LMFDB - Embedded newform 8550.2.a.ct.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
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			1.1
Root			$3.35386$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +4.89449 q^{11} -1.89449 q^{13} -2.35386 q^{14} +1.00000 q^{16} -2.35386 q^{17} -1.00000 q^{19} +4.89449 q^{22} -6.16707 q^{23} -1.89449 q^{26} -2.35386 q^{28} +4.27258 q^{29} +7.70771 q^{31} +1.00000 q^{32} -2.35386 q^{34} +7.78899 q^{37} -1.00000 q^{38} +3.54064 q^{41} -1.73194 q^{43} +4.89449 q^{44} -6.16707 q^{46} -5.32962 q^{47} -1.45936 q^{49} -1.89449 q^{52} -1.97577 q^{53} -2.35386 q^{56} +4.27258 q^{58} +6.81322 q^{59} +3.97577 q^{61} +7.70771 q^{62} +1.00000 q^{64} +5.19131 q^{67} -2.35386 q^{68} -4.14284 q^{71} +4.37809 q^{73} +7.78899 q^{74} -1.00000 q^{76} -11.5209 q^{77} +9.49670 q^{79} +3.54064 q^{82} -4.43513 q^{83} -1.73194 q^{86} +4.89449 q^{88} +3.62191 q^{89} +4.45936 q^{91} -6.16707 q^{92} -5.32962 q^{94} +7.06157 q^{97} -1.45936 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.35386	−0.889674	−0.444837	−	0.895612i	$-0.646738\pi$
$7$	−2.35386	−0.889674	−0.444837	+	0.895612i	$0.646738\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	4.89449	1.47575	0.737873	−	0.674940i	$-0.235831\pi$
$11$	4.89449	1.47575	0.737873	+	0.674940i	$0.235831\pi$
$12$	0	0
$13$	−1.89449	−0.525438	−0.262719	−	0.964872i	$-0.584619\pi$
$13$	−1.89449	−0.525438	−0.262719	+	0.964872i	$0.584619\pi$
$14$	−2.35386	−0.629094
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.35386	−0.570894	−0.285447	−	0.958395i	$-0.592142\pi$
$17$	−2.35386	−0.570894	−0.285447	+	0.958395i	$0.592142\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	4.89449	1.04351
$23$	−6.16707	−1.28592	−0.642962	−	0.765898i	$-0.722295\pi$
$23$	−6.16707	−1.28592	−0.642962	+	0.765898i	$0.722295\pi$
$24$	0	0
$25$	0	0
$26$	−1.89449	−0.371541
$27$	0	0
$28$	−2.35386	−0.444837
$29$	4.27258	0.793398	0.396699	−	0.917949i	$-0.370156\pi$
$29$	4.27258	0.793398	0.396699	+	0.917949i	$0.370156\pi$
$30$	0	0
$31$	7.70771	1.38435	0.692173	−	0.721732i	$-0.256654\pi$
$31$	7.70771	1.38435	0.692173	+	0.721732i	$0.256654\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.35386	−0.403683
$35$	0	0
$36$	0	0
$37$	7.78899	1.28050	0.640251	−	0.768166i	$-0.278830\pi$
$37$	7.78899	1.28050	0.640251	+	0.768166i	$0.278830\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	3.54064	0.552955	0.276477	−	0.961020i	$-0.410833\pi$
$41$	3.54064	0.552955	0.276477	+	0.961020i	$0.410833\pi$
$42$	0	0
$43$	−1.73194	−0.264119	−0.132060	−	0.991242i	$-0.542159\pi$
$43$	−1.73194	−0.264119	−0.132060	+	0.991242i	$0.542159\pi$
$44$	4.89449	0.737873
$45$	0	0
$46$	−6.16707	−0.909286
$47$	−5.32962	−0.777405	−0.388703	−	0.921363i	$-0.627077\pi$
$47$	−5.32962	−0.777405	−0.388703	+	0.921363i	$0.627077\pi$
$48$	0	0
$49$	−1.45936	−0.208480
$50$	0	0
$51$	0	0
$52$	−1.89449	−0.262719
$53$	−1.97577	−0.271392	−0.135696	−	0.990750i	$-0.543327\pi$
$53$	−1.97577	−0.271392	−0.135696	+	0.990750i	$0.543327\pi$
$54$	0	0
$55$	0	0
$56$	−2.35386	−0.314547
$57$	0	0
$58$	4.27258	0.561017
$59$	6.81322	0.887006	0.443503	−	0.896273i	$-0.353736\pi$
$59$	6.81322	0.887006	0.443503	+	0.896273i	$0.353736\pi$
$60$	0	0
$61$	3.97577	0.509045	0.254522	−	0.967067i	$-0.418082\pi$
$61$	3.97577	0.509045	0.254522	+	0.967067i	$0.418082\pi$
$62$	7.70771	0.978880
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.19131	0.634219	0.317110	−	0.948389i	$-0.397288\pi$
$67$	5.19131	0.634219	0.317110	+	0.948389i	$0.397288\pi$
$68$	−2.35386	−0.285447
$69$	0	0
$70$	0	0
$71$	−4.14284	−0.491665	−0.245832	−	0.969312i	$-0.579061\pi$
$71$	−4.14284	−0.491665	−0.245832	+	0.969312i	$0.579061\pi$
$72$	0	0
$73$	4.37809	0.512417	0.256208	−	0.966622i	$-0.417527\pi$
$73$	4.37809	0.512417	0.256208	+	0.966622i	$0.417527\pi$
$74$	7.78899	0.905451
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−11.5209	−1.31293
$78$	0	0
$79$	9.49670	1.06846	0.534231	−	0.845339i	$-0.320601\pi$
$79$	9.49670	1.06846	0.534231	+	0.845339i	$0.320601\pi$
$80$	0	0
$81$	0	0
$82$	3.54064	0.390998
$83$	−4.43513	−0.486819	−0.243409	−	0.969924i	$-0.578266\pi$
$83$	−4.43513	−0.486819	−0.243409	+	0.969924i	$0.578266\pi$
$84$	0	0
$85$	0	0
$86$	−1.73194	−0.186760
$87$	0	0
$88$	4.89449	0.521755
$89$	3.62191	0.383922	0.191961	−	0.981403i	$-0.438515\pi$
$89$	3.62191	0.383922	0.191961	+	0.981403i	$0.438515\pi$
$90$	0	0
$91$	4.45936	0.467468
$92$	−6.16707	−0.642962
$93$	0	0
$94$	−5.32962	−0.549709
$95$	0	0
$96$	0	0
$97$	7.06157	0.716994	0.358497	−	0.933531i	$-0.383289\pi$
$97$	7.06157	0.716994	0.358497	+	0.933531i	$0.383289\pi$
$98$	−1.45936	−0.147418
$99$	0	0
$100$	0	0
$101$	6.10551	0.607521	0.303760	−	0.952748i	$-0.401758\pi$
$101$	6.10551	0.607521	0.303760	+	0.952748i	$0.401758\pi$
$102$	0	0
$103$	11.2483	1.10833	0.554166	−	0.832406i	$-0.313037\pi$
$103$	11.2483	1.10833	0.554166	+	0.832406i	$0.313037\pi$
$104$	−1.89449	−0.185770
$105$	0	0
$106$	−1.97577	−0.191903
$107$	15.3099	1.48007	0.740033	−	0.672571i	$-0.234810\pi$
$107$	15.3099	1.48007	0.740033	+	0.672571i	$0.234810\pi$
$108$	0	0
$109$	−14.7077	−1.40874	−0.704372	−	0.709831i	$-0.748771\pi$
$109$	−14.7077	−1.40874	−0.704372	+	0.709831i	$0.748771\pi$
$110$	0	0
$111$	0	0
$112$	−2.35386	−0.222418
$113$	−3.29681	−0.310138	−0.155069	−	0.987904i	$-0.549560\pi$
$113$	−3.29681	−0.310138	−0.155069	+	0.987904i	$0.549560\pi$
$114$	0	0
$115$	0	0
$116$	4.27258	0.396699
$117$	0	0
$118$	6.81322	0.627208
$119$	5.54064	0.507909
$120$	0	0
$121$	12.9561	1.17782
$122$	3.97577	0.359949
$123$	0	0
$124$	7.70771	0.692173
$125$	0	0
$126$	0	0
$127$	−6.62644	−0.588001	−0.294001	−	0.955805i	$-0.594987\pi$
$127$	−6.62644	−0.588001	−0.294001	+	0.955805i	$0.594987\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	16.9803	1.48358	0.741788	−	0.670635i	$-0.233978\pi$
$131$	16.9803	1.48358	0.741788	+	0.670635i	$0.233978\pi$
$132$	0	0
$133$	2.35386	0.204105
$134$	5.19131	0.448461
$135$	0	0
$136$	−2.35386	−0.201841
$137$	12.8703	1.09958	0.549790	−	0.835303i	$-0.314708\pi$
$137$	12.8703	1.09958	0.549790	+	0.835303i	$0.314708\pi$
$138$	0	0
$139$	9.30992	0.789657	0.394828	−	0.918755i	$-0.370804\pi$
$139$	9.30992	0.789657	0.394828	+	0.918755i	$0.370804\pi$
$140$	0	0
$141$	0	0
$142$	−4.14284	−0.347660
$143$	−9.27258	−0.775412
$144$	0	0
$145$	0	0
$146$	4.37809	0.362333
$147$	0	0
$148$	7.78899	0.640251
$149$	−16.1231	−1.32086	−0.660429	−	0.750888i	$-0.729626\pi$
$149$	−16.1231	−1.32086	−0.660429	+	0.750888i	$0.729626\pi$
$150$	0	0
$151$	−15.6638	−1.27470	−0.637350	−	0.770575i	$-0.719969\pi$
$151$	−15.6638	−1.27470	−0.637350	+	0.770575i	$0.719969\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−11.5209	−0.928383
$155$	0	0
$156$	0	0
$157$	−14.2483	−1.13714	−0.568571	−	0.822634i	$-0.692504\pi$
$157$	−14.2483	−1.13714	−0.568571	+	0.822634i	$0.692504\pi$
$158$	9.49670	0.755517
$159$	0	0
$160$	0	0
$161$	14.5164	1.14405
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	3.54064	0.276477
$165$	0	0
$166$	−4.43513	−0.344233
$167$	0.756178	0.0585148	0.0292574	−	0.999572i	$-0.490686\pi$
$167$	0.756178	0.0585148	0.0292574	+	0.999572i	$0.490686\pi$
$168$	0	0
$169$	−9.41090	−0.723915
$170$	0	0
$171$	0	0
$172$	−1.73194	−0.132060
$173$	13.6880	1.04068	0.520340	−	0.853959i	$-0.325805\pi$
$173$	13.6880	1.04068	0.520340	+	0.853959i	$0.325805\pi$
$174$	0	0
$175$	0	0
$176$	4.89449	0.368936
$177$	0	0
$178$	3.62191	0.271474
$179$	23.2044	1.73438	0.867189	−	0.497978i	$-0.165924\pi$
$179$	23.2044	1.73438	0.867189	+	0.497978i	$0.165924\pi$
$180$	0	0
$181$	6.65067	0.494340	0.247170	−	0.968972i	$-0.420499\pi$
$181$	6.65067	0.494340	0.247170	+	0.968972i	$0.420499\pi$
$182$	4.45936	0.330550
$183$	0	0
$184$	−6.16707	−0.454643
$185$	0	0
$186$	0	0
$187$	−11.5209	−0.842494
$188$	−5.32962	−0.388703
$189$	0	0
$190$	0	0
$191$	14.7890	1.07009	0.535047	−	0.844822i	$-0.320294\pi$
$191$	14.7890	1.07009	0.535047	+	0.844822i	$0.320294\pi$
$192$	0	0
$193$	11.5977	0.834819	0.417410	−	0.908718i	$-0.362938\pi$
$193$	11.5977	0.834819	0.417410	+	0.908718i	$0.362938\pi$
$194$	7.06157	0.506991
$195$	0	0
$196$	−1.45936	−0.104240
$197$	12.6022	0.897870	0.448935	−	0.893564i	$-0.351803\pi$
$197$	12.6022	0.897870	0.448935	+	0.893564i	$0.351803\pi$
$198$	0	0
$199$	−23.7405	−1.68292	−0.841460	−	0.540319i	$-0.818304\pi$
$199$	−23.7405	−1.68292	−0.841460	+	0.540319i	$0.818304\pi$
$200$	0	0
$201$	0	0
$202$	6.10551	0.429582
$203$	−10.0570	−0.705866
$204$	0	0
$205$	0	0
$206$	11.2483	0.783710
$207$	0	0
$208$	−1.89449	−0.131359
$209$	−4.89449	−0.338559
$210$	0	0
$211$	19.9848	1.37581	0.687906	−	0.725800i	$-0.258530\pi$
$211$	19.9848	1.37581	0.687906	+	0.725800i	$0.258530\pi$
$212$	−1.97577	−0.135696
$213$	0	0
$214$	15.3099	1.04656
$215$	0	0
$216$	0	0
$217$	−18.1428	−1.23162
$218$	−14.7077	−0.996132
$219$	0	0
$220$	0	0
$221$	4.45936	0.299969
$222$	0	0
$223$	18.7122	1.25306	0.626532	−	0.779396i	$-0.284474\pi$
$223$	18.7122	1.25306	0.626532	+	0.779396i	$0.284474\pi$
$224$	−2.35386	−0.157274
$225$	0	0
$226$	−3.29681	−0.219301
$227$	13.0813	0.868235	0.434117	−	0.900856i	$-0.357060\pi$
$227$	13.0813	0.868235	0.434117	+	0.900856i	$0.357060\pi$
$228$	0	0
$229$	−9.89902	−0.654146	−0.327073	−	0.944999i	$-0.606062\pi$
$229$	−9.89902	−0.654146	−0.327073	+	0.944999i	$0.606062\pi$
$230$	0	0
$231$	0	0
$232$	4.27258	0.280509
$233$	12.6395	0.828044	0.414022	−	0.910267i	$-0.364124\pi$
$233$	12.6395	0.828044	0.414022	+	0.910267i	$0.364124\pi$
$234$	0	0
$235$	0	0
$236$	6.81322	0.443503
$237$	0	0
$238$	5.54064	0.359146
$239$	−20.1605	−1.30407	−0.652036	−	0.758188i	$-0.726085\pi$
$239$	−20.1605	−1.30407	−0.652036	+	0.758188i	$0.726085\pi$
$240$	0	0
$241$	−11.7693	−0.758126	−0.379063	−	0.925371i	$-0.623754\pi$
$241$	−11.7693	−0.758126	−0.379063	+	0.925371i	$0.623754\pi$
$242$	12.9561	0.832847
$243$	0	0
$244$	3.97577	0.254522
$245$	0	0
$246$	0	0
$247$	1.89449	0.120544
$248$	7.70771	0.489440
$249$	0	0
$250$	0	0
$251$	−18.3715	−1.15960	−0.579799	−	0.814760i	$-0.696869\pi$
$251$	−18.3715	−1.15960	−0.579799	+	0.814760i	$0.696869\pi$
$252$	0	0
$253$	−30.1847	−1.89770
$254$	−6.62644	−0.415780
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.83293	0.239091	0.119546	−	0.992829i	$-0.461856\pi$
$257$	3.83293	0.239091	0.119546	+	0.992829i	$0.461856\pi$
$258$	0	0
$259$	−18.3341	−1.13923
$260$	0	0
$261$	0	0
$262$	16.9803	1.04905
$263$	−21.2483	−1.31023	−0.655115	−	0.755529i	$-0.727380\pi$
$263$	−21.2483	−1.31023	−0.655115	+	0.755529i	$0.727380\pi$
$264$	0	0
$265$	0	0
$266$	2.35386	0.144324
$267$	0	0
$268$	5.19131	0.317110
$269$	5.78899	0.352961	0.176480	−	0.984304i	$-0.443529\pi$
$269$	5.78899	0.352961	0.176480	+	0.984304i	$0.443529\pi$
$270$	0	0
$271$	4.01971	0.244180	0.122090	−	0.992519i	$-0.461040\pi$
$271$	4.01971	0.244180	0.122090	+	0.992519i	$0.461040\pi$
$272$	−2.35386	−0.142723
$273$	0	0
$274$	12.8703	0.777521
$275$	0	0
$276$	0	0
$277$	12.0989	0.726953	0.363476	−	0.931603i	$-0.381590\pi$
$277$	12.0989	0.726953	0.363476	+	0.931603i	$0.381590\pi$
$278$	9.30992	0.558372
$279$	0	0
$280$	0	0
$281$	9.21101	0.549483	0.274742	−	0.961518i	$-0.411408\pi$
$281$	9.21101	0.549483	0.274742	+	0.961518i	$0.411408\pi$
$282$	0	0
$283$	−8.49670	−0.505076	−0.252538	−	0.967587i	$-0.581265\pi$
$283$	−8.49670	−0.505076	−0.252538	+	0.967587i	$0.581265\pi$
$284$	−4.14284	−0.245832
$285$	0	0
$286$	−9.27258	−0.548299
$287$	−8.33415	−0.491949
$288$	0	0
$289$	−11.4594	−0.674080
$290$	0	0
$291$	0	0
$292$	4.37809	0.256208
$293$	12.0989	0.706825	0.353413	−	0.935468i	$-0.385021\pi$
$293$	12.0989	0.706825	0.353413	+	0.935468i	$0.385021\pi$
$294$	0	0
$295$	0	0
$296$	7.78899	0.452726
$297$	0	0
$298$	−16.1231	−0.933988
$299$	11.6835	0.675673
$300$	0	0
$301$	4.07675	0.234980
$302$	−15.6638	−0.901349
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−20.9318	−1.19464	−0.597321	−	0.802002i	$-0.703768\pi$
$307$	−20.9318	−1.19464	−0.597321	+	0.802002i	$0.703768\pi$
$308$	−11.5209	−0.656466
$309$	0	0
$310$	0	0
$311$	14.9561	0.848080	0.424040	−	0.905643i	$-0.360612\pi$
$311$	14.9561	0.848080	0.424040	+	0.905643i	$0.360612\pi$
$312$	0	0
$313$	−32.2902	−1.82515	−0.912575	−	0.408909i	$-0.865909\pi$
$313$	−32.2902	−1.82515	−0.912575	+	0.408909i	$0.865909\pi$
$314$	−14.2483	−0.804081
$315$	0	0
$316$	9.49670	0.534231
$317$	24.2726	1.36328	0.681642	−	0.731686i	$-0.261266\pi$
$317$	24.2726	1.36328	0.681642	+	0.731686i	$0.261266\pi$
$318$	0	0
$319$	20.9121	1.17085
$320$	0	0
$321$	0	0
$322$	14.5164	0.808968
$323$	2.35386	0.130972
$324$	0	0
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	3.54064	0.195499
$329$	12.5452	0.691637
$330$	0	0
$331$	6.80869	0.374240	0.187120	−	0.982337i	$-0.440085\pi$
$331$	6.80869	0.374240	0.187120	+	0.982337i	$0.440085\pi$
$332$	−4.43513	−0.243409
$333$	0	0
$334$	0.756178	0.0413762
$335$	0	0
$336$	0	0
$337$	16.1231	0.878283	0.439142	−	0.898418i	$-0.355283\pi$
$337$	16.1231	0.878283	0.439142	+	0.898418i	$0.355283\pi$
$338$	−9.41090	−0.511885
$339$	0	0
$340$	0	0
$341$	37.7253	2.04294
$342$	0	0
$343$	19.9121	1.07515
$344$	−1.73194	−0.0933802
$345$	0	0
$346$	13.6880	0.735872
$347$	23.3670	1.25440	0.627202	−	0.778857i	$-0.284200\pi$
$347$	23.3670	1.25440	0.627202	+	0.778857i	$0.284200\pi$
$348$	0	0
$349$	16.5977	0.888453	0.444227	−	0.895914i	$-0.353478\pi$
$349$	16.5977	0.888453	0.444227	+	0.895914i	$0.353478\pi$
$350$	0	0
$351$	0	0
$352$	4.89449	0.260877
$353$	−5.32105	−0.283211	−0.141605	−	0.989923i	$-0.545226\pi$
$353$	−5.32105	−0.283211	−0.141605	+	0.989923i	$0.545226\pi$
$354$	0	0
$355$	0	0
$356$	3.62191	0.191961
$357$	0	0
$358$	23.2044	1.22639
$359$	6.67038	0.352049	0.176025	−	0.984386i	$-0.443676\pi$
$359$	6.67038	0.352049	0.176025	+	0.984386i	$0.443676\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.65067	0.349551
$363$	0	0
$364$	4.45936	0.233734
$365$	0	0
$366$	0	0
$367$	8.14284	0.425053	0.212526	−	0.977155i	$-0.431831\pi$
$367$	8.14284	0.425053	0.212526	+	0.977155i	$0.431831\pi$
$368$	−6.16707	−0.321481
$369$	0	0
$370$	0	0
$371$	4.65067	0.241451
$372$	0	0
$373$	3.94296	0.204159	0.102079	−	0.994776i	$-0.467450\pi$
$373$	3.94296	0.204159	0.102079	+	0.994776i	$0.467450\pi$
$374$	−11.5209	−0.595733
$375$	0	0
$376$	−5.32962	−0.274854
$377$	−8.09438	−0.416882
$378$	0	0
$379$	−33.2044	−1.70560	−0.852798	−	0.522241i	$-0.825096\pi$
$379$	−33.2044	−1.70560	−0.852798	+	0.522241i	$0.825096\pi$
$380$	0	0
$381$	0	0
$382$	14.7890	0.756670
$383$	−19.3867	−0.990612	−0.495306	−	0.868719i	$-0.664944\pi$
$383$	−19.3867	−0.990612	−0.495306	+	0.868719i	$0.664944\pi$
$384$	0	0
$385$	0	0
$386$	11.5977	0.590306
$387$	0	0
$388$	7.06157	0.358497
$389$	−6.87026	−0.348336	−0.174168	−	0.984716i	$-0.555724\pi$
$389$	−6.87026	−0.348336	−0.174168	+	0.984716i	$0.555724\pi$
$390$	0	0
$391$	14.5164	0.734126
$392$	−1.45936	−0.0737090
$393$	0	0
$394$	12.6022	0.634890
$395$	0	0
$396$	0	0
$397$	21.8879	1.09852	0.549261	−	0.835651i	$-0.314909\pi$
$397$	21.8879	1.09852	0.549261	+	0.835651i	$0.314909\pi$
$398$	−23.7405	−1.19000
$399$	0	0
$400$	0	0
$401$	7.49670	0.374367	0.187184	−	0.982325i	$-0.440064\pi$
$401$	7.49670	0.374367	0.187184	+	0.982325i	$0.440064\pi$
$402$	0	0
$403$	−14.6022	−0.727388
$404$	6.10551	0.303760
$405$	0	0
$406$	−10.0570	−0.499123
$407$	38.1231	1.88969
$408$	0	0
$409$	4.21101	0.208221	0.104111	−	0.994566i	$-0.466800\pi$
$409$	4.21101	0.208221	0.104111	+	0.994566i	$0.466800\pi$
$410$	0	0
$411$	0	0
$412$	11.2483	0.554166
$413$	−16.0373	−0.789146
$414$	0	0
$415$	0	0
$416$	−1.89449	−0.0928852
$417$	0	0
$418$	−4.89449	−0.239397
$419$	10.8703	0.531047	0.265523	−	0.964104i	$-0.414455\pi$
$419$	10.8703	0.531047	0.265523	+	0.964104i	$0.414455\pi$
$420$	0	0
$421$	33.7496	1.64485	0.822427	−	0.568871i	$-0.192620\pi$
$421$	33.7496	1.64485	0.822427	+	0.568871i	$0.192620\pi$
$422$	19.9848	0.972846
$423$	0	0
$424$	−1.97577	−0.0959517
$425$	0	0
$426$	0	0
$427$	−9.35838	−0.452884
$428$	15.3099	0.740033
$429$	0	0
$430$	0	0
$431$	−15.7011	−0.756296	−0.378148	−	0.925745i	$-0.623439\pi$
$431$	−15.7011	−0.756296	−0.378148	+	0.925745i	$0.623439\pi$
$432$	0	0
$433$	15.1847	0.729730	0.364865	−	0.931060i	$-0.381115\pi$
$433$	15.1847	0.729730	0.364865	+	0.931060i	$0.381115\pi$
$434$	−18.1428	−0.870884
$435$	0	0
$436$	−14.7077	−0.704372
$437$	6.16707	0.295011
$438$	0	0
$439$	−25.4482	−1.21458	−0.607289	−	0.794481i	$-0.707743\pi$
$439$	−25.4482	−1.21458	−0.607289	+	0.794481i	$0.707743\pi$
$440$	0	0
$441$	0	0
$442$	4.45936	0.212110
$443$	−24.4351	−1.16095	−0.580474	−	0.814279i	$-0.697133\pi$
$443$	−24.4351	−1.16095	−0.580474	+	0.814279i	$0.697133\pi$
$444$	0	0
$445$	0	0
$446$	18.7122	0.886050
$447$	0	0
$448$	−2.35386	−0.111209
$449$	26.5012	1.25067	0.625335	−	0.780356i	$-0.284962\pi$
$449$	26.5012	1.25067	0.625335	+	0.780356i	$0.284962\pi$
$450$	0	0
$451$	17.3296	0.816020
$452$	−3.29681	−0.155069
$453$	0	0
$454$	13.0813	0.613935
$455$	0	0
$456$	0	0
$457$	−36.3230	−1.69912	−0.849560	−	0.527493i	$-0.823132\pi$
$457$	−36.3230	−1.69912	−0.849560	+	0.527493i	$0.823132\pi$
$458$	−9.89902	−0.462551
$459$	0	0
$460$	0	0
$461$	23.0813	1.07500	0.537501	−	0.843263i	$-0.319368\pi$
$461$	23.0813	1.07500	0.537501	+	0.843263i	$0.319368\pi$
$462$	0	0
$463$	31.0419	1.44264	0.721319	−	0.692603i	$-0.243536\pi$
$463$	31.0419	1.44264	0.721319	+	0.692603i	$0.243536\pi$
$464$	4.27258	0.198350
$465$	0	0
$466$	12.6395	0.585515
$467$	−16.6925	−0.772438	−0.386219	−	0.922407i	$-0.626219\pi$
$467$	−16.6925	−0.772438	−0.386219	+	0.922407i	$0.626219\pi$
$468$	0	0
$469$	−12.2196	−0.564248
$470$	0	0
$471$	0	0
$472$	6.81322	0.313604
$473$	−8.47699	−0.389772
$474$	0	0
$475$	0	0
$476$	5.54064	0.253955
$477$	0	0
$478$	−20.1605	−0.922118
$479$	−12.5891	−0.575211	−0.287605	−	0.957749i	$-0.592859\pi$
$479$	−12.5891	−0.575211	−0.287605	+	0.957749i	$0.592859\pi$
$480$	0	0
$481$	−14.7562	−0.672824
$482$	−11.7693	−0.536076
$483$	0	0
$484$	12.9561	0.588912
$485$	0	0
$486$	0	0
$487$	20.7077	0.938356	0.469178	−	0.883104i	$-0.344550\pi$
$487$	20.7077	0.938356	0.469178	+	0.883104i	$0.344550\pi$
$488$	3.97577	0.179975
$489$	0	0
$490$	0	0
$491$	20.0747	0.905957	0.452979	−	0.891521i	$-0.350361\pi$
$491$	20.0747	0.905957	0.452979	+	0.891521i	$0.350361\pi$
$492$	0	0
$493$	−10.0570	−0.452946
$494$	1.89449	0.0852373
$495$	0	0
$496$	7.70771	0.346086
$497$	9.75165	0.437421
$498$	0	0
$499$	41.5956	1.86207	0.931037	−	0.364924i	$-0.118905\pi$
$499$	41.5956	1.86207	0.931037	+	0.364924i	$0.118905\pi$
$500$	0	0
$501$	0	0
$502$	−18.3715	−0.819959
$503$	−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
$506$	−30.1847	−1.34187
$507$	0	0
$508$	−6.62644	−0.294001
$509$	25.8879	1.14746	0.573730	−	0.819044i	$-0.305496\pi$
$509$	25.8879	1.14746	0.573730	+	0.819044i	$0.305496\pi$
$510$	0	0
$511$	−10.3054	−0.455884
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.83293	0.169063
$515$	0	0
$516$	0	0
$517$	−26.0858	−1.14725
$518$	−18.3341	−0.805556
$519$	0	0
$520$	0	0
$521$	−24.5386	−1.07505	−0.537527	−	0.843247i	$-0.680641\pi$
$521$	−24.5386	−1.07505	−0.537527	+	0.843247i	$0.680641\pi$
$522$	0	0
$523$	−4.21101	−0.184135	−0.0920674	−	0.995753i	$-0.529348\pi$
$523$	−4.21101	−0.184135	−0.0920674	+	0.995753i	$0.529348\pi$
$524$	16.9803	0.741788
$525$	0	0
$526$	−21.2483	−0.926472
$527$	−18.1428	−0.790315
$528$	0	0
$529$	15.0328	0.653600
$530$	0	0
$531$	0	0
$532$	2.35386	0.102053
$533$	−6.70771	−0.290543
$534$	0	0
$535$	0	0
$536$	5.19131	0.224230
$537$	0	0
$538$	5.78899	0.249581
$539$	−7.14284	−0.307664
$540$	0	0
$541$	−12.7208	−0.546910	−0.273455	−	0.961885i	$-0.588167\pi$
$541$	−12.7208	−0.546910	−0.273455	+	0.961885i	$0.588167\pi$
$542$	4.01971	0.172661
$543$	0	0
$544$	−2.35386	−0.100921
$545$	0	0
$546$	0	0
$547$	−5.97577	−0.255505	−0.127753	−	0.991806i	$-0.540776\pi$
$547$	−5.97577	−0.255505	−0.127753	+	0.991806i	$0.540776\pi$
$548$	12.8703	0.549790
$549$	0	0
$550$	0	0
$551$	−4.27258	−0.182018
$552$	0	0
$553$	−22.3539	−0.950583
$554$	12.0989	0.514033
$555$	0	0
$556$	9.30992	0.394828
$557$	29.6441	1.25606	0.628030	−	0.778189i	$-0.283862\pi$
$557$	29.6441	1.25606	0.628030	+	0.778189i	$0.283862\pi$
$558$	0	0
$559$	3.28116	0.138778
$560$	0	0
$561$	0	0
$562$	9.21101	0.388543
$563$	24.3427	1.02592	0.512962	−	0.858411i	$-0.328548\pi$
$563$	24.3427	1.02592	0.512962	+	0.858411i	$0.328548\pi$
$564$	0	0
$565$	0	0
$566$	−8.49670	−0.357143
$567$	0	0
$568$	−4.14284	−0.173830
$569$	−23.6264	−0.990472	−0.495236	−	0.868759i	$-0.664918\pi$
$569$	−23.6264	−0.990472	−0.495236	+	0.868759i	$0.664918\pi$
$570$	0	0
$571$	8.81322	0.368822	0.184411	−	0.982849i	$-0.440962\pi$
$571$	8.81322	0.368822	0.184411	+	0.982849i	$0.440962\pi$
$572$	−9.27258	−0.387706
$573$	0	0
$574$	−8.33415	−0.347861
$575$	0	0
$576$	0	0
$577$	−36.9121	−1.53667	−0.768336	−	0.640047i	$-0.778915\pi$
$577$	−36.9121	−1.53667	−0.768336	+	0.640047i	$0.778915\pi$
$578$	−11.4594	−0.476647
$579$	0	0
$580$	0	0
$581$	10.4397	0.433110
$582$	0	0
$583$	−9.67038	−0.400506
$584$	4.37809	0.181167
$585$	0	0
$586$	12.0989	0.499801
$587$	−15.9737	−0.659305	−0.329652	−	0.944102i	$-0.606932\pi$
$587$	−15.9737	−0.659305	−0.329652	+	0.944102i	$0.606932\pi$
$588$	0	0
$589$	−7.70771	−0.317591
$590$	0	0
$591$	0	0
$592$	7.78899	0.320125
$593$	4.22007	0.173297	0.0866487	−	0.996239i	$-0.472384\pi$
$593$	4.22007	0.173297	0.0866487	+	0.996239i	$0.472384\pi$
$594$	0	0
$595$	0	0
$596$	−16.1231	−0.660429
$597$	0	0
$598$	11.6835	0.477773
$599$	−21.6552	−0.884807	−0.442404	−	0.896816i	$-0.645874\pi$
$599$	−21.6552	−0.884807	−0.442404	+	0.896816i	$0.645874\pi$
$600$	0	0
$601$	−30.0944	−1.22758	−0.613788	−	0.789471i	$-0.710355\pi$
$601$	−30.0944	−1.22758	−0.613788	+	0.789471i	$0.710355\pi$
$602$	4.07675	0.166156
$603$	0	0
$604$	−15.6638	−0.637350
$605$	0	0
$606$	0	0
$607$	−14.9979	−0.608747	−0.304373	−	0.952553i	$-0.598447\pi$
$607$	−14.9979	−0.608747	−0.304373	+	0.952553i	$0.598447\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	10.0969	0.408478
$612$	0	0
$613$	−16.7562	−0.676776	−0.338388	−	0.941007i	$-0.609882\pi$
$613$	−16.7562	−0.676776	−0.338388	+	0.941007i	$0.609882\pi$
$614$	−20.9318	−0.844740
$615$	0	0
$616$	−11.5209	−0.464192
$617$	18.1141	0.729245	0.364623	−	0.931155i	$-0.381198\pi$
$617$	18.1141	0.729245	0.364623	+	0.931155i	$0.381198\pi$
$618$	0	0
$619$	−36.0176	−1.44767	−0.723835	−	0.689973i	$-0.757622\pi$
$619$	−36.0176	−1.44767	−0.723835	+	0.689973i	$0.757622\pi$
$620$	0	0
$621$	0	0
$622$	14.9561	0.599683
$623$	−8.52546	−0.341565
$624$	0	0
$625$	0	0
$626$	−32.2902	−1.29058
$627$	0	0
$628$	−14.2483	−0.568571
$629$	−18.3341	−0.731030
$630$	0	0
$631$	17.6749	0.703627	0.351813	−	0.936070i	$-0.385565\pi$
$631$	17.6749	0.703627	0.351813	+	0.936070i	$0.385565\pi$
$632$	9.49670	0.377758
$633$	0	0
$634$	24.2726	0.963987
$635$	0	0
$636$	0	0
$637$	2.76475	0.109543
$638$	20.9121	0.827919
$639$	0	0
$640$	0	0
$641$	−2.08580	−0.0823842	−0.0411921	−	0.999151i	$-0.513116\pi$
$641$	−2.08580	−0.0823842	−0.0411921	+	0.999151i	$0.513116\pi$
$642$	0	0
$643$	37.5295	1.48002	0.740010	−	0.672596i	$-0.234821\pi$
$643$	37.5295	1.48002	0.740010	+	0.672596i	$0.234821\pi$
$644$	14.5164	0.572026
$645$	0	0
$646$	2.35386	0.0926112
$647$	−9.12521	−0.358749	−0.179375	−	0.983781i	$-0.557407\pi$
$647$	−9.12521	−0.358749	−0.179375	+	0.983781i	$0.557407\pi$
$648$	0	0
$649$	33.3473	1.30899
$650$	0	0
$651$	0	0
$652$	6.00000	0.234978
$653$	−13.2529	−0.518625	−0.259313	−	0.965793i	$-0.583496\pi$
$653$	−13.2529	−0.518625	−0.259313	+	0.965793i	$0.583496\pi$
$654$	0	0
$655$	0	0
$656$	3.54064	0.138239
$657$	0	0
$658$	12.5452	0.489061
$659$	24.9364	0.971382	0.485691	−	0.874130i	$-0.338568\pi$
$659$	24.9364	0.971382	0.485691	+	0.874130i	$0.338568\pi$
$660$	0	0
$661$	−29.9606	−1.16533	−0.582666	−	0.812712i	$-0.697990\pi$
$661$	−29.9606	−1.16533	−0.582666	+	0.812712i	$0.697990\pi$
$662$	6.80869	0.264627
$663$	0	0
$664$	−4.43513	−0.172116
$665$	0	0
$666$	0	0
$667$	−26.3493	−1.02025
$668$	0.756178	0.0292574
$669$	0	0
$670$	0	0
$671$	19.4594	0.751220
$672$	0	0
$673$	2.85055	0.109881	0.0549404	−	0.998490i	$-0.482503\pi$
$673$	2.85055	0.109881	0.0549404	+	0.998490i	$0.482503\pi$
$674$	16.1231	0.621040
$675$	0	0
$676$	−9.41090	−0.361958
$677$	42.5562	1.63557	0.817784	−	0.575526i	$-0.195203\pi$
$677$	42.5562	1.63557	0.817784	+	0.575526i	$0.195203\pi$
$678$	0	0
$679$	−16.6219	−0.637890
$680$	0	0
$681$	0	0
$682$	37.7253	1.44458
$683$	12.2286	0.467916	0.233958	−	0.972247i	$-0.424832\pi$
$683$	12.2286	0.467916	0.233958	+	0.972247i	$0.424832\pi$
$684$	0	0
$685$	0	0
$686$	19.9121	0.760248
$687$	0	0
$688$	−1.73194	−0.0660298
$689$	3.74308	0.142600
$690$	0	0
$691$	21.7581	0.827719	0.413859	−	0.910341i	$-0.364180\pi$
$691$	21.7581	0.827719	0.413859	+	0.910341i	$0.364180\pi$
$692$	13.6880	0.520340
$693$	0	0
$694$	23.3670	0.886998
$695$	0	0
$696$	0	0
$697$	−8.33415	−0.315678
$698$	16.5977	0.628231
$699$	0	0
$700$	0	0
$701$	28.6592	1.08244	0.541222	−	0.840880i	$-0.317962\pi$
$701$	28.6592	1.08244	0.541222	+	0.840880i	$0.317962\pi$
$702$	0	0
$703$	−7.78899	−0.293767
$704$	4.89449	0.184468
$705$	0	0
$706$	−5.32105	−0.200260
$707$	−14.3715	−0.540495
$708$	0	0
$709$	6.69461	0.251421	0.125711	−	0.992067i	$-0.459879\pi$
$709$	6.69461	0.251421	0.125711	+	0.992067i	$0.459879\pi$
$710$	0	0
$711$	0	0
$712$	3.62191	0.135737
$713$	−47.5340	−1.78016
$714$	0	0
$715$	0	0
$716$	23.2044	0.867189
$717$	0	0
$718$	6.67038	0.248936
$719$	24.7980	0.924811	0.462405	−	0.886669i	$-0.346986\pi$
$719$	24.7980	0.924811	0.462405	+	0.886669i	$0.346986\pi$
$720$	0	0
$721$	−26.4770	−0.986055
$722$	1.00000	0.0372161
$723$	0	0
$724$	6.65067	0.247170
$725$	0	0
$726$	0	0
$727$	−33.0328	−1.22512	−0.612560	−	0.790424i	$-0.709860\pi$
$727$	−33.0328	−1.22512	−0.612560	+	0.790424i	$0.709860\pi$
$728$	4.45936	0.165275
$729$	0	0
$730$	0	0
$731$	4.07675	0.150784
$732$	0	0
$733$	−42.5098	−1.57014	−0.785068	−	0.619410i	$-0.787372\pi$
$733$	−42.5098	−1.57014	−0.785068	+	0.619410i	$0.787372\pi$
$734$	8.14284	0.300558
$735$	0	0
$736$	−6.16707	−0.227321
$737$	25.4088	0.935946
$738$	0	0
$739$	1.20441	0.0443049	0.0221524	−	0.999755i	$-0.492948\pi$
$739$	1.20441	0.0443049	0.0221524	+	0.999755i	$0.492948\pi$
$740$	0	0
$741$	0	0
$742$	4.65067	0.170731
$743$	−21.5870	−0.791951	−0.395976	−	0.918261i	$-0.629594\pi$
$743$	−21.5870	−0.791951	−0.395976	+	0.918261i	$0.629594\pi$
$744$	0	0
$745$	0	0
$746$	3.94296	0.144362
$747$	0	0
$748$	−11.5209	−0.421247
$749$	−36.0373	−1.31678
$750$	0	0
$751$	−42.2089	−1.54023	−0.770113	−	0.637908i	$-0.779800\pi$
$751$	−42.2089	−1.54023	−0.770113	+	0.637908i	$0.779800\pi$
$752$	−5.32962	−0.194351
$753$	0	0
$754$	−8.09438	−0.294780
$755$	0	0
$756$	0	0
$757$	−32.3867	−1.17711	−0.588557	−	0.808456i	$-0.700304\pi$
$757$	−32.3867	−1.17711	−0.588557	+	0.808456i	$0.700304\pi$
$758$	−33.2044	−1.20604
$759$	0	0
$760$	0	0
$761$	−43.1362	−1.56369	−0.781844	−	0.623475i	$-0.785721\pi$
$761$	−43.1362	−1.56369	−0.781844	+	0.623475i	$0.785721\pi$
$762$	0	0
$763$	34.6198	1.25332
$764$	14.7890	0.535047
$765$	0	0
$766$	−19.3867	−0.700469
$767$	−12.9076	−0.466066
$768$	0	0
$769$	−33.7541	−1.21720	−0.608602	−	0.793476i	$-0.708269\pi$
$769$	−33.7541	−1.21720	−0.608602	+	0.793476i	$0.708269\pi$
$770$	0	0
$771$	0	0
$772$	11.5977	0.417410
$773$	−44.9055	−1.61514	−0.807570	−	0.589772i	$-0.799217\pi$
$773$	−44.9055	−1.61514	−0.807570	+	0.589772i	$0.799217\pi$
$774$	0	0
$775$	0	0
$776$	7.06157	0.253495
$777$	0	0
$778$	−6.87026	−0.246311
$779$	−3.54064	−0.126856
$780$	0	0
$781$	−20.2771	−0.725572
$782$	14.5164	0.519106
$783$	0	0
$784$	−1.45936	−0.0521201
$785$	0	0
$786$	0	0
$787$	23.0550	0.821821	0.410910	−	0.911676i	$-0.365211\pi$
$787$	23.0550	0.821821	0.410910	+	0.911676i	$0.365211\pi$
$788$	12.6022	0.448935
$789$	0	0
$790$	0	0
$791$	7.76023	0.275922
$792$	0	0
$793$	−7.53206	−0.267471
$794$	21.8879	0.776772
$795$	0	0
$796$	−23.7405	−0.841460
$797$	−35.7384	−1.26592	−0.632960	−	0.774184i	$-0.718160\pi$
$797$	−35.7384	−1.26592	−0.632960	+	0.774184i	$0.718160\pi$
$798$	0	0
$799$	12.5452	0.443816
$800$	0	0
$801$	0	0
$802$	7.49670	0.264718
$803$	21.4285	0.756196
$804$	0	0
$805$	0	0
$806$	−14.6022	−0.514341
$807$	0	0
$808$	6.10551	0.214791
$809$	21.0525	0.740167	0.370084	−	0.928998i	$-0.379329\pi$
$809$	21.0525	0.740167	0.370084	+	0.928998i	$0.379329\pi$
$810$	0	0
$811$	17.3912	0.610687	0.305344	−	0.952242i	$-0.401229\pi$
$811$	17.3912	0.610687	0.305344	+	0.952242i	$0.401229\pi$
$812$	−10.0570	−0.352933
$813$	0	0
$814$	38.1231	1.33622
$815$	0	0
$816$	0	0
$817$	1.73194	0.0605931
$818$	4.21101	0.147235
$819$	0	0
$820$	0	0
$821$	6.43966	0.224746	0.112373	−	0.993666i	$-0.464155\pi$
$821$	6.43966	0.224746	0.112373	+	0.993666i	$0.464155\pi$
$822$	0	0
$823$	−22.8308	−0.795833	−0.397917	−	0.917422i	$-0.630267\pi$
$823$	−22.8308	−0.795833	−0.397917	+	0.917422i	$0.630267\pi$
$824$	11.2483	0.391855
$825$	0	0
$826$	−16.0373	−0.558010
$827$	29.2705	1.01784	0.508918	−	0.860815i	$-0.330046\pi$
$827$	29.2705	1.01784	0.508918	+	0.860815i	$0.330046\pi$
$828$	0	0
$829$	−22.5053	−0.781640	−0.390820	−	0.920467i	$-0.627809\pi$
$829$	−22.5053	−0.781640	−0.390820	+	0.920467i	$0.627809\pi$
$830$	0	0
$831$	0	0
$832$	−1.89449	−0.0656797
$833$	3.43513	0.119020
$834$	0	0
$835$	0	0
$836$	−4.89449	−0.169280
$837$	0	0
$838$	10.8703	0.375507
$839$	−25.2241	−0.870833	−0.435417	−	0.900229i	$-0.643399\pi$
$839$	−25.2241	−0.870833	−0.435417	+	0.900229i	$0.643399\pi$
$840$	0	0
$841$	−10.7450	−0.370519
$842$	33.7496	1.16309
$843$	0	0
$844$	19.9848	0.687906
$845$	0	0
$846$	0	0
$847$	−30.4967	−1.04788
$848$	−1.97577	−0.0678481
$849$	0	0
$850$	0	0
$851$	−48.0353	−1.64663
$852$	0	0
$853$	−19.4548	−0.666121	−0.333060	−	0.942905i	$-0.608081\pi$
$853$	−19.4548	−0.666121	−0.333060	+	0.942905i	$0.608081\pi$
$854$	−9.35838	−0.320237
$855$	0	0
$856$	15.3099	0.523282
$857$	−50.2351	−1.71600	−0.858000	−	0.513650i	$-0.828293\pi$
$857$	−50.2351	−1.71600	−0.858000	+	0.513650i	$0.828293\pi$
$858$	0	0
$859$	−5.82840	−0.198862	−0.0994312	−	0.995044i	$-0.531702\pi$
$859$	−5.82840	−0.198862	−0.0994312	+	0.995044i	$0.531702\pi$
$860$	0	0
$861$	0	0
$862$	−15.7011	−0.534782
$863$	−8.37356	−0.285039	−0.142520	−	0.989792i	$-0.545520\pi$
$863$	−8.37356	−0.285039	−0.142520	+	0.989792i	$0.545520\pi$
$864$	0	0
$865$	0	0
$866$	15.1847	0.515997
$867$	0	0
$868$	−18.1428	−0.615808
$869$	46.4815	1.57678
$870$	0	0
$871$	−9.83490	−0.333243
$872$	−14.7077	−0.498066
$873$	0	0
$874$	6.16707	0.208604
$875$	0	0
$876$	0	0
$877$	−23.9031	−0.807149	−0.403575	−	0.914947i	$-0.632232\pi$
$877$	−23.9031	−0.807149	−0.403575	+	0.914947i	$0.632232\pi$
$878$	−25.4482	−0.858836
$879$	0	0
$880$	0	0
$881$	25.0419	0.843682	0.421841	−	0.906670i	$-0.361384\pi$
$881$	25.0419	0.843682	0.421841	+	0.906670i	$0.361384\pi$
$882$	0	0
$883$	−9.51188	−0.320100	−0.160050	−	0.987109i	$-0.551166\pi$
$883$	−9.51188	−0.320100	−0.160050	+	0.987109i	$0.551166\pi$
$884$	4.45936	0.149985
$885$	0	0
$886$	−24.4351	−0.820914
$887$	−40.7824	−1.36934	−0.684669	−	0.728854i	$-0.740053\pi$
$887$	−40.7824	−1.36934	−0.684669	+	0.728854i	$0.740053\pi$
$888$	0	0
$889$	15.5977	0.523129
$890$	0	0
$891$	0	0
$892$	18.7122	0.626532
$893$	5.32962	0.178349
$894$	0	0
$895$	0	0
$896$	−2.35386	−0.0786368
$897$	0	0
$898$	26.5012	0.884357
$899$	32.9318	1.09834
$900$	0	0
$901$	4.65067	0.154936
$902$	17.3296	0.577013
$903$	0	0
$904$	−3.29681	−0.109650
$905$	0	0
$906$	0	0
$907$	−9.05299	−0.300600	−0.150300	−	0.988640i	$-0.548024\pi$
$907$	−9.05299	−0.300600	−0.150300	+	0.988640i	$0.548024\pi$
$908$	13.0813	0.434117
$909$	0	0
$910$	0	0
$911$	28.2947	0.937446	0.468723	−	0.883345i	$-0.344714\pi$
$911$	28.2947	0.937446	0.468723	+	0.883345i	$0.344714\pi$
$912$	0	0
$913$	−21.7077	−0.718420
$914$	−36.3230	−1.20146
$915$	0	0
$916$	−9.89902	−0.327073
$917$	−39.9692	−1.31990
$918$	0	0
$919$	42.3801	1.39799	0.698995	−	0.715127i	$-0.253631\pi$
$919$	42.3801	1.39799	0.698995	+	0.715127i	$0.253631\pi$
$920$	0	0
$921$	0	0
$922$	23.0813	0.760141
$923$	7.84858	0.258339
$924$	0	0
$925$	0	0
$926$	31.0419	1.02010
$927$	0	0
$928$	4.27258	0.140254
$929$	−38.7430	−1.27112	−0.635558	−	0.772053i	$-0.719230\pi$
$929$	−38.7430	−1.27112	−0.635558	+	0.772053i	$0.719230\pi$
$930$	0	0
$931$	1.45936	0.0478287
$932$	12.6395	0.414022
$933$	0	0
$934$	−16.6925	−0.546196
$935$	0	0
$936$	0	0
$937$	3.20441	0.104683	0.0523417	−	0.998629i	$-0.483331\pi$
$937$	3.20441	0.104683	0.0523417	+	0.998629i	$0.483331\pi$
$938$	−12.2196	−0.398984
$939$	0	0
$940$	0	0
$941$	−3.14492	−0.102521	−0.0512607	−	0.998685i	$-0.516324\pi$
$941$	−3.14492	−0.102521	−0.0512607	+	0.998685i	$0.516324\pi$
$942$	0	0
$943$	−21.8354	−0.711058
$944$	6.81322	0.221751
$945$	0	0
$946$	−8.47699	−0.275611
$947$	−29.0509	−0.944028	−0.472014	−	0.881591i	$-0.656473\pi$
$947$	−29.0509	−0.944028	−0.472014	+	0.881591i	$0.656473\pi$
$948$	0	0
$949$	−8.29426	−0.269243
$950$	0	0
$951$	0	0
$952$	5.54064	0.179573
$953$	−11.5452	−0.373985	−0.186992	−	0.982361i	$-0.559874\pi$
$953$	−11.5452	−0.373985	−0.186992	+	0.982361i	$0.559874\pi$
$954$	0	0
$955$	0	0
$956$	−20.1605	−0.652036
$957$	0	0
$958$	−12.5891	−0.406735
$959$	−30.2947	−0.978268
$960$	0	0
$961$	28.4088	0.916413
$962$	−14.7562	−0.475758
$963$	0	0
$964$	−11.7693	−0.379063
$965$	0	0
$966$	0	0
$967$	−17.2044	−0.553256	−0.276628	−	0.960977i	$-0.589217\pi$
$967$	−17.2044	−0.553256	−0.276628	+	0.960977i	$0.589217\pi$
$968$	12.9561	0.416424
$969$	0	0
$970$	0	0
$971$	−38.2771	−1.22837	−0.614185	−	0.789162i	$-0.710515\pi$
$971$	−38.2771	−1.22837	−0.614185	+	0.789162i	$0.710515\pi$
$972$	0	0
$973$	−21.9142	−0.702537
$974$	20.7077	0.663518
$975$	0	0
$976$	3.97577	0.127261
$977$	16.5340	0.528971	0.264485	−	0.964390i	$-0.414798\pi$
$977$	16.5340	0.528971	0.264485	+	0.964390i	$0.414798\pi$
$978$	0	0
$979$	17.7274	0.566571
$980$	0	0
$981$	0	0
$982$	20.0747	0.640608
$983$	41.7783	1.33252	0.666261	−	0.745719i	$-0.267894\pi$
$983$	41.7783	1.33252	0.666261	+	0.745719i	$0.267894\pi$
$984$	0	0
$985$	0	0
$986$	−10.0570	−0.320281
$987$	0	0
$988$	1.89449	0.0602718
$989$	10.6810	0.339637
$990$	0	0
$991$	27.7845	0.882602	0.441301	−	0.897359i	$-0.354517\pi$
$991$	27.7845	0.882602	0.441301	+	0.897359i	$0.354517\pi$
$992$	7.70771	0.244720
$993$	0	0
$994$	9.75165	0.309304
$995$	0	0
$996$	0	0
$997$	16.9297	0.536170	0.268085	−	0.963395i	$-0.413609\pi$
$997$	16.9297	0.536170	0.268085	+	0.963395i	$0.413609\pi$
$998$	41.5956	1.31669
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +4.89449 q^{11} -1.89449 q^{13} -2.35386 q^{14} +1.00000 q^{16} -2.35386 q^{17} -1.00000 q^{19} +4.89449 q^{22} -6.16707 q^{23} -1.89449 q^{26} -2.35386 q^{28} +4.27258 q^{29} +7.70771 q^{31} +1.00000 q^{32} -2.35386 q^{34} +7.78899 q^{37} -1.00000 q^{38} +3.54064 q^{41} -1.73194 q^{43} +4.89449 q^{44} -6.16707 q^{46} -5.32962 q^{47} -1.45936 q^{49} -1.89449 q^{52} -1.97577 q^{53} -2.35386 q^{56} +4.27258 q^{58} +6.81322 q^{59} +3.97577 q^{61} +7.70771 q^{62} +1.00000 q^{64} +5.19131 q^{67} -2.35386 q^{68} -4.14284 q^{71} +4.37809 q^{73} +7.78899 q^{74} -1.00000 q^{76} -11.5209 q^{77} +9.49670 q^{79} +3.54064 q^{82} -4.43513 q^{83} -1.73194 q^{86} +4.89449 q^{88} +3.62191 q^{89} +4.45936 q^{91} -6.16707 q^{92} -5.32962 q^{94} +7.06157 q^{97} -1.45936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 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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