LMFDB - Embedded newform 1900.4.a.l.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1900,4,Mod(1,1900)] 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("1900.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$112.103629011$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 158x^{10} + 9117x^{8} - 255920x^{6} + 3734252x^{4} - 26837136x^{2} + 73273600 $ x^12 - 158x^10 + 9117x^8 - 255920x^6 + 3734252x^4 - 26837136*x^2 + 73273600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.64108$ of defining polynomial
Character	$\chi$	$=$	1900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64108 q^{3} -2.80379 q^{7} -20.0247 q^{9} +15.7130 q^{11} -36.9826 q^{13} -26.1961 q^{17} +19.0000 q^{19} +7.40504 q^{21} +204.806 q^{23} +124.196 q^{27} -103.968 q^{29} +97.5589 q^{31} -41.4993 q^{33} +277.344 q^{37} +97.6741 q^{39} +51.2288 q^{41} -9.31058 q^{43} -15.7447 q^{47} -335.139 q^{49} +69.1860 q^{51} -496.668 q^{53} -50.1805 q^{57} -6.35314 q^{59} +339.061 q^{61} +56.1451 q^{63} +140.091 q^{67} -540.909 q^{69} +114.538 q^{71} -491.107 q^{73} -44.0560 q^{77} -534.706 q^{79} +212.655 q^{81} +755.413 q^{83} +274.589 q^{87} -315.138 q^{89} +103.692 q^{91} -257.661 q^{93} +38.7369 q^{97} -314.648 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9} - 110 q^{11} + 228 q^{19} - 8 q^{21} + 76 q^{29} + 80 q^{31} - 60 q^{39} - 648 q^{41} + 886 q^{49} - 328 q^{51} - 984 q^{59} - 1970 q^{61} - 364 q^{69} - 3060 q^{71} - 552 q^{79} - 3388 q^{81}+ \cdots - 2366 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 - 110 * q^11 + 228 * q^19 - 8 * q^21 + 76 * q^29 + 80 * q^31 - 60 * q^39 - 648 * q^41 + 886 * q^49 - 328 * q^51 - 984 * q^59 - 1970 * q^61 - 364 * q^69 - 3060 * q^71 - 552 * q^79 - 3388 * q^81 - 2428 * q^89 - 3468 * q^91 - 2366 * 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
$3$	−2.64108	−0.508276	−0.254138	−	0.967168i	$-0.581792\pi$
$3$	−2.64108	−0.508276	−0.254138	+	0.967168i	$0.581792\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.80379	−0.151391	−0.0756953	−	0.997131i	$-0.524118\pi$
$7$	−2.80379	−0.151391	−0.0756953	+	0.997131i	$0.524118\pi$
$8$	0	0
$9$	−20.0247	−0.741655
$10$	0	0
$11$	15.7130	0.430695	0.215348	−	0.976537i	$-0.430912\pi$
$11$	15.7130	0.430695	0.215348	+	0.976537i	$0.430912\pi$
$12$	0	0
$13$	−36.9826	−0.789010	−0.394505	−	0.918894i	$-0.629084\pi$
$13$	−36.9826	−0.789010	−0.394505	+	0.918894i	$0.629084\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.1961	−0.373734	−0.186867	−	0.982385i	$-0.559833\pi$
$17$	−26.1961	−0.373734	−0.186867	+	0.982385i	$0.559833\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	7.40504	0.0769482
$22$	0	0
$23$	204.806	1.85674	0.928369	−	0.371659i	$-0.121211\pi$
$23$	204.806	1.85674	0.928369	+	0.371659i	$0.121211\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	124.196	0.885242
$28$	0	0
$29$	−103.968	−0.665739	−0.332870	−	0.942973i	$-0.608017\pi$
$29$	−103.968	−0.665739	−0.332870	+	0.942973i	$0.608017\pi$
$30$	0	0
$31$	97.5589	0.565229	0.282614	−	0.959234i	$-0.408798\pi$
$31$	97.5589	0.565229	0.282614	+	0.959234i	$0.408798\pi$
$32$	0	0
$33$	−41.4993	−0.218912
$34$	0	0
$35$	0	0
$36$	0	0
$37$	277.344	1.23230	0.616149	−	0.787630i	$-0.288692\pi$
$37$	277.344	1.23230	0.616149	+	0.787630i	$0.288692\pi$
$38$	0	0
$39$	97.6741	0.401035
$40$	0	0
$41$	51.2288	0.195136	0.0975682	−	0.995229i	$-0.468894\pi$
$41$	51.2288	0.195136	0.0975682	+	0.995229i	$0.468894\pi$
$42$	0	0
$43$	−9.31058	−0.0330198	−0.0165099	−	0.999864i	$-0.505255\pi$
$43$	−9.31058	−0.0330198	−0.0165099	+	0.999864i	$0.505255\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−15.7447	−0.0488637	−0.0244318	−	0.999701i	$-0.507778\pi$
$47$	−15.7447	−0.0488637	−0.0244318	+	0.999701i	$0.507778\pi$
$48$	0	0
$49$	−335.139	−0.977081
$50$	0	0
$51$	69.1860	0.189960
$52$	0	0
$53$	−496.668	−1.28722	−0.643609	−	0.765354i	$-0.722564\pi$
$53$	−496.668	−1.28722	−0.643609	+	0.765354i	$0.722564\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−50.1805	−0.116607
$58$	0	0
$59$	−6.35314	−0.0140188	−0.00700940	−	0.999975i	$-0.502231\pi$
$59$	−6.35314	−0.0140188	−0.00700940	+	0.999975i	$0.502231\pi$
$60$	0	0
$61$	339.061	0.711676	0.355838	−	0.934548i	$-0.384195\pi$
$61$	339.061	0.711676	0.355838	+	0.934548i	$0.384195\pi$
$62$	0	0
$63$	56.1451	0.112280
$64$	0	0
$65$	0	0
$66$	0	0
$67$	140.091	0.255446	0.127723	−	0.991810i	$-0.459233\pi$
$67$	140.091	0.255446	0.127723	+	0.991810i	$0.459233\pi$
$68$	0	0
$69$	−540.909	−0.943736
$70$	0	0
$71$	114.538	0.191453	0.0957263	−	0.995408i	$-0.469483\pi$
$71$	114.538	0.191453	0.0957263	+	0.995408i	$0.469483\pi$
$72$	0	0
$73$	−491.107	−0.787394	−0.393697	−	0.919240i	$-0.628804\pi$
$73$	−491.107	−0.787394	−0.393697	+	0.919240i	$0.628804\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−44.0560	−0.0652032
$78$	0	0
$79$	−534.706	−0.761508	−0.380754	−	0.924676i	$-0.624335\pi$
$79$	−534.706	−0.761508	−0.380754	+	0.924676i	$0.624335\pi$
$80$	0	0
$81$	212.655	0.291708
$82$	0	0
$83$	755.413	0.999004	0.499502	−	0.866313i	$-0.333516\pi$
$83$	755.413	0.999004	0.499502	+	0.866313i	$0.333516\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	274.589	0.338379
$88$	0	0
$89$	−315.138	−0.375332	−0.187666	−	0.982233i	$-0.560092\pi$
$89$	−315.138	−0.375332	−0.187666	+	0.982233i	$0.560092\pi$
$90$	0	0
$91$	103.692	0.119449
$92$	0	0
$93$	−257.661	−0.287292
$94$	0	0
$95$	0	0
$96$	0	0
$97$	38.7369	0.0405478	0.0202739	−	0.999794i	$-0.493546\pi$
$97$	38.7369	0.0405478	0.0202739	+	0.999794i	$0.493546\pi$
$98$	0	0
$99$	−314.648	−0.319427
$100$	0	0
$101$	−1198.94	−1.18118	−0.590588	−	0.806973i	$-0.701104\pi$
$101$	−1198.94	−1.18118	−0.590588	+	0.806973i	$0.701104\pi$
$102$	0	0
$103$	318.071	0.304276	0.152138	−	0.988359i	$-0.451384\pi$
$103$	318.071	0.304276	0.152138	+	0.988359i	$0.451384\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	185.473	0.167574	0.0837868	−	0.996484i	$-0.473299\pi$
$107$	185.473	0.167574	0.0837868	+	0.996484i	$0.473299\pi$
$108$	0	0
$109$	4.01175	0.00352528	0.00176264	−	0.999998i	$-0.499439\pi$
$109$	4.01175	0.00352528	0.00176264	+	0.999998i	$0.499439\pi$
$110$	0	0
$111$	−732.487	−0.626348
$112$	0	0
$113$	1782.80	1.48417	0.742087	−	0.670304i	$-0.233836\pi$
$113$	1782.80	1.48417	0.742087	+	0.670304i	$0.233836\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	740.566	0.585174
$118$	0	0
$119$	73.4484	0.0565798
$120$	0	0
$121$	−1084.10	−0.814502
$122$	0	0
$123$	−135.299	−0.0991832
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−358.901	−0.250766	−0.125383	−	0.992108i	$-0.540016\pi$
$127$	−358.901	−0.250766	−0.125383	+	0.992108i	$0.540016\pi$
$128$	0	0
$129$	24.5900	0.0167832
$130$	0	0
$131$	−1888.31	−1.25941	−0.629703	−	0.776836i	$-0.716824\pi$
$131$	−1888.31	−1.25941	−0.629703	+	0.776836i	$0.716824\pi$
$132$	0	0
$133$	−53.2721	−0.0347314
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2027.41	1.26433	0.632164	−	0.774835i	$-0.282167\pi$
$137$	2027.41	1.26433	0.632164	+	0.774835i	$0.282167\pi$
$138$	0	0
$139$	1606.99	0.980596	0.490298	−	0.871555i	$-0.336888\pi$
$139$	1606.99	0.980596	0.490298	+	0.871555i	$0.336888\pi$
$140$	0	0
$141$	41.5829	0.0248363
$142$	0	0
$143$	−581.108	−0.339823
$144$	0	0
$145$	0	0
$146$	0	0
$147$	885.129	0.496627
$148$	0	0
$149$	−330.544	−0.181740	−0.0908699	−	0.995863i	$-0.528965\pi$
$149$	−330.544	−0.181740	−0.0908699	+	0.995863i	$0.528965\pi$
$150$	0	0
$151$	−1147.31	−0.618325	−0.309163	−	0.951009i	$-0.600049\pi$
$151$	−1147.31	−0.618325	−0.309163	+	0.951009i	$0.600049\pi$
$152$	0	0
$153$	524.568	0.277182
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1410.52	−0.717018	−0.358509	−	0.933526i	$-0.616715\pi$
$157$	−1410.52	−0.717018	−0.358509	+	0.933526i	$0.616715\pi$
$158$	0	0
$159$	1311.74	0.654263
$160$	0	0
$161$	−574.233	−0.281093
$162$	0	0
$163$	−2843.93	−1.36659	−0.683293	−	0.730144i	$-0.739453\pi$
$163$	−2843.93	−1.36659	−0.683293	+	0.730144i	$0.739453\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3450.67	−1.59893	−0.799464	−	0.600714i	$-0.794883\pi$
$167$	−3450.67	−1.59893	−0.799464	+	0.600714i	$0.794883\pi$
$168$	0	0
$169$	−829.285	−0.377463
$170$	0	0
$171$	−380.469	−0.170147
$172$	0	0
$173$	2769.02	1.21691	0.608454	−	0.793590i	$-0.291790\pi$
$173$	2769.02	1.21691	0.608454	+	0.793590i	$0.291790\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.7792	0.00712542
$178$	0	0
$179$	830.302	0.346702	0.173351	−	0.984860i	$-0.444540\pi$
$179$	830.302	0.346702	0.173351	+	0.984860i	$0.444540\pi$
$180$	0	0
$181$	−3497.10	−1.43612	−0.718059	−	0.695982i	$-0.754969\pi$
$181$	−3497.10	−1.43612	−0.718059	+	0.695982i	$0.754969\pi$
$182$	0	0
$183$	−895.487	−0.361728
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−411.619	−0.160966
$188$	0	0
$189$	−348.220	−0.134017
$190$	0	0
$191$	−3703.44	−1.40299	−0.701496	−	0.712674i	$-0.747484\pi$
$191$	−3703.44	−1.40299	−0.701496	+	0.712674i	$0.747484\pi$
$192$	0	0
$193$	−3419.00	−1.27516	−0.637578	−	0.770386i	$-0.720063\pi$
$193$	−3419.00	−1.27516	−0.637578	+	0.770386i	$0.720063\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1155.99	0.418076	0.209038	−	0.977908i	$-0.432967\pi$
$197$	1155.99	0.418076	0.209038	+	0.977908i	$0.432967\pi$
$198$	0	0
$199$	763.802	0.272083	0.136041	−	0.990703i	$-0.456562\pi$
$199$	763.802	0.272083	0.136041	+	0.990703i	$0.456562\pi$
$200$	0	0
$201$	−369.993	−0.129837
$202$	0	0
$203$	291.506	0.100787
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4101.18	−1.37706
$208$	0	0
$209$	298.547	0.0988083
$210$	0	0
$211$	−5515.79	−1.79963	−0.899816	−	0.436269i	$-0.856299\pi$
$211$	−5515.79	−1.79963	−0.899816	+	0.436269i	$0.856299\pi$
$212$	0	0
$213$	−302.504	−0.0973108
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−273.535	−0.0855703
$218$	0	0
$219$	1297.05	0.400214
$220$	0	0
$221$	968.800	0.294880
$222$	0	0
$223$	6391.64	1.91935	0.959677	−	0.281106i	$-0.0907013\pi$
$223$	6391.64	1.91935	0.959677	+	0.281106i	$0.0907013\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4365.76	1.27650	0.638250	−	0.769829i	$-0.279659\pi$
$227$	4365.76	1.27650	0.638250	+	0.769829i	$0.279659\pi$
$228$	0	0
$229$	1203.88	0.347401	0.173701	−	0.984799i	$-0.444427\pi$
$229$	1203.88	0.347401	0.173701	+	0.984799i	$0.444427\pi$
$230$	0	0
$231$	116.355	0.0331412
$232$	0	0
$233$	4240.55	1.19231	0.596154	−	0.802870i	$-0.296695\pi$
$233$	4240.55	1.19231	0.596154	+	0.802870i	$0.296695\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1412.20	0.387056
$238$	0	0
$239$	285.645	0.0773090	0.0386545	−	0.999253i	$-0.487693\pi$
$239$	285.645	0.0773090	0.0386545	+	0.999253i	$0.487693\pi$
$240$	0	0
$241$	−1537.73	−0.411012	−0.205506	−	0.978656i	$-0.565884\pi$
$241$	−1537.73	−0.411012	−0.205506	+	0.978656i	$0.565884\pi$
$242$	0	0
$243$	−3914.93	−1.03351
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−702.670	−0.181011
$248$	0	0
$249$	−1995.11	−0.507770
$250$	0	0
$251$	−1527.59	−0.384147	−0.192073	−	0.981381i	$-0.561521\pi$
$251$	−1527.59	−0.384147	−0.192073	+	0.981381i	$0.561521\pi$
$252$	0	0
$253$	3218.12	0.799689
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4175.82	−1.01354	−0.506772	−	0.862080i	$-0.669161\pi$
$257$	−4175.82	−1.01354	−0.506772	+	0.862080i	$0.669161\pi$
$258$	0	0
$259$	−777.614	−0.186558
$260$	0	0
$261$	2081.93	0.493749
$262$	0	0
$263$	−6977.87	−1.63602	−0.818011	−	0.575203i	$-0.804923\pi$
$263$	−6977.87	−1.63602	−0.818011	+	0.575203i	$0.804923\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	832.305	0.190772
$268$	0	0
$269$	−7020.19	−1.59118	−0.795592	−	0.605832i	$-0.792840\pi$
$269$	−7020.19	−1.59118	−0.795592	+	0.605832i	$0.792840\pi$
$270$	0	0
$271$	−5481.46	−1.22869	−0.614345	−	0.789037i	$-0.710580\pi$
$271$	−5481.46	−1.22869	−0.614345	+	0.789037i	$0.710580\pi$
$272$	0	0
$273$	−273.858	−0.0607130
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4019.52	−0.871875	−0.435937	−	0.899977i	$-0.643583\pi$
$277$	−4019.52	−0.871875	−0.435937	+	0.899977i	$0.643583\pi$
$278$	0	0
$279$	−1953.59	−0.419205
$280$	0	0
$281$	2778.19	0.589798	0.294899	−	0.955528i	$-0.404714\pi$
$281$	2778.19	0.589798	0.294899	+	0.955528i	$0.404714\pi$
$282$	0	0
$283$	3560.40	0.747857	0.373929	−	0.927457i	$-0.378011\pi$
$283$	3560.40	0.747857	0.373929	+	0.927457i	$0.378011\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−143.635	−0.0295418
$288$	0	0
$289$	−4226.77	−0.860323
$290$	0	0
$291$	−102.307	−0.0206095
$292$	0	0
$293$	−5175.66	−1.03196	−0.515981	−	0.856600i	$-0.672573\pi$
$293$	−5175.66	−1.03196	−0.515981	+	0.856600i	$0.672573\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1951.49	0.381270
$298$	0	0
$299$	−7574.26	−1.46499
$300$	0	0
$301$	26.1049	0.00499888
$302$	0	0
$303$	3166.49	0.600364
$304$	0	0
$305$	0	0
$306$	0	0
$307$	243.963	0.0453541	0.0226771	−	0.999743i	$-0.492781\pi$
$307$	243.963	0.0453541	0.0226771	+	0.999743i	$0.492781\pi$
$308$	0	0
$309$	−840.050	−0.154656
$310$	0	0
$311$	3072.67	0.560242	0.280121	−	0.959965i	$-0.409625\pi$
$311$	3072.67	0.560242	0.280121	+	0.959965i	$0.409625\pi$
$312$	0	0
$313$	−6417.23	−1.15886	−0.579430	−	0.815022i	$-0.696725\pi$
$313$	−6417.23	−1.15886	−0.579430	+	0.815022i	$0.696725\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2486.50	0.440554	0.220277	−	0.975437i	$-0.429304\pi$
$317$	2486.50	0.440554	0.220277	+	0.975437i	$0.429304\pi$
$318$	0	0
$319$	−1633.65	−0.286731
$320$	0	0
$321$	−489.850	−0.0851737
$322$	0	0
$323$	−497.725	−0.0857405
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.5954	−0.00179182
$328$	0	0
$329$	44.1447	0.00739750
$330$	0	0
$331$	−10978.2	−1.82302	−0.911509	−	0.411281i	$-0.865081\pi$
$331$	−10978.2	−1.82302	−0.911509	+	0.411281i	$0.865081\pi$
$332$	0	0
$333$	−5553.72	−0.913940
$334$	0	0
$335$	0	0
$336$	0	0
$337$	822.597	0.132966	0.0664832	−	0.997788i	$-0.478822\pi$
$337$	822.597	0.132966	0.0664832	+	0.997788i	$0.478822\pi$
$338$	0	0
$339$	−4708.52	−0.754370
$340$	0	0
$341$	1532.94	0.243441
$342$	0	0
$343$	1901.36	0.299311
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6067.83	−0.938727	−0.469363	−	0.883005i	$-0.655517\pi$
$347$	−6067.83	−0.938727	−0.469363	+	0.883005i	$0.655517\pi$
$348$	0	0
$349$	1517.92	0.232815	0.116407	−	0.993202i	$-0.462862\pi$
$349$	1517.92	0.232815	0.116407	+	0.993202i	$0.462862\pi$
$350$	0	0
$351$	−4593.10	−0.698465
$352$	0	0
$353$	−11.9177	−0.00179693	−0.000898467	−	1.00000i	$-0.500286\pi$
$353$	−11.9177	−0.00179693	−0.000898467		1.00000i	$0.500286\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−193.983	−0.0287582
$358$	0	0
$359$	−2994.81	−0.440278	−0.220139	−	0.975468i	$-0.570651\pi$
$359$	−2994.81	−0.440278	−0.220139	+	0.975468i	$0.570651\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	2863.20	0.413992
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1119.41	−0.159217	−0.0796087	−	0.996826i	$-0.525367\pi$
$367$	−1119.41	−0.159217	−0.0796087	+	0.996826i	$0.525367\pi$
$368$	0	0
$369$	−1025.84	−0.144724
$370$	0	0
$371$	1392.55	0.194873
$372$	0	0
$373$	9031.40	1.25370	0.626848	−	0.779142i	$-0.284345\pi$
$373$	9031.40	1.25370	0.626848	+	0.779142i	$0.284345\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3845.02	0.525275
$378$	0	0
$379$	448.079	0.0607290	0.0303645	−	0.999539i	$-0.490333\pi$
$379$	448.079	0.0607290	0.0303645	+	0.999539i	$0.490333\pi$
$380$	0	0
$381$	947.887	0.127459
$382$	0	0
$383$	−10925.1	−1.45757	−0.728783	−	0.684745i	$-0.759914\pi$
$383$	−10925.1	−1.45757	−0.728783	+	0.684745i	$0.759914\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	186.442	0.0244893
$388$	0	0
$389$	−6071.65	−0.791375	−0.395688	−	0.918385i	$-0.629494\pi$
$389$	−6071.65	−0.791375	−0.395688	+	0.918385i	$0.629494\pi$
$390$	0	0
$391$	−5365.11	−0.693927
$392$	0	0
$393$	4987.18	0.640127
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8500.97	1.07469	0.537345	−	0.843363i	$-0.319427\pi$
$397$	8500.97	1.07469	0.537345	+	0.843363i	$0.319427\pi$
$398$	0	0
$399$	140.696	0.0176531
$400$	0	0
$401$	−5471.73	−0.681410	−0.340705	−	0.940170i	$-0.610666\pi$
$401$	−5471.73	−0.681410	−0.340705	+	0.940170i	$0.610666\pi$
$402$	0	0
$403$	−3607.98	−0.445971
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4357.90	0.530745
$408$	0	0
$409$	10339.8	1.25005	0.625024	−	0.780605i	$-0.285089\pi$
$409$	10339.8	1.25005	0.625024	+	0.780605i	$0.285089\pi$
$410$	0	0
$411$	−5354.54	−0.642628
$412$	0	0
$413$	17.8129	0.00212231
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4244.18	−0.498414
$418$	0	0
$419$	−8994.38	−1.04870	−0.524349	−	0.851504i	$-0.675691\pi$
$419$	−8994.38	−1.04870	−0.524349	+	0.851504i	$0.675691\pi$
$420$	0	0
$421$	−782.094	−0.0905390	−0.0452695	−	0.998975i	$-0.514415\pi$
$421$	−782.094	−0.0905390	−0.0452695	+	0.998975i	$0.514415\pi$
$422$	0	0
$423$	315.282	0.0362400
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−950.656	−0.107741
$428$	0	0
$429$	1534.75	0.172724
$430$	0	0
$431$	7304.49	0.816345	0.408173	−	0.912905i	$-0.366166\pi$
$431$	7304.49	0.816345	0.408173	+	0.912905i	$0.366166\pi$
$432$	0	0
$433$	3763.24	0.417667	0.208833	−	0.977951i	$-0.433033\pi$
$433$	3763.24	0.417667	0.208833	+	0.977951i	$0.433033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3891.31	0.425965
$438$	0	0
$439$	−1.31513	−0.000142979 0	−7.14894e−5	−	1.00000i	$-0.500023\pi$
$439$	−1.31513	−0.000142979 0	−7.14894e−5		1.00000i	$0.500023\pi$
$440$	0	0
$441$	6711.05	0.724657
$442$	0	0
$443$	−8098.13	−0.868518	−0.434259	−	0.900788i	$-0.642990\pi$
$443$	−8098.13	−0.868518	−0.434259	+	0.900788i	$0.642990\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	872.994	0.0923741
$448$	0	0
$449$	15697.2	1.64988	0.824939	−	0.565222i	$-0.191209\pi$
$449$	15697.2	1.64988	0.824939	+	0.565222i	$0.191209\pi$
$450$	0	0
$451$	804.958	0.0840443
$452$	0	0
$453$	3030.15	0.314280
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17279.6	−1.76872	−0.884362	−	0.466802i	$-0.845406\pi$
$457$	−17279.6	−1.76872	−0.884362	+	0.466802i	$0.845406\pi$
$458$	0	0
$459$	−3253.45	−0.330845
$460$	0	0
$461$	−4204.09	−0.424738	−0.212369	−	0.977190i	$-0.568118\pi$
$461$	−4204.09	−0.424738	−0.212369	+	0.977190i	$0.568118\pi$
$462$	0	0
$463$	−6582.02	−0.660675	−0.330337	−	0.943863i	$-0.607163\pi$
$463$	−6582.02	−0.660675	−0.330337	+	0.943863i	$0.607163\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5162.51	−0.511547	−0.255774	−	0.966737i	$-0.582330\pi$
$467$	−5162.51	−0.511547	−0.255774	+	0.966737i	$0.582330\pi$
$468$	0	0
$469$	−392.787	−0.0386721
$470$	0	0
$471$	3725.30	0.364443
$472$	0	0
$473$	−146.297	−0.0142215
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9945.62	0.954672
$478$	0	0
$479$	−2068.94	−0.197354	−0.0986768	−	0.995120i	$-0.531461\pi$
$479$	−2068.94	−0.197354	−0.0986768	+	0.995120i	$0.531461\pi$
$480$	0	0
$481$	−10256.9	−0.972296
$482$	0	0
$483$	1516.60	0.142873
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7366.54	−0.685441	−0.342720	−	0.939437i	$-0.611348\pi$
$487$	−7366.54	−0.685441	−0.342720	+	0.939437i	$0.611348\pi$
$488$	0	0
$489$	7511.04	0.694603
$490$	0	0
$491$	−4604.18	−0.423185	−0.211592	−	0.977358i	$-0.567865\pi$
$491$	−4604.18	−0.423185	−0.211592	+	0.977358i	$0.567865\pi$
$492$	0	0
$493$	2723.56	0.248809
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−321.140	−0.0289841
$498$	0	0
$499$	10912.7	0.978996	0.489498	−	0.872004i	$-0.337180\pi$
$499$	10912.7	0.978996	0.489498	+	0.872004i	$0.337180\pi$
$500$	0	0
$501$	9113.50	0.812697
$502$	0	0
$503$	16722.4	1.48233	0.741167	−	0.671320i	$-0.234272\pi$
$503$	16722.4	1.48233	0.741167	+	0.671320i	$0.234272\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2190.21	0.191855
$508$	0	0
$509$	−12980.5	−1.13035	−0.565175	−	0.824971i	$-0.691191\pi$
$509$	−12980.5	−1.13035	−0.565175	+	0.824971i	$0.691191\pi$
$510$	0	0
$511$	1376.96	0.119204
$512$	0	0
$513$	2359.72	0.203088
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−247.396	−0.0210454
$518$	0	0
$519$	−7313.21	−0.618525
$520$	0	0
$521$	−13033.9	−1.09602	−0.548010	−	0.836472i	$-0.684614\pi$
$521$	−13033.9	−1.09602	−0.548010	+	0.836472i	$0.684614\pi$
$522$	0	0
$523$	−134.248	−0.0112242	−0.00561208	−	0.999984i	$-0.501786\pi$
$523$	−134.248	−0.0112242	−0.00561208	+	0.999984i	$0.501786\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2555.66	−0.211245
$528$	0	0
$529$	29778.5	2.44748
$530$	0	0
$531$	127.220	0.0103971
$532$	0	0
$533$	−1894.57	−0.153965
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2192.90	−0.176221
$538$	0	0
$539$	−5266.03	−0.420824
$540$	0	0
$541$	−7526.09	−0.598100	−0.299050	−	0.954238i	$-0.596670\pi$
$541$	−7526.09	−0.598100	−0.299050	+	0.954238i	$0.596670\pi$
$542$	0	0
$543$	9236.13	0.729945
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21280.4	1.66341	0.831704	−	0.555220i	$-0.187366\pi$
$547$	21280.4	1.66341	0.831704	+	0.555220i	$0.187366\pi$
$548$	0	0
$549$	−6789.58	−0.527818
$550$	0	0
$551$	−1975.40	−0.152731
$552$	0	0
$553$	1499.20	0.115285
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8335.03	0.634051	0.317026	−	0.948417i	$-0.397316\pi$
$557$	8335.03	0.634051	0.317026	+	0.948417i	$0.397316\pi$
$558$	0	0
$559$	344.330	0.0260529
$560$	0	0
$561$	1087.12	0.0818150
$562$	0	0
$563$	6696.25	0.501267	0.250633	−	0.968082i	$-0.419361\pi$
$563$	6696.25	0.501267	0.250633	+	0.968082i	$0.419361\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−596.240	−0.0441618
$568$	0	0
$569$	22543.7	1.66095	0.830476	−	0.557055i	$-0.188069\pi$
$569$	22543.7	1.66095	0.830476	+	0.557055i	$0.188069\pi$
$570$	0	0
$571$	−4166.28	−0.305347	−0.152674	−	0.988277i	$-0.548788\pi$
$571$	−4166.28	−0.305347	−0.152674	+	0.988277i	$0.548788\pi$
$572$	0	0
$573$	9781.08	0.713107
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5629.47	−0.406166	−0.203083	−	0.979162i	$-0.565096\pi$
$577$	−5629.47	−0.406166	−0.203083	+	0.979162i	$0.565096\pi$
$578$	0	0
$579$	9029.86	0.648131
$580$	0	0
$581$	−2118.02	−0.151240
$582$	0	0
$583$	−7804.14	−0.554399
$584$	0	0
$585$	0	0
$586$	0	0
$587$	69.9606	0.00491922	0.00245961	−	0.999997i	$-0.499217\pi$
$587$	69.9606	0.00491922	0.00245961	+	0.999997i	$0.499217\pi$
$588$	0	0
$589$	1853.62	0.129672
$590$	0	0
$591$	−3053.06	−0.212498
$592$	0	0
$593$	2172.73	0.150461	0.0752306	−	0.997166i	$-0.476031\pi$
$593$	2172.73	0.150461	0.0752306	+	0.997166i	$0.476031\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2017.26	−0.138293
$598$	0	0
$599$	−9736.23	−0.664126	−0.332063	−	0.943257i	$-0.607745\pi$
$599$	−9736.23	−0.664126	−0.332063	+	0.943257i	$0.607745\pi$
$600$	0	0
$601$	−25060.2	−1.70088	−0.850439	−	0.526074i	$-0.823664\pi$
$601$	−25060.2	−1.70088	−0.850439	+	0.526074i	$0.823664\pi$
$602$	0	0
$603$	−2805.29	−0.189453
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14464.8	0.967230	0.483615	−	0.875281i	$-0.339324\pi$
$607$	14464.8	0.967230	0.483615	+	0.875281i	$0.339324\pi$
$608$	0	0
$609$	−769.890	−0.0512275
$610$	0	0
$611$	582.279	0.0385540
$612$	0	0
$613$	−24069.6	−1.58591	−0.792956	−	0.609279i	$-0.791459\pi$
$613$	−24069.6	−1.58591	−0.792956	+	0.609279i	$0.791459\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5258.43	−0.343106	−0.171553	−	0.985175i	$-0.554878\pi$
$617$	−5258.43	−0.343106	−0.171553	+	0.985175i	$0.554878\pi$
$618$	0	0
$619$	12110.3	0.786353	0.393177	−	0.919463i	$-0.371376\pi$
$619$	12110.3	0.786353	0.393177	+	0.919463i	$0.371376\pi$
$620$	0	0
$621$	25436.1	1.64366
$622$	0	0
$623$	883.582	0.0568217
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−788.487	−0.0502219
$628$	0	0
$629$	−7265.32	−0.460552
$630$	0	0
$631$	13737.8	0.866710	0.433355	−	0.901223i	$-0.357330\pi$
$631$	13737.8	0.866710	0.433355	+	0.901223i	$0.357330\pi$
$632$	0	0
$633$	14567.6	0.914710
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12394.3	0.770927
$638$	0	0
$639$	−2293.58	−0.141992
$640$	0	0
$641$	−3604.21	−0.222087	−0.111043	−	0.993816i	$-0.535419\pi$
$641$	−3604.21	−0.222087	−0.111043	+	0.993816i	$0.535419\pi$
$642$	0	0
$643$	15515.2	0.951567	0.475784	−	0.879562i	$-0.342164\pi$
$643$	15515.2	0.951567	0.475784	+	0.879562i	$0.342164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26283.6	1.59709	0.798544	−	0.601936i	$-0.205604\pi$
$647$	26283.6	1.59709	0.798544	+	0.601936i	$0.205604\pi$
$648$	0	0
$649$	−99.8269	−0.00603783
$650$	0	0
$651$	722.428	0.0434934
$652$	0	0
$653$	−16344.2	−0.979477	−0.489739	−	0.871869i	$-0.662908\pi$
$653$	−16344.2	−0.979477	−0.489739	+	0.871869i	$0.662908\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9834.27	0.583975
$658$	0	0
$659$	−21530.5	−1.27270	−0.636349	−	0.771401i	$-0.719556\pi$
$659$	−21530.5	−1.27270	−0.636349	+	0.771401i	$0.719556\pi$
$660$	0	0
$661$	21799.0	1.28273	0.641364	−	0.767237i	$-0.278369\pi$
$661$	21799.0	1.28273	0.641364	+	0.767237i	$0.278369\pi$
$662$	0	0
$663$	−2558.68	−0.149881
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−21293.3	−1.23610
$668$	0	0
$669$	−16880.8	−0.975562
$670$	0	0
$671$	5327.66	0.306516
$672$	0	0
$673$	−16189.5	−0.927280	−0.463640	−	0.886024i	$-0.653457\pi$
$673$	−16189.5	−0.927280	−0.463640	+	0.886024i	$0.653457\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14467.0	−0.821289	−0.410644	−	0.911796i	$-0.634696\pi$
$677$	−14467.0	−0.821289	−0.410644	+	0.911796i	$0.634696\pi$
$678$	0	0
$679$	−108.610	−0.00613855
$680$	0	0
$681$	−11530.3	−0.648815
$682$	0	0
$683$	−17336.3	−0.971240	−0.485620	−	0.874170i	$-0.661406\pi$
$683$	−17336.3	−0.971240	−0.485620	+	0.874170i	$0.661406\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3179.56	−0.176576
$688$	0	0
$689$	18368.1	1.01563
$690$	0	0
$691$	22585.3	1.24339	0.621696	−	0.783258i	$-0.286444\pi$
$691$	22585.3	1.24339	0.621696	+	0.783258i	$0.286444\pi$
$692$	0	0
$693$	882.208	0.0483583
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1341.99	−0.0729291
$698$	0	0
$699$	−11199.6	−0.606022
$700$	0	0
$701$	19433.5	1.04707	0.523533	−	0.852006i	$-0.324614\pi$
$701$	19433.5	1.04707	0.523533	+	0.852006i	$0.324614\pi$
$702$	0	0
$703$	5269.53	0.282709
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3361.57	0.178819
$708$	0	0
$709$	11765.3	0.623209	0.311604	−	0.950212i	$-0.399134\pi$
$709$	11765.3	0.623209	0.311604	+	0.950212i	$0.399134\pi$
$710$	0	0
$711$	10707.3	0.564776
$712$	0	0
$713$	19980.6	1.04948
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−754.412	−0.0392943
$718$	0	0
$719$	801.432	0.0415693	0.0207847	−	0.999784i	$-0.493384\pi$
$719$	801.432	0.0415693	0.0207847	+	0.999784i	$0.493384\pi$
$720$	0	0
$721$	−891.804	−0.0460645
$722$	0	0
$723$	4061.26	0.208907
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16487.4	−0.841106	−0.420553	−	0.907268i	$-0.638164\pi$
$727$	−16487.4	−0.841106	−0.420553	+	0.907268i	$0.638164\pi$
$728$	0	0
$729$	4597.97	0.233601
$730$	0	0
$731$	243.901	0.0123406
$732$	0	0
$733$	32237.6	1.62445	0.812227	−	0.583342i	$-0.198255\pi$
$733$	32237.6	1.62445	0.812227	+	0.583342i	$0.198255\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2201.25	0.110019
$738$	0	0
$739$	−20124.5	−1.00175	−0.500873	−	0.865521i	$-0.666988\pi$
$739$	−20124.5	−1.00175	−0.500873	+	0.865521i	$0.666988\pi$
$740$	0	0
$741$	1855.81	0.0920038
$742$	0	0
$743$	−10623.3	−0.524536	−0.262268	−	0.964995i	$-0.584470\pi$
$743$	−10623.3	−0.524536	−0.262268	+	0.964995i	$0.584470\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−15126.9	−0.740916
$748$	0	0
$749$	−520.029	−0.0253691
$750$	0	0
$751$	5960.18	0.289600	0.144800	−	0.989461i	$-0.453746\pi$
$751$	5960.18	0.289600	0.144800	+	0.989461i	$0.453746\pi$
$752$	0	0
$753$	4034.50	0.195253
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17526.5	−0.841495	−0.420747	−	0.907178i	$-0.638232\pi$
$757$	−17526.5	−0.841495	−0.420747	+	0.907178i	$0.638232\pi$
$758$	0	0
$759$	−8499.31	−0.406463
$760$	0	0
$761$	25844.4	1.23109	0.615545	−	0.788102i	$-0.288936\pi$
$761$	25844.4	1.23109	0.615545	+	0.788102i	$0.288936\pi$
$762$	0	0
$763$	−11.2481	−0.000533695 0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	234.956	0.0110610
$768$	0	0
$769$	28361.0	1.32994	0.664970	−	0.746870i	$-0.268444\pi$
$769$	28361.0	1.32994	0.664970	+	0.746870i	$0.268444\pi$
$770$	0	0
$771$	11028.7	0.515160
$772$	0	0
$773$	−21479.4	−0.999430	−0.499715	−	0.866190i	$-0.666562\pi$
$773$	−21479.4	−0.999430	−0.499715	+	0.866190i	$0.666562\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2053.74	0.0948232
$778$	0	0
$779$	973.347	0.0447674
$780$	0	0
$781$	1799.73	0.0824577
$782$	0	0
$783$	−12912.5	−0.589340
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4347.40	−0.196910	−0.0984551	−	0.995141i	$-0.531390\pi$
$787$	−4347.40	−0.196910	−0.0984551	+	0.995141i	$0.531390\pi$
$788$	0	0
$789$	18429.1	0.831551
$790$	0	0
$791$	−4998.60	−0.224690
$792$	0	0
$793$	−12539.4	−0.561520
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31172.1	−1.38541	−0.692706	−	0.721220i	$-0.743582\pi$
$797$	−31172.1	−1.38541	−0.692706	+	0.721220i	$0.743582\pi$
$798$	0	0
$799$	412.448	0.0182620
$800$	0	0
$801$	6310.54	0.278367
$802$	0	0
$803$	−7716.77	−0.339127
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18540.9	0.808761
$808$	0	0
$809$	19641.6	0.853601	0.426800	−	0.904346i	$-0.359641\pi$
$809$	19641.6	0.853601	0.426800	+	0.904346i	$0.359641\pi$
$810$	0	0
$811$	42726.3	1.84996	0.924982	−	0.380010i	$-0.124079\pi$
$811$	42726.3	1.84996	0.924982	+	0.380010i	$0.124079\pi$
$812$	0	0
$813$	14477.0	0.624514
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−176.901	−0.00757526
$818$	0	0
$819$	−2076.39	−0.0885898
$820$	0	0
$821$	40929.7	1.73990	0.869950	−	0.493140i	$-0.164151\pi$
$821$	40929.7	1.73990	0.869950	+	0.493140i	$0.164151\pi$
$822$	0	0
$823$	−32450.5	−1.37443	−0.687213	−	0.726456i	$-0.741166\pi$
$823$	−32450.5	−1.37443	−0.687213	+	0.726456i	$0.741166\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10197.9	0.428798	0.214399	−	0.976746i	$-0.431221\pi$
$827$	10197.9	0.428798	0.214399	+	0.976746i	$0.431221\pi$
$828$	0	0
$829$	32760.9	1.37254	0.686268	−	0.727349i	$-0.259248\pi$
$829$	32760.9	1.37254	0.686268	+	0.727349i	$0.259248\pi$
$830$	0	0
$831$	10615.9	0.443153
$832$	0	0
$833$	8779.32	0.365169
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12116.4	0.500364
$838$	0	0
$839$	−42309.2	−1.74097	−0.870485	−	0.492194i	$-0.836195\pi$
$839$	−42309.2	−1.74097	−0.870485	+	0.492194i	$0.836195\pi$
$840$	0	0
$841$	−13579.6	−0.556791
$842$	0	0
$843$	−7337.43	−0.299780
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3039.60	0.123308
$848$	0	0
$849$	−9403.30	−0.380118
$850$	0	0
$851$	56801.7	2.28806
$852$	0	0
$853$	−13240.6	−0.531475	−0.265737	−	0.964045i	$-0.585615\pi$
$853$	−13240.6	−0.531475	−0.265737	+	0.964045i	$0.585615\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14609.4	0.582319	0.291160	−	0.956674i	$-0.405959\pi$
$857$	14609.4	0.582319	0.291160	+	0.956674i	$0.405959\pi$
$858$	0	0
$859$	−25437.6	−1.01039	−0.505193	−	0.863007i	$-0.668579\pi$
$859$	−25437.6	−1.01039	−0.505193	+	0.863007i	$0.668579\pi$
$860$	0	0
$861$	379.351	0.0150154
$862$	0	0
$863$	25010.3	0.986513	0.493256	−	0.869884i	$-0.335806\pi$
$863$	25010.3	0.986513	0.493256	+	0.869884i	$0.335806\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11163.2	0.437282
$868$	0	0
$869$	−8401.83	−0.327978
$870$	0	0
$871$	−5180.95	−0.201550
$872$	0	0
$873$	−775.694	−0.0300725
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17530.7	0.674994	0.337497	−	0.941327i	$-0.390420\pi$
$877$	17530.7	0.674994	0.337497	+	0.941327i	$0.390420\pi$
$878$	0	0
$879$	13669.3	0.524522
$880$	0	0
$881$	18030.5	0.689516	0.344758	−	0.938692i	$-0.387961\pi$
$881$	18030.5	0.689516	0.344758	+	0.938692i	$0.387961\pi$
$882$	0	0
$883$	1786.25	0.0680769	0.0340385	−	0.999421i	$-0.489163\pi$
$883$	1786.25	0.0680769	0.0340385	+	0.999421i	$0.489163\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14135.9	−0.535104	−0.267552	−	0.963543i	$-0.586215\pi$
$887$	−14135.9	−0.535104	−0.267552	+	0.963543i	$0.586215\pi$
$888$	0	0
$889$	1006.28	0.0379637
$890$	0	0
$891$	3341.44	0.125637
$892$	0	0
$893$	−299.148	−0.0112101
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20004.2	0.744618
$898$	0	0
$899$	−10143.0	−0.376295
$900$	0	0
$901$	13010.8	0.481078
$902$	0	0
$903$	−68.9453	−0.00254081
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1816.18	−0.0664887	−0.0332444	−	0.999447i	$-0.510584\pi$
$907$	−1816.18	−0.0664887	−0.0332444	+	0.999447i	$0.510584\pi$
$908$	0	0
$909$	24008.4	0.876025
$910$	0	0
$911$	27773.9	1.01009	0.505043	−	0.863094i	$-0.331477\pi$
$911$	27773.9	1.01009	0.505043	+	0.863094i	$0.331477\pi$
$912$	0	0
$913$	11869.8	0.430266
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5294.43	0.190662
$918$	0	0
$919$	−14210.2	−0.510068	−0.255034	−	0.966932i	$-0.582087\pi$
$919$	−14210.2	−0.510068	−0.255034	+	0.966932i	$0.582087\pi$
$920$	0	0
$921$	−644.327	−0.0230524
$922$	0	0
$923$	−4235.91	−0.151058
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6369.27	−0.225668
$928$	0	0
$929$	−11420.1	−0.403316	−0.201658	−	0.979456i	$-0.564633\pi$
$929$	−11420.1	−0.403316	−0.201658	+	0.979456i	$0.564633\pi$
$930$	0	0
$931$	−6367.64	−0.224158
$932$	0	0
$933$	−8115.18	−0.284758
$934$	0	0
$935$	0	0
$936$	0	0
$937$	23134.0	0.806570	0.403285	−	0.915074i	$-0.367868\pi$
$937$	23134.0	0.806570	0.403285	+	0.915074i	$0.367868\pi$
$938$	0	0
$939$	16948.4	0.589021
$940$	0	0
$941$	−11078.8	−0.383803	−0.191901	−	0.981414i	$-0.561465\pi$
$941$	−11078.8	−0.383803	−0.191901	+	0.981414i	$0.561465\pi$
$942$	0	0
$943$	10492.0	0.362317
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6730.30	−0.230946	−0.115473	−	0.993311i	$-0.536838\pi$
$947$	−6730.30	−0.230946	−0.115473	+	0.993311i	$0.536838\pi$
$948$	0	0
$949$	18162.4	0.621262
$950$	0	0
$951$	−6567.05	−0.223923
$952$	0	0
$953$	−21372.4	−0.726464	−0.363232	−	0.931699i	$-0.618327\pi$
$953$	−21372.4	−0.726464	−0.363232	+	0.931699i	$0.618327\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4314.61	0.145738
$958$	0	0
$959$	−5684.43	−0.191407
$960$	0	0
$961$	−20273.3	−0.680516
$962$	0	0
$963$	−3714.05	−0.124282
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1549.89	0.0515418	0.0257709	−	0.999668i	$-0.491796\pi$
$967$	1549.89	0.0515418	0.0257709	+	0.999668i	$0.491796\pi$
$968$	0	0
$969$	1314.53	0.0435799
$970$	0	0
$971$	−33222.8	−1.09801	−0.549006	−	0.835819i	$-0.684993\pi$
$971$	−33222.8	−1.09801	−0.549006	+	0.835819i	$0.684993\pi$
$972$	0	0
$973$	−4505.66	−0.148453
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53383.0	−1.74808	−0.874040	−	0.485854i	$-0.838509\pi$
$977$	−53383.0	−1.74808	−0.874040	+	0.485854i	$0.838509\pi$
$978$	0	0
$979$	−4951.76	−0.161654
$980$	0	0
$981$	−80.3340	−0.00261455
$982$	0	0
$983$	−3934.88	−0.127674	−0.0638368	−	0.997960i	$-0.520334\pi$
$983$	−3934.88	−0.127674	−0.0638368	+	0.997960i	$0.520334\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−116.590	−0.00375998
$988$	0	0
$989$	−1906.86	−0.0613091
$990$	0	0
$991$	40122.9	1.28612	0.643060	−	0.765816i	$-0.277665\pi$
$991$	40122.9	1.28612	0.643060	+	0.765816i	$0.277665\pi$
$992$	0	0
$993$	28994.4	0.926596
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−55361.4	−1.75859	−0.879295	−	0.476278i	$-0.841986\pi$
$997$	−55361.4	−1.75859	−0.879295	+	0.476278i	$0.841986\pi$
$998$	0	0
$999$	34445.0	1.09088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.a.l.1.6		12
5.2	odd	4		380.4.c.a.229.7	yes	12
5.3	odd	4		380.4.c.a.229.6	✓	12
5.4	even	2	inner	1900.4.a.l.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.4.c.a.229.6	✓	12	5.3	odd	4
380.4.c.a.229.7	yes	12	5.2	odd	4
1900.4.a.l.1.6		12	1.1	even	1	trivial
1900.4.a.l.1.7		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64108 q^{3} -2.80379 q^{7} -20.0247 q^{9} +15.7130 q^{11} -36.9826 q^{13} -26.1961 q^{17} +19.0000 q^{19} +7.40504 q^{21} +204.806 q^{23} +124.196 q^{27} -103.968 q^{29} +97.5589 q^{31} -41.4993 q^{33} +277.344 q^{37} +97.6741 q^{39} +51.2288 q^{41} -9.31058 q^{43} -15.7447 q^{47} -335.139 q^{49} +69.1860 q^{51} -496.668 q^{53} -50.1805 q^{57} -6.35314 q^{59} +339.061 q^{61} +56.1451 q^{63} +140.091 q^{67} -540.909 q^{69} +114.538 q^{71} -491.107 q^{73} -44.0560 q^{77} -534.706 q^{79} +212.655 q^{81} +755.413 q^{83} +274.589 q^{87} -315.138 q^{89} +103.692 q^{91} -257.661 q^{93} +38.7369 q^{97} -314.648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9} - 110 q^{11} + 228 q^{19} - 8 q^{21} + 76 q^{29} + 80 q^{31} - 60 q^{39} - 648 q^{41} + 886 q^{49} - 328 q^{51} - 984 q^{59} - 1970 q^{61} - 364 q^{69} - 3060 q^{71} - 552 q^{79} - 3388 q^{81}+ \cdots - 2366 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 - 110 * q^11 + 228 * q^19 - 8 * q^21 + 76 * q^29 + 80 * q^31 - 60 * q^39 - 648 * q^41 + 886 * q^49 - 328 * q^51 - 984 * q^59 - 1970 * q^61 - 364 * q^69 - 3060 * q^71 - 552 * q^79 - 3388 * q^81 - 2428 * q^89 - 3468 * q^91 - 2366 * q^99

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