LMFDB - Embedded newform 8550.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.24482 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -4.24482 q^{7} -1.00000 q^{8} +1.42801 q^{11} +6.91764 q^{13} +4.24482 q^{14} +1.00000 q^{16} +5.10083 q^{17} +1.00000 q^{19} -1.42801 q^{22} -3.67282 q^{23} -6.91764 q^{26} -4.24482 q^{28} +8.10083 q^{29} -1.28797 q^{31} -1.00000 q^{32} -5.10083 q^{34} +0.856013 q^{37} -1.00000 q^{38} -8.01847 q^{41} -3.57199 q^{43} +1.42801 q^{44} +3.67282 q^{46} +3.81681 q^{47} +11.0185 q^{49} +6.91764 q^{52} -9.06163 q^{53} +4.24482 q^{56} -8.10083 q^{58} +12.3496 q^{59} +8.20561 q^{61} +1.28797 q^{62} +1.00000 q^{64} -4.38880 q^{67} +5.10083 q^{68} +11.1008 q^{71} -5.38485 q^{73} -0.856013 q^{74} +1.00000 q^{76} -6.06163 q^{77} +2.14399 q^{79} +8.01847 q^{82} -1.04316 q^{83} +3.57199 q^{86} -1.42801 q^{88} +16.7305 q^{89} -29.3641 q^{91} -3.67282 q^{92} -3.81681 q^{94} -6.81286 q^{97} -11.0185 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 5 q^{11} + 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 5 q^{22} - q^{23} - 2 q^{28} + 15 q^{29} - q^{31} - 3 q^{32} - 6 q^{34} + 4 q^{37} - 3 q^{38} + 6 q^{41} - 10 q^{43} + 5 q^{44} + q^{46} + 3 q^{49} - 5 q^{53} + 2 q^{56} - 15 q^{58} + 12 q^{59} + q^{61} + q^{62} + 3 q^{64} - q^{67} + 6 q^{68} + 24 q^{71} - 9 q^{73} - 4 q^{74} + 3 q^{76} + 4 q^{77} + 5 q^{79} - 6 q^{82} - 11 q^{83} + 10 q^{86} - 5 q^{88} + 23 q^{89} - 38 q^{91} - q^{92} - 14 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 5 * q^11 + 2 * q^14 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 5 * q^22 - q^23 - 2 * q^28 + 15 * q^29 - q^31 - 3 * q^32 - 6 * q^34 + 4 * q^37 - 3 * q^38 + 6 * q^41 - 10 * q^43 + 5 * q^44 + q^46 + 3 * q^49 - 5 * q^53 + 2 * q^56 - 15 * q^58 + 12 * q^59 + q^61 + q^62 + 3 * q^64 - q^67 + 6 * q^68 + 24 * q^71 - 9 * q^73 - 4 * q^74 + 3 * q^76 + 4 * q^77 + 5 * q^79 - 6 * q^82 - 11 * q^83 + 10 * q^86 - 5 * q^88 + 23 * q^89 - 38 * q^91 - q^92 - 14 * q^97 - 3 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.24482	−1.60439	−0.802195	−	0.597062i	$-0.796335\pi$
$7$	−4.24482	−1.60439	−0.802195	+	0.597062i	$0.796335\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.42801	0.430560	0.215280	−	0.976552i	$-0.430934\pi$
$11$	1.42801	0.430560	0.215280	+	0.976552i	$0.430934\pi$
$12$	0	0
$13$	6.91764	1.91861	0.959304	−	0.282375i	$-0.0911222\pi$
$13$	6.91764	1.91861	0.959304	+	0.282375i	$0.0911222\pi$
$14$	4.24482	1.13448
$15$	0	0
$16$	1.00000	0.250000
$17$	5.10083	1.23713	0.618567	−	0.785732i	$-0.287714\pi$
$17$	5.10083	1.23713	0.618567	+	0.785732i	$0.287714\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−1.42801	−0.304452
$23$	−3.67282	−0.765837	−0.382918	−	0.923782i	$-0.625081\pi$
$23$	−3.67282	−0.765837	−0.382918	+	0.923782i	$0.625081\pi$
$24$	0	0
$25$	0	0
$26$	−6.91764	−1.35666
$27$	0	0
$28$	−4.24482	−0.802195
$29$	8.10083	1.50429	0.752143	−	0.659000i	$-0.229020\pi$
$29$	8.10083	1.50429	0.752143	+	0.659000i	$0.229020\pi$
$30$	0	0
$31$	−1.28797	−0.231327	−0.115663	−	0.993288i	$-0.536899\pi$
$31$	−1.28797	−0.231327	−0.115663	+	0.993288i	$0.536899\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.10083	−0.874785
$35$	0	0
$36$	0	0
$37$	0.856013	0.140728	0.0703639	−	0.997521i	$-0.477584\pi$
$37$	0.856013	0.140728	0.0703639	+	0.997521i	$0.477584\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−8.01847	−1.25227	−0.626137	−	0.779713i	$-0.715365\pi$
$41$	−8.01847	−1.25227	−0.626137	+	0.779713i	$0.715365\pi$
$42$	0	0
$43$	−3.57199	−0.544724	−0.272362	−	0.962195i	$-0.587805\pi$
$43$	−3.57199	−0.544724	−0.272362	+	0.962195i	$0.587805\pi$
$44$	1.42801	0.215280
$45$	0	0
$46$	3.67282	0.541528
$47$	3.81681	0.556739	0.278369	−	0.960474i	$-0.410206\pi$
$47$	3.81681	0.556739	0.278369	+	0.960474i	$0.410206\pi$
$48$	0	0
$49$	11.0185	1.57407
$50$	0	0
$51$	0	0
$52$	6.91764	0.959304
$53$	−9.06163	−1.24471	−0.622355	−	0.782735i	$-0.713824\pi$
$53$	−9.06163	−1.24471	−0.622355	+	0.782735i	$0.713824\pi$
$54$	0	0
$55$	0	0
$56$	4.24482	0.567238
$57$	0	0
$58$	−8.10083	−1.06369
$59$	12.3496	1.60778	0.803891	−	0.594777i	$-0.202760\pi$
$59$	12.3496	1.60778	0.803891	+	0.594777i	$0.202760\pi$
$60$	0	0
$61$	8.20561	1.05062	0.525311	−	0.850911i	$-0.323949\pi$
$61$	8.20561	1.05062	0.525311	+	0.850911i	$0.323949\pi$
$62$	1.28797	0.163573
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.38880	−0.536178	−0.268089	−	0.963394i	$-0.586392\pi$
$67$	−4.38880	−0.536178	−0.268089	+	0.963394i	$0.586392\pi$
$68$	5.10083	0.618567
$69$	0	0
$70$	0	0
$71$	11.1008	1.31743	0.658713	−	0.752394i	$-0.271101\pi$
$71$	11.1008	1.31743	0.658713	+	0.752394i	$0.271101\pi$
$72$	0	0
$73$	−5.38485	−0.630249	−0.315125	−	0.949050i	$-0.602046\pi$
$73$	−5.38485	−0.630249	−0.315125	+	0.949050i	$0.602046\pi$
$74$	−0.856013	−0.0995095
$75$	0	0
$76$	1.00000	0.114708
$77$	−6.06163	−0.690787
$78$	0	0
$79$	2.14399	0.241217	0.120609	−	0.992700i	$-0.461515\pi$
$79$	2.14399	0.241217	0.120609	+	0.992700i	$0.461515\pi$
$80$	0	0
$81$	0	0
$82$	8.01847	0.885492
$83$	−1.04316	−0.114501	−0.0572506	−	0.998360i	$-0.518233\pi$
$83$	−1.04316	−0.114501	−0.0572506	+	0.998360i	$0.518233\pi$
$84$	0	0
$85$	0	0
$86$	3.57199	0.385178
$87$	0	0
$88$	−1.42801	−0.152226
$89$	16.7305	1.77343	0.886715	−	0.462317i	$-0.152982\pi$
$89$	16.7305	1.77343	0.886715	+	0.462317i	$0.152982\pi$
$90$	0	0
$91$	−29.3641	−3.07820
$92$	−3.67282	−0.382918
$93$	0	0
$94$	−3.81681	−0.393674
$95$	0	0
$96$	0	0
$97$	−6.81286	−0.691741	−0.345870	−	0.938282i	$-0.612416\pi$
$97$	−6.81286	−0.691741	−0.345870	+	0.938282i	$0.612416\pi$
$98$	−11.0185	−1.11303
$99$	0	0
$100$	0	0
$101$	11.7737	1.17152	0.585761	−	0.810484i	$-0.300796\pi$
$101$	11.7737	1.17152	0.585761	+	0.810484i	$0.300796\pi$
$102$	0	0
$103$	−4.32718	−0.426369	−0.213185	−	0.977012i	$-0.568384\pi$
$103$	−4.32718	−0.426369	−0.213185	+	0.977012i	$0.568384\pi$
$104$	−6.91764	−0.678330
$105$	0	0
$106$	9.06163	0.880143
$107$	4.14003	0.400232	0.200116	−	0.979772i	$-0.435868\pi$
$107$	4.14003	0.400232	0.200116	+	0.979772i	$0.435868\pi$
$108$	0	0
$109$	−2.48963	−0.238464	−0.119232	−	0.992866i	$-0.538043\pi$
$109$	−2.48963	−0.238464	−0.119232	+	0.992866i	$0.538043\pi$
$110$	0	0
$111$	0	0
$112$	−4.24482	−0.401098
$113$	−11.0185	−1.03653	−0.518265	−	0.855220i	$-0.673422\pi$
$113$	−11.0185	−1.03653	−0.518265	+	0.855220i	$0.673422\pi$
$114$	0	0
$115$	0	0
$116$	8.10083	0.752143
$117$	0	0
$118$	−12.3496	−1.13687
$119$	−21.6521	−1.98484
$120$	0	0
$121$	−8.96080	−0.814618
$122$	−8.20561	−0.742901
$123$	0	0
$124$	−1.28797	−0.115663
$125$	0	0
$126$	0	0
$127$	−6.05767	−0.537532	−0.268766	−	0.963206i	$-0.586616\pi$
$127$	−6.05767	−0.537532	−0.268766	+	0.963206i	$0.586616\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−9.44648	−0.825343	−0.412671	−	0.910880i	$-0.635404\pi$
$131$	−9.44648	−0.825343	−0.412671	+	0.910880i	$0.635404\pi$
$132$	0	0
$133$	−4.24482	−0.368072
$134$	4.38880	0.379135
$135$	0	0
$136$	−5.10083	−0.437393
$137$	−4.20166	−0.358972	−0.179486	−	0.983761i	$-0.557443\pi$
$137$	−4.20166	−0.358972	−0.179486	+	0.983761i	$0.557443\pi$
$138$	0	0
$139$	−5.40727	−0.458639	−0.229320	−	0.973351i	$-0.573650\pi$
$139$	−5.40727	−0.458639	−0.229320	+	0.973351i	$0.573650\pi$
$140$	0	0
$141$	0	0
$142$	−11.1008	−0.931561
$143$	9.87844	0.826076
$144$	0	0
$145$	0	0
$146$	5.38485	0.445653
$147$	0	0
$148$	0.856013	0.0703639
$149$	1.51037	0.123734	0.0618670	−	0.998084i	$-0.480295\pi$
$149$	1.51037	0.123734	0.0618670	+	0.998084i	$0.480295\pi$
$150$	0	0
$151$	−15.4504	−1.25734	−0.628669	−	0.777673i	$-0.716400\pi$
$151$	−15.4504	−1.25734	−0.628669	+	0.777673i	$0.716400\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	6.06163	0.488460
$155$	0	0
$156$	0	0
$157$	9.16246	0.731244	0.365622	−	0.930763i	$-0.380856\pi$
$157$	9.16246	0.731244	0.365622	+	0.930763i	$0.380856\pi$
$158$	−2.14399	−0.170566
$159$	0	0
$160$	0	0
$161$	15.5905	1.22870
$162$	0	0
$163$	12.6913	0.994059	0.497029	−	0.867734i	$-0.334424\pi$
$163$	12.6913	0.994059	0.497029	+	0.867734i	$0.334424\pi$
$164$	−8.01847	−0.626137
$165$	0	0
$166$	1.04316	0.0809646
$167$	−16.3249	−1.26326	−0.631630	−	0.775270i	$-0.717614\pi$
$167$	−16.3249	−1.26326	−0.631630	+	0.775270i	$0.717614\pi$
$168$	0	0
$169$	34.8538	2.68106
$170$	0	0
$171$	0	0
$172$	−3.57199	−0.272362
$173$	−3.32322	−0.252660	−0.126330	−	0.991988i	$-0.540320\pi$
$173$	−3.32322	−0.252660	−0.126330	+	0.991988i	$0.540320\pi$
$174$	0	0
$175$	0	0
$176$	1.42801	0.107640
$177$	0	0
$178$	−16.7305	−1.25400
$179$	−5.54731	−0.414625	−0.207313	−	0.978275i	$-0.566472\pi$
$179$	−5.54731	−0.414625	−0.207313	+	0.978275i	$0.566472\pi$
$180$	0	0
$181$	−17.2840	−1.28471	−0.642356	−	0.766407i	$-0.722043\pi$
$181$	−17.2840	−1.28471	−0.642356	+	0.766407i	$0.722043\pi$
$182$	29.3641	2.17661
$183$	0	0
$184$	3.67282	0.270764
$185$	0	0
$186$	0	0
$187$	7.28402	0.532660
$188$	3.81681	0.278369
$189$	0	0
$190$	0	0
$191$	22.6336	1.63771	0.818856	−	0.573999i	$-0.194609\pi$
$191$	22.6336	1.63771	0.818856	+	0.573999i	$0.194609\pi$
$192$	0	0
$193$	−4.24482	−0.305549	−0.152774	−	0.988261i	$-0.548821\pi$
$193$	−4.24482	−0.305549	−0.152774	+	0.988261i	$0.548821\pi$
$194$	6.81286	0.489135
$195$	0	0
$196$	11.0185	0.787034
$197$	5.00395	0.356517	0.178258	−	0.983984i	$-0.442954\pi$
$197$	5.00395	0.356517	0.178258	+	0.983984i	$0.442954\pi$
$198$	0	0
$199$	−27.3826	−1.94110	−0.970550	−	0.240899i	$-0.922558\pi$
$199$	−27.3826	−1.94110	−0.970550	+	0.240899i	$0.922558\pi$
$200$	0	0
$201$	0	0
$202$	−11.7737	−0.828391
$203$	−34.3865	−2.41346
$204$	0	0
$205$	0	0
$206$	4.32718	0.301489
$207$	0	0
$208$	6.91764	0.479652
$209$	1.42801	0.0987773
$210$	0	0
$211$	−5.14003	−0.353855	−0.176927	−	0.984224i	$-0.556616\pi$
$211$	−5.14003	−0.353855	−0.176927	+	0.984224i	$0.556616\pi$
$212$	−9.06163	−0.622355
$213$	0	0
$214$	−4.14003	−0.283007
$215$	0	0
$216$	0	0
$217$	5.46721	0.371138
$218$	2.48963	0.168619
$219$	0	0
$220$	0	0
$221$	35.2857	2.37357
$222$	0	0
$223$	−10.1625	−0.680528	−0.340264	−	0.940330i	$-0.610517\pi$
$223$	−10.1625	−0.680528	−0.340264	+	0.940330i	$0.610517\pi$
$224$	4.24482	0.283619
$225$	0	0
$226$	11.0185	0.732938
$227$	7.92159	0.525775	0.262887	−	0.964827i	$-0.415325\pi$
$227$	7.92159	0.525775	0.262887	+	0.964827i	$0.415325\pi$
$228$	0	0
$229$	−7.44648	−0.492077	−0.246039	−	0.969260i	$-0.579129\pi$
$229$	−7.44648	−0.492077	−0.246039	+	0.969260i	$0.579129\pi$
$230$	0	0
$231$	0	0
$232$	−8.10083	−0.531846
$233$	28.4834	1.86601	0.933005	−	0.359862i	$-0.117176\pi$
$233$	28.4834	1.86601	0.933005	+	0.359862i	$0.117176\pi$
$234$	0	0
$235$	0	0
$236$	12.3496	0.803891
$237$	0	0
$238$	21.6521	1.40350
$239$	3.24086	0.209634	0.104817	−	0.994492i	$-0.466574\pi$
$239$	3.24086	0.209634	0.104817	+	0.994492i	$0.466574\pi$
$240$	0	0
$241$	25.9938	1.67441	0.837203	−	0.546892i	$-0.184189\pi$
$241$	25.9938	1.67441	0.837203	+	0.546892i	$0.184189\pi$
$242$	8.96080	0.576022
$243$	0	0
$244$	8.20561	0.525311
$245$	0	0
$246$	0	0
$247$	6.91764	0.440159
$248$	1.28797	0.0817864
$249$	0	0
$250$	0	0
$251$	3.98153	0.251312	0.125656	−	0.992074i	$-0.459896\pi$
$251$	3.98153	0.251312	0.125656	+	0.992074i	$0.459896\pi$
$252$	0	0
$253$	−5.24482	−0.329739
$254$	6.05767	0.380092
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.1625	0.883430	0.441715	−	0.897155i	$-0.354370\pi$
$257$	14.1625	0.883430	0.441715	+	0.897155i	$0.354370\pi$
$258$	0	0
$259$	−3.63362	−0.225782
$260$	0	0
$261$	0	0
$262$	9.44648	0.583605
$263$	21.3434	1.31609	0.658045	−	0.752979i	$-0.271384\pi$
$263$	21.3434	1.31609	0.658045	+	0.752979i	$0.271384\pi$
$264$	0	0
$265$	0	0
$266$	4.24482	0.260266
$267$	0	0
$268$	−4.38880	−0.268089
$269$	24.4112	1.48838	0.744189	−	0.667969i	$-0.232836\pi$
$269$	24.4112	1.48838	0.744189	+	0.667969i	$0.232836\pi$
$270$	0	0
$271$	−14.8129	−0.899817	−0.449909	−	0.893075i	$-0.648543\pi$
$271$	−14.8129	−0.899817	−0.449909	+	0.893075i	$0.648543\pi$
$272$	5.10083	0.309283
$273$	0	0
$274$	4.20166	0.253832
$275$	0	0
$276$	0	0
$277$	−15.2633	−0.917082	−0.458541	−	0.888673i	$-0.651628\pi$
$277$	−15.2633	−0.917082	−0.458541	+	0.888673i	$0.651628\pi$
$278$	5.40727	0.324307
$279$	0	0
$280$	0	0
$281$	−2.83528	−0.169139	−0.0845694	−	0.996418i	$-0.526951\pi$
$281$	−2.83528	−0.169139	−0.0845694	+	0.996418i	$0.526951\pi$
$282$	0	0
$283$	−0.942326	−0.0560154	−0.0280077	−	0.999608i	$-0.508916\pi$
$283$	−0.942326	−0.0560154	−0.0280077	+	0.999608i	$0.508916\pi$
$284$	11.1008	0.658713
$285$	0	0
$286$	−9.87844	−0.584124
$287$	34.0369	2.00914
$288$	0	0
$289$	9.01847	0.530498
$290$	0	0
$291$	0	0
$292$	−5.38485	−0.315125
$293$	13.5513	0.791673	0.395837	−	0.918321i	$-0.370455\pi$
$293$	13.5513	0.791673	0.395837	+	0.918321i	$0.370455\pi$
$294$	0	0
$295$	0	0
$296$	−0.856013	−0.0497548
$297$	0	0
$298$	−1.51037	−0.0874932
$299$	−25.4073	−1.46934
$300$	0	0
$301$	15.1625	0.873950
$302$	15.4504	0.889072
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	26.7921	1.52911	0.764554	−	0.644560i	$-0.222959\pi$
$307$	26.7921	1.52911	0.764554	+	0.644560i	$0.222959\pi$
$308$	−6.06163	−0.345393
$309$	0	0
$310$	0	0
$311$	5.03920	0.285747	0.142873	−	0.989741i	$-0.454366\pi$
$311$	5.03920	0.285747	0.142873	+	0.989741i	$0.454366\pi$
$312$	0	0
$313$	15.0185	0.848894	0.424447	−	0.905453i	$-0.360468\pi$
$313$	15.0185	0.848894	0.424447	+	0.905453i	$0.360468\pi$
$314$	−9.16246	−0.517067
$315$	0	0
$316$	2.14399	0.120609
$317$	20.1008	1.12898	0.564488	−	0.825442i	$-0.309074\pi$
$317$	20.1008	1.12898	0.564488	+	0.825442i	$0.309074\pi$
$318$	0	0
$319$	11.5680	0.647686
$320$	0	0
$321$	0	0
$322$	−15.5905	−0.868823
$323$	5.10083	0.283818
$324$	0	0
$325$	0	0
$326$	−12.6913	−0.702906
$327$	0	0
$328$	8.01847	0.442746
$329$	−16.2017	−0.893226
$330$	0	0
$331$	−25.5697	−1.40544	−0.702720	−	0.711467i	$-0.748031\pi$
$331$	−25.5697	−1.40544	−0.702720	+	0.711467i	$0.748031\pi$
$332$	−1.04316	−0.0572506
$333$	0	0
$334$	16.3249	0.893260
$335$	0	0
$336$	0	0
$337$	13.9137	0.757927	0.378963	−	0.925412i	$-0.376281\pi$
$337$	13.9137	0.757927	0.378963	+	0.925412i	$0.376281\pi$
$338$	−34.8538	−1.89579
$339$	0	0
$340$	0	0
$341$	−1.83923	−0.0996001
$342$	0	0
$343$	−17.0577	−0.921028
$344$	3.57199	0.192589
$345$	0	0
$346$	3.32322	0.178658
$347$	2.56804	0.137860	0.0689298	−	0.997622i	$-0.478042\pi$
$347$	2.56804	0.137860	0.0689298	+	0.997622i	$0.478042\pi$
$348$	0	0
$349$	−5.23691	−0.280325	−0.140163	−	0.990128i	$-0.544763\pi$
$349$	−5.23691	−0.280325	−0.140163	+	0.990128i	$0.544763\pi$
$350$	0	0
$351$	0	0
$352$	−1.42801	−0.0761130
$353$	12.2818	0.653692	0.326846	−	0.945078i	$-0.394014\pi$
$353$	12.2818	0.653692	0.326846	+	0.945078i	$0.394014\pi$
$354$	0	0
$355$	0	0
$356$	16.7305	0.886715
$357$	0	0
$358$	5.54731	0.293184
$359$	33.7753	1.78259	0.891297	−	0.453419i	$-0.149796\pi$
$359$	33.7753	1.78259	0.891297	+	0.453419i	$0.149796\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	17.2840	0.908428
$363$	0	0
$364$	−29.3641	−1.53910
$365$	0	0
$366$	0	0
$367$	30.3681	1.58520	0.792600	−	0.609742i	$-0.208727\pi$
$367$	30.3681	1.58520	0.792600	+	0.609742i	$0.208727\pi$
$368$	−3.67282	−0.191459
$369$	0	0
$370$	0	0
$371$	38.4649	1.99700
$372$	0	0
$373$	−25.7322	−1.33236	−0.666181	−	0.745790i	$-0.732072\pi$
$373$	−25.7322	−1.33236	−0.666181	+	0.745790i	$0.732072\pi$
$374$	−7.28402	−0.376648
$375$	0	0
$376$	−3.81681	−0.196837
$377$	56.0386	2.88614
$378$	0	0
$379$	37.2593	1.91388	0.956942	−	0.290280i	$-0.0937485\pi$
$379$	37.2593	1.91388	0.956942	+	0.290280i	$0.0937485\pi$
$380$	0	0
$381$	0	0
$382$	−22.6336	−1.15804
$383$	−31.9154	−1.63080	−0.815400	−	0.578898i	$-0.803483\pi$
$383$	−31.9154	−1.63080	−0.815400	+	0.578898i	$0.803483\pi$
$384$	0	0
$385$	0	0
$386$	4.24482	0.216055
$387$	0	0
$388$	−6.81286	−0.345870
$389$	−4.48963	−0.227633	−0.113817	−	0.993502i	$-0.536308\pi$
$389$	−4.48963	−0.227633	−0.113817	+	0.993502i	$0.536308\pi$
$390$	0	0
$391$	−18.7345	−0.947442
$392$	−11.0185	−0.556517
$393$	0	0
$394$	−5.00395	−0.252096
$395$	0	0
$396$	0	0
$397$	34.6873	1.74091	0.870454	−	0.492250i	$-0.163825\pi$
$397$	34.6873	1.74091	0.870454	+	0.492250i	$0.163825\pi$
$398$	27.3826	1.37257
$399$	0	0
$400$	0	0
$401$	−7.03694	−0.351408	−0.175704	−	0.984443i	$-0.556220\pi$
$401$	−7.03694	−0.351408	−0.175704	+	0.984443i	$0.556220\pi$
$402$	0	0
$403$	−8.90973	−0.443826
$404$	11.7737	0.585761
$405$	0	0
$406$	34.3865	1.70658
$407$	1.22239	0.0605918
$408$	0	0
$409$	26.4112	1.30595	0.652976	−	0.757379i	$-0.273520\pi$
$409$	26.4112	1.30595	0.652976	+	0.757379i	$0.273520\pi$
$410$	0	0
$411$	0	0
$412$	−4.32718	−0.213185
$413$	−52.4218	−2.57951
$414$	0	0
$415$	0	0
$416$	−6.91764	−0.339165
$417$	0	0
$418$	−1.42801	−0.0698461
$419$	−16.4896	−0.805571	−0.402786	−	0.915294i	$-0.631958\pi$
$419$	−16.4896	−0.805571	−0.402786	+	0.915294i	$0.631958\pi$
$420$	0	0
$421$	39.0162	1.90153	0.950767	−	0.309907i	$-0.100298\pi$
$421$	39.0162	1.90153	0.950767	+	0.309907i	$0.100298\pi$
$422$	5.14003	0.250213
$423$	0	0
$424$	9.06163	0.440072
$425$	0	0
$426$	0	0
$427$	−34.8313	−1.68561
$428$	4.14003	0.200116
$429$	0	0
$430$	0	0
$431$	−39.8722	−1.92058	−0.960289	−	0.279008i	$-0.909994\pi$
$431$	−39.8722	−1.92058	−0.960289	+	0.279008i	$0.909994\pi$
$432$	0	0
$433$	30.3681	1.45940	0.729698	−	0.683769i	$-0.239661\pi$
$433$	30.3681	1.45940	0.729698	+	0.683769i	$0.239661\pi$
$434$	−5.46721	−0.262434
$435$	0	0
$436$	−2.48963	−0.119232
$437$	−3.67282	−0.175695
$438$	0	0
$439$	19.4033	0.926070	0.463035	−	0.886340i	$-0.346760\pi$
$439$	19.4033	0.926070	0.463035	+	0.886340i	$0.346760\pi$
$440$	0	0
$441$	0	0
$442$	−35.2857	−1.67837
$443$	−20.8705	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$443$	−20.8705	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$444$	0	0
$445$	0	0
$446$	10.1625	0.481206
$447$	0	0
$448$	−4.24482	−0.200549
$449$	−30.7753	−1.45238	−0.726189	−	0.687495i	$-0.758710\pi$
$449$	−30.7753	−1.45238	−0.726189	+	0.687495i	$0.758710\pi$
$450$	0	0
$451$	−11.4504	−0.539180
$452$	−11.0185	−0.518265
$453$	0	0
$454$	−7.92159	−0.371779
$455$	0	0
$456$	0	0
$457$	−27.5658	−1.28947	−0.644736	−	0.764405i	$-0.723033\pi$
$457$	−27.5658	−1.28947	−0.644736	+	0.764405i	$0.723033\pi$
$458$	7.44648	0.347951
$459$	0	0
$460$	0	0
$461$	31.3042	1.45798	0.728991	−	0.684524i	$-0.239990\pi$
$461$	31.3042	1.45798	0.728991	+	0.684524i	$0.239990\pi$
$462$	0	0
$463$	−11.1440	−0.517905	−0.258952	−	0.965890i	$-0.583377\pi$
$463$	−11.1440	−0.517905	−0.258952	+	0.965890i	$0.583377\pi$
$464$	8.10083	0.376072
$465$	0	0
$466$	−28.4834	−1.31947
$467$	36.3289	1.68110	0.840550	−	0.541734i	$-0.182232\pi$
$467$	36.3289	1.68110	0.840550	+	0.541734i	$0.182232\pi$
$468$	0	0
$469$	18.6297	0.860238
$470$	0	0
$471$	0	0
$472$	−12.3496	−0.568436
$473$	−5.10083	−0.234536
$474$	0	0
$475$	0	0
$476$	−21.6521	−0.992422
$477$	0	0
$478$	−3.24086	−0.148234
$479$	17.0185	0.777594	0.388797	−	0.921323i	$-0.372891\pi$
$479$	17.0185	0.777594	0.388797	+	0.921323i	$0.372891\pi$
$480$	0	0
$481$	5.92159	0.270001
$482$	−25.9938	−1.18398
$483$	0	0
$484$	−8.96080	−0.407309
$485$	0	0
$486$	0	0
$487$	27.3536	1.23951	0.619754	−	0.784796i	$-0.287232\pi$
$487$	27.3536	1.23951	0.619754	+	0.784796i	$0.287232\pi$
$488$	−8.20561	−0.371451
$489$	0	0
$490$	0	0
$491$	12.7697	0.576289	0.288144	−	0.957587i	$-0.406962\pi$
$491$	12.7697	0.576289	0.288144	+	0.957587i	$0.406962\pi$
$492$	0	0
$493$	41.3210	1.86100
$494$	−6.91764	−0.311239
$495$	0	0
$496$	−1.28797	−0.0578317
$497$	−47.1210	−2.11367
$498$	0	0
$499$	−33.8969	−1.51743	−0.758717	−	0.651420i	$-0.774173\pi$
$499$	−33.8969	−1.51743	−0.758717	+	0.651420i	$0.774173\pi$
$500$	0	0
$501$	0	0
$502$	−3.98153	−0.177704
$503$	35.6706	1.59047	0.795236	−	0.606300i	$-0.207347\pi$
$503$	35.6706	1.59047	0.795236	+	0.606300i	$0.207347\pi$
$504$	0	0
$505$	0	0
$506$	5.24482	0.233161
$507$	0	0
$508$	−6.05767	−0.268766
$509$	24.2347	1.07418	0.537091	−	0.843524i	$-0.319523\pi$
$509$	24.2347	1.07418	0.537091	+	0.843524i	$0.319523\pi$
$510$	0	0
$511$	22.8577	1.01117
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.1625	−0.624679
$515$	0	0
$516$	0	0
$517$	5.45043	0.239710
$518$	3.63362	0.159652
$519$	0	0
$520$	0	0
$521$	34.4689	1.51011	0.755055	−	0.655661i	$-0.227610\pi$
$521$	34.4689	1.51011	0.755055	+	0.655661i	$0.227610\pi$
$522$	0	0
$523$	−35.3905	−1.54752	−0.773759	−	0.633480i	$-0.781626\pi$
$523$	−35.3905	−1.54752	−0.773759	+	0.633480i	$0.781626\pi$
$524$	−9.44648	−0.412671
$525$	0	0
$526$	−21.3434	−0.930616
$527$	−6.56973	−0.286182
$528$	0	0
$529$	−9.51037	−0.413494
$530$	0	0
$531$	0	0
$532$	−4.24482	−0.184036
$533$	−55.4689	−2.40262
$534$	0	0
$535$	0	0
$536$	4.38880	0.189567
$537$	0	0
$538$	−24.4112	−1.05244
$539$	15.7345	0.677731
$540$	0	0
$541$	−21.6560	−0.931066	−0.465533	−	0.885031i	$-0.654137\pi$
$541$	−21.6560	−0.931066	−0.465533	+	0.885031i	$0.654137\pi$
$542$	14.8129	0.636267
$543$	0	0
$544$	−5.10083	−0.218696
$545$	0	0
$546$	0	0
$547$	3.21844	0.137611	0.0688053	−	0.997630i	$-0.478081\pi$
$547$	3.21844	0.137611	0.0688053	+	0.997630i	$0.478081\pi$
$548$	−4.20166	−0.179486
$549$	0	0
$550$	0	0
$551$	8.10083	0.345107
$552$	0	0
$553$	−9.10083	−0.387007
$554$	15.2633	0.648475
$555$	0	0
$556$	−5.40727	−0.229320
$557$	−40.3865	−1.71123	−0.855616	−	0.517611i	$-0.826822\pi$
$557$	−40.3865	−1.71123	−0.855616	+	0.517611i	$0.826822\pi$
$558$	0	0
$559$	−24.7098	−1.04511
$560$	0	0
$561$	0	0
$562$	2.83528	0.119599
$563$	−9.65831	−0.407049	−0.203525	−	0.979070i	$-0.565240\pi$
$563$	−9.65831	−0.407049	−0.203525	+	0.979070i	$0.565240\pi$
$564$	0	0
$565$	0	0
$566$	0.942326	0.0396089
$567$	0	0
$568$	−11.1008	−0.465780
$569$	18.6992	0.783911	0.391956	−	0.919984i	$-0.371799\pi$
$569$	18.6992	0.783911	0.391956	+	0.919984i	$0.371799\pi$
$570$	0	0
$571$	4.34169	0.181694	0.0908471	−	0.995865i	$-0.471043\pi$
$571$	4.34169	0.181694	0.0908471	+	0.995865i	$0.471043\pi$
$572$	9.87844	0.413038
$573$	0	0
$574$	−34.0369	−1.42067
$575$	0	0
$576$	0	0
$577$	−3.03694	−0.126430	−0.0632148	−	0.998000i	$-0.520135\pi$
$577$	−3.03694	−0.126430	−0.0632148	+	0.998000i	$0.520135\pi$
$578$	−9.01847	−0.375119
$579$	0	0
$580$	0	0
$581$	4.42801	0.183705
$582$	0	0
$583$	−12.9401	−0.535923
$584$	5.38485	0.222827
$585$	0	0
$586$	−13.5513	−0.559797
$587$	33.7345	1.39237	0.696185	−	0.717863i	$-0.254879\pi$
$587$	33.7345	1.39237	0.696185	+	0.717863i	$0.254879\pi$
$588$	0	0
$589$	−1.28797	−0.0530700
$590$	0	0
$591$	0	0
$592$	0.856013	0.0351819
$593$	−34.7361	−1.42644	−0.713221	−	0.700939i	$-0.752764\pi$
$593$	−34.7361	−1.42644	−0.713221	+	0.700939i	$0.752764\pi$
$594$	0	0
$595$	0	0
$596$	1.51037	0.0618670
$597$	0	0
$598$	25.4073	1.03898
$599$	−6.28176	−0.256666	−0.128333	−	0.991731i	$-0.540963\pi$
$599$	−6.28176	−0.256666	−0.128333	+	0.991731i	$0.540963\pi$
$600$	0	0
$601$	18.4544	0.752770	0.376385	−	0.926463i	$-0.377167\pi$
$601$	18.4544	0.752770	0.376385	+	0.926463i	$0.377167\pi$
$602$	−15.1625	−0.617976
$603$	0	0
$604$	−15.4504	−0.628669
$605$	0	0
$606$	0	0
$607$	20.5367	0.833561	0.416780	−	0.909007i	$-0.363158\pi$
$607$	20.5367	0.833561	0.416780	+	0.909007i	$0.363158\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	26.4033	1.06816
$612$	0	0
$613$	11.9216	0.481509	0.240754	−	0.970586i	$-0.422605\pi$
$613$	11.9216	0.481509	0.240754	+	0.970586i	$0.422605\pi$
$614$	−26.7921	−1.08124
$615$	0	0
$616$	6.06163	0.244230
$617$	−17.5473	−0.706428	−0.353214	−	0.935543i	$-0.614911\pi$
$617$	−17.5473	−0.706428	−0.353214	+	0.935543i	$0.614911\pi$
$618$	0	0
$619$	−33.7322	−1.35581	−0.677906	−	0.735149i	$-0.737112\pi$
$619$	−33.7322	−1.35581	−0.677906	+	0.735149i	$0.737112\pi$
$620$	0	0
$621$	0	0
$622$	−5.03920	−0.202054
$623$	−71.0179	−2.84527
$624$	0	0
$625$	0	0
$626$	−15.0185	−0.600259
$627$	0	0
$628$	9.16246	0.365622
$629$	4.36638	0.174099
$630$	0	0
$631$	22.4033	0.891862	0.445931	−	0.895067i	$-0.352873\pi$
$631$	22.4033	0.891862	0.445931	+	0.895067i	$0.352873\pi$
$632$	−2.14399	−0.0852832
$633$	0	0
$634$	−20.1008	−0.798306
$635$	0	0
$636$	0	0
$637$	76.2218	3.02002
$638$	−11.5680	−0.457983
$639$	0	0
$640$	0	0
$641$	−25.2778	−0.998413	−0.499207	−	0.866483i	$-0.666375\pi$
$641$	−25.2778	−0.998413	−0.499207	+	0.866483i	$0.666375\pi$
$642$	0	0
$643$	13.4320	0.529705	0.264852	−	0.964289i	$-0.414677\pi$
$643$	13.4320	0.529705	0.264852	+	0.964289i	$0.414677\pi$
$644$	15.5905	0.614350
$645$	0	0
$646$	−5.10083	−0.200689
$647$	−5.95289	−0.234032	−0.117016	−	0.993130i	$-0.537333\pi$
$647$	−5.95289	−0.234032	−0.117016	+	0.993130i	$0.537333\pi$
$648$	0	0
$649$	17.6353	0.692247
$650$	0	0
$651$	0	0
$652$	12.6913	0.497029
$653$	−35.3826	−1.38463	−0.692314	−	0.721597i	$-0.743409\pi$
$653$	−35.3826	−1.38463	−0.692314	+	0.721597i	$0.743409\pi$
$654$	0	0
$655$	0	0
$656$	−8.01847	−0.313069
$657$	0	0
$658$	16.2017	0.631606
$659$	46.2218	1.80055	0.900273	−	0.435325i	$-0.143367\pi$
$659$	46.2218	1.80055	0.900273	+	0.435325i	$0.143367\pi$
$660$	0	0
$661$	−37.5843	−1.46186	−0.730929	−	0.682454i	$-0.760913\pi$
$661$	−37.5843	−1.46186	−0.730929	+	0.682454i	$0.760913\pi$
$662$	25.5697	0.993796
$663$	0	0
$664$	1.04316	0.0404823
$665$	0	0
$666$	0	0
$667$	−29.7529	−1.15204
$668$	−16.3249	−0.631630
$669$	0	0
$670$	0	0
$671$	11.7177	0.452356
$672$	0	0
$673$	43.4257	1.67394	0.836970	−	0.547249i	$-0.184325\pi$
$673$	43.4257	1.67394	0.836970	+	0.547249i	$0.184325\pi$
$674$	−13.9137	−0.535935
$675$	0	0
$676$	34.8538	1.34053
$677$	−25.2218	−0.969353	−0.484677	−	0.874693i	$-0.661063\pi$
$677$	−25.2218	−0.969353	−0.484677	+	0.874693i	$0.661063\pi$
$678$	0	0
$679$	28.9193	1.10982
$680$	0	0
$681$	0	0
$682$	1.83923	0.0704279
$683$	18.0616	0.691109	0.345554	−	0.938399i	$-0.387691\pi$
$683$	18.0616	0.691109	0.345554	+	0.938399i	$0.387691\pi$
$684$	0	0
$685$	0	0
$686$	17.0577	0.651265
$687$	0	0
$688$	−3.57199	−0.136181
$689$	−62.6851	−2.38811
$690$	0	0
$691$	−9.36243	−0.356163	−0.178082	−	0.984016i	$-0.556989\pi$
$691$	−9.36243	−0.356163	−0.178082	+	0.984016i	$0.556989\pi$
$692$	−3.32322	−0.126330
$693$	0	0
$694$	−2.56804	−0.0974815
$695$	0	0
$696$	0	0
$697$	−40.9009	−1.54923
$698$	5.23691	0.198220
$699$	0	0
$700$	0	0
$701$	−52.5714	−1.98560	−0.992798	−	0.119803i	$-0.961774\pi$
$701$	−52.5714	−1.98560	−0.992798	+	0.119803i	$0.961774\pi$
$702$	0	0
$703$	0.856013	0.0322852
$704$	1.42801	0.0538200
$705$	0	0
$706$	−12.2818	−0.462230
$707$	−49.9770	−1.87958
$708$	0	0
$709$	14.3025	0.537141	0.268571	−	0.963260i	$-0.413449\pi$
$709$	14.3025	0.537141	0.268571	+	0.963260i	$0.413449\pi$
$710$	0	0
$711$	0	0
$712$	−16.7305	−0.627002
$713$	4.73050	0.177159
$714$	0	0
$715$	0	0
$716$	−5.54731	−0.207313
$717$	0	0
$718$	−33.7753	−1.26048
$719$	50.6706	1.88969	0.944847	−	0.327513i	$-0.106211\pi$
$719$	50.6706	1.88969	0.944847	+	0.327513i	$0.106211\pi$
$720$	0	0
$721$	18.3681	0.684063
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−17.2840	−0.642356
$725$	0	0
$726$	0	0
$727$	43.0083	1.59509	0.797545	−	0.603260i	$-0.206132\pi$
$727$	43.0083	1.59509	0.797545	+	0.603260i	$0.206132\pi$
$728$	29.3641	1.08831
$729$	0	0
$730$	0	0
$731$	−18.2201	−0.673896
$732$	0	0
$733$	1.77592	0.0655949	0.0327975	−	0.999462i	$-0.489558\pi$
$733$	1.77592	0.0655949	0.0327975	+	0.999462i	$0.489558\pi$
$734$	−30.3681	−1.12091
$735$	0	0
$736$	3.67282	0.135382
$737$	−6.26724	−0.230857
$738$	0	0
$739$	5.43196	0.199818	0.0999089	−	0.994997i	$-0.468145\pi$
$739$	5.43196	0.199818	0.0999089	+	0.994997i	$0.468145\pi$
$740$	0	0
$741$	0	0
$742$	−38.4649	−1.41209
$743$	0.769701	0.0282376	0.0141188	−	0.999900i	$-0.495506\pi$
$743$	0.769701	0.0282376	0.0141188	+	0.999900i	$0.495506\pi$
$744$	0	0
$745$	0	0
$746$	25.7322	0.942122
$747$	0	0
$748$	7.28402	0.266330
$749$	−17.5737	−0.642128
$750$	0	0
$751$	42.6683	1.55699	0.778494	−	0.627652i	$-0.215984\pi$
$751$	42.6683	1.55699	0.778494	+	0.627652i	$0.215984\pi$
$752$	3.81681	0.139185
$753$	0	0
$754$	−56.0386	−2.04081
$755$	0	0
$756$	0	0
$757$	39.2448	1.42638	0.713189	−	0.700972i	$-0.247250\pi$
$757$	39.2448	1.42638	0.713189	+	0.700972i	$0.247250\pi$
$758$	−37.2593	−1.35332
$759$	0	0
$760$	0	0
$761$	1.38090	0.0500575	0.0250288	−	0.999687i	$-0.492032\pi$
$761$	1.38090	0.0500575	0.0250288	+	0.999687i	$0.492032\pi$
$762$	0	0
$763$	10.5680	0.382589
$764$	22.6336	0.818856
$765$	0	0
$766$	31.9154	1.15315
$767$	85.4301	3.08470
$768$	0	0
$769$	32.9691	1.18890	0.594448	−	0.804134i	$-0.297371\pi$
$769$	32.9691	1.18890	0.594448	+	0.804134i	$0.297371\pi$
$770$	0	0
$771$	0	0
$772$	−4.24482	−0.152774
$773$	32.4975	1.16886	0.584428	−	0.811446i	$-0.301319\pi$
$773$	32.4975	1.16886	0.584428	+	0.811446i	$0.301319\pi$
$774$	0	0
$775$	0	0
$776$	6.81286	0.244567
$777$	0	0
$778$	4.48963	0.160961
$779$	−8.01847	−0.287292
$780$	0	0
$781$	15.8521	0.567231
$782$	18.7345	0.669943
$783$	0	0
$784$	11.0185	0.393517
$785$	0	0
$786$	0	0
$787$	−0.187143	−0.00667091	−0.00333546	−	0.999994i	$-0.501062\pi$
$787$	−0.187143	−0.00667091	−0.00333546	+	0.999994i	$0.501062\pi$
$788$	5.00395	0.178258
$789$	0	0
$790$	0	0
$791$	46.7714	1.66300
$792$	0	0
$793$	56.7635	2.01573
$794$	−34.6873	−1.23101
$795$	0	0
$796$	−27.3826	−0.970550
$797$	33.9691	1.20325	0.601624	−	0.798780i	$-0.294521\pi$
$797$	33.9691	1.20325	0.601624	+	0.798780i	$0.294521\pi$
$798$	0	0
$799$	19.4689	0.688760
$800$	0	0
$801$	0	0
$802$	7.03694	0.248483
$803$	−7.68960	−0.271360
$804$	0	0
$805$	0	0
$806$	8.90973	0.313832
$807$	0	0
$808$	−11.7737	−0.414196
$809$	34.2897	1.20556	0.602780	−	0.797907i	$-0.294060\pi$
$809$	34.2897	1.20556	0.602780	+	0.797907i	$0.294060\pi$
$810$	0	0
$811$	−18.7737	−0.659232	−0.329616	−	0.944115i	$-0.606919\pi$
$811$	−18.7737	−0.659232	−0.329616	+	0.944115i	$0.606919\pi$
$812$	−34.3865	−1.20673
$813$	0	0
$814$	−1.22239	−0.0428449
$815$	0	0
$816$	0	0
$817$	−3.57199	−0.124968
$818$	−26.4112	−0.923447
$819$	0	0
$820$	0	0
$821$	1.16077	0.0405110	0.0202555	−	0.999795i	$-0.493552\pi$
$821$	1.16077	0.0405110	0.0202555	+	0.999795i	$0.493552\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	4.32718	0.150744
$825$	0	0
$826$	52.4218	1.82399
$827$	−35.4857	−1.23396	−0.616979	−	0.786980i	$-0.711644\pi$
$827$	−35.4857	−1.23396	−0.616979	+	0.786980i	$0.711644\pi$
$828$	0	0
$829$	−10.9546	−0.380468	−0.190234	−	0.981739i	$-0.560925\pi$
$829$	−10.9546	−0.380468	−0.190234	+	0.981739i	$0.560925\pi$
$830$	0	0
$831$	0	0
$832$	6.91764	0.239826
$833$	56.2034	1.94733
$834$	0	0
$835$	0	0
$836$	1.42801	0.0493886
$837$	0	0
$838$	16.4896	0.569625
$839$	−0.417441	−0.0144117	−0.00720584	−	0.999974i	$-0.502294\pi$
$839$	−0.417441	−0.0144117	−0.00720584	+	0.999974i	$0.502294\pi$
$840$	0	0
$841$	36.6235	1.26288
$842$	−39.0162	−1.34459
$843$	0	0
$844$	−5.14003	−0.176927
$845$	0	0
$846$	0	0
$847$	38.0369	1.30696
$848$	−9.06163	−0.311178
$849$	0	0
$850$	0	0
$851$	−3.14399	−0.107774
$852$	0	0
$853$	44.7776	1.53316	0.766578	−	0.642151i	$-0.221958\pi$
$853$	44.7776	1.53316	0.766578	+	0.642151i	$0.221958\pi$
$854$	34.8313	1.19190
$855$	0	0
$856$	−4.14003	−0.141503
$857$	43.3720	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$857$	43.3720	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$858$	0	0
$859$	−2.36638	−0.0807398	−0.0403699	−	0.999185i	$-0.512854\pi$
$859$	−2.36638	−0.0807398	−0.0403699	+	0.999185i	$0.512854\pi$
$860$	0	0
$861$	0	0
$862$	39.8722	1.35805
$863$	19.9921	0.680539	0.340269	−	0.940328i	$-0.389482\pi$
$863$	19.9921	0.680539	0.340269	+	0.940328i	$0.389482\pi$
$864$	0	0
$865$	0	0
$866$	−30.3681	−1.03195
$867$	0	0
$868$	5.46721	0.185569
$869$	3.06163	0.103859
$870$	0	0
$871$	−30.3602	−1.02871
$872$	2.48963	0.0843096
$873$	0	0
$874$	3.67282	0.124235
$875$	0	0
$876$	0	0
$877$	−19.0946	−0.644779	−0.322390	−	0.946607i	$-0.604486\pi$
$877$	−19.0946	−0.644779	−0.322390	+	0.946607i	$0.604486\pi$
$878$	−19.4033	−0.654830
$879$	0	0
$880$	0	0
$881$	18.1233	0.610588	0.305294	−	0.952258i	$-0.401245\pi$
$881$	18.1233	0.610588	0.305294	+	0.952258i	$0.401245\pi$
$882$	0	0
$883$	−19.7737	−0.665436	−0.332718	−	0.943026i	$-0.607966\pi$
$883$	−19.7737	−0.665436	−0.332718	+	0.943026i	$0.607966\pi$
$884$	35.2857	1.18679
$885$	0	0
$886$	20.8705	0.701159
$887$	−35.5473	−1.19356	−0.596781	−	0.802404i	$-0.703554\pi$
$887$	−35.5473	−1.19356	−0.596781	+	0.802404i	$0.703554\pi$
$888$	0	0
$889$	25.7137	0.862410
$890$	0	0
$891$	0	0
$892$	−10.1625	−0.340264
$893$	3.81681	0.127725
$894$	0	0
$895$	0	0
$896$	4.24482	0.141809
$897$	0	0
$898$	30.7753	1.02699
$899$	−10.4337	−0.347982
$900$	0	0
$901$	−46.2218	−1.53987
$902$	11.4504	0.381258
$903$	0	0
$904$	11.0185	0.366469
$905$	0	0
$906$	0	0
$907$	−32.0554	−1.06438	−0.532191	−	0.846624i	$-0.678631\pi$
$907$	−32.0554	−1.06438	−0.532191	+	0.846624i	$0.678631\pi$
$908$	7.92159	0.262887
$909$	0	0
$910$	0	0
$911$	−16.9714	−0.562286	−0.281143	−	0.959666i	$-0.590714\pi$
$911$	−16.9714	−0.562286	−0.281143	+	0.959666i	$0.590714\pi$
$912$	0	0
$913$	−1.48963	−0.0492997
$914$	27.5658	0.911795
$915$	0	0
$916$	−7.44648	−0.246039
$917$	40.0986	1.32417
$918$	0	0
$919$	32.4465	1.07031	0.535155	−	0.844754i	$-0.320253\pi$
$919$	32.4465	1.07031	0.535155	+	0.844754i	$0.320253\pi$
$920$	0	0
$921$	0	0
$922$	−31.3042	−1.03095
$923$	76.7916	2.52762
$924$	0	0
$925$	0	0
$926$	11.1440	0.366214
$927$	0	0
$928$	−8.10083	−0.265923
$929$	−13.9631	−0.458113	−0.229057	−	0.973413i	$-0.573564\pi$
$929$	−13.9631	−0.458113	−0.229057	+	0.973413i	$0.573564\pi$
$930$	0	0
$931$	11.0185	0.361116
$932$	28.4834	0.933005
$933$	0	0
$934$	−36.3289	−1.18872
$935$	0	0
$936$	0	0
$937$	−19.5058	−0.637228	−0.318614	−	0.947885i	$-0.603217\pi$
$937$	−19.5058	−0.637228	−0.318614	+	0.947885i	$0.603217\pi$
$938$	−18.6297	−0.608280
$939$	0	0
$940$	0	0
$941$	−8.48568	−0.276625	−0.138313	−	0.990389i	$-0.544168\pi$
$941$	−8.48568	−0.276625	−0.138313	+	0.990389i	$0.544168\pi$
$942$	0	0
$943$	29.4504	0.959038
$944$	12.3496	0.401945
$945$	0	0
$946$	5.10083	0.165842
$947$	5.06558	0.164609	0.0823046	−	0.996607i	$-0.473772\pi$
$947$	5.06558	0.164609	0.0823046	+	0.996607i	$0.473772\pi$
$948$	0	0
$949$	−37.2505	−1.20920
$950$	0	0
$951$	0	0
$952$	21.6521	0.701748
$953$	−4.46890	−0.144762	−0.0723810	−	0.997377i	$-0.523060\pi$
$953$	−4.46890	−0.144762	−0.0723810	+	0.997377i	$0.523060\pi$
$954$	0	0
$955$	0	0
$956$	3.24086	0.104817
$957$	0	0
$958$	−17.0185	−0.549842
$959$	17.8353	0.575931
$960$	0	0
$961$	−29.3411	−0.946488
$962$	−5.92159	−0.190920
$963$	0	0
$964$	25.9938	0.837203
$965$	0	0
$966$	0	0
$967$	10.9872	0.353324	0.176662	−	0.984272i	$-0.443470\pi$
$967$	10.9872	0.353324	0.176662	+	0.984272i	$0.443470\pi$
$968$	8.96080	0.288011
$969$	0	0
$970$	0	0
$971$	−11.8969	−0.381790	−0.190895	−	0.981610i	$-0.561139\pi$
$971$	−11.8969	−0.381790	−0.190895	+	0.981610i	$0.561139\pi$
$972$	0	0
$973$	22.9529	0.735836
$974$	−27.3536	−0.876464
$975$	0	0
$976$	8.20561	0.262655
$977$	−29.9031	−0.956686	−0.478343	−	0.878173i	$-0.658762\pi$
$977$	−29.9031	−0.956686	−0.478343	+	0.878173i	$0.658762\pi$
$978$	0	0
$979$	23.8913	0.763568
$980$	0	0
$981$	0	0
$982$	−12.7697	−0.407498
$983$	−6.89917	−0.220049	−0.110025	−	0.993929i	$-0.535093\pi$
$983$	−6.89917	−0.220049	−0.110025	+	0.993929i	$0.535093\pi$
$984$	0	0
$985$	0	0
$986$	−41.3210	−1.31593
$987$	0	0
$988$	6.91764	0.220079
$989$	13.1193	0.417170
$990$	0	0
$991$	12.3720	0.393010	0.196505	−	0.980503i	$-0.437041\pi$
$991$	12.3720	0.393010	0.196505	+	0.980503i	$0.437041\pi$
$992$	1.28797	0.0408932
$993$	0	0
$994$	47.1210	1.49459
$995$	0	0
$996$	0	0
$997$	2.73671	0.0866725	0.0433363	−	0.999061i	$-0.486201\pi$
$997$	2.73671	0.0866725	0.0433363	+	0.999061i	$0.486201\pi$
$998$	33.8969	1.07299
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cd.1.1	✓	3
3.2	odd	2		8550.2.a.cn.1.1	yes	3
5.4	even	2		8550.2.a.cs.1.3	yes	3
15.14	odd	2		8550.2.a.cg.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8550.2.a.cd.1.1	✓	3	1.1	even	1	trivial
8550.2.a.cg.1.3	yes	3	15.14	odd	2
8550.2.a.cn.1.1	yes	3	3.2	odd	2
8550.2.a.cs.1.3	yes	3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.24482 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.24482 q^{7} -1.00000 q^{8} +1.42801 q^{11} +6.91764 q^{13} +4.24482 q^{14} +1.00000 q^{16} +5.10083 q^{17} +1.00000 q^{19} -1.42801 q^{22} -3.67282 q^{23} -6.91764 q^{26} -4.24482 q^{28} +8.10083 q^{29} -1.28797 q^{31} -1.00000 q^{32} -5.10083 q^{34} +0.856013 q^{37} -1.00000 q^{38} -8.01847 q^{41} -3.57199 q^{43} +1.42801 q^{44} +3.67282 q^{46} +3.81681 q^{47} +11.0185 q^{49} +6.91764 q^{52} -9.06163 q^{53} +4.24482 q^{56} -8.10083 q^{58} +12.3496 q^{59} +8.20561 q^{61} +1.28797 q^{62} +1.00000 q^{64} -4.38880 q^{67} +5.10083 q^{68} +11.1008 q^{71} -5.38485 q^{73} -0.856013 q^{74} +1.00000 q^{76} -6.06163 q^{77} +2.14399 q^{79} +8.01847 q^{82} -1.04316 q^{83} +3.57199 q^{86} -1.42801 q^{88} +16.7305 q^{89} -29.3641 q^{91} -3.67282 q^{92} -3.81681 q^{94} -6.81286 q^{97} -11.0185 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 5 q^{11} + 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 5 q^{22} - q^{23} - 2 q^{28} + 15 q^{29} - q^{31} - 3 q^{32} - 6 q^{34} + 4 q^{37} - 3 q^{38} + 6 q^{41} - 10 q^{43} + 5 q^{44} + q^{46} + 3 q^{49} - 5 q^{53} + 2 q^{56} - 15 q^{58} + 12 q^{59} + q^{61} + q^{62} + 3 q^{64} - q^{67} + 6 q^{68} + 24 q^{71} - 9 q^{73} - 4 q^{74} + 3 q^{76} + 4 q^{77} + 5 q^{79} - 6 q^{82} - 11 q^{83} + 10 q^{86} - 5 q^{88} + 23 q^{89} - 38 q^{91} - q^{92} - 14 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 5 * q^11 + 2 * q^14 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 5 * q^22 - q^23 - 2 * q^28 + 15 * q^29 - q^31 - 3 * q^32 - 6 * q^34 + 4 * q^37 - 3 * q^38 + 6 * q^41 - 10 * q^43 + 5 * q^44 + q^46 + 3 * q^49 - 5 * q^53 + 2 * q^56 - 15 * q^58 + 12 * q^59 + q^61 + q^62 + 3 * q^64 - q^67 + 6 * q^68 + 24 * q^71 - 9 * q^73 - 4 * q^74 + 3 * q^76 + 4 * q^77 + 5 * q^79 - 6 * q^82 - 11 * q^83 + 10 * q^86 - 5 * q^88 + 23 * q^89 - 38 * q^91 - q^92 - 14 * q^97 - 3 * q^98

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