LMFDB - Embedded newform 8700.2.a.bj.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8700,2,Mod(1,8700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8700, 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("8700.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8700.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,0,0,6,0,6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.4698497585$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.71480896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 10x - 6 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 10*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.511965$ of defining polynomial
Character	$\chi$	$=$	8700.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.0863317 q^{7} +1.00000 q^{9} +1.70184 q^{11} +2.93760 q^{13} +7.61055 q^{17} +2.40053 q^{19} -0.0863317 q^{21} -1.94053 q^{23} -1.00000 q^{27} +1.00000 q^{29} +9.38449 q^{31} -1.70184 q^{33} +9.79606 q^{37} -2.93760 q^{39} -1.11691 q^{41} +8.39126 q^{43} +0.945743 q^{47} -6.99255 q^{49} -7.61055 q^{51} -2.71154 q^{53} -2.40053 q^{57} -7.86705 q^{59} -10.2279 q^{61} +0.0863317 q^{63} +2.92156 q^{67} +1.94053 q^{69} +8.31213 q^{71} -10.2197 q^{73} +0.146923 q^{77} +15.1831 q^{79} +1.00000 q^{81} +3.85101 q^{83} -1.00000 q^{87} -7.36238 q^{89} +0.253608 q^{91} -9.38449 q^{93} -1.01524 q^{97} +1.70184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 10 q^{19} - 6 q^{21} + 10 q^{23} - 6 q^{27} + 6 q^{29} - 14 q^{31} - 4 q^{33} + 2 q^{37} - 2 q^{39} - 12 q^{41} + 32 q^{43} + 10 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 10 * q^19 - 6 * q^21 + 10 * q^23 - 6 * q^27 + 6 * q^29 - 14 * q^31 - 4 * q^33 + 2 * q^37 - 2 * q^39 - 12 * q^41 + 32 * q^43 + 10 * q^47 + 14 * q^49 - 2 * q^51 + 26 * q^53 + 10 * q^57 + 4 * q^59 - 26 * q^61 + 6 * q^63 + 38 * q^67 - 10 * q^69 + 6 * q^71 - 6 * q^73 + 2 * q^77 + 30 * q^79 + 6 * q^81 + 8 * q^83 - 6 * q^87 - 4 * q^89 - 24 * q^91 + 14 * q^93 - 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0863317	0.0326303	0.0163152	−	0.999867i	$-0.494806\pi$
$7$	0.0863317	0.0326303	0.0163152	+	0.999867i	$0.494806\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.70184	0.513124	0.256562	−	0.966528i	$-0.417410\pi$
$11$	1.70184	0.513124	0.256562	+	0.966528i	$0.417410\pi$
$12$	0	0
$13$	2.93760	0.814743	0.407371	−	0.913263i	$-0.366445\pi$
$13$	2.93760	0.814743	0.407371	+	0.913263i	$0.366445\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.61055	1.84583	0.922915	−	0.385005i	$-0.125800\pi$
$17$	7.61055	1.84583	0.922915	+	0.385005i	$0.125800\pi$
$18$	0	0
$19$	2.40053	0.550719	0.275360	−	0.961341i	$-0.411203\pi$
$19$	2.40053	0.550719	0.275360	+	0.961341i	$0.411203\pi$
$20$	0	0
$21$	−0.0863317	−0.0188391
$22$	0	0
$23$	−1.94053	−0.404628	−0.202314	−	0.979321i	$-0.564846\pi$
$23$	−1.94053	−0.404628	−0.202314	+	0.979321i	$0.564846\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.38449	1.68550	0.842752	−	0.538301i	$-0.180934\pi$
$31$	9.38449	1.68550	0.842752	+	0.538301i	$0.180934\pi$
$32$	0	0
$33$	−1.70184	−0.296252
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.79606	1.61046	0.805231	−	0.592961i	$-0.202041\pi$
$37$	9.79606	1.61046	0.805231	+	0.592961i	$0.202041\pi$
$38$	0	0
$39$	−2.93760	−0.470392
$40$	0	0
$41$	−1.11691	−0.174433	−0.0872163	−	0.996189i	$-0.527797\pi$
$41$	−1.11691	−0.174433	−0.0872163	+	0.996189i	$0.527797\pi$
$42$	0	0
$43$	8.39126	1.27966	0.639828	−	0.768518i	$-0.279006\pi$
$43$	8.39126	1.27966	0.639828	+	0.768518i	$0.279006\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.945743	0.137951	0.0689754	−	0.997618i	$-0.478027\pi$
$47$	0.945743	0.137951	0.0689754	+	0.997618i	$0.478027\pi$
$48$	0	0
$49$	−6.99255	−0.998935
$50$	0	0
$51$	−7.61055	−1.06569
$52$	0	0
$53$	−2.71154	−0.372459	−0.186230	−	0.982506i	$-0.559627\pi$
$53$	−2.71154	−0.372459	−0.186230	+	0.982506i	$0.559627\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.40053	−0.317958
$58$	0	0
$59$	−7.86705	−1.02420	−0.512101	−	0.858925i	$-0.671133\pi$
$59$	−7.86705	−1.02420	−0.512101	+	0.858925i	$0.671133\pi$
$60$	0	0
$61$	−10.2279	−1.30954	−0.654772	−	0.755826i	$-0.727235\pi$
$61$	−10.2279	−1.30954	−0.654772	+	0.755826i	$0.727235\pi$
$62$	0	0
$63$	0.0863317	0.0108768
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.92156	0.356925	0.178463	−	0.983947i	$-0.442888\pi$
$67$	2.92156	0.356925	0.178463	+	0.983947i	$0.442888\pi$
$68$	0	0
$69$	1.94053	0.233612
$70$	0	0
$71$	8.31213	0.986469	0.493234	−	0.869896i	$-0.335815\pi$
$71$	8.31213	0.986469	0.493234	+	0.869896i	$0.335815\pi$
$72$	0	0
$73$	−10.2197	−1.19613	−0.598064	−	0.801448i	$-0.704063\pi$
$73$	−10.2197	−1.19613	−0.598064	+	0.801448i	$0.704063\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.146923	0.0167434
$78$	0	0
$79$	15.1831	1.70823	0.854114	−	0.520086i	$-0.174100\pi$
$79$	15.1831	1.70823	0.854114	+	0.520086i	$0.174100\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.85101	0.422703	0.211352	−	0.977410i	$-0.432213\pi$
$83$	3.85101	0.422703	0.211352	+	0.977410i	$0.432213\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−7.36238	−0.780410	−0.390205	−	0.920728i	$-0.627596\pi$
$89$	−7.36238	−0.780410	−0.390205	+	0.920728i	$0.627596\pi$
$90$	0	0
$91$	0.253608	0.0265853
$92$	0	0
$93$	−9.38449	−0.973127
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.01524	−0.103082	−0.0515408	−	0.998671i	$-0.516413\pi$
$97$	−1.01524	−0.103082	−0.0515408	+	0.998671i	$0.516413\pi$
$98$	0	0
$99$	1.70184	0.171041
$100$	0	0
$101$	−10.7661	−1.07126	−0.535631	−	0.844452i	$-0.679926\pi$
$101$	−10.7661	−1.07126	−0.535631	+	0.844452i	$0.679926\pi$
$102$	0	0
$103$	−3.21014	−0.316304	−0.158152	−	0.987415i	$-0.550554\pi$
$103$	−3.21014	−0.316304	−0.158152	+	0.987415i	$0.550554\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.24849	0.507391	0.253695	−	0.967284i	$-0.418354\pi$
$107$	5.24849	0.507391	0.253695	+	0.967284i	$0.418354\pi$
$108$	0	0
$109$	−7.96830	−0.763225	−0.381612	−	0.924322i	$-0.624631\pi$
$109$	−7.96830	−0.763225	−0.381612	+	0.924322i	$0.624631\pi$
$110$	0	0
$111$	−9.79606	−0.929801
$112$	0	0
$113$	10.3452	0.973196	0.486598	−	0.873626i	$-0.338238\pi$
$113$	10.3452	0.973196	0.486598	+	0.873626i	$0.338238\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.93760	0.271581
$118$	0	0
$119$	0.657032	0.0602300
$120$	0	0
$121$	−8.10374	−0.736704
$122$	0	0
$123$	1.11691	0.100709
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.8138	−1.58072	−0.790359	−	0.612644i	$-0.790106\pi$
$127$	−17.8138	−1.58072	−0.790359	+	0.612644i	$0.790106\pi$
$128$	0	0
$129$	−8.39126	−0.738810
$130$	0	0
$131$	10.4514	0.913145	0.456572	−	0.889686i	$-0.349077\pi$
$131$	10.4514	0.913145	0.456572	+	0.889686i	$0.349077\pi$
$132$	0	0
$133$	0.207242	0.0179702
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.7048	1.17088	0.585439	−	0.810717i	$-0.300922\pi$
$137$	13.7048	1.17088	0.585439	+	0.810717i	$0.300922\pi$
$138$	0	0
$139$	9.32998	0.791359	0.395679	−	0.918389i	$-0.370509\pi$
$139$	9.32998	0.791359	0.395679	+	0.918389i	$0.370509\pi$
$140$	0	0
$141$	−0.945743	−0.0796459
$142$	0	0
$143$	4.99932	0.418064
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.99255	0.576736
$148$	0	0
$149$	−13.8670	−1.13603	−0.568016	−	0.823017i	$-0.692289\pi$
$149$	−13.8670	−1.13603	−0.568016	+	0.823017i	$0.692289\pi$
$150$	0	0
$151$	1.58843	0.129265	0.0646323	−	0.997909i	$-0.479413\pi$
$151$	1.58843	0.129265	0.0646323	+	0.997909i	$0.479413\pi$
$152$	0	0
$153$	7.61055	0.615276
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0766	0.963820	0.481910	−	0.876221i	$-0.339943\pi$
$157$	12.0766	0.963820	0.481910	+	0.876221i	$0.339943\pi$
$158$	0	0
$159$	2.71154	0.215039
$160$	0	0
$161$	−0.167529	−0.0132032
$162$	0	0
$163$	17.9098	1.40281	0.701403	−	0.712765i	$-0.252557\pi$
$163$	17.9098	1.40281	0.701403	+	0.712765i	$0.252557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.20506	0.557544	0.278772	−	0.960357i	$-0.410073\pi$
$167$	7.20506	0.557544	0.278772	+	0.960357i	$0.410073\pi$
$168$	0	0
$169$	−4.37052	−0.336194
$170$	0	0
$171$	2.40053	0.183573
$172$	0	0
$173$	5.40954	0.411280	0.205640	−	0.978628i	$-0.434072\pi$
$173$	5.40954	0.411280	0.205640	+	0.978628i	$0.434072\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.86705	0.591324
$178$	0	0
$179$	−9.11319	−0.681152	−0.340576	−	0.940217i	$-0.610622\pi$
$179$	−9.11319	−0.681152	−0.340576	+	0.940217i	$0.610622\pi$
$180$	0	0
$181$	−17.9948	−1.33754	−0.668771	−	0.743468i	$-0.733179\pi$
$181$	−17.9948	−1.33754	−0.668771	+	0.743468i	$0.733179\pi$
$182$	0	0
$183$	10.2279	0.756066
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.9519	0.947139
$188$	0	0
$189$	−0.0863317	−0.00627971
$190$	0	0
$191$	−18.9574	−1.37171	−0.685856	−	0.727737i	$-0.740572\pi$
$191$	−18.9574	−1.37171	−0.685856	+	0.727737i	$0.740572\pi$
$192$	0	0
$193$	−5.61204	−0.403964	−0.201982	−	0.979389i	$-0.564738\pi$
$193$	−5.61204	−0.403964	−0.201982	+	0.979389i	$0.564738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.44387	0.459106	0.229553	−	0.973296i	$-0.426273\pi$
$197$	6.44387	0.459106	0.229553	+	0.973296i	$0.426273\pi$
$198$	0	0
$199$	−7.26374	−0.514913	−0.257456	−	0.966290i	$-0.582884\pi$
$199$	−7.26374	−0.514913	−0.257456	+	0.966290i	$0.582884\pi$
$200$	0	0
$201$	−2.92156	−0.206071
$202$	0	0
$203$	0.0863317	0.00605930
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.94053	−0.134876
$208$	0	0
$209$	4.08532	0.282587
$210$	0	0
$211$	−18.5216	−1.27508	−0.637540	−	0.770417i	$-0.720048\pi$
$211$	−18.5216	−1.27508	−0.637540	+	0.770417i	$0.720048\pi$
$212$	0	0
$213$	−8.31213	−0.569538
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.810179	0.0549986
$218$	0	0
$219$	10.2197	0.690585
$220$	0	0
$221$	22.3567	1.50388
$222$	0	0
$223$	−6.47792	−0.433793	−0.216897	−	0.976195i	$-0.569593\pi$
$223$	−6.47792	−0.433793	−0.216897	+	0.976195i	$0.569593\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.3062	−1.67963	−0.839815	−	0.542873i	$-0.817337\pi$
$227$	−25.3062	−1.67963	−0.839815	+	0.542873i	$0.817337\pi$
$228$	0	0
$229$	11.3881	0.752548	0.376274	−	0.926508i	$-0.377205\pi$
$229$	11.3881	0.752548	0.376274	+	0.926508i	$0.377205\pi$
$230$	0	0
$231$	−0.146923	−0.00966680
$232$	0	0
$233$	−12.1884	−0.798492	−0.399246	−	0.916844i	$-0.630728\pi$
$233$	−12.1884	−0.798492	−0.399246	+	0.916844i	$0.630728\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.1831	−0.986246
$238$	0	0
$239$	0.738073	0.0477420	0.0238710	−	0.999715i	$-0.492401\pi$
$239$	0.738073	0.0477420	0.0238710	+	0.999715i	$0.492401\pi$
$240$	0	0
$241$	17.7064	1.14057	0.570283	−	0.821448i	$-0.306833\pi$
$241$	17.7064	1.14057	0.570283	+	0.821448i	$0.306833\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.05179	0.448695
$248$	0	0
$249$	−3.85101	−0.244048
$250$	0	0
$251$	7.94712	0.501618	0.250809	−	0.968037i	$-0.419303\pi$
$251$	7.94712	0.501618	0.250809	+	0.968037i	$0.419303\pi$
$252$	0	0
$253$	−3.30247	−0.207624
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.9020	−1.86524	−0.932618	−	0.360865i	$-0.882481\pi$
$257$	−29.9020	−1.86524	−0.932618	+	0.360865i	$0.882481\pi$
$258$	0	0
$259$	0.845711	0.0525499
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	17.1721	1.05888	0.529438	−	0.848348i	$-0.322403\pi$
$263$	17.1721	1.05888	0.529438	+	0.848348i	$0.322403\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.36238	0.450570
$268$	0	0
$269$	−11.7897	−0.718832	−0.359416	−	0.933177i	$-0.617024\pi$
$269$	−11.7897	−0.718832	−0.359416	+	0.933177i	$0.617024\pi$
$270$	0	0
$271$	−11.2152	−0.681277	−0.340638	−	0.940194i	$-0.610643\pi$
$271$	−11.2152	−0.681277	−0.340638	+	0.940194i	$0.610643\pi$
$272$	0	0
$273$	−0.253608	−0.0153490
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.15029	−0.549788	−0.274894	−	0.961475i	$-0.588643\pi$
$277$	−9.15029	−0.549788	−0.274894	+	0.961475i	$0.588643\pi$
$278$	0	0
$279$	9.38449	0.561835
$280$	0	0
$281$	11.1974	0.667981	0.333991	−	0.942576i	$-0.391605\pi$
$281$	11.1974	0.667981	0.333991	+	0.942576i	$0.391605\pi$
$282$	0	0
$283$	22.5134	1.33828	0.669142	−	0.743135i	$-0.266662\pi$
$283$	22.5134	1.33828	0.669142	+	0.743135i	$0.266662\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.0964251	−0.00569179
$288$	0	0
$289$	40.9204	2.40708
$290$	0	0
$291$	1.01524	0.0595142
$292$	0	0
$293$	25.3153	1.47894	0.739468	−	0.673191i	$-0.235077\pi$
$293$	25.3153	1.47894	0.739468	+	0.673191i	$0.235077\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.70184	−0.0987507
$298$	0	0
$299$	−5.70049	−0.329668
$300$	0	0
$301$	0.724432	0.0417556
$302$	0	0
$303$	10.7661	0.618494
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.77753	0.558033	0.279017	−	0.960286i	$-0.409992\pi$
$307$	9.77753	0.558033	0.279017	+	0.960286i	$0.409992\pi$
$308$	0	0
$309$	3.21014	0.182618
$310$	0	0
$311$	24.9181	1.41298	0.706488	−	0.707725i	$-0.250279\pi$
$311$	24.9181	1.41298	0.706488	+	0.707725i	$0.250279\pi$
$312$	0	0
$313$	32.8771	1.85832	0.929161	−	0.369675i	$-0.120531\pi$
$313$	32.8771	1.85832	0.929161	+	0.369675i	$0.120531\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.58630	−0.313758	−0.156879	−	0.987618i	$-0.550143\pi$
$317$	−5.58630	−0.313758	−0.156879	+	0.987618i	$0.550143\pi$
$318$	0	0
$319$	1.70184	0.0952847
$320$	0	0
$321$	−5.24849	−0.292942
$322$	0	0
$323$	18.2694	1.01653
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.96830	0.440648
$328$	0	0
$329$	0.0816476	0.00450138
$330$	0	0
$331$	21.3330	1.17257	0.586284	−	0.810105i	$-0.300590\pi$
$331$	21.3330	1.17257	0.586284	+	0.810105i	$0.300590\pi$
$332$	0	0
$333$	9.79606	0.536821
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.296030	0.0161258	0.00806289	−	0.999967i	$-0.497433\pi$
$337$	0.296030	0.0161258	0.00806289	+	0.999967i	$0.497433\pi$
$338$	0	0
$339$	−10.3452	−0.561875
$340$	0	0
$341$	15.9709	0.864873
$342$	0	0
$343$	−1.20800	−0.0652259
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.2059	−1.72890	−0.864452	−	0.502715i	$-0.832335\pi$
$347$	−32.2059	−1.72890	−0.864452	+	0.502715i	$0.832335\pi$
$348$	0	0
$349$	5.98147	0.320181	0.160090	−	0.987102i	$-0.448821\pi$
$349$	5.98147	0.320181	0.160090	+	0.987102i	$0.448821\pi$
$350$	0	0
$351$	−2.93760	−0.156797
$352$	0	0
$353$	11.1076	0.591200	0.295600	−	0.955312i	$-0.404480\pi$
$353$	11.1076	0.591200	0.295600	+	0.955312i	$0.404480\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.657032	−0.0347738
$358$	0	0
$359$	−9.24108	−0.487726	−0.243863	−	0.969810i	$-0.578415\pi$
$359$	−9.24108	−0.487726	−0.243863	+	0.969810i	$0.578415\pi$
$360$	0	0
$361$	−13.2375	−0.696708
$362$	0	0
$363$	8.10374	0.425336
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.2721	0.692796	0.346398	−	0.938088i	$-0.387405\pi$
$367$	13.2721	0.692796	0.346398	+	0.938088i	$0.387405\pi$
$368$	0	0
$369$	−1.11691	−0.0581442
$370$	0	0
$371$	−0.234092	−0.0121535
$372$	0	0
$373$	29.9886	1.55275	0.776374	−	0.630272i	$-0.217057\pi$
$373$	29.9886	1.55275	0.776374	+	0.630272i	$0.217057\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.93760	0.151294
$378$	0	0
$379$	29.7555	1.52844	0.764219	−	0.644956i	$-0.223125\pi$
$379$	29.7555	1.52844	0.764219	+	0.644956i	$0.223125\pi$
$380$	0	0
$381$	17.8138	0.912628
$382$	0	0
$383$	33.6277	1.71829	0.859147	−	0.511729i	$-0.170995\pi$
$383$	33.6277	1.71829	0.859147	+	0.511729i	$0.170995\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.39126	0.426552
$388$	0	0
$389$	9.97346	0.505674	0.252837	−	0.967509i	$-0.418636\pi$
$389$	9.97346	0.505674	0.252837	+	0.967509i	$0.418636\pi$
$390$	0	0
$391$	−14.7685	−0.746875
$392$	0	0
$393$	−10.4514	−0.527204
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.85376	−0.143226	−0.0716130	−	0.997432i	$-0.522815\pi$
$397$	−2.85376	−0.143226	−0.0716130	+	0.997432i	$0.522815\pi$
$398$	0	0
$399$	−0.207242	−0.0103751
$400$	0	0
$401$	−23.2457	−1.16083	−0.580417	−	0.814319i	$-0.697110\pi$
$401$	−23.2457	−1.16083	−0.580417	+	0.814319i	$0.697110\pi$
$402$	0	0
$403$	27.5679	1.37325
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.6713	0.826367
$408$	0	0
$409$	16.6924	0.825384	0.412692	−	0.910871i	$-0.364589\pi$
$409$	16.6924	0.825384	0.412692	+	0.910871i	$0.364589\pi$
$410$	0	0
$411$	−13.7048	−0.676007
$412$	0	0
$413$	−0.679176	−0.0334201
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.32998	−0.456891
$418$	0	0
$419$	−3.97632	−0.194256	−0.0971279	−	0.995272i	$-0.530966\pi$
$419$	−3.97632	−0.194256	−0.0971279	+	0.995272i	$0.530966\pi$
$420$	0	0
$421$	28.7014	1.39882	0.699410	−	0.714720i	$-0.253446\pi$
$421$	28.7014	1.39882	0.699410	+	0.714720i	$0.253446\pi$
$422$	0	0
$423$	0.945743	0.0459836
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.882989	−0.0427308
$428$	0	0
$429$	−4.99932	−0.241369
$430$	0	0
$431$	−22.8175	−1.09908	−0.549540	−	0.835468i	$-0.685197\pi$
$431$	−22.8175	−1.09908	−0.549540	+	0.835468i	$0.685197\pi$
$432$	0	0
$433$	−33.2710	−1.59890	−0.799450	−	0.600733i	$-0.794876\pi$
$433$	−33.2710	−1.59890	−0.799450	+	0.600733i	$0.794876\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.65830	−0.222837
$438$	0	0
$439$	14.4826	0.691217	0.345608	−	0.938379i	$-0.387673\pi$
$439$	14.4826	0.691217	0.345608	+	0.938379i	$0.387673\pi$
$440$	0	0
$441$	−6.99255	−0.332978
$442$	0	0
$443$	−12.6821	−0.602544	−0.301272	−	0.953538i	$-0.597411\pi$
$443$	−12.6821	−0.602544	−0.301272	+	0.953538i	$0.597411\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.8670	0.655889
$448$	0	0
$449$	−40.5628	−1.91428	−0.957140	−	0.289627i	$-0.906469\pi$
$449$	−40.5628	−1.91428	−0.957140	+	0.289627i	$0.906469\pi$
$450$	0	0
$451$	−1.90081	−0.0895055
$452$	0	0
$453$	−1.58843	−0.0746310
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.30406	0.435225	0.217613	−	0.976035i	$-0.430173\pi$
$457$	9.30406	0.435225	0.217613	+	0.976035i	$0.430173\pi$
$458$	0	0
$459$	−7.61055	−0.355230
$460$	0	0
$461$	−42.8411	−1.99531	−0.997654	−	0.0684524i	$-0.978194\pi$
$461$	−42.8411	−1.99531	−0.997654	+	0.0684524i	$0.978194\pi$
$462$	0	0
$463$	36.6769	1.70452	0.852261	−	0.523117i	$-0.175231\pi$
$463$	36.6769	1.70452	0.852261	+	0.523117i	$0.175231\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.4216	−1.36147	−0.680735	−	0.732529i	$-0.738340\pi$
$467$	−29.4216	−1.36147	−0.680735	+	0.732529i	$0.738340\pi$
$468$	0	0
$469$	0.252223	0.0116466
$470$	0	0
$471$	−12.0766	−0.556462
$472$	0	0
$473$	14.2806	0.656622
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.71154	−0.124153
$478$	0	0
$479$	−21.8388	−0.997840	−0.498920	−	0.866648i	$-0.666270\pi$
$479$	−21.8388	−0.997840	−0.498920	+	0.866648i	$0.666270\pi$
$480$	0	0
$481$	28.7769	1.31211
$482$	0	0
$483$	0.167529	0.00762284
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.9371	−0.812809	−0.406404	−	0.913693i	$-0.633218\pi$
$487$	−17.9371	−0.812809	−0.406404	+	0.913693i	$0.633218\pi$
$488$	0	0
$489$	−17.9098	−0.809910
$490$	0	0
$491$	33.4680	1.51039	0.755194	−	0.655501i	$-0.227543\pi$
$491$	33.4680	1.51039	0.755194	+	0.655501i	$0.227543\pi$
$492$	0	0
$493$	7.61055	0.342762
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.717601	0.0321888
$498$	0	0
$499$	−2.09063	−0.0935893	−0.0467947	−	0.998905i	$-0.514901\pi$
$499$	−2.09063	−0.0935893	−0.0467947	+	0.998905i	$0.514901\pi$
$500$	0	0
$501$	−7.20506	−0.321898
$502$	0	0
$503$	18.2783	0.814991	0.407495	−	0.913207i	$-0.366402\pi$
$503$	18.2783	0.814991	0.407495	+	0.913207i	$0.366402\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.37052	0.194102
$508$	0	0
$509$	−26.1018	−1.15694	−0.578470	−	0.815704i	$-0.696350\pi$
$509$	−26.1018	−1.15694	−0.578470	+	0.815704i	$0.696350\pi$
$510$	0	0
$511$	−0.882286	−0.0390300
$512$	0	0
$513$	−2.40053	−0.105986
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.60950	0.0707858
$518$	0	0
$519$	−5.40954	−0.237453
$520$	0	0
$521$	5.90027	0.258496	0.129248	−	0.991612i	$-0.458744\pi$
$521$	5.90027	0.258496	0.129248	+	0.991612i	$0.458744\pi$
$522$	0	0
$523$	−19.5206	−0.853578	−0.426789	−	0.904351i	$-0.640355\pi$
$523$	−19.5206	−0.853578	−0.426789	+	0.904351i	$0.640355\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	71.4211	3.11115
$528$	0	0
$529$	−19.2343	−0.836276
$530$	0	0
$531$	−7.86705	−0.341401
$532$	0	0
$533$	−3.28104	−0.142118
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.11319	0.393263
$538$	0	0
$539$	−11.9002	−0.512577
$540$	0	0
$541$	37.5741	1.61544	0.807718	−	0.589569i	$-0.200702\pi$
$541$	37.5741	1.61544	0.807718	+	0.589569i	$0.200702\pi$
$542$	0	0
$543$	17.9948	0.772230
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.81738	−0.377004	−0.188502	−	0.982073i	$-0.560363\pi$
$547$	−8.81738	−0.377004	−0.188502	+	0.982073i	$0.560363\pi$
$548$	0	0
$549$	−10.2279	−0.436515
$550$	0	0
$551$	2.40053	0.102266
$552$	0	0
$553$	1.31078	0.0557400
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.60084	−0.364429	−0.182215	−	0.983259i	$-0.558327\pi$
$557$	−8.60084	−0.364429	−0.182215	+	0.983259i	$0.558327\pi$
$558$	0	0
$559$	24.6502	1.04259
$560$	0	0
$561$	−12.9519	−0.546831
$562$	0	0
$563$	−7.44311	−0.313690	−0.156845	−	0.987623i	$-0.550132\pi$
$563$	−7.44311	−0.313690	−0.156845	+	0.987623i	$0.550132\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0863317	0.00362559
$568$	0	0
$569$	−2.93373	−0.122988	−0.0614942	−	0.998107i	$-0.519587\pi$
$569$	−2.93373	−0.122988	−0.0614942	+	0.998107i	$0.519587\pi$
$570$	0	0
$571$	−35.0057	−1.46494	−0.732471	−	0.680799i	$-0.761633\pi$
$571$	−35.0057	−1.46494	−0.732471	+	0.680799i	$0.761633\pi$
$572$	0	0
$573$	18.9574	0.791958
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.8811	0.452987	0.226494	−	0.974013i	$-0.427274\pi$
$577$	10.8811	0.452987	0.226494	+	0.974013i	$0.427274\pi$
$578$	0	0
$579$	5.61204	0.233228
$580$	0	0
$581$	0.332464	0.0137929
$582$	0	0
$583$	−4.61461	−0.191118
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.22269	0.0504659	0.0252329	−	0.999682i	$-0.491967\pi$
$587$	1.22269	0.0504659	0.0252329	+	0.999682i	$0.491967\pi$
$588$	0	0
$589$	22.5278	0.928240
$590$	0	0
$591$	−6.44387	−0.265065
$592$	0	0
$593$	−20.5341	−0.843235	−0.421617	−	0.906774i	$-0.638537\pi$
$593$	−20.5341	−0.843235	−0.421617	+	0.906774i	$0.638537\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.26374	0.297285
$598$	0	0
$599$	38.3997	1.56897	0.784485	−	0.620147i	$-0.212927\pi$
$599$	38.3997	1.56897	0.784485	+	0.620147i	$0.212927\pi$
$600$	0	0
$601$	39.8975	1.62745	0.813726	−	0.581248i	$-0.197435\pi$
$601$	39.8975	1.62745	0.813726	+	0.581248i	$0.197435\pi$
$602$	0	0
$603$	2.92156	0.118975
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.4957	−0.710127	−0.355063	−	0.934842i	$-0.615541\pi$
$607$	−17.4957	−0.710127	−0.355063	+	0.934842i	$0.615541\pi$
$608$	0	0
$609$	−0.0863317	−0.00349834
$610$	0	0
$611$	2.77821	0.112394
$612$	0	0
$613$	−20.8587	−0.842475	−0.421237	−	0.906950i	$-0.638404\pi$
$613$	−20.8587	−0.842475	−0.421237	+	0.906950i	$0.638404\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.8938	0.519085	0.259542	−	0.965732i	$-0.416428\pi$
$617$	12.8938	0.519085	0.259542	+	0.965732i	$0.416428\pi$
$618$	0	0
$619$	−9.09969	−0.365748	−0.182874	−	0.983136i	$-0.558540\pi$
$619$	−9.09969	−0.365748	−0.182874	+	0.983136i	$0.558540\pi$
$620$	0	0
$621$	1.94053	0.0778708
$622$	0	0
$623$	−0.635607	−0.0254650
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.08532	−0.163152
$628$	0	0
$629$	74.5534	2.97264
$630$	0	0
$631$	−9.92630	−0.395160	−0.197580	−	0.980287i	$-0.563308\pi$
$631$	−9.92630	−0.395160	−0.197580	+	0.980287i	$0.563308\pi$
$632$	0	0
$633$	18.5216	0.736168
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.5413	−0.813875
$638$	0	0
$639$	8.31213	0.328823
$640$	0	0
$641$	−25.1936	−0.995086	−0.497543	−	0.867439i	$-0.665764\pi$
$641$	−25.1936	−0.995086	−0.497543	+	0.867439i	$0.665764\pi$
$642$	0	0
$643$	−4.84807	−0.191189	−0.0955946	−	0.995420i	$-0.530475\pi$
$643$	−4.84807	−0.191189	−0.0955946	+	0.995420i	$0.530475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.03185	0.355079	0.177539	−	0.984114i	$-0.443186\pi$
$647$	9.03185	0.355079	0.177539	+	0.984114i	$0.443186\pi$
$648$	0	0
$649$	−13.3885	−0.525543
$650$	0	0
$651$	−0.810179	−0.0317534
$652$	0	0
$653$	32.1899	1.25969	0.629843	−	0.776722i	$-0.283119\pi$
$653$	32.1899	1.25969	0.629843	+	0.776722i	$0.283119\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.2197	−0.398709
$658$	0	0
$659$	−9.66702	−0.376573	−0.188287	−	0.982114i	$-0.560293\pi$
$659$	−9.66702	−0.376573	−0.188287	+	0.982114i	$0.560293\pi$
$660$	0	0
$661$	24.0476	0.935345	0.467672	−	0.883902i	$-0.345093\pi$
$661$	24.0476	0.935345	0.467672	+	0.883902i	$0.345093\pi$
$662$	0	0
$663$	−22.3567	−0.868263
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.94053	−0.0751376
$668$	0	0
$669$	6.47792	0.250451
$670$	0	0
$671$	−17.4062	−0.671958
$672$	0	0
$673$	−34.6367	−1.33514	−0.667572	−	0.744545i	$-0.732667\pi$
$673$	−34.6367	−1.33514	−0.667572	+	0.744545i	$0.732667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.1443	1.81190	0.905951	−	0.423384i	$-0.139158\pi$
$677$	47.1443	1.81190	0.905951	+	0.423384i	$0.139158\pi$
$678$	0	0
$679$	−0.0876471	−0.00336359
$680$	0	0
$681$	25.3062	0.969735
$682$	0	0
$683$	25.6738	0.982382	0.491191	−	0.871052i	$-0.336562\pi$
$683$	25.6738	0.982382	0.491191	+	0.871052i	$0.336562\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−11.3881	−0.434484
$688$	0	0
$689$	−7.96542	−0.303458
$690$	0	0
$691$	−26.7700	−1.01838	−0.509189	−	0.860654i	$-0.670055\pi$
$691$	−26.7700	−1.01838	−0.509189	+	0.860654i	$0.670055\pi$
$692$	0	0
$693$	0.146923	0.00558113
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.50033	−0.321973
$698$	0	0
$699$	12.1884	0.461009
$700$	0	0
$701$	−46.8954	−1.77122	−0.885608	−	0.464434i	$-0.846258\pi$
$701$	−46.8954	−1.77122	−0.885608	+	0.464434i	$0.846258\pi$
$702$	0	0
$703$	23.5157	0.886913
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.929452	−0.0349556
$708$	0	0
$709$	7.80324	0.293057	0.146528	−	0.989206i	$-0.453190\pi$
$709$	7.80324	0.293057	0.146528	+	0.989206i	$0.453190\pi$
$710$	0	0
$711$	15.1831	0.569409
$712$	0	0
$713$	−18.2109	−0.682003
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.738073	−0.0275638
$718$	0	0
$719$	3.42899	0.127880	0.0639399	−	0.997954i	$-0.479633\pi$
$719$	3.42899	0.127880	0.0639399	+	0.997954i	$0.479633\pi$
$720$	0	0
$721$	−0.277137	−0.0103211
$722$	0	0
$723$	−17.7064	−0.658507
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.50489	−0.204165	−0.102083	−	0.994776i	$-0.532551\pi$
$727$	−5.50489	−0.204165	−0.102083	+	0.994776i	$0.532551\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	63.8621	2.36203
$732$	0	0
$733$	11.0645	0.408678	0.204339	−	0.978900i	$-0.434496\pi$
$733$	11.0645	0.408678	0.204339	+	0.978900i	$0.434496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.97202	0.183147
$738$	0	0
$739$	30.3879	1.11784	0.558918	−	0.829223i	$-0.311217\pi$
$739$	30.3879	1.11784	0.558918	+	0.829223i	$0.311217\pi$
$740$	0	0
$741$	−7.05179	−0.259054
$742$	0	0
$743$	−26.3463	−0.966552	−0.483276	−	0.875468i	$-0.660553\pi$
$743$	−26.3463	−0.966552	−0.483276	+	0.875468i	$0.660553\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.85101	0.140901
$748$	0	0
$749$	0.453111	0.0165563
$750$	0	0
$751$	12.9716	0.473341	0.236670	−	0.971590i	$-0.423944\pi$
$751$	12.9716	0.473341	0.236670	+	0.971590i	$0.423944\pi$
$752$	0	0
$753$	−7.94712	−0.289609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.50097	0.127245	0.0636225	−	0.997974i	$-0.479735\pi$
$757$	3.50097	0.127245	0.0636225	+	0.997974i	$0.479735\pi$
$758$	0	0
$759$	3.30247	0.119872
$760$	0	0
$761$	13.3077	0.482404	0.241202	−	0.970475i	$-0.422458\pi$
$761$	13.3077	0.482404	0.241202	+	0.970475i	$0.422458\pi$
$762$	0	0
$763$	−0.687917	−0.0249043
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.1102	−0.834462
$768$	0	0
$769$	−31.5112	−1.13632	−0.568162	−	0.822917i	$-0.692345\pi$
$769$	−31.5112	−1.13632	−0.568162	+	0.822917i	$0.692345\pi$
$770$	0	0
$771$	29.9020	1.07689
$772$	0	0
$773$	13.8301	0.497434	0.248717	−	0.968576i	$-0.419991\pi$
$773$	13.8301	0.497434	0.248717	+	0.968576i	$0.419991\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.845711	−0.0303397
$778$	0	0
$779$	−2.68119	−0.0960635
$780$	0	0
$781$	14.1459	0.506181
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.8094	1.63293	0.816465	−	0.577395i	$-0.195931\pi$
$787$	45.8094	1.63293	0.816465	+	0.577395i	$0.195931\pi$
$788$	0	0
$789$	−17.1721	−0.611343
$790$	0	0
$791$	0.893120	0.0317557
$792$	0	0
$793$	−30.0454	−1.06694
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.2005	−0.821805	−0.410903	−	0.911679i	$-0.634786\pi$
$797$	−23.2005	−0.821805	−0.410903	+	0.911679i	$0.634786\pi$
$798$	0	0
$799$	7.19762	0.254634
$800$	0	0
$801$	−7.36238	−0.260137
$802$	0	0
$803$	−17.3923	−0.613762
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.7897	0.415018
$808$	0	0
$809$	−12.6305	−0.444066	−0.222033	−	0.975039i	$-0.571269\pi$
$809$	−12.6305	−0.444066	−0.222033	+	0.975039i	$0.571269\pi$
$810$	0	0
$811$	49.7822	1.74809	0.874044	−	0.485846i	$-0.161489\pi$
$811$	49.7822	1.74809	0.874044	+	0.485846i	$0.161489\pi$
$812$	0	0
$813$	11.2152	0.393335
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.1435	0.704731
$818$	0	0
$819$	0.253608	0.00886177
$820$	0	0
$821$	−14.6057	−0.509744	−0.254872	−	0.966975i	$-0.582033\pi$
$821$	−14.6057	−0.509744	−0.254872	+	0.966975i	$0.582033\pi$
$822$	0	0
$823$	40.1288	1.39880	0.699400	−	0.714730i	$-0.253451\pi$
$823$	40.1288	1.39880	0.699400	+	0.714730i	$0.253451\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.4889	1.51226	0.756129	−	0.654423i	$-0.227088\pi$
$827$	43.4889	1.51226	0.756129	+	0.654423i	$0.227088\pi$
$828$	0	0
$829$	22.2210	0.771768	0.385884	−	0.922547i	$-0.373896\pi$
$829$	22.2210	0.771768	0.385884	+	0.922547i	$0.373896\pi$
$830$	0	0
$831$	9.15029	0.317420
$832$	0	0
$833$	−53.2171	−1.84386
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.38449	−0.324376
$838$	0	0
$839$	−45.8662	−1.58348	−0.791738	−	0.610861i	$-0.790824\pi$
$839$	−45.8662	−1.58348	−0.791738	+	0.610861i	$0.790824\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−11.1974	−0.385659
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.699610	−0.0240389
$848$	0	0
$849$	−22.5134	−0.772658
$850$	0	0
$851$	−19.0095	−0.651639
$852$	0	0
$853$	42.0910	1.44117	0.720585	−	0.693367i	$-0.243873\pi$
$853$	42.0910	1.44117	0.720585	+	0.693367i	$0.243873\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.2236	0.451711	0.225855	−	0.974161i	$-0.427482\pi$
$857$	13.2236	0.451711	0.225855	+	0.974161i	$0.427482\pi$
$858$	0	0
$859$	53.5925	1.82855	0.914277	−	0.405090i	$-0.132760\pi$
$859$	53.5925	1.82855	0.914277	+	0.405090i	$0.132760\pi$
$860$	0	0
$861$	0.0964251	0.00328616
$862$	0	0
$863$	28.0920	0.956263	0.478132	−	0.878288i	$-0.341314\pi$
$863$	28.0920	0.956263	0.478132	+	0.878288i	$0.341314\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−40.9204	−1.38973
$868$	0	0
$869$	25.8391	0.876532
$870$	0	0
$871$	8.58237	0.290802
$872$	0	0
$873$	−1.01524	−0.0343606
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.3854	−1.02604	−0.513020	−	0.858376i	$-0.671473\pi$
$877$	−30.3854	−1.02604	−0.513020	+	0.858376i	$0.671473\pi$
$878$	0	0
$879$	−25.3153	−0.853865
$880$	0	0
$881$	3.46133	0.116615	0.0583076	−	0.998299i	$-0.481430\pi$
$881$	3.46133	0.116615	0.0583076	+	0.998299i	$0.481430\pi$
$882$	0	0
$883$	−11.3866	−0.383190	−0.191595	−	0.981474i	$-0.561366\pi$
$883$	−11.3866	−0.383190	−0.191595	+	0.981474i	$0.561366\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.3582	1.59013	0.795066	−	0.606523i	$-0.207436\pi$
$887$	47.3582	1.59013	0.795066	+	0.606523i	$0.207436\pi$
$888$	0	0
$889$	−1.53790	−0.0515794
$890$	0	0
$891$	1.70184	0.0570138
$892$	0	0
$893$	2.27028	0.0759722
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.70049	0.190334
$898$	0	0
$899$	9.38449	0.312990
$900$	0	0
$901$	−20.6363	−0.687496
$902$	0	0
$903$	−0.724432	−0.0241076
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22.7370	0.754970	0.377485	−	0.926016i	$-0.376789\pi$
$907$	22.7370	0.754970	0.377485	+	0.926016i	$0.376789\pi$
$908$	0	0
$909$	−10.7661	−0.357087
$910$	0	0
$911$	−18.7161	−0.620092	−0.310046	−	0.950722i	$-0.600344\pi$
$911$	−18.7161	−0.620092	−0.310046	+	0.950722i	$0.600344\pi$
$912$	0	0
$913$	6.55380	0.216899
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.902289	0.0297962
$918$	0	0
$919$	2.80576	0.0925536	0.0462768	−	0.998929i	$-0.485264\pi$
$919$	2.80576	0.0925536	0.0462768	+	0.998929i	$0.485264\pi$
$920$	0	0
$921$	−9.77753	−0.322181
$922$	0	0
$923$	24.4177	0.803718
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.21014	−0.105435
$928$	0	0
$929$	24.1779	0.793251	0.396625	−	0.917981i	$-0.370181\pi$
$929$	24.1779	0.793251	0.396625	+	0.917981i	$0.370181\pi$
$930$	0	0
$931$	−16.7858	−0.550133
$932$	0	0
$933$	−24.9181	−0.815782
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−47.8220	−1.56228	−0.781139	−	0.624357i	$-0.785361\pi$
$937$	−47.8220	−1.56228	−0.781139	+	0.624357i	$0.785361\pi$
$938$	0	0
$939$	−32.8771	−1.07290
$940$	0	0
$941$	−3.25697	−0.106174	−0.0530870	−	0.998590i	$-0.516906\pi$
$941$	−3.25697	−0.106174	−0.0530870	+	0.998590i	$0.516906\pi$
$942$	0	0
$943$	2.16740	0.0705804
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.6584	0.898778	0.449389	−	0.893336i	$-0.351642\pi$
$947$	27.6584	0.898778	0.449389	+	0.893336i	$0.351642\pi$
$948$	0	0
$949$	−30.0214	−0.974537
$950$	0	0
$951$	5.58630	0.181148
$952$	0	0
$953$	−34.3048	−1.11124	−0.555620	−	0.831436i	$-0.687519\pi$
$953$	−34.3048	−1.11124	−0.555620	+	0.831436i	$0.687519\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.70184	−0.0550126
$958$	0	0
$959$	1.18316	0.0382061
$960$	0	0
$961$	57.0687	1.84093
$962$	0	0
$963$	5.24849	0.169130
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.4858	0.819568	0.409784	−	0.912183i	$-0.365604\pi$
$967$	25.4858	0.819568	0.409784	+	0.912183i	$0.365604\pi$
$968$	0	0
$969$	−18.2694	−0.586896
$970$	0	0
$971$	0.399988	0.0128362	0.00641812	−	0.999979i	$-0.497957\pi$
$971$	0.399988	0.0128362	0.00641812	+	0.999979i	$0.497957\pi$
$972$	0	0
$973$	0.805473	0.0258223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.428515	0.0137094	0.00685470	−	0.999977i	$-0.497818\pi$
$977$	0.428515	0.0137094	0.00685470	+	0.999977i	$0.497818\pi$
$978$	0	0
$979$	−12.5296	−0.400447
$980$	0	0
$981$	−7.96830	−0.254408
$982$	0	0
$983$	−37.2259	−1.18732	−0.593662	−	0.804715i	$-0.702318\pi$
$983$	−37.2259	−1.18732	−0.593662	+	0.804715i	$0.702318\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.0816476	−0.00259887
$988$	0	0
$989$	−16.2835	−0.517785
$990$	0	0
$991$	7.11721	0.226086	0.113043	−	0.993590i	$-0.463940\pi$
$991$	7.11721	0.226086	0.113043	+	0.993590i	$0.463940\pi$
$992$	0	0
$993$	−21.3330	−0.676983
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.6348	0.368479	0.184239	−	0.982881i	$-0.441018\pi$
$997$	11.6348	0.368479	0.184239	+	0.982881i	$0.441018\pi$
$998$	0	0
$999$	−9.79606	−0.309934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8700.2.a.bj.1.3		6
5.2	odd	4		1740.2.g.d.349.11	yes	12
5.3	odd	4		1740.2.g.d.349.5	✓	12
5.4	even	2		8700.2.a.bk.1.4		6
15.2	even	4		5220.2.g.e.2089.3		12
15.8	even	4		5220.2.g.e.2089.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.g.d.349.5	✓	12	5.3	odd	4
1740.2.g.d.349.11	yes	12	5.2	odd	4
5220.2.g.e.2089.3		12	15.2	even	4
5220.2.g.e.2089.4		12	15.8	even	4
8700.2.a.bj.1.3		6	1.1	even	1	trivial
8700.2.a.bk.1.4		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8700.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.0863317 q^{7} +1.00000 q^{9} +1.70184 q^{11} +2.93760 q^{13} +7.61055 q^{17} +2.40053 q^{19} -0.0863317 q^{21} -1.94053 q^{23} -1.00000 q^{27} +1.00000 q^{29} +9.38449 q^{31} -1.70184 q^{33} +9.79606 q^{37} -2.93760 q^{39} -1.11691 q^{41} +8.39126 q^{43} +0.945743 q^{47} -6.99255 q^{49} -7.61055 q^{51} -2.71154 q^{53} -2.40053 q^{57} -7.86705 q^{59} -10.2279 q^{61} +0.0863317 q^{63} +2.92156 q^{67} +1.94053 q^{69} +8.31213 q^{71} -10.2197 q^{73} +0.146923 q^{77} +15.1831 q^{79} +1.00000 q^{81} +3.85101 q^{83} -1.00000 q^{87} -7.36238 q^{89} +0.253608 q^{91} -9.38449 q^{93} -1.01524 q^{97} +1.70184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 10 q^{19} - 6 q^{21} + 10 q^{23} - 6 q^{27} + 6 q^{29} - 14 q^{31} - 4 q^{33} + 2 q^{37} - 2 q^{39} - 12 q^{41} + 32 q^{43} + 10 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 10 * q^19 - 6 * q^21 + 10 * q^23 - 6 * q^27 + 6 * q^29 - 14 * q^31 - 4 * q^33 + 2 * q^37 - 2 * q^39 - 12 * q^41 + 32 * q^43 + 10 * q^47 + 14 * q^49 - 2 * q^51 + 26 * q^53 + 10 * q^57 + 4 * q^59 - 26 * q^61 + 6 * q^63 + 38 * q^67 - 10 * q^69 + 6 * q^71 - 6 * q^73 + 2 * q^77 + 30 * q^79 + 6 * q^81 + 8 * q^83 - 6 * q^87 - 4 * q^89 - 24 * q^91 + 14 * q^93 - 6 * q^97 + 4 * q^99

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