LMFDB - Embedded newform 891.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("891.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.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:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.52892 q^{2} +4.39543 q^{4} -0.528918 q^{5} +0.133492 q^{7} +6.05784 q^{8} -1.33759 q^{10} +1.00000 q^{11} +2.52892 q^{13} +0.337590 q^{14} +6.52892 q^{16} +7.92434 q^{17} -3.66241 q^{19} -2.32482 q^{20} +2.52892 q^{22} -4.92434 q^{23} -4.72025 q^{25} +6.39543 q^{26} +0.586754 q^{28} +3.60457 q^{29} -7.39543 q^{31} +4.39543 q^{32} +20.0400 q^{34} -0.0706063 q^{35} +7.79085 q^{37} -9.26193 q^{38} -3.20410 q^{40} +3.39543 q^{41} +2.13349 q^{43} +4.39543 q^{44} -12.4533 q^{46} +5.66241 q^{47} -6.98218 q^{49} -11.9371 q^{50} +11.1157 q^{52} -13.2441 q^{53} -0.528918 q^{55} +0.808672 q^{56} +9.11567 q^{58} -12.7730 q^{59} +0.866508 q^{61} -18.7024 q^{62} -1.94216 q^{64} -1.33759 q^{65} -9.79590 q^{67} +34.8309 q^{68} -0.178557 q^{70} +1.12844 q^{71} -5.52387 q^{73} +19.7024 q^{74} -16.0979 q^{76} +0.133492 q^{77} -13.3776 q^{79} -3.45326 q^{80} +8.58675 q^{82} +6.00000 q^{83} -4.19133 q^{85} +5.39543 q^{86} +6.05784 q^{88} +14.3954 q^{89} +0.337590 q^{91} -21.6446 q^{92} +14.3198 q^{94} +1.93711 q^{95} -11.8487 q^{97} -17.6574 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 6 q^{5} + 3 q^{8} - 12 q^{10} + 3 q^{11} + 9 q^{14} + 12 q^{16} + 9 q^{17} - 3 q^{19} + 9 q^{20} + 9 q^{25} + 12 q^{26} - 21 q^{28} + 18 q^{29} - 15 q^{31} + 6 q^{32} + 15 q^{34} - 9 q^{35}+ \cdots - 39 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 + 6 * q^5 + 3 * q^8 - 12 * q^10 + 3 * q^11 + 9 * q^14 + 12 * q^16 + 9 * q^17 - 3 * q^19 + 9 * q^20 + 9 * q^25 + 12 * q^26 - 21 * q^28 + 18 * q^29 - 15 * q^31 + 6 * q^32 + 15 * q^34 - 9 * q^35 + 9 * q^37 - 21 * q^38 - 18 * q^40 + 3 * q^41 + 6 * q^43 + 6 * q^44 - 15 * q^46 + 9 * q^47 + 9 * q^49 - 45 * q^50 + 3 * q^52 - 3 * q^53 + 6 * q^55 + 18 * q^56 - 3 * q^58 + 6 * q^59 + 3 * q^61 - 3 * q^62 - 21 * q^64 - 12 * q^65 - 21 * q^67 + 45 * q^68 + 39 * q^70 - 3 * q^71 - 3 * q^73 + 6 * q^74 + 12 * q^76 - 3 * q^79 + 12 * q^80 + 3 * q^82 + 18 * q^83 + 3 * q^85 + 9 * q^86 + 3 * q^88 + 36 * q^89 + 9 * q^91 - 27 * q^92 + 21 * q^94 + 15 * q^95 - 6 * q^97 - 39 * 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 $

$2$	2.52892	1.78822	0.894108	−	0.447852i	$-0.147811\pi$
$2$	2.52892	1.78822	0.894108	+	0.447852i	$0.147811\pi$
$3$	0	0
$3$	0	0
$4$	4.39543	2.19771
$5$	−0.528918	−0.236539	−0.118270	−	0.992982i	$-0.537735\pi$
$5$	−0.528918	−0.236539	−0.118270	+	0.992982i	$0.537735\pi$
$6$	0	0
$7$	0.133492	0.0504552	0.0252276	−	0.999682i	$-0.491969\pi$
$7$	0.133492	0.0504552	0.0252276	+	0.999682i	$0.491969\pi$
$8$	6.05784	2.14177
$9$	0	0
$10$	−1.33759	−0.422983
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.52892	0.701396	0.350698	−	0.936489i	$-0.385944\pi$
$13$	2.52892	0.701396	0.350698	+	0.936489i	$0.385944\pi$
$14$	0.337590	0.0902248
$15$	0	0
$16$	6.52892	1.63223
$17$	7.92434	1.92194	0.960968	−	0.276660i	$-0.0892276\pi$
$17$	7.92434	1.92194	0.960968	+	0.276660i	$0.0892276\pi$
$18$	0	0
$19$	−3.66241	−0.840214	−0.420107	−	0.907474i	$-0.638008\pi$
$19$	−3.66241	−0.840214	−0.420107	+	0.907474i	$0.638008\pi$
$20$	−2.32482	−0.519846
$21$	0	0
$22$	2.52892	0.539167
$23$	−4.92434	−1.02680	−0.513398	−	0.858150i	$-0.671614\pi$
$23$	−4.92434	−1.02680	−0.513398	+	0.858150i	$0.671614\pi$
$24$	0	0
$25$	−4.72025	−0.944049
$26$	6.39543	1.25425
$27$	0	0
$28$	0.586754	0.110886
$29$	3.60457	0.669353	0.334676	−	0.942333i	$-0.391373\pi$
$29$	3.60457	0.669353	0.334676	+	0.942333i	$0.391373\pi$
$30$	0	0
$31$	−7.39543	−1.32826	−0.664129	−	0.747618i	$-0.731197\pi$
$31$	−7.39543	−1.32826	−0.664129	+	0.747618i	$0.731197\pi$
$32$	4.39543	0.777009
$33$	0	0
$34$	20.0400	3.43683
$35$	−0.0706063	−0.0119346
$36$	0	0
$37$	7.79085	1.28081	0.640404	−	0.768038i	$-0.278767\pi$
$37$	7.79085	1.28081	0.640404	+	0.768038i	$0.278767\pi$
$38$	−9.26193	−1.50248
$39$	0	0
$40$	−3.20410	−0.506612
$41$	3.39543	0.530276	0.265138	−	0.964210i	$-0.414582\pi$
$41$	3.39543	0.530276	0.265138	+	0.964210i	$0.414582\pi$
$42$	0	0
$43$	2.13349	0.325354	0.162677	−	0.986679i	$-0.447987\pi$
$43$	2.13349	0.325354	0.162677	+	0.986679i	$0.447987\pi$
$44$	4.39543	0.662635
$45$	0	0
$46$	−12.4533	−1.83613
$47$	5.66241	0.825947	0.412974	−	0.910743i	$-0.364490\pi$
$47$	5.66241	0.825947	0.412974	+	0.910743i	$0.364490\pi$
$48$	0	0
$49$	−6.98218	−0.997454
$50$	−11.9371	−1.68816
$51$	0	0
$52$	11.1157	1.54147
$53$	−13.2441	−1.81922	−0.909609	−	0.415464i	$-0.863619\pi$
$53$	−13.2441	−1.81922	−0.909609	+	0.415464i	$0.863619\pi$
$54$	0	0
$55$	−0.528918	−0.0713193
$56$	0.808672	0.108063
$57$	0	0
$58$	9.11567	1.19695
$59$	−12.7730	−1.66291	−0.831454	−	0.555594i	$-0.812491\pi$
$59$	−12.7730	−1.66291	−0.831454	+	0.555594i	$0.812491\pi$
$60$	0	0
$61$	0.866508	0.110945	0.0554725	−	0.998460i	$-0.482333\pi$
$61$	0.866508	0.110945	0.0554725	+	0.998460i	$0.482333\pi$
$62$	−18.7024	−2.37521
$63$	0	0
$64$	−1.94216	−0.242771
$65$	−1.33759	−0.165908
$66$	0	0
$67$	−9.79590	−1.19676	−0.598380	−	0.801212i	$-0.704189\pi$
$67$	−9.79590	−1.19676	−0.598380	+	0.801212i	$0.704189\pi$
$68$	34.8309	4.22386
$69$	0	0
$70$	−0.178557	−0.0213417
$71$	1.12844	0.133921	0.0669607	−	0.997756i	$-0.478670\pi$
$71$	1.12844	0.133921	0.0669607	+	0.997756i	$0.478670\pi$
$72$	0	0
$73$	−5.52387	−0.646520	−0.323260	−	0.946310i	$-0.604779\pi$
$73$	−5.52387	−0.646520	−0.323260	+	0.946310i	$0.604779\pi$
$74$	19.7024	2.29036
$75$	0	0
$76$	−16.0979	−1.84655
$77$	0.133492	0.0152128
$78$	0	0
$79$	−13.3776	−1.50510	−0.752549	−	0.658536i	$-0.771176\pi$
$79$	−13.3776	−1.50510	−0.752549	+	0.658536i	$0.771176\pi$
$80$	−3.45326	−0.386086
$81$	0	0
$82$	8.58675	0.948248
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−4.19133	−0.454613
$86$	5.39543	0.581804
$87$	0	0
$88$	6.05784	0.645767
$89$	14.3954	1.52591	0.762956	−	0.646450i	$-0.223747\pi$
$89$	14.3954	1.52591	0.762956	+	0.646450i	$0.223747\pi$
$90$	0	0
$91$	0.337590	0.0353891
$92$	−21.6446	−2.25660
$93$	0	0
$94$	14.3198	1.47697
$95$	1.93711	0.198744
$96$	0	0
$97$	−11.8487	−1.20305	−0.601526	−	0.798853i	$-0.705440\pi$
$97$	−11.8487	−1.20305	−0.601526	+	0.798853i	$0.705440\pi$
$98$	−17.6574	−1.78366
$99$	0	0
$100$	−20.7475	−2.07475
$101$	9.44821	0.940132	0.470066	−	0.882631i	$-0.344230\pi$
$101$	9.44821	0.940132	0.470066	+	0.882631i	$0.344230\pi$
$102$	0	0
$103$	10.6574	1.05010	0.525050	−	0.851071i	$-0.324046\pi$
$103$	10.6574	1.05010	0.525050	+	0.851071i	$0.324046\pi$
$104$	15.3198	1.50223
$105$	0	0
$106$	−33.4933	−3.25315
$107$	3.05784	0.295612	0.147806	−	0.989016i	$-0.452779\pi$
$107$	3.05784	0.295612	0.147806	+	0.989016i	$0.452779\pi$
$108$	0	0
$109$	−11.1157	−1.06469	−0.532344	−	0.846528i	$-0.678689\pi$
$109$	−11.1157	−1.06469	−0.532344	+	0.846528i	$0.678689\pi$
$110$	−1.33759	−0.127534
$111$	0	0
$112$	0.871558	0.0823545
$113$	2.19133	0.206143	0.103071	−	0.994674i	$-0.467133\pi$
$113$	2.19133	0.206143	0.103071	+	0.994674i	$0.467133\pi$
$114$	0	0
$115$	2.60457	0.242878
$116$	15.8436	1.47104
$117$	0	0
$118$	−32.3019	−2.97364
$119$	1.05784	0.0969717
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.19133	0.198394
$123$	0	0
$124$	−32.5060	−2.91913
$125$	5.14121	0.459844
$126$	0	0
$127$	15.2441	1.35270	0.676348	−	0.736582i	$-0.263561\pi$
$127$	15.2441	1.35270	0.676348	+	0.736582i	$0.263561\pi$
$128$	−13.7024	−1.21113
$129$	0	0
$130$	−3.38266	−0.296679
$131$	6.31977	0.552161	0.276080	−	0.961135i	$-0.410964\pi$
$131$	6.31977	0.552161	0.276080	+	0.961135i	$0.410964\pi$
$132$	0	0
$133$	−0.488902	−0.0423932
$134$	−24.7730	−2.14006
$135$	0	0
$136$	48.0044	4.11634
$137$	15.0979	1.28990	0.644948	−	0.764226i	$-0.276879\pi$
$137$	15.0979	1.28990	0.644948	+	0.764226i	$0.276879\pi$
$138$	0	0
$139$	−10.3376	−0.876823	−0.438411	−	0.898774i	$-0.644459\pi$
$139$	−10.3376	−0.876823	−0.438411	+	0.898774i	$0.644459\pi$
$140$	−0.310345	−0.0262289
$141$	0	0
$142$	2.85374	0.239480
$143$	2.52892	0.211479
$144$	0	0
$145$	−1.90652	−0.158328
$146$	−13.9694	−1.15612
$147$	0	0
$148$	34.2441	2.81485
$149$	−9.85374	−0.807250	−0.403625	−	0.914925i	$-0.632250\pi$
$149$	−9.85374	−0.807250	−0.403625	+	0.914925i	$0.632250\pi$
$150$	0	0
$151$	15.3598	1.24996	0.624981	−	0.780640i	$-0.285107\pi$
$151$	15.3598	1.24996	0.624981	+	0.780640i	$0.285107\pi$
$152$	−22.1863	−1.79954
$153$	0	0
$154$	0.337590	0.0272038
$155$	3.91157	0.314185
$156$	0	0
$157$	−9.13349	−0.728932	−0.364466	−	0.931217i	$-0.618748\pi$
$157$	−9.13349	−0.728932	−0.364466	+	0.931217i	$0.618748\pi$
$158$	−33.8309	−2.69144
$159$	0	0
$160$	−2.32482	−0.183793
$161$	−0.657360	−0.0518072
$162$	0	0
$163$	14.9872	1.17389	0.586945	−	0.809627i	$-0.300330\pi$
$163$	14.9872	1.17389	0.586945	+	0.809627i	$0.300330\pi$
$164$	14.9243	1.16540
$165$	0	0
$166$	15.1735	1.17769
$167$	9.86651	0.763493	0.381747	−	0.924267i	$-0.375323\pi$
$167$	9.86651	0.763493	0.381747	+	0.924267i	$0.375323\pi$
$168$	0	0
$169$	−6.60457	−0.508044
$170$	−10.5995	−0.812946
$171$	0	0
$172$	9.37761	0.715036
$173$	8.07566	0.613981	0.306990	−	0.951713i	$-0.400678\pi$
$173$	8.07566	0.613981	0.306990	+	0.951713i	$0.400678\pi$
$174$	0	0
$175$	−0.630115	−0.0476322
$176$	6.52892	0.492136
$177$	0	0
$178$	36.4049	2.72866
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	9.79590	0.728124	0.364062	−	0.931375i	$-0.381390\pi$
$181$	9.79590	0.728124	0.364062	+	0.931375i	$0.381390\pi$
$182$	0.853738	0.0632832
$183$	0	0
$184$	−29.8309	−2.19916
$185$	−4.12072	−0.302961
$186$	0	0
$187$	7.92434	0.579485
$188$	24.8887	1.81520
$189$	0	0
$190$	4.89880	0.355397
$191$	−17.7603	−1.28509	−0.642544	−	0.766249i	$-0.722121\pi$
$191$	−17.7603	−1.28509	−0.642544	+	0.766249i	$0.722121\pi$
$192$	0	0
$193$	−4.91157	−0.353543	−0.176771	−	0.984252i	$-0.556565\pi$
$193$	−4.91157	−0.353543	−0.176771	+	0.984252i	$0.556565\pi$
$194$	−29.9644	−2.15132
$195$	0	0
$196$	−30.6897	−2.19212
$197$	−5.11567	−0.364477	−0.182238	−	0.983254i	$-0.558334\pi$
$197$	−5.11567	−0.364477	−0.182238	+	0.983254i	$0.558334\pi$
$198$	0	0
$199$	−0.924344	−0.0655250	−0.0327625	−	0.999463i	$-0.510430\pi$
$199$	−0.924344	−0.0655250	−0.0327625	+	0.999463i	$0.510430\pi$
$200$	−28.5945	−2.02193
$201$	0	0
$202$	23.8938	1.68116
$203$	0.481182	0.0337723
$204$	0	0
$205$	−1.79590	−0.125431
$206$	26.9516	1.87781
$207$	0	0
$208$	16.5111	1.14484
$209$	−3.66241	−0.253334
$210$	0	0
$211$	−25.9644	−1.78746	−0.893730	−	0.448605i	$-0.851921\pi$
$211$	−25.9644	−1.78746	−0.893730	+	0.448605i	$0.851921\pi$
$212$	−58.2135	−3.99812
$213$	0	0
$214$	7.73302	0.528618
$215$	−1.12844	−0.0769591
$216$	0	0
$217$	−0.987230	−0.0670175
$218$	−28.1106	−1.90389
$219$	0	0
$220$	−2.32482	−0.156739
$221$	20.0400	1.34804
$222$	0	0
$223$	6.25688	0.418992	0.209496	−	0.977810i	$-0.432818\pi$
$223$	6.25688	0.418992	0.209496	+	0.977810i	$0.432818\pi$
$224$	0.586754	0.0392041
$225$	0	0
$226$	5.54169	0.368628
$227$	2.45326	0.162829	0.0814144	−	0.996680i	$-0.474056\pi$
$227$	2.45326	0.162829	0.0814144	+	0.996680i	$0.474056\pi$
$228$	0	0
$229$	22.2364	1.46942	0.734711	−	0.678380i	$-0.237318\pi$
$229$	22.2364	1.46942	0.734711	+	0.678380i	$0.237318\pi$
$230$	6.58675	0.434318
$231$	0	0
$232$	21.8359	1.43360
$233$	−5.94216	−0.389284	−0.194642	−	0.980874i	$-0.562355\pi$
$233$	−5.94216	−0.389284	−0.194642	+	0.980874i	$0.562355\pi$
$234$	0	0
$235$	−2.99495	−0.195369
$236$	−56.1429	−3.65459
$237$	0	0
$238$	2.67518	0.173406
$239$	−6.48890	−0.419732	−0.209866	−	0.977730i	$-0.567303\pi$
$239$	−6.48890	−0.419732	−0.209866	+	0.977730i	$0.567303\pi$
$240$	0	0
$241$	−15.9872	−1.02983	−0.514914	−	0.857242i	$-0.672176\pi$
$241$	−15.9872	−1.02983	−0.514914	+	0.857242i	$0.672176\pi$
$242$	2.52892	0.162565
$243$	0	0
$244$	3.80867	0.243825
$245$	3.69300	0.235937
$246$	0	0
$247$	−9.26193	−0.589323
$248$	−44.8003	−2.84482
$249$	0	0
$250$	13.0017	0.822300
$251$	21.0222	1.32691	0.663455	−	0.748217i	$-0.269090\pi$
$251$	21.0222	1.32691	0.663455	+	0.748217i	$0.269090\pi$
$252$	0	0
$253$	−4.92434	−0.309591
$254$	38.5511	2.41891
$255$	0	0
$256$	−30.7680	−1.92300
$257$	1.28480	0.0801439	0.0400719	−	0.999197i	$-0.487241\pi$
$257$	1.28480	0.0801439	0.0400719	+	0.999197i	$0.487241\pi$
$258$	0	0
$259$	1.04002	0.0646234
$260$	−5.87928	−0.364617
$261$	0	0
$262$	15.9822	0.987382
$263$	−17.9294	−1.10557	−0.552787	−	0.833323i	$-0.686436\pi$
$263$	−17.9294	−1.10557	−0.552787	+	0.833323i	$0.686436\pi$
$264$	0	0
$265$	7.00505	0.430317
$266$	−1.23639	−0.0758081
$267$	0	0
$268$	−43.0572	−2.63013
$269$	31.2313	1.90421	0.952104	−	0.305773i	$-0.0989150\pi$
$269$	31.2313	1.90421	0.952104	+	0.305773i	$0.0989150\pi$
$270$	0	0
$271$	20.2391	1.22944	0.614718	−	0.788747i	$-0.289270\pi$
$271$	20.2391	1.22944	0.614718	+	0.788747i	$0.289270\pi$
$272$	51.7374	3.13704
$273$	0	0
$274$	38.1812	2.30661
$275$	−4.72025	−0.284642
$276$	0	0
$277$	22.5161	1.35286	0.676432	−	0.736505i	$-0.263525\pi$
$277$	22.5161	1.35286	0.676432	+	0.736505i	$0.263525\pi$
$278$	−26.1429	−1.56795
$279$	0	0
$280$	−0.427721	−0.0255612
$281$	−10.8487	−0.647178	−0.323589	−	0.946198i	$-0.604890\pi$
$281$	−10.8487	−0.647178	−0.323589	+	0.946198i	$0.604890\pi$
$282$	0	0
$283$	18.4882	1.09901	0.549506	−	0.835490i	$-0.314816\pi$
$283$	18.4882	1.09901	0.549506	+	0.835490i	$0.314816\pi$
$284$	4.95998	0.294321
$285$	0	0
$286$	6.39543	0.378169
$287$	0.453262	0.0267552
$288$	0	0
$289$	45.7952	2.69384
$290$	−4.82144	−0.283125
$291$	0	0
$292$	−24.2798	−1.42087
$293$	12.4933	0.729865	0.364933	−	0.931034i	$-0.381092\pi$
$293$	12.4933	0.729865	0.364933	+	0.931034i	$0.381092\pi$
$294$	0	0
$295$	6.75589	0.393343
$296$	47.1957	2.74319
$297$	0	0
$298$	−24.9193	−1.44354
$299$	−12.4533	−0.720191
$300$	0	0
$301$	0.284804	0.0164158
$302$	38.8436	2.23520
$303$	0	0
$304$	−23.9116	−1.37142
$305$	−0.458312	−0.0262429
$306$	0	0
$307$	−23.5639	−1.34486	−0.672431	−	0.740160i	$-0.734750\pi$
$307$	−23.5639	−1.34486	−0.672431	+	0.740160i	$0.734750\pi$
$308$	0.586754	0.0334334
$309$	0	0
$310$	9.89205	0.561831
$311$	−8.73807	−0.495490	−0.247745	−	0.968825i	$-0.579690\pi$
$311$	−8.73807	−0.495490	−0.247745	+	0.968825i	$0.579690\pi$
$312$	0	0
$313$	−10.8665	−0.614211	−0.307106	−	0.951675i	$-0.599361\pi$
$313$	−10.8665	−0.614211	−0.307106	+	0.951675i	$0.599361\pi$
$314$	−23.0979	−1.30349
$315$	0	0
$316$	−58.8003	−3.30777
$317$	13.4005	0.752646	0.376323	−	0.926489i	$-0.377188\pi$
$317$	13.4005	0.752646	0.376323	+	0.926489i	$0.377188\pi$
$318$	0	0
$319$	3.60457	0.201817
$320$	1.02725	0.0574248
$321$	0	0
$322$	−1.66241	−0.0926425
$323$	−29.0222	−1.61484
$324$	0	0
$325$	−11.9371	−0.662152
$326$	37.9015	2.09917
$327$	0	0
$328$	20.5689	1.13573
$329$	0.755886	0.0416733
$330$	0	0
$331$	4.40048	0.241872	0.120936	−	0.992660i	$-0.461410\pi$
$331$	4.40048	0.241872	0.120936	+	0.992660i	$0.461410\pi$
$332$	26.3726	1.44738
$333$	0	0
$334$	24.9516	1.36529
$335$	5.18123	0.283081
$336$	0	0
$337$	32.9593	1.79541	0.897704	−	0.440599i	$-0.145234\pi$
$337$	32.9593	1.79541	0.897704	+	0.440599i	$0.145234\pi$
$338$	−16.7024	−0.908492
$339$	0	0
$340$	−18.4227	−0.999110
$341$	−7.39543	−0.400485
$342$	0	0
$343$	−1.86651	−0.100782
$344$	12.9243	0.696834
$345$	0	0
$346$	20.4227	1.09793
$347$	29.0750	1.56083	0.780413	−	0.625264i	$-0.215009\pi$
$347$	29.0750	1.56083	0.780413	+	0.625264i	$0.215009\pi$
$348$	0	0
$349$	4.79085	0.256448	0.128224	−	0.991745i	$-0.459072\pi$
$349$	4.79085	0.256448	0.128224	+	0.991745i	$0.459072\pi$
$350$	−1.59351	−0.0851766
$351$	0	0
$352$	4.39543	0.234277
$353$	4.66746	0.248424	0.124212	−	0.992256i	$-0.460360\pi$
$353$	4.66746	0.248424	0.124212	+	0.992256i	$0.460360\pi$
$354$	0	0
$355$	−0.596853	−0.0316777
$356$	63.2740	3.35352
$357$	0	0
$358$	−15.1735	−0.801945
$359$	−15.6446	−0.825690	−0.412845	−	0.910801i	$-0.635465\pi$
$359$	−15.6446	−0.825690	−0.412845	+	0.910801i	$0.635465\pi$
$360$	0	0
$361$	−5.58675	−0.294040
$362$	24.7730	1.30204
$363$	0	0
$364$	1.48385	0.0777750
$365$	2.92167	0.152927
$366$	0	0
$367$	6.01782	0.314128	0.157064	−	0.987588i	$-0.449797\pi$
$367$	6.01782	0.314128	0.157064	+	0.987588i	$0.449797\pi$
$368$	−32.1506	−1.67597
$369$	0	0
$370$	−10.4210	−0.541760
$371$	−1.76798	−0.0917891
$372$	0	0
$373$	22.3547	1.15748	0.578742	−	0.815511i	$-0.303544\pi$
$373$	22.3547	1.15748	0.578742	+	0.815511i	$0.303544\pi$
$374$	20.0400	1.03624
$375$	0	0
$376$	34.3019	1.76899
$377$	9.11567	0.469481
$378$	0	0
$379$	18.9694	0.974393	0.487197	−	0.873292i	$-0.338020\pi$
$379$	18.9694	0.974393	0.487197	+	0.873292i	$0.338020\pi$
$380$	8.51444	0.436782
$381$	0	0
$382$	−44.9142	−2.29801
$383$	3.44049	0.175801	0.0879005	−	0.996129i	$-0.471984\pi$
$383$	3.44049	0.175801	0.0879005	+	0.996129i	$0.471984\pi$
$384$	0	0
$385$	−0.0706063	−0.00359843
$386$	−12.4210	−0.632211
$387$	0	0
$388$	−52.0800	−2.64396
$389$	−6.51615	−0.330382	−0.165191	−	0.986262i	$-0.552824\pi$
$389$	−6.51615	−0.330382	−0.165191	+	0.986262i	$0.552824\pi$
$390$	0	0
$391$	−39.0222	−1.97344
$392$	−42.2969	−2.13632
$393$	0	0
$394$	−12.9371	−0.651762
$395$	7.07566	0.356015
$396$	0	0
$397$	12.9465	0.649768	0.324884	−	0.945754i	$-0.394675\pi$
$397$	12.9465	0.649768	0.324884	+	0.945754i	$0.394675\pi$
$398$	−2.33759	−0.117173
$399$	0	0
$400$	−30.8181	−1.54090
$401$	26.4583	1.32127	0.660633	−	0.750709i	$-0.270288\pi$
$401$	26.4583	1.32127	0.660633	+	0.750709i	$0.270288\pi$
$402$	0	0
$403$	−18.7024	−0.931634
$404$	41.5289	2.06614
$405$	0	0
$406$	1.21687	0.0603922
$407$	7.79085	0.386178
$408$	0	0
$409$	8.59180	0.424837	0.212419	−	0.977179i	$-0.431866\pi$
$409$	8.59180	0.424837	0.212419	+	0.977179i	$0.431866\pi$
$410$	−4.54169	−0.224298
$411$	0	0
$412$	46.8436	2.30782
$413$	−1.70510	−0.0839023
$414$	0	0
$415$	−3.17351	−0.155781
$416$	11.1157	0.544991
$417$	0	0
$418$	−9.26193	−0.453016
$419$	22.8887	1.11819	0.559093	−	0.829105i	$-0.311149\pi$
$419$	22.8887	1.11819	0.559093	+	0.829105i	$0.311149\pi$
$420$	0	0
$421$	−19.5767	−0.954108	−0.477054	−	0.878874i	$-0.658295\pi$
$421$	−19.5767	−0.954108	−0.477054	+	0.878874i	$0.658295\pi$
$422$	−65.6617	−3.19636
$423$	0	0
$424$	−80.2307	−3.89635
$425$	−37.4049	−1.81440
$426$	0	0
$427$	0.115672	0.00559775
$428$	13.4405	0.649671
$429$	0	0
$430$	−2.85374	−0.137619
$431$	20.5767	0.991143	0.495571	−	0.868567i	$-0.334959\pi$
$431$	20.5767	0.991143	0.495571	+	0.868567i	$0.334959\pi$
$432$	0	0
$433$	−36.2714	−1.74309	−0.871545	−	0.490315i	$-0.836882\pi$
$433$	−36.2714	−1.74309	−0.871545	+	0.490315i	$0.836882\pi$
$434$	−2.49662	−0.119842
$435$	0	0
$436$	−48.8581	−2.33988
$437$	18.0350	0.862729
$438$	0	0
$439$	32.6140	1.55658	0.778291	−	0.627904i	$-0.216087\pi$
$439$	32.6140	1.55658	0.778291	+	0.627904i	$0.216087\pi$
$440$	−3.20410	−0.152749
$441$	0	0
$442$	50.6796	2.41058
$443$	−23.1812	−1.10137	−0.550687	−	0.834712i	$-0.685634\pi$
$443$	−23.1812	−1.10137	−0.550687	+	0.834712i	$0.685634\pi$
$444$	0	0
$445$	−7.61400	−0.360938
$446$	15.8231	0.749248
$447$	0	0
$448$	−0.259263	−0.0122490
$449$	−11.2091	−0.528992	−0.264496	−	0.964387i	$-0.585206\pi$
$449$	−11.2091	−0.528992	−0.264496	+	0.964387i	$0.585206\pi$
$450$	0	0
$451$	3.39543	0.159884
$452$	9.63182	0.453043
$453$	0	0
$454$	6.20410	0.291173
$455$	−0.178557	−0.00837090
$456$	0	0
$457$	−17.2569	−0.807243	−0.403622	−	0.914926i	$-0.632249\pi$
$457$	−17.2569	−0.807243	−0.403622	+	0.914926i	$0.632249\pi$
$458$	56.2340	2.62764
$459$	0	0
$460$	11.4482	0.533776
$461$	−18.3547	−0.854865	−0.427433	−	0.904047i	$-0.640582\pi$
$461$	−18.3547	−0.854865	−0.427433	+	0.904047i	$0.640582\pi$
$462$	0	0
$463$	−38.8709	−1.80648	−0.903242	−	0.429132i	$-0.858819\pi$
$463$	−38.8709	−1.80648	−0.903242	+	0.429132i	$0.858819\pi$
$464$	23.5340	1.09254
$465$	0	0
$466$	−15.0272	−0.696124
$467$	−36.9516	−1.70992	−0.854958	−	0.518698i	$-0.826417\pi$
$467$	−36.9516	−1.70992	−0.854958	+	0.518698i	$0.826417\pi$
$468$	0	0
$469$	−1.30767	−0.0603828
$470$	−7.57398	−0.349362
$471$	0	0
$472$	−77.3769	−3.56156
$473$	2.13349	0.0980981
$474$	0	0
$475$	17.2875	0.793204
$476$	4.64964	0.213116
$477$	0	0
$478$	−16.4099	−0.750571
$479$	1.88433	0.0860972	0.0430486	−	0.999073i	$-0.486293\pi$
$479$	1.88433	0.0860972	0.0430486	+	0.999073i	$0.486293\pi$
$480$	0	0
$481$	19.7024	0.898353
$482$	−40.4304	−1.84155
$483$	0	0
$484$	4.39543	0.199792
$485$	6.26698	0.284569
$486$	0	0
$487$	−18.8537	−0.854344	−0.427172	−	0.904170i	$-0.640490\pi$
$487$	−18.8537	−0.854344	−0.427172	+	0.904170i	$0.640490\pi$
$488$	5.24916	0.237618
$489$	0	0
$490$	9.33929	0.421906
$491$	−28.6574	−1.29329	−0.646644	−	0.762792i	$-0.723828\pi$
$491$	−28.6574	−1.29329	−0.646644	+	0.762792i	$0.723828\pi$
$492$	0	0
$493$	28.5639	1.28645
$494$	−23.4227	−1.05384
$495$	0	0
$496$	−48.2841	−2.16802
$497$	0.150638	0.00675703
$498$	0	0
$499$	−4.93206	−0.220790	−0.110395	−	0.993888i	$-0.535212\pi$
$499$	−4.93206	−0.220790	−0.110395	+	0.993888i	$0.535212\pi$
$500$	22.5978	1.01061
$501$	0	0
$502$	53.1634	2.37280
$503$	−42.9065	−1.91311	−0.956554	−	0.291556i	$-0.905827\pi$
$503$	−42.9065	−1.91311	−0.956554	+	0.291556i	$0.905827\pi$
$504$	0	0
$505$	−4.99733	−0.222378
$506$	−12.4533	−0.553615
$507$	0	0
$508$	67.0044	2.97284
$509$	34.9065	1.54720	0.773602	−	0.633672i	$-0.218453\pi$
$509$	34.9065	1.54720	0.773602	+	0.633672i	$0.218453\pi$
$510$	0	0
$511$	−0.737392	−0.0326203
$512$	−50.4049	−2.22760
$513$	0	0
$514$	3.24916	0.143314
$515$	−5.63687	−0.248390
$516$	0	0
$517$	5.66241	0.249033
$518$	2.63011	0.115561
$519$	0	0
$520$	−8.10290	−0.355336
$521$	−18.5588	−0.813077	−0.406539	−	0.913634i	$-0.633264\pi$
$521$	−18.5588	−0.813077	−0.406539	+	0.913634i	$0.633264\pi$
$522$	0	0
$523$	35.0851	1.53416	0.767082	−	0.641549i	$-0.221708\pi$
$523$	35.0851	1.53416	0.767082	+	0.641549i	$0.221708\pi$
$524$	27.7781	1.21349
$525$	0	0
$526$	−45.3420	−1.97700
$527$	−58.6039	−2.55283
$528$	0	0
$529$	1.24916	0.0543115
$530$	17.7152	0.769499
$531$	0	0
$532$	−2.14893	−0.0931681
$533$	8.58675	0.371934
$534$	0	0
$535$	−1.61734	−0.0699239
$536$	−59.3420	−2.56318
$537$	0	0
$538$	78.9815	3.40513
$539$	−6.98218	−0.300744
$540$	0	0
$541$	2.32482	0.0999518	0.0499759	−	0.998750i	$-0.484086\pi$
$541$	2.32482	0.0999518	0.0499759	+	0.998750i	$0.484086\pi$
$542$	51.1829	2.19850
$543$	0	0
$544$	34.8309	1.49336
$545$	5.87928	0.251841
$546$	0	0
$547$	8.40048	0.359178	0.179589	−	0.983742i	$-0.442523\pi$
$547$	8.40048	0.359178	0.179589	+	0.983742i	$0.442523\pi$
$548$	66.3615	2.83482
$549$	0	0
$550$	−11.9371	−0.509000
$551$	−13.2014	−0.562400
$552$	0	0
$553$	−1.78580	−0.0759400
$554$	56.9415	2.41921
$555$	0	0
$556$	−45.4381	−1.92701
$557$	21.1685	0.896936	0.448468	−	0.893799i	$-0.351970\pi$
$557$	21.1685	0.896936	0.448468	+	0.893799i	$0.351970\pi$
$558$	0	0
$559$	5.39543	0.228202
$560$	−0.460983	−0.0194801
$561$	0	0
$562$	−27.4354	−1.15729
$563$	−17.9015	−0.754457	−0.377229	−	0.926120i	$-0.623123\pi$
$563$	−17.9015	−0.754457	−0.377229	+	0.926120i	$0.623123\pi$
$564$	0	0
$565$	−1.15903	−0.0487609
$566$	46.7552	1.96527
$567$	0	0
$568$	6.83592	0.286829
$569$	24.2542	1.01679	0.508395	−	0.861124i	$-0.330239\pi$
$569$	24.2542	1.01679	0.508395	+	0.861124i	$0.330239\pi$
$570$	0	0
$571$	34.4882	1.44329	0.721644	−	0.692265i	$-0.243387\pi$
$571$	34.4882	1.44329	0.721644	+	0.692265i	$0.243387\pi$
$572$	11.1157	0.464770
$573$	0	0
$574$	1.14626	0.0478441
$575$	23.2441	0.969347
$576$	0	0
$577$	34.3248	1.42896	0.714480	−	0.699655i	$-0.246663\pi$
$577$	34.3248	1.42896	0.714480	+	0.699655i	$0.246663\pi$
$578$	115.812	4.81716
$579$	0	0
$580$	−8.37998	−0.347960
$581$	0.800952	0.0332291
$582$	0	0
$583$	−13.2441	−0.548515
$584$	−33.4627	−1.38470
$585$	0	0
$586$	31.5945	1.30516
$587$	−2.88938	−0.119257	−0.0596287	−	0.998221i	$-0.518992\pi$
$587$	−2.88938	−0.119257	−0.0596287	+	0.998221i	$0.518992\pi$
$588$	0	0
$589$	27.0851	1.11602
$590$	17.0851	0.703382
$591$	0	0
$592$	50.8658	2.09057
$593$	−7.19638	−0.295520	−0.147760	−	0.989023i	$-0.547206\pi$
$593$	−7.19638	−0.295520	−0.147760	+	0.989023i	$0.547206\pi$
$594$	0	0
$595$	−0.559508	−0.0229376
$596$	−43.3114	−1.77410
$597$	0	0
$598$	−31.4933	−1.28786
$599$	4.19638	0.171459	0.0857297	−	0.996318i	$-0.472678\pi$
$599$	4.19638	0.171459	0.0857297	+	0.996318i	$0.472678\pi$
$600$	0	0
$601$	0.733016	0.0299004	0.0149502	−	0.999888i	$-0.495241\pi$
$601$	0.733016	0.0299004	0.0149502	+	0.999888i	$0.495241\pi$
$602$	0.720246	0.0293550
$603$	0	0
$604$	67.5128	2.74706
$605$	−0.528918	−0.0215036
$606$	0	0
$607$	46.2307	1.87644	0.938222	−	0.346033i	$-0.112471\pi$
$607$	46.2307	1.87644	0.938222	+	0.346033i	$0.112471\pi$
$608$	−16.0979	−0.652854
$609$	0	0
$610$	−1.15903	−0.0469279
$611$	14.3198	0.579316
$612$	0	0
$613$	−12.6224	−0.509814	−0.254907	−	0.966966i	$-0.582045\pi$
$613$	−12.6224	−0.509814	−0.254907	+	0.966966i	$0.582045\pi$
$614$	−59.5911	−2.40490
$615$	0	0
$616$	0.808672	0.0325823
$617$	−17.2414	−0.694114	−0.347057	−	0.937844i	$-0.612819\pi$
$617$	−17.2414	−0.694114	−0.347057	+	0.937844i	$0.612819\pi$
$618$	0	0
$619$	19.0299	0.764877	0.382438	−	0.923981i	$-0.375084\pi$
$619$	19.0299	0.764877	0.382438	+	0.923981i	$0.375084\pi$
$620$	17.1930	0.690489
$621$	0	0
$622$	−22.0979	−0.886043
$623$	1.92167	0.0769902
$624$	0	0
$625$	20.8819	0.835278
$626$	−27.4805	−1.09834
$627$	0	0
$628$	−40.1456	−1.60198
$629$	61.7374	2.46163
$630$	0	0
$631$	−41.4856	−1.65151	−0.825757	−	0.564026i	$-0.809252\pi$
$631$	−41.4856	−1.65151	−0.825757	+	0.564026i	$0.809252\pi$
$632$	−81.0393	−3.22357
$633$	0	0
$634$	33.8887	1.34589
$635$	−8.06289	−0.319966
$636$	0	0
$637$	−17.6574	−0.699610
$638$	9.11567	0.360893
$639$	0	0
$640$	7.24746	0.286481
$641$	27.7552	1.09626	0.548132	−	0.836392i	$-0.315339\pi$
$641$	27.7552	1.09626	0.548132	+	0.836392i	$0.315339\pi$
$642$	0	0
$643$	−10.4354	−0.411534	−0.205767	−	0.978601i	$-0.565969\pi$
$643$	−10.4354	−0.411534	−0.205767	+	0.978601i	$0.565969\pi$
$644$	−2.88938	−0.113857
$645$	0	0
$646$	−73.3948	−2.88768
$647$	−39.1379	−1.53867	−0.769334	−	0.638847i	$-0.779412\pi$
$647$	−39.1379	−1.53867	−0.769334	+	0.638847i	$0.779412\pi$
$648$	0	0
$649$	−12.7730	−0.501385
$650$	−30.1880	−1.18407
$651$	0	0
$652$	65.8753	2.57987
$653$	7.86651	0.307840	0.153920	−	0.988083i	$-0.450810\pi$
$653$	7.86651	0.307840	0.153920	+	0.988083i	$0.450810\pi$
$654$	0	0
$655$	−3.34264	−0.130608
$656$	22.1685	0.865533
$657$	0	0
$658$	1.91157	0.0745209
$659$	−44.6318	−1.73861	−0.869304	−	0.494277i	$-0.835433\pi$
$659$	−44.6318	−1.73861	−0.869304	+	0.494277i	$0.835433\pi$
$660$	0	0
$661$	−17.7125	−0.688937	−0.344469	−	0.938798i	$-0.611941\pi$
$661$	−17.7125	−0.688937	−0.344469	+	0.938798i	$0.611941\pi$
$662$	11.1284	0.432519
$663$	0	0
$664$	36.3470	1.41054
$665$	0.258589	0.0100277
$666$	0	0
$667$	−17.7502	−0.687289
$668$	43.3675	1.67794
$669$	0	0
$670$	13.1029	0.506209
$671$	0.866508	0.0334512
$672$	0	0
$673$	−42.9193	−1.65442	−0.827209	−	0.561895i	$-0.810073\pi$
$673$	−42.9193	−1.65442	−0.827209	+	0.561895i	$0.810073\pi$
$674$	83.3514	3.21058
$675$	0	0
$676$	−29.0299	−1.11654
$677$	−8.97713	−0.345019	−0.172510	−	0.985008i	$-0.555188\pi$
$677$	−8.97713	−0.345019	−0.172510	+	0.985008i	$0.555188\pi$
$678$	0	0
$679$	−1.58170	−0.0607002
$680$	−25.3904	−0.973676
$681$	0	0
$682$	−18.7024	−0.716153
$683$	20.9243	0.800648	0.400324	−	0.916374i	$-0.368898\pi$
$683$	20.9243	0.800648	0.400324	+	0.916374i	$0.368898\pi$
$684$	0	0
$685$	−7.98552	−0.305111
$686$	−4.72025	−0.180220
$687$	0	0
$688$	13.9294	0.531053
$689$	−33.4933	−1.27599
$690$	0	0
$691$	10.3376	0.393260	0.196630	−	0.980478i	$-0.437000\pi$
$691$	10.3376	0.393260	0.196630	+	0.980478i	$0.437000\pi$
$692$	35.4959	1.34935
$693$	0	0
$694$	73.5282	2.79109
$695$	5.46774	0.207403
$696$	0	0
$697$	26.9065	1.01916
$698$	12.1157	0.458585
$699$	0	0
$700$	−2.76962	−0.104682
$701$	−4.58170	−0.173049	−0.0865243	−	0.996250i	$-0.527576\pi$
$701$	−4.58170	−0.173049	−0.0865243	+	0.996250i	$0.527576\pi$
$702$	0	0
$703$	−28.5333	−1.07615
$704$	−1.94216	−0.0731981
$705$	0	0
$706$	11.8036	0.444235
$707$	1.26126	0.0474346
$708$	0	0
$709$	1.24916	0.0469133	0.0234567	−	0.999725i	$-0.492533\pi$
$709$	1.24916	0.0469133	0.0234567	+	0.999725i	$0.492533\pi$
$710$	−1.50939	−0.0566465
$711$	0	0
$712$	87.2051	3.26815
$713$	36.4176	1.36385
$714$	0	0
$715$	−1.33759	−0.0500230
$716$	−26.3726	−0.985589
$717$	0	0
$718$	−39.5639	−1.47651
$719$	3.05012	0.113750	0.0568751	−	0.998381i	$-0.481886\pi$
$719$	3.05012	0.113750	0.0568751	+	0.998381i	$0.481886\pi$
$720$	0	0
$721$	1.42267	0.0529831
$722$	−14.1284	−0.525806
$723$	0	0
$724$	43.0572	1.60021
$725$	−17.0145	−0.631902
$726$	0	0
$727$	6.84364	0.253816	0.126908	−	0.991914i	$-0.459495\pi$
$727$	6.84364	0.253816	0.126908	+	0.991914i	$0.459495\pi$
$728$	2.04507	0.0757952
$729$	0	0
$730$	7.38867	0.273467
$731$	16.9065	0.625310
$732$	0	0
$733$	−24.4432	−0.902829	−0.451414	−	0.892314i	$-0.649080\pi$
$733$	−24.4432	−0.902829	−0.451414	+	0.892314i	$0.649080\pi$
$734$	15.2186	0.561728
$735$	0	0
$736$	−21.6446	−0.797830
$737$	−9.79590	−0.360837
$738$	0	0
$739$	−3.76094	−0.138348	−0.0691741	−	0.997605i	$-0.522036\pi$
$739$	−3.76094	−0.138348	−0.0691741	+	0.997605i	$0.522036\pi$
$740$	−18.1123	−0.665822
$741$	0	0
$742$	−4.47108	−0.164139
$743$	22.4704	0.824359	0.412180	−	0.911103i	$-0.364768\pi$
$743$	22.4704	0.824359	0.412180	+	0.911103i	$0.364768\pi$
$744$	0	0
$745$	5.21182	0.190946
$746$	56.5333	2.06983
$747$	0	0
$748$	34.8309	1.27354
$749$	0.408196	0.0149152
$750$	0	0
$751$	8.07061	0.294501	0.147250	−	0.989099i	$-0.452958\pi$
$751$	8.07061	0.294501	0.147250	+	0.989099i	$0.452958\pi$
$752$	36.9694	1.34814
$753$	0	0
$754$	23.0528	0.839533
$755$	−8.12407	−0.295665
$756$	0	0
$757$	−13.9822	−0.508191	−0.254095	−	0.967179i	$-0.581778\pi$
$757$	−13.9822	−0.508191	−0.254095	+	0.967179i	$0.581778\pi$
$758$	47.9721	1.74242
$759$	0	0
$760$	11.7347	0.425663
$761$	−8.43039	−0.305601	−0.152801	−	0.988257i	$-0.548829\pi$
$761$	−8.43039	−0.305601	−0.152801	+	0.988257i	$0.548829\pi$
$762$	0	0
$763$	−1.48385	−0.0537191
$764$	−78.0639	−2.82425
$765$	0	0
$766$	8.70072	0.314370
$767$	−32.3019	−1.16636
$768$	0	0
$769$	−5.71015	−0.205913	−0.102957	−	0.994686i	$-0.532830\pi$
$769$	−5.71015	−0.205913	−0.102957	+	0.994686i	$0.532830\pi$
$770$	−0.178557	−0.00643476
$771$	0	0
$772$	−21.5885	−0.776986
$773$	−9.10795	−0.327590	−0.163795	−	0.986494i	$-0.552374\pi$
$773$	−9.10795	−0.327590	−0.163795	+	0.986494i	$0.552374\pi$
$774$	0	0
$775$	34.9082	1.25394
$776$	−71.7774	−2.57666
$777$	0	0
$778$	−16.4788	−0.590794
$779$	−12.4354	−0.445546
$780$	0	0
$781$	1.12844	0.0403788
$782$	−98.6839	−3.52893
$783$	0	0
$784$	−45.5861	−1.62807
$785$	4.83087	0.172421
$786$	0	0
$787$	−42.6140	−1.51903	−0.759513	−	0.650493i	$-0.774562\pi$
$787$	−42.6140	−1.51903	−0.759513	+	0.650493i	$0.774562\pi$
$788$	−22.4856	−0.801015
$789$	0	0
$790$	17.8938	0.636631
$791$	0.292525	0.0104010
$792$	0	0
$793$	2.19133	0.0778163
$794$	32.7407	1.16193
$795$	0	0
$796$	−4.06289	−0.144005
$797$	−20.8137	−0.737260	−0.368630	−	0.929576i	$-0.620173\pi$
$797$	−20.8137	−0.737260	−0.368630	+	0.929576i	$0.620173\pi$
$798$	0	0
$799$	44.8709	1.58742
$800$	−20.7475	−0.733535
$801$	0	0
$802$	66.9109	2.36271
$803$	−5.52387	−0.194933
$804$	0	0
$805$	0.347690	0.0122544
$806$	−47.2969	−1.66596
$807$	0	0
$808$	57.2357	2.01355
$809$	1.27470	0.0448162	0.0224081	−	0.999749i	$-0.492867\pi$
$809$	1.27470	0.0448162	0.0224081	+	0.999749i	$0.492867\pi$
$810$	0	0
$811$	−31.4227	−1.10340	−0.551700	−	0.834043i	$-0.686021\pi$
$811$	−31.4227	−1.10340	−0.551700	+	0.834043i	$0.686021\pi$
$812$	2.11500	0.0742219
$813$	0	0
$814$	19.7024	0.690570
$815$	−7.92701	−0.277671
$816$	0	0
$817$	−7.81372	−0.273368
$818$	21.7280	0.759701
$819$	0	0
$820$	−7.89375	−0.275662
$821$	35.8988	1.25288	0.626438	−	0.779471i	$-0.284512\pi$
$821$	35.8988	1.25288	0.626438	+	0.779471i	$0.284512\pi$
$822$	0	0
$823$	−27.5282	−0.959574	−0.479787	−	0.877385i	$-0.659286\pi$
$823$	−27.5282	−0.959574	−0.479787	+	0.877385i	$0.659286\pi$
$824$	64.5605	2.24907
$825$	0	0
$826$	−4.31205	−0.150035
$827$	−26.6217	−0.925728	−0.462864	−	0.886429i	$-0.653178\pi$
$827$	−26.6217	−0.925728	−0.462864	+	0.886429i	$0.653178\pi$
$828$	0	0
$829$	−30.7451	−1.06782	−0.533911	−	0.845541i	$-0.679278\pi$
$829$	−30.7451	−1.06782	−0.533911	+	0.845541i	$0.679278\pi$
$830$	−8.02554	−0.278571
$831$	0	0
$832$	−4.91157	−0.170278
$833$	−55.3292	−1.91704
$834$	0	0
$835$	−5.21857	−0.180596
$836$	−16.0979	−0.556756
$837$	0	0
$838$	57.8837	1.99956
$839$	27.0145	0.932643	0.466322	−	0.884615i	$-0.345579\pi$
$839$	27.0145	0.932643	0.466322	+	0.884615i	$0.345579\pi$
$840$	0	0
$841$	−16.0070	−0.551967
$842$	−49.5078	−1.70615
$843$	0	0
$844$	−114.124	−3.92832
$845$	3.49328	0.120172
$846$	0	0
$847$	0.133492	0.00458684
$848$	−86.4697	−2.96938
$849$	0	0
$850$	−94.5938	−3.24454
$851$	−38.3648	−1.31513
$852$	0	0
$853$	−13.6547	−0.467528	−0.233764	−	0.972293i	$-0.575104\pi$
$853$	−13.6547	−0.467528	−0.233764	+	0.972293i	$0.575104\pi$
$854$	0.292525	0.0100100
$855$	0	0
$856$	18.5239	0.633133
$857$	−25.8010	−0.881344	−0.440672	−	0.897668i	$-0.645260\pi$
$857$	−25.8010	−0.881344	−0.440672	+	0.897668i	$0.645260\pi$
$858$	0	0
$859$	14.5340	0.495893	0.247946	−	0.968774i	$-0.420244\pi$
$859$	14.5340	0.495893	0.247946	+	0.968774i	$0.420244\pi$
$860$	−4.95998	−0.169134
$861$	0	0
$862$	52.0367	1.77238
$863$	−8.54169	−0.290762	−0.145381	−	0.989376i	$-0.546441\pi$
$863$	−8.54169	−0.290762	−0.145381	+	0.989376i	$0.546441\pi$
$864$	0	0
$865$	−4.27136	−0.145231
$866$	−91.7273	−3.11702
$867$	0	0
$868$	−4.33929	−0.147285
$869$	−13.3776	−0.453804
$870$	0	0
$871$	−24.7730	−0.839402
$872$	−67.3369	−2.28032
$873$	0	0
$874$	45.6089	1.54275
$875$	0.686310	0.0232015
$876$	0	0
$877$	38.0222	1.28392	0.641959	−	0.766739i	$-0.278122\pi$
$877$	38.0222	1.28392	0.641959	+	0.766739i	$0.278122\pi$
$878$	82.4781	2.78350
$879$	0	0
$880$	−3.45326	−0.116409
$881$	−23.2518	−0.783374	−0.391687	−	0.920098i	$-0.628108\pi$
$881$	−23.2518	−0.783374	−0.391687	+	0.920098i	$0.628108\pi$
$882$	0	0
$883$	21.2798	0.716121	0.358060	−	0.933698i	$-0.383438\pi$
$883$	21.2798	0.716121	0.358060	+	0.933698i	$0.383438\pi$
$884$	88.0844	2.96260
$885$	0	0
$886$	−58.6234	−1.96949
$887$	39.8759	1.33890	0.669451	−	0.742856i	$-0.266529\pi$
$887$	39.8759	1.33890	0.669451	+	0.742856i	$0.266529\pi$
$888$	0	0
$889$	2.03497	0.0682506
$890$	−19.2552	−0.645435
$891$	0	0
$892$	27.5017	0.920824
$893$	−20.7381	−0.693973
$894$	0	0
$895$	3.17351	0.106079
$896$	−1.82916	−0.0611081
$897$	0	0
$898$	−28.3470	−0.945952
$899$	−26.6574	−0.889073
$900$	0	0
$901$	−104.951	−3.49642
$902$	8.58675	0.285908
$903$	0	0
$904$	13.2747	0.441510
$905$	−5.18123	−0.172230
$906$	0	0
$907$	5.88433	0.195386	0.0976930	−	0.995217i	$-0.468854\pi$
$907$	5.88433	0.195386	0.0976930	+	0.995217i	$0.468854\pi$
$908$	10.7831	0.357851
$909$	0	0
$910$	−0.451557	−0.0149690
$911$	33.3776	1.10585	0.552925	−	0.833231i	$-0.313512\pi$
$911$	33.3776	1.10585	0.552925	+	0.833231i	$0.313512\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−43.6412	−1.44352
$915$	0	0
$916$	97.7384	3.22937
$917$	0.843638	0.0278594
$918$	0	0
$919$	−30.4906	−1.00579	−0.502896	−	0.864347i	$-0.667732\pi$
$919$	−30.4906	−1.00579	−0.502896	+	0.864347i	$0.667732\pi$
$920$	15.7781	0.520188
$921$	0	0
$922$	−46.4176	−1.52868
$923$	2.85374	0.0939319
$924$	0	0
$925$	−36.7747	−1.20915
$926$	−98.3013	−3.23038
$927$	0	0
$928$	15.8436	0.520093
$929$	4.73302	0.155285	0.0776426	−	0.996981i	$-0.475261\pi$
$929$	4.73302	0.155285	0.0776426	+	0.996981i	$0.475261\pi$
$930$	0	0
$931$	25.5716	0.838075
$932$	−26.1183	−0.855535
$933$	0	0
$934$	−93.4475	−3.05770
$935$	−4.19133	−0.137071
$936$	0	0
$937$	−5.39810	−0.176348	−0.0881741	−	0.996105i	$-0.528103\pi$
$937$	−5.39810	−0.176348	−0.0881741	+	0.996105i	$0.528103\pi$
$938$	−3.30700	−0.107977
$939$	0	0
$940$	−13.1641	−0.429365
$941$	30.5663	0.996432	0.498216	−	0.867053i	$-0.333989\pi$
$941$	30.5663	0.996432	0.498216	+	0.867053i	$0.333989\pi$
$942$	0	0
$943$	−16.7202	−0.544486
$944$	−83.3941	−2.71425
$945$	0	0
$946$	5.39543	0.175420
$947$	18.7280	0.608577	0.304289	−	0.952580i	$-0.401581\pi$
$947$	18.7280	0.608577	0.304289	+	0.952580i	$0.401581\pi$
$948$	0	0
$949$	−13.9694	−0.453466
$950$	43.7186	1.41842
$951$	0	0
$952$	6.40820	0.207691
$953$	18.4839	0.598751	0.299375	−	0.954135i	$-0.403222\pi$
$953$	18.4839	0.598751	0.299375	+	0.954135i	$0.403222\pi$
$954$	0	0
$955$	9.39372	0.303974
$956$	−28.5215	−0.922451
$957$	0	0
$958$	4.76531	0.153960
$959$	2.01544	0.0650820
$960$	0	0
$961$	23.6923	0.764269
$962$	49.8258	1.60645
$963$	0	0
$964$	−70.2707	−2.26327
$965$	2.59782	0.0836268
$966$	0	0
$967$	5.84869	0.188081	0.0940406	−	0.995568i	$-0.470022\pi$
$967$	5.84869	0.188081	0.0940406	+	0.995568i	$0.470022\pi$
$968$	6.05784	0.194706
$969$	0	0
$970$	15.8487	0.508871
$971$	−10.1234	−0.324875	−0.162438	−	0.986719i	$-0.551936\pi$
$971$	−10.1234	−0.324875	−0.162438	+	0.986719i	$0.551936\pi$
$972$	0	0
$973$	−1.37998	−0.0442403
$974$	−47.6796	−1.52775
$975$	0	0
$976$	5.65736	0.181088
$977$	−19.5212	−0.624538	−0.312269	−	0.949994i	$-0.601089\pi$
$977$	−19.5212	−0.624538	−0.312269	+	0.949994i	$0.601089\pi$
$978$	0	0
$979$	14.3954	0.460080
$980$	16.2323	0.518522
$981$	0	0
$982$	−72.4721	−2.31268
$983$	−39.0044	−1.24405	−0.622023	−	0.782999i	$-0.713689\pi$
$983$	−39.0044	−1.24405	−0.622023	+	0.782999i	$0.713689\pi$
$984$	0	0
$985$	2.70577	0.0862130
$986$	72.2357	2.30045
$987$	0	0
$988$	−40.7101	−1.29516
$989$	−10.5060	−0.334073
$990$	0	0
$991$	33.1913	1.05436	0.527179	−	0.849754i	$-0.323250\pi$
$991$	33.1913	1.05436	0.527179	+	0.849754i	$0.323250\pi$
$992$	−32.5060	−1.03207
$993$	0	0
$994$	0.380951	0.0120830
$995$	0.488902	0.0154992
$996$	0	0
$997$	−1.23134	−0.0389970	−0.0194985	−	0.999810i	$-0.506207\pi$
$997$	−1.23134	−0.0389970	−0.0194985	+	0.999810i	$0.506207\pi$
$998$	−12.4728	−0.394819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.n.1.3	yes	3
3.2	odd	2		891.2.a.m.1.1	✓	3
9.2	odd	6		891.2.e.s.595.3		6
9.4	even	3		891.2.e.r.298.1		6
9.5	odd	6		891.2.e.s.298.3		6
9.7	even	3		891.2.e.r.595.1		6
11.10	odd	2		9801.2.a.bg.1.1		3
33.32	even	2		9801.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
891.2.a.m.1.1	✓	3	3.2	odd	2
891.2.a.n.1.3	yes	3	1.1	even	1	trivial
891.2.e.r.298.1		6	9.4	even	3
891.2.e.r.595.1		6	9.7	even	3
891.2.e.s.298.3		6	9.5	odd	6
891.2.e.s.595.3		6	9.2	odd	6
9801.2.a.bf.1.3		3	33.32	even	2
9801.2.a.bg.1.1		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52892 q^{2} +4.39543 q^{4} -0.528918 q^{5} +0.133492 q^{7} +6.05784 q^{8} -1.33759 q^{10} +1.00000 q^{11} +2.52892 q^{13} +0.337590 q^{14} +6.52892 q^{16} +7.92434 q^{17} -3.66241 q^{19} -2.32482 q^{20} +2.52892 q^{22} -4.92434 q^{23} -4.72025 q^{25} +6.39543 q^{26} +0.586754 q^{28} +3.60457 q^{29} -7.39543 q^{31} +4.39543 q^{32} +20.0400 q^{34} -0.0706063 q^{35} +7.79085 q^{37} -9.26193 q^{38} -3.20410 q^{40} +3.39543 q^{41} +2.13349 q^{43} +4.39543 q^{44} -12.4533 q^{46} +5.66241 q^{47} -6.98218 q^{49} -11.9371 q^{50} +11.1157 q^{52} -13.2441 q^{53} -0.528918 q^{55} +0.808672 q^{56} +9.11567 q^{58} -12.7730 q^{59} +0.866508 q^{61} -18.7024 q^{62} -1.94216 q^{64} -1.33759 q^{65} -9.79590 q^{67} +34.8309 q^{68} -0.178557 q^{70} +1.12844 q^{71} -5.52387 q^{73} +19.7024 q^{74} -16.0979 q^{76} +0.133492 q^{77} -13.3776 q^{79} -3.45326 q^{80} +8.58675 q^{82} +6.00000 q^{83} -4.19133 q^{85} +5.39543 q^{86} +6.05784 q^{88} +14.3954 q^{89} +0.337590 q^{91} -21.6446 q^{92} +14.3198 q^{94} +1.93711 q^{95} -11.8487 q^{97} -17.6574 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 6 q^{5} + 3 q^{8} - 12 q^{10} + 3 q^{11} + 9 q^{14} + 12 q^{16} + 9 q^{17} - 3 q^{19} + 9 q^{20} + 9 q^{25} + 12 q^{26} - 21 q^{28} + 18 q^{29} - 15 q^{31} + 6 q^{32} + 15 q^{34} - 9 q^{35}+ \cdots - 39 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 + 6 * q^5 + 3 * q^8 - 12 * q^10 + 3 * q^11 + 9 * q^14 + 12 * q^16 + 9 * q^17 - 3 * q^19 + 9 * q^20 + 9 * q^25 + 12 * q^26 - 21 * q^28 + 18 * q^29 - 15 * q^31 + 6 * q^32 + 15 * q^34 - 9 * q^35 + 9 * q^37 - 21 * q^38 - 18 * q^40 + 3 * q^41 + 6 * q^43 + 6 * q^44 - 15 * q^46 + 9 * q^47 + 9 * q^49 - 45 * q^50 + 3 * q^52 - 3 * q^53 + 6 * q^55 + 18 * q^56 - 3 * q^58 + 6 * q^59 + 3 * q^61 - 3 * q^62 - 21 * q^64 - 12 * q^65 - 21 * q^67 + 45 * q^68 + 39 * q^70 - 3 * q^71 - 3 * q^73 + 6 * q^74 + 12 * q^76 - 3 * q^79 + 12 * q^80 + 3 * q^82 + 18 * q^83 + 3 * q^85 + 9 * q^86 + 3 * q^88 + 36 * q^89 + 9 * q^91 - 27 * q^92 + 21 * q^94 + 15 * q^95 - 6 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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