LMFDB - Embedded newform 1100.4.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,4,Mod(1,1100)] 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("1100.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1100.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.9021010063$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	1100.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.35890 q^{3} -8.71780 q^{7} -8.00000 q^{9} -11.0000 q^{11} +69.7424 q^{13} -26.1534 q^{17} -68.0000 q^{19} -38.0000 q^{21} +117.690 q^{23} -152.561 q^{27} +260.000 q^{29} +175.000 q^{31} -47.9479 q^{33} -169.997 q^{37} +304.000 q^{39} -380.000 q^{41} +305.123 q^{43} +305.123 q^{47} -267.000 q^{49} -114.000 q^{51} +453.325 q^{53} -296.405 q^{57} -143.000 q^{59} +676.000 q^{61} +69.7424 q^{63} +527.427 q^{67} +513.000 q^{69} +1035.00 q^{71} +331.276 q^{73} +95.8958 q^{77} +218.000 q^{79} -449.000 q^{81} -758.448 q^{83} +1133.31 q^{87} +1279.00 q^{89} -608.000 q^{91} +762.807 q^{93} +771.525 q^{97} +88.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{9} - 22 q^{11} - 136 q^{19} - 76 q^{21} + 520 q^{29} + 350 q^{31} + 608 q^{39} - 760 q^{41} - 534 q^{49} - 228 q^{51} - 286 q^{59} + 1352 q^{61} + 1026 q^{69} + 2070 q^{71} + 436 q^{79} - 898 q^{81}+ \cdots + 176 q^{99}+O(q^{100}) $ 2 * q - 16 * q^9 - 22 * q^11 - 136 * q^19 - 76 * q^21 + 520 * q^29 + 350 * q^31 + 608 * q^39 - 760 * q^41 - 534 * q^49 - 228 * q^51 - 286 * q^59 + 1352 * q^61 + 1026 * q^69 + 2070 * q^71 + 436 * q^79 - 898 * q^81 + 2558 * q^89 - 1216 * q^91 + 176 * 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
$3$	4.35890	0.838870	0.419435	−	0.907785i	$-0.362228\pi$
$3$	4.35890	0.838870	0.419435	+	0.907785i	$0.362228\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.71780	−0.470717	−0.235358	−	0.971909i	$-0.575626\pi$
$7$	−8.71780	−0.470717	−0.235358	+	0.971909i	$0.575626\pi$
$8$	0	0
$9$	−8.00000	−0.296296
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	69.7424	1.48793	0.743964	−	0.668220i	$-0.232944\pi$
$13$	69.7424	1.48793	0.743964	+	0.668220i	$0.232944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.1534	−0.373125	−0.186563	−	0.982443i	$-0.559735\pi$
$17$	−26.1534	−0.373125	−0.186563	+	0.982443i	$0.559735\pi$
$18$	0	0
$19$	−68.0000	−0.821067	−0.410533	−	0.911846i	$-0.634657\pi$
$19$	−68.0000	−0.821067	−0.410533	+	0.911846i	$0.634657\pi$
$20$	0	0
$21$	−38.0000	−0.394870
$22$	0	0
$23$	117.690	1.06696	0.533481	−	0.845812i	$-0.320884\pi$
$23$	117.690	1.06696	0.533481	+	0.845812i	$0.320884\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−152.561	−1.08742
$28$	0	0
$29$	260.000	1.66485	0.832427	−	0.554134i	$-0.186951\pi$
$29$	260.000	1.66485	0.832427	+	0.554134i	$0.186951\pi$
$30$	0	0
$31$	175.000	1.01390	0.506950	−	0.861975i	$-0.330773\pi$
$31$	175.000	1.01390	0.506950	+	0.861975i	$0.330773\pi$
$32$	0	0
$33$	−47.9479	−0.252929
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−169.997	−0.755334	−0.377667	−	0.925942i	$-0.623274\pi$
$37$	−169.997	−0.755334	−0.377667	+	0.925942i	$0.623274\pi$
$38$	0	0
$39$	304.000	1.24818
$40$	0	0
$41$	−380.000	−1.44746	−0.723732	−	0.690081i	$-0.757575\pi$
$41$	−380.000	−1.44746	−0.723732	+	0.690081i	$0.757575\pi$
$42$	0	0
$43$	305.123	1.08211	0.541056	−	0.840987i	$-0.318025\pi$
$43$	305.123	1.08211	0.541056	+	0.840987i	$0.318025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	305.123	0.946952	0.473476	−	0.880807i	$-0.342999\pi$
$47$	305.123	0.946952	0.473476	+	0.880807i	$0.342999\pi$
$48$	0	0
$49$	−267.000	−0.778426
$50$	0	0
$51$	−114.000	−0.313004
$52$	0	0
$53$	453.325	1.17489	0.587444	−	0.809265i	$-0.300134\pi$
$53$	453.325	1.17489	0.587444	+	0.809265i	$0.300134\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−296.405	−0.688769
$58$	0	0
$59$	−143.000	−0.315543	−0.157771	−	0.987476i	$-0.550431\pi$
$59$	−143.000	−0.315543	−0.157771	+	0.987476i	$0.550431\pi$
$60$	0	0
$61$	676.000	1.41890	0.709450	−	0.704756i	$-0.248943\pi$
$61$	676.000	1.41890	0.709450	+	0.704756i	$0.248943\pi$
$62$	0	0
$63$	69.7424	0.139472
$64$	0	0
$65$	0	0
$66$	0	0
$67$	527.427	0.961723	0.480861	−	0.876797i	$-0.340324\pi$
$67$	527.427	0.961723	0.480861	+	0.876797i	$0.340324\pi$
$68$	0	0
$69$	513.000	0.895043
$70$	0	0
$71$	1035.00	1.73003	0.865013	−	0.501749i	$-0.167310\pi$
$71$	1035.00	1.73003	0.865013	+	0.501749i	$0.167310\pi$
$72$	0	0
$73$	331.276	0.531136	0.265568	−	0.964092i	$-0.414440\pi$
$73$	331.276	0.531136	0.265568	+	0.964092i	$0.414440\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	95.8958	0.141926
$78$	0	0
$79$	218.000	0.310467	0.155234	−	0.987878i	$-0.450387\pi$
$79$	218.000	0.310467	0.155234	+	0.987878i	$0.450387\pi$
$80$	0	0
$81$	−449.000	−0.615912
$82$	0	0
$83$	−758.448	−1.00302	−0.501509	−	0.865152i	$-0.667222\pi$
$83$	−758.448	−1.00302	−0.501509	+	0.865152i	$0.667222\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1133.31	1.39660
$88$	0	0
$89$	1279.00	1.52330	0.761650	−	0.647988i	$-0.224390\pi$
$89$	1279.00	1.52330	0.761650	+	0.647988i	$0.224390\pi$
$90$	0	0
$91$	−608.000	−0.700393
$92$	0	0
$93$	762.807	0.850532
$94$	0	0
$95$	0	0
$96$	0	0
$97$	771.525	0.807593	0.403796	−	0.914849i	$-0.367690\pi$
$97$	771.525	0.807593	0.403796	+	0.914849i	$0.367690\pi$
$98$	0	0
$99$	88.0000	0.0893367
$100$	0	0
$101$	638.000	0.628548	0.314274	−	0.949332i	$-0.398239\pi$
$101$	638.000	0.628548	0.314274	+	0.949332i	$0.398239\pi$
$102$	0	0
$103$	−531.786	−0.508722	−0.254361	−	0.967109i	$-0.581865\pi$
$103$	−531.786	−0.508722	−0.254361	+	0.967109i	$0.581865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	61.0246	0.0551352	0.0275676	−	0.999620i	$-0.491224\pi$
$107$	61.0246	0.0551352	0.0275676	+	0.999620i	$0.491224\pi$
$108$	0	0
$109$	−142.000	−0.124781	−0.0623905	−	0.998052i	$-0.519872\pi$
$109$	−142.000	−0.124781	−0.0623905	+	0.998052i	$0.519872\pi$
$110$	0	0
$111$	−741.000	−0.633627
$112$	0	0
$113$	−2332.01	−1.94139	−0.970695	−	0.240314i	$-0.922750\pi$
$113$	−2332.01	−1.94139	−0.970695	+	0.240314i	$0.922750\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−557.939	−0.440867
$118$	0	0
$119$	228.000	0.175636
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1656.38	−1.21423
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−976.393	−0.682212	−0.341106	−	0.940025i	$-0.610801\pi$
$127$	−976.393	−0.682212	−0.341106	+	0.940025i	$0.610801\pi$
$128$	0	0
$129$	1330.00	0.907752
$130$	0	0
$131$	774.000	0.516219	0.258110	−	0.966116i	$-0.416900\pi$
$131$	774.000	0.516219	0.258110	+	0.966116i	$0.416900\pi$
$132$	0	0
$133$	592.810	0.386490
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−39.2301	−0.0244646	−0.0122323	−	0.999925i	$-0.503894\pi$
$137$	−39.2301	−0.0244646	−0.0122323	+	0.999925i	$0.503894\pi$
$138$	0	0
$139$	2986.00	1.82208	0.911040	−	0.412317i	$-0.135280\pi$
$139$	2986.00	1.82208	0.911040	+	0.412317i	$0.135280\pi$
$140$	0	0
$141$	1330.00	0.794370
$142$	0	0
$143$	−767.166	−0.448627
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1163.83	−0.652998
$148$	0	0
$149$	−1546.00	−0.850022	−0.425011	−	0.905188i	$-0.639730\pi$
$149$	−1546.00	−0.850022	−0.425011	+	0.905188i	$0.639730\pi$
$150$	0	0
$151$	3150.00	1.69764	0.848819	−	0.528683i	$-0.177314\pi$
$151$	3150.00	1.69764	0.848819	+	0.528683i	$0.177314\pi$
$152$	0	0
$153$	209.227	0.110556
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−501.273	−0.254815	−0.127408	−	0.991850i	$-0.540666\pi$
$157$	−501.273	−0.254815	−0.127408	+	0.991850i	$0.540666\pi$
$158$	0	0
$159$	1976.00	0.985579
$160$	0	0
$161$	−1026.00	−0.502237
$162$	0	0
$163$	−932.804	−0.448239	−0.224119	−	0.974562i	$-0.571951\pi$
$163$	−932.804	−0.448239	−0.224119	+	0.974562i	$0.571951\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1952.79	−0.904857	−0.452429	−	0.891801i	$-0.649442\pi$
$167$	−1952.79	−0.904857	−0.452429	+	0.891801i	$0.649442\pi$
$168$	0	0
$169$	2667.00	1.21393
$170$	0	0
$171$	544.000	0.243279
$172$	0	0
$173$	2345.09	1.03060	0.515300	−	0.857010i	$-0.327681\pi$
$173$	2345.09	1.03060	0.515300	+	0.857010i	$0.327681\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−623.323	−0.264699
$178$	0	0
$179$	699.000	0.291875	0.145938	−	0.989294i	$-0.453380\pi$
$179$	699.000	0.291875	0.145938	+	0.989294i	$0.453380\pi$
$180$	0	0
$181$	−2603.00	−1.06895	−0.534474	−	0.845185i	$-0.679490\pi$
$181$	−2603.00	−1.06895	−0.534474	+	0.845185i	$0.679490\pi$
$182$	0	0
$183$	2946.62	1.19027
$184$	0	0
$185$	0	0
$186$	0	0
$187$	287.687	0.112502
$188$	0	0
$189$	1330.00	0.511869
$190$	0	0
$191$	1329.00	0.503472	0.251736	−	0.967796i	$-0.418999\pi$
$191$	1329.00	0.503472	0.251736	+	0.967796i	$0.418999\pi$
$192$	0	0
$193$	1394.85	0.520225	0.260112	−	0.965578i	$-0.416240\pi$
$193$	1394.85	0.520225	0.260112	+	0.965578i	$0.416240\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2327.65	0.841819	0.420909	−	0.907103i	$-0.361711\pi$
$197$	2327.65	0.841819	0.420909	+	0.907103i	$0.361711\pi$
$198$	0	0
$199$	−8.00000	−0.00284977	−0.00142489	−	0.999999i	$-0.500454\pi$
$199$	−8.00000	−0.00284977	−0.00142489	+	0.999999i	$0.500454\pi$
$200$	0	0
$201$	2299.00	0.806761
$202$	0	0
$203$	−2266.63	−0.783675
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−941.522	−0.316137
$208$	0	0
$209$	748.000	0.247561
$210$	0	0
$211$	−2840.00	−0.926605	−0.463303	−	0.886200i	$-0.653336\pi$
$211$	−2840.00	−0.926605	−0.463303	+	0.886200i	$0.653336\pi$
$212$	0	0
$213$	4511.46	1.45127
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1525.61	−0.477260
$218$	0	0
$219$	1444.00	0.445555
$220$	0	0
$221$	−1824.00	−0.555183
$222$	0	0
$223$	−4154.03	−1.24742	−0.623710	−	0.781656i	$-0.714375\pi$
$223$	−4154.03	−1.24742	−0.623710	+	0.781656i	$0.714375\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2467.14	−0.721364	−0.360682	−	0.932689i	$-0.617456\pi$
$227$	−2467.14	−0.721364	−0.360682	+	0.932689i	$0.617456\pi$
$228$	0	0
$229$	5813.00	1.67744	0.838720	−	0.544563i	$-0.183304\pi$
$229$	5813.00	1.67744	0.838720	+	0.544563i	$0.183304\pi$
$230$	0	0
$231$	418.000	0.119058
$232$	0	0
$233$	−2022.53	−0.568671	−0.284335	−	0.958725i	$-0.591773\pi$
$233$	−2022.53	−0.568671	−0.284335	+	0.958725i	$0.591773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	950.240	0.260442
$238$	0	0
$239$	−246.000	−0.0665792	−0.0332896	−	0.999446i	$-0.510598\pi$
$239$	−246.000	−0.0665792	−0.0332896	+	0.999446i	$0.510598\pi$
$240$	0	0
$241$	3388.00	0.905561	0.452781	−	0.891622i	$-0.350432\pi$
$241$	3388.00	0.905561	0.452781	+	0.891622i	$0.350432\pi$
$242$	0	0
$243$	2162.01	0.570754
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4742.48	−1.22169
$248$	0	0
$249$	−3306.00	−0.841403
$250$	0	0
$251$	1091.00	0.274356	0.137178	−	0.990546i	$-0.456197\pi$
$251$	1091.00	0.274356	0.137178	+	0.990546i	$0.456197\pi$
$252$	0	0
$253$	−1294.59	−0.321701
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4132.24	1.00296	0.501482	−	0.865168i	$-0.332788\pi$
$257$	4132.24	1.00296	0.501482	+	0.865168i	$0.332788\pi$
$258$	0	0
$259$	1482.00	0.355548
$260$	0	0
$261$	−2080.00	−0.493290
$262$	0	0
$263$	−6982.96	−1.63721	−0.818607	−	0.574353i	$-0.805254\pi$
$263$	−6982.96	−1.63721	−0.818607	+	0.574353i	$0.805254\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5575.03	1.27785
$268$	0	0
$269$	−314.000	−0.0711707	−0.0355853	−	0.999367i	$-0.511330\pi$
$269$	−314.000	−0.0711707	−0.0355853	+	0.999367i	$0.511330\pi$
$270$	0	0
$271$	3180.00	0.712809	0.356405	−	0.934332i	$-0.384003\pi$
$271$	3180.00	0.712809	0.356405	+	0.934332i	$0.384003\pi$
$272$	0	0
$273$	−2650.21	−0.587539
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3879.42	−0.841487	−0.420743	−	0.907180i	$-0.638231\pi$
$277$	−3879.42	−0.841487	−0.420743	+	0.907180i	$0.638231\pi$
$278$	0	0
$279$	−1400.00	−0.300415
$280$	0	0
$281$	−4218.00	−0.895462	−0.447731	−	0.894168i	$-0.647768\pi$
$281$	−4218.00	−0.895462	−0.447731	+	0.894168i	$0.647768\pi$
$282$	0	0
$283$	8351.65	1.75425	0.877127	−	0.480258i	$-0.159457\pi$
$283$	8351.65	1.75425	0.877127	+	0.480258i	$0.159457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3312.76	0.681346
$288$	0	0
$289$	−4229.00	−0.860778
$290$	0	0
$291$	3363.00	0.677466
$292$	0	0
$293$	6363.99	1.26890	0.634451	−	0.772963i	$-0.281226\pi$
$293$	6363.99	1.26890	0.634451	+	0.772963i	$0.281226\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1678.18	0.327871
$298$	0	0
$299$	8208.00	1.58756
$300$	0	0
$301$	−2660.00	−0.509368
$302$	0	0
$303$	2780.98	0.527271
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6608.09	1.22848	0.614240	−	0.789119i	$-0.289462\pi$
$307$	6608.09	1.22848	0.614240	+	0.789119i	$0.289462\pi$
$308$	0	0
$309$	−2318.00	−0.426752
$310$	0	0
$311$	5812.00	1.05971	0.529853	−	0.848090i	$-0.322247\pi$
$311$	5812.00	1.05971	0.529853	+	0.848090i	$0.322247\pi$
$312$	0	0
$313$	−5888.87	−1.06345	−0.531723	−	0.846918i	$-0.678455\pi$
$313$	−5888.87	−1.06345	−0.531723	+	0.846918i	$0.678455\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8966.26	1.58863	0.794314	−	0.607507i	$-0.207831\pi$
$317$	8966.26	1.58863	0.794314	+	0.607507i	$0.207831\pi$
$318$	0	0
$319$	−2860.00	−0.501973
$320$	0	0
$321$	266.000	0.0462513
$322$	0	0
$323$	1778.43	0.306361
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−618.964	−0.104675
$328$	0	0
$329$	−2660.00	−0.445746
$330$	0	0
$331$	8683.00	1.44188	0.720938	−	0.693000i	$-0.243711\pi$
$331$	8683.00	1.44188	0.720938	+	0.693000i	$0.243711\pi$
$332$	0	0
$333$	1359.98	0.223803
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5152.22	−0.832817	−0.416408	−	0.909178i	$-0.636711\pi$
$337$	−5152.22	−0.832817	−0.416408	+	0.909178i	$0.636711\pi$
$338$	0	0
$339$	−10165.0	−1.62858
$340$	0	0
$341$	−1925.00	−0.305703
$342$	0	0
$343$	5317.86	0.837135
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−52.3068	−0.00809215	−0.00404607	−	0.999992i	$-0.501288\pi$
$347$	−52.3068	−0.00809215	−0.00404607	+	0.999992i	$0.501288\pi$
$348$	0	0
$349$	2126.00	0.326081	0.163040	−	0.986619i	$-0.447870\pi$
$349$	2126.00	0.326081	0.163040	+	0.986619i	$0.447870\pi$
$350$	0	0
$351$	−10640.0	−1.61801
$352$	0	0
$353$	−7588.84	−1.14423	−0.572115	−	0.820173i	$-0.693877\pi$
$353$	−7588.84	−1.14423	−0.572115	+	0.820173i	$0.693877\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	993.829	0.147336
$358$	0	0
$359$	4156.00	0.610990	0.305495	−	0.952194i	$-0.401178\pi$
$359$	4156.00	0.610990	0.305495	+	0.952194i	$0.401178\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	527.427	0.0762610
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13299.0	−1.89156	−0.945780	−	0.324809i	$-0.894700\pi$
$367$	−13299.0	−1.89156	−0.945780	+	0.324809i	$0.894700\pi$
$368$	0	0
$369$	3040.00	0.428878
$370$	0	0
$371$	−3952.00	−0.553039
$372$	0	0
$373$	−5622.98	−0.780555	−0.390277	−	0.920697i	$-0.627621\pi$
$373$	−5622.98	−0.780555	−0.390277	+	0.920697i	$0.627621\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18133.0	2.47718
$378$	0	0
$379$	631.000	0.0855206	0.0427603	−	0.999085i	$-0.486385\pi$
$379$	631.000	0.0855206	0.0427603	+	0.999085i	$0.486385\pi$
$380$	0	0
$381$	−4256.00	−0.572287
$382$	0	0
$383$	−7091.93	−0.946164	−0.473082	−	0.881019i	$-0.656858\pi$
$383$	−7091.93	−0.946164	−0.473082	+	0.881019i	$0.656858\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2440.98	−0.320626
$388$	0	0
$389$	−5613.00	−0.731595	−0.365797	−	0.930694i	$-0.619204\pi$
$389$	−5613.00	−0.731595	−0.365797	+	0.930694i	$0.619204\pi$
$390$	0	0
$391$	−3078.00	−0.398110
$392$	0	0
$393$	3373.79	0.433041
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14018.2	−1.77218	−0.886088	−	0.463516i	$-0.846588\pi$
$397$	−14018.2	−1.77218	−0.886088	+	0.463516i	$0.846588\pi$
$398$	0	0
$399$	2584.00	0.324215
$400$	0	0
$401$	−162.000	−0.0201743	−0.0100871	−	0.999949i	$-0.503211\pi$
$401$	−162.000	−0.0201743	−0.0100871	+	0.999949i	$0.503211\pi$
$402$	0	0
$403$	12204.9	1.50861
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1869.97	0.227742
$408$	0	0
$409$	−14142.0	−1.70972	−0.854862	−	0.518856i	$-0.826358\pi$
$409$	−14142.0	−1.70972	−0.854862	+	0.518856i	$0.826358\pi$
$410$	0	0
$411$	−171.000	−0.0205226
$412$	0	0
$413$	1246.65	0.148531
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13015.7	1.52849
$418$	0	0
$419$	−11532.0	−1.34457	−0.672285	−	0.740292i	$-0.734687\pi$
$419$	−11532.0	−1.34457	−0.672285	+	0.740292i	$0.734687\pi$
$420$	0	0
$421$	−3430.00	−0.397074	−0.198537	−	0.980093i	$-0.563619\pi$
$421$	−3430.00	−0.397074	−0.198537	+	0.980093i	$0.563619\pi$
$422$	0	0
$423$	−2440.98	−0.280578
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5893.23	−0.667900
$428$	0	0
$429$	−3344.00	−0.376340
$430$	0	0
$431$	8658.00	0.967613	0.483806	−	0.875175i	$-0.339254\pi$
$431$	8658.00	0.967613	0.483806	+	0.875175i	$0.339254\pi$
$432$	0	0
$433$	−745.372	−0.0827258	−0.0413629	−	0.999144i	$-0.513170\pi$
$433$	−745.372	−0.0827258	−0.0413629	+	0.999144i	$0.513170\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8002.94	−0.876047
$438$	0	0
$439$	4532.00	0.492712	0.246356	−	0.969179i	$-0.420767\pi$
$439$	4532.00	0.492712	0.246356	+	0.969179i	$0.420767\pi$
$440$	0	0
$441$	2136.00	0.230645
$442$	0	0
$443$	4310.95	0.462346	0.231173	−	0.972913i	$-0.425744\pi$
$443$	4310.95	0.462346	0.231173	+	0.972913i	$0.425744\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6738.86	−0.713058
$448$	0	0
$449$	−2333.00	−0.245214	−0.122607	−	0.992455i	$-0.539125\pi$
$449$	−2333.00	−0.245214	−0.122607	+	0.992455i	$0.539125\pi$
$450$	0	0
$451$	4180.00	0.436427
$452$	0	0
$453$	13730.5	1.42410
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6921.93	0.708521	0.354261	−	0.935147i	$-0.384733\pi$
$457$	6921.93	0.708521	0.354261	+	0.935147i	$0.384733\pi$
$458$	0	0
$459$	3990.00	0.405746
$460$	0	0
$461$	5332.00	0.538690	0.269345	−	0.963044i	$-0.413193\pi$
$461$	5332.00	0.538690	0.269345	+	0.963044i	$0.413193\pi$
$462$	0	0
$463$	−9314.97	−0.934996	−0.467498	−	0.883994i	$-0.654845\pi$
$463$	−9314.97	−0.934996	−0.467498	+	0.883994i	$0.654845\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17623.0	−1.74625	−0.873123	−	0.487501i	$-0.837909\pi$
$467$	−17623.0	−1.74625	−0.873123	+	0.487501i	$0.837909\pi$
$468$	0	0
$469$	−4598.00	−0.452699
$470$	0	0
$471$	−2185.00	−0.213757
$472$	0	0
$473$	−3356.35	−0.326269
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3626.60	−0.348115
$478$	0	0
$479$	−13748.0	−1.31140	−0.655702	−	0.755020i	$-0.727627\pi$
$479$	−13748.0	−1.31140	−0.655702	+	0.755020i	$0.727627\pi$
$480$	0	0
$481$	−11856.0	−1.12388
$482$	0	0
$483$	−4472.23	−0.421312
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4101.72	0.381657	0.190828	−	0.981623i	$-0.438883\pi$
$487$	4101.72	0.381657	0.190828	+	0.981623i	$0.438883\pi$
$488$	0	0
$489$	−4066.00	−0.376014
$490$	0	0
$491$	−4016.00	−0.369123	−0.184562	−	0.982821i	$-0.559087\pi$
$491$	−4016.00	−0.369123	−0.184562	+	0.982821i	$0.559087\pi$
$492$	0	0
$493$	−6799.88	−0.621199
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9022.92	−0.814353
$498$	0	0
$499$	−14236.0	−1.27714	−0.638568	−	0.769565i	$-0.720473\pi$
$499$	−14236.0	−1.27714	−0.638568	+	0.769565i	$0.720473\pi$
$500$	0	0
$501$	−8512.00	−0.759058
$502$	0	0
$503$	18089.4	1.60351	0.801757	−	0.597650i	$-0.203899\pi$
$503$	18089.4	1.60351	0.801757	+	0.597650i	$0.203899\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11625.2	1.01833
$508$	0	0
$509$	−8379.00	−0.729652	−0.364826	−	0.931076i	$-0.618871\pi$
$509$	−8379.00	−0.729652	−0.364826	+	0.931076i	$0.618871\pi$
$510$	0	0
$511$	−2888.00	−0.250015
$512$	0	0
$513$	10374.2	0.892848
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3356.35	−0.285517
$518$	0	0
$519$	10222.0	0.864539
$520$	0	0
$521$	−20277.0	−1.70509	−0.852545	−	0.522654i	$-0.824942\pi$
$521$	−20277.0	−1.70509	−0.852545	+	0.522654i	$0.824942\pi$
$522$	0	0
$523$	12152.6	1.01605	0.508027	−	0.861341i	$-0.330375\pi$
$523$	12152.6	1.01605	0.508027	+	0.861341i	$0.330375\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4576.84	−0.378312
$528$	0	0
$529$	1684.00	0.138407
$530$	0	0
$531$	1144.00	0.0934941
$532$	0	0
$533$	−26502.1	−2.15372
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3046.87	0.244846
$538$	0	0
$539$	2937.00	0.234704
$540$	0	0
$541$	−4796.00	−0.381139	−0.190569	−	0.981674i	$-0.561033\pi$
$541$	−4796.00	−0.381139	−0.190569	+	0.981674i	$0.561033\pi$
$542$	0	0
$543$	−11346.2	−0.896708
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16790.5	1.31245	0.656224	−	0.754566i	$-0.272153\pi$
$547$	16790.5	1.31245	0.656224	+	0.754566i	$0.272153\pi$
$548$	0	0
$549$	−5408.00	−0.420415
$550$	0	0
$551$	−17680.0	−1.36696
$552$	0	0
$553$	−1900.48	−0.146142
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9859.83	−0.750044	−0.375022	−	0.927016i	$-0.622365\pi$
$557$	−9859.83	−0.750044	−0.375022	+	0.927016i	$0.622365\pi$
$558$	0	0
$559$	21280.0	1.61010
$560$	0	0
$561$	1254.00	0.0943742
$562$	0	0
$563$	−10095.2	−0.755706	−0.377853	−	0.925866i	$-0.623338\pi$
$563$	−10095.2	−0.755706	−0.377853	+	0.925866i	$0.623338\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3914.29	0.289920
$568$	0	0
$569$	−12240.0	−0.901806	−0.450903	−	0.892573i	$-0.648898\pi$
$569$	−12240.0	−0.901806	−0.450903	+	0.892573i	$0.648898\pi$
$570$	0	0
$571$	21224.0	1.55551	0.777755	−	0.628567i	$-0.216358\pi$
$571$	21224.0	1.55551	0.777755	+	0.628567i	$0.216358\pi$
$572$	0	0
$573$	5792.98	0.422347
$574$	0	0
$575$	0	0
$576$	0	0
$577$	972.034	0.0701323	0.0350661	−	0.999385i	$-0.488836\pi$
$577$	972.034	0.0701323	0.0350661	+	0.999385i	$0.488836\pi$
$578$	0	0
$579$	6080.00	0.436401
$580$	0	0
$581$	6612.00	0.472138
$582$	0	0
$583$	−4986.58	−0.354242
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7662.94	−0.538814	−0.269407	−	0.963026i	$-0.586828\pi$
$587$	−7662.94	−0.538814	−0.269407	+	0.963026i	$0.586828\pi$
$588$	0	0
$589$	−11900.0	−0.832480
$590$	0	0
$591$	10146.0	0.706177
$592$	0	0
$593$	−14541.3	−1.00698	−0.503490	−	0.864001i	$-0.667951\pi$
$593$	−14541.3	−1.00698	−0.503490	+	0.864001i	$0.667951\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−34.8712	−0.00239059
$598$	0	0
$599$	−20520.0	−1.39971	−0.699853	−	0.714286i	$-0.746751\pi$
$599$	−20520.0	−1.39971	−0.699853	+	0.714286i	$0.746751\pi$
$600$	0	0
$601$	12726.0	0.863734	0.431867	−	0.901937i	$-0.357855\pi$
$601$	12726.0	0.863734	0.431867	+	0.901937i	$0.357855\pi$
$602$	0	0
$603$	−4219.41	−0.284955
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3338.92	0.223266	0.111633	−	0.993750i	$-0.464392\pi$
$607$	3338.92	0.223266	0.111633	+	0.993750i	$0.464392\pi$
$608$	0	0
$609$	−9880.00	−0.657402
$610$	0	0
$611$	21280.0	1.40900
$612$	0	0
$613$	5457.34	0.359576	0.179788	−	0.983705i	$-0.442459\pi$
$613$	5457.34	0.359576	0.179788	+	0.983705i	$0.442459\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1272.80	0.0830485	0.0415243	−	0.999137i	$-0.486779\pi$
$617$	1272.80	0.0830485	0.0415243	+	0.999137i	$0.486779\pi$
$618$	0	0
$619$	−17307.0	−1.12379	−0.561896	−	0.827208i	$-0.689928\pi$
$619$	−17307.0	−1.12379	−0.561896	+	0.827208i	$0.689928\pi$
$620$	0	0
$621$	−17955.0	−1.16024
$622$	0	0
$623$	−11150.1	−0.717043
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3260.46	0.207672
$628$	0	0
$629$	4446.00	0.281834
$630$	0	0
$631$	24977.0	1.57578	0.787891	−	0.615814i	$-0.211173\pi$
$631$	24977.0	1.57578	0.787891	+	0.615814i	$0.211173\pi$
$632$	0	0
$633$	−12379.3	−0.777302
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18621.2	−1.15824
$638$	0	0
$639$	−8280.00	−0.512601
$640$	0	0
$641$	−23151.0	−1.42654	−0.713268	−	0.700891i	$-0.752786\pi$
$641$	−23151.0	−1.42654	−0.713268	+	0.700891i	$0.752786\pi$
$642$	0	0
$643$	−710.501	−0.0435761	−0.0217880	−	0.999763i	$-0.506936\pi$
$643$	−710.501	−0.0435761	−0.0217880	+	0.999763i	$0.506936\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3081.74	0.187258	0.0936289	−	0.995607i	$-0.470153\pi$
$647$	3081.74	0.187258	0.0936289	+	0.995607i	$0.470153\pi$
$648$	0	0
$649$	1573.00	0.0951397
$650$	0	0
$651$	−6650.00	−0.400360
$652$	0	0
$653$	579.734	0.0347423	0.0173712	−	0.999849i	$-0.494470\pi$
$653$	579.734	0.0347423	0.0173712	+	0.999849i	$0.494470\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2650.21	−0.157374
$658$	0	0
$659$	−3458.00	−0.204408	−0.102204	−	0.994763i	$-0.532589\pi$
$659$	−3458.00	−0.204408	−0.102204	+	0.994763i	$0.532589\pi$
$660$	0	0
$661$	−12983.0	−0.763964	−0.381982	−	0.924170i	$-0.624758\pi$
$661$	−12983.0	−0.763964	−0.381982	+	0.924170i	$0.624758\pi$
$662$	0	0
$663$	−7950.63	−0.465727
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30599.5	1.77634
$668$	0	0
$669$	−18107.0	−1.04642
$670$	0	0
$671$	−7436.00	−0.427815
$672$	0	0
$673$	−31357.9	−1.79608	−0.898038	−	0.439918i	$-0.855007\pi$
$673$	−31357.9	−1.79608	−0.898038	+	0.439918i	$0.855007\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8796.26	−0.499361	−0.249681	−	0.968328i	$-0.580326\pi$
$677$	−8796.26	−0.499361	−0.249681	+	0.968328i	$0.580326\pi$
$678$	0	0
$679$	−6726.00	−0.380148
$680$	0	0
$681$	−10754.0	−0.605131
$682$	0	0
$683$	6355.27	0.356044	0.178022	−	0.984027i	$-0.443030\pi$
$683$	6355.27	0.356044	0.178022	+	0.984027i	$0.443030\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25338.3	1.40716
$688$	0	0
$689$	31616.0	1.74815
$690$	0	0
$691$	11819.0	0.650674	0.325337	−	0.945598i	$-0.394522\pi$
$691$	11819.0	0.650674	0.325337	+	0.945598i	$0.394522\pi$
$692$	0	0
$693$	−767.166	−0.0420523
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9938.29	0.540085
$698$	0	0
$699$	−8816.00	−0.477041
$700$	0	0
$701$	6978.00	0.375971	0.187985	−	0.982172i	$-0.439804\pi$
$701$	6978.00	0.375971	0.187985	+	0.982172i	$0.439804\pi$
$702$	0	0
$703$	11559.8	0.620179
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5561.96	−0.295868
$708$	0	0
$709$	17947.0	0.950654	0.475327	−	0.879809i	$-0.342330\pi$
$709$	17947.0	0.950654	0.475327	+	0.879809i	$0.342330\pi$
$710$	0	0
$711$	−1744.00	−0.0919903
$712$	0	0
$713$	20595.8	1.08179
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1072.29	−0.0558513
$718$	0	0
$719$	905.000	0.0469413	0.0234707	−	0.999725i	$-0.492528\pi$
$719$	905.000	0.0469413	0.0234707	+	0.999725i	$0.492528\pi$
$720$	0	0
$721$	4636.00	0.239464
$722$	0	0
$723$	14767.9	0.759649
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30961.3	1.57949	0.789745	−	0.613435i	$-0.210213\pi$
$727$	30961.3	1.57949	0.789745	+	0.613435i	$0.210213\pi$
$728$	0	0
$729$	21547.0	1.09470
$730$	0	0
$731$	−7980.00	−0.403763
$732$	0	0
$733$	23520.6	1.18520	0.592602	−	0.805496i	$-0.298101\pi$
$733$	23520.6	1.18520	0.592602	+	0.805496i	$0.298101\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5801.69	−0.289970
$738$	0	0
$739$	30654.0	1.52588	0.762940	−	0.646469i	$-0.223755\pi$
$739$	30654.0	1.52588	0.762940	+	0.646469i	$0.223755\pi$
$740$	0	0
$741$	−20672.0	−1.02484
$742$	0	0
$743$	40154.2	1.98266	0.991328	−	0.131408i	$-0.0419499\pi$
$743$	40154.2	1.98266	0.991328	+	0.131408i	$0.0419499\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6067.59	0.297191
$748$	0	0
$749$	−532.000	−0.0259531
$750$	0	0
$751$	19735.0	0.958909	0.479454	−	0.877567i	$-0.340835\pi$
$751$	19735.0	0.958909	0.479454	+	0.877567i	$0.340835\pi$
$752$	0	0
$753$	4755.56	0.230149
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10583.4	−0.508138	−0.254069	−	0.967186i	$-0.581769\pi$
$757$	−10583.4	−0.508138	−0.254069	+	0.967186i	$0.581769\pi$
$758$	0	0
$759$	−5643.00	−0.269866
$760$	0	0
$761$	6876.00	0.327536	0.163768	−	0.986499i	$-0.447635\pi$
$761$	6876.00	0.327536	0.163768	+	0.986499i	$0.447635\pi$
$762$	0	0
$763$	1237.93	0.0587365
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9973.16	−0.469505
$768$	0	0
$769$	16956.0	0.795122	0.397561	−	0.917576i	$-0.369857\pi$
$769$	16956.0	0.795122	0.397561	+	0.917576i	$0.369857\pi$
$770$	0	0
$771$	18012.0	0.841357
$772$	0	0
$773$	29954.4	1.39377	0.696884	−	0.717184i	$-0.254569\pi$
$773$	29954.4	1.39377	0.696884	+	0.717184i	$0.254569\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6459.89	0.298259
$778$	0	0
$779$	25840.0	1.18846
$780$	0	0
$781$	−11385.0	−0.521623
$782$	0	0
$783$	−39666.0	−1.81040
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1098.44	−0.0497525	−0.0248763	−	0.999691i	$-0.507919\pi$
$787$	−1098.44	−0.0497525	−0.0248763	+	0.999691i	$0.507919\pi$
$788$	0	0
$789$	−30438.0	−1.37341
$790$	0	0
$791$	20330.0	0.913845
$792$	0	0
$793$	47145.9	2.11122
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14118.5	−0.627481	−0.313740	−	0.949509i	$-0.601582\pi$
$797$	−14118.5	−0.627481	−0.313740	+	0.949509i	$0.601582\pi$
$798$	0	0
$799$	−7980.00	−0.353332
$800$	0	0
$801$	−10232.0	−0.451348
$802$	0	0
$803$	−3644.04	−0.160144
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1368.69	−0.0597030
$808$	0	0
$809$	−30076.0	−1.30707	−0.653533	−	0.756898i	$-0.726714\pi$
$809$	−30076.0	−1.30707	−0.653533	+	0.756898i	$0.726714\pi$
$810$	0	0
$811$	−7062.00	−0.305771	−0.152886	−	0.988244i	$-0.548857\pi$
$811$	−7062.00	−0.305771	−0.152886	+	0.988244i	$0.548857\pi$
$812$	0	0
$813$	13861.3	0.597954
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20748.4	−0.888486
$818$	0	0
$819$	4864.00	0.207524
$820$	0	0
$821$	−14090.0	−0.598958	−0.299479	−	0.954103i	$-0.596813\pi$
$821$	−14090.0	−0.598958	−0.299479	+	0.954103i	$0.596813\pi$
$822$	0	0
$823$	−10300.1	−0.436255	−0.218128	−	0.975920i	$-0.569995\pi$
$823$	−10300.1	−0.436255	−0.218128	+	0.975920i	$0.569995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16110.5	0.677408	0.338704	−	0.940893i	$-0.390011\pi$
$827$	16110.5	0.677408	0.338704	+	0.940893i	$0.390011\pi$
$828$	0	0
$829$	−14611.0	−0.612136	−0.306068	−	0.952010i	$-0.599014\pi$
$829$	−14611.0	−0.612136	−0.306068	+	0.952010i	$0.599014\pi$
$830$	0	0
$831$	−16910.0	−0.705898
$832$	0	0
$833$	6982.96	0.290450
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26698.3	−1.10254
$838$	0	0
$839$	37259.0	1.53316	0.766581	−	0.642147i	$-0.221956\pi$
$839$	37259.0	1.53316	0.766581	+	0.642147i	$0.221956\pi$
$840$	0	0
$841$	43211.0	1.77174
$842$	0	0
$843$	−18385.8	−0.751177
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1054.85	−0.0427924
$848$	0	0
$849$	36404.0	1.47159
$850$	0	0
$851$	−20007.0	−0.805912
$852$	0	0
$853$	−5239.40	−0.210309	−0.105154	−	0.994456i	$-0.533534\pi$
$853$	−5239.40	−0.210309	−0.105154	+	0.994456i	$0.533534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26781.1	−1.06747	−0.533736	−	0.845651i	$-0.679213\pi$
$857$	−26781.1	−1.06747	−0.533736	+	0.845651i	$0.679213\pi$
$858$	0	0
$859$	29955.0	1.18982	0.594908	−	0.803794i	$-0.297189\pi$
$859$	29955.0	1.18982	0.594908	+	0.803794i	$0.297189\pi$
$860$	0	0
$861$	14440.0	0.571561
$862$	0	0
$863$	14462.8	0.570475	0.285238	−	0.958457i	$-0.407927\pi$
$863$	14462.8	0.570475	0.285238	+	0.958457i	$0.407927\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−18433.8	−0.722081
$868$	0	0
$869$	−2398.00	−0.0936094
$870$	0	0
$871$	36784.0	1.43097
$872$	0	0
$873$	−6172.20	−0.239287
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24898.0	0.958662	0.479331	−	0.877634i	$-0.340879\pi$
$877$	24898.0	0.958662	0.479331	+	0.877634i	$0.340879\pi$
$878$	0	0
$879$	27740.0	1.06444
$880$	0	0
$881$	−19987.0	−0.764335	−0.382168	−	0.924093i	$-0.624822\pi$
$881$	−19987.0	−0.764335	−0.382168	+	0.924093i	$0.624822\pi$
$882$	0	0
$883$	−5466.06	−0.208321	−0.104161	−	0.994560i	$-0.533216\pi$
$883$	−5466.06	−0.208321	−0.104161	+	0.994560i	$0.533216\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25586.7	−0.968567	−0.484283	−	0.874911i	$-0.660920\pi$
$887$	−25586.7	−0.968567	−0.484283	+	0.874911i	$0.660920\pi$
$888$	0	0
$889$	8512.00	0.321129
$890$	0	0
$891$	4939.00	0.185705
$892$	0	0
$893$	−20748.4	−0.777511
$894$	0	0
$895$	0	0
$896$	0	0
$897$	35777.8	1.33176
$898$	0	0
$899$	45500.0	1.68800
$900$	0	0
$901$	−11856.0	−0.438380
$902$	0	0
$903$	−11594.7	−0.427294
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1368.69	0.0501067	0.0250533	−	0.999686i	$-0.492024\pi$
$907$	1368.69	0.0501067	0.0250533	+	0.999686i	$0.492024\pi$
$908$	0	0
$909$	−5104.00	−0.186237
$910$	0	0
$911$	−20068.0	−0.729838	−0.364919	−	0.931039i	$-0.618903\pi$
$911$	−20068.0	−0.729838	−0.364919	+	0.931039i	$0.618903\pi$
$912$	0	0
$913$	8342.93	0.302421
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6747.58	−0.242993
$918$	0	0
$919$	−10946.0	−0.392900	−0.196450	−	0.980514i	$-0.562941\pi$
$919$	−10946.0	−0.392900	−0.196450	+	0.980514i	$0.562941\pi$
$920$	0	0
$921$	28804.0	1.03054
$922$	0	0
$923$	72183.4	2.57415
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4254.29	0.150733
$928$	0	0
$929$	−33338.0	−1.17738	−0.588689	−	0.808360i	$-0.700356\pi$
$929$	−33338.0	−1.17738	−0.588689	+	0.808360i	$0.700356\pi$
$930$	0	0
$931$	18156.0	0.639139
$932$	0	0
$933$	25333.9	0.888955
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51487.3	−1.79511	−0.897555	−	0.440903i	$-0.854658\pi$
$937$	−51487.3	−1.79511	−0.897555	+	0.440903i	$0.854658\pi$
$938$	0	0
$939$	−25669.0	−0.892094
$940$	0	0
$941$	−21590.0	−0.747942	−0.373971	−	0.927440i	$-0.622004\pi$
$941$	−21590.0	−0.747942	−0.373971	+	0.927440i	$0.622004\pi$
$942$	0	0
$943$	−44722.3	−1.54439
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23420.4	0.803653	0.401827	−	0.915716i	$-0.368375\pi$
$947$	23420.4	0.803653	0.401827	+	0.915716i	$0.368375\pi$
$948$	0	0
$949$	23104.0	0.790292
$950$	0	0
$951$	39083.0	1.33265
$952$	0	0
$953$	−22221.7	−0.755331	−0.377665	−	0.925942i	$-0.623273\pi$
$953$	−22221.7	−0.755331	−0.377665	+	0.925942i	$0.623273\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12466.5	−0.421090
$958$	0	0
$959$	342.000	0.0115159
$960$	0	0
$961$	834.000	0.0279950
$962$	0	0
$963$	−488.197	−0.0163364
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15883.8	−0.528221	−0.264110	−	0.964492i	$-0.585078\pi$
$967$	−15883.8	−0.528221	−0.264110	+	0.964492i	$0.585078\pi$
$968$	0	0
$969$	7752.00	0.256997
$970$	0	0
$971$	−13965.0	−0.461543	−0.230771	−	0.973008i	$-0.574125\pi$
$971$	−13965.0	−0.461543	−0.230771	+	0.973008i	$0.574125\pi$
$972$	0	0
$973$	−26031.3	−0.857684
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36418.6	1.19256	0.596282	−	0.802775i	$-0.296644\pi$
$977$	36418.6	1.19256	0.596282	+	0.802775i	$0.296644\pi$
$978$	0	0
$979$	−14069.0	−0.459292
$980$	0	0
$981$	1136.00	0.0369722
$982$	0	0
$983$	−49844.0	−1.61727	−0.808635	−	0.588310i	$-0.799793\pi$
$983$	−49844.0	−1.61727	−0.808635	+	0.588310i	$0.799793\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−11594.7	−0.373923
$988$	0	0
$989$	35910.0	1.15457
$990$	0	0
$991$	55024.0	1.76377	0.881884	−	0.471466i	$-0.156275\pi$
$991$	55024.0	1.76377	0.881884	+	0.471466i	$0.156275\pi$
$992$	0	0
$993$	37848.3	1.20955
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51740.1	1.64356	0.821779	−	0.569807i	$-0.192982\pi$
$997$	51740.1	1.64356	0.821779	+	0.569807i	$0.192982\pi$
$998$	0	0
$999$	25935.0	0.821368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.a.f.1.2		2
5.2	odd	4		220.4.b.a.89.1	✓	2
5.3	odd	4		220.4.b.a.89.2	yes	2
5.4	even	2	inner	1100.4.a.f.1.1		2
15.2	even	4		1980.4.c.a.1189.1		2
15.8	even	4		1980.4.c.a.1189.2		2
20.3	even	4		880.4.b.b.529.1		2
20.7	even	4		880.4.b.b.529.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.b.a.89.1	✓	2	5.2	odd	4
220.4.b.a.89.2	yes	2	5.3	odd	4
880.4.b.b.529.1		2	20.3	even	4
880.4.b.b.529.2		2	20.7	even	4
1100.4.a.f.1.1		2	5.4	even	2	inner
1100.4.a.f.1.2		2	1.1	even	1	trivial
1980.4.c.a.1189.1		2	15.2	even	4
1980.4.c.a.1189.2		2	15.8	even	4

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q+4.35890 q^{3} -8.71780 q^{7} -8.00000 q^{9} -11.0000 q^{11} +69.7424 q^{13} -26.1534 q^{17} -68.0000 q^{19} -38.0000 q^{21} +117.690 q^{23} -152.561 q^{27} +260.000 q^{29} +175.000 q^{31} -47.9479 q^{33} -169.997 q^{37} +304.000 q^{39} -380.000 q^{41} +305.123 q^{43} +305.123 q^{47} -267.000 q^{49} -114.000 q^{51} +453.325 q^{53} -296.405 q^{57} -143.000 q^{59} +676.000 q^{61} +69.7424 q^{63} +527.427 q^{67} +513.000 q^{69} +1035.00 q^{71} +331.276 q^{73} +95.8958 q^{77} +218.000 q^{79} -449.000 q^{81} -758.448 q^{83} +1133.31 q^{87} +1279.00 q^{89} -608.000 q^{91} +762.807 q^{93} +771.525 q^{97} +88.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{9} - 22 q^{11} - 136 q^{19} - 76 q^{21} + 520 q^{29} + 350 q^{31} + 608 q^{39} - 760 q^{41} - 534 q^{49} - 228 q^{51} - 286 q^{59} + 1352 q^{61} + 1026 q^{69} + 2070 q^{71} + 436 q^{79} - 898 q^{81}+ \cdots + 176 q^{99}+O(q^{100}) \) 2 * q - 16 * q^9 - 22 * q^11 - 136 * q^19 - 76 * q^21 + 520 * q^29 + 350 * q^31 + 608 * q^39 - 760 * q^41 - 534 * q^49 - 228 * q^51 - 286 * q^59 + 1352 * q^61 + 1026 * q^69 + 2070 * q^71 + 436 * q^79 - 898 * q^81 + 2558 * q^89 - 1216 * q^91 + 176 * q^99

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