LMFDB - Embedded newform 9200.2.a.ce.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,0,0,2,0,10,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.07912$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.07912 q^{3} -2.07912 q^{7} +6.48097 q^{9} -5.07912 q^{11} +3.48097 q^{13} -6.48097 q^{17} +7.48097 q^{19} -6.40185 q^{21} +1.00000 q^{23} +10.7183 q^{27} -1.56009 q^{29} -0.0791189 q^{31} -15.6392 q^{33} +9.71833 q^{37} +10.7183 q^{39} -0.480973 q^{41} +8.00000 q^{43} +6.96195 q^{47} -2.67726 q^{49} -19.9557 q^{51} -11.7183 q^{53} +23.0348 q^{57} +11.5601 q^{59} +7.88283 q^{61} -13.4747 q^{63} -9.71833 q^{67} +3.07912 q^{69} +9.67726 q^{71} +13.2784 q^{73} +10.5601 q^{77} +12.3165 q^{79} +13.5601 q^{81} -4.59815 q^{83} -4.80371 q^{87} +8.31648 q^{89} -7.23736 q^{91} -0.243616 q^{93} -7.23736 q^{97} -32.9176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} - 10 q^{17} + 13 q^{19} - 18 q^{21} + 3 q^{23} - 2 q^{27} + 13 q^{29} + 8 q^{31} - 21 q^{33} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} + 2 q^{47}+ \cdots - 21 q^{99}+O(q^{100}) $ 3 * q + q^3 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 - 10 * q^17 + 13 * q^19 - 18 * q^21 + 3 * q^23 - 2 * q^27 + 13 * q^29 + 8 * q^31 - 21 * q^33 - 5 * q^37 - 2 * q^39 + 8 * q^41 + 24 * q^43 + 2 * q^47 - q^49 - q^51 - q^53 + 2 * q^57 + 17 * q^59 + 13 * q^61 + 9 * q^63 + 5 * q^67 + q^69 + 22 * q^71 - 12 * q^73 + 14 * q^77 + 4 * q^79 + 23 * q^81 - 15 * q^83 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 + 3 * q^97 - 21 * 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
$3$	3.07912	1.77773	0.888865	−	0.458169i	$-0.151495\pi$
$3$	3.07912	1.77773	0.888865	+	0.458169i	$0.151495\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.07912	−0.785833	−0.392917	−	0.919574i	$-0.628534\pi$
$7$	−2.07912	−0.785833	−0.392917	+	0.919574i	$0.628534\pi$
$8$	0	0
$9$	6.48097	2.16032
$10$	0	0
$11$	−5.07912	−1.53141	−0.765706	−	0.643191i	$-0.777610\pi$
$11$	−5.07912	−1.53141	−0.765706	+	0.643191i	$0.777610\pi$
$12$	0	0
$13$	3.48097	0.965448	0.482724	−	0.875772i	$-0.339647\pi$
$13$	3.48097	0.965448	0.482724	+	0.875772i	$0.339647\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.48097	−1.57187	−0.785933	−	0.618311i	$-0.787817\pi$
$17$	−6.48097	−1.57187	−0.785933	+	0.618311i	$0.787817\pi$
$18$	0	0
$19$	7.48097	1.71625	0.858126	−	0.513438i	$-0.171629\pi$
$19$	7.48097	1.71625	0.858126	+	0.513438i	$0.171629\pi$
$20$	0	0
$21$	−6.40185	−1.39700
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10.7183	2.06274
$28$	0	0
$29$	−1.56009	−0.289702	−0.144851	−	0.989453i	$-0.546270\pi$
$29$	−1.56009	−0.289702	−0.144851	+	0.989453i	$0.546270\pi$
$30$	0	0
$31$	−0.0791189	−0.0142102	−0.00710508	−	0.999975i	$-0.502262\pi$
$31$	−0.0791189	−0.0142102	−0.00710508	+	0.999975i	$0.502262\pi$
$32$	0	0
$33$	−15.6392	−2.72244
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.71833	1.59768	0.798842	−	0.601541i	$-0.205446\pi$
$37$	9.71833	1.59768	0.798842	+	0.601541i	$0.205446\pi$
$38$	0	0
$39$	10.7183	1.71631
$40$	0	0
$41$	−0.480973	−0.0751154	−0.0375577	−	0.999294i	$-0.511958\pi$
$41$	−0.480973	−0.0751154	−0.0375577	+	0.999294i	$0.511958\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.96195	1.01550	0.507752	−	0.861503i	$-0.330477\pi$
$47$	6.96195	1.01550	0.507752	+	0.861503i	$0.330477\pi$
$48$	0	0
$49$	−2.67726	−0.382466
$50$	0	0
$51$	−19.9557	−2.79435
$52$	0	0
$53$	−11.7183	−1.60964	−0.804818	−	0.593521i	$-0.797737\pi$
$53$	−11.7183	−1.60964	−0.804818	+	0.593521i	$0.797737\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	23.0348	3.05103
$58$	0	0
$59$	11.5601	1.50500	0.752498	−	0.658595i	$-0.228849\pi$
$59$	11.5601	1.50500	0.752498	+	0.658595i	$0.228849\pi$
$60$	0	0
$61$	7.88283	1.00929	0.504646	−	0.863326i	$-0.331623\pi$
$61$	7.88283	1.00929	0.504646	+	0.863326i	$0.331623\pi$
$62$	0	0
$63$	−13.4747	−1.69765
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.71833	−1.18728	−0.593641	−	0.804730i	$-0.702310\pi$
$67$	−9.71833	−1.18728	−0.593641	+	0.804730i	$0.702310\pi$
$68$	0	0
$69$	3.07912	0.370682
$70$	0	0
$71$	9.67726	1.14848	0.574240	−	0.818687i	$-0.305298\pi$
$71$	9.67726	1.14848	0.574240	+	0.818687i	$0.305298\pi$
$72$	0	0
$73$	13.2784	1.55412	0.777061	−	0.629425i	$-0.216710\pi$
$73$	13.2784	1.55412	0.777061	+	0.629425i	$0.216710\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5601	1.20343
$78$	0	0
$79$	12.3165	1.38571	0.692856	−	0.721076i	$-0.256352\pi$
$79$	12.3165	1.38571	0.692856	+	0.721076i	$0.256352\pi$
$80$	0	0
$81$	13.5601	1.50668
$82$	0	0
$83$	−4.59815	−0.504712	−0.252356	−	0.967634i	$-0.581205\pi$
$83$	−4.59815	−0.504712	−0.252356	+	0.967634i	$0.581205\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.80371	−0.515012
$88$	0	0
$89$	8.31648	0.881545	0.440772	−	0.897619i	$-0.354705\pi$
$89$	8.31648	0.881545	0.440772	+	0.897619i	$0.354705\pi$
$90$	0	0
$91$	−7.23736	−0.758681
$92$	0	0
$93$	−0.243616	−0.0252618
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.23736	−0.734842	−0.367421	−	0.930055i	$-0.619759\pi$
$97$	−7.23736	−0.734842	−0.367421	+	0.930055i	$0.619759\pi$
$98$	0	0
$99$	−32.9176	−3.30835
$100$	0	0
$101$	2.43991	0.242780	0.121390	−	0.992605i	$-0.461265\pi$
$101$	2.43991	0.242780	0.121390	+	0.992605i	$0.461265\pi$
$102$	0	0
$103$	7.88283	0.776718	0.388359	−	0.921508i	$-0.373042\pi$
$103$	7.88283	0.776718	0.388359	+	0.921508i	$0.373042\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.6803	−1.61254	−0.806272	−	0.591546i	$-0.798518\pi$
$107$	−16.6803	−1.61254	−0.806272	+	0.591546i	$0.798518\pi$
$108$	0	0
$109$	5.32274	0.509826	0.254913	−	0.966964i	$-0.417953\pi$
$109$	5.32274	0.509826	0.254913	+	0.966964i	$0.417953\pi$
$110$	0	0
$111$	29.9239	2.84025
$112$	0	0
$113$	2.20556	0.207482	0.103741	−	0.994604i	$-0.466919\pi$
$113$	2.20556	0.207482	0.103741	+	0.994604i	$0.466919\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	22.5601	2.08568
$118$	0	0
$119$	13.4747	1.23522
$120$	0	0
$121$	14.7974	1.34522
$122$	0	0
$123$	−1.48097	−0.133535
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.15824	0.546455	0.273228	−	0.961949i	$-0.411909\pi$
$127$	6.15824	0.546455	0.273228	+	0.961949i	$0.411909\pi$
$128$	0	0
$129$	24.6330	2.16881
$130$	0	0
$131$	7.19629	0.628743	0.314371	−	0.949300i	$-0.398206\pi$
$131$	7.19629	0.628743	0.314371	+	0.949300i	$0.398206\pi$
$132$	0	0
$133$	−15.5538	−1.34869
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.6012	−1.07659	−0.538295	−	0.842757i	$-0.680931\pi$
$137$	−12.6012	−1.07659	−0.538295	+	0.842757i	$0.680931\pi$
$138$	0	0
$139$	8.75638	0.742707	0.371353	−	0.928492i	$-0.378894\pi$
$139$	8.75638	0.742707	0.371353	+	0.928492i	$0.378894\pi$
$140$	0	0
$141$	21.4367	1.80529
$142$	0	0
$143$	−17.6803	−1.47850
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.24362	−0.679922
$148$	0	0
$149$	1.79745	0.147253	0.0736264	−	0.997286i	$-0.476543\pi$
$149$	1.79745	0.147253	0.0736264	+	0.997286i	$0.476543\pi$
$150$	0	0
$151$	19.3956	1.57839	0.789196	−	0.614142i	$-0.210498\pi$
$151$	19.3956	1.57839	0.789196	+	0.614142i	$0.210498\pi$
$152$	0	0
$153$	−42.0030	−3.39574
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.36380	−0.348269	−0.174135	−	0.984722i	$-0.555713\pi$
$157$	−4.36380	−0.348269	−0.174135	+	0.984722i	$0.555713\pi$
$158$	0	0
$159$	−36.0821	−2.86150
$160$	0	0
$161$	−2.07912	−0.163858
$162$	0	0
$163$	5.32274	0.416909	0.208454	−	0.978032i	$-0.433157\pi$
$163$	5.32274	0.416909	0.208454	+	0.978032i	$0.433157\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−0.882827	−0.0679098
$170$	0	0
$171$	48.4840	3.70766
$172$	0	0
$173$	9.23736	0.702303	0.351152	−	0.936319i	$-0.385790\pi$
$173$	9.23736	0.702303	0.351152	+	0.936319i	$0.385790\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35.5949	2.67548
$178$	0	0
$179$	−2.96195	−0.221386	−0.110693	−	0.993855i	$-0.535307\pi$
$179$	−2.96195	−0.221386	−0.110693	+	0.993855i	$0.535307\pi$
$180$	0	0
$181$	10.8355	0.805397	0.402698	−	0.915333i	$-0.368072\pi$
$181$	10.8355	0.805397	0.402698	+	0.915333i	$0.368072\pi$
$182$	0	0
$183$	24.2722	1.79425
$184$	0	0
$185$	0	0
$186$	0	0
$187$	32.9176	2.40718
$188$	0	0
$189$	−22.2847	−1.62097
$190$	0	0
$191$	−7.76565	−0.561903	−0.280952	−	0.959722i	$-0.590650\pi$
$191$	−7.76565	−0.561903	−0.280952	+	0.959722i	$0.590650\pi$
$192$	0	0
$193$	2.15824	0.155353	0.0776767	−	0.996979i	$-0.475250\pi$
$193$	2.15824	0.155353	0.0776767	+	0.996979i	$0.475250\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0411	−0.857890	−0.428945	−	0.903331i	$-0.641115\pi$
$197$	−12.0411	−0.857890	−0.428945	+	0.903331i	$0.641115\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−29.9239	−2.11067
$202$	0	0
$203$	3.24362	0.227657
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.48097	0.450459
$208$	0	0
$209$	−37.9968	−2.62829
$210$	0	0
$211$	−3.40185	−0.234193	−0.117097	−	0.993121i	$-0.537359\pi$
$211$	−3.40185	−0.234193	−0.117097	+	0.993121i	$0.537359\pi$
$212$	0	0
$213$	29.7974	2.04169
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.164498	0.0111668
$218$	0	0
$219$	40.8858	2.76281
$220$	0	0
$221$	−22.5601	−1.51756
$222$	0	0
$223$	5.19629	0.347969	0.173985	−	0.984748i	$-0.444336\pi$
$223$	5.19629	0.347969	0.173985	+	0.984748i	$0.444336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.35453	0.355393	0.177696	−	0.984085i	$-0.443136\pi$
$227$	5.35453	0.355393	0.177696	+	0.984085i	$0.443136\pi$
$228$	0	0
$229$	−0.803708	−0.0531105	−0.0265553	−	0.999647i	$-0.508454\pi$
$229$	−0.803708	−0.0531105	−0.0265553	+	0.999647i	$0.508454\pi$
$230$	0	0
$231$	32.5158	2.13938
$232$	0	0
$233$	−14.1582	−0.927537	−0.463768	−	0.885956i	$-0.653503\pi$
$233$	−14.1582	−0.927537	−0.463768	+	0.885956i	$0.653503\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	37.9239	2.46342
$238$	0	0
$239$	−10.9146	−0.706008	−0.353004	−	0.935622i	$-0.614840\pi$
$239$	−10.9146	−0.706008	−0.353004	+	0.935622i	$0.614840\pi$
$240$	0	0
$241$	−27.4367	−1.76735	−0.883675	−	0.468100i	$-0.844939\pi$
$241$	−27.4367	−1.76735	−0.883675	+	0.468100i	$0.844939\pi$
$242$	0	0
$243$	9.59815	0.615721
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.0411	1.65695
$248$	0	0
$249$	−14.1582	−0.897242
$250$	0	0
$251$	16.5190	1.04267	0.521336	−	0.853352i	$-0.325434\pi$
$251$	16.5190	1.04267	0.521336	+	0.853352i	$0.325434\pi$
$252$	0	0
$253$	−5.07912	−0.319321
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.7657	−1.23295	−0.616474	−	0.787375i	$-0.711439\pi$
$257$	−19.7657	−1.23295	−0.616474	+	0.787375i	$0.711439\pi$
$258$	0	0
$259$	−20.2056	−1.25551
$260$	0	0
$261$	−10.1109	−0.625850
$262$	0	0
$263$	7.44292	0.458950	0.229475	−	0.973315i	$-0.426299\pi$
$263$	7.44292	0.458950	0.229475	+	0.973315i	$0.426299\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.6074	1.56715
$268$	0	0
$269$	−22.7564	−1.38748	−0.693741	−	0.720225i	$-0.744039\pi$
$269$	−22.7564	−1.38748	−0.693741	+	0.720225i	$0.744039\pi$
$270$	0	0
$271$	19.0411	1.15666	0.578331	−	0.815802i	$-0.303704\pi$
$271$	19.0411	1.15666	0.578331	+	0.815802i	$0.303704\pi$
$272$	0	0
$273$	−22.2847	−1.34873
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.3165	1.10053	0.550265	−	0.834990i	$-0.314527\pi$
$277$	18.3165	1.10053	0.550265	+	0.834990i	$0.314527\pi$
$278$	0	0
$279$	−0.512767	−0.0306986
$280$	0	0
$281$	26.0821	1.55593	0.777965	−	0.628308i	$-0.216252\pi$
$281$	26.0821	1.55593	0.777965	+	0.628308i	$0.216252\pi$
$282$	0	0
$283$	−25.6422	−1.52427	−0.762136	−	0.647417i	$-0.775849\pi$
$283$	−25.6422	−1.52427	−0.762136	+	0.647417i	$0.775849\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	25.0030	1.47077
$290$	0	0
$291$	−22.2847	−1.30635
$292$	0	0
$293$	−30.8385	−1.80161	−0.900803	−	0.434229i	$-0.857021\pi$
$293$	−30.8385	−1.80161	−0.900803	+	0.434229i	$0.857021\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−54.4397	−3.15891
$298$	0	0
$299$	3.48097	0.201310
$300$	0	0
$301$	−16.6330	−0.958707
$302$	0	0
$303$	7.51277	0.431597
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.88283	−0.107459	−0.0537293	−	0.998556i	$-0.517111\pi$
$307$	−1.88283	−0.107459	−0.0537293	+	0.998556i	$0.517111\pi$
$308$	0	0
$309$	24.2722	1.38080
$310$	0	0
$311$	−1.60742	−0.0911482	−0.0455741	−	0.998961i	$-0.514512\pi$
$311$	−1.60742	−0.0911482	−0.0455741	+	0.998961i	$0.514512\pi$
$312$	0	0
$313$	1.68654	0.0953286	0.0476643	−	0.998863i	$-0.484822\pi$
$313$	1.68654	0.0953286	0.0476643	+	0.998863i	$0.484822\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6773	0.712026	0.356013	−	0.934481i	$-0.384136\pi$
$317$	12.6773	0.712026	0.356013	+	0.934481i	$0.384136\pi$
$318$	0	0
$319$	7.92389	0.443653
$320$	0	0
$321$	−51.3606	−2.86667
$322$	0	0
$323$	−48.4840	−2.69772
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.3893	0.906332
$328$	0	0
$329$	−14.4747	−0.798017
$330$	0	0
$331$	−33.3893	−1.83524	−0.917622	−	0.397454i	$-0.869894\pi$
$331$	−33.3893	−1.83524	−0.917622	+	0.397454i	$0.869894\pi$
$332$	0	0
$333$	62.9842	3.45151
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.6392	0.960869	0.480435	−	0.877031i	$-0.340479\pi$
$337$	17.6392	0.960869	0.480435	+	0.877031i	$0.340479\pi$
$338$	0	0
$339$	6.79119	0.368847
$340$	0	0
$341$	0.401854	0.0217616
$342$	0	0
$343$	20.1202	1.08639
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.7121	1.05820	0.529100	−	0.848560i	$-0.322530\pi$
$347$	19.7121	1.05820	0.529100	+	0.848560i	$0.322530\pi$
$348$	0	0
$349$	−16.0348	−0.858323	−0.429162	−	0.903228i	$-0.641191\pi$
$349$	−16.0348	−0.858323	−0.429162	+	0.903228i	$0.641191\pi$
$350$	0	0
$351$	37.3102	1.99147
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	41.4902	2.19590
$358$	0	0
$359$	13.1202	0.692457	0.346228	−	0.938150i	$-0.387462\pi$
$359$	13.1202	0.692457	0.346228	+	0.938150i	$0.387462\pi$
$360$	0	0
$361$	36.9650	1.94552
$362$	0	0
$363$	45.5631	2.39144
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.47796	0.181548	0.0907741	−	0.995872i	$-0.471066\pi$
$367$	3.47796	0.181548	0.0907741	+	0.995872i	$0.471066\pi$
$368$	0	0
$369$	−3.11717	−0.162274
$370$	0	0
$371$	24.3638	1.26491
$372$	0	0
$373$	−21.5949	−1.11814	−0.559071	−	0.829120i	$-0.688842\pi$
$373$	−21.5949	−1.11814	−0.559071	+	0.829120i	$0.688842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.43064	−0.279692
$378$	0	0
$379$	−7.80070	−0.400695	−0.200347	−	0.979725i	$-0.564207\pi$
$379$	−7.80070	−0.400695	−0.200347	+	0.979725i	$0.564207\pi$
$380$	0	0
$381$	18.9619	0.971450
$382$	0	0
$383$	32.4274	1.65696	0.828481	−	0.560017i	$-0.189205\pi$
$383$	32.4274	1.65696	0.828481	+	0.560017i	$0.189205\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	51.8478	2.63557
$388$	0	0
$389$	23.5538	1.19423	0.597113	−	0.802157i	$-0.296314\pi$
$389$	23.5538	1.19423	0.597113	+	0.802157i	$0.296314\pi$
$390$	0	0
$391$	−6.48097	−0.327757
$392$	0	0
$393$	22.1582	1.11774
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.370060	0.0185728	0.00928639	−	0.999957i	$-0.497044\pi$
$397$	0.370060	0.0185728	0.00928639	+	0.999957i	$0.497044\pi$
$398$	0	0
$399$	−47.8921	−2.39760
$400$	0	0
$401$	11.3545	0.567018	0.283509	−	0.958970i	$-0.408501\pi$
$401$	11.3545	0.567018	0.283509	+	0.958970i	$0.408501\pi$
$402$	0	0
$403$	−0.275411	−0.0137192
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−49.3606	−2.44671
$408$	0	0
$409$	4.88283	0.241440	0.120720	−	0.992687i	$-0.461480\pi$
$409$	4.88283	0.241440	0.120720	+	0.992687i	$0.461480\pi$
$410$	0	0
$411$	−38.8005	−1.91389
$412$	0	0
$413$	−24.0348	−1.18268
$414$	0	0
$415$	0	0
$416$	0	0
$417$	26.9619	1.32033
$418$	0	0
$419$	−1.51277	−0.0739035	−0.0369518	−	0.999317i	$-0.511765\pi$
$419$	−1.51277	−0.0739035	−0.0369518	+	0.999317i	$0.511765\pi$
$420$	0	0
$421$	22.2722	1.08548	0.542739	−	0.839901i	$-0.317387\pi$
$421$	22.2722	1.08548	0.542739	+	0.839901i	$0.317387\pi$
$422$	0	0
$423$	45.1202	2.19382
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.3893	−0.793135
$428$	0	0
$429$	−54.4397	−2.62837
$430$	0	0
$431$	16.8858	0.813362	0.406681	−	0.913570i	$-0.366686\pi$
$431$	16.8858	0.813362	0.406681	+	0.913570i	$0.366686\pi$
$432$	0	0
$433$	−24.3956	−1.17238	−0.586189	−	0.810175i	$-0.699372\pi$
$433$	−24.3956	−1.17238	−0.586189	+	0.810175i	$0.699372\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.48097	0.357863
$438$	0	0
$439$	−40.0378	−1.91090	−0.955450	−	0.295152i	$-0.904630\pi$
$439$	−40.0378	−1.91090	−0.955450	+	0.295152i	$0.904630\pi$
$440$	0	0
$441$	−17.3513	−0.826251
$442$	0	0
$443$	−5.22809	−0.248394	−0.124197	−	0.992258i	$-0.539635\pi$
$443$	−5.22809	−0.248394	−0.124197	+	0.992258i	$0.539635\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.53456	0.261776
$448$	0	0
$449$	4.72459	0.222967	0.111484	−	0.993766i	$-0.464440\pi$
$449$	4.72459	0.222967	0.111484	+	0.993766i	$0.464440\pi$
$450$	0	0
$451$	2.44292	0.115033
$452$	0	0
$453$	59.7213	2.80595
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.2311	−1.55449	−0.777243	−	0.629201i	$-0.783382\pi$
$457$	−33.2311	−1.55449	−0.777243	+	0.629201i	$0.783382\pi$
$458$	0	0
$459$	−69.4652	−3.24236
$460$	0	0
$461$	−28.9619	−1.34889	−0.674446	−	0.738324i	$-0.735618\pi$
$461$	−28.9619	−1.34889	−0.674446	+	0.738324i	$0.735618\pi$
$462$	0	0
$463$	4.39258	0.204141	0.102070	−	0.994777i	$-0.467453\pi$
$463$	4.39258	0.204141	0.102070	+	0.994777i	$0.467453\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0729	0.790038	0.395019	−	0.918673i	$-0.370738\pi$
$467$	17.0729	0.790038	0.395019	+	0.918673i	$0.370738\pi$
$468$	0	0
$469$	20.2056	0.933006
$470$	0	0
$471$	−13.4367	−0.619129
$472$	0	0
$473$	−40.6330	−1.86831
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−75.9462	−3.47734
$478$	0	0
$479$	−27.0441	−1.23568	−0.617838	−	0.786306i	$-0.711991\pi$
$479$	−27.0441	−1.23568	−0.617838	+	0.786306i	$0.711991\pi$
$480$	0	0
$481$	33.8292	1.54248
$482$	0	0
$483$	−6.40185	−0.291294
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−28.0821	−1.27252	−0.636261	−	0.771474i	$-0.719520\pi$
$487$	−28.0821	−1.27252	−0.636261	+	0.771474i	$0.719520\pi$
$488$	0	0
$489$	16.3893	0.741151
$490$	0	0
$491$	7.48398	0.337747	0.168874	−	0.985638i	$-0.445987\pi$
$491$	7.48398	0.337747	0.168874	+	0.985638i	$0.445987\pi$
$492$	0	0
$493$	10.1109	0.455373
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.1202	−0.902514
$498$	0	0
$499$	−21.7183	−0.972246	−0.486123	−	0.873890i	$-0.661589\pi$
$499$	−21.7183	−0.972246	−0.486123	+	0.873890i	$0.661589\pi$
$500$	0	0
$501$	−24.6330	−1.10052
$502$	0	0
$503$	−7.27541	−0.324395	−0.162197	−	0.986758i	$-0.551858\pi$
$503$	−7.27541	−0.324395	−0.162197	+	0.986758i	$0.551858\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.71833	−0.120725
$508$	0	0
$509$	22.7151	1.00683	0.503414	−	0.864045i	$-0.332077\pi$
$509$	22.7151	1.00683	0.503414	+	0.864045i	$0.332077\pi$
$510$	0	0
$511$	−27.6074	−1.22128
$512$	0	0
$513$	80.1835	3.54019
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−35.3606	−1.55516
$518$	0	0
$519$	28.4429	1.24851
$520$	0	0
$521$	−16.5568	−0.725368	−0.362684	−	0.931912i	$-0.618140\pi$
$521$	−16.5568	−0.725368	−0.362684	+	0.931912i	$0.618140\pi$
$522$	0	0
$523$	24.8037	1.08459	0.542295	−	0.840188i	$-0.317555\pi$
$523$	24.8037	1.08459	0.542295	+	0.840188i	$0.317555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.512767	0.0223365
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	74.9206	3.25128
$532$	0	0
$533$	−1.67425	−0.0725200
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.12018	−0.393565
$538$	0	0
$539$	13.5981	0.585714
$540$	0	0
$541$	−21.8292	−0.938512	−0.469256	−	0.883062i	$-0.655478\pi$
$541$	−21.8292	−0.938512	−0.469256	+	0.883062i	$0.655478\pi$
$542$	0	0
$543$	33.3638	1.43178
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.0759	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$547$	−29.0759	−1.24319	−0.621597	+	0.783337i	$0.713516\pi$
$548$	0	0
$549$	51.0884	2.18040
$550$	0	0
$551$	−11.6710	−0.497202
$552$	0	0
$553$	−25.6074	−1.08894
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0348	0.933645	0.466822	−	0.884351i	$-0.345399\pi$
$557$	22.0348	0.933645	0.466822	+	0.884351i	$0.345399\pi$
$558$	0	0
$559$	27.8478	1.17784
$560$	0	0
$561$	101.357	4.27931
$562$	0	0
$563$	1.40185	0.0590811	0.0295406	−	0.999564i	$-0.490596\pi$
$563$	1.40185	0.0590811	0.0295406	+	0.999564i	$0.490596\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−28.1930	−1.18400
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	18.3575	0.768239	0.384120	−	0.923283i	$-0.374505\pi$
$571$	18.3575	0.768239	0.384120	+	0.923283i	$0.374505\pi$
$572$	0	0
$573$	−23.9114	−0.998912
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.5128	0.479283	0.239641	−	0.970861i	$-0.422970\pi$
$577$	11.5128	0.479283	0.239641	+	0.970861i	$0.422970\pi$
$578$	0	0
$579$	6.64547	0.276176
$580$	0	0
$581$	9.56009	0.396619
$582$	0	0
$583$	59.5188	2.46502
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.1232	1.65606	0.828031	−	0.560683i	$-0.189461\pi$
$587$	40.1232	1.65606	0.828031	+	0.560683i	$0.189461\pi$
$588$	0	0
$589$	−0.591886	−0.0243882
$590$	0	0
$591$	−37.0759	−1.52510
$592$	0	0
$593$	−18.7912	−0.771662	−0.385831	−	0.922570i	$-0.626085\pi$
$593$	−18.7912	−0.771662	−0.385831	+	0.922570i	$0.626085\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	43.1077	1.76428
$598$	0	0
$599$	27.4777	1.12271	0.561355	−	0.827575i	$-0.310280\pi$
$599$	27.4777	1.12271	0.561355	+	0.827575i	$0.310280\pi$
$600$	0	0
$601$	−19.5283	−0.796576	−0.398288	−	0.917260i	$-0.630396\pi$
$601$	−19.5283	−0.796576	−0.398288	+	0.917260i	$0.630396\pi$
$602$	0	0
$603$	−62.9842	−2.56492
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.9239	1.86399	0.931997	−	0.362467i	$-0.118065\pi$
$607$	45.9239	1.86399	0.931997	+	0.362467i	$0.118065\pi$
$608$	0	0
$609$	9.98748	0.404713
$610$	0	0
$611$	24.2343	0.980417
$612$	0	0
$613$	−12.7091	−0.513314	−0.256657	−	0.966503i	$-0.582621\pi$
$613$	−12.7091	−0.513314	−0.256657	+	0.966503i	$0.582621\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.3102	−1.94490	−0.972448	−	0.233120i	$-0.925107\pi$
$617$	−48.3102	−1.94490	−0.972448	+	0.233120i	$0.925107\pi$
$618$	0	0
$619$	−0.677265	−0.0272216	−0.0136108	−	0.999907i	$-0.504333\pi$
$619$	−0.677265	−0.0272216	−0.0136108	+	0.999907i	$0.504333\pi$
$620$	0	0
$621$	10.7183	0.430112
$622$	0	0
$623$	−17.2909	−0.692747
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−116.997	−4.67239
$628$	0	0
$629$	−62.9842	−2.51135
$630$	0	0
$631$	−44.3986	−1.76748	−0.883740	−	0.467978i	$-0.844983\pi$
$631$	−44.3986	−1.76748	−0.883740	+	0.467978i	$0.844983\pi$
$632$	0	0
$633$	−10.4747	−0.416332
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.31949	−0.369251
$638$	0	0
$639$	62.7181	2.48109
$640$	0	0
$641$	16.8798	0.666713	0.333356	−	0.942801i	$-0.391819\pi$
$641$	16.8798	0.666713	0.333356	+	0.942801i	$0.391819\pi$
$642$	0	0
$643$	−26.7564	−1.05517	−0.527584	−	0.849503i	$-0.676902\pi$
$643$	−26.7564	−1.05517	−0.527584	+	0.849503i	$0.676902\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.5568	1.27994	0.639971	−	0.768399i	$-0.278946\pi$
$647$	32.5568	1.27994	0.639971	+	0.768399i	$0.278946\pi$
$648$	0	0
$649$	−58.7151	−2.30477
$650$	0	0
$651$	0.506507	0.0198516
$652$	0	0
$653$	9.87356	0.386382	0.193191	−	0.981161i	$-0.438116\pi$
$653$	9.87356	0.386382	0.193191	+	0.981161i	$0.438116\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	86.0571	3.35741
$658$	0	0
$659$	13.5128	0.526383	0.263191	−	0.964744i	$-0.415225\pi$
$659$	13.5128	0.526383	0.263191	+	0.964744i	$0.415225\pi$
$660$	0	0
$661$	−26.7687	−1.04118	−0.520590	−	0.853807i	$-0.674288\pi$
$661$	−26.7687	−1.04118	−0.520590	+	0.853807i	$0.674288\pi$
$662$	0	0
$663$	−69.4652	−2.69780
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.56009	−0.0604070
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−40.0378	−1.54564
$672$	0	0
$673$	0.803708	0.0309807	0.0154903	−	0.999880i	$-0.495069\pi$
$673$	0.803708	0.0309807	0.0154903	+	0.999880i	$0.495069\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.6422	1.29298	0.646488	−	0.762924i	$-0.276237\pi$
$677$	33.6422	1.29298	0.646488	+	0.762924i	$0.276237\pi$
$678$	0	0
$679$	15.0473	0.577463
$680$	0	0
$681$	16.4872	0.631792
$682$	0	0
$683$	−11.8736	−0.454329	−0.227165	−	0.973856i	$-0.572946\pi$
$683$	−11.8736	−0.454329	−0.227165	+	0.973856i	$0.572946\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.47471	−0.0944162
$688$	0	0
$689$	−40.7912	−1.55402
$690$	0	0
$691$	−27.9114	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$691$	−27.9114	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$692$	0	0
$693$	68.4397	2.59981
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.11717	0.118071
$698$	0	0
$699$	−43.5949	−1.64891
$700$	0	0
$701$	−26.3668	−0.995861	−0.497930	−	0.867217i	$-0.665906\pi$
$701$	−26.3668	−0.995861	−0.497930	+	0.867217i	$0.665906\pi$
$702$	0	0
$703$	72.7026	2.74203
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.07286	−0.190785
$708$	0	0
$709$	11.8007	0.443184	0.221592	−	0.975139i	$-0.428875\pi$
$709$	11.8007	0.443184	0.221592	+	0.975139i	$0.428875\pi$
$710$	0	0
$711$	79.8227	2.99359
$712$	0	0
$713$	−0.0791189	−0.00296302
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−33.6074	−1.25509
$718$	0	0
$719$	12.9589	0.483287	0.241643	−	0.970365i	$-0.422314\pi$
$719$	12.9589	0.483287	0.241643	+	0.970365i	$0.422314\pi$
$720$	0	0
$721$	−16.3893	−0.610371
$722$	0	0
$723$	−84.4807	−3.14187
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.0318	−0.705850	−0.352925	−	0.935652i	$-0.614813\pi$
$727$	−19.0318	−0.705850	−0.352925	+	0.935652i	$0.614813\pi$
$728$	0	0
$729$	−11.1264	−0.412090
$730$	0	0
$731$	−51.8478	−1.91766
$732$	0	0
$733$	30.5220	1.12736	0.563679	−	0.825994i	$-0.309386\pi$
$733$	30.5220	1.12736	0.563679	+	0.825994i	$0.309386\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.3606	1.81822
$738$	0	0
$739$	6.18702	0.227593	0.113797	−	0.993504i	$-0.463699\pi$
$739$	6.18702	0.227593	0.113797	+	0.993504i	$0.463699\pi$
$740$	0	0
$741$	80.1835	2.94562
$742$	0	0
$743$	16.2722	0.596968	0.298484	−	0.954415i	$-0.403519\pi$
$743$	16.2722	0.596968	0.298484	+	0.954415i	$0.403519\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−29.8005	−1.09034
$748$	0	0
$749$	34.6803	1.26719
$750$	0	0
$751$	17.5949	0.642047	0.321023	−	0.947071i	$-0.395973\pi$
$751$	17.5949	0.642047	0.321023	+	0.947071i	$0.395973\pi$
$752$	0	0
$753$	50.8640	1.85359
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.2496	−1.49924	−0.749622	−	0.661866i	$-0.769765\pi$
$757$	−41.2496	−1.49924	−0.749622	+	0.661866i	$0.769765\pi$
$758$	0	0
$759$	−15.6392	−0.567667
$760$	0	0
$761$	26.8067	0.971743	0.485871	−	0.874030i	$-0.338502\pi$
$761$	26.8067	0.971743	0.485871	+	0.874030i	$0.338502\pi$
$762$	0	0
$763$	−11.0666	−0.400638
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.2404	1.45300
$768$	0	0
$769$	−26.8798	−0.969311	−0.484655	−	0.874705i	$-0.661055\pi$
$769$	−26.8798	−0.969311	−0.484655	+	0.874705i	$0.661055\pi$
$770$	0	0
$771$	−60.8608	−2.19185
$772$	0	0
$773$	−41.0441	−1.47625	−0.738126	−	0.674662i	$-0.764289\pi$
$773$	−41.0441	−1.47625	−0.738126	+	0.674662i	$0.764289\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−62.2153	−2.23196
$778$	0	0
$779$	−3.59815	−0.128917
$780$	0	0
$781$	−49.1520	−1.75880
$782$	0	0
$783$	−16.7216	−0.597580
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−21.4840	−0.765821	−0.382911	−	0.923785i	$-0.625078\pi$
$787$	−21.4840	−0.765821	−0.382911	+	0.923785i	$0.625078\pi$
$788$	0	0
$789$	22.9176	0.815889
$790$	0	0
$791$	−4.58563	−0.163046
$792$	0	0
$793$	27.4399	0.974420
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.2692	−1.35556	−0.677781	−	0.735263i	$-0.737058\pi$
$797$	−38.2692	−1.35556	−0.677781	+	0.735263i	$0.737058\pi$
$798$	0	0
$799$	−45.1202	−1.59624
$800$	0	0
$801$	53.8989	1.90442
$802$	0	0
$803$	−67.4427	−2.38000
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−70.0696	−2.46657
$808$	0	0
$809$	−27.2026	−0.956391	−0.478195	−	0.878253i	$-0.658709\pi$
$809$	−27.2026	−0.956391	−0.478195	+	0.878253i	$0.658709\pi$
$810$	0	0
$811$	−38.5095	−1.35225	−0.676126	−	0.736786i	$-0.736343\pi$
$811$	−38.5095	−1.35225	−0.676126	+	0.736786i	$0.736343\pi$
$812$	0	0
$813$	58.6297	2.05623
$814$	0	0
$815$	0	0
$816$	0	0
$817$	59.8478	2.09381
$818$	0	0
$819$	−46.9051	−1.63900
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	40.1457	1.39939	0.699696	−	0.714441i	$-0.253319\pi$
$823$	40.1457	1.39939	0.699696	+	0.714441i	$0.253319\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.851033	0.0295933	0.0147967	−	0.999891i	$-0.495290\pi$
$827$	0.851033	0.0295933	0.0147967	+	0.999891i	$0.495290\pi$
$828$	0	0
$829$	−2.83851	−0.0985856	−0.0492928	−	0.998784i	$-0.515697\pi$
$829$	−2.83851	−0.0985856	−0.0492928	+	0.998784i	$0.515697\pi$
$830$	0	0
$831$	56.3986	1.95645
$832$	0	0
$833$	17.3513	0.601186
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.848022	−0.0293119
$838$	0	0
$839$	10.2343	0.353329	0.176664	−	0.984271i	$-0.443469\pi$
$839$	10.2343	0.353329	0.176664	+	0.984271i	$0.443469\pi$
$840$	0	0
$841$	−26.5661	−0.916073
$842$	0	0
$843$	80.3100	2.76602
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.7657	−1.05712
$848$	0	0
$849$	−78.9554	−2.70974
$850$	0	0
$851$	9.71833	0.333140
$852$	0	0
$853$	−17.7213	−0.606767	−0.303384	−	0.952869i	$-0.598116\pi$
$853$	−17.7213	−0.606767	−0.303384	+	0.952869i	$0.598116\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.6835	−0.604058	−0.302029	−	0.953299i	$-0.597664\pi$
$857$	−17.6835	−0.604058	−0.302029	+	0.953299i	$0.597664\pi$
$858$	0	0
$859$	26.6042	0.907722	0.453861	−	0.891072i	$-0.350046\pi$
$859$	26.6042	0.907722	0.453861	+	0.891072i	$0.350046\pi$
$860$	0	0
$861$	3.07912	0.104936
$862$	0	0
$863$	−53.5949	−1.82439	−0.912196	−	0.409755i	$-0.865614\pi$
$863$	−53.5949	−1.82439	−0.912196	+	0.409755i	$0.865614\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	76.9872	2.61462
$868$	0	0
$869$	−62.5568	−2.12210
$870$	0	0
$871$	−33.8292	−1.14626
$872$	0	0
$873$	−46.9051	−1.58750
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.9650	−0.876774	−0.438387	−	0.898786i	$-0.644450\pi$
$877$	−25.9650	−0.876774	−0.438387	+	0.898786i	$0.644450\pi$
$878$	0	0
$879$	−94.9554	−3.20277
$880$	0	0
$881$	29.9239	1.00816	0.504081	−	0.863657i	$-0.331831\pi$
$881$	29.9239	1.00816	0.504081	+	0.863657i	$0.331831\pi$
$882$	0	0
$883$	−4.75939	−0.160166	−0.0800832	−	0.996788i	$-0.525519\pi$
$883$	−4.75939	−0.160166	−0.0800832	+	0.996788i	$0.525519\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.8037	1.30290	0.651451	−	0.758691i	$-0.274161\pi$
$887$	38.8037	1.30290	0.651451	+	0.758691i	$0.274161\pi$
$888$	0	0
$889$	−12.8037	−0.429423
$890$	0	0
$891$	−68.8733	−2.30734
$892$	0	0
$893$	52.0821	1.74286
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.7183	0.357875
$898$	0	0
$899$	0.123433	0.00411671
$900$	0	0
$901$	75.9462	2.53013
$902$	0	0
$903$	−51.2148	−1.70432
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−51.0729	−1.69585	−0.847923	−	0.530119i	$-0.822147\pi$
$907$	−51.0729	−1.69585	−0.847923	+	0.530119i	$0.822147\pi$
$908$	0	0
$909$	15.8130	0.524483
$910$	0	0
$911$	30.4933	1.01029	0.505143	−	0.863035i	$-0.331440\pi$
$911$	30.4933	1.01029	0.505143	+	0.863035i	$0.331440\pi$
$912$	0	0
$913$	23.3545	0.772922
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.9619	−0.494087
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−5.79745	−0.191032
$922$	0	0
$923$	33.6863	1.10880
$924$	0	0
$925$	0	0
$926$	0	0
$927$	51.0884	1.67796
$928$	0	0
$929$	−17.3257	−0.568439	−0.284220	−	0.958759i	$-0.591734\pi$
$929$	−17.3257	−0.568439	−0.284220	+	0.958759i	$0.591734\pi$
$930$	0	0
$931$	−20.0285	−0.656409
$932$	0	0
$933$	−4.94943	−0.162037
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47.5631	1.55382	0.776909	−	0.629612i	$-0.216786\pi$
$937$	47.5631	1.55382	0.776909	+	0.629612i	$0.216786\pi$
$938$	0	0
$939$	5.19304	0.169469
$940$	0	0
$941$	14.2754	0.465365	0.232683	−	0.972553i	$-0.425250\pi$
$941$	14.2754	0.465365	0.232683	+	0.972553i	$0.425250\pi$
$942$	0	0
$943$	−0.480973	−0.0156626
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.1900	1.17602	0.588009	−	0.808854i	$-0.299912\pi$
$947$	36.1900	1.17602	0.588009	+	0.808854i	$0.299912\pi$
$948$	0	0
$949$	46.2218	1.50042
$950$	0	0
$951$	39.0348	1.26579
$952$	0	0
$953$	−37.7942	−1.22427	−0.612137	−	0.790752i	$-0.709690\pi$
$953$	−37.7942	−1.22427	−0.612137	+	0.790752i	$0.709690\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.3986	0.788695
$958$	0	0
$959$	26.1993	0.846020
$960$	0	0
$961$	−30.9937	−0.999798
$962$	0	0
$963$	−108.104	−3.48362
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.6390	1.62844	0.814220	−	0.580557i	$-0.197165\pi$
$967$	50.6390	1.62844	0.814220	+	0.580557i	$0.197165\pi$
$968$	0	0
$969$	−149.288	−4.79582
$970$	0	0
$971$	−2.82623	−0.0906981	−0.0453490	−	0.998971i	$-0.514440\pi$
$971$	−2.82623	−0.0906981	−0.0453490	+	0.998971i	$0.514440\pi$
$972$	0	0
$973$	−18.2056	−0.583644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.8448	−0.826848	−0.413424	−	0.910539i	$-0.635667\pi$
$977$	−25.8448	−0.826848	−0.413424	+	0.910539i	$0.635667\pi$
$978$	0	0
$979$	−42.2404	−1.35001
$980$	0	0
$981$	34.4965	1.10139
$982$	0	0
$983$	−41.3668	−1.31940	−0.659698	−	0.751531i	$-0.729316\pi$
$983$	−41.3668	−1.31940	−0.659698	+	0.751531i	$0.729316\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−44.5694	−1.41866
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−54.4745	−1.73044	−0.865219	−	0.501394i	$-0.832821\pi$
$991$	−54.4745	−1.73044	−0.865219	+	0.501394i	$0.832821\pi$
$992$	0	0
$993$	−102.810	−3.26257
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.5128	0.491294	0.245647	−	0.969359i	$-0.421000\pi$
$997$	15.5128	0.491294	0.245647	+	0.969359i	$0.421000\pi$
$998$	0	0
$999$	104.164	3.29561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ce.1.3		3
4.3	odd	2		4600.2.a.x.1.1		3
5.4	even	2		1840.2.a.s.1.1		3
20.3	even	4		4600.2.e.p.4049.1		6
20.7	even	4		4600.2.e.p.4049.6		6
20.19	odd	2		920.2.a.h.1.3	✓	3
40.19	odd	2		7360.2.a.by.1.1		3
40.29	even	2		7360.2.a.cc.1.3		3
60.59	even	2		8280.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.3	✓	3	20.19	odd	2
1840.2.a.s.1.1		3	5.4	even	2
4600.2.a.x.1.1		3	4.3	odd	2
4600.2.e.p.4049.1		6	20.3	even	4
4600.2.e.p.4049.6		6	20.7	even	4
7360.2.a.by.1.1		3	40.19	odd	2
7360.2.a.cc.1.3		3	40.29	even	2
8280.2.a.bj.1.1		3	60.59	even	2
9200.2.a.ce.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+3.07912 q^{3} -2.07912 q^{7} +6.48097 q^{9} -5.07912 q^{11} +3.48097 q^{13} -6.48097 q^{17} +7.48097 q^{19} -6.40185 q^{21} +1.00000 q^{23} +10.7183 q^{27} -1.56009 q^{29} -0.0791189 q^{31} -15.6392 q^{33} +9.71833 q^{37} +10.7183 q^{39} -0.480973 q^{41} +8.00000 q^{43} +6.96195 q^{47} -2.67726 q^{49} -19.9557 q^{51} -11.7183 q^{53} +23.0348 q^{57} +11.5601 q^{59} +7.88283 q^{61} -13.4747 q^{63} -9.71833 q^{67} +3.07912 q^{69} +9.67726 q^{71} +13.2784 q^{73} +10.5601 q^{77} +12.3165 q^{79} +13.5601 q^{81} -4.59815 q^{83} -4.80371 q^{87} +8.31648 q^{89} -7.23736 q^{91} -0.243616 q^{93} -7.23736 q^{97} -32.9176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} - 10 q^{17} + 13 q^{19} - 18 q^{21} + 3 q^{23} - 2 q^{27} + 13 q^{29} + 8 q^{31} - 21 q^{33} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} + 2 q^{47}+ \cdots - 21 q^{99}+O(q^{100}) \) 3 * q + q^3 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 - 10 * q^17 + 13 * q^19 - 18 * q^21 + 3 * q^23 - 2 * q^27 + 13 * q^29 + 8 * q^31 - 21 * q^33 - 5 * q^37 - 2 * q^39 + 8 * q^41 + 24 * q^43 + 2 * q^47 - q^49 - q^51 - q^53 + 2 * q^57 + 17 * q^59 + 13 * q^61 + 9 * q^63 + 5 * q^67 + q^69 + 22 * q^71 - 12 * q^73 + 14 * q^77 + 4 * q^79 + 23 * q^81 - 15 * q^83 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 + 3 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more