LMFDB - Embedded newform 8550.2.a.ct.1.3

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:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.87740$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.87740 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.87740 q^{7} +1.00000 q^{8} +2.40205 q^{11} +0.597951 q^{13} +2.87740 q^{14} +1.00000 q^{16} +2.87740 q^{17} -1.00000 q^{19} +2.40205 q^{22} +7.03426 q^{23} +0.597951 q^{26} +2.87740 q^{28} -6.43631 q^{29} -2.75481 q^{31} +1.00000 q^{32} +2.87740 q^{34} +2.80410 q^{37} -1.00000 q^{38} +6.27945 q^{41} +11.7158 q^{43} +2.40205 q^{44} +7.03426 q^{46} -3.08355 q^{47} +1.27945 q^{49} +0.597951 q^{52} -4.96095 q^{53} +2.87740 q^{56} -6.43631 q^{58} -1.15686 q^{59} +6.96095 q^{61} -2.75481 q^{62} +1.00000 q^{64} -10.9952 q^{67} +2.87740 q^{68} +6.07331 q^{71} -3.83836 q^{73} +2.80410 q^{74} -1.00000 q^{76} +6.91166 q^{77} -5.95071 q^{79} +6.27945 q^{82} -4.68150 q^{83} +11.7158 q^{86} +2.40205 q^{88} +11.8384 q^{89} +1.72055 q^{91} +7.03426 q^{92} -3.08355 q^{94} -8.63221 q^{97} +1.27945 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} + 5 q^{22} - q^{23} + 4 q^{26} + 2 q^{28} + 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{37} - 3 q^{38} + 10 q^{41} + 2 q^{43} + 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} + 4 q^{52} + 5 q^{53} + 2 q^{56} + 5 q^{58} + 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{68} + 16 q^{71} + 15 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} + 10 q^{82} - 3 q^{83} + 2 q^{86} + 5 q^{88} + 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} - 6 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 + 5 * q^11 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 + 5 * q^22 - q^23 + 4 * q^26 + 2 * q^28 + 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^37 - 3 * q^38 + 10 * q^41 + 2 * q^43 + 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 + 4 * q^52 + 5 * q^53 + 2 * q^56 + 5 * q^58 + 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^68 + 16 * q^71 + 15 * q^73 + 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 + 10 * q^82 - 3 * q^83 + 2 * q^86 + 5 * q^88 + 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 - 6 * q^97 - 5 * 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
$7$	2.87740	1.08756	0.543778	−	0.839229i	$-0.316993\pi$
$7$	2.87740	1.08756	0.543778	+	0.839229i	$0.316993\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.40205	0.724245	0.362122	−	0.932131i	$-0.382052\pi$
$11$	2.40205	0.724245	0.362122	+	0.932131i	$0.382052\pi$
$12$	0	0
$13$	0.597951	0.165842	0.0829209	−	0.996556i	$-0.473575\pi$
$13$	0.597951	0.165842	0.0829209	+	0.996556i	$0.473575\pi$
$14$	2.87740	0.769018
$15$	0	0
$16$	1.00000	0.250000
$17$	2.87740	0.697873	0.348936	−	0.937146i	$-0.386543\pi$
$17$	2.87740	0.697873	0.348936	+	0.937146i	$0.386543\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	2.40205	0.512118
$23$	7.03426	1.46674	0.733372	−	0.679827i	$-0.237945\pi$
$23$	7.03426	1.46674	0.733372	+	0.679827i	$0.237945\pi$
$24$	0	0
$25$	0	0
$26$	0.597951	0.117268
$27$	0	0
$28$	2.87740	0.543778
$29$	−6.43631	−1.19519	−0.597596	−	0.801797i	$-0.703877\pi$
$29$	−6.43631	−1.19519	−0.597596	+	0.801797i	$0.703877\pi$
$30$	0	0
$31$	−2.75481	−0.494778	−0.247389	−	0.968916i	$-0.579573\pi$
$31$	−2.75481	−0.494778	−0.247389	+	0.968916i	$0.579573\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.87740	0.493471
$35$	0	0
$36$	0	0
$37$	2.80410	0.460991	0.230495	−	0.973073i	$-0.425965\pi$
$37$	2.80410	0.460991	0.230495	+	0.973073i	$0.425965\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	6.27945	0.980686	0.490343	−	0.871530i	$-0.336872\pi$
$41$	6.27945	0.980686	0.490343	+	0.871530i	$0.336872\pi$
$42$	0	0
$43$	11.7158	1.78664	0.893318	−	0.449424i	$-0.148371\pi$
$43$	11.7158	1.78664	0.893318	+	0.449424i	$0.148371\pi$
$44$	2.40205	0.362122
$45$	0	0
$46$	7.03426	1.03715
$47$	−3.08355	−0.449782	−0.224891	−	0.974384i	$-0.572203\pi$
$47$	−3.08355	−0.449782	−0.224891	+	0.974384i	$0.572203\pi$
$48$	0	0
$49$	1.27945	0.182779
$50$	0	0
$51$	0	0
$52$	0.597951	0.0829209
$53$	−4.96095	−0.681439	−0.340720	−	0.940165i	$-0.610671\pi$
$53$	−4.96095	−0.681439	−0.340720	+	0.940165i	$0.610671\pi$
$54$	0	0
$55$	0	0
$56$	2.87740	0.384509
$57$	0	0
$58$	−6.43631	−0.845129
$59$	−1.15686	−0.150610	−0.0753049	−	0.997161i	$-0.523993\pi$
$59$	−1.15686	−0.150610	−0.0753049	+	0.997161i	$0.523993\pi$
$60$	0	0
$61$	6.96095	0.891259	0.445629	−	0.895218i	$-0.352980\pi$
$61$	6.96095	0.891259	0.445629	+	0.895218i	$0.352980\pi$
$62$	−2.75481	−0.349861
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−10.9952	−1.34328	−0.671640	−	0.740878i	$-0.734410\pi$
$67$	−10.9952	−1.34328	−0.671640	+	0.740878i	$0.734410\pi$
$68$	2.87740	0.348936
$69$	0	0
$70$	0	0
$71$	6.07331	0.720769	0.360384	−	0.932804i	$-0.382646\pi$
$71$	6.07331	0.720769	0.360384	+	0.932804i	$0.382646\pi$
$72$	0	0
$73$	−3.83836	−0.449246	−0.224623	−	0.974446i	$-0.572115\pi$
$73$	−3.83836	−0.449246	−0.224623	+	0.974446i	$0.572115\pi$
$74$	2.80410	0.325970
$75$	0	0
$76$	−1.00000	−0.114708
$77$	6.91166	0.787657
$78$	0	0
$79$	−5.95071	−0.669507	−0.334754	−	0.942306i	$-0.608653\pi$
$79$	−5.95071	−0.669507	−0.334754	+	0.942306i	$0.608653\pi$
$80$	0	0
$81$	0	0
$82$	6.27945	0.693450
$83$	−4.68150	−0.513861	−0.256931	−	0.966430i	$-0.582711\pi$
$83$	−4.68150	−0.513861	−0.256931	+	0.966430i	$0.582711\pi$
$84$	0	0
$85$	0	0
$86$	11.7158	1.26334
$87$	0	0
$88$	2.40205	0.256059
$89$	11.8384	1.25486	0.627432	−	0.778672i	$-0.284106\pi$
$89$	11.8384	1.25486	0.627432	+	0.778672i	$0.284106\pi$
$90$	0	0
$91$	1.72055	0.180362
$92$	7.03426	0.733372
$93$	0	0
$94$	−3.08355	−0.318044
$95$	0	0
$96$	0	0
$97$	−8.63221	−0.876468	−0.438234	−	0.898861i	$-0.644396\pi$
$97$	−8.63221	−0.876468	−0.438234	+	0.898861i	$0.644396\pi$
$98$	1.27945	0.129244
$99$	0	0
$100$	0	0
$101$	8.59795	0.855528	0.427764	−	0.903890i	$-0.359301\pi$
$101$	8.59795	0.855528	0.427764	+	0.903890i	$0.359301\pi$
$102$	0	0
$103$	3.52464	0.347294	0.173647	−	0.984808i	$-0.444445\pi$
$103$	3.52464	0.347294	0.173647	+	0.984808i	$0.444445\pi$
$104$	0.597951	0.0586340
$105$	0	0
$106$	−4.96095	−0.481850
$107$	−8.10757	−0.783788	−0.391894	−	0.920010i	$-0.628180\pi$
$107$	−8.10757	−0.783788	−0.391894	+	0.920010i	$0.628180\pi$
$108$	0	0
$109$	−4.24519	−0.406616	−0.203308	−	0.979115i	$-0.565169\pi$
$109$	−4.24519	−0.406616	−0.203308	+	0.979115i	$0.565169\pi$
$110$	0	0
$111$	0	0
$112$	2.87740	0.271889
$113$	10.3973	0.978092	0.489046	−	0.872258i	$-0.337345\pi$
$113$	10.3973	0.978092	0.489046	+	0.872258i	$0.337345\pi$
$114$	0	0
$115$	0	0
$116$	−6.43631	−0.597596
$117$	0	0
$118$	−1.15686	−0.106497
$119$	8.27945	0.758976
$120$	0	0
$121$	−5.23016	−0.475469
$122$	6.96095	0.630215
$123$	0	0
$124$	−2.75481	−0.247389
$125$	0	0
$126$	0	0
$127$	9.31371	0.826458	0.413229	−	0.910627i	$-0.364401\pi$
$127$	9.31371	0.826458	0.413229	+	0.910627i	$0.364401\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−4.19112	−0.366179	−0.183090	−	0.983096i	$-0.558610\pi$
$131$	−4.19112	−0.366179	−0.183090	+	0.983096i	$0.558610\pi$
$132$	0	0
$133$	−2.87740	−0.249503
$134$	−10.9952	−0.949842
$135$	0	0
$136$	2.87740	0.246735
$137$	13.3630	1.14168	0.570839	−	0.821062i	$-0.306618\pi$
$137$	13.3630	1.14168	0.570839	+	0.821062i	$0.306618\pi$
$138$	0	0
$139$	−14.1076	−1.19659	−0.598294	−	0.801277i	$-0.704155\pi$
$139$	−14.1076	−1.19659	−0.598294	+	0.801277i	$0.704155\pi$
$140$	0	0
$141$	0	0
$142$	6.07331	0.509661
$143$	1.43631	0.120110
$144$	0	0
$145$	0	0
$146$	−3.83836	−0.317665
$147$	0	0
$148$	2.80410	0.230495
$149$	15.2644	1.25051	0.625255	−	0.780420i	$-0.284995\pi$
$149$	15.2644	1.25051	0.625255	+	0.780420i	$0.284995\pi$
$150$	0	0
$151$	12.9850	1.05670	0.528351	−	0.849026i	$-0.322811\pi$
$151$	12.9850	1.05670	0.528351	+	0.849026i	$0.322811\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	6.91166	0.556958
$155$	0	0
$156$	0	0
$157$	−6.52464	−0.520723	−0.260362	−	0.965511i	$-0.583842\pi$
$157$	−6.52464	−0.520723	−0.260362	+	0.965511i	$0.583842\pi$
$158$	−5.95071	−0.473413
$159$	0	0
$160$	0	0
$161$	20.2404	1.59517
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	6.27945	0.490343
$165$	0	0
$166$	−4.68150	−0.363355
$167$	−15.6767	−1.21310	−0.606550	−	0.795045i	$-0.707447\pi$
$167$	−15.6767	−1.21310	−0.606550	+	0.795045i	$0.707447\pi$
$168$	0	0
$169$	−12.6425	−0.972496
$170$	0	0
$171$	0	0
$172$	11.7158	0.893318
$173$	−17.9459	−1.36440	−0.682202	−	0.731164i	$-0.738977\pi$
$173$	−17.9459	−1.36440	−0.682202	+	0.731164i	$0.738977\pi$
$174$	0	0
$175$	0	0
$176$	2.40205	0.181061
$177$	0	0
$178$	11.8384	0.887322
$179$	−2.70552	−0.202220	−0.101110	−	0.994875i	$-0.532239\pi$
$179$	−2.70552	−0.202220	−0.101110	+	0.994875i	$0.532239\pi$
$180$	0	0
$181$	−12.2747	−0.912369	−0.456184	−	0.889885i	$-0.650784\pi$
$181$	−12.2747	−0.912369	−0.456184	+	0.889885i	$0.650784\pi$
$182$	1.72055	0.127535
$183$	0	0
$184$	7.03426	0.518573
$185$	0	0
$186$	0	0
$187$	6.91166	0.505431
$188$	−3.08355	−0.224891
$189$	0	0
$190$	0	0
$191$	9.80410	0.709400	0.354700	−	0.934980i	$-0.384583\pi$
$191$	9.80410	0.709400	0.354700	+	0.934980i	$0.384583\pi$
$192$	0	0
$193$	22.7993	1.64113	0.820565	−	0.571553i	$-0.193659\pi$
$193$	22.7993	1.64113	0.820565	+	0.571553i	$0.193659\pi$
$194$	−8.63221	−0.619757
$195$	0	0
$196$	1.27945	0.0913895
$197$	−0.352759	−0.0251330	−0.0125665	−	0.999921i	$-0.504000\pi$
$197$	−0.352759	−0.0251330	−0.0125665	+	0.999921i	$0.504000\pi$
$198$	0	0
$199$	−24.7260	−1.75278	−0.876390	−	0.481602i	$-0.840055\pi$
$199$	−24.7260	−1.75278	−0.876390	+	0.481602i	$0.840055\pi$
$200$	0	0
$201$	0	0
$202$	8.59795	0.604950
$203$	−18.5199	−1.29984
$204$	0	0
$205$	0	0
$206$	3.52464	0.245574
$207$	0	0
$208$	0.597951	0.0414605
$209$	−2.40205	−0.166153
$210$	0	0
$211$	−25.3432	−1.74470	−0.872348	−	0.488885i	$-0.837404\pi$
$211$	−25.3432	−1.74470	−0.872348	+	0.488885i	$0.837404\pi$
$212$	−4.96095	−0.340720
$213$	0	0
$214$	−8.10757	−0.554222
$215$	0	0
$216$	0	0
$217$	−7.92669	−0.538099
$218$	−4.24519	−0.287521
$219$	0	0
$220$	0	0
$221$	1.72055	0.115737
$222$	0	0
$223$	−15.9069	−1.06520	−0.532602	−	0.846366i	$-0.678786\pi$
$223$	−15.9069	−1.06520	−0.532602	+	0.846366i	$0.678786\pi$
$224$	2.87740	0.192255
$225$	0	0
$226$	10.3973	0.691616
$227$	18.5589	1.23180	0.615899	−	0.787825i	$-0.288793\pi$
$227$	18.5589	1.23180	0.615899	+	0.787825i	$0.288793\pi$
$228$	0	0
$229$	16.7500	1.10687	0.553436	−	0.832892i	$-0.313316\pi$
$229$	16.7500	1.10687	0.553436	+	0.832892i	$0.313316\pi$
$230$	0	0
$231$	0	0
$232$	−6.43631	−0.422564
$233$	−13.0240	−0.853232	−0.426616	−	0.904433i	$-0.640294\pi$
$233$	−13.0240	−0.853232	−0.426616	+	0.904433i	$0.640294\pi$
$234$	0	0
$235$	0	0
$236$	−1.15686	−0.0753049
$237$	0	0
$238$	8.27945	0.536677
$239$	23.9357	1.54827	0.774135	−	0.633020i	$-0.218185\pi$
$239$	23.9357	1.54827	0.774135	+	0.633020i	$0.218185\pi$
$240$	0	0
$241$	14.3870	0.926749	0.463375	−	0.886162i	$-0.346638\pi$
$241$	14.3870	0.926749	0.463375	+	0.886162i	$0.346638\pi$
$242$	−5.23016	−0.336208
$243$	0	0
$244$	6.96095	0.445629
$245$	0	0
$246$	0	0
$247$	−0.597951	−0.0380467
$248$	−2.75481	−0.174930
$249$	0	0
$250$	0	0
$251$	20.7398	1.30908	0.654542	−	0.756026i	$-0.272862\pi$
$251$	20.7398	1.30908	0.654542	+	0.756026i	$0.272862\pi$
$252$	0	0
$253$	16.8966	1.06228
$254$	9.31371	0.584394
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.0343	1.06257	0.531284	−	0.847194i	$-0.321710\pi$
$257$	17.0343	1.06257	0.531284	+	0.847194i	$0.321710\pi$
$258$	0	0
$259$	8.06852	0.501353
$260$	0	0
$261$	0	0
$262$	−4.19112	−0.258928
$263$	−13.5246	−0.833965	−0.416983	−	0.908914i	$-0.636912\pi$
$263$	−13.5246	−0.833965	−0.416983	+	0.908914i	$0.636912\pi$
$264$	0	0
$265$	0	0
$266$	−2.87740	−0.176425
$267$	0	0
$268$	−10.9952	−0.671640
$269$	0.804097	0.0490267	0.0245133	−	0.999700i	$-0.492196\pi$
$269$	0.804097	0.0490267	0.0245133	+	0.999700i	$0.492196\pi$
$270$	0	0
$271$	25.1911	1.53025	0.765126	−	0.643881i	$-0.222677\pi$
$271$	25.1911	1.53025	0.765126	+	0.643881i	$0.222677\pi$
$272$	2.87740	0.174468
$273$	0	0
$274$	13.3630	0.807288
$275$	0	0
$276$	0	0
$277$	−16.3035	−0.979581	−0.489790	−	0.871840i	$-0.662927\pi$
$277$	−16.3035	−0.979581	−0.489790	+	0.871840i	$0.662927\pi$
$278$	−14.1076	−0.846116
$279$	0	0
$280$	0	0
$281$	14.1959	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$281$	14.1959	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$282$	0	0
$283$	6.95071	0.413177	0.206588	−	0.978428i	$-0.433764\pi$
$283$	6.95071	0.413177	0.206588	+	0.978428i	$0.433764\pi$
$284$	6.07331	0.360384
$285$	0	0
$286$	1.43631	0.0849307
$287$	18.0685	1.06655
$288$	0	0
$289$	−8.72055	−0.512973
$290$	0	0
$291$	0	0
$292$	−3.83836	−0.224623
$293$	−16.3035	−0.952459	−0.476229	−	0.879321i	$-0.657997\pi$
$293$	−16.3035	−0.952459	−0.476229	+	0.879321i	$0.657997\pi$
$294$	0	0
$295$	0	0
$296$	2.80410	0.162985
$297$	0	0
$298$	15.2644	0.884244
$299$	4.20615	0.243248
$300$	0	0
$301$	33.7110	1.94307
$302$	12.9850	0.747201
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−5.73079	−0.327073	−0.163537	−	0.986537i	$-0.552290\pi$
$307$	−5.73079	−0.327073	−0.163537	+	0.986537i	$0.552290\pi$
$308$	6.91166	0.393829
$309$	0	0
$310$	0	0
$311$	−3.23016	−0.183166	−0.0915829	−	0.995797i	$-0.529193\pi$
$311$	−3.23016	−0.183166	−0.0915829	+	0.995797i	$0.529193\pi$
$312$	0	0
$313$	12.2987	0.695163	0.347581	−	0.937650i	$-0.387003\pi$
$313$	12.2987	0.695163	0.347581	+	0.937650i	$0.387003\pi$
$314$	−6.52464	−0.368207
$315$	0	0
$316$	−5.95071	−0.334754
$317$	13.5637	0.761813	0.380906	−	0.924614i	$-0.375612\pi$
$317$	13.5637	0.761813	0.380906	+	0.924614i	$0.375612\pi$
$318$	0	0
$319$	−15.4603	−0.865612
$320$	0	0
$321$	0	0
$322$	20.2404	1.12795
$323$	−2.87740	−0.160103
$324$	0	0
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	6.27945	0.346725
$329$	−8.87262	−0.489163
$330$	0	0
$331$	22.9952	1.26393	0.631966	−	0.774996i	$-0.282248\pi$
$331$	22.9952	1.26393	0.631966	+	0.774996i	$0.282248\pi$
$332$	−4.68150	−0.256931
$333$	0	0
$334$	−15.6767	−0.857792
$335$	0	0
$336$	0	0
$337$	−15.2644	−0.831506	−0.415753	−	0.909478i	$-0.636482\pi$
$337$	−15.2644	−0.831506	−0.415753	+	0.909478i	$0.636482\pi$
$338$	−12.6425	−0.687659
$339$	0	0
$340$	0	0
$341$	−6.61718	−0.358340
$342$	0	0
$343$	−16.4603	−0.888774
$344$	11.7158	0.631671
$345$	0	0
$346$	−17.9459	−0.964779
$347$	8.41229	0.451595	0.225798	−	0.974174i	$-0.427501\pi$
$347$	8.41229	0.451595	0.225798	+	0.974174i	$0.427501\pi$
$348$	0	0
$349$	27.7993	1.48806	0.744031	−	0.668145i	$-0.232911\pi$
$349$	27.7993	1.48806	0.744031	+	0.668145i	$0.232911\pi$
$350$	0	0
$351$	0	0
$352$	2.40205	0.128030
$353$	11.3582	0.604537	0.302268	−	0.953223i	$-0.402256\pi$
$353$	11.3582	0.604537	0.302268	+	0.953223i	$0.402256\pi$
$354$	0	0
$355$	0	0
$356$	11.8384	0.627432
$357$	0	0
$358$	−2.70552	−0.142991
$359$	8.91645	0.470592	0.235296	−	0.971924i	$-0.424394\pi$
$359$	8.91645	0.470592	0.235296	+	0.971924i	$0.424394\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.2747	−0.645142
$363$	0	0
$364$	1.72055	0.0901812
$365$	0	0
$366$	0	0
$367$	−2.07331	−0.108226	−0.0541129	−	0.998535i	$-0.517233\pi$
$367$	−2.07331	−0.108226	−0.0541129	+	0.998535i	$0.517233\pi$
$368$	7.03426	0.366686
$369$	0	0
$370$	0	0
$371$	−14.2747	−0.741104
$372$	0	0
$373$	−4.51986	−0.234029	−0.117015	−	0.993130i	$-0.537332\pi$
$373$	−4.51986	−0.234029	−0.117015	+	0.993130i	$0.537332\pi$
$374$	6.91166	0.357394
$375$	0	0
$376$	−3.08355	−0.159022
$377$	−3.84860	−0.198213
$378$	0	0
$379$	−7.29448	−0.374692	−0.187346	−	0.982294i	$-0.559989\pi$
$379$	−7.29448	−0.374692	−0.187346	+	0.982294i	$0.559989\pi$
$380$	0	0
$381$	0	0
$382$	9.80410	0.501621
$383$	−25.6034	−1.30827	−0.654136	−	0.756377i	$-0.726968\pi$
$383$	−25.6034	−1.30827	−0.654136	+	0.756377i	$0.726968\pi$
$384$	0	0
$385$	0	0
$386$	22.7993	1.16045
$387$	0	0
$388$	−8.63221	−0.438234
$389$	−7.36300	−0.373319	−0.186660	−	0.982425i	$-0.559766\pi$
$389$	−7.36300	−0.373319	−0.186660	+	0.982425i	$0.559766\pi$
$390$	0	0
$391$	20.2404	1.02360
$392$	1.27945	0.0646221
$393$	0	0
$394$	−0.352759	−0.0177717
$395$	0	0
$396$	0	0
$397$	−11.4994	−0.577137	−0.288568	−	0.957459i	$-0.593179\pi$
$397$	−11.4994	−0.577137	−0.288568	+	0.957459i	$0.593179\pi$
$398$	−24.7260	−1.23940
$399$	0	0
$400$	0	0
$401$	−7.95071	−0.397040	−0.198520	−	0.980097i	$-0.563613\pi$
$401$	−7.95071	−0.397040	−0.198520	+	0.980097i	$0.563613\pi$
$402$	0	0
$403$	−1.64724	−0.0820549
$404$	8.59795	0.427764
$405$	0	0
$406$	−18.5199	−0.919125
$407$	6.73558	0.333870
$408$	0	0
$409$	9.19590	0.454708	0.227354	−	0.973812i	$-0.426993\pi$
$409$	9.19590	0.454708	0.227354	+	0.973812i	$0.426993\pi$
$410$	0	0
$411$	0	0
$412$	3.52464	0.173647
$413$	−3.32874	−0.163797
$414$	0	0
$415$	0	0
$416$	0.597951	0.0293170
$417$	0	0
$418$	−2.40205	−0.117488
$419$	11.3630	0.555119	0.277559	−	0.960708i	$-0.410474\pi$
$419$	11.3630	0.555119	0.277559	+	0.960708i	$0.410474\pi$
$420$	0	0
$421$	−13.5781	−0.661758	−0.330879	−	0.943673i	$-0.607345\pi$
$421$	−13.5781	−0.661758	−0.330879	+	0.943673i	$0.607345\pi$
$422$	−25.3432	−1.23369
$423$	0	0
$424$	−4.96095	−0.240925
$425$	0	0
$426$	0	0
$427$	20.0295	0.969294
$428$	−8.10757	−0.391894
$429$	0	0
$430$	0	0
$431$	25.6562	1.23582	0.617909	−	0.786250i	$-0.287980\pi$
$431$	25.6562	1.23582	0.617909	+	0.786250i	$0.287980\pi$
$432$	0	0
$433$	−31.8966	−1.53285	−0.766427	−	0.642331i	$-0.777967\pi$
$433$	−31.8966	−1.53285	−0.766427	+	0.642331i	$0.777967\pi$
$434$	−7.92669	−0.380493
$435$	0	0
$436$	−4.24519	−0.203308
$437$	−7.03426	−0.336494
$438$	0	0
$439$	−15.9712	−0.762264	−0.381132	−	0.924521i	$-0.624466\pi$
$439$	−15.9712	−0.762264	−0.381132	+	0.924521i	$0.624466\pi$
$440$	0	0
$441$	0	0
$442$	1.72055	0.0818381
$443$	−24.6815	−1.17265	−0.586327	−	0.810075i	$-0.699427\pi$
$443$	−24.6815	−1.17265	−0.586327	+	0.810075i	$0.699427\pi$
$444$	0	0
$445$	0	0
$446$	−15.9069	−0.753212
$447$	0	0
$448$	2.87740	0.135945
$449$	−13.1028	−0.618358	−0.309179	−	0.951004i	$-0.600054\pi$
$449$	−13.1028	−0.618358	−0.309179	+	0.951004i	$0.600054\pi$
$450$	0	0
$451$	15.0835	0.710257
$452$	10.3973	0.489046
$453$	0	0
$454$	18.5589	0.871013
$455$	0	0
$456$	0	0
$457$	−3.18213	−0.148854	−0.0744269	−	0.997226i	$-0.523713\pi$
$457$	−3.18213	−0.148854	−0.0744269	+	0.997226i	$0.523713\pi$
$458$	16.7500	0.782677
$459$	0	0
$460$	0	0
$461$	28.5589	1.33012	0.665060	−	0.746790i	$-0.268406\pi$
$461$	28.5589	1.33012	0.665060	+	0.746790i	$0.268406\pi$
$462$	0	0
$463$	−5.82333	−0.270633	−0.135316	−	0.990802i	$-0.543205\pi$
$463$	−5.82333	−0.270633	−0.135316	+	0.990802i	$0.543205\pi$
$464$	−6.43631	−0.298798
$465$	0	0
$466$	−13.0240	−0.603326
$467$	39.0980	1.80924	0.904620	−	0.426220i	$-0.140155\pi$
$467$	39.0980	1.80924	0.904620	+	0.426220i	$0.140155\pi$
$468$	0	0
$469$	−31.6377	−1.46089
$470$	0	0
$471$	0	0
$472$	−1.15686	−0.0532486
$473$	28.1418	1.29396
$474$	0	0
$475$	0	0
$476$	8.27945	0.379488
$477$	0	0
$478$	23.9357	1.09479
$479$	−9.35755	−0.427557	−0.213779	−	0.976882i	$-0.568577\pi$
$479$	−9.35755	−0.427557	−0.213779	+	0.976882i	$0.568577\pi$
$480$	0	0
$481$	1.67671	0.0764516
$482$	14.3870	0.655311
$483$	0	0
$484$	−5.23016	−0.237735
$485$	0	0
$486$	0	0
$487$	10.2452	0.464254	0.232127	−	0.972685i	$-0.425432\pi$
$487$	10.2452	0.464254	0.232127	+	0.972685i	$0.425432\pi$
$488$	6.96095	0.315108
$489$	0	0
$490$	0	0
$491$	−5.34252	−0.241104	−0.120552	−	0.992707i	$-0.538467\pi$
$491$	−5.34252	−0.241104	−0.120552	+	0.992707i	$0.538467\pi$
$492$	0	0
$493$	−18.5199	−0.834092
$494$	−0.597951	−0.0269031
$495$	0	0
$496$	−2.75481	−0.123695
$497$	17.4754	0.783877
$498$	0	0
$499$	−2.25418	−0.100911	−0.0504555	−	0.998726i	$-0.516067\pi$
$499$	−2.25418	−0.100911	−0.0504555	+	0.998726i	$0.516067\pi$
$500$	0	0
$501$	0	0
$502$	20.7398	0.925662
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	16.8966	0.751147
$507$	0	0
$508$	9.31371	0.413229
$509$	−7.49937	−0.332404	−0.166202	−	0.986092i	$-0.553150\pi$
$509$	−7.49937	−0.332404	−0.166202	+	0.986092i	$0.553150\pi$
$510$	0	0
$511$	−11.0445	−0.488580
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.0343	0.751349
$515$	0	0
$516$	0	0
$517$	−7.40684	−0.325752
$518$	8.06852	0.354510
$519$	0	0
$520$	0	0
$521$	27.7740	1.21680	0.608401	−	0.793630i	$-0.291811\pi$
$521$	27.7740	1.21680	0.608401	+	0.793630i	$0.291811\pi$
$522$	0	0
$523$	−9.19590	−0.402109	−0.201054	−	0.979580i	$-0.564437\pi$
$523$	−9.19590	−0.402109	−0.201054	+	0.979580i	$0.564437\pi$
$524$	−4.19112	−0.183090
$525$	0	0
$526$	−13.5246	−0.589703
$527$	−7.92669	−0.345292
$528$	0	0
$529$	26.4808	1.15134
$530$	0	0
$531$	0	0
$532$	−2.87740	−0.124751
$533$	3.75481	0.162639
$534$	0	0
$535$	0	0
$536$	−10.9952	−0.474921
$537$	0	0
$538$	0.804097	0.0346671
$539$	3.07331	0.132377
$540$	0	0
$541$	7.46511	0.320950	0.160475	−	0.987040i	$-0.448697\pi$
$541$	7.46511	0.320950	0.160475	+	0.987040i	$0.448697\pi$
$542$	25.1911	1.08205
$543$	0	0
$544$	2.87740	0.123368
$545$	0	0
$546$	0	0
$547$	−8.96095	−0.383143	−0.191571	−	0.981479i	$-0.561358\pi$
$547$	−8.96095	−0.383143	−0.191571	+	0.981479i	$0.561358\pi$
$548$	13.3630	0.570839
$549$	0	0
$550$	0	0
$551$	6.43631	0.274196
$552$	0	0
$553$	−17.1226	−0.728127
$554$	−16.3035	−0.692668
$555$	0	0
$556$	−14.1076	−0.598294
$557$	−20.1761	−0.854888	−0.427444	−	0.904042i	$-0.640586\pi$
$557$	−20.1761	−0.854888	−0.427444	+	0.904042i	$0.640586\pi$
$558$	0	0
$559$	7.00546	0.296299
$560$	0	0
$561$	0	0
$562$	14.1959	0.598818
$563$	12.3732	0.521470	0.260735	−	0.965410i	$-0.416035\pi$
$563$	12.3732	0.521470	0.260735	+	0.965410i	$0.416035\pi$
$564$	0	0
$565$	0	0
$566$	6.95071	0.292160
$567$	0	0
$568$	6.07331	0.254830
$569$	−7.68629	−0.322226	−0.161113	−	0.986936i	$-0.551508\pi$
$569$	−7.68629	−0.322226	−0.161113	+	0.986936i	$0.551508\pi$
$570$	0	0
$571$	0.843144	0.0352845	0.0176422	−	0.999844i	$-0.494384\pi$
$571$	0.843144	0.0352845	0.0176422	+	0.999844i	$0.494384\pi$
$572$	1.43631	0.0600551
$573$	0	0
$574$	18.0685	0.754165
$575$	0	0
$576$	0	0
$577$	−0.539675	−0.0224670	−0.0112335	−	0.999937i	$-0.503576\pi$
$577$	−0.539675	−0.0224670	−0.0112335	+	0.999937i	$0.503576\pi$
$578$	−8.72055	−0.362727
$579$	0	0
$580$	0	0
$581$	−13.4706	−0.558853
$582$	0	0
$583$	−11.9165	−0.493529
$584$	−3.83836	−0.158832
$585$	0	0
$586$	−16.3035	−0.673490
$587$	36.0925	1.48970	0.744849	−	0.667233i	$-0.232521\pi$
$587$	36.0925	1.48970	0.744849	+	0.667233i	$0.232521\pi$
$588$	0	0
$589$	2.75481	0.113510
$590$	0	0
$591$	0	0
$592$	2.80410	0.115248
$593$	−39.1082	−1.60598	−0.802991	−	0.595991i	$-0.796760\pi$
$593$	−39.1082	−1.60598	−0.802991	+	0.595991i	$0.796760\pi$
$594$	0	0
$595$	0	0
$596$	15.2644	0.625255
$597$	0	0
$598$	4.20615	0.172002
$599$	21.4267	0.875473	0.437736	−	0.899103i	$-0.355780\pi$
$599$	21.4267	0.875473	0.437736	+	0.899103i	$0.355780\pi$
$600$	0	0
$601$	−25.8486	−1.05439	−0.527193	−	0.849745i	$-0.676756\pi$
$601$	−25.8486	−1.05439	−0.527193	+	0.849745i	$0.676756\pi$
$602$	33.7110	1.37396
$603$	0	0
$604$	12.9850	0.528351
$605$	0	0
$606$	0	0
$607$	40.0535	1.62572	0.812860	−	0.582458i	$-0.197909\pi$
$607$	40.0535	1.62572	0.812860	+	0.582458i	$0.197909\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−1.84381	−0.0745927
$612$	0	0
$613$	−0.323286	−0.0130574	−0.00652870	−	0.999979i	$-0.502078\pi$
$613$	−0.323286	−0.0130574	−0.00652870	+	0.999979i	$0.502078\pi$
$614$	−5.73079	−0.231276
$615$	0	0
$616$	6.91166	0.278479
$617$	35.0397	1.41065	0.705323	−	0.708886i	$-0.250802\pi$
$617$	35.0397	1.41065	0.705323	+	0.708886i	$0.250802\pi$
$618$	0	0
$619$	−2.13763	−0.0859184	−0.0429592	−	0.999077i	$-0.513679\pi$
$619$	−2.13763	−0.0859184	−0.0429592	+	0.999077i	$0.513679\pi$
$620$	0	0
$621$	0	0
$622$	−3.23016	−0.129518
$623$	34.0637	1.36473
$624$	0	0
$625$	0	0
$626$	12.2987	0.491554
$627$	0	0
$628$	−6.52464	−0.260362
$629$	8.06852	0.321713
$630$	0	0
$631$	−4.23562	−0.168617	−0.0843087	−	0.996440i	$-0.526868\pi$
$631$	−4.23562	−0.168617	−0.0843087	+	0.996440i	$0.526868\pi$
$632$	−5.95071	−0.236707
$633$	0	0
$634$	13.5637	0.538683
$635$	0	0
$636$	0	0
$637$	0.765050	0.0303124
$638$	−15.4603	−0.612080
$639$	0	0
$640$	0	0
$641$	16.5932	0.655391	0.327695	−	0.944783i	$-0.393728\pi$
$641$	16.5932	0.655391	0.327695	+	0.944783i	$0.393728\pi$
$642$	0	0
$643$	33.5301	1.32230	0.661149	−	0.750255i	$-0.270069\pi$
$643$	33.5301	1.32230	0.661149	+	0.750255i	$0.270069\pi$
$644$	20.2404	0.797584
$645$	0	0
$646$	−2.87740	−0.113210
$647$	−32.7891	−1.28907	−0.644536	−	0.764574i	$-0.722949\pi$
$647$	−32.7891	−1.28907	−0.644536	+	0.764574i	$0.722949\pi$
$648$	0	0
$649$	−2.77882	−0.109078
$650$	0	0
$651$	0	0
$652$	6.00000	0.234978
$653$	18.6274	0.728947	0.364474	−	0.931214i	$-0.381249\pi$
$653$	18.6274	0.728947	0.364474	+	0.931214i	$0.381249\pi$
$654$	0	0
$655$	0	0
$656$	6.27945	0.245171
$657$	0	0
$658$	−8.87262	−0.345891
$659$	−14.4213	−0.561773	−0.280887	−	0.959741i	$-0.590628\pi$
$659$	−14.4213	−0.561773	−0.280887	+	0.959741i	$0.590628\pi$
$660$	0	0
$661$	12.3822	0.481613	0.240806	−	0.970573i	$-0.422588\pi$
$661$	12.3822	0.481613	0.240806	+	0.970573i	$0.422588\pi$
$662$	22.9952	0.893734
$663$	0	0
$664$	−4.68150	−0.181677
$665$	0	0
$666$	0	0
$667$	−45.2747	−1.75304
$668$	−15.6767	−0.606550
$669$	0	0
$670$	0	0
$671$	16.7205	0.645490
$672$	0	0
$673$	−17.8281	−0.687223	−0.343612	−	0.939112i	$-0.611650\pi$
$673$	−17.8281	−0.687223	−0.343612	+	0.939112i	$0.611650\pi$
$674$	−15.2644	−0.587964
$675$	0	0
$676$	−12.6425	−0.486248
$677$	−43.6364	−1.67708	−0.838542	−	0.544837i	$-0.816591\pi$
$677$	−43.6364	−1.67708	−0.838542	+	0.544837i	$0.816591\pi$
$678$	0	0
$679$	−24.8384	−0.953209
$680$	0	0
$681$	0	0
$682$	−6.61718	−0.253385
$683$	−16.6665	−0.637725	−0.318862	−	0.947801i	$-0.603301\pi$
$683$	−16.6665	−0.637725	−0.318862	+	0.947801i	$0.603301\pi$
$684$	0	0
$685$	0	0
$686$	−16.4603	−0.628458
$687$	0	0
$688$	11.7158	0.446659
$689$	−2.96641	−0.113011
$690$	0	0
$691$	−11.1364	−0.423647	−0.211824	−	0.977308i	$-0.567940\pi$
$691$	−11.1364	−0.423647	−0.211824	+	0.977308i	$0.567940\pi$
$692$	−17.9459	−0.682202
$693$	0	0
$694$	8.41229	0.319326
$695$	0	0
$696$	0	0
$697$	18.0685	0.684394
$698$	27.7993	1.05222
$699$	0	0
$700$	0	0
$701$	24.1671	0.912779	0.456389	−	0.889780i	$-0.349142\pi$
$701$	24.1671	0.912779	0.456389	+	0.889780i	$0.349142\pi$
$702$	0	0
$703$	−2.80410	−0.105759
$704$	2.40205	0.0905306
$705$	0	0
$706$	11.3582	0.427472
$707$	24.7398	0.930435
$708$	0	0
$709$	5.95550	0.223663	0.111832	−	0.993727i	$-0.464328\pi$
$709$	5.95550	0.223663	0.111832	+	0.993727i	$0.464328\pi$
$710$	0	0
$711$	0	0
$712$	11.8384	0.443661
$713$	−19.3780	−0.725713
$714$	0	0
$715$	0	0
$716$	−2.70552	−0.101110
$717$	0	0
$718$	8.91645	0.332759
$719$	−28.5000	−1.06287	−0.531436	−	0.847098i	$-0.678347\pi$
$719$	−28.5000	−1.06287	−0.531436	+	0.847098i	$0.678347\pi$
$720$	0	0
$721$	10.1418	0.377701
$722$	1.00000	0.0372161
$723$	0	0
$724$	−12.2747	−0.456184
$725$	0	0
$726$	0	0
$727$	−44.4808	−1.64970	−0.824851	−	0.565350i	$-0.808741\pi$
$727$	−44.4808	−1.64970	−0.824851	+	0.565350i	$0.808741\pi$
$728$	1.72055	0.0637677
$729$	0	0
$730$	0	0
$731$	33.7110	1.24685
$732$	0	0
$733$	−17.3390	−0.640430	−0.320215	−	0.947345i	$-0.603755\pi$
$733$	−17.3390	−0.640430	−0.320215	+	0.947345i	$0.603755\pi$
$734$	−2.07331	−0.0765271
$735$	0	0
$736$	7.03426	0.259286
$737$	−26.4110	−0.972863
$738$	0	0
$739$	−24.7055	−0.908807	−0.454404	−	0.890796i	$-0.650148\pi$
$739$	−24.7055	−0.908807	−0.454404	+	0.890796i	$0.650148\pi$
$740$	0	0
$741$	0	0
$742$	−14.2747	−0.524039
$743$	36.6959	1.34624	0.673122	−	0.739532i	$-0.264953\pi$
$743$	36.6959	1.34624	0.673122	+	0.739532i	$0.264953\pi$
$744$	0	0
$745$	0	0
$746$	−4.51986	−0.165484
$747$	0	0
$748$	6.91166	0.252715
$749$	−23.3287	−0.852414
$750$	0	0
$751$	7.85759	0.286727	0.143364	−	0.989670i	$-0.454208\pi$
$751$	7.85759	0.286727	0.143364	+	0.989670i	$0.454208\pi$
$752$	−3.08355	−0.112445
$753$	0	0
$754$	−3.84860	−0.140158
$755$	0	0
$756$	0	0
$757$	−38.6034	−1.40306	−0.701532	−	0.712638i	$-0.747500\pi$
$757$	−38.6034	−1.40306	−0.701532	+	0.712638i	$0.747500\pi$
$758$	−7.29448	−0.264948
$759$	0	0
$760$	0	0
$761$	−2.02527	−0.0734161	−0.0367080	−	0.999326i	$-0.511687\pi$
$761$	−2.02527	−0.0734161	−0.0367080	+	0.999326i	$0.511687\pi$
$762$	0	0
$763$	−12.2151	−0.442217
$764$	9.80410	0.354700
$765$	0	0
$766$	−25.6034	−0.925089
$767$	−0.691744	−0.0249774
$768$	0	0
$769$	37.7302	1.36059	0.680293	−	0.732940i	$-0.261853\pi$
$769$	37.7302	1.36059	0.680293	+	0.732940i	$0.261853\pi$
$770$	0	0
$771$	0	0
$772$	22.7993	0.820565
$773$	22.3617	0.804296	0.402148	−	0.915575i	$-0.368264\pi$
$773$	22.3617	0.804296	0.402148	+	0.915575i	$0.368264\pi$
$774$	0	0
$775$	0	0
$776$	−8.63221	−0.309878
$777$	0	0
$778$	−7.36300	−0.263976
$779$	−6.27945	−0.224985
$780$	0	0
$781$	14.5884	0.522013
$782$	20.2404	0.723795
$783$	0	0
$784$	1.27945	0.0456947
$785$	0	0
$786$	0	0
$787$	−23.5336	−0.838883	−0.419442	−	0.907782i	$-0.637774\pi$
$787$	−23.5336	−0.838883	−0.419442	+	0.907782i	$0.637774\pi$
$788$	−0.352759	−0.0125665
$789$	0	0
$790$	0	0
$791$	29.9171	1.06373
$792$	0	0
$793$	4.16231	0.147808
$794$	−11.4994	−0.408097
$795$	0	0
$796$	−24.7260	−0.876390
$797$	18.3275	0.649193	0.324596	−	0.945853i	$-0.394772\pi$
$797$	18.3275	0.649193	0.324596	+	0.945853i	$0.394772\pi$
$798$	0	0
$799$	−8.87262	−0.313891
$800$	0	0
$801$	0	0
$802$	−7.95071	−0.280749
$803$	−9.21992	−0.325364
$804$	0	0
$805$	0	0
$806$	−1.64724	−0.0580216
$807$	0	0
$808$	8.59795	0.302475
$809$	53.6719	1.88700	0.943502	−	0.331366i	$-0.107510\pi$
$809$	53.6719	1.88700	0.943502	+	0.331366i	$0.107510\pi$
$810$	0	0
$811$	−0.548662	−0.0192661	−0.00963306	−	0.999954i	$-0.503066\pi$
$811$	−0.548662	−0.0192661	−0.00963306	+	0.999954i	$0.503066\pi$
$812$	−18.5199	−0.649920
$813$	0	0
$814$	6.73558	0.236082
$815$	0	0
$816$	0	0
$817$	−11.7158	−0.409883
$818$	9.19590	0.321527
$819$	0	0
$820$	0	0
$821$	−17.4706	−0.609727	−0.304864	−	0.952396i	$-0.598611\pi$
$821$	−17.4706	−0.609727	−0.304864	+	0.952396i	$0.598611\pi$
$822$	0	0
$823$	19.0192	0.662969	0.331484	−	0.943461i	$-0.392451\pi$
$823$	19.0192	0.662969	0.331484	+	0.943461i	$0.392451\pi$
$824$	3.52464	0.122787
$825$	0	0
$826$	−3.32874	−0.115822
$827$	−36.4898	−1.26887	−0.634437	−	0.772974i	$-0.718768\pi$
$827$	−36.4898	−1.26887	−0.634437	+	0.772974i	$0.718768\pi$
$828$	0	0
$829$	−21.4911	−0.746415	−0.373208	−	0.927748i	$-0.621742\pi$
$829$	−21.4911	−0.746415	−0.373208	+	0.927748i	$0.621742\pi$
$830$	0	0
$831$	0	0
$832$	0.597951	0.0207302
$833$	3.68150	0.127556
$834$	0	0
$835$	0	0
$836$	−2.40205	−0.0830766
$837$	0	0
$838$	11.3630	0.392528
$839$	−20.4856	−0.707241	−0.353621	−	0.935389i	$-0.615050\pi$
$839$	−20.4856	−0.707241	−0.353621	+	0.935389i	$0.615050\pi$
$840$	0	0
$841$	12.4261	0.428485
$842$	−13.5781	−0.467933
$843$	0	0
$844$	−25.3432	−0.872348
$845$	0	0
$846$	0	0
$847$	−15.0493	−0.517100
$848$	−4.96095	−0.170360
$849$	0	0
$850$	0	0
$851$	19.7247	0.676156
$852$	0	0
$853$	−40.8726	−1.39945	−0.699726	−	0.714411i	$-0.746695\pi$
$853$	−40.8726	−1.39945	−0.699726	+	0.714411i	$0.746695\pi$
$854$	20.0295	0.685394
$855$	0	0
$856$	−8.10757	−0.277111
$857$	19.2782	0.658531	0.329265	−	0.944237i	$-0.393199\pi$
$857$	19.2782	0.658531	0.329265	+	0.944237i	$0.393199\pi$
$858$	0	0
$859$	−43.1863	−1.47350	−0.736749	−	0.676166i	$-0.763640\pi$
$859$	−43.1863	−1.47350	−0.736749	+	0.676166i	$0.763640\pi$
$860$	0	0
$861$	0	0
$862$	25.6562	0.873855
$863$	−24.3137	−0.827648	−0.413824	−	0.910357i	$-0.635807\pi$
$863$	−24.3137	−0.827648	−0.413824	+	0.910357i	$0.635807\pi$
$864$	0	0
$865$	0	0
$866$	−31.8966	−1.08389
$867$	0	0
$868$	−7.92669	−0.269050
$869$	−14.2939	−0.484887
$870$	0	0
$871$	−6.57460	−0.222772
$872$	−4.24519	−0.143760
$873$	0	0
$874$	−7.03426	−0.237937
$875$	0	0
$876$	0	0
$877$	−35.8438	−1.21036	−0.605180	−	0.796089i	$-0.706899\pi$
$877$	−35.8438	−1.21036	−0.605180	+	0.796089i	$0.706899\pi$
$878$	−15.9712	−0.539002
$879$	0	0
$880$	0	0
$881$	−11.8233	−0.398338	−0.199169	−	0.979965i	$-0.563824\pi$
$881$	−11.8233	−0.398338	−0.199169	+	0.979965i	$0.563824\pi$
$882$	0	0
$883$	−39.3925	−1.32566	−0.662831	−	0.748769i	$-0.730645\pi$
$883$	−39.3925	−1.32566	−0.662831	+	0.748769i	$0.730645\pi$
$884$	1.72055	0.0578683
$885$	0	0
$886$	−24.6815	−0.829191
$887$	−4.90268	−0.164616	−0.0823079	−	0.996607i	$-0.526229\pi$
$887$	−4.90268	−0.164616	−0.0823079	+	0.996607i	$0.526229\pi$
$888$	0	0
$889$	26.7993	0.898820
$890$	0	0
$891$	0	0
$892$	−15.9069	−0.532602
$893$	3.08355	0.103187
$894$	0	0
$895$	0	0
$896$	2.87740	0.0961273
$897$	0	0
$898$	−13.1028	−0.437245
$899$	17.7308	0.591355
$900$	0	0
$901$	−14.2747	−0.475558
$902$	15.0835	0.502227
$903$	0	0
$904$	10.3973	0.345808
$905$	0	0
$906$	0	0
$907$	21.0740	0.699750	0.349875	−	0.936796i	$-0.386224\pi$
$907$	21.0740	0.699750	0.349875	+	0.936796i	$0.386224\pi$
$908$	18.5589	0.615899
$909$	0	0
$910$	0	0
$911$	−40.4508	−1.34019	−0.670097	−	0.742274i	$-0.733747\pi$
$911$	−40.4508	−1.34019	−0.670097	+	0.742274i	$0.733747\pi$
$912$	0	0
$913$	−11.2452	−0.372162
$914$	−3.18213	−0.105255
$915$	0	0
$916$	16.7500	0.553436
$917$	−12.0595	−0.398241
$918$	0	0
$919$	17.7020	0.583935	0.291967	−	0.956428i	$-0.405690\pi$
$919$	17.7020	0.583935	0.291967	+	0.956428i	$0.405690\pi$
$920$	0	0
$921$	0	0
$922$	28.5589	0.940537
$923$	3.63154	0.119534
$924$	0	0
$925$	0	0
$926$	−5.82333	−0.191366
$927$	0	0
$928$	−6.43631	−0.211282
$929$	39.4796	1.29528	0.647641	−	0.761946i	$-0.275756\pi$
$929$	39.4796	1.29528	0.647641	+	0.761946i	$0.275756\pi$
$930$	0	0
$931$	−1.27945	−0.0419324
$932$	−13.0240	−0.426616
$933$	0	0
$934$	39.0980	1.27933
$935$	0	0
$936$	0	0
$937$	−22.7055	−0.741757	−0.370878	−	0.928681i	$-0.620943\pi$
$937$	−22.7055	−0.741757	−0.370878	+	0.928681i	$0.620943\pi$
$938$	−31.6377	−1.03301
$939$	0	0
$940$	0	0
$941$	−47.9802	−1.56411	−0.782055	−	0.623210i	$-0.785828\pi$
$941$	−47.9802	−1.56411	−0.782055	+	0.623210i	$0.785828\pi$
$942$	0	0
$943$	44.1713	1.43842
$944$	−1.15686	−0.0376525
$945$	0	0
$946$	28.1418	0.914970
$947$	56.1275	1.82390	0.911949	−	0.410304i	$-0.134577\pi$
$947$	56.1275	1.82390	0.911949	+	0.410304i	$0.134577\pi$
$948$	0	0
$949$	−2.29515	−0.0745038
$950$	0	0
$951$	0	0
$952$	8.27945	0.268339
$953$	9.87262	0.319805	0.159903	−	0.987133i	$-0.448882\pi$
$953$	9.87262	0.319805	0.159903	+	0.987133i	$0.448882\pi$
$954$	0	0
$955$	0	0
$956$	23.9357	0.774135
$957$	0	0
$958$	−9.35755	−0.302329
$959$	38.4508	1.24164
$960$	0	0
$961$	−23.4110	−0.755195
$962$	1.67671	0.0540594
$963$	0	0
$964$	14.3870	0.463375
$965$	0	0
$966$	0	0
$967$	8.70552	0.279951	0.139975	−	0.990155i	$-0.455298\pi$
$967$	8.70552	0.279951	0.139975	+	0.990155i	$0.455298\pi$
$968$	−5.23016	−0.168104
$969$	0	0
$970$	0	0
$971$	−3.41162	−0.109484	−0.0547421	−	0.998501i	$-0.517434\pi$
$971$	−3.41162	−0.109484	−0.0547421	+	0.998501i	$0.517434\pi$
$972$	0	0
$973$	−40.5932	−1.30136
$974$	10.2452	0.328277
$975$	0	0
$976$	6.96095	0.222815
$977$	−11.6220	−0.371820	−0.185910	−	0.982567i	$-0.559523\pi$
$977$	−11.6220	−0.371820	−0.185910	+	0.982567i	$0.559523\pi$
$978$	0	0
$979$	28.4363	0.908828
$980$	0	0
$981$	0	0
$982$	−5.34252	−0.170487
$983$	−32.6912	−1.04269	−0.521343	−	0.853347i	$-0.674569\pi$
$983$	−32.6912	−1.04269	−0.521343	+	0.853347i	$0.674569\pi$
$984$	0	0
$985$	0	0
$986$	−18.5199	−0.589792
$987$	0	0
$988$	−0.597951	−0.0190234
$989$	82.4117	2.62054
$990$	0	0
$991$	46.9562	1.49161	0.745806	−	0.666163i	$-0.232065\pi$
$991$	46.9562	1.49161	0.745806	+	0.666163i	$0.232065\pi$
$992$	−2.75481	−0.0874652
$993$	0	0
$994$	17.4754	0.554285
$995$	0	0
$996$	0	0
$997$	−53.3227	−1.68875	−0.844373	−	0.535755i	$-0.820027\pi$
$997$	−53.3227	−1.68875	−0.844373	+	0.535755i	$0.820027\pi$
$998$	−2.25418	−0.0713548
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.ct.1.3	yes	3
3.2	odd	2		8550.2.a.ch.1.3	yes	3
5.4	even	2		8550.2.a.cc.1.1	✓	3
15.14	odd	2		8550.2.a.cm.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8550.2.a.cc.1.1	✓	3	5.4	even	2
8550.2.a.ch.1.3	yes	3	3.2	odd	2
8550.2.a.cm.1.1	yes	3	15.14	odd	2
8550.2.a.ct.1.3	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.87740 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.87740 q^{7} +1.00000 q^{8} +2.40205 q^{11} +0.597951 q^{13} +2.87740 q^{14} +1.00000 q^{16} +2.87740 q^{17} -1.00000 q^{19} +2.40205 q^{22} +7.03426 q^{23} +0.597951 q^{26} +2.87740 q^{28} -6.43631 q^{29} -2.75481 q^{31} +1.00000 q^{32} +2.87740 q^{34} +2.80410 q^{37} -1.00000 q^{38} +6.27945 q^{41} +11.7158 q^{43} +2.40205 q^{44} +7.03426 q^{46} -3.08355 q^{47} +1.27945 q^{49} +0.597951 q^{52} -4.96095 q^{53} +2.87740 q^{56} -6.43631 q^{58} -1.15686 q^{59} +6.96095 q^{61} -2.75481 q^{62} +1.00000 q^{64} -10.9952 q^{67} +2.87740 q^{68} +6.07331 q^{71} -3.83836 q^{73} +2.80410 q^{74} -1.00000 q^{76} +6.91166 q^{77} -5.95071 q^{79} +6.27945 q^{82} -4.68150 q^{83} +11.7158 q^{86} +2.40205 q^{88} +11.8384 q^{89} +1.72055 q^{91} +7.03426 q^{92} -3.08355 q^{94} -8.63221 q^{97} +1.27945 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8} + 5 q^{11} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} + 5 q^{22} - q^{23} + 4 q^{26} + 2 q^{28} + 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{37} - 3 q^{38} + 10 q^{41} + 2 q^{43} + 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} + 4 q^{52} + 5 q^{53} + 2 q^{56} + 5 q^{58} + 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{68} + 16 q^{71} + 15 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} + 10 q^{82} - 3 q^{83} + 2 q^{86} + 5 q^{88} + 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} - 6 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 + 5 * q^11 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 + 5 * q^22 - q^23 + 4 * q^26 + 2 * q^28 + 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^37 - 3 * q^38 + 10 * q^41 + 2 * q^43 + 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 + 4 * q^52 + 5 * q^53 + 2 * q^56 + 5 * q^58 + 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^68 + 16 * q^71 + 15 * q^73 + 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 + 10 * q^82 - 3 * q^83 + 2 * q^86 + 5 * q^88 + 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 - 6 * q^97 - 5 * q^98

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