LMFDB - Embedded newform 1472.4.a.bk.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,0,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 147x^{7} - 97x^{6} + 4561x^{5} + 7383x^{4} - 31427x^{3} - 43981x^{2} + 17596x + 12306 $ x^9 - 3x^8 - 147x^7 - 97x^6 + 4561x^5 + 7383x^4 - 31427x^3 - 43981x^2 + 17596x + 12306
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$5.55395$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.02965 q^{3} +0.275790 q^{5} -36.2914 q^{7} -17.8212 q^{9} -58.7296 q^{11} +11.0440 q^{13} +0.835548 q^{15} -133.818 q^{17} +74.5690 q^{19} -109.950 q^{21} +23.0000 q^{23} -124.924 q^{25} -135.793 q^{27} +153.539 q^{29} +211.443 q^{31} -177.930 q^{33} -10.0088 q^{35} -124.725 q^{37} +33.4594 q^{39} -238.404 q^{41} +355.165 q^{43} -4.91492 q^{45} +138.046 q^{47} +974.063 q^{49} -405.421 q^{51} -563.486 q^{53} -16.1971 q^{55} +225.918 q^{57} -238.749 q^{59} +419.333 q^{61} +646.756 q^{63} +3.04582 q^{65} -1.30367 q^{67} +69.6820 q^{69} +20.8426 q^{71} +730.873 q^{73} -378.476 q^{75} +2131.38 q^{77} +331.744 q^{79} +69.7684 q^{81} -321.794 q^{83} -36.9056 q^{85} +465.170 q^{87} -360.677 q^{89} -400.801 q^{91} +640.600 q^{93} +20.5654 q^{95} -87.8723 q^{97} +1046.63 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 42 q^{7} + 71 q^{9} - 66 q^{11} - 48 q^{13} + 90 q^{15} - 32 q^{17} - 6 q^{21} + 207 q^{23} + 135 q^{25} + 88 q^{29} + 556 q^{31} - 230 q^{33} - 368 q^{35} + 368 q^{37} + 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) $ 9 * q + 42 * q^7 + 71 * q^9 - 66 * q^11 - 48 * q^13 + 90 * q^15 - 32 * q^17 - 6 * q^21 + 207 * q^23 + 135 * q^25 + 88 * q^29 + 556 * q^31 - 230 * q^33 - 368 * q^35 + 368 * q^37 + 468 * q^39 - 216 * q^41 + 552 * q^43 + 206 * q^45 + 1300 * q^47 + 1041 * q^49 - 1658 * q^51 - 510 * q^53 + 1948 * q^55 - 804 * q^57 - 1444 * q^59 + 882 * q^61 + 3128 * q^63 + 334 * q^65 - 1186 * q^67 + 2440 * q^71 - 368 * q^73 - 1636 * q^75 - 556 * q^77 + 2576 * q^79 + 1069 * q^81 - 1710 * q^83 - 640 * q^85 + 4844 * q^87 - 474 * q^89 - 2790 * q^91 - 1178 * q^93 + 4564 * q^95 - 884 * q^97 - 552 * 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
$3$	3.02965	0.583057	0.291528	−	0.956562i	$-0.405836\pi$
$3$	3.02965	0.583057	0.291528	+	0.956562i	$0.405836\pi$
$4$	0	0
$5$	0.275790	0.0246674	0.0123337	−	0.999924i	$-0.496074\pi$
$5$	0.275790	0.0246674	0.0123337	+	0.999924i	$0.496074\pi$
$6$	0	0
$7$	−36.2914	−1.95955	−0.979775	−	0.200105i	$-0.935872\pi$
$7$	−36.2914	−1.95955	−0.979775	+	0.200105i	$0.935872\pi$
$8$	0	0
$9$	−17.8212	−0.660045
$10$	0	0
$11$	−58.7296	−1.60979	−0.804893	−	0.593420i	$-0.797777\pi$
$11$	−58.7296	−1.60979	−0.804893	+	0.593420i	$0.797777\pi$
$12$	0	0
$13$	11.0440	0.235619	0.117810	−	0.993036i	$-0.462413\pi$
$13$	11.0440	0.235619	0.117810	+	0.993036i	$0.462413\pi$
$14$	0	0
$15$	0.835548	0.0143825
$16$	0	0
$17$	−133.818	−1.90915	−0.954575	−	0.297971i	$-0.903690\pi$
$17$	−133.818	−1.90915	−0.954575	+	0.297971i	$0.903690\pi$
$18$	0	0
$19$	74.5690	0.900384	0.450192	−	0.892932i	$-0.351356\pi$
$19$	74.5690	0.900384	0.450192	+	0.892932i	$0.351356\pi$
$20$	0	0
$21$	−109.950	−1.14253
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−124.924	−0.999392
$26$	0	0
$27$	−135.793	−0.967900
$28$	0	0
$29$	153.539	0.983156	0.491578	−	0.870834i	$-0.336420\pi$
$29$	153.539	0.983156	0.491578	+	0.870834i	$0.336420\pi$
$30$	0	0
$31$	211.443	1.22504	0.612522	−	0.790454i	$-0.290155\pi$
$31$	211.443	1.22504	0.612522	+	0.790454i	$0.290155\pi$
$32$	0	0
$33$	−177.930	−0.938597
$34$	0	0
$35$	−10.0088	−0.0483370
$36$	0	0
$37$	−124.725	−0.554179	−0.277089	−	0.960844i	$-0.589370\pi$
$37$	−124.725	−0.554179	−0.277089	+	0.960844i	$0.589370\pi$
$38$	0	0
$39$	33.4594	0.137379
$40$	0	0
$41$	−238.404	−0.908110	−0.454055	−	0.890974i	$-0.650023\pi$
$41$	−238.404	−0.908110	−0.454055	+	0.890974i	$0.650023\pi$
$42$	0	0
$43$	355.165	1.25958	0.629792	−	0.776764i	$-0.283140\pi$
$43$	355.165	1.25958	0.629792	+	0.776764i	$0.283140\pi$
$44$	0	0
$45$	−4.91492	−0.0162816
$46$	0	0
$47$	138.046	0.428429	0.214214	−	0.976787i	$-0.431281\pi$
$47$	138.046	0.428429	0.214214	+	0.976787i	$0.431281\pi$
$48$	0	0
$49$	974.063	2.83983
$50$	0	0
$51$	−405.421	−1.11314
$52$	0	0
$53$	−563.486	−1.46039	−0.730196	−	0.683238i	$-0.760571\pi$
$53$	−563.486	−1.46039	−0.730196	+	0.683238i	$0.760571\pi$
$54$	0	0
$55$	−16.1971	−0.0397093
$56$	0	0
$57$	225.918	0.524975
$58$	0	0
$59$	−238.749	−0.526822	−0.263411	−	0.964684i	$-0.584848\pi$
$59$	−238.749	−0.526822	−0.263411	+	0.964684i	$0.584848\pi$
$60$	0	0
$61$	419.333	0.880166	0.440083	−	0.897957i	$-0.354949\pi$
$61$	419.333	0.880166	0.440083	+	0.897957i	$0.354949\pi$
$62$	0	0
$63$	646.756	1.29339
$64$	0	0
$65$	3.04582	0.00581212
$66$	0	0
$67$	−1.30367	−0.00237715	−0.00118857	−	0.999999i	$-0.500378\pi$
$67$	−1.30367	−0.00237715	−0.00118857	+	0.999999i	$0.500378\pi$
$68$	0	0
$69$	69.6820	0.121576
$70$	0	0
$71$	20.8426	0.0348390	0.0174195	−	0.999848i	$-0.494455\pi$
$71$	20.8426	0.0348390	0.0174195	+	0.999848i	$0.494455\pi$
$72$	0	0
$73$	730.873	1.17181	0.585905	−	0.810380i	$-0.300739\pi$
$73$	730.873	1.17181	0.585905	+	0.810380i	$0.300739\pi$
$74$	0	0
$75$	−378.476	−0.582702
$76$	0	0
$77$	2131.38	3.15445
$78$	0	0
$79$	331.744	0.472458	0.236229	−	0.971697i	$-0.424089\pi$
$79$	331.744	0.472458	0.236229	+	0.971697i	$0.424089\pi$
$80$	0	0
$81$	69.7684	0.0957042
$82$	0	0
$83$	−321.794	−0.425560	−0.212780	−	0.977100i	$-0.568252\pi$
$83$	−321.794	−0.425560	−0.212780	+	0.977100i	$0.568252\pi$
$84$	0	0
$85$	−36.9056	−0.0470938
$86$	0	0
$87$	465.170	0.573236
$88$	0	0
$89$	−360.677	−0.429569	−0.214785	−	0.976661i	$-0.568905\pi$
$89$	−360.677	−0.429569	−0.214785	+	0.976661i	$0.568905\pi$
$90$	0	0
$91$	−400.801	−0.461707
$92$	0	0
$93$	640.600	0.714270
$94$	0	0
$95$	20.5654	0.0222102
$96$	0	0
$97$	−87.8723	−0.0919802	−0.0459901	−	0.998942i	$-0.514644\pi$
$97$	−87.8723	−0.0919802	−0.0459901	+	0.998942i	$0.514644\pi$
$98$	0	0
$99$	1046.63	1.06253
$100$	0	0
$101$	652.719	0.643049	0.321525	−	0.946901i	$-0.395805\pi$
$101$	652.719	0.643049	0.321525	+	0.946901i	$0.395805\pi$
$102$	0	0
$103$	1249.22	1.19504	0.597521	−	0.801854i	$-0.296153\pi$
$103$	1249.22	1.19504	0.597521	+	0.801854i	$0.296153\pi$
$104$	0	0
$105$	−30.3232	−0.0281832
$106$	0	0
$107$	1179.09	1.06530	0.532648	−	0.846337i	$-0.321197\pi$
$107$	1179.09	1.06530	0.532648	+	0.846337i	$0.321197\pi$
$108$	0	0
$109$	−1518.41	−1.33429	−0.667143	−	0.744930i	$-0.732483\pi$
$109$	−1518.41	−1.33429	−0.667143	+	0.744930i	$0.732483\pi$
$110$	0	0
$111$	−377.872	−0.323118
$112$	0	0
$113$	−1806.94	−1.50428	−0.752138	−	0.659006i	$-0.770977\pi$
$113$	−1806.94	−1.50428	−0.752138	+	0.659006i	$0.770977\pi$
$114$	0	0
$115$	6.34318	0.00514352
$116$	0	0
$117$	−196.817	−0.155519
$118$	0	0
$119$	4856.42	3.74107
$120$	0	0
$121$	2118.17	1.59141
$122$	0	0
$123$	−722.282	−0.529479
$124$	0	0
$125$	−68.9266	−0.0493199
$126$	0	0
$127$	1251.01	0.874086	0.437043	−	0.899441i	$-0.356026\pi$
$127$	1251.01	0.874086	0.437043	+	0.899441i	$0.356026\pi$
$128$	0	0
$129$	1076.03	0.734409
$130$	0	0
$131$	−2190.58	−1.46100	−0.730502	−	0.682910i	$-0.760714\pi$
$131$	−2190.58	−1.46100	−0.730502	+	0.682910i	$0.760714\pi$
$132$	0	0
$133$	−2706.21	−1.76435
$134$	0	0
$135$	−37.4503	−0.0238756
$136$	0	0
$137$	72.8940	0.0454580	0.0227290	−	0.999742i	$-0.492765\pi$
$137$	72.8940	0.0454580	0.0227290	+	0.999742i	$0.492765\pi$
$138$	0	0
$139$	−2432.82	−1.48453	−0.742264	−	0.670108i	$-0.766248\pi$
$139$	−2432.82	−1.48453	−0.742264	+	0.670108i	$0.766248\pi$
$140$	0	0
$141$	418.233	0.249798
$142$	0	0
$143$	−648.609	−0.379296
$144$	0	0
$145$	42.3446	0.0242519
$146$	0	0
$147$	2951.07	1.65578
$148$	0	0
$149$	−2580.10	−1.41859	−0.709295	−	0.704912i	$-0.750987\pi$
$149$	−2580.10	−1.41859	−0.709295	+	0.704912i	$0.750987\pi$
$150$	0	0
$151$	−1335.90	−0.719960	−0.359980	−	0.932960i	$-0.617216\pi$
$151$	−1335.90	−0.719960	−0.359980	+	0.932960i	$0.617216\pi$
$152$	0	0
$153$	2384.79	1.26012
$154$	0	0
$155$	58.3140	0.0302187
$156$	0	0
$157$	1331.65	0.676925	0.338462	−	0.940980i	$-0.390093\pi$
$157$	1331.65	0.676925	0.338462	+	0.940980i	$0.390093\pi$
$158$	0	0
$159$	−1707.17	−0.851491
$160$	0	0
$161$	−834.701	−0.408594
$162$	0	0
$163$	−41.7783	−0.0200756	−0.0100378	−	0.999950i	$-0.503195\pi$
$163$	−41.7783	−0.0200756	−0.0100378	+	0.999950i	$0.503195\pi$
$164$	0	0
$165$	−49.0714	−0.0231528
$166$	0	0
$167$	−743.454	−0.344492	−0.172246	−	0.985054i	$-0.555102\pi$
$167$	−743.454	−0.344492	−0.172246	+	0.985054i	$0.555102\pi$
$168$	0	0
$169$	−2075.03	−0.944484
$170$	0	0
$171$	−1328.91	−0.594294
$172$	0	0
$173$	−1975.08	−0.867993	−0.433997	−	0.900914i	$-0.642897\pi$
$173$	−1975.08	−0.867993	−0.433997	+	0.900914i	$0.642897\pi$
$174$	0	0
$175$	4533.66	1.95836
$176$	0	0
$177$	−723.327	−0.307167
$178$	0	0
$179$	1367.19	0.570885	0.285443	−	0.958396i	$-0.407859\pi$
$179$	1367.19	0.570885	0.285443	+	0.958396i	$0.407859\pi$
$180$	0	0
$181$	−3138.15	−1.28871	−0.644356	−	0.764726i	$-0.722874\pi$
$181$	−3138.15	−1.28871	−0.644356	+	0.764726i	$0.722874\pi$
$182$	0	0
$183$	1270.43	0.513186
$184$	0	0
$185$	−34.3979	−0.0136702
$186$	0	0
$187$	7859.06	3.07332
$188$	0	0
$189$	4928.10	1.89665
$190$	0	0
$191$	−301.122	−0.114076	−0.0570378	−	0.998372i	$-0.518166\pi$
$191$	−301.122	−0.114076	−0.0570378	+	0.998372i	$0.518166\pi$
$192$	0	0
$193$	−1789.03	−0.667239	−0.333619	−	0.942708i	$-0.608270\pi$
$193$	−1789.03	−0.667239	−0.333619	+	0.942708i	$0.608270\pi$
$194$	0	0
$195$	9.22778	0.00338879
$196$	0	0
$197$	4543.96	1.64337	0.821684	−	0.569943i	$-0.193035\pi$
$197$	4543.96	1.64337	0.821684	+	0.569943i	$0.193035\pi$
$198$	0	0
$199$	2342.65	0.834504	0.417252	−	0.908791i	$-0.362993\pi$
$199$	2342.65	0.834504	0.417252	+	0.908791i	$0.362993\pi$
$200$	0	0
$201$	−3.94967	−0.00138601
$202$	0	0
$203$	−5572.15	−1.92654
$204$	0	0
$205$	−65.7496	−0.0224007
$206$	0	0
$207$	−409.888	−0.137629
$208$	0	0
$209$	−4379.41	−1.44943
$210$	0	0
$211$	−757.113	−0.247023	−0.123511	−	0.992343i	$-0.539416\pi$
$211$	−757.113	−0.247023	−0.123511	+	0.992343i	$0.539416\pi$
$212$	0	0
$213$	63.1460	0.0203131
$214$	0	0
$215$	97.9510	0.0310707
$216$	0	0
$217$	−7673.57	−2.40053
$218$	0	0
$219$	2214.29	0.683232
$220$	0	0
$221$	−1477.88	−0.449832
$222$	0	0
$223$	4913.72	1.47555	0.737773	−	0.675049i	$-0.235878\pi$
$223$	4913.72	1.47555	0.737773	+	0.675049i	$0.235878\pi$
$224$	0	0
$225$	2226.30	0.659643
$226$	0	0
$227$	4162.00	1.21692	0.608462	−	0.793583i	$-0.291787\pi$
$227$	4162.00	1.21692	0.608462	+	0.793583i	$0.291787\pi$
$228$	0	0
$229$	−1523.65	−0.439674	−0.219837	−	0.975537i	$-0.570553\pi$
$229$	−1523.65	−0.439674	−0.219837	+	0.975537i	$0.570553\pi$
$230$	0	0
$231$	6457.33	1.83923
$232$	0	0
$233$	738.594	0.207669	0.103834	−	0.994595i	$-0.466889\pi$
$233$	738.594	0.207669	0.103834	+	0.994595i	$0.466889\pi$
$234$	0	0
$235$	38.0719	0.0105682
$236$	0	0
$237$	1005.07	0.275470
$238$	0	0
$239$	−5561.08	−1.50509	−0.752545	−	0.658540i	$-0.771174\pi$
$239$	−5561.08	−1.50509	−0.752545	+	0.658540i	$0.771174\pi$
$240$	0	0
$241$	−3194.91	−0.853952	−0.426976	−	0.904263i	$-0.640421\pi$
$241$	−3194.91	−0.853952	−0.426976	+	0.904263i	$0.640421\pi$
$242$	0	0
$243$	3877.78	1.02370
$244$	0	0
$245$	268.637	0.0700514
$246$	0	0
$247$	823.538	0.212148
$248$	0	0
$249$	−974.924	−0.248126
$250$	0	0
$251$	1224.38	0.307898	0.153949	−	0.988079i	$-0.450801\pi$
$251$	1224.38	0.307898	0.153949	+	0.988079i	$0.450801\pi$
$252$	0	0
$253$	−1350.78	−0.335664
$254$	0	0
$255$	−111.811	−0.0274584
$256$	0	0
$257$	2333.45	0.566369	0.283185	−	0.959065i	$-0.408609\pi$
$257$	2333.45	0.566369	0.283185	+	0.959065i	$0.408609\pi$
$258$	0	0
$259$	4526.43	1.08594
$260$	0	0
$261$	−2736.26	−0.648927
$262$	0	0
$263$	−4321.81	−1.01329	−0.506643	−	0.862156i	$-0.669114\pi$
$263$	−4321.81	−1.01329	−0.506643	+	0.862156i	$0.669114\pi$
$264$	0	0
$265$	−155.404	−0.0360241
$266$	0	0
$267$	−1092.72	−0.250463
$268$	0	0
$269$	−4535.26	−1.02795	−0.513977	−	0.857804i	$-0.671828\pi$
$269$	−4535.26	−1.02795	−0.513977	+	0.857804i	$0.671828\pi$
$270$	0	0
$271$	−2711.86	−0.607874	−0.303937	−	0.952692i	$-0.598301\pi$
$271$	−2711.86	−0.607874	−0.303937	+	0.952692i	$0.598301\pi$
$272$	0	0
$273$	−1214.29	−0.269201
$274$	0	0
$275$	7336.74	1.60881
$276$	0	0
$277$	8317.49	1.80415	0.902075	−	0.431578i	$-0.142043\pi$
$277$	8317.49	1.80415	0.902075	+	0.431578i	$0.142043\pi$
$278$	0	0
$279$	−3768.18	−0.808584
$280$	0	0
$281$	6004.12	1.27465	0.637324	−	0.770596i	$-0.280041\pi$
$281$	6004.12	1.27465	0.637324	+	0.770596i	$0.280041\pi$
$282$	0	0
$283$	8202.04	1.72283	0.861415	−	0.507902i	$-0.169579\pi$
$283$	8202.04	1.72283	0.861415	+	0.507902i	$0.169579\pi$
$284$	0	0
$285$	62.3060	0.0129498
$286$	0	0
$287$	8652.02	1.77949
$288$	0	0
$289$	12994.2	2.64485
$290$	0	0
$291$	−266.222	−0.0536296
$292$	0	0
$293$	2324.09	0.463395	0.231698	−	0.972788i	$-0.425572\pi$
$293$	2324.09	0.463395	0.231698	+	0.972788i	$0.425572\pi$
$294$	0	0
$295$	−65.8447	−0.0129954
$296$	0	0
$297$	7975.05	1.55811
$298$	0	0
$299$	254.011	0.0491300
$300$	0	0
$301$	−12889.4	−2.46822
$302$	0	0
$303$	1977.51	0.374934
$304$	0	0
$305$	115.648	0.0217114
$306$	0	0
$307$	−3462.82	−0.643758	−0.321879	−	0.946781i	$-0.604314\pi$
$307$	−3462.82	−0.643758	−0.321879	+	0.946781i	$0.604314\pi$
$308$	0	0
$309$	3784.70	0.696777
$310$	0	0
$311$	6510.92	1.18714	0.593570	−	0.804782i	$-0.297718\pi$
$311$	6510.92	1.18714	0.593570	+	0.804782i	$0.297718\pi$
$312$	0	0
$313$	−9089.42	−1.64142	−0.820710	−	0.571345i	$-0.806422\pi$
$313$	−9089.42	−1.64142	−0.820710	+	0.571345i	$0.806422\pi$
$314$	0	0
$315$	178.369	0.0319046
$316$	0	0
$317$	7467.44	1.32307	0.661535	−	0.749914i	$-0.269905\pi$
$317$	7467.44	1.32307	0.661535	+	0.749914i	$0.269905\pi$
$318$	0	0
$319$	−9017.30	−1.58267
$320$	0	0
$321$	3572.22	0.621128
$322$	0	0
$323$	−9978.64	−1.71897
$324$	0	0
$325$	−1379.66	−0.235476
$326$	0	0
$327$	−4600.25	−0.777964
$328$	0	0
$329$	−5009.89	−0.839527
$330$	0	0
$331$	11210.4	1.86157	0.930785	−	0.365568i	$-0.119125\pi$
$331$	11210.4	1.86157	0.930785	+	0.365568i	$0.119125\pi$
$332$	0	0
$333$	2222.75	0.365783
$334$	0	0
$335$	−0.359540	−5.86381e−5 0
$336$	0	0
$337$	−4895.19	−0.791271	−0.395635	−	0.918408i	$-0.629476\pi$
$337$	−4895.19	−0.791271	−0.395635	+	0.918408i	$0.629476\pi$
$338$	0	0
$339$	−5474.41	−0.877078
$340$	0	0
$341$	−12418.0	−1.97206
$342$	0	0
$343$	−22902.1	−3.60524
$344$	0	0
$345$	19.2176	0.00299896
$346$	0	0
$347$	4287.33	0.663274	0.331637	−	0.943407i	$-0.392399\pi$
$347$	4287.33	0.663274	0.331637	+	0.943407i	$0.392399\pi$
$348$	0	0
$349$	1058.89	0.162409	0.0812047	−	0.996697i	$-0.474123\pi$
$349$	1058.89	0.162409	0.0812047	+	0.996697i	$0.474123\pi$
$350$	0	0
$351$	−1499.69	−0.228056
$352$	0	0
$353$	8788.65	1.32514	0.662568	−	0.749002i	$-0.269467\pi$
$353$	8788.65	1.32514	0.662568	+	0.749002i	$0.269467\pi$
$354$	0	0
$355$	5.74820	0.000859388 0
$356$	0	0
$357$	14713.3	2.18126
$358$	0	0
$359$	9208.23	1.35374	0.676869	−	0.736104i	$-0.263336\pi$
$359$	9208.23	1.35374	0.676869	+	0.736104i	$0.263336\pi$
$360$	0	0
$361$	−1298.47	−0.189309
$362$	0	0
$363$	6417.31	0.927883
$364$	0	0
$365$	201.568	0.0289056
$366$	0	0
$367$	429.163	0.0610413	0.0305206	−	0.999534i	$-0.490283\pi$
$367$	429.163	0.0610413	0.0305206	+	0.999534i	$0.490283\pi$
$368$	0	0
$369$	4248.65	0.599393
$370$	0	0
$371$	20449.7	2.86171
$372$	0	0
$373$	−2126.51	−0.295192	−0.147596	−	0.989048i	$-0.547154\pi$
$373$	−2126.51	−0.295192	−0.147596	+	0.989048i	$0.547154\pi$
$374$	0	0
$375$	−208.824	−0.0287563
$376$	0	0
$377$	1695.68	0.231650
$378$	0	0
$379$	−7544.01	−1.02245	−0.511227	−	0.859446i	$-0.670809\pi$
$379$	−7544.01	−1.02245	−0.511227	+	0.859446i	$0.670809\pi$
$380$	0	0
$381$	3790.11	0.509642
$382$	0	0
$383$	2150.83	0.286951	0.143476	−	0.989654i	$-0.454172\pi$
$383$	2150.83	0.286951	0.143476	+	0.989654i	$0.454172\pi$
$384$	0	0
$385$	587.813	0.0778123
$386$	0	0
$387$	−6329.46	−0.831382
$388$	0	0
$389$	−7353.64	−0.958469	−0.479235	−	0.877687i	$-0.659086\pi$
$389$	−7353.64	−0.958469	−0.479235	+	0.877687i	$0.659086\pi$
$390$	0	0
$391$	−3077.81	−0.398085
$392$	0	0
$393$	−6636.68	−0.851848
$394$	0	0
$395$	91.4919	0.0116543
$396$	0	0
$397$	−968.391	−0.122424	−0.0612118	−	0.998125i	$-0.519497\pi$
$397$	−968.391	−0.122424	−0.0612118	+	0.998125i	$0.519497\pi$
$398$	0	0
$399$	−8198.87	−1.02871
$400$	0	0
$401$	10728.6	1.33606	0.668032	−	0.744132i	$-0.267137\pi$
$401$	10728.6	1.33606	0.668032	+	0.744132i	$0.267137\pi$
$402$	0	0
$403$	2335.18	0.288644
$404$	0	0
$405$	19.2414	0.00236078
$406$	0	0
$407$	7325.03	0.892109
$408$	0	0
$409$	−4611.10	−0.557468	−0.278734	−	0.960368i	$-0.589915\pi$
$409$	−4611.10	−0.557468	−0.278734	+	0.960368i	$0.589915\pi$
$410$	0	0
$411$	220.843	0.0265046
$412$	0	0
$413$	8664.54	1.03233
$414$	0	0
$415$	−88.7477	−0.0104975
$416$	0	0
$417$	−7370.61	−0.865564
$418$	0	0
$419$	2791.09	0.325426	0.162713	−	0.986673i	$-0.447975\pi$
$419$	2791.09	0.325426	0.162713	+	0.986673i	$0.447975\pi$
$420$	0	0
$421$	8033.51	0.929998	0.464999	−	0.885311i	$-0.346055\pi$
$421$	8033.51	0.929998	0.464999	+	0.885311i	$0.346055\pi$
$422$	0	0
$423$	−2460.15	−0.282782
$424$	0	0
$425$	16717.0	1.90799
$426$	0	0
$427$	−15218.2	−1.72473
$428$	0	0
$429$	−1965.06	−0.221151
$430$	0	0
$431$	3217.53	0.359589	0.179795	−	0.983704i	$-0.442457\pi$
$431$	3217.53	0.359589	0.179795	+	0.983704i	$0.442457\pi$
$432$	0	0
$433$	−2296.56	−0.254886	−0.127443	−	0.991846i	$-0.540677\pi$
$433$	−2296.56	−0.254886	−0.127443	+	0.991846i	$0.540677\pi$
$434$	0	0
$435$	128.290	0.0141403
$436$	0	0
$437$	1715.09	0.187743
$438$	0	0
$439$	16907.4	1.83815	0.919074	−	0.394085i	$-0.128938\pi$
$439$	16907.4	1.83815	0.919074	+	0.394085i	$0.128938\pi$
$440$	0	0
$441$	−17359.0	−1.87442
$442$	0	0
$443$	−14784.6	−1.58564	−0.792821	−	0.609455i	$-0.791388\pi$
$443$	−14784.6	−1.58564	−0.792821	+	0.609455i	$0.791388\pi$
$444$	0	0
$445$	−99.4711	−0.0105964
$446$	0	0
$447$	−7816.80	−0.827119
$448$	0	0
$449$	9675.76	1.01699	0.508494	−	0.861066i	$-0.330203\pi$
$449$	9675.76	1.01699	0.508494	+	0.861066i	$0.330203\pi$
$450$	0	0
$451$	14001.4	1.46186
$452$	0	0
$453$	−4047.31	−0.419777
$454$	0	0
$455$	−110.537	−0.0113891
$456$	0	0
$457$	−10213.4	−1.04543	−0.522714	−	0.852508i	$-0.675081\pi$
$457$	−10213.4	−1.04543	−0.522714	+	0.852508i	$0.675081\pi$
$458$	0	0
$459$	18171.5	1.84787
$460$	0	0
$461$	−15680.5	−1.58419	−0.792097	−	0.610395i	$-0.791011\pi$
$461$	−15680.5	−1.58419	−0.792097	+	0.610395i	$0.791011\pi$
$462$	0	0
$463$	652.571	0.0655022	0.0327511	−	0.999464i	$-0.489573\pi$
$463$	652.571	0.0655022	0.0327511	+	0.999464i	$0.489573\pi$
$464$	0	0
$465$	176.671	0.0176192
$466$	0	0
$467$	11175.8	1.10740	0.553700	−	0.832716i	$-0.313216\pi$
$467$	11175.8	1.10740	0.553700	+	0.832716i	$0.313216\pi$
$468$	0	0
$469$	47.3120	0.00465813
$470$	0	0
$471$	4034.43	0.394686
$472$	0	0
$473$	−20858.7	−2.02766
$474$	0	0
$475$	−9315.45	−0.899836
$476$	0	0
$477$	10042.0	0.963924
$478$	0	0
$479$	−7598.05	−0.724768	−0.362384	−	0.932029i	$-0.618037\pi$
$479$	−7598.05	−0.724768	−0.362384	+	0.932029i	$0.618037\pi$
$480$	0	0
$481$	−1377.46	−0.130575
$482$	0	0
$483$	−2528.85	−0.238234
$484$	0	0
$485$	−24.2343	−0.00226891
$486$	0	0
$487$	6187.04	0.575691	0.287846	−	0.957677i	$-0.407061\pi$
$487$	6187.04	0.575691	0.287846	+	0.957677i	$0.407061\pi$
$488$	0	0
$489$	−126.574	−0.0117052
$490$	0	0
$491$	−8467.85	−0.778307	−0.389153	−	0.921173i	$-0.627232\pi$
$491$	−8467.85	−0.778307	−0.389153	+	0.921173i	$0.627232\pi$
$492$	0	0
$493$	−20546.3	−1.87699
$494$	0	0
$495$	288.651	0.0262099
$496$	0	0
$497$	−756.408	−0.0682687
$498$	0	0
$499$	12653.8	1.13519	0.567597	−	0.823307i	$-0.307873\pi$
$499$	12653.8	1.13519	0.567597	+	0.823307i	$0.307873\pi$
$500$	0	0
$501$	−2252.41	−0.200859
$502$	0	0
$503$	12749.6	1.13017	0.565086	−	0.825032i	$-0.308843\pi$
$503$	12749.6	1.13017	0.565086	+	0.825032i	$0.308843\pi$
$504$	0	0
$505$	180.014	0.0158624
$506$	0	0
$507$	−6286.62	−0.550687
$508$	0	0
$509$	10596.7	0.922771	0.461386	−	0.887200i	$-0.347352\pi$
$509$	10596.7	0.922771	0.461386	+	0.887200i	$0.347352\pi$
$510$	0	0
$511$	−26524.4	−2.29622
$512$	0	0
$513$	−10125.9	−0.871482
$514$	0	0
$515$	344.523	0.0294786
$516$	0	0
$517$	−8107.41	−0.689678
$518$	0	0
$519$	−5983.82	−0.506089
$520$	0	0
$521$	19725.1	1.65868	0.829340	−	0.558744i	$-0.188717\pi$
$521$	19725.1	1.65868	0.829340	+	0.558744i	$0.188717\pi$
$522$	0	0
$523$	−5389.96	−0.450643	−0.225322	−	0.974284i	$-0.572343\pi$
$523$	−5389.96	−0.450643	−0.225322	+	0.974284i	$0.572343\pi$
$524$	0	0
$525$	13735.4	1.14183
$526$	0	0
$527$	−28294.9	−2.33879
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	4254.80	0.347726
$532$	0	0
$533$	−2632.93	−0.213968
$534$	0	0
$535$	325.181	0.0262781
$536$	0	0
$537$	4142.11	0.332859
$538$	0	0
$539$	−57206.3	−4.57152
$540$	0	0
$541$	−1979.56	−0.157316	−0.0786580	−	0.996902i	$-0.525063\pi$
$541$	−1979.56	−0.157316	−0.0786580	+	0.996902i	$0.525063\pi$
$542$	0	0
$543$	−9507.50	−0.751392
$544$	0	0
$545$	−418.762	−0.0329134
$546$	0	0
$547$	−8174.95	−0.639005	−0.319502	−	0.947586i	$-0.603516\pi$
$547$	−8174.95	−0.639005	−0.319502	+	0.947586i	$0.603516\pi$
$548$	0	0
$549$	−7473.02	−0.580949
$550$	0	0
$551$	11449.3	0.885218
$552$	0	0
$553$	−12039.5	−0.925804
$554$	0	0
$555$	−104.214	−0.00797048
$556$	0	0
$557$	−7290.39	−0.554585	−0.277292	−	0.960786i	$-0.589437\pi$
$557$	−7290.39	−0.554585	−0.277292	+	0.960786i	$0.589437\pi$
$558$	0	0
$559$	3922.43	0.296782
$560$	0	0
$561$	23810.2	1.79192
$562$	0	0
$563$	−21688.3	−1.62354	−0.811771	−	0.583976i	$-0.801496\pi$
$563$	−21688.3	−1.62354	−0.811771	+	0.583976i	$0.801496\pi$
$564$	0	0
$565$	−498.338	−0.0371066
$566$	0	0
$567$	−2531.99	−0.187537
$568$	0	0
$569$	5643.16	0.415771	0.207885	−	0.978153i	$-0.433342\pi$
$569$	5643.16	0.415771	0.207885	+	0.978153i	$0.433342\pi$
$570$	0	0
$571$	4777.47	0.350141	0.175071	−	0.984556i	$-0.443985\pi$
$571$	4777.47	0.350141	0.175071	+	0.984556i	$0.443985\pi$
$572$	0	0
$573$	−912.296	−0.0665126
$574$	0	0
$575$	−2873.25	−0.208388
$576$	0	0
$577$	22322.7	1.61058	0.805291	−	0.592879i	$-0.202009\pi$
$577$	22322.7	1.61058	0.805291	+	0.592879i	$0.202009\pi$
$578$	0	0
$579$	−5420.13	−0.389038
$580$	0	0
$581$	11678.3	0.833906
$582$	0	0
$583$	33093.3	2.35092
$584$	0	0
$585$	−54.2802	−0.00383626
$586$	0	0
$587$	11591.2	0.815022	0.407511	−	0.913200i	$-0.366397\pi$
$587$	11591.2	0.815022	0.407511	+	0.913200i	$0.366397\pi$
$588$	0	0
$589$	15767.1	1.10301
$590$	0	0
$591$	13766.6	0.958177
$592$	0	0
$593$	−16207.8	−1.12238	−0.561192	−	0.827686i	$-0.689657\pi$
$593$	−16207.8	−1.12238	−0.561192	+	0.827686i	$0.689657\pi$
$594$	0	0
$595$	1339.35	0.0922827
$596$	0	0
$597$	7097.42	0.486563
$598$	0	0
$599$	−11167.4	−0.761748	−0.380874	−	0.924627i	$-0.624377\pi$
$599$	−11167.4	−0.761748	−0.380874	+	0.924627i	$0.624377\pi$
$600$	0	0
$601$	508.029	0.0344807	0.0172404	−	0.999851i	$-0.494512\pi$
$601$	508.029	0.0344807	0.0172404	+	0.999851i	$0.494512\pi$
$602$	0	0
$603$	23.2330	0.00156902
$604$	0	0
$605$	584.170	0.0392560
$606$	0	0
$607$	24285.6	1.62392	0.811962	−	0.583711i	$-0.198400\pi$
$607$	24285.6	1.62392	0.811962	+	0.583711i	$0.198400\pi$
$608$	0	0
$609$	−16881.7	−1.12328
$610$	0	0
$611$	1524.58	0.100946
$612$	0	0
$613$	9400.66	0.619395	0.309697	−	0.950835i	$-0.399772\pi$
$613$	9400.66	0.619395	0.309697	+	0.950835i	$0.399772\pi$
$614$	0	0
$615$	−199.198	−0.0130609
$616$	0	0
$617$	6351.99	0.414460	0.207230	−	0.978292i	$-0.433555\pi$
$617$	6351.99	0.414460	0.207230	+	0.978292i	$0.433555\pi$
$618$	0	0
$619$	−25037.4	−1.62575	−0.812874	−	0.582440i	$-0.802098\pi$
$619$	−25037.4	−1.62575	−0.812874	+	0.582440i	$0.802098\pi$
$620$	0	0
$621$	−3123.23	−0.201821
$622$	0	0
$623$	13089.4	0.841762
$624$	0	0
$625$	15596.5	0.998175
$626$	0	0
$627$	−13268.1	−0.845097
$628$	0	0
$629$	16690.4	1.05801
$630$	0	0
$631$	−21260.7	−1.34132	−0.670662	−	0.741763i	$-0.733990\pi$
$631$	−21260.7	−1.34132	−0.670662	+	0.741763i	$0.733990\pi$
$632$	0	0
$633$	−2293.79	−0.144028
$634$	0	0
$635$	345.016	0.0215615
$636$	0	0
$637$	10757.5	0.669119
$638$	0	0
$639$	−371.441	−0.0229953
$640$	0	0
$641$	−27569.8	−1.69882	−0.849410	−	0.527734i	$-0.823042\pi$
$641$	−27569.8	−1.69882	−0.849410	+	0.527734i	$0.823042\pi$
$642$	0	0
$643$	11037.8	0.676964	0.338482	−	0.940973i	$-0.390086\pi$
$643$	11037.8	0.676964	0.338482	+	0.940973i	$0.390086\pi$
$644$	0	0
$645$	296.757	0.0181160
$646$	0	0
$647$	−74.1252	−0.00450412	−0.00225206	−	0.999997i	$-0.500717\pi$
$647$	−74.1252	−0.00450412	−0.00225206	+	0.999997i	$0.500717\pi$
$648$	0	0
$649$	14021.7	0.848071
$650$	0	0
$651$	−23248.2	−1.39965
$652$	0	0
$653$	6413.33	0.384338	0.192169	−	0.981362i	$-0.438448\pi$
$653$	6413.33	0.384338	0.192169	+	0.981362i	$0.438448\pi$
$654$	0	0
$655$	−604.140	−0.0360392
$656$	0	0
$657$	−13025.0	−0.773447
$658$	0	0
$659$	−13664.1	−0.807708	−0.403854	−	0.914824i	$-0.632330\pi$
$659$	−13664.1	−0.807708	−0.403854	+	0.914824i	$0.632330\pi$
$660$	0	0
$661$	−25788.6	−1.51749	−0.758743	−	0.651390i	$-0.774186\pi$
$661$	−25788.6	−1.51749	−0.758743	+	0.651390i	$0.774186\pi$
$662$	0	0
$663$	−4477.46	−0.262278
$664$	0	0
$665$	−746.346	−0.0435219
$666$	0	0
$667$	3531.40	0.205002
$668$	0	0
$669$	14886.8	0.860327
$670$	0	0
$671$	−24627.3	−1.41688
$672$	0	0
$673$	8273.47	0.473876	0.236938	−	0.971525i	$-0.423856\pi$
$673$	8273.47	0.473876	0.236938	+	0.971525i	$0.423856\pi$
$674$	0	0
$675$	16963.8	0.967311
$676$	0	0
$677$	23309.3	1.32326	0.661632	−	0.749829i	$-0.269864\pi$
$677$	23309.3	1.32326	0.661632	+	0.749829i	$0.269864\pi$
$678$	0	0
$679$	3189.00	0.180240
$680$	0	0
$681$	12609.4	0.709535
$682$	0	0
$683$	−11157.4	−0.625073	−0.312536	−	0.949906i	$-0.601179\pi$
$683$	−11157.4	−0.625073	−0.312536	+	0.949906i	$0.601179\pi$
$684$	0	0
$685$	20.1034	0.00112133
$686$	0	0
$687$	−4616.11	−0.256355
$688$	0	0
$689$	−6223.13	−0.344096
$690$	0	0
$691$	−11339.6	−0.624282	−0.312141	−	0.950036i	$-0.601046\pi$
$691$	−11339.6	−0.624282	−0.312141	+	0.950036i	$0.601046\pi$
$692$	0	0
$693$	−37983.7	−2.08208
$694$	0	0
$695$	−670.949	−0.0366195
$696$	0	0
$697$	31902.7	1.73372
$698$	0	0
$699$	2237.68	0.121083
$700$	0	0
$701$	24605.3	1.32572	0.662859	−	0.748744i	$-0.269343\pi$
$701$	24605.3	1.32572	0.662859	+	0.748744i	$0.269343\pi$
$702$	0	0
$703$	−9300.59	−0.498974
$704$	0	0
$705$	115.344	0.00616188
$706$	0	0
$707$	−23688.1	−1.26009
$708$	0	0
$709$	18362.1	0.972644	0.486322	−	0.873780i	$-0.338338\pi$
$709$	18362.1	0.972644	0.486322	+	0.873780i	$0.338338\pi$
$710$	0	0
$711$	−5912.09	−0.311843
$712$	0	0
$713$	4863.20	0.255439
$714$	0	0
$715$	−178.880	−0.00935627
$716$	0	0
$717$	−16848.1	−0.877553
$718$	0	0
$719$	7706.37	0.399721	0.199860	−	0.979824i	$-0.435951\pi$
$719$	7706.37	0.399721	0.199860	+	0.979824i	$0.435951\pi$
$720$	0	0
$721$	−45335.9	−2.34174
$722$	0	0
$723$	−9679.47	−0.497902
$724$	0	0
$725$	−19180.7	−0.982558
$726$	0	0
$727$	9051.76	0.461776	0.230888	−	0.972980i	$-0.425837\pi$
$727$	9051.76	0.461776	0.230888	+	0.972980i	$0.425837\pi$
$728$	0	0
$729$	9864.56	0.501172
$730$	0	0
$731$	−47527.3	−2.40473
$732$	0	0
$733$	−9636.50	−0.485583	−0.242791	−	0.970079i	$-0.578063\pi$
$733$	−9636.50	−0.485583	−0.242791	+	0.970079i	$0.578063\pi$
$734$	0	0
$735$	813.876	0.0408439
$736$	0	0
$737$	76.5641	0.00382670
$738$	0	0
$739$	2264.22	0.112707	0.0563537	−	0.998411i	$-0.482053\pi$
$739$	2264.22	0.112707	0.0563537	+	0.998411i	$0.482053\pi$
$740$	0	0
$741$	2495.03	0.123694
$742$	0	0
$743$	4566.17	0.225460	0.112730	−	0.993626i	$-0.464041\pi$
$743$	4566.17	0.225460	0.112730	+	0.993626i	$0.464041\pi$
$744$	0	0
$745$	−711.566	−0.0349930
$746$	0	0
$747$	5734.76	0.280889
$748$	0	0
$749$	−42790.7	−2.08750
$750$	0	0
$751$	7425.66	0.360807	0.180404	−	0.983593i	$-0.442260\pi$
$751$	7425.66	0.360807	0.180404	+	0.983593i	$0.442260\pi$
$752$	0	0
$753$	3709.46	0.179522
$754$	0	0
$755$	−368.428	−0.0177596
$756$	0	0
$757$	−5702.50	−0.273792	−0.136896	−	0.990585i	$-0.543713\pi$
$757$	−5702.50	−0.273792	−0.136896	+	0.990585i	$0.543713\pi$
$758$	0	0
$759$	−4092.40	−0.195711
$760$	0	0
$761$	−2718.52	−0.129496	−0.0647479	−	0.997902i	$-0.520624\pi$
$761$	−2718.52	−0.129496	−0.0647479	+	0.997902i	$0.520624\pi$
$762$	0	0
$763$	55105.1	2.61460
$764$	0	0
$765$	657.703	0.0310840
$766$	0	0
$767$	−2636.74	−0.124129
$768$	0	0
$769$	−22348.6	−1.04800	−0.524000	−	0.851718i	$-0.675561\pi$
$769$	−22348.6	−1.04800	−0.524000	+	0.851718i	$0.675561\pi$
$770$	0	0
$771$	7069.55	0.330225
$772$	0	0
$773$	−29249.0	−1.36095	−0.680473	−	0.732773i	$-0.738226\pi$
$773$	−29249.0	−1.36095	−0.680473	+	0.732773i	$0.738226\pi$
$774$	0	0
$775$	−26414.3	−1.22430
$776$	0	0
$777$	13713.5	0.633165
$778$	0	0
$779$	−17777.6	−0.817647
$780$	0	0
$781$	−1224.08	−0.0560833
$782$	0	0
$783$	−20849.5	−0.951597
$784$	0	0
$785$	367.256	0.0166980
$786$	0	0
$787$	13674.7	0.619378	0.309689	−	0.950838i	$-0.399775\pi$
$787$	13674.7	0.619378	0.309689	+	0.950838i	$0.399775\pi$
$788$	0	0
$789$	−13093.6	−0.590804
$790$	0	0
$791$	65576.5	2.94770
$792$	0	0
$793$	4631.10	0.207384
$794$	0	0
$795$	−470.820	−0.0210041
$796$	0	0
$797$	−27180.4	−1.20800	−0.604002	−	0.796983i	$-0.706428\pi$
$797$	−27180.4	−1.20800	−0.604002	+	0.796983i	$0.706428\pi$
$798$	0	0
$799$	−18473.0	−0.817934
$800$	0	0
$801$	6427.69	0.283535
$802$	0	0
$803$	−42923.9	−1.88636
$804$	0	0
$805$	−230.202	−0.0100790
$806$	0	0
$807$	−13740.3	−0.599356
$808$	0	0
$809$	5525.23	0.240119	0.120060	−	0.992767i	$-0.461691\pi$
$809$	5525.23	0.240119	0.120060	+	0.992767i	$0.461691\pi$
$810$	0	0
$811$	−33085.9	−1.43256	−0.716278	−	0.697815i	$-0.754156\pi$
$811$	−33085.9	−1.43256	−0.716278	+	0.697815i	$0.754156\pi$
$812$	0	0
$813$	−8215.99	−0.354425
$814$	0	0
$815$	−11.5220	−0.000495214 0
$816$	0	0
$817$	26484.3	1.13411
$818$	0	0
$819$	7142.76	0.304747
$820$	0	0
$821$	−22886.6	−0.972896	−0.486448	−	0.873710i	$-0.661708\pi$
$821$	−22886.6	−0.972896	−0.486448	+	0.873710i	$0.661708\pi$
$822$	0	0
$823$	29113.5	1.23309	0.616545	−	0.787319i	$-0.288532\pi$
$823$	29113.5	1.23309	0.616545	+	0.787319i	$0.288532\pi$
$824$	0	0
$825$	22227.8	0.938025
$826$	0	0
$827$	−28107.4	−1.18185	−0.590925	−	0.806726i	$-0.701237\pi$
$827$	−28107.4	−1.18185	−0.590925	+	0.806726i	$0.701237\pi$
$828$	0	0
$829$	−28553.2	−1.19625	−0.598126	−	0.801402i	$-0.704088\pi$
$829$	−28553.2	−1.19625	−0.598126	+	0.801402i	$0.704088\pi$
$830$	0	0
$831$	25199.1	1.05192
$832$	0	0
$833$	−130347.	−5.42167
$834$	0	0
$835$	−205.037	−0.00849774
$836$	0	0
$837$	−28712.5	−1.18572
$838$	0	0
$839$	2444.15	0.100574	0.0502869	−	0.998735i	$-0.483986\pi$
$839$	2444.15	0.100574	0.0502869	+	0.998735i	$0.483986\pi$
$840$	0	0
$841$	−814.691	−0.0334040
$842$	0	0
$843$	18190.4	0.743192
$844$	0	0
$845$	−572.273	−0.0232980
$846$	0	0
$847$	−76871.2	−3.11845
$848$	0	0
$849$	24849.3	1.00451
$850$	0	0
$851$	−2868.67	−0.115554
$852$	0	0
$853$	18469.0	0.741344	0.370672	−	0.928764i	$-0.379127\pi$
$853$	18469.0	0.741344	0.370672	+	0.928764i	$0.379127\pi$
$854$	0	0
$855$	−366.500	−0.0146597
$856$	0	0
$857$	−9906.35	−0.394860	−0.197430	−	0.980317i	$-0.563259\pi$
$857$	−9906.35	−0.394860	−0.197430	+	0.980317i	$0.563259\pi$
$858$	0	0
$859$	15351.1	0.609748	0.304874	−	0.952393i	$-0.401386\pi$
$859$	15351.1	0.609748	0.304874	+	0.952393i	$0.401386\pi$
$860$	0	0
$861$	26212.6	1.03754
$862$	0	0
$863$	−6654.73	−0.262491	−0.131245	−	0.991350i	$-0.541898\pi$
$863$	−6654.73	−0.262491	−0.131245	+	0.991350i	$0.541898\pi$
$864$	0	0
$865$	−544.709	−0.0214112
$866$	0	0
$867$	39367.8	1.54210
$868$	0	0
$869$	−19483.2	−0.760556
$870$	0	0
$871$	−14.3977	−0.000560101 0
$872$	0	0
$873$	1565.99	0.0607110
$874$	0	0
$875$	2501.44	0.0966447
$876$	0	0
$877$	−3982.77	−0.153351	−0.0766754	−	0.997056i	$-0.524431\pi$
$877$	−3982.77	−0.153351	−0.0766754	+	0.997056i	$0.524431\pi$
$878$	0	0
$879$	7041.18	0.270186
$880$	0	0
$881$	−15109.1	−0.577795	−0.288897	−	0.957360i	$-0.593289\pi$
$881$	−15109.1	−0.577795	−0.288897	+	0.957360i	$0.593289\pi$
$882$	0	0
$883$	−7041.73	−0.268373	−0.134186	−	0.990956i	$-0.542842\pi$
$883$	−7041.73	−0.268373	−0.134186	+	0.990956i	$0.542842\pi$
$884$	0	0
$885$	−199.487	−0.00757703
$886$	0	0
$887$	−7742.14	−0.293073	−0.146537	−	0.989205i	$-0.546813\pi$
$887$	−7742.14	−0.293073	−0.146537	+	0.989205i	$0.546813\pi$
$888$	0	0
$889$	−45400.7	−1.71281
$890$	0	0
$891$	−4097.47	−0.154063
$892$	0	0
$893$	10294.0	0.385750
$894$	0	0
$895$	377.057	0.0140823
$896$	0	0
$897$	769.566	0.0286456
$898$	0	0
$899$	32464.9	1.20441
$900$	0	0
$901$	75404.4	2.78811
$902$	0	0
$903$	−39050.4	−1.43911
$904$	0	0
$905$	−865.471	−0.0317892
$906$	0	0
$907$	6151.24	0.225191	0.112596	−	0.993641i	$-0.464084\pi$
$907$	6151.24	0.225191	0.112596	+	0.993641i	$0.464084\pi$
$908$	0	0
$909$	−11632.2	−0.424441
$910$	0	0
$911$	10999.0	0.400013	0.200006	−	0.979795i	$-0.435904\pi$
$911$	10999.0	0.400013	0.200006	+	0.979795i	$0.435904\pi$
$912$	0	0
$913$	18898.8	0.685061
$914$	0	0
$915$	350.373	0.0126590
$916$	0	0
$917$	79499.0	2.86291
$918$	0	0
$919$	35572.5	1.27686	0.638428	−	0.769682i	$-0.279585\pi$
$919$	35572.5	1.27686	0.638428	+	0.769682i	$0.279585\pi$
$920$	0	0
$921$	−10491.1	−0.375347
$922$	0	0
$923$	230.186	0.00820873
$924$	0	0
$925$	15581.1	0.553841
$926$	0	0
$927$	−22262.6	−0.788781
$928$	0	0
$929$	−29849.4	−1.05417	−0.527087	−	0.849811i	$-0.676716\pi$
$929$	−29849.4	−1.05417	−0.527087	+	0.849811i	$0.676716\pi$
$930$	0	0
$931$	72634.8	2.55694
$932$	0	0
$933$	19725.8	0.692170
$934$	0	0
$935$	2167.45	0.0758110
$936$	0	0
$937$	10376.2	0.361765	0.180883	−	0.983505i	$-0.442105\pi$
$937$	10376.2	0.361765	0.180883	+	0.983505i	$0.442105\pi$
$938$	0	0
$939$	−27537.8	−0.957041
$940$	0	0
$941$	45307.1	1.56957	0.784787	−	0.619766i	$-0.212772\pi$
$941$	45307.1	1.56957	0.784787	+	0.619766i	$0.212772\pi$
$942$	0	0
$943$	−5483.30	−0.189354
$944$	0	0
$945$	1359.12	0.0467854
$946$	0	0
$947$	17054.2	0.585203	0.292601	−	0.956235i	$-0.405479\pi$
$947$	17054.2	0.585203	0.292601	+	0.956235i	$0.405479\pi$
$948$	0	0
$949$	8071.74	0.276101
$950$	0	0
$951$	22623.7	0.771425
$952$	0	0
$953$	38930.2	1.32327	0.661633	−	0.749828i	$-0.269864\pi$
$953$	38930.2	1.32327	0.661633	+	0.749828i	$0.269864\pi$
$954$	0	0
$955$	−83.0466	−0.00281395
$956$	0	0
$957$	−27319.3	−0.922787
$958$	0	0
$959$	−2645.42	−0.0890773
$960$	0	0
$961$	14917.3	0.500732
$962$	0	0
$963$	−21012.8	−0.703143
$964$	0	0
$965$	−493.397	−0.0164591
$966$	0	0
$967$	−34227.9	−1.13826	−0.569129	−	0.822248i	$-0.692719\pi$
$967$	−34227.9	−1.13826	−0.569129	+	0.822248i	$0.692719\pi$
$968$	0	0
$969$	−30231.8	−1.00226
$970$	0	0
$971$	−44221.4	−1.46152	−0.730759	−	0.682636i	$-0.760834\pi$
$971$	−44221.4	−1.46152	−0.730759	+	0.682636i	$0.760834\pi$
$972$	0	0
$973$	88290.4	2.90900
$974$	0	0
$975$	−4179.88	−0.137296
$976$	0	0
$977$	35205.3	1.15283	0.576416	−	0.817157i	$-0.304451\pi$
$977$	35205.3	1.15283	0.576416	+	0.817157i	$0.304451\pi$
$978$	0	0
$979$	21182.4	0.691514
$980$	0	0
$981$	27059.9	0.880688
$982$	0	0
$983$	−37390.7	−1.21320	−0.606601	−	0.795006i	$-0.707468\pi$
$983$	−37390.7	−1.21320	−0.606601	+	0.795006i	$0.707468\pi$
$984$	0	0
$985$	1253.18	0.0405377
$986$	0	0
$987$	−15178.2	−0.489492
$988$	0	0
$989$	8168.79	0.262641
$990$	0	0
$991$	49921.9	1.60022	0.800112	−	0.599850i	$-0.204773\pi$
$991$	49921.9	1.60022	0.800112	+	0.599850i	$0.204773\pi$
$992$	0	0
$993$	33963.6	1.08540
$994$	0	0
$995$	646.081	0.0205851
$996$	0	0
$997$	27936.3	0.887414	0.443707	−	0.896172i	$-0.353663\pi$
$997$	27936.3	0.887414	0.443707	+	0.896172i	$0.353663\pi$
$998$	0	0
$999$	16936.7	0.536390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bk.1.6		9
4.3	odd	2		1472.4.a.bj.1.4		9
8.3	odd	2		736.4.a.h.1.6	✓	9
8.5	even	2		736.4.a.i.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.h.1.6	✓	9	8.3	odd	2
736.4.a.i.1.4	yes	9	8.5	even	2
1472.4.a.bj.1.4		9	4.3	odd	2
1472.4.a.bk.1.6		9	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+3.02965 q^{3} +0.275790 q^{5} -36.2914 q^{7} -17.8212 q^{9} -58.7296 q^{11} +11.0440 q^{13} +0.835548 q^{15} -133.818 q^{17} +74.5690 q^{19} -109.950 q^{21} +23.0000 q^{23} -124.924 q^{25} -135.793 q^{27} +153.539 q^{29} +211.443 q^{31} -177.930 q^{33} -10.0088 q^{35} -124.725 q^{37} +33.4594 q^{39} -238.404 q^{41} +355.165 q^{43} -4.91492 q^{45} +138.046 q^{47} +974.063 q^{49} -405.421 q^{51} -563.486 q^{53} -16.1971 q^{55} +225.918 q^{57} -238.749 q^{59} +419.333 q^{61} +646.756 q^{63} +3.04582 q^{65} -1.30367 q^{67} +69.6820 q^{69} +20.8426 q^{71} +730.873 q^{73} -378.476 q^{75} +2131.38 q^{77} +331.744 q^{79} +69.7684 q^{81} -321.794 q^{83} -36.9056 q^{85} +465.170 q^{87} -360.677 q^{89} -400.801 q^{91} +640.600 q^{93} +20.5654 q^{95} -87.8723 q^{97} +1046.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 42 q^{7} + 71 q^{9} - 66 q^{11} - 48 q^{13} + 90 q^{15} - 32 q^{17} - 6 q^{21} + 207 q^{23} + 135 q^{25} + 88 q^{29} + 556 q^{31} - 230 q^{33} - 368 q^{35} + 368 q^{37} + 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) \) 9 * q + 42 * q^7 + 71 * q^9 - 66 * q^11 - 48 * q^13 + 90 * q^15 - 32 * q^17 - 6 * q^21 + 207 * q^23 + 135 * q^25 + 88 * q^29 + 556 * q^31 - 230 * q^33 - 368 * q^35 + 368 * q^37 + 468 * q^39 - 216 * q^41 + 552 * q^43 + 206 * q^45 + 1300 * q^47 + 1041 * q^49 - 1658 * q^51 - 510 * q^53 + 1948 * q^55 - 804 * q^57 - 1444 * q^59 + 882 * q^61 + 3128 * q^63 + 334 * q^65 - 1186 * q^67 + 2440 * q^71 - 368 * q^73 - 1636 * q^75 - 556 * q^77 + 2576 * q^79 + 1069 * q^81 - 1710 * q^83 - 640 * q^85 + 4844 * q^87 - 474 * q^89 - 2790 * q^91 - 1178 * q^93 + 4564 * q^95 - 884 * q^97 - 552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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