LMFDB - Embedded newform 2888.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] 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("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,0,0,-3,0,-3,0,3,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87939 q^{3} +0.347296 q^{5} -4.75877 q^{7} +0.532089 q^{9} +4.75877 q^{11} -1.87939 q^{13} +0.652704 q^{15} +1.53209 q^{17} -8.94356 q^{21} -6.10607 q^{23} -4.87939 q^{25} -4.63816 q^{27} -1.98545 q^{29} -3.36959 q^{31} +8.94356 q^{33} -1.65270 q^{35} -3.38919 q^{37} -3.53209 q^{39} -4.49020 q^{41} +2.10607 q^{43} +0.184793 q^{45} -4.92127 q^{47} +15.6459 q^{49} +2.87939 q^{51} -12.5594 q^{53} +1.65270 q^{55} +5.66044 q^{59} +9.99319 q^{61} -2.53209 q^{63} -0.652704 q^{65} +10.5963 q^{67} -11.4757 q^{69} +11.4953 q^{71} -14.6186 q^{73} -9.17024 q^{75} -22.6459 q^{77} +2.36184 q^{79} -10.3131 q^{81} -11.5371 q^{83} +0.532089 q^{85} -3.73143 q^{87} +5.63816 q^{89} +8.94356 q^{91} -6.33275 q^{93} +7.66044 q^{97} +2.53209 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * 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
$3$	1.87939	1.08506	0.542532	−	0.840035i	$-0.317466\pi$
$3$	1.87939	1.08506	0.542532	+	0.840035i	$0.317466\pi$
$4$	0	0
$5$	0.347296	0.155316	0.0776578	−	0.996980i	$-0.475256\pi$
$5$	0.347296	0.155316	0.0776578	+	0.996980i	$0.475256\pi$
$6$	0	0
$7$	−4.75877	−1.79865	−0.899323	−	0.437285i	$-0.855940\pi$
$7$	−4.75877	−1.79865	−0.899323	+	0.437285i	$0.855940\pi$
$8$	0	0
$9$	0.532089	0.177363
$10$	0	0
$11$	4.75877	1.43482	0.717412	−	0.696650i	$-0.245327\pi$
$11$	4.75877	1.43482	0.717412	+	0.696650i	$0.245327\pi$
$12$	0	0
$13$	−1.87939	−0.521248	−0.260624	−	0.965440i	$-0.583928\pi$
$13$	−1.87939	−0.521248	−0.260624	+	0.965440i	$0.583928\pi$
$14$	0	0
$15$	0.652704	0.168527
$16$	0	0
$17$	1.53209	0.371586	0.185793	−	0.982589i	$-0.440515\pi$
$17$	1.53209	0.371586	0.185793	+	0.982589i	$0.440515\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−8.94356	−1.95165
$22$	0	0
$23$	−6.10607	−1.27320	−0.636601	−	0.771193i	$-0.719660\pi$
$23$	−6.10607	−1.27320	−0.636601	+	0.771193i	$0.719660\pi$
$24$	0	0
$25$	−4.87939	−0.975877
$26$	0	0
$27$	−4.63816	−0.892613
$28$	0	0
$29$	−1.98545	−0.368689	−0.184345	−	0.982862i	$-0.559016\pi$
$29$	−1.98545	−0.368689	−0.184345	+	0.982862i	$0.559016\pi$
$30$	0	0
$31$	−3.36959	−0.605195	−0.302598	−	0.953118i	$-0.597854\pi$
$31$	−3.36959	−0.605195	−0.302598	+	0.953118i	$0.597854\pi$
$32$	0	0
$33$	8.94356	1.55687
$34$	0	0
$35$	−1.65270	−0.279358
$36$	0	0
$37$	−3.38919	−0.557179	−0.278589	−	0.960410i	$-0.589867\pi$
$37$	−3.38919	−0.557179	−0.278589	+	0.960410i	$0.589867\pi$
$38$	0	0
$39$	−3.53209	−0.565587
$40$	0	0
$41$	−4.49020	−0.701251	−0.350626	−	0.936516i	$-0.614031\pi$
$41$	−4.49020	−0.701251	−0.350626	+	0.936516i	$0.614031\pi$
$42$	0	0
$43$	2.10607	0.321172	0.160586	−	0.987022i	$-0.448662\pi$
$43$	2.10607	0.321172	0.160586	+	0.987022i	$0.448662\pi$
$44$	0	0
$45$	0.184793	0.0275472
$46$	0	0
$47$	−4.92127	−0.717842	−0.358921	−	0.933368i	$-0.616855\pi$
$47$	−4.92127	−0.717842	−0.358921	+	0.933368i	$0.616855\pi$
$48$	0	0
$49$	15.6459	2.23513
$50$	0	0
$51$	2.87939	0.403195
$52$	0	0
$53$	−12.5594	−1.72517	−0.862585	−	0.505912i	$-0.831156\pi$
$53$	−12.5594	−1.72517	−0.862585	+	0.505912i	$0.831156\pi$
$54$	0	0
$55$	1.65270	0.222851
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.66044	0.736927	0.368464	−	0.929642i	$-0.379884\pi$
$59$	5.66044	0.736927	0.368464	+	0.929642i	$0.379884\pi$
$60$	0	0
$61$	9.99319	1.27950	0.639749	−	0.768584i	$-0.279038\pi$
$61$	9.99319	1.27950	0.639749	+	0.768584i	$0.279038\pi$
$62$	0	0
$63$	−2.53209	−0.319013
$64$	0	0
$65$	−0.652704	−0.0809579
$66$	0	0
$67$	10.5963	1.29454	0.647270	−	0.762261i	$-0.275911\pi$
$67$	10.5963	1.29454	0.647270	+	0.762261i	$0.275911\pi$
$68$	0	0
$69$	−11.4757	−1.38151
$70$	0	0
$71$	11.4953	1.36424	0.682118	−	0.731242i	$-0.261059\pi$
$71$	11.4953	1.36424	0.682118	+	0.731242i	$0.261059\pi$
$72$	0	0
$73$	−14.6186	−1.71097	−0.855486	−	0.517826i	$-0.826742\pi$
$73$	−14.6186	−1.71097	−0.855486	+	0.517826i	$0.826742\pi$
$74$	0	0
$75$	−9.17024	−1.05889
$76$	0	0
$77$	−22.6459	−2.58074
$78$	0	0
$79$	2.36184	0.265728	0.132864	−	0.991134i	$-0.457583\pi$
$79$	2.36184	0.265728	0.132864	+	0.991134i	$0.457583\pi$
$80$	0	0
$81$	−10.3131	−1.14591
$82$	0	0
$83$	−11.5371	−1.26637	−0.633183	−	0.774002i	$-0.718252\pi$
$83$	−11.5371	−1.26637	−0.633183	+	0.774002i	$0.718252\pi$
$84$	0	0
$85$	0.532089	0.0577131
$86$	0	0
$87$	−3.73143	−0.400051
$88$	0	0
$89$	5.63816	0.597643	0.298822	−	0.954309i	$-0.403406\pi$
$89$	5.63816	0.597643	0.298822	+	0.954309i	$0.403406\pi$
$90$	0	0
$91$	8.94356	0.937540
$92$	0	0
$93$	−6.33275	−0.656675
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.66044	0.777800	0.388900	−	0.921280i	$-0.372855\pi$
$97$	7.66044	0.777800	0.388900	+	0.921280i	$0.372855\pi$
$98$	0	0
$99$	2.53209	0.254485
$100$	0	0
$101$	−4.49020	−0.446792	−0.223396	−	0.974728i	$-0.571714\pi$
$101$	−4.49020	−0.446792	−0.223396	+	0.974728i	$0.571714\pi$
$102$	0	0
$103$	−15.0155	−1.47952	−0.739760	−	0.672871i	$-0.765061\pi$
$103$	−15.0155	−1.47952	−0.739760	+	0.672871i	$0.765061\pi$
$104$	0	0
$105$	−3.10607	−0.303121
$106$	0	0
$107$	−10.8871	−1.05250	−0.526249	−	0.850330i	$-0.676402\pi$
$107$	−10.8871	−1.05250	−0.526249	+	0.850330i	$0.676402\pi$
$108$	0	0
$109$	−7.65270	−0.732996	−0.366498	−	0.930419i	$-0.619443\pi$
$109$	−7.65270	−0.732996	−0.366498	+	0.930419i	$0.619443\pi$
$110$	0	0
$111$	−6.36959	−0.604574
$112$	0	0
$113$	0.128356	0.0120747	0.00603734	−	0.999982i	$-0.498078\pi$
$113$	0.128356	0.0120747	0.00603734	+	0.999982i	$0.498078\pi$
$114$	0	0
$115$	−2.12061	−0.197748
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−7.29086	−0.668352
$120$	0	0
$121$	11.6459	1.05872
$122$	0	0
$123$	−8.43882	−0.760902
$124$	0	0
$125$	−3.43107	−0.306885
$126$	0	0
$127$	19.2841	1.71118	0.855591	−	0.517652i	$-0.173194\pi$
$127$	19.2841	1.71118	0.855591	+	0.517652i	$0.173194\pi$
$128$	0	0
$129$	3.95811	0.348492
$130$	0	0
$131$	−17.6313	−1.54046	−0.770229	−	0.637767i	$-0.779858\pi$
$131$	−17.6313	−1.54046	−0.770229	+	0.637767i	$0.779858\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.61081	−0.138637
$136$	0	0
$137$	−19.6527	−1.67904	−0.839522	−	0.543326i	$-0.817165\pi$
$137$	−19.6527	−1.67904	−0.839522	+	0.543326i	$0.817165\pi$
$138$	0	0
$139$	−10.3618	−0.878880	−0.439440	−	0.898272i	$-0.644823\pi$
$139$	−10.3618	−0.878880	−0.439440	+	0.898272i	$0.644823\pi$
$140$	0	0
$141$	−9.24897	−0.778904
$142$	0	0
$143$	−8.94356	−0.747898
$144$	0	0
$145$	−0.689540	−0.0572632
$146$	0	0
$147$	29.4047	2.42526
$148$	0	0
$149$	−0.00774079	−0.000634150 0	−0.000317075	−	1.00000i	$-0.500101\pi$
$149$	−0.00774079	−0.000634150 0	−0.000317075		1.00000i	$0.500101\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0.815207	0.0659056
$154$	0	0
$155$	−1.17024	−0.0939963
$156$	0	0
$157$	5.73648	0.457821	0.228911	−	0.973447i	$-0.426484\pi$
$157$	5.73648	0.457821	0.228911	+	0.973447i	$0.426484\pi$
$158$	0	0
$159$	−23.6040	−1.87192
$160$	0	0
$161$	29.0574	2.29004
$162$	0	0
$163$	−2.59121	−0.202960	−0.101480	−	0.994838i	$-0.532358\pi$
$163$	−2.59121	−0.202960	−0.101480	+	0.994838i	$0.532358\pi$
$164$	0	0
$165$	3.10607	0.241807
$166$	0	0
$167$	18.1506	1.40454	0.702270	−	0.711911i	$-0.252170\pi$
$167$	18.1506	1.40454	0.702270	+	0.711911i	$0.252170\pi$
$168$	0	0
$169$	−9.46791	−0.728301
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.07873	−0.690243	−0.345121	−	0.938558i	$-0.612162\pi$
$173$	−9.07873	−0.690243	−0.345121	+	0.938558i	$0.612162\pi$
$174$	0	0
$175$	23.2199	1.75526
$176$	0	0
$177$	10.6382	0.799613
$178$	0	0
$179$	6.63041	0.495580	0.247790	−	0.968814i	$-0.420296\pi$
$179$	6.63041	0.495580	0.247790	+	0.968814i	$0.420296\pi$
$180$	0	0
$181$	−5.07873	−0.377499	−0.188749	−	0.982025i	$-0.560443\pi$
$181$	−5.07873	−0.377499	−0.188749	+	0.982025i	$0.560443\pi$
$182$	0	0
$183$	18.7811	1.38834
$184$	0	0
$185$	−1.17705	−0.0865386
$186$	0	0
$187$	7.29086	0.533160
$188$	0	0
$189$	22.0719	1.60550
$190$	0	0
$191$	8.90673	0.644468	0.322234	−	0.946660i	$-0.395566\pi$
$191$	8.90673	0.644468	0.322234	+	0.946660i	$0.395566\pi$
$192$	0	0
$193$	19.2841	1.38810	0.694048	−	0.719929i	$-0.255825\pi$
$193$	19.2841	1.38810	0.694048	+	0.719929i	$0.255825\pi$
$194$	0	0
$195$	−1.22668	−0.0878445
$196$	0	0
$197$	−17.9067	−1.27580	−0.637901	−	0.770119i	$-0.720197\pi$
$197$	−17.9067	−1.27580	−0.637901	+	0.770119i	$0.720197\pi$
$198$	0	0
$199$	8.72462	0.618472	0.309236	−	0.950985i	$-0.399927\pi$
$199$	8.72462	0.618472	0.309236	+	0.950985i	$0.399927\pi$
$200$	0	0
$201$	19.9145	1.40466
$202$	0	0
$203$	9.44831	0.663141
$204$	0	0
$205$	−1.55943	−0.108915
$206$	0	0
$207$	−3.24897	−0.225819
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.753718	0.0518881	0.0259440	−	0.999663i	$-0.491741\pi$
$211$	0.753718	0.0518881	0.0259440	+	0.999663i	$0.491741\pi$
$212$	0	0
$213$	21.6040	1.48028
$214$	0	0
$215$	0.731429	0.0498831
$216$	0	0
$217$	16.0351	1.08853
$218$	0	0
$219$	−27.4739	−1.85651
$220$	0	0
$221$	−2.87939	−0.193688
$222$	0	0
$223$	−4.76146	−0.318851	−0.159425	−	0.987210i	$-0.550964\pi$
$223$	−4.76146	−0.318851	−0.159425	+	0.987210i	$0.550964\pi$
$224$	0	0
$225$	−2.59627	−0.173084
$226$	0	0
$227$	28.4243	1.88658	0.943292	−	0.331963i	$-0.107711\pi$
$227$	28.4243	1.88658	0.943292	+	0.331963i	$0.107711\pi$
$228$	0	0
$229$	−15.6459	−1.03391	−0.516955	−	0.856013i	$-0.672935\pi$
$229$	−15.6459	−1.03391	−0.516955	+	0.856013i	$0.672935\pi$
$230$	0	0
$231$	−42.5604	−2.80027
$232$	0	0
$233$	0.347296	0.0227521	0.0113761	−	0.999935i	$-0.496379\pi$
$233$	0.347296	0.0227521	0.0113761	+	0.999935i	$0.496379\pi$
$234$	0	0
$235$	−1.70914	−0.111492
$236$	0	0
$237$	4.43882	0.288332
$238$	0	0
$239$	6.40467	0.414283	0.207142	−	0.978311i	$-0.433584\pi$
$239$	6.40467	0.414283	0.207142	+	0.978311i	$0.433584\pi$
$240$	0	0
$241$	16.4165	1.05748	0.528741	−	0.848783i	$-0.322664\pi$
$241$	16.4165	1.05748	0.528741	+	0.848783i	$0.322664\pi$
$242$	0	0
$243$	−5.46791	−0.350767
$244$	0	0
$245$	5.43376	0.347150
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−21.6827	−1.37409
$250$	0	0
$251$	0.672304	0.0424355	0.0212177	−	0.999775i	$-0.493246\pi$
$251$	0.672304	0.0424355	0.0212177	+	0.999775i	$0.493246\pi$
$252$	0	0
$253$	−29.0574	−1.82682
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−9.33544	−0.582329	−0.291164	−	0.956673i	$-0.594043\pi$
$257$	−9.33544	−0.582329	−0.291164	+	0.956673i	$0.594043\pi$
$258$	0	0
$259$	16.1284	1.00217
$260$	0	0
$261$	−1.05644	−0.0653918
$262$	0	0
$263$	11.3577	0.700347	0.350174	−	0.936685i	$-0.386123\pi$
$263$	11.3577	0.700347	0.350174	+	0.936685i	$0.386123\pi$
$264$	0	0
$265$	−4.36184	−0.267946
$266$	0	0
$267$	10.5963	0.648481
$268$	0	0
$269$	18.5449	1.13070	0.565351	−	0.824851i	$-0.308741\pi$
$269$	18.5449	1.13070	0.565351	+	0.824851i	$0.308741\pi$
$270$	0	0
$271$	21.4561	1.30336	0.651681	−	0.758493i	$-0.274064\pi$
$271$	21.4561	1.30336	0.651681	+	0.758493i	$0.274064\pi$
$272$	0	0
$273$	16.8084	1.01729
$274$	0	0
$275$	−23.2199	−1.40021
$276$	0	0
$277$	−16.0351	−0.963455	−0.481727	−	0.876321i	$-0.659990\pi$
$277$	−16.0351	−0.963455	−0.481727	+	0.876321i	$0.659990\pi$
$278$	0	0
$279$	−1.79292	−0.107339
$280$	0	0
$281$	18.7324	1.11748	0.558740	−	0.829343i	$-0.311285\pi$
$281$	18.7324	1.11748	0.558740	+	0.829343i	$0.311285\pi$
$282$	0	0
$283$	7.04963	0.419057	0.209528	−	0.977803i	$-0.432807\pi$
$283$	7.04963	0.419057	0.209528	+	0.977803i	$0.432807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.3678	1.26130
$288$	0	0
$289$	−14.6527	−0.861924
$290$	0	0
$291$	14.3969	0.843963
$292$	0	0
$293$	−8.26083	−0.482603	−0.241301	−	0.970450i	$-0.577574\pi$
$293$	−8.26083	−0.482603	−0.241301	+	0.970450i	$0.577574\pi$
$294$	0	0
$295$	1.96585	0.114456
$296$	0	0
$297$	−22.0719	−1.28074
$298$	0	0
$299$	11.4757	0.663654
$300$	0	0
$301$	−10.0223	−0.577675
$302$	0	0
$303$	−8.43882	−0.484797
$304$	0	0
$305$	3.47060	0.198726
$306$	0	0
$307$	−4.73143	−0.270037	−0.135018	−	0.990843i	$-0.543109\pi$
$307$	−4.73143	−0.270037	−0.135018	+	0.990843i	$0.543109\pi$
$308$	0	0
$309$	−28.2199	−1.60537
$310$	0	0
$311$	11.2412	0.637432	0.318716	−	0.947850i	$-0.396748\pi$
$311$	11.2412	0.637432	0.318716	+	0.947850i	$0.396748\pi$
$312$	0	0
$313$	6.92127	0.391214	0.195607	−	0.980682i	$-0.437332\pi$
$313$	6.92127	0.391214	0.195607	+	0.980682i	$0.437332\pi$
$314$	0	0
$315$	−0.879385	−0.0495477
$316$	0	0
$317$	−14.6186	−0.821060	−0.410530	−	0.911847i	$-0.634656\pi$
$317$	−14.6186	−0.821060	−0.410530	+	0.911847i	$0.634656\pi$
$318$	0	0
$319$	−9.44831	−0.529004
$320$	0	0
$321$	−20.4611	−1.14203
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9.17024	0.508674
$326$	0	0
$327$	−14.3824	−0.795347
$328$	0	0
$329$	23.4192	1.29114
$330$	0	0
$331$	−3.66550	−0.201474	−0.100737	−	0.994913i	$-0.532120\pi$
$331$	−3.66550	−0.201474	−0.100737	+	0.994913i	$0.532120\pi$
$332$	0	0
$333$	−1.80335	−0.0988229
$334$	0	0
$335$	3.68004	0.201062
$336$	0	0
$337$	−3.33780	−0.181822	−0.0909108	−	0.995859i	$-0.528978\pi$
$337$	−3.33780	−0.181822	−0.0909108	+	0.995859i	$0.528978\pi$
$338$	0	0
$339$	0.241230	0.0131018
$340$	0	0
$341$	−16.0351	−0.868348
$342$	0	0
$343$	−41.1438	−2.22156
$344$	0	0
$345$	−3.98545	−0.214570
$346$	0	0
$347$	8.97359	0.481728	0.240864	−	0.970559i	$-0.422569\pi$
$347$	8.97359	0.481728	0.240864	+	0.970559i	$0.422569\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	8.71688	0.465273
$352$	0	0
$353$	−23.5526	−1.25358	−0.626790	−	0.779188i	$-0.715632\pi$
$353$	−23.5526	−1.25358	−0.626790	+	0.779188i	$0.715632\pi$
$354$	0	0
$355$	3.99226	0.211887
$356$	0	0
$357$	−13.7023	−0.725204
$358$	0	0
$359$	23.2172	1.22536	0.612678	−	0.790333i	$-0.290092\pi$
$359$	23.2172	1.22536	0.612678	+	0.790333i	$0.290092\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	21.8871	1.14878
$364$	0	0
$365$	−5.07697	−0.265741
$366$	0	0
$367$	8.98814	0.469177	0.234589	−	0.972095i	$-0.424626\pi$
$367$	8.98814	0.469177	0.234589	+	0.972095i	$0.424626\pi$
$368$	0	0
$369$	−2.38919	−0.124376
$370$	0	0
$371$	59.7674	3.10297
$372$	0	0
$373$	20.6459	1.06900	0.534502	−	0.845167i	$-0.320499\pi$
$373$	20.6459	1.06900	0.534502	+	0.845167i	$0.320499\pi$
$374$	0	0
$375$	−6.44831	−0.332989
$376$	0	0
$377$	3.73143	0.192178
$378$	0	0
$379$	0.650015	0.0333890	0.0166945	−	0.999861i	$-0.494686\pi$
$379$	0.650015	0.0333890	0.0166945	+	0.999861i	$0.494686\pi$
$380$	0	0
$381$	36.2422	1.85674
$382$	0	0
$383$	18.6186	0.951364	0.475682	−	0.879617i	$-0.342201\pi$
$383$	18.6186	0.951364	0.475682	+	0.879617i	$0.342201\pi$
$384$	0	0
$385$	−7.86484	−0.400829
$386$	0	0
$387$	1.12061	0.0569640
$388$	0	0
$389$	0.625362	0.0317071	0.0158536	−	0.999874i	$-0.494953\pi$
$389$	0.625362	0.0317071	0.0158536	+	0.999874i	$0.494953\pi$
$390$	0	0
$391$	−9.35504	−0.473105
$392$	0	0
$393$	−33.1361	−1.67149
$394$	0	0
$395$	0.820260	0.0412718
$396$	0	0
$397$	−7.85710	−0.394336	−0.197168	−	0.980370i	$-0.563175\pi$
$397$	−7.85710	−0.394336	−0.197168	+	0.980370i	$0.563175\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.4097	−0.519837	−0.259918	−	0.965631i	$-0.583696\pi$
$401$	−10.4097	−0.519837	−0.259918	+	0.965631i	$0.583696\pi$
$402$	0	0
$403$	6.33275	0.315457
$404$	0	0
$405$	−3.58172	−0.177977
$406$	0	0
$407$	−16.1284	−0.799453
$408$	0	0
$409$	14.6955	0.726647	0.363324	−	0.931663i	$-0.381642\pi$
$409$	14.6955	0.726647	0.363324	+	0.931663i	$0.381642\pi$
$410$	0	0
$411$	−36.9350	−1.82187
$412$	0	0
$413$	−26.9368	−1.32547
$414$	0	0
$415$	−4.00681	−0.196686
$416$	0	0
$417$	−19.4739	−0.953641
$418$	0	0
$419$	7.03508	0.343686	0.171843	−	0.985124i	$-0.445028\pi$
$419$	7.03508	0.343686	0.171843	+	0.985124i	$0.445028\pi$
$420$	0	0
$421$	35.5408	1.73215	0.866075	−	0.499913i	$-0.166635\pi$
$421$	35.5408	1.73215	0.866075	+	0.499913i	$0.166635\pi$
$422$	0	0
$423$	−2.61856	−0.127319
$424$	0	0
$425$	−7.47565	−0.362622
$426$	0	0
$427$	−47.5553	−2.30136
$428$	0	0
$429$	−16.8084	−0.811517
$430$	0	0
$431$	20.5057	0.987724	0.493862	−	0.869540i	$-0.335585\pi$
$431$	20.5057	0.987724	0.493862	+	0.869540i	$0.335585\pi$
$432$	0	0
$433$	−20.1661	−0.969122	−0.484561	−	0.874757i	$-0.661021\pi$
$433$	−20.1661	−0.969122	−0.484561	+	0.874757i	$0.661021\pi$
$434$	0	0
$435$	−1.29591	−0.0621342
$436$	0	0
$437$	0	0
$438$	0	0
$439$	3.10782	0.148328	0.0741641	−	0.997246i	$-0.476371\pi$
$439$	3.10782	0.148328	0.0741641	+	0.997246i	$0.476371\pi$
$440$	0	0
$441$	8.32501	0.396429
$442$	0	0
$443$	−8.74691	−0.415578	−0.207789	−	0.978174i	$-0.566627\pi$
$443$	−8.74691	−0.415578	−0.207789	+	0.978174i	$0.566627\pi$
$444$	0	0
$445$	1.95811	0.0928234
$446$	0	0
$447$	−0.0145479	−0.000688093 0
$448$	0	0
$449$	−38.8135	−1.83172	−0.915860	−	0.401498i	$-0.868490\pi$
$449$	−38.8135	−1.83172	−0.915860	+	0.401498i	$0.868490\pi$
$450$	0	0
$451$	−21.3678	−1.00617
$452$	0	0
$453$	−30.0702	−1.41282
$454$	0	0
$455$	3.10607	0.145615
$456$	0	0
$457$	13.0933	0.612478	0.306239	−	0.951955i	$-0.400929\pi$
$457$	13.0933	0.612478	0.306239	+	0.951955i	$0.400929\pi$
$458$	0	0
$459$	−7.10607	−0.331683
$460$	0	0
$461$	−11.7419	−0.546873	−0.273436	−	0.961890i	$-0.588160\pi$
$461$	−11.7419	−0.546873	−0.273436	+	0.961890i	$0.588160\pi$
$462$	0	0
$463$	−15.1438	−0.703794	−0.351897	−	0.936039i	$-0.614463\pi$
$463$	−15.1438	−0.703794	−0.351897	+	0.936039i	$0.614463\pi$
$464$	0	0
$465$	−2.19934	−0.101992
$466$	0	0
$467$	31.3696	1.45161	0.725806	−	0.687900i	$-0.241467\pi$
$467$	31.3696	1.45161	0.725806	+	0.687900i	$0.241467\pi$
$468$	0	0
$469$	−50.4252	−2.32842
$470$	0	0
$471$	10.7811	0.496765
$472$	0	0
$473$	10.0223	0.460825
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.68273	−0.305981
$478$	0	0
$479$	−15.2695	−0.697681	−0.348841	−	0.937182i	$-0.613425\pi$
$479$	−15.2695	−0.697681	−0.348841	+	0.937182i	$0.613425\pi$
$480$	0	0
$481$	6.36959	0.290428
$482$	0	0
$483$	54.6100	2.48484
$484$	0	0
$485$	2.66044	0.120805
$486$	0	0
$487$	37.0856	1.68051	0.840256	−	0.542191i	$-0.182405\pi$
$487$	37.0856	1.68051	0.840256	+	0.542191i	$0.182405\pi$
$488$	0	0
$489$	−4.86989	−0.220224
$490$	0	0
$491$	28.1753	1.27153	0.635767	−	0.771881i	$-0.280684\pi$
$491$	28.1753	1.27153	0.635767	+	0.771881i	$0.280684\pi$
$492$	0	0
$493$	−3.04189	−0.137000
$494$	0	0
$495$	0.879385	0.0395254
$496$	0	0
$497$	−54.7033	−2.45378
$498$	0	0
$499$	12.9736	0.580778	0.290389	−	0.956909i	$-0.406215\pi$
$499$	12.9736	0.580778	0.290389	+	0.956909i	$0.406215\pi$
$500$	0	0
$501$	34.1121	1.52401
$502$	0	0
$503$	−10.8821	−0.485208	−0.242604	−	0.970125i	$-0.578002\pi$
$503$	−10.8821	−0.485208	−0.242604	+	0.970125i	$0.578002\pi$
$504$	0	0
$505$	−1.55943	−0.0693937
$506$	0	0
$507$	−17.7939	−0.790253
$508$	0	0
$509$	−11.4851	−0.509070	−0.254535	−	0.967064i	$-0.581922\pi$
$509$	−11.4851	−0.509070	−0.254535	+	0.967064i	$0.581922\pi$
$510$	0	0
$511$	69.5663	3.07743
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.21482	−0.229793
$516$	0	0
$517$	−23.4192	−1.02998
$518$	0	0
$519$	−17.0624	−0.748957
$520$	0	0
$521$	−6.87164	−0.301052	−0.150526	−	0.988606i	$-0.548097\pi$
$521$	−6.87164	−0.301052	−0.150526	+	0.988606i	$0.548097\pi$
$522$	0	0
$523$	−12.4388	−0.543911	−0.271956	−	0.962310i	$-0.587670\pi$
$523$	−12.4388	−0.543911	−0.271956	+	0.962310i	$0.587670\pi$
$524$	0	0
$525$	43.6391	1.90457
$526$	0	0
$527$	−5.16250	−0.224882
$528$	0	0
$529$	14.2841	0.621046
$530$	0	0
$531$	3.01186	0.130704
$532$	0	0
$533$	8.43882	0.365526
$534$	0	0
$535$	−3.78106	−0.163469
$536$	0	0
$537$	12.4611	0.537736
$538$	0	0
$539$	74.4552	3.20701
$540$	0	0
$541$	−12.3705	−0.531850	−0.265925	−	0.963994i	$-0.585677\pi$
$541$	−12.3705	−0.531850	−0.265925	+	0.963994i	$0.585677\pi$
$542$	0	0
$543$	−9.54488	−0.409610
$544$	0	0
$545$	−2.65776	−0.113846
$546$	0	0
$547$	−22.9344	−0.980604	−0.490302	−	0.871553i	$-0.663113\pi$
$547$	−22.9344	−0.980604	−0.490302	+	0.871553i	$0.663113\pi$
$548$	0	0
$549$	5.31727	0.226935
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.2395	−0.477951
$554$	0	0
$555$	−2.21213	−0.0938998
$556$	0	0
$557$	21.3233	0.903495	0.451748	−	0.892146i	$-0.350801\pi$
$557$	21.3233	0.903495	0.451748	+	0.892146i	$0.350801\pi$
$558$	0	0
$559$	−3.95811	−0.167410
$560$	0	0
$561$	13.7023	0.578513
$562$	0	0
$563$	18.7588	0.790588	0.395294	−	0.918555i	$-0.370643\pi$
$563$	18.7588	0.790588	0.395294	+	0.918555i	$0.370643\pi$
$564$	0	0
$565$	0.0445774	0.00187539
$566$	0	0
$567$	49.0779	2.06108
$568$	0	0
$569$	6.16756	0.258557	0.129279	−	0.991608i	$-0.458734\pi$
$569$	6.16756	0.258557	0.129279	+	0.991608i	$0.458734\pi$
$570$	0	0
$571$	22.2175	0.929774	0.464887	−	0.885370i	$-0.346095\pi$
$571$	22.2175	0.929774	0.464887	+	0.885370i	$0.346095\pi$
$572$	0	0
$573$	16.7392	0.699289
$574$	0	0
$575$	29.7939	1.24249
$576$	0	0
$577$	−37.7701	−1.57239	−0.786196	−	0.617978i	$-0.787952\pi$
$577$	−37.7701	−1.57239	−0.786196	+	0.617978i	$0.787952\pi$
$578$	0	0
$579$	36.2422	1.50617
$580$	0	0
$581$	54.9026	2.27774
$582$	0	0
$583$	−59.7674	−2.47531
$584$	0	0
$585$	−0.347296	−0.0143589
$586$	0	0
$587$	−4.30035	−0.177495	−0.0887473	−	0.996054i	$-0.528286\pi$
$587$	−4.30035	−0.177495	−0.0887473	+	0.996054i	$0.528286\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−33.6536	−1.38433
$592$	0	0
$593$	7.44057	0.305548	0.152774	−	0.988261i	$-0.451179\pi$
$593$	7.44057	0.305548	0.152774	+	0.988261i	$0.451179\pi$
$594$	0	0
$595$	−2.53209	−0.103806
$596$	0	0
$597$	16.3969	0.671082
$598$	0	0
$599$	0.635467	0.0259645	0.0129822	−	0.999916i	$-0.495868\pi$
$599$	0.635467	0.0259645	0.0129822	+	0.999916i	$0.495868\pi$
$600$	0	0
$601$	−2.64590	−0.107928	−0.0539642	−	0.998543i	$-0.517186\pi$
$601$	−2.64590	−0.107928	−0.0539642	+	0.998543i	$0.517186\pi$
$602$	0	0
$603$	5.63816	0.229603
$604$	0	0
$605$	4.04458	0.164435
$606$	0	0
$607$	18.3851	0.746227	0.373113	−	0.927786i	$-0.378290\pi$
$607$	18.3851	0.746227	0.373113	+	0.927786i	$0.378290\pi$
$608$	0	0
$609$	17.7570	0.719551
$610$	0	0
$611$	9.24897	0.374173
$612$	0	0
$613$	18.0823	0.730339	0.365170	−	0.930941i	$-0.381011\pi$
$613$	18.0823	0.730339	0.365170	+	0.930941i	$0.381011\pi$
$614$	0	0
$615$	−2.93077	−0.118180
$616$	0	0
$617$	18.6064	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$617$	18.6064	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$618$	0	0
$619$	−10.8479	−0.436015	−0.218007	−	0.975947i	$-0.569956\pi$
$619$	−10.8479	−0.436015	−0.218007	+	0.975947i	$0.569956\pi$
$620$	0	0
$621$	28.3209	1.13648
$622$	0	0
$623$	−26.8307	−1.07495
$624$	0	0
$625$	23.2053	0.928213
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.19253	−0.207040
$630$	0	0
$631$	−15.1019	−0.601199	−0.300600	−	0.953750i	$-0.597187\pi$
$631$	−15.1019	−0.601199	−0.300600	+	0.953750i	$0.597187\pi$
$632$	0	0
$633$	1.41653	0.0563019
$634$	0	0
$635$	6.69728	0.265773
$636$	0	0
$637$	−29.4047	−1.16506
$638$	0	0
$639$	6.11650	0.241965
$640$	0	0
$641$	−46.2945	−1.82852	−0.914261	−	0.405126i	$-0.867228\pi$
$641$	−46.2945	−1.82852	−0.914261	+	0.405126i	$0.867228\pi$
$642$	0	0
$643$	−13.5800	−0.535542	−0.267771	−	0.963483i	$-0.586287\pi$
$643$	−13.5800	−0.535542	−0.267771	+	0.963483i	$0.586287\pi$
$644$	0	0
$645$	1.37464	0.0541263
$646$	0	0
$647$	33.8135	1.32934	0.664672	−	0.747135i	$-0.268571\pi$
$647$	33.8135	1.32934	0.664672	+	0.747135i	$0.268571\pi$
$648$	0	0
$649$	26.9368	1.05736
$650$	0	0
$651$	30.1361	1.18113
$652$	0	0
$653$	6.74329	0.263885	0.131943	−	0.991257i	$-0.457879\pi$
$653$	6.74329	0.263885	0.131943	+	0.991257i	$0.457879\pi$
$654$	0	0
$655$	−6.12330	−0.239257
$656$	0	0
$657$	−7.77837	−0.303463
$658$	0	0
$659$	0.474718	0.0184924	0.00924620	−	0.999957i	$-0.497057\pi$
$659$	0.474718	0.0184924	0.00924620	+	0.999957i	$0.497057\pi$
$660$	0	0
$661$	3.86484	0.150325	0.0751624	−	0.997171i	$-0.476052\pi$
$661$	3.86484	0.150325	0.0751624	+	0.997171i	$0.476052\pi$
$662$	0	0
$663$	−5.41147	−0.210164
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.1233	0.469416
$668$	0	0
$669$	−8.94862	−0.345973
$670$	0	0
$671$	47.5553	1.83585
$672$	0	0
$673$	4.22163	0.162732	0.0813659	−	0.996684i	$-0.474072\pi$
$673$	4.22163	0.162732	0.0813659	+	0.996684i	$0.474072\pi$
$674$	0	0
$675$	22.6313	0.871081
$676$	0	0
$677$	18.1242	0.696571	0.348286	−	0.937389i	$-0.386764\pi$
$677$	18.1242	0.696571	0.348286	+	0.937389i	$0.386764\pi$
$678$	0	0
$679$	−36.4543	−1.39899
$680$	0	0
$681$	53.4201	2.04706
$682$	0	0
$683$	27.2608	1.04311	0.521553	−	0.853219i	$-0.325353\pi$
$683$	27.2608	1.04311	0.521553	+	0.853219i	$0.325353\pi$
$684$	0	0
$685$	−6.82531	−0.260782
$686$	0	0
$687$	−29.4047	−1.12186
$688$	0	0
$689$	23.6040	0.899241
$690$	0	0
$691$	−26.2371	−0.998107	−0.499053	−	0.866571i	$-0.666319\pi$
$691$	−26.2371	−0.998107	−0.499053	+	0.866571i	$0.666319\pi$
$692$	0	0
$693$	−12.0496	−0.457728
$694$	0	0
$695$	−3.59863	−0.136504
$696$	0	0
$697$	−6.87939	−0.260575
$698$	0	0
$699$	0.652704	0.0246875
$700$	0	0
$701$	24.5672	0.927889	0.463945	−	0.885864i	$-0.346434\pi$
$701$	24.5672	0.927889	0.463945	+	0.885864i	$0.346434\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−3.21213	−0.120976
$706$	0	0
$707$	21.3678	0.803620
$708$	0	0
$709$	−10.9537	−0.411374	−0.205687	−	0.978618i	$-0.565943\pi$
$709$	−10.9537	−0.411374	−0.205687	+	0.978618i	$0.565943\pi$
$710$	0	0
$711$	1.25671	0.0471303
$712$	0	0
$713$	20.5749	0.770536
$714$	0	0
$715$	−3.10607	−0.116160
$716$	0	0
$717$	12.0368	0.449524
$718$	0	0
$719$	−42.5449	−1.58666	−0.793328	−	0.608794i	$-0.791654\pi$
$719$	−42.5449	−1.58666	−0.793328	+	0.608794i	$0.791654\pi$
$720$	0	0
$721$	71.4552	2.66113
$722$	0	0
$723$	30.8530	1.14743
$724$	0	0
$725$	9.68779	0.359795
$726$	0	0
$727$	−39.5398	−1.46645	−0.733226	−	0.679986i	$-0.761986\pi$
$727$	−39.5398	−1.46645	−0.733226	+	0.679986i	$0.761986\pi$
$728$	0	0
$729$	20.6631	0.765301
$730$	0	0
$731$	3.22668	0.119343
$732$	0	0
$733$	−44.2918	−1.63595	−0.817977	−	0.575250i	$-0.804905\pi$
$733$	−44.2918	−1.63595	−0.817977	+	0.575250i	$0.804905\pi$
$734$	0	0
$735$	10.2121	0.376680
$736$	0	0
$737$	50.4252	1.85744
$738$	0	0
$739$	−53.7989	−1.97902	−0.989512	−	0.144448i	$-0.953859\pi$
$739$	−53.7989	−1.97902	−0.989512	+	0.144448i	$0.953859\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.8179	−0.433557	−0.216778	−	0.976221i	$-0.569555\pi$
$743$	−11.8179	−0.433557	−0.216778	+	0.976221i	$0.569555\pi$
$744$	0	0
$745$	−0.00268835	−9.84934e−5 0
$746$	0	0
$747$	−6.13878	−0.224606
$748$	0	0
$749$	51.8093	1.89307
$750$	0	0
$751$	−37.6604	−1.37425	−0.687125	−	0.726540i	$-0.741127\pi$
$751$	−37.6604	−1.37425	−0.687125	+	0.726540i	$0.741127\pi$
$752$	0	0
$753$	1.26352	0.0460452
$754$	0	0
$755$	−5.55674	−0.202231
$756$	0	0
$757$	7.59896	0.276189	0.138094	−	0.990419i	$-0.455902\pi$
$757$	7.59896	0.276189	0.138094	+	0.990419i	$0.455902\pi$
$758$	0	0
$759$	−54.6100	−1.98222
$760$	0	0
$761$	−1.51754	−0.0550108	−0.0275054	−	0.999622i	$-0.508756\pi$
$761$	−1.51754	−0.0550108	−0.0275054	+	0.999622i	$0.508756\pi$
$762$	0	0
$763$	36.4175	1.31840
$764$	0	0
$765$	0.283119	0.0102362
$766$	0	0
$767$	−10.6382	−0.384122
$768$	0	0
$769$	−17.0787	−0.615875	−0.307937	−	0.951407i	$-0.599639\pi$
$769$	−17.0787	−0.615875	−0.307937	+	0.951407i	$0.599639\pi$
$770$	0	0
$771$	−17.5449	−0.631863
$772$	0	0
$773$	0.731429	0.0263077	0.0131538	−	0.999913i	$-0.495813\pi$
$773$	0.731429	0.0263077	0.0131538	+	0.999913i	$0.495813\pi$
$774$	0	0
$775$	16.4415	0.590596
$776$	0	0
$777$	30.3114	1.08742
$778$	0	0
$779$	0	0
$780$	0	0
$781$	54.7033	1.95744
$782$	0	0
$783$	9.20884	0.329097
$784$	0	0
$785$	1.99226	0.0711068
$786$	0	0
$787$	24.9573	0.889631	0.444816	−	0.895622i	$-0.353269\pi$
$787$	24.9573	0.889631	0.444816	+	0.895622i	$0.353269\pi$
$788$	0	0
$789$	21.3455	0.759921
$790$	0	0
$791$	−0.610815	−0.0217181
$792$	0	0
$793$	−18.7811	−0.666935
$794$	0	0
$795$	−8.19759	−0.290738
$796$	0	0
$797$	−46.2877	−1.63959	−0.819797	−	0.572655i	$-0.805914\pi$
$797$	−46.2877	−1.63959	−0.819797	+	0.572655i	$0.805914\pi$
$798$	0	0
$799$	−7.53983	−0.266740
$800$	0	0
$801$	3.00000	0.106000
$802$	0	0
$803$	−69.5663	−2.45494
$804$	0	0
$805$	10.0915	0.355679
$806$	0	0
$807$	34.8530	1.22688
$808$	0	0
$809$	−52.3809	−1.84162	−0.920808	−	0.390016i	$-0.872469\pi$
$809$	−52.3809	−1.84162	−0.920808	+	0.390016i	$0.872469\pi$
$810$	0	0
$811$	−47.8711	−1.68098	−0.840492	−	0.541824i	$-0.817734\pi$
$811$	−47.8711	−1.68098	−0.840492	+	0.541824i	$0.817734\pi$
$812$	0	0
$813$	40.3242	1.41423
$814$	0	0
$815$	−0.899919	−0.0315228
$816$	0	0
$817$	0	0
$818$	0	0
$819$	4.75877	0.166285
$820$	0	0
$821$	−33.8120	−1.18005	−0.590024	−	0.807386i	$-0.700882\pi$
$821$	−33.8120	−1.18005	−0.590024	+	0.807386i	$0.700882\pi$
$822$	0	0
$823$	6.04221	0.210618	0.105309	−	0.994440i	$-0.466417\pi$
$823$	6.04221	0.210618	0.105309	+	0.994440i	$0.466417\pi$
$824$	0	0
$825$	−43.6391	−1.51932
$826$	0	0
$827$	−24.3604	−0.847095	−0.423547	−	0.905874i	$-0.639215\pi$
$827$	−24.3604	−0.847095	−0.423547	+	0.905874i	$0.639215\pi$
$828$	0	0
$829$	23.9067	0.830315	0.415157	−	0.909750i	$-0.363726\pi$
$829$	23.9067	0.830315	0.415157	+	0.909750i	$0.363726\pi$
$830$	0	0
$831$	−30.1361	−1.04541
$832$	0	0
$833$	23.9709	0.830543
$834$	0	0
$835$	6.30365	0.218147
$836$	0	0
$837$	15.6287	0.540206
$838$	0	0
$839$	−15.8449	−0.547027	−0.273514	−	0.961868i	$-0.588186\pi$
$839$	−15.8449	−0.547027	−0.273514	+	0.961868i	$0.588186\pi$
$840$	0	0
$841$	−25.0580	−0.864068
$842$	0	0
$843$	35.2053	1.21254
$844$	0	0
$845$	−3.28817	−0.113117
$846$	0	0
$847$	−55.4201	−1.90426
$848$	0	0
$849$	13.2490	0.454703
$850$	0	0
$851$	20.6946	0.709402
$852$	0	0
$853$	1.21719	0.0416757	0.0208378	−	0.999783i	$-0.493367\pi$
$853$	1.21719	0.0416757	0.0208378	+	0.999783i	$0.493367\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.3806	1.58433	0.792166	−	0.610306i	$-0.208953\pi$
$857$	46.3806	1.58433	0.792166	+	0.610306i	$0.208953\pi$
$858$	0	0
$859$	15.3141	0.522510	0.261255	−	0.965270i	$-0.415864\pi$
$859$	15.3141	0.522510	0.261255	+	0.965270i	$0.415864\pi$
$860$	0	0
$861$	40.1584	1.36859
$862$	0	0
$863$	−45.7357	−1.55686	−0.778430	−	0.627731i	$-0.783984\pi$
$863$	−45.7357	−1.55686	−0.778430	+	0.627731i	$0.783984\pi$
$864$	0	0
$865$	−3.15301	−0.107205
$866$	0	0
$867$	−27.5381	−0.935242
$868$	0	0
$869$	11.2395	0.381273
$870$	0	0
$871$	−19.9145	−0.674776
$872$	0	0
$873$	4.07604	0.137953
$874$	0	0
$875$	16.3277	0.551977
$876$	0	0
$877$	−33.1712	−1.12011	−0.560056	−	0.828455i	$-0.689220\pi$
$877$	−33.1712	−1.12011	−0.560056	+	0.828455i	$0.689220\pi$
$878$	0	0
$879$	−15.5253	−0.523655
$880$	0	0
$881$	43.8485	1.47729	0.738647	−	0.674092i	$-0.235465\pi$
$881$	43.8485	1.47729	0.738647	+	0.674092i	$0.235465\pi$
$882$	0	0
$883$	−49.8590	−1.67789	−0.838944	−	0.544218i	$-0.816826\pi$
$883$	−49.8590	−1.67789	−0.838944	+	0.544218i	$0.816826\pi$
$884$	0	0
$885$	3.69459	0.124192
$886$	0	0
$887$	34.0934	1.14474	0.572372	−	0.819994i	$-0.306023\pi$
$887$	34.0934	1.14474	0.572372	+	0.819994i	$0.306023\pi$
$888$	0	0
$889$	−91.7684	−3.07781
$890$	0	0
$891$	−49.0779	−1.64417
$892$	0	0
$893$	0	0
$894$	0	0
$895$	2.30272	0.0769714
$896$	0	0
$897$	21.5672	0.720107
$898$	0	0
$899$	6.69015	0.223129
$900$	0	0
$901$	−19.2422	−0.641049
$902$	0	0
$903$	−18.8357	−0.626814
$904$	0	0
$905$	−1.76382	−0.0586315
$906$	0	0
$907$	46.4519	1.54241	0.771206	−	0.636586i	$-0.219654\pi$
$907$	46.4519	1.54241	0.771206	+	0.636586i	$0.219654\pi$
$908$	0	0
$909$	−2.38919	−0.0792443
$910$	0	0
$911$	−19.8527	−0.657748	−0.328874	−	0.944374i	$-0.606669\pi$
$911$	−19.8527	−0.657748	−0.328874	+	0.944374i	$0.606669\pi$
$912$	0	0
$913$	−54.9026	−1.81701
$914$	0	0
$915$	6.52259	0.215630
$916$	0	0
$917$	83.9035	2.77074
$918$	0	0
$919$	25.1129	0.828397	0.414199	−	0.910187i	$-0.364062\pi$
$919$	25.1129	0.828397	0.414199	+	0.910187i	$0.364062\pi$
$920$	0	0
$921$	−8.89218	−0.293007
$922$	0	0
$923$	−21.6040	−0.711105
$924$	0	0
$925$	16.5371	0.543738
$926$	0	0
$927$	−7.98957	−0.262412
$928$	0	0
$929$	−54.0164	−1.77222	−0.886111	−	0.463474i	$-0.846603\pi$
$929$	−54.0164	−1.77222	−0.886111	+	0.463474i	$0.846603\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.1266	0.691654
$934$	0	0
$935$	2.53209	0.0828082
$936$	0	0
$937$	−35.4361	−1.15765	−0.578824	−	0.815453i	$-0.696488\pi$
$937$	−35.4361	−1.15765	−0.578824	+	0.815453i	$0.696488\pi$
$938$	0	0
$939$	13.0077	0.424492
$940$	0	0
$941$	28.3996	0.925801	0.462900	−	0.886410i	$-0.346809\pi$
$941$	28.3996	0.925801	0.462900	+	0.886410i	$0.346809\pi$
$942$	0	0
$943$	27.4175	0.892835
$944$	0	0
$945$	7.66550	0.249359
$946$	0	0
$947$	41.2294	1.33977	0.669887	−	0.742463i	$-0.266342\pi$
$947$	41.2294	1.33977	0.669887	+	0.742463i	$0.266342\pi$
$948$	0	0
$949$	27.4739	0.891840
$950$	0	0
$951$	−27.4739	−0.890902
$952$	0	0
$953$	−25.3387	−0.820802	−0.410401	−	0.911905i	$-0.634611\pi$
$953$	−25.3387	−0.820802	−0.410401	+	0.911905i	$0.634611\pi$
$954$	0	0
$955$	3.09327	0.100096
$956$	0	0
$957$	−17.7570	−0.574003
$958$	0	0
$959$	93.5227	3.02001
$960$	0	0
$961$	−19.6459	−0.633739
$962$	0	0
$963$	−5.79292	−0.186674
$964$	0	0
$965$	6.69728	0.215593
$966$	0	0
$967$	−3.78693	−0.121780	−0.0608898	−	0.998144i	$-0.519394\pi$
$967$	−3.78693	−0.121780	−0.0608898	+	0.998144i	$0.519394\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.20708	−0.295469	−0.147735	−	0.989027i	$-0.547198\pi$
$971$	−9.20708	−0.295469	−0.147735	+	0.989027i	$0.547198\pi$
$972$	0	0
$973$	49.3096	1.58079
$974$	0	0
$975$	17.2344	0.551943
$976$	0	0
$977$	50.4593	1.61434	0.807169	−	0.590321i	$-0.200999\pi$
$977$	50.4593	1.61434	0.807169	+	0.590321i	$0.200999\pi$
$978$	0	0
$979$	26.8307	0.857513
$980$	0	0
$981$	−4.07192	−0.130006
$982$	0	0
$983$	−48.6141	−1.55055	−0.775275	−	0.631624i	$-0.782389\pi$
$983$	−48.6141	−1.55055	−0.775275	+	0.631624i	$0.782389\pi$
$984$	0	0
$985$	−6.21894	−0.198152
$986$	0	0
$987$	44.0137	1.40097
$988$	0	0
$989$	−12.8598	−0.408917
$990$	0	0
$991$	37.1130	1.17893	0.589466	−	0.807793i	$-0.299338\pi$
$991$	37.1130	1.17893	0.589466	+	0.807793i	$0.299338\pi$
$992$	0	0
$993$	−6.88888	−0.218612
$994$	0	0
$995$	3.03003	0.0960584
$996$	0	0
$997$	−41.3354	−1.30911	−0.654553	−	0.756016i	$-0.727143\pi$
$997$	−41.3354	−1.30911	−0.654553	+	0.756016i	$0.727143\pi$
$998$	0	0
$999$	15.7196	0.497345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.q.1.3		3
4.3	odd	2		5776.2.a.bl.1.1		3
19.4	even	9		152.2.q.a.73.1	yes	6
19.5	even	9		152.2.q.a.25.1	✓	6
19.18	odd	2		2888.2.a.p.1.1		3
76.23	odd	18		304.2.u.d.225.1		6
76.43	odd	18		304.2.u.d.177.1		6
76.75	even	2		5776.2.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.a.25.1	✓	6	19.5	even	9
152.2.q.a.73.1	yes	6	19.4	even	9
304.2.u.d.177.1		6	76.43	odd	18
304.2.u.d.225.1		6	76.23	odd	18
2888.2.a.p.1.1		3	19.18	odd	2
2888.2.a.q.1.3		3	1.1	even	1	trivial
5776.2.a.bl.1.1		3	4.3	odd	2
5776.2.a.bm.1.3		3	76.75	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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87939 q^{3} +0.347296 q^{5} -4.75877 q^{7} +0.532089 q^{9} +4.75877 q^{11} -1.87939 q^{13} +0.652704 q^{15} +1.53209 q^{17} -8.94356 q^{21} -6.10607 q^{23} -4.87939 q^{25} -4.63816 q^{27} -1.98545 q^{29} -3.36959 q^{31} +8.94356 q^{33} -1.65270 q^{35} -3.38919 q^{37} -3.53209 q^{39} -4.49020 q^{41} +2.10607 q^{43} +0.184793 q^{45} -4.92127 q^{47} +15.6459 q^{49} +2.87939 q^{51} -12.5594 q^{53} +1.65270 q^{55} +5.66044 q^{59} +9.99319 q^{61} -2.53209 q^{63} -0.652704 q^{65} +10.5963 q^{67} -11.4757 q^{69} +11.4953 q^{71} -14.6186 q^{73} -9.17024 q^{75} -22.6459 q^{77} +2.36184 q^{79} -10.3131 q^{81} -11.5371 q^{83} +0.532089 q^{85} -3.73143 q^{87} +5.63816 q^{89} +8.94356 q^{91} -6.33275 q^{93} +7.66044 q^{97} +2.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * q^99

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