LMFDB - Embedded newform 6897.2.a.bc.1.3

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 = [8,-1,-8,9,-2,1,-6,-6,8,-3,0,-9,-1] 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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 9x^{5} + 40x^{4} - 22x^{3} - 28x^{2} + 12x - 1 $ x^8 - x^7 - 12x^6 + 9x^5 + 40x^4 - 22x^3 - 28x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.07029$ of defining polynomial
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)

$f(q)$	$=$	\(q-1.07029 q^{2} -1.00000 q^{3} -0.854469 q^{4} -4.21558 q^{5} +1.07029 q^{6} -3.04581 q^{7} +3.05512 q^{8} +1.00000 q^{9} +4.51192 q^{10} +0.854469 q^{12} -1.07029 q^{13} +3.25991 q^{14} +4.21558 q^{15} -1.56095 q^{16} -6.33447 q^{17} -1.07029 q^{18} +1.00000 q^{19} +3.60209 q^{20} +3.04581 q^{21} -7.29424 q^{23} -3.05512 q^{24} +12.7712 q^{25} +1.14553 q^{26} -1.00000 q^{27} +2.60255 q^{28} -1.89962 q^{29} -4.51192 q^{30} -9.84443 q^{31} -4.43957 q^{32} +6.77975 q^{34} +12.8399 q^{35} -0.854469 q^{36} -5.95117 q^{37} -1.07029 q^{38} +1.07029 q^{39} -12.8791 q^{40} +0.672341 q^{41} -3.25991 q^{42} +11.5029 q^{43} -4.21558 q^{45} +7.80698 q^{46} +6.95445 q^{47} +1.56095 q^{48} +2.27694 q^{49} -13.6689 q^{50} +6.33447 q^{51} +0.914534 q^{52} -4.85314 q^{53} +1.07029 q^{54} -9.30532 q^{56} -1.00000 q^{57} +2.03315 q^{58} -0.840641 q^{59} -3.60209 q^{60} +10.8024 q^{61} +10.5364 q^{62} -3.04581 q^{63} +7.87355 q^{64} +4.51192 q^{65} -8.73056 q^{67} +5.41261 q^{68} +7.29424 q^{69} -13.7424 q^{70} +14.1862 q^{71} +3.05512 q^{72} +9.15644 q^{73} +6.36950 q^{74} -12.7712 q^{75} -0.854469 q^{76} -1.14553 q^{78} -15.0178 q^{79} +6.58030 q^{80} +1.00000 q^{81} -0.719603 q^{82} +5.58016 q^{83} -2.60255 q^{84} +26.7035 q^{85} -12.3115 q^{86} +1.89962 q^{87} +5.06617 q^{89} +4.51192 q^{90} +3.25991 q^{91} +6.23270 q^{92} +9.84443 q^{93} -7.44331 q^{94} -4.21558 q^{95} +4.43957 q^{96} +13.7339 q^{97} -2.43700 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) $ 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * 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 $

$2$	−1.07029	−0.756813	−0.378406	−	0.925640i	$-0.623528\pi$
$2$	−1.07029	−0.756813	−0.378406	+	0.925640i	$0.623528\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.854469	−0.427234
$5$	−4.21558	−1.88527	−0.942633	−	0.333830i	$-0.891659\pi$
$5$	−4.21558	−1.88527	−0.942633	+	0.333830i	$0.891659\pi$
$6$	1.07029	0.436946
$7$	−3.04581	−1.15121	−0.575603	−	0.817729i	$-0.695233\pi$
$7$	−3.04581	−1.15121	−0.575603	+	0.817729i	$0.695233\pi$
$8$	3.05512	1.08015
$9$	1.00000	0.333333
$10$	4.51192	1.42679
$11$	0	0
$11$	0	0
$12$	0.854469	0.246664
$13$	−1.07029	−0.296846	−0.148423	−	0.988924i	$-0.547420\pi$
$13$	−1.07029	−0.296846	−0.148423	+	0.988924i	$0.547420\pi$
$14$	3.25991	0.871248
$15$	4.21558	1.08846
$16$	−1.56095	−0.390236
$17$	−6.33447	−1.53633	−0.768167	−	0.640249i	$-0.778831\pi$
$17$	−6.33447	−1.53633	−0.768167	+	0.640249i	$0.778831\pi$
$18$	−1.07029	−0.252271
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.60209	0.805451
$21$	3.04581	0.664650
$22$	0	0
$23$	−7.29424	−1.52095	−0.760477	−	0.649365i	$-0.775035\pi$
$23$	−7.29424	−1.52095	−0.760477	+	0.649365i	$0.775035\pi$
$24$	−3.05512	−0.623624
$25$	12.7712	2.55423
$26$	1.14553	0.224657
$27$	−1.00000	−0.192450
$28$	2.60255	0.491835
$29$	−1.89962	−0.352751	−0.176375	−	0.984323i	$-0.556437\pi$
$29$	−1.89962	−0.352751	−0.176375	+	0.984323i	$0.556437\pi$
$30$	−4.51192	−0.823760
$31$	−9.84443	−1.76811	−0.884056	−	0.467380i	$-0.845198\pi$
$31$	−9.84443	−1.76811	−0.884056	+	0.467380i	$0.845198\pi$
$32$	−4.43957	−0.784813
$33$	0	0
$34$	6.77975	1.16272
$35$	12.8399	2.17033
$36$	−0.854469	−0.142411
$37$	−5.95117	−0.978366	−0.489183	−	0.872181i	$-0.662705\pi$
$37$	−5.95117	−0.978366	−0.489183	+	0.872181i	$0.662705\pi$
$38$	−1.07029	−0.173625
$39$	1.07029	0.171384
$40$	−12.8791	−2.03637
$41$	0.672341	0.105002	0.0525010	−	0.998621i	$-0.483281\pi$
$41$	0.672341	0.105002	0.0525010	+	0.998621i	$0.483281\pi$
$42$	−3.25991	−0.503015
$43$	11.5029	1.75418	0.877091	−	0.480324i	$-0.159481\pi$
$43$	11.5029	1.75418	0.877091	+	0.480324i	$0.159481\pi$
$44$	0	0
$45$	−4.21558	−0.628422
$46$	7.80698	1.15108
$47$	6.95445	1.01441	0.507206	−	0.861825i	$-0.330678\pi$
$47$	6.95445	1.01441	0.507206	+	0.861825i	$0.330678\pi$
$48$	1.56095	0.225303
$49$	2.27694	0.325277
$50$	−13.6689	−1.93307
$51$	6.33447	0.887003
$52$	0.914534	0.126823
$53$	−4.85314	−0.666630	−0.333315	−	0.942816i	$-0.608167\pi$
$53$	−4.85314	−0.666630	−0.333315	+	0.942816i	$0.608167\pi$
$54$	1.07029	0.145649
$55$	0	0
$56$	−9.30532	−1.24348
$57$	−1.00000	−0.132453
$58$	2.03315	0.266966
$59$	−0.840641	−0.109442	−0.0547211	−	0.998502i	$-0.517427\pi$
$59$	−0.840641	−0.109442	−0.0547211	+	0.998502i	$0.517427\pi$
$60$	−3.60209	−0.465027
$61$	10.8024	1.38311	0.691554	−	0.722325i	$-0.256926\pi$
$61$	10.8024	1.38311	0.691554	+	0.722325i	$0.256926\pi$
$62$	10.5364	1.33813
$63$	−3.04581	−0.383736
$64$	7.87355	0.984193
$65$	4.51192	0.559635
$66$	0	0
$67$	−8.73056	−1.06661	−0.533304	−	0.845924i	$-0.679050\pi$
$67$	−8.73056	−1.06661	−0.533304	+	0.845924i	$0.679050\pi$
$68$	5.41261	0.656375
$69$	7.29424	0.878123
$70$	−13.7424	−1.64253
$71$	14.1862	1.68359	0.841793	−	0.539800i	$-0.181500\pi$
$71$	14.1862	1.68359	0.841793	+	0.539800i	$0.181500\pi$
$72$	3.05512	0.360050
$73$	9.15644	1.07168	0.535840	−	0.844320i	$-0.319995\pi$
$73$	9.15644	1.07168	0.535840	+	0.844320i	$0.319995\pi$
$74$	6.36950	0.740440
$75$	−12.7712	−1.47469
$76$	−0.854469	−0.0980143
$77$	0	0
$78$	−1.14553	−0.129706
$79$	−15.0178	−1.68964	−0.844819	−	0.535052i	$-0.820292\pi$
$79$	−15.0178	−1.68964	−0.844819	+	0.535052i	$0.820292\pi$
$80$	6.58030	0.735700
$81$	1.00000	0.111111
$82$	−0.719603	−0.0794669
$83$	5.58016	0.612502	0.306251	−	0.951951i	$-0.400925\pi$
$83$	5.58016	0.612502	0.306251	+	0.951951i	$0.400925\pi$
$84$	−2.60255	−0.283961
$85$	26.7035	2.89640
$86$	−12.3115	−1.32759
$87$	1.89962	0.203661
$88$	0	0
$89$	5.06617	0.537013	0.268506	−	0.963278i	$-0.413470\pi$
$89$	5.06617	0.537013	0.268506	+	0.963278i	$0.413470\pi$
$90$	4.51192	0.475598
$91$	3.25991	0.341732
$92$	6.23270	0.649804
$93$	9.84443	1.02082
$94$	−7.44331	−0.767719
$95$	−4.21558	−0.432510
$96$	4.43957	0.453112
$97$	13.7339	1.39447	0.697233	−	0.716844i	$-0.254414\pi$
$97$	13.7339	1.39447	0.697233	+	0.716844i	$0.254414\pi$
$98$	−2.43700	−0.246174
$99$	0	0
$100$	−10.9126	−1.09126
$101$	−10.3717	−1.03202	−0.516010	−	0.856583i	$-0.672583\pi$
$101$	−10.3717	−1.03202	−0.516010	+	0.856583i	$0.672583\pi$
$102$	−6.77975	−0.671296
$103$	9.72187	0.957924	0.478962	−	0.877836i	$-0.341013\pi$
$103$	9.72187	0.957924	0.478962	+	0.877836i	$0.341013\pi$
$104$	−3.26988	−0.320638
$105$	−12.8399	−1.25304
$106$	5.19429	0.504514
$107$	−13.6762	−1.32213	−0.661066	−	0.750328i	$-0.729896\pi$
$107$	−13.6762	−1.32213	−0.661066	+	0.750328i	$0.729896\pi$
$108$	0.854469	0.0822213
$109$	12.7021	1.21664	0.608320	−	0.793692i	$-0.291844\pi$
$109$	12.7021	1.21664	0.608320	+	0.793692i	$0.291844\pi$
$110$	0	0
$111$	5.95117	0.564860
$112$	4.75434	0.449243
$113$	10.8536	1.02102	0.510511	−	0.859871i	$-0.329456\pi$
$113$	10.8536	1.02102	0.510511	+	0.859871i	$0.329456\pi$
$114$	1.07029	0.100242
$115$	30.7495	2.86740
$116$	1.62317	0.150707
$117$	−1.07029	−0.0989488
$118$	0.899734	0.0828272
$119$	19.2936	1.76864
$120$	12.8791	1.17570
$121$	0	0
$122$	−11.5618	−1.04675
$123$	−0.672341	−0.0606230
$124$	8.41176	0.755398
$125$	−32.7600	−2.93014
$126$	3.25991	0.290416
$127$	6.04791	0.536665	0.268333	−	0.963326i	$-0.413527\pi$
$127$	6.04791	0.536665	0.268333	+	0.963326i	$0.413527\pi$
$128$	0.452133	0.0399633
$129$	−11.5029	−1.01278
$130$	−4.82908	−0.423539
$131$	8.86309	0.774372	0.387186	−	0.922002i	$-0.373447\pi$
$131$	8.86309	0.774372	0.387186	+	0.922002i	$0.373447\pi$
$132$	0	0
$133$	−3.04581	−0.264105
$134$	9.34427	0.807222
$135$	4.21558	0.362820
$136$	−19.3526	−1.65947
$137$	3.19996	0.273391	0.136695	−	0.990613i	$-0.456352\pi$
$137$	3.19996	0.273391	0.136695	+	0.990613i	$0.456352\pi$
$138$	−7.80698	−0.664575
$139$	−2.57273	−0.218216	−0.109108	−	0.994030i	$-0.534799\pi$
$139$	−2.57273	−0.218216	−0.109108	+	0.994030i	$0.534799\pi$
$140$	−10.9713	−0.927240
$141$	−6.95445	−0.585671
$142$	−15.1834	−1.27416
$143$	0	0
$144$	−1.56095	−0.130079
$145$	8.00801	0.665029
$146$	−9.80009	−0.811061
$147$	−2.27694	−0.187799
$148$	5.08509	0.417991
$149$	4.85084	0.397396	0.198698	−	0.980061i	$-0.436329\pi$
$149$	4.85084	0.397396	0.198698	+	0.980061i	$0.436329\pi$
$150$	13.6689	1.11606
$151$	−2.02828	−0.165059	−0.0825296	−	0.996589i	$-0.526300\pi$
$151$	−2.02828	−0.165059	−0.0825296	+	0.996589i	$0.526300\pi$
$152$	3.05512	0.247803
$153$	−6.33447	−0.512112
$154$	0	0
$155$	41.5000	3.33336
$156$	−0.914534	−0.0732213
$157$	−4.22450	−0.337152	−0.168576	−	0.985689i	$-0.553917\pi$
$157$	−4.22450	−0.337152	−0.168576	+	0.985689i	$0.553917\pi$
$158$	16.0735	1.27874
$159$	4.85314	0.384879
$160$	18.7154	1.47958
$161$	22.2168	1.75093
$162$	−1.07029	−0.0840903
$163$	18.7353	1.46746	0.733730	−	0.679441i	$-0.237778\pi$
$163$	18.7353	1.46746	0.733730	+	0.679441i	$0.237778\pi$
$164$	−0.574495	−0.0448605
$165$	0	0
$166$	−5.97241	−0.463549
$167$	−12.6973	−0.982547	−0.491273	−	0.871005i	$-0.663468\pi$
$167$	−12.6973	−0.982547	−0.491273	+	0.871005i	$0.663468\pi$
$168$	9.30532	0.717921
$169$	−11.8545	−0.911882
$170$	−28.5806	−2.19203
$171$	1.00000	0.0764719
$172$	−9.82891	−0.749447
$173$	−6.92230	−0.526293	−0.263146	−	0.964756i	$-0.584760\pi$
$173$	−6.92230	−0.526293	−0.263146	+	0.964756i	$0.584760\pi$
$174$	−2.03315	−0.154133
$175$	−38.8985	−2.94045
$176$	0	0
$177$	0.840641	0.0631865
$178$	−5.42229	−0.406418
$179$	12.0605	0.901442	0.450721	−	0.892665i	$-0.351167\pi$
$179$	12.0605	0.901442	0.450721	+	0.892665i	$0.351167\pi$
$180$	3.60209	0.268484
$181$	−5.60525	−0.416635	−0.208317	−	0.978061i	$-0.566799\pi$
$181$	−5.60525	−0.416635	−0.208317	+	0.978061i	$0.566799\pi$
$182$	−3.48907	−0.258627
$183$	−10.8024	−0.798538
$184$	−22.2848	−1.64286
$185$	25.0876	1.84448
$186$	−10.5364	−0.772570
$187$	0	0
$188$	−5.94236	−0.433391
$189$	3.04581	0.221550
$190$	4.51192	0.327329
$191$	22.6045	1.63560	0.817802	−	0.575499i	$-0.195192\pi$
$191$	22.6045	1.63560	0.817802	+	0.575499i	$0.195192\pi$
$192$	−7.87355	−0.568224
$193$	−1.49251	−0.107433	−0.0537166	−	0.998556i	$-0.517107\pi$
$193$	−1.49251	−0.107433	−0.0537166	+	0.998556i	$0.517107\pi$
$194$	−14.6993	−1.05535
$195$	−4.51192	−0.323105
$196$	−1.94557	−0.138970
$197$	−13.0698	−0.931187	−0.465594	−	0.884999i	$-0.654159\pi$
$197$	−13.0698	−0.931187	−0.465594	+	0.884999i	$0.654159\pi$
$198$	0	0
$199$	−7.70873	−0.546458	−0.273229	−	0.961949i	$-0.588092\pi$
$199$	−7.70873	−0.546458	−0.273229	+	0.961949i	$0.588092\pi$
$200$	39.0175	2.75895
$201$	8.73056	0.615806
$202$	11.1007	0.781046
$203$	5.78588	0.406089
$204$	−5.41261	−0.378958
$205$	−2.83431	−0.197957
$206$	−10.4053	−0.724969
$207$	−7.29424	−0.506984
$208$	1.67067	0.115840
$209$	0	0
$210$	13.7424	0.948318
$211$	16.0870	1.10748	0.553739	−	0.832690i	$-0.313201\pi$
$211$	16.0870	1.10748	0.553739	+	0.832690i	$0.313201\pi$
$212$	4.14685	0.284807
$213$	−14.1862	−0.972019
$214$	14.6376	1.00061
$215$	−48.4916	−3.30710
$216$	−3.05512	−0.207875
$217$	29.9842	2.03546
$218$	−13.5950	−0.920769
$219$	−9.15644	−0.618735
$220$	0	0
$221$	6.77975	0.456056
$222$	−6.36950	−0.427493
$223$	2.12052	0.142000	0.0710001	−	0.997476i	$-0.477381\pi$
$223$	2.12052	0.142000	0.0710001	+	0.997476i	$0.477381\pi$
$224$	13.5221	0.903482
$225$	12.7712	0.851410
$226$	−11.6166	−0.772723
$227$	−11.4989	−0.763212	−0.381606	−	0.924325i	$-0.624629\pi$
$227$	−11.4989	−0.763212	−0.381606	+	0.924325i	$0.624629\pi$
$228$	0.854469	0.0565886
$229$	21.9950	1.45347	0.726737	−	0.686916i	$-0.241036\pi$
$229$	21.9950	1.45347	0.726737	+	0.686916i	$0.241036\pi$
$230$	−32.9110	−2.17009
$231$	0	0
$232$	−5.80358	−0.381023
$233$	−0.990018	−0.0648582	−0.0324291	−	0.999474i	$-0.510324\pi$
$233$	−0.990018	−0.0648582	−0.0324291	+	0.999474i	$0.510324\pi$
$234$	1.14553	0.0748857
$235$	−29.3171	−1.91244
$236$	0.718302	0.0467575
$237$	15.0178	0.975513
$238$	−20.6498	−1.33853
$239$	11.6106	0.751029	0.375515	−	0.926816i	$-0.377466\pi$
$239$	11.6106	0.751029	0.375515	+	0.926816i	$0.377466\pi$
$240$	−6.58030	−0.424756
$241$	11.0505	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$241$	11.0505	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−9.23033	−0.590911
$245$	−9.59863	−0.613234
$246$	0.719603	0.0458802
$247$	−1.07029	−0.0681012
$248$	−30.0760	−1.90983
$249$	−5.58016	−0.353628
$250$	35.0628	2.21757
$251$	−12.2910	−0.775800	−0.387900	−	0.921701i	$-0.626799\pi$
$251$	−12.2910	−0.775800	−0.387900	+	0.921701i	$0.626799\pi$
$252$	2.60255	0.163945
$253$	0	0
$254$	−6.47305	−0.406155
$255$	−26.7035	−1.67224
$256$	−16.2310	−1.01444
$257$	−16.8226	−1.04937	−0.524683	−	0.851298i	$-0.675816\pi$
$257$	−16.8226	−1.04937	−0.524683	+	0.851298i	$0.675816\pi$
$258$	12.3115	0.766483
$259$	18.1261	1.12630
$260$	−3.85529	−0.239095
$261$	−1.89962	−0.117584
$262$	−9.48612	−0.586055
$263$	−11.3136	−0.697629	−0.348814	−	0.937192i	$-0.613416\pi$
$263$	−11.3136	−0.697629	−0.348814	+	0.937192i	$0.613416\pi$
$264$	0	0
$265$	20.4588	1.25677
$266$	3.25991	0.199878
$267$	−5.06617	−0.310044
$268$	7.45999	0.455691
$269$	9.25786	0.564462	0.282231	−	0.959347i	$-0.408926\pi$
$269$	9.25786	0.564462	0.282231	+	0.959347i	$0.408926\pi$
$270$	−4.51192	−0.274587
$271$	9.51383	0.577924	0.288962	−	0.957341i	$-0.406690\pi$
$271$	9.51383	0.577924	0.288962	+	0.957341i	$0.406690\pi$
$272$	9.88777	0.599534
$273$	−3.25991	−0.197299
$274$	−3.42490	−0.206906
$275$	0	0
$276$	−6.23270	−0.375164
$277$	−20.8566	−1.25315	−0.626577	−	0.779359i	$-0.715545\pi$
$277$	−20.8566	−1.25315	−0.626577	+	0.779359i	$0.715545\pi$
$278$	2.75358	0.165149
$279$	−9.84443	−0.589371
$280$	39.2273	2.34428
$281$	10.5486	0.629274	0.314637	−	0.949212i	$-0.398117\pi$
$281$	10.5486	0.629274	0.314637	+	0.949212i	$0.398117\pi$
$282$	7.44331	0.443243
$283$	−5.03751	−0.299449	−0.149725	−	0.988728i	$-0.547839\pi$
$283$	−5.03751	−0.299449	−0.149725	+	0.988728i	$0.547839\pi$
$284$	−12.1216	−0.719286
$285$	4.21558	0.249710
$286$	0	0
$287$	−2.04782	−0.120879
$288$	−4.43957	−0.261604
$289$	23.1255	1.36033
$290$	−8.57093	−0.503303
$291$	−13.7339	−0.805096
$292$	−7.82389	−0.457859
$293$	−24.4532	−1.42857	−0.714286	−	0.699854i	$-0.753249\pi$
$293$	−24.4532	−1.42857	−0.714286	+	0.699854i	$0.753249\pi$
$294$	2.43700	0.142129
$295$	3.54379	0.206328
$296$	−18.1815	−1.05678
$297$	0	0
$298$	−5.19182	−0.300754
$299$	7.80698	0.451490
$300$	10.9126	0.630036
$301$	−35.0357	−2.01943
$302$	2.17086	0.124919
$303$	10.3717	0.595837
$304$	−1.56095	−0.0895264
$305$	−45.5385	−2.60753
$306$	6.77975	0.387573
$307$	−6.17580	−0.352472	−0.176236	−	0.984348i	$-0.556392\pi$
$307$	−6.17580	−0.352472	−0.176236	+	0.984348i	$0.556392\pi$
$308$	0	0
$309$	−9.72187	−0.553058
$310$	−44.4173	−2.52273
$311$	−0.839114	−0.0475818	−0.0237909	−	0.999717i	$-0.507574\pi$
$311$	−0.839114	−0.0475818	−0.0237909	+	0.999717i	$0.507574\pi$
$312$	3.26988	0.185121
$313$	6.34941	0.358890	0.179445	−	0.983768i	$-0.442570\pi$
$313$	6.34941	0.358890	0.179445	+	0.983768i	$0.442570\pi$
$314$	4.52146	0.255161
$315$	12.8399	0.723444
$316$	12.8323	0.721871
$317$	−7.62744	−0.428400	−0.214200	−	0.976790i	$-0.568714\pi$
$317$	−7.62744	−0.428400	−0.214200	+	0.976790i	$0.568714\pi$
$318$	−5.19429	−0.291281
$319$	0	0
$320$	−33.1916	−1.85547
$321$	13.6762	0.763333
$322$	−23.7786	−1.32513
$323$	−6.33447	−0.352459
$324$	−0.854469	−0.0474705
$325$	−13.6689	−0.758214
$326$	−20.0523	−1.11059
$327$	−12.7021	−0.702428
$328$	2.05409	0.113418
$329$	−21.1819	−1.16780
$330$	0	0
$331$	11.7060	0.643419	0.321710	−	0.946838i	$-0.395742\pi$
$331$	11.7060	0.643419	0.321710	+	0.946838i	$0.395742\pi$
$332$	−4.76807	−0.261682
$333$	−5.95117	−0.326122
$334$	13.5899	0.743604
$335$	36.8044	2.01084
$336$	−4.75434	−0.259370
$337$	24.5905	1.33953	0.669766	−	0.742572i	$-0.266394\pi$
$337$	24.5905	1.33953	0.669766	+	0.742572i	$0.266394\pi$
$338$	12.6878	0.690124
$339$	−10.8536	−0.589488
$340$	−22.8173	−1.23744
$341$	0	0
$342$	−1.07029	−0.0578749
$343$	14.3855	0.776746
$344$	35.1429	1.89478
$345$	−30.7495	−1.65550
$346$	7.40890	0.398305
$347$	−22.0875	−1.18572	−0.592859	−	0.805306i	$-0.702001\pi$
$347$	−22.0875	−1.18572	−0.592859	+	0.805306i	$0.702001\pi$
$348$	−1.62317	−0.0870109
$349$	9.93617	0.531871	0.265935	−	0.963991i	$-0.414319\pi$
$349$	9.93617	0.531871	0.265935	+	0.963991i	$0.414319\pi$
$350$	41.6328	2.22537
$351$	1.07029	0.0571281
$352$	0	0
$353$	−25.5724	−1.36108	−0.680541	−	0.732710i	$-0.738255\pi$
$353$	−25.5724	−1.36108	−0.680541	+	0.732710i	$0.738255\pi$
$354$	−0.899734	−0.0478203
$355$	−59.8029	−3.17401
$356$	−4.32888	−0.229430
$357$	−19.2936	−1.02112
$358$	−12.9083	−0.682223
$359$	−25.3784	−1.33942	−0.669710	−	0.742623i	$-0.733582\pi$
$359$	−25.3784	−1.33942	−0.669710	+	0.742623i	$0.733582\pi$
$360$	−12.8791	−0.678790
$361$	1.00000	0.0526316
$362$	5.99927	0.315315
$363$	0	0
$364$	−2.78549	−0.145999
$365$	−38.5997	−2.02040
$366$	11.5618	0.604343
$367$	10.0040	0.522202	0.261101	−	0.965311i	$-0.415914\pi$
$367$	10.0040	0.522202	0.261101	+	0.965311i	$0.415914\pi$
$368$	11.3859	0.593531
$369$	0.672341	0.0350007
$370$	−26.8512	−1.39593
$371$	14.7817	0.767429
$372$	−8.41176	−0.436130
$373$	6.36741	0.329692	0.164846	−	0.986319i	$-0.447287\pi$
$373$	6.36741	0.329692	0.164846	+	0.986319i	$0.447287\pi$
$374$	0	0
$375$	32.7600	1.69172
$376$	21.2467	1.09572
$377$	2.03315	0.104713
$378$	−3.25991	−0.167672
$379$	−27.4995	−1.41255	−0.706277	−	0.707936i	$-0.749626\pi$
$379$	−27.4995	−1.41255	−0.706277	+	0.707936i	$0.749626\pi$
$380$	3.60209	0.184783
$381$	−6.04791	−0.309844
$382$	−24.1935	−1.23785
$383$	−16.3072	−0.833261	−0.416630	−	0.909076i	$-0.636789\pi$
$383$	−16.3072	−0.833261	−0.416630	+	0.909076i	$0.636789\pi$
$384$	−0.452133	−0.0230728
$385$	0	0
$386$	1.59743	0.0813069
$387$	11.5029	0.584728
$388$	−11.7352	−0.595764
$389$	14.6918	0.744904	0.372452	−	0.928051i	$-0.378517\pi$
$389$	14.6918	0.744904	0.372452	+	0.928051i	$0.378517\pi$
$390$	4.82908	0.244530
$391$	46.2051	2.33669
$392$	6.95633	0.351348
$393$	−8.86309	−0.447084
$394$	13.9886	0.704734
$395$	63.3089	3.18542
$396$	0	0
$397$	−1.46425	−0.0734887	−0.0367443	−	0.999325i	$-0.511699\pi$
$397$	−1.46425	−0.0734887	−0.0367443	+	0.999325i	$0.511699\pi$
$398$	8.25062	0.413566
$399$	3.04581	0.152481
$400$	−19.9351	−0.996754
$401$	−10.5825	−0.528466	−0.264233	−	0.964459i	$-0.585119\pi$
$401$	−10.5825	−0.528466	−0.264233	+	0.964459i	$0.585119\pi$
$402$	−9.34427	−0.466050
$403$	10.5364	0.524858
$404$	8.86227	0.440914
$405$	−4.21558	−0.209474
$406$	−6.19260	−0.307333
$407$	0	0
$408$	19.3526	0.958096
$409$	−30.8444	−1.52516	−0.762578	−	0.646896i	$-0.776067\pi$
$409$	−30.8444	−1.52516	−0.762578	+	0.646896i	$0.776067\pi$
$410$	3.03355	0.149816
$411$	−3.19996	−0.157842
$412$	−8.30703	−0.409258
$413$	2.56043	0.125991
$414$	7.80698	0.383692
$415$	−23.5236	−1.15473
$416$	4.75165	0.232969
$417$	2.57273	0.125987
$418$	0	0
$419$	4.42597	0.216223	0.108112	−	0.994139i	$-0.465520\pi$
$419$	4.42597	0.216223	0.108112	+	0.994139i	$0.465520\pi$
$420$	10.9713	0.535342
$421$	33.2729	1.62162	0.810812	−	0.585306i	$-0.199026\pi$
$421$	33.2729	1.62162	0.810812	+	0.585306i	$0.199026\pi$
$422$	−17.2179	−0.838154
$423$	6.95445	0.338137
$424$	−14.8269	−0.720060
$425$	−80.8985	−3.92415
$426$	15.1834	0.735637
$427$	−32.9021	−1.59224
$428$	11.6859	0.564860
$429$	0	0
$430$	51.9004	2.50286
$431$	−28.3120	−1.36374	−0.681871	−	0.731472i	$-0.738833\pi$
$431$	−28.3120	−1.36374	−0.681871	+	0.731472i	$0.738833\pi$
$432$	1.56095	0.0751010
$433$	−2.67693	−0.128645	−0.0643225	−	0.997929i	$-0.520489\pi$
$433$	−2.67693	−0.128645	−0.0643225	+	0.997929i	$0.520489\pi$
$434$	−32.0920	−1.54046
$435$	−8.00801	−0.383955
$436$	−10.8535	−0.519791
$437$	−7.29424	−0.348931
$438$	9.80009	0.468266
$439$	−11.1827	−0.533722	−0.266861	−	0.963735i	$-0.585986\pi$
$439$	−11.1827	−0.533722	−0.266861	+	0.963735i	$0.585986\pi$
$440$	0	0
$441$	2.27694	0.108426
$442$	−7.25634	−0.345149
$443$	−22.2122	−1.05533	−0.527667	−	0.849451i	$-0.676933\pi$
$443$	−22.2122	−1.05533	−0.527667	+	0.849451i	$0.676933\pi$
$444$	−5.08509	−0.241327
$445$	−21.3569	−1.01241
$446$	−2.26958	−0.107468
$447$	−4.85084	−0.229437
$448$	−23.9813	−1.13301
$449$	−25.0743	−1.18333	−0.591666	−	0.806183i	$-0.701529\pi$
$449$	−25.0743	−1.18333	−0.591666	+	0.806183i	$0.701529\pi$
$450$	−13.6689	−0.644358
$451$	0	0
$452$	−9.27408	−0.436216
$453$	2.02828	0.0952970
$454$	12.3073	0.577608
$455$	−13.7424	−0.644255
$456$	−3.05512	−0.143069
$457$	4.68423	0.219119	0.109560	−	0.993980i	$-0.465056\pi$
$457$	4.68423	0.219119	0.109560	+	0.993980i	$0.465056\pi$
$458$	−23.5412	−1.10001
$459$	6.33447	0.295668
$460$	−26.2745	−1.22505
$461$	−25.4417	−1.18494	−0.592468	−	0.805594i	$-0.701846\pi$
$461$	−25.4417	−1.18494	−0.592468	+	0.805594i	$0.701846\pi$
$462$	0	0
$463$	3.94757	0.183459	0.0917295	−	0.995784i	$-0.470760\pi$
$463$	3.94757	0.183459	0.0917295	+	0.995784i	$0.470760\pi$
$464$	2.96520	0.137656
$465$	−41.5000	−1.92452
$466$	1.05961	0.0490855
$467$	21.1481	0.978617	0.489309	−	0.872111i	$-0.337249\pi$
$467$	21.1481	0.978617	0.489309	+	0.872111i	$0.337249\pi$
$468$	0.914534	0.0422743
$469$	26.5916	1.22789
$470$	31.3779	1.44736
$471$	4.22450	0.194655
$472$	−2.56826	−0.118214
$473$	0	0
$474$	−16.0735	−0.738281
$475$	12.7712	0.585981
$476$	−16.4858	−0.755623
$477$	−4.85314	−0.222210
$478$	−12.4268	−0.568388
$479$	−19.6529	−0.897963	−0.448982	−	0.893541i	$-0.648213\pi$
$479$	−19.6529	−0.897963	−0.448982	+	0.893541i	$0.648213\pi$
$480$	−18.7154	−0.854237
$481$	6.36950	0.290424
$482$	−11.8273	−0.538719
$483$	−22.2168	−1.01090
$484$	0	0
$485$	−57.8964	−2.62894
$486$	1.07029	0.0485496
$487$	20.3848	0.923723	0.461862	−	0.886952i	$-0.347182\pi$
$487$	20.3848	0.923723	0.461862	+	0.886952i	$0.347182\pi$
$488$	33.0027	1.49396
$489$	−18.7353	−0.847238
$490$	10.2734	0.464103
$491$	23.0605	1.04070	0.520352	−	0.853952i	$-0.325801\pi$
$491$	23.0605	1.04070	0.520352	+	0.853952i	$0.325801\pi$
$492$	0.574495	0.0259002
$493$	12.0331	0.541943
$494$	1.14553	0.0515399
$495$	0	0
$496$	15.3666	0.689982
$497$	−43.2083	−1.93816
$498$	5.97241	0.267630
$499$	32.5621	1.45768	0.728840	−	0.684684i	$-0.240060\pi$
$499$	32.5621	1.45768	0.728840	+	0.684684i	$0.240060\pi$
$500$	27.9924	1.25186
$501$	12.6973	0.567274
$502$	13.1550	0.587136
$503$	−17.4466	−0.777904	−0.388952	−	0.921258i	$-0.627163\pi$
$503$	−17.4466	−0.777904	−0.388952	+	0.921258i	$0.627163\pi$
$504$	−9.30532	−0.414492
$505$	43.7227	1.94563
$506$	0	0
$507$	11.8545	0.526475
$508$	−5.16775	−0.229282
$509$	42.7689	1.89570	0.947849	−	0.318719i	$-0.103253\pi$
$509$	42.7689	1.89570	0.947849	+	0.318719i	$0.103253\pi$
$510$	28.5806	1.26557
$511$	−27.8887	−1.23373
$512$	16.4677	0.727776
$513$	−1.00000	−0.0441511
$514$	18.0052	0.794173
$515$	−40.9834	−1.80594
$516$	9.82891	0.432693
$517$	0	0
$518$	−19.4003	−0.852399
$519$	6.92230	0.303855
$520$	13.7845	0.604489
$521$	32.6794	1.43171	0.715856	−	0.698248i	$-0.246037\pi$
$521$	32.6794	1.43171	0.715856	+	0.698248i	$0.246037\pi$
$522$	2.03315	0.0889887
$523$	−14.0045	−0.612372	−0.306186	−	0.951972i	$-0.599053\pi$
$523$	−14.0045	−0.612372	−0.306186	+	0.951972i	$0.599053\pi$
$524$	−7.57324	−0.330838
$525$	38.8985	1.69767
$526$	12.1089	0.527974
$527$	62.3593	2.71641
$528$	0	0
$529$	30.2059	1.31330
$530$	−21.8970	−0.951143
$531$	−0.840641	−0.0364807
$532$	2.60255	0.112835
$533$	−0.719603	−0.0311695
$534$	5.42229	0.234646
$535$	57.6533	2.49257
$536$	−26.6729	−1.15210
$537$	−12.0605	−0.520448
$538$	−9.90864	−0.427192
$539$	0	0
$540$	−3.60209	−0.155009
$541$	−25.3730	−1.09087	−0.545435	−	0.838153i	$-0.683635\pi$
$541$	−25.3730	−1.09087	−0.545435	+	0.838153i	$0.683635\pi$
$542$	−10.1826	−0.437380
$543$	5.60525	0.240544
$544$	28.1224	1.20574
$545$	−53.5468	−2.29369
$546$	3.48907	0.149318
$547$	−11.0470	−0.472336	−0.236168	−	0.971712i	$-0.575892\pi$
$547$	−11.0470	−0.472336	−0.236168	+	0.971712i	$0.575892\pi$
$548$	−2.73426	−0.116802
$549$	10.8024	0.461036
$550$	0	0
$551$	−1.89962	−0.0809266
$552$	22.2848	0.948504
$553$	45.7414	1.94512
$554$	22.3228	0.948403
$555$	−25.0876	−1.06491
$556$	2.19832	0.0932294
$557$	6.40282	0.271296	0.135648	−	0.990757i	$-0.456688\pi$
$557$	6.40282	0.271296	0.135648	+	0.990757i	$0.456688\pi$
$558$	10.5364	0.446043
$559$	−12.3115	−0.520723
$560$	−20.0423	−0.846943
$561$	0	0
$562$	−11.2901	−0.476242
$563$	6.70403	0.282541	0.141271	−	0.989971i	$-0.454881\pi$
$563$	6.70403	0.282541	0.141271	+	0.989971i	$0.454881\pi$
$564$	5.94236	0.250219
$565$	−45.7543	−1.92490
$566$	5.39163	0.226627
$567$	−3.04581	−0.127912
$568$	43.3405	1.81853
$569$	10.1066	0.423693	0.211846	−	0.977303i	$-0.432052\pi$
$569$	10.1066	0.423693	0.211846	+	0.977303i	$0.432052\pi$
$570$	−4.51192	−0.188983
$571$	1.04160	0.0435898	0.0217949	−	0.999762i	$-0.493062\pi$
$571$	1.04160	0.0435898	0.0217949	+	0.999762i	$0.493062\pi$
$572$	0	0
$573$	−22.6045	−0.944317
$574$	2.19177	0.0914828
$575$	−93.1558	−3.88487
$576$	7.87355	0.328064
$577$	18.2008	0.757711	0.378855	−	0.925456i	$-0.376318\pi$
$577$	18.2008	0.757711	0.378855	+	0.925456i	$0.376318\pi$
$578$	−24.7511	−1.02951
$579$	1.49251	0.0620266
$580$	−6.84260	−0.284123
$581$	−16.9961	−0.705116
$582$	14.6993	0.609307
$583$	0	0
$584$	27.9740	1.15757
$585$	4.51192	0.186545
$586$	26.1722	1.08116
$587$	−13.6535	−0.563539	−0.281770	−	0.959482i	$-0.590921\pi$
$587$	−13.6535	−0.563539	−0.281770	+	0.959482i	$0.590921\pi$
$588$	1.94557	0.0802341
$589$	−9.84443	−0.405633
$590$	−3.79291	−0.156151
$591$	13.0698	0.537621
$592$	9.28945	0.381794
$593$	39.7578	1.63266	0.816329	−	0.577588i	$-0.196006\pi$
$593$	39.7578	1.63266	0.816329	+	0.577588i	$0.196006\pi$
$594$	0	0
$595$	−81.3337	−3.33436
$596$	−4.14489	−0.169781
$597$	7.70873	0.315498
$598$	−8.35578	−0.341693
$599$	−1.67283	−0.0683502	−0.0341751	−	0.999416i	$-0.510880\pi$
$599$	−1.67283	−0.0683502	−0.0341751	+	0.999416i	$0.510880\pi$
$600$	−39.0175	−1.59288
$601$	−30.8491	−1.25836	−0.629180	−	0.777260i	$-0.716609\pi$
$601$	−30.8491	−1.25836	−0.629180	+	0.777260i	$0.716609\pi$
$602$	37.4986	1.52833
$603$	−8.73056	−0.355536
$604$	1.73310	0.0705190
$605$	0	0
$606$	−11.1007	−0.450937
$607$	−17.2206	−0.698963	−0.349481	−	0.936943i	$-0.613642\pi$
$607$	−17.2206	−0.698963	−0.349481	+	0.936943i	$0.613642\pi$
$608$	−4.43957	−0.180049
$609$	−5.78588	−0.234456
$610$	48.7396	1.97341
$611$	−7.44331	−0.301124
$612$	5.41261	0.218792
$613$	27.9892	1.13047	0.565236	−	0.824929i	$-0.308785\pi$
$613$	27.9892	1.13047	0.565236	+	0.824929i	$0.308785\pi$
$614$	6.60993	0.266755
$615$	2.83431	0.114290
$616$	0	0
$617$	−29.8719	−1.20260	−0.601299	−	0.799024i	$-0.705350\pi$
$617$	−29.8719	−1.20260	−0.601299	+	0.799024i	$0.705350\pi$
$618$	10.4053	0.418561
$619$	37.8137	1.51986	0.759931	−	0.650004i	$-0.225233\pi$
$619$	37.8137	1.51986	0.759931	+	0.650004i	$0.225233\pi$
$620$	−35.4605	−1.42413
$621$	7.29424	0.292708
$622$	0.898099	0.0360105
$623$	−15.4306	−0.618212
$624$	−1.67067	−0.0668804
$625$	74.2466	2.96986
$626$	−6.79575	−0.271613
$627$	0	0
$628$	3.60971	0.144043
$629$	37.6975	1.50310
$630$	−13.7424	−0.547512
$631$	48.7418	1.94038	0.970190	−	0.242347i	$-0.0779171\pi$
$631$	48.7418	1.94038	0.970190	+	0.242347i	$0.0779171\pi$
$632$	−45.8813	−1.82506
$633$	−16.0870	−0.639403
$634$	8.16361	0.324219
$635$	−25.4955	−1.01176
$636$	−4.14685	−0.164433
$637$	−2.43700	−0.0965573
$638$	0	0
$639$	14.1862	0.561196
$640$	−1.90600	−0.0753414
$641$	1.00552	0.0397158	0.0198579	−	0.999803i	$-0.493679\pi$
$641$	1.00552	0.0397158	0.0198579	+	0.999803i	$0.493679\pi$
$642$	−14.6376	−0.577700
$643$	38.7682	1.52887	0.764435	−	0.644701i	$-0.223018\pi$
$643$	38.7682	1.52887	0.764435	+	0.644701i	$0.223018\pi$
$644$	−18.9836	−0.748058
$645$	48.4916	1.90936
$646$	6.77975	0.266746
$647$	28.7318	1.12956	0.564782	−	0.825240i	$-0.308960\pi$
$647$	28.7318	1.12956	0.564782	+	0.825240i	$0.308960\pi$
$648$	3.05512	0.120017
$649$	0	0
$650$	14.6298	0.573826
$651$	−29.9842	−1.17518
$652$	−16.0087	−0.626949
$653$	12.4522	0.487291	0.243646	−	0.969864i	$-0.421657\pi$
$653$	12.4522	0.487291	0.243646	+	0.969864i	$0.421657\pi$
$654$	13.5950	0.531606
$655$	−37.3631	−1.45990
$656$	−1.04949	−0.0409756
$657$	9.15644	0.357227
$658$	22.6709	0.883804
$659$	25.5435	0.995035	0.497518	−	0.867454i	$-0.334245\pi$
$659$	25.5435	0.995035	0.497518	+	0.867454i	$0.334245\pi$
$660$	0	0
$661$	1.69425	0.0658987	0.0329493	−	0.999457i	$-0.489510\pi$
$661$	1.69425	0.0658987	0.0329493	+	0.999457i	$0.489510\pi$
$662$	−12.5289	−0.486948
$663$	−6.77975	−0.263304
$664$	17.0481	0.661593
$665$	12.8399	0.497908
$666$	6.36950	0.246813
$667$	13.8563	0.536517
$668$	10.8494	0.419778
$669$	−2.12052	−0.0819839
$670$	−39.3916	−1.52183
$671$	0	0
$672$	−13.5221	−0.521626
$673$	27.1021	1.04471	0.522354	−	0.852729i	$-0.325054\pi$
$673$	27.1021	1.04471	0.522354	+	0.852729i	$0.325054\pi$
$674$	−26.3191	−1.01378
$675$	−12.7712	−0.491562
$676$	10.1293	0.389587
$677$	−10.9519	−0.420914	−0.210457	−	0.977603i	$-0.567495\pi$
$677$	−10.9519	−0.420914	−0.210457	+	0.977603i	$0.567495\pi$
$678$	11.6166	0.446132
$679$	−41.8308	−1.60532
$680$	81.5825	3.12855
$681$	11.4989	0.440640
$682$	0	0
$683$	−3.69762	−0.141485	−0.0707427	−	0.997495i	$-0.522537\pi$
$683$	−3.69762	−0.141485	−0.0707427	+	0.997495i	$0.522537\pi$
$684$	−0.854469	−0.0326714
$685$	−13.4897	−0.515415
$686$	−15.3968	−0.587851
$687$	−21.9950	−0.839163
$688$	−17.9555	−0.684546
$689$	5.19429	0.197887
$690$	32.9110	1.25290
$691$	27.3492	1.04041	0.520206	−	0.854041i	$-0.325855\pi$
$691$	27.3492	1.04041	0.520206	+	0.854041i	$0.325855\pi$
$692$	5.91489	0.224850
$693$	0	0
$694$	23.6401	0.897367
$695$	10.8456	0.411396
$696$	5.80358	0.219984
$697$	−4.25893	−0.161318
$698$	−10.6346	−0.402527
$699$	0.990018	0.0374459
$700$	33.2375	1.25626
$701$	35.5698	1.34345	0.671727	−	0.740799i	$-0.265553\pi$
$701$	35.5698	1.34345	0.671727	+	0.740799i	$0.265553\pi$
$702$	−1.14553	−0.0432353
$703$	−5.95117	−0.224453
$704$	0	0
$705$	29.3171	1.10415
$706$	27.3700	1.03008
$707$	31.5901	1.18807
$708$	−0.718302	−0.0269954
$709$	40.0327	1.50346	0.751730	−	0.659471i	$-0.229219\pi$
$709$	40.0327	1.50346	0.751730	+	0.659471i	$0.229219\pi$
$710$	64.0068	2.40213
$711$	−15.0178	−0.563213
$712$	15.4778	0.580054
$713$	71.8076	2.68922
$714$	20.6498	0.772800
$715$	0	0
$716$	−10.3053	−0.385127
$717$	−11.6106	−0.433607
$718$	27.1624	1.01369
$719$	−9.35452	−0.348865	−0.174432	−	0.984669i	$-0.555809\pi$
$719$	−9.35452	−0.348865	−0.174432	+	0.984669i	$0.555809\pi$
$720$	6.58030	0.245233
$721$	−29.6109	−1.10277
$722$	−1.07029	−0.0398323
$723$	−11.0505	−0.410973
$724$	4.78951	0.178001
$725$	−24.2603	−0.901007
$726$	0	0
$727$	−19.4936	−0.722979	−0.361489	−	0.932376i	$-0.617732\pi$
$727$	−19.4936	−0.722979	−0.361489	+	0.932376i	$0.617732\pi$
$728$	9.95943	0.369121
$729$	1.00000	0.0370370
$730$	41.3131	1.52907
$731$	−72.8651	−2.69501
$732$	9.23033	0.341163
$733$	1.42990	0.0528146	0.0264073	−	0.999651i	$-0.491593\pi$
$733$	1.42990	0.0528146	0.0264073	+	0.999651i	$0.491593\pi$
$734$	−10.7072	−0.395209
$735$	9.59863	0.354051
$736$	32.3833	1.19366
$737$	0	0
$738$	−0.719603	−0.0264890
$739$	−36.8322	−1.35489	−0.677447	−	0.735572i	$-0.736914\pi$
$739$	−36.8322	−1.35489	−0.677447	+	0.735572i	$0.736914\pi$
$740$	−21.4366	−0.788025
$741$	1.07029	0.0393183
$742$	−15.8208	−0.580800
$743$	2.32178	0.0851777	0.0425889	−	0.999093i	$-0.486439\pi$
$743$	2.32178	0.0851777	0.0425889	+	0.999093i	$0.486439\pi$
$744$	30.0760	1.10264
$745$	−20.4491	−0.749197
$746$	−6.81501	−0.249515
$747$	5.58016	0.204167
$748$	0	0
$749$	41.6552	1.52205
$750$	−35.0628	−1.28031
$751$	12.5420	0.457663	0.228832	−	0.973466i	$-0.426509\pi$
$751$	12.5420	0.457663	0.228832	+	0.973466i	$0.426509\pi$
$752$	−10.8555	−0.395860
$753$	12.2910	0.447909
$754$	−2.17607	−0.0792480
$755$	8.55039	0.311181
$756$	−2.60255	−0.0946537
$757$	−39.5566	−1.43771	−0.718854	−	0.695161i	$-0.755333\pi$
$757$	−39.5566	−1.43771	−0.718854	+	0.695161i	$0.755333\pi$
$758$	29.4325	1.06904
$759$	0	0
$760$	−12.8791	−0.467175
$761$	−5.23236	−0.189673	−0.0948365	−	0.995493i	$-0.530233\pi$
$761$	−5.23236	−0.189673	−0.0948365	+	0.995493i	$0.530233\pi$
$762$	6.47305	0.234494
$763$	−38.6881	−1.40060
$764$	−19.3148	−0.698786
$765$	26.7035	0.965467
$766$	17.4536	0.630623
$767$	0.899734	0.0324875
$768$	16.2310	0.585686
$769$	−15.5919	−0.562257	−0.281128	−	0.959670i	$-0.590709\pi$
$769$	−15.5919	−0.562257	−0.281128	+	0.959670i	$0.590709\pi$
$770$	0	0
$771$	16.8226	0.605852
$772$	1.27530	0.0458992
$773$	−12.2154	−0.439358	−0.219679	−	0.975572i	$-0.570501\pi$
$773$	−12.2154	−0.439358	−0.219679	+	0.975572i	$0.570501\pi$
$774$	−12.3115	−0.442529
$775$	−125.725	−4.51617
$776$	41.9588	1.50623
$777$	−18.1261	−0.650270
$778$	−15.7246	−0.563753
$779$	0.672341	0.0240891
$780$	3.85529	0.138042
$781$	0	0
$782$	−49.4531	−1.76844
$783$	1.89962	0.0678869
$784$	−3.55418	−0.126935
$785$	17.8088	0.635622
$786$	9.48612	0.338359
$787$	−14.9375	−0.532463	−0.266232	−	0.963909i	$-0.585779\pi$
$787$	−14.9375	−0.532463	−0.266232	+	0.963909i	$0.585779\pi$
$788$	11.1678	0.397835
$789$	11.3136	0.402776
$790$	−67.7592	−2.41077
$791$	−33.0580	−1.17541
$792$	0	0
$793$	−11.5618	−0.410571
$794$	1.56718	0.0556172
$795$	−20.4588	−0.725599
$796$	6.58687	0.233466
$797$	41.1159	1.45640	0.728199	−	0.685365i	$-0.240358\pi$
$797$	41.1159	1.45640	0.728199	+	0.685365i	$0.240358\pi$
$798$	−3.25991	−0.115400
$799$	−44.0528	−1.55848
$800$	−56.6985	−2.00459
$801$	5.06617	0.179004
$802$	11.3264	0.399950
$803$	0	0
$804$	−7.45999	−0.263093
$805$	−93.6570	−3.30097
$806$	−11.2771	−0.397219
$807$	−9.25786	−0.325892
$808$	−31.6867	−1.11474
$809$	37.6861	1.32497	0.662487	−	0.749074i	$-0.269501\pi$
$809$	37.6861	1.32497	0.662487	+	0.749074i	$0.269501\pi$
$810$	4.51192	0.158533
$811$	−30.6436	−1.07604	−0.538021	−	0.842932i	$-0.680828\pi$
$811$	−30.6436	−1.07604	−0.538021	+	0.842932i	$0.680828\pi$
$812$	−4.94385	−0.173495
$813$	−9.51383	−0.333665
$814$	0	0
$815$	−78.9801	−2.76655
$816$	−9.88777	−0.346141
$817$	11.5029	0.402437
$818$	33.0126	1.15426
$819$	3.25991	0.113911
$820$	2.42183	0.0845740
$821$	−34.1029	−1.19020	−0.595100	−	0.803652i	$-0.702888\pi$
$821$	−34.1029	−1.19020	−0.595100	+	0.803652i	$0.702888\pi$
$822$	3.42490	0.119457
$823$	31.2177	1.08818	0.544090	−	0.839027i	$-0.316875\pi$
$823$	31.2177	1.08818	0.544090	+	0.839027i	$0.316875\pi$
$824$	29.7015	1.03470
$825$	0	0
$826$	−2.74042	−0.0953513
$827$	19.6677	0.683911	0.341956	−	0.939716i	$-0.388911\pi$
$827$	19.6677	0.683911	0.341956	+	0.939716i	$0.388911\pi$
$828$	6.23270	0.216601
$829$	−34.5316	−1.19933	−0.599665	−	0.800251i	$-0.704700\pi$
$829$	−34.5316	−1.19933	−0.599665	+	0.800251i	$0.704700\pi$
$830$	25.1772	0.873914
$831$	20.8566	0.723509
$832$	−8.42702	−0.292154
$833$	−14.4232	−0.499734
$834$	−2.75358	−0.0953487
$835$	53.5266	1.85236
$836$	0	0
$837$	9.84443	0.340273
$838$	−4.73710	−0.163640
$839$	−56.2637	−1.94244	−0.971220	−	0.238185i	$-0.923448\pi$
$839$	−56.2637	−1.94244	−0.971220	+	0.238185i	$0.923448\pi$
$840$	−39.2273	−1.35347
$841$	−25.3914	−0.875567
$842$	−35.6119	−1.22727
$843$	−10.5486	−0.363311
$844$	−13.7459	−0.473153
$845$	49.9735	1.71914
$846$	−7.44331	−0.255906
$847$	0	0
$848$	7.57548	0.260143
$849$	5.03751	0.172887
$850$	86.5853	2.96985
$851$	43.4092	1.48805
$852$	12.1216	0.415280
$853$	−24.3569	−0.833963	−0.416981	−	0.908915i	$-0.636912\pi$
$853$	−24.3569	−0.833963	−0.416981	+	0.908915i	$0.636912\pi$
$854$	35.2149	1.20503
$855$	−4.21558	−0.144170
$856$	−41.7826	−1.42810
$857$	−6.01678	−0.205529	−0.102765	−	0.994706i	$-0.532769\pi$
$857$	−6.01678	−0.205529	−0.102765	+	0.994706i	$0.532769\pi$
$858$	0	0
$859$	33.6877	1.14941	0.574705	−	0.818361i	$-0.305117\pi$
$859$	33.6877	1.14941	0.574705	+	0.818361i	$0.305117\pi$
$860$	41.4346	1.41291
$861$	2.04782	0.0697896
$862$	30.3022	1.03210
$863$	−35.5089	−1.20874	−0.604368	−	0.796705i	$-0.706574\pi$
$863$	−35.5089	−1.20874	−0.604368	+	0.796705i	$0.706574\pi$
$864$	4.43957	0.151037
$865$	29.1815	0.992202
$866$	2.86510	0.0973602
$867$	−23.1255	−0.785384
$868$	−25.6206	−0.869620
$869$	0	0
$870$	8.57093	0.290582
$871$	9.34427	0.316619
$872$	38.8065	1.31415
$873$	13.7339	0.464822
$874$	7.80698	0.264075
$875$	99.7805	3.37320
$876$	7.82389	0.264345
$877$	24.6083	0.830963	0.415482	−	0.909602i	$-0.363613\pi$
$877$	24.6083	0.830963	0.415482	+	0.909602i	$0.363613\pi$
$878$	11.9688	0.403927
$879$	24.4532	0.824787
$880$	0	0
$881$	−0.590267	−0.0198866	−0.00994330	−	0.999951i	$-0.503165\pi$
$881$	−0.590267	−0.0198866	−0.00994330	+	0.999951i	$0.503165\pi$
$882$	−2.43700	−0.0820579
$883$	−41.7797	−1.40600	−0.703000	−	0.711190i	$-0.748156\pi$
$883$	−41.7797	−1.40600	−0.703000	+	0.711190i	$0.748156\pi$
$884$	−5.79309	−0.194843
$885$	−3.54379	−0.119123
$886$	23.7736	0.798691
$887$	−41.2641	−1.38551	−0.692756	−	0.721172i	$-0.743604\pi$
$887$	−41.2641	−1.38551	−0.692756	+	0.721172i	$0.743604\pi$
$888$	18.1815	0.610133
$889$	−18.4208	−0.617813
$890$	22.8581	0.766206
$891$	0	0
$892$	−1.81192	−0.0606674
$893$	6.95445	0.232722
$894$	5.19182	0.173641
$895$	−50.8419	−1.69946
$896$	−1.37711	−0.0460060
$897$	−7.80698	−0.260668
$898$	26.8369	0.895560
$899$	18.7007	0.623703
$900$	−10.9126	−0.363752
$901$	30.7421	1.02417
$902$	0	0
$903$	35.0357	1.16592
$904$	33.1591	1.10286
$905$	23.6294	0.785468
$906$	−2.17086	−0.0721220
$907$	−28.2681	−0.938627	−0.469313	−	0.883032i	$-0.655499\pi$
$907$	−28.2681	−0.938627	−0.469313	+	0.883032i	$0.655499\pi$
$908$	9.82549	0.326070
$909$	−10.3717	−0.344007
$910$	14.7085	0.487581
$911$	22.2316	0.736565	0.368283	−	0.929714i	$-0.379946\pi$
$911$	22.2316	0.736565	0.368283	+	0.929714i	$0.379946\pi$
$912$	1.56095	0.0516881
$913$	0	0
$914$	−5.01351	−0.165832
$915$	45.5385	1.50546
$916$	−18.7941	−0.620974
$917$	−26.9953	−0.891462
$918$	−6.77975	−0.223765
$919$	−3.91472	−0.129135	−0.0645674	−	0.997913i	$-0.520567\pi$
$919$	−3.91472	−0.129135	−0.0645674	+	0.997913i	$0.520567\pi$
$920$	93.9434	3.09722
$921$	6.17580	0.203500
$922$	27.2301	0.896775
$923$	−15.1834	−0.499767
$924$	0	0
$925$	−76.0033	−2.49897
$926$	−4.22506	−0.138844
$927$	9.72187	0.319308
$928$	8.43351	0.276843
$929$	−48.0505	−1.57648	−0.788242	−	0.615365i	$-0.789009\pi$
$929$	−48.0505	−1.57648	−0.788242	+	0.615365i	$0.789009\pi$
$930$	44.4173	1.45650
$931$	2.27694	0.0746237
$932$	0.845939	0.0277097
$933$	0.839114	0.0274713
$934$	−22.6347	−0.740630
$935$	0	0
$936$	−3.26988	−0.106879
$937$	−52.3215	−1.70927	−0.854635	−	0.519229i	$-0.826219\pi$
$937$	−52.3215	−1.70927	−0.854635	+	0.519229i	$0.826219\pi$
$938$	−28.4608	−0.929279
$939$	−6.34941	−0.207205
$940$	25.0505	0.817058
$941$	−46.3936	−1.51239	−0.756194	−	0.654347i	$-0.772943\pi$
$941$	−46.3936	−1.51239	−0.756194	+	0.654347i	$0.772943\pi$
$942$	−4.52146	−0.147317
$943$	−4.90422	−0.159703
$944$	1.31220	0.0427083
$945$	−12.8399	−0.417681
$946$	0	0
$947$	21.6085	0.702182	0.351091	−	0.936341i	$-0.385811\pi$
$947$	21.6085	0.702182	0.351091	+	0.936341i	$0.385811\pi$
$948$	−12.8323	−0.416773
$949$	−9.80009	−0.318124
$950$	−13.6689	−0.443478
$951$	7.62744	0.247337
$952$	58.9443	1.91039
$953$	36.8365	1.19325	0.596625	−	0.802520i	$-0.296508\pi$
$953$	36.8365	1.19325	0.596625	+	0.802520i	$0.296508\pi$
$954$	5.19429	0.168171
$955$	−95.2912	−3.08355
$956$	−9.92092	−0.320865
$957$	0	0
$958$	21.0344	0.679590
$959$	−9.74646	−0.314729
$960$	33.1916	1.07125
$961$	65.9129	2.12622
$962$	−6.81725	−0.219797
$963$	−13.6762	−0.440711
$964$	−9.44232	−0.304117
$965$	6.29181	0.202540
$966$	23.7786	0.765063
$967$	9.21555	0.296352	0.148176	−	0.988961i	$-0.452660\pi$
$967$	9.21555	0.296352	0.148176	+	0.988961i	$0.452660\pi$
$968$	0	0
$969$	6.33447	0.203493
$970$	61.9662	1.98962
$971$	−13.7384	−0.440885	−0.220442	−	0.975400i	$-0.570750\pi$
$971$	−13.7384	−0.440885	−0.220442	+	0.975400i	$0.570750\pi$
$972$	0.854469	0.0274071
$973$	7.83604	0.251212
$974$	−21.8177	−0.699086
$975$	13.6689	0.437755
$976$	−16.8620	−0.539739
$977$	−35.6361	−1.14010	−0.570051	−	0.821610i	$-0.693076\pi$
$977$	−35.6361	−1.14010	−0.570051	+	0.821610i	$0.693076\pi$
$978$	20.0523	0.641201
$979$	0	0
$980$	8.20173	0.261995
$981$	12.7021	0.405547
$982$	−24.6815	−0.787618
$983$	−30.3992	−0.969583	−0.484792	−	0.874630i	$-0.661105\pi$
$983$	−30.3992	−0.969583	−0.484792	+	0.874630i	$0.661105\pi$
$984$	−2.05409	−0.0654819
$985$	55.0970	1.75554
$986$	−12.8790	−0.410150
$987$	21.1819	0.674228
$988$	0.914534	0.0290952
$989$	−83.9052	−2.66803
$990$	0	0
$991$	9.49878	0.301739	0.150869	−	0.988554i	$-0.451793\pi$
$991$	9.49878	0.301739	0.150869	+	0.988554i	$0.451793\pi$
$992$	43.7051	1.38764
$993$	−11.7060	−0.371478
$994$	46.2456	1.46682
$995$	32.4968	1.03022
$996$	4.76807	0.151082
$997$	7.53774	0.238723	0.119361	−	0.992851i	$-0.461915\pi$
$997$	7.53774	0.238723	0.119361	+	0.992851i	$0.461915\pi$
$998$	−34.8511	−1.10319
$999$	5.95117	0.188287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.bc.1.3	✓	8
11.10	odd	2		6897.2.a.bd.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6897.2.a.bc.1.3	✓	8	1.1	even	1	trivial
6897.2.a.bd.1.6	yes	8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.07029 q^{2} -1.00000 q^{3} -0.854469 q^{4} -4.21558 q^{5} +1.07029 q^{6} -3.04581 q^{7} +3.05512 q^{8} +1.00000 q^{9} +4.51192 q^{10} +0.854469 q^{12} -1.07029 q^{13} +3.25991 q^{14} +4.21558 q^{15} -1.56095 q^{16} -6.33447 q^{17} -1.07029 q^{18} +1.00000 q^{19} +3.60209 q^{20} +3.04581 q^{21} -7.29424 q^{23} -3.05512 q^{24} +12.7712 q^{25} +1.14553 q^{26} -1.00000 q^{27} +2.60255 q^{28} -1.89962 q^{29} -4.51192 q^{30} -9.84443 q^{31} -4.43957 q^{32} +6.77975 q^{34} +12.8399 q^{35} -0.854469 q^{36} -5.95117 q^{37} -1.07029 q^{38} +1.07029 q^{39} -12.8791 q^{40} +0.672341 q^{41} -3.25991 q^{42} +11.5029 q^{43} -4.21558 q^{45} +7.80698 q^{46} +6.95445 q^{47} +1.56095 q^{48} +2.27694 q^{49} -13.6689 q^{50} +6.33447 q^{51} +0.914534 q^{52} -4.85314 q^{53} +1.07029 q^{54} -9.30532 q^{56} -1.00000 q^{57} +2.03315 q^{58} -0.840641 q^{59} -3.60209 q^{60} +10.8024 q^{61} +10.5364 q^{62} -3.04581 q^{63} +7.87355 q^{64} +4.51192 q^{65} -8.73056 q^{67} +5.41261 q^{68} +7.29424 q^{69} -13.7424 q^{70} +14.1862 q^{71} +3.05512 q^{72} +9.15644 q^{73} +6.36950 q^{74} -12.7712 q^{75} -0.854469 q^{76} -1.14553 q^{78} -15.0178 q^{79} +6.58030 q^{80} +1.00000 q^{81} -0.719603 q^{82} +5.58016 q^{83} -2.60255 q^{84} +26.7035 q^{85} -12.3115 q^{86} +1.89962 q^{87} +5.06617 q^{89} +4.51192 q^{90} +3.25991 q^{91} +6.23270 q^{92} +9.84443 q^{93} -7.44331 q^{94} -4.21558 q^{95} +4.43957 q^{96} +13.7339 q^{97} -2.43700 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) \) 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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