LMFDB - Embedded newform 2070.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.30278 q^{11} -0.302776 q^{13} -0.302776 q^{14} +1.00000 q^{16} +3.90833 q^{17} -4.90833 q^{19} -1.00000 q^{20} +5.30278 q^{22} +1.00000 q^{23} +1.00000 q^{25} -0.302776 q^{26} -0.302776 q^{28} -4.60555 q^{29} +2.90833 q^{31} +1.00000 q^{32} +3.90833 q^{34} +0.302776 q^{35} +8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{40} +9.90833 q^{41} +5.21110 q^{43} +5.30278 q^{44} +1.00000 q^{46} -4.60555 q^{47} -6.90833 q^{49} +1.00000 q^{50} -0.302776 q^{52} -3.21110 q^{53} -5.30278 q^{55} -0.302776 q^{56} -4.60555 q^{58} +10.6056 q^{59} -6.51388 q^{61} +2.90833 q^{62} +1.00000 q^{64} +0.302776 q^{65} -4.00000 q^{67} +3.90833 q^{68} +0.302776 q^{70} +12.6972 q^{71} +15.8167 q^{73} +8.00000 q^{74} -4.90833 q^{76} -1.60555 q^{77} +14.4222 q^{79} -1.00000 q^{80} +9.90833 q^{82} +3.21110 q^{83} -3.90833 q^{85} +5.21110 q^{86} +5.30278 q^{88} +0.0916731 q^{91} +1.00000 q^{92} -4.60555 q^{94} +4.90833 q^{95} +2.69722 q^{97} -6.90833 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 - 2 * q^10 + 7 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 + q^19 - 2 * q^20 + 7 * q^22 + 2 * q^23 + 2 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^29 - 5 * q^31 + 2 * q^32 - 3 * q^34 - 3 * q^35 + 16 * q^37 + q^38 - 2 * q^40 + 9 * q^41 - 4 * q^43 + 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^50 + 3 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^59 + 5 * q^61 - 5 * q^62 + 2 * q^64 - 3 * q^65 - 8 * q^67 - 3 * q^68 - 3 * q^70 + 29 * q^71 + 10 * q^73 + 16 * q^74 + q^76 + 4 * q^77 - 2 * q^80 + 9 * q^82 - 8 * q^83 + 3 * q^85 - 4 * q^86 + 7 * q^88 + 11 * q^91 + 2 * q^92 - 2 * q^94 - q^95 + 9 * q^97 - 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.302776	−0.114438	−0.0572192	−	0.998362i	$-0.518223\pi$
$7$	−0.302776	−0.114438	−0.0572192	+	0.998362i	$0.518223\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	5.30278	1.59885	0.799424	−	0.600768i	$-0.205138\pi$
$11$	5.30278	1.59885	0.799424	+	0.600768i	$0.205138\pi$
$12$	0	0
$13$	−0.302776	−0.0839749	−0.0419874	−	0.999118i	$-0.513369\pi$
$13$	−0.302776	−0.0839749	−0.0419874	+	0.999118i	$0.513369\pi$
$14$	−0.302776	−0.0809202
$15$	0	0
$16$	1.00000	0.250000
$17$	3.90833	0.947909	0.473954	−	0.880549i	$-0.342826\pi$
$17$	3.90833	0.947909	0.473954	+	0.880549i	$0.342826\pi$
$18$	0	0
$19$	−4.90833	−1.12605	−0.563024	−	0.826441i	$-0.690362\pi$
$19$	−4.90833	−1.12605	−0.563024	+	0.826441i	$0.690362\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.30278	1.13056
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.302776	−0.0593792
$27$	0	0
$28$	−0.302776	−0.0572192
$29$	−4.60555	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$29$	−4.60555	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$30$	0	0
$31$	2.90833	0.522351	0.261175	−	0.965291i	$-0.415890\pi$
$31$	2.90833	0.522351	0.261175	+	0.965291i	$0.415890\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.90833	0.670273
$35$	0.302776	0.0511784
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.90833	−0.796236
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.90833	1.54742	0.773710	−	0.633540i	$-0.218399\pi$
$41$	9.90833	1.54742	0.773710	+	0.633540i	$0.218399\pi$
$42$	0	0
$43$	5.21110	0.794686	0.397343	−	0.917670i	$-0.369932\pi$
$43$	5.21110	0.794686	0.397343	+	0.917670i	$0.369932\pi$
$44$	5.30278	0.799424
$45$	0	0
$46$	1.00000	0.147442
$47$	−4.60555	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$47$	−4.60555	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	1.00000	0.141421
$51$	0	0
$52$	−0.302776	−0.0419874
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	0	0
$55$	−5.30278	−0.715026
$56$	−0.302776	−0.0404601
$57$	0	0
$58$	−4.60555	−0.604739
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	−6.51388	−0.834017	−0.417008	−	0.908903i	$-0.636921\pi$
$61$	−6.51388	−0.834017	−0.417008	+	0.908903i	$0.636921\pi$
$62$	2.90833	0.369358
$63$	0	0
$64$	1.00000	0.125000
$65$	0.302776	0.0375547
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.90833	0.473954
$69$	0	0
$70$	0.302776	0.0361886
$71$	12.6972	1.50688	0.753442	−	0.657515i	$-0.228392\pi$
$71$	12.6972	1.50688	0.753442	+	0.657515i	$0.228392\pi$
$72$	0	0
$73$	15.8167	1.85120	0.925600	−	0.378504i	$-0.123561\pi$
$73$	15.8167	1.85120	0.925600	+	0.378504i	$0.123561\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.90833	−0.563024
$77$	−1.60555	−0.182970
$78$	0	0
$79$	14.4222	1.62262	0.811312	−	0.584613i	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	9.90833	1.09419
$83$	3.21110	0.352464	0.176232	−	0.984349i	$-0.443609\pi$
$83$	3.21110	0.352464	0.176232	+	0.984349i	$0.443609\pi$
$84$	0	0
$85$	−3.90833	−0.423918
$86$	5.21110	0.561928
$87$	0	0
$88$	5.30278	0.565278
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0.0916731	0.00960995
$92$	1.00000	0.104257
$93$	0	0
$94$	−4.60555	−0.475026
$95$	4.90833	0.503584
$96$	0	0
$97$	2.69722	0.273862	0.136931	−	0.990581i	$-0.456276\pi$
$97$	2.69722	0.273862	0.136931	+	0.990581i	$0.456276\pi$
$98$	−6.90833	−0.697846
$99$	0	0
$100$	1.00000	0.100000
$101$	4.60555	0.458269	0.229135	−	0.973395i	$-0.426410\pi$
$101$	4.60555	0.458269	0.229135	+	0.973395i	$0.426410\pi$
$102$	0	0
$103$	−17.1194	−1.68683	−0.843414	−	0.537265i	$-0.819458\pi$
$103$	−17.1194	−1.68683	−0.843414	+	0.537265i	$0.819458\pi$
$104$	−0.302776	−0.0296896
$105$	0	0
$106$	−3.21110	−0.311890
$107$	−4.60555	−0.445235	−0.222618	−	0.974906i	$-0.571460\pi$
$107$	−4.60555	−0.445235	−0.222618	+	0.974906i	$0.571460\pi$
$108$	0	0
$109$	19.5139	1.86909	0.934545	−	0.355844i	$-0.115807\pi$
$109$	19.5139	1.86909	0.934545	+	0.355844i	$0.115807\pi$
$110$	−5.30278	−0.505600
$111$	0	0
$112$	−0.302776	−0.0286096
$113$	−12.4222	−1.16858	−0.584291	−	0.811544i	$-0.698627\pi$
$113$	−12.4222	−1.16858	−0.584291	+	0.811544i	$0.698627\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−4.60555	−0.427615
$117$	0	0
$118$	10.6056	0.976320
$119$	−1.18335	−0.108477
$120$	0	0
$121$	17.1194	1.55631
$122$	−6.51388	−0.589739
$123$	0	0
$124$	2.90833	0.261175
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.8167	−1.04856	−0.524279	−	0.851546i	$-0.675665\pi$
$127$	−11.8167	−1.04856	−0.524279	+	0.851546i	$0.675665\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.302776	0.0265552
$131$	−3.21110	−0.280555	−0.140278	−	0.990112i	$-0.544800\pi$
$131$	−3.21110	−0.280555	−0.140278	+	0.990112i	$0.544800\pi$
$132$	0	0
$133$	1.48612	0.128863
$134$	−4.00000	−0.345547
$135$	0	0
$136$	3.90833	0.335136
$137$	6.90833	0.590218	0.295109	−	0.955464i	$-0.404644\pi$
$137$	6.90833	0.590218	0.295109	+	0.955464i	$0.404644\pi$
$138$	0	0
$139$	−5.39445	−0.457551	−0.228776	−	0.973479i	$-0.573472\pi$
$139$	−5.39445	−0.457551	−0.228776	+	0.973479i	$0.573472\pi$
$140$	0.302776	0.0255892
$141$	0	0
$142$	12.6972	1.06553
$143$	−1.60555	−0.134263
$144$	0	0
$145$	4.60555	0.382470
$146$	15.8167	1.30900
$147$	0	0
$148$	8.00000	0.657596
$149$	−9.69722	−0.794428	−0.397214	−	0.917726i	$-0.630023\pi$
$149$	−9.69722	−0.794428	−0.397214	+	0.917726i	$0.630023\pi$
$150$	0	0
$151$	−1.90833	−0.155297	−0.0776487	−	0.996981i	$-0.524741\pi$
$151$	−1.90833	−0.155297	−0.0776487	+	0.996981i	$0.524741\pi$
$152$	−4.90833	−0.398118
$153$	0	0
$154$	−1.60555	−0.129379
$155$	−2.90833	−0.233602
$156$	0	0
$157$	−11.3944	−0.909376	−0.454688	−	0.890651i	$-0.650249\pi$
$157$	−11.3944	−0.909376	−0.454688	+	0.890651i	$0.650249\pi$
$158$	14.4222	1.14737
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−0.302776	−0.0238621
$162$	0	0
$163$	5.69722	0.446241	0.223121	−	0.974791i	$-0.428376\pi$
$163$	5.69722	0.446241	0.223121	+	0.974791i	$0.428376\pi$
$164$	9.90833	0.773710
$165$	0	0
$166$	3.21110	0.249230
$167$	−21.2111	−1.64136	−0.820682	−	0.571385i	$-0.806406\pi$
$167$	−21.2111	−1.64136	−0.820682	+	0.571385i	$0.806406\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	−3.90833	−0.299755
$171$	0	0
$172$	5.21110	0.397343
$173$	−23.3028	−1.77168	−0.885839	−	0.463993i	$-0.846416\pi$
$173$	−23.3028	−1.77168	−0.885839	+	0.463993i	$0.846416\pi$
$174$	0	0
$175$	−0.302776	−0.0228877
$176$	5.30278	0.399712
$177$	0	0
$178$	0	0
$179$	−16.6056	−1.24116	−0.620579	−	0.784144i	$-0.713102\pi$
$179$	−16.6056	−1.24116	−0.620579	+	0.784144i	$0.713102\pi$
$180$	0	0
$181$	−8.11943	−0.603512	−0.301756	−	0.953385i	$-0.597573\pi$
$181$	−8.11943	−0.603512	−0.301756	+	0.953385i	$0.597573\pi$
$182$	0.0916731	0.00679526
$183$	0	0
$184$	1.00000	0.0737210
$185$	−8.00000	−0.588172
$186$	0	0
$187$	20.7250	1.51556
$188$	−4.60555	−0.335894
$189$	0	0
$190$	4.90833	0.356087
$191$	1.39445	0.100899	0.0504494	−	0.998727i	$-0.483935\pi$
$191$	1.39445	0.100899	0.0504494	+	0.998727i	$0.483935\pi$
$192$	0	0
$193$	3.81665	0.274729	0.137364	−	0.990521i	$-0.456137\pi$
$193$	3.81665	0.274729	0.137364	+	0.990521i	$0.456137\pi$
$194$	2.69722	0.193649
$195$	0	0
$196$	−6.90833	−0.493452
$197$	−0.697224	−0.0496752	−0.0248376	−	0.999691i	$-0.507907\pi$
$197$	−0.697224	−0.0496752	−0.0248376	+	0.999691i	$0.507907\pi$
$198$	0	0
$199$	8.42221	0.597034	0.298517	−	0.954404i	$-0.403508\pi$
$199$	8.42221	0.597034	0.298517	+	0.954404i	$0.403508\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	4.60555	0.324045
$203$	1.39445	0.0978711
$204$	0	0
$205$	−9.90833	−0.692028
$206$	−17.1194	−1.19277
$207$	0	0
$208$	−0.302776	−0.0209937
$209$	−26.0278	−1.80038
$210$	0	0
$211$	−7.21110	−0.496433	−0.248216	−	0.968705i	$-0.579844\pi$
$211$	−7.21110	−0.496433	−0.248216	+	0.968705i	$0.579844\pi$
$212$	−3.21110	−0.220539
$213$	0	0
$214$	−4.60555	−0.314829
$215$	−5.21110	−0.355394
$216$	0	0
$217$	−0.880571	−0.0597770
$218$	19.5139	1.32165
$219$	0	0
$220$	−5.30278	−0.357513
$221$	−1.18335	−0.0796005
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−0.302776	−0.0202300
$225$	0	0
$226$	−12.4222	−0.826313
$227$	7.39445	0.490787	0.245393	−	0.969424i	$-0.421083\pi$
$227$	7.39445	0.490787	0.245393	+	0.969424i	$0.421083\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−4.60555	−0.302369
$233$	−4.18335	−0.274060	−0.137030	−	0.990567i	$-0.543756\pi$
$233$	−4.18335	−0.274060	−0.137030	+	0.990567i	$0.543756\pi$
$234$	0	0
$235$	4.60555	0.300433
$236$	10.6056	0.690363
$237$	0	0
$238$	−1.18335	−0.0767049
$239$	9.21110	0.595817	0.297908	−	0.954594i	$-0.403711\pi$
$239$	9.21110	0.595817	0.297908	+	0.954594i	$0.403711\pi$
$240$	0	0
$241$	14.4222	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$241$	14.4222	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$242$	17.1194	1.10048
$243$	0	0
$244$	−6.51388	−0.417008
$245$	6.90833	0.441357
$246$	0	0
$247$	1.48612	0.0945597
$248$	2.90833	0.184679
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	5.51388	0.348033	0.174016	−	0.984743i	$-0.444325\pi$
$251$	5.51388	0.348033	0.174016	+	0.984743i	$0.444325\pi$
$252$	0	0
$253$	5.30278	0.333383
$254$	−11.8167	−0.741443
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.8167	−1.23613	−0.618064	−	0.786127i	$-0.712083\pi$
$257$	−19.8167	−1.23613	−0.618064	+	0.786127i	$0.712083\pi$
$258$	0	0
$259$	−2.42221	−0.150509
$260$	0.302776	0.0187773
$261$	0	0
$262$	−3.21110	−0.198383
$263$	14.5139	0.894964	0.447482	−	0.894293i	$-0.352321\pi$
$263$	14.5139	0.894964	0.447482	+	0.894293i	$0.352321\pi$
$264$	0	0
$265$	3.21110	0.197256
$266$	1.48612	0.0911200
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−25.8167	−1.57407	−0.787035	−	0.616909i	$-0.788385\pi$
$269$	−25.8167	−1.57407	−0.787035	+	0.616909i	$0.788385\pi$
$270$	0	0
$271$	−6.30278	−0.382866	−0.191433	−	0.981506i	$-0.561314\pi$
$271$	−6.30278	−0.382866	−0.191433	+	0.981506i	$0.561314\pi$
$272$	3.90833	0.236977
$273$	0	0
$274$	6.90833	0.417347
$275$	5.30278	0.319769
$276$	0	0
$277$	−12.7889	−0.768410	−0.384205	−	0.923248i	$-0.625524\pi$
$277$	−12.7889	−0.768410	−0.384205	+	0.923248i	$0.625524\pi$
$278$	−5.39445	−0.323538
$279$	0	0
$280$	0.302776	0.0180943
$281$	19.3944	1.15698	0.578488	−	0.815691i	$-0.303643\pi$
$281$	19.3944	1.15698	0.578488	+	0.815691i	$0.303643\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	12.6972	0.753442
$285$	0	0
$286$	−1.60555	−0.0949382
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−1.72498	−0.101469
$290$	4.60555	0.270447
$291$	0	0
$292$	15.8167	0.925600
$293$	−8.78890	−0.513453	−0.256726	−	0.966484i	$-0.582644\pi$
$293$	−8.78890	−0.513453	−0.256726	+	0.966484i	$0.582644\pi$
$294$	0	0
$295$	−10.6056	−0.617479
$296$	8.00000	0.464991
$297$	0	0
$298$	−9.69722	−0.561745
$299$	−0.302776	−0.0175100
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	−1.90833	−0.109812
$303$	0	0
$304$	−4.90833	−0.281512
$305$	6.51388	0.372984
$306$	0	0
$307$	−15.3028	−0.873376	−0.436688	−	0.899613i	$-0.643849\pi$
$307$	−15.3028	−0.873376	−0.436688	+	0.899613i	$0.643849\pi$
$308$	−1.60555	−0.0914848
$309$	0	0
$310$	−2.90833	−0.165182
$311$	6.42221	0.364170	0.182085	−	0.983283i	$-0.441715\pi$
$311$	6.42221	0.364170	0.182085	+	0.983283i	$0.441715\pi$
$312$	0	0
$313$	−12.7250	−0.719258	−0.359629	−	0.933095i	$-0.617097\pi$
$313$	−12.7250	−0.719258	−0.359629	+	0.933095i	$0.617097\pi$
$314$	−11.3944	−0.643026
$315$	0	0
$316$	14.4222	0.811312
$317$	14.7250	0.827037	0.413519	−	0.910496i	$-0.364300\pi$
$317$	14.7250	0.827037	0.413519	+	0.910496i	$0.364300\pi$
$318$	0	0
$319$	−24.4222	−1.36738
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.302776	−0.0168730
$323$	−19.1833	−1.06739
$324$	0	0
$325$	−0.302776	−0.0167950
$326$	5.69722	0.315540
$327$	0	0
$328$	9.90833	0.547096
$329$	1.39445	0.0768784
$330$	0	0
$331$	9.39445	0.516366	0.258183	−	0.966096i	$-0.416876\pi$
$331$	9.39445	0.516366	0.258183	+	0.966096i	$0.416876\pi$
$332$	3.21110	0.176232
$333$	0	0
$334$	−21.2111	−1.16062
$335$	4.00000	0.218543
$336$	0	0
$337$	−4.48612	−0.244375	−0.122187	−	0.992507i	$-0.538991\pi$
$337$	−4.48612	−0.244375	−0.122187	+	0.992507i	$0.538991\pi$
$338$	−12.9083	−0.702120
$339$	0	0
$340$	−3.90833	−0.211959
$341$	15.4222	0.835159
$342$	0	0
$343$	4.21110	0.227378
$344$	5.21110	0.280964
$345$	0	0
$346$	−23.3028	−1.25276
$347$	25.5416	1.37115	0.685573	−	0.728004i	$-0.259552\pi$
$347$	25.5416	1.37115	0.685573	+	0.728004i	$0.259552\pi$
$348$	0	0
$349$	−12.7889	−0.684574	−0.342287	−	0.939595i	$-0.611202\pi$
$349$	−12.7889	−0.684574	−0.342287	+	0.939595i	$0.611202\pi$
$350$	−0.302776	−0.0161840
$351$	0	0
$352$	5.30278	0.282639
$353$	−18.4222	−0.980515	−0.490258	−	0.871578i	$-0.663097\pi$
$353$	−18.4222	−0.980515	−0.490258	+	0.871578i	$0.663097\pi$
$354$	0	0
$355$	−12.6972	−0.673899
$356$	0	0
$357$	0	0
$358$	−16.6056	−0.877631
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	−8.11943	−0.426748
$363$	0	0
$364$	0.0916731	0.00480498
$365$	−15.8167	−0.827881
$366$	0	0
$367$	29.2111	1.52481	0.762404	−	0.647102i	$-0.224019\pi$
$367$	29.2111	1.52481	0.762404	+	0.647102i	$0.224019\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−8.00000	−0.415900
$371$	0.972244	0.0504764
$372$	0	0
$373$	−2.60555	−0.134910	−0.0674552	−	0.997722i	$-0.521488\pi$
$373$	−2.60555	−0.134910	−0.0674552	+	0.997722i	$0.521488\pi$
$374$	20.7250	1.07166
$375$	0	0
$376$	−4.60555	−0.237513
$377$	1.39445	0.0718178
$378$	0	0
$379$	4.09167	0.210175	0.105088	−	0.994463i	$-0.466488\pi$
$379$	4.09167	0.210175	0.105088	+	0.994463i	$0.466488\pi$
$380$	4.90833	0.251792
$381$	0	0
$382$	1.39445	0.0713462
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	1.60555	0.0818265
$386$	3.81665	0.194263
$387$	0	0
$388$	2.69722	0.136931
$389$	−20.9361	−1.06150	−0.530751	−	0.847528i	$-0.678090\pi$
$389$	−20.9361	−1.06150	−0.530751	+	0.847528i	$0.678090\pi$
$390$	0	0
$391$	3.90833	0.197653
$392$	−6.90833	−0.348923
$393$	0	0
$394$	−0.697224	−0.0351257
$395$	−14.4222	−0.725660
$396$	0	0
$397$	−21.7250	−1.09035	−0.545173	−	0.838324i	$-0.683536\pi$
$397$	−21.7250	−1.09035	−0.545173	+	0.838324i	$0.683536\pi$
$398$	8.42221	0.422167
$399$	0	0
$400$	1.00000	0.0500000
$401$	1.39445	0.0696354	0.0348177	−	0.999394i	$-0.488915\pi$
$401$	1.39445	0.0696354	0.0348177	+	0.999394i	$0.488915\pi$
$402$	0	0
$403$	−0.880571	−0.0438643
$404$	4.60555	0.229135
$405$	0	0
$406$	1.39445	0.0692053
$407$	42.4222	2.10279
$408$	0	0
$409$	−15.0917	−0.746235	−0.373118	−	0.927784i	$-0.621711\pi$
$409$	−15.0917	−0.746235	−0.373118	+	0.927784i	$0.621711\pi$
$410$	−9.90833	−0.489337
$411$	0	0
$412$	−17.1194	−0.843414
$413$	−3.21110	−0.158008
$414$	0	0
$415$	−3.21110	−0.157627
$416$	−0.302776	−0.0148448
$417$	0	0
$418$	−26.0278	−1.27306
$419$	39.6333	1.93621	0.968107	−	0.250538i	$-0.0806073\pi$
$419$	39.6333	1.93621	0.968107	+	0.250538i	$0.0806073\pi$
$420$	0	0
$421$	34.3028	1.67181	0.835907	−	0.548870i	$-0.184942\pi$
$421$	34.3028	1.67181	0.835907	+	0.548870i	$0.184942\pi$
$422$	−7.21110	−0.351031
$423$	0	0
$424$	−3.21110	−0.155945
$425$	3.90833	0.189582
$426$	0	0
$427$	1.97224	0.0954436
$428$	−4.60555	−0.222618
$429$	0	0
$430$	−5.21110	−0.251302
$431$	−20.2389	−0.974872	−0.487436	−	0.873159i	$-0.662068\pi$
$431$	−20.2389	−0.974872	−0.487436	+	0.873159i	$0.662068\pi$
$432$	0	0
$433$	−34.9083	−1.67759	−0.838794	−	0.544450i	$-0.816739\pi$
$433$	−34.9083	−1.67759	−0.838794	+	0.544450i	$0.816739\pi$
$434$	−0.880571	−0.0422687
$435$	0	0
$436$	19.5139	0.934545
$437$	−4.90833	−0.234797
$438$	0	0
$439$	−18.3028	−0.873544	−0.436772	−	0.899572i	$-0.643878\pi$
$439$	−18.3028	−0.873544	−0.436772	+	0.899572i	$0.643878\pi$
$440$	−5.30278	−0.252800
$441$	0	0
$442$	−1.18335	−0.0562860
$443$	−35.5139	−1.68732	−0.843658	−	0.536882i	$-0.819602\pi$
$443$	−35.5139	−1.68732	−0.843658	+	0.536882i	$0.819602\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−0.302776	−0.0143048
$449$	12.9083	0.609182	0.304591	−	0.952483i	$-0.401480\pi$
$449$	12.9083	0.609182	0.304591	+	0.952483i	$0.401480\pi$
$450$	0	0
$451$	52.5416	2.47409
$452$	−12.4222	−0.584291
$453$	0	0
$454$	7.39445	0.347039
$455$	−0.0916731	−0.00429770
$456$	0	0
$457$	−3.57779	−0.167362	−0.0836811	−	0.996493i	$-0.526668\pi$
$457$	−3.57779	−0.167362	−0.0836811	+	0.996493i	$0.526668\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−31.8167	−1.48185	−0.740925	−	0.671588i	$-0.765612\pi$
$461$	−31.8167	−1.48185	−0.740925	+	0.671588i	$0.765612\pi$
$462$	0	0
$463$	−25.6333	−1.19128	−0.595640	−	0.803251i	$-0.703102\pi$
$463$	−25.6333	−1.19128	−0.595640	+	0.803251i	$0.703102\pi$
$464$	−4.60555	−0.213807
$465$	0	0
$466$	−4.18335	−0.193790
$467$	−19.8167	−0.917005	−0.458503	−	0.888693i	$-0.651614\pi$
$467$	−19.8167	−0.917005	−0.458503	+	0.888693i	$0.651614\pi$
$468$	0	0
$469$	1.21110	0.0559235
$470$	4.60555	0.212438
$471$	0	0
$472$	10.6056	0.488160
$473$	27.6333	1.27058
$474$	0	0
$475$	−4.90833	−0.225209
$476$	−1.18335	−0.0542386
$477$	0	0
$478$	9.21110	0.421306
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−2.42221	−0.110443
$482$	14.4222	0.656913
$483$	0	0
$484$	17.1194	0.778156
$485$	−2.69722	−0.122475
$486$	0	0
$487$	−11.8167	−0.535464	−0.267732	−	0.963493i	$-0.586274\pi$
$487$	−11.8167	−0.535464	−0.267732	+	0.963493i	$0.586274\pi$
$488$	−6.51388	−0.294869
$489$	0	0
$490$	6.90833	0.312086
$491$	25.8167	1.16509	0.582545	−	0.812799i	$-0.302057\pi$
$491$	25.8167	1.16509	0.582545	+	0.812799i	$0.302057\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	1.48612	0.0668638
$495$	0	0
$496$	2.90833	0.130588
$497$	−3.84441	−0.172445
$498$	0	0
$499$	11.6333	0.520778	0.260389	−	0.965504i	$-0.416149\pi$
$499$	11.6333	0.520778	0.260389	+	0.965504i	$0.416149\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	5.51388	0.246096
$503$	2.72498	0.121501	0.0607504	−	0.998153i	$-0.480651\pi$
$503$	2.72498	0.121501	0.0607504	+	0.998153i	$0.480651\pi$
$504$	0	0
$505$	−4.60555	−0.204944
$506$	5.30278	0.235737
$507$	0	0
$508$	−11.8167	−0.524279
$509$	−29.4500	−1.30535	−0.652673	−	0.757639i	$-0.726353\pi$
$509$	−29.4500	−1.30535	−0.652673	+	0.757639i	$0.726353\pi$
$510$	0	0
$511$	−4.78890	−0.211848
$512$	1.00000	0.0441942
$513$	0	0
$514$	−19.8167	−0.874075
$515$	17.1194	0.754372
$516$	0	0
$517$	−24.4222	−1.07409
$518$	−2.42221	−0.106426
$519$	0	0
$520$	0.302776	0.0132776
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	8.42221	0.368277	0.184139	−	0.982900i	$-0.441050\pi$
$523$	8.42221	0.368277	0.184139	+	0.982900i	$0.441050\pi$
$524$	−3.21110	−0.140278
$525$	0	0
$526$	14.5139	0.632835
$527$	11.3667	0.495141
$528$	0	0
$529$	1.00000	0.0434783
$530$	3.21110	0.139481
$531$	0	0
$532$	1.48612	0.0644316
$533$	−3.00000	−0.129944
$534$	0	0
$535$	4.60555	0.199115
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−25.8167	−1.11303
$539$	−36.6333	−1.57791
$540$	0	0
$541$	−28.8444	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$541$	−28.8444	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$542$	−6.30278	−0.270727
$543$	0	0
$544$	3.90833	0.167568
$545$	−19.5139	−0.835883
$546$	0	0
$547$	7.51388	0.321270	0.160635	−	0.987014i	$-0.448646\pi$
$547$	7.51388	0.321270	0.160635	+	0.987014i	$0.448646\pi$
$548$	6.90833	0.295109
$549$	0	0
$550$	5.30278	0.226111
$551$	22.6056	0.963029
$552$	0	0
$553$	−4.36669	−0.185691
$554$	−12.7889	−0.543348
$555$	0	0
$556$	−5.39445	−0.228776
$557$	6.42221	0.272118	0.136059	−	0.990701i	$-0.456556\pi$
$557$	6.42221	0.272118	0.136059	+	0.990701i	$0.456556\pi$
$558$	0	0
$559$	−1.57779	−0.0667336
$560$	0.302776	0.0127946
$561$	0	0
$562$	19.3944	0.818105
$563$	−39.6333	−1.67034	−0.835172	−	0.549988i	$-0.814632\pi$
$563$	−39.6333	−1.67034	−0.835172	+	0.549988i	$0.814632\pi$
$564$	0	0
$565$	12.4222	0.522606
$566$	2.00000	0.0840663
$567$	0	0
$568$	12.6972	0.532764
$569$	0.422205	0.0176998	0.00884988	−	0.999961i	$-0.497183\pi$
$569$	0.422205	0.0176998	0.00884988	+	0.999961i	$0.497183\pi$
$570$	0	0
$571$	9.11943	0.381636	0.190818	−	0.981625i	$-0.438886\pi$
$571$	9.11943	0.381636	0.190818	+	0.981625i	$0.438886\pi$
$572$	−1.60555	−0.0671315
$573$	0	0
$574$	−3.00000	−0.125218
$575$	1.00000	0.0417029
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−1.72498	−0.0717497
$579$	0	0
$580$	4.60555	0.191235
$581$	−0.972244	−0.0403355
$582$	0	0
$583$	−17.0278	−0.705218
$584$	15.8167	0.654498
$585$	0	0
$586$	−8.78890	−0.363066
$587$	37.5416	1.54951	0.774755	−	0.632262i	$-0.217873\pi$
$587$	37.5416	1.54951	0.774755	+	0.632262i	$0.217873\pi$
$588$	0	0
$589$	−14.2750	−0.588192
$590$	−10.6056	−0.436624
$591$	0	0
$592$	8.00000	0.328798
$593$	−19.8167	−0.813772	−0.406886	−	0.913479i	$-0.633385\pi$
$593$	−19.8167	−0.813772	−0.406886	+	0.913479i	$0.633385\pi$
$594$	0	0
$595$	1.18335	0.0485125
$596$	−9.69722	−0.397214
$597$	0	0
$598$	−0.302776	−0.0123814
$599$	−4.33053	−0.176941	−0.0884704	−	0.996079i	$-0.528198\pi$
$599$	−4.33053	−0.176941	−0.0884704	+	0.996079i	$0.528198\pi$
$600$	0	0
$601$	−3.93608	−0.160556	−0.0802781	−	0.996773i	$-0.525581\pi$
$601$	−3.93608	−0.160556	−0.0802781	+	0.996773i	$0.525581\pi$
$602$	−1.57779	−0.0643061
$603$	0	0
$604$	−1.90833	−0.0776487
$605$	−17.1194	−0.696004
$606$	0	0
$607$	−26.0555	−1.05756	−0.528780	−	0.848759i	$-0.677350\pi$
$607$	−26.0555	−1.05756	−0.528780	+	0.848759i	$0.677350\pi$
$608$	−4.90833	−0.199059
$609$	0	0
$610$	6.51388	0.263739
$611$	1.39445	0.0564134
$612$	0	0
$613$	32.4222	1.30952	0.654760	−	0.755837i	$-0.272770\pi$
$613$	32.4222	1.30952	0.654760	+	0.755837i	$0.272770\pi$
$614$	−15.3028	−0.617570
$615$	0	0
$616$	−1.60555	−0.0646895
$617$	−8.09167	−0.325758	−0.162879	−	0.986646i	$-0.552078\pi$
$617$	−8.09167	−0.325758	−0.162879	+	0.986646i	$0.552078\pi$
$618$	0	0
$619$	27.3305	1.09851	0.549253	−	0.835656i	$-0.314912\pi$
$619$	27.3305	1.09851	0.549253	+	0.835656i	$0.314912\pi$
$620$	−2.90833	−0.116801
$621$	0	0
$622$	6.42221	0.257507
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.7250	−0.508593
$627$	0	0
$628$	−11.3944	−0.454688
$629$	31.2666	1.24668
$630$	0	0
$631$	30.6056	1.21839	0.609194	−	0.793021i	$-0.291493\pi$
$631$	30.6056	1.21839	0.609194	+	0.793021i	$0.291493\pi$
$632$	14.4222	0.573685
$633$	0	0
$634$	14.7250	0.584804
$635$	11.8167	0.468930
$636$	0	0
$637$	2.09167	0.0828751
$638$	−24.4222	−0.966884
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	16.2389	0.640398	0.320199	−	0.947350i	$-0.396250\pi$
$643$	16.2389	0.640398	0.320199	+	0.947350i	$0.396250\pi$
$644$	−0.302776	−0.0119310
$645$	0	0
$646$	−19.1833	−0.754759
$647$	30.8444	1.21262	0.606309	−	0.795229i	$-0.292649\pi$
$647$	30.8444	1.21262	0.606309	+	0.795229i	$0.292649\pi$
$648$	0	0
$649$	56.2389	2.20757
$650$	−0.302776	−0.0118758
$651$	0	0
$652$	5.69722	0.223121
$653$	−9.27502	−0.362960	−0.181480	−	0.983395i	$-0.558089\pi$
$653$	−9.27502	−0.362960	−0.181480	+	0.983395i	$0.558089\pi$
$654$	0	0
$655$	3.21110	0.125468
$656$	9.90833	0.386855
$657$	0	0
$658$	1.39445	0.0543613
$659$	27.6333	1.07644	0.538220	−	0.842804i	$-0.319097\pi$
$659$	27.6333	1.07644	0.538220	+	0.842804i	$0.319097\pi$
$660$	0	0
$661$	−24.0917	−0.937057	−0.468529	−	0.883448i	$-0.655216\pi$
$661$	−24.0917	−0.937057	−0.468529	+	0.883448i	$0.655216\pi$
$662$	9.39445	0.365126
$663$	0	0
$664$	3.21110	0.124615
$665$	−1.48612	−0.0576293
$666$	0	0
$667$	−4.60555	−0.178328
$668$	−21.2111	−0.820682
$669$	0	0
$670$	4.00000	0.154533
$671$	−34.5416	−1.33347
$672$	0	0
$673$	5.63331	0.217148	0.108574	−	0.994088i	$-0.465372\pi$
$673$	5.63331	0.217148	0.108574	+	0.994088i	$0.465372\pi$
$674$	−4.48612	−0.172799
$675$	0	0
$676$	−12.9083	−0.496474
$677$	12.4222	0.477424	0.238712	−	0.971090i	$-0.423275\pi$
$677$	12.4222	0.477424	0.238712	+	0.971090i	$0.423275\pi$
$678$	0	0
$679$	−0.816654	−0.0313403
$680$	−3.90833	−0.149877
$681$	0	0
$682$	15.4222	0.590547
$683$	32.7250	1.25219	0.626093	−	0.779748i	$-0.284653\pi$
$683$	32.7250	1.25219	0.626093	+	0.779748i	$0.284653\pi$
$684$	0	0
$685$	−6.90833	−0.263954
$686$	4.21110	0.160781
$687$	0	0
$688$	5.21110	0.198671
$689$	0.972244	0.0370395
$690$	0	0
$691$	30.1833	1.14823	0.574114	−	0.818775i	$-0.305347\pi$
$691$	30.1833	1.14823	0.574114	+	0.818775i	$0.305347\pi$
$692$	−23.3028	−0.885839
$693$	0	0
$694$	25.5416	0.969547
$695$	5.39445	0.204623
$696$	0	0
$697$	38.7250	1.46681
$698$	−12.7889	−0.484067
$699$	0	0
$700$	−0.302776	−0.0114438
$701$	−42.9083	−1.62063	−0.810313	−	0.585998i	$-0.800703\pi$
$701$	−42.9083	−1.62063	−0.810313	+	0.585998i	$0.800703\pi$
$702$	0	0
$703$	−39.2666	−1.48097
$704$	5.30278	0.199856
$705$	0	0
$706$	−18.4222	−0.693329
$707$	−1.39445	−0.0524436
$708$	0	0
$709$	−41.1194	−1.54427	−0.772136	−	0.635457i	$-0.780812\pi$
$709$	−41.1194	−1.54427	−0.772136	+	0.635457i	$0.780812\pi$
$710$	−12.6972	−0.476518
$711$	0	0
$712$	0	0
$713$	2.90833	0.108918
$714$	0	0
$715$	1.60555	0.0600442
$716$	−16.6056	−0.620579
$717$	0	0
$718$	3.21110	0.119837
$719$	−14.3028	−0.533404	−0.266702	−	0.963779i	$-0.585934\pi$
$719$	−14.3028	−0.533404	−0.266702	+	0.963779i	$0.585934\pi$
$720$	0	0
$721$	5.18335	0.193038
$722$	5.09167	0.189492
$723$	0	0
$724$	−8.11943	−0.301756
$725$	−4.60555	−0.171046
$726$	0	0
$727$	−7.90833	−0.293304	−0.146652	−	0.989188i	$-0.546850\pi$
$727$	−7.90833	−0.293304	−0.146652	+	0.989188i	$0.546850\pi$
$728$	0.0916731	0.00339763
$729$	0	0
$730$	−15.8167	−0.585401
$731$	20.3667	0.753289
$732$	0	0
$733$	−13.6333	−0.503558	−0.251779	−	0.967785i	$-0.581016\pi$
$733$	−13.6333	−0.503558	−0.251779	+	0.967785i	$0.581016\pi$
$734$	29.2111	1.07820
$735$	0	0
$736$	1.00000	0.0368605
$737$	−21.2111	−0.781321
$738$	0	0
$739$	−7.63331	−0.280796	−0.140398	−	0.990095i	$-0.544838\pi$
$739$	−7.63331	−0.280796	−0.140398	+	0.990095i	$0.544838\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0.972244	0.0356922
$743$	−7.33053	−0.268931	−0.134466	−	0.990918i	$-0.542932\pi$
$743$	−7.33053	−0.268931	−0.134466	+	0.990918i	$0.542932\pi$
$744$	0	0
$745$	9.69722	0.355279
$746$	−2.60555	−0.0953960
$747$	0	0
$748$	20.7250	0.757780
$749$	1.39445	0.0509520
$750$	0	0
$751$	0.183346	0.00669040	0.00334520	−	0.999994i	$-0.498935\pi$
$751$	0.183346	0.00669040	0.00334520	+	0.999994i	$0.498935\pi$
$752$	−4.60555	−0.167947
$753$	0	0
$754$	1.39445	0.0507828
$755$	1.90833	0.0694511
$756$	0	0
$757$	−1.21110	−0.0440183	−0.0220091	−	0.999758i	$-0.507006\pi$
$757$	−1.21110	−0.0440183	−0.0220091	+	0.999758i	$0.507006\pi$
$758$	4.09167	0.148616
$759$	0	0
$760$	4.90833	0.178044
$761$	−4.54163	−0.164634	−0.0823171	−	0.996606i	$-0.526232\pi$
$761$	−4.54163	−0.164634	−0.0823171	+	0.996606i	$0.526232\pi$
$762$	0	0
$763$	−5.90833	−0.213896
$764$	1.39445	0.0504494
$765$	0	0
$766$	0	0
$767$	−3.21110	−0.115946
$768$	0	0
$769$	−41.2666	−1.48811	−0.744056	−	0.668117i	$-0.767100\pi$
$769$	−41.2666	−1.48811	−0.744056	+	0.668117i	$0.767100\pi$
$770$	1.60555	0.0578601
$771$	0	0
$772$	3.81665	0.137364
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	2.90833	0.104470
$776$	2.69722	0.0968247
$777$	0	0
$778$	−20.9361	−0.750595
$779$	−48.6333	−1.74247
$780$	0	0
$781$	67.3305	2.40928
$782$	3.90833	0.139761
$783$	0	0
$784$	−6.90833	−0.246726
$785$	11.3944	0.406685
$786$	0	0
$787$	−27.4500	−0.978485	−0.489243	−	0.872148i	$-0.662727\pi$
$787$	−27.4500	−0.978485	−0.489243	+	0.872148i	$0.662727\pi$
$788$	−0.697224	−0.0248376
$789$	0	0
$790$	−14.4222	−0.513119
$791$	3.76114	0.133731
$792$	0	0
$793$	1.97224	0.0700364
$794$	−21.7250	−0.770991
$795$	0	0
$796$	8.42221	0.298517
$797$	19.8167	0.701942	0.350971	−	0.936386i	$-0.385852\pi$
$797$	19.8167	0.701942	0.350971	+	0.936386i	$0.385852\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	1.00000	0.0353553
$801$	0	0
$802$	1.39445	0.0492397
$803$	83.8722	2.95978
$804$	0	0
$805$	0.302776	0.0106714
$806$	−0.880571	−0.0310168
$807$	0	0
$808$	4.60555	0.162023
$809$	−18.2750	−0.642515	−0.321258	−	0.946992i	$-0.604106\pi$
$809$	−18.2750	−0.642515	−0.321258	+	0.946992i	$0.604106\pi$
$810$	0	0
$811$	−4.97224	−0.174599	−0.0872995	−	0.996182i	$-0.527824\pi$
$811$	−4.97224	−0.174599	−0.0872995	+	0.996182i	$0.527824\pi$
$812$	1.39445	0.0489356
$813$	0	0
$814$	42.4222	1.48690
$815$	−5.69722	−0.199565
$816$	0	0
$817$	−25.5778	−0.894854
$818$	−15.0917	−0.527668
$819$	0	0
$820$	−9.90833	−0.346014
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−0.788897	−0.0274992	−0.0137496	−	0.999905i	$-0.504377\pi$
$823$	−0.788897	−0.0274992	−0.0137496	+	0.999905i	$0.504377\pi$
$824$	−17.1194	−0.596384
$825$	0	0
$826$	−3.21110	−0.111729
$827$	−35.4500	−1.23272	−0.616358	−	0.787466i	$-0.711393\pi$
$827$	−35.4500	−1.23272	−0.616358	+	0.787466i	$0.711393\pi$
$828$	0	0
$829$	16.7889	0.583103	0.291551	−	0.956555i	$-0.405829\pi$
$829$	16.7889	0.583103	0.291551	+	0.956555i	$0.405829\pi$
$830$	−3.21110	−0.111459
$831$	0	0
$832$	−0.302776	−0.0104969
$833$	−27.0000	−0.935495
$834$	0	0
$835$	21.2111	0.734040
$836$	−26.0278	−0.900189
$837$	0	0
$838$	39.6333	1.36911
$839$	22.1833	0.765854	0.382927	−	0.923779i	$-0.374916\pi$
$839$	22.1833	0.765854	0.382927	+	0.923779i	$0.374916\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	34.3028	1.18215
$843$	0	0
$844$	−7.21110	−0.248216
$845$	12.9083	0.444060
$846$	0	0
$847$	−5.18335	−0.178102
$848$	−3.21110	−0.110270
$849$	0	0
$850$	3.90833	0.134055
$851$	8.00000	0.274236
$852$	0	0
$853$	10.7250	0.367216	0.183608	−	0.983000i	$-0.441222\pi$
$853$	10.7250	0.367216	0.183608	+	0.983000i	$0.441222\pi$
$854$	1.97224	0.0674888
$855$	0	0
$856$	−4.60555	−0.157415
$857$	33.6333	1.14889	0.574446	−	0.818543i	$-0.305218\pi$
$857$	33.6333	1.14889	0.574446	+	0.818543i	$0.305218\pi$
$858$	0	0
$859$	−14.1833	−0.483930	−0.241965	−	0.970285i	$-0.577792\pi$
$859$	−14.1833	−0.483930	−0.241965	+	0.970285i	$0.577792\pi$
$860$	−5.21110	−0.177697
$861$	0	0
$862$	−20.2389	−0.689338
$863$	−23.4500	−0.798246	−0.399123	−	0.916897i	$-0.630685\pi$
$863$	−23.4500	−0.798246	−0.399123	+	0.916897i	$0.630685\pi$
$864$	0	0
$865$	23.3028	0.792318
$866$	−34.9083	−1.18623
$867$	0	0
$868$	−0.880571	−0.0298885
$869$	76.4777	2.59433
$870$	0	0
$871$	1.21110	0.0410366
$872$	19.5139	0.660823
$873$	0	0
$874$	−4.90833	−0.166027
$875$	0.302776	0.0102357
$876$	0	0
$877$	49.1749	1.66052	0.830260	−	0.557376i	$-0.188192\pi$
$877$	49.1749	1.66052	0.830260	+	0.557376i	$0.188192\pi$
$878$	−18.3028	−0.617689
$879$	0	0
$880$	−5.30278	−0.178757
$881$	31.2666	1.05340	0.526700	−	0.850052i	$-0.323429\pi$
$881$	31.2666	1.05340	0.526700	+	0.850052i	$0.323429\pi$
$882$	0	0
$883$	40.7250	1.37050	0.685252	−	0.728306i	$-0.259692\pi$
$883$	40.7250	1.37050	0.685252	+	0.728306i	$0.259692\pi$
$884$	−1.18335	−0.0398002
$885$	0	0
$886$	−35.5139	−1.19311
$887$	15.6333	0.524915	0.262458	−	0.964944i	$-0.415467\pi$
$887$	15.6333	0.524915	0.262458	+	0.964944i	$0.415467\pi$
$888$	0	0
$889$	3.57779	0.119995
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	22.6056	0.756466
$894$	0	0
$895$	16.6056	0.555062
$896$	−0.302776	−0.0101150
$897$	0	0
$898$	12.9083	0.430756
$899$	−13.3944	−0.446730
$900$	0	0
$901$	−12.5500	−0.418102
$902$	52.5416	1.74945
$903$	0	0
$904$	−12.4222	−0.413156
$905$	8.11943	0.269899
$906$	0	0
$907$	−30.6611	−1.01808	−0.509042	−	0.860742i	$-0.670000\pi$
$907$	−30.6611	−1.01808	−0.509042	+	0.860742i	$0.670000\pi$
$908$	7.39445	0.245393
$909$	0	0
$910$	−0.0916731	−0.00303893
$911$	25.8167	0.855344	0.427672	−	0.903934i	$-0.359334\pi$
$911$	25.8167	0.855344	0.427672	+	0.903934i	$0.359334\pi$
$912$	0	0
$913$	17.0278	0.563536
$914$	−3.57779	−0.118343
$915$	0	0
$916$	2.00000	0.0660819
$917$	0.972244	0.0321063
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−31.8167	−1.04783
$923$	−3.84441	−0.126540
$924$	0	0
$925$	8.00000	0.263038
$926$	−25.6333	−0.842363
$927$	0	0
$928$	−4.60555	−0.151185
$929$	57.6333	1.89089	0.945444	−	0.325785i	$-0.105629\pi$
$929$	57.6333	1.89089	0.945444	+	0.325785i	$0.105629\pi$
$930$	0	0
$931$	33.9083	1.11130
$932$	−4.18335	−0.137030
$933$	0	0
$934$	−19.8167	−0.648421
$935$	−20.7250	−0.677779
$936$	0	0
$937$	−44.9638	−1.46890	−0.734452	−	0.678660i	$-0.762561\pi$
$937$	−44.9638	−1.46890	−0.734452	+	0.678660i	$0.762561\pi$
$938$	1.21110	0.0395439
$939$	0	0
$940$	4.60555	0.150217
$941$	20.9361	0.682497	0.341248	−	0.939973i	$-0.389150\pi$
$941$	20.9361	0.682497	0.341248	+	0.939973i	$0.389150\pi$
$942$	0	0
$943$	9.90833	0.322660
$944$	10.6056	0.345181
$945$	0	0
$946$	27.6333	0.898436
$947$	−41.9361	−1.36274	−0.681370	−	0.731939i	$-0.738615\pi$
$947$	−41.9361	−1.36274	−0.681370	+	0.731939i	$0.738615\pi$
$948$	0	0
$949$	−4.78890	−0.155454
$950$	−4.90833	−0.159247
$951$	0	0
$952$	−1.18335	−0.0383525
$953$	1.66947	0.0540794	0.0270397	−	0.999634i	$-0.491392\pi$
$953$	1.66947	0.0540794	0.0270397	+	0.999634i	$0.491392\pi$
$954$	0	0
$955$	−1.39445	−0.0451233
$956$	9.21110	0.297908
$957$	0	0
$958$	30.0000	0.969256
$959$	−2.09167	−0.0675436
$960$	0	0
$961$	−22.5416	−0.727150
$962$	−2.42221	−0.0780950
$963$	0	0
$964$	14.4222	0.464508
$965$	−3.81665	−0.122862
$966$	0	0
$967$	−5.39445	−0.173474	−0.0867369	−	0.996231i	$-0.527644\pi$
$967$	−5.39445	−0.173474	−0.0867369	+	0.996231i	$0.527644\pi$
$968$	17.1194	0.550239
$969$	0	0
$970$	−2.69722	−0.0866027
$971$	27.9083	0.895621	0.447810	−	0.894129i	$-0.352204\pi$
$971$	27.9083	0.895621	0.447810	+	0.894129i	$0.352204\pi$
$972$	0	0
$973$	1.63331	0.0523614
$974$	−11.8167	−0.378630
$975$	0	0
$976$	−6.51388	−0.208504
$977$	11.5139	0.368362	0.184181	−	0.982892i	$-0.441037\pi$
$977$	11.5139	0.368362	0.184181	+	0.982892i	$0.441037\pi$
$978$	0	0
$979$	0	0
$980$	6.90833	0.220678
$981$	0	0
$982$	25.8167	0.823843
$983$	19.5416	0.623281	0.311641	−	0.950200i	$-0.399121\pi$
$983$	19.5416	0.623281	0.311641	+	0.950200i	$0.399121\pi$
$984$	0	0
$985$	0.697224	0.0222154
$986$	−18.0000	−0.573237
$987$	0	0
$988$	1.48612	0.0472798
$989$	5.21110	0.165703
$990$	0	0
$991$	24.3305	0.772885	0.386442	−	0.922314i	$-0.373704\pi$
$991$	24.3305	0.772885	0.386442	+	0.922314i	$0.373704\pi$
$992$	2.90833	0.0923395
$993$	0	0
$994$	−3.84441	−0.121937
$995$	−8.42221	−0.267002
$996$	0	0
$997$	−31.2111	−0.988466	−0.494233	−	0.869330i	$-0.664551\pi$
$997$	−31.2111	−0.988466	−0.494233	+	0.869330i	$0.664551\pi$
$998$	11.6333	0.368246
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.w.1.1		2
3.2	odd	2		230.2.a.b.1.2	✓	2
12.11	even	2		1840.2.a.j.1.1		2
15.2	even	4		1150.2.b.f.599.1		4
15.8	even	4		1150.2.b.f.599.4		4
15.14	odd	2		1150.2.a.m.1.1		2
24.5	odd	2		7360.2.a.bc.1.1		2
24.11	even	2		7360.2.a.bu.1.2		2
60.59	even	2		9200.2.a.ca.1.2		2
69.68	even	2		5290.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.b.1.2	✓	2	3.2	odd	2
1150.2.a.m.1.1		2	15.14	odd	2
1150.2.b.f.599.1		4	15.2	even	4
1150.2.b.f.599.4		4	15.8	even	4
1840.2.a.j.1.1		2	12.11	even	2
2070.2.a.w.1.1		2	1.1	even	1	trivial
5290.2.a.j.1.2		2	69.68	even	2
7360.2.a.bc.1.1		2	24.5	odd	2
7360.2.a.bu.1.2		2	24.11	even	2
9200.2.a.ca.1.2		2	60.59	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.302776 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.30278 q^{11} -0.302776 q^{13} -0.302776 q^{14} +1.00000 q^{16} +3.90833 q^{17} -4.90833 q^{19} -1.00000 q^{20} +5.30278 q^{22} +1.00000 q^{23} +1.00000 q^{25} -0.302776 q^{26} -0.302776 q^{28} -4.60555 q^{29} +2.90833 q^{31} +1.00000 q^{32} +3.90833 q^{34} +0.302776 q^{35} +8.00000 q^{37} -4.90833 q^{38} -1.00000 q^{40} +9.90833 q^{41} +5.21110 q^{43} +5.30278 q^{44} +1.00000 q^{46} -4.60555 q^{47} -6.90833 q^{49} +1.00000 q^{50} -0.302776 q^{52} -3.21110 q^{53} -5.30278 q^{55} -0.302776 q^{56} -4.60555 q^{58} +10.6056 q^{59} -6.51388 q^{61} +2.90833 q^{62} +1.00000 q^{64} +0.302776 q^{65} -4.00000 q^{67} +3.90833 q^{68} +0.302776 q^{70} +12.6972 q^{71} +15.8167 q^{73} +8.00000 q^{74} -4.90833 q^{76} -1.60555 q^{77} +14.4222 q^{79} -1.00000 q^{80} +9.90833 q^{82} +3.21110 q^{83} -3.90833 q^{85} +5.21110 q^{86} +5.30278 q^{88} +0.0916731 q^{91} +1.00000 q^{92} -4.60555 q^{94} +4.90833 q^{95} +2.69722 q^{97} -6.90833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 - 2 * q^10 + 7 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 + q^19 - 2 * q^20 + 7 * q^22 + 2 * q^23 + 2 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^29 - 5 * q^31 + 2 * q^32 - 3 * q^34 - 3 * q^35 + 16 * q^37 + q^38 - 2 * q^40 + 9 * q^41 - 4 * q^43 + 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^50 + 3 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^59 + 5 * q^61 - 5 * q^62 + 2 * q^64 - 3 * q^65 - 8 * q^67 - 3 * q^68 - 3 * q^70 + 29 * q^71 + 10 * q^73 + 16 * q^74 + q^76 + 4 * q^77 - 2 * q^80 + 9 * q^82 - 8 * q^83 + 3 * q^85 - 4 * q^86 + 7 * q^88 + 11 * q^91 + 2 * q^92 - 2 * q^94 - q^95 + 9 * q^97 - 3 * q^98

Introduction

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