LMFDB - Embedded newform 1452.2.a.i.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.85410 q^{5} -4.23607 q^{7} +1.00000 q^{9} +1.76393 q^{13} -2.85410 q^{15} -4.61803 q^{17} -6.09017 q^{19} +4.23607 q^{21} +4.23607 q^{23} +3.14590 q^{25} -1.00000 q^{27} +4.47214 q^{29} -8.61803 q^{31} -12.0902 q^{35} -8.23607 q^{37} -1.76393 q^{39} -0.527864 q^{41} -0.527864 q^{43} +2.85410 q^{45} +1.38197 q^{47} +10.9443 q^{49} +4.61803 q^{51} -13.5623 q^{53} +6.09017 q^{57} +8.85410 q^{59} +0.381966 q^{61} -4.23607 q^{63} +5.03444 q^{65} -6.85410 q^{67} -4.23607 q^{69} -3.61803 q^{71} +1.23607 q^{73} -3.14590 q^{75} -9.76393 q^{79} +1.00000 q^{81} -6.52786 q^{83} -13.1803 q^{85} -4.47214 q^{87} +1.00000 q^{89} -7.47214 q^{91} +8.61803 q^{93} -17.3820 q^{95} +6.09017 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	−2.85410	−0.736926
$16$	0	0
$17$	−4.61803	−1.12004	−0.560019	−	0.828480i	$-0.689206\pi$
$17$	−4.61803	−1.12004	−0.560019	+	0.828480i	$0.689206\pi$
$18$	0	0
$19$	−6.09017	−1.39718	−0.698590	−	0.715522i	$-0.746189\pi$
$19$	−6.09017	−1.39718	−0.698590	+	0.715522i	$0.746189\pi$
$20$	0	0
$21$	4.23607	0.924386
$22$	0	0
$23$	4.23607	0.883281	0.441641	−	0.897192i	$-0.354397\pi$
$23$	4.23607	0.883281	0.441641	+	0.897192i	$0.354397\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−8.61803	−1.54784	−0.773922	−	0.633281i	$-0.781708\pi$
$31$	−8.61803	−1.54784	−0.773922	+	0.633281i	$0.781708\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.0902	−2.04361
$36$	0	0
$37$	−8.23607	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$37$	−8.23607	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$38$	0	0
$39$	−1.76393	−0.282455
$40$	0	0
$41$	−0.527864	−0.0824385	−0.0412193	−	0.999150i	$-0.513124\pi$
$41$	−0.527864	−0.0824385	−0.0412193	+	0.999150i	$0.513124\pi$
$42$	0	0
$43$	−0.527864	−0.0804985	−0.0402493	−	0.999190i	$-0.512815\pi$
$43$	−0.527864	−0.0804985	−0.0402493	+	0.999190i	$0.512815\pi$
$44$	0	0
$45$	2.85410	0.425464
$46$	0	0
$47$	1.38197	0.201580	0.100790	−	0.994908i	$-0.467863\pi$
$47$	1.38197	0.201580	0.100790	+	0.994908i	$0.467863\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	4.61803	0.646654
$52$	0	0
$53$	−13.5623	−1.86293	−0.931463	−	0.363836i	$-0.881467\pi$
$53$	−13.5623	−1.86293	−0.931463	+	0.363836i	$0.881467\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.09017	0.806663
$58$	0	0
$59$	8.85410	1.15271	0.576353	−	0.817201i	$-0.304475\pi$
$59$	8.85410	1.15271	0.576353	+	0.817201i	$0.304475\pi$
$60$	0	0
$61$	0.381966	0.0489057	0.0244529	−	0.999701i	$-0.492216\pi$
$61$	0.381966	0.0489057	0.0244529	+	0.999701i	$0.492216\pi$
$62$	0	0
$63$	−4.23607	−0.533694
$64$	0	0
$65$	5.03444	0.624446
$66$	0	0
$67$	−6.85410	−0.837362	−0.418681	−	0.908133i	$-0.637507\pi$
$67$	−6.85410	−0.837362	−0.418681	+	0.908133i	$0.637507\pi$
$68$	0	0
$69$	−4.23607	−0.509963
$70$	0	0
$71$	−3.61803	−0.429382	−0.214691	−	0.976682i	$-0.568874\pi$
$71$	−3.61803	−0.429382	−0.214691	+	0.976682i	$0.568874\pi$
$72$	0	0
$73$	1.23607	0.144671	0.0723354	−	0.997380i	$-0.476955\pi$
$73$	1.23607	0.144671	0.0723354	+	0.997380i	$0.476955\pi$
$74$	0	0
$75$	−3.14590	−0.363257
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.76393	−1.09853	−0.549264	−	0.835649i	$-0.685092\pi$
$79$	−9.76393	−1.09853	−0.549264	+	0.835649i	$0.685092\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.52786	−0.716526	−0.358263	−	0.933621i	$-0.616631\pi$
$83$	−6.52786	−0.716526	−0.358263	+	0.933621i	$0.616631\pi$
$84$	0	0
$85$	−13.1803	−1.42961
$86$	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
$91$	−7.47214	−0.783293
$92$	0	0
$93$	8.61803	0.893648
$94$	0	0
$95$	−17.3820	−1.78335
$96$	0	0
$97$	6.09017	0.618363	0.309182	−	0.951003i	$-0.399945\pi$
$97$	6.09017	0.618363	0.309182	+	0.951003i	$0.399945\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.9443	−1.18850	−0.594250	−	0.804281i	$-0.702551\pi$
$101$	−11.9443	−1.18850	−0.594250	+	0.804281i	$0.702551\pi$
$102$	0	0
$103$	3.52786	0.347611	0.173805	−	0.984780i	$-0.444394\pi$
$103$	3.52786	0.347611	0.173805	+	0.984780i	$0.444394\pi$
$104$	0	0
$105$	12.0902	1.17988
$106$	0	0
$107$	−15.4721	−1.49575	−0.747874	−	0.663841i	$-0.768925\pi$
$107$	−15.4721	−1.49575	−0.747874	+	0.663841i	$0.768925\pi$
$108$	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
$111$	8.23607	0.781733
$112$	0	0
$113$	7.76393	0.730369	0.365185	−	0.930935i	$-0.381006\pi$
$113$	7.76393	0.730369	0.365185	+	0.930935i	$0.381006\pi$
$114$	0	0
$115$	12.0902	1.12741
$116$	0	0
$117$	1.76393	0.163076
$118$	0	0
$119$	19.5623	1.79327
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0.527864	0.0475959
$124$	0	0
$125$	−5.29180	−0.473313
$126$	0	0
$127$	−2.29180	−0.203364	−0.101682	−	0.994817i	$-0.532422\pi$
$127$	−2.29180	−0.203364	−0.101682	+	0.994817i	$0.532422\pi$
$128$	0	0
$129$	0.527864	0.0464758
$130$	0	0
$131$	−11.5623	−1.01020	−0.505102	−	0.863060i	$-0.668545\pi$
$131$	−11.5623	−1.01020	−0.505102	+	0.863060i	$0.668545\pi$
$132$	0	0
$133$	25.7984	2.23700
$134$	0	0
$135$	−2.85410	−0.245642
$136$	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
$139$	20.7984	1.76410	0.882048	−	0.471160i	$-0.156165\pi$
$139$	20.7984	1.76410	0.882048	+	0.471160i	$0.156165\pi$
$140$	0	0
$141$	−1.38197	−0.116383
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.7639	1.05999
$146$	0	0
$147$	−10.9443	−0.902668
$148$	0	0
$149$	−18.8885	−1.54741	−0.773705	−	0.633546i	$-0.781599\pi$
$149$	−18.8885	−1.54741	−0.773705	+	0.633546i	$0.781599\pi$
$150$	0	0
$151$	4.47214	0.363937	0.181969	−	0.983304i	$-0.441753\pi$
$151$	4.47214	0.363937	0.181969	+	0.983304i	$0.441753\pi$
$152$	0	0
$153$	−4.61803	−0.373346
$154$	0	0
$155$	−24.5967	−1.97566
$156$	0	0
$157$	−12.7639	−1.01867	−0.509336	−	0.860568i	$-0.670109\pi$
$157$	−12.7639	−1.01867	−0.509336	+	0.860568i	$0.670109\pi$
$158$	0	0
$159$	13.5623	1.07556
$160$	0	0
$161$	−17.9443	−1.41421
$162$	0	0
$163$	1.85410	0.145224	0.0726122	−	0.997360i	$-0.476866\pi$
$163$	1.85410	0.145224	0.0726122	+	0.997360i	$0.476866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.61803	−0.125207	−0.0626036	−	0.998038i	$-0.519940\pi$
$167$	−1.61803	−0.125207	−0.0626036	+	0.998038i	$0.519940\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	−6.09017	−0.465727
$172$	0	0
$173$	14.5623	1.10715	0.553576	−	0.832799i	$-0.313263\pi$
$173$	14.5623	1.10715	0.553576	+	0.832799i	$0.313263\pi$
$174$	0	0
$175$	−13.3262	−1.00737
$176$	0	0
$177$	−8.85410	−0.665515
$178$	0	0
$179$	−0.0557281	−0.00416531	−0.00208266	−	0.999998i	$-0.500663\pi$
$179$	−0.0557281	−0.00416531	−0.00208266	+	0.999998i	$0.500663\pi$
$180$	0	0
$181$	−11.9443	−0.887811	−0.443905	−	0.896074i	$-0.646407\pi$
$181$	−11.9443	−0.887811	−0.443905	+	0.896074i	$0.646407\pi$
$182$	0	0
$183$	−0.381966	−0.0282357
$184$	0	0
$185$	−23.5066	−1.72824
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.23607	0.308129
$190$	0	0
$191$	9.65248	0.698429	0.349214	−	0.937043i	$-0.386449\pi$
$191$	9.65248	0.698429	0.349214	+	0.937043i	$0.386449\pi$
$192$	0	0
$193$	0.326238	0.0234831	0.0117416	−	0.999931i	$-0.496262\pi$
$193$	0.326238	0.0234831	0.0117416	+	0.999931i	$0.496262\pi$
$194$	0	0
$195$	−5.03444	−0.360524
$196$	0	0
$197$	17.0902	1.21762	0.608812	−	0.793314i	$-0.291646\pi$
$197$	17.0902	1.21762	0.608812	+	0.793314i	$0.291646\pi$
$198$	0	0
$199$	16.4164	1.16373	0.581864	−	0.813286i	$-0.302324\pi$
$199$	16.4164	1.16373	0.581864	+	0.813286i	$0.302324\pi$
$200$	0	0
$201$	6.85410	0.483451
$202$	0	0
$203$	−18.9443	−1.32963
$204$	0	0
$205$	−1.50658	−0.105224
$206$	0	0
$207$	4.23607	0.294427
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.7426	1.63451	0.817256	−	0.576275i	$-0.195494\pi$
$211$	23.7426	1.63451	0.817256	+	0.576275i	$0.195494\pi$
$212$	0	0
$213$	3.61803	0.247904
$214$	0	0
$215$	−1.50658	−0.102748
$216$	0	0
$217$	36.5066	2.47823
$218$	0	0
$219$	−1.23607	−0.0835257
$220$	0	0
$221$	−8.14590	−0.547952
$222$	0	0
$223$	1.47214	0.0985815	0.0492908	−	0.998784i	$-0.484304\pi$
$223$	1.47214	0.0985815	0.0492908	+	0.998784i	$0.484304\pi$
$224$	0	0
$225$	3.14590	0.209727
$226$	0	0
$227$	0.708204	0.0470051	0.0235026	−	0.999724i	$-0.492518\pi$
$227$	0.708204	0.0470051	0.0235026	+	0.999724i	$0.492518\pi$
$228$	0	0
$229$	−6.94427	−0.458890	−0.229445	−	0.973322i	$-0.573691\pi$
$229$	−6.94427	−0.458890	−0.229445	+	0.973322i	$0.573691\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.7984	−1.69011	−0.845054	−	0.534681i	$-0.820432\pi$
$233$	−25.7984	−1.69011	−0.845054	+	0.534681i	$0.820432\pi$
$234$	0	0
$235$	3.94427	0.257296
$236$	0	0
$237$	9.76393	0.634236
$238$	0	0
$239$	20.5066	1.32646	0.663230	−	0.748416i	$-0.269185\pi$
$239$	20.5066	1.32646	0.663230	+	0.748416i	$0.269185\pi$
$240$	0	0
$241$	10.2918	0.662953	0.331476	−	0.943463i	$-0.392453\pi$
$241$	10.2918	0.662953	0.331476	+	0.943463i	$0.392453\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	31.2361	1.99560
$246$	0	0
$247$	−10.7426	−0.683538
$248$	0	0
$249$	6.52786	0.413687
$250$	0	0
$251$	12.9098	0.814861	0.407431	−	0.913236i	$-0.366425\pi$
$251$	12.9098	0.814861	0.407431	+	0.913236i	$0.366425\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	13.1803	0.825385
$256$	0	0
$257$	21.7984	1.35975	0.679873	−	0.733330i	$-0.262035\pi$
$257$	21.7984	1.35975	0.679873	+	0.733330i	$0.262035\pi$
$258$	0	0
$259$	34.8885	2.16787
$260$	0	0
$261$	4.47214	0.276818
$262$	0	0
$263$	−17.1459	−1.05726	−0.528631	−	0.848852i	$-0.677294\pi$
$263$	−17.1459	−1.05726	−0.528631	+	0.848852i	$0.677294\pi$
$264$	0	0
$265$	−38.7082	−2.37783
$266$	0	0
$267$	−1.00000	−0.0611990
$268$	0	0
$269$	19.5279	1.19063	0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279	1.19063	0.595317	+	0.803491i	$0.297026\pi$
$270$	0	0
$271$	22.5623	1.37056	0.685281	−	0.728279i	$-0.259679\pi$
$271$	22.5623	1.37056	0.685281	+	0.728279i	$0.259679\pi$
$272$	0	0
$273$	7.47214	0.452234
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.5623	1.05522	0.527608	−	0.849488i	$-0.323089\pi$
$277$	17.5623	1.05522	0.527608	+	0.849488i	$0.323089\pi$
$278$	0	0
$279$	−8.61803	−0.515948
$280$	0	0
$281$	23.1246	1.37950	0.689749	−	0.724048i	$-0.257721\pi$
$281$	23.1246	1.37950	0.689749	+	0.724048i	$0.257721\pi$
$282$	0	0
$283$	−18.7639	−1.11540	−0.557700	−	0.830043i	$-0.688316\pi$
$283$	−18.7639	−1.11540	−0.557700	+	0.830043i	$0.688316\pi$
$284$	0	0
$285$	17.3820	1.02962
$286$	0	0
$287$	2.23607	0.131991
$288$	0	0
$289$	4.32624	0.254485
$290$	0	0
$291$	−6.09017	−0.357012
$292$	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
$295$	25.2705	1.47131
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.47214	0.432125
$300$	0	0
$301$	2.23607	0.128885
$302$	0	0
$303$	11.9443	0.686180
$304$	0	0
$305$	1.09017	0.0624229
$306$	0	0
$307$	16.3262	0.931788	0.465894	−	0.884841i	$-0.345733\pi$
$307$	16.3262	0.931788	0.465894	+	0.884841i	$0.345733\pi$
$308$	0	0
$309$	−3.52786	−0.200693
$310$	0	0
$311$	−6.81966	−0.386707	−0.193354	−	0.981129i	$-0.561936\pi$
$311$	−6.81966	−0.386707	−0.193354	+	0.981129i	$0.561936\pi$
$312$	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
$315$	−12.0902	−0.681204
$316$	0	0
$317$	17.4721	0.981333	0.490666	−	0.871347i	$-0.336753\pi$
$317$	17.4721	0.981333	0.490666	+	0.871347i	$0.336753\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	15.4721	0.863570
$322$	0	0
$323$	28.1246	1.56490
$324$	0	0
$325$	5.54915	0.307811
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	−5.85410	−0.322747
$330$	0	0
$331$	−5.94427	−0.326727	−0.163363	−	0.986566i	$-0.552234\pi$
$331$	−5.94427	−0.326727	−0.163363	+	0.986566i	$0.552234\pi$
$332$	0	0
$333$	−8.23607	−0.451334
$334$	0	0
$335$	−19.5623	−1.06880
$336$	0	0
$337$	−31.5967	−1.72118	−0.860592	−	0.509295i	$-0.829906\pi$
$337$	−31.5967	−1.72118	−0.860592	+	0.509295i	$0.829906\pi$
$338$	0	0
$339$	−7.76393	−0.421679
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	−12.0902	−0.650913
$346$	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
$349$	23.7639	1.27205	0.636027	−	0.771667i	$-0.280577\pi$
$349$	23.7639	1.27205	0.636027	+	0.771667i	$0.280577\pi$
$350$	0	0
$351$	−1.76393	−0.0941517
$352$	0	0
$353$	−1.52786	−0.0813200	−0.0406600	−	0.999173i	$-0.512946\pi$
$353$	−1.52786	−0.0813200	−0.0406600	+	0.999173i	$0.512946\pi$
$354$	0	0
$355$	−10.3262	−0.548060
$356$	0	0
$357$	−19.5623	−1.03535
$358$	0	0
$359$	18.7639	0.990322	0.495161	−	0.868801i	$-0.335109\pi$
$359$	18.7639	0.990322	0.495161	+	0.868801i	$0.335109\pi$
$360$	0	0
$361$	18.0902	0.952114
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.52786	0.184657
$366$	0	0
$367$	−7.61803	−0.397658	−0.198829	−	0.980034i	$-0.563714\pi$
$367$	−7.61803	−0.397658	−0.198829	+	0.980034i	$0.563714\pi$
$368$	0	0
$369$	−0.527864	−0.0274795
$370$	0	0
$371$	57.4508	2.98270
$372$	0	0
$373$	−3.94427	−0.204227	−0.102113	−	0.994773i	$-0.532560\pi$
$373$	−3.94427	−0.204227	−0.102113	+	0.994773i	$0.532560\pi$
$374$	0	0
$375$	5.29180	0.273267
$376$	0	0
$377$	7.88854	0.406281
$378$	0	0
$379$	0.708204	0.0363780	0.0181890	−	0.999835i	$-0.494210\pi$
$379$	0.708204	0.0363780	0.0181890	+	0.999835i	$0.494210\pi$
$380$	0	0
$381$	2.29180	0.117412
$382$	0	0
$383$	5.76393	0.294523	0.147262	−	0.989098i	$-0.452954\pi$
$383$	5.76393	0.294523	0.147262	+	0.989098i	$0.452954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.527864	−0.0268328
$388$	0	0
$389$	4.90983	0.248938	0.124469	−	0.992223i	$-0.460277\pi$
$389$	4.90983	0.248938	0.124469	+	0.992223i	$0.460277\pi$
$390$	0	0
$391$	−19.5623	−0.989308
$392$	0	0
$393$	11.5623	0.583241
$394$	0	0
$395$	−27.8673	−1.40215
$396$	0	0
$397$	4.81966	0.241892	0.120946	−	0.992659i	$-0.461407\pi$
$397$	4.81966	0.241892	0.120946	+	0.992659i	$0.461407\pi$
$398$	0	0
$399$	−25.7984	−1.29153
$400$	0	0
$401$	3.85410	0.192465	0.0962323	−	0.995359i	$-0.469321\pi$
$401$	3.85410	0.192465	0.0962323	+	0.995359i	$0.469321\pi$
$402$	0	0
$403$	−15.2016	−0.757247
$404$	0	0
$405$	2.85410	0.141821
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.8885	1.28011	0.640053	−	0.768331i	$-0.278912\pi$
$409$	25.8885	1.28011	0.640053	+	0.768331i	$0.278912\pi$
$410$	0	0
$411$	21.0000	1.03585
$412$	0	0
$413$	−37.5066	−1.84558
$414$	0	0
$415$	−18.6312	−0.914569
$416$	0	0
$417$	−20.7984	−1.01850
$418$	0	0
$419$	11.1459	0.544513	0.272256	−	0.962225i	$-0.412230\pi$
$419$	11.1459	0.544513	0.272256	+	0.962225i	$0.412230\pi$
$420$	0	0
$421$	24.2705	1.18287	0.591436	−	0.806352i	$-0.298561\pi$
$421$	24.2705	1.18287	0.591436	+	0.806352i	$0.298561\pi$
$422$	0	0
$423$	1.38197	0.0671935
$424$	0	0
$425$	−14.5279	−0.704705
$426$	0	0
$427$	−1.61803	−0.0783022
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.32624	0.112051	0.0560255	−	0.998429i	$-0.482157\pi$
$431$	2.32624	0.112051	0.0560255	+	0.998429i	$0.482157\pi$
$432$	0	0
$433$	2.94427	0.141493	0.0707463	−	0.997494i	$-0.477462\pi$
$433$	2.94427	0.141493	0.0707463	+	0.997494i	$0.477462\pi$
$434$	0	0
$435$	−12.7639	−0.611984
$436$	0	0
$437$	−25.7984	−1.23410
$438$	0	0
$439$	−25.3607	−1.21040	−0.605200	−	0.796074i	$-0.706907\pi$
$439$	−25.3607	−1.21040	−0.605200	+	0.796074i	$0.706907\pi$
$440$	0	0
$441$	10.9443	0.521156
$442$	0	0
$443$	−22.1803	−1.05382	−0.526910	−	0.849921i	$-0.676649\pi$
$443$	−22.1803	−1.05382	−0.526910	+	0.849921i	$0.676649\pi$
$444$	0	0
$445$	2.85410	0.135297
$446$	0	0
$447$	18.8885	0.893397
$448$	0	0
$449$	−16.4721	−0.777368	−0.388684	−	0.921371i	$-0.627070\pi$
$449$	−16.4721	−0.777368	−0.388684	+	0.921371i	$0.627070\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4.47214	−0.210119
$454$	0	0
$455$	−21.3262	−0.999789
$456$	0	0
$457$	−7.85410	−0.367399	−0.183700	−	0.982982i	$-0.558807\pi$
$457$	−7.85410	−0.367399	−0.183700	+	0.982982i	$0.558807\pi$
$458$	0	0
$459$	4.61803	0.215551
$460$	0	0
$461$	0.562306	0.0261892	0.0130946	−	0.999914i	$-0.495832\pi$
$461$	0.562306	0.0261892	0.0130946	+	0.999914i	$0.495832\pi$
$462$	0	0
$463$	−30.2148	−1.40420	−0.702100	−	0.712078i	$-0.747754\pi$
$463$	−30.2148	−1.40420	−0.702100	+	0.712078i	$0.747754\pi$
$464$	0	0
$465$	24.5967	1.14065
$466$	0	0
$467$	−33.8328	−1.56560	−0.782798	−	0.622276i	$-0.786208\pi$
$467$	−33.8328	−1.56560	−0.782798	+	0.622276i	$0.786208\pi$
$468$	0	0
$469$	29.0344	1.34069
$470$	0	0
$471$	12.7639	0.588131
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−19.1591	−0.879078
$476$	0	0
$477$	−13.5623	−0.620975
$478$	0	0
$479$	2.32624	0.106289	0.0531443	−	0.998587i	$-0.483076\pi$
$479$	2.32624	0.106289	0.0531443	+	0.998587i	$0.483076\pi$
$480$	0	0
$481$	−14.5279	−0.662414
$482$	0	0
$483$	17.9443	0.816493
$484$	0	0
$485$	17.3820	0.789274
$486$	0	0
$487$	−17.8328	−0.808082	−0.404041	−	0.914741i	$-0.632395\pi$
$487$	−17.8328	−0.808082	−0.404041	+	0.914741i	$0.632395\pi$
$488$	0	0
$489$	−1.85410	−0.0838454
$490$	0	0
$491$	11.8541	0.534968	0.267484	−	0.963562i	$-0.413808\pi$
$491$	11.8541	0.534968	0.267484	+	0.963562i	$0.413808\pi$
$492$	0	0
$493$	−20.6525	−0.930141
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.3262	0.687476
$498$	0	0
$499$	−18.2705	−0.817900	−0.408950	−	0.912557i	$-0.634105\pi$
$499$	−18.2705	−0.817900	−0.408950	+	0.912557i	$0.634105\pi$
$500$	0	0
$501$	1.61803	0.0722884
$502$	0	0
$503$	−12.7639	−0.569116	−0.284558	−	0.958659i	$-0.591847\pi$
$503$	−12.7639	−0.569116	−0.284558	+	0.958659i	$0.591847\pi$
$504$	0	0
$505$	−34.0902	−1.51699
$506$	0	0
$507$	9.88854	0.439166
$508$	0	0
$509$	−29.7426	−1.31832	−0.659160	−	0.752003i	$-0.729088\pi$
$509$	−29.7426	−1.31832	−0.659160	+	0.752003i	$0.729088\pi$
$510$	0	0
$511$	−5.23607	−0.231630
$512$	0	0
$513$	6.09017	0.268888
$514$	0	0
$515$	10.0689	0.443688
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.5623	−0.639214
$520$	0	0
$521$	−38.8328	−1.70130	−0.850648	−	0.525735i	$-0.823790\pi$
$521$	−38.8328	−1.70130	−0.850648	+	0.525735i	$0.823790\pi$
$522$	0	0
$523$	−2.79837	−0.122364	−0.0611822	−	0.998127i	$-0.519487\pi$
$523$	−2.79837	−0.122364	−0.0611822	+	0.998127i	$0.519487\pi$
$524$	0	0
$525$	13.3262	0.581605
$526$	0	0
$527$	39.7984	1.73364
$528$	0	0
$529$	−5.05573	−0.219814
$530$	0	0
$531$	8.85410	0.384235
$532$	0	0
$533$	−0.931116	−0.0403311
$534$	0	0
$535$	−44.1591	−1.90916
$536$	0	0
$537$	0.0557281	0.00240484
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.32624	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$541$	−9.32624	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$542$	0	0
$543$	11.9443	0.512578
$544$	0	0
$545$	22.8328	0.978050
$546$	0	0
$547$	2.32624	0.0994628	0.0497314	−	0.998763i	$-0.484163\pi$
$547$	2.32624	0.0994628	0.0497314	+	0.998763i	$0.484163\pi$
$548$	0	0
$549$	0.381966	0.0163019
$550$	0	0
$551$	−27.2361	−1.16030
$552$	0	0
$553$	41.3607	1.75884
$554$	0	0
$555$	23.5066	0.997799
$556$	0	0
$557$	6.09017	0.258049	0.129024	−	0.991641i	$-0.458815\pi$
$557$	6.09017	0.258049	0.129024	+	0.991641i	$0.458815\pi$
$558$	0	0
$559$	−0.931116	−0.0393820
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.2918	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$563$	−27.2918	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$564$	0	0
$565$	22.1591	0.932238
$566$	0	0
$567$	−4.23607	−0.177898
$568$	0	0
$569$	−45.1246	−1.89172	−0.945861	−	0.324572i	$-0.894780\pi$
$569$	−45.1246	−1.89172	−0.945861	+	0.324572i	$0.894780\pi$
$570$	0	0
$571$	−8.67376	−0.362986	−0.181493	−	0.983392i	$-0.558093\pi$
$571$	−8.67376	−0.362986	−0.181493	+	0.983392i	$0.558093\pi$
$572$	0	0
$573$	−9.65248	−0.403238
$574$	0	0
$575$	13.3262	0.555743
$576$	0	0
$577$	24.7639	1.03094	0.515468	−	0.856909i	$-0.327618\pi$
$577$	24.7639	1.03094	0.515468	+	0.856909i	$0.327618\pi$
$578$	0	0
$579$	−0.326238	−0.0135580
$580$	0	0
$581$	27.6525	1.14722
$582$	0	0
$583$	0	0
$584$	0	0
$585$	5.03444	0.208149
$586$	0	0
$587$	30.2361	1.24798	0.623988	−	0.781434i	$-0.285511\pi$
$587$	30.2361	1.24798	0.623988	+	0.781434i	$0.285511\pi$
$588$	0	0
$589$	52.4853	2.16262
$590$	0	0
$591$	−17.0902	−0.702996
$592$	0	0
$593$	36.4508	1.49686	0.748428	−	0.663215i	$-0.230809\pi$
$593$	36.4508	1.49686	0.748428	+	0.663215i	$0.230809\pi$
$594$	0	0
$595$	55.8328	2.28892
$596$	0	0
$597$	−16.4164	−0.671879
$598$	0	0
$599$	9.34752	0.381929	0.190965	−	0.981597i	$-0.438838\pi$
$599$	9.34752	0.381929	0.190965	+	0.981597i	$0.438838\pi$
$600$	0	0
$601$	−2.05573	−0.0838549	−0.0419274	−	0.999121i	$-0.513350\pi$
$601$	−2.05573	−0.0838549	−0.0419274	+	0.999121i	$0.513350\pi$
$602$	0	0
$603$	−6.85410	−0.279121
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.3262	−0.987372	−0.493686	−	0.869640i	$-0.664351\pi$
$607$	−24.3262	−0.987372	−0.493686	+	0.869640i	$0.664351\pi$
$608$	0	0
$609$	18.9443	0.767661
$610$	0	0
$611$	2.43769	0.0986185
$612$	0	0
$613$	23.1246	0.933994	0.466997	−	0.884259i	$-0.345336\pi$
$613$	23.1246	0.933994	0.466997	+	0.884259i	$0.345336\pi$
$614$	0	0
$615$	1.50658	0.0607511
$616$	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
$619$	1.47214	0.0591701	0.0295851	−	0.999562i	$-0.490581\pi$
$619$	1.47214	0.0591701	0.0295851	+	0.999562i	$0.490581\pi$
$620$	0	0
$621$	−4.23607	−0.169988
$622$	0	0
$623$	−4.23607	−0.169714
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38.0344	1.51653
$630$	0	0
$631$	41.2705	1.64295	0.821477	−	0.570242i	$-0.193151\pi$
$631$	41.2705	1.64295	0.821477	+	0.570242i	$0.193151\pi$
$632$	0	0
$633$	−23.7426	−0.943685
$634$	0	0
$635$	−6.54102	−0.259572
$636$	0	0
$637$	19.3050	0.764890
$638$	0	0
$639$	−3.61803	−0.143127
$640$	0	0
$641$	−21.9787	−0.868107	−0.434054	−	0.900887i	$-0.642917\pi$
$641$	−21.9787	−0.868107	−0.434054	+	0.900887i	$0.642917\pi$
$642$	0	0
$643$	29.1591	1.14992	0.574960	−	0.818181i	$-0.305017\pi$
$643$	29.1591	1.14992	0.574960	+	0.818181i	$0.305017\pi$
$644$	0	0
$645$	1.50658	0.0593214
$646$	0	0
$647$	−24.6869	−0.970543	−0.485271	−	0.874364i	$-0.661279\pi$
$647$	−24.6869	−0.970543	−0.485271	+	0.874364i	$0.661279\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−36.5066	−1.43081
$652$	0	0
$653$	−27.8541	−1.09002	−0.545008	−	0.838431i	$-0.683473\pi$
$653$	−27.8541	−1.09002	−0.545008	+	0.838431i	$0.683473\pi$
$654$	0	0
$655$	−33.0000	−1.28942
$656$	0	0
$657$	1.23607	0.0482236
$658$	0	0
$659$	3.70820	0.144451	0.0722256	−	0.997388i	$-0.476990\pi$
$659$	3.70820	0.144451	0.0722256	+	0.997388i	$0.476990\pi$
$660$	0	0
$661$	2.32624	0.0904802	0.0452401	−	0.998976i	$-0.485595\pi$
$661$	2.32624	0.0904802	0.0452401	+	0.998976i	$0.485595\pi$
$662$	0	0
$663$	8.14590	0.316360
$664$	0	0
$665$	73.6312	2.85530
$666$	0	0
$667$	18.9443	0.733525
$668$	0	0
$669$	−1.47214	−0.0569161
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−26.5279	−1.02257	−0.511287	−	0.859410i	$-0.670831\pi$
$673$	−26.5279	−1.02257	−0.511287	+	0.859410i	$0.670831\pi$
$674$	0	0
$675$	−3.14590	−0.121086
$676$	0	0
$677$	6.47214	0.248744	0.124372	−	0.992236i	$-0.460308\pi$
$677$	6.47214	0.248744	0.124372	+	0.992236i	$0.460308\pi$
$678$	0	0
$679$	−25.7984	−0.990051
$680$	0	0
$681$	−0.708204	−0.0271384
$682$	0	0
$683$	−6.34752	−0.242881	−0.121441	−	0.992599i	$-0.538751\pi$
$683$	−6.34752	−0.242881	−0.121441	+	0.992599i	$0.538751\pi$
$684$	0	0
$685$	−59.9361	−2.29004
$686$	0	0
$687$	6.94427	0.264940
$688$	0	0
$689$	−23.9230	−0.911393
$690$	0	0
$691$	−34.5410	−1.31400	−0.657001	−	0.753890i	$-0.728175\pi$
$691$	−34.5410	−1.31400	−0.657001	+	0.753890i	$0.728175\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	59.3607	2.25168
$696$	0	0
$697$	2.43769	0.0923342
$698$	0	0
$699$	25.7984	0.975784
$700$	0	0
$701$	−9.50658	−0.359058	−0.179529	−	0.983753i	$-0.557457\pi$
$701$	−9.50658	−0.359058	−0.179529	+	0.983753i	$0.557457\pi$
$702$	0	0
$703$	50.1591	1.89178
$704$	0	0
$705$	−3.94427	−0.148550
$706$	0	0
$707$	50.5967	1.90289
$708$	0	0
$709$	−25.7426	−0.966785	−0.483393	−	0.875404i	$-0.660596\pi$
$709$	−25.7426	−0.966785	−0.483393	+	0.875404i	$0.660596\pi$
$710$	0	0
$711$	−9.76393	−0.366176
$712$	0	0
$713$	−36.5066	−1.36718
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.5066	−0.765832
$718$	0	0
$719$	−39.0689	−1.45702	−0.728512	−	0.685033i	$-0.759788\pi$
$719$	−39.0689	−1.45702	−0.728512	+	0.685033i	$0.759788\pi$
$720$	0	0
$721$	−14.9443	−0.556554
$722$	0	0
$723$	−10.2918	−0.382756
$724$	0	0
$725$	14.0689	0.522505
$726$	0	0
$727$	−2.14590	−0.0795870	−0.0397935	−	0.999208i	$-0.512670\pi$
$727$	−2.14590	−0.0795870	−0.0397935	+	0.999208i	$0.512670\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.43769	0.0901614
$732$	0	0
$733$	22.5410	0.832572	0.416286	−	0.909234i	$-0.363332\pi$
$733$	22.5410	0.832572	0.416286	+	0.909234i	$0.363332\pi$
$734$	0	0
$735$	−31.2361	−1.15216
$736$	0	0
$737$	0	0
$738$	0	0
$739$	35.1803	1.29413	0.647065	−	0.762435i	$-0.275996\pi$
$739$	35.1803	1.29413	0.647065	+	0.762435i	$0.275996\pi$
$740$	0	0
$741$	10.7426	0.394641
$742$	0	0
$743$	−2.52786	−0.0927383	−0.0463692	−	0.998924i	$-0.514765\pi$
$743$	−2.52786	−0.0927383	−0.0463692	+	0.998924i	$0.514765\pi$
$744$	0	0
$745$	−53.9098	−1.97510
$746$	0	0
$747$	−6.52786	−0.238842
$748$	0	0
$749$	65.5410	2.39482
$750$	0	0
$751$	36.6312	1.33669	0.668346	−	0.743851i	$-0.267003\pi$
$751$	36.6312	1.33669	0.668346	+	0.743851i	$0.267003\pi$
$752$	0	0
$753$	−12.9098	−0.470460
$754$	0	0
$755$	12.7639	0.464527
$756$	0	0
$757$	16.4164	0.596664	0.298332	−	0.954462i	$-0.403570\pi$
$757$	16.4164	0.596664	0.298332	+	0.954462i	$0.403570\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−51.2492	−1.85778	−0.928891	−	0.370352i	$-0.879237\pi$
$761$	−51.2492	−1.85778	−0.928891	+	0.370352i	$0.879237\pi$
$762$	0	0
$763$	−33.8885	−1.22685
$764$	0	0
$765$	−13.1803	−0.476536
$766$	0	0
$767$	15.6180	0.563935
$768$	0	0
$769$	13.4377	0.484576	0.242288	−	0.970204i	$-0.422102\pi$
$769$	13.4377	0.484576	0.242288	+	0.970204i	$0.422102\pi$
$770$	0	0
$771$	−21.7984	−0.785049
$772$	0	0
$773$	−15.3607	−0.552485	−0.276243	−	0.961088i	$-0.589089\pi$
$773$	−15.3607	−0.552485	−0.276243	+	0.961088i	$0.589089\pi$
$774$	0	0
$775$	−27.1115	−0.973872
$776$	0	0
$777$	−34.8885	−1.25162
$778$	0	0
$779$	3.21478	0.115182
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.47214	−0.159821
$784$	0	0
$785$	−36.4296	−1.30023
$786$	0	0
$787$	−49.2361	−1.75508	−0.877538	−	0.479507i	$-0.840816\pi$
$787$	−49.2361	−1.75508	−0.877538	+	0.479507i	$0.840816\pi$
$788$	0	0
$789$	17.1459	0.610410
$790$	0	0
$791$	−32.8885	−1.16938
$792$	0	0
$793$	0.673762	0.0239260
$794$	0	0
$795$	38.7082	1.37284
$796$	0	0
$797$	27.0132	0.956855	0.478428	−	0.878127i	$-0.341207\pi$
$797$	27.0132	0.956855	0.478428	+	0.878127i	$0.341207\pi$
$798$	0	0
$799$	−6.38197	−0.225778
$800$	0	0
$801$	1.00000	0.0353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−51.2148	−1.80508
$806$	0	0
$807$	−19.5279	−0.687413
$808$	0	0
$809$	−1.72949	−0.0608056	−0.0304028	−	0.999538i	$-0.509679\pi$
$809$	−1.72949	−0.0608056	−0.0304028	+	0.999538i	$0.509679\pi$
$810$	0	0
$811$	26.7984	0.941018	0.470509	−	0.882395i	$-0.344070\pi$
$811$	26.7984	0.941018	0.470509	+	0.882395i	$0.344070\pi$
$812$	0	0
$813$	−22.5623	−0.791295
$814$	0	0
$815$	5.29180	0.185364
$816$	0	0
$817$	3.21478	0.112471
$818$	0	0
$819$	−7.47214	−0.261098
$820$	0	0
$821$	17.3607	0.605892	0.302946	−	0.953008i	$-0.402030\pi$
$821$	17.3607	0.605892	0.302946	+	0.953008i	$0.402030\pi$
$822$	0	0
$823$	0.819660	0.0285716	0.0142858	−	0.999898i	$-0.495453\pi$
$823$	0.819660	0.0285716	0.0142858	+	0.999898i	$0.495453\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.875388	0.0304402	0.0152201	−	0.999884i	$-0.495155\pi$
$827$	0.875388	0.0304402	0.0152201	+	0.999884i	$0.495155\pi$
$828$	0	0
$829$	−37.7426	−1.31086	−0.655428	−	0.755258i	$-0.727512\pi$
$829$	−37.7426	−1.31086	−0.655428	+	0.755258i	$0.727512\pi$
$830$	0	0
$831$	−17.5623	−0.609230
$832$	0	0
$833$	−50.5410	−1.75114
$834$	0	0
$835$	−4.61803	−0.159814
$836$	0	0
$837$	8.61803	0.297883
$838$	0	0
$839$	−15.8328	−0.546610	−0.273305	−	0.961927i	$-0.588117\pi$
$839$	−15.8328	−0.546610	−0.273305	+	0.961927i	$0.588117\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	−23.1246	−0.796454
$844$	0	0
$845$	−28.2229	−0.970898
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18.7639	0.643976
$850$	0	0
$851$	−34.8885	−1.19596
$852$	0	0
$853$	−55.3607	−1.89551	−0.947757	−	0.318994i	$-0.896655\pi$
$853$	−55.3607	−1.89551	−0.947757	+	0.318994i	$0.896655\pi$
$854$	0	0
$855$	−17.3820	−0.594451
$856$	0	0
$857$	−48.1935	−1.64626	−0.823129	−	0.567854i	$-0.807774\pi$
$857$	−48.1935	−1.64626	−0.823129	+	0.567854i	$0.807774\pi$
$858$	0	0
$859$	31.1803	1.06386	0.531930	−	0.846788i	$-0.321467\pi$
$859$	31.1803	1.06386	0.531930	+	0.846788i	$0.321467\pi$
$860$	0	0
$861$	−2.23607	−0.0762050
$862$	0	0
$863$	30.3607	1.03349	0.516745	−	0.856139i	$-0.327144\pi$
$863$	30.3607	1.03349	0.516745	+	0.856139i	$0.327144\pi$
$864$	0	0
$865$	41.5623	1.41316
$866$	0	0
$867$	−4.32624	−0.146927
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.0902	−0.409660
$872$	0	0
$873$	6.09017	0.206121
$874$	0	0
$875$	22.4164	0.757813
$876$	0	0
$877$	34.8885	1.17810	0.589051	−	0.808096i	$-0.299502\pi$
$877$	34.8885	1.17810	0.589051	+	0.808096i	$0.299502\pi$
$878$	0	0
$879$	−5.00000	−0.168646
$880$	0	0
$881$	20.4377	0.688563	0.344282	−	0.938866i	$-0.388122\pi$
$881$	20.4377	0.688563	0.344282	+	0.938866i	$0.388122\pi$
$882$	0	0
$883$	−2.94427	−0.0990826	−0.0495413	−	0.998772i	$-0.515776\pi$
$883$	−2.94427	−0.0990826	−0.0495413	+	0.998772i	$0.515776\pi$
$884$	0	0
$885$	−25.2705	−0.849459
$886$	0	0
$887$	46.7082	1.56831	0.784154	−	0.620566i	$-0.213097\pi$
$887$	46.7082	1.56831	0.784154	+	0.620566i	$0.213097\pi$
$888$	0	0
$889$	9.70820	0.325603
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.41641	−0.281644
$894$	0	0
$895$	−0.159054	−0.00531658
$896$	0	0
$897$	−7.47214	−0.249487
$898$	0	0
$899$	−38.5410	−1.28541
$900$	0	0
$901$	62.6312	2.08655
$902$	0	0
$903$	−2.23607	−0.0744117
$904$	0	0
$905$	−34.0902	−1.13320
$906$	0	0
$907$	39.8673	1.32377	0.661885	−	0.749605i	$-0.269757\pi$
$907$	39.8673	1.32377	0.661885	+	0.749605i	$0.269757\pi$
$908$	0	0
$909$	−11.9443	−0.396166
$910$	0	0
$911$	−41.5410	−1.37632	−0.688158	−	0.725561i	$-0.741580\pi$
$911$	−41.5410	−1.37632	−0.688158	+	0.725561i	$0.741580\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.09017	−0.0360399
$916$	0	0
$917$	48.9787	1.61742
$918$	0	0
$919$	−37.8328	−1.24799	−0.623995	−	0.781429i	$-0.714491\pi$
$919$	−37.8328	−1.24799	−0.623995	+	0.781429i	$0.714491\pi$
$920$	0	0
$921$	−16.3262	−0.537968
$922$	0	0
$923$	−6.38197	−0.210065
$924$	0	0
$925$	−25.9098	−0.851910
$926$	0	0
$927$	3.52786	0.115870
$928$	0	0
$929$	46.4164	1.52287	0.761436	−	0.648240i	$-0.224494\pi$
$929$	46.4164	1.52287	0.761436	+	0.648240i	$0.224494\pi$
$930$	0	0
$931$	−66.6525	−2.18445
$932$	0	0
$933$	6.81966	0.223266
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.2361	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$937$	−24.2361	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	0	0
$941$	−17.5623	−0.572515	−0.286257	−	0.958153i	$-0.592411\pi$
$941$	−17.5623	−0.572515	−0.286257	+	0.958153i	$0.592411\pi$
$942$	0	0
$943$	−2.23607	−0.0728164
$944$	0	0
$945$	12.0902	0.393293
$946$	0	0
$947$	21.7984	0.708352	0.354176	−	0.935179i	$-0.384761\pi$
$947$	21.7984	0.708352	0.354176	+	0.935179i	$0.384761\pi$
$948$	0	0
$949$	2.18034	0.0707768
$950$	0	0
$951$	−17.4721	−0.566573
$952$	0	0
$953$	−39.2361	−1.27098	−0.635490	−	0.772109i	$-0.719202\pi$
$953$	−39.2361	−1.27098	−0.635490	+	0.772109i	$0.719202\pi$
$954$	0	0
$955$	27.5492	0.891470
$956$	0	0
$957$	0	0
$958$	0	0
$959$	88.9574	2.87259
$960$	0	0
$961$	43.2705	1.39582
$962$	0	0
$963$	−15.4721	−0.498583
$964$	0	0
$965$	0.931116	0.0299737
$966$	0	0
$967$	−48.2705	−1.55227	−0.776137	−	0.630564i	$-0.782824\pi$
$967$	−48.2705	−1.55227	−0.776137	+	0.630564i	$0.782824\pi$
$968$	0	0
$969$	−28.1246	−0.903493
$970$	0	0
$971$	−42.3607	−1.35942	−0.679709	−	0.733481i	$-0.737894\pi$
$971$	−42.3607	−1.35942	−0.679709	+	0.733481i	$0.737894\pi$
$972$	0	0
$973$	−88.1033	−2.82446
$974$	0	0
$975$	−5.54915	−0.177715
$976$	0	0
$977$	2.88854	0.0924127	0.0462064	−	0.998932i	$-0.485287\pi$
$977$	2.88854	0.0924127	0.0462064	+	0.998932i	$0.485287\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8.00000	0.255420
$982$	0	0
$983$	16.8328	0.536883	0.268442	−	0.963296i	$-0.413491\pi$
$983$	16.8328	0.536883	0.268442	+	0.963296i	$0.413491\pi$
$984$	0	0
$985$	48.7771	1.55417
$986$	0	0
$987$	5.85410	0.186338
$988$	0	0
$989$	−2.23607	−0.0711028
$990$	0	0
$991$	9.72949	0.309067	0.154534	−	0.987988i	$-0.450612\pi$
$991$	9.72949	0.309067	0.154534	+	0.987988i	$0.450612\pi$
$992$	0	0
$993$	5.94427	0.188636
$994$	0	0
$995$	46.8541	1.48537
$996$	0	0
$997$	4.27051	0.135248	0.0676242	−	0.997711i	$-0.478458\pi$
$997$	4.27051	0.135248	0.0676242	+	0.997711i	$0.478458\pi$
$998$	0	0
$999$	8.23607	0.260578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.2.a.i.1.2		2
3.2	odd	2		4356.2.a.s.1.1		2
4.3	odd	2		5808.2.a.cf.1.2		2
11.2	odd	10		132.2.i.b.37.1	yes	4
11.3	even	5		1452.2.i.p.493.1		4
11.4	even	5		1452.2.i.p.1237.1		4
11.5	even	5		1452.2.i.j.1213.1		4
11.6	odd	10		132.2.i.b.25.1	✓	4
11.7	odd	10		1452.2.i.o.1237.1		4
11.8	odd	10		1452.2.i.o.493.1		4
11.9	even	5		1452.2.i.j.565.1		4
11.10	odd	2		1452.2.a.j.1.2		2
33.2	even	10		396.2.j.c.37.1		4
33.17	even	10		396.2.j.c.289.1		4
33.32	even	2		4356.2.a.v.1.1		2
44.35	even	10		528.2.y.a.433.1		4
44.39	even	10		528.2.y.a.289.1		4
44.43	even	2		5808.2.a.cc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.i.b.25.1	✓	4	11.6	odd	10
132.2.i.b.37.1	yes	4	11.2	odd	10
396.2.j.c.37.1		4	33.2	even	10
396.2.j.c.289.1		4	33.17	even	10
528.2.y.a.289.1		4	44.39	even	10
528.2.y.a.433.1		4	44.35	even	10
1452.2.a.i.1.2		2	1.1	even	1	trivial
1452.2.a.j.1.2		2	11.10	odd	2
1452.2.i.j.565.1		4	11.9	even	5
1452.2.i.j.1213.1		4	11.5	even	5
1452.2.i.o.493.1		4	11.8	odd	10
1452.2.i.o.1237.1		4	11.7	odd	10
1452.2.i.p.493.1		4	11.3	even	5
1452.2.i.p.1237.1		4	11.4	even	5
4356.2.a.s.1.1		2	3.2	odd	2
4356.2.a.v.1.1		2	33.32	even	2
5808.2.a.cc.1.2		2	44.43	even	2
5808.2.a.cf.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.85410 q^{5} -4.23607 q^{7} +1.00000 q^{9} +1.76393 q^{13} -2.85410 q^{15} -4.61803 q^{17} -6.09017 q^{19} +4.23607 q^{21} +4.23607 q^{23} +3.14590 q^{25} -1.00000 q^{27} +4.47214 q^{29} -8.61803 q^{31} -12.0902 q^{35} -8.23607 q^{37} -1.76393 q^{39} -0.527864 q^{41} -0.527864 q^{43} +2.85410 q^{45} +1.38197 q^{47} +10.9443 q^{49} +4.61803 q^{51} -13.5623 q^{53} +6.09017 q^{57} +8.85410 q^{59} +0.381966 q^{61} -4.23607 q^{63} +5.03444 q^{65} -6.85410 q^{67} -4.23607 q^{69} -3.61803 q^{71} +1.23607 q^{73} -3.14590 q^{75} -9.76393 q^{79} +1.00000 q^{81} -6.52786 q^{83} -13.1803 q^{85} -4.47214 q^{87} +1.00000 q^{89} -7.47214 q^{91} +8.61803 q^{93} -17.3820 q^{95} +6.09017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 4 q^{7} + 2 q^{9} + 8 q^{13} + q^{15} - 7 q^{17} - q^{19} + 4 q^{21} + 4 q^{23} + 13 q^{25} - 2 q^{27} - 15 q^{31} - 13 q^{35} - 12 q^{37} - 8 q^{39} - 10 q^{41} - 10 q^{43} - q^{45}+ \cdots + q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more