LMFDB - Embedded newform 1872.4.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$110.451575531$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.561553 q^{5} -18.1771 q^{7} +O(q^{10})$	$q-0.561553 q^{5} -18.1771 q^{7} +64.7386 q^{11} -13.0000 q^{13} +25.5464 q^{17} +107.970 q^{19} +73.2614 q^{23} -124.685 q^{25} -175.909 q^{29} +113.093 q^{31} +10.2074 q^{35} +114.808 q^{37} +69.6458 q^{41} -438.302 q^{43} -31.9479 q^{47} -12.5937 q^{49} -2.84658 q^{53} -36.3542 q^{55} +71.6325 q^{59} -920.695 q^{61} +7.30019 q^{65} +444.280 q^{67} -541.719 q^{71} +764.004 q^{73} -1176.76 q^{77} +421.538 q^{79} +603.797 q^{83} -14.3457 q^{85} +1159.88 q^{89} +236.302 q^{91} -60.6307 q^{95} +583.269 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} + 9 q^{7}+O(q^{10}) $ 2 * q + 3 * q^5 + 9 * q^7	$ 2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 + 9 * q^7 + 80 * q^11 - 26 * q^13 - 19 * q^17 + 84 * q^19 + 196 * q^23 - 237 * q^25 + 44 * q^29 + 86 * q^31 + 107 * q^35 + 209 * q^37 + 230 * q^41 - 287 * q^43 + 435 * q^47 + 383 * q^49 + 118 * q^53 + 18 * q^55 - 368 * q^59 - 1058 * q^61 - 39 * q^65 - 68 * q^67 - 131 * q^71 + 456 * q^73 - 762 * q^77 + 1008 * q^79 + 1958 * q^83 - 173 * q^85 + 720 * q^89 - 117 * q^91 - 146 * q^95 - 928 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.561553	−0.0502268	−0.0251134	−	0.999685i	$-0.507995\pi$
$5$	−0.561553	−0.0502268	−0.0251134	+	0.999685i	$0.507995\pi$
$6$	0	0
$7$	−18.1771	−0.981470	−0.490735	−	0.871309i	$-0.663272\pi$
$7$	−18.1771	−0.981470	−0.490735	+	0.871309i	$0.663272\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	64.7386	1.77449	0.887247	−	0.461295i	$-0.152615\pi$
$11$	64.7386	1.77449	0.887247	+	0.461295i	$0.152615\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25.5464	0.364465	0.182233	−	0.983255i	$-0.441668\pi$
$17$	25.5464	0.364465	0.182233	+	0.983255i	$0.441668\pi$
$18$	0	0
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	73.2614	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$23$	73.2614	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$24$	0	0
$25$	−124.685	−0.997477
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−175.909	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$29$	−175.909	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$30$	0	0
$31$	113.093	0.655228	0.327614	−	0.944812i	$-0.393755\pi$
$31$	113.093	0.655228	0.327614	+	0.944812i	$0.393755\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.2074	0.0492961
$36$	0	0
$37$	114.808	0.510116	0.255058	−	0.966926i	$-0.417905\pi$
$37$	114.808	0.510116	0.255058	+	0.966926i	$0.417905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	69.6458	0.265289	0.132645	−	0.991164i	$-0.457653\pi$
$41$	69.6458	0.265289	0.132645	+	0.991164i	$0.457653\pi$
$42$	0	0
$43$	−438.302	−1.55443	−0.777214	−	0.629236i	$-0.783368\pi$
$43$	−438.302	−1.55443	−0.777214	+	0.629236i	$0.783368\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−31.9479	−0.0991506	−0.0495753	−	0.998770i	$-0.515787\pi$
$47$	−31.9479	−0.0991506	−0.0495753	+	0.998770i	$0.515787\pi$
$48$	0	0
$49$	−12.5937	−0.0367164
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.84658	−0.00737752	−0.00368876	−	0.999993i	$-0.501174\pi$
$53$	−2.84658	−0.00737752	−0.00368876	+	0.999993i	$0.501174\pi$
$54$	0	0
$55$	−36.3542	−0.0891272
$56$	0	0
$57$	0	0
$58$	0	0
$59$	71.6325	0.158064	0.0790319	−	0.996872i	$-0.474817\pi$
$59$	71.6325	0.158064	0.0790319	+	0.996872i	$0.474817\pi$
$60$	0	0
$61$	−920.695	−1.93251	−0.966253	−	0.257593i	$-0.917071\pi$
$61$	−920.695	−1.93251	−0.966253	+	0.257593i	$0.917071\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.30019	0.0139304
$66$	0	0
$67$	444.280	0.810112	0.405056	−	0.914292i	$-0.367252\pi$
$67$	444.280	0.810112	0.405056	+	0.914292i	$0.367252\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−541.719	−0.905496	−0.452748	−	0.891639i	$-0.649556\pi$
$71$	−541.719	−0.905496	−0.452748	+	0.891639i	$0.649556\pi$
$72$	0	0
$73$	764.004	1.22493	0.612465	−	0.790498i	$-0.290178\pi$
$73$	764.004	1.22493	0.612465	+	0.790498i	$0.290178\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1176.76	−1.74161
$78$	0	0
$79$	421.538	0.600338	0.300169	−	0.953886i	$-0.402957\pi$
$79$	421.538	0.600338	0.300169	+	0.953886i	$0.402957\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	603.797	0.798498	0.399249	−	0.916842i	$-0.369271\pi$
$83$	603.797	0.798498	0.399249	+	0.916842i	$0.369271\pi$
$84$	0	0
$85$	−14.3457	−0.0183059
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1159.88	1.38143	0.690715	−	0.723127i	$-0.257296\pi$
$89$	1159.88	1.38143	0.690715	+	0.723127i	$0.257296\pi$
$90$	0	0
$91$	236.302	0.272211
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−60.6307	−0.0654798
$96$	0	0
$97$	583.269	0.610536	0.305268	−	0.952267i	$-0.401254\pi$
$97$	583.269	0.610536	0.305268	+	0.952267i	$0.401254\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−921.740	−0.908085	−0.454043	−	0.890980i	$-0.650019\pi$
$101$	−921.740	−0.908085	−0.454043	+	0.890980i	$0.650019\pi$
$102$	0	0
$103$	930.712	0.890347	0.445174	−	0.895444i	$-0.353142\pi$
$103$	930.712	0.890347	0.445174	+	0.895444i	$0.353142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	857.383	0.774638	0.387319	−	0.921946i	$-0.373401\pi$
$107$	857.383	0.774638	0.387319	+	0.921946i	$0.373401\pi$
$108$	0	0
$109$	671.853	0.590384	0.295192	−	0.955438i	$-0.404616\pi$
$109$	671.853	0.590384	0.295192	+	0.955438i	$0.404616\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−641.474	−0.534024	−0.267012	−	0.963693i	$-0.586036\pi$
$113$	−641.474	−0.534024	−0.267012	+	0.963693i	$0.586036\pi$
$114$	0	0
$115$	−41.1401	−0.0333594
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−464.359	−0.357712
$120$	0	0
$121$	2860.09	2.14883
$122$	0	0
$123$	0	0
$124$	0	0
$125$	140.211	0.100327
$126$	0	0
$127$	553.174	0.386506	0.193253	−	0.981149i	$-0.438096\pi$
$127$	553.174	0.386506	0.193253	+	0.981149i	$0.438096\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2056.40	1.37152	0.685758	−	0.727830i	$-0.259471\pi$
$131$	2056.40	1.37152	0.685758	+	0.727830i	$0.259471\pi$
$132$	0	0
$133$	−1962.57	−1.27952
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1808.57	1.12786	0.563928	−	0.825824i	$-0.309290\pi$
$137$	1808.57	1.12786	0.563928	+	0.825824i	$0.309290\pi$
$138$	0	0
$139$	−1493.64	−0.911428	−0.455714	−	0.890126i	$-0.650616\pi$
$139$	−1493.64	−0.911428	−0.455714	+	0.890126i	$0.650616\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−841.602	−0.492156
$144$	0	0
$145$	98.7822	0.0565753
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2759.02	1.51696	0.758482	−	0.651694i	$-0.225941\pi$
$149$	2759.02	1.51696	0.758482	+	0.651694i	$0.225941\pi$
$150$	0	0
$151$	976.355	0.526190	0.263095	−	0.964770i	$-0.415257\pi$
$151$	976.355	0.526190	0.263095	+	0.964770i	$0.415257\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−63.5076	−0.0329100
$156$	0	0
$157$	−564.875	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$157$	−564.875	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1331.68	−0.651869
$162$	0	0
$163$	−1508.53	−0.724892	−0.362446	−	0.932005i	$-0.618058\pi$
$163$	−1508.53	−0.724892	−0.362446	+	0.932005i	$0.618058\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	592.521	0.274555	0.137277	−	0.990533i	$-0.456165\pi$
$167$	592.521	0.274555	0.137277	+	0.990533i	$0.456165\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4495.57	1.97568	0.987838	−	0.155488i	$-0.0496952\pi$
$173$	4495.57	1.97568	0.987838	+	0.155488i	$0.0496952\pi$
$174$	0	0
$175$	2266.40	0.978994
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−154.285	−0.0644235	−0.0322117	−	0.999481i	$-0.510255\pi$
$179$	−154.285	−0.0644235	−0.0322117	+	0.999481i	$0.510255\pi$
$180$	0	0
$181$	1071.35	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$181$	1071.35	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−64.4706	−0.0256215
$186$	0	0
$187$	1653.84	0.646742
$188$	0	0
$189$	0	0
$190$	0	0
$191$	677.203	0.256548	0.128274	−	0.991739i	$-0.459056\pi$
$191$	677.203	0.256548	0.128274	+	0.991739i	$0.459056\pi$
$192$	0	0
$193$	1321.68	0.492936	0.246468	−	0.969151i	$-0.420730\pi$
$193$	1321.68	0.492936	0.246468	+	0.969151i	$0.420730\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1267.37	−0.458356	−0.229178	−	0.973385i	$-0.573604\pi$
$197$	−1267.37	−0.458356	−0.229178	+	0.973385i	$0.573604\pi$
$198$	0	0
$199$	−2396.24	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$199$	−2396.24	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3197.51	1.10552
$204$	0	0
$205$	−39.1098	−0.0133246
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6989.81	2.31337
$210$	0	0
$211$	91.5539	0.0298712	0.0149356	−	0.999888i	$-0.495246\pi$
$211$	91.5539	0.0298712	0.0149356	+	0.999888i	$0.495246\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	246.130	0.0780740
$216$	0	0
$217$	−2055.70	−0.643087
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−332.103	−0.101085
$222$	0	0
$223$	−1235.42	−0.370985	−0.185493	−	0.982646i	$-0.559388\pi$
$223$	−1235.42	−0.370985	−0.185493	+	0.982646i	$0.559388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3301.66	0.965370	0.482685	−	0.875794i	$-0.339662\pi$
$227$	3301.66	0.965370	0.482685	+	0.875794i	$0.339662\pi$
$228$	0	0
$229$	211.283	0.0609694	0.0304847	−	0.999535i	$-0.490295\pi$
$229$	211.283	0.0609694	0.0304847	+	0.999535i	$0.490295\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	256.724	0.0721827	0.0360913	−	0.999348i	$-0.488509\pi$
$233$	256.724	0.0721827	0.0360913	+	0.999348i	$0.488509\pi$
$234$	0	0
$235$	17.9404	0.00498002
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3549.62	−0.960694	−0.480347	−	0.877078i	$-0.659489\pi$
$239$	−3549.62	−0.960694	−0.480347	+	0.877078i	$0.659489\pi$
$240$	0	0
$241$	−5030.10	−1.34447	−0.672235	−	0.740338i	$-0.734665\pi$
$241$	−5030.10	−1.34447	−0.672235	+	0.740338i	$0.734665\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.07204	0.00184415
$246$	0	0
$247$	−1403.61	−0.361576
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−718.784	−0.180754	−0.0903770	−	0.995908i	$-0.528807\pi$
$251$	−718.784	−0.180754	−0.0903770	+	0.995908i	$0.528807\pi$
$252$	0	0
$253$	4742.84	1.17858
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1280.79	−0.310871	−0.155435	−	0.987846i	$-0.549678\pi$
$257$	−1280.79	−0.310871	−0.155435	+	0.987846i	$0.549678\pi$
$258$	0	0
$259$	−2086.87	−0.500663
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5225.55	1.22517	0.612587	−	0.790403i	$-0.290129\pi$
$263$	5225.55	1.22517	0.612587	+	0.790403i	$0.290129\pi$
$264$	0	0
$265$	1.59851	0.000370549 0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6443.80	−1.46054	−0.730270	−	0.683158i	$-0.760606\pi$
$269$	−6443.80	−1.46054	−0.730270	+	0.683158i	$0.760606\pi$
$270$	0	0
$271$	3929.93	0.880909	0.440455	−	0.897775i	$-0.354817\pi$
$271$	3929.93	0.880909	0.440455	+	0.897775i	$0.354817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8071.91	−1.77002
$276$	0	0
$277$	−5884.40	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$277$	−5884.40	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3529.79	−0.749358	−0.374679	−	0.927155i	$-0.622247\pi$
$281$	−3529.79	−0.749358	−0.374679	+	0.927155i	$0.622247\pi$
$282$	0	0
$283$	2611.00	0.548438	0.274219	−	0.961667i	$-0.411581\pi$
$283$	2611.00	0.548438	0.274219	+	0.961667i	$0.411581\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1265.96	−0.260373
$288$	0	0
$289$	−4260.38	−0.867165
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5491.03	1.09484	0.547422	−	0.836857i	$-0.315609\pi$
$293$	5491.03	1.09484	0.547422	+	0.836857i	$0.315609\pi$
$294$	0	0
$295$	−40.2255	−0.00793904
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−952.398	−0.184209
$300$	0	0
$301$	7967.05	1.52563
$302$	0	0
$303$	0	0
$304$	0	0
$305$	517.019	0.0970637
$306$	0	0
$307$	7307.59	1.35852	0.679261	−	0.733897i	$-0.262300\pi$
$307$	7307.59	1.35852	0.679261	+	0.733897i	$0.262300\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7904.92	1.44131	0.720654	−	0.693295i	$-0.243842\pi$
$311$	7904.92	1.44131	0.720654	+	0.693295i	$0.243842\pi$
$312$	0	0
$313$	10002.4	1.80629	0.903145	−	0.429336i	$-0.141252\pi$
$313$	10002.4	1.80629	0.903145	+	0.429336i	$0.141252\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6230.81	1.10397	0.551983	−	0.833856i	$-0.313871\pi$
$317$	6230.81	1.10397	0.551983	+	0.833856i	$0.313871\pi$
$318$	0	0
$319$	−11388.1	−1.99878
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2758.24	0.475147
$324$	0	0
$325$	1620.90	0.276650
$326$	0	0
$327$	0	0
$328$	0	0
$329$	580.719	0.0973134
$330$	0	0
$331$	4634.51	0.769594	0.384797	−	0.923001i	$-0.374271\pi$
$331$	4634.51	0.769594	0.384797	+	0.923001i	$0.374271\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−249.487	−0.0406893
$336$	0	0
$337$	3029.82	0.489747	0.244874	−	0.969555i	$-0.421254\pi$
$337$	3029.82	0.489747	0.244874	+	0.969555i	$0.421254\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7321.47	1.16270
$342$	0	0
$343$	6463.66	1.01751
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2841.60	0.439611	0.219805	−	0.975544i	$-0.429458\pi$
$347$	2841.60	0.439611	0.219805	+	0.975544i	$0.429458\pi$
$348$	0	0
$349$	7565.68	1.16040	0.580202	−	0.814472i	$-0.302973\pi$
$349$	7565.68	1.16040	0.580202	+	0.814472i	$0.302973\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2339.44	0.352736	0.176368	−	0.984324i	$-0.443565\pi$
$353$	2339.44	0.352736	0.176368	+	0.984324i	$0.443565\pi$
$354$	0	0
$355$	304.204	0.0454802
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2531.68	−0.372192	−0.186096	−	0.982532i	$-0.559583\pi$
$359$	−2531.68	−0.372192	−0.186096	+	0.982532i	$0.559583\pi$
$360$	0	0
$361$	4798.45	0.699585
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−429.028	−0.0615243
$366$	0	0
$367$	−6577.81	−0.935583	−0.467792	−	0.883839i	$-0.654950\pi$
$367$	−6577.81	−0.935583	−0.467792	+	0.883839i	$0.654950\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	51.7426	0.00724081
$372$	0	0
$373$	2902.72	0.402942	0.201471	−	0.979495i	$-0.435428\pi$
$373$	2902.72	0.402942	0.201471	+	0.979495i	$0.435428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2286.82	0.312406
$378$	0	0
$379$	1865.73	0.252866	0.126433	−	0.991975i	$-0.459647\pi$
$379$	1865.73	0.252866	0.126433	+	0.991975i	$0.459647\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10836.0	1.44567	0.722837	−	0.691019i	$-0.242838\pi$
$383$	10836.0	1.44567	0.722837	+	0.691019i	$0.242838\pi$
$384$	0	0
$385$	660.813	0.0874757
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9520.34	1.24088	0.620438	−	0.784256i	$-0.286955\pi$
$389$	9520.34	1.24088	0.620438	+	0.784256i	$0.286955\pi$
$390$	0	0
$391$	1871.56	0.242069
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−236.716	−0.0301531
$396$	0	0
$397$	−10108.8	−1.27796	−0.638978	−	0.769225i	$-0.720642\pi$
$397$	−10108.8	−1.27796	−0.638978	+	0.769225i	$0.720642\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2084.38	−0.259573	−0.129787	−	0.991542i	$-0.541429\pi$
$401$	−2084.38	−0.259573	−0.129787	+	0.991542i	$0.541429\pi$
$402$	0	0
$403$	−1470.21	−0.181728
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7432.50	0.905197
$408$	0	0
$409$	−9716.53	−1.17470	−0.587349	−	0.809334i	$-0.699828\pi$
$409$	−9716.53	−1.17470	−0.587349	+	0.809334i	$0.699828\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1302.07	−0.155135
$414$	0	0
$415$	−339.064	−0.0401060
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13381.9	1.56026	0.780129	−	0.625619i	$-0.215153\pi$
$419$	13381.9	1.56026	0.780129	+	0.625619i	$0.215153\pi$
$420$	0	0
$421$	−9463.37	−1.09553	−0.547763	−	0.836633i	$-0.684521\pi$
$421$	−9463.37	−1.09553	−0.547763	+	0.836633i	$0.684521\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3185.24	−0.363546
$426$	0	0
$427$	16735.5	1.89670
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4852.28	0.542288	0.271144	−	0.962539i	$-0.412598\pi$
$431$	4852.28	0.542288	0.271144	+	0.962539i	$0.412598\pi$
$432$	0	0
$433$	−8208.00	−0.910973	−0.455486	−	0.890243i	$-0.650535\pi$
$433$	−8208.00	−0.910973	−0.455486	+	0.890243i	$0.650535\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7910.01	0.865874
$438$	0	0
$439$	2993.80	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$439$	2993.80	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9743.67	1.04500	0.522501	−	0.852639i	$-0.324999\pi$
$443$	9743.67	1.04500	0.522501	+	0.852639i	$0.324999\pi$
$444$	0	0
$445$	−651.335	−0.0693848
$446$	0	0
$447$	0	0
$448$	0	0
$449$	561.459	0.0590131	0.0295065	−	0.999565i	$-0.490606\pi$
$449$	561.459	0.0590131	0.0295065	+	0.999565i	$0.490606\pi$
$450$	0	0
$451$	4508.78	0.470754
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−132.696	−0.0136723
$456$	0	0
$457$	13758.4	1.40830	0.704148	−	0.710054i	$-0.251329\pi$
$457$	13758.4	1.40830	0.704148	+	0.710054i	$0.251329\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12009.2	−1.21329	−0.606644	−	0.794974i	$-0.707485\pi$
$461$	−12009.2	−1.21329	−0.606644	+	0.794974i	$0.707485\pi$
$462$	0	0
$463$	−13635.7	−1.36870	−0.684348	−	0.729156i	$-0.739913\pi$
$463$	−13635.7	−1.36870	−0.684348	+	0.729156i	$0.739913\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8821.95	0.874157	0.437079	−	0.899423i	$-0.356013\pi$
$467$	8821.95	0.874157	0.437079	+	0.899423i	$0.356013\pi$
$468$	0	0
$469$	−8075.72	−0.795100
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28375.1	−2.75832
$474$	0	0
$475$	−13462.2	−1.30039
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14620.0	−1.39459	−0.697293	−	0.716786i	$-0.745612\pi$
$479$	−14620.0	−1.39459	−0.697293	+	0.716786i	$0.745612\pi$
$480$	0	0
$481$	−1492.50	−0.141481
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−327.536	−0.0306653
$486$	0	0
$487$	9798.86	0.911763	0.455882	−	0.890040i	$-0.349324\pi$
$487$	9798.86	0.911763	0.455882	+	0.890040i	$0.349324\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10836.1	−0.995977	−0.497989	−	0.867184i	$-0.665928\pi$
$491$	−10836.1	−0.995977	−0.497989	+	0.867184i	$0.665928\pi$
$492$	0	0
$493$	−4493.84	−0.410532
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9846.86	0.888717
$498$	0	0
$499$	−2589.96	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$499$	−2589.96	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17067.5	−1.51292	−0.756462	−	0.654038i	$-0.773074\pi$
$503$	−17067.5	−1.51292	−0.756462	+	0.654038i	$0.773074\pi$
$504$	0	0
$505$	517.606	0.0456102
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1012.89	0.0882038	0.0441019	−	0.999027i	$-0.485957\pi$
$509$	1012.89	0.0882038	0.0441019	+	0.999027i	$0.485957\pi$
$510$	0	0
$511$	−13887.4	−1.20223
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−522.644	−0.0447193
$516$	0	0
$517$	−2068.26	−0.175942
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14367.7	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$521$	14367.7	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$522$	0	0
$523$	16219.9	1.35611	0.678057	−	0.735010i	$-0.262822\pi$
$523$	16219.9	1.35611	0.678057	+	0.735010i	$0.262822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2889.11	0.238808
$528$	0	0
$529$	−6799.77	−0.558870
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−905.396	−0.0735780
$534$	0	0
$535$	−481.466	−0.0389076
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−815.301	−0.0651530
$540$	0	0
$541$	17592.2	1.39806	0.699029	−	0.715094i	$-0.253616\pi$
$541$	17592.2	1.39806	0.699029	+	0.715094i	$0.253616\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−377.281	−0.0296531
$546$	0	0
$547$	−10504.6	−0.821103	−0.410552	−	0.911837i	$-0.634664\pi$
$547$	−10504.6	−0.821103	−0.410552	+	0.911837i	$0.634664\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−18992.8	−1.46846
$552$	0	0
$553$	−7662.33	−0.589214
$554$	0	0
$555$	0	0
$556$	0	0
$557$	507.558	0.0386102	0.0193051	−	0.999814i	$-0.493855\pi$
$557$	507.558	0.0386102	0.0193051	+	0.999814i	$0.493855\pi$
$558$	0	0
$559$	5697.93	0.431121
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3443.14	−0.257746	−0.128873	−	0.991661i	$-0.541136\pi$
$563$	−3443.14	−0.257746	−0.128873	+	0.991661i	$0.541136\pi$
$564$	0	0
$565$	360.221	0.0268223
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23972.2	−1.76620	−0.883098	−	0.469189i	$-0.844546\pi$
$569$	−23972.2	−1.76620	−0.883098	+	0.469189i	$0.844546\pi$
$570$	0	0
$571$	7458.32	0.546622	0.273311	−	0.961926i	$-0.411881\pi$
$571$	7458.32	0.546622	0.273311	+	0.961926i	$0.411881\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9134.57	−0.662501
$576$	0	0
$577$	5669.57	0.409059	0.204530	−	0.978860i	$-0.434434\pi$
$577$	5669.57	0.409059	0.204530	+	0.978860i	$0.434434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10975.3	−0.783702
$582$	0	0
$583$	−184.284	−0.0130914
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1017.39	0.0715371	0.0357685	−	0.999360i	$-0.488612\pi$
$587$	1017.39	0.0715371	0.0357685	+	0.999360i	$0.488612\pi$
$588$	0	0
$589$	12210.6	0.854208
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10198.2	0.706221	0.353111	−	0.935582i	$-0.385124\pi$
$593$	10198.2	0.706221	0.353111	+	0.935582i	$0.385124\pi$
$594$	0	0
$595$	260.762	0.0179667
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12516.3	0.853763	0.426881	−	0.904308i	$-0.359612\pi$
$599$	12516.3	0.853763	0.426881	+	0.904308i	$0.359612\pi$
$600$	0	0
$601$	9627.46	0.653431	0.326716	−	0.945123i	$-0.394058\pi$
$601$	9627.46	0.653431	0.326716	+	0.945123i	$0.394058\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1606.09	−0.107929
$606$	0	0
$607$	−6667.20	−0.445821	−0.222910	−	0.974839i	$-0.571556\pi$
$607$	−6667.20	−0.445821	−0.222910	+	0.974839i	$0.571556\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	415.323	0.0274994
$612$	0	0
$613$	−23085.4	−1.52106	−0.760530	−	0.649302i	$-0.775061\pi$
$613$	−23085.4	−1.52106	−0.760530	+	0.649302i	$0.775061\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3049.24	−0.198959	−0.0994796	−	0.995040i	$-0.531718\pi$
$617$	−3049.24	−0.198959	−0.0994796	+	0.995040i	$0.531718\pi$
$618$	0	0
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21083.3	−1.35583
$624$	0	0
$625$	15506.8	0.992438
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2932.92	0.185920
$630$	0	0
$631$	23829.5	1.50339	0.751694	−	0.659512i	$-0.229237\pi$
$631$	23829.5	1.50339	0.751694	+	0.659512i	$0.229237\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−310.637	−0.0194130
$636$	0	0
$637$	163.718	0.0101833
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13405.3	−0.826016	−0.413008	−	0.910727i	$-0.635522\pi$
$641$	−13405.3	−0.826016	−0.413008	+	0.910727i	$0.635522\pi$
$642$	0	0
$643$	−5251.51	−0.322083	−0.161042	−	0.986948i	$-0.551485\pi$
$643$	−5251.51	−0.322083	−0.161042	+	0.986948i	$0.551485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21611.4	1.31319	0.656595	−	0.754244i	$-0.271996\pi$
$647$	21611.4	1.31319	0.656595	+	0.754244i	$0.271996\pi$
$648$	0	0
$649$	4637.39	0.280483
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21595.8	1.29420	0.647099	−	0.762406i	$-0.275982\pi$
$653$	21595.8	1.29420	0.647099	+	0.762406i	$0.275982\pi$
$654$	0	0
$655$	−1154.78	−0.0688869
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16642.6	−0.983768	−0.491884	−	0.870661i	$-0.663692\pi$
$659$	−16642.6	−0.983768	−0.491884	+	0.870661i	$0.663692\pi$
$660$	0	0
$661$	26981.1	1.58766	0.793831	−	0.608139i	$-0.208084\pi$
$661$	26981.1	1.58766	0.793831	+	0.608139i	$0.208084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1102.09	0.0642664
$666$	0	0
$667$	−12887.3	−0.748126
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−59604.5	−3.42922
$672$	0	0
$673$	11149.2	0.638591	0.319296	−	0.947655i	$-0.396554\pi$
$673$	11149.2	0.638591	0.319296	+	0.947655i	$0.396554\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3314.33	−0.188154	−0.0940769	−	0.995565i	$-0.529990\pi$
$677$	−3314.33	−0.188154	−0.0940769	+	0.995565i	$0.529990\pi$
$678$	0	0
$679$	−10602.1	−0.599223
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24505.2	1.37287	0.686433	−	0.727193i	$-0.259176\pi$
$683$	24505.2	1.37287	0.686433	+	0.727193i	$0.259176\pi$
$684$	0	0
$685$	−1015.61	−0.0566486
$686$	0	0
$687$	0	0
$688$	0	0
$689$	37.0056	0.00204616
$690$	0	0
$691$	21752.8	1.19756	0.598782	−	0.800912i	$-0.295652\pi$
$691$	21752.8	1.19756	0.598782	+	0.800912i	$0.295652\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	838.755	0.0457781
$696$	0	0
$697$	1779.20	0.0966887
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34250.9	−1.84542	−0.922709	−	0.385496i	$-0.874030\pi$
$701$	−34250.9	−1.84542	−0.922709	+	0.385496i	$0.874030\pi$
$702$	0	0
$703$	12395.8	0.665028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16754.6	0.891259
$708$	0	0
$709$	−5527.11	−0.292771	−0.146386	−	0.989228i	$-0.546764\pi$
$709$	−5527.11	−0.292771	−0.146386	+	0.989228i	$0.546764\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8285.33	0.435187
$714$	0	0
$715$	472.604	0.0247194
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3777.78	−0.195949	−0.0979745	−	0.995189i	$-0.531236\pi$
$719$	−3777.78	−0.195949	−0.0979745	+	0.995189i	$0.531236\pi$
$720$	0	0
$721$	−16917.6	−0.873849
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21933.2	1.12355
$726$	0	0
$727$	−19076.8	−0.973204	−0.486602	−	0.873624i	$-0.661764\pi$
$727$	−19076.8	−0.973204	−0.486602	+	0.873624i	$0.661764\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11197.0	−0.566535
$732$	0	0
$733$	7997.30	0.402984	0.201492	−	0.979490i	$-0.435421\pi$
$733$	7997.30	0.402984	0.201492	+	0.979490i	$0.435421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28762.1	1.43754
$738$	0	0
$739$	−28983.6	−1.44273	−0.721367	−	0.692553i	$-0.756486\pi$
$739$	−28983.6	−1.44273	−0.721367	+	0.692553i	$0.756486\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19145.4	−0.945324	−0.472662	−	0.881244i	$-0.656707\pi$
$743$	−19145.4	−0.945324	−0.472662	+	0.881244i	$0.656707\pi$
$744$	0	0
$745$	−1549.33	−0.0761923
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15584.7	−0.760284
$750$	0	0
$751$	25516.9	1.23985	0.619923	−	0.784663i	$-0.287164\pi$
$751$	25516.9	1.23985	0.619923	+	0.784663i	$0.287164\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−548.275	−0.0264288
$756$	0	0
$757$	−17230.6	−0.827289	−0.413645	−	0.910438i	$-0.635744\pi$
$757$	−17230.6	−0.827289	−0.413645	+	0.910438i	$0.635744\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2343.06	0.111611	0.0558053	−	0.998442i	$-0.482227\pi$
$761$	2343.06	0.111611	0.0558053	+	0.998442i	$0.482227\pi$
$762$	0	0
$763$	−12212.3	−0.579444
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−931.223	−0.0438390
$768$	0	0
$769$	−7100.18	−0.332950	−0.166475	−	0.986046i	$-0.553239\pi$
$769$	−7100.18	−0.332950	−0.166475	+	0.986046i	$0.553239\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12270.4	−0.570940	−0.285470	−	0.958388i	$-0.592150\pi$
$773$	−12270.4	−0.570940	−0.285470	+	0.958388i	$0.592150\pi$
$774$	0	0
$775$	−14100.9	−0.653575
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7519.64	0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	0	0
$783$	0	0
$784$	0	0
$785$	317.207	0.0144224
$786$	0	0
$787$	−3425.04	−0.155133	−0.0775663	−	0.996987i	$-0.524715\pi$
$787$	−3425.04	−0.155133	−0.0775663	+	0.996987i	$0.524715\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11660.1	0.524129
$792$	0	0
$793$	11969.0	0.535981
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11781.1	0.523600	0.261800	−	0.965122i	$-0.415684\pi$
$797$	11781.1	0.523600	0.261800	+	0.965122i	$0.415684\pi$
$798$	0	0
$799$	−816.154	−0.0361370
$800$	0	0
$801$	0	0
$802$	0	0
$803$	49460.6	2.17363
$804$	0	0
$805$	747.807	0.0327413
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18910.1	−0.821810	−0.410905	−	0.911678i	$-0.634787\pi$
$809$	−18910.1	−0.821810	−0.410905	+	0.911678i	$0.634787\pi$
$810$	0	0
$811$	−12803.3	−0.554359	−0.277180	−	0.960818i	$-0.589400\pi$
$811$	−12803.3	−0.554359	−0.277180	+	0.960818i	$0.589400\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	847.121	0.0364090
$816$	0	0
$817$	−47323.3	−2.02648
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19335.1	−0.821923	−0.410962	−	0.911653i	$-0.634807\pi$
$821$	−19335.1	−0.821923	−0.410962	+	0.911653i	$0.634807\pi$
$822$	0	0
$823$	2125.90	0.0900417	0.0450209	−	0.998986i	$-0.485665\pi$
$823$	2125.90	0.0900417	0.0450209	+	0.998986i	$0.485665\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6989.24	−0.293881	−0.146941	−	0.989145i	$-0.546943\pi$
$827$	−6989.24	−0.293881	−0.146941	+	0.989145i	$0.546943\pi$
$828$	0	0
$829$	−32649.7	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$829$	−32649.7	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−321.724	−0.0133819
$834$	0	0
$835$	−332.732	−0.0137900
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4038.23	−0.166168	−0.0830841	−	0.996543i	$-0.526477\pi$
$839$	−4038.23	−0.166168	−0.0830841	+	0.996543i	$0.526477\pi$
$840$	0	0
$841$	6555.00	0.268769
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−94.9024	−0.00386360
$846$	0	0
$847$	−51988.1	−2.10901
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8410.97	0.338807
$852$	0	0
$853$	8114.12	0.325700	0.162850	−	0.986651i	$-0.447931\pi$
$853$	8114.12	0.325700	0.162850	+	0.986651i	$0.447931\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22298.1	0.888786	0.444393	−	0.895832i	$-0.353419\pi$
$857$	22298.1	0.888786	0.444393	+	0.895832i	$0.353419\pi$
$858$	0	0
$859$	−33550.5	−1.33263	−0.666315	−	0.745670i	$-0.732130\pi$
$859$	−33550.5	−1.33263	−0.666315	+	0.745670i	$0.732130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14120.5	−0.556972	−0.278486	−	0.960440i	$-0.589833\pi$
$863$	−14120.5	−0.556972	−0.278486	+	0.960440i	$0.589833\pi$
$864$	0	0
$865$	−2524.50	−0.0992319
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27289.8	1.06530
$870$	0	0
$871$	−5775.64	−0.224685
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2548.63	−0.0984679
$876$	0	0
$877$	−1941.69	−0.0747619	−0.0373809	−	0.999301i	$-0.511901\pi$
$877$	−1941.69	−0.0747619	−0.0373809	+	0.999301i	$0.511901\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	790.231	0.0302197	0.0151099	−	0.999886i	$-0.495190\pi$
$881$	790.231	0.0302197	0.0151099	+	0.999886i	$0.495190\pi$
$882$	0	0
$883$	36638.6	1.39636	0.698180	−	0.715922i	$-0.253993\pi$
$883$	36638.6	1.39636	0.698180	+	0.715922i	$0.253993\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40686.3	−1.54015	−0.770075	−	0.637954i	$-0.779781\pi$
$887$	−40686.3	−1.54015	−0.770075	+	0.637954i	$0.779781\pi$
$888$	0	0
$889$	−10055.1	−0.379344
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3449.40	−0.129261
$894$	0	0
$895$	86.6392	0.00323579
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19894.0	−0.738046
$900$	0	0
$901$	−72.7200	−0.00268885
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−601.618	−0.0220977
$906$	0	0
$907$	10464.4	0.383093	0.191547	−	0.981484i	$-0.438650\pi$
$907$	10464.4	0.383093	0.191547	+	0.981484i	$0.438650\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35611.5	−1.29513	−0.647563	−	0.762011i	$-0.724212\pi$
$911$	−35611.5	−1.29513	−0.647563	+	0.762011i	$0.724212\pi$
$912$	0	0
$913$	39089.0	1.41693
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37379.4	−1.34610
$918$	0	0
$919$	−1077.25	−0.0386674	−0.0193337	−	0.999813i	$-0.506154\pi$
$919$	−1077.25	−0.0386674	−0.0193337	+	0.999813i	$0.506154\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7042.34	0.251139
$924$	0	0
$925$	−14314.8	−0.508829
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−55733.8	−1.96832	−0.984159	−	0.177290i	$-0.943267\pi$
$929$	−55733.8	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$930$	0	0
$931$	−1359.74	−0.0478665
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−928.718	−0.0324838
$936$	0	0
$937$	−3198.60	−0.111519	−0.0557596	−	0.998444i	$-0.517758\pi$
$937$	−3198.60	−0.111519	−0.0557596	+	0.998444i	$0.517758\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8823.35	−0.305667	−0.152834	−	0.988252i	$-0.548840\pi$
$941$	−8823.35	−0.305667	−0.152834	+	0.988252i	$0.548840\pi$
$942$	0	0
$943$	5102.35	0.176199
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28290.4	0.970766	0.485383	−	0.874301i	$-0.338680\pi$
$947$	28290.4	0.970766	0.485383	+	0.874301i	$0.338680\pi$
$948$	0	0
$949$	−9932.05	−0.339734
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12399.0	0.421452	0.210726	−	0.977545i	$-0.432417\pi$
$953$	12399.0	0.421452	0.210726	+	0.977545i	$0.432417\pi$
$954$	0	0
$955$	−380.285	−0.0128856
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32874.5	−1.10696
$960$	0	0
$961$	−17001.0	−0.570676
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−742.193	−0.0247586
$966$	0	0
$967$	26667.1	0.886820	0.443410	−	0.896319i	$-0.353769\pi$
$967$	26667.1	0.886820	0.443410	+	0.896319i	$0.353769\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49420.7	1.63335	0.816676	−	0.577096i	$-0.195814\pi$
$971$	49420.7	1.63335	0.816676	+	0.577096i	$0.195814\pi$
$972$	0	0
$973$	27149.9	0.894539
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−778.759	−0.0255012	−0.0127506	−	0.999919i	$-0.504059\pi$
$977$	−778.759	−0.0255012	−0.0127506	+	0.999919i	$0.504059\pi$
$978$	0	0
$979$	75089.2	2.45134
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5997.90	0.194612	0.0973059	−	0.995255i	$-0.468977\pi$
$983$	5997.90	0.194612	0.0973059	+	0.995255i	$0.468977\pi$
$984$	0	0
$985$	711.693	0.0230218
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32110.6	−1.03241
$990$	0	0
$991$	−8974.94	−0.287688	−0.143844	−	0.989600i	$-0.545946\pi$
$991$	−8974.94	−0.287688	−0.143844	+	0.989600i	$0.545946\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1345.62	0.0428732
$996$	0	0
$997$	28530.2	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$997$	28530.2	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bb.1.1		2
3.2	odd	2		208.4.a.h.1.2		2
4.3	odd	2		117.4.a.d.1.1		2
12.11	even	2		13.4.a.b.1.2	✓	2
24.5	odd	2		832.4.a.z.1.1		2
24.11	even	2		832.4.a.s.1.2		2
52.51	odd	2		1521.4.a.r.1.2		2
60.23	odd	4		325.4.b.e.274.1		4
60.47	odd	4		325.4.b.e.274.4		4
60.59	even	2		325.4.a.f.1.1		2
84.83	odd	2		637.4.a.b.1.2		2
132.131	odd	2		1573.4.a.b.1.1		2
156.11	odd	12		169.4.e.f.147.1		8
156.23	even	6		169.4.c.j.22.2		4
156.35	even	6		169.4.c.g.146.1		4
156.47	odd	4		169.4.b.f.168.4		4
156.59	odd	12		169.4.e.f.23.4		8
156.71	odd	12		169.4.e.f.23.1		8
156.83	odd	4		169.4.b.f.168.1		4
156.95	even	6		169.4.c.j.146.2		4
156.107	even	6		169.4.c.g.22.1		4
156.119	odd	12		169.4.e.f.147.4		8
156.155	even	2		169.4.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	12.11	even	2
117.4.a.d.1.1		2	4.3	odd	2
169.4.a.g.1.1		2	156.155	even	2
169.4.b.f.168.1		4	156.83	odd	4
169.4.b.f.168.4		4	156.47	odd	4
169.4.c.g.22.1		4	156.107	even	6
169.4.c.g.146.1		4	156.35	even	6
169.4.c.j.22.2		4	156.23	even	6
169.4.c.j.146.2		4	156.95	even	6
169.4.e.f.23.1		8	156.71	odd	12
169.4.e.f.23.4		8	156.59	odd	12
169.4.e.f.147.1		8	156.11	odd	12
169.4.e.f.147.4		8	156.119	odd	12
208.4.a.h.1.2		2	3.2	odd	2
325.4.a.f.1.1		2	60.59	even	2
325.4.b.e.274.1		4	60.23	odd	4
325.4.b.e.274.4		4	60.47	odd	4
637.4.a.b.1.2		2	84.83	odd	2
832.4.a.s.1.2		2	24.11	even	2
832.4.a.z.1.1		2	24.5	odd	2
1521.4.a.r.1.2		2	52.51	odd	2
1573.4.a.b.1.1		2	132.131	odd	2
1872.4.a.bb.1.1		2	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q-0.561553 q^{5} -18.1771 q^{7} +O(q^{10})\)	\(q-0.561553 q^{5} -18.1771 q^{7} +64.7386 q^{11} -13.0000 q^{13} +25.5464 q^{17} +107.970 q^{19} +73.2614 q^{23} -124.685 q^{25} -175.909 q^{29} +113.093 q^{31} +10.2074 q^{35} +114.808 q^{37} +69.6458 q^{41} -438.302 q^{43} -31.9479 q^{47} -12.5937 q^{49} -2.84658 q^{53} -36.3542 q^{55} +71.6325 q^{59} -920.695 q^{61} +7.30019 q^{65} +444.280 q^{67} -541.719 q^{71} +764.004 q^{73} -1176.76 q^{77} +421.538 q^{79} +603.797 q^{83} -14.3457 q^{85} +1159.88 q^{89} +236.302 q^{91} -60.6307 q^{95} +583.269 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} + 9 q^{7}+O(q^{10}) \) 2 * q + 3 * q^5 + 9 * q^7	\( 2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 + 9 * q^7 + 80 * q^11 - 26 * q^13 - 19 * q^17 + 84 * q^19 + 196 * q^23 - 237 * q^25 + 44 * q^29 + 86 * q^31 + 107 * q^35 + 209 * q^37 + 230 * q^41 - 287 * q^43 + 435 * q^47 + 383 * q^49 + 118 * q^53 + 18 * q^55 - 368 * q^59 - 1058 * q^61 - 39 * q^65 - 68 * q^67 - 131 * q^71 + 456 * q^73 - 762 * q^77 + 1008 * q^79 + 1958 * q^83 - 173 * q^85 + 720 * q^89 - 117 * q^91 - 146 * q^95 - 928 * q^97

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