LMFDB - Embedded newform 1200.4.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(1,1200)] 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("1200.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,0,0,-26,0,18,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +4.43560 q^{7} +9.00000 q^{9} +3.43560 q^{11} +78.7424 q^{13} +53.1780 q^{17} -20.4356 q^{19} -13.3068 q^{21} -118.307 q^{23} -27.0000 q^{27} +168.049 q^{29} +61.0492 q^{31} -10.3068 q^{33} -246.614 q^{37} -236.227 q^{39} +422.663 q^{41} -362.436 q^{43} +170.515 q^{47} -323.325 q^{49} -159.534 q^{51} -546.049 q^{53} +61.3068 q^{57} +216.970 q^{59} +130.902 q^{61} +39.9204 q^{63} +614.890 q^{67} +354.920 q^{69} -324.822 q^{71} -88.8712 q^{73} +15.2389 q^{77} +1137.42 q^{79} +81.0000 q^{81} +758.909 q^{83} -504.148 q^{87} +195.681 q^{89} +349.269 q^{91} -183.148 q^{93} +521.000 q^{97} +30.9204 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9} - 28 q^{11} + 18 q^{13} - 68 q^{17} - 6 q^{19} + 78 q^{21} - 132 q^{23} - 54 q^{27} + 92 q^{29} - 122 q^{31} + 84 q^{33} - 284 q^{37} - 54 q^{39} + 392 q^{41} - 690 q^{43}+ \cdots - 252 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 26 * q^7 + 18 * q^9 - 28 * q^11 + 18 * q^13 - 68 * q^17 - 6 * q^19 + 78 * q^21 - 132 * q^23 - 54 * q^27 + 92 * q^29 - 122 * q^31 + 84 * q^33 - 284 * q^37 - 54 * q^39 + 392 * q^41 - 690 * q^43 + 620 * q^47 + 260 * q^49 + 204 * q^51 - 848 * q^53 + 18 * q^57 - 124 * q^59 + 750 * q^61 - 234 * q^63 + 358 * q^67 + 396 * q^69 - 824 * q^71 - 108 * q^73 + 972 * q^77 + 880 * q^79 + 162 * q^81 - 156 * q^83 - 276 * q^87 - 864 * q^89 + 2198 * q^91 + 366 * q^93 + 1042 * q^97 - 252 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.43560	0.239500	0.119750	−	0.992804i	$-0.461791\pi$
$7$	4.43560	0.239500	0.119750	+	0.992804i	$0.461791\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	3.43560	0.0941701	0.0470851	−	0.998891i	$-0.485007\pi$
$11$	3.43560	0.0941701	0.0470851	+	0.998891i	$0.485007\pi$
$12$	0	0
$13$	78.7424	1.67994	0.839970	−	0.542634i	$-0.182573\pi$
$13$	78.7424	1.67994	0.839970	+	0.542634i	$0.182573\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	53.1780	0.758680	0.379340	−	0.925257i	$-0.376151\pi$
$17$	53.1780	0.758680	0.379340	+	0.925257i	$0.376151\pi$
$18$	0	0
$19$	−20.4356	−0.246750	−0.123375	−	0.992360i	$-0.539372\pi$
$19$	−20.4356	−0.246750	−0.123375	+	0.992360i	$0.539372\pi$
$20$	0	0
$21$	−13.3068	−0.138275
$22$	0	0
$23$	−118.307	−1.07255	−0.536275	−	0.844043i	$-0.680169\pi$
$23$	−118.307	−1.07255	−0.536275	+	0.844043i	$0.680169\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	168.049	1.07607	0.538034	−	0.842923i	$-0.319167\pi$
$29$	168.049	1.07607	0.538034	+	0.842923i	$0.319167\pi$
$30$	0	0
$31$	61.0492	0.353702	0.176851	−	0.984238i	$-0.443409\pi$
$31$	61.0492	0.353702	0.176851	+	0.984238i	$0.443409\pi$
$32$	0	0
$33$	−10.3068	−0.0543691
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−246.614	−1.09576	−0.547879	−	0.836558i	$-0.684564\pi$
$37$	−246.614	−1.09576	−0.547879	+	0.836558i	$0.684564\pi$
$38$	0	0
$39$	−236.227	−0.969913
$40$	0	0
$41$	422.663	1.60997	0.804986	−	0.593294i	$-0.202173\pi$
$41$	422.663	1.60997	0.804986	+	0.593294i	$0.202173\pi$
$42$	0	0
$43$	−362.436	−1.28537	−0.642685	−	0.766131i	$-0.722180\pi$
$43$	−362.436	−1.28537	−0.642685	+	0.766131i	$0.722180\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	170.515	0.529196	0.264598	−	0.964359i	$-0.414761\pi$
$47$	170.515	0.529196	0.264598	+	0.964359i	$0.414761\pi$
$48$	0	0
$49$	−323.325	−0.942640
$50$	0	0
$51$	−159.534	−0.438024
$52$	0	0
$53$	−546.049	−1.41520	−0.707600	−	0.706613i	$-0.750222\pi$
$53$	−546.049	−1.41520	−0.707600	+	0.706613i	$0.750222\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	61.3068	0.142461
$58$	0	0
$59$	216.970	0.478763	0.239382	−	0.970926i	$-0.423055\pi$
$59$	216.970	0.478763	0.239382	+	0.970926i	$0.423055\pi$
$60$	0	0
$61$	130.902	0.274758	0.137379	−	0.990519i	$-0.456132\pi$
$61$	130.902	0.274758	0.137379	+	0.990519i	$0.456132\pi$
$62$	0	0
$63$	39.9204	0.0798332
$64$	0	0
$65$	0	0
$66$	0	0
$67$	614.890	1.12121	0.560603	−	0.828085i	$-0.310570\pi$
$67$	614.890	1.12121	0.560603	+	0.828085i	$0.310570\pi$
$68$	0	0
$69$	354.920	0.619238
$70$	0	0
$71$	−324.822	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$71$	−324.822	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$72$	0	0
$73$	−88.8712	−0.142487	−0.0712437	−	0.997459i	$-0.522697\pi$
$73$	−88.8712	−0.142487	−0.0712437	+	0.997459i	$0.522697\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2389	0.0225537
$78$	0	0
$79$	1137.42	1.61988	0.809938	−	0.586516i	$-0.199501\pi$
$79$	1137.42	1.61988	0.809938	+	0.586516i	$0.199501\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	758.909	1.00363	0.501813	−	0.864976i	$-0.332666\pi$
$83$	758.909	1.00363	0.501813	+	0.864976i	$0.332666\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−504.148	−0.621268
$88$	0	0
$89$	195.681	0.233058	0.116529	−	0.993187i	$-0.462823\pi$
$89$	195.681	0.233058	0.116529	+	0.993187i	$0.462823\pi$
$90$	0	0
$91$	349.269	0.402345
$92$	0	0
$93$	−183.148	−0.204210
$94$	0	0
$95$	0	0
$96$	0	0
$97$	521.000	0.545356	0.272678	−	0.962105i	$-0.412091\pi$
$97$	521.000	0.545356	0.272678	+	0.962105i	$0.412091\pi$
$98$	0	0
$99$	30.9204	0.0313900
$100$	0	0
$101$	660.920	0.651129	0.325565	−	0.945520i	$-0.394446\pi$
$101$	660.920	0.651129	0.325565	+	0.945520i	$0.394446\pi$
$102$	0	0
$103$	−1530.75	−1.46436	−0.732181	−	0.681110i	$-0.761497\pi$
$103$	−1530.75	−1.46436	−0.732181	+	0.681110i	$0.761497\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	264.625	0.239087	0.119543	−	0.992829i	$-0.461857\pi$
$107$	264.625	0.239087	0.119543	+	0.992829i	$0.461857\pi$
$108$	0	0
$109$	1117.61	0.982091	0.491046	−	0.871134i	$-0.336615\pi$
$109$	1117.61	0.982091	0.491046	+	0.871134i	$0.336615\pi$
$110$	0	0
$111$	739.841	0.632636
$112$	0	0
$113$	−934.061	−0.777602	−0.388801	−	0.921322i	$-0.627111\pi$
$113$	−934.061	−0.777602	−0.388801	+	0.921322i	$0.627111\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	708.681	0.559980
$118$	0	0
$119$	235.876	0.181704
$120$	0	0
$121$	−1319.20	−0.991132
$122$	0	0
$123$	−1267.99	−0.929517
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−630.356	−0.440433	−0.220217	−	0.975451i	$-0.570676\pi$
$127$	−630.356	−0.440433	−0.220217	+	0.975451i	$0.570676\pi$
$128$	0	0
$129$	1087.31	0.742109
$130$	0	0
$131$	2163.06	1.44265	0.721325	−	0.692597i	$-0.243534\pi$
$131$	2163.06	1.44265	0.721325	+	0.692597i	$0.243534\pi$
$132$	0	0
$133$	−90.6440	−0.0590965
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1118.61	0.697588	0.348794	−	0.937199i	$-0.386591\pi$
$137$	1118.61	0.697588	0.348794	+	0.937199i	$0.386591\pi$
$138$	0	0
$139$	166.478	0.101586	0.0507930	−	0.998709i	$-0.483825\pi$
$139$	166.478	0.101586	0.0507930	+	0.998709i	$0.483825\pi$
$140$	0	0
$141$	−511.546	−0.305531
$142$	0	0
$143$	270.527	0.158200
$144$	0	0
$145$	0	0
$146$	0	0
$147$	969.976	0.544233
$148$	0	0
$149$	−653.143	−0.359111	−0.179555	−	0.983748i	$-0.557466\pi$
$149$	−653.143	−0.359111	−0.179555	+	0.983748i	$0.557466\pi$
$150$	0	0
$151$	1929.38	1.03981	0.519903	−	0.854225i	$-0.325968\pi$
$151$	1929.38	1.03981	0.519903	+	0.854225i	$0.325968\pi$
$152$	0	0
$153$	478.602	0.252893
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2169.75	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$157$	2169.75	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$158$	0	0
$159$	1638.15	0.817066
$160$	0	0
$161$	−524.761	−0.256876
$162$	0	0
$163$	763.738	0.366997	0.183499	−	0.983020i	$-0.441258\pi$
$163$	763.738	0.366997	0.183499	+	0.983020i	$0.441258\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2564.28	1.18820	0.594102	−	0.804389i	$-0.297507\pi$
$167$	2564.28	1.18820	0.594102	+	0.804389i	$0.297507\pi$
$168$	0	0
$169$	4003.36	1.82220
$170$	0	0
$171$	−183.920	−0.0822500
$172$	0	0
$173$	51.8290	0.0227774	0.0113887	−	0.999935i	$-0.496375\pi$
$173$	51.8290	0.0227774	0.0113887	+	0.999935i	$0.496375\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−650.909	−0.276414
$178$	0	0
$179$	3956.63	1.65214	0.826068	−	0.563571i	$-0.190573\pi$
$179$	3956.63	1.65214	0.826068	+	0.563571i	$0.190573\pi$
$180$	0	0
$181$	1804.04	0.740848	0.370424	−	0.928863i	$-0.379212\pi$
$181$	1804.04	0.740848	0.370424	+	0.928863i	$0.379212\pi$
$182$	0	0
$183$	−392.705	−0.158632
$184$	0	0
$185$	0	0
$186$	0	0
$187$	182.698	0.0714449
$188$	0	0
$189$	−119.761	−0.0460917
$190$	0	0
$191$	−3666.75	−1.38909	−0.694547	−	0.719448i	$-0.744395\pi$
$191$	−3666.75	−1.38909	−0.694547	+	0.719448i	$0.744395\pi$
$192$	0	0
$193$	−2716.98	−1.01333	−0.506664	−	0.862144i	$-0.669121\pi$
$193$	−2716.98	−1.01333	−0.506664	+	0.862144i	$0.669121\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2034.30	0.735723	0.367862	−	0.929881i	$-0.380090\pi$
$197$	2034.30	0.735723	0.367862	+	0.929881i	$0.380090\pi$
$198$	0	0
$199$	1551.27	0.552596	0.276298	−	0.961072i	$-0.410892\pi$
$199$	1551.27	0.552596	0.276298	+	0.961072i	$0.410892\pi$
$200$	0	0
$201$	−1844.67	−0.647328
$202$	0	0
$203$	745.398	0.257718
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1064.76	−0.357517
$208$	0	0
$209$	−70.2084	−0.0232365
$210$	0	0
$211$	−3192.51	−1.04162	−0.520809	−	0.853673i	$-0.674370\pi$
$211$	−3192.51	−1.04162	−0.520809	+	0.853673i	$0.674370\pi$
$212$	0	0
$213$	974.466	0.313471
$214$	0	0
$215$	0	0
$216$	0	0
$217$	270.789	0.0847115
$218$	0	0
$219$	266.614	0.0822652
$220$	0	0
$221$	4187.36	1.27454
$222$	0	0
$223$	1555.55	0.467120	0.233560	−	0.972342i	$-0.424963\pi$
$223$	1555.55	0.467120	0.233560	+	0.972342i	$0.424963\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6206.86	1.81482	0.907409	−	0.420248i	$-0.138057\pi$
$227$	6206.86	1.81482	0.907409	+	0.420248i	$0.138057\pi$
$228$	0	0
$229$	4679.51	1.35035	0.675176	−	0.737657i	$-0.264068\pi$
$229$	4679.51	1.35035	0.675176	+	0.737657i	$0.264068\pi$
$230$	0	0
$231$	−45.7167	−0.0130214
$232$	0	0
$233$	−3244.53	−0.912259	−0.456129	−	0.889913i	$-0.650765\pi$
$233$	−3244.53	−0.912259	−0.456129	+	0.889913i	$0.650765\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3412.27	−0.935236
$238$	0	0
$239$	3658.62	0.990193	0.495097	−	0.868838i	$-0.335133\pi$
$239$	3658.62	0.990193	0.495097	+	0.868838i	$0.335133\pi$
$240$	0	0
$241$	1931.01	0.516131	0.258065	−	0.966127i	$-0.416915\pi$
$241$	1931.01	0.516131	0.258065	+	0.966127i	$0.416915\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1609.15	−0.414525
$248$	0	0
$249$	−2276.73	−0.579444
$250$	0	0
$251$	5843.34	1.46944	0.734718	−	0.678373i	$-0.237315\pi$
$251$	5843.34	1.46944	0.734718	+	0.678373i	$0.237315\pi$
$252$	0	0
$253$	−406.454	−0.101002
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4506.11	1.09371	0.546855	−	0.837227i	$-0.315825\pi$
$257$	4506.11	1.09371	0.546855	+	0.837227i	$0.315825\pi$
$258$	0	0
$259$	−1093.88	−0.262434
$260$	0	0
$261$	1512.44	0.358689
$262$	0	0
$263$	−5340.16	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$263$	−5340.16	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−587.044	−0.134556
$268$	0	0
$269$	2809.79	0.636863	0.318431	−	0.947946i	$-0.396844\pi$
$269$	2809.79	0.636863	0.318431	+	0.947946i	$0.396844\pi$
$270$	0	0
$271$	−3102.95	−0.695537	−0.347769	−	0.937580i	$-0.613060\pi$
$271$	−3102.95	−0.695537	−0.347769	+	0.937580i	$0.613060\pi$
$272$	0	0
$273$	−1047.81	−0.232294
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4598.93	−0.997555	−0.498777	−	0.866730i	$-0.666217\pi$
$277$	−4598.93	−0.997555	−0.498777	+	0.866730i	$0.666217\pi$
$278$	0	0
$279$	549.443	0.117901
$280$	0	0
$281$	2571.83	0.545987	0.272994	−	0.962016i	$-0.411986\pi$
$281$	2571.83	0.545987	0.272994	+	0.962016i	$0.411986\pi$
$282$	0	0
$283$	−5575.31	−1.17109	−0.585544	−	0.810641i	$-0.699119\pi$
$283$	−5575.31	−1.17109	−0.585544	+	0.810641i	$0.699119\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1874.76	0.385588
$288$	0	0
$289$	−2085.10	−0.424405
$290$	0	0
$291$	−1563.00	−0.314861
$292$	0	0
$293$	5794.27	1.15531	0.577654	−	0.816282i	$-0.303968\pi$
$293$	5794.27	1.15531	0.577654	+	0.816282i	$0.303968\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−92.7611	−0.0181230
$298$	0	0
$299$	−9315.76	−1.80182
$300$	0	0
$301$	−1607.62	−0.307846
$302$	0	0
$303$	−1982.76	−0.375930
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1404.47	0.261099	0.130550	−	0.991442i	$-0.458326\pi$
$307$	1404.47	0.261099	0.130550	+	0.991442i	$0.458326\pi$
$308$	0	0
$309$	4592.25	0.845449
$310$	0	0
$311$	4096.75	0.746963	0.373481	−	0.927638i	$-0.378164\pi$
$311$	4096.75	0.746963	0.373481	+	0.927638i	$0.378164\pi$
$312$	0	0
$313$	974.611	0.176001	0.0880004	−	0.996120i	$-0.471952\pi$
$313$	974.611	0.176001	0.0880004	+	0.996120i	$0.471952\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2071.69	−0.367058	−0.183529	−	0.983014i	$-0.558752\pi$
$317$	−2071.69	−0.367058	−0.183529	+	0.983014i	$0.558752\pi$
$318$	0	0
$319$	577.349	0.101333
$320$	0	0
$321$	−793.876	−0.138037
$322$	0	0
$323$	−1086.72	−0.187204
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3352.84	−0.567011
$328$	0	0
$329$	756.337	0.126742
$330$	0	0
$331$	6159.17	1.02278	0.511388	−	0.859350i	$-0.329132\pi$
$331$	6159.17	1.02278	0.511388	+	0.859350i	$0.329132\pi$
$332$	0	0
$333$	−2219.52	−0.365252
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2791.26	0.451186	0.225593	−	0.974222i	$-0.427568\pi$
$337$	2791.26	0.451186	0.225593	+	0.974222i	$0.427568\pi$
$338$	0	0
$339$	2802.18	0.448949
$340$	0	0
$341$	209.740	0.0333081
$342$	0	0
$343$	−2955.55	−0.465262
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−940.848	−0.145554	−0.0727772	−	0.997348i	$-0.523186\pi$
$347$	−940.848	−0.145554	−0.0727772	+	0.997348i	$0.523186\pi$
$348$	0	0
$349$	−3519.62	−0.539831	−0.269915	−	0.962884i	$-0.586996\pi$
$349$	−3519.62	−0.539831	−0.269915	+	0.962884i	$0.586996\pi$
$350$	0	0
$351$	−2126.04	−0.323304
$352$	0	0
$353$	5021.60	0.757147	0.378573	−	0.925571i	$-0.376415\pi$
$353$	5021.60	0.757147	0.378573	+	0.925571i	$0.376415\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−707.628	−0.104907
$358$	0	0
$359$	−6811.99	−1.00146	−0.500728	−	0.865604i	$-0.666934\pi$
$359$	−6811.99	−1.00146	−0.500728	+	0.865604i	$0.666934\pi$
$360$	0	0
$361$	−6441.39	−0.939115
$362$	0	0
$363$	3957.59	0.572230
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3748.07	−0.533099	−0.266550	−	0.963821i	$-0.585884\pi$
$367$	−3748.07	−0.533099	−0.266550	+	0.963821i	$0.585884\pi$
$368$	0	0
$369$	3803.96	0.536657
$370$	0	0
$371$	−2422.05	−0.338940
$372$	0	0
$373$	−898.302	−0.124698	−0.0623489	−	0.998054i	$-0.519859\pi$
$373$	−898.302	−0.124698	−0.0623489	+	0.998054i	$0.519859\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13232.6	1.80773
$378$	0	0
$379$	9378.99	1.27115	0.635576	−	0.772038i	$-0.280763\pi$
$379$	9378.99	1.27115	0.635576	+	0.772038i	$0.280763\pi$
$380$	0	0
$381$	1891.07	0.254284
$382$	0	0
$383$	9446.29	1.26027	0.630134	−	0.776486i	$-0.283000\pi$
$383$	9446.29	1.26027	0.630134	+	0.776486i	$0.283000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3261.92	−0.428457
$388$	0	0
$389$	−7643.23	−0.996214	−0.498107	−	0.867116i	$-0.665971\pi$
$389$	−7643.23	−0.996214	−0.498107	+	0.867116i	$0.665971\pi$
$390$	0	0
$391$	−6291.32	−0.813723
$392$	0	0
$393$	−6489.17	−0.832914
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12013.6	−1.51876	−0.759378	−	0.650650i	$-0.774497\pi$
$397$	−12013.6	−1.51876	−0.759378	+	0.650650i	$0.774497\pi$
$398$	0	0
$399$	271.932	0.0341194
$400$	0	0
$401$	−8538.51	−1.06332	−0.531662	−	0.846957i	$-0.678432\pi$
$401$	−8538.51	−1.06332	−0.531662	+	0.846957i	$0.678432\pi$
$402$	0	0
$403$	4807.16	0.594197
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−847.265	−0.103188
$408$	0	0
$409$	−12267.6	−1.48312	−0.741558	−	0.670889i	$-0.765913\pi$
$409$	−12267.6	−1.48312	−0.741558	+	0.670889i	$0.765913\pi$
$410$	0	0
$411$	−3355.84	−0.402753
$412$	0	0
$413$	962.389	0.114664
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−499.433	−0.0586508
$418$	0	0
$419$	−15493.0	−1.80641	−0.903204	−	0.429212i	$-0.858791\pi$
$419$	−15493.0	−1.80641	−0.903204	+	0.429212i	$0.858791\pi$
$420$	0	0
$421$	7510.67	0.869472	0.434736	−	0.900558i	$-0.356842\pi$
$421$	7510.67	0.869472	0.434736	+	0.900558i	$0.356842\pi$
$422$	0	0
$423$	1534.64	0.176399
$424$	0	0
$425$	0	0
$426$	0	0
$427$	580.627	0.0658045
$428$	0	0
$429$	−811.581	−0.0913368
$430$	0	0
$431$	15675.3	1.75186	0.875932	−	0.482434i	$-0.160247\pi$
$431$	15675.3	1.75186	0.875932	+	0.482434i	$0.160247\pi$
$432$	0	0
$433$	−9604.78	−1.06600	−0.532998	−	0.846117i	$-0.678935\pi$
$433$	−9604.78	−1.06600	−0.532998	+	0.846117i	$0.678935\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2417.67	0.264652
$438$	0	0
$439$	6362.06	0.691673	0.345837	−	0.938295i	$-0.387595\pi$
$439$	6362.06	0.691673	0.345837	+	0.938295i	$0.387595\pi$
$440$	0	0
$441$	−2909.93	−0.314213
$442$	0	0
$443$	−931.658	−0.0999196	−0.0499598	−	0.998751i	$-0.515909\pi$
$443$	−931.658	−0.0999196	−0.0499598	+	0.998751i	$0.515909\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1959.43	0.207333
$448$	0	0
$449$	−18684.1	−1.96383	−0.981914	−	0.189329i	$-0.939369\pi$
$449$	−18684.1	−1.96383	−0.981914	+	0.189329i	$0.939369\pi$
$450$	0	0
$451$	1452.10	0.151611
$452$	0	0
$453$	−5788.14	−0.600333
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11565.9	−1.18387	−0.591936	−	0.805985i	$-0.701636\pi$
$457$	−11565.9	−1.18387	−0.591936	+	0.805985i	$0.701636\pi$
$458$	0	0
$459$	−1435.81	−0.146008
$460$	0	0
$461$	19401.0	1.96008	0.980039	−	0.198806i	$-0.0637062\pi$
$461$	19401.0	1.96008	0.980039	+	0.198806i	$0.0637062\pi$
$462$	0	0
$463$	−1576.28	−0.158220	−0.0791099	−	0.996866i	$-0.525208\pi$
$463$	−1576.28	−0.158220	−0.0791099	+	0.996866i	$0.525208\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3256.55	−0.322687	−0.161344	−	0.986898i	$-0.551583\pi$
$467$	−3256.55	−0.322687	−0.161344	+	0.986898i	$0.551583\pi$
$468$	0	0
$469$	2727.40	0.268528
$470$	0	0
$471$	−6509.25	−0.636795
$472$	0	0
$473$	−1245.18	−0.121043
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4914.44	−0.471733
$478$	0	0
$479$	−8291.59	−0.790924	−0.395462	−	0.918482i	$-0.629415\pi$
$479$	−8291.59	−0.790924	−0.395462	+	0.918482i	$0.629415\pi$
$480$	0	0
$481$	−19418.9	−1.84081
$482$	0	0
$483$	1574.28	0.148307
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4758.55	−0.442773	−0.221387	−	0.975186i	$-0.571058\pi$
$487$	−4758.55	−0.442773	−0.221387	+	0.975186i	$0.571058\pi$
$488$	0	0
$489$	−2291.21	−0.211886
$490$	0	0
$491$	−3906.46	−0.359055	−0.179528	−	0.983753i	$-0.557457\pi$
$491$	−3906.46	−0.359055	−0.179528	+	0.983753i	$0.557457\pi$
$492$	0	0
$493$	8936.52	0.816390
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1440.78	−0.130036
$498$	0	0
$499$	3093.31	0.277506	0.138753	−	0.990327i	$-0.455691\pi$
$499$	3093.31	0.277506	0.138753	+	0.990327i	$0.455691\pi$
$500$	0	0
$501$	−7692.85	−0.686010
$502$	0	0
$503$	18153.9	1.60923	0.804616	−	0.593796i	$-0.202371\pi$
$503$	18153.9	1.60923	0.804616	+	0.593796i	$0.202371\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12010.1	−1.05204
$508$	0	0
$509$	2281.32	0.198660	0.0993298	−	0.995055i	$-0.468330\pi$
$509$	2281.32	0.198660	0.0993298	+	0.995055i	$0.468330\pi$
$510$	0	0
$511$	−394.197	−0.0341257
$512$	0	0
$513$	551.761	0.0474870
$514$	0	0
$515$	0	0
$516$	0	0
$517$	585.821	0.0498344
$518$	0	0
$519$	−155.487	−0.0131505
$520$	0	0
$521$	−16691.9	−1.40362	−0.701809	−	0.712366i	$-0.747624\pi$
$521$	−16691.9	−1.40362	−0.701809	+	0.712366i	$0.747624\pi$
$522$	0	0
$523$	17090.4	1.42889	0.714446	−	0.699690i	$-0.246679\pi$
$523$	17090.4	1.42889	0.714446	+	0.699690i	$0.246679\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3246.47	0.268346
$528$	0	0
$529$	1829.50	0.150365
$530$	0	0
$531$	1952.73	0.159588
$532$	0	0
$533$	33281.5	2.70465
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11869.9	−0.953861
$538$	0	0
$539$	−1110.82	−0.0887685
$540$	0	0
$541$	5271.40	0.418919	0.209459	−	0.977817i	$-0.432830\pi$
$541$	5271.40	0.418919	0.209459	+	0.977817i	$0.432830\pi$
$542$	0	0
$543$	−5412.13	−0.427729
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15182.2	−1.18673	−0.593366	−	0.804933i	$-0.702201\pi$
$547$	−15182.2	−1.18673	−0.593366	+	0.804933i	$0.702201\pi$
$548$	0	0
$549$	1178.11	0.0915860
$550$	0	0
$551$	−3434.18	−0.265519
$552$	0	0
$553$	5045.15	0.387960
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12241.2	0.931198	0.465599	−	0.884996i	$-0.345839\pi$
$557$	12241.2	0.931198	0.465599	+	0.884996i	$0.345839\pi$
$558$	0	0
$559$	−28539.0	−2.15934
$560$	0	0
$561$	−548.094	−0.0412488
$562$	0	0
$563$	−14196.4	−1.06271	−0.531355	−	0.847149i	$-0.678317\pi$
$563$	−14196.4	−1.06271	−0.531355	+	0.847149i	$0.678317\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	359.283	0.0266111
$568$	0	0
$569$	9150.05	0.674148	0.337074	−	0.941478i	$-0.390563\pi$
$569$	9150.05	0.674148	0.337074	+	0.941478i	$0.390563\pi$
$570$	0	0
$571$	−23582.1	−1.72833	−0.864167	−	0.503206i	$-0.832154\pi$
$571$	−23582.1	−1.72833	−0.864167	+	0.503206i	$0.832154\pi$
$572$	0	0
$573$	11000.3	0.801993
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3906.22	0.281834	0.140917	−	0.990021i	$-0.454995\pi$
$577$	3906.22	0.281834	0.140917	+	0.990021i	$0.454995\pi$
$578$	0	0
$579$	8150.93	0.585045
$580$	0	0
$581$	3366.21	0.240368
$582$	0	0
$583$	−1876.00	−0.133270
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25938.0	1.82381	0.911905	−	0.410401i	$-0.134611\pi$
$587$	25938.0	1.82381	0.911905	+	0.410401i	$0.134611\pi$
$588$	0	0
$589$	−1247.58	−0.0872759
$590$	0	0
$591$	−6102.89	−0.424770
$592$	0	0
$593$	1908.23	0.132145	0.0660723	−	0.997815i	$-0.478953\pi$
$593$	1908.23	0.132145	0.0660723	+	0.997815i	$0.478953\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4653.81	−0.319041
$598$	0	0
$599$	3495.41	0.238429	0.119214	−	0.992869i	$-0.461962\pi$
$599$	3495.41	0.238429	0.119214	+	0.992869i	$0.461962\pi$
$600$	0	0
$601$	−18267.2	−1.23983	−0.619913	−	0.784671i	$-0.712832\pi$
$601$	−18267.2	−1.23983	−0.619913	+	0.784671i	$0.712832\pi$
$602$	0	0
$603$	5534.01	0.373735
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11538.2	−0.771534	−0.385767	−	0.922596i	$-0.626063\pi$
$607$	−11538.2	−0.771534	−0.385767	+	0.922596i	$0.626063\pi$
$608$	0	0
$609$	−2236.19	−0.148793
$610$	0	0
$611$	13426.8	0.889017
$612$	0	0
$613$	21136.9	1.39268	0.696340	−	0.717713i	$-0.254811\pi$
$613$	21136.9	1.39268	0.696340	+	0.717713i	$0.254811\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15673.2	1.02266	0.511329	−	0.859385i	$-0.329153\pi$
$617$	15673.2	1.02266	0.511329	+	0.859385i	$0.329153\pi$
$618$	0	0
$619$	−22923.7	−1.48850	−0.744249	−	0.667902i	$-0.767193\pi$
$619$	−22923.7	−1.48850	−0.744249	+	0.667902i	$0.767193\pi$
$620$	0	0
$621$	3194.28	0.206413
$622$	0	0
$623$	867.964	0.0558174
$624$	0	0
$625$	0	0
$626$	0	0
$627$	210.625	0.0134156
$628$	0	0
$629$	−13114.4	−0.831329
$630$	0	0
$631$	−9108.23	−0.574632	−0.287316	−	0.957836i	$-0.592763\pi$
$631$	−9108.23	−0.574632	−0.287316	+	0.957836i	$0.592763\pi$
$632$	0	0
$633$	9577.53	0.601379
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−25459.4	−1.58358
$638$	0	0
$639$	−2923.40	−0.180983
$640$	0	0
$641$	20103.5	1.23875	0.619375	−	0.785095i	$-0.287386\pi$
$641$	20103.5	1.23875	0.619375	+	0.785095i	$0.287386\pi$
$642$	0	0
$643$	−5934.92	−0.363997	−0.181999	−	0.983299i	$-0.558257\pi$
$643$	−5934.92	−0.363997	−0.181999	+	0.983299i	$0.558257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14193.7	0.862460	0.431230	−	0.902242i	$-0.358080\pi$
$647$	14193.7	0.862460	0.431230	+	0.902242i	$0.358080\pi$
$648$	0	0
$649$	745.420	0.0450852
$650$	0	0
$651$	−812.368	−0.0489082
$652$	0	0
$653$	−4795.80	−0.287403	−0.143701	−	0.989621i	$-0.545900\pi$
$653$	−4795.80	−0.287403	−0.143701	+	0.989621i	$0.545900\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−799.841	−0.0474958
$658$	0	0
$659$	−4399.57	−0.260065	−0.130032	−	0.991510i	$-0.541508\pi$
$659$	−4399.57	−0.260065	−0.130032	+	0.991510i	$0.541508\pi$
$660$	0	0
$661$	24096.0	1.41789	0.708945	−	0.705263i	$-0.249171\pi$
$661$	24096.0	1.41789	0.708945	+	0.705263i	$0.249171\pi$
$662$	0	0
$663$	−12562.1	−0.735853
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19881.4	−1.15414
$668$	0	0
$669$	−4666.66	−0.269692
$670$	0	0
$671$	449.725	0.0258740
$672$	0	0
$673$	−27648.3	−1.58360	−0.791800	−	0.610781i	$-0.790856\pi$
$673$	−27648.3	−1.58360	−0.791800	+	0.610781i	$0.790856\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27605.5	−1.56716	−0.783580	−	0.621292i	$-0.786608\pi$
$677$	−27605.5	−1.56716	−0.783580	+	0.621292i	$0.786608\pi$
$678$	0	0
$679$	2310.95	0.130613
$680$	0	0
$681$	−18620.6	−1.04779
$682$	0	0
$683$	14949.4	0.837513	0.418756	−	0.908099i	$-0.362466\pi$
$683$	14949.4	0.837513	0.418756	+	0.908099i	$0.362466\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14038.5	−0.779626
$688$	0	0
$689$	−42997.2	−2.37745
$690$	0	0
$691$	−8884.30	−0.489110	−0.244555	−	0.969635i	$-0.578642\pi$
$691$	−8884.30	−0.489110	−0.244555	+	0.969635i	$0.578642\pi$
$692$	0	0
$693$	137.150	0.00751790
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22476.4	1.22145
$698$	0	0
$699$	9733.59	0.526693
$700$	0	0
$701$	10556.9	0.568798	0.284399	−	0.958706i	$-0.408206\pi$
$701$	10556.9	0.568798	0.284399	+	0.958706i	$0.408206\pi$
$702$	0	0
$703$	5039.70	0.270378
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2931.58	0.155945
$708$	0	0
$709$	25351.9	1.34289	0.671445	−	0.741055i	$-0.265674\pi$
$709$	25351.9	1.34289	0.671445	+	0.741055i	$0.265674\pi$
$710$	0	0
$711$	10236.8	0.539959
$712$	0	0
$713$	−7222.53	−0.379363
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10975.8	−0.571688
$718$	0	0
$719$	−9719.94	−0.504162	−0.252081	−	0.967706i	$-0.581115\pi$
$719$	−9719.94	−0.504162	−0.252081	+	0.967706i	$0.581115\pi$
$720$	0	0
$721$	−6789.79	−0.350714
$722$	0	0
$723$	−5793.04	−0.297988
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27509.3	1.40339	0.701694	−	0.712479i	$-0.252428\pi$
$727$	27509.3	1.40339	0.701694	+	0.712479i	$0.252428\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−19273.6	−0.975184
$732$	0	0
$733$	−7240.49	−0.364848	−0.182424	−	0.983220i	$-0.558394\pi$
$733$	−7240.49	−0.364848	−0.182424	+	0.983220i	$0.558394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2112.51	0.105584
$738$	0	0
$739$	15875.3	0.790234	0.395117	−	0.918631i	$-0.370704\pi$
$739$	15875.3	0.790234	0.395117	+	0.918631i	$0.370704\pi$
$740$	0	0
$741$	4827.44	0.239326
$742$	0	0
$743$	25714.3	1.26967	0.634836	−	0.772647i	$-0.281068\pi$
$743$	25714.3	1.26967	0.634836	+	0.772647i	$0.281068\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6830.18	0.334542
$748$	0	0
$749$	1173.77	0.0572612
$750$	0	0
$751$	9709.09	0.471757	0.235879	−	0.971783i	$-0.424203\pi$
$751$	9709.09	0.471757	0.235879	+	0.971783i	$0.424203\pi$
$752$	0	0
$753$	−17530.0	−0.848379
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9567.13	−0.459344	−0.229672	−	0.973268i	$-0.573765\pi$
$757$	−9567.13	−0.459344	−0.229672	+	0.973268i	$0.573765\pi$
$758$	0	0
$759$	1219.36	0.0583137
$760$	0	0
$761$	−12322.5	−0.586980	−0.293490	−	0.955962i	$-0.594817\pi$
$761$	−12322.5	−0.586980	−0.293490	+	0.955962i	$0.594817\pi$
$762$	0	0
$763$	4957.28	0.235211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17084.7	0.804293
$768$	0	0
$769$	2575.56	0.120776	0.0603881	−	0.998175i	$-0.480766\pi$
$769$	2575.56	0.120776	0.0603881	+	0.998175i	$0.480766\pi$
$770$	0	0
$771$	−13518.3	−0.631454
$772$	0	0
$773$	−6606.23	−0.307386	−0.153693	−	0.988119i	$-0.549117\pi$
$773$	−6606.23	−0.307386	−0.153693	+	0.988119i	$0.549117\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3281.63	0.151516
$778$	0	0
$779$	−8637.37	−0.397260
$780$	0	0
$781$	−1115.96	−0.0511294
$782$	0	0
$783$	−4537.33	−0.207089
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16417.0	−0.743587	−0.371793	−	0.928315i	$-0.621257\pi$
$787$	−16417.0	−0.743587	−0.371793	+	0.928315i	$0.621257\pi$
$788$	0	0
$789$	16020.5	0.722870
$790$	0	0
$791$	−4143.12	−0.186235
$792$	0	0
$793$	10307.5	0.461577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3944.19	−0.175295	−0.0876477	−	0.996152i	$-0.527935\pi$
$797$	−3944.19	−0.175295	−0.0876477	+	0.996152i	$0.527935\pi$
$798$	0	0
$799$	9067.66	0.401490
$800$	0	0
$801$	1761.13	0.0776861
$802$	0	0
$803$	−305.325	−0.0134181
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8429.37	−0.367693
$808$	0	0
$809$	−17960.7	−0.780549	−0.390275	−	0.920699i	$-0.627620\pi$
$809$	−17960.7	−0.780549	−0.390275	+	0.920699i	$0.627620\pi$
$810$	0	0
$811$	13162.5	0.569912	0.284956	−	0.958541i	$-0.408021\pi$
$811$	13162.5	0.569912	0.284956	+	0.958541i	$0.408021\pi$
$812$	0	0
$813$	9308.84	0.401569
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7406.59	0.317165
$818$	0	0
$819$	3143.42	0.134115
$820$	0	0
$821$	26502.4	1.12660	0.563302	−	0.826251i	$-0.309531\pi$
$821$	26502.4	1.12660	0.563302	+	0.826251i	$0.309531\pi$
$822$	0	0
$823$	−6937.86	−0.293850	−0.146925	−	0.989148i	$-0.546938\pi$
$823$	−6937.86	−0.293850	−0.146925	+	0.989148i	$0.546938\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41197.9	−1.73228	−0.866138	−	0.499805i	$-0.833405\pi$
$827$	−41197.9	−1.73228	−0.866138	+	0.499805i	$0.833405\pi$
$828$	0	0
$829$	−693.324	−0.0290472	−0.0145236	−	0.999895i	$-0.504623\pi$
$829$	−693.324	−0.0290472	−0.0145236	+	0.999895i	$0.504623\pi$
$830$	0	0
$831$	13796.8	0.575938
$832$	0	0
$833$	−17193.8	−0.715162
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1648.33	−0.0680699
$838$	0	0
$839$	6491.28	0.267108	0.133554	−	0.991042i	$-0.457361\pi$
$839$	6491.28	0.267108	0.133554	+	0.991042i	$0.457361\pi$
$840$	0	0
$841$	3851.52	0.157921
$842$	0	0
$843$	−7715.49	−0.315226
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5851.42	−0.237376
$848$	0	0
$849$	16725.9	0.676128
$850$	0	0
$851$	29176.1	1.17526
$852$	0	0
$853$	−1116.68	−0.0448233	−0.0224117	−	0.999749i	$-0.507134\pi$
$853$	−1116.68	−0.0448233	−0.0224117	+	0.999749i	$0.507134\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44383.9	1.76911	0.884554	−	0.466438i	$-0.154463\pi$
$857$	44383.9	1.76911	0.884554	+	0.466438i	$0.154463\pi$
$858$	0	0
$859$	25579.3	1.01601	0.508006	−	0.861354i	$-0.330383\pi$
$859$	25579.3	1.01601	0.508006	+	0.861354i	$0.330383\pi$
$860$	0	0
$861$	−5624.28	−0.222619
$862$	0	0
$863$	11194.8	0.441570	0.220785	−	0.975323i	$-0.429138\pi$
$863$	11194.8	0.441570	0.220785	+	0.975323i	$0.429138\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6255.31	0.245030
$868$	0	0
$869$	3907.73	0.152544
$870$	0	0
$871$	48417.9	1.88356
$872$	0	0
$873$	4689.00	0.181785
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5721.75	−0.220308	−0.110154	−	0.993915i	$-0.535134\pi$
$877$	−5721.75	−0.220308	−0.110154	+	0.993915i	$0.535134\pi$
$878$	0	0
$879$	−17382.8	−0.667017
$880$	0	0
$881$	−34682.8	−1.32633	−0.663163	−	0.748475i	$-0.730786\pi$
$881$	−34682.8	−1.32633	−0.663163	+	0.748475i	$0.730786\pi$
$882$	0	0
$883$	−37990.4	−1.44788	−0.723941	−	0.689862i	$-0.757671\pi$
$883$	−37990.4	−1.44788	−0.723941	+	0.689862i	$0.757671\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28299.0	1.07124	0.535618	−	0.844460i	$-0.320079\pi$
$887$	28299.0	1.07124	0.535618	+	0.844460i	$0.320079\pi$
$888$	0	0
$889$	−2796.00	−0.105484
$890$	0	0
$891$	278.283	0.0104633
$892$	0	0
$893$	−3484.58	−0.130579
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27947.3	1.04028
$898$	0	0
$899$	10259.3	0.380607
$900$	0	0
$901$	−29037.8	−1.07368
$902$	0	0
$903$	4822.85	0.177735
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17388.0	0.636559	0.318280	−	0.947997i	$-0.396895\pi$
$907$	17388.0	0.636559	0.318280	+	0.947997i	$0.396895\pi$
$908$	0	0
$909$	5948.28	0.217043
$910$	0	0
$911$	−23555.3	−0.856663	−0.428332	−	0.903622i	$-0.640899\pi$
$911$	−23555.3	−0.856663	−0.428332	+	0.903622i	$0.640899\pi$
$912$	0	0
$913$	2607.30	0.0945117
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9594.44	0.345514
$918$	0	0
$919$	5983.09	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$919$	5983.09	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$920$	0	0
$921$	−4213.42	−0.150746
$922$	0	0
$923$	−25577.3	−0.912119
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−13776.7	−0.488120
$928$	0	0
$929$	20576.7	0.726694	0.363347	−	0.931654i	$-0.381634\pi$
$929$	20576.7	0.726694	0.363347	+	0.931654i	$0.381634\pi$
$930$	0	0
$931$	6607.35	0.232596
$932$	0	0
$933$	−12290.2	−0.431259
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−11228.6	−0.391485	−0.195743	−	0.980655i	$-0.562712\pi$
$937$	−11228.6	−0.391485	−0.195743	+	0.980655i	$0.562712\pi$
$938$	0	0
$939$	−2923.83	−0.101614
$940$	0	0
$941$	38567.6	1.33610	0.668049	−	0.744118i	$-0.267130\pi$
$941$	38567.6	1.33610	0.668049	+	0.744118i	$0.267130\pi$
$942$	0	0
$943$	−50003.9	−1.72678
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4606.17	−0.158057	−0.0790287	−	0.996872i	$-0.525182\pi$
$947$	−4606.17	−0.158057	−0.0790287	+	0.996872i	$0.525182\pi$
$948$	0	0
$949$	−6997.93	−0.239370
$950$	0	0
$951$	6215.06	0.211921
$952$	0	0
$953$	−25559.7	−0.868795	−0.434397	−	0.900721i	$-0.643039\pi$
$953$	−25559.7	−0.868795	−0.434397	+	0.900721i	$0.643039\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1732.05	−0.0585048
$958$	0	0
$959$	4961.72	0.167072
$960$	0	0
$961$	−26064.0	−0.874895
$962$	0	0
$963$	2381.63	0.0796956
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37895.8	−1.26023	−0.630117	−	0.776500i	$-0.716993\pi$
$967$	−37895.8	−1.26023	−0.630117	+	0.776500i	$0.716993\pi$
$968$	0	0
$969$	3260.17	0.108082
$970$	0	0
$971$	46761.0	1.54545	0.772726	−	0.634740i	$-0.218893\pi$
$971$	46761.0	1.54545	0.772726	+	0.634740i	$0.218893\pi$
$972$	0	0
$973$	738.428	0.0243298
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3070.29	0.100540	0.0502698	−	0.998736i	$-0.483992\pi$
$977$	3070.29	0.100540	0.0502698	+	0.998736i	$0.483992\pi$
$978$	0	0
$979$	672.282	0.0219471
$980$	0	0
$981$	10058.5	0.327364
$982$	0	0
$983$	−16319.0	−0.529498	−0.264749	−	0.964317i	$-0.585289\pi$
$983$	−16319.0	−0.529498	−0.264749	+	0.964317i	$0.585289\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2269.01	−0.0731747
$988$	0	0
$989$	42878.6	1.37862
$990$	0	0
$991$	−5105.79	−0.163664	−0.0818319	−	0.996646i	$-0.526077\pi$
$991$	−5105.79	−0.163664	−0.0818319	+	0.996646i	$0.526077\pi$
$992$	0	0
$993$	−18477.5	−0.590500
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7206.97	0.228934	0.114467	−	0.993427i	$-0.463484\pi$
$997$	7206.97	0.228934	0.114467	+	0.993427i	$0.463484\pi$
$998$	0	0
$999$	6658.57	0.210879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bl.1.2		2
4.3	odd	2		75.4.a.d.1.1	✓	2
5.2	odd	4		1200.4.f.v.49.4		4
5.3	odd	4		1200.4.f.v.49.1		4
5.4	even	2		1200.4.a.bu.1.1		2
12.11	even	2		225.4.a.n.1.2		2
20.3	even	4		75.4.b.c.49.4		4
20.7	even	4		75.4.b.c.49.1		4
20.19	odd	2		75.4.a.e.1.2	yes	2
60.23	odd	4		225.4.b.h.199.1		4
60.47	odd	4		225.4.b.h.199.4		4
60.59	even	2		225.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.1	✓	2	4.3	odd	2
75.4.a.e.1.2	yes	2	20.19	odd	2
75.4.b.c.49.1		4	20.7	even	4
75.4.b.c.49.4		4	20.3	even	4
225.4.a.j.1.1		2	60.59	even	2
225.4.a.n.1.2		2	12.11	even	2
225.4.b.h.199.1		4	60.23	odd	4
225.4.b.h.199.4		4	60.47	odd	4
1200.4.a.bl.1.2		2	1.1	even	1	trivial
1200.4.a.bu.1.1		2	5.4	even	2
1200.4.f.v.49.1		4	5.3	odd	4
1200.4.f.v.49.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +4.43560 q^{7} +9.00000 q^{9} +3.43560 q^{11} +78.7424 q^{13} +53.1780 q^{17} -20.4356 q^{19} -13.3068 q^{21} -118.307 q^{23} -27.0000 q^{27} +168.049 q^{29} +61.0492 q^{31} -10.3068 q^{33} -246.614 q^{37} -236.227 q^{39} +422.663 q^{41} -362.436 q^{43} +170.515 q^{47} -323.325 q^{49} -159.534 q^{51} -546.049 q^{53} +61.3068 q^{57} +216.970 q^{59} +130.902 q^{61} +39.9204 q^{63} +614.890 q^{67} +354.920 q^{69} -324.822 q^{71} -88.8712 q^{73} +15.2389 q^{77} +1137.42 q^{79} +81.0000 q^{81} +758.909 q^{83} -504.148 q^{87} +195.681 q^{89} +349.269 q^{91} -183.148 q^{93} +521.000 q^{97} +30.9204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9} - 28 q^{11} + 18 q^{13} - 68 q^{17} - 6 q^{19} + 78 q^{21} - 132 q^{23} - 54 q^{27} + 92 q^{29} - 122 q^{31} + 84 q^{33} - 284 q^{37} - 54 q^{39} + 392 q^{41} - 690 q^{43}+ \cdots - 252 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 26 * q^7 + 18 * q^9 - 28 * q^11 + 18 * q^13 - 68 * q^17 - 6 * q^19 + 78 * q^21 - 132 * q^23 - 54 * q^27 + 92 * q^29 - 122 * q^31 + 84 * q^33 - 284 * q^37 - 54 * q^39 + 392 * q^41 - 690 * q^43 + 620 * q^47 + 260 * q^49 + 204 * q^51 - 848 * q^53 + 18 * q^57 - 124 * q^59 + 750 * q^61 - 234 * q^63 + 358 * q^67 + 396 * q^69 - 824 * q^71 - 108 * q^73 + 972 * q^77 + 880 * q^79 + 162 * q^81 - 156 * q^83 - 276 * q^87 - 864 * q^89 + 2198 * q^91 + 366 * q^93 + 1042 * q^97 - 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more