LMFDB - Embedded newform 1088.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,2,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1088.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:	$8.68772373992$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70928 q^{3} +4.34017 q^{5} -0.630898 q^{7} -0.0783777 q^{9} +O(q^{10})$	$q+1.70928 q^{3} +4.34017 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -5.70928 q^{11} +3.07838 q^{13} +7.41855 q^{15} +1.00000 q^{17} +3.41855 q^{19} -1.07838 q^{21} +5.89269 q^{23} +13.8371 q^{25} -5.26180 q^{27} -0.340173 q^{29} +6.78765 q^{31} -9.75872 q^{33} -2.73820 q^{35} -0.340173 q^{37} +5.26180 q^{39} +4.15676 q^{41} +5.26180 q^{43} -0.340173 q^{45} -8.68035 q^{47} -6.60197 q^{49} +1.70928 q^{51} -0.156755 q^{53} -24.7792 q^{55} +5.84324 q^{57} -12.0989 q^{59} -7.17727 q^{61} +0.0494483 q^{63} +13.3607 q^{65} -6.83710 q^{67} +10.0722 q^{69} +3.95055 q^{71} -4.83710 q^{73} +23.6514 q^{75} +3.60197 q^{77} -8.94441 q^{79} -8.75872 q^{81} -5.26180 q^{83} +4.34017 q^{85} -0.581449 q^{87} +9.60197 q^{89} -1.94214 q^{91} +11.6020 q^{93} +14.8371 q^{95} -12.8371 q^{97} +0.447480 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 10 q^{11} + 6 q^{13} + 8 q^{15} + 3 q^{17} - 4 q^{19} + 6 q^{23} + 13 q^{25} - 8 q^{27} + 10 q^{29} + 10 q^{31} - 4 q^{33} - 16 q^{35} + 10 q^{37} + 8 q^{39} + 6 q^{41} + 8 q^{43} + 10 q^{45} - 4 q^{47} - q^{49} - 2 q^{51} + 6 q^{53} - 16 q^{55} + 24 q^{57} + 18 q^{61} - 18 q^{63} - 4 q^{65} + 8 q^{67} - 8 q^{69} + 30 q^{71} + 14 q^{73} + 34 q^{75} - 8 q^{77} - 10 q^{79} - q^{81} - 8 q^{83} + 2 q^{85} - 16 q^{87} + 10 q^{89} + 24 q^{91} + 16 q^{93} + 16 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 10 * q^11 + 6 * q^13 + 8 * q^15 + 3 * q^17 - 4 * q^19 + 6 * q^23 + 13 * q^25 - 8 * q^27 + 10 * q^29 + 10 * q^31 - 4 * q^33 - 16 * q^35 + 10 * q^37 + 8 * q^39 + 6 * q^41 + 8 * q^43 + 10 * q^45 - 4 * q^47 - q^49 - 2 * q^51 + 6 * q^53 - 16 * q^55 + 24 * q^57 + 18 * q^61 - 18 * q^63 - 4 * q^65 + 8 * q^67 - 8 * q^69 + 30 * q^71 + 14 * q^73 + 34 * q^75 - 8 * q^77 - 10 * q^79 - q^81 - 8 * q^83 + 2 * q^85 - 16 * q^87 + 10 * q^89 + 24 * q^91 + 16 * q^93 + 16 * q^95 - 10 * q^97 + 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	0	0
$5$	4.34017	1.94098	0.970492	−	0.241133i	$-0.0775189\pi$
$5$	4.34017	1.94098	0.970492	+	0.241133i	$0.0775189\pi$
$6$	0	0
$7$	−0.630898	−0.238457	−0.119228	−	0.992867i	$-0.538042\pi$
$7$	−0.630898	−0.238457	−0.119228	+	0.992867i	$0.538042\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	−5.70928	−1.72141	−0.860706	−	0.509103i	$-0.829977\pi$
$11$	−5.70928	−1.72141	−0.860706	+	0.509103i	$0.829977\pi$
$12$	0	0
$13$	3.07838	0.853788	0.426894	−	0.904302i	$-0.359608\pi$
$13$	3.07838	0.853788	0.426894	+	0.904302i	$0.359608\pi$
$14$	0	0
$15$	7.41855	1.91546
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.41855	0.784269	0.392135	−	0.919908i	$-0.371737\pi$
$19$	3.41855	0.784269	0.392135	+	0.919908i	$0.371737\pi$
$20$	0	0
$21$	−1.07838	−0.235321
$22$	0	0
$23$	5.89269	1.22871	0.614356	−	0.789029i	$-0.289416\pi$
$23$	5.89269	1.22871	0.614356	+	0.789029i	$0.289416\pi$
$24$	0	0
$25$	13.8371	2.76742
$26$	0	0
$27$	−5.26180	−1.01263
$28$	0	0
$29$	−0.340173	−0.0631685	−0.0315843	−	0.999501i	$-0.510055\pi$
$29$	−0.340173	−0.0631685	−0.0315843	+	0.999501i	$0.510055\pi$
$30$	0	0
$31$	6.78765	1.21910	0.609549	−	0.792748i	$-0.291350\pi$
$31$	6.78765	1.21910	0.609549	+	0.792748i	$0.291350\pi$
$32$	0	0
$33$	−9.75872	−1.69878
$34$	0	0
$35$	−2.73820	−0.462841
$36$	0	0
$37$	−0.340173	−0.0559241	−0.0279620	−	0.999609i	$-0.508902\pi$
$37$	−0.340173	−0.0559241	−0.0279620	+	0.999609i	$0.508902\pi$
$38$	0	0
$39$	5.26180	0.842562
$40$	0	0
$41$	4.15676	0.649176	0.324588	−	0.945855i	$-0.394774\pi$
$41$	4.15676	0.649176	0.324588	+	0.945855i	$0.394774\pi$
$42$	0	0
$43$	5.26180	0.802416	0.401208	−	0.915987i	$-0.368590\pi$
$43$	5.26180	0.802416	0.401208	+	0.915987i	$0.368590\pi$
$44$	0	0
$45$	−0.340173	−0.0507100
$46$	0	0
$47$	−8.68035	−1.26616	−0.633079	−	0.774087i	$-0.718209\pi$
$47$	−8.68035	−1.26616	−0.633079	+	0.774087i	$0.718209\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	1.70928	0.239346
$52$	0	0
$53$	−0.156755	−0.0215320	−0.0107660	−	0.999942i	$-0.503427\pi$
$53$	−0.156755	−0.0215320	−0.0107660	+	0.999942i	$0.503427\pi$
$54$	0	0
$55$	−24.7792	−3.34123
$56$	0	0
$57$	5.84324	0.773957
$58$	0	0
$59$	−12.0989	−1.57514	−0.787571	−	0.616224i	$-0.788662\pi$
$59$	−12.0989	−1.57514	−0.787571	+	0.616224i	$0.788662\pi$
$60$	0	0
$61$	−7.17727	−0.918956	−0.459478	−	0.888189i	$-0.651963\pi$
$61$	−7.17727	−0.918956	−0.459478	+	0.888189i	$0.651963\pi$
$62$	0	0
$63$	0.0494483	0.00622990
$64$	0	0
$65$	13.3607	1.65719
$66$	0	0
$67$	−6.83710	−0.835285	−0.417642	−	0.908611i	$-0.637144\pi$
$67$	−6.83710	−0.835285	−0.417642	+	0.908611i	$0.637144\pi$
$68$	0	0
$69$	10.0722	1.21255
$70$	0	0
$71$	3.95055	0.468844	0.234422	−	0.972135i	$-0.424680\pi$
$71$	3.95055	0.468844	0.234422	+	0.972135i	$0.424680\pi$
$72$	0	0
$73$	−4.83710	−0.566140	−0.283070	−	0.959099i	$-0.591353\pi$
$73$	−4.83710	−0.566140	−0.283070	+	0.959099i	$0.591353\pi$
$74$	0	0
$75$	23.6514	2.73103
$76$	0	0
$77$	3.60197	0.410482
$78$	0	0
$79$	−8.94441	−1.00632	−0.503162	−	0.864192i	$-0.667830\pi$
$79$	−8.94441	−1.00632	−0.503162	+	0.864192i	$0.667830\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	−5.26180	−0.577557	−0.288779	−	0.957396i	$-0.593249\pi$
$83$	−5.26180	−0.577557	−0.288779	+	0.957396i	$0.593249\pi$
$84$	0	0
$85$	4.34017	0.470758
$86$	0	0
$87$	−0.581449	−0.0623379
$88$	0	0
$89$	9.60197	1.01781	0.508903	−	0.860824i	$-0.330051\pi$
$89$	9.60197	1.01781	0.508903	+	0.860824i	$0.330051\pi$
$90$	0	0
$91$	−1.94214	−0.203592
$92$	0	0
$93$	11.6020	1.20307
$94$	0	0
$95$	14.8371	1.52225
$96$	0	0
$97$	−12.8371	−1.30341	−0.651705	−	0.758472i	$-0.725946\pi$
$97$	−12.8371	−1.30341	−0.651705	+	0.758472i	$0.725946\pi$
$98$	0	0
$99$	0.447480	0.0449734
$100$	0	0
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−4.68035	−0.456755
$106$	0	0
$107$	−17.8082	−1.72158	−0.860790	−	0.508959i	$-0.830030\pi$
$107$	−17.8082	−1.72158	−0.860790	+	0.508959i	$0.830030\pi$
$108$	0	0
$109$	1.81658	0.173997	0.0869985	−	0.996208i	$-0.472272\pi$
$109$	1.81658	0.173997	0.0869985	+	0.996208i	$0.472272\pi$
$110$	0	0
$111$	−0.581449	−0.0551887
$112$	0	0
$113$	10.9939	1.03422	0.517108	−	0.855920i	$-0.327009\pi$
$113$	10.9939	1.03422	0.517108	+	0.855920i	$0.327009\pi$
$114$	0	0
$115$	25.5753	2.38491
$116$	0	0
$117$	−0.241276	−0.0223060
$118$	0	0
$119$	−0.630898	−0.0578343
$120$	0	0
$121$	21.5958	1.96326
$122$	0	0
$123$	7.10504	0.640640
$124$	0	0
$125$	38.3545	3.43054
$126$	0	0
$127$	−6.73820	−0.597919	−0.298959	−	0.954266i	$-0.596640\pi$
$127$	−6.73820	−0.597919	−0.298959	+	0.954266i	$0.596640\pi$
$128$	0	0
$129$	8.99386	0.791865
$130$	0	0
$131$	6.38962	0.558264	0.279132	−	0.960253i	$-0.409953\pi$
$131$	6.38962	0.558264	0.279132	+	0.960253i	$0.409953\pi$
$132$	0	0
$133$	−2.15676	−0.187014
$134$	0	0
$135$	−22.8371	−1.96550
$136$	0	0
$137$	−14.5958	−1.24701	−0.623503	−	0.781821i	$-0.714291\pi$
$137$	−14.5958	−1.24701	−0.623503	+	0.781821i	$0.714291\pi$
$138$	0	0
$139$	−6.60424	−0.560164	−0.280082	−	0.959976i	$-0.590362\pi$
$139$	−6.60424	−0.560164	−0.280082	+	0.959976i	$0.590362\pi$
$140$	0	0
$141$	−14.8371	−1.24951
$142$	0	0
$143$	−17.5753	−1.46972
$144$	0	0
$145$	−1.47641	−0.122609
$146$	0	0
$147$	−11.2846	−0.930737
$148$	0	0
$149$	−23.3607	−1.91378	−0.956891	−	0.290447i	$-0.906196\pi$
$149$	−23.3607	−1.91378	−0.956891	+	0.290447i	$0.906196\pi$
$150$	0	0
$151$	16.0989	1.31011	0.655055	−	0.755581i	$-0.272646\pi$
$151$	16.0989	1.31011	0.655055	+	0.755581i	$0.272646\pi$
$152$	0	0
$153$	−0.0783777	−0.00633647
$154$	0	0
$155$	29.4596	2.36625
$156$	0	0
$157$	12.5236	0.999491	0.499746	−	0.866172i	$-0.333427\pi$
$157$	12.5236	0.999491	0.499746	+	0.866172i	$0.333427\pi$
$158$	0	0
$159$	−0.267938	−0.0212489
$160$	0	0
$161$	−3.71769	−0.292995
$162$	0	0
$163$	−3.55252	−0.278255	−0.139127	−	0.990274i	$-0.544430\pi$
$163$	−3.55252	−0.278255	−0.139127	+	0.990274i	$0.544430\pi$
$164$	0	0
$165$	−42.3545	−3.29730
$166$	0	0
$167$	1.79380	0.138808	0.0694041	−	0.997589i	$-0.477890\pi$
$167$	1.79380	0.138808	0.0694041	+	0.997589i	$0.477890\pi$
$168$	0	0
$169$	−3.52359	−0.271045
$170$	0	0
$171$	−0.267938	−0.0204898
$172$	0	0
$173$	−15.1773	−1.15391	−0.576953	−	0.816777i	$-0.695759\pi$
$173$	−15.1773	−1.15391	−0.576953	+	0.816777i	$0.695759\pi$
$174$	0	0
$175$	−8.72979	−0.659910
$176$	0	0
$177$	−20.6803	−1.55443
$178$	0	0
$179$	10.2557	0.766543	0.383272	−	0.923636i	$-0.374797\pi$
$179$	10.2557	0.766543	0.383272	+	0.923636i	$0.374797\pi$
$180$	0	0
$181$	−15.1773	−1.12812	−0.564059	−	0.825735i	$-0.690761\pi$
$181$	−15.1773	−1.12812	−0.564059	+	0.825735i	$0.690761\pi$
$182$	0	0
$183$	−12.2679	−0.906872
$184$	0	0
$185$	−1.47641	−0.108548
$186$	0	0
$187$	−5.70928	−0.417504
$188$	0	0
$189$	3.31965	0.241469
$190$	0	0
$191$	16.6803	1.20695	0.603474	−	0.797383i	$-0.293783\pi$
$191$	16.6803	1.20695	0.603474	+	0.797383i	$0.293783\pi$
$192$	0	0
$193$	7.84324	0.564569	0.282285	−	0.959331i	$-0.408908\pi$
$193$	7.84324	0.564569	0.282285	+	0.959331i	$0.408908\pi$
$194$	0	0
$195$	22.8371	1.63540
$196$	0	0
$197$	−13.0205	−0.927674	−0.463837	−	0.885921i	$-0.653528\pi$
$197$	−13.0205	−0.927674	−0.463837	+	0.885921i	$0.653528\pi$
$198$	0	0
$199$	18.3051	1.29761	0.648807	−	0.760953i	$-0.275268\pi$
$199$	18.3051	1.29761	0.648807	+	0.760953i	$0.275268\pi$
$200$	0	0
$201$	−11.6865	−0.824301
$202$	0	0
$203$	0.214614	0.0150630
$204$	0	0
$205$	18.0410	1.26004
$206$	0	0
$207$	−0.461856	−0.0321012
$208$	0	0
$209$	−19.5174	−1.35005
$210$	0	0
$211$	−6.38962	−0.439880	−0.219940	−	0.975513i	$-0.570586\pi$
$211$	−6.38962	−0.439880	−0.219940	+	0.975513i	$0.570586\pi$
$212$	0	0
$213$	6.75258	0.462679
$214$	0	0
$215$	22.8371	1.55748
$216$	0	0
$217$	−4.28231	−0.290702
$218$	0	0
$219$	−8.26794	−0.558695
$220$	0	0
$221$	3.07838	0.207074
$222$	0	0
$223$	15.6163	1.04575	0.522874	−	0.852410i	$-0.324860\pi$
$223$	15.6163	1.04575	0.522874	+	0.852410i	$0.324860\pi$
$224$	0	0
$225$	−1.08452	−0.0723014
$226$	0	0
$227$	2.65756	0.176388	0.0881942	−	0.996103i	$-0.471890\pi$
$227$	2.65756	0.176388	0.0881942	+	0.996103i	$0.471890\pi$
$228$	0	0
$229$	5.60197	0.370188	0.185094	−	0.982721i	$-0.440741\pi$
$229$	5.60197	0.370188	0.185094	+	0.982721i	$0.440741\pi$
$230$	0	0
$231$	6.15676	0.405085
$232$	0	0
$233$	−21.2039	−1.38912	−0.694558	−	0.719437i	$-0.744400\pi$
$233$	−21.2039	−1.38912	−0.694558	+	0.719437i	$0.744400\pi$
$234$	0	0
$235$	−37.6742	−2.45759
$236$	0	0
$237$	−15.2885	−0.993092
$238$	0	0
$239$	22.1568	1.43320	0.716601	−	0.697484i	$-0.245697\pi$
$239$	22.1568	1.43320	0.716601	+	0.697484i	$0.245697\pi$
$240$	0	0
$241$	6.68035	0.430319	0.215159	−	0.976579i	$-0.430973\pi$
$241$	6.68035	0.430319	0.215159	+	0.976579i	$0.430973\pi$
$242$	0	0
$243$	0.814315	0.0522383
$244$	0	0
$245$	−28.6537	−1.83062
$246$	0	0
$247$	10.5236	0.669600
$248$	0	0
$249$	−8.99386	−0.569963
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−33.6430	−2.11512
$254$	0	0
$255$	7.41855	0.464568
$256$	0	0
$257$	4.92162	0.307002	0.153501	−	0.988148i	$-0.450945\pi$
$257$	4.92162	0.307002	0.153501	+	0.988148i	$0.450945\pi$
$258$	0	0
$259$	0.214614	0.0133355
$260$	0	0
$261$	0.0266620	0.00165034
$262$	0	0
$263$	−8.09890	−0.499399	−0.249700	−	0.968323i	$-0.580332\pi$
$263$	−8.09890	−0.499399	−0.249700	+	0.968323i	$0.580332\pi$
$264$	0	0
$265$	−0.680346	−0.0417933
$266$	0	0
$267$	16.4124	1.00442
$268$	0	0
$269$	20.5380	1.25222	0.626111	−	0.779734i	$-0.284646\pi$
$269$	20.5380	1.25222	0.626111	+	0.779734i	$0.284646\pi$
$270$	0	0
$271$	−7.51745	−0.456652	−0.228326	−	0.973585i	$-0.573325\pi$
$271$	−7.51745	−0.456652	−0.228326	+	0.973585i	$0.573325\pi$
$272$	0	0
$273$	−3.31965	−0.200915
$274$	0	0
$275$	−78.9998	−4.76387
$276$	0	0
$277$	25.6475	1.54101	0.770506	−	0.637433i	$-0.220004\pi$
$277$	25.6475	1.54101	0.770506	+	0.637433i	$0.220004\pi$
$278$	0	0
$279$	−0.532001	−0.0318501
$280$	0	0
$281$	−16.5236	−0.985715	−0.492857	−	0.870110i	$-0.664048\pi$
$281$	−16.5236	−0.985715	−0.492857	+	0.870110i	$0.664048\pi$
$282$	0	0
$283$	19.6514	1.16816	0.584078	−	0.811698i	$-0.301456\pi$
$283$	19.6514	1.16816	0.584078	+	0.811698i	$0.301456\pi$
$284$	0	0
$285$	25.3607	1.50224
$286$	0	0
$287$	−2.62249	−0.154801
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−21.9421	−1.28627
$292$	0	0
$293$	−9.68649	−0.565891	−0.282945	−	0.959136i	$-0.591312\pi$
$293$	−9.68649	−0.565891	−0.282945	+	0.959136i	$0.591312\pi$
$294$	0	0
$295$	−52.5113	−3.05733
$296$	0	0
$297$	30.0410	1.74316
$298$	0	0
$299$	18.1399	1.04906
$300$	0	0
$301$	−3.31965	−0.191342
$302$	0	0
$303$	−8.41241	−0.483280
$304$	0	0
$305$	−31.1506	−1.78368
$306$	0	0
$307$	−22.1568	−1.26455	−0.632276	−	0.774743i	$-0.717879\pi$
$307$	−22.1568	−1.26455	−0.632276	+	0.774743i	$0.717879\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.6309	0.943052	0.471526	−	0.881852i	$-0.343703\pi$
$311$	16.6309	0.943052	0.471526	+	0.881852i	$0.343703\pi$
$312$	0	0
$313$	19.7275	1.11507	0.557533	−	0.830155i	$-0.311748\pi$
$313$	19.7275	1.11507	0.557533	+	0.830155i	$0.311748\pi$
$314$	0	0
$315$	0.214614	0.0120921
$316$	0	0
$317$	20.3402	1.14242	0.571209	−	0.820805i	$-0.306475\pi$
$317$	20.3402	1.14242	0.571209	+	0.820805i	$0.306475\pi$
$318$	0	0
$319$	1.94214	0.108739
$320$	0	0
$321$	−30.4391	−1.69894
$322$	0	0
$323$	3.41855	0.190213
$324$	0	0
$325$	42.5958	2.36279
$326$	0	0
$327$	3.10504	0.171709
$328$	0	0
$329$	5.47641	0.301924
$330$	0	0
$331$	8.94828	0.491842	0.245921	−	0.969290i	$-0.420910\pi$
$331$	8.94828	0.491842	0.245921	+	0.969290i	$0.420910\pi$
$332$	0	0
$333$	0.0266620	0.00146107
$334$	0	0
$335$	−29.6742	−1.62127
$336$	0	0
$337$	12.5236	0.682203	0.341102	−	0.940026i	$-0.389200\pi$
$337$	12.5236	0.682203	0.341102	+	0.940026i	$0.389200\pi$
$338$	0	0
$339$	18.7915	1.02062
$340$	0	0
$341$	−38.7526	−2.09857
$342$	0	0
$343$	8.58145	0.463355
$344$	0	0
$345$	43.7152	2.35355
$346$	0	0
$347$	−27.7503	−1.48971	−0.744857	−	0.667224i	$-0.767483\pi$
$347$	−27.7503	−1.48971	−0.744857	+	0.667224i	$0.767483\pi$
$348$	0	0
$349$	−8.15676	−0.436621	−0.218311	−	0.975879i	$-0.570055\pi$
$349$	−8.15676	−0.436621	−0.218311	+	0.975879i	$0.570055\pi$
$350$	0	0
$351$	−16.1978	−0.864574
$352$	0	0
$353$	16.8371	0.896148	0.448074	−	0.893996i	$-0.352110\pi$
$353$	16.8371	0.896148	0.448074	+	0.893996i	$0.352110\pi$
$354$	0	0
$355$	17.1461	0.910019
$356$	0	0
$357$	−1.07838	−0.0570738
$358$	0	0
$359$	9.20847	0.486005	0.243002	−	0.970026i	$-0.421868\pi$
$359$	9.20847	0.486005	0.243002	+	0.970026i	$0.421868\pi$
$360$	0	0
$361$	−7.31351	−0.384922
$362$	0	0
$363$	36.9132	1.93744
$364$	0	0
$365$	−20.9939	−1.09887
$366$	0	0
$367$	20.0494	1.04657	0.523286	−	0.852157i	$-0.324706\pi$
$367$	20.0494	1.04657	0.523286	+	0.852157i	$0.324706\pi$
$368$	0	0
$369$	−0.325797	−0.0169603
$370$	0	0
$371$	0.0988967	0.00513446
$372$	0	0
$373$	−5.54864	−0.287298	−0.143649	−	0.989629i	$-0.545884\pi$
$373$	−5.54864	−0.287298	−0.143649	+	0.989629i	$0.545884\pi$
$374$	0	0
$375$	65.5585	3.38543
$376$	0	0
$377$	−1.04718	−0.0539326
$378$	0	0
$379$	1.70928	0.0877996	0.0438998	−	0.999036i	$-0.486022\pi$
$379$	1.70928	0.0877996	0.0438998	+	0.999036i	$0.486022\pi$
$380$	0	0
$381$	−11.5174	−0.590057
$382$	0	0
$383$	−10.4247	−0.532677	−0.266338	−	0.963880i	$-0.585814\pi$
$383$	−10.4247	−0.532677	−0.266338	+	0.963880i	$0.585814\pi$
$384$	0	0
$385$	15.6332	0.796740
$386$	0	0
$387$	−0.412408	−0.0209639
$388$	0	0
$389$	22.9627	1.16425	0.582127	−	0.813098i	$-0.302221\pi$
$389$	22.9627	1.16425	0.582127	+	0.813098i	$0.302221\pi$
$390$	0	0
$391$	5.89269	0.298006
$392$	0	0
$393$	10.9216	0.550923
$394$	0	0
$395$	−38.8203	−1.95326
$396$	0	0
$397$	−1.70086	−0.0853640	−0.0426820	−	0.999089i	$-0.513590\pi$
$397$	−1.70086	−0.0853640	−0.0426820	+	0.999089i	$0.513590\pi$
$398$	0	0
$399$	−3.68649	−0.184555
$400$	0	0
$401$	−22.9939	−1.14826	−0.574129	−	0.818765i	$-0.694659\pi$
$401$	−22.9939	−1.14826	−0.574129	+	0.818765i	$0.694659\pi$
$402$	0	0
$403$	20.8950	1.04085
$404$	0	0
$405$	−38.0144	−1.88895
$406$	0	0
$407$	1.94214	0.0962684
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−24.9483	−1.23061
$412$	0	0
$413$	7.63317	0.375603
$414$	0	0
$415$	−22.8371	−1.12103
$416$	0	0
$417$	−11.2885	−0.552798
$418$	0	0
$419$	−14.7565	−0.720900	−0.360450	−	0.932779i	$-0.617377\pi$
$419$	−14.7565	−0.720900	−0.360450	+	0.932779i	$0.617377\pi$
$420$	0	0
$421$	35.0784	1.70962	0.854808	−	0.518945i	$-0.173675\pi$
$421$	35.0784	1.70962	0.854808	+	0.518945i	$0.173675\pi$
$422$	0	0
$423$	0.680346	0.0330796
$424$	0	0
$425$	13.8371	0.671198
$426$	0	0
$427$	4.52813	0.219131
$428$	0	0
$429$	−30.0410	−1.45039
$430$	0	0
$431$	−26.3051	−1.26707	−0.633536	−	0.773713i	$-0.718397\pi$
$431$	−26.3051	−1.26707	−0.633536	+	0.773713i	$0.718397\pi$
$432$	0	0
$433$	−14.9627	−0.719060	−0.359530	−	0.933134i	$-0.617063\pi$
$433$	−14.9627	−0.719060	−0.359530	+	0.933134i	$0.617063\pi$
$434$	0	0
$435$	−2.52359	−0.120997
$436$	0	0
$437$	20.1445	0.963641
$438$	0	0
$439$	−17.0966	−0.815978	−0.407989	−	0.912987i	$-0.633770\pi$
$439$	−17.0966	−0.815978	−0.407989	+	0.912987i	$0.633770\pi$
$440$	0	0
$441$	0.517447	0.0246404
$442$	0	0
$443$	2.52359	0.119899	0.0599497	−	0.998201i	$-0.480906\pi$
$443$	2.52359	0.119899	0.0599497	+	0.998201i	$0.480906\pi$
$444$	0	0
$445$	41.6742	1.97555
$446$	0	0
$447$	−39.9299	−1.88862
$448$	0	0
$449$	−27.6742	−1.30603	−0.653013	−	0.757347i	$-0.726495\pi$
$449$	−27.6742	−1.30603	−0.653013	+	0.757347i	$0.726495\pi$
$450$	0	0
$451$	−23.7321	−1.11750
$452$	0	0
$453$	27.5174	1.29288
$454$	0	0
$455$	−8.42923	−0.395168
$456$	0	0
$457$	16.8059	0.786147	0.393074	−	0.919507i	$-0.371412\pi$
$457$	16.8059	0.786147	0.393074	+	0.919507i	$0.371412\pi$
$458$	0	0
$459$	−5.26180	−0.245600
$460$	0	0
$461$	−29.2039	−1.36016	−0.680081	−	0.733137i	$-0.738056\pi$
$461$	−29.2039	−1.36016	−0.680081	+	0.733137i	$0.738056\pi$
$462$	0	0
$463$	8.19779	0.380984	0.190492	−	0.981689i	$-0.438992\pi$
$463$	8.19779	0.380984	0.190492	+	0.981689i	$0.438992\pi$
$464$	0	0
$465$	50.3545	2.33514
$466$	0	0
$467$	30.5692	1.41457	0.707286	−	0.706927i	$-0.249919\pi$
$467$	30.5692	1.41457	0.707286	+	0.706927i	$0.249919\pi$
$468$	0	0
$469$	4.31351	0.199179
$470$	0	0
$471$	21.4063	0.986349
$472$	0	0
$473$	−30.0410	−1.38129
$474$	0	0
$475$	47.3028	2.17040
$476$	0	0
$477$	0.0122861	0.000562544 0
$478$	0	0
$479$	37.9916	1.73588	0.867940	−	0.496669i	$-0.165444\pi$
$479$	37.9916	1.73588	0.867940	+	0.496669i	$0.165444\pi$
$480$	0	0
$481$	−1.04718	−0.0477473
$482$	0	0
$483$	−6.35455	−0.289142
$484$	0	0
$485$	−55.7152	−2.52990
$486$	0	0
$487$	15.0023	0.679818	0.339909	−	0.940458i	$-0.389604\pi$
$487$	15.0023	0.679818	0.339909	+	0.940458i	$0.389604\pi$
$488$	0	0
$489$	−6.07223	−0.274596
$490$	0	0
$491$	0.465732	0.0210182	0.0105091	−	0.999945i	$-0.496655\pi$
$491$	0.465732	0.0210182	0.0105091	+	0.999945i	$0.496655\pi$
$492$	0	0
$493$	−0.340173	−0.0153206
$494$	0	0
$495$	1.94214	0.0872928
$496$	0	0
$497$	−2.49239	−0.111799
$498$	0	0
$499$	30.1217	1.34843	0.674216	−	0.738534i	$-0.264482\pi$
$499$	30.1217	1.34843	0.674216	+	0.738534i	$0.264482\pi$
$500$	0	0
$501$	3.06609	0.136983
$502$	0	0
$503$	−39.8348	−1.77615	−0.888074	−	0.459701i	$-0.847957\pi$
$503$	−39.8348	−1.77615	−0.888074	+	0.459701i	$0.847957\pi$
$504$	0	0
$505$	−21.3607	−0.950538
$506$	0	0
$507$	−6.02279	−0.267481
$508$	0	0
$509$	−19.6742	−0.872044	−0.436022	−	0.899936i	$-0.643613\pi$
$509$	−19.6742	−0.872044	−0.436022	+	0.899936i	$0.643613\pi$
$510$	0	0
$511$	3.05172	0.135000
$512$	0	0
$513$	−17.9877	−0.794177
$514$	0	0
$515$	0	0
$516$	0	0
$517$	49.5585	2.17958
$518$	0	0
$519$	−25.9421	−1.13873
$520$	0	0
$521$	20.8904	0.915226	0.457613	−	0.889151i	$-0.348704\pi$
$521$	20.8904	0.915226	0.457613	+	0.889151i	$0.348704\pi$
$522$	0	0
$523$	20.3135	0.888248	0.444124	−	0.895965i	$-0.353515\pi$
$523$	20.3135	0.888248	0.444124	+	0.895965i	$0.353515\pi$
$524$	0	0
$525$	−14.9216	−0.651233
$526$	0	0
$527$	6.78765	0.295675
$528$	0	0
$529$	11.7238	0.509732
$530$	0	0
$531$	0.948284	0.0411520
$532$	0	0
$533$	12.7961	0.554259
$534$	0	0
$535$	−77.2905	−3.34156
$536$	0	0
$537$	17.5297	0.756464
$538$	0	0
$539$	37.6925	1.62353
$540$	0	0
$541$	−5.18956	−0.223117	−0.111558	−	0.993758i	$-0.535584\pi$
$541$	−5.18956	−0.223117	−0.111558	+	0.993758i	$0.535584\pi$
$542$	0	0
$543$	−25.9421	−1.11328
$544$	0	0
$545$	7.88428	0.337726
$546$	0	0
$547$	29.2267	1.24964	0.624822	−	0.780767i	$-0.285171\pi$
$547$	29.2267	1.24964	0.624822	+	0.780767i	$0.285171\pi$
$548$	0	0
$549$	0.562539	0.0240086
$550$	0	0
$551$	−1.16290	−0.0495411
$552$	0	0
$553$	5.64301	0.239965
$554$	0	0
$555$	−2.52359	−0.107120
$556$	0	0
$557$	−13.5486	−0.574074	−0.287037	−	0.957919i	$-0.592670\pi$
$557$	−13.5486	−0.574074	−0.287037	+	0.957919i	$0.592670\pi$
$558$	0	0
$559$	16.1978	0.685094
$560$	0	0
$561$	−9.75872	−0.412014
$562$	0	0
$563$	46.0866	1.94232	0.971160	−	0.238431i	$-0.0766330\pi$
$563$	46.0866	1.94232	0.971160	+	0.238431i	$0.0766330\pi$
$564$	0	0
$565$	47.7152	2.00740
$566$	0	0
$567$	5.52586	0.232064
$568$	0	0
$569$	−43.8720	−1.83921	−0.919605	−	0.392845i	$-0.871491\pi$
$569$	−43.8720	−1.83921	−0.919605	+	0.392845i	$0.871491\pi$
$570$	0	0
$571$	−16.3486	−0.684167	−0.342083	−	0.939670i	$-0.611133\pi$
$571$	−16.3486	−0.684167	−0.342083	+	0.939670i	$0.611133\pi$
$572$	0	0
$573$	28.5113	1.19108
$574$	0	0
$575$	81.5378	3.40036
$576$	0	0
$577$	19.3919	0.807295	0.403647	−	0.914915i	$-0.367742\pi$
$577$	19.3919	0.807295	0.403647	+	0.914915i	$0.367742\pi$
$578$	0	0
$579$	13.4063	0.557145
$580$	0	0
$581$	3.31965	0.137722
$582$	0	0
$583$	0.894960	0.0370655
$584$	0	0
$585$	−1.04718	−0.0432956
$586$	0	0
$587$	20.5814	0.849487	0.424744	−	0.905314i	$-0.360364\pi$
$587$	20.5814	0.849487	0.424744	+	0.905314i	$0.360364\pi$
$588$	0	0
$589$	23.2039	0.956102
$590$	0	0
$591$	−22.2557	−0.915475
$592$	0	0
$593$	−4.21008	−0.172887	−0.0864436	−	0.996257i	$-0.527550\pi$
$593$	−4.21008	−0.172887	−0.0864436	+	0.996257i	$0.527550\pi$
$594$	0	0
$595$	−2.73820	−0.112255
$596$	0	0
$597$	31.2885	1.28055
$598$	0	0
$599$	6.21008	0.253737	0.126868	−	0.991920i	$-0.459507\pi$
$599$	6.21008	0.253737	0.126868	+	0.991920i	$0.459507\pi$
$600$	0	0
$601$	4.95282	0.202030	0.101015	−	0.994885i	$-0.467791\pi$
$601$	4.95282	0.202030	0.101015	+	0.994885i	$0.467791\pi$
$602$	0	0
$603$	0.535877	0.0218226
$604$	0	0
$605$	93.7296	3.81065
$606$	0	0
$607$	4.26406	0.173073	0.0865365	−	0.996249i	$-0.472420\pi$
$607$	4.26406	0.173073	0.0865365	+	0.996249i	$0.472420\pi$
$608$	0	0
$609$	0.366835	0.0148649
$610$	0	0
$611$	−26.7214	−1.08103
$612$	0	0
$613$	−16.7214	−0.675370	−0.337685	−	0.941259i	$-0.609644\pi$
$613$	−16.7214	−0.675370	−0.337685	+	0.941259i	$0.609644\pi$
$614$	0	0
$615$	30.8371	1.24347
$616$	0	0
$617$	4.15676	0.167345	0.0836723	−	0.996493i	$-0.473335\pi$
$617$	4.15676	0.167345	0.0836723	+	0.996493i	$0.473335\pi$
$618$	0	0
$619$	19.2846	0.775113	0.387556	−	0.921846i	$-0.373319\pi$
$619$	19.2846	0.775113	0.387556	+	0.921846i	$0.373319\pi$
$620$	0	0
$621$	−31.0061	−1.24423
$622$	0	0
$623$	−6.05786	−0.242703
$624$	0	0
$625$	97.2799	3.89119
$626$	0	0
$627$	−33.3607	−1.33230
$628$	0	0
$629$	−0.340173	−0.0135636
$630$	0	0
$631$	13.6286	0.542547	0.271274	−	0.962502i	$-0.412555\pi$
$631$	13.6286	0.542547	0.271274	+	0.962502i	$0.412555\pi$
$632$	0	0
$633$	−10.9216	−0.434096
$634$	0	0
$635$	−29.2450	−1.16055
$636$	0	0
$637$	−20.3234	−0.805241
$638$	0	0
$639$	−0.309635	−0.0122490
$640$	0	0
$641$	31.2450	1.23410	0.617051	−	0.786923i	$-0.288327\pi$
$641$	31.2450	1.23410	0.617051	+	0.786923i	$0.288327\pi$
$642$	0	0
$643$	37.0577	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$643$	37.0577	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$644$	0	0
$645$	39.0349	1.53700
$646$	0	0
$647$	−49.5052	−1.94625	−0.973124	−	0.230280i	$-0.926036\pi$
$647$	−49.5052	−1.94625	−0.973124	+	0.230280i	$0.926036\pi$
$648$	0	0
$649$	69.0759	2.71147
$650$	0	0
$651$	−7.31965	−0.286880
$652$	0	0
$653$	−1.50307	−0.0588197	−0.0294099	−	0.999567i	$-0.509363\pi$
$653$	−1.50307	−0.0588197	−0.0294099	+	0.999567i	$0.509363\pi$
$654$	0	0
$655$	27.7321	1.08358
$656$	0	0
$657$	0.379121	0.0147909
$658$	0	0
$659$	29.7275	1.15802	0.579010	−	0.815320i	$-0.303439\pi$
$659$	29.7275	1.15802	0.579010	+	0.815320i	$0.303439\pi$
$660$	0	0
$661$	−1.94668	−0.0757169	−0.0378585	−	0.999283i	$-0.512054\pi$
$661$	−1.94668	−0.0757169	−0.0378585	+	0.999283i	$0.512054\pi$
$662$	0	0
$663$	5.26180	0.204351
$664$	0	0
$665$	−9.36069	−0.362992
$666$	0	0
$667$	−2.00453	−0.0776159
$668$	0	0
$669$	26.6926	1.03200
$670$	0	0
$671$	40.9770	1.58190
$672$	0	0
$673$	8.63931	0.333021	0.166510	−	0.986040i	$-0.446750\pi$
$673$	8.63931	0.333021	0.166510	+	0.986040i	$0.446750\pi$
$674$	0	0
$675$	−72.8080	−2.80238
$676$	0	0
$677$	−37.2183	−1.43042	−0.715208	−	0.698912i	$-0.753668\pi$
$677$	−37.2183	−1.43042	−0.715208	+	0.698912i	$0.753668\pi$
$678$	0	0
$679$	8.09890	0.310807
$680$	0	0
$681$	4.54250	0.174069
$682$	0	0
$683$	−24.8143	−0.949493	−0.474747	−	0.880122i	$-0.657460\pi$
$683$	−24.8143	−0.949493	−0.474747	+	0.880122i	$0.657460\pi$
$684$	0	0
$685$	−63.3484	−2.42042
$686$	0	0
$687$	9.57531	0.365321
$688$	0	0
$689$	−0.482553	−0.0183838
$690$	0	0
$691$	−24.2329	−0.921862	−0.460931	−	0.887436i	$-0.652484\pi$
$691$	−24.2329	−0.921862	−0.460931	+	0.887436i	$0.652484\pi$
$692$	0	0
$693$	−0.282314	−0.0107242
$694$	0	0
$695$	−28.6635	−1.08727
$696$	0	0
$697$	4.15676	0.157448
$698$	0	0
$699$	−36.2434	−1.37085
$700$	0	0
$701$	−22.2823	−0.841591	−0.420796	−	0.907155i	$-0.638249\pi$
$701$	−22.2823	−0.841591	−0.420796	+	0.907155i	$0.638249\pi$
$702$	0	0
$703$	−1.16290	−0.0438595
$704$	0	0
$705$	−64.3956	−2.42528
$706$	0	0
$707$	3.10504	0.116777
$708$	0	0
$709$	−31.8043	−1.19444	−0.597218	−	0.802079i	$-0.703727\pi$
$709$	−31.8043	−1.19444	−0.597218	+	0.802079i	$0.703727\pi$
$710$	0	0
$711$	0.701043	0.0262912
$712$	0	0
$713$	39.9976	1.49792
$714$	0	0
$715$	−76.2799	−2.85271
$716$	0	0
$717$	37.8720	1.41436
$718$	0	0
$719$	47.2990	1.76395	0.881977	−	0.471293i	$-0.156213\pi$
$719$	47.2990	1.76395	0.881977	+	0.471293i	$0.156213\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.4186	0.424660
$724$	0	0
$725$	−4.70701	−0.174814
$726$	0	0
$727$	0.680346	0.0252326	0.0126163	−	0.999920i	$-0.495984\pi$
$727$	0.680346	0.0252326	0.0126163	+	0.999920i	$0.495984\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	5.26180	0.194615
$732$	0	0
$733$	24.0410	0.887976	0.443988	−	0.896033i	$-0.353563\pi$
$733$	24.0410	0.887976	0.443988	+	0.896033i	$0.353563\pi$
$734$	0	0
$735$	−48.9770	−1.80655
$736$	0	0
$737$	39.0349	1.43787
$738$	0	0
$739$	−31.1050	−1.14422	−0.572109	−	0.820178i	$-0.693874\pi$
$739$	−31.1050	−1.14422	−0.572109	+	0.820178i	$0.693874\pi$
$740$	0	0
$741$	17.9877	0.660795
$742$	0	0
$743$	−18.5730	−0.681379	−0.340689	−	0.940176i	$-0.610660\pi$
$743$	−18.5730	−0.681379	−0.340689	+	0.940176i	$0.610660\pi$
$744$	0	0
$745$	−101.389	−3.71462
$746$	0	0
$747$	0.412408	0.0150892
$748$	0	0
$749$	11.2351	0.410523
$750$	0	0
$751$	−15.9337	−0.581430	−0.290715	−	0.956810i	$-0.593893\pi$
$751$	−15.9337	−0.581430	−0.290715	+	0.956810i	$0.593893\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	69.8720	2.54290
$756$	0	0
$757$	−9.43293	−0.342846	−0.171423	−	0.985198i	$-0.554836\pi$
$757$	−9.43293	−0.342846	−0.171423	+	0.985198i	$0.554836\pi$
$758$	0	0
$759$	−57.5052	−2.08731
$760$	0	0
$761$	−42.2823	−1.53273	−0.766366	−	0.642404i	$-0.777937\pi$
$761$	−42.2823	−1.53273	−0.766366	+	0.642404i	$0.777937\pi$
$762$	0	0
$763$	−1.14608	−0.0414908
$764$	0	0
$765$	−0.340173	−0.0122990
$766$	0	0
$767$	−37.2450	−1.34484
$768$	0	0
$769$	−1.54864	−0.0558455	−0.0279228	−	0.999610i	$-0.508889\pi$
$769$	−1.54864	−0.0558455	−0.0279228	+	0.999610i	$0.508889\pi$
$770$	0	0
$771$	8.41241	0.302965
$772$	0	0
$773$	5.60197	0.201489	0.100744	−	0.994912i	$-0.467878\pi$
$773$	5.60197	0.201489	0.100744	+	0.994912i	$0.467878\pi$
$774$	0	0
$775$	93.9214	3.37376
$776$	0	0
$777$	0.366835	0.0131601
$778$	0	0
$779$	14.2101	0.509129
$780$	0	0
$781$	−22.5548	−0.807074
$782$	0	0
$783$	1.78992	0.0639665
$784$	0	0
$785$	54.3545	1.94000
$786$	0	0
$787$	−49.3256	−1.75827	−0.879134	−	0.476574i	$-0.841878\pi$
$787$	−49.3256	−1.75827	−0.879134	+	0.476574i	$0.841878\pi$
$788$	0	0
$789$	−13.8432	−0.492833
$790$	0	0
$791$	−6.93600	−0.246616
$792$	0	0
$793$	−22.0944	−0.784594
$794$	0	0
$795$	−1.16290	−0.0412438
$796$	0	0
$797$	−35.3074	−1.25065	−0.625326	−	0.780364i	$-0.715034\pi$
$797$	−35.3074	−1.25065	−0.625326	+	0.780364i	$0.715034\pi$
$798$	0	0
$799$	−8.68035	−0.307089
$800$	0	0
$801$	−0.752581	−0.0265911
$802$	0	0
$803$	27.6163	0.974560
$804$	0	0
$805$	−16.1354	−0.568698
$806$	0	0
$807$	35.1050	1.23576
$808$	0	0
$809$	45.1506	1.58741	0.793705	−	0.608302i	$-0.208149\pi$
$809$	45.1506	1.58741	0.793705	+	0.608302i	$0.208149\pi$
$810$	0	0
$811$	19.0166	0.667765	0.333882	−	0.942615i	$-0.391641\pi$
$811$	19.0166	0.667765	0.333882	+	0.942615i	$0.391641\pi$
$812$	0	0
$813$	−12.8494	−0.450648
$814$	0	0
$815$	−15.4186	−0.540088
$816$	0	0
$817$	17.9877	0.629310
$818$	0	0
$819$	0.152221	0.00531902
$820$	0	0
$821$	50.2122	1.75242	0.876208	−	0.481932i	$-0.160065\pi$
$821$	50.2122	1.75242	0.876208	+	0.481932i	$0.160065\pi$
$822$	0	0
$823$	−2.30510	−0.0803508	−0.0401754	−	0.999193i	$-0.512792\pi$
$823$	−2.30510	−0.0803508	−0.0401754	+	0.999193i	$0.512792\pi$
$824$	0	0
$825$	−135.032	−4.70123
$826$	0	0
$827$	−9.34244	−0.324868	−0.162434	−	0.986719i	$-0.551935\pi$
$827$	−9.34244	−0.324868	−0.162434	+	0.986719i	$0.551935\pi$
$828$	0	0
$829$	12.5236	0.434962	0.217481	−	0.976065i	$-0.430216\pi$
$829$	12.5236	0.434962	0.217481	+	0.976065i	$0.430216\pi$
$830$	0	0
$831$	43.8387	1.52075
$832$	0	0
$833$	−6.60197	−0.228745
$834$	0	0
$835$	7.78539	0.269424
$836$	0	0
$837$	−35.7152	−1.23450
$838$	0	0
$839$	23.5669	0.813620	0.406810	−	0.913513i	$-0.366641\pi$
$839$	23.5669	0.813620	0.406810	+	0.913513i	$0.366641\pi$
$840$	0	0
$841$	−28.8843	−0.996010
$842$	0	0
$843$	−28.2434	−0.972753
$844$	0	0
$845$	−15.2930	−0.526095
$846$	0	0
$847$	−13.6248	−0.468152
$848$	0	0
$849$	33.5897	1.15279
$850$	0	0
$851$	−2.00453	−0.0687146
$852$	0	0
$853$	−4.28685	−0.146779	−0.0733895	−	0.997303i	$-0.523382\pi$
$853$	−4.28685	−0.146779	−0.0733895	+	0.997303i	$0.523382\pi$
$854$	0	0
$855$	−1.16290	−0.0397703
$856$	0	0
$857$	14.6803	0.501471	0.250736	−	0.968056i	$-0.419328\pi$
$857$	14.6803	0.501471	0.250736	+	0.968056i	$0.419328\pi$
$858$	0	0
$859$	−49.1461	−1.67684	−0.838421	−	0.545023i	$-0.816521\pi$
$859$	−49.1461	−1.67684	−0.838421	+	0.545023i	$0.816521\pi$
$860$	0	0
$861$	−4.48255	−0.152765
$862$	0	0
$863$	−27.8843	−0.949192	−0.474596	−	0.880204i	$-0.657406\pi$
$863$	−27.8843	−0.949192	−0.474596	+	0.880204i	$0.657406\pi$
$864$	0	0
$865$	−65.8720	−2.23972
$866$	0	0
$867$	1.70928	0.0580500
$868$	0	0
$869$	51.0661	1.73230
$870$	0	0
$871$	−21.0472	−0.713157
$872$	0	0
$873$	1.00614	0.0340528
$874$	0	0
$875$	−24.1978	−0.818035
$876$	0	0
$877$	−9.86991	−0.333283	−0.166642	−	0.986018i	$-0.553292\pi$
$877$	−9.86991	−0.333283	−0.166642	+	0.986018i	$0.553292\pi$
$878$	0	0
$879$	−16.5569	−0.558450
$880$	0	0
$881$	−46.8248	−1.57757	−0.788784	−	0.614670i	$-0.789289\pi$
$881$	−46.8248	−1.57757	−0.788784	+	0.614670i	$0.789289\pi$
$882$	0	0
$883$	−4.84939	−0.163195	−0.0815974	−	0.996665i	$-0.526002\pi$
$883$	−4.84939	−0.163195	−0.0815974	+	0.996665i	$0.526002\pi$
$884$	0	0
$885$	−89.7563	−3.01712
$886$	0	0
$887$	42.5029	1.42711	0.713554	−	0.700600i	$-0.247084\pi$
$887$	42.5029	1.42711	0.713554	+	0.700600i	$0.247084\pi$
$888$	0	0
$889$	4.25112	0.142578
$890$	0	0
$891$	50.0060	1.67526
$892$	0	0
$893$	−29.6742	−0.993009
$894$	0	0
$895$	44.5113	1.48785
$896$	0	0
$897$	31.0061	1.03526
$898$	0	0
$899$	−2.30898	−0.0770087
$900$	0	0
$901$	−0.156755	−0.00522228
$902$	0	0
$903$	−5.67420	−0.188826
$904$	0	0
$905$	−65.8720	−2.18966
$906$	0	0
$907$	−36.0105	−1.19571	−0.597855	−	0.801605i	$-0.703980\pi$
$907$	−36.0105	−1.19571	−0.597855	+	0.801605i	$0.703980\pi$
$908$	0	0
$909$	0.385746	0.0127944
$910$	0	0
$911$	−43.7815	−1.45055	−0.725273	−	0.688461i	$-0.758286\pi$
$911$	−43.7815	−1.45055	−0.725273	+	0.688461i	$0.758286\pi$
$912$	0	0
$913$	30.0410	0.994213
$914$	0	0
$915$	−53.2450	−1.76022
$916$	0	0
$917$	−4.03120	−0.133122
$918$	0	0
$919$	−13.1917	−0.435152	−0.217576	−	0.976043i	$-0.569815\pi$
$919$	−13.1917	−0.435152	−0.217576	+	0.976043i	$0.569815\pi$
$920$	0	0
$921$	−37.8720	−1.24792
$922$	0	0
$923$	12.1613	0.400294
$924$	0	0
$925$	−4.70701	−0.154765
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.68035	0.219175	0.109588	−	0.993977i	$-0.465047\pi$
$929$	6.68035	0.219175	0.109588	+	0.993977i	$0.465047\pi$
$930$	0	0
$931$	−22.5692	−0.739674
$932$	0	0
$933$	28.4268	0.930651
$934$	0	0
$935$	−24.7792	−0.810368
$936$	0	0
$937$	−26.3135	−0.859625	−0.429812	−	0.902918i	$-0.641420\pi$
$937$	−26.3135	−0.859625	−0.429812	+	0.902918i	$0.641420\pi$
$938$	0	0
$939$	33.7198	1.10040
$940$	0	0
$941$	−5.44975	−0.177657	−0.0888283	−	0.996047i	$-0.528312\pi$
$941$	−5.44975	−0.177657	−0.0888283	+	0.996047i	$0.528312\pi$
$942$	0	0
$943$	24.4945	0.797650
$944$	0	0
$945$	14.4079	0.468688
$946$	0	0
$947$	−28.2784	−0.918926	−0.459463	−	0.888197i	$-0.651958\pi$
$947$	−28.2784	−0.918926	−0.459463	+	0.888197i	$0.651958\pi$
$948$	0	0
$949$	−14.8904	−0.483364
$950$	0	0
$951$	34.7670	1.12740
$952$	0	0
$953$	−2.71154	−0.0878355	−0.0439177	−	0.999035i	$-0.513984\pi$
$953$	−2.71154	−0.0878355	−0.0439177	+	0.999035i	$0.513984\pi$
$954$	0	0
$955$	72.3956	2.34267
$956$	0	0
$957$	3.31965	0.107309
$958$	0	0
$959$	9.20847	0.297357
$960$	0	0
$961$	15.0722	0.486201
$962$	0	0
$963$	1.39576	0.0449779
$964$	0	0
$965$	34.0410	1.09582
$966$	0	0
$967$	48.9770	1.57500	0.787498	−	0.616318i	$-0.211376\pi$
$967$	48.9770	1.57500	0.787498	+	0.616318i	$0.211376\pi$
$968$	0	0
$969$	5.84324	0.187712
$970$	0	0
$971$	3.47187	0.111418	0.0557089	−	0.998447i	$-0.482258\pi$
$971$	3.47187	0.111418	0.0557089	+	0.998447i	$0.482258\pi$
$972$	0	0
$973$	4.16660	0.133575
$974$	0	0
$975$	72.8080	2.33172
$976$	0	0
$977$	18.8248	0.602259	0.301130	−	0.953583i	$-0.402636\pi$
$977$	18.8248	0.602259	0.301130	+	0.953583i	$0.402636\pi$
$978$	0	0
$979$	−54.8203	−1.75206
$980$	0	0
$981$	−0.142380	−0.00454583
$982$	0	0
$983$	−11.3158	−0.360917	−0.180459	−	0.983583i	$-0.557758\pi$
$983$	−11.3158	−0.360917	−0.180459	+	0.983583i	$0.557758\pi$
$984$	0	0
$985$	−56.5113	−1.80060
$986$	0	0
$987$	9.36069	0.297954
$988$	0	0
$989$	31.0061	0.985938
$990$	0	0
$991$	40.9977	1.30234	0.651168	−	0.758934i	$-0.274279\pi$
$991$	40.9977	1.30234	0.651168	+	0.758934i	$0.274279\pi$
$992$	0	0
$993$	15.2951	0.485375
$994$	0	0
$995$	79.4473	2.51865
$996$	0	0
$997$	48.6537	1.54088	0.770439	−	0.637514i	$-0.220037\pi$
$997$	48.6537	1.54088	0.770439	+	0.637514i	$0.220037\pi$
$998$	0	0
$999$	1.78992	0.0566306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.2.a.u.1.3		3
3.2	odd	2		9792.2.a.dd.1.1		3
4.3	odd	2		1088.2.a.v.1.1		3
8.3	odd	2		544.2.a.i.1.3	✓	3
8.5	even	2		544.2.a.j.1.1	yes	3
12.11	even	2		9792.2.a.dc.1.1		3
24.5	odd	2		4896.2.a.bf.1.3		3
24.11	even	2		4896.2.a.be.1.3		3
136.67	odd	2		9248.2.a.u.1.1		3
136.101	even	2		9248.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.a.i.1.3	✓	3	8.3	odd	2
544.2.a.j.1.1	yes	3	8.5	even	2
1088.2.a.u.1.3		3	1.1	even	1	trivial
1088.2.a.v.1.1		3	4.3	odd	2
4896.2.a.be.1.3		3	24.11	even	2
4896.2.a.bf.1.3		3	24.5	odd	2
9248.2.a.t.1.3		3	136.101	even	2
9248.2.a.u.1.1		3	136.67	odd	2
9792.2.a.dc.1.1		3	12.11	even	2
9792.2.a.dd.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70928 q^{3} +4.34017 q^{5} -0.630898 q^{7} -0.0783777 q^{9} +O(q^{10})\)	\(q+1.70928 q^{3} +4.34017 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -5.70928 q^{11} +3.07838 q^{13} +7.41855 q^{15} +1.00000 q^{17} +3.41855 q^{19} -1.07838 q^{21} +5.89269 q^{23} +13.8371 q^{25} -5.26180 q^{27} -0.340173 q^{29} +6.78765 q^{31} -9.75872 q^{33} -2.73820 q^{35} -0.340173 q^{37} +5.26180 q^{39} +4.15676 q^{41} +5.26180 q^{43} -0.340173 q^{45} -8.68035 q^{47} -6.60197 q^{49} +1.70928 q^{51} -0.156755 q^{53} -24.7792 q^{55} +5.84324 q^{57} -12.0989 q^{59} -7.17727 q^{61} +0.0494483 q^{63} +13.3607 q^{65} -6.83710 q^{67} +10.0722 q^{69} +3.95055 q^{71} -4.83710 q^{73} +23.6514 q^{75} +3.60197 q^{77} -8.94441 q^{79} -8.75872 q^{81} -5.26180 q^{83} +4.34017 q^{85} -0.581449 q^{87} +9.60197 q^{89} -1.94214 q^{91} +11.6020 q^{93} +14.8371 q^{95} -12.8371 q^{97} +0.447480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 10 q^{11} + 6 q^{13} + 8 q^{15} + 3 q^{17} - 4 q^{19} + 6 q^{23} + 13 q^{25} - 8 q^{27} + 10 q^{29} + 10 q^{31} - 4 q^{33} - 16 q^{35} + 10 q^{37} + 8 q^{39} + 6 q^{41} + 8 q^{43} + 10 q^{45} - 4 q^{47} - q^{49} - 2 q^{51} + 6 q^{53} - 16 q^{55} + 24 q^{57} + 18 q^{61} - 18 q^{63} - 4 q^{65} + 8 q^{67} - 8 q^{69} + 30 q^{71} + 14 q^{73} + 34 q^{75} - 8 q^{77} - 10 q^{79} - q^{81} - 8 q^{83} + 2 q^{85} - 16 q^{87} + 10 q^{89} + 24 q^{91} + 16 q^{93} + 16 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 10 * q^11 + 6 * q^13 + 8 * q^15 + 3 * q^17 - 4 * q^19 + 6 * q^23 + 13 * q^25 - 8 * q^27 + 10 * q^29 + 10 * q^31 - 4 * q^33 - 16 * q^35 + 10 * q^37 + 8 * q^39 + 6 * q^41 + 8 * q^43 + 10 * q^45 - 4 * q^47 - q^49 - 2 * q^51 + 6 * q^53 - 16 * q^55 + 24 * q^57 + 18 * q^61 - 18 * q^63 - 4 * q^65 + 8 * q^67 - 8 * q^69 + 30 * q^71 + 14 * q^73 + 34 * q^75 - 8 * q^77 - 10 * q^79 - q^81 - 8 * q^83 + 2 * q^85 - 16 * q^87 + 10 * q^89 + 24 * q^91 + 16 * q^93 + 16 * q^95 - 10 * q^97 + 2 * q^99

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