LMFDB - Embedded newform 6897.2.a.bq.1.15

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			1.15
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)

$f(q)$	$=$	\(q+1.27260 q^{2} +1.00000 q^{3} -0.380477 q^{4} -0.389881 q^{5} +1.27260 q^{6} -3.02973 q^{7} -3.02941 q^{8} +1.00000 q^{9} -0.496164 q^{10} -0.380477 q^{12} -0.780173 q^{13} -3.85565 q^{14} -0.389881 q^{15} -3.09428 q^{16} +3.23952 q^{17} +1.27260 q^{18} +1.00000 q^{19} +0.148341 q^{20} -3.02973 q^{21} -8.08948 q^{23} -3.02941 q^{24} -4.84799 q^{25} -0.992851 q^{26} +1.00000 q^{27} +1.15274 q^{28} +4.61676 q^{29} -0.496164 q^{30} +10.3448 q^{31} +2.12101 q^{32} +4.12263 q^{34} +1.18123 q^{35} -0.380477 q^{36} -0.658789 q^{37} +1.27260 q^{38} -0.780173 q^{39} +1.18111 q^{40} -7.65258 q^{41} -3.85565 q^{42} +5.34992 q^{43} -0.389881 q^{45} -10.2947 q^{46} +1.72384 q^{47} -3.09428 q^{48} +2.17926 q^{49} -6.16958 q^{50} +3.23952 q^{51} +0.296838 q^{52} +7.36492 q^{53} +1.27260 q^{54} +9.17828 q^{56} +1.00000 q^{57} +5.87531 q^{58} +0.778211 q^{59} +0.148341 q^{60} -7.55016 q^{61} +13.1648 q^{62} -3.02973 q^{63} +8.88778 q^{64} +0.304174 q^{65} +14.4474 q^{67} -1.23256 q^{68} -8.08948 q^{69} +1.50324 q^{70} +0.118679 q^{71} -3.02941 q^{72} -3.17072 q^{73} -0.838378 q^{74} -4.84799 q^{75} -0.380477 q^{76} -0.992851 q^{78} -3.21242 q^{79} +1.20640 q^{80} +1.00000 q^{81} -9.73871 q^{82} +7.71818 q^{83} +1.15274 q^{84} -1.26303 q^{85} +6.80833 q^{86} +4.61676 q^{87} +4.87723 q^{89} -0.496164 q^{90} +2.36371 q^{91} +3.07786 q^{92} +10.3448 q^{93} +2.19377 q^{94} -0.389881 q^{95} +2.12101 q^{96} +8.68858 q^{97} +2.77333 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.27260	0.899867	0.449934	−	0.893062i	$-0.351448\pi$
$2$	1.27260	0.899867	0.449934	+	0.893062i	$0.351448\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.380477	−0.190239
$5$	−0.389881	−0.174360	−0.0871800	−	0.996193i	$-0.527786\pi$
$5$	−0.389881	−0.174360	−0.0871800	+	0.996193i	$0.527786\pi$
$6$	1.27260	0.519539
$7$	−3.02973	−1.14513	−0.572565	−	0.819859i	$-0.694052\pi$
$7$	−3.02973	−1.14513	−0.572565	+	0.819859i	$0.694052\pi$
$8$	−3.02941	−1.07106
$9$	1.00000	0.333333
$10$	−0.496164	−0.156901
$11$	0	0
$11$	0	0
$12$	−0.380477	−0.109834
$13$	−0.780173	−0.216381	−0.108190	−	0.994130i	$-0.534506\pi$
$13$	−0.780173	−0.216381	−0.108190	+	0.994130i	$0.534506\pi$
$14$	−3.85565	−1.03047
$15$	−0.389881	−0.100667
$16$	−3.09428	−0.773571
$17$	3.23952	0.785700	0.392850	−	0.919603i	$-0.371489\pi$
$17$	3.23952	0.785700	0.392850	+	0.919603i	$0.371489\pi$
$18$	1.27260	0.299956
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.148341	0.0331700
$21$	−3.02973	−0.661141
$22$	0	0
$23$	−8.08948	−1.68677	−0.843387	−	0.537307i	$-0.819442\pi$
$23$	−8.08948	−1.68677	−0.843387	+	0.537307i	$0.819442\pi$
$24$	−3.02941	−0.618375
$25$	−4.84799	−0.969599
$26$	−0.992851	−0.194714
$27$	1.00000	0.192450
$28$	1.15274	0.217848
$29$	4.61676	0.857311	0.428655	−	0.903468i	$-0.358987\pi$
$29$	4.61676	0.857311	0.428655	+	0.903468i	$0.358987\pi$
$30$	−0.496164	−0.0905867
$31$	10.3448	1.85797	0.928987	−	0.370111i	$-0.120680\pi$
$31$	10.3448	1.85797	0.928987	+	0.370111i	$0.120680\pi$
$32$	2.12101	0.374946
$33$	0	0
$34$	4.12263	0.707026
$35$	1.18123	0.199665
$36$	−0.380477	−0.0634129
$37$	−0.658789	−0.108304	−0.0541521	−	0.998533i	$-0.517246\pi$
$37$	−0.658789	−0.108304	−0.0541521	+	0.998533i	$0.517246\pi$
$38$	1.27260	0.206444
$39$	−0.780173	−0.124928
$40$	1.18111	0.186749
$41$	−7.65258	−1.19513	−0.597566	−	0.801820i	$-0.703865\pi$
$41$	−7.65258	−1.19513	−0.597566	+	0.801820i	$0.703865\pi$
$42$	−3.85565	−0.594939
$43$	5.34992	0.815855	0.407927	−	0.913014i	$-0.366252\pi$
$43$	5.34992	0.815855	0.407927	+	0.913014i	$0.366252\pi$
$44$	0	0
$45$	−0.389881	−0.0581200
$46$	−10.2947	−1.51787
$47$	1.72384	0.251448	0.125724	−	0.992065i	$-0.459875\pi$
$47$	1.72384	0.251448	0.125724	+	0.992065i	$0.459875\pi$
$48$	−3.09428	−0.446621
$49$	2.17926	0.311322
$50$	−6.16958	−0.872510
$51$	3.23952	0.453624
$52$	0.296838	0.0411640
$53$	7.36492	1.01165	0.505825	−	0.862636i	$-0.331188\pi$
$53$	7.36492	1.01165	0.505825	+	0.862636i	$0.331188\pi$
$54$	1.27260	0.173180
$55$	0	0
$56$	9.17828	1.22650
$57$	1.00000	0.132453
$58$	5.87531	0.771466
$59$	0.778211	0.101314	0.0506572	−	0.998716i	$-0.483868\pi$
$59$	0.778211	0.101314	0.0506572	+	0.998716i	$0.483868\pi$
$60$	0.148341	0.0191507
$61$	−7.55016	−0.966699	−0.483349	−	0.875428i	$-0.660580\pi$
$61$	−7.55016	−0.966699	−0.483349	+	0.875428i	$0.660580\pi$
$62$	13.1648	1.67193
$63$	−3.02973	−0.381710
$64$	8.88778	1.11097
$65$	0.304174	0.0377282
$66$	0	0
$67$	14.4474	1.76503	0.882513	−	0.470288i	$-0.155850\pi$
$67$	14.4474	1.76503	0.882513	+	0.470288i	$0.155850\pi$
$68$	−1.23256	−0.149470
$69$	−8.08948	−0.973859
$70$	1.50324	0.179672
$71$	0.118679	0.0140846	0.00704232	−	0.999975i	$-0.497758\pi$
$71$	0.118679	0.0140846	0.00704232	+	0.999975i	$0.497758\pi$
$72$	−3.02941	−0.357019
$73$	−3.17072	−0.371104	−0.185552	−	0.982634i	$-0.559407\pi$
$73$	−3.17072	−0.371104	−0.185552	+	0.982634i	$0.559407\pi$
$74$	−0.838378	−0.0974595
$75$	−4.84799	−0.559798
$76$	−0.380477	−0.0436437
$77$	0	0
$78$	−0.992851	−0.112418
$79$	−3.21242	−0.361426	−0.180713	−	0.983536i	$-0.557840\pi$
$79$	−3.21242	−0.361426	−0.180713	+	0.983536i	$0.557840\pi$
$80$	1.20640	0.134880
$81$	1.00000	0.111111
$82$	−9.73871	−1.07546
$83$	7.71818	0.847181	0.423590	−	0.905854i	$-0.360770\pi$
$83$	7.71818	0.847181	0.423590	+	0.905854i	$0.360770\pi$
$84$	1.15274	0.125775
$85$	−1.26303	−0.136995
$86$	6.80833	0.734161
$87$	4.61676	0.494968
$88$	0	0
$89$	4.87723	0.516985	0.258492	−	0.966013i	$-0.416774\pi$
$89$	4.87723	0.516985	0.258492	+	0.966013i	$0.416774\pi$
$90$	−0.496164	−0.0523003
$91$	2.36371	0.247784
$92$	3.07786	0.320889
$93$	10.3448	1.07270
$94$	2.19377	0.226270
$95$	−0.389881	−0.0400009
$96$	2.12101	0.216475
$97$	8.68858	0.882192	0.441096	−	0.897460i	$-0.354590\pi$
$97$	8.68858	0.882192	0.441096	+	0.897460i	$0.354590\pi$
$98$	2.77333	0.280149
$99$	0	0
$100$	1.84455	0.184455
$101$	0.371480	0.0369636	0.0184818	−	0.999829i	$-0.494117\pi$
$101$	0.371480	0.0369636	0.0184818	+	0.999829i	$0.494117\pi$
$102$	4.12263	0.408201
$103$	11.5303	1.13612	0.568059	−	0.822987i	$-0.307694\pi$
$103$	11.5303	1.13612	0.568059	+	0.822987i	$0.307694\pi$
$104$	2.36346	0.231756
$105$	1.18123	0.115276
$106$	9.37263	0.910350
$107$	−14.0638	−1.35959	−0.679797	−	0.733400i	$-0.737932\pi$
$107$	−14.0638	−1.35959	−0.679797	+	0.733400i	$0.737932\pi$
$108$	−0.380477	−0.0366114
$109$	14.1880	1.35897	0.679484	−	0.733690i	$-0.262204\pi$
$109$	14.1880	1.35897	0.679484	+	0.733690i	$0.262204\pi$
$110$	0	0
$111$	−0.658789	−0.0625295
$112$	9.37484	0.885839
$113$	16.2049	1.52443	0.762214	−	0.647325i	$-0.224112\pi$
$113$	16.2049	1.52443	0.762214	+	0.647325i	$0.224112\pi$
$114$	1.27260	0.119190
$115$	3.15393	0.294106
$116$	−1.75657	−0.163094
$117$	−0.780173	−0.0721270
$118$	0.990355	0.0911696
$119$	−9.81488	−0.899728
$120$	1.18111	0.107820
$121$	0	0
$122$	−9.60837	−0.869901
$123$	−7.65258	−0.690010
$124$	−3.93595	−0.353458
$125$	3.83954	0.343419
$126$	−3.85565	−0.343488
$127$	6.72317	0.596585	0.298292	−	0.954475i	$-0.403583\pi$
$127$	6.72317	0.596585	0.298292	+	0.954475i	$0.403583\pi$
$128$	7.06860	0.624782
$129$	5.34992	0.471034
$130$	0.387094	0.0339504
$131$	12.6273	1.10325	0.551626	−	0.834092i	$-0.314008\pi$
$131$	12.6273	1.10325	0.551626	+	0.834092i	$0.314008\pi$
$132$	0	0
$133$	−3.02973	−0.262711
$134$	18.3858	1.58829
$135$	−0.389881	−0.0335556
$136$	−9.81383	−0.841529
$137$	11.3911	0.973205	0.486602	−	0.873624i	$-0.338236\pi$
$137$	11.3911	0.973205	0.486602	+	0.873624i	$0.338236\pi$
$138$	−10.2947	−0.876344
$139$	−4.05559	−0.343991	−0.171995	−	0.985098i	$-0.555021\pi$
$139$	−4.05559	−0.343991	−0.171995	+	0.985098i	$0.555021\pi$
$140$	−0.449432	−0.0379839
$141$	1.72384	0.145174
$142$	0.151032	0.0126743
$143$	0	0
$144$	−3.09428	−0.257857
$145$	−1.79998	−0.149481
$146$	−4.03507	−0.333945
$147$	2.17926	0.179742
$148$	0.250654	0.0206037
$149$	18.2465	1.49481	0.747406	−	0.664368i	$-0.231299\pi$
$149$	18.2465	1.49481	0.747406	+	0.664368i	$0.231299\pi$
$150$	−6.16958	−0.503744
$151$	−18.0942	−1.47249	−0.736244	−	0.676716i	$-0.763403\pi$
$151$	−18.0942	−1.47249	−0.736244	+	0.676716i	$0.763403\pi$
$152$	−3.02941	−0.245717
$153$	3.23952	0.261900
$154$	0	0
$155$	−4.03322	−0.323956
$156$	0.296838	0.0237660
$157$	4.93005	0.393461	0.196730	−	0.980458i	$-0.436968\pi$
$157$	4.93005	0.393461	0.196730	+	0.980458i	$0.436968\pi$
$158$	−4.08815	−0.325235
$159$	7.36492	0.584076
$160$	−0.826942	−0.0653755
$161$	24.5089	1.93157
$162$	1.27260	0.0999853
$163$	−11.9824	−0.938536	−0.469268	−	0.883056i	$-0.655482\pi$
$163$	−11.9824	−0.938536	−0.469268	+	0.883056i	$0.655482\pi$
$164$	2.91163	0.227360
$165$	0	0
$166$	9.82220	0.762350
$167$	−24.5677	−1.90110	−0.950552	−	0.310564i	$-0.899482\pi$
$167$	−24.5677	−1.90110	−0.950552	+	0.310564i	$0.899482\pi$
$168$	9.17828	0.708120
$169$	−12.3913	−0.953179
$170$	−1.60733	−0.123277
$171$	1.00000	0.0764719
$172$	−2.03552	−0.155207
$173$	19.4060	1.47541	0.737707	−	0.675122i	$-0.235909\pi$
$173$	19.4060	1.47541	0.737707	+	0.675122i	$0.235909\pi$
$174$	5.87531	0.445406
$175$	14.6881	1.11032
$176$	0	0
$177$	0.778211	0.0584939
$178$	6.20678	0.465218
$179$	−1.15824	−0.0865712	−0.0432856	−	0.999063i	$-0.513783\pi$
$179$	−1.15824	−0.0865712	−0.0432856	+	0.999063i	$0.513783\pi$
$180$	0.148341	0.0110567
$181$	8.96032	0.666015	0.333008	−	0.942924i	$-0.391937\pi$
$181$	8.96032	0.666015	0.333008	+	0.942924i	$0.391937\pi$
$182$	3.00807	0.222973
$183$	−7.55016	−0.558124
$184$	24.5063	1.80663
$185$	0.256849	0.0188839
$186$	13.1648	0.965290
$187$	0	0
$188$	−0.655883	−0.0478352
$189$	−3.02973	−0.220380
$190$	−0.496164	−0.0359955
$191$	11.9769	0.866619	0.433309	−	0.901245i	$-0.357346\pi$
$191$	11.9769	0.866619	0.433309	+	0.901245i	$0.357346\pi$
$192$	8.88778	0.641420
$193$	−10.9102	−0.785336	−0.392668	−	0.919680i	$-0.628448\pi$
$193$	−10.9102	−0.785336	−0.392668	+	0.919680i	$0.628448\pi$
$194$	11.0571	0.793856
$195$	0.304174	0.0217824
$196$	−0.829157	−0.0592255
$197$	−16.2198	−1.15561	−0.577805	−	0.816175i	$-0.696090\pi$
$197$	−16.2198	−1.15561	−0.577805	+	0.816175i	$0.696090\pi$
$198$	0	0
$199$	16.5439	1.17277	0.586384	−	0.810033i	$-0.300551\pi$
$199$	16.5439	1.17277	0.586384	+	0.810033i	$0.300551\pi$
$200$	14.6865	1.03850
$201$	14.4474	1.01904
$202$	0.472747	0.0332624
$203$	−13.9875	−0.981732
$204$	−1.23256	−0.0862968
$205$	2.98359	0.208383
$206$	14.6736	1.02236
$207$	−8.08948	−0.562258
$208$	2.41407	0.167386
$209$	0	0
$210$	1.50324	0.103734
$211$	−9.68164	−0.666511	−0.333256	−	0.942837i	$-0.608147\pi$
$211$	−9.68164	−0.666511	−0.333256	+	0.942837i	$0.608147\pi$
$212$	−2.80218	−0.192455
$213$	0.118679	0.00813177
$214$	−17.8976	−1.22345
$215$	−2.08583	−0.142252
$216$	−3.02941	−0.206125
$217$	−31.3418	−2.12762
$218$	18.0558	1.22289
$219$	−3.17072	−0.214257
$220$	0	0
$221$	−2.52739	−0.170010
$222$	−0.838378	−0.0562683
$223$	−2.69887	−0.180730	−0.0903649	−	0.995909i	$-0.528803\pi$
$223$	−2.69887	−0.180730	−0.0903649	+	0.995909i	$0.528803\pi$
$224$	−6.42610	−0.429362
$225$	−4.84799	−0.323200
$226$	20.6224	1.37178
$227$	11.0762	0.735153	0.367577	−	0.929993i	$-0.380188\pi$
$227$	11.0762	0.735153	0.367577	+	0.929993i	$0.380188\pi$
$228$	−0.380477	−0.0251977
$229$	14.3726	0.949770	0.474885	−	0.880048i	$-0.342490\pi$
$229$	14.3726	0.949770	0.474885	+	0.880048i	$0.342490\pi$
$230$	4.01371	0.264656
$231$	0	0
$232$	−13.9860	−0.918228
$233$	−12.1148	−0.793666	−0.396833	−	0.917891i	$-0.629891\pi$
$233$	−12.1148	−0.793666	−0.396833	+	0.917891i	$0.629891\pi$
$234$	−0.992851	−0.0649047
$235$	−0.672093	−0.0438425
$236$	−0.296092	−0.0192739
$237$	−3.21242	−0.208669
$238$	−12.4905	−0.809636
$239$	13.1867	0.852976	0.426488	−	0.904493i	$-0.359751\pi$
$239$	13.1867	0.852976	0.426488	+	0.904493i	$0.359751\pi$
$240$	1.20640	0.0778729
$241$	12.2879	0.791535	0.395768	−	0.918351i	$-0.370479\pi$
$241$	12.2879	0.791535	0.395768	+	0.918351i	$0.370479\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	2.87266	0.183903
$245$	−0.849650	−0.0542821
$246$	−9.73871	−0.620917
$247$	−0.780173	−0.0496412
$248$	−31.3385	−1.99000
$249$	7.71818	0.489120
$250$	4.88622	0.309032
$251$	−7.52213	−0.474792	−0.237396	−	0.971413i	$-0.576294\pi$
$251$	−7.52213	−0.474792	−0.237396	+	0.971413i	$0.576294\pi$
$252$	1.15274	0.0726160
$253$	0	0
$254$	8.55594	0.536847
$255$	−1.26303	−0.0790938
$256$	−8.78002	−0.548751
$257$	−21.3035	−1.32888	−0.664438	−	0.747343i	$-0.731329\pi$
$257$	−21.3035	−1.32888	−0.664438	+	0.747343i	$0.731329\pi$
$258$	6.80833	0.423868
$259$	1.99595	0.124022
$260$	−0.115731	−0.00717735
$261$	4.61676	0.285770
$262$	16.0696	0.992781
$263$	18.6850	1.15217	0.576084	−	0.817391i	$-0.304580\pi$
$263$	18.6850	1.15217	0.576084	+	0.817391i	$0.304580\pi$
$264$	0	0
$265$	−2.87144	−0.176391
$266$	−3.85565	−0.236405
$267$	4.87723	0.298481
$268$	−5.49689	−0.335776
$269$	10.4499	0.637141	0.318570	−	0.947899i	$-0.396797\pi$
$269$	10.4499	0.637141	0.318570	+	0.947899i	$0.396797\pi$
$270$	−0.496164	−0.0301956
$271$	10.8103	0.656677	0.328338	−	0.944560i	$-0.393511\pi$
$271$	10.8103	0.656677	0.328338	+	0.944560i	$0.393511\pi$
$272$	−10.0240	−0.607794
$273$	2.36371	0.143058
$274$	14.4963	0.875755
$275$	0	0
$276$	3.07786	0.185266
$277$	14.0870	0.846407	0.423204	−	0.906035i	$-0.360905\pi$
$277$	14.0870	0.846407	0.423204	+	0.906035i	$0.360905\pi$
$278$	−5.16117	−0.309546
$279$	10.3448	0.619325
$280$	−3.57843	−0.213852
$281$	−28.9479	−1.72689	−0.863444	−	0.504445i	$-0.831697\pi$
$281$	−28.9479	−1.72689	−0.863444	+	0.504445i	$0.831697\pi$
$282$	2.19377	0.130637
$283$	3.46552	0.206004	0.103002	−	0.994681i	$-0.467155\pi$
$283$	3.46552	0.206004	0.103002	+	0.994681i	$0.467155\pi$
$284$	−0.0451547	−0.00267944
$285$	−0.389881	−0.0230945
$286$	0	0
$287$	23.1852	1.36858
$288$	2.12101	0.124982
$289$	−6.50549	−0.382676
$290$	−2.29067	−0.134513
$291$	8.68858	0.509334
$292$	1.20639	0.0705984
$293$	11.5329	0.673762	0.336881	−	0.941547i	$-0.390628\pi$
$293$	11.5329	0.673762	0.336881	+	0.941547i	$0.390628\pi$
$294$	2.77333	0.161744
$295$	−0.303409	−0.0176652
$296$	1.99574	0.116000
$297$	0	0
$298$	23.2206	1.34513
$299$	6.31119	0.364986
$300$	1.84455	0.106495
$301$	−16.2088	−0.934260
$302$	−23.0268	−1.32504
$303$	0.371480	0.0213410
$304$	−3.09428	−0.177469
$305$	2.94366	0.168554
$306$	4.12263	0.235675
$307$	−12.0356	−0.686908	−0.343454	−	0.939169i	$-0.611597\pi$
$307$	−12.0356	−0.686908	−0.343454	+	0.939169i	$0.611597\pi$
$308$	0	0
$309$	11.5303	0.655939
$310$	−5.13270	−0.291518
$311$	−11.7129	−0.664176	−0.332088	−	0.943248i	$-0.607753\pi$
$311$	−11.7129	−0.664176	−0.332088	+	0.943248i	$0.607753\pi$
$312$	2.36346	0.133805
$313$	27.6594	1.56340	0.781701	−	0.623654i	$-0.214352\pi$
$313$	27.6594	1.56340	0.781701	+	0.623654i	$0.214352\pi$
$314$	6.27400	0.354063
$315$	1.18123	0.0665549
$316$	1.22225	0.0687571
$317$	24.2852	1.36399	0.681996	−	0.731356i	$-0.261112\pi$
$317$	24.2852	1.36399	0.681996	+	0.731356i	$0.261112\pi$
$318$	9.37263	0.525591
$319$	0	0
$320$	−3.46517	−0.193709
$321$	−14.0638	−0.784962
$322$	31.1902	1.73816
$323$	3.23952	0.180252
$324$	−0.380477	−0.0211376
$325$	3.78227	0.209803
$326$	−15.2489	−0.844558
$327$	14.1880	0.784600
$328$	23.1828	1.28005
$329$	−5.22278	−0.287941
$330$	0	0
$331$	32.3549	1.77839	0.889194	−	0.457531i	$-0.151266\pi$
$331$	32.3549	1.77839	0.889194	+	0.457531i	$0.151266\pi$
$332$	−2.93659	−0.161166
$333$	−0.658789	−0.0361014
$334$	−31.2650	−1.71074
$335$	−5.63275	−0.307750
$336$	9.37484	0.511439
$337$	−17.3896	−0.947270	−0.473635	−	0.880721i	$-0.657058\pi$
$337$	−17.3896	−0.947270	−0.473635	+	0.880721i	$0.657058\pi$
$338$	−15.7693	−0.857735
$339$	16.2049	0.880129
$340$	0.480553	0.0260617
$341$	0	0
$342$	1.27260	0.0688146
$343$	14.6055	0.788625
$344$	−16.2071	−0.873827
$345$	3.15393	0.169802
$346$	24.6962	1.32768
$347$	9.23820	0.495932	0.247966	−	0.968769i	$-0.420238\pi$
$347$	9.23820	0.495932	0.247966	+	0.968769i	$0.420238\pi$
$348$	−1.75657	−0.0941621
$349$	−20.2834	−1.08575	−0.542873	−	0.839815i	$-0.682664\pi$
$349$	−20.2834	−1.08575	−0.542873	+	0.839815i	$0.682664\pi$
$350$	18.6922	0.999137
$351$	−0.780173	−0.0416425
$352$	0	0
$353$	37.1556	1.97759	0.988795	−	0.149277i	$-0.0476946\pi$
$353$	37.1556	1.97759	0.988795	+	0.149277i	$0.0476946\pi$
$354$	0.990355	0.0526368
$355$	−0.0462707	−0.00245580
$356$	−1.85567	−0.0983505
$357$	−9.81488	−0.519458
$358$	−1.47399	−0.0779026
$359$	2.05786	0.108610	0.0543048	−	0.998524i	$-0.482706\pi$
$359$	2.05786	0.108610	0.0543048	+	0.998524i	$0.482706\pi$
$360$	1.18111	0.0622498
$361$	1.00000	0.0526316
$362$	11.4029	0.599325
$363$	0	0
$364$	−0.899338	−0.0471381
$365$	1.23620	0.0647057
$366$	−9.60837	−0.502237
$367$	36.0637	1.88251	0.941256	−	0.337694i	$-0.109647\pi$
$367$	36.0637	1.88251	0.941256	+	0.337694i	$0.109647\pi$
$368$	25.0311	1.30484
$369$	−7.65258	−0.398377
$370$	0.326867	0.0169930
$371$	−22.3137	−1.15847
$372$	−3.93595	−0.204069
$373$	−10.1008	−0.523001	−0.261500	−	0.965203i	$-0.584217\pi$
$373$	−10.1008	−0.523001	−0.261500	+	0.965203i	$0.584217\pi$
$374$	0	0
$375$	3.83954	0.198273
$376$	−5.22222	−0.269315
$377$	−3.60187	−0.185506
$378$	−3.85565	−0.198313
$379$	17.8412	0.916439	0.458219	−	0.888839i	$-0.348487\pi$
$379$	17.8412	0.916439	0.458219	+	0.888839i	$0.348487\pi$
$380$	0.148341	0.00760972
$381$	6.72317	0.344438
$382$	15.2419	0.779842
$383$	−18.9908	−0.970385	−0.485193	−	0.874407i	$-0.661250\pi$
$383$	−18.9908	−0.970385	−0.485193	+	0.874407i	$0.661250\pi$
$384$	7.06860	0.360718
$385$	0	0
$386$	−13.8844	−0.706699
$387$	5.34992	0.271952
$388$	−3.30581	−0.167827
$389$	29.7741	1.50961	0.754803	−	0.655951i	$-0.227732\pi$
$389$	29.7741	1.50961	0.754803	+	0.655951i	$0.227732\pi$
$390$	0.387094	0.0196012
$391$	−26.2061	−1.32530
$392$	−6.60185	−0.333444
$393$	12.6273	0.636963
$394$	−20.6413	−1.03990
$395$	1.25246	0.0630182
$396$	0	0
$397$	−29.1609	−1.46354	−0.731772	−	0.681549i	$-0.761306\pi$
$397$	−29.1609	−1.46354	−0.731772	+	0.681549i	$0.761306\pi$
$398$	21.0539	1.05534
$399$	−3.02973	−0.151676
$400$	15.0011	0.750053
$401$	−34.9173	−1.74369	−0.871843	−	0.489786i	$-0.837075\pi$
$401$	−34.9173	−1.74369	−0.871843	+	0.489786i	$0.837075\pi$
$402$	18.3858	0.916999
$403$	−8.07070	−0.402030
$404$	−0.141340	−0.00703191
$405$	−0.389881	−0.0193733
$406$	−17.8006	−0.883429
$407$	0	0
$408$	−9.81383	−0.485857
$409$	−34.2886	−1.69546	−0.847730	−	0.530427i	$-0.822031\pi$
$409$	−34.2886	−1.69546	−0.847730	+	0.530427i	$0.822031\pi$
$410$	3.79693	0.187517
$411$	11.3911	0.561880
$412$	−4.38703	−0.216134
$413$	−2.35777	−0.116018
$414$	−10.2947	−0.505957
$415$	−3.00917	−0.147714
$416$	−1.65476	−0.0811311
$417$	−4.05559	−0.198603
$418$	0	0
$419$	−2.16294	−0.105666	−0.0528332	−	0.998603i	$-0.516825\pi$
$419$	−2.16294	−0.105666	−0.0528332	+	0.998603i	$0.516825\pi$
$420$	−0.449432	−0.0219300
$421$	15.7171	0.766006	0.383003	−	0.923747i	$-0.374890\pi$
$421$	15.7171	0.766006	0.383003	+	0.923747i	$0.374890\pi$
$422$	−12.3209	−0.599772
$423$	1.72384	0.0838161
$424$	−22.3113	−1.08353
$425$	−15.7052	−0.761813
$426$	0.151032	0.00731751
$427$	22.8749	1.10700
$428$	5.35094	0.258647
$429$	0	0
$430$	−2.65444	−0.128008
$431$	24.3019	1.17058	0.585290	−	0.810824i	$-0.300981\pi$
$431$	24.3019	1.17058	0.585290	+	0.810824i	$0.300981\pi$
$432$	−3.09428	−0.148874
$433$	−39.3047	−1.88887	−0.944433	−	0.328705i	$-0.893388\pi$
$433$	−39.3047	−1.88887	−0.944433	+	0.328705i	$0.893388\pi$
$434$	−39.8858	−1.91458
$435$	−1.79998	−0.0863027
$436$	−5.39823	−0.258528
$437$	−8.08948	−0.386972
$438$	−4.03507	−0.192803
$439$	−30.3803	−1.44997	−0.724985	−	0.688764i	$-0.758153\pi$
$439$	−30.3803	−1.44997	−0.724985	+	0.688764i	$0.758153\pi$
$440$	0	0
$441$	2.17926	0.103774
$442$	−3.21637	−0.152987
$443$	−19.0075	−0.903076	−0.451538	−	0.892252i	$-0.649124\pi$
$443$	−19.0075	−0.903076	−0.451538	+	0.892252i	$0.649124\pi$
$444$	0.250654	0.0118955
$445$	−1.90154	−0.0901414
$446$	−3.43460	−0.162633
$447$	18.2465	0.863030
$448$	−26.9276	−1.27221
$449$	0.390034	0.0184069	0.00920343	−	0.999958i	$-0.497070\pi$
$449$	0.390034	0.0184069	0.00920343	+	0.999958i	$0.497070\pi$
$450$	−6.16958	−0.290837
$451$	0	0
$452$	−6.16559	−0.290005
$453$	−18.0942	−0.850142
$454$	14.0956	0.661541
$455$	−0.921565	−0.0432037
$456$	−3.02941	−0.141865
$457$	−28.7775	−1.34615	−0.673077	−	0.739572i	$-0.735028\pi$
$457$	−28.7775	−1.34615	−0.673077	+	0.739572i	$0.735028\pi$
$458$	18.2907	0.854667
$459$	3.23952	0.151208
$460$	−1.20000	−0.0559502
$461$	−28.9897	−1.35018	−0.675092	−	0.737734i	$-0.735896\pi$
$461$	−28.9897	−1.35018	−0.675092	+	0.737734i	$0.735896\pi$
$462$	0	0
$463$	−18.3888	−0.854602	−0.427301	−	0.904109i	$-0.640536\pi$
$463$	−18.3888	−0.854602	−0.427301	+	0.904109i	$0.640536\pi$
$464$	−14.2856	−0.663190
$465$	−4.03322	−0.187036
$466$	−15.4173	−0.714194
$467$	−18.6109	−0.861212	−0.430606	−	0.902540i	$-0.641700\pi$
$467$	−18.6109	−0.861212	−0.430606	+	0.902540i	$0.641700\pi$
$468$	0.296838	0.0137213
$469$	−43.7716	−2.02118
$470$	−0.855309	−0.0394525
$471$	4.93005	0.227165
$472$	−2.35752	−0.108514
$473$	0	0
$474$	−4.08815	−0.187775
$475$	−4.84799	−0.222441
$476$	3.73434	0.171163
$477$	7.36492	0.337216
$478$	16.7814	0.767565
$479$	1.22677	0.0560527	0.0280264	−	0.999607i	$-0.491078\pi$
$479$	1.22677	0.0560527	0.0280264	+	0.999607i	$0.491078\pi$
$480$	−0.826942	−0.0377446
$481$	0.513969	0.0234350
$482$	15.6377	0.712277
$483$	24.5089	1.11520
$484$	0	0
$485$	−3.38751	−0.153819
$486$	1.27260	0.0577265
$487$	24.6184	1.11557	0.557784	−	0.829986i	$-0.311652\pi$
$487$	24.6184	1.11557	0.557784	+	0.829986i	$0.311652\pi$
$488$	22.8725	1.03539
$489$	−11.9824	−0.541864
$490$	−1.08127	−0.0488467
$491$	−4.14965	−0.187271	−0.0936355	−	0.995607i	$-0.529849\pi$
$491$	−4.14965	−0.187271	−0.0936355	+	0.995607i	$0.529849\pi$
$492$	2.91163	0.131266
$493$	14.9561	0.673589
$494$	−0.992851	−0.0446705
$495$	0	0
$496$	−32.0096	−1.43727
$497$	−0.359566	−0.0161287
$498$	9.82220	0.440143
$499$	−0.374054	−0.0167450	−0.00837248	−	0.999965i	$-0.502665\pi$
$499$	−0.374054	−0.0167450	−0.00837248	+	0.999965i	$0.502665\pi$
$500$	−1.46086	−0.0653316
$501$	−24.5677	−1.09760
$502$	−9.57269	−0.427250
$503$	10.2965	0.459097	0.229549	−	0.973297i	$-0.426275\pi$
$503$	10.2965	0.459097	0.229549	+	0.973297i	$0.426275\pi$
$504$	9.17828	0.408833
$505$	−0.144833	−0.00644498
$506$	0	0
$507$	−12.3913	−0.550318
$508$	−2.55801	−0.113493
$509$	−40.6131	−1.80014	−0.900072	−	0.435742i	$-0.856486\pi$
$509$	−40.6131	−1.80014	−0.900072	+	0.435742i	$0.856486\pi$
$510$	−1.60733	−0.0711740
$511$	9.60641	0.424963
$512$	−25.3107	−1.11859
$513$	1.00000	0.0441511
$514$	−27.1110	−1.19581
$515$	−4.49546	−0.198094
$516$	−2.03552	−0.0896088
$517$	0	0
$518$	2.54006	0.111604
$519$	19.4060	0.851830
$520$	−0.921467	−0.0404090
$521$	−11.6176	−0.508978	−0.254489	−	0.967076i	$-0.581907\pi$
$521$	−11.6176	−0.508978	−0.254489	+	0.967076i	$0.581907\pi$
$522$	5.87531	0.257155
$523$	15.9253	0.696365	0.348183	−	0.937427i	$-0.386799\pi$
$523$	15.9253	0.696365	0.348183	+	0.937427i	$0.386799\pi$
$524$	−4.80440	−0.209881
$525$	14.6881	0.641041
$526$	23.7786	1.03680
$527$	33.5121	1.45981
$528$	0	0
$529$	42.4397	1.84520
$530$	−3.65421	−0.158729
$531$	0.778211	0.0337715
$532$	1.15274	0.0499777
$533$	5.97033	0.258604
$534$	6.20678	0.268594
$535$	5.48319	0.237059
$536$	−43.7669	−1.89044
$537$	−1.15824	−0.0499819
$538$	13.2986	0.573342
$539$	0	0
$540$	0.148341	0.00638357
$541$	11.5567	0.496862	0.248431	−	0.968650i	$-0.420085\pi$
$541$	11.5567	0.496862	0.248431	+	0.968650i	$0.420085\pi$
$542$	13.7572	0.590922
$543$	8.96032	0.384524
$544$	6.87107	0.294595
$545$	−5.53164	−0.236950
$546$	3.00807	0.128734
$547$	0.00782129	0.000334414 0	0.000167207	−	1.00000i	$-0.499947\pi$
$547$	0.00782129	0.000334414 0	0.000167207		1.00000i	$0.499947\pi$
$548$	−4.33404	−0.185141
$549$	−7.55016	−0.322233
$550$	0	0
$551$	4.61676	0.196681
$552$	24.5063	1.04306
$553$	9.73277	0.413880
$554$	17.9272	0.761654
$555$	0.256849	0.0109026
$556$	1.54306	0.0654404
$557$	33.2618	1.40935	0.704674	−	0.709532i	$-0.251093\pi$
$557$	33.2618	1.40935	0.704674	+	0.709532i	$0.251093\pi$
$558$	13.1648	0.557310
$559$	−4.17386	−0.176535
$560$	−3.65507	−0.154455
$561$	0	0
$562$	−36.8392	−1.55397
$563$	−31.8907	−1.34403	−0.672017	−	0.740536i	$-0.734572\pi$
$563$	−31.8907	−1.34403	−0.672017	+	0.740536i	$0.734572\pi$
$564$	−0.655883	−0.0276177
$565$	−6.31797	−0.265799
$566$	4.41024	0.185376
$567$	−3.02973	−0.127237
$568$	−0.359528	−0.0150854
$569$	−16.3590	−0.685804	−0.342902	−	0.939371i	$-0.611410\pi$
$569$	−16.3590	−0.685804	−0.342902	+	0.939371i	$0.611410\pi$
$570$	−0.496164	−0.0207820
$571$	21.4345	0.897006	0.448503	−	0.893781i	$-0.351957\pi$
$571$	21.4345	0.897006	0.448503	+	0.893781i	$0.351957\pi$
$572$	0	0
$573$	11.9769	0.500342
$574$	29.5056	1.23154
$575$	39.2177	1.63549
$576$	8.88778	0.370324
$577$	−30.4851	−1.26911	−0.634556	−	0.772876i	$-0.718817\pi$
$577$	−30.4851	−1.26911	−0.634556	+	0.772876i	$0.718817\pi$
$578$	−8.27892	−0.344358
$579$	−10.9102	−0.453414
$580$	0.684853	0.0284370
$581$	−23.3840	−0.970132
$582$	11.0571	0.458333
$583$	0	0
$584$	9.60539	0.397474
$585$	0.304174	0.0125761
$586$	14.6769	0.606296
$587$	−19.0030	−0.784339	−0.392169	−	0.919893i	$-0.628275\pi$
$587$	−19.0030	−0.784339	−0.392169	+	0.919893i	$0.628275\pi$
$588$	−0.829157	−0.0341939
$589$	10.3448	0.426249
$590$	−0.386120	−0.0158963
$591$	−16.2198	−0.667191
$592$	2.03848	0.0837810
$593$	−7.19833	−0.295600	−0.147800	−	0.989017i	$-0.547219\pi$
$593$	−7.19833	−0.295600	−0.147800	+	0.989017i	$0.547219\pi$
$594$	0	0
$595$	3.82663	0.156877
$596$	−6.94237	−0.284371
$597$	16.5439	0.677098
$598$	8.03165	0.328439
$599$	−1.23101	−0.0502977	−0.0251489	−	0.999684i	$-0.508006\pi$
$599$	−1.23101	−0.0502977	−0.0251489	+	0.999684i	$0.508006\pi$
$600$	14.6865	0.599576
$601$	33.3511	1.36042	0.680210	−	0.733017i	$-0.261889\pi$
$601$	33.3511	1.36042	0.680210	+	0.733017i	$0.261889\pi$
$602$	−20.6274	−0.840710
$603$	14.4474	0.588342
$604$	6.88445	0.280124
$605$	0	0
$606$	0.472747	0.0192040
$607$	−10.7441	−0.436089	−0.218044	−	0.975939i	$-0.569968\pi$
$607$	−10.7441	−0.436089	−0.218044	+	0.975939i	$0.569968\pi$
$608$	2.12101	0.0860185
$609$	−13.9875	−0.566803
$610$	3.74612	0.151676
$611$	−1.34490	−0.0544086
$612$	−1.23256	−0.0498235
$613$	−0.804073	−0.0324762	−0.0162381	−	0.999868i	$-0.505169\pi$
$613$	−0.804073	−0.0324762	−0.0162381	+	0.999868i	$0.505169\pi$
$614$	−15.3166	−0.618126
$615$	2.98359	0.120310
$616$	0	0
$617$	−0.583005	−0.0234709	−0.0117354	−	0.999931i	$-0.503736\pi$
$617$	−0.583005	−0.0234709	−0.0117354	+	0.999931i	$0.503736\pi$
$618$	14.6736	0.590258
$619$	8.18216	0.328869	0.164434	−	0.986388i	$-0.447420\pi$
$619$	8.18216	0.328869	0.164434	+	0.986388i	$0.447420\pi$
$620$	1.53455	0.0616290
$621$	−8.08948	−0.324620
$622$	−14.9059	−0.597671
$623$	−14.7767	−0.592015
$624$	2.41407	0.0966403
$625$	22.7430	0.909720
$626$	35.1995	1.40685
$627$	0	0
$628$	−1.87577	−0.0748514
$629$	−2.13416	−0.0850946
$630$	1.50324	0.0598906
$631$	−7.05581	−0.280887	−0.140444	−	0.990089i	$-0.544853\pi$
$631$	−7.05581	−0.280887	−0.140444	+	0.990089i	$0.544853\pi$
$632$	9.73174	0.387108
$633$	−9.68164	−0.384810
$634$	30.9054	1.22741
$635$	−2.62123	−0.104020
$636$	−2.80218	−0.111114
$637$	−1.70020	−0.0673642
$638$	0	0
$639$	0.118679	0.00469488
$640$	−2.75591	−0.108937
$641$	7.55486	0.298399	0.149199	−	0.988807i	$-0.452330\pi$
$641$	7.55486	0.298399	0.149199	+	0.988807i	$0.452330\pi$
$642$	−17.8976	−0.706362
$643$	−5.72776	−0.225881	−0.112940	−	0.993602i	$-0.536027\pi$
$643$	−5.72776	−0.225881	−0.112940	+	0.993602i	$0.536027\pi$
$644$	−9.32509	−0.367460
$645$	−2.08583	−0.0821295
$646$	4.12263	0.162203
$647$	−25.6313	−1.00767	−0.503835	−	0.863800i	$-0.668078\pi$
$647$	−25.6313	−1.00767	−0.503835	+	0.863800i	$0.668078\pi$
$648$	−3.02941	−0.119006
$649$	0	0
$650$	4.81334	0.188795
$651$	−31.3418	−1.22838
$652$	4.55904	0.178546
$653$	47.6540	1.86484	0.932422	−	0.361372i	$-0.117692\pi$
$653$	47.6540	1.86484	0.932422	+	0.361372i	$0.117692\pi$
$654$	18.0558	0.706036
$655$	−4.92314	−0.192363
$656$	23.6792	0.924519
$657$	−3.17072	−0.123701
$658$	−6.64653	−0.259109
$659$	1.67437	0.0652243	0.0326122	−	0.999468i	$-0.489617\pi$
$659$	1.67437	0.0652243	0.0326122	+	0.999468i	$0.489617\pi$
$660$	0	0
$661$	−3.97399	−0.154570	−0.0772852	−	0.997009i	$-0.524625\pi$
$661$	−3.97399	−0.154570	−0.0772852	+	0.997009i	$0.524625\pi$
$662$	41.1750	1.60031
$663$	−2.52739	−0.0981556
$664$	−23.3815	−0.907379
$665$	1.18123	0.0458062
$666$	−0.838378	−0.0324865
$667$	−37.3472	−1.44609
$668$	9.34744	0.361663
$669$	−2.69887	−0.104344
$670$	−7.16826	−0.276934
$671$	0	0
$672$	−6.42610	−0.247892
$673$	27.4802	1.05928	0.529642	−	0.848221i	$-0.322326\pi$
$673$	27.4802	1.05928	0.529642	+	0.848221i	$0.322326\pi$
$674$	−22.1300	−0.852417
$675$	−4.84799	−0.186599
$676$	4.71462	0.181331
$677$	14.2739	0.548591	0.274295	−	0.961645i	$-0.411555\pi$
$677$	14.2739	0.548591	0.274295	+	0.961645i	$0.411555\pi$
$678$	20.6224	0.791999
$679$	−26.3241	−1.01022
$680$	3.82622	0.146729
$681$	11.0762	0.424441
$682$	0	0
$683$	22.1806	0.848718	0.424359	−	0.905494i	$-0.360500\pi$
$683$	22.1806	0.848718	0.424359	+	0.905494i	$0.360500\pi$
$684$	−0.380477	−0.0145479
$685$	−4.44116	−0.169688
$686$	18.5871	0.709658
$687$	14.3726	0.548350
$688$	−16.5542	−0.631121
$689$	−5.74591	−0.218902
$690$	4.01371	0.152799
$691$	30.4440	1.15815	0.579073	−	0.815276i	$-0.303415\pi$
$691$	30.4440	1.15815	0.579073	+	0.815276i	$0.303415\pi$
$692$	−7.38355	−0.280680
$693$	0	0
$694$	11.7566	0.446273
$695$	1.58120	0.0599782
$696$	−13.9860	−0.530139
$697$	−24.7907	−0.939015
$698$	−25.8128	−0.977028
$699$	−12.1148	−0.458223
$700$	−5.58849	−0.211225
$701$	24.2419	0.915603	0.457802	−	0.889054i	$-0.348637\pi$
$701$	24.2419	0.915603	0.457802	+	0.889054i	$0.348637\pi$
$702$	−0.992851	−0.0374728
$703$	−0.658789	−0.0248467
$704$	0	0
$705$	−0.672093	−0.0253125
$706$	47.2843	1.77957
$707$	−1.12548	−0.0423282
$708$	−0.296092	−0.0111278
$709$	−21.1620	−0.794757	−0.397378	−	0.917655i	$-0.630080\pi$
$709$	−21.1620	−0.794757	−0.397378	+	0.917655i	$0.630080\pi$
$710$	−0.0588844	−0.00220989
$711$	−3.21242	−0.120475
$712$	−14.7751	−0.553720
$713$	−83.6838	−3.13398
$714$	−12.4905	−0.467444
$715$	0	0
$716$	0.440685	0.0164692
$717$	13.1867	0.492466
$718$	2.61884	0.0977342
$719$	24.1280	0.899823	0.449911	−	0.893073i	$-0.351456\pi$
$719$	24.1280	0.899823	0.449911	+	0.893073i	$0.351456\pi$
$720$	1.20640	0.0449599
$721$	−34.9338	−1.30100
$722$	1.27260	0.0473614
$723$	12.2879	0.456993
$724$	−3.40920	−0.126702
$725$	−22.3820	−0.831247
$726$	0	0
$727$	−24.0622	−0.892418	−0.446209	−	0.894929i	$-0.647226\pi$
$727$	−24.0622	−0.892418	−0.446209	+	0.894929i	$0.647226\pi$
$728$	−7.16064	−0.265391
$729$	1.00000	0.0370370
$730$	1.57320	0.0582266
$731$	17.3312	0.641017
$732$	2.87266	0.106177
$733$	5.11191	0.188813	0.0944063	−	0.995534i	$-0.469905\pi$
$733$	5.11191	0.188813	0.0944063	+	0.995534i	$0.469905\pi$
$734$	45.8949	1.69401
$735$	−0.849650	−0.0313398
$736$	−17.1579	−0.632449
$737$	0	0
$738$	−9.73871	−0.358487
$739$	−23.7086	−0.872137	−0.436068	−	0.899914i	$-0.643629\pi$
$739$	−23.7086	−0.872137	−0.436068	+	0.899914i	$0.643629\pi$
$740$	−0.0977252	−0.00359245
$741$	−0.780173	−0.0286604
$742$	−28.3965	−1.04247
$743$	0.259004	0.00950193	0.00475096	−	0.999989i	$-0.498488\pi$
$743$	0.259004	0.00950193	0.00475096	+	0.999989i	$0.498488\pi$
$744$	−31.3385	−1.14893
$745$	−7.11396	−0.260635
$746$	−12.8544	−0.470631
$747$	7.71818	0.282394
$748$	0	0
$749$	42.6094	1.55691
$750$	4.88622	0.178420
$751$	22.0617	0.805044	0.402522	−	0.915410i	$-0.368134\pi$
$751$	22.0617	0.805044	0.402522	+	0.915410i	$0.368134\pi$
$752$	−5.33406	−0.194513
$753$	−7.52213	−0.274122
$754$	−4.58376	−0.166931
$755$	7.05459	0.256743
$756$	1.15274	0.0419248
$757$	26.7946	0.973867	0.486934	−	0.873439i	$-0.338115\pi$
$757$	26.7946	0.973867	0.486934	+	0.873439i	$0.338115\pi$
$758$	22.7047	0.824674
$759$	0	0
$760$	1.18111	0.0428433
$761$	23.0817	0.836711	0.418356	−	0.908283i	$-0.362607\pi$
$761$	23.0817	0.836711	0.418356	+	0.908283i	$0.362607\pi$
$762$	8.55594	0.309949
$763$	−42.9859	−1.55619
$764$	−4.55694	−0.164864
$765$	−1.26303	−0.0456649
$766$	−24.1678	−0.873218
$767$	−0.607139	−0.0219225
$768$	−8.78002	−0.316822
$769$	−10.7457	−0.387500	−0.193750	−	0.981051i	$-0.562065\pi$
$769$	−10.7457	−0.387500	−0.193750	+	0.981051i	$0.562065\pi$
$770$	0	0
$771$	−21.3035	−0.767227
$772$	4.15110	0.149401
$773$	−19.6228	−0.705781	−0.352891	−	0.935665i	$-0.614801\pi$
$773$	−19.6228	−0.705781	−0.352891	+	0.935665i	$0.614801\pi$
$774$	6.80833	0.244720
$775$	−50.1514	−1.80149
$776$	−26.3213	−0.944878
$777$	1.99595	0.0716044
$778$	37.8907	1.35845
$779$	−7.65258	−0.274182
$780$	−0.115731	−0.00414385
$781$	0	0
$782$	−33.3500	−1.19259
$783$	4.61676	0.164989
$784$	−6.74323	−0.240830
$785$	−1.92213	−0.0686038
$786$	16.0696	0.573182
$787$	13.8423	0.493425	0.246713	−	0.969089i	$-0.420650\pi$
$787$	13.8423	0.493425	0.246713	+	0.969089i	$0.420650\pi$
$788$	6.17124	0.219841
$789$	18.6850	0.665204
$790$	1.59389	0.0567080
$791$	−49.0964	−1.74567
$792$	0	0
$793$	5.89043	0.209175
$794$	−37.1103	−1.31700
$795$	−2.87144	−0.101839
$796$	−6.29459	−0.223106
$797$	0.166848	0.00591006	0.00295503	−	0.999996i	$-0.499059\pi$
$797$	0.166848	0.00591006	0.00295503	+	0.999996i	$0.499059\pi$
$798$	−3.85565	−0.136488
$799$	5.58443	0.197563
$800$	−10.2827	−0.363547
$801$	4.87723	0.172328
$802$	−44.4359	−1.56909
$803$	0	0
$804$	−5.49689	−0.193860
$805$	−9.55556	−0.336789
$806$	−10.2708	−0.361774
$807$	10.4499	0.367853
$808$	−1.12536	−0.0395902
$809$	5.95252	0.209280	0.104640	−	0.994510i	$-0.466631\pi$
$809$	5.95252	0.209280	0.104640	+	0.994510i	$0.466631\pi$
$810$	−0.496164	−0.0174334
$811$	56.2039	1.97359	0.986793	−	0.161984i	$-0.0517893\pi$
$811$	56.2039	1.97359	0.986793	+	0.161984i	$0.0517893\pi$
$812$	5.32193	0.186763
$813$	10.8103	0.379132
$814$	0	0
$815$	4.67172	0.163643
$816$	−10.0240	−0.350910
$817$	5.34992	0.187170
$818$	−43.6358	−1.52569
$819$	2.36371	0.0825948
$820$	−1.13519	−0.0396425
$821$	−32.6126	−1.13819	−0.569094	−	0.822273i	$-0.692706\pi$
$821$	−32.6126	−1.13819	−0.569094	+	0.822273i	$0.692706\pi$
$822$	14.4963	0.505617
$823$	−39.9071	−1.39107	−0.695537	−	0.718490i	$-0.744833\pi$
$823$	−39.9071	−1.39107	−0.695537	+	0.718490i	$0.744833\pi$
$824$	−34.9301	−1.21685
$825$	0	0
$826$	−3.00051	−0.104401
$827$	33.4075	1.16169	0.580846	−	0.814014i	$-0.302722\pi$
$827$	33.4075	1.16169	0.580846	+	0.814014i	$0.302722\pi$
$828$	3.07786	0.106963
$829$	34.7158	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$829$	34.7158	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$830$	−3.82948	−0.132923
$831$	14.0870	0.488674
$832$	−6.93400	−0.240393
$833$	7.05975	0.244606
$834$	−5.16117	−0.178717
$835$	9.57847	0.331477
$836$	0	0
$837$	10.3448	0.357567
$838$	−2.75256	−0.0950857
$839$	−12.0360	−0.415530	−0.207765	−	0.978179i	$-0.566619\pi$
$839$	−12.0360	−0.415530	−0.207765	+	0.978179i	$0.566619\pi$
$840$	−3.57843	−0.123468
$841$	−7.68554	−0.265019
$842$	20.0017	0.689304
$843$	−28.9479	−0.997019
$844$	3.68364	0.126796
$845$	4.83114	0.166196
$846$	2.19377	0.0754234
$847$	0	0
$848$	−22.7891	−0.782582
$849$	3.46552	0.118936
$850$	−19.9865	−0.685531
$851$	5.32926	0.182685
$852$	−0.0451547	−0.00154698
$853$	47.5442	1.62788	0.813940	−	0.580948i	$-0.197318\pi$
$853$	47.5442	1.62788	0.813940	+	0.580948i	$0.197318\pi$
$854$	29.1107	0.996149
$855$	−0.389881	−0.0133336
$856$	42.6048	1.45620
$857$	−42.0106	−1.43505	−0.717527	−	0.696531i	$-0.754726\pi$
$857$	−42.0106	−1.43505	−0.717527	+	0.696531i	$0.754726\pi$
$858$	0	0
$859$	1.02089	0.0348324	0.0174162	−	0.999848i	$-0.494456\pi$
$859$	1.02089	0.0348324	0.0174162	+	0.999848i	$0.494456\pi$
$860$	0.793610	0.0270619
$861$	23.1852	0.790151
$862$	30.9267	1.05337
$863$	−29.4599	−1.00283	−0.501414	−	0.865208i	$-0.667187\pi$
$863$	−29.4599	−1.00283	−0.501414	+	0.865208i	$0.667187\pi$
$864$	2.12101	0.0721584
$865$	−7.56603	−0.257253
$866$	−50.0194	−1.69973
$867$	−6.50549	−0.220938
$868$	11.9249	0.404756
$869$	0	0
$870$	−2.29067	−0.0776610
$871$	−11.2714	−0.381918
$872$	−42.9813	−1.45553
$873$	8.68858	0.294064
$874$	−10.2947	−0.348224
$875$	−11.6328	−0.393259
$876$	1.20639	0.0407600
$877$	−10.0319	−0.338753	−0.169376	−	0.985551i	$-0.554175\pi$
$877$	−10.0319	−0.338753	−0.169376	+	0.985551i	$0.554175\pi$
$878$	−38.6621	−1.30478
$879$	11.5329	0.388997
$880$	0	0
$881$	4.15852	0.140104	0.0700521	−	0.997543i	$-0.477683\pi$
$881$	4.15852	0.140104	0.0700521	+	0.997543i	$0.477683\pi$
$882$	2.77333	0.0933829
$883$	40.4459	1.36111	0.680557	−	0.732695i	$-0.261738\pi$
$883$	40.4459	1.36111	0.680557	+	0.732695i	$0.261738\pi$
$884$	0.961613	0.0323426
$885$	−0.303409	−0.0101990
$886$	−24.1891	−0.812648
$887$	−0.418438	−0.0140498	−0.00702489	−	0.999975i	$-0.502236\pi$
$887$	−0.418438	−0.0140498	−0.00702489	+	0.999975i	$0.502236\pi$
$888$	1.99574	0.0669727
$889$	−20.3694	−0.683167
$890$	−2.41990	−0.0811153
$891$	0	0
$892$	1.02686	0.0343818
$893$	1.72384	0.0576862
$894$	23.2206	0.776612
$895$	0.451577	0.0150945
$896$	−21.4159	−0.715456
$897$	6.31119	0.210725
$898$	0.496359	0.0165637
$899$	47.7593	1.59286
$900$	1.84455	0.0614850
$901$	23.8588	0.794853
$902$	0	0
$903$	−16.2088	−0.539395
$904$	−49.0912	−1.63275
$905$	−3.49345	−0.116126
$906$	−23.0268	−0.765015
$907$	41.8633	1.39005	0.695024	−	0.718987i	$-0.255394\pi$
$907$	41.8633	1.39005	0.695024	+	0.718987i	$0.255394\pi$
$908$	−4.21424	−0.139855
$909$	0.371480	0.0123212
$910$	−1.17279	−0.0388776
$911$	−19.9497	−0.660964	−0.330482	−	0.943812i	$-0.607211\pi$
$911$	−19.9497	−0.660964	−0.330482	+	0.943812i	$0.607211\pi$
$912$	−3.09428	−0.102462
$913$	0	0
$914$	−36.6224	−1.21136
$915$	2.94366	0.0973144
$916$	−5.46846	−0.180683
$917$	−38.2573	−1.26337
$918$	4.12263	0.136067
$919$	15.3347	0.505845	0.252923	−	0.967486i	$-0.418608\pi$
$919$	15.3347	0.505845	0.252923	+	0.967486i	$0.418608\pi$
$920$	−9.55454	−0.315004
$921$	−12.0356	−0.396587
$922$	−36.8924	−1.21499
$923$	−0.0925903	−0.00304765
$924$	0	0
$925$	3.19381	0.105012
$926$	−23.4017	−0.769028
$927$	11.5303	0.378706
$928$	9.79221	0.321445
$929$	−27.3688	−0.897941	−0.448971	−	0.893546i	$-0.648209\pi$
$929$	−27.3688	−0.897941	−0.448971	+	0.893546i	$0.648209\pi$
$930$	−5.13270	−0.168308
$931$	2.17926	0.0714222
$932$	4.60940	0.150986
$933$	−11.7129	−0.383462
$934$	−23.6844	−0.774976
$935$	0	0
$936$	2.36346	0.0772521
$937$	−35.3896	−1.15613	−0.578064	−	0.815991i	$-0.696192\pi$
$937$	−35.3896	−1.15613	−0.578064	+	0.815991i	$0.696192\pi$
$938$	−55.7039	−1.81880
$939$	27.6594	0.902630
$940$	0.255716	0.00834054
$941$	17.6662	0.575902	0.287951	−	0.957645i	$-0.407026\pi$
$941$	17.6662	0.575902	0.287951	+	0.957645i	$0.407026\pi$
$942$	6.27400	0.204418
$943$	61.9054	2.01592
$944$	−2.40801	−0.0783739
$945$	1.18123	0.0384255
$946$	0	0
$947$	42.1818	1.37072	0.685362	−	0.728202i	$-0.259644\pi$
$947$	42.1818	1.37072	0.685362	+	0.728202i	$0.259644\pi$
$948$	1.22225	0.0396970
$949$	2.47371	0.0802999
$950$	−6.16958	−0.200168
$951$	24.2852	0.787501
$952$	29.7333	0.963660
$953$	−3.75469	−0.121626	−0.0608131	−	0.998149i	$-0.519369\pi$
$953$	−3.75469	−0.121626	−0.0608131	+	0.998149i	$0.519369\pi$
$954$	9.37263	0.303450
$955$	−4.66956	−0.151104
$956$	−5.01723	−0.162269
$957$	0	0
$958$	1.56120	0.0504400
$959$	−34.5118	−1.11445
$960$	−3.46517	−0.111838
$961$	76.0142	2.45207
$962$	0.654080	0.0210884
$963$	−14.0638	−0.453198
$964$	−4.67528	−0.150581
$965$	4.25369	0.136931
$966$	31.1902	1.00353
$967$	−20.2051	−0.649753	−0.324877	−	0.945756i	$-0.605323\pi$
$967$	−20.2051	−0.649753	−0.324877	+	0.945756i	$0.605323\pi$
$968$	0	0
$969$	3.23952	0.104068
$970$	−4.31096	−0.138417
$971$	25.3697	0.814152	0.407076	−	0.913394i	$-0.366548\pi$
$971$	25.3697	0.814152	0.407076	+	0.913394i	$0.366548\pi$
$972$	−0.380477	−0.0122038
$973$	12.2873	0.393914
$974$	31.3295	1.00386
$975$	3.78227	0.121130
$976$	23.3623	0.747810
$977$	28.9196	0.925221	0.462611	−	0.886562i	$-0.346913\pi$
$977$	28.9196	0.925221	0.462611	+	0.886562i	$0.346913\pi$
$978$	−15.2489	−0.487606
$979$	0	0
$980$	0.323272	0.0103266
$981$	14.1880	0.452989
$982$	−5.28086	−0.168519
$983$	20.0642	0.639948	0.319974	−	0.947426i	$-0.396326\pi$
$983$	20.0642	0.639948	0.319974	+	0.947426i	$0.396326\pi$
$984$	23.1828	0.739040
$985$	6.32377	0.201492
$986$	19.0332	0.606141
$987$	−5.22278	−0.166243
$988$	0.296838	0.00944367
$989$	−43.2781	−1.37616
$990$	0	0
$991$	8.64344	0.274568	0.137284	−	0.990532i	$-0.456163\pi$
$991$	8.64344	0.274568	0.137284	+	0.990532i	$0.456163\pi$
$992$	21.9414	0.696640
$993$	32.3549	1.02675
$994$	−0.457585	−0.0145137
$995$	−6.45016	−0.204484
$996$	−2.93659	−0.0930495
$997$	−35.7350	−1.13174	−0.565870	−	0.824495i	$-0.691460\pi$
$997$	−35.7350	−1.13174	−0.565870	+	0.824495i	$0.691460\pi$
$998$	−0.476023	−0.0150682
$999$	−0.658789	−0.0208432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.bq.1.15		24
11.7	odd	10		627.2.j.d.115.5	✓	48
11.8	odd	10		627.2.j.d.229.5	yes	48
11.10	odd	2		6897.2.a.bp.1.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.j.d.115.5	✓	48	11.7	odd	10
627.2.j.d.229.5	yes	48	11.8	odd	10
6897.2.a.bp.1.10		24	11.10	odd	2
6897.2.a.bq.1.15		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)

\(f(q)\)	\(=\)	\(q+1.27260 q^{2} +1.00000 q^{3} -0.380477 q^{4} -0.389881 q^{5} +1.27260 q^{6} -3.02973 q^{7} -3.02941 q^{8} +1.00000 q^{9} -0.496164 q^{10} -0.380477 q^{12} -0.780173 q^{13} -3.85565 q^{14} -0.389881 q^{15} -3.09428 q^{16} +3.23952 q^{17} +1.27260 q^{18} +1.00000 q^{19} +0.148341 q^{20} -3.02973 q^{21} -8.08948 q^{23} -3.02941 q^{24} -4.84799 q^{25} -0.992851 q^{26} +1.00000 q^{27} +1.15274 q^{28} +4.61676 q^{29} -0.496164 q^{30} +10.3448 q^{31} +2.12101 q^{32} +4.12263 q^{34} +1.18123 q^{35} -0.380477 q^{36} -0.658789 q^{37} +1.27260 q^{38} -0.780173 q^{39} +1.18111 q^{40} -7.65258 q^{41} -3.85565 q^{42} +5.34992 q^{43} -0.389881 q^{45} -10.2947 q^{46} +1.72384 q^{47} -3.09428 q^{48} +2.17926 q^{49} -6.16958 q^{50} +3.23952 q^{51} +0.296838 q^{52} +7.36492 q^{53} +1.27260 q^{54} +9.17828 q^{56} +1.00000 q^{57} +5.87531 q^{58} +0.778211 q^{59} +0.148341 q^{60} -7.55016 q^{61} +13.1648 q^{62} -3.02973 q^{63} +8.88778 q^{64} +0.304174 q^{65} +14.4474 q^{67} -1.23256 q^{68} -8.08948 q^{69} +1.50324 q^{70} +0.118679 q^{71} -3.02941 q^{72} -3.17072 q^{73} -0.838378 q^{74} -4.84799 q^{75} -0.380477 q^{76} -0.992851 q^{78} -3.21242 q^{79} +1.20640 q^{80} +1.00000 q^{81} -9.73871 q^{82} +7.71818 q^{83} +1.15274 q^{84} -1.26303 q^{85} +6.80833 q^{86} +4.61676 q^{87} +4.87723 q^{89} -0.496164 q^{90} +2.36371 q^{91} +3.07786 q^{92} +10.3448 q^{93} +2.19377 q^{94} -0.389881 q^{95} +2.12101 q^{96} +8.68858 q^{97} +2.77333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 24 q^{3} + 37 q^{4} + 9 q^{5} + q^{6} + 3 q^{7} + 24 q^{9} - q^{10} + 37 q^{12} + 3 q^{13} - 2 q^{14} + 9 q^{15} + 55 q^{16} - 3 q^{17} + q^{18} + 24 q^{19} + 12 q^{20} + 3 q^{21} + 30 q^{23}+ \cdots + 59 q^{98}+O(q^{100}) \) 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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