LMFDB - Embedded newform 1098.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1098,2,Mod(1,1098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1098, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1098 = 2 \cdot 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1098.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:	$8.76757414194$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 122)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	1098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.39821 q^{5} -3.18421 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.39821 q^{5} -3.18421 q^{7} -1.00000 q^{8} +1.39821 q^{10} -0.323404 q^{11} +2.32340 q^{13} +3.18421 q^{14} +1.00000 q^{16} -1.72161 q^{17} -2.86081 q^{19} -1.39821 q^{20} +0.323404 q^{22} +6.50761 q^{23} -3.04502 q^{25} -2.32340 q^{26} -3.18421 q^{28} -0.0643910 q^{29} +5.10941 q^{31} -1.00000 q^{32} +1.72161 q^{34} +4.45219 q^{35} +8.98062 q^{37} +2.86081 q^{38} +1.39821 q^{40} +2.65722 q^{41} +11.4432 q^{43} -0.323404 q^{44} -6.50761 q^{46} +4.79641 q^{47} +3.13919 q^{49} +3.04502 q^{50} +2.32340 q^{52} -12.9598 q^{53} +0.452186 q^{55} +3.18421 q^{56} +0.0643910 q^{58} +6.60179 q^{59} -1.00000 q^{61} -5.10941 q^{62} +1.00000 q^{64} -3.24860 q^{65} -4.69182 q^{67} -1.72161 q^{68} -4.45219 q^{70} -3.41758 q^{71} +13.9806 q^{73} -8.98062 q^{74} -2.86081 q^{76} +1.02979 q^{77} +4.19462 q^{79} -1.39821 q^{80} -2.65722 q^{82} +16.1544 q^{83} +2.40717 q^{85} -11.4432 q^{86} +0.323404 q^{88} -2.51803 q^{89} -7.39821 q^{91} +6.50761 q^{92} -4.79641 q^{94} +4.00000 q^{95} -6.18421 q^{97} -3.13919 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - q^5 + 4 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8} + q^{10} + 7 q^{11} - q^{13} - 4 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{19} - q^{20} - 7 q^{22} - 2 q^{23} + 10 q^{25} + q^{26} + 4 q^{28} - q^{29} - 3 q^{31} - 3 q^{32} - 6 q^{34} + 7 q^{35} + 7 q^{37} + 3 q^{38} + q^{40} - 4 q^{41} + 12 q^{43} + 7 q^{44} + 2 q^{46} + 8 q^{47} + 15 q^{49} - 10 q^{50} - q^{52} - 11 q^{53} - 5 q^{55} - 4 q^{56} + q^{58} + 23 q^{59} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 3 q^{65} + 21 q^{67} + 6 q^{68} - 7 q^{70} - 27 q^{71} + 22 q^{73} - 7 q^{74} - 3 q^{76} + 27 q^{77} + 3 q^{79} - q^{80} + 4 q^{82} + 11 q^{83} + 20 q^{85} - 12 q^{86} - 7 q^{88} + 10 q^{89} - 19 q^{91} - 2 q^{92} - 8 q^{94} + 12 q^{95} - 5 q^{97} - 15 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - q^5 + 4 * q^7 - 3 * q^8 + q^10 + 7 * q^11 - q^13 - 4 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^19 - q^20 - 7 * q^22 - 2 * q^23 + 10 * q^25 + q^26 + 4 * q^28 - q^29 - 3 * q^31 - 3 * q^32 - 6 * q^34 + 7 * q^35 + 7 * q^37 + 3 * q^38 + q^40 - 4 * q^41 + 12 * q^43 + 7 * q^44 + 2 * q^46 + 8 * q^47 + 15 * q^49 - 10 * q^50 - q^52 - 11 * q^53 - 5 * q^55 - 4 * q^56 + q^58 + 23 * q^59 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 3 * q^65 + 21 * q^67 + 6 * q^68 - 7 * q^70 - 27 * q^71 + 22 * q^73 - 7 * q^74 - 3 * q^76 + 27 * q^77 + 3 * q^79 - q^80 + 4 * q^82 + 11 * q^83 + 20 * q^85 - 12 * q^86 - 7 * q^88 + 10 * q^89 - 19 * q^91 - 2 * q^92 - 8 * q^94 + 12 * q^95 - 5 * q^97 - 15 * 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
$5$	−1.39821	−0.625297	−0.312649	−	0.949869i	$-0.601216\pi$
$5$	−1.39821	−0.625297	−0.312649	+	0.949869i	$0.601216\pi$
$6$	0	0
$7$	−3.18421	−1.20352	−0.601759	−	0.798678i	$-0.705533\pi$
$7$	−3.18421	−1.20352	−0.601759	+	0.798678i	$0.705533\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.39821	0.442152
$11$	−0.323404	−0.0975101	−0.0487550	−	0.998811i	$-0.515525\pi$
$11$	−0.323404	−0.0975101	−0.0487550	+	0.998811i	$0.515525\pi$
$12$	0	0
$13$	2.32340	0.644396	0.322198	−	0.946672i	$-0.395578\pi$
$13$	2.32340	0.644396	0.322198	+	0.946672i	$0.395578\pi$
$14$	3.18421	0.851016
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.72161	−0.417552	−0.208776	−	0.977963i	$-0.566948\pi$
$17$	−1.72161	−0.417552	−0.208776	+	0.977963i	$0.566948\pi$
$18$	0	0
$19$	−2.86081	−0.656314	−0.328157	−	0.944623i	$-0.606427\pi$
$19$	−2.86081	−0.656314	−0.328157	+	0.944623i	$0.606427\pi$
$20$	−1.39821	−0.312649
$21$	0	0
$22$	0.323404	0.0689500
$23$	6.50761	1.35693	0.678466	−	0.734632i	$-0.262645\pi$
$23$	6.50761	1.35693	0.678466	+	0.734632i	$0.262645\pi$
$24$	0	0
$25$	−3.04502	−0.609003
$26$	−2.32340	−0.455657
$27$	0	0
$28$	−3.18421	−0.601759
$29$	−0.0643910	−0.0119571	−0.00597855	−	0.999982i	$-0.501903\pi$
$29$	−0.0643910	−0.0119571	−0.00597855	+	0.999982i	$0.501903\pi$
$30$	0	0
$31$	5.10941	0.917677	0.458838	−	0.888520i	$-0.348266\pi$
$31$	5.10941	0.917677	0.458838	+	0.888520i	$0.348266\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.72161	0.295254
$35$	4.45219	0.752557
$36$	0	0
$37$	8.98062	1.47641	0.738203	−	0.674579i	$-0.235675\pi$
$37$	8.98062	1.47641	0.738203	+	0.674579i	$0.235675\pi$
$38$	2.86081	0.464084
$39$	0	0
$40$	1.39821	0.221076
$41$	2.65722	0.414988	0.207494	−	0.978236i	$-0.433469\pi$
$41$	2.65722	0.414988	0.207494	+	0.978236i	$0.433469\pi$
$42$	0	0
$43$	11.4432	1.74508	0.872538	−	0.488547i	$-0.162473\pi$
$43$	11.4432	1.74508	0.872538	+	0.488547i	$0.162473\pi$
$44$	−0.323404	−0.0487550
$45$	0	0
$46$	−6.50761	−0.959495
$47$	4.79641	0.699629	0.349815	−	0.936819i	$-0.386245\pi$
$47$	4.79641	0.699629	0.349815	+	0.936819i	$0.386245\pi$
$48$	0	0
$49$	3.13919	0.448456
$50$	3.04502	0.430630
$51$	0	0
$52$	2.32340	0.322198
$53$	−12.9598	−1.78017	−0.890083	−	0.455799i	$-0.849354\pi$
$53$	−12.9598	−1.78017	−0.890083	+	0.455799i	$0.849354\pi$
$54$	0	0
$55$	0.452186	0.0609728
$56$	3.18421	0.425508
$57$	0	0
$58$	0.0643910	0.00845495
$59$	6.60179	0.859480	0.429740	−	0.902953i	$-0.358605\pi$
$59$	6.60179	0.859480	0.429740	+	0.902953i	$0.358605\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−5.10941	−0.648895
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.24860	−0.402939
$66$	0	0
$67$	−4.69182	−0.573198	−0.286599	−	0.958051i	$-0.592525\pi$
$67$	−4.69182	−0.573198	−0.286599	+	0.958051i	$0.592525\pi$
$68$	−1.72161	−0.208776
$69$	0	0
$70$	−4.45219	−0.532138
$71$	−3.41758	−0.405592	−0.202796	−	0.979221i	$-0.565003\pi$
$71$	−3.41758	−0.405592	−0.202796	+	0.979221i	$0.565003\pi$
$72$	0	0
$73$	13.9806	1.63631	0.818154	−	0.574999i	$-0.194998\pi$
$73$	13.9806	1.63631	0.818154	+	0.574999i	$0.194998\pi$
$74$	−8.98062	−1.04398
$75$	0	0
$76$	−2.86081	−0.328157
$77$	1.02979	0.117355
$78$	0	0
$79$	4.19462	0.471932	0.235966	−	0.971761i	$-0.424175\pi$
$79$	4.19462	0.471932	0.235966	+	0.971761i	$0.424175\pi$
$80$	−1.39821	−0.156324
$81$	0	0
$82$	−2.65722	−0.293441
$83$	16.1544	1.77318	0.886589	−	0.462558i	$-0.153068\pi$
$83$	16.1544	1.77318	0.886589	+	0.462558i	$0.153068\pi$
$84$	0	0
$85$	2.40717	0.261094
$86$	−11.4432	−1.23395
$87$	0	0
$88$	0.323404	0.0344750
$89$	−2.51803	−0.266910	−0.133455	−	0.991055i	$-0.542607\pi$
$89$	−2.51803	−0.266910	−0.133455	+	0.991055i	$0.542607\pi$
$90$	0	0
$91$	−7.39821	−0.775543
$92$	6.50761	0.678466
$93$	0	0
$94$	−4.79641	−0.494712
$95$	4.00000	0.410391
$96$	0	0
$97$	−6.18421	−0.627911	−0.313956	−	0.949438i	$-0.601654\pi$
$97$	−6.18421	−0.627911	−0.313956	+	0.949438i	$0.601654\pi$
$98$	−3.13919	−0.317106
$99$	0	0
$100$	−3.04502	−0.304502
$101$	4.31299	0.429159	0.214579	−	0.976707i	$-0.431162\pi$
$101$	4.31299	0.429159	0.214579	+	0.976707i	$0.431162\pi$
$102$	0	0
$103$	11.5720	1.14022	0.570112	−	0.821567i	$-0.306900\pi$
$103$	11.5720	1.14022	0.570112	+	0.821567i	$0.306900\pi$
$104$	−2.32340	−0.227829
$105$	0	0
$106$	12.9598	1.25877
$107$	2.34278	0.226485	0.113243	−	0.993567i	$-0.463876\pi$
$107$	2.34278	0.226485	0.113243	+	0.993567i	$0.463876\pi$
$108$	0	0
$109$	−2.32340	−0.222542	−0.111271	−	0.993790i	$-0.535492\pi$
$109$	−2.32340	−0.222542	−0.111271	+	0.993790i	$0.535492\pi$
$110$	−0.452186	−0.0431143
$111$	0	0
$112$	−3.18421	−0.300880
$113$	4.90582	0.461501	0.230750	−	0.973013i	$-0.425882\pi$
$113$	4.90582	0.461501	0.230750	+	0.973013i	$0.425882\pi$
$114$	0	0
$115$	−9.09899	−0.848486
$116$	−0.0643910	−0.00597855
$117$	0	0
$118$	−6.60179	−0.607744
$119$	5.48197	0.502532
$120$	0	0
$121$	−10.8954	−0.990492
$122$	1.00000	0.0905357
$123$	0	0
$124$	5.10941	0.458838
$125$	11.2486	1.00611
$126$	0	0
$127$	1.87122	0.166044	0.0830219	−	0.996548i	$-0.473543\pi$
$127$	1.87122	0.166044	0.0830219	+	0.996548i	$0.473543\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.24860	0.284921
$131$	6.14961	0.537294	0.268647	−	0.963239i	$-0.413424\pi$
$131$	6.14961	0.537294	0.268647	+	0.963239i	$0.413424\pi$
$132$	0	0
$133$	9.10941	0.789886
$134$	4.69182	0.405312
$135$	0	0
$136$	1.72161	0.147627
$137$	16.5076	1.41034	0.705170	−	0.709038i	$-0.250871\pi$
$137$	16.5076	1.41034	0.705170	+	0.709038i	$0.250871\pi$
$138$	0	0
$139$	−10.9702	−0.930481	−0.465241	−	0.885184i	$-0.654032\pi$
$139$	−10.9702	−0.930481	−0.465241	+	0.885184i	$0.654032\pi$
$140$	4.45219	0.376278
$141$	0	0
$142$	3.41758	0.286797
$143$	−0.751399	−0.0628351
$144$	0	0
$145$	0.0900320	0.00747675
$146$	−13.9806	−1.15704
$147$	0	0
$148$	8.98062	0.738203
$149$	21.2099	1.73758	0.868789	−	0.495182i	$-0.164899\pi$
$149$	21.2099	1.73758	0.868789	+	0.495182i	$0.164899\pi$
$150$	0	0
$151$	21.0048	1.70935	0.854674	−	0.519165i	$-0.173757\pi$
$151$	21.0048	1.70935	0.854674	+	0.519165i	$0.173757\pi$
$152$	2.86081	0.232042
$153$	0	0
$154$	−1.02979	−0.0829826
$155$	−7.14401	−0.573821
$156$	0	0
$157$	−5.13919	−0.410152	−0.205076	−	0.978746i	$-0.565744\pi$
$157$	−5.13919	−0.410152	−0.205076	+	0.978746i	$0.565744\pi$
$158$	−4.19462	−0.333706
$159$	0	0
$160$	1.39821	0.110538
$161$	−20.7216	−1.63309
$162$	0	0
$163$	10.9806	0.860069	0.430034	−	0.902812i	$-0.358501\pi$
$163$	10.9806	0.860069	0.430034	+	0.902812i	$0.358501\pi$
$164$	2.65722	0.207494
$165$	0	0
$166$	−16.1544	−1.25383
$167$	−24.6081	−1.90423	−0.952114	−	0.305742i	$-0.901095\pi$
$167$	−24.6081	−1.90423	−0.952114	+	0.305742i	$0.901095\pi$
$168$	0	0
$169$	−7.60179	−0.584753
$170$	−2.40717	−0.184622
$171$	0	0
$172$	11.4432	0.872538
$173$	−1.65722	−0.125996	−0.0629981	−	0.998014i	$-0.520066\pi$
$173$	−1.65722	−0.125996	−0.0629981	+	0.998014i	$0.520066\pi$
$174$	0	0
$175$	9.69597	0.732946
$176$	−0.323404	−0.0243775
$177$	0	0
$178$	2.51803	0.188734
$179$	−5.75622	−0.430240	−0.215120	−	0.976588i	$-0.569014\pi$
$179$	−5.75622	−0.430240	−0.215120	+	0.976588i	$0.569014\pi$
$180$	0	0
$181$	−25.8760	−1.92335	−0.961675	−	0.274191i	$-0.911590\pi$
$181$	−25.8760	−1.92335	−0.961675	+	0.274191i	$0.911590\pi$
$182$	7.39821	0.548392
$183$	0	0
$184$	−6.50761	−0.479748
$185$	−12.5568	−0.923193
$186$	0	0
$187$	0.556777	0.0407155
$188$	4.79641	0.349815
$189$	0	0
$190$	−4.00000	−0.290191
$191$	2.13919	0.154787	0.0773933	−	0.997001i	$-0.475340\pi$
$191$	2.13919	0.154787	0.0773933	+	0.997001i	$0.475340\pi$
$192$	0	0
$193$	25.4224	1.82994	0.914972	−	0.403517i	$-0.132212\pi$
$193$	25.4224	1.82994	0.914972	+	0.403517i	$0.132212\pi$
$194$	6.18421	0.444000
$195$	0	0
$196$	3.13919	0.224228
$197$	−22.6918	−1.61673	−0.808363	−	0.588685i	$-0.799646\pi$
$197$	−22.6918	−1.61673	−0.808363	+	0.588685i	$0.799646\pi$
$198$	0	0
$199$	−23.6620	−1.67736	−0.838679	−	0.544627i	$-0.816671\pi$
$199$	−23.6620	−1.67736	−0.838679	+	0.544627i	$0.816671\pi$
$200$	3.04502	0.215315
$201$	0	0
$202$	−4.31299	−0.303461
$203$	0.205034	0.0143906
$204$	0	0
$205$	−3.71535	−0.259491
$206$	−11.5720	−0.806260
$207$	0	0
$208$	2.32340	0.161099
$209$	0.925197	0.0639972
$210$	0	0
$211$	9.72161	0.669263	0.334632	−	0.942349i	$-0.391388\pi$
$211$	9.72161	0.669263	0.334632	+	0.942349i	$0.391388\pi$
$212$	−12.9598	−0.890083
$213$	0	0
$214$	−2.34278	−0.160149
$215$	−16.0000	−1.09119
$216$	0	0
$217$	−16.2694	−1.10444
$218$	2.32340	0.157361
$219$	0	0
$220$	0.452186	0.0304864
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−16.2847	−1.09050	−0.545251	−	0.838273i	$-0.683565\pi$
$223$	−16.2847	−1.09050	−0.545251	+	0.838273i	$0.683565\pi$
$224$	3.18421	0.212754
$225$	0	0
$226$	−4.90582	−0.326330
$227$	14.4134	0.956653	0.478327	−	0.878182i	$-0.341243\pi$
$227$	14.4134	0.956653	0.478327	+	0.878182i	$0.341243\pi$
$228$	0	0
$229$	−1.37738	−0.0910200	−0.0455100	−	0.998964i	$-0.514491\pi$
$229$	−1.37738	−0.0910200	−0.0455100	+	0.998964i	$0.514491\pi$
$230$	9.09899	0.599970
$231$	0	0
$232$	0.0643910	0.00422748
$233$	26.1801	1.71511	0.857557	−	0.514390i	$-0.171982\pi$
$233$	26.1801	1.71511	0.857557	+	0.514390i	$0.171982\pi$
$234$	0	0
$235$	−6.70638	−0.437476
$236$	6.60179	0.429740
$237$	0	0
$238$	−5.48197	−0.355344
$239$	−4.02082	−0.260086	−0.130043	−	0.991508i	$-0.541511\pi$
$239$	−4.02082	−0.260086	−0.130043	+	0.991508i	$0.541511\pi$
$240$	0	0
$241$	−18.3282	−1.18062	−0.590312	−	0.807175i	$-0.700995\pi$
$241$	−18.3282	−1.18062	−0.590312	+	0.807175i	$0.700995\pi$
$242$	10.8954	0.700383
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	−4.38924	−0.280419
$246$	0	0
$247$	−6.64681	−0.422926
$248$	−5.10941	−0.324448
$249$	0	0
$250$	−11.2486	−0.711424
$251$	19.5124	1.23161	0.615807	−	0.787897i	$-0.288830\pi$
$251$	19.5124	1.23161	0.615807	+	0.787897i	$0.288830\pi$
$252$	0	0
$253$	−2.10459	−0.132314
$254$	−1.87122	−0.117411
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.9508	1.18212	0.591060	−	0.806627i	$-0.298709\pi$
$257$	18.9508	1.18212	0.591060	+	0.806627i	$0.298709\pi$
$258$	0	0
$259$	−28.5962	−1.77688
$260$	−3.24860	−0.201470
$261$	0	0
$262$	−6.14961	−0.379924
$263$	−28.1801	−1.73766	−0.868829	−	0.495113i	$-0.835127\pi$
$263$	−28.1801	−1.73766	−0.868829	+	0.495113i	$0.835127\pi$
$264$	0	0
$265$	18.1205	1.11313
$266$	−9.10941	−0.558534
$267$	0	0
$268$	−4.69182	−0.286599
$269$	−28.0900	−1.71268	−0.856340	−	0.516413i	$-0.827267\pi$
$269$	−28.0900	−1.71268	−0.856340	+	0.516413i	$0.827267\pi$
$270$	0	0
$271$	6.73684	0.409234	0.204617	−	0.978842i	$-0.434405\pi$
$271$	6.73684	0.409234	0.204617	+	0.978842i	$0.434405\pi$
$272$	−1.72161	−0.104388
$273$	0	0
$274$	−16.5076	−0.997261
$275$	0.984771	0.0593839
$276$	0	0
$277$	0.0643910	0.00386888	0.00193444	−	0.999998i	$-0.499384\pi$
$277$	0.0643910	0.00386888	0.00193444	+	0.999998i	$0.499384\pi$
$278$	10.9702	0.657950
$279$	0	0
$280$	−4.45219	−0.266069
$281$	−4.55678	−0.271835	−0.135917	−	0.990720i	$-0.543398\pi$
$281$	−4.55678	−0.271835	−0.135917	+	0.990720i	$0.543398\pi$
$282$	0	0
$283$	18.3130	1.08859	0.544297	−	0.838892i	$-0.316796\pi$
$283$	18.3130	1.08859	0.544297	+	0.838892i	$0.316796\pi$
$284$	−3.41758	−0.202796
$285$	0	0
$286$	0.751399	0.0444311
$287$	−8.46115	−0.499446
$288$	0	0
$289$	−14.0361	−0.825650
$290$	−0.0900320	−0.00528686
$291$	0	0
$292$	13.9806	0.818154
$293$	−13.4432	−0.785361	−0.392681	−	0.919675i	$-0.628452\pi$
$293$	−13.4432	−0.785361	−0.392681	+	0.919675i	$0.628452\pi$
$294$	0	0
$295$	−9.23068	−0.537431
$296$	−8.98062	−0.521988
$297$	0	0
$298$	−21.2099	−1.22865
$299$	15.1198	0.874402
$300$	0	0
$301$	−36.4376	−2.10023
$302$	−21.0048	−1.20869
$303$	0	0
$304$	−2.86081	−0.164078
$305$	1.39821	0.0800611
$306$	0	0
$307$	17.8954	1.02134	0.510672	−	0.859775i	$-0.329396\pi$
$307$	17.8954	1.02134	0.510672	+	0.859775i	$0.329396\pi$
$308$	1.02979	0.0586776
$309$	0	0
$310$	7.14401	0.405753
$311$	−19.2742	−1.09294	−0.546471	−	0.837478i	$-0.684029\pi$
$311$	−19.2742	−1.09294	−0.546471	+	0.837478i	$0.684029\pi$
$312$	0	0
$313$	−11.2936	−0.638353	−0.319176	−	0.947695i	$-0.603406\pi$
$313$	−11.2936	−0.638353	−0.319176	+	0.947695i	$0.603406\pi$
$314$	5.13919	0.290022
$315$	0	0
$316$	4.19462	0.235966
$317$	−3.05398	−0.171529	−0.0857643	−	0.996315i	$-0.527333\pi$
$317$	−3.05398	−0.171529	−0.0857643	+	0.996315i	$0.527333\pi$
$318$	0	0
$319$	0.0208243	0.00116594
$320$	−1.39821	−0.0781622
$321$	0	0
$322$	20.7216	1.15477
$323$	4.92520	0.274045
$324$	0	0
$325$	−7.07480	−0.392439
$326$	−10.9806	−0.608161
$327$	0	0
$328$	−2.65722	−0.146720
$329$	−15.2728	−0.842016
$330$	0	0
$331$	11.4674	0.630306	0.315153	−	0.949041i	$-0.397944\pi$
$331$	11.4674	0.630306	0.315153	+	0.949041i	$0.397944\pi$
$332$	16.1544	0.886589
$333$	0	0
$334$	24.6081	1.34649
$335$	6.56014	0.358419
$336$	0	0
$337$	29.1440	1.58758	0.793788	−	0.608195i	$-0.208106\pi$
$337$	29.1440	1.58758	0.793788	+	0.608195i	$0.208106\pi$
$338$	7.60179	0.413483
$339$	0	0
$340$	2.40717	0.130547
$341$	−1.65240	−0.0894827
$342$	0	0
$343$	12.2936	0.663793
$344$	−11.4432	−0.616977
$345$	0	0
$346$	1.65722	0.0890927
$347$	−4.29921	−0.230794	−0.115397	−	0.993319i	$-0.536814\pi$
$347$	−4.29921	−0.230794	−0.115397	+	0.993319i	$0.536814\pi$
$348$	0	0
$349$	−8.53885	−0.457074	−0.228537	−	0.973535i	$-0.573394\pi$
$349$	−8.53885	−0.457074	−0.228537	+	0.973535i	$0.573394\pi$
$350$	−9.69597	−0.518271
$351$	0	0
$352$	0.323404	0.0172375
$353$	−25.9002	−1.37853	−0.689265	−	0.724509i	$-0.742066\pi$
$353$	−25.9002	−1.37853	−0.689265	+	0.724509i	$0.742066\pi$
$354$	0	0
$355$	4.77849	0.253616
$356$	−2.51803	−0.133455
$357$	0	0
$358$	5.75622	0.304225
$359$	15.4689	0.816415	0.408208	−	0.912889i	$-0.366154\pi$
$359$	15.4689	0.816415	0.408208	+	0.912889i	$0.366154\pi$
$360$	0	0
$361$	−10.8158	−0.569252
$362$	25.8760	1.36001
$363$	0	0
$364$	−7.39821	−0.387771
$365$	−19.5478	−1.02318
$366$	0	0
$367$	11.2936	0.589522	0.294761	−	0.955571i	$-0.404760\pi$
$367$	11.2936	0.589522	0.294761	+	0.955571i	$0.404760\pi$
$368$	6.50761	0.339233
$369$	0	0
$370$	12.5568	0.652796
$371$	41.2667	2.14246
$372$	0	0
$373$	10.4834	0.542811	0.271406	−	0.962465i	$-0.412512\pi$
$373$	10.4834	0.542811	0.271406	+	0.962465i	$0.412512\pi$
$374$	−0.556777	−0.0287902
$375$	0	0
$376$	−4.79641	−0.247356
$377$	−0.149606	−0.00770512
$378$	0	0
$379$	3.25005	0.166944	0.0834719	−	0.996510i	$-0.473399\pi$
$379$	3.25005	0.166944	0.0834719	+	0.996510i	$0.473399\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−2.13919	−0.109451
$383$	13.3338	0.681326	0.340663	−	0.940185i	$-0.389348\pi$
$383$	13.3338	0.681326	0.340663	+	0.940185i	$0.389348\pi$
$384$	0	0
$385$	−1.43986	−0.0733819
$386$	−25.4224	−1.29397
$387$	0	0
$388$	−6.18421	−0.313956
$389$	10.9508	0.555230	0.277615	−	0.960692i	$-0.410456\pi$
$389$	10.9508	0.555230	0.277615	+	0.960692i	$0.410456\pi$
$390$	0	0
$391$	−11.2036	−0.566590
$392$	−3.13919	−0.158553
$393$	0	0
$394$	22.6918	1.14320
$395$	−5.86495	−0.295098
$396$	0	0
$397$	−24.3345	−1.22131	−0.610656	−	0.791896i	$-0.709094\pi$
$397$	−24.3345	−1.22131	−0.610656	+	0.791896i	$0.709094\pi$
$398$	23.6620	1.18607
$399$	0	0
$400$	−3.04502	−0.152251
$401$	6.51803	0.325495	0.162747	−	0.986668i	$-0.447964\pi$
$401$	6.51803	0.325495	0.162747	+	0.986668i	$0.447964\pi$
$402$	0	0
$403$	11.8712	0.591347
$404$	4.31299	0.214579
$405$	0	0
$406$	−0.205034	−0.0101757
$407$	−2.90437	−0.143964
$408$	0	0
$409$	36.0305	1.78159	0.890796	−	0.454404i	$-0.150148\pi$
$409$	36.0305	1.78159	0.890796	+	0.454404i	$0.150148\pi$
$410$	3.71535	0.183488
$411$	0	0
$412$	11.5720	0.570112
$413$	−21.0215	−1.03440
$414$	0	0
$415$	−22.5872	−1.10876
$416$	−2.32340	−0.113914
$417$	0	0
$418$	−0.925197	−0.0452529
$419$	−24.3505	−1.18960	−0.594800	−	0.803874i	$-0.702769\pi$
$419$	−24.3505	−1.18960	−0.594800	+	0.803874i	$0.702769\pi$
$420$	0	0
$421$	9.87603	0.481328	0.240664	−	0.970608i	$-0.422635\pi$
$421$	9.87603	0.481328	0.240664	+	0.970608i	$0.422635\pi$
$422$	−9.72161	−0.473241
$423$	0	0
$424$	12.9598	0.629384
$425$	5.24234	0.254291
$426$	0	0
$427$	3.18421	0.154095
$428$	2.34278	0.113243
$429$	0	0
$430$	16.0000	0.771589
$431$	−15.4224	−0.742871	−0.371435	−	0.928459i	$-0.621134\pi$
$431$	−15.4224	−0.742871	−0.371435	+	0.928459i	$0.621134\pi$
$432$	0	0
$433$	−23.5124	−1.12994	−0.564968	−	0.825113i	$-0.691111\pi$
$433$	−23.5124	−1.12994	−0.564968	+	0.825113i	$0.691111\pi$
$434$	16.2694	0.780957
$435$	0	0
$436$	−2.32340	−0.111271
$437$	−18.6170	−0.890573
$438$	0	0
$439$	1.05398	0.0503037	0.0251518	−	0.999684i	$-0.491993\pi$
$439$	1.05398	0.0503037	0.0251518	+	0.999684i	$0.491993\pi$
$440$	−0.452186	−0.0215571
$441$	0	0
$442$	4.00000	0.190261
$443$	39.2590	1.86525	0.932626	−	0.360844i	$-0.117511\pi$
$443$	39.2590	1.86525	0.932626	+	0.360844i	$0.117511\pi$
$444$	0	0
$445$	3.52072	0.166898
$446$	16.2847	0.771101
$447$	0	0
$448$	−3.18421	−0.150440
$449$	−14.9356	−0.704855	−0.352427	−	0.935839i	$-0.614644\pi$
$449$	−14.9356	−0.704855	−0.352427	+	0.935839i	$0.614644\pi$
$450$	0	0
$451$	−0.859357	−0.0404655
$452$	4.90582	0.230750
$453$	0	0
$454$	−14.4134	−0.676456
$455$	10.3442	0.484945
$456$	0	0
$457$	−8.55678	−0.400269	−0.200135	−	0.979768i	$-0.564138\pi$
$457$	−8.55678	−0.400269	−0.200135	+	0.979768i	$0.564138\pi$
$458$	1.37738	0.0643609
$459$	0	0
$460$	−9.09899	−0.424243
$461$	−3.66204	−0.170558	−0.0852790	−	0.996357i	$-0.527178\pi$
$461$	−3.66204	−0.170558	−0.0852790	+	0.996357i	$0.527178\pi$
$462$	0	0
$463$	−19.6829	−0.914740	−0.457370	−	0.889276i	$-0.651209\pi$
$463$	−19.6829	−0.914740	−0.457370	+	0.889276i	$0.651209\pi$
$464$	−0.0643910	−0.00298928
$465$	0	0
$466$	−26.1801	−1.21277
$467$	−18.2251	−0.843356	−0.421678	−	0.906746i	$-0.638559\pi$
$467$	−18.2251	−0.843356	−0.421678	+	0.906746i	$0.638559\pi$
$468$	0	0
$469$	14.9398	0.689854
$470$	6.70638	0.309342
$471$	0	0
$472$	−6.60179	−0.303872
$473$	−3.70079	−0.170162
$474$	0	0
$475$	8.71120	0.399697
$476$	5.48197	0.251266
$477$	0	0
$478$	4.02082	0.183908
$479$	31.2936	1.42984	0.714921	−	0.699205i	$-0.246463\pi$
$479$	31.2936	1.42984	0.714921	+	0.699205i	$0.246463\pi$
$480$	0	0
$481$	20.8656	0.951390
$482$	18.3282	0.834828
$483$	0	0
$484$	−10.8954	−0.495246
$485$	8.64681	0.392631
$486$	0	0
$487$	28.6773	1.29949	0.649745	−	0.760152i	$-0.274875\pi$
$487$	28.6773	1.29949	0.649745	+	0.760152i	$0.274875\pi$
$488$	1.00000	0.0452679
$489$	0	0
$490$	4.38924	0.198286
$491$	9.43841	0.425949	0.212975	−	0.977058i	$-0.431685\pi$
$491$	9.43841	0.425949	0.212975	+	0.977058i	$0.431685\pi$
$492$	0	0
$493$	0.110856	0.00499272
$494$	6.64681	0.299054
$495$	0	0
$496$	5.10941	0.229419
$497$	10.8823	0.488138
$498$	0	0
$499$	35.9162	1.60783	0.803916	−	0.594743i	$-0.202746\pi$
$499$	35.9162	1.60783	0.803916	+	0.594743i	$0.202746\pi$
$500$	11.2486	0.503053
$501$	0	0
$502$	−19.5124	−0.870882
$503$	14.3684	0.640656	0.320328	−	0.947307i	$-0.396207\pi$
$503$	14.3684	0.640656	0.320328	+	0.947307i	$0.396207\pi$
$504$	0	0
$505$	−6.03046	−0.268352
$506$	2.10459	0.0935605
$507$	0	0
$508$	1.87122	0.0830219
$509$	32.3047	1.43188	0.715940	−	0.698161i	$-0.245998\pi$
$509$	32.3047	1.43188	0.715940	+	0.698161i	$0.245998\pi$
$510$	0	0
$511$	−44.5172	−1.96933
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.9508	−0.835886
$515$	−16.1801	−0.712979
$516$	0	0
$517$	−1.55118	−0.0682209
$518$	28.5962	1.25644
$519$	0	0
$520$	3.24860	0.142461
$521$	19.8809	0.870996	0.435498	−	0.900190i	$-0.356572\pi$
$521$	19.8809	0.870996	0.435498	+	0.900190i	$0.356572\pi$
$522$	0	0
$523$	−2.08377	−0.0911167	−0.0455584	−	0.998962i	$-0.514507\pi$
$523$	−2.08377	−0.0911167	−0.0455584	+	0.998962i	$0.514507\pi$
$524$	6.14961	0.268647
$525$	0	0
$526$	28.1801	1.22871
$527$	−8.79641	−0.383178
$528$	0	0
$529$	19.3490	0.841263
$530$	−18.1205	−0.787104
$531$	0	0
$532$	9.10941	0.394943
$533$	6.17380	0.267417
$534$	0	0
$535$	−3.27569	−0.141620
$536$	4.69182	0.202656
$537$	0	0
$538$	28.0900	1.21105
$539$	−1.01523	−0.0437290
$540$	0	0
$541$	24.0554	1.03422	0.517112	−	0.855918i	$-0.327007\pi$
$541$	24.0554	1.03422	0.517112	+	0.855918i	$0.327007\pi$
$542$	−6.73684	−0.289372
$543$	0	0
$544$	1.72161	0.0738135
$545$	3.24860	0.139155
$546$	0	0
$547$	12.0242	0.514117	0.257059	−	0.966396i	$-0.417247\pi$
$547$	12.0242	0.514117	0.257059	+	0.966396i	$0.417247\pi$
$548$	16.5076	0.705170
$549$	0	0
$550$	−0.984771	−0.0419908
$551$	0.184210	0.00784762
$552$	0	0
$553$	−13.3566	−0.567979
$554$	−0.0643910	−0.00273571
$555$	0	0
$556$	−10.9702	−0.465241
$557$	−10.3476	−0.438442	−0.219221	−	0.975675i	$-0.570352\pi$
$557$	−10.3476	−0.438442	−0.219221	+	0.975675i	$0.570352\pi$
$558$	0	0
$559$	26.5872	1.12452
$560$	4.45219	0.188139
$561$	0	0
$562$	4.55678	0.192216
$563$	0.0900320	0.00379439	0.00189720	−	0.999998i	$-0.499396\pi$
$563$	0.0900320	0.00379439	0.00189720	+	0.999998i	$0.499396\pi$
$564$	0	0
$565$	−6.85936	−0.288575
$566$	−18.3130	−0.769752
$567$	0	0
$568$	3.41758	0.143399
$569$	−24.5243	−1.02811	−0.514056	−	0.857757i	$-0.671858\pi$
$569$	−24.5243	−1.02811	−0.514056	+	0.857757i	$0.671858\pi$
$570$	0	0
$571$	−21.7979	−0.912212	−0.456106	−	0.889925i	$-0.650756\pi$
$571$	−21.7979	−0.912212	−0.456106	+	0.889925i	$0.650756\pi$
$572$	−0.751399	−0.0314176
$573$	0	0
$574$	8.46115	0.353162
$575$	−19.8158	−0.826376
$576$	0	0
$577$	34.3088	1.42830	0.714148	−	0.699995i	$-0.246814\pi$
$577$	34.3088	1.42830	0.714148	+	0.699995i	$0.246814\pi$
$578$	14.0361	0.583823
$579$	0	0
$580$	0.0900320	0.00373837
$581$	−51.4391	−2.13405
$582$	0	0
$583$	4.19125	0.173584
$584$	−13.9806	−0.578522
$585$	0	0
$586$	13.4432	0.555334
$587$	32.8477	1.35577	0.677885	−	0.735168i	$-0.262897\pi$
$587$	32.8477	1.35577	0.677885	+	0.735168i	$0.262897\pi$
$588$	0	0
$589$	−14.6170	−0.602284
$590$	9.23068	0.380021
$591$	0	0
$592$	8.98062	0.369101
$593$	−15.4640	−0.635032	−0.317516	−	0.948253i	$-0.602849\pi$
$593$	−15.4640	−0.635032	−0.317516	+	0.948253i	$0.602849\pi$
$594$	0	0
$595$	−7.66494	−0.314232
$596$	21.2099	0.868789
$597$	0	0
$598$	−15.1198	−0.618295
$599$	25.7458	1.05195	0.525973	−	0.850502i	$-0.323701\pi$
$599$	25.7458	1.05195	0.525973	+	0.850502i	$0.323701\pi$
$600$	0	0
$601$	7.56159	0.308444	0.154222	−	0.988036i	$-0.450713\pi$
$601$	7.56159	0.308444	0.154222	+	0.988036i	$0.450713\pi$
$602$	36.4376	1.48509
$603$	0	0
$604$	21.0048	0.854674
$605$	15.2340	0.619352
$606$	0	0
$607$	20.7160	0.840837	0.420419	−	0.907330i	$-0.361883\pi$
$607$	20.7160	0.840837	0.420419	+	0.907330i	$0.361883\pi$
$608$	2.86081	0.116021
$609$	0	0
$610$	−1.39821	−0.0566118
$611$	11.1440	0.450838
$612$	0	0
$613$	2.05957	0.0831854	0.0415927	−	0.999135i	$-0.486757\pi$
$613$	2.05957	0.0831854	0.0415927	+	0.999135i	$0.486757\pi$
$614$	−17.8954	−0.722200
$615$	0	0
$616$	−1.02979	−0.0414913
$617$	−34.6773	−1.39605	−0.698027	−	0.716071i	$-0.745939\pi$
$617$	−34.6773	−1.39605	−0.698027	+	0.716071i	$0.745939\pi$
$618$	0	0
$619$	20.6635	0.830536	0.415268	−	0.909699i	$-0.363688\pi$
$619$	20.6635	0.830536	0.415268	+	0.909699i	$0.363688\pi$
$620$	−7.14401	−0.286910
$621$	0	0
$622$	19.2742	0.772827
$623$	8.01793	0.321231
$624$	0	0
$625$	−0.502798	−0.0201119
$626$	11.2936	0.451384
$627$	0	0
$628$	−5.13919	−0.205076
$629$	−15.4611	−0.616476
$630$	0	0
$631$	−17.5914	−0.700302	−0.350151	−	0.936693i	$-0.613870\pi$
$631$	−17.5914	−0.700302	−0.350151	+	0.936693i	$0.613870\pi$
$632$	−4.19462	−0.166853
$633$	0	0
$634$	3.05398	0.121289
$635$	−2.61635	−0.103827
$636$	0	0
$637$	7.29362	0.288984
$638$	−0.0208243	−0.000824443 0
$639$	0	0
$640$	1.39821	0.0552690
$641$	−22.8477	−0.902430	−0.451215	−	0.892415i	$-0.649009\pi$
$641$	−22.8477	−0.902430	−0.451215	+	0.892415i	$0.649009\pi$
$642$	0	0
$643$	31.0748	1.22547	0.612735	−	0.790288i	$-0.290069\pi$
$643$	31.0748	1.22547	0.612735	+	0.790288i	$0.290069\pi$
$644$	−20.7216	−0.816546
$645$	0	0
$646$	−4.92520	−0.193779
$647$	−27.5187	−1.08187	−0.540936	−	0.841064i	$-0.681930\pi$
$647$	−27.5187	−1.08187	−0.540936	+	0.841064i	$0.681930\pi$
$648$	0	0
$649$	−2.13505	−0.0838080
$650$	7.07480	0.277497
$651$	0	0
$652$	10.9806	0.430034
$653$	30.7804	1.20453	0.602265	−	0.798296i	$-0.294265\pi$
$653$	30.7804	1.20453	0.602265	+	0.798296i	$0.294265\pi$
$654$	0	0
$655$	−8.59843	−0.335968
$656$	2.65722	0.103747
$657$	0	0
$658$	15.2728	0.595395
$659$	22.4238	0.873509	0.436755	−	0.899581i	$-0.356128\pi$
$659$	22.4238	0.873509	0.436755	+	0.899581i	$0.356128\pi$
$660$	0	0
$661$	15.1094	0.587688	0.293844	−	0.955853i	$-0.405065\pi$
$661$	15.1094	0.587688	0.293844	+	0.955853i	$0.405065\pi$
$662$	−11.4674	−0.445694
$663$	0	0
$664$	−16.1544	−0.626913
$665$	−12.7368	−0.493913
$666$	0	0
$667$	−0.419032	−0.0162250
$668$	−24.6081	−0.952114
$669$	0	0
$670$	−6.56014	−0.253440
$671$	0.323404	0.0124849
$672$	0	0
$673$	−16.9073	−0.651727	−0.325864	−	0.945417i	$-0.605655\pi$
$673$	−16.9073	−0.651727	−0.325864	+	0.945417i	$0.605655\pi$
$674$	−29.1440	−1.12259
$675$	0	0
$676$	−7.60179	−0.292377
$677$	40.6773	1.56335	0.781677	−	0.623683i	$-0.214364\pi$
$677$	40.6773	1.56335	0.781677	+	0.623683i	$0.214364\pi$
$678$	0	0
$679$	19.6918	0.755703
$680$	−2.40717	−0.0923108
$681$	0	0
$682$	1.65240	0.0632738
$683$	26.9508	1.03125	0.515623	−	0.856816i	$-0.327561\pi$
$683$	26.9508	1.03125	0.515623	+	0.856816i	$0.327561\pi$
$684$	0	0
$685$	−23.0811	−0.881882
$686$	−12.2936	−0.469372
$687$	0	0
$688$	11.4432	0.436269
$689$	−30.1109	−1.14713
$690$	0	0
$691$	−20.8518	−0.793241	−0.396621	−	0.917983i	$-0.629817\pi$
$691$	−20.8518	−0.793241	−0.396621	+	0.917983i	$0.629817\pi$
$692$	−1.65722	−0.0629981
$693$	0	0
$694$	4.29921	0.163196
$695$	15.3386	0.581828
$696$	0	0
$697$	−4.57470	−0.173279
$698$	8.53885	0.323200
$699$	0	0
$700$	9.69597	0.366473
$701$	−18.2951	−0.690995	−0.345498	−	0.938420i	$-0.612290\pi$
$701$	−18.2951	−0.690995	−0.345498	+	0.938420i	$0.612290\pi$
$702$	0	0
$703$	−25.6918	−0.968986
$704$	−0.323404	−0.0121888
$705$	0	0
$706$	25.9002	0.974768
$707$	−13.7335	−0.516500
$708$	0	0
$709$	41.3241	1.55196	0.775979	−	0.630759i	$-0.217256\pi$
$709$	41.3241	1.55196	0.775979	+	0.630759i	$0.217256\pi$
$710$	−4.77849	−0.179334
$711$	0	0
$712$	2.51803	0.0943670
$713$	33.2501	1.24522
$714$	0	0
$715$	1.05061	0.0392906
$716$	−5.75622	−0.215120
$717$	0	0
$718$	−15.4689	−0.577293
$719$	26.9765	1.00605	0.503026	−	0.864271i	$-0.332220\pi$
$719$	26.9765	1.00605	0.503026	+	0.864271i	$0.332220\pi$
$720$	0	0
$721$	−36.8477	−1.37228
$722$	10.8158	0.402522
$723$	0	0
$724$	−25.8760	−0.961675
$725$	0.196072	0.00728192
$726$	0	0
$727$	−1.85039	−0.0686273	−0.0343137	−	0.999411i	$-0.510925\pi$
$727$	−1.85039	−0.0686273	−0.0343137	+	0.999411i	$0.510925\pi$
$728$	7.39821	0.274196
$729$	0	0
$730$	19.5478	0.723497
$731$	−19.7008	−0.728660
$732$	0	0
$733$	−42.7639	−1.57952	−0.789761	−	0.613415i	$-0.789795\pi$
$733$	−42.7639	−1.57952	−0.789761	+	0.613415i	$0.789795\pi$
$734$	−11.2936	−0.416855
$735$	0	0
$736$	−6.50761	−0.239874
$737$	1.51736	0.0558925
$738$	0	0
$739$	−9.74580	−0.358505	−0.179253	−	0.983803i	$-0.557368\pi$
$739$	−9.74580	−0.358505	−0.179253	+	0.983803i	$0.557368\pi$
$740$	−12.5568	−0.461596
$741$	0	0
$742$	−41.2667	−1.51495
$743$	33.5526	1.23093	0.615463	−	0.788166i	$-0.288969\pi$
$743$	33.5526	1.23093	0.615463	+	0.788166i	$0.288969\pi$
$744$	0	0
$745$	−29.6558	−1.08650
$746$	−10.4834	−0.383825
$747$	0	0
$748$	0.556777	0.0203578
$749$	−7.45990	−0.272579
$750$	0	0
$751$	−32.7756	−1.19600	−0.597999	−	0.801497i	$-0.704037\pi$
$751$	−32.7756	−1.19600	−0.597999	+	0.801497i	$0.704037\pi$
$752$	4.79641	0.174907
$753$	0	0
$754$	0.149606	0.00544834
$755$	−29.3691	−1.06885
$756$	0	0
$757$	6.28465	0.228420	0.114210	−	0.993457i	$-0.463566\pi$
$757$	6.28465	0.228420	0.114210	+	0.993457i	$0.463566\pi$
$758$	−3.25005	−0.118047
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−2.86562	−0.103879	−0.0519394	−	0.998650i	$-0.516540\pi$
$761$	−2.86562	−0.103879	−0.0519394	+	0.998650i	$0.516540\pi$
$762$	0	0
$763$	7.39821	0.267833
$764$	2.13919	0.0773933
$765$	0	0
$766$	−13.3338	−0.481770
$767$	15.3386	0.553846
$768$	0	0
$769$	17.7812	0.641206	0.320603	−	0.947214i	$-0.396114\pi$
$769$	17.7812	0.641206	0.320603	+	0.947214i	$0.396114\pi$
$770$	1.43986	0.0518888
$771$	0	0
$772$	25.4224	0.914972
$773$	−20.1080	−0.723233	−0.361616	−	0.932327i	$-0.617775\pi$
$773$	−20.1080	−0.723233	−0.361616	+	0.932327i	$0.617775\pi$
$774$	0	0
$775$	−15.5582	−0.558868
$776$	6.18421	0.222000
$777$	0	0
$778$	−10.9508	−0.392607
$779$	−7.60179	−0.272362
$780$	0	0
$781$	1.10526	0.0395493
$782$	11.2036	0.400639
$783$	0	0
$784$	3.13919	0.112114
$785$	7.18566	0.256467
$786$	0	0
$787$	−27.0665	−0.964817	−0.482408	−	0.875946i	$-0.660238\pi$
$787$	−27.0665	−0.964817	−0.482408	+	0.875946i	$0.660238\pi$
$788$	−22.6918	−0.808363
$789$	0	0
$790$	5.86495	0.208666
$791$	−15.6212	−0.555425
$792$	0	0
$793$	−2.32340	−0.0825065
$794$	24.3345	0.863599
$795$	0	0
$796$	−23.6620	−0.838679
$797$	44.3330	1.57036	0.785178	−	0.619270i	$-0.212571\pi$
$797$	44.3330	1.57036	0.785178	+	0.619270i	$0.212571\pi$
$798$	0	0
$799$	−8.25756	−0.292132
$800$	3.04502	0.107658
$801$	0	0
$802$	−6.51803	−0.230160
$803$	−4.52139	−0.159557
$804$	0	0
$805$	28.9731	1.02117
$806$	−11.8712	−0.418146
$807$	0	0
$808$	−4.31299	−0.151731
$809$	5.21467	0.183338	0.0916690	−	0.995790i	$-0.470780\pi$
$809$	5.21467	0.183338	0.0916690	+	0.995790i	$0.470780\pi$
$810$	0	0
$811$	50.0963	1.75912	0.879559	−	0.475789i	$-0.157837\pi$
$811$	50.0963	1.75912	0.879559	+	0.475789i	$0.157837\pi$
$812$	0.205034	0.00719530
$813$	0	0
$814$	2.90437	0.101798
$815$	−15.3532	−0.537799
$816$	0	0
$817$	−32.7368	−1.14532
$818$	−36.0305	−1.25978
$819$	0	0
$820$	−3.71535	−0.129746
$821$	23.2984	0.813121	0.406560	−	0.913624i	$-0.366728\pi$
$821$	23.2984	0.813121	0.406560	+	0.913624i	$0.366728\pi$
$822$	0	0
$823$	−10.2624	−0.357724	−0.178862	−	0.983874i	$-0.557242\pi$
$823$	−10.2624	−0.357724	−0.178862	+	0.983874i	$0.557242\pi$
$824$	−11.5720	−0.403130
$825$	0	0
$826$	21.0215	0.731431
$827$	1.77704	0.0617937	0.0308969	−	0.999523i	$-0.490164\pi$
$827$	1.77704	0.0617937	0.0308969	+	0.999523i	$0.490164\pi$
$828$	0	0
$829$	10.3539	0.359604	0.179802	−	0.983703i	$-0.442454\pi$
$829$	10.3539	0.359604	0.179802	+	0.983703i	$0.442454\pi$
$830$	22.5872	0.784014
$831$	0	0
$832$	2.32340	0.0805496
$833$	−5.40447	−0.187254
$834$	0	0
$835$	34.4072	1.19071
$836$	0.925197	0.0319986
$837$	0	0
$838$	24.3505	0.841174
$839$	−49.4529	−1.70730	−0.853651	−	0.520845i	$-0.825617\pi$
$839$	−49.4529	−1.70730	−0.853651	+	0.520845i	$0.825617\pi$
$840$	0	0
$841$	−28.9959	−0.999857
$842$	−9.87603	−0.340351
$843$	0	0
$844$	9.72161	0.334632
$845$	10.6289	0.365645
$846$	0	0
$847$	34.6933	1.19207
$848$	−12.9598	−0.445041
$849$	0	0
$850$	−5.24234	−0.179811
$851$	58.4424	2.00338
$852$	0	0
$853$	3.07817	0.105395	0.0526973	−	0.998611i	$-0.483218\pi$
$853$	3.07817	0.105395	0.0526973	+	0.998611i	$0.483218\pi$
$854$	−3.18421	−0.108961
$855$	0	0
$856$	−2.34278	−0.0800745
$857$	38.7658	1.32422	0.662108	−	0.749408i	$-0.269662\pi$
$857$	38.7658	1.32422	0.662108	+	0.749408i	$0.269662\pi$
$858$	0	0
$859$	44.5318	1.51941	0.759703	−	0.650270i	$-0.225344\pi$
$859$	44.5318	1.51941	0.759703	+	0.650270i	$0.225344\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	15.4224	0.525289
$863$	−6.34760	−0.216075	−0.108037	−	0.994147i	$-0.534457\pi$
$863$	−6.34760	−0.216075	−0.108037	+	0.994147i	$0.534457\pi$
$864$	0	0
$865$	2.31714	0.0787851
$866$	23.5124	0.798985
$867$	0	0
$868$	−16.2694	−0.552220
$869$	−1.35656	−0.0460181
$870$	0	0
$871$	−10.9010	−0.369366
$872$	2.32340	0.0786804
$873$	0	0
$874$	18.6170	0.629730
$875$	−35.8179	−1.21087
$876$	0	0
$877$	−6.47446	−0.218627	−0.109313	−	0.994007i	$-0.534865\pi$
$877$	−6.47446	−0.218627	−0.109313	+	0.994007i	$0.534865\pi$
$878$	−1.05398	−0.0355701
$879$	0	0
$880$	0.452186	0.0152432
$881$	−17.4209	−0.586927	−0.293463	−	0.955970i	$-0.594808\pi$
$881$	−17.4209	−0.586927	−0.293463	+	0.955970i	$0.594808\pi$
$882$	0	0
$883$	28.9798	0.975249	0.487625	−	0.873053i	$-0.337863\pi$
$883$	28.9798	0.975249	0.487625	+	0.873053i	$0.337863\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−39.2590	−1.31893
$887$	−38.4072	−1.28959	−0.644793	−	0.764357i	$-0.723057\pi$
$887$	−38.4072	−1.28959	−0.644793	+	0.764357i	$0.723057\pi$
$888$	0	0
$889$	−5.95835	−0.199837
$890$	−3.52072	−0.118015
$891$	0	0
$892$	−16.2847	−0.545251
$893$	−13.7216	−0.459176
$894$	0	0
$895$	8.04838	0.269028
$896$	3.18421	0.106377
$897$	0	0
$898$	14.9356	0.498408
$899$	−0.329000	−0.0109728
$900$	0	0
$901$	22.3117	0.743312
$902$	0.859357	0.0286134
$903$	0	0
$904$	−4.90582	−0.163165
$905$	36.1801	1.20267
$906$	0	0
$907$	−10.8269	−0.359500	−0.179750	−	0.983712i	$-0.557529\pi$
$907$	−10.8269	−0.359500	−0.179750	+	0.983712i	$0.557529\pi$
$908$	14.4134	0.478327
$909$	0	0
$910$	−10.3442	−0.342908
$911$	−31.9792	−1.05952	−0.529759	−	0.848148i	$-0.677718\pi$
$911$	−31.9792	−1.05952	−0.529759	+	0.848148i	$0.677718\pi$
$912$	0	0
$913$	−5.22441	−0.172903
$914$	8.55678	0.283033
$915$	0	0
$916$	−1.37738	−0.0455100
$917$	−19.5816	−0.646643
$918$	0	0
$919$	−18.2188	−0.600983	−0.300492	−	0.953784i	$-0.597151\pi$
$919$	−18.2188	−0.600983	−0.300492	+	0.953784i	$0.597151\pi$
$920$	9.09899	0.299985
$921$	0	0
$922$	3.66204	0.120603
$923$	−7.94043	−0.261362
$924$	0	0
$925$	−27.3461	−0.899136
$926$	19.6829	0.646819
$927$	0	0
$928$	0.0643910	0.00211374
$929$	−47.3691	−1.55413	−0.777065	−	0.629421i	$-0.783292\pi$
$929$	−47.3691	−1.55413	−0.777065	+	0.629421i	$0.783292\pi$
$930$	0	0
$931$	−8.98062	−0.294328
$932$	26.1801	0.857557
$933$	0	0
$934$	18.2251	0.596343
$935$	−0.778489	−0.0254593
$936$	0	0
$937$	−33.0830	−1.08077	−0.540387	−	0.841417i	$-0.681722\pi$
$937$	−33.0830	−1.08077	−0.540387	+	0.841417i	$0.681722\pi$
$938$	−14.9398	−0.487800
$939$	0	0
$940$	−6.70638	−0.218738
$941$	−10.8560	−0.353895	−0.176948	−	0.984220i	$-0.556622\pi$
$941$	−10.8560	−0.353895	−0.176948	+	0.984220i	$0.556622\pi$
$942$	0	0
$943$	17.2922	0.563110
$944$	6.60179	0.214870
$945$	0	0
$946$	3.70079	0.120323
$947$	−31.9646	−1.03871	−0.519355	−	0.854558i	$-0.673828\pi$
$947$	−31.9646	−1.03871	−0.519355	+	0.854558i	$0.673828\pi$
$948$	0	0
$949$	32.4826	1.05443
$950$	−8.71120	−0.282629
$951$	0	0
$952$	−5.48197	−0.177672
$953$	−0.646809	−0.0209522	−0.0104761	−	0.999945i	$-0.503335\pi$
$953$	−0.646809	−0.0209522	−0.0104761	+	0.999945i	$0.503335\pi$
$954$	0	0
$955$	−2.99104	−0.0967877
$956$	−4.02082	−0.130043
$957$	0	0
$958$	−31.2936	−1.01105
$959$	−52.5637	−1.69737
$960$	0	0
$961$	−4.89396	−0.157870
$962$	−20.8656	−0.672735
$963$	0	0
$964$	−18.3282	−0.590312
$965$	−35.5458	−1.14426
$966$	0	0
$967$	−34.5068	−1.10967	−0.554833	−	0.831962i	$-0.687218\pi$
$967$	−34.5068	−1.10967	−0.554833	+	0.831962i	$0.687218\pi$
$968$	10.8954	0.350192
$969$	0	0
$970$	−8.64681	−0.277632
$971$	−0.504247	−0.0161821	−0.00809103	−	0.999967i	$-0.502575\pi$
$971$	−0.504247	−0.0161821	−0.00809103	+	0.999967i	$0.502575\pi$
$972$	0	0
$973$	34.9315	1.11985
$974$	−28.6773	−0.918879
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−17.9959	−0.575738	−0.287869	−	0.957670i	$-0.592947\pi$
$977$	−17.9959	−0.575738	−0.287869	+	0.957670i	$0.592947\pi$
$978$	0	0
$979$	0.814341	0.0260264
$980$	−4.38924	−0.140209
$981$	0	0
$982$	−9.43841	−0.301192
$983$	25.8462	0.824367	0.412184	−	0.911101i	$-0.364766\pi$
$983$	25.8462	0.824367	0.412184	+	0.911101i	$0.364766\pi$
$984$	0	0
$985$	31.7279	1.01093
$986$	−0.110856	−0.00353038
$987$	0	0
$988$	−6.64681	−0.211463
$989$	74.4681	2.36795
$990$	0	0
$991$	14.6052	0.463948	0.231974	−	0.972722i	$-0.425482\pi$
$991$	14.6052	0.463948	0.231974	+	0.972722i	$0.425482\pi$
$992$	−5.10941	−0.162224
$993$	0	0
$994$	−10.8823	−0.345166
$995$	33.0844	1.04885
$996$	0	0
$997$	−6.90023	−0.218532	−0.109266	−	0.994013i	$-0.534850\pi$
$997$	−6.90023	−0.218532	−0.109266	+	0.994013i	$0.534850\pi$
$998$	−35.9162	−1.13691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1098.2.a.p.1.2		3
3.2	odd	2		122.2.a.c.1.2	✓	3
4.3	odd	2		8784.2.a.bm.1.2		3
12.11	even	2		976.2.a.g.1.2		3
15.2	even	4		3050.2.b.k.1099.5		6
15.8	even	4		3050.2.b.k.1099.2		6
15.14	odd	2		3050.2.a.t.1.2		3
21.20	even	2		5978.2.a.q.1.2		3
24.5	odd	2		3904.2.a.u.1.2		3
24.11	even	2		3904.2.a.t.1.2		3
183.182	odd	2		7442.2.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
122.2.a.c.1.2	✓	3	3.2	odd	2
976.2.a.g.1.2		3	12.11	even	2
1098.2.a.p.1.2		3	1.1	even	1	trivial
3050.2.a.t.1.2		3	15.14	odd	2
3050.2.b.k.1099.2		6	15.8	even	4
3050.2.b.k.1099.5		6	15.2	even	4
3904.2.a.t.1.2		3	24.11	even	2
3904.2.a.u.1.2		3	24.5	odd	2
5978.2.a.q.1.2		3	21.20	even	2
7442.2.a.j.1.2		3	183.182	odd	2
8784.2.a.bm.1.2		3	4.3	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:	\( N \)	\(=\)	\( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1098.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.39821 q^{5} -3.18421 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.39821 q^{5} -3.18421 q^{7} -1.00000 q^{8} +1.39821 q^{10} -0.323404 q^{11} +2.32340 q^{13} +3.18421 q^{14} +1.00000 q^{16} -1.72161 q^{17} -2.86081 q^{19} -1.39821 q^{20} +0.323404 q^{22} +6.50761 q^{23} -3.04502 q^{25} -2.32340 q^{26} -3.18421 q^{28} -0.0643910 q^{29} +5.10941 q^{31} -1.00000 q^{32} +1.72161 q^{34} +4.45219 q^{35} +8.98062 q^{37} +2.86081 q^{38} +1.39821 q^{40} +2.65722 q^{41} +11.4432 q^{43} -0.323404 q^{44} -6.50761 q^{46} +4.79641 q^{47} +3.13919 q^{49} +3.04502 q^{50} +2.32340 q^{52} -12.9598 q^{53} +0.452186 q^{55} +3.18421 q^{56} +0.0643910 q^{58} +6.60179 q^{59} -1.00000 q^{61} -5.10941 q^{62} +1.00000 q^{64} -3.24860 q^{65} -4.69182 q^{67} -1.72161 q^{68} -4.45219 q^{70} -3.41758 q^{71} +13.9806 q^{73} -8.98062 q^{74} -2.86081 q^{76} +1.02979 q^{77} +4.19462 q^{79} -1.39821 q^{80} -2.65722 q^{82} +16.1544 q^{83} +2.40717 q^{85} -11.4432 q^{86} +0.323404 q^{88} -2.51803 q^{89} -7.39821 q^{91} +6.50761 q^{92} -4.79641 q^{94} +4.00000 q^{95} -6.18421 q^{97} -3.13919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - q^5 + 4 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - q^{5} + 4 q^{7} - 3 q^{8} + q^{10} + 7 q^{11} - q^{13} - 4 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{19} - q^{20} - 7 q^{22} - 2 q^{23} + 10 q^{25} + q^{26} + 4 q^{28} - q^{29} - 3 q^{31} - 3 q^{32} - 6 q^{34} + 7 q^{35} + 7 q^{37} + 3 q^{38} + q^{40} - 4 q^{41} + 12 q^{43} + 7 q^{44} + 2 q^{46} + 8 q^{47} + 15 q^{49} - 10 q^{50} - q^{52} - 11 q^{53} - 5 q^{55} - 4 q^{56} + q^{58} + 23 q^{59} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 3 q^{65} + 21 q^{67} + 6 q^{68} - 7 q^{70} - 27 q^{71} + 22 q^{73} - 7 q^{74} - 3 q^{76} + 27 q^{77} + 3 q^{79} - q^{80} + 4 q^{82} + 11 q^{83} + 20 q^{85} - 12 q^{86} - 7 q^{88} + 10 q^{89} - 19 q^{91} - 2 q^{92} - 8 q^{94} + 12 q^{95} - 5 q^{97} - 15 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - q^5 + 4 * q^7 - 3 * q^8 + q^10 + 7 * q^11 - q^13 - 4 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^19 - q^20 - 7 * q^22 - 2 * q^23 + 10 * q^25 + q^26 + 4 * q^28 - q^29 - 3 * q^31 - 3 * q^32 - 6 * q^34 + 7 * q^35 + 7 * q^37 + 3 * q^38 + q^40 - 4 * q^41 + 12 * q^43 + 7 * q^44 + 2 * q^46 + 8 * q^47 + 15 * q^49 - 10 * q^50 - q^52 - 11 * q^53 - 5 * q^55 - 4 * q^56 + q^58 + 23 * q^59 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 3 * q^65 + 21 * q^67 + 6 * q^68 - 7 * q^70 - 27 * q^71 + 22 * q^73 - 7 * q^74 - 3 * q^76 + 27 * q^77 + 3 * q^79 - q^80 + 4 * q^82 + 11 * q^83 + 20 * q^85 - 12 * q^86 - 7 * q^88 + 10 * q^89 - 19 * q^91 - 2 * q^92 - 8 * q^94 + 12 * q^95 - 5 * q^97 - 15 * q^98

Introduction

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