LMFDB - Embedded newform 6897.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6897,2,Mod(1,6897)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$55.0728222741$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 627)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.69963$ 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)

$f(q)$	$=$	\(q+2.69963 q^{2} +1.00000 q^{3} +5.28799 q^{4} -0.888736 q^{5} +2.69963 q^{6} +0.411636 q^{7} +8.87636 q^{8} +1.00000 q^{9} -2.39926 q^{10} +5.28799 q^{12} +2.30037 q^{13} +1.11126 q^{14} -0.888736 q^{15} +13.3869 q^{16} +8.09888 q^{17} +2.69963 q^{18} +1.00000 q^{19} -4.69963 q^{20} +0.411636 q^{21} -5.00000 q^{23} +8.87636 q^{24} -4.21015 q^{25} +6.21015 q^{26} +1.00000 q^{27} +2.17673 q^{28} -7.38688 q^{29} -2.39926 q^{30} -1.98762 q^{31} +18.3869 q^{32} +21.8640 q^{34} -0.365836 q^{35} +5.28799 q^{36} +9.35346 q^{37} +2.69963 q^{38} +2.30037 q^{39} -7.88874 q^{40} -1.69963 q^{41} +1.11126 q^{42} +7.28799 q^{43} -0.888736 q^{45} -13.4981 q^{46} -4.47710 q^{47} +13.3869 q^{48} -6.83056 q^{49} -11.3658 q^{50} +8.09888 q^{51} +12.1643 q^{52} -0.333792 q^{53} +2.69963 q^{54} +3.65383 q^{56} +1.00000 q^{57} -19.9418 q^{58} -4.76509 q^{59} -4.69963 q^{60} +10.5105 q^{61} -5.36584 q^{62} +0.411636 q^{63} +22.8640 q^{64} -2.04442 q^{65} +10.0531 q^{67} +42.8268 q^{68} -5.00000 q^{69} -0.987620 q^{70} -5.30037 q^{71} +8.87636 q^{72} -2.47710 q^{73} +25.2509 q^{74} -4.21015 q^{75} +5.28799 q^{76} +6.21015 q^{78} +16.0741 q^{79} -11.8974 q^{80} +1.00000 q^{81} -4.58836 q^{82} -8.73305 q^{83} +2.17673 q^{84} -7.19777 q^{85} +19.6749 q^{86} -7.38688 q^{87} +1.98762 q^{89} -2.39926 q^{90} +0.946916 q^{91} -26.4400 q^{92} -1.98762 q^{93} -12.0865 q^{94} -0.888736 q^{95} +18.3869 q^{96} -15.6094 q^{97} -18.4400 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} + 7 q^{7} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 4 q^{12} + 13 q^{13} + 3 q^{14} - 3 q^{15} + 10 q^{16} + 6 q^{17} + 2 q^{18} + 3 q^{19} - 8 q^{20} + 7 q^{21}+ \cdots + 4 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 3 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 + 7 * q^7 + 9 * q^8 + 3 * q^9 + 5 * q^10 + 4 * q^12 + 13 * q^13 + 3 * q^14 - 3 * q^15 + 10 * q^16 + 6 * q^17 + 2 * q^18 + 3 * q^19 - 8 * q^20 + 7 * q^21 - 15 * q^23 + 9 * q^24 + 6 * q^25 + 3 * q^27 - 5 * q^28 + 8 * q^29 + 5 * q^30 + 12 * q^31 + 25 * q^32 + 30 * q^34 + 4 * q^35 + 4 * q^36 + 5 * q^37 + 2 * q^38 + 13 * q^39 - 24 * q^40 + q^41 + 3 * q^42 + 10 * q^43 - 3 * q^45 - 10 * q^46 - 8 * q^47 + 10 * q^48 + 8 * q^49 - 29 * q^50 + 6 * q^51 + 7 * q^52 + 2 * q^54 - 6 * q^56 + 3 * q^57 - 31 * q^58 + 3 * q^59 - 8 * q^60 + 19 * q^61 - 11 * q^62 + 7 * q^63 + 33 * q^64 - 20 * q^65 + q^67 + 39 * q^68 - 15 * q^69 + 15 * q^70 - 22 * q^71 + 9 * q^72 - 2 * q^73 + 10 * q^74 + 6 * q^75 + 4 * q^76 - 6 * q^79 + 7 * q^80 + 3 * q^81 - 8 * q^82 - 13 * q^83 - 5 * q^84 + 15 * q^85 + 17 * q^86 + 8 * q^87 - 12 * q^89 + 5 * q^90 + 32 * q^91 - 20 * q^92 + 12 * q^93 - 3 * q^95 + 25 * q^96 - 16 * q^97 + 4 * 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 $

$2$	2.69963	1.90893	0.954463	−	0.298330i	$-0.0964297\pi$
$2$	2.69963	1.90893	0.954463	+	0.298330i	$0.0964297\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.28799	2.64400
$5$	−0.888736	−0.397455	−0.198727	−	0.980055i	$-0.563681\pi$
$5$	−0.888736	−0.397455	−0.198727	+	0.980055i	$0.563681\pi$
$6$	2.69963	1.10212
$7$	0.411636	0.155584	0.0777919	−	0.996970i	$-0.475213\pi$
$7$	0.411636	0.155584	0.0777919	+	0.996970i	$0.475213\pi$
$8$	8.87636	3.13827
$9$	1.00000	0.333333
$10$	−2.39926	−0.758711
$11$	0	0
$11$	0	0
$12$	5.28799	1.52651
$13$	2.30037	0.638008	0.319004	−	0.947753i	$-0.396652\pi$
$13$	2.30037	0.638008	0.319004	+	0.947753i	$0.396652\pi$
$14$	1.11126	0.296998
$15$	−0.888736	−0.229471
$16$	13.3869	3.34672
$17$	8.09888	1.96427	0.982134	−	0.188183i	$-0.0602598\pi$
$17$	8.09888	1.96427	0.982134	+	0.188183i	$0.0602598\pi$
$18$	2.69963	0.636308
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.69963	−1.05087
$21$	0.411636	0.0898263
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	8.87636	1.81188
$25$	−4.21015	−0.842030
$26$	6.21015	1.21791
$27$	1.00000	0.192450
$28$	2.17673	0.411363
$29$	−7.38688	−1.37171	−0.685854	−	0.727739i	$-0.740571\pi$
$29$	−7.38688	−1.37171	−0.685854	+	0.727739i	$0.740571\pi$
$30$	−2.39926	−0.438042
$31$	−1.98762	−0.356987	−0.178494	−	0.983941i	$-0.557122\pi$
$31$	−1.98762	−0.356987	−0.178494	+	0.983941i	$0.557122\pi$
$32$	18.3869	3.25037
$33$	0	0
$34$	21.8640	3.74964
$35$	−0.365836	−0.0618375
$36$	5.28799	0.881332
$37$	9.35346	1.53770	0.768849	−	0.639430i	$-0.220830\pi$
$37$	9.35346	1.53770	0.768849	+	0.639430i	$0.220830\pi$
$38$	2.69963	0.437938
$39$	2.30037	0.368354
$40$	−7.88874	−1.24732
$41$	−1.69963	−0.265437	−0.132719	−	0.991154i	$-0.542371\pi$
$41$	−1.69963	−0.265437	−0.132719	+	0.991154i	$0.542371\pi$
$42$	1.11126	0.171472
$43$	7.28799	1.11141	0.555704	−	0.831380i	$-0.312449\pi$
$43$	7.28799	1.11141	0.555704	+	0.831380i	$0.312449\pi$
$44$	0	0
$45$	−0.888736	−0.132485
$46$	−13.4981	−1.99019
$47$	−4.47710	−0.653052	−0.326526	−	0.945188i	$-0.605878\pi$
$47$	−4.47710	−0.653052	−0.326526	+	0.945188i	$0.605878\pi$
$48$	13.3869	1.93223
$49$	−6.83056	−0.975794
$50$	−11.3658	−1.60737
$51$	8.09888	1.13407
$52$	12.1643	1.68689
$53$	−0.333792	−0.0458499	−0.0229250	−	0.999737i	$-0.507298\pi$
$53$	−0.333792	−0.0458499	−0.0229250	+	0.999737i	$0.507298\pi$
$54$	2.69963	0.367373
$55$	0	0
$56$	3.65383	0.488263
$57$	1.00000	0.132453
$58$	−19.9418	−2.61849
$59$	−4.76509	−0.620362	−0.310181	−	0.950677i	$-0.600390\pi$
$59$	−4.76509	−0.620362	−0.310181	+	0.950677i	$0.600390\pi$
$60$	−4.69963	−0.606719
$61$	10.5105	1.34573	0.672867	−	0.739763i	$-0.265063\pi$
$61$	10.5105	1.34573	0.672867	+	0.739763i	$0.265063\pi$
$62$	−5.36584	−0.681462
$63$	0.411636	0.0518613
$64$	22.8640	2.85800
$65$	−2.04442	−0.253579
$66$	0	0
$67$	10.0531	1.22818	0.614090	−	0.789236i	$-0.289523\pi$
$67$	10.0531	1.22818	0.614090	+	0.789236i	$0.289523\pi$
$68$	42.8268	5.19352
$69$	−5.00000	−0.601929
$70$	−0.987620	−0.118043
$71$	−5.30037	−0.629038	−0.314519	−	0.949251i	$-0.601843\pi$
$71$	−5.30037	−0.629038	−0.314519	+	0.949251i	$0.601843\pi$
$72$	8.87636	1.04609
$73$	−2.47710	−0.289923	−0.144961	−	0.989437i	$-0.546306\pi$
$73$	−2.47710	−0.289923	−0.144961	+	0.989437i	$0.546306\pi$
$74$	25.2509	2.93535
$75$	−4.21015	−0.486146
$76$	5.28799	0.606574
$77$	0	0
$78$	6.21015	0.703161
$79$	16.0741	1.80848	0.904240	−	0.427024i	$-0.140438\pi$
$79$	16.0741	1.80848	0.904240	+	0.427024i	$0.140438\pi$
$80$	−11.8974	−1.33017
$81$	1.00000	0.111111
$82$	−4.58836	−0.506700
$83$	−8.73305	−0.958577	−0.479288	−	0.877658i	$-0.659105\pi$
$83$	−8.73305	−0.958577	−0.479288	+	0.877658i	$0.659105\pi$
$84$	2.17673	0.237500
$85$	−7.19777	−0.780708
$86$	19.6749	2.12160
$87$	−7.38688	−0.791956
$88$	0	0
$89$	1.98762	0.210687	0.105344	−	0.994436i	$-0.466406\pi$
$89$	1.98762	0.210687	0.105344	+	0.994436i	$0.466406\pi$
$90$	−2.39926	−0.252904
$91$	0.946916	0.0992638
$92$	−26.4400	−2.75656
$93$	−1.98762	−0.206107
$94$	−12.0865	−1.24663
$95$	−0.888736	−0.0911824
$96$	18.3869	1.87660
$97$	−15.6094	−1.58489	−0.792447	−	0.609940i	$-0.791193\pi$
$97$	−15.6094	−1.58489	−0.792447	+	0.609940i	$0.791193\pi$
$98$	−18.4400	−1.86272
$99$	0	0
$100$	−22.2632	−2.22632
$101$	−0.888736	−0.0884325	−0.0442163	−	0.999022i	$-0.514079\pi$
$101$	−0.888736	−0.0884325	−0.0442163	+	0.999022i	$0.514079\pi$
$102$	21.8640	2.16486
$103$	−11.7193	−1.15474	−0.577368	−	0.816484i	$-0.695920\pi$
$103$	−11.7193	−1.15474	−0.577368	+	0.816484i	$0.695920\pi$
$104$	20.4189	2.00224
$105$	−0.365836	−0.0357019
$106$	−0.901116	−0.0875241
$107$	6.18911	0.598324	0.299162	−	0.954202i	$-0.403293\pi$
$107$	6.18911	0.598324	0.299162	+	0.954202i	$0.403293\pi$
$108$	5.28799	0.508837
$109$	11.7651	1.12689	0.563446	−	0.826153i	$-0.309475\pi$
$109$	11.7651	1.12689	0.563446	+	0.826153i	$0.309475\pi$
$110$	0	0
$111$	9.35346	0.887791
$112$	5.51052	0.520695
$113$	−1.84294	−0.173369	−0.0866844	−	0.996236i	$-0.527627\pi$
$113$	−1.84294	−0.173369	−0.0866844	+	0.996236i	$0.527627\pi$
$114$	2.69963	0.252843
$115$	4.44368	0.414375
$116$	−39.0617	−3.62679
$117$	2.30037	0.212669
$118$	−12.8640	−1.18423
$119$	3.33379	0.305608
$120$	−7.88874	−0.720140
$121$	0	0
$122$	28.3745	2.56891
$123$	−1.69963	−0.153250
$124$	−10.5105	−0.943873
$125$	8.18539	0.732123
$126$	1.11126	0.0989993
$127$	−9.43268	−0.837015	−0.418507	−	0.908213i	$-0.637447\pi$
$127$	−9.43268	−0.837015	−0.418507	+	0.908213i	$0.637447\pi$
$128$	24.9505	2.20533
$129$	7.28799	0.641672
$130$	−5.51918	−0.484064
$131$	−9.24357	−0.807614	−0.403807	−	0.914844i	$-0.632313\pi$
$131$	−9.24357	−0.807614	−0.403807	+	0.914844i	$0.632313\pi$
$132$	0	0
$133$	0.411636	0.0356934
$134$	27.1396	2.34450
$135$	−0.888736	−0.0764902
$136$	71.8886	6.16440
$137$	−1.15706	−0.0988547	−0.0494273	−	0.998778i	$-0.515740\pi$
$137$	−1.15706	−0.0988547	−0.0494273	+	0.998778i	$0.515740\pi$
$138$	−13.4981	−1.14904
$139$	4.25457	0.360868	0.180434	−	0.983587i	$-0.442250\pi$
$139$	4.25457	0.360868	0.180434	+	0.983587i	$0.442250\pi$
$140$	−1.93454	−0.163498
$141$	−4.47710	−0.377040
$142$	−14.3090	−1.20079
$143$	0	0
$144$	13.3869	1.11557
$145$	6.56498	0.545192
$146$	−6.68725	−0.553441
$147$	−6.83056	−0.563375
$148$	49.4610	4.06567
$149$	−8.30037	−0.679993	−0.339996	−	0.940427i	$-0.610426\pi$
$149$	−8.30037	−0.679993	−0.339996	+	0.940427i	$0.610426\pi$
$150$	−11.3658	−0.928017
$151$	−21.4189	−1.74305	−0.871523	−	0.490354i	$-0.836868\pi$
$151$	−21.4189	−1.74305	−0.871523	+	0.490354i	$0.836868\pi$
$152$	8.87636	0.719968
$153$	8.09888	0.654756
$154$	0	0
$155$	1.76647	0.141886
$156$	12.1643	0.973927
$157$	−5.85532	−0.467305	−0.233653	−	0.972320i	$-0.575068\pi$
$157$	−5.85532	−0.467305	−0.233653	+	0.972320i	$0.575068\pi$
$158$	43.3942	3.45225
$159$	−0.333792	−0.0264715
$160$	−16.3411	−1.29188
$161$	−2.05818	−0.162207
$162$	2.69963	0.212103
$163$	18.6094	1.45760	0.728801	−	0.684726i	$-0.240078\pi$
$163$	18.6094	1.45760	0.728801	+	0.684726i	$0.240078\pi$
$164$	−8.98762	−0.701815
$165$	0	0
$166$	−23.5760	−1.82985
$167$	1.11126	0.0859922	0.0429961	−	0.999075i	$-0.486310\pi$
$167$	1.11126	0.0859922	0.0429961	+	0.999075i	$0.486310\pi$
$168$	3.65383	0.281899
$169$	−7.70829	−0.592945
$170$	−19.4313	−1.49031
$171$	1.00000	0.0764719
$172$	38.5388	2.93856
$173$	9.66621	0.734908	0.367454	−	0.930042i	$-0.380230\pi$
$173$	9.66621	0.734908	0.367454	+	0.930042i	$0.380230\pi$
$174$	−19.9418	−1.51179
$175$	−1.73305	−0.131006
$176$	0	0
$177$	−4.76509	−0.358166
$178$	5.36584	0.402186
$179$	14.3658	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$179$	14.3658	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$180$	−4.69963	−0.350290
$181$	4.14468	0.308072	0.154036	−	0.988065i	$-0.450773\pi$
$181$	4.14468	0.308072	0.154036	+	0.988065i	$0.450773\pi$
$182$	2.55632	0.189487
$183$	10.5105	0.776960
$184$	−44.3818	−3.27187
$185$	−8.31275	−0.611166
$186$	−5.36584	−0.393442
$187$	0	0
$188$	−23.6749	−1.72667
$189$	0.411636	0.0299421
$190$	−2.39926	−0.174060
$191$	−18.2101	−1.31764	−0.658820	−	0.752301i	$-0.728944\pi$
$191$	−18.2101	−1.31764	−0.658820	+	0.752301i	$0.728944\pi$
$192$	22.8640	1.65007
$193$	−24.0989	−1.73468	−0.867338	−	0.497720i	$-0.834171\pi$
$193$	−24.0989	−1.73468	−0.867338	+	0.497720i	$0.834171\pi$
$194$	−42.1396	−3.02545
$195$	−2.04442	−0.146404
$196$	−36.1199	−2.57999
$197$	16.5512	1.17923	0.589613	−	0.807686i	$-0.299280\pi$
$197$	16.5512	1.17923	0.589613	+	0.807686i	$0.299280\pi$
$198$	0	0
$199$	1.53528	0.108833	0.0544166	−	0.998518i	$-0.482670\pi$
$199$	1.53528	0.108833	0.0544166	+	0.998518i	$0.482670\pi$
$200$	−37.3708	−2.64251
$201$	10.0531	0.709090
$202$	−2.39926	−0.168811
$203$	−3.04070	−0.213416
$204$	42.8268	2.99848
$205$	1.51052	0.105499
$206$	−31.6377	−2.20431
$207$	−5.00000	−0.347524
$208$	30.7948	2.13523
$209$	0	0
$210$	−0.987620	−0.0681523
$211$	−9.01966	−0.620939	−0.310470	−	0.950583i	$-0.600486\pi$
$211$	−9.01966	−0.620939	−0.310470	+	0.950583i	$0.600486\pi$
$212$	−1.76509	−0.121227
$213$	−5.30037	−0.363175
$214$	16.7083	1.14216
$215$	−6.47710	−0.441735
$216$	8.87636	0.603960
$217$	−0.818176	−0.0555414
$218$	31.7614	2.15115
$219$	−2.47710	−0.167387
$220$	0	0
$221$	18.6304	1.25322
$222$	25.2509	1.69473
$223$	11.3535	0.760284	0.380142	−	0.924928i	$-0.375875\pi$
$223$	11.3535	0.760284	0.380142	+	0.924928i	$0.375875\pi$
$224$	7.56870	0.505705
$225$	−4.21015	−0.280677
$226$	−4.97524	−0.330948
$227$	13.7651	0.913621	0.456811	−	0.889564i	$-0.348992\pi$
$227$	13.7651	0.913621	0.456811	+	0.889564i	$0.348992\pi$
$228$	5.28799	0.350206
$229$	−23.5512	−1.55631	−0.778154	−	0.628073i	$-0.783844\pi$
$229$	−23.5512	−1.55631	−0.778154	+	0.628073i	$0.783844\pi$
$230$	11.9963	0.791011
$231$	0	0
$232$	−65.5685	−4.30479
$233$	−27.2632	−1.78607	−0.893037	−	0.449983i	$-0.851430\pi$
$233$	−27.2632	−1.78607	−0.893037	+	0.449983i	$0.851430\pi$
$234$	6.21015	0.405970
$235$	3.97896	0.259559
$236$	−25.1978	−1.64024
$237$	16.0741	1.04413
$238$	9.00000	0.583383
$239$	−0.320035	−0.0207014	−0.0103507	−	0.999946i	$-0.503295\pi$
$239$	−0.320035	−0.0207014	−0.0103507	+	0.999946i	$0.503295\pi$
$240$	−11.8974	−0.767974
$241$	−26.8392	−1.72887	−0.864433	−	0.502748i	$-0.832322\pi$
$241$	−26.8392	−1.72887	−0.864433	+	0.502748i	$0.832322\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	55.5795	3.55812
$245$	6.07056	0.387834
$246$	−4.58836	−0.292543
$247$	2.30037	0.146369
$248$	−17.6428	−1.12032
$249$	−8.73305	−0.553434
$250$	22.0975	1.39757
$251$	27.5709	1.74026	0.870130	−	0.492823i	$-0.164035\pi$
$251$	27.5709	1.74026	0.870130	+	0.492823i	$0.164035\pi$
$252$	2.17673	0.137121
$253$	0	0
$254$	−25.4647	−1.59780
$255$	−7.19777	−0.450742
$256$	21.6291	1.35182
$257$	1.19049	0.0742604	0.0371302	−	0.999310i	$-0.488178\pi$
$257$	1.19049	0.0742604	0.0371302	+	0.999310i	$0.488178\pi$
$258$	19.6749	1.22490
$259$	3.85022	0.239241
$260$	−10.8109	−0.670463
$261$	−7.38688	−0.457236
$262$	−24.9542	−1.54168
$263$	19.0334	1.17365	0.586825	−	0.809713i	$-0.300377\pi$
$263$	19.0334	1.17365	0.586825	+	0.809713i	$0.300377\pi$
$264$	0	0
$265$	0.296653	0.0182233
$266$	1.11126	0.0681360
$267$	1.98762	0.121640
$268$	53.1606	3.24730
$269$	11.7069	0.713783	0.356892	−	0.934146i	$-0.383837\pi$
$269$	11.7069	0.713783	0.356892	+	0.934146i	$0.383837\pi$
$270$	−2.39926	−0.146014
$271$	15.4523	0.938663	0.469331	−	0.883022i	$-0.344495\pi$
$271$	15.4523	0.938663	0.469331	+	0.883022i	$0.344495\pi$
$272$	108.419	6.57385
$273$	0.946916	0.0573100
$274$	−3.12364	−0.188706
$275$	0	0
$276$	−26.4400	−1.59150
$277$	20.9876	1.26102	0.630512	−	0.776180i	$-0.282845\pi$
$277$	20.9876	1.26102	0.630512	+	0.776180i	$0.282845\pi$
$278$	11.4858	0.688870
$279$	−1.98762	−0.118996
$280$	−3.24729	−0.194063
$281$	−9.68725	−0.577893	−0.288946	−	0.957345i	$-0.593305\pi$
$281$	−9.68725	−0.577893	−0.288946	+	0.957345i	$0.593305\pi$
$282$	−12.0865	−0.719741
$283$	−13.0865	−0.777912	−0.388956	−	0.921256i	$-0.627164\pi$
$283$	−13.0865	−0.777912	−0.388956	+	0.921256i	$0.627164\pi$
$284$	−28.0283	−1.66318
$285$	−0.888736	−0.0526442
$286$	0	0
$287$	−0.699628	−0.0412977
$288$	18.3869	1.08346
$289$	48.5919	2.85835
$290$	17.7230	1.04073
$291$	−15.6094	−0.915040
$292$	−13.0989	−0.766554
$293$	19.1447	1.11844	0.559222	−	0.829018i	$-0.311100\pi$
$293$	19.1447	1.11844	0.559222	+	0.829018i	$0.311100\pi$
$294$	−18.4400	−1.07544
$295$	4.23491	0.246566
$296$	83.0246	4.82571
$297$	0	0
$298$	−22.4079	−1.29806
$299$	−11.5019	−0.665170
$300$	−22.2632	−1.28537
$301$	3.00000	0.172917
$302$	−57.8231	−3.32735
$303$	−0.888736	−0.0510565
$304$	13.3869	0.767790
$305$	−9.34108	−0.534868
$306$	21.8640	1.24988
$307$	3.28799	0.187656	0.0938278	−	0.995588i	$-0.470090\pi$
$307$	3.28799	0.187656	0.0938278	+	0.995588i	$0.470090\pi$
$308$	0	0
$309$	−11.7193	−0.666687
$310$	4.76881	0.270850
$311$	9.83056	0.557440	0.278720	−	0.960372i	$-0.410090\pi$
$311$	9.83056	0.557440	0.278720	+	0.960372i	$0.410090\pi$
$312$	20.4189	1.15599
$313$	−29.8516	−1.68731	−0.843656	−	0.536884i	$-0.819601\pi$
$313$	−29.8516	−1.68731	−0.843656	+	0.536884i	$0.819601\pi$
$314$	−15.8072	−0.892050
$315$	−0.365836	−0.0206125
$316$	84.9998	4.78161
$317$	−9.66249	−0.542699	−0.271350	−	0.962481i	$-0.587470\pi$
$317$	−9.66249	−0.542699	−0.271350	+	0.962481i	$0.587470\pi$
$318$	−0.901116	−0.0505321
$319$	0	0
$320$	−20.3200	−1.13592
$321$	6.18911	0.345442
$322$	−5.55632	−0.309642
$323$	8.09888	0.450634
$324$	5.28799	0.293777
$325$	−9.68491	−0.537222
$326$	50.2385	2.78245
$327$	11.7651	0.650611
$328$	−15.0865	−0.833013
$329$	−1.84294	−0.101604
$330$	0	0
$331$	−0.333792	−0.0183469	−0.00917345	−	0.999958i	$-0.502920\pi$
$331$	−0.333792	−0.0183469	−0.00917345	+	0.999958i	$0.502920\pi$
$332$	−46.1803	−2.53447
$333$	9.35346	0.512566
$334$	3.00000	0.164153
$335$	−8.93454	−0.488146
$336$	5.51052	0.300624
$337$	−22.8306	−1.24366	−0.621830	−	0.783152i	$-0.713610\pi$
$337$	−22.8306	−1.24366	−0.621830	+	0.783152i	$0.713610\pi$
$338$	−20.8095	−1.13189
$339$	−1.84294	−0.100095
$340$	−38.0617	−2.06419
$341$	0	0
$342$	2.69963	0.145979
$343$	−5.69315	−0.307401
$344$	64.6908	3.48789
$345$	4.44368	0.239240
$346$	26.0952	1.40288
$347$	−26.5054	−1.42289	−0.711443	−	0.702744i	$-0.751958\pi$
$347$	−26.5054	−1.42289	−0.711443	+	0.702744i	$0.751958\pi$
$348$	−39.0617	−2.09393
$349$	11.9469	0.639504	0.319752	−	0.947501i	$-0.396400\pi$
$349$	11.9469	0.639504	0.319752	+	0.947501i	$0.396400\pi$
$350$	−4.67859	−0.250081
$351$	2.30037	0.122785
$352$	0	0
$353$	28.2312	1.50259	0.751297	−	0.659964i	$-0.229428\pi$
$353$	28.2312	1.50259	0.751297	+	0.659964i	$0.229428\pi$
$354$	−12.8640	−0.683713
$355$	4.71063	0.250014
$356$	10.5105	0.557056
$357$	3.33379	0.176443
$358$	38.7824	2.04971
$359$	−18.3039	−0.966045	−0.483022	−	0.875608i	$-0.660461\pi$
$359$	−18.3039	−0.966045	−0.483022	+	0.875608i	$0.660461\pi$
$360$	−7.88874	−0.415773
$361$	1.00000	0.0526316
$362$	11.1891	0.588086
$363$	0	0
$364$	5.00728	0.262453
$365$	2.20149	0.115231
$366$	28.3745	1.48316
$367$	30.7600	1.60566	0.802829	−	0.596209i	$-0.203327\pi$
$367$	30.7600	1.60566	0.802829	+	0.596209i	$0.203327\pi$
$368$	−66.9344	−3.48920
$369$	−1.69963	−0.0884791
$370$	−22.4413	−1.16667
$371$	−0.137401	−0.00713350
$372$	−10.5105	−0.544945
$373$	2.22115	0.115007	0.0575034	−	0.998345i	$-0.481686\pi$
$373$	2.22115	0.115007	0.0575034	+	0.998345i	$0.481686\pi$
$374$	0	0
$375$	8.18539	0.422692
$376$	−39.7403	−2.04945
$377$	−16.9926	−0.875162
$378$	1.11126	0.0571573
$379$	−8.05308	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$379$	−8.05308	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$380$	−4.69963	−0.241086
$381$	−9.43268	−0.483251
$382$	−49.1606	−2.51528
$383$	−6.59703	−0.337092	−0.168546	−	0.985694i	$-0.553907\pi$
$383$	−6.59703	−0.337092	−0.168546	+	0.985694i	$0.553907\pi$
$384$	24.9505	1.27325
$385$	0	0
$386$	−65.0580	−3.31137
$387$	7.28799	0.370469
$388$	−82.5424	−4.19046
$389$	−25.1199	−1.27363	−0.636815	−	0.771016i	$-0.719749\pi$
$389$	−25.1199	−1.27363	−0.636815	+	0.771016i	$0.719749\pi$
$390$	−5.51918	−0.279475
$391$	−40.4944	−2.04789
$392$	−60.6304	−3.06230
$393$	−9.24357	−0.466276
$394$	44.6822	2.25105
$395$	−14.2857	−0.718789
$396$	0	0
$397$	−31.5068	−1.58128	−0.790641	−	0.612281i	$-0.790252\pi$
$397$	−31.5068	−1.58128	−0.790641	+	0.612281i	$0.790252\pi$
$398$	4.14468	0.207754
$399$	0.411636	0.0206076
$400$	−56.3607	−2.81804
$401$	−34.1643	−1.70609	−0.853043	−	0.521841i	$-0.825246\pi$
$401$	−34.1643	−1.70609	−0.853043	+	0.521841i	$0.825246\pi$
$402$	27.1396	1.35360
$403$	−4.57227	−0.227761
$404$	−4.69963	−0.233815
$405$	−0.888736	−0.0441616
$406$	−8.20877	−0.407394
$407$	0	0
$408$	71.8886	3.55902
$409$	−0.642826	−0.0317857	−0.0158928	−	0.999874i	$-0.505059\pi$
$409$	−0.642826	−0.0317857	−0.0158928	+	0.999874i	$0.505059\pi$
$410$	4.07784	0.201390
$411$	−1.15706	−0.0570738
$412$	−61.9715	−3.05312
$413$	−1.96148	−0.0965183
$414$	−13.4981	−0.663397
$415$	7.76137	0.380991
$416$	42.2967	2.07376
$417$	4.25457	0.208347
$418$	0	0
$419$	20.8502	1.01860	0.509300	−	0.860589i	$-0.329904\pi$
$419$	20.8502	1.01860	0.509300	+	0.860589i	$0.329904\pi$
$420$	−1.93454	−0.0943957
$421$	−10.3425	−0.504060	−0.252030	−	0.967719i	$-0.581098\pi$
$421$	−10.3425	−0.504060	−0.252030	+	0.967719i	$0.581098\pi$
$422$	−24.3497	−1.18533
$423$	−4.47710	−0.217684
$424$	−2.96286	−0.143889
$425$	−34.0975	−1.65397
$426$	−14.3090	−0.693275
$427$	4.32651	0.209374
$428$	32.7280	1.58197
$429$	0	0
$430$	−17.4858	−0.843238
$431$	13.9615	0.672501	0.336250	−	0.941773i	$-0.390841\pi$
$431$	13.9615	0.672501	0.336250	+	0.941773i	$0.390841\pi$
$432$	13.3869	0.644076
$433$	5.53156	0.265830	0.132915	−	0.991127i	$-0.457566\pi$
$433$	5.53156	0.265830	0.132915	+	0.991127i	$0.457566\pi$
$434$	−2.20877	−0.106024
$435$	6.56498	0.314767
$436$	62.2137	2.97950
$437$	−5.00000	−0.239182
$438$	−6.68725	−0.319529
$439$	20.2509	0.966520	0.483260	−	0.875477i	$-0.339452\pi$
$439$	20.2509	0.966520	0.483260	+	0.875477i	$0.339452\pi$
$440$	0	0
$441$	−6.83056	−0.325265
$442$	50.2953	2.39230
$443$	−30.6080	−1.45423	−0.727116	−	0.686515i	$-0.759140\pi$
$443$	−30.6080	−1.45423	−0.727116	+	0.686515i	$0.759140\pi$
$444$	49.4610	2.34732
$445$	−1.76647	−0.0837387
$446$	30.6501	1.45132
$447$	−8.30037	−0.392594
$448$	9.41164	0.444658
$449$	−3.75643	−0.177277	−0.0886385	−	0.996064i	$-0.528252\pi$
$449$	−3.75643	−0.177277	−0.0886385	+	0.996064i	$0.528252\pi$
$450$	−11.3658	−0.535791
$451$	0	0
$452$	−9.74543	−0.458386
$453$	−21.4189	−1.00635
$454$	37.1606	1.74403
$455$	−0.841558	−0.0394529
$456$	8.87636	0.415673
$457$	2.41026	0.112747	0.0563736	−	0.998410i	$-0.482046\pi$
$457$	2.41026	0.112747	0.0563736	+	0.998410i	$0.482046\pi$
$458$	−63.5795	−2.97088
$459$	8.09888	0.378024
$460$	23.4981	1.09561
$461$	−4.35855	−0.202998	−0.101499	−	0.994836i	$-0.532364\pi$
$461$	−4.35855	−0.202998	−0.101499	+	0.994836i	$0.532364\pi$
$462$	0	0
$463$	38.6167	1.79467	0.897335	−	0.441350i	$-0.145500\pi$
$463$	38.6167	1.79467	0.897335	+	0.441350i	$0.145500\pi$
$464$	−98.8872	−4.59072
$465$	1.76647	0.0819181
$466$	−73.6006	−3.40948
$467$	−16.1767	−0.748570	−0.374285	−	0.927314i	$-0.622112\pi$
$467$	−16.1767	−0.748570	−0.374285	+	0.927314i	$0.622112\pi$
$468$	12.1643	0.562297
$469$	4.13821	0.191085
$470$	10.7417	0.495478
$471$	−5.85532	−0.269799
$472$	−42.2967	−1.94686
$473$	0	0
$474$	43.3942	1.99316
$475$	−4.21015	−0.193175
$476$	17.6291	0.808027
$477$	−0.333792	−0.0152833
$478$	−0.863976	−0.0395174
$479$	28.9825	1.32425	0.662123	−	0.749395i	$-0.269656\pi$
$479$	28.9825	1.32425	0.662123	+	0.749395i	$0.269656\pi$
$480$	−16.3411	−0.745865
$481$	21.5164	0.981065
$482$	−72.4559	−3.30028
$483$	−2.05818	−0.0936504
$484$	0	0
$485$	13.8726	0.629924
$486$	2.69963	0.122458
$487$	21.2670	0.963698	0.481849	−	0.876254i	$-0.339965\pi$
$487$	21.2670	0.963698	0.481849	+	0.876254i	$0.339965\pi$
$488$	93.2951	4.22327
$489$	18.6094	0.841546
$490$	16.3883	0.740346
$491$	3.55494	0.160432	0.0802162	−	0.996777i	$-0.474439\pi$
$491$	3.55494	0.160432	0.0802162	+	0.996777i	$0.474439\pi$
$492$	−8.98762	−0.405193
$493$	−59.8255	−2.69440
$494$	6.21015	0.279408
$495$	0	0
$496$	−26.6080	−1.19474
$497$	−2.18182	−0.0978682
$498$	−23.5760	−1.05647
$499$	−23.0952	−1.03388	−0.516941	−	0.856021i	$-0.672929\pi$
$499$	−23.0952	−1.03388	−0.516941	+	0.856021i	$0.672929\pi$
$500$	43.2843	1.93573
$501$	1.11126	0.0496476
$502$	74.4311	3.32202
$503$	−36.4930	−1.62714	−0.813572	−	0.581464i	$-0.802480\pi$
$503$	−36.4930	−1.62714	−0.813572	+	0.581464i	$0.802480\pi$
$504$	3.65383	0.162754
$505$	0.789851	0.0351479
$506$	0	0
$507$	−7.70829	−0.342337
$508$	−49.8799	−2.21306
$509$	−25.0036	−1.10826	−0.554132	−	0.832429i	$-0.686950\pi$
$509$	−25.0036	−1.10826	−0.554132	+	0.832429i	$0.686950\pi$
$510$	−19.4313	−0.860432
$511$	−1.01966	−0.0451073
$512$	8.48948	0.375186
$513$	1.00000	0.0441511
$514$	3.21387	0.141758
$515$	10.4154	0.458955
$516$	38.5388	1.69658
$517$	0	0
$518$	10.3942	0.456693
$519$	9.66621	0.424299
$520$	−18.1470	−0.795800
$521$	−43.8355	−1.92047	−0.960234	−	0.279196i	$-0.909932\pi$
$521$	−43.8355	−1.92047	−0.960234	+	0.279196i	$0.909932\pi$
$522$	−19.9418	−0.872830
$523$	−23.2829	−1.01809	−0.509045	−	0.860740i	$-0.670001\pi$
$523$	−23.2829	−1.01809	−0.509045	+	0.860740i	$0.670001\pi$
$524$	−48.8799	−2.13533
$525$	−1.73305	−0.0756364
$526$	51.3832	2.24041
$527$	−16.0975	−0.701218
$528$	0	0
$529$	2.00000	0.0869565
$530$	0.800854	0.0347869
$531$	−4.76509	−0.206787
$532$	2.17673	0.0943731
$533$	−3.90978	−0.169351
$534$	5.36584	0.232202
$535$	−5.50048	−0.237807
$536$	89.2348	3.85435
$537$	14.3658	0.619932
$538$	31.6043	1.36256
$539$	0	0
$540$	−4.69963	−0.202240
$541$	−29.9556	−1.28789	−0.643945	−	0.765071i	$-0.722704\pi$
$541$	−29.9556	−1.28789	−0.643945	+	0.765071i	$0.722704\pi$
$542$	41.7156	1.79184
$543$	4.14468	0.177865
$544$	148.913	6.38460
$545$	−10.4561	−0.447888
$546$	2.55632	0.109400
$547$	−45.7700	−1.95699	−0.978493	−	0.206282i	$-0.933864\pi$
$547$	−45.7700	−1.95699	−0.978493	+	0.206282i	$0.933864\pi$
$548$	−6.11855	−0.261371
$549$	10.5105	0.448578
$550$	0	0
$551$	−7.38688	−0.314692
$552$	−44.3818	−1.88901
$553$	6.61669	0.281370
$554$	56.6588	2.40720
$555$	−8.31275	−0.352857
$556$	22.4981	0.954134
$557$	−0.987620	−0.0418468	−0.0209234	−	0.999781i	$-0.506661\pi$
$557$	−0.987620	−0.0418468	−0.0209234	+	0.999781i	$0.506661\pi$
$558$	−5.36584	−0.227154
$559$	16.7651	0.709088
$560$	−4.89740	−0.206953
$561$	0	0
$562$	−26.1520	−1.10315
$563$	−5.97896	−0.251983	−0.125992	−	0.992031i	$-0.540211\pi$
$563$	−5.97896	−0.251983	−0.125992	+	0.992031i	$0.540211\pi$
$564$	−23.6749	−0.996892
$565$	1.63788	0.0689062
$566$	−35.3287	−1.48498
$567$	0.411636	0.0172871
$568$	−47.0480	−1.97409
$569$	−29.8799	−1.25263	−0.626316	−	0.779569i	$-0.715438\pi$
$569$	−29.8799	−1.25263	−0.626316	+	0.779569i	$0.715438\pi$
$570$	−2.39926	−0.100494
$571$	15.4785	0.647754	0.323877	−	0.946099i	$-0.395014\pi$
$571$	15.4785	0.647754	0.323877	+	0.946099i	$0.395014\pi$
$572$	0	0
$573$	−18.2101	−0.760740
$574$	−1.88874	−0.0788343
$575$	21.0507	0.877877
$576$	22.8640	0.952666
$577$	−4.14840	−0.172700	−0.0863501	−	0.996265i	$-0.527520\pi$
$577$	−4.14840	−0.172700	−0.0863501	+	0.996265i	$0.527520\pi$
$578$	131.180	5.45637
$579$	−24.0989	−1.00152
$580$	34.7156	1.44149
$581$	−3.59484	−0.149139
$582$	−42.1396	−1.74674
$583$	0	0
$584$	−21.9876	−0.909854
$585$	−2.04442	−0.0845265
$586$	51.6835	2.13503
$587$	−23.2522	−0.959722	−0.479861	−	0.877344i	$-0.659313\pi$
$587$	−23.2522	−0.959722	−0.479861	+	0.877344i	$0.659313\pi$
$588$	−36.1199	−1.48956
$589$	−1.98762	−0.0818985
$590$	11.4327	0.470676
$591$	16.5512	0.680826
$592$	125.214	5.14625
$593$	17.2225	0.707244	0.353622	−	0.935388i	$-0.384950\pi$
$593$	17.2225	0.707244	0.353622	+	0.935388i	$0.384950\pi$
$594$	0	0
$595$	−2.96286	−0.121465
$596$	−43.8923	−1.79790
$597$	1.53528	0.0628348
$598$	−31.0507	−1.26976
$599$	−15.8268	−0.646667	−0.323334	−	0.946285i	$-0.604804\pi$
$599$	−15.8268	−0.646667	−0.323334	+	0.946285i	$0.604804\pi$
$600$	−37.3708	−1.52566
$601$	37.5599	1.53210	0.766050	−	0.642781i	$-0.222220\pi$
$601$	37.5599	1.53210	0.766050	+	0.642781i	$0.222220\pi$
$602$	8.09888	0.330086
$603$	10.0531	0.409393
$604$	−113.263	−4.60861
$605$	0	0
$606$	−2.39926	−0.0974631
$607$	−1.07413	−0.0435974	−0.0217987	−	0.999762i	$-0.506939\pi$
$607$	−1.07413	−0.0435974	−0.0217987	+	0.999762i	$0.506939\pi$
$608$	18.3869	0.745686
$609$	−3.04070	−0.123216
$610$	−25.2174	−1.02102
$611$	−10.2990	−0.416653
$612$	42.8268	1.73117
$613$	−19.4771	−0.786673	−0.393336	−	0.919395i	$-0.628679\pi$
$613$	−19.4771	−0.786673	−0.393336	+	0.919395i	$0.628679\pi$
$614$	8.87636	0.358221
$615$	1.51052	0.0609101
$616$	0	0
$617$	−44.3845	−1.78685	−0.893427	−	0.449208i	$-0.851706\pi$
$617$	−44.3845	−1.78685	−0.893427	+	0.449208i	$0.851706\pi$
$618$	−31.6377	−1.27266
$619$	39.5636	1.59020	0.795098	−	0.606481i	$-0.207419\pi$
$619$	39.5636	1.59020	0.795098	+	0.606481i	$0.207419\pi$
$620$	9.34108	0.375147
$621$	−5.00000	−0.200643
$622$	26.5388	1.06411
$623$	0.818176	0.0327795
$624$	30.7948	1.23278
$625$	13.7761	0.551044
$626$	−80.5882	−3.22095
$627$	0	0
$628$	−30.9629	−1.23555
$629$	75.7526	3.02045
$630$	−0.987620	−0.0393477
$631$	13.4203	0.534254	0.267127	−	0.963661i	$-0.413926\pi$
$631$	13.4203	0.534254	0.267127	+	0.963661i	$0.413926\pi$
$632$	142.680	5.67549
$633$	−9.01966	−0.358499
$634$	−26.0851	−1.03597
$635$	8.38316	0.332675
$636$	−1.76509	−0.0699904
$637$	−15.7128	−0.622565
$638$	0	0
$639$	−5.30037	−0.209679
$640$	−22.1744	−0.876520
$641$	−23.9381	−0.945498	−0.472749	−	0.881197i	$-0.656738\pi$
$641$	−23.9381	−0.945498	−0.472749	+	0.881197i	$0.656738\pi$
$642$	16.7083	0.659424
$643$	−11.5956	−0.457288	−0.228644	−	0.973510i	$-0.573429\pi$
$643$	−11.5956	−0.457288	−0.228644	+	0.973510i	$0.573429\pi$
$644$	−10.8836	−0.428875
$645$	−6.47710	−0.255036
$646$	21.8640	0.860227
$647$	−48.9998	−1.92638	−0.963191	−	0.268817i	$-0.913367\pi$
$647$	−48.9998	−1.92638	−0.963191	+	0.268817i	$0.913367\pi$
$648$	8.87636	0.348696
$649$	0	0
$650$	−26.1456	−1.02552
$651$	−0.818176	−0.0320668
$652$	98.4064	3.85389
$653$	−31.6996	−1.24050	−0.620251	−	0.784403i	$-0.712969\pi$
$653$	−31.6996	−1.24050	−0.620251	+	0.784403i	$0.712969\pi$
$654$	31.7614	1.24197
$655$	8.21509	0.320990
$656$	−22.7527	−0.888344
$657$	−2.47710	−0.0966409
$658$	−4.97524	−0.193955
$659$	−8.70101	−0.338943	−0.169472	−	0.985535i	$-0.554206\pi$
$659$	−8.70101	−0.338943	−0.169472	+	0.985535i	$0.554206\pi$
$660$	0	0
$661$	23.4647	0.912672	0.456336	−	0.889808i	$-0.349162\pi$
$661$	23.4647	0.912672	0.456336	+	0.889808i	$0.349162\pi$
$662$	−0.901116	−0.0350229
$663$	18.6304	0.723547
$664$	−77.5177	−3.00827
$665$	−0.365836	−0.0141865
$666$	25.2509	0.978451
$667$	36.9344	1.43011
$668$	5.87636	0.227363
$669$	11.3535	0.438950
$670$	−24.1199	−0.931834
$671$	0	0
$672$	7.56870	0.291969
$673$	37.7861	1.45655	0.728274	−	0.685286i	$-0.240323\pi$
$673$	37.7861	1.45655	0.728274	+	0.685286i	$0.240323\pi$
$674$	−61.6340	−2.37405
$675$	−4.21015	−0.162049
$676$	−40.7614	−1.56775
$677$	41.2705	1.58615	0.793077	−	0.609121i	$-0.208478\pi$
$677$	41.2705	1.58615	0.793077	+	0.609121i	$0.208478\pi$
$678$	−4.97524	−0.191073
$679$	−6.42539	−0.246584
$680$	−63.8900	−2.45007
$681$	13.7651	0.527479
$682$	0	0
$683$	36.3374	1.39041	0.695205	−	0.718811i	$-0.255314\pi$
$683$	36.3374	1.39041	0.695205	+	0.718811i	$0.255314\pi$
$684$	5.28799	0.202191
$685$	1.02832	0.0392903
$686$	−15.3694	−0.586806
$687$	−23.5512	−0.898535
$688$	97.5635	3.71957
$689$	−0.767847	−0.0292526
$690$	11.9963	0.456691
$691$	35.8777	1.36485	0.682427	−	0.730954i	$-0.260925\pi$
$691$	35.8777	1.36485	0.682427	+	0.730954i	$0.260925\pi$
$692$	51.1148	1.94309
$693$	0	0
$694$	−71.5548	−2.71618
$695$	−3.78119	−0.143429
$696$	−65.5685	−2.48537
$697$	−13.7651	−0.521390
$698$	32.2522	1.22076
$699$	−27.2632	−1.03119
$700$	−9.16435	−0.346380
$701$	−26.2632	−0.991949	−0.495974	−	0.868337i	$-0.665189\pi$
$701$	−26.2632	−0.991949	−0.495974	+	0.868337i	$0.665189\pi$
$702$	6.21015	0.234387
$703$	9.35346	0.352772
$704$	0	0
$705$	3.97896	0.149856
$706$	76.2137	2.86834
$707$	−0.365836	−0.0137587
$708$	−25.1978	−0.946990
$709$	33.7824	1.26873	0.634363	−	0.773036i	$-0.281263\pi$
$709$	33.7824	1.26873	0.634363	+	0.773036i	$0.281263\pi$
$710$	12.7170	0.477259
$711$	16.0741	0.602827
$712$	17.6428	0.661193
$713$	9.93810	0.372185
$714$	9.00000	0.336817
$715$	0	0
$716$	75.9664	2.83900
$717$	−0.320035	−0.0119519
$718$	−49.4138	−1.84411
$719$	−17.8777	−0.666727	−0.333363	−	0.942798i	$-0.608184\pi$
$719$	−17.8777	−0.666727	−0.333363	+	0.942798i	$0.608184\pi$
$720$	−11.8974	−0.443390
$721$	−4.82408	−0.179658
$722$	2.69963	0.100470
$723$	−26.8392	−0.998161
$724$	21.9171	0.814541
$725$	31.0998	1.15502
$726$	0	0
$727$	20.8502	0.773292	0.386646	−	0.922228i	$-0.373634\pi$
$727$	20.8502	0.773292	0.386646	+	0.922228i	$0.373634\pi$
$728$	8.40516	0.311516
$729$	1.00000	0.0370370
$730$	5.94320	0.219968
$731$	59.0246	2.18310
$732$	55.5795	2.05428
$733$	39.2756	1.45068	0.725339	−	0.688392i	$-0.241683\pi$
$733$	39.2756	1.45068	0.725339	+	0.688392i	$0.241683\pi$
$734$	83.0406	3.06508
$735$	6.07056	0.223916
$736$	−91.9344	−3.38875
$737$	0	0
$738$	−4.58836	−0.168900
$739$	43.7366	1.60888	0.804439	−	0.594036i	$-0.202466\pi$
$739$	43.7366	1.60888	0.804439	+	0.594036i	$0.202466\pi$
$740$	−43.9578	−1.61592
$741$	2.30037	0.0845063
$742$	−0.370932	−0.0136173
$743$	11.2312	0.412032	0.206016	−	0.978549i	$-0.433950\pi$
$743$	11.2312	0.412032	0.206016	+	0.978549i	$0.433950\pi$
$744$	−17.6428	−0.646817
$745$	7.37684	0.270266
$746$	5.99628	0.219539
$747$	−8.73305	−0.319526
$748$	0	0
$749$	2.54766	0.0930895
$750$	22.0975	0.806887
$751$	−11.2742	−0.411403	−0.205701	−	0.978615i	$-0.565948\pi$
$751$	−11.2742	−0.411403	−0.205701	+	0.978615i	$0.565948\pi$
$752$	−59.9344	−2.18558
$753$	27.5709	1.00474
$754$	−45.8736	−1.67062
$755$	19.0358	0.692782
$756$	2.17673	0.0791668
$757$	13.9257	0.506139	0.253069	−	0.967448i	$-0.418560\pi$
$757$	13.9257	0.506139	0.253069	+	0.967448i	$0.418560\pi$
$758$	−21.7403	−0.789644
$759$	0	0
$760$	−7.88874	−0.286155
$761$	−20.0952	−0.728449	−0.364225	−	0.931311i	$-0.618666\pi$
$761$	−20.0952	−0.728449	−0.364225	+	0.931311i	$0.618666\pi$
$762$	−25.4647	−0.922489
$763$	4.84294	0.175326
$764$	−96.2951	−3.48384
$765$	−7.19777	−0.260236
$766$	−17.8095	−0.643484
$767$	−10.9615	−0.395796
$768$	21.6291	0.780472
$769$	39.9098	1.43918	0.719592	−	0.694397i	$-0.244329\pi$
$769$	39.9098	1.43918	0.719592	+	0.694397i	$0.244329\pi$
$770$	0	0
$771$	1.19049	0.0428743
$772$	−127.435	−4.58648
$773$	38.6464	1.39001	0.695007	−	0.719003i	$-0.255401\pi$
$773$	38.6464	1.39001	0.695007	+	0.719003i	$0.255401\pi$
$774$	19.6749	0.707199
$775$	8.36818	0.300594
$776$	−138.555	−4.97382
$777$	3.85022	0.138126
$778$	−67.8145	−2.43127
$779$	−1.69963	−0.0608955
$780$	−10.8109	−0.387092
$781$	0	0
$782$	−109.320	−3.90927
$783$	−7.38688	−0.263985
$784$	−91.4398	−3.26571
$785$	5.20383	0.185733
$786$	−24.9542	−0.890087
$787$	−28.8988	−1.03013	−0.515065	−	0.857151i	$-0.672232\pi$
$787$	−28.8988	−1.03013	−0.515065	+	0.857151i	$0.672232\pi$
$788$	87.5227	3.11787
$789$	19.0334	0.677608
$790$	−38.5659	−1.37211
$791$	−0.758619	−0.0269734
$792$	0	0
$793$	24.1781	0.858590
$794$	−85.0566	−3.01855
$795$	0.296653	0.0105212
$796$	8.11855	0.287754
$797$	8.62907	0.305657	0.152829	−	0.988253i	$-0.451162\pi$
$797$	8.62907	0.305657	0.152829	+	0.988253i	$0.451162\pi$
$798$	1.11126	0.0393383
$799$	−36.2595	−1.28277
$800$	−77.4115	−2.73691
$801$	1.98762	0.0702291
$802$	−92.2310	−3.25679
$803$	0	0
$804$	53.1606	1.87483
$805$	1.82918	0.0644701
$806$	−12.3434	−0.434778
$807$	11.7069	0.412103
$808$	−7.88874	−0.277525
$809$	21.1162	0.742406	0.371203	−	0.928552i	$-0.378945\pi$
$809$	21.1162	0.742406	0.371203	+	0.928552i	$0.378945\pi$
$810$	−2.39926	−0.0843013
$811$	0.634164	0.0222685	0.0111343	−	0.999938i	$-0.496456\pi$
$811$	0.634164	0.0222685	0.0111343	+	0.999938i	$0.496456\pi$
$812$	−16.0792	−0.564270
$813$	15.4523	0.541937
$814$	0	0
$815$	−16.5388	−0.579330
$816$	108.419	3.79542
$817$	7.28799	0.254975
$818$	−1.73539	−0.0606765
$819$	0.946916	0.0330879
$820$	7.98762	0.278940
$821$	−5.01238	−0.174933	−0.0874666	−	0.996167i	$-0.527877\pi$
$821$	−5.01238	−0.174933	−0.0874666	+	0.996167i	$0.527877\pi$
$822$	−3.12364	−0.108950
$823$	−5.65383	−0.197080	−0.0985400	−	0.995133i	$-0.531417\pi$
$823$	−5.65383	−0.197080	−0.0985400	+	0.995133i	$0.531417\pi$
$824$	−104.025	−3.62387
$825$	0	0
$826$	−5.29528	−0.184246
$827$	10.8726	0.378079	0.189039	−	0.981970i	$-0.439463\pi$
$827$	10.8726	0.378079	0.189039	+	0.981970i	$0.439463\pi$
$828$	−26.4400	−0.918852
$829$	8.45234	0.293562	0.146781	−	0.989169i	$-0.453109\pi$
$829$	8.45234	0.293562	0.146781	+	0.989169i	$0.453109\pi$
$830$	20.9528	0.727283
$831$	20.9876	0.728052
$832$	52.5956	1.82343
$833$	−55.3199	−1.91672
$834$	11.4858	0.397719
$835$	−0.987620	−0.0341780
$836$	0	0
$837$	−1.98762	−0.0687022
$838$	56.2878	1.94443
$839$	−29.6800	−1.02467	−0.512333	−	0.858787i	$-0.671219\pi$
$839$	−29.6800	−1.02467	−0.512333	+	0.858787i	$0.671219\pi$
$840$	−3.24729	−0.112042
$841$	25.5659	0.881584
$842$	−27.9208	−0.962214
$843$	−9.68725	−0.333647
$844$	−47.6959	−1.64176
$845$	6.85063	0.235669
$846$	−12.0865	−0.415543
$847$	0	0
$848$	−4.46844	−0.153447
$849$	−13.0865	−0.449128
$850$	−92.0506	−3.15731
$851$	−46.7673	−1.60316
$852$	−28.0283	−0.960235
$853$	−39.3584	−1.34761	−0.673803	−	0.738911i	$-0.735340\pi$
$853$	−39.3584	−1.34761	−0.673803	+	0.738911i	$0.735340\pi$
$854$	11.6800	0.399680
$855$	−0.888736	−0.0303941
$856$	54.9367	1.87770
$857$	34.4079	1.17535	0.587676	−	0.809096i	$-0.300043\pi$
$857$	34.4079	1.17535	0.587676	+	0.809096i	$0.300043\pi$
$858$	0	0
$859$	−21.7293	−0.741395	−0.370698	−	0.928754i	$-0.620881\pi$
$859$	−21.7293	−0.741395	−0.370698	+	0.928754i	$0.620881\pi$
$860$	−34.2509	−1.16794
$861$	−0.699628	−0.0238433
$862$	37.6908	1.28375
$863$	−4.26461	−0.145169	−0.0725845	−	0.997362i	$-0.523125\pi$
$863$	−4.26461	−0.145169	−0.0725845	+	0.997362i	$0.523125\pi$
$864$	18.3869	0.625534
$865$	−8.59071	−0.292093
$866$	14.9332	0.507449
$867$	48.5919	1.65027
$868$	−4.32651	−0.146851
$869$	0	0
$870$	17.7230	0.600866
$871$	23.1258	0.783589
$872$	104.431	3.53648
$873$	−15.6094	−0.528298
$874$	−13.4981	−0.456581
$875$	3.36940	0.113907
$876$	−13.0989	−0.442570
$877$	23.6908	0.799982	0.399991	−	0.916519i	$-0.369013\pi$
$877$	23.6908	0.799982	0.399991	+	0.916519i	$0.369013\pi$
$878$	54.6698	1.84502
$879$	19.1447	0.645734
$880$	0	0
$881$	−10.4276	−0.351314	−0.175657	−	0.984451i	$-0.556205\pi$
$881$	−10.4276	−0.351314	−0.175657	+	0.984451i	$0.556205\pi$
$882$	−18.4400	−0.620906
$883$	−16.2349	−0.546348	−0.273174	−	0.961965i	$-0.588074\pi$
$883$	−16.2349	−0.546348	−0.273174	+	0.961965i	$0.588074\pi$
$884$	98.5177	3.31351
$885$	4.23491	0.142355
$886$	−82.6303	−2.77602
$887$	−31.1991	−1.04756	−0.523782	−	0.851852i	$-0.675480\pi$
$887$	−31.1991	−1.04756	−0.523782	+	0.851852i	$0.675480\pi$
$888$	83.0246	2.78612
$889$	−3.88283	−0.130226
$890$	−4.76881	−0.159851
$891$	0	0
$892$	60.0370	2.01019
$893$	−4.47710	−0.149820
$894$	−22.4079	−0.749433
$895$	−12.7674	−0.426768
$896$	10.2705	0.343114
$897$	−11.5019	−0.384036
$898$	−10.1410	−0.338408
$899$	14.6823	0.489682
$900$	−22.2632	−0.742108
$901$	−2.70335	−0.0900615
$902$	0	0
$903$	3.00000	0.0998337
$904$	−16.3586	−0.544077
$905$	−3.68353	−0.122445
$906$	−57.8231	−1.92104
$907$	9.56498	0.317600	0.158800	−	0.987311i	$-0.449237\pi$
$907$	9.56498	0.317600	0.158800	+	0.987311i	$0.449237\pi$
$908$	72.7897	2.41561
$909$	−0.888736	−0.0294775
$910$	−2.27189	−0.0753125
$911$	32.8589	1.08866	0.544332	−	0.838870i	$-0.316783\pi$
$911$	32.8589	1.08866	0.544332	+	0.838870i	$0.316783\pi$
$912$	13.3869	0.443284
$913$	0	0
$914$	6.50680	0.215226
$915$	−9.34108	−0.308806
$916$	−124.539	−4.11487
$917$	−3.80499	−0.125652
$918$	21.8640	0.721619
$919$	−52.0755	−1.71781	−0.858906	−	0.512133i	$-0.828855\pi$
$919$	−52.0755	−1.71781	−0.858906	+	0.512133i	$0.828855\pi$
$920$	39.4437	1.30042
$921$	3.28799	0.108343
$922$	−11.7665	−0.387508
$923$	−12.1928	−0.401332
$924$	0	0
$925$	−39.3794	−1.29479
$926$	104.251	3.42589
$927$	−11.7193	−0.384912
$928$	−135.822	−4.45856
$929$	−6.73305	−0.220904	−0.110452	−	0.993881i	$-0.535230\pi$
$929$	−6.73305	−0.220904	−0.110452	+	0.993881i	$0.535230\pi$
$930$	4.76881	0.156375
$931$	−6.83056	−0.223862
$932$	−144.168	−4.72237
$933$	9.83056	0.321838
$934$	−43.6712	−1.42896
$935$	0	0
$936$	20.4189	0.667413
$937$	13.1557	0.429778	0.214889	−	0.976639i	$-0.431061\pi$
$937$	13.1557	0.429778	0.214889	+	0.976639i	$0.431061\pi$
$938$	11.1716	0.364767
$939$	−29.8516	−0.974170
$940$	21.0407	0.686272
$941$	18.5105	0.603426	0.301713	−	0.953399i	$-0.402442\pi$
$941$	18.5105	0.603426	0.301713	+	0.953399i	$0.402442\pi$
$942$	−15.8072	−0.515026
$943$	8.49814	0.276738
$944$	−63.7897	−2.07618
$945$	−0.365836	−0.0119006
$946$	0	0
$947$	2.67349	0.0868768	0.0434384	−	0.999056i	$-0.486169\pi$
$947$	2.67349	0.0868768	0.0434384	+	0.999056i	$0.486169\pi$
$948$	84.9998	2.76067
$949$	−5.69825	−0.184973
$950$	−11.3658	−0.368756
$951$	−9.66249	−0.313328
$952$	29.5919	0.959080
$953$	−45.3336	−1.46850	−0.734250	−	0.678879i	$-0.762466\pi$
$953$	−45.3336	−1.46850	−0.734250	+	0.678879i	$0.762466\pi$
$954$	−0.901116	−0.0291747
$955$	16.1840	0.523702
$956$	−1.69234	−0.0547343
$957$	0	0
$958$	78.2420	2.52789
$959$	−0.476289	−0.0153802
$960$	−20.3200	−0.655826
$961$	−27.0494	−0.872560
$962$	58.0864	1.87278
$963$	6.18911	0.199441
$964$	−141.926	−4.57111
$965$	21.4175	0.689455
$966$	−5.55632	−0.178772
$967$	49.7797	1.60081	0.800403	−	0.599462i	$-0.204619\pi$
$967$	49.7797	1.60081	0.800403	+	0.599462i	$0.204619\pi$
$968$	0	0
$969$	8.09888	0.260174
$970$	37.4510	1.20248
$971$	−5.46982	−0.175535	−0.0877674	−	0.996141i	$-0.527973\pi$
$971$	−5.46982	−0.175535	−0.0877674	+	0.996141i	$0.527973\pi$
$972$	5.28799	0.169612
$973$	1.75133	0.0561452
$974$	57.4129	1.83963
$975$	−9.68491	−0.310165
$976$	140.703	4.50379
$977$	33.5933	1.07475	0.537373	−	0.843345i	$-0.319417\pi$
$977$	33.5933	1.07475	0.537373	+	0.843345i	$0.319417\pi$
$978$	50.2385	1.60645
$979$	0	0
$980$	32.1011	1.02543
$981$	11.7651	0.375630
$982$	9.59703	0.306253
$983$	49.0122	1.56325	0.781624	−	0.623750i	$-0.214392\pi$
$983$	49.0122	1.56325	0.781624	+	0.623750i	$0.214392\pi$
$984$	−15.0865	−0.480940
$985$	−14.7097	−0.468689
$986$	−161.506	−5.14341
$987$	−1.84294	−0.0586613
$988$	12.1643	0.386999
$989$	−36.4400	−1.15872
$990$	0	0
$991$	27.0159	0.858190	0.429095	−	0.903259i	$-0.358833\pi$
$991$	27.0159	0.858190	0.429095	+	0.903259i	$0.358833\pi$
$992$	−36.5461	−1.16034
$993$	−0.333792	−0.0105926
$994$	−5.89011	−0.186823
$995$	−1.36446	−0.0432562
$996$	−46.1803	−1.46328
$997$	16.4981	0.522501	0.261251	−	0.965271i	$-0.415865\pi$
$997$	16.4981	0.522501	0.261251	+	0.965271i	$0.415865\pi$
$998$	−62.3484	−1.97360
$999$	9.35346	0.295930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.p.1.3		3
11.10	odd	2		627.2.a.e.1.1	✓	3
33.32	even	2		1881.2.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.e.1.1	✓	3	11.10	odd	2
1881.2.a.i.1.3		3	33.32	even	2
6897.2.a.p.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.69963 q^{2} +1.00000 q^{3} +5.28799 q^{4} -0.888736 q^{5} +2.69963 q^{6} +0.411636 q^{7} +8.87636 q^{8} +1.00000 q^{9} -2.39926 q^{10} +5.28799 q^{12} +2.30037 q^{13} +1.11126 q^{14} -0.888736 q^{15} +13.3869 q^{16} +8.09888 q^{17} +2.69963 q^{18} +1.00000 q^{19} -4.69963 q^{20} +0.411636 q^{21} -5.00000 q^{23} +8.87636 q^{24} -4.21015 q^{25} +6.21015 q^{26} +1.00000 q^{27} +2.17673 q^{28} -7.38688 q^{29} -2.39926 q^{30} -1.98762 q^{31} +18.3869 q^{32} +21.8640 q^{34} -0.365836 q^{35} +5.28799 q^{36} +9.35346 q^{37} +2.69963 q^{38} +2.30037 q^{39} -7.88874 q^{40} -1.69963 q^{41} +1.11126 q^{42} +7.28799 q^{43} -0.888736 q^{45} -13.4981 q^{46} -4.47710 q^{47} +13.3869 q^{48} -6.83056 q^{49} -11.3658 q^{50} +8.09888 q^{51} +12.1643 q^{52} -0.333792 q^{53} +2.69963 q^{54} +3.65383 q^{56} +1.00000 q^{57} -19.9418 q^{58} -4.76509 q^{59} -4.69963 q^{60} +10.5105 q^{61} -5.36584 q^{62} +0.411636 q^{63} +22.8640 q^{64} -2.04442 q^{65} +10.0531 q^{67} +42.8268 q^{68} -5.00000 q^{69} -0.987620 q^{70} -5.30037 q^{71} +8.87636 q^{72} -2.47710 q^{73} +25.2509 q^{74} -4.21015 q^{75} +5.28799 q^{76} +6.21015 q^{78} +16.0741 q^{79} -11.8974 q^{80} +1.00000 q^{81} -4.58836 q^{82} -8.73305 q^{83} +2.17673 q^{84} -7.19777 q^{85} +19.6749 q^{86} -7.38688 q^{87} +1.98762 q^{89} -2.39926 q^{90} +0.946916 q^{91} -26.4400 q^{92} -1.98762 q^{93} -12.0865 q^{94} -0.888736 q^{95} +18.3869 q^{96} -15.6094 q^{97} -18.4400 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} + 7 q^{7} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 4 q^{12} + 13 q^{13} + 3 q^{14} - 3 q^{15} + 10 q^{16} + 6 q^{17} + 2 q^{18} + 3 q^{19} - 8 q^{20} + 7 q^{21}+ \cdots + 4 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 3 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 + 7 * q^7 + 9 * q^8 + 3 * q^9 + 5 * q^10 + 4 * q^12 + 13 * q^13 + 3 * q^14 - 3 * q^15 + 10 * q^16 + 6 * q^17 + 2 * q^18 + 3 * q^19 - 8 * q^20 + 7 * q^21 - 15 * q^23 + 9 * q^24 + 6 * q^25 + 3 * q^27 - 5 * q^28 + 8 * q^29 + 5 * q^30 + 12 * q^31 + 25 * q^32 + 30 * q^34 + 4 * q^35 + 4 * q^36 + 5 * q^37 + 2 * q^38 + 13 * q^39 - 24 * q^40 + q^41 + 3 * q^42 + 10 * q^43 - 3 * q^45 - 10 * q^46 - 8 * q^47 + 10 * q^48 + 8 * q^49 - 29 * q^50 + 6 * q^51 + 7 * q^52 + 2 * q^54 - 6 * q^56 + 3 * q^57 - 31 * q^58 + 3 * q^59 - 8 * q^60 + 19 * q^61 - 11 * q^62 + 7 * q^63 + 33 * q^64 - 20 * q^65 + q^67 + 39 * q^68 - 15 * q^69 + 15 * q^70 - 22 * q^71 + 9 * q^72 - 2 * q^73 + 10 * q^74 + 6 * q^75 + 4 * q^76 - 6 * q^79 + 7 * q^80 + 3 * q^81 - 8 * q^82 - 13 * q^83 - 5 * q^84 + 15 * q^85 + 17 * q^86 + 8 * q^87 - 12 * q^89 + 5 * q^90 + 32 * q^91 - 20 * q^92 + 12 * q^93 - 3 * q^95 + 25 * q^96 - 16 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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