LMFDB - Embedded newform 9072.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59358 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.59358 q^{5} +1.00000 q^{7} +4.51459 q^{11} +1.00000 q^{13} +0.945916 q^{17} +4.05408 q^{19} -0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{29} -2.32743 q^{31} +2.59358 q^{35} +1.78074 q^{37} +6.40642 q^{41} +10.4356 q^{43} -12.1623 q^{47} +1.00000 q^{49} +6.27335 q^{53} +11.7089 q^{55} -2.72665 q^{59} -2.27335 q^{61} +2.59358 q^{65} +15.8171 q^{67} +3.27335 q^{71} -1.50739 q^{73} +4.51459 q^{77} -14.7089 q^{79} -0.945916 q^{83} +2.45331 q^{85} +14.3566 q^{89} +1.00000 q^{91} +10.5146 q^{95} -11.4897 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + 5 * q^5 + 3 * q^7	$ 3 q + 5 q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 12 q^{17} + 3 q^{19} + 6 q^{25} - q^{29} + 3 q^{31} + 5 q^{35} - 3 q^{37} + 22 q^{41} + 3 q^{43} - 9 q^{47} + 3 q^{49} + 18 q^{53} + 6 q^{55} - 9 q^{59} - 6 q^{61} + 5 q^{65} + 9 q^{71} + 3 q^{73} - 2 q^{77} - 15 q^{79} - 12 q^{83} + 9 q^{85} + 2 q^{89} + 3 q^{91} + 16 q^{95} + 3 q^{97}+O(q^{100}) $ 3 * q + 5 * q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 12 * q^17 + 3 * q^19 + 6 * q^25 - q^29 + 3 * q^31 + 5 * q^35 - 3 * q^37 + 22 * q^41 + 3 * q^43 - 9 * q^47 + 3 * q^49 + 18 * q^53 + 6 * q^55 - 9 * q^59 - 6 * q^61 + 5 * q^65 + 9 * q^71 + 3 * q^73 - 2 * q^77 - 15 * q^79 - 12 * q^83 + 9 * q^85 + 2 * q^89 + 3 * q^91 + 16 * q^95 + 3 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.59358	1.15988	0.579942	−	0.814658i	$-0.303075\pi$
$5$	2.59358	1.15988	0.579942	+	0.814658i	$0.303075\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.51459	1.36120	0.680600	−	0.732655i	$-0.261719\pi$
$11$	4.51459	1.36120	0.680600	+	0.732655i	$0.261719\pi$
$12$	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
$17$	0.945916	0.229418	0.114709	−	0.993399i	$-0.463406\pi$
$17$	0.945916	0.229418	0.114709	+	0.993399i	$0.463406\pi$
$18$	0	0
$19$	4.05408	0.930071	0.465035	−	0.885292i	$-0.346042\pi$
$19$	4.05408	0.930071	0.465035	+	0.885292i	$0.346042\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.273346	−0.0569966	−0.0284983	−	0.999594i	$-0.509073\pi$
$23$	−0.273346	−0.0569966	−0.0284983	+	0.999594i	$0.509073\pi$
$24$	0	0
$25$	1.72665	0.345331
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.46050	−0.456904	−0.228452	−	0.973555i	$-0.573366\pi$
$29$	−2.46050	−0.456904	−0.228452	+	0.973555i	$0.573366\pi$
$30$	0	0
$31$	−2.32743	−0.418019	−0.209009	−	0.977914i	$-0.567024\pi$
$31$	−2.32743	−0.418019	−0.209009	+	0.977914i	$0.567024\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.59358	0.438395
$36$	0	0
$37$	1.78074	0.292752	0.146376	−	0.989229i	$-0.453239\pi$
$37$	1.78074	0.292752	0.146376	+	0.989229i	$0.453239\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.40642	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$41$	6.40642	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$42$	0	0
$43$	10.4356	1.59141	0.795707	−	0.605682i	$-0.207100\pi$
$43$	10.4356	1.59141	0.795707	+	0.605682i	$0.207100\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.1623	−1.77405	−0.887023	−	0.461724i	$-0.847231\pi$
$47$	−12.1623	−1.77405	−0.887023	+	0.461724i	$0.847231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.27335	0.861710	0.430855	−	0.902421i	$-0.358212\pi$
$53$	6.27335	0.861710	0.430855	+	0.902421i	$0.358212\pi$
$54$	0	0
$55$	11.7089	1.57883
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.72665	−0.354980	−0.177490	−	0.984123i	$-0.556798\pi$
$59$	−2.72665	−0.354980	−0.177490	+	0.984123i	$0.556798\pi$
$60$	0	0
$61$	−2.27335	−0.291072	−0.145536	−	0.989353i	$-0.546491\pi$
$61$	−2.27335	−0.291072	−0.145536	+	0.989353i	$0.546491\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.59358	0.321694
$66$	0	0
$67$	15.8171	1.93237	0.966184	−	0.257854i	$-0.0830152\pi$
$67$	15.8171	1.93237	0.966184	+	0.257854i	$0.0830152\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.27335	0.388475	0.194237	−	0.980955i	$-0.437777\pi$
$71$	3.27335	0.388475	0.194237	+	0.980955i	$0.437777\pi$
$72$	0	0
$73$	−1.50739	−0.176427	−0.0882134	−	0.996102i	$-0.528116\pi$
$73$	−1.50739	−0.176427	−0.0882134	+	0.996102i	$0.528116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.51459	0.514485
$78$	0	0
$79$	−14.7089	−1.65489	−0.827443	−	0.561550i	$-0.810205\pi$
$79$	−14.7089	−1.65489	−0.827443	+	0.561550i	$0.810205\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.945916	−0.103828	−0.0519139	−	0.998652i	$-0.516532\pi$
$83$	−0.945916	−0.103828	−0.0519139	+	0.998652i	$0.516532\pi$
$84$	0	0
$85$	2.45331	0.266099
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.3566	1.52180	0.760899	−	0.648871i	$-0.224758\pi$
$89$	14.3566	1.52180	0.760899	+	0.648871i	$0.224758\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.5146	1.07877
$96$	0	0
$97$	−11.4897	−1.16660	−0.583300	−	0.812257i	$-0.698239\pi$
$97$	−11.4897	−1.16660	−0.583300	+	0.812257i	$0.698239\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.67977	0.366150	0.183075	−	0.983099i	$-0.441395\pi$
$101$	3.67977	0.366150	0.183075	+	0.983099i	$0.441395\pi$
$102$	0	0
$103$	9.72665	0.958396	0.479198	−	0.877707i	$-0.340928\pi$
$103$	9.72665	0.958396	0.479198	+	0.877707i	$0.340928\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.37432	−0.132860	−0.0664301	−	0.997791i	$-0.521161\pi$
$107$	−1.37432	−0.132860	−0.0664301	+	0.997791i	$0.521161\pi$
$108$	0	0
$109$	−3.39922	−0.325587	−0.162793	−	0.986660i	$-0.552050\pi$
$109$	−3.39922	−0.325587	−0.162793	+	0.986660i	$0.552050\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.3887	−0.977288	−0.488644	−	0.872483i	$-0.662508\pi$
$113$	−10.3887	−0.977288	−0.488644	+	0.872483i	$0.662508\pi$
$114$	0	0
$115$	−0.708945	−0.0661095
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.945916	0.0867120
$120$	0	0
$121$	9.38151	0.852865
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.48968	−0.759340
$126$	0	0
$127$	−0.672570	−0.0596809	−0.0298405	−	0.999555i	$-0.509500\pi$
$127$	−0.672570	−0.0596809	−0.0298405	+	0.999555i	$0.509500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.91381	0.691433	0.345717	−	0.938339i	$-0.387636\pi$
$131$	7.91381	0.691433	0.345717	+	0.938339i	$0.387636\pi$
$132$	0	0
$133$	4.05408	0.351534
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.67257	0.313769	0.156884	−	0.987617i	$-0.449855\pi$
$137$	3.67257	0.313769	0.156884	+	0.987617i	$0.449855\pi$
$138$	0	0
$139$	2.05408	0.174225	0.0871126	−	0.996198i	$-0.472236\pi$
$139$	2.05408	0.174225	0.0871126	+	0.996198i	$0.472236\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.51459	0.377529
$144$	0	0
$145$	−6.38151	−0.529956
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.5438	1.10955	0.554774	−	0.832001i	$-0.312805\pi$
$149$	13.5438	1.10955	0.554774	+	0.832001i	$0.312805\pi$
$150$	0	0
$151$	−9.92821	−0.807946	−0.403973	−	0.914771i	$-0.632371\pi$
$151$	−9.92821	−0.807946	−0.403973	+	0.914771i	$0.632371\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.03638	−0.484853
$156$	0	0
$157$	6.05408	0.483169	0.241584	−	0.970380i	$-0.422333\pi$
$157$	6.05408	0.483169	0.241584	+	0.970380i	$0.422333\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.273346	−0.0215427
$162$	0	0
$163$	−17.8171	−1.39554	−0.697772	−	0.716320i	$-0.745825\pi$
$163$	−17.8171	−1.39554	−0.697772	+	0.716320i	$0.745825\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.46770	−0.655250	−0.327625	−	0.944808i	$-0.606248\pi$
$167$	−8.46770	−0.655250	−0.327625	+	0.944808i	$0.606248\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.3566	−1.31960	−0.659799	−	0.751442i	$-0.729359\pi$
$173$	−17.3566	−1.31960	−0.659799	+	0.751442i	$0.729359\pi$
$174$	0	0
$175$	1.72665	0.130523
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.3494	−0.848295	−0.424147	−	0.905593i	$-0.639426\pi$
$179$	−11.3494	−0.848295	−0.424147	+	0.905593i	$0.639426\pi$
$180$	0	0
$181$	21.8889	1.62699	0.813495	−	0.581572i	$-0.197562\pi$
$181$	21.8889	1.62699	0.813495	+	0.581572i	$0.197562\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.61849	0.339558
$186$	0	0
$187$	4.27042	0.312284
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.701748	−0.0507767	−0.0253883	−	0.999678i	$-0.508082\pi$
$191$	−0.701748	−0.0507767	−0.0253883	+	0.999678i	$0.508082\pi$
$192$	0	0
$193$	12.1445	0.874183	0.437092	−	0.899417i	$-0.356009\pi$
$193$	12.1445	0.874183	0.437092	+	0.899417i	$0.356009\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.4107	1.16921	0.584607	−	0.811317i	$-0.301249\pi$
$197$	16.4107	1.16921	0.584607	+	0.811317i	$0.301249\pi$
$198$	0	0
$199$	−22.7060	−1.60959	−0.804794	−	0.593555i	$-0.797724\pi$
$199$	−22.7060	−1.60959	−0.804794	+	0.593555i	$0.797724\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.46050	−0.172694
$204$	0	0
$205$	16.6156	1.16048
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.3025	1.26601
$210$	0	0
$211$	−4.56148	−0.314025	−0.157012	−	0.987597i	$-0.550186\pi$
$211$	−4.56148	−0.314025	−0.157012	+	0.987597i	$0.550186\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.0656	1.84586
$216$	0	0
$217$	−2.32743	−0.157996
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.945916	0.0636292
$222$	0	0
$223$	−13.3245	−0.892275	−0.446137	−	0.894964i	$-0.647201\pi$
$223$	−13.3245	−0.892275	−0.446137	+	0.894964i	$0.647201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.38151	0.0916943	0.0458472	−	0.998948i	$-0.485401\pi$
$227$	1.38151	0.0916943	0.0458472	+	0.998948i	$0.485401\pi$
$228$	0	0
$229$	−17.9794	−1.18811	−0.594055	−	0.804424i	$-0.702474\pi$
$229$	−17.9794	−1.18811	−0.594055	+	0.804424i	$0.702474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.9823	1.24357	0.621786	−	0.783187i	$-0.286408\pi$
$233$	18.9823	1.24357	0.621786	+	0.783187i	$0.286408\pi$
$234$	0	0
$235$	−31.5438	−2.05769
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.89183	0.316426	0.158213	−	0.987405i	$-0.449427\pi$
$239$	4.89183	0.316426	0.158213	+	0.987405i	$0.449427\pi$
$240$	0	0
$241$	−26.1593	−1.68507	−0.842535	−	0.538641i	$-0.818938\pi$
$241$	−26.1593	−1.68507	−0.842535	+	0.538641i	$0.818938\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.59358	0.165698
$246$	0	0
$247$	4.05408	0.257955
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.4576	1.16503	0.582516	−	0.812819i	$-0.302068\pi$
$251$	18.4576	1.16503	0.582516	+	0.812819i	$0.302068\pi$
$252$	0	0
$253$	−1.23405	−0.0775838
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.7339	0.731938	0.365969	−	0.930627i	$-0.380738\pi$
$257$	11.7339	0.731938	0.365969	+	0.930627i	$0.380738\pi$
$258$	0	0
$259$	1.78074	0.110650
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.52179	−0.463813	−0.231907	−	0.972738i	$-0.574496\pi$
$263$	−7.52179	−0.463813	−0.231907	+	0.972738i	$0.574496\pi$
$264$	0	0
$265$	16.2704	0.999484
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.8348	1.14838	0.574190	−	0.818722i	$-0.305317\pi$
$269$	18.8348	1.14838	0.574190	+	0.818722i	$0.305317\pi$
$270$	0	0
$271$	23.9823	1.45682	0.728410	−	0.685141i	$-0.240260\pi$
$271$	23.9823	1.45682	0.728410	+	0.685141i	$0.240260\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.79513	0.470064
$276$	0	0
$277$	7.16225	0.430338	0.215169	−	0.976577i	$-0.430970\pi$
$277$	7.16225	0.430338	0.215169	+	0.976577i	$0.430970\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.8817	−0.887768	−0.443884	−	0.896084i	$-0.646400\pi$
$281$	−14.8817	−0.887768	−0.443884	+	0.896084i	$0.646400\pi$
$282$	0	0
$283$	−19.9971	−1.18870	−0.594351	−	0.804205i	$-0.702591\pi$
$283$	−19.9971	−1.18870	−0.594351	+	0.804205i	$0.702591\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.40642	0.378159
$288$	0	0
$289$	−16.1052	−0.947367
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.0656	−0.880139	−0.440070	−	0.897964i	$-0.645046\pi$
$293$	−15.0656	−0.880139	−0.440070	+	0.897964i	$0.645046\pi$
$294$	0	0
$295$	−7.07179	−0.411736
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.273346	−0.0158080
$300$	0	0
$301$	10.4356	0.601498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.89610	−0.337610
$306$	0	0
$307$	27.2704	1.55641	0.778203	−	0.628013i	$-0.216132\pi$
$307$	27.2704	1.55641	0.778203	+	0.628013i	$0.216132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.9823	−0.906273	−0.453136	−	0.891441i	$-0.649695\pi$
$311$	−15.9823	−0.906273	−0.453136	+	0.891441i	$0.649695\pi$
$312$	0	0
$313$	11.5979	0.655549	0.327775	−	0.944756i	$-0.393701\pi$
$313$	11.5979	0.655549	0.327775	+	0.944756i	$0.393701\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.01771	0.113326	0.0566629	−	0.998393i	$-0.481954\pi$
$317$	2.01771	0.113326	0.0566629	+	0.998393i	$0.481954\pi$
$318$	0	0
$319$	−11.1082	−0.621938
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.83482	0.213375
$324$	0	0
$325$	1.72665	0.0957775
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.1623	−0.670527
$330$	0	0
$331$	19.7089	1.08330	0.541651	−	0.840604i	$-0.317800\pi$
$331$	19.7089	1.08330	0.541651	+	0.840604i	$0.317800\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	41.0229	2.24132
$336$	0	0
$337$	−29.0512	−1.58252	−0.791259	−	0.611481i	$-0.790574\pi$
$337$	−29.0512	−1.58252	−0.791259	+	0.611481i	$0.790574\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.5074	−0.569007
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.0833	1.56127	0.780636	−	0.624986i	$-0.214895\pi$
$347$	29.0833	1.56127	0.780636	+	0.624986i	$0.214895\pi$
$348$	0	0
$349$	24.7630	1.32553	0.662767	−	0.748825i	$-0.269382\pi$
$349$	24.7630	1.32553	0.662767	+	0.748825i	$0.269382\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	33.3025	1.77251	0.886257	−	0.463193i	$-0.153296\pi$
$353$	33.3025	1.77251	0.886257	+	0.463193i	$0.153296\pi$
$354$	0	0
$355$	8.48968	0.450586
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.5366	1.34777	0.673884	−	0.738837i	$-0.264625\pi$
$359$	25.5366	1.34777	0.673884	+	0.738837i	$0.264625\pi$
$360$	0	0
$361$	−2.56440	−0.134968
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.90954	−0.204635
$366$	0	0
$367$	−27.4504	−1.43290	−0.716449	−	0.697639i	$-0.754234\pi$
$367$	−27.4504	−1.43290	−0.716449	+	0.697639i	$0.754234\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.27335	0.325696
$372$	0	0
$373$	16.3274	0.845402	0.422701	−	0.906269i	$-0.361082\pi$
$373$	16.3274	0.845402	0.422701	+	0.906269i	$0.361082\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.46050	−0.126722
$378$	0	0
$379$	−12.0364	−0.618267	−0.309134	−	0.951019i	$-0.600039\pi$
$379$	−12.0364	−0.618267	−0.309134	+	0.951019i	$0.600039\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.4356	−0.635429	−0.317715	−	0.948186i	$-0.602915\pi$
$383$	−12.4356	−0.635429	−0.317715	+	0.948186i	$0.602915\pi$
$384$	0	0
$385$	11.7089	0.596743
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.6008	−1.04450	−0.522250	−	0.852792i	$-0.674907\pi$
$389$	−20.6008	−1.04450	−0.522250	+	0.852792i	$0.674907\pi$
$390$	0	0
$391$	−0.258562	−0.0130761
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−38.1488	−1.91948
$396$	0	0
$397$	−23.6372	−1.18631	−0.593157	−	0.805087i	$-0.702119\pi$
$397$	−23.6372	−1.18631	−0.593157	+	0.805087i	$0.702119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.56440	0.128060	0.0640300	−	0.997948i	$-0.479605\pi$
$401$	2.56440	0.128060	0.0640300	+	0.997948i	$0.479605\pi$
$402$	0	0
$403$	−2.32743	−0.115938
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.03930	0.398493
$408$	0	0
$409$	−34.3245	−1.69724	−0.848619	−	0.529005i	$-0.822565\pi$
$409$	−34.3245	−1.69724	−0.848619	+	0.529005i	$0.822565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.72665	−0.134170
$414$	0	0
$415$	−2.45331	−0.120428
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.05701	0.198198	0.0990989	−	0.995078i	$-0.468404\pi$
$419$	4.05701	0.198198	0.0990989	+	0.995078i	$0.468404\pi$
$420$	0	0
$421$	−21.0689	−1.02683	−0.513417	−	0.858139i	$-0.671621\pi$
$421$	−21.0689	−1.02683	−0.513417	+	0.858139i	$0.671621\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.63327	0.0792252
$426$	0	0
$427$	−2.27335	−0.110015
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.6185	1.08949	0.544747	−	0.838600i	$-0.316626\pi$
$431$	22.6185	1.08949	0.544747	+	0.838600i	$0.316626\pi$
$432$	0	0
$433$	2.41789	0.116196	0.0580982	−	0.998311i	$-0.481496\pi$
$433$	2.41789	0.116196	0.0580982	+	0.998311i	$0.481496\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.10817	−0.0530109
$438$	0	0
$439$	23.4897	1.12110	0.560551	−	0.828120i	$-0.310589\pi$
$439$	23.4897	1.12110	0.560551	+	0.828120i	$0.310589\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.4179	−0.637503	−0.318752	−	0.947838i	$-0.603264\pi$
$443$	−13.4179	−0.637503	−0.318752	+	0.947838i	$0.603264\pi$
$444$	0	0
$445$	37.2350	1.76511
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.16225	0.432393	0.216197	−	0.976350i	$-0.430635\pi$
$449$	9.16225	0.432393	0.216197	+	0.976350i	$0.430635\pi$
$450$	0	0
$451$	28.9224	1.36190
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.59358	0.121589
$456$	0	0
$457$	8.81711	0.412447	0.206224	−	0.978505i	$-0.433883\pi$
$457$	8.81711	0.412447	0.206224	+	0.978505i	$0.433883\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.65913	0.263572	0.131786	−	0.991278i	$-0.457929\pi$
$461$	5.65913	0.263572	0.131786	+	0.991278i	$0.457929\pi$
$462$	0	0
$463$	−15.7267	−0.730880	−0.365440	−	0.930835i	$-0.619081\pi$
$463$	−15.7267	−0.730880	−0.365440	+	0.930835i	$0.619081\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.9971	−1.01790	−0.508952	−	0.860795i	$-0.669967\pi$
$467$	−21.9971	−1.01790	−0.508952	+	0.860795i	$0.669967\pi$
$468$	0	0
$469$	15.8171	0.730366
$470$	0	0
$471$	0	0
$472$	0	0
$473$	47.1124	2.16623
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.9751	1.14114	0.570571	−	0.821249i	$-0.306722\pi$
$479$	24.9751	1.14114	0.570571	+	0.821249i	$0.306722\pi$
$480$	0	0
$481$	1.78074	0.0811947
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−29.7994	−1.35312
$486$	0	0
$487$	17.5979	0.797435	0.398717	−	0.917074i	$-0.369455\pi$
$487$	17.5979	0.797435	0.398717	+	0.917074i	$0.369455\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.7951	0.622566	0.311283	−	0.950317i	$-0.399241\pi$
$491$	13.7951	0.622566	0.311283	+	0.950317i	$0.399241\pi$
$492$	0	0
$493$	−2.32743	−0.104822
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.27335	0.146830
$498$	0	0
$499$	−13.0875	−0.585879	−0.292939	−	0.956131i	$-0.594633\pi$
$499$	−13.0875	−0.585879	−0.292939	+	0.956131i	$0.594633\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.3068	−0.994611	−0.497305	−	0.867576i	$-0.665677\pi$
$503$	−22.3068	−0.994611	−0.497305	+	0.867576i	$0.665677\pi$
$504$	0	0
$505$	9.54377	0.424692
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8932	0.704453	0.352226	−	0.935915i	$-0.385425\pi$
$509$	15.8932	0.704453	0.352226	+	0.935915i	$0.385425\pi$
$510$	0	0
$511$	−1.50739	−0.0666831
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.2268	1.11163
$516$	0	0
$517$	−54.9076	−2.41483
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.41789	−0.193551	−0.0967756	−	0.995306i	$-0.530853\pi$
$521$	−4.41789	−0.193551	−0.0967756	+	0.995306i	$0.530853\pi$
$522$	0	0
$523$	−25.2733	−1.10513	−0.552563	−	0.833471i	$-0.686350\pi$
$523$	−25.2733	−1.10513	−0.552563	+	0.833471i	$0.686350\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.20155	−0.0959012
$528$	0	0
$529$	−22.9253	−0.996751
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.40642	0.277493
$534$	0	0
$535$	−3.56440	−0.154103
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.51459	0.194457
$540$	0	0
$541$	−3.43852	−0.147834	−0.0739168	−	0.997264i	$-0.523550\pi$
$541$	−3.43852	−0.147834	−0.0739168	+	0.997264i	$0.523550\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.81616	−0.377643
$546$	0	0
$547$	6.92821	0.296229	0.148114	−	0.988970i	$-0.452680\pi$
$547$	6.92821	0.296229	0.148114	+	0.988970i	$0.452680\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.97509	−0.424953
$552$	0	0
$553$	−14.7089	−0.625488
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.5835	−1.42298	−0.711488	−	0.702698i	$-0.751979\pi$
$557$	−33.5835	−1.42298	−0.711488	+	0.702698i	$0.751979\pi$
$558$	0	0
$559$	10.4356	0.441379
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.4792	1.79028	0.895142	−	0.445781i	$-0.147074\pi$
$563$	42.4792	1.79028	0.895142	+	0.445781i	$0.147074\pi$
$564$	0	0
$565$	−26.9439	−1.13354
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.4035	−0.436137	−0.218069	−	0.975933i	$-0.569976\pi$
$569$	−10.4035	−0.436137	−0.218069	+	0.975933i	$0.569976\pi$
$570$	0	0
$571$	−17.8496	−0.746983	−0.373491	−	0.927634i	$-0.621839\pi$
$571$	−17.8496	−0.746983	−0.373491	+	0.927634i	$0.621839\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.471974	−0.0196827
$576$	0	0
$577$	11.9430	0.497193	0.248597	−	0.968607i	$-0.420031\pi$
$577$	11.9430	0.497193	0.248597	+	0.968607i	$0.420031\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.945916	−0.0392432
$582$	0	0
$583$	28.3216	1.17296
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.8597	0.984796	0.492398	−	0.870370i	$-0.336120\pi$
$587$	23.8597	0.984796	0.492398	+	0.870370i	$0.336120\pi$
$588$	0	0
$589$	−9.43560	−0.388787
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.5801	−0.804060	−0.402030	−	0.915626i	$-0.631695\pi$
$593$	−19.5801	−0.804060	−0.402030	+	0.915626i	$0.631695\pi$
$594$	0	0
$595$	2.45331	0.100576
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.5467	0.757797	0.378899	−	0.925438i	$-0.376303\pi$
$599$	18.5467	0.757797	0.378899	+	0.925438i	$0.376303\pi$
$600$	0	0
$601$	−18.1986	−0.742338	−0.371169	−	0.928565i	$-0.621043\pi$
$601$	−18.1986	−0.742338	−0.371169	+	0.928565i	$0.621043\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.3317	0.989224
$606$	0	0
$607$	22.3097	0.905524	0.452762	−	0.891631i	$-0.350439\pi$
$607$	22.3097	0.905524	0.452762	+	0.891631i	$0.350439\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.1623	−0.492032
$612$	0	0
$613$	10.2370	0.413467	0.206734	−	0.978397i	$-0.433717\pi$
$613$	10.2370	0.413467	0.206734	+	0.978397i	$0.433717\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.3274	0.456025	0.228013	−	0.973658i	$-0.426777\pi$
$617$	11.3274	0.456025	0.228013	+	0.973658i	$0.426777\pi$
$618$	0	0
$619$	−8.63327	−0.347000	−0.173500	−	0.984834i	$-0.555508\pi$
$619$	−8.63327	−0.347000	−0.173500	+	0.984834i	$0.555508\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.3566	0.575185
$624$	0	0
$625$	−30.6519	−1.22608
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.68443	0.0671626
$630$	0	0
$631$	14.8535	0.591308	0.295654	−	0.955295i	$-0.404462\pi$
$631$	14.8535	0.591308	0.295654	+	0.955295i	$0.404462\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.74436	−0.0692229
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.1593	1.34921	0.674606	−	0.738178i	$-0.264313\pi$
$641$	34.1593	1.34921	0.674606	+	0.738178i	$0.264313\pi$
$642$	0	0
$643$	10.8348	0.427284	0.213642	−	0.976912i	$-0.431467\pi$
$643$	10.8348	0.427284	0.213642	+	0.976912i	$0.431467\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.9692	1.29615	0.648077	−	0.761575i	$-0.275573\pi$
$647$	32.9692	1.29615	0.648077	+	0.761575i	$0.275573\pi$
$648$	0	0
$649$	−12.3097	−0.483199
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.93113	0.153837	0.0769185	−	0.997037i	$-0.475492\pi$
$653$	3.93113	0.153837	0.0769185	+	0.997037i	$0.475492\pi$
$654$	0	0
$655$	20.5251	0.801982
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.8171	0.655102	0.327551	−	0.944834i	$-0.393777\pi$
$659$	16.8171	0.655102	0.327551	+	0.944834i	$0.393777\pi$
$660$	0	0
$661$	−17.0216	−0.662063	−0.331032	−	0.943620i	$-0.607397\pi$
$661$	−17.0216	−0.662063	−0.331032	+	0.943620i	$0.607397\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.5146	0.407738
$666$	0	0
$667$	0.672570	0.0260420
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.2632	−0.396207
$672$	0	0
$673$	28.7453	1.10805	0.554025	−	0.832500i	$-0.313091\pi$
$673$	28.7453	1.10805	0.554025	+	0.832500i	$0.313091\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.03638	0.231997	0.115998	−	0.993249i	$-0.462993\pi$
$677$	6.03638	0.231997	0.115998	+	0.993249i	$0.462993\pi$
$678$	0	0
$679$	−11.4897	−0.440934
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.5113	0.784842	0.392421	−	0.919786i	$-0.371638\pi$
$683$	20.5113	0.784842	0.392421	+	0.919786i	$0.371638\pi$
$684$	0	0
$685$	9.52510	0.363935
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.27335	0.238995
$690$	0	0
$691$	15.0029	0.570738	0.285369	−	0.958418i	$-0.407884\pi$
$691$	15.0029	0.570738	0.285369	+	0.958418i	$0.407884\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.32743	0.202081
$696$	0	0
$697$	6.05993	0.229536
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.5113	−1.45455	−0.727275	−	0.686346i	$-0.759214\pi$
$701$	−38.5113	−1.45455	−0.727275	+	0.686346i	$0.759214\pi$
$702$	0	0
$703$	7.21926	0.272280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.67977	0.138392
$708$	0	0
$709$	7.64008	0.286929	0.143465	−	0.989655i	$-0.454176\pi$
$709$	7.64008	0.286929	0.143465	+	0.989655i	$0.454176\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.636194	0.0238257
$714$	0	0
$715$	11.7089	0.437890
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.0364	−1.12017	−0.560084	−	0.828436i	$-0.689231\pi$
$719$	−30.0364	−1.12017	−0.560084	+	0.828436i	$0.689231\pi$
$720$	0	0
$721$	9.72665	0.362240
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24844	−0.157783
$726$	0	0
$727$	−3.45623	−0.128185	−0.0640923	−	0.997944i	$-0.520415\pi$
$727$	−3.45623	−0.128185	−0.0640923	+	0.997944i	$0.520415\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.87120	0.365099
$732$	0	0
$733$	38.5261	1.42299	0.711496	−	0.702690i	$-0.248018\pi$
$733$	38.5261	1.42299	0.711496	+	0.702690i	$0.248018\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	71.4078	2.63034
$738$	0	0
$739$	−45.1239	−1.65991	−0.829955	−	0.557830i	$-0.811634\pi$
$739$	−45.1239	−1.65991	−0.829955	+	0.557830i	$0.811634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.48676	0.348035	0.174018	−	0.984743i	$-0.444325\pi$
$743$	9.48676	0.348035	0.174018	+	0.984743i	$0.444325\pi$
$744$	0	0
$745$	35.1268	1.28695
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.37432	−0.0502165
$750$	0	0
$751$	9.83190	0.358771	0.179386	−	0.983779i	$-0.442589\pi$
$751$	9.83190	0.358771	0.179386	+	0.983779i	$0.442589\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−25.7496	−0.937124
$756$	0	0
$757$	−41.8171	−1.51987	−0.759934	−	0.650000i	$-0.774769\pi$
$757$	−41.8171	−1.51987	−0.759934	+	0.650000i	$0.774769\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.9794	−0.833001	−0.416501	−	0.909135i	$-0.636744\pi$
$761$	−22.9794	−0.833001	−0.416501	+	0.909135i	$0.636744\pi$
$762$	0	0
$763$	−3.39922	−0.123060
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.72665	−0.0984538
$768$	0	0
$769$	6.08658	0.219488	0.109744	−	0.993960i	$-0.464997\pi$
$769$	6.08658	0.219488	0.109744	+	0.993960i	$0.464997\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41.8214	1.50421	0.752105	−	0.659043i	$-0.229038\pi$
$773$	41.8214	1.50421	0.752105	+	0.659043i	$0.229038\pi$
$774$	0	0
$775$	−4.01867	−0.144355
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.9722	0.930550
$780$	0	0
$781$	14.7778	0.528792
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.7017	0.560419
$786$	0	0
$787$	−32.2920	−1.15109	−0.575543	−	0.817772i	$-0.695209\pi$
$787$	−32.2920	−1.15109	−0.575543	+	0.817772i	$0.695209\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.3887	−0.369380
$792$	0	0
$793$	−2.27335	−0.0807289
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.5657	−1.64944	−0.824722	−	0.565539i	$-0.808668\pi$
$797$	−46.5657	−1.64944	−0.824722	+	0.565539i	$0.808668\pi$
$798$	0	0
$799$	−11.5045	−0.406999
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.80525	−0.240152
$804$	0	0
$805$	−0.708945	−0.0249870
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.8023	0.379790	0.189895	−	0.981804i	$-0.439185\pi$
$809$	10.8023	0.379790	0.189895	+	0.981804i	$0.439185\pi$
$810$	0	0
$811$	−5.58307	−0.196048	−0.0980240	−	0.995184i	$-0.531252\pi$
$811$	−5.58307	−0.196048	−0.0980240	+	0.995184i	$0.531252\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−46.2101	−1.61867
$816$	0	0
$817$	42.3068	1.48013
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.7879	1.10941	0.554703	−	0.832048i	$-0.312832\pi$
$821$	31.7879	1.10941	0.554703	+	0.832048i	$0.312832\pi$
$822$	0	0
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.9224	0.553675	0.276837	−	0.960917i	$-0.410714\pi$
$827$	15.9224	0.553675	0.276837	+	0.960917i	$0.410714\pi$
$828$	0	0
$829$	35.4720	1.23199	0.615996	−	0.787749i	$-0.288754\pi$
$829$	35.4720	1.23199	0.615996	+	0.787749i	$0.288754\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.945916	0.0327740
$834$	0	0
$835$	−21.9617	−0.760014
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.6782	−1.88770	−0.943850	−	0.330373i	$-0.892825\pi$
$839$	−54.6782	−1.88770	−0.943850	+	0.330373i	$0.892825\pi$
$840$	0	0
$841$	−22.9459	−0.791238
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.1230	−1.07066
$846$	0	0
$847$	9.38151	0.322353
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.486758	−0.0166858
$852$	0	0
$853$	−2.19767	−0.0752468	−0.0376234	−	0.999292i	$-0.511979\pi$
$853$	−2.19767	−0.0752468	−0.0376234	+	0.999292i	$0.511979\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.7765	0.538914	0.269457	−	0.963012i	$-0.413156\pi$
$857$	15.7765	0.538914	0.269457	+	0.963012i	$0.413156\pi$
$858$	0	0
$859$	−5.57626	−0.190260	−0.0951298	−	0.995465i	$-0.530327\pi$
$859$	−5.57626	−0.190260	−0.0951298	+	0.995465i	$0.530327\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.1268	0.787247	0.393623	−	0.919272i	$-0.371221\pi$
$863$	23.1268	0.787247	0.393623	+	0.919272i	$0.371221\pi$
$864$	0	0
$865$	−45.0157	−1.53058
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−66.4048	−2.25263
$870$	0	0
$871$	15.8171	0.535942
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.48968	−0.287004
$876$	0	0
$877$	3.92528	0.132547	0.0662737	−	0.997801i	$-0.478889\pi$
$877$	3.92528	0.132547	0.0662737	+	0.997801i	$0.478889\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.1986	−0.916345	−0.458173	−	0.888863i	$-0.651496\pi$
$881$	−27.1986	−0.916345	−0.458173	+	0.888863i	$0.651496\pi$
$882$	0	0
$883$	−8.21341	−0.276403	−0.138202	−	0.990404i	$-0.544132\pi$
$883$	−8.21341	−0.276403	−0.138202	+	0.990404i	$0.544132\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.48114	−0.217615	−0.108808	−	0.994063i	$-0.534703\pi$
$887$	−6.48114	−0.217615	−0.108808	+	0.994063i	$0.534703\pi$
$888$	0	0
$889$	−0.672570	−0.0225573
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−49.3068	−1.64999
$894$	0	0
$895$	−29.4356	−0.983924
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.72665	0.190995
$900$	0	0
$901$	5.93406	0.197692
$902$	0	0
$903$	0	0
$904$	0	0
$905$	56.7706	1.88712
$906$	0	0
$907$	−10.1288	−0.336321	−0.168161	−	0.985760i	$-0.553783\pi$
$907$	−10.1288	−0.336321	−0.168161	+	0.985760i	$0.553783\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.9224	1.52148	0.760738	−	0.649059i	$-0.224837\pi$
$911$	45.9224	1.52148	0.760738	+	0.649059i	$0.224837\pi$
$912$	0	0
$913$	−4.27042	−0.141330
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.91381	0.261337
$918$	0	0
$919$	4.92432	0.162438	0.0812192	−	0.996696i	$-0.474119\pi$
$919$	4.92432	0.162438	0.0812192	+	0.996696i	$0.474119\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.27335	0.107744
$924$	0	0
$925$	3.07472	0.101096
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.00758649	−0.000248905 0	−0.000124452	−	1.00000i	$-0.500040\pi$
$929$	−0.00758649	−0.000248905 0	−0.000124452		1.00000i	$0.500040\pi$
$930$	0	0
$931$	4.05408	0.132867
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.0757	0.362213
$936$	0	0
$937$	21.1623	0.691341	0.345670	−	0.938356i	$-0.387652\pi$
$937$	21.1623	0.691341	0.345670	+	0.938356i	$0.387652\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.55816	−0.148592	−0.0742959	−	0.997236i	$-0.523671\pi$
$941$	−4.55816	−0.148592	−0.0742959	+	0.997236i	$0.523671\pi$
$942$	0	0
$943$	−1.75117	−0.0570260
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.7352	0.446334	0.223167	−	0.974780i	$-0.428360\pi$
$947$	13.7352	0.446334	0.223167	+	0.974780i	$0.428360\pi$
$948$	0	0
$949$	−1.50739	−0.0489320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.80699	−0.285286	−0.142643	−	0.989774i	$-0.545560\pi$
$953$	−8.80699	−0.285286	−0.142643	+	0.989774i	$0.545560\pi$
$954$	0	0
$955$	−1.82004	−0.0588951
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.67257	0.118593
$960$	0	0
$961$	−25.5831	−0.825260
$962$	0	0
$963$	0	0
$964$	0	0
$965$	31.4978	1.01395
$966$	0	0
$967$	−38.3284	−1.23256	−0.616279	−	0.787528i	$-0.711361\pi$
$967$	−38.3284	−1.23256	−0.616279	+	0.787528i	$0.711361\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.0187	−0.995436	−0.497718	−	0.867339i	$-0.665829\pi$
$971$	−31.0187	−0.995436	−0.497718	+	0.867339i	$0.665829\pi$
$972$	0	0
$973$	2.05408	0.0658509
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.7424	1.68738	0.843689	−	0.536832i	$-0.180379\pi$
$977$	52.7424	1.68738	0.843689	+	0.536832i	$0.180379\pi$
$978$	0	0
$979$	64.8142	2.07147
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.3029	−0.583772	−0.291886	−	0.956453i	$-0.594283\pi$
$983$	−18.3029	−0.583772	−0.291886	+	0.956453i	$0.594283\pi$
$984$	0	0
$985$	42.5624	1.35615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.85253	−0.0907052
$990$	0	0
$991$	12.6008	0.400277	0.200138	−	0.979768i	$-0.435861\pi$
$991$	12.6008	0.400277	0.200138	+	0.979768i	$0.435861\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−58.8899	−1.86693
$996$	0	0
$997$	11.7424	0.371885	0.185943	−	0.982561i	$-0.440466\pi$
$997$	11.7424	0.371885	0.185943	+	0.982561i	$0.440466\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.cd.1.2		3
3.2	odd	2		9072.2.a.bq.1.2		3
4.3	odd	2		567.2.a.g.1.3		3
9.2	odd	6		1008.2.r.k.337.3		6
9.4	even	3		3024.2.r.g.2017.2		6
9.5	odd	6		1008.2.r.k.673.3		6
9.7	even	3		3024.2.r.g.1009.2		6
12.11	even	2		567.2.a.d.1.1		3
28.27	even	2		3969.2.a.p.1.3		3
36.7	odd	6		189.2.f.a.64.1		6
36.11	even	6		63.2.f.b.22.3	✓	6
36.23	even	6		63.2.f.b.43.3	yes	6
36.31	odd	6		189.2.f.a.127.1		6
84.83	odd	2		3969.2.a.m.1.1		3
252.11	even	6		441.2.h.c.373.1		6
252.23	even	6		441.2.h.c.214.1		6
252.31	even	6		1323.2.g.b.667.1		6
252.47	odd	6		441.2.g.d.67.3		6
252.59	odd	6		441.2.g.d.79.3		6
252.67	odd	6		1323.2.g.c.667.1		6
252.79	odd	6		1323.2.g.c.361.1		6
252.83	odd	6		441.2.f.d.148.3		6
252.95	even	6		441.2.g.e.79.3		6
252.103	even	6		1323.2.h.e.802.3		6
252.115	even	6		1323.2.h.e.226.3		6
252.131	odd	6		441.2.h.b.214.1		6
252.139	even	6		1323.2.f.c.883.1		6
252.151	odd	6		1323.2.h.d.226.3		6
252.167	odd	6		441.2.f.d.295.3		6
252.187	even	6		1323.2.g.b.361.1		6
252.191	even	6		441.2.g.e.67.3		6
252.223	even	6		1323.2.f.c.442.1		6
252.227	odd	6		441.2.h.b.373.1		6
252.247	odd	6		1323.2.h.d.802.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.b.22.3	✓	6	36.11	even	6
63.2.f.b.43.3	yes	6	36.23	even	6
189.2.f.a.64.1		6	36.7	odd	6
189.2.f.a.127.1		6	36.31	odd	6
441.2.f.d.148.3		6	252.83	odd	6
441.2.f.d.295.3		6	252.167	odd	6
441.2.g.d.67.3		6	252.47	odd	6
441.2.g.d.79.3		6	252.59	odd	6
441.2.g.e.67.3		6	252.191	even	6
441.2.g.e.79.3		6	252.95	even	6
441.2.h.b.214.1		6	252.131	odd	6
441.2.h.b.373.1		6	252.227	odd	6
441.2.h.c.214.1		6	252.23	even	6
441.2.h.c.373.1		6	252.11	even	6
567.2.a.d.1.1		3	12.11	even	2
567.2.a.g.1.3		3	4.3	odd	2
1008.2.r.k.337.3		6	9.2	odd	6
1008.2.r.k.673.3		6	9.5	odd	6
1323.2.f.c.442.1		6	252.223	even	6
1323.2.f.c.883.1		6	252.139	even	6
1323.2.g.b.361.1		6	252.187	even	6
1323.2.g.b.667.1		6	252.31	even	6
1323.2.g.c.361.1		6	252.79	odd	6
1323.2.g.c.667.1		6	252.67	odd	6
1323.2.h.d.226.3		6	252.151	odd	6
1323.2.h.d.802.3		6	252.247	odd	6
1323.2.h.e.226.3		6	252.115	even	6
1323.2.h.e.802.3		6	252.103	even	6
3024.2.r.g.1009.2		6	9.7	even	3
3024.2.r.g.2017.2		6	9.4	even	3
3969.2.a.m.1.1		3	84.83	odd	2
3969.2.a.p.1.3		3	28.27	even	2
9072.2.a.bq.1.2		3	3.2	odd	2
9072.2.a.cd.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59358 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.59358 q^{5} +1.00000 q^{7} +4.51459 q^{11} +1.00000 q^{13} +0.945916 q^{17} +4.05408 q^{19} -0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{29} -2.32743 q^{31} +2.59358 q^{35} +1.78074 q^{37} +6.40642 q^{41} +10.4356 q^{43} -12.1623 q^{47} +1.00000 q^{49} +6.27335 q^{53} +11.7089 q^{55} -2.72665 q^{59} -2.27335 q^{61} +2.59358 q^{65} +15.8171 q^{67} +3.27335 q^{71} -1.50739 q^{73} +4.51459 q^{77} -14.7089 q^{79} -0.945916 q^{83} +2.45331 q^{85} +14.3566 q^{89} +1.00000 q^{91} +10.5146 q^{95} -11.4897 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + 5 * q^5 + 3 * q^7	\( 3 q + 5 q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 12 q^{17} + 3 q^{19} + 6 q^{25} - q^{29} + 3 q^{31} + 5 q^{35} - 3 q^{37} + 22 q^{41} + 3 q^{43} - 9 q^{47} + 3 q^{49} + 18 q^{53} + 6 q^{55} - 9 q^{59} - 6 q^{61} + 5 q^{65} + 9 q^{71} + 3 q^{73} - 2 q^{77} - 15 q^{79} - 12 q^{83} + 9 q^{85} + 2 q^{89} + 3 q^{91} + 16 q^{95} + 3 q^{97}+O(q^{100}) \) 3 * q + 5 * q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 12 * q^17 + 3 * q^19 + 6 * q^25 - q^29 + 3 * q^31 + 5 * q^35 - 3 * q^37 + 22 * q^41 + 3 * q^43 - 9 * q^47 + 3 * q^49 + 18 * q^53 + 6 * q^55 - 9 * q^59 - 6 * q^61 + 5 * q^65 + 9 * q^71 + 3 * q^73 - 2 * q^77 - 15 * q^79 - 12 * q^83 + 9 * q^85 + 2 * q^89 + 3 * q^91 + 16 * q^95 + 3 * q^97

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