LMFDB - Embedded newform 891.2.a.p.1.1

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:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.33866$ 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)

$f(q)$	$=$	\(q-2.46934 q^{2} +4.09762 q^{4} +2.43628 q^{5} +2.33866 q^{7} -5.17972 q^{8} -6.01598 q^{10} -1.00000 q^{11} +4.71038 q^{13} -5.77494 q^{14} +4.59522 q^{16} +3.20799 q^{17} +7.77494 q^{19} +9.98292 q^{20} +2.46934 q^{22} -2.75895 q^{23} +0.935443 q^{25} -11.6315 q^{26} +9.58293 q^{28} +2.37172 q^{29} -1.37172 q^{31} -0.987711 q^{32} -7.92159 q^{34} +5.69762 q^{35} -8.47256 q^{37} -19.1989 q^{38} -12.6192 q^{40} -3.54665 q^{41} -7.46934 q^{43} -4.09762 q^{44} +6.81278 q^{46} -0.207987 q^{47} -1.53066 q^{49} -2.30992 q^{50} +19.3013 q^{52} +9.11360 q^{53} -2.43628 q^{55} -12.1136 q^{56} -5.85657 q^{58} +0.241045 q^{59} +1.66134 q^{61} +3.38724 q^{62} -6.75145 q^{64} +11.4758 q^{65} -7.68055 q^{67} +13.1451 q^{68} -14.0693 q^{70} -1.07731 q^{71} -2.37495 q^{73} +20.9216 q^{74} +31.8587 q^{76} -2.33866 q^{77} +12.7136 q^{79} +11.1952 q^{80} +8.75786 q^{82} -10.5008 q^{83} +7.81554 q^{85} +18.4443 q^{86} +5.17972 q^{88} +14.2933 q^{89} +11.0160 q^{91} -11.3051 q^{92} +0.513589 q^{94} +18.9419 q^{95} +8.93388 q^{97} +3.77972 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 11 q^{4} - 4 q^{5} + q^{7} + q^{10} - 4 q^{11} + 7 q^{13} - q^{14} + 17 q^{16} + 5 q^{17} + 9 q^{19} + 10 q^{20} + q^{22} - 14 q^{23} + 14 q^{25} - 22 q^{26} - q^{28} + 6 q^{29} - 2 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) $ 4 * q - q^2 + 11 * q^4 - 4 * q^5 + q^7 + q^10 - 4 * q^11 + 7 * q^13 - q^14 + 17 * q^16 + 5 * q^17 + 9 * q^19 + 10 * q^20 + q^22 - 14 * q^23 + 14 * q^25 - 22 * q^26 - q^28 + 6 * q^29 - 2 * q^31 + 34 * q^32 + 16 * q^34 + 8 * q^35 + 3 * q^37 - 3 * q^38 + 12 * q^40 + 2 * q^41 - 21 * q^43 - 11 * q^44 + 2 * q^46 + 7 * q^47 - 15 * q^49 - 23 * q^50 - 10 * q^52 + 6 * q^53 + 4 * q^55 - 18 * q^56 - 21 * q^58 - 2 * q^59 + 15 * q^61 + 20 * q^62 + 16 * q^64 - 19 * q^65 + 14 * q^67 + 7 * q^68 - 38 * q^70 + 3 * q^71 + 22 * q^73 + 36 * q^74 + 42 * q^76 - q^77 + 11 * q^79 + 34 * q^80 - 17 * q^82 - 18 * q^83 + 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 - 19 * q^94 + 30 * q^95 + 26 * q^97 - 15 * 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 $

$2$	−2.46934	−1.74608	−0.873042	−	0.487645i	$-0.837856\pi$
$2$	−2.46934	−1.74608	−0.873042	+	0.487645i	$0.837856\pi$
$3$	0	0
$3$	0	0
$4$	4.09762	2.04881
$5$	2.43628	1.08954	0.544768	−	0.838587i	$-0.316618\pi$
$5$	2.43628	1.08954	0.544768	+	0.838587i	$0.316618\pi$
$6$	0	0
$7$	2.33866	0.883931	0.441965	−	0.897032i	$-0.354281\pi$
$7$	2.33866	0.883931	0.441965	+	0.897032i	$0.354281\pi$
$8$	−5.17972	−1.83131
$9$	0	0
$10$	−6.01598	−1.90242
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.71038	1.30642	0.653212	−	0.757175i	$-0.273421\pi$
$13$	4.71038	1.30642	0.653212	+	0.757175i	$0.273421\pi$
$14$	−5.77494	−1.54342
$15$	0	0
$16$	4.59522	1.14881
$17$	3.20799	0.778051	0.389026	−	0.921227i	$-0.372812\pi$
$17$	3.20799	0.778051	0.389026	+	0.921227i	$0.372812\pi$
$18$	0	0
$19$	7.77494	1.78369	0.891846	−	0.452338i	$-0.149410\pi$
$19$	7.77494	1.78369	0.891846	+	0.452338i	$0.149410\pi$
$20$	9.98292	2.23225
$21$	0	0
$22$	2.46934	0.526464
$23$	−2.75895	−0.575282	−0.287641	−	0.957738i	$-0.592871\pi$
$23$	−2.75895	−0.575282	−0.287641	+	0.957738i	$0.592871\pi$
$24$	0	0
$25$	0.935443	0.187089
$26$	−11.6315	−2.28113
$27$	0	0
$28$	9.58293	1.81100
$29$	2.37172	0.440417	0.220209	−	0.975453i	$-0.429326\pi$
$29$	2.37172	0.440417	0.220209	+	0.975453i	$0.429326\pi$
$30$	0	0
$31$	−1.37172	−0.246368	−0.123184	−	0.992384i	$-0.539311\pi$
$31$	−1.37172	−0.246368	−0.123184	+	0.992384i	$0.539311\pi$
$32$	−0.987711	−0.174604
$33$	0	0
$34$	−7.92159	−1.35854
$35$	5.69762	0.963074
$36$	0	0
$37$	−8.47256	−1.39288	−0.696440	−	0.717615i	$-0.745234\pi$
$37$	−8.47256	−1.39288	−0.696440	+	0.717615i	$0.745234\pi$
$38$	−19.1989	−3.11448
$39$	0	0
$40$	−12.6192	−1.99527
$41$	−3.54665	−0.553893	−0.276947	−	0.960885i	$-0.589323\pi$
$41$	−3.54665	−0.553893	−0.276947	+	0.960885i	$0.589323\pi$
$42$	0	0
$43$	−7.46934	−1.13906	−0.569531	−	0.821970i	$-0.692875\pi$
$43$	−7.46934	−1.13906	−0.569531	+	0.821970i	$0.692875\pi$
$44$	−4.09762	−0.617739
$45$	0	0
$46$	6.81278	1.00449
$47$	−0.207987	−0.0303380	−0.0151690	−	0.999885i	$-0.504829\pi$
$47$	−0.207987	−0.0303380	−0.0151690	+	0.999885i	$0.504829\pi$
$48$	0	0
$49$	−1.53066	−0.218666
$50$	−2.30992	−0.326672
$51$	0	0
$52$	19.3013	2.67661
$53$	9.11360	1.25185	0.625925	−	0.779884i	$-0.284722\pi$
$53$	9.11360	1.25185	0.625925	+	0.779884i	$0.284722\pi$
$54$	0	0
$55$	−2.43628	−0.328507
$56$	−12.1136	−1.61875
$57$	0	0
$58$	−5.85657	−0.769005
$59$	0.241045	0.0313814	0.0156907	−	0.999877i	$-0.495005\pi$
$59$	0.241045	0.0313814	0.0156907	+	0.999877i	$0.495005\pi$
$60$	0	0
$61$	1.66134	0.212713	0.106356	−	0.994328i	$-0.466082\pi$
$61$	1.66134	0.212713	0.106356	+	0.994328i	$0.466082\pi$
$62$	3.38724	0.430179
$63$	0	0
$64$	−6.75145	−0.843932
$65$	11.4758	1.42340
$66$	0	0
$67$	−7.68055	−0.938328	−0.469164	−	0.883111i	$-0.655445\pi$
$67$	−7.68055	−0.938328	−0.469164	+	0.883111i	$0.655445\pi$
$68$	13.1451	1.59408
$69$	0	0
$70$	−14.0693	−1.68161
$71$	−1.07731	−0.127854	−0.0639268	−	0.997955i	$-0.520362\pi$
$71$	−1.07731	−0.127854	−0.0639268	+	0.997955i	$0.520362\pi$
$72$	0	0
$73$	−2.37495	−0.277966	−0.138983	−	0.990295i	$-0.544383\pi$
$73$	−2.37495	−0.277966	−0.138983	+	0.990295i	$0.544383\pi$
$74$	20.9216	2.43209
$75$	0	0
$76$	31.8587	3.65444
$77$	−2.33866	−0.266515
$78$	0	0
$79$	12.7136	1.43039	0.715196	−	0.698924i	$-0.246337\pi$
$79$	12.7136	1.43039	0.715196	+	0.698924i	$0.246337\pi$
$80$	11.1952	1.25166
$81$	0	0
$82$	8.75786	0.967144
$83$	−10.5008	−1.15262	−0.576308	−	0.817233i	$-0.695507\pi$
$83$	−10.5008	−1.15262	−0.576308	+	0.817233i	$0.695507\pi$
$84$	0	0
$85$	7.81554	0.847715
$86$	18.4443	1.98890
$87$	0	0
$88$	5.17972	0.552160
$89$	14.2933	1.51509	0.757544	−	0.652784i	$-0.226399\pi$
$89$	14.2933	1.51509	0.757544	+	0.652784i	$0.226399\pi$
$90$	0	0
$91$	11.0160	1.15479
$92$	−11.3051	−1.17864
$93$	0	0
$94$	0.513589	0.0529727
$95$	18.9419	1.94340
$96$	0	0
$97$	8.93388	0.907098	0.453549	−	0.891231i	$-0.350158\pi$
$97$	8.93388	0.907098	0.453549	+	0.891231i	$0.350158\pi$
$98$	3.77972	0.381810
$99$	0	0
$100$	3.83309	0.383309
$101$	4.87687	0.485267	0.242633	−	0.970118i	$-0.421989\pi$
$101$	4.87687	0.485267	0.242633	+	0.970118i	$0.421989\pi$
$102$	0	0
$103$	−10.1702	−1.00210	−0.501049	−	0.865419i	$-0.667052\pi$
$103$	−10.1702	−1.00210	−0.501049	+	0.865419i	$0.667052\pi$
$104$	−24.3984	−2.39246
$105$	0	0
$106$	−22.5045	−2.18583
$107$	−9.59845	−0.927917	−0.463959	−	0.885857i	$-0.653571\pi$
$107$	−9.59845	−0.927917	−0.463959	+	0.885857i	$0.653571\pi$
$108$	0	0
$109$	−1.95143	−0.186913	−0.0934564	−	0.995623i	$-0.529792\pi$
$109$	−1.95143	−0.186913	−0.0934564	+	0.995623i	$0.529792\pi$
$110$	6.01598	0.573601
$111$	0	0
$112$	10.7467	1.01546
$113$	7.90561	0.743697	0.371849	−	0.928293i	$-0.378724\pi$
$113$	7.90561	0.743697	0.371849	+	0.928293i	$0.378724\pi$
$114$	0	0
$115$	−6.72158	−0.626790
$116$	9.71839	0.902330
$117$	0	0
$118$	−0.595222	−0.0547946
$119$	7.50239	0.687743
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.10240	−0.371414
$123$	0	0
$124$	−5.62078	−0.504761
$125$	−9.90238	−0.885696
$126$	0	0
$127$	−0.943412	−0.0837142	−0.0418571	−	0.999124i	$-0.513327\pi$
$127$	−0.943412	−0.0837142	−0.0418571	+	0.999124i	$0.513327\pi$
$128$	18.6470	1.64818
$129$	0	0
$130$	−28.3376	−2.48537
$131$	3.06133	0.267470	0.133735	−	0.991017i	$-0.457303\pi$
$131$	3.06133	0.267470	0.133735	+	0.991017i	$0.457303\pi$
$132$	0	0
$133$	18.1829	1.57666
$134$	18.9658	1.63840
$135$	0	0
$136$	−16.6165	−1.42485
$137$	20.4560	1.74767	0.873835	−	0.486223i	$-0.161626\pi$
$137$	20.4560	1.74767	0.873835	+	0.486223i	$0.161626\pi$
$138$	0	0
$139$	12.9088	1.09491	0.547457	−	0.836834i	$-0.315596\pi$
$139$	12.9088	1.09491	0.547457	+	0.836834i	$0.315596\pi$
$140$	23.3467	1.97315
$141$	0	0
$142$	2.66025	0.223243
$143$	−4.71038	−0.393902
$144$	0	0
$145$	5.77816	0.479850
$146$	5.86454	0.485353
$147$	0	0
$148$	−34.7173	−2.85374
$149$	−10.0805	−0.825830	−0.412915	−	0.910770i	$-0.635489\pi$
$149$	−10.0805	−0.825830	−0.412915	+	0.910770i	$0.635489\pi$
$150$	0	0
$151$	4.72590	0.384588	0.192294	−	0.981337i	$-0.438407\pi$
$151$	4.72590	0.384588	0.192294	+	0.981337i	$0.438407\pi$
$152$	−40.2720	−3.26649
$153$	0	0
$154$	5.77494	0.465358
$155$	−3.34189	−0.268427
$156$	0	0
$157$	−4.95465	−0.395424	−0.197712	−	0.980260i	$-0.563351\pi$
$157$	−4.95465	−0.395424	−0.197712	+	0.980260i	$0.563351\pi$
$158$	−31.3942	−2.49758
$159$	0	0
$160$	−2.40634	−0.190238
$161$	−6.45226	−0.508509
$162$	0	0
$163$	−8.60277	−0.673821	−0.336910	−	0.941537i	$-0.609382\pi$
$163$	−8.60277	−0.673821	−0.336910	+	0.941537i	$0.609382\pi$
$164$	−14.5328	−1.13482
$165$	0	0
$166$	25.9301	2.01256
$167$	−7.17816	−0.555462	−0.277731	−	0.960659i	$-0.589582\pi$
$167$	−7.17816	−0.555462	−0.277731	+	0.960659i	$0.589582\pi$
$168$	0	0
$169$	9.18768	0.706745
$170$	−19.2992	−1.48018
$171$	0	0
$172$	−30.6065	−2.33372
$173$	−7.25182	−0.551346	−0.275673	−	0.961252i	$-0.588901\pi$
$173$	−7.25182	−0.551346	−0.275673	+	0.961252i	$0.588901\pi$
$174$	0	0
$175$	2.18768	0.165373
$176$	−4.59522	−0.346378
$177$	0	0
$178$	−35.2950	−2.64547
$179$	7.29009	0.544887	0.272443	−	0.962172i	$-0.412168\pi$
$179$	7.29009	0.544887	0.272443	+	0.962172i	$0.412168\pi$
$180$	0	0
$181$	13.4235	0.997762	0.498881	−	0.866670i	$-0.333744\pi$
$181$	13.4235	0.997762	0.498881	+	0.866670i	$0.333744\pi$
$182$	−27.2022	−2.01636
$183$	0	0
$184$	14.2906	1.05352
$185$	−20.6415	−1.51759
$186$	0	0
$187$	−3.20799	−0.234591
$188$	−0.852250	−0.0621567
$189$	0	0
$190$	−46.7739	−3.39333
$191$	2.30082	0.166481	0.0832406	−	0.996529i	$-0.473473\pi$
$191$	2.30082	0.166481	0.0832406	+	0.996529i	$0.473473\pi$
$192$	0	0
$193$	14.0981	1.01480	0.507401	−	0.861710i	$-0.330606\pi$
$193$	14.0981	1.01480	0.507401	+	0.861710i	$0.330606\pi$
$194$	−22.0607	−1.58387
$195$	0	0
$196$	−6.27208	−0.448005
$197$	−1.31362	−0.0935913	−0.0467957	−	0.998904i	$-0.514901\pi$
$197$	−1.31362	−0.0935913	−0.0467957	+	0.998904i	$0.514901\pi$
$198$	0	0
$199$	15.8491	1.12351	0.561755	−	0.827303i	$-0.310126\pi$
$199$	15.8491	1.12351	0.561755	+	0.827303i	$0.310126\pi$
$200$	−4.84533	−0.342616
$201$	0	0
$202$	−12.0426	−0.847317
$203$	5.54665	0.389298
$204$	0	0
$205$	−8.64061	−0.603487
$206$	25.1136	1.74975
$207$	0	0
$208$	21.6452	1.50083
$209$	−7.77494	−0.537804
$210$	0	0
$211$	−8.58449	−0.590981	−0.295490	−	0.955346i	$-0.595483\pi$
$211$	−8.58449	−0.590981	−0.295490	+	0.955346i	$0.595483\pi$
$212$	37.3440	2.56480
$213$	0	0
$214$	23.7018	1.62022
$215$	−18.1974	−1.24105
$216$	0	0
$217$	−3.20799	−0.217772
$218$	4.81872	0.326365
$219$	0	0
$220$	−9.98292	−0.673049
$221$	15.1108	1.01647
$222$	0	0
$223$	−26.5184	−1.77580	−0.887901	−	0.460035i	$-0.847837\pi$
$223$	−26.5184	−1.77580	−0.887901	+	0.460035i	$0.847837\pi$
$224$	−2.30992	−0.154338
$225$	0	0
$226$	−19.5216	−1.29856
$227$	−23.6853	−1.57205	−0.786025	−	0.618194i	$-0.787865\pi$
$227$	−23.6853	−1.57205	−0.786025	+	0.618194i	$0.787865\pi$
$228$	0	0
$229$	−11.9909	−0.792384	−0.396192	−	0.918168i	$-0.629668\pi$
$229$	−11.9909	−0.792384	−0.396192	+	0.918168i	$0.629668\pi$
$230$	16.5978	1.09443
$231$	0	0
$232$	−12.2848	−0.806539
$233$	−18.9188	−1.23941	−0.619707	−	0.784833i	$-0.712749\pi$
$233$	−18.9188	−1.23941	−0.619707	+	0.784833i	$0.712749\pi$
$234$	0	0
$235$	−0.506713	−0.0330543
$236$	0.987711	0.0642945
$237$	0	0
$238$	−18.5259	−1.20086
$239$	16.5659	1.07156	0.535778	−	0.844359i	$-0.320018\pi$
$239$	16.5659	1.07156	0.535778	+	0.844359i	$0.320018\pi$
$240$	0	0
$241$	−16.2890	−1.04927	−0.524633	−	0.851328i	$-0.675798\pi$
$241$	−16.2890	−1.04927	−0.524633	+	0.851328i	$0.675798\pi$
$242$	−2.46934	−0.158735
$243$	0	0
$244$	6.80753	0.435807
$245$	−3.72912	−0.238245
$246$	0	0
$247$	36.6229	2.33026
$248$	7.10512	0.451175
$249$	0	0
$250$	24.4523	1.54650
$251$	15.1248	0.954671	0.477336	−	0.878721i	$-0.341603\pi$
$251$	15.1248	0.954671	0.477336	+	0.878721i	$0.341603\pi$
$252$	0	0
$253$	2.75895	0.173454
$254$	2.32960	0.146172
$255$	0	0
$256$	−32.5428	−2.03393
$257$	−19.5056	−1.21673	−0.608364	−	0.793658i	$-0.708174\pi$
$257$	−19.5056	−1.21673	−0.608364	+	0.793658i	$0.708174\pi$
$258$	0	0
$259$	−19.8145	−1.23121
$260$	47.0234	2.91627
$261$	0	0
$262$	−7.55945	−0.467024
$263$	−9.43950	−0.582065	−0.291032	−	0.956713i	$-0.593999\pi$
$263$	−9.43950	−0.582065	−0.291032	+	0.956713i	$0.593999\pi$
$264$	0	0
$265$	22.2032	1.36393
$266$	−44.8998	−2.75298
$267$	0	0
$268$	−31.4719	−1.92245
$269$	−7.33069	−0.446960	−0.223480	−	0.974708i	$-0.571742\pi$
$269$	−7.33069	−0.446960	−0.223480	+	0.974708i	$0.571742\pi$
$270$	0	0
$271$	−2.50286	−0.152038	−0.0760190	−	0.997106i	$-0.524221\pi$
$271$	−2.50286	−0.152038	−0.0760190	+	0.997106i	$0.524221\pi$
$272$	14.7414	0.893829
$273$	0	0
$274$	−50.5126	−3.05158
$275$	−0.935443	−0.0564093
$276$	0	0
$277$	21.9889	1.32119	0.660593	−	0.750744i	$-0.270305\pi$
$277$	21.9889	1.32119	0.660593	+	0.750744i	$0.270305\pi$
$278$	−31.8762	−1.91181
$279$	0	0
$280$	−29.5121	−1.76368
$281$	16.9937	1.01376	0.506880	−	0.862017i	$-0.330799\pi$
$281$	16.9937	1.01376	0.506880	+	0.862017i	$0.330799\pi$
$282$	0	0
$283$	3.77816	0.224589	0.112294	−	0.993675i	$-0.464180\pi$
$283$	3.77816	0.224589	0.112294	+	0.993675i	$0.464180\pi$
$284$	−4.41441	−0.261947
$285$	0	0
$286$	11.6315	0.687785
$287$	−8.29441	−0.489603
$288$	0	0
$289$	−6.70882	−0.394636
$290$	−14.2682	−0.837859
$291$	0	0
$292$	−9.73162	−0.569500
$293$	23.3754	1.36561	0.682803	−	0.730602i	$-0.260761\pi$
$293$	23.3754	1.36561	0.682803	+	0.730602i	$0.260761\pi$
$294$	0	0
$295$	0.587253	0.0341912
$296$	43.8855	2.55079
$297$	0	0
$298$	24.8922	1.44197
$299$	−12.9957	−0.751562
$300$	0	0
$301$	−17.4682	−1.00685
$302$	−11.6698	−0.671523
$303$	0	0
$304$	35.7276	2.04912
$305$	4.04748	0.231758
$306$	0	0
$307$	15.5574	0.887909	0.443954	−	0.896049i	$-0.353575\pi$
$307$	15.5574	0.887909	0.443954	+	0.896049i	$0.353575\pi$
$308$	−9.58293	−0.546038
$309$	0	0
$310$	8.25224	0.468696
$311$	−24.2838	−1.37701	−0.688504	−	0.725233i	$-0.741732\pi$
$311$	−24.2838	−1.37701	−0.688504	+	0.725233i	$0.741732\pi$
$312$	0	0
$313$	7.80368	0.441090	0.220545	−	0.975377i	$-0.429216\pi$
$313$	7.80368	0.441090	0.220545	+	0.975377i	$0.429216\pi$
$314$	12.2347	0.690444
$315$	0	0
$316$	52.0955	2.93060
$317$	−13.5750	−0.762446	−0.381223	−	0.924483i	$-0.624497\pi$
$317$	−13.5750	−0.762446	−0.381223	+	0.924483i	$0.624497\pi$
$318$	0	0
$319$	−2.37172	−0.132791
$320$	−16.4484	−0.919494
$321$	0	0
$322$	15.9328	0.887900
$323$	24.9419	1.38780
$324$	0	0
$325$	4.40629	0.244417
$326$	21.2431	1.17655
$327$	0	0
$328$	18.3706	1.01435
$329$	−0.486411	−0.0268167
$330$	0	0
$331$	20.7910	1.14277	0.571387	−	0.820680i	$-0.306405\pi$
$331$	20.7910	1.14277	0.571387	+	0.820680i	$0.306405\pi$
$332$	−43.0284	−2.36149
$333$	0	0
$334$	17.7253	0.969884
$335$	−18.7119	−1.02234
$336$	0	0
$337$	6.66503	0.363068	0.181534	−	0.983385i	$-0.441894\pi$
$337$	6.66503	0.363068	0.181534	+	0.983385i	$0.441894\pi$
$338$	−22.6875	−1.23404
$339$	0	0
$340$	32.0251	1.73680
$341$	1.37172	0.0742828
$342$	0	0
$343$	−19.9503	−1.07722
$344$	38.6890	2.08597
$345$	0	0
$346$	17.9072	0.962696
$347$	6.91467	0.371199	0.185600	−	0.982625i	$-0.440577\pi$
$347$	6.91467	0.371199	0.185600	+	0.982625i	$0.440577\pi$
$348$	0	0
$349$	5.67566	0.303811	0.151905	−	0.988395i	$-0.451459\pi$
$349$	5.67566	0.303811	0.151905	+	0.988395i	$0.451459\pi$
$350$	−5.40213	−0.288756
$351$	0	0
$352$	0.987711	0.0526452
$353$	7.56648	0.402723	0.201362	−	0.979517i	$-0.435463\pi$
$353$	7.56648	0.402723	0.201362	+	0.979517i	$0.435463\pi$
$354$	0	0
$355$	−2.62463	−0.139301
$356$	58.5685	3.10412
$357$	0	0
$358$	−18.0017	−0.951418
$359$	−29.1504	−1.53850	−0.769248	−	0.638950i	$-0.779369\pi$
$359$	−29.1504	−1.53850	−0.769248	+	0.638950i	$0.779369\pi$
$360$	0	0
$361$	41.4497	2.18156
$362$	−33.1472	−1.74218
$363$	0	0
$364$	45.1393	2.36594
$365$	−5.78603	−0.302854
$366$	0	0
$367$	−34.4544	−1.79851	−0.899254	−	0.437428i	$-0.855890\pi$
$367$	−34.4544	−1.79851	−0.899254	+	0.437428i	$0.855890\pi$
$368$	−12.6780	−0.660887
$369$	0	0
$370$	50.9708	2.64985
$371$	21.3136	1.10655
$372$	0	0
$373$	3.90557	0.202223	0.101111	−	0.994875i	$-0.467760\pi$
$373$	3.90557	0.202223	0.101111	+	0.994875i	$0.467760\pi$
$374$	7.92159	0.409616
$375$	0	0
$376$	1.07731	0.0555582
$377$	11.1717	0.575372
$378$	0	0
$379$	−10.2129	−0.524600	−0.262300	−	0.964986i	$-0.584481\pi$
$379$	−10.2129	−0.524600	−0.262300	+	0.964986i	$0.584481\pi$
$380$	77.6166	3.98165
$381$	0	0
$382$	−5.68148	−0.290690
$383$	−14.1239	−0.721698	−0.360849	−	0.932624i	$-0.617513\pi$
$383$	−14.1239	−0.721698	−0.360849	+	0.932624i	$0.617513\pi$
$384$	0	0
$385$	−5.69762	−0.290378
$386$	−34.8129	−1.77193
$387$	0	0
$388$	36.6076	1.85847
$389$	−14.2277	−0.721371	−0.360686	−	0.932687i	$-0.617457\pi$
$389$	−14.2277	−0.721371	−0.360686	+	0.932687i	$0.617457\pi$
$390$	0	0
$391$	−8.85069	−0.447599
$392$	7.92841	0.400445
$393$	0	0
$394$	3.24376	0.163418
$395$	30.9739	1.55846
$396$	0	0
$397$	−23.5328	−1.18108	−0.590539	−	0.807009i	$-0.701085\pi$
$397$	−23.5328	−1.18108	−0.590539	+	0.807009i	$0.701085\pi$
$398$	−39.1367	−1.96174
$399$	0	0
$400$	4.29857	0.214928
$401$	36.5308	1.82426	0.912131	−	0.409899i	$-0.134436\pi$
$401$	36.5308	1.82426	0.912131	+	0.409899i	$0.134436\pi$
$402$	0	0
$403$	−6.46132	−0.321861
$404$	19.9835	0.994219
$405$	0	0
$406$	−13.6965	−0.679747
$407$	8.47256	0.419969
$408$	0	0
$409$	16.7873	0.830077	0.415039	−	0.909804i	$-0.363768\pi$
$409$	16.7873	0.830077	0.415039	+	0.909804i	$0.363768\pi$
$410$	21.3366	1.05374
$411$	0	0
$412$	−41.6735	−2.05311
$413$	0.563724	0.0277390
$414$	0	0
$415$	−25.5829	−1.25582
$416$	−4.65250	−0.228107
$417$	0	0
$418$	19.1989	0.939050
$419$	−8.67456	−0.423780	−0.211890	−	0.977294i	$-0.567962\pi$
$419$	−8.67456	−0.423780	−0.211890	+	0.977294i	$0.567962\pi$
$420$	0	0
$421$	20.3654	0.992550	0.496275	−	0.868165i	$-0.334701\pi$
$421$	20.3654	0.992550	0.496275	+	0.868165i	$0.334701\pi$
$422$	21.1980	1.03190
$423$	0	0
$424$	−47.2058	−2.29252
$425$	3.00089	0.145564
$426$	0	0
$427$	3.88531	0.188023
$428$	−39.3308	−1.90112
$429$	0	0
$430$	44.9354	2.16698
$431$	30.7365	1.48053	0.740263	−	0.672318i	$-0.234701\pi$
$431$	30.7365	1.48053	0.740263	+	0.672318i	$0.234701\pi$
$432$	0	0
$433$	−35.7978	−1.72033	−0.860167	−	0.510012i	$-0.829641\pi$
$433$	−35.7978	−1.72033	−0.860167	+	0.510012i	$0.829641\pi$
$434$	7.92159	0.380249
$435$	0	0
$436$	−7.99619	−0.382948
$437$	−21.4507	−1.02613
$438$	0	0
$439$	−18.3084	−0.873813	−0.436906	−	0.899507i	$-0.643926\pi$
$439$	−18.3084	−0.873813	−0.436906	+	0.899507i	$0.643926\pi$
$440$	12.6192	0.601598
$441$	0	0
$442$	−37.3137	−1.77483
$443$	−1.05332	−0.0500445	−0.0250223	−	0.999687i	$-0.507966\pi$
$443$	−1.05332	−0.0500445	−0.0250223	+	0.999687i	$0.507966\pi$
$444$	0	0
$445$	34.8225	1.65074
$446$	65.4828	3.10070
$447$	0	0
$448$	−15.7894	−0.745977
$449$	−29.3702	−1.38607	−0.693033	−	0.720906i	$-0.743726\pi$
$449$	−29.3702	−1.38607	−0.693033	+	0.720906i	$0.743726\pi$
$450$	0	0
$451$	3.54665	0.167005
$452$	32.3942	1.52369
$453$	0	0
$454$	58.4870	2.74493
$455$	26.8380	1.25818
$456$	0	0
$457$	5.31674	0.248706	0.124353	−	0.992238i	$-0.460314\pi$
$457$	5.31674	0.248706	0.124353	+	0.992238i	$0.460314\pi$
$458$	29.6096	1.38357
$459$	0	0
$460$	−27.5424	−1.28417
$461$	6.86016	0.319509	0.159755	−	0.987157i	$-0.448930\pi$
$461$	6.86016	0.319509	0.159755	+	0.987157i	$0.448930\pi$
$462$	0	0
$463$	−25.3061	−1.17607	−0.588036	−	0.808834i	$-0.700099\pi$
$463$	−25.3061	−1.17607	−0.588036	+	0.808834i	$0.700099\pi$
$464$	10.8986	0.505954
$465$	0	0
$466$	46.7169	2.16412
$467$	−40.1018	−1.85569	−0.927844	−	0.372967i	$-0.878340\pi$
$467$	−40.1018	−1.85569	−0.927844	+	0.372967i	$0.878340\pi$
$468$	0	0
$469$	−17.9622	−0.829417
$470$	1.25125	0.0577156
$471$	0	0
$472$	−1.24855	−0.0574690
$473$	7.46934	0.343440
$474$	0	0
$475$	7.27301	0.333709
$476$	30.7419	1.40905
$477$	0	0
$478$	−40.9067	−1.87103
$479$	16.3754	0.748210	0.374105	−	0.927386i	$-0.377950\pi$
$479$	16.3754	0.748210	0.374105	+	0.927386i	$0.377950\pi$
$480$	0	0
$481$	−39.9090	−1.81969
$482$	40.2230	1.83211
$483$	0	0
$484$	4.09762	0.186255
$485$	21.7654	0.988316
$486$	0	0
$487$	14.9456	0.677249	0.338625	−	0.940922i	$-0.390038\pi$
$487$	14.9456	0.677249	0.338625	+	0.940922i	$0.390038\pi$
$488$	−8.60526	−0.389542
$489$	0	0
$490$	9.20845	0.415996
$491$	−27.8683	−1.25768	−0.628839	−	0.777536i	$-0.716470\pi$
$491$	−27.8683	−1.25768	−0.628839	+	0.777536i	$0.716470\pi$
$492$	0	0
$493$	7.60845	0.342667
$494$	−90.4342	−4.06883
$495$	0	0
$496$	−6.30336	−0.283029
$497$	−2.51947	−0.113014
$498$	0	0
$499$	−33.1552	−1.48423	−0.742116	−	0.670271i	$-0.766178\pi$
$499$	−33.1552	−1.48423	−0.742116	+	0.670271i	$0.766178\pi$
$500$	−40.5762	−1.81462
$501$	0	0
$502$	−37.3483	−1.66694
$503$	−8.90129	−0.396889	−0.198444	−	0.980112i	$-0.563589\pi$
$503$	−8.90129	−0.396889	−0.198444	+	0.980112i	$0.563589\pi$
$504$	0	0
$505$	11.8814	0.528716
$506$	−6.81278	−0.302865
$507$	0	0
$508$	−3.86574	−0.171514
$509$	−40.7883	−1.80791	−0.903955	−	0.427627i	$-0.859350\pi$
$509$	−40.7883	−1.80791	−0.903955	+	0.427627i	$0.859350\pi$
$510$	0	0
$511$	−5.55419	−0.245703
$512$	43.0651	1.90323
$513$	0	0
$514$	48.1659	2.12451
$515$	−24.7774	−1.09182
$516$	0	0
$517$	0.207987	0.00914725
$518$	48.9285	2.14980
$519$	0	0
$520$	−59.4413	−2.60667
$521$	8.54191	0.374228	0.187114	−	0.982338i	$-0.440087\pi$
$521$	8.54191	0.374228	0.187114	+	0.982338i	$0.440087\pi$
$522$	0	0
$523$	26.9328	1.17769	0.588845	−	0.808246i	$-0.299583\pi$
$523$	26.9328	1.17769	0.588845	+	0.808246i	$0.299583\pi$
$524$	12.5442	0.547994
$525$	0	0
$526$	23.3093	1.01633
$527$	−4.40046	−0.191687
$528$	0	0
$529$	−15.3882	−0.669051
$530$	−54.8273	−2.38154
$531$	0	0
$532$	74.5067	3.23028
$533$	−16.7061	−0.723620
$534$	0	0
$535$	−23.3845	−1.01100
$536$	39.7831	1.71837
$537$	0	0
$538$	18.1019	0.780430
$539$	1.53066	0.0659304
$540$	0	0
$541$	44.5039	1.91337	0.956686	−	0.291121i	$-0.0940284\pi$
$541$	44.5039	1.91337	0.956686	+	0.291121i	$0.0940284\pi$
$542$	6.18040	0.265471
$543$	0	0
$544$	−3.16857	−0.135851
$545$	−4.75421	−0.203648
$546$	0	0
$547$	9.69450	0.414507	0.207254	−	0.978287i	$-0.433547\pi$
$547$	9.69450	0.414507	0.207254	+	0.978287i	$0.433547\pi$
$548$	83.8206	3.58064
$549$	0	0
$550$	2.30992	0.0984954
$551$	18.4400	0.785569
$552$	0	0
$553$	29.7328	1.26437
$554$	−54.2980	−2.30690
$555$	0	0
$556$	52.8955	2.24327
$557$	−44.5990	−1.88972	−0.944861	−	0.327473i	$-0.893803\pi$
$557$	−44.5990	−1.88972	−0.944861	+	0.327473i	$0.893803\pi$
$558$	0	0
$559$	−35.1834	−1.48810
$560$	26.1818	1.10639
$561$	0	0
$562$	−41.9631	−1.77011
$563$	27.8101	1.17206	0.586029	−	0.810290i	$-0.300691\pi$
$563$	27.8101	1.17206	0.586029	+	0.810290i	$0.300691\pi$
$564$	0	0
$565$	19.2603	0.810285
$566$	−9.32955	−0.392150
$567$	0	0
$568$	5.58017	0.234139
$569$	−37.0112	−1.55159	−0.775796	−	0.630984i	$-0.782651\pi$
$569$	−37.0112	−1.55159	−0.775796	+	0.630984i	$0.782651\pi$
$570$	0	0
$571$	−27.9279	−1.16875	−0.584374	−	0.811484i	$-0.698660\pi$
$571$	−27.9279	−1.16875	−0.584374	+	0.811484i	$0.698660\pi$
$572$	−19.3013	−0.807029
$573$	0	0
$574$	20.4817	0.854888
$575$	−2.58084	−0.107629
$576$	0	0
$577$	24.6587	1.02656	0.513278	−	0.858222i	$-0.328431\pi$
$577$	24.6587	1.02656	0.513278	+	0.858222i	$0.328431\pi$
$578$	16.5663	0.689068
$579$	0	0
$580$	23.6767	0.983121
$581$	−24.5579	−1.01883
$582$	0	0
$583$	−9.11360	−0.377447
$584$	12.3015	0.509042
$585$	0	0
$586$	−57.7217	−2.38446
$587$	26.2463	1.08330	0.541650	−	0.840604i	$-0.317800\pi$
$587$	26.2463	1.08330	0.541650	+	0.840604i	$0.317800\pi$
$588$	0	0
$589$	−10.6650	−0.439445
$590$	−1.45013	−0.0597007
$591$	0	0
$592$	−38.9333	−1.60015
$593$	−11.8551	−0.486829	−0.243414	−	0.969922i	$-0.578267\pi$
$593$	−11.8551	−0.486829	−0.243414	+	0.969922i	$0.578267\pi$
$594$	0	0
$595$	18.2779	0.749321
$596$	−41.3062	−1.69197
$597$	0	0
$598$	32.0908	1.31229
$599$	−30.5759	−1.24930	−0.624648	−	0.780907i	$-0.714757\pi$
$599$	−30.5759	−1.24930	−0.624648	+	0.780907i	$0.714757\pi$
$600$	0	0
$601$	27.8235	1.13494	0.567472	−	0.823392i	$-0.307921\pi$
$601$	27.8235	1.13494	0.567472	+	0.823392i	$0.307921\pi$
$602$	43.1349	1.75805
$603$	0	0
$604$	19.3649	0.787947
$605$	2.43628	0.0990487
$606$	0	0
$607$	−9.19087	−0.373046	−0.186523	−	0.982451i	$-0.559722\pi$
$607$	−9.19087	−0.373046	−0.186523	+	0.982451i	$0.559722\pi$
$608$	−7.67939	−0.311441
$609$	0	0
$610$	−9.99459	−0.404669
$611$	−0.979697	−0.0396343
$612$	0	0
$613$	−12.7376	−0.514467	−0.257234	−	0.966349i	$-0.582811\pi$
$613$	−12.7376	−0.514467	−0.257234	+	0.966349i	$0.582811\pi$
$614$	−38.4165	−1.55036
$615$	0	0
$616$	12.1136	0.488071
$617$	−13.8225	−0.556471	−0.278236	−	0.960513i	$-0.589750\pi$
$617$	−13.8225	−0.556471	−0.278236	+	0.960513i	$0.589750\pi$
$618$	0	0
$619$	45.3456	1.82259	0.911297	−	0.411749i	$-0.135082\pi$
$619$	45.3456	1.82259	0.911297	+	0.411749i	$0.135082\pi$
$620$	−13.6938	−0.549955
$621$	0	0
$622$	59.9648	2.40437
$623$	33.4272	1.33923
$624$	0	0
$625$	−28.8022	−1.15209
$626$	−19.2699	−0.770180
$627$	0	0
$628$	−20.3023	−0.810148
$629$	−27.1799	−1.08373
$630$	0	0
$631$	−48.0257	−1.91187	−0.955936	−	0.293576i	$-0.905155\pi$
$631$	−48.0257	−1.91187	−0.955936	+	0.293576i	$0.905155\pi$
$632$	−65.8529	−2.61949
$633$	0	0
$634$	33.5211	1.33129
$635$	−2.29841	−0.0912097
$636$	0	0
$637$	−7.21001	−0.285671
$638$	5.85657	0.231864
$639$	0	0
$640$	45.4293	1.79575
$641$	14.7462	0.582442	0.291221	−	0.956656i	$-0.405939\pi$
$641$	14.7462	0.582442	0.291221	+	0.956656i	$0.405939\pi$
$642$	0	0
$643$	−45.6656	−1.80088	−0.900438	−	0.434985i	$-0.856754\pi$
$643$	−45.6656	−1.80088	−0.900438	+	0.434985i	$0.856754\pi$
$644$	−26.4389	−1.04184
$645$	0	0
$646$	−61.5899	−2.42322
$647$	−18.8503	−0.741081	−0.370540	−	0.928816i	$-0.620827\pi$
$647$	−18.8503	−0.741081	−0.370540	+	0.928816i	$0.620827\pi$
$648$	0	0
$649$	−0.241045	−0.00946186
$650$	−10.8806	−0.426773
$651$	0	0
$652$	−35.2508	−1.38053
$653$	20.1558	0.788756	0.394378	−	0.918948i	$-0.370960\pi$
$653$	20.1558	0.788756	0.394378	+	0.918948i	$0.370960\pi$
$654$	0	0
$655$	7.45825	0.291418
$656$	−16.2976	−0.636316
$657$	0	0
$658$	1.20111	0.0468242
$659$	−9.67566	−0.376910	−0.188455	−	0.982082i	$-0.560348\pi$
$659$	−9.67566	−0.376910	−0.188455	+	0.982082i	$0.560348\pi$
$660$	0	0
$661$	25.3471	0.985890	0.492945	−	0.870061i	$-0.335920\pi$
$661$	25.3471	0.985890	0.492945	+	0.870061i	$0.335920\pi$
$662$	−51.3399	−1.99538
$663$	0	0
$664$	54.3913	2.11079
$665$	44.2987	1.71783
$666$	0	0
$667$	−6.54347	−0.253364
$668$	−29.4133	−1.13804
$669$	0	0
$670$	46.2061	1.78510
$671$	−1.66134	−0.0641353
$672$	0	0
$673$	−1.53342	−0.0591092	−0.0295546	−	0.999563i	$-0.509409\pi$
$673$	−1.53342	−0.0591092	−0.0295546	+	0.999563i	$0.509409\pi$
$674$	−16.4582	−0.633946
$675$	0	0
$676$	37.6476	1.44798
$677$	4.65967	0.179086	0.0895429	−	0.995983i	$-0.471459\pi$
$677$	4.65967	0.179086	0.0895429	+	0.995983i	$0.471459\pi$
$678$	0	0
$679$	20.8933	0.801812
$680$	−40.4823	−1.55242
$681$	0	0
$682$	−3.38724	−0.129704
$683$	25.9322	0.992269	0.496134	−	0.868246i	$-0.334752\pi$
$683$	25.9322	0.992269	0.496134	+	0.868246i	$0.334752\pi$
$684$	0	0
$685$	49.8364	1.90415
$686$	49.2641	1.88091
$687$	0	0
$688$	−34.3233	−1.30856
$689$	42.9285	1.63545
$690$	0	0
$691$	−27.2639	−1.03717	−0.518584	−	0.855027i	$-0.673541\pi$
$691$	−27.2639	−1.03717	−0.518584	+	0.855027i	$0.673541\pi$
$692$	−29.7152	−1.12960
$693$	0	0
$694$	−17.0746	−0.648145
$695$	31.4495	1.19295
$696$	0	0
$697$	−11.3776	−0.430957
$698$	−14.0151	−0.530479
$699$	0	0
$700$	8.96429	0.338818
$701$	1.77806	0.0671563	0.0335782	−	0.999436i	$-0.489310\pi$
$701$	1.77806	0.0671563	0.0335782	+	0.999436i	$0.489310\pi$
$702$	0	0
$703$	−65.8736	−2.48447
$704$	6.75145	0.254455
$705$	0	0
$706$	−18.6842	−0.703188
$707$	11.4054	0.428942
$708$	0	0
$709$	14.6001	0.548317	0.274158	−	0.961685i	$-0.411601\pi$
$709$	14.6001	0.548317	0.274158	+	0.961685i	$0.411601\pi$
$710$	6.48110	0.243231
$711$	0	0
$712$	−74.0353	−2.77459
$713$	3.78451	0.141731
$714$	0	0
$715$	−11.4758	−0.429170
$716$	29.8720	1.11637
$717$	0	0
$718$	71.9820	2.68634
$719$	−19.7642	−0.737081	−0.368540	−	0.929612i	$-0.620142\pi$
$719$	−19.7642	−0.737081	−0.368540	+	0.929612i	$0.620142\pi$
$720$	0	0
$721$	−23.7846	−0.885785
$722$	−102.353	−3.80919
$723$	0	0
$724$	55.0044	2.04422
$725$	2.21861	0.0823971
$726$	0	0
$727$	−18.8112	−0.697670	−0.348835	−	0.937184i	$-0.613423\pi$
$727$	−18.8112	−0.697670	−0.348835	+	0.937184i	$0.613423\pi$
$728$	−57.0597	−2.11477
$729$	0	0
$730$	14.2876	0.528809
$731$	−23.9615	−0.886249
$732$	0	0
$733$	46.6603	1.72344	0.861719	−	0.507386i	$-0.169388\pi$
$733$	46.6603	1.72344	0.861719	+	0.507386i	$0.169388\pi$
$734$	85.0796	3.14034
$735$	0	0
$736$	2.72505	0.100447
$737$	7.68055	0.282917
$738$	0	0
$739$	−6.50838	−0.239415	−0.119707	−	0.992809i	$-0.538196\pi$
$739$	−6.50838	−0.239415	−0.119707	+	0.992809i	$0.538196\pi$
$740$	−84.5809	−3.10926
$741$	0	0
$742$	−52.6305	−1.93212
$743$	43.7996	1.60685	0.803425	−	0.595406i	$-0.203009\pi$
$743$	43.7996	1.60685	0.803425	+	0.595406i	$0.203009\pi$
$744$	0	0
$745$	−24.5590	−0.899771
$746$	−9.64415	−0.353097
$747$	0	0
$748$	−13.1451	−0.480632
$749$	−22.4475	−0.820214
$750$	0	0
$751$	19.6993	0.718837	0.359419	−	0.933176i	$-0.382975\pi$
$751$	19.6993	0.718837	0.359419	+	0.933176i	$0.382975\pi$
$752$	−0.955746	−0.0348525
$753$	0	0
$754$	−27.5867	−1.00465
$755$	11.5136	0.419022
$756$	0	0
$757$	23.1719	0.842195	0.421098	−	0.907015i	$-0.361645\pi$
$757$	23.1719	0.842195	0.421098	+	0.907015i	$0.361645\pi$
$758$	25.2190	0.915996
$759$	0	0
$760$	−98.1136	−3.55896
$761$	−13.9371	−0.505220	−0.252610	−	0.967568i	$-0.581289\pi$
$761$	−13.9371	−0.505220	−0.252610	+	0.967568i	$0.581289\pi$
$762$	0	0
$763$	−4.56372	−0.165218
$764$	9.42786	0.341088
$765$	0	0
$766$	34.8767	1.26014
$767$	1.13542	0.0409975
$768$	0	0
$769$	−5.09642	−0.183781	−0.0918907	−	0.995769i	$-0.529291\pi$
$769$	−5.09642	−0.183781	−0.0918907	+	0.995769i	$0.529291\pi$
$770$	14.0693	0.507024
$771$	0	0
$772$	57.7685	2.07913
$773$	20.5798	0.740202	0.370101	−	0.928991i	$-0.379323\pi$
$773$	20.5798	0.740202	0.370101	+	0.928991i	$0.379323\pi$
$774$	0	0
$775$	−1.28317	−0.0460927
$776$	−46.2750	−1.66117
$777$	0	0
$778$	35.1329	1.25957
$779$	−27.5750	−0.987976
$780$	0	0
$781$	1.07731	0.0385493
$782$	21.8553	0.781545
$783$	0	0
$784$	−7.03375	−0.251205
$785$	−12.0709	−0.430829
$786$	0	0
$787$	24.2544	0.864577	0.432288	−	0.901735i	$-0.357706\pi$
$787$	24.2544	0.864577	0.432288	+	0.901735i	$0.357706\pi$
$788$	−5.38270	−0.191751
$789$	0	0
$790$	−76.4848	−2.72121
$791$	18.4885	0.657377
$792$	0	0
$793$	7.82554	0.277893
$794$	58.1104	2.06226
$795$	0	0
$796$	64.9434	2.30186
$797$	−39.1356	−1.38625	−0.693126	−	0.720816i	$-0.743767\pi$
$797$	−39.1356	−1.38625	−0.693126	+	0.720816i	$0.743767\pi$
$798$	0	0
$799$	−0.667219	−0.0236045
$800$	−0.923948	−0.0326665
$801$	0	0
$802$	−90.2068	−3.18531
$803$	2.37495	0.0838100
$804$	0	0
$805$	−15.7195	−0.554039
$806$	15.9552	0.561997
$807$	0	0
$808$	−25.2608	−0.888672
$809$	32.9926	1.15996	0.579979	−	0.814631i	$-0.303061\pi$
$809$	32.9926	1.15996	0.579979	+	0.814631i	$0.303061\pi$
$810$	0	0
$811$	−32.0811	−1.12652	−0.563259	−	0.826280i	$-0.690453\pi$
$811$	−32.0811	−1.12652	−0.563259	+	0.826280i	$0.690453\pi$
$812$	22.7280	0.797597
$813$	0	0
$814$	−20.9216	−0.733302
$815$	−20.9587	−0.734152
$816$	0	0
$817$	−58.0736	−2.03174
$818$	−41.4534	−1.44938
$819$	0	0
$820$	−35.4059	−1.23643
$821$	−42.7568	−1.49222	−0.746111	−	0.665822i	$-0.768081\pi$
$821$	−42.7568	−1.49222	−0.746111	+	0.665822i	$0.768081\pi$
$822$	0	0
$823$	15.8022	0.550829	0.275414	−	0.961326i	$-0.411185\pi$
$823$	15.8022	0.550829	0.275414	+	0.961326i	$0.411185\pi$
$824$	52.6787	1.83515
$825$	0	0
$826$	−1.39202	−0.0484346
$827$	23.8246	0.828461	0.414230	−	0.910172i	$-0.364051\pi$
$827$	23.8246	0.828461	0.414230	+	0.910172i	$0.364051\pi$
$828$	0	0
$829$	−8.75791	−0.304175	−0.152087	−	0.988367i	$-0.548600\pi$
$829$	−8.75791	−0.304175	−0.152087	+	0.988367i	$0.548600\pi$
$830$	63.1728	2.19276
$831$	0	0
$832$	−31.8019	−1.10253
$833$	−4.91035	−0.170134
$834$	0	0
$835$	−17.4880	−0.605196
$836$	−31.8587	−1.10186
$837$	0	0
$838$	21.4204	0.739955
$839$	−19.4801	−0.672528	−0.336264	−	0.941768i	$-0.609163\pi$
$839$	−19.4801	−0.672528	−0.336264	+	0.941768i	$0.609163\pi$
$840$	0	0
$841$	−23.3749	−0.806033
$842$	−50.2890	−1.73307
$843$	0	0
$844$	−35.1760	−1.21081
$845$	22.3837	0.770024
$846$	0	0
$847$	2.33866	0.0803573
$848$	41.8790	1.43813
$849$	0	0
$850$	−7.41020	−0.254168
$851$	23.3754	0.801299
$852$	0	0
$853$	52.2795	1.79002	0.895008	−	0.446050i	$-0.147170\pi$
$853$	52.2795	1.79002	0.895008	+	0.446050i	$0.147170\pi$
$854$	−9.59413	−0.328304
$855$	0	0
$856$	49.7172	1.69930
$857$	54.8348	1.87312	0.936560	−	0.350508i	$-0.113991\pi$
$857$	54.8348	1.87312	0.936560	+	0.350508i	$0.113991\pi$
$858$	0	0
$859$	−19.4123	−0.662340	−0.331170	−	0.943571i	$-0.607443\pi$
$859$	−19.4123	−0.662340	−0.331170	+	0.943571i	$0.607443\pi$
$860$	−74.5658	−2.54267
$861$	0	0
$862$	−75.8987	−2.58512
$863$	−2.89484	−0.0985414	−0.0492707	−	0.998785i	$-0.515690\pi$
$863$	−2.89484	−0.0985414	−0.0492707	+	0.998785i	$0.515690\pi$
$864$	0	0
$865$	−17.6674	−0.600711
$866$	88.3969	3.00385
$867$	0	0
$868$	−13.1451	−0.446174
$869$	−12.7136	−0.431280
$870$	0	0
$871$	−36.1783	−1.22586
$872$	10.1078	0.342294
$873$	0	0
$874$	52.9690	1.79170
$875$	−23.1583	−0.782894
$876$	0	0
$877$	−10.4730	−0.353649	−0.176825	−	0.984242i	$-0.556583\pi$
$877$	−10.4730	−0.353649	−0.176825	+	0.984242i	$0.556583\pi$
$878$	45.2096	1.52575
$879$	0	0
$880$	−11.1952	−0.377391
$881$	−31.5232	−1.06204	−0.531022	−	0.847358i	$-0.678192\pi$
$881$	−31.5232	−1.06204	−0.531022	+	0.847358i	$0.678192\pi$
$882$	0	0
$883$	5.25391	0.176808	0.0884040	−	0.996085i	$-0.471823\pi$
$883$	5.25391	0.176808	0.0884040	+	0.996085i	$0.471823\pi$
$884$	61.9184	2.08254
$885$	0	0
$886$	2.60099	0.0873819
$887$	−20.8619	−0.700474	−0.350237	−	0.936661i	$-0.613899\pi$
$887$	−20.8619	−0.700474	−0.350237	+	0.936661i	$0.613899\pi$
$888$	0	0
$889$	−2.20632	−0.0739976
$890$	−85.9883	−2.88234
$891$	0	0
$892$	−108.662	−3.63828
$893$	−1.61708	−0.0541137
$894$	0	0
$895$	17.7607	0.593674
$896$	43.6091	1.45688
$897$	0	0
$898$	72.5249	2.42019
$899$	−3.25333	−0.108505
$900$	0	0
$901$	29.2363	0.974002
$902$	−8.75786	−0.291605
$903$	0	0
$904$	−40.9488	−1.36194
$905$	32.7034	1.08710
$906$	0	0
$907$	22.0009	0.730529	0.365265	−	0.930904i	$-0.380979\pi$
$907$	22.0009	0.730529	0.365265	+	0.930904i	$0.380979\pi$
$908$	−97.0534	−3.22083
$909$	0	0
$910$	−66.2720	−2.19689
$911$	−4.66134	−0.154437	−0.0772185	−	0.997014i	$-0.524604\pi$
$911$	−4.66134	−0.154437	−0.0772185	+	0.997014i	$0.524604\pi$
$912$	0	0
$913$	10.5008	0.347527
$914$	−13.1288	−0.434262
$915$	0	0
$916$	−49.1343	−1.62344
$917$	7.15941	0.236425
$918$	0	0
$919$	−6.91941	−0.228250	−0.114125	−	0.993466i	$-0.536407\pi$
$919$	−6.91941	−0.228250	−0.114125	+	0.993466i	$0.536407\pi$
$920$	34.8159	1.14784
$921$	0	0
$922$	−16.9400	−0.557890
$923$	−5.07455	−0.167031
$924$	0	0
$925$	−7.92560	−0.260592
$926$	62.4892	2.05352
$927$	0	0
$928$	−2.34257	−0.0768988
$929$	1.27838	0.0419422	0.0209711	−	0.999780i	$-0.493324\pi$
$929$	1.27838	0.0419422	0.0209711	+	0.999780i	$0.493324\pi$
$930$	0	0
$931$	−11.9008	−0.390034
$932$	−77.5221	−2.53932
$933$	0	0
$934$	99.0247	3.24019
$935$	−7.81554	−0.255596
$936$	0	0
$937$	−47.7193	−1.55892	−0.779461	−	0.626450i	$-0.784507\pi$
$937$	−47.7193	−1.55892	−0.779461	+	0.626450i	$0.784507\pi$
$938$	44.3547	1.44823
$939$	0	0
$940$	−2.07632	−0.0677220
$941$	8.13978	0.265349	0.132675	−	0.991160i	$-0.457643\pi$
$941$	8.13978	0.265349	0.132675	+	0.991160i	$0.457643\pi$
$942$	0	0
$943$	9.78504	0.318645
$944$	1.10766	0.0360512
$945$	0	0
$946$	−18.4443	−0.599676
$947$	27.9690	0.908871	0.454435	−	0.890780i	$-0.349841\pi$
$947$	27.9690	0.908871	0.454435	+	0.890780i	$0.349841\pi$
$948$	0	0
$949$	−11.1869	−0.363142
$950$	−17.9595	−0.582683
$951$	0	0
$952$	−38.8603	−1.25947
$953$	42.5121	1.37710	0.688551	−	0.725188i	$-0.258247\pi$
$953$	42.5121	1.37710	0.688551	+	0.725188i	$0.258247\pi$
$954$	0	0
$955$	5.60542	0.181387
$956$	67.8805	2.19541
$957$	0	0
$958$	−40.4363	−1.30644
$959$	47.8395	1.54482
$960$	0	0
$961$	−29.1184	−0.939303
$962$	98.5487	3.17734
$963$	0	0
$964$	−66.7460	−2.14975
$965$	34.3468	1.10566
$966$	0	0
$967$	−22.0229	−0.708209	−0.354104	−	0.935206i	$-0.615214\pi$
$967$	−22.0229	−0.708209	−0.354104	+	0.935206i	$0.615214\pi$
$968$	−5.17972	−0.166482
$969$	0	0
$970$	−53.7461	−1.72568
$971$	29.0719	0.932963	0.466482	−	0.884531i	$-0.345521\pi$
$971$	29.0719	0.932963	0.466482	+	0.884531i	$0.345521\pi$
$972$	0	0
$973$	30.1894	0.967828
$974$	−36.9057	−1.18253
$975$	0	0
$976$	7.63422	0.244365
$977$	43.2481	1.38363	0.691815	−	0.722075i	$-0.256811\pi$
$977$	43.2481	1.38363	0.691815	+	0.722075i	$0.256811\pi$
$978$	0	0
$979$	−14.2933	−0.456816
$980$	−15.2805	−0.488118
$981$	0	0
$982$	68.8161	2.19601
$983$	32.2186	1.02761	0.513807	−	0.857906i	$-0.328235\pi$
$983$	32.2186	1.02761	0.513807	+	0.857906i	$0.328235\pi$
$984$	0	0
$985$	−3.20033	−0.101971
$986$	−18.7878	−0.598325
$987$	0	0
$988$	150.067	4.77426
$989$	20.6076	0.655282
$990$	0	0
$991$	38.9476	1.23721	0.618605	−	0.785702i	$-0.287698\pi$
$991$	38.9476	1.23721	0.618605	+	0.785702i	$0.287698\pi$
$992$	1.35486	0.0430169
$993$	0	0
$994$	6.22141	0.197331
$995$	38.6127	1.22411
$996$	0	0
$997$	−36.7131	−1.16272	−0.581358	−	0.813648i	$-0.697479\pi$
$997$	−36.7131	−1.16272	−0.581358	+	0.813648i	$0.697479\pi$
$998$	81.8714	2.59159
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.p.1.1		4
3.2	odd	2		891.2.a.q.1.4		4
9.2	odd	6		99.2.e.e.67.1	yes	8
9.4	even	3		297.2.e.e.100.4		8
9.5	odd	6		99.2.e.e.34.1	✓	8
9.7	even	3		297.2.e.e.199.4		8
11.10	odd	2		9801.2.a.bl.1.4		4
33.32	even	2		9801.2.a.bi.1.1		4
99.32	even	6		1089.2.e.i.727.4		8
99.65	even	6		1089.2.e.i.364.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.1	✓	8	9.5	odd	6
99.2.e.e.67.1	yes	8	9.2	odd	6
297.2.e.e.100.4		8	9.4	even	3
297.2.e.e.199.4		8	9.7	even	3
891.2.a.p.1.1		4	1.1	even	1	trivial
891.2.a.q.1.4		4	3.2	odd	2
1089.2.e.i.364.4		8	99.65	even	6
1089.2.e.i.727.4		8	99.32	even	6
9801.2.a.bi.1.1		4	33.32	even	2
9801.2.a.bl.1.4		4	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)

\(f(q)\)	\(=\)	\(q-2.46934 q^{2} +4.09762 q^{4} +2.43628 q^{5} +2.33866 q^{7} -5.17972 q^{8} -6.01598 q^{10} -1.00000 q^{11} +4.71038 q^{13} -5.77494 q^{14} +4.59522 q^{16} +3.20799 q^{17} +7.77494 q^{19} +9.98292 q^{20} +2.46934 q^{22} -2.75895 q^{23} +0.935443 q^{25} -11.6315 q^{26} +9.58293 q^{28} +2.37172 q^{29} -1.37172 q^{31} -0.987711 q^{32} -7.92159 q^{34} +5.69762 q^{35} -8.47256 q^{37} -19.1989 q^{38} -12.6192 q^{40} -3.54665 q^{41} -7.46934 q^{43} -4.09762 q^{44} +6.81278 q^{46} -0.207987 q^{47} -1.53066 q^{49} -2.30992 q^{50} +19.3013 q^{52} +9.11360 q^{53} -2.43628 q^{55} -12.1136 q^{56} -5.85657 q^{58} +0.241045 q^{59} +1.66134 q^{61} +3.38724 q^{62} -6.75145 q^{64} +11.4758 q^{65} -7.68055 q^{67} +13.1451 q^{68} -14.0693 q^{70} -1.07731 q^{71} -2.37495 q^{73} +20.9216 q^{74} +31.8587 q^{76} -2.33866 q^{77} +12.7136 q^{79} +11.1952 q^{80} +8.75786 q^{82} -10.5008 q^{83} +7.81554 q^{85} +18.4443 q^{86} +5.17972 q^{88} +14.2933 q^{89} +11.0160 q^{91} -11.3051 q^{92} +0.513589 q^{94} +18.9419 q^{95} +8.93388 q^{97} +3.77972 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 11 q^{4} - 4 q^{5} + q^{7} + q^{10} - 4 q^{11} + 7 q^{13} - q^{14} + 17 q^{16} + 5 q^{17} + 9 q^{19} + 10 q^{20} + q^{22} - 14 q^{23} + 14 q^{25} - 22 q^{26} - q^{28} + 6 q^{29} - 2 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 4 * q - q^2 + 11 * q^4 - 4 * q^5 + q^7 + q^10 - 4 * q^11 + 7 * q^13 - q^14 + 17 * q^16 + 5 * q^17 + 9 * q^19 + 10 * q^20 + q^22 - 14 * q^23 + 14 * q^25 - 22 * q^26 - q^28 + 6 * q^29 - 2 * q^31 + 34 * q^32 + 16 * q^34 + 8 * q^35 + 3 * q^37 - 3 * q^38 + 12 * q^40 + 2 * q^41 - 21 * q^43 - 11 * q^44 + 2 * q^46 + 7 * q^47 - 15 * q^49 - 23 * q^50 - 10 * q^52 + 6 * q^53 + 4 * q^55 - 18 * q^56 - 21 * q^58 - 2 * q^59 + 15 * q^61 + 20 * q^62 + 16 * q^64 - 19 * q^65 + 14 * q^67 + 7 * q^68 - 38 * q^70 + 3 * q^71 + 22 * q^73 + 36 * q^74 + 42 * q^76 - q^77 + 11 * q^79 + 34 * q^80 - 17 * q^82 - 18 * q^83 + 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 - 19 * q^94 + 30 * q^95 + 26 * q^97 - 15 * q^98

Introduction

L-functions

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