LMFDB - Embedded newform 549.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [549,2,Mod(1,549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(549, 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("549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 549 = 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	549.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:	$4.38378707097$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 61)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009 q^{2} +2.70928 q^{4} +1.63090 q^{5} -0.460811 q^{7} -1.53919 q^{8} +O(q^{10})$	\(q-2.17009 q^{2} +2.70928 q^{4} +1.63090 q^{5} -0.460811 q^{7} -1.53919 q^{8} -3.53919 q^{10} -6.17009 q^{11} -4.07838 q^{13} +1.00000 q^{14} -2.07838 q^{16} -0.630898 q^{17} +7.12783 q^{19} +4.41855 q^{20} +13.3896 q^{22} +0.170086 q^{23} -2.34017 q^{25} +8.85043 q^{26} -1.24846 q^{28} -2.63090 q^{29} -10.3896 q^{31} +7.58864 q^{32} +1.36910 q^{34} -0.751536 q^{35} +5.12783 q^{37} -15.4680 q^{38} -2.51026 q^{40} -3.15676 q^{41} -3.36910 q^{43} -16.7165 q^{44} -0.369102 q^{46} -0.183417 q^{47} -6.78765 q^{49} +5.07838 q^{50} -11.0494 q^{52} +4.34017 q^{53} -10.0628 q^{55} +0.709275 q^{56} +5.70928 q^{58} -1.78047 q^{59} +1.00000 q^{61} +22.5464 q^{62} -12.3112 q^{64} -6.65142 q^{65} -8.55971 q^{67} -1.70928 q^{68} +1.63090 q^{70} -14.3896 q^{71} -0.552520 q^{73} -11.1278 q^{74} +19.3112 q^{76} +2.84324 q^{77} -7.00719 q^{79} -3.38962 q^{80} +6.85043 q^{82} -6.83710 q^{83} -1.02893 q^{85} +7.31124 q^{86} +9.49693 q^{88} -4.49693 q^{89} +1.87936 q^{91} +0.460811 q^{92} +0.398032 q^{94} +11.6248 q^{95} +8.52359 q^{97} +14.7298 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8	$ 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8} - 9 q^{10} - 13 q^{11} - 9 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} + 11 q^{22} - 5 q^{23} + 4 q^{25} - q^{26} + 5 q^{28} - 4 q^{29} - 2 q^{31} + 3 q^{32} + 8 q^{34} - 11 q^{35} - 6 q^{37} - 14 q^{38} + 9 q^{40} - 3 q^{41} - 14 q^{43} - 9 q^{44} - 5 q^{46} + 4 q^{47} - 10 q^{49} + 12 q^{50} - 15 q^{52} + 2 q^{53} - 13 q^{55} - 5 q^{56} + 10 q^{58} - 29 q^{59} + 3 q^{61} + 32 q^{62} - 11 q^{64} + 17 q^{65} + 9 q^{67} + 2 q^{68} + q^{70} - 14 q^{71} - q^{73} - 12 q^{74} + 32 q^{76} + 15 q^{77} + 13 q^{79} + 19 q^{80} - 7 q^{82} + 8 q^{83} - 18 q^{85} - 4 q^{86} + 11 q^{88} + 4 q^{89} - 7 q^{91} + 3 q^{92} + 20 q^{94} - 4 q^{95} + 10 q^{97} + 4 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8 - 9 * q^10 - 13 * q^11 - 9 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 - q^20 + 11 * q^22 - 5 * q^23 + 4 * q^25 - q^26 + 5 * q^28 - 4 * q^29 - 2 * q^31 + 3 * q^32 + 8 * q^34 - 11 * q^35 - 6 * q^37 - 14 * q^38 + 9 * q^40 - 3 * q^41 - 14 * q^43 - 9 * q^44 - 5 * q^46 + 4 * q^47 - 10 * q^49 + 12 * q^50 - 15 * q^52 + 2 * q^53 - 13 * q^55 - 5 * q^56 + 10 * q^58 - 29 * q^59 + 3 * q^61 + 32 * q^62 - 11 * q^64 + 17 * q^65 + 9 * q^67 + 2 * q^68 + q^70 - 14 * q^71 - q^73 - 12 * q^74 + 32 * q^76 + 15 * q^77 + 13 * q^79 + 19 * q^80 - 7 * q^82 + 8 * q^83 - 18 * q^85 - 4 * q^86 + 11 * q^88 + 4 * q^89 - 7 * q^91 + 3 * q^92 + 20 * q^94 - 4 * q^95 + 10 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	1.63090	0.729360	0.364680	−	0.931133i	$-0.381178\pi$
$5$	1.63090	0.729360	0.364680	+	0.931133i	$0.381178\pi$
$6$	0	0
$7$	−0.460811	−0.174170	−0.0870851	−	0.996201i	$-0.527755\pi$
$7$	−0.460811	−0.174170	−0.0870851	+	0.996201i	$0.527755\pi$
$8$	−1.53919	−0.544185
$9$	0	0
$10$	−3.53919	−1.11919
$11$	−6.17009	−1.86035	−0.930176	−	0.367115i	$-0.880346\pi$
$11$	−6.17009	−1.86035	−0.930176	+	0.367115i	$0.880346\pi$
$12$	0	0
$13$	−4.07838	−1.13114	−0.565569	−	0.824701i	$-0.691343\pi$
$13$	−4.07838	−1.13114	−0.565569	+	0.824701i	$0.691343\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−2.07838	−0.519594
$17$	−0.630898	−0.153015	−0.0765076	−	0.997069i	$-0.524377\pi$
$17$	−0.630898	−0.153015	−0.0765076	+	0.997069i	$0.524377\pi$
$18$	0	0
$19$	7.12783	1.63524	0.817618	−	0.575761i	$-0.195294\pi$
$19$	7.12783	1.63524	0.817618	+	0.575761i	$0.195294\pi$
$20$	4.41855	0.988018
$21$	0	0
$22$	13.3896	2.85468
$23$	0.170086	0.0354655	0.0177327	−	0.999843i	$-0.494355\pi$
$23$	0.170086	0.0354655	0.0177327	+	0.999843i	$0.494355\pi$
$24$	0	0
$25$	−2.34017	−0.468035
$26$	8.85043	1.73571
$27$	0	0
$28$	−1.24846	−0.235938
$29$	−2.63090	−0.488545	−0.244273	−	0.969707i	$-0.578549\pi$
$29$	−2.63090	−0.488545	−0.244273	+	0.969707i	$0.578549\pi$
$30$	0	0
$31$	−10.3896	−1.86603	−0.933016	−	0.359836i	$-0.882833\pi$
$31$	−10.3896	−1.86603	−0.933016	+	0.359836i	$0.882833\pi$
$32$	7.58864	1.34149
$33$	0	0
$34$	1.36910	0.234799
$35$	−0.751536	−0.127033
$36$	0	0
$37$	5.12783	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$37$	5.12783	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$38$	−15.4680	−2.50924
$39$	0	0
$40$	−2.51026	−0.396907
$41$	−3.15676	−0.493002	−0.246501	−	0.969142i	$-0.579281\pi$
$41$	−3.15676	−0.493002	−0.246501	+	0.969142i	$0.579281\pi$
$42$	0	0
$43$	−3.36910	−0.513783	−0.256892	−	0.966440i	$-0.582698\pi$
$43$	−3.36910	−0.513783	−0.256892	+	0.966440i	$0.582698\pi$
$44$	−16.7165	−2.52010
$45$	0	0
$46$	−0.369102	−0.0544212
$47$	−0.183417	−0.0267542	−0.0133771	−	0.999911i	$-0.504258\pi$
$47$	−0.183417	−0.0267542	−0.0133771	+	0.999911i	$0.504258\pi$
$48$	0	0
$49$	−6.78765	−0.969665
$50$	5.07838	0.718191
$51$	0	0
$52$	−11.0494	−1.53228
$53$	4.34017	0.596169	0.298084	−	0.954540i	$-0.403652\pi$
$53$	4.34017	0.596169	0.298084	+	0.954540i	$0.403652\pi$
$54$	0	0
$55$	−10.0628	−1.35686
$56$	0.709275	0.0947809
$57$	0	0
$58$	5.70928	0.749665
$59$	−1.78047	−0.231797	−0.115898	−	0.993261i	$-0.536975\pi$
$59$	−1.78047	−0.231797	−0.115898	+	0.993261i	$0.536975\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	22.5464	2.86339
$63$	0	0
$64$	−12.3112	−1.53891
$65$	−6.65142	−0.825007
$66$	0	0
$67$	−8.55971	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$67$	−8.55971	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$68$	−1.70928	−0.207280
$69$	0	0
$70$	1.63090	0.194930
$71$	−14.3896	−1.70773	−0.853867	−	0.520491i	$-0.825749\pi$
$71$	−14.3896	−1.70773	−0.853867	+	0.520491i	$0.825749\pi$
$72$	0	0
$73$	−0.552520	−0.0646676	−0.0323338	−	0.999477i	$-0.510294\pi$
$73$	−0.552520	−0.0646676	−0.0323338	+	0.999477i	$0.510294\pi$
$74$	−11.1278	−1.29358
$75$	0	0
$76$	19.3112	2.21515
$77$	2.84324	0.324018
$78$	0	0
$79$	−7.00719	−0.788370	−0.394185	−	0.919031i	$-0.628973\pi$
$79$	−7.00719	−0.788370	−0.394185	+	0.919031i	$0.628973\pi$
$80$	−3.38962	−0.378971
$81$	0	0
$82$	6.85043	0.756504
$83$	−6.83710	−0.750469	−0.375235	−	0.926930i	$-0.622438\pi$
$83$	−6.83710	−0.750469	−0.375235	+	0.926930i	$0.622438\pi$
$84$	0	0
$85$	−1.02893	−0.111603
$86$	7.31124	0.788392
$87$	0	0
$88$	9.49693	1.01238
$89$	−4.49693	−0.476673	−0.238337	−	0.971183i	$-0.576602\pi$
$89$	−4.49693	−0.476673	−0.238337	+	0.971183i	$0.576602\pi$
$90$	0	0
$91$	1.87936	0.197011
$92$	0.460811	0.0480429
$93$	0	0
$94$	0.398032	0.0410538
$95$	11.6248	1.19267
$96$	0	0
$97$	8.52359	0.865439	0.432720	−	0.901528i	$-0.357554\pi$
$97$	8.52359	0.865439	0.432720	+	0.901528i	$0.357554\pi$
$98$	14.7298	1.48793
$99$	0	0
$100$	−6.34017	−0.634017
$101$	6.78765	0.675397	0.337698	−	0.941254i	$-0.390352\pi$
$101$	6.78765	0.675397	0.337698	+	0.941254i	$0.390352\pi$
$102$	0	0
$103$	11.1278	1.09646	0.548229	−	0.836328i	$-0.315302\pi$
$103$	11.1278	1.09646	0.548229	+	0.836328i	$0.315302\pi$
$104$	6.27739	0.615549
$105$	0	0
$106$	−9.41855	−0.914811
$107$	−15.5753	−1.50572	−0.752861	−	0.658180i	$-0.771327\pi$
$107$	−15.5753	−1.50572	−0.752861	+	0.658180i	$0.771327\pi$
$108$	0	0
$109$	11.6803	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$109$	11.6803	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$110$	21.8371	2.08209
$111$	0	0
$112$	0.957740	0.0904979
$113$	12.9421	1.21749	0.608747	−	0.793364i	$-0.291672\pi$
$113$	12.9421	1.21749	0.608747	+	0.793364i	$0.291672\pi$
$114$	0	0
$115$	0.277394	0.0258671
$116$	−7.12783	−0.661802
$117$	0	0
$118$	3.86376	0.355688
$119$	0.290725	0.0266507
$120$	0	0
$121$	27.0700	2.46091
$122$	−2.17009	−0.196470
$123$	0	0
$124$	−28.1483	−2.52780
$125$	−11.9711	−1.07073
$126$	0	0
$127$	−1.02893	−0.0913027	−0.0456514	−	0.998957i	$-0.514536\pi$
$127$	−1.02893	−0.0913027	−0.0456514	+	0.998957i	$0.514536\pi$
$128$	11.5392	1.01993
$129$	0	0
$130$	14.4341	1.26596
$131$	−2.44748	−0.213837	−0.106919	−	0.994268i	$-0.534098\pi$
$131$	−2.44748	−0.213837	−0.106919	+	0.994268i	$0.534098\pi$
$132$	0	0
$133$	−3.28458	−0.284809
$134$	18.5753	1.60466
$135$	0	0
$136$	0.971071	0.0832686
$137$	16.0700	1.37295	0.686475	−	0.727153i	$-0.259157\pi$
$137$	16.0700	1.37295	0.686475	+	0.727153i	$0.259157\pi$
$138$	0	0
$139$	−1.53919	−0.130552	−0.0652761	−	0.997867i	$-0.520793\pi$
$139$	−1.53919	−0.130552	−0.0652761	+	0.997867i	$0.520793\pi$
$140$	−2.03612	−0.172083
$141$	0	0
$142$	31.2267	2.62049
$143$	25.1639	2.10431
$144$	0	0
$145$	−4.29072	−0.356325
$146$	1.19902	0.0992313
$147$	0	0
$148$	13.8927	1.14197
$149$	−1.92162	−0.157425	−0.0787127	−	0.996897i	$-0.525081\pi$
$149$	−1.92162	−0.157425	−0.0787127	+	0.996897i	$0.525081\pi$
$150$	0	0
$151$	9.32457	0.758823	0.379412	−	0.925228i	$-0.376126\pi$
$151$	9.32457	0.758823	0.379412	+	0.925228i	$0.376126\pi$
$152$	−10.9711	−0.889871
$153$	0	0
$154$	−6.17009	−0.497200
$155$	−16.9444	−1.36101
$156$	0	0
$157$	−13.4947	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$157$	−13.4947	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$158$	15.2062	1.20974
$159$	0	0
$160$	12.3763	0.978432
$161$	−0.0783777	−0.00617703
$162$	0	0
$163$	−10.7031	−0.838334	−0.419167	−	0.907909i	$-0.637678\pi$
$163$	−10.7031	−0.838334	−0.419167	+	0.907909i	$0.637678\pi$
$164$	−8.55252	−0.667840
$165$	0	0
$166$	14.8371	1.15158
$167$	−4.88655	−0.378133	−0.189066	−	0.981964i	$-0.560546\pi$
$167$	−4.88655	−0.378133	−0.189066	+	0.981964i	$0.560546\pi$
$168$	0	0
$169$	3.63317	0.279474
$170$	2.23287	0.171253
$171$	0	0
$172$	−9.12783	−0.695990
$173$	0.738205	0.0561247	0.0280623	−	0.999606i	$-0.491066\pi$
$173$	0.738205	0.0561247	0.0280623	+	0.999606i	$0.491066\pi$
$174$	0	0
$175$	1.07838	0.0815177
$176$	12.8238	0.966628
$177$	0	0
$178$	9.75872	0.731447
$179$	−1.90110	−0.142095	−0.0710476	−	0.997473i	$-0.522634\pi$
$179$	−1.90110	−0.142095	−0.0710476	+	0.997473i	$0.522634\pi$
$180$	0	0
$181$	−21.3607	−1.58773	−0.793864	−	0.608096i	$-0.791934\pi$
$181$	−21.3607	−1.58773	−0.793864	+	0.608096i	$0.791934\pi$
$182$	−4.07838	−0.302309
$183$	0	0
$184$	−0.261795	−0.0192998
$185$	8.36296	0.614857
$186$	0	0
$187$	3.89269	0.284662
$188$	−0.496928	−0.0362422
$189$	0	0
$190$	−25.2267	−1.83014
$191$	−2.88550	−0.208788	−0.104394	−	0.994536i	$-0.533290\pi$
$191$	−2.88550	−0.208788	−0.104394	+	0.994536i	$0.533290\pi$
$192$	0	0
$193$	0.680346	0.0489724	0.0244862	−	0.999700i	$-0.492205\pi$
$193$	0.680346	0.0489724	0.0244862	+	0.999700i	$0.492205\pi$
$194$	−18.4969	−1.32800
$195$	0	0
$196$	−18.3896	−1.31354
$197$	19.3896	1.38145	0.690727	−	0.723116i	$-0.257291\pi$
$197$	19.3896	1.38145	0.690727	+	0.723116i	$0.257291\pi$
$198$	0	0
$199$	−21.4186	−1.51832	−0.759160	−	0.650904i	$-0.774390\pi$
$199$	−21.4186	−1.51832	−0.759160	+	0.650904i	$0.774390\pi$
$200$	3.60197	0.254698
$201$	0	0
$202$	−14.7298	−1.03638
$203$	1.21235	0.0850901
$204$	0	0
$205$	−5.14834	−0.359576
$206$	−24.1483	−1.68249
$207$	0	0
$208$	8.47641	0.587733
$209$	−43.9793	−3.04211
$210$	0	0
$211$	5.39576	0.371460	0.185730	−	0.982601i	$-0.440535\pi$
$211$	5.39576	0.371460	0.185730	+	0.982601i	$0.440535\pi$
$212$	11.7587	0.807592
$213$	0	0
$214$	33.7998	2.31050
$215$	−5.49466	−0.374733
$216$	0	0
$217$	4.78765	0.325007
$218$	−25.3474	−1.71674
$219$	0	0
$220$	−27.2628	−1.83806
$221$	2.57304	0.173081
$222$	0	0
$223$	12.4257	0.832089	0.416045	−	0.909344i	$-0.363416\pi$
$223$	12.4257	0.832089	0.416045	+	0.909344i	$0.363416\pi$
$224$	−3.49693	−0.233648
$225$	0	0
$226$	−28.0856	−1.86822
$227$	22.3679	1.48461	0.742304	−	0.670063i	$-0.233733\pi$
$227$	22.3679	1.48461	0.742304	+	0.670063i	$0.233733\pi$
$228$	0	0
$229$	−0.185685	−0.0122704	−0.00613520	−	0.999981i	$-0.501953\pi$
$229$	−0.185685	−0.0122704	−0.00613520	+	0.999981i	$0.501953\pi$
$230$	−0.601968	−0.0396926
$231$	0	0
$232$	4.04945	0.265859
$233$	17.8660	1.17044	0.585221	−	0.810874i	$-0.301008\pi$
$233$	17.8660	1.17044	0.585221	+	0.810874i	$0.301008\pi$
$234$	0	0
$235$	−0.299135	−0.0195134
$236$	−4.82377	−0.314001
$237$	0	0
$238$	−0.630898	−0.0408950
$239$	22.3896	1.44826	0.724132	−	0.689661i	$-0.242241\pi$
$239$	22.3896	1.44826	0.724132	+	0.689661i	$0.242241\pi$
$240$	0	0
$241$	20.0433	1.29110	0.645551	−	0.763717i	$-0.276628\pi$
$241$	20.0433	1.29110	0.645551	+	0.763717i	$0.276628\pi$
$242$	−58.7442	−3.77622
$243$	0	0
$244$	2.70928	0.173444
$245$	−11.0700	−0.707234
$246$	0	0
$247$	−29.0700	−1.84968
$248$	15.9916	1.01547
$249$	0	0
$250$	25.9783	1.64301
$251$	−7.05559	−0.445345	−0.222672	−	0.974893i	$-0.571478\pi$
$251$	−7.05559	−0.445345	−0.222672	+	0.974893i	$0.571478\pi$
$252$	0	0
$253$	−1.04945	−0.0659783
$254$	2.23287	0.140102
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	−3.26180	−0.203465	−0.101733	−	0.994812i	$-0.532439\pi$
$257$	−3.26180	−0.203465	−0.101733	+	0.994812i	$0.532439\pi$
$258$	0	0
$259$	−2.36296	−0.146827
$260$	−18.0205	−1.11759
$261$	0	0
$262$	5.31124	0.328130
$263$	20.5730	1.26859	0.634294	−	0.773092i	$-0.281291\pi$
$263$	20.5730	1.26859	0.634294	+	0.773092i	$0.281291\pi$
$264$	0	0
$265$	7.07838	0.434821
$266$	7.12783	0.437035
$267$	0	0
$268$	−23.1906	−1.41659
$269$	3.97334	0.242259	0.121129	−	0.992637i	$-0.461348\pi$
$269$	3.97334	0.242259	0.121129	+	0.992637i	$0.461348\pi$
$270$	0	0
$271$	−4.99773	−0.303591	−0.151795	−	0.988412i	$-0.548505\pi$
$271$	−4.99773	−0.303591	−0.151795	+	0.988412i	$0.548505\pi$
$272$	1.31124	0.0795058
$273$	0	0
$274$	−34.8732	−2.10677
$275$	14.4391	0.870709
$276$	0	0
$277$	17.9649	1.07941	0.539704	−	0.841855i	$-0.318536\pi$
$277$	17.9649	1.07941	0.539704	+	0.841855i	$0.318536\pi$
$278$	3.34017	0.200330
$279$	0	0
$280$	1.15676	0.0691294
$281$	−15.6020	−0.930735	−0.465368	−	0.885117i	$-0.654078\pi$
$281$	−15.6020	−0.930735	−0.465368	+	0.885117i	$0.654078\pi$
$282$	0	0
$283$	24.7298	1.47003	0.735017	−	0.678049i	$-0.237174\pi$
$283$	24.7298	1.47003	0.735017	+	0.678049i	$0.237174\pi$
$284$	−38.9854	−2.31336
$285$	0	0
$286$	−54.6079	−3.22903
$287$	1.45467	0.0858663
$288$	0	0
$289$	−16.6020	−0.976586
$290$	9.31124	0.546775
$291$	0	0
$292$	−1.49693	−0.0876011
$293$	−24.1711	−1.41209	−0.706046	−	0.708166i	$-0.749523\pi$
$293$	−24.1711	−1.41209	−0.706046	+	0.708166i	$0.749523\pi$
$294$	0	0
$295$	−2.90376	−0.169063
$296$	−7.89269	−0.458753
$297$	0	0
$298$	4.17009	0.241567
$299$	−0.693677	−0.0401164
$300$	0	0
$301$	1.55252	0.0894858
$302$	−20.2351	−1.16440
$303$	0	0
$304$	−14.8143	−0.849659
$305$	1.63090	0.0933849
$306$	0	0
$307$	19.5041	1.11316	0.556579	−	0.830794i	$-0.312114\pi$
$307$	19.5041	1.11316	0.556579	+	0.830794i	$0.312114\pi$
$308$	7.70313	0.438927
$309$	0	0
$310$	36.7708	2.08844
$311$	−8.35350	−0.473684	−0.236842	−	0.971548i	$-0.576112\pi$
$311$	−8.35350	−0.473684	−0.236842	+	0.971548i	$0.576112\pi$
$312$	0	0
$313$	2.09890	0.118637	0.0593183	−	0.998239i	$-0.481107\pi$
$313$	2.09890	0.118637	0.0593183	+	0.998239i	$0.481107\pi$
$314$	29.2846	1.65262
$315$	0	0
$316$	−18.9844	−1.06796
$317$	−27.7587	−1.55909	−0.779543	−	0.626349i	$-0.784548\pi$
$317$	−27.7587	−1.55909	−0.779543	+	0.626349i	$0.784548\pi$
$318$	0	0
$319$	16.2329	0.908866
$320$	−20.0784	−1.12242
$321$	0	0
$322$	0.170086	0.00947855
$323$	−4.49693	−0.250216
$324$	0	0
$325$	9.54411	0.529412
$326$	23.2267	1.28641
$327$	0	0
$328$	4.85884	0.268285
$329$	0.0845208	0.00465978
$330$	0	0
$331$	−1.19902	−0.0659039	−0.0329519	−	0.999457i	$-0.510491\pi$
$331$	−1.19902	−0.0659039	−0.0329519	+	0.999457i	$0.510491\pi$
$332$	−18.5236	−1.01661
$333$	0	0
$334$	10.6042	0.580238
$335$	−13.9600	−0.762717
$336$	0	0
$337$	20.9672	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$337$	20.9672	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$338$	−7.88428	−0.428848
$339$	0	0
$340$	−2.78765	−0.151182
$341$	64.1049	3.47147
$342$	0	0
$343$	6.35350	0.343057
$344$	5.18568	0.279593
$345$	0	0
$346$	−1.60197	−0.0861223
$347$	−20.3896	−1.09457	−0.547286	−	0.836946i	$-0.684339\pi$
$347$	−20.3896	−1.09457	−0.547286	+	0.836946i	$0.684339\pi$
$348$	0	0
$349$	−7.65142	−0.409571	−0.204785	−	0.978807i	$-0.565650\pi$
$349$	−7.65142	−0.409571	−0.204785	+	0.978807i	$0.565650\pi$
$350$	−2.34017	−0.125088
$351$	0	0
$352$	−46.8225	−2.49565
$353$	19.6765	1.04727	0.523636	−	0.851942i	$-0.324575\pi$
$353$	19.6765	1.04727	0.523636	+	0.851942i	$0.324575\pi$
$354$	0	0
$355$	−23.4680	−1.24555
$356$	−12.1834	−0.645720
$357$	0	0
$358$	4.12556	0.218043
$359$	−20.8599	−1.10094	−0.550471	−	0.834854i	$-0.685552\pi$
$359$	−20.8599	−1.10094	−0.550471	+	0.834854i	$0.685552\pi$
$360$	0	0
$361$	31.8059	1.67399
$362$	46.3545	2.43634
$363$	0	0
$364$	5.09171	0.266878
$365$	−0.901103	−0.0471659
$366$	0	0
$367$	−5.31965	−0.277684	−0.138842	−	0.990315i	$-0.544338\pi$
$367$	−5.31965	−0.277684	−0.138842	+	0.990315i	$0.544338\pi$
$368$	−0.353504	−0.0184277
$369$	0	0
$370$	−18.1483	−0.943488
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−20.0905	−1.04025	−0.520123	−	0.854091i	$-0.674114\pi$
$373$	−20.0905	−1.04025	−0.520123	+	0.854091i	$0.674114\pi$
$374$	−8.44748	−0.436809
$375$	0	0
$376$	0.282314	0.0145592
$377$	10.7298	0.552613
$378$	0	0
$379$	−26.4040	−1.35628	−0.678141	−	0.734932i	$-0.737214\pi$
$379$	−26.4040	−1.35628	−0.678141	+	0.734932i	$0.737214\pi$
$380$	31.4947	1.61564
$381$	0	0
$382$	6.26180	0.320381
$383$	10.9350	0.558750	0.279375	−	0.960182i	$-0.409873\pi$
$383$	10.9350	0.558750	0.279375	+	0.960182i	$0.409873\pi$
$384$	0	0
$385$	4.63704	0.236325
$386$	−1.47641	−0.0751473
$387$	0	0
$388$	23.0928	1.17236
$389$	5.50307	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$389$	5.50307	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$390$	0	0
$391$	−0.107307	−0.00542676
$392$	10.4475	0.527677
$393$	0	0
$394$	−42.0772	−2.11982
$395$	−11.4280	−0.575005
$396$	0	0
$397$	14.7565	0.740605	0.370303	−	0.928911i	$-0.379254\pi$
$397$	14.7565	0.740605	0.370303	+	0.928911i	$0.379254\pi$
$398$	46.4801	2.32984
$399$	0	0
$400$	4.86376	0.243188
$401$	17.7587	0.886828	0.443414	−	0.896317i	$-0.353767\pi$
$401$	17.7587	0.886828	0.443414	+	0.896317i	$0.353767\pi$
$402$	0	0
$403$	42.3728	2.11074
$404$	18.3896	0.914918
$405$	0	0
$406$	−2.63090	−0.130569
$407$	−31.6391	−1.56829
$408$	0	0
$409$	−16.4391	−0.812860	−0.406430	−	0.913682i	$-0.633226\pi$
$409$	−16.4391	−0.812860	−0.406430	+	0.913682i	$0.633226\pi$
$410$	11.1724	0.551763
$411$	0	0
$412$	30.1483	1.48530
$413$	0.820458	0.0403721
$414$	0	0
$415$	−11.1506	−0.547362
$416$	−30.9493	−1.51742
$417$	0	0
$418$	95.4389	4.66807
$419$	−32.0183	−1.56419	−0.782097	−	0.623157i	$-0.785850\pi$
$419$	−32.0183	−1.56419	−0.782097	+	0.623157i	$0.785850\pi$
$420$	0	0
$421$	−0.630898	−0.0307481	−0.0153740	−	0.999882i	$-0.504894\pi$
$421$	−0.630898	−0.0307481	−0.0153740	+	0.999882i	$0.504894\pi$
$422$	−11.7093	−0.569999
$423$	0	0
$424$	−6.68035	−0.324426
$425$	1.47641	0.0716164
$426$	0	0
$427$	−0.460811	−0.0223002
$428$	−42.1978	−2.03971
$429$	0	0
$430$	11.9239	0.575021
$431$	16.6225	0.800677	0.400339	−	0.916367i	$-0.368893\pi$
$431$	16.6225	0.800677	0.400339	+	0.916367i	$0.368893\pi$
$432$	0	0
$433$	−20.4969	−0.985020	−0.492510	−	0.870307i	$-0.663920\pi$
$433$	−20.4969	−0.985020	−0.492510	+	0.870307i	$0.663920\pi$
$434$	−10.3896	−0.498718
$435$	0	0
$436$	31.6453	1.51553
$437$	1.21235	0.0579944
$438$	0	0
$439$	−2.34858	−0.112092	−0.0560459	−	0.998428i	$-0.517849\pi$
$439$	−2.34858	−0.112092	−0.0560459	+	0.998428i	$0.517849\pi$
$440$	15.4885	0.738386
$441$	0	0
$442$	−5.58372	−0.265590
$443$	−16.5958	−0.788491	−0.394246	−	0.919005i	$-0.628994\pi$
$443$	−16.5958	−0.788491	−0.394246	+	0.919005i	$0.628994\pi$
$444$	0	0
$445$	−7.33403	−0.347666
$446$	−26.9649	−1.27683
$447$	0	0
$448$	5.67316	0.268032
$449$	20.4680	0.965945	0.482972	−	0.875636i	$-0.339557\pi$
$449$	20.4680	0.965945	0.482972	+	0.875636i	$0.339557\pi$
$450$	0	0
$451$	19.4775	0.917158
$452$	35.0638	1.64926
$453$	0	0
$454$	−48.5402	−2.27811
$455$	3.06505	0.143692
$456$	0	0
$457$	23.9506	1.12036	0.560180	−	0.828371i	$-0.310732\pi$
$457$	23.9506	1.12036	0.560180	+	0.828371i	$0.310732\pi$
$458$	0.402952	0.0188287
$459$	0	0
$460$	0.751536	0.0350405
$461$	26.5113	1.23475	0.617377	−	0.786667i	$-0.288195\pi$
$461$	26.5113	1.23475	0.617377	+	0.786667i	$0.288195\pi$
$462$	0	0
$463$	34.0410	1.58202	0.791011	−	0.611802i	$-0.209555\pi$
$463$	34.0410	1.58202	0.791011	+	0.611802i	$0.209555\pi$
$464$	5.46800	0.253845
$465$	0	0
$466$	−38.7708	−1.79602
$467$	−4.55971	−0.210998	−0.105499	−	0.994419i	$-0.533644\pi$
$467$	−4.55971	−0.210998	−0.105499	+	0.994419i	$0.533644\pi$
$468$	0	0
$469$	3.94441	0.182136
$470$	0.649149	0.0299430
$471$	0	0
$472$	2.74047	0.126140
$473$	20.7877	0.955817
$474$	0	0
$475$	−16.6803	−0.765347
$476$	0.787653	0.0361020
$477$	0	0
$478$	−48.5874	−2.22234
$479$	0.653684	0.0298676	0.0149338	−	0.999888i	$-0.495246\pi$
$479$	0.653684	0.0298676	0.0149338	+	0.999888i	$0.495246\pi$
$480$	0	0
$481$	−20.9132	−0.953560
$482$	−43.4957	−1.98118
$483$	0	0
$484$	73.3400	3.33364
$485$	13.9011	0.631217
$486$	0	0
$487$	−0.0227863	−0.00103255	−0.000516274	−	1.00000i	$-0.500164\pi$
$487$	−0.0227863	−0.00103255	−0.000516274		1.00000i	$0.500164\pi$
$488$	−1.53919	−0.0696758
$489$	0	0
$490$	24.0228	1.08524
$491$	2.94441	0.132879	0.0664396	−	0.997790i	$-0.478836\pi$
$491$	2.94441	0.132879	0.0664396	+	0.997790i	$0.478836\pi$
$492$	0	0
$493$	1.65983	0.0747548
$494$	63.0843	2.83830
$495$	0	0
$496$	21.5936	0.969579
$497$	6.63090	0.297436
$498$	0	0
$499$	23.0878	1.03355	0.516777	−	0.856120i	$-0.327132\pi$
$499$	23.0878	1.03355	0.516777	+	0.856120i	$0.327132\pi$
$500$	−32.4329	−1.45044
$501$	0	0
$502$	15.3112	0.683374
$503$	−5.62475	−0.250795	−0.125398	−	0.992107i	$-0.540021\pi$
$503$	−5.62475	−0.250795	−0.125398	+	0.992107i	$0.540021\pi$
$504$	0	0
$505$	11.0700	0.492607
$506$	2.27739	0.101242
$507$	0	0
$508$	−2.78765	−0.123682
$509$	19.5525	0.866650	0.433325	−	0.901238i	$-0.357340\pi$
$509$	19.5525	0.866650	0.433325	+	0.901238i	$0.357340\pi$
$510$	0	0
$511$	0.254607	0.0112632
$512$	−22.1701	−0.979789
$513$	0	0
$514$	7.07838	0.312214
$515$	18.1483	0.799712
$516$	0	0
$517$	1.13170	0.0497722
$518$	5.12783	0.225304
$519$	0	0
$520$	10.2378	0.448957
$521$	−16.7442	−0.733575	−0.366788	−	0.930305i	$-0.619542\pi$
$521$	−16.7442	−0.733575	−0.366788	+	0.930305i	$0.619542\pi$
$522$	0	0
$523$	−1.69982	−0.0743279	−0.0371640	−	0.999309i	$-0.511832\pi$
$523$	−1.69982	−0.0743279	−0.0371640	+	0.999309i	$0.511832\pi$
$524$	−6.63090	−0.289672
$525$	0	0
$526$	−44.6453	−1.94663
$527$	6.55479	0.285531
$528$	0	0
$529$	−22.9711	−0.998742
$530$	−15.3607	−0.667226
$531$	0	0
$532$	−8.89884	−0.385813
$533$	12.8744	0.557654
$534$	0	0
$535$	−25.4017	−1.09821
$536$	13.1750	0.569074
$537$	0	0
$538$	−8.62249	−0.371742
$539$	41.8804	1.80392
$540$	0	0
$541$	13.3074	0.572128	0.286064	−	0.958210i	$-0.407653\pi$
$541$	13.3074	0.572128	0.286064	+	0.958210i	$0.407653\pi$
$542$	10.8455	0.465855
$543$	0	0
$544$	−4.78765	−0.205269
$545$	19.0494	0.815989
$546$	0	0
$547$	−24.2690	−1.03767	−0.518833	−	0.854875i	$-0.673634\pi$
$547$	−24.2690	−1.03767	−0.518833	+	0.854875i	$0.673634\pi$
$548$	43.5380	1.85985
$549$	0	0
$550$	−31.3340	−1.33609
$551$	−18.7526	−0.798887
$552$	0	0
$553$	3.22899	0.137311
$554$	−38.9854	−1.65633
$555$	0	0
$556$	−4.17009	−0.176851
$557$	−1.36069	−0.0576544	−0.0288272	−	0.999584i	$-0.509177\pi$
$557$	−1.36069	−0.0576544	−0.0288272	+	0.999584i	$0.509177\pi$
$558$	0	0
$559$	13.7405	0.581160
$560$	1.56198	0.0660055
$561$	0	0
$562$	33.8576	1.42820
$563$	−14.4885	−0.610618	−0.305309	−	0.952253i	$-0.598760\pi$
$563$	−14.4885	−0.610618	−0.305309	+	0.952253i	$0.598760\pi$
$564$	0	0
$565$	21.1073	0.887991
$566$	−53.6658	−2.25574
$567$	0	0
$568$	22.1483	0.929324
$569$	3.78765	0.158787	0.0793933	−	0.996843i	$-0.474702\pi$
$569$	3.78765	0.158787	0.0793933	+	0.996843i	$0.474702\pi$
$570$	0	0
$571$	−3.87444	−0.162140	−0.0810702	−	0.996708i	$-0.525834\pi$
$571$	−3.87444	−0.162140	−0.0810702	+	0.996708i	$0.525834\pi$
$572$	68.1761	2.85058
$573$	0	0
$574$	−3.15676	−0.131760
$575$	−0.398032	−0.0165991
$576$	0	0
$577$	−23.9071	−0.995264	−0.497632	−	0.867388i	$-0.665797\pi$
$577$	−23.9071	−0.995264	−0.497632	+	0.867388i	$0.665797\pi$
$578$	36.0277	1.49856
$579$	0	0
$580$	−11.6248	−0.482692
$581$	3.15061	0.130709
$582$	0	0
$583$	−26.7792	−1.10908
$584$	0.850432	0.0351911
$585$	0	0
$586$	52.4534	2.16683
$587$	−12.3318	−0.508986	−0.254493	−	0.967075i	$-0.581909\pi$
$587$	−12.3318	−0.508986	−0.254493	+	0.967075i	$0.581909\pi$
$588$	0	0
$589$	−74.0554	−3.05140
$590$	6.30140	0.259425
$591$	0	0
$592$	−10.6576	−0.438023
$593$	−11.1278	−0.456965	−0.228483	−	0.973548i	$-0.573376\pi$
$593$	−11.1278	−0.456965	−0.228483	+	0.973548i	$0.573376\pi$
$594$	0	0
$595$	0.474142	0.0194379
$596$	−5.20620	−0.213254
$597$	0	0
$598$	1.50534	0.0615579
$599$	−23.0878	−0.943343	−0.471672	−	0.881774i	$-0.656349\pi$
$599$	−23.0878	−0.943343	−0.471672	+	0.881774i	$0.656349\pi$
$600$	0	0
$601$	−10.3340	−0.421534	−0.210767	−	0.977536i	$-0.567596\pi$
$601$	−10.3340	−0.421534	−0.210767	+	0.977536i	$0.567596\pi$
$602$	−3.36910	−0.137314
$603$	0	0
$604$	25.2628	1.02793
$605$	44.1483	1.79489
$606$	0	0
$607$	−24.8371	−1.00811	−0.504053	−	0.863672i	$-0.668159\pi$
$607$	−24.8371	−1.00811	−0.504053	+	0.863672i	$0.668159\pi$
$608$	54.0905	2.19366
$609$	0	0
$610$	−3.53919	−0.143298
$611$	0.748046	0.0302627
$612$	0	0
$613$	−23.9877	−0.968855	−0.484427	−	0.874832i	$-0.660972\pi$
$613$	−23.9877	−0.968855	−0.484427	+	0.874832i	$0.660972\pi$
$614$	−42.3256	−1.70812
$615$	0	0
$616$	−4.37629	−0.176326
$617$	−21.6020	−0.869662	−0.434831	−	0.900512i	$-0.643192\pi$
$617$	−21.6020	−0.869662	−0.434831	+	0.900512i	$0.643192\pi$
$618$	0	0
$619$	−46.2388	−1.85850	−0.929248	−	0.369457i	$-0.879544\pi$
$619$	−46.2388	−1.85850	−0.929248	+	0.369457i	$0.879544\pi$
$620$	−45.9071	−1.84367
$621$	0	0
$622$	18.1278	0.726860
$623$	2.07223	0.0830223
$624$	0	0
$625$	−7.82273	−0.312909
$626$	−4.55479	−0.182046
$627$	0	0
$628$	−36.5608	−1.45893
$629$	−3.23513	−0.128993
$630$	0	0
$631$	33.2628	1.32417	0.662086	−	0.749428i	$-0.269671\pi$
$631$	33.2628	1.32417	0.662086	+	0.749428i	$0.269671\pi$
$632$	10.7854	0.429020
$633$	0	0
$634$	60.2388	2.39239
$635$	−1.67808	−0.0665925
$636$	0	0
$637$	27.6826	1.09683
$638$	−35.2267	−1.39464
$639$	0	0
$640$	18.8192	0.743896
$641$	−33.7009	−1.33110	−0.665552	−	0.746351i	$-0.731804\pi$
$641$	−33.7009	−1.33110	−0.665552	+	0.746351i	$0.731804\pi$
$642$	0	0
$643$	−6.97107	−0.274912	−0.137456	−	0.990508i	$-0.543893\pi$
$643$	−6.97107	−0.274912	−0.137456	+	0.990508i	$0.543893\pi$
$644$	−0.212347	−0.00836764
$645$	0	0
$646$	9.75872	0.383952
$647$	16.0049	0.629218	0.314609	−	0.949221i	$-0.398127\pi$
$647$	16.0049	0.629218	0.314609	+	0.949221i	$0.398127\pi$
$648$	0	0
$649$	10.9856	0.431223
$650$	−20.7115	−0.812374
$651$	0	0
$652$	−28.9977	−1.13564
$653$	−43.2762	−1.69353	−0.846764	−	0.531969i	$-0.821452\pi$
$653$	−43.2762	−1.69353	−0.846764	+	0.531969i	$0.821452\pi$
$654$	0	0
$655$	−3.99159	−0.155964
$656$	6.56093	0.256161
$657$	0	0
$658$	−0.183417	−0.00715036
$659$	16.9276	0.659405	0.329703	−	0.944085i	$-0.393052\pi$
$659$	16.9276	0.659405	0.329703	+	0.944085i	$0.393052\pi$
$660$	0	0
$661$	15.6781	0.609807	0.304903	−	0.952383i	$-0.401376\pi$
$661$	15.6781	0.609807	0.304903	+	0.952383i	$0.401376\pi$
$662$	2.60197	0.101128
$663$	0	0
$664$	10.5236	0.408395
$665$	−5.35682	−0.207728
$666$	0	0
$667$	−0.447480	−0.0173265
$668$	−13.2390	−0.512233
$669$	0	0
$670$	30.2944	1.17038
$671$	−6.17009	−0.238194
$672$	0	0
$673$	−50.6330	−1.95176	−0.975879	−	0.218312i	$-0.929945\pi$
$673$	−50.6330	−1.95176	−0.975879	+	0.218312i	$0.929945\pi$
$674$	−45.5006	−1.75262
$675$	0	0
$676$	9.84324	0.378586
$677$	−38.1711	−1.46704	−0.733518	−	0.679670i	$-0.762123\pi$
$677$	−38.1711	−1.46704	−0.733518	+	0.679670i	$0.762123\pi$
$678$	0	0
$679$	−3.92777	−0.150734
$680$	1.58372	0.0607328
$681$	0	0
$682$	−139.113	−5.32692
$683$	−36.6309	−1.40164	−0.700821	−	0.713337i	$-0.747183\pi$
$683$	−36.6309	−1.40164	−0.700821	+	0.713337i	$0.747183\pi$
$684$	0	0
$685$	26.2085	1.00137
$686$	−13.7877	−0.526415
$687$	0	0
$688$	7.00227	0.266959
$689$	−17.7009	−0.674349
$690$	0	0
$691$	−17.4764	−0.664834	−0.332417	−	0.943133i	$-0.607864\pi$
$691$	−17.4764	−0.664834	−0.332417	+	0.943133i	$0.607864\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	44.2472	1.67960
$695$	−2.51026	−0.0952196
$696$	0	0
$697$	1.99159	0.0754368
$698$	16.6042	0.628480
$699$	0	0
$700$	2.92162	0.110427
$701$	9.83096	0.371310	0.185655	−	0.982615i	$-0.440559\pi$
$701$	9.83096	0.371310	0.185655	+	0.982615i	$0.440559\pi$
$702$	0	0
$703$	36.5503	1.37852
$704$	75.9614	2.86290
$705$	0	0
$706$	−42.6996	−1.60702
$707$	−3.12783	−0.117634
$708$	0	0
$709$	35.9214	1.34906	0.674529	−	0.738248i	$-0.264347\pi$
$709$	35.9214	1.34906	0.674529	+	0.738248i	$0.264347\pi$
$710$	50.9276	1.91128
$711$	0	0
$712$	6.92162	0.259399
$713$	−1.76713	−0.0661797
$714$	0	0
$715$	41.0398	1.53480
$716$	−5.15061	−0.192487
$717$	0	0
$718$	45.2678	1.68938
$719$	20.3773	0.759946	0.379973	−	0.924998i	$-0.375933\pi$
$719$	20.3773	0.759946	0.379973	+	0.924998i	$0.375933\pi$
$720$	0	0
$721$	−5.12783	−0.190970
$722$	−69.0216	−2.56872
$723$	0	0
$724$	−57.8720	−2.15080
$725$	6.15676	0.228656
$726$	0	0
$727$	30.0722	1.11532	0.557659	−	0.830070i	$-0.311700\pi$
$727$	30.0722	1.11532	0.557659	+	0.830070i	$0.311700\pi$
$728$	−2.89269	−0.107210
$729$	0	0
$730$	1.95547	0.0723753
$731$	2.12556	0.0786166
$732$	0	0
$733$	1.44360	0.0533207	0.0266604	−	0.999645i	$-0.491513\pi$
$733$	1.44360	0.0533207	0.0266604	+	0.999645i	$0.491513\pi$
$734$	11.5441	0.426101
$735$	0	0
$736$	1.29072	0.0475767
$737$	52.8141	1.94543
$738$	0	0
$739$	−16.1122	−0.592698	−0.296349	−	0.955080i	$-0.595769\pi$
$739$	−16.1122	−0.592698	−0.296349	+	0.955080i	$0.595769\pi$
$740$	22.6576	0.832908
$741$	0	0
$742$	4.34017	0.159333
$743$	−37.1100	−1.36143	−0.680716	−	0.732547i	$-0.738331\pi$
$743$	−37.1100	−1.36143	−0.680716	+	0.732547i	$0.738331\pi$
$744$	0	0
$745$	−3.13397	−0.114820
$746$	43.5981	1.59624
$747$	0	0
$748$	10.5464	0.385614
$749$	7.17727	0.262252
$750$	0	0
$751$	15.6430	0.570821	0.285411	−	0.958405i	$-0.407870\pi$
$751$	15.6430	0.570821	0.285411	+	0.958405i	$0.407870\pi$
$752$	0.381211	0.0139013
$753$	0	0
$754$	−23.2846	−0.847974
$755$	15.2074	0.553455
$756$	0	0
$757$	−41.3728	−1.50372	−0.751860	−	0.659323i	$-0.770843\pi$
$757$	−41.3728	−1.50372	−0.751860	+	0.659323i	$0.770843\pi$
$758$	57.2990	2.08119
$759$	0	0
$760$	−17.8927	−0.649036
$761$	39.4947	1.43168	0.715840	−	0.698264i	$-0.246044\pi$
$761$	39.4947	1.43168	0.715840	+	0.698264i	$0.246044\pi$
$762$	0	0
$763$	−5.38243	−0.194857
$764$	−7.81763	−0.282832
$765$	0	0
$766$	−23.7298	−0.857392
$767$	7.26141	0.262194
$768$	0	0
$769$	−34.7031	−1.25143	−0.625713	−	0.780053i	$-0.715192\pi$
$769$	−34.7031	−1.25143	−0.625713	+	0.780053i	$0.715192\pi$
$770$	−10.0628	−0.362637
$771$	0	0
$772$	1.84324	0.0663398
$773$	−3.34632	−0.120359	−0.0601793	−	0.998188i	$-0.519167\pi$
$773$	−3.34632	−0.120359	−0.0601793	+	0.998188i	$0.519167\pi$
$774$	0	0
$775$	24.3135	0.873367
$776$	−13.1194	−0.470960
$777$	0	0
$778$	−11.9421	−0.428147
$779$	−22.5008	−0.806175
$780$	0	0
$781$	88.7852	3.17698
$782$	0.232866	0.00832726
$783$	0	0
$784$	14.1073	0.503832
$785$	−22.0084	−0.785514
$786$	0	0
$787$	−32.0950	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$787$	−32.0950	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$788$	52.5318	1.87137
$789$	0	0
$790$	24.7998	0.882336
$791$	−5.96388	−0.212051
$792$	0	0
$793$	−4.07838	−0.144827
$794$	−32.0228	−1.13645
$795$	0	0
$796$	−58.0288	−2.05677
$797$	−38.3979	−1.36012	−0.680061	−	0.733156i	$-0.738047\pi$
$797$	−38.3979	−1.36012	−0.680061	+	0.733156i	$0.738047\pi$
$798$	0	0
$799$	0.115718	0.00409380
$800$	−17.7587	−0.627866
$801$	0	0
$802$	−38.5380	−1.36082
$803$	3.40910	0.120304
$804$	0	0
$805$	−0.127826	−0.00450528
$806$	−91.9526	−3.23889
$807$	0	0
$808$	−10.4475	−0.367541
$809$	22.0577	0.775507	0.387753	−	0.921763i	$-0.373251\pi$
$809$	22.0577	0.775507	0.387753	+	0.921763i	$0.373251\pi$
$810$	0	0
$811$	−31.1100	−1.09242	−0.546209	−	0.837649i	$-0.683930\pi$
$811$	−31.1100	−1.09242	−0.546209	+	0.837649i	$0.683930\pi$
$812$	3.28458	0.115266
$813$	0	0
$814$	68.6596	2.40652
$815$	−17.4557	−0.611447
$816$	0	0
$817$	−24.0144	−0.840157
$818$	35.6742	1.24732
$819$	0	0
$820$	−13.9483	−0.487095
$821$	35.4641	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$821$	35.4641	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$822$	0	0
$823$	−4.89884	−0.170763	−0.0853813	−	0.996348i	$-0.527211\pi$
$823$	−4.89884	−0.170763	−0.0853813	+	0.996348i	$0.527211\pi$
$824$	−17.1278	−0.596676
$825$	0	0
$826$	−1.78047	−0.0619503
$827$	−13.9506	−0.485108	−0.242554	−	0.970138i	$-0.577985\pi$
$827$	−13.9506	−0.485108	−0.242554	+	0.970138i	$0.577985\pi$
$828$	0	0
$829$	42.4863	1.47561	0.737804	−	0.675015i	$-0.235863\pi$
$829$	42.4863	1.47561	0.737804	+	0.675015i	$0.235863\pi$
$830$	24.1978	0.839918
$831$	0	0
$832$	50.2099	1.74072
$833$	4.28231	0.148373
$834$	0	0
$835$	−7.96946	−0.275795
$836$	−119.152	−4.12096
$837$	0	0
$838$	69.4824	2.40023
$839$	11.9421	0.412288	0.206144	−	0.978522i	$-0.433908\pi$
$839$	11.9421	0.412288	0.206144	+	0.978522i	$0.433908\pi$
$840$	0	0
$841$	−22.0784	−0.761323
$842$	1.36910	0.0471824
$843$	0	0
$844$	14.6186	0.503193
$845$	5.92532	0.203837
$846$	0	0
$847$	−12.4741	−0.428617
$848$	−9.02052	−0.309766
$849$	0	0
$850$	−3.20394	−0.109894
$851$	0.872174	0.0298977
$852$	0	0
$853$	16.2800	0.557418	0.278709	−	0.960376i	$-0.410093\pi$
$853$	16.2800	0.557418	0.278709	+	0.960376i	$0.410093\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	23.9733	0.819392
$857$	−26.2085	−0.895264	−0.447632	−	0.894218i	$-0.647733\pi$
$857$	−26.2085	−0.895264	−0.447632	+	0.894218i	$0.647733\pi$
$858$	0	0
$859$	−0.0350725	−0.00119666	−0.000598329	−	1.00000i	$-0.500190\pi$
$859$	−0.0350725	−0.00119666	−0.000598329		1.00000i	$0.500190\pi$
$860$	−14.8865	−0.507627
$861$	0	0
$862$	−36.0722	−1.22863
$863$	−10.8515	−0.369389	−0.184694	−	0.982796i	$-0.559129\pi$
$863$	−10.8515	−0.369389	−0.184694	+	0.982796i	$0.559129\pi$
$864$	0	0
$865$	1.20394	0.0409351
$866$	44.4801	1.51150
$867$	0	0
$868$	12.9711	0.440267
$869$	43.2350	1.46665
$870$	0	0
$871$	34.9097	1.18287
$872$	−17.9783	−0.608821
$873$	0	0
$874$	−2.63090	−0.0889914
$875$	5.51640	0.186488
$876$	0	0
$877$	0.665970	0.0224882	0.0112441	−	0.999937i	$-0.496421\pi$
$877$	0.665970	0.0224882	0.0112441	+	0.999937i	$0.496421\pi$
$878$	5.09663	0.172003
$879$	0	0
$880$	20.9143	0.705019
$881$	−14.4101	−0.485490	−0.242745	−	0.970090i	$-0.578048\pi$
$881$	−14.4101	−0.485490	−0.242745	+	0.970090i	$0.578048\pi$
$882$	0	0
$883$	57.8937	1.94828	0.974140	−	0.225947i	$-0.0725475\pi$
$883$	57.8937	1.94828	0.974140	+	0.225947i	$0.0725475\pi$
$884$	6.97107	0.234462
$885$	0	0
$886$	36.0144	1.20993
$887$	38.4040	1.28948	0.644740	−	0.764402i	$-0.276966\pi$
$887$	38.4040	1.28948	0.644740	+	0.764402i	$0.276966\pi$
$888$	0	0
$889$	0.474142	0.0159022
$890$	15.9155	0.533488
$891$	0	0
$892$	33.6647	1.12718
$893$	−1.30737	−0.0437494
$894$	0	0
$895$	−3.10050	−0.103638
$896$	−5.31739	−0.177641
$897$	0	0
$898$	−44.4173	−1.48223
$899$	27.3340	0.911641
$900$	0	0
$901$	−2.73820	−0.0912228
$902$	−42.2678	−1.40736
$903$	0	0
$904$	−19.9204	−0.662543
$905$	−34.8371	−1.15802
$906$	0	0
$907$	−42.6407	−1.41586	−0.707931	−	0.706281i	$-0.750371\pi$
$907$	−42.6407	−1.41586	−0.707931	+	0.706281i	$0.750371\pi$
$908$	60.6007	2.01111
$909$	0	0
$910$	−6.65142	−0.220492
$911$	−18.6042	−0.616386	−0.308193	−	0.951324i	$-0.599724\pi$
$911$	−18.6042	−0.616386	−0.308193	+	0.951324i	$0.599724\pi$
$912$	0	0
$913$	42.1855	1.39614
$914$	−51.9748	−1.71917
$915$	0	0
$916$	−0.503072	−0.0166220
$917$	1.12783	0.0372441
$918$	0	0
$919$	33.2306	1.09618	0.548088	−	0.836421i	$-0.315356\pi$
$919$	33.2306	1.09618	0.548088	+	0.836421i	$0.315356\pi$
$920$	−0.426961	−0.0140765
$921$	0	0
$922$	−57.5318	−1.89471
$923$	58.6863	1.93168
$924$	0	0
$925$	−12.0000	−0.394558
$926$	−73.8720	−2.42758
$927$	0	0
$928$	−19.9649	−0.655381
$929$	−13.0989	−0.429761	−0.214880	−	0.976640i	$-0.568936\pi$
$929$	−13.0989	−0.429761	−0.214880	+	0.976640i	$0.568936\pi$
$930$	0	0
$931$	−48.3812	−1.58563
$932$	48.4040	1.58553
$933$	0	0
$934$	9.89496	0.323773
$935$	6.34858	0.207621
$936$	0	0
$937$	−21.0144	−0.686510	−0.343255	−	0.939242i	$-0.611529\pi$
$937$	−21.0144	−0.686510	−0.343255	+	0.939242i	$0.611529\pi$
$938$	−8.55971	−0.279484
$939$	0	0
$940$	−0.810439	−0.0264336
$941$	37.3523	1.21765	0.608825	−	0.793305i	$-0.291641\pi$
$941$	37.3523	1.21765	0.608825	+	0.793305i	$0.291641\pi$
$942$	0	0
$943$	−0.536921	−0.0174846
$944$	3.70048	0.120440
$945$	0	0
$946$	−45.1110	−1.46669
$947$	32.2729	1.04873	0.524363	−	0.851495i	$-0.324303\pi$
$947$	32.2729	1.04873	0.524363	+	0.851495i	$0.324303\pi$
$948$	0	0
$949$	2.25338	0.0731480
$950$	36.1978	1.17441
$951$	0	0
$952$	−0.447480	−0.0145029
$953$	−51.3400	−1.66307	−0.831533	−	0.555476i	$-0.812536\pi$
$953$	−51.3400	−1.66307	−0.831533	+	0.555476i	$0.812536\pi$
$954$	0	0
$955$	−4.70596	−0.152281
$956$	60.6596	1.96187
$957$	0	0
$958$	−1.41855	−0.0458313
$959$	−7.40522	−0.239127
$960$	0	0
$961$	76.9442	2.48207
$962$	45.3835	1.46322
$963$	0	0
$964$	54.3028	1.74898
$965$	1.10957	0.0357185
$966$	0	0
$967$	−0.746615	−0.0240095	−0.0120048	−	0.999928i	$-0.503821\pi$
$967$	−0.746615	−0.0240095	−0.0120048	+	0.999928i	$0.503821\pi$
$968$	−41.6658	−1.33919
$969$	0	0
$970$	−30.1666	−0.968591
$971$	−58.1276	−1.86541	−0.932703	−	0.360647i	$-0.882556\pi$
$971$	−58.1276	−1.86541	−0.932703	+	0.360647i	$0.882556\pi$
$972$	0	0
$973$	0.709275	0.0227383
$974$	0.0494483	0.00158443
$975$	0	0
$976$	−2.07838	−0.0665273
$977$	−6.86376	−0.219591	−0.109796	−	0.993954i	$-0.535020\pi$
$977$	−6.86376	−0.219591	−0.109796	+	0.993954i	$0.535020\pi$
$978$	0	0
$979$	27.7464	0.886780
$980$	−29.9916	−0.958046
$981$	0	0
$982$	−6.38962	−0.203901
$983$	27.8660	0.888788	0.444394	−	0.895831i	$-0.353419\pi$
$983$	27.8660	0.888788	0.444394	+	0.895831i	$0.353419\pi$
$984$	0	0
$985$	31.6225	1.00758
$986$	−3.60197	−0.114710
$987$	0	0
$988$	−78.7585	−2.50564
$989$	−0.573039	−0.0182216
$990$	0	0
$991$	−20.0312	−0.636312	−0.318156	−	0.948038i	$-0.603064\pi$
$991$	−20.0312	−0.636312	−0.318156	+	0.948038i	$0.603064\pi$
$992$	−78.8431	−2.50327
$993$	0	0
$994$	−14.3896	−0.456411
$995$	−34.9315	−1.10740
$996$	0	0
$997$	0.500804	0.0158606	0.00793031	−	0.999969i	$-0.497476\pi$
$997$	0.500804	0.0158606	0.00793031	+	0.999969i	$0.497476\pi$
$998$	−50.1026	−1.58597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	549.2.a.g.1.1		3
3.2	odd	2		61.2.a.b.1.3	✓	3
4.3	odd	2		8784.2.a.bn.1.2		3
12.11	even	2		976.2.a.f.1.3		3
15.2	even	4		1525.2.b.b.1099.6		6
15.8	even	4		1525.2.b.b.1099.1		6
15.14	odd	2		1525.2.a.d.1.1		3
21.20	even	2		2989.2.a.i.1.3		3
24.5	odd	2		3904.2.a.r.1.3		3
24.11	even	2		3904.2.a.w.1.1		3
33.32	even	2		7381.2.a.f.1.1		3
183.182	odd	2		3721.2.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.a.b.1.3	✓	3	3.2	odd	2
549.2.a.g.1.1		3	1.1	even	1	trivial
976.2.a.f.1.3		3	12.11	even	2
1525.2.a.d.1.1		3	15.14	odd	2
1525.2.b.b.1099.1		6	15.8	even	4
1525.2.b.b.1099.6		6	15.2	even	4
2989.2.a.i.1.3		3	21.20	even	2
3721.2.a.c.1.1		3	183.182	odd	2
3904.2.a.r.1.3		3	24.5	odd	2
3904.2.a.w.1.1		3	24.11	even	2
7381.2.a.f.1.1		3	33.32	even	2
8784.2.a.bn.1.2		3	4.3	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:	\( N \)	\(=\)	\( 549 = 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	549.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +2.70928 q^{4} +1.63090 q^{5} -0.460811 q^{7} -1.53919 q^{8} +O(q^{10})\)	\(q-2.17009 q^{2} +2.70928 q^{4} +1.63090 q^{5} -0.460811 q^{7} -1.53919 q^{8} -3.53919 q^{10} -6.17009 q^{11} -4.07838 q^{13} +1.00000 q^{14} -2.07838 q^{16} -0.630898 q^{17} +7.12783 q^{19} +4.41855 q^{20} +13.3896 q^{22} +0.170086 q^{23} -2.34017 q^{25} +8.85043 q^{26} -1.24846 q^{28} -2.63090 q^{29} -10.3896 q^{31} +7.58864 q^{32} +1.36910 q^{34} -0.751536 q^{35} +5.12783 q^{37} -15.4680 q^{38} -2.51026 q^{40} -3.15676 q^{41} -3.36910 q^{43} -16.7165 q^{44} -0.369102 q^{46} -0.183417 q^{47} -6.78765 q^{49} +5.07838 q^{50} -11.0494 q^{52} +4.34017 q^{53} -10.0628 q^{55} +0.709275 q^{56} +5.70928 q^{58} -1.78047 q^{59} +1.00000 q^{61} +22.5464 q^{62} -12.3112 q^{64} -6.65142 q^{65} -8.55971 q^{67} -1.70928 q^{68} +1.63090 q^{70} -14.3896 q^{71} -0.552520 q^{73} -11.1278 q^{74} +19.3112 q^{76} +2.84324 q^{77} -7.00719 q^{79} -3.38962 q^{80} +6.85043 q^{82} -6.83710 q^{83} -1.02893 q^{85} +7.31124 q^{86} +9.49693 q^{88} -4.49693 q^{89} +1.87936 q^{91} +0.460811 q^{92} +0.398032 q^{94} +11.6248 q^{95} +8.52359 q^{97} +14.7298 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8	\( 3 q - q^{2} + q^{4} + q^{5} - 3 q^{7} - 3 q^{8} - 9 q^{10} - 13 q^{11} - 9 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} + 11 q^{22} - 5 q^{23} + 4 q^{25} - q^{26} + 5 q^{28} - 4 q^{29} - 2 q^{31} + 3 q^{32} + 8 q^{34} - 11 q^{35} - 6 q^{37} - 14 q^{38} + 9 q^{40} - 3 q^{41} - 14 q^{43} - 9 q^{44} - 5 q^{46} + 4 q^{47} - 10 q^{49} + 12 q^{50} - 15 q^{52} + 2 q^{53} - 13 q^{55} - 5 q^{56} + 10 q^{58} - 29 q^{59} + 3 q^{61} + 32 q^{62} - 11 q^{64} + 17 q^{65} + 9 q^{67} + 2 q^{68} + q^{70} - 14 q^{71} - q^{73} - 12 q^{74} + 32 q^{76} + 15 q^{77} + 13 q^{79} + 19 q^{80} - 7 q^{82} + 8 q^{83} - 18 q^{85} - 4 q^{86} + 11 q^{88} + 4 q^{89} - 7 q^{91} + 3 q^{92} + 20 q^{94} - 4 q^{95} + 10 q^{97} + 4 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + q^5 - 3 * q^7 - 3 * q^8 - 9 * q^10 - 13 * q^11 - 9 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 - q^20 + 11 * q^22 - 5 * q^23 + 4 * q^25 - q^26 + 5 * q^28 - 4 * q^29 - 2 * q^31 + 3 * q^32 + 8 * q^34 - 11 * q^35 - 6 * q^37 - 14 * q^38 + 9 * q^40 - 3 * q^41 - 14 * q^43 - 9 * q^44 - 5 * q^46 + 4 * q^47 - 10 * q^49 + 12 * q^50 - 15 * q^52 + 2 * q^53 - 13 * q^55 - 5 * q^56 + 10 * q^58 - 29 * q^59 + 3 * q^61 + 32 * q^62 - 11 * q^64 + 17 * q^65 + 9 * q^67 + 2 * q^68 + q^70 - 14 * q^71 - q^73 - 12 * q^74 + 32 * q^76 + 15 * q^77 + 13 * q^79 + 19 * q^80 - 7 * q^82 + 8 * q^83 - 18 * q^85 - 4 * q^86 + 11 * q^88 + 4 * q^89 - 7 * q^91 + 3 * q^92 + 20 * q^94 - 4 * q^95 + 10 * q^97 + 4 * q^98

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