LMFDB - Embedded newform 3584.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.12836864.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 22x^{2} - 8x - 4 $ x^6 - 10x^4 + 22x^2 - 8*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.36086$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36086 q^{3} -0.711757 q^{5} -1.00000 q^{7} +2.57364 q^{9} +O(q^{10})$	$q-2.36086 q^{3} -0.711757 q^{5} -1.00000 q^{7} +2.57364 q^{9} +2.98785 q^{11} +6.85246 q^{13} +1.68036 q^{15} +3.63038 q^{17} +1.35428 q^{19} +2.36086 q^{21} +1.07739 q^{23} -4.49340 q^{25} +1.00658 q^{27} +9.17376 q^{29} -8.57079 q^{31} -7.05389 q^{33} +0.711757 q^{35} -6.30788 q^{37} -16.1777 q^{39} -4.55764 q^{41} -8.12198 q^{43} -1.83181 q^{45} -0.349902 q^{47} +1.00000 q^{49} -8.57079 q^{51} +2.37638 q^{53} -2.12662 q^{55} -3.19726 q^{57} -4.30257 q^{59} +4.92519 q^{61} -2.57364 q^{63} -4.87728 q^{65} +2.16618 q^{67} -2.54356 q^{69} -3.44999 q^{71} +7.06704 q^{73} +10.6083 q^{75} -2.98785 q^{77} +13.7182 q^{79} -10.0973 q^{81} +14.7807 q^{83} -2.58394 q^{85} -21.6579 q^{87} +8.41460 q^{89} -6.85246 q^{91} +20.2344 q^{93} -0.963918 q^{95} +15.3981 q^{97} +7.68966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^5 - 6 * q^7 + 2 * q^9	$ 6 q + 4 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} + 12 q^{13} + 8 q^{15} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 4 q^{33} - 4 q^{35} + 16 q^{37} + 4 q^{41} - 12 q^{43} + 20 q^{45} - 8 q^{47} + 6 q^{49} - 8 q^{51} + 8 q^{53} + 16 q^{55} - 12 q^{57} + 44 q^{61} - 2 q^{63} - 20 q^{67} + 24 q^{69} + 8 q^{71} - 8 q^{73} + 40 q^{75} + 4 q^{77} + 24 q^{79} - 14 q^{81} + 16 q^{83} + 40 q^{85} - 8 q^{87} + 8 q^{89} - 12 q^{91} + 16 q^{93} - 16 q^{95} + 36 q^{99}+O(q^{100}) $ 6 * q + 4 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 + 12 * q^13 + 8 * q^15 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 4 * q^33 - 4 * q^35 + 16 * q^37 + 4 * q^41 - 12 * q^43 + 20 * q^45 - 8 * q^47 + 6 * q^49 - 8 * q^51 + 8 * q^53 + 16 * q^55 - 12 * q^57 + 44 * q^61 - 2 * q^63 - 20 * q^67 + 24 * q^69 + 8 * q^71 - 8 * q^73 + 40 * q^75 + 4 * q^77 + 24 * q^79 - 14 * q^81 + 16 * q^83 + 40 * q^85 - 8 * q^87 + 8 * q^89 - 12 * q^91 + 16 * q^93 - 16 * q^95 + 36 * 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
$3$	−2.36086	−1.36304	−0.681520	−	0.731799i	$-0.738681\pi$
$3$	−2.36086	−1.36304	−0.681520	+	0.731799i	$0.738681\pi$
$4$	0	0
$5$	−0.711757	−0.318307	−0.159154	−	0.987254i	$-0.550877\pi$
$5$	−0.711757	−0.318307	−0.159154	+	0.987254i	$0.550877\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.57364	0.857880
$10$	0	0
$11$	2.98785	0.900872	0.450436	−	0.892809i	$-0.351269\pi$
$11$	2.98785	0.900872	0.450436	+	0.892809i	$0.351269\pi$
$12$	0	0
$13$	6.85246	1.90053	0.950265	−	0.311442i	$-0.100812\pi$
$13$	6.85246	1.90053	0.950265	+	0.311442i	$0.100812\pi$
$14$	0	0
$15$	1.68036	0.433866
$16$	0	0
$17$	3.63038	0.880495	0.440248	−	0.897876i	$-0.354891\pi$
$17$	3.63038	0.880495	0.440248	+	0.897876i	$0.354891\pi$
$18$	0	0
$19$	1.35428	0.310693	0.155347	−	0.987860i	$-0.450351\pi$
$19$	1.35428	0.310693	0.155347	+	0.987860i	$0.450351\pi$
$20$	0	0
$21$	2.36086	0.515181
$22$	0	0
$23$	1.07739	0.224651	0.112326	−	0.993671i	$-0.464170\pi$
$23$	1.07739	0.224651	0.112326	+	0.993671i	$0.464170\pi$
$24$	0	0
$25$	−4.49340	−0.898680
$26$	0	0
$27$	1.00658	0.193716
$28$	0	0
$29$	9.17376	1.70352	0.851762	−	0.523929i	$-0.175534\pi$
$29$	9.17376	1.70352	0.851762	+	0.523929i	$0.175534\pi$
$30$	0	0
$31$	−8.57079	−1.53936	−0.769680	−	0.638430i	$-0.779584\pi$
$31$	−8.57079	−1.53936	−0.769680	+	0.638430i	$0.779584\pi$
$32$	0	0
$33$	−7.05389	−1.22792
$34$	0	0
$35$	0.711757	0.120309
$36$	0	0
$37$	−6.30788	−1.03701	−0.518505	−	0.855075i	$-0.673511\pi$
$37$	−6.30788	−1.03701	−0.518505	+	0.855075i	$0.673511\pi$
$38$	0	0
$39$	−16.1777	−2.59050
$40$	0	0
$41$	−4.55764	−0.711784	−0.355892	−	0.934527i	$-0.615823\pi$
$41$	−4.55764	−0.711784	−0.355892	+	0.934527i	$0.615823\pi$
$42$	0	0
$43$	−8.12198	−1.23859	−0.619295	−	0.785158i	$-0.712582\pi$
$43$	−8.12198	−1.23859	−0.619295	+	0.785158i	$0.712582\pi$
$44$	0	0
$45$	−1.83181	−0.273069
$46$	0	0
$47$	−0.349902	−0.0510385	−0.0255192	−	0.999674i	$-0.508124\pi$
$47$	−0.349902	−0.0510385	−0.0255192	+	0.999674i	$0.508124\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.57079	−1.20015
$52$	0	0
$53$	2.37638	0.326421	0.163211	−	0.986591i	$-0.447815\pi$
$53$	2.37638	0.326421	0.163211	+	0.986591i	$0.447815\pi$
$54$	0	0
$55$	−2.12662	−0.286754
$56$	0	0
$57$	−3.19726	−0.423487
$58$	0	0
$59$	−4.30257	−0.560148	−0.280074	−	0.959978i	$-0.590359\pi$
$59$	−4.30257	−0.560148	−0.280074	+	0.959978i	$0.590359\pi$
$60$	0	0
$61$	4.92519	0.630606	0.315303	−	0.948991i	$-0.397894\pi$
$61$	4.92519	0.630606	0.315303	+	0.948991i	$0.397894\pi$
$62$	0	0
$63$	−2.57364	−0.324248
$64$	0	0
$65$	−4.87728	−0.604953
$66$	0	0
$67$	2.16618	0.264641	0.132321	−	0.991207i	$-0.457757\pi$
$67$	2.16618	0.264641	0.132321	+	0.991207i	$0.457757\pi$
$68$	0	0
$69$	−2.54356	−0.306209
$70$	0	0
$71$	−3.44999	−0.409439	−0.204719	−	0.978821i	$-0.565628\pi$
$71$	−3.44999	−0.409439	−0.204719	+	0.978821i	$0.565628\pi$
$72$	0	0
$73$	7.06704	0.827135	0.413567	−	0.910474i	$-0.364283\pi$
$73$	7.06704	0.827135	0.413567	+	0.910474i	$0.364283\pi$
$74$	0	0
$75$	10.6083	1.22494
$76$	0	0
$77$	−2.98785	−0.340497
$78$	0	0
$79$	13.7182	1.54342	0.771712	−	0.635973i	$-0.219401\pi$
$79$	13.7182	1.54342	0.771712	+	0.635973i	$0.219401\pi$
$80$	0	0
$81$	−10.0973	−1.12192
$82$	0	0
$83$	14.7807	1.62240	0.811198	−	0.584772i	$-0.198816\pi$
$83$	14.7807	1.62240	0.811198	+	0.584772i	$0.198816\pi$
$84$	0	0
$85$	−2.58394	−0.280268
$86$	0	0
$87$	−21.6579	−2.32197
$88$	0	0
$89$	8.41460	0.891946	0.445973	−	0.895046i	$-0.352858\pi$
$89$	8.41460	0.891946	0.445973	+	0.895046i	$0.352858\pi$
$90$	0	0
$91$	−6.85246	−0.718333
$92$	0	0
$93$	20.2344	2.09821
$94$	0	0
$95$	−0.963918	−0.0988959
$96$	0	0
$97$	15.3981	1.56344	0.781720	−	0.623630i	$-0.214343\pi$
$97$	15.3981	1.56344	0.781720	+	0.623630i	$0.214343\pi$
$98$	0	0
$99$	7.68966	0.772839
$100$	0	0
$101$	11.6052	1.15476	0.577381	−	0.816475i	$-0.304075\pi$
$101$	11.6052	1.15476	0.577381	+	0.816475i	$0.304075\pi$
$102$	0	0
$103$	−14.0581	−1.38519	−0.692594	−	0.721327i	$-0.743532\pi$
$103$	−14.0581	−1.38519	−0.692594	+	0.721327i	$0.743532\pi$
$104$	0	0
$105$	−1.68036	−0.163986
$106$	0	0
$107$	2.39821	0.231844	0.115922	−	0.993258i	$-0.463018\pi$
$107$	2.39821	0.231844	0.115922	+	0.993258i	$0.463018\pi$
$108$	0	0
$109$	−1.28723	−0.123294	−0.0616471	−	0.998098i	$-0.519635\pi$
$109$	−1.28723	−0.123294	−0.0616471	+	0.998098i	$0.519635\pi$
$110$	0	0
$111$	14.8920	1.41349
$112$	0	0
$113$	−4.85411	−0.456636	−0.228318	−	0.973587i	$-0.573323\pi$
$113$	−4.85411	−0.456636	−0.228318	+	0.973587i	$0.573323\pi$
$114$	0	0
$115$	−0.766840	−0.0715082
$116$	0	0
$117$	17.6358	1.63043
$118$	0	0
$119$	−3.63038	−0.332796
$120$	0	0
$121$	−2.07274	−0.188431
$122$	0	0
$123$	10.7599	0.970190
$124$	0	0
$125$	6.75699	0.604364
$126$	0	0
$127$	−3.92442	−0.348236	−0.174118	−	0.984725i	$-0.555707\pi$
$127$	−3.92442	−0.348236	−0.174118	+	0.984725i	$0.555707\pi$
$128$	0	0
$129$	19.1748	1.68825
$130$	0	0
$131$	−13.4518	−1.17529	−0.587646	−	0.809118i	$-0.699945\pi$
$131$	−13.4518	−1.17529	−0.587646	+	0.809118i	$0.699945\pi$
$132$	0	0
$133$	−1.35428	−0.117431
$134$	0	0
$135$	−0.716437	−0.0616611
$136$	0	0
$137$	−18.6239	−1.59115	−0.795575	−	0.605855i	$-0.792831\pi$
$137$	−18.6239	−1.59115	−0.795575	+	0.605855i	$0.792831\pi$
$138$	0	0
$139$	−7.14960	−0.606421	−0.303211	−	0.952924i	$-0.598059\pi$
$139$	−7.14960	−0.606421	−0.303211	+	0.952924i	$0.598059\pi$
$140$	0	0
$141$	0.826069	0.0695675
$142$	0	0
$143$	20.4741	1.71213
$144$	0	0
$145$	−6.52948	−0.542244
$146$	0	0
$147$	−2.36086	−0.194720
$148$	0	0
$149$	0.489493	0.0401008	0.0200504	−	0.999799i	$-0.493617\pi$
$149$	0.489493	0.0401008	0.0200504	+	0.999799i	$0.493617\pi$
$150$	0	0
$151$	22.4722	1.82876	0.914382	−	0.404853i	$-0.132677\pi$
$151$	22.4722	1.82876	0.914382	+	0.404853i	$0.132677\pi$
$152$	0	0
$153$	9.34328	0.755359
$154$	0	0
$155$	6.10032	0.489989
$156$	0	0
$157$	16.2478	1.29672	0.648359	−	0.761335i	$-0.275456\pi$
$157$	16.2478	1.29672	0.648359	+	0.761335i	$0.275456\pi$
$158$	0	0
$159$	−5.61029	−0.444925
$160$	0	0
$161$	−1.07739	−0.0849102
$162$	0	0
$163$	−4.84263	−0.379304	−0.189652	−	0.981851i	$-0.560736\pi$
$163$	−4.84263	−0.379304	−0.189652	+	0.981851i	$0.560736\pi$
$164$	0	0
$165$	5.02065	0.390857
$166$	0	0
$167$	9.49149	0.734473	0.367237	−	0.930128i	$-0.380304\pi$
$167$	9.49149	0.734473	0.367237	+	0.930128i	$0.380304\pi$
$168$	0	0
$169$	33.9562	2.61202
$170$	0	0
$171$	3.48543	0.266537
$172$	0	0
$173$	−7.89792	−0.600467	−0.300234	−	0.953866i	$-0.597065\pi$
$173$	−7.89792	−0.600467	−0.300234	+	0.953866i	$0.597065\pi$
$174$	0	0
$175$	4.49340	0.339669
$176$	0	0
$177$	10.1578	0.763504
$178$	0	0
$179$	−22.5817	−1.68783	−0.843916	−	0.536475i	$-0.819756\pi$
$179$	−22.5817	−1.68783	−0.843916	+	0.536475i	$0.819756\pi$
$180$	0	0
$181$	13.7043	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$181$	13.7043	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$182$	0	0
$183$	−11.6277	−0.859541
$184$	0	0
$185$	4.48968	0.330088
$186$	0	0
$187$	10.8470	0.793213
$188$	0	0
$189$	−1.00658	−0.0732176
$190$	0	0
$191$	19.1266	1.38395	0.691975	−	0.721921i	$-0.256741\pi$
$191$	19.1266	1.38395	0.691975	+	0.721921i	$0.256741\pi$
$192$	0	0
$193$	−8.35271	−0.601241	−0.300620	−	0.953744i	$-0.597194\pi$
$193$	−8.35271	−0.601241	−0.300620	+	0.953744i	$0.597194\pi$
$194$	0	0
$195$	11.5146	0.824575
$196$	0	0
$197$	−1.38203	−0.0984657	−0.0492328	−	0.998787i	$-0.515678\pi$
$197$	−1.38203	−0.0984657	−0.0492328	+	0.998787i	$0.515678\pi$
$198$	0	0
$199$	2.47653	0.175556	0.0877782	−	0.996140i	$-0.472023\pi$
$199$	2.47653	0.175556	0.0877782	+	0.996140i	$0.472023\pi$
$200$	0	0
$201$	−5.11404	−0.360717
$202$	0	0
$203$	−9.17376	−0.643872
$204$	0	0
$205$	3.24393	0.226566
$206$	0	0
$207$	2.77281	0.192724
$208$	0	0
$209$	4.04639	0.279894
$210$	0	0
$211$	9.71501	0.668809	0.334405	−	0.942430i	$-0.391465\pi$
$211$	9.71501	0.668809	0.334405	+	0.942430i	$0.391465\pi$
$212$	0	0
$213$	8.14493	0.558082
$214$	0	0
$215$	5.78087	0.394252
$216$	0	0
$217$	8.57079	0.581823
$218$	0	0
$219$	−16.6843	−1.12742
$220$	0	0
$221$	24.8770	1.67341
$222$	0	0
$223$	2.70540	0.181167	0.0905835	−	0.995889i	$-0.471127\pi$
$223$	2.70540	0.181167	0.0905835	+	0.995889i	$0.471127\pi$
$224$	0	0
$225$	−11.5644	−0.770960
$226$	0	0
$227$	16.4716	1.09326	0.546628	−	0.837375i	$-0.315911\pi$
$227$	16.4716	1.09326	0.546628	+	0.837375i	$0.315911\pi$
$228$	0	0
$229$	22.0757	1.45880	0.729401	−	0.684086i	$-0.239799\pi$
$229$	22.0757	1.45880	0.729401	+	0.684086i	$0.239799\pi$
$230$	0	0
$231$	7.05389	0.464112
$232$	0	0
$233$	−18.2673	−1.19673	−0.598365	−	0.801224i	$-0.704183\pi$
$233$	−18.2673	−1.19673	−0.598365	+	0.801224i	$0.704183\pi$
$234$	0	0
$235$	0.249045	0.0162459
$236$	0	0
$237$	−32.3868	−2.10375
$238$	0	0
$239$	13.4441	0.869627	0.434814	−	0.900520i	$-0.356814\pi$
$239$	13.4441	0.869627	0.434814	+	0.900520i	$0.356814\pi$
$240$	0	0
$241$	8.34893	0.537802	0.268901	−	0.963168i	$-0.413340\pi$
$241$	8.34893	0.537802	0.268901	+	0.963168i	$0.413340\pi$
$242$	0	0
$243$	20.8185	1.33551
$244$	0	0
$245$	−0.711757	−0.0454725
$246$	0	0
$247$	9.28014	0.590481
$248$	0	0
$249$	−34.8952	−2.21139
$250$	0	0
$251$	−5.33731	−0.336888	−0.168444	−	0.985711i	$-0.553874\pi$
$251$	−5.33731	−0.336888	−0.168444	+	0.985711i	$0.553874\pi$
$252$	0	0
$253$	3.21908	0.202382
$254$	0	0
$255$	6.10032	0.382017
$256$	0	0
$257$	−25.4346	−1.58657	−0.793285	−	0.608851i	$-0.791631\pi$
$257$	−25.4346	−1.58657	−0.793285	+	0.608851i	$0.791631\pi$
$258$	0	0
$259$	6.30788	0.391953
$260$	0	0
$261$	23.6099	1.46142
$262$	0	0
$263$	−13.2187	−0.815102	−0.407551	−	0.913182i	$-0.633617\pi$
$263$	−13.2187	−0.815102	−0.407551	+	0.913182i	$0.633617\pi$
$264$	0	0
$265$	−1.69141	−0.103902
$266$	0	0
$267$	−19.8657	−1.21576
$268$	0	0
$269$	−17.5765	−1.07166	−0.535830	−	0.844326i	$-0.680001\pi$
$269$	−17.5765	−1.07166	−0.535830	+	0.844326i	$0.680001\pi$
$270$	0	0
$271$	20.5468	1.24813	0.624064	−	0.781373i	$-0.285480\pi$
$271$	20.5468	1.24813	0.624064	+	0.781373i	$0.285480\pi$
$272$	0	0
$273$	16.1777	0.979117
$274$	0	0
$275$	−13.4256	−0.809596
$276$	0	0
$277$	22.0715	1.32615	0.663073	−	0.748555i	$-0.269252\pi$
$277$	22.0715	1.32615	0.663073	+	0.748555i	$0.269252\pi$
$278$	0	0
$279$	−22.0581	−1.32059
$280$	0	0
$281$	−17.0269	−1.01574	−0.507869	−	0.861434i	$-0.669567\pi$
$281$	−17.0269	−1.01574	−0.507869	+	0.861434i	$0.669567\pi$
$282$	0	0
$283$	−15.0782	−0.896304	−0.448152	−	0.893957i	$-0.647918\pi$
$283$	−15.0782	−0.896304	−0.448152	+	0.893957i	$0.647918\pi$
$284$	0	0
$285$	2.27567	0.134799
$286$	0	0
$287$	4.55764	0.269029
$288$	0	0
$289$	−3.82037	−0.224728
$290$	0	0
$291$	−36.3527	−2.13103
$292$	0	0
$293$	7.28162	0.425397	0.212699	−	0.977118i	$-0.431775\pi$
$293$	7.28162	0.425397	0.212699	+	0.977118i	$0.431775\pi$
$294$	0	0
$295$	3.06239	0.178299
$296$	0	0
$297$	3.00750	0.174513
$298$	0	0
$299$	7.38277	0.426957
$300$	0	0
$301$	8.12198	0.468143
$302$	0	0
$303$	−27.3983	−1.57399
$304$	0	0
$305$	−3.50554	−0.200726
$306$	0	0
$307$	8.84254	0.504670	0.252335	−	0.967640i	$-0.418801\pi$
$307$	8.84254	0.504670	0.252335	+	0.967640i	$0.418801\pi$
$308$	0	0
$309$	33.1892	1.88807
$310$	0	0
$311$	31.9075	1.80931	0.904654	−	0.426147i	$-0.140129\pi$
$311$	31.9075	1.80931	0.904654	+	0.426147i	$0.140129\pi$
$312$	0	0
$313$	9.06377	0.512314	0.256157	−	0.966635i	$-0.417544\pi$
$313$	9.06377	0.512314	0.256157	+	0.966635i	$0.417544\pi$
$314$	0	0
$315$	1.83181	0.103211
$316$	0	0
$317$	−29.5991	−1.66245	−0.831225	−	0.555936i	$-0.812360\pi$
$317$	−29.5991	−1.66245	−0.831225	+	0.555936i	$0.812360\pi$
$318$	0	0
$319$	27.4098	1.53466
$320$	0	0
$321$	−5.66184	−0.316013
$322$	0	0
$323$	4.91654	0.273564
$324$	0	0
$325$	−30.7909	−1.70797
$326$	0	0
$327$	3.03896	0.168055
$328$	0	0
$329$	0.349902	0.0192907
$330$	0	0
$331$	15.4620	0.849868	0.424934	−	0.905224i	$-0.360297\pi$
$331$	15.4620	0.849868	0.424934	+	0.905224i	$0.360297\pi$
$332$	0	0
$333$	−16.2342	−0.889630
$334$	0	0
$335$	−1.54179	−0.0842373
$336$	0	0
$337$	2.21951	0.120904	0.0604522	−	0.998171i	$-0.480746\pi$
$337$	2.21951	0.120904	0.0604522	+	0.998171i	$0.480746\pi$
$338$	0	0
$339$	11.4599	0.622414
$340$	0	0
$341$	−25.6083	−1.38677
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.81040	0.0974685
$346$	0	0
$347$	13.5674	0.728336	0.364168	−	0.931333i	$-0.381353\pi$
$347$	13.5674	0.728336	0.364168	+	0.931333i	$0.381353\pi$
$348$	0	0
$349$	32.4963	1.73949	0.869743	−	0.493504i	$-0.164284\pi$
$349$	32.4963	1.73949	0.869743	+	0.493504i	$0.164284\pi$
$350$	0	0
$351$	6.89752	0.368162
$352$	0	0
$353$	12.1266	0.645435	0.322718	−	0.946495i	$-0.395404\pi$
$353$	12.1266	0.645435	0.322718	+	0.946495i	$0.395404\pi$
$354$	0	0
$355$	2.45556	0.130327
$356$	0	0
$357$	8.57079	0.453614
$358$	0	0
$359$	−14.1927	−0.749060	−0.374530	−	0.927215i	$-0.622196\pi$
$359$	−14.1927	−0.749060	−0.374530	+	0.927215i	$0.622196\pi$
$360$	0	0
$361$	−17.1659	−0.903470
$362$	0	0
$363$	4.89343	0.256838
$364$	0	0
$365$	−5.03002	−0.263283
$366$	0	0
$367$	−25.2817	−1.31969	−0.659847	−	0.751400i	$-0.729379\pi$
$367$	−25.2817	−1.31969	−0.659847	+	0.751400i	$0.729379\pi$
$368$	0	0
$369$	−11.7297	−0.610625
$370$	0	0
$371$	−2.37638	−0.123376
$372$	0	0
$373$	25.1188	1.30060	0.650300	−	0.759677i	$-0.274643\pi$
$373$	25.1188	1.30060	0.650300	+	0.759677i	$0.274643\pi$
$374$	0	0
$375$	−15.9523	−0.823773
$376$	0	0
$377$	62.8628	3.23760
$378$	0	0
$379$	19.4994	1.00162	0.500809	−	0.865558i	$-0.333036\pi$
$379$	19.4994	1.00162	0.500809	+	0.865558i	$0.333036\pi$
$380$	0	0
$381$	9.26498	0.474659
$382$	0	0
$383$	33.8525	1.72978	0.864891	−	0.501960i	$-0.167387\pi$
$383$	33.8525	1.72978	0.864891	+	0.501960i	$0.167387\pi$
$384$	0	0
$385$	2.12662	0.108383
$386$	0	0
$387$	−20.9030	−1.06256
$388$	0	0
$389$	−30.2154	−1.53198	−0.765990	−	0.642852i	$-0.777751\pi$
$389$	−30.2154	−1.53198	−0.765990	+	0.642852i	$0.777751\pi$
$390$	0	0
$391$	3.91133	0.197804
$392$	0	0
$393$	31.7579	1.60197
$394$	0	0
$395$	−9.76405	−0.491283
$396$	0	0
$397$	−9.05698	−0.454557	−0.227278	−	0.973830i	$-0.572983\pi$
$397$	−9.05698	−0.454557	−0.227278	+	0.973830i	$0.572983\pi$
$398$	0	0
$399$	3.19726	0.160063
$400$	0	0
$401$	3.78707	0.189117	0.0945587	−	0.995519i	$-0.469856\pi$
$401$	3.78707	0.189117	0.0945587	+	0.995519i	$0.469856\pi$
$402$	0	0
$403$	−58.7310	−2.92560
$404$	0	0
$405$	7.18682	0.357116
$406$	0	0
$407$	−18.8470	−0.934212
$408$	0	0
$409$	−23.7532	−1.17452	−0.587260	−	0.809398i	$-0.699793\pi$
$409$	−23.7532	−1.17452	−0.587260	+	0.809398i	$0.699793\pi$
$410$	0	0
$411$	43.9684	2.16880
$412$	0	0
$413$	4.30257	0.211716
$414$	0	0
$415$	−10.5203	−0.516420
$416$	0	0
$417$	16.8792	0.826577
$418$	0	0
$419$	30.9658	1.51278	0.756388	−	0.654123i	$-0.226962\pi$
$419$	30.9658	1.51278	0.756388	+	0.654123i	$0.226962\pi$
$420$	0	0
$421$	−15.9523	−0.777467	−0.388733	−	0.921350i	$-0.627087\pi$
$421$	−15.9523	−0.777467	−0.388733	+	0.921350i	$0.627087\pi$
$422$	0	0
$423$	−0.900522	−0.0437849
$424$	0	0
$425$	−16.3127	−0.791284
$426$	0	0
$427$	−4.92519	−0.238347
$428$	0	0
$429$	−48.3365	−2.33371
$430$	0	0
$431$	−9.50425	−0.457804	−0.228902	−	0.973449i	$-0.573514\pi$
$431$	−9.50425	−0.457804	−0.228902	+	0.973449i	$0.573514\pi$
$432$	0	0
$433$	5.41658	0.260304	0.130152	−	0.991494i	$-0.458453\pi$
$433$	5.41658	0.260304	0.130152	+	0.991494i	$0.458453\pi$
$434$	0	0
$435$	15.4152	0.739101
$436$	0	0
$437$	1.45909	0.0697976
$438$	0	0
$439$	29.7082	1.41790	0.708948	−	0.705261i	$-0.249170\pi$
$439$	29.7082	1.41790	0.708948	+	0.705261i	$0.249170\pi$
$440$	0	0
$441$	2.57364	0.122554
$442$	0	0
$443$	−17.0290	−0.809071	−0.404535	−	0.914522i	$-0.632567\pi$
$443$	−17.0290	−0.809071	−0.404535	+	0.914522i	$0.632567\pi$
$444$	0	0
$445$	−5.98915	−0.283913
$446$	0	0
$447$	−1.15562	−0.0546590
$448$	0	0
$449$	41.6242	1.96437	0.982183	−	0.187929i	$-0.0601773\pi$
$449$	41.6242	1.96437	0.982183	+	0.187929i	$0.0601773\pi$
$450$	0	0
$451$	−13.6176	−0.641226
$452$	0	0
$453$	−53.0537	−2.49268
$454$	0	0
$455$	4.87728	0.228651
$456$	0	0
$457$	−29.5834	−1.38385	−0.691925	−	0.721969i	$-0.743237\pi$
$457$	−29.5834	−1.38385	−0.691925	+	0.721969i	$0.743237\pi$
$458$	0	0
$459$	3.65425	0.170566
$460$	0	0
$461$	−0.815920	−0.0380012	−0.0190006	−	0.999819i	$-0.506048\pi$
$461$	−0.815920	−0.0380012	−0.0190006	+	0.999819i	$0.506048\pi$
$462$	0	0
$463$	−22.9633	−1.06719	−0.533597	−	0.845739i	$-0.679160\pi$
$463$	−22.9633	−1.06719	−0.533597	+	0.845739i	$0.679160\pi$
$464$	0	0
$465$	−14.4020	−0.667876
$466$	0	0
$467$	11.5170	0.532945	0.266473	−	0.963842i	$-0.414142\pi$
$467$	11.5170	0.532945	0.266473	+	0.963842i	$0.414142\pi$
$468$	0	0
$469$	−2.16618	−0.100025
$470$	0	0
$471$	−38.3588	−1.76748
$472$	0	0
$473$	−24.2673	−1.11581
$474$	0	0
$475$	−6.08532	−0.279214
$476$	0	0
$477$	6.11595	0.280030
$478$	0	0
$479$	14.9202	0.681719	0.340860	−	0.940114i	$-0.389282\pi$
$479$	14.9202	0.681719	0.340860	+	0.940114i	$0.389282\pi$
$480$	0	0
$481$	−43.2245	−1.97087
$482$	0	0
$483$	2.54356	0.115736
$484$	0	0
$485$	−10.9597	−0.497654
$486$	0	0
$487$	36.1278	1.63711	0.818554	−	0.574429i	$-0.194776\pi$
$487$	36.1278	1.63711	0.818554	+	0.574429i	$0.194776\pi$
$488$	0	0
$489$	11.4327	0.517007
$490$	0	0
$491$	33.6926	1.52052	0.760262	−	0.649616i	$-0.225070\pi$
$491$	33.6926	1.52052	0.760262	+	0.649616i	$0.225070\pi$
$492$	0	0
$493$	33.3042	1.49994
$494$	0	0
$495$	−5.47316	−0.246000
$496$	0	0
$497$	3.44999	0.154753
$498$	0	0
$499$	16.1230	0.721766	0.360883	−	0.932611i	$-0.382475\pi$
$499$	16.1230	0.721766	0.360883	+	0.932611i	$0.382475\pi$
$500$	0	0
$501$	−22.4080	−1.00112
$502$	0	0
$503$	11.3607	0.506549	0.253275	−	0.967394i	$-0.418492\pi$
$503$	11.3607	0.506549	0.253275	+	0.967394i	$0.418492\pi$
$504$	0	0
$505$	−8.26010	−0.367569
$506$	0	0
$507$	−80.1657	−3.56028
$508$	0	0
$509$	36.1459	1.60214	0.801069	−	0.598572i	$-0.204265\pi$
$509$	36.1459	1.60214	0.801069	+	0.598572i	$0.204265\pi$
$510$	0	0
$511$	−7.06704	−0.312628
$512$	0	0
$513$	1.36319	0.0601861
$514$	0	0
$515$	10.0060	0.440916
$516$	0	0
$517$	−1.04546	−0.0459791
$518$	0	0
$519$	18.6458	0.818461
$520$	0	0
$521$	36.6860	1.60724	0.803622	−	0.595140i	$-0.202903\pi$
$521$	36.6860	1.60724	0.803622	+	0.595140i	$0.202903\pi$
$522$	0	0
$523$	32.5558	1.42357	0.711783	−	0.702399i	$-0.247888\pi$
$523$	32.5558	1.42357	0.711783	+	0.702399i	$0.247888\pi$
$524$	0	0
$525$	−10.6083	−0.462983
$526$	0	0
$527$	−31.1152	−1.35540
$528$	0	0
$529$	−21.8392	−0.949532
$530$	0	0
$531$	−11.0733	−0.480539
$532$	0	0
$533$	−31.2310	−1.35277
$534$	0	0
$535$	−1.70695	−0.0737977
$536$	0	0
$537$	53.3120	2.30058
$538$	0	0
$539$	2.98785	0.128696
$540$	0	0
$541$	33.9654	1.46029	0.730144	−	0.683293i	$-0.239453\pi$
$541$	33.9654	1.46029	0.730144	+	0.683293i	$0.239453\pi$
$542$	0	0
$543$	−32.3538	−1.38843
$544$	0	0
$545$	0.916195	0.0392455
$546$	0	0
$547$	2.41390	0.103211	0.0516055	−	0.998668i	$-0.483566\pi$
$547$	2.41390	0.103211	0.0516055	+	0.998668i	$0.483566\pi$
$548$	0	0
$549$	12.6757	0.540984
$550$	0	0
$551$	12.4238	0.529273
$552$	0	0
$553$	−13.7182	−0.583359
$554$	0	0
$555$	−10.5995	−0.449923
$556$	0	0
$557$	11.0466	0.468060	0.234030	−	0.972229i	$-0.424809\pi$
$557$	11.0466	0.468060	0.234030	+	0.972229i	$0.424809\pi$
$558$	0	0
$559$	−55.6555	−2.35398
$560$	0	0
$561$	−25.6083	−1.08118
$562$	0	0
$563$	12.5465	0.528773	0.264387	−	0.964417i	$-0.414830\pi$
$563$	12.5465	0.528773	0.264387	+	0.964417i	$0.414830\pi$
$564$	0	0
$565$	3.45495	0.145351
$566$	0	0
$567$	10.0973	0.424047
$568$	0	0
$569$	−27.7545	−1.16353	−0.581764	−	0.813358i	$-0.697637\pi$
$569$	−27.7545	−1.16353	−0.581764	+	0.813358i	$0.697637\pi$
$570$	0	0
$571$	4.50930	0.188708	0.0943541	−	0.995539i	$-0.469921\pi$
$571$	4.50930	0.188708	0.0943541	+	0.995539i	$0.469921\pi$
$572$	0	0
$573$	−45.1551	−1.88638
$574$	0	0
$575$	−4.84115	−0.201890
$576$	0	0
$577$	−2.11497	−0.0880475	−0.0440238	−	0.999030i	$-0.514018\pi$
$577$	−2.11497	−0.0880475	−0.0440238	+	0.999030i	$0.514018\pi$
$578$	0	0
$579$	19.7195	0.819516
$580$	0	0
$581$	−14.7807	−0.613208
$582$	0	0
$583$	7.10028	0.294063
$584$	0	0
$585$	−12.5524	−0.518977
$586$	0	0
$587$	−44.9688	−1.85606	−0.928030	−	0.372505i	$-0.878499\pi$
$587$	−44.9688	−1.85606	−0.928030	+	0.372505i	$0.878499\pi$
$588$	0	0
$589$	−11.6072	−0.478268
$590$	0	0
$591$	3.26278	0.134213
$592$	0	0
$593$	−24.5463	−1.00800	−0.503998	−	0.863705i	$-0.668138\pi$
$593$	−24.5463	−1.00800	−0.503998	+	0.863705i	$0.668138\pi$
$594$	0	0
$595$	2.58394	0.105931
$596$	0	0
$597$	−5.84672	−0.239290
$598$	0	0
$599$	−30.0946	−1.22963	−0.614817	−	0.788670i	$-0.710770\pi$
$599$	−30.0946	−1.22963	−0.614817	+	0.788670i	$0.710770\pi$
$600$	0	0
$601$	1.02003	0.0416078	0.0208039	−	0.999784i	$-0.493377\pi$
$601$	1.02003	0.0416078	0.0208039	+	0.999784i	$0.493377\pi$
$602$	0	0
$603$	5.57497	0.227030
$604$	0	0
$605$	1.47528	0.0599788
$606$	0	0
$607$	−19.1251	−0.776263	−0.388132	−	0.921604i	$-0.626879\pi$
$607$	−19.1251	−0.776263	−0.388132	+	0.921604i	$0.626879\pi$
$608$	0	0
$609$	21.6579	0.877623
$610$	0	0
$611$	−2.39769	−0.0970002
$612$	0	0
$613$	25.0418	1.01143	0.505715	−	0.862701i	$-0.331229\pi$
$613$	25.0418	1.01143	0.505715	+	0.862701i	$0.331229\pi$
$614$	0	0
$615$	−7.65845	−0.308819
$616$	0	0
$617$	43.1687	1.73790	0.868952	−	0.494896i	$-0.164794\pi$
$617$	43.1687	1.73790	0.868952	+	0.494896i	$0.164794\pi$
$618$	0	0
$619$	12.7180	0.511180	0.255590	−	0.966785i	$-0.417730\pi$
$619$	12.7180	0.511180	0.255590	+	0.966785i	$0.417730\pi$
$620$	0	0
$621$	1.08448	0.0435185
$622$	0	0
$623$	−8.41460	−0.337124
$624$	0	0
$625$	17.6577	0.706307
$626$	0	0
$627$	−9.55294	−0.381508
$628$	0	0
$629$	−22.9000	−0.913082
$630$	0	0
$631$	33.4744	1.33260	0.666298	−	0.745685i	$-0.267878\pi$
$631$	33.4744	1.33260	0.666298	+	0.745685i	$0.267878\pi$
$632$	0	0
$633$	−22.9357	−0.911614
$634$	0	0
$635$	2.79323	0.110846
$636$	0	0
$637$	6.85246	0.271504
$638$	0	0
$639$	−8.87904	−0.351249
$640$	0	0
$641$	−10.3148	−0.407408	−0.203704	−	0.979033i	$-0.565298\pi$
$641$	−10.3148	−0.407408	−0.203704	+	0.979033i	$0.565298\pi$
$642$	0	0
$643$	−20.7404	−0.817920	−0.408960	−	0.912552i	$-0.634108\pi$
$643$	−20.7404	−0.817920	−0.408960	+	0.912552i	$0.634108\pi$
$644$	0	0
$645$	−13.6478	−0.537382
$646$	0	0
$647$	−15.0656	−0.592291	−0.296146	−	0.955143i	$-0.595701\pi$
$647$	−15.0656	−0.592291	−0.296146	+	0.955143i	$0.595701\pi$
$648$	0	0
$649$	−12.8555	−0.504621
$650$	0	0
$651$	−20.2344	−0.793049
$652$	0	0
$653$	21.9568	0.859236	0.429618	−	0.903011i	$-0.358648\pi$
$653$	21.9568	0.859236	0.429618	+	0.903011i	$0.358648\pi$
$654$	0	0
$655$	9.57444	0.374104
$656$	0	0
$657$	18.1880	0.709582
$658$	0	0
$659$	29.8657	1.16340	0.581700	−	0.813403i	$-0.302388\pi$
$659$	29.8657	1.16340	0.581700	+	0.813403i	$0.302388\pi$
$660$	0	0
$661$	−14.5152	−0.564575	−0.282287	−	0.959330i	$-0.591093\pi$
$661$	−14.5152	−0.564575	−0.282287	+	0.959330i	$0.591093\pi$
$662$	0	0
$663$	−58.7310	−2.28092
$664$	0	0
$665$	0.963918	0.0373791
$666$	0	0
$667$	9.88371	0.382699
$668$	0	0
$669$	−6.38706	−0.246938
$670$	0	0
$671$	14.7157	0.568095
$672$	0	0
$673$	16.1998	0.624456	0.312228	−	0.950007i	$-0.398925\pi$
$673$	16.1998	0.624456	0.312228	+	0.950007i	$0.398925\pi$
$674$	0	0
$675$	−4.52295	−0.174088
$676$	0	0
$677$	10.7247	0.412185	0.206092	−	0.978533i	$-0.433925\pi$
$677$	10.7247	0.412185	0.206092	+	0.978533i	$0.433925\pi$
$678$	0	0
$679$	−15.3981	−0.590925
$680$	0	0
$681$	−38.8870	−1.49015
$682$	0	0
$683$	−22.7354	−0.869946	−0.434973	−	0.900443i	$-0.643242\pi$
$683$	−22.7354	−0.869946	−0.434973	+	0.900443i	$0.643242\pi$
$684$	0	0
$685$	13.2557	0.506475
$686$	0	0
$687$	−52.1175	−1.98841
$688$	0	0
$689$	16.2841	0.620373
$690$	0	0
$691$	23.6241	0.898705	0.449352	−	0.893355i	$-0.351655\pi$
$691$	23.6241	0.898705	0.449352	+	0.893355i	$0.351655\pi$
$692$	0	0
$693$	−7.68966	−0.292106
$694$	0	0
$695$	5.08878	0.193028
$696$	0	0
$697$	−16.5459	−0.626722
$698$	0	0
$699$	43.1264	1.63119
$700$	0	0
$701$	−1.12363	−0.0424387	−0.0212194	−	0.999775i	$-0.506755\pi$
$701$	−1.12363	−0.0424387	−0.0212194	+	0.999775i	$0.506755\pi$
$702$	0	0
$703$	−8.54264	−0.322192
$704$	0	0
$705$	−0.587960	−0.0221439
$706$	0	0
$707$	−11.6052	−0.436459
$708$	0	0
$709$	−16.1324	−0.605865	−0.302933	−	0.953012i	$-0.597966\pi$
$709$	−16.1324	−0.605865	−0.302933	+	0.953012i	$0.597966\pi$
$710$	0	0
$711$	35.3058	1.32407
$712$	0	0
$713$	−9.23409	−0.345819
$714$	0	0
$715$	−14.5726	−0.544985
$716$	0	0
$717$	−31.7396	−1.18534
$718$	0	0
$719$	−46.6686	−1.74044	−0.870222	−	0.492660i	$-0.836025\pi$
$719$	−46.6686	−1.74044	−0.870222	+	0.492660i	$0.836025\pi$
$720$	0	0
$721$	14.0581	0.523552
$722$	0	0
$723$	−19.7106	−0.733046
$724$	0	0
$725$	−41.2214	−1.53092
$726$	0	0
$727$	10.4515	0.387625	0.193812	−	0.981039i	$-0.437915\pi$
$727$	10.4515	0.387625	0.193812	+	0.981039i	$0.437915\pi$
$728$	0	0
$729$	−18.8577	−0.698432
$730$	0	0
$731$	−29.4858	−1.09057
$732$	0	0
$733$	−15.5723	−0.575176	−0.287588	−	0.957754i	$-0.592853\pi$
$733$	−15.5723	−0.575176	−0.287588	+	0.957754i	$0.592853\pi$
$734$	0	0
$735$	1.68036	0.0619808
$736$	0	0
$737$	6.47223	0.238408
$738$	0	0
$739$	−44.0571	−1.62067	−0.810334	−	0.585969i	$-0.800714\pi$
$739$	−44.0571	−1.62067	−0.810334	+	0.585969i	$0.800714\pi$
$740$	0	0
$741$	−21.9091	−0.804850
$742$	0	0
$743$	−27.0389	−0.991960	−0.495980	−	0.868334i	$-0.665191\pi$
$743$	−27.0389	−0.991960	−0.495980	+	0.868334i	$0.665191\pi$
$744$	0	0
$745$	−0.348400	−0.0127644
$746$	0	0
$747$	38.0403	1.39182
$748$	0	0
$749$	−2.39821	−0.0876288
$750$	0	0
$751$	0.0575811	0.00210116	0.00105058	−	0.999999i	$-0.499666\pi$
$751$	0.0575811	0.00210116	0.00105058	+	0.999999i	$0.499666\pi$
$752$	0	0
$753$	12.6006	0.459192
$754$	0	0
$755$	−15.9948	−0.582109
$756$	0	0
$757$	−18.1685	−0.660345	−0.330172	−	0.943921i	$-0.607107\pi$
$757$	−18.1685	−0.660345	−0.330172	+	0.943921i	$0.607107\pi$
$758$	0	0
$759$	−7.59979	−0.275855
$760$	0	0
$761$	21.8713	0.792835	0.396418	−	0.918070i	$-0.370253\pi$
$761$	21.8713	0.792835	0.396418	+	0.918070i	$0.370253\pi$
$762$	0	0
$763$	1.28723	0.0466008
$764$	0	0
$765$	−6.65014	−0.240436
$766$	0	0
$767$	−29.4832	−1.06458
$768$	0	0
$769$	−41.3141	−1.48982	−0.744912	−	0.667163i	$-0.767508\pi$
$769$	−41.3141	−1.48982	−0.744912	+	0.667163i	$0.767508\pi$
$770$	0	0
$771$	60.0475	2.16256
$772$	0	0
$773$	−30.2428	−1.08776	−0.543879	−	0.839164i	$-0.683045\pi$
$773$	−30.2428	−1.08776	−0.543879	+	0.839164i	$0.683045\pi$
$774$	0	0
$775$	38.5120	1.38339
$776$	0	0
$777$	−14.8920	−0.534248
$778$	0	0
$779$	−6.17232	−0.221146
$780$	0	0
$781$	−10.3081	−0.368852
$782$	0	0
$783$	9.23409	0.329999
$784$	0	0
$785$	−11.5645	−0.412755
$786$	0	0
$787$	16.3754	0.583721	0.291861	−	0.956461i	$-0.405726\pi$
$787$	16.3754	0.583721	0.291861	+	0.956461i	$0.405726\pi$
$788$	0	0
$789$	31.2075	1.11102
$790$	0	0
$791$	4.85411	0.172592
$792$	0	0
$793$	33.7497	1.19849
$794$	0	0
$795$	3.99316	0.141623
$796$	0	0
$797$	−17.9392	−0.635440	−0.317720	−	0.948185i	$-0.602917\pi$
$797$	−17.9392	−0.635440	−0.317720	+	0.948185i	$0.602917\pi$
$798$	0	0
$799$	−1.27028	−0.0449391
$800$	0	0
$801$	21.6561	0.765182
$802$	0	0
$803$	21.1153	0.745142
$804$	0	0
$805$	0.766840	0.0270275
$806$	0	0
$807$	41.4956	1.46072
$808$	0	0
$809$	33.9919	1.19509	0.597545	−	0.801835i	$-0.296143\pi$
$809$	33.9919	1.19509	0.597545	+	0.801835i	$0.296143\pi$
$810$	0	0
$811$	−17.5862	−0.617534	−0.308767	−	0.951138i	$-0.599916\pi$
$811$	−17.5862	−0.617534	−0.308767	+	0.951138i	$0.599916\pi$
$812$	0	0
$813$	−48.5080	−1.70125
$814$	0	0
$815$	3.44677	0.120735
$816$	0	0
$817$	−10.9994	−0.384821
$818$	0	0
$819$	−17.6358	−0.616243
$820$	0	0
$821$	−36.7270	−1.28178	−0.640891	−	0.767632i	$-0.721435\pi$
$821$	−36.7270	−1.28178	−0.640891	+	0.767632i	$0.721435\pi$
$822$	0	0
$823$	37.5706	1.30963	0.654814	−	0.755790i	$-0.272747\pi$
$823$	37.5706	1.30963	0.654814	+	0.755790i	$0.272747\pi$
$824$	0	0
$825$	31.6960	1.10351
$826$	0	0
$827$	32.4315	1.12775	0.563877	−	0.825859i	$-0.309309\pi$
$827$	32.4315	1.12775	0.563877	+	0.825859i	$0.309309\pi$
$828$	0	0
$829$	−20.3050	−0.705222	−0.352611	−	0.935770i	$-0.614706\pi$
$829$	−20.3050	−0.705222	−0.352611	+	0.935770i	$0.614706\pi$
$830$	0	0
$831$	−52.1075	−1.80759
$832$	0	0
$833$	3.63038	0.125785
$834$	0	0
$835$	−6.75563	−0.233788
$836$	0	0
$837$	−8.62716	−0.298198
$838$	0	0
$839$	−24.6124	−0.849713	−0.424857	−	0.905261i	$-0.639675\pi$
$839$	−24.6124	−0.849713	−0.424857	+	0.905261i	$0.639675\pi$
$840$	0	0
$841$	55.1578	1.90199
$842$	0	0
$843$	40.1980	1.38449
$844$	0	0
$845$	−24.1686	−0.831424
$846$	0	0
$847$	2.07274	0.0712200
$848$	0	0
$849$	35.5974	1.22170
$850$	0	0
$851$	−6.79605	−0.232966
$852$	0	0
$853$	−32.0447	−1.09719	−0.548595	−	0.836088i	$-0.684837\pi$
$853$	−32.0447	−1.09719	−0.548595	+	0.836088i	$0.684837\pi$
$854$	0	0
$855$	−2.48078	−0.0848408
$856$	0	0
$857$	−29.0599	−0.992668	−0.496334	−	0.868132i	$-0.665321\pi$
$857$	−29.0599	−0.992668	−0.496334	+	0.868132i	$0.665321\pi$
$858$	0	0
$859$	−0.261668	−0.00892799	−0.00446399	−	0.999990i	$-0.501421\pi$
$859$	−0.261668	−0.00892799	−0.00446399	+	0.999990i	$0.501421\pi$
$860$	0	0
$861$	−10.7599	−0.366697
$862$	0	0
$863$	17.9690	0.611673	0.305836	−	0.952084i	$-0.401064\pi$
$863$	17.9690	0.611673	0.305836	+	0.952084i	$0.401064\pi$
$864$	0	0
$865$	5.62140	0.191133
$866$	0	0
$867$	9.01935	0.306313
$868$	0	0
$869$	40.9881	1.39043
$870$	0	0
$871$	14.8437	0.502959
$872$	0	0
$873$	39.6291	1.34124
$874$	0	0
$875$	−6.75699	−0.228428
$876$	0	0
$877$	−4.35104	−0.146924	−0.0734621	−	0.997298i	$-0.523405\pi$
$877$	−4.35104	−0.146924	−0.0734621	+	0.997298i	$0.523405\pi$
$878$	0	0
$879$	−17.1909	−0.579833
$880$	0	0
$881$	11.2303	0.378357	0.189178	−	0.981943i	$-0.439418\pi$
$881$	11.2303	0.378357	0.189178	+	0.981943i	$0.439418\pi$
$882$	0	0
$883$	−39.7974	−1.33929	−0.669645	−	0.742681i	$-0.733554\pi$
$883$	−39.7974	−1.33929	−0.669645	+	0.742681i	$0.733554\pi$
$884$	0	0
$885$	−7.22985	−0.243029
$886$	0	0
$887$	−11.8390	−0.397515	−0.198757	−	0.980049i	$-0.563691\pi$
$887$	−11.8390	−0.397515	−0.198757	+	0.980049i	$0.563691\pi$
$888$	0	0
$889$	3.92442	0.131621
$890$	0	0
$891$	−30.1692	−1.01071
$892$	0	0
$893$	−0.473865	−0.0158573
$894$	0	0
$895$	16.0726	0.537249
$896$	0	0
$897$	−17.4297	−0.581959
$898$	0	0
$899$	−78.6264	−2.62234
$900$	0	0
$901$	8.62716	0.287412
$902$	0	0
$903$	−19.1748	−0.638098
$904$	0	0
$905$	−9.75410	−0.324237
$906$	0	0
$907$	−2.14412	−0.0711943	−0.0355972	−	0.999366i	$-0.511333\pi$
$907$	−2.14412	−0.0711943	−0.0355972	+	0.999366i	$0.511333\pi$
$908$	0	0
$909$	29.8677	0.990648
$910$	0	0
$911$	54.1211	1.79311	0.896557	−	0.442929i	$-0.146061\pi$
$911$	54.1211	1.79311	0.896557	+	0.442929i	$0.146061\pi$
$912$	0	0
$913$	44.1626	1.46157
$914$	0	0
$915$	8.27607	0.273598
$916$	0	0
$917$	13.4518	0.444219
$918$	0	0
$919$	47.0290	1.55134	0.775672	−	0.631136i	$-0.217411\pi$
$919$	47.0290	1.55134	0.775672	+	0.631136i	$0.217411\pi$
$920$	0	0
$921$	−20.8760	−0.687886
$922$	0	0
$923$	−23.6409	−0.778151
$924$	0	0
$925$	28.3439	0.931940
$926$	0	0
$927$	−36.1805	−1.18833
$928$	0	0
$929$	−53.9999	−1.77168	−0.885839	−	0.463993i	$-0.846416\pi$
$929$	−53.9999	−1.77168	−0.885839	+	0.463993i	$0.846416\pi$
$930$	0	0
$931$	1.35428	0.0443847
$932$	0	0
$933$	−75.3290	−2.46616
$934$	0	0
$935$	−7.72045	−0.252486
$936$	0	0
$937$	−10.4089	−0.340044	−0.170022	−	0.985440i	$-0.554384\pi$
$937$	−10.4089	−0.340044	−0.170022	+	0.985440i	$0.554384\pi$
$938$	0	0
$939$	−21.3982	−0.698305
$940$	0	0
$941$	−32.6610	−1.06472	−0.532359	−	0.846518i	$-0.678694\pi$
$941$	−32.6610	−1.06472	−0.532359	+	0.846518i	$0.678694\pi$
$942$	0	0
$943$	−4.91036	−0.159903
$944$	0	0
$945$	0.716437	0.0233057
$946$	0	0
$947$	−25.1457	−0.817126	−0.408563	−	0.912730i	$-0.633970\pi$
$947$	−25.1457	−0.817126	−0.408563	+	0.912730i	$0.633970\pi$
$948$	0	0
$949$	48.4266	1.57199
$950$	0	0
$951$	69.8792	2.26599
$952$	0	0
$953$	−43.5674	−1.41129	−0.705643	−	0.708567i	$-0.749342\pi$
$953$	−43.5674	−1.41129	−0.705643	+	0.708567i	$0.749342\pi$
$954$	0	0
$955$	−13.6135	−0.440522
$956$	0	0
$957$	−64.7107	−2.09180
$958$	0	0
$959$	18.6239	0.601398
$960$	0	0
$961$	42.4585	1.36963
$962$	0	0
$963$	6.17214	0.198894
$964$	0	0
$965$	5.94510	0.191379
$966$	0	0
$967$	−8.97147	−0.288503	−0.144252	−	0.989541i	$-0.546077\pi$
$967$	−8.97147	−0.288503	−0.144252	+	0.989541i	$0.546077\pi$
$968$	0	0
$969$	−11.6072	−0.372879
$970$	0	0
$971$	53.5414	1.71822	0.859112	−	0.511787i	$-0.171016\pi$
$971$	53.5414	1.71822	0.859112	+	0.511787i	$0.171016\pi$
$972$	0	0
$973$	7.14960	0.229206
$974$	0	0
$975$	72.6928	2.32803
$976$	0	0
$977$	37.3412	1.19465	0.597326	−	0.801998i	$-0.296230\pi$
$977$	37.3412	1.19465	0.597326	+	0.801998i	$0.296230\pi$
$978$	0	0
$979$	25.1416	0.803529
$980$	0	0
$981$	−3.31286	−0.105772
$982$	0	0
$983$	36.2225	1.15532	0.577659	−	0.816278i	$-0.303966\pi$
$983$	36.2225	1.15532	0.577659	+	0.816278i	$0.303966\pi$
$984$	0	0
$985$	0.983671	0.0313424
$986$	0	0
$987$	−0.826069	−0.0262941
$988$	0	0
$989$	−8.75054	−0.278251
$990$	0	0
$991$	11.4960	0.365182	0.182591	−	0.983189i	$-0.441552\pi$
$991$	11.4960	0.365182	0.182591	+	0.983189i	$0.441552\pi$
$992$	0	0
$993$	−36.5035	−1.15840
$994$	0	0
$995$	−1.76268	−0.0558809
$996$	0	0
$997$	20.5111	0.649594	0.324797	−	0.945784i	$-0.394704\pi$
$997$	20.5111	0.649594	0.324797	+	0.945784i	$0.394704\pi$
$998$	0	0
$999$	−6.34937	−0.200885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.i.1.1	yes	6
4.3	odd	2		3584.2.a.j.1.6	yes	6
8.3	odd	2		3584.2.a.h.1.1	yes	6
8.5	even	2		3584.2.a.g.1.6	✓	6
16.3	odd	4		3584.2.b.j.1793.11		12
16.5	even	4		3584.2.b.l.1793.11		12
16.11	odd	4		3584.2.b.j.1793.2		12
16.13	even	4		3584.2.b.l.1793.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.g.1.6	✓	6	8.5	even	2
3584.2.a.h.1.1	yes	6	8.3	odd	2
3584.2.a.i.1.1	yes	6	1.1	even	1	trivial
3584.2.a.j.1.6	yes	6	4.3	odd	2
3584.2.b.j.1793.2		12	16.11	odd	4
3584.2.b.j.1793.11		12	16.3	odd	4
3584.2.b.l.1793.2		12	16.13	even	4
3584.2.b.l.1793.11		12	16.5	even	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36086 q^{3} -0.711757 q^{5} -1.00000 q^{7} +2.57364 q^{9} +O(q^{10})\)	\(q-2.36086 q^{3} -0.711757 q^{5} -1.00000 q^{7} +2.57364 q^{9} +2.98785 q^{11} +6.85246 q^{13} +1.68036 q^{15} +3.63038 q^{17} +1.35428 q^{19} +2.36086 q^{21} +1.07739 q^{23} -4.49340 q^{25} +1.00658 q^{27} +9.17376 q^{29} -8.57079 q^{31} -7.05389 q^{33} +0.711757 q^{35} -6.30788 q^{37} -16.1777 q^{39} -4.55764 q^{41} -8.12198 q^{43} -1.83181 q^{45} -0.349902 q^{47} +1.00000 q^{49} -8.57079 q^{51} +2.37638 q^{53} -2.12662 q^{55} -3.19726 q^{57} -4.30257 q^{59} +4.92519 q^{61} -2.57364 q^{63} -4.87728 q^{65} +2.16618 q^{67} -2.54356 q^{69} -3.44999 q^{71} +7.06704 q^{73} +10.6083 q^{75} -2.98785 q^{77} +13.7182 q^{79} -10.0973 q^{81} +14.7807 q^{83} -2.58394 q^{85} -21.6579 q^{87} +8.41460 q^{89} -6.85246 q^{91} +20.2344 q^{93} -0.963918 q^{95} +15.3981 q^{97} +7.68966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^5 - 6 * q^7 + 2 * q^9	\( 6 q + 4 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} + 12 q^{13} + 8 q^{15} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 4 q^{33} - 4 q^{35} + 16 q^{37} + 4 q^{41} - 12 q^{43} + 20 q^{45} - 8 q^{47} + 6 q^{49} - 8 q^{51} + 8 q^{53} + 16 q^{55} - 12 q^{57} + 44 q^{61} - 2 q^{63} - 20 q^{67} + 24 q^{69} + 8 q^{71} - 8 q^{73} + 40 q^{75} + 4 q^{77} + 24 q^{79} - 14 q^{81} + 16 q^{83} + 40 q^{85} - 8 q^{87} + 8 q^{89} - 12 q^{91} + 16 q^{93} - 16 q^{95} + 36 q^{99}+O(q^{100}) \) 6 * q + 4 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 + 12 * q^13 + 8 * q^15 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 4 * q^33 - 4 * q^35 + 16 * q^37 + 4 * q^41 - 12 * q^43 + 20 * q^45 - 8 * q^47 + 6 * q^49 - 8 * q^51 + 8 * q^53 + 16 * q^55 - 12 * q^57 + 44 * q^61 - 2 * q^63 - 20 * q^67 + 24 * q^69 + 8 * q^71 - 8 * q^73 + 40 * q^75 + 4 * q^77 + 24 * q^79 - 14 * q^81 + 16 * q^83 + 40 * q^85 - 8 * q^87 + 8 * q^89 - 12 * q^91 + 16 * q^93 - 16 * q^95 + 36 * q^99

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