LMFDB - Embedded newform 891.2.a.l.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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
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			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	\(q+1.53209 q^{2} +0.347296 q^{4} -0.532089 q^{5} -2.87939 q^{7} -2.53209 q^{8} -0.815207 q^{10} -1.00000 q^{11} +0.0641778 q^{13} -4.41147 q^{14} -4.57398 q^{16} +1.22668 q^{17} +0.411474 q^{19} -0.184793 q^{20} -1.53209 q^{22} -4.22668 q^{23} -4.71688 q^{25} +0.0983261 q^{26} -1.00000 q^{28} -8.33275 q^{29} -7.94356 q^{31} -1.94356 q^{32} +1.87939 q^{34} +1.53209 q^{35} -8.94356 q^{37} +0.630415 q^{38} +1.34730 q^{40} +8.92902 q^{41} +11.4388 q^{43} -0.347296 q^{44} -6.47565 q^{46} +3.85710 q^{47} +1.29086 q^{49} -7.22668 q^{50} +0.0222887 q^{52} +0.448311 q^{53} +0.532089 q^{55} +7.29086 q^{56} -12.7665 q^{58} +13.6800 q^{59} -5.12061 q^{61} -12.1702 q^{62} +6.17024 q^{64} -0.0341483 q^{65} +9.84524 q^{67} +0.426022 q^{68} +2.34730 q^{70} +4.49020 q^{71} -8.96585 q^{73} -13.7023 q^{74} +0.142903 q^{76} +2.87939 q^{77} -2.65270 q^{79} +2.43376 q^{80} +13.6800 q^{82} +5.63816 q^{83} -0.652704 q^{85} +17.5253 q^{86} +2.53209 q^{88} +7.97090 q^{89} -0.184793 q^{91} -1.46791 q^{92} +5.90941 q^{94} -0.218941 q^{95} -9.90673 q^{97} +1.97771 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} - 3 q^{17} - 9 q^{19} + 3 q^{20} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 6 q^{29} - 9 q^{31} + 9 q^{32} - 12 q^{37}+ \cdots + 12 q^{98}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 - 6 * q^10 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 - 3 * q^17 - 9 * q^19 + 3 * q^20 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 6 * q^29 - 9 * q^31 + 9 * q^32 - 12 * q^37 + 9 * q^38 + 3 * q^40 - 6 * q^41 + 3 * q^43 + 12 * q^47 - 12 * q^49 - 15 * q^50 - 6 * q^52 + 3 * q^53 - 3 * q^55 + 6 * q^56 - 3 * q^58 + 21 * q^59 - 21 * q^61 - 15 * q^62 - 3 * q^64 - 21 * q^65 + 3 * q^67 + 9 * q^68 + 6 * q^70 + 12 * q^71 - 6 * q^73 - 15 * q^74 + 3 * q^77 - 9 * q^79 - 9 * q^80 + 21 * q^82 - 3 * q^85 + 6 * q^86 + 3 * q^88 - 12 * q^89 + 3 * q^91 - 9 * q^92 - 18 * q^94 - 18 * q^95 - 3 * q^97 + 12 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.53209	1.08335	0.541675	−	0.840588i	$-0.317790\pi$
$2$	1.53209	1.08335	0.541675	+	0.840588i	$0.317790\pi$
$3$	0	0
$3$	0	0
$4$	0.347296	0.173648
$5$	−0.532089	−0.237957	−0.118979	−	0.992897i	$-0.537962\pi$
$5$	−0.532089	−0.237957	−0.118979	+	0.992897i	$0.537962\pi$
$6$	0	0
$7$	−2.87939	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$7$	−2.87939	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$8$	−2.53209	−0.895229
$9$	0	0
$10$	−0.815207	−0.257791
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.0641778	0.0177997	0.00889986	−	0.999960i	$-0.497167\pi$
$13$	0.0641778	0.0177997	0.00889986	+	0.999960i	$0.497167\pi$
$14$	−4.41147	−1.17902
$15$	0	0
$16$	−4.57398	−1.14349
$17$	1.22668	0.297514	0.148757	−	0.988874i	$-0.452473\pi$
$17$	1.22668	0.297514	0.148757	+	0.988874i	$0.452473\pi$
$18$	0	0
$19$	0.411474	0.0943986	0.0471993	−	0.998885i	$-0.484970\pi$
$19$	0.411474	0.0943986	0.0471993	+	0.998885i	$0.484970\pi$
$20$	−0.184793	−0.0413209
$21$	0	0
$22$	−1.53209	−0.326642
$23$	−4.22668	−0.881324	−0.440662	−	0.897673i	$-0.645256\pi$
$23$	−4.22668	−0.881324	−0.440662	+	0.897673i	$0.645256\pi$
$24$	0	0
$25$	−4.71688	−0.943376
$26$	0.0983261	0.0192833
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.33275	−1.54735	−0.773676	−	0.633581i	$-0.781584\pi$
$29$	−8.33275	−1.54735	−0.773676	+	0.633581i	$0.781584\pi$
$30$	0	0
$31$	−7.94356	−1.42671	−0.713353	−	0.700805i	$-0.752824\pi$
$31$	−7.94356	−1.42671	−0.713353	+	0.700805i	$0.752824\pi$
$32$	−1.94356	−0.343577
$33$	0	0
$34$	1.87939	0.322312
$35$	1.53209	0.258970
$36$	0	0
$37$	−8.94356	−1.47031	−0.735156	−	0.677898i	$-0.762891\pi$
$37$	−8.94356	−1.47031	−0.735156	+	0.677898i	$0.762891\pi$
$38$	0.630415	0.102267
$39$	0	0
$40$	1.34730	0.213026
$41$	8.92902	1.39448	0.697239	−	0.716839i	$-0.254412\pi$
$41$	8.92902	1.39448	0.697239	+	0.716839i	$0.254412\pi$
$42$	0	0
$43$	11.4388	1.74440	0.872201	−	0.489147i	$-0.162692\pi$
$43$	11.4388	1.74440	0.872201	+	0.489147i	$0.162692\pi$
$44$	−0.347296	−0.0523569
$45$	0	0
$46$	−6.47565	−0.954783
$47$	3.85710	0.562615	0.281308	−	0.959618i	$-0.409232\pi$
$47$	3.85710	0.562615	0.281308	+	0.959618i	$0.409232\pi$
$48$	0	0
$49$	1.29086	0.184408
$50$	−7.22668	−1.02201
$51$	0	0
$52$	0.0222887	0.00309089
$53$	0.448311	0.0615802	0.0307901	−	0.999526i	$-0.490198\pi$
$53$	0.448311	0.0615802	0.0307901	+	0.999526i	$0.490198\pi$
$54$	0	0
$55$	0.532089	0.0717469
$56$	7.29086	0.974282
$57$	0	0
$58$	−12.7665	−1.67633
$59$	13.6800	1.78099	0.890495	−	0.454993i	$-0.150358\pi$
$59$	13.6800	1.78099	0.890495	+	0.454993i	$0.150358\pi$
$60$	0	0
$61$	−5.12061	−0.655628	−0.327814	−	0.944742i	$-0.606312\pi$
$61$	−5.12061	−0.655628	−0.327814	+	0.944742i	$0.606312\pi$
$62$	−12.1702	−1.54562
$63$	0	0
$64$	6.17024	0.771281
$65$	−0.0341483	−0.00423557
$66$	0	0
$67$	9.84524	1.20279	0.601394	−	0.798953i	$-0.294612\pi$
$67$	9.84524	1.20279	0.601394	+	0.798953i	$0.294612\pi$
$68$	0.426022	0.0516628
$69$	0	0
$70$	2.34730	0.280556
$71$	4.49020	0.532889	0.266444	−	0.963850i	$-0.414151\pi$
$71$	4.49020	0.532889	0.266444	+	0.963850i	$0.414151\pi$
$72$	0	0
$73$	−8.96585	−1.04937	−0.524687	−	0.851295i	$-0.675818\pi$
$73$	−8.96585	−1.04937	−0.524687	+	0.851295i	$0.675818\pi$
$74$	−13.7023	−1.59286
$75$	0	0
$76$	0.142903	0.0163922
$77$	2.87939	0.328136
$78$	0	0
$79$	−2.65270	−0.298452	−0.149226	−	0.988803i	$-0.547678\pi$
$79$	−2.65270	−0.298452	−0.149226	+	0.988803i	$0.547678\pi$
$80$	2.43376	0.272103
$81$	0	0
$82$	13.6800	1.51071
$83$	5.63816	0.618868	0.309434	−	0.950921i	$-0.399860\pi$
$83$	5.63816	0.618868	0.309434	+	0.950921i	$0.399860\pi$
$84$	0	0
$85$	−0.652704	−0.0707957
$86$	17.5253	1.88980
$87$	0	0
$88$	2.53209	0.269922
$89$	7.97090	0.844914	0.422457	−	0.906383i	$-0.361168\pi$
$89$	7.97090	0.844914	0.422457	+	0.906383i	$0.361168\pi$
$90$	0	0
$91$	−0.184793	−0.0193715
$92$	−1.46791	−0.153040
$93$	0	0
$94$	5.90941	0.609510
$95$	−0.218941	−0.0224629
$96$	0	0
$97$	−9.90673	−1.00588	−0.502938	−	0.864323i	$-0.667748\pi$
$97$	−9.90673	−1.00588	−0.502938	+	0.864323i	$0.667748\pi$
$98$	1.97771	0.199779
$99$	0	0
$100$	−1.63816	−0.163816
$101$	8.32770	0.828637	0.414318	−	0.910132i	$-0.364020\pi$
$101$	8.32770	0.828637	0.414318	+	0.910132i	$0.364020\pi$
$102$	0	0
$103$	−13.2986	−1.31035	−0.655175	−	0.755477i	$-0.727405\pi$
$103$	−13.2986	−1.31035	−0.655175	+	0.755477i	$0.727405\pi$
$104$	−0.162504	−0.0159348
$105$	0	0
$106$	0.686852	0.0667130
$107$	−4.61587	−0.446233	−0.223116	−	0.974792i	$-0.571623\pi$
$107$	−4.61587	−0.446233	−0.223116	+	0.974792i	$0.571623\pi$
$108$	0	0
$109$	−18.8280	−1.80340	−0.901698	−	0.432367i	$-0.857678\pi$
$109$	−18.8280	−1.80340	−0.901698	+	0.432367i	$0.857678\pi$
$110$	0.815207	0.0777270
$111$	0	0
$112$	13.1702	1.24447
$113$	−2.69459	−0.253486	−0.126743	−	0.991936i	$-0.540452\pi$
$113$	−2.69459	−0.253486	−0.126743	+	0.991936i	$0.540452\pi$
$114$	0	0
$115$	2.24897	0.209718
$116$	−2.89393	−0.268695
$117$	0	0
$118$	20.9590	1.92944
$119$	−3.53209	−0.323786
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.84524	−0.710274
$123$	0	0
$124$	−2.75877	−0.247745
$125$	5.17024	0.462441
$126$	0	0
$127$	−5.70233	−0.506000	−0.253000	−	0.967466i	$-0.581417\pi$
$127$	−5.70233	−0.506000	−0.253000	+	0.967466i	$0.581417\pi$
$128$	13.3405	1.17914
$129$	0	0
$130$	−0.0523182	−0.00458861
$131$	−9.84255	−0.859947	−0.429974	−	0.902841i	$-0.641477\pi$
$131$	−9.84255	−0.859947	−0.429974	+	0.902841i	$0.641477\pi$
$132$	0	0
$133$	−1.18479	−0.102735
$134$	15.0838	1.30304
$135$	0	0
$136$	−3.10607	−0.266343
$137$	9.58853	0.819203	0.409602	−	0.912265i	$-0.365668\pi$
$137$	9.58853	0.819203	0.409602	+	0.912265i	$0.365668\pi$
$138$	0	0
$139$	−11.6040	−0.984239	−0.492120	−	0.870528i	$-0.663778\pi$
$139$	−11.6040	−0.984239	−0.492120	+	0.870528i	$0.663778\pi$
$140$	0.532089	0.0449697
$141$	0	0
$142$	6.87939	0.577305
$143$	−0.0641778	−0.00536681
$144$	0	0
$145$	4.43376	0.368204
$146$	−13.7365	−1.13684
$147$	0	0
$148$	−3.10607	−0.255317
$149$	−20.5476	−1.68332	−0.841661	−	0.540006i	$-0.818422\pi$
$149$	−20.5476	−1.68332	−0.841661	+	0.540006i	$0.818422\pi$
$150$	0	0
$151$	16.7050	1.35944	0.679718	−	0.733474i	$-0.262102\pi$
$151$	16.7050	1.35944	0.679718	+	0.733474i	$0.262102\pi$
$152$	−1.04189	−0.0845084
$153$	0	0
$154$	4.41147	0.355487
$155$	4.22668	0.339495
$156$	0	0
$157$	11.1557	0.890322	0.445161	−	0.895451i	$-0.353147\pi$
$157$	11.1557	0.890322	0.445161	+	0.895451i	$0.353147\pi$
$158$	−4.06418	−0.323329
$159$	0	0
$160$	1.03415	0.0817566
$161$	12.1702	0.959150
$162$	0	0
$163$	17.9145	1.40317	0.701585	−	0.712586i	$-0.252476\pi$
$163$	17.9145	1.40317	0.701585	+	0.712586i	$0.252476\pi$
$164$	3.10101	0.242149
$165$	0	0
$166$	8.63816	0.670451
$167$	−3.15570	−0.244195	−0.122098	−	0.992518i	$-0.538962\pi$
$167$	−3.15570	−0.244195	−0.122098	+	0.992518i	$0.538962\pi$
$168$	0	0
$169$	−12.9959	−0.999683
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	3.97266	0.302912
$173$	−8.10101	−0.615909	−0.307954	−	0.951401i	$-0.599644\pi$
$173$	−8.10101	−0.615909	−0.307954	+	0.951401i	$0.599644\pi$
$174$	0	0
$175$	13.5817	1.02668
$176$	4.57398	0.344777
$177$	0	0
$178$	12.2121	0.915338
$179$	−11.1506	−0.833438	−0.416719	−	0.909035i	$-0.636820\pi$
$179$	−11.1506	−0.833438	−0.416719	+	0.909035i	$0.636820\pi$
$180$	0	0
$181$	−1.50206	−0.111647	−0.0558236	−	0.998441i	$-0.517778\pi$
$181$	−1.50206	−0.111647	−0.0558236	+	0.998441i	$0.517778\pi$
$182$	−0.283119	−0.0209861
$183$	0	0
$184$	10.7023	0.788986
$185$	4.75877	0.349872
$186$	0	0
$187$	−1.22668	−0.0897038
$188$	1.33956	0.0976971
$189$	0	0
$190$	−0.335437	−0.0243351
$191$	12.6382	0.914465	0.457232	−	0.889347i	$-0.348841\pi$
$191$	12.6382	0.914465	0.457232	+	0.889347i	$0.348841\pi$
$192$	0	0
$193$	1.63041	0.117360	0.0586799	−	0.998277i	$-0.481311\pi$
$193$	1.63041	0.117360	0.0586799	+	0.998277i	$0.481311\pi$
$194$	−15.1780	−1.08972
$195$	0	0
$196$	0.448311	0.0320222
$197$	−7.44831	−0.530670	−0.265335	−	0.964156i	$-0.585483\pi$
$197$	−7.44831	−0.530670	−0.265335	+	0.964156i	$0.585483\pi$
$198$	0	0
$199$	12.8229	0.908995	0.454497	−	0.890748i	$-0.349819\pi$
$199$	12.8229	0.908995	0.454497	+	0.890748i	$0.349819\pi$
$200$	11.9436	0.844537
$201$	0	0
$202$	12.7588	0.897704
$203$	23.9932	1.68399
$204$	0	0
$205$	−4.75103	−0.331826
$206$	−20.3746	−1.41957
$207$	0	0
$208$	−0.293548	−0.0203539
$209$	−0.411474	−0.0284623
$210$	0	0
$211$	−2.82026	−0.194155	−0.0970773	−	0.995277i	$-0.530949\pi$
$211$	−2.82026	−0.194155	−0.0970773	+	0.995277i	$0.530949\pi$
$212$	0.155697	0.0106933
$213$	0	0
$214$	−7.07192	−0.483426
$215$	−6.08647	−0.415094
$216$	0	0
$217$	22.8726	1.55269
$218$	−28.8462	−1.95371
$219$	0	0
$220$	0.184793	0.0124587
$221$	0.0787257	0.00529566
$222$	0	0
$223$	−10.3824	−0.695255	−0.347628	−	0.937633i	$-0.613013\pi$
$223$	−10.3824	−0.695255	−0.347628	+	0.937633i	$0.613013\pi$
$224$	5.59627	0.373916
$225$	0	0
$226$	−4.12836	−0.274614
$227$	−29.6682	−1.96915	−0.984573	−	0.174973i	$-0.944016\pi$
$227$	−29.6682	−1.96915	−0.984573	+	0.174973i	$0.944016\pi$
$228$	0	0
$229$	−9.77332	−0.645839	−0.322920	−	0.946426i	$-0.604664\pi$
$229$	−9.77332	−0.645839	−0.322920	+	0.946426i	$0.604664\pi$
$230$	3.44562	0.227198
$231$	0	0
$232$	21.0993	1.38523
$233$	−3.57398	−0.234139	−0.117070	−	0.993124i	$-0.537350\pi$
$233$	−3.57398	−0.234139	−0.117070	+	0.993124i	$0.537350\pi$
$234$	0	0
$235$	−2.05232	−0.133878
$236$	4.75103	0.309266
$237$	0	0
$238$	−5.41147	−0.350774
$239$	−14.9240	−0.965351	−0.482675	−	0.875799i	$-0.660335\pi$
$239$	−14.9240	−0.965351	−0.482675	+	0.875799i	$0.660335\pi$
$240$	0	0
$241$	−21.2344	−1.36783	−0.683915	−	0.729562i	$-0.739724\pi$
$241$	−21.2344	−1.36783	−0.683915	+	0.729562i	$0.739724\pi$
$242$	1.53209	0.0984864
$243$	0	0
$244$	−1.77837	−0.113849
$245$	−0.686852	−0.0438814
$246$	0	0
$247$	0.0264075	0.00168027
$248$	20.1138	1.27723
$249$	0	0
$250$	7.92127	0.500985
$251$	−20.6955	−1.30629	−0.653145	−	0.757233i	$-0.726551\pi$
$251$	−20.6955	−1.30629	−0.653145	+	0.757233i	$0.726551\pi$
$252$	0	0
$253$	4.22668	0.265729
$254$	−8.73648	−0.548176
$255$	0	0
$256$	8.09833	0.506145
$257$	−21.8503	−1.36298	−0.681492	−	0.731826i	$-0.738668\pi$
$257$	−21.8503	−1.36298	−0.681492	+	0.731826i	$0.738668\pi$
$258$	0	0
$259$	25.7520	1.60015
$260$	−0.0118596	−0.000735499 0
$261$	0	0
$262$	−15.0797	−0.931624
$263$	14.7023	0.906585	0.453292	−	0.891362i	$-0.350249\pi$
$263$	14.7023	0.906585	0.453292	+	0.891362i	$0.350249\pi$
$264$	0	0
$265$	−0.238541	−0.0146535
$266$	−1.81521	−0.111298
$267$	0	0
$268$	3.41921	0.208862
$269$	17.8298	1.08710	0.543550	−	0.839377i	$-0.317080\pi$
$269$	17.8298	1.08710	0.543550	+	0.839377i	$0.317080\pi$
$270$	0	0
$271$	2.15570	0.130949	0.0654746	−	0.997854i	$-0.479144\pi$
$271$	2.15570	0.130949	0.0654746	+	0.997854i	$0.479144\pi$
$272$	−5.61081	−0.340206
$273$	0	0
$274$	14.6905	0.887484
$275$	4.71688	0.284439
$276$	0	0
$277$	21.9368	1.31805	0.659026	−	0.752120i	$-0.270969\pi$
$277$	21.9368	1.31805	0.659026	+	0.752120i	$0.270969\pi$
$278$	−17.7784	−1.06628
$279$	0	0
$280$	−3.87939	−0.231838
$281$	−21.9659	−1.31037	−0.655186	−	0.755467i	$-0.727410\pi$
$281$	−21.9659	−1.31037	−0.655186	+	0.755467i	$0.727410\pi$
$282$	0	0
$283$	−11.2121	−0.666492	−0.333246	−	0.942840i	$-0.608144\pi$
$283$	−11.2121	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$284$	1.55943	0.0925351
$285$	0	0
$286$	−0.0983261	−0.00581414
$287$	−25.7101	−1.51762
$288$	0	0
$289$	−15.4953	−0.911485
$290$	6.79292	0.398894
$291$	0	0
$292$	−3.11381	−0.182222
$293$	2.96316	0.173110	0.0865549	−	0.996247i	$-0.472414\pi$
$293$	2.96316	0.173110	0.0865549	+	0.996247i	$0.472414\pi$
$294$	0	0
$295$	−7.27900	−0.423800
$296$	22.6459	1.31627
$297$	0	0
$298$	−31.4807	−1.82363
$299$	−0.271259	−0.0156873
$300$	0	0
$301$	−32.9368	−1.89844
$302$	25.5936	1.47274
$303$	0	0
$304$	−1.88207	−0.107944
$305$	2.72462	0.156011
$306$	0	0
$307$	1.19665	0.0682965	0.0341483	−	0.999417i	$-0.489128\pi$
$307$	1.19665	0.0682965	0.0341483	+	0.999417i	$0.489128\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	6.47565	0.367792
$311$	17.2003	0.975338	0.487669	−	0.873029i	$-0.337847\pi$
$311$	17.2003	0.975338	0.487669	+	0.873029i	$0.337847\pi$
$312$	0	0
$313$	15.9513	0.901621	0.450810	−	0.892620i	$-0.351135\pi$
$313$	15.9513	0.901621	0.450810	+	0.892620i	$0.351135\pi$
$314$	17.0915	0.964530
$315$	0	0
$316$	−0.921274	−0.0518257
$317$	29.8726	1.67781	0.838906	−	0.544277i	$-0.183196\pi$
$317$	29.8726	1.67781	0.838906	+	0.544277i	$0.183196\pi$
$318$	0	0
$319$	8.33275	0.466544
$320$	−3.28312	−0.183532
$321$	0	0
$322$	18.6459	1.03910
$323$	0.504748	0.0280849
$324$	0	0
$325$	−0.302719	−0.0167918
$326$	27.4466	1.52012
$327$	0	0
$328$	−22.6091	−1.24838
$329$	−11.1061	−0.612297
$330$	0	0
$331$	12.1506	0.667860	0.333930	−	0.942598i	$-0.391625\pi$
$331$	12.1506	0.667860	0.333930	+	0.942598i	$0.391625\pi$
$332$	1.95811	0.107465
$333$	0	0
$334$	−4.83481	−0.264549
$335$	−5.23854	−0.286212
$336$	0	0
$337$	2.36184	0.128658	0.0643289	−	0.997929i	$-0.479509\pi$
$337$	2.36184	0.128658	0.0643289	+	0.997929i	$0.479509\pi$
$338$	−19.9108	−1.08301
$339$	0	0
$340$	−0.226682	−0.0122935
$341$	7.94356	0.430168
$342$	0	0
$343$	16.4388	0.887613
$344$	−28.9641	−1.56164
$345$	0	0
$346$	−12.4115	−0.667245
$347$	−2.19160	−0.117651	−0.0588256	−	0.998268i	$-0.518736\pi$
$347$	−2.19160	−0.117651	−0.0588256	+	0.998268i	$0.518736\pi$
$348$	0	0
$349$	−17.3892	−0.930822	−0.465411	−	0.885095i	$-0.654093\pi$
$349$	−17.3892	−0.930822	−0.465411	+	0.885095i	$0.654093\pi$
$350$	20.8084	1.11226
$351$	0	0
$352$	1.94356	0.103592
$353$	−16.0051	−0.851863	−0.425931	−	0.904755i	$-0.640054\pi$
$353$	−16.0051	−0.851863	−0.425931	+	0.904755i	$0.640054\pi$
$354$	0	0
$355$	−2.38919	−0.126805
$356$	2.76827	0.146718
$357$	0	0
$358$	−17.0838	−0.902906
$359$	−8.38413	−0.442498	−0.221249	−	0.975217i	$-0.571013\pi$
$359$	−8.38413	−0.442498	−0.221249	+	0.975217i	$0.571013\pi$
$360$	0	0
$361$	−18.8307	−0.991089
$362$	−2.30129	−0.120953
$363$	0	0
$364$	−0.0641778	−0.00336383
$365$	4.77063	0.249706
$366$	0	0
$367$	−24.3037	−1.26864	−0.634320	−	0.773070i	$-0.718720\pi$
$367$	−24.3037	−1.26864	−0.634320	+	0.773070i	$0.718720\pi$
$368$	19.3327	1.00779
$369$	0	0
$370$	7.29086	0.379034
$371$	−1.29086	−0.0670181
$372$	0	0
$373$	5.11112	0.264644	0.132322	−	0.991207i	$-0.457757\pi$
$373$	5.11112	0.264644	0.132322	+	0.991207i	$0.457757\pi$
$374$	−1.87939	−0.0971807
$375$	0	0
$376$	−9.76651	−0.503669
$377$	−0.534777	−0.0275424
$378$	0	0
$379$	−10.5253	−0.540647	−0.270324	−	0.962769i	$-0.587131\pi$
$379$	−10.5253	−0.540647	−0.270324	+	0.962769i	$0.587131\pi$
$380$	−0.0760373	−0.00390063
$381$	0	0
$382$	19.3628	0.990686
$383$	2.80241	0.143197	0.0715983	−	0.997434i	$-0.477190\pi$
$383$	2.80241	0.143197	0.0715983	+	0.997434i	$0.477190\pi$
$384$	0	0
$385$	−1.53209	−0.0780825
$386$	2.49794	0.127142
$387$	0	0
$388$	−3.44057	−0.174668
$389$	0.829755	0.0420703	0.0210351	−	0.999779i	$-0.493304\pi$
$389$	0.829755	0.0420703	0.0210351	+	0.999779i	$0.493304\pi$
$390$	0	0
$391$	−5.18479	−0.262206
$392$	−3.26857	−0.165088
$393$	0	0
$394$	−11.4115	−0.574902
$395$	1.41147	0.0710190
$396$	0	0
$397$	13.0232	0.653617	0.326808	−	0.945091i	$-0.394027\pi$
$397$	13.0232	0.653617	0.326808	+	0.945091i	$0.394027\pi$
$398$	19.6459	0.984760
$399$	0	0
$400$	21.5749	1.07875
$401$	24.0642	1.20171	0.600854	−	0.799359i	$-0.294827\pi$
$401$	24.0642	1.20171	0.600854	+	0.799359i	$0.294827\pi$
$402$	0	0
$403$	−0.509800	−0.0253950
$404$	2.89218	0.143891
$405$	0	0
$406$	36.7597	1.82435
$407$	8.94356	0.443316
$408$	0	0
$409$	3.01279	0.148973	0.0744865	−	0.997222i	$-0.476268\pi$
$409$	3.01279	0.148973	0.0744865	+	0.997222i	$0.476268\pi$
$410$	−7.27900	−0.359484
$411$	0	0
$412$	−4.61856	−0.227540
$413$	−39.3901	−1.93826
$414$	0	0
$415$	−3.00000	−0.147264
$416$	−0.124734	−0.00611557
$417$	0	0
$418$	−0.630415	−0.0308346
$419$	−3.87164	−0.189142	−0.0945711	−	0.995518i	$-0.530148\pi$
$419$	−3.87164	−0.189142	−0.0945711	+	0.995518i	$0.530148\pi$
$420$	0	0
$421$	−5.48515	−0.267330	−0.133665	−	0.991027i	$-0.542675\pi$
$421$	−5.48515	−0.267330	−0.133665	+	0.991027i	$0.542675\pi$
$422$	−4.32089	−0.210338
$423$	0	0
$424$	−1.13516	−0.0551284
$425$	−5.78611	−0.280668
$426$	0	0
$427$	14.7442	0.713523
$428$	−1.60307	−0.0774875
$429$	0	0
$430$	−9.32501	−0.449692
$431$	27.1361	1.30710	0.653550	−	0.756883i	$-0.273279\pi$
$431$	27.1361	1.30710	0.653550	+	0.756883i	$0.273279\pi$
$432$	0	0
$433$	14.4361	0.693756	0.346878	−	0.937910i	$-0.387242\pi$
$433$	14.4361	0.693756	0.346878	+	0.937910i	$0.387242\pi$
$434$	35.0428	1.68211
$435$	0	0
$436$	−6.53890	−0.313156
$437$	−1.73917	−0.0831958
$438$	0	0
$439$	−30.2276	−1.44269	−0.721343	−	0.692578i	$-0.756475\pi$
$439$	−30.2276	−1.44269	−0.721343	+	0.692578i	$0.756475\pi$
$440$	−1.34730	−0.0642298
$441$	0	0
$442$	0.120615	0.00573706
$443$	13.8999	0.660405	0.330203	−	0.943910i	$-0.392883\pi$
$443$	13.8999	0.660405	0.330203	+	0.943910i	$0.392883\pi$
$444$	0	0
$445$	−4.24123	−0.201054
$446$	−15.9067	−0.753205
$447$	0	0
$448$	−17.7665	−0.839389
$449$	−6.46017	−0.304874	−0.152437	−	0.988313i	$-0.548712\pi$
$449$	−6.46017	−0.304874	−0.152437	+	0.988313i	$0.548712\pi$
$450$	0	0
$451$	−8.92902	−0.420451
$452$	−0.935822	−0.0440174
$453$	0	0
$454$	−45.4543	−2.13328
$455$	0.0983261	0.00460960
$456$	0	0
$457$	−5.91716	−0.276793	−0.138396	−	0.990377i	$-0.544195\pi$
$457$	−5.91716	−0.276793	−0.138396	+	0.990377i	$0.544195\pi$
$458$	−14.9736	−0.699670
$459$	0	0
$460$	0.781059	0.0364171
$461$	17.3455	0.807862	0.403931	−	0.914789i	$-0.367644\pi$
$461$	17.3455	0.807862	0.403931	+	0.914789i	$0.367644\pi$
$462$	0	0
$463$	16.5868	0.770853	0.385426	−	0.922739i	$-0.374054\pi$
$463$	16.5868	0.770853	0.385426	+	0.922739i	$0.374054\pi$
$464$	38.1138	1.76939
$465$	0	0
$466$	−5.47565	−0.253655
$467$	14.6946	0.679985	0.339992	−	0.940428i	$-0.389576\pi$
$467$	14.6946	0.679985	0.339992	+	0.940428i	$0.389576\pi$
$468$	0	0
$469$	−28.3482	−1.30900
$470$	−3.14433	−0.145037
$471$	0	0
$472$	−34.6391	−1.59439
$473$	−11.4388	−0.525957
$474$	0	0
$475$	−1.94087	−0.0890534
$476$	−1.22668	−0.0562249
$477$	0	0
$478$	−22.8648	−1.04581
$479$	2.08915	0.0954559	0.0477280	−	0.998860i	$-0.484802\pi$
$479$	2.08915	0.0954559	0.0477280	+	0.998860i	$0.484802\pi$
$480$	0	0
$481$	−0.573978	−0.0261711
$482$	−32.5330	−1.48184
$483$	0	0
$484$	0.347296	0.0157862
$485$	5.27126	0.239356
$486$	0	0
$487$	7.74186	0.350817	0.175409	−	0.984496i	$-0.443875\pi$
$487$	7.74186	0.350817	0.175409	+	0.984496i	$0.443875\pi$
$488$	12.9659	0.586937
$489$	0	0
$490$	−1.05232	−0.0475389
$491$	6.08471	0.274599	0.137300	−	0.990530i	$-0.456158\pi$
$491$	6.08471	0.274599	0.137300	+	0.990530i	$0.456158\pi$
$492$	0	0
$493$	−10.2216	−0.460359
$494$	0.0404586	0.00182032
$495$	0	0
$496$	36.3337	1.63143
$497$	−12.9290	−0.579946
$498$	0	0
$499$	1.77063	0.0792643	0.0396321	−	0.999214i	$-0.487381\pi$
$499$	1.77063	0.0792643	0.0396321	+	0.999214i	$0.487381\pi$
$500$	1.79561	0.0803020
$501$	0	0
$502$	−31.7074	−1.41517
$503$	2.96822	0.132346	0.0661731	−	0.997808i	$-0.478921\pi$
$503$	2.96822	0.132346	0.0661731	+	0.997808i	$0.478921\pi$
$504$	0	0
$505$	−4.43107	−0.197180
$506$	6.47565	0.287878
$507$	0	0
$508$	−1.98040	−0.0878660
$509$	−37.4201	−1.65862	−0.829309	−	0.558790i	$-0.811266\pi$
$509$	−37.4201	−1.65862	−0.829309	+	0.558790i	$0.811266\pi$
$510$	0	0
$511$	25.8161	1.14204
$512$	−14.2736	−0.630811
$513$	0	0
$514$	−33.4766	−1.47659
$515$	7.07604	0.311807
$516$	0	0
$517$	−3.85710	−0.169635
$518$	39.4543	1.73352
$519$	0	0
$520$	0.0864665	0.00379181
$521$	32.1739	1.40956	0.704781	−	0.709425i	$-0.251045\pi$
$521$	32.1739	1.40956	0.704781	+	0.709425i	$0.251045\pi$
$522$	0	0
$523$	−20.3773	−0.891038	−0.445519	−	0.895272i	$-0.646981\pi$
$523$	−20.3773	−0.891038	−0.445519	+	0.895272i	$0.646981\pi$
$524$	−3.41828	−0.149328
$525$	0	0
$526$	22.5253	0.982149
$527$	−9.74422	−0.424465
$528$	0	0
$529$	−5.13516	−0.223268
$530$	−0.365466	−0.0158748
$531$	0	0
$532$	−0.411474	−0.0178397
$533$	0.573044	0.0248213
$534$	0	0
$535$	2.45605	0.106184
$536$	−24.9290	−1.07677
$537$	0	0
$538$	27.3168	1.17771
$539$	−1.29086	−0.0556012
$540$	0	0
$541$	−15.0591	−0.647442	−0.323721	−	0.946153i	$-0.604934\pi$
$541$	−15.0591	−0.647442	−0.323721	+	0.946153i	$0.604934\pi$
$542$	3.30272	0.141864
$543$	0	0
$544$	−2.38413	−0.102219
$545$	10.0182	0.429131
$546$	0	0
$547$	24.8476	1.06241	0.531203	−	0.847244i	$-0.321740\pi$
$547$	24.8476	1.06241	0.531203	+	0.847244i	$0.321740\pi$
$548$	3.33006	0.142253
$549$	0	0
$550$	7.22668	0.308147
$551$	−3.42871	−0.146068
$552$	0	0
$553$	7.63816	0.324807
$554$	33.6091	1.42791
$555$	0	0
$556$	−4.03003	−0.170911
$557$	17.3746	0.736187	0.368094	−	0.929789i	$-0.380011\pi$
$557$	17.3746	0.736187	0.368094	+	0.929789i	$0.380011\pi$
$558$	0	0
$559$	0.734118	0.0310499
$560$	−7.00774	−0.296131
$561$	0	0
$562$	−33.6536	−1.41959
$563$	20.6168	0.868895	0.434447	−	0.900697i	$-0.356944\pi$
$563$	20.6168	0.868895	0.434447	+	0.900697i	$0.356944\pi$
$564$	0	0
$565$	1.43376	0.0603189
$566$	−17.1780	−0.722045
$567$	0	0
$568$	−11.3696	−0.477057
$569$	−8.79385	−0.368657	−0.184329	−	0.982865i	$-0.559011\pi$
$569$	−8.79385	−0.368657	−0.184329	+	0.982865i	$0.559011\pi$
$570$	0	0
$571$	−22.1685	−0.927722	−0.463861	−	0.885908i	$-0.653536\pi$
$571$	−22.1685	−0.927722	−0.463861	+	0.885908i	$0.653536\pi$
$572$	−0.0222887	−0.000931938 0
$573$	0	0
$574$	−39.3901	−1.64411
$575$	19.9368	0.831420
$576$	0	0
$577$	3.41147	0.142022	0.0710108	−	0.997476i	$-0.477378\pi$
$577$	3.41147	0.142022	0.0710108	+	0.997476i	$0.477378\pi$
$578$	−23.7401	−0.987458
$579$	0	0
$580$	1.53983	0.0639379
$581$	−16.2344	−0.673517
$582$	0	0
$583$	−0.448311	−0.0185671
$584$	22.7023	0.939429
$585$	0	0
$586$	4.53983	0.187539
$587$	−30.6441	−1.26482	−0.632410	−	0.774634i	$-0.717934\pi$
$587$	−30.6441	−1.26482	−0.632410	+	0.774634i	$0.717934\pi$
$588$	0	0
$589$	−3.26857	−0.134679
$590$	−11.1521	−0.459124
$591$	0	0
$592$	40.9077	1.68129
$593$	−26.5904	−1.09194	−0.545968	−	0.837806i	$-0.683838\pi$
$593$	−26.5904	−1.09194	−0.545968	+	0.837806i	$0.683838\pi$
$594$	0	0
$595$	1.87939	0.0770473
$596$	−7.13610	−0.292306
$597$	0	0
$598$	−0.415593	−0.0169949
$599$	−27.8999	−1.13996	−0.569980	−	0.821659i	$-0.693049\pi$
$599$	−27.8999	−1.13996	−0.569980	+	0.821659i	$0.693049\pi$
$600$	0	0
$601$	39.7820	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$601$	39.7820	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$602$	−50.4620	−2.05668
$603$	0	0
$604$	5.80159	0.236063
$605$	−0.532089	−0.0216325
$606$	0	0
$607$	−20.6955	−0.840006	−0.420003	−	0.907523i	$-0.637971\pi$
$607$	−20.6955	−0.840006	−0.420003	+	0.907523i	$0.637971\pi$
$608$	−0.799726	−0.0324332
$609$	0	0
$610$	4.17436	0.169015
$611$	0.247540	0.0100144
$612$	0	0
$613$	−8.05232	−0.325230	−0.162615	−	0.986690i	$-0.551993\pi$
$613$	−8.05232	−0.325230	−0.162615	+	0.986690i	$0.551993\pi$
$614$	1.83338	0.0739891
$615$	0	0
$616$	−7.29086	−0.293757
$617$	−0.591214	−0.0238014	−0.0119007	−	0.999929i	$-0.503788\pi$
$617$	−0.591214	−0.0238014	−0.0119007	+	0.999929i	$0.503788\pi$
$618$	0	0
$619$	11.3378	0.455705	0.227852	−	0.973696i	$-0.426830\pi$
$619$	11.3378	0.455705	0.227852	+	0.973696i	$0.426830\pi$
$620$	1.46791	0.0589527
$621$	0	0
$622$	26.3523	1.05663
$623$	−22.9513	−0.919525
$624$	0	0
$625$	20.8334	0.833335
$626$	24.4388	0.976772
$627$	0	0
$628$	3.87433	0.154603
$629$	−10.9709	−0.437439
$630$	0	0
$631$	−16.3259	−0.649925	−0.324963	−	0.945727i	$-0.605352\pi$
$631$	−16.3259	−0.649925	−0.324963	+	0.945727i	$0.605352\pi$
$632$	6.71688	0.267183
$633$	0	0
$634$	45.7674	1.81766
$635$	3.03415	0.120407
$636$	0	0
$637$	0.0828445	0.00328242
$638$	12.7665	0.505431
$639$	0	0
$640$	−7.09833	−0.280586
$641$	−1.58347	−0.0625434	−0.0312717	−	0.999511i	$-0.509956\pi$
$641$	−1.58347	−0.0625434	−0.0312717	+	0.999511i	$0.509956\pi$
$642$	0	0
$643$	39.2253	1.54689	0.773446	−	0.633862i	$-0.218531\pi$
$643$	39.2253	1.54689	0.773446	+	0.633862i	$0.218531\pi$
$644$	4.22668	0.166555
$645$	0	0
$646$	0.773318	0.0304258
$647$	44.3729	1.74448	0.872239	−	0.489080i	$-0.162667\pi$
$647$	44.3729	1.74448	0.872239	+	0.489080i	$0.162667\pi$
$648$	0	0
$649$	−13.6800	−0.536989
$650$	−0.463792	−0.0181914
$651$	0	0
$652$	6.22163	0.243658
$653$	18.3969	0.719927	0.359964	−	0.932966i	$-0.382789\pi$
$653$	18.3969	0.719927	0.359964	+	0.932966i	$0.382789\pi$
$654$	0	0
$655$	5.23711	0.204631
$656$	−40.8411	−1.59458
$657$	0	0
$658$	−17.0155	−0.663333
$659$	−42.6418	−1.66109	−0.830544	−	0.556953i	$-0.811970\pi$
$659$	−42.6418	−1.66109	−0.830544	+	0.556953i	$0.811970\pi$
$660$	0	0
$661$	−14.3678	−0.558844	−0.279422	−	0.960168i	$-0.590143\pi$
$661$	−14.3678	−0.558844	−0.279422	+	0.960168i	$0.590143\pi$
$662$	18.6159	0.723526
$663$	0	0
$664$	−14.2763	−0.554028
$665$	0.630415	0.0244464
$666$	0	0
$667$	35.2199	1.36372
$668$	−1.09596	−0.0424040
$669$	0	0
$670$	−8.02591	−0.310068
$671$	5.12061	0.197679
$672$	0	0
$673$	−2.13753	−0.0823956	−0.0411978	−	0.999151i	$-0.513117\pi$
$673$	−2.13753	−0.0823956	−0.0411978	+	0.999151i	$0.513117\pi$
$674$	3.61856	0.139382
$675$	0	0
$676$	−4.51342	−0.173593
$677$	−34.2449	−1.31614	−0.658068	−	0.752958i	$-0.728626\pi$
$677$	−34.2449	−1.31614	−0.658068	+	0.752958i	$0.728626\pi$
$678$	0	0
$679$	28.5253	1.09470
$680$	1.65270	0.0633783
$681$	0	0
$682$	12.1702	0.466023
$683$	−34.0898	−1.30441	−0.652204	−	0.758043i	$-0.726156\pi$
$683$	−34.0898	−1.30441	−0.652204	+	0.758043i	$0.726156\pi$
$684$	0	0
$685$	−5.10195	−0.194935
$686$	25.1857	0.961595
$687$	0	0
$688$	−52.3209	−1.99472
$689$	0.0287716	0.00109611
$690$	0	0
$691$	31.4516	1.19648	0.598238	−	0.801319i	$-0.295868\pi$
$691$	31.4516	1.19648	0.598238	+	0.801319i	$0.295868\pi$
$692$	−2.81345	−0.106951
$693$	0	0
$694$	−3.35773	−0.127458
$695$	6.17436	0.234207
$696$	0	0
$697$	10.9531	0.414877
$698$	−26.6418	−1.00841
$699$	0	0
$700$	4.71688	0.178281
$701$	2.15476	0.0813843	0.0406921	−	0.999172i	$-0.487044\pi$
$701$	2.15476	0.0813843	0.0406921	+	0.999172i	$0.487044\pi$
$702$	0	0
$703$	−3.68004	−0.138796
$704$	−6.17024	−0.232550
$705$	0	0
$706$	−24.5212	−0.922866
$707$	−23.9786	−0.901810
$708$	0	0
$709$	−37.5371	−1.40974	−0.704868	−	0.709338i	$-0.748994\pi$
$709$	−37.5371	−1.40974	−0.704868	+	0.709338i	$0.748994\pi$
$710$	−3.66044	−0.137374
$711$	0	0
$712$	−20.1830	−0.756391
$713$	33.5749	1.25739
$714$	0	0
$715$	0.0341483	0.00127707
$716$	−3.87258	−0.144725
$717$	0	0
$718$	−12.8452	−0.479380
$719$	−50.2063	−1.87238	−0.936189	−	0.351498i	$-0.885673\pi$
$719$	−50.2063	−1.87238	−0.936189	+	0.351498i	$0.885673\pi$
$720$	0	0
$721$	38.2918	1.42606
$722$	−28.8503	−1.07370
$723$	0	0
$724$	−0.521660	−0.0193873
$725$	39.3046	1.45974
$726$	0	0
$727$	−9.80933	−0.363808	−0.181904	−	0.983316i	$-0.558226\pi$
$727$	−9.80933	−0.363808	−0.181904	+	0.983316i	$0.558226\pi$
$728$	0.467911	0.0173419
$729$	0	0
$730$	7.30903	0.270519
$731$	14.0318	0.518984
$732$	0	0
$733$	−14.2371	−0.525860	−0.262930	−	0.964815i	$-0.584689\pi$
$733$	−14.2371	−0.525860	−0.262930	+	0.964815i	$0.584689\pi$
$734$	−37.2354	−1.37438
$735$	0	0
$736$	8.21482	0.302802
$737$	−9.84524	−0.362654
$738$	0	0
$739$	26.1557	0.962153	0.481076	−	0.876679i	$-0.340246\pi$
$739$	26.1557	0.962153	0.481076	+	0.876679i	$0.340246\pi$
$740$	1.65270	0.0607546
$741$	0	0
$742$	−1.97771	−0.0726041
$743$	−20.5871	−0.755267	−0.377634	−	0.925955i	$-0.623262\pi$
$743$	−20.5871	−0.755267	−0.377634	+	0.925955i	$0.623262\pi$
$744$	0	0
$745$	10.9331	0.400559
$746$	7.83069	0.286702
$747$	0	0
$748$	−0.426022	−0.0155769
$749$	13.2909	0.485637
$750$	0	0
$751$	42.4448	1.54883	0.774416	−	0.632676i	$-0.218044\pi$
$751$	42.4448	1.54883	0.774416	+	0.632676i	$0.218044\pi$
$752$	−17.6423	−0.643348
$753$	0	0
$754$	−0.819326	−0.0298381
$755$	−8.88856	−0.323488
$756$	0	0
$757$	−18.1762	−0.660626	−0.330313	−	0.943871i	$-0.607154\pi$
$757$	−18.1762	−0.660626	−0.330313	+	0.943871i	$0.607154\pi$
$758$	−16.1257	−0.585711
$759$	0	0
$760$	0.554378	0.0201094
$761$	16.8316	0.610146	0.305073	−	0.952329i	$-0.401319\pi$
$761$	16.8316	0.610146	0.305073	+	0.952329i	$0.401319\pi$
$762$	0	0
$763$	54.2131	1.96264
$764$	4.38919	0.158795
$765$	0	0
$766$	4.29355	0.155132
$767$	0.877955	0.0317011
$768$	0	0
$769$	−9.31078	−0.335755	−0.167878	−	0.985808i	$-0.553691\pi$
$769$	−9.31078	−0.335755	−0.167878	+	0.985808i	$0.553691\pi$
$770$	−2.34730	−0.0845907
$771$	0	0
$772$	0.566237	0.0203793
$773$	−52.1976	−1.87742	−0.938708	−	0.344712i	$-0.887976\pi$
$773$	−52.1976	−1.87742	−0.938708	+	0.344712i	$0.887976\pi$
$774$	0	0
$775$	37.4688	1.34592
$776$	25.0847	0.900489
$777$	0	0
$778$	1.27126	0.0455768
$779$	3.67406	0.131637
$780$	0	0
$781$	−4.49020	−0.160672
$782$	−7.94356	−0.284061
$783$	0	0
$784$	−5.90436	−0.210870
$785$	−5.93582	−0.211859
$786$	0	0
$787$	36.6982	1.30815	0.654075	−	0.756430i	$-0.273058\pi$
$787$	36.6982	1.30815	0.654075	+	0.756430i	$0.273058\pi$
$788$	−2.58677	−0.0921499
$789$	0	0
$790$	2.16250	0.0769384
$791$	7.75877	0.275870
$792$	0	0
$793$	−0.328630	−0.0116700
$794$	19.9527	0.708096
$795$	0	0
$796$	4.45336	0.157845
$797$	15.2175	0.539032	0.269516	−	0.962996i	$-0.413136\pi$
$797$	15.2175	0.539032	0.269516	+	0.962996i	$0.413136\pi$
$798$	0	0
$799$	4.73143	0.167386
$800$	9.16756	0.324122
$801$	0	0
$802$	36.8685	1.30187
$803$	8.96585	0.316398
$804$	0	0
$805$	−6.47565	−0.228237
$806$	−0.781059	−0.0275116
$807$	0	0
$808$	−21.0865	−0.741819
$809$	23.1334	0.813327	0.406664	−	0.913578i	$-0.366692\pi$
$809$	23.1334	0.813327	0.406664	+	0.913578i	$0.366692\pi$
$810$	0	0
$811$	−31.5681	−1.10851	−0.554253	−	0.832348i	$-0.686996\pi$
$811$	−31.5681	−1.10851	−0.554253	+	0.832348i	$0.686996\pi$
$812$	8.33275	0.292422
$813$	0	0
$814$	13.7023	0.480267
$815$	−9.53209	−0.333895
$816$	0	0
$817$	4.70678	0.164669
$818$	4.61587	0.161390
$819$	0	0
$820$	−1.65002	−0.0576210
$821$	−3.85710	−0.134614	−0.0673068	−	0.997732i	$-0.521441\pi$
$821$	−3.85710	−0.134614	−0.0673068	+	0.997732i	$0.521441\pi$
$822$	0	0
$823$	9.12330	0.318018	0.159009	−	0.987277i	$-0.449170\pi$
$823$	9.12330	0.318018	0.159009	+	0.987277i	$0.449170\pi$
$824$	33.6732	1.17306
$825$	0	0
$826$	−60.3492	−2.09982
$827$	18.6117	0.647194	0.323597	−	0.946195i	$-0.395108\pi$
$827$	18.6117	0.647194	0.323597	+	0.946195i	$0.395108\pi$
$828$	0	0
$829$	16.8958	0.586815	0.293408	−	0.955987i	$-0.405211\pi$
$829$	16.8958	0.586815	0.293408	+	0.955987i	$0.405211\pi$
$830$	−4.59627	−0.159539
$831$	0	0
$832$	0.395993	0.0137286
$833$	1.58347	0.0548641
$834$	0	0
$835$	1.67911	0.0581080
$836$	−0.142903	−0.00494242
$837$	0	0
$838$	−5.93170	−0.204907
$839$	34.8907	1.20456	0.602281	−	0.798284i	$-0.294259\pi$
$839$	34.8907	1.20456	0.602281	+	0.798284i	$0.294259\pi$
$840$	0	0
$841$	40.4347	1.39430
$842$	−8.40373	−0.289612
$843$	0	0
$844$	−0.979466	−0.0337146
$845$	6.91496	0.237882
$846$	0	0
$847$	−2.87939	−0.0989368
$848$	−2.05056	−0.0704166
$849$	0	0
$850$	−8.86484	−0.304061
$851$	37.8016	1.29582
$852$	0	0
$853$	−44.6647	−1.52929	−0.764645	−	0.644452i	$-0.777086\pi$
$853$	−44.6647	−1.52929	−0.764645	+	0.644452i	$0.777086\pi$
$854$	22.5895	0.772995
$855$	0	0
$856$	11.6878	0.399480
$857$	−25.6382	−0.875783	−0.437891	−	0.899028i	$-0.644275\pi$
$857$	−25.6382	−0.875783	−0.437891	+	0.899028i	$0.644275\pi$
$858$	0	0
$859$	7.45573	0.254386	0.127193	−	0.991878i	$-0.459403\pi$
$859$	7.45573	0.254386	0.127193	+	0.991878i	$0.459403\pi$
$860$	−2.11381	−0.0720802
$861$	0	0
$862$	41.5749	1.41605
$863$	46.3201	1.57675	0.788377	−	0.615193i	$-0.210922\pi$
$863$	46.3201	1.57675	0.788377	+	0.615193i	$0.210922\pi$
$864$	0	0
$865$	4.31046	0.146560
$866$	22.1174	0.751581
$867$	0	0
$868$	7.94356	0.269622
$869$	2.65270	0.0899868
$870$	0	0
$871$	0.631845	0.0214093
$872$	47.6742	1.61445
$873$	0	0
$874$	−2.66456	−0.0901302
$875$	−14.8871	−0.503277
$876$	0	0
$877$	−48.6272	−1.64202	−0.821012	−	0.570910i	$-0.806590\pi$
$877$	−48.6272	−1.64202	−0.821012	+	0.570910i	$0.806590\pi$
$878$	−46.3114	−1.56293
$879$	0	0
$880$	−2.43376	−0.0820421
$881$	21.4320	0.722063	0.361031	−	0.932554i	$-0.382425\pi$
$881$	21.4320	0.722063	0.361031	+	0.932554i	$0.382425\pi$
$882$	0	0
$883$	38.6932	1.30213	0.651064	−	0.759023i	$-0.274323\pi$
$883$	38.6932	1.30213	0.651064	+	0.759023i	$0.274323\pi$
$884$	0.0273411	0.000919582 0
$885$	0	0
$886$	21.2959	0.715450
$887$	−11.4243	−0.383589	−0.191795	−	0.981435i	$-0.561431\pi$
$887$	−11.4243	−0.383589	−0.191795	+	0.981435i	$0.561431\pi$
$888$	0	0
$889$	16.4192	0.550683
$890$	−6.49794	−0.217811
$891$	0	0
$892$	−3.60576	−0.120730
$893$	1.58710	0.0531101
$894$	0	0
$895$	5.93313	0.198323
$896$	−38.4124	−1.28327
$897$	0	0
$898$	−9.89756	−0.330286
$899$	66.1917	2.20762
$900$	0	0
$901$	0.549935	0.0183210
$902$	−13.6800	−0.455496
$903$	0	0
$904$	6.82295	0.226928
$905$	0.799229	0.0265673
$906$	0	0
$907$	7.65951	0.254330	0.127165	−	0.991882i	$-0.459412\pi$
$907$	7.65951	0.254330	0.127165	+	0.991882i	$0.459412\pi$
$908$	−10.3037	−0.341939
$909$	0	0
$910$	0.150644	0.00499381
$911$	47.1611	1.56252	0.781258	−	0.624208i	$-0.214578\pi$
$911$	47.1611	1.56252	0.781258	+	0.624208i	$0.214578\pi$
$912$	0	0
$913$	−5.63816	−0.186596
$914$	−9.06561	−0.299864
$915$	0	0
$916$	−3.39424	−0.112149
$917$	28.3405	0.935885
$918$	0	0
$919$	43.3378	1.42958	0.714791	−	0.699338i	$-0.246522\pi$
$919$	43.3378	1.42958	0.714791	+	0.699338i	$0.246522\pi$
$920$	−5.69459	−0.187745
$921$	0	0
$922$	26.5749	0.875198
$923$	0.288171	0.00948526
$924$	0	0
$925$	42.1857	1.38706
$926$	25.4124	0.835104
$927$	0	0
$928$	16.1952	0.531634
$929$	36.9614	1.21266	0.606332	−	0.795211i	$-0.292640\pi$
$929$	36.9614	1.21266	0.606332	+	0.795211i	$0.292640\pi$
$930$	0	0
$931$	0.531155	0.0174079
$932$	−1.24123	−0.0406578
$933$	0	0
$934$	22.5134	0.736662
$935$	0.652704	0.0213457
$936$	0	0
$937$	19.1506	0.625624	0.312812	−	0.949815i	$-0.398729\pi$
$937$	19.1506	0.625624	0.312812	+	0.949815i	$0.398729\pi$
$938$	−43.4320	−1.41811
$939$	0	0
$940$	−0.712763	−0.0232478
$941$	23.7155	0.773102	0.386551	−	0.922268i	$-0.373666\pi$
$941$	23.7155	0.773102	0.386551	+	0.922268i	$0.373666\pi$
$942$	0	0
$943$	−37.7401	−1.22899
$944$	−62.5722	−2.03655
$945$	0	0
$946$	−17.5253	−0.569796
$947$	5.12660	0.166592	0.0832961	−	0.996525i	$-0.473455\pi$
$947$	5.12660	0.166592	0.0832961	+	0.996525i	$0.473455\pi$
$948$	0	0
$949$	−0.575408	−0.0186785
$950$	−2.97359	−0.0964761
$951$	0	0
$952$	8.94356	0.289863
$953$	−5.37969	−0.174265	−0.0871326	−	0.996197i	$-0.527770\pi$
$953$	−5.37969	−0.174265	−0.0871326	+	0.996197i	$0.527770\pi$
$954$	0	0
$955$	−6.72462	−0.217604
$956$	−5.18304	−0.167631
$957$	0	0
$958$	3.20077	0.103412
$959$	−27.6091	−0.891543
$960$	0	0
$961$	32.1002	1.03549
$962$	−0.879385	−0.0283525
$963$	0	0
$964$	−7.37464	−0.237521
$965$	−0.867526	−0.0279266
$966$	0	0
$967$	12.2267	0.393184	0.196592	−	0.980485i	$-0.437013\pi$
$967$	12.2267	0.393184	0.196592	+	0.980485i	$0.437013\pi$
$968$	−2.53209	−0.0813844
$969$	0	0
$970$	8.07604	0.259306
$971$	−33.5262	−1.07591	−0.537954	−	0.842974i	$-0.680802\pi$
$971$	−33.5262	−1.07591	−0.537954	+	0.842974i	$0.680802\pi$
$972$	0	0
$973$	33.4124	1.07115
$974$	11.8612	0.380058
$975$	0	0
$976$	23.4216	0.749706
$977$	42.3783	1.35580	0.677900	−	0.735154i	$-0.262890\pi$
$977$	42.3783	1.35580	0.677900	+	0.735154i	$0.262890\pi$
$978$	0	0
$979$	−7.97090	−0.254751
$980$	−0.238541	−0.00761992
$981$	0	0
$982$	9.32232	0.297487
$983$	4.87527	0.155497	0.0777484	−	0.996973i	$-0.475227\pi$
$983$	4.87527	0.155497	0.0777484	+	0.996973i	$0.475227\pi$
$984$	0	0
$985$	3.96316	0.126277
$986$	−15.6604	−0.498730
$987$	0	0
$988$	0.00917123	0.000291776 0
$989$	−48.3482	−1.53738
$990$	0	0
$991$	8.03952	0.255384	0.127692	−	0.991814i	$-0.459243\pi$
$991$	8.03952	0.255384	0.127692	+	0.991814i	$0.459243\pi$
$992$	15.4388	0.490183
$993$	0	0
$994$	−19.8084	−0.628284
$995$	−6.82295	−0.216302
$996$	0	0
$997$	37.4056	1.18465	0.592324	−	0.805700i	$-0.298211\pi$
$997$	37.4056	1.18465	0.592324	+	0.805700i	$0.298211\pi$
$998$	2.71276	0.0858710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.l.1.3		3
3.2	odd	2		891.2.a.k.1.1		3
9.2	odd	6		297.2.e.d.199.3		6
9.4	even	3		99.2.e.d.34.1	✓	6
9.5	odd	6		297.2.e.d.100.3		6
9.7	even	3		99.2.e.d.67.1	yes	6
11.10	odd	2		9801.2.a.be.1.1		3
33.32	even	2		9801.2.a.bd.1.3		3
99.43	odd	6		1089.2.e.h.364.3		6
99.76	odd	6		1089.2.e.h.727.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.d.34.1	✓	6	9.4	even	3
99.2.e.d.67.1	yes	6	9.7	even	3
297.2.e.d.100.3		6	9.5	odd	6
297.2.e.d.199.3		6	9.2	odd	6
891.2.a.k.1.1		3	3.2	odd	2
891.2.a.l.1.3		3	1.1	even	1	trivial
1089.2.e.h.364.3		6	99.43	odd	6
1089.2.e.h.727.3		6	99.76	odd	6
9801.2.a.bd.1.3		3	33.32	even	2
9801.2.a.be.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)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +0.347296 q^{4} -0.532089 q^{5} -2.87939 q^{7} -2.53209 q^{8} -0.815207 q^{10} -1.00000 q^{11} +0.0641778 q^{13} -4.41147 q^{14} -4.57398 q^{16} +1.22668 q^{17} +0.411474 q^{19} -0.184793 q^{20} -1.53209 q^{22} -4.22668 q^{23} -4.71688 q^{25} +0.0983261 q^{26} -1.00000 q^{28} -8.33275 q^{29} -7.94356 q^{31} -1.94356 q^{32} +1.87939 q^{34} +1.53209 q^{35} -8.94356 q^{37} +0.630415 q^{38} +1.34730 q^{40} +8.92902 q^{41} +11.4388 q^{43} -0.347296 q^{44} -6.47565 q^{46} +3.85710 q^{47} +1.29086 q^{49} -7.22668 q^{50} +0.0222887 q^{52} +0.448311 q^{53} +0.532089 q^{55} +7.29086 q^{56} -12.7665 q^{58} +13.6800 q^{59} -5.12061 q^{61} -12.1702 q^{62} +6.17024 q^{64} -0.0341483 q^{65} +9.84524 q^{67} +0.426022 q^{68} +2.34730 q^{70} +4.49020 q^{71} -8.96585 q^{73} -13.7023 q^{74} +0.142903 q^{76} +2.87939 q^{77} -2.65270 q^{79} +2.43376 q^{80} +13.6800 q^{82} +5.63816 q^{83} -0.652704 q^{85} +17.5253 q^{86} +2.53209 q^{88} +7.97090 q^{89} -0.184793 q^{91} -1.46791 q^{92} +5.90941 q^{94} -0.218941 q^{95} -9.90673 q^{97} +1.97771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} - 3 q^{17} - 9 q^{19} + 3 q^{20} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 6 q^{29} - 9 q^{31} + 9 q^{32} - 12 q^{37}+ \cdots + 12 q^{98}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - 3 * q^8 - 6 * q^10 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 - 3 * q^17 - 9 * q^19 + 3 * q^20 - 6 * q^23 - 6 * q^25 + 12 * q^26 - 3 * q^28 - 6 * q^29 - 9 * q^31 + 9 * q^32 - 12 * q^37 + 9 * q^38 + 3 * q^40 - 6 * q^41 + 3 * q^43 + 12 * q^47 - 12 * q^49 - 15 * q^50 - 6 * q^52 + 3 * q^53 - 3 * q^55 + 6 * q^56 - 3 * q^58 + 21 * q^59 - 21 * q^61 - 15 * q^62 - 3 * q^64 - 21 * q^65 + 3 * q^67 + 9 * q^68 + 6 * q^70 + 12 * q^71 - 6 * q^73 - 15 * q^74 + 3 * q^77 - 9 * q^79 - 9 * q^80 + 21 * q^82 - 3 * q^85 + 6 * q^86 + 3 * q^88 - 12 * q^89 + 3 * q^91 - 9 * q^92 - 18 * q^94 - 18 * q^95 - 3 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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