LMFDB - Embedded newform 2352.4.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.7361$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -15.7361 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -15.7361 q^{5} +9.00000 q^{9} -58.6805 q^{11} +20.7361 q^{13} +47.2083 q^{15} -42.9444 q^{17} +137.153 q^{19} -29.0556 q^{23} +122.625 q^{25} -27.0000 q^{27} +8.68051 q^{29} -202.472 q^{31} +176.042 q^{33} -15.2639 q^{37} -62.2083 q^{39} -117.250 q^{41} +101.875 q^{43} -141.625 q^{45} +588.500 q^{47} +128.833 q^{51} +404.681 q^{53} +923.403 q^{55} -411.458 q^{57} +10.8748 q^{59} +894.611 q^{61} -326.305 q^{65} +703.735 q^{67} +87.1668 q^{69} -1161.94 q^{71} +1116.37 q^{73} -367.875 q^{75} -138.333 q^{79} +81.0000 q^{81} -894.319 q^{83} +675.778 q^{85} -26.0415 q^{87} +681.806 q^{89} +607.417 q^{93} -2158.25 q^{95} -246.514 q^{97} -528.125 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 9 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9} - 5 q^{11} + 19 q^{13} + 27 q^{15} + 4 q^{17} + 117 q^{19} - 148 q^{23} + 43 q^{25} - 54 q^{27} - 95 q^{29} - 360 q^{31} + 15 q^{33} - 53 q^{37} - 57 q^{39} + 170 q^{41} - 403 q^{43} - 81 q^{45} + 368 q^{47} - 12 q^{51} + 697 q^{53} + 1285 q^{55} - 351 q^{57} - 585 q^{59} + 1160 q^{61} - 338 q^{65} - 233 q^{67} + 444 q^{69} - 616 q^{71} + 817 q^{73} - 129 q^{75} + 802 q^{79} + 162 q^{81} - 283 q^{83} + 992 q^{85} + 285 q^{87} + 1858 q^{89} + 1080 q^{93} - 2294 q^{95} - 1729 q^{97} - 45 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 9 * q^5 + 18 * q^9 - 5 * q^11 + 19 * q^13 + 27 * q^15 + 4 * q^17 + 117 * q^19 - 148 * q^23 + 43 * q^25 - 54 * q^27 - 95 * q^29 - 360 * q^31 + 15 * q^33 - 53 * q^37 - 57 * q^39 + 170 * q^41 - 403 * q^43 - 81 * q^45 + 368 * q^47 - 12 * q^51 + 697 * q^53 + 1285 * q^55 - 351 * q^57 - 585 * q^59 + 1160 * q^61 - 338 * q^65 - 233 * q^67 + 444 * q^69 - 616 * q^71 + 817 * q^73 - 129 * q^75 + 802 * q^79 + 162 * q^81 - 283 * q^83 + 992 * q^85 + 285 * q^87 + 1858 * q^89 + 1080 * q^93 - 2294 * q^95 - 1729 * q^97 - 45 * 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
$5$	−15.7361	−1.40748	−0.703740	−	0.710458i	$-0.748488\pi$
$5$	−15.7361	−1.40748	−0.703740	+	0.710458i	$0.748488\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−58.6805	−1.60844	−0.804220	−	0.594332i	$-0.797417\pi$
$11$	−58.6805	−1.60844	−0.804220	+	0.594332i	$0.797417\pi$
$12$	0	0
$13$	20.7361	0.442397	0.221198	−	0.975229i	$-0.429003\pi$
$13$	20.7361	0.442397	0.221198	+	0.975229i	$0.429003\pi$
$14$	0	0
$15$	47.2083	0.812609
$16$	0	0
$17$	−42.9444	−0.612679	−0.306340	−	0.951922i	$-0.599104\pi$
$17$	−42.9444	−0.612679	−0.306340	+	0.951922i	$0.599104\pi$
$18$	0	0
$19$	137.153	1.65605	0.828026	−	0.560690i	$-0.189464\pi$
$19$	137.153	1.65605	0.828026	+	0.560690i	$0.189464\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−29.0556	−0.263413	−0.131707	−	0.991289i	$-0.542046\pi$
$23$	−29.0556	−0.263413	−0.131707	+	0.991289i	$0.542046\pi$
$24$	0	0
$25$	122.625	0.980999
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	8.68051	0.0555838	0.0277919	−	0.999614i	$-0.491152\pi$
$29$	8.68051	0.0555838	0.0277919	+	0.999614i	$0.491152\pi$
$30$	0	0
$31$	−202.472	−1.17307	−0.586534	−	0.809925i	$-0.699508\pi$
$31$	−202.472	−1.17307	−0.586534	+	0.809925i	$0.699508\pi$
$32$	0	0
$33$	176.042	0.928633
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−15.2639	−0.0678208	−0.0339104	−	0.999425i	$-0.510796\pi$
$37$	−15.2639	−0.0678208	−0.0339104	+	0.999425i	$0.510796\pi$
$38$	0	0
$39$	−62.2083	−0.255418
$40$	0	0
$41$	−117.250	−0.446618	−0.223309	−	0.974748i	$-0.571686\pi$
$41$	−117.250	−0.446618	−0.223309	+	0.974748i	$0.571686\pi$
$42$	0	0
$43$	101.875	0.361297	0.180648	−	0.983548i	$-0.442180\pi$
$43$	101.875	0.361297	0.180648	+	0.983548i	$0.442180\pi$
$44$	0	0
$45$	−141.625	−0.469160
$46$	0	0
$47$	588.500	1.82641	0.913207	−	0.407495i	$-0.133598\pi$
$47$	588.500	1.82641	0.913207	+	0.407495i	$0.133598\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	128.833	0.353731
$52$	0	0
$53$	404.681	1.04881	0.524407	−	0.851468i	$-0.324287\pi$
$53$	404.681	1.04881	0.524407	+	0.851468i	$0.324287\pi$
$54$	0	0
$55$	923.403	2.26385
$56$	0	0
$57$	−411.458	−0.956122
$58$	0	0
$59$	10.8748	0.0239962	0.0119981	−	0.999928i	$-0.496181\pi$
$59$	10.8748	0.0239962	0.0119981	+	0.999928i	$0.496181\pi$
$60$	0	0
$61$	894.611	1.87776	0.938879	−	0.344249i	$-0.111866\pi$
$61$	894.611	1.87776	0.938879	+	0.344249i	$0.111866\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−326.305	−0.622665
$66$	0	0
$67$	703.735	1.28321	0.641604	−	0.767036i	$-0.278269\pi$
$67$	703.735	1.28321	0.641604	+	0.767036i	$0.278269\pi$
$68$	0	0
$69$	87.1668	0.152082
$70$	0	0
$71$	−1161.94	−1.94222	−0.971108	−	0.238640i	$-0.923298\pi$
$71$	−1161.94	−1.94222	−0.971108	+	0.238640i	$0.923298\pi$
$72$	0	0
$73$	1116.37	1.78989	0.894943	−	0.446180i	$-0.147216\pi$
$73$	1116.37	1.78989	0.894943	+	0.446180i	$0.147216\pi$
$74$	0	0
$75$	−367.875	−0.566380
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−138.333	−0.197008	−0.0985042	−	0.995137i	$-0.531406\pi$
$79$	−138.333	−0.197008	−0.0985042	+	0.995137i	$0.531406\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−894.319	−1.18270	−0.591351	−	0.806414i	$-0.701405\pi$
$83$	−894.319	−1.18270	−0.591351	+	0.806414i	$0.701405\pi$
$84$	0	0
$85$	675.778	0.862334
$86$	0	0
$87$	−26.0415	−0.0320913
$88$	0	0
$89$	681.806	0.812037	0.406018	−	0.913865i	$-0.366917\pi$
$89$	681.806	0.812037	0.406018	+	0.913865i	$0.366917\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	607.417	0.677271
$94$	0	0
$95$	−2158.25	−2.33086
$96$	0	0
$97$	−246.514	−0.258039	−0.129019	−	0.991642i	$-0.541183\pi$
$97$	−246.514	−0.258039	−0.129019	+	0.991642i	$0.541183\pi$
$98$	0	0
$99$	−528.125	−0.536147
$100$	0	0
$101$	1089.94	1.07380	0.536898	−	0.843647i	$-0.319596\pi$
$101$	1089.94	1.07380	0.536898	+	0.843647i	$0.319596\pi$
$102$	0	0
$103$	−30.2364	−0.0289251	−0.0144625	−	0.999895i	$-0.504604\pi$
$103$	−30.2364	−0.0289251	−0.0144625	+	0.999895i	$0.504604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1887.62	−1.70545	−0.852727	−	0.522357i	$-0.825053\pi$
$107$	−1887.62	−1.70545	−0.852727	+	0.522357i	$0.825053\pi$
$108$	0	0
$109$	−964.597	−0.847630	−0.423815	−	0.905749i	$-0.639309\pi$
$109$	−964.597	−0.847630	−0.423815	+	0.905749i	$0.639309\pi$
$110$	0	0
$111$	45.7917	0.0391564
$112$	0	0
$113$	119.806	0.0997378	0.0498689	−	0.998756i	$-0.484120\pi$
$113$	119.806	0.0997378	0.0498689	+	0.998756i	$0.484120\pi$
$114$	0	0
$115$	457.222	0.370749
$116$	0	0
$117$	186.625	0.147466
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2112.40	1.58708
$122$	0	0
$123$	351.750	0.257855
$124$	0	0
$125$	37.3745	0.0267430
$126$	0	0
$127$	−683.722	−0.477721	−0.238860	−	0.971054i	$-0.576774\pi$
$127$	−683.722	−0.477721	−0.238860	+	0.971054i	$0.576774\pi$
$128$	0	0
$129$	−305.624	−0.208595
$130$	0	0
$131$	260.237	0.173565	0.0867825	−	0.996227i	$-0.472341\pi$
$131$	260.237	0.173565	0.0867825	+	0.996227i	$0.472341\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	424.875	0.270870
$136$	0	0
$137$	−731.695	−0.456298	−0.228149	−	0.973626i	$-0.573267\pi$
$137$	−731.695	−0.456298	−0.228149	+	0.973626i	$0.573267\pi$
$138$	0	0
$139$	2287.74	1.39599	0.697997	−	0.716101i	$-0.254075\pi$
$139$	2287.74	1.39599	0.697997	+	0.716101i	$0.254075\pi$
$140$	0	0
$141$	−1765.50	−1.05448
$142$	0	0
$143$	−1216.81	−0.711569
$144$	0	0
$145$	−136.597	−0.0782331
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3439.19	1.89094	0.945469	−	0.325712i	$-0.105604\pi$
$149$	3439.19	1.89094	0.945469	+	0.325712i	$0.105604\pi$
$150$	0	0
$151$	−2851.76	−1.53691	−0.768455	−	0.639904i	$-0.778974\pi$
$151$	−2851.76	−1.53691	−0.768455	+	0.639904i	$0.778974\pi$
$152$	0	0
$153$	−386.500	−0.204226
$154$	0	0
$155$	3186.12	1.65107
$156$	0	0
$157$	−2440.25	−1.24046	−0.620232	−	0.784418i	$-0.712962\pi$
$157$	−2440.25	−1.24046	−0.620232	+	0.784418i	$0.712962\pi$
$158$	0	0
$159$	−1214.04	−0.605533
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−241.390	−0.115995	−0.0579974	−	0.998317i	$-0.518472\pi$
$163$	−241.390	−0.115995	−0.0579974	+	0.998317i	$0.518472\pi$
$164$	0	0
$165$	−2770.21	−1.30703
$166$	0	0
$167$	326.445	0.151264	0.0756319	−	0.997136i	$-0.475903\pi$
$167$	326.445	0.151264	0.0756319	+	0.997136i	$0.475903\pi$
$168$	0	0
$169$	−1767.01	−0.804285
$170$	0	0
$171$	1234.37	0.552017
$172$	0	0
$173$	1595.69	0.701262	0.350631	−	0.936514i	$-0.385967\pi$
$173$	1595.69	0.701262	0.350631	+	0.936514i	$0.385967\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−32.6243	−0.0138542
$178$	0	0
$179$	−2882.86	−1.20377	−0.601886	−	0.798582i	$-0.705584\pi$
$179$	−2882.86	−1.20377	−0.601886	+	0.798582i	$0.705584\pi$
$180$	0	0
$181$	−1128.60	−0.463470	−0.231735	−	0.972779i	$-0.574440\pi$
$181$	−1128.60	−0.463470	−0.231735	+	0.972779i	$0.574440\pi$
$182$	0	0
$183$	−2683.83	−1.08412
$184$	0	0
$185$	240.194	0.0954564
$186$	0	0
$187$	2520.00	0.985458
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2205.25	−0.835426	−0.417713	−	0.908579i	$-0.637168\pi$
$191$	−2205.25	−0.835426	−0.417713	+	0.908579i	$0.637168\pi$
$192$	0	0
$193$	3795.36	1.41552	0.707761	−	0.706451i	$-0.249705\pi$
$193$	3795.36	1.41552	0.707761	+	0.706451i	$0.249705\pi$
$194$	0	0
$195$	978.916	0.359496
$196$	0	0
$197$	3701.25	1.33859	0.669297	−	0.742995i	$-0.266595\pi$
$197$	3701.25	1.33859	0.669297	+	0.742995i	$0.266595\pi$
$198$	0	0
$199$	−3600.39	−1.28254	−0.641268	−	0.767317i	$-0.721591\pi$
$199$	−3600.39	−1.28254	−0.641268	+	0.767317i	$0.721591\pi$
$200$	0	0
$201$	−2111.21	−0.740861
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1845.06	0.628606
$206$	0	0
$207$	−261.500	−0.0878045
$208$	0	0
$209$	−8048.19	−2.66366
$210$	0	0
$211$	4137.64	1.34998	0.674992	−	0.737825i	$-0.264147\pi$
$211$	4137.64	1.34998	0.674992	+	0.737825i	$0.264147\pi$
$212$	0	0
$213$	3485.83	1.12134
$214$	0	0
$215$	−1603.11	−0.508518
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3349.12	−1.03339
$220$	0	0
$221$	−890.500	−0.271047
$222$	0	0
$223$	4261.62	1.27973	0.639864	−	0.768488i	$-0.278991\pi$
$223$	4261.62	1.27973	0.639864	+	0.768488i	$0.278991\pi$
$224$	0	0
$225$	1103.62	0.327000
$226$	0	0
$227$	−1534.13	−0.448562	−0.224281	−	0.974525i	$-0.572003\pi$
$227$	−1534.13	−0.448562	−0.224281	+	0.974525i	$0.572003\pi$
$228$	0	0
$229$	574.903	0.165898	0.0829491	−	0.996554i	$-0.473566\pi$
$229$	574.903	0.165898	0.0829491	+	0.996554i	$0.473566\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4152.14	−1.16745	−0.583724	−	0.811952i	$-0.698405\pi$
$233$	−4152.14	−1.16745	−0.583724	+	0.811952i	$0.698405\pi$
$234$	0	0
$235$	−9260.69	−2.57064
$236$	0	0
$237$	414.999	0.113743
$238$	0	0
$239$	−4548.36	−1.23100	−0.615500	−	0.788137i	$-0.711046\pi$
$239$	−4548.36	−1.23100	−0.615500	+	0.788137i	$0.711046\pi$
$240$	0	0
$241$	−7008.79	−1.87334	−0.936672	−	0.350208i	$-0.886111\pi$
$241$	−7008.79	−1.87334	−0.936672	+	0.350208i	$0.886111\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2844.01	0.732632
$248$	0	0
$249$	2682.96	0.682833
$250$	0	0
$251$	682.819	0.171710	0.0858548	−	0.996308i	$-0.472638\pi$
$251$	682.819	0.171710	0.0858548	+	0.996308i	$0.472638\pi$
$252$	0	0
$253$	1705.00	0.423685
$254$	0	0
$255$	−2027.33	−0.497869
$256$	0	0
$257$	−1702.55	−0.413238	−0.206619	−	0.978421i	$-0.566246\pi$
$257$	−1702.55	−0.413238	−0.206619	+	0.978421i	$0.566246\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	78.1246	0.0185279
$262$	0	0
$263$	2924.25	0.685615	0.342808	−	0.939406i	$-0.388622\pi$
$263$	2924.25	0.685615	0.342808	+	0.939406i	$0.388622\pi$
$264$	0	0
$265$	−6368.09	−1.47618
$266$	0	0
$267$	−2045.42	−0.468830
$268$	0	0
$269$	−1373.82	−0.311388	−0.155694	−	0.987805i	$-0.549761\pi$
$269$	−1373.82	−0.311388	−0.155694	+	0.987805i	$0.549761\pi$
$270$	0	0
$271$	−781.511	−0.175179	−0.0875894	−	0.996157i	$-0.527916\pi$
$271$	−781.511	−0.175179	−0.0875894	+	0.996157i	$0.527916\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7195.69	−1.57788
$276$	0	0
$277$	−2151.62	−0.466709	−0.233355	−	0.972392i	$-0.574970\pi$
$277$	−2151.62	−0.466709	−0.233355	+	0.972392i	$0.574970\pi$
$278$	0	0
$279$	−1822.25	−0.391022
$280$	0	0
$281$	−2388.25	−0.507014	−0.253507	−	0.967334i	$-0.581584\pi$
$281$	−2388.25	−0.507014	−0.253507	+	0.967334i	$0.581584\pi$
$282$	0	0
$283$	−3294.65	−0.692038	−0.346019	−	0.938228i	$-0.612467\pi$
$283$	−3294.65	−0.692038	−0.346019	+	0.938228i	$0.612467\pi$
$284$	0	0
$285$	6474.75	1.34572
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3068.78	−0.624624
$290$	0	0
$291$	739.543	0.148979
$292$	0	0
$293$	4355.37	0.868408	0.434204	−	0.900815i	$-0.357030\pi$
$293$	4355.37	0.868408	0.434204	+	0.900815i	$0.357030\pi$
$294$	0	0
$295$	−171.126	−0.0337741
$296$	0	0
$297$	1584.37	0.309544
$298$	0	0
$299$	−602.500	−0.116533
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3269.83	−0.619957
$304$	0	0
$305$	−14077.7	−2.64291
$306$	0	0
$307$	−8189.87	−1.52254	−0.761271	−	0.648434i	$-0.775424\pi$
$307$	−8189.87	−1.52254	−0.761271	+	0.648434i	$0.775424\pi$
$308$	0	0
$309$	90.7092	0.0166999
$310$	0	0
$311$	1165.30	0.212470	0.106235	−	0.994341i	$-0.466120\pi$
$311$	1165.30	0.212470	0.106235	+	0.994341i	$0.466120\pi$
$312$	0	0
$313$	5784.52	1.04460	0.522301	−	0.852761i	$-0.325074\pi$
$313$	5784.52	1.04460	0.522301	+	0.852761i	$0.325074\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4567.37	−0.809240	−0.404620	−	0.914485i	$-0.632596\pi$
$317$	−4567.37	−0.809240	−0.404620	+	0.914485i	$0.632596\pi$
$318$	0	0
$319$	−509.377	−0.0894032
$320$	0	0
$321$	5662.87	0.984644
$322$	0	0
$323$	−5889.94	−1.01463
$324$	0	0
$325$	2542.76	0.433991
$326$	0	0
$327$	2893.79	0.489379
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3540.62	−0.587946	−0.293973	−	0.955814i	$-0.594978\pi$
$331$	−3540.62	−0.587946	−0.293973	+	0.955814i	$0.594978\pi$
$332$	0	0
$333$	−137.375	−0.0226069
$334$	0	0
$335$	−11074.1	−1.80609
$336$	0	0
$337$	−9183.50	−1.48444	−0.742221	−	0.670155i	$-0.766227\pi$
$337$	−9183.50	−1.48444	−0.742221	+	0.670155i	$0.766227\pi$
$338$	0	0
$339$	−359.417	−0.0575837
$340$	0	0
$341$	11881.2	1.88681
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1371.67	−0.214052
$346$	0	0
$347$	5955.00	0.921271	0.460636	−	0.887589i	$-0.347622\pi$
$347$	5955.00	0.921271	0.460636	+	0.887589i	$0.347622\pi$
$348$	0	0
$349$	−2816.11	−0.431929	−0.215964	−	0.976401i	$-0.569289\pi$
$349$	−2816.11	−0.431929	−0.215964	+	0.976401i	$0.569289\pi$
$350$	0	0
$351$	−559.875	−0.0851393
$352$	0	0
$353$	381.306	0.0574926	0.0287463	−	0.999587i	$-0.490849\pi$
$353$	381.306	0.0574926	0.0287463	+	0.999587i	$0.490849\pi$
$354$	0	0
$355$	18284.5	2.73363
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2994.75	0.440270	0.220135	−	0.975469i	$-0.429350\pi$
$359$	2994.75	0.440270	0.220135	+	0.975469i	$0.429350\pi$
$360$	0	0
$361$	11951.9	1.74251
$362$	0	0
$363$	−6337.21	−0.916301
$364$	0	0
$365$	−17567.4	−2.51923
$366$	0	0
$367$	−4650.03	−0.661388	−0.330694	−	0.943738i	$-0.607283\pi$
$367$	−4650.03	−0.661388	−0.330694	+	0.943738i	$0.607283\pi$
$368$	0	0
$369$	−1055.25	−0.148873
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4828.34	0.670247	0.335123	−	0.942174i	$-0.391222\pi$
$373$	4828.34	0.670247	0.335123	+	0.942174i	$0.391222\pi$
$374$	0	0
$375$	−112.123	−0.0154401
$376$	0	0
$377$	180.000	0.0245901
$378$	0	0
$379$	−13230.5	−1.79315	−0.896577	−	0.442887i	$-0.853954\pi$
$379$	−13230.5	−1.79315	−0.896577	+	0.442887i	$0.853954\pi$
$380$	0	0
$381$	2051.17	0.275812
$382$	0	0
$383$	−4409.75	−0.588323	−0.294161	−	0.955756i	$-0.595040\pi$
$383$	−4409.75	−0.588323	−0.294161	+	0.955756i	$0.595040\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	916.873	0.120432
$388$	0	0
$389$	575.383	0.0749950	0.0374975	−	0.999297i	$-0.488061\pi$
$389$	575.383	0.0749950	0.0374975	+	0.999297i	$0.488061\pi$
$390$	0	0
$391$	1247.78	0.161388
$392$	0	0
$393$	−780.711	−0.100208
$394$	0	0
$395$	2176.82	0.277285
$396$	0	0
$397$	−7241.62	−0.915482	−0.457741	−	0.889086i	$-0.651341\pi$
$397$	−7241.62	−0.915482	−0.457741	+	0.889086i	$0.651341\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2247.00	0.279825	0.139912	−	0.990164i	$-0.455318\pi$
$401$	2247.00	0.279825	0.139912	+	0.990164i	$0.455318\pi$
$402$	0	0
$403$	−4198.48	−0.518961
$404$	0	0
$405$	−1274.62	−0.156387
$406$	0	0
$407$	895.693	0.109086
$408$	0	0
$409$	1650.64	0.199557	0.0997784	−	0.995010i	$-0.468187\pi$
$409$	1650.64	0.199557	0.0997784	+	0.995010i	$0.468187\pi$
$410$	0	0
$411$	2195.08	0.263444
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14073.1	1.66463
$416$	0	0
$417$	−6863.21	−0.805978
$418$	0	0
$419$	−3178.64	−0.370613	−0.185307	−	0.982681i	$-0.559328\pi$
$419$	−3178.64	−0.370613	−0.185307	+	0.982681i	$0.559328\pi$
$420$	0	0
$421$	8781.12	1.01655	0.508273	−	0.861196i	$-0.330284\pi$
$421$	8781.12	1.01655	0.508273	+	0.861196i	$0.330284\pi$
$422$	0	0
$423$	5296.50	0.608805
$424$	0	0
$425$	−5266.05	−0.601038
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3650.42	0.410825
$430$	0	0
$431$	−9086.69	−1.01552	−0.507762	−	0.861498i	$-0.669527\pi$
$431$	−9086.69	−1.01552	−0.507762	+	0.861498i	$0.669527\pi$
$432$	0	0
$433$	−3150.88	−0.349703	−0.174852	−	0.984595i	$-0.555945\pi$
$433$	−3150.88	−0.349703	−0.174852	+	0.984595i	$0.555945\pi$
$434$	0	0
$435$	409.792	0.0451679
$436$	0	0
$437$	−3985.05	−0.436226
$438$	0	0
$439$	10331.4	1.12321	0.561605	−	0.827405i	$-0.310184\pi$
$439$	10331.4	1.12321	0.561605	+	0.827405i	$0.310184\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8272.07	0.887173	0.443587	−	0.896231i	$-0.353706\pi$
$443$	8272.07	0.887173	0.443587	+	0.896231i	$0.353706\pi$
$444$	0	0
$445$	−10729.0	−1.14293
$446$	0	0
$447$	−10317.6	−1.09173
$448$	0	0
$449$	10111.1	1.06275	0.531374	−	0.847137i	$-0.321676\pi$
$449$	10111.1	1.06275	0.531374	+	0.847137i	$0.321676\pi$
$450$	0	0
$451$	6880.28	0.718359
$452$	0	0
$453$	8555.29	0.887335
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17840.7	−1.82616	−0.913080	−	0.407781i	$-0.866303\pi$
$457$	−17840.7	−1.82616	−0.913080	+	0.407781i	$0.866303\pi$
$458$	0	0
$459$	1159.50	0.117910
$460$	0	0
$461$	15787.7	1.59503	0.797515	−	0.603299i	$-0.206148\pi$
$461$	15787.7	1.59503	0.797515	+	0.603299i	$0.206148\pi$
$462$	0	0
$463$	5960.87	0.598326	0.299163	−	0.954202i	$-0.403293\pi$
$463$	5960.87	0.598326	0.299163	+	0.954202i	$0.403293\pi$
$464$	0	0
$465$	−9558.37	−0.953245
$466$	0	0
$467$	−11058.3	−1.09575	−0.547877	−	0.836559i	$-0.684564\pi$
$467$	−11058.3	−1.09575	−0.547877	+	0.836559i	$0.684564\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7320.74	0.716183
$472$	0	0
$473$	−5978.06	−0.581124
$474$	0	0
$475$	16818.3	1.62459
$476$	0	0
$477$	3642.12	0.349605
$478$	0	0
$479$	−11863.8	−1.13167	−0.565836	−	0.824518i	$-0.691446\pi$
$479$	−11863.8	−1.13167	−0.565836	+	0.824518i	$0.691446\pi$
$480$	0	0
$481$	−316.514	−0.0300037
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3879.18	0.363184
$486$	0	0
$487$	5924.97	0.551306	0.275653	−	0.961257i	$-0.411106\pi$
$487$	5924.97	0.551306	0.275653	+	0.961257i	$0.411106\pi$
$488$	0	0
$489$	724.171	0.0669696
$490$	0	0
$491$	9203.12	0.845888	0.422944	−	0.906156i	$-0.360997\pi$
$491$	9203.12	0.845888	0.422944	+	0.906156i	$0.360997\pi$
$492$	0	0
$493$	−372.779	−0.0340551
$494$	0	0
$495$	8310.62	0.754616
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9807.60	0.879857	0.439928	−	0.898033i	$-0.355004\pi$
$499$	9807.60	0.879857	0.439928	+	0.898033i	$0.355004\pi$
$500$	0	0
$501$	−979.334	−0.0873322
$502$	0	0
$503$	10462.6	0.927446	0.463723	−	0.885980i	$-0.346513\pi$
$503$	10462.6	0.927446	0.463723	+	0.885980i	$0.346513\pi$
$504$	0	0
$505$	−17151.5	−1.51135
$506$	0	0
$507$	5301.04	0.464354
$508$	0	0
$509$	−5341.83	−0.465171	−0.232586	−	0.972576i	$-0.574719\pi$
$509$	−5341.83	−0.465171	−0.232586	+	0.972576i	$0.574719\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3703.12	−0.318707
$514$	0	0
$515$	475.803	0.0407115
$516$	0	0
$517$	−34533.5	−2.93768
$518$	0	0
$519$	−4787.08	−0.404874
$520$	0	0
$521$	−11553.3	−0.971511	−0.485755	−	0.874095i	$-0.661455\pi$
$521$	−11553.3	−0.971511	−0.485755	+	0.874095i	$0.661455\pi$
$522$	0	0
$523$	5251.51	0.439068	0.219534	−	0.975605i	$-0.429546\pi$
$523$	5251.51	0.439068	0.219534	+	0.975605i	$0.429546\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8695.05	0.718714
$528$	0	0
$529$	−11322.8	−0.930613
$530$	0	0
$531$	97.8729	0.00799872
$532$	0	0
$533$	−2431.30	−0.197583
$534$	0	0
$535$	29703.9	2.40039
$536$	0	0
$537$	8648.58	0.694998
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20617.6	−1.63848	−0.819241	−	0.573449i	$-0.805605\pi$
$541$	−20617.6	−1.63848	−0.819241	+	0.573449i	$0.805605\pi$
$542$	0	0
$543$	3385.79	0.267584
$544$	0	0
$545$	15179.0	1.19302
$546$	0	0
$547$	−10669.4	−0.833985	−0.416993	−	0.908910i	$-0.636916\pi$
$547$	−10669.4	−0.833985	−0.416993	+	0.908910i	$0.636916\pi$
$548$	0	0
$549$	8051.50	0.625919
$550$	0	0
$551$	1190.56	0.0920497
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−720.583	−0.0551118
$556$	0	0
$557$	−5377.87	−0.409098	−0.204549	−	0.978856i	$-0.565573\pi$
$557$	−5377.87	−0.409098	−0.204549	+	0.978856i	$0.565573\pi$
$558$	0	0
$559$	2112.49	0.159837
$560$	0	0
$561$	−7560.00	−0.568954
$562$	0	0
$563$	−3518.87	−0.263415	−0.131708	−	0.991289i	$-0.542046\pi$
$563$	−3518.87	−0.263415	−0.131708	+	0.991289i	$0.542046\pi$
$564$	0	0
$565$	−1885.28	−0.140379
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2550.30	0.187898	0.0939492	−	0.995577i	$-0.470051\pi$
$569$	2550.30	0.187898	0.0939492	+	0.995577i	$0.470051\pi$
$570$	0	0
$571$	−13655.3	−1.00080	−0.500401	−	0.865794i	$-0.666814\pi$
$571$	−13655.3	−1.00080	−0.500401	+	0.865794i	$0.666814\pi$
$572$	0	0
$573$	6615.75	0.482333
$574$	0	0
$575$	−3562.94	−0.258408
$576$	0	0
$577$	16695.9	1.20461	0.602304	−	0.798267i	$-0.294250\pi$
$577$	16695.9	1.20461	0.602304	+	0.798267i	$0.294250\pi$
$578$	0	0
$579$	−11386.1	−0.817253
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−23746.9	−1.68695
$584$	0	0
$585$	−2936.75	−0.207555
$586$	0	0
$587$	−16341.4	−1.14903	−0.574517	−	0.818493i	$-0.694810\pi$
$587$	−16341.4	−1.14903	−0.574517	+	0.818493i	$0.694810\pi$
$588$	0	0
$589$	−27769.6	−1.94266
$590$	0	0
$591$	−11103.7	−0.772838
$592$	0	0
$593$	−19925.2	−1.37981	−0.689906	−	0.723899i	$-0.742348\pi$
$593$	−19925.2	−1.37981	−0.689906	+	0.723899i	$0.742348\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10801.2	0.740472
$598$	0	0
$599$	−14781.7	−1.00829	−0.504143	−	0.863620i	$-0.668191\pi$
$599$	−14781.7	−1.00829	−0.504143	+	0.863620i	$0.668191\pi$
$600$	0	0
$601$	20487.7	1.39054	0.695268	−	0.718751i	$-0.255286\pi$
$601$	20487.7	1.39054	0.695268	+	0.718751i	$0.255286\pi$
$602$	0	0
$603$	6333.62	0.427736
$604$	0	0
$605$	−33241.0	−2.23378
$606$	0	0
$607$	7524.04	0.503116	0.251558	−	0.967842i	$-0.419057\pi$
$607$	7524.04	0.503116	0.251558	+	0.967842i	$0.419057\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12203.2	0.808000
$612$	0	0
$613$	3414.64	0.224985	0.112493	−	0.993653i	$-0.464117\pi$
$613$	3414.64	0.224985	0.112493	+	0.993653i	$0.464117\pi$
$614$	0	0
$615$	−5535.17	−0.362926
$616$	0	0
$617$	22219.4	1.44979	0.724895	−	0.688859i	$-0.241888\pi$
$617$	22219.4	1.44979	0.724895	+	0.688859i	$0.241888\pi$
$618$	0	0
$619$	21863.5	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$619$	21863.5	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$620$	0	0
$621$	784.501	0.0506939
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−15916.2	−1.01864
$626$	0	0
$627$	24144.6	1.53787
$628$	0	0
$629$	655.499	0.0415524
$630$	0	0
$631$	16080.8	1.01452	0.507262	−	0.861792i	$-0.330658\pi$
$631$	16080.8	1.01452	0.507262	+	0.861792i	$0.330658\pi$
$632$	0	0
$633$	−12412.9	−0.779414
$634$	0	0
$635$	10759.1	0.672382
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10457.5	−0.647405
$640$	0	0
$641$	−18463.8	−1.13772	−0.568858	−	0.822436i	$-0.692615\pi$
$641$	−18463.8	−1.13772	−0.568858	+	0.822436i	$0.692615\pi$
$642$	0	0
$643$	−3110.87	−0.190794	−0.0953971	−	0.995439i	$-0.530412\pi$
$643$	−3110.87	−0.190794	−0.0953971	+	0.995439i	$0.530412\pi$
$644$	0	0
$645$	4809.34	0.293593
$646$	0	0
$647$	−24729.0	−1.50263	−0.751313	−	0.659946i	$-0.770579\pi$
$647$	−24729.0	−1.50263	−0.751313	+	0.659946i	$0.770579\pi$
$648$	0	0
$649$	−638.137	−0.0385964
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7052.63	0.422651	0.211325	−	0.977416i	$-0.432222\pi$
$653$	7052.63	0.422651	0.211325	+	0.977416i	$0.432222\pi$
$654$	0	0
$655$	−4095.12	−0.244289
$656$	0	0
$657$	10047.4	0.596629
$658$	0	0
$659$	−7895.55	−0.466718	−0.233359	−	0.972391i	$-0.574972\pi$
$659$	−7895.55	−0.466718	−0.233359	+	0.972391i	$0.574972\pi$
$660$	0	0
$661$	13101.1	0.770914	0.385457	−	0.922726i	$-0.374044\pi$
$661$	13101.1	0.770914	0.385457	+	0.922726i	$0.374044\pi$
$662$	0	0
$663$	2671.50	0.156489
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−252.217	−0.0146415
$668$	0	0
$669$	−12784.9	−0.738852
$670$	0	0
$671$	−52496.2	−3.02026
$672$	0	0
$673$	−9128.25	−0.522836	−0.261418	−	0.965226i	$-0.584190\pi$
$673$	−9128.25	−0.522836	−0.261418	+	0.965226i	$0.584190\pi$
$674$	0	0
$675$	−3310.87	−0.188793
$676$	0	0
$677$	−16029.9	−0.910011	−0.455006	−	0.890489i	$-0.650363\pi$
$677$	−16029.9	−0.910011	−0.455006	+	0.890489i	$0.650363\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4602.38	0.258977
$682$	0	0
$683$	22176.4	1.24239	0.621197	−	0.783655i	$-0.286647\pi$
$683$	22176.4	1.24239	0.621197	+	0.783655i	$0.286647\pi$
$684$	0	0
$685$	11514.0	0.642231
$686$	0	0
$687$	−1724.71	−0.0957814
$688$	0	0
$689$	8391.50	0.463992
$690$	0	0
$691$	−1184.14	−0.0651909	−0.0325955	−	0.999469i	$-0.510377\pi$
$691$	−1184.14	−0.0651909	−0.0325955	+	0.999469i	$0.510377\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−36000.0	−1.96483
$696$	0	0
$697$	5035.23	0.273634
$698$	0	0
$699$	12456.4	0.674027
$700$	0	0
$701$	31064.6	1.67374	0.836870	−	0.547401i	$-0.184383\pi$
$701$	31064.6	1.67374	0.836870	+	0.547401i	$0.184383\pi$
$702$	0	0
$703$	−2093.49	−0.112315
$704$	0	0
$705$	27782.1	1.48416
$706$	0	0
$707$	0	0
$708$	0	0
$709$	575.891	0.0305050	0.0152525	−	0.999884i	$-0.495145\pi$
$709$	575.891	0.0305050	0.0152525	+	0.999884i	$0.495145\pi$
$710$	0	0
$711$	−1245.00	−0.0656695
$712$	0	0
$713$	5882.95	0.309002
$714$	0	0
$715$	19147.8	1.00152
$716$	0	0
$717$	13645.1	0.710718
$718$	0	0
$719$	4835.54	0.250814	0.125407	−	0.992105i	$-0.459976\pi$
$719$	4835.54	0.250814	0.125407	+	0.992105i	$0.459976\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21026.4	1.08158
$724$	0	0
$725$	1064.45	0.0545277
$726$	0	0
$727$	−17999.8	−0.918259	−0.459129	−	0.888369i	$-0.651839\pi$
$727$	−17999.8	−0.918259	−0.459129	+	0.888369i	$0.651839\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4374.95	−0.221359
$732$	0	0
$733$	−5113.40	−0.257664	−0.128832	−	0.991666i	$-0.541123\pi$
$733$	−5113.40	−0.257664	−0.128832	+	0.991666i	$0.541123\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−41295.6	−2.06396
$738$	0	0
$739$	−13056.1	−0.649902	−0.324951	−	0.945731i	$-0.605348\pi$
$739$	−13056.1	−0.649902	−0.324951	+	0.945731i	$0.605348\pi$
$740$	0	0
$741$	−8532.04	−0.422986
$742$	0	0
$743$	28402.9	1.40242	0.701212	−	0.712953i	$-0.252643\pi$
$743$	28402.9	1.40242	0.701212	+	0.712953i	$0.252643\pi$
$744$	0	0
$745$	−54119.5	−2.66146
$746$	0	0
$747$	−8048.87	−0.394234
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33270.3	1.61658	0.808288	−	0.588787i	$-0.200394\pi$
$751$	33270.3	1.61658	0.808288	+	0.588787i	$0.200394\pi$
$752$	0	0
$753$	−2048.46	−0.0991366
$754$	0	0
$755$	44875.7	2.16317
$756$	0	0
$757$	18331.2	0.880129	0.440064	−	0.897966i	$-0.354956\pi$
$757$	18331.2	0.880129	0.440064	+	0.897966i	$0.354956\pi$
$758$	0	0
$759$	−5114.99	−0.244614
$760$	0	0
$761$	−4997.01	−0.238031	−0.119015	−	0.992892i	$-0.537974\pi$
$761$	−4997.01	−0.238031	−0.119015	+	0.992892i	$0.537974\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6082.00	0.287445
$766$	0	0
$767$	225.500	0.0106158
$768$	0	0
$769$	9834.47	0.461171	0.230585	−	0.973052i	$-0.425936\pi$
$769$	9834.47	0.461171	0.230585	+	0.973052i	$0.425936\pi$
$770$	0	0
$771$	5107.66	0.238583
$772$	0	0
$773$	13983.0	0.650625	0.325312	−	0.945607i	$-0.394531\pi$
$773$	13983.0	0.650625	0.325312	+	0.945607i	$0.394531\pi$
$774$	0	0
$775$	−24828.1	−1.15078
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16081.1	−0.739623
$780$	0	0
$781$	68183.5	3.12394
$782$	0	0
$783$	−234.374	−0.0106971
$784$	0	0
$785$	38400.0	1.74593
$786$	0	0
$787$	−31278.6	−1.41672	−0.708362	−	0.705849i	$-0.750566\pi$
$787$	−31278.6	−1.41672	−0.708362	+	0.705849i	$0.750566\pi$
$788$	0	0
$789$	−8772.74	−0.395840
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18550.7	0.830714
$794$	0	0
$795$	19104.3	0.852276
$796$	0	0
$797$	−30546.9	−1.35762	−0.678811	−	0.734313i	$-0.737505\pi$
$797$	−30546.9	−1.35762	−0.678811	+	0.734313i	$0.737505\pi$
$798$	0	0
$799$	−25272.8	−1.11901
$800$	0	0
$801$	6136.25	0.270679
$802$	0	0
$803$	−65509.4	−2.87893
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4121.46	0.179780
$808$	0	0
$809$	−29572.7	−1.28519	−0.642596	−	0.766205i	$-0.722142\pi$
$809$	−29572.7	−1.28519	−0.642596	+	0.766205i	$0.722142\pi$
$810$	0	0
$811$	45069.0	1.95140	0.975701	−	0.219107i	$-0.0703144\pi$
$811$	45069.0	1.95140	0.975701	+	0.219107i	$0.0703144\pi$
$812$	0	0
$813$	2344.53	0.101139
$814$	0	0
$815$	3798.54	0.163260
$816$	0	0
$817$	13972.4	0.598326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9400.93	0.399628	0.199814	−	0.979834i	$-0.435966\pi$
$821$	9400.93	0.399628	0.199814	+	0.979834i	$0.435966\pi$
$822$	0	0
$823$	28008.9	1.18630	0.593152	−	0.805091i	$-0.297883\pi$
$823$	28008.9	1.18630	0.593152	+	0.805091i	$0.297883\pi$
$824$	0	0
$825$	21587.1	0.910989
$826$	0	0
$827$	28650.8	1.20470	0.602349	−	0.798232i	$-0.294231\pi$
$827$	28650.8	1.20470	0.602349	+	0.798232i	$0.294231\pi$
$828$	0	0
$829$	4249.64	0.178041	0.0890205	−	0.996030i	$-0.471626\pi$
$829$	4249.64	0.178041	0.0890205	+	0.996030i	$0.471626\pi$
$830$	0	0
$831$	6454.87	0.269455
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5136.97	−0.212901
$836$	0	0
$837$	5466.75	0.225757
$838$	0	0
$839$	9265.50	0.381264	0.190632	−	0.981662i	$-0.438946\pi$
$839$	9265.50	0.381264	0.190632	+	0.981662i	$0.438946\pi$
$840$	0	0
$841$	−24313.6	−0.996910
$842$	0	0
$843$	7164.75	0.292725
$844$	0	0
$845$	27805.9	1.13201
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9883.96	0.399548
$850$	0	0
$851$	443.502	0.0178649
$852$	0	0
$853$	−36321.1	−1.45793	−0.728964	−	0.684552i	$-0.759998\pi$
$853$	−36321.1	−1.45793	−0.728964	+	0.684552i	$0.759998\pi$
$854$	0	0
$855$	−19424.2	−0.776953
$856$	0	0
$857$	23676.2	0.943716	0.471858	−	0.881675i	$-0.343583\pi$
$857$	23676.2	0.943716	0.471858	+	0.881675i	$0.343583\pi$
$858$	0	0
$859$	27412.3	1.08882	0.544410	−	0.838819i	$-0.316753\pi$
$859$	27412.3	1.08882	0.544410	+	0.838819i	$0.316753\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22725.5	0.896390	0.448195	−	0.893936i	$-0.352067\pi$
$863$	22725.5	0.896390	0.448195	+	0.893936i	$0.352067\pi$
$864$	0	0
$865$	−25110.0	−0.987013
$866$	0	0
$867$	9206.33	0.360627
$868$	0	0
$869$	8117.45	0.316876
$870$	0	0
$871$	14592.7	0.567688
$872$	0	0
$873$	−2218.63	−0.0860129
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11981.9	0.461344	0.230672	−	0.973032i	$-0.425908\pi$
$877$	11981.9	0.461344	0.230672	+	0.973032i	$0.425908\pi$
$878$	0	0
$879$	−13066.1	−0.501376
$880$	0	0
$881$	22891.8	0.875419	0.437709	−	0.899117i	$-0.355790\pi$
$881$	22891.8	0.875419	0.437709	+	0.899117i	$0.355790\pi$
$882$	0	0
$883$	−36489.6	−1.39068	−0.695341	−	0.718680i	$-0.744747\pi$
$883$	−36489.6	−1.39068	−0.695341	+	0.718680i	$0.744747\pi$
$884$	0	0
$885$	513.379	0.0194995
$886$	0	0
$887$	24105.8	0.912506	0.456253	−	0.889850i	$-0.349191\pi$
$887$	24105.8	0.912506	0.456253	+	0.889850i	$0.349191\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4753.12	−0.178716
$892$	0	0
$893$	80714.3	3.02464
$894$	0	0
$895$	45365.0	1.69428
$896$	0	0
$897$	1807.50	0.0672805
$898$	0	0
$899$	−1757.56	−0.0652036
$900$	0	0
$901$	−17378.8	−0.642587
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17759.7	0.652324
$906$	0	0
$907$	−22363.3	−0.818699	−0.409350	−	0.912378i	$-0.634244\pi$
$907$	−22363.3	−0.818699	−0.409350	+	0.912378i	$0.634244\pi$
$908$	0	0
$909$	9809.49	0.357932
$910$	0	0
$911$	21426.3	0.779238	0.389619	−	0.920976i	$-0.372607\pi$
$911$	21426.3	0.779238	0.389619	+	0.920976i	$0.372607\pi$
$912$	0	0
$913$	52479.1	1.90230
$914$	0	0
$915$	42233.1	1.52588
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30526.1	1.09572	0.547858	−	0.836571i	$-0.315443\pi$
$919$	30526.1	1.09572	0.547858	+	0.836571i	$0.315443\pi$
$920$	0	0
$921$	24569.6	0.879040
$922$	0	0
$923$	−24094.2	−0.859231
$924$	0	0
$925$	−1871.73	−0.0665322
$926$	0	0
$927$	−272.128	−0.00964169
$928$	0	0
$929$	−4638.90	−0.163829	−0.0819146	−	0.996639i	$-0.526103\pi$
$929$	−4638.90	−0.163829	−0.0819146	+	0.996639i	$0.526103\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3495.90	−0.122670
$934$	0	0
$935$	−39655.0	−1.38701
$936$	0	0
$937$	7713.38	0.268928	0.134464	−	0.990919i	$-0.457069\pi$
$937$	7713.38	0.268928	0.134464	+	0.990919i	$0.457069\pi$
$938$	0	0
$939$	−17353.6	−0.603102
$940$	0	0
$941$	7490.63	0.259498	0.129749	−	0.991547i	$-0.458583\pi$
$941$	7490.63	0.259498	0.129749	+	0.991547i	$0.458583\pi$
$942$	0	0
$943$	3406.76	0.117645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25139.1	−0.862632	−0.431316	−	0.902201i	$-0.641951\pi$
$947$	−25139.1	−0.862632	−0.431316	+	0.902201i	$0.641951\pi$
$948$	0	0
$949$	23149.3	0.791840
$950$	0	0
$951$	13702.1	0.467215
$952$	0	0
$953$	−2505.29	−0.0851568	−0.0425784	−	0.999093i	$-0.513557\pi$
$953$	−2505.29	−0.0851568	−0.0425784	+	0.999093i	$0.513557\pi$
$954$	0	0
$955$	34702.0	1.17584
$956$	0	0
$957$	1528.13	0.0516170
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11204.0	0.376087
$962$	0	0
$963$	−16988.6	−0.568485
$964$	0	0
$965$	−59724.2	−1.99232
$966$	0	0
$967$	9331.61	0.310325	0.155162	−	0.987889i	$-0.450410\pi$
$967$	9331.61	0.310325	0.155162	+	0.987889i	$0.450410\pi$
$968$	0	0
$969$	17669.8	0.585796
$970$	0	0
$971$	−41985.9	−1.38763	−0.693817	−	0.720152i	$-0.744072\pi$
$971$	−41985.9	−1.38763	−0.693817	+	0.720152i	$0.744072\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7628.29	−0.250565
$976$	0	0
$977$	−7950.95	−0.260362	−0.130181	−	0.991490i	$-0.541556\pi$
$977$	−7950.95	−0.260362	−0.130181	+	0.991490i	$0.541556\pi$
$978$	0	0
$979$	−40008.7	−1.30611
$980$	0	0
$981$	−8681.37	−0.282543
$982$	0	0
$983$	10539.5	0.341971	0.170986	−	0.985274i	$-0.445305\pi$
$983$	10539.5	0.341971	0.170986	+	0.985274i	$0.445305\pi$
$984$	0	0
$985$	−58243.2	−1.88404
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2960.03	−0.0951704
$990$	0	0
$991$	4431.86	0.142061	0.0710306	−	0.997474i	$-0.477371\pi$
$991$	4431.86	0.142061	0.0710306	+	0.997474i	$0.477371\pi$
$992$	0	0
$993$	10621.9	0.339451
$994$	0	0
$995$	56656.0	1.80514
$996$	0	0
$997$	39415.6	1.25206	0.626031	−	0.779799i	$-0.284679\pi$
$997$	39415.6	1.25206	0.626031	+	0.779799i	$0.284679\pi$
$998$	0	0
$999$	412.125	0.0130521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bo.1.1		2
4.3	odd	2		1176.4.a.u.1.1		2
7.3	odd	6		336.4.q.g.289.1		4
7.5	odd	6		336.4.q.g.193.1		4
7.6	odd	2		2352.4.a.cc.1.2		2
28.3	even	6		168.4.q.d.121.1	yes	4
28.19	even	6		168.4.q.d.25.1	✓	4
28.27	even	2		1176.4.a.r.1.2		2
84.47	odd	6		504.4.s.f.361.2		4
84.59	odd	6		504.4.s.f.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.d.25.1	✓	4	28.19	even	6
168.4.q.d.121.1	yes	4	28.3	even	6
336.4.q.g.193.1		4	7.5	odd	6
336.4.q.g.289.1		4	7.3	odd	6
504.4.s.f.289.2		4	84.59	odd	6
504.4.s.f.361.2		4	84.47	odd	6
1176.4.a.r.1.2		2	28.27	even	2
1176.4.a.u.1.1		2	4.3	odd	2
2352.4.a.bo.1.1		2	1.1	even	1	trivial
2352.4.a.cc.1.2		2	7.6	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -15.7361 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -15.7361 q^{5} +9.00000 q^{9} -58.6805 q^{11} +20.7361 q^{13} +47.2083 q^{15} -42.9444 q^{17} +137.153 q^{19} -29.0556 q^{23} +122.625 q^{25} -27.0000 q^{27} +8.68051 q^{29} -202.472 q^{31} +176.042 q^{33} -15.2639 q^{37} -62.2083 q^{39} -117.250 q^{41} +101.875 q^{43} -141.625 q^{45} +588.500 q^{47} +128.833 q^{51} +404.681 q^{53} +923.403 q^{55} -411.458 q^{57} +10.8748 q^{59} +894.611 q^{61} -326.305 q^{65} +703.735 q^{67} +87.1668 q^{69} -1161.94 q^{71} +1116.37 q^{73} -367.875 q^{75} -138.333 q^{79} +81.0000 q^{81} -894.319 q^{83} +675.778 q^{85} -26.0415 q^{87} +681.806 q^{89} +607.417 q^{93} -2158.25 q^{95} -246.514 q^{97} -528.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 9 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9} - 5 q^{11} + 19 q^{13} + 27 q^{15} + 4 q^{17} + 117 q^{19} - 148 q^{23} + 43 q^{25} - 54 q^{27} - 95 q^{29} - 360 q^{31} + 15 q^{33} - 53 q^{37} - 57 q^{39} + 170 q^{41} - 403 q^{43} - 81 q^{45} + 368 q^{47} - 12 q^{51} + 697 q^{53} + 1285 q^{55} - 351 q^{57} - 585 q^{59} + 1160 q^{61} - 338 q^{65} - 233 q^{67} + 444 q^{69} - 616 q^{71} + 817 q^{73} - 129 q^{75} + 802 q^{79} + 162 q^{81} - 283 q^{83} + 992 q^{85} + 285 q^{87} + 1858 q^{89} + 1080 q^{93} - 2294 q^{95} - 1729 q^{97} - 45 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 9 * q^5 + 18 * q^9 - 5 * q^11 + 19 * q^13 + 27 * q^15 + 4 * q^17 + 117 * q^19 - 148 * q^23 + 43 * q^25 - 54 * q^27 - 95 * q^29 - 360 * q^31 + 15 * q^33 - 53 * q^37 - 57 * q^39 + 170 * q^41 - 403 * q^43 - 81 * q^45 + 368 * q^47 - 12 * q^51 + 697 * q^53 + 1285 * q^55 - 351 * q^57 - 585 * q^59 + 1160 * q^61 - 338 * q^65 - 233 * q^67 + 444 * q^69 - 616 * q^71 + 817 * q^73 - 129 * q^75 + 802 * q^79 + 162 * q^81 - 283 * q^83 + 992 * q^85 + 285 * q^87 + 1858 * q^89 + 1080 * q^93 - 2294 * q^95 - 1729 * q^97 - 45 * q^99

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