LMFDB - Embedded newform 8550.2.a.bp.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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -5.12311 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -5.12311 q^{7} -1.00000 q^{8} +2.00000 q^{11} -4.00000 q^{13} +5.12311 q^{14} +1.00000 q^{16} +5.12311 q^{17} -1.00000 q^{19} -2.00000 q^{22} +4.00000 q^{26} -5.12311 q^{28} -2.00000 q^{29} +0.876894 q^{31} -1.00000 q^{32} -5.12311 q^{34} +1.00000 q^{38} +3.12311 q^{41} +6.24621 q^{43} +2.00000 q^{44} +6.24621 q^{47} +19.2462 q^{49} -4.00000 q^{52} -12.2462 q^{53} +5.12311 q^{56} +2.00000 q^{58} +5.12311 q^{59} -4.24621 q^{61} -0.876894 q^{62} +1.00000 q^{64} +10.2462 q^{67} +5.12311 q^{68} -10.2462 q^{71} -6.00000 q^{73} -1.00000 q^{76} -10.2462 q^{77} +0.876894 q^{79} -3.12311 q^{82} -15.1231 q^{83} -6.24621 q^{86} -2.00000 q^{88} +13.3693 q^{89} +20.4924 q^{91} -6.24621 q^{94} +6.00000 q^{97} -19.2462 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{11} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{22} + 8 q^{26} - 2 q^{28} - 4 q^{29} + 10 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 2 q^{41} - 4 q^{43} + 4 q^{44} - 4 q^{47} + 22 q^{49} - 8 q^{52} - 8 q^{53} + 2 q^{56} + 4 q^{58} + 2 q^{59} + 8 q^{61} - 10 q^{62} + 2 q^{64} + 4 q^{67} + 2 q^{68} - 4 q^{71} - 12 q^{73} - 2 q^{76} - 4 q^{77} + 10 q^{79} + 2 q^{82} - 22 q^{83} + 4 q^{86} - 4 q^{88} + 2 q^{89} + 8 q^{91} + 4 q^{94} + 12 q^{97} - 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^11 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^22 + 8 * q^26 - 2 * q^28 - 4 * q^29 + 10 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 2 * q^41 - 4 * q^43 + 4 * q^44 - 4 * q^47 + 22 * q^49 - 8 * q^52 - 8 * q^53 + 2 * q^56 + 4 * q^58 + 2 * q^59 + 8 * q^61 - 10 * q^62 + 2 * q^64 + 4 * q^67 + 2 * q^68 - 4 * q^71 - 12 * q^73 - 2 * q^76 - 4 * q^77 + 10 * q^79 + 2 * q^82 - 22 * q^83 + 4 * q^86 - 4 * q^88 + 2 * q^89 + 8 * q^91 + 4 * q^94 + 12 * q^97 - 22 * 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
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	5.12311	1.36921
$15$	0	0
$16$	1.00000	0.250000
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	−5.12311	−0.968176
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0.876894	0.157495	0.0787474	−	0.996895i	$-0.474908\pi$
$31$	0.876894	0.157495	0.0787474	+	0.996895i	$0.474908\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.12311	−0.878605
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	3.12311	0.487747	0.243874	−	0.969807i	$-0.421582\pi$
$41$	3.12311	0.487747	0.243874	+	0.969807i	$0.421582\pi$
$42$	0	0
$43$	6.24621	0.952538	0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621	0.952538	0.476269	+	0.879300i	$0.341989\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	0	0
$47$	6.24621	0.911104	0.455552	−	0.890209i	$-0.349442\pi$
$47$	6.24621	0.911104	0.455552	+	0.890209i	$0.349442\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	0	0
$56$	5.12311	0.684604
$57$	0	0
$58$	2.00000	0.262613
$59$	5.12311	0.666972	0.333486	−	0.942755i	$-0.391775\pi$
$59$	5.12311	0.666972	0.333486	+	0.942755i	$0.391775\pi$
$60$	0	0
$61$	−4.24621	−0.543672	−0.271836	−	0.962344i	$-0.587631\pi$
$61$	−4.24621	−0.543672	−0.271836	+	0.962344i	$0.587631\pi$
$62$	−0.876894	−0.111366
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.2462	1.25177	0.625887	−	0.779914i	$-0.284737\pi$
$67$	10.2462	1.25177	0.625887	+	0.779914i	$0.284737\pi$
$68$	5.12311	0.621268
$69$	0	0
$70$	0	0
$71$	−10.2462	−1.21600	−0.608001	−	0.793936i	$-0.708028\pi$
$71$	−10.2462	−1.21600	−0.608001	+	0.793936i	$0.708028\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−10.2462	−1.16766
$78$	0	0
$79$	0.876894	0.0986583	0.0493292	−	0.998783i	$-0.484292\pi$
$79$	0.876894	0.0986583	0.0493292	+	0.998783i	$0.484292\pi$
$80$	0	0
$81$	0	0
$82$	−3.12311	−0.344889
$83$	−15.1231	−1.65998	−0.829988	−	0.557781i	$-0.811653\pi$
$83$	−15.1231	−1.65998	−0.829988	+	0.557781i	$0.811653\pi$
$84$	0	0
$85$	0	0
$86$	−6.24621	−0.673546
$87$	0	0
$88$	−2.00000	−0.213201
$89$	13.3693	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$89$	13.3693	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$90$	0	0
$91$	20.4924	2.14819
$92$	0	0
$93$	0	0
$94$	−6.24621	−0.644247
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−19.2462	−1.94416
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−8.24621	−0.812523	−0.406262	−	0.913757i	$-0.633168\pi$
$103$	−8.24621	−0.812523	−0.406262	+	0.913757i	$0.633168\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	12.2462	1.18946
$107$	2.24621	0.217149	0.108575	−	0.994088i	$-0.465371\pi$
$107$	2.24621	0.217149	0.108575	+	0.994088i	$0.465371\pi$
$108$	0	0
$109$	13.1231	1.25697	0.628483	−	0.777824i	$-0.283676\pi$
$109$	13.1231	1.25697	0.628483	+	0.777824i	$0.283676\pi$
$110$	0	0
$111$	0	0
$112$	−5.12311	−0.484088
$113$	12.2462	1.15203	0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462	1.15203	0.576013	+	0.817440i	$0.304608\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−5.12311	−0.471620
$119$	−26.2462	−2.40599
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.24621	0.384434
$123$	0	0
$124$	0.876894	0.0787474
$125$	0	0
$126$	0	0
$127$	20.2462	1.79656	0.898280	−	0.439423i	$-0.144817\pi$
$127$	20.2462	1.79656	0.898280	+	0.439423i	$0.144817\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	10.4924	0.916727	0.458364	−	0.888765i	$-0.348436\pi$
$131$	10.4924	0.916727	0.458364	+	0.888765i	$0.348436\pi$
$132$	0	0
$133$	5.12311	0.444230
$134$	−10.2462	−0.885138
$135$	0	0
$136$	−5.12311	−0.439303
$137$	10.8769	0.929276	0.464638	−	0.885501i	$-0.346184\pi$
$137$	10.8769	0.929276	0.464638	+	0.885501i	$0.346184\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	0	0
$142$	10.2462	0.859843
$143$	−8.00000	−0.668994
$144$	0	0
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	11.1231	0.905185	0.452593	−	0.891717i	$-0.350499\pi$
$151$	11.1231	0.905185	0.452593	+	0.891717i	$0.350499\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	10.2462	0.825663
$155$	0	0
$156$	0	0
$157$	−7.12311	−0.568486	−0.284243	−	0.958752i	$-0.591742\pi$
$157$	−7.12311	−0.568486	−0.284243	+	0.958752i	$0.591742\pi$
$158$	−0.876894	−0.0697620
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	3.12311	0.243874
$165$	0	0
$166$	15.1231	1.17378
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	6.24621	0.476269
$173$	8.24621	0.626948	0.313474	−	0.949597i	$-0.398507\pi$
$173$	8.24621	0.626948	0.313474	+	0.949597i	$0.398507\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	−13.3693	−1.00207
$179$	−17.1231	−1.27984	−0.639921	−	0.768441i	$-0.721033\pi$
$179$	−17.1231	−1.27984	−0.639921	+	0.768441i	$0.721033\pi$
$180$	0	0
$181$	−15.3693	−1.14239	−0.571196	−	0.820814i	$-0.693520\pi$
$181$	−15.3693	−1.14239	−0.571196	+	0.820814i	$0.693520\pi$
$182$	−20.4924	−1.51900
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.2462	0.749277
$188$	6.24621	0.455552
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	14.4924	1.04319	0.521594	−	0.853194i	$-0.325338\pi$
$193$	14.4924	1.04319	0.521594	+	0.853194i	$0.325338\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	19.2462	1.37473
$197$	−8.24621	−0.587518	−0.293759	−	0.955879i	$-0.594906\pi$
$197$	−8.24621	−0.587518	−0.293759	+	0.955879i	$0.594906\pi$
$198$	0	0
$199$	−6.24621	−0.442782	−0.221391	−	0.975185i	$-0.571060\pi$
$199$	−6.24621	−0.442782	−0.221391	+	0.975185i	$0.571060\pi$
$200$	0	0
$201$	0	0
$202$	10.0000	0.703598
$203$	10.2462	0.719143
$204$	0	0
$205$	0	0
$206$	8.24621	0.574541
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−12.2462	−0.841073
$213$	0	0
$214$	−2.24621	−0.153548
$215$	0	0
$216$	0	0
$217$	−4.49242	−0.304966
$218$	−13.1231	−0.888809
$219$	0	0
$220$	0	0
$221$	−20.4924	−1.37847
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	5.12311	0.342302
$225$	0	0
$226$	−12.2462	−0.814606
$227$	−12.4924	−0.829151	−0.414576	−	0.910015i	$-0.636070\pi$
$227$	−12.4924	−0.829151	−0.414576	+	0.910015i	$0.636070\pi$
$228$	0	0
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	0	0
$231$	0	0
$232$	2.00000	0.131306
$233$	−23.3693	−1.53097	−0.765487	−	0.643451i	$-0.777502\pi$
$233$	−23.3693	−1.53097	−0.765487	+	0.643451i	$0.777502\pi$
$234$	0	0
$235$	0	0
$236$	5.12311	0.333486
$237$	0	0
$238$	26.2462	1.70129
$239$	−18.2462	−1.18025	−0.590125	−	0.807312i	$-0.700921\pi$
$239$	−18.2462	−1.18025	−0.590125	+	0.807312i	$0.700921\pi$
$240$	0	0
$241$	−16.2462	−1.04651	−0.523255	−	0.852176i	$-0.675283\pi$
$241$	−16.2462	−1.04651	−0.523255	+	0.852176i	$0.675283\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−4.24621	−0.271836
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−0.876894	−0.0556828
$249$	0	0
$250$	0	0
$251$	−0.246211	−0.0155407	−0.00777036	−	0.999970i	$-0.502473\pi$
$251$	−0.246211	−0.0155407	−0.00777036	+	0.999970i	$0.502473\pi$
$252$	0	0
$253$	0	0
$254$	−20.2462	−1.27036
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−10.4924	−0.648224
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	0	0
$266$	−5.12311	−0.314118
$267$	0	0
$268$	10.2462	0.625887
$269$	−28.7386	−1.75223	−0.876113	−	0.482106i	$-0.839872\pi$
$269$	−28.7386	−1.75223	−0.876113	+	0.482106i	$0.839872\pi$
$270$	0	0
$271$	−14.2462	−0.865396	−0.432698	−	0.901539i	$-0.642438\pi$
$271$	−14.2462	−0.865396	−0.432698	+	0.901539i	$0.642438\pi$
$272$	5.12311	0.310634
$273$	0	0
$274$	−10.8769	−0.657097
$275$	0	0
$276$	0	0
$277$	−31.6155	−1.89959	−0.949796	−	0.312868i	$-0.898710\pi$
$277$	−31.6155	−1.89959	−0.949796	+	0.312868i	$0.898710\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	0	0
$281$	−5.36932	−0.320307	−0.160153	−	0.987092i	$-0.551199\pi$
$281$	−5.36932	−0.320307	−0.160153	+	0.987092i	$0.551199\pi$
$282$	0	0
$283$	24.4924	1.45592	0.727962	−	0.685618i	$-0.240468\pi$
$283$	24.4924	1.45592	0.727962	+	0.685618i	$0.240468\pi$
$284$	−10.2462	−0.608001
$285$	0	0
$286$	8.00000	0.473050
$287$	−16.0000	−0.944450
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−14.4924	−0.846656	−0.423328	−	0.905976i	$-0.639138\pi$
$293$	−14.4924	−0.846656	−0.423328	+	0.905976i	$0.639138\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	22.0000	1.27443
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	−11.1231	−0.640063
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	22.2462	1.26966	0.634829	−	0.772653i	$-0.281070\pi$
$307$	22.2462	1.26966	0.634829	+	0.772653i	$0.281070\pi$
$308$	−10.2462	−0.583832
$309$	0	0
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	−24.7386	−1.39831	−0.699155	−	0.714970i	$-0.746440\pi$
$313$	−24.7386	−1.39831	−0.699155	+	0.714970i	$0.746440\pi$
$314$	7.12311	0.401980
$315$	0	0
$316$	0.876894	0.0493292
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.12311	−0.285057
$324$	0	0
$325$	0	0
$326$	12.0000	0.664619
$327$	0	0
$328$	−3.12311	−0.172445
$329$	−32.0000	−1.76422
$330$	0	0
$331$	8.49242	0.466786	0.233393	−	0.972383i	$-0.425017\pi$
$331$	8.49242	0.466786	0.233393	+	0.972383i	$0.425017\pi$
$332$	−15.1231	−0.829988
$333$	0	0
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	0	0
$341$	1.75379	0.0949730
$342$	0	0
$343$	−62.7386	−3.38757
$344$	−6.24621	−0.336773
$345$	0	0
$346$	−8.24621	−0.443319
$347$	−33.3693	−1.79136	−0.895679	−	0.444700i	$-0.853310\pi$
$347$	−33.3693	−1.79136	−0.895679	+	0.444700i	$0.853310\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	31.8617	1.69583	0.847915	−	0.530133i	$-0.177858\pi$
$353$	31.8617	1.69583	0.847915	+	0.530133i	$0.177858\pi$
$354$	0	0
$355$	0	0
$356$	13.3693	0.708572
$357$	0	0
$358$	17.1231	0.904984
$359$	6.24621	0.329662	0.164831	−	0.986322i	$-0.447292\pi$
$359$	6.24621	0.329662	0.164831	+	0.986322i	$0.447292\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.3693	0.807793
$363$	0	0
$364$	20.4924	1.07409
$365$	0	0
$366$	0	0
$367$	9.12311	0.476222	0.238111	−	0.971238i	$-0.423472\pi$
$367$	9.12311	0.476222	0.238111	+	0.971238i	$0.423472\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	62.7386	3.25723
$372$	0	0
$373$	−2.24621	−0.116304	−0.0581522	−	0.998308i	$-0.518521\pi$
$373$	−2.24621	−0.116304	−0.0581522	+	0.998308i	$0.518521\pi$
$374$	−10.2462	−0.529819
$375$	0	0
$376$	−6.24621	−0.322124
$377$	8.00000	0.412021
$378$	0	0
$379$	−24.4924	−1.25809	−0.629046	−	0.777368i	$-0.716554\pi$
$379$	−24.4924	−1.25809	−0.629046	+	0.777368i	$0.716554\pi$
$380$	0	0
$381$	0	0
$382$	4.00000	0.204658
$383$	−3.50758	−0.179229	−0.0896144	−	0.995977i	$-0.528563\pi$
$383$	−3.50758	−0.179229	−0.0896144	+	0.995977i	$0.528563\pi$
$384$	0	0
$385$	0	0
$386$	−14.4924	−0.737645
$387$	0	0
$388$	6.00000	0.304604
$389$	−1.50758	−0.0764372	−0.0382186	−	0.999269i	$-0.512168\pi$
$389$	−1.50758	−0.0764372	−0.0382186	+	0.999269i	$0.512168\pi$
$390$	0	0
$391$	0	0
$392$	−19.2462	−0.972080
$393$	0	0
$394$	8.24621	0.415438
$395$	0	0
$396$	0	0
$397$	−4.87689	−0.244764	−0.122382	−	0.992483i	$-0.539053\pi$
$397$	−4.87689	−0.244764	−0.122382	+	0.992483i	$0.539053\pi$
$398$	6.24621	0.313094
$399$	0	0
$400$	0	0
$401$	25.8617	1.29147	0.645737	−	0.763560i	$-0.276550\pi$
$401$	25.8617	1.29147	0.645737	+	0.763560i	$0.276550\pi$
$402$	0	0
$403$	−3.50758	−0.174725
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−10.2462	−0.508511
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	−8.24621	−0.406262
$413$	−26.2462	−1.29149
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	2.00000	0.0978232
$419$	4.24621	0.207441	0.103720	−	0.994606i	$-0.466925\pi$
$419$	4.24621	0.207441	0.103720	+	0.994606i	$0.466925\pi$
$420$	0	0
$421$	−6.87689	−0.335159	−0.167580	−	0.985859i	$-0.553595\pi$
$421$	−6.87689	−0.335159	−0.167580	+	0.985859i	$0.553595\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	12.2462	0.594729
$425$	0	0
$426$	0	0
$427$	21.7538	1.05274
$428$	2.24621	0.108575
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	4.49242	0.215643
$435$	0	0
$436$	13.1231	0.628483
$437$	0	0
$438$	0	0
$439$	−19.1231	−0.912696	−0.456348	−	0.889801i	$-0.650843\pi$
$439$	−19.1231	−0.912696	−0.456348	+	0.889801i	$0.650843\pi$
$440$	0	0
$441$	0	0
$442$	20.4924	0.974725
$443$	−37.8617	−1.79887	−0.899433	−	0.437059i	$-0.856020\pi$
$443$	−37.8617	−1.79887	−0.899433	+	0.437059i	$0.856020\pi$
$444$	0	0
$445$	0	0
$446$	10.0000	0.473514
$447$	0	0
$448$	−5.12311	−0.242044
$449$	−29.3693	−1.38602	−0.693012	−	0.720926i	$-0.743717\pi$
$449$	−29.3693	−1.38602	−0.693012	+	0.720926i	$0.743717\pi$
$450$	0	0
$451$	6.24621	0.294123
$452$	12.2462	0.576013
$453$	0	0
$454$	12.4924	0.586298
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	14.4924	0.677186
$459$	0	0
$460$	0	0
$461$	−10.4924	−0.488681	−0.244340	−	0.969690i	$-0.578571\pi$
$461$	−10.4924	−0.488681	−0.244340	+	0.969690i	$0.578571\pi$
$462$	0	0
$463$	14.8769	0.691388	0.345694	−	0.938347i	$-0.387644\pi$
$463$	14.8769	0.691388	0.345694	+	0.938347i	$0.387644\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	23.3693	1.08256
$467$	−39.1231	−1.81040	−0.905201	−	0.424984i	$-0.860280\pi$
$467$	−39.1231	−1.81040	−0.905201	+	0.424984i	$0.860280\pi$
$468$	0	0
$469$	−52.4924	−2.42387
$470$	0	0
$471$	0	0
$472$	−5.12311	−0.235810
$473$	12.4924	0.574402
$474$	0	0
$475$	0	0
$476$	−26.2462	−1.20299
$477$	0	0
$478$	18.2462	0.834562
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	0	0
$482$	16.2462	0.739995
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	32.2462	1.46122	0.730608	−	0.682798i	$-0.239237\pi$
$487$	32.2462	1.46122	0.730608	+	0.682798i	$0.239237\pi$
$488$	4.24621	0.192217
$489$	0	0
$490$	0	0
$491$	10.4924	0.473516	0.236758	−	0.971569i	$-0.423915\pi$
$491$	10.4924	0.473516	0.236758	+	0.971569i	$0.423915\pi$
$492$	0	0
$493$	−10.2462	−0.461466
$494$	−4.00000	−0.179969
$495$	0	0
$496$	0.876894	0.0393737
$497$	52.4924	2.35461
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	0.246211	0.0109889
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	20.2462	0.898280
$509$	9.50758	0.421416	0.210708	−	0.977549i	$-0.432423\pi$
$509$	9.50758	0.421416	0.210708	+	0.977549i	$0.432423\pi$
$510$	0	0
$511$	30.7386	1.35980
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.0000	0.441081
$515$	0	0
$516$	0	0
$517$	12.4924	0.549416
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.6155	−1.20986	−0.604929	−	0.796279i	$-0.706799\pi$
$521$	−27.6155	−1.20986	−0.604929	+	0.796279i	$0.706799\pi$
$522$	0	0
$523$	6.73863	0.294660	0.147330	−	0.989087i	$-0.452932\pi$
$523$	6.73863	0.294660	0.147330	+	0.989087i	$0.452932\pi$
$524$	10.4924	0.458364
$525$	0	0
$526$	−20.4924	−0.893512
$527$	4.49242	0.195693
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	5.12311	0.222115
$533$	−12.4924	−0.541107
$534$	0	0
$535$	0	0
$536$	−10.2462	−0.442569
$537$	0	0
$538$	28.7386	1.23901
$539$	38.4924	1.65799
$540$	0	0
$541$	−34.4924	−1.48295	−0.741473	−	0.670983i	$-0.765872\pi$
$541$	−34.4924	−1.48295	−0.741473	+	0.670983i	$0.765872\pi$
$542$	14.2462	0.611927
$543$	0	0
$544$	−5.12311	−0.219651
$545$	0	0
$546$	0	0
$547$	14.2462	0.609124	0.304562	−	0.952493i	$-0.401490\pi$
$547$	14.2462	0.609124	0.304562	+	0.952493i	$0.401490\pi$
$548$	10.8769	0.464638
$549$	0	0
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	−4.49242	−0.191037
$554$	31.6155	1.34322
$555$	0	0
$556$	16.4924	0.699435
$557$	0.246211	0.0104323	0.00521615	−	0.999986i	$-0.498340\pi$
$557$	0.246211	0.0104323	0.00521615	+	0.999986i	$0.498340\pi$
$558$	0	0
$559$	−24.9848	−1.05675
$560$	0	0
$561$	0	0
$562$	5.36932	0.226491
$563$	−39.2311	−1.65339	−0.826696	−	0.562649i	$-0.809782\pi$
$563$	−39.2311	−1.65339	−0.826696	+	0.562649i	$0.809782\pi$
$564$	0	0
$565$	0	0
$566$	−24.4924	−1.02949
$567$	0	0
$568$	10.2462	0.429921
$569$	−32.8769	−1.37827	−0.689136	−	0.724632i	$-0.742010\pi$
$569$	−32.8769	−1.37827	−0.689136	+	0.724632i	$0.742010\pi$
$570$	0	0
$571$	−0.492423	−0.0206072	−0.0103036	−	0.999947i	$-0.503280\pi$
$571$	−0.492423	−0.0206072	−0.0103036	+	0.999947i	$0.503280\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	16.0000	0.667827
$575$	0	0
$576$	0	0
$577$	24.7386	1.02988	0.514941	−	0.857225i	$-0.327814\pi$
$577$	24.7386	1.02988	0.514941	+	0.857225i	$0.327814\pi$
$578$	−9.24621	−0.384592
$579$	0	0
$580$	0	0
$581$	77.4773	3.21430
$582$	0	0
$583$	−24.4924	−1.01437
$584$	6.00000	0.248282
$585$	0	0
$586$	14.4924	0.598676
$587$	10.6307	0.438775	0.219388	−	0.975638i	$-0.429594\pi$
$587$	10.6307	0.438775	0.219388	+	0.975638i	$0.429594\pi$
$588$	0	0
$589$	−0.876894	−0.0361318
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.6155	−1.05190	−0.525952	−	0.850514i	$-0.676291\pi$
$593$	−25.6155	−1.05190	−0.525952	+	0.850514i	$0.676291\pi$
$594$	0	0
$595$	0	0
$596$	−22.0000	−0.901155
$597$	0	0
$598$	0	0
$599$	20.4924	0.837298	0.418649	−	0.908148i	$-0.362504\pi$
$599$	20.4924	0.837298	0.418649	+	0.908148i	$0.362504\pi$
$600$	0	0
$601$	−12.7386	−0.519620	−0.259810	−	0.965660i	$-0.583660\pi$
$601$	−12.7386	−0.519620	−0.259810	+	0.965660i	$0.583660\pi$
$602$	32.0000	1.30422
$603$	0	0
$604$	11.1231	0.452593
$605$	0	0
$606$	0	0
$607$	−12.2462	−0.497058	−0.248529	−	0.968624i	$-0.579947\pi$
$607$	−12.2462	−0.497058	−0.248529	+	0.968624i	$0.579947\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−24.9848	−1.01078
$612$	0	0
$613$	−41.3693	−1.67089	−0.835445	−	0.549573i	$-0.814790\pi$
$613$	−41.3693	−1.67089	−0.835445	+	0.549573i	$0.814790\pi$
$614$	−22.2462	−0.897784
$615$	0	0
$616$	10.2462	0.412832
$617$	−9.12311	−0.367282	−0.183641	−	0.982993i	$-0.558788\pi$
$617$	−9.12311	−0.367282	−0.183641	+	0.982993i	$0.558788\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	0	0
$622$	−20.0000	−0.801927
$623$	−68.4924	−2.74409
$624$	0	0
$625$	0	0
$626$	24.7386	0.988755
$627$	0	0
$628$	−7.12311	−0.284243
$629$	0	0
$630$	0	0
$631$	28.9848	1.15387	0.576934	−	0.816791i	$-0.304249\pi$
$631$	28.9848	1.15387	0.576934	+	0.816791i	$0.304249\pi$
$632$	−0.876894	−0.0348810
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	−76.9848	−3.05025
$638$	4.00000	0.158362
$639$	0	0
$640$	0	0
$641$	−45.8617	−1.81143	−0.905715	−	0.423887i	$-0.860665\pi$
$641$	−45.8617	−1.81143	−0.905715	+	0.423887i	$0.860665\pi$
$642$	0	0
$643$	−0.492423	−0.0194192	−0.00970962	−	0.999953i	$-0.503091\pi$
$643$	−0.492423	−0.0194192	−0.00970962	+	0.999953i	$0.503091\pi$
$644$	0	0
$645$	0	0
$646$	5.12311	0.201566
$647$	28.4924	1.12015	0.560076	−	0.828441i	$-0.310772\pi$
$647$	28.4924	1.12015	0.560076	+	0.828441i	$0.310772\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	38.9848	1.52559	0.762797	−	0.646638i	$-0.223825\pi$
$653$	38.9848	1.52559	0.762797	+	0.646638i	$0.223825\pi$
$654$	0	0
$655$	0	0
$656$	3.12311	0.121937
$657$	0	0
$658$	32.0000	1.24749
$659$	−41.6155	−1.62111	−0.810555	−	0.585662i	$-0.800835\pi$
$659$	−41.6155	−1.62111	−0.810555	+	0.585662i	$0.800835\pi$
$660$	0	0
$661$	9.61553	0.374001	0.187000	−	0.982360i	$-0.440123\pi$
$661$	9.61553	0.374001	0.187000	+	0.982360i	$0.440123\pi$
$662$	−8.49242	−0.330067
$663$	0	0
$664$	15.1231	0.586890
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	0	0
$670$	0	0
$671$	−8.49242	−0.327846
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	3.00000	0.115385
$677$	30.9848	1.19084	0.595422	−	0.803413i	$-0.296985\pi$
$677$	30.9848	1.19084	0.595422	+	0.803413i	$0.296985\pi$
$678$	0	0
$679$	−30.7386	−1.17964
$680$	0	0
$681$	0	0
$682$	−1.75379	−0.0671560
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	0	0
$686$	62.7386	2.39537
$687$	0	0
$688$	6.24621	0.238135
$689$	48.9848	1.86617
$690$	0	0
$691$	−32.4924	−1.23607	−0.618035	−	0.786151i	$-0.712071\pi$
$691$	−32.4924	−1.23607	−0.618035	+	0.786151i	$0.712071\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	33.3693	1.26668
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	−22.4924	−0.851351
$699$	0	0
$700$	0	0
$701$	30.9848	1.17028	0.585141	−	0.810932i	$-0.301039\pi$
$701$	30.9848	1.17028	0.585141	+	0.810932i	$0.301039\pi$
$702$	0	0
$703$	0	0
$704$	2.00000	0.0753778
$705$	0	0
$706$	−31.8617	−1.19913
$707$	51.2311	1.92674
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	−13.3693	−0.501036
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−17.1231	−0.639921
$717$	0	0
$718$	−6.24621	−0.233107
$719$	−6.24621	−0.232944	−0.116472	−	0.993194i	$-0.537159\pi$
$719$	−6.24621	−0.232944	−0.116472	+	0.993194i	$0.537159\pi$
$720$	0	0
$721$	42.2462	1.57333
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−15.3693	−0.571196
$725$	0	0
$726$	0	0
$727$	−31.3693	−1.16342	−0.581712	−	0.813395i	$-0.697617\pi$
$727$	−31.3693	−1.16342	−0.581712	+	0.813395i	$0.697617\pi$
$728$	−20.4924	−0.759500
$729$	0	0
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	7.61553	0.281286	0.140643	−	0.990060i	$-0.455083\pi$
$733$	7.61553	0.281286	0.140643	+	0.990060i	$0.455083\pi$
$734$	−9.12311	−0.336740
$735$	0	0
$736$	0	0
$737$	20.4924	0.754848
$738$	0	0
$739$	16.4924	0.606684	0.303342	−	0.952882i	$-0.401898\pi$
$739$	16.4924	0.606684	0.303342	+	0.952882i	$0.401898\pi$
$740$	0	0
$741$	0	0
$742$	−62.7386	−2.30321
$743$	40.9848	1.50359	0.751794	−	0.659398i	$-0.229189\pi$
$743$	40.9848	1.50359	0.751794	+	0.659398i	$0.229189\pi$
$744$	0	0
$745$	0	0
$746$	2.24621	0.0822396
$747$	0	0
$748$	10.2462	0.374639
$749$	−11.5076	−0.420478
$750$	0	0
$751$	49.8617	1.81948	0.909740	−	0.415178i	$-0.136281\pi$
$751$	49.8617	1.81948	0.909740	+	0.415178i	$0.136281\pi$
$752$	6.24621	0.227776
$753$	0	0
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	−21.8617	−0.794578	−0.397289	−	0.917693i	$-0.630049\pi$
$757$	−21.8617	−0.794578	−0.397289	+	0.917693i	$0.630049\pi$
$758$	24.4924	0.889605
$759$	0	0
$760$	0	0
$761$	−29.7538	−1.07857	−0.539287	−	0.842122i	$-0.681306\pi$
$761$	−29.7538	−1.07857	−0.539287	+	0.842122i	$0.681306\pi$
$762$	0	0
$763$	−67.2311	−2.43393
$764$	−4.00000	−0.144715
$765$	0	0
$766$	3.50758	0.126734
$767$	−20.4924	−0.739938
$768$	0	0
$769$	−40.2462	−1.45132	−0.725658	−	0.688056i	$-0.758464\pi$
$769$	−40.2462	−1.45132	−0.725658	+	0.688056i	$0.758464\pi$
$770$	0	0
$771$	0	0
$772$	14.4924	0.521594
$773$	−48.2462	−1.73530	−0.867648	−	0.497179i	$-0.834369\pi$
$773$	−48.2462	−1.73530	−0.867648	+	0.497179i	$0.834369\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	1.50758	0.0540493
$779$	−3.12311	−0.111897
$780$	0	0
$781$	−20.4924	−0.733277
$782$	0	0
$783$	0	0
$784$	19.2462	0.687365
$785$	0	0
$786$	0	0
$787$	8.49242	0.302722	0.151361	−	0.988479i	$-0.451634\pi$
$787$	8.49242	0.302722	0.151361	+	0.988479i	$0.451634\pi$
$788$	−8.24621	−0.293759
$789$	0	0
$790$	0	0
$791$	−62.7386	−2.23073
$792$	0	0
$793$	16.9848	0.603150
$794$	4.87689	0.173075
$795$	0	0
$796$	−6.24621	−0.221391
$797$	30.4924	1.08010	0.540049	−	0.841634i	$-0.318406\pi$
$797$	30.4924	1.08010	0.540049	+	0.841634i	$0.318406\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	0	0
$802$	−25.8617	−0.913210
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0	0
$806$	3.50758	0.123549
$807$	0	0
$808$	10.0000	0.351799
$809$	14.2462	0.500870	0.250435	−	0.968133i	$-0.419426\pi$
$809$	14.2462	0.500870	0.250435	+	0.968133i	$0.419426\pi$
$810$	0	0
$811$	20.9848	0.736878	0.368439	−	0.929652i	$-0.379892\pi$
$811$	20.9848	0.736878	0.368439	+	0.929652i	$0.379892\pi$
$812$	10.2462	0.359572
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.24621	−0.218527
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	14.0000	0.488603	0.244302	−	0.969699i	$-0.421441\pi$
$821$	14.0000	0.488603	0.244302	+	0.969699i	$0.421441\pi$
$822$	0	0
$823$	55.8617	1.94722	0.973609	−	0.228223i	$-0.0732915\pi$
$823$	55.8617	1.94722	0.973609	+	0.228223i	$0.0732915\pi$
$824$	8.24621	0.287270
$825$	0	0
$826$	26.2462	0.913222
$827$	−30.2462	−1.05176	−0.525882	−	0.850558i	$-0.676265\pi$
$827$	−30.2462	−1.05176	−0.525882	+	0.850558i	$0.676265\pi$
$828$	0	0
$829$	30.1080	1.04569	0.522846	−	0.852427i	$-0.324870\pi$
$829$	30.1080	1.04569	0.522846	+	0.852427i	$0.324870\pi$
$830$	0	0
$831$	0	0
$832$	−4.00000	−0.138675
$833$	98.6004	3.41630
$834$	0	0
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	−4.24621	−0.146683
$839$	40.9848	1.41495	0.707477	−	0.706736i	$-0.249833\pi$
$839$	40.9848	1.41495	0.707477	+	0.706736i	$0.249833\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.87689	0.236993
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	35.8617	1.23222
$848$	−12.2462	−0.420537
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	9.36932	0.320799	0.160400	−	0.987052i	$-0.448722\pi$
$853$	9.36932	0.320799	0.160400	+	0.987052i	$0.448722\pi$
$854$	−21.7538	−0.744399
$855$	0	0
$856$	−2.24621	−0.0767739
$857$	−43.4773	−1.48516	−0.742578	−	0.669760i	$-0.766397\pi$
$857$	−43.4773	−1.48516	−0.742578	+	0.669760i	$0.766397\pi$
$858$	0	0
$859$	32.4924	1.10863	0.554314	−	0.832308i	$-0.312981\pi$
$859$	32.4924	1.10863	0.554314	+	0.832308i	$0.312981\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−4.49242	−0.152924	−0.0764619	−	0.997073i	$-0.524362\pi$
$863$	−4.49242	−0.152924	−0.0764619	+	0.997073i	$0.524362\pi$
$864$	0	0
$865$	0	0
$866$	−18.0000	−0.611665
$867$	0	0
$868$	−4.49242	−0.152483
$869$	1.75379	0.0594932
$870$	0	0
$871$	−40.9848	−1.38872
$872$	−13.1231	−0.444404
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21.7538	−0.734573	−0.367287	−	0.930108i	$-0.619713\pi$
$877$	−21.7538	−0.734573	−0.367287	+	0.930108i	$0.619713\pi$
$878$	19.1231	0.645374
$879$	0	0
$880$	0	0
$881$	0.492423	0.0165901	0.00829507	−	0.999966i	$-0.497360\pi$
$881$	0.492423	0.0165901	0.00829507	+	0.999966i	$0.497360\pi$
$882$	0	0
$883$	−7.50758	−0.252650	−0.126325	−	0.991989i	$-0.540318\pi$
$883$	−7.50758	−0.252650	−0.126325	+	0.991989i	$0.540318\pi$
$884$	−20.4924	−0.689235
$885$	0	0
$886$	37.8617	1.27199
$887$	−56.9848	−1.91336	−0.956682	−	0.291135i	$-0.905967\pi$
$887$	−56.9848	−1.91336	−0.956682	+	0.291135i	$0.905967\pi$
$888$	0	0
$889$	−103.723	−3.47877
$890$	0	0
$891$	0	0
$892$	−10.0000	−0.334825
$893$	−6.24621	−0.209021
$894$	0	0
$895$	0	0
$896$	5.12311	0.171151
$897$	0	0
$898$	29.3693	0.980067
$899$	−1.75379	−0.0584921
$900$	0	0
$901$	−62.7386	−2.09013
$902$	−6.24621	−0.207976
$903$	0	0
$904$	−12.2462	−0.407303
$905$	0	0
$906$	0	0
$907$	−48.9848	−1.62652	−0.813258	−	0.581904i	$-0.802308\pi$
$907$	−48.9848	−1.62652	−0.813258	+	0.581904i	$0.802308\pi$
$908$	−12.4924	−0.414576
$909$	0	0
$910$	0	0
$911$	56.9848	1.88799	0.943996	−	0.329957i	$-0.107034\pi$
$911$	56.9848	1.88799	0.943996	+	0.329957i	$0.107034\pi$
$912$	0	0
$913$	−30.2462	−1.00100
$914$	26.0000	0.860004
$915$	0	0
$916$	−14.4924	−0.478843
$917$	−53.7538	−1.77511
$918$	0	0
$919$	−16.4924	−0.544035	−0.272017	−	0.962292i	$-0.587691\pi$
$919$	−16.4924	−0.544035	−0.272017	+	0.962292i	$0.587691\pi$
$920$	0	0
$921$	0	0
$922$	10.4924	0.345550
$923$	40.9848	1.34903
$924$	0	0
$925$	0	0
$926$	−14.8769	−0.488885
$927$	0	0
$928$	2.00000	0.0656532
$929$	−16.4924	−0.541099	−0.270549	−	0.962706i	$-0.587205\pi$
$929$	−16.4924	−0.541099	−0.270549	+	0.962706i	$0.587205\pi$
$930$	0	0
$931$	−19.2462	−0.630769
$932$	−23.3693	−0.765487
$933$	0	0
$934$	39.1231	1.28015
$935$	0	0
$936$	0	0
$937$	−48.2462	−1.57614	−0.788068	−	0.615589i	$-0.788918\pi$
$937$	−48.2462	−1.57614	−0.788068	+	0.615589i	$0.788918\pi$
$938$	52.4924	1.71394
$939$	0	0
$940$	0	0
$941$	−45.2311	−1.47449	−0.737245	−	0.675625i	$-0.763874\pi$
$941$	−45.2311	−1.47449	−0.737245	+	0.675625i	$0.763874\pi$
$942$	0	0
$943$	0	0
$944$	5.12311	0.166743
$945$	0	0
$946$	−12.4924	−0.406164
$947$	−31.6155	−1.02737	−0.513683	−	0.857980i	$-0.671719\pi$
$947$	−31.6155	−1.02737	−0.513683	+	0.857980i	$0.671719\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	26.2462	0.850645
$953$	−39.4773	−1.27879	−0.639397	−	0.768877i	$-0.720816\pi$
$953$	−39.4773	−1.27879	−0.639397	+	0.768877i	$0.720816\pi$
$954$	0	0
$955$	0	0
$956$	−18.2462	−0.590125
$957$	0	0
$958$	−4.00000	−0.129234
$959$	−55.7235	−1.79940
$960$	0	0
$961$	−30.2311	−0.975195
$962$	0	0
$963$	0	0
$964$	−16.2462	−0.523255
$965$	0	0
$966$	0	0
$967$	42.8769	1.37883	0.689414	−	0.724368i	$-0.257868\pi$
$967$	42.8769	1.37883	0.689414	+	0.724368i	$0.257868\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	−1.61553	−0.0518448	−0.0259224	−	0.999664i	$-0.508252\pi$
$971$	−1.61553	−0.0518448	−0.0259224	+	0.999664i	$0.508252\pi$
$972$	0	0
$973$	−84.4924	−2.70870
$974$	−32.2462	−1.03324
$975$	0	0
$976$	−4.24621	−0.135918
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	0	0
$979$	26.7386	0.854570
$980$	0	0
$981$	0	0
$982$	−10.4924	−0.334827
$983$	3.50758	0.111874	0.0559372	−	0.998434i	$-0.482185\pi$
$983$	3.50758	0.111874	0.0559372	+	0.998434i	$0.482185\pi$
$984$	0	0
$985$	0	0
$986$	10.2462	0.326306
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	39.6155	1.25843	0.629214	−	0.777232i	$-0.283377\pi$
$991$	39.6155	1.25843	0.629214	+	0.777232i	$0.283377\pi$
$992$	−0.876894	−0.0278414
$993$	0	0
$994$	−52.4924	−1.66496
$995$	0	0
$996$	0	0
$997$	−10.6307	−0.336677	−0.168339	−	0.985729i	$-0.553840\pi$
$997$	−10.6307	−0.336677	−0.168339	+	0.985729i	$0.553840\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bp.1.1		2
3.2	odd	2		8550.2.a.bx.1.1		2
5.4	even	2		1710.2.a.x.1.2	yes	2
15.14	odd	2		1710.2.a.v.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.a.v.1.2	✓	2	15.14	odd	2
1710.2.a.x.1.2	yes	2	5.4	even	2
8550.2.a.bp.1.1		2	1.1	even	1	trivial
8550.2.a.bx.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -5.12311 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -5.12311 q^{7} -1.00000 q^{8} +2.00000 q^{11} -4.00000 q^{13} +5.12311 q^{14} +1.00000 q^{16} +5.12311 q^{17} -1.00000 q^{19} -2.00000 q^{22} +4.00000 q^{26} -5.12311 q^{28} -2.00000 q^{29} +0.876894 q^{31} -1.00000 q^{32} -5.12311 q^{34} +1.00000 q^{38} +3.12311 q^{41} +6.24621 q^{43} +2.00000 q^{44} +6.24621 q^{47} +19.2462 q^{49} -4.00000 q^{52} -12.2462 q^{53} +5.12311 q^{56} +2.00000 q^{58} +5.12311 q^{59} -4.24621 q^{61} -0.876894 q^{62} +1.00000 q^{64} +10.2462 q^{67} +5.12311 q^{68} -10.2462 q^{71} -6.00000 q^{73} -1.00000 q^{76} -10.2462 q^{77} +0.876894 q^{79} -3.12311 q^{82} -15.1231 q^{83} -6.24621 q^{86} -2.00000 q^{88} +13.3693 q^{89} +20.4924 q^{91} -6.24621 q^{94} +6.00000 q^{97} -19.2462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{11} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{22} + 8 q^{26} - 2 q^{28} - 4 q^{29} + 10 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 2 q^{41} - 4 q^{43} + 4 q^{44} - 4 q^{47} + 22 q^{49} - 8 q^{52} - 8 q^{53} + 2 q^{56} + 4 q^{58} + 2 q^{59} + 8 q^{61} - 10 q^{62} + 2 q^{64} + 4 q^{67} + 2 q^{68} - 4 q^{71} - 12 q^{73} - 2 q^{76} - 4 q^{77} + 10 q^{79} + 2 q^{82} - 22 q^{83} + 4 q^{86} - 4 q^{88} + 2 q^{89} + 8 q^{91} + 4 q^{94} + 12 q^{97} - 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 4 * q^11 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^22 + 8 * q^26 - 2 * q^28 - 4 * q^29 + 10 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 2 * q^41 - 4 * q^43 + 4 * q^44 - 4 * q^47 + 22 * q^49 - 8 * q^52 - 8 * q^53 + 2 * q^56 + 4 * q^58 + 2 * q^59 + 8 * q^61 - 10 * q^62 + 2 * q^64 + 4 * q^67 + 2 * q^68 - 4 * q^71 - 12 * q^73 - 2 * q^76 - 4 * q^77 + 10 * q^79 + 2 * q^82 - 22 * q^83 + 4 * q^86 - 4 * q^88 + 2 * q^89 + 8 * q^91 + 4 * q^94 + 12 * q^97 - 22 * q^98

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