LMFDB - Embedded newform 6900.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, 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("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
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			1.1
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -4.77200 q^{11} -1.00000 q^{13} -1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{27} +2.77200 q^{29} +5.77200 q^{31} -4.77200 q^{33} +10.0000 q^{37} -1.00000 q^{39} +6.77200 q^{41} -10.5440 q^{43} -3.54400 q^{47} -6.00000 q^{49} -4.00000 q^{53} -1.00000 q^{57} +11.5440 q^{59} -11.3160 q^{61} -1.00000 q^{63} +1.77200 q^{67} +1.00000 q^{69} +4.00000 q^{71} +16.7720 q^{73} +4.77200 q^{77} -16.3160 q^{79} +1.00000 q^{81} +16.7720 q^{83} +2.77200 q^{87} -7.54400 q^{89} +1.00000 q^{91} +5.77200 q^{93} +11.7720 q^{97} -4.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{13} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{27} - 3 q^{29} + 3 q^{31} - q^{33} + 20 q^{37} - 2 q^{39} + 5 q^{41} - 4 q^{43} + 10 q^{47} - 12 q^{49} - 8 q^{53} - 2 q^{57} + 6 q^{59} + 3 q^{61} - 2 q^{63} - 5 q^{67} + 2 q^{69} + 8 q^{71} + 25 q^{73} + q^{77} - 7 q^{79} + 2 q^{81} + 25 q^{83} - 3 q^{87} + 2 q^{89} + 2 q^{91} + 3 q^{93} + 15 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - q^11 - 2 * q^13 - 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^27 - 3 * q^29 + 3 * q^31 - q^33 + 20 * q^37 - 2 * q^39 + 5 * q^41 - 4 * q^43 + 10 * q^47 - 12 * q^49 - 8 * q^53 - 2 * q^57 + 6 * q^59 + 3 * q^61 - 2 * q^63 - 5 * q^67 + 2 * q^69 + 8 * q^71 + 25 * q^73 + q^77 - 7 * q^79 + 2 * q^81 + 25 * q^83 - 3 * q^87 + 2 * q^89 + 2 * q^91 + 3 * q^93 + 15 * 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	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.77200	−1.43881	−0.719406	−	0.694589i	$-0.755586\pi$
$11$	−4.77200	−1.43881	−0.719406	+	0.694589i	$0.755586\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.77200	0.514748	0.257374	−	0.966312i	$-0.417143\pi$
$29$	2.77200	0.514748	0.257374	+	0.966312i	$0.417143\pi$
$30$	0	0
$31$	5.77200	1.03668	0.518341	−	0.855174i	$-0.326550\pi$
$31$	5.77200	1.03668	0.518341	+	0.855174i	$0.326550\pi$
$32$	0	0
$33$	−4.77200	−0.830699
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.77200	1.05761	0.528805	−	0.848744i	$-0.322640\pi$
$41$	6.77200	1.05761	0.528805	+	0.848744i	$0.322640\pi$
$42$	0	0
$43$	−10.5440	−1.60795	−0.803973	−	0.594666i	$-0.797284\pi$
$43$	−10.5440	−1.60795	−0.803973	+	0.594666i	$0.797284\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.54400	−0.516946	−0.258473	−	0.966018i	$-0.583219\pi$
$47$	−3.54400	−0.516946	−0.258473	+	0.966018i	$0.583219\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	11.5440	1.50290	0.751451	−	0.659789i	$-0.229354\pi$
$59$	11.5440	1.50290	0.751451	+	0.659789i	$0.229354\pi$
$60$	0	0
$61$	−11.3160	−1.44887	−0.724433	−	0.689345i	$-0.757898\pi$
$61$	−11.3160	−1.44887	−0.724433	+	0.689345i	$0.757898\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.77200	0.216484	0.108242	−	0.994125i	$-0.465478\pi$
$67$	1.77200	0.216484	0.108242	+	0.994125i	$0.465478\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	16.7720	1.96301	0.981507	−	0.191425i	$-0.0613110\pi$
$73$	16.7720	1.96301	0.981507	+	0.191425i	$0.0613110\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.77200	0.543820
$78$	0	0
$79$	−16.3160	−1.83569	−0.917847	−	0.396934i	$-0.870074\pi$
$79$	−16.3160	−1.83569	−0.917847	+	0.396934i	$0.870074\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.7720	1.84097	0.920483	−	0.390782i	$-0.127795\pi$
$83$	16.7720	1.84097	0.920483	+	0.390782i	$0.127795\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.77200	0.297190
$88$	0	0
$89$	−7.54400	−0.799663	−0.399831	−	0.916589i	$-0.630931\pi$
$89$	−7.54400	−0.799663	−0.399831	+	0.916589i	$0.630931\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	5.77200	0.598529
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.7720	1.19527	0.597633	−	0.801770i	$-0.296108\pi$
$97$	11.7720	1.19527	0.597633	+	0.801770i	$0.296108\pi$
$98$	0	0
$99$	−4.77200	−0.479604
$100$	0	0
$101$	17.5440	1.74569	0.872847	−	0.487994i	$-0.162271\pi$
$101$	17.5440	1.74569	0.872847	+	0.487994i	$0.162271\pi$
$102$	0	0
$103$	2.77200	0.273133	0.136567	−	0.990631i	$-0.456393\pi$
$103$	2.77200	0.273133	0.136567	+	0.990631i	$0.456393\pi$
$104$	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
$109$	1.77200	0.169727	0.0848635	−	0.996393i	$-0.472955\pi$
$109$	1.77200	0.169727	0.0848635	+	0.996393i	$0.472955\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	5.54400	0.521536	0.260768	−	0.965401i	$-0.416024\pi$
$113$	5.54400	0.521536	0.260768	+	0.965401i	$0.416024\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.7720	1.07018
$122$	0	0
$123$	6.77200	0.610611
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.45600	0.217934	0.108967	−	0.994045i	$-0.465246\pi$
$127$	2.45600	0.217934	0.108967	+	0.994045i	$0.465246\pi$
$128$	0	0
$129$	−10.5440	−0.928348
$130$	0	0
$131$	17.5440	1.53283	0.766413	−	0.642348i	$-0.222039\pi$
$131$	17.5440	1.53283	0.766413	+	0.642348i	$0.222039\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5440	1.66976	0.834878	−	0.550434i	$-0.185538\pi$
$137$	19.5440	1.66976	0.834878	+	0.550434i	$0.185538\pi$
$138$	0	0
$139$	17.5440	1.48806	0.744031	−	0.668145i	$-0.232911\pi$
$139$	17.5440	1.48806	0.744031	+	0.668145i	$0.232911\pi$
$140$	0	0
$141$	−3.54400	−0.298459
$142$	0	0
$143$	4.77200	0.399055
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	11.0880	0.908365	0.454182	−	0.890909i	$-0.349931\pi$
$149$	11.0880	0.908365	0.454182	+	0.890909i	$0.349931\pi$
$150$	0	0
$151$	13.3160	1.08364	0.541821	−	0.840494i	$-0.317735\pi$
$151$	13.3160	1.08364	0.541821	+	0.840494i	$0.317735\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.7720	1.41836	0.709180	−	0.705027i	$-0.249065\pi$
$157$	17.7720	1.41836	0.709180	+	0.705027i	$0.249065\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−7.77200	−0.608750	−0.304375	−	0.952552i	$-0.598448\pi$
$163$	−7.77200	−0.608750	−0.304375	+	0.952552i	$0.598448\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.54400	0.274243	0.137122	−	0.990554i	$-0.456215\pi$
$167$	3.54400	0.274243	0.137122	+	0.990554i	$0.456215\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	13.2280	1.00571	0.502853	−	0.864372i	$-0.332284\pi$
$173$	13.2280	1.00571	0.502853	+	0.864372i	$0.332284\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.5440	0.867700
$178$	0	0
$179$	−25.0880	−1.87517	−0.937583	−	0.347762i	$-0.886942\pi$
$179$	−25.0880	−1.87517	−0.937583	+	0.347762i	$0.886942\pi$
$180$	0	0
$181$	−13.3160	−0.989771	−0.494885	−	0.868958i	$-0.664790\pi$
$181$	−13.3160	−0.989771	−0.494885	+	0.868958i	$0.664790\pi$
$182$	0	0
$183$	−11.3160	−0.836503
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−0.316006	−0.0228654	−0.0114327	−	0.999935i	$-0.503639\pi$
$191$	−0.316006	−0.0228654	−0.0114327	+	0.999935i	$0.503639\pi$
$192$	0	0
$193$	7.77200	0.559441	0.279720	−	0.960081i	$-0.409758\pi$
$193$	7.77200	0.559441	0.279720	+	0.960081i	$0.409758\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.3160	−1.58995	−0.794975	−	0.606642i	$-0.792516\pi$
$197$	−22.3160	−1.58995	−0.794975	+	0.606642i	$0.792516\pi$
$198$	0	0
$199$	−3.45600	−0.244989	−0.122495	−	0.992469i	$-0.539089\pi$
$199$	−3.45600	−0.244989	−0.122495	+	0.992469i	$0.539089\pi$
$200$	0	0
$201$	1.77200	0.124987
$202$	0	0
$203$	−2.77200	−0.194556
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	4.77200	0.330086
$210$	0	0
$211$	9.31601	0.641340	0.320670	−	0.947191i	$-0.396092\pi$
$211$	9.31601	0.641340	0.320670	+	0.947191i	$0.396092\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.77200	−0.391829
$218$	0	0
$219$	16.7720	1.13335
$220$	0	0
$221$	0	0
$222$	0	0
$223$	21.3160	1.42743	0.713713	−	0.700439i	$-0.247012\pi$
$223$	21.3160	1.42743	0.713713	+	0.700439i	$0.247012\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.455996	−0.0302655	−0.0151328	−	0.999885i	$-0.504817\pi$
$227$	−0.455996	−0.0302655	−0.0151328	+	0.999885i	$0.504817\pi$
$228$	0	0
$229$	−22.8600	−1.51063	−0.755316	−	0.655361i	$-0.772517\pi$
$229$	−22.8600	−1.51063	−0.755316	+	0.655361i	$0.772517\pi$
$230$	0	0
$231$	4.77200	0.313975
$232$	0	0
$233$	5.22800	0.342498	0.171249	−	0.985228i	$-0.445220\pi$
$233$	5.22800	0.342498	0.171249	+	0.985228i	$0.445220\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.3160	−1.05984
$238$	0	0
$239$	−25.0880	−1.62281	−0.811404	−	0.584486i	$-0.801296\pi$
$239$	−25.0880	−1.62281	−0.811404	+	0.584486i	$0.801296\pi$
$240$	0	0
$241$	13.7720	0.887133	0.443566	−	0.896242i	$-0.353713\pi$
$241$	13.7720	0.887133	0.443566	+	0.896242i	$0.353713\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	0	0
$249$	16.7720	1.06288
$250$	0	0
$251$	8.45600	0.533738	0.266869	−	0.963733i	$-0.414011\pi$
$251$	8.45600	0.533738	0.266869	+	0.963733i	$0.414011\pi$
$252$	0	0
$253$	−4.77200	−0.300013
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	2.77200	0.171583
$262$	0	0
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.54400	−0.461686
$268$	0	0
$269$	26.3160	1.60452	0.802258	−	0.596978i	$-0.203632\pi$
$269$	26.3160	1.60452	0.802258	+	0.596978i	$0.203632\pi$
$270$	0	0
$271$	−11.0880	−0.673548	−0.336774	−	0.941585i	$-0.609336\pi$
$271$	−11.0880	−0.673548	−0.336774	+	0.941585i	$0.609336\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.5440	−0.633528	−0.316764	−	0.948504i	$-0.602596\pi$
$277$	−10.5440	−0.633528	−0.316764	+	0.948504i	$0.602596\pi$
$278$	0	0
$279$	5.77200	0.345561
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	6.22800	0.370216	0.185108	−	0.982718i	$-0.440736\pi$
$283$	6.22800	0.370216	0.185108	+	0.982718i	$0.440736\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.77200	−0.399739
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	11.7720	0.690087
$292$	0	0
$293$	−5.08801	−0.297245	−0.148622	−	0.988894i	$-0.547484\pi$
$293$	−5.08801	−0.297245	−0.148622	+	0.988894i	$0.547484\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.77200	−0.276900
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	10.5440	0.607746
$302$	0	0
$303$	17.5440	1.00788
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.77200	−0.443572	−0.221786	−	0.975095i	$-0.571189\pi$
$307$	−7.77200	−0.443572	−0.221786	+	0.975095i	$0.571189\pi$
$308$	0	0
$309$	2.77200	0.157694
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	1.77200	0.100159	0.0500797	−	0.998745i	$-0.484052\pi$
$313$	1.77200	0.100159	0.0500797	+	0.998745i	$0.484052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.3160	1.81505	0.907524	−	0.420001i	$-0.137970\pi$
$317$	32.3160	1.81505	0.907524	+	0.420001i	$0.137970\pi$
$318$	0	0
$319$	−13.2280	−0.740626
$320$	0	0
$321$	−15.5440	−0.867582
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.77200	0.0979919
$328$	0	0
$329$	3.54400	0.195387
$330$	0	0
$331$	−15.0880	−0.829312	−0.414656	−	0.909978i	$-0.636098\pi$
$331$	−15.0880	−0.829312	−0.414656	+	0.909978i	$0.636098\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.3160	1.37905	0.689525	−	0.724262i	$-0.257819\pi$
$337$	25.3160	1.37905	0.689525	+	0.724262i	$0.257819\pi$
$338$	0	0
$339$	5.54400	0.301109
$340$	0	0
$341$	−27.5440	−1.49159
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.08801	0.487870	0.243935	−	0.969792i	$-0.421562\pi$
$347$	9.08801	0.487870	0.243935	+	0.969792i	$0.421562\pi$
$348$	0	0
$349$	15.8600	0.848967	0.424483	−	0.905436i	$-0.360456\pi$
$349$	15.8600	0.848967	0.424483	+	0.905436i	$0.360456\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	10.7720	0.573336	0.286668	−	0.958030i	$-0.407452\pi$
$353$	10.7720	0.573336	0.286668	+	0.958030i	$0.407452\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.3160	1.49446	0.747231	−	0.664565i	$-0.231383\pi$
$359$	28.3160	1.49446	0.747231	+	0.664565i	$0.231383\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	11.7720	0.617870
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.5440	0.550393	0.275196	−	0.961388i	$-0.411257\pi$
$367$	10.5440	0.550393	0.275196	+	0.961388i	$0.411257\pi$
$368$	0	0
$369$	6.77200	0.352536
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−32.4040	−1.67782	−0.838908	−	0.544273i	$-0.816806\pi$
$373$	−32.4040	−1.67782	−0.838908	+	0.544273i	$0.816806\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.77200	−0.142765
$378$	0	0
$379$	−25.7720	−1.32382	−0.661909	−	0.749584i	$-0.730254\pi$
$379$	−25.7720	−1.32382	−0.661909	+	0.749584i	$0.730254\pi$
$380$	0	0
$381$	2.45600	0.125824
$382$	0	0
$383$	−8.31601	−0.424928	−0.212464	−	0.977169i	$-0.568149\pi$
$383$	−8.31601	−0.424928	−0.212464	+	0.977169i	$0.568149\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.5440	−0.535982
$388$	0	0
$389$	−30.6320	−1.55310	−0.776552	−	0.630053i	$-0.783033\pi$
$389$	−30.6320	−1.55310	−0.776552	+	0.630053i	$0.783033\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	17.5440	0.884978
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.31601	−0.367180	−0.183590	−	0.983003i	$-0.558772\pi$
$397$	−7.31601	−0.367180	−0.183590	+	0.983003i	$0.558772\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−5.77200	−0.287524
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−47.7200	−2.36539
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	19.5440	0.964035
$412$	0	0
$413$	−11.5440	−0.568043
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.5440	0.859134
$418$	0	0
$419$	−21.8600	−1.06793	−0.533966	−	0.845506i	$-0.679299\pi$
$419$	−21.8600	−1.06793	−0.533966	+	0.845506i	$0.679299\pi$
$420$	0	0
$421$	22.6320	1.10302	0.551508	−	0.834169i	$-0.314052\pi$
$421$	22.6320	1.10302	0.551508	+	0.834169i	$0.314052\pi$
$422$	0	0
$423$	−3.54400	−0.172315
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.3160	0.547620
$428$	0	0
$429$	4.77200	0.230394
$430$	0	0
$431$	3.54400	0.170709	0.0853543	−	0.996351i	$-0.472798\pi$
$431$	3.54400	0.170709	0.0853543	+	0.996351i	$0.472798\pi$
$432$	0	0
$433$	4.86001	0.233557	0.116779	−	0.993158i	$-0.462743\pi$
$433$	4.86001	0.233557	0.116779	+	0.993158i	$0.462743\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	−10.2280	−0.488156	−0.244078	−	0.969756i	$-0.578485\pi$
$439$	−10.2280	−0.488156	−0.244078	+	0.969756i	$0.578485\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−6.45600	−0.306734	−0.153367	−	0.988169i	$-0.549012\pi$
$443$	−6.45600	−0.306734	−0.153367	+	0.988169i	$0.549012\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.0880	0.524445
$448$	0	0
$449$	−21.0880	−0.995205	−0.497602	−	0.867405i	$-0.665786\pi$
$449$	−21.0880	−0.995205	−0.497602	+	0.867405i	$0.665786\pi$
$450$	0	0
$451$	−32.3160	−1.52170
$452$	0	0
$453$	13.3160	0.625641
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.5440	1.28845	0.644227	−	0.764834i	$-0.277179\pi$
$457$	27.5440	1.28845	0.644227	+	0.764834i	$0.277179\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.2280	1.08184	0.540918	−	0.841075i	$-0.318077\pi$
$461$	23.2280	1.08184	0.540918	+	0.841075i	$0.318077\pi$
$462$	0	0
$463$	−24.6320	−1.14475	−0.572373	−	0.819993i	$-0.693977\pi$
$463$	−24.6320	−1.14475	−0.572373	+	0.819993i	$0.693977\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.3160	0.662466	0.331233	−	0.943549i	$-0.392535\pi$
$467$	14.3160	0.662466	0.331233	+	0.943549i	$0.392535\pi$
$468$	0	0
$469$	−1.77200	−0.0818234
$470$	0	0
$471$	17.7720	0.818891
$472$	0	0
$473$	50.3160	2.31353
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−5.22800	−0.238873	−0.119437	−	0.992842i	$-0.538109\pi$
$479$	−5.22800	−0.238873	−0.119437	+	0.992842i	$0.538109\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.7720	−0.533440	−0.266720	−	0.963774i	$-0.585940\pi$
$487$	−11.7720	−0.533440	−0.266720	+	0.963774i	$0.585940\pi$
$488$	0	0
$489$	−7.77200	−0.351462
$490$	0	0
$491$	−9.08801	−0.410136	−0.205068	−	0.978748i	$-0.565742\pi$
$491$	−9.08801	−0.410136	−0.205068	+	0.978748i	$0.565742\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	2.86001	0.128032	0.0640158	−	0.997949i	$-0.479609\pi$
$499$	2.86001	0.128032	0.0640158	+	0.997949i	$0.479609\pi$
$500$	0	0
$501$	3.54400	0.158334
$502$	0	0
$503$	36.7720	1.63958	0.819791	−	0.572662i	$-0.194089\pi$
$503$	36.7720	1.63958	0.819791	+	0.572662i	$0.194089\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−28.6320	−1.26909	−0.634546	−	0.772885i	$-0.718813\pi$
$509$	−28.6320	−1.26909	−0.634546	+	0.772885i	$0.718813\pi$
$510$	0	0
$511$	−16.7720	−0.741950
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.9120	0.743789
$518$	0	0
$519$	13.2280	0.580645
$520$	0	0
$521$	−2.91199	−0.127577	−0.0637884	−	0.997963i	$-0.520318\pi$
$521$	−2.91199	−0.127577	−0.0637884	+	0.997963i	$0.520318\pi$
$522$	0	0
$523$	21.6320	0.945902	0.472951	−	0.881089i	$-0.343189\pi$
$523$	21.6320	0.945902	0.472951	+	0.881089i	$0.343189\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.5440	0.500967
$532$	0	0
$533$	−6.77200	−0.293328
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−25.0880	−1.08263
$538$	0	0
$539$	28.6320	1.23327
$540$	0	0
$541$	11.4560	0.492532	0.246266	−	0.969202i	$-0.420796\pi$
$541$	11.4560	0.492532	0.246266	+	0.969202i	$0.420796\pi$
$542$	0	0
$543$	−13.3160	−0.571444
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.08801	−0.303061	−0.151531	−	0.988453i	$-0.548420\pi$
$547$	−7.08801	−0.303061	−0.151531	+	0.988453i	$0.548420\pi$
$548$	0	0
$549$	−11.3160	−0.482955
$550$	0	0
$551$	−2.77200	−0.118091
$552$	0	0
$553$	16.3160	0.693827
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.6320	1.04369	0.521846	−	0.853040i	$-0.325244\pi$
$557$	24.6320	1.04369	0.521846	+	0.853040i	$0.325244\pi$
$558$	0	0
$559$	10.5440	0.445964
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.3160	0.856218	0.428109	−	0.903727i	$-0.359180\pi$
$563$	20.3160	0.856218	0.428109	+	0.903727i	$0.359180\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−3.54400	−0.148572	−0.0742862	−	0.997237i	$-0.523668\pi$
$569$	−3.54400	−0.148572	−0.0742862	+	0.997237i	$0.523668\pi$
$570$	0	0
$571$	24.8600	1.04036	0.520180	−	0.854057i	$-0.325865\pi$
$571$	24.8600	1.04036	0.520180	+	0.854057i	$0.325865\pi$
$572$	0	0
$573$	−0.316006	−0.0132013
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−42.7200	−1.77846	−0.889229	−	0.457463i	$-0.848758\pi$
$577$	−42.7200	−1.77846	−0.889229	+	0.457463i	$0.848758\pi$
$578$	0	0
$579$	7.77200	0.322993
$580$	0	0
$581$	−16.7720	−0.695820
$582$	0	0
$583$	19.0880	0.790544
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.6320	0.438830	0.219415	−	0.975632i	$-0.429585\pi$
$587$	10.6320	0.438830	0.219415	+	0.975632i	$0.429585\pi$
$588$	0	0
$589$	−5.77200	−0.237831
$590$	0	0
$591$	−22.3160	−0.917958
$592$	0	0
$593$	−14.3160	−0.587888	−0.293944	−	0.955823i	$-0.594968\pi$
$593$	−14.3160	−0.587888	−0.293944	+	0.955823i	$0.594968\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.45600	−0.141445
$598$	0	0
$599$	23.0880	0.943350	0.471675	−	0.881772i	$-0.343649\pi$
$599$	23.0880	0.943350	0.471675	+	0.881772i	$0.343649\pi$
$600$	0	0
$601$	3.31601	0.135263	0.0676313	−	0.997710i	$-0.478456\pi$
$601$	3.31601	0.135263	0.0676313	+	0.997710i	$0.478456\pi$
$602$	0	0
$603$	1.77200	0.0721615
$604$	0	0
$605$	0	0
$606$	0	0
$607$	43.7200	1.77454	0.887270	−	0.461250i	$-0.152599\pi$
$607$	43.7200	1.77454	0.887270	+	0.461250i	$0.152599\pi$
$608$	0	0
$609$	−2.77200	−0.112327
$610$	0	0
$611$	3.54400	0.143375
$612$	0	0
$613$	17.0880	0.690178	0.345089	−	0.938570i	$-0.387849\pi$
$613$	17.0880	0.690178	0.345089	+	0.938570i	$0.387849\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.0880	−1.33207	−0.666037	−	0.745919i	$-0.732011\pi$
$617$	−33.0880	−1.33207	−0.666037	+	0.745919i	$0.732011\pi$
$618$	0	0
$619$	−49.3160	−1.98218	−0.991089	−	0.133203i	$-0.957474\pi$
$619$	−49.3160	−1.98218	−0.991089	+	0.133203i	$0.957474\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	7.54400	0.302244
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.77200	0.190575
$628$	0	0
$629$	0	0
$630$	0	0
$631$	38.5440	1.53441	0.767206	−	0.641400i	$-0.221646\pi$
$631$	38.5440	1.53441	0.767206	+	0.641400i	$0.221646\pi$
$632$	0	0
$633$	9.31601	0.370278
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−9.22800	−0.363917	−0.181958	−	0.983306i	$-0.558244\pi$
$643$	−9.22800	−0.363917	−0.181958	+	0.983306i	$0.558244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.0880	0.986311	0.493156	−	0.869941i	$-0.335843\pi$
$647$	25.0880	0.986311	0.493156	+	0.869941i	$0.335843\pi$
$648$	0	0
$649$	−55.0880	−2.16239
$650$	0	0
$651$	−5.77200	−0.226223
$652$	0	0
$653$	−15.8600	−0.620650	−0.310325	−	0.950631i	$-0.600438\pi$
$653$	−15.8600	−0.620650	−0.310325	+	0.950631i	$0.600438\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16.7720	0.654338
$658$	0	0
$659$	−26.3160	−1.02513	−0.512563	−	0.858650i	$-0.671304\pi$
$659$	−26.3160	−1.02513	−0.512563	+	0.858650i	$0.671304\pi$
$660$	0	0
$661$	−23.5440	−0.915756	−0.457878	−	0.889015i	$-0.651390\pi$
$661$	−23.5440	−0.915756	−0.457878	+	0.889015i	$0.651390\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.77200	0.107332
$668$	0	0
$669$	21.3160	0.824124
$670$	0	0
$671$	54.0000	2.08465
$672$	0	0
$673$	−11.8600	−0.457170	−0.228585	−	0.973524i	$-0.573410\pi$
$673$	−11.8600	−0.457170	−0.228585	+	0.973524i	$0.573410\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.54400	0.136207	0.0681036	−	0.997678i	$-0.478305\pi$
$677$	3.54400	0.136207	0.0681036	+	0.997678i	$0.478305\pi$
$678$	0	0
$679$	−11.7720	−0.451768
$680$	0	0
$681$	−0.455996	−0.0174738
$682$	0	0
$683$	9.08801	0.347743	0.173871	−	0.984768i	$-0.444372\pi$
$683$	9.08801	0.347743	0.173871	+	0.984768i	$0.444372\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.8600	−0.872164
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	0	0
$693$	4.77200	0.181273
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	5.22800	0.197741
$700$	0	0
$701$	16.6320	0.628182	0.314091	−	0.949393i	$-0.398300\pi$
$701$	16.6320	0.628182	0.314091	+	0.949393i	$0.398300\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.5440	−0.659810
$708$	0	0
$709$	1.77200	0.0665489	0.0332745	−	0.999446i	$-0.489406\pi$
$709$	1.77200	0.0665489	0.0332745	+	0.999446i	$0.489406\pi$
$710$	0	0
$711$	−16.3160	−0.611898
$712$	0	0
$713$	5.77200	0.216163
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.0880	−0.936929
$718$	0	0
$719$	29.5440	1.10181	0.550903	−	0.834569i	$-0.314284\pi$
$719$	29.5440	1.10181	0.550903	+	0.834569i	$0.314284\pi$
$720$	0	0
$721$	−2.77200	−0.103235
$722$	0	0
$723$	13.7720	0.512186
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.86001	0.180248	0.0901239	−	0.995931i	$-0.471274\pi$
$727$	4.86001	0.180248	0.0901239	+	0.995931i	$0.471274\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.08801	0.0401865	0.0200932	−	0.999798i	$-0.493604\pi$
$733$	1.08801	0.0401865	0.0200932	+	0.999798i	$0.493604\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.45600	−0.311481
$738$	0	0
$739$	−47.7200	−1.75541	−0.877705	−	0.479202i	$-0.840926\pi$
$739$	−47.7200	−1.75541	−0.877705	+	0.479202i	$0.840926\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	25.8600	0.948712	0.474356	−	0.880333i	$-0.342681\pi$
$743$	25.8600	0.948712	0.474356	+	0.880333i	$0.342681\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.7720	0.613655
$748$	0	0
$749$	15.5440	0.567966
$750$	0	0
$751$	28.3160	1.03327	0.516633	−	0.856207i	$-0.327185\pi$
$751$	28.3160	1.03327	0.516633	+	0.856207i	$0.327185\pi$
$752$	0	0
$753$	8.45600	0.308154
$754$	0	0
$755$	0	0
$756$	0	0
$757$	52.4040	1.90466	0.952328	−	0.305076i	$-0.0986817\pi$
$757$	52.4040	1.90466	0.952328	+	0.305076i	$0.0986817\pi$
$758$	0	0
$759$	−4.77200	−0.173213
$760$	0	0
$761$	−38.3160	−1.38895	−0.694477	−	0.719515i	$-0.744364\pi$
$761$	−38.3160	−1.38895	−0.694477	+	0.719515i	$0.744364\pi$
$762$	0	0
$763$	−1.77200	−0.0641508
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.5440	−0.416830
$768$	0	0
$769$	−12.2280	−0.440953	−0.220476	−	0.975392i	$-0.570761\pi$
$769$	−12.2280	−0.440953	−0.220476	+	0.975392i	$0.570761\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	−6.77200	−0.242632
$780$	0	0
$781$	−19.0880	−0.683023
$782$	0	0
$783$	2.77200	0.0990633
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−51.6320	−1.84048	−0.920241	−	0.391352i	$-0.872008\pi$
$787$	−51.6320	−1.84048	−0.920241	+	0.391352i	$0.872008\pi$
$788$	0	0
$789$	20.0000	0.712019
$790$	0	0
$791$	−5.54400	−0.197122
$792$	0	0
$793$	11.3160	0.401843
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.6320	−0.801667	−0.400833	−	0.916151i	$-0.631279\pi$
$797$	−22.6320	−0.801667	−0.400833	+	0.916151i	$0.631279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−7.54400	−0.266554
$802$	0	0
$803$	−80.0360	−2.82441
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.3160	0.926367
$808$	0	0
$809$	15.8600	0.557608	0.278804	−	0.960348i	$-0.410062\pi$
$809$	15.8600	0.557608	0.278804	+	0.960348i	$0.410062\pi$
$810$	0	0
$811$	−47.3160	−1.66149	−0.830745	−	0.556653i	$-0.812085\pi$
$811$	−47.3160	−1.66149	−0.830745	+	0.556653i	$0.812085\pi$
$812$	0	0
$813$	−11.0880	−0.388873
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.5440	0.368888
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	8.31601	0.290231	0.145115	−	0.989415i	$-0.453645\pi$
$821$	8.31601	0.290231	0.145115	+	0.989415i	$0.453645\pi$
$822$	0	0
$823$	−36.8600	−1.28486	−0.642430	−	0.766345i	$-0.722073\pi$
$823$	−36.8600	−1.28486	−0.642430	+	0.766345i	$0.722073\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.3160	−0.706457	−0.353228	−	0.935537i	$-0.614916\pi$
$827$	−20.3160	−0.706457	−0.353228	+	0.935537i	$0.614916\pi$
$828$	0	0
$829$	19.8600	0.689767	0.344883	−	0.938646i	$-0.387918\pi$
$829$	19.8600	0.689767	0.344883	+	0.938646i	$0.387918\pi$
$830$	0	0
$831$	−10.5440	−0.365767
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.77200	0.199510
$838$	0	0
$839$	21.8600	0.754691	0.377346	−	0.926072i	$-0.376837\pi$
$839$	21.8600	0.754691	0.377346	+	0.926072i	$0.376837\pi$
$840$	0	0
$841$	−21.3160	−0.735035
$842$	0	0
$843$	2.00000	0.0688837
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.7720	−0.404491
$848$	0	0
$849$	6.22800	0.213744
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	−36.5440	−1.25124	−0.625621	−	0.780127i	$-0.715155\pi$
$853$	−36.5440	−1.25124	−0.625621	+	0.780127i	$0.715155\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.4560	1.17699	0.588497	−	0.808499i	$-0.299720\pi$
$857$	34.4560	1.17699	0.588497	+	0.808499i	$0.299720\pi$
$858$	0	0
$859$	26.1760	0.893114	0.446557	−	0.894755i	$-0.352650\pi$
$859$	26.1760	0.893114	0.446557	+	0.894755i	$0.352650\pi$
$860$	0	0
$861$	−6.77200	−0.230789
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	77.8600	2.64122
$870$	0	0
$871$	−1.77200	−0.0600420
$872$	0	0
$873$	11.7720	0.398422
$874$	0	0
$875$	0	0
$876$	0	0
$877$	43.7720	1.47808	0.739038	−	0.673664i	$-0.235280\pi$
$877$	43.7720	1.47808	0.739038	+	0.673664i	$0.235280\pi$
$878$	0	0
$879$	−5.08801	−0.171614
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	−17.7720	−0.598075	−0.299038	−	0.954241i	$-0.596666\pi$
$883$	−17.7720	−0.598075	−0.299038	+	0.954241i	$0.596666\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.4560	−0.351078	−0.175539	−	0.984472i	$-0.556167\pi$
$887$	−10.4560	−0.351078	−0.175539	+	0.984472i	$0.556167\pi$
$888$	0	0
$889$	−2.45600	−0.0823715
$890$	0	0
$891$	−4.77200	−0.159868
$892$	0	0
$893$	3.54400	0.118596
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	16.0000	0.533630
$900$	0	0
$901$	0	0
$902$	0	0
$903$	10.5440	0.350882
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−52.9480	−1.75811	−0.879055	−	0.476720i	$-0.841825\pi$
$907$	−52.9480	−1.75811	−0.879055	+	0.476720i	$0.841825\pi$
$908$	0	0
$909$	17.5440	0.581898
$910$	0	0
$911$	32.3160	1.07068	0.535339	−	0.844638i	$-0.320184\pi$
$911$	32.3160	1.07068	0.535339	+	0.844638i	$0.320184\pi$
$912$	0	0
$913$	−80.0360	−2.64881
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.5440	−0.579354
$918$	0	0
$919$	−48.4040	−1.59670	−0.798351	−	0.602193i	$-0.794294\pi$
$919$	−48.4040	−1.59670	−0.798351	+	0.602193i	$0.794294\pi$
$920$	0	0
$921$	−7.77200	−0.256096
$922$	0	0
$923$	−4.00000	−0.131662
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.77200	0.0910445
$928$	0	0
$929$	54.7720	1.79701	0.898506	−	0.438962i	$-0.144654\pi$
$929$	54.7720	1.79701	0.898506	+	0.438962i	$0.144654\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.8600	0.420118	0.210059	−	0.977689i	$-0.432634\pi$
$937$	12.8600	0.420118	0.210059	+	0.977689i	$0.432634\pi$
$938$	0	0
$939$	1.77200	0.0578271
$940$	0	0
$941$	15.5440	0.506720	0.253360	−	0.967372i	$-0.418464\pi$
$941$	15.5440	0.506720	0.253360	+	0.967372i	$0.418464\pi$
$942$	0	0
$943$	6.77200	0.220527
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.6320	−0.865424	−0.432712	−	0.901532i	$-0.642443\pi$
$947$	−26.6320	−0.865424	−0.432712	+	0.901532i	$0.642443\pi$
$948$	0	0
$949$	−16.7720	−0.544442
$950$	0	0
$951$	32.3160	1.04792
$952$	0	0
$953$	37.0880	1.20140	0.600699	−	0.799475i	$-0.294889\pi$
$953$	37.0880	1.20140	0.600699	+	0.799475i	$0.294889\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.2280	−0.427600
$958$	0	0
$959$	−19.5440	−0.631109
$960$	0	0
$961$	2.31601	0.0747099
$962$	0	0
$963$	−15.5440	−0.500899
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.5440	−1.33597	−0.667983	−	0.744177i	$-0.732842\pi$
$967$	−41.5440	−1.33597	−0.667983	+	0.744177i	$0.732842\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.8600	−0.701521	−0.350761	−	0.936465i	$-0.614077\pi$
$971$	−21.8600	−0.701521	−0.350761	+	0.936465i	$0.614077\pi$
$972$	0	0
$973$	−17.5440	−0.562435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.6320	−0.468120	−0.234060	−	0.972222i	$-0.575201\pi$
$977$	−14.6320	−0.468120	−0.234060	+	0.972222i	$0.575201\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	1.77200	0.0565757
$982$	0	0
$983$	−23.2280	−0.740858	−0.370429	−	0.928861i	$-0.620789\pi$
$983$	−23.2280	−0.740858	−0.370429	+	0.928861i	$0.620789\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.54400	0.112807
$988$	0	0
$989$	−10.5440	−0.335280
$990$	0	0
$991$	3.31601	0.105336	0.0526682	−	0.998612i	$-0.483227\pi$
$991$	3.31601	0.105336	0.0526682	+	0.998612i	$0.483227\pi$
$992$	0	0
$993$	−15.0880	−0.478803
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34.3160	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$997$	34.3160	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.t.1.1	yes	2
5.2	odd	4		6900.2.f.j.6349.1		4
5.3	odd	4		6900.2.f.j.6349.3		4
5.4	even	2		6900.2.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.n.1.1	✓	2	5.4	even	2
6900.2.a.t.1.1	yes	2	1.1	even	1	trivial
6900.2.f.j.6349.1		4	5.2	odd	4
6900.2.f.j.6349.3		4	5.3	odd	4

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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -4.77200 q^{11} -1.00000 q^{13} -1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{27} +2.77200 q^{29} +5.77200 q^{31} -4.77200 q^{33} +10.0000 q^{37} -1.00000 q^{39} +6.77200 q^{41} -10.5440 q^{43} -3.54400 q^{47} -6.00000 q^{49} -4.00000 q^{53} -1.00000 q^{57} +11.5440 q^{59} -11.3160 q^{61} -1.00000 q^{63} +1.77200 q^{67} +1.00000 q^{69} +4.00000 q^{71} +16.7720 q^{73} +4.77200 q^{77} -16.3160 q^{79} +1.00000 q^{81} +16.7720 q^{83} +2.77200 q^{87} -7.54400 q^{89} +1.00000 q^{91} +5.77200 q^{93} +11.7720 q^{97} -4.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - q^{11} - 2 q^{13} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{27} - 3 q^{29} + 3 q^{31} - q^{33} + 20 q^{37} - 2 q^{39} + 5 q^{41} - 4 q^{43} + 10 q^{47} - 12 q^{49} - 8 q^{53} - 2 q^{57} + 6 q^{59} + 3 q^{61} - 2 q^{63} - 5 q^{67} + 2 q^{69} + 8 q^{71} + 25 q^{73} + q^{77} - 7 q^{79} + 2 q^{81} + 25 q^{83} - 3 q^{87} + 2 q^{89} + 2 q^{91} + 3 q^{93} + 15 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - q^11 - 2 * q^13 - 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^27 - 3 * q^29 + 3 * q^31 - q^33 + 20 * q^37 - 2 * q^39 + 5 * q^41 - 4 * q^43 + 10 * q^47 - 12 * q^49 - 8 * q^53 - 2 * q^57 + 6 * q^59 + 3 * q^61 - 2 * q^63 - 5 * q^67 + 2 * q^69 + 8 * q^71 + 25 * q^73 + q^77 - 7 * q^79 + 2 * q^81 + 25 * q^83 - 3 * q^87 + 2 * q^89 + 2 * q^91 + 3 * q^93 + 15 * q^97 - q^99

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