LMFDB - Embedded newform 6897.2.a.bq.1.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6897,2,Mod(1,6897)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6897 = 3 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6897.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,1,24,37,9,1,3,0,24,-1,0,37,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.0728222741$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 627)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6897.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.61883 q^{2} +1.00000 q^{3} +0.620622 q^{4} -3.40265 q^{5} +1.61883 q^{6} +4.29833 q^{7} -2.23298 q^{8} +1.00000 q^{9} -5.50832 q^{10} +0.620622 q^{12} -6.30293 q^{13} +6.95829 q^{14} -3.40265 q^{15} -4.85607 q^{16} +3.03943 q^{17} +1.61883 q^{18} +1.00000 q^{19} -2.11176 q^{20} +4.29833 q^{21} +7.60064 q^{23} -2.23298 q^{24} +6.57801 q^{25} -10.2034 q^{26} +1.00000 q^{27} +2.66764 q^{28} +0.482994 q^{29} -5.50832 q^{30} +0.850012 q^{31} -3.39520 q^{32} +4.92033 q^{34} -14.6257 q^{35} +0.620622 q^{36} -0.380554 q^{37} +1.61883 q^{38} -6.30293 q^{39} +7.59806 q^{40} +5.41273 q^{41} +6.95829 q^{42} -1.20988 q^{43} -3.40265 q^{45} +12.3042 q^{46} -6.56066 q^{47} -4.85607 q^{48} +11.4757 q^{49} +10.6487 q^{50} +3.03943 q^{51} -3.91174 q^{52} -12.7236 q^{53} +1.61883 q^{54} -9.59811 q^{56} +1.00000 q^{57} +0.781886 q^{58} +0.187988 q^{59} -2.11176 q^{60} +14.8977 q^{61} +1.37603 q^{62} +4.29833 q^{63} +4.21587 q^{64} +21.4467 q^{65} +4.83822 q^{67} +1.88634 q^{68} +7.60064 q^{69} -23.6766 q^{70} +12.5140 q^{71} -2.23298 q^{72} -4.09126 q^{73} -0.616053 q^{74} +6.57801 q^{75} +0.620622 q^{76} -10.2034 q^{78} +13.0069 q^{79} +16.5235 q^{80} +1.00000 q^{81} +8.76230 q^{82} +6.43241 q^{83} +2.66764 q^{84} -10.3421 q^{85} -1.95860 q^{86} +0.482994 q^{87} +3.64434 q^{89} -5.50832 q^{90} -27.0921 q^{91} +4.71712 q^{92} +0.850012 q^{93} -10.6206 q^{94} -3.40265 q^{95} -3.39520 q^{96} +0.830401 q^{97} +18.5772 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 24 q^{3} + 37 q^{4} + 9 q^{5} + q^{6} + 3 q^{7} + 24 q^{9} - q^{10} + 37 q^{12} + 3 q^{13} - 2 q^{14} + 9 q^{15} + 55 q^{16} - 3 q^{17} + q^{18} + 24 q^{19} + 12 q^{20} + 3 q^{21} + 30 q^{23}+ \cdots + 59 q^{98}+O(q^{100}) $ 24 * q + q^2 + 24 * q^3 + 37 * q^4 + 9 * q^5 + q^6 + 3 * q^7 + 24 * q^9 - q^10 + 37 * q^12 + 3 * q^13 - 2 * q^14 + 9 * q^15 + 55 * q^16 - 3 * q^17 + q^18 + 24 * q^19 + 12 * q^20 + 3 * q^21 + 30 * q^23 + 51 * q^25 - 6 * q^26 + 24 * q^27 - 17 * q^28 + 13 * q^29 - q^30 + 47 * q^31 + 16 * q^32 + 23 * q^34 - 26 * q^35 + 37 * q^36 + 56 * q^37 + q^38 + 3 * q^39 + 38 * q^40 - 5 * q^41 - 2 * q^42 - q^43 + 9 * q^45 - 23 * q^46 + 49 * q^47 + 55 * q^48 + 59 * q^49 - 28 * q^50 - 3 * q^51 + 5 * q^52 + 14 * q^53 + q^54 + 4 * q^56 + 24 * q^57 + 53 * q^58 + 17 * q^59 + 12 * q^60 - 8 * q^61 + 17 * q^62 + 3 * q^63 + 96 * q^64 + 30 * q^65 + 46 * q^67 - 66 * q^68 + 30 * q^69 + 23 * q^70 + 21 * q^71 - 13 * q^73 - 15 * q^74 + 51 * q^75 + 37 * q^76 - 6 * q^78 - 3 * q^79 + 36 * q^80 + 24 * q^81 + 38 * q^82 + 10 * q^83 - 17 * q^84 - 12 * q^85 - 17 * q^86 + 13 * q^87 + 21 * q^89 - q^90 + 20 * q^91 + 47 * q^92 + 47 * q^93 - 27 * q^94 + 9 * q^95 + 16 * q^96 + 70 * q^97 + 59 * q^98

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 $

$2$	1.61883	1.14469	0.572344	−	0.820014i	$-0.306034\pi$
$2$	1.61883	1.14469	0.572344	+	0.820014i	$0.306034\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.620622	0.310311
$5$	−3.40265	−1.52171	−0.760855	−	0.648922i	$-0.775220\pi$
$5$	−3.40265	−1.52171	−0.760855	+	0.648922i	$0.775220\pi$
$6$	1.61883	0.660886
$7$	4.29833	1.62462	0.812309	−	0.583228i	$-0.198210\pi$
$7$	4.29833	1.62462	0.812309	+	0.583228i	$0.198210\pi$
$8$	−2.23298	−0.789479
$9$	1.00000	0.333333
$10$	−5.50832	−1.74188
$11$	0	0
$11$	0	0
$12$	0.620622	0.179158
$13$	−6.30293	−1.74812	−0.874060	−	0.485819i	$-0.838522\pi$
$13$	−6.30293	−1.74812	−0.874060	+	0.485819i	$0.838522\pi$
$14$	6.95829	1.85968
$15$	−3.40265	−0.878560
$16$	−4.85607	−1.21402
$17$	3.03943	0.737170	0.368585	−	0.929594i	$-0.379842\pi$
$17$	3.03943	0.737170	0.368585	+	0.929594i	$0.379842\pi$
$18$	1.61883	0.381563
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.11176	−0.472203
$21$	4.29833	0.937973
$22$	0	0
$23$	7.60064	1.58484	0.792422	−	0.609974i	$-0.208820\pi$
$23$	7.60064	1.58484	0.792422	+	0.609974i	$0.208820\pi$
$24$	−2.23298	−0.455806
$25$	6.57801	1.31560
$26$	−10.2034	−2.00105
$27$	1.00000	0.192450
$28$	2.66764	0.504137
$29$	0.482994	0.0896897	0.0448448	−	0.998994i	$-0.485721\pi$
$29$	0.482994	0.0896897	0.0448448	+	0.998994i	$0.485721\pi$
$30$	−5.50832	−1.00568
$31$	0.850012	0.152667	0.0763334	−	0.997082i	$-0.475679\pi$
$31$	0.850012	0.152667	0.0763334	+	0.997082i	$0.475679\pi$
$32$	−3.39520	−0.600193
$33$	0	0
$34$	4.92033	0.843830
$35$	−14.6257	−2.47220
$36$	0.620622	0.103437
$37$	−0.380554	−0.0625627	−0.0312813	−	0.999511i	$-0.509959\pi$
$37$	−0.380554	−0.0625627	−0.0312813	+	0.999511i	$0.509959\pi$
$38$	1.61883	0.262609
$39$	−6.30293	−1.00928
$40$	7.59806	1.20136
$41$	5.41273	0.845326	0.422663	−	0.906287i	$-0.361095\pi$
$41$	5.41273	0.845326	0.422663	+	0.906287i	$0.361095\pi$
$42$	6.95829	1.07369
$43$	−1.20988	−0.184505	−0.0922526	−	0.995736i	$-0.529407\pi$
$43$	−1.20988	−0.184505	−0.0922526	+	0.995736i	$0.529407\pi$
$44$	0	0
$45$	−3.40265	−0.507237
$46$	12.3042	1.81415
$47$	−6.56066	−0.956971	−0.478485	−	0.878096i	$-0.658814\pi$
$47$	−6.56066	−0.956971	−0.478485	+	0.878096i	$0.658814\pi$
$48$	−4.85607	−0.700914
$49$	11.4757	1.63938
$50$	10.6487	1.50595
$51$	3.03943	0.425606
$52$	−3.91174	−0.542460
$53$	−12.7236	−1.74772	−0.873858	−	0.486181i	$-0.838390\pi$
$53$	−12.7236	−1.74772	−0.873858	+	0.486181i	$0.838390\pi$
$54$	1.61883	0.220295
$55$	0	0
$56$	−9.59811	−1.28260
$57$	1.00000	0.132453
$58$	0.781886	0.102667
$59$	0.187988	0.0244739	0.0122370	−	0.999925i	$-0.496105\pi$
$59$	0.187988	0.0244739	0.0122370	+	0.999925i	$0.496105\pi$
$60$	−2.11176	−0.272627
$61$	14.8977	1.90745	0.953725	−	0.300682i	$-0.0972141\pi$
$61$	14.8977	1.90745	0.953725	+	0.300682i	$0.0972141\pi$
$62$	1.37603	0.174756
$63$	4.29833	0.541539
$64$	4.21587	0.526984
$65$	21.4467	2.66013
$66$	0	0
$67$	4.83822	0.591082	0.295541	−	0.955330i	$-0.404500\pi$
$67$	4.83822	0.591082	0.295541	+	0.955330i	$0.404500\pi$
$68$	1.88634	0.228752
$69$	7.60064	0.915010
$70$	−23.6766	−2.82989
$71$	12.5140	1.48514	0.742571	−	0.669768i	$-0.233606\pi$
$71$	12.5140	1.48514	0.742571	+	0.669768i	$0.233606\pi$
$72$	−2.23298	−0.263160
$73$	−4.09126	−0.478846	−0.239423	−	0.970915i	$-0.576958\pi$
$73$	−4.09126	−0.478846	−0.239423	+	0.970915i	$0.576958\pi$
$74$	−0.616053	−0.0716148
$75$	6.57801	0.759563
$76$	0.620622	0.0711902
$77$	0	0
$78$	−10.2034	−1.15531
$79$	13.0069	1.46339	0.731693	−	0.681635i	$-0.238731\pi$
$79$	13.0069	1.46339	0.731693	+	0.681635i	$0.238731\pi$
$80$	16.5235	1.84738
$81$	1.00000	0.111111
$82$	8.76230	0.967635
$83$	6.43241	0.706048	0.353024	−	0.935614i	$-0.385153\pi$
$83$	6.43241	0.706048	0.353024	+	0.935614i	$0.385153\pi$
$84$	2.66764	0.291063
$85$	−10.3421	−1.12176
$86$	−1.95860	−0.211201
$87$	0.482994	0.0517824
$88$	0	0
$89$	3.64434	0.386299	0.193150	−	0.981169i	$-0.438130\pi$
$89$	3.64434	0.386299	0.193150	+	0.981169i	$0.438130\pi$
$90$	−5.50832	−0.580628
$91$	−27.0921	−2.84003
$92$	4.71712	0.491794
$93$	0.850012	0.0881422
$94$	−10.6206	−1.09543
$95$	−3.40265	−0.349104
$96$	−3.39520	−0.346522
$97$	0.830401	0.0843145	0.0421572	−	0.999111i	$-0.486577\pi$
$97$	0.830401	0.0843145	0.0421572	+	0.999111i	$0.486577\pi$
$98$	18.5772	1.87658
$99$	0	0
$100$	4.08245	0.408245
$101$	−10.7206	−1.06674	−0.533370	−	0.845882i	$-0.679075\pi$
$101$	−10.7206	−1.06674	−0.533370	+	0.845882i	$0.679075\pi$
$102$	4.92033	0.487186
$103$	13.4613	1.32638	0.663191	−	0.748450i	$-0.269202\pi$
$103$	13.4613	1.32638	0.663191	+	0.748450i	$0.269202\pi$
$104$	14.0743	1.38010
$105$	−14.6257	−1.42732
$106$	−20.5973	−2.00059
$107$	15.5046	1.49888	0.749441	−	0.662071i	$-0.230322\pi$
$107$	15.5046	1.49888	0.749441	+	0.662071i	$0.230322\pi$
$108$	0.620622	0.0597194
$109$	−5.64576	−0.540766	−0.270383	−	0.962753i	$-0.587150\pi$
$109$	−5.64576	−0.540766	−0.270383	+	0.962753i	$0.587150\pi$
$110$	0	0
$111$	−0.380554	−0.0361206
$112$	−20.8730	−1.97232
$113$	−4.35151	−0.409355	−0.204678	−	0.978829i	$-0.565615\pi$
$113$	−4.35151	−0.409355	−0.204678	+	0.978829i	$0.565615\pi$
$114$	1.61883	0.151618
$115$	−25.8623	−2.41167
$116$	0.299756	0.0278317
$117$	−6.30293	−0.582706
$118$	0.304321	0.0280150
$119$	13.0645	1.19762
$120$	7.59806	0.693604
$121$	0	0
$122$	24.1168	2.18343
$123$	5.41273	0.488049
$124$	0.527536	0.0473741
$125$	−5.36940	−0.480254
$126$	6.95829	0.619894
$127$	17.1349	1.52047	0.760236	−	0.649646i	$-0.225083\pi$
$127$	17.1349	1.52047	0.760236	+	0.649646i	$0.225083\pi$
$128$	13.6152	1.20343
$129$	−1.20988	−0.106524
$130$	34.7186	3.04502
$131$	14.6778	1.28241	0.641204	−	0.767371i	$-0.278435\pi$
$131$	14.6778	1.28241	0.641204	+	0.767371i	$0.278435\pi$
$132$	0	0
$133$	4.29833	0.372713
$134$	7.83227	0.676605
$135$	−3.40265	−0.292853
$136$	−6.78700	−0.581981
$137$	−17.7825	−1.51926	−0.759630	−	0.650356i	$-0.774620\pi$
$137$	−17.7825	−1.51926	−0.759630	+	0.650356i	$0.774620\pi$
$138$	12.3042	1.04740
$139$	12.0482	1.02192	0.510959	−	0.859605i	$-0.329290\pi$
$139$	12.0482	1.02192	0.510959	+	0.859605i	$0.329290\pi$
$140$	−9.07704	−0.767150
$141$	−6.56066	−0.552507
$142$	20.2581	1.70002
$143$	0	0
$144$	−4.85607	−0.404673
$145$	−1.64346	−0.136482
$146$	−6.62307	−0.548129
$147$	11.4757	0.946498
$148$	−0.236180	−0.0194139
$149$	−9.83653	−0.805840	−0.402920	−	0.915235i	$-0.632005\pi$
$149$	−9.83653	−0.805840	−0.402920	+	0.915235i	$0.632005\pi$
$150$	10.6487	0.869463
$151$	−10.2538	−0.834444	−0.417222	−	0.908805i	$-0.636996\pi$
$151$	−10.2538	−0.834444	−0.417222	+	0.908805i	$0.636996\pi$
$152$	−2.23298	−0.181119
$153$	3.03943	0.245723
$154$	0	0
$155$	−2.89229	−0.232314
$156$	−3.91174	−0.313190
$157$	3.25955	0.260140	0.130070	−	0.991505i	$-0.458480\pi$
$157$	3.25955	0.260140	0.130070	+	0.991505i	$0.458480\pi$
$158$	21.0559	1.67512
$159$	−12.7236	−1.00904
$160$	11.5527	0.913320
$161$	32.6701	2.57476
$162$	1.61883	0.127188
$163$	0.267410	0.0209451	0.0104726	−	0.999945i	$-0.496666\pi$
$163$	0.267410	0.0209451	0.0104726	+	0.999945i	$0.496666\pi$
$164$	3.35926	0.262314
$165$	0	0
$166$	10.4130	0.808205
$167$	11.7560	0.909704	0.454852	−	0.890567i	$-0.349692\pi$
$167$	11.7560	0.909704	0.454852	+	0.890567i	$0.349692\pi$
$168$	−9.59811	−0.740510
$169$	26.7270	2.05592
$170$	−16.7422	−1.28406
$171$	1.00000	0.0764719
$172$	−0.750879	−0.0572540
$173$	−0.604841	−0.0459852	−0.0229926	−	0.999736i	$-0.507319\pi$
$173$	−0.604841	−0.0459852	−0.0229926	+	0.999736i	$0.507319\pi$
$174$	0.781886	0.0592746
$175$	28.2745	2.13735
$176$	0	0
$177$	0.187988	0.0141300
$178$	5.89958	0.442192
$179$	−0.304566	−0.0227643	−0.0113822	−	0.999935i	$-0.503623\pi$
$179$	−0.304566	−0.0227643	−0.0113822	+	0.999935i	$0.503623\pi$
$180$	−2.11176	−0.157401
$181$	10.4909	0.779783	0.389892	−	0.920861i	$-0.372512\pi$
$181$	10.4909	0.779783	0.389892	+	0.920861i	$0.372512\pi$
$182$	−43.8576	−3.25094
$183$	14.8977	1.10127
$184$	−16.9721	−1.25120
$185$	1.29489	0.0952023
$186$	1.37603	0.100895
$187$	0	0
$188$	−4.07169	−0.296958
$189$	4.29833	0.312658
$190$	−5.50832	−0.399615
$191$	−9.78243	−0.707832	−0.353916	−	0.935277i	$-0.615150\pi$
$191$	−9.78243	−0.707832	−0.353916	+	0.935277i	$0.615150\pi$
$192$	4.21587	0.304254
$193$	−9.09080	−0.654370	−0.327185	−	0.944960i	$-0.606100\pi$
$193$	−9.09080	−0.654370	−0.327185	+	0.944960i	$0.606100\pi$
$194$	1.34428	0.0965138
$195$	21.4467	1.53583
$196$	7.12206	0.508718
$197$	14.6543	1.04408	0.522040	−	0.852921i	$-0.325171\pi$
$197$	14.6543	1.04408	0.522040	+	0.852921i	$0.325171\pi$
$198$	0	0
$199$	11.3370	0.803661	0.401831	−	0.915714i	$-0.368374\pi$
$199$	11.3370	0.803661	0.401831	+	0.915714i	$0.368374\pi$
$200$	−14.6886	−1.03864
$201$	4.83822	0.341261
$202$	−17.3549	−1.22109
$203$	2.07607	0.145711
$204$	1.88634	0.132070
$205$	−18.4176	−1.28634
$206$	21.7916	1.51829
$207$	7.60064	0.528281
$208$	30.6075	2.12225
$209$	0	0
$210$	−23.6766	−1.63384
$211$	−11.0746	−0.762404	−0.381202	−	0.924492i	$-0.624490\pi$
$211$	−11.0746	−0.762404	−0.381202	+	0.924492i	$0.624490\pi$
$212$	−7.89652	−0.542336
$213$	12.5140	0.857447
$214$	25.0993	1.71575
$215$	4.11680	0.280763
$216$	−2.23298	−0.151935
$217$	3.65364	0.248025
$218$	−9.13954	−0.619008
$219$	−4.09126	−0.276462
$220$	0	0
$221$	−19.1573	−1.28866
$222$	−0.616053	−0.0413468
$223$	25.6691	1.71893	0.859467	−	0.511192i	$-0.170796\pi$
$223$	25.6691	1.71893	0.859467	+	0.511192i	$0.170796\pi$
$224$	−14.5937	−0.975084
$225$	6.57801	0.438534
$226$	−7.04436	−0.468584
$227$	−23.4800	−1.55842	−0.779210	−	0.626764i	$-0.784379\pi$
$227$	−23.4800	−1.55842	−0.779210	+	0.626764i	$0.784379\pi$
$228$	0.620622	0.0411017
$229$	13.7327	0.907481	0.453741	−	0.891134i	$-0.350089\pi$
$229$	13.7327	0.907481	0.453741	+	0.891134i	$0.350089\pi$
$230$	−41.8668	−2.76061
$231$	0	0
$232$	−1.07852	−0.0708081
$233$	−4.64661	−0.304410	−0.152205	−	0.988349i	$-0.548637\pi$
$233$	−4.64661	−0.304410	−0.152205	+	0.988349i	$0.548637\pi$
$234$	−10.2034	−0.667017
$235$	22.3236	1.45623
$236$	0.116669	0.00759452
$237$	13.0069	0.844886
$238$	21.1492	1.37090
$239$	−6.86628	−0.444143	−0.222071	−	0.975030i	$-0.571282\pi$
$239$	−6.86628	−0.444143	−0.222071	+	0.975030i	$0.571282\pi$
$240$	16.5235	1.06659
$241$	−4.81072	−0.309886	−0.154943	−	0.987923i	$-0.549519\pi$
$241$	−4.81072	−0.309886	−0.154943	+	0.987923i	$0.549519\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	9.24581	0.591902
$245$	−39.0477	−2.49466
$246$	8.76230	0.558664
$247$	−6.30293	−0.401046
$248$	−1.89806	−0.120527
$249$	6.43241	0.407637
$250$	−8.69217	−0.549741
$251$	−24.9376	−1.57405	−0.787023	−	0.616923i	$-0.788379\pi$
$251$	−24.9376	−1.57405	−0.787023	+	0.616923i	$0.788379\pi$
$252$	2.66764	0.168046
$253$	0	0
$254$	27.7385	1.74047
$255$	−10.3421	−0.647648
$256$	13.6090	0.850563
$257$	4.05550	0.252975	0.126487	−	0.991968i	$-0.459630\pi$
$257$	4.05550	0.252975	0.126487	+	0.991968i	$0.459630\pi$
$258$	−1.95860	−0.121937
$259$	−1.63575	−0.101640
$260$	13.3103	0.825468
$261$	0.482994	0.0298966
$262$	23.7610	1.46796
$263$	5.92290	0.365222	0.182611	−	0.983185i	$-0.441545\pi$
$263$	5.92290	0.365222	0.182611	+	0.983185i	$0.441545\pi$
$264$	0	0
$265$	43.2938	2.65952
$266$	6.95829	0.426640
$267$	3.64434	0.223030
$268$	3.00270	0.183419
$269$	14.5610	0.887801	0.443900	−	0.896076i	$-0.353594\pi$
$269$	14.5610	0.887801	0.443900	+	0.896076i	$0.353594\pi$
$270$	−5.50832	−0.335226
$271$	8.16105	0.495748	0.247874	−	0.968792i	$-0.420268\pi$
$271$	8.16105	0.495748	0.247874	+	0.968792i	$0.420268\pi$
$272$	−14.7597	−0.894938
$273$	−27.0921	−1.63969
$274$	−28.7869	−1.73908
$275$	0	0
$276$	4.71712	0.283938
$277$	4.53940	0.272746	0.136373	−	0.990658i	$-0.456455\pi$
$277$	4.53940	0.272746	0.136373	+	0.990658i	$0.456455\pi$
$278$	19.5041	1.16978
$279$	0.850012	0.0508889
$280$	32.6590	1.95175
$281$	−29.6230	−1.76716	−0.883579	−	0.468282i	$-0.844873\pi$
$281$	−29.6230	−1.76716	−0.883579	+	0.468282i	$0.844873\pi$
$282$	−10.6206	−0.632449
$283$	−9.09551	−0.540672	−0.270336	−	0.962766i	$-0.587135\pi$
$283$	−9.09551	−0.540672	−0.270336	+	0.962766i	$0.587135\pi$
$284$	7.76647	0.460855
$285$	−3.40265	−0.201555
$286$	0	0
$287$	23.2657	1.37333
$288$	−3.39520	−0.200064
$289$	−7.76186	−0.456580
$290$	−2.66048	−0.156229
$291$	0.830401	0.0486790
$292$	−2.53912	−0.148591
$293$	25.8607	1.51080	0.755400	−	0.655264i	$-0.227443\pi$
$293$	25.8607	1.51080	0.755400	+	0.655264i	$0.227443\pi$
$294$	18.5772	1.08344
$295$	−0.639656	−0.0372422
$296$	0.849771	0.0493919
$297$	0	0
$298$	−15.9237	−0.922435
$299$	−47.9063	−2.77050
$300$	4.08245	0.235701
$301$	−5.20048	−0.299750
$302$	−16.5992	−0.955178
$303$	−10.7206	−0.615883
$304$	−4.85607	−0.278515
$305$	−50.6915	−2.90258
$306$	4.92033	0.281277
$307$	−13.6516	−0.779140	−0.389570	−	0.920997i	$-0.627376\pi$
$307$	−13.6516	−0.779140	−0.389570	+	0.920997i	$0.627376\pi$
$308$	0	0
$309$	13.4613	0.765787
$310$	−4.68214	−0.265928
$311$	−5.06909	−0.287442	−0.143721	−	0.989618i	$-0.545907\pi$
$311$	−5.06909	−0.287442	−0.143721	+	0.989618i	$0.545907\pi$
$312$	14.0743	0.796803
$313$	15.3606	0.868231	0.434116	−	0.900857i	$-0.357061\pi$
$313$	15.3606	0.868231	0.434116	+	0.900857i	$0.357061\pi$
$314$	5.27666	0.297779
$315$	−14.6257	−0.824066
$316$	8.07234	0.454104
$317$	1.01520	0.0570194	0.0285097	−	0.999594i	$-0.490924\pi$
$317$	1.01520	0.0570194	0.0285097	+	0.999594i	$0.490924\pi$
$318$	−20.5973	−1.15504
$319$	0	0
$320$	−14.3451	−0.801917
$321$	15.5046	0.865380
$322$	52.8875	2.94730
$323$	3.03943	0.169118
$324$	0.620622	0.0344790
$325$	−41.4607	−2.29983
$326$	0.432892	0.0239757
$327$	−5.64576	−0.312211
$328$	−12.0865	−0.667367
$329$	−28.1999	−1.55471
$330$	0	0
$331$	−7.51641	−0.413139	−0.206570	−	0.978432i	$-0.566230\pi$
$331$	−7.51641	−0.413139	−0.206570	+	0.978432i	$0.566230\pi$
$332$	3.99209	0.219094
$333$	−0.380554	−0.0208542
$334$	19.0310	1.04133
$335$	−16.4627	−0.899456
$336$	−20.8730	−1.13872
$337$	−19.8591	−1.08179	−0.540897	−	0.841089i	$-0.681915\pi$
$337$	−19.8591	−1.08179	−0.540897	+	0.841089i	$0.681915\pi$
$338$	43.2665	2.35339
$339$	−4.35151	−0.236341
$340$	−6.41854	−0.348094
$341$	0	0
$342$	1.61883	0.0875365
$343$	19.2380	1.03875
$344$	2.70165	0.145663
$345$	−25.8623	−1.39238
$346$	−0.979137	−0.0526388
$347$	−16.0320	−0.860643	−0.430322	−	0.902676i	$-0.641600\pi$
$347$	−16.0320	−0.860643	−0.430322	+	0.902676i	$0.641600\pi$
$348$	0.299756	0.0160686
$349$	−8.91102	−0.476996	−0.238498	−	0.971143i	$-0.576655\pi$
$349$	−8.91102	−0.476996	−0.238498	+	0.971143i	$0.576655\pi$
$350$	45.7717	2.44660
$351$	−6.30293	−0.336426
$352$	0	0
$353$	5.45154	0.290156	0.145078	−	0.989420i	$-0.453657\pi$
$353$	5.45154	0.290156	0.145078	+	0.989420i	$0.453657\pi$
$354$	0.304321	0.0161745
$355$	−42.5808	−2.25995
$356$	2.26176	0.119873
$357$	13.0645	0.691446
$358$	−0.493042	−0.0260581
$359$	2.61761	0.138152	0.0690760	−	0.997611i	$-0.477995\pi$
$359$	2.61761	0.138152	0.0690760	+	0.997611i	$0.477995\pi$
$360$	7.59806	0.400453
$361$	1.00000	0.0526316
$362$	16.9830	0.892608
$363$	0	0
$364$	−16.8140	−0.881291
$365$	13.9211	0.728664
$366$	24.1168	1.26061
$367$	−24.5721	−1.28265	−0.641326	−	0.767268i	$-0.721616\pi$
$367$	−24.5721	−1.28265	−0.641326	+	0.767268i	$0.721616\pi$
$368$	−36.9093	−1.92403
$369$	5.41273	0.281775
$370$	2.09621	0.108977
$371$	−54.6902	−2.83937
$372$	0.527536	0.0273515
$373$	15.2655	0.790417	0.395208	−	0.918591i	$-0.370672\pi$
$373$	15.2655	0.790417	0.395208	+	0.918591i	$0.370672\pi$
$374$	0	0
$375$	−5.36940	−0.277275
$376$	14.6498	0.755508
$377$	−3.04428	−0.156788
$378$	6.95829	0.357896
$379$	13.2369	0.679932	0.339966	−	0.940438i	$-0.389584\pi$
$379$	13.2369	0.679932	0.339966	+	0.940438i	$0.389584\pi$
$380$	−2.11176	−0.108331
$381$	17.1349	0.877845
$382$	−15.8361	−0.810247
$383$	15.6941	0.801932	0.400966	−	0.916093i	$-0.368674\pi$
$383$	15.6941	0.801932	0.400966	+	0.916093i	$0.368674\pi$
$384$	13.6152	0.694798
$385$	0	0
$386$	−14.7165	−0.749050
$387$	−1.20988	−0.0615017
$388$	0.515365	0.0261637
$389$	−12.9049	−0.654305	−0.327153	−	0.944972i	$-0.606089\pi$
$389$	−12.9049	−0.654305	−0.327153	+	0.944972i	$0.606089\pi$
$390$	34.7186	1.75804
$391$	23.1016	1.16830
$392$	−25.6250	−1.29426
$393$	14.6778	0.740399
$394$	23.7230	1.19514
$395$	−44.2577	−2.22685
$396$	0	0
$397$	24.3055	1.21986	0.609930	−	0.792455i	$-0.291198\pi$
$397$	24.3055	1.21986	0.609930	+	0.792455i	$0.291198\pi$
$398$	18.3528	0.919941
$399$	4.29833	0.215186
$400$	−31.9433	−1.59716
$401$	10.5381	0.526250	0.263125	−	0.964762i	$-0.415247\pi$
$401$	10.5381	0.526250	0.263125	+	0.964762i	$0.415247\pi$
$402$	7.83227	0.390638
$403$	−5.35757	−0.266880
$404$	−6.65345	−0.331021
$405$	−3.40265	−0.169079
$406$	3.36081	0.166794
$407$	0	0
$408$	−6.78700	−0.336007
$409$	34.0436	1.68335	0.841674	−	0.539986i	$-0.181571\pi$
$409$	34.0436	1.68335	0.841674	+	0.539986i	$0.181571\pi$
$410$	−29.8150	−1.47246
$411$	−17.7825	−0.877145
$412$	8.35438	0.411591
$413$	0.808034	0.0397608
$414$	12.3042	0.604717
$415$	−21.8872	−1.07440
$416$	21.3998	1.04921
$417$	12.0482	0.590005
$418$	0	0
$419$	−22.7546	−1.11163	−0.555817	−	0.831305i	$-0.687595\pi$
$419$	−22.7546	−1.11163	−0.555817	+	0.831305i	$0.687595\pi$
$420$	−9.07704	−0.442914
$421$	25.5640	1.24591	0.622957	−	0.782256i	$-0.285931\pi$
$421$	25.5640	1.24591	0.622957	+	0.782256i	$0.285931\pi$
$422$	−17.9279	−0.872714
$423$	−6.56066	−0.318990
$424$	28.4115	1.37979
$425$	19.9934	0.969822
$426$	20.2581	0.981509
$427$	64.0351	3.09888
$428$	9.62247	0.465119
$429$	0	0
$430$	6.66441	0.321387
$431$	−19.3171	−0.930473	−0.465237	−	0.885186i	$-0.654031\pi$
$431$	−19.3171	−0.930473	−0.465237	+	0.885186i	$0.654031\pi$
$432$	−4.85607	−0.233638
$433$	−6.43654	−0.309320	−0.154660	−	0.987968i	$-0.549428\pi$
$433$	−6.43654	−0.309320	−0.154660	+	0.987968i	$0.549428\pi$
$434$	5.91463	0.283911
$435$	−1.64346	−0.0787977
$436$	−3.50388	−0.167805
$437$	7.60064	0.363588
$438$	−6.62307	−0.316462
$439$	−28.2963	−1.35051	−0.675255	−	0.737585i	$-0.735966\pi$
$439$	−28.2963	−1.35051	−0.675255	+	0.737585i	$0.735966\pi$
$440$	0	0
$441$	11.4757	0.546461
$442$	−31.0125	−1.47512
$443$	−39.6011	−1.88151	−0.940754	−	0.339091i	$-0.889881\pi$
$443$	−39.6011	−1.88151	−0.940754	+	0.339091i	$0.889881\pi$
$444$	−0.236180	−0.0112086
$445$	−12.4004	−0.587835
$446$	41.5541	1.96764
$447$	−9.83653	−0.465252
$448$	18.1212	0.856148
$449$	−0.108917	−0.00514010	−0.00257005	−	0.999997i	$-0.500818\pi$
$449$	−0.108917	−0.00514010	−0.00257005	+	0.999997i	$0.500818\pi$
$450$	10.6487	0.501984
$451$	0	0
$452$	−2.70064	−0.127027
$453$	−10.2538	−0.481766
$454$	−38.0101	−1.78390
$455$	92.1849	4.32170
$456$	−2.23298	−0.104569
$457$	−3.71701	−0.173874	−0.0869371	−	0.996214i	$-0.527708\pi$
$457$	−3.71701	−0.173874	−0.0869371	+	0.996214i	$0.527708\pi$
$458$	22.2309	1.03878
$459$	3.03943	0.141869
$460$	−16.0507	−0.748368
$461$	6.87357	0.320134	0.160067	−	0.987106i	$-0.448829\pi$
$461$	6.87357	0.320134	0.160067	+	0.987106i	$0.448829\pi$
$462$	0	0
$463$	30.4807	1.41656	0.708278	−	0.705933i	$-0.249472\pi$
$463$	30.4807	1.41656	0.708278	+	0.705933i	$0.249472\pi$
$464$	−2.34545	−0.108885
$465$	−2.89229	−0.134127
$466$	−7.52209	−0.348454
$467$	23.4716	1.08614	0.543068	−	0.839689i	$-0.317262\pi$
$467$	23.4716	1.08614	0.543068	+	0.839689i	$0.317262\pi$
$468$	−3.91174	−0.180820
$469$	20.7963	0.960283
$470$	36.1382	1.66693
$471$	3.25955	0.150192
$472$	−0.419774	−0.0193216
$473$	0	0
$474$	21.0559	0.967131
$475$	6.57801	0.301820
$476$	8.10811	0.371635
$477$	−12.7236	−0.582572
$478$	−11.1154	−0.508405
$479$	−11.2954	−0.516099	−0.258049	−	0.966132i	$-0.583080\pi$
$479$	−11.2954	−0.516099	−0.258049	+	0.966132i	$0.583080\pi$
$480$	11.5527	0.527305
$481$	2.39861	0.109367
$482$	−7.78775	−0.354722
$483$	32.6701	1.48654
$484$	0	0
$485$	−2.82556	−0.128302
$486$	1.61883	0.0734318
$487$	39.7896	1.80304	0.901519	−	0.432739i	$-0.142453\pi$
$487$	39.7896	1.80304	0.901519	+	0.432739i	$0.142453\pi$
$488$	−33.2662	−1.50589
$489$	0.267410	0.0120927
$490$	−63.2117	−2.85561
$491$	−35.3165	−1.59381	−0.796905	−	0.604105i	$-0.793531\pi$
$491$	−35.3165	−1.59381	−0.796905	+	0.604105i	$0.793531\pi$
$492$	3.35926	0.151447
$493$	1.46803	0.0661166
$494$	−10.2034	−0.459073
$495$	0	0
$496$	−4.12772	−0.185340
$497$	53.7894	2.41279
$498$	10.4130	0.466617
$499$	−8.18525	−0.366422	−0.183211	−	0.983074i	$-0.558649\pi$
$499$	−8.18525	−0.366422	−0.183211	+	0.983074i	$0.558649\pi$
$500$	−3.33237	−0.149028
$501$	11.7560	0.525218
$502$	−40.3698	−1.80179
$503$	−8.33435	−0.371610	−0.185805	−	0.982587i	$-0.559489\pi$
$503$	−8.33435	−0.371610	−0.185805	+	0.982587i	$0.559489\pi$
$504$	−9.59811	−0.427534
$505$	36.4785	1.62327
$506$	0	0
$507$	26.7270	1.18699
$508$	10.6343	0.471819
$509$	−1.31934	−0.0584788	−0.0292394	−	0.999572i	$-0.509309\pi$
$509$	−1.31934	−0.0584788	−0.0292394	+	0.999572i	$0.509309\pi$
$510$	−16.7422	−0.741355
$511$	−17.5856	−0.777941
$512$	−5.19970	−0.229796
$513$	1.00000	0.0441511
$514$	6.56517	0.289577
$515$	−45.8041	−2.01837
$516$	−0.750879	−0.0330556
$517$	0	0
$518$	−2.64800	−0.116347
$519$	−0.604841	−0.0265496
$520$	−47.8900	−2.10012
$521$	−11.2800	−0.494186	−0.247093	−	0.968992i	$-0.579475\pi$
$521$	−11.2800	−0.494186	−0.247093	+	0.968992i	$0.579475\pi$
$522$	0.781886	0.0342222
$523$	27.2021	1.18946	0.594732	−	0.803924i	$-0.297258\pi$
$523$	27.2021	1.18946	0.594732	+	0.803924i	$0.297258\pi$
$524$	9.10938	0.397945
$525$	28.2745	1.23400
$526$	9.58819	0.418065
$527$	2.58355	0.112541
$528$	0	0
$529$	34.7698	1.51173
$530$	70.0855	3.04432
$531$	0.187988	0.00815797
$532$	2.66764	0.115657
$533$	−34.1161	−1.47773
$534$	5.89958	0.255300
$535$	−52.7565	−2.28086
$536$	−10.8037	−0.466647
$537$	−0.304566	−0.0131430
$538$	23.5719	1.01625
$539$	0	0
$540$	−2.11176	−0.0908755
$541$	−25.1418	−1.08093	−0.540465	−	0.841367i	$-0.681752\pi$
$541$	−25.1418	−1.08093	−0.540465	+	0.841367i	$0.681752\pi$
$542$	13.2114	0.567477
$543$	10.4909	0.450208
$544$	−10.3195	−0.442445
$545$	19.2105	0.822888
$546$	−43.8576	−1.87693
$547$	−26.7928	−1.14558	−0.572788	−	0.819704i	$-0.694138\pi$
$547$	−26.7928	−1.14558	−0.572788	+	0.819704i	$0.694138\pi$
$548$	−11.0362	−0.471443
$549$	14.8977	0.635816
$550$	0	0
$551$	0.482994	0.0205762
$552$	−16.9721	−0.722381
$553$	55.9078	2.37744
$554$	7.34853	0.312209
$555$	1.29489	0.0549650
$556$	7.47740	0.317112
$557$	44.4127	1.88183	0.940913	−	0.338647i	$-0.109969\pi$
$557$	44.4127	1.88183	0.940913	+	0.338647i	$0.109969\pi$
$558$	1.37603	0.0582519
$559$	7.62580	0.322537
$560$	71.0235	3.00129
$561$	0	0
$562$	−47.9547	−2.02285
$563$	−16.4171	−0.691899	−0.345949	−	0.938253i	$-0.612443\pi$
$563$	−16.4171	−0.691899	−0.345949	+	0.938253i	$0.612443\pi$
$564$	−4.07169	−0.171449
$565$	14.8066	0.622920
$566$	−14.7241	−0.618901
$567$	4.29833	0.180513
$568$	−27.9436	−1.17249
$569$	29.8588	1.25175	0.625873	−	0.779925i	$-0.284743\pi$
$569$	29.8588	1.25175	0.625873	+	0.779925i	$0.284743\pi$
$570$	−5.50832	−0.230718
$571$	1.14808	0.0480458	0.0240229	−	0.999711i	$-0.492353\pi$
$571$	1.14808	0.0480458	0.0240229	+	0.999711i	$0.492353\pi$
$572$	0	0
$573$	−9.78243	−0.408667
$574$	37.6633	1.57204
$575$	49.9971	2.08502
$576$	4.21587	0.175661
$577$	−13.0927	−0.545055	−0.272527	−	0.962148i	$-0.587860\pi$
$577$	−13.0927	−0.545055	−0.272527	+	0.962148i	$0.587860\pi$
$578$	−12.5652	−0.522641
$579$	−9.09080	−0.377801
$580$	−1.01997	−0.0423517
$581$	27.6486	1.14706
$582$	1.34428	0.0557223
$583$	0	0
$584$	9.13571	0.378039
$585$	21.4467	0.886710
$586$	41.8642	1.72939
$587$	−30.3647	−1.25329	−0.626643	−	0.779306i	$-0.715572\pi$
$587$	−30.3647	−1.25329	−0.626643	+	0.779306i	$0.715572\pi$
$588$	7.12206	0.293709
$589$	0.850012	0.0350241
$590$	−1.03550	−0.0426307
$591$	14.6543	0.602799
$592$	1.84800	0.0759522
$593$	−2.44837	−0.100543	−0.0502713	−	0.998736i	$-0.516009\pi$
$593$	−2.44837	−0.100543	−0.0502713	+	0.998736i	$0.516009\pi$
$594$	0	0
$595$	−44.4539	−1.82243
$596$	−6.10476	−0.250061
$597$	11.3370	0.463994
$598$	−77.5524	−3.17135
$599$	−19.7952	−0.808808	−0.404404	−	0.914580i	$-0.632521\pi$
$599$	−19.7952	−0.808808	−0.404404	+	0.914580i	$0.632521\pi$
$600$	−14.6886	−0.599659
$601$	−15.5014	−0.632317	−0.316159	−	0.948706i	$-0.602393\pi$
$601$	−15.5014	−0.632317	−0.316159	+	0.948706i	$0.602393\pi$
$602$	−8.41870	−0.343121
$603$	4.83822	0.197027
$604$	−6.36374	−0.258937
$605$	0	0
$606$	−17.3549	−0.704994
$607$	−29.4019	−1.19339	−0.596693	−	0.802470i	$-0.703519\pi$
$607$	−29.4019	−1.19339	−0.596693	+	0.802470i	$0.703519\pi$
$608$	−3.39520	−0.137694
$609$	2.07607	0.0841265
$610$	−82.0610	−3.32255
$611$	41.3514	1.67290
$612$	1.88634	0.0762507
$613$	1.74886	0.0706359	0.0353180	−	0.999376i	$-0.488756\pi$
$613$	1.74886	0.0706359	0.0353180	+	0.999376i	$0.488756\pi$
$614$	−22.0997	−0.891872
$615$	−18.4176	−0.742669
$616$	0	0
$617$	12.9992	0.523329	0.261665	−	0.965159i	$-0.415729\pi$
$617$	12.9992	0.523329	0.261665	+	0.965159i	$0.415729\pi$
$618$	21.7916	0.876587
$619$	35.6811	1.43414	0.717072	−	0.696999i	$-0.245482\pi$
$619$	35.6811	1.43414	0.717072	+	0.696999i	$0.245482\pi$
$620$	−1.79502	−0.0720897
$621$	7.60064	0.305003
$622$	−8.20601	−0.329031
$623$	15.6646	0.627588
$624$	30.6075	1.22528
$625$	−14.6199	−0.584794
$626$	24.8662	0.993854
$627$	0	0
$628$	2.02295	0.0807243
$629$	−1.15667	−0.0461194
$630$	−23.6766	−0.943298
$631$	−23.4314	−0.932791	−0.466395	−	0.884576i	$-0.654448\pi$
$631$	−23.4314	−0.932791	−0.466395	+	0.884576i	$0.654448\pi$
$632$	−29.0441	−1.15531
$633$	−11.0746	−0.440174
$634$	1.64344	0.0652694
$635$	−58.3039	−2.31372
$636$	−7.89652	−0.313118
$637$	−72.3304	−2.86584
$638$	0	0
$639$	12.5140	0.495047
$640$	−46.3277	−1.83126
$641$	22.6263	0.893685	0.446843	−	0.894613i	$-0.352548\pi$
$641$	22.6263	0.893685	0.446843	+	0.894613i	$0.352548\pi$
$642$	25.0993	0.990590
$643$	37.7505	1.48873	0.744367	−	0.667771i	$-0.232751\pi$
$643$	37.7505	1.48873	0.744367	+	0.667771i	$0.232751\pi$
$644$	20.2758	0.798978
$645$	4.11680	0.162099
$646$	4.92033	0.193588
$647$	−31.1079	−1.22298	−0.611489	−	0.791253i	$-0.709429\pi$
$647$	−31.1079	−1.22298	−0.611489	+	0.791253i	$0.709429\pi$
$648$	−2.23298	−0.0877199
$649$	0	0
$650$	−67.1180	−2.63259
$651$	3.65364	0.143197
$652$	0.165960	0.00649950
$653$	−8.14725	−0.318826	−0.159413	−	0.987212i	$-0.550960\pi$
$653$	−8.14725	−0.318826	−0.159413	+	0.987212i	$0.550960\pi$
$654$	−9.13954	−0.357384
$655$	−49.9435	−1.95145
$656$	−26.2846	−1.02624
$657$	−4.09126	−0.159615
$658$	−45.6510	−1.77966
$659$	−4.80144	−0.187038	−0.0935188	−	0.995618i	$-0.529812\pi$
$659$	−4.80144	−0.187038	−0.0935188	+	0.995618i	$0.529812\pi$
$660$	0	0
$661$	8.02763	0.312239	0.156119	−	0.987738i	$-0.450102\pi$
$661$	8.02763	0.312239	0.156119	+	0.987738i	$0.450102\pi$
$662$	−12.1678	−0.472916
$663$	−19.1573	−0.744009
$664$	−14.3635	−0.557410
$665$	−14.6257	−0.567161
$666$	−0.616053	−0.0238716
$667$	3.67106	0.142144
$668$	7.29601	0.282291
$669$	25.6691	0.992427
$670$	−26.6504	−1.02960
$671$	0	0
$672$	−14.5937	−0.562965
$673$	2.50991	0.0967501	0.0483750	−	0.998829i	$-0.484596\pi$
$673$	2.50991	0.0967501	0.0483750	+	0.998829i	$0.484596\pi$
$674$	−32.1486	−1.23832
$675$	6.57801	0.253188
$676$	16.5873	0.637975
$677$	2.38010	0.0914748	0.0457374	−	0.998953i	$-0.485436\pi$
$677$	2.38010	0.0914748	0.0457374	+	0.998953i	$0.485436\pi$
$678$	−7.04436	−0.270537
$679$	3.56934	0.136979
$680$	23.0938	0.885606
$681$	−23.4800	−0.899754
$682$	0	0
$683$	14.4781	0.553988	0.276994	−	0.960872i	$-0.410662\pi$
$683$	14.4781	0.553988	0.276994	+	0.960872i	$0.410662\pi$
$684$	0.620622	0.0237301
$685$	60.5075	2.31187
$686$	31.1431	1.18905
$687$	13.7327	0.523935
$688$	5.87527	0.223993
$689$	80.1958	3.05522
$690$	−41.8668	−1.59384
$691$	−35.9981	−1.36943	−0.684717	−	0.728809i	$-0.740074\pi$
$691$	−35.9981	−1.36943	−0.684717	+	0.728809i	$0.740074\pi$
$692$	−0.375378	−0.0142697
$693$	0	0
$694$	−25.9531	−0.985168
$695$	−40.9959	−1.55506
$696$	−1.07852	−0.0408811
$697$	16.4516	0.623149
$698$	−14.4255	−0.546012
$699$	−4.64661	−0.175751
$700$	17.5478	0.663243
$701$	−1.23499	−0.0466448	−0.0233224	−	0.999728i	$-0.507424\pi$
$701$	−1.23499	−0.0466448	−0.0233224	+	0.999728i	$0.507424\pi$
$702$	−10.2034	−0.385103
$703$	−0.380554	−0.0143529
$704$	0	0
$705$	22.3236	0.840756
$706$	8.82514	0.332139
$707$	−46.0808	−1.73305
$708$	0.116669	0.00438470
$709$	−10.7414	−0.403400	−0.201700	−	0.979447i	$-0.564647\pi$
$709$	−10.7414	−0.403400	−0.201700	+	0.979447i	$0.564647\pi$
$710$	−68.9312	−2.58694
$711$	13.0069	0.487795
$712$	−8.13775	−0.304975
$713$	6.46064	0.241953
$714$	21.1492	0.791490
$715$	0	0
$716$	−0.189020	−0.00706402
$717$	−6.86628	−0.256426
$718$	4.23747	0.158141
$719$	11.4775	0.428039	0.214020	−	0.976829i	$-0.431344\pi$
$719$	11.4775	0.428039	0.214020	+	0.976829i	$0.431344\pi$
$720$	16.5235	0.615794
$721$	57.8612	2.15486
$722$	1.61883	0.0602467
$723$	−4.81072	−0.178913
$724$	6.51089	0.241975
$725$	3.17714	0.117996
$726$	0	0
$727$	32.7108	1.21318	0.606588	−	0.795016i	$-0.292538\pi$
$727$	32.7108	1.21318	0.606588	+	0.795016i	$0.292538\pi$
$728$	60.4963	2.24214
$729$	1.00000	0.0370370
$730$	22.5360	0.834093
$731$	−3.67735	−0.136012
$732$	9.24581	0.341735
$733$	−25.3791	−0.937398	−0.468699	−	0.883358i	$-0.655277\pi$
$733$	−25.3791	−0.937398	−0.468699	+	0.883358i	$0.655277\pi$
$734$	−39.7781	−1.46824
$735$	−39.0477	−1.44030
$736$	−25.8057	−0.951212
$737$	0	0
$738$	8.76230	0.322545
$739$	50.7555	1.86707	0.933535	−	0.358486i	$-0.116707\pi$
$739$	50.7555	1.86707	0.933535	+	0.358486i	$0.116707\pi$
$740$	0.803637	0.0295423
$741$	−6.30293	−0.231544
$742$	−88.5343	−3.25019
$743$	17.8336	0.654252	0.327126	−	0.944981i	$-0.393920\pi$
$743$	17.8336	0.654252	0.327126	+	0.944981i	$0.393920\pi$
$744$	−1.89806	−0.0695864
$745$	33.4702	1.22625
$746$	24.7123	0.904781
$747$	6.43241	0.235349
$748$	0	0
$749$	66.6438	2.43511
$750$	−8.69217	−0.317393
$751$	−33.9006	−1.23705	−0.618524	−	0.785766i	$-0.712269\pi$
$751$	−33.9006	−1.23705	−0.618524	+	0.785766i	$0.712269\pi$
$752$	31.8590	1.16178
$753$	−24.9376	−0.908776
$754$	−4.92818	−0.179474
$755$	34.8901	1.26978
$756$	2.66764	0.0970211
$757$	−16.1516	−0.587040	−0.293520	−	0.955953i	$-0.594827\pi$
$757$	−16.1516	−0.587040	−0.293520	+	0.955953i	$0.594827\pi$
$758$	21.4283	0.778310
$759$	0	0
$760$	7.59806	0.275610
$761$	11.2925	0.409354	0.204677	−	0.978830i	$-0.434386\pi$
$761$	11.2925	0.409354	0.204677	+	0.978830i	$0.434386\pi$
$762$	27.7385	1.00486
$763$	−24.2674	−0.878537
$764$	−6.07119	−0.219648
$765$	−10.3421	−0.373920
$766$	25.4062	0.917963
$767$	−1.18487	−0.0427833
$768$	13.6090	0.491073
$769$	42.2894	1.52500	0.762498	−	0.646990i	$-0.223973\pi$
$769$	42.2894	1.52500	0.762498	+	0.646990i	$0.223973\pi$
$770$	0	0
$771$	4.05550	0.146055
$772$	−5.64195	−0.203058
$773$	27.0257	0.972048	0.486024	−	0.873945i	$-0.338447\pi$
$773$	27.0257	0.972048	0.486024	+	0.873945i	$0.338447\pi$
$774$	−1.95860	−0.0704003
$775$	5.59139	0.200849
$776$	−1.85427	−0.0665645
$777$	−1.63575	−0.0586821
$778$	−20.8909	−0.748975
$779$	5.41273	0.193931
$780$	13.3103	0.476584
$781$	0	0
$782$	37.3977	1.33734
$783$	0.482994	0.0172608
$784$	−55.7267	−1.99024
$785$	−11.0911	−0.395858
$786$	23.7610	0.847525
$787$	9.33805	0.332866	0.166433	−	0.986053i	$-0.446775\pi$
$787$	9.33805	0.332866	0.166433	+	0.986053i	$0.446775\pi$
$788$	9.09481	0.323989
$789$	5.92290	0.210861
$790$	−71.6459	−2.54905
$791$	−18.7042	−0.665046
$792$	0	0
$793$	−93.8989	−3.33445
$794$	39.3466	1.39636
$795$	43.2938	1.53547
$796$	7.03601	0.249385
$797$	19.7735	0.700414	0.350207	−	0.936672i	$-0.386111\pi$
$797$	19.7735	0.700414	0.350207	+	0.936672i	$0.386111\pi$
$798$	6.95829	0.246321
$799$	−19.9407	−0.705450
$800$	−22.3337	−0.789615
$801$	3.64434	0.128766
$802$	17.0595	0.602392
$803$	0	0
$804$	3.00270	0.105897
$805$	−111.165	−3.91805
$806$	−8.67302	−0.305494
$807$	14.5610	0.512572
$808$	23.9390	0.842170
$809$	−47.4270	−1.66745	−0.833723	−	0.552183i	$-0.813795\pi$
$809$	−47.4270	−1.66745	−0.833723	+	0.552183i	$0.813795\pi$
$810$	−5.50832	−0.193543
$811$	7.40695	0.260093	0.130047	−	0.991508i	$-0.458487\pi$
$811$	7.40695	0.260093	0.130047	+	0.991508i	$0.458487\pi$
$812$	1.28845	0.0452158
$813$	8.16105	0.286220
$814$	0	0
$815$	−0.909901	−0.0318724
$816$	−14.7597	−0.516693
$817$	−1.20988	−0.0423284
$818$	55.1109	1.92691
$819$	−27.0921	−0.946675
$820$	−11.4304	−0.399166
$821$	−54.3694	−1.89751	−0.948753	−	0.316019i	$-0.897654\pi$
$821$	−54.3694	−1.89751	−0.948753	+	0.316019i	$0.897654\pi$
$822$	−28.7869	−1.00406
$823$	26.0284	0.907292	0.453646	−	0.891182i	$-0.350123\pi$
$823$	26.0284	0.907292	0.453646	+	0.891182i	$0.350123\pi$
$824$	−30.0589	−1.04715
$825$	0	0
$826$	1.30807	0.0455137
$827$	−23.7981	−0.827542	−0.413771	−	0.910381i	$-0.635789\pi$
$827$	−23.7981	−0.827542	−0.413771	+	0.910381i	$0.635789\pi$
$828$	4.71712	0.163931
$829$	−3.64388	−0.126557	−0.0632785	−	0.997996i	$-0.520156\pi$
$829$	−3.64388	−0.126557	−0.0632785	+	0.997996i	$0.520156\pi$
$830$	−35.4317	−1.22985
$831$	4.53940	0.157470
$832$	−26.5724	−0.921231
$833$	34.8795	1.20850
$834$	19.5041	0.675372
$835$	−40.0014	−1.38431
$836$	0	0
$837$	0.850012	0.0293807
$838$	−36.8359	−1.27247
$839$	43.2108	1.49180	0.745901	−	0.666057i	$-0.232019\pi$
$839$	43.2108	1.49180	0.745901	+	0.666057i	$0.232019\pi$
$840$	32.6590	1.12684
$841$	−28.7667	−0.991956
$842$	41.3839	1.42618
$843$	−29.6230	−1.02027
$844$	−6.87311	−0.236582
$845$	−90.9425	−3.12852
$846$	−10.6206	−0.365144
$847$	0	0
$848$	61.7866	2.12176
$849$	−9.09551	−0.312157
$850$	32.3660	1.11014
$851$	−2.89245	−0.0991521
$852$	7.76647	0.266075
$853$	−43.3987	−1.48594	−0.742971	−	0.669323i	$-0.766584\pi$
$853$	−43.3987	−1.48594	−0.742971	+	0.669323i	$0.766584\pi$
$854$	103.662	3.54725
$855$	−3.40265	−0.116368
$856$	−34.6214	−1.18334
$857$	−28.5773	−0.976181	−0.488091	−	0.872793i	$-0.662306\pi$
$857$	−28.5773	−0.976181	−0.488091	+	0.872793i	$0.662306\pi$
$858$	0	0
$859$	5.13029	0.175043	0.0875215	−	0.996163i	$-0.472105\pi$
$859$	5.13029	0.175043	0.0875215	+	0.996163i	$0.472105\pi$
$860$	2.55498	0.0871239
$861$	23.2657	0.792893
$862$	−31.2712	−1.06510
$863$	16.8331	0.573007	0.286503	−	0.958079i	$-0.407507\pi$
$863$	16.8331	0.573007	0.286503	+	0.958079i	$0.407507\pi$
$864$	−3.39520	−0.115507
$865$	2.05806	0.0699762
$866$	−10.4197	−0.354075
$867$	−7.76186	−0.263606
$868$	2.26753	0.0769649
$869$	0	0
$870$	−2.66048	−0.0901988
$871$	−30.4950	−1.03328
$872$	12.6069	0.426923
$873$	0.830401	0.0281048
$874$	12.3042	0.416195
$875$	−23.0795	−0.780229
$876$	−2.53912	−0.0857891
$877$	25.8834	0.874021	0.437011	−	0.899456i	$-0.356037\pi$
$877$	25.8834	0.874021	0.437011	+	0.899456i	$0.356037\pi$
$878$	−45.8070	−1.54591
$879$	25.8607	0.872260
$880$	0	0
$881$	−26.6715	−0.898585	−0.449292	−	0.893385i	$-0.648324\pi$
$881$	−26.6715	−0.898585	−0.449292	+	0.893385i	$0.648324\pi$
$882$	18.5772	0.625527
$883$	−42.0700	−1.41577	−0.707884	−	0.706329i	$-0.750350\pi$
$883$	−42.0700	−1.41577	−0.707884	+	0.706329i	$0.750350\pi$
$884$	−11.8895	−0.399886
$885$	−0.639656	−0.0215018
$886$	−64.1077	−2.15374
$887$	−37.1891	−1.24869	−0.624344	−	0.781149i	$-0.714634\pi$
$887$	−37.1891	−1.24869	−0.624344	+	0.781149i	$0.714634\pi$
$888$	0.849771	0.0285164
$889$	73.6514	2.47019
$890$	−20.0742	−0.672888
$891$	0	0
$892$	15.9308	0.533404
$893$	−6.56066	−0.219544
$894$	−15.9237	−0.532568
$895$	1.03633	0.0346407
$896$	58.5227	1.95511
$897$	−47.9063	−1.59955
$898$	−0.176318	−0.00588381
$899$	0.410551	0.0136926
$900$	4.08245	0.136082
$901$	−38.6724	−1.28837
$902$	0	0
$903$	−5.20048	−0.173061
$904$	9.71684	0.323177
$905$	−35.6969	−1.18660
$906$	−16.5992	−0.551472
$907$	−2.40581	−0.0798836	−0.0399418	−	0.999202i	$-0.512717\pi$
$907$	−2.40581	−0.0798836	−0.0399418	+	0.999202i	$0.512717\pi$
$908$	−14.5722	−0.483594
$909$	−10.7206	−0.355580
$910$	149.232	4.94699
$911$	−35.6645	−1.18162	−0.590809	−	0.806812i	$-0.701191\pi$
$911$	−35.6645	−1.18162	−0.590809	+	0.806812i	$0.701191\pi$
$912$	−4.85607	−0.160801
$913$	0	0
$914$	−6.01721	−0.199032
$915$	−50.6915	−1.67581
$916$	8.52280	0.281601
$917$	63.0902	2.08342
$918$	4.92033	0.162395
$919$	11.5249	0.380171	0.190085	−	0.981768i	$-0.439124\pi$
$919$	11.5249	0.380171	0.190085	+	0.981768i	$0.439124\pi$
$920$	57.7501	1.90396
$921$	−13.6516	−0.449836
$922$	11.1272	0.366454
$923$	−78.8750	−2.59620
$924$	0	0
$925$	−2.50329	−0.0823076
$926$	49.3431	1.62152
$927$	13.4613	0.442127
$928$	−1.63986	−0.0538311
$929$	−5.55193	−0.182153	−0.0910764	−	0.995844i	$-0.529031\pi$
$929$	−5.55193	−0.182153	−0.0910764	+	0.995844i	$0.529031\pi$
$930$	−4.68214	−0.153533
$931$	11.4757	0.376100
$932$	−2.88379	−0.0944616
$933$	−5.06909	−0.165954
$934$	37.9966	1.24329
$935$	0	0
$936$	14.0743	0.460034
$937$	−54.6356	−1.78487	−0.892434	−	0.451178i	$-0.851004\pi$
$937$	−54.6356	−1.78487	−0.892434	+	0.451178i	$0.851004\pi$
$938$	33.6657	1.09922
$939$	15.3606	0.501273
$940$	13.8545	0.451885
$941$	−5.19429	−0.169329	−0.0846644	−	0.996410i	$-0.526982\pi$
$941$	−5.19429	−0.169329	−0.0846644	+	0.996410i	$0.526982\pi$
$942$	5.27666	0.171923
$943$	41.1402	1.33971
$944$	−0.912882	−0.0297118
$945$	−14.6257	−0.475775
$946$	0	0
$947$	24.4580	0.794777	0.397388	−	0.917651i	$-0.369917\pi$
$947$	24.4580	0.794777	0.397388	+	0.917651i	$0.369917\pi$
$948$	8.07234	0.262177
$949$	25.7869	0.837079
$950$	10.6487	0.345489
$951$	1.01520	0.0329202
$952$	−29.1728	−0.945496
$953$	−53.5669	−1.73520	−0.867601	−	0.497260i	$-0.834339\pi$
$953$	−53.5669	−1.73520	−0.867601	+	0.497260i	$0.834339\pi$
$954$	−20.5973	−0.666864
$955$	33.2862	1.07712
$956$	−4.26136	−0.137822
$957$	0	0
$958$	−18.2853	−0.590772
$959$	−76.4350	−2.46822
$960$	−14.3451	−0.462987
$961$	−30.2775	−0.976693
$962$	3.88294	0.125191
$963$	15.5046	0.499627
$964$	−2.98564	−0.0961609
$965$	30.9328	0.995761
$966$	52.8875	1.70163
$967$	−43.3890	−1.39530	−0.697648	−	0.716440i	$-0.745770\pi$
$967$	−43.3890	−1.39530	−0.697648	+	0.716440i	$0.745770\pi$
$968$	0	0
$969$	3.03943	0.0976406
$970$	−4.57412	−0.146866
$971$	16.1868	0.519459	0.259729	−	0.965681i	$-0.416367\pi$
$971$	16.1868	0.519459	0.259729	+	0.965681i	$0.416367\pi$
$972$	0.620622	0.0199065
$973$	51.7874	1.66023
$974$	64.4127	2.06392
$975$	−41.4607	−1.32781
$976$	−72.3441	−2.31568
$977$	−5.00884	−0.160247	−0.0801235	−	0.996785i	$-0.525531\pi$
$977$	−5.00884	−0.160247	−0.0801235	+	0.996785i	$0.525531\pi$
$978$	0.432892	0.0138423
$979$	0	0
$980$	−24.2338	−0.774122
$981$	−5.64576	−0.180255
$982$	−57.1715	−1.82442
$983$	59.5077	1.89800	0.949000	−	0.315276i	$-0.102097\pi$
$983$	59.5077	1.89800	0.949000	+	0.315276i	$0.102097\pi$
$984$	−12.0865	−0.385305
$985$	−49.8636	−1.58879
$986$	2.37649	0.0756829
$987$	−28.1999	−0.897613
$988$	−3.91174	−0.124449
$989$	−9.19588	−0.292412
$990$	0	0
$991$	−11.0714	−0.351695	−0.175847	−	0.984417i	$-0.556267\pi$
$991$	−11.0714	−0.351695	−0.175847	+	0.984417i	$0.556267\pi$
$992$	−2.88597	−0.0916295
$993$	−7.51641	−0.238526
$994$	87.0762	2.76189
$995$	−38.5759	−1.22294
$996$	3.99209	0.126494
$997$	−42.7111	−1.35268	−0.676338	−	0.736592i	$-0.736434\pi$
$997$	−42.7111	−1.35268	−0.676338	+	0.736592i	$0.736434\pi$
$998$	−13.2506	−0.419439
$999$	−0.380554	−0.0120402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.bq.1.19		24
11.7	odd	10		627.2.j.d.115.3	✓	48
11.8	odd	10		627.2.j.d.229.3	yes	48
11.10	odd	2		6897.2.a.bp.1.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.j.d.115.3	✓	48	11.7	odd	10
627.2.j.d.229.3	yes	48	11.8	odd	10
6897.2.a.bp.1.6		24	11.10	odd	2
6897.2.a.bq.1.19		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6897.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61883 q^{2} +1.00000 q^{3} +0.620622 q^{4} -3.40265 q^{5} +1.61883 q^{6} +4.29833 q^{7} -2.23298 q^{8} +1.00000 q^{9} -5.50832 q^{10} +0.620622 q^{12} -6.30293 q^{13} +6.95829 q^{14} -3.40265 q^{15} -4.85607 q^{16} +3.03943 q^{17} +1.61883 q^{18} +1.00000 q^{19} -2.11176 q^{20} +4.29833 q^{21} +7.60064 q^{23} -2.23298 q^{24} +6.57801 q^{25} -10.2034 q^{26} +1.00000 q^{27} +2.66764 q^{28} +0.482994 q^{29} -5.50832 q^{30} +0.850012 q^{31} -3.39520 q^{32} +4.92033 q^{34} -14.6257 q^{35} +0.620622 q^{36} -0.380554 q^{37} +1.61883 q^{38} -6.30293 q^{39} +7.59806 q^{40} +5.41273 q^{41} +6.95829 q^{42} -1.20988 q^{43} -3.40265 q^{45} +12.3042 q^{46} -6.56066 q^{47} -4.85607 q^{48} +11.4757 q^{49} +10.6487 q^{50} +3.03943 q^{51} -3.91174 q^{52} -12.7236 q^{53} +1.61883 q^{54} -9.59811 q^{56} +1.00000 q^{57} +0.781886 q^{58} +0.187988 q^{59} -2.11176 q^{60} +14.8977 q^{61} +1.37603 q^{62} +4.29833 q^{63} +4.21587 q^{64} +21.4467 q^{65} +4.83822 q^{67} +1.88634 q^{68} +7.60064 q^{69} -23.6766 q^{70} +12.5140 q^{71} -2.23298 q^{72} -4.09126 q^{73} -0.616053 q^{74} +6.57801 q^{75} +0.620622 q^{76} -10.2034 q^{78} +13.0069 q^{79} +16.5235 q^{80} +1.00000 q^{81} +8.76230 q^{82} +6.43241 q^{83} +2.66764 q^{84} -10.3421 q^{85} -1.95860 q^{86} +0.482994 q^{87} +3.64434 q^{89} -5.50832 q^{90} -27.0921 q^{91} +4.71712 q^{92} +0.850012 q^{93} -10.6206 q^{94} -3.40265 q^{95} -3.39520 q^{96} +0.830401 q^{97} +18.5772 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 24 q^{3} + 37 q^{4} + 9 q^{5} + q^{6} + 3 q^{7} + 24 q^{9} - q^{10} + 37 q^{12} + 3 q^{13} - 2 q^{14} + 9 q^{15} + 55 q^{16} - 3 q^{17} + q^{18} + 24 q^{19} + 12 q^{20} + 3 q^{21} + 30 q^{23}+ \cdots + 59 q^{98}+O(q^{100}) \) 24 * q + q^2 + 24 * q^3 + 37 * q^4 + 9 * q^5 + q^6 + 3 * q^7 + 24 * q^9 - q^10 + 37 * q^12 + 3 * q^13 - 2 * q^14 + 9 * q^15 + 55 * q^16 - 3 * q^17 + q^18 + 24 * q^19 + 12 * q^20 + 3 * q^21 + 30 * q^23 + 51 * q^25 - 6 * q^26 + 24 * q^27 - 17 * q^28 + 13 * q^29 - q^30 + 47 * q^31 + 16 * q^32 + 23 * q^34 - 26 * q^35 + 37 * q^36 + 56 * q^37 + q^38 + 3 * q^39 + 38 * q^40 - 5 * q^41 - 2 * q^42 - q^43 + 9 * q^45 - 23 * q^46 + 49 * q^47 + 55 * q^48 + 59 * q^49 - 28 * q^50 - 3 * q^51 + 5 * q^52 + 14 * q^53 + q^54 + 4 * q^56 + 24 * q^57 + 53 * q^58 + 17 * q^59 + 12 * q^60 - 8 * q^61 + 17 * q^62 + 3 * q^63 + 96 * q^64 + 30 * q^65 + 46 * q^67 - 66 * q^68 + 30 * q^69 + 23 * q^70 + 21 * q^71 - 13 * q^73 - 15 * q^74 + 51 * q^75 + 37 * q^76 - 6 * q^78 - 3 * q^79 + 36 * q^80 + 24 * q^81 + 38 * q^82 + 10 * q^83 - 17 * q^84 - 12 * q^85 - 17 * q^86 + 13 * q^87 + 21 * q^89 - q^90 + 20 * q^91 + 47 * q^92 + 47 * q^93 - 27 * q^94 + 9 * q^95 + 16 * q^96 + 70 * q^97 + 59 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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