LMFDB - Embedded newform 1100.6.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$176.422201794$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 $ x^8 - x^7 - 1155x^6 + 1122x^5 + 365994x^4 - 476334x^3 - 28296355x^2 + 54177943x + 132029745
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.42686$ 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)

$f(q)$	$=$	$q+0.426864 q^{3} +59.7938 q^{7} -242.818 q^{9} -121.000 q^{11} +1125.47 q^{13} -1949.47 q^{17} -1244.75 q^{19} +25.5238 q^{21} +4642.40 q^{23} -207.378 q^{27} +1000.22 q^{29} -2478.21 q^{31} -51.6505 q^{33} +10827.6 q^{37} +480.423 q^{39} -11727.8 q^{41} -6900.79 q^{43} +4951.00 q^{47} -13231.7 q^{49} -832.158 q^{51} -8177.38 q^{53} -531.337 q^{57} +36965.0 q^{59} -11885.0 q^{61} -14519.0 q^{63} +825.191 q^{67} +1981.67 q^{69} -11961.1 q^{71} +59009.3 q^{73} -7235.05 q^{77} -2638.60 q^{79} +58916.2 q^{81} -108055. q^{83} +426.958 q^{87} +32135.2 q^{89} +67296.2 q^{91} -1057.86 q^{93} -165772. q^{97} +29381.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) $ 8 * q - 9 * q^3 - 175 * q^7 + 377 * q^9 - 968 * q^11 + 1072 * q^13 - 37 * q^17 - 1356 * q^19 + 5462 * q^21 - 1327 * q^23 - 2670 * q^27 - 11785 * q^29 + 3532 * q^31 + 1089 * q^33 + 8176 * q^37 - 16831 * q^39 + 15846 * q^41 + 530 * q^43 + 9252 * q^47 + 45745 * q^49 + 47733 * q^51 + 24991 * q^53 - 41957 * q^57 + 1440 * q^59 - 54447 * q^61 - 108968 * q^63 - 109232 * q^67 + 134200 * q^69 - 45026 * q^71 + 2221 * q^73 + 21175 * q^77 + 74369 * q^79 - 1132 * q^81 - 5443 * q^83 - 18665 * q^87 + 106151 * q^89 - 230723 * q^91 + 111315 * q^93 - 75273 * q^97 - 45617 * 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
$3$	0.426864	0.0273833	0.0136917	−	0.999906i	$-0.495642\pi$
$3$	0.426864	0.0273833	0.0136917	+	0.999906i	$0.495642\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	59.7938	0.461223	0.230611	−	0.973046i	$-0.425927\pi$
$7$	59.7938	0.461223	0.230611	+	0.973046i	$0.425927\pi$
$8$	0	0
$9$	−242.818	−0.999250
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	1125.47	1.84704	0.923521	−	0.383549i	$-0.125298\pi$
$13$	1125.47	1.84704	0.923521	+	0.383549i	$0.125298\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1949.47	−1.63604	−0.818021	−	0.575188i	$-0.804929\pi$
$17$	−1949.47	−1.63604	−0.818021	+	0.575188i	$0.804929\pi$
$18$	0	0
$19$	−1244.75	−0.791037	−0.395518	−	0.918458i	$-0.629435\pi$
$19$	−1244.75	−0.791037	−0.395518	+	0.918458i	$0.629435\pi$
$20$	0	0
$21$	25.5238	0.0126298
$22$	0	0
$23$	4642.40	1.82988	0.914941	−	0.403587i	$-0.132237\pi$
$23$	4642.40	1.82988	0.914941	+	0.403587i	$0.132237\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−207.378	−0.0547461
$28$	0	0
$29$	1000.22	0.220852	0.110426	−	0.993884i	$-0.464778\pi$
$29$	1000.22	0.220852	0.110426	+	0.993884i	$0.464778\pi$
$30$	0	0
$31$	−2478.21	−0.463164	−0.231582	−	0.972815i	$-0.574390\pi$
$31$	−2478.21	−0.463164	−0.231582	+	0.972815i	$0.574390\pi$
$32$	0	0
$33$	−51.6505	−0.00825638
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10827.6	1.30025	0.650127	−	0.759825i	$-0.274716\pi$
$37$	10827.6	1.30025	0.650127	+	0.759825i	$0.274716\pi$
$38$	0	0
$39$	480.423	0.0505781
$40$	0	0
$41$	−11727.8	−1.08958	−0.544789	−	0.838573i	$-0.683390\pi$
$41$	−11727.8	−1.08958	−0.544789	+	0.838573i	$0.683390\pi$
$42$	0	0
$43$	−6900.79	−0.569151	−0.284576	−	0.958654i	$-0.591853\pi$
$43$	−6900.79	−0.569151	−0.284576	+	0.958654i	$0.591853\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4951.00	0.326925	0.163463	−	0.986550i	$-0.447734\pi$
$47$	4951.00	0.326925	0.163463	+	0.986550i	$0.447734\pi$
$48$	0	0
$49$	−13231.7	−0.787273
$50$	0	0
$51$	−832.158	−0.0448003
$52$	0	0
$53$	−8177.38	−0.399875	−0.199938	−	0.979809i	$-0.564074\pi$
$53$	−8177.38	−0.399875	−0.199938	+	0.979809i	$0.564074\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−531.337	−0.0216612
$58$	0	0
$59$	36965.0	1.38249	0.691244	−	0.722622i	$-0.257063\pi$
$59$	36965.0	1.38249	0.691244	+	0.722622i	$0.257063\pi$
$60$	0	0
$61$	−11885.0	−0.408954	−0.204477	−	0.978871i	$-0.565549\pi$
$61$	−11885.0	−0.408954	−0.204477	+	0.978871i	$0.565549\pi$
$62$	0	0
$63$	−14519.0	−0.460877
$64$	0	0
$65$	0	0
$66$	0	0
$67$	825.191	0.0224578	0.0112289	−	0.999937i	$-0.496426\pi$
$67$	825.191	0.0224578	0.0112289	+	0.999937i	$0.496426\pi$
$68$	0	0
$69$	1981.67	0.0501082
$70$	0	0
$71$	−11961.1	−0.281594	−0.140797	−	0.990038i	$-0.544967\pi$
$71$	−11961.1	−0.281594	−0.140797	+	0.990038i	$0.544967\pi$
$72$	0	0
$73$	59009.3	1.29602	0.648012	−	0.761630i	$-0.275601\pi$
$73$	59009.3	1.29602	0.648012	+	0.761630i	$0.275601\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7235.05	−0.139064
$78$	0	0
$79$	−2638.60	−0.0475671	−0.0237835	−	0.999717i	$-0.507571\pi$
$79$	−2638.60	−0.0475671	−0.0237835	+	0.999717i	$0.507571\pi$
$80$	0	0
$81$	58916.2	0.997751
$82$	0	0
$83$	−108055.	−1.72167	−0.860836	−	0.508882i	$-0.830059\pi$
$83$	−108055.	−1.72167	−0.860836	+	0.508882i	$0.830059\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	426.958	0.00604766
$88$	0	0
$89$	32135.2	0.430038	0.215019	−	0.976610i	$-0.431019\pi$
$89$	32135.2	0.430038	0.215019	+	0.976610i	$0.431019\pi$
$90$	0	0
$91$	67296.2	0.851898
$92$	0	0
$93$	−1057.86	−0.0126830
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−165772.	−1.78888	−0.894440	−	0.447187i	$-0.852426\pi$
$97$	−165772.	−1.78888	−0.894440	+	0.447187i	$0.852426\pi$
$98$	0	0
$99$	29381.0	0.301285
$100$	0	0
$101$	−41646.8	−0.406236	−0.203118	−	0.979154i	$-0.565107\pi$
$101$	−41646.8	−0.406236	−0.203118	+	0.979154i	$0.565107\pi$
$102$	0	0
$103$	29129.0	0.270541	0.135270	−	0.990809i	$-0.456810\pi$
$103$	29129.0	0.270541	0.135270	+	0.990809i	$0.456810\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	97108.3	0.819968	0.409984	−	0.912093i	$-0.365534\pi$
$107$	97108.3	0.819968	0.409984	+	0.912093i	$0.365534\pi$
$108$	0	0
$109$	−157166.	−1.26705	−0.633524	−	0.773723i	$-0.718392\pi$
$109$	−157166.	−1.26705	−0.633524	+	0.773723i	$0.718392\pi$
$110$	0	0
$111$	4621.91	0.0356053
$112$	0	0
$113$	55971.5	0.412354	0.206177	−	0.978515i	$-0.433898\pi$
$113$	55971.5	0.412354	0.206177	+	0.978515i	$0.433898\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−273285.	−1.84566
$118$	0	0
$119$	−116566.	−0.754580
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−5006.19	−0.0298363
$124$	0	0
$125$	0	0
$126$	0	0
$127$	123758.	0.680872	0.340436	−	0.940268i	$-0.389425\pi$
$127$	123758.	0.680872	0.340436	+	0.940268i	$0.389425\pi$
$128$	0	0
$129$	−2945.70	−0.0155853
$130$	0	0
$131$	−231335.	−1.17778	−0.588889	−	0.808214i	$-0.700434\pi$
$131$	−231335.	−1.17778	−0.588889	+	0.808214i	$0.700434\pi$
$132$	0	0
$133$	−74428.0	−0.364844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13429.0	−0.0611283	−0.0305642	−	0.999533i	$-0.509730\pi$
$137$	−13429.0	−0.0611283	−0.0305642	+	0.999533i	$0.509730\pi$
$138$	0	0
$139$	−137339.	−0.602916	−0.301458	−	0.953479i	$-0.597473\pi$
$139$	−137339.	−0.602916	−0.301458	+	0.953479i	$0.597473\pi$
$140$	0	0
$141$	2113.40	0.00895229
$142$	0	0
$143$	−136182.	−0.556904
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5648.13	−0.0215582
$148$	0	0
$149$	456905.	1.68601	0.843005	−	0.537906i	$-0.180785\pi$
$149$	456905.	1.68601	0.843005	+	0.537906i	$0.180785\pi$
$150$	0	0
$151$	−479355.	−1.71086	−0.855431	−	0.517917i	$-0.826708\pi$
$151$	−479355.	−1.71086	−0.855431	+	0.517917i	$0.826708\pi$
$152$	0	0
$153$	473366.	1.63482
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−70972.6	−0.229796	−0.114898	−	0.993377i	$-0.536654\pi$
$157$	−70972.6	−0.229796	−0.114898	+	0.993377i	$0.536654\pi$
$158$	0	0
$159$	−3490.62	−0.0109499
$160$	0	0
$161$	277587.	0.843983
$162$	0	0
$163$	56957.6	0.167912	0.0839561	−	0.996469i	$-0.473244\pi$
$163$	56957.6	0.167912	0.0839561	+	0.996469i	$0.473244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	35650.3	0.0989172	0.0494586	−	0.998776i	$-0.484250\pi$
$167$	35650.3	0.0989172	0.0494586	+	0.998776i	$0.484250\pi$
$168$	0	0
$169$	895396.	2.41156
$170$	0	0
$171$	302246.	0.790444
$172$	0	0
$173$	−395809.	−1.00547	−0.502736	−	0.864440i	$-0.667673\pi$
$173$	−395809.	−1.00547	−0.502736	+	0.864440i	$0.667673\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15779.0	0.0378571
$178$	0	0
$179$	852131.	1.98780	0.993902	−	0.110264i	$-0.0351696\pi$
$179$	852131.	1.98780	0.993902	+	0.110264i	$0.0351696\pi$
$180$	0	0
$181$	−423532.	−0.960927	−0.480463	−	0.877015i	$-0.659531\pi$
$181$	−423532.	−0.960927	−0.480463	+	0.877015i	$0.659531\pi$
$182$	0	0
$183$	−5073.27	−0.0111985
$184$	0	0
$185$	0	0
$186$	0	0
$187$	235886.	0.493285
$188$	0	0
$189$	−12399.9	−0.0252502
$190$	0	0
$191$	647068.	1.28341	0.641706	−	0.766951i	$-0.278227\pi$
$191$	647068.	1.28341	0.641706	+	0.766951i	$0.278227\pi$
$192$	0	0
$193$	−527210.	−1.01880	−0.509402	−	0.860529i	$-0.670133\pi$
$193$	−527210.	−1.01880	−0.509402	+	0.860529i	$0.670133\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−590667.	−1.08437	−0.542184	−	0.840260i	$-0.682403\pi$
$197$	−590667.	−1.08437	−0.542184	+	0.840260i	$0.682403\pi$
$198$	0	0
$199$	−891266.	−1.59542	−0.797709	−	0.603042i	$-0.793955\pi$
$199$	−891266.	−1.59542	−0.797709	+	0.603042i	$0.793955\pi$
$200$	0	0
$201$	352.244	0.000614970 0
$202$	0	0
$203$	59807.0	0.101862
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.12726e6	−1.82851
$208$	0	0
$209$	150614.	0.238507
$210$	0	0
$211$	−587022.	−0.907712	−0.453856	−	0.891075i	$-0.649952\pi$
$211$	−587022.	−0.907712	−0.453856	+	0.891075i	$0.649952\pi$
$212$	0	0
$213$	−5105.74	−0.00771098
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−148182.	−0.213622
$218$	0	0
$219$	25188.9	0.0354894
$220$	0	0
$221$	−2.19408e6	−3.02184
$222$	0	0
$223$	−126468.	−0.170301	−0.0851505	−	0.996368i	$-0.527137\pi$
$223$	−126468.	−0.170301	−0.0851505	+	0.996368i	$0.527137\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−442024.	−0.569353	−0.284676	−	0.958624i	$-0.591886\pi$
$227$	−442024.	−0.569353	−0.284676	+	0.958624i	$0.591886\pi$
$228$	0	0
$229$	−1.50388e6	−1.89507	−0.947533	−	0.319659i	$-0.896432\pi$
$229$	−1.50388e6	−1.89507	−0.947533	+	0.319659i	$0.896432\pi$
$230$	0	0
$231$	−3088.38	−0.00380803
$232$	0	0
$233$	−925184.	−1.11645	−0.558224	−	0.829691i	$-0.688517\pi$
$233$	−925184.	−1.11645	−0.558224	+	0.829691i	$0.688517\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1126.32	−0.00130254
$238$	0	0
$239$	142193.	0.161022	0.0805109	−	0.996754i	$-0.474345\pi$
$239$	142193.	0.161022	0.0805109	+	0.996754i	$0.474345\pi$
$240$	0	0
$241$	−749243.	−0.830960	−0.415480	−	0.909602i	$-0.636386\pi$
$241$	−749243.	−0.830960	−0.415480	+	0.909602i	$0.636386\pi$
$242$	0	0
$243$	75542.0	0.0820678
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.40093e6	−1.46108
$248$	0	0
$249$	−46124.8	−0.0471451
$250$	0	0
$251$	1.06930e6	1.07131	0.535655	−	0.844437i	$-0.320065\pi$
$251$	1.06930e6	1.07131	0.535655	+	0.844437i	$0.320065\pi$
$252$	0	0
$253$	−561731.	−0.551730
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.18220e6	−1.11650	−0.558249	−	0.829673i	$-0.688527\pi$
$257$	−1.18220e6	−1.11650	−0.558249	+	0.829673i	$0.688527\pi$
$258$	0	0
$259$	647424.	0.599707
$260$	0	0
$261$	−242872.	−0.220686
$262$	0	0
$263$	571345.	0.509342	0.254671	−	0.967028i	$-0.418033\pi$
$263$	571345.	0.509342	0.254671	+	0.967028i	$0.418033\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13717.4	0.0117759
$268$	0	0
$269$	−483732.	−0.407591	−0.203795	−	0.979013i	$-0.565328\pi$
$269$	−483732.	−0.407591	−0.203795	+	0.979013i	$0.565328\pi$
$270$	0	0
$271$	−582909.	−0.482145	−0.241072	−	0.970507i	$-0.577499\pi$
$271$	−582909.	−0.482145	−0.241072	+	0.970507i	$0.577499\pi$
$272$	0	0
$273$	28726.3	0.0233278
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.21640e6	0.952528	0.476264	−	0.879302i	$-0.341991\pi$
$277$	1.21640e6	0.952528	0.476264	+	0.879302i	$0.341991\pi$
$278$	0	0
$279$	601754.	0.462816
$280$	0	0
$281$	791954.	0.598321	0.299160	−	0.954203i	$-0.403293\pi$
$281$	791954.	0.598321	0.299160	+	0.954203i	$0.403293\pi$
$282$	0	0
$283$	−2.01071e6	−1.49239	−0.746196	−	0.665726i	$-0.768122\pi$
$283$	−2.01071e6	−1.49239	−0.746196	+	0.665726i	$0.768122\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−701252.	−0.502538
$288$	0	0
$289$	2.38058e6	1.67663
$290$	0	0
$291$	−70762.0	−0.0489855
$292$	0	0
$293$	554775.	0.377527	0.188763	−	0.982023i	$-0.439552\pi$
$293$	554775.	0.377527	0.188763	+	0.982023i	$0.439552\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	25092.7	0.0165066
$298$	0	0
$299$	5.22490e6	3.37987
$300$	0	0
$301$	−412624.	−0.262506
$302$	0	0
$303$	−17777.5	−0.0111241
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.93272e6	−1.17037	−0.585185	−	0.810900i	$-0.698978\pi$
$307$	−1.93272e6	−1.17037	−0.585185	+	0.810900i	$0.698978\pi$
$308$	0	0
$309$	12434.1	0.00740830
$310$	0	0
$311$	582591.	0.341557	0.170778	−	0.985309i	$-0.445372\pi$
$311$	582591.	0.341557	0.170778	+	0.985309i	$0.445372\pi$
$312$	0	0
$313$	−240474.	−0.138742	−0.0693710	−	0.997591i	$-0.522099\pi$
$313$	−240474.	−0.138742	−0.0693710	+	0.997591i	$0.522099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.07237e6	−1.15829	−0.579146	−	0.815223i	$-0.696614\pi$
$317$	−2.07237e6	−1.15829	−0.579146	+	0.815223i	$0.696614\pi$
$318$	0	0
$319$	−121027.	−0.0665894
$320$	0	0
$321$	41452.0	0.0224534
$322$	0	0
$323$	2.42660e6	1.29417
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−67088.6	−0.0346960
$328$	0	0
$329$	296039.	0.150785
$330$	0	0
$331$	−2.74738e6	−1.37832	−0.689158	−	0.724611i	$-0.742020\pi$
$331$	−2.74738e6	−1.37832	−0.689158	+	0.724611i	$0.742020\pi$
$332$	0	0
$333$	−2.62914e6	−1.29928
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.79072e6	−0.858920	−0.429460	−	0.903086i	$-0.641296\pi$
$337$	−1.79072e6	−0.858920	−0.429460	+	0.903086i	$0.641296\pi$
$338$	0	0
$339$	23892.2	0.0112916
$340$	0	0
$341$	299864.	0.139649
$342$	0	0
$343$	−1.79613e6	−0.824331
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.52763e6	−0.681073	−0.340537	−	0.940231i	$-0.610609\pi$
$347$	−1.52763e6	−0.681073	−0.340537	+	0.940231i	$0.610609\pi$
$348$	0	0
$349$	−946310.	−0.415882	−0.207941	−	0.978141i	$-0.566676\pi$
$349$	−946310.	−0.415882	−0.207941	+	0.978141i	$0.566676\pi$
$350$	0	0
$351$	−233398.	−0.101118
$352$	0	0
$353$	2.53382e6	1.08228	0.541140	−	0.840933i	$-0.317993\pi$
$353$	2.53382e6	1.08228	0.541140	+	0.840933i	$0.317993\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−49757.9	−0.0206629
$358$	0	0
$359$	2.70662e6	1.10839	0.554194	−	0.832388i	$-0.313027\pi$
$359$	2.70662e6	1.10839	0.554194	+	0.832388i	$0.313027\pi$
$360$	0	0
$361$	−926707.	−0.374261
$362$	0	0
$363$	6249.71	0.00248939
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.22091e6	−1.24828	−0.624142	−	0.781311i	$-0.714551\pi$
$367$	−3.22091e6	−1.24828	−0.624142	+	0.781311i	$0.714551\pi$
$368$	0	0
$369$	2.84773e6	1.08876
$370$	0	0
$371$	−488956.	−0.184432
$372$	0	0
$373$	3.10550e6	1.15574	0.577869	−	0.816130i	$-0.303885\pi$
$373$	3.10550e6	1.15574	0.577869	+	0.816130i	$0.303885\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.12572e6	0.407923
$378$	0	0
$379$	−3.22941e6	−1.15485	−0.577424	−	0.816445i	$-0.695942\pi$
$379$	−3.22941e6	−1.15485	−0.577424	+	0.816445i	$0.695942\pi$
$380$	0	0
$381$	52828.0	0.0186445
$382$	0	0
$383$	−5.12412e6	−1.78493	−0.892467	−	0.451113i	$-0.851027\pi$
$383$	−5.12412e6	−1.78493	−0.892467	+	0.451113i	$0.851027\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.67564e6	0.568725
$388$	0	0
$389$	−2.69794e6	−0.903978	−0.451989	−	0.892024i	$-0.649285\pi$
$389$	−2.69794e6	−0.903978	−0.451989	+	0.892024i	$0.649285\pi$
$390$	0	0
$391$	−9.05023e6	−2.99376
$392$	0	0
$393$	−98748.5	−0.0322514
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.53717e6	−1.44480	−0.722402	−	0.691473i	$-0.756962\pi$
$397$	−4.53717e6	−1.44480	−0.722402	+	0.691473i	$0.756962\pi$
$398$	0	0
$399$	−31770.6	−0.00999064
$400$	0	0
$401$	−4.16130e6	−1.29231	−0.646157	−	0.763204i	$-0.723625\pi$
$401$	−4.16130e6	−1.29231	−0.646157	+	0.763204i	$0.723625\pi$
$402$	0	0
$403$	−2.78916e6	−0.855482
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.31014e6	−0.392041
$408$	0	0
$409$	4.47771e6	1.32357	0.661786	−	0.749693i	$-0.269799\pi$
$409$	4.47771e6	1.32357	0.661786	+	0.749693i	$0.269799\pi$
$410$	0	0
$411$	−5732.35	−0.00167390
$412$	0	0
$413$	2.21028e6	0.637635
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−58625.0	−0.0165098
$418$	0	0
$419$	−4.80723e6	−1.33770	−0.668852	−	0.743396i	$-0.733214\pi$
$419$	−4.80723e6	−1.33770	−0.668852	+	0.743396i	$0.733214\pi$
$420$	0	0
$421$	967052.	0.265916	0.132958	−	0.991122i	$-0.457552\pi$
$421$	967052.	0.265916	0.132958	+	0.991122i	$0.457552\pi$
$422$	0	0
$423$	−1.20219e6	−0.326680
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−710649.	−0.188619
$428$	0	0
$429$	−58131.2	−0.0152499
$430$	0	0
$431$	777588.	0.201630	0.100815	−	0.994905i	$-0.467855\pi$
$431$	777588.	0.201630	0.100815	+	0.994905i	$0.467855\pi$
$432$	0	0
$433$	−2.57482e6	−0.659973	−0.329987	−	0.943986i	$-0.607044\pi$
$433$	−2.57482e6	−0.659973	−0.329987	+	0.943986i	$0.607044\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.77861e6	−1.44750
$438$	0	0
$439$	5.54606e6	1.37348	0.686742	−	0.726902i	$-0.259040\pi$
$439$	5.54606e6	1.37348	0.686742	+	0.726902i	$0.259040\pi$
$440$	0	0
$441$	3.21289e6	0.786683
$442$	0	0
$443$	−3.53071e6	−0.854777	−0.427389	−	0.904068i	$-0.640566\pi$
$443$	−3.53071e6	−0.854777	−0.427389	+	0.904068i	$0.640566\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	195036.	0.0461685
$448$	0	0
$449$	3.25993e6	0.763119	0.381560	−	0.924344i	$-0.375387\pi$
$449$	3.25993e6	0.763119	0.381560	+	0.924344i	$0.375387\pi$
$450$	0	0
$451$	1.41907e6	0.328520
$452$	0	0
$453$	−204619.	−0.0468491
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.68918e6	0.378344	0.189172	−	0.981944i	$-0.439420\pi$
$457$	1.68918e6	0.378344	0.189172	+	0.981944i	$0.439420\pi$
$458$	0	0
$459$	404277.	0.0895669
$460$	0	0
$461$	−4.56288e6	−0.999969	−0.499985	−	0.866034i	$-0.666661\pi$
$461$	−4.56288e6	−0.999969	−0.499985	+	0.866034i	$0.666661\pi$
$462$	0	0
$463$	4.27642e6	0.927103	0.463551	−	0.886070i	$-0.346575\pi$
$463$	4.27642e6	0.927103	0.463551	+	0.886070i	$0.346575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.53049e6	−0.961286	−0.480643	−	0.876916i	$-0.659597\pi$
$467$	−4.53049e6	−0.961286	−0.480643	+	0.876916i	$0.659597\pi$
$468$	0	0
$469$	49341.3	0.0103581
$470$	0	0
$471$	−30295.6	−0.00629257
$472$	0	0
$473$	834996.	0.171606
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.98561e6	0.399575
$478$	0	0
$479$	−5.10458e6	−1.01653	−0.508267	−	0.861200i	$-0.669714\pi$
$479$	−5.10458e6	−1.01653	−0.508267	+	0.861200i	$0.669714\pi$
$480$	0	0
$481$	1.21862e7	2.40162
$482$	0	0
$483$	118492.	0.0231111
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.12419e6	−0.405854	−0.202927	−	0.979194i	$-0.565045\pi$
$487$	−2.12419e6	−0.405854	−0.202927	+	0.979194i	$0.565045\pi$
$488$	0	0
$489$	24313.1	0.00459799
$490$	0	0
$491$	−328018.	−0.0614036	−0.0307018	−	0.999529i	$-0.509774\pi$
$491$	−328018.	−0.0614036	−0.0307018	+	0.999529i	$0.509774\pi$
$492$	0	0
$493$	−1.94990e6	−0.361323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−715197.	−0.129878
$498$	0	0
$499$	−1.23315e6	−0.221699	−0.110850	−	0.993837i	$-0.535357\pi$
$499$	−1.23315e6	−0.221699	−0.110850	+	0.993837i	$0.535357\pi$
$500$	0	0
$501$	15217.8	0.00270868
$502$	0	0
$503$	−6.31783e6	−1.11339	−0.556696	−	0.830716i	$-0.687931\pi$
$503$	−6.31783e6	−1.11339	−0.556696	+	0.830716i	$0.687931\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	382212.	0.0660365
$508$	0	0
$509$	−2.21357e6	−0.378702	−0.189351	−	0.981909i	$-0.560638\pi$
$509$	−2.21357e6	−0.378702	−0.189351	+	0.981909i	$0.560638\pi$
$510$	0	0
$511$	3.52839e6	0.597756
$512$	0	0
$513$	258133.	0.0433062
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−599071.	−0.0985716
$518$	0	0
$519$	−168956.	−0.0275332
$520$	0	0
$521$	−7.47689e6	−1.20678	−0.603388	−	0.797448i	$-0.706183\pi$
$521$	−7.47689e6	−1.20678	−0.603388	+	0.797448i	$0.706183\pi$
$522$	0	0
$523$	3.94795e6	0.631127	0.315564	−	0.948904i	$-0.397806\pi$
$523$	3.94795e6	0.631127	0.315564	+	0.948904i	$0.397806\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.83121e6	0.757755
$528$	0	0
$529$	1.51155e7	2.34847
$530$	0	0
$531$	−8.97577e6	−1.38145
$532$	0	0
$533$	−1.31994e7	−2.01249
$534$	0	0
$535$	0	0
$536$	0	0
$537$	363744.	0.0544327
$538$	0	0
$539$	1.60104e6	0.237372
$540$	0	0
$541$	−3.36070e6	−0.493670	−0.246835	−	0.969057i	$-0.579391\pi$
$541$	−3.36070e6	−0.493670	−0.246835	+	0.969057i	$0.579391\pi$
$542$	0	0
$543$	−180791.	−0.0263134
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.96941e6	−0.710127	−0.355064	−	0.934842i	$-0.615541\pi$
$547$	−4.96941e6	−0.710127	−0.355064	+	0.934842i	$0.615541\pi$
$548$	0	0
$549$	2.88589e6	0.408647
$550$	0	0
$551$	−1.24502e6	−0.174702
$552$	0	0
$553$	−157772.	−0.0219390
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.69093e6	0.504079	0.252039	−	0.967717i	$-0.418899\pi$
$557$	3.69093e6	0.504079	0.252039	+	0.967717i	$0.418899\pi$
$558$	0	0
$559$	−7.76665e6	−1.05125
$560$	0	0
$561$	100691.	0.0135078
$562$	0	0
$563$	1.15522e7	1.53600	0.768002	−	0.640447i	$-0.221251\pi$
$563$	1.15522e7	1.53600	0.768002	+	0.640447i	$0.221251\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.52282e6	0.460186
$568$	0	0
$569$	4.16683e6	0.539541	0.269771	−	0.962925i	$-0.413052\pi$
$569$	4.16683e6	0.539541	0.269771	+	0.962925i	$0.413052\pi$
$570$	0	0
$571$	1.39436e7	1.78972	0.894860	−	0.446346i	$-0.147275\pi$
$571$	1.39436e7	1.78972	0.894860	+	0.446346i	$0.147275\pi$
$572$	0	0
$573$	276210.	0.0351441
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.67114e6	1.20931	0.604656	−	0.796487i	$-0.293311\pi$
$577$	9.67114e6	1.20931	0.604656	+	0.796487i	$0.293311\pi$
$578$	0	0
$579$	−225047.	−0.0278982
$580$	0	0
$581$	−6.46103e6	−0.794075
$582$	0	0
$583$	989463.	0.120567
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.06654e6	−0.127756	−0.0638779	−	0.997958i	$-0.520347\pi$
$587$	−1.06654e6	−0.127756	−0.0638779	+	0.997958i	$0.520347\pi$
$588$	0	0
$589$	3.08475e6	0.366379
$590$	0	0
$591$	−252134.	−0.0296936
$592$	0	0
$593$	1.18923e7	1.38876	0.694382	−	0.719607i	$-0.255678\pi$
$593$	1.18923e7	1.38876	0.694382	+	0.719607i	$0.255678\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−380449.	−0.0436878
$598$	0	0
$599$	3.67655e6	0.418672	0.209336	−	0.977844i	$-0.432870\pi$
$599$	3.67655e6	0.418672	0.209336	+	0.977844i	$0.432870\pi$
$600$	0	0
$601$	−9.72241e6	−1.09796	−0.548981	−	0.835835i	$-0.684984\pi$
$601$	−9.72241e6	−1.09796	−0.548981	+	0.835835i	$0.684984\pi$
$602$	0	0
$603$	−200371.	−0.0224410
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.36077e7	−1.49904	−0.749521	−	0.661981i	$-0.769716\pi$
$607$	−1.36077e7	−1.49904	−0.749521	+	0.661981i	$0.769716\pi$
$608$	0	0
$609$	25529.4	0.00278932
$610$	0	0
$611$	5.57221e6	0.603844
$612$	0	0
$613$	1.39912e7	1.50385	0.751923	−	0.659251i	$-0.229126\pi$
$613$	1.39912e7	1.50385	0.751923	+	0.659251i	$0.229126\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.17574e7	1.24336	0.621681	−	0.783270i	$-0.286450\pi$
$617$	1.17574e7	1.24336	0.621681	+	0.783270i	$0.286450\pi$
$618$	0	0
$619$	−1.28405e7	−1.34696	−0.673482	−	0.739203i	$-0.735202\pi$
$619$	−1.28405e7	−1.34696	−0.673482	+	0.739203i	$0.735202\pi$
$620$	0	0
$621$	−962732.	−0.100179
$622$	0	0
$623$	1.92149e6	0.198343
$624$	0	0
$625$	0	0
$626$	0	0
$627$	64291.7	0.00653110
$628$	0	0
$629$	−2.11081e7	−2.12727
$630$	0	0
$631$	9.84645e6	0.984479	0.492239	−	0.870460i	$-0.336179\pi$
$631$	9.84645e6	0.984479	0.492239	+	0.870460i	$0.336179\pi$
$632$	0	0
$633$	−250578.	−0.0248562
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.48919e7	−1.45413
$638$	0	0
$639$	2.90436e6	0.281383
$640$	0	0
$641$	−8.11995e6	−0.780563	−0.390282	−	0.920696i	$-0.627622\pi$
$641$	−8.11995e6	−0.780563	−0.390282	+	0.920696i	$0.627622\pi$
$642$	0	0
$643$	5.61124e6	0.535219	0.267610	−	0.963527i	$-0.413766\pi$
$643$	5.61124e6	0.535219	0.267610	+	0.963527i	$0.413766\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.85057e6	0.831210	0.415605	−	0.909545i	$-0.363570\pi$
$647$	8.85057e6	0.831210	0.415605	+	0.909545i	$0.363570\pi$
$648$	0	0
$649$	−4.47277e6	−0.416836
$650$	0	0
$651$	−63253.4	−0.00584967
$652$	0	0
$653$	−1.55563e6	−0.142765	−0.0713827	−	0.997449i	$-0.522741\pi$
$653$	−1.55563e6	−0.142765	−0.0713827	+	0.997449i	$0.522741\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.43285e7	−1.29505
$658$	0	0
$659$	3.19066e6	0.286198	0.143099	−	0.989708i	$-0.454293\pi$
$659$	3.19066e6	0.286198	0.143099	+	0.989708i	$0.454293\pi$
$660$	0	0
$661$	9.58530e6	0.853301	0.426650	−	0.904417i	$-0.359693\pi$
$661$	9.58530e6	0.853301	0.426650	+	0.904417i	$0.359693\pi$
$662$	0	0
$663$	−936571.	−0.0827479
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.64343e6	0.404133
$668$	0	0
$669$	−53984.4	−0.00466340
$670$	0	0
$671$	1.43808e6	0.123304
$672$	0	0
$673$	1.39002e7	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$673$	1.39002e7	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.61469e6	0.806239	0.403119	−	0.915147i	$-0.367926\pi$
$677$	9.61469e6	0.806239	0.403119	+	0.915147i	$0.367926\pi$
$678$	0	0
$679$	−9.91212e6	−0.825073
$680$	0	0
$681$	−188684.	−0.0155908
$682$	0	0
$683$	−1.83927e7	−1.50867	−0.754333	−	0.656492i	$-0.772039\pi$
$683$	−1.83927e7	−1.50867	−0.754333	+	0.656492i	$0.772039\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−641951.	−0.0518932
$688$	0	0
$689$	−9.20341e6	−0.738586
$690$	0	0
$691$	1.95250e7	1.55560	0.777798	−	0.628514i	$-0.216337\pi$
$691$	1.95250e7	1.55560	0.777798	+	0.628514i	$0.216337\pi$
$692$	0	0
$693$	1.75680e6	0.138960
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.28631e7	1.78260
$698$	0	0
$699$	−394927.	−0.0305720
$700$	0	0
$701$	1.50246e7	1.15480	0.577402	−	0.816460i	$-0.304066\pi$
$701$	1.50246e7	1.15480	0.577402	+	0.816460i	$0.304066\pi$
$702$	0	0
$703$	−1.34776e7	−1.02855
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.49022e6	−0.187365
$708$	0	0
$709$	1.25193e7	0.935329	0.467664	−	0.883906i	$-0.345096\pi$
$709$	1.25193e7	0.935329	0.467664	+	0.883906i	$0.345096\pi$
$710$	0	0
$711$	640700.	0.0475314
$712$	0	0
$713$	−1.15049e7	−0.847535
$714$	0	0
$715$	0	0
$716$	0	0
$717$	60697.2	0.00440931
$718$	0	0
$719$	2.65725e7	1.91694	0.958472	−	0.285186i	$-0.0920555\pi$
$719$	2.65725e7	1.91694	0.958472	+	0.285186i	$0.0920555\pi$
$720$	0	0
$721$	1.74173e6	0.124780
$722$	0	0
$723$	−319825.	−0.0227544
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.16513e7	1.51931	0.759657	−	0.650324i	$-0.225367\pi$
$727$	2.16513e7	1.51931	0.759657	+	0.650324i	$0.225367\pi$
$728$	0	0
$729$	−1.42844e7	−0.995504
$730$	0	0
$731$	1.34529e7	0.931156
$732$	0	0
$733$	−1.64375e6	−0.112999	−0.0564996	−	0.998403i	$-0.517994\pi$
$733$	−1.64375e6	−0.112999	−0.0564996	+	0.998403i	$0.517994\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−99848.2	−0.00677129
$738$	0	0
$739$	−2.56963e7	−1.73085	−0.865425	−	0.501039i	$-0.832951\pi$
$739$	−2.56963e7	−1.73085	−0.865425	+	0.501039i	$0.832951\pi$
$740$	0	0
$741$	−598005.	−0.0400091
$742$	0	0
$743$	−9.65474e6	−0.641606	−0.320803	−	0.947146i	$-0.603953\pi$
$743$	−9.65474e6	−0.641606	−0.320803	+	0.947146i	$0.603953\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.62377e7	1.72038
$748$	0	0
$749$	5.80647e6	0.378188
$750$	0	0
$751$	7.87594e6	0.509569	0.254784	−	0.966998i	$-0.417996\pi$
$751$	7.87594e6	0.509569	0.254784	+	0.966998i	$0.417996\pi$
$752$	0	0
$753$	456445.	0.0293360
$754$	0	0
$755$	0	0
$756$	0	0
$757$	666629.	0.0422809	0.0211405	−	0.999777i	$-0.493270\pi$
$757$	666629.	0.0422809	0.0211405	+	0.999777i	$0.493270\pi$
$758$	0	0
$759$	−239782.	−0.0151082
$760$	0	0
$761$	−1.51451e7	−0.948007	−0.474004	−	0.880523i	$-0.657192\pi$
$761$	−1.51451e7	−0.948007	−0.474004	+	0.880523i	$0.657192\pi$
$762$	0	0
$763$	−9.39757e6	−0.584392
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.16031e7	2.55351
$768$	0	0
$769$	−2.55800e6	−0.155986	−0.0779930	−	0.996954i	$-0.524851\pi$
$769$	−2.55800e6	−0.155986	−0.0779930	+	0.996954i	$0.524851\pi$
$770$	0	0
$771$	−504638.	−0.0305734
$772$	0	0
$773$	2.30532e7	1.38766	0.693831	−	0.720138i	$-0.255922\pi$
$773$	2.30532e7	1.38766	0.693831	+	0.720138i	$0.255922\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	276362.	0.0164220
$778$	0	0
$779$	1.45982e7	0.861896
$780$	0	0
$781$	1.44729e6	0.0849039
$782$	0	0
$783$	−207424.	−0.0120908
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.09486e7	−0.630116	−0.315058	−	0.949072i	$-0.602024\pi$
$787$	−1.09486e7	−0.630116	−0.315058	+	0.949072i	$0.602024\pi$
$788$	0	0
$789$	243886.	0.0139475
$790$	0	0
$791$	3.34675e6	0.190187
$792$	0	0
$793$	−1.33762e7	−0.755355
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.68503e7	1.49728	0.748640	−	0.662976i	$-0.230707\pi$
$797$	2.68503e7	1.49728	0.748640	+	0.662976i	$0.230707\pi$
$798$	0	0
$799$	−9.65183e6	−0.534863
$800$	0	0
$801$	−7.80301e6	−0.429715
$802$	0	0
$803$	−7.14012e6	−0.390766
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−206488.	−0.0111612
$808$	0	0
$809$	−9.94941e6	−0.534474	−0.267237	−	0.963631i	$-0.586111\pi$
$809$	−9.94941e6	−0.534474	−0.267237	+	0.963631i	$0.586111\pi$
$810$	0	0
$811$	−3.48705e7	−1.86168	−0.930841	−	0.365425i	$-0.880924\pi$
$811$	−3.48705e7	−1.86168	−0.930841	+	0.365425i	$0.880924\pi$
$812$	0	0
$813$	−248823.	−0.0132027
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.58973e6	0.450220
$818$	0	0
$819$	−1.63407e7	−0.851259
$820$	0	0
$821$	2.58268e7	1.33725	0.668626	−	0.743599i	$-0.266883\pi$
$821$	2.58268e7	1.33725	0.668626	+	0.743599i	$0.266883\pi$
$822$	0	0
$823$	2.41044e7	1.24050	0.620251	−	0.784403i	$-0.287031\pi$
$823$	2.41044e7	1.24050	0.620251	+	0.784403i	$0.287031\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.11476e7	−1.07522	−0.537610	−	0.843193i	$-0.680673\pi$
$827$	−2.11476e7	−1.07522	−0.537610	+	0.843193i	$0.680673\pi$
$828$	0	0
$829$	−2.79313e7	−1.41158	−0.705789	−	0.708422i	$-0.749407\pi$
$829$	−2.79313e7	−1.41158	−0.705789	+	0.708422i	$0.749407\pi$
$830$	0	0
$831$	519238.	0.0260834
$832$	0	0
$833$	2.57948e7	1.28801
$834$	0	0
$835$	0	0
$836$	0	0
$837$	513927.	0.0253564
$838$	0	0
$839$	−2.08785e7	−1.02399	−0.511994	−	0.858989i	$-0.671093\pi$
$839$	−2.08785e7	−1.02399	−0.511994	+	0.858989i	$0.671093\pi$
$840$	0	0
$841$	−1.95107e7	−0.951224
$842$	0	0
$843$	338056.	0.0163840
$844$	0	0
$845$	0	0
$846$	0	0
$847$	875441.	0.0419294
$848$	0	0
$849$	−858298.	−0.0408666
$850$	0	0
$851$	5.02661e7	2.37931
$852$	0	0
$853$	7.61166e6	0.358185	0.179092	−	0.983832i	$-0.442684\pi$
$853$	7.61166e6	0.358185	0.179092	+	0.983832i	$0.442684\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.51792e7	−0.705988	−0.352994	−	0.935626i	$-0.614836\pi$
$857$	−1.51792e7	−0.705988	−0.352994	+	0.935626i	$0.614836\pi$
$858$	0	0
$859$	−6.25178e6	−0.289082	−0.144541	−	0.989499i	$-0.546170\pi$
$859$	−6.25178e6	−0.289082	−0.144541	+	0.989499i	$0.546170\pi$
$860$	0	0
$861$	−299339.	−0.0137612
$862$	0	0
$863$	2.05078e7	0.937331	0.468665	−	0.883376i	$-0.344735\pi$
$863$	2.05078e7	0.937331	0.468665	+	0.883376i	$0.344735\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.01618e6	0.0459118
$868$	0	0
$869$	319271.	0.0143420
$870$	0	0
$871$	928730.	0.0414805
$872$	0	0
$873$	4.02524e7	1.78754
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.50576e7	1.10012	0.550062	−	0.835124i	$-0.314604\pi$
$877$	2.50576e7	1.10012	0.550062	+	0.835124i	$0.314604\pi$
$878$	0	0
$879$	236813.	0.0103379
$880$	0	0
$881$	1.29790e7	0.563380	0.281690	−	0.959505i	$-0.409105\pi$
$881$	1.29790e7	0.563380	0.281690	+	0.959505i	$0.409105\pi$
$882$	0	0
$883$	3.40078e7	1.46783	0.733916	−	0.679240i	$-0.237691\pi$
$883$	3.40078e7	1.46783	0.733916	+	0.679240i	$0.237691\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.23924e7	−1.38240	−0.691200	−	0.722664i	$-0.742918\pi$
$887$	−3.23924e7	−1.38240	−0.691200	+	0.722664i	$0.742918\pi$
$888$	0	0
$889$	7.39999e6	0.314034
$890$	0	0
$891$	−7.12886e6	−0.300833
$892$	0	0
$893$	−6.16274e6	−0.258610
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.23032e6	0.0925520
$898$	0	0
$899$	−2.47876e6	−0.102291
$900$	0	0
$901$	1.59416e7	0.654212
$902$	0	0
$903$	−176134.	−0.00718827
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.26335e7	0.913551	0.456775	−	0.889582i	$-0.349004\pi$
$907$	2.26335e7	0.913551	0.456775	+	0.889582i	$0.349004\pi$
$908$	0	0
$909$	1.01126e7	0.405931
$910$	0	0
$911$	1.28865e7	0.514447	0.257223	−	0.966352i	$-0.417192\pi$
$911$	1.28865e7	0.514447	0.257223	+	0.966352i	$0.417192\pi$
$912$	0	0
$913$	1.30747e7	0.519104
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.38324e7	−0.543218
$918$	0	0
$919$	−4.65224e7	−1.81708	−0.908539	−	0.417801i	$-0.862801\pi$
$919$	−4.65224e7	−1.81708	−0.908539	+	0.417801i	$0.862801\pi$
$920$	0	0
$921$	−825008.	−0.0320486
$922$	0	0
$923$	−1.34618e7	−0.520116
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.07304e6	−0.270338
$928$	0	0
$929$	1.51531e7	0.576053	0.288026	−	0.957623i	$-0.407001\pi$
$929$	1.51531e7	0.576053	0.288026	+	0.957623i	$0.407001\pi$
$930$	0	0
$931$	1.64701e7	0.622762
$932$	0	0
$933$	248687.	0.00935295
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.58125e7	1.70465	0.852325	−	0.523013i	$-0.175192\pi$
$937$	4.58125e7	1.70465	0.852325	+	0.523013i	$0.175192\pi$
$938$	0	0
$939$	−102650.	−0.00379921
$940$	0	0
$941$	1.22426e7	0.450712	0.225356	−	0.974276i	$-0.427645\pi$
$941$	1.22426e7	0.450712	0.225356	+	0.974276i	$0.427645\pi$
$942$	0	0
$943$	−5.44453e7	−1.99380
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.18830e7	0.430577	0.215289	−	0.976550i	$-0.430931\pi$
$947$	1.18830e7	0.430577	0.215289	+	0.976550i	$0.430931\pi$
$948$	0	0
$949$	6.64133e7	2.39381
$950$	0	0
$951$	−884618.	−0.0317179
$952$	0	0
$953$	3.26639e7	1.16503	0.582513	−	0.812821i	$-0.302069\pi$
$953$	3.26639e7	1.16503	0.582513	+	0.812821i	$0.302069\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−51661.9	−0.00182344
$958$	0	0
$959$	−802971.	−0.0281938
$960$	0	0
$961$	−2.24876e7	−0.785479
$962$	0	0
$963$	−2.35796e7	−0.819353
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.06332e7	0.709579	0.354790	−	0.934946i	$-0.384553\pi$
$967$	2.06332e7	0.709579	0.354790	+	0.934946i	$0.384553\pi$
$968$	0	0
$969$	1.03583e6	0.0354386
$970$	0	0
$971$	2.69684e7	0.917926	0.458963	−	0.888455i	$-0.348221\pi$
$971$	2.69684e7	0.917926	0.458963	+	0.888455i	$0.348221\pi$
$972$	0	0
$973$	−8.21201e6	−0.278079
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.57039e7	1.19668	0.598342	−	0.801241i	$-0.295826\pi$
$977$	3.57039e7	1.19668	0.598342	+	0.801241i	$0.295826\pi$
$978$	0	0
$979$	−3.88836e6	−0.129661
$980$	0	0
$981$	3.81628e7	1.26610
$982$	0	0
$983$	1.97214e7	0.650958	0.325479	−	0.945549i	$-0.394474\pi$
$983$	1.97214e7	0.650958	0.325479	+	0.945549i	$0.394474\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	126368.	0.00412900
$988$	0	0
$989$	−3.20363e7	−1.04148
$990$	0	0
$991$	−1.93715e7	−0.626582	−0.313291	−	0.949657i	$-0.601432\pi$
$991$	−1.93715e7	−0.626582	−0.313291	+	0.949657i	$0.601432\pi$
$992$	0	0
$993$	−1.17276e6	−0.0377429
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.92775e7	1.25143	0.625713	−	0.780053i	$-0.284808\pi$
$997$	3.92775e7	1.25143	0.625713	+	0.780053i	$0.284808\pi$
$998$	0	0
$999$	−2.24541e6	−0.0711838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.i.1.5	✓	8
5.2	odd	4		1100.6.b.h.749.8		16
5.3	odd	4		1100.6.b.h.749.9		16
5.4	even	2		1100.6.a.j.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.6.a.i.1.5	✓	8	1.1	even	1	trivial
1100.6.a.j.1.4	yes	8	5.4	even	2
1100.6.b.h.749.8		16	5.2	odd	4
1100.6.b.h.749.9		16	5.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q+0.426864 q^{3} +59.7938 q^{7} -242.818 q^{9} -121.000 q^{11} +1125.47 q^{13} -1949.47 q^{17} -1244.75 q^{19} +25.5238 q^{21} +4642.40 q^{23} -207.378 q^{27} +1000.22 q^{29} -2478.21 q^{31} -51.6505 q^{33} +10827.6 q^{37} +480.423 q^{39} -11727.8 q^{41} -6900.79 q^{43} +4951.00 q^{47} -13231.7 q^{49} -832.158 q^{51} -8177.38 q^{53} -531.337 q^{57} +36965.0 q^{59} -11885.0 q^{61} -14519.0 q^{63} +825.191 q^{67} +1981.67 q^{69} -11961.1 q^{71} +59009.3 q^{73} -7235.05 q^{77} -2638.60 q^{79} +58916.2 q^{81} -108055. q^{83} +426.958 q^{87} +32135.2 q^{89} +67296.2 q^{91} -1057.86 q^{93} -165772. q^{97} +29381.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) \) 8 * q - 9 * q^3 - 175 * q^7 + 377 * q^9 - 968 * q^11 + 1072 * q^13 - 37 * q^17 - 1356 * q^19 + 5462 * q^21 - 1327 * q^23 - 2670 * q^27 - 11785 * q^29 + 3532 * q^31 + 1089 * q^33 + 8176 * q^37 - 16831 * q^39 + 15846 * q^41 + 530 * q^43 + 9252 * q^47 + 45745 * q^49 + 47733 * q^51 + 24991 * q^53 - 41957 * q^57 + 1440 * q^59 - 54447 * q^61 - 108968 * q^63 - 109232 * q^67 + 134200 * q^69 - 45026 * q^71 + 2221 * q^73 + 21175 * q^77 + 74369 * q^79 - 1132 * q^81 - 5443 * q^83 - 18665 * q^87 + 106151 * q^89 - 230723 * q^91 + 111315 * q^93 - 75273 * q^97 - 45617 * q^99

Introduction

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