LMFDB - Embedded newform 3648.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3648,2,Mod(1,3648)] 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("3648.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,3,0,1,0,2,0,-1] 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:	$29.1294266574$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1824)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3648.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +1.56155 q^{11} +5.12311 q^{13} -3.56155 q^{15} +7.56155 q^{17} +1.00000 q^{19} +1.56155 q^{21} +7.68466 q^{25} -1.00000 q^{27} -5.12311 q^{29} +7.12311 q^{31} -1.56155 q^{33} -5.56155 q^{35} -1.12311 q^{37} -5.12311 q^{39} -1.12311 q^{41} -12.6847 q^{43} +3.56155 q^{45} +5.56155 q^{47} -4.56155 q^{49} -7.56155 q^{51} -8.24621 q^{53} +5.56155 q^{55} -1.00000 q^{57} -0.876894 q^{59} +7.56155 q^{61} -1.56155 q^{63} +18.2462 q^{65} +13.3693 q^{67} +4.87689 q^{71} -14.6847 q^{73} -7.68466 q^{75} -2.43845 q^{77} +7.12311 q^{79} +1.00000 q^{81} -10.2462 q^{83} +26.9309 q^{85} +5.12311 q^{87} +2.00000 q^{89} -8.00000 q^{91} -7.12311 q^{93} +3.56155 q^{95} +13.1231 q^{97} +1.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} + q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{15} + 11 q^{17} + 2 q^{19} - q^{21} + 3 q^{25} - 2 q^{27} - 2 q^{29} + 6 q^{31} + q^{33} - 7 q^{35} + 6 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 + q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^15 + 11 * q^17 + 2 * q^19 - q^21 + 3 * q^25 - 2 * q^27 - 2 * q^29 + 6 * q^31 + q^33 - 7 * q^35 + 6 * q^37 - 2 * q^39 + 6 * q^41 - 13 * q^43 + 3 * q^45 + 7 * q^47 - 5 * q^49 - 11 * q^51 + 7 * q^55 - 2 * q^57 - 10 * q^59 + 11 * q^61 + q^63 + 20 * q^65 + 2 * q^67 + 18 * q^71 - 17 * q^73 - 3 * q^75 - 9 * q^77 + 6 * q^79 + 2 * q^81 - 4 * q^83 + 25 * q^85 + 2 * q^87 + 4 * q^89 - 16 * q^91 - 6 * q^93 + 3 * q^95 + 18 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	−3.56155	−0.919589
$16$	0	0
$17$	7.56155	1.83395	0.916973	−	0.398949i	$-0.130625\pi$
$17$	7.56155	1.83395	0.916973	+	0.398949i	$0.130625\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.12311	−0.951337	−0.475668	−	0.879625i	$-0.657794\pi$
$29$	−5.12311	−0.951337	−0.475668	+	0.879625i	$0.657794\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	−5.56155	−0.940074
$36$	0	0
$37$	−1.12311	−0.184637	−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311	−0.184637	−0.0923187	+	0.995730i	$0.529428\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	−12.6847	−1.93439	−0.967196	−	0.254031i	$-0.918244\pi$
$43$	−12.6847	−1.93439	−0.967196	+	0.254031i	$0.918244\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	5.56155	0.811236	0.405618	−	0.914043i	$-0.367056\pi$
$47$	5.56155	0.811236	0.405618	+	0.914043i	$0.367056\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−7.56155	−1.05883
$52$	0	0
$53$	−8.24621	−1.13270	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	−8.24621	−1.13270	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	5.56155	0.749920
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−0.876894	−0.114162	−0.0570810	−	0.998370i	$-0.518179\pi$
$59$	−0.876894	−0.114162	−0.0570810	+	0.998370i	$0.518179\pi$
$60$	0	0
$61$	7.56155	0.968158	0.484079	−	0.875024i	$-0.339155\pi$
$61$	7.56155	0.968158	0.484079	+	0.875024i	$0.339155\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	18.2462	2.26316
$66$	0	0
$67$	13.3693	1.63332	0.816661	−	0.577118i	$-0.195823\pi$
$67$	13.3693	1.63332	0.816661	+	0.577118i	$0.195823\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.87689	0.578781	0.289390	−	0.957211i	$-0.406547\pi$
$71$	4.87689	0.578781	0.289390	+	0.957211i	$0.406547\pi$
$72$	0	0
$73$	−14.6847	−1.71871	−0.859355	−	0.511380i	$-0.829134\pi$
$73$	−14.6847	−1.71871	−0.859355	+	0.511380i	$0.829134\pi$
$74$	0	0
$75$	−7.68466	−0.887348
$76$	0	0
$77$	−2.43845	−0.277887
$78$	0	0
$79$	7.12311	0.801412	0.400706	−	0.916207i	$-0.368765\pi$
$79$	7.12311	0.801412	0.400706	+	0.916207i	$0.368765\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.2462	−1.12467	−0.562334	−	0.826910i	$-0.690096\pi$
$83$	−10.2462	−1.12467	−0.562334	+	0.826910i	$0.690096\pi$
$84$	0	0
$85$	26.9309	2.92106
$86$	0	0
$87$	5.12311	0.549255
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−7.12311	−0.738632
$94$	0	0
$95$	3.56155	0.365408
$96$	0	0
$97$	13.1231	1.33245	0.666225	−	0.745751i	$-0.267909\pi$
$97$	13.1231	1.33245	0.666225	+	0.745751i	$0.267909\pi$
$98$	0	0
$99$	1.56155	0.156942
$100$	0	0
$101$	−8.24621	−0.820529	−0.410264	−	0.911967i	$-0.634564\pi$
$101$	−8.24621	−0.820529	−0.410264	+	0.911967i	$0.634564\pi$
$102$	0	0
$103$	13.3693	1.31732	0.658659	−	0.752442i	$-0.271124\pi$
$103$	13.3693	1.31732	0.658659	+	0.752442i	$0.271124\pi$
$104$	0	0
$105$	5.56155	0.542752
$106$	0	0
$107$	−0.876894	−0.0847726	−0.0423863	−	0.999101i	$-0.513496\pi$
$107$	−0.876894	−0.0847726	−0.0423863	+	0.999101i	$0.513496\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	1.12311	0.106600
$112$	0	0
$113$	−1.12311	−0.105653	−0.0528264	−	0.998604i	$-0.516823\pi$
$113$	−1.12311	−0.105653	−0.0528264	+	0.998604i	$0.516823\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.12311	0.473631
$118$	0	0
$119$	−11.8078	−1.08242
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	1.12311	0.101267
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−2.24621	−0.199319	−0.0996595	−	0.995022i	$-0.531775\pi$
$127$	−2.24621	−0.199319	−0.0996595	+	0.995022i	$0.531775\pi$
$128$	0	0
$129$	12.6847	1.11682
$130$	0	0
$131$	−22.0540	−1.92686	−0.963432	−	0.267952i	$-0.913653\pi$
$131$	−22.0540	−1.92686	−0.963432	+	0.267952i	$0.913653\pi$
$132$	0	0
$133$	−1.56155	−0.135404
$134$	0	0
$135$	−3.56155	−0.306530
$136$	0	0
$137$	−12.9309	−1.10476	−0.552379	−	0.833593i	$-0.686280\pi$
$137$	−12.9309	−1.10476	−0.552379	+	0.833593i	$0.686280\pi$
$138$	0	0
$139$	−20.6847	−1.75445	−0.877225	−	0.480080i	$-0.840608\pi$
$139$	−20.6847	−1.75445	−0.877225	+	0.480080i	$0.840608\pi$
$140$	0	0
$141$	−5.56155	−0.468367
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−18.2462	−1.51527
$146$	0	0
$147$	4.56155	0.376231
$148$	0	0
$149$	14.6847	1.20301	0.601507	−	0.798867i	$-0.294567\pi$
$149$	14.6847	1.20301	0.601507	+	0.798867i	$0.294567\pi$
$150$	0	0
$151$	−7.12311	−0.579670	−0.289835	−	0.957077i	$-0.593600\pi$
$151$	−7.12311	−0.579670	−0.289835	+	0.957077i	$0.593600\pi$
$152$	0	0
$153$	7.56155	0.611315
$154$	0	0
$155$	25.3693	2.03771
$156$	0	0
$157$	3.75379	0.299585	0.149792	−	0.988717i	$-0.452139\pi$
$157$	3.75379	0.299585	0.149792	+	0.988717i	$0.452139\pi$
$158$	0	0
$159$	8.24621	0.653967
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.4924	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$163$	16.4924	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$164$	0	0
$165$	−5.56155	−0.432966
$166$	0	0
$167$	14.2462	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$167$	14.2462	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	0.876894	0.0659114
$178$	0	0
$179$	18.2462	1.36379	0.681893	−	0.731452i	$-0.261157\pi$
$179$	18.2462	1.36379	0.681893	+	0.731452i	$0.261157\pi$
$180$	0	0
$181$	−10.4924	−0.779896	−0.389948	−	0.920837i	$-0.627507\pi$
$181$	−10.4924	−0.779896	−0.389948	+	0.920837i	$0.627507\pi$
$182$	0	0
$183$	−7.56155	−0.558966
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	11.8078	0.863469
$188$	0	0
$189$	1.56155	0.113586
$190$	0	0
$191$	24.6847	1.78612	0.893060	−	0.449938i	$-0.148554\pi$
$191$	24.6847	1.78612	0.893060	+	0.449938i	$0.148554\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−18.2462	−1.30664
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−9.56155	−0.677801	−0.338900	−	0.940822i	$-0.610055\pi$
$199$	−9.56155	−0.677801	−0.338900	+	0.940822i	$0.610055\pi$
$200$	0	0
$201$	−13.3693	−0.942999
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.56155	0.108015
$210$	0	0
$211$	0.876894	0.0603679	0.0301839	−	0.999544i	$-0.490391\pi$
$211$	0.876894	0.0603679	0.0301839	+	0.999544i	$0.490391\pi$
$212$	0	0
$213$	−4.87689	−0.334159
$214$	0	0
$215$	−45.1771	−3.08105
$216$	0	0
$217$	−11.1231	−0.755086
$218$	0	0
$219$	14.6847	0.992297
$220$	0	0
$221$	38.7386	2.60584
$222$	0	0
$223$	−2.24621	−0.150417	−0.0752087	−	0.997168i	$-0.523962\pi$
$223$	−2.24621	−0.150417	−0.0752087	+	0.997168i	$0.523962\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−16.0540	−1.06088	−0.530438	−	0.847724i	$-0.677973\pi$
$229$	−16.0540	−1.06088	−0.530438	+	0.847724i	$0.677973\pi$
$230$	0	0
$231$	2.43845	0.160438
$232$	0	0
$233$	2.68466	0.175878	0.0879389	−	0.996126i	$-0.471972\pi$
$233$	2.68466	0.175878	0.0879389	+	0.996126i	$0.471972\pi$
$234$	0	0
$235$	19.8078	1.29212
$236$	0	0
$237$	−7.12311	−0.462695
$238$	0	0
$239$	8.68466	0.561764	0.280882	−	0.959742i	$-0.409373\pi$
$239$	8.68466	0.561764	0.280882	+	0.959742i	$0.409373\pi$
$240$	0	0
$241$	13.1231	0.845334	0.422667	−	0.906285i	$-0.361094\pi$
$241$	13.1231	0.845334	0.422667	+	0.906285i	$0.361094\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−16.2462	−1.03793
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	10.2462	0.649327
$250$	0	0
$251$	−22.0540	−1.39203	−0.696017	−	0.718025i	$-0.745046\pi$
$251$	−22.0540	−1.39203	−0.696017	+	0.718025i	$0.745046\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−26.9309	−1.68648
$256$	0	0
$257$	14.8769	0.927995	0.463998	−	0.885836i	$-0.346415\pi$
$257$	14.8769	0.927995	0.463998	+	0.885836i	$0.346415\pi$
$258$	0	0
$259$	1.75379	0.108975
$260$	0	0
$261$	−5.12311	−0.317112
$262$	0	0
$263$	11.8078	0.728098	0.364049	−	0.931380i	$-0.381394\pi$
$263$	11.8078	0.728098	0.364049	+	0.931380i	$0.381394\pi$
$264$	0	0
$265$	−29.3693	−1.80414
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	18.4924	1.12750	0.563751	−	0.825944i	$-0.309358\pi$
$269$	18.4924	1.12750	0.563751	+	0.825944i	$0.309358\pi$
$270$	0	0
$271$	−10.2462	−0.622413	−0.311207	−	0.950342i	$-0.600733\pi$
$271$	−10.2462	−0.622413	−0.311207	+	0.950342i	$0.600733\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	6.19224	0.372055	0.186028	−	0.982545i	$-0.440439\pi$
$277$	6.19224	0.372055	0.186028	+	0.982545i	$0.440439\pi$
$278$	0	0
$279$	7.12311	0.426449
$280$	0	0
$281$	11.7538	0.701172	0.350586	−	0.936530i	$-0.385982\pi$
$281$	11.7538	0.701172	0.350586	+	0.936530i	$0.385982\pi$
$282$	0	0
$283$	1.56155	0.0928247	0.0464123	−	0.998922i	$-0.485221\pi$
$283$	1.56155	0.0928247	0.0464123	+	0.998922i	$0.485221\pi$
$284$	0	0
$285$	−3.56155	−0.210968
$286$	0	0
$287$	1.75379	0.103523
$288$	0	0
$289$	40.1771	2.36336
$290$	0	0
$291$	−13.1231	−0.769290
$292$	0	0
$293$	2.87689	0.168070	0.0840350	−	0.996463i	$-0.473219\pi$
$293$	2.87689	0.168070	0.0840350	+	0.996463i	$0.473219\pi$
$294$	0	0
$295$	−3.12311	−0.181834
$296$	0	0
$297$	−1.56155	−0.0906105
$298$	0	0
$299$	0	0
$300$	0	0
$301$	19.8078	1.14170
$302$	0	0
$303$	8.24621	0.473732
$304$	0	0
$305$	26.9309	1.54206
$306$	0	0
$307$	−33.8617	−1.93259	−0.966296	−	0.257434i	$-0.917123\pi$
$307$	−33.8617	−1.93259	−0.966296	+	0.257434i	$0.917123\pi$
$308$	0	0
$309$	−13.3693	−0.760554
$310$	0	0
$311$	5.56155	0.315367	0.157683	−	0.987490i	$-0.449597\pi$
$311$	5.56155	0.315367	0.157683	+	0.987490i	$0.449597\pi$
$312$	0	0
$313$	0.246211	0.0139167	0.00695834	−	0.999976i	$-0.497785\pi$
$313$	0.246211	0.0139167	0.00695834	+	0.999976i	$0.497785\pi$
$314$	0	0
$315$	−5.56155	−0.313358
$316$	0	0
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0.876894	0.0489435
$322$	0	0
$323$	7.56155	0.420736
$324$	0	0
$325$	39.3693	2.18382
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−8.68466	−0.478801
$330$	0	0
$331$	−19.6155	−1.07817	−0.539083	−	0.842252i	$-0.681229\pi$
$331$	−19.6155	−1.07817	−0.539083	+	0.842252i	$0.681229\pi$
$332$	0	0
$333$	−1.12311	−0.0615458
$334$	0	0
$335$	47.6155	2.60151
$336$	0	0
$337$	−21.6155	−1.17747	−0.588736	−	0.808325i	$-0.700374\pi$
$337$	−21.6155	−1.17747	−0.588736	+	0.808325i	$0.700374\pi$
$338$	0	0
$339$	1.12311	0.0609987
$340$	0	0
$341$	11.1231	0.602350
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.192236	0.0103198	0.00515988	−	0.999987i	$-0.498358\pi$
$347$	0.192236	0.0103198	0.00515988	+	0.999987i	$0.498358\pi$
$348$	0	0
$349$	−13.3153	−0.712754	−0.356377	−	0.934342i	$-0.615988\pi$
$349$	−13.3153	−0.712754	−0.356377	+	0.934342i	$0.615988\pi$
$350$	0	0
$351$	−5.12311	−0.273451
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	17.3693	0.921868
$356$	0	0
$357$	11.8078	0.624933
$358$	0	0
$359$	8.68466	0.458359	0.229179	−	0.973384i	$-0.426396\pi$
$359$	8.68466	0.458359	0.229179	+	0.973384i	$0.426396\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.56155	0.449365
$364$	0	0
$365$	−52.3002	−2.73752
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	12.8769	0.668535
$372$	0	0
$373$	33.6155	1.74055	0.870273	−	0.492570i	$-0.163942\pi$
$373$	33.6155	1.74055	0.870273	+	0.492570i	$0.163942\pi$
$374$	0	0
$375$	−9.56155	−0.493756
$376$	0	0
$377$	−26.2462	−1.35175
$378$	0	0
$379$	−27.6155	−1.41851	−0.709257	−	0.704950i	$-0.750970\pi$
$379$	−27.6155	−1.41851	−0.709257	+	0.704950i	$0.750970\pi$
$380$	0	0
$381$	2.24621	0.115077
$382$	0	0
$383$	−33.3693	−1.70509	−0.852546	−	0.522652i	$-0.824943\pi$
$383$	−33.3693	−1.70509	−0.852546	+	0.522652i	$0.824943\pi$
$384$	0	0
$385$	−8.68466	−0.442611
$386$	0	0
$387$	−12.6847	−0.644797
$388$	0	0
$389$	25.8078	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$389$	25.8078	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	22.0540	1.11248
$394$	0	0
$395$	25.3693	1.27647
$396$	0	0
$397$	2.68466	0.134739	0.0673696	−	0.997728i	$-0.478539\pi$
$397$	2.68466	0.134739	0.0673696	+	0.997728i	$0.478539\pi$
$398$	0	0
$399$	1.56155	0.0781754
$400$	0	0
$401$	14.4924	0.723717	0.361859	−	0.932233i	$-0.382142\pi$
$401$	14.4924	0.723717	0.361859	+	0.932233i	$0.382142\pi$
$402$	0	0
$403$	36.4924	1.81782
$404$	0	0
$405$	3.56155	0.176975
$406$	0	0
$407$	−1.75379	−0.0869321
$408$	0	0
$409$	0.246211	0.0121744	0.00608718	−	0.999981i	$-0.498062\pi$
$409$	0.246211	0.0121744	0.00608718	+	0.999981i	$0.498062\pi$
$410$	0	0
$411$	12.9309	0.637833
$412$	0	0
$413$	1.36932	0.0673797
$414$	0	0
$415$	−36.4924	−1.79134
$416$	0	0
$417$	20.6847	1.01293
$418$	0	0
$419$	36.9848	1.80683	0.903414	−	0.428769i	$-0.141053\pi$
$419$	36.9848	1.80683	0.903414	+	0.428769i	$0.141053\pi$
$420$	0	0
$421$	24.2462	1.18169	0.590844	−	0.806786i	$-0.298795\pi$
$421$	24.2462	1.18169	0.590844	+	0.806786i	$0.298795\pi$
$422$	0	0
$423$	5.56155	0.270412
$424$	0	0
$425$	58.1080	2.81865
$426$	0	0
$427$	−11.8078	−0.571418
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−32.9848	−1.58882	−0.794412	−	0.607379i	$-0.792221\pi$
$431$	−32.9848	−1.58882	−0.794412	+	0.607379i	$0.792221\pi$
$432$	0	0
$433$	−5.61553	−0.269865	−0.134933	−	0.990855i	$-0.543082\pi$
$433$	−5.61553	−0.269865	−0.134933	+	0.990855i	$0.543082\pi$
$434$	0	0
$435$	18.2462	0.874839
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.49242	0.405321	0.202661	−	0.979249i	$-0.435041\pi$
$439$	8.49242	0.405321	0.202661	+	0.979249i	$0.435041\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	12.6847	0.602666	0.301333	−	0.953519i	$-0.402568\pi$
$443$	12.6847	0.602666	0.301333	+	0.953519i	$0.402568\pi$
$444$	0	0
$445$	7.12311	0.337668
$446$	0	0
$447$	−14.6847	−0.694561
$448$	0	0
$449$	3.36932	0.159008	0.0795039	−	0.996835i	$-0.474666\pi$
$449$	3.36932	0.159008	0.0795039	+	0.996835i	$0.474666\pi$
$450$	0	0
$451$	−1.75379	−0.0825827
$452$	0	0
$453$	7.12311	0.334673
$454$	0	0
$455$	−28.4924	−1.33575
$456$	0	0
$457$	15.1771	0.709954	0.354977	−	0.934875i	$-0.384489\pi$
$457$	15.1771	0.709954	0.354977	+	0.934875i	$0.384489\pi$
$458$	0	0
$459$	−7.56155	−0.352943
$460$	0	0
$461$	6.68466	0.311336	0.155668	−	0.987809i	$-0.450247\pi$
$461$	6.68466	0.311336	0.155668	+	0.987809i	$0.450247\pi$
$462$	0	0
$463$	−38.4384	−1.78639	−0.893193	−	0.449673i	$-0.851540\pi$
$463$	−38.4384	−1.78639	−0.893193	+	0.449673i	$0.851540\pi$
$464$	0	0
$465$	−25.3693	−1.17647
$466$	0	0
$467$	−7.80776	−0.361300	−0.180650	−	0.983547i	$-0.557820\pi$
$467$	−7.80776	−0.361300	−0.180650	+	0.983547i	$0.557820\pi$
$468$	0	0
$469$	−20.8769	−0.964005
$470$	0	0
$471$	−3.75379	−0.172965
$472$	0	0
$473$	−19.8078	−0.910762
$474$	0	0
$475$	7.68466	0.352596
$476$	0	0
$477$	−8.24621	−0.377568
$478$	0	0
$479$	36.4924	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$479$	36.4924	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$480$	0	0
$481$	−5.75379	−0.262350
$482$	0	0
$483$	0	0
$484$	0	0
$485$	46.7386	2.12229
$486$	0	0
$487$	−7.12311	−0.322779	−0.161389	−	0.986891i	$-0.551597\pi$
$487$	−7.12311	−0.322779	−0.161389	+	0.986891i	$0.551597\pi$
$488$	0	0
$489$	−16.4924	−0.745813
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−38.7386	−1.74470
$494$	0	0
$495$	5.56155	0.249973
$496$	0	0
$497$	−7.61553	−0.341603
$498$	0	0
$499$	−39.8078	−1.78204	−0.891020	−	0.453964i	$-0.850010\pi$
$499$	−39.8078	−1.78204	−0.891020	+	0.453964i	$0.850010\pi$
$500$	0	0
$501$	−14.2462	−0.636474
$502$	0	0
$503$	12.4924	0.557010	0.278505	−	0.960435i	$-0.410161\pi$
$503$	12.4924	0.557010	0.278505	+	0.960435i	$0.410161\pi$
$504$	0	0
$505$	−29.3693	−1.30692
$506$	0	0
$507$	−13.2462	−0.588285
$508$	0	0
$509$	−33.6155	−1.48998	−0.744991	−	0.667074i	$-0.767546\pi$
$509$	−33.6155	−1.48998	−0.744991	+	0.667074i	$0.767546\pi$
$510$	0	0
$511$	22.9309	1.01440
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	47.6155	2.09819
$516$	0	0
$517$	8.68466	0.381951
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	16.8769	0.737975	0.368988	−	0.929434i	$-0.379705\pi$
$523$	16.8769	0.737975	0.368988	+	0.929434i	$0.379705\pi$
$524$	0	0
$525$	12.0000	0.523723
$526$	0	0
$527$	53.8617	2.34625
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−0.876894	−0.0380540
$532$	0	0
$533$	−5.75379	−0.249224
$534$	0	0
$535$	−3.12311	−0.135024
$536$	0	0
$537$	−18.2462	−0.787382
$538$	0	0
$539$	−7.12311	−0.306814
$540$	0	0
$541$	−35.1771	−1.51238	−0.756190	−	0.654352i	$-0.772942\pi$
$541$	−35.1771	−1.51238	−0.756190	+	0.654352i	$0.772942\pi$
$542$	0	0
$543$	10.4924	0.450273
$544$	0	0
$545$	35.6155	1.52560
$546$	0	0
$547$	−5.75379	−0.246014	−0.123007	−	0.992406i	$-0.539254\pi$
$547$	−5.75379	−0.246014	−0.123007	+	0.992406i	$0.539254\pi$
$548$	0	0
$549$	7.56155	0.322719
$550$	0	0
$551$	−5.12311	−0.218252
$552$	0	0
$553$	−11.1231	−0.473003
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−34.3002	−1.45335	−0.726673	−	0.686984i	$-0.758934\pi$
$557$	−34.3002	−1.45335	−0.726673	+	0.686984i	$0.758934\pi$
$558$	0	0
$559$	−64.9848	−2.74857
$560$	0	0
$561$	−11.8078	−0.498524
$562$	0	0
$563$	26.2462	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$563$	26.2462	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	−1.12311	−0.0470830	−0.0235415	−	0.999723i	$-0.507494\pi$
$569$	−1.12311	−0.0470830	−0.0235415	+	0.999723i	$0.507494\pi$
$570$	0	0
$571$	10.2462	0.428791	0.214395	−	0.976747i	$-0.431222\pi$
$571$	10.2462	0.428791	0.214395	+	0.976747i	$0.431222\pi$
$572$	0	0
$573$	−24.6847	−1.03122
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.1771	1.63096	0.815482	−	0.578783i	$-0.196472\pi$
$577$	39.1771	1.63096	0.815482	+	0.578783i	$0.196472\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−12.8769	−0.533306
$584$	0	0
$585$	18.2462	0.754388
$586$	0	0
$587$	23.4233	0.966783	0.483391	−	0.875404i	$-0.339405\pi$
$587$	23.4233	0.966783	0.483391	+	0.875404i	$0.339405\pi$
$588$	0	0
$589$	7.12311	0.293502
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−45.2311	−1.85742	−0.928708	−	0.370811i	$-0.879080\pi$
$593$	−45.2311	−1.85742	−0.928708	+	0.370811i	$0.879080\pi$
$594$	0	0
$595$	−42.0540	−1.72404
$596$	0	0
$597$	9.56155	0.391328
$598$	0	0
$599$	39.6155	1.61865	0.809323	−	0.587363i	$-0.199834\pi$
$599$	39.6155	1.61865	0.809323	+	0.587363i	$0.199834\pi$
$600$	0	0
$601$	−31.7538	−1.29526	−0.647632	−	0.761953i	$-0.724241\pi$
$601$	−31.7538	−1.29526	−0.647632	+	0.761953i	$0.724241\pi$
$602$	0	0
$603$	13.3693	0.544441
$604$	0	0
$605$	−30.4924	−1.23969
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	28.4924	1.15268
$612$	0	0
$613$	42.6847	1.72402	0.862009	−	0.506894i	$-0.169206\pi$
$613$	42.6847	1.72402	0.862009	+	0.506894i	$0.169206\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	32.5464	1.31027	0.655134	−	0.755512i	$-0.272612\pi$
$617$	32.5464	1.31027	0.655134	+	0.755512i	$0.272612\pi$
$618$	0	0
$619$	−40.4924	−1.62753	−0.813764	−	0.581196i	$-0.802585\pi$
$619$	−40.4924	−1.62753	−0.813764	+	0.581196i	$0.802585\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.12311	−0.125125
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−1.56155	−0.0623624
$628$	0	0
$629$	−8.49242	−0.338615
$630$	0	0
$631$	−14.0540	−0.559480	−0.279740	−	0.960076i	$-0.590248\pi$
$631$	−14.0540	−0.559480	−0.279740	+	0.960076i	$0.590248\pi$
$632$	0	0
$633$	−0.876894	−0.0348534
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−23.3693	−0.925926
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	−15.3693	−0.607052	−0.303526	−	0.952823i	$-0.598164\pi$
$641$	−15.3693	−0.607052	−0.303526	+	0.952823i	$0.598164\pi$
$642$	0	0
$643$	−4.68466	−0.184745	−0.0923724	−	0.995725i	$-0.529445\pi$
$643$	−4.68466	−0.184745	−0.0923724	+	0.995725i	$0.529445\pi$
$644$	0	0
$645$	45.1771	1.77885
$646$	0	0
$647$	24.6847	0.970454	0.485227	−	0.874388i	$-0.338737\pi$
$647$	24.6847	0.970454	0.485227	+	0.874388i	$0.338737\pi$
$648$	0	0
$649$	−1.36932	−0.0537504
$650$	0	0
$651$	11.1231	0.435949
$652$	0	0
$653$	−21.8078	−0.853404	−0.426702	−	0.904392i	$-0.640325\pi$
$653$	−21.8078	−0.853404	−0.426702	+	0.904392i	$0.640325\pi$
$654$	0	0
$655$	−78.5464	−3.06906
$656$	0	0
$657$	−14.6847	−0.572903
$658$	0	0
$659$	−11.6155	−0.452477	−0.226238	−	0.974072i	$-0.572643\pi$
$659$	−11.6155	−0.452477	−0.226238	+	0.974072i	$0.572643\pi$
$660$	0	0
$661$	−26.8769	−1.04539	−0.522695	−	0.852520i	$-0.675073\pi$
$661$	−26.8769	−1.04539	−0.522695	+	0.852520i	$0.675073\pi$
$662$	0	0
$663$	−38.7386	−1.50448
$664$	0	0
$665$	−5.56155	−0.215668
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.24621	0.0868435
$670$	0	0
$671$	11.8078	0.455834
$672$	0	0
$673$	−26.4924	−1.02121	−0.510604	−	0.859816i	$-0.670578\pi$
$673$	−26.4924	−1.02121	−0.510604	+	0.859816i	$0.670578\pi$
$674$	0	0
$675$	−7.68466	−0.295783
$676$	0	0
$677$	−23.8617	−0.917081	−0.458541	−	0.888673i	$-0.651628\pi$
$677$	−23.8617	−0.917081	−0.458541	+	0.888673i	$0.651628\pi$
$678$	0	0
$679$	−20.4924	−0.786427
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	−16.4924	−0.631065	−0.315533	−	0.948915i	$-0.602183\pi$
$683$	−16.4924	−0.631065	−0.315533	+	0.948915i	$0.602183\pi$
$684$	0	0
$685$	−46.0540	−1.75963
$686$	0	0
$687$	16.0540	0.612497
$688$	0	0
$689$	−42.2462	−1.60945
$690$	0	0
$691$	9.17708	0.349113	0.174556	−	0.984647i	$-0.444151\pi$
$691$	9.17708	0.349113	0.174556	+	0.984647i	$0.444151\pi$
$692$	0	0
$693$	−2.43845	−0.0926289
$694$	0	0
$695$	−73.6695	−2.79444
$696$	0	0
$697$	−8.49242	−0.321673
$698$	0	0
$699$	−2.68466	−0.101543
$700$	0	0
$701$	−32.2462	−1.21792	−0.608961	−	0.793200i	$-0.708414\pi$
$701$	−32.2462	−1.21792	−0.608961	+	0.793200i	$0.708414\pi$
$702$	0	0
$703$	−1.12311	−0.0423587
$704$	0	0
$705$	−19.8078	−0.746004
$706$	0	0
$707$	12.8769	0.484285
$708$	0	0
$709$	22.4924	0.844721	0.422360	−	0.906428i	$-0.361202\pi$
$709$	22.4924	0.844721	0.422360	+	0.906428i	$0.361202\pi$
$710$	0	0
$711$	7.12311	0.267137
$712$	0	0
$713$	0	0
$714$	0	0
$715$	28.4924	1.06556
$716$	0	0
$717$	−8.68466	−0.324335
$718$	0	0
$719$	−2.05398	−0.0766004	−0.0383002	−	0.999266i	$-0.512194\pi$
$719$	−2.05398	−0.0766004	−0.0383002	+	0.999266i	$0.512194\pi$
$720$	0	0
$721$	−20.8769	−0.777496
$722$	0	0
$723$	−13.1231	−0.488054
$724$	0	0
$725$	−39.3693	−1.46214
$726$	0	0
$727$	41.5616	1.54143	0.770716	−	0.637178i	$-0.219899\pi$
$727$	41.5616	1.54143	0.770716	+	0.637178i	$0.219899\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−95.9157	−3.54757
$732$	0	0
$733$	24.2462	0.895554	0.447777	−	0.894145i	$-0.352216\pi$
$733$	24.2462	0.895554	0.447777	+	0.894145i	$0.352216\pi$
$734$	0	0
$735$	16.2462	0.599251
$736$	0	0
$737$	20.8769	0.769010
$738$	0	0
$739$	−50.9309	−1.87352	−0.936761	−	0.349969i	$-0.886192\pi$
$739$	−50.9309	−1.87352	−0.936761	+	0.349969i	$0.886192\pi$
$740$	0	0
$741$	−5.12311	−0.188202
$742$	0	0
$743$	21.8617	0.802029	0.401015	−	0.916072i	$-0.368658\pi$
$743$	21.8617	0.802029	0.401015	+	0.916072i	$0.368658\pi$
$744$	0	0
$745$	52.3002	1.91613
$746$	0	0
$747$	−10.2462	−0.374889
$748$	0	0
$749$	1.36932	0.0500337
$750$	0	0
$751$	24.4924	0.893741	0.446871	−	0.894599i	$-0.352538\pi$
$751$	24.4924	0.893741	0.446871	+	0.894599i	$0.352538\pi$
$752$	0	0
$753$	22.0540	0.803692
$754$	0	0
$755$	−25.3693	−0.923284
$756$	0	0
$757$	4.05398	0.147344	0.0736721	−	0.997283i	$-0.476528\pi$
$757$	4.05398	0.147344	0.0736721	+	0.997283i	$0.476528\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.9309	−1.04874	−0.524372	−	0.851490i	$-0.675700\pi$
$761$	−28.9309	−1.04874	−0.524372	+	0.851490i	$0.675700\pi$
$762$	0	0
$763$	−15.6155	−0.565320
$764$	0	0
$765$	26.9309	0.973688
$766$	0	0
$767$	−4.49242	−0.162212
$768$	0	0
$769$	−8.05398	−0.290434	−0.145217	−	0.989400i	$-0.546388\pi$
$769$	−8.05398	−0.290434	−0.145217	+	0.989400i	$0.546388\pi$
$770$	0	0
$771$	−14.8769	−0.535778
$772$	0	0
$773$	−8.24621	−0.296596	−0.148298	−	0.988943i	$-0.547379\pi$
$773$	−8.24621	−0.296596	−0.148298	+	0.988943i	$0.547379\pi$
$774$	0	0
$775$	54.7386	1.96627
$776$	0	0
$777$	−1.75379	−0.0629168
$778$	0	0
$779$	−1.12311	−0.0402394
$780$	0	0
$781$	7.61553	0.272505
$782$	0	0
$783$	5.12311	0.183085
$784$	0	0
$785$	13.3693	0.477171
$786$	0	0
$787$	−15.5076	−0.552785	−0.276393	−	0.961045i	$-0.589139\pi$
$787$	−15.5076	−0.552785	−0.276393	+	0.961045i	$0.589139\pi$
$788$	0	0
$789$	−11.8078	−0.420368
$790$	0	0
$791$	1.75379	0.0623575
$792$	0	0
$793$	38.7386	1.37565
$794$	0	0
$795$	29.3693	1.04162
$796$	0	0
$797$	22.9848	0.814165	0.407082	−	0.913391i	$-0.366546\pi$
$797$	22.9848	0.814165	0.407082	+	0.913391i	$0.366546\pi$
$798$	0	0
$799$	42.0540	1.48776
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−22.9309	−0.809213
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.4924	−0.650964
$808$	0	0
$809$	−21.3153	−0.749408	−0.374704	−	0.927145i	$-0.622256\pi$
$809$	−21.3153	−0.749408	−0.374704	+	0.927145i	$0.622256\pi$
$810$	0	0
$811$	−32.4924	−1.14096	−0.570482	−	0.821310i	$-0.693243\pi$
$811$	−32.4924	−1.14096	−0.570482	+	0.821310i	$0.693243\pi$
$812$	0	0
$813$	10.2462	0.359350
$814$	0	0
$815$	58.7386	2.05752
$816$	0	0
$817$	−12.6847	−0.443780
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−41.3153	−1.44192	−0.720958	−	0.692979i	$-0.756298\pi$
$821$	−41.3153	−1.44192	−0.720958	+	0.692979i	$0.756298\pi$
$822$	0	0
$823$	−30.0540	−1.04762	−0.523808	−	0.851836i	$-0.675489\pi$
$823$	−30.0540	−1.04762	−0.523808	+	0.851836i	$0.675489\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−25.8617	−0.899301	−0.449650	−	0.893205i	$-0.648451\pi$
$827$	−25.8617	−0.899301	−0.449650	+	0.893205i	$0.648451\pi$
$828$	0	0
$829$	−45.6155	−1.58429	−0.792146	−	0.610331i	$-0.791036\pi$
$829$	−45.6155	−1.58429	−0.792146	+	0.610331i	$0.791036\pi$
$830$	0	0
$831$	−6.19224	−0.214806
$832$	0	0
$833$	−34.4924	−1.19509
$834$	0	0
$835$	50.7386	1.75588
$836$	0	0
$837$	−7.12311	−0.246211
$838$	0	0
$839$	−7.61553	−0.262917	−0.131459	−	0.991322i	$-0.541966\pi$
$839$	−7.61553	−0.262917	−0.131459	+	0.991322i	$0.541966\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	−11.7538	−0.404822
$844$	0	0
$845$	47.1771	1.62294
$846$	0	0
$847$	13.3693	0.459375
$848$	0	0
$849$	−1.56155	−0.0535924
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−23.7538	−0.813314	−0.406657	−	0.913581i	$-0.633306\pi$
$853$	−23.7538	−0.813314	−0.406657	+	0.913581i	$0.633306\pi$
$854$	0	0
$855$	3.56155	0.121803
$856$	0	0
$857$	−46.9848	−1.60497	−0.802486	−	0.596671i	$-0.796490\pi$
$857$	−46.9848	−1.60497	−0.802486	+	0.596671i	$0.796490\pi$
$858$	0	0
$859$	15.8078	0.539354	0.269677	−	0.962951i	$-0.413083\pi$
$859$	15.8078	0.539354	0.269677	+	0.962951i	$0.413083\pi$
$860$	0	0
$861$	−1.75379	−0.0597690
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−7.12311	−0.242193
$866$	0	0
$867$	−40.1771	−1.36449
$868$	0	0
$869$	11.1231	0.377326
$870$	0	0
$871$	68.4924	2.32078
$872$	0	0
$873$	13.1231	0.444150
$874$	0	0
$875$	−14.9309	−0.504756
$876$	0	0
$877$	−39.7538	−1.34239	−0.671195	−	0.741281i	$-0.734219\pi$
$877$	−39.7538	−1.34239	−0.671195	+	0.741281i	$0.734219\pi$
$878$	0	0
$879$	−2.87689	−0.0970352
$880$	0	0
$881$	−38.6847	−1.30332	−0.651660	−	0.758512i	$-0.725927\pi$
$881$	−38.6847	−1.30332	−0.651660	+	0.758512i	$0.725927\pi$
$882$	0	0
$883$	17.5616	0.590993	0.295497	−	0.955344i	$-0.404515\pi$
$883$	17.5616	0.590993	0.295497	+	0.955344i	$0.404515\pi$
$884$	0	0
$885$	3.12311	0.104982
$886$	0	0
$887$	12.8769	0.432364	0.216182	−	0.976353i	$-0.430640\pi$
$887$	12.8769	0.432364	0.216182	+	0.976353i	$0.430640\pi$
$888$	0	0
$889$	3.50758	0.117640
$890$	0	0
$891$	1.56155	0.0523140
$892$	0	0
$893$	5.56155	0.186110
$894$	0	0
$895$	64.9848	2.17220
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−36.4924	−1.21709
$900$	0	0
$901$	−62.3542	−2.07732
$902$	0	0
$903$	−19.8078	−0.659161
$904$	0	0
$905$	−37.3693	−1.24220
$906$	0	0
$907$	−2.63068	−0.0873504	−0.0436752	−	0.999046i	$-0.513907\pi$
$907$	−2.63068	−0.0873504	−0.0436752	+	0.999046i	$0.513907\pi$
$908$	0	0
$909$	−8.24621	−0.273510
$910$	0	0
$911$	−13.8617	−0.459260	−0.229630	−	0.973278i	$-0.573752\pi$
$911$	−13.8617	−0.459260	−0.229630	+	0.973278i	$0.573752\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	−26.9309	−0.890307
$916$	0	0
$917$	34.4384	1.13726
$918$	0	0
$919$	−1.26137	−0.0416086	−0.0208043	−	0.999784i	$-0.506623\pi$
$919$	−1.26137	−0.0416086	−0.0208043	+	0.999784i	$0.506623\pi$
$920$	0	0
$921$	33.8617	1.11578
$922$	0	0
$923$	24.9848	0.822386
$924$	0	0
$925$	−8.63068	−0.283775
$926$	0	0
$927$	13.3693	0.439106
$928$	0	0
$929$	19.7538	0.648101	0.324050	−	0.946040i	$-0.394955\pi$
$929$	19.7538	0.648101	0.324050	+	0.946040i	$0.394955\pi$
$930$	0	0
$931$	−4.56155	−0.149499
$932$	0	0
$933$	−5.56155	−0.182077
$934$	0	0
$935$	42.0540	1.37531
$936$	0	0
$937$	1.31534	0.0429703	0.0214852	−	0.999769i	$-0.493161\pi$
$937$	1.31534	0.0429703	0.0214852	+	0.999769i	$0.493161\pi$
$938$	0	0
$939$	−0.246211	−0.00803480
$940$	0	0
$941$	17.5076	0.570731	0.285365	−	0.958419i	$-0.407885\pi$
$941$	17.5076	0.570731	0.285365	+	0.958419i	$0.407885\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.56155	0.180917
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	−75.2311	−2.44210
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	50.9848	1.65156	0.825781	−	0.563992i	$-0.190735\pi$
$953$	50.9848	1.65156	0.825781	+	0.563992i	$0.190735\pi$
$954$	0	0
$955$	87.9157	2.84489
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	20.1922	0.652041
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	−0.876894	−0.0282575
$964$	0	0
$965$	7.12311	0.229301
$966$	0	0
$967$	−55.7235	−1.79195	−0.895973	−	0.444108i	$-0.853521\pi$
$967$	−55.7235	−1.79195	−0.895973	+	0.444108i	$0.853521\pi$
$968$	0	0
$969$	−7.56155	−0.242912
$970$	0	0
$971$	36.9848	1.18690	0.593450	−	0.804871i	$-0.297765\pi$
$971$	36.9848	1.18690	0.593450	+	0.804871i	$0.297765\pi$
$972$	0	0
$973$	32.3002	1.03550
$974$	0	0
$975$	−39.3693	−1.26083
$976$	0	0
$977$	14.4924	0.463654	0.231827	−	0.972757i	$-0.425530\pi$
$977$	14.4924	0.463654	0.231827	+	0.972757i	$0.425530\pi$
$978$	0	0
$979$	3.12311	0.0998149
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−5.26137	−0.167812	−0.0839058	−	0.996474i	$-0.526739\pi$
$983$	−5.26137	−0.167812	−0.0839058	+	0.996474i	$0.526739\pi$
$984$	0	0
$985$	−7.12311	−0.226961
$986$	0	0
$987$	8.68466	0.276436
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.4924	−0.523899	−0.261950	−	0.965082i	$-0.584365\pi$
$991$	−16.4924	−0.523899	−0.261950	+	0.965082i	$0.584365\pi$
$992$	0	0
$993$	19.6155	0.622480
$994$	0	0
$995$	−34.0540	−1.07958
$996$	0	0
$997$	32.9309	1.04293	0.521466	−	0.853272i	$-0.325386\pi$
$997$	32.9309	1.04293	0.521466	+	0.853272i	$0.325386\pi$
$998$	0	0
$999$	1.12311	0.0355335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.a.bp.1.2		2
4.3	odd	2		3648.2.a.bu.1.2		2
8.3	odd	2		1824.2.a.m.1.1	✓	2
8.5	even	2		1824.2.a.q.1.1	yes	2
24.5	odd	2		5472.2.a.bl.1.2		2
24.11	even	2		5472.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.a.m.1.1	✓	2	8.3	odd	2
1824.2.a.q.1.1	yes	2	8.5	even	2
3648.2.a.bp.1.2		2	1.1	even	1	trivial
3648.2.a.bu.1.2		2	4.3	odd	2
5472.2.a.bi.1.2		2	24.11	even	2
5472.2.a.bl.1.2		2	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +1.56155 q^{11} +5.12311 q^{13} -3.56155 q^{15} +7.56155 q^{17} +1.00000 q^{19} +1.56155 q^{21} +7.68466 q^{25} -1.00000 q^{27} -5.12311 q^{29} +7.12311 q^{31} -1.56155 q^{33} -5.56155 q^{35} -1.12311 q^{37} -5.12311 q^{39} -1.12311 q^{41} -12.6847 q^{43} +3.56155 q^{45} +5.56155 q^{47} -4.56155 q^{49} -7.56155 q^{51} -8.24621 q^{53} +5.56155 q^{55} -1.00000 q^{57} -0.876894 q^{59} +7.56155 q^{61} -1.56155 q^{63} +18.2462 q^{65} +13.3693 q^{67} +4.87689 q^{71} -14.6847 q^{73} -7.68466 q^{75} -2.43845 q^{77} +7.12311 q^{79} +1.00000 q^{81} -10.2462 q^{83} +26.9309 q^{85} +5.12311 q^{87} +2.00000 q^{89} -8.00000 q^{91} -7.12311 q^{93} +3.56155 q^{95} +13.1231 q^{97} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} + q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{15} + 11 q^{17} + 2 q^{19} - q^{21} + 3 q^{25} - 2 q^{27} - 2 q^{29} + 6 q^{31} + q^{33} - 7 q^{35} + 6 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 + q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^15 + 11 * q^17 + 2 * q^19 - q^21 + 3 * q^25 - 2 * q^27 - 2 * q^29 + 6 * q^31 + q^33 - 7 * q^35 + 6 * q^37 - 2 * q^39 + 6 * q^41 - 13 * q^43 + 3 * q^45 + 7 * q^47 - 5 * q^49 - 11 * q^51 + 7 * q^55 - 2 * q^57 - 10 * q^59 + 11 * q^61 + q^63 + 20 * q^65 + 2 * q^67 + 18 * q^71 - 17 * q^73 - 3 * q^75 - 9 * q^77 + 6 * q^79 + 2 * q^81 - 4 * q^83 + 25 * q^85 + 2 * q^87 + 4 * q^89 - 16 * q^91 - 6 * q^93 + 3 * q^95 + 18 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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