LMFDB - Embedded newform 2888.2.a.r.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 = [3,0,1,0,1,0,-2,0,6,0,4,0,-1,0,5] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$0.470683$ 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)

$f(q)$	$=$	$q+3.24914 q^{3} +0.470683 q^{5} -3.30777 q^{7} +7.55691 q^{9} +1.47068 q^{11} -2.83709 q^{13} +1.52932 q^{15} +6.71982 q^{17} -10.7474 q^{21} +0.470683 q^{23} -4.77846 q^{25} +14.8061 q^{27} +4.96896 q^{29} +6.74742 q^{31} +4.77846 q^{33} -1.55691 q^{35} +4.74742 q^{37} -9.21811 q^{39} -0.501719 q^{41} +3.89572 q^{43} +3.55691 q^{45} +11.9103 q^{47} +3.94137 q^{49} +21.8337 q^{51} -4.71982 q^{53} +0.692226 q^{55} +7.24914 q^{59} -12.5259 q^{61} -24.9966 q^{63} -1.33537 q^{65} -0.633593 q^{67} +1.52932 q^{69} -3.77846 q^{71} -3.94137 q^{73} -15.5259 q^{75} -4.86469 q^{77} -8.83709 q^{79} +25.4362 q^{81} +9.14486 q^{83} +3.16291 q^{85} +16.1449 q^{87} +2.95436 q^{89} +9.38445 q^{91} +21.9233 q^{93} +0.0586332 q^{97} +11.1138 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) $ 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

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
$3$	3.24914	1.87589	0.937946	−	0.346781i	$-0.112725\pi$
$3$	3.24914	1.87589	0.937946	+	0.346781i	$0.112725\pi$
$4$	0	0
$5$	0.470683	0.210496	0.105248	−	0.994446i	$-0.466436\pi$
$5$	0.470683	0.210496	0.105248	+	0.994446i	$0.466436\pi$
$6$	0	0
$7$	−3.30777	−1.25022	−0.625110	−	0.780536i	$-0.714946\pi$
$7$	−3.30777	−1.25022	−0.625110	+	0.780536i	$0.714946\pi$
$8$	0	0
$9$	7.55691	2.51897
$10$	0	0
$11$	1.47068	0.443428	0.221714	−	0.975112i	$-0.428835\pi$
$11$	1.47068	0.443428	0.221714	+	0.975112i	$0.428835\pi$
$12$	0	0
$13$	−2.83709	−0.786867	−0.393434	−	0.919353i	$-0.628713\pi$
$13$	−2.83709	−0.786867	−0.393434	+	0.919353i	$0.628713\pi$
$14$	0	0
$15$	1.52932	0.394868
$16$	0	0
$17$	6.71982	1.62980	0.814898	−	0.579604i	$-0.196793\pi$
$17$	6.71982	1.62980	0.814898	+	0.579604i	$0.196793\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−10.7474	−2.34528
$22$	0	0
$23$	0.470683	0.0981443	0.0490721	−	0.998795i	$-0.484374\pi$
$23$	0.470683	0.0981443	0.0490721	+	0.998795i	$0.484374\pi$
$24$	0	0
$25$	−4.77846	−0.955691
$26$	0	0
$27$	14.8061	2.84943
$28$	0	0
$29$	4.96896	0.922714	0.461357	−	0.887215i	$-0.347363\pi$
$29$	4.96896	0.922714	0.461357	+	0.887215i	$0.347363\pi$
$30$	0	0
$31$	6.74742	1.21187	0.605936	−	0.795513i	$-0.292799\pi$
$31$	6.74742	1.21187	0.605936	+	0.795513i	$0.292799\pi$
$32$	0	0
$33$	4.77846	0.831823
$34$	0	0
$35$	−1.55691	−0.263167
$36$	0	0
$37$	4.74742	0.780471	0.390236	−	0.920715i	$-0.372394\pi$
$37$	4.74742	0.780471	0.390236	+	0.920715i	$0.372394\pi$
$38$	0	0
$39$	−9.21811	−1.47608
$40$	0	0
$41$	−0.501719	−0.0783553	−0.0391777	−	0.999232i	$-0.512474\pi$
$41$	−0.501719	−0.0783553	−0.0391777	+	0.999232i	$0.512474\pi$
$42$	0	0
$43$	3.89572	0.594092	0.297046	−	0.954863i	$-0.403998\pi$
$43$	3.89572	0.594092	0.297046	+	0.954863i	$0.403998\pi$
$44$	0	0
$45$	3.55691	0.530233
$46$	0	0
$47$	11.9103	1.73730	0.868650	−	0.495426i	$-0.164988\pi$
$47$	11.9103	1.73730	0.868650	+	0.495426i	$0.164988\pi$
$48$	0	0
$49$	3.94137	0.563052
$50$	0	0
$51$	21.8337	3.05732
$52$	0	0
$53$	−4.71982	−0.648318	−0.324159	−	0.946003i	$-0.605081\pi$
$53$	−4.71982	−0.648318	−0.324159	+	0.946003i	$0.605081\pi$
$54$	0	0
$55$	0.692226	0.0933398
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.24914	0.943758	0.471879	−	0.881663i	$-0.343576\pi$
$59$	7.24914	0.943758	0.471879	+	0.881663i	$0.343576\pi$
$60$	0	0
$61$	−12.5259	−1.60377	−0.801887	−	0.597475i	$-0.796171\pi$
$61$	−12.5259	−1.60377	−0.801887	+	0.597475i	$0.796171\pi$
$62$	0	0
$63$	−24.9966	−3.14927
$64$	0	0
$65$	−1.33537	−0.165632
$66$	0	0
$67$	−0.633593	−0.0774057	−0.0387029	−	0.999251i	$-0.512323\pi$
$67$	−0.633593	−0.0774057	−0.0387029	+	0.999251i	$0.512323\pi$
$68$	0	0
$69$	1.52932	0.184108
$70$	0	0
$71$	−3.77846	−0.448420	−0.224210	−	0.974541i	$-0.571980\pi$
$71$	−3.77846	−0.448420	−0.224210	+	0.974541i	$0.571980\pi$
$72$	0	0
$73$	−3.94137	−0.461302	−0.230651	−	0.973037i	$-0.574086\pi$
$73$	−3.94137	−0.461302	−0.230651	+	0.973037i	$0.574086\pi$
$74$	0	0
$75$	−15.5259	−1.79277
$76$	0	0
$77$	−4.86469	−0.554383
$78$	0	0
$79$	−8.83709	−0.994250	−0.497125	−	0.867679i	$-0.665611\pi$
$79$	−8.83709	−0.994250	−0.497125	+	0.867679i	$0.665611\pi$
$80$	0	0
$81$	25.4362	2.82625
$82$	0	0
$83$	9.14486	1.00378	0.501890	−	0.864932i	$-0.332638\pi$
$83$	9.14486	1.00378	0.501890	+	0.864932i	$0.332638\pi$
$84$	0	0
$85$	3.16291	0.343066
$86$	0	0
$87$	16.1449	1.73091
$88$	0	0
$89$	2.95436	0.313161	0.156581	−	0.987665i	$-0.449953\pi$
$89$	2.95436	0.313161	0.156581	+	0.987665i	$0.449953\pi$
$90$	0	0
$91$	9.38445	0.983758
$92$	0	0
$93$	21.9233	2.27334
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.0586332	0.00595330	0.00297665	−	0.999996i	$-0.499053\pi$
$97$	0.0586332	0.00595330	0.00297665	+	0.999996i	$0.499053\pi$
$98$	0	0
$99$	11.1138	1.11698
$100$	0	0
$101$	11.5845	1.15270	0.576351	−	0.817202i	$-0.304476\pi$
$101$	11.5845	1.15270	0.576351	+	0.817202i	$0.304476\pi$
$102$	0	0
$103$	−8.17246	−0.805257	−0.402628	−	0.915364i	$-0.631903\pi$
$103$	−8.17246	−0.805257	−0.402628	+	0.915364i	$0.631903\pi$
$104$	0	0
$105$	−5.05863	−0.493672
$106$	0	0
$107$	0.498281	0.0481706	0.0240853	−	0.999710i	$-0.492333\pi$
$107$	0.498281	0.0481706	0.0240853	+	0.999710i	$0.492333\pi$
$108$	0	0
$109$	−8.39400	−0.804000	−0.402000	−	0.915640i	$-0.631685\pi$
$109$	−8.39400	−0.804000	−0.402000	+	0.915640i	$0.631685\pi$
$110$	0	0
$111$	15.4250	1.46408
$112$	0	0
$113$	4.83365	0.454712	0.227356	−	0.973812i	$-0.426992\pi$
$113$	4.83365	0.454712	0.227356	+	0.973812i	$0.426992\pi$
$114$	0	0
$115$	0.221543	0.0206590
$116$	0	0
$117$	−21.4396	−1.98210
$118$	0	0
$119$	−22.2277	−2.03761
$120$	0	0
$121$	−8.83709	−0.803372
$122$	0	0
$123$	−1.63016	−0.146986
$124$	0	0
$125$	−4.60256	−0.411665
$126$	0	0
$127$	−18.4492	−1.63710	−0.818551	−	0.574434i	$-0.805222\pi$
$127$	−18.4492	−1.63710	−0.818551	+	0.574434i	$0.805222\pi$
$128$	0	0
$129$	12.6578	1.11445
$130$	0	0
$131$	0.868126	0.0758485	0.0379243	−	0.999281i	$-0.487925\pi$
$131$	0.868126	0.0758485	0.0379243	+	0.999281i	$0.487925\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.96896	0.599793
$136$	0	0
$137$	4.55691	0.389323	0.194662	−	0.980870i	$-0.437639\pi$
$137$	4.55691	0.389323	0.194662	+	0.980870i	$0.437639\pi$
$138$	0	0
$139$	−16.3078	−1.38321	−0.691604	−	0.722277i	$-0.743095\pi$
$139$	−16.3078	−1.38321	−0.691604	+	0.722277i	$0.743095\pi$
$140$	0	0
$141$	38.6983	3.25899
$142$	0	0
$143$	−4.17246	−0.348919
$144$	0	0
$145$	2.33881	0.194228
$146$	0	0
$147$	12.8061	1.05623
$148$	0	0
$149$	20.9069	1.71276	0.856380	−	0.516347i	$-0.172708\pi$
$149$	20.9069	1.71276	0.856380	+	0.516347i	$0.172708\pi$
$150$	0	0
$151$	1.75086	0.142483	0.0712415	−	0.997459i	$-0.477304\pi$
$151$	1.75086	0.142483	0.0712415	+	0.997459i	$0.477304\pi$
$152$	0	0
$153$	50.7811	4.10541
$154$	0	0
$155$	3.17590	0.255094
$156$	0	0
$157$	−0.159472	−0.0127272	−0.00636362	−	0.999980i	$-0.502026\pi$
$157$	−0.159472	−0.0127272	−0.00636362	+	0.999980i	$0.502026\pi$
$158$	0	0
$159$	−15.3354	−1.21617
$160$	0	0
$161$	−1.55691	−0.122702
$162$	0	0
$163$	−0.0862308	−0.00675412	−0.00337706	−	0.999994i	$-0.501075\pi$
$163$	−0.0862308	−0.00675412	−0.00337706	+	0.999994i	$0.501075\pi$
$164$	0	0
$165$	2.24914	0.175095
$166$	0	0
$167$	−6.27674	−0.485709	−0.242854	−	0.970063i	$-0.578084\pi$
$167$	−6.27674	−0.485709	−0.242854	+	0.970063i	$0.578084\pi$
$168$	0	0
$169$	−4.95092	−0.380840
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.8371	−0.823929	−0.411964	−	0.911200i	$-0.635157\pi$
$173$	−10.8371	−0.823929	−0.411964	+	0.911200i	$0.635157\pi$
$174$	0	0
$175$	15.8061	1.19483
$176$	0	0
$177$	23.5535	1.77039
$178$	0	0
$179$	7.08967	0.529907	0.264953	−	0.964261i	$-0.414643\pi$
$179$	7.08967	0.529907	0.264953	+	0.964261i	$0.414643\pi$
$180$	0	0
$181$	−12.3534	−0.918222	−0.459111	−	0.888379i	$-0.651832\pi$
$181$	−12.3534	−0.918222	−0.459111	+	0.888379i	$0.651832\pi$
$182$	0	0
$183$	−40.6983	−3.00851
$184$	0	0
$185$	2.23453	0.164286
$186$	0	0
$187$	9.88273	0.722697
$188$	0	0
$189$	−48.9751	−3.56241
$190$	0	0
$191$	8.99656	0.650968	0.325484	−	0.945547i	$-0.394473\pi$
$191$	8.99656	0.650968	0.325484	+	0.945547i	$0.394473\pi$
$192$	0	0
$193$	−3.16291	−0.227671	−0.113836	−	0.993500i	$-0.536314\pi$
$193$	−3.16291	−0.227671	−0.113836	+	0.993500i	$0.536314\pi$
$194$	0	0
$195$	−4.33881	−0.310709
$196$	0	0
$197$	−21.2457	−1.51369	−0.756847	−	0.653592i	$-0.773261\pi$
$197$	−21.2457	−1.51369	−0.756847	+	0.653592i	$0.773261\pi$
$198$	0	0
$199$	9.89572	0.701489	0.350745	−	0.936471i	$-0.385929\pi$
$199$	9.89572	0.701489	0.350745	+	0.936471i	$0.385929\pi$
$200$	0	0
$201$	−2.05863	−0.145205
$202$	0	0
$203$	−16.4362	−1.15360
$204$	0	0
$205$	−0.236151	−0.0164935
$206$	0	0
$207$	3.55691	0.247223
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.1008	1.31496	0.657478	−	0.753474i	$-0.271623\pi$
$211$	19.1008	1.31496	0.657478	+	0.753474i	$0.271623\pi$
$212$	0	0
$213$	−12.2767	−0.841188
$214$	0	0
$215$	1.83365	0.125054
$216$	0	0
$217$	−22.3189	−1.51511
$218$	0	0
$219$	−12.8061	−0.865353
$220$	0	0
$221$	−19.0647	−1.28243
$222$	0	0
$223$	−16.9690	−1.13633	−0.568163	−	0.822916i	$-0.692346\pi$
$223$	−16.9690	−1.13633	−0.568163	+	0.822916i	$0.692346\pi$
$224$	0	0
$225$	−36.1104	−2.40736
$226$	0	0
$227$	0.646583	0.0429152	0.0214576	−	0.999770i	$-0.493169\pi$
$227$	0.646583	0.0429152	0.0214576	+	0.999770i	$0.493169\pi$
$228$	0	0
$229$	−8.55348	−0.565230	−0.282615	−	0.959233i	$-0.591202\pi$
$229$	−8.55348	−0.565230	−0.282615	+	0.959233i	$0.591202\pi$
$230$	0	0
$231$	−15.8061	−1.03996
$232$	0	0
$233$	−3.70683	−0.242843	−0.121421	−	0.992601i	$-0.538745\pi$
$233$	−3.70683	−0.242843	−0.121421	+	0.992601i	$0.538745\pi$
$234$	0	0
$235$	5.60600	0.365695
$236$	0	0
$237$	−28.7129	−1.86511
$238$	0	0
$239$	15.7655	1.01978	0.509892	−	0.860239i	$-0.329685\pi$
$239$	15.7655	1.01978	0.509892	+	0.860239i	$0.329685\pi$
$240$	0	0
$241$	−16.1138	−1.03798	−0.518991	−	0.854779i	$-0.673692\pi$
$241$	−16.1138	−1.03798	−0.518991	+	0.854779i	$0.673692\pi$
$242$	0	0
$243$	38.2277	2.45231
$244$	0	0
$245$	1.85514	0.118520
$246$	0	0
$247$	0	0
$248$	0	0
$249$	29.7129	1.88298
$250$	0	0
$251$	−15.6561	−0.988206	−0.494103	−	0.869403i	$-0.664504\pi$
$251$	−15.6561	−0.988206	−0.494103	+	0.869403i	$0.664504\pi$
$252$	0	0
$253$	0.692226	0.0435199
$254$	0	0
$255$	10.2767	0.643554
$256$	0	0
$257$	−27.8172	−1.73519	−0.867595	−	0.497271i	$-0.834336\pi$
$257$	−27.8172	−1.73519	−0.867595	+	0.497271i	$0.834336\pi$
$258$	0	0
$259$	−15.7034	−0.975762
$260$	0	0
$261$	37.5500	2.32429
$262$	0	0
$263$	16.8517	1.03912	0.519560	−	0.854434i	$-0.326096\pi$
$263$	16.8517	1.03912	0.519560	+	0.854434i	$0.326096\pi$
$264$	0	0
$265$	−2.22154	−0.136468
$266$	0	0
$267$	9.59912	0.587457
$268$	0	0
$269$	−29.6251	−1.80627	−0.903137	−	0.429352i	$-0.858742\pi$
$269$	−29.6251	−1.80627	−0.903137	+	0.429352i	$0.858742\pi$
$270$	0	0
$271$	14.4707	0.879031	0.439516	−	0.898235i	$-0.355150\pi$
$271$	14.4707	0.879031	0.439516	+	0.898235i	$0.355150\pi$
$272$	0	0
$273$	30.4914	1.84542
$274$	0	0
$275$	−7.02760	−0.423780
$276$	0	0
$277$	−12.4216	−0.746342	−0.373171	−	0.927763i	$-0.621729\pi$
$277$	−12.4216	−0.746342	−0.373171	+	0.927763i	$0.621729\pi$
$278$	0	0
$279$	50.9897	3.05267
$280$	0	0
$281$	−10.2932	−0.614039	−0.307019	−	0.951703i	$-0.599332\pi$
$281$	−10.2932	−0.614039	−0.307019	+	0.951703i	$0.599332\pi$
$282$	0	0
$283$	−15.8647	−0.943058	−0.471529	−	0.881851i	$-0.656298\pi$
$283$	−15.8647	−0.943058	−0.471529	+	0.881851i	$0.656298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.65957	0.0979615
$288$	0	0
$289$	28.1560	1.65624
$290$	0	0
$291$	0.190507	0.0111677
$292$	0	0
$293$	−12.1579	−0.710269	−0.355135	−	0.934815i	$-0.615565\pi$
$293$	−12.1579	−0.710269	−0.355135	+	0.934815i	$0.615565\pi$
$294$	0	0
$295$	3.41205	0.198657
$296$	0	0
$297$	21.7750	1.26351
$298$	0	0
$299$	−1.33537	−0.0772265
$300$	0	0
$301$	−12.8862	−0.742747
$302$	0	0
$303$	37.6397	2.16234
$304$	0	0
$305$	−5.89572	−0.337588
$306$	0	0
$307$	−12.3009	−0.702049	−0.351025	−	0.936366i	$-0.614167\pi$
$307$	−12.3009	−0.702049	−0.351025	+	0.936366i	$0.614167\pi$
$308$	0	0
$309$	−26.5535	−1.51057
$310$	0	0
$311$	−0.519765	−0.0294732	−0.0147366	−	0.999891i	$-0.504691\pi$
$311$	−0.519765	−0.0294732	−0.0147366	+	0.999891i	$0.504691\pi$
$312$	0	0
$313$	18.6121	1.05202	0.526009	−	0.850479i	$-0.323688\pi$
$313$	18.6121	1.05202	0.526009	+	0.850479i	$0.323688\pi$
$314$	0	0
$315$	−11.7655	−0.662909
$316$	0	0
$317$	−2.83709	−0.159347	−0.0796734	−	0.996821i	$-0.525388\pi$
$317$	−2.83709	−0.159347	−0.0796734	+	0.996821i	$0.525388\pi$
$318$	0	0
$319$	7.30777	0.409157
$320$	0	0
$321$	1.61899	0.0903629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	13.5569	0.752002
$326$	0	0
$327$	−27.2733	−1.50822
$328$	0	0
$329$	−39.3967	−2.17201
$330$	0	0
$331$	−19.0276	−1.04585	−0.522926	−	0.852378i	$-0.675159\pi$
$331$	−19.0276	−1.04585	−0.522926	+	0.852378i	$0.675159\pi$
$332$	0	0
$333$	35.8759	1.96598
$334$	0	0
$335$	−0.298222	−0.0162936
$336$	0	0
$337$	−14.7294	−0.802360	−0.401180	−	0.915999i	$-0.631400\pi$
$337$	−14.7294	−0.802360	−0.401180	+	0.915999i	$0.631400\pi$
$338$	0	0
$339$	15.7052	0.852990
$340$	0	0
$341$	9.92332	0.537378
$342$	0	0
$343$	10.1173	0.546281
$344$	0	0
$345$	0.719824	0.0387540
$346$	0	0
$347$	−10.1353	−0.544092	−0.272046	−	0.962284i	$-0.587700\pi$
$347$	−10.1353	−0.544092	−0.272046	+	0.962284i	$0.587700\pi$
$348$	0	0
$349$	2.56035	0.137053	0.0685263	−	0.997649i	$-0.478170\pi$
$349$	2.56035	0.137053	0.0685263	+	0.997649i	$0.478170\pi$
$350$	0	0
$351$	−42.0061	−2.24212
$352$	0	0
$353$	20.4458	1.08822	0.544109	−	0.839015i	$-0.316868\pi$
$353$	20.4458	1.08822	0.544109	+	0.839015i	$0.316868\pi$
$354$	0	0
$355$	−1.77846	−0.0943907
$356$	0	0
$357$	−72.2208	−3.82233
$358$	0	0
$359$	−14.5879	−0.769923	−0.384961	−	0.922933i	$-0.625785\pi$
$359$	−14.5879	−0.769923	−0.384961	+	0.922933i	$0.625785\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−28.7129	−1.50704
$364$	0	0
$365$	−1.85514	−0.0971023
$366$	0	0
$367$	29.0794	1.51793	0.758965	−	0.651131i	$-0.225705\pi$
$367$	29.0794	1.51793	0.758965	+	0.651131i	$0.225705\pi$
$368$	0	0
$369$	−3.79145	−0.197375
$370$	0	0
$371$	15.6121	0.810540
$372$	0	0
$373$	20.7880	1.07636	0.538181	−	0.842829i	$-0.319112\pi$
$373$	20.7880	1.07636	0.538181	+	0.842829i	$0.319112\pi$
$374$	0	0
$375$	−14.9544	−0.772240
$376$	0	0
$377$	−14.0974	−0.726053
$378$	0	0
$379$	19.7034	1.01210	0.506048	−	0.862505i	$-0.331106\pi$
$379$	19.7034	1.01210	0.506048	+	0.862505i	$0.331106\pi$
$380$	0	0
$381$	−59.9440	−3.07103
$382$	0	0
$383$	28.4638	1.45443	0.727216	−	0.686408i	$-0.240814\pi$
$383$	28.4638	1.45443	0.727216	+	0.686408i	$0.240814\pi$
$384$	0	0
$385$	−2.28973	−0.116695
$386$	0	0
$387$	29.4396	1.49650
$388$	0	0
$389$	3.71639	0.188428	0.0942141	−	0.995552i	$-0.469966\pi$
$389$	3.71639	0.188428	0.0942141	+	0.995552i	$0.469966\pi$
$390$	0	0
$391$	3.16291	0.159955
$392$	0	0
$393$	2.82066	0.142284
$394$	0	0
$395$	−4.15947	−0.209286
$396$	0	0
$397$	17.6397	0.885312	0.442656	−	0.896692i	$-0.354036\pi$
$397$	17.6397	0.885312	0.442656	+	0.896692i	$0.354036\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.5569	−1.52594	−0.762970	−	0.646434i	$-0.776259\pi$
$401$	−30.5569	−1.52594	−0.762970	+	0.646434i	$0.776259\pi$
$402$	0	0
$403$	−19.1430	−0.953583
$404$	0	0
$405$	11.9724	0.594913
$406$	0	0
$407$	6.98195	0.346083
$408$	0	0
$409$	−31.2277	−1.54411	−0.772054	−	0.635557i	$-0.780771\pi$
$409$	−31.2277	−1.54411	−0.772054	+	0.635557i	$0.780771\pi$
$410$	0	0
$411$	14.8061	0.730329
$412$	0	0
$413$	−23.9785	−1.17991
$414$	0	0
$415$	4.30434	0.211292
$416$	0	0
$417$	−52.9862	−2.59475
$418$	0	0
$419$	−28.9966	−1.41657	−0.708287	−	0.705924i	$-0.750532\pi$
$419$	−28.9966	−1.41657	−0.708287	+	0.705924i	$0.750532\pi$
$420$	0	0
$421$	31.2587	1.52346	0.761728	−	0.647897i	$-0.224351\pi$
$421$	31.2587	1.52346	0.761728	+	0.647897i	$0.224351\pi$
$422$	0	0
$423$	90.0054	4.37621
$424$	0	0
$425$	−32.1104	−1.55758
$426$	0	0
$427$	41.4328	2.00507
$428$	0	0
$429$	−13.5569	−0.654534
$430$	0	0
$431$	−14.8663	−0.716085	−0.358042	−	0.933705i	$-0.616556\pi$
$431$	−14.8663	−0.716085	−0.358042	+	0.933705i	$0.616556\pi$
$432$	0	0
$433$	39.5630	1.90128	0.950639	−	0.310299i	$-0.100429\pi$
$433$	39.5630	1.90128	0.950639	+	0.310299i	$0.100429\pi$
$434$	0	0
$435$	7.59912	0.364350
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−11.5845	−0.552899	−0.276449	−	0.961029i	$-0.589158\pi$
$439$	−11.5845	−0.552899	−0.276449	+	0.961029i	$0.589158\pi$
$440$	0	0
$441$	29.7846	1.41831
$442$	0	0
$443$	28.9233	1.37419	0.687094	−	0.726568i	$-0.258886\pi$
$443$	28.9233	1.37419	0.687094	+	0.726568i	$0.258886\pi$
$444$	0	0
$445$	1.39057	0.0659192
$446$	0	0
$447$	67.9294	3.21295
$448$	0	0
$449$	31.2147	1.47311	0.736556	−	0.676377i	$-0.236451\pi$
$449$	31.2147	1.47311	0.736556	+	0.676377i	$0.236451\pi$
$450$	0	0
$451$	−0.737870	−0.0347449
$452$	0	0
$453$	5.68879	0.267283
$454$	0	0
$455$	4.41711	0.207077
$456$	0	0
$457$	17.2215	0.805590	0.402795	−	0.915290i	$-0.368039\pi$
$457$	17.2215	0.805590	0.402795	+	0.915290i	$0.368039\pi$
$458$	0	0
$459$	99.4941	4.64399
$460$	0	0
$461$	−32.8302	−1.52906	−0.764528	−	0.644591i	$-0.777028\pi$
$461$	−32.8302	−1.52906	−0.764528	+	0.644591i	$0.777028\pi$
$462$	0	0
$463$	6.28973	0.292308	0.146154	−	0.989262i	$-0.453310\pi$
$463$	6.28973	0.292308	0.146154	+	0.989262i	$0.453310\pi$
$464$	0	0
$465$	10.3189	0.478530
$466$	0	0
$467$	−9.47068	−0.438251	−0.219125	−	0.975697i	$-0.570320\pi$
$467$	−9.47068	−0.438251	−0.219125	+	0.975697i	$0.570320\pi$
$468$	0	0
$469$	2.09578	0.0967743
$470$	0	0
$471$	−0.518147	−0.0238749
$472$	0	0
$473$	5.72938	0.263437
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−35.6673	−1.63309
$478$	0	0
$479$	26.0682	1.19109	0.595543	−	0.803324i	$-0.296937\pi$
$479$	26.0682	1.19109	0.595543	+	0.803324i	$0.296937\pi$
$480$	0	0
$481$	−13.4689	−0.614127
$482$	0	0
$483$	−5.05863	−0.230176
$484$	0	0
$485$	0.0275977	0.00125315
$486$	0	0
$487$	16.8939	0.765536	0.382768	−	0.923845i	$-0.374971\pi$
$487$	16.8939	0.765536	0.382768	+	0.923845i	$0.374971\pi$
$488$	0	0
$489$	−0.280176	−0.0126700
$490$	0	0
$491$	17.8888	0.807312	0.403656	−	0.914911i	$-0.367739\pi$
$491$	17.8888	0.807312	0.403656	+	0.914911i	$0.367739\pi$
$492$	0	0
$493$	33.3906	1.50384
$494$	0	0
$495$	5.23109	0.235120
$496$	0	0
$497$	12.4983	0.560625
$498$	0	0
$499$	−29.4216	−1.31709	−0.658546	−	0.752541i	$-0.728828\pi$
$499$	−29.4216	−1.31709	−0.658546	+	0.752541i	$0.728828\pi$
$500$	0	0
$501$	−20.3940	−0.911137
$502$	0	0
$503$	9.07936	0.404828	0.202414	−	0.979300i	$-0.435121\pi$
$503$	9.07936	0.404828	0.202414	+	0.979300i	$0.435121\pi$
$504$	0	0
$505$	5.45264	0.242639
$506$	0	0
$507$	−16.0862	−0.714415
$508$	0	0
$509$	−20.3940	−0.903948	−0.451974	−	0.892031i	$-0.649280\pi$
$509$	−20.3940	−0.903948	−0.451974	+	0.892031i	$0.649280\pi$
$510$	0	0
$511$	13.0371	0.576730
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.84664	−0.169503
$516$	0	0
$517$	17.5163	0.770367
$518$	0	0
$519$	−35.2112	−1.54560
$520$	0	0
$521$	−14.8697	−0.651455	−0.325728	−	0.945464i	$-0.605609\pi$
$521$	−14.8697	−0.651455	−0.325728	+	0.945464i	$0.605609\pi$
$522$	0	0
$523$	−10.8992	−0.476587	−0.238294	−	0.971193i	$-0.576588\pi$
$523$	−10.8992	−0.476587	−0.238294	+	0.971193i	$0.576588\pi$
$524$	0	0
$525$	51.3561	2.24136
$526$	0	0
$527$	45.3415	1.97511
$528$	0	0
$529$	−22.7785	−0.990368
$530$	0	0
$531$	54.7811	2.37730
$532$	0	0
$533$	1.42342	0.0616552
$534$	0	0
$535$	0.234533	0.0101397
$536$	0	0
$537$	23.0353	0.994048
$538$	0	0
$539$	5.79650	0.249673
$540$	0	0
$541$	11.9509	0.513810	0.256905	−	0.966437i	$-0.417297\pi$
$541$	11.9509	0.513810	0.256905	+	0.966437i	$0.417297\pi$
$542$	0	0
$543$	−40.1380	−1.72249
$544$	0	0
$545$	−3.95092	−0.169239
$546$	0	0
$547$	−4.00611	−0.171289	−0.0856445	−	0.996326i	$-0.527295\pi$
$547$	−4.00611	−0.171289	−0.0856445	+	0.996326i	$0.527295\pi$
$548$	0	0
$549$	−94.6570	−4.03986
$550$	0	0
$551$	0	0
$552$	0	0
$553$	29.2311	1.24303
$554$	0	0
$555$	7.26031	0.308183
$556$	0	0
$557$	10.7198	0.454214	0.227107	−	0.973870i	$-0.427073\pi$
$557$	10.7198	0.454214	0.227107	+	0.973870i	$0.427073\pi$
$558$	0	0
$559$	−11.0525	−0.467472
$560$	0	0
$561$	32.1104	1.35570
$562$	0	0
$563$	7.18096	0.302641	0.151321	−	0.988485i	$-0.451647\pi$
$563$	7.18096	0.302641	0.151321	+	0.988485i	$0.451647\pi$
$564$	0	0
$565$	2.27512	0.0957150
$566$	0	0
$567$	−84.1372	−3.53343
$568$	0	0
$569$	31.4328	1.31773	0.658865	−	0.752261i	$-0.271037\pi$
$569$	31.4328	1.31773	0.658865	+	0.752261i	$0.271037\pi$
$570$	0	0
$571$	8.08623	0.338398	0.169199	−	0.985582i	$-0.445882\pi$
$571$	8.08623	0.338398	0.169199	+	0.985582i	$0.445882\pi$
$572$	0	0
$573$	29.2311	1.22115
$574$	0	0
$575$	−2.24914	−0.0937956
$576$	0	0
$577$	4.95779	0.206396	0.103198	−	0.994661i	$-0.467093\pi$
$577$	4.95779	0.206396	0.103198	+	0.994661i	$0.467093\pi$
$578$	0	0
$579$	−10.2767	−0.427087
$580$	0	0
$581$	−30.2491	−1.25495
$582$	0	0
$583$	−6.94137	−0.287482
$584$	0	0
$585$	−10.0913	−0.417223
$586$	0	0
$587$	23.7716	0.981158	0.490579	−	0.871397i	$-0.336785\pi$
$587$	23.7716	0.981158	0.490579	+	0.871397i	$0.336785\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−69.0303	−2.83953
$592$	0	0
$593$	−42.4036	−1.74131	−0.870653	−	0.491898i	$-0.836303\pi$
$593$	−42.4036	−1.74131	−0.870653	+	0.491898i	$0.836303\pi$
$594$	0	0
$595$	−10.4622	−0.428908
$596$	0	0
$597$	32.1526	1.31592
$598$	0	0
$599$	42.8517	1.75087	0.875436	−	0.483333i	$-0.160574\pi$
$599$	42.8517	1.75087	0.875436	+	0.483333i	$0.160574\pi$
$600$	0	0
$601$	−15.1855	−0.619427	−0.309714	−	0.950830i	$-0.600233\pi$
$601$	−15.1855	−0.619427	−0.309714	+	0.950830i	$0.600233\pi$
$602$	0	0
$603$	−4.78801	−0.194983
$604$	0	0
$605$	−4.15947	−0.169107
$606$	0	0
$607$	−32.3336	−1.31238	−0.656189	−	0.754596i	$-0.727833\pi$
$607$	−32.3336	−1.31238	−0.656189	+	0.754596i	$0.727833\pi$
$608$	0	0
$609$	−53.4036	−2.16402
$610$	0	0
$611$	−33.7907	−1.36702
$612$	0	0
$613$	−7.83365	−0.316398	−0.158199	−	0.987407i	$-0.550569\pi$
$613$	−7.83365	−0.316398	−0.158199	+	0.987407i	$0.550569\pi$
$614$	0	0
$615$	−0.767287	−0.0309400
$616$	0	0
$617$	−4.26719	−0.171790	−0.0858952	−	0.996304i	$-0.527375\pi$
$617$	−4.26719	−0.171790	−0.0858952	+	0.996304i	$0.527375\pi$
$618$	0	0
$619$	42.8363	1.72174	0.860869	−	0.508827i	$-0.169921\pi$
$619$	42.8363	1.72174	0.860869	+	0.508827i	$0.169921\pi$
$620$	0	0
$621$	6.96896	0.279655
$622$	0	0
$623$	−9.77234	−0.391521
$624$	0	0
$625$	21.7259	0.869038
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.9018	1.27201
$630$	0	0
$631$	−36.3104	−1.44550	−0.722748	−	0.691112i	$-0.757121\pi$
$631$	−36.3104	−1.44550	−0.722748	+	0.691112i	$0.757121\pi$
$632$	0	0
$633$	62.0613	2.46672
$634$	0	0
$635$	−8.68373	−0.344603
$636$	0	0
$637$	−11.1820	−0.443048
$638$	0	0
$639$	−28.5535	−1.12956
$640$	0	0
$641$	−49.6087	−1.95942	−0.979712	−	0.200409i	$-0.935773\pi$
$641$	−49.6087	−1.95942	−0.979712	+	0.200409i	$0.935773\pi$
$642$	0	0
$643$	−4.98539	−0.196605	−0.0983023	−	0.995157i	$-0.531341\pi$
$643$	−4.98539	−0.196605	−0.0983023	+	0.995157i	$0.531341\pi$
$644$	0	0
$645$	5.95779	0.234588
$646$	0	0
$647$	−43.2863	−1.70176	−0.850880	−	0.525360i	$-0.823931\pi$
$647$	−43.2863	−1.70176	−0.850880	+	0.525360i	$0.823931\pi$
$648$	0	0
$649$	10.6612	0.418488
$650$	0	0
$651$	−72.5174	−2.84218
$652$	0	0
$653$	−5.43965	−0.212870	−0.106435	−	0.994320i	$-0.533944\pi$
$653$	−5.43965	−0.212870	−0.106435	+	0.994320i	$0.533944\pi$
$654$	0	0
$655$	0.408612	0.0159658
$656$	0	0
$657$	−29.7846	−1.16201
$658$	0	0
$659$	−5.74924	−0.223959	−0.111979	−	0.993711i	$-0.535719\pi$
$659$	−5.74924	−0.223959	−0.111979	+	0.993711i	$0.535719\pi$
$660$	0	0
$661$	−10.2553	−0.398883	−0.199442	−	0.979910i	$-0.563913\pi$
$661$	−10.2553	−0.398883	−0.199442	+	0.979910i	$0.563913\pi$
$662$	0	0
$663$	−61.9440	−2.40571
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.33881	0.0905591
$668$	0	0
$669$	−55.1346	−2.13162
$670$	0	0
$671$	−18.4216	−0.711158
$672$	0	0
$673$	−29.9379	−1.15402	−0.577011	−	0.816736i	$-0.695781\pi$
$673$	−29.9379	−1.15402	−0.577011	+	0.816736i	$0.695781\pi$
$674$	0	0
$675$	−70.7501	−2.72317
$676$	0	0
$677$	−11.8827	−0.456691	−0.228345	−	0.973580i	$-0.573332\pi$
$677$	−11.8827	−0.456691	−0.228345	+	0.973580i	$0.573332\pi$
$678$	0	0
$679$	−0.193945	−0.00744293
$680$	0	0
$681$	2.10084	0.0805043
$682$	0	0
$683$	6.35180	0.243045	0.121522	−	0.992589i	$-0.461222\pi$
$683$	6.35180	0.243045	0.121522	+	0.992589i	$0.461222\pi$
$684$	0	0
$685$	2.14486	0.0819510
$686$	0	0
$687$	−27.7914	−1.06031
$688$	0	0
$689$	13.3906	0.510140
$690$	0	0
$691$	28.9966	1.10308	0.551541	−	0.834148i	$-0.314040\pi$
$691$	28.9966	1.10308	0.551541	+	0.834148i	$0.314040\pi$
$692$	0	0
$693$	−36.7620	−1.39647
$694$	0	0
$695$	−7.67580	−0.291160
$696$	0	0
$697$	−3.37146	−0.127703
$698$	0	0
$699$	−12.0440	−0.455547
$700$	0	0
$701$	4.00162	0.151139	0.0755695	−	0.997141i	$-0.475923\pi$
$701$	4.00162	0.151139	0.0755695	+	0.997141i	$0.475923\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	18.2147	0.686004
$706$	0	0
$707$	−38.3189	−1.44113
$708$	0	0
$709$	−16.4638	−0.618311	−0.309156	−	0.951011i	$-0.600046\pi$
$709$	−16.4638	−0.618311	−0.309156	+	0.951011i	$0.600046\pi$
$710$	0	0
$711$	−66.7811	−2.50449
$712$	0	0
$713$	3.17590	0.118938
$714$	0	0
$715$	−1.96391	−0.0734460
$716$	0	0
$717$	51.2242	1.91300
$718$	0	0
$719$	19.2112	0.716458	0.358229	−	0.933634i	$-0.383381\pi$
$719$	19.2112	0.716458	0.358229	+	0.933634i	$0.383381\pi$
$720$	0	0
$721$	27.0327	1.00675
$722$	0	0
$723$	−52.3561	−1.94714
$724$	0	0
$725$	−23.7440	−0.881829
$726$	0	0
$727$	18.1663	0.673753	0.336876	−	0.941549i	$-0.390630\pi$
$727$	18.1663	0.673753	0.336876	+	0.941549i	$0.390630\pi$
$728$	0	0
$729$	47.8984	1.77401
$730$	0	0
$731$	26.1786	0.968250
$732$	0	0
$733$	50.2423	1.85574	0.927870	−	0.372903i	$-0.121638\pi$
$733$	50.2423	1.85574	0.927870	+	0.372903i	$0.121638\pi$
$734$	0	0
$735$	6.02760	0.222331
$736$	0	0
$737$	−0.931815	−0.0343238
$738$	0	0
$739$	−29.7766	−1.09535	−0.547676	−	0.836691i	$-0.684487\pi$
$739$	−29.7766	−1.09535	−0.547676	+	0.836691i	$0.684487\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.5156	1.44969	0.724843	−	0.688914i	$-0.241912\pi$
$743$	39.5156	1.44969	0.724843	+	0.688914i	$0.241912\pi$
$744$	0	0
$745$	9.84053	0.360529
$746$	0	0
$747$	69.1070	2.52849
$748$	0	0
$749$	−1.64820	−0.0602240
$750$	0	0
$751$	−18.1043	−0.660634	−0.330317	−	0.943870i	$-0.607156\pi$
$751$	−18.1043	−0.660634	−0.330317	+	0.943870i	$0.607156\pi$
$752$	0	0
$753$	−50.8690	−1.85377
$754$	0	0
$755$	0.824101	0.0299921
$756$	0	0
$757$	−23.5991	−0.857725	−0.428862	−	0.903370i	$-0.641085\pi$
$757$	−23.5991	−0.857725	−0.428862	+	0.903370i	$0.641085\pi$
$758$	0	0
$759$	2.24914	0.0816386
$760$	0	0
$761$	−1.69061	−0.0612845	−0.0306422	−	0.999530i	$-0.509755\pi$
$761$	−1.69061	−0.0612845	−0.0306422	+	0.999530i	$0.509755\pi$
$762$	0	0
$763$	27.7655	1.00518
$764$	0	0
$765$	23.9018	0.864173
$766$	0	0
$767$	−20.5665	−0.742612
$768$	0	0
$769$	−52.1165	−1.87937	−0.939685	−	0.342042i	$-0.888881\pi$
$769$	−52.1165	−1.87937	−0.939685	+	0.342042i	$0.888881\pi$
$770$	0	0
$771$	−90.3821	−3.25503
$772$	0	0
$773$	45.0794	1.62139	0.810696	−	0.585468i	$-0.199089\pi$
$773$	45.0794	1.62139	0.810696	+	0.585468i	$0.199089\pi$
$774$	0	0
$775$	−32.2423	−1.15818
$776$	0	0
$777$	−51.0225	−1.83042
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−5.55691	−0.198842
$782$	0	0
$783$	73.5708	2.62920
$784$	0	0
$785$	−0.0750608	−0.00267903
$786$	0	0
$787$	−38.1414	−1.35960	−0.679798	−	0.733400i	$-0.737932\pi$
$787$	−38.1414	−1.35960	−0.679798	+	0.733400i	$0.737932\pi$
$788$	0	0
$789$	54.7535	1.94928
$790$	0	0
$791$	−15.9886	−0.568490
$792$	0	0
$793$	35.5370	1.26196
$794$	0	0
$795$	−7.21811	−0.256000
$796$	0	0
$797$	−13.7148	−0.485802	−0.242901	−	0.970051i	$-0.578099\pi$
$797$	−13.7148	−0.485802	−0.242901	+	0.970051i	$0.578099\pi$
$798$	0	0
$799$	80.0353	2.83145
$800$	0	0
$801$	22.3258	0.788844
$802$	0	0
$803$	−5.79650	−0.204554
$804$	0	0
$805$	−0.732814	−0.0258283
$806$	0	0
$807$	−96.2561	−3.38838
$808$	0	0
$809$	−21.6285	−0.760419	−0.380209	−	0.924900i	$-0.624148\pi$
$809$	−21.6285	−0.760419	−0.380209	+	0.924900i	$0.624148\pi$
$810$	0	0
$811$	−6.54049	−0.229668	−0.114834	−	0.993385i	$-0.536634\pi$
$811$	−6.54049	−0.229668	−0.114834	+	0.993385i	$0.536634\pi$
$812$	0	0
$813$	47.0173	1.64897
$814$	0	0
$815$	−0.0405874	−0.00142172
$816$	0	0
$817$	0	0
$818$	0	0
$819$	70.9175	2.47806
$820$	0	0
$821$	41.6182	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$821$	41.6182	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$822$	0	0
$823$	−5.45264	−0.190067	−0.0950335	−	0.995474i	$-0.530296\pi$
$823$	−5.45264	−0.190067	−0.0950335	+	0.995474i	$0.530296\pi$
$824$	0	0
$825$	−22.8337	−0.794966
$826$	0	0
$827$	−2.68879	−0.0934983	−0.0467492	−	0.998907i	$-0.514886\pi$
$827$	−2.68879	−0.0934983	−0.0467492	+	0.998907i	$0.514886\pi$
$828$	0	0
$829$	−27.4328	−0.952780	−0.476390	−	0.879234i	$-0.658055\pi$
$829$	−27.4328	−0.952780	−0.476390	+	0.879234i	$0.658055\pi$
$830$	0	0
$831$	−40.3595	−1.40006
$832$	0	0
$833$	26.4853	0.917661
$834$	0	0
$835$	−2.95436	−0.102240
$836$	0	0
$837$	99.9027	3.45314
$838$	0	0
$839$	−27.4413	−0.947378	−0.473689	−	0.880692i	$-0.657078\pi$
$839$	−27.4413	−0.947378	−0.473689	+	0.880692i	$0.657078\pi$
$840$	0	0
$841$	−4.30939	−0.148600
$842$	0	0
$843$	−33.4439	−1.15187
$844$	0	0
$845$	−2.33032	−0.0801653
$846$	0	0
$847$	29.2311	1.00439
$848$	0	0
$849$	−51.5466	−1.76907
$850$	0	0
$851$	2.23453	0.0765988
$852$	0	0
$853$	36.3871	1.24587	0.622936	−	0.782273i	$-0.285940\pi$
$853$	36.3871	1.24587	0.622936	+	0.782273i	$0.285940\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.1295	−1.67823	−0.839116	−	0.543953i	$-0.816927\pi$
$857$	−49.1295	−1.67823	−0.839116	+	0.543953i	$0.816927\pi$
$858$	0	0
$859$	51.9130	1.77125	0.885624	−	0.464402i	$-0.153731\pi$
$859$	51.9130	1.77125	0.885624	+	0.464402i	$0.153731\pi$
$860$	0	0
$861$	5.39218	0.183765
$862$	0	0
$863$	27.9785	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$863$	27.9785	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$864$	0	0
$865$	−5.10084	−0.173434
$866$	0	0
$867$	91.4829	3.10692
$868$	0	0
$869$	−12.9966	−0.440878
$870$	0	0
$871$	1.79756	0.0609080
$872$	0	0
$873$	0.443086	0.0149962
$874$	0	0
$875$	15.2242	0.514673
$876$	0	0
$877$	−12.4017	−0.418777	−0.209388	−	0.977833i	$-0.567147\pi$
$877$	−12.4017	−0.418777	−0.209388	+	0.977833i	$0.567147\pi$
$878$	0	0
$879$	−39.5026	−1.33239
$880$	0	0
$881$	17.8923	0.602806	0.301403	−	0.953497i	$-0.402545\pi$
$881$	17.8923	0.602806	0.301403	+	0.953497i	$0.402545\pi$
$882$	0	0
$883$	−11.0337	−0.371314	−0.185657	−	0.982615i	$-0.559441\pi$
$883$	−11.0337	−0.371314	−0.185657	+	0.982615i	$0.559441\pi$
$884$	0	0
$885$	11.0862	0.372660
$886$	0	0
$887$	−40.3940	−1.35630	−0.678149	−	0.734924i	$-0.737218\pi$
$887$	−40.3940	−1.35630	−0.678149	+	0.734924i	$0.737218\pi$
$888$	0	0
$889$	61.0258	2.04674
$890$	0	0
$891$	37.4086	1.25324
$892$	0	0
$893$	0	0
$894$	0	0
$895$	3.33699	0.111543
$896$	0	0
$897$	−4.33881	−0.144869
$898$	0	0
$899$	33.5277	1.11821
$900$	0	0
$901$	−31.7164	−1.05663
$902$	0	0
$903$	−41.8690	−1.39331
$904$	0	0
$905$	−5.81455	−0.193282
$906$	0	0
$907$	−39.1510	−1.29999	−0.649993	−	0.759940i	$-0.725228\pi$
$907$	−39.1510	−1.29999	−0.649993	+	0.759940i	$0.725228\pi$
$908$	0	0
$909$	87.5432	2.90362
$910$	0	0
$911$	−27.3561	−0.906348	−0.453174	−	0.891422i	$-0.649708\pi$
$911$	−27.3561	−0.906348	−0.453174	+	0.891422i	$0.649708\pi$
$912$	0	0
$913$	13.4492	0.445104
$914$	0	0
$915$	−19.1560	−0.633279
$916$	0	0
$917$	−2.87156	−0.0948274
$918$	0	0
$919$	−0.948243	−0.0312796	−0.0156398	−	0.999878i	$-0.504979\pi$
$919$	−0.948243	−0.0312796	−0.0156398	+	0.999878i	$0.504979\pi$
$920$	0	0
$921$	−39.9673	−1.31697
$922$	0	0
$923$	10.7198	0.352847
$924$	0	0
$925$	−22.6854	−0.745890
$926$	0	0
$927$	−61.7586	−2.02842
$928$	0	0
$929$	5.26051	0.172592	0.0862959	−	0.996270i	$-0.472497\pi$
$929$	5.26051	0.172592	0.0862959	+	0.996270i	$0.472497\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.68879	−0.0552885
$934$	0	0
$935$	4.65164	0.152125
$936$	0	0
$937$	45.9276	1.50039	0.750195	−	0.661217i	$-0.229960\pi$
$937$	45.9276	1.50039	0.750195	+	0.661217i	$0.229960\pi$
$938$	0	0
$939$	60.4734	1.97347
$940$	0	0
$941$	−44.5957	−1.45378	−0.726889	−	0.686755i	$-0.759034\pi$
$941$	−44.5957	−1.45378	−0.726889	+	0.686755i	$0.759034\pi$
$942$	0	0
$943$	−0.236151	−0.00769013
$944$	0	0
$945$	−23.0518	−0.749874
$946$	0	0
$947$	−14.0162	−0.455466	−0.227733	−	0.973724i	$-0.573131\pi$
$947$	−14.0162	−0.455466	−0.227733	+	0.973724i	$0.573131\pi$
$948$	0	0
$949$	11.1820	0.362984
$950$	0	0
$951$	−9.21811	−0.298918
$952$	0	0
$953$	2.84664	0.0922118	0.0461059	−	0.998937i	$-0.485319\pi$
$953$	2.84664	0.0922118	0.0461059	+	0.998937i	$0.485319\pi$
$954$	0	0
$955$	4.23453	0.137026
$956$	0	0
$957$	23.7440	0.767534
$958$	0	0
$959$	−15.0732	−0.486740
$960$	0	0
$961$	14.5277	0.468635
$962$	0	0
$963$	3.76547	0.121340
$964$	0	0
$965$	−1.48873	−0.0479239
$966$	0	0
$967$	−1.66119	−0.0534203	−0.0267101	−	0.999643i	$-0.508503\pi$
$967$	−1.66119	−0.0534203	−0.0267101	+	0.999643i	$0.508503\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.4182	1.29708	0.648540	−	0.761180i	$-0.275380\pi$
$971$	40.4182	1.29708	0.648540	+	0.761180i	$0.275380\pi$
$972$	0	0
$973$	53.9424	1.72931
$974$	0	0
$975$	44.0483	1.41068
$976$	0	0
$977$	10.5439	0.337330	0.168665	−	0.985673i	$-0.446054\pi$
$977$	10.5439	0.337330	0.168665	+	0.985673i	$0.446054\pi$
$978$	0	0
$979$	4.34492	0.138864
$980$	0	0
$981$	−63.4328	−2.02525
$982$	0	0
$983$	−9.83365	−0.313645	−0.156822	−	0.987627i	$-0.550125\pi$
$983$	−9.83365	−0.313645	−0.156822	+	0.987627i	$0.550125\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	−128.005	−4.07446
$988$	0	0
$989$	1.83365	0.0583068
$990$	0	0
$991$	20.2699	0.643893	0.321947	−	0.946758i	$-0.395663\pi$
$991$	20.2699	0.643893	0.321947	+	0.946758i	$0.395663\pi$
$992$	0	0
$993$	−61.8233	−1.96190
$994$	0	0
$995$	4.65775	0.147661
$996$	0	0
$997$	15.3208	0.485213	0.242607	−	0.970125i	$-0.421998\pi$
$997$	15.3208	0.485213	0.242607	+	0.970125i	$0.421998\pi$
$998$	0	0
$999$	70.2906	2.22390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.r.1.3		3
4.3	odd	2		5776.2.a.bk.1.1		3
19.7	even	3		152.2.i.c.49.1	✓	6
19.11	even	3		152.2.i.c.121.1	yes	6
19.18	odd	2		2888.2.a.n.1.1		3
57.11	odd	6		1368.2.s.k.577.2		6
57.26	odd	6		1368.2.s.k.505.2		6
76.7	odd	6		304.2.i.f.49.3		6
76.11	odd	6		304.2.i.f.273.3		6
76.75	even	2		5776.2.a.bq.1.3		3
152.11	odd	6		1216.2.i.m.577.1		6
152.45	even	6		1216.2.i.n.961.3		6
152.83	odd	6		1216.2.i.m.961.1		6
152.125	even	6		1216.2.i.n.577.3		6
228.11	even	6		2736.2.s.y.577.2		6
228.83	even	6		2736.2.s.y.1873.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.i.c.49.1	✓	6	19.7	even	3
152.2.i.c.121.1	yes	6	19.11	even	3
304.2.i.f.49.3		6	76.7	odd	6
304.2.i.f.273.3		6	76.11	odd	6
1216.2.i.m.577.1		6	152.11	odd	6
1216.2.i.m.961.1		6	152.83	odd	6
1216.2.i.n.577.3		6	152.125	even	6
1216.2.i.n.961.3		6	152.45	even	6
1368.2.s.k.505.2		6	57.26	odd	6
1368.2.s.k.577.2		6	57.11	odd	6
2736.2.s.y.577.2		6	228.11	even	6
2736.2.s.y.1873.2		6	228.83	even	6
2888.2.a.n.1.1		3	19.18	odd	2
2888.2.a.r.1.3		3	1.1	even	1	trivial
5776.2.a.bk.1.1		3	4.3	odd	2
5776.2.a.bq.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)

\(f(q)\)	\(=\)	\(q+3.24914 q^{3} +0.470683 q^{5} -3.30777 q^{7} +7.55691 q^{9} +1.47068 q^{11} -2.83709 q^{13} +1.52932 q^{15} +6.71982 q^{17} -10.7474 q^{21} +0.470683 q^{23} -4.77846 q^{25} +14.8061 q^{27} +4.96896 q^{29} +6.74742 q^{31} +4.77846 q^{33} -1.55691 q^{35} +4.74742 q^{37} -9.21811 q^{39} -0.501719 q^{41} +3.89572 q^{43} +3.55691 q^{45} +11.9103 q^{47} +3.94137 q^{49} +21.8337 q^{51} -4.71982 q^{53} +0.692226 q^{55} +7.24914 q^{59} -12.5259 q^{61} -24.9966 q^{63} -1.33537 q^{65} -0.633593 q^{67} +1.52932 q^{69} -3.77846 q^{71} -3.94137 q^{73} -15.5259 q^{75} -4.86469 q^{77} -8.83709 q^{79} +25.4362 q^{81} +9.14486 q^{83} +3.16291 q^{85} +16.1449 q^{87} +2.95436 q^{89} +9.38445 q^{91} +21.9233 q^{93} +0.0586332 q^{97} +11.1138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) \) 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

Introduction

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