LMFDB - Embedded newform 1089.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.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:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} +0.618034 q^{5} -1.00000 q^{7} +7.47214 q^{8} +1.61803 q^{10} +0.236068 q^{13} -2.61803 q^{14} +9.85410 q^{16} +1.14590 q^{17} +5.85410 q^{19} +3.00000 q^{20} -0.236068 q^{23} -4.61803 q^{25} +0.618034 q^{26} -4.85410 q^{28} +6.00000 q^{29} -6.09017 q^{31} +10.8541 q^{32} +3.00000 q^{34} -0.618034 q^{35} -6.23607 q^{37} +15.3262 q^{38} +4.61803 q^{40} -0.236068 q^{41} -6.70820 q^{43} -0.618034 q^{46} +10.0902 q^{47} -6.00000 q^{49} -12.0902 q^{50} +1.14590 q^{52} +0.381966 q^{53} -7.47214 q^{56} +15.7082 q^{58} -7.38197 q^{59} -11.5623 q^{61} -15.9443 q^{62} +8.70820 q^{64} +0.145898 q^{65} +1.85410 q^{67} +5.56231 q^{68} -1.61803 q^{70} -10.3262 q^{71} -5.70820 q^{73} -16.3262 q^{74} +28.4164 q^{76} +11.0000 q^{79} +6.09017 q^{80} -0.618034 q^{82} -1.47214 q^{83} +0.708204 q^{85} -17.5623 q^{86} +8.23607 q^{89} -0.236068 q^{91} -1.14590 q^{92} +26.4164 q^{94} +3.61803 q^{95} +7.85410 q^{97} -15.7082 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8} + q^{10} - 4 q^{13} - 3 q^{14} + 13 q^{16} + 9 q^{17} + 5 q^{19} + 6 q^{20} + 4 q^{23} - 7 q^{25} - q^{26} - 3 q^{28} + 12 q^{29} - q^{31} + 15 q^{32}+ \cdots - 18 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - q^5 - 2 * q^7 + 6 * q^8 + q^10 - 4 * q^13 - 3 * q^14 + 13 * q^16 + 9 * q^17 + 5 * q^19 + 6 * q^20 + 4 * q^23 - 7 * q^25 - q^26 - 3 * q^28 + 12 * q^29 - q^31 + 15 * q^32 + 6 * q^34 + q^35 - 8 * q^37 + 15 * q^38 + 7 * q^40 + 4 * q^41 + q^46 + 9 * q^47 - 12 * q^49 - 13 * q^50 + 9 * q^52 + 3 * q^53 - 6 * q^56 + 18 * q^58 - 17 * q^59 - 3 * q^61 - 14 * q^62 + 4 * q^64 + 7 * q^65 - 3 * q^67 - 9 * q^68 - q^70 - 5 * q^71 + 2 * q^73 - 17 * q^74 + 30 * q^76 + 22 * q^79 + q^80 + q^82 + 6 * q^83 - 12 * q^85 - 15 * q^86 + 12 * q^89 + 4 * q^91 - 9 * q^92 + 26 * q^94 + 5 * q^95 + 9 * q^97 - 18 * 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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	7.47214	2.64180
$9$	0	0
$10$	1.61803	0.511667
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	−2.61803	−0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	1.14590	0.277921	0.138961	−	0.990298i	$-0.455624\pi$
$17$	1.14590	0.277921	0.138961	+	0.990298i	$0.455624\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	0	0
$23$	−0.236068	−0.0492236	−0.0246118	−	0.999697i	$-0.507835\pi$
$23$	−0.236068	−0.0492236	−0.0246118	+	0.999697i	$0.507835\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0.618034	0.121206
$27$	0	0
$28$	−4.85410	−0.917339
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−6.09017	−1.09383	−0.546913	−	0.837189i	$-0.684197\pi$
$31$	−6.09017	−1.09383	−0.546913	+	0.837189i	$0.684197\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	3.00000	0.514496
$35$	−0.618034	−0.104467
$36$	0	0
$37$	−6.23607	−1.02520	−0.512602	−	0.858627i	$-0.671318\pi$
$37$	−6.23607	−1.02520	−0.512602	+	0.858627i	$0.671318\pi$
$38$	15.3262	2.48624
$39$	0	0
$40$	4.61803	0.730175
$41$	−0.236068	−0.0368676	−0.0184338	−	0.999830i	$-0.505868\pi$
$41$	−0.236068	−0.0368676	−0.0184338	+	0.999830i	$0.505868\pi$
$42$	0	0
$43$	−6.70820	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$43$	−6.70820	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$44$	0	0
$45$	0	0
$46$	−0.618034	−0.0911241
$47$	10.0902	1.47180	0.735901	−	0.677089i	$-0.236759\pi$
$47$	10.0902	1.47180	0.735901	+	0.677089i	$0.236759\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−12.0902	−1.70981
$51$	0	0
$52$	1.14590	0.158907
$53$	0.381966	0.0524671	0.0262335	−	0.999656i	$-0.491649\pi$
$53$	0.381966	0.0524671	0.0262335	+	0.999656i	$0.491649\pi$
$54$	0	0
$55$	0	0
$56$	−7.47214	−0.998506
$57$	0	0
$58$	15.7082	2.06259
$59$	−7.38197	−0.961050	−0.480525	−	0.876981i	$-0.659554\pi$
$59$	−7.38197	−0.961050	−0.480525	+	0.876981i	$0.659554\pi$
$60$	0	0
$61$	−11.5623	−1.48040	−0.740201	−	0.672386i	$-0.765270\pi$
$61$	−11.5623	−1.48040	−0.740201	+	0.672386i	$0.765270\pi$
$62$	−15.9443	−2.02492
$63$	0	0
$64$	8.70820	1.08853
$65$	0.145898	0.0180964
$66$	0	0
$67$	1.85410	0.226515	0.113257	−	0.993566i	$-0.463872\pi$
$67$	1.85410	0.226515	0.113257	+	0.993566i	$0.463872\pi$
$68$	5.56231	0.674529
$69$	0	0
$70$	−1.61803	−0.193392
$71$	−10.3262	−1.22550	−0.612749	−	0.790277i	$-0.709937\pi$
$71$	−10.3262	−1.22550	−0.612749	+	0.790277i	$0.709937\pi$
$72$	0	0
$73$	−5.70820	−0.668095	−0.334047	−	0.942556i	$-0.608415\pi$
$73$	−5.70820	−0.668095	−0.334047	+	0.942556i	$0.608415\pi$
$74$	−16.3262	−1.89789
$75$	0	0
$76$	28.4164	3.25959
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	6.09017	0.680902
$81$	0	0
$82$	−0.618034	−0.0682504
$83$	−1.47214	−0.161588	−0.0807940	−	0.996731i	$-0.525746\pi$
$83$	−1.47214	−0.161588	−0.0807940	+	0.996731i	$0.525746\pi$
$84$	0	0
$85$	0.708204	0.0768155
$86$	−17.5623	−1.89379
$87$	0	0
$88$	0	0
$89$	8.23607	0.873021	0.436511	−	0.899699i	$-0.356214\pi$
$89$	8.23607	0.873021	0.436511	+	0.899699i	$0.356214\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	−1.14590	−0.119468
$93$	0	0
$94$	26.4164	2.72464
$95$	3.61803	0.371202
$96$	0	0
$97$	7.85410	0.797463	0.398732	−	0.917068i	$-0.369451\pi$
$97$	7.85410	0.797463	0.398732	+	0.917068i	$0.369451\pi$
$98$	−15.7082	−1.58677
$99$	0	0
$100$	−22.4164	−2.24164
$101$	10.2361	1.01853	0.509263	−	0.860611i	$-0.329918\pi$
$101$	10.2361	1.01853	0.509263	+	0.860611i	$0.329918\pi$
$102$	0	0
$103$	−10.9443	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$103$	−10.9443	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$104$	1.76393	0.172968
$105$	0	0
$106$	1.00000	0.0971286
$107$	11.4721	1.10905	0.554527	−	0.832166i	$-0.312899\pi$
$107$	11.4721	1.10905	0.554527	+	0.832166i	$0.312899\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	−9.85410	−0.931125
$113$	−13.4721	−1.26735	−0.633676	−	0.773599i	$-0.718455\pi$
$113$	−13.4721	−1.26735	−0.633676	+	0.773599i	$0.718455\pi$
$114$	0	0
$115$	−0.145898	−0.0136051
$116$	29.1246	2.70415
$117$	0	0
$118$	−19.3262	−1.77912
$119$	−1.14590	−0.105044
$120$	0	0
$121$	0	0
$122$	−30.2705	−2.74056
$123$	0	0
$124$	−29.5623	−2.65477
$125$	−5.94427	−0.531672
$126$	0	0
$127$	7.70820	0.683992	0.341996	−	0.939701i	$-0.388897\pi$
$127$	7.70820	0.683992	0.341996	+	0.939701i	$0.388897\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	0.381966	0.0335006
$131$	11.7984	1.03083	0.515414	−	0.856941i	$-0.327638\pi$
$131$	11.7984	1.03083	0.515414	+	0.856941i	$0.327638\pi$
$132$	0	0
$133$	−5.85410	−0.507615
$134$	4.85410	0.419331
$135$	0	0
$136$	8.56231	0.734212
$137$	−9.76393	−0.834189	−0.417095	−	0.908863i	$-0.636952\pi$
$137$	−9.76393	−0.834189	−0.417095	+	0.908863i	$0.636952\pi$
$138$	0	0
$139$	14.5623	1.23516	0.617579	−	0.786509i	$-0.288113\pi$
$139$	14.5623	1.23516	0.617579	+	0.786509i	$0.288113\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−27.0344	−2.26868
$143$	0	0
$144$	0	0
$145$	3.70820	0.307950
$146$	−14.9443	−1.23680
$147$	0	0
$148$	−30.2705	−2.48822
$149$	4.23607	0.347032	0.173516	−	0.984831i	$-0.444487\pi$
$149$	4.23607	0.347032	0.173516	+	0.984831i	$0.444487\pi$
$150$	0	0
$151$	1.05573	0.0859139	0.0429570	−	0.999077i	$-0.486322\pi$
$151$	1.05573	0.0859139	0.0429570	+	0.999077i	$0.486322\pi$
$152$	43.7426	3.54800
$153$	0	0
$154$	0	0
$155$	−3.76393	−0.302326
$156$	0	0
$157$	15.7082	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$157$	15.7082	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$158$	28.7984	2.29108
$159$	0	0
$160$	6.70820	0.530330
$161$	0.236068	0.0186048
$162$	0	0
$163$	−5.14590	−0.403058	−0.201529	−	0.979483i	$-0.564591\pi$
$163$	−5.14590	−0.403058	−0.201529	+	0.979483i	$0.564591\pi$
$164$	−1.14590	−0.0894796
$165$	0	0
$166$	−3.85410	−0.299136
$167$	−12.0344	−0.931253	−0.465627	−	0.884981i	$-0.654171\pi$
$167$	−12.0344	−0.931253	−0.465627	+	0.884981i	$0.654171\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	1.85410	0.142203
$171$	0	0
$172$	−32.5623	−2.48285
$173$	18.0344	1.37113	0.685567	−	0.728010i	$-0.259555\pi$
$173$	18.0344	1.37113	0.685567	+	0.728010i	$0.259555\pi$
$174$	0	0
$175$	4.61803	0.349091
$176$	0	0
$177$	0	0
$178$	21.5623	1.61616
$179$	8.52786	0.637402	0.318701	−	0.947855i	$-0.396753\pi$
$179$	8.52786	0.637402	0.318701	+	0.947855i	$0.396753\pi$
$180$	0	0
$181$	2.52786	0.187895	0.0939473	−	0.995577i	$-0.470051\pi$
$181$	2.52786	0.187895	0.0939473	+	0.995577i	$0.470051\pi$
$182$	−0.618034	−0.0458117
$183$	0	0
$184$	−1.76393	−0.130039
$185$	−3.85410	−0.283359
$186$	0	0
$187$	0	0
$188$	48.9787	3.57214
$189$	0	0
$190$	9.47214	0.687181
$191$	0.819660	0.0593085	0.0296543	−	0.999560i	$-0.490559\pi$
$191$	0.819660	0.0593085	0.0296543	+	0.999560i	$0.490559\pi$
$192$	0	0
$193$	−3.14590	−0.226447	−0.113223	−	0.993570i	$-0.536118\pi$
$193$	−3.14590	−0.226447	−0.113223	+	0.993570i	$0.536118\pi$
$194$	20.5623	1.47629
$195$	0	0
$196$	−29.1246	−2.08033
$197$	−13.0344	−0.928666	−0.464333	−	0.885661i	$-0.653706\pi$
$197$	−13.0344	−0.928666	−0.464333	+	0.885661i	$0.653706\pi$
$198$	0	0
$199$	6.70820	0.475532	0.237766	−	0.971322i	$-0.423585\pi$
$199$	6.70820	0.475532	0.237766	+	0.971322i	$0.423585\pi$
$200$	−34.5066	−2.43998
$201$	0	0
$202$	26.7984	1.88553
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−0.145898	−0.0101900
$206$	−28.6525	−1.99631
$207$	0	0
$208$	2.32624	0.161296
$209$	0	0
$210$	0	0
$211$	3.61803	0.249076	0.124538	−	0.992215i	$-0.460255\pi$
$211$	3.61803	0.249076	0.124538	+	0.992215i	$0.460255\pi$
$212$	1.85410	0.127340
$213$	0	0
$214$	30.0344	2.05311
$215$	−4.14590	−0.282748
$216$	0	0
$217$	6.09017	0.413428
$218$	−31.4164	−2.12779
$219$	0	0
$220$	0	0
$221$	0.270510	0.0181965
$222$	0	0
$223$	7.18034	0.480831	0.240416	−	0.970670i	$-0.422716\pi$
$223$	7.18034	0.480831	0.240416	+	0.970670i	$0.422716\pi$
$224$	−10.8541	−0.725220
$225$	0	0
$226$	−35.2705	−2.34616
$227$	−13.1803	−0.874810	−0.437405	−	0.899265i	$-0.644102\pi$
$227$	−13.1803	−0.874810	−0.437405	+	0.899265i	$0.644102\pi$
$228$	0	0
$229$	0.472136	0.0311996	0.0155998	−	0.999878i	$-0.495034\pi$
$229$	0.472136	0.0311996	0.0155998	+	0.999878i	$0.495034\pi$
$230$	−0.381966	−0.0251861
$231$	0	0
$232$	44.8328	2.94342
$233$	4.14590	0.271607	0.135803	−	0.990736i	$-0.456638\pi$
$233$	4.14590	0.271607	0.135803	+	0.990736i	$0.456638\pi$
$234$	0	0
$235$	6.23607	0.406796
$236$	−35.8328	−2.33252
$237$	0	0
$238$	−3.00000	−0.194461
$239$	−0.381966	−0.0247073	−0.0123537	−	0.999924i	$-0.503932\pi$
$239$	−0.381966	−0.0247073	−0.0123537	+	0.999924i	$0.503932\pi$
$240$	0	0
$241$	8.29180	0.534122	0.267061	−	0.963680i	$-0.413948\pi$
$241$	8.29180	0.534122	0.267061	+	0.963680i	$0.413948\pi$
$242$	0	0
$243$	0	0
$244$	−56.1246	−3.59301
$245$	−3.70820	−0.236908
$246$	0	0
$247$	1.38197	0.0879324
$248$	−45.5066	−2.88967
$249$	0	0
$250$	−15.5623	−0.984247
$251$	21.9787	1.38728	0.693642	−	0.720320i	$-0.256005\pi$
$251$	21.9787	1.38728	0.693642	+	0.720320i	$0.256005\pi$
$252$	0	0
$253$	0	0
$254$	20.1803	1.26623
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−29.7426	−1.85530	−0.927648	−	0.373457i	$-0.878172\pi$
$257$	−29.7426	−1.85530	−0.927648	+	0.373457i	$0.878172\pi$
$258$	0	0
$259$	6.23607	0.387490
$260$	0.708204	0.0439209
$261$	0	0
$262$	30.8885	1.90830
$263$	15.2705	0.941620	0.470810	−	0.882235i	$-0.343962\pi$
$263$	15.2705	0.941620	0.470810	+	0.882235i	$0.343962\pi$
$264$	0	0
$265$	0.236068	0.0145015
$266$	−15.3262	−0.939712
$267$	0	0
$268$	9.00000	0.549762
$269$	25.4164	1.54967	0.774833	−	0.632166i	$-0.217834\pi$
$269$	25.4164	1.54967	0.774833	+	0.632166i	$0.217834\pi$
$270$	0	0
$271$	18.6180	1.13097	0.565483	−	0.824760i	$-0.308690\pi$
$271$	18.6180	1.13097	0.565483	+	0.824760i	$0.308690\pi$
$272$	11.2918	0.684666
$273$	0	0
$274$	−25.5623	−1.54428
$275$	0	0
$276$	0	0
$277$	−29.2148	−1.75535	−0.877673	−	0.479260i	$-0.840905\pi$
$277$	−29.2148	−1.75535	−0.877673	+	0.479260i	$0.840905\pi$
$278$	38.1246	2.28656
$279$	0	0
$280$	−4.61803	−0.275980
$281$	−24.7639	−1.47729	−0.738646	−	0.674093i	$-0.764535\pi$
$281$	−24.7639	−1.47729	−0.738646	+	0.674093i	$0.764535\pi$
$282$	0	0
$283$	−5.70820	−0.339318	−0.169659	−	0.985503i	$-0.554267\pi$
$283$	−5.70820	−0.339318	−0.169659	+	0.985503i	$0.554267\pi$
$284$	−50.1246	−2.97435
$285$	0	0
$286$	0	0
$287$	0.236068	0.0139347
$288$	0	0
$289$	−15.6869	−0.922760
$290$	9.70820	0.570085
$291$	0	0
$292$	−27.7082	−1.62150
$293$	21.6525	1.26495	0.632476	−	0.774580i	$-0.282039\pi$
$293$	21.6525	1.26495	0.632476	+	0.774580i	$0.282039\pi$
$294$	0	0
$295$	−4.56231	−0.265628
$296$	−46.5967	−2.70838
$297$	0	0
$298$	11.0902	0.642436
$299$	−0.0557281	−0.00322284
$300$	0	0
$301$	6.70820	0.386654
$302$	2.76393	0.159046
$303$	0	0
$304$	57.6869	3.30857
$305$	−7.14590	−0.409173
$306$	0	0
$307$	27.9787	1.59683	0.798415	−	0.602108i	$-0.205672\pi$
$307$	27.9787	1.59683	0.798415	+	0.602108i	$0.205672\pi$
$308$	0	0
$309$	0	0
$310$	−9.85410	−0.559675
$311$	−11.6525	−0.660751	−0.330376	−	0.943850i	$-0.607175\pi$
$311$	−11.6525	−0.660751	−0.330376	+	0.943850i	$0.607175\pi$
$312$	0	0
$313$	−2.52786	−0.142883	−0.0714417	−	0.997445i	$-0.522760\pi$
$313$	−2.52786	−0.142883	−0.0714417	+	0.997445i	$0.522760\pi$
$314$	41.1246	2.32080
$315$	0	0
$316$	53.3951	3.00371
$317$	6.81966	0.383030	0.191515	−	0.981490i	$-0.438660\pi$
$317$	6.81966	0.383030	0.191515	+	0.981490i	$0.438660\pi$
$318$	0	0
$319$	0	0
$320$	5.38197	0.300861
$321$	0	0
$322$	0.618034	0.0344417
$323$	6.70820	0.373254
$324$	0	0
$325$	−1.09017	−0.0604717
$326$	−13.4721	−0.746153
$327$	0	0
$328$	−1.76393	−0.0973969
$329$	−10.0902	−0.556289
$330$	0	0
$331$	16.7082	0.918366	0.459183	−	0.888342i	$-0.348142\pi$
$331$	16.7082	0.918366	0.459183	+	0.888342i	$0.348142\pi$
$332$	−7.14590	−0.392182
$333$	0	0
$334$	−31.5066	−1.72396
$335$	1.14590	0.0626071
$336$	0	0
$337$	−18.1803	−0.990346	−0.495173	−	0.868794i	$-0.664895\pi$
$337$	−18.1803	−0.990346	−0.495173	+	0.868794i	$0.664895\pi$
$338$	−33.8885	−1.84329
$339$	0	0
$340$	3.43769	0.186435
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−50.1246	−2.70254
$345$	0	0
$346$	47.2148	2.53828
$347$	1.52786	0.0820200	0.0410100	−	0.999159i	$-0.486942\pi$
$347$	1.52786	0.0820200	0.0410100	+	0.999159i	$0.486942\pi$
$348$	0	0
$349$	12.7082	0.680255	0.340127	−	0.940379i	$-0.389530\pi$
$349$	12.7082	0.680255	0.340127	+	0.940379i	$0.389530\pi$
$350$	12.0902	0.646247
$351$	0	0
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	−6.38197	−0.338720
$356$	39.9787	2.11887
$357$	0	0
$358$	22.3262	1.17998
$359$	9.70820	0.512379	0.256190	−	0.966627i	$-0.417533\pi$
$359$	9.70820	0.512379	0.256190	+	0.966627i	$0.417533\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	6.61803	0.347836
$363$	0	0
$364$	−1.14590	−0.0600614
$365$	−3.52786	−0.184657
$366$	0	0
$367$	22.1459	1.15601	0.578003	−	0.816034i	$-0.303832\pi$
$367$	22.1459	1.15601	0.578003	+	0.816034i	$0.303832\pi$
$368$	−2.32624	−0.121264
$369$	0	0
$370$	−10.0902	−0.524563
$371$	−0.381966	−0.0198307
$372$	0	0
$373$	0.888544	0.0460071	0.0230035	−	0.999735i	$-0.492677\pi$
$373$	0.888544	0.0460071	0.0230035	+	0.999735i	$0.492677\pi$
$374$	0	0
$375$	0	0
$376$	75.3951	3.88821
$377$	1.41641	0.0729487
$378$	0	0
$379$	−24.8885	−1.27844	−0.639219	−	0.769024i	$-0.720742\pi$
$379$	−24.8885	−1.27844	−0.639219	+	0.769024i	$0.720742\pi$
$380$	17.5623	0.900927
$381$	0	0
$382$	2.14590	0.109794
$383$	12.7082	0.649359	0.324679	−	0.945824i	$-0.394744\pi$
$383$	12.7082	0.649359	0.324679	+	0.945824i	$0.394744\pi$
$384$	0	0
$385$	0	0
$386$	−8.23607	−0.419205
$387$	0	0
$388$	38.1246	1.93548
$389$	36.7426	1.86293	0.931463	−	0.363836i	$-0.118533\pi$
$389$	36.7426	1.86293	0.931463	+	0.363836i	$0.118533\pi$
$390$	0	0
$391$	−0.270510	−0.0136803
$392$	−44.8328	−2.26440
$393$	0	0
$394$	−34.1246	−1.71917
$395$	6.79837	0.342063
$396$	0	0
$397$	−18.7082	−0.938938	−0.469469	−	0.882949i	$-0.655555\pi$
$397$	−18.7082	−0.938938	−0.469469	+	0.882949i	$0.655555\pi$
$398$	17.5623	0.880319
$399$	0	0
$400$	−45.5066	−2.27533
$401$	31.6869	1.58237	0.791185	−	0.611577i	$-0.209465\pi$
$401$	31.6869	1.58237	0.791185	+	0.611577i	$0.209465\pi$
$402$	0	0
$403$	−1.43769	−0.0716166
$404$	49.6869	2.47202
$405$	0	0
$406$	−15.7082	−0.779585
$407$	0	0
$408$	0	0
$409$	−6.47214	−0.320027	−0.160013	−	0.987115i	$-0.551154\pi$
$409$	−6.47214	−0.320027	−0.160013	+	0.987115i	$0.551154\pi$
$410$	−0.381966	−0.0188640
$411$	0	0
$412$	−53.1246	−2.61726
$413$	7.38197	0.363243
$414$	0	0
$415$	−0.909830	−0.0446618
$416$	2.56231	0.125627
$417$	0	0
$418$	0	0
$419$	−31.4508	−1.53647	−0.768237	−	0.640165i	$-0.778866\pi$
$419$	−31.4508	−1.53647	−0.768237	+	0.640165i	$0.778866\pi$
$420$	0	0
$421$	10.5066	0.512059	0.256030	−	0.966669i	$-0.417586\pi$
$421$	10.5066	0.512059	0.256030	+	0.966669i	$0.417586\pi$
$422$	9.47214	0.461096
$423$	0	0
$424$	2.85410	0.138607
$425$	−5.29180	−0.256690
$426$	0	0
$427$	11.5623	0.559539
$428$	55.6869	2.69173
$429$	0	0
$430$	−10.8541	−0.523431
$431$	5.90983	0.284666	0.142333	−	0.989819i	$-0.454540\pi$
$431$	5.90983	0.284666	0.142333	+	0.989819i	$0.454540\pi$
$432$	0	0
$433$	35.3050	1.69665	0.848324	−	0.529478i	$-0.177612\pi$
$433$	35.3050	1.69665	0.848324	+	0.529478i	$0.177612\pi$
$434$	15.9443	0.765350
$435$	0	0
$436$	−58.2492	−2.78963
$437$	−1.38197	−0.0661084
$438$	0	0
$439$	23.2918	1.11166	0.555828	−	0.831297i	$-0.312401\pi$
$439$	23.2918	1.11166	0.555828	+	0.831297i	$0.312401\pi$
$440$	0	0
$441$	0	0
$442$	0.708204	0.0336858
$443$	31.5967	1.50121	0.750603	−	0.660753i	$-0.229763\pi$
$443$	31.5967	1.50121	0.750603	+	0.660753i	$0.229763\pi$
$444$	0	0
$445$	5.09017	0.241297
$446$	18.7984	0.890129
$447$	0	0
$448$	−8.70820	−0.411424
$449$	9.05573	0.427366	0.213683	−	0.976903i	$-0.431454\pi$
$449$	9.05573	0.427366	0.213683	+	0.976903i	$0.431454\pi$
$450$	0	0
$451$	0	0
$452$	−65.3951	−3.07593
$453$	0	0
$454$	−34.5066	−1.61947
$455$	−0.145898	−0.00683981
$456$	0	0
$457$	23.9787	1.12168	0.560838	−	0.827925i	$-0.310479\pi$
$457$	23.9787	1.12168	0.560838	+	0.827925i	$0.310479\pi$
$458$	1.23607	0.0577577
$459$	0	0
$460$	−0.708204	−0.0330202
$461$	−9.27051	−0.431771	−0.215885	−	0.976419i	$-0.569264\pi$
$461$	−9.27051	−0.431771	−0.215885	+	0.976419i	$0.569264\pi$
$462$	0	0
$463$	1.72949	0.0803762	0.0401881	−	0.999192i	$-0.487204\pi$
$463$	1.72949	0.0803762	0.0401881	+	0.999192i	$0.487204\pi$
$464$	59.1246	2.74479
$465$	0	0
$466$	10.8541	0.502807
$467$	−20.8885	−0.966607	−0.483303	−	0.875453i	$-0.660563\pi$
$467$	−20.8885	−0.966607	−0.483303	+	0.875453i	$0.660563\pi$
$468$	0	0
$469$	−1.85410	−0.0856145
$470$	16.3262	0.753073
$471$	0	0
$472$	−55.1591	−2.53890
$473$	0	0
$474$	0	0
$475$	−27.0344	−1.24043
$476$	−5.56231	−0.254948
$477$	0	0
$478$	−1.00000	−0.0457389
$479$	28.3820	1.29681	0.648403	−	0.761298i	$-0.275437\pi$
$479$	28.3820	1.29681	0.648403	+	0.761298i	$0.275437\pi$
$480$	0	0
$481$	−1.47214	−0.0671236
$482$	21.7082	0.988782
$483$	0	0
$484$	0	0
$485$	4.85410	0.220413
$486$	0	0
$487$	−12.7082	−0.575864	−0.287932	−	0.957651i	$-0.592968\pi$
$487$	−12.7082	−0.575864	−0.287932	+	0.957651i	$0.592968\pi$
$488$	−86.3951	−3.91092
$489$	0	0
$490$	−9.70820	−0.438572
$491$	17.9098	0.808259	0.404130	−	0.914702i	$-0.367574\pi$
$491$	17.9098	0.808259	0.404130	+	0.914702i	$0.367574\pi$
$492$	0	0
$493$	6.87539	0.309652
$494$	3.61803	0.162783
$495$	0	0
$496$	−60.0132	−2.69467
$497$	10.3262	0.463195
$498$	0	0
$499$	−18.1459	−0.812322	−0.406161	−	0.913802i	$-0.633133\pi$
$499$	−18.1459	−0.812322	−0.406161	+	0.913802i	$0.633133\pi$
$500$	−28.8541	−1.29039
$501$	0	0
$502$	57.5410	2.56818
$503$	−8.65248	−0.385795	−0.192897	−	0.981219i	$-0.561788\pi$
$503$	−8.65248	−0.385795	−0.192897	+	0.981219i	$0.561788\pi$
$504$	0	0
$505$	6.32624	0.281514
$506$	0	0
$507$	0	0
$508$	37.4164	1.66008
$509$	−38.7426	−1.71724	−0.858619	−	0.512615i	$-0.828677\pi$
$509$	−38.7426	−1.71724	−0.858619	+	0.512615i	$0.828677\pi$
$510$	0	0
$511$	5.70820	0.252516
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−77.8673	−3.43458
$515$	−6.76393	−0.298054
$516$	0	0
$517$	0	0
$518$	16.3262	0.717334
$519$	0	0
$520$	1.09017	0.0478071
$521$	−8.94427	−0.391856	−0.195928	−	0.980618i	$-0.562772\pi$
$521$	−8.94427	−0.391856	−0.195928	+	0.980618i	$0.562772\pi$
$522$	0	0
$523$	18.2705	0.798914	0.399457	−	0.916752i	$-0.369199\pi$
$523$	18.2705	0.798914	0.399457	+	0.916752i	$0.369199\pi$
$524$	57.2705	2.50187
$525$	0	0
$526$	39.9787	1.74315
$527$	−6.97871	−0.303998
$528$	0	0
$529$	−22.9443	−0.997577
$530$	0.618034	0.0268457
$531$	0	0
$532$	−28.4164	−1.23201
$533$	−0.0557281	−0.00241385
$534$	0	0
$535$	7.09017	0.306535
$536$	13.8541	0.598406
$537$	0	0
$538$	66.5410	2.86879
$539$	0	0
$540$	0	0
$541$	−7.49342	−0.322167	−0.161084	−	0.986941i	$-0.551499\pi$
$541$	−7.49342	−0.322167	−0.161084	+	0.986941i	$0.551499\pi$
$542$	48.7426	2.09368
$543$	0	0
$544$	12.4377	0.533262
$545$	−7.41641	−0.317684
$546$	0	0
$547$	−30.7426	−1.31446	−0.657230	−	0.753690i	$-0.728272\pi$
$547$	−30.7426	−1.31446	−0.657230	+	0.753690i	$0.728272\pi$
$548$	−47.3951	−2.02462
$549$	0	0
$550$	0	0
$551$	35.1246	1.49636
$552$	0	0
$553$	−11.0000	−0.467768
$554$	−76.4853	−3.24955
$555$	0	0
$556$	70.6869	2.99779
$557$	−37.6312	−1.59448	−0.797242	−	0.603659i	$-0.793709\pi$
$557$	−37.6312	−1.59448	−0.797242	+	0.603659i	$0.793709\pi$
$558$	0	0
$559$	−1.58359	−0.0669788
$560$	−6.09017	−0.257357
$561$	0	0
$562$	−64.8328	−2.73481
$563$	−40.5967	−1.71095	−0.855474	−	0.517845i	$-0.826734\pi$
$563$	−40.5967	−1.71095	−0.855474	+	0.517845i	$0.826734\pi$
$564$	0	0
$565$	−8.32624	−0.350287
$566$	−14.9443	−0.628155
$567$	0	0
$568$	−77.1591	−3.23752
$569$	−34.1803	−1.43291	−0.716457	−	0.697631i	$-0.754237\pi$
$569$	−34.1803	−1.43291	−0.716457	+	0.697631i	$0.754237\pi$
$570$	0	0
$571$	−9.09017	−0.380412	−0.190206	−	0.981744i	$-0.560916\pi$
$571$	−9.09017	−0.380412	−0.190206	+	0.981744i	$0.560916\pi$
$572$	0	0
$573$	0	0
$574$	0.618034	0.0257962
$575$	1.09017	0.0454632
$576$	0	0
$577$	31.7082	1.32003	0.660015	−	0.751253i	$-0.270550\pi$
$577$	31.7082	1.32003	0.660015	+	0.751253i	$0.270550\pi$
$578$	−41.0689	−1.70824
$579$	0	0
$580$	18.0000	0.747409
$581$	1.47214	0.0610745
$582$	0	0
$583$	0	0
$584$	−42.6525	−1.76497
$585$	0	0
$586$	56.6869	2.34171
$587$	2.12461	0.0876921	0.0438461	−	0.999038i	$-0.486039\pi$
$587$	2.12461	0.0876921	0.0438461	+	0.999038i	$0.486039\pi$
$588$	0	0
$589$	−35.6525	−1.46903
$590$	−11.9443	−0.491738
$591$	0	0
$592$	−61.4508	−2.52561
$593$	14.0344	0.576325	0.288163	−	0.957581i	$-0.406956\pi$
$593$	14.0344	0.576325	0.288163	+	0.957581i	$0.406956\pi$
$594$	0	0
$595$	−0.708204	−0.0290335
$596$	20.5623	0.842265
$597$	0	0
$598$	−0.145898	−0.00596621
$599$	−12.6525	−0.516966	−0.258483	−	0.966016i	$-0.583223\pi$
$599$	−12.6525	−0.516966	−0.258483	+	0.966016i	$0.583223\pi$
$600$	0	0
$601$	−6.88854	−0.280990	−0.140495	−	0.990081i	$-0.544869\pi$
$601$	−6.88854	−0.280990	−0.140495	+	0.990081i	$0.544869\pi$
$602$	17.5623	0.715786
$603$	0	0
$604$	5.12461	0.208517
$605$	0	0
$606$	0	0
$607$	−16.5623	−0.672243	−0.336122	−	0.941819i	$-0.609115\pi$
$607$	−16.5623	−0.672243	−0.336122	+	0.941819i	$0.609115\pi$
$608$	63.5410	2.57693
$609$	0	0
$610$	−18.7082	−0.757473
$611$	2.38197	0.0963640
$612$	0	0
$613$	−14.2918	−0.577240	−0.288620	−	0.957444i	$-0.593196\pi$
$613$	−14.2918	−0.577240	−0.288620	+	0.957444i	$0.593196\pi$
$614$	73.2492	2.95610
$615$	0	0
$616$	0	0
$617$	−11.1803	−0.450104	−0.225052	−	0.974347i	$-0.572255\pi$
$617$	−11.1803	−0.450104	−0.225052	+	0.974347i	$0.572255\pi$
$618$	0	0
$619$	24.1246	0.969650	0.484825	−	0.874611i	$-0.338883\pi$
$619$	24.1246	0.969650	0.484825	+	0.874611i	$0.338883\pi$
$620$	−18.2705	−0.733761
$621$	0	0
$622$	−30.5066	−1.22320
$623$	−8.23607	−0.329971
$624$	0	0
$625$	19.4164	0.776656
$626$	−6.61803	−0.264510
$627$	0	0
$628$	76.2492	3.04268
$629$	−7.14590	−0.284926
$630$	0	0
$631$	19.2148	0.764928	0.382464	−	0.923970i	$-0.375076\pi$
$631$	19.2148	0.764928	0.382464	+	0.923970i	$0.375076\pi$
$632$	82.1935	3.26948
$633$	0	0
$634$	17.8541	0.709077
$635$	4.76393	0.189051
$636$	0	0
$637$	−1.41641	−0.0561201
$638$	0	0
$639$	0	0
$640$	0.673762	0.0266328
$641$	25.0902	0.991002	0.495501	−	0.868607i	$-0.334984\pi$
$641$	25.0902	0.991002	0.495501	+	0.868607i	$0.334984\pi$
$642$	0	0
$643$	20.8541	0.822406	0.411203	−	0.911544i	$-0.365109\pi$
$643$	20.8541	0.822406	0.411203	+	0.911544i	$0.365109\pi$
$644$	1.14590	0.0451547
$645$	0	0
$646$	17.5623	0.690980
$647$	−45.0344	−1.77049	−0.885243	−	0.465128i	$-0.846008\pi$
$647$	−45.0344	−1.77049	−0.885243	+	0.465128i	$0.846008\pi$
$648$	0	0
$649$	0	0
$650$	−2.85410	−0.111947
$651$	0	0
$652$	−24.9787	−0.978242
$653$	5.61803	0.219851	0.109925	−	0.993940i	$-0.464939\pi$
$653$	5.61803	0.219851	0.109925	+	0.993940i	$0.464939\pi$
$654$	0	0
$655$	7.29180	0.284914
$656$	−2.32624	−0.0908243
$657$	0	0
$658$	−26.4164	−1.02982
$659$	41.1246	1.60199	0.800994	−	0.598673i	$-0.204305\pi$
$659$	41.1246	1.60199	0.800994	+	0.598673i	$0.204305\pi$
$660$	0	0
$661$	36.5623	1.42211	0.711054	−	0.703137i	$-0.248218\pi$
$661$	36.5623	1.42211	0.711054	+	0.703137i	$0.248218\pi$
$662$	43.7426	1.70011
$663$	0	0
$664$	−11.0000	−0.426883
$665$	−3.61803	−0.140301
$666$	0	0
$667$	−1.41641	−0.0548435
$668$	−58.4164	−2.26020
$669$	0	0
$670$	3.00000	0.115900
$671$	0	0
$672$	0	0
$673$	35.8328	1.38125	0.690627	−	0.723211i	$-0.257335\pi$
$673$	35.8328	1.38125	0.690627	+	0.723211i	$0.257335\pi$
$674$	−47.5967	−1.83336
$675$	0	0
$676$	−62.8328	−2.41665
$677$	−13.5279	−0.519918	−0.259959	−	0.965620i	$-0.583709\pi$
$677$	−13.5279	−0.519918	−0.259959	+	0.965620i	$0.583709\pi$
$678$	0	0
$679$	−7.85410	−0.301413
$680$	5.29180	0.202931
$681$	0	0
$682$	0	0
$683$	−9.06888	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$683$	−9.06888	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$684$	0	0
$685$	−6.03444	−0.230564
$686$	34.0344	1.29944
$687$	0	0
$688$	−66.1033	−2.52017
$689$	0.0901699	0.00343520
$690$	0	0
$691$	1.34752	0.0512622	0.0256311	−	0.999671i	$-0.491840\pi$
$691$	1.34752	0.0512622	0.0256311	+	0.999671i	$0.491840\pi$
$692$	87.5410	3.32781
$693$	0	0
$694$	4.00000	0.151838
$695$	9.00000	0.341389
$696$	0	0
$697$	−0.270510	−0.0102463
$698$	33.2705	1.25931
$699$	0	0
$700$	22.4164	0.847261
$701$	34.7984	1.31432	0.657158	−	0.753753i	$-0.271758\pi$
$701$	34.7984	1.31432	0.657158	+	0.753753i	$0.271758\pi$
$702$	0	0
$703$	−36.5066	−1.37687
$704$	0	0
$705$	0	0
$706$	31.4164	1.18237
$707$	−10.2361	−0.384967
$708$	0	0
$709$	11.2148	0.421180	0.210590	−	0.977574i	$-0.432462\pi$
$709$	11.2148	0.421180	0.210590	+	0.977574i	$0.432462\pi$
$710$	−16.7082	−0.627048
$711$	0	0
$712$	61.5410	2.30635
$713$	1.43769	0.0538421
$714$	0	0
$715$	0	0
$716$	41.3951	1.54701
$717$	0	0
$718$	25.4164	0.948532
$719$	38.4853	1.43526	0.717630	−	0.696425i	$-0.245227\pi$
$719$	38.4853	1.43526	0.717630	+	0.696425i	$0.245227\pi$
$720$	0	0
$721$	10.9443	0.407586
$722$	39.9787	1.48785
$723$	0	0
$724$	12.2705	0.456030
$725$	−27.7082	−1.02906
$726$	0	0
$727$	−9.14590	−0.339203	−0.169601	−	0.985513i	$-0.554248\pi$
$727$	−9.14590	−0.339203	−0.169601	+	0.985513i	$0.554248\pi$
$728$	−1.76393	−0.0653757
$729$	0	0
$730$	−9.23607	−0.341842
$731$	−7.68692	−0.284311
$732$	0	0
$733$	−0.403252	−0.0148945	−0.00744723	−	0.999972i	$-0.502371\pi$
$733$	−0.403252	−0.0148945	−0.00744723	+	0.999972i	$0.502371\pi$
$734$	57.9787	2.14003
$735$	0	0
$736$	−2.56231	−0.0944478
$737$	0	0
$738$	0	0
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	−18.7082	−0.687727
$741$	0	0
$742$	−1.00000	−0.0367112
$743$	42.8885	1.57343	0.786714	−	0.617318i	$-0.211781\pi$
$743$	42.8885	1.57343	0.786714	+	0.617318i	$0.211781\pi$
$744$	0	0
$745$	2.61803	0.0959173
$746$	2.32624	0.0851696
$747$	0	0
$748$	0	0
$749$	−11.4721	−0.419183
$750$	0	0
$751$	16.1459	0.589172	0.294586	−	0.955625i	$-0.404818\pi$
$751$	16.1459	0.589172	0.294586	+	0.955625i	$0.404818\pi$
$752$	99.4296	3.62582
$753$	0	0
$754$	3.70820	0.135045
$755$	0.652476	0.0237460
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	−65.1591	−2.36668
$759$	0	0
$760$	27.0344	0.980642
$761$	−29.2918	−1.06183	−0.530913	−	0.847426i	$-0.678151\pi$
$761$	−29.2918	−1.06183	−0.530913	+	0.847426i	$0.678151\pi$
$762$	0	0
$763$	12.0000	0.434429
$764$	3.97871	0.143945
$765$	0	0
$766$	33.2705	1.20211
$767$	−1.74265	−0.0629233
$768$	0	0
$769$	−34.5066	−1.24434	−0.622170	−	0.782883i	$-0.713749\pi$
$769$	−34.5066	−1.24434	−0.622170	+	0.782883i	$0.713749\pi$
$770$	0	0
$771$	0	0
$772$	−15.2705	−0.549598
$773$	−27.1803	−0.977609	−0.488804	−	0.872393i	$-0.662567\pi$
$773$	−27.1803	−0.977609	−0.488804	+	0.872393i	$0.662567\pi$
$774$	0	0
$775$	28.1246	1.01027
$776$	58.6869	2.10674
$777$	0	0
$778$	96.1935	3.44870
$779$	−1.38197	−0.0495141
$780$	0	0
$781$	0	0
$782$	−0.708204	−0.0253253
$783$	0	0
$784$	−59.1246	−2.11159
$785$	9.70820	0.346501
$786$	0	0
$787$	−9.70820	−0.346060	−0.173030	−	0.984917i	$-0.555356\pi$
$787$	−9.70820	−0.346060	−0.173030	+	0.984917i	$0.555356\pi$
$788$	−63.2705	−2.25392
$789$	0	0
$790$	17.7984	0.633238
$791$	13.4721	0.479014
$792$	0	0
$793$	−2.72949	−0.0969270
$794$	−48.9787	−1.73819
$795$	0	0
$796$	32.5623	1.15414
$797$	−18.5410	−0.656757	−0.328378	−	0.944546i	$-0.606502\pi$
$797$	−18.5410	−0.656757	−0.328378	+	0.944546i	$0.606502\pi$
$798$	0	0
$799$	11.5623	0.409045
$800$	−50.1246	−1.77217
$801$	0	0
$802$	82.9574	2.92933
$803$	0	0
$804$	0	0
$805$	0.145898	0.00514223
$806$	−3.76393	−0.132579
$807$	0	0
$808$	76.4853	2.69074
$809$	−27.0902	−0.952440	−0.476220	−	0.879326i	$-0.657993\pi$
$809$	−27.0902	−0.952440	−0.476220	+	0.879326i	$0.657993\pi$
$810$	0	0
$811$	36.5623	1.28388	0.641938	−	0.766756i	$-0.278131\pi$
$811$	36.5623	1.28388	0.641938	+	0.766756i	$0.278131\pi$
$812$	−29.1246	−1.02207
$813$	0	0
$814$	0	0
$815$	−3.18034	−0.111402
$816$	0	0
$817$	−39.2705	−1.37390
$818$	−16.9443	−0.592443
$819$	0	0
$820$	−0.708204	−0.0247316
$821$	−40.5967	−1.41684	−0.708418	−	0.705793i	$-0.750591\pi$
$821$	−40.5967	−1.41684	−0.708418	+	0.705793i	$0.750591\pi$
$822$	0	0
$823$	−27.8328	−0.970191	−0.485095	−	0.874461i	$-0.661215\pi$
$823$	−27.8328	−0.970191	−0.485095	+	0.874461i	$0.661215\pi$
$824$	−81.7771	−2.84884
$825$	0	0
$826$	19.3262	0.672446
$827$	10.6525	0.370423	0.185211	−	0.982699i	$-0.440703\pi$
$827$	10.6525	0.370423	0.185211	+	0.982699i	$0.440703\pi$
$828$	0	0
$829$	−31.3951	−1.09040	−0.545199	−	0.838307i	$-0.683546\pi$
$829$	−31.3951	−1.09040	−0.545199	+	0.838307i	$0.683546\pi$
$830$	−2.38197	−0.0826792
$831$	0	0
$832$	2.05573	0.0712695
$833$	−6.87539	−0.238218
$834$	0	0
$835$	−7.43769	−0.257392
$836$	0	0
$837$	0	0
$838$	−82.3394	−2.84437
$839$	17.8328	0.615657	0.307829	−	0.951442i	$-0.400398\pi$
$839$	17.8328	0.615657	0.307829	+	0.951442i	$0.400398\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	27.5066	0.947939
$843$	0	0
$844$	17.5623	0.604519
$845$	−8.00000	−0.275208
$846$	0	0
$847$	0	0
$848$	3.76393	0.129254
$849$	0	0
$850$	−13.8541	−0.475192
$851$	1.47214	0.0504642
$852$	0	0
$853$	−55.8328	−1.91168	−0.955840	−	0.293889i	$-0.905050\pi$
$853$	−55.8328	−1.91168	−0.955840	+	0.293889i	$0.905050\pi$
$854$	30.2705	1.03584
$855$	0	0
$856$	85.7214	2.92990
$857$	−27.7639	−0.948398	−0.474199	−	0.880418i	$-0.657262\pi$
$857$	−27.7639	−0.948398	−0.474199	+	0.880418i	$0.657262\pi$
$858$	0	0
$859$	−34.4164	−1.17427	−0.587136	−	0.809488i	$-0.699745\pi$
$859$	−34.4164	−1.17427	−0.587136	+	0.809488i	$0.699745\pi$
$860$	−20.1246	−0.686244
$861$	0	0
$862$	15.4721	0.526983
$863$	−0.111456	−0.00379401	−0.00189701	−	0.999998i	$-0.500604\pi$
$863$	−0.111456	−0.00379401	−0.00189701	+	0.999998i	$0.500604\pi$
$864$	0	0
$865$	11.1459	0.378972
$866$	92.4296	3.14088
$867$	0	0
$868$	29.5623	1.00341
$869$	0	0
$870$	0	0
$871$	0.437694	0.0148307
$872$	−89.6656	−3.03646
$873$	0	0
$874$	−3.61803	−0.122382
$875$	5.94427	0.200953
$876$	0	0
$877$	−57.9443	−1.95664	−0.978320	−	0.207101i	$-0.933597\pi$
$877$	−57.9443	−1.95664	−0.978320	+	0.207101i	$0.933597\pi$
$878$	60.9787	2.05793
$879$	0	0
$880$	0	0
$881$	−6.20163	−0.208938	−0.104469	−	0.994528i	$-0.533314\pi$
$881$	−6.20163	−0.208938	−0.104469	+	0.994528i	$0.533314\pi$
$882$	0	0
$883$	1.05573	0.0355281	0.0177640	−	0.999842i	$-0.494345\pi$
$883$	1.05573	0.0355281	0.0177640	+	0.999842i	$0.494345\pi$
$884$	1.31308	0.0441637
$885$	0	0
$886$	82.7214	2.77908
$887$	54.4853	1.82944	0.914719	−	0.404092i	$-0.132412\pi$
$887$	54.4853	1.82944	0.914719	+	0.404092i	$0.132412\pi$
$888$	0	0
$889$	−7.70820	−0.258525
$890$	13.3262	0.446697
$891$	0	0
$892$	34.8541	1.16700
$893$	59.0689	1.97666
$894$	0	0
$895$	5.27051	0.176174
$896$	−1.09017	−0.0364200
$897$	0	0
$898$	23.7082	0.791153
$899$	−36.5410	−1.21871
$900$	0	0
$901$	0.437694	0.0145817
$902$	0	0
$903$	0	0
$904$	−100.666	−3.34809
$905$	1.56231	0.0519328
$906$	0	0
$907$	−42.3951	−1.40771	−0.703853	−	0.710345i	$-0.748539\pi$
$907$	−42.3951	−1.40771	−0.703853	+	0.710345i	$0.748539\pi$
$908$	−63.9787	−2.12321
$909$	0	0
$910$	−0.381966	−0.0126620
$911$	−38.5967	−1.27877	−0.639384	−	0.768888i	$-0.720810\pi$
$911$	−38.5967	−1.27877	−0.639384	+	0.768888i	$0.720810\pi$
$912$	0	0
$913$	0	0
$914$	62.7771	2.07648
$915$	0	0
$916$	2.29180	0.0757231
$917$	−11.7984	−0.389617
$918$	0	0
$919$	25.6525	0.846197	0.423099	−	0.906084i	$-0.360942\pi$
$919$	25.6525	0.846197	0.423099	+	0.906084i	$0.360942\pi$
$920$	−1.09017	−0.0359418
$921$	0	0
$922$	−24.2705	−0.799307
$923$	−2.43769	−0.0802377
$924$	0	0
$925$	28.7984	0.946885
$926$	4.52786	0.148795
$927$	0	0
$928$	65.1246	2.13782
$929$	12.7082	0.416943	0.208471	−	0.978028i	$-0.433151\pi$
$929$	12.7082	0.416943	0.208471	+	0.978028i	$0.433151\pi$
$930$	0	0
$931$	−35.1246	−1.15116
$932$	20.1246	0.659204
$933$	0	0
$934$	−54.6869	−1.78941
$935$	0	0
$936$	0	0
$937$	41.6525	1.36073	0.680364	−	0.732875i	$-0.261822\pi$
$937$	41.6525	1.36073	0.680364	+	0.732875i	$0.261822\pi$
$938$	−4.85410	−0.158492
$939$	0	0
$940$	30.2705	0.987315
$941$	14.4508	0.471084	0.235542	−	0.971864i	$-0.424313\pi$
$941$	14.4508	0.471084	0.235542	+	0.971864i	$0.424313\pi$
$942$	0	0
$943$	0.0557281	0.00181476
$944$	−72.7426	−2.36757
$945$	0	0
$946$	0	0
$947$	32.3951	1.05270	0.526350	−	0.850268i	$-0.323560\pi$
$947$	32.3951	1.05270	0.526350	+	0.850268i	$0.323560\pi$
$948$	0	0
$949$	−1.34752	−0.0437425
$950$	−70.7771	−2.29631
$951$	0	0
$952$	−8.56231	−0.277506
$953$	−11.3475	−0.367582	−0.183791	−	0.982965i	$-0.558837\pi$
$953$	−11.3475	−0.367582	−0.183791	+	0.982965i	$0.558837\pi$
$954$	0	0
$955$	0.506578	0.0163925
$956$	−1.85410	−0.0599659
$957$	0	0
$958$	74.3050	2.40068
$959$	9.76393	0.315294
$960$	0	0
$961$	6.09017	0.196457
$962$	−3.85410	−0.124261
$963$	0	0
$964$	40.2492	1.29634
$965$	−1.94427	−0.0625883
$966$	0	0
$967$	43.9230	1.41247	0.706234	−	0.707978i	$-0.250393\pi$
$967$	43.9230	1.41247	0.706234	+	0.707978i	$0.250393\pi$
$968$	0	0
$969$	0	0
$970$	12.7082	0.408036
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	−14.5623	−0.466846
$974$	−33.2705	−1.06606
$975$	0	0
$976$	−113.936	−3.64701
$977$	0.596748	0.0190917	0.00954583	−	0.999954i	$-0.496961\pi$
$977$	0.596748	0.0190917	0.00954583	+	0.999954i	$0.496961\pi$
$978$	0	0
$979$	0	0
$980$	−18.0000	−0.574989
$981$	0	0
$982$	46.8885	1.49627
$983$	8.11146	0.258715	0.129358	−	0.991598i	$-0.458708\pi$
$983$	8.11146	0.258715	0.129358	+	0.991598i	$0.458708\pi$
$984$	0	0
$985$	−8.05573	−0.256677
$986$	18.0000	0.573237
$987$	0	0
$988$	6.70820	0.213416
$989$	1.58359	0.0503553
$990$	0	0
$991$	3.74265	0.118889	0.0594445	−	0.998232i	$-0.481067\pi$
$991$	3.74265	0.118889	0.0594445	+	0.998232i	$0.481067\pi$
$992$	−66.1033	−2.09878
$993$	0	0
$994$	27.0344	0.857480
$995$	4.14590	0.131434
$996$	0	0
$997$	−21.2016	−0.671462	−0.335731	−	0.941958i	$-0.608983\pi$
$997$	−21.2016	−0.671462	−0.335731	+	0.941958i	$0.608983\pi$
$998$	−47.5066	−1.50379
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.t.1.2		2
3.2	odd	2		363.2.a.d.1.1		2
11.3	even	5		99.2.f.a.64.1		4
11.4	even	5		99.2.f.a.82.1		4
11.10	odd	2		1089.2.a.l.1.1		2
12.11	even	2		5808.2.a.cj.1.1		2
15.14	odd	2		9075.2.a.cb.1.2		2
33.2	even	10		363.2.e.b.202.1		4
33.5	odd	10		363.2.e.k.124.1		4
33.8	even	10		363.2.e.f.130.1		4
33.14	odd	10		33.2.e.b.31.1	yes	4
33.17	even	10		363.2.e.b.124.1		4
33.20	odd	10		363.2.e.k.202.1		4
33.26	odd	10		33.2.e.b.16.1	✓	4
33.29	even	10		363.2.e.f.148.1		4
33.32	even	2		363.2.a.i.1.2		2
99.4	even	15		891.2.n.b.379.1		8
99.14	odd	30		891.2.n.c.460.1		8
99.25	even	15		891.2.n.b.757.1		8
99.47	odd	30		891.2.n.c.757.1		8
99.58	even	15		891.2.n.b.460.1		8
99.59	odd	30		891.2.n.c.379.1		8
99.70	even	15		891.2.n.b.676.1		8
99.92	odd	30		891.2.n.c.676.1		8
132.47	even	10		528.2.y.b.97.1		4
132.59	even	10		528.2.y.b.49.1		4
132.131	odd	2		5808.2.a.ci.1.1		2
165.14	odd	10		825.2.n.c.526.1		4
165.47	even	20		825.2.bx.d.724.1		8
165.59	odd	10		825.2.n.c.676.1		4
165.92	even	20		825.2.bx.d.49.2		8
165.113	even	20		825.2.bx.d.724.2		8
165.158	even	20		825.2.bx.d.49.1		8
165.164	even	2		9075.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.b.16.1	✓	4	33.26	odd	10
33.2.e.b.31.1	yes	4	33.14	odd	10
99.2.f.a.64.1		4	11.3	even	5
99.2.f.a.82.1		4	11.4	even	5
363.2.a.d.1.1		2	3.2	odd	2
363.2.a.i.1.2		2	33.32	even	2
363.2.e.b.124.1		4	33.17	even	10
363.2.e.b.202.1		4	33.2	even	10
363.2.e.f.130.1		4	33.8	even	10
363.2.e.f.148.1		4	33.29	even	10
363.2.e.k.124.1		4	33.5	odd	10
363.2.e.k.202.1		4	33.20	odd	10
528.2.y.b.49.1		4	132.59	even	10
528.2.y.b.97.1		4	132.47	even	10
825.2.n.c.526.1		4	165.14	odd	10
825.2.n.c.676.1		4	165.59	odd	10
825.2.bx.d.49.1		8	165.158	even	20
825.2.bx.d.49.2		8	165.92	even	20
825.2.bx.d.724.1		8	165.47	even	20
825.2.bx.d.724.2		8	165.113	even	20
891.2.n.b.379.1		8	99.4	even	15
891.2.n.b.460.1		8	99.58	even	15
891.2.n.b.676.1		8	99.70	even	15
891.2.n.b.757.1		8	99.25	even	15
891.2.n.c.379.1		8	99.59	odd	30
891.2.n.c.460.1		8	99.14	odd	30
891.2.n.c.676.1		8	99.92	odd	30
891.2.n.c.757.1		8	99.47	odd	30
1089.2.a.l.1.1		2	11.10	odd	2
1089.2.a.t.1.2		2	1.1	even	1	trivial
5808.2.a.ci.1.1		2	132.131	odd	2
5808.2.a.cj.1.1		2	12.11	even	2
9075.2.a.u.1.1		2	165.164	even	2
9075.2.a.cb.1.2		2	15.14	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} +0.618034 q^{5} -1.00000 q^{7} +7.47214 q^{8} +1.61803 q^{10} +0.236068 q^{13} -2.61803 q^{14} +9.85410 q^{16} +1.14590 q^{17} +5.85410 q^{19} +3.00000 q^{20} -0.236068 q^{23} -4.61803 q^{25} +0.618034 q^{26} -4.85410 q^{28} +6.00000 q^{29} -6.09017 q^{31} +10.8541 q^{32} +3.00000 q^{34} -0.618034 q^{35} -6.23607 q^{37} +15.3262 q^{38} +4.61803 q^{40} -0.236068 q^{41} -6.70820 q^{43} -0.618034 q^{46} +10.0902 q^{47} -6.00000 q^{49} -12.0902 q^{50} +1.14590 q^{52} +0.381966 q^{53} -7.47214 q^{56} +15.7082 q^{58} -7.38197 q^{59} -11.5623 q^{61} -15.9443 q^{62} +8.70820 q^{64} +0.145898 q^{65} +1.85410 q^{67} +5.56231 q^{68} -1.61803 q^{70} -10.3262 q^{71} -5.70820 q^{73} -16.3262 q^{74} +28.4164 q^{76} +11.0000 q^{79} +6.09017 q^{80} -0.618034 q^{82} -1.47214 q^{83} +0.708204 q^{85} -17.5623 q^{86} +8.23607 q^{89} -0.236068 q^{91} -1.14590 q^{92} +26.4164 q^{94} +3.61803 q^{95} +7.85410 q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8} + q^{10} - 4 q^{13} - 3 q^{14} + 13 q^{16} + 9 q^{17} + 5 q^{19} + 6 q^{20} + 4 q^{23} - 7 q^{25} - q^{26} - 3 q^{28} + 12 q^{29} - q^{31} + 15 q^{32}+ \cdots - 18 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - q^5 - 2 * q^7 + 6 * q^8 + q^10 - 4 * q^13 - 3 * q^14 + 13 * q^16 + 9 * q^17 + 5 * q^19 + 6 * q^20 + 4 * q^23 - 7 * q^25 - q^26 - 3 * q^28 + 12 * q^29 - q^31 + 15 * q^32 + 6 * q^34 + q^35 - 8 * q^37 + 15 * q^38 + 7 * q^40 + 4 * q^41 + q^46 + 9 * q^47 - 12 * q^49 - 13 * q^50 + 9 * q^52 + 3 * q^53 - 6 * q^56 + 18 * q^58 - 17 * q^59 - 3 * q^61 - 14 * q^62 + 4 * q^64 + 7 * q^65 - 3 * q^67 - 9 * q^68 - q^70 - 5 * q^71 + 2 * q^73 - 17 * q^74 + 30 * q^76 + 22 * q^79 + q^80 + q^82 + 6 * q^83 - 12 * q^85 - 15 * q^86 + 12 * q^89 + 4 * q^91 - 9 * q^92 + 26 * q^94 + 5 * q^95 + 9 * q^97 - 18 * q^98

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