LMFDB - Embedded newform 2800.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +6.12311 q^{11} -2.00000 q^{13} -1.56155 q^{17} -3.56155 q^{19} -1.56155 q^{21} +1.43845 q^{23} +5.56155 q^{27} +3.43845 q^{29} +9.12311 q^{31} -9.56155 q^{33} -8.80776 q^{37} +3.12311 q^{39} -2.43845 q^{41} +6.56155 q^{43} -8.24621 q^{47} +1.00000 q^{49} +2.43845 q^{51} +1.12311 q^{53} +5.56155 q^{57} -11.3693 q^{59} +11.1231 q^{61} -0.561553 q^{63} +7.87689 q^{67} -2.24621 q^{69} -1.68466 q^{71} -6.43845 q^{73} +6.12311 q^{77} +5.68466 q^{79} -7.00000 q^{81} -1.31534 q^{83} -5.36932 q^{87} +9.80776 q^{89} -2.00000 q^{91} -14.2462 q^{93} -6.00000 q^{97} -3.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} + 3 q^{37} - 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} + 9 q^{51} - 6 q^{53} + 7 q^{57} + 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} + 9 q^{71} - 17 q^{73} + 4 q^{77} - q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} - 4 q^{91} - 12 q^{93} - 12 q^{97} - 11 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 + 3 * q^37 - 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 + 9 * q^51 - 6 * q^53 + 7 * q^57 + 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 + 9 * q^71 - 17 * q^73 + 4 * q^77 - q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 - 4 * q^91 - 12 * q^93 - 12 * q^97 - 11 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	6.12311	1.84619	0.923093	−	0.384577i	$-0.125653\pi$
$11$	6.12311	1.84619	0.923093	+	0.384577i	$0.125653\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.56155	−0.378732	−0.189366	−	0.981907i	$-0.560643\pi$
$17$	−1.56155	−0.378732	−0.189366	+	0.981907i	$0.560643\pi$
$18$	0	0
$19$	−3.56155	−0.817076	−0.408538	−	0.912741i	$-0.633961\pi$
$19$	−3.56155	−0.817076	−0.408538	+	0.912741i	$0.633961\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	3.43845	0.638504	0.319252	−	0.947670i	$-0.396568\pi$
$29$	3.43845	0.638504	0.319252	+	0.947670i	$0.396568\pi$
$30$	0	0
$31$	9.12311	1.63856	0.819279	−	0.573395i	$-0.194374\pi$
$31$	9.12311	1.63856	0.819279	+	0.573395i	$0.194374\pi$
$32$	0	0
$33$	−9.56155	−1.66445
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.80776	−1.44799	−0.723994	−	0.689807i	$-0.757696\pi$
$37$	−8.80776	−1.44799	−0.723994	+	0.689807i	$0.757696\pi$
$38$	0	0
$39$	3.12311	0.500097
$40$	0	0
$41$	−2.43845	−0.380821	−0.190411	−	0.981705i	$-0.560982\pi$
$41$	−2.43845	−0.380821	−0.190411	+	0.981705i	$0.560982\pi$
$42$	0	0
$43$	6.56155	1.00063	0.500314	−	0.865844i	$-0.333218\pi$
$43$	6.56155	1.00063	0.500314	+	0.865844i	$0.333218\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.24621	−1.20283	−0.601417	−	0.798935i	$-0.705397\pi$
$47$	−8.24621	−1.20283	−0.601417	+	0.798935i	$0.705397\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.43845	0.341451
$52$	0	0
$53$	1.12311	0.154270	0.0771352	−	0.997021i	$-0.475423\pi$
$53$	1.12311	0.154270	0.0771352	+	0.997021i	$0.475423\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.56155	0.736646
$58$	0	0
$59$	−11.3693	−1.48016	−0.740079	−	0.672519i	$-0.765212\pi$
$59$	−11.3693	−1.48016	−0.740079	+	0.672519i	$0.765212\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	−0.561553	−0.0707490
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.87689	0.962316	0.481158	−	0.876634i	$-0.340216\pi$
$67$	7.87689	0.962316	0.481158	+	0.876634i	$0.340216\pi$
$68$	0	0
$69$	−2.24621	−0.270412
$70$	0	0
$71$	−1.68466	−0.199932	−0.0999661	−	0.994991i	$-0.531873\pi$
$71$	−1.68466	−0.199932	−0.0999661	+	0.994991i	$0.531873\pi$
$72$	0	0
$73$	−6.43845	−0.753563	−0.376782	−	0.926302i	$-0.622969\pi$
$73$	−6.43845	−0.753563	−0.376782	+	0.926302i	$0.622969\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.12311	0.697793
$78$	0	0
$79$	5.68466	0.639574	0.319787	−	0.947489i	$-0.396389\pi$
$79$	5.68466	0.639574	0.319787	+	0.947489i	$0.396389\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−1.31534	−0.144377	−0.0721887	−	0.997391i	$-0.522998\pi$
$83$	−1.31534	−0.144377	−0.0721887	+	0.997391i	$0.522998\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.36932	−0.575651
$88$	0	0
$89$	9.80776	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$89$	9.80776	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−14.2462	−1.47726
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−3.43845	−0.345577
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−1.12311	−0.110663	−0.0553314	−	0.998468i	$-0.517622\pi$
$103$	−1.12311	−0.110663	−0.0553314	+	0.998468i	$0.517622\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.80776	0.754805	0.377403	−	0.926049i	$-0.376817\pi$
$107$	7.80776	0.754805	0.377403	+	0.926049i	$0.376817\pi$
$108$	0	0
$109$	19.9309	1.90903	0.954516	−	0.298161i	$-0.0963733\pi$
$109$	19.9309	1.90903	0.954516	+	0.298161i	$0.0963733\pi$
$110$	0	0
$111$	13.7538	1.30545
$112$	0	0
$113$	−17.2462	−1.62239	−0.811194	−	0.584778i	$-0.801182\pi$
$113$	−17.2462	−1.62239	−0.811194	+	0.584778i	$0.801182\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.12311	0.103831
$118$	0	0
$119$	−1.56155	−0.143147
$120$	0	0
$121$	26.4924	2.40840
$122$	0	0
$123$	3.80776	0.343335
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.8078	0.959034	0.479517	−	0.877533i	$-0.340812\pi$
$127$	10.8078	0.959034	0.479517	+	0.877533i	$0.340812\pi$
$128$	0	0
$129$	−10.2462	−0.902129
$130$	0	0
$131$	18.2462	1.59418	0.797089	−	0.603861i	$-0.206372\pi$
$131$	18.2462	1.59418	0.797089	+	0.603861i	$0.206372\pi$
$132$	0	0
$133$	−3.56155	−0.308826
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.6847	1.25460	0.627298	−	0.778780i	$-0.284161\pi$
$137$	14.6847	1.25460	0.627298	+	0.778780i	$0.284161\pi$
$138$	0	0
$139$	7.31534	0.620479	0.310240	−	0.950658i	$-0.399591\pi$
$139$	7.31534	0.620479	0.310240	+	0.950658i	$0.399591\pi$
$140$	0	0
$141$	12.8769	1.08443
$142$	0	0
$143$	−12.2462	−1.02408
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.56155	−0.128795
$148$	0	0
$149$	13.4384	1.10092	0.550460	−	0.834861i	$-0.314452\pi$
$149$	13.4384	1.10092	0.550460	+	0.834861i	$0.314452\pi$
$150$	0	0
$151$	1.43845	0.117059	0.0585296	−	0.998286i	$-0.481359\pi$
$151$	1.43845	0.117059	0.0585296	+	0.998286i	$0.481359\pi$
$152$	0	0
$153$	0.876894	0.0708927
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.49242	0.677769	0.338885	−	0.940828i	$-0.389950\pi$
$157$	8.49242	0.677769	0.338885	+	0.940828i	$0.389950\pi$
$158$	0	0
$159$	−1.75379	−0.139084
$160$	0	0
$161$	1.43845	0.113366
$162$	0	0
$163$	14.9309	1.16948	0.584738	−	0.811222i	$-0.301197\pi$
$163$	14.9309	1.16948	0.584738	+	0.811222i	$0.301197\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.6155	1.67266	0.836330	−	0.548227i	$-0.184697\pi$
$167$	21.6155	1.67266	0.836330	+	0.548227i	$0.184697\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	18.4924	1.40595	0.702976	−	0.711213i	$-0.251854\pi$
$173$	18.4924	1.40595	0.702976	+	0.711213i	$0.251854\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.7538	1.33446
$178$	0	0
$179$	20.6847	1.54604	0.773022	−	0.634379i	$-0.218744\pi$
$179$	20.6847	1.54604	0.773022	+	0.634379i	$0.218744\pi$
$180$	0	0
$181$	−11.1231	−0.826774	−0.413387	−	0.910555i	$-0.635654\pi$
$181$	−11.1231	−0.826774	−0.413387	+	0.910555i	$0.635654\pi$
$182$	0	0
$183$	−17.3693	−1.28398
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.56155	−0.699210
$188$	0	0
$189$	5.56155	0.404543
$190$	0	0
$191$	−25.3693	−1.83566	−0.917830	−	0.396974i	$-0.870060\pi$
$191$	−25.3693	−1.83566	−0.917830	+	0.396974i	$0.870060\pi$
$192$	0	0
$193$	20.6155	1.48394	0.741969	−	0.670434i	$-0.233892\pi$
$193$	20.6155	1.48394	0.741969	+	0.670434i	$0.233892\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.561553	−0.0400090	−0.0200045	−	0.999800i	$-0.506368\pi$
$197$	−0.561553	−0.0400090	−0.0200045	+	0.999800i	$0.506368\pi$
$198$	0	0
$199$	4.87689	0.345714	0.172857	−	0.984947i	$-0.444700\pi$
$199$	4.87689	0.345714	0.172857	+	0.984947i	$0.444700\pi$
$200$	0	0
$201$	−12.3002	−0.867588
$202$	0	0
$203$	3.43845	0.241332
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.807764	−0.0561435
$208$	0	0
$209$	−21.8078	−1.50847
$210$	0	0
$211$	−6.43845	−0.443241	−0.221620	−	0.975133i	$-0.571135\pi$
$211$	−6.43845	−0.443241	−0.221620	+	0.975133i	$0.571135\pi$
$212$	0	0
$213$	2.63068	0.180251
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.12311	0.619317
$218$	0	0
$219$	10.0540	0.679385
$220$	0	0
$221$	3.12311	0.210083
$222$	0	0
$223$	−25.3693	−1.69886	−0.849428	−	0.527705i	$-0.823053\pi$
$223$	−25.3693	−1.69886	−0.849428	+	0.527705i	$0.823053\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3693	0.754608	0.377304	−	0.926089i	$-0.376851\pi$
$227$	11.3693	0.754608	0.377304	+	0.926089i	$0.376851\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−9.56155	−0.629104
$232$	0	0
$233$	0.561553	0.0367885	0.0183943	−	0.999831i	$-0.494145\pi$
$233$	0.561553	0.0367885	0.0183943	+	0.999831i	$0.494145\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.87689	−0.576616
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−12.0540	−0.776465	−0.388232	−	0.921562i	$-0.626914\pi$
$241$	−12.0540	−0.776465	−0.388232	+	0.921562i	$0.626914\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.12311	0.453232
$248$	0	0
$249$	2.05398	0.130165
$250$	0	0
$251$	24.6847	1.55808	0.779041	−	0.626973i	$-0.215706\pi$
$251$	24.6847	1.55808	0.779041	+	0.626973i	$0.215706\pi$
$252$	0	0
$253$	8.80776	0.553739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.2462	−0.763898	−0.381949	−	0.924183i	$-0.624747\pi$
$257$	−12.2462	−0.763898	−0.381949	+	0.924183i	$0.624747\pi$
$258$	0	0
$259$	−8.80776	−0.547288
$260$	0	0
$261$	−1.93087	−0.119518
$262$	0	0
$263$	1.68466	0.103880	0.0519402	−	0.998650i	$-0.483459\pi$
$263$	1.68466	0.103880	0.0519402	+	0.998650i	$0.483459\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.3153	−0.937284
$268$	0	0
$269$	28.2462	1.72220	0.861101	−	0.508434i	$-0.169775\pi$
$269$	28.2462	1.72220	0.861101	+	0.508434i	$0.169775\pi$
$270$	0	0
$271$	0.246211	0.0149563	0.00747813	−	0.999972i	$-0.497620\pi$
$271$	0.246211	0.0149563	0.00747813	+	0.999972i	$0.497620\pi$
$272$	0	0
$273$	3.12311	0.189019
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.4924	−1.35144	−0.675719	−	0.737159i	$-0.736167\pi$
$277$	−22.4924	−1.35144	−0.675719	+	0.737159i	$0.736167\pi$
$278$	0	0
$279$	−5.12311	−0.306712
$280$	0	0
$281$	7.43845	0.443741	0.221870	−	0.975076i	$-0.428784\pi$
$281$	7.43845	0.443741	0.221870	+	0.975076i	$0.428784\pi$
$282$	0	0
$283$	−1.31534	−0.0781889	−0.0390945	−	0.999236i	$-0.512447\pi$
$283$	−1.31534	−0.0781889	−0.0390945	+	0.999236i	$0.512447\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.43845	−0.143937
$288$	0	0
$289$	−14.5616	−0.856562
$290$	0	0
$291$	9.36932	0.549239
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	34.0540	1.97601
$298$	0	0
$299$	−2.87689	−0.166375
$300$	0	0
$301$	6.56155	0.378202
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.6847	0.952244	0.476122	−	0.879379i	$-0.342042\pi$
$307$	16.6847	0.952244	0.476122	+	0.879379i	$0.342042\pi$
$308$	0	0
$309$	1.75379	0.0997696
$310$	0	0
$311$	−11.1231	−0.630733	−0.315367	−	0.948970i	$-0.602128\pi$
$311$	−11.1231	−0.630733	−0.315367	+	0.948970i	$0.602128\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.31534	0.467036	0.233518	−	0.972352i	$-0.424976\pi$
$317$	8.31534	0.467036	0.233518	+	0.972352i	$0.424976\pi$
$318$	0	0
$319$	21.0540	1.17880
$320$	0	0
$321$	−12.1922	−0.680504
$322$	0	0
$323$	5.56155	0.309453
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−31.1231	−1.72111
$328$	0	0
$329$	−8.24621	−0.454628
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	4.94602	0.271040
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.68466	−0.364137	−0.182068	−	0.983286i	$-0.558279\pi$
$337$	−6.68466	−0.364137	−0.182068	+	0.983286i	$0.558279\pi$
$338$	0	0
$339$	26.9309	1.46268
$340$	0	0
$341$	55.8617	3.02508
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.6155	−0.999334	−0.499667	−	0.866218i	$-0.666544\pi$
$347$	−18.6155	−0.999334	−0.499667	+	0.866218i	$0.666544\pi$
$348$	0	0
$349$	−21.1231	−1.13069	−0.565347	−	0.824853i	$-0.691258\pi$
$349$	−21.1231	−1.13069	−0.565347	+	0.824853i	$0.691258\pi$
$350$	0	0
$351$	−11.1231	−0.593707
$352$	0	0
$353$	28.2462	1.50339	0.751697	−	0.659509i	$-0.229236\pi$
$353$	28.2462	1.50339	0.751697	+	0.659509i	$0.229236\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.43845	0.129056
$358$	0	0
$359$	2.31534	0.122199	0.0610995	−	0.998132i	$-0.480539\pi$
$359$	2.31534	0.122199	0.0610995	+	0.998132i	$0.480539\pi$
$360$	0	0
$361$	−6.31534	−0.332386
$362$	0	0
$363$	−41.3693	−2.17133
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.2462	1.78764	0.893819	−	0.448428i	$-0.148016\pi$
$367$	34.2462	1.78764	0.893819	+	0.448428i	$0.148016\pi$
$368$	0	0
$369$	1.36932	0.0712838
$370$	0	0
$371$	1.12311	0.0583087
$372$	0	0
$373$	23.0540	1.19369	0.596845	−	0.802357i	$-0.296421\pi$
$373$	23.0540	1.19369	0.596845	+	0.802357i	$0.296421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.87689	−0.354178
$378$	0	0
$379$	−32.8617	−1.68799	−0.843997	−	0.536348i	$-0.819804\pi$
$379$	−32.8617	−1.68799	−0.843997	+	0.536348i	$0.819804\pi$
$380$	0	0
$381$	−16.8769	−0.864629
$382$	0	0
$383$	−3.75379	−0.191810	−0.0959048	−	0.995391i	$-0.530574\pi$
$383$	−3.75379	−0.191810	−0.0959048	+	0.995391i	$0.530574\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.68466	−0.187302
$388$	0	0
$389$	6.56155	0.332684	0.166342	−	0.986068i	$-0.446804\pi$
$389$	6.56155	0.332684	0.166342	+	0.986068i	$0.446804\pi$
$390$	0	0
$391$	−2.24621	−0.113596
$392$	0	0
$393$	−28.4924	−1.43725
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.50758	−0.276417	−0.138209	−	0.990403i	$-0.544134\pi$
$397$	−5.50758	−0.276417	−0.138209	+	0.990403i	$0.544134\pi$
$398$	0	0
$399$	5.56155	0.278426
$400$	0	0
$401$	1.49242	0.0745280	0.0372640	−	0.999305i	$-0.488136\pi$
$401$	1.49242	0.0745280	0.0372640	+	0.999305i	$0.488136\pi$
$402$	0	0
$403$	−18.2462	−0.908909
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−53.9309	−2.67325
$408$	0	0
$409$	−25.8078	−1.27611	−0.638056	−	0.769990i	$-0.720261\pi$
$409$	−25.8078	−1.27611	−0.638056	+	0.769990i	$0.720261\pi$
$410$	0	0
$411$	−22.9309	−1.13110
$412$	0	0
$413$	−11.3693	−0.559448
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.4233	−0.559401
$418$	0	0
$419$	−1.80776	−0.0883151	−0.0441575	−	0.999025i	$-0.514060\pi$
$419$	−1.80776	−0.0883151	−0.0441575	+	0.999025i	$0.514060\pi$
$420$	0	0
$421$	33.9309	1.65369	0.826845	−	0.562430i	$-0.190134\pi$
$421$	33.9309	1.65369	0.826845	+	0.562430i	$0.190134\pi$
$422$	0	0
$423$	4.63068	0.225152
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.1231	0.538285
$428$	0	0
$429$	19.1231	0.923272
$430$	0	0
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	−34.5464	−1.66019	−0.830097	−	0.557619i	$-0.811715\pi$
$433$	−34.5464	−1.66019	−0.830097	+	0.557619i	$0.811715\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.12311	−0.245071
$438$	0	0
$439$	−1.12311	−0.0536029	−0.0268015	−	0.999641i	$-0.508532\pi$
$439$	−1.12311	−0.0536029	−0.0268015	+	0.999641i	$0.508532\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	18.4384	0.876037	0.438019	−	0.898966i	$-0.355680\pi$
$443$	18.4384	0.876037	0.438019	+	0.898966i	$0.355680\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−20.9848	−0.992549
$448$	0	0
$449$	−14.3693	−0.678130	−0.339065	−	0.940763i	$-0.610111\pi$
$449$	−14.3693	−0.678130	−0.339065	+	0.940763i	$0.610111\pi$
$450$	0	0
$451$	−14.9309	−0.703067
$452$	0	0
$453$	−2.24621	−0.105536
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.1080	1.73584	0.867918	−	0.496707i	$-0.165458\pi$
$457$	37.1080	1.73584	0.867918	+	0.496707i	$0.165458\pi$
$458$	0	0
$459$	−8.68466	−0.405365
$460$	0	0
$461$	−26.4924	−1.23388	−0.616938	−	0.787012i	$-0.711627\pi$
$461$	−26.4924	−1.23388	−0.616938	+	0.787012i	$0.711627\pi$
$462$	0	0
$463$	12.4924	0.580572	0.290286	−	0.956940i	$-0.406250\pi$
$463$	12.4924	0.580572	0.290286	+	0.956940i	$0.406250\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.1231	−1.53275	−0.766377	−	0.642391i	$-0.777943\pi$
$467$	−33.1231	−1.53275	−0.766377	+	0.642391i	$0.777943\pi$
$468$	0	0
$469$	7.87689	0.363721
$470$	0	0
$471$	−13.2614	−0.611052
$472$	0	0
$473$	40.1771	1.84734
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.630683	−0.0288770
$478$	0	0
$479$	−8.73863	−0.399278	−0.199639	−	0.979869i	$-0.563977\pi$
$479$	−8.73863	−0.399278	−0.199639	+	0.979869i	$0.563977\pi$
$480$	0	0
$481$	17.6155	0.803199
$482$	0	0
$483$	−2.24621	−0.102206
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.4384	1.42461	0.712306	−	0.701869i	$-0.247651\pi$
$487$	31.4384	1.42461	0.712306	+	0.701869i	$0.247651\pi$
$488$	0	0
$489$	−23.3153	−1.05436
$490$	0	0
$491$	−8.31534	−0.375266	−0.187633	−	0.982239i	$-0.560082\pi$
$491$	−8.31534	−0.375266	−0.187633	+	0.982239i	$0.560082\pi$
$492$	0	0
$493$	−5.36932	−0.241822
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.68466	−0.0755673
$498$	0	0
$499$	−16.4924	−0.738302	−0.369151	−	0.929369i	$-0.620352\pi$
$499$	−16.4924	−0.738302	−0.369151	+	0.929369i	$0.620352\pi$
$500$	0	0
$501$	−33.7538	−1.50801
$502$	0	0
$503$	36.2462	1.61614	0.808069	−	0.589087i	$-0.200513\pi$
$503$	36.2462	1.61614	0.808069	+	0.589087i	$0.200513\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.0540	0.624159
$508$	0	0
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−6.43845	−0.284820
$512$	0	0
$513$	−19.8078	−0.874534
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−50.4924	−2.22065
$518$	0	0
$519$	−28.8769	−1.26755
$520$	0	0
$521$	−31.5616	−1.38274	−0.691368	−	0.722502i	$-0.742992\pi$
$521$	−31.5616	−1.38274	−0.691368	+	0.722502i	$0.742992\pi$
$522$	0	0
$523$	40.6847	1.77902	0.889508	−	0.456920i	$-0.151047\pi$
$523$	40.6847	1.77902	0.889508	+	0.456920i	$0.151047\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.2462	−0.620575
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	6.38447	0.277062
$532$	0	0
$533$	4.87689	0.211242
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−32.3002	−1.39386
$538$	0	0
$539$	6.12311	0.263741
$540$	0	0
$541$	−20.4233	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$541$	−20.4233	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$542$	0	0
$543$	17.3693	0.745389
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15.7386	−0.672935	−0.336468	−	0.941695i	$-0.609232\pi$
$547$	−15.7386	−0.672935	−0.336468	+	0.941695i	$0.609232\pi$
$548$	0	0
$549$	−6.24621	−0.266582
$550$	0	0
$551$	−12.2462	−0.521706
$552$	0	0
$553$	5.68466	0.241736
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.43845	−0.230434	−0.115217	−	0.993340i	$-0.536756\pi$
$557$	−5.43845	−0.230434	−0.115217	+	0.993340i	$0.536756\pi$
$558$	0	0
$559$	−13.1231	−0.555048
$560$	0	0
$561$	14.9309	0.630382
$562$	0	0
$563$	1.12311	0.0473333	0.0236666	−	0.999720i	$-0.492466\pi$
$563$	1.12311	0.0473333	0.0236666	+	0.999720i	$0.492466\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	45.9848	1.92778	0.963892	−	0.266292i	$-0.0857984\pi$
$569$	45.9848	1.92778	0.963892	+	0.266292i	$0.0857984\pi$
$570$	0	0
$571$	−39.6847	−1.66075	−0.830376	−	0.557204i	$-0.811874\pi$
$571$	−39.6847	−1.66075	−0.830376	+	0.557204i	$0.811874\pi$
$572$	0	0
$573$	39.6155	1.65496
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.5464	−1.02188	−0.510940	−	0.859616i	$-0.670703\pi$
$577$	−24.5464	−1.02188	−0.510940	+	0.859616i	$0.670703\pi$
$578$	0	0
$579$	−32.1922	−1.33786
$580$	0	0
$581$	−1.31534	−0.0545696
$582$	0	0
$583$	6.87689	0.284812
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.1771	−1.53446	−0.767231	−	0.641371i	$-0.778366\pi$
$587$	−37.1771	−1.53446	−0.767231	+	0.641371i	$0.778366\pi$
$588$	0	0
$589$	−32.4924	−1.33883
$590$	0	0
$591$	0.876894	0.0360706
$592$	0	0
$593$	−24.6847	−1.01368	−0.506839	−	0.862041i	$-0.669186\pi$
$593$	−24.6847	−1.01368	−0.506839	+	0.862041i	$0.669186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.61553	−0.311683
$598$	0	0
$599$	29.3002	1.19717	0.598587	−	0.801058i	$-0.295729\pi$
$599$	29.3002	1.19717	0.598587	+	0.801058i	$0.295729\pi$
$600$	0	0
$601$	−42.0540	−1.71542	−0.857709	−	0.514136i	$-0.828113\pi$
$601$	−42.0540	−1.71542	−0.857709	+	0.514136i	$0.828113\pi$
$602$	0	0
$603$	−4.42329	−0.180130
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.6307	−0.512664	−0.256332	−	0.966589i	$-0.582514\pi$
$607$	−12.6307	−0.512664	−0.256332	+	0.966589i	$0.582514\pi$
$608$	0	0
$609$	−5.36932	−0.217576
$610$	0	0
$611$	16.4924	0.667212
$612$	0	0
$613$	−5.68466	−0.229601	−0.114801	−	0.993389i	$-0.536623\pi$
$613$	−5.68466	−0.229601	−0.114801	+	0.993389i	$0.536623\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.80776	0.113036	0.0565182	−	0.998402i	$-0.482000\pi$
$617$	2.80776	0.113036	0.0565182	+	0.998402i	$0.482000\pi$
$618$	0	0
$619$	−4.63068	−0.186123	−0.0930614	−	0.995660i	$-0.529665\pi$
$619$	−4.63068	−0.186123	−0.0930614	+	0.995660i	$0.529665\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	9.80776	0.392940
$624$	0	0
$625$	0	0
$626$	0	0
$627$	34.0540	1.35998
$628$	0	0
$629$	13.7538	0.548399
$630$	0	0
$631$	3.05398	0.121577	0.0607884	−	0.998151i	$-0.480639\pi$
$631$	3.05398	0.121577	0.0607884	+	0.998151i	$0.480639\pi$
$632$	0	0
$633$	10.0540	0.399610
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0.946025	0.0374242
$640$	0	0
$641$	−20.5616	−0.812133	−0.406066	−	0.913844i	$-0.633100\pi$
$641$	−20.5616	−0.812133	−0.406066	+	0.913844i	$0.633100\pi$
$642$	0	0
$643$	−46.7386	−1.84319	−0.921596	−	0.388151i	$-0.873114\pi$
$643$	−46.7386	−1.84319	−0.921596	+	0.388151i	$0.873114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.630683	−0.0247947	−0.0123974	−	0.999923i	$-0.503946\pi$
$647$	−0.630683	−0.0247947	−0.0123974	+	0.999923i	$0.503946\pi$
$648$	0	0
$649$	−69.6155	−2.73265
$650$	0	0
$651$	−14.2462	−0.558353
$652$	0	0
$653$	23.8617	0.933782	0.466891	−	0.884315i	$-0.345374\pi$
$653$	23.8617	0.933782	0.466891	+	0.884315i	$0.345374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.61553	0.141055
$658$	0	0
$659$	−14.4384	−0.562442	−0.281221	−	0.959643i	$-0.590739\pi$
$659$	−14.4384	−0.562442	−0.281221	+	0.959643i	$0.590739\pi$
$660$	0	0
$661$	10.4924	0.408108	0.204054	−	0.978960i	$-0.434588\pi$
$661$	10.4924	0.408108	0.204054	+	0.978960i	$0.434588\pi$
$662$	0	0
$663$	−4.87689	−0.189403
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.94602	0.191511
$668$	0	0
$669$	39.6155	1.53162
$670$	0	0
$671$	68.1080	2.62928
$672$	0	0
$673$	−13.5076	−0.520679	−0.260339	−	0.965517i	$-0.583834\pi$
$673$	−13.5076	−0.520679	−0.260339	+	0.965517i	$0.583834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.3693	−1.51309	−0.756543	−	0.653944i	$-0.773113\pi$
$677$	−39.3693	−1.51309	−0.756543	+	0.653944i	$0.773113\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−17.7538	−0.680327
$682$	0	0
$683$	25.8769	0.990152	0.495076	−	0.868850i	$-0.335140\pi$
$683$	25.8769	0.990152	0.495076	+	0.868850i	$0.335140\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.8617	0.834077
$688$	0	0
$689$	−2.24621	−0.0855738
$690$	0	0
$691$	21.5616	0.820240	0.410120	−	0.912032i	$-0.365487\pi$
$691$	21.5616	0.820240	0.410120	+	0.912032i	$0.365487\pi$
$692$	0	0
$693$	−3.43845	−0.130616
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.80776	0.144229
$698$	0	0
$699$	−0.876894	−0.0331672
$700$	0	0
$701$	−30.1080	−1.13716	−0.568581	−	0.822627i	$-0.692507\pi$
$701$	−30.1080	−1.13716	−0.568581	+	0.822627i	$0.692507\pi$
$702$	0	0
$703$	31.3693	1.18312
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.6307	0.474355	0.237178	−	0.971466i	$-0.423778\pi$
$709$	12.6307	0.474355	0.237178	+	0.971466i	$0.423778\pi$
$710$	0	0
$711$	−3.19224	−0.119718
$712$	0	0
$713$	13.1231	0.491464
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.24621	−0.233269
$718$	0	0
$719$	−21.7538	−0.811279	−0.405640	−	0.914033i	$-0.632951\pi$
$719$	−21.7538	−0.811279	−0.405640	+	0.914033i	$0.632951\pi$
$720$	0	0
$721$	−1.12311	−0.0418266
$722$	0	0
$723$	18.8229	0.700032
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.6155	1.46926	0.734629	−	0.678469i	$-0.237356\pi$
$727$	39.6155	1.46926	0.734629	+	0.678469i	$0.237356\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−10.2462	−0.378970
$732$	0	0
$733$	−5.75379	−0.212521	−0.106261	−	0.994338i	$-0.533888\pi$
$733$	−5.75379	−0.212521	−0.106261	+	0.994338i	$0.533888\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.2311	1.77661
$738$	0	0
$739$	28.1771	1.03651	0.518255	−	0.855226i	$-0.326582\pi$
$739$	28.1771	1.03651	0.518255	+	0.855226i	$0.326582\pi$
$740$	0	0
$741$	−11.1231	−0.408617
$742$	0	0
$743$	30.7386	1.12769	0.563846	−	0.825880i	$-0.309321\pi$
$743$	30.7386	1.12769	0.563846	+	0.825880i	$0.309321\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.738634	0.0270252
$748$	0	0
$749$	7.80776	0.285289
$750$	0	0
$751$	−2.63068	−0.0959950	−0.0479975	−	0.998847i	$-0.515284\pi$
$751$	−2.63068	−0.0959950	−0.0479975	+	0.998847i	$0.515284\pi$
$752$	0	0
$753$	−38.5464	−1.40471
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.6847	−0.933525	−0.466762	−	0.884383i	$-0.654580\pi$
$757$	−25.6847	−0.933525	−0.466762	+	0.884383i	$0.654580\pi$
$758$	0	0
$759$	−13.7538	−0.499231
$760$	0	0
$761$	−27.4233	−0.994094	−0.497047	−	0.867724i	$-0.665582\pi$
$761$	−27.4233	−0.994094	−0.497047	+	0.867724i	$0.665582\pi$
$762$	0	0
$763$	19.9309	0.721546
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.7386	0.821044
$768$	0	0
$769$	20.3002	0.732043	0.366022	−	0.930606i	$-0.380720\pi$
$769$	20.3002	0.732043	0.366022	+	0.930606i	$0.380720\pi$
$770$	0	0
$771$	19.1231	0.688702
$772$	0	0
$773$	−29.6155	−1.06520	−0.532598	−	0.846368i	$-0.678784\pi$
$773$	−29.6155	−1.06520	−0.532598	+	0.846368i	$0.678784\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	13.7538	0.493414
$778$	0	0
$779$	8.68466	0.311160
$780$	0	0
$781$	−10.3153	−0.369112
$782$	0	0
$783$	19.1231	0.683404
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.75379	0.347685	0.173843	−	0.984773i	$-0.444382\pi$
$787$	9.75379	0.347685	0.173843	+	0.984773i	$0.444382\pi$
$788$	0	0
$789$	−2.63068	−0.0936548
$790$	0	0
$791$	−17.2462	−0.613205
$792$	0	0
$793$	−22.2462	−0.789986
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.384472	−0.0136187	−0.00680935	−	0.999977i	$-0.502167\pi$
$797$	−0.384472	−0.0136187	−0.00680935	+	0.999977i	$0.502167\pi$
$798$	0	0
$799$	12.8769	0.455552
$800$	0	0
$801$	−5.50758	−0.194601
$802$	0	0
$803$	−39.4233	−1.39122
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−44.1080	−1.55267
$808$	0	0
$809$	−12.4233	−0.436780	−0.218390	−	0.975862i	$-0.570080\pi$
$809$	−12.4233	−0.436780	−0.218390	+	0.975862i	$0.570080\pi$
$810$	0	0
$811$	−2.38447	−0.0837301	−0.0418651	−	0.999123i	$-0.513330\pi$
$811$	−2.38447	−0.0837301	−0.0418651	+	0.999123i	$0.513330\pi$
$812$	0	0
$813$	−0.384472	−0.0134840
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−23.3693	−0.817589
$818$	0	0
$819$	1.12311	0.0392445
$820$	0	0
$821$	−11.8617	−0.413978	−0.206989	−	0.978343i	$-0.566366\pi$
$821$	−11.8617	−0.413978	−0.206989	+	0.978343i	$0.566366\pi$
$822$	0	0
$823$	16.5616	0.577299	0.288650	−	0.957435i	$-0.406794\pi$
$823$	16.5616	0.577299	0.288650	+	0.957435i	$0.406794\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.24621	0.251975	0.125988	−	0.992032i	$-0.459790\pi$
$827$	7.24621	0.251975	0.125988	+	0.992032i	$0.459790\pi$
$828$	0	0
$829$	14.2462	0.494791	0.247396	−	0.968915i	$-0.420425\pi$
$829$	14.2462	0.494791	0.247396	+	0.968915i	$0.420425\pi$
$830$	0	0
$831$	35.1231	1.21841
$832$	0	0
$833$	−1.56155	−0.0541046
$834$	0	0
$835$	0	0
$836$	0	0
$837$	50.7386	1.75378
$838$	0	0
$839$	−7.12311	−0.245917	−0.122958	−	0.992412i	$-0.539238\pi$
$839$	−7.12311	−0.245917	−0.122958	+	0.992412i	$0.539238\pi$
$840$	0	0
$841$	−17.1771	−0.592313
$842$	0	0
$843$	−11.6155	−0.400060
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.4924	0.910290
$848$	0	0
$849$	2.05398	0.0704923
$850$	0	0
$851$	−12.6695	−0.434305
$852$	0	0
$853$	−19.1231	−0.654763	−0.327381	−	0.944892i	$-0.606166\pi$
$853$	−19.1231	−0.654763	−0.327381	+	0.944892i	$0.606166\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0540	−1.43654	−0.718268	−	0.695766i	$-0.755065\pi$
$857$	−42.0540	−1.43654	−0.718268	+	0.695766i	$0.755065\pi$
$858$	0	0
$859$	11.1771	0.381357	0.190679	−	0.981653i	$-0.438931\pi$
$859$	11.1771	0.381357	0.190679	+	0.981653i	$0.438931\pi$
$860$	0	0
$861$	3.80776	0.129768
$862$	0	0
$863$	−3.19224	−0.108665	−0.0543325	−	0.998523i	$-0.517303\pi$
$863$	−3.19224	−0.108665	−0.0543325	+	0.998523i	$0.517303\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	22.7386	0.772244
$868$	0	0
$869$	34.8078	1.18077
$870$	0	0
$871$	−15.7538	−0.533797
$872$	0	0
$873$	3.36932	0.114034
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.6155	0.594834	0.297417	−	0.954748i	$-0.403875\pi$
$877$	17.6155	0.594834	0.297417	+	0.954748i	$0.403875\pi$
$878$	0	0
$879$	−46.8466	−1.58010
$880$	0	0
$881$	8.73863	0.294412	0.147206	−	0.989106i	$-0.452972\pi$
$881$	8.73863	0.294412	0.147206	+	0.989106i	$0.452972\pi$
$882$	0	0
$883$	31.4924	1.05980	0.529902	−	0.848059i	$-0.322229\pi$
$883$	31.4924	1.05980	0.529902	+	0.848059i	$0.322229\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.38447	−0.281523	−0.140762	−	0.990044i	$-0.544955\pi$
$887$	−8.38447	−0.281523	−0.140762	+	0.990044i	$0.544955\pi$
$888$	0	0
$889$	10.8078	0.362481
$890$	0	0
$891$	−42.8617	−1.43592
$892$	0	0
$893$	29.3693	0.982807
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.49242	0.149998
$898$	0	0
$899$	31.3693	1.04623
$900$	0	0
$901$	−1.75379	−0.0584272
$902$	0	0
$903$	−10.2462	−0.340973
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.4233	−0.742917	−0.371458	−	0.928450i	$-0.621142\pi$
$911$	−22.4233	−0.742917	−0.371458	+	0.928450i	$0.621142\pi$
$912$	0	0
$913$	−8.05398	−0.266548
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.2462	0.602543
$918$	0	0
$919$	−38.8078	−1.28015	−0.640075	−	0.768312i	$-0.721097\pi$
$919$	−38.8078	−1.28015	−0.640075	+	0.768312i	$0.721097\pi$
$920$	0	0
$921$	−26.0540	−0.858508
$922$	0	0
$923$	3.36932	0.110902
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.630683	0.0207144
$928$	0	0
$929$	−34.3542	−1.12712	−0.563562	−	0.826074i	$-0.690569\pi$
$929$	−34.3542	−1.12712	−0.563562	+	0.826074i	$0.690569\pi$
$930$	0	0
$931$	−3.56155	−0.116725
$932$	0	0
$933$	17.3693	0.568646
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.3002	1.05520	0.527601	−	0.849493i	$-0.323092\pi$
$937$	32.3002	1.05520	0.527601	+	0.849493i	$0.323092\pi$
$938$	0	0
$939$	34.3542	1.12111
$940$	0	0
$941$	23.3693	0.761818	0.380909	−	0.924613i	$-0.375611\pi$
$941$	23.3693	0.761818	0.380909	+	0.924613i	$0.375611\pi$
$942$	0	0
$943$	−3.50758	−0.114222
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.4924	−0.535932	−0.267966	−	0.963428i	$-0.586351\pi$
$947$	−16.4924	−0.535932	−0.267966	+	0.963428i	$0.586351\pi$
$948$	0	0
$949$	12.8769	0.418002
$950$	0	0
$951$	−12.9848	−0.421062
$952$	0	0
$953$	36.1231	1.17014	0.585071	−	0.810982i	$-0.301067\pi$
$953$	36.1231	1.17014	0.585071	+	0.810982i	$0.301067\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−32.8769	−1.06276
$958$	0	0
$959$	14.6847	0.474192
$960$	0	0
$961$	52.2311	1.68487
$962$	0	0
$963$	−4.38447	−0.141288
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.3693	−1.33035	−0.665174	−	0.746689i	$-0.731643\pi$
$967$	−41.3693	−1.33035	−0.665174	+	0.746689i	$0.731643\pi$
$968$	0	0
$969$	−8.68466	−0.278991
$970$	0	0
$971$	14.6847	0.471253	0.235627	−	0.971844i	$-0.424286\pi$
$971$	14.6847	0.471253	0.235627	+	0.971844i	$0.424286\pi$
$972$	0	0
$973$	7.31534	0.234519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.7538	0.536001	0.268001	−	0.963419i	$-0.413637\pi$
$977$	16.7538	0.536001	0.268001	+	0.963419i	$0.413637\pi$
$978$	0	0
$979$	60.0540	1.91933
$980$	0	0
$981$	−11.1922	−0.357341
$982$	0	0
$983$	28.3542	0.904357	0.452179	−	0.891927i	$-0.350647\pi$
$983$	28.3542	0.904357	0.452179	+	0.891927i	$0.350647\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.8769	0.409876
$988$	0	0
$989$	9.43845	0.300125
$990$	0	0
$991$	−48.6695	−1.54604	−0.773019	−	0.634383i	$-0.781254\pi$
$991$	−48.6695	−1.54604	−0.773019	+	0.634383i	$0.781254\pi$
$992$	0	0
$993$	42.1619	1.33797
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.24621	0.134479	0.0672394	−	0.997737i	$-0.478581\pi$
$997$	4.24621	0.134479	0.0672394	+	0.997737i	$0.478581\pi$
$998$	0	0
$999$	−48.9848	−1.54981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bo.1.1		2
4.3	odd	2		1400.2.a.o.1.2	✓	2
5.2	odd	4		2800.2.g.v.449.3		4
5.3	odd	4		2800.2.g.v.449.2		4
5.4	even	2		2800.2.a.bj.1.2		2
20.3	even	4		1400.2.g.j.449.3		4
20.7	even	4		1400.2.g.j.449.2		4
20.19	odd	2		1400.2.a.q.1.1	yes	2
28.27	even	2		9800.2.a.bx.1.1		2
140.139	even	2		9800.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.o.1.2	✓	2	4.3	odd	2
1400.2.a.q.1.1	yes	2	20.19	odd	2
1400.2.g.j.449.2		4	20.7	even	4
1400.2.g.j.449.3		4	20.3	even	4
2800.2.a.bj.1.2		2	5.4	even	2
2800.2.a.bo.1.1		2	1.1	even	1	trivial
2800.2.g.v.449.2		4	5.3	odd	4
2800.2.g.v.449.3		4	5.2	odd	4
9800.2.a.bt.1.2		2	140.139	even	2
9800.2.a.bx.1.1		2	28.27	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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +6.12311 q^{11} -2.00000 q^{13} -1.56155 q^{17} -3.56155 q^{19} -1.56155 q^{21} +1.43845 q^{23} +5.56155 q^{27} +3.43845 q^{29} +9.12311 q^{31} -9.56155 q^{33} -8.80776 q^{37} +3.12311 q^{39} -2.43845 q^{41} +6.56155 q^{43} -8.24621 q^{47} +1.00000 q^{49} +2.43845 q^{51} +1.12311 q^{53} +5.56155 q^{57} -11.3693 q^{59} +11.1231 q^{61} -0.561553 q^{63} +7.87689 q^{67} -2.24621 q^{69} -1.68466 q^{71} -6.43845 q^{73} +6.12311 q^{77} +5.68466 q^{79} -7.00000 q^{81} -1.31534 q^{83} -5.36932 q^{87} +9.80776 q^{89} -2.00000 q^{91} -14.2462 q^{93} -6.00000 q^{97} -3.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} + 3 q^{37} - 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} + 9 q^{51} - 6 q^{53} + 7 q^{57} + 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} + 9 q^{71} - 17 q^{73} + 4 q^{77} - q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} - 4 q^{91} - 12 q^{93} - 12 q^{97} - 11 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 + 3 * q^37 - 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 + 9 * q^51 - 6 * q^53 + 7 * q^57 + 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 + 9 * q^71 - 17 * q^73 + 4 * q^77 - q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 - 4 * q^91 - 12 * q^93 - 12 * q^97 - 11 * q^99

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