LMFDB - Embedded newform 8550.2.a.cg.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:	$1$
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			$-2.08613$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} -4.08613 q^{11} +3.79001 q^{13} +2.43807 q^{14} +1.00000 q^{16} +3.73419 q^{17} +1.00000 q^{19} +4.08613 q^{22} +0.351939 q^{23} -3.79001 q^{26} -2.43807 q^{28} -6.73419 q^{29} +9.34452 q^{31} -1.00000 q^{32} -3.73419 q^{34} -6.17226 q^{37} -1.00000 q^{38} -4.05582 q^{41} +0.913870 q^{43} -4.08613 q^{44} -0.351939 q^{46} -5.52420 q^{47} -1.05582 q^{49} +3.79001 q^{52} +6.96227 q^{53} +2.43807 q^{56} +6.73419 q^{58} +14.3068 q^{59} -13.1345 q^{61} -9.34452 q^{62} +1.00000 q^{64} -7.61033 q^{67} +3.73419 q^{68} -9.73419 q^{71} +11.9926 q^{73} +6.17226 q^{74} +1.00000 q^{76} +9.96227 q^{77} -3.17226 q^{79} +4.05582 q^{82} +2.90645 q^{83} -0.913870 q^{86} +4.08613 q^{88} -15.2887 q^{89} -9.24030 q^{91} +0.351939 q^{92} +5.52420 q^{94} +16.0787 q^{97} +1.05582 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.43807	−0.921504	−0.460752	−	0.887529i	$-0.652420\pi$
$7$	−2.43807	−0.921504	−0.460752	+	0.887529i	$0.652420\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.08613	−1.23201	−0.616007	−	0.787740i	$-0.711251\pi$
$11$	−4.08613	−1.23201	−0.616007	+	0.787740i	$0.711251\pi$
$12$	0	0
$13$	3.79001	1.05116	0.525580	−	0.850744i	$-0.323848\pi$
$13$	3.79001	1.05116	0.525580	+	0.850744i	$0.323848\pi$
$14$	2.43807	0.651601
$15$	0	0
$16$	1.00000	0.250000
$17$	3.73419	0.905674	0.452837	−	0.891593i	$-0.350412\pi$
$17$	3.73419	0.905674	0.452837	+	0.891593i	$0.350412\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	4.08613	0.871166
$23$	0.351939	0.0733844	0.0366922	−	0.999327i	$-0.488318\pi$
$23$	0.351939	0.0733844	0.0366922	+	0.999327i	$0.488318\pi$
$24$	0	0
$25$	0	0
$26$	−3.79001	−0.743282
$27$	0	0
$28$	−2.43807	−0.460752
$29$	−6.73419	−1.25051	−0.625254	−	0.780421i	$-0.715005\pi$
$29$	−6.73419	−1.25051	−0.625254	+	0.780421i	$0.715005\pi$
$30$	0	0
$31$	9.34452	1.67833	0.839163	−	0.543880i	$-0.183045\pi$
$31$	9.34452	1.67833	0.839163	+	0.543880i	$0.183045\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.73419	−0.640408
$35$	0	0
$36$	0	0
$37$	−6.17226	−1.01471	−0.507357	−	0.861736i	$-0.669377\pi$
$37$	−6.17226	−1.01471	−0.507357	+	0.861736i	$0.669377\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−4.05582	−0.633412	−0.316706	−	0.948524i	$-0.602577\pi$
$41$	−4.05582	−0.633412	−0.316706	+	0.948524i	$0.602577\pi$
$42$	0	0
$43$	0.913870	0.139364	0.0696819	−	0.997569i	$-0.477802\pi$
$43$	0.913870	0.139364	0.0696819	+	0.997569i	$0.477802\pi$
$44$	−4.08613	−0.616007
$45$	0	0
$46$	−0.351939	−0.0518906
$47$	−5.52420	−0.805787	−0.402894	−	0.915247i	$-0.631996\pi$
$47$	−5.52420	−0.805787	−0.402894	+	0.915247i	$0.631996\pi$
$48$	0	0
$49$	−1.05582	−0.150831
$50$	0	0
$51$	0	0
$52$	3.79001	0.525580
$53$	6.96227	0.956341	0.478171	−	0.878267i	$-0.341300\pi$
$53$	6.96227	0.956341	0.478171	+	0.878267i	$0.341300\pi$
$54$	0	0
$55$	0	0
$56$	2.43807	0.325801
$57$	0	0
$58$	6.73419	0.884243
$59$	14.3068	1.86259	0.931293	−	0.364272i	$-0.118682\pi$
$59$	14.3068	1.86259	0.931293	+	0.364272i	$0.118682\pi$
$60$	0	0
$61$	−13.1345	−1.68170	−0.840852	−	0.541265i	$-0.817946\pi$
$61$	−13.1345	−1.68170	−0.840852	+	0.541265i	$0.817946\pi$
$62$	−9.34452	−1.18676
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−7.61033	−0.929750	−0.464875	−	0.885376i	$-0.653901\pi$
$67$	−7.61033	−0.929750	−0.464875	+	0.885376i	$0.653901\pi$
$68$	3.73419	0.452837
$69$	0	0
$70$	0	0
$71$	−9.73419	−1.15524	−0.577618	−	0.816307i	$-0.696018\pi$
$71$	−9.73419	−1.15524	−0.577618	+	0.816307i	$0.696018\pi$
$72$	0	0
$73$	11.9926	1.40363	0.701813	−	0.712361i	$-0.252374\pi$
$73$	11.9926	1.40363	0.701813	+	0.712361i	$0.252374\pi$
$74$	6.17226	0.717511
$75$	0	0
$76$	1.00000	0.114708
$77$	9.96227	1.13531
$78$	0	0
$79$	−3.17226	−0.356907	−0.178454	−	0.983948i	$-0.557109\pi$
$79$	−3.17226	−0.356907	−0.178454	+	0.983948i	$0.557109\pi$
$80$	0	0
$81$	0	0
$82$	4.05582	0.447890
$83$	2.90645	0.319024	0.159512	−	0.987196i	$-0.449008\pi$
$83$	2.90645	0.319024	0.159512	+	0.987196i	$0.449008\pi$
$84$	0	0
$85$	0	0
$86$	−0.913870	−0.0985451
$87$	0	0
$88$	4.08613	0.435583
$89$	−15.2887	−1.62060	−0.810300	−	0.586016i	$-0.800696\pi$
$89$	−15.2887	−1.62060	−0.810300	+	0.586016i	$0.800696\pi$
$90$	0	0
$91$	−9.24030	−0.968647
$92$	0.351939	0.0366922
$93$	0	0
$94$	5.52420	0.569778
$95$	0	0
$96$	0	0
$97$	16.0787	1.63255	0.816273	−	0.577666i	$-0.196037\pi$
$97$	16.0787	1.63255	0.816273	+	0.577666i	$0.196037\pi$
$98$	1.05582	0.106654
$99$	0	0
$100$	0	0
$101$	−6.38225	−0.635058	−0.317529	−	0.948249i	$-0.602853\pi$
$101$	−6.38225	−0.635058	−0.317529	+	0.948249i	$0.602853\pi$
$102$	0	0
$103$	8.35194	0.822941	0.411471	−	0.911423i	$-0.365015\pi$
$103$	8.35194	0.822941	0.411471	+	0.911423i	$0.365015\pi$
$104$	−3.79001	−0.371641
$105$	0	0
$106$	−6.96227	−0.676235
$107$	17.4307	1.68508	0.842542	−	0.538630i	$-0.181058\pi$
$107$	17.4307	1.68508	0.842542	+	0.538630i	$0.181058\pi$
$108$	0	0
$109$	10.8761	1.04175	0.520873	−	0.853634i	$-0.325607\pi$
$109$	10.8761	1.04175	0.520873	+	0.853634i	$0.325607\pi$
$110$	0	0
$111$	0	0
$112$	−2.43807	−0.230376
$113$	1.05582	0.0993230	0.0496615	−	0.998766i	$-0.484186\pi$
$113$	1.05582	0.0993230	0.0496615	+	0.998766i	$0.484186\pi$
$114$	0	0
$115$	0	0
$116$	−6.73419	−0.625254
$117$	0	0
$118$	−14.3068	−1.31705
$119$	−9.10422	−0.834582
$120$	0	0
$121$	5.69646	0.517860
$122$	13.1345	1.18914
$123$	0	0
$124$	9.34452	0.839163
$125$	0	0
$126$	0	0
$127$	8.64064	0.766733	0.383367	−	0.923596i	$-0.374765\pi$
$127$	8.64064	0.766733	0.383367	+	0.923596i	$0.374765\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	0.0303126	0.00264842	0.00132421	−	0.999999i	$-0.499578\pi$
$131$	0.0303126	0.00264842	0.00132421	+	0.999999i	$0.499578\pi$
$132$	0	0
$133$	−2.43807	−0.211407
$134$	7.61033	0.657432
$135$	0	0
$136$	−3.73419	−0.320204
$137$	−1.46838	−0.125452	−0.0627262	−	0.998031i	$-0.519979\pi$
$137$	−1.46838	−0.125452	−0.0627262	+	0.998031i	$0.519979\pi$
$138$	0	0
$139$	18.6661	1.58324	0.791621	−	0.611012i	$-0.209237\pi$
$139$	18.6661	1.58324	0.791621	+	0.611012i	$0.209237\pi$
$140$	0	0
$141$	0	0
$142$	9.73419	0.816875
$143$	−15.4865	−1.29504
$144$	0	0
$145$	0	0
$146$	−11.9926	−0.992513
$147$	0	0
$148$	−6.17226	−0.507357
$149$	−14.8761	−1.21870	−0.609350	−	0.792901i	$-0.708570\pi$
$149$	−14.8761	−1.21870	−0.609350	+	0.792901i	$0.708570\pi$
$150$	0	0
$151$	12.5726	1.02314	0.511572	−	0.859241i	$-0.329063\pi$
$151$	12.5726	1.02314	0.511572	+	0.859241i	$0.329063\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−9.96227	−0.802783
$155$	0	0
$156$	0	0
$157$	8.22808	0.656672	0.328336	−	0.944561i	$-0.393512\pi$
$157$	8.22808	0.656672	0.328336	+	0.944561i	$0.393512\pi$
$158$	3.17226	0.252371
$159$	0	0
$160$	0	0
$161$	−0.858052	−0.0676240
$162$	0	0
$163$	3.40776	0.266916	0.133458	−	0.991054i	$-0.457392\pi$
$163$	3.40776	0.266916	0.133458	+	0.991054i	$0.457392\pi$
$164$	−4.05582	−0.316706
$165$	0	0
$166$	−2.90645	−0.225584
$167$	18.4562	1.42818	0.714090	−	0.700054i	$-0.246841\pi$
$167$	18.4562	1.42818	0.714090	+	0.700054i	$0.246841\pi$
$168$	0	0
$169$	1.36417	0.104936
$170$	0	0
$171$	0	0
$172$	0.913870	0.0696819
$173$	−25.9549	−1.97331	−0.986655	−	0.162823i	$-0.947940\pi$
$173$	−25.9549	−1.97331	−0.986655	+	0.162823i	$0.947940\pi$
$174$	0	0
$175$	0	0
$176$	−4.08613	−0.308004
$177$	0	0
$178$	15.2887	1.14594
$179$	−5.23550	−0.391319	−0.195660	−	0.980672i	$-0.562685\pi$
$179$	−5.23550	−0.391319	−0.195660	+	0.980672i	$0.562685\pi$
$180$	0	0
$181$	−25.2584	−1.87744	−0.938721	−	0.344679i	$-0.887988\pi$
$181$	−25.2584	−1.87744	−0.938721	+	0.344679i	$0.887988\pi$
$182$	9.24030	0.684937
$183$	0	0
$184$	−0.351939	−0.0259453
$185$	0	0
$186$	0	0
$187$	−15.2584	−1.11580
$188$	−5.52420	−0.402894
$189$	0	0
$190$	0	0
$191$	−3.95160	−0.285928	−0.142964	−	0.989728i	$-0.545663\pi$
$191$	−3.95160	−0.285928	−0.142964	+	0.989728i	$0.545663\pi$
$192$	0	0
$193$	−2.43807	−0.175496	−0.0877480	−	0.996143i	$-0.527967\pi$
$193$	−2.43807	−0.175496	−0.0877480	+	0.996143i	$0.527967\pi$
$194$	−16.0787	−1.15438
$195$	0	0
$196$	−1.05582	−0.0754155
$197$	−13.6029	−0.969167	−0.484584	−	0.874745i	$-0.661029\pi$
$197$	−13.6029	−0.969167	−0.484584	+	0.874745i	$0.661029\pi$
$198$	0	0
$199$	4.81551	0.341363	0.170681	−	0.985326i	$-0.445403\pi$
$199$	4.81551	0.341363	0.170681	+	0.985326i	$0.445403\pi$
$200$	0	0
$201$	0	0
$202$	6.38225	0.449054
$203$	16.4184	1.15235
$204$	0	0
$205$	0	0
$206$	−8.35194	−0.581907
$207$	0	0
$208$	3.79001	0.262790
$209$	−4.08613	−0.282644
$210$	0	0
$211$	−18.4307	−1.26882	−0.634409	−	0.772997i	$-0.718757\pi$
$211$	−18.4307	−1.26882	−0.634409	+	0.772997i	$0.718757\pi$
$212$	6.96227	0.478171
$213$	0	0
$214$	−17.4307	−1.19153
$215$	0	0
$216$	0	0
$217$	−22.7826	−1.54658
$218$	−10.8761	−0.736625
$219$	0	0
$220$	0	0
$221$	14.1526	0.952008
$222$	0	0
$223$	−7.22808	−0.484028	−0.242014	−	0.970273i	$-0.577808\pi$
$223$	−7.22808	−0.484028	−0.242014	+	0.970273i	$0.577808\pi$
$224$	2.43807	0.162900
$225$	0	0
$226$	−1.05582	−0.0702319
$227$	−21.3929	−1.41990	−0.709949	−	0.704253i	$-0.751282\pi$
$227$	−21.3929	−1.41990	−0.709949	+	0.704253i	$0.751282\pi$
$228$	0	0
$229$	1.96969	0.130161	0.0650803	−	0.997880i	$-0.479270\pi$
$229$	1.96969	0.130161	0.0650803	+	0.997880i	$0.479270\pi$
$230$	0	0
$231$	0	0
$232$	6.73419	0.442121
$233$	−5.08132	−0.332889	−0.166444	−	0.986051i	$-0.553229\pi$
$233$	−5.08132	−0.332889	−0.166444	+	0.986051i	$0.553229\pi$
$234$	0	0
$235$	0	0
$236$	14.3068	0.931293
$237$	0	0
$238$	9.10422	0.590139
$239$	−15.1648	−0.980932	−0.490466	−	0.871460i	$-0.663173\pi$
$239$	−15.1648	−0.980932	−0.490466	+	0.871460i	$0.663173\pi$
$240$	0	0
$241$	5.79482	0.373277	0.186638	−	0.982429i	$-0.440241\pi$
$241$	5.79482	0.373277	0.186638	+	0.982429i	$0.440241\pi$
$242$	−5.69646	−0.366182
$243$	0	0
$244$	−13.1345	−0.840852
$245$	0	0
$246$	0	0
$247$	3.79001	0.241152
$248$	−9.34452	−0.593378
$249$	0	0
$250$	0	0
$251$	−16.0558	−1.01343	−0.506717	−	0.862112i	$-0.669141\pi$
$251$	−16.0558	−1.01343	−0.506717	+	0.862112i	$0.669141\pi$
$252$	0	0
$253$	−1.43807	−0.0904106
$254$	−8.64064	−0.542162
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.22808	−0.201362	−0.100681	−	0.994919i	$-0.532102\pi$
$257$	−3.22808	−0.201362	−0.100681	+	0.994919i	$0.532102\pi$
$258$	0	0
$259$	15.0484	0.935062
$260$	0	0
$261$	0	0
$262$	−0.0303126	−0.00187272
$263$	−25.5120	−1.57314	−0.786568	−	0.617504i	$-0.788144\pi$
$263$	−25.5120	−1.57314	−0.786568	+	0.617504i	$0.788144\pi$
$264$	0	0
$265$	0	0
$266$	2.43807	0.149488
$267$	0	0
$268$	−7.61033	−0.464875
$269$	18.2691	1.11388	0.556942	−	0.830551i	$-0.311975\pi$
$269$	18.2691	1.11388	0.556942	+	0.830551i	$0.311975\pi$
$270$	0	0
$271$	−24.0787	−1.46268	−0.731339	−	0.682014i	$-0.761104\pi$
$271$	−24.0787	−1.46268	−0.731339	+	0.682014i	$0.761104\pi$
$272$	3.73419	0.226419
$273$	0	0
$274$	1.46838	0.0887082
$275$	0	0
$276$	0	0
$277$	−3.49389	−0.209927	−0.104964	−	0.994476i	$-0.533473\pi$
$277$	−3.49389	−0.209927	−0.104964	+	0.994476i	$0.533473\pi$
$278$	−18.6661	−1.11952
$279$	0	0
$280$	0	0
$281$	−18.5800	−1.10839	−0.554195	−	0.832387i	$-0.686974\pi$
$281$	−18.5800	−1.10839	−0.554195	+	0.832387i	$0.686974\pi$
$282$	0	0
$283$	−1.64064	−0.0975261	−0.0487630	−	0.998810i	$-0.515528\pi$
$283$	−1.64064	−0.0975261	−0.0487630	+	0.998810i	$0.515528\pi$
$284$	−9.73419	−0.577618
$285$	0	0
$286$	15.4865	0.915734
$287$	9.88836	0.583692
$288$	0	0
$289$	−3.05582	−0.179754
$290$	0	0
$291$	0	0
$292$	11.9926	0.701813
$293$	−15.8384	−0.925290	−0.462645	−	0.886544i	$-0.653099\pi$
$293$	−15.8384	−0.925290	−0.462645	+	0.886544i	$0.653099\pi$
$294$	0	0
$295$	0	0
$296$	6.17226	0.358755
$297$	0	0
$298$	14.8761	0.861752
$299$	1.33385	0.0771387
$300$	0	0
$301$	−2.22808	−0.128424
$302$	−12.5726	−0.723472
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−9.32643	−0.532288	−0.266144	−	0.963933i	$-0.585750\pi$
$307$	−9.32643	−0.532288	−0.266144	+	0.963933i	$0.585750\pi$
$308$	9.96227	0.567653
$309$	0	0
$310$	0	0
$311$	−19.6965	−1.11688	−0.558442	−	0.829544i	$-0.688601\pi$
$311$	−19.6965	−1.11688	−0.558442	+	0.829544i	$0.688601\pi$
$312$	0	0
$313$	−2.94418	−0.166415	−0.0832075	−	0.996532i	$-0.526516\pi$
$313$	−2.94418	−0.166415	−0.0832075	+	0.996532i	$0.526516\pi$
$314$	−8.22808	−0.464337
$315$	0	0
$316$	−3.17226	−0.178454
$317$	18.7342	1.05222	0.526108	−	0.850417i	$-0.323651\pi$
$317$	18.7342	1.05222	0.526108	+	0.850417i	$0.323651\pi$
$318$	0	0
$319$	27.5168	1.54064
$320$	0	0
$321$	0	0
$322$	0.858052	0.0478174
$323$	3.73419	0.207776
$324$	0	0
$325$	0	0
$326$	−3.40776	−0.188738
$327$	0	0
$328$	4.05582	0.223945
$329$	13.4684	0.742536
$330$	0	0
$331$	15.8942	0.873626	0.436813	−	0.899552i	$-0.356107\pi$
$331$	15.8942	0.873626	0.436813	+	0.899552i	$0.356107\pi$
$332$	2.90645	0.159512
$333$	0	0
$334$	−18.4562	−1.00988
$335$	0	0
$336$	0	0
$337$	−21.8129	−1.18822	−0.594112	−	0.804382i	$-0.702497\pi$
$337$	−21.8129	−1.18822	−0.594112	+	0.804382i	$0.702497\pi$
$338$	−1.36417	−0.0742008
$339$	0	0
$340$	0	0
$341$	−38.1829	−2.06772
$342$	0	0
$343$	19.6406	1.06050
$344$	−0.913870	−0.0492726
$345$	0	0
$346$	25.9549	1.39534
$347$	18.5168	0.994033	0.497016	−	0.867741i	$-0.334429\pi$
$347$	18.5168	0.994033	0.497016	+	0.867741i	$0.334429\pi$
$348$	0	0
$349$	−35.7678	−1.91460	−0.957302	−	0.289090i	$-0.906647\pi$
$349$	−35.7678	−1.91460	−0.957302	+	0.289090i	$0.906647\pi$
$350$	0	0
$351$	0	0
$352$	4.08613	0.217791
$353$	−18.5497	−0.987301	−0.493651	−	0.869660i	$-0.664338\pi$
$353$	−18.5497	−0.987301	−0.493651	+	0.869660i	$0.664338\pi$
$354$	0	0
$355$	0	0
$356$	−15.2887	−0.810300
$357$	0	0
$358$	5.23550	0.276705
$359$	29.0288	1.53208	0.766040	−	0.642793i	$-0.222225\pi$
$359$	29.0288	1.53208	0.766040	+	0.642793i	$0.222225\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.2584	1.32755
$363$	0	0
$364$	−9.24030	−0.484324
$365$	0	0
$366$	0	0
$367$	8.36261	0.436525	0.218262	−	0.975890i	$-0.429961\pi$
$367$	8.36261	0.436525	0.218262	+	0.975890i	$0.429961\pi$
$368$	0.351939	0.0183461
$369$	0	0
$370$	0	0
$371$	−16.9745	−0.881272
$372$	0	0
$373$	−33.1223	−1.71501	−0.857504	−	0.514477i	$-0.827986\pi$
$373$	−33.1223	−1.71501	−0.857504	+	0.514477i	$0.827986\pi$
$374$	15.2584	0.788993
$375$	0	0
$376$	5.52420	0.284889
$377$	−25.5226	−1.31448
$378$	0	0
$379$	37.1090	1.90616	0.953081	−	0.302715i	$-0.0978929\pi$
$379$	37.1090	1.90616	0.953081	+	0.302715i	$0.0978929\pi$
$380$	0	0
$381$	0	0
$382$	3.95160	0.202181
$383$	17.5981	0.899221	0.449611	−	0.893225i	$-0.351563\pi$
$383$	17.5981	0.899221	0.449611	+	0.893225i	$0.351563\pi$
$384$	0	0
$385$	0	0
$386$	2.43807	0.124094
$387$	0	0
$388$	16.0787	0.816273
$389$	−8.87614	−0.450038	−0.225019	−	0.974354i	$-0.572244\pi$
$389$	−8.87614	−0.450038	−0.225019	+	0.974354i	$0.572244\pi$
$390$	0	0
$391$	1.31421	0.0664624
$392$	1.05582	0.0533268
$393$	0	0
$394$	13.6029	0.685305
$395$	0	0
$396$	0	0
$397$	−37.1952	−1.86677	−0.933386	−	0.358875i	$-0.883160\pi$
$397$	−37.1952	−1.86677	−0.933386	+	0.358875i	$0.883160\pi$
$398$	−4.81551	−0.241380
$399$	0	0
$400$	0	0
$401$	−17.1116	−0.854514	−0.427257	−	0.904130i	$-0.640520\pi$
$401$	−17.1116	−0.854514	−0.427257	+	0.904130i	$0.640520\pi$
$402$	0	0
$403$	35.4158	1.76419
$404$	−6.38225	−0.317529
$405$	0	0
$406$	−16.4184	−0.814833
$407$	25.2207	1.25014
$408$	0	0
$409$	−16.2691	−0.804453	−0.402227	−	0.915540i	$-0.631764\pi$
$409$	−16.2691	−0.804453	−0.402227	+	0.915540i	$0.631764\pi$
$410$	0	0
$411$	0	0
$412$	8.35194	0.411471
$413$	−34.8809	−1.71638
$414$	0	0
$415$	0	0
$416$	−3.79001	−0.185820
$417$	0	0
$418$	4.08613	0.199859
$419$	3.12386	0.152611	0.0763053	−	0.997085i	$-0.475688\pi$
$419$	3.12386	0.152611	0.0763053	+	0.997085i	$0.475688\pi$
$420$	0	0
$421$	−11.8639	−0.578212	−0.289106	−	0.957297i	$-0.593358\pi$
$421$	−11.8639	−0.578212	−0.289106	+	0.957297i	$0.593358\pi$
$422$	18.4307	0.897190
$423$	0	0
$424$	−6.96227	−0.338118
$425$	0	0
$426$	0	0
$427$	32.0229	1.54970
$428$	17.4307	0.842542
$429$	0	0
$430$	0	0
$431$	−5.69165	−0.274157	−0.137079	−	0.990560i	$-0.543771\pi$
$431$	−5.69165	−0.274157	−0.137079	+	0.990560i	$0.543771\pi$
$432$	0	0
$433$	8.36261	0.401881	0.200941	−	0.979603i	$-0.435600\pi$
$433$	8.36261	0.401881	0.200941	+	0.979603i	$0.435600\pi$
$434$	22.7826	1.09360
$435$	0	0
$436$	10.8761	0.520873
$437$	0.351939	0.0168355
$438$	0	0
$439$	13.9368	0.665165	0.332583	−	0.943074i	$-0.392080\pi$
$439$	13.9368	0.665165	0.332583	+	0.943074i	$0.392080\pi$
$440$	0	0
$441$	0	0
$442$	−14.1526	−0.673171
$443$	−32.7194	−1.55454	−0.777272	−	0.629165i	$-0.783397\pi$
$443$	−32.7194	−1.55454	−0.777272	+	0.629165i	$0.783397\pi$
$444$	0	0
$445$	0	0
$446$	7.22808	0.342259
$447$	0	0
$448$	−2.43807	−0.115188
$449$	−32.0288	−1.51153	−0.755765	−	0.654843i	$-0.772735\pi$
$449$	−32.0288	−1.51153	−0.755765	+	0.654843i	$0.772735\pi$
$450$	0	0
$451$	16.5726	0.780373
$452$	1.05582	0.0496615
$453$	0	0
$454$	21.3929	1.00402
$455$	0	0
$456$	0	0
$457$	4.70869	0.220263	0.110132	−	0.993917i	$-0.464873\pi$
$457$	4.70869	0.220263	0.110132	+	0.993917i	$0.464873\pi$
$458$	−1.96969	−0.0920374
$459$	0	0
$460$	0	0
$461$	30.2084	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$461$	30.2084	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$462$	0	0
$463$	5.82774	0.270838	0.135419	−	0.990788i	$-0.456762\pi$
$463$	5.82774	0.270838	0.135419	+	0.990788i	$0.456762\pi$
$464$	−6.73419	−0.312627
$465$	0	0
$466$	5.08132	0.235388
$467$	−17.0591	−0.789399	−0.394700	−	0.918810i	$-0.629151\pi$
$467$	−17.0591	−0.789399	−0.394700	+	0.918810i	$0.629151\pi$
$468$	0	0
$469$	18.5545	0.856768
$470$	0	0
$471$	0	0
$472$	−14.3068	−0.658523
$473$	−3.73419	−0.171698
$474$	0	0
$475$	0	0
$476$	−9.10422	−0.417291
$477$	0	0
$478$	15.1648	0.693624
$479$	−4.94418	−0.225905	−0.112953	−	0.993600i	$-0.536031\pi$
$479$	−4.94418	−0.225905	−0.112953	+	0.993600i	$0.536031\pi$
$480$	0	0
$481$	−23.3929	−1.06663
$482$	−5.79482	−0.263947
$483$	0	0
$484$	5.69646	0.258930
$485$	0	0
$486$	0	0
$487$	17.9097	0.811566	0.405783	−	0.913969i	$-0.366999\pi$
$487$	17.9097	0.811566	0.405783	+	0.913969i	$0.366999\pi$
$488$	13.1345	0.594572
$489$	0	0
$490$	0	0
$491$	−25.9852	−1.17269	−0.586347	−	0.810060i	$-0.699434\pi$
$491$	−25.9852	−1.17269	−0.586347	+	0.810060i	$0.699434\pi$
$492$	0	0
$493$	−25.1468	−1.13255
$494$	−3.79001	−0.170521
$495$	0	0
$496$	9.34452	0.419581
$497$	23.7326	1.06455
$498$	0	0
$499$	3.54229	0.158575	0.0792873	−	0.996852i	$-0.474736\pi$
$499$	3.54229	0.158575	0.0792873	+	0.996852i	$0.474736\pi$
$500$	0	0
$501$	0	0
$502$	16.0558	0.716606
$503$	−7.16003	−0.319250	−0.159625	−	0.987178i	$-0.551029\pi$
$503$	−7.16003	−0.319250	−0.159625	+	0.987178i	$0.551029\pi$
$504$	0	0
$505$	0	0
$506$	1.43807	0.0639300
$507$	0	0
$508$	8.64064	0.383367
$509$	−15.9597	−0.707399	−0.353700	−	0.935359i	$-0.615077\pi$
$509$	−15.9597	−0.707399	−0.353700	+	0.935359i	$0.615077\pi$
$510$	0	0
$511$	−29.2387	−1.29345
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.22808	0.142384
$515$	0	0
$516$	0	0
$517$	22.5726	0.992742
$518$	−15.0484	−0.661189
$519$	0	0
$520$	0	0
$521$	5.62842	0.246585	0.123293	−	0.992370i	$-0.460655\pi$
$521$	5.62842	0.246585	0.123293	+	0.992370i	$0.460655\pi$
$522$	0	0
$523$	−34.0213	−1.48765	−0.743825	−	0.668375i	$-0.766990\pi$
$523$	−34.0213	−1.48765	−0.743825	+	0.668375i	$0.766990\pi$
$524$	0.0303126	0.00132421
$525$	0	0
$526$	25.5120	1.11237
$527$	34.8942	1.52002
$528$	0	0
$529$	−22.8761	−0.994615
$530$	0	0
$531$	0	0
$532$	−2.43807	−0.105704
$533$	−15.3716	−0.665817
$534$	0	0
$535$	0	0
$536$	7.61033	0.328716
$537$	0	0
$538$	−18.2691	−0.787635
$539$	4.31421	0.185826
$540$	0	0
$541$	27.7071	1.19122	0.595611	−	0.803273i	$-0.296910\pi$
$541$	27.7071	1.19122	0.595611	+	0.803273i	$0.296910\pi$
$542$	24.0787	1.03427
$543$	0	0
$544$	−3.73419	−0.160102
$545$	0	0
$546$	0	0
$547$	−45.8236	−1.95927	−0.979637	−	0.200776i	$-0.935654\pi$
$547$	−45.8236	−1.95927	−0.979637	+	0.200776i	$0.935654\pi$
$548$	−1.46838	−0.0627262
$549$	0	0
$550$	0	0
$551$	−6.73419	−0.286886
$552$	0	0
$553$	7.73419	0.328891
$554$	3.49389	0.148441
$555$	0	0
$556$	18.6661	0.791621
$557$	10.4184	0.441443	0.220721	−	0.975337i	$-0.429159\pi$
$557$	10.4184	0.441443	0.220721	+	0.975337i	$0.429159\pi$
$558$	0	0
$559$	3.46357	0.146494
$560$	0	0
$561$	0	0
$562$	18.5800	0.783751
$563$	0.899033	0.0378897	0.0189449	−	0.999821i	$-0.493969\pi$
$563$	0.899033	0.0378897	0.0189449	+	0.999821i	$0.493969\pi$
$564$	0	0
$565$	0	0
$566$	1.64064	0.0689613
$567$	0	0
$568$	9.73419	0.408438
$569$	34.6136	1.45108	0.725538	−	0.688182i	$-0.241591\pi$
$569$	34.6136	1.45108	0.725538	+	0.688182i	$0.241591\pi$
$570$	0	0
$571$	14.8990	0.623505	0.311753	−	0.950163i	$-0.399084\pi$
$571$	14.8990	0.623505	0.311753	+	0.950163i	$0.399084\pi$
$572$	−15.4865	−0.647522
$573$	0	0
$574$	−9.88836	−0.412732
$575$	0	0
$576$	0	0
$577$	−21.1116	−0.878889	−0.439444	−	0.898270i	$-0.644825\pi$
$577$	−21.1116	−0.878889	−0.439444	+	0.898270i	$0.644825\pi$
$578$	3.05582	0.127105
$579$	0	0
$580$	0	0
$581$	−7.08613	−0.293982
$582$	0	0
$583$	−28.4487	−1.17823
$584$	−11.9926	−0.496257
$585$	0	0
$586$	15.8384	0.654279
$587$	13.6858	0.564873	0.282437	−	0.959286i	$-0.408857\pi$
$587$	13.6858	0.564873	0.282437	+	0.959286i	$0.408857\pi$
$588$	0	0
$589$	9.34452	0.385034
$590$	0	0
$591$	0	0
$592$	−6.17226	−0.253678
$593$	42.7252	1.75451	0.877257	−	0.480021i	$-0.159371\pi$
$593$	42.7252	1.75451	0.877257	+	0.480021i	$0.159371\pi$
$594$	0	0
$595$	0	0
$596$	−14.8761	−0.609350
$597$	0	0
$598$	−1.33385	−0.0545453
$599$	−24.5497	−1.00307	−0.501537	−	0.865136i	$-0.667232\pi$
$599$	−24.5497	−1.00307	−0.501537	+	0.865136i	$0.667232\pi$
$600$	0	0
$601$	−28.1755	−1.14930	−0.574652	−	0.818398i	$-0.694862\pi$
$601$	−28.1755	−1.14930	−0.574652	+	0.818398i	$0.694862\pi$
$602$	2.22808	0.0908097
$603$	0	0
$604$	12.5726	0.511572
$605$	0	0
$606$	0	0
$607$	15.3855	0.624478	0.312239	−	0.950004i	$-0.398921\pi$
$607$	15.3855	0.624478	0.312239	+	0.950004i	$0.398921\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−20.9368	−0.847011
$612$	0	0
$613$	17.3929	0.702493	0.351247	−	0.936283i	$-0.385758\pi$
$613$	17.3929	0.702493	0.351247	+	0.936283i	$0.385758\pi$
$614$	9.32643	0.376384
$615$	0	0
$616$	−9.96227	−0.401391
$617$	−6.76450	−0.272329	−0.136164	−	0.990686i	$-0.543478\pi$
$617$	−6.76450	−0.272329	−0.136164	+	0.990686i	$0.543478\pi$
$618$	0	0
$619$	25.1223	1.00975	0.504875	−	0.863192i	$-0.331538\pi$
$619$	25.1223	1.00975	0.504875	+	0.863192i	$0.331538\pi$
$620$	0	0
$621$	0	0
$622$	19.6965	0.789756
$623$	37.2749	1.49339
$624$	0	0
$625$	0	0
$626$	2.94418	0.117673
$627$	0	0
$628$	8.22808	0.328336
$629$	−23.0484	−0.919000
$630$	0	0
$631$	16.9368	0.674242	0.337121	−	0.941461i	$-0.390547\pi$
$631$	16.9368	0.674242	0.337121	+	0.941461i	$0.390547\pi$
$632$	3.17226	0.126186
$633$	0	0
$634$	−18.7342	−0.744030
$635$	0	0
$636$	0	0
$637$	−4.00156	−0.158547
$638$	−27.5168	−1.08940
$639$	0	0
$640$	0	0
$641$	13.0532	0.515571	0.257785	−	0.966202i	$-0.417007\pi$
$641$	13.0532	0.515571	0.257785	+	0.966202i	$0.417007\pi$
$642$	0	0
$643$	2.51678	0.0992522	0.0496261	−	0.998768i	$-0.484197\pi$
$643$	2.51678	0.0992522	0.0496261	+	0.998768i	$0.484197\pi$
$644$	−0.858052	−0.0338120
$645$	0	0
$646$	−3.73419	−0.146920
$647$	−28.5094	−1.12082	−0.560409	−	0.828216i	$-0.689356\pi$
$647$	−28.5094	−1.12082	−0.560409	+	0.828216i	$0.689356\pi$
$648$	0	0
$649$	−58.4594	−2.29473
$650$	0	0
$651$	0	0
$652$	3.40776	0.133458
$653$	−3.18449	−0.124619	−0.0623093	−	0.998057i	$-0.519847\pi$
$653$	−3.18449	−0.124619	−0.0623093	+	0.998057i	$0.519847\pi$
$654$	0	0
$655$	0	0
$656$	−4.05582	−0.158353
$657$	0	0
$658$	−13.4684	−0.525052
$659$	25.9984	1.01276	0.506378	−	0.862312i	$-0.330984\pi$
$659$	25.9984	1.01276	0.506378	+	0.862312i	$0.330984\pi$
$660$	0	0
$661$	−2.65287	−0.103185	−0.0515923	−	0.998668i	$-0.516430\pi$
$661$	−2.65287	−0.103185	−0.0515923	+	0.998668i	$0.516430\pi$
$662$	−15.8942	−0.617747
$663$	0	0
$664$	−2.90645	−0.112792
$665$	0	0
$666$	0	0
$667$	−2.37003	−0.0917678
$668$	18.4562	0.714090
$669$	0	0
$670$	0	0
$671$	53.6694	2.07188
$672$	0	0
$673$	−7.27803	−0.280548	−0.140274	−	0.990113i	$-0.544798\pi$
$673$	−7.27803	−0.280548	−0.140274	+	0.990113i	$0.544798\pi$
$674$	21.8129	0.840202
$675$	0	0
$676$	1.36417	0.0524679
$677$	46.9984	1.80630	0.903148	−	0.429328i	$-0.141250\pi$
$677$	46.9984	1.80630	0.903148	+	0.429328i	$0.141250\pi$
$678$	0	0
$679$	−39.2010	−1.50440
$680$	0	0
$681$	0	0
$682$	38.1829	1.46210
$683$	2.03773	0.0779716	0.0389858	−	0.999240i	$-0.487587\pi$
$683$	2.03773	0.0779716	0.0389858	+	0.999240i	$0.487587\pi$
$684$	0	0
$685$	0	0
$686$	−19.6406	−0.749883
$687$	0	0
$688$	0.913870	0.0348410
$689$	26.3871	1.00527
$690$	0	0
$691$	−46.6513	−1.77470	−0.887350	−	0.461097i	$-0.847456\pi$
$691$	−46.6513	−1.77470	−0.887350	+	0.461097i	$0.847456\pi$
$692$	−25.9549	−0.986655
$693$	0	0
$694$	−18.5168	−0.702887
$695$	0	0
$696$	0	0
$697$	−15.1452	−0.573665
$698$	35.7678	1.35383
$699$	0	0
$700$	0	0
$701$	−46.3052	−1.74892	−0.874462	−	0.485094i	$-0.838786\pi$
$701$	−46.3052	−1.74892	−0.874462	+	0.485094i	$0.838786\pi$
$702$	0	0
$703$	−6.17226	−0.232791
$704$	−4.08613	−0.154002
$705$	0	0
$706$	18.5497	0.698127
$707$	15.5604	0.585208
$708$	0	0
$709$	10.2026	0.383166	0.191583	−	0.981476i	$-0.438638\pi$
$709$	10.2026	0.383166	0.191583	+	0.981476i	$0.438638\pi$
$710$	0	0
$711$	0	0
$712$	15.2887	0.572968
$713$	3.28870	0.123163
$714$	0	0
$715$	0	0
$716$	−5.23550	−0.195660
$717$	0	0
$718$	−29.0288	−1.08334
$719$	−7.83997	−0.292381	−0.146191	−	0.989256i	$-0.546701\pi$
$719$	−7.83997	−0.292381	−0.146191	+	0.989256i	$0.546701\pi$
$720$	0	0
$721$	−20.3626	−0.758343
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−25.2584	−0.938721
$725$	0	0
$726$	0	0
$727$	−29.3419	−1.08823	−0.544116	−	0.839010i	$-0.683135\pi$
$727$	−29.3419	−1.08823	−0.544116	+	0.839010i	$0.683135\pi$
$728$	9.24030	0.342468
$729$	0	0
$730$	0	0
$731$	3.41256	0.126218
$732$	0	0
$733$	−35.1903	−1.29979	−0.649893	−	0.760026i	$-0.725186\pi$
$733$	−35.1903	−1.29979	−0.649893	+	0.760026i	$0.725186\pi$
$734$	−8.36261	−0.308669
$735$	0	0
$736$	−0.351939	−0.0129727
$737$	31.0968	1.14547
$738$	0	0
$739$	−10.5168	−0.386866	−0.193433	−	0.981113i	$-0.561962\pi$
$739$	−10.5168	−0.386866	−0.193433	+	0.981113i	$0.561962\pi$
$740$	0	0
$741$	0	0
$742$	16.9745	0.623153
$743$	13.9852	0.513066	0.256533	−	0.966535i	$-0.417420\pi$
$743$	13.9852	0.513066	0.256533	+	0.966535i	$0.417420\pi$
$744$	0	0
$745$	0	0
$746$	33.1223	1.21269
$747$	0	0
$748$	−15.2584	−0.557902
$749$	−42.4971	−1.55281
$750$	0	0
$751$	−38.9681	−1.42197	−0.710984	−	0.703209i	$-0.751750\pi$
$751$	−38.9681	−1.42197	−0.710984	+	0.703209i	$0.751750\pi$
$752$	−5.52420	−0.201447
$753$	0	0
$754$	25.5226	0.929480
$755$	0	0
$756$	0	0
$757$	−32.5619	−1.18348	−0.591742	−	0.806128i	$-0.701559\pi$
$757$	−32.5619	−1.18348	−0.591742	+	0.806128i	$0.701559\pi$
$758$	−37.1090	−1.34786
$759$	0	0
$760$	0	0
$761$	−26.5955	−0.964086	−0.482043	−	0.876148i	$-0.660105\pi$
$761$	−26.5955	−0.964086	−0.482043	+	0.876148i	$0.660105\pi$
$762$	0	0
$763$	−26.5168	−0.959972
$764$	−3.95160	−0.142964
$765$	0	0
$766$	−17.5981	−0.635845
$767$	54.2229	1.95787
$768$	0	0
$769$	4.64545	0.167519	0.0837596	−	0.996486i	$-0.473307\pi$
$769$	4.64545	0.167519	0.0837596	+	0.996486i	$0.473307\pi$
$770$	0	0
$771$	0	0
$772$	−2.43807	−0.0877480
$773$	−18.0820	−0.650363	−0.325181	−	0.945652i	$-0.605425\pi$
$773$	−18.0820	−0.650363	−0.325181	+	0.945652i	$0.605425\pi$
$774$	0	0
$775$	0	0
$776$	−16.0787	−0.577192
$777$	0	0
$778$	8.87614	0.318225
$779$	−4.05582	−0.145315
$780$	0	0
$781$	39.7752	1.42327
$782$	−1.31421	−0.0469960
$783$	0	0
$784$	−1.05582	−0.0377078
$785$	0	0
$786$	0	0
$787$	−9.07871	−0.323621	−0.161811	−	0.986822i	$-0.551733\pi$
$787$	−9.07871	−0.323621	−0.161811	+	0.986822i	$0.551733\pi$
$788$	−13.6029	−0.484584
$789$	0	0
$790$	0	0
$791$	−2.57416	−0.0915265
$792$	0	0
$793$	−49.7800	−1.76774
$794$	37.1952	1.32001
$795$	0	0
$796$	4.81551	0.170681
$797$	5.64545	0.199972	0.0999860	−	0.994989i	$-0.468120\pi$
$797$	5.64545	0.199972	0.0999860	+	0.994989i	$0.468120\pi$
$798$	0	0
$799$	−20.6284	−0.729781
$800$	0	0
$801$	0	0
$802$	17.1116	0.604233
$803$	−49.0032	−1.72929
$804$	0	0
$805$	0	0
$806$	−35.4158	−1.24747
$807$	0	0
$808$	6.38225	0.224527
$809$	33.7555	1.18678	0.593391	−	0.804915i	$-0.297789\pi$
$809$	33.7555	1.18678	0.593391	+	0.804915i	$0.297789\pi$
$810$	0	0
$811$	−13.3823	−0.469914	−0.234957	−	0.972006i	$-0.575495\pi$
$811$	−13.3823	−0.469914	−0.234957	+	0.972006i	$0.575495\pi$
$812$	16.4184	0.576174
$813$	0	0
$814$	−25.2207	−0.883984
$815$	0	0
$816$	0	0
$817$	0.913870	0.0319723
$818$	16.2691	0.568834
$819$	0	0
$820$	0	0
$821$	−41.1829	−1.43729	−0.718647	−	0.695375i	$-0.755238\pi$
$821$	−41.1829	−1.43729	−0.718647	+	0.695375i	$0.755238\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−8.35194	−0.290954
$825$	0	0
$826$	34.8809	1.21366
$827$	−40.7268	−1.41621	−0.708104	−	0.706108i	$-0.750449\pi$
$827$	−40.7268	−1.41621	−0.708104	+	0.706108i	$0.750449\pi$
$828$	0	0
$829$	23.9016	0.830138	0.415069	−	0.909790i	$-0.363757\pi$
$829$	23.9016	0.830138	0.415069	+	0.909790i	$0.363757\pi$
$830$	0	0
$831$	0	0
$832$	3.79001	0.131395
$833$	−3.94262	−0.136604
$834$	0	0
$835$	0	0
$836$	−4.08613	−0.141322
$837$	0	0
$838$	−3.12386	−0.107912
$839$	−22.0639	−0.761730	−0.380865	−	0.924631i	$-0.624374\pi$
$839$	−22.0639	−0.761730	−0.380865	+	0.924631i	$0.624374\pi$
$840$	0	0
$841$	16.3493	0.563770
$842$	11.8639	0.408857
$843$	0	0
$844$	−18.4307	−0.634409
$845$	0	0
$846$	0	0
$847$	−13.8884	−0.477210
$848$	6.96227	0.239085
$849$	0	0
$850$	0	0
$851$	−2.17226	−0.0744641
$852$	0	0
$853$	−20.7793	−0.711471	−0.355736	−	0.934587i	$-0.615770\pi$
$853$	−20.7793	−0.711471	−0.355736	+	0.934587i	$0.615770\pi$
$854$	−32.0229	−1.09580
$855$	0	0
$856$	−17.4307	−0.595767
$857$	−13.9655	−0.477053	−0.238527	−	0.971136i	$-0.576664\pi$
$857$	−13.9655	−0.477053	−0.238527	+	0.971136i	$0.576664\pi$
$858$	0	0
$859$	−21.0484	−0.718162	−0.359081	−	0.933306i	$-0.616910\pi$
$859$	−21.0484	−0.718162	−0.359081	+	0.933306i	$0.616910\pi$
$860$	0	0
$861$	0	0
$862$	5.69165	0.193858
$863$	57.2058	1.94731	0.973654	−	0.228029i	$-0.0732280\pi$
$863$	57.2058	1.94731	0.973654	+	0.228029i	$0.0732280\pi$
$864$	0	0
$865$	0	0
$866$	−8.36261	−0.284173
$867$	0	0
$868$	−22.7826	−0.773291
$869$	12.9623	0.439715
$870$	0	0
$871$	−28.8432	−0.977315
$872$	−10.8761	−0.368313
$873$	0	0
$874$	−0.351939	−0.0119045
$875$	0	0
$876$	0	0
$877$	−2.47099	−0.0834395	−0.0417197	−	0.999129i	$-0.513284\pi$
$877$	−2.47099	−0.0834395	−0.0417197	+	0.999129i	$0.513284\pi$
$878$	−13.9368	−0.470343
$879$	0	0
$880$	0	0
$881$	13.9245	0.469130	0.234565	−	0.972100i	$-0.424633\pi$
$881$	13.9245	0.469130	0.234565	+	0.972100i	$0.424633\pi$
$882$	0	0
$883$	14.3823	0.484001	0.242001	−	0.970276i	$-0.422196\pi$
$883$	14.3823	0.484001	0.242001	+	0.970276i	$0.422196\pi$
$884$	14.1526	0.476004
$885$	0	0
$886$	32.7194	1.09923
$887$	−24.7645	−0.831511	−0.415755	−	0.909477i	$-0.636483\pi$
$887$	−24.7645	−0.831511	−0.415755	+	0.909477i	$0.636483\pi$
$888$	0	0
$889$	−21.0665	−0.706547
$890$	0	0
$891$	0	0
$892$	−7.22808	−0.242014
$893$	−5.52420	−0.184860
$894$	0	0
$895$	0	0
$896$	2.43807	0.0814502
$897$	0	0
$898$	32.0288	1.06881
$899$	−62.9278	−2.09876
$900$	0	0
$901$	25.9984	0.866134
$902$	−16.5726	−0.551807
$903$	0	0
$904$	−1.05582	−0.0351160
$905$	0	0
$906$	0	0
$907$	−4.16745	−0.138378	−0.0691890	−	0.997604i	$-0.522041\pi$
$907$	−4.16745	−0.138378	−0.0691890	+	0.997604i	$0.522041\pi$
$908$	−21.3929	−0.709949
$909$	0	0
$910$	0	0
$911$	27.4535	0.909577	0.454788	−	0.890600i	$-0.349715\pi$
$911$	27.4535	0.909577	0.454788	+	0.890600i	$0.349715\pi$
$912$	0	0
$913$	−11.8761	−0.393043
$914$	−4.70869	−0.155749
$915$	0	0
$916$	1.96969	0.0650803
$917$	−0.0739042	−0.00244053
$918$	0	0
$919$	23.0303	0.759700	0.379850	−	0.925048i	$-0.375976\pi$
$919$	23.0303	0.759700	0.379850	+	0.925048i	$0.375976\pi$
$920$	0	0
$921$	0	0
$922$	−30.2084	−0.994862
$923$	−36.8927	−1.21434
$924$	0	0
$925$	0	0
$926$	−5.82774	−0.191511
$927$	0	0
$928$	6.73419	0.221061
$929$	38.1116	1.25040	0.625201	−	0.780464i	$-0.285017\pi$
$929$	38.1116	1.25040	0.625201	+	0.780464i	$0.285017\pi$
$930$	0	0
$931$	−1.05582	−0.0346030
$932$	−5.08132	−0.166444
$933$	0	0
$934$	17.0591	0.558190
$935$	0	0
$936$	0	0
$937$	−44.7401	−1.46159	−0.730797	−	0.682595i	$-0.760851\pi$
$937$	−44.7401	−1.46159	−0.730797	+	0.682595i	$0.760851\pi$
$938$	−18.5545	−0.605826
$939$	0	0
$940$	0	0
$941$	13.7268	0.447480	0.223740	−	0.974649i	$-0.428173\pi$
$941$	13.7268	0.447480	0.223740	+	0.974649i	$0.428173\pi$
$942$	0	0
$943$	−1.42740	−0.0464826
$944$	14.3068	0.465646
$945$	0	0
$946$	3.73419	0.121409
$947$	−29.5652	−0.960739	−0.480370	−	0.877066i	$-0.659497\pi$
$947$	−29.5652	−0.960739	−0.480370	+	0.877066i	$0.659497\pi$
$948$	0	0
$949$	45.4520	1.47543
$950$	0	0
$951$	0	0
$952$	9.10422	0.295069
$953$	35.6284	1.15412	0.577059	−	0.816703i	$-0.304200\pi$
$953$	35.6284	1.15412	0.577059	+	0.816703i	$0.304200\pi$
$954$	0	0
$955$	0	0
$956$	−15.1648	−0.490466
$957$	0	0
$958$	4.94418	0.159739
$959$	3.58002	0.115605
$960$	0	0
$961$	56.3201	1.81678
$962$	23.3929	0.754218
$963$	0	0
$964$	5.79482	0.186638
$965$	0	0
$966$	0	0
$967$	52.9581	1.70302	0.851509	−	0.524340i	$-0.175688\pi$
$967$	52.9581	1.70302	0.851509	+	0.524340i	$0.175688\pi$
$968$	−5.69646	−0.183091
$969$	0	0
$970$	0	0
$971$	−25.5423	−0.819691	−0.409845	−	0.912155i	$-0.634417\pi$
$971$	−25.5423	−0.819691	−0.409845	+	0.912155i	$0.634417\pi$
$972$	0	0
$973$	−45.5094	−1.45896
$974$	−17.9097	−0.573864
$975$	0	0
$976$	−13.1345	−0.420426
$977$	−12.6629	−0.405122	−0.202561	−	0.979270i	$-0.564926\pi$
$977$	−12.6629	−0.405122	−0.202561	+	0.979270i	$0.564926\pi$
$978$	0	0
$979$	62.4716	1.99660
$980$	0	0
$981$	0	0
$982$	25.9852	0.829220
$983$	−8.26581	−0.263638	−0.131819	−	0.991274i	$-0.542082\pi$
$983$	−8.26581	−0.263638	−0.131819	+	0.991274i	$0.542082\pi$
$984$	0	0
$985$	0	0
$986$	25.1468	0.800836
$987$	0	0
$988$	3.79001	0.120576
$989$	0.321627	0.0102271
$990$	0	0
$991$	−44.9655	−1.42838	−0.714188	−	0.699954i	$-0.753204\pi$
$991$	−44.9655	−1.42838	−0.714188	+	0.699954i	$0.753204\pi$
$992$	−9.34452	−0.296689
$993$	0	0
$994$	−23.7326	−0.752753
$995$	0	0
$996$	0	0
$997$	−21.4939	−0.680718	−0.340359	−	0.940296i	$-0.610549\pi$
$997$	−21.4939	−0.680718	−0.340359	+	0.940296i	$0.610549\pi$
$998$	−3.54229	−0.112129
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.43807 q^{7} -1.00000 q^{8} -4.08613 q^{11} +3.79001 q^{13} +2.43807 q^{14} +1.00000 q^{16} +3.73419 q^{17} +1.00000 q^{19} +4.08613 q^{22} +0.351939 q^{23} -3.79001 q^{26} -2.43807 q^{28} -6.73419 q^{29} +9.34452 q^{31} -1.00000 q^{32} -3.73419 q^{34} -6.17226 q^{37} -1.00000 q^{38} -4.05582 q^{41} +0.913870 q^{43} -4.08613 q^{44} -0.351939 q^{46} -5.52420 q^{47} -1.05582 q^{49} +3.79001 q^{52} +6.96227 q^{53} +2.43807 q^{56} +6.73419 q^{58} +14.3068 q^{59} -13.1345 q^{61} -9.34452 q^{62} +1.00000 q^{64} -7.61033 q^{67} +3.73419 q^{68} -9.73419 q^{71} +11.9926 q^{73} +6.17226 q^{74} +1.00000 q^{76} +9.96227 q^{77} -3.17226 q^{79} +4.05582 q^{82} +2.90645 q^{83} -0.913870 q^{86} +4.08613 q^{88} -15.2887 q^{89} -9.24030 q^{91} +0.351939 q^{92} +5.52420 q^{94} +16.0787 q^{97} +1.05582 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) 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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