LMFDB - Embedded newform 3087.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(1,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, 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("3087.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.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:	$24.6498191040$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1029)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	3087.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24698 q^{2} -0.445042 q^{4} +2.00000 q^{5} +3.04892 q^{8} +O(q^{10})$	$q-1.24698 q^{2} -0.445042 q^{4} +2.00000 q^{5} +3.04892 q^{8} -2.49396 q^{10} +1.24698 q^{11} -1.10992 q^{13} -2.91185 q^{16} +6.09783 q^{17} -6.00000 q^{19} -0.890084 q^{20} -1.55496 q^{22} -6.51573 q^{23} -1.00000 q^{25} +1.38404 q^{26} -0.198062 q^{29} -6.27413 q^{31} -2.46681 q^{32} -7.60388 q^{34} -7.46681 q^{37} +7.48188 q^{38} +6.09783 q^{40} -8.59179 q^{41} +9.25667 q^{43} -0.554958 q^{44} +8.12498 q^{46} +3.28621 q^{47} +1.24698 q^{50} +0.493959 q^{52} +2.26875 q^{53} +2.49396 q^{55} +0.246980 q^{58} +1.01208 q^{59} -10.7681 q^{61} +7.82371 q^{62} +8.89977 q^{64} -2.21983 q^{65} -6.69202 q^{67} -2.71379 q^{68} -0.127375 q^{71} -7.82371 q^{73} +9.31096 q^{74} +2.67025 q^{76} +6.93900 q^{79} -5.82371 q^{80} +10.7138 q^{82} +11.3840 q^{83} +12.1957 q^{85} -11.5429 q^{86} +3.80194 q^{88} -14.8659 q^{89} +2.89977 q^{92} -4.09783 q^{94} -12.0000 q^{95} +7.03146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 6 q^{5}+O(q^{10}) $ 3 * q + q^2 - q^4 + 6 * q^5	$ 3 q + q^{2} - q^{4} + 6 q^{5} + 2 q^{10} - q^{11} - 4 q^{13} - 5 q^{16} - 18 q^{19} - 2 q^{20} - 5 q^{22} - 7 q^{23} - 3 q^{25} - 6 q^{26} - 5 q^{29} - 8 q^{31} - 4 q^{32} - 14 q^{34} - 19 q^{37} - 6 q^{38} + 2 q^{41} + q^{43} - 2 q^{44} + 18 q^{47} - q^{50} - 8 q^{52} - q^{53} - 2 q^{55} - 4 q^{58} + 22 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} - 8 q^{65} - 15 q^{67} - 17 q^{71} - 16 q^{73} - 11 q^{74} + 6 q^{76} + 11 q^{79} - 10 q^{80} + 24 q^{82} + 24 q^{83} - 16 q^{86} + 7 q^{88} - 6 q^{89} - 14 q^{92} + 6 q^{94} - 36 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q + q^2 - q^4 + 6 * q^5 + 2 * q^10 - q^11 - 4 * q^13 - 5 * q^16 - 18 * q^19 - 2 * q^20 - 5 * q^22 - 7 * q^23 - 3 * q^25 - 6 * q^26 - 5 * q^29 - 8 * q^31 - 4 * q^32 - 14 * q^34 - 19 * q^37 - 6 * q^38 + 2 * q^41 + q^43 - 2 * q^44 + 18 * q^47 - q^50 - 8 * q^52 - q^53 - 2 * q^55 - 4 * q^58 + 22 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 - 8 * q^65 - 15 * q^67 - 17 * q^71 - 16 * q^73 - 11 * q^74 + 6 * q^76 + 11 * q^79 - 10 * q^80 + 24 * q^82 + 24 * q^83 - 16 * q^86 + 7 * q^88 - 6 * q^89 - 14 * q^92 + 6 * q^94 - 36 * q^95 - 4 * q^97

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 $

$2$	−1.24698	−0.881748	−0.440874	−	0.897569i	$-0.645331\pi$
$2$	−1.24698	−0.881748	−0.440874	+	0.897569i	$0.645331\pi$
$3$	0	0
$3$	0	0
$4$	−0.445042	−0.222521
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.04892	1.07796
$9$	0	0
$10$	−2.49396	−0.788659
$11$	1.24698	0.375978	0.187989	−	0.982171i	$-0.439803\pi$
$11$	1.24698	0.375978	0.187989	+	0.982171i	$0.439803\pi$
$12$	0	0
$13$	−1.10992	−0.307835	−0.153918	−	0.988084i	$-0.549189\pi$
$13$	−1.10992	−0.307835	−0.153918	+	0.988084i	$0.549189\pi$
$14$	0	0
$15$	0	0
$16$	−2.91185	−0.727963
$17$	6.09783	1.47894	0.739471	−	0.673188i	$-0.235076\pi$
$17$	6.09783	1.47894	0.739471	+	0.673188i	$0.235076\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−0.890084	−0.199029
$21$	0	0
$22$	−1.55496	−0.331518
$23$	−6.51573	−1.35862	−0.679312	−	0.733850i	$-0.737722\pi$
$23$	−6.51573	−1.35862	−0.679312	+	0.733850i	$0.737722\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.38404	0.271433
$27$	0	0
$28$	0	0
$29$	−0.198062	−0.0367792	−0.0183896	−	0.999831i	$-0.505854\pi$
$29$	−0.198062	−0.0367792	−0.0183896	+	0.999831i	$0.505854\pi$
$30$	0	0
$31$	−6.27413	−1.12687	−0.563433	−	0.826162i	$-0.690520\pi$
$31$	−6.27413	−1.12687	−0.563433	+	0.826162i	$0.690520\pi$
$32$	−2.46681	−0.436075
$33$	0	0
$34$	−7.60388	−1.30405
$35$	0	0
$36$	0	0
$37$	−7.46681	−1.22754	−0.613768	−	0.789486i	$-0.710347\pi$
$37$	−7.46681	−1.22754	−0.613768	+	0.789486i	$0.710347\pi$
$38$	7.48188	1.21372
$39$	0	0
$40$	6.09783	0.964152
$41$	−8.59179	−1.34181	−0.670906	−	0.741542i	$-0.734095\pi$
$41$	−8.59179	−1.34181	−0.670906	+	0.741542i	$0.734095\pi$
$42$	0	0
$43$	9.25667	1.41163	0.705814	−	0.708397i	$-0.250581\pi$
$43$	9.25667	1.41163	0.705814	+	0.708397i	$0.250581\pi$
$44$	−0.554958	−0.0836631
$45$	0	0
$46$	8.12498	1.19796
$47$	3.28621	0.479343	0.239671	−	0.970854i	$-0.422960\pi$
$47$	3.28621	0.479343	0.239671	+	0.970854i	$0.422960\pi$
$48$	0	0
$49$	0	0
$50$	1.24698	0.176350
$51$	0	0
$52$	0.493959	0.0684998
$53$	2.26875	0.311637	0.155818	−	0.987786i	$-0.450199\pi$
$53$	2.26875	0.311637	0.155818	+	0.987786i	$0.450199\pi$
$54$	0	0
$55$	2.49396	0.336285
$56$	0	0
$57$	0	0
$58$	0.246980	0.0324300
$59$	1.01208	0.131762	0.0658809	−	0.997827i	$-0.479014\pi$
$59$	1.01208	0.131762	0.0658809	+	0.997827i	$0.479014\pi$
$60$	0	0
$61$	−10.7681	−1.37871	−0.689356	−	0.724423i	$-0.742106\pi$
$61$	−10.7681	−1.37871	−0.689356	+	0.724423i	$0.742106\pi$
$62$	7.82371	0.993612
$63$	0	0
$64$	8.89977	1.11247
$65$	−2.21983	−0.275336
$66$	0	0
$67$	−6.69202	−0.817561	−0.408780	−	0.912633i	$-0.634046\pi$
$67$	−6.69202	−0.817561	−0.408780	+	0.912633i	$0.634046\pi$
$68$	−2.71379	−0.329096
$69$	0	0
$70$	0	0
$71$	−0.127375	−0.0151166	−0.00755830	−	0.999971i	$-0.502406\pi$
$71$	−0.127375	−0.0151166	−0.00755830	+	0.999971i	$0.502406\pi$
$72$	0	0
$73$	−7.82371	−0.915696	−0.457848	−	0.889031i	$-0.651380\pi$
$73$	−7.82371	−0.915696	−0.457848	+	0.889031i	$0.651380\pi$
$74$	9.31096	1.08238
$75$	0	0
$76$	2.67025	0.306299
$77$	0	0
$78$	0	0
$79$	6.93900	0.780699	0.390349	−	0.920667i	$-0.372354\pi$
$79$	6.93900	0.780699	0.390349	+	0.920667i	$0.372354\pi$
$80$	−5.82371	−0.651110
$81$	0	0
$82$	10.7138	1.18314
$83$	11.3840	1.24956	0.624781	−	0.780800i	$-0.285188\pi$
$83$	11.3840	1.24956	0.624781	+	0.780800i	$0.285188\pi$
$84$	0	0
$85$	12.1957	1.32281
$86$	−11.5429	−1.24470
$87$	0	0
$88$	3.80194	0.405288
$89$	−14.8659	−1.57578	−0.787892	−	0.615813i	$-0.788828\pi$
$89$	−14.8659	−1.57578	−0.787892	+	0.615813i	$0.788828\pi$
$90$	0	0
$91$	0	0
$92$	2.89977	0.302322
$93$	0	0
$94$	−4.09783	−0.422659
$95$	−12.0000	−1.23117
$96$	0	0
$97$	7.03146	0.713936	0.356968	−	0.934117i	$-0.383810\pi$
$97$	7.03146	0.713936	0.356968	+	0.934117i	$0.383810\pi$
$98$	0	0
$99$	0	0
$100$	0.445042	0.0445042
$101$	−7.26205	−0.722600	−0.361300	−	0.932450i	$-0.617667\pi$
$101$	−7.26205	−0.722600	−0.361300	+	0.932450i	$0.617667\pi$
$102$	0	0
$103$	12.8659	1.26772	0.633858	−	0.773449i	$-0.281470\pi$
$103$	12.8659	1.26772	0.633858	+	0.773449i	$0.281470\pi$
$104$	−3.38404	−0.331833
$105$	0	0
$106$	−2.82908	−0.274785
$107$	−15.8998	−1.53709	−0.768545	−	0.639796i	$-0.779019\pi$
$107$	−15.8998	−1.53709	−0.768545	+	0.639796i	$0.779019\pi$
$108$	0	0
$109$	−10.7627	−1.03088	−0.515440	−	0.856926i	$-0.672372\pi$
$109$	−10.7627	−1.03088	−0.515440	+	0.856926i	$0.672372\pi$
$110$	−3.10992	−0.296519
$111$	0	0
$112$	0	0
$113$	5.29052	0.497690	0.248845	−	0.968543i	$-0.419949\pi$
$113$	5.29052	0.497690	0.248845	+	0.968543i	$0.419949\pi$
$114$	0	0
$115$	−13.0315	−1.21519
$116$	0.0881460	0.00818415
$117$	0	0
$118$	−1.26205	−0.116181
$119$	0	0
$120$	0	0
$121$	−9.44504	−0.858640
$122$	13.4276	1.21568
$123$	0	0
$124$	2.79225	0.250751
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−7.61894	−0.676072	−0.338036	−	0.941133i	$-0.609763\pi$
$127$	−7.61894	−0.676072	−0.338036	+	0.941133i	$0.609763\pi$
$128$	−6.16421	−0.544844
$129$	0	0
$130$	2.76809	0.242777
$131$	19.1293	1.67133	0.835667	−	0.549236i	$-0.185081\pi$
$131$	19.1293	1.67133	0.835667	+	0.549236i	$0.185081\pi$
$132$	0	0
$133$	0	0
$134$	8.34481	0.720882
$135$	0	0
$136$	18.5918	1.59423
$137$	4.35019	0.371662	0.185831	−	0.982582i	$-0.440502\pi$
$137$	4.35019	0.371662	0.185831	+	0.982582i	$0.440502\pi$
$138$	0	0
$139$	18.0737	1.53299	0.766494	−	0.642251i	$-0.221999\pi$
$139$	18.0737	1.53299	0.766494	+	0.642251i	$0.221999\pi$
$140$	0	0
$141$	0	0
$142$	0.158834	0.0133290
$143$	−1.38404	−0.115739
$144$	0	0
$145$	−0.396125	−0.0328964
$146$	9.75600	0.807413
$147$	0	0
$148$	3.32304	0.273153
$149$	−7.34481	−0.601711	−0.300855	−	0.953670i	$-0.597272\pi$
$149$	−7.34481	−0.601711	−0.300855	+	0.953670i	$0.597272\pi$
$150$	0	0
$151$	11.7845	0.959007	0.479504	−	0.877540i	$-0.340817\pi$
$151$	11.7845	0.959007	0.479504	+	0.877540i	$0.340817\pi$
$152$	−18.2935	−1.48380
$153$	0	0
$154$	0	0
$155$	−12.5483	−1.00790
$156$	0	0
$157$	17.7560	1.41708	0.708542	−	0.705669i	$-0.249353\pi$
$157$	17.7560	1.41708	0.708542	+	0.705669i	$0.249353\pi$
$158$	−8.65279	−0.688379
$159$	0	0
$160$	−4.93362	−0.390037
$161$	0	0
$162$	0	0
$163$	−9.59850	−0.751812	−0.375906	−	0.926658i	$-0.622669\pi$
$163$	−9.59850	−0.751812	−0.375906	+	0.926658i	$0.622669\pi$
$164$	3.82371	0.298581
$165$	0	0
$166$	−14.1957	−1.10180
$167$	−1.37329	−0.106268	−0.0531342	−	0.998587i	$-0.516921\pi$
$167$	−1.37329	−0.106268	−0.0531342	+	0.998587i	$0.516921\pi$
$168$	0	0
$169$	−11.7681	−0.905237
$170$	−15.2078	−1.16638
$171$	0	0
$172$	−4.11960	−0.314117
$173$	2.68963	0.204489	0.102244	−	0.994759i	$-0.467398\pi$
$173$	2.68963	0.204489	0.102244	+	0.994759i	$0.467398\pi$
$174$	0	0
$175$	0	0
$176$	−3.63102	−0.273699
$177$	0	0
$178$	18.5375	1.38944
$179$	−0.762709	−0.0570076	−0.0285038	−	0.999594i	$-0.509074\pi$
$179$	−0.762709	−0.0570076	−0.0285038	+	0.999594i	$0.509074\pi$
$180$	0	0
$181$	−8.67025	−0.644455	−0.322227	−	0.946662i	$-0.604432\pi$
$181$	−8.67025	−0.644455	−0.322227	+	0.946662i	$0.604432\pi$
$182$	0	0
$183$	0	0
$184$	−19.8659	−1.46454
$185$	−14.9336	−1.09794
$186$	0	0
$187$	7.60388	0.556050
$188$	−1.46250	−0.106664
$189$	0	0
$190$	14.9638	1.08558
$191$	9.88231	0.715059	0.357530	−	0.933902i	$-0.383619\pi$
$191$	9.88231	0.715059	0.357530	+	0.933902i	$0.383619\pi$
$192$	0	0
$193$	−23.5821	−1.69748	−0.848739	−	0.528813i	$-0.822637\pi$
$193$	−23.5821	−1.69748	−0.848739	+	0.528813i	$0.822637\pi$
$194$	−8.76809	−0.629512
$195$	0	0
$196$	0	0
$197$	−3.99761	−0.284818	−0.142409	−	0.989808i	$-0.545485\pi$
$197$	−3.99761	−0.284818	−0.142409	+	0.989808i	$0.545485\pi$
$198$	0	0
$199$	23.7211	1.68154	0.840772	−	0.541390i	$-0.182102\pi$
$199$	23.7211	1.68154	0.840772	+	0.541390i	$0.182102\pi$
$200$	−3.04892	−0.215591
$201$	0	0
$202$	9.05562	0.637151
$203$	0	0
$204$	0	0
$205$	−17.1836	−1.20015
$206$	−16.0435	−1.11781
$207$	0	0
$208$	3.23191	0.224093
$209$	−7.48188	−0.517532
$210$	0	0
$211$	−27.0398	−1.86150	−0.930749	−	0.365659i	$-0.880844\pi$
$211$	−27.0398	−1.86150	−0.930749	+	0.365659i	$0.880844\pi$
$212$	−1.00969	−0.0693457
$213$	0	0
$214$	19.8267	1.35532
$215$	18.5133	1.26260
$216$	0	0
$217$	0	0
$218$	13.4209	0.908977
$219$	0	0
$220$	−1.10992	−0.0748305
$221$	−6.76809	−0.455271
$222$	0	0
$223$	−14.1172	−0.945358	−0.472679	−	0.881235i	$-0.656713\pi$
$223$	−14.1172	−0.945358	−0.472679	+	0.881235i	$0.656713\pi$
$224$	0	0
$225$	0	0
$226$	−6.59717	−0.438837
$227$	−16.7332	−1.11062	−0.555310	−	0.831644i	$-0.687400\pi$
$227$	−16.7332	−1.11062	−0.555310	+	0.831644i	$0.687400\pi$
$228$	0	0
$229$	−25.9409	−1.71422	−0.857112	−	0.515130i	$-0.827744\pi$
$229$	−25.9409	−1.71422	−0.857112	+	0.515130i	$0.827744\pi$
$230$	16.2500	1.07149
$231$	0	0
$232$	−0.603875	−0.0396464
$233$	−28.2446	−1.85036	−0.925182	−	0.379523i	$-0.876088\pi$
$233$	−28.2446	−1.85036	−0.925182	+	0.379523i	$0.876088\pi$
$234$	0	0
$235$	6.57242	0.428737
$236$	−0.450419	−0.0293198
$237$	0	0
$238$	0	0
$239$	8.49157	0.549274	0.274637	−	0.961548i	$-0.411442\pi$
$239$	8.49157	0.549274	0.274637	+	0.961548i	$0.411442\pi$
$240$	0	0
$241$	−13.0315	−0.839430	−0.419715	−	0.907656i	$-0.637870\pi$
$241$	−13.0315	−0.839430	−0.419715	+	0.907656i	$0.637870\pi$
$242$	11.7778	0.757104
$243$	0	0
$244$	4.79225	0.306792
$245$	0	0
$246$	0	0
$247$	6.65950	0.423734
$248$	−19.1293	−1.21471
$249$	0	0
$250$	14.9638	0.946391
$251$	15.9758	1.00839	0.504193	−	0.863591i	$-0.331790\pi$
$251$	15.9758	1.00839	0.504193	+	0.863591i	$0.331790\pi$
$252$	0	0
$253$	−8.12498	−0.510813
$254$	9.50066	0.596125
$255$	0	0
$256$	−10.1129	−0.632056
$257$	29.2379	1.82381	0.911904	−	0.410403i	$-0.134612\pi$
$257$	29.2379	1.82381	0.911904	+	0.410403i	$0.134612\pi$
$258$	0	0
$259$	0	0
$260$	0.987918	0.0612681
$261$	0	0
$262$	−23.8538	−1.47370
$263$	16.9245	1.04361	0.521806	−	0.853064i	$-0.325259\pi$
$263$	16.9245	1.04361	0.521806	+	0.853064i	$0.325259\pi$
$264$	0	0
$265$	4.53750	0.278736
$266$	0	0
$267$	0	0
$268$	2.97823	0.181924
$269$	8.37196	0.510447	0.255224	−	0.966882i	$-0.417851\pi$
$269$	8.37196	0.510447	0.255224	+	0.966882i	$0.417851\pi$
$270$	0	0
$271$	−18.3370	−1.11390	−0.556948	−	0.830547i	$-0.688028\pi$
$271$	−18.3370	−1.11390	−0.556948	+	0.830547i	$0.688028\pi$
$272$	−17.7560	−1.07662
$273$	0	0
$274$	−5.42460	−0.327712
$275$	−1.24698	−0.0751957
$276$	0	0
$277$	−3.19269	−0.191830	−0.0959149	−	0.995390i	$-0.530578\pi$
$277$	−3.19269	−0.191830	−0.0959149	+	0.995390i	$0.530578\pi$
$278$	−22.5375	−1.35171
$279$	0	0
$280$	0	0
$281$	−26.6310	−1.58867	−0.794337	−	0.607478i	$-0.792181\pi$
$281$	−26.6310	−1.58867	−0.794337	+	0.607478i	$0.792181\pi$
$282$	0	0
$283$	3.76941	0.224068	0.112034	−	0.993704i	$-0.464263\pi$
$283$	3.76941	0.224068	0.112034	+	0.993704i	$0.464263\pi$
$284$	0.0566871	0.00336376
$285$	0	0
$286$	1.72587	0.102053
$287$	0	0
$288$	0	0
$289$	20.1836	1.18727
$290$	0.493959	0.0290063
$291$	0	0
$292$	3.48188	0.203761
$293$	5.32975	0.311367	0.155684	−	0.987807i	$-0.450242\pi$
$293$	5.32975	0.311367	0.155684	+	0.987807i	$0.450242\pi$
$294$	0	0
$295$	2.02416	0.117851
$296$	−22.7657	−1.32323
$297$	0	0
$298$	9.15883	0.530557
$299$	7.23191	0.418232
$300$	0	0
$301$	0	0
$302$	−14.6950	−0.845603
$303$	0	0
$304$	17.4711	1.00204
$305$	−21.5362	−1.23316
$306$	0	0
$307$	−28.2935	−1.61480	−0.807398	−	0.590007i	$-0.799125\pi$
$307$	−28.2935	−1.61480	−0.807398	+	0.590007i	$0.799125\pi$
$308$	0	0
$309$	0	0
$310$	15.6474	0.888714
$311$	0.977165	0.0554099	0.0277050	−	0.999616i	$-0.491180\pi$
$311$	0.977165	0.0554099	0.0277050	+	0.999616i	$0.491180\pi$
$312$	0	0
$313$	−14.1715	−0.801021	−0.400510	−	0.916292i	$-0.631167\pi$
$313$	−14.1715	−0.801021	−0.400510	+	0.916292i	$0.631167\pi$
$314$	−22.1414	−1.24951
$315$	0	0
$316$	−3.08815	−0.173722
$317$	25.5948	1.43755	0.718773	−	0.695245i	$-0.244704\pi$
$317$	25.5948	1.43755	0.718773	+	0.695245i	$0.244704\pi$
$318$	0	0
$319$	−0.246980	−0.0138282
$320$	17.7995	0.995025
$321$	0	0
$322$	0	0
$323$	−36.5870	−2.03576
$324$	0	0
$325$	1.10992	0.0615671
$326$	11.9691	0.662909
$327$	0	0
$328$	−26.1957	−1.44641
$329$	0	0
$330$	0	0
$331$	−6.53511	−0.359202	−0.179601	−	0.983740i	$-0.557481\pi$
$331$	−6.53511	−0.359202	−0.179601	+	0.983740i	$0.557481\pi$
$332$	−5.06638	−0.278053
$333$	0	0
$334$	1.71246	0.0937018
$335$	−13.3840	−0.731248
$336$	0	0
$337$	−11.4668	−0.624637	−0.312319	−	0.949977i	$-0.601106\pi$
$337$	−11.4668	−0.624637	−0.312319	+	0.949977i	$0.601106\pi$
$338$	14.6746	0.798191
$339$	0	0
$340$	−5.42758	−0.294352
$341$	−7.82371	−0.423678
$342$	0	0
$343$	0	0
$344$	28.2228	1.52167
$345$	0	0
$346$	−3.35391	−0.180307
$347$	−5.34050	−0.286693	−0.143347	−	0.989673i	$-0.545786\pi$
$347$	−5.34050	−0.286693	−0.143347	+	0.989673i	$0.545786\pi$
$348$	0	0
$349$	17.9215	0.959318	0.479659	−	0.877455i	$-0.340760\pi$
$349$	17.9215	0.959318	0.479659	+	0.877455i	$0.340760\pi$
$350$	0	0
$351$	0	0
$352$	−3.07606	−0.163955
$353$	−3.28621	−0.174907	−0.0874536	−	0.996169i	$-0.527873\pi$
$353$	−3.28621	−0.174907	−0.0874536	+	0.996169i	$0.527873\pi$
$354$	0	0
$355$	−0.254749	−0.0135207
$356$	6.61596	0.350645
$357$	0	0
$358$	0.951083	0.0502663
$359$	−21.8189	−1.15156	−0.575779	−	0.817605i	$-0.695301\pi$
$359$	−21.8189	−1.15156	−0.575779	+	0.817605i	$0.695301\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	10.8116	0.568246
$363$	0	0
$364$	0	0
$365$	−15.6474	−0.819023
$366$	0	0
$367$	17.7995	0.929129	0.464564	−	0.885539i	$-0.346211\pi$
$367$	17.7995	0.929129	0.464564	+	0.885539i	$0.346211\pi$
$368$	18.9729	0.989028
$369$	0	0
$370$	18.6219	0.968108
$371$	0	0
$372$	0	0
$373$	17.4929	0.905748	0.452874	−	0.891575i	$-0.350399\pi$
$373$	17.4929	0.905748	0.452874	+	0.891575i	$0.350399\pi$
$374$	−9.48188	−0.490296
$375$	0	0
$376$	10.0194	0.516710
$377$	0.219833	0.0113220
$378$	0	0
$379$	−12.4397	−0.638983	−0.319491	−	0.947589i	$-0.603512\pi$
$379$	−12.4397	−0.638983	−0.319491	+	0.947589i	$0.603512\pi$
$380$	5.34050	0.273962
$381$	0	0
$382$	−12.3230	−0.630502
$383$	−30.2392	−1.54515	−0.772576	−	0.634923i	$-0.781032\pi$
$383$	−30.2392	−1.54515	−0.772576	+	0.634923i	$0.781032\pi$
$384$	0	0
$385$	0	0
$386$	29.4064	1.49675
$387$	0	0
$388$	−3.12929	−0.158866
$389$	−34.4644	−1.74742	−0.873708	−	0.486451i	$-0.838291\pi$
$389$	−34.4644	−1.74742	−0.873708	+	0.486451i	$0.838291\pi$
$390$	0	0
$391$	−39.7318	−2.00933
$392$	0	0
$393$	0	0
$394$	4.98493	0.251137
$395$	13.8780	0.698278
$396$	0	0
$397$	1.25129	0.0628005	0.0314003	−	0.999507i	$-0.490003\pi$
$397$	1.25129	0.0628005	0.0314003	+	0.999507i	$0.490003\pi$
$398$	−29.5797	−1.48270
$399$	0	0
$400$	2.91185	0.145593
$401$	18.1226	0.904999	0.452499	−	0.891765i	$-0.350532\pi$
$401$	18.1226	0.904999	0.452499	+	0.891765i	$0.350532\pi$
$402$	0	0
$403$	6.96376	0.346889
$404$	3.23191	0.160794
$405$	0	0
$406$	0	0
$407$	−9.31096	−0.461527
$408$	0	0
$409$	0.846543	0.0418589	0.0209294	−	0.999781i	$-0.493337\pi$
$409$	0.846543	0.0418589	0.0209294	+	0.999781i	$0.493337\pi$
$410$	21.4276	1.05823
$411$	0	0
$412$	−5.72587	−0.282094
$413$	0	0
$414$	0	0
$415$	22.7681	1.11764
$416$	2.73795	0.134239
$417$	0	0
$418$	9.32975	0.456333
$419$	12.8358	0.627069	0.313535	−	0.949577i	$-0.398487\pi$
$419$	12.8358	0.627069	0.313535	+	0.949577i	$0.398487\pi$
$420$	0	0
$421$	−20.2010	−0.984539	−0.492269	−	0.870443i	$-0.663833\pi$
$421$	−20.2010	−0.984539	−0.492269	+	0.870443i	$0.663833\pi$
$422$	33.7181	1.64137
$423$	0	0
$424$	6.91723	0.335930
$425$	−6.09783	−0.295788
$426$	0	0
$427$	0	0
$428$	7.07606	0.342034
$429$	0	0
$430$	−23.0858	−1.11329
$431$	−2.20105	−0.106021	−0.0530103	−	0.998594i	$-0.516882\pi$
$431$	−2.20105	−0.106021	−0.0530103	+	0.998594i	$0.516882\pi$
$432$	0	0
$433$	24.2064	1.16329	0.581643	−	0.813444i	$-0.302410\pi$
$433$	24.2064	1.16329	0.581643	+	0.813444i	$0.302410\pi$
$434$	0	0
$435$	0	0
$436$	4.78986	0.229393
$437$	39.0944	1.87014
$438$	0	0
$439$	21.5120	1.02671	0.513356	−	0.858176i	$-0.328402\pi$
$439$	21.5120	1.02671	0.513356	+	0.858176i	$0.328402\pi$
$440$	7.60388	0.362501
$441$	0	0
$442$	8.43967	0.401434
$443$	36.3043	1.72487	0.862434	−	0.506170i	$-0.168939\pi$
$443$	36.3043	1.72487	0.862434	+	0.506170i	$0.168939\pi$
$444$	0	0
$445$	−29.7318	−1.40942
$446$	17.6039	0.833568
$447$	0	0
$448$	0	0
$449$	−23.0019	−1.08553	−0.542764	−	0.839885i	$-0.682622\pi$
$449$	−23.0019	−1.08553	−0.542764	+	0.839885i	$0.682622\pi$
$450$	0	0
$451$	−10.7138	−0.504493
$452$	−2.35450	−0.110747
$453$	0	0
$454$	20.8659	0.979286
$455$	0	0
$456$	0	0
$457$	−19.8649	−0.929239	−0.464619	−	0.885510i	$-0.653809\pi$
$457$	−19.8649	−0.929239	−0.464619	+	0.885510i	$0.653809\pi$
$458$	32.3478	1.51151
$459$	0	0
$460$	5.79954	0.270405
$461$	14.2306	0.662784	0.331392	−	0.943493i	$-0.392482\pi$
$461$	14.2306	0.662784	0.331392	+	0.943493i	$0.392482\pi$
$462$	0	0
$463$	−0.302602	−0.0140631	−0.00703155	−	0.999975i	$-0.502238\pi$
$463$	−0.302602	−0.0140631	−0.00703155	+	0.999975i	$0.502238\pi$
$464$	0.576728	0.0267739
$465$	0	0
$466$	35.2204	1.63155
$467$	−16.3913	−0.758501	−0.379250	−	0.925294i	$-0.623818\pi$
$467$	−16.3913	−0.758501	−0.379250	+	0.925294i	$0.623818\pi$
$468$	0	0
$469$	0	0
$470$	−8.19567	−0.378038
$471$	0	0
$472$	3.08575	0.142033
$473$	11.5429	0.530742
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	−10.5888	−0.484321
$479$	13.8345	0.632113	0.316056	−	0.948740i	$-0.397641\pi$
$479$	13.8345	0.632113	0.316056	+	0.948740i	$0.397641\pi$
$480$	0	0
$481$	8.28754	0.377879
$482$	16.2500	0.740166
$483$	0	0
$484$	4.20344	0.191065
$485$	14.0629	0.638564
$486$	0	0
$487$	−38.1715	−1.72972	−0.864858	−	0.502017i	$-0.832592\pi$
$487$	−38.1715	−1.72972	−0.864858	+	0.502017i	$0.832592\pi$
$488$	−32.8310	−1.48619
$489$	0	0
$490$	0	0
$491$	−7.27605	−0.328363	−0.164182	−	0.986430i	$-0.552498\pi$
$491$	−7.27605	−0.328363	−0.164182	+	0.986430i	$0.552498\pi$
$492$	0	0
$493$	−1.20775	−0.0543944
$494$	−8.30426	−0.373626
$495$	0	0
$496$	18.2693	0.820318
$497$	0	0
$498$	0	0
$499$	−12.2155	−0.546842	−0.273421	−	0.961894i	$-0.588155\pi$
$499$	−12.2155	−0.546842	−0.273421	+	0.961894i	$0.588155\pi$
$500$	5.34050	0.238835
$501$	0	0
$502$	−19.9215	−0.889142
$503$	29.8189	1.32956	0.664780	−	0.747039i	$-0.268525\pi$
$503$	29.8189	1.32956	0.664780	+	0.747039i	$0.268525\pi$
$504$	0	0
$505$	−14.5241	−0.646314
$506$	10.1317	0.450408
$507$	0	0
$508$	3.39075	0.150440
$509$	−5.54958	−0.245981	−0.122990	−	0.992408i	$-0.539248\pi$
$509$	−5.54958	−0.245981	−0.122990	+	0.992408i	$0.539248\pi$
$510$	0	0
$511$	0	0
$512$	24.9390	1.10216
$513$	0	0
$514$	−36.4590	−1.60814
$515$	25.7318	1.13388
$516$	0	0
$517$	4.09783	0.180223
$518$	0	0
$519$	0	0
$520$	−6.76809	−0.296800
$521$	11.1099	0.486734	0.243367	−	0.969934i	$-0.421748\pi$
$521$	11.1099	0.486734	0.243367	+	0.969934i	$0.421748\pi$
$522$	0	0
$523$	31.3056	1.36890	0.684449	−	0.729061i	$-0.260043\pi$
$523$	31.3056	1.36890	0.684449	+	0.729061i	$0.260043\pi$
$524$	−8.51334	−0.371907
$525$	0	0
$526$	−21.1045	−0.920202
$527$	−38.2586	−1.66657
$528$	0	0
$529$	19.4547	0.845858
$530$	−5.65817	−0.245775
$531$	0	0
$532$	0	0
$533$	9.53617	0.413057
$534$	0	0
$535$	−31.7995	−1.37481
$536$	−20.4034	−0.881294
$537$	0	0
$538$	−10.4397	−0.450086
$539$	0	0
$540$	0	0
$541$	14.7463	0.633994	0.316997	−	0.948427i	$-0.397326\pi$
$541$	14.7463	0.633994	0.316997	+	0.948427i	$0.397326\pi$
$542$	22.8659	0.982175
$543$	0	0
$544$	−15.0422	−0.644930
$545$	−21.5254	−0.922048
$546$	0	0
$547$	−22.5042	−0.962212	−0.481106	−	0.876662i	$-0.659765\pi$
$547$	−22.5042	−0.962212	−0.481106	+	0.876662i	$0.659765\pi$
$548$	−1.93602	−0.0827026
$549$	0	0
$550$	1.55496	0.0663036
$551$	1.18837	0.0506264
$552$	0	0
$553$	0	0
$554$	3.98121	0.169146
$555$	0	0
$556$	−8.04354	−0.341122
$557$	3.94033	0.166957	0.0834785	−	0.996510i	$-0.473397\pi$
$557$	3.94033	0.166957	0.0834785	+	0.996510i	$0.473397\pi$
$558$	0	0
$559$	−10.2741	−0.434549
$560$	0	0
$561$	0	0
$562$	33.2083	1.40081
$563$	1.68233	0.0709019	0.0354509	−	0.999371i	$-0.488713\pi$
$563$	1.68233	0.0709019	0.0354509	+	0.999371i	$0.488713\pi$
$564$	0	0
$565$	10.5810	0.445148
$566$	−4.70038	−0.197572
$567$	0	0
$568$	−0.388355	−0.0162950
$569$	38.8528	1.62879	0.814397	−	0.580309i	$-0.197068\pi$
$569$	38.8528	1.62879	0.814397	+	0.580309i	$0.197068\pi$
$570$	0	0
$571$	−36.1226	−1.51168	−0.755842	−	0.654754i	$-0.772772\pi$
$571$	−36.1226	−1.51168	−0.755842	+	0.654754i	$0.772772\pi$
$572$	0.615957	0.0257545
$573$	0	0
$574$	0	0
$575$	6.51573	0.271725
$576$	0	0
$577$	−14.4590	−0.601938	−0.300969	−	0.953634i	$-0.597310\pi$
$577$	−14.4590	−0.601938	−0.300969	+	0.953634i	$0.597310\pi$
$578$	−25.1685	−1.04687
$579$	0	0
$580$	0.176292	0.00732013
$581$	0	0
$582$	0	0
$583$	2.82908	0.117169
$584$	−23.8538	−0.987079
$585$	0	0
$586$	−6.64609	−0.274547
$587$	29.2513	1.20733	0.603665	−	0.797238i	$-0.293706\pi$
$587$	29.2513	1.20733	0.603665	+	0.797238i	$0.293706\pi$
$588$	0	0
$589$	37.6448	1.55113
$590$	−2.52409	−0.103915
$591$	0	0
$592$	21.7423	0.893602
$593$	−12.0871	−0.496357	−0.248178	−	0.968714i	$-0.579832\pi$
$593$	−12.0871	−0.496357	−0.248178	+	0.968714i	$0.579832\pi$
$594$	0	0
$595$	0	0
$596$	3.26875	0.133893
$597$	0	0
$598$	−9.01805	−0.368775
$599$	42.8068	1.74904	0.874520	−	0.484989i	$-0.161177\pi$
$599$	42.8068	1.74904	0.874520	+	0.484989i	$0.161177\pi$
$600$	0	0
$601$	30.8310	1.25762	0.628811	−	0.777558i	$-0.283542\pi$
$601$	30.8310	1.25762	0.628811	+	0.777558i	$0.283542\pi$
$602$	0	0
$603$	0	0
$604$	−5.24459	−0.213399
$605$	−18.8901	−0.767991
$606$	0	0
$607$	−10.0301	−0.407110	−0.203555	−	0.979063i	$-0.565250\pi$
$607$	−10.0301	−0.407110	−0.203555	+	0.979063i	$0.565250\pi$
$608$	14.8009	0.600255
$609$	0	0
$610$	26.8552	1.08733
$611$	−3.64742	−0.147559
$612$	0	0
$613$	32.8159	1.32542	0.662712	−	0.748875i	$-0.269406\pi$
$613$	32.8159	1.32542	0.662712	+	0.748875i	$0.269406\pi$
$614$	35.2814	1.42384
$615$	0	0
$616$	0	0
$617$	15.5114	0.624466	0.312233	−	0.950006i	$-0.398923\pi$
$617$	15.5114	0.624466	0.312233	+	0.950006i	$0.398923\pi$
$618$	0	0
$619$	−12.8901	−0.518096	−0.259048	−	0.965864i	$-0.583409\pi$
$619$	−12.8901	−0.518096	−0.259048	+	0.965864i	$0.583409\pi$
$620$	5.58450	0.224279
$621$	0	0
$622$	−1.21850	−0.0488576
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	17.6716	0.706298
$627$	0	0
$628$	−7.90217	−0.315331
$629$	−45.5314	−1.81546
$630$	0	0
$631$	38.2040	1.52088	0.760439	−	0.649409i	$-0.224984\pi$
$631$	38.2040	1.52088	0.760439	+	0.649409i	$0.224984\pi$
$632$	21.1564	0.841558
$633$	0	0
$634$	−31.9162	−1.26755
$635$	−15.2379	−0.604697
$636$	0	0
$637$	0	0
$638$	0.307979	0.0121930
$639$	0	0
$640$	−12.3284	−0.487324
$641$	−41.4902	−1.63877	−0.819383	−	0.573247i	$-0.805684\pi$
$641$	−41.4902	−1.63877	−0.819383	+	0.573247i	$0.805684\pi$
$642$	0	0
$643$	−40.8504	−1.61098	−0.805491	−	0.592608i	$-0.798098\pi$
$643$	−40.8504	−1.61098	−0.805491	+	0.592608i	$0.798098\pi$
$644$	0	0
$645$	0	0
$646$	45.6233	1.79502
$647$	−31.4470	−1.23631	−0.618154	−	0.786057i	$-0.712119\pi$
$647$	−31.4470	−1.23631	−0.618154	+	0.786057i	$0.712119\pi$
$648$	0	0
$649$	1.26205	0.0495396
$650$	−1.38404	−0.0542866
$651$	0	0
$652$	4.27173	0.167294
$653$	11.7362	0.459271	0.229636	−	0.973277i	$-0.426247\pi$
$653$	11.7362	0.459271	0.229636	+	0.973277i	$0.426247\pi$
$654$	0	0
$655$	38.2586	1.49489
$656$	25.0180	0.976791
$657$	0	0
$658$	0	0
$659$	21.9239	0.854035	0.427018	−	0.904243i	$-0.359564\pi$
$659$	21.9239	0.854035	0.427018	+	0.904243i	$0.359564\pi$
$660$	0	0
$661$	34.3612	1.33650	0.668248	−	0.743939i	$-0.267044\pi$
$661$	34.3612	1.33650	0.668248	+	0.743939i	$0.267044\pi$
$662$	8.14914	0.316726
$663$	0	0
$664$	34.7090	1.34697
$665$	0	0
$666$	0	0
$667$	1.29052	0.0499691
$668$	0.611171	0.0236469
$669$	0	0
$670$	16.6896	0.644777
$671$	−13.4276	−0.518366
$672$	0	0
$673$	8.64071	0.333075	0.166537	−	0.986035i	$-0.446741\pi$
$673$	8.64071	0.333075	0.166537	+	0.986035i	$0.446741\pi$
$674$	14.2989	0.550772
$675$	0	0
$676$	5.23729	0.201434
$677$	15.5362	0.597104	0.298552	−	0.954393i	$-0.403496\pi$
$677$	15.5362	0.597104	0.298552	+	0.954393i	$0.403496\pi$
$678$	0	0
$679$	0	0
$680$	37.1836	1.42593
$681$	0	0
$682$	9.75600	0.373577
$683$	−17.6297	−0.674582	−0.337291	−	0.941401i	$-0.609511\pi$
$683$	−17.6297	−0.674582	−0.337291	+	0.941401i	$0.609511\pi$
$684$	0	0
$685$	8.70038	0.332425
$686$	0	0
$687$	0	0
$688$	−26.9541	−1.02761
$689$	−2.51812	−0.0959328
$690$	0	0
$691$	−24.1414	−0.918381	−0.459191	−	0.888338i	$-0.651861\pi$
$691$	−24.1414	−0.918381	−0.459191	+	0.888338i	$0.651861\pi$
$692$	−1.19700	−0.0455030
$693$	0	0
$694$	6.65950	0.252791
$695$	36.1473	1.37115
$696$	0	0
$697$	−52.3913	−1.98446
$698$	−22.3478	−0.845877
$699$	0	0
$700$	0	0
$701$	−39.8146	−1.50378	−0.751889	−	0.659290i	$-0.770857\pi$
$701$	−39.8146	−1.50378	−0.751889	+	0.659290i	$0.770857\pi$
$702$	0	0
$703$	44.8009	1.68970
$704$	11.0978	0.418265
$705$	0	0
$706$	4.09783	0.154224
$707$	0	0
$708$	0	0
$709$	32.5924	1.22403	0.612016	−	0.790845i	$-0.290359\pi$
$709$	32.5924	1.22403	0.612016	+	0.790845i	$0.290359\pi$
$710$	0.317667	0.0119218
$711$	0	0
$712$	−45.3250	−1.69862
$713$	40.8805	1.53099
$714$	0	0
$715$	−2.76809	−0.103521
$716$	0.339437	0.0126854
$717$	0	0
$718$	27.2078	1.01538
$719$	19.6125	0.731423	0.365711	−	0.930728i	$-0.380826\pi$
$719$	19.6125	0.731423	0.365711	+	0.930728i	$0.380826\pi$
$720$	0	0
$721$	0	0
$722$	−21.1987	−0.788932
$723$	0	0
$724$	3.85862	0.143405
$725$	0.198062	0.00735585
$726$	0	0
$727$	−6.53750	−0.242462	−0.121231	−	0.992624i	$-0.538684\pi$
$727$	−6.53750	−0.242462	−0.121231	+	0.992624i	$0.538684\pi$
$728$	0	0
$729$	0	0
$730$	19.5120	0.722172
$731$	56.4456	2.08772
$732$	0	0
$733$	43.2573	1.59774	0.798872	−	0.601502i	$-0.205431\pi$
$733$	43.2573	1.59774	0.798872	+	0.601502i	$0.205431\pi$
$734$	−22.1957	−0.819257
$735$	0	0
$736$	16.0731	0.592462
$737$	−8.34481	−0.307385
$738$	0	0
$739$	19.2838	0.709367	0.354683	−	0.934986i	$-0.384589\pi$
$739$	19.2838	0.709367	0.354683	+	0.934986i	$0.384589\pi$
$740$	6.64609	0.244315
$741$	0	0
$742$	0	0
$743$	−45.7706	−1.67916	−0.839580	−	0.543236i	$-0.817199\pi$
$743$	−45.7706	−1.67916	−0.839580	+	0.543236i	$0.817199\pi$
$744$	0	0
$745$	−14.6896	−0.538186
$746$	−21.8133	−0.798641
$747$	0	0
$748$	−3.38404	−0.123733
$749$	0	0
$750$	0	0
$751$	−5.96556	−0.217686	−0.108843	−	0.994059i	$-0.534715\pi$
$751$	−5.96556	−0.217686	−0.108843	+	0.994059i	$0.534715\pi$
$752$	−9.56896	−0.348944
$753$	0	0
$754$	−0.274127	−0.00998310
$755$	23.5690	0.857762
$756$	0	0
$757$	−4.80923	−0.174795	−0.0873973	−	0.996174i	$-0.527855\pi$
$757$	−4.80923	−0.174795	−0.0873973	+	0.996174i	$0.527855\pi$
$758$	15.5120	0.563422
$759$	0	0
$760$	−36.5870	−1.32715
$761$	−3.10992	−0.112734	−0.0563672	−	0.998410i	$-0.517952\pi$
$761$	−3.10992	−0.112734	−0.0563672	+	0.998410i	$0.517952\pi$
$762$	0	0
$763$	0	0
$764$	−4.39804	−0.159116
$765$	0	0
$766$	37.7077	1.36243
$767$	−1.12333	−0.0405609
$768$	0	0
$769$	35.6039	1.28391	0.641954	−	0.766743i	$-0.278124\pi$
$769$	35.6039	1.28391	0.641954	+	0.766743i	$0.278124\pi$
$770$	0	0
$771$	0	0
$772$	10.4950	0.377724
$773$	−4.74871	−0.170799	−0.0853996	−	0.996347i	$-0.527217\pi$
$773$	−4.74871	−0.170799	−0.0853996	+	0.996347i	$0.527217\pi$
$774$	0	0
$775$	6.27413	0.225373
$776$	21.4383	0.769591
$777$	0	0
$778$	42.9764	1.54078
$779$	51.5508	1.84700
$780$	0	0
$781$	−0.158834	−0.00568351
$782$	49.5448	1.77172
$783$	0	0
$784$	0	0
$785$	35.5120	1.26748
$786$	0	0
$787$	−31.3551	−1.11769	−0.558844	−	0.829273i	$-0.688755\pi$
$787$	−31.3551	−1.11769	−0.558844	+	0.829273i	$0.688755\pi$
$788$	1.77910	0.0633779
$789$	0	0
$790$	−17.3056	−0.615705
$791$	0	0
$792$	0	0
$793$	11.9517	0.424416
$794$	−1.56033	−0.0553742
$795$	0	0
$796$	−10.5569	−0.374179
$797$	−51.0858	−1.80955	−0.904775	−	0.425890i	$-0.859961\pi$
$797$	−51.0858	−1.80955	−0.904775	+	0.425890i	$0.859961\pi$
$798$	0	0
$799$	20.0388	0.708920
$800$	2.46681	0.0872150
$801$	0	0
$802$	−22.5985	−0.797981
$803$	−9.75600	−0.344282
$804$	0	0
$805$	0	0
$806$	−8.68366	−0.305869
$807$	0	0
$808$	−22.1414	−0.778931
$809$	28.7638	1.01128	0.505640	−	0.862744i	$-0.331256\pi$
$809$	28.7638	1.01128	0.505640	+	0.862744i	$0.331256\pi$
$810$	0	0
$811$	−19.0180	−0.667814	−0.333907	−	0.942606i	$-0.608367\pi$
$811$	−19.0180	−0.667814	−0.333907	+	0.942606i	$0.608367\pi$
$812$	0	0
$813$	0	0
$814$	11.6106	0.406951
$815$	−19.1970	−0.672441
$816$	0	0
$817$	−55.5400	−1.94310
$818$	−1.05562	−0.0369090
$819$	0	0
$820$	7.64742	0.267059
$821$	16.1575	0.563901	0.281950	−	0.959429i	$-0.409019\pi$
$821$	16.1575	0.563901	0.281950	+	0.959429i	$0.409019\pi$
$822$	0	0
$823$	17.3545	0.604940	0.302470	−	0.953159i	$-0.402189\pi$
$823$	17.3545	0.604940	0.302470	+	0.953159i	$0.402189\pi$
$824$	39.2271	1.36654
$825$	0	0
$826$	0	0
$827$	2.13275	0.0741630	0.0370815	−	0.999312i	$-0.488194\pi$
$827$	2.13275	0.0741630	0.0370815	+	0.999312i	$0.488194\pi$
$828$	0	0
$829$	6.23921	0.216697	0.108348	−	0.994113i	$-0.465444\pi$
$829$	6.23921	0.216697	0.108348	+	0.994113i	$0.465444\pi$
$830$	−28.3913	−0.985478
$831$	0	0
$832$	−9.87800	−0.342458
$833$	0	0
$834$	0	0
$835$	−2.74658	−0.0950493
$836$	3.32975	0.115162
$837$	0	0
$838$	−16.0060	−0.552917
$839$	14.7332	0.508645	0.254323	−	0.967119i	$-0.418147\pi$
$839$	14.7332	0.508645	0.254323	+	0.967119i	$0.418147\pi$
$840$	0	0
$841$	−28.9608	−0.998647
$842$	25.1903	0.868115
$843$	0	0
$844$	12.0339	0.414222
$845$	−23.5362	−0.809669
$846$	0	0
$847$	0	0
$848$	−6.60627	−0.226860
$849$	0	0
$850$	7.60388	0.260811
$851$	48.6517	1.66776
$852$	0	0
$853$	−5.26205	−0.180169	−0.0900845	−	0.995934i	$-0.528714\pi$
$853$	−5.26205	−0.180169	−0.0900845	+	0.995934i	$0.528714\pi$
$854$	0	0
$855$	0	0
$856$	−48.4771	−1.65691
$857$	−45.6775	−1.56032	−0.780158	−	0.625583i	$-0.784861\pi$
$857$	−45.6775	−1.56032	−0.780158	+	0.625583i	$0.784861\pi$
$858$	0	0
$859$	14.9396	0.509732	0.254866	−	0.966976i	$-0.417969\pi$
$859$	14.9396	0.509732	0.254866	+	0.966976i	$0.417969\pi$
$860$	−8.23921	−0.280955
$861$	0	0
$862$	2.74466	0.0934835
$863$	24.7627	0.842932	0.421466	−	0.906844i	$-0.361516\pi$
$863$	24.7627	0.842932	0.421466	+	0.906844i	$0.361516\pi$
$864$	0	0
$865$	5.37926	0.182900
$866$	−30.1849	−1.02573
$867$	0	0
$868$	0	0
$869$	8.65279	0.293526
$870$	0	0
$871$	7.42758	0.251674
$872$	−32.8146	−1.11124
$873$	0	0
$874$	−48.7499	−1.64899
$875$	0	0
$876$	0	0
$877$	44.2562	1.49443	0.747213	−	0.664585i	$-0.231392\pi$
$877$	44.2562	1.49443	0.747213	+	0.664585i	$0.231392\pi$
$878$	−26.8250	−0.905301
$879$	0	0
$880$	−7.26205	−0.244803
$881$	1.71725	0.0578556	0.0289278	−	0.999582i	$-0.490791\pi$
$881$	1.71725	0.0578556	0.0289278	+	0.999582i	$0.490791\pi$
$882$	0	0
$883$	7.78448	0.261969	0.130984	−	0.991384i	$-0.458186\pi$
$883$	7.78448	0.261969	0.130984	+	0.991384i	$0.458186\pi$
$884$	3.01208	0.101307
$885$	0	0
$886$	−45.2707	−1.52090
$887$	−12.2789	−0.412286	−0.206143	−	0.978522i	$-0.566091\pi$
$887$	−12.2789	−0.412286	−0.206143	+	0.978522i	$0.566091\pi$
$888$	0	0
$889$	0	0
$890$	37.0750	1.24276
$891$	0	0
$892$	6.28275	0.210362
$893$	−19.7172	−0.659813
$894$	0	0
$895$	−1.52542	−0.0509891
$896$	0	0
$897$	0	0
$898$	28.6829	0.957162
$899$	1.24267	0.0414453
$900$	0	0
$901$	13.8345	0.460893
$902$	13.3599	0.444835
$903$	0	0
$904$	16.1304	0.536488
$905$	−17.3405	−0.576418
$906$	0	0
$907$	−0.576728	−0.0191500	−0.00957498	−	0.999954i	$-0.503048\pi$
$907$	−0.576728	−0.0191500	−0.00957498	+	0.999954i	$0.503048\pi$
$908$	7.44696	0.247136
$909$	0	0
$910$	0	0
$911$	1.13839	0.0377166	0.0188583	−	0.999822i	$-0.493997\pi$
$911$	1.13839	0.0377166	0.0188583	+	0.999822i	$0.493997\pi$
$912$	0	0
$913$	14.1957	0.469808
$914$	24.7711	0.819354
$915$	0	0
$916$	11.5448	0.381451
$917$	0	0
$918$	0	0
$919$	12.9879	0.428432	0.214216	−	0.976786i	$-0.431280\pi$
$919$	12.9879	0.428432	0.214216	+	0.976786i	$0.431280\pi$
$920$	−39.7318	−1.30992
$921$	0	0
$922$	−17.7453	−0.584409
$923$	0.141375	0.00465342
$924$	0	0
$925$	7.46681	0.245507
$926$	0.377338	0.0124001
$927$	0	0
$928$	0.488582	0.0160385
$929$	−21.8103	−0.715573	−0.357786	−	0.933803i	$-0.616468\pi$
$929$	−21.8103	−0.715573	−0.357786	+	0.933803i	$0.616468\pi$
$930$	0	0
$931$	0	0
$932$	12.5700	0.411745
$933$	0	0
$934$	20.4397	0.668806
$935$	15.2078	0.497347
$936$	0	0
$937$	30.3478	0.991419	0.495710	−	0.868488i	$-0.334908\pi$
$937$	30.3478	0.991419	0.495710	+	0.868488i	$0.334908\pi$
$938$	0	0
$939$	0	0
$940$	−2.92500	−0.0954030
$941$	4.23921	0.138194	0.0690971	−	0.997610i	$-0.477988\pi$
$941$	4.23921	0.138194	0.0690971	+	0.997610i	$0.477988\pi$
$942$	0	0
$943$	55.9818	1.82302
$944$	−2.94703	−0.0959178
$945$	0	0
$946$	−14.3937	−0.467981
$947$	23.4330	0.761469	0.380734	−	0.924684i	$-0.375671\pi$
$947$	23.4330	0.761469	0.380734	+	0.924684i	$0.375671\pi$
$948$	0	0
$949$	8.68366	0.281884
$950$	−7.48188	−0.242744
$951$	0	0
$952$	0	0
$953$	26.8267	0.869002	0.434501	−	0.900671i	$-0.356925\pi$
$953$	26.8267	0.869002	0.434501	+	0.900671i	$0.356925\pi$
$954$	0	0
$955$	19.7646	0.639568
$956$	−3.77910	−0.122225
$957$	0	0
$958$	−17.2513	−0.557364
$959$	0	0
$960$	0	0
$961$	8.36467	0.269828
$962$	−10.3344	−0.333194
$963$	0	0
$964$	5.79954	0.186791
$965$	−47.1642	−1.51827
$966$	0	0
$967$	8.03252	0.258309	0.129154	−	0.991625i	$-0.458774\pi$
$967$	8.03252	0.258309	0.129154	+	0.991625i	$0.458774\pi$
$968$	−28.7972	−0.925576
$969$	0	0
$970$	−17.5362	−0.563053
$971$	42.0167	1.34838	0.674190	−	0.738558i	$-0.264493\pi$
$971$	42.0167	1.34838	0.674190	+	0.738558i	$0.264493\pi$
$972$	0	0
$973$	0	0
$974$	47.5991	1.52517
$975$	0	0
$976$	31.3551	1.00365
$977$	−59.4620	−1.90236	−0.951179	−	0.308639i	$-0.900127\pi$
$977$	−59.4620	−1.90236	−0.951179	+	0.308639i	$0.900127\pi$
$978$	0	0
$979$	−18.5375	−0.592461
$980$	0	0
$981$	0	0
$982$	9.07308	0.289534
$983$	−7.70171	−0.245646	−0.122823	−	0.992429i	$-0.539195\pi$
$983$	−7.70171	−0.245646	−0.122823	+	0.992429i	$0.539195\pi$
$984$	0	0
$985$	−7.99521	−0.254749
$986$	1.50604	0.0479621
$987$	0	0
$988$	−2.96376	−0.0942896
$989$	−60.3139	−1.91787
$990$	0	0
$991$	23.0183	0.731201	0.365600	−	0.930772i	$-0.380864\pi$
$991$	23.0183	0.731201	0.365600	+	0.930772i	$0.380864\pi$
$992$	15.4771	0.491398
$993$	0	0
$994$	0	0
$995$	47.4422	1.50402
$996$	0	0
$997$	49.4965	1.56757	0.783784	−	0.621033i	$-0.213287\pi$
$997$	49.4965	1.56757	0.783784	+	0.621033i	$0.213287\pi$
$998$	15.2325	0.482177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.a.e.1.1		3
3.2	odd	2		1029.2.a.c.1.3	yes	3
7.6	odd	2		3087.2.a.b.1.1		3
21.2	odd	6		1029.2.e.b.361.1		6
21.5	even	6		1029.2.e.c.361.1		6
21.11	odd	6		1029.2.e.b.667.1		6
21.17	even	6		1029.2.e.c.667.1		6
21.20	even	2		1029.2.a.b.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1029.2.a.b.1.3	✓	3	21.20	even	2
1029.2.a.c.1.3	yes	3	3.2	odd	2
1029.2.e.b.361.1		6	21.2	odd	6
1029.2.e.b.667.1		6	21.11	odd	6
1029.2.e.c.361.1		6	21.5	even	6
1029.2.e.c.667.1		6	21.17	even	6
3087.2.a.b.1.1		3	7.6	odd	2
3087.2.a.e.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.a (trivial)

\(f(q)\)	\(=\)	\(q-1.24698 q^{2} -0.445042 q^{4} +2.00000 q^{5} +3.04892 q^{8} +O(q^{10})\)	\(q-1.24698 q^{2} -0.445042 q^{4} +2.00000 q^{5} +3.04892 q^{8} -2.49396 q^{10} +1.24698 q^{11} -1.10992 q^{13} -2.91185 q^{16} +6.09783 q^{17} -6.00000 q^{19} -0.890084 q^{20} -1.55496 q^{22} -6.51573 q^{23} -1.00000 q^{25} +1.38404 q^{26} -0.198062 q^{29} -6.27413 q^{31} -2.46681 q^{32} -7.60388 q^{34} -7.46681 q^{37} +7.48188 q^{38} +6.09783 q^{40} -8.59179 q^{41} +9.25667 q^{43} -0.554958 q^{44} +8.12498 q^{46} +3.28621 q^{47} +1.24698 q^{50} +0.493959 q^{52} +2.26875 q^{53} +2.49396 q^{55} +0.246980 q^{58} +1.01208 q^{59} -10.7681 q^{61} +7.82371 q^{62} +8.89977 q^{64} -2.21983 q^{65} -6.69202 q^{67} -2.71379 q^{68} -0.127375 q^{71} -7.82371 q^{73} +9.31096 q^{74} +2.67025 q^{76} +6.93900 q^{79} -5.82371 q^{80} +10.7138 q^{82} +11.3840 q^{83} +12.1957 q^{85} -11.5429 q^{86} +3.80194 q^{88} -14.8659 q^{89} +2.89977 q^{92} -4.09783 q^{94} -12.0000 q^{95} +7.03146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 6 q^{5}+O(q^{10}) \) 3 * q + q^2 - q^4 + 6 * q^5	\( 3 q + q^{2} - q^{4} + 6 q^{5} + 2 q^{10} - q^{11} - 4 q^{13} - 5 q^{16} - 18 q^{19} - 2 q^{20} - 5 q^{22} - 7 q^{23} - 3 q^{25} - 6 q^{26} - 5 q^{29} - 8 q^{31} - 4 q^{32} - 14 q^{34} - 19 q^{37} - 6 q^{38} + 2 q^{41} + q^{43} - 2 q^{44} + 18 q^{47} - q^{50} - 8 q^{52} - q^{53} - 2 q^{55} - 4 q^{58} + 22 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} - 8 q^{65} - 15 q^{67} - 17 q^{71} - 16 q^{73} - 11 q^{74} + 6 q^{76} + 11 q^{79} - 10 q^{80} + 24 q^{82} + 24 q^{83} - 16 q^{86} + 7 q^{88} - 6 q^{89} - 14 q^{92} + 6 q^{94} - 36 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q + q^2 - q^4 + 6 * q^5 + 2 * q^10 - q^11 - 4 * q^13 - 5 * q^16 - 18 * q^19 - 2 * q^20 - 5 * q^22 - 7 * q^23 - 3 * q^25 - 6 * q^26 - 5 * q^29 - 8 * q^31 - 4 * q^32 - 14 * q^34 - 19 * q^37 - 6 * q^38 + 2 * q^41 + q^43 - 2 * q^44 + 18 * q^47 - q^50 - 8 * q^52 - q^53 - 2 * q^55 - 4 * q^58 + 22 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 - 8 * q^65 - 15 * q^67 - 17 * q^71 - 16 * q^73 - 11 * q^74 + 6 * q^76 + 11 * q^79 - 10 * q^80 + 24 * q^82 + 24 * q^83 - 16 * q^86 + 7 * q^88 - 6 * q^89 - 14 * q^92 + 6 * q^94 - 36 * q^95 - 4 * q^97

Introduction

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