LMFDB - Embedded newform 2400.4.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23109.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 37x + 10 $ x^3 - x^2 - 37*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.47542$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +11.6634 q^{7} +9.00000 q^{9} -41.5651 q^{11} -1.76173 q^{13} +63.3879 q^{17} -25.3463 q^{19} +34.9903 q^{21} -181.906 q^{23} +27.0000 q^{27} +204.916 q^{29} +61.8228 q^{31} -124.695 q^{33} -365.452 q^{37} -5.28520 q^{39} +130.634 q^{41} -135.252 q^{43} +380.524 q^{47} -206.964 q^{49} +190.164 q^{51} -352.853 q^{53} -76.0388 q^{57} -556.502 q^{59} +88.5048 q^{61} +104.971 q^{63} +708.780 q^{67} -545.718 q^{69} -750.078 q^{71} -57.1465 q^{73} -484.792 q^{77} +544.449 q^{79} +81.0000 q^{81} -1414.64 q^{83} +614.747 q^{87} +533.702 q^{89} -20.5479 q^{91} +185.468 q^{93} +300.461 q^{97} -374.086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} + 32 q^{7} + 27 q^{9} - 48 q^{11} - 76 q^{13} - 16 q^{17} - 88 q^{19} + 96 q^{21} + 20 q^{23} + 81 q^{27} + 58 q^{29} + 56 q^{31} - 144 q^{33} - 436 q^{37} - 228 q^{39} + 362 q^{41} + 148 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 + 32 * q^7 + 27 * q^9 - 48 * q^11 - 76 * q^13 - 16 * q^17 - 88 * q^19 + 96 * q^21 + 20 * q^23 + 81 * q^27 + 58 * q^29 + 56 * q^31 - 144 * q^33 - 436 * q^37 - 228 * q^39 + 362 * q^41 + 148 * q^43 + 52 * q^47 - 109 * q^49 - 48 * q^51 - 940 * q^53 - 264 * q^57 - 816 * q^59 - 842 * q^61 + 288 * q^63 - 780 * q^67 + 60 * q^69 - 304 * q^71 - 512 * q^73 - 1536 * q^77 + 440 * q^79 + 243 * q^81 - 1212 * q^83 + 174 * q^87 - 590 * q^89 - 944 * q^91 + 168 * q^93 - 1752 * q^97 - 432 * 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
$7$	11.6634	0.629766	0.314883	−	0.949130i	$-0.398035\pi$
$7$	11.6634	0.629766	0.314883	+	0.949130i	$0.398035\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−41.5651	−1.13931	−0.569653	−	0.821886i	$-0.692922\pi$
$11$	−41.5651	−1.13931	−0.569653	+	0.821886i	$0.692922\pi$
$12$	0	0
$13$	−1.76173	−0.0375859	−0.0187930	−	0.999823i	$-0.505982\pi$
$13$	−1.76173	−0.0375859	−0.0187930	+	0.999823i	$0.505982\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	63.3879	0.904343	0.452172	−	0.891931i	$-0.350649\pi$
$17$	63.3879	0.904343	0.452172	+	0.891931i	$0.350649\pi$
$18$	0	0
$19$	−25.3463	−0.306044	−0.153022	−	0.988223i	$-0.548901\pi$
$19$	−25.3463	−0.306044	−0.153022	+	0.988223i	$0.548901\pi$
$20$	0	0
$21$	34.9903	0.363596
$22$	0	0
$23$	−181.906	−1.64913	−0.824566	−	0.565765i	$-0.808581\pi$
$23$	−181.906	−1.64913	−0.824566	+	0.565765i	$0.808581\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	204.916	1.31213	0.656067	−	0.754702i	$-0.272219\pi$
$29$	204.916	1.31213	0.656067	+	0.754702i	$0.272219\pi$
$30$	0	0
$31$	61.8228	0.358184	0.179092	−	0.983832i	$-0.442684\pi$
$31$	61.8228	0.358184	0.179092	+	0.983832i	$0.442684\pi$
$32$	0	0
$33$	−124.695	−0.657778
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−365.452	−1.62378	−0.811891	−	0.583809i	$-0.801562\pi$
$37$	−365.452	−1.62378	−0.811891	+	0.583809i	$0.801562\pi$
$38$	0	0
$39$	−5.28520	−0.0217003
$40$	0	0
$41$	130.634	0.497601	0.248801	−	0.968555i	$-0.419964\pi$
$41$	130.634	0.497601	0.248801	+	0.968555i	$0.419964\pi$
$42$	0	0
$43$	−135.252	−0.479670	−0.239835	−	0.970814i	$-0.577093\pi$
$43$	−135.252	−0.479670	−0.239835	+	0.970814i	$0.577093\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	380.524	1.18096	0.590480	−	0.807052i	$-0.298938\pi$
$47$	380.524	1.18096	0.590480	+	0.807052i	$0.298938\pi$
$48$	0	0
$49$	−206.964	−0.603395
$50$	0	0
$51$	190.164	0.522123
$52$	0	0
$53$	−352.853	−0.914493	−0.457246	−	0.889340i	$-0.651164\pi$
$53$	−352.853	−0.914493	−0.457246	+	0.889340i	$0.651164\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−76.0388	−0.176694
$58$	0	0
$59$	−556.502	−1.22797	−0.613986	−	0.789317i	$-0.710435\pi$
$59$	−556.502	−1.22797	−0.613986	+	0.789317i	$0.710435\pi$
$60$	0	0
$61$	88.5048	0.185768	0.0928842	−	0.995677i	$-0.470391\pi$
$61$	88.5048	0.185768	0.0928842	+	0.995677i	$0.470391\pi$
$62$	0	0
$63$	104.971	0.209922
$64$	0	0
$65$	0	0
$66$	0	0
$67$	708.780	1.29241	0.646203	−	0.763165i	$-0.276356\pi$
$67$	708.780	1.29241	0.646203	+	0.763165i	$0.276356\pi$
$68$	0	0
$69$	−545.718	−0.952127
$70$	0	0
$71$	−750.078	−1.25377	−0.626887	−	0.779110i	$-0.715671\pi$
$71$	−750.078	−1.25377	−0.626887	+	0.779110i	$0.715671\pi$
$72$	0	0
$73$	−57.1465	−0.0916232	−0.0458116	−	0.998950i	$-0.514587\pi$
$73$	−57.1465	−0.0916232	−0.0458116	+	0.998950i	$0.514587\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−484.792	−0.717496
$78$	0	0
$79$	544.449	0.775384	0.387692	−	0.921789i	$-0.373272\pi$
$79$	544.449	0.775384	0.387692	+	0.921789i	$0.373272\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1414.64	−1.87081	−0.935406	−	0.353576i	$-0.884966\pi$
$83$	−1414.64	−1.87081	−0.935406	+	0.353576i	$0.884966\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	614.747	0.757561
$88$	0	0
$89$	533.702	0.635644	0.317822	−	0.948150i	$-0.397049\pi$
$89$	533.702	0.635644	0.317822	+	0.948150i	$0.397049\pi$
$90$	0	0
$91$	−20.5479	−0.0236704
$92$	0	0
$93$	185.468	0.206798
$94$	0	0
$95$	0	0
$96$	0	0
$97$	300.461	0.314507	0.157254	−	0.987558i	$-0.449736\pi$
$97$	300.461	0.314507	0.157254	+	0.987558i	$0.449736\pi$
$98$	0	0
$99$	−374.086	−0.379768
$100$	0	0
$101$	−1723.37	−1.69784	−0.848920	−	0.528521i	$-0.822747\pi$
$101$	−1723.37	−1.69784	−0.848920	+	0.528521i	$0.822747\pi$
$102$	0	0
$103$	−613.463	−0.586857	−0.293428	−	0.955981i	$-0.594796\pi$
$103$	−613.463	−0.586857	−0.293428	+	0.955981i	$0.594796\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−544.039	−0.491535	−0.245767	−	0.969329i	$-0.579040\pi$
$107$	−544.039	−0.491535	−0.245767	+	0.969329i	$0.579040\pi$
$108$	0	0
$109$	−291.638	−0.256274	−0.128137	−	0.991757i	$-0.540900\pi$
$109$	−291.638	−0.256274	−0.128137	+	0.991757i	$0.540900\pi$
$110$	0	0
$111$	−1096.36	−0.937491
$112$	0	0
$113$	−107.737	−0.0896908	−0.0448454	−	0.998994i	$-0.514280\pi$
$113$	−107.737	−0.0896908	−0.0448454	+	0.998994i	$0.514280\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.8556	−0.0125286
$118$	0	0
$119$	739.321	0.569525
$120$	0	0
$121$	396.660	0.298017
$122$	0	0
$123$	391.903	0.287290
$124$	0	0
$125$	0	0
$126$	0	0
$127$	218.699	0.152806	0.0764031	−	0.997077i	$-0.475656\pi$
$127$	218.699	0.152806	0.0764031	+	0.997077i	$0.475656\pi$
$128$	0	0
$129$	−405.757	−0.276937
$130$	0	0
$131$	1311.65	0.874806	0.437403	−	0.899266i	$-0.355898\pi$
$131$	1311.65	0.874806	0.437403	+	0.899266i	$0.355898\pi$
$132$	0	0
$133$	−295.624	−0.192736
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1920.23	−1.19749	−0.598747	−	0.800939i	$-0.704334\pi$
$137$	−1920.23	−1.19749	−0.598747	+	0.800939i	$0.704334\pi$
$138$	0	0
$139$	−1151.69	−0.702772	−0.351386	−	0.936231i	$-0.614290\pi$
$139$	−1151.69	−0.702772	−0.351386	+	0.936231i	$0.614290\pi$
$140$	0	0
$141$	1141.57	0.681828
$142$	0	0
$143$	73.2267	0.0428219
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−620.893	−0.348370
$148$	0	0
$149$	2650.40	1.45724	0.728621	−	0.684917i	$-0.240162\pi$
$149$	2650.40	1.45724	0.728621	+	0.684917i	$0.240162\pi$
$150$	0	0
$151$	−2382.66	−1.28409	−0.642046	−	0.766666i	$-0.721914\pi$
$151$	−2382.66	−1.28409	−0.642046	+	0.766666i	$0.721914\pi$
$152$	0	0
$153$	570.491	0.301448
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−664.035	−0.337553	−0.168776	−	0.985654i	$-0.553982\pi$
$157$	−664.035	−0.337553	−0.168776	+	0.985654i	$0.553982\pi$
$158$	0	0
$159$	−1058.56	−0.527983
$160$	0	0
$161$	−2121.65	−1.03857
$162$	0	0
$163$	2309.27	1.10967	0.554835	−	0.831960i	$-0.312781\pi$
$163$	2309.27	1.10967	0.554835	+	0.831960i	$0.312781\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	134.026	0.0621032	0.0310516	−	0.999518i	$-0.490114\pi$
$167$	134.026	0.0621032	0.0310516	+	0.999518i	$0.490114\pi$
$168$	0	0
$169$	−2193.90	−0.998587
$170$	0	0
$171$	−228.116	−0.102015
$172$	0	0
$173$	713.912	0.313744	0.156872	−	0.987619i	$-0.449859\pi$
$173$	713.912	0.313744	0.156872	+	0.987619i	$0.449859\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1669.51	−0.708970
$178$	0	0
$179$	−2212.94	−0.924037	−0.462018	−	0.886870i	$-0.652875\pi$
$179$	−2212.94	−0.924037	−0.462018	+	0.886870i	$0.652875\pi$
$180$	0	0
$181$	2249.65	0.923841	0.461920	−	0.886921i	$-0.347161\pi$
$181$	2249.65	0.923841	0.461920	+	0.886921i	$0.347161\pi$
$182$	0	0
$183$	265.514	0.107253
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2634.73	−1.03032
$188$	0	0
$189$	314.913	0.121199
$190$	0	0
$191$	−4830.48	−1.82995	−0.914977	−	0.403507i	$-0.867791\pi$
$191$	−4830.48	−1.82995	−0.914977	+	0.403507i	$0.867791\pi$
$192$	0	0
$193$	859.268	0.320474	0.160237	−	0.987079i	$-0.448774\pi$
$193$	859.268	0.320474	0.160237	+	0.987079i	$0.448774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3478.85	1.25816	0.629080	−	0.777340i	$-0.283432\pi$
$197$	3478.85	1.25816	0.629080	+	0.777340i	$0.283432\pi$
$198$	0	0
$199$	3586.28	1.27751	0.638756	−	0.769409i	$-0.279449\pi$
$199$	3586.28	1.27751	0.638756	+	0.769409i	$0.279449\pi$
$200$	0	0
$201$	2126.34	0.746171
$202$	0	0
$203$	2390.02	0.826338
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1637.16	−0.549711
$208$	0	0
$209$	1053.52	0.348677
$210$	0	0
$211$	−3571.10	−1.16514	−0.582570	−	0.812780i	$-0.697953\pi$
$211$	−3571.10	−1.16514	−0.582570	+	0.812780i	$0.697953\pi$
$212$	0	0
$213$	−2250.24	−0.723867
$214$	0	0
$215$	0	0
$216$	0	0
$217$	721.066	0.225572
$218$	0	0
$219$	−171.440	−0.0528987
$220$	0	0
$221$	−111.673	−0.0339906
$222$	0	0
$223$	3283.29	0.985944	0.492972	−	0.870045i	$-0.335910\pi$
$223$	3283.29	0.985944	0.492972	+	0.870045i	$0.335910\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3940.04	−1.15203	−0.576013	−	0.817441i	$-0.695392\pi$
$227$	−3940.04	−1.15203	−0.576013	+	0.817441i	$0.695392\pi$
$228$	0	0
$229$	633.256	0.182737	0.0913684	−	0.995817i	$-0.470876\pi$
$229$	633.256	0.182737	0.0913684	+	0.995817i	$0.470876\pi$
$230$	0	0
$231$	−1454.38	−0.414246
$232$	0	0
$233$	−1925.29	−0.541331	−0.270666	−	0.962673i	$-0.587244\pi$
$233$	−1925.29	−0.541331	−0.270666	+	0.962673i	$0.587244\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1633.35	0.447668
$238$	0	0
$239$	−3800.09	−1.02848	−0.514241	−	0.857646i	$-0.671926\pi$
$239$	−3800.09	−1.02848	−0.514241	+	0.857646i	$0.671926\pi$
$240$	0	0
$241$	3265.31	0.872769	0.436384	−	0.899760i	$-0.356259\pi$
$241$	3265.31	0.872769	0.436384	+	0.899760i	$0.356259\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	44.6534	0.0115029
$248$	0	0
$249$	−4243.93	−1.08011
$250$	0	0
$251$	−6918.67	−1.73985	−0.869926	−	0.493182i	$-0.835833\pi$
$251$	−6918.67	−1.73985	−0.869926	+	0.493182i	$0.835833\pi$
$252$	0	0
$253$	7560.95	1.87887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1318.70	0.320071	0.160036	−	0.987111i	$-0.448839\pi$
$257$	1318.70	0.320071	0.160036	+	0.987111i	$0.448839\pi$
$258$	0	0
$259$	−4262.42	−1.02260
$260$	0	0
$261$	1844.24	0.437378
$262$	0	0
$263$	7131.42	1.67202	0.836012	−	0.548711i	$-0.184881\pi$
$263$	7131.42	1.67202	0.836012	+	0.548711i	$0.184881\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1601.11	0.366989
$268$	0	0
$269$	1924.07	0.436107	0.218054	−	0.975937i	$-0.430029\pi$
$269$	1924.07	0.436107	0.218054	+	0.975937i	$0.430029\pi$
$270$	0	0
$271$	1384.60	0.310364	0.155182	−	0.987886i	$-0.450404\pi$
$271$	1384.60	0.310364	0.155182	+	0.987886i	$0.450404\pi$
$272$	0	0
$273$	−61.6436	−0.0136661
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5198.50	−1.12761	−0.563805	−	0.825908i	$-0.690663\pi$
$277$	−5198.50	−1.12761	−0.563805	+	0.825908i	$0.690663\pi$
$278$	0	0
$279$	556.405	0.119395
$280$	0	0
$281$	−9293.25	−1.97291	−0.986457	−	0.164021i	$-0.947554\pi$
$281$	−9293.25	−1.97291	−0.986457	+	0.164021i	$0.947554\pi$
$282$	0	0
$283$	−5127.94	−1.07712	−0.538559	−	0.842588i	$-0.681031\pi$
$283$	−5127.94	−1.07712	−0.538559	+	0.842588i	$0.681031\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1523.64	0.313372
$288$	0	0
$289$	−894.970	−0.182164
$290$	0	0
$291$	901.383	0.181581
$292$	0	0
$293$	−7146.94	−1.42501	−0.712506	−	0.701666i	$-0.752440\pi$
$293$	−7146.94	−1.42501	−0.712506	+	0.701666i	$0.752440\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1122.26	−0.219259
$298$	0	0
$299$	320.470	0.0619842
$300$	0	0
$301$	−1577.51	−0.302080
$302$	0	0
$303$	−5170.12	−0.980249
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6428.75	−1.19514	−0.597570	−	0.801817i	$-0.703867\pi$
$307$	−6428.75	−1.19514	−0.597570	+	0.801817i	$0.703867\pi$
$308$	0	0
$309$	−1840.39	−0.338822
$310$	0	0
$311$	−7273.31	−1.32615	−0.663073	−	0.748555i	$-0.730748\pi$
$311$	−7273.31	−1.32615	−0.663073	+	0.748555i	$0.730748\pi$
$312$	0	0
$313$	−6757.12	−1.22024	−0.610120	−	0.792309i	$-0.708879\pi$
$313$	−6757.12	−1.22024	−0.610120	+	0.792309i	$0.708879\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2773.99	−0.491492	−0.245746	−	0.969334i	$-0.579033\pi$
$317$	−2773.99	−0.491492	−0.245746	+	0.969334i	$0.579033\pi$
$318$	0	0
$319$	−8517.35	−1.49492
$320$	0	0
$321$	−1632.12	−0.283788
$322$	0	0
$323$	−1606.65	−0.276769
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−874.913	−0.147960
$328$	0	0
$329$	4438.22	0.743729
$330$	0	0
$331$	−11223.4	−1.86374	−0.931868	−	0.362798i	$-0.881822\pi$
$331$	−11223.4	−1.86374	−0.931868	+	0.362798i	$0.881822\pi$
$332$	0	0
$333$	−3289.07	−0.541260
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11193.3	1.80932	0.904659	−	0.426135i	$-0.140125\pi$
$337$	11193.3	1.80932	0.904659	+	0.426135i	$0.140125\pi$
$338$	0	0
$339$	−323.211	−0.0517830
$340$	0	0
$341$	−2569.67	−0.408081
$342$	0	0
$343$	−6414.47	−1.00976
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3275.24	0.506698	0.253349	−	0.967375i	$-0.418468\pi$
$347$	3275.24	0.506698	0.253349	+	0.967375i	$0.418468\pi$
$348$	0	0
$349$	−59.2693	−0.00909059	−0.00454529	−	0.999990i	$-0.501447\pi$
$349$	−59.2693	−0.00909059	−0.00454529	+	0.999990i	$0.501447\pi$
$350$	0	0
$351$	−47.5668	−0.00723342
$352$	0	0
$353$	6885.71	1.03821	0.519107	−	0.854709i	$-0.326265\pi$
$353$	6885.71	1.03821	0.519107	+	0.854709i	$0.326265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2217.96	0.328815
$358$	0	0
$359$	−1089.97	−0.160241	−0.0801207	−	0.996785i	$-0.525531\pi$
$359$	−1089.97	−0.160241	−0.0801207	+	0.996785i	$0.525531\pi$
$360$	0	0
$361$	−6216.57	−0.906337
$362$	0	0
$363$	1189.98	0.172060
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11974.8	1.70321	0.851607	−	0.524181i	$-0.175629\pi$
$367$	11974.8	1.70321	0.851607	+	0.524181i	$0.175629\pi$
$368$	0	0
$369$	1175.71	0.165867
$370$	0	0
$371$	−4115.48	−0.575917
$372$	0	0
$373$	11363.2	1.57739	0.788695	−	0.614784i	$-0.210757\pi$
$373$	11363.2	1.57739	0.788695	+	0.614784i	$0.210757\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−361.007	−0.0493178
$378$	0	0
$379$	9179.37	1.24410	0.622048	−	0.782979i	$-0.286301\pi$
$379$	9179.37	1.24410	0.622048	+	0.782979i	$0.286301\pi$
$380$	0	0
$381$	656.097	0.0882227
$382$	0	0
$383$	610.812	0.0814910	0.0407455	−	0.999170i	$-0.487027\pi$
$383$	610.812	0.0814910	0.0407455	+	0.999170i	$0.487027\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1217.27	−0.159890
$388$	0	0
$389$	5645.96	0.735890	0.367945	−	0.929847i	$-0.380061\pi$
$389$	5645.96	0.735890	0.367945	+	0.929847i	$0.380061\pi$
$390$	0	0
$391$	−11530.7	−1.49138
$392$	0	0
$393$	3934.96	0.505069
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8028.62	1.01497	0.507487	−	0.861659i	$-0.330575\pi$
$397$	8028.62	1.01497	0.507487	+	0.861659i	$0.330575\pi$
$398$	0	0
$399$	−886.873	−0.111276
$400$	0	0
$401$	11847.6	1.47542	0.737709	−	0.675119i	$-0.235908\pi$
$401$	11847.6	1.47542	0.737709	+	0.675119i	$0.235908\pi$
$402$	0	0
$403$	−108.915	−0.0134627
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15190.1	1.84998
$408$	0	0
$409$	−13465.2	−1.62790	−0.813949	−	0.580937i	$-0.802686\pi$
$409$	−13465.2	−1.62790	−0.813949	+	0.580937i	$0.802686\pi$
$410$	0	0
$411$	−5760.70	−0.691373
$412$	0	0
$413$	−6490.72	−0.773336
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3455.08	−0.405746
$418$	0	0
$419$	2570.65	0.299724	0.149862	−	0.988707i	$-0.452117\pi$
$419$	2570.65	0.299724	0.149862	+	0.988707i	$0.452117\pi$
$420$	0	0
$421$	8360.03	0.967798	0.483899	−	0.875124i	$-0.339220\pi$
$421$	8360.03	0.967798	0.483899	+	0.875124i	$0.339220\pi$
$422$	0	0
$423$	3424.72	0.393654
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1032.27	0.116991
$428$	0	0
$429$	219.680	0.0247232
$430$	0	0
$431$	10432.8	1.16597	0.582984	−	0.812484i	$-0.301885\pi$
$431$	10432.8	1.16597	0.582984	+	0.812484i	$0.301885\pi$
$432$	0	0
$433$	2439.39	0.270738	0.135369	−	0.990795i	$-0.456778\pi$
$433$	2439.39	0.270738	0.135369	+	0.990795i	$0.456778\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4610.64	0.504707
$438$	0	0
$439$	−13826.6	−1.50320	−0.751601	−	0.659618i	$-0.770718\pi$
$439$	−13826.6	−1.50320	−0.751601	+	0.659618i	$0.770718\pi$
$440$	0	0
$441$	−1862.68	−0.201132
$442$	0	0
$443$	−17520.3	−1.87904	−0.939518	−	0.342500i	$-0.888726\pi$
$443$	−17520.3	−1.87904	−0.939518	+	0.342500i	$0.888726\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7951.19	0.841339
$448$	0	0
$449$	−1353.24	−0.142234	−0.0711171	−	0.997468i	$-0.522656\pi$
$449$	−1353.24	−0.142234	−0.0711171	+	0.997468i	$0.522656\pi$
$450$	0	0
$451$	−5429.83	−0.566920
$452$	0	0
$453$	−7147.97	−0.741371
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11601.7	1.18754	0.593771	−	0.804634i	$-0.297639\pi$
$457$	11601.7	1.18754	0.593771	+	0.804634i	$0.297639\pi$
$458$	0	0
$459$	1711.47	0.174041
$460$	0	0
$461$	5113.70	0.516635	0.258318	−	0.966060i	$-0.416832\pi$
$461$	5113.70	0.516635	0.258318	+	0.966060i	$0.416832\pi$
$462$	0	0
$463$	2761.40	0.277178	0.138589	−	0.990350i	$-0.455743\pi$
$463$	2761.40	0.277178	0.138589	+	0.990350i	$0.455743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9663.24	−0.957519	−0.478760	−	0.877946i	$-0.658913\pi$
$467$	−9663.24	−0.957519	−0.478760	+	0.877946i	$0.658913\pi$
$468$	0	0
$469$	8266.81	0.813914
$470$	0	0
$471$	−1992.10	−0.194886
$472$	0	0
$473$	5621.78	0.546490
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3175.68	−0.304831
$478$	0	0
$479$	−11386.1	−1.08610	−0.543052	−	0.839699i	$-0.682731\pi$
$479$	−11386.1	−1.08610	−0.543052	+	0.839699i	$0.682731\pi$
$480$	0	0
$481$	643.829	0.0610314
$482$	0	0
$483$	−6364.95	−0.599617
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2106.69	0.196023	0.0980113	−	0.995185i	$-0.468752\pi$
$487$	2106.69	0.196023	0.0980113	+	0.995185i	$0.468752\pi$
$488$	0	0
$489$	6927.82	0.640669
$490$	0	0
$491$	−15897.6	−1.46119	−0.730597	−	0.682809i	$-0.760758\pi$
$491$	−15897.6	−1.46119	−0.730597	+	0.682809i	$0.760758\pi$
$492$	0	0
$493$	12989.2	1.18662
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8748.49	−0.789584
$498$	0	0
$499$	−18439.4	−1.65423	−0.827115	−	0.562032i	$-0.810020\pi$
$499$	−18439.4	−1.65423	−0.827115	+	0.562032i	$0.810020\pi$
$500$	0	0
$501$	402.078	0.0358553
$502$	0	0
$503$	−3305.82	−0.293040	−0.146520	−	0.989208i	$-0.546807\pi$
$503$	−3305.82	−0.293040	−0.146520	+	0.989208i	$0.546807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6581.69	−0.576535
$508$	0	0
$509$	−16712.0	−1.45530	−0.727650	−	0.685949i	$-0.759387\pi$
$509$	−16712.0	−1.45530	−0.727650	+	0.685949i	$0.759387\pi$
$510$	0	0
$511$	−666.525	−0.0577012
$512$	0	0
$513$	−684.349	−0.0588982
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15816.5	−1.34547
$518$	0	0
$519$	2141.74	0.181140
$520$	0	0
$521$	13038.2	1.09638	0.548188	−	0.836355i	$-0.315318\pi$
$521$	13038.2	1.09638	0.548188	+	0.836355i	$0.315318\pi$
$522$	0	0
$523$	−7503.18	−0.627326	−0.313663	−	0.949534i	$-0.601556\pi$
$523$	−7503.18	−0.627326	−0.313663	+	0.949534i	$0.601556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3918.82	0.323921
$528$	0	0
$529$	20922.8	1.71964
$530$	0	0
$531$	−5008.52	−0.409324
$532$	0	0
$533$	−230.143	−0.0187028
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6638.81	−0.533493
$538$	0	0
$539$	8602.50	0.687451
$540$	0	0
$541$	−6604.05	−0.524825	−0.262412	−	0.964956i	$-0.584518\pi$
$541$	−6604.05	−0.524825	−0.262412	+	0.964956i	$0.584518\pi$
$542$	0	0
$543$	6748.95	0.533380
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7065.78	0.552305	0.276153	−	0.961114i	$-0.410940\pi$
$547$	7065.78	0.552305	0.276153	+	0.961114i	$0.410940\pi$
$548$	0	0
$549$	796.543	0.0619228
$550$	0	0
$551$	−5193.85	−0.401571
$552$	0	0
$553$	6350.15	0.488310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12082.2	0.919098	0.459549	−	0.888152i	$-0.348011\pi$
$557$	12082.2	0.919098	0.459549	+	0.888152i	$0.348011\pi$
$558$	0	0
$559$	238.279	0.0180288
$560$	0	0
$561$	−7904.18	−0.594857
$562$	0	0
$563$	−3893.48	−0.291458	−0.145729	−	0.989325i	$-0.546553\pi$
$563$	−3893.48	−0.291458	−0.145729	+	0.989325i	$0.546553\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	944.738	0.0699740
$568$	0	0
$569$	−17381.1	−1.28059	−0.640294	−	0.768130i	$-0.721188\pi$
$569$	−17381.1	−1.28059	−0.640294	+	0.768130i	$0.721188\pi$
$570$	0	0
$571$	−5199.76	−0.381091	−0.190546	−	0.981678i	$-0.561026\pi$
$571$	−5199.76	−0.381091	−0.190546	+	0.981678i	$0.561026\pi$
$572$	0	0
$573$	−14491.4	−1.05652
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−431.725	−0.0311490	−0.0155745	−	0.999879i	$-0.504958\pi$
$577$	−431.725	−0.0311490	−0.0155745	+	0.999879i	$0.504958\pi$
$578$	0	0
$579$	2577.80	0.185026
$580$	0	0
$581$	−16499.6	−1.17817
$582$	0	0
$583$	14666.4	1.04189
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25072.0	−1.76292	−0.881459	−	0.472261i	$-0.843438\pi$
$587$	−25072.0	−1.76292	−0.881459	+	0.472261i	$0.843438\pi$
$588$	0	0
$589$	−1566.98	−0.109620
$590$	0	0
$591$	10436.5	0.726399
$592$	0	0
$593$	−20774.7	−1.43864	−0.719322	−	0.694676i	$-0.755548\pi$
$593$	−20774.7	−1.43864	−0.719322	+	0.694676i	$0.755548\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10758.9	0.737572
$598$	0	0
$599$	12128.5	0.827308	0.413654	−	0.910434i	$-0.364252\pi$
$599$	12128.5	0.827308	0.413654	+	0.910434i	$0.364252\pi$
$600$	0	0
$601$	−5384.63	−0.365463	−0.182732	−	0.983163i	$-0.558494\pi$
$601$	−5384.63	−0.365463	−0.182732	+	0.983163i	$0.558494\pi$
$602$	0	0
$603$	6379.02	0.430802
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2112.00	−0.141225	−0.0706125	−	0.997504i	$-0.522495\pi$
$607$	−2112.00	−0.141225	−0.0706125	+	0.997504i	$0.522495\pi$
$608$	0	0
$609$	7170.07	0.477087
$610$	0	0
$611$	−670.383	−0.0443875
$612$	0	0
$613$	24806.7	1.63448	0.817239	−	0.576299i	$-0.195504\pi$
$613$	24806.7	1.63448	0.817239	+	0.576299i	$0.195504\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9265.71	−0.604576	−0.302288	−	0.953217i	$-0.597750\pi$
$617$	−9265.71	−0.604576	−0.302288	+	0.953217i	$0.597750\pi$
$618$	0	0
$619$	18334.2	1.19049	0.595246	−	0.803544i	$-0.297055\pi$
$619$	18334.2	1.19049	0.595246	+	0.803544i	$0.297055\pi$
$620$	0	0
$621$	−4911.47	−0.317376
$622$	0	0
$623$	6224.80	0.400307
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3160.56	0.201309
$628$	0	0
$629$	−23165.2	−1.46846
$630$	0	0
$631$	−17943.0	−1.13201	−0.566005	−	0.824402i	$-0.691512\pi$
$631$	−17943.0	−1.13201	−0.566005	+	0.824402i	$0.691512\pi$
$632$	0	0
$633$	−10713.3	−0.672694
$634$	0	0
$635$	0	0
$636$	0	0
$637$	364.616	0.0226792
$638$	0	0
$639$	−6750.71	−0.417925
$640$	0	0
$641$	20712.2	1.27626	0.638130	−	0.769928i	$-0.279708\pi$
$641$	20712.2	1.27626	0.638130	+	0.769928i	$0.279708\pi$
$642$	0	0
$643$	14215.3	0.871845	0.435923	−	0.899984i	$-0.356422\pi$
$643$	14215.3	0.871845	0.435923	+	0.899984i	$0.356422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6049.75	0.367605	0.183802	−	0.982963i	$-0.441159\pi$
$647$	6049.75	0.367605	0.183802	+	0.982963i	$0.441159\pi$
$648$	0	0
$649$	23131.1	1.39904
$650$	0	0
$651$	2163.20	0.130234
$652$	0	0
$653$	−8493.91	−0.509023	−0.254512	−	0.967070i	$-0.581915\pi$
$653$	−8493.91	−0.509023	−0.254512	+	0.967070i	$0.581915\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−514.319	−0.0305411
$658$	0	0
$659$	−22783.1	−1.34674	−0.673372	−	0.739304i	$-0.735155\pi$
$659$	−22783.1	−1.34674	−0.673372	+	0.739304i	$0.735155\pi$
$660$	0	0
$661$	−28896.4	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$661$	−28896.4	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$662$	0	0
$663$	−335.018	−0.0196245
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−37275.4	−2.16388
$668$	0	0
$669$	9849.88	0.569235
$670$	0	0
$671$	−3678.71	−0.211647
$672$	0	0
$673$	17060.8	0.977184	0.488592	−	0.872512i	$-0.337511\pi$
$673$	17060.8	0.977184	0.488592	+	0.872512i	$0.337511\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14151.4	−0.803370	−0.401685	−	0.915778i	$-0.631575\pi$
$677$	−14151.4	−0.803370	−0.401685	+	0.915778i	$0.631575\pi$
$678$	0	0
$679$	3504.41	0.198066
$680$	0	0
$681$	−11820.1	−0.665122
$682$	0	0
$683$	14040.6	0.786600	0.393300	−	0.919410i	$-0.371333\pi$
$683$	14040.6	0.786600	0.393300	+	0.919410i	$0.371333\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1899.77	0.105503
$688$	0	0
$689$	621.634	0.0343721
$690$	0	0
$691$	31165.8	1.71578	0.857890	−	0.513833i	$-0.171775\pi$
$691$	31165.8	1.71578	0.857890	+	0.513833i	$0.171775\pi$
$692$	0	0
$693$	−4363.13	−0.239165
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8280.64	0.450002
$698$	0	0
$699$	−5775.88	−0.312538
$700$	0	0
$701$	12236.2	0.659277	0.329639	−	0.944107i	$-0.393073\pi$
$701$	12236.2	0.659277	0.329639	+	0.944107i	$0.393073\pi$
$702$	0	0
$703$	9262.84	0.496948
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20100.4	−1.06924
$708$	0	0
$709$	−14852.5	−0.786736	−0.393368	−	0.919381i	$-0.628690\pi$
$709$	−14852.5	−0.786736	−0.393368	+	0.919381i	$0.628690\pi$
$710$	0	0
$711$	4900.04	0.258461
$712$	0	0
$713$	−11245.9	−0.590693
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11400.3	−0.593794
$718$	0	0
$719$	−4359.05	−0.226099	−0.113049	−	0.993589i	$-0.536062\pi$
$719$	−4359.05	−0.226099	−0.113049	+	0.993589i	$0.536062\pi$
$720$	0	0
$721$	−7155.08	−0.369583
$722$	0	0
$723$	9795.94	0.503893
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27492.5	1.40253	0.701267	−	0.712899i	$-0.252618\pi$
$727$	27492.5	1.40253	0.701267	+	0.712899i	$0.252618\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−8573.37	−0.433786
$732$	0	0
$733$	−7621.45	−0.384045	−0.192022	−	0.981391i	$-0.561505\pi$
$733$	−7621.45	−0.384045	−0.192022	+	0.981391i	$0.561505\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29460.5	−1.47245
$738$	0	0
$739$	−9833.06	−0.489466	−0.244733	−	0.969591i	$-0.578700\pi$
$739$	−9833.06	−0.489466	−0.244733	+	0.969591i	$0.578700\pi$
$740$	0	0
$741$	133.960	0.00664123
$742$	0	0
$743$	14571.3	0.719474	0.359737	−	0.933054i	$-0.382866\pi$
$743$	14571.3	0.719474	0.359737	+	0.933054i	$0.382866\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12731.8	−0.623604
$748$	0	0
$749$	−6345.36	−0.309552
$750$	0	0
$751$	11234.6	0.545882	0.272941	−	0.962031i	$-0.412004\pi$
$751$	11234.6	0.545882	0.272941	+	0.962031i	$0.412004\pi$
$752$	0	0
$753$	−20756.0	−1.00450
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6495.40	0.311862	0.155931	−	0.987768i	$-0.450162\pi$
$757$	6495.40	0.311862	0.155931	+	0.987768i	$0.450162\pi$
$758$	0	0
$759$	22682.9	1.08476
$760$	0	0
$761$	10465.3	0.498510	0.249255	−	0.968438i	$-0.419814\pi$
$761$	10465.3	0.498510	0.249255	+	0.968438i	$0.419814\pi$
$762$	0	0
$763$	−3401.50	−0.161392
$764$	0	0
$765$	0	0
$766$	0	0
$767$	980.409	0.0461545
$768$	0	0
$769$	−22255.7	−1.04364	−0.521822	−	0.853054i	$-0.674748\pi$
$769$	−22255.7	−1.04364	−0.521822	+	0.853054i	$0.674748\pi$
$770$	0	0
$771$	3956.10	0.184793
$772$	0	0
$773$	−16941.8	−0.788299	−0.394150	−	0.919046i	$-0.628961\pi$
$773$	−16941.8	−0.788299	−0.394150	+	0.919046i	$0.628961\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12787.3	−0.590400
$778$	0	0
$779$	−3311.09	−0.152288
$780$	0	0
$781$	31177.1	1.42843
$782$	0	0
$783$	5532.73	0.252520
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−32648.1	−1.47875	−0.739377	−	0.673291i	$-0.764880\pi$
$787$	−32648.1	−1.47875	−0.739377	+	0.673291i	$0.764880\pi$
$788$	0	0
$789$	21394.3	0.965344
$790$	0	0
$791$	−1256.59	−0.0564842
$792$	0	0
$793$	−155.922	−0.00698228
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36971.6	−1.64316	−0.821581	−	0.570091i	$-0.806908\pi$
$797$	−36971.6	−1.64316	−0.821581	+	0.570091i	$0.806908\pi$
$798$	0	0
$799$	24120.6	1.06799
$800$	0	0
$801$	4803.32	0.211881
$802$	0	0
$803$	2375.30	0.104387
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5772.22	0.251787
$808$	0	0
$809$	12381.0	0.538062	0.269031	−	0.963132i	$-0.413297\pi$
$809$	12381.0	0.538062	0.269031	+	0.963132i	$0.413297\pi$
$810$	0	0
$811$	23483.4	1.01679	0.508393	−	0.861125i	$-0.330240\pi$
$811$	23483.4	1.01679	0.508393	+	0.861125i	$0.330240\pi$
$812$	0	0
$813$	4153.81	0.179189
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3428.14	0.146800
$818$	0	0
$819$	−184.931	−0.00789012
$820$	0	0
$821$	24594.8	1.04551	0.522756	−	0.852482i	$-0.324904\pi$
$821$	24594.8	1.04551	0.522756	+	0.852482i	$0.324904\pi$
$822$	0	0
$823$	6881.39	0.291458	0.145729	−	0.989325i	$-0.453447\pi$
$823$	6881.39	0.291458	0.145729	+	0.989325i	$0.453447\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14527.6	0.610850	0.305425	−	0.952216i	$-0.401201\pi$
$827$	14527.6	0.610850	0.305425	+	0.952216i	$0.401201\pi$
$828$	0	0
$829$	23060.0	0.966112	0.483056	−	0.875590i	$-0.339527\pi$
$829$	23060.0	0.966112	0.483056	+	0.875590i	$0.339527\pi$
$830$	0	0
$831$	−15595.5	−0.651026
$832$	0	0
$833$	−13119.0	−0.545676
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1669.22	0.0689325
$838$	0	0
$839$	43802.7	1.80243	0.901215	−	0.433373i	$-0.142677\pi$
$839$	43802.7	1.80243	0.901215	+	0.433373i	$0.142677\pi$
$840$	0	0
$841$	17601.5	0.721698
$842$	0	0
$843$	−27879.7	−1.13906
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4626.42	0.187681
$848$	0	0
$849$	−15383.8	−0.621874
$850$	0	0
$851$	66477.9	2.67783
$852$	0	0
$853$	31828.7	1.27760	0.638801	−	0.769372i	$-0.279431\pi$
$853$	31828.7	1.27760	0.638801	+	0.769372i	$0.279431\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25320.5	1.00925	0.504627	−	0.863338i	$-0.331630\pi$
$857$	25320.5	1.00925	0.504627	+	0.863338i	$0.331630\pi$
$858$	0	0
$859$	21700.3	0.861937	0.430968	−	0.902367i	$-0.358172\pi$
$859$	21700.3	0.861937	0.430968	+	0.902367i	$0.358172\pi$
$860$	0	0
$861$	4570.93	0.180926
$862$	0	0
$863$	−23951.2	−0.944736	−0.472368	−	0.881401i	$-0.656601\pi$
$863$	−23951.2	−0.944736	−0.472368	+	0.881401i	$0.656601\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2684.91	−0.105172
$868$	0	0
$869$	−22630.1	−0.883399
$870$	0	0
$871$	−1248.68	−0.0485763
$872$	0	0
$873$	2704.15	0.104836
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23481.3	0.904115	0.452057	−	0.891989i	$-0.350690\pi$
$877$	23481.3	0.904115	0.452057	+	0.891989i	$0.350690\pi$
$878$	0	0
$879$	−21440.8	−0.822731
$880$	0	0
$881$	7375.38	0.282046	0.141023	−	0.990006i	$-0.454961\pi$
$881$	7375.38	0.282046	0.141023	+	0.990006i	$0.454961\pi$
$882$	0	0
$883$	−31736.5	−1.20954	−0.604768	−	0.796402i	$-0.706734\pi$
$883$	−31736.5	−1.20954	−0.604768	+	0.796402i	$0.706734\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27836.3	1.05372	0.526862	−	0.849951i	$-0.323369\pi$
$887$	27836.3	1.05372	0.526862	+	0.849951i	$0.323369\pi$
$888$	0	0
$889$	2550.78	0.0962322
$890$	0	0
$891$	−3366.78	−0.126589
$892$	0	0
$893$	−9644.87	−0.361426
$894$	0	0
$895$	0	0
$896$	0	0
$897$	961.411	0.0357866
$898$	0	0
$899$	12668.5	0.469986
$900$	0	0
$901$	−22366.6	−0.827015
$902$	0	0
$903$	−4732.52	−0.174406
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5961.53	0.218246	0.109123	−	0.994028i	$-0.465196\pi$
$907$	5961.53	0.218246	0.109123	+	0.994028i	$0.465196\pi$
$908$	0	0
$909$	−15510.3	−0.565947
$910$	0	0
$911$	12068.3	0.438903	0.219452	−	0.975623i	$-0.429573\pi$
$911$	12068.3	0.438903	0.219452	+	0.975623i	$0.429573\pi$
$912$	0	0
$913$	58799.9	2.13143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15298.4	0.550923
$918$	0	0
$919$	13799.2	0.495316	0.247658	−	0.968848i	$-0.420339\pi$
$919$	13799.2	0.495316	0.247658	+	0.968848i	$0.420339\pi$
$920$	0	0
$921$	−19286.3	−0.690015
$922$	0	0
$923$	1321.44	0.0471243
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5521.16	−0.195619
$928$	0	0
$929$	37148.8	1.31196	0.655980	−	0.754778i	$-0.272256\pi$
$929$	37148.8	1.31196	0.655980	+	0.754778i	$0.272256\pi$
$930$	0	0
$931$	5245.77	0.184665
$932$	0	0
$933$	−21819.9	−0.765651
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21470.1	0.748556	0.374278	−	0.927316i	$-0.377891\pi$
$937$	21470.1	0.748556	0.374278	+	0.927316i	$0.377891\pi$
$938$	0	0
$939$	−20271.4	−0.704506
$940$	0	0
$941$	−38843.0	−1.34564	−0.672820	−	0.739806i	$-0.734917\pi$
$941$	−38843.0	−1.34564	−0.672820	+	0.739806i	$0.734917\pi$
$942$	0	0
$943$	−23763.2	−0.820611
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9323.86	0.319942	0.159971	−	0.987122i	$-0.448860\pi$
$947$	9323.86	0.319942	0.159971	+	0.987122i	$0.448860\pi$
$948$	0	0
$949$	100.677	0.00344375
$950$	0	0
$951$	−8321.98	−0.283763
$952$	0	0
$953$	36337.8	1.23515	0.617574	−	0.786512i	$-0.288115\pi$
$953$	36337.8	1.23515	0.617574	+	0.786512i	$0.288115\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−25552.1	−0.863094
$958$	0	0
$959$	−22396.5	−0.754141
$960$	0	0
$961$	−25968.9	−0.871704
$962$	0	0
$963$	−4896.35	−0.163845
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35267.8	1.17284	0.586420	−	0.810007i	$-0.300537\pi$
$967$	35267.8	1.17284	0.586420	+	0.810007i	$0.300537\pi$
$968$	0	0
$969$	−4819.94	−0.159792
$970$	0	0
$971$	50634.5	1.67347	0.836735	−	0.547608i	$-0.184461\pi$
$971$	50634.5	1.67347	0.836735	+	0.547608i	$0.184461\pi$
$972$	0	0
$973$	−13432.7	−0.442582
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36423.9	1.19274	0.596368	−	0.802711i	$-0.296610\pi$
$977$	36423.9	1.19274	0.596368	+	0.802711i	$0.296610\pi$
$978$	0	0
$979$	−22183.4	−0.724193
$980$	0	0
$981$	−2624.74	−0.0854245
$982$	0	0
$983$	24006.8	0.778941	0.389470	−	0.921039i	$-0.372658\pi$
$983$	24006.8	0.778941	0.389470	+	0.921039i	$0.372658\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13314.7	0.429392
$988$	0	0
$989$	24603.2	0.791039
$990$	0	0
$991$	35440.5	1.13603	0.568014	−	0.823019i	$-0.307712\pi$
$991$	35440.5	1.13603	0.568014	+	0.823019i	$0.307712\pi$
$992$	0	0
$993$	−33670.3	−1.07603
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12930.8	0.410754	0.205377	−	0.978683i	$-0.434158\pi$
$997$	12930.8	0.410754	0.205377	+	0.978683i	$0.434158\pi$
$998$	0	0
$999$	−9867.20	−0.312497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bw.1.2		3
4.3	odd	2		2400.4.a.bf.1.2		3
5.2	odd	4		480.4.f.d.289.2	✓	6
5.3	odd	4		480.4.f.d.289.5	yes	6
5.4	even	2		2400.4.a.be.1.2		3
15.2	even	4		1440.4.f.j.289.4		6
15.8	even	4		1440.4.f.j.289.3		6
20.3	even	4		480.4.f.e.289.2	yes	6
20.7	even	4		480.4.f.e.289.5	yes	6
20.19	odd	2		2400.4.a.bx.1.2		3
40.3	even	4		960.4.f.r.769.5		6
40.13	odd	4		960.4.f.s.769.2		6
40.27	even	4		960.4.f.r.769.2		6
40.37	odd	4		960.4.f.s.769.5		6
60.23	odd	4		1440.4.f.i.289.3		6
60.47	odd	4		1440.4.f.i.289.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.d.289.2	✓	6	5.2	odd	4
480.4.f.d.289.5	yes	6	5.3	odd	4
480.4.f.e.289.2	yes	6	20.3	even	4
480.4.f.e.289.5	yes	6	20.7	even	4
960.4.f.r.769.2		6	40.27	even	4
960.4.f.r.769.5		6	40.3	even	4
960.4.f.s.769.2		6	40.13	odd	4
960.4.f.s.769.5		6	40.37	odd	4
1440.4.f.i.289.3		6	60.23	odd	4
1440.4.f.i.289.4		6	60.47	odd	4
1440.4.f.j.289.3		6	15.8	even	4
1440.4.f.j.289.4		6	15.2	even	4
2400.4.a.be.1.2		3	5.4	even	2
2400.4.a.bf.1.2		3	4.3	odd	2
2400.4.a.bw.1.2		3	1.1	even	1	trivial
2400.4.a.bx.1.2		3	20.19	odd	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +11.6634 q^{7} +9.00000 q^{9} -41.5651 q^{11} -1.76173 q^{13} +63.3879 q^{17} -25.3463 q^{19} +34.9903 q^{21} -181.906 q^{23} +27.0000 q^{27} +204.916 q^{29} +61.8228 q^{31} -124.695 q^{33} -365.452 q^{37} -5.28520 q^{39} +130.634 q^{41} -135.252 q^{43} +380.524 q^{47} -206.964 q^{49} +190.164 q^{51} -352.853 q^{53} -76.0388 q^{57} -556.502 q^{59} +88.5048 q^{61} +104.971 q^{63} +708.780 q^{67} -545.718 q^{69} -750.078 q^{71} -57.1465 q^{73} -484.792 q^{77} +544.449 q^{79} +81.0000 q^{81} -1414.64 q^{83} +614.747 q^{87} +533.702 q^{89} -20.5479 q^{91} +185.468 q^{93} +300.461 q^{97} -374.086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} + 32 q^{7} + 27 q^{9} - 48 q^{11} - 76 q^{13} - 16 q^{17} - 88 q^{19} + 96 q^{21} + 20 q^{23} + 81 q^{27} + 58 q^{29} + 56 q^{31} - 144 q^{33} - 436 q^{37} - 228 q^{39} + 362 q^{41} + 148 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 + 32 * q^7 + 27 * q^9 - 48 * q^11 - 76 * q^13 - 16 * q^17 - 88 * q^19 + 96 * q^21 + 20 * q^23 + 81 * q^27 + 58 * q^29 + 56 * q^31 - 144 * q^33 - 436 * q^37 - 228 * q^39 + 362 * q^41 + 148 * q^43 + 52 * q^47 - 109 * q^49 - 48 * q^51 - 940 * q^53 - 264 * q^57 - 816 * q^59 - 842 * q^61 + 288 * q^63 - 780 * q^67 + 60 * q^69 - 304 * q^71 - 512 * q^73 - 1536 * q^77 + 440 * q^79 + 243 * q^81 - 1212 * q^83 + 174 * q^87 - 590 * q^89 - 944 * q^91 + 168 * q^93 - 1752 * q^97 - 432 * q^99

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