LMFDB - Embedded newform 2800.2.a.bq.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ 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)

$f(q)$	$=$	$q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} +O(q^{10})$	$q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} -2.48486 q^{11} -4.15516 q^{13} +5.76491 q^{17} +1.60975 q^{19} -3.12489 q^{21} -7.28005 q^{23} -11.7649 q^{27} +1.45459 q^{29} +2.24977 q^{31} +7.76491 q^{33} -6.00000 q^{37} +12.9844 q^{39} +11.2800 q^{41} +5.28005 q^{43} -3.45459 q^{47} +1.00000 q^{49} -18.0147 q^{51} -9.21949 q^{53} -5.03028 q^{57} +5.92007 q^{59} +5.35998 q^{61} +6.76491 q^{63} +7.52982 q^{67} +22.7493 q^{69} +4.24977 q^{71} +7.28005 q^{73} -2.48486 q^{77} -16.9844 q^{79} +16.4693 q^{81} +10.1093 q^{83} -4.54541 q^{87} -11.4693 q^{89} -4.15516 q^{91} -7.03028 q^{93} -2.73463 q^{97} -16.8099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + 3 * q^7 + 4 * q^9	$ 3 q - q^{3} + 3 q^{7} + 4 q^{9} - 7 q^{11} - 5 q^{13} + q^{17} - 4 q^{19} - q^{21} - 6 q^{23} - 19 q^{27} + 3 q^{29} - 10 q^{31} + 7 q^{33} - 18 q^{37} + 5 q^{39} + 18 q^{41} - 9 q^{47} + 3 q^{49} - 21 q^{51} - 10 q^{53} - 16 q^{57} - 6 q^{59} + 24 q^{61} + 4 q^{63} - 10 q^{67} + 18 q^{69} - 4 q^{71} + 6 q^{73} - 7 q^{77} - 17 q^{79} + 15 q^{81} - 12 q^{83} - 15 q^{87} - 5 q^{91} - 22 q^{93} + 9 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - q^3 + 3 * q^7 + 4 * q^9 - 7 * q^11 - 5 * q^13 + q^17 - 4 * q^19 - q^21 - 6 * q^23 - 19 * q^27 + 3 * q^29 - 10 * q^31 + 7 * q^33 - 18 * q^37 + 5 * q^39 + 18 * q^41 - 9 * q^47 + 3 * q^49 - 21 * q^51 - 10 * q^53 - 16 * q^57 - 6 * q^59 + 24 * q^61 + 4 * q^63 - 10 * q^67 + 18 * q^69 - 4 * q^71 + 6 * q^73 - 7 * q^77 - 17 * q^79 + 15 * q^81 - 12 * q^83 - 15 * q^87 - 5 * q^91 - 22 * q^93 + 9 * q^97 - 2 * 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
$3$	−3.12489	−1.80415	−0.902077	−	0.431576i	$-0.857958\pi$
$3$	−3.12489	−1.80415	−0.902077	+	0.431576i	$0.857958\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.76491	2.25497
$10$	0	0
$11$	−2.48486	−0.749214	−0.374607	−	0.927184i	$-0.622222\pi$
$11$	−2.48486	−0.749214	−0.374607	+	0.927184i	$0.622222\pi$
$12$	0	0
$13$	−4.15516	−1.15243	−0.576217	−	0.817297i	$-0.695472\pi$
$13$	−4.15516	−1.15243	−0.576217	+	0.817297i	$0.695472\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.76491	1.39820	0.699098	−	0.715026i	$-0.253585\pi$
$17$	5.76491	1.39820	0.699098	+	0.715026i	$0.253585\pi$
$18$	0	0
$19$	1.60975	0.369301	0.184651	−	0.982804i	$-0.440885\pi$
$19$	1.60975	0.369301	0.184651	+	0.982804i	$0.440885\pi$
$20$	0	0
$21$	−3.12489	−0.681906
$22$	0	0
$23$	−7.28005	−1.51799	−0.758997	−	0.651094i	$-0.774310\pi$
$23$	−7.28005	−1.51799	−0.758997	+	0.651094i	$0.774310\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−11.7649	−2.26416
$28$	0	0
$29$	1.45459	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$29$	1.45459	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$30$	0	0
$31$	2.24977	0.404071	0.202035	−	0.979378i	$-0.435244\pi$
$31$	2.24977	0.404071	0.202035	+	0.979378i	$0.435244\pi$
$32$	0	0
$33$	7.76491	1.35170
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	12.9844	2.07917
$40$	0	0
$41$	11.2800	1.76165	0.880824	−	0.473444i	$-0.156990\pi$
$41$	11.2800	1.76165	0.880824	+	0.473444i	$0.156990\pi$
$42$	0	0
$43$	5.28005	0.805200	0.402600	−	0.915376i	$-0.368107\pi$
$43$	5.28005	0.805200	0.402600	+	0.915376i	$0.368107\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.45459	−0.503903	−0.251952	−	0.967740i	$-0.581072\pi$
$47$	−3.45459	−0.503903	−0.251952	+	0.967740i	$0.581072\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−18.0147	−2.52256
$52$	0	0
$53$	−9.21949	−1.26639	−0.633197	−	0.773990i	$-0.718258\pi$
$53$	−9.21949	−1.26639	−0.633197	+	0.773990i	$0.718258\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.03028	−0.666276
$58$	0	0
$59$	5.92007	0.770728	0.385364	−	0.922765i	$-0.374076\pi$
$59$	5.92007	0.770728	0.385364	+	0.922765i	$0.374076\pi$
$60$	0	0
$61$	5.35998	0.686275	0.343137	−	0.939285i	$-0.388510\pi$
$61$	5.35998	0.686275	0.343137	+	0.939285i	$0.388510\pi$
$62$	0	0
$63$	6.76491	0.852298
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.52982	0.919914	0.459957	−	0.887941i	$-0.347865\pi$
$67$	7.52982	0.919914	0.459957	+	0.887941i	$0.347865\pi$
$68$	0	0
$69$	22.7493	2.73870
$70$	0	0
$71$	4.24977	0.504355	0.252178	−	0.967681i	$-0.418853\pi$
$71$	4.24977	0.504355	0.252178	+	0.967681i	$0.418853\pi$
$72$	0	0
$73$	7.28005	0.852065	0.426033	−	0.904708i	$-0.359911\pi$
$73$	7.28005	0.852065	0.426033	+	0.904708i	$0.359911\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.48486	−0.283176
$78$	0	0
$79$	−16.9844	−1.91089	−0.955447	−	0.295162i	$-0.904627\pi$
$79$	−16.9844	−1.91089	−0.955447	+	0.295162i	$0.904627\pi$
$80$	0	0
$81$	16.4693	1.82992
$82$	0	0
$83$	10.1093	1.10964	0.554819	−	0.831971i	$-0.312787\pi$
$83$	10.1093	1.10964	0.554819	+	0.831971i	$0.312787\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.54541	−0.487320
$88$	0	0
$89$	−11.4693	−1.21574	−0.607870	−	0.794037i	$-0.707976\pi$
$89$	−11.4693	−1.21574	−0.607870	+	0.794037i	$0.707976\pi$
$90$	0	0
$91$	−4.15516	−0.435579
$92$	0	0
$93$	−7.03028	−0.729006
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.73463	−0.277660	−0.138830	−	0.990316i	$-0.544334\pi$
$97$	−2.73463	−0.277660	−0.138830	+	0.990316i	$0.544334\pi$
$98$	0	0
$99$	−16.8099	−1.68945
$100$	0	0
$101$	−4.57947	−0.455674	−0.227837	−	0.973699i	$-0.573165\pi$
$101$	−4.57947	−0.455674	−0.227837	+	0.973699i	$0.573165\pi$
$102$	0	0
$103$	2.48486	0.244841	0.122420	−	0.992478i	$-0.460934\pi$
$103$	2.48486	0.244841	0.122420	+	0.992478i	$0.460934\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−9.95413	−0.953433	−0.476716	−	0.879057i	$-0.658173\pi$
$109$	−9.95413	−0.953433	−0.476716	+	0.879057i	$0.658173\pi$
$110$	0	0
$111$	18.7493	1.77961
$112$	0	0
$113$	−18.4995	−1.74029	−0.870145	−	0.492795i	$-0.835975\pi$
$113$	−18.4995	−1.74029	−0.870145	+	0.492795i	$0.835975\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−28.1093	−2.59870
$118$	0	0
$119$	5.76491	0.528468
$120$	0	0
$121$	−4.82546	−0.438678
$122$	0	0
$123$	−35.2489	−3.17828
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.15894	0.635253	0.317627	−	0.948216i	$-0.397114\pi$
$127$	7.15894	0.635253	0.317627	+	0.948216i	$0.397114\pi$
$128$	0	0
$129$	−16.4995	−1.45270
$130$	0	0
$131$	−7.85952	−0.686689	−0.343345	−	0.939209i	$-0.611560\pi$
$131$	−7.85952	−0.686689	−0.343345	+	0.939209i	$0.611560\pi$
$132$	0	0
$133$	1.60975	0.139583
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.5601	−0.902210	−0.451105	−	0.892471i	$-0.648970\pi$
$137$	−10.5601	−0.902210	−0.451105	+	0.892471i	$0.648970\pi$
$138$	0	0
$139$	−9.79897	−0.831137	−0.415569	−	0.909562i	$-0.636417\pi$
$139$	−9.79897	−0.831137	−0.415569	+	0.909562i	$0.636417\pi$
$140$	0	0
$141$	10.7952	0.909119
$142$	0	0
$143$	10.3250	0.863420
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.12489	−0.257736
$148$	0	0
$149$	2.49954	0.204770	0.102385	−	0.994745i	$-0.467353\pi$
$149$	2.49954	0.204770	0.102385	+	0.994745i	$0.467353\pi$
$150$	0	0
$151$	3.76491	0.306384	0.153192	−	0.988196i	$-0.451045\pi$
$151$	3.76491	0.306384	0.153192	+	0.988196i	$0.451045\pi$
$152$	0	0
$153$	38.9991	3.15289
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.67030	−0.452539	−0.226270	−	0.974065i	$-0.572653\pi$
$157$	−5.67030	−0.452539	−0.226270	+	0.974065i	$0.572653\pi$
$158$	0	0
$159$	28.8099	2.28477
$160$	0	0
$161$	−7.28005	−0.573748
$162$	0	0
$163$	10.2498	0.802824	0.401412	−	0.915898i	$-0.368520\pi$
$163$	10.2498	0.802824	0.401412	+	0.915898i	$0.368520\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.95504	−0.228668	−0.114334	−	0.993442i	$-0.536473\pi$
$167$	−2.95504	−0.228668	−0.114334	+	0.993442i	$0.536473\pi$
$168$	0	0
$169$	4.26537	0.328105
$170$	0	0
$171$	10.8898	0.832763
$172$	0	0
$173$	−22.4049	−1.70342	−0.851708	−	0.524017i	$-0.824433\pi$
$173$	−22.4049	−1.70342	−0.851708	+	0.524017i	$0.824433\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.4995	−1.39051
$178$	0	0
$179$	−14.5601	−1.08827	−0.544136	−	0.838997i	$-0.683143\pi$
$179$	−14.5601	−1.08827	−0.544136	+	0.838997i	$0.683143\pi$
$180$	0	0
$181$	−9.67030	−0.718788	−0.359394	−	0.933186i	$-0.617017\pi$
$181$	−9.67030	−0.718788	−0.359394	+	0.933186i	$0.617017\pi$
$182$	0	0
$183$	−16.7493	−1.23814
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−14.3250	−1.04755
$188$	0	0
$189$	−11.7649	−0.855771
$190$	0	0
$191$	−8.01468	−0.579922	−0.289961	−	0.957038i	$-0.593642\pi$
$191$	−8.01468	−0.579922	−0.289961	+	0.957038i	$0.593642\pi$
$192$	0	0
$193$	−3.46927	−0.249723	−0.124862	−	0.992174i	$-0.539849\pi$
$193$	−3.46927	−0.249723	−0.124862	+	0.992174i	$0.539849\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2791	1.15984	0.579920	−	0.814673i	$-0.303084\pi$
$197$	16.2791	1.15984	0.579920	+	0.814673i	$0.303084\pi$
$198$	0	0
$199$	−26.8704	−1.90479	−0.952397	−	0.304862i	$-0.901390\pi$
$199$	−26.8704	−1.90479	−0.952397	+	0.304862i	$0.901390\pi$
$200$	0	0
$201$	−23.5298	−1.65967
$202$	0	0
$203$	1.45459	0.102092
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−49.2489	−3.42303
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−17.2342	−1.18645	−0.593225	−	0.805037i	$-0.702145\pi$
$211$	−17.2342	−1.18645	−0.593225	+	0.805037i	$0.702145\pi$
$212$	0	0
$213$	−13.2800	−0.909934
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.24977	0.152724
$218$	0	0
$219$	−22.7493	−1.53726
$220$	0	0
$221$	−23.9541	−1.61133
$222$	0	0
$223$	−0.235091	−0.0157429	−0.00787143	−	0.999969i	$-0.502506\pi$
$223$	−0.235091	−0.0157429	−0.00787143	+	0.999969i	$0.502506\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.564792	−0.0374865	−0.0187433	−	0.999824i	$-0.505967\pi$
$227$	−0.564792	−0.0374865	−0.0187433	+	0.999824i	$0.505967\pi$
$228$	0	0
$229$	7.11021	0.469856	0.234928	−	0.972013i	$-0.424515\pi$
$229$	7.11021	0.469856	0.234928	+	0.972013i	$0.424515\pi$
$230$	0	0
$231$	7.76491	0.510893
$232$	0	0
$233$	−8.49954	−0.556823	−0.278412	−	0.960462i	$-0.589808\pi$
$233$	−8.49954	−0.556823	−0.278412	+	0.960462i	$0.589808\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	53.0743	3.44755
$238$	0	0
$239$	7.26537	0.469958	0.234979	−	0.972001i	$-0.424498\pi$
$239$	7.26537	0.469958	0.234979	+	0.972001i	$0.424498\pi$
$240$	0	0
$241$	7.28005	0.468949	0.234475	−	0.972122i	$-0.424663\pi$
$241$	7.28005	0.468949	0.234475	+	0.972122i	$0.424663\pi$
$242$	0	0
$243$	−16.1698	−1.03730
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.68876	−0.425596
$248$	0	0
$249$	−31.5904	−2.00196
$250$	0	0
$251$	−22.9192	−1.44664	−0.723322	−	0.690511i	$-0.757386\pi$
$251$	−22.9192	−1.44664	−0.723322	+	0.690511i	$0.757386\pi$
$252$	0	0
$253$	18.0899	1.13730
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.719953	−0.0449094	−0.0224547	−	0.999748i	$-0.507148\pi$
$257$	−0.719953	−0.0449094	−0.0224547	+	0.999748i	$0.507148\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	9.84014	0.609089
$262$	0	0
$263$	−18.5601	−1.14446	−0.572232	−	0.820092i	$-0.693922\pi$
$263$	−18.5601	−1.14446	−0.572232	+	0.820092i	$0.693922\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	35.8401	2.19338
$268$	0	0
$269$	−22.1698	−1.35172	−0.675860	−	0.737030i	$-0.736227\pi$
$269$	−22.1698	−1.35172	−0.675860	+	0.737030i	$0.736227\pi$
$270$	0	0
$271$	7.87890	0.478609	0.239304	−	0.970945i	$-0.423081\pi$
$271$	7.87890	0.478609	0.239304	+	0.970945i	$0.423081\pi$
$272$	0	0
$273$	12.9844	0.785852
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.78051	0.167064	0.0835322	−	0.996505i	$-0.473380\pi$
$277$	2.78051	0.167064	0.0835322	+	0.996505i	$0.473380\pi$
$278$	0	0
$279$	15.2195	0.911167
$280$	0	0
$281$	−23.7044	−1.41408	−0.707042	−	0.707172i	$-0.749971\pi$
$281$	−23.7044	−1.41408	−0.707042	+	0.707172i	$0.749971\pi$
$282$	0	0
$283$	26.8439	1.59571	0.797853	−	0.602852i	$-0.205969\pi$
$283$	26.8439	1.59571	0.797853	+	0.602852i	$0.205969\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.2800	0.665840
$288$	0	0
$289$	16.2342	0.954951
$290$	0	0
$291$	8.54541	0.500941
$292$	0	0
$293$	9.93475	0.580394	0.290197	−	0.956967i	$-0.406279\pi$
$293$	9.93475	0.580394	0.290197	+	0.956967i	$0.406279\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	29.2342	1.69634
$298$	0	0
$299$	30.2498	1.74939
$300$	0	0
$301$	5.28005	0.304337
$302$	0	0
$303$	14.3103	0.822107
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.0946	−1.83174	−0.915868	−	0.401479i	$-0.868496\pi$
$307$	−32.0946	−1.83174	−0.915868	+	0.401479i	$0.868496\pi$
$308$	0	0
$309$	−7.76491	−0.441730
$310$	0	0
$311$	10.0606	0.570482	0.285241	−	0.958456i	$-0.407926\pi$
$311$	10.0606	0.570482	0.285241	+	0.958456i	$0.407926\pi$
$312$	0	0
$313$	−20.8245	−1.17707	−0.588536	−	0.808471i	$-0.700296\pi$
$313$	−20.8245	−1.17707	−0.588536	+	0.808471i	$0.700296\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.7502	−0.884621	−0.442311	−	0.896862i	$-0.645841\pi$
$317$	−15.7502	−0.884621	−0.442311	+	0.896862i	$0.645841\pi$
$318$	0	0
$319$	−3.61445	−0.202370
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.28005	0.516356
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.1055	1.72014
$328$	0	0
$329$	−3.45459	−0.190457
$330$	0	0
$331$	25.7796	1.41697	0.708487	−	0.705724i	$-0.249378\pi$
$331$	25.7796	1.41697	0.708487	+	0.705724i	$0.249378\pi$
$332$	0	0
$333$	−40.5895	−2.22429
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.0605522	−0.00329849	−0.00164924	−	0.999999i	$-0.500525\pi$
$337$	−0.0605522	−0.00329849	−0.00164924	+	0.999999i	$0.500525\pi$
$338$	0	0
$339$	57.8089	3.13975
$340$	0	0
$341$	−5.59037	−0.302736
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9310	−1.12363	−0.561817	−	0.827262i	$-0.689897\pi$
$347$	−20.9310	−1.12363	−0.561817	+	0.827262i	$0.689897\pi$
$348$	0	0
$349$	5.48108	0.293396	0.146698	−	0.989181i	$-0.453136\pi$
$349$	5.48108	0.293396	0.146698	+	0.989181i	$0.453136\pi$
$350$	0	0
$351$	48.8851	2.60929
$352$	0	0
$353$	26.9239	1.43301	0.716506	−	0.697581i	$-0.245740\pi$
$353$	26.9239	1.43301	0.716506	+	0.697581i	$0.245740\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.0147	−0.953438
$358$	0	0
$359$	−6.06055	−0.319864	−0.159932	−	0.987128i	$-0.551127\pi$
$359$	−6.06055	−0.319864	−0.159932	+	0.987128i	$0.551127\pi$
$360$	0	0
$361$	−16.4087	−0.863616
$362$	0	0
$363$	15.0790	0.791443
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.7034	1.60271	0.801353	−	0.598191i	$-0.204114\pi$
$367$	30.7034	1.60271	0.801353	+	0.598191i	$0.204114\pi$
$368$	0	0
$369$	76.3085	3.97246
$370$	0	0
$371$	−9.21949	−0.478652
$372$	0	0
$373$	10.8099	0.559714	0.279857	−	0.960042i	$-0.409713\pi$
$373$	10.8099	0.559714	0.279857	+	0.960042i	$0.409713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.04404	−0.311284
$378$	0	0
$379$	9.28005	0.476684	0.238342	−	0.971181i	$-0.423396\pi$
$379$	9.28005	0.476684	0.238342	+	0.971181i	$0.423396\pi$
$380$	0	0
$381$	−22.3709	−1.14609
$382$	0	0
$383$	−5.28005	−0.269798	−0.134899	−	0.990859i	$-0.543071\pi$
$383$	−5.28005	−0.269798	−0.134899	+	0.990859i	$0.543071\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	35.7190	1.81570
$388$	0	0
$389$	5.48395	0.278047	0.139024	−	0.990289i	$-0.455604\pi$
$389$	5.48395	0.278047	0.139024	+	0.990289i	$0.455604\pi$
$390$	0	0
$391$	−41.9688	−2.12245
$392$	0	0
$393$	24.5601	1.23889
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.40493	−0.321454	−0.160727	−	0.986999i	$-0.551384\pi$
$397$	−6.40493	−0.321454	−0.160727	+	0.986999i	$0.551384\pi$
$398$	0	0
$399$	−5.03028	−0.251829
$400$	0	0
$401$	−1.20482	−0.0601656	−0.0300828	−	0.999547i	$-0.509577\pi$
$401$	−1.20482	−0.0601656	−0.0300828	+	0.999547i	$0.509577\pi$
$402$	0	0
$403$	−9.34816	−0.465665
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.9092	0.739020
$408$	0	0
$409$	−2.31032	−0.114238	−0.0571191	−	0.998367i	$-0.518191\pi$
$409$	−2.31032	−0.114238	−0.0571191	+	0.998367i	$0.518191\pi$
$410$	0	0
$411$	32.9991	1.62772
$412$	0	0
$413$	5.92007	0.291308
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.6206	1.49950
$418$	0	0
$419$	19.7384	0.964285	0.482142	−	0.876093i	$-0.339859\pi$
$419$	19.7384	0.964285	0.482142	+	0.876093i	$0.339859\pi$
$420$	0	0
$421$	30.1433	1.46910	0.734548	−	0.678556i	$-0.237394\pi$
$421$	30.1433	1.46910	0.734548	+	0.678556i	$0.237394\pi$
$422$	0	0
$423$	−23.3700	−1.13629
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.35998	0.259387
$428$	0	0
$429$	−32.2645	−1.55774
$430$	0	0
$431$	4.61353	0.222226	0.111113	−	0.993808i	$-0.464558\pi$
$431$	4.61353	0.222226	0.111113	+	0.993808i	$0.464558\pi$
$432$	0	0
$433$	11.4399	0.549767	0.274883	−	0.961478i	$-0.411361\pi$
$433$	11.4399	0.549767	0.274883	+	0.961478i	$0.411361\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.7190	−0.560598
$438$	0	0
$439$	−12.0294	−0.574130	−0.287065	−	0.957911i	$-0.592680\pi$
$439$	−12.0294	−0.574130	−0.287065	+	0.957911i	$0.592680\pi$
$440$	0	0
$441$	6.76491	0.322139
$442$	0	0
$443$	−14.4390	−0.686017	−0.343009	−	0.939332i	$-0.611446\pi$
$443$	−14.4390	−0.686017	−0.343009	+	0.939332i	$0.611446\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.81078	−0.369437
$448$	0	0
$449$	19.2342	0.907717	0.453858	−	0.891074i	$-0.350047\pi$
$449$	19.2342	0.907717	0.453858	+	0.891074i	$0.350047\pi$
$450$	0	0
$451$	−28.0294	−1.31985
$452$	0	0
$453$	−11.7649	−0.552764
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.8780	−1.63152	−0.815762	−	0.578388i	$-0.803682\pi$
$457$	−34.8780	−1.63152	−0.815762	+	0.578388i	$0.803682\pi$
$458$	0	0
$459$	−67.8236	−3.16574
$460$	0	0
$461$	−15.2607	−0.710760	−0.355380	−	0.934722i	$-0.615649\pi$
$461$	−15.2607	−0.710760	−0.355380	+	0.934722i	$0.615649\pi$
$462$	0	0
$463$	24.9991	1.16181	0.580903	−	0.813973i	$-0.302700\pi$
$463$	24.9991	1.16181	0.580903	+	0.813973i	$0.302700\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.06433	0.234349	0.117175	−	0.993111i	$-0.462616\pi$
$467$	5.06433	0.234349	0.117175	+	0.993111i	$0.462616\pi$
$468$	0	0
$469$	7.52982	0.347695
$470$	0	0
$471$	17.7190	0.816450
$472$	0	0
$473$	−13.1202	−0.603267
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−62.3690	−2.85568
$478$	0	0
$479$	−41.8089	−1.91030	−0.955150	−	0.296123i	$-0.904306\pi$
$479$	−41.8089	−1.91030	−0.955150	+	0.296123i	$0.904306\pi$
$480$	0	0
$481$	24.9310	1.13675
$482$	0	0
$483$	22.7493	1.03513
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21.3406	0.967035	0.483517	−	0.875335i	$-0.339359\pi$
$487$	21.3406	0.967035	0.483517	+	0.875335i	$0.339359\pi$
$488$	0	0
$489$	−32.0294	−1.44842
$490$	0	0
$491$	−8.35620	−0.377110	−0.188555	−	0.982063i	$-0.560380\pi$
$491$	−8.35620	−0.377110	−0.188555	+	0.982063i	$0.560380\pi$
$492$	0	0
$493$	8.38555	0.377666
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.24977	0.190628
$498$	0	0
$499$	38.8539	1.73934	0.869670	−	0.493634i	$-0.164332\pi$
$499$	38.8539	1.73934	0.869670	+	0.493634i	$0.164332\pi$
$500$	0	0
$501$	9.23417	0.412552
$502$	0	0
$503$	−17.7044	−0.789398	−0.394699	−	0.918810i	$-0.629151\pi$
$503$	−17.7044	−0.789398	−0.394699	+	0.918810i	$0.629151\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−13.3288	−0.591952
$508$	0	0
$509$	7.11021	0.315154	0.157577	−	0.987507i	$-0.449632\pi$
$509$	7.11021	0.315154	0.157577	+	0.987507i	$0.449632\pi$
$510$	0	0
$511$	7.28005	0.322050
$512$	0	0
$513$	−18.9385	−0.836157
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.58417	0.377531
$518$	0	0
$519$	70.0128	3.07322
$520$	0	0
$521$	18.1892	0.796884	0.398442	−	0.917194i	$-0.369551\pi$
$521$	18.1892	0.796884	0.398442	+	0.917194i	$0.369551\pi$
$522$	0	0
$523$	−13.9882	−0.611661	−0.305830	−	0.952086i	$-0.598934\pi$
$523$	−13.9882	−0.611661	−0.305830	+	0.952086i	$0.598934\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.9697	0.564970
$528$	0	0
$529$	29.9991	1.30431
$530$	0	0
$531$	40.0487	1.73797
$532$	0	0
$533$	−46.8704	−2.03018
$534$	0	0
$535$	0	0
$536$	0	0
$537$	45.4986	1.96341
$538$	0	0
$539$	−2.48486	−0.107031
$540$	0	0
$541$	−20.8245	−0.895317	−0.447659	−	0.894205i	$-0.647742\pi$
$541$	−20.8245	−0.895317	−0.447659	+	0.894205i	$0.647742\pi$
$542$	0	0
$543$	30.2186	1.29680
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.09839	−0.132478	−0.0662388	−	0.997804i	$-0.521100\pi$
$547$	−3.09839	−0.132478	−0.0662388	+	0.997804i	$0.521100\pi$
$548$	0	0
$549$	36.2598	1.54753
$550$	0	0
$551$	2.34152	0.0997519
$552$	0	0
$553$	−16.9844	−0.722250
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.8099	−0.797000	−0.398500	−	0.917168i	$-0.630469\pi$
$557$	−18.8099	−0.797000	−0.398500	+	0.917168i	$0.630469\pi$
$558$	0	0
$559$	−21.9394	−0.927940
$560$	0	0
$561$	44.7640	1.88994
$562$	0	0
$563$	−28.9503	−1.22011	−0.610056	−	0.792358i	$-0.708853\pi$
$563$	−28.9503	−1.22011	−0.610056	+	0.792358i	$0.708853\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16.4693	0.691644
$568$	0	0
$569$	−29.9688	−1.25636	−0.628179	−	0.778069i	$-0.716199\pi$
$569$	−29.9688	−1.25636	−0.628179	+	0.778069i	$0.716199\pi$
$570$	0	0
$571$	−0.280964	−0.0117580	−0.00587898	−	0.999983i	$-0.501871\pi$
$571$	−0.280964	−0.0117580	−0.00587898	+	0.999983i	$0.501871\pi$
$572$	0	0
$573$	25.0450	1.04627
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.9541	1.08048	0.540242	−	0.841510i	$-0.318333\pi$
$577$	25.9541	1.08048	0.540242	+	0.841510i	$0.318333\pi$
$578$	0	0
$579$	10.8411	0.450539
$580$	0	0
$581$	10.1093	0.419404
$582$	0	0
$583$	22.9092	0.948801
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.2304	0.917547	0.458773	−	0.888553i	$-0.348289\pi$
$587$	22.2304	0.917547	0.458773	+	0.888553i	$0.348289\pi$
$588$	0	0
$589$	3.62156	0.149224
$590$	0	0
$591$	−50.8704	−2.09253
$592$	0	0
$593$	35.0743	1.44033	0.720165	−	0.693803i	$-0.244066\pi$
$593$	35.0743	1.44033	0.720165	+	0.693803i	$0.244066\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	83.9670	3.43654
$598$	0	0
$599$	21.7262	0.887707	0.443853	−	0.896099i	$-0.353611\pi$
$599$	21.7262	0.887707	0.443853	+	0.896099i	$0.353611\pi$
$600$	0	0
$601$	−23.9688	−0.977708	−0.488854	−	0.872366i	$-0.662585\pi$
$601$	−23.9688	−0.977708	−0.488854	+	0.872366i	$0.662585\pi$
$602$	0	0
$603$	50.9385	2.07438
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−43.3241	−1.75847	−0.879235	−	0.476388i	$-0.841946\pi$
$607$	−43.3241	−1.75847	−0.879235	+	0.476388i	$0.841946\pi$
$608$	0	0
$609$	−4.54541	−0.184189
$610$	0	0
$611$	14.3544	0.580715
$612$	0	0
$613$	−8.90917	−0.359838	−0.179919	−	0.983681i	$-0.557584\pi$
$613$	−8.90917	−0.359838	−0.179919	+	0.983681i	$0.557584\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.2413	1.09669	0.548347	−	0.836251i	$-0.315257\pi$
$617$	27.2413	1.09669	0.548347	+	0.836251i	$0.315257\pi$
$618$	0	0
$619$	−24.1405	−0.970288	−0.485144	−	0.874434i	$-0.661233\pi$
$619$	−24.1405	−0.970288	−0.485144	+	0.874434i	$0.661233\pi$
$620$	0	0
$621$	85.6491	3.43698
$622$	0	0
$623$	−11.4693	−0.459506
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.4995	0.499184
$628$	0	0
$629$	−34.5895	−1.37917
$630$	0	0
$631$	−8.01468	−0.319059	−0.159530	−	0.987193i	$-0.550998\pi$
$631$	−8.01468	−0.319059	−0.159530	+	0.987193i	$0.550998\pi$
$632$	0	0
$633$	53.8548	2.14054
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.15516	−0.164633
$638$	0	0
$639$	28.7493	1.13731
$640$	0	0
$641$	−49.1495	−1.94129	−0.970645	−	0.240516i	$-0.922683\pi$
$641$	−49.1495	−1.94129	−0.970645	+	0.240516i	$0.922683\pi$
$642$	0	0
$643$	−7.24599	−0.285754	−0.142877	−	0.989740i	$-0.545635\pi$
$643$	−7.24599	−0.285754	−0.142877	+	0.989740i	$0.545635\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.2791	−1.19040	−0.595198	−	0.803579i	$-0.702926\pi$
$647$	−30.2791	−1.19040	−0.595198	+	0.803579i	$0.702926\pi$
$648$	0	0
$649$	−14.7106	−0.577440
$650$	0	0
$651$	−7.03028	−0.275538
$652$	0	0
$653$	7.96881	0.311844	0.155922	−	0.987769i	$-0.450165\pi$
$653$	7.96881	0.311844	0.155922	+	0.987769i	$0.450165\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	49.2489	1.92138
$658$	0	0
$659$	−20.9239	−0.815078	−0.407539	−	0.913188i	$-0.633613\pi$
$659$	−20.9239	−0.815078	−0.407539	+	0.913188i	$0.633613\pi$
$660$	0	0
$661$	46.1992	1.79694	0.898470	−	0.439034i	$-0.144679\pi$
$661$	46.1992	1.79694	0.898470	+	0.439034i	$0.144679\pi$
$662$	0	0
$663$	74.8539	2.90708
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.5895	−0.410025
$668$	0	0
$669$	0.734633	0.0284025
$670$	0	0
$671$	−13.3188	−0.514167
$672$	0	0
$673$	40.3784	1.55647	0.778237	−	0.627970i	$-0.216114\pi$
$673$	40.3784	1.55647	0.778237	+	0.627970i	$0.216114\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.3737	−1.70542	−0.852711	−	0.522383i	$-0.825043\pi$
$677$	−44.3737	−1.70542	−0.852711	+	0.522383i	$0.825043\pi$
$678$	0	0
$679$	−2.73463	−0.104946
$680$	0	0
$681$	1.76491	0.0676315
$682$	0	0
$683$	−17.9394	−0.686434	−0.343217	−	0.939256i	$-0.611517\pi$
$683$	−17.9394	−0.686434	−0.343217	+	0.939256i	$0.611517\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.2186	−0.847692
$688$	0	0
$689$	38.3085	1.45944
$690$	0	0
$691$	12.7905	0.486573	0.243287	−	0.969954i	$-0.421774\pi$
$691$	12.7905	0.486573	0.243287	+	0.969954i	$0.421774\pi$
$692$	0	0
$693$	−16.8099	−0.638554
$694$	0	0
$695$	0	0
$696$	0	0
$697$	65.0284	2.46313
$698$	0	0
$699$	26.5601	1.00460
$700$	0	0
$701$	−19.4234	−0.733611	−0.366806	−	0.930298i	$-0.619549\pi$
$701$	−19.4234	−0.733611	−0.366806	+	0.930298i	$0.619549\pi$
$702$	0	0
$703$	−9.65848	−0.364277
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.57947	−0.172229
$708$	0	0
$709$	−13.4839	−0.506400	−0.253200	−	0.967414i	$-0.581483\pi$
$709$	−13.4839	−0.506400	−0.253200	+	0.967414i	$0.581483\pi$
$710$	0	0
$711$	−114.898	−4.30901
$712$	0	0
$713$	−16.3784	−0.613377
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.7034	−0.847876
$718$	0	0
$719$	−15.3700	−0.573203	−0.286601	−	0.958050i	$-0.592526\pi$
$719$	−15.3700	−0.573203	−0.286601	+	0.958050i	$0.592526\pi$
$720$	0	0
$721$	2.48486	0.0925411
$722$	0	0
$723$	−22.7493	−0.846056
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.1589	0.488038	0.244019	−	0.969770i	$-0.421534\pi$
$727$	13.1589	0.488038	0.244019	+	0.969770i	$0.421534\pi$
$728$	0	0
$729$	1.12110	0.0415224
$730$	0	0
$731$	30.4390	1.12583
$732$	0	0
$733$	10.0265	0.370337	0.185169	−	0.982707i	$-0.440717\pi$
$733$	10.0265	0.370337	0.185169	+	0.982707i	$0.440717\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.7106	−0.689212
$738$	0	0
$739$	−19.2266	−0.707262	−0.353631	−	0.935385i	$-0.615053\pi$
$739$	−19.2266	−0.707262	−0.353631	+	0.935385i	$0.615053\pi$
$740$	0	0
$741$	20.9016	0.767840
$742$	0	0
$743$	−5.21949	−0.191485	−0.0957423	−	0.995406i	$-0.530522\pi$
$743$	−5.21949	−0.191485	−0.0957423	+	0.995406i	$0.530522\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	68.3884	2.50220
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.8548	−1.08942	−0.544709	−	0.838625i	$-0.683360\pi$
$751$	−29.8548	−1.08942	−0.544709	+	0.838625i	$0.683360\pi$
$752$	0	0
$753$	71.6197	2.60997
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0294	0.655288	0.327644	−	0.944801i	$-0.393745\pi$
$757$	18.0294	0.655288	0.327644	+	0.944801i	$0.393745\pi$
$758$	0	0
$759$	−56.5289	−2.05187
$760$	0	0
$761$	8.22041	0.297990	0.148995	−	0.988838i	$-0.452396\pi$
$761$	8.22041	0.297990	0.148995	+	0.988838i	$0.452396\pi$
$762$	0	0
$763$	−9.95413	−0.360364
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.5988	−0.888213
$768$	0	0
$769$	50.6888	1.82788	0.913942	−	0.405845i	$-0.133023\pi$
$769$	50.6888	1.82788	0.913942	+	0.405845i	$0.133023\pi$
$770$	0	0
$771$	2.24977	0.0810235
$772$	0	0
$773$	−6.99622	−0.251637	−0.125818	−	0.992053i	$-0.540156\pi$
$773$	−6.99622	−0.251637	−0.125818	+	0.992053i	$0.540156\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18.7493	0.672628
$778$	0	0
$779$	18.1580	0.650579
$780$	0	0
$781$	−10.5601	−0.377870
$782$	0	0
$783$	−17.1131	−0.611571
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.3444	−0.511322	−0.255661	−	0.966767i	$-0.582293\pi$
$787$	−14.3444	−0.511322	−0.255661	+	0.966767i	$0.582293\pi$
$788$	0	0
$789$	57.9982	2.06479
$790$	0	0
$791$	−18.4995	−0.657768
$792$	0	0
$793$	−22.2716	−0.790887
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.9348	0.493594	0.246797	−	0.969067i	$-0.420622\pi$
$797$	13.9348	0.493594	0.246797	+	0.969067i	$0.420622\pi$
$798$	0	0
$799$	−19.9154	−0.704555
$800$	0	0
$801$	−77.5885	−2.74146
$802$	0	0
$803$	−18.0899	−0.638379
$804$	0	0
$805$	0	0
$806$	0	0
$807$	69.2782	2.43871
$808$	0	0
$809$	−5.58325	−0.196297	−0.0981483	−	0.995172i	$-0.531292\pi$
$809$	−5.58325	−0.196297	−0.0981483	+	0.995172i	$0.531292\pi$
$810$	0	0
$811$	6.57947	0.231036	0.115518	−	0.993305i	$-0.463147\pi$
$811$	6.57947	0.231036	0.115518	+	0.993305i	$0.463147\pi$
$812$	0	0
$813$	−24.6206	−0.863484
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.49954	0.297361
$818$	0	0
$819$	−28.1093	−0.982218
$820$	0	0
$821$	16.3250	0.569747	0.284873	−	0.958565i	$-0.408048\pi$
$821$	16.3250	0.569747	0.284873	+	0.958565i	$0.408048\pi$
$822$	0	0
$823$	12.3491	0.430462	0.215231	−	0.976563i	$-0.430950\pi$
$823$	12.3491	0.430462	0.215231	+	0.976563i	$0.430950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.40963	0.222885	0.111442	−	0.993771i	$-0.464453\pi$
$827$	6.40963	0.222885	0.111442	+	0.993771i	$0.464453\pi$
$828$	0	0
$829$	−26.1698	−0.908916	−0.454458	−	0.890768i	$-0.650167\pi$
$829$	−26.1698	−0.908916	−0.454458	+	0.890768i	$0.650167\pi$
$830$	0	0
$831$	−8.68876	−0.301410
$832$	0	0
$833$	5.76491	0.199742
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26.4683	−0.914880
$838$	0	0
$839$	1.12958	0.0389975	0.0194988	−	0.999810i	$-0.493793\pi$
$839$	1.12958	0.0389975	0.0194988	+	0.999810i	$0.493793\pi$
$840$	0	0
$841$	−26.8842	−0.927041
$842$	0	0
$843$	74.0734	2.55122
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.82546	−0.165805
$848$	0	0
$849$	−83.8842	−2.87890
$850$	0	0
$851$	43.6803	1.49734
$852$	0	0
$853$	−6.29095	−0.215398	−0.107699	−	0.994184i	$-0.534348\pi$
$853$	−6.29095	−0.215398	−0.107699	+	0.994184i	$0.534348\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.0596	0.856021	0.428010	−	0.903774i	$-0.359215\pi$
$857$	25.0596	0.856021	0.428010	+	0.903774i	$0.359215\pi$
$858$	0	0
$859$	−53.7896	−1.83528	−0.917638	−	0.397417i	$-0.869907\pi$
$859$	−53.7896	−1.83528	−0.917638	+	0.397417i	$0.869907\pi$
$860$	0	0
$861$	−35.2489	−1.20128
$862$	0	0
$863$	−58.0899	−1.97740	−0.988702	−	0.149896i	$-0.952106\pi$
$863$	−58.0899	−1.97740	−0.988702	+	0.149896i	$0.952106\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−50.7299	−1.72288
$868$	0	0
$869$	42.2039	1.43167
$870$	0	0
$871$	−31.2876	−1.06014
$872$	0	0
$873$	−18.4995	−0.626115
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.0899	0.813459	0.406729	−	0.913549i	$-0.366669\pi$
$877$	24.0899	0.813459	0.406729	+	0.913549i	$0.366669\pi$
$878$	0	0
$879$	−31.0450	−1.04712
$880$	0	0
$881$	40.9679	1.38024	0.690122	−	0.723693i	$-0.257557\pi$
$881$	40.9679	1.38024	0.690122	+	0.723693i	$0.257557\pi$
$882$	0	0
$883$	−43.8014	−1.47403	−0.737017	−	0.675874i	$-0.763766\pi$
$883$	−43.8014	−1.47403	−0.737017	+	0.675874i	$0.763766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.77959	−0.0597527	−0.0298764	−	0.999554i	$-0.509511\pi$
$887$	−1.77959	−0.0597527	−0.0298764	+	0.999554i	$0.509511\pi$
$888$	0	0
$889$	7.15894	0.240103
$890$	0	0
$891$	−40.9239	−1.37100
$892$	0	0
$893$	−5.56101	−0.186092
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−94.5271	−3.15617
$898$	0	0
$899$	3.27248	0.109143
$900$	0	0
$901$	−53.1495	−1.77067
$902$	0	0
$903$	−16.4995	−0.549070
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.0284	1.76078	0.880390	−	0.474250i	$-0.157281\pi$
$907$	53.0284	1.76078	0.880390	+	0.474250i	$0.157281\pi$
$908$	0	0
$909$	−30.9797	−1.02753
$910$	0	0
$911$	−28.1798	−0.933639	−0.466820	−	0.884353i	$-0.654600\pi$
$911$	−28.1798	−0.933639	−0.466820	+	0.884353i	$0.654600\pi$
$912$	0	0
$913$	−25.1202	−0.831357
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.85952	−0.259544
$918$	0	0
$919$	−38.4243	−1.26750	−0.633751	−	0.773538i	$-0.718485\pi$
$919$	−38.4243	−1.26750	−0.633751	+	0.773538i	$0.718485\pi$
$920$	0	0
$921$	100.292	3.30473
$922$	0	0
$923$	−17.6585	−0.581236
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.8099	0.552108
$928$	0	0
$929$	−36.6576	−1.20270	−0.601348	−	0.798987i	$-0.705369\pi$
$929$	−36.6576	−1.20270	−0.601348	+	0.798987i	$0.705369\pi$
$930$	0	0
$931$	1.60975	0.0527573
$932$	0	0
$933$	−31.4381	−1.02924
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.83302	−0.190557	−0.0952783	−	0.995451i	$-0.530374\pi$
$937$	−5.83302	−0.190557	−0.0952783	+	0.995451i	$0.530374\pi$
$938$	0	0
$939$	65.0743	2.12362
$940$	0	0
$941$	−2.23796	−0.0729553	−0.0364776	−	0.999334i	$-0.511614\pi$
$941$	−2.23796	−0.0729553	−0.0364776	+	0.999334i	$0.511614\pi$
$942$	0	0
$943$	−82.1193	−2.67417
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.7502	−0.576805	−0.288402	−	0.957509i	$-0.593124\pi$
$947$	−17.7502	−0.576805	−0.288402	+	0.957509i	$0.593124\pi$
$948$	0	0
$949$	−30.2498	−0.981949
$950$	0	0
$951$	49.2177	1.59599
$952$	0	0
$953$	32.0294	1.03753	0.518766	−	0.854916i	$-0.326391\pi$
$953$	32.0294	1.03753	0.518766	+	0.854916i	$0.326391\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.2947	0.365107
$958$	0	0
$959$	−10.5601	−0.341003
$960$	0	0
$961$	−25.9385	−0.836727
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.4305	1.39663	0.698316	−	0.715790i	$-0.253933\pi$
$967$	43.4305	1.39663	0.698316	+	0.715790i	$0.253933\pi$
$968$	0	0
$969$	−28.9991	−0.931585
$970$	0	0
$971$	1.79897	0.0577316	0.0288658	−	0.999583i	$-0.490810\pi$
$971$	1.79897	0.0577316	0.0288658	+	0.999583i	$0.490810\pi$
$972$	0	0
$973$	−9.79897	−0.314140
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.4390	0.717887	0.358943	−	0.933359i	$-0.383137\pi$
$977$	22.4390	0.717887	0.358943	+	0.933359i	$0.383137\pi$
$978$	0	0
$979$	28.4995	0.910849
$980$	0	0
$981$	−67.3388	−2.14996
$982$	0	0
$983$	37.6950	1.20228	0.601141	−	0.799143i	$-0.294713\pi$
$983$	37.6950	1.20228	0.601141	+	0.799143i	$0.294713\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10.7952	0.343615
$988$	0	0
$989$	−38.4390	−1.22229
$990$	0	0
$991$	−13.5686	−0.431020	−0.215510	−	0.976502i	$-0.569141\pi$
$991$	−13.5686	−0.431020	−0.215510	+	0.976502i	$0.569141\pi$
$992$	0	0
$993$	−80.5583	−2.55644
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.2838	−0.452373	−0.226187	−	0.974084i	$-0.572626\pi$
$997$	−14.2838	−0.452373	−0.226187	+	0.974084i	$0.572626\pi$
$998$	0	0
$999$	70.5895	2.23335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bq.1.1		3
4.3	odd	2		1400.2.a.t.1.3		3
5.2	odd	4		560.2.g.f.449.6		6
5.3	odd	4		560.2.g.f.449.1		6
5.4	even	2		2800.2.a.br.1.3		3
15.2	even	4		5040.2.t.y.1009.2		6
15.8	even	4		5040.2.t.y.1009.1		6
20.3	even	4		280.2.g.b.169.6	yes	6
20.7	even	4		280.2.g.b.169.1	✓	6
20.19	odd	2		1400.2.a.s.1.1		3
28.27	even	2		9800.2.a.cd.1.1		3
40.3	even	4		2240.2.g.l.449.1		6
40.13	odd	4		2240.2.g.m.449.6		6
40.27	even	4		2240.2.g.l.449.6		6
40.37	odd	4		2240.2.g.m.449.1		6
60.23	odd	4		2520.2.t.g.1009.1		6
60.47	odd	4		2520.2.t.g.1009.2		6
140.27	odd	4		1960.2.g.c.1569.6		6
140.83	odd	4		1960.2.g.c.1569.1		6
140.139	even	2		9800.2.a.cg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.g.b.169.1	✓	6	20.7	even	4
280.2.g.b.169.6	yes	6	20.3	even	4
560.2.g.f.449.1		6	5.3	odd	4
560.2.g.f.449.6		6	5.2	odd	4
1400.2.a.s.1.1		3	20.19	odd	2
1400.2.a.t.1.3		3	4.3	odd	2
1960.2.g.c.1569.1		6	140.83	odd	4
1960.2.g.c.1569.6		6	140.27	odd	4
2240.2.g.l.449.1		6	40.3	even	4
2240.2.g.l.449.6		6	40.27	even	4
2240.2.g.m.449.1		6	40.37	odd	4
2240.2.g.m.449.6		6	40.13	odd	4
2520.2.t.g.1009.1		6	60.23	odd	4
2520.2.t.g.1009.2		6	60.47	odd	4
2800.2.a.bq.1.1		3	1.1	even	1	trivial
2800.2.a.br.1.3		3	5.4	even	2
5040.2.t.y.1009.1		6	15.8	even	4
5040.2.t.y.1009.2		6	15.2	even	4
9800.2.a.cd.1.1		3	28.27	even	2
9800.2.a.cg.1.3		3	140.139	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)

\(f(q)\)	\(=\)	\(q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} +O(q^{10})\)	\(q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} -2.48486 q^{11} -4.15516 q^{13} +5.76491 q^{17} +1.60975 q^{19} -3.12489 q^{21} -7.28005 q^{23} -11.7649 q^{27} +1.45459 q^{29} +2.24977 q^{31} +7.76491 q^{33} -6.00000 q^{37} +12.9844 q^{39} +11.2800 q^{41} +5.28005 q^{43} -3.45459 q^{47} +1.00000 q^{49} -18.0147 q^{51} -9.21949 q^{53} -5.03028 q^{57} +5.92007 q^{59} +5.35998 q^{61} +6.76491 q^{63} +7.52982 q^{67} +22.7493 q^{69} +4.24977 q^{71} +7.28005 q^{73} -2.48486 q^{77} -16.9844 q^{79} +16.4693 q^{81} +10.1093 q^{83} -4.54541 q^{87} -11.4693 q^{89} -4.15516 q^{91} -7.03028 q^{93} -2.73463 q^{97} -16.8099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + 3 * q^7 + 4 * q^9	\( 3 q - q^{3} + 3 q^{7} + 4 q^{9} - 7 q^{11} - 5 q^{13} + q^{17} - 4 q^{19} - q^{21} - 6 q^{23} - 19 q^{27} + 3 q^{29} - 10 q^{31} + 7 q^{33} - 18 q^{37} + 5 q^{39} + 18 q^{41} - 9 q^{47} + 3 q^{49} - 21 q^{51} - 10 q^{53} - 16 q^{57} - 6 q^{59} + 24 q^{61} + 4 q^{63} - 10 q^{67} + 18 q^{69} - 4 q^{71} + 6 q^{73} - 7 q^{77} - 17 q^{79} + 15 q^{81} - 12 q^{83} - 15 q^{87} - 5 q^{91} - 22 q^{93} + 9 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - q^3 + 3 * q^7 + 4 * q^9 - 7 * q^11 - 5 * q^13 + q^17 - 4 * q^19 - q^21 - 6 * q^23 - 19 * q^27 + 3 * q^29 - 10 * q^31 + 7 * q^33 - 18 * q^37 + 5 * q^39 + 18 * q^41 - 9 * q^47 + 3 * q^49 - 21 * q^51 - 10 * q^53 - 16 * q^57 - 6 * q^59 + 24 * q^61 + 4 * q^63 - 10 * q^67 + 18 * q^69 - 4 * q^71 + 6 * q^73 - 7 * q^77 - 17 * q^79 + 15 * q^81 - 12 * q^83 - 15 * q^87 - 5 * q^91 - 22 * q^93 + 9 * q^97 - 2 * q^99

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