	0.110126
$130$	0	0
$131$	4.82339	0.421422	0.210711	−	0.977548i	$-0.432422\pi$
$131$	4.82339	0.421422	0.210711	+	0.977548i	$0.432422\pi$
$132$	0	0
$133$	−7.09808	−0.615482
$134$	0	0
$135$	1.09926	0.0946093
$136$	0	0
$137$	−4.74001	−0.404967	−0.202483	−	0.979286i	$-0.564901\pi$
$137$	−4.74001	−0.404967	−0.202483	+	0.979286i	$0.564901\pi$
$138$	0	0
$139$	20.2187	1.71492	0.857462	−	0.514547i	$-0.172040\pi$
$139$	20.2187	1.71492	0.857462	+	0.514547i	$0.172040\pi$
$140$	0	0
$141$	6.41448	0.540197
$142$	0	0
$143$	−16.4621	−1.37663
$144$	0	0
$145$	−1.09926	−0.0912886
$146$	0	0
$147$	−4.36263	−0.359824
$148$	0	0
$149$	−1.49552	−0.122518	−0.0612588	−	0.998122i	$-0.519512\pi$
$149$	−1.49552	−0.122518	−0.0612588	+	0.998122i	$0.519512\pi$
$150$	0	0
$151$	13.0744	1.06398	0.531988	−	0.846752i	$-0.321445\pi$
$151$	13.0744	1.06398	0.531988	+	0.846752i	$0.321445\pi$
$152$	0	0
$153$	−2.36393	−0.191112
$154$	0	0
$155$	−8.89828	−0.714727
$156$	0	0
$157$	−12.6257	−1.00764	−0.503819	−	0.863809i	$-0.668072\pi$
$157$	−12.6257	−1.00764	−0.503819	+	0.863809i	$0.668072\pi$
$158$	0	0
$159$	−3.55412	−0.281860
$160$	0	0
$161$	1.62400	0.127989
$162$	0	0
$163$	−5.89587	−0.461800	−0.230900	−	0.972977i	$-0.574167\pi$
$163$	−5.89587	−0.461800	−0.230900	+	0.972977i	$0.574167\pi$
$164$	0	0
$165$	−4.57734	−0.356346
$166$	0	0
$167$	−23.7013	−1.83406	−0.917032	−	0.398813i	$-0.869422\pi$
$167$	−23.7013	−1.83406	−0.917032	+	0.398813i	$0.869422\pi$
$168$	0	0
$169$	2.62958	0.202275
$170$	0	0
$171$	4.37074	0.334239
$172$	0	0
$173$	2.81905	0.214328	0.107164	−	0.994241i	$-0.465823\pi$
$173$	2.81905	0.214328	0.107164	+	0.994241i	$0.465823\pi$
$174$	0	0
$175$	6.15759	0.465470
$176$	0	0
$177$	5.96232	0.448155
$178$	0	0
$179$	12.0776	0.902722	0.451361	−	0.892342i	$-0.350939\pi$
$179$	12.0776	0.902722	0.451361	+	0.892342i	$0.350939\pi$
$180$	0	0
$181$	23.8023	1.76921	0.884606	−	0.466340i	$-0.154428\pi$
$181$	23.8023	1.76921	0.884606	+	0.466340i	$0.154428\pi$
$182$	0	0
$183$	−12.3425	−0.912386
$184$	0	0
$185$	3.88004	0.285266
$186$	0	0
$187$	9.84345	0.719824
$188$	0	0
$189$	−1.62400	−0.118129
$190$	0	0
$191$	3.52735	0.255230	0.127615	−	0.991824i	$-0.459268\pi$
$191$	3.52735	0.255230	0.127615	+	0.991824i	$0.459268\pi$
$192$	0	0
$193$	−26.2610	−1.89031	−0.945153	−	0.326627i	$-0.894088\pi$
$193$	−26.2610	−1.89031	−0.945153	+	0.326627i	$0.894088\pi$
$194$	0	0
$195$	4.34585	0.311212
$196$	0	0
$197$	−16.7606	−1.19414	−0.597072	−	0.802187i	$-0.703669\pi$
$197$	−16.7606	−1.19414	−0.597072	+	0.802187i	$0.703669\pi$
$198$	0	0
$199$	−10.9678	−0.777489	−0.388744	−	0.921346i	$-0.627091\pi$
$199$	−10.9678	−0.777489	−0.388744	+	0.921346i	$0.627091\pi$
$200$	0	0
$201$	−12.8312	−0.905044
$202$	0	0
$203$	1.62400	0.113982
$204$	0	0
$205$	−4.19340	−0.292880
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−18.1999	−1.25891
$210$	0	0
$211$	1.67146	0.115068	0.0575341	−	0.998344i	$-0.481676\pi$
$211$	1.67146	0.115068	0.0575341	+	0.998344i	$0.481676\pi$
$212$	0	0
$213$	8.30445	0.569011
$214$	0	0
$215$	1.37495	0.0937707
$216$	0	0
$217$	13.1459	0.892403
$218$	0	0
$219$	−2.26071	−0.152765
$220$	0	0
$221$	−9.34562	−0.628654
$222$	0	0
$223$	−11.4523	−0.766902	−0.383451	−	0.923561i	$-0.625264\pi$
$223$	−11.4523	−0.766902	−0.383451	+	0.923561i	$0.625264\pi$
$224$	0	0
$225$	−3.79163	−0.252775
$226$	0	0
$227$	10.3236	0.685201	0.342601	−	0.939481i	$-0.388692\pi$
$227$	10.3236	0.685201	0.342601	+	0.939481i	$0.388692\pi$
$228$	0	0
$229$	−20.4908	−1.35407	−0.677037	−	0.735949i	$-0.736736\pi$
$229$	−20.4908	−1.35407	−0.677037	+	0.735949i	$0.736736\pi$
$230$	0	0
$231$	6.76236	0.444931
$232$	0	0
$233$	−16.3015	−1.06795	−0.533974	−	0.845501i	$-0.679302\pi$
$233$	−16.3015	−1.06795	−0.533974	+	0.845501i	$0.679302\pi$
$234$	0	0
$235$	7.05119	0.459969
$236$	0	0
$237$	−14.7456	−0.957827
$238$	0	0
$239$	−24.9115	−1.61139	−0.805695	−	0.592331i	$-0.798208\pi$
$239$	−24.9115	−1.61139	−0.805695	+	0.592331i	$0.798208\pi$
$240$	0	0
$241$	−20.5384	−1.32300	−0.661498	−	0.749947i	$-0.730079\pi$
$241$	−20.5384	−1.32300	−0.661498	+	0.749947i	$0.730079\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.79567	−0.306384
$246$	0	0
$247$	17.2794	1.09946
$248$	0	0
$249$	9.49232	0.601551
$250$	0	0
$251$	5.15343	0.325281	0.162641	−	0.986685i	$-0.447999\pi$
$251$	5.15343	0.325281	0.162641	+	0.986685i	$0.447999\pi$
$252$	0	0
$253$	4.16402	0.261790
$254$	0	0
$255$	−2.59857	−0.162729
$256$	0	0
$257$	12.3581	0.770878	0.385439	−	0.922733i	$-0.374050\pi$
$257$	12.3581	0.770878	0.385439	+	0.922733i	$0.374050\pi$
$258$	0	0
$259$	−5.73219	−0.356181
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−5.61176	−0.346036	−0.173018	−	0.984919i	$-0.555352\pi$
$263$	−5.61176	−0.346036	−0.173018	+	0.984919i	$0.555352\pi$
$264$	0	0
$265$	−3.90690	−0.239999
$266$	0	0
$267$	−4.52312	−0.276811
$268$	0	0
$269$	2.50357	0.152645	0.0763226	−	0.997083i	$-0.475682\pi$
$269$	2.50357	0.152645	0.0763226	+	0.997083i	$0.475682\pi$
$270$	0	0
$271$	8.01951	0.487151	0.243575	−	0.969882i	$-0.421680\pi$
$271$	8.01951	0.487151	0.243575	+	0.969882i	$0.421680\pi$
$272$	0	0
$273$	−6.42036	−0.388578
$274$	0	0
$275$	15.7884	0.952077
$276$	0	0
$277$	−15.3260	−0.920850	−0.460425	−	0.887699i	$-0.652303\pi$
$277$	−15.3260	−0.920850	−0.460425	+	0.887699i	$0.652303\pi$
$278$	0	0
$279$	−8.09478	−0.484622
$280$	0	0
$281$	20.6963	1.23464	0.617319	−	0.786713i	$-0.288219\pi$
$281$	20.6963	1.23464	0.617319	+	0.786713i	$0.288219\pi$
$282$	0	0
$283$	−12.2936	−0.730780	−0.365390	−	0.930855i	$-0.619064\pi$
$283$	−12.2936	−0.730780	−0.365390	+	0.930855i	$0.619064\pi$
$284$	0	0
$285$	4.80459	0.284599
$286$	0	0
$287$	6.19515	0.365688
$288$	0	0
$289$	−11.4118	−0.671285
$290$	0	0
$291$	−2.79546	−0.163873
$292$	0	0
$293$	−26.5827	−1.55298	−0.776489	−	0.630131i	$-0.783001\pi$
$293$	−26.5827	−1.55298	−0.776489	+	0.630131i	$0.783001\pi$
$294$	0	0
$295$	6.55414	0.381597
$296$	0	0
$297$	−4.16402	−0.241621
$298$	0	0
$299$	−3.95343	−0.228633
$300$	0	0
$301$	−2.03129	−0.117081
$302$	0	0
$303$	−2.71767	−0.156126
$304$	0	0
$305$	−13.5677	−0.776882
$306$	0	0
$307$	−3.50926	−0.200284	−0.100142	−	0.994973i	$-0.531930\pi$
$307$	−3.50926	−0.200284	−0.100142	+	0.994973i	$0.531930\pi$
$308$	0	0
$309$	0.577486	0.0328520
$310$	0	0
$311$	−7.88485	−0.447109	−0.223554	−	0.974691i	$-0.571766\pi$
$311$	−7.88485	−0.447109	−0.223554	+	0.974691i	$0.571766\pi$
$312$	0	0
$313$	−27.5619	−1.55789	−0.778946	−	0.627091i	$-0.784246\pi$
$313$	−27.5619	−1.55789	−0.778946	+	0.627091i	$0.784246\pi$
$314$	0	0
$315$	−1.78520	−0.100585
$316$	0	0
$317$	5.95975	0.334733	0.167366	−	0.985895i	$-0.446474\pi$
$317$	5.95975	0.334733	0.167366	+	0.985895i	$0.446474\pi$
$318$	0	0
$319$	4.16402	0.233140
$320$	0	0
$321$	7.55434	0.421642
$322$	0	0
$323$	−10.3321	−0.574895
$324$	0	0
$325$	−14.9899	−0.831491
$326$	0	0
$327$	−5.65806	−0.312891
$328$	0	0
$329$	−10.4171	−0.574314
$330$	0	0
$331$	−10.9771	−0.603356	−0.301678	−	0.953410i	$-0.597547\pi$
$331$	−10.9771	−0.603356	−0.301678	+	0.953410i	$0.597547\pi$
$332$	0	0
$333$	3.52968	0.193425
$334$	0	0
$335$	−14.1049	−0.770631
$336$	0	0
$337$	−22.2823	−1.21379	−0.606897	−	0.794781i	$-0.707586\pi$
$337$	−22.2823	−1.21379	−0.606897	+	0.794781i	$0.707586\pi$
$338$	0	0
$339$	−10.3321	−0.561161
$340$	0	0
$341$	33.7068	1.82533
$342$	0	0
$343$	18.4529	0.996363
$344$	0	0
$345$	−1.09926	−0.0591822
$346$	0	0
$347$	18.0078	0.966709	0.483355	−	0.875425i	$-0.339418\pi$
$347$	18.0078	0.966709	0.483355	+	0.875425i	$0.339418\pi$
$348$	0	0
$349$	23.3714	1.25104	0.625521	−	0.780208i	$-0.284887\pi$
$349$	23.3714	1.25104	0.625521	+	0.780208i	$0.284887\pi$
$350$	0	0
$351$	3.95343	0.211018
$352$	0	0
$353$	−7.85756	−0.418216	−0.209108	−	0.977893i	$-0.567056\pi$
$353$	−7.85756	−0.418216	−0.209108	+	0.977893i	$0.567056\pi$
$354$	0	0
$355$	9.12875	0.484504
$356$	0	0
$357$	3.83902	0.203182
$358$	0	0
$359$	25.9850	1.37143	0.685717	−	0.727868i	$-0.259489\pi$
$359$	25.9850	1.37143	0.685717	+	0.727868i	$0.259489\pi$
$360$	0	0
$361$	0.103389	0.00544152
$362$	0	0
$363$	6.33907	0.332715
$364$	0	0
$365$	−2.48511	−0.130077
$366$	0	0
$367$	14.7725	0.771117	0.385559	−	0.922683i	$-0.374009\pi$
$367$	14.7725	0.771117	0.385559	+	0.922683i	$0.374009\pi$
$368$	0	0
$369$	−3.81475	−0.198588
$370$	0	0
$371$	5.77189	0.299661
$372$	0	0
$373$	27.5103	1.42443	0.712215	−	0.701962i	$-0.247692\pi$
$373$	27.5103	1.42443	0.712215	+	0.701962i	$0.247692\pi$
$374$	0	0
$375$	−9.66429	−0.499062
$376$	0	0
$377$	−3.95343	−0.203612
$378$	0	0
$379$	24.8568	1.27681	0.638404	−	0.769701i	$-0.279595\pi$
$379$	24.8568	1.27681	0.638404	+	0.769701i	$0.279595\pi$
$380$	0	0
$381$	5.87590	0.301032
$382$	0	0
$383$	−20.3583	−1.04026	−0.520131	−	0.854087i	$-0.674117\pi$
$383$	−20.3583	−1.04026	−0.520131	+	0.854087i	$0.674117\pi$
$384$	0	0
$385$	7.43360	0.378851
$386$	0	0
$387$	1.25079	0.0635814
$388$	0	0
$389$	−6.10949	−0.309764	−0.154882	−	0.987933i	$-0.549500\pi$
$389$	−6.10949	−0.309764	−0.154882	+	0.987933i	$0.549500\pi$
$390$	0	0
$391$	2.36393	0.119549
$392$	0	0
$393$	4.82339	0.243308
$394$	0	0
$395$	−16.2092	−0.815574
$396$	0	0
$397$	14.6577	0.735650	0.367825	−	0.929895i	$-0.380103\pi$
$397$	14.6577	0.735650	0.367825	+	0.929895i	$0.380103\pi$
$398$	0	0
$399$	−7.09808	−0.355348
$400$	0	0
$401$	16.7090	0.834410	0.417205	−	0.908812i	$-0.363010\pi$
$401$	16.7090	0.834410	0.417205	+	0.908812i	$0.363010\pi$
$402$	0	0
$403$	−32.0021	−1.59414
$404$	0	0
$405$	1.09926	0.0546227
$406$	0	0
$407$	−14.6977	−0.728536
$408$	0	0
$409$	−5.84895	−0.289212	−0.144606	−	0.989489i	$-0.546191\pi$
$409$	−5.84895	−0.289212	−0.144606	+	0.989489i	$0.546191\pi$
$410$	0	0
$411$	−4.74001	−0.233808
$412$	0	0
$413$	−9.68280	−0.476459
$414$	0	0
$415$	10.4345	0.512211
$416$	0	0
$417$	20.2187	0.990112
$418$	0	0
$419$	21.8906	1.06943	0.534714	−	0.845033i	$-0.320419\pi$
$419$	21.8906	1.06943	0.534714	+	0.845033i	$0.320419\pi$
$420$	0	0
$421$	28.1392	1.37142	0.685710	−	0.727875i	$-0.259492\pi$
$421$	28.1392	1.37142	0.685710	+	0.727875i	$0.259492\pi$
$422$	0	0
$423$	6.41448	0.311883
$424$	0	0
$425$	8.96313	0.434776
$426$	0	0
$427$	20.0442	0.970009
$428$	0	0
$429$	−16.4621	−0.794800
$430$	0	0
$431$	−14.0315	−0.675873	−0.337937	−	0.941169i	$-0.609729\pi$
$431$	−14.0315	−0.675873	−0.337937	+	0.941169i	$0.609729\pi$
$432$	0	0
$433$	2.07861	0.0998917	0.0499459	−	0.998752i	$-0.484095\pi$
$433$	2.07861	0.0998917	0.0499459	+	0.998752i	$0.484095\pi$
$434$	0	0
$435$	−1.09926	−0.0527055
$436$	0	0
$437$	−4.37074	−0.209081
$438$	0	0
$439$	−27.1515	−1.29587	−0.647935	−	0.761695i	$-0.724367\pi$
$439$	−27.1515	−1.29587	−0.647935	+	0.761695i	$0.724367\pi$
$440$	0	0
$441$	−4.36263	−0.207744
$442$	0	0
$443$	21.9145	1.04119	0.520595	−	0.853804i	$-0.325710\pi$
$443$	21.9145	1.04119	0.520595	+	0.853804i	$0.325710\pi$
$444$	0	0
$445$	−4.97209	−0.235700
$446$	0	0
$447$	−1.49552	−0.0707356
$448$	0	0
$449$	11.5385	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$449$	11.5385	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$450$	0	0
$451$	15.8847	0.747981
$452$	0	0
$453$	13.0744	0.614287
$454$	0	0
$455$	−7.05765	−0.330868
$456$	0	0
$457$	−15.7409	−0.736326	−0.368163	−	0.929761i	$-0.620013\pi$
$457$	−15.7409	−0.736326	−0.368163	+	0.929761i	$0.620013\pi$
$458$	0	0
$459$	−2.36393	−0.110339
$460$	0	0
$461$	30.0594	1.40001	0.700003	−	0.714140i	$-0.253182\pi$
$461$	30.0594	1.40001	0.700003	+	0.714140i	$0.253182\pi$
$462$	0	0
$463$	12.3524	0.574064	0.287032	−	0.957921i	$-0.407331\pi$
$463$	12.3524	0.574064	0.287032	+	0.957921i	$0.407331\pi$
$464$	0	0
$465$	−8.89828	−0.412648
$466$	0	0
$467$	9.95785	0.460794	0.230397	−	0.973097i	$-0.425997\pi$
$467$	9.95785	0.460794	0.230397	+	0.973097i	$0.425997\pi$
$468$	0	0
$469$	20.8379	0.962204
$470$	0	0
$471$	−12.6257	−0.581761
$472$	0	0
$473$	−5.20833	−0.239479
$474$	0	0
$475$	−16.5722	−0.760386
$476$	0	0
$477$	−3.55412	−0.162732
$478$	0	0
$479$	−29.1940	−1.33391	−0.666953	−	0.745099i	$-0.732402\pi$
$479$	−29.1940	−1.33391	−0.666953	+	0.745099i	$0.732402\pi$
$480$	0	0
$481$	13.9543	0.636262
$482$	0	0
$483$	1.62400	0.0738945
$484$	0	0
$485$	−3.07294	−0.139535
$486$	0	0
$487$	−14.9615	−0.677969	−0.338985	−	0.940792i	$-0.610083\pi$
$487$	−14.9615	−0.677969	−0.338985	+	0.940792i	$0.610083\pi$
$488$	0	0
$489$	−5.89587	−0.266621
$490$	0	0
$491$	−19.4089	−0.875910	−0.437955	−	0.898997i	$-0.644297\pi$
$491$	−19.4089	−0.875910	−0.437955	+	0.898997i	$0.644297\pi$
$492$	0	0
$493$	2.36393	0.106466
$494$	0	0
$495$	−4.57734	−0.205736
$496$	0	0
$497$	−13.4864	−0.604948
$498$	0	0
$499$	13.6996	0.613278	0.306639	−	0.951826i	$-0.400796\pi$
$499$	13.6996	0.613278	0.306639	+	0.951826i	$0.400796\pi$
$500$	0	0
$501$	−23.7013	−1.05890
$502$	0	0
$503$	13.8413	0.617152	0.308576	−	0.951200i	$-0.400148\pi$
$503$	13.8413	0.617152	0.308576	+	0.951200i	$0.400148\pi$
$504$	0	0
$505$	−2.98743	−0.132939
$506$	0	0
$507$	2.62958	0.116784
$508$	0	0
$509$	23.8823	1.05857	0.529283	−	0.848446i	$-0.322461\pi$
$509$	23.8823	1.05857	0.529283	+	0.848446i	$0.322461\pi$
$510$	0	0
$511$	3.67139	0.162413
$512$	0	0
$513$	4.37074	0.192973
$514$	0	0
$515$	0.634808	0.0279730
$516$	0	0
$517$	−26.7100	−1.17471
$518$	0	0
$519$	2.81905	0.123743
$520$	0	0
$521$	−22.9572	−1.00577	−0.502887	−	0.864352i	$-0.667729\pi$
$521$	−22.9572	−1.00577	−0.502887	+	0.864352i	$0.667729\pi$
$522$	0	0
$523$	−4.15498	−0.181685	−0.0908423	−	0.995865i	$-0.528956\pi$
$523$	−4.15498	−0.181685	−0.0908423	+	0.995865i	$0.528956\pi$
$524$	0	0
$525$	6.15759	0.268739
$526$	0	0
$527$	19.1355	0.833555
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.96232	0.258743
$532$	0	0
$533$	−15.0813	−0.653245
$534$	0	0
$535$	8.30419	0.359021
$536$	0	0
$537$	12.0776	0.521187
$538$	0	0
$539$	18.1661	0.782468
$540$	0	0
$541$	−15.5952	−0.670492	−0.335246	−	0.942131i	$-0.608819\pi$
$541$	−15.5952	−0.670492	−0.335246	+	0.942131i	$0.608819\pi$
$542$	0	0
$543$	23.8023	1.02145
$544$	0	0
$545$	−6.21968	−0.266422
$546$	0	0
$547$	−12.3173	−0.526650	−0.263325	−	0.964707i	$-0.584819\pi$
$547$	−12.3173	−0.526650	−0.263325	+	0.964707i	$0.584819\pi$
$548$	0	0
$549$	−12.3425	−0.526766
$550$	0	0
$551$	−4.37074	−0.186200
$552$	0	0
$553$	23.9468	1.01832
$554$	0	0
$555$	3.88004	0.164698
$556$	0	0
$557$	29.4030	1.24584	0.622922	−	0.782284i	$-0.285946\pi$
$557$	29.4030	1.24584	0.622922	+	0.782284i	$0.285946\pi$
$558$	0	0
$559$	4.94492	0.209148
$560$	0	0
$561$	9.84345	0.415591
$562$	0	0
$563$	−7.67982	−0.323666	−0.161833	−	0.986818i	$-0.551741\pi$
$563$	−7.67982	−0.323666	−0.161833	+	0.986818i	$0.551741\pi$
$564$	0	0
$565$	−11.3576	−0.477819
$566$	0	0
$567$	−1.62400	−0.0682015
$568$	0	0
$569$	−46.7619	−1.96036	−0.980180	−	0.198109i	$-0.936520\pi$
$569$	−46.7619	−1.96036	−0.980180	+	0.198109i	$0.936520\pi$
$570$	0	0
$571$	−22.7497	−0.952044	−0.476022	−	0.879433i	$-0.657922\pi$
$571$	−22.7497	−0.952044	−0.476022	+	0.879433i	$0.657922\pi$
$572$	0	0
$573$	3.52735	0.147357
$574$	0	0
$575$	3.79163	0.158122
$576$	0	0
$577$	32.4667	1.35161	0.675803	−	0.737082i	$-0.263797\pi$
$577$	32.4667	1.35161	0.675803	+	0.737082i	$0.263797\pi$
$578$	0	0
$579$	−26.2610	−1.09137
$580$	0	0
$581$	−15.4155	−0.639543
$582$	0	0
$583$	14.7994	0.612930
$584$	0	0
$585$	4.34585	0.179679
$586$	0	0
$587$	44.9305	1.85448	0.927240	−	0.374467i	$-0.122174\pi$
$587$	44.9305	1.85448	0.927240	+	0.374467i	$0.122174\pi$
$588$	0	0
$589$	−35.3802	−1.45782
$590$	0	0
$591$	−16.7606	−0.689440
$592$	0	0
$593$	−19.3818	−0.795914	−0.397957	−	0.917404i	$-0.630281\pi$
$593$	−19.3818	−0.795914	−0.397957	+	0.917404i	$0.630281\pi$
$594$	0	0
$595$	4.22008	0.173006
$596$	0	0
$597$	−10.9678	−0.448883
$598$	0	0
$599$	−13.0213	−0.532035	−0.266018	−	0.963968i	$-0.585708\pi$
$599$	−13.0213	−0.532035	−0.266018	+	0.963968i	$0.585708\pi$
$600$	0	0
$601$	35.0563	1.42997	0.714987	−	0.699137i	$-0.246432\pi$
$601$	35.0563	1.42997	0.714987	+	0.699137i	$0.246432\pi$
$602$	0	0
$603$	−12.8312	−0.522528
$604$	0	0
$605$	6.96829	0.283301
$606$	0	0
$607$	−8.13500	−0.330190	−0.165095	−	0.986278i	$-0.552793\pi$
$607$	−8.13500	−0.330190	−0.165095	+	0.986278i	$0.552793\pi$
$608$	0	0
$609$	1.62400	0.0658077
$610$	0	0
$611$	25.3592	1.02592
$612$	0	0
$613$	−34.2074	−1.38162	−0.690812	−	0.723034i	$-0.742747\pi$
$613$	−34.2074	−1.38162	−0.690812	+	0.723034i	$0.742747\pi$
$614$	0	0
$615$	−4.19340	−0.169094
$616$	0	0
$617$	−13.6354	−0.548940	−0.274470	−	0.961596i	$-0.588502\pi$
$617$	−13.6354	−0.548940	−0.274470	+	0.961596i	$0.588502\pi$
$618$	0	0
$619$	−24.8104	−0.997213	−0.498606	−	0.866829i	$-0.666155\pi$
$619$	−24.8104	−0.997213	−0.498606	+	0.866829i	$0.666155\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	7.34554	0.294293
$624$	0	0
$625$	8.33456	0.333382
$626$	0	0
$627$	−18.1999	−0.726832
$628$	0	0
$629$	−8.34391	−0.332693
$630$	0	0
$631$	6.51667	0.259425	0.129712	−	0.991552i	$-0.458595\pi$
$631$	6.51667	0.259425	0.129712	+	0.991552i	$0.458595\pi$
$632$	0	0
$633$	1.67146	0.0664346
$634$	0	0
$635$	6.45915	0.256323
$636$	0	0
$637$	−17.2473	−0.683364
$638$	0	0
$639$	8.30445	0.328519
$640$	0	0
$641$	32.7254	1.29258	0.646289	−	0.763093i	$-0.276320\pi$
$641$	32.7254	1.29258	0.646289	+	0.763093i	$0.276320\pi$
$642$	0	0
$643$	−11.9951	−0.473039	−0.236519	−	0.971627i	$-0.576007\pi$
$643$	−11.9951	−0.473039	−0.236519	+	0.971627i	$0.576007\pi$
$644$	0	0
$645$	1.37495	0.0541385
$646$	0	0
$647$	−5.83271	−0.229308	−0.114654	−	0.993406i	$-0.536576\pi$
$647$	−5.83271	−0.229308	−0.114654	+	0.993406i	$0.536576\pi$
$648$	0	0
$649$	−24.8272	−0.974554
$650$	0	0
$651$	13.1459	0.515229
$652$	0	0
$653$	−15.8443	−0.620034	−0.310017	−	0.950731i	$-0.600335\pi$
$653$	−15.8443	−0.620034	−0.310017	+	0.950731i	$0.600335\pi$
$654$	0	0
$655$	5.30217	0.207173
$656$	0	0
$657$	−2.26071	−0.0881986
$658$	0	0
$659$	−3.25099	−0.126641	−0.0633203	−	0.997993i	$-0.520169\pi$
$659$	−3.25099	−0.126641	−0.0633203	+	0.997993i	$0.520169\pi$
$660$	0	0
$661$	−2.29424	−0.0892357	−0.0446179	−	0.999004i	$-0.514207\pi$
$661$	−2.29424	−0.0892357	−0.0446179	+	0.999004i	$0.514207\pi$
$662$	0	0
$663$	−9.34562	−0.362954
$664$	0	0
$665$	−7.80264	−0.302573
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−11.4523	−0.442771
$670$	0	0
$671$	51.3945	1.98406
$672$	0	0
$673$	23.3973	0.901901	0.450951	−	0.892549i	$-0.351085\pi$
$673$	23.3973	0.901901	0.450951	+	0.892549i	$0.351085\pi$
$674$	0	0
$675$	−3.79163	−0.145940
$676$	0	0
$677$	41.0661	1.57830	0.789150	−	0.614201i	$-0.210522\pi$
$677$	41.0661	1.57830	0.789150	+	0.614201i	$0.210522\pi$
$678$	0	0
$679$	4.53982	0.174222
$680$	0	0
$681$	10.3236	0.395601
$682$	0	0
$683$	0.0729842	0.00279266	0.00139633	−	0.999999i	$-0.499556\pi$
$683$	0.0729842	0.00279266	0.00139633	+	0.999999i	$0.499556\pi$
$684$	0	0
$685$	−5.21051	−0.199083
$686$	0	0
$687$	−20.4908	−0.781774
$688$	0	0
$689$	−14.0510	−0.535299
$690$	0	0
$691$	−24.0332	−0.914264	−0.457132	−	0.889399i	$-0.651123\pi$
$691$	−24.0332	−0.914264	−0.457132	+	0.889399i	$0.651123\pi$
$692$	0	0
$693$	6.76236	0.256881
$694$	0	0
$695$	22.2256	0.843065
$696$	0	0
$697$	9.01780	0.341573
$698$	0	0
$699$	−16.3015	−0.616580
$700$	0	0
$701$	32.4959	1.22735	0.613676	−	0.789558i	$-0.289690\pi$
$701$	32.4959	1.22735	0.613676	+	0.789558i	$0.289690\pi$
$702$	0	0
$703$	15.4273	0.581852
$704$	0	0
$705$	7.05119	0.265563
$706$	0	0
$707$	4.41349	0.165986
$708$	0	0
$709$	8.31742	0.312367	0.156184	−	0.987728i	$-0.450081\pi$
$709$	8.31742	0.312367	0.156184	+	0.987728i	$0.450081\pi$
$710$	0	0
$711$	−14.7456	−0.553001
$712$	0	0
$713$	8.09478	0.303152
$714$	0	0
$715$	−18.0962	−0.676759
$716$	0	0
$717$	−24.9115	−0.930336
$718$	0	0
$719$	4.29581	0.160207	0.0801034	−	0.996787i	$-0.474475\pi$
$719$	4.29581	0.160207	0.0801034	+	0.996787i	$0.474475\pi$
$720$	0	0
$721$	−0.937837	−0.0349269
$722$	0	0
$723$	−20.5384	−0.763833
$724$	0	0
$725$	3.79163	0.140817
$726$	0	0
$727$	−19.2693	−0.714658	−0.357329	−	0.933979i	$-0.616312\pi$
$727$	−19.2693	−0.714658	−0.357329	+	0.933979i	$0.616312\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.95679	−0.109361
$732$	0	0
$733$	20.0166	0.739332	0.369666	−	0.929165i	$-0.379472\pi$
$733$	20.0166	0.739332	0.369666	+	0.929165i	$0.379472\pi$
$734$	0	0
$735$	−4.79567	−0.176891
$736$	0	0
$737$	53.4295	1.96810
$738$	0	0
$739$	−20.9829	−0.771868	−0.385934	−	0.922526i	$-0.626121\pi$
$739$	−20.9829	−0.771868	−0.385934	+	0.922526i	$0.626121\pi$
$740$	0	0
$741$	17.2794	0.634775
$742$	0	0
$743$	−14.9474	−0.548366	−0.274183	−	0.961678i	$-0.588407\pi$
$743$	−14.9474	−0.548366	−0.274183	+	0.961678i	$0.588407\pi$
$744$	0	0
$745$	−1.64397	−0.0602302
$746$	0	0
$747$	9.49232	0.347306
$748$	0	0
$749$	−12.2682	−0.448272
$750$	0	0
$751$	−44.1871	−1.61241	−0.806205	−	0.591636i	$-0.798482\pi$
$751$	−44.1871	−1.61241	−0.806205	+	0.591636i	$0.798482\pi$
$752$	0	0
$753$	5.15343	0.187801
$754$	0	0
$755$	14.3721	0.523056
$756$	0	0
$757$	16.5638	0.602021	0.301011	−	0.953621i	$-0.402676\pi$
$757$	16.5638	0.602021	0.301011	+	0.953621i	$0.402676\pi$
$758$	0	0
$759$	4.16402	0.151144
$760$	0	0
$761$	7.04978	0.255554	0.127777	−	0.991803i	$-0.459216\pi$
$761$	7.04978	0.255554	0.127777	+	0.991803i	$0.459216\pi$
$762$	0	0
$763$	9.18868	0.332653
$764$	0	0
$765$	−2.59857	−0.0939516
$766$	0	0
$767$	23.5716	0.851121
$768$	0	0
$769$	−33.0490	−1.19178	−0.595889	−	0.803067i	$-0.703200\pi$
$769$	−33.0490	−1.19178	−0.595889	+	0.803067i	$0.703200\pi$
$770$	0	0
$771$	12.3581	0.445067
$772$	0	0
$773$	42.7305	1.53691	0.768454	−	0.639904i	$-0.221026\pi$
$773$	42.7305	1.53691	0.768454	+	0.639904i	$0.221026\pi$
$774$	0	0
$775$	30.6924	1.10250
$776$	0	0
$777$	−5.73219	−0.205641
$778$	0	0
$779$	−16.6733	−0.597383
$780$	0	0
$781$	−34.5799	−1.23737
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−13.8789	−0.495360
$786$	0	0
$787$	−39.7655	−1.41749	−0.708744	−	0.705466i	$-0.750738\pi$
$787$	−39.7655	−1.41749	−0.708744	+	0.705466i	$0.750738\pi$
$788$	0	0
$789$	−5.61176	−0.199784
$790$	0	0
$791$	16.7793	0.596602
$792$	0	0
$793$	−48.7953	−1.73277
$794$	0	0
$795$	−3.90690	−0.138564
$796$	0	0
$797$	−9.66395	−0.342315	−0.171157	−	0.985244i	$-0.554751\pi$
$797$	−9.66395	−0.342315	−0.171157	+	0.985244i	$0.554751\pi$
$798$	0	0
$799$	−15.1634	−0.536442
$800$	0	0
$801$	−4.52312	−0.159817
$802$	0	0
$803$	9.41364	0.332200
$804$	0	0
$805$	1.78520	0.0629200
$806$	0	0
$807$	2.50357	0.0881297
$808$	0	0
$809$	7.85384	0.276126	0.138063	−	0.990423i	$-0.455912\pi$
$809$	7.85384	0.276126	0.138063	+	0.990423i	$0.455912\pi$
$810$	0	0
$811$	−27.7017	−0.972738	−0.486369	−	0.873753i	$-0.661679\pi$
$811$	−27.7017	−0.972738	−0.486369	+	0.873753i	$0.661679\pi$
$812$	0	0
$813$	8.01951	0.281257
$814$	0	0
$815$	−6.48110	−0.227023
$816$	0	0
$817$	5.46689	0.191262
$818$	0	0
$819$	−6.42036	−0.224345
$820$	0	0
$821$	−28.4164	−0.991738	−0.495869	−	0.868397i	$-0.665150\pi$
$821$	−28.4164	−0.991738	−0.495869	+	0.868397i	$0.665150\pi$
$822$	0	0
$823$	18.9282	0.659796	0.329898	−	0.944017i	$-0.392986\pi$
$823$	18.9282	0.659796	0.329898	+	0.944017i	$0.392986\pi$
$824$	0	0
$825$	15.7884	0.549682
$826$	0	0
$827$	12.4745	0.433779	0.216890	−	0.976196i	$-0.430409\pi$
$827$	12.4745	0.433779	0.216890	+	0.976196i	$0.430409\pi$
$828$	0	0
$829$	−7.06835	−0.245494	−0.122747	−	0.992438i	$-0.539170\pi$
$829$	−7.06835	−0.245494	−0.122747	+	0.992438i	$0.539170\pi$
$830$	0	0
$831$	−15.3260	−0.531653
$832$	0	0
$833$	10.3129	0.357322
$834$	0	0
$835$	−26.0540	−0.901634
$836$	0	0
$837$	−8.09478	−0.279797
$838$	0	0
$839$	8.30055	0.286567	0.143283	−	0.989682i	$-0.454234\pi$
$839$	8.30055	0.286567	0.143283	+	0.989682i	$0.454234\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	20.6963	0.712818
$844$	0	0
$845$	2.89059	0.0994394
$846$	0	0
$847$	−10.2946	−0.353728
$848$	0	0
$849$	−12.2936	−0.421916
$850$	0	0
$851$	−3.52968	−0.120996
$852$	0	0
$853$	48.3189	1.65441	0.827205	−	0.561901i	$-0.189930\pi$
$853$	48.3189	1.65441	0.827205	+	0.561901i	$0.189930\pi$
$854$	0	0
$855$	4.80459	0.164313
$856$	0	0
$857$	35.7504	1.22121	0.610606	−	0.791935i	$-0.290926\pi$
$857$	35.7504	1.22121	0.610606	+	0.791935i	$0.290926\pi$
$858$	0	0
$859$	−41.9841	−1.43248	−0.716239	−	0.697855i	$-0.754138\pi$
$859$	−41.9841	−1.43248	−0.716239	+	0.697855i	$0.754138\pi$
$860$	0	0
$861$	6.19515	0.211130
$862$	0	0
$863$	31.7225	1.07984	0.539922	−	0.841715i	$-0.318454\pi$
$863$	31.7225	1.07984	0.539922	+	0.841715i	$0.318454\pi$
$864$	0	0
$865$	3.09887	0.105365
$866$	0	0
$867$	−11.4118	−0.387566
$868$	0	0
$869$	61.4008	2.08288
$870$	0	0
$871$	−50.7273	−1.71883
$872$	0	0
$873$	−2.79546	−0.0946120
$874$	0	0
$875$	15.6948	0.530581
$876$	0	0
$877$	−18.7023	−0.631531	−0.315765	−	0.948837i	$-0.602261\pi$
$877$	−18.7023	−0.631531	−0.315765	+	0.948837i	$0.602261\pi$
$878$	0	0
$879$	−26.5827	−0.896612
$880$	0	0
$881$	−27.4113	−0.923510	−0.461755	−	0.887008i	$-0.652780\pi$
$881$	−27.4113	−0.923510	−0.461755	+	0.887008i	$0.652780\pi$
$882$	0	0
$883$	5.94240	0.199978	0.0999889	−	0.994989i	$-0.468119\pi$
$883$	5.94240	0.199978	0.0999889	+	0.994989i	$0.468119\pi$
$884$	0	0
$885$	6.55414	0.220315
$886$	0	0
$887$	−41.6929	−1.39991	−0.699956	−	0.714186i	$-0.746797\pi$
$887$	−41.6929	−1.39991	−0.699956	+	0.714186i	$0.746797\pi$
$888$	0	0
$889$	−9.54246	−0.320044
$890$	0	0
$891$	−4.16402	−0.139500
$892$	0	0
$893$	28.0361	0.938191
$894$	0	0
$895$	13.2764	0.443782
$896$	0	0
$897$	−3.95343	−0.132001
$898$	0	0
$899$	8.09478	0.269976
$900$	0	0
$901$	8.40169	0.279901
$902$	0	0
$903$	−2.03129	−0.0675970
$904$	0	0
$905$	26.1649	0.869752
$906$	0	0
$907$	47.3811	1.57326	0.786632	−	0.617422i	$-0.211823\pi$
$907$	47.3811	1.57326	0.786632	+	0.617422i	$0.211823\pi$
$908$	0	0
$909$	−2.71767	−0.0901394
$910$	0	0
$911$	23.9539	0.793627	0.396814	−	0.917899i	$-0.370116\pi$
$911$	23.9539	0.793627	0.396814	+	0.917899i	$0.370116\pi$
$912$	0	0
$913$	−39.5262	−1.30813
$914$	0	0
$915$	−13.5677	−0.448533
$916$	0	0
$917$	−7.83319	−0.258675
$918$	0	0
$919$	50.7326	1.67351	0.836757	−	0.547574i	$-0.184449\pi$
$919$	50.7326	1.67351	0.836757	+	0.547574i	$0.184449\pi$
$920$	0	0
$921$	−3.50926	−0.115634
$922$	0	0
$923$	32.8310	1.08065
$924$	0	0
$925$	−13.3832	−0.440038
$926$	0	0
$927$	0.577486	0.0189671
$928$	0	0
$929$	−4.26851	−0.140045	−0.0700226	−	0.997545i	$-0.522307\pi$
$929$	−4.26851	−0.140045	−0.0700226	+	0.997545i	$0.522307\pi$
$930$	0	0
$931$	−19.0679	−0.624926
$932$	0	0
$933$	−7.88485	−0.258138
$934$	0	0
$935$	10.8205	0.353869
$936$	0	0
$937$	10.9332	0.357171	0.178586	−	0.983924i	$-0.442848\pi$
$937$	10.9332	0.357171	0.178586	+	0.983924i	$0.442848\pi$
$938$	0	0
$939$	−27.5619	−0.899450
$940$	0	0
$941$	31.1352	1.01498	0.507489	−	0.861658i	$-0.330574\pi$
$941$	31.1352	1.01498	0.507489	+	0.861658i	$0.330574\pi$
$942$	0	0
$943$	3.81475	0.124225
$944$	0	0
$945$	−1.78520	−0.0580725
$946$	0	0
$947$	4.71474	0.153209	0.0766043	−	0.997062i	$-0.475592\pi$
$947$	4.71474	0.153209	0.0766043	+	0.997062i	$0.475592\pi$
$948$	0	0
$949$	−8.93754	−0.290125
$950$	0	0
$951$	5.95975	0.193258
$952$	0	0
$953$	55.1516	1.78653	0.893267	−	0.449526i	$-0.148407\pi$
$953$	55.1516	1.78653	0.893267	+	0.449526i	$0.148407\pi$
$954$	0	0
$955$	3.87747	0.125472
$956$	0	0
$957$	4.16402	0.134604
$958$	0	0
$959$	7.69777	0.248574
$960$	0	0
$961$	34.5255	1.11373
$962$	0	0
$963$	7.55434	0.243435
$964$	0	0
$965$	−28.8677	−0.929283
$966$	0	0
$967$	−23.9274	−0.769453	−0.384727	−	0.923031i	$-0.625704\pi$
$967$	−23.9274	−0.769453	−0.384727	+	0.923031i	$0.625704\pi$
$968$	0	0
$969$	−10.3321	−0.331916
$970$	0	0
$971$	−32.7441	−1.05081	−0.525404	−	0.850853i	$-0.676086\pi$
$971$	−32.7441	−1.05081	−0.525404	+	0.850853i	$0.676086\pi$
$972$	0	0
$973$	−32.8351	−1.05264
$974$	0	0
$975$	−14.9899	−0.480061
$976$	0	0
$977$	−32.5643	−1.04183	−0.520913	−	0.853610i	$-0.674408\pi$
$977$	−32.5643	−1.04183	−0.520913	+	0.853610i	$0.674408\pi$
$978$	0	0
$979$	18.8344	0.601949
$980$	0	0
$981$	−5.65806	−0.180648
$982$	0	0
$983$	−61.7387	−1.96916	−0.984580	−	0.174935i	$-0.944029\pi$
$983$	−61.7387	−1.96916	−0.984580	+	0.174935i	$0.944029\pi$
$984$	0	0
$985$	−18.4243	−0.587047
$986$	0	0
$987$	−10.4171	−0.331580
$988$	0	0
$989$	−1.25079	−0.0397729
$990$	0	0
$991$	26.3446	0.836864	0.418432	−	0.908248i	$-0.362580\pi$
$991$	26.3446	0.836864	0.418432	+	0.908248i	$0.362580\pi$
$992$	0	0
$993$	−10.9771	−0.348348
$994$	0	0
$995$	−12.0565	−0.382217
$996$	0	0
$997$	−3.02186	−0.0957032	−0.0478516	−	0.998854i	$-0.515237\pi$
$997$	−3.02186	−0.0957032	−0.0478516	+	0.998854i	$0.515237\pi$
$998$	0	0
$999$	3.52968	0.111674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.7	✓	9	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.09926 q^{5} -1.62400 q^{7} +1.00000 q^{9} -4.16402 q^{11} +3.95343 q^{13} +1.09926 q^{15} -2.36393 q^{17} +4.37074 q^{19} -1.62400 q^{21} -1.00000 q^{23} -3.79163 q^{25} +1.00000 q^{27} -1.00000 q^{29} -8.09478 q^{31} -4.16402 q^{33} -1.78520 q^{35} +3.52968 q^{37} +3.95343 q^{39} -3.81475 q^{41} +1.25079 q^{43} +1.09926 q^{45} +6.41448 q^{47} -4.36263 q^{49} -2.36393 q^{51} -3.55412 q^{53} -4.57734 q^{55} +4.37074 q^{57} +5.96232 q^{59} -12.3425 q^{61} -1.62400 q^{63} +4.34585 q^{65} -12.8312 q^{67} -1.00000 q^{69} +8.30445 q^{71} -2.26071 q^{73} -3.79163 q^{75} +6.76236 q^{77} -14.7456 q^{79} +1.00000 q^{81} +9.49232 q^{83} -2.59857 q^{85} -1.00000 q^{87} -4.52312 q^{89} -6.42036 q^{91} -8.09478 q^{93} +4.80459 q^{95} -2.79546 q^{97} -4.16402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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