LMFDB - Embedded newform 1200.4.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17891$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +33.0735 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +33.0735 q^{7} +9.00000 q^{9} -48.3578 q^{11} -60.3578 q^{13} -17.7891 q^{17} +130.863 q^{19} +99.2204 q^{21} -70.8625 q^{23} +27.0000 q^{27} +104.505 q^{29} +210.441 q^{31} -145.073 q^{33} +300.945 q^{37} -181.073 q^{39} +240.147 q^{41} +108.000 q^{43} +278.991 q^{47} +750.853 q^{49} -53.3673 q^{51} +328.358 q^{53} +392.588 q^{57} -889.533 q^{59} -241.450 q^{61} +297.661 q^{63} -103.834 q^{67} -212.588 q^{69} +277.597 q^{71} +274.403 q^{73} -1599.36 q^{77} -366.991 q^{79} +81.0000 q^{81} -57.7251 q^{83} +313.514 q^{87} -203.175 q^{89} -1996.24 q^{91} +631.322 q^{93} +1283.45 q^{97} -435.220 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 74 q^{11} - 98 q^{13} + 78 q^{17} + 80 q^{19} - 6 q^{21} + 40 q^{23} + 54 q^{27} + 50 q^{29} + 12 q^{31} - 222 q^{33} + 34 q^{37} - 294 q^{39} + 344 q^{41} + 216 q^{43} + 876 q^{47} + 1638 q^{49} + 234 q^{51} + 634 q^{53} + 240 q^{57} - 666 q^{59} + 244 q^{61} - 18 q^{63} - 980 q^{67} + 120 q^{69} - 308 q^{71} + 1412 q^{73} - 700 q^{77} - 1052 q^{79} + 162 q^{81} + 248 q^{83} + 150 q^{87} + 684 q^{89} - 676 q^{91} + 36 q^{93} + 1840 q^{97} - 666 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 - 74 * q^11 - 98 * q^13 + 78 * q^17 + 80 * q^19 - 6 * q^21 + 40 * q^23 + 54 * q^27 + 50 * q^29 + 12 * q^31 - 222 * q^33 + 34 * q^37 - 294 * q^39 + 344 * q^41 + 216 * q^43 + 876 * q^47 + 1638 * q^49 + 234 * q^51 + 634 * q^53 + 240 * q^57 - 666 * q^59 + 244 * q^61 - 18 * q^63 - 980 * q^67 + 120 * q^69 - 308 * q^71 + 1412 * q^73 - 700 * q^77 - 1052 * q^79 + 162 * q^81 + 248 * q^83 + 150 * q^87 + 684 * q^89 - 676 * q^91 + 36 * q^93 + 1840 * q^97 - 666 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	33.0735	1.78580	0.892899	−	0.450257i	$-0.148667\pi$
$7$	33.0735	1.78580	0.892899	+	0.450257i	$0.148667\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−48.3578	−1.32549	−0.662747	−	0.748844i	$-0.730609\pi$
$11$	−48.3578	−1.32549	−0.662747	+	0.748844i	$0.730609\pi$
$12$	0	0
$13$	−60.3578	−1.28771	−0.643856	−	0.765147i	$-0.722666\pi$
$13$	−60.3578	−1.28771	−0.643856	+	0.765147i	$0.722666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.7891	−0.253793	−0.126897	−	0.991916i	$-0.540502\pi$
$17$	−17.7891	−0.253793	−0.126897	+	0.991916i	$0.540502\pi$
$18$	0	0
$19$	130.863	1.58010	0.790051	−	0.613042i	$-0.210054\pi$
$19$	130.863	1.58010	0.790051	+	0.613042i	$0.210054\pi$
$20$	0	0
$21$	99.2204	1.03103
$22$	0	0
$23$	−70.8625	−0.642429	−0.321214	−	0.947007i	$-0.604091\pi$
$23$	−70.8625	−0.642429	−0.321214	+	0.947007i	$0.604091\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	104.505	0.669174	0.334587	−	0.942365i	$-0.391403\pi$
$29$	104.505	0.669174	0.334587	+	0.942365i	$0.391403\pi$
$30$	0	0
$31$	210.441	1.21923	0.609617	−	0.792696i	$-0.291323\pi$
$31$	210.441	1.21923	0.609617	+	0.792696i	$0.291323\pi$
$32$	0	0
$33$	−145.073	−0.765274
$34$	0	0
$35$	0	0
$36$	0	0
$37$	300.945	1.33717	0.668583	−	0.743638i	$-0.266901\pi$
$37$	300.945	1.33717	0.668583	+	0.743638i	$0.266901\pi$
$38$	0	0
$39$	−181.073	−0.743460
$40$	0	0
$41$	240.147	0.914747	0.457374	−	0.889275i	$-0.348790\pi$
$41$	240.147	0.914747	0.457374	+	0.889275i	$0.348790\pi$
$42$	0	0
$43$	108.000	0.383020	0.191510	−	0.981491i	$-0.438662\pi$
$43$	108.000	0.383020	0.191510	+	0.981491i	$0.438662\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	278.991	0.865850	0.432925	−	0.901430i	$-0.357481\pi$
$47$	278.991	0.865850	0.432925	+	0.901430i	$0.357481\pi$
$48$	0	0
$49$	750.853	2.18908
$50$	0	0
$51$	−53.3673	−0.146528
$52$	0	0
$53$	328.358	0.851008	0.425504	−	0.904957i	$-0.360097\pi$
$53$	328.358	0.851008	0.425504	+	0.904957i	$0.360097\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	392.588	0.912272
$58$	0	0
$59$	−889.533	−1.96284	−0.981418	−	0.191882i	$-0.938541\pi$
$59$	−889.533	−1.96284	−0.981418	+	0.191882i	$0.938541\pi$
$60$	0	0
$61$	−241.450	−0.506795	−0.253398	−	0.967362i	$-0.581548\pi$
$61$	−241.450	−0.506795	−0.253398	+	0.967362i	$0.581548\pi$
$62$	0	0
$63$	297.661	0.595266
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−103.834	−0.189334	−0.0946669	−	0.995509i	$-0.530179\pi$
$67$	−103.834	−0.189334	−0.0946669	+	0.995509i	$0.530179\pi$
$68$	0	0
$69$	−212.588	−0.370906
$70$	0	0
$71$	277.597	0.464010	0.232005	−	0.972715i	$-0.425471\pi$
$71$	277.597	0.464010	0.232005	+	0.972715i	$0.425471\pi$
$72$	0	0
$73$	274.403	0.439951	0.219976	−	0.975505i	$-0.429402\pi$
$73$	274.403	0.439951	0.219976	+	0.975505i	$0.429402\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1599.36	−2.36706
$78$	0	0
$79$	−366.991	−0.522654	−0.261327	−	0.965250i	$-0.584160\pi$
$79$	−366.991	−0.522654	−0.261327	+	0.965250i	$0.584160\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−57.7251	−0.0763391	−0.0381696	−	0.999271i	$-0.512153\pi$
$83$	−57.7251	−0.0763391	−0.0381696	+	0.999271i	$0.512153\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	313.514	0.386348
$88$	0	0
$89$	−203.175	−0.241983	−0.120992	−	0.992654i	$-0.538607\pi$
$89$	−203.175	−0.241983	−0.120992	+	0.992654i	$0.538607\pi$
$90$	0	0
$91$	−1996.24	−2.29959
$92$	0	0
$93$	631.322	0.703925
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1283.45	1.34345	0.671725	−	0.740801i	$-0.265554\pi$
$97$	1283.45	1.34345	0.671725	+	0.740801i	$0.265554\pi$
$98$	0	0
$99$	−435.220	−0.441831
$100$	0	0
$101$	886.908	0.873768	0.436884	−	0.899518i	$-0.356082\pi$
$101$	886.908	0.873768	0.436884	+	0.899518i	$0.356082\pi$
$102$	0	0
$103$	783.055	0.749094	0.374547	−	0.927208i	$-0.377798\pi$
$103$	783.055	0.749094	0.374547	+	0.927208i	$0.377798\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1301.51	−1.17590	−0.587950	−	0.808897i	$-0.700065\pi$
$107$	−1301.51	−1.17590	−0.587950	+	0.808897i	$0.700065\pi$
$108$	0	0
$109$	1161.91	1.02102	0.510508	−	0.859873i	$-0.329457\pi$
$109$	1161.91	1.02102	0.510508	+	0.859873i	$0.329457\pi$
$110$	0	0
$111$	902.836	0.772013
$112$	0	0
$113$	507.808	0.422748	0.211374	−	0.977405i	$-0.432206\pi$
$113$	507.808	0.422748	0.211374	+	0.977405i	$0.432206\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−543.220	−0.429237
$118$	0	0
$119$	−588.346	−0.453224
$120$	0	0
$121$	1007.48	0.756933
$122$	0	0
$123$	720.441	0.528130
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−796.064	−0.556215	−0.278107	−	0.960550i	$-0.589707\pi$
$127$	−796.064	−0.556215	−0.278107	+	0.960550i	$0.589707\pi$
$128$	0	0
$129$	324.000	0.221137
$130$	0	0
$131$	91.4764	0.0610102	0.0305051	−	0.999535i	$-0.490288\pi$
$131$	91.4764	0.0610102	0.0305051	+	0.999535i	$0.490288\pi$
$132$	0	0
$133$	4328.08	2.82174
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2273.28	1.41766	0.708829	−	0.705380i	$-0.249224\pi$
$137$	2273.28	1.41766	0.708829	+	0.705380i	$0.249224\pi$
$138$	0	0
$139$	738.735	0.450782	0.225391	−	0.974268i	$-0.427634\pi$
$139$	738.735	0.450782	0.225391	+	0.974268i	$0.427634\pi$
$140$	0	0
$141$	836.972	0.499899
$142$	0	0
$143$	2918.77	1.70685
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2252.56	1.26386
$148$	0	0
$149$	1507.15	0.828661	0.414330	−	0.910127i	$-0.364016\pi$
$149$	1507.15	0.828661	0.414330	+	0.910127i	$0.364016\pi$
$150$	0	0
$151$	−154.365	−0.0831925	−0.0415962	−	0.999135i	$-0.513244\pi$
$151$	−154.365	−0.0831925	−0.0415962	+	0.999135i	$0.513244\pi$
$152$	0	0
$153$	−160.102	−0.0845978
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−315.514	−0.160387	−0.0801935	−	0.996779i	$-0.525554\pi$
$157$	−315.514	−0.160387	−0.0801935	+	0.996779i	$0.525554\pi$
$158$	0	0
$159$	985.073	0.491330
$160$	0	0
$161$	−2343.67	−1.14725
$162$	0	0
$163$	1457.85	0.700539	0.350270	−	0.936649i	$-0.386090\pi$
$163$	1457.85	0.700539	0.350270	+	0.936649i	$0.386090\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2198.61	1.01876	0.509381	−	0.860541i	$-0.329874\pi$
$167$	2198.61	1.01876	0.509381	+	0.860541i	$0.329874\pi$
$168$	0	0
$169$	1446.07	0.658200
$170$	0	0
$171$	1177.76	0.526700
$172$	0	0
$173$	−2030.49	−0.892343	−0.446171	−	0.894948i	$-0.647213\pi$
$173$	−2030.49	−0.892343	−0.446171	+	0.894948i	$0.647213\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2668.60	−1.13324
$178$	0	0
$179$	−427.220	−0.178391	−0.0891954	−	0.996014i	$-0.528430\pi$
$179$	−427.220	−0.178391	−0.0891954	+	0.996014i	$0.528430\pi$
$180$	0	0
$181$	−3779.97	−1.55228	−0.776140	−	0.630561i	$-0.782825\pi$
$181$	−3779.97	−1.55228	−0.776140	+	0.630561i	$0.782825\pi$
$182$	0	0
$183$	−724.350	−0.292598
$184$	0	0
$185$	0	0
$186$	0	0
$187$	860.241	0.336401
$188$	0	0
$189$	892.983	0.343677
$190$	0	0
$191$	−1565.60	−0.593103	−0.296551	−	0.955017i	$-0.595837\pi$
$191$	−1565.60	−0.593103	−0.296551	+	0.955017i	$0.595837\pi$
$192$	0	0
$193$	−2642.63	−0.985599	−0.492800	−	0.870143i	$-0.664026\pi$
$193$	−2642.63	−0.985599	−0.492800	+	0.870143i	$0.664026\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	98.4522	0.0356062	0.0178031	−	0.999842i	$-0.494333\pi$
$197$	98.4522	0.0356062	0.0178031	+	0.999842i	$0.494333\pi$
$198$	0	0
$199$	−1131.62	−0.403106	−0.201553	−	0.979478i	$-0.564599\pi$
$199$	−1131.62	−0.403106	−0.201553	+	0.979478i	$0.564599\pi$
$200$	0	0
$201$	−311.503	−0.109312
$202$	0	0
$203$	3456.33	1.19501
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−637.763	−0.214143
$208$	0	0
$209$	−6328.23	−2.09441
$210$	0	0
$211$	1902.85	0.620841	0.310420	−	0.950599i	$-0.399530\pi$
$211$	1902.85	0.620841	0.310420	+	0.950599i	$0.399530\pi$
$212$	0	0
$213$	832.791	0.267896
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6960.00	2.17731
$218$	0	0
$219$	823.209	0.254006
$220$	0	0
$221$	1073.71	0.326813
$222$	0	0
$223$	−4855.59	−1.45809	−0.729046	−	0.684465i	$-0.760036\pi$
$223$	−4855.59	−1.45809	−0.729046	+	0.684465i	$0.760036\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6536.19	1.91111	0.955555	−	0.294812i	$-0.0952571\pi$
$227$	6536.19	1.91111	0.955555	+	0.294812i	$0.0952571\pi$
$228$	0	0
$229$	−5510.35	−1.59011	−0.795053	−	0.606539i	$-0.792557\pi$
$229$	−5510.35	−1.59011	−0.795053	+	0.606539i	$0.792557\pi$
$230$	0	0
$231$	−4798.08	−1.36663
$232$	0	0
$233$	5915.65	1.66329	0.831646	−	0.555306i	$-0.187399\pi$
$233$	5915.65	1.66329	0.831646	+	0.555306i	$0.187399\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1100.97	−0.301754
$238$	0	0
$239$	−2263.25	−0.612543	−0.306272	−	0.951944i	$-0.599082\pi$
$239$	−2263.25	−0.612543	−0.306272	+	0.951944i	$0.599082\pi$
$240$	0	0
$241$	772.493	0.206476	0.103238	−	0.994657i	$-0.467080\pi$
$241$	772.493	0.206476	0.103238	+	0.994657i	$0.467080\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7898.58	−2.03471
$248$	0	0
$249$	−173.175	−0.0440744
$250$	0	0
$251$	−596.192	−0.149926	−0.0749628	−	0.997186i	$-0.523884\pi$
$251$	−596.192	−0.149926	−0.0749628	+	0.997186i	$0.523884\pi$
$252$	0	0
$253$	3426.76	0.851535
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4139.19	1.00465	0.502326	−	0.864678i	$-0.332478\pi$
$257$	4139.19	1.00465	0.502326	+	0.864678i	$0.332478\pi$
$258$	0	0
$259$	9953.30	2.38791
$260$	0	0
$261$	940.542	0.223058
$262$	0	0
$263$	1611.34	0.377792	0.188896	−	0.981997i	$-0.439509\pi$
$263$	1611.34	0.377792	0.188896	+	0.981997i	$0.439509\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−609.526	−0.139709
$268$	0	0
$269$	−7031.33	−1.59371	−0.796855	−	0.604171i	$-0.793504\pi$
$269$	−7031.33	−1.59371	−0.796855	+	0.604171i	$0.793504\pi$
$270$	0	0
$271$	5441.27	1.21968	0.609840	−	0.792524i	$-0.291234\pi$
$271$	5441.27	1.21968	0.609840	+	0.792524i	$0.291234\pi$
$272$	0	0
$273$	−5988.72	−1.32767
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1080.43	0.234355	0.117178	−	0.993111i	$-0.462615\pi$
$277$	1080.43	0.234355	0.117178	+	0.993111i	$0.462615\pi$
$278$	0	0
$279$	1893.97	0.406411
$280$	0	0
$281$	−1602.99	−0.340308	−0.170154	−	0.985417i	$-0.554427\pi$
$281$	−1602.99	−0.340308	−0.170154	+	0.985417i	$0.554427\pi$
$282$	0	0
$283$	−334.810	−0.0703265	−0.0351632	−	0.999382i	$-0.511195\pi$
$283$	−334.810	−0.0703265	−0.0351632	+	0.999382i	$0.511195\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7942.49	1.63355
$288$	0	0
$289$	−4596.55	−0.935589
$290$	0	0
$291$	3850.35	0.775641
$292$	0	0
$293$	539.250	0.107520	0.0537599	−	0.998554i	$-0.482879\pi$
$293$	539.250	0.107520	0.0537599	+	0.998554i	$0.482879\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1305.66	−0.255091
$298$	0	0
$299$	4277.11	0.827263
$300$	0	0
$301$	3571.93	0.683996
$302$	0	0
$303$	2660.72	0.504470
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8477.58	−1.57603	−0.788015	−	0.615656i	$-0.788891\pi$
$307$	−8477.58	−1.57603	−0.788015	+	0.615656i	$0.788891\pi$
$308$	0	0
$309$	2349.16	0.432490
$310$	0	0
$311$	−3646.92	−0.664945	−0.332473	−	0.943113i	$-0.607883\pi$
$311$	−3646.92	−0.664945	−0.332473	+	0.943113i	$0.607883\pi$
$312$	0	0
$313$	7537.05	1.36108	0.680542	−	0.732709i	$-0.261745\pi$
$313$	7537.05	1.36108	0.680542	+	0.732709i	$0.261745\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10721.2	−1.89956	−0.949781	−	0.312916i	$-0.898694\pi$
$317$	−10721.2	−1.89956	−0.949781	+	0.312916i	$0.898694\pi$
$318$	0	0
$319$	−5053.62	−0.886986
$320$	0	0
$321$	−3904.52	−0.678907
$322$	0	0
$323$	−2327.92	−0.401019
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3485.73	0.589484
$328$	0	0
$329$	9227.18	1.54623
$330$	0	0
$331$	4405.14	0.731505	0.365753	−	0.930712i	$-0.380812\pi$
$331$	4405.14	0.731505	0.365753	+	0.930712i	$0.380812\pi$
$332$	0	0
$333$	2708.51	0.445722
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−186.117	−0.0300844	−0.0150422	−	0.999887i	$-0.504788\pi$
$337$	−186.117	−0.0300844	−0.0150422	+	0.999887i	$0.504788\pi$
$338$	0	0
$339$	1523.42	0.244074
$340$	0	0
$341$	−10176.5	−1.61609
$342$	0	0
$343$	13489.1	2.12345
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5547.38	−0.858211	−0.429105	−	0.903254i	$-0.641171\pi$
$347$	−5547.38	−0.858211	−0.429105	+	0.903254i	$0.641171\pi$
$348$	0	0
$349$	9078.82	1.39249	0.696244	−	0.717805i	$-0.254853\pi$
$349$	9078.82	1.39249	0.696244	+	0.717805i	$0.254853\pi$
$350$	0	0
$351$	−1629.66	−0.247820
$352$	0	0
$353$	10678.1	1.61002	0.805009	−	0.593262i	$-0.202160\pi$
$353$	10678.1	1.61002	0.805009	+	0.593262i	$0.202160\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1765.04	−0.261669
$358$	0	0
$359$	265.733	0.0390665	0.0195332	−	0.999809i	$-0.493782\pi$
$359$	265.733	0.0390665	0.0195332	+	0.999809i	$0.493782\pi$
$360$	0	0
$361$	10266.0	1.49672
$362$	0	0
$363$	3022.44	0.437016
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5854.21	0.832663	0.416332	−	0.909213i	$-0.363316\pi$
$367$	5854.21	0.832663	0.416332	+	0.909213i	$0.363316\pi$
$368$	0	0
$369$	2161.32	0.304916
$370$	0	0
$371$	10859.9	1.51973
$372$	0	0
$373$	−10134.5	−1.40682	−0.703408	−	0.710787i	$-0.748339\pi$
$373$	−10134.5	−1.40682	−0.703408	+	0.710787i	$0.748339\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6307.68	−0.861703
$378$	0	0
$379$	4235.70	0.574072	0.287036	−	0.957920i	$-0.407330\pi$
$379$	4235.70	0.574072	0.287036	+	0.957920i	$0.407330\pi$
$380$	0	0
$381$	−2388.19	−0.321131
$382$	0	0
$383$	−8100.84	−1.08077	−0.540383	−	0.841419i	$-0.681721\pi$
$383$	−8100.84	−1.08077	−0.540383	+	0.841419i	$0.681721\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	972.000	0.127673
$388$	0	0
$389$	−13820.2	−1.80131	−0.900656	−	0.434532i	$-0.856914\pi$
$389$	−13820.2	−1.80131	−0.900656	+	0.434532i	$0.856914\pi$
$390$	0	0
$391$	1260.58	0.163044
$392$	0	0
$393$	274.429	0.0352242
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11523.9	1.45685	0.728425	−	0.685125i	$-0.240252\pi$
$397$	11523.9	1.45685	0.728425	+	0.685125i	$0.240252\pi$
$398$	0	0
$399$	12984.2	1.62913
$400$	0	0
$401$	700.915	0.0872869	0.0436434	−	0.999047i	$-0.486103\pi$
$401$	700.915	0.0872869	0.0436434	+	0.999047i	$0.486103\pi$
$402$	0	0
$403$	−12701.7	−1.57002
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14553.1	−1.77240
$408$	0	0
$409$	6650.69	0.804047	0.402024	−	0.915629i	$-0.368307\pi$
$409$	6650.69	0.804047	0.402024	+	0.915629i	$0.368307\pi$
$410$	0	0
$411$	6819.83	0.818485
$412$	0	0
$413$	−29419.9	−3.50523
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2216.20	0.260259
$418$	0	0
$419$	−13844.4	−1.61418	−0.807091	−	0.590426i	$-0.798960\pi$
$419$	−13844.4	−1.61418	−0.807091	+	0.590426i	$0.798960\pi$
$420$	0	0
$421$	12576.3	1.45590	0.727949	−	0.685631i	$-0.240474\pi$
$421$	12576.3	1.45590	0.727949	+	0.685631i	$0.240474\pi$
$422$	0	0
$423$	2510.92	0.288617
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7985.59	−0.905035
$428$	0	0
$429$	8756.32	0.985452
$430$	0	0
$431$	−13440.6	−1.50212	−0.751059	−	0.660236i	$-0.770456\pi$
$431$	−13440.6	−1.50212	−0.751059	+	0.660236i	$0.770456\pi$
$432$	0	0
$433$	3012.87	0.334387	0.167193	−	0.985924i	$-0.446530\pi$
$433$	3012.87	0.334387	0.167193	+	0.985924i	$0.446530\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9273.25	−1.01510
$438$	0	0
$439$	−10675.0	−1.16057	−0.580283	−	0.814415i	$-0.697058\pi$
$439$	−10675.0	−1.16057	−0.580283	+	0.814415i	$0.697058\pi$
$440$	0	0
$441$	6757.68	0.729692
$442$	0	0
$443$	−125.868	−0.0134992	−0.00674962	−	0.999977i	$-0.502148\pi$
$443$	−125.868	−0.0134992	−0.00674962	+	0.999977i	$0.502148\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4521.45	0.478428
$448$	0	0
$449$	−9707.21	−1.02029	−0.510146	−	0.860088i	$-0.670409\pi$
$449$	−9707.21	−1.02029	−0.510146	+	0.860088i	$0.670409\pi$
$450$	0	0
$451$	−11613.0	−1.21249
$452$	0	0
$453$	−463.096	−0.0480312
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1279.97	−0.131016	−0.0655082	−	0.997852i	$-0.520867\pi$
$457$	−1279.97	−0.131016	−0.0655082	+	0.997852i	$0.520867\pi$
$458$	0	0
$459$	−480.305	−0.0488425
$460$	0	0
$461$	3080.48	0.311220	0.155610	−	0.987819i	$-0.450266\pi$
$461$	3080.48	0.311220	0.155610	+	0.987819i	$0.450266\pi$
$462$	0	0
$463$	−18017.3	−1.80850	−0.904248	−	0.427008i	$-0.859568\pi$
$463$	−18017.3	−1.80850	−0.904248	+	0.427008i	$0.859568\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7236.77	0.717083	0.358541	−	0.933514i	$-0.383274\pi$
$467$	7236.77	0.717083	0.358541	+	0.933514i	$0.383274\pi$
$468$	0	0
$469$	−3434.16	−0.338112
$470$	0	0
$471$	−946.542	−0.0925995
$472$	0	0
$473$	−5222.64	−0.507690
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2955.22	0.283669
$478$	0	0
$479$	−4932.78	−0.470532	−0.235266	−	0.971931i	$-0.575596\pi$
$479$	−4932.78	−0.470532	−0.235266	+	0.971931i	$0.575596\pi$
$480$	0	0
$481$	−18164.4	−1.72188
$482$	0	0
$483$	−7031.01	−0.662364
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5937.91	0.552510	0.276255	−	0.961084i	$-0.410907\pi$
$487$	5937.91	0.552510	0.276255	+	0.961084i	$0.410907\pi$
$488$	0	0
$489$	4373.56	0.404456
$490$	0	0
$491$	−15703.5	−1.44336	−0.721678	−	0.692229i	$-0.756629\pi$
$491$	−15703.5	−1.44336	−0.721678	+	0.692229i	$0.756629\pi$
$492$	0	0
$493$	−1859.04	−0.169832
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9181.09	0.828628
$498$	0	0
$499$	5656.77	0.507478	0.253739	−	0.967273i	$-0.418340\pi$
$499$	5656.77	0.507478	0.253739	+	0.967273i	$0.418340\pi$
$500$	0	0
$501$	6595.82	0.588183
$502$	0	0
$503$	−290.441	−0.0257457	−0.0128729	−	0.999917i	$-0.504098\pi$
$503$	−290.441	−0.0257457	−0.0128729	+	0.999917i	$0.504098\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4338.20	0.380012
$508$	0	0
$509$	17330.6	1.50916	0.754582	−	0.656206i	$-0.227840\pi$
$509$	17330.6	1.50916	0.754582	+	0.656206i	$0.227840\pi$
$510$	0	0
$511$	9075.45	0.785664
$512$	0	0
$513$	3533.29	0.304091
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13491.4	−1.14768
$518$	0	0
$519$	−6091.47	−0.515194
$520$	0	0
$521$	6174.36	0.519201	0.259601	−	0.965716i	$-0.416409\pi$
$521$	6174.36	0.519201	0.259601	+	0.965716i	$0.416409\pi$
$522$	0	0
$523$	13389.4	1.11946	0.559730	−	0.828675i	$-0.310905\pi$
$523$	13389.4	1.11946	0.559730	+	0.828675i	$0.310905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3743.55	−0.309434
$528$	0	0
$529$	−7145.50	−0.587285
$530$	0	0
$531$	−8005.80	−0.654279
$532$	0	0
$533$	−14494.7	−1.17793
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1281.66	−0.102994
$538$	0	0
$539$	−36309.6	−2.90161
$540$	0	0
$541$	−14355.5	−1.14084	−0.570418	−	0.821354i	$-0.693219\pi$
$541$	−14355.5	−1.14084	−0.570418	+	0.821354i	$0.693219\pi$
$542$	0	0
$543$	−11339.9	−0.896209
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21133.9	1.65195	0.825977	−	0.563704i	$-0.190624\pi$
$547$	21133.9	1.65195	0.825977	+	0.563704i	$0.190624\pi$
$548$	0	0
$549$	−2173.05	−0.168932
$550$	0	0
$551$	13675.8	1.05736
$552$	0	0
$553$	−12137.6	−0.933355
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3098.60	0.235713	0.117856	−	0.993031i	$-0.462398\pi$
$557$	3098.60	0.235713	0.117856	+	0.993031i	$0.462398\pi$
$558$	0	0
$559$	−6518.64	−0.493219
$560$	0	0
$561$	2580.72	0.194221
$562$	0	0
$563$	−7908.77	−0.592033	−0.296017	−	0.955183i	$-0.595658\pi$
$563$	−7908.77	−0.592033	−0.296017	+	0.955183i	$0.595658\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2678.95	0.198422
$568$	0	0
$569$	−10740.7	−0.791345	−0.395673	−	0.918392i	$-0.629488\pi$
$569$	−10740.7	−0.791345	−0.395673	+	0.918392i	$0.629488\pi$
$570$	0	0
$571$	−14701.2	−1.07745	−0.538725	−	0.842482i	$-0.681094\pi$
$571$	−14701.2	−1.07745	−0.538725	+	0.842482i	$0.681094\pi$
$572$	0	0
$573$	−4696.79	−0.342428
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15788.1	1.13911	0.569554	−	0.821954i	$-0.307116\pi$
$577$	15788.1	1.13911	0.569554	+	0.821954i	$0.307116\pi$
$578$	0	0
$579$	−7927.89	−0.569036
$580$	0	0
$581$	−1909.17	−0.136326
$582$	0	0
$583$	−15878.7	−1.12801
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14778.5	−1.03913	−0.519567	−	0.854430i	$-0.673907\pi$
$587$	−14778.5	−1.03913	−0.519567	+	0.854430i	$0.673907\pi$
$588$	0	0
$589$	27538.8	1.92651
$590$	0	0
$591$	295.356	0.0205573
$592$	0	0
$593$	11143.9	0.771713	0.385857	−	0.922559i	$-0.373906\pi$
$593$	11143.9	0.771713	0.385857	+	0.922559i	$0.373906\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3394.85	−0.232733
$598$	0	0
$599$	7707.87	0.525768	0.262884	−	0.964827i	$-0.415326\pi$
$599$	7707.87	0.525768	0.262884	+	0.964827i	$0.415326\pi$
$600$	0	0
$601$	−13681.4	−0.928580	−0.464290	−	0.885683i	$-0.653691\pi$
$601$	−13681.4	−0.928580	−0.464290	+	0.885683i	$0.653691\pi$
$602$	0	0
$603$	−934.508	−0.0631113
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11552.4	−0.772481	−0.386241	−	0.922398i	$-0.626227\pi$
$607$	−11552.4	−0.772481	−0.386241	+	0.922398i	$0.626227\pi$
$608$	0	0
$609$	10369.0	0.689939
$610$	0	0
$611$	−16839.3	−1.11496
$612$	0	0
$613$	−9904.66	−0.652603	−0.326301	−	0.945266i	$-0.605802\pi$
$613$	−9904.66	−0.652603	−0.326301	+	0.945266i	$0.605802\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20323.5	1.32608	0.663042	−	0.748582i	$-0.269265\pi$
$617$	20323.5	1.32608	0.663042	+	0.748582i	$0.269265\pi$
$618$	0	0
$619$	9223.99	0.598940	0.299470	−	0.954106i	$-0.403190\pi$
$619$	9223.99	0.598940	0.299470	+	0.954106i	$0.403190\pi$
$620$	0	0
$621$	−1913.29	−0.123635
$622$	0	0
$623$	−6719.70	−0.432134
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18984.7	−1.20921
$628$	0	0
$629$	−5353.54	−0.339364
$630$	0	0
$631$	16916.0	1.06722	0.533610	−	0.845730i	$-0.320835\pi$
$631$	16916.0	1.06722	0.533610	+	0.845730i	$0.320835\pi$
$632$	0	0
$633$	5708.54	0.358443
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−45319.9	−2.81890
$638$	0	0
$639$	2498.37	0.154670
$640$	0	0
$641$	−5811.35	−0.358088	−0.179044	−	0.983841i	$-0.557300\pi$
$641$	−5811.35	−0.358088	−0.179044	+	0.983841i	$0.557300\pi$
$642$	0	0
$643$	−27931.7	−1.71309	−0.856547	−	0.516069i	$-0.827395\pi$
$643$	−27931.7	−1.71309	−0.856547	+	0.516069i	$0.827395\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25437.8	−1.54569	−0.772845	−	0.634595i	$-0.781167\pi$
$647$	−25437.8	−1.54569	−0.772845	+	0.634595i	$0.781167\pi$
$648$	0	0
$649$	43015.9	2.60173
$650$	0	0
$651$	20880.0	1.25707
$652$	0	0
$653$	22647.7	1.35723	0.678617	−	0.734493i	$-0.262580\pi$
$653$	22647.7	1.35723	0.678617	+	0.734493i	$0.262580\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2469.63	0.146650
$658$	0	0
$659$	−25158.5	−1.48716	−0.743579	−	0.668649i	$-0.766873\pi$
$659$	−25158.5	−1.48716	−0.743579	+	0.668649i	$0.766873\pi$
$660$	0	0
$661$	23441.0	1.37935	0.689675	−	0.724119i	$-0.257753\pi$
$661$	23441.0	1.37935	0.689675	+	0.724119i	$0.257753\pi$
$662$	0	0
$663$	3221.13	0.188685
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7405.47	−0.429896
$668$	0	0
$669$	−14566.8	−0.841829
$670$	0	0
$671$	11676.0	0.671754
$672$	0	0
$673$	−3145.33	−0.180154	−0.0900770	−	0.995935i	$-0.528711\pi$
$673$	−3145.33	−0.180154	−0.0900770	+	0.995935i	$0.528711\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10606.2	−0.602109	−0.301054	−	0.953607i	$-0.597339\pi$
$677$	−10606.2	−0.602109	−0.301054	+	0.953607i	$0.597339\pi$
$678$	0	0
$679$	42448.1	2.39913
$680$	0	0
$681$	19608.6	1.10338
$682$	0	0
$683$	7825.06	0.438386	0.219193	−	0.975682i	$-0.429658\pi$
$683$	7825.06	0.438386	0.219193	+	0.975682i	$0.429658\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16531.1	−0.918049
$688$	0	0
$689$	−19819.0	−1.09585
$690$	0	0
$691$	−22750.9	−1.25251	−0.626256	−	0.779618i	$-0.715413\pi$
$691$	−22750.9	−1.25251	−0.626256	+	0.779618i	$0.715413\pi$
$692$	0	0
$693$	−14394.2	−0.789022
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4271.99	−0.232157
$698$	0	0
$699$	17747.0	0.960302
$700$	0	0
$701$	−133.598	−0.00719817	−0.00359908	−	0.999994i	$-0.501146\pi$
$701$	−133.598	−0.00719817	−0.00359908	+	0.999994i	$0.501146\pi$
$702$	0	0
$703$	39382.5	2.11286
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29333.1	1.56037
$708$	0	0
$709$	6886.26	0.364766	0.182383	−	0.983228i	$-0.441619\pi$
$709$	6886.26	0.364766	0.182383	+	0.983228i	$0.441619\pi$
$710$	0	0
$711$	−3302.92	−0.174218
$712$	0	0
$713$	−14912.4	−0.783271
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6789.76	−0.353652
$718$	0	0
$719$	1570.18	0.0814436	0.0407218	−	0.999171i	$-0.487034\pi$
$719$	1570.18	0.0814436	0.0407218	+	0.999171i	$0.487034\pi$
$720$	0	0
$721$	25898.3	1.33773
$722$	0	0
$723$	2317.48	0.119209
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3399.18	0.173409	0.0867047	−	0.996234i	$-0.472366\pi$
$727$	3399.18	0.173409	0.0867047	+	0.996234i	$0.472366\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1921.22	−0.0972078
$732$	0	0
$733$	−14152.1	−0.713125	−0.356562	−	0.934272i	$-0.616051\pi$
$733$	−14152.1	−0.713125	−0.356562	+	0.934272i	$0.616051\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5021.20	0.250961
$738$	0	0
$739$	14919.5	0.742655	0.371328	−	0.928502i	$-0.378903\pi$
$739$	14919.5	0.742655	0.371328	+	0.928502i	$0.378903\pi$
$740$	0	0
$741$	−23695.7	−1.17474
$742$	0	0
$743$	−7287.29	−0.359818	−0.179909	−	0.983683i	$-0.557580\pi$
$743$	−7287.29	−0.359818	−0.179909	+	0.983683i	$0.557580\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−519.526	−0.0254464
$748$	0	0
$749$	−43045.3	−2.09992
$750$	0	0
$751$	−18783.4	−0.912670	−0.456335	−	0.889808i	$-0.650838\pi$
$751$	−18783.4	−0.912670	−0.456335	+	0.889808i	$0.650838\pi$
$752$	0	0
$753$	−1788.58	−0.0865595
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30614.3	−1.46988	−0.734939	−	0.678134i	$-0.762789\pi$
$757$	−30614.3	−1.46988	−0.734939	+	0.678134i	$0.762789\pi$
$758$	0	0
$759$	10280.3	0.491634
$760$	0	0
$761$	−15277.2	−0.727723	−0.363861	−	0.931453i	$-0.618542\pi$
$761$	−15277.2	−0.727723	−0.363861	+	0.931453i	$0.618542\pi$
$762$	0	0
$763$	38428.4	1.82333
$764$	0	0
$765$	0	0
$766$	0	0
$767$	53690.3	2.52757
$768$	0	0
$769$	−17700.1	−0.830016	−0.415008	−	0.909818i	$-0.636221\pi$
$769$	−17700.1	−0.830016	−0.415008	+	0.909818i	$0.636221\pi$
$770$	0	0
$771$	12417.6	0.580036
$772$	0	0
$773$	29362.5	1.36623	0.683116	−	0.730310i	$-0.260625\pi$
$773$	29362.5	1.36623	0.683116	+	0.730310i	$0.260625\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	29859.9	1.37866
$778$	0	0
$779$	31426.2	1.44539
$780$	0	0
$781$	−13424.0	−0.615042
$782$	0	0
$783$	2821.63	0.128783
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2816.92	0.127589	0.0637943	−	0.997963i	$-0.479680\pi$
$787$	2816.92	0.127589	0.0637943	+	0.997963i	$0.479680\pi$
$788$	0	0
$789$	4834.01	0.218118
$790$	0	0
$791$	16795.0	0.754943
$792$	0	0
$793$	14573.4	0.652606
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18433.9	−0.819276	−0.409638	−	0.912248i	$-0.634345\pi$
$797$	−18433.9	−0.819276	−0.409638	+	0.912248i	$0.634345\pi$
$798$	0	0
$799$	−4962.99	−0.219747
$800$	0	0
$801$	−1828.58	−0.0806612
$802$	0	0
$803$	−13269.5	−0.583153
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21094.0	−0.920128
$808$	0	0
$809$	−13276.6	−0.576983	−0.288492	−	0.957482i	$-0.593154\pi$
$809$	−13276.6	−0.576983	−0.288492	+	0.957482i	$0.593154\pi$
$810$	0	0
$811$	18842.5	0.815845	0.407923	−	0.913016i	$-0.366253\pi$
$811$	18842.5	0.815845	0.407923	+	0.913016i	$0.366253\pi$
$812$	0	0
$813$	16323.8	0.704183
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14133.2	0.605210
$818$	0	0
$819$	−17966.2	−0.766531
$820$	0	0
$821$	5863.62	0.249259	0.124629	−	0.992203i	$-0.460226\pi$
$821$	5863.62	0.249259	0.124629	+	0.992203i	$0.460226\pi$
$822$	0	0
$823$	−5018.31	−0.212548	−0.106274	−	0.994337i	$-0.533892\pi$
$823$	−5018.31	−0.212548	−0.106274	+	0.994337i	$0.533892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20303.9	−0.853730	−0.426865	−	0.904315i	$-0.640382\pi$
$827$	−20303.9	−0.853730	−0.426865	+	0.904315i	$0.640382\pi$
$828$	0	0
$829$	−38156.2	−1.59857	−0.799287	−	0.600949i	$-0.794789\pi$
$829$	−38156.2	−1.59857	−0.799287	+	0.600949i	$0.794789\pi$
$830$	0	0
$831$	3241.28	0.135305
$832$	0	0
$833$	−13357.0	−0.555573
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5681.90	0.234642
$838$	0	0
$839$	−41004.6	−1.68729	−0.843645	−	0.536901i	$-0.819595\pi$
$839$	−41004.6	−1.68729	−0.843645	+	0.536901i	$0.819595\pi$
$840$	0	0
$841$	−13467.8	−0.552206
$842$	0	0
$843$	−4808.98	−0.196477
$844$	0	0
$845$	0	0
$846$	0	0
$847$	33320.8	1.35173
$848$	0	0
$849$	−1004.43	−0.0406030
$850$	0	0
$851$	−21325.8	−0.859033
$852$	0	0
$853$	−22451.8	−0.901214	−0.450607	−	0.892722i	$-0.648792\pi$
$853$	−22451.8	−0.901214	−0.450607	+	0.892722i	$0.648792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39858.0	1.58871	0.794354	−	0.607455i	$-0.207809\pi$
$857$	39858.0	1.58871	0.794354	+	0.607455i	$0.207809\pi$
$858$	0	0
$859$	−32585.0	−1.29428	−0.647139	−	0.762372i	$-0.724035\pi$
$859$	−32585.0	−1.29428	−0.647139	+	0.762372i	$0.724035\pi$
$860$	0	0
$861$	23827.5	0.943133
$862$	0	0
$863$	15593.6	0.615078	0.307539	−	0.951535i	$-0.400495\pi$
$863$	15593.6	0.615078	0.307539	+	0.951535i	$0.400495\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13789.6	−0.540163
$868$	0	0
$869$	17746.9	0.692775
$870$	0	0
$871$	6267.21	0.243807
$872$	0	0
$873$	11551.1	0.447817
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1804.75	−0.0694892	−0.0347446	−	0.999396i	$-0.511062\pi$
$877$	−1804.75	−0.0694892	−0.0347446	+	0.999396i	$0.511062\pi$
$878$	0	0
$879$	1617.75	0.0620766
$880$	0	0
$881$	−14788.0	−0.565515	−0.282758	−	0.959191i	$-0.591249\pi$
$881$	−14788.0	−0.565515	−0.282758	+	0.959191i	$0.591249\pi$
$882$	0	0
$883$	−1999.24	−0.0761947	−0.0380973	−	0.999274i	$-0.512130\pi$
$883$	−1999.24	−0.0761947	−0.0380973	+	0.999274i	$0.512130\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32474.4	−1.22929	−0.614647	−	0.788802i	$-0.710701\pi$
$887$	−32474.4	−1.22929	−0.614647	+	0.788802i	$0.710701\pi$
$888$	0	0
$889$	−26328.6	−0.993287
$890$	0	0
$891$	−3916.98	−0.147277
$892$	0	0
$893$	36509.4	1.36813
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12831.3	0.477620
$898$	0	0
$899$	21992.0	0.815880
$900$	0	0
$901$	−5841.18	−0.215980
$902$	0	0
$903$	10715.8	0.394905
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27597.9	1.01033	0.505167	−	0.863022i	$-0.331431\pi$
$907$	27597.9	1.01033	0.505167	+	0.863022i	$0.331431\pi$
$908$	0	0
$909$	7982.17	0.291256
$910$	0	0
$911$	19861.4	0.722325	0.361163	−	0.932503i	$-0.382380\pi$
$911$	19861.4	0.722325	0.361163	+	0.932503i	$0.382380\pi$
$912$	0	0
$913$	2791.46	0.101187
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3025.44	0.108952
$918$	0	0
$919$	20886.0	0.749691	0.374845	−	0.927087i	$-0.377696\pi$
$919$	20886.0	0.749691	0.374845	+	0.927087i	$0.377696\pi$
$920$	0	0
$921$	−25432.7	−0.909921
$922$	0	0
$923$	−16755.2	−0.597511
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7047.49	0.249698
$928$	0	0
$929$	37461.2	1.32299	0.661497	−	0.749948i	$-0.269922\pi$
$929$	37461.2	1.32299	0.661497	+	0.749948i	$0.269922\pi$
$930$	0	0
$931$	98258.5	3.45896
$932$	0	0
$933$	−10940.8	−0.383906
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27149.0	0.946552	0.473276	−	0.880914i	$-0.343071\pi$
$937$	27149.0	0.946552	0.473276	+	0.880914i	$0.343071\pi$
$938$	0	0
$939$	22611.1	0.785822
$940$	0	0
$941$	−43068.3	−1.49202	−0.746008	−	0.665937i	$-0.768032\pi$
$941$	−43068.3	−1.49202	−0.746008	+	0.665937i	$0.768032\pi$
$942$	0	0
$943$	−17017.4	−0.587660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30021.1	−1.03015	−0.515076	−	0.857145i	$-0.672236\pi$
$947$	−30021.1	−1.03015	−0.515076	+	0.857145i	$0.672236\pi$
$948$	0	0
$949$	−16562.4	−0.566530
$950$	0	0
$951$	−32163.5	−1.09671
$952$	0	0
$953$	−53820.4	−1.82939	−0.914697	−	0.404140i	$-0.867571\pi$
$953$	−53820.4	−1.82939	−0.914697	+	0.404140i	$0.867571\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−15160.9	−0.512101
$958$	0	0
$959$	75185.1	2.53165
$960$	0	0
$961$	14494.3	0.486532
$962$	0	0
$963$	−11713.6	−0.391967
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32210.2	1.07116	0.535580	−	0.844485i	$-0.320093\pi$
$967$	32210.2	1.07116	0.535580	+	0.844485i	$0.320093\pi$
$968$	0	0
$969$	−6983.77	−0.231529
$970$	0	0
$971$	21616.1	0.714411	0.357206	−	0.934026i	$-0.383730\pi$
$971$	21616.1	0.714411	0.357206	+	0.934026i	$0.383730\pi$
$972$	0	0
$973$	24432.5	0.805005
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46002.8	1.50641	0.753204	−	0.657787i	$-0.228507\pi$
$977$	46002.8	1.50641	0.753204	+	0.657787i	$0.228507\pi$
$978$	0	0
$979$	9825.11	0.320748
$980$	0	0
$981$	10457.2	0.340339
$982$	0	0
$983$	−3926.43	−0.127400	−0.0636998	−	0.997969i	$-0.520290\pi$
$983$	−3926.43	−0.127400	−0.0636998	+	0.997969i	$0.520290\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	27681.5	0.892719
$988$	0	0
$989$	−7653.15	−0.246063
$990$	0	0
$991$	−55802.3	−1.78872	−0.894359	−	0.447351i	$-0.852368\pi$
$991$	−55802.3	−1.78872	−0.894359	+	0.447351i	$0.852368\pi$
$992$	0	0
$993$	13215.4	0.422335
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−43880.8	−1.39390	−0.696951	−	0.717119i	$-0.745460\pi$
$997$	−43880.8	−1.39390	−0.696951	+	0.717119i	$0.745460\pi$
$998$	0	0
$999$	8125.53	0.257338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bq.1.2		2
4.3	odd	2		600.4.a.t.1.1		2
5.2	odd	4		240.4.f.g.49.2		4
5.3	odd	4		240.4.f.g.49.4		4
5.4	even	2		1200.4.a.bo.1.1		2
12.11	even	2		1800.4.a.bn.1.1		2
15.2	even	4		720.4.f.i.289.1		4
15.8	even	4		720.4.f.i.289.2		4
20.3	even	4		120.4.f.d.49.2	✓	4
20.7	even	4		120.4.f.d.49.4	yes	4
20.19	odd	2		600.4.a.v.1.2		2
40.3	even	4		960.4.f.n.769.3		4
40.13	odd	4		960.4.f.o.769.1		4
40.27	even	4		960.4.f.n.769.1		4
40.37	odd	4		960.4.f.o.769.3		4
60.23	odd	4		360.4.f.d.289.2		4
60.47	odd	4		360.4.f.d.289.1		4
60.59	even	2		1800.4.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.d.49.2	✓	4	20.3	even	4
120.4.f.d.49.4	yes	4	20.7	even	4
240.4.f.g.49.2		4	5.2	odd	4
240.4.f.g.49.4		4	5.3	odd	4
360.4.f.d.289.1		4	60.47	odd	4
360.4.f.d.289.2		4	60.23	odd	4
600.4.a.t.1.1		2	4.3	odd	2
600.4.a.v.1.2		2	20.19	odd	2
720.4.f.i.289.1		4	15.2	even	4
720.4.f.i.289.2		4	15.8	even	4
960.4.f.n.769.1		4	40.27	even	4
960.4.f.n.769.3		4	40.3	even	4
960.4.f.o.769.1		4	40.13	odd	4
960.4.f.o.769.3		4	40.37	odd	4
1200.4.a.bo.1.1		2	5.4	even	2
1200.4.a.bq.1.2		2	1.1	even	1	trivial
1800.4.a.bl.1.2		2	60.59	even	2
1800.4.a.bn.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +33.0735 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +33.0735 q^{7} +9.00000 q^{9} -48.3578 q^{11} -60.3578 q^{13} -17.7891 q^{17} +130.863 q^{19} +99.2204 q^{21} -70.8625 q^{23} +27.0000 q^{27} +104.505 q^{29} +210.441 q^{31} -145.073 q^{33} +300.945 q^{37} -181.073 q^{39} +240.147 q^{41} +108.000 q^{43} +278.991 q^{47} +750.853 q^{49} -53.3673 q^{51} +328.358 q^{53} +392.588 q^{57} -889.533 q^{59} -241.450 q^{61} +297.661 q^{63} -103.834 q^{67} -212.588 q^{69} +277.597 q^{71} +274.403 q^{73} -1599.36 q^{77} -366.991 q^{79} +81.0000 q^{81} -57.7251 q^{83} +313.514 q^{87} -203.175 q^{89} -1996.24 q^{91} +631.322 q^{93} +1283.45 q^{97} -435.220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 74 q^{11} - 98 q^{13} + 78 q^{17} + 80 q^{19} - 6 q^{21} + 40 q^{23} + 54 q^{27} + 50 q^{29} + 12 q^{31} - 222 q^{33} + 34 q^{37} - 294 q^{39} + 344 q^{41} + 216 q^{43} + 876 q^{47} + 1638 q^{49} + 234 q^{51} + 634 q^{53} + 240 q^{57} - 666 q^{59} + 244 q^{61} - 18 q^{63} - 980 q^{67} + 120 q^{69} - 308 q^{71} + 1412 q^{73} - 700 q^{77} - 1052 q^{79} + 162 q^{81} + 248 q^{83} + 150 q^{87} + 684 q^{89} - 676 q^{91} + 36 q^{93} + 1840 q^{97} - 666 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 - 74 * q^11 - 98 * q^13 + 78 * q^17 + 80 * q^19 - 6 * q^21 + 40 * q^23 + 54 * q^27 + 50 * q^29 + 12 * q^31 - 222 * q^33 + 34 * q^37 - 294 * q^39 + 344 * q^41 + 216 * q^43 + 876 * q^47 + 1638 * q^49 + 234 * q^51 + 634 * q^53 + 240 * q^57 - 666 * q^59 + 244 * q^61 - 18 * q^63 - 980 * q^67 + 120 * q^69 - 308 * q^71 + 1412 * q^73 - 700 * q^77 - 1052 * q^79 + 162 * q^81 + 248 * q^83 + 150 * q^87 + 684 * q^89 - 676 * q^91 + 36 * q^93 + 1840 * q^97 - 666 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more