LMFDB - Embedded newform 8550.2.a.cn.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:	$1$
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			1.3
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	$-0.347580\pi$
$7$	2.43807	0.921504	0.460752	+	0.887529i	$0.347580\pi$
$8$	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	$-0.676152\pi$
$13$	−3.79001	−1.05116	−0.525580	+	0.850744i	$0.676152\pi$
$14$	2.43807	0.651601
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.73419	−0.905674	−0.452837	−	0.891593i	$-0.649588\pi$
$17$	−3.73419	−0.905674	−0.452837	+	0.891593i	$0.649588\pi$
$18$	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	$-0.511682\pi$
$23$	−0.351939	−0.0733844	−0.0366922	+	0.999327i	$0.511682\pi$
$24$	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	$-0.330623\pi$
$37$	6.17226	1.01471	0.507357	+	0.861736i	$0.330623\pi$
$38$	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	$-0.522198\pi$
$43$	−0.913870	−0.139364	−0.0696819	+	0.997569i	$0.522198\pi$
$44$	−4.08613	−0.616007
$45$	0	0
$46$	−0.351939	−0.0518906
$47$	5.52420	0.805787	0.402894	−	0.915247i	$-0.368004\pi$
$47$	5.52420	0.805787	0.402894	+	0.915247i	$0.368004\pi$
$48$	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	$-0.658700\pi$
$53$	−6.96227	−0.956341	−0.478171	+	0.878267i	$0.658700\pi$
$54$	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	$-0.346099\pi$
$67$	7.61033	0.929750	0.464875	+	0.885376i	$0.346099\pi$
$68$	−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	$-0.747626\pi$
$73$	−11.9926	−1.40363	−0.701813	+	0.712361i	$0.747626\pi$
$74$	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	$-0.550992\pi$
$83$	−2.90645	−0.319024	−0.159512	+	0.987196i	$0.550992\pi$
$84$	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	$-0.803963\pi$
$97$	−16.0787	−1.63255	−0.816273	+	0.577666i	$0.803963\pi$
$98$	−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	$-0.634985\pi$
$103$	−8.35194	−0.822941	−0.411471	+	0.911423i	$0.634985\pi$
$104$	−3.79001	−0.371641
$105$	0	0
$106$	−6.96227	−0.676235
$107$	−17.4307	−1.68508	−0.842542	−	0.538630i	$-0.818942\pi$
$107$	−17.4307	−1.68508	−0.842542	+	0.538630i	$0.818942\pi$
$108$	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	$-0.515814\pi$
$113$	−1.05582	−0.0993230	−0.0496615	+	0.998766i	$0.515814\pi$
$114$	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	$-0.625235\pi$
$127$	−8.64064	−0.766733	−0.383367	+	0.923596i	$0.625235\pi$
$128$	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	$-0.480021\pi$
$137$	1.46838	0.125452	0.0627262	+	0.998031i	$0.480021\pi$
$138$	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	$-0.606488\pi$
$157$	−8.22808	−0.656672	−0.328336	+	0.944561i	$0.606488\pi$
$158$	−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	$-0.542608\pi$
$163$	−3.40776	−0.266916	−0.133458	+	0.991054i	$0.542608\pi$
$164$	−4.05582	−0.316706
$165$	0	0
$166$	−2.90645	−0.225584
$167$	−18.4562	−1.42818	−0.714090	−	0.700054i	$-0.753159\pi$
$167$	−18.4562	−1.42818	−0.714090	+	0.700054i	$0.753159\pi$
$168$	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	$-0.0520598\pi$
$173$	25.9549	1.97331	0.986655	+	0.162823i	$0.0520598\pi$
$174$	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	$-0.472033\pi$
$193$	2.43807	0.175496	0.0877480	+	0.996143i	$0.472033\pi$
$194$	−16.0787	−1.15438
$195$	0	0
$196$	−1.05582	−0.0754155
$197$	13.6029	0.969167	0.484584	−	0.874745i	$-0.338971\pi$
$197$	13.6029	0.969167	0.484584	+	0.874745i	$0.338971\pi$
$198$	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	$-0.422192\pi$
$223$	7.22808	0.484028	0.242014	+	0.970273i	$0.422192\pi$
$224$	2.43807	0.162900
$225$	0	0
$226$	−1.05582	−0.0702319
$227$	21.3929	1.41990	0.709949	−	0.704253i	$-0.248718\pi$
$227$	21.3929	1.41990	0.709949	+	0.704253i	$0.248718\pi$
$228$	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	$-0.446771\pi$
$233$	5.08132	0.332889	0.166444	+	0.986051i	$0.446771\pi$
$234$	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	$-0.467898\pi$
$257$	3.22808	0.201362	0.100681	+	0.994919i	$0.467898\pi$
$258$	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	$-0.211856\pi$
$263$	25.5120	1.57314	0.786568	+	0.617504i	$0.211856\pi$
$264$	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	$-0.466527\pi$
$277$	3.49389	0.209927	0.104964	+	0.994476i	$0.466527\pi$
$278$	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	$-0.484472\pi$
$283$	1.64064	0.0975261	0.0487630	+	0.998810i	$0.484472\pi$
$284$	−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	$-0.346901\pi$
$293$	15.8384	0.925290	0.462645	+	0.886544i	$0.346901\pi$
$294$	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	$-0.414250\pi$
$307$	9.32643	0.532288	0.266144	+	0.963933i	$0.414250\pi$
$308$	−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	$-0.473484\pi$
$313$	2.94418	0.166415	0.0832075	+	0.996532i	$0.473484\pi$
$314$	−8.22808	−0.464337
$315$	0	0
$316$	−3.17226	−0.178454
$317$	−18.7342	−1.05222	−0.526108	−	0.850417i	$-0.676349\pi$
$317$	−18.7342	−1.05222	−0.526108	+	0.850417i	$0.676349\pi$
$318$	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	$-0.297503\pi$
$337$	21.8129	1.18822	0.594112	+	0.804382i	$0.297503\pi$
$338$	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	$-0.665571\pi$
$347$	−18.5168	−0.994033	−0.497016	+	0.867741i	$0.665571\pi$
$348$	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	$-0.335662\pi$
$353$	18.5497	0.987301	0.493651	+	0.869660i	$0.335662\pi$
$354$	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	$-0.570039\pi$
$367$	−8.36261	−0.436525	−0.218262	+	0.975890i	$0.570039\pi$
$368$	−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	$-0.172014\pi$
$373$	33.1223	1.71501	0.857504	+	0.514477i	$0.172014\pi$
$374$	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	$-0.648437\pi$
$383$	−17.5981	−0.899221	−0.449611	+	0.893225i	$0.648437\pi$
$384$	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	$-0.116840\pi$
$397$	37.1952	1.86677	0.933386	+	0.358875i	$0.116840\pi$
$398$	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	$-0.564400\pi$
$433$	−8.36261	−0.401881	−0.200941	+	0.979603i	$0.564400\pi$
$434$	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	$-0.216603\pi$
$443$	32.7194	1.55454	0.777272	+	0.629165i	$0.216603\pi$
$444$	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	$-0.535127\pi$
$457$	−4.70869	−0.220263	−0.110132	+	0.993917i	$0.535127\pi$
$458$	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	$-0.543238\pi$
$463$	−5.82774	−0.270838	−0.135419	+	0.990788i	$0.543238\pi$
$464$	−6.73419	−0.312627
$465$	0	0
$466$	5.08132	0.235388
$467$	17.0591	0.789399	0.394700	−	0.918810i	$-0.370849\pi$
$467$	17.0591	0.789399	0.394700	+	0.918810i	$0.370849\pi$
$468$	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	$-0.633001\pi$
$487$	−17.9097	−0.811566	−0.405783	+	0.913969i	$0.633001\pi$
$488$	−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	$-0.448971\pi$
$503$	7.16003	0.319250	0.159625	+	0.987178i	$0.448971\pi$
$504$	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	$-0.233010\pi$
$523$	34.0213	1.48765	0.743825	+	0.668375i	$0.233010\pi$
$524$	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	$-0.0643463\pi$
$547$	45.8236	1.95927	0.979637	+	0.200776i	$0.0643463\pi$
$548$	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	$-0.570841\pi$
$557$	−10.4184	−0.441443	−0.220721	+	0.975337i	$0.570841\pi$
$558$	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	$-0.506031\pi$
$563$	−0.899033	−0.0378897	−0.0189449	+	0.999821i	$0.506031\pi$
$564$	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	$-0.355175\pi$
$577$	21.1116	0.878889	0.439444	+	0.898270i	$0.355175\pi$
$578$	−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	$-0.591143\pi$
$587$	−13.6858	−0.564873	−0.282437	+	0.959286i	$0.591143\pi$
$588$	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	$-0.840629\pi$
$593$	−42.7252	−1.75451	−0.877257	+	0.480021i	$0.840629\pi$
$594$	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	$-0.601079\pi$
$607$	−15.3855	−0.624478	−0.312239	+	0.950004i	$0.601079\pi$
$608$	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	$-0.614242\pi$
$613$	−17.3929	−0.702493	−0.351247	+	0.936283i	$0.614242\pi$
$614$	9.32643	0.376384
$615$	0	0
$616$	−9.96227	−0.401391
$617$	6.76450	0.272329	0.136164	−	0.990686i	$-0.456522\pi$
$617$	6.76450	0.272329	0.136164	+	0.990686i	$0.456522\pi$
$618$	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	$-0.515803\pi$
$643$	−2.51678	−0.0992522	−0.0496261	+	0.998768i	$0.515803\pi$
$644$	−0.858052	−0.0338120
$645$	0	0
$646$	−3.73419	−0.146920
$647$	28.5094	1.12082	0.560409	−	0.828216i	$-0.310644\pi$
$647$	28.5094	1.12082	0.560409	+	0.828216i	$0.310644\pi$
$648$	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	$-0.480153\pi$
$653$	3.18449	0.124619	0.0623093	+	0.998057i	$0.480153\pi$
$654$	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	$-0.455202\pi$
$673$	7.27803	0.280548	0.140274	+	0.990113i	$0.455202\pi$
$674$	21.8129	0.840202
$675$	0	0
$676$	1.36417	0.0524679
$677$	−46.9984	−1.80630	−0.903148	−	0.429328i	$-0.858750\pi$
$677$	−46.9984	−1.80630	−0.903148	+	0.429328i	$0.858750\pi$
$678$	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	$-0.512413\pi$
$683$	−2.03773	−0.0779716	−0.0389858	+	0.999240i	$0.512413\pi$
$684$	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	$-0.316865\pi$
$727$	29.3419	1.08823	0.544116	+	0.839010i	$0.316865\pi$
$728$	−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	$-0.274814\pi$
$733$	35.1903	1.29979	0.649893	+	0.760026i	$0.274814\pi$
$734$	−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	$-0.582580\pi$
$743$	−13.9852	−0.513066	−0.256533	+	0.966535i	$0.582580\pi$
$744$	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	$-0.298441\pi$
$757$	32.5619	1.18348	0.591742	+	0.806128i	$0.298441\pi$
$758$	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	$-0.394575\pi$
$773$	18.0820	0.650363	0.325181	+	0.945652i	$0.394575\pi$
$774$	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	$-0.448267\pi$
$787$	9.07871	0.323621	0.161811	+	0.986822i	$0.448267\pi$
$788$	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	$-0.531880\pi$
$797$	−5.64545	−0.199972	−0.0999860	+	0.994989i	$0.531880\pi$
$798$	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	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−8.35194	−0.290954
$825$	0	0
$826$	34.8809	1.21366
$827$	40.7268	1.41621	0.708104	−	0.706108i	$-0.249551\pi$
$827$	40.7268	1.41621	0.708104	+	0.706108i	$0.249551\pi$
$828$	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	$-0.384230\pi$
$853$	20.7793	0.711471	0.355736	+	0.934587i	$0.384230\pi$
$854$	−32.0229	−1.09580
$855$	0	0
$856$	−17.4307	−0.595767
$857$	13.9655	0.477053	0.238527	−	0.971136i	$-0.423336\pi$
$857$	13.9655	0.477053	0.238527	+	0.971136i	$0.423336\pi$
$858$	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	$-0.926772\pi$
$863$	−57.2058	−1.94731	−0.973654	+	0.228029i	$0.926772\pi$
$864$	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	$-0.486716\pi$
$877$	2.47099	0.0834395	0.0417197	+	0.999129i	$0.486716\pi$
$878$	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	$-0.577804\pi$
$883$	−14.3823	−0.484001	−0.242001	+	0.970276i	$0.577804\pi$
$884$	14.1526	0.476004
$885$	0	0
$886$	32.7194	1.09923
$887$	24.7645	0.831511	0.415755	−	0.909477i	$-0.363517\pi$
$887$	24.7645	0.831511	0.415755	+	0.909477i	$0.363517\pi$
$888$	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	$-0.477959\pi$
$907$	4.16745	0.138378	0.0691890	+	0.997604i	$0.477959\pi$
$908$	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	$-0.239149\pi$
$937$	44.7401	1.46159	0.730797	+	0.682595i	$0.239149\pi$
$938$	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	$-0.340503\pi$
$947$	29.5652	0.960739	0.480370	+	0.877066i	$0.340503\pi$
$948$	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	$-0.695800\pi$
$953$	−35.6284	−1.15412	−0.577059	+	0.816703i	$0.695800\pi$
$954$	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	$-0.824312\pi$
$967$	−52.9581	−1.70302	−0.851509	+	0.524340i	$0.824312\pi$
$968$	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	$-0.435074\pi$
$977$	12.6629	0.405122	0.202561	+	0.979270i	$0.435074\pi$
$978$	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	$-0.457918\pi$
$983$	8.26581	0.263638	0.131819	+	0.991274i	$0.457918\pi$
$984$	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	$-0.389451\pi$
$997$	21.4939	0.680718	0.340359	+	0.940296i	$0.389451\pi$
$998$	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.cn.1.3	yes	3
3.2	odd	2		8550.2.a.cd.1.3	✓	3
5.4	even	2		8550.2.a.cg.1.1	yes	3
15.14	odd	2		8550.2.a.cs.1.1	yes	3

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