LMFDB - Embedded newform 6900.2.a.n.1.2

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:	$1$
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	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +3.77200 q^{11} +1.00000 q^{13} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} -5.77200 q^{29} -2.77200 q^{31} -3.77200 q^{33} -10.0000 q^{37} -1.00000 q^{39} -1.77200 q^{41} -6.54400 q^{43} -13.5440 q^{47} -6.00000 q^{49} +4.00000 q^{53} +1.00000 q^{57} -5.54400 q^{59} +14.3160 q^{61} +1.00000 q^{63} +6.77200 q^{67} +1.00000 q^{69} +4.00000 q^{71} -8.22800 q^{73} +3.77200 q^{77} +9.31601 q^{79} +1.00000 q^{81} -8.22800 q^{83} +5.77200 q^{87} +9.54400 q^{89} +1.00000 q^{91} +2.77200 q^{93} -3.22800 q^{97} +3.77200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.77200	1.13730	0.568651	−	0.822579i	$-0.307466\pi$
$11$	3.77200	1.13730	0.568651	+	0.822579i	$0.307466\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	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
$29$	−5.77200	−1.07183	−0.535917	−	0.844271i	$-0.680034\pi$
$29$	−5.77200	−1.07183	−0.535917	+	0.844271i	$0.680034\pi$
$30$	0	0
$31$	−2.77200	−0.497866	−0.248933	−	0.968521i	$-0.580080\pi$
$31$	−2.77200	−0.497866	−0.248933	+	0.968521i	$0.580080\pi$
$32$	0	0
$33$	−3.77200	−0.656621
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.77200	−0.276740	−0.138370	−	0.990381i	$-0.544186\pi$
$41$	−1.77200	−0.276740	−0.138370	+	0.990381i	$0.544186\pi$
$42$	0	0
$43$	−6.54400	−0.997951	−0.498976	−	0.866616i	$-0.666290\pi$
$43$	−6.54400	−0.997951	−0.498976	+	0.866616i	$0.666290\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.5440	−1.97560	−0.987798	−	0.155741i	$-0.950224\pi$
$47$	−13.5440	−1.97560	−0.987798	+	0.155741i	$0.950224\pi$
$48$	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	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−5.54400	−0.721768	−0.360884	−	0.932611i	$-0.617525\pi$
$59$	−5.54400	−0.721768	−0.360884	+	0.932611i	$0.617525\pi$
$60$	0	0
$61$	14.3160	1.83298	0.916488	−	0.400061i	$-0.131011\pi$
$61$	14.3160	1.83298	0.916488	+	0.400061i	$0.131011\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.77200	0.827332	0.413666	−	0.910429i	$-0.364248\pi$
$67$	6.77200	0.827332	0.413666	+	0.910429i	$0.364248\pi$
$68$	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
$73$	−8.22800	−0.963014	−0.481507	−	0.876442i	$-0.659910\pi$
$73$	−8.22800	−0.963014	−0.481507	+	0.876442i	$0.659910\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.77200	0.429860
$78$	0	0
$79$	9.31601	1.04813	0.524066	−	0.851677i	$-0.324414\pi$
$79$	9.31601	1.04813	0.524066	+	0.851677i	$0.324414\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.22800	−0.903140	−0.451570	−	0.892236i	$-0.649136\pi$
$83$	−8.22800	−0.903140	−0.451570	+	0.892236i	$0.649136\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.77200	0.618824
$88$	0	0
$89$	9.54400	1.01166	0.505831	−	0.862632i	$-0.331186\pi$
$89$	9.54400	1.01166	0.505831	+	0.862632i	$0.331186\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.77200	0.287443
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.22800	−0.327754	−0.163877	−	0.986481i	$-0.552400\pi$
$97$	−3.22800	−0.327754	−0.163877	+	0.986481i	$0.552400\pi$
$98$	0	0
$99$	3.77200	0.379100
$100$	0	0
$101$	0.455996	0.0453733	0.0226867	−	0.999743i	$-0.492778\pi$
$101$	0.455996	0.0453733	0.0226867	+	0.999743i	$0.492778\pi$
$102$	0	0
$103$	5.77200	0.568732	0.284366	−	0.958716i	$-0.408217\pi$
$103$	5.77200	0.568732	0.284366	+	0.958716i	$0.408217\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.54400	−0.149264	−0.0746322	−	0.997211i	$-0.523778\pi$
$107$	−1.54400	−0.149264	−0.0746322	+	0.997211i	$0.523778\pi$
$108$	0	0
$109$	−6.77200	−0.648640	−0.324320	−	0.945947i	$-0.605135\pi$
$109$	−6.77200	−0.648640	−0.324320	+	0.945947i	$0.605135\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	11.5440	1.08597	0.542984	−	0.839743i	$-0.317294\pi$
$113$	11.5440	1.08597	0.542984	+	0.839743i	$0.317294\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.22800	0.293454
$122$	0	0
$123$	1.77200	0.159776
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.5440	−1.73425	−0.867125	−	0.498091i	$-0.834035\pi$
$127$	−19.5440	−1.73425	−0.867125	+	0.498091i	$0.834035\pi$
$128$	0	0
$129$	6.54400	0.576167
$130$	0	0
$131$	0.455996	0.0398406	0.0199203	−	0.999802i	$-0.493659\pi$
$131$	0.455996	0.0398406	0.0199203	+	0.999802i	$0.493659\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.45600	−0.209830	−0.104915	−	0.994481i	$-0.533457\pi$
$137$	−2.45600	−0.209830	−0.104915	+	0.994481i	$0.533457\pi$
$138$	0	0
$139$	0.455996	0.0386771	0.0193385	−	0.999813i	$-0.493844\pi$
$139$	0.455996	0.0386771	0.0193385	+	0.999813i	$0.493844\pi$
$140$	0	0
$141$	13.5440	1.14061
$142$	0	0
$143$	3.77200	0.315431
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−23.0880	−1.89144	−0.945722	−	0.324978i	$-0.894643\pi$
$149$	−23.0880	−1.89144	−0.945722	+	0.324978i	$0.894643\pi$
$150$	0	0
$151$	−12.3160	−1.00226	−0.501131	−	0.865371i	$-0.667083\pi$
$151$	−12.3160	−1.00226	−0.501131	+	0.865371i	$0.667083\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.22800	−0.736474	−0.368237	−	0.929732i	$-0.620039\pi$
$157$	−9.22800	−0.736474	−0.368237	+	0.929732i	$0.620039\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−0.772002	−0.0604679	−0.0302339	−	0.999543i	$-0.509625\pi$
$163$	−0.772002	−0.0604679	−0.0302339	+	0.999543i	$0.509625\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.5440	1.04807	0.524033	−	0.851698i	$-0.324427\pi$
$167$	13.5440	1.04807	0.524033	+	0.851698i	$0.324427\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−21.7720	−1.65529	−0.827647	−	0.561249i	$-0.810321\pi$
$173$	−21.7720	−1.65529	−0.827647	+	0.561249i	$0.810321\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.54400	0.416713
$178$	0	0
$179$	9.08801	0.679270	0.339635	−	0.940557i	$-0.389697\pi$
$179$	9.08801	0.679270	0.339635	+	0.940557i	$0.389697\pi$
$180$	0	0
$181$	12.3160	0.915441	0.457721	−	0.889096i	$-0.348666\pi$
$181$	12.3160	0.915441	0.457721	+	0.889096i	$0.348666\pi$
$182$	0	0
$183$	−14.3160	−1.05827
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	25.3160	1.83180	0.915901	−	0.401404i	$-0.131478\pi$
$191$	25.3160	1.83180	0.915901	+	0.401404i	$0.131478\pi$
$192$	0	0
$193$	0.772002	0.0555699	0.0277850	−	0.999614i	$-0.491155\pi$
$193$	0.772002	0.0555699	0.0277850	+	0.999614i	$0.491155\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.31601	−0.236256	−0.118128	−	0.992998i	$-0.537689\pi$
$197$	−3.31601	−0.236256	−0.118128	+	0.992998i	$0.537689\pi$
$198$	0	0
$199$	−20.5440	−1.45633	−0.728163	−	0.685404i	$-0.759626\pi$
$199$	−20.5440	−1.45633	−0.728163	+	0.685404i	$0.759626\pi$
$200$	0	0
$201$	−6.77200	−0.477660
$202$	0	0
$203$	−5.77200	−0.405115
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.77200	−0.260915
$210$	0	0
$211$	−16.3160	−1.12324	−0.561620	−	0.827395i	$-0.689822\pi$
$211$	−16.3160	−1.12324	−0.561620	+	0.827395i	$0.689822\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.77200	−0.188176
$218$	0	0
$219$	8.22800	0.555997
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.31601	0.289021	0.144511	−	0.989503i	$-0.453839\pi$
$223$	4.31601	0.289021	0.144511	+	0.989503i	$0.453839\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.5440	1.16444	0.582218	−	0.813033i	$-0.302185\pi$
$227$	17.5440	1.16444	0.582218	+	0.813033i	$0.302185\pi$
$228$	0	0
$229$	19.8600	1.31239	0.656193	−	0.754593i	$-0.272166\pi$
$229$	19.8600	1.31239	0.656193	+	0.754593i	$0.272166\pi$
$230$	0	0
$231$	−3.77200	−0.248180
$232$	0	0
$233$	−13.7720	−0.902234	−0.451117	−	0.892465i	$-0.648974\pi$
$233$	−13.7720	−0.902234	−0.451117	+	0.892465i	$0.648974\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.31601	−0.605140
$238$	0	0
$239$	9.08801	0.587854	0.293927	−	0.955828i	$-0.405038\pi$
$239$	9.08801	0.587854	0.293927	+	0.955828i	$0.405038\pi$
$240$	0	0
$241$	5.22800	0.336765	0.168382	−	0.985722i	$-0.446146\pi$
$241$	5.22800	0.336765	0.168382	+	0.985722i	$0.446146\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	8.22800	0.521428
$250$	0	0
$251$	25.5440	1.61232	0.806162	−	0.591695i	$-0.201541\pi$
$251$	25.5440	1.61232	0.806162	+	0.591695i	$0.201541\pi$
$252$	0	0
$253$	−3.77200	−0.237144
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−5.77200	−0.357278
$262$	0	0
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.54400	−0.584084
$268$	0	0
$269$	0.683994	0.0417039	0.0208519	−	0.999783i	$-0.493362\pi$
$269$	0.683994	0.0417039	0.0208519	+	0.999783i	$0.493362\pi$
$270$	0	0
$271$	23.0880	1.40250	0.701248	−	0.712917i	$-0.252626\pi$
$271$	23.0880	1.40250	0.701248	+	0.712917i	$0.252626\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.54400	−0.393191	−0.196596	−	0.980485i	$-0.562989\pi$
$277$	−6.54400	−0.393191	−0.196596	+	0.980485i	$0.562989\pi$
$278$	0	0
$279$	−2.77200	−0.165955
$280$	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
$283$	−14.7720	−0.878104	−0.439052	−	0.898462i	$-0.644686\pi$
$283$	−14.7720	−0.878104	−0.439052	+	0.898462i	$0.644686\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.77200	−0.104598
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	3.22800	0.189229
$292$	0	0
$293$	−29.0880	−1.69934	−0.849670	−	0.527315i	$-0.823199\pi$
$293$	−29.0880	−1.69934	−0.849670	+	0.527315i	$0.823199\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.77200	−0.218874
$298$	0	0
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	−6.54400	−0.377190
$302$	0	0
$303$	−0.455996	−0.0261963
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.772002	−0.0440605	−0.0220302	−	0.999757i	$-0.507013\pi$
$307$	−0.772002	−0.0440605	−0.0220302	+	0.999757i	$0.507013\pi$
$308$	0	0
$309$	−5.77200	−0.328358
$310$	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
$313$	6.77200	0.382776	0.191388	−	0.981514i	$-0.438701\pi$
$313$	6.77200	0.382776	0.191388	+	0.981514i	$0.438701\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.68399	−0.375411	−0.187705	−	0.982225i	$-0.560105\pi$
$317$	−6.68399	−0.375411	−0.187705	+	0.982225i	$0.560105\pi$
$318$	0	0
$319$	−21.7720	−1.21900
$320$	0	0
$321$	1.54400	0.0861779
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.77200	0.374493
$328$	0	0
$329$	−13.5440	−0.746705
$330$	0	0
$331$	19.0880	1.04917	0.524586	−	0.851358i	$-0.324220\pi$
$331$	19.0880	1.04917	0.524586	+	0.851358i	$0.324220\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.316006	0.0172139	0.00860696	−	0.999963i	$-0.497260\pi$
$337$	0.316006	0.0172139	0.00860696	+	0.999963i	$0.497260\pi$
$338$	0	0
$339$	−11.5440	−0.626984
$340$	0	0
$341$	−10.4560	−0.566224
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.0880	1.34679	0.673397	−	0.739281i	$-0.264834\pi$
$347$	25.0880	1.34679	0.673397	+	0.739281i	$0.264834\pi$
$348$	0	0
$349$	−26.8600	−1.43778	−0.718892	−	0.695122i	$-0.755350\pi$
$349$	−26.8600	−1.43778	−0.718892	+	0.695122i	$0.755350\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−2.22800	−0.118584	−0.0592922	−	0.998241i	$-0.518884\pi$
$353$	−2.22800	−0.118584	−0.0592922	+	0.998241i	$0.518884\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.68399	0.141656	0.0708279	−	0.997489i	$-0.477436\pi$
$359$	2.68399	0.141656	0.0708279	+	0.997489i	$0.477436\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−3.22800	−0.169426
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.54400	0.341594	0.170797	−	0.985306i	$-0.445366\pi$
$367$	6.54400	0.341594	0.170797	+	0.985306i	$0.445366\pi$
$368$	0	0
$369$	−1.77200	−0.0922467
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−27.4040	−1.41893	−0.709463	−	0.704743i	$-0.751062\pi$
$373$	−27.4040	−1.41893	−0.709463	+	0.704743i	$0.751062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.77200	−0.297273
$378$	0	0
$379$	−17.2280	−0.884943	−0.442471	−	0.896783i	$-0.645898\pi$
$379$	−17.2280	−0.884943	−0.442471	+	0.896783i	$0.645898\pi$
$380$	0	0
$381$	19.5440	1.00127
$382$	0	0
$383$	−17.3160	−0.884807	−0.442403	−	0.896816i	$-0.645874\pi$
$383$	−17.3160	−0.884807	−0.442403	+	0.896816i	$0.645874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.54400	−0.332650
$388$	0	0
$389$	20.6320	1.04608	0.523042	−	0.852307i	$-0.324797\pi$
$389$	20.6320	1.04608	0.523042	+	0.852307i	$0.324797\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−0.455996	−0.0230020
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.3160	−0.919254	−0.459627	−	0.888112i	$-0.652017\pi$
$397$	−18.3160	−0.919254	−0.459627	+	0.888112i	$0.652017\pi$
$398$	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
$403$	−2.77200	−0.138083
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−37.7200	−1.86971
$408$	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
$411$	2.45600	0.121145
$412$	0	0
$413$	−5.54400	−0.272803
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.455996	−0.0223302
$418$	0	0
$419$	20.8600	1.01908	0.509539	−	0.860448i	$-0.329816\pi$
$419$	20.8600	1.01908	0.509539	+	0.860448i	$0.329816\pi$
$420$	0	0
$421$	−28.6320	−1.39544	−0.697719	−	0.716371i	$-0.745802\pi$
$421$	−28.6320	−1.39544	−0.697719	+	0.716371i	$0.745802\pi$
$422$	0	0
$423$	−13.5440	−0.658532
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.3160	0.692800
$428$	0	0
$429$	−3.77200	−0.182114
$430$	0	0
$431$	−13.5440	−0.652392	−0.326196	−	0.945302i	$-0.605767\pi$
$431$	−13.5440	−0.652392	−0.326196	+	0.945302i	$0.605767\pi$
$432$	0	0
$433$	37.8600	1.81944	0.909718	−	0.415227i	$-0.136298\pi$
$433$	37.8600	1.81944	0.909718	+	0.415227i	$0.136298\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00000	0.0478365
$438$	0	0
$439$	−18.7720	−0.895939	−0.447969	−	0.894049i	$-0.647853\pi$
$439$	−18.7720	−0.895939	−0.447969	+	0.894049i	$0.647853\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	23.5440	1.11861	0.559305	−	0.828962i	$-0.311068\pi$
$443$	23.5440	1.11861	0.559305	+	0.828962i	$0.311068\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.0880	1.09203
$448$	0	0
$449$	13.0880	0.617661	0.308831	−	0.951117i	$-0.400062\pi$
$449$	13.0880	0.617661	0.308831	+	0.951117i	$0.400062\pi$
$450$	0	0
$451$	−6.68399	−0.314737
$452$	0	0
$453$	12.3160	0.578656
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.4560	−0.489111	−0.244555	−	0.969635i	$-0.578642\pi$
$457$	−10.4560	−0.489111	−0.244555	+	0.969635i	$0.578642\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.7720	1.47977	0.739885	−	0.672734i	$-0.234880\pi$
$461$	31.7720	1.47977	0.739885	+	0.672734i	$0.234880\pi$
$462$	0	0
$463$	−26.6320	−1.23769	−0.618847	−	0.785511i	$-0.712400\pi$
$463$	−26.6320	−1.23769	−0.618847	+	0.785511i	$0.712400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.3160	0.523642	0.261821	−	0.965116i	$-0.415677\pi$
$467$	11.3160	0.523642	0.261821	+	0.965116i	$0.415677\pi$
$468$	0	0
$469$	6.77200	0.312702
$470$	0	0
$471$	9.22800	0.425204
$472$	0	0
$473$	−24.6840	−1.13497
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	−13.7720	−0.629259	−0.314629	−	0.949215i	$-0.601880\pi$
$479$	−13.7720	−0.629259	−0.314629	+	0.949215i	$0.601880\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.22800	0.146275	0.0731373	−	0.997322i	$-0.476699\pi$
$487$	3.22800	0.146275	0.0731373	+	0.997322i	$0.476699\pi$
$488$	0	0
$489$	0.772002	0.0349111
$490$	0	0
$491$	25.0880	1.13221	0.566103	−	0.824335i	$-0.308450\pi$
$491$	25.0880	1.13221	0.566103	+	0.824335i	$0.308450\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	−39.8600	−1.78438	−0.892190	−	0.451661i	$-0.850832\pi$
$499$	−39.8600	−1.78438	−0.892190	+	0.451661i	$0.850832\pi$
$500$	0	0
$501$	−13.5440	−0.605101
$502$	0	0
$503$	−28.2280	−1.25862	−0.629312	−	0.777153i	$-0.716663\pi$
$503$	−28.2280	−1.25862	−0.629312	+	0.777153i	$0.716663\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	22.6320	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$509$	22.6320	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$510$	0	0
$511$	−8.22800	−0.363985
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−51.0880	−2.24685
$518$	0	0
$519$	21.7720	0.955685
$520$	0	0
$521$	−37.0880	−1.62486	−0.812428	−	0.583062i	$-0.801854\pi$
$521$	−37.0880	−1.62486	−0.812428	+	0.583062i	$0.801854\pi$
$522$	0	0
$523$	29.6320	1.29572	0.647859	−	0.761761i	$-0.275665\pi$
$523$	29.6320	1.29572	0.647859	+	0.761761i	$0.275665\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.54400	−0.240589
$532$	0	0
$533$	−1.77200	−0.0767539
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.08801	−0.392176
$538$	0	0
$539$	−22.6320	−0.974830
$540$	0	0
$541$	28.5440	1.22720	0.613601	−	0.789616i	$-0.289720\pi$
$541$	28.5440	1.22720	0.613601	+	0.789616i	$0.289720\pi$
$542$	0	0
$543$	−12.3160	−0.528530
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−27.0880	−1.15820	−0.579100	−	0.815257i	$-0.696596\pi$
$547$	−27.0880	−1.15820	−0.579100	+	0.815257i	$0.696596\pi$
$548$	0	0
$549$	14.3160	0.610992
$550$	0	0
$551$	5.77200	0.245896
$552$	0	0
$553$	9.31601	0.396157
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.6320	1.12843	0.564217	−	0.825626i	$-0.309178\pi$
$557$	26.6320	1.12843	0.564217	+	0.825626i	$0.309178\pi$
$558$	0	0
$559$	−6.54400	−0.276782
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.31601	0.224043	0.112021	−	0.993706i	$-0.464267\pi$
$563$	5.31601	0.224043	0.112021	+	0.993706i	$0.464267\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	13.5440	0.567794	0.283897	−	0.958855i	$-0.408373\pi$
$569$	13.5440	0.567794	0.283897	+	0.958855i	$0.408373\pi$
$570$	0	0
$571$	−17.8600	−0.747418	−0.373709	−	0.927546i	$-0.621914\pi$
$571$	−17.8600	−0.747418	−0.373709	+	0.927546i	$0.621914\pi$
$572$	0	0
$573$	−25.3160	−1.05759
$574$	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
$579$	−0.772002	−0.0320833
$580$	0	0
$581$	−8.22800	−0.341355
$582$	0	0
$583$	15.0880	0.624881
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.6320	1.67706	0.838531	−	0.544853i	$-0.183415\pi$
$587$	40.6320	1.67706	0.838531	+	0.544853i	$0.183415\pi$
$588$	0	0
$589$	2.77200	0.114218
$590$	0	0
$591$	3.31601	0.136402
$592$	0	0
$593$	−11.3160	−0.464693	−0.232346	−	0.972633i	$-0.574640\pi$
$593$	−11.3160	−0.464693	−0.232346	+	0.972633i	$0.574640\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.5440	0.840810
$598$	0	0
$599$	−11.0880	−0.453044	−0.226522	−	0.974006i	$-0.572735\pi$
$599$	−11.0880	−0.453044	−0.226522	+	0.974006i	$0.572735\pi$
$600$	0	0
$601$	−22.3160	−0.910289	−0.455144	−	0.890418i	$-0.650412\pi$
$601$	−22.3160	−0.910289	−0.455144	+	0.890418i	$0.650412\pi$
$602$	0	0
$603$	6.77200	0.275777
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.7200	1.69336	0.846682	−	0.532100i	$-0.178597\pi$
$607$	41.7200	1.69336	0.846682	+	0.532100i	$0.178597\pi$
$608$	0	0
$609$	5.77200	0.233893
$610$	0	0
$611$	−13.5440	−0.547932
$612$	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
$617$	−1.08801	−0.0438015	−0.0219008	−	0.999760i	$-0.506972\pi$
$617$	−1.08801	−0.0438015	−0.0219008	+	0.999760i	$0.506972\pi$
$618$	0	0
$619$	−23.6840	−0.951940	−0.475970	−	0.879461i	$-0.657903\pi$
$619$	−23.6840	−0.951940	−0.475970	+	0.879461i	$0.657903\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	9.54400	0.382372
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.77200	0.150639
$628$	0	0
$629$	0	0
$630$	0	0
$631$	21.4560	0.854150	0.427075	−	0.904216i	$-0.359544\pi$
$631$	21.4560	0.854150	0.427075	+	0.904216i	$0.359544\pi$
$632$	0	0
$633$	16.3160	0.648503
$634$	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
$643$	17.7720	0.700859	0.350430	−	0.936589i	$-0.386036\pi$
$643$	17.7720	0.700859	0.350430	+	0.936589i	$0.386036\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.08801	0.357286	0.178643	−	0.983914i	$-0.442829\pi$
$647$	9.08801	0.357286	0.178643	+	0.983914i	$0.442829\pi$
$648$	0	0
$649$	−20.9120	−0.820868
$650$	0	0
$651$	2.77200	0.108643
$652$	0	0
$653$	−26.8600	−1.05111	−0.525557	−	0.850759i	$-0.676143\pi$
$653$	−26.8600	−1.05111	−0.525557	+	0.850759i	$0.676143\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.22800	−0.321005
$658$	0	0
$659$	−0.683994	−0.0266446	−0.0133223	−	0.999911i	$-0.504241\pi$
$659$	−0.683994	−0.0266446	−0.0133223	+	0.999911i	$0.504241\pi$
$660$	0	0
$661$	−6.45600	−0.251109	−0.125555	−	0.992087i	$-0.540071\pi$
$661$	−6.45600	−0.251109	−0.125555	+	0.992087i	$0.540071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.77200	0.223493
$668$	0	0
$669$	−4.31601	−0.166866
$670$	0	0
$671$	54.0000	2.08465
$672$	0	0
$673$	−30.8600	−1.18957	−0.594783	−	0.803886i	$-0.702762\pi$
$673$	−30.8600	−1.18957	−0.594783	+	0.803886i	$0.702762\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.5440	0.520538	0.260269	−	0.965536i	$-0.416189\pi$
$677$	13.5440	0.520538	0.260269	+	0.965536i	$0.416189\pi$
$678$	0	0
$679$	−3.22800	−0.123879
$680$	0	0
$681$	−17.5440	−0.672288
$682$	0	0
$683$	25.0880	0.959966	0.479983	−	0.877278i	$-0.340643\pi$
$683$	25.0880	0.959966	0.479983	+	0.877278i	$0.340643\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−19.8600	−0.757707
$688$	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
$693$	3.77200	0.143287
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	13.7720	0.520905
$700$	0	0
$701$	−34.6320	−1.30803	−0.654017	−	0.756480i	$-0.726917\pi$
$701$	−34.6320	−1.30803	−0.654017	+	0.756480i	$0.726917\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.455996	0.0171495
$708$	0	0
$709$	−6.77200	−0.254328	−0.127164	−	0.991882i	$-0.540587\pi$
$709$	−6.77200	−0.254328	−0.127164	+	0.991882i	$0.540587\pi$
$710$	0	0
$711$	9.31601	0.349378
$712$	0	0
$713$	2.77200	0.103812
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.08801	−0.339398
$718$	0	0
$719$	12.4560	0.464530	0.232265	−	0.972653i	$-0.425386\pi$
$719$	12.4560	0.464530	0.232265	+	0.972653i	$0.425386\pi$
$720$	0	0
$721$	5.77200	0.214961
$722$	0	0
$723$	−5.22800	−0.194431
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.8600	1.40415	0.702075	−	0.712103i	$-0.252257\pi$
$727$	37.8600	1.40415	0.702075	+	0.712103i	$0.252257\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.0880	1.22213	0.611067	−	0.791579i	$-0.290741\pi$
$733$	33.0880	1.22213	0.611067	+	0.791579i	$0.290741\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.5440	0.940925
$738$	0	0
$739$	37.7200	1.38755	0.693777	−	0.720190i	$-0.255945\pi$
$739$	37.7200	1.38755	0.693777	+	0.720190i	$0.255945\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	16.8600	0.618534	0.309267	−	0.950975i	$-0.399916\pi$
$743$	16.8600	0.618534	0.309267	+	0.950975i	$0.399916\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.22800	−0.301047
$748$	0	0
$749$	−1.54400	−0.0564167
$750$	0	0
$751$	2.68399	0.0979403	0.0489702	−	0.998800i	$-0.484406\pi$
$751$	2.68399	0.0979403	0.0489702	+	0.998800i	$0.484406\pi$
$752$	0	0
$753$	−25.5440	−0.930875
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.40401	0.269103	0.134552	−	0.990907i	$-0.457041\pi$
$757$	7.40401	0.269103	0.134552	+	0.990907i	$0.457041\pi$
$758$	0	0
$759$	3.77200	0.136915
$760$	0	0
$761$	−12.6840	−0.459794	−0.229897	−	0.973215i	$-0.573839\pi$
$761$	−12.6840	−0.459794	−0.229897	+	0.973215i	$0.573839\pi$
$762$	0	0
$763$	−6.77200	−0.245163
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.54400	−0.200182
$768$	0	0
$769$	−20.7720	−0.749058	−0.374529	−	0.927215i	$-0.622196\pi$
$769$	−20.7720	−0.749058	−0.374529	+	0.927215i	$0.622196\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	1.77200	0.0634886
$780$	0	0
$781$	15.0880	0.539891
$782$	0	0
$783$	5.77200	0.206275
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.367989	0.0131174	0.00655869	−	0.999978i	$-0.497912\pi$
$787$	0.367989	0.0131174	0.00655869	+	0.999978i	$0.497912\pi$
$788$	0	0
$789$	20.0000	0.712019
$790$	0	0
$791$	11.5440	0.410458
$792$	0	0
$793$	14.3160	0.508376
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.6320	−1.01420	−0.507099	−	0.861888i	$-0.669282\pi$
$797$	−28.6320	−1.01420	−0.507099	+	0.861888i	$0.669282\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	9.54400	0.337221
$802$	0	0
$803$	−31.0360	−1.09524
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.683994	−0.0240777
$808$	0	0
$809$	−26.8600	−0.944348	−0.472174	−	0.881505i	$-0.656531\pi$
$809$	−26.8600	−0.944348	−0.472174	+	0.881505i	$0.656531\pi$
$810$	0	0
$811$	−21.6840	−0.761428	−0.380714	−	0.924693i	$-0.624322\pi$
$811$	−21.6840	−0.761428	−0.380714	+	0.924693i	$0.624322\pi$
$812$	0	0
$813$	−23.0880	−0.809732
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.54400	0.228946
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−17.3160	−0.604333	−0.302166	−	0.953255i	$-0.597710\pi$
$821$	−17.3160	−0.604333	−0.302166	+	0.953255i	$0.597710\pi$
$822$	0	0
$823$	−5.86001	−0.204267	−0.102134	−	0.994771i	$-0.532567\pi$
$823$	−5.86001	−0.204267	−0.102134	+	0.994771i	$0.532567\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.31601	−0.184856	−0.0924278	−	0.995719i	$-0.529463\pi$
$827$	−5.31601	−0.184856	−0.0924278	+	0.995719i	$0.529463\pi$
$828$	0	0
$829$	−22.8600	−0.793961	−0.396980	−	0.917827i	$-0.629942\pi$
$829$	−22.8600	−0.793961	−0.396980	+	0.917827i	$0.629942\pi$
$830$	0	0
$831$	6.54400	0.227009
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.77200	0.0958144
$838$	0	0
$839$	−20.8600	−0.720168	−0.360084	−	0.932920i	$-0.617252\pi$
$839$	−20.8600	−0.720168	−0.360084	+	0.932920i	$0.617252\pi$
$840$	0	0
$841$	4.31601	0.148828
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.22800	0.110915
$848$	0	0
$849$	14.7720	0.506974
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	19.4560	0.666161	0.333080	−	0.942898i	$-0.391912\pi$
$853$	19.4560	0.666161	0.333080	+	0.942898i	$0.391912\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.5440	−1.76071	−0.880355	−	0.474316i	$-0.842696\pi$
$857$	−51.5440	−1.76071	−0.880355	+	0.474316i	$0.842696\pi$
$858$	0	0
$859$	−42.1760	−1.43903	−0.719514	−	0.694478i	$-0.755635\pi$
$859$	−42.1760	−1.43903	−0.719514	+	0.694478i	$0.755635\pi$
$860$	0	0
$861$	1.77200	0.0603897
$862$	0	0
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	35.1400	1.19204
$870$	0	0
$871$	6.77200	0.229461
$872$	0	0
$873$	−3.22800	−0.109251
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.2280	−1.18956	−0.594782	−	0.803887i	$-0.702762\pi$
$877$	−35.2280	−1.18956	−0.594782	+	0.803887i	$0.702762\pi$
$878$	0	0
$879$	29.0880	0.981114
$880$	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
$883$	9.22800	0.310547	0.155273	−	0.987872i	$-0.450374\pi$
$883$	9.22800	0.310547	0.155273	+	0.987872i	$0.450374\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.5440	0.924837	0.462419	−	0.886662i	$-0.346982\pi$
$887$	27.5440	0.924837	0.462419	+	0.886662i	$0.346982\pi$
$888$	0	0
$889$	−19.5440	−0.655485
$890$	0	0
$891$	3.77200	0.126367
$892$	0	0
$893$	13.5440	0.453233
$894$	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
$903$	6.54400	0.217771
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.9480	−0.795181	−0.397590	−	0.917563i	$-0.630154\pi$
$907$	−23.9480	−0.795181	−0.397590	+	0.917563i	$0.630154\pi$
$908$	0	0
$909$	0.455996	0.0151244
$910$	0	0
$911$	6.68399	0.221451	0.110725	−	0.993851i	$-0.464683\pi$
$911$	6.68399	0.221451	0.110725	+	0.993851i	$0.464683\pi$
$912$	0	0
$913$	−31.0360	−1.02714
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.455996	0.0150583
$918$	0	0
$919$	11.4040	0.376184	0.188092	−	0.982151i	$-0.439770\pi$
$919$	11.4040	0.376184	0.188092	+	0.982151i	$0.439770\pi$
$920$	0	0
$921$	0.772002	0.0254383
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.77200	0.189577
$928$	0	0
$929$	46.2280	1.51669	0.758346	−	0.651853i	$-0.226008\pi$
$929$	46.2280	1.51669	0.758346	+	0.651853i	$0.226008\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.8600	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$937$	29.8600	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$938$	0	0
$939$	−6.77200	−0.220996
$940$	0	0
$941$	−1.54400	−0.0503331	−0.0251665	−	0.999683i	$-0.508012\pi$
$941$	−1.54400	−0.0503331	−0.0251665	+	0.999683i	$0.508012\pi$
$942$	0	0
$943$	1.77200	0.0577043
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.6320	−0.800433	−0.400216	−	0.916421i	$-0.631065\pi$
$947$	−24.6320	−0.800433	−0.400216	+	0.916421i	$0.631065\pi$
$948$	0	0
$949$	−8.22800	−0.267092
$950$	0	0
$951$	6.68399	0.216743
$952$	0	0
$953$	−2.91199	−0.0943287	−0.0471643	−	0.998887i	$-0.515018\pi$
$953$	−2.91199	−0.0943287	−0.0471643	+	0.998887i	$0.515018\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	21.7720	0.703789
$958$	0	0
$959$	−2.45600	−0.0793083
$960$	0	0
$961$	−23.3160	−0.752129
$962$	0	0
$963$	−1.54400	−0.0497548
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.4560	0.786452	0.393226	−	0.919442i	$-0.371359\pi$
$967$	24.4560	0.786452	0.393226	+	0.919442i	$0.371359\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.8600	0.669430	0.334715	−	0.942319i	$-0.391360\pi$
$971$	20.8600	0.669430	0.334715	+	0.942319i	$0.391360\pi$
$972$	0	0
$973$	0.455996	0.0146186
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.6320	−1.17196	−0.585981	−	0.810325i	$-0.699291\pi$
$977$	−36.6320	−1.17196	−0.585981	+	0.810325i	$0.699291\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	−6.77200	−0.216213
$982$	0	0
$983$	31.7720	1.01337	0.506685	−	0.862131i	$-0.330871\pi$
$983$	31.7720	1.01337	0.506685	+	0.862131i	$0.330871\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.5440	0.431110
$988$	0	0
$989$	6.54400	0.208087
$990$	0	0
$991$	−22.3160	−0.708891	−0.354446	−	0.935077i	$-0.615330\pi$
$991$	−22.3160	−0.708891	−0.354446	+	0.935077i	$0.615330\pi$
$992$	0	0
$993$	−19.0880	−0.605740
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.68399	−0.275025	−0.137512	−	0.990500i	$-0.543911\pi$
$997$	−8.68399	−0.275025	−0.137512	+	0.990500i	$0.543911\pi$
$998$	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.n.1.2	✓	2
5.2	odd	4		6900.2.f.j.6349.4		4
5.3	odd	4		6900.2.f.j.6349.2		4
5.4	even	2		6900.2.a.t.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.n.1.2	✓	2	1.1	even	1	trivial
6900.2.a.t.1.2	yes	2	5.4	even	2
6900.2.f.j.6349.2		4	5.3	odd	4
6900.2.f.j.6349.4		4	5.2	odd	4

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +3.77200 q^{11} +1.00000 q^{13} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} -5.77200 q^{29} -2.77200 q^{31} -3.77200 q^{33} -10.0000 q^{37} -1.00000 q^{39} -1.77200 q^{41} -6.54400 q^{43} -13.5440 q^{47} -6.00000 q^{49} +4.00000 q^{53} +1.00000 q^{57} -5.54400 q^{59} +14.3160 q^{61} +1.00000 q^{63} +6.77200 q^{67} +1.00000 q^{69} +4.00000 q^{71} -8.22800 q^{73} +3.77200 q^{77} +9.31601 q^{79} +1.00000 q^{81} -8.22800 q^{83} +5.77200 q^{87} +9.54400 q^{89} +1.00000 q^{91} +2.77200 q^{93} -3.22800 q^{97} +3.77200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more