LMFDB - Embedded newform 891.2.a.l.1.2

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			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	\(q+0.347296 q^{2} -1.87939 q^{4} +0.652704 q^{5} +0.532089 q^{7} -1.34730 q^{8} +0.226682 q^{10} -1.00000 q^{11} -2.30541 q^{13} +0.184793 q^{14} +3.29086 q^{16} -4.41147 q^{17} -4.18479 q^{19} -1.22668 q^{20} -0.347296 q^{22} +1.41147 q^{23} -4.57398 q^{25} -0.800660 q^{26} -1.00000 q^{28} +6.35504 q^{29} -2.16250 q^{31} +3.83750 q^{32} -1.53209 q^{34} +0.347296 q^{35} -3.16250 q^{37} -1.45336 q^{38} -0.879385 q^{40} -9.31315 q^{41} -12.2986 q^{43} +1.87939 q^{44} +0.490200 q^{46} -3.86484 q^{47} -6.71688 q^{49} -1.58853 q^{50} +4.33275 q^{52} +12.6236 q^{53} -0.652704 q^{55} -0.716881 q^{56} +2.20708 q^{58} -3.23442 q^{59} -8.53209 q^{61} -0.751030 q^{62} -5.24897 q^{64} -1.50475 q^{65} +4.96316 q^{67} +8.29086 q^{68} +0.120615 q^{70} +9.98545 q^{71} -7.49525 q^{73} -1.09833 q^{74} +7.86484 q^{76} -0.532089 q^{77} -4.87939 q^{79} +2.14796 q^{80} -3.23442 q^{82} -4.59627 q^{83} -2.87939 q^{85} -4.27126 q^{86} +1.34730 q^{88} -16.9513 q^{89} -1.22668 q^{91} -2.65270 q^{92} -1.34224 q^{94} -2.73143 q^{95} +12.6459 q^{97} -2.33275 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.347296	0.245576	0.122788	−	0.992433i	$-0.460817\pi$
$2$	0.347296	0.245576	0.122788	+	0.992433i	$0.460817\pi$
$3$	0	0
$3$	0	0
$4$	−1.87939	−0.939693
$5$	0.652704	0.291898	0.145949	−	0.989292i	$-0.453376\pi$
$5$	0.652704	0.291898	0.145949	+	0.989292i	$0.453376\pi$
$6$	0	0
$7$	0.532089	0.201111	0.100555	−	0.994931i	$-0.467938\pi$
$7$	0.532089	0.201111	0.100555	+	0.994931i	$0.467938\pi$
$8$	−1.34730	−0.476341
$9$	0	0
$10$	0.226682	0.0716830
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.30541	−0.639405	−0.319702	−	0.947518i	$-0.603583\pi$
$13$	−2.30541	−0.639405	−0.319702	+	0.947518i	$0.603583\pi$
$14$	0.184793	0.0493879
$15$	0	0
$16$	3.29086	0.822715
$17$	−4.41147	−1.06994	−0.534970	−	0.844871i	$-0.679677\pi$
$17$	−4.41147	−1.06994	−0.534970	+	0.844871i	$0.679677\pi$
$18$	0	0
$19$	−4.18479	−0.960057	−0.480029	−	0.877253i	$-0.659374\pi$
$19$	−4.18479	−0.960057	−0.480029	+	0.877253i	$0.659374\pi$
$20$	−1.22668	−0.274294
$21$	0	0
$22$	−0.347296	−0.0740438
$23$	1.41147	0.294313	0.147156	−	0.989113i	$-0.452988\pi$
$23$	1.41147	0.294313	0.147156	+	0.989113i	$0.452988\pi$
$24$	0	0
$25$	−4.57398	−0.914796
$26$	−0.800660	−0.157022
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.35504	1.18010	0.590050	−	0.807366i	$-0.299108\pi$
$29$	6.35504	1.18010	0.590050	+	0.807366i	$0.299108\pi$
$30$	0	0
$31$	−2.16250	−0.388397	−0.194199	−	0.980962i	$-0.562211\pi$
$31$	−2.16250	−0.388397	−0.194199	+	0.980962i	$0.562211\pi$
$32$	3.83750	0.678380
$33$	0	0
$34$	−1.53209	−0.262751
$35$	0.347296	0.0587038
$36$	0	0
$37$	−3.16250	−0.519912	−0.259956	−	0.965620i	$-0.583708\pi$
$37$	−3.16250	−0.519912	−0.259956	+	0.965620i	$0.583708\pi$
$38$	−1.45336	−0.235767
$39$	0	0
$40$	−0.879385	−0.139043
$41$	−9.31315	−1.45447	−0.727235	−	0.686389i	$-0.759195\pi$
$41$	−9.31315	−1.45447	−0.727235	+	0.686389i	$0.759195\pi$
$42$	0	0
$43$	−12.2986	−1.87552	−0.937759	−	0.347285i	$-0.887104\pi$
$43$	−12.2986	−1.87552	−0.937759	+	0.347285i	$0.887104\pi$
$44$	1.87939	0.283328
$45$	0	0
$46$	0.490200	0.0722760
$47$	−3.86484	−0.563744	−0.281872	−	0.959452i	$-0.590955\pi$
$47$	−3.86484	−0.563744	−0.281872	+	0.959452i	$0.590955\pi$
$48$	0	0
$49$	−6.71688	−0.959554
$50$	−1.58853	−0.224651
$51$	0	0
$52$	4.33275	0.600844
$53$	12.6236	1.73399	0.866993	−	0.498320i	$-0.166050\pi$
$53$	12.6236	1.73399	0.866993	+	0.498320i	$0.166050\pi$
$54$	0	0
$55$	−0.652704	−0.0880105
$56$	−0.716881	−0.0957973
$57$	0	0
$58$	2.20708	0.289804
$59$	−3.23442	−0.421086	−0.210543	−	0.977585i	$-0.567523\pi$
$59$	−3.23442	−0.421086	−0.210543	+	0.977585i	$0.567523\pi$
$60$	0	0
$61$	−8.53209	−1.09242	−0.546211	−	0.837648i	$-0.683930\pi$
$61$	−8.53209	−1.09242	−0.546211	+	0.837648i	$0.683930\pi$
$62$	−0.751030	−0.0953809
$63$	0	0
$64$	−5.24897	−0.656121
$65$	−1.50475	−0.186641
$66$	0	0
$67$	4.96316	0.606347	0.303173	−	0.952935i	$-0.401954\pi$
$67$	4.96316	0.606347	0.303173	+	0.952935i	$0.401954\pi$
$68$	8.29086	1.00541
$69$	0	0
$70$	0.120615	0.0144162
$71$	9.98545	1.18506	0.592528	−	0.805550i	$-0.298130\pi$
$71$	9.98545	1.18506	0.592528	+	0.805550i	$0.298130\pi$
$72$	0	0
$73$	−7.49525	−0.877253	−0.438626	−	0.898669i	$-0.644535\pi$
$73$	−7.49525	−0.877253	−0.438626	+	0.898669i	$0.644535\pi$
$74$	−1.09833	−0.127678
$75$	0	0
$76$	7.86484	0.902159
$77$	−0.532089	−0.0606372
$78$	0	0
$79$	−4.87939	−0.548974	−0.274487	−	0.961591i	$-0.588508\pi$
$79$	−4.87939	−0.548974	−0.274487	+	0.961591i	$0.588508\pi$
$80$	2.14796	0.240149
$81$	0	0
$82$	−3.23442	−0.357182
$83$	−4.59627	−0.504506	−0.252253	−	0.967661i	$-0.581171\pi$
$83$	−4.59627	−0.504506	−0.252253	+	0.967661i	$0.581171\pi$
$84$	0	0
$85$	−2.87939	−0.312313
$86$	−4.27126	−0.460582
$87$	0	0
$88$	1.34730	0.143622
$89$	−16.9513	−1.79683	−0.898417	−	0.439143i	$-0.855282\pi$
$89$	−16.9513	−1.79683	−0.898417	+	0.439143i	$0.855282\pi$
$90$	0	0
$91$	−1.22668	−0.128591
$92$	−2.65270	−0.276563
$93$	0	0
$94$	−1.34224	−0.138442
$95$	−2.73143	−0.280239
$96$	0	0
$97$	12.6459	1.28400	0.641998	−	0.766706i	$-0.278106\pi$
$97$	12.6459	1.28400	0.641998	+	0.766706i	$0.278106\pi$
$98$	−2.33275	−0.235643
$99$	0	0
$100$	8.59627	0.859627
$101$	17.0915	1.70067	0.850335	−	0.526242i	$-0.176399\pi$
$101$	17.0915	1.70067	0.850335	+	0.526242i	$0.176399\pi$
$102$	0	0
$103$	2.85978	0.281783	0.140891	−	0.990025i	$-0.455003\pi$
$103$	2.85978	0.281783	0.140891	+	0.990025i	$0.455003\pi$
$104$	3.10607	0.304575
$105$	0	0
$106$	4.38413	0.425825
$107$	9.92902	0.959874	0.479937	−	0.877303i	$-0.340659\pi$
$107$	9.92902	0.959874	0.479937	+	0.877303i	$0.340659\pi$
$108$	0	0
$109$	13.8161	1.32335	0.661673	−	0.749792i	$-0.269847\pi$
$109$	13.8161	1.32335	0.661673	+	0.749792i	$0.269847\pi$
$110$	−0.226682	−0.0216132
$111$	0	0
$112$	1.75103	0.165457
$113$	1.75877	0.165451	0.0827256	−	0.996572i	$-0.473637\pi$
$113$	1.75877	0.165451	0.0827256	+	0.996572i	$0.473637\pi$
$114$	0	0
$115$	0.921274	0.0859093
$116$	−11.9436	−1.10893
$117$	0	0
$118$	−1.12330	−0.103408
$119$	−2.34730	−0.215176
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.96316	−0.268272
$123$	0	0
$124$	4.06418	0.364974
$125$	−6.24897	−0.558925
$126$	0	0
$127$	6.90167	0.612425	0.306212	−	0.951963i	$-0.400938\pi$
$127$	6.90167	0.612425	0.306212	+	0.951963i	$0.400938\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	−0.522593	−0.0458345
$131$	10.3405	0.903453	0.451726	−	0.892157i	$-0.350808\pi$
$131$	10.3405	0.903453	0.451726	+	0.892157i	$0.350808\pi$
$132$	0	0
$133$	−2.22668	−0.193078
$134$	1.72369	0.148904
$135$	0	0
$136$	5.94356	0.509656
$137$	14.1848	1.21189	0.605944	−	0.795507i	$-0.292795\pi$
$137$	14.1848	1.21189	0.605944	+	0.795507i	$0.292795\pi$
$138$	0	0
$139$	0.101014	0.00856793	0.00428396	−	0.999991i	$-0.498636\pi$
$139$	0.101014	0.00856793	0.00428396	+	0.999991i	$0.498636\pi$
$140$	−0.652704	−0.0551635
$141$	0	0
$142$	3.46791	0.291021
$143$	2.30541	0.192788
$144$	0	0
$145$	4.14796	0.344469
$146$	−2.60307	−0.215432
$147$	0	0
$148$	5.94356	0.488558
$149$	−3.06149	−0.250807	−0.125403	−	0.992106i	$-0.540023\pi$
$149$	−3.06149	−0.250807	−0.125403	+	0.992106i	$0.540023\pi$
$150$	0	0
$151$	19.4020	1.57891	0.789455	−	0.613808i	$-0.210363\pi$
$151$	19.4020	1.57891	0.789455	+	0.613808i	$0.210363\pi$
$152$	5.63816	0.457315
$153$	0	0
$154$	−0.184793	−0.0148910
$155$	−1.41147	−0.113372
$156$	0	0
$157$	−12.7246	−1.01554	−0.507768	−	0.861494i	$-0.669529\pi$
$157$	−12.7246	−1.01554	−0.507768	+	0.861494i	$0.669529\pi$
$158$	−1.69459	−0.134815
$159$	0	0
$160$	2.50475	0.198018
$161$	0.751030	0.0591894
$162$	0	0
$163$	−12.7888	−1.00170	−0.500848	−	0.865535i	$-0.666978\pi$
$163$	−12.7888	−1.00170	−0.500848	+	0.865535i	$0.666978\pi$
$164$	17.5030	1.36675
$165$	0	0
$166$	−1.59627	−0.123894
$167$	20.7246	1.60372	0.801860	−	0.597512i	$-0.203844\pi$
$167$	20.7246	1.60372	0.801860	+	0.597512i	$0.203844\pi$
$168$	0	0
$169$	−7.68510	−0.591161
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	23.1138	1.76241
$173$	−22.5030	−1.71087	−0.855435	−	0.517909i	$-0.826710\pi$
$173$	−22.5030	−1.71087	−0.855435	+	0.517909i	$0.826710\pi$
$174$	0	0
$175$	−2.43376	−0.183975
$176$	−3.29086	−0.248058
$177$	0	0
$178$	−5.88713	−0.441259
$179$	−10.7219	−0.801395	−0.400697	−	0.916210i	$-0.631232\pi$
$179$	−10.7219	−0.801395	−0.400697	+	0.916210i	$0.631232\pi$
$180$	0	0
$181$	−4.15745	−0.309021	−0.154510	−	0.987991i	$-0.549380\pi$
$181$	−4.15745	−0.309021	−0.154510	+	0.987991i	$0.549380\pi$
$182$	−0.426022	−0.0315789
$183$	0	0
$184$	−1.90167	−0.140193
$185$	−2.06418	−0.151761
$186$	0	0
$187$	4.41147	0.322599
$188$	7.26352	0.529747
$189$	0	0
$190$	−0.948615	−0.0688198
$191$	2.40373	0.173928	0.0869640	−	0.996211i	$-0.472283\pi$
$191$	2.40373	0.173928	0.0869640	+	0.996211i	$0.472283\pi$
$192$	0	0
$193$	−0.453363	−0.0326338	−0.0163169	−	0.999867i	$-0.505194\pi$
$193$	−0.453363	−0.0326338	−0.0163169	+	0.999867i	$0.505194\pi$
$194$	4.39187	0.315318
$195$	0	0
$196$	12.6236	0.901686
$197$	−19.6236	−1.39812	−0.699062	−	0.715061i	$-0.746399\pi$
$197$	−19.6236	−1.39812	−0.699062	+	0.715061i	$0.746399\pi$
$198$	0	0
$199$	3.63041	0.257353	0.128677	−	0.991687i	$-0.458927\pi$
$199$	3.63041	0.257353	0.128677	+	0.991687i	$0.458927\pi$
$200$	6.16250	0.435755
$201$	0	0
$202$	5.93582	0.417643
$203$	3.38144	0.237331
$204$	0	0
$205$	−6.07873	−0.424557
$206$	0.993193	0.0691990
$207$	0	0
$208$	−7.58677	−0.526048
$209$	4.18479	0.289468
$210$	0	0
$211$	21.6732	1.49205	0.746024	−	0.665919i	$-0.231961\pi$
$211$	21.6732	1.49205	0.746024	+	0.665919i	$0.231961\pi$
$212$	−23.7246	−1.62941
$213$	0	0
$214$	3.44831	0.235722
$215$	−8.02734	−0.547460
$216$	0	0
$217$	−1.15064	−0.0781108
$218$	4.79830	0.324982
$219$	0	0
$220$	1.22668	0.0827029
$221$	10.1702	0.684125
$222$	0	0
$223$	19.1361	1.28145	0.640724	−	0.767771i	$-0.278634\pi$
$223$	19.1361	1.28145	0.640724	+	0.767771i	$0.278634\pi$
$224$	2.04189	0.136429
$225$	0	0
$226$	0.610815	0.0406308
$227$	−15.5936	−1.03498	−0.517491	−	0.855689i	$-0.673134\pi$
$227$	−15.5936	−1.03498	−0.517491	+	0.855689i	$0.673134\pi$
$228$	0	0
$229$	−15.4115	−1.01842	−0.509209	−	0.860643i	$-0.670062\pi$
$229$	−15.4115	−1.01842	−0.509209	+	0.860643i	$0.670062\pi$
$230$	0.319955	0.0210972
$231$	0	0
$232$	−8.56212	−0.562131
$233$	4.29086	0.281104	0.140552	−	0.990073i	$-0.455112\pi$
$233$	4.29086	0.281104	0.140552	+	0.990073i	$0.455112\pi$
$234$	0	0
$235$	−2.52259	−0.164556
$236$	6.07873	0.395691
$237$	0	0
$238$	−0.815207	−0.0528421
$239$	−20.1334	−1.30232	−0.651161	−	0.758940i	$-0.725718\pi$
$239$	−20.1334	−1.30232	−0.651161	+	0.758940i	$0.725718\pi$
$240$	0	0
$241$	−7.44562	−0.479615	−0.239807	−	0.970820i	$-0.577084\pi$
$241$	−7.44562	−0.479615	−0.239807	+	0.970820i	$0.577084\pi$
$242$	0.347296	0.0223251
$243$	0	0
$244$	16.0351	1.02654
$245$	−4.38413	−0.280092
$246$	0	0
$247$	9.64765	0.613865
$248$	2.91353	0.185010
$249$	0	0
$250$	−2.17024	−0.137258
$251$	12.5202	0.790270	0.395135	−	0.918623i	$-0.370698\pi$
$251$	12.5202	0.790270	0.395135	+	0.918623i	$0.370698\pi$
$252$	0	0
$253$	−1.41147	−0.0887386
$254$	2.39693	0.150397
$255$	0	0
$256$	7.19934	0.449959
$257$	6.48339	0.404423	0.202211	−	0.979342i	$-0.435187\pi$
$257$	6.48339	0.404423	0.202211	+	0.979342i	$0.435187\pi$
$258$	0	0
$259$	−1.68273	−0.104560
$260$	2.82800	0.175385
$261$	0	0
$262$	3.59121	0.221866
$263$	2.09833	0.129388	0.0646942	−	0.997905i	$-0.479393\pi$
$263$	2.09833	0.129388	0.0646942	+	0.997905i	$0.479393\pi$
$264$	0	0
$265$	8.23947	0.506147
$266$	−0.773318	−0.0474152
$267$	0	0
$268$	−9.32770	−0.569780
$269$	29.2490	1.78334	0.891671	−	0.452685i	$-0.149534\pi$
$269$	29.2490	1.78334	0.891671	+	0.452685i	$0.149534\pi$
$270$	0	0
$271$	−21.7246	−1.31968	−0.659838	−	0.751408i	$-0.729375\pi$
$271$	−21.7246	−1.31968	−0.659838	+	0.751408i	$0.729375\pi$
$272$	−14.5175	−0.880255
$273$	0	0
$274$	4.92633	0.297610
$275$	4.57398	0.275821
$276$	0	0
$277$	−4.45605	−0.267738	−0.133869	−	0.990999i	$-0.542740\pi$
$277$	−4.45605	−0.267738	−0.133869	+	0.990999i	$0.542740\pi$
$278$	0.0350819	0.00210407
$279$	0	0
$280$	−0.467911	−0.0279630
$281$	−20.4953	−1.22264	−0.611322	−	0.791382i	$-0.709362\pi$
$281$	−20.4953	−1.22264	−0.611322	+	0.791382i	$0.709362\pi$
$282$	0	0
$283$	6.88713	0.409397	0.204699	−	0.978825i	$-0.434379\pi$
$283$	6.88713	0.409397	0.204699	+	0.978825i	$0.434379\pi$
$284$	−18.7665	−1.11359
$285$	0	0
$286$	0.800660	0.0473440
$287$	−4.95542	−0.292509
$288$	0	0
$289$	2.46110	0.144771
$290$	1.44057	0.0845932
$291$	0	0
$292$	14.0865	0.824348
$293$	−13.8084	−0.806695	−0.403348	−	0.915047i	$-0.632153\pi$
$293$	−13.8084	−0.806695	−0.403348	+	0.915047i	$0.632153\pi$
$294$	0	0
$295$	−2.11112	−0.122914
$296$	4.26083	0.247656
$297$	0	0
$298$	−1.06324	−0.0615921
$299$	−3.25402	−0.188185
$300$	0	0
$301$	−6.54395	−0.377187
$302$	6.73824	0.387742
$303$	0	0
$304$	−13.7716	−0.789853
$305$	−5.56893	−0.318876
$306$	0	0
$307$	−0.601319	−0.0343191	−0.0171595	−	0.999853i	$-0.505462\pi$
$307$	−0.601319	−0.0343191	−0.0171595	+	0.999853i	$0.505462\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−0.490200	−0.0278415
$311$	1.94087	0.110057	0.0550285	−	0.998485i	$-0.482475\pi$
$311$	1.94087	0.110057	0.0550285	+	0.998485i	$0.482475\pi$
$312$	0	0
$313$	2.01960	0.114155	0.0570773	−	0.998370i	$-0.481822\pi$
$313$	2.01960	0.114155	0.0570773	+	0.998370i	$0.481822\pi$
$314$	−4.41921	−0.249391
$315$	0	0
$316$	9.17024	0.515867
$317$	5.84936	0.328533	0.164266	−	0.986416i	$-0.447474\pi$
$317$	5.84936	0.328533	0.164266	+	0.986416i	$0.447474\pi$
$318$	0	0
$319$	−6.35504	−0.355814
$320$	−3.42602	−0.191520
$321$	0	0
$322$	0.260830	0.0145355
$323$	18.4611	1.02720
$324$	0	0
$325$	10.5449	0.584925
$326$	−4.44150	−0.245992
$327$	0	0
$328$	12.5476	0.692824
$329$	−2.05644	−0.113375
$330$	0	0
$331$	11.7219	0.644296	0.322148	−	0.946689i	$-0.395595\pi$
$331$	11.7219	0.644296	0.322148	+	0.946689i	$0.395595\pi$
$332$	8.63816	0.474080
$333$	0	0
$334$	7.19759	0.393834
$335$	3.23947	0.176991
$336$	0	0
$337$	12.5963	0.686162	0.343081	−	0.939306i	$-0.388529\pi$
$337$	12.5963	0.686162	0.343081	+	0.939306i	$0.388529\pi$
$338$	−2.66901	−0.145175
$339$	0	0
$340$	5.41147	0.293478
$341$	2.16250	0.117106
$342$	0	0
$343$	−7.29860	−0.394087
$344$	16.5699	0.893387
$345$	0	0
$346$	−7.81521	−0.420148
$347$	−23.8452	−1.28008	−0.640040	−	0.768342i	$-0.721082\pi$
$347$	−23.8452	−1.28008	−0.640040	+	0.768342i	$0.721082\pi$
$348$	0	0
$349$	−8.48246	−0.454056	−0.227028	−	0.973888i	$-0.572901\pi$
$349$	−8.48246	−0.454056	−0.227028	+	0.973888i	$0.572901\pi$
$350$	−0.845237	−0.0451798
$351$	0	0
$352$	−3.83750	−0.204539
$353$	7.44656	0.396340	0.198170	−	0.980168i	$-0.436500\pi$
$353$	7.44656	0.396340	0.198170	+	0.980168i	$0.436500\pi$
$354$	0	0
$355$	6.51754	0.345915
$356$	31.8580	1.68847
$357$	0	0
$358$	−3.72369	−0.196803
$359$	−22.9290	−1.21015	−0.605074	−	0.796170i	$-0.706856\pi$
$359$	−22.9290	−1.21015	−0.605074	+	0.796170i	$0.706856\pi$
$360$	0	0
$361$	−1.48751	−0.0782901
$362$	−1.44387	−0.0758880
$363$	0	0
$364$	2.30541	0.120836
$365$	−4.89218	−0.256068
$366$	0	0
$367$	15.3063	0.798984	0.399492	−	0.916737i	$-0.369186\pi$
$367$	15.3063	0.798984	0.399492	+	0.916737i	$0.369186\pi$
$368$	4.64496	0.242135
$369$	0	0
$370$	−0.716881	−0.0372689
$371$	6.71688	0.348723
$372$	0	0
$373$	−27.3901	−1.41821	−0.709103	−	0.705105i	$-0.750900\pi$
$373$	−27.3901	−1.41821	−0.709103	+	0.705105i	$0.750900\pi$
$374$	1.53209	0.0792224
$375$	0	0
$376$	5.20708	0.268535
$377$	−14.6509	−0.754562
$378$	0	0
$379$	11.2713	0.578966	0.289483	−	0.957183i	$-0.406517\pi$
$379$	11.2713	0.578966	0.289483	+	0.957183i	$0.406517\pi$
$380$	5.13341	0.263338
$381$	0	0
$382$	0.834808	0.0427125
$383$	33.3628	1.70476	0.852379	−	0.522924i	$-0.175159\pi$
$383$	33.3628	1.70476	0.852379	+	0.522924i	$0.175159\pi$
$384$	0	0
$385$	−0.347296	−0.0176999
$386$	−0.157451	−0.00801406
$387$	0	0
$388$	−23.7665	−1.20656
$389$	12.2490	0.621047	0.310524	−	0.950566i	$-0.399496\pi$
$389$	12.2490	0.621047	0.310524	+	0.950566i	$0.399496\pi$
$390$	0	0
$391$	−6.22668	−0.314897
$392$	9.04963	0.457075
$393$	0	0
$394$	−6.81521	−0.343345
$395$	−3.18479	−0.160244
$396$	0	0
$397$	−11.4287	−0.573591	−0.286795	−	0.957992i	$-0.592590\pi$
$397$	−11.4287	−0.573591	−0.286795	+	0.957992i	$0.592590\pi$
$398$	1.26083	0.0631997
$399$	0	0
$400$	−15.0523	−0.752616
$401$	21.6946	1.08338	0.541688	−	0.840580i	$-0.317785\pi$
$401$	21.6946	1.08338	0.541688	+	0.840580i	$0.317785\pi$
$402$	0	0
$403$	4.98545	0.248343
$404$	−32.1215	−1.59811
$405$	0	0
$406$	1.17436	0.0582827
$407$	3.16250	0.156759
$408$	0	0
$409$	−28.5895	−1.41366	−0.706829	−	0.707385i	$-0.749875\pi$
$409$	−28.5895	−1.41366	−0.706829	+	0.707385i	$0.749875\pi$
$410$	−2.11112	−0.104261
$411$	0	0
$412$	−5.37464	−0.264789
$413$	−1.72100	−0.0846849
$414$	0	0
$415$	−3.00000	−0.147264
$416$	−8.84699	−0.433759
$417$	0	0
$418$	1.45336	0.0710863
$419$	−8.61081	−0.420666	−0.210333	−	0.977630i	$-0.567455\pi$
$419$	−8.61081	−0.420666	−0.210333	+	0.977630i	$0.567455\pi$
$420$	0	0
$421$	−34.4320	−1.67811	−0.839057	−	0.544044i	$-0.816892\pi$
$421$	−34.4320	−1.67811	−0.839057	+	0.544044i	$0.816892\pi$
$422$	7.52704	0.366410
$423$	0	0
$424$	−17.0077	−0.825969
$425$	20.1780	0.978776
$426$	0	0
$427$	−4.53983	−0.219698
$428$	−18.6604	−0.901987
$429$	0	0
$430$	−2.78787	−0.134443
$431$	14.2463	0.686219	0.343110	−	0.939295i	$-0.388520\pi$
$431$	14.2463	0.686219	0.343110	+	0.939295i	$0.388520\pi$
$432$	0	0
$433$	−24.6023	−1.18231	−0.591154	−	0.806558i	$-0.701328\pi$
$433$	−24.6023	−1.18231	−0.591154	+	0.806558i	$0.701328\pi$
$434$	−0.399615	−0.0191821
$435$	0	0
$436$	−25.9659	−1.24354
$437$	−5.90673	−0.282557
$438$	0	0
$439$	4.17293	0.199163	0.0995816	−	0.995029i	$-0.468250\pi$
$439$	4.17293	0.199163	0.0995816	+	0.995029i	$0.468250\pi$
$440$	0.879385	0.0419230
$441$	0	0
$442$	3.53209	0.168004
$443$	−29.2645	−1.39040	−0.695198	−	0.718818i	$-0.744683\pi$
$443$	−29.2645	−1.39040	−0.695198	+	0.718818i	$0.744683\pi$
$444$	0	0
$445$	−11.0642	−0.524492
$446$	6.64590	0.314692
$447$	0	0
$448$	−2.79292	−0.131953
$449$	−15.7956	−0.745441	−0.372720	−	0.927944i	$-0.621575\pi$
$449$	−15.7956	−0.745441	−0.372720	+	0.927944i	$0.621575\pi$
$450$	0	0
$451$	9.31315	0.438539
$452$	−3.30541	−0.155473
$453$	0	0
$454$	−5.41559	−0.254166
$455$	−0.800660	−0.0375355
$456$	0	0
$457$	9.48515	0.443696	0.221848	−	0.975081i	$-0.428791\pi$
$457$	9.48515	0.443696	0.221848	+	0.975081i	$0.428791\pi$
$458$	−5.35235	−0.250099
$459$	0	0
$460$	−1.73143	−0.0807283
$461$	−28.9445	−1.34808	−0.674040	−	0.738695i	$-0.735442\pi$
$461$	−28.9445	−1.34808	−0.674040	+	0.738695i	$0.735442\pi$
$462$	0	0
$463$	−22.8803	−1.06334	−0.531669	−	0.846952i	$-0.678435\pi$
$463$	−22.8803	−1.06334	−0.531669	+	0.846952i	$0.678435\pi$
$464$	20.9135	0.970886
$465$	0	0
$466$	1.49020	0.0690322
$467$	10.2412	0.473908	0.236954	−	0.971521i	$-0.423851\pi$
$467$	10.2412	0.473908	0.236954	+	0.971521i	$0.423851\pi$
$468$	0	0
$469$	2.64084	0.121943
$470$	−0.876087	−0.0404109
$471$	0	0
$472$	4.35773	0.200581
$473$	12.2986	0.565490
$474$	0	0
$475$	19.1411	0.878256
$476$	4.41147	0.202200
$477$	0	0
$478$	−6.99226	−0.319818
$479$	19.3310	0.883256	0.441628	−	0.897198i	$-0.354401\pi$
$479$	19.3310	0.883256	0.441628	+	0.897198i	$0.354401\pi$
$480$	0	0
$481$	7.29086	0.332435
$482$	−2.58584	−0.117782
$483$	0	0
$484$	−1.87939	−0.0854266
$485$	8.25402	0.374796
$486$	0	0
$487$	27.2104	1.23302	0.616510	−	0.787347i	$-0.288546\pi$
$487$	27.2104	1.23302	0.616510	+	0.787347i	$0.288546\pi$
$488$	11.4953	0.520366
$489$	0	0
$490$	−1.52259	−0.0687838
$491$	−36.0378	−1.62636	−0.813181	−	0.582011i	$-0.802266\pi$
$491$	−36.0378	−1.62636	−0.813181	+	0.582011i	$0.802266\pi$
$492$	0	0
$493$	−28.0351	−1.26264
$494$	3.35059	0.150750
$495$	0	0
$496$	−7.11650	−0.319540
$497$	5.31315	0.238327
$498$	0	0
$499$	−7.89218	−0.353302	−0.176651	−	0.984274i	$-0.556526\pi$
$499$	−7.89218	−0.353302	−0.176651	+	0.984274i	$0.556526\pi$
$500$	11.7442	0.525218
$501$	0	0
$502$	4.34823	0.194071
$503$	−37.2550	−1.66112	−0.830558	−	0.556932i	$-0.811978\pi$
$503$	−37.2550	−1.66112	−0.830558	+	0.556932i	$0.811978\pi$
$504$	0	0
$505$	11.1557	0.496422
$506$	−0.490200	−0.0217920
$507$	0	0
$508$	−12.9709	−0.575491
$509$	4.08915	0.181249	0.0906243	−	0.995885i	$-0.471114\pi$
$509$	4.08915	0.181249	0.0906243	+	0.995885i	$0.471114\pi$
$510$	0	0
$511$	−3.98814	−0.176425
$512$	21.4962	0.950006
$513$	0	0
$514$	2.25166	0.0993164
$515$	1.86659	0.0822519
$516$	0	0
$517$	3.86484	0.169975
$518$	−0.584407	−0.0256774
$519$	0	0
$520$	2.02734	0.0889048
$521$	7.29322	0.319522	0.159761	−	0.987156i	$-0.448928\pi$
$521$	7.29322	0.319522	0.159761	+	0.987156i	$0.448928\pi$
$522$	0	0
$523$	−14.3105	−0.625753	−0.312876	−	0.949794i	$-0.601293\pi$
$523$	−14.3105	−0.625753	−0.312876	+	0.949794i	$0.601293\pi$
$524$	−19.4338	−0.848968
$525$	0	0
$526$	0.728741	0.0317746
$527$	9.53983	0.415562
$528$	0	0
$529$	−21.0077	−0.913380
$530$	2.86154	0.124297
$531$	0	0
$532$	4.18479	0.181434
$533$	21.4706	0.929995
$534$	0	0
$535$	6.48070	0.280185
$536$	−6.68685	−0.288828
$537$	0	0
$538$	10.1581	0.437945
$539$	6.71688	0.289317
$540$	0	0
$541$	−36.1411	−1.55383	−0.776915	−	0.629606i	$-0.783216\pi$
$541$	−36.1411	−1.55383	−0.776915	+	0.629606i	$0.783216\pi$
$542$	−7.54488	−0.324080
$543$	0	0
$544$	−16.9290	−0.725826
$545$	9.01785	0.386282
$546$	0	0
$547$	−18.7870	−0.803276	−0.401638	−	0.915799i	$-0.631559\pi$
$547$	−18.7870	−0.803276	−0.401638	+	0.915799i	$0.631559\pi$
$548$	−26.6587	−1.13880
$549$	0	0
$550$	1.58853	0.0677350
$551$	−26.5945	−1.13296
$552$	0	0
$553$	−2.59627	−0.110404
$554$	−1.54757	−0.0657500
$555$	0	0
$556$	−0.189845	−0.00805122
$557$	−3.99319	−0.169197	−0.0845985	−	0.996415i	$-0.526961\pi$
$557$	−3.99319	−0.169197	−0.0845985	+	0.996415i	$0.526961\pi$
$558$	0	0
$559$	28.3533	1.19922
$560$	1.14290	0.0482965
$561$	0	0
$562$	−7.11793	−0.300252
$563$	−22.6905	−0.956289	−0.478145	−	0.878281i	$-0.658691\pi$
$563$	−22.6905	−0.956289	−0.478145	+	0.878281i	$0.658691\pi$
$564$	0	0
$565$	1.14796	0.0482949
$566$	2.39187	0.100538
$567$	0	0
$568$	−13.4534	−0.564491
$569$	25.3209	1.06151	0.530753	−	0.847526i	$-0.321909\pi$
$569$	25.3209	1.06151	0.530753	+	0.847526i	$0.321909\pi$
$570$	0	0
$571$	33.3141	1.39415	0.697075	−	0.716998i	$-0.254484\pi$
$571$	33.3141	1.39415	0.697075	+	0.716998i	$0.254484\pi$
$572$	−4.33275	−0.181161
$573$	0	0
$574$	−1.72100	−0.0718332
$575$	−6.45605	−0.269236
$576$	0	0
$577$	−1.18479	−0.0493236	−0.0246618	−	0.999696i	$-0.507851\pi$
$577$	−1.18479	−0.0493236	−0.0246618	+	0.999696i	$0.507851\pi$
$578$	0.854732	0.0355522
$579$	0	0
$580$	−7.79561	−0.323695
$581$	−2.44562	−0.101462
$582$	0	0
$583$	−12.6236	−0.522816
$584$	10.0983	0.417872
$585$	0	0
$586$	−4.79561	−0.198105
$587$	31.8043	1.31270	0.656352	−	0.754455i	$-0.272099\pi$
$587$	31.8043	1.31270	0.656352	+	0.754455i	$0.272099\pi$
$588$	0	0
$589$	9.04963	0.372884
$590$	−0.733184	−0.0301847
$591$	0	0
$592$	−10.4074	−0.427740
$593$	26.3381	1.08158	0.540789	−	0.841159i	$-0.318126\pi$
$593$	26.3381	1.08158	0.540789	+	0.841159i	$0.318126\pi$
$594$	0	0
$595$	−1.53209	−0.0628095
$596$	5.75372	0.235681
$597$	0	0
$598$	−1.13011	−0.0462136
$599$	15.2645	0.623689	0.311844	−	0.950133i	$-0.399053\pi$
$599$	15.2645	0.623689	0.311844	+	0.950133i	$0.399053\pi$
$600$	0	0
$601$	8.50711	0.347012	0.173506	−	0.984833i	$-0.444490\pi$
$601$	8.50711	0.347012	0.173506	+	0.984833i	$0.444490\pi$
$602$	−2.27269	−0.0926279
$603$	0	0
$604$	−36.4638	−1.48369
$605$	0.652704	0.0265362
$606$	0	0
$607$	12.5202	0.508180	0.254090	−	0.967181i	$-0.418224\pi$
$607$	12.5202	0.508180	0.254090	+	0.967181i	$0.418224\pi$
$608$	−16.0591	−0.651284
$609$	0	0
$610$	−1.93407	−0.0783081
$611$	8.91002	0.360461
$612$	0	0
$613$	−8.52259	−0.344224	−0.172112	−	0.985077i	$-0.555059\pi$
$613$	−8.52259	−0.344224	−0.172112	+	0.985077i	$0.555059\pi$
$614$	−0.208836	−0.00842793
$615$	0	0
$616$	0.716881	0.0288840
$617$	−20.4884	−0.824834	−0.412417	−	0.910995i	$-0.635315\pi$
$617$	−20.4884	−0.824834	−0.412417	+	0.910995i	$0.635315\pi$
$618$	0	0
$619$	−26.8016	−1.07725	−0.538623	−	0.842547i	$-0.681055\pi$
$619$	−26.8016	−1.07725	−0.538623	+	0.842547i	$0.681055\pi$
$620$	2.65270	0.106535
$621$	0	0
$622$	0.674059	0.0270273
$623$	−9.01960	−0.361363
$624$	0	0
$625$	18.7912	0.751647
$626$	0.701400	0.0280336
$627$	0	0
$628$	23.9145	0.954291
$629$	13.9513	0.556275
$630$	0	0
$631$	18.9736	0.755327	0.377663	−	0.925943i	$-0.376728\pi$
$631$	18.9736	0.755327	0.377663	+	0.925943i	$0.376728\pi$
$632$	6.57398	0.261499
$633$	0	0
$634$	2.03146	0.0806796
$635$	4.50475	0.178765
$636$	0	0
$637$	15.4851	0.613544
$638$	−2.20708	−0.0873792
$639$	0	0
$640$	−6.19934	−0.245050
$641$	−29.6313	−1.17037	−0.585184	−	0.810901i	$-0.698978\pi$
$641$	−29.6313	−1.17037	−0.585184	+	0.810901i	$0.698978\pi$
$642$	0	0
$643$	43.5773	1.71852	0.859260	−	0.511539i	$-0.170924\pi$
$643$	43.5773	1.71852	0.859260	+	0.511539i	$0.170924\pi$
$644$	−1.41147	−0.0556199
$645$	0	0
$646$	6.41147	0.252256
$647$	−21.0583	−0.827887	−0.413944	−	0.910302i	$-0.635849\pi$
$647$	−21.0583	−0.827887	−0.413944	+	0.910302i	$0.635849\pi$
$648$	0	0
$649$	3.23442	0.126962
$650$	3.66220	0.143643
$651$	0	0
$652$	24.0351	0.941286
$653$	1.33956	0.0524209	0.0262104	−	0.999656i	$-0.491656\pi$
$653$	1.33956	0.0524209	0.0262104	+	0.999656i	$0.491656\pi$
$654$	0	0
$655$	6.74928	0.263716
$656$	−30.6483	−1.19661
$657$	0	0
$658$	−0.714193	−0.0278421
$659$	−18.9459	−0.738029	−0.369014	−	0.929424i	$-0.620305\pi$
$659$	−18.9459	−0.738029	−0.369014	+	0.929424i	$0.620305\pi$
$660$	0	0
$661$	27.6117	1.07397	0.536986	−	0.843591i	$-0.319563\pi$
$661$	27.6117	1.07397	0.536986	+	0.843591i	$0.319563\pi$
$662$	4.07098	0.158223
$663$	0	0
$664$	6.19253	0.240317
$665$	−1.45336	−0.0563590
$666$	0	0
$667$	8.96997	0.347319
$668$	−38.9495	−1.50700
$669$	0	0
$670$	1.12506	0.0434648
$671$	8.53209	0.329378
$672$	0	0
$673$	20.7425	0.799563	0.399782	−	0.916610i	$-0.369086\pi$
$673$	20.7425	0.799563	0.399782	+	0.916610i	$0.369086\pi$
$674$	4.37464	0.168505
$675$	0	0
$676$	14.4433	0.555510
$677$	−27.6064	−1.06100	−0.530500	−	0.847685i	$-0.677996\pi$
$677$	−27.6064	−1.06100	−0.530500	+	0.847685i	$0.677996\pi$
$678$	0	0
$679$	6.72874	0.258225
$680$	3.87939	0.148768
$681$	0	0
$682$	0.751030	0.0287584
$683$	31.4843	1.20471	0.602357	−	0.798227i	$-0.294228\pi$
$683$	31.4843	1.20471	0.602357	+	0.798227i	$0.294228\pi$
$684$	0	0
$685$	9.25847	0.353748
$686$	−2.53478	−0.0967782
$687$	0	0
$688$	−40.4730	−1.54302
$689$	−29.1026	−1.10872
$690$	0	0
$691$	−23.8881	−0.908745	−0.454372	−	0.890812i	$-0.650136\pi$
$691$	−23.8881	−0.908745	−0.454372	+	0.890812i	$0.650136\pi$
$692$	42.2918	1.60769
$693$	0	0
$694$	−8.28136	−0.314356
$695$	0.0659325	0.00250096
$696$	0	0
$697$	41.0847	1.55619
$698$	−2.94593	−0.111505
$699$	0	0
$700$	4.57398	0.172880
$701$	7.03684	0.265778	0.132889	−	0.991131i	$-0.457575\pi$
$701$	7.03684	0.265778	0.132889	+	0.991131i	$0.457575\pi$
$702$	0	0
$703$	13.2344	0.499146
$704$	5.24897	0.197828
$705$	0	0
$706$	2.58616	0.0973315
$707$	9.09421	0.342023
$708$	0	0
$709$	−12.9007	−0.484497	−0.242249	−	0.970214i	$-0.577885\pi$
$709$	−12.9007	−0.484497	−0.242249	+	0.970214i	$0.577885\pi$
$710$	2.26352	0.0849483
$711$	0	0
$712$	22.8384	0.855906
$713$	−3.05232	−0.114310
$714$	0	0
$715$	1.50475	0.0562744
$716$	20.1506	0.753065
$717$	0	0
$718$	−7.96316	−0.297183
$719$	17.2671	0.643956	0.321978	−	0.946747i	$-0.395652\pi$
$719$	17.2671	0.643956	0.321978	+	0.946747i	$0.395652\pi$
$720$	0	0
$721$	1.52166	0.0566696
$722$	−0.516607	−0.0192261
$723$	0	0
$724$	7.81345	0.290385
$725$	−29.0678	−1.07955
$726$	0	0
$727$	40.6067	1.50602	0.753009	−	0.658010i	$-0.228601\pi$
$727$	40.6067	1.50602	0.753009	+	0.658010i	$0.228601\pi$
$728$	1.65270	0.0612533
$729$	0	0
$730$	−1.69904	−0.0628841
$731$	54.2550	2.00669
$732$	0	0
$733$	−15.7493	−0.581713	−0.290856	−	0.956767i	$-0.593940\pi$
$733$	−15.7493	−0.581713	−0.290856	+	0.956767i	$0.593940\pi$
$734$	5.31584	0.196211
$735$	0	0
$736$	5.41653	0.199656
$737$	−4.96316	−0.182820
$738$	0	0
$739$	2.27538	0.0837011	0.0418506	−	0.999124i	$-0.486675\pi$
$739$	2.27538	0.0837011	0.0418506	+	0.999124i	$0.486675\pi$
$740$	3.87939	0.142609
$741$	0	0
$742$	2.33275	0.0856379
$743$	−35.1735	−1.29039	−0.645196	−	0.764017i	$-0.723224\pi$
$743$	−35.1735	−1.29039	−0.645196	+	0.764017i	$0.723224\pi$
$744$	0	0
$745$	−1.99825	−0.0732100
$746$	−9.51249	−0.348277
$747$	0	0
$748$	−8.29086	−0.303144
$749$	5.28312	0.193041
$750$	0	0
$751$	−33.5066	−1.22267	−0.611337	−	0.791370i	$-0.709368\pi$
$751$	−33.5066	−1.22267	−0.611337	+	0.791370i	$0.709368\pi$
$752$	−12.7186	−0.463801
$753$	0	0
$754$	−5.08822	−0.185302
$755$	12.6637	0.460881
$756$	0	0
$757$	45.4570	1.65216	0.826081	−	0.563551i	$-0.190565\pi$
$757$	45.4570	1.65216	0.826081	+	0.563551i	$0.190565\pi$
$758$	3.91447	0.142180
$759$	0	0
$760$	3.68004	0.133489
$761$	−29.2739	−1.06118	−0.530590	−	0.847629i	$-0.678030\pi$
$761$	−29.2739	−1.06118	−0.530590	+	0.847629i	$0.678030\pi$
$762$	0	0
$763$	7.35142	0.266139
$764$	−4.51754	−0.163439
$765$	0	0
$766$	11.5868	0.418647
$767$	7.45666	0.269244
$768$	0	0
$769$	−44.3661	−1.59988	−0.799941	−	0.600079i	$-0.795136\pi$
$769$	−44.3661	−1.59988	−0.799941	+	0.600079i	$0.795136\pi$
$770$	−0.120615	−0.00434665
$771$	0	0
$772$	0.852044	0.0306657
$773$	−21.6372	−0.778237	−0.389118	−	0.921188i	$-0.627220\pi$
$773$	−21.6372	−0.778237	−0.389118	+	0.921188i	$0.627220\pi$
$774$	0	0
$775$	9.89124	0.355304
$776$	−17.0378	−0.611620
$777$	0	0
$778$	4.25402	0.152514
$779$	38.9736	1.39637
$780$	0	0
$781$	−9.98545	−0.357308
$782$	−2.16250	−0.0773310
$783$	0	0
$784$	−22.1043	−0.789440
$785$	−8.30541	−0.296433
$786$	0	0
$787$	18.7834	0.669557	0.334778	−	0.942297i	$-0.391339\pi$
$787$	18.7834	0.669557	0.334778	+	0.942297i	$0.391339\pi$
$788$	36.8803	1.31381
$789$	0	0
$790$	−1.10607	−0.0393521
$791$	0.935822	0.0332740
$792$	0	0
$793$	19.6699	0.698500
$794$	−3.96915	−0.140860
$795$	0	0
$796$	−6.82295	−0.241833
$797$	27.7202	0.981899	0.490950	−	0.871188i	$-0.336650\pi$
$797$	27.7202	0.981899	0.490950	+	0.871188i	$0.336650\pi$
$798$	0	0
$799$	17.0496	0.603173
$800$	−17.5526	−0.620579
$801$	0	0
$802$	7.53445	0.266051
$803$	7.49525	0.264502
$804$	0	0
$805$	0.490200	0.0172773
$806$	1.73143	0.0609870
$807$	0	0
$808$	−23.0273	−0.810099
$809$	−5.05737	−0.177808	−0.0889038	−	0.996040i	$-0.528336\pi$
$809$	−5.05737	−0.177808	−0.0889038	+	0.996040i	$0.528336\pi$
$810$	0	0
$811$	25.6709	0.901426	0.450713	−	0.892669i	$-0.351170\pi$
$811$	25.6709	0.901426	0.450713	+	0.892669i	$0.351170\pi$
$812$	−6.35504	−0.223018
$813$	0	0
$814$	1.09833	0.0384963
$815$	−8.34730	−0.292393
$816$	0	0
$817$	51.4671	1.80061
$818$	−9.92902	−0.347160
$819$	0	0
$820$	11.4243	0.398953
$821$	3.86484	0.134884	0.0674419	−	0.997723i	$-0.478516\pi$
$821$	3.86484	0.134884	0.0674419	+	0.997723i	$0.478516\pi$
$822$	0	0
$823$	27.8357	0.970293	0.485146	−	0.874433i	$-0.338766\pi$
$823$	27.8357	0.970293	0.485146	+	0.874433i	$0.338766\pi$
$824$	−3.85298	−0.134225
$825$	0	0
$826$	−0.597697	−0.0207965
$827$	−1.24392	−0.0432553	−0.0216276	−	0.999766i	$-0.506885\pi$
$827$	−1.24392	−0.0432553	−0.0216276	+	0.999766i	$0.506885\pi$
$828$	0	0
$829$	−31.5794	−1.09680	−0.548398	−	0.836217i	$-0.684762\pi$
$829$	−31.5794	−1.09680	−0.548398	+	0.836217i	$0.684762\pi$
$830$	−1.04189	−0.0361645
$831$	0	0
$832$	12.1010	0.419527
$833$	29.6313	1.02667
$834$	0	0
$835$	13.5270	0.468122
$836$	−7.86484	−0.272011
$837$	0	0
$838$	−2.99050	−0.103305
$839$	9.86720	0.340654	0.170327	−	0.985388i	$-0.445518\pi$
$839$	9.86720	0.340654	0.170327	+	0.985388i	$0.445518\pi$
$840$	0	0
$841$	11.3865	0.392638
$842$	−11.9581	−0.412104
$843$	0	0
$844$	−40.7324	−1.40207
$845$	−5.01609	−0.172559
$846$	0	0
$847$	0.532089	0.0182828
$848$	41.5425	1.42658
$849$	0	0
$850$	7.00774	0.240364
$851$	−4.46379	−0.153017
$852$	0	0
$853$	57.5366	1.97002	0.985009	−	0.172505i	$-0.0551862\pi$
$853$	57.5366	1.97002	0.985009	+	0.172505i	$0.0551862\pi$
$854$	−1.57667	−0.0539524
$855$	0	0
$856$	−13.3773	−0.457228
$857$	−15.4037	−0.526182	−0.263091	−	0.964771i	$-0.584742\pi$
$857$	−15.4037	−0.526182	−0.263091	+	0.964771i	$0.584742\pi$
$858$	0	0
$859$	−42.5732	−1.45258	−0.726289	−	0.687390i	$-0.758756\pi$
$859$	−42.5732	−1.45258	−0.726289	+	0.687390i	$0.758756\pi$
$860$	15.0865	0.514444
$861$	0	0
$862$	4.94768	0.168519
$863$	−38.3536	−1.30557	−0.652786	−	0.757542i	$-0.726400\pi$
$863$	−38.3536	−1.30557	−0.652786	+	0.757542i	$0.726400\pi$
$864$	0	0
$865$	−14.6878	−0.499400
$866$	−8.54427	−0.290346
$867$	0	0
$868$	2.16250	0.0734002
$869$	4.87939	0.165522
$870$	0	0
$871$	−11.4421	−0.387701
$872$	−18.6144	−0.630364
$873$	0	0
$874$	−2.05138	−0.0693891
$875$	−3.32501	−0.112406
$876$	0	0
$877$	−12.4703	−0.421091	−0.210546	−	0.977584i	$-0.567524\pi$
$877$	−12.4703	−0.421091	−0.210546	+	0.977584i	$0.567524\pi$
$878$	1.44924	0.0489096
$879$	0	0
$880$	−2.14796	−0.0724076
$881$	−22.9172	−0.772099	−0.386049	−	0.922478i	$-0.626161\pi$
$881$	−22.9172	−0.772099	−0.386049	+	0.922478i	$0.626161\pi$
$882$	0	0
$883$	44.2300	1.48846	0.744229	−	0.667925i	$-0.232817\pi$
$883$	44.2300	1.48846	0.744229	+	0.667925i	$0.232817\pi$
$884$	−19.1138	−0.642867
$885$	0	0
$886$	−10.1634	−0.341447
$887$	24.7743	0.831838	0.415919	−	0.909402i	$-0.363460\pi$
$887$	24.7743	0.831838	0.415919	+	0.909402i	$0.363460\pi$
$888$	0	0
$889$	3.67230	0.123165
$890$	−3.84255	−0.128803
$891$	0	0
$892$	−35.9641	−1.20417
$893$	16.1735	0.541227
$894$	0	0
$895$	−6.99825	−0.233926
$896$	−5.05375	−0.168834
$897$	0	0
$898$	−5.48576	−0.183062
$899$	−13.7428	−0.458348
$900$	0	0
$901$	−55.6887	−1.85526
$902$	3.23442	0.107694
$903$	0	0
$904$	−2.36959	−0.0788112
$905$	−2.71358	−0.0902026
$906$	0	0
$907$	30.4979	1.01267	0.506334	−	0.862338i	$-0.331000\pi$
$907$	30.4979	1.01267	0.506334	+	0.862338i	$0.331000\pi$
$908$	29.3063	0.972565
$909$	0	0
$910$	−0.278066	−0.00921780
$911$	53.8827	1.78521	0.892606	−	0.450837i	$-0.148874\pi$
$911$	53.8827	1.78521	0.892606	+	0.450837i	$0.148874\pi$
$912$	0	0
$913$	4.59627	0.152114
$914$	3.29416	0.108961
$915$	0	0
$916$	28.9641	0.957001
$917$	5.50206	0.181694
$918$	0	0
$919$	5.19841	0.171480	0.0857398	−	0.996318i	$-0.472675\pi$
$919$	5.19841	0.171480	0.0857398	+	0.996318i	$0.472675\pi$
$920$	−1.24123	−0.0409221
$921$	0	0
$922$	−10.0523	−0.331055
$923$	−23.0205	−0.757730
$924$	0	0
$925$	14.4652	0.475614
$926$	−7.94625	−0.261130
$927$	0	0
$928$	24.3874	0.800557
$929$	−23.8735	−0.783265	−0.391632	−	0.920122i	$-0.628089\pi$
$929$	−23.8735	−0.783265	−0.391632	+	0.920122i	$0.628089\pi$
$930$	0	0
$931$	28.1088	0.921227
$932$	−8.06418	−0.264151
$933$	0	0
$934$	3.55674	0.116380
$935$	2.87939	0.0941660
$936$	0	0
$937$	18.7219	0.611619	0.305809	−	0.952093i	$-0.401073\pi$
$937$	18.7219	0.611619	0.305809	+	0.952093i	$0.401073\pi$
$938$	0.917156	0.0299462
$939$	0	0
$940$	4.74092	0.154632
$941$	33.5627	1.09411	0.547057	−	0.837095i	$-0.315748\pi$
$941$	33.5627	1.09411	0.547057	+	0.837095i	$0.315748\pi$
$942$	0	0
$943$	−13.1453	−0.428069
$944$	−10.6440	−0.346434
$945$	0	0
$946$	4.27126	0.138871
$947$	−43.6759	−1.41928	−0.709638	−	0.704566i	$-0.751142\pi$
$947$	−43.6759	−1.41928	−0.709638	+	0.704566i	$0.751142\pi$
$948$	0	0
$949$	17.2796	0.560920
$950$	6.64765	0.215678
$951$	0	0
$952$	3.16250	0.102497
$953$	39.4397	1.27758	0.638789	−	0.769382i	$-0.279436\pi$
$953$	39.4397	1.27758	0.638789	+	0.769382i	$0.279436\pi$
$954$	0	0
$955$	1.56893	0.0507692
$956$	37.8384	1.22378
$957$	0	0
$958$	6.71358	0.216906
$959$	7.54757	0.243724
$960$	0	0
$961$	−26.3236	−0.849148
$962$	2.53209	0.0816378
$963$	0	0
$964$	13.9932	0.450690
$965$	−0.295912	−0.00952574
$966$	0	0
$967$	6.58853	0.211873	0.105936	−	0.994373i	$-0.466216\pi$
$967$	6.58853	0.211873	0.105936	+	0.994373i	$0.466216\pi$
$968$	−1.34730	−0.0433037
$969$	0	0
$970$	2.86659	0.0920407
$971$	17.0327	0.546606	0.273303	−	0.961928i	$-0.411884\pi$
$971$	17.0327	0.546606	0.273303	+	0.961928i	$0.411884\pi$
$972$	0	0
$973$	0.0537486	0.00172310
$974$	9.45007	0.302800
$975$	0	0
$976$	−28.0779	−0.898752
$977$	7.54900	0.241514	0.120757	−	0.992682i	$-0.461468\pi$
$977$	7.54900	0.241514	0.120757	+	0.992682i	$0.461468\pi$
$978$	0	0
$979$	16.9513	0.541766
$980$	8.23947	0.263200
$981$	0	0
$982$	−12.5158	−0.399395
$983$	−3.84699	−0.122700	−0.0613500	−	0.998116i	$-0.519541\pi$
$983$	−3.84699	−0.122700	−0.0613500	+	0.998116i	$0.519541\pi$
$984$	0	0
$985$	−12.8084	−0.408110
$986$	−9.73648	−0.310073
$987$	0	0
$988$	−18.1317	−0.576845
$989$	−17.3592	−0.551989
$990$	0	0
$991$	40.1121	1.27420	0.637101	−	0.770781i	$-0.280134\pi$
$991$	40.1121	1.27420	0.637101	+	0.770781i	$0.280134\pi$
$992$	−8.29860	−0.263481
$993$	0	0
$994$	1.84524	0.0585274
$995$	2.36959	0.0751209
$996$	0	0
$997$	−16.5648	−0.524613	−0.262306	−	0.964985i	$-0.584483\pi$
$997$	−16.5648	−0.524613	−0.262306	+	0.964985i	$0.584483\pi$
$998$	−2.74092	−0.0867625
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.l.1.2		3
3.2	odd	2		891.2.a.k.1.2		3
9.2	odd	6		297.2.e.d.199.2		6
9.4	even	3		99.2.e.d.34.2	✓	6
9.5	odd	6		297.2.e.d.100.2		6
9.7	even	3		99.2.e.d.67.2	yes	6
11.10	odd	2		9801.2.a.be.1.2		3
33.32	even	2		9801.2.a.bd.1.2		3
99.43	odd	6		1089.2.e.h.364.2		6
99.76	odd	6		1089.2.e.h.727.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.d.34.2	✓	6	9.4	even	3
99.2.e.d.67.2	yes	6	9.7	even	3
297.2.e.d.100.2		6	9.5	odd	6
297.2.e.d.199.2		6	9.2	odd	6
891.2.a.k.1.2		3	3.2	odd	2
891.2.a.l.1.2		3	1.1	even	1	trivial
1089.2.e.h.364.2		6	99.43	odd	6
1089.2.e.h.727.2		6	99.76	odd	6
9801.2.a.bd.1.2		3	33.32	even	2
9801.2.a.be.1.2		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)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{4} +0.652704 q^{5} +0.532089 q^{7} -1.34730 q^{8} +0.226682 q^{10} -1.00000 q^{11} -2.30541 q^{13} +0.184793 q^{14} +3.29086 q^{16} -4.41147 q^{17} -4.18479 q^{19} -1.22668 q^{20} -0.347296 q^{22} +1.41147 q^{23} -4.57398 q^{25} -0.800660 q^{26} -1.00000 q^{28} +6.35504 q^{29} -2.16250 q^{31} +3.83750 q^{32} -1.53209 q^{34} +0.347296 q^{35} -3.16250 q^{37} -1.45336 q^{38} -0.879385 q^{40} -9.31315 q^{41} -12.2986 q^{43} +1.87939 q^{44} +0.490200 q^{46} -3.86484 q^{47} -6.71688 q^{49} -1.58853 q^{50} +4.33275 q^{52} +12.6236 q^{53} -0.652704 q^{55} -0.716881 q^{56} +2.20708 q^{58} -3.23442 q^{59} -8.53209 q^{61} -0.751030 q^{62} -5.24897 q^{64} -1.50475 q^{65} +4.96316 q^{67} +8.29086 q^{68} +0.120615 q^{70} +9.98545 q^{71} -7.49525 q^{73} -1.09833 q^{74} +7.86484 q^{76} -0.532089 q^{77} -4.87939 q^{79} +2.14796 q^{80} -3.23442 q^{82} -4.59627 q^{83} -2.87939 q^{85} -4.27126 q^{86} +1.34730 q^{88} -16.9513 q^{89} -1.22668 q^{91} -2.65270 q^{92} -1.34224 q^{94} -2.73143 q^{95} +12.6459 q^{97} -2.33275 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} - 3 q^{17} - 9 q^{19} + 3 q^{20} - 6 q^{23} - 6 q^{25} + 12 q^{26} - 3 q^{28} - 6 q^{29} - 9 q^{31} + 9 q^{32} - 12 q^{37}+ \cdots + 12 q^{98}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more