LMFDB - Embedded newform 4200.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.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:	$33.5371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.77200$ of defining polynomial
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +4.77200 q^{11} +5.77200 q^{13} -3.77200 q^{17} +1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{27} +3.00000 q^{29} -3.77200 q^{31} +4.77200 q^{33} +0.772002 q^{37} +5.77200 q^{39} +5.77200 q^{41} -4.54400 q^{43} +1.00000 q^{49} -3.77200 q^{51} -3.77200 q^{53} +5.77200 q^{59} -1.77200 q^{61} +1.00000 q^{63} -1.22800 q^{67} +3.00000 q^{69} +1.22800 q^{71} +2.00000 q^{73} +4.77200 q^{77} -14.3160 q^{79} +1.00000 q^{81} +15.7720 q^{83} +3.00000 q^{87} +0.455996 q^{89} +5.77200 q^{91} -3.77200 q^{93} -10.0000 q^{97} +4.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + q^{17} + 2 q^{21} + 6 q^{23} + 2 q^{27} + 6 q^{29} + q^{31} + q^{33} - 7 q^{37} + 3 q^{39} + 3 q^{41} + 8 q^{43} + 2 q^{49} + q^{51} + q^{53} + 3 q^{59} + 5 q^{61} + 2 q^{63} - 11 q^{67} + 6 q^{69} + 11 q^{71} + 4 q^{73} + q^{77} - 3 q^{79} + 2 q^{81} + 23 q^{83} + 6 q^{87} + 18 q^{89} + 3 q^{91} + q^{93} - 20 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + q^17 + 2 * q^21 + 6 * q^23 + 2 * q^27 + 6 * q^29 + q^31 + q^33 - 7 * q^37 + 3 * q^39 + 3 * q^41 + 8 * q^43 + 2 * q^49 + q^51 + q^53 + 3 * q^59 + 5 * q^61 + 2 * q^63 - 11 * q^67 + 6 * q^69 + 11 * q^71 + 4 * q^73 + q^77 - 3 * q^79 + 2 * q^81 + 23 * q^83 + 6 * q^87 + 18 * q^89 + 3 * q^91 + q^93 - 20 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.77200	1.43881	0.719406	−	0.694589i	$-0.244414\pi$
$11$	4.77200	1.43881	0.719406	+	0.694589i	$0.244414\pi$
$12$	0	0
$13$	5.77200	1.60087	0.800433	−	0.599423i	$-0.204603\pi$
$13$	5.77200	1.60087	0.800433	+	0.599423i	$0.204603\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.77200	−0.914845	−0.457422	−	0.889250i	$-0.651227\pi$
$17$	−3.77200	−0.914845	−0.457422	+	0.889250i	$0.651227\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−3.77200	−0.677472	−0.338736	−	0.940882i	$-0.609999\pi$
$31$	−3.77200	−0.677472	−0.338736	+	0.940882i	$0.609999\pi$
$32$	0	0
$33$	4.77200	0.830699
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.772002	0.126916	0.0634582	−	0.997984i	$-0.479787\pi$
$37$	0.772002	0.126916	0.0634582	+	0.997984i	$0.479787\pi$
$38$	0	0
$39$	5.77200	0.924260
$40$	0	0
$41$	5.77200	0.901435	0.450718	−	0.892667i	$-0.351168\pi$
$41$	5.77200	0.901435	0.450718	+	0.892667i	$0.351168\pi$
$42$	0	0
$43$	−4.54400	−0.692954	−0.346477	−	0.938058i	$-0.612622\pi$
$43$	−4.54400	−0.692954	−0.346477	+	0.938058i	$0.612622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.77200	−0.528186
$52$	0	0
$53$	−3.77200	−0.518124	−0.259062	−	0.965861i	$-0.583413\pi$
$53$	−3.77200	−0.518124	−0.259062	+	0.965861i	$0.583413\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.77200	0.751451	0.375725	−	0.926731i	$-0.377394\pi$
$59$	5.77200	0.751451	0.375725	+	0.926731i	$0.377394\pi$
$60$	0	0
$61$	−1.77200	−0.226882	−0.113441	−	0.993545i	$-0.536187\pi$
$61$	−1.77200	−0.226882	−0.113441	+	0.993545i	$0.536187\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.22800	−0.150024	−0.0750119	−	0.997183i	$-0.523899\pi$
$67$	−1.22800	−0.150024	−0.0750119	+	0.997183i	$0.523899\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	1.22800	0.145737	0.0728683	−	0.997342i	$-0.476785\pi$
$71$	1.22800	0.145737	0.0728683	+	0.997342i	$0.476785\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.77200	0.543820
$78$	0	0
$79$	−14.3160	−1.61068	−0.805338	−	0.592816i	$-0.798016\pi$
$79$	−14.3160	−1.61068	−0.805338	+	0.592816i	$0.798016\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.7720	1.73120	0.865601	−	0.500734i	$-0.166937\pi$
$83$	15.7720	1.73120	0.865601	+	0.500734i	$0.166937\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	0.455996	0.0483355	0.0241678	−	0.999708i	$-0.492306\pi$
$89$	0.455996	0.0483355	0.0241678	+	0.999708i	$0.492306\pi$
$90$	0	0
$91$	5.77200	0.605070
$92$	0	0
$93$	−3.77200	−0.391138
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	4.77200	0.479604
$100$	0	0
$101$	−9.54400	−0.949664	−0.474832	−	0.880076i	$-0.657491\pi$
$101$	−9.54400	−0.949664	−0.474832	+	0.880076i	$0.657491\pi$
$102$	0	0
$103$	2.22800	0.219531	0.109766	−	0.993958i	$-0.464990\pi$
$103$	2.22800	0.219531	0.109766	+	0.993958i	$0.464990\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.5440	−1.50270	−0.751348	−	0.659906i	$-0.770596\pi$
$107$	−15.5440	−1.50270	−0.751348	+	0.659906i	$0.770596\pi$
$108$	0	0
$109$	10.7720	1.03177	0.515885	−	0.856658i	$-0.327463\pi$
$109$	10.7720	1.03177	0.515885	+	0.856658i	$0.327463\pi$
$110$	0	0
$111$	0.772002	0.0732752
$112$	0	0
$113$	−14.3160	−1.34674	−0.673368	−	0.739307i	$-0.735153\pi$
$113$	−14.3160	−1.34674	−0.673368	+	0.739307i	$0.735153\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.77200	0.533622
$118$	0	0
$119$	−3.77200	−0.345779
$120$	0	0
$121$	11.7720	1.07018
$122$	0	0
$123$	5.77200	0.520444
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.77200	−0.778389	−0.389195	−	0.921156i	$-0.627247\pi$
$127$	−8.77200	−0.778389	−0.389195	+	0.921156i	$0.627247\pi$
$128$	0	0
$129$	−4.54400	−0.400077
$130$	0	0
$131$	13.5440	1.18335	0.591673	−	0.806178i	$-0.298468\pi$
$131$	13.5440	1.18335	0.591673	+	0.806178i	$0.298468\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.54400	−0.815399	−0.407700	−	0.913116i	$-0.633669\pi$
$137$	−9.54400	−0.815399	−0.407700	+	0.913116i	$0.633669\pi$
$138$	0	0
$139$	1.54400	0.130961	0.0654803	−	0.997854i	$-0.479142\pi$
$139$	1.54400	0.130961	0.0654803	+	0.997854i	$0.479142\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	27.5440	2.30335
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	20.5440	1.68303	0.841515	−	0.540233i	$-0.181664\pi$
$149$	20.5440	1.68303	0.841515	+	0.540233i	$0.181664\pi$
$150$	0	0
$151$	−4.77200	−0.388340	−0.194170	−	0.980968i	$-0.562201\pi$
$151$	−4.77200	−0.388340	−0.194170	+	0.980968i	$0.562201\pi$
$152$	0	0
$153$	−3.77200	−0.304948
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.5440	−1.87902	−0.939508	−	0.342527i	$-0.888717\pi$
$157$	−23.5440	−1.87902	−0.939508	+	0.342527i	$0.888717\pi$
$158$	0	0
$159$	−3.77200	−0.299139
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	24.8600	1.94719	0.973593	−	0.228290i	$-0.0733135\pi$
$163$	24.8600	1.94719	0.973593	+	0.228290i	$0.0733135\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.54400	0.583773	0.291886	−	0.956453i	$-0.405717\pi$
$167$	7.54400	0.583773	0.291886	+	0.956453i	$0.405717\pi$
$168$	0	0
$169$	20.3160	1.56277
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.0880	0.843006	0.421503	−	0.906827i	$-0.361503\pi$
$173$	11.0880	0.843006	0.421503	+	0.906827i	$0.361503\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.77200	0.433850
$178$	0	0
$179$	8.45600	0.632031	0.316015	−	0.948754i	$-0.397655\pi$
$179$	8.45600	0.632031	0.316015	+	0.948754i	$0.397655\pi$
$180$	0	0
$181$	25.0880	1.86478	0.932388	−	0.361458i	$-0.117721\pi$
$181$	25.0880	1.86478	0.932388	+	0.361458i	$0.117721\pi$
$182$	0	0
$183$	−1.77200	−0.130990
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−18.0000	−1.31629
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−1.77200	−0.128218	−0.0641088	−	0.997943i	$-0.520420\pi$
$191$	−1.77200	−0.128218	−0.0641088	+	0.997943i	$0.520420\pi$
$192$	0	0
$193$	−20.7720	−1.49520	−0.747601	−	0.664148i	$-0.768794\pi$
$193$	−20.7720	−1.49520	−0.747601	+	0.664148i	$0.768794\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0880	1.28872	0.644359	−	0.764723i	$-0.277124\pi$
$197$	18.0880	1.28872	0.644359	+	0.764723i	$0.277124\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−1.22800	−0.0866163
$202$	0	0
$203$	3.00000	0.210559
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	25.7720	1.77422	0.887109	−	0.461560i	$-0.152710\pi$
$211$	25.7720	1.77422	0.887109	+	0.461560i	$0.152710\pi$
$212$	0	0
$213$	1.22800	0.0841410
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.77200	−0.256060
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−21.7720	−1.46454
$222$	0	0
$223$	19.7720	1.32403	0.662016	−	0.749490i	$-0.269701\pi$
$223$	19.7720	1.32403	0.662016	+	0.749490i	$0.269701\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.77200	0.117612	0.0588059	−	0.998269i	$-0.481271\pi$
$227$	1.77200	0.117612	0.0588059	+	0.998269i	$0.481271\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	4.77200	0.313975
$232$	0	0
$233$	−13.2280	−0.866595	−0.433297	−	0.901251i	$-0.642650\pi$
$233$	−13.2280	−0.866595	−0.433297	+	0.901251i	$0.642650\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.3160	−0.929924
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−7.54400	−0.485952	−0.242976	−	0.970032i	$-0.578124\pi$
$241$	−7.54400	−0.485952	−0.242976	+	0.970032i	$0.578124\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	15.7720	0.999510
$250$	0	0
$251$	−22.8600	−1.44291	−0.721455	−	0.692461i	$-0.756527\pi$
$251$	−22.8600	−1.44291	−0.721455	+	0.692461i	$0.756527\pi$
$252$	0	0
$253$	14.3160	0.900040
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.22800	0.513248	0.256624	−	0.966511i	$-0.417390\pi$
$257$	8.22800	0.513248	0.256624	+	0.966511i	$0.417390\pi$
$258$	0	0
$259$	0.772002	0.0479699
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	1.45600	0.0897806	0.0448903	−	0.998992i	$-0.485706\pi$
$263$	1.45600	0.0897806	0.0448903	+	0.998992i	$0.485706\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.455996	0.0279065
$268$	0	0
$269$	25.0880	1.52964	0.764821	−	0.644242i	$-0.222827\pi$
$269$	25.0880	1.52964	0.764821	+	0.644242i	$0.222827\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	5.77200	0.349337
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	−3.77200	−0.225824
$280$	0	0
$281$	−10.7720	−0.642604	−0.321302	−	0.946977i	$-0.604120\pi$
$281$	−10.7720	−0.642604	−0.321302	+	0.946977i	$0.604120\pi$
$282$	0	0
$283$	−24.6320	−1.46422	−0.732111	−	0.681186i	$-0.761465\pi$
$283$	−24.6320	−1.46422	−0.732111	+	0.681186i	$0.761465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.77200	0.340710
$288$	0	0
$289$	−2.77200	−0.163059
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−11.0880	−0.647768	−0.323884	−	0.946097i	$-0.604989\pi$
$293$	−11.0880	−0.647768	−0.323884	+	0.946097i	$0.604989\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.77200	0.276900
$298$	0	0
$299$	17.3160	1.00141
$300$	0	0
$301$	−4.54400	−0.261912
$302$	0	0
$303$	−9.54400	−0.548289
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	2.22800	0.126746
$310$	0	0
$311$	25.0880	1.42261	0.711305	−	0.702883i	$-0.248104\pi$
$311$	25.0880	1.42261	0.711305	+	0.702883i	$0.248104\pi$
$312$	0	0
$313$	−27.5440	−1.55688	−0.778440	−	0.627720i	$-0.783988\pi$
$313$	−27.5440	−1.55688	−0.778440	+	0.627720i	$0.783988\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	14.3160	0.801542
$320$	0	0
$321$	−15.5440	−0.867582
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.7720	0.595693
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.5440	1.45899	0.729495	−	0.683986i	$-0.239755\pi$
$331$	26.5440	1.45899	0.729495	+	0.683986i	$0.239755\pi$
$332$	0	0
$333$	0.772002	0.0423054
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.3160	−1.48800	−0.743999	−	0.668181i	$-0.767073\pi$
$337$	−27.3160	−1.48800	−0.743999	+	0.668181i	$0.767073\pi$
$338$	0	0
$339$	−14.3160	−0.777539
$340$	0	0
$341$	−18.0000	−0.974755
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.31601	0.339061	0.169530	−	0.985525i	$-0.445775\pi$
$347$	6.31601	0.339061	0.169530	+	0.985525i	$0.445775\pi$
$348$	0	0
$349$	−24.4040	−1.30632	−0.653158	−	0.757221i	$-0.726556\pi$
$349$	−24.4040	−1.30632	−0.653158	+	0.757221i	$0.726556\pi$
$350$	0	0
$351$	5.77200	0.308087
$352$	0	0
$353$	21.0880	1.12240	0.561201	−	0.827680i	$-0.310340\pi$
$353$	21.0880	1.12240	0.561201	+	0.827680i	$0.310340\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.77200	−0.199636
$358$	0	0
$359$	34.5440	1.82316	0.911581	−	0.411120i	$-0.134862\pi$
$359$	34.5440	1.82316	0.911581	+	0.411120i	$0.134862\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.7720	0.617870
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.7720	0.927691	0.463845	−	0.885916i	$-0.346469\pi$
$367$	17.7720	0.927691	0.463845	+	0.885916i	$0.346469\pi$
$368$	0	0
$369$	5.77200	0.300478
$370$	0	0
$371$	−3.77200	−0.195833
$372$	0	0
$373$	−32.3160	−1.67326	−0.836630	−	0.547769i	$-0.815477\pi$
$373$	−32.3160	−1.67326	−0.836630	+	0.547769i	$0.815477\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.3160	0.891820
$378$	0	0
$379$	−32.0880	−1.64825	−0.824125	−	0.566408i	$-0.808333\pi$
$379$	−32.0880	−1.64825	−0.824125	+	0.566408i	$0.808333\pi$
$380$	0	0
$381$	−8.77200	−0.449403
$382$	0	0
$383$	−15.0880	−0.770961	−0.385481	−	0.922716i	$-0.625964\pi$
$383$	−15.0880	−0.770961	−0.385481	+	0.922716i	$0.625964\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.54400	−0.230985
$388$	0	0
$389$	−25.4040	−1.28803	−0.644017	−	0.765011i	$-0.722734\pi$
$389$	−25.4040	−1.28803	−0.644017	+	0.765011i	$0.722734\pi$
$390$	0	0
$391$	−11.3160	−0.572275
$392$	0	0
$393$	13.5440	0.683205
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.2280	−0.613706	−0.306853	−	0.951757i	$-0.599276\pi$
$397$	−12.2280	−0.613706	−0.306853	+	0.951757i	$0.599276\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.31601	0.415282	0.207641	−	0.978205i	$-0.433422\pi$
$401$	8.31601	0.415282	0.207641	+	0.978205i	$0.433422\pi$
$402$	0	0
$403$	−21.7720	−1.08454
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.68399	0.182609
$408$	0	0
$409$	−23.5440	−1.16418	−0.582088	−	0.813126i	$-0.697764\pi$
$409$	−23.5440	−1.16418	−0.582088	+	0.813126i	$0.697764\pi$
$410$	0	0
$411$	−9.54400	−0.470771
$412$	0	0
$413$	5.77200	0.284022
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.54400	0.0756102
$418$	0	0
$419$	−23.3160	−1.13906	−0.569531	−	0.821970i	$-0.692875\pi$
$419$	−23.3160	−1.13906	−0.569531	+	0.821970i	$0.692875\pi$
$420$	0	0
$421$	8.77200	0.427521	0.213761	−	0.976886i	$-0.431429\pi$
$421$	8.77200	0.427521	0.213761	+	0.976886i	$0.431429\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.77200	−0.0857532
$428$	0	0
$429$	27.5440	1.32984
$430$	0	0
$431$	1.31601	0.0633898	0.0316949	−	0.999498i	$-0.489910\pi$
$431$	1.31601	0.0633898	0.0316949	+	0.999498i	$0.489910\pi$
$432$	0	0
$433$	19.5440	0.939225	0.469612	−	0.882873i	$-0.344394\pi$
$433$	19.5440	0.939225	0.469612	+	0.882873i	$0.344394\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	5.77200	0.275483	0.137741	−	0.990468i	$-0.456016\pi$
$439$	5.77200	0.275483	0.137741	+	0.990468i	$0.456016\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−3.08801	−0.146716	−0.0733578	−	0.997306i	$-0.523372\pi$
$443$	−3.08801	−0.146716	−0.0733578	+	0.997306i	$0.523372\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.5440	0.971698
$448$	0	0
$449$	−20.3160	−0.958772	−0.479386	−	0.877604i	$-0.659141\pi$
$449$	−20.3160	−0.958772	−0.479386	+	0.877604i	$0.659141\pi$
$450$	0	0
$451$	27.5440	1.29700
$452$	0	0
$453$	−4.77200	−0.224208
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.08801	0.191229	0.0956145	−	0.995418i	$-0.469518\pi$
$457$	4.08801	0.191229	0.0956145	+	0.995418i	$0.469518\pi$
$458$	0	0
$459$	−3.77200	−0.176062
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	34.1760	1.58829	0.794147	−	0.607726i	$-0.207918\pi$
$463$	34.1760	1.58829	0.794147	+	0.607726i	$0.207918\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.3160	−1.07894	−0.539468	−	0.842006i	$-0.681375\pi$
$467$	−23.3160	−1.07894	−0.539468	+	0.842006i	$0.681375\pi$
$468$	0	0
$469$	−1.22800	−0.0567037
$470$	0	0
$471$	−23.5440	−1.08485
$472$	0	0
$473$	−21.6840	−0.997031
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.77200	−0.172708
$478$	0	0
$479$	15.0880	0.689389	0.344694	−	0.938715i	$-0.387983\pi$
$479$	15.0880	0.689389	0.344694	+	0.938715i	$0.387983\pi$
$480$	0	0
$481$	4.45600	0.203176
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−10.3160	−0.467463	−0.233731	−	0.972301i	$-0.575094\pi$
$487$	−10.3160	−0.467463	−0.233731	+	0.972301i	$0.575094\pi$
$488$	0	0
$489$	24.8600	1.12421
$490$	0	0
$491$	9.68399	0.437033	0.218516	−	0.975833i	$-0.429878\pi$
$491$	9.68399	0.437033	0.218516	+	0.975833i	$0.429878\pi$
$492$	0	0
$493$	−11.3160	−0.509647
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.22800	0.0550832
$498$	0	0
$499$	−10.6840	−0.478281	−0.239141	−	0.970985i	$-0.576866\pi$
$499$	−10.6840	−0.478281	−0.239141	+	0.970985i	$0.576866\pi$
$500$	0	0
$501$	7.54400	0.337041
$502$	0	0
$503$	34.0000	1.51599	0.757993	−	0.652263i	$-0.226180\pi$
$503$	34.0000	1.51599	0.757993	+	0.652263i	$0.226180\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.3160	0.902265
$508$	0	0
$509$	15.0880	0.668764	0.334382	−	0.942438i	$-0.391472\pi$
$509$	15.0880	0.668764	0.334382	+	0.942438i	$0.391472\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	11.0880	0.486710
$520$	0	0
$521$	33.9480	1.48729	0.743645	−	0.668575i	$-0.233095\pi$
$521$	33.9480	1.48729	0.743645	+	0.668575i	$0.233095\pi$
$522$	0	0
$523$	−0.911993	−0.0398786	−0.0199393	−	0.999801i	$-0.506347\pi$
$523$	−0.911993	−0.0398786	−0.0199393	+	0.999801i	$0.506347\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.2280	0.619781
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	5.77200	0.250484
$532$	0	0
$533$	33.3160	1.44308
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.45600	0.364903
$538$	0	0
$539$	4.77200	0.205545
$540$	0	0
$541$	5.22800	0.224769	0.112385	−	0.993665i	$-0.464151\pi$
$541$	5.22800	0.224769	0.112385	+	0.993665i	$0.464151\pi$
$542$	0	0
$543$	25.0880	1.07663
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.6320	1.01043	0.505216	−	0.862993i	$-0.331413\pi$
$547$	23.6320	1.01043	0.505216	+	0.862993i	$0.331413\pi$
$548$	0	0
$549$	−1.77200	−0.0756272
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−14.3160	−0.608778
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.3160	−1.62350	−0.811751	−	0.584004i	$-0.801485\pi$
$557$	−38.3160	−1.62350	−0.811751	+	0.584004i	$0.801485\pi$
$558$	0	0
$559$	−26.2280	−1.10933
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	38.8600	1.63775	0.818877	−	0.573969i	$-0.194597\pi$
$563$	38.8600	1.63775	0.818877	+	0.573969i	$0.194597\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−7.22800	−0.303013	−0.151507	−	0.988456i	$-0.548413\pi$
$569$	−7.22800	−0.303013	−0.151507	+	0.988456i	$0.548413\pi$
$570$	0	0
$571$	−24.0880	−1.00805	−0.504026	−	0.863689i	$-0.668148\pi$
$571$	−24.0880	−1.00805	−0.504026	+	0.863689i	$0.668148\pi$
$572$	0	0
$573$	−1.77200	−0.0740264
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.6320	0.609139	0.304569	−	0.952490i	$-0.401487\pi$
$577$	14.6320	0.609139	0.304569	+	0.952490i	$0.401487\pi$
$578$	0	0
$579$	−20.7720	−0.863255
$580$	0	0
$581$	15.7720	0.654333
$582$	0	0
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.3160	−0.797257	−0.398628	−	0.917113i	$-0.630514\pi$
$587$	−19.3160	−0.797257	−0.398628	+	0.917113i	$0.630514\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.0880	0.744041
$592$	0	0
$593$	−39.5440	−1.62388	−0.811939	−	0.583743i	$-0.801588\pi$
$593$	−39.5440	−1.62388	−0.811939	+	0.583743i	$0.801588\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.0000	1.34834	0.674172	−	0.738575i	$-0.264501\pi$
$599$	33.0000	1.34834	0.674172	+	0.738575i	$0.264501\pi$
$600$	0	0
$601$	−19.0880	−0.778616	−0.389308	−	0.921108i	$-0.627286\pi$
$601$	−19.0880	−0.778616	−0.389308	+	0.921108i	$0.627286\pi$
$602$	0	0
$603$	−1.22800	−0.0500079
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.13999	−0.167213	−0.0836063	−	0.996499i	$-0.526644\pi$
$613$	−4.13999	−0.167213	−0.0836063	+	0.996499i	$0.526644\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.2280	0.774090	0.387045	−	0.922061i	$-0.373496\pi$
$617$	19.2280	0.774090	0.387045	+	0.922061i	$0.373496\pi$
$618$	0	0
$619$	−47.5440	−1.91095	−0.955477	−	0.295064i	$-0.904659\pi$
$619$	−47.5440	−1.91095	−0.955477	+	0.295064i	$0.904659\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	0.455996	0.0182691
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.91199	−0.116109
$630$	0	0
$631$	36.7720	1.46387	0.731935	−	0.681374i	$-0.238617\pi$
$631$	36.7720	1.46387	0.731935	+	0.681374i	$0.238617\pi$
$632$	0	0
$633$	25.7720	1.02435
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.77200	0.228695
$638$	0	0
$639$	1.22800	0.0485789
$640$	0	0
$641$	−49.4040	−1.95134	−0.975671	−	0.219242i	$-0.929642\pi$
$641$	−49.4040	−1.95134	−0.975671	+	0.219242i	$0.929642\pi$
$642$	0	0
$643$	−6.63201	−0.261541	−0.130770	−	0.991413i	$-0.541745\pi$
$643$	−6.63201	−0.261541	−0.130770	+	0.991413i	$0.541745\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.9120	0.586251	0.293125	−	0.956074i	$-0.405305\pi$
$647$	14.9120	0.586251	0.293125	+	0.956074i	$0.405305\pi$
$648$	0	0
$649$	27.5440	1.08120
$650$	0	0
$651$	−3.77200	−0.147836
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−38.6320	−1.50489	−0.752445	−	0.658655i	$-0.771126\pi$
$659$	−38.6320	−1.50489	−0.752445	+	0.658655i	$0.771126\pi$
$660$	0	0
$661$	15.5440	0.604592	0.302296	−	0.953214i	$-0.402247\pi$
$661$	15.5440	0.604592	0.302296	+	0.953214i	$0.402247\pi$
$662$	0	0
$663$	−21.7720	−0.845554
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	19.7720	0.764430
$670$	0	0
$671$	−8.45600	−0.326440
$672$	0	0
$673$	4.68399	0.180555	0.0902774	−	0.995917i	$-0.471225\pi$
$673$	4.68399	0.180555	0.0902774	+	0.995917i	$0.471225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.4560	0.401857	0.200928	−	0.979606i	$-0.435604\pi$
$677$	10.4560	0.401857	0.200928	+	0.979606i	$0.435604\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	1.77200	0.0679033
$682$	0	0
$683$	−40.7720	−1.56010	−0.780049	−	0.625719i	$-0.784806\pi$
$683$	−40.7720	−1.56010	−0.780049	+	0.625719i	$0.784806\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	−21.7720	−0.829447
$690$	0	0
$691$	15.5440	0.591322	0.295661	−	0.955293i	$-0.404460\pi$
$691$	15.5440	0.591322	0.295661	+	0.955293i	$0.404460\pi$
$692$	0	0
$693$	4.77200	0.181273
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21.7720	−0.824673
$698$	0	0
$699$	−13.2280	−0.500329
$700$	0	0
$701$	−12.6840	−0.479068	−0.239534	−	0.970888i	$-0.576995\pi$
$701$	−12.6840	−0.479068	−0.239534	+	0.970888i	$0.576995\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.54400	−0.358939
$708$	0	0
$709$	−6.45600	−0.242460	−0.121230	−	0.992624i	$-0.538684\pi$
$709$	−6.45600	−0.242460	−0.121230	+	0.992624i	$0.538684\pi$
$710$	0	0
$711$	−14.3160	−0.536892
$712$	0	0
$713$	−11.3160	−0.423788
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	−26.6320	−0.993206	−0.496603	−	0.867978i	$-0.665420\pi$
$719$	−26.6320	−0.993206	−0.496603	+	0.867978i	$0.665420\pi$
$720$	0	0
$721$	2.22800	0.0829750
$722$	0	0
$723$	−7.54400	−0.280565
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.4040	−0.830919	−0.415459	−	0.909612i	$-0.636379\pi$
$727$	−22.4040	−0.830919	−0.415459	+	0.909612i	$0.636379\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.1400	0.633945
$732$	0	0
$733$	−47.9480	−1.77100	−0.885500	−	0.464639i	$-0.846184\pi$
$733$	−47.9480	−1.77100	−0.885500	+	0.464639i	$0.846184\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.86001	−0.215856
$738$	0	0
$739$	−26.5440	−0.976437	−0.488218	−	0.872721i	$-0.662353\pi$
$739$	−26.5440	−0.976437	−0.488218	+	0.872721i	$0.662353\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.3160	0.635263	0.317631	−	0.948214i	$-0.397113\pi$
$743$	17.3160	0.635263	0.317631	+	0.948214i	$0.397113\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.7720	0.577067
$748$	0	0
$749$	−15.5440	−0.567966
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−22.8600	−0.833065
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.22800	−0.262706	−0.131353	−	0.991336i	$-0.541932\pi$
$757$	−7.22800	−0.262706	−0.131353	+	0.991336i	$0.541932\pi$
$758$	0	0
$759$	14.3160	0.519638
$760$	0	0
$761$	−23.5440	−0.853469	−0.426735	−	0.904377i	$-0.640336\pi$
$761$	−23.5440	−0.853469	−0.426735	+	0.904377i	$0.640336\pi$
$762$	0	0
$763$	10.7720	0.389973
$764$	0	0
$765$	0	0
$766$	0	0
$767$	33.3160	1.20297
$768$	0	0
$769$	−0.911993	−0.0328873	−0.0164436	−	0.999865i	$-0.505234\pi$
$769$	−0.911993	−0.0328873	−0.0164436	+	0.999865i	$0.505234\pi$
$770$	0	0
$771$	8.22800	0.296324
$772$	0	0
$773$	−32.6320	−1.17369	−0.586846	−	0.809699i	$-0.699631\pi$
$773$	−32.6320	−1.17369	−0.586846	+	0.809699i	$0.699631\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.772002	0.0276954
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.86001	0.209688
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.0880	0.466537	0.233269	−	0.972412i	$-0.425058\pi$
$787$	13.0880	0.466537	0.233269	+	0.972412i	$0.425058\pi$
$788$	0	0
$789$	1.45600	0.0518348
$790$	0	0
$791$	−14.3160	−0.509019
$792$	0	0
$793$	−10.2280	−0.363207
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.9120	0.669897	0.334949	−	0.942236i	$-0.391281\pi$
$797$	18.9120	0.669897	0.334949	+	0.942236i	$0.391281\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.455996	0.0161118
$802$	0	0
$803$	9.54400	0.336801
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.0880	0.883140
$808$	0	0
$809$	13.8600	0.487292	0.243646	−	0.969864i	$-0.421656\pi$
$809$	13.8600	0.487292	0.243646	+	0.969864i	$0.421656\pi$
$810$	0	0
$811$	44.1760	1.55123	0.775615	−	0.631206i	$-0.217440\pi$
$811$	44.1760	1.55123	0.775615	+	0.631206i	$0.217440\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	5.77200	0.201690
$820$	0	0
$821$	−13.0880	−0.456775	−0.228387	−	0.973570i	$-0.573345\pi$
$821$	−13.0880	−0.456775	−0.228387	+	0.973570i	$0.573345\pi$
$822$	0	0
$823$	−44.7720	−1.56065	−0.780327	−	0.625372i	$-0.784947\pi$
$823$	−44.7720	−1.56065	−0.780327	+	0.625372i	$0.784947\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.4040	1.16157	0.580786	−	0.814057i	$-0.302745\pi$
$827$	33.4040	1.16157	0.580786	+	0.814057i	$0.302745\pi$
$828$	0	0
$829$	−35.7720	−1.24241	−0.621206	−	0.783647i	$-0.713357\pi$
$829$	−35.7720	−1.24241	−0.621206	+	0.783647i	$0.713357\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−3.77200	−0.130692
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.77200	−0.130379
$838$	0	0
$839$	−21.5440	−0.743782	−0.371891	−	0.928276i	$-0.621290\pi$
$839$	−21.5440	−0.743782	−0.371891	+	0.928276i	$0.621290\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−10.7720	−0.371007
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.7720	0.404491
$848$	0	0
$849$	−24.6320	−0.845368
$850$	0	0
$851$	2.31601	0.0793917
$852$	0	0
$853$	52.4040	1.79428	0.897140	−	0.441747i	$-0.145641\pi$
$853$	52.4040	1.79428	0.897140	+	0.441747i	$0.145641\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.5440	0.530973	0.265487	−	0.964115i	$-0.414467\pi$
$857$	15.5440	0.530973	0.265487	+	0.964115i	$0.414467\pi$
$858$	0	0
$859$	−2.45600	−0.0837975	−0.0418988	−	0.999122i	$-0.513341\pi$
$859$	−2.45600	−0.0837975	−0.0418988	+	0.999122i	$0.513341\pi$
$860$	0	0
$861$	5.77200	0.196709
$862$	0	0
$863$	10.7720	0.366683	0.183342	−	0.983049i	$-0.441309\pi$
$863$	10.7720	0.366683	0.183342	+	0.983049i	$0.441309\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.77200	−0.0941421
$868$	0	0
$869$	−68.3160	−2.31746
$870$	0	0
$871$	−7.08801	−0.240168
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.6320	−0.831764	−0.415882	−	0.909419i	$-0.636527\pi$
$877$	−24.6320	−0.831764	−0.415882	+	0.909419i	$0.636527\pi$
$878$	0	0
$879$	−11.0880	−0.373989
$880$	0	0
$881$	−10.8600	−0.365883	−0.182942	−	0.983124i	$-0.558562\pi$
$881$	−10.8600	−0.365883	−0.182942	+	0.983124i	$0.558562\pi$
$882$	0	0
$883$	−18.5440	−0.624055	−0.312028	−	0.950073i	$-0.601008\pi$
$883$	−18.5440	−0.624055	−0.312028	+	0.950073i	$0.601008\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.7200	1.53513	0.767564	−	0.640972i	$-0.221469\pi$
$887$	45.7200	1.53513	0.767564	+	0.640972i	$0.221469\pi$
$888$	0	0
$889$	−8.77200	−0.294204
$890$	0	0
$891$	4.77200	0.159868
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.3160	0.578165
$898$	0	0
$899$	−11.3160	−0.377410
$900$	0	0
$901$	14.2280	0.474003
$902$	0	0
$903$	−4.54400	−0.151215
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.4040	−1.07596	−0.537979	−	0.842958i	$-0.680812\pi$
$907$	−32.4040	−1.07596	−0.537979	+	0.842958i	$0.680812\pi$
$908$	0	0
$909$	−9.54400	−0.316555
$910$	0	0
$911$	39.6320	1.31307	0.656534	−	0.754297i	$-0.272022\pi$
$911$	39.6320	1.31307	0.656534	+	0.754297i	$0.272022\pi$
$912$	0	0
$913$	75.2640	2.49088
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.5440	0.447262
$918$	0	0
$919$	−29.8600	−0.984991	−0.492495	−	0.870315i	$-0.663915\pi$
$919$	−29.8600	−0.984991	−0.492495	+	0.870315i	$0.663915\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	7.08801	0.233305
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.22800	0.0731771
$928$	0	0
$929$	−27.7720	−0.911170	−0.455585	−	0.890192i	$-0.650570\pi$
$929$	−27.7720	−0.911170	−0.455585	+	0.890192i	$0.650570\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	25.0880	0.821344
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.6320	1.00070	0.500352	−	0.865822i	$-0.333204\pi$
$937$	30.6320	1.00070	0.500352	+	0.865822i	$0.333204\pi$
$938$	0	0
$939$	−27.5440	−0.898865
$940$	0	0
$941$	−45.5440	−1.48469	−0.742346	−	0.670017i	$-0.766287\pi$
$941$	−45.5440	−1.48469	−0.742346	+	0.670017i	$0.766287\pi$
$942$	0	0
$943$	17.3160	0.563887
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.1760	1.11057	0.555286	−	0.831660i	$-0.312609\pi$
$947$	34.1760	1.11057	0.555286	+	0.831660i	$0.312609\pi$
$948$	0	0
$949$	11.5440	0.374734
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	−2.77200	−0.0897939	−0.0448970	−	0.998992i	$-0.514296\pi$
$953$	−2.77200	−0.0897939	−0.0448970	+	0.998992i	$0.514296\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.3160	0.462771
$958$	0	0
$959$	−9.54400	−0.308192
$960$	0	0
$961$	−16.7720	−0.541032
$962$	0	0
$963$	−15.5440	−0.500899
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	1.54400	0.0494985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.4920	1.80734	0.903670	−	0.428230i	$-0.140863\pi$
$977$	56.4920	1.80734	0.903670	+	0.428230i	$0.140863\pi$
$978$	0	0
$979$	2.17601	0.0695457
$980$	0	0
$981$	10.7720	0.343924
$982$	0	0
$983$	34.4560	1.09898	0.549488	−	0.835502i	$-0.314823\pi$
$983$	34.4560	1.09898	0.549488	+	0.835502i	$0.314823\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.6320	−0.433473
$990$	0	0
$991$	−31.6840	−1.00648	−0.503238	−	0.864148i	$-0.667858\pi$
$991$	−31.6840	−1.00648	−0.503238	+	0.864148i	$0.667858\pi$
$992$	0	0
$993$	26.5440	0.842348
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.7200	−1.06792	−0.533962	−	0.845509i	$-0.679297\pi$
$997$	−33.7200	−1.06792	−0.533962	+	0.845509i	$0.679297\pi$
$998$	0	0
$999$	0.772002	0.0244251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bl.1.2	yes	2
4.3	odd	2		8400.2.a.cv.1.1		2
5.2	odd	4		4200.2.t.v.1849.2		4
5.3	odd	4		4200.2.t.v.1849.4		4
5.4	even	2		4200.2.a.bh.1.2	✓	2
20.19	odd	2		8400.2.a.dd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4200.2.a.bh.1.2	✓	2	5.4	even	2
4200.2.a.bl.1.2	yes	2	1.1	even	1	trivial
4200.2.t.v.1849.2		4	5.2	odd	4
4200.2.t.v.1849.4		4	5.3	odd	4
8400.2.a.cv.1.1		2	4.3	odd	2
8400.2.a.dd.1.1		2	20.19	odd	2

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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:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +4.77200 q^{11} +5.77200 q^{13} -3.77200 q^{17} +1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{27} +3.00000 q^{29} -3.77200 q^{31} +4.77200 q^{33} +0.772002 q^{37} +5.77200 q^{39} +5.77200 q^{41} -4.54400 q^{43} +1.00000 q^{49} -3.77200 q^{51} -3.77200 q^{53} +5.77200 q^{59} -1.77200 q^{61} +1.00000 q^{63} -1.22800 q^{67} +3.00000 q^{69} +1.22800 q^{71} +2.00000 q^{73} +4.77200 q^{77} -14.3160 q^{79} +1.00000 q^{81} +15.7720 q^{83} +3.00000 q^{87} +0.455996 q^{89} +5.77200 q^{91} -3.77200 q^{93} -10.0000 q^{97} +4.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + q^{17} + 2 q^{21} + 6 q^{23} + 2 q^{27} + 6 q^{29} + q^{31} + q^{33} - 7 q^{37} + 3 q^{39} + 3 q^{41} + 8 q^{43} + 2 q^{49} + q^{51} + q^{53} + 3 q^{59} + 5 q^{61} + 2 q^{63} - 11 q^{67} + 6 q^{69} + 11 q^{71} + 4 q^{73} + q^{77} - 3 q^{79} + 2 q^{81} + 23 q^{83} + 6 q^{87} + 18 q^{89} + 3 q^{91} + q^{93} - 20 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + q^17 + 2 * q^21 + 6 * q^23 + 2 * q^27 + 6 * q^29 + q^31 + q^33 - 7 * q^37 + 3 * q^39 + 3 * q^41 + 8 * q^43 + 2 * q^49 + q^51 + q^53 + 3 * q^59 + 5 * q^61 + 2 * q^63 - 11 * q^67 + 6 * q^69 + 11 * q^71 + 4 * q^73 + q^77 - 3 * q^79 + 2 * q^81 + 23 * q^83 + 6 * q^87 + 18 * q^89 + 3 * q^91 + q^93 - 20 * q^97 + q^99

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