LMFDB - Embedded newform 9072.2.a.ce.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.519120$ 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)

$f(q)$	$=$	$q+0.936586 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+0.936586 q^{5} -1.00000 q^{7} -4.97483 q^{11} +1.24431 q^{13} +5.22446 q^{17} -5.18622 q^{19} +2.00532 q^{23} -4.12281 q^{25} +6.87849 q^{29} +5.73051 q^{31} -0.936586 q^{35} +9.73051 q^{37} -11.4741 q^{41} -9.60369 q^{43} +1.96951 q^{47} +1.00000 q^{49} -7.63418 q^{53} -4.65935 q^{55} -4.87849 q^{59} -3.04356 q^{61} +1.16541 q^{65} +1.14798 q^{67} +8.83749 q^{71} +6.10698 q^{73} +4.97483 q^{77} +12.1083 q^{79} +0.862663 q^{83} +4.89316 q^{85} -10.8480 q^{89} -1.24431 q^{91} -4.85734 q^{95} +7.57043 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 - 4 * q^7	$ 4 q - 4 q^{5} - 4 q^{7} - 6 q^{11} + 3 q^{13} - 8 q^{17} + 2 q^{19} - 5 q^{23} + 14 q^{25} - q^{29} + 11 q^{31} + 4 q^{35} + 27 q^{37} - 2 q^{41} - 11 q^{43} + 7 q^{47} + 4 q^{49} - 4 q^{53} - 6 q^{55} + 9 q^{59} + 7 q^{61} + 9 q^{65} - 12 q^{67} + 12 q^{71} + 13 q^{73} + 6 q^{77} - 22 q^{79} - 6 q^{83} + 11 q^{85} - 14 q^{89} - 3 q^{91} - 23 q^{95} + q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 6 * q^11 + 3 * q^13 - 8 * q^17 + 2 * q^19 - 5 * q^23 + 14 * q^25 - q^29 + 11 * q^31 + 4 * q^35 + 27 * q^37 - 2 * q^41 - 11 * q^43 + 7 * q^47 + 4 * q^49 - 4 * q^53 - 6 * q^55 + 9 * q^59 + 7 * q^61 + 9 * q^65 - 12 * q^67 + 12 * q^71 + 13 * q^73 + 6 * q^77 - 22 * q^79 - 6 * q^83 + 11 * q^85 - 14 * q^89 - 3 * q^91 - 23 * q^95 + 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
$5$	0.936586	0.418854	0.209427	−	0.977824i	$-0.432840\pi$
$5$	0.936586	0.418854	0.209427	+	0.977824i	$0.432840\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.97483	−1.49997	−0.749983	−	0.661457i	$-0.769939\pi$
$11$	−4.97483	−1.49997	−0.749983	+	0.661457i	$0.769939\pi$
$12$	0	0
$13$	1.24431	0.345110	0.172555	−	0.985000i	$-0.444798\pi$
$13$	1.24431	0.345110	0.172555	+	0.985000i	$0.444798\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.22446	1.26712	0.633559	−	0.773694i	$-0.281593\pi$
$17$	5.22446	1.26712	0.633559	+	0.773694i	$0.281593\pi$
$18$	0	0
$19$	−5.18622	−1.18980	−0.594900	−	0.803800i	$-0.702808\pi$
$19$	−5.18622	−1.18980	−0.594900	+	0.803800i	$0.702808\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00532	0.418138	0.209069	−	0.977901i	$-0.432957\pi$
$23$	2.00532	0.418138	0.209069	+	0.977901i	$0.432957\pi$
$24$	0	0
$25$	−4.12281	−0.824561
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.87849	1.27730	0.638652	−	0.769496i	$-0.279492\pi$
$29$	6.87849	1.27730	0.638652	+	0.769496i	$0.279492\pi$
$30$	0	0
$31$	5.73051	1.02923	0.514615	−	0.857421i	$-0.327935\pi$
$31$	5.73051	1.02923	0.514615	+	0.857421i	$0.327935\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.936586	−0.158312
$36$	0	0
$37$	9.73051	1.59969	0.799843	−	0.600209i	$-0.204916\pi$
$37$	9.73051	1.59969	0.799843	+	0.600209i	$0.204916\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.4741	−1.79195	−0.895976	−	0.444102i	$-0.853523\pi$
$41$	−11.4741	−1.79195	−0.895976	+	0.444102i	$0.853523\pi$
$42$	0	0
$43$	−9.60369	−1.46455	−0.732274	−	0.681010i	$-0.761541\pi$
$43$	−9.60369	−1.46455	−0.732274	+	0.681010i	$0.761541\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.96951	0.287282	0.143641	−	0.989630i	$-0.454119\pi$
$47$	1.96951	0.287282	0.143641	+	0.989630i	$0.454119\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.63418	−1.04864	−0.524318	−	0.851523i	$-0.675680\pi$
$53$	−7.63418	−1.04864	−0.524318	+	0.851523i	$0.675680\pi$
$54$	0	0
$55$	−4.65935	−0.628267
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.87849	−0.635126	−0.317563	−	0.948237i	$-0.602864\pi$
$59$	−4.87849	−0.635126	−0.317563	+	0.948237i	$0.602864\pi$
$60$	0	0
$61$	−3.04356	−0.389688	−0.194844	−	0.980834i	$-0.562420\pi$
$61$	−3.04356	−0.389688	−0.194844	+	0.980834i	$0.562420\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.16541	0.144551
$66$	0	0
$67$	1.14798	0.140248	0.0701240	−	0.997538i	$-0.477660\pi$
$67$	1.14798	0.140248	0.0701240	+	0.997538i	$0.477660\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.83749	1.04882	0.524409	−	0.851467i	$-0.324286\pi$
$71$	8.83749	1.04882	0.524409	+	0.851467i	$0.324286\pi$
$72$	0	0
$73$	6.10698	0.714768	0.357384	−	0.933958i	$-0.383669\pi$
$73$	6.10698	0.714768	0.357384	+	0.933958i	$0.383669\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.97483	0.566934
$78$	0	0
$79$	12.1083	1.36229	0.681144	−	0.732150i	$-0.261483\pi$
$79$	12.1083	1.36229	0.681144	+	0.732150i	$0.261483\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.862663	0.0946896	0.0473448	−	0.998879i	$-0.484924\pi$
$83$	0.862663	0.0946896	0.0473448	+	0.998879i	$0.484924\pi$
$84$	0	0
$85$	4.89316	0.530737
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.8480	−1.14989	−0.574943	−	0.818194i	$-0.694976\pi$
$89$	−10.8480	−1.14989	−0.574943	+	0.818194i	$0.694976\pi$
$90$	0	0
$91$	−1.24431	−0.130439
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.85734	−0.498353
$96$	0	0
$97$	7.57043	0.768661	0.384330	−	0.923196i	$-0.374432\pi$
$97$	7.57043	0.768661	0.384330	+	0.923196i	$0.374432\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.05939	0.503428	0.251714	−	0.967802i	$-0.419006\pi$
$101$	5.05939	0.503428	0.251714	+	0.967802i	$0.419006\pi$
$102$	0	0
$103$	−0.238992	−0.0235485	−0.0117743	−	0.999931i	$-0.503748\pi$
$103$	−0.238992	−0.0235485	−0.0117743	+	0.999931i	$0.503748\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.25496	−0.894710	−0.447355	−	0.894356i	$-0.647634\pi$
$107$	−9.25496	−0.894710	−0.447355	+	0.894356i	$0.647634\pi$
$108$	0	0
$109$	−10.5453	−1.01005	−0.505026	−	0.863104i	$-0.668517\pi$
$109$	−10.5453	−1.01005	−0.505026	+	0.863104i	$0.668517\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.1281	−0.952774	−0.476387	−	0.879236i	$-0.658054\pi$
$113$	−10.1281	−0.952774	−0.476387	+	0.879236i	$0.658054\pi$
$114$	0	0
$115$	1.87816	0.175139
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.22446	−0.478926
$120$	0	0
$121$	13.7489	1.24990
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.54429	−0.764225
$126$	0	0
$127$	−6.52720	−0.579196	−0.289598	−	0.957148i	$-0.593522\pi$
$127$	−6.52720	−0.579196	−0.289598	+	0.957148i	$0.593522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.53495	0.308850	0.154425	−	0.988005i	$-0.450647\pi$
$131$	3.53495	0.308850	0.154425	+	0.988005i	$0.450647\pi$
$132$	0	0
$133$	5.18622	0.449702
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.97206	0.424792	0.212396	−	0.977184i	$-0.431873\pi$
$137$	4.97206	0.424792	0.212396	+	0.977184i	$0.431873\pi$
$138$	0	0
$139$	−9.75133	−0.827097	−0.413548	−	0.910482i	$-0.635711\pi$
$139$	−9.75133	−0.827097	−0.413548	+	0.910482i	$0.635711\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.19024	−0.517654
$144$	0	0
$145$	6.44230	0.535004
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.7822	−1.04716	−0.523578	−	0.851978i	$-0.675403\pi$
$149$	−12.7822	−1.04716	−0.523578	+	0.851978i	$0.675403\pi$
$150$	0	0
$151$	−22.9563	−1.86816	−0.934078	−	0.357070i	$-0.883776\pi$
$151$	−22.9563	−1.86816	−0.934078	+	0.357070i	$0.883776\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.36712	0.431097
$156$	0	0
$157$	16.5098	1.31762	0.658812	−	0.752308i	$-0.271059\pi$
$157$	16.5098	1.31762	0.658812	+	0.752308i	$0.271059\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00532	−0.158041
$162$	0	0
$163$	−17.2245	−1.34912	−0.674562	−	0.738218i	$-0.735667\pi$
$163$	−17.2245	−1.34912	−0.674562	+	0.738218i	$0.735667\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1281	−1.01589	−0.507943	−	0.861391i	$-0.669594\pi$
$167$	−13.1281	−1.01589	−0.507943	+	0.861391i	$0.669594\pi$
$168$	0	0
$169$	−11.4517	−0.880899
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.9882	−1.44365	−0.721824	−	0.692076i	$-0.756696\pi$
$173$	−18.9882	−1.44365	−0.721824	+	0.692076i	$0.756696\pi$
$174$	0	0
$175$	4.12281	0.311655
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.61835	−0.569422	−0.284711	−	0.958613i	$-0.591898\pi$
$179$	−7.61835	−0.569422	−0.284711	+	0.958613i	$0.591898\pi$
$180$	0	0
$181$	−9.27737	−0.689581	−0.344791	−	0.938680i	$-0.612050\pi$
$181$	−9.27737	−0.689581	−0.344791	+	0.938680i	$0.612050\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.11346	0.670035
$186$	0	0
$187$	−25.9908	−1.90063
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.60239	−0.694804	−0.347402	−	0.937716i	$-0.612936\pi$
$191$	−9.60239	−0.694804	−0.347402	+	0.937716i	$0.612936\pi$
$192$	0	0
$193$	−15.0831	−1.08571	−0.542853	−	0.839828i	$-0.682656\pi$
$193$	−15.0831	−1.08571	−0.542853	+	0.839828i	$0.682656\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.0712	−0.788788	−0.394394	−	0.918942i	$-0.629045\pi$
$197$	−11.0712	−0.788788	−0.394394	+	0.918942i	$0.629045\pi$
$198$	0	0
$199$	−18.3368	−1.29986	−0.649929	−	0.759995i	$-0.725201\pi$
$199$	−18.3368	−1.29986	−0.649929	+	0.759995i	$0.725201\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.87849	−0.482776
$204$	0	0
$205$	−10.7465	−0.750567
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.8005	1.78466
$210$	0	0
$211$	−2.61822	−0.180245	−0.0901227	−	0.995931i	$-0.528726\pi$
$211$	−2.61822	−0.180245	−0.0901227	+	0.995931i	$0.528726\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.99468	−0.613432
$216$	0	0
$217$	−5.73051	−0.389013
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.50087	0.437296
$222$	0	0
$223$	25.0884	1.68005	0.840023	−	0.542551i	$-0.182542\pi$
$223$	25.0884	1.68005	0.840023	+	0.542551i	$0.182542\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.5562	1.69623	0.848113	−	0.529815i	$-0.177738\pi$
$227$	25.5562	1.69623	0.848113	+	0.529815i	$0.177738\pi$
$228$	0	0
$229$	5.47313	0.361675	0.180837	−	0.983513i	$-0.442119\pi$
$229$	5.47313	0.361675	0.180837	+	0.983513i	$0.442119\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.6774	0.699500	0.349750	−	0.936843i	$-0.386267\pi$
$233$	10.6774	0.699500	0.349750	+	0.936843i	$0.386267\pi$
$234$	0	0
$235$	1.84461	0.120329
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	20.3368	1.31001	0.655003	−	0.755626i	$-0.272667\pi$
$241$	20.3368	1.31001	0.655003	+	0.755626i	$0.272667\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.936586	0.0598363
$246$	0	0
$247$	−6.45328	−0.410612
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.8151	−1.25072	−0.625358	−	0.780338i	$-0.715047\pi$
$251$	−19.8151	−1.25072	−0.625358	+	0.780338i	$0.715047\pi$
$252$	0	0
$253$	−9.97613	−0.627194
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.93901	−0.120952	−0.0604761	−	0.998170i	$-0.519262\pi$
$257$	−1.93901	−0.120952	−0.0604761	+	0.998170i	$0.519262\pi$
$258$	0	0
$259$	−9.73051	−0.604625
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.3554	0.823526	0.411763	−	0.911291i	$-0.364913\pi$
$263$	13.3554	0.823526	0.411763	+	0.911291i	$0.364913\pi$
$264$	0	0
$265$	−7.15007	−0.439225
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.15040	0.253055	0.126527	−	0.991963i	$-0.459617\pi$
$269$	4.15040	0.253055	0.126527	+	0.991963i	$0.459617\pi$
$270$	0	0
$271$	16.6050	1.00868	0.504341	−	0.863505i	$-0.331736\pi$
$271$	16.6050	1.00868	0.504341	+	0.863505i	$0.331736\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.5102	1.23681
$276$	0	0
$277$	17.1083	1.02794	0.513968	−	0.857809i	$-0.328175\pi$
$277$	17.1083	1.02794	0.513968	+	0.857809i	$0.328175\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.9801	−0.893641	−0.446820	−	0.894624i	$-0.647444\pi$
$281$	−14.9801	−0.893641	−0.446820	+	0.894624i	$0.647444\pi$
$282$	0	0
$283$	18.0116	1.07068	0.535339	−	0.844637i	$-0.320184\pi$
$283$	18.0116	1.07068	0.535339	+	0.844637i	$0.320184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.4741	0.677294
$288$	0	0
$289$	10.2950	0.605588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.6724	−1.61664	−0.808320	−	0.588743i	$-0.799623\pi$
$293$	−27.6724	−1.61664	−0.808320	+	0.588743i	$0.799623\pi$
$294$	0	0
$295$	−4.56913	−0.266025
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.49525	0.144304
$300$	0	0
$301$	9.60369	0.553547
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.85056	−0.163222
$306$	0	0
$307$	6.10040	0.348168	0.174084	−	0.984731i	$-0.444304\pi$
$307$	6.10040	0.348168	0.174084	+	0.984731i	$0.444304\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.6553	−1.51149	−0.755743	−	0.654869i	$-0.772724\pi$
$311$	−26.6553	−1.51149	−0.755743	+	0.654869i	$0.772724\pi$
$312$	0	0
$313$	24.1362	1.36426	0.682130	−	0.731231i	$-0.261054\pi$
$313$	24.1362	1.36426	0.682130	+	0.731231i	$0.261054\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.6009	−0.988566	−0.494283	−	0.869301i	$-0.664569\pi$
$317$	−17.6009	−0.988566	−0.494283	+	0.869301i	$0.664569\pi$
$318$	0	0
$319$	−34.2193	−1.91591
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−27.0952	−1.50762
$324$	0	0
$325$	−5.13006	−0.284565
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.96951	−0.108582
$330$	0	0
$331$	−24.2995	−1.33562	−0.667810	−	0.744332i	$-0.732768\pi$
$331$	−24.2995	−1.33562	−0.667810	+	0.744332i	$0.732768\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.07518	0.0587435
$336$	0	0
$337$	22.9443	1.24986	0.624929	−	0.780682i	$-0.285128\pi$
$337$	22.9443	1.24986	0.624929	+	0.780682i	$0.285128\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−28.5083	−1.54381
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.7386	0.683844	0.341922	−	0.939728i	$-0.388922\pi$
$347$	12.7386	0.683844	0.341922	+	0.939728i	$0.388922\pi$
$348$	0	0
$349$	−7.98936	−0.427660	−0.213830	−	0.976871i	$-0.568594\pi$
$349$	−7.98936	−0.427660	−0.213830	+	0.976871i	$0.568594\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.9563	−1.16861	−0.584307	−	0.811532i	$-0.698634\pi$
$353$	−21.9563	−1.16861	−0.584307	+	0.811532i	$0.698634\pi$
$354$	0	0
$355$	8.27707	0.439301
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.4673	−1.08022	−0.540112	−	0.841593i	$-0.681618\pi$
$359$	−20.4673	−1.08022	−0.540112	+	0.841593i	$0.681618\pi$
$360$	0	0
$361$	7.89688	0.415625
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.71971	0.299383
$366$	0	0
$367$	19.8835	1.03791	0.518955	−	0.854802i	$-0.326321\pi$
$367$	19.8835	1.03791	0.518955	+	0.854802i	$0.326321\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.63418	0.396347
$372$	0	0
$373$	34.2876	1.77534	0.887672	−	0.460477i	$-0.152322\pi$
$373$	34.2876	1.77534	0.887672	+	0.460477i	$0.152322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.55900	0.440811
$378$	0	0
$379$	−18.6248	−0.956694	−0.478347	−	0.878171i	$-0.658764\pi$
$379$	−18.6248	−0.956694	−0.478347	+	0.878171i	$0.658764\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.8720	−1.67968	−0.839842	−	0.542832i	$-0.817352\pi$
$383$	−32.8720	−1.67968	−0.839842	+	0.542832i	$0.817352\pi$
$384$	0	0
$385$	4.65935	0.237463
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−21.6500	−1.09770	−0.548850	−	0.835921i	$-0.684934\pi$
$389$	−21.6500	−1.09770	−0.548850	+	0.835921i	$0.684934\pi$
$390$	0	0
$391$	10.4767	0.529831
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.3404	0.570600
$396$	0	0
$397$	16.2721	0.816673	0.408336	−	0.912832i	$-0.366109\pi$
$397$	16.2721	0.816673	0.408336	+	0.912832i	$0.366109\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.14266	−0.456563	−0.228281	−	0.973595i	$-0.573311\pi$
$401$	−9.14266	−0.456563	−0.228281	+	0.973595i	$0.573311\pi$
$402$	0	0
$403$	7.13055	0.355198
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−48.4076	−2.39948
$408$	0	0
$409$	−22.1953	−1.09749	−0.548743	−	0.835991i	$-0.684893\pi$
$409$	−22.1953	−1.09749	−0.548743	+	0.835991i	$0.684893\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.87849	0.240055
$414$	0	0
$415$	0.807958	0.0396611
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.9201	−1.36399	−0.681994	−	0.731358i	$-0.738887\pi$
$419$	−27.9201	−1.36399	−0.681994	+	0.731358i	$0.738887\pi$
$420$	0	0
$421$	24.0883	1.17399	0.586996	−	0.809590i	$-0.300311\pi$
$421$	24.0883	1.17399	0.586996	+	0.809590i	$0.300311\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−21.5394	−1.04482
$426$	0	0
$427$	3.04356	0.147288
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.1623	1.26020	0.630098	−	0.776516i	$-0.283015\pi$
$431$	26.1623	1.26020	0.630098	+	0.776516i	$0.283015\pi$
$432$	0	0
$433$	−18.5630	−0.892082	−0.446041	−	0.895013i	$-0.647166\pi$
$433$	−18.5630	−0.892082	−0.446041	+	0.895013i	$0.647166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.4000	−0.497501
$438$	0	0
$439$	−7.70790	−0.367878	−0.183939	−	0.982938i	$-0.558885\pi$
$439$	−7.70790	−0.367878	−0.183939	+	0.982938i	$0.558885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.0408	−1.56982	−0.784909	−	0.619611i	$-0.787290\pi$
$443$	−33.0408	−1.56982	−0.784909	+	0.619611i	$0.787290\pi$
$444$	0	0
$445$	−10.1601	−0.481634
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.9283	0.940477	0.470238	−	0.882540i	$-0.344168\pi$
$449$	19.9283	0.940477	0.470238	+	0.882540i	$0.344168\pi$
$450$	0	0
$451$	57.0816	2.68787
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.16541	−0.0546351
$456$	0	0
$457$	−33.8653	−1.58415	−0.792075	−	0.610424i	$-0.790999\pi$
$457$	−33.8653	−1.58415	−0.792075	+	0.610424i	$0.790999\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.5070	−0.675659	−0.337830	−	0.941207i	$-0.609693\pi$
$461$	−14.5070	−0.675659	−0.337830	+	0.941207i	$0.609693\pi$
$462$	0	0
$463$	−25.0697	−1.16509	−0.582544	−	0.812799i	$-0.697943\pi$
$463$	−25.0697	−1.16509	−0.582544	+	0.812799i	$0.697943\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.102955	0.00476419	0.00238209	−	0.999997i	$-0.499242\pi$
$467$	0.102955	0.00476419	0.00238209	+	0.999997i	$0.499242\pi$
$468$	0	0
$469$	−1.14798	−0.0530088
$470$	0	0
$471$	0	0
$472$	0	0
$473$	47.7767	2.19677
$474$	0	0
$475$	21.3818	0.981064
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.25336	−0.240032	−0.120016	−	0.992772i	$-0.538295\pi$
$479$	−5.25336	−0.240032	−0.120016	+	0.992772i	$0.538295\pi$
$480$	0	0
$481$	12.1078	0.552068
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.09036	0.321957
$486$	0	0
$487$	30.7319	1.39260	0.696299	−	0.717752i	$-0.254829\pi$
$487$	30.7319	1.39260	0.696299	+	0.717752i	$0.254829\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.71841	0.212939	0.106469	−	0.994316i	$-0.466045\pi$
$491$	4.71841	0.212939	0.106469	+	0.994316i	$0.466045\pi$
$492$	0	0
$493$	35.9364	1.61850
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.83749	−0.396416
$498$	0	0
$499$	−24.7132	−1.10632	−0.553158	−	0.833076i	$-0.686577\pi$
$499$	−24.7132	−1.10632	−0.553158	+	0.833076i	$0.686577\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.4083	0.865371	0.432686	−	0.901545i	$-0.357566\pi$
$503$	19.4083	0.865371	0.432686	+	0.901545i	$0.357566\pi$
$504$	0	0
$505$	4.73856	0.210863
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.45701	0.241878	0.120939	−	0.992660i	$-0.461410\pi$
$509$	5.45701	0.241878	0.120939	+	0.992660i	$0.461410\pi$
$510$	0	0
$511$	−6.10698	−0.270157
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.223836	−0.00986340
$516$	0	0
$517$	−9.79795	−0.430913
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.7683	−1.08512	−0.542558	−	0.840018i	$-0.682544\pi$
$521$	−24.7683	−1.08512	−0.542558	+	0.840018i	$0.682544\pi$
$522$	0	0
$523$	32.7530	1.43219	0.716094	−	0.698004i	$-0.245928\pi$
$523$	32.7530	1.43219	0.716094	+	0.698004i	$0.245928\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.9388	1.30416
$528$	0	0
$529$	−18.9787	−0.825160
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14.2774	−0.618421
$534$	0	0
$535$	−8.66806	−0.374753
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.97483	−0.214281
$540$	0	0
$541$	37.8575	1.62762	0.813811	−	0.581130i	$-0.197389\pi$
$541$	37.8575	1.62762	0.813811	+	0.581130i	$0.197389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.87654	−0.423064
$546$	0	0
$547$	−3.49671	−0.149509	−0.0747543	−	0.997202i	$-0.523817\pi$
$547$	−3.49671	−0.149509	−0.0747543	+	0.997202i	$0.523817\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−35.6734	−1.51974
$552$	0	0
$553$	−12.1083	−0.514896
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.1226	−1.44582	−0.722911	−	0.690941i	$-0.757197\pi$
$557$	−34.1226	−1.44582	−0.722911	+	0.690941i	$0.757197\pi$
$558$	0	0
$559$	−11.9500	−0.505431
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.3014	−0.644878	−0.322439	−	0.946590i	$-0.604503\pi$
$563$	−15.3014	−0.644878	−0.322439	+	0.946590i	$0.604503\pi$
$564$	0	0
$565$	−9.48586	−0.399073
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.9907	1.38304	0.691520	−	0.722357i	$-0.256941\pi$
$569$	32.9907	1.38304	0.691520	+	0.722357i	$0.256941\pi$
$570$	0	0
$571$	21.4300	0.896819	0.448410	−	0.893828i	$-0.351991\pi$
$571$	21.4300	0.896819	0.448410	+	0.893828i	$0.351991\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.26755	−0.344781
$576$	0	0
$577$	−38.4188	−1.59939	−0.799697	−	0.600404i	$-0.795007\pi$
$577$	−38.4188	−1.59939	−0.799697	+	0.600404i	$0.795007\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.862663	−0.0357893
$582$	0	0
$583$	37.9787	1.57292
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.8754	0.696524	0.348262	−	0.937397i	$-0.386772\pi$
$587$	16.8754	0.696524	0.348262	+	0.937397i	$0.386772\pi$
$588$	0	0
$589$	−29.7197	−1.22458
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.7817	0.894467	0.447233	−	0.894417i	$-0.352409\pi$
$593$	21.7817	0.894467	0.447233	+	0.894417i	$0.352409\pi$
$594$	0	0
$595$	−4.89316	−0.200600
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.7555	1.09320	0.546601	−	0.837393i	$-0.315922\pi$
$599$	26.7555	1.09320	0.546601	+	0.837393i	$0.315922\pi$
$600$	0	0
$601$	−25.1229	−1.02479	−0.512393	−	0.858751i	$-0.671241\pi$
$601$	−25.1229	−1.02479	−0.512393	+	0.858751i	$0.671241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.8770	0.523526
$606$	0	0
$607$	−8.63950	−0.350667	−0.175333	−	0.984509i	$-0.556100\pi$
$607$	−8.63950	−0.350667	−0.175333	+	0.984509i	$0.556100\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.45068	0.0991440
$612$	0	0
$613$	46.1789	1.86515	0.932575	−	0.360977i	$-0.117557\pi$
$613$	46.1789	1.86515	0.932575	+	0.360977i	$0.117557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.5244	−0.463957	−0.231978	−	0.972721i	$-0.574520\pi$
$617$	−11.5244	−0.463957	−0.231978	+	0.972721i	$0.574520\pi$
$618$	0	0
$619$	−39.2650	−1.57819	−0.789096	−	0.614270i	$-0.789451\pi$
$619$	−39.2650	−1.57819	−0.789096	+	0.614270i	$0.789451\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.8480	0.434616
$624$	0	0
$625$	12.6116	0.504463
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.8367	2.02699
$630$	0	0
$631$	−22.8387	−0.909193	−0.454596	−	0.890698i	$-0.650216\pi$
$631$	−22.8387	−0.909193	−0.454596	+	0.890698i	$0.650216\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.11329	−0.242598
$636$	0	0
$637$	1.24431	0.0493015
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0980	−0.714827	−0.357413	−	0.933946i	$-0.616341\pi$
$641$	−18.0980	−0.714827	−0.357413	+	0.933946i	$0.616341\pi$
$642$	0	0
$643$	30.0833	1.18637	0.593184	−	0.805067i	$-0.297871\pi$
$643$	30.0833	1.18637	0.593184	+	0.805067i	$0.297871\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.1323	0.988054	0.494027	−	0.869447i	$-0.335524\pi$
$647$	25.1323	0.988054	0.494027	+	0.869447i	$0.335524\pi$
$648$	0	0
$649$	24.2697	0.952668
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.4977	0.410806	0.205403	−	0.978678i	$-0.434150\pi$
$653$	10.4977	0.410806	0.205403	+	0.978678i	$0.434150\pi$
$654$	0	0
$655$	3.31079	0.129363
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.08277	0.314860	0.157430	−	0.987530i	$-0.449679\pi$
$659$	8.08277	0.314860	0.157430	+	0.987530i	$0.449679\pi$
$660$	0	0
$661$	−26.9059	−1.04652	−0.523260	−	0.852173i	$-0.675284\pi$
$661$	−26.9059	−1.04652	−0.523260	+	0.852173i	$0.675284\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.85734	0.188360
$666$	0	0
$667$	13.7936	0.534090
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.1412	0.584519
$672$	0	0
$673$	−12.2232	−0.471170	−0.235585	−	0.971854i	$-0.575701\pi$
$673$	−12.2232	−0.471170	−0.235585	+	0.971854i	$0.575701\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.68030	0.256745	0.128372	−	0.991726i	$-0.459025\pi$
$677$	6.68030	0.256745	0.128372	+	0.991726i	$0.459025\pi$
$678$	0	0
$679$	−7.57043	−0.290526
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.2125	−0.735146	−0.367573	−	0.929995i	$-0.619811\pi$
$683$	−19.2125	−0.735146	−0.367573	+	0.929995i	$0.619811\pi$
$684$	0	0
$685$	4.65677	0.177926
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.49931	−0.361895
$690$	0	0
$691$	11.4541	0.435735	0.217867	−	0.975978i	$-0.430090\pi$
$691$	11.4541	0.435735	0.217867	+	0.975978i	$0.430090\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.13296	−0.346433
$696$	0	0
$697$	−59.9460	−2.27062
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.29596	−0.162256	−0.0811281	−	0.996704i	$-0.525852\pi$
$701$	−4.29596	−0.162256	−0.0811281	+	0.996704i	$0.525852\pi$
$702$	0	0
$703$	−50.4646	−1.90331
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.05939	−0.190278
$708$	0	0
$709$	12.8901	0.484099	0.242049	−	0.970264i	$-0.422180\pi$
$709$	12.8901	0.484099	0.242049	+	0.970264i	$0.422180\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.4915	0.430361
$714$	0	0
$715$	−5.79769	−0.216821
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.46489	−0.203806	−0.101903	−	0.994794i	$-0.532493\pi$
$719$	−5.46489	−0.203806	−0.101903	+	0.994794i	$0.532493\pi$
$720$	0	0
$721$	0.238992	0.00890051
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−28.3587	−1.05322
$726$	0	0
$727$	49.8599	1.84920	0.924601	−	0.380936i	$-0.124398\pi$
$727$	49.8599	1.84920	0.924601	+	0.380936i	$0.124398\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−50.1741	−1.85576
$732$	0	0
$733$	−35.2246	−1.30105	−0.650525	−	0.759485i	$-0.725451\pi$
$733$	−35.2246	−1.30105	−0.650525	+	0.759485i	$0.725451\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.71100	−0.210367
$738$	0	0
$739$	35.7209	1.31401	0.657007	−	0.753885i	$-0.271822\pi$
$739$	35.7209	1.31401	0.657007	+	0.753885i	$0.271822\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.0609	−0.479156	−0.239578	−	0.970877i	$-0.577009\pi$
$743$	−13.0609	−0.479156	−0.239578	+	0.970877i	$0.577009\pi$
$744$	0	0
$745$	−11.9716	−0.438605
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.25496	0.338169
$750$	0	0
$751$	2.22446	0.0811717	0.0405859	−	0.999176i	$-0.487078\pi$
$751$	2.22446	0.0811717	0.0405859	+	0.999176i	$0.487078\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.5005	−0.782484
$756$	0	0
$757$	4.27335	0.155317	0.0776587	−	0.996980i	$-0.475256\pi$
$757$	4.27335	0.155317	0.0776587	+	0.996980i	$0.475256\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.4411	1.82849	0.914245	−	0.405162i	$-0.132785\pi$
$761$	50.4411	1.82849	0.914245	+	0.405162i	$0.132785\pi$
$762$	0	0
$763$	10.5453	0.381764
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.07037	−0.219188
$768$	0	0
$769$	−2.06986	−0.0746411	−0.0373205	−	0.999303i	$-0.511882\pi$
$769$	−2.06986	−0.0746411	−0.0373205	+	0.999303i	$0.511882\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.3739	−1.02054	−0.510269	−	0.860015i	$-0.670454\pi$
$773$	−28.3739	−1.02054	−0.510269	+	0.860015i	$0.670454\pi$
$774$	0	0
$775$	−23.6258	−0.848664
$776$	0	0
$777$	0	0
$778$	0	0
$779$	59.5072	2.13207
$780$	0	0
$781$	−43.9650	−1.57319
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.4628	0.551892
$786$	0	0
$787$	3.45072	0.123005	0.0615025	−	0.998107i	$-0.480411\pi$
$787$	3.45072	0.123005	0.0615025	+	0.998107i	$0.480411\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.1281	0.360115
$792$	0	0
$793$	−3.78714	−0.134485
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.4460	−0.795079	−0.397540	−	0.917585i	$-0.630136\pi$
$797$	−22.4460	−0.795079	−0.397540	+	0.917585i	$0.630136\pi$
$798$	0	0
$799$	10.2896	0.364020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30.3812	−1.07213
$804$	0	0
$805$	−1.87816	−0.0661963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.2366	0.641164	0.320582	−	0.947221i	$-0.396122\pi$
$809$	18.2366	0.641164	0.320582	+	0.947221i	$0.396122\pi$
$810$	0	0
$811$	−16.8280	−0.590910	−0.295455	−	0.955357i	$-0.595471\pi$
$811$	−16.8280	−0.590910	−0.295455	+	0.955357i	$0.595471\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.1322	−0.565086
$816$	0	0
$817$	49.8068	1.74252
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.2298	−1.29933	−0.649665	−	0.760221i	$-0.725091\pi$
$821$	−37.2298	−1.29933	−0.649665	+	0.760221i	$0.725091\pi$
$822$	0	0
$823$	12.2749	0.427878	0.213939	−	0.976847i	$-0.431371\pi$
$823$	12.2749	0.427878	0.213939	+	0.976847i	$0.431371\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.60355	0.299175	0.149587	−	0.988749i	$-0.452205\pi$
$827$	8.60355	0.299175	0.149587	+	0.988749i	$0.452205\pi$
$828$	0	0
$829$	−44.2887	−1.53821	−0.769105	−	0.639123i	$-0.779298\pi$
$829$	−44.2887	−1.53821	−0.769105	+	0.639123i	$0.779298\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.22446	0.181017
$834$	0	0
$835$	−12.2956	−0.425508
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−47.2307	−1.63059	−0.815293	−	0.579048i	$-0.803424\pi$
$839$	−47.2307	−1.63059	−0.815293	+	0.579048i	$0.803424\pi$
$840$	0	0
$841$	18.3137	0.631506
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.7255	−0.368968
$846$	0	0
$847$	−13.7489	−0.472418
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.5128	0.668890
$852$	0	0
$853$	−23.9650	−0.820546	−0.410273	−	0.911963i	$-0.634567\pi$
$853$	−23.9650	−0.820546	−0.410273	+	0.911963i	$0.634567\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.0181	0.444688	0.222344	−	0.974968i	$-0.428629\pi$
$857$	13.0181	0.444688	0.222344	+	0.974968i	$0.428629\pi$
$858$	0	0
$859$	0.0947371	0.00323239	0.00161619	−	0.999999i	$-0.499486\pi$
$859$	0.0947371	0.00323239	0.00161619	+	0.999999i	$0.499486\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.8729	0.880723	0.440361	−	0.897821i	$-0.354850\pi$
$863$	25.8729	0.880723	0.440361	+	0.897821i	$0.354850\pi$
$864$	0	0
$865$	−17.7841	−0.604678
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.2366	−2.04339
$870$	0	0
$871$	1.42845	0.0484010
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.54429	0.288850
$876$	0	0
$877$	−53.8083	−1.81698	−0.908489	−	0.417908i	$-0.862763\pi$
$877$	−53.8083	−1.81698	−0.908489	+	0.417908i	$0.862763\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.8689	−1.44429	−0.722144	−	0.691743i	$-0.756843\pi$
$881$	−42.8689	−1.44429	−0.722144	+	0.691743i	$0.756843\pi$
$882$	0	0
$883$	−34.5967	−1.16427	−0.582136	−	0.813092i	$-0.697783\pi$
$883$	−34.5967	−1.16427	−0.582136	+	0.813092i	$0.697783\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.4555	0.686830	0.343415	−	0.939184i	$-0.388416\pi$
$887$	20.4555	0.686830	0.343415	+	0.939184i	$0.388416\pi$
$888$	0	0
$889$	6.52720	0.218915
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.2143	−0.341808
$894$	0	0
$895$	−7.13524	−0.238505
$896$	0	0
$897$	0	0
$898$	0	0
$899$	39.4173	1.31464
$900$	0	0
$901$	−39.8845	−1.32874
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.68905	−0.288834
$906$	0	0
$907$	7.84912	0.260626	0.130313	−	0.991473i	$-0.458402\pi$
$907$	7.84912	0.260626	0.130313	+	0.991473i	$0.458402\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.74379	0.256563	0.128282	−	0.991738i	$-0.459054\pi$
$911$	7.74379	0.256563	0.128282	+	0.991738i	$0.459054\pi$
$912$	0	0
$913$	−4.29160	−0.142031
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.53495	−0.116734
$918$	0	0
$919$	−19.3702	−0.638963	−0.319482	−	0.947592i	$-0.603509\pi$
$919$	−19.3702	−0.638963	−0.319482	+	0.947592i	$0.603509\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.9966	0.361958
$924$	0	0
$925$	−40.1170	−1.31904
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.55979	−0.116793	−0.0583964	−	0.998293i	$-0.518599\pi$
$929$	−3.55979	−0.116793	−0.0583964	+	0.998293i	$0.518599\pi$
$930$	0	0
$931$	−5.18622	−0.169972
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.3426	−0.796089
$936$	0	0
$937$	−34.2230	−1.11802	−0.559008	−	0.829162i	$-0.688818\pi$
$937$	−34.2230	−1.11802	−0.559008	+	0.829162i	$0.688818\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.0407	−0.392514	−0.196257	−	0.980552i	$-0.562879\pi$
$941$	−12.0407	−0.392514	−0.196257	+	0.980552i	$0.562879\pi$
$942$	0	0
$943$	−23.0093	−0.749284
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.7247	0.738453	0.369227	−	0.929339i	$-0.379623\pi$
$947$	22.7247	0.738453	0.369227	+	0.929339i	$0.379623\pi$
$948$	0	0
$949$	7.59899	0.246674
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.74488	−0.218488	−0.109244	−	0.994015i	$-0.534843\pi$
$953$	−6.74488	−0.218488	−0.109244	+	0.994015i	$0.534843\pi$
$954$	0	0
$955$	−8.99346	−0.291022
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.97206	−0.160556
$960$	0	0
$961$	1.83879	0.0593158
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.1266	−0.454752
$966$	0	0
$967$	25.6534	0.824958	0.412479	−	0.910967i	$-0.364663\pi$
$967$	25.6534	0.824958	0.412479	+	0.910967i	$0.364663\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.9298	−1.31350	−0.656750	−	0.754108i	$-0.728069\pi$
$971$	−40.9298	−1.31350	−0.656750	+	0.754108i	$0.728069\pi$
$972$	0	0
$973$	9.75133	0.312613
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.4730	1.42282	0.711408	−	0.702779i	$-0.248058\pi$
$977$	44.4730	1.42282	0.711408	+	0.702779i	$0.248058\pi$
$978$	0	0
$979$	53.9669	1.72479
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56.9100	1.81515	0.907573	−	0.419893i	$-0.137933\pi$
$983$	56.9100	1.81515	0.907573	+	0.419893i	$0.137933\pi$
$984$	0	0
$985$	−10.3691	−0.330387
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.2585	−0.612384
$990$	0	0
$991$	44.7331	1.42099	0.710496	−	0.703701i	$-0.248470\pi$
$991$	44.7331	1.42099	0.710496	+	0.703701i	$0.248470\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.1740	−0.544451
$996$	0	0
$997$	−18.7336	−0.593300	−0.296650	−	0.954986i	$-0.595869\pi$
$997$	−18.7336	−0.593300	−0.296650	+	0.954986i	$0.595869\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.ce.1.3		4
3.2	odd	2		9072.2.a.cl.1.2		4
4.3	odd	2		4536.2.a.x.1.3		4
9.2	odd	6		3024.2.r.l.1009.3		8
9.4	even	3		1008.2.r.m.673.3		8
9.5	odd	6		3024.2.r.l.2017.3		8
9.7	even	3		1008.2.r.m.337.3		8
12.11	even	2		4536.2.a.ba.1.2		4
36.7	odd	6		504.2.r.d.337.2	yes	8
36.11	even	6		1512.2.r.d.1009.3		8
36.23	even	6		1512.2.r.d.505.3		8
36.31	odd	6		504.2.r.d.169.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.d.169.2	✓	8	36.31	odd	6
504.2.r.d.337.2	yes	8	36.7	odd	6
1008.2.r.m.337.3		8	9.7	even	3
1008.2.r.m.673.3		8	9.4	even	3
1512.2.r.d.505.3		8	36.23	even	6
1512.2.r.d.1009.3		8	36.11	even	6
3024.2.r.l.1009.3		8	9.2	odd	6
3024.2.r.l.2017.3		8	9.5	odd	6
4536.2.a.x.1.3		4	4.3	odd	2
4536.2.a.ba.1.2		4	12.11	even	2
9072.2.a.ce.1.3		4	1.1	even	1	trivial
9072.2.a.cl.1.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.936586 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+0.936586 q^{5} -1.00000 q^{7} -4.97483 q^{11} +1.24431 q^{13} +5.22446 q^{17} -5.18622 q^{19} +2.00532 q^{23} -4.12281 q^{25} +6.87849 q^{29} +5.73051 q^{31} -0.936586 q^{35} +9.73051 q^{37} -11.4741 q^{41} -9.60369 q^{43} +1.96951 q^{47} +1.00000 q^{49} -7.63418 q^{53} -4.65935 q^{55} -4.87849 q^{59} -3.04356 q^{61} +1.16541 q^{65} +1.14798 q^{67} +8.83749 q^{71} +6.10698 q^{73} +4.97483 q^{77} +12.1083 q^{79} +0.862663 q^{83} +4.89316 q^{85} -10.8480 q^{89} -1.24431 q^{91} -4.85734 q^{95} +7.57043 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 - 4 * q^7	\( 4 q - 4 q^{5} - 4 q^{7} - 6 q^{11} + 3 q^{13} - 8 q^{17} + 2 q^{19} - 5 q^{23} + 14 q^{25} - q^{29} + 11 q^{31} + 4 q^{35} + 27 q^{37} - 2 q^{41} - 11 q^{43} + 7 q^{47} + 4 q^{49} - 4 q^{53} - 6 q^{55} + 9 q^{59} + 7 q^{61} + 9 q^{65} - 12 q^{67} + 12 q^{71} + 13 q^{73} + 6 q^{77} - 22 q^{79} - 6 q^{83} + 11 q^{85} - 14 q^{89} - 3 q^{91} - 23 q^{95} + q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 6 * q^11 + 3 * q^13 - 8 * q^17 + 2 * q^19 - 5 * q^23 + 14 * q^25 - q^29 + 11 * q^31 + 4 * q^35 + 27 * q^37 - 2 * q^41 - 11 * q^43 + 7 * q^47 + 4 * q^49 - 4 * q^53 - 6 * q^55 + 9 * q^59 + 7 * q^61 + 9 * q^65 - 12 * q^67 + 12 * q^71 + 13 * q^73 + 6 * q^77 - 22 * q^79 - 6 * q^83 + 11 * q^85 - 14 * q^89 - 3 * q^91 - 23 * q^95 + q^97

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