LMFDB - Embedded newform 1452.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,2,Mod(1,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1452.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-1,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.5942783735$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1452.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.85410 q^{5} +0.236068 q^{7} +1.00000 q^{9} +6.23607 q^{13} +3.85410 q^{15} -2.38197 q^{17} +5.09017 q^{19} -0.236068 q^{21} -0.236068 q^{23} +9.85410 q^{25} -1.00000 q^{27} -4.47214 q^{29} -6.38197 q^{31} -0.909830 q^{35} -3.76393 q^{37} -6.23607 q^{39} -9.47214 q^{41} -9.47214 q^{43} -3.85410 q^{45} +3.61803 q^{47} -6.94427 q^{49} +2.38197 q^{51} +6.56231 q^{53} -5.09017 q^{57} +2.14590 q^{59} +2.61803 q^{61} +0.236068 q^{63} -24.0344 q^{65} -0.145898 q^{67} +0.236068 q^{69} -1.38197 q^{71} -3.23607 q^{73} -9.85410 q^{75} -14.2361 q^{79} +1.00000 q^{81} -15.4721 q^{83} +9.18034 q^{85} +4.47214 q^{87} +1.00000 q^{89} +1.47214 q^{91} +6.38197 q^{93} -19.6180 q^{95} -5.09017 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9} + 8 q^{13} + q^{15} - 7 q^{17} - q^{19} + 4 q^{21} + 4 q^{23} + 13 q^{25} - 2 q^{27} - 15 q^{31} - 13 q^{35} - 12 q^{37} - 8 q^{39} - 10 q^{41} - 10 q^{43} - q^{45}+ \cdots + q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9 + 8 * q^13 + q^15 - 7 * q^17 - q^19 + 4 * q^21 + 4 * q^23 + 13 * q^25 - 2 * q^27 - 15 * q^31 - 13 * q^35 - 12 * q^37 - 8 * q^39 - 10 * q^41 - 10 * q^43 - q^45 + 5 * q^47 + 4 * q^49 + 7 * q^51 - 7 * q^53 + q^57 + 11 * q^59 + 3 * q^61 - 4 * q^63 - 19 * q^65 - 7 * q^67 - 4 * q^69 - 5 * q^71 - 2 * q^73 - 13 * q^75 - 24 * q^79 + 2 * q^81 - 22 * q^83 - 4 * q^85 + 2 * q^89 - 6 * q^91 + 15 * q^93 - 37 * q^95 + q^97

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
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.23607	1.72957	0.864787	−	0.502139i	$-0.167453\pi$
$13$	6.23607	1.72957	0.864787	+	0.502139i	$0.167453\pi$
$14$	0	0
$15$	3.85410	0.995125
$16$	0	0
$17$	−2.38197	−0.577712	−0.288856	−	0.957373i	$-0.593275\pi$
$17$	−2.38197	−0.577712	−0.288856	+	0.957373i	$0.593275\pi$
$18$	0	0
$19$	5.09017	1.16777	0.583883	−	0.811838i	$-0.301533\pi$
$19$	5.09017	1.16777	0.583883	+	0.811838i	$0.301533\pi$
$20$	0	0
$21$	−0.236068	−0.0515143
$22$	0	0
$23$	−0.236068	−0.0492236	−0.0246118	−	0.999697i	$-0.507835\pi$
$23$	−0.236068	−0.0492236	−0.0246118	+	0.999697i	$0.507835\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−6.38197	−1.14623	−0.573117	−	0.819473i	$-0.694266\pi$
$31$	−6.38197	−1.14623	−0.573117	+	0.819473i	$0.694266\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.909830	−0.153789
$36$	0	0
$37$	−3.76393	−0.618787	−0.309393	−	0.950934i	$-0.600126\pi$
$37$	−3.76393	−0.618787	−0.309393	+	0.950934i	$0.600126\pi$
$38$	0	0
$39$	−6.23607	−0.998570
$40$	0	0
$41$	−9.47214	−1.47930	−0.739650	−	0.672992i	$-0.765009\pi$
$41$	−9.47214	−1.47930	−0.739650	+	0.672992i	$0.765009\pi$
$42$	0	0
$43$	−9.47214	−1.44449	−0.722244	−	0.691639i	$-0.756889\pi$
$43$	−9.47214	−1.44449	−0.722244	+	0.691639i	$0.756889\pi$
$44$	0	0
$45$	−3.85410	−0.574536
$46$	0	0
$47$	3.61803	0.527744	0.263872	−	0.964558i	$-0.415000\pi$
$47$	3.61803	0.527744	0.263872	+	0.964558i	$0.415000\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	2.38197	0.333542
$52$	0	0
$53$	6.56231	0.901402	0.450701	−	0.892675i	$-0.351174\pi$
$53$	6.56231	0.901402	0.450701	+	0.892675i	$0.351174\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.09017	−0.674209
$58$	0	0
$59$	2.14590	0.279372	0.139686	−	0.990196i	$-0.455391\pi$
$59$	2.14590	0.279372	0.139686	+	0.990196i	$0.455391\pi$
$60$	0	0
$61$	2.61803	0.335205	0.167602	−	0.985855i	$-0.446398\pi$
$61$	2.61803	0.335205	0.167602	+	0.985855i	$0.446398\pi$
$62$	0	0
$63$	0.236068	0.0297418
$64$	0	0
$65$	−24.0344	−2.98111
$66$	0	0
$67$	−0.145898	−0.0178243	−0.00891214	−	0.999960i	$-0.502837\pi$
$67$	−0.145898	−0.0178243	−0.00891214	+	0.999960i	$0.502837\pi$
$68$	0	0
$69$	0.236068	0.0284192
$70$	0	0
$71$	−1.38197	−0.164009	−0.0820046	−	0.996632i	$-0.526132\pi$
$71$	−1.38197	−0.164009	−0.0820046	+	0.996632i	$0.526132\pi$
$72$	0	0
$73$	−3.23607	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$73$	−3.23607	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$74$	0	0
$75$	−9.85410	−1.13785
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.2361	−1.60168	−0.800841	−	0.598877i	$-0.795614\pi$
$79$	−14.2361	−1.60168	−0.800841	+	0.598877i	$0.795614\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.4721	−1.69829	−0.849144	−	0.528162i	$-0.822881\pi$
$83$	−15.4721	−1.69829	−0.849144	+	0.528162i	$0.822881\pi$
$84$	0	0
$85$	9.18034	0.995748
$86$	0	0
$87$	4.47214	0.479463
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	1.47214	0.154322
$92$	0	0
$93$	6.38197	0.661779
$94$	0	0
$95$	−19.6180	−2.01277
$96$	0	0
$97$	−5.09017	−0.516828	−0.258414	−	0.966034i	$-0.583200\pi$
$97$	−5.09017	−0.516828	−0.258414	+	0.966034i	$0.583200\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.94427	0.591477	0.295739	−	0.955269i	$-0.404434\pi$
$101$	5.94427	0.591477	0.295739	+	0.955269i	$0.404434\pi$
$102$	0	0
$103$	12.4721	1.22892	0.614458	−	0.788950i	$-0.289375\pi$
$103$	12.4721	1.22892	0.614458	+	0.788950i	$0.289375\pi$
$104$	0	0
$105$	0.909830	0.0887903
$106$	0	0
$107$	−6.52786	−0.631072	−0.315536	−	0.948914i	$-0.602184\pi$
$107$	−6.52786	−0.631072	−0.315536	+	0.948914i	$0.602184\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	3.76393	0.357257
$112$	0	0
$113$	12.2361	1.15107	0.575536	−	0.817776i	$-0.304793\pi$
$113$	12.2361	1.15107	0.575536	+	0.817776i	$0.304793\pi$
$114$	0	0
$115$	0.909830	0.0848421
$116$	0	0
$117$	6.23607	0.576525
$118$	0	0
$119$	−0.562306	−0.0515465
$120$	0	0
$121$	0	0
$122$	0	0
$123$	9.47214	0.854074
$124$	0	0
$125$	−18.7082	−1.67331
$126$	0	0
$127$	−15.7082	−1.39388	−0.696939	−	0.717131i	$-0.745455\pi$
$127$	−15.7082	−1.39388	−0.696939	+	0.717131i	$0.745455\pi$
$128$	0	0
$129$	9.47214	0.833975
$130$	0	0
$131$	8.56231	0.748092	0.374046	−	0.927410i	$-0.377970\pi$
$131$	8.56231	0.748092	0.374046	+	0.927410i	$0.377970\pi$
$132$	0	0
$133$	1.20163	0.104194
$134$	0	0
$135$	3.85410	0.331708
$136$	0	0
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	−3.79837	−0.322174	−0.161087	−	0.986940i	$-0.551500\pi$
$139$	−3.79837	−0.322174	−0.161087	+	0.986940i	$0.551500\pi$
$140$	0	0
$141$	−3.61803	−0.304693
$142$	0	0
$143$	0	0
$144$	0	0
$145$	17.2361	1.43138
$146$	0	0
$147$	6.94427	0.572754
$148$	0	0
$149$	16.8885	1.38356	0.691782	−	0.722107i	$-0.256826\pi$
$149$	16.8885	1.38356	0.691782	+	0.722107i	$0.256826\pi$
$150$	0	0
$151$	−4.47214	−0.363937	−0.181969	−	0.983304i	$-0.558247\pi$
$151$	−4.47214	−0.363937	−0.181969	+	0.983304i	$0.558247\pi$
$152$	0	0
$153$	−2.38197	−0.192571
$154$	0	0
$155$	24.5967	1.97566
$156$	0	0
$157$	−17.2361	−1.37559	−0.687794	−	0.725906i	$-0.741421\pi$
$157$	−17.2361	−1.37559	−0.687794	+	0.725906i	$0.741421\pi$
$158$	0	0
$159$	−6.56231	−0.520425
$160$	0	0
$161$	−0.0557281	−0.00439199
$162$	0	0
$163$	−4.85410	−0.380203	−0.190101	−	0.981764i	$-0.560882\pi$
$163$	−4.85410	−0.380203	−0.190101	+	0.981764i	$0.560882\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.618034	0.0478249	0.0239125	−	0.999714i	$-0.492388\pi$
$167$	0.618034	0.0478249	0.0239125	+	0.999714i	$0.492388\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	5.09017	0.389255
$172$	0	0
$173$	−5.56231	−0.422894	−0.211447	−	0.977389i	$-0.567818\pi$
$173$	−5.56231	−0.422894	−0.211447	+	0.977389i	$0.567818\pi$
$174$	0	0
$175$	2.32624	0.175847
$176$	0	0
$177$	−2.14590	−0.161296
$178$	0	0
$179$	−17.9443	−1.34122	−0.670609	−	0.741811i	$-0.733967\pi$
$179$	−17.9443	−1.34122	−0.670609	+	0.741811i	$0.733967\pi$
$180$	0	0
$181$	5.94427	0.441834	0.220917	−	0.975293i	$-0.429095\pi$
$181$	5.94427	0.441834	0.220917	+	0.975293i	$0.429095\pi$
$182$	0	0
$183$	−2.61803	−0.193531
$184$	0	0
$185$	14.5066	1.06654
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.236068	−0.0171714
$190$	0	0
$191$	−21.6525	−1.56672	−0.783359	−	0.621569i	$-0.786495\pi$
$191$	−21.6525	−1.56672	−0.783359	+	0.621569i	$0.786495\pi$
$192$	0	0
$193$	−15.3262	−1.10321	−0.551603	−	0.834107i	$-0.685984\pi$
$193$	−15.3262	−1.10321	−0.551603	+	0.834107i	$0.685984\pi$
$194$	0	0
$195$	24.0344	1.72114
$196$	0	0
$197$	5.90983	0.421058	0.210529	−	0.977588i	$-0.432481\pi$
$197$	5.90983	0.421058	0.210529	+	0.977588i	$0.432481\pi$
$198$	0	0
$199$	−10.4164	−0.738400	−0.369200	−	0.929350i	$-0.620368\pi$
$199$	−10.4164	−0.738400	−0.369200	+	0.929350i	$0.620368\pi$
$200$	0	0
$201$	0.145898	0.0102909
$202$	0	0
$203$	−1.05573	−0.0740976
$204$	0	0
$205$	36.5066	2.54973
$206$	0	0
$207$	−0.236068	−0.0164079
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−18.7426	−1.29030	−0.645148	−	0.764057i	$-0.723204\pi$
$211$	−18.7426	−1.29030	−0.645148	+	0.764057i	$0.723204\pi$
$212$	0	0
$213$	1.38197	0.0946908
$214$	0	0
$215$	36.5066	2.48973
$216$	0	0
$217$	−1.50658	−0.102273
$218$	0	0
$219$	3.23607	0.218673
$220$	0	0
$221$	−14.8541	−0.999195
$222$	0	0
$223$	−7.47214	−0.500371	−0.250186	−	0.968198i	$-0.580492\pi$
$223$	−7.47214	−0.500371	−0.250186	+	0.968198i	$0.580492\pi$
$224$	0	0
$225$	9.85410	0.656940
$226$	0	0
$227$	−12.7082	−0.843473	−0.421737	−	0.906718i	$-0.638579\pi$
$227$	−12.7082	−0.843473	−0.421737	+	0.906718i	$0.638579\pi$
$228$	0	0
$229$	10.9443	0.723218	0.361609	−	0.932330i	$-0.382228\pi$
$229$	10.9443	0.723218	0.361609	+	0.932330i	$0.382228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.20163	−0.0787211	−0.0393606	−	0.999225i	$-0.512532\pi$
$233$	−1.20163	−0.0787211	−0.0393606	+	0.999225i	$0.512532\pi$
$234$	0	0
$235$	−13.9443	−0.909624
$236$	0	0
$237$	14.2361	0.924732
$238$	0	0
$239$	−17.5066	−1.13241	−0.566203	−	0.824266i	$-0.691588\pi$
$239$	−17.5066	−1.13241	−0.566203	+	0.824266i	$0.691588\pi$
$240$	0	0
$241$	23.7082	1.52718	0.763590	−	0.645702i	$-0.223435\pi$
$241$	23.7082	1.52718	0.763590	+	0.645702i	$0.223435\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	26.7639	1.70988
$246$	0	0
$247$	31.7426	2.01974
$248$	0	0
$249$	15.4721	0.980507
$250$	0	0
$251$	24.0902	1.52056	0.760279	−	0.649597i	$-0.225062\pi$
$251$	24.0902	1.52056	0.760279	+	0.649597i	$0.225062\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.18034	−0.574895
$256$	0	0
$257$	−2.79837	−0.174558	−0.0872789	−	0.996184i	$-0.527817\pi$
$257$	−2.79837	−0.174558	−0.0872789	+	0.996184i	$0.527817\pi$
$258$	0	0
$259$	−0.888544	−0.0552114
$260$	0	0
$261$	−4.47214	−0.276818
$262$	0	0
$263$	−23.8541	−1.47091	−0.735453	−	0.677575i	$-0.763031\pi$
$263$	−23.8541	−1.47091	−0.735453	+	0.677575i	$0.763031\pi$
$264$	0	0
$265$	−25.2918	−1.55366
$266$	0	0
$267$	−1.00000	−0.0611990
$268$	0	0
$269$	28.4721	1.73598	0.867988	−	0.496584i	$-0.165413\pi$
$269$	28.4721	1.73598	0.867988	+	0.496584i	$0.165413\pi$
$270$	0	0
$271$	2.43769	0.148079	0.0740397	−	0.997255i	$-0.476411\pi$
$271$	2.43769	0.148079	0.0740397	+	0.997255i	$0.476411\pi$
$272$	0	0
$273$	−1.47214	−0.0890977
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.56231	−0.153954	−0.0769770	−	0.997033i	$-0.524527\pi$
$277$	−2.56231	−0.153954	−0.0769770	+	0.997033i	$0.524527\pi$
$278$	0	0
$279$	−6.38197	−0.382078
$280$	0	0
$281$	−17.1246	−1.02157	−0.510784	−	0.859709i	$-0.670645\pi$
$281$	−17.1246	−1.02157	−0.510784	+	0.859709i	$0.670645\pi$
$282$	0	0
$283$	−23.2361	−1.38124	−0.690620	−	0.723217i	$-0.742662\pi$
$283$	−23.2361	−1.38124	−0.690620	+	0.723217i	$0.742662\pi$
$284$	0	0
$285$	19.6180	1.16207
$286$	0	0
$287$	−2.23607	−0.131991
$288$	0	0
$289$	−11.3262	−0.666249
$290$	0	0
$291$	5.09017	0.298391
$292$	0	0
$293$	5.00000	0.292103	0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000	0.292103	0.146052	+	0.989277i	$0.453343\pi$
$294$	0	0
$295$	−8.27051	−0.481528
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.47214	−0.0851358
$300$	0	0
$301$	−2.23607	−0.128885
$302$	0	0
$303$	−5.94427	−0.341489
$304$	0	0
$305$	−10.0902	−0.577761
$306$	0	0
$307$	0.673762	0.0384536	0.0192268	−	0.999815i	$-0.493880\pi$
$307$	0.673762	0.0384536	0.0192268	+	0.999815i	$0.493880\pi$
$308$	0	0
$309$	−12.4721	−0.709515
$310$	0	0
$311$	−29.1803	−1.65467	−0.827333	−	0.561712i	$-0.810143\pi$
$311$	−29.1803	−1.65467	−0.827333	+	0.561712i	$0.810143\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	−0.909830	−0.0512631
$316$	0	0
$317$	8.52786	0.478973	0.239486	−	0.970900i	$-0.423021\pi$
$317$	8.52786	0.478973	0.239486	+	0.970900i	$0.423021\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	6.52786	0.364350
$322$	0	0
$323$	−12.1246	−0.674631
$324$	0	0
$325$	61.4508	3.40868
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	0.854102	0.0470882
$330$	0	0
$331$	11.9443	0.656517	0.328258	−	0.944588i	$-0.393538\pi$
$331$	11.9443	0.656517	0.328258	+	0.944588i	$0.393538\pi$
$332$	0	0
$333$	−3.76393	−0.206262
$334$	0	0
$335$	0.562306	0.0307221
$336$	0	0
$337$	17.5967	0.958556	0.479278	−	0.877663i	$-0.340899\pi$
$337$	17.5967	0.958556	0.479278	+	0.877663i	$0.340899\pi$
$338$	0	0
$339$	−12.2361	−0.664572
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	−0.909830	−0.0489836
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	28.2361	1.51144	0.755721	−	0.654894i	$-0.227287\pi$
$349$	28.2361	1.51144	0.755721	+	0.654894i	$0.227287\pi$
$350$	0	0
$351$	−6.23607	−0.332857
$352$	0	0
$353$	−10.4721	−0.557376	−0.278688	−	0.960382i	$-0.589899\pi$
$353$	−10.4721	−0.557376	−0.278688	+	0.960382i	$0.589899\pi$
$354$	0	0
$355$	5.32624	0.282687
$356$	0	0
$357$	0.562306	0.0297604
$358$	0	0
$359$	23.2361	1.22635	0.613176	−	0.789946i	$-0.289891\pi$
$359$	23.2361	1.22635	0.613176	+	0.789946i	$0.289891\pi$
$360$	0	0
$361$	6.90983	0.363675
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.4721	0.652821
$366$	0	0
$367$	−5.38197	−0.280936	−0.140468	−	0.990085i	$-0.544861\pi$
$367$	−5.38197	−0.280936	−0.140468	+	0.990085i	$0.544861\pi$
$368$	0	0
$369$	−9.47214	−0.493100
$370$	0	0
$371$	1.54915	0.0804279
$372$	0	0
$373$	13.9443	0.722007	0.361004	−	0.932564i	$-0.382434\pi$
$373$	13.9443	0.722007	0.361004	+	0.932564i	$0.382434\pi$
$374$	0	0
$375$	18.7082	0.966087
$376$	0	0
$377$	−27.8885	−1.43633
$378$	0	0
$379$	−12.7082	−0.652777	−0.326388	−	0.945236i	$-0.605832\pi$
$379$	−12.7082	−0.652777	−0.326388	+	0.945236i	$0.605832\pi$
$380$	0	0
$381$	15.7082	0.804756
$382$	0	0
$383$	10.2361	0.523039	0.261519	−	0.965198i	$-0.415776\pi$
$383$	10.2361	0.523039	0.261519	+	0.965198i	$0.415776\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.47214	−0.481496
$388$	0	0
$389$	16.0902	0.815804	0.407902	−	0.913026i	$-0.366261\pi$
$389$	16.0902	0.815804	0.407902	+	0.913026i	$0.366261\pi$
$390$	0	0
$391$	0.562306	0.0284370
$392$	0	0
$393$	−8.56231	−0.431911
$394$	0	0
$395$	54.8673	2.76067
$396$	0	0
$397$	27.1803	1.36414	0.682071	−	0.731286i	$-0.261079\pi$
$397$	27.1803	1.36414	0.682071	+	0.731286i	$0.261079\pi$
$398$	0	0
$399$	−1.20163	−0.0601565
$400$	0	0
$401$	−2.85410	−0.142527	−0.0712635	−	0.997458i	$-0.522703\pi$
$401$	−2.85410	−0.142527	−0.0712635	+	0.997458i	$0.522703\pi$
$402$	0	0
$403$	−39.7984	−1.98250
$404$	0	0
$405$	−3.85410	−0.191512
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9.88854	−0.488957	−0.244479	−	0.969655i	$-0.578617\pi$
$409$	−9.88854	−0.488957	−0.244479	+	0.969655i	$0.578617\pi$
$410$	0	0
$411$	21.0000	1.03585
$412$	0	0
$413$	0.506578	0.0249271
$414$	0	0
$415$	59.6312	2.92718
$416$	0	0
$417$	3.79837	0.186007
$418$	0	0
$419$	17.8541	0.872230	0.436115	−	0.899891i	$-0.356354\pi$
$419$	17.8541	0.872230	0.436115	+	0.899891i	$0.356354\pi$
$420$	0	0
$421$	−9.27051	−0.451817	−0.225909	−	0.974149i	$-0.572535\pi$
$421$	−9.27051	−0.451817	−0.225909	+	0.974149i	$0.572535\pi$
$422$	0	0
$423$	3.61803	0.175915
$424$	0	0
$425$	−23.4721	−1.13857
$426$	0	0
$427$	0.618034	0.0299088
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.3262	−0.641902	−0.320951	−	0.947096i	$-0.604003\pi$
$431$	−13.3262	−0.641902	−0.320951	+	0.947096i	$0.604003\pi$
$432$	0	0
$433$	−14.9443	−0.718176	−0.359088	−	0.933304i	$-0.616912\pi$
$433$	−14.9443	−0.718176	−0.359088	+	0.933304i	$0.616912\pi$
$434$	0	0
$435$	−17.2361	−0.826406
$436$	0	0
$437$	−1.20163	−0.0574816
$438$	0	0
$439$	19.3607	0.924035	0.462017	−	0.886871i	$-0.347126\pi$
$439$	19.3607	0.924035	0.462017	+	0.886871i	$0.347126\pi$
$440$	0	0
$441$	−6.94427	−0.330680
$442$	0	0
$443$	0.180340	0.00856821	0.00428410	−	0.999991i	$-0.498636\pi$
$443$	0.180340	0.00856821	0.00428410	+	0.999991i	$0.498636\pi$
$444$	0	0
$445$	−3.85410	−0.182702
$446$	0	0
$447$	−16.8885	−0.798801
$448$	0	0
$449$	−7.52786	−0.355262	−0.177631	−	0.984097i	$-0.556843\pi$
$449$	−7.52786	−0.355262	−0.177631	+	0.984097i	$0.556843\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4.47214	0.210119
$454$	0	0
$455$	−5.67376	−0.265990
$456$	0	0
$457$	−1.14590	−0.0536028	−0.0268014	−	0.999641i	$-0.508532\pi$
$457$	−1.14590	−0.0536028	−0.0268014	+	0.999641i	$0.508532\pi$
$458$	0	0
$459$	2.38197	0.111181
$460$	0	0
$461$	−19.5623	−0.911107	−0.455554	−	0.890208i	$-0.650559\pi$
$461$	−19.5623	−0.911107	−0.455554	+	0.890208i	$0.650559\pi$
$462$	0	0
$463$	21.2148	0.985935	0.492967	−	0.870048i	$-0.335912\pi$
$463$	21.2148	0.985935	0.492967	+	0.870048i	$0.335912\pi$
$464$	0	0
$465$	−24.5967	−1.14065
$466$	0	0
$467$	19.8328	0.917753	0.458877	−	0.888500i	$-0.348252\pi$
$467$	19.8328	0.917753	0.458877	+	0.888500i	$0.348252\pi$
$468$	0	0
$469$	−0.0344419	−0.00159038
$470$	0	0
$471$	17.2361	0.794196
$472$	0	0
$473$	0	0
$474$	0	0
$475$	50.1591	2.30146
$476$	0	0
$477$	6.56231	0.300467
$478$	0	0
$479$	−13.3262	−0.608891	−0.304446	−	0.952530i	$-0.598471\pi$
$479$	−13.3262	−0.608891	−0.304446	+	0.952530i	$0.598471\pi$
$480$	0	0
$481$	−23.4721	−1.07024
$482$	0	0
$483$	0.0557281	0.00253572
$484$	0	0
$485$	19.6180	0.890809
$486$	0	0
$487$	35.8328	1.62374	0.811870	−	0.583838i	$-0.198450\pi$
$487$	35.8328	1.62374	0.811870	+	0.583838i	$0.198450\pi$
$488$	0	0
$489$	4.85410	0.219510
$490$	0	0
$491$	5.14590	0.232231	0.116116	−	0.993236i	$-0.462956\pi$
$491$	5.14590	0.232231	0.116116	+	0.993236i	$0.462956\pi$
$492$	0	0
$493$	10.6525	0.479763
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.326238	−0.0146338
$498$	0	0
$499$	15.2705	0.683602	0.341801	−	0.939772i	$-0.388963\pi$
$499$	15.2705	0.683602	0.341801	+	0.939772i	$0.388963\pi$
$500$	0	0
$501$	−0.618034	−0.0276117
$502$	0	0
$503$	−17.2361	−0.768518	−0.384259	−	0.923225i	$-0.625543\pi$
$503$	−17.2361	−0.768518	−0.384259	+	0.923225i	$0.625543\pi$
$504$	0	0
$505$	−22.9098	−1.01947
$506$	0	0
$507$	−25.8885	−1.14975
$508$	0	0
$509$	12.7426	0.564808	0.282404	−	0.959296i	$-0.408868\pi$
$509$	12.7426	0.564808	0.282404	+	0.959296i	$0.408868\pi$
$510$	0	0
$511$	−0.763932	−0.0337944
$512$	0	0
$513$	−5.09017	−0.224736
$514$	0	0
$515$	−48.0689	−2.11817
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5.56231	0.244158
$520$	0	0
$521$	14.8328	0.649837	0.324919	−	0.945742i	$-0.394663\pi$
$521$	14.8328	0.649837	0.324919	+	0.945742i	$0.394663\pi$
$522$	0	0
$523$	21.7984	0.953176	0.476588	−	0.879127i	$-0.341873\pi$
$523$	21.7984	0.953176	0.476588	+	0.879127i	$0.341873\pi$
$524$	0	0
$525$	−2.32624	−0.101525
$526$	0	0
$527$	15.2016	0.662193
$528$	0	0
$529$	−22.9443	−0.997577
$530$	0	0
$531$	2.14590	0.0931240
$532$	0	0
$533$	−59.0689	−2.55856
$534$	0	0
$535$	25.1591	1.08772
$536$	0	0
$537$	17.9443	0.774352
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.32624	0.271986	0.135993	−	0.990710i	$-0.456577\pi$
$541$	6.32624	0.271986	0.135993	+	0.990710i	$0.456577\pi$
$542$	0	0
$543$	−5.94427	−0.255093
$544$	0	0
$545$	−30.8328	−1.32073
$546$	0	0
$547$	−13.3262	−0.569789	−0.284894	−	0.958559i	$-0.591959\pi$
$547$	−13.3262	−0.569789	−0.284894	+	0.958559i	$0.591959\pi$
$548$	0	0
$549$	2.61803	0.111735
$550$	0	0
$551$	−22.7639	−0.969776
$552$	0	0
$553$	−3.36068	−0.142911
$554$	0	0
$555$	−14.5066	−0.615770
$556$	0	0
$557$	−5.09017	−0.215677	−0.107839	−	0.994168i	$-0.534393\pi$
$557$	−5.09017	−0.215677	−0.107839	+	0.994168i	$0.534393\pi$
$558$	0	0
$559$	−59.0689	−2.49835
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−40.7082	−1.71565	−0.857823	−	0.513945i	$-0.828184\pi$
$563$	−40.7082	−1.71565	−0.857823	+	0.513945i	$0.828184\pi$
$564$	0	0
$565$	−47.1591	−1.98400
$566$	0	0
$567$	0.236068	0.00991392
$568$	0	0
$569$	−4.87539	−0.204387	−0.102193	−	0.994765i	$-0.532586\pi$
$569$	−4.87539	−0.204387	−0.102193	+	0.994765i	$0.532586\pi$
$570$	0	0
$571$	−24.3262	−1.01802	−0.509011	−	0.860760i	$-0.669989\pi$
$571$	−24.3262	−1.01802	−0.509011	+	0.860760i	$0.669989\pi$
$572$	0	0
$573$	21.6525	0.904545
$574$	0	0
$575$	−2.32624	−0.0970108
$576$	0	0
$577$	29.2361	1.21711	0.608557	−	0.793510i	$-0.291749\pi$
$577$	29.2361	1.21711	0.608557	+	0.793510i	$0.291749\pi$
$578$	0	0
$579$	15.3262	0.636937
$580$	0	0
$581$	−3.65248	−0.151530
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−24.0344	−0.993702
$586$	0	0
$587$	25.7639	1.06339	0.531696	−	0.846936i	$-0.321555\pi$
$587$	25.7639	1.06339	0.531696	+	0.846936i	$0.321555\pi$
$588$	0	0
$589$	−32.4853	−1.33853
$590$	0	0
$591$	−5.90983	−0.243098
$592$	0	0
$593$	−19.4508	−0.798751	−0.399375	−	0.916788i	$-0.630773\pi$
$593$	−19.4508	−0.798751	−0.399375	+	0.916788i	$0.630773\pi$
$594$	0	0
$595$	2.16718	0.0888459
$596$	0	0
$597$	10.4164	0.426315
$598$	0	0
$599$	40.6525	1.66102	0.830508	−	0.557007i	$-0.188050\pi$
$599$	40.6525	1.66102	0.830508	+	0.557007i	$0.188050\pi$
$600$	0	0
$601$	−19.9443	−0.813544	−0.406772	−	0.913530i	$-0.633346\pi$
$601$	−19.9443	−0.813544	−0.406772	+	0.913530i	$0.633346\pi$
$602$	0	0
$603$	−0.145898	−0.00594143
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.67376	−0.352057	−0.176029	−	0.984385i	$-0.556325\pi$
$607$	−8.67376	−0.352057	−0.176029	+	0.984385i	$0.556325\pi$
$608$	0	0
$609$	1.05573	0.0427803
$610$	0	0
$611$	22.5623	0.912773
$612$	0	0
$613$	−17.1246	−0.691657	−0.345828	−	0.938298i	$-0.612402\pi$
$613$	−17.1246	−0.691657	−0.345828	+	0.938298i	$0.612402\pi$
$614$	0	0
$615$	−36.5066	−1.47209
$616$	0	0
$617$	−7.00000	−0.281809	−0.140905	−	0.990023i	$-0.545001\pi$
$617$	−7.00000	−0.281809	−0.140905	+	0.990023i	$0.545001\pi$
$618$	0	0
$619$	−7.47214	−0.300331	−0.150165	−	0.988661i	$-0.547981\pi$
$619$	−7.47214	−0.300331	−0.150165	+	0.988661i	$0.547981\pi$
$620$	0	0
$621$	0.236068	0.00947308
$622$	0	0
$623$	0.236068	0.00945786
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.96556	0.357480
$630$	0	0
$631$	7.72949	0.307706	0.153853	−	0.988094i	$-0.450832\pi$
$631$	7.72949	0.307706	0.153853	+	0.988094i	$0.450832\pi$
$632$	0	0
$633$	18.7426	0.744953
$634$	0	0
$635$	60.5410	2.40250
$636$	0	0
$637$	−43.3050	−1.71580
$638$	0	0
$639$	−1.38197	−0.0546697
$640$	0	0
$641$	24.9787	0.986600	0.493300	−	0.869859i	$-0.335791\pi$
$641$	24.9787	0.986600	0.493300	+	0.869859i	$0.335791\pi$
$642$	0	0
$643$	−40.1591	−1.58372	−0.791859	−	0.610704i	$-0.790887\pi$
$643$	−40.1591	−1.58372	−0.791859	+	0.610704i	$0.790887\pi$
$644$	0	0
$645$	−36.5066	−1.43745
$646$	0	0
$647$	35.6869	1.40300	0.701499	−	0.712671i	$-0.252515\pi$
$647$	35.6869	1.40300	0.701499	+	0.712671i	$0.252515\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.50658	0.0590474
$652$	0	0
$653$	−21.1459	−0.827503	−0.413751	−	0.910390i	$-0.635782\pi$
$653$	−21.1459	−0.827503	−0.413751	+	0.910390i	$0.635782\pi$
$654$	0	0
$655$	−33.0000	−1.28942
$656$	0	0
$657$	−3.23607	−0.126251
$658$	0	0
$659$	−9.70820	−0.378178	−0.189089	−	0.981960i	$-0.560553\pi$
$659$	−9.70820	−0.378178	−0.189089	+	0.981960i	$0.560553\pi$
$660$	0	0
$661$	−13.3262	−0.518331	−0.259165	−	0.965833i	$-0.583447\pi$
$661$	−13.3262	−0.518331	−0.259165	+	0.965833i	$0.583447\pi$
$662$	0	0
$663$	14.8541	0.576886
$664$	0	0
$665$	−4.63119	−0.179590
$666$	0	0
$667$	1.05573	0.0408780
$668$	0	0
$669$	7.47214	0.288889
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−35.4721	−1.36735	−0.683675	−	0.729786i	$-0.739619\pi$
$673$	−35.4721	−1.36735	−0.683675	+	0.729786i	$0.739619\pi$
$674$	0	0
$675$	−9.85410	−0.379285
$676$	0	0
$677$	−2.47214	−0.0950119	−0.0475060	−	0.998871i	$-0.515127\pi$
$677$	−2.47214	−0.0950119	−0.0475060	+	0.998871i	$0.515127\pi$
$678$	0	0
$679$	−1.20163	−0.0461142
$680$	0	0
$681$	12.7082	0.486979
$682$	0	0
$683$	−37.6525	−1.44073	−0.720366	−	0.693594i	$-0.756026\pi$
$683$	−37.6525	−1.44073	−0.720366	+	0.693594i	$0.756026\pi$
$684$	0	0
$685$	80.9361	3.09241
$686$	0	0
$687$	−10.9443	−0.417550
$688$	0	0
$689$	40.9230	1.55904
$690$	0	0
$691$	32.5410	1.23792	0.618959	−	0.785423i	$-0.287555\pi$
$691$	32.5410	1.23792	0.618959	+	0.785423i	$0.287555\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.6393	0.555301
$696$	0	0
$697$	22.5623	0.854608
$698$	0	0
$699$	1.20163	0.0454497
$700$	0	0
$701$	28.5066	1.07668	0.538339	−	0.842728i	$-0.319052\pi$
$701$	28.5066	1.07668	0.538339	+	0.842728i	$0.319052\pi$
$702$	0	0
$703$	−19.1591	−0.722597
$704$	0	0
$705$	13.9443	0.525172
$706$	0	0
$707$	1.40325	0.0527747
$708$	0	0
$709$	16.7426	0.628783	0.314392	−	0.949293i	$-0.398199\pi$
$709$	16.7426	0.628783	0.314392	+	0.949293i	$0.398199\pi$
$710$	0	0
$711$	−14.2361	−0.533894
$712$	0	0
$713$	1.50658	0.0564218
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.5066	0.653795
$718$	0	0
$719$	19.0689	0.711149	0.355575	−	0.934648i	$-0.384285\pi$
$719$	19.0689	0.711149	0.355575	+	0.934648i	$0.384285\pi$
$720$	0	0
$721$	2.94427	0.109650
$722$	0	0
$723$	−23.7082	−0.881718
$724$	0	0
$725$	−44.0689	−1.63668
$726$	0	0
$727$	−8.85410	−0.328380	−0.164190	−	0.986429i	$-0.552501\pi$
$727$	−8.85410	−0.328380	−0.164190	+	0.986429i	$0.552501\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.5623	0.834497
$732$	0	0
$733$	−44.5410	−1.64516	−0.822580	−	0.568649i	$-0.807466\pi$
$733$	−44.5410	−1.64516	−0.822580	+	0.568649i	$0.807466\pi$
$734$	0	0
$735$	−26.7639	−0.987203
$736$	0	0
$737$	0	0
$738$	0	0
$739$	12.8197	0.471579	0.235789	−	0.971804i	$-0.424232\pi$
$739$	12.8197	0.471579	0.235789	+	0.971804i	$0.424232\pi$
$740$	0	0
$741$	−31.7426	−1.16610
$742$	0	0
$743$	−11.4721	−0.420872	−0.210436	−	0.977608i	$-0.567488\pi$
$743$	−11.4721	−0.420872	−0.210436	+	0.977608i	$0.567488\pi$
$744$	0	0
$745$	−65.0902	−2.38472
$746$	0	0
$747$	−15.4721	−0.566096
$748$	0	0
$749$	−1.54102	−0.0563076
$750$	0	0
$751$	−41.6312	−1.51914	−0.759572	−	0.650423i	$-0.774591\pi$
$751$	−41.6312	−1.51914	−0.759572	+	0.650423i	$0.774591\pi$
$752$	0	0
$753$	−24.0902	−0.877895
$754$	0	0
$755$	17.2361	0.627285
$756$	0	0
$757$	−10.4164	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$757$	−10.4164	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.2492	1.06028	0.530142	−	0.847909i	$-0.322139\pi$
$761$	29.2492	1.06028	0.530142	+	0.847909i	$0.322139\pi$
$762$	0	0
$763$	1.88854	0.0683699
$764$	0	0
$765$	9.18034	0.331916
$766$	0	0
$767$	13.3820	0.483195
$768$	0	0
$769$	33.5623	1.21029	0.605144	−	0.796116i	$-0.293116\pi$
$769$	33.5623	1.21029	0.605144	+	0.796116i	$0.293116\pi$
$770$	0	0
$771$	2.79837	0.100781
$772$	0	0
$773$	29.3607	1.05603	0.528015	−	0.849235i	$-0.322936\pi$
$773$	29.3607	1.05603	0.528015	+	0.849235i	$0.322936\pi$
$774$	0	0
$775$	−62.8885	−2.25902
$776$	0	0
$777$	0.888544	0.0318763
$778$	0	0
$779$	−48.2148	−1.72747
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.47214	0.159821
$784$	0	0
$785$	66.4296	2.37097
$786$	0	0
$787$	−44.7639	−1.59566	−0.797831	−	0.602881i	$-0.794019\pi$
$787$	−44.7639	−1.59566	−0.797831	+	0.602881i	$0.794019\pi$
$788$	0	0
$789$	23.8541	0.849229
$790$	0	0
$791$	2.88854	0.102705
$792$	0	0
$793$	16.3262	0.579762
$794$	0	0
$795$	25.2918	0.897008
$796$	0	0
$797$	−49.0132	−1.73614	−0.868068	−	0.496446i	$-0.834638\pi$
$797$	−49.0132	−1.73614	−0.868068	+	0.496446i	$0.834638\pi$
$798$	0	0
$799$	−8.61803	−0.304884
$800$	0	0
$801$	1.00000	0.0353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0.214782	0.00757006
$806$	0	0
$807$	−28.4721	−1.00227
$808$	0	0
$809$	−35.2705	−1.24005	−0.620023	−	0.784584i	$-0.712877\pi$
$809$	−35.2705	−1.24005	−0.620023	+	0.784584i	$0.712877\pi$
$810$	0	0
$811$	2.20163	0.0773095	0.0386548	−	0.999253i	$-0.487693\pi$
$811$	2.20163	0.0773095	0.0386548	+	0.999253i	$0.487693\pi$
$812$	0	0
$813$	−2.43769	−0.0854937
$814$	0	0
$815$	18.7082	0.655320
$816$	0	0
$817$	−48.2148	−1.68682
$818$	0	0
$819$	1.47214	0.0514406
$820$	0	0
$821$	−27.3607	−0.954894	−0.477447	−	0.878660i	$-0.658438\pi$
$821$	−27.3607	−0.954894	−0.477447	+	0.878660i	$0.658438\pi$
$822$	0	0
$823$	23.1803	0.808016	0.404008	−	0.914755i	$-0.367617\pi$
$823$	23.1803	0.808016	0.404008	+	0.914755i	$0.367617\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.1246	1.43004	0.715021	−	0.699103i	$-0.246417\pi$
$827$	41.1246	1.43004	0.715021	+	0.699103i	$0.246417\pi$
$828$	0	0
$829$	4.74265	0.164719	0.0823594	−	0.996603i	$-0.473754\pi$
$829$	4.74265	0.164719	0.0823594	+	0.996603i	$0.473754\pi$
$830$	0	0
$831$	2.56231	0.0888854
$832$	0	0
$833$	16.5410	0.573112
$834$	0	0
$835$	−2.38197	−0.0824313
$836$	0	0
$837$	6.38197	0.220593
$838$	0	0
$839$	37.8328	1.30613	0.653067	−	0.757300i	$-0.273482\pi$
$839$	37.8328	1.30613	0.653067	+	0.757300i	$0.273482\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	17.1246	0.589803
$844$	0	0
$845$	−99.7771	−3.43244
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23.2361	0.797460
$850$	0	0
$851$	0.888544	0.0304589
$852$	0	0
$853$	−10.6393	−0.364283	−0.182142	−	0.983272i	$-0.558303\pi$
$853$	−10.6393	−0.364283	−0.182142	+	0.983272i	$0.558303\pi$
$854$	0	0
$855$	−19.6180	−0.670923
$856$	0	0
$857$	50.1935	1.71458	0.857289	−	0.514836i	$-0.172147\pi$
$857$	50.1935	1.71458	0.857289	+	0.514836i	$0.172147\pi$
$858$	0	0
$859$	8.81966	0.300923	0.150461	−	0.988616i	$-0.451924\pi$
$859$	8.81966	0.300923	0.150461	+	0.988616i	$0.451924\pi$
$860$	0	0
$861$	2.23607	0.0762050
$862$	0	0
$863$	−14.3607	−0.488843	−0.244422	−	0.969669i	$-0.578598\pi$
$863$	−14.3607	−0.488843	−0.244422	+	0.969669i	$0.578598\pi$
$864$	0	0
$865$	21.4377	0.728903
$866$	0	0
$867$	11.3262	0.384659
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.909830	−0.0308284
$872$	0	0
$873$	−5.09017	−0.172276
$874$	0	0
$875$	−4.41641	−0.149302
$876$	0	0
$877$	−0.888544	−0.0300040	−0.0150020	−	0.999887i	$-0.504775\pi$
$877$	−0.888544	−0.0300040	−0.0150020	+	0.999887i	$0.504775\pi$
$878$	0	0
$879$	−5.00000	−0.168646
$880$	0	0
$881$	40.5623	1.36658	0.683289	−	0.730148i	$-0.260549\pi$
$881$	40.5623	1.36658	0.683289	+	0.730148i	$0.260549\pi$
$882$	0	0
$883$	14.9443	0.502915	0.251457	−	0.967868i	$-0.419090\pi$
$883$	14.9443	0.502915	0.251457	+	0.967868i	$0.419090\pi$
$884$	0	0
$885$	8.27051	0.278010
$886$	0	0
$887$	33.2918	1.11783	0.558915	−	0.829225i	$-0.311218\pi$
$887$	33.2918	1.11783	0.558915	+	0.829225i	$0.311218\pi$
$888$	0	0
$889$	−3.70820	−0.124369
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.4164	0.616282
$894$	0	0
$895$	69.1591	2.31173
$896$	0	0
$897$	1.47214	0.0491532
$898$	0	0
$899$	28.5410	0.951896
$900$	0	0
$901$	−15.6312	−0.520750
$902$	0	0
$903$	2.23607	0.0744117
$904$	0	0
$905$	−22.9098	−0.761549
$906$	0	0
$907$	−42.8673	−1.42338	−0.711692	−	0.702492i	$-0.752071\pi$
$907$	−42.8673	−1.42338	−0.711692	+	0.702492i	$0.752071\pi$
$908$	0	0
$909$	5.94427	0.197159
$910$	0	0
$911$	25.5410	0.846212	0.423106	−	0.906080i	$-0.360940\pi$
$911$	25.5410	0.846212	0.423106	+	0.906080i	$0.360940\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	10.0902	0.333571
$916$	0	0
$917$	2.02129	0.0667488
$918$	0	0
$919$	15.8328	0.522276	0.261138	−	0.965301i	$-0.415902\pi$
$919$	15.8328	0.522276	0.261138	+	0.965301i	$0.415902\pi$
$920$	0	0
$921$	−0.673762	−0.0222012
$922$	0	0
$923$	−8.61803	−0.283666
$924$	0	0
$925$	−37.0902	−1.21952
$926$	0	0
$927$	12.4721	0.409639
$928$	0	0
$929$	19.5836	0.642517	0.321258	−	0.946992i	$-0.395894\pi$
$929$	19.5836	0.642517	0.321258	+	0.946992i	$0.395894\pi$
$930$	0	0
$931$	−35.3475	−1.15847
$932$	0	0
$933$	29.1803	0.955321
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.7639	−0.645660	−0.322830	−	0.946457i	$-0.604634\pi$
$937$	−19.7639	−0.645660	−0.322830	+	0.946457i	$0.604634\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	0	0
$941$	2.56231	0.0835288	0.0417644	−	0.999127i	$-0.486702\pi$
$941$	2.56231	0.0835288	0.0417644	+	0.999127i	$0.486702\pi$
$942$	0	0
$943$	2.23607	0.0728164
$944$	0	0
$945$	0.909830	0.0295968
$946$	0	0
$947$	−2.79837	−0.0909349	−0.0454675	−	0.998966i	$-0.514478\pi$
$947$	−2.79837	−0.0909349	−0.0454675	+	0.998966i	$0.514478\pi$
$948$	0	0
$949$	−20.1803	−0.655082
$950$	0	0
$951$	−8.52786	−0.276535
$952$	0	0
$953$	−34.7639	−1.12611	−0.563057	−	0.826418i	$-0.690375\pi$
$953$	−34.7639	−1.12611	−0.563057	+	0.826418i	$0.690375\pi$
$954$	0	0
$955$	83.4508	2.70041
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.95743	−0.160084
$960$	0	0
$961$	9.72949	0.313855
$962$	0	0
$963$	−6.52786	−0.210357
$964$	0	0
$965$	59.0689	1.90149
$966$	0	0
$967$	−14.7295	−0.473668	−0.236834	−	0.971550i	$-0.576110\pi$
$967$	−14.7295	−0.473668	−0.236834	+	0.971550i	$0.576110\pi$
$968$	0	0
$969$	12.1246	0.389499
$970$	0	0
$971$	2.36068	0.0757578	0.0378789	−	0.999282i	$-0.487940\pi$
$971$	2.36068	0.0757578	0.0378789	+	0.999282i	$0.487940\pi$
$972$	0	0
$973$	−0.896674	−0.0287461
$974$	0	0
$975$	−61.4508	−1.96800
$976$	0	0
$977$	−32.8885	−1.05220	−0.526099	−	0.850423i	$-0.676346\pi$
$977$	−32.8885	−1.05220	−0.526099	+	0.850423i	$0.676346\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8.00000	0.255420
$982$	0	0
$983$	−36.8328	−1.17478	−0.587392	−	0.809302i	$-0.699845\pi$
$983$	−36.8328	−1.17478	−0.587392	+	0.809302i	$0.699845\pi$
$984$	0	0
$985$	−22.7771	−0.725738
$986$	0	0
$987$	−0.854102	−0.0271864
$988$	0	0
$989$	2.23607	0.0711028
$990$	0	0
$991$	43.2705	1.37453	0.687267	−	0.726405i	$-0.258810\pi$
$991$	43.2705	1.37453	0.687267	+	0.726405i	$0.258810\pi$
$992$	0	0
$993$	−11.9443	−0.379040
$994$	0	0
$995$	40.1459	1.27271
$996$	0	0
$997$	−29.2705	−0.927006	−0.463503	−	0.886095i	$-0.653408\pi$
$997$	−29.2705	−0.927006	−0.463503	+	0.886095i	$0.653408\pi$
$998$	0	0
$999$	3.76393	0.119086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.2.a.i.1.1		2
3.2	odd	2		4356.2.a.s.1.2		2
4.3	odd	2		5808.2.a.cf.1.1		2
11.2	odd	10		1452.2.i.o.565.1		4
11.3	even	5		1452.2.i.j.493.1		4
11.4	even	5		1452.2.i.j.1237.1		4
11.5	even	5		1452.2.i.p.1213.1		4
11.6	odd	10		1452.2.i.o.1213.1		4
11.7	odd	10		132.2.i.b.49.1	✓	4
11.8	odd	10		132.2.i.b.97.1	yes	4
11.9	even	5		1452.2.i.p.565.1		4
11.10	odd	2		1452.2.a.j.1.1		2
33.8	even	10		396.2.j.c.361.1		4
33.29	even	10		396.2.j.c.181.1		4
33.32	even	2		4356.2.a.v.1.2		2
44.7	even	10		528.2.y.a.49.1		4
44.19	even	10		528.2.y.a.97.1		4
44.43	even	2		5808.2.a.cc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.i.b.49.1	✓	4	11.7	odd	10
132.2.i.b.97.1	yes	4	11.8	odd	10
396.2.j.c.181.1		4	33.29	even	10
396.2.j.c.361.1		4	33.8	even	10
528.2.y.a.49.1		4	44.7	even	10
528.2.y.a.97.1		4	44.19	even	10
1452.2.a.i.1.1		2	1.1	even	1	trivial
1452.2.a.j.1.1		2	11.10	odd	2
1452.2.i.j.493.1		4	11.3	even	5
1452.2.i.j.1237.1		4	11.4	even	5
1452.2.i.o.565.1		4	11.2	odd	10
1452.2.i.o.1213.1		4	11.6	odd	10
1452.2.i.p.565.1		4	11.9	even	5
1452.2.i.p.1213.1		4	11.5	even	5
4356.2.a.s.1.2		2	3.2	odd	2
4356.2.a.v.1.2		2	33.32	even	2
5808.2.a.cc.1.1		2	44.43	even	2
5808.2.a.cf.1.1		2	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1452.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.85410 q^{5} +0.236068 q^{7} +1.00000 q^{9} +6.23607 q^{13} +3.85410 q^{15} -2.38197 q^{17} +5.09017 q^{19} -0.236068 q^{21} -0.236068 q^{23} +9.85410 q^{25} -1.00000 q^{27} -4.47214 q^{29} -6.38197 q^{31} -0.909830 q^{35} -3.76393 q^{37} -6.23607 q^{39} -9.47214 q^{41} -9.47214 q^{43} -3.85410 q^{45} +3.61803 q^{47} -6.94427 q^{49} +2.38197 q^{51} +6.56231 q^{53} -5.09017 q^{57} +2.14590 q^{59} +2.61803 q^{61} +0.236068 q^{63} -24.0344 q^{65} -0.145898 q^{67} +0.236068 q^{69} -1.38197 q^{71} -3.23607 q^{73} -9.85410 q^{75} -14.2361 q^{79} +1.00000 q^{81} -15.4721 q^{83} +9.18034 q^{85} +4.47214 q^{87} +1.00000 q^{89} +1.47214 q^{91} +6.38197 q^{93} -19.6180 q^{95} -5.09017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9} + 8 q^{13} + q^{15} - 7 q^{17} - q^{19} + 4 q^{21} + 4 q^{23} + 13 q^{25} - 2 q^{27} - 15 q^{31} - 13 q^{35} - 12 q^{37} - 8 q^{39} - 10 q^{41} - 10 q^{43} - q^{45}+ \cdots + q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 4 * q^7 + 2 * q^9 + 8 * q^13 + q^15 - 7 * q^17 - q^19 + 4 * q^21 + 4 * q^23 + 13 * q^25 - 2 * q^27 - 15 * q^31 - 13 * q^35 - 12 * q^37 - 8 * q^39 - 10 * q^41 - 10 * q^43 - q^45 + 5 * q^47 + 4 * q^49 + 7 * q^51 - 7 * q^53 + q^57 + 11 * q^59 + 3 * q^61 - 4 * q^63 - 19 * q^65 - 7 * q^67 - 4 * q^69 - 5 * q^71 - 2 * q^73 - 13 * q^75 - 24 * q^79 + 2 * q^81 - 22 * q^83 - 4 * q^85 + 2 * q^89 - 6 * q^91 + 15 * q^93 - 37 * q^95 + q^97

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