LMFDB - Embedded newform 1341.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1341,2,Mod(1,1341)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1341, 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("1341.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1341 = 3^{2} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1341.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:	$10.7079389111$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 $ x^9 - x^8 - 15x^7 + 12x^6 + 75x^5 - 48x^4 - 137x^3 + 76x^2 + 68*x - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 149)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.59227$ of defining polynomial
Character	$\chi$	$=$	1341.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59227 q^{2} +0.535312 q^{4} +4.15709 q^{5} +0.321851 q^{7} +2.33217 q^{8} +O(q^{10})$	\(q-1.59227 q^{2} +0.535312 q^{4} +4.15709 q^{5} +0.321851 q^{7} +2.33217 q^{8} -6.61920 q^{10} -1.64360 q^{11} -0.142836 q^{13} -0.512473 q^{14} -4.78407 q^{16} +1.92222 q^{17} +2.74852 q^{19} +2.22534 q^{20} +2.61706 q^{22} +8.74674 q^{23} +12.2814 q^{25} +0.227434 q^{26} +0.172291 q^{28} -6.14379 q^{29} +8.99153 q^{31} +2.95316 q^{32} -3.06069 q^{34} +1.33797 q^{35} -1.11565 q^{37} -4.37638 q^{38} +9.69506 q^{40} -9.98634 q^{41} -11.0924 q^{43} -0.879841 q^{44} -13.9271 q^{46} +3.41903 q^{47} -6.89641 q^{49} -19.5553 q^{50} -0.0764621 q^{52} +3.24131 q^{53} -6.83262 q^{55} +0.750613 q^{56} +9.78255 q^{58} -1.71358 q^{59} +8.51115 q^{61} -14.3169 q^{62} +4.86591 q^{64} -0.593784 q^{65} +12.7225 q^{67} +1.02899 q^{68} -2.13040 q^{70} +4.58467 q^{71} -7.01909 q^{73} +1.77641 q^{74} +1.47132 q^{76} -0.528996 q^{77} -6.79375 q^{79} -19.8878 q^{80} +15.9009 q^{82} -0.389142 q^{83} +7.99085 q^{85} +17.6621 q^{86} -3.83317 q^{88} +7.03497 q^{89} -0.0459721 q^{91} +4.68223 q^{92} -5.44400 q^{94} +11.4259 q^{95} -4.37428 q^{97} +10.9809 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * 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 $

$2$	−1.59227	−1.12590	−0.562951	−	0.826490i	$-0.690334\pi$
$2$	−1.59227	−1.12590	−0.562951	+	0.826490i	$0.690334\pi$
$3$	0	0
$3$	0	0
$4$	0.535312	0.267656
$5$	4.15709	1.85911	0.929555	−	0.368685i	$-0.120192\pi$
$5$	4.15709	1.85911	0.929555	+	0.368685i	$0.120192\pi$
$6$	0	0
$7$	0.321851	0.121648	0.0608242	−	0.998148i	$-0.480627\pi$
$7$	0.321851	0.121648	0.0608242	+	0.998148i	$0.480627\pi$
$8$	2.33217	0.824548
$9$	0	0
$10$	−6.61920	−2.09318
$11$	−1.64360	−0.495565	−0.247783	−	0.968816i	$-0.579702\pi$
$11$	−1.64360	−0.495565	−0.247783	+	0.968816i	$0.579702\pi$
$12$	0	0
$13$	−0.142836	−0.0396157	−0.0198078	−	0.999804i	$-0.506305\pi$
$13$	−0.142836	−0.0396157	−0.0198078	+	0.999804i	$0.506305\pi$
$14$	−0.512473	−0.136964
$15$	0	0
$16$	−4.78407	−1.19602
$17$	1.92222	0.466207	0.233103	−	0.972452i	$-0.425112\pi$
$17$	1.92222	0.466207	0.233103	+	0.972452i	$0.425112\pi$
$18$	0	0
$19$	2.74852	0.630554	0.315277	−	0.949000i	$-0.397903\pi$
$19$	2.74852	0.630554	0.315277	+	0.949000i	$0.397903\pi$
$20$	2.22534	0.497602
$21$	0	0
$22$	2.61706	0.557958
$23$	8.74674	1.82382	0.911910	−	0.410390i	$-0.134607\pi$
$23$	8.74674	1.82382	0.911910	+	0.410390i	$0.134607\pi$
$24$	0	0
$25$	12.2814	2.45629
$26$	0.227434	0.0446034
$27$	0	0
$28$	0.172291	0.0325599
$29$	−6.14379	−1.14087	−0.570436	−	0.821342i	$-0.693226\pi$
$29$	−6.14379	−1.14087	−0.570436	+	0.821342i	$0.693226\pi$
$30$	0	0
$31$	8.99153	1.61493	0.807463	−	0.589918i	$-0.200840\pi$
$31$	8.99153	1.61493	0.807463	+	0.589918i	$0.200840\pi$
$32$	2.95316	0.522050
$33$	0	0
$34$	−3.06069	−0.524903
$35$	1.33797	0.226158
$36$	0	0
$37$	−1.11565	−0.183412	−0.0917059	−	0.995786i	$-0.529232\pi$
$37$	−1.11565	−0.183412	−0.0917059	+	0.995786i	$0.529232\pi$
$38$	−4.37638	−0.709942
$39$	0	0
$40$	9.69506	1.53292
$41$	−9.98634	−1.55960	−0.779802	−	0.626026i	$-0.784680\pi$
$41$	−9.98634	−1.55960	−0.779802	+	0.626026i	$0.784680\pi$
$42$	0	0
$43$	−11.0924	−1.69158	−0.845789	−	0.533518i	$-0.820870\pi$
$43$	−11.0924	−1.69158	−0.845789	+	0.533518i	$0.820870\pi$
$44$	−0.879841	−0.132641
$45$	0	0
$46$	−13.9271	−2.05344
$47$	3.41903	0.498716	0.249358	−	0.968411i	$-0.419780\pi$
$47$	3.41903	0.498716	0.249358	+	0.968411i	$0.419780\pi$
$48$	0	0
$49$	−6.89641	−0.985202
$50$	−19.5553	−2.76554
$51$	0	0
$52$	−0.0764621	−0.0106034
$53$	3.24131	0.445228	0.222614	−	0.974907i	$-0.428541\pi$
$53$	3.24131	0.445228	0.222614	+	0.974907i	$0.428541\pi$
$54$	0	0
$55$	−6.83262	−0.921310
$56$	0.750613	0.100305
$57$	0	0
$58$	9.78255	1.28451
$59$	−1.71358	−0.223089	−0.111544	−	0.993759i	$-0.535580\pi$
$59$	−1.71358	−0.223089	−0.111544	+	0.993759i	$0.535580\pi$
$60$	0	0
$61$	8.51115	1.08974	0.544871	−	0.838520i	$-0.316579\pi$
$61$	8.51115	1.08974	0.544871	+	0.838520i	$0.316579\pi$
$62$	−14.3169	−1.81825
$63$	0	0
$64$	4.86591	0.608239
$65$	−0.593784	−0.0736499
$66$	0	0
$67$	12.7225	1.55430	0.777149	−	0.629317i	$-0.216665\pi$
$67$	12.7225	1.55430	0.777149	+	0.629317i	$0.216665\pi$
$68$	1.02899	0.124783
$69$	0	0
$70$	−2.13040	−0.254631
$71$	4.58467	0.544101	0.272050	−	0.962283i	$-0.412298\pi$
$71$	4.58467	0.544101	0.272050	+	0.962283i	$0.412298\pi$
$72$	0	0
$73$	−7.01909	−0.821522	−0.410761	−	0.911743i	$-0.634737\pi$
$73$	−7.01909	−0.821522	−0.410761	+	0.911743i	$0.634737\pi$
$74$	1.77641	0.206504
$75$	0	0
$76$	1.47132	0.168772
$77$	−0.528996	−0.0602847
$78$	0	0
$79$	−6.79375	−0.764357	−0.382178	−	0.924089i	$-0.624826\pi$
$79$	−6.79375	−0.764357	−0.382178	+	0.924089i	$0.624826\pi$
$80$	−19.8878	−2.22352
$81$	0	0
$82$	15.9009	1.75596
$83$	−0.389142	−0.0427139	−0.0213570	−	0.999772i	$-0.506799\pi$
$83$	−0.389142	−0.0427139	−0.0213570	+	0.999772i	$0.506799\pi$
$84$	0	0
$85$	7.99085	0.866729
$86$	17.6621	1.90455
$87$	0	0
$88$	−3.83317	−0.408617
$89$	7.03497	0.745706	0.372853	−	0.927891i	$-0.378380\pi$
$89$	7.03497	0.745706	0.372853	+	0.927891i	$0.378380\pi$
$90$	0	0
$91$	−0.0459721	−0.00481918
$92$	4.68223	0.488157
$93$	0	0
$94$	−5.44400	−0.561506
$95$	11.4259	1.17227
$96$	0	0
$97$	−4.37428	−0.444141	−0.222071	−	0.975031i	$-0.571282\pi$
$97$	−4.37428	−0.444141	−0.222071	+	0.975031i	$0.571282\pi$
$98$	10.9809	1.10924
$99$	0	0
$100$	6.57440	0.657440
$101$	−0.0262645	−0.00261341	−0.00130671	−	0.999999i	$-0.500416\pi$
$101$	−0.0262645	−0.00261341	−0.00130671	+	0.999999i	$0.500416\pi$
$102$	0	0
$103$	8.55969	0.843411	0.421706	−	0.906733i	$-0.361432\pi$
$103$	8.55969	0.843411	0.421706	+	0.906733i	$0.361432\pi$
$104$	−0.333119	−0.0326650
$105$	0	0
$106$	−5.16103	−0.501284
$107$	9.85395	0.952618	0.476309	−	0.879278i	$-0.341974\pi$
$107$	9.85395	0.952618	0.476309	+	0.879278i	$0.341974\pi$
$108$	0	0
$109$	−1.62408	−0.155559	−0.0777795	−	0.996971i	$-0.524783\pi$
$109$	−1.62408	−0.155559	−0.0777795	+	0.996971i	$0.524783\pi$
$110$	10.8793	1.03731
$111$	0	0
$112$	−1.53976	−0.145493
$113$	9.28527	0.873485	0.436743	−	0.899587i	$-0.356132\pi$
$113$	9.28527	0.873485	0.436743	+	0.899587i	$0.356132\pi$
$114$	0	0
$115$	36.3610	3.39068
$116$	−3.28884	−0.305362
$117$	0	0
$118$	2.72847	0.251176
$119$	0.618669	0.0567133
$120$	0	0
$121$	−8.29857	−0.754415
$122$	−13.5520	−1.22694
$123$	0	0
$124$	4.81327	0.432245
$125$	30.2696	2.70740
$126$	0	0
$127$	−1.24392	−0.110380	−0.0551898	−	0.998476i	$-0.517576\pi$
$127$	−1.24392	−0.110380	−0.0551898	+	0.998476i	$0.517576\pi$
$128$	−13.6541	−1.20687
$129$	0	0
$130$	0.945463	0.0829226
$131$	4.06287	0.354975	0.177487	−	0.984123i	$-0.443203\pi$
$131$	4.06287	0.354975	0.177487	+	0.984123i	$0.443203\pi$
$132$	0	0
$133$	0.884615	0.0767059
$134$	−20.2576	−1.74999
$135$	0	0
$136$	4.48295	0.384410
$137$	−15.8558	−1.35465	−0.677325	−	0.735684i	$-0.736861\pi$
$137$	−15.8558	−1.35465	−0.677325	+	0.735684i	$0.736861\pi$
$138$	0	0
$139$	8.61602	0.730801	0.365400	−	0.930850i	$-0.380932\pi$
$139$	8.61602	0.730801	0.365400	+	0.930850i	$0.380932\pi$
$140$	0.716230	0.0605325
$141$	0	0
$142$	−7.30002	−0.612604
$143$	0.234766	0.0196322
$144$	0	0
$145$	−25.5403	−2.12101
$146$	11.1763	0.924954
$147$	0	0
$148$	−0.597221	−0.0490913
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	18.0970	1.47272	0.736358	−	0.676592i	$-0.236544\pi$
$151$	18.0970	1.47272	0.736358	+	0.676592i	$0.236544\pi$
$152$	6.41003	0.519922
$153$	0	0
$154$	0.842303	0.0678747
$155$	37.3786	3.00232
$156$	0	0
$157$	9.91735	0.791491	0.395745	−	0.918360i	$-0.370486\pi$
$157$	9.91735	0.791491	0.395745	+	0.918360i	$0.370486\pi$
$158$	10.8175	0.860591
$159$	0	0
$160$	12.2766	0.970548
$161$	2.81515	0.221865
$162$	0	0
$163$	−4.57678	−0.358481	−0.179241	−	0.983805i	$-0.557364\pi$
$163$	−4.57678	−0.358481	−0.179241	+	0.983805i	$0.557364\pi$
$164$	−5.34581	−0.417438
$165$	0	0
$166$	0.619618	0.0480917
$167$	−1.52465	−0.117981	−0.0589907	−	0.998259i	$-0.518788\pi$
$167$	−1.52465	−0.117981	−0.0589907	+	0.998259i	$0.518788\pi$
$168$	0	0
$169$	−12.9796	−0.998431
$170$	−12.7236	−0.975852
$171$	0	0
$172$	−5.93790	−0.452761
$173$	0.223807	0.0170157	0.00850787	−	0.999964i	$-0.497292\pi$
$173$	0.223807	0.0170157	0.00850787	+	0.999964i	$0.497292\pi$
$174$	0	0
$175$	3.95280	0.298803
$176$	7.86311	0.592704
$177$	0	0
$178$	−11.2016	−0.839592
$179$	−6.76844	−0.505897	−0.252948	−	0.967480i	$-0.581400\pi$
$179$	−6.76844	−0.505897	−0.252948	+	0.967480i	$0.581400\pi$
$180$	0	0
$181$	3.79368	0.281982	0.140991	−	0.990011i	$-0.454971\pi$
$181$	3.79368	0.281982	0.140991	+	0.990011i	$0.454971\pi$
$182$	0.0731998	0.00542593
$183$	0	0
$184$	20.3989	1.50383
$185$	−4.63787	−0.340983
$186$	0	0
$187$	−3.15937	−0.231036
$188$	1.83025	0.133484
$189$	0	0
$190$	−18.1930	−1.31986
$191$	1.79940	0.130200	0.0651001	−	0.997879i	$-0.479263\pi$
$191$	1.79940	0.130200	0.0651001	+	0.997879i	$0.479263\pi$
$192$	0	0
$193$	−9.54269	−0.686898	−0.343449	−	0.939171i	$-0.611595\pi$
$193$	−9.54269	−0.686898	−0.343449	+	0.939171i	$0.611595\pi$
$194$	6.96503	0.500060
$195$	0	0
$196$	−3.69173	−0.263695
$197$	8.53396	0.608020	0.304010	−	0.952669i	$-0.401674\pi$
$197$	8.53396	0.608020	0.304010	+	0.952669i	$0.401674\pi$
$198$	0	0
$199$	−18.8914	−1.33918	−0.669589	−	0.742732i	$-0.733530\pi$
$199$	−18.8914	−1.33918	−0.669589	+	0.742732i	$0.733530\pi$
$200$	28.6424	2.02533
$201$	0	0
$202$	0.0418200	0.00294245
$203$	−1.97739	−0.138785
$204$	0	0
$205$	−41.5141	−2.89947
$206$	−13.6293	−0.949598
$207$	0	0
$208$	0.683339	0.0473810
$209$	−4.51748	−0.312481
$210$	0	0
$211$	25.1726	1.73295	0.866477	−	0.499216i	$-0.166379\pi$
$211$	25.1726	1.73295	0.866477	+	0.499216i	$0.166379\pi$
$212$	1.73511	0.119168
$213$	0	0
$214$	−15.6901	−1.07255
$215$	−46.1122	−3.14483
$216$	0	0
$217$	2.89393	0.196453
$218$	2.58597	0.175144
$219$	0	0
$220$	−3.65758	−0.246594
$221$	−0.274563	−0.0184691
$222$	0	0
$223$	8.82877	0.591218	0.295609	−	0.955309i	$-0.404477\pi$
$223$	8.82877	0.591218	0.295609	+	0.955309i	$0.404477\pi$
$224$	0.950479	0.0635065
$225$	0	0
$226$	−14.7846	−0.983459
$227$	1.58942	0.105494	0.0527468	−	0.998608i	$-0.483202\pi$
$227$	1.58942	0.105494	0.0527468	+	0.998608i	$0.483202\pi$
$228$	0	0
$229$	14.8449	0.980980	0.490490	−	0.871447i	$-0.336818\pi$
$229$	14.8449	0.980980	0.490490	+	0.871447i	$0.336818\pi$
$230$	−57.8964	−3.81758
$231$	0	0
$232$	−14.3284	−0.940704
$233$	−15.3183	−1.00354	−0.501769	−	0.865002i	$-0.667317\pi$
$233$	−15.3183	−1.00354	−0.501769	+	0.865002i	$0.667317\pi$
$234$	0	0
$235$	14.2132	0.927168
$236$	−0.917298	−0.0597110
$237$	0	0
$238$	−0.985086	−0.0638536
$239$	18.4338	1.19239	0.596193	−	0.802841i	$-0.296679\pi$
$239$	18.4338	1.19239	0.596193	+	0.802841i	$0.296679\pi$
$240$	0	0
$241$	12.1838	0.784830	0.392415	−	0.919788i	$-0.371640\pi$
$241$	12.1838	0.784830	0.392415	+	0.919788i	$0.371640\pi$
$242$	13.2135	0.849398
$243$	0	0
$244$	4.55613	0.291676
$245$	−28.6690	−1.83160
$246$	0	0
$247$	−0.392589	−0.0249798
$248$	20.9698	1.33158
$249$	0	0
$250$	−48.1973	−3.04826
$251$	−13.3382	−0.841899	−0.420949	−	0.907084i	$-0.638303\pi$
$251$	−13.3382	−0.841899	−0.420949	+	0.907084i	$0.638303\pi$
$252$	0	0
$253$	−14.3762	−0.903822
$254$	1.98064	0.124277
$255$	0	0
$256$	12.0092	0.750576
$257$	−22.9691	−1.43277	−0.716386	−	0.697704i	$-0.754205\pi$
$257$	−22.9691	−1.43277	−0.716386	+	0.697704i	$0.754205\pi$
$258$	0	0
$259$	−0.359074	−0.0223118
$260$	−0.317860	−0.0197128
$261$	0	0
$262$	−6.46917	−0.399667
$263$	20.3931	1.25749	0.628747	−	0.777610i	$-0.283568\pi$
$263$	20.3931	1.25749	0.628747	+	0.777610i	$0.283568\pi$
$264$	0	0
$265$	13.4744	0.827728
$266$	−1.40854	−0.0863633
$267$	0	0
$268$	6.81050	0.416017
$269$	5.85699	0.357107	0.178553	−	0.983930i	$-0.442858\pi$
$269$	5.85699	0.357107	0.178553	+	0.983930i	$0.442858\pi$
$270$	0	0
$271$	0.355506	0.0215954	0.0107977	−	0.999942i	$-0.496563\pi$
$271$	0.355506	0.0215954	0.0107977	+	0.999942i	$0.496563\pi$
$272$	−9.19602	−0.557591
$273$	0	0
$274$	25.2466	1.52520
$275$	−20.1858	−1.21725
$276$	0	0
$277$	11.2279	0.674620	0.337310	−	0.941394i	$-0.390483\pi$
$277$	11.2279	0.674620	0.337310	+	0.941394i	$0.390483\pi$
$278$	−13.7190	−0.822811
$279$	0	0
$280$	3.12037	0.186478
$281$	23.5233	1.40328	0.701640	−	0.712532i	$-0.252452\pi$
$281$	23.5233	1.40328	0.701640	+	0.712532i	$0.252452\pi$
$282$	0	0
$283$	−24.2000	−1.43854	−0.719271	−	0.694730i	$-0.755524\pi$
$283$	−24.2000	−1.43854	−0.719271	+	0.694730i	$0.755524\pi$
$284$	2.45423	0.145632
$285$	0	0
$286$	−0.373811	−0.0221039
$287$	−3.21412	−0.189723
$288$	0	0
$289$	−13.3051	−0.782651
$290$	40.6670	2.38805
$291$	0	0
$292$	−3.75741	−0.219886
$293$	−7.79038	−0.455119	−0.227559	−	0.973764i	$-0.573075\pi$
$293$	−7.79038	−0.455119	−0.227559	+	0.973764i	$0.573075\pi$
$294$	0	0
$295$	−7.12350	−0.414746
$296$	−2.60189	−0.151232
$297$	0	0
$298$	1.59227	0.0922375
$299$	−1.24935	−0.0722519
$300$	0	0
$301$	−3.57011	−0.205778
$302$	−28.8153	−1.65813
$303$	0	0
$304$	−13.1491	−0.754153
$305$	35.3817	2.02595
$306$	0	0
$307$	−2.54370	−0.145177	−0.0725884	−	0.997362i	$-0.523126\pi$
$307$	−2.54370	−0.145177	−0.0725884	+	0.997362i	$0.523126\pi$
$308$	−0.283178	−0.0161356
$309$	0	0
$310$	−59.5167	−3.38032
$311$	−7.56137	−0.428766	−0.214383	−	0.976750i	$-0.568774\pi$
$311$	−7.56137	−0.428766	−0.214383	+	0.976750i	$0.568774\pi$
$312$	0	0
$313$	4.93985	0.279217	0.139608	−	0.990207i	$-0.455416\pi$
$313$	4.93985	0.279217	0.139608	+	0.990207i	$0.455416\pi$
$314$	−15.7911	−0.891141
$315$	0	0
$316$	−3.63678	−0.204585
$317$	−11.2965	−0.634476	−0.317238	−	0.948346i	$-0.602755\pi$
$317$	−11.2965	−0.634476	−0.317238	+	0.948346i	$0.602755\pi$
$318$	0	0
$319$	10.0980	0.565377
$320$	20.2281	1.13078
$321$	0	0
$322$	−4.48247	−0.249798
$323$	5.28326	0.293968
$324$	0	0
$325$	−1.75424	−0.0973075
$326$	7.28746	0.403615
$327$	0	0
$328$	−23.2899	−1.28597
$329$	1.10042	0.0606680
$330$	0	0
$331$	8.22195	0.451919	0.225960	−	0.974137i	$-0.427448\pi$
$331$	8.22195	0.451919	0.225960	+	0.974137i	$0.427448\pi$
$332$	−0.208313	−0.0114326
$333$	0	0
$334$	2.42766	0.132835
$335$	52.8885	2.88961
$336$	0	0
$337$	−23.5381	−1.28220	−0.641102	−	0.767456i	$-0.721522\pi$
$337$	−23.5381	−1.28220	−0.641102	+	0.767456i	$0.721522\pi$
$338$	20.6670	1.12414
$339$	0	0
$340$	4.27760	0.231985
$341$	−14.7785	−0.800301
$342$	0	0
$343$	−4.47258	−0.241497
$344$	−25.8694	−1.39479
$345$	0	0
$346$	−0.356360	−0.0191581
$347$	−21.6025	−1.15968	−0.579841	−	0.814730i	$-0.696885\pi$
$347$	−21.6025	−1.15968	−0.579841	+	0.814730i	$0.696885\pi$
$348$	0	0
$349$	19.2191	1.02878	0.514388	−	0.857558i	$-0.328019\pi$
$349$	19.2191	1.02878	0.514388	+	0.857558i	$0.328019\pi$
$350$	−6.29390	−0.336423
$351$	0	0
$352$	−4.85383	−0.258710
$353$	−28.9234	−1.53943	−0.769717	−	0.638385i	$-0.779603\pi$
$353$	−28.9234	−1.53943	−0.769717	+	0.638385i	$0.779603\pi$
$354$	0	0
$355$	19.0589	1.01154
$356$	3.76591	0.199593
$357$	0	0
$358$	10.7772	0.569590
$359$	−32.0665	−1.69241	−0.846204	−	0.532860i	$-0.821117\pi$
$359$	−32.0665	−1.69241	−0.846204	+	0.532860i	$0.821117\pi$
$360$	0	0
$361$	−11.4456	−0.602402
$362$	−6.04055	−0.317484
$363$	0	0
$364$	−0.0246094	−0.00128988
$365$	−29.1790	−1.52730
$366$	0	0
$367$	−4.87186	−0.254309	−0.127154	−	0.991883i	$-0.540584\pi$
$367$	−4.87186	−0.254309	−0.127154	+	0.991883i	$0.540584\pi$
$368$	−41.8450	−2.18132
$369$	0	0
$370$	7.38472	0.383913
$371$	1.04322	0.0541613
$372$	0	0
$373$	−14.3830	−0.744721	−0.372361	−	0.928088i	$-0.621452\pi$
$373$	−14.3830	−0.744721	−0.372361	+	0.928088i	$0.621452\pi$
$374$	5.03056	0.260124
$375$	0	0
$376$	7.97376	0.411215
$377$	0.877556	0.0451964
$378$	0	0
$379$	−0.936090	−0.0480837	−0.0240418	−	0.999711i	$-0.507653\pi$
$379$	−0.936090	−0.0480837	−0.0240418	+	0.999711i	$0.507653\pi$
$380$	6.11640	0.313765
$381$	0	0
$382$	−2.86513	−0.146593
$383$	−13.8045	−0.705375	−0.352687	−	0.935741i	$-0.614732\pi$
$383$	−13.8045	−0.705375	−0.352687	+	0.935741i	$0.614732\pi$
$384$	0	0
$385$	−2.19909	−0.112076
$386$	15.1945	0.773380
$387$	0	0
$388$	−2.34161	−0.118877
$389$	25.3195	1.28375	0.641875	−	0.766809i	$-0.278157\pi$
$389$	25.3195	1.28375	0.641875	+	0.766809i	$0.278157\pi$
$390$	0	0
$391$	16.8131	0.850277
$392$	−16.0836	−0.812346
$393$	0	0
$394$	−13.5883	−0.684571
$395$	−28.2423	−1.42102
$396$	0	0
$397$	2.79670	0.140363	0.0701813	−	0.997534i	$-0.477642\pi$
$397$	2.79670	0.140363	0.0701813	+	0.997534i	$0.477642\pi$
$398$	30.0802	1.50778
$399$	0	0
$400$	−58.7552	−2.93776
$401$	7.00757	0.349941	0.174971	−	0.984574i	$-0.444017\pi$
$401$	7.00757	0.349941	0.174971	+	0.984574i	$0.444017\pi$
$402$	0	0
$403$	−1.28432	−0.0639764
$404$	−0.0140597	−0.000699496 0
$405$	0	0
$406$	3.14853	0.156259
$407$	1.83369	0.0908925
$408$	0	0
$409$	−30.4717	−1.50673	−0.753365	−	0.657602i	$-0.771571\pi$
$409$	−30.4717	−1.50673	−0.753365	+	0.657602i	$0.771571\pi$
$410$	66.1016	3.26452
$411$	0	0
$412$	4.58211	0.225744
$413$	−0.551517	−0.0271384
$414$	0	0
$415$	−1.61770	−0.0794098
$416$	−0.421819	−0.0206814
$417$	0	0
$418$	7.19303	0.351823
$419$	−12.1683	−0.594460	−0.297230	−	0.954806i	$-0.596063\pi$
$419$	−12.1683	−0.594460	−0.297230	+	0.954806i	$0.596063\pi$
$420$	0	0
$421$	−16.1210	−0.785688	−0.392844	−	0.919605i	$-0.628509\pi$
$421$	−16.1210	−0.785688	−0.392844	+	0.919605i	$0.628509\pi$
$422$	−40.0815	−1.95114
$423$	0	0
$424$	7.55930	0.367112
$425$	23.6076	1.14514
$426$	0	0
$427$	2.73933	0.132565
$428$	5.27494	0.254974
$429$	0	0
$430$	73.4229	3.54077
$431$	−0.715869	−0.0344822	−0.0172411	−	0.999851i	$-0.505488\pi$
$431$	−0.715869	−0.0344822	−0.0172411	+	0.999851i	$0.505488\pi$
$432$	0	0
$433$	24.6916	1.18660	0.593302	−	0.804980i	$-0.297824\pi$
$433$	24.6916	1.18660	0.593302	+	0.804980i	$0.297824\pi$
$434$	−4.60791	−0.221187
$435$	0	0
$436$	−0.869392	−0.0416363
$437$	24.0406	1.15002
$438$	0	0
$439$	14.8500	0.708754	0.354377	−	0.935103i	$-0.384693\pi$
$439$	14.8500	0.708754	0.354377	+	0.935103i	$0.384693\pi$
$440$	−15.9348	−0.759664
$441$	0	0
$442$	0.437177	0.0207944
$443$	−20.9532	−0.995515	−0.497758	−	0.867316i	$-0.665843\pi$
$443$	−20.9532	−0.995515	−0.497758	+	0.867316i	$0.665843\pi$
$444$	0	0
$445$	29.2450	1.38635
$446$	−14.0578	−0.665654
$447$	0	0
$448$	1.56610	0.0739913
$449$	−9.59397	−0.452768	−0.226384	−	0.974038i	$-0.572690\pi$
$449$	−9.59397	−0.452768	−0.226384	+	0.974038i	$0.572690\pi$
$450$	0	0
$451$	16.4136	0.772885
$452$	4.97052	0.233794
$453$	0	0
$454$	−2.53078	−0.118776
$455$	−0.191110	−0.00895939
$456$	0	0
$457$	−14.7445	−0.689721	−0.344860	−	0.938654i	$-0.612074\pi$
$457$	−14.7445	−0.689721	−0.344860	+	0.938654i	$0.612074\pi$
$458$	−23.6371	−1.10449
$459$	0	0
$460$	19.4645	0.907537
$461$	−7.04932	−0.328319	−0.164160	−	0.986434i	$-0.552491\pi$
$461$	−7.04932	−0.328319	−0.164160	+	0.986434i	$0.552491\pi$
$462$	0	0
$463$	1.47493	0.0685457	0.0342728	−	0.999413i	$-0.489088\pi$
$463$	1.47493	0.0685457	0.0342728	+	0.999413i	$0.489088\pi$
$464$	29.3923	1.36450
$465$	0	0
$466$	24.3909	1.12989
$467$	2.43720	0.112780	0.0563900	−	0.998409i	$-0.482041\pi$
$467$	2.43720	0.112780	0.0563900	+	0.998409i	$0.482041\pi$
$468$	0	0
$469$	4.09475	0.189078
$470$	−22.6312	−1.04390
$471$	0	0
$472$	−3.99636	−0.183947
$473$	18.2315	0.838287
$474$	0	0
$475$	33.7558	1.54882
$476$	0.331181	0.0151797
$477$	0	0
$478$	−29.3516	−1.34251
$479$	−30.2467	−1.38201	−0.691004	−	0.722851i	$-0.742831\pi$
$479$	−30.2467	−1.38201	−0.691004	+	0.722851i	$0.742831\pi$
$480$	0	0
$481$	0.159356	0.00726599
$482$	−19.3999	−0.883641
$483$	0	0
$484$	−4.44232	−0.201924
$485$	−18.1843	−0.825707
$486$	0	0
$487$	−26.0228	−1.17921	−0.589604	−	0.807693i	$-0.700716\pi$
$487$	−26.0228	−1.17921	−0.589604	+	0.807693i	$0.700716\pi$
$488$	19.8495	0.898544
$489$	0	0
$490$	45.6487	2.06220
$491$	−16.7894	−0.757695	−0.378847	−	0.925459i	$-0.623679\pi$
$491$	−16.7894	−0.757695	−0.378847	+	0.925459i	$0.623679\pi$
$492$	0	0
$493$	−11.8097	−0.531882
$494$	0.625106	0.0281248
$495$	0	0
$496$	−43.0160	−1.93148
$497$	1.47558	0.0661890
$498$	0	0
$499$	12.1034	0.541824	0.270912	−	0.962604i	$-0.412675\pi$
$499$	12.1034	0.541824	0.270912	+	0.962604i	$0.412675\pi$
$500$	16.2037	0.724651
$501$	0	0
$502$	21.2379	0.947896
$503$	14.6848	0.654764	0.327382	−	0.944892i	$-0.393834\pi$
$503$	14.6848	0.654764	0.327382	+	0.944892i	$0.393834\pi$
$504$	0	0
$505$	−0.109184	−0.00485862
$506$	22.8907	1.01762
$507$	0	0
$508$	−0.665883	−0.0295438
$509$	−22.6319	−1.00314	−0.501570	−	0.865117i	$-0.667244\pi$
$509$	−22.6319	−1.00314	−0.501570	+	0.865117i	$0.667244\pi$
$510$	0	0
$511$	−2.25910	−0.0999369
$512$	8.18642	0.361792
$513$	0	0
$514$	36.5729	1.61316
$515$	35.5834	1.56799
$516$	0	0
$517$	−5.61952	−0.247146
$518$	0.571741	0.0251209
$519$	0	0
$520$	−1.38481	−0.0607278
$521$	30.8063	1.34965	0.674823	−	0.737979i	$-0.264220\pi$
$521$	30.8063	1.34965	0.674823	+	0.737979i	$0.264220\pi$
$522$	0	0
$523$	32.7495	1.43203	0.716017	−	0.698083i	$-0.245963\pi$
$523$	32.7495	1.43203	0.716017	+	0.698083i	$0.245963\pi$
$524$	2.17490	0.0950111
$525$	0	0
$526$	−32.4713	−1.41582
$527$	17.2837	0.752889
$528$	0	0
$529$	53.5054	2.32632
$530$	−21.4549	−0.931941
$531$	0	0
$532$	0.473545	0.0205308
$533$	1.42641	0.0617848
$534$	0	0
$535$	40.9638	1.77102
$536$	29.6710	1.28159
$537$	0	0
$538$	−9.32588	−0.402067
$539$	11.3350	0.488232
$540$	0	0
$541$	−42.6164	−1.83222	−0.916111	−	0.400925i	$-0.868689\pi$
$541$	−42.6164	−1.83222	−0.916111	+	0.400925i	$0.868689\pi$
$542$	−0.566060	−0.0243143
$543$	0	0
$544$	5.67662	0.243383
$545$	−6.75147	−0.289201
$546$	0	0
$547$	−16.4163	−0.701912	−0.350956	−	0.936392i	$-0.614143\pi$
$547$	−16.4163	−0.701912	−0.350956	+	0.936392i	$0.614143\pi$
$548$	−8.48778	−0.362580
$549$	0	0
$550$	32.1412	1.37051
$551$	−16.8863	−0.719382
$552$	0	0
$553$	−2.18658	−0.0929828
$554$	−17.8778	−0.759557
$555$	0	0
$556$	4.61226	0.195603
$557$	−29.6975	−1.25832	−0.629161	−	0.777275i	$-0.716602\pi$
$557$	−29.6975	−1.25832	−0.629161	+	0.777275i	$0.716602\pi$
$558$	0	0
$559$	1.58440	0.0670130
$560$	−6.40092	−0.270488
$561$	0	0
$562$	−37.4553	−1.57996
$563$	7.99627	0.337003	0.168501	−	0.985701i	$-0.446107\pi$
$563$	7.99627	0.337003	0.168501	+	0.985701i	$0.446107\pi$
$564$	0	0
$565$	38.5998	1.62390
$566$	38.5329	1.61966
$567$	0	0
$568$	10.6923	0.448637
$569$	4.88286	0.204700	0.102350	−	0.994748i	$-0.467364\pi$
$569$	4.88286	0.204700	0.102350	+	0.994748i	$0.467364\pi$
$570$	0	0
$571$	−24.3232	−1.01789	−0.508947	−	0.860798i	$-0.669965\pi$
$571$	−24.3232	−1.01789	−0.508947	+	0.860798i	$0.669965\pi$
$572$	0.125673	0.00525467
$573$	0	0
$574$	5.11773	0.213610
$575$	107.422	4.47983
$576$	0	0
$577$	−47.5978	−1.98152	−0.990761	−	0.135619i	$-0.956698\pi$
$577$	−47.5978	−1.98152	−0.990761	+	0.135619i	$0.956698\pi$
$578$	21.1852	0.881189
$579$	0	0
$580$	−13.6720	−0.567700
$581$	−0.125246	−0.00519608
$582$	0	0
$583$	−5.32743	−0.220640
$584$	−16.3697	−0.677384
$585$	0	0
$586$	12.4044	0.512419
$587$	−16.6275	−0.686290	−0.343145	−	0.939282i	$-0.611492\pi$
$587$	−16.6275	−0.686290	−0.343145	+	0.939282i	$0.611492\pi$
$588$	0	0
$589$	24.7134	1.01830
$590$	11.3425	0.466964
$591$	0	0
$592$	5.33735	0.219364
$593$	−4.85791	−0.199491	−0.0997453	−	0.995013i	$-0.531803\pi$
$593$	−4.85791	−0.199491	−0.0997453	+	0.995013i	$0.531803\pi$
$594$	0	0
$595$	2.57187	0.105436
$596$	−0.535312	−0.0219272
$597$	0	0
$598$	1.98930	0.0813486
$599$	11.2171	0.458320	0.229160	−	0.973389i	$-0.426402\pi$
$599$	11.2171	0.458320	0.229160	+	0.973389i	$0.426402\pi$
$600$	0	0
$601$	−19.0626	−0.777581	−0.388790	−	0.921326i	$-0.627107\pi$
$601$	−19.0626	−0.777581	−0.388790	+	0.921326i	$0.627107\pi$
$602$	5.68456	0.231686
$603$	0	0
$604$	9.68757	0.394182
$605$	−34.4979	−1.40254
$606$	0	0
$607$	−30.1342	−1.22311	−0.611556	−	0.791201i	$-0.709456\pi$
$607$	−30.1342	−1.22311	−0.611556	+	0.791201i	$0.709456\pi$
$608$	8.11682	0.329181
$609$	0	0
$610$	−56.3371	−2.28102
$611$	−0.488361	−0.0197570
$612$	0	0
$613$	24.8769	1.00477	0.502385	−	0.864644i	$-0.332456\pi$
$613$	24.8769	1.00477	0.502385	+	0.864644i	$0.332456\pi$
$614$	4.05025	0.163455
$615$	0	0
$616$	−1.23371	−0.0497076
$617$	−25.3870	−1.02204	−0.511021	−	0.859568i	$-0.670732\pi$
$617$	−25.3870	−1.02204	−0.511021	+	0.859568i	$0.670732\pi$
$618$	0	0
$619$	11.8401	0.475892	0.237946	−	0.971278i	$-0.423526\pi$
$619$	11.8401	0.475892	0.237946	+	0.971278i	$0.423526\pi$
$620$	20.0092	0.803590
$621$	0	0
$622$	12.0397	0.482749
$623$	2.26422	0.0907139
$624$	0	0
$625$	64.4265	2.57706
$626$	−7.86555	−0.314371
$627$	0	0
$628$	5.30888	0.211847
$629$	−2.14453	−0.0855078
$630$	0	0
$631$	4.15003	0.165210	0.0826050	−	0.996582i	$-0.473676\pi$
$631$	4.15003	0.165210	0.0826050	+	0.996582i	$0.473676\pi$
$632$	−15.8442	−0.630249
$633$	0	0
$634$	17.9871	0.714358
$635$	−5.17107	−0.205208
$636$	0	0
$637$	0.985058	0.0390294
$638$	−16.0786	−0.636559
$639$	0	0
$640$	−56.7616	−2.24370
$641$	20.0368	0.791407	0.395703	−	0.918378i	$-0.370501\pi$
$641$	20.0368	0.791407	0.395703	+	0.918378i	$0.370501\pi$
$642$	0	0
$643$	−20.7612	−0.818744	−0.409372	−	0.912368i	$-0.634252\pi$
$643$	−20.7612	−0.818744	−0.409372	+	0.912368i	$0.634252\pi$
$644$	1.50698	0.0593835
$645$	0	0
$646$	−8.41236	−0.330980
$647$	−32.0717	−1.26087	−0.630434	−	0.776243i	$-0.717123\pi$
$647$	−32.0717	−1.26087	−0.630434	+	0.776243i	$0.717123\pi$
$648$	0	0
$649$	2.81644	0.110555
$650$	2.79321	0.109559
$651$	0	0
$652$	−2.45001	−0.0959497
$653$	−8.68126	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$653$	−8.68126	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$654$	0	0
$655$	16.8897	0.659936
$656$	47.7753	1.86531
$657$	0	0
$658$	−1.75216	−0.0683062
$659$	−15.1610	−0.590590	−0.295295	−	0.955406i	$-0.595418\pi$
$659$	−15.1610	−0.590590	−0.295295	+	0.955406i	$0.595418\pi$
$660$	0	0
$661$	5.87836	0.228642	0.114321	−	0.993444i	$-0.463531\pi$
$661$	5.87836	0.228642	0.114321	+	0.993444i	$0.463531\pi$
$662$	−13.0915	−0.508817
$663$	0	0
$664$	−0.907547	−0.0352197
$665$	3.67743	0.142605
$666$	0	0
$667$	−53.7381	−2.08075
$668$	−0.816166	−0.0315784
$669$	0	0
$670$	−84.2126	−3.25342
$671$	−13.9890	−0.540038
$672$	0	0
$673$	−28.5135	−1.09911	−0.549557	−	0.835456i	$-0.685203\pi$
$673$	−28.5135	−1.09911	−0.549557	+	0.835456i	$0.685203\pi$
$674$	37.4790	1.44364
$675$	0	0
$676$	−6.94814	−0.267236
$677$	−11.6456	−0.447577	−0.223789	−	0.974638i	$-0.571843\pi$
$677$	−11.6456	−0.447577	−0.223789	+	0.974638i	$0.571843\pi$
$678$	0	0
$679$	−1.40787	−0.0540291
$680$	18.6360	0.714660
$681$	0	0
$682$	23.5313	0.901061
$683$	−13.5645	−0.519031	−0.259515	−	0.965739i	$-0.583563\pi$
$683$	−13.5645	−0.519031	−0.259515	+	0.965739i	$0.583563\pi$
$684$	0	0
$685$	−65.9139	−2.51844
$686$	7.12154	0.271902
$687$	0	0
$688$	53.0668	2.02315
$689$	−0.462977	−0.0176380
$690$	0	0
$691$	−27.3852	−1.04178	−0.520891	−	0.853623i	$-0.674400\pi$
$691$	−27.3852	−1.04178	−0.520891	+	0.853623i	$0.674400\pi$
$692$	0.119807	0.00455437
$693$	0	0
$694$	34.3969	1.30569
$695$	35.8176	1.35864
$696$	0	0
$697$	−19.1959	−0.727098
$698$	−30.6020	−1.15830
$699$	0	0
$700$	2.11598	0.0799765
$701$	40.8822	1.54410	0.772049	−	0.635563i	$-0.219232\pi$
$701$	40.8822	1.54410	0.772049	+	0.635563i	$0.219232\pi$
$702$	0	0
$703$	−3.06639	−0.115651
$704$	−7.99763	−0.301422
$705$	0	0
$706$	46.0537	1.73325
$707$	−0.00845326	−0.000317917 0
$708$	0	0
$709$	38.2012	1.43468	0.717338	−	0.696725i	$-0.245360\pi$
$709$	38.2012	1.43468	0.717338	+	0.696725i	$0.245360\pi$
$710$	−30.3469	−1.13890
$711$	0	0
$712$	16.4068	0.614870
$713$	78.6465	2.94533
$714$	0	0
$715$	0.975946	0.0364983
$716$	−3.62323	−0.135406
$717$	0	0
$718$	51.0585	1.90549
$719$	40.6443	1.51577	0.757887	−	0.652385i	$-0.226232\pi$
$719$	40.6443	1.51577	0.757887	+	0.652385i	$0.226232\pi$
$720$	0	0
$721$	2.75495	0.102600
$722$	18.2245	0.678246
$723$	0	0
$724$	2.03080	0.0754742
$725$	−75.4545	−2.80231
$726$	0	0
$727$	−8.92739	−0.331099	−0.165549	−	0.986202i	$-0.552940\pi$
$727$	−8.92739	−0.331099	−0.165549	+	0.986202i	$0.552940\pi$
$728$	−0.107215	−0.00397365
$729$	0	0
$730$	46.4608	1.71959
$731$	−21.3221	−0.788625
$732$	0	0
$733$	−27.2514	−1.00655	−0.503277	−	0.864125i	$-0.667873\pi$
$733$	−27.2514	−1.00655	−0.503277	+	0.864125i	$0.667873\pi$
$734$	7.75730	0.286327
$735$	0	0
$736$	25.8305	0.952125
$737$	−20.9107	−0.770256
$738$	0	0
$739$	−25.9682	−0.955254	−0.477627	−	0.878563i	$-0.658503\pi$
$739$	−25.9682	−0.955254	−0.477627	+	0.878563i	$0.658503\pi$
$740$	−2.48271	−0.0912661
$741$	0	0
$742$	−1.66108	−0.0609803
$743$	37.1301	1.36217	0.681086	−	0.732203i	$-0.261508\pi$
$743$	37.1301	1.36217	0.681086	+	0.732203i	$0.261508\pi$
$744$	0	0
$745$	−4.15709	−0.152304
$746$	22.9015	0.838483
$747$	0	0
$748$	−1.69125	−0.0618382
$749$	3.17151	0.115884
$750$	0	0
$751$	41.7793	1.52455	0.762274	−	0.647254i	$-0.224083\pi$
$751$	41.7793	1.52455	0.762274	+	0.647254i	$0.224083\pi$
$752$	−16.3568	−0.596473
$753$	0	0
$754$	−1.39730	−0.0508868
$755$	75.2311	2.73794
$756$	0	0
$757$	44.8211	1.62905	0.814525	−	0.580129i	$-0.196998\pi$
$757$	44.8211	1.62905	0.814525	+	0.580129i	$0.196998\pi$
$758$	1.49050	0.0541375
$759$	0	0
$760$	26.6471	0.966591
$761$	34.5170	1.25124	0.625621	−	0.780127i	$-0.284846\pi$
$761$	34.5170	1.25124	0.625621	+	0.780127i	$0.284846\pi$
$762$	0	0
$763$	−0.522713	−0.0189235
$764$	0.963242	0.0348489
$765$	0	0
$766$	21.9804	0.794183
$767$	0.244761	0.00883781
$768$	0	0
$769$	−44.4538	−1.60304	−0.801522	−	0.597965i	$-0.795976\pi$
$769$	−44.4538	−1.60304	−0.801522	+	0.597965i	$0.795976\pi$
$770$	3.50153	0.126186
$771$	0	0
$772$	−5.10832	−0.183852
$773$	25.9955	0.934994	0.467497	−	0.883995i	$-0.345156\pi$
$773$	25.9955	0.934994	0.467497	+	0.883995i	$0.345156\pi$
$774$	0	0
$775$	110.429	3.96672
$776$	−10.2016	−0.366216
$777$	0	0
$778$	−40.3154	−1.44538
$779$	−27.4477	−0.983414
$780$	0	0
$781$	−7.53539	−0.269637
$782$	−26.7710	−0.957329
$783$	0	0
$784$	32.9929	1.17832
$785$	41.2274	1.47147
$786$	0	0
$787$	−5.13042	−0.182880	−0.0914399	−	0.995811i	$-0.529147\pi$
$787$	−5.13042	−0.182880	−0.0914399	+	0.995811i	$0.529147\pi$
$788$	4.56834	0.162740
$789$	0	0
$790$	44.9692	1.59993
$791$	2.98848	0.106258
$792$	0	0
$793$	−1.21570	−0.0431709
$794$	−4.45310	−0.158034
$795$	0	0
$796$	−10.1128	−0.358439
$797$	29.3048	1.03803	0.519014	−	0.854766i	$-0.326299\pi$
$797$	29.3048	1.03803	0.519014	+	0.854766i	$0.326299\pi$
$798$	0	0
$799$	6.57212	0.232505
$800$	36.2690	1.28230
$801$	0	0
$802$	−11.1579	−0.394000
$803$	11.5366	0.407118
$804$	0	0
$805$	11.7028	0.412471
$806$	2.04497	0.0720312
$807$	0	0
$808$	−0.0612533	−0.00215488
$809$	26.3107	0.925034	0.462517	−	0.886610i	$-0.346946\pi$
$809$	26.3107	0.925034	0.462517	+	0.886610i	$0.346946\pi$
$810$	0	0
$811$	20.2846	0.712288	0.356144	−	0.934431i	$-0.384091\pi$
$811$	20.2846	0.712288	0.356144	+	0.934431i	$0.384091\pi$
$812$	−1.05852	−0.0371467
$813$	0	0
$814$	−2.91972	−0.102336
$815$	−19.0261	−0.666456
$816$	0	0
$817$	−30.4877	−1.06663
$818$	48.5191	1.69643
$819$	0	0
$820$	−22.2230	−0.776062
$821$	50.3158	1.75603	0.878017	−	0.478630i	$-0.158867\pi$
$821$	50.3158	1.75603	0.878017	+	0.478630i	$0.158867\pi$
$822$	0	0
$823$	32.3327	1.12705	0.563523	−	0.826101i	$-0.309446\pi$
$823$	32.3327	1.12705	0.563523	+	0.826101i	$0.309446\pi$
$824$	19.9627	0.695433
$825$	0	0
$826$	0.878162	0.0305552
$827$	−27.5246	−0.957126	−0.478563	−	0.878053i	$-0.658842\pi$
$827$	−27.5246	−0.957126	−0.478563	+	0.878053i	$0.658842\pi$
$828$	0	0
$829$	−30.2929	−1.05212	−0.526059	−	0.850448i	$-0.676331\pi$
$829$	−30.2929	−1.05212	−0.526059	+	0.850448i	$0.676331\pi$
$830$	2.57581	0.0894077
$831$	0	0
$832$	−0.695029	−0.0240958
$833$	−13.2564	−0.459308
$834$	0	0
$835$	−6.33813	−0.219340
$836$	−2.41826	−0.0836373
$837$	0	0
$838$	19.3752	0.669304
$839$	−44.8239	−1.54749	−0.773746	−	0.633496i	$-0.781619\pi$
$839$	−44.8239	−1.54749	−0.773746	+	0.633496i	$0.781619\pi$
$840$	0	0
$841$	8.74612	0.301590
$842$	25.6689	0.884608
$843$	0	0
$844$	13.4752	0.463836
$845$	−53.9574	−1.85619
$846$	0	0
$847$	−2.67090	−0.0917734
$848$	−15.5066	−0.532500
$849$	0	0
$850$	−37.5896	−1.28931
$851$	−9.75830	−0.334510
$852$	0	0
$853$	−29.7202	−1.01760	−0.508799	−	0.860885i	$-0.669910\pi$
$853$	−29.7202	−1.01760	−0.508799	+	0.860885i	$0.669910\pi$
$854$	−4.36174	−0.149256
$855$	0	0
$856$	22.9811	0.785479
$857$	−15.3029	−0.522737	−0.261369	−	0.965239i	$-0.584174\pi$
$857$	−15.3029	−0.522737	−0.261369	+	0.965239i	$0.584174\pi$
$858$	0	0
$859$	−51.5003	−1.75717	−0.878584	−	0.477589i	$-0.841511\pi$
$859$	−51.5003	−1.75717	−0.878584	+	0.477589i	$0.841511\pi$
$860$	−24.6844	−0.841732
$861$	0	0
$862$	1.13985	0.0388236
$863$	−33.0337	−1.12448	−0.562239	−	0.826975i	$-0.690060\pi$
$863$	−33.0337	−1.12448	−0.562239	+	0.826975i	$0.690060\pi$
$864$	0	0
$865$	0.930387	0.0316341
$866$	−39.3157	−1.33600
$867$	0	0
$868$	1.54916	0.0525819
$869$	11.1662	0.378789
$870$	0	0
$871$	−1.81723	−0.0615746
$872$	−3.78764	−0.128266
$873$	0	0
$874$	−38.2790	−1.29481
$875$	9.74232	0.329350
$876$	0	0
$877$	34.3964	1.16148	0.580742	−	0.814088i	$-0.302763\pi$
$877$	34.3964	1.16148	0.580742	+	0.814088i	$0.302763\pi$
$878$	−23.6452	−0.797988
$879$	0	0
$880$	32.6877	1.10190
$881$	−39.3672	−1.32631	−0.663157	−	0.748481i	$-0.730784\pi$
$881$	−39.3672	−1.32631	−0.663157	+	0.748481i	$0.730784\pi$
$882$	0	0
$883$	10.5579	0.355300	0.177650	−	0.984094i	$-0.443150\pi$
$883$	10.5579	0.355300	0.177650	+	0.984094i	$0.443150\pi$
$884$	−0.146977	−0.00494337
$885$	0	0
$886$	33.3630	1.12085
$887$	39.6149	1.33014	0.665070	−	0.746781i	$-0.268402\pi$
$887$	39.6149	1.33014	0.665070	+	0.746781i	$0.268402\pi$
$888$	0	0
$889$	−0.400356	−0.0134275
$890$	−46.5659	−1.56089
$891$	0	0
$892$	4.72615	0.158243
$893$	9.39726	0.314467
$894$	0	0
$895$	−28.1370	−0.940517
$896$	−4.39461	−0.146814
$897$	0	0
$898$	15.2762	0.509772
$899$	−55.2420	−1.84242
$900$	0	0
$901$	6.23051	0.207568
$902$	−26.1348	−0.870194
$903$	0	0
$904$	21.6549	0.720230
$905$	15.7707	0.524235
$906$	0	0
$907$	0.498983	0.0165685	0.00828423	−	0.999966i	$-0.497363\pi$
$907$	0.498983	0.0165685	0.00828423	+	0.999966i	$0.497363\pi$
$908$	0.850837	0.0282360
$909$	0	0
$910$	0.304299	0.0100874
$911$	43.8735	1.45359	0.726797	−	0.686852i	$-0.241008\pi$
$911$	43.8735	1.45359	0.726797	+	0.686852i	$0.241008\pi$
$912$	0	0
$913$	0.639596	0.0211675
$914$	23.4772	0.776558
$915$	0	0
$916$	7.94667	0.262565
$917$	1.30764	0.0431821
$918$	0	0
$919$	18.6796	0.616183	0.308092	−	0.951357i	$-0.400310\pi$
$919$	18.6796	0.616183	0.308092	+	0.951357i	$0.400310\pi$
$920$	84.8002	2.79578
$921$	0	0
$922$	11.2244	0.369656
$923$	−0.654858	−0.0215549
$924$	0	0
$925$	−13.7018	−0.450512
$926$	−2.34848	−0.0771758
$927$	0	0
$928$	−18.1436	−0.595592
$929$	−25.6598	−0.841871	−0.420936	−	0.907091i	$-0.638298\pi$
$929$	−25.6598	−0.841871	−0.420936	+	0.907091i	$0.638298\pi$
$930$	0	0
$931$	−18.9549	−0.621223
$932$	−8.20010	−0.268603
$933$	0	0
$934$	−3.88067	−0.126979
$935$	−13.1338	−0.429521
$936$	0	0
$937$	−15.3080	−0.500089	−0.250045	−	0.968234i	$-0.580445\pi$
$937$	−15.3080	−0.500089	−0.250045	+	0.968234i	$0.580445\pi$
$938$	−6.51993	−0.212883
$939$	0	0
$940$	7.60851	0.248162
$941$	35.9503	1.17195	0.585973	−	0.810331i	$-0.300713\pi$
$941$	35.9503	1.17195	0.585973	+	0.810331i	$0.300713\pi$
$942$	0	0
$943$	−87.3478	−2.84444
$944$	8.19786	0.266818
$945$	0	0
$946$	−29.0295	−0.943829
$947$	−54.4299	−1.76873	−0.884367	−	0.466792i	$-0.845410\pi$
$947$	−54.4299	−1.76873	−0.884367	+	0.466792i	$0.845410\pi$
$948$	0	0
$949$	1.00258	0.0325452
$950$	−53.7482	−1.74382
$951$	0	0
$952$	1.44284	0.0467628
$953$	−23.3944	−0.757819	−0.378909	−	0.925434i	$-0.623701\pi$
$953$	−23.3944	−0.757819	−0.378909	+	0.925434i	$0.623701\pi$
$954$	0	0
$955$	7.48028	0.242056
$956$	9.86786	0.319149
$957$	0	0
$958$	48.1608	1.55601
$959$	−5.10320	−0.164791
$960$	0	0
$961$	49.8475	1.60798
$962$	−0.253736	−0.00818079
$963$	0	0
$964$	6.52215	0.210064
$965$	−39.6698	−1.27702
$966$	0	0
$967$	15.1383	0.486814	0.243407	−	0.969924i	$-0.421735\pi$
$967$	15.1383	0.486814	0.243407	+	0.969924i	$0.421735\pi$
$968$	−19.3537	−0.622051
$969$	0	0
$970$	28.9543	0.929666
$971$	−38.6152	−1.23922	−0.619611	−	0.784909i	$-0.712710\pi$
$971$	−38.6152	−1.23922	−0.619611	+	0.784909i	$0.712710\pi$
$972$	0	0
$973$	2.77308	0.0889008
$974$	41.4353	1.32767
$975$	0	0
$976$	−40.7179	−1.30335
$977$	13.1025	0.419187	0.209593	−	0.977789i	$-0.432786\pi$
$977$	13.1025	0.419187	0.209593	+	0.977789i	$0.432786\pi$
$978$	0	0
$979$	−11.5627	−0.369546
$980$	−15.3469	−0.490238
$981$	0	0
$982$	26.7332	0.853090
$983$	−25.9247	−0.826871	−0.413435	−	0.910533i	$-0.635671\pi$
$983$	−25.9247	−0.826871	−0.413435	+	0.910533i	$0.635671\pi$
$984$	0	0
$985$	35.4765	1.13038
$986$	18.8042	0.598848
$987$	0	0
$988$	−0.210158	−0.00668600
$989$	−97.0224	−3.08513
$990$	0	0
$991$	−28.5641	−0.907368	−0.453684	−	0.891163i	$-0.649890\pi$
$991$	−28.5641	−0.907368	−0.453684	+	0.891163i	$0.649890\pi$
$992$	26.5534	0.843072
$993$	0	0
$994$	−2.34952	−0.0745223
$995$	−78.5334	−2.48968
$996$	0	0
$997$	−51.0347	−1.61628	−0.808142	−	0.588988i	$-0.799527\pi$
$997$	−51.0347	−1.61628	−0.808142	+	0.588988i	$0.799527\pi$
$998$	−19.2719	−0.610041
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.e.1.3		9
3.2	odd	2		149.2.a.b.1.7	✓	9
12.11	even	2		2384.2.a.j.1.2		9
15.14	odd	2		3725.2.a.c.1.3		9
21.20	even	2		7301.2.a.j.1.7		9
24.5	odd	2		9536.2.a.v.1.2		9
24.11	even	2		9536.2.a.w.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.a.b.1.7	✓	9	3.2	odd	2
1341.2.a.e.1.3		9	1.1	even	1	trivial
2384.2.a.j.1.2		9	12.11	even	2
3725.2.a.c.1.3		9	15.14	odd	2
7301.2.a.j.1.7		9	21.20	even	2
9536.2.a.v.1.2		9	24.5	odd	2
9536.2.a.w.1.8		9	24.11	even	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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q-1.59227 q^{2} +0.535312 q^{4} +4.15709 q^{5} +0.321851 q^{7} +2.33217 q^{8} +O(q^{10})\)	\(q-1.59227 q^{2} +0.535312 q^{4} +4.15709 q^{5} +0.321851 q^{7} +2.33217 q^{8} -6.61920 q^{10} -1.64360 q^{11} -0.142836 q^{13} -0.512473 q^{14} -4.78407 q^{16} +1.92222 q^{17} +2.74852 q^{19} +2.22534 q^{20} +2.61706 q^{22} +8.74674 q^{23} +12.2814 q^{25} +0.227434 q^{26} +0.172291 q^{28} -6.14379 q^{29} +8.99153 q^{31} +2.95316 q^{32} -3.06069 q^{34} +1.33797 q^{35} -1.11565 q^{37} -4.37638 q^{38} +9.69506 q^{40} -9.98634 q^{41} -11.0924 q^{43} -0.879841 q^{44} -13.9271 q^{46} +3.41903 q^{47} -6.89641 q^{49} -19.5553 q^{50} -0.0764621 q^{52} +3.24131 q^{53} -6.83262 q^{55} +0.750613 q^{56} +9.78255 q^{58} -1.71358 q^{59} +8.51115 q^{61} -14.3169 q^{62} +4.86591 q^{64} -0.593784 q^{65} +12.7225 q^{67} +1.02899 q^{68} -2.13040 q^{70} +4.58467 q^{71} -7.01909 q^{73} +1.77641 q^{74} +1.47132 q^{76} -0.528996 q^{77} -6.79375 q^{79} -19.8878 q^{80} +15.9009 q^{82} -0.389142 q^{83} +7.99085 q^{85} +17.6621 q^{86} -3.83317 q^{88} +7.03497 q^{89} -0.0459721 q^{91} +4.68223 q^{92} -5.44400 q^{94} +11.4259 q^{95} -4.37428 q^{97} +10.9809 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * q^98

Introduction

L-functions

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Database

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