LMFDB - Embedded newform 1144.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1144,2,Mod(1,1144)] 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("1144.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1144 = 2^{3} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1144.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,1,0,-4,0,-1,0,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.13488599123$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	1144.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86081 q^{3} -2.32340 q^{5} -0.537402 q^{7} +0.462598 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.32340 q^{15} -8.04502 q^{17} -1.53740 q^{19} -1.00000 q^{21} -1.13919 q^{23} +0.398207 q^{25} -4.72161 q^{27} -0.796415 q^{29} +8.36842 q^{31} -1.86081 q^{33} +1.24860 q^{35} +2.60179 q^{37} -1.86081 q^{39} -3.78600 q^{41} -4.60179 q^{43} -1.07480 q^{45} +5.11982 q^{47} -6.71120 q^{49} -14.9702 q^{51} +0.333816 q^{53} +2.32340 q^{55} -2.86081 q^{57} -9.57201 q^{59} +2.64681 q^{61} -0.248601 q^{63} +2.32340 q^{65} +1.05398 q^{67} -2.11982 q^{69} -3.52699 q^{71} +9.10941 q^{73} +0.740987 q^{75} +0.537402 q^{77} -5.20359 q^{79} -10.1738 q^{81} +5.37883 q^{83} +18.6918 q^{85} -1.48197 q^{87} -13.4224 q^{89} +0.537402 q^{91} +15.5720 q^{93} +3.57201 q^{95} +4.49720 q^{97} -0.462598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} - 4 q^{7} - q^{9} - 3 q^{11} - 3 q^{13} - 5 q^{15} - 5 q^{17} - 7 q^{19} - 3 q^{21} - 9 q^{23} - 2 q^{25} - 3 q^{27} + 4 q^{29} - 2 q^{31} - 9 q^{35} + 11 q^{37} - q^{41} - 17 q^{43} - 8 q^{45}+ \cdots + q^{99}+O(q^{100}) $ 3 * q + q^5 - 4 * q^7 - q^9 - 3 * q^11 - 3 * q^13 - 5 * q^15 - 5 * q^17 - 7 * q^19 - 3 * q^21 - 9 * q^23 - 2 * q^25 - 3 * q^27 + 4 * q^29 - 2 * q^31 - 9 * q^35 + 11 * q^37 - q^41 - 17 * q^43 - 8 * q^45 + q^47 - 5 * q^49 - 21 * q^51 - 3 * q^53 - q^55 - 3 * q^57 - 8 * q^59 - 8 * q^61 + 12 * q^63 - q^65 + 8 * q^69 - 9 * q^71 + 9 * q^73 + 11 * q^75 + 4 * q^77 - 22 * q^79 - 13 * q^81 - 7 * q^83 + 21 * q^85 - 22 * q^87 - 10 * q^89 + 4 * q^91 + 26 * q^93 - 10 * q^95 - 12 * q^97 + q^99

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
$3$	1.86081	1.07434	0.537168	−	0.843475i	$-0.319494\pi$
$3$	1.86081	1.07434	0.537168	+	0.843475i	$0.319494\pi$
$4$	0	0
$5$	−2.32340	−1.03906	−0.519529	−	0.854453i	$-0.673893\pi$
$5$	−2.32340	−1.03906	−0.519529	+	0.854453i	$0.673893\pi$
$6$	0	0
$7$	−0.537402	−0.203119	−0.101559	−	0.994829i	$-0.532383\pi$
$7$	−0.537402	−0.203119	−0.101559	+	0.994829i	$0.532383\pi$
$8$	0	0
$9$	0.462598	0.154199
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.32340	−1.11630
$16$	0	0
$17$	−8.04502	−1.95120	−0.975601	−	0.219549i	$-0.929541\pi$
$17$	−8.04502	−1.95120	−0.975601	+	0.219549i	$0.929541\pi$
$18$	0	0
$19$	−1.53740	−0.352704	−0.176352	−	0.984327i	$-0.556430\pi$
$19$	−1.53740	−0.352704	−0.176352	+	0.984327i	$0.556430\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.13919	−0.237538	−0.118769	−	0.992922i	$-0.537895\pi$
$23$	−1.13919	−0.237538	−0.118769	+	0.992922i	$0.537895\pi$
$24$	0	0
$25$	0.398207	0.0796415
$26$	0	0
$27$	−4.72161	−0.908675
$28$	0	0
$29$	−0.796415	−0.147891	−0.0739453	−	0.997262i	$-0.523559\pi$
$29$	−0.796415	−0.147891	−0.0739453	+	0.997262i	$0.523559\pi$
$30$	0	0
$31$	8.36842	1.50301	0.751506	−	0.659726i	$-0.229328\pi$
$31$	8.36842	1.50301	0.751506	+	0.659726i	$0.229328\pi$
$32$	0	0
$33$	−1.86081	−0.323925
$34$	0	0
$35$	1.24860	0.211052
$36$	0	0
$37$	2.60179	0.427732	0.213866	−	0.976863i	$-0.431394\pi$
$37$	2.60179	0.427732	0.213866	+	0.976863i	$0.431394\pi$
$38$	0	0
$39$	−1.86081	−0.297967
$40$	0	0
$41$	−3.78600	−0.591274	−0.295637	−	0.955300i	$-0.595532\pi$
$41$	−3.78600	−0.591274	−0.295637	+	0.955300i	$0.595532\pi$
$42$	0	0
$43$	−4.60179	−0.701767	−0.350883	−	0.936419i	$-0.614119\pi$
$43$	−4.60179	−0.701767	−0.350883	+	0.936419i	$0.614119\pi$
$44$	0	0
$45$	−1.07480	−0.160222
$46$	0	0
$47$	5.11982	0.746802	0.373401	−	0.927670i	$-0.378191\pi$
$47$	5.11982	0.746802	0.373401	+	0.927670i	$0.378191\pi$
$48$	0	0
$49$	−6.71120	−0.958743
$50$	0	0
$51$	−14.9702	−2.09625
$52$	0	0
$53$	0.333816	0.0458532	0.0229266	−	0.999737i	$-0.492702\pi$
$53$	0.333816	0.0458532	0.0229266	+	0.999737i	$0.492702\pi$
$54$	0	0
$55$	2.32340	0.313288
$56$	0	0
$57$	−2.86081	−0.378923
$58$	0	0
$59$	−9.57201	−1.24617	−0.623084	−	0.782155i	$-0.714121\pi$
$59$	−9.57201	−1.24617	−0.623084	+	0.782155i	$0.714121\pi$
$60$	0	0
$61$	2.64681	0.338889	0.169445	−	0.985540i	$-0.445803\pi$
$61$	2.64681	0.338889	0.169445	+	0.985540i	$0.445803\pi$
$62$	0	0
$63$	−0.248601	−0.0313208
$64$	0	0
$65$	2.32340	0.288183
$66$	0	0
$67$	1.05398	0.128764	0.0643820	−	0.997925i	$-0.479492\pi$
$67$	1.05398	0.128764	0.0643820	+	0.997925i	$0.479492\pi$
$68$	0	0
$69$	−2.11982	−0.255196
$70$	0	0
$71$	−3.52699	−0.418577	−0.209288	−	0.977854i	$-0.567115\pi$
$71$	−3.52699	−0.418577	−0.209288	+	0.977854i	$0.567115\pi$
$72$	0	0
$73$	9.10941	1.06618	0.533088	−	0.846060i	$-0.321032\pi$
$73$	9.10941	1.06618	0.533088	+	0.846060i	$0.321032\pi$
$74$	0	0
$75$	0.740987	0.0855618
$76$	0	0
$77$	0.537402	0.0612426
$78$	0	0
$79$	−5.20359	−0.585449	−0.292725	−	0.956197i	$-0.594562\pi$
$79$	−5.20359	−0.585449	−0.292725	+	0.956197i	$0.594562\pi$
$80$	0	0
$81$	−10.1738	−1.13042
$82$	0	0
$83$	5.37883	0.590404	0.295202	−	0.955435i	$-0.404613\pi$
$83$	5.37883	0.590404	0.295202	+	0.955435i	$0.404613\pi$
$84$	0	0
$85$	18.6918	2.02741
$86$	0	0
$87$	−1.48197	−0.158884
$88$	0	0
$89$	−13.4224	−1.42277	−0.711386	−	0.702802i	$-0.751932\pi$
$89$	−13.4224	−1.42277	−0.711386	+	0.702802i	$0.751932\pi$
$90$	0	0
$91$	0.537402	0.0563350
$92$	0	0
$93$	15.5720	1.61474
$94$	0	0
$95$	3.57201	0.366480
$96$	0	0
$97$	4.49720	0.456622	0.228311	−	0.973588i	$-0.426680\pi$
$97$	4.49720	0.456622	0.228311	+	0.973588i	$0.426680\pi$
$98$	0	0
$99$	−0.462598	−0.0464929
$100$	0	0
$101$	5.57201	0.554435	0.277218	−	0.960807i	$-0.410588\pi$
$101$	5.57201	0.554435	0.277218	+	0.960807i	$0.410588\pi$
$102$	0	0
$103$	−10.9508	−1.07902	−0.539509	−	0.841980i	$-0.681390\pi$
$103$	−10.9508	−1.07902	−0.539509	+	0.841980i	$0.681390\pi$
$104$	0	0
$105$	2.32340	0.226741
$106$	0	0
$107$	−3.44322	−0.332869	−0.166434	−	0.986053i	$-0.553225\pi$
$107$	−3.44322	−0.332869	−0.166434	+	0.986053i	$0.553225\pi$
$108$	0	0
$109$	13.0256	1.24763	0.623815	−	0.781572i	$-0.285582\pi$
$109$	13.0256	1.24763	0.623815	+	0.781572i	$0.285582\pi$
$110$	0	0
$111$	4.84143	0.459528
$112$	0	0
$113$	15.4778	1.45603	0.728016	−	0.685560i	$-0.240443\pi$
$113$	15.4778	1.45603	0.728016	+	0.685560i	$0.240443\pi$
$114$	0	0
$115$	2.64681	0.246816
$116$	0	0
$117$	−0.462598	−0.0427672
$118$	0	0
$119$	4.32340	0.396326
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.04502	−0.635228
$124$	0	0
$125$	10.6918	0.956306
$126$	0	0
$127$	−21.2936	−1.88950	−0.944751	−	0.327787i	$-0.893697\pi$
$127$	−21.2936	−1.88950	−0.944751	+	0.327787i	$0.893697\pi$
$128$	0	0
$129$	−8.56304	−0.753934
$130$	0	0
$131$	8.82061	0.770660	0.385330	−	0.922779i	$-0.374088\pi$
$131$	8.82061	0.770660	0.385330	+	0.922779i	$0.374088\pi$
$132$	0	0
$133$	0.826202	0.0716408
$134$	0	0
$135$	10.9702	0.944166
$136$	0	0
$137$	14.3684	1.22758	0.613788	−	0.789471i	$-0.289645\pi$
$137$	14.3684	1.22758	0.613788	+	0.789471i	$0.289645\pi$
$138$	0	0
$139$	−10.9910	−0.932248	−0.466124	−	0.884720i	$-0.654350\pi$
$139$	−10.9910	−0.932248	−0.466124	+	0.884720i	$0.654350\pi$
$140$	0	0
$141$	9.52699	0.802317
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.85039	0.153667
$146$	0	0
$147$	−12.4882	−1.03001
$148$	0	0
$149$	−20.3040	−1.66337	−0.831685	−	0.555247i	$-0.812624\pi$
$149$	−20.3040	−1.66337	−0.831685	+	0.555247i	$0.812624\pi$
$150$	0	0
$151$	0.751399	0.0611480	0.0305740	−	0.999533i	$-0.490266\pi$
$151$	0.751399	0.0611480	0.0305740	+	0.999533i	$0.490266\pi$
$152$	0	0
$153$	−3.72161	−0.300874
$154$	0	0
$155$	−19.4432	−1.56172
$156$	0	0
$157$	19.8969	1.58794	0.793971	−	0.607955i	$-0.208010\pi$
$157$	19.8969	1.58794	0.793971	+	0.607955i	$0.208010\pi$
$158$	0	0
$159$	0.621168	0.0492618
$160$	0	0
$161$	0.612205	0.0482485
$162$	0	0
$163$	12.5180	0.980488	0.490244	−	0.871585i	$-0.336908\pi$
$163$	12.5180	0.980488	0.490244	+	0.871585i	$0.336908\pi$
$164$	0	0
$165$	4.32340	0.336577
$166$	0	0
$167$	−8.98062	−0.694942	−0.347471	−	0.937691i	$-0.612959\pi$
$167$	−8.98062	−0.694942	−0.347471	+	0.937691i	$0.612959\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.711200	−0.0543868
$172$	0	0
$173$	−10.2188	−0.776922	−0.388461	−	0.921465i	$-0.626993\pi$
$173$	−10.2188	−0.776922	−0.388461	+	0.921465i	$0.626993\pi$
$174$	0	0
$175$	−0.213997	−0.0161767
$176$	0	0
$177$	−17.8116	−1.33881
$178$	0	0
$179$	−19.8954	−1.48705	−0.743526	−	0.668707i	$-0.766848\pi$
$179$	−19.8954	−1.48705	−0.743526	+	0.668707i	$0.766848\pi$
$180$	0	0
$181$	8.89059	0.660833	0.330416	−	0.943835i	$-0.392811\pi$
$181$	8.89059	0.660833	0.330416	+	0.943835i	$0.392811\pi$
$182$	0	0
$183$	4.92520	0.364081
$184$	0	0
$185$	−6.04502	−0.444438
$186$	0	0
$187$	8.04502	0.588310
$188$	0	0
$189$	2.53740	0.184569
$190$	0	0
$191$	−5.77704	−0.418012	−0.209006	−	0.977914i	$-0.567023\pi$
$191$	−5.77704	−0.418012	−0.209006	+	0.977914i	$0.567023\pi$
$192$	0	0
$193$	−4.95084	−0.356369	−0.178185	−	0.983997i	$-0.557022\pi$
$193$	−4.95084	−0.356369	−0.178185	+	0.983997i	$0.557022\pi$
$194$	0	0
$195$	4.32340	0.309605
$196$	0	0
$197$	0.935609	0.0666594	0.0333297	−	0.999444i	$-0.489389\pi$
$197$	0.935609	0.0666594	0.0333297	+	0.999444i	$0.489389\pi$
$198$	0	0
$199$	22.1246	1.56837	0.784187	−	0.620525i	$-0.213080\pi$
$199$	22.1246	1.56837	0.784187	+	0.620525i	$0.213080\pi$
$200$	0	0
$201$	1.96125	0.138336
$202$	0	0
$203$	0.427995	0.0300393
$204$	0	0
$205$	8.79641	0.614368
$206$	0	0
$207$	−0.526989	−0.0366283
$208$	0	0
$209$	1.53740	0.106344
$210$	0	0
$211$	2.60179	0.179115	0.0895574	−	0.995982i	$-0.471455\pi$
$211$	2.60179	0.179115	0.0895574	+	0.995982i	$0.471455\pi$
$212$	0	0
$213$	−6.56304	−0.449692
$214$	0	0
$215$	10.6918	0.729176
$216$	0	0
$217$	−4.49720	−0.305290
$218$	0	0
$219$	16.9508	1.14543
$220$	0	0
$221$	8.04502	0.541166
$222$	0	0
$223$	−12.1350	−0.812623	−0.406311	−	0.913735i	$-0.633185\pi$
$223$	−12.1350	−0.812623	−0.406311	+	0.913735i	$0.633185\pi$
$224$	0	0
$225$	0.184210	0.0122807
$226$	0	0
$227$	−1.53740	−0.102041	−0.0510205	−	0.998698i	$-0.516247\pi$
$227$	−1.53740	−0.102041	−0.0510205	+	0.998698i	$0.516247\pi$
$228$	0	0
$229$	5.15857	0.340888	0.170444	−	0.985367i	$-0.445480\pi$
$229$	5.15857	0.340888	0.170444	+	0.985367i	$0.445480\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−20.9702	−1.37380	−0.686902	−	0.726750i	$-0.741030\pi$
$233$	−20.9702	−1.37380	−0.686902	+	0.726750i	$0.741030\pi$
$234$	0	0
$235$	−11.8954	−0.775971
$236$	0	0
$237$	−9.68286	−0.628969
$238$	0	0
$239$	−0.806827	−0.0521893	−0.0260947	−	0.999659i	$-0.508307\pi$
$239$	−0.806827	−0.0521893	−0.0260947	+	0.999659i	$0.508307\pi$
$240$	0	0
$241$	−0.770774	−0.0496499	−0.0248250	−	0.999692i	$-0.507903\pi$
$241$	−0.770774	−0.0496499	−0.0248250	+	0.999692i	$0.507903\pi$
$242$	0	0
$243$	−4.76663	−0.305779
$244$	0	0
$245$	15.5928	0.996189
$246$	0	0
$247$	1.53740	0.0978225
$248$	0	0
$249$	10.0090	0.634292
$250$	0	0
$251$	26.8608	1.69544	0.847720	−	0.530445i	$-0.177975\pi$
$251$	26.8608	1.69544	0.847720	+	0.530445i	$0.177975\pi$
$252$	0	0
$253$	1.13919	0.0716205
$254$	0	0
$255$	34.7819	2.17812
$256$	0	0
$257$	−0.168981	−0.0105408	−0.00527038	−	0.999986i	$-0.501678\pi$
$257$	−0.168981	−0.0105408	−0.00527038	+	0.999986i	$0.501678\pi$
$258$	0	0
$259$	−1.39821	−0.0868804
$260$	0	0
$261$	−0.368420	−0.0228046
$262$	0	0
$263$	−28.4376	−1.75354	−0.876770	−	0.480911i	$-0.840306\pi$
$263$	−28.4376	−1.75354	−0.876770	+	0.480911i	$0.840306\pi$
$264$	0	0
$265$	−0.775591	−0.0476441
$266$	0	0
$267$	−24.9765	−1.52854
$268$	0	0
$269$	−19.5076	−1.18940	−0.594700	−	0.803948i	$-0.702729\pi$
$269$	−19.5076	−1.18940	−0.594700	+	0.803948i	$0.702729\pi$
$270$	0	0
$271$	−19.2832	−1.17137	−0.585686	−	0.810538i	$-0.699175\pi$
$271$	−19.2832	−1.17137	−0.585686	+	0.810538i	$0.699175\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−0.398207	−0.0240128
$276$	0	0
$277$	−21.2549	−1.27708	−0.638541	−	0.769588i	$-0.720462\pi$
$277$	−21.2549	−1.27708	−0.638541	+	0.769588i	$0.720462\pi$
$278$	0	0
$279$	3.87122	0.231764
$280$	0	0
$281$	4.55263	0.271587	0.135794	−	0.990737i	$-0.456642\pi$
$281$	4.55263	0.271587	0.135794	+	0.990737i	$0.456642\pi$
$282$	0	0
$283$	27.9404	1.66089	0.830443	−	0.557104i	$-0.188087\pi$
$283$	27.9404	1.66089	0.830443	+	0.557104i	$0.188087\pi$
$284$	0	0
$285$	6.64681	0.393723
$286$	0	0
$287$	2.03460	0.120099
$288$	0	0
$289$	47.7223	2.80719
$290$	0	0
$291$	8.36842	0.490565
$292$	0	0
$293$	2.75140	0.160738	0.0803692	−	0.996765i	$-0.474390\pi$
$293$	2.75140	0.160738	0.0803692	+	0.996765i	$0.474390\pi$
$294$	0	0
$295$	22.2396	1.29484
$296$	0	0
$297$	4.72161	0.273976
$298$	0	0
$299$	1.13919	0.0658813
$300$	0	0
$301$	2.47301	0.142542
$302$	0	0
$303$	10.3684	0.595650
$304$	0	0
$305$	−6.14961	−0.352125
$306$	0	0
$307$	−10.7964	−0.616184	−0.308092	−	0.951357i	$-0.599690\pi$
$307$	−10.7964	−0.616184	−0.308092	+	0.951357i	$0.599690\pi$
$308$	0	0
$309$	−20.3774	−1.15923
$310$	0	0
$311$	14.7714	0.837612	0.418806	−	0.908076i	$-0.362449\pi$
$311$	14.7714	0.837612	0.418806	+	0.908076i	$0.362449\pi$
$312$	0	0
$313$	26.3282	1.48816	0.744080	−	0.668091i	$-0.232888\pi$
$313$	26.3282	1.48816	0.744080	+	0.668091i	$0.232888\pi$
$314$	0	0
$315$	0.577601	0.0325441
$316$	0	0
$317$	26.6081	1.49446	0.747229	−	0.664567i	$-0.231384\pi$
$317$	26.6081	1.49446	0.747229	+	0.664567i	$0.231384\pi$
$318$	0	0
$319$	0.796415	0.0445907
$320$	0	0
$321$	−6.40717	−0.357613
$322$	0	0
$323$	12.3684	0.688197
$324$	0	0
$325$	−0.398207	−0.0220886
$326$	0	0
$327$	24.2382	1.34037
$328$	0	0
$329$	−2.75140	−0.151690
$330$	0	0
$331$	−6.25756	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$331$	−6.25756	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$332$	0	0
$333$	1.20359	0.0659561
$334$	0	0
$335$	−2.44882	−0.133793
$336$	0	0
$337$	−21.3595	−1.16352	−0.581762	−	0.813359i	$-0.697636\pi$
$337$	−21.3595	−1.16352	−0.581762	+	0.813359i	$0.697636\pi$
$338$	0	0
$339$	28.8012	1.56427
$340$	0	0
$341$	−8.36842	−0.453175
$342$	0	0
$343$	7.36842	0.397857
$344$	0	0
$345$	4.92520	0.265164
$346$	0	0
$347$	5.78745	0.310687	0.155343	−	0.987861i	$-0.450352\pi$
$347$	5.78745	0.310687	0.155343	+	0.987861i	$0.450352\pi$
$348$	0	0
$349$	21.3130	1.14086	0.570429	−	0.821347i	$-0.306777\pi$
$349$	21.3130	1.14086	0.570429	+	0.821347i	$0.306777\pi$
$350$	0	0
$351$	4.72161	0.252021
$352$	0	0
$353$	−9.22441	−0.490966	−0.245483	−	0.969401i	$-0.578947\pi$
$353$	−9.22441	−0.490966	−0.245483	+	0.969401i	$0.578947\pi$
$354$	0	0
$355$	8.19462	0.434925
$356$	0	0
$357$	8.04502	0.425787
$358$	0	0
$359$	28.2653	1.49178	0.745892	−	0.666067i	$-0.232023\pi$
$359$	28.2653	1.49178	0.745892	+	0.666067i	$0.232023\pi$
$360$	0	0
$361$	−16.6364	−0.875600
$362$	0	0
$363$	1.86081	0.0976670
$364$	0	0
$365$	−21.1648	−1.10782
$366$	0	0
$367$	15.2382	0.795427	0.397713	−	0.917510i	$-0.369804\pi$
$367$	15.2382	0.795427	0.397713	+	0.917510i	$0.369804\pi$
$368$	0	0
$369$	−1.75140	−0.0911742
$370$	0	0
$371$	−0.179393	−0.00931365
$372$	0	0
$373$	2.12878	0.110224	0.0551121	−	0.998480i	$-0.482448\pi$
$373$	2.12878	0.110224	0.0551121	+	0.998480i	$0.482448\pi$
$374$	0	0
$375$	19.8954	1.02739
$376$	0	0
$377$	0.796415	0.0410175
$378$	0	0
$379$	−25.6025	−1.31511	−0.657555	−	0.753406i	$-0.728409\pi$
$379$	−25.6025	−1.31511	−0.657555	+	0.753406i	$0.728409\pi$
$380$	0	0
$381$	−39.6233	−2.02996
$382$	0	0
$383$	−14.1142	−0.721203	−0.360602	−	0.932720i	$-0.617429\pi$
$383$	−14.1142	−0.721203	−0.360602	+	0.932720i	$0.617429\pi$
$384$	0	0
$385$	−1.24860	−0.0636346
$386$	0	0
$387$	−2.12878	−0.108212
$388$	0	0
$389$	−3.59698	−0.182374	−0.0911870	−	0.995834i	$-0.529066\pi$
$389$	−3.59698	−0.182374	−0.0911870	+	0.995834i	$0.529066\pi$
$390$	0	0
$391$	9.16484	0.463486
$392$	0	0
$393$	16.4134	0.827948
$394$	0	0
$395$	12.0900	0.608316
$396$	0	0
$397$	25.9404	1.30191	0.650956	−	0.759115i	$-0.274368\pi$
$397$	25.9404	1.30191	0.650956	+	0.759115i	$0.274368\pi$
$398$	0	0
$399$	1.53740	0.0769663
$400$	0	0
$401$	−8.11086	−0.405037	−0.202518	−	0.979278i	$-0.564913\pi$
$401$	−8.11086	−0.405037	−0.202518	+	0.979278i	$0.564913\pi$
$402$	0	0
$403$	−8.36842	−0.416861
$404$	0	0
$405$	23.6378	1.17457
$406$	0	0
$407$	−2.60179	−0.128966
$408$	0	0
$409$	−34.6468	−1.71317	−0.856587	−	0.516002i	$-0.827420\pi$
$409$	−34.6468	−1.71317	−0.856587	+	0.516002i	$0.827420\pi$
$410$	0	0
$411$	26.7368	1.31883
$412$	0	0
$413$	5.14401	0.253120
$414$	0	0
$415$	−12.4972	−0.613464
$416$	0	0
$417$	−20.4522	−1.00155
$418$	0	0
$419$	−4.38780	−0.214358	−0.107179	−	0.994240i	$-0.534182\pi$
$419$	−4.38780	−0.214358	−0.107179	+	0.994240i	$0.534182\pi$
$420$	0	0
$421$	3.09899	0.151036	0.0755179	−	0.997144i	$-0.475939\pi$
$421$	3.09899	0.151036	0.0755179	+	0.997144i	$0.475939\pi$
$422$	0	0
$423$	2.36842	0.115157
$424$	0	0
$425$	−3.20359	−0.155397
$426$	0	0
$427$	−1.42240	−0.0688347
$428$	0	0
$429$	1.86081	0.0898406
$430$	0	0
$431$	−19.1246	−0.921201	−0.460601	−	0.887607i	$-0.652366\pi$
$431$	−19.1246	−0.921201	−0.460601	+	0.887607i	$0.652366\pi$
$432$	0	0
$433$	−17.1544	−0.824389	−0.412194	−	0.911096i	$-0.635238\pi$
$433$	−17.1544	−0.824389	−0.412194	+	0.911096i	$0.635238\pi$
$434$	0	0
$435$	3.44322	0.165090
$436$	0	0
$437$	1.75140	0.0837808
$438$	0	0
$439$	−29.8325	−1.42383	−0.711913	−	0.702268i	$-0.752171\pi$
$439$	−29.8325	−1.42383	−0.711913	+	0.702268i	$0.752171\pi$
$440$	0	0
$441$	−3.10459	−0.147838
$442$	0	0
$443$	7.61220	0.361667	0.180833	−	0.983514i	$-0.442121\pi$
$443$	7.61220	0.361667	0.180833	+	0.983514i	$0.442121\pi$
$444$	0	0
$445$	31.1857	1.47834
$446$	0	0
$447$	−37.7819	−1.78702
$448$	0	0
$449$	−13.8116	−0.651812	−0.325906	−	0.945402i	$-0.605669\pi$
$449$	−13.8116	−0.651812	−0.325906	+	0.945402i	$0.605669\pi$
$450$	0	0
$451$	3.78600	0.178276
$452$	0	0
$453$	1.39821	0.0656935
$454$	0	0
$455$	−1.24860	−0.0585353
$456$	0	0
$457$	13.7979	0.645437	0.322718	−	0.946495i	$-0.395403\pi$
$457$	13.7979	0.645437	0.322718	+	0.946495i	$0.395403\pi$
$458$	0	0
$459$	37.9854	1.77301
$460$	0	0
$461$	−26.4868	−1.23361	−0.616806	−	0.787115i	$-0.711574\pi$
$461$	−26.4868	−1.23361	−0.616806	+	0.787115i	$0.711574\pi$
$462$	0	0
$463$	−9.79082	−0.455018	−0.227509	−	0.973776i	$-0.573058\pi$
$463$	−9.79082	−0.455018	−0.227509	+	0.973776i	$0.573058\pi$
$464$	0	0
$465$	−36.1801	−1.67781
$466$	0	0
$467$	−42.3297	−1.95878	−0.979392	−	0.201970i	$-0.935266\pi$
$467$	−42.3297	−1.95878	−0.979392	+	0.201970i	$0.935266\pi$
$468$	0	0
$469$	−0.566410	−0.0261544
$470$	0	0
$471$	37.0242	1.70598
$472$	0	0
$473$	4.60179	0.211591
$474$	0	0
$475$	−0.612205	−0.0280899
$476$	0	0
$477$	0.154423	0.00707054
$478$	0	0
$479$	−22.0450	−1.00726	−0.503631	−	0.863919i	$-0.668003\pi$
$479$	−22.0450	−1.00726	−0.503631	+	0.863919i	$0.668003\pi$
$480$	0	0
$481$	−2.60179	−0.118632
$482$	0	0
$483$	1.13919	0.0518351
$484$	0	0
$485$	−10.4488	−0.474456
$486$	0	0
$487$	−15.6829	−0.710658	−0.355329	−	0.934741i	$-0.615631\pi$
$487$	−15.6829	−0.710658	−0.355329	+	0.934741i	$0.615631\pi$
$488$	0	0
$489$	23.2936	1.05337
$490$	0	0
$491$	3.46742	0.156482	0.0782411	−	0.996934i	$-0.475070\pi$
$491$	3.46742	0.156482	0.0782411	+	0.996934i	$0.475070\pi$
$492$	0	0
$493$	6.40717	0.288564
$494$	0	0
$495$	1.07480	0.0483088
$496$	0	0
$497$	1.89541	0.0850207
$498$	0	0
$499$	13.6620	0.611597	0.305798	−	0.952096i	$-0.401077\pi$
$499$	13.6620	0.611597	0.305798	+	0.952096i	$0.401077\pi$
$500$	0	0
$501$	−16.7112	−0.746601
$502$	0	0
$503$	−9.23404	−0.411726	−0.205863	−	0.978581i	$-0.566000\pi$
$503$	−9.23404	−0.411726	−0.205863	+	0.978581i	$0.566000\pi$
$504$	0	0
$505$	−12.9460	−0.576090
$506$	0	0
$507$	1.86081	0.0826413
$508$	0	0
$509$	−29.1261	−1.29099	−0.645496	−	0.763764i	$-0.723349\pi$
$509$	−29.1261	−1.29099	−0.645496	+	0.763764i	$0.723349\pi$
$510$	0	0
$511$	−4.89541	−0.216560
$512$	0	0
$513$	7.25901	0.320493
$514$	0	0
$515$	25.4432	1.12116
$516$	0	0
$517$	−5.11982	−0.225169
$518$	0	0
$519$	−19.0152	−0.834676
$520$	0	0
$521$	−2.35734	−0.103277	−0.0516384	−	0.998666i	$-0.516444\pi$
$521$	−2.35734	−0.103277	−0.0516384	+	0.998666i	$0.516444\pi$
$522$	0	0
$523$	44.4376	1.94312	0.971561	−	0.236790i	$-0.0760953\pi$
$523$	44.4376	1.94312	0.971561	+	0.236790i	$0.0760953\pi$
$524$	0	0
$525$	−0.398207	−0.0173792
$526$	0	0
$527$	−67.3241	−2.93268
$528$	0	0
$529$	−21.7022	−0.943576
$530$	0	0
$531$	−4.42799	−0.192159
$532$	0	0
$533$	3.78600	0.163990
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−37.0215	−1.59760
$538$	0	0
$539$	6.71120	0.289072
$540$	0	0
$541$	−28.0290	−1.20506	−0.602531	−	0.798096i	$-0.705841\pi$
$541$	−28.0290	−1.20506	−0.602531	+	0.798096i	$0.705841\pi$
$542$	0	0
$543$	16.5437	0.709957
$544$	0	0
$545$	−30.2638	−1.29636
$546$	0	0
$547$	−39.5783	−1.69224	−0.846122	−	0.532989i	$-0.821069\pi$
$547$	−39.5783	−1.69224	−0.846122	+	0.532989i	$0.821069\pi$
$548$	0	0
$549$	1.22441	0.0522565
$550$	0	0
$551$	1.22441	0.0521616
$552$	0	0
$553$	2.79641	0.118916
$554$	0	0
$555$	−11.2486	−0.477477
$556$	0	0
$557$	35.1246	1.48828	0.744139	−	0.668024i	$-0.232860\pi$
$557$	35.1246	1.48828	0.744139	+	0.668024i	$0.232860\pi$
$558$	0	0
$559$	4.60179	0.194635
$560$	0	0
$561$	14.9702	0.632043
$562$	0	0
$563$	13.3595	0.563034	0.281517	−	0.959556i	$-0.409162\pi$
$563$	13.3595	0.563034	0.281517	+	0.959556i	$0.409162\pi$
$564$	0	0
$565$	−35.9612	−1.51290
$566$	0	0
$567$	5.46742	0.229610
$568$	0	0
$569$	8.15587	0.341912	0.170956	−	0.985279i	$-0.445314\pi$
$569$	8.15587	0.341912	0.170956	+	0.985279i	$0.445314\pi$
$570$	0	0
$571$	−17.3082	−0.724325	−0.362162	−	0.932115i	$-0.617961\pi$
$571$	−17.3082	−0.724325	−0.362162	+	0.932115i	$0.617961\pi$
$572$	0	0
$573$	−10.7499	−0.449086
$574$	0	0
$575$	−0.453636	−0.0189179
$576$	0	0
$577$	−17.3836	−0.723691	−0.361845	−	0.932238i	$-0.617853\pi$
$577$	−17.3836	−0.723691	−0.361845	+	0.932238i	$0.617853\pi$
$578$	0	0
$579$	−9.21255	−0.382860
$580$	0	0
$581$	−2.89059	−0.119922
$582$	0	0
$583$	−0.333816	−0.0138253
$584$	0	0
$585$	1.07480	0.0444376
$586$	0	0
$587$	6.23674	0.257418	0.128709	−	0.991682i	$-0.458917\pi$
$587$	6.23674	0.257418	0.128709	+	0.991682i	$0.458917\pi$
$588$	0	0
$589$	−12.8656	−0.530119
$590$	0	0
$591$	1.74099	0.0716146
$592$	0	0
$593$	−43.2805	−1.77732	−0.888659	−	0.458569i	$-0.848362\pi$
$593$	−43.2805	−1.77732	−0.888659	+	0.458569i	$0.848362\pi$
$594$	0	0
$595$	−10.0450	−0.411806
$596$	0	0
$597$	41.1697	1.68496
$598$	0	0
$599$	6.31299	0.257942	0.128971	−	0.991648i	$-0.458833\pi$
$599$	6.31299	0.257942	0.128971	+	0.991648i	$0.458833\pi$
$600$	0	0
$601$	9.55745	0.389856	0.194928	−	0.980818i	$-0.437553\pi$
$601$	9.55745	0.389856	0.194928	+	0.980818i	$0.437553\pi$
$602$	0	0
$603$	0.487569	0.0198553
$604$	0	0
$605$	−2.32340	−0.0944598
$606$	0	0
$607$	−27.7521	−1.12642	−0.563211	−	0.826313i	$-0.690434\pi$
$607$	−27.7521	−1.12642	−0.563211	+	0.826313i	$0.690434\pi$
$608$	0	0
$609$	0.796415	0.0322724
$610$	0	0
$611$	−5.11982	−0.207126
$612$	0	0
$613$	20.4626	0.826477	0.413238	−	0.910623i	$-0.364398\pi$
$613$	20.4626	0.826477	0.413238	+	0.910623i	$0.364398\pi$
$614$	0	0
$615$	16.3684	0.660038
$616$	0	0
$617$	−32.1496	−1.29429	−0.647147	−	0.762365i	$-0.724038\pi$
$617$	−32.1496	−1.29429	−0.647147	+	0.762365i	$0.724038\pi$
$618$	0	0
$619$	−19.3144	−0.776313	−0.388156	−	0.921593i	$-0.626888\pi$
$619$	−19.3144	−0.776313	−0.388156	+	0.921593i	$0.626888\pi$
$620$	0	0
$621$	5.37883	0.215845
$622$	0	0
$623$	7.21322	0.288991
$624$	0	0
$625$	−26.8325	−1.07330
$626$	0	0
$627$	2.86081	0.114250
$628$	0	0
$629$	−20.9315	−0.834592
$630$	0	0
$631$	−26.0963	−1.03888	−0.519439	−	0.854508i	$-0.673859\pi$
$631$	−26.0963	−1.03888	−0.519439	+	0.854508i	$0.673859\pi$
$632$	0	0
$633$	4.84143	0.192430
$634$	0	0
$635$	49.4737	1.96330
$636$	0	0
$637$	6.71120	0.265907
$638$	0	0
$639$	−1.63158	−0.0645443
$640$	0	0
$641$	−33.1876	−1.31083	−0.655415	−	0.755269i	$-0.727506\pi$
$641$	−33.1876	−1.31083	−0.655415	+	0.755269i	$0.727506\pi$
$642$	0	0
$643$	19.6620	0.775395	0.387698	−	0.921787i	$-0.373271\pi$
$643$	19.6620	0.775395	0.387698	+	0.921787i	$0.373271\pi$
$644$	0	0
$645$	19.8954	0.783381
$646$	0	0
$647$	−18.1455	−0.713372	−0.356686	−	0.934224i	$-0.616093\pi$
$647$	−18.1455	−0.713372	−0.356686	+	0.934224i	$0.616093\pi$
$648$	0	0
$649$	9.57201	0.375734
$650$	0	0
$651$	−8.36842	−0.327984
$652$	0	0
$653$	23.4737	0.918596	0.459298	−	0.888282i	$-0.348101\pi$
$653$	23.4737	0.918596	0.459298	+	0.888282i	$0.348101\pi$
$654$	0	0
$655$	−20.4938	−0.800760
$656$	0	0
$657$	4.21400	0.164404
$658$	0	0
$659$	−28.3684	−1.10508	−0.552538	−	0.833487i	$-0.686341\pi$
$659$	−28.3684	−1.10508	−0.552538	+	0.833487i	$0.686341\pi$
$660$	0	0
$661$	21.6412	0.841746	0.420873	−	0.907120i	$-0.361724\pi$
$661$	21.6412	0.841746	0.420873	+	0.907120i	$0.361724\pi$
$662$	0	0
$663$	14.9702	0.581395
$664$	0	0
$665$	−1.91960	−0.0744389
$666$	0	0
$667$	0.907271	0.0351297
$668$	0	0
$669$	−22.5810	−0.873031
$670$	0	0
$671$	−2.64681	−0.102179
$672$	0	0
$673$	−15.4370	−0.595051	−0.297525	−	0.954714i	$-0.596161\pi$
$673$	−15.4370	−0.595051	−0.297525	+	0.954714i	$0.596161\pi$
$674$	0	0
$675$	−1.88018	−0.0723682
$676$	0	0
$677$	−21.9821	−0.844840	−0.422420	−	0.906400i	$-0.638819\pi$
$677$	−21.9821	−0.844840	−0.422420	+	0.906400i	$0.638819\pi$
$678$	0	0
$679$	−2.41680	−0.0927484
$680$	0	0
$681$	−2.86081	−0.109626
$682$	0	0
$683$	28.0692	1.07404	0.537019	−	0.843570i	$-0.319550\pi$
$683$	28.0692	1.07404	0.537019	+	0.843570i	$0.319550\pi$
$684$	0	0
$685$	−33.3836	−1.27552
$686$	0	0
$687$	9.59910	0.366228
$688$	0	0
$689$	−0.333816	−0.0127174
$690$	0	0
$691$	−16.9944	−0.646498	−0.323249	−	0.946314i	$-0.604775\pi$
$691$	−16.9944	−0.646498	−0.323249	+	0.946314i	$0.604775\pi$
$692$	0	0
$693$	0.248601	0.00944358
$694$	0	0
$695$	25.5366	0.968659
$696$	0	0
$697$	30.4585	1.15370
$698$	0	0
$699$	−39.0215	−1.47593
$700$	0	0
$701$	3.18566	0.120321	0.0601603	−	0.998189i	$-0.480839\pi$
$701$	3.18566	0.120321	0.0601603	+	0.998189i	$0.480839\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	−22.1350	−0.833654
$706$	0	0
$707$	−2.99440	−0.112616
$708$	0	0
$709$	8.53595	0.320574	0.160287	−	0.987070i	$-0.448758\pi$
$709$	8.53595	0.320574	0.160287	+	0.987070i	$0.448758\pi$
$710$	0	0
$711$	−2.40717	−0.0902759
$712$	0	0
$713$	−9.53326	−0.357023
$714$	0	0
$715$	−2.32340	−0.0868904
$716$	0	0
$717$	−1.50135	−0.0560689
$718$	0	0
$719$	7.09563	0.264622	0.132311	−	0.991208i	$-0.457760\pi$
$719$	7.09563	0.264622	0.132311	+	0.991208i	$0.457760\pi$
$720$	0	0
$721$	5.88500	0.219169
$722$	0	0
$723$	−1.43426	−0.0533407
$724$	0	0
$725$	−0.317138	−0.0117782
$726$	0	0
$727$	3.72643	0.138206	0.0691028	−	0.997610i	$-0.477986\pi$
$727$	3.72643	0.138206	0.0691028	+	0.997610i	$0.477986\pi$
$728$	0	0
$729$	21.6516	0.801912
$730$	0	0
$731$	37.0215	1.36929
$732$	0	0
$733$	37.9598	1.40208	0.701039	−	0.713123i	$-0.252720\pi$
$733$	37.9598	1.40208	0.701039	+	0.713123i	$0.252720\pi$
$734$	0	0
$735$	29.0152	1.07024
$736$	0	0
$737$	−1.05398	−0.0388238
$738$	0	0
$739$	4.78041	0.175850	0.0879251	−	0.996127i	$-0.471976\pi$
$739$	4.78041	0.175850	0.0879251	+	0.996127i	$0.471976\pi$
$740$	0	0
$741$	2.86081	0.105094
$742$	0	0
$743$	20.2847	0.744172	0.372086	−	0.928198i	$-0.378643\pi$
$743$	20.2847	0.744172	0.372086	+	0.928198i	$0.378643\pi$
$744$	0	0
$745$	47.1745	1.72834
$746$	0	0
$747$	2.48824	0.0910399
$748$	0	0
$749$	1.85039	0.0676119
$750$	0	0
$751$	32.6427	1.19115	0.595574	−	0.803301i	$-0.296925\pi$
$751$	32.6427	1.19115	0.595574	+	0.803301i	$0.296925\pi$
$752$	0	0
$753$	49.9827	1.82147
$754$	0	0
$755$	−1.74580	−0.0635363
$756$	0	0
$757$	−17.4480	−0.634160	−0.317080	−	0.948399i	$-0.602702\pi$
$757$	−17.4480	−0.634160	−0.317080	+	0.948399i	$0.602702\pi$
$758$	0	0
$759$	2.11982	0.0769446
$760$	0	0
$761$	41.2597	1.49566	0.747831	−	0.663889i	$-0.231095\pi$
$761$	41.2597	1.49566	0.747831	+	0.663889i	$0.231095\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	0	0
$765$	8.64681	0.312626
$766$	0	0
$767$	9.57201	0.345625
$768$	0	0
$769$	26.4924	0.955340	0.477670	−	0.878539i	$-0.341481\pi$
$769$	26.4924	0.955340	0.477670	+	0.878539i	$0.341481\pi$
$770$	0	0
$771$	−0.314441	−0.0113243
$772$	0	0
$773$	−38.3747	−1.38024	−0.690121	−	0.723694i	$-0.742443\pi$
$773$	−38.3747	−1.38024	−0.690121	+	0.723694i	$0.742443\pi$
$774$	0	0
$775$	3.33237	0.119702
$776$	0	0
$777$	−2.60179	−0.0933388
$778$	0	0
$779$	5.82061	0.208545
$780$	0	0
$781$	3.52699	0.126206
$782$	0	0
$783$	3.76036	0.134384
$784$	0	0
$785$	−46.2284	−1.64996
$786$	0	0
$787$	18.7625	0.668810	0.334405	−	0.942429i	$-0.391465\pi$
$787$	18.7625	0.668810	0.334405	+	0.942429i	$0.391465\pi$
$788$	0	0
$789$	−52.9169	−1.88389
$790$	0	0
$791$	−8.31781	−0.295747
$792$	0	0
$793$	−2.64681	−0.0939909
$794$	0	0
$795$	−1.44322	−0.0511859
$796$	0	0
$797$	53.7133	1.90262	0.951312	−	0.308231i	$-0.0997368\pi$
$797$	53.7133	1.90262	0.951312	+	0.308231i	$0.0997368\pi$
$798$	0	0
$799$	−41.1890	−1.45716
$800$	0	0
$801$	−6.20918	−0.219391
$802$	0	0
$803$	−9.10941	−0.321464
$804$	0	0
$805$	−1.42240	−0.0501330
$806$	0	0
$807$	−36.2999	−1.27782
$808$	0	0
$809$	−33.6170	−1.18191	−0.590956	−	0.806704i	$-0.701249\pi$
$809$	−33.6170	−1.18191	−0.590956	+	0.806704i	$0.701249\pi$
$810$	0	0
$811$	−32.6516	−1.14655	−0.573277	−	0.819362i	$-0.694328\pi$
$811$	−32.6516	−1.14655	−0.573277	+	0.819362i	$0.694328\pi$
$812$	0	0
$813$	−35.8823	−1.25845
$814$	0	0
$815$	−29.0844	−1.01878
$816$	0	0
$817$	7.07480	0.247516
$818$	0	0
$819$	0.248601	0.00868683
$820$	0	0
$821$	48.9619	1.70878	0.854391	−	0.519630i	$-0.173930\pi$
$821$	48.9619	1.70878	0.854391	+	0.519630i	$0.173930\pi$
$822$	0	0
$823$	28.7112	1.00081	0.500405	−	0.865792i	$-0.333185\pi$
$823$	28.7112	1.00081	0.500405	+	0.865792i	$0.333185\pi$
$824$	0	0
$825$	−0.740987	−0.0257978
$826$	0	0
$827$	−40.2839	−1.40081	−0.700404	−	0.713747i	$-0.746997\pi$
$827$	−40.2839	−1.40081	−0.700404	+	0.713747i	$0.746997\pi$
$828$	0	0
$829$	50.4424	1.75194	0.875969	−	0.482367i	$-0.160223\pi$
$829$	50.4424	1.75194	0.875969	+	0.482367i	$0.160223\pi$
$830$	0	0
$831$	−39.5512	−1.37202
$832$	0	0
$833$	53.9917	1.87070
$834$	0	0
$835$	20.8656	0.722085
$836$	0	0
$837$	−39.5124	−1.36575
$838$	0	0
$839$	−26.2701	−0.906944	−0.453472	−	0.891270i	$-0.649815\pi$
$839$	−26.2701	−0.906944	−0.453472	+	0.891270i	$0.649815\pi$
$840$	0	0
$841$	−28.3657	−0.978128
$842$	0	0
$843$	8.47156	0.291776
$844$	0	0
$845$	−2.32340	−0.0799275
$846$	0	0
$847$	−0.537402	−0.0184653
$848$	0	0
$849$	51.9917	1.78435
$850$	0	0
$851$	−2.96395	−0.101603
$852$	0	0
$853$	−6.02005	−0.206122	−0.103061	−	0.994675i	$-0.532864\pi$
$853$	−6.02005	−0.206122	−0.103061	+	0.994675i	$0.532864\pi$
$854$	0	0
$855$	1.65240	0.0565110
$856$	0	0
$857$	6.29921	0.215177	0.107589	−	0.994196i	$-0.465687\pi$
$857$	6.29921	0.215177	0.107589	+	0.994196i	$0.465687\pi$
$858$	0	0
$859$	8.37257	0.285668	0.142834	−	0.989747i	$-0.454378\pi$
$859$	8.37257	0.285668	0.142834	+	0.989747i	$0.454378\pi$
$860$	0	0
$861$	3.78600	0.129027
$862$	0	0
$863$	−26.0063	−0.885264	−0.442632	−	0.896703i	$-0.645955\pi$
$863$	−26.0063	−0.885264	−0.442632	+	0.896703i	$0.645955\pi$
$864$	0	0
$865$	23.7424	0.807267
$866$	0	0
$867$	88.8019	3.01587
$868$	0	0
$869$	5.20359	0.176520
$870$	0	0
$871$	−1.05398	−0.0357127
$872$	0	0
$873$	2.08040	0.0704108
$874$	0	0
$875$	−5.74580	−0.194244
$876$	0	0
$877$	39.7029	1.34067	0.670336	−	0.742058i	$-0.266150\pi$
$877$	39.7029	1.34067	0.670336	+	0.742058i	$0.266150\pi$
$878$	0	0
$879$	5.11982	0.172687
$880$	0	0
$881$	−16.2501	−0.547478	−0.273739	−	0.961804i	$-0.588260\pi$
$881$	−16.2501	−0.547478	−0.273739	+	0.961804i	$0.588260\pi$
$882$	0	0
$883$	−22.3968	−0.753711	−0.376855	−	0.926272i	$-0.622995\pi$
$883$	−22.3968	−0.753711	−0.376855	+	0.926272i	$0.622995\pi$
$884$	0	0
$885$	41.3836	1.39110
$886$	0	0
$887$	−7.70369	−0.258664	−0.129332	−	0.991601i	$-0.541283\pi$
$887$	−7.70369	−0.258664	−0.129332	+	0.991601i	$0.541283\pi$
$888$	0	0
$889$	11.4432	0.383793
$890$	0	0
$891$	10.1738	0.340835
$892$	0	0
$893$	−7.87122	−0.263400
$894$	0	0
$895$	46.2251	1.54513
$896$	0	0
$897$	2.11982	0.0707787
$898$	0	0
$899$	−6.66473	−0.222281
$900$	0	0
$901$	−2.68556	−0.0894689
$902$	0	0
$903$	4.60179	0.153138
$904$	0	0
$905$	−20.6564	−0.686643
$906$	0	0
$907$	−28.7354	−0.954143	−0.477072	−	0.878864i	$-0.658302\pi$
$907$	−28.7354	−0.954143	−0.477072	+	0.878864i	$0.658302\pi$
$908$	0	0
$909$	2.57760	0.0854936
$910$	0	0
$911$	−33.2292	−1.10093	−0.550467	−	0.834857i	$-0.685550\pi$
$911$	−33.2292	−1.10093	−0.550467	+	0.834857i	$0.685550\pi$
$912$	0	0
$913$	−5.37883	−0.178013
$914$	0	0
$915$	−11.4432	−0.378301
$916$	0	0
$917$	−4.74021	−0.156535
$918$	0	0
$919$	44.5097	1.46824	0.734120	−	0.679019i	$-0.237595\pi$
$919$	44.5097	1.46824	0.734120	+	0.679019i	$0.237595\pi$
$920$	0	0
$921$	−20.0900	−0.661989
$922$	0	0
$923$	3.52699	0.116092
$924$	0	0
$925$	1.03605	0.0340652
$926$	0	0
$927$	−5.06584	−0.166384
$928$	0	0
$929$	−34.4889	−1.13154	−0.565772	−	0.824562i	$-0.691422\pi$
$929$	−34.4889	−1.13154	−0.565772	+	0.824562i	$0.691422\pi$
$930$	0	0
$931$	10.3178	0.338153
$932$	0	0
$933$	27.4868	0.899877
$934$	0	0
$935$	−18.6918	−0.611288
$936$	0	0
$937$	−22.4280	−0.732691	−0.366345	−	0.930479i	$-0.619391\pi$
$937$	−22.4280	−0.732691	−0.366345	+	0.930479i	$0.619391\pi$
$938$	0	0
$939$	48.9917	1.59878
$940$	0	0
$941$	44.0713	1.43668	0.718342	−	0.695690i	$-0.244901\pi$
$941$	44.0713	1.43668	0.718342	+	0.695690i	$0.244901\pi$
$942$	0	0
$943$	4.31299	0.140450
$944$	0	0
$945$	−5.89541	−0.191778
$946$	0	0
$947$	−42.1384	−1.36931	−0.684657	−	0.728865i	$-0.740048\pi$
$947$	−42.1384	−1.36931	−0.684657	+	0.728865i	$0.740048\pi$
$948$	0	0
$949$	−9.10941	−0.295704
$950$	0	0
$951$	49.5124	1.60555
$952$	0	0
$953$	−25.9162	−0.839509	−0.419755	−	0.907638i	$-0.637884\pi$
$953$	−25.9162	−0.839509	−0.419755	+	0.907638i	$0.637884\pi$
$954$	0	0
$955$	13.4224	0.434339
$956$	0	0
$957$	1.48197	0.0479054
$958$	0	0
$959$	−7.72161	−0.249344
$960$	0	0
$961$	39.0305	1.25905
$962$	0	0
$963$	−1.59283	−0.0513282
$964$	0	0
$965$	11.5028	0.370288
$966$	0	0
$967$	−57.7446	−1.85694	−0.928470	−	0.371408i	$-0.878875\pi$
$967$	−57.7446	−1.85694	−0.928470	+	0.371408i	$0.878875\pi$
$968$	0	0
$969$	23.0152	0.739356
$970$	0	0
$971$	−13.4357	−0.431172	−0.215586	−	0.976485i	$-0.569166\pi$
$971$	−13.4357	−0.431172	−0.215586	+	0.976485i	$0.569166\pi$
$972$	0	0
$973$	5.90660	0.189357
$974$	0	0
$975$	−0.740987	−0.0237306
$976$	0	0
$977$	21.2936	0.681243	0.340622	−	0.940200i	$-0.389363\pi$
$977$	21.2936	0.681243	0.340622	+	0.940200i	$0.389363\pi$
$978$	0	0
$979$	13.4224	0.428982
$980$	0	0
$981$	6.02564	0.192384
$982$	0	0
$983$	16.5810	0.528851	0.264425	−	0.964406i	$-0.414818\pi$
$983$	16.5810	0.528851	0.264425	+	0.964406i	$0.414818\pi$
$984$	0	0
$985$	−2.17380	−0.0692630
$986$	0	0
$987$	−5.11982	−0.162966
$988$	0	0
$989$	5.24234	0.166697
$990$	0	0
$991$	9.10941	0.289370	0.144685	−	0.989478i	$-0.453783\pi$
$991$	9.10941	0.289370	0.144685	+	0.989478i	$0.453783\pi$
$992$	0	0
$993$	−11.6441	−0.369515
$994$	0	0
$995$	−51.4045	−1.62963
$996$	0	0
$997$	1.62329	0.0514100	0.0257050	−	0.999670i	$-0.491817\pi$
$997$	1.62329	0.0514100	0.0257050	+	0.999670i	$0.491817\pi$
$998$	0	0
$999$	−12.2847	−0.388669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1144.2.a.i.1.3	✓	3
4.3	odd	2		2288.2.a.v.1.1		3
8.3	odd	2		9152.2.a.by.1.3		3
8.5	even	2		9152.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1144.2.a.i.1.3	✓	3	1.1	even	1	trivial
2288.2.a.v.1.1		3	4.3	odd	2
9152.2.a.bx.1.1		3	8.5	even	2
9152.2.a.by.1.3		3	8.3	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:	\( N \)	\(=\)	\( 1144 = 2^{3} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1144.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86081 q^{3} -2.32340 q^{5} -0.537402 q^{7} +0.462598 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.32340 q^{15} -8.04502 q^{17} -1.53740 q^{19} -1.00000 q^{21} -1.13919 q^{23} +0.398207 q^{25} -4.72161 q^{27} -0.796415 q^{29} +8.36842 q^{31} -1.86081 q^{33} +1.24860 q^{35} +2.60179 q^{37} -1.86081 q^{39} -3.78600 q^{41} -4.60179 q^{43} -1.07480 q^{45} +5.11982 q^{47} -6.71120 q^{49} -14.9702 q^{51} +0.333816 q^{53} +2.32340 q^{55} -2.86081 q^{57} -9.57201 q^{59} +2.64681 q^{61} -0.248601 q^{63} +2.32340 q^{65} +1.05398 q^{67} -2.11982 q^{69} -3.52699 q^{71} +9.10941 q^{73} +0.740987 q^{75} +0.537402 q^{77} -5.20359 q^{79} -10.1738 q^{81} +5.37883 q^{83} +18.6918 q^{85} -1.48197 q^{87} -13.4224 q^{89} +0.537402 q^{91} +15.5720 q^{93} +3.57201 q^{95} +4.49720 q^{97} -0.462598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} - 4 q^{7} - q^{9} - 3 q^{11} - 3 q^{13} - 5 q^{15} - 5 q^{17} - 7 q^{19} - 3 q^{21} - 9 q^{23} - 2 q^{25} - 3 q^{27} + 4 q^{29} - 2 q^{31} - 9 q^{35} + 11 q^{37} - q^{41} - 17 q^{43} - 8 q^{45}+ \cdots + q^{99}+O(q^{100}) \) 3 * q + q^5 - 4 * q^7 - q^9 - 3 * q^11 - 3 * q^13 - 5 * q^15 - 5 * q^17 - 7 * q^19 - 3 * q^21 - 9 * q^23 - 2 * q^25 - 3 * q^27 + 4 * q^29 - 2 * q^31 - 9 * q^35 + 11 * q^37 - q^41 - 17 * q^43 - 8 * q^45 + q^47 - 5 * q^49 - 21 * q^51 - 3 * q^53 - q^55 - 3 * q^57 - 8 * q^59 - 8 * q^61 + 12 * q^63 - q^65 + 8 * q^69 - 9 * q^71 + 9 * q^73 + 11 * q^75 + 4 * q^77 - 22 * q^79 - 13 * q^81 - 7 * q^83 + 21 * q^85 - 22 * q^87 - 10 * q^89 + 4 * q^91 + 26 * q^93 - 10 * q^95 - 12 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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