LMFDB - Embedded newform 1100.6.a.m.1.2

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2272 x^{12} + 1983198 x^{10} - 827062096 x^{8} + 165415157329 x^{6} - 13843733383152 x^{4} + \cdots - 27\!\cdots\!24 $ x^14 - 2272x^12 + 1983198x^10 - 827062096x^8 + 165415157329x^6 - 13843733383152x^4 + 402818574022128x^2 - 2766293773242624
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{2}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-24.5916$ 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)

$f(q)$	$=$	$q-24.5916 q^{3} -68.6167 q^{7} +361.745 q^{9} +121.000 q^{11} -322.943 q^{13} -1078.94 q^{17} -2901.39 q^{19} +1687.39 q^{21} +1136.77 q^{23} -2920.12 q^{27} +732.481 q^{29} +6971.80 q^{31} -2975.58 q^{33} -3293.91 q^{37} +7941.68 q^{39} +15016.5 q^{41} -15377.2 q^{43} +6002.58 q^{47} -12098.7 q^{49} +26532.8 q^{51} -35344.7 q^{53} +71349.6 q^{57} +20735.0 q^{59} -44901.6 q^{61} -24821.7 q^{63} -67912.8 q^{67} -27954.9 q^{69} +11219.8 q^{71} -51188.9 q^{73} -8302.62 q^{77} -49679.5 q^{79} -16093.8 q^{81} -1551.85 q^{83} -18012.8 q^{87} -103366. q^{89} +22159.3 q^{91} -171447. q^{93} +541.081 q^{97} +43771.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 1142 q^{9} + 1694 q^{11} + 4540 q^{19} + 3824 q^{21} + 9972 q^{29} + 19076 q^{31} + 13616 q^{39} + 15052 q^{41} + 55346 q^{49} - 13380 q^{51} + 2108 q^{59} + 11660 q^{61} - 1620 q^{69} - 89284 q^{71}+ \cdots + 138182 q^{99}+O(q^{100}) $ 14 * q + 1142 * q^9 + 1694 * q^11 + 4540 * q^19 + 3824 * q^21 + 9972 * q^29 + 19076 * q^31 + 13616 * q^39 + 15052 * q^41 + 55346 * q^49 - 13380 * q^51 + 2108 * q^59 + 11660 * q^61 - 1620 * q^69 - 89284 * q^71 - 19432 * q^79 - 94714 * q^81 + 38836 * q^89 + 39632 * q^91 + 138182 * 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
$3$	−24.5916	−1.57755	−0.788775	−	0.614683i	$-0.789284\pi$
$3$	−24.5916	−1.57755	−0.788775	+	0.614683i	$0.789284\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−68.6167	−0.529279	−0.264640	−	0.964347i	$-0.585253\pi$
$7$	−68.6167	−0.529279	−0.264640	+	0.964347i	$0.585253\pi$
$8$	0	0
$9$	361.745	1.48866
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−322.943	−0.529991	−0.264995	−	0.964250i	$-0.585370\pi$
$13$	−322.943	−0.529991	−0.264995	+	0.964250i	$0.585370\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1078.94	−0.905471	−0.452736	−	0.891645i	$-0.649552\pi$
$17$	−1078.94	−0.905471	−0.452736	+	0.891645i	$0.649552\pi$
$18$	0	0
$19$	−2901.39	−1.84383	−0.921917	−	0.387387i	$-0.873378\pi$
$19$	−2901.39	−1.84383	−0.921917	+	0.387387i	$0.873378\pi$
$20$	0	0
$21$	1687.39	0.834964
$22$	0	0
$23$	1136.77	0.448076	0.224038	−	0.974580i	$-0.428076\pi$
$23$	1136.77	0.448076	0.224038	+	0.974580i	$0.428076\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2920.12	−0.770887
$28$	0	0
$29$	732.481	0.161734	0.0808670	−	0.996725i	$-0.474231\pi$
$29$	732.481	0.161734	0.0808670	+	0.996725i	$0.474231\pi$
$30$	0	0
$31$	6971.80	1.30299	0.651495	−	0.758653i	$-0.274142\pi$
$31$	6971.80	1.30299	0.651495	+	0.758653i	$0.274142\pi$
$32$	0	0
$33$	−2975.58	−0.475649
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3293.91	−0.395556	−0.197778	−	0.980247i	$-0.563373\pi$
$37$	−3293.91	−0.395556	−0.197778	+	0.980247i	$0.563373\pi$
$38$	0	0
$39$	7941.68	0.836086
$40$	0	0
$41$	15016.5	1.39511	0.697555	−	0.716531i	$-0.254271\pi$
$41$	15016.5	1.39511	0.697555	+	0.716531i	$0.254271\pi$
$42$	0	0
$43$	−15377.2	−1.26825	−0.634127	−	0.773229i	$-0.718640\pi$
$43$	−15377.2	−1.26825	−0.634127	+	0.773229i	$0.718640\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6002.58	0.396363	0.198182	−	0.980165i	$-0.436496\pi$
$47$	6002.58	0.396363	0.198182	+	0.980165i	$0.436496\pi$
$48$	0	0
$49$	−12098.7	−0.719864
$50$	0	0
$51$	26532.8	1.42842
$52$	0	0
$53$	−35344.7	−1.72836	−0.864180	−	0.503183i	$-0.832162\pi$
$53$	−35344.7	−1.72836	−0.864180	+	0.503183i	$0.832162\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	71349.6	2.90874
$58$	0	0
$59$	20735.0	0.775486	0.387743	−	0.921768i	$-0.373255\pi$
$59$	20735.0	0.775486	0.387743	+	0.921768i	$0.373255\pi$
$60$	0	0
$61$	−44901.6	−1.54503	−0.772516	−	0.634995i	$-0.781002\pi$
$61$	−44901.6	−1.54503	−0.772516	+	0.634995i	$0.781002\pi$
$62$	0	0
$63$	−24821.7	−0.787917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−67912.8	−1.84827	−0.924133	−	0.382071i	$-0.875211\pi$
$67$	−67912.8	−1.84827	−0.924133	+	0.382071i	$0.875211\pi$
$68$	0	0
$69$	−27954.9	−0.706862
$70$	0	0
$71$	11219.8	0.264142	0.132071	−	0.991240i	$-0.457837\pi$
$71$	11219.8	0.264142	0.132071	+	0.991240i	$0.457837\pi$
$72$	0	0
$73$	−51188.9	−1.12426	−0.562132	−	0.827047i	$-0.690019\pi$
$73$	−51188.9	−1.12426	−0.562132	+	0.827047i	$0.690019\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8302.62	−0.159584
$78$	0	0
$79$	−49679.5	−0.895591	−0.447795	−	0.894136i	$-0.647791\pi$
$79$	−49679.5	−0.895591	−0.447795	+	0.894136i	$0.647791\pi$
$80$	0	0
$81$	−16093.8	−0.272549
$82$	0	0
$83$	−1551.85	−0.0247260	−0.0123630	−	0.999924i	$-0.503935\pi$
$83$	−1551.85	−0.0247260	−0.0123630	+	0.999924i	$0.503935\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18012.8	−0.255143
$88$	0	0
$89$	−103366.	−1.38325	−0.691627	−	0.722255i	$-0.743106\pi$
$89$	−103366.	−1.38325	−0.691627	+	0.722255i	$0.743106\pi$
$90$	0	0
$91$	22159.3	0.280513
$92$	0	0
$93$	−171447.	−2.05553
$94$	0	0
$95$	0	0
$96$	0	0
$97$	541.081	0.00583893	0.00291946	−	0.999996i	$-0.499071\pi$
$97$	541.081	0.00583893	0.00291946	+	0.999996i	$0.499071\pi$
$98$	0	0
$99$	43771.1	0.448848
$100$	0	0
$101$	6404.26	0.0624691	0.0312346	−	0.999512i	$-0.490056\pi$
$101$	6404.26	0.0624691	0.0312346	+	0.999512i	$0.490056\pi$
$102$	0	0
$103$	45740.8	0.424826	0.212413	−	0.977180i	$-0.431868\pi$
$103$	45740.8	0.424826	0.212413	+	0.977180i	$0.431868\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−95191.4	−0.803782	−0.401891	−	0.915687i	$-0.631647\pi$
$107$	−95191.4	−0.803782	−0.401891	+	0.915687i	$0.631647\pi$
$108$	0	0
$109$	−55413.3	−0.446733	−0.223366	−	0.974735i	$-0.571705\pi$
$109$	−55413.3	−0.446733	−0.223366	+	0.974735i	$0.571705\pi$
$110$	0	0
$111$	81002.5	0.624009
$112$	0	0
$113$	−93724.8	−0.690492	−0.345246	−	0.938512i	$-0.612204\pi$
$113$	−93724.8	−0.690492	−0.345246	+	0.938512i	$0.612204\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−116823.	−0.788976
$118$	0	0
$119$	74033.2	0.479247
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−369279.	−2.20085
$124$	0	0
$125$	0	0
$126$	0	0
$127$	29470.4	0.162135	0.0810675	−	0.996709i	$-0.474167\pi$
$127$	29470.4	0.162135	0.0810675	+	0.996709i	$0.474167\pi$
$128$	0	0
$129$	378149.	2.00073
$130$	0	0
$131$	−39243.8	−0.199799	−0.0998993	−	0.994998i	$-0.531852\pi$
$131$	−39243.8	−0.199799	−0.0998993	+	0.994998i	$0.531852\pi$
$132$	0	0
$133$	199084.	0.975903
$134$	0	0
$135$	0	0
$136$	0	0
$137$	337986.	1.53850	0.769249	−	0.638949i	$-0.220631\pi$
$137$	337986.	1.53850	0.769249	+	0.638949i	$0.220631\pi$
$138$	0	0
$139$	18346.5	0.0805407	0.0402704	−	0.999189i	$-0.487178\pi$
$139$	18346.5	0.0805407	0.0402704	+	0.999189i	$0.487178\pi$
$140$	0	0
$141$	−147613.	−0.625282
$142$	0	0
$143$	−39076.2	−0.159798
$144$	0	0
$145$	0	0
$146$	0	0
$147$	297527.	1.13562
$148$	0	0
$149$	146424.	0.540314	0.270157	−	0.962816i	$-0.412924\pi$
$149$	146424.	0.540314	0.270157	+	0.962816i	$0.412924\pi$
$150$	0	0
$151$	290139.	1.03553	0.517766	−	0.855522i	$-0.326764\pi$
$151$	290139.	1.03553	0.517766	+	0.855522i	$0.326764\pi$
$152$	0	0
$153$	−390300.	−1.34794
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−597096.	−1.93328	−0.966640	−	0.256138i	$-0.917550\pi$
$157$	−597096.	−1.93328	−0.966640	+	0.256138i	$0.917550\pi$
$158$	0	0
$159$	869180.	2.72657
$160$	0	0
$161$	−78001.2	−0.237157
$162$	0	0
$163$	112140.	0.330591	0.165296	−	0.986244i	$-0.447142\pi$
$163$	112140.	0.330591	0.165296	+	0.986244i	$0.447142\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−504282.	−1.39921	−0.699604	−	0.714530i	$-0.746640\pi$
$167$	−504282.	−1.39921	−0.699604	+	0.714530i	$0.746640\pi$
$168$	0	0
$169$	−267001.	−0.719110
$170$	0	0
$171$	−1.04956e6	−2.74484
$172$	0	0
$173$	143639.	0.364886	0.182443	−	0.983216i	$-0.441600\pi$
$173$	143639.	0.364886	0.182443	+	0.983216i	$0.441600\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−509906.	−1.22337
$178$	0	0
$179$	429976.	1.00302	0.501512	−	0.865151i	$-0.332777\pi$
$179$	429976.	1.00302	0.501512	+	0.865151i	$0.332777\pi$
$180$	0	0
$181$	491559.	1.11527	0.557634	−	0.830087i	$-0.311709\pi$
$181$	491559.	1.11527	0.557634	+	0.830087i	$0.311709\pi$
$182$	0	0
$183$	1.10420e6	2.43736
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−130552.	−0.273010
$188$	0	0
$189$	200369.	0.408014
$190$	0	0
$191$	514282.	1.02004	0.510021	−	0.860162i	$-0.329638\pi$
$191$	514282.	1.02004	0.510021	+	0.860162i	$0.329638\pi$
$192$	0	0
$193$	−484037.	−0.935374	−0.467687	−	0.883894i	$-0.654913\pi$
$193$	−484037.	−0.935374	−0.467687	+	0.883894i	$0.654913\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	44622.0	0.0819188	0.0409594	−	0.999161i	$-0.486959\pi$
$197$	44622.0	0.0819188	0.0409594	+	0.999161i	$0.486959\pi$
$198$	0	0
$199$	−550206.	−0.984900	−0.492450	−	0.870341i	$-0.663899\pi$
$199$	−550206.	−0.984900	−0.492450	+	0.870341i	$0.663899\pi$
$200$	0	0
$201$	1.67008e6	2.91573
$202$	0	0
$203$	−50260.4	−0.0856024
$204$	0	0
$205$	0	0
$206$	0	0
$207$	411219.	0.667034
$208$	0	0
$209$	−351068.	−0.555937
$210$	0	0
$211$	−1.17040e6	−1.80979	−0.904894	−	0.425636i	$-0.860050\pi$
$211$	−1.17040e6	−1.80979	−0.904894	+	0.425636i	$0.860050\pi$
$212$	0	0
$213$	−275911.	−0.416697
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−478382.	−0.689645
$218$	0	0
$219$	1.25881e6	1.77358
$220$	0	0
$221$	348436.	0.479891
$222$	0	0
$223$	957130.	1.28887	0.644434	−	0.764660i	$-0.277093\pi$
$223$	957130.	1.28887	0.644434	+	0.764660i	$0.277093\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−411895.	−0.530544	−0.265272	−	0.964174i	$-0.585462\pi$
$227$	−411895.	−0.530544	−0.265272	+	0.964174i	$0.585462\pi$
$228$	0	0
$229$	−304437.	−0.383627	−0.191813	−	0.981431i	$-0.561437\pi$
$229$	−304437.	−0.383627	−0.191813	+	0.981431i	$0.561437\pi$
$230$	0	0
$231$	204174.	0.251751
$232$	0	0
$233$	−15539.4	−0.0187519	−0.00937593	−	0.999956i	$-0.502984\pi$
$233$	−15539.4	−0.0187519	−0.00937593	+	0.999956i	$0.502984\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.22170e6	1.41284
$238$	0	0
$239$	794336.	0.899517	0.449758	−	0.893150i	$-0.351510\pi$
$239$	794336.	0.899517	0.449758	+	0.893150i	$0.351510\pi$
$240$	0	0
$241$	−1.42354e6	−1.57880	−0.789400	−	0.613879i	$-0.789608\pi$
$241$	−1.42354e6	−1.57880	−0.789400	+	0.613879i	$0.789608\pi$
$242$	0	0
$243$	1.10536e6	1.20085
$244$	0	0
$245$	0	0
$246$	0	0
$247$	936984.	0.977215
$248$	0	0
$249$	38162.4	0.0390065
$250$	0	0
$251$	45704.8	0.0457908	0.0228954	−	0.999738i	$-0.492712\pi$
$251$	45704.8	0.0457908	0.0228954	+	0.999738i	$0.492712\pi$
$252$	0	0
$253$	137549.	0.135100
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−258018.	−0.243679	−0.121839	−	0.992550i	$-0.538879\pi$
$257$	−258018.	−0.243679	−0.121839	+	0.992550i	$0.538879\pi$
$258$	0	0
$259$	226018.	0.209360
$260$	0	0
$261$	264971.	0.240767
$262$	0	0
$263$	523132.	0.466360	0.233180	−	0.972434i	$-0.425087\pi$
$263$	523132.	0.466360	0.233180	+	0.972434i	$0.425087\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.54193e6	2.18215
$268$	0	0
$269$	1.02600e6	0.864503	0.432251	−	0.901753i	$-0.357719\pi$
$269$	1.02600e6	0.864503	0.432251	+	0.901753i	$0.357719\pi$
$270$	0	0
$271$	1.55354e6	1.28499	0.642494	−	0.766291i	$-0.277900\pi$
$271$	1.55354e6	1.28499	0.642494	+	0.766291i	$0.277900\pi$
$272$	0	0
$273$	−544932.	−0.442523
$274$	0	0
$275$	0	0
$276$	0	0
$277$	742412.	0.581361	0.290680	−	0.956820i	$-0.406118\pi$
$277$	742412.	0.581361	0.290680	+	0.956820i	$0.406118\pi$
$278$	0	0
$279$	2.52201e6	1.93971
$280$	0	0
$281$	−1.47286e6	−1.11274	−0.556372	−	0.830933i	$-0.687807\pi$
$281$	−1.47286e6	−1.11274	−0.556372	+	0.830933i	$0.687807\pi$
$282$	0	0
$283$	−1.26344e6	−0.937756	−0.468878	−	0.883263i	$-0.655342\pi$
$283$	−1.26344e6	−0.937756	−0.468878	+	0.883263i	$0.655342\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.03038e6	−0.738403
$288$	0	0
$289$	−255748.	−0.180122
$290$	0	0
$291$	−13306.0	−0.00921119
$292$	0	0
$293$	1.50653e6	1.02520	0.512600	−	0.858627i	$-0.328682\pi$
$293$	1.50653e6	1.02520	0.512600	+	0.858627i	$0.328682\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−353334.	−0.232431
$298$	0	0
$299$	−367112.	−0.237476
$300$	0	0
$301$	1.05513e6	0.671260
$302$	0	0
$303$	−157491.	−0.0985481
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.16775e6	1.31269	0.656346	−	0.754460i	$-0.272101\pi$
$307$	2.16775e6	1.31269	0.656346	+	0.754460i	$0.272101\pi$
$308$	0	0
$309$	−1.12484e6	−0.670184
$310$	0	0
$311$	1.82214e6	1.06827	0.534135	−	0.845399i	$-0.320637\pi$
$311$	1.82214e6	1.06827	0.534135	+	0.845399i	$0.320637\pi$
$312$	0	0
$313$	−475864.	−0.274550	−0.137275	−	0.990533i	$-0.543834\pi$
$313$	−475864.	−0.274550	−0.137275	+	0.990533i	$0.543834\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	763607.	0.426797	0.213399	−	0.976965i	$-0.431547\pi$
$317$	763607.	0.426797	0.213399	+	0.976965i	$0.431547\pi$
$318$	0	0
$319$	88630.2	0.0487646
$320$	0	0
$321$	2.34091e6	1.26801
$322$	0	0
$323$	3.13042e6	1.66954
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.36270e6	0.704743
$328$	0	0
$329$	−411877.	−0.209787
$330$	0	0
$331$	1.13499e6	0.569407	0.284703	−	0.958616i	$-0.408105\pi$
$331$	1.13499e6	0.569407	0.284703	+	0.958616i	$0.408105\pi$
$332$	0	0
$333$	−1.19156e6	−0.588849
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.08011e6	−0.518078	−0.259039	−	0.965867i	$-0.583406\pi$
$337$	−1.08011e6	−0.518078	−0.259039	+	0.965867i	$0.583406\pi$
$338$	0	0
$339$	2.30484e6	1.08928
$340$	0	0
$341$	843588.	0.392866
$342$	0	0
$343$	1.98342e6	0.910288
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.31228e6	1.03090	0.515449	−	0.856920i	$-0.327625\pi$
$347$	2.31228e6	1.03090	0.515449	+	0.856920i	$0.327625\pi$
$348$	0	0
$349$	3.32947e6	1.46323	0.731614	−	0.681719i	$-0.238767\pi$
$349$	3.32947e6	1.46323	0.731614	+	0.681719i	$0.238767\pi$
$350$	0	0
$351$	943032.	0.408563
$352$	0	0
$353$	3.35633e6	1.43360	0.716801	−	0.697278i	$-0.245606\pi$
$353$	3.35633e6	1.43360	0.716801	+	0.697278i	$0.245606\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.82059e6	−0.756035
$358$	0	0
$359$	600243.	0.245805	0.122903	−	0.992419i	$-0.460780\pi$
$359$	600243.	0.245805	0.122903	+	0.992419i	$0.460780\pi$
$360$	0	0
$361$	5.94195e6	2.39972
$362$	0	0
$363$	−360045.	−0.143414
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.60152e6	−1.00824	−0.504118	−	0.863635i	$-0.668182\pi$
$367$	−2.60152e6	−1.00824	−0.504118	+	0.863635i	$0.668182\pi$
$368$	0	0
$369$	5.43213e6	2.07685
$370$	0	0
$371$	2.42523e6	0.914784
$372$	0	0
$373$	−4.00605e6	−1.49089	−0.745443	−	0.666570i	$-0.767762\pi$
$373$	−4.00605e6	−1.49089	−0.745443	+	0.666570i	$0.767762\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−236550.	−0.0857175
$378$	0	0
$379$	1.27812e6	0.457059	0.228529	−	0.973537i	$-0.426608\pi$
$379$	1.27812e6	0.457059	0.228529	+	0.973537i	$0.426608\pi$
$380$	0	0
$381$	−724723.	−0.255776
$382$	0	0
$383$	1.93797e6	0.675073	0.337536	−	0.941312i	$-0.390406\pi$
$383$	1.93797e6	0.675073	0.337536	+	0.941312i	$0.390406\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.56262e6	−1.88800
$388$	0	0
$389$	−1.01122e6	−0.338823	−0.169412	−	0.985545i	$-0.554187\pi$
$389$	−1.01122e6	−0.338823	−0.169412	+	0.985545i	$0.554187\pi$
$390$	0	0
$391$	−1.22650e6	−0.405720
$392$	0	0
$393$	965065.	0.315192
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.31935e6	0.738566	0.369283	−	0.929317i	$-0.379603\pi$
$397$	2.31935e6	0.738566	0.369283	+	0.929317i	$0.379603\pi$
$398$	0	0
$399$	−4.89578e6	−1.53953
$400$	0	0
$401$	159220.	0.0494467	0.0247233	−	0.999694i	$-0.492130\pi$
$401$	159220.	0.0494467	0.0247233	+	0.999694i	$0.492130\pi$
$402$	0	0
$403$	−2.25150e6	−0.690572
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−398564.	−0.119265
$408$	0	0
$409$	5.97006e6	1.76470	0.882348	−	0.470597i	$-0.155961\pi$
$409$	5.97006e6	1.76470	0.882348	+	0.470597i	$0.155961\pi$
$410$	0	0
$411$	−8.31159e6	−2.42706
$412$	0	0
$413$	−1.42277e6	−0.410448
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−451168.	−0.127057
$418$	0	0
$419$	5.36018e6	1.49157	0.745786	−	0.666186i	$-0.232074\pi$
$419$	5.36018e6	1.49157	0.745786	+	0.666186i	$0.232074\pi$
$420$	0	0
$421$	−4.23113e6	−1.16346	−0.581729	−	0.813383i	$-0.697624\pi$
$421$	−4.23113e6	−1.16346	−0.581729	+	0.813383i	$0.697624\pi$
$422$	0	0
$423$	2.17140e6	0.590050
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.08100e6	0.817753
$428$	0	0
$429$	960944.	0.252089
$430$	0	0
$431$	−7.37436e6	−1.91219	−0.956096	−	0.293055i	$-0.905328\pi$
$431$	−7.37436e6	−1.91219	−0.956096	+	0.293055i	$0.905328\pi$
$432$	0	0
$433$	−4.04144e6	−1.03590	−0.517948	−	0.855412i	$-0.673304\pi$
$433$	−4.04144e6	−1.03590	−0.517948	+	0.855412i	$0.673304\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.29820e6	−0.826178
$438$	0	0
$439$	−3.79231e6	−0.939165	−0.469583	−	0.882889i	$-0.655596\pi$
$439$	−3.79231e6	−0.939165	−0.469583	+	0.882889i	$0.655596\pi$
$440$	0	0
$441$	−4.37666e6	−1.07163
$442$	0	0
$443$	−5.95941e6	−1.44276	−0.721380	−	0.692540i	$-0.756492\pi$
$443$	−5.95941e6	−1.44276	−0.721380	+	0.692540i	$0.756492\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.60079e6	−0.852372
$448$	0	0
$449$	5.90973e6	1.38341	0.691706	−	0.722179i	$-0.256859\pi$
$449$	5.90973e6	1.38341	0.691706	+	0.722179i	$0.256859\pi$
$450$	0	0
$451$	1.81699e6	0.420641
$452$	0	0
$453$	−7.13497e6	−1.63360
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.15000e6	−0.705537	−0.352769	−	0.935711i	$-0.614760\pi$
$457$	−3.15000e6	−0.705537	−0.352769	+	0.935711i	$0.614760\pi$
$458$	0	0
$459$	3.15063e6	0.698016
$460$	0	0
$461$	−8.43560e6	−1.84869	−0.924344	−	0.381561i	$-0.875387\pi$
$461$	−8.43560e6	−1.84869	−0.924344	+	0.381561i	$0.875387\pi$
$462$	0	0
$463$	−4.81421e6	−1.04369	−0.521846	−	0.853040i	$-0.674756\pi$
$463$	−4.81421e6	−1.04369	−0.521846	+	0.853040i	$0.674756\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.34298e6	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$467$	1.34298e6	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$468$	0	0
$469$	4.65995e6	0.978248
$470$	0	0
$471$	1.46835e7	3.04984
$472$	0	0
$473$	−1.86064e6	−0.382393
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.27857e7	−2.57294
$478$	0	0
$479$	−2.57179e6	−0.512149	−0.256074	−	0.966657i	$-0.582429\pi$
$479$	−2.57179e6	−0.512149	−0.256074	+	0.966657i	$0.582429\pi$
$480$	0	0
$481$	1.06375e6	0.209641
$482$	0	0
$483$	1.91817e6	0.374127
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.80826e6	1.10975	0.554874	−	0.831935i	$-0.312767\pi$
$487$	5.80826e6	1.10975	0.554874	+	0.831935i	$0.312767\pi$
$488$	0	0
$489$	−2.75770e6	−0.521524
$490$	0	0
$491$	−7.13394e6	−1.33544	−0.667722	−	0.744411i	$-0.732730\pi$
$491$	−7.13394e6	−1.33544	−0.667722	+	0.744411i	$0.732730\pi$
$492$	0	0
$493$	−790302.	−0.146445
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−769863.	−0.139805
$498$	0	0
$499$	5.11035e6	0.918755	0.459377	−	0.888241i	$-0.348073\pi$
$499$	5.11035e6	0.918755	0.459377	+	0.888241i	$0.348073\pi$
$500$	0	0
$501$	1.24011e7	2.20732
$502$	0	0
$503$	9.08121e6	1.60038	0.800191	−	0.599745i	$-0.204732\pi$
$503$	9.08121e6	1.60038	0.800191	+	0.599745i	$0.204732\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.56596e6	1.13443
$508$	0	0
$509$	8.26612e6	1.41419	0.707094	−	0.707119i	$-0.250006\pi$
$509$	8.26612e6	1.41419	0.707094	+	0.707119i	$0.250006\pi$
$510$	0	0
$511$	3.51241e6	0.595050
$512$	0	0
$513$	8.47239e6	1.42139
$514$	0	0
$515$	0	0
$516$	0	0
$517$	726312.	0.119508
$518$	0	0
$519$	−3.53230e6	−0.575625
$520$	0	0
$521$	2.07813e6	0.335412	0.167706	−	0.985837i	$-0.446364\pi$
$521$	2.07813e6	0.335412	0.167706	+	0.985837i	$0.446364\pi$
$522$	0	0
$523$	1.94810e6	0.311427	0.155714	−	0.987802i	$-0.450232\pi$
$523$	1.94810e6	0.311427	0.155714	+	0.987802i	$0.450232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.52215e6	−1.17982
$528$	0	0
$529$	−5.14410e6	−0.799228
$530$	0	0
$531$	7.50077e6	1.15444
$532$	0	0
$533$	−4.84947e6	−0.739395
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.05738e7	−1.58232
$538$	0	0
$539$	−1.46395e6	−0.217047
$540$	0	0
$541$	−8.56903e6	−1.25875	−0.629374	−	0.777103i	$-0.716688\pi$
$541$	−8.56903e6	−1.25875	−0.629374	+	0.777103i	$0.716688\pi$
$542$	0	0
$543$	−1.20882e7	−1.75939
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.59536e6	0.799577	0.399788	−	0.916607i	$-0.369084\pi$
$547$	5.59536e6	0.799577	0.399788	+	0.916607i	$0.369084\pi$
$548$	0	0
$549$	−1.62429e7	−2.30003
$550$	0	0
$551$	−2.12521e6	−0.298211
$552$	0	0
$553$	3.40884e6	0.474017
$554$	0	0
$555$	0	0
$556$	0	0
$557$	774159.	0.105728	0.0528642	−	0.998602i	$-0.483165\pi$
$557$	774159.	0.105728	0.0528642	+	0.998602i	$0.483165\pi$
$558$	0	0
$559$	4.96597e6	0.672163
$560$	0	0
$561$	3.21047e6	0.430686
$562$	0	0
$563$	7.62044e6	1.01323	0.506616	−	0.862172i	$-0.330896\pi$
$563$	7.62044e6	1.01323	0.506616	+	0.862172i	$0.330896\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.10430e6	0.144255
$568$	0	0
$569$	592534.	0.0767243	0.0383622	−	0.999264i	$-0.487786\pi$
$569$	592534.	0.0767243	0.0383622	+	0.999264i	$0.487786\pi$
$570$	0	0
$571$	1.30108e7	1.66999	0.834995	−	0.550257i	$-0.185470\pi$
$571$	1.30108e7	1.66999	0.834995	+	0.550257i	$0.185470\pi$
$572$	0	0
$573$	−1.26470e7	−1.60917
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.70946e6	−0.838973	−0.419486	−	0.907762i	$-0.637790\pi$
$577$	−6.70946e6	−0.838973	−0.419486	+	0.907762i	$0.637790\pi$
$578$	0	0
$579$	1.19032e7	1.47560
$580$	0	0
$581$	106483.	0.0130870
$582$	0	0
$583$	−4.27670e6	−0.521120
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.19616e7	1.43283	0.716415	−	0.697674i	$-0.245782\pi$
$587$	1.19616e7	1.43283	0.716415	+	0.697674i	$0.245782\pi$
$588$	0	0
$589$	−2.02279e7	−2.40250
$590$	0	0
$591$	−1.09732e6	−0.129231
$592$	0	0
$593$	−1.36302e7	−1.59172	−0.795860	−	0.605481i	$-0.792981\pi$
$593$	−1.36302e7	−1.59172	−0.795860	+	0.605481i	$0.792981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.35304e7	1.55373
$598$	0	0
$599$	3.87186e6	0.440913	0.220456	−	0.975397i	$-0.429245\pi$
$599$	3.87186e6	0.440913	0.220456	+	0.975397i	$0.429245\pi$
$600$	0	0
$601$	−2.64964e6	−0.299227	−0.149613	−	0.988745i	$-0.547803\pi$
$601$	−2.64964e6	−0.299227	−0.149613	+	0.988745i	$0.547803\pi$
$602$	0	0
$603$	−2.45671e7	−2.75144
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.36476e7	1.50343	0.751717	−	0.659486i	$-0.229226\pi$
$607$	1.36476e7	1.50343	0.751717	+	0.659486i	$0.229226\pi$
$608$	0	0
$609$	1.23598e6	0.135042
$610$	0	0
$611$	−1.93849e6	−0.210069
$612$	0	0
$613$	1.60364e7	1.72367	0.861837	−	0.507185i	$-0.169314\pi$
$613$	1.60364e7	1.72367	0.861837	+	0.507185i	$0.169314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.42285e6	0.784979	0.392489	−	0.919757i	$-0.371614\pi$
$617$	7.42285e6	0.784979	0.392489	+	0.919757i	$0.371614\pi$
$618$	0	0
$619$	−1.37805e7	−1.44557	−0.722784	−	0.691074i	$-0.757138\pi$
$619$	−1.37805e7	−1.44557	−0.722784	+	0.691074i	$0.757138\pi$
$620$	0	0
$621$	−3.31949e6	−0.345416
$622$	0	0
$623$	7.09262e6	0.732128
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.63331e6	0.877018
$628$	0	0
$629$	3.55393e6	0.358165
$630$	0	0
$631$	2.22868e6	0.222830	0.111415	−	0.993774i	$-0.464462\pi$
$631$	2.22868e6	0.222830	0.111415	+	0.993774i	$0.464462\pi$
$632$	0	0
$633$	2.87819e7	2.85503
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.90721e6	0.381521
$638$	0	0
$639$	4.05869e6	0.393218
$640$	0	0
$641$	1.75304e6	0.168519	0.0842593	−	0.996444i	$-0.473148\pi$
$641$	1.75304e6	0.168519	0.0842593	+	0.996444i	$0.473148\pi$
$642$	0	0
$643$	−1.68507e7	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$643$	−1.68507e7	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.86231e6	0.174901	0.0874505	−	0.996169i	$-0.472128\pi$
$647$	1.86231e6	0.174901	0.0874505	+	0.996169i	$0.472128\pi$
$648$	0	0
$649$	2.50893e6	0.233818
$650$	0	0
$651$	1.17642e7	1.08795
$652$	0	0
$653$	1.64223e7	1.50713	0.753563	−	0.657375i	$-0.228333\pi$
$653$	1.64223e7	1.50713	0.753563	+	0.657375i	$0.228333\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.85173e7	−1.67365
$658$	0	0
$659$	−6.16058e6	−0.552596	−0.276298	−	0.961072i	$-0.589108\pi$
$659$	−6.16058e6	−0.552596	−0.276298	+	0.961072i	$0.589108\pi$
$660$	0	0
$661$	−2.06032e7	−1.83413	−0.917065	−	0.398737i	$-0.869449\pi$
$661$	−2.06032e7	−1.83413	−0.917065	+	0.398737i	$0.869449\pi$
$662$	0	0
$663$	−8.56859e6	−0.757052
$664$	0	0
$665$	0	0
$666$	0	0
$667$	832660.	0.0724692
$668$	0	0
$669$	−2.35373e7	−2.03325
$670$	0	0
$671$	−5.43310e6	−0.465845
$672$	0	0
$673$	−1.14267e7	−0.972488	−0.486244	−	0.873823i	$-0.661633\pi$
$673$	−1.14267e7	−0.972488	−0.486244	+	0.873823i	$0.661633\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	679572.	0.0569854	0.0284927	−	0.999594i	$-0.490929\pi$
$677$	679572.	0.0569854	0.0284927	+	0.999594i	$0.490929\pi$
$678$	0	0
$679$	−37127.2	−0.00309042
$680$	0	0
$681$	1.01291e7	0.836959
$682$	0	0
$683$	6.88710e6	0.564917	0.282458	−	0.959280i	$-0.408850\pi$
$683$	6.88710e6	0.564917	0.282458	+	0.959280i	$0.408850\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.48659e6	0.605190
$688$	0	0
$689$	1.14143e7	0.916014
$690$	0	0
$691$	4.64486e6	0.370065	0.185032	−	0.982732i	$-0.440761\pi$
$691$	4.64486e6	0.370065	0.185032	+	0.982732i	$0.440761\pi$
$692$	0	0
$693$	−3.00343e6	−0.237566
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.62019e7	−1.26323
$698$	0	0
$699$	382138.	0.0295820
$700$	0	0
$701$	−3.04696e6	−0.234192	−0.117096	−	0.993121i	$-0.537358\pi$
$701$	−3.04696e6	−0.234192	−0.117096	+	0.993121i	$0.537358\pi$
$702$	0	0
$703$	9.55692e6	0.729340
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−439439.	−0.0330636
$708$	0	0
$709$	−1.50817e7	−1.12677	−0.563385	−	0.826194i	$-0.690501\pi$
$709$	−1.50817e7	−1.12677	−0.563385	+	0.826194i	$0.690501\pi$
$710$	0	0
$711$	−1.79713e7	−1.33323
$712$	0	0
$713$	7.92532e6	0.583839
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.95339e7	−1.41903
$718$	0	0
$719$	1.20848e7	0.871803	0.435901	−	0.899994i	$-0.356430\pi$
$719$	1.20848e7	0.871803	0.435901	+	0.899994i	$0.356430\pi$
$720$	0	0
$721$	−3.13858e6	−0.224851
$722$	0	0
$723$	3.50071e7	2.49063
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.52669e7	1.77303	0.886516	−	0.462697i	$-0.153118\pi$
$727$	2.52669e7	1.77303	0.886516	+	0.462697i	$0.153118\pi$
$728$	0	0
$729$	−2.32717e7	−1.62185
$730$	0	0
$731$	1.65911e7	1.14837
$732$	0	0
$733$	3.21976e6	0.221342	0.110671	−	0.993857i	$-0.464700\pi$
$733$	3.21976e6	0.221342	0.110671	+	0.993857i	$0.464700\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.21744e6	−0.557273
$738$	0	0
$739$	−1.83282e7	−1.23455	−0.617274	−	0.786748i	$-0.711763\pi$
$739$	−1.83282e7	−1.23455	−0.617274	+	0.786748i	$0.711763\pi$
$740$	0	0
$741$	−2.30419e7	−1.54160
$742$	0	0
$743$	2.56985e7	1.70779	0.853896	−	0.520444i	$-0.174233\pi$
$743$	2.56985e7	1.70779	0.853896	+	0.520444i	$0.174233\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−561373.	−0.0368086
$748$	0	0
$749$	6.53172e6	0.425425
$750$	0	0
$751$	1.44321e7	0.933748	0.466874	−	0.884324i	$-0.345380\pi$
$751$	1.44321e7	0.933748	0.466874	+	0.884324i	$0.345380\pi$
$752$	0	0
$753$	−1.12395e6	−0.0722372
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.39198e7	1.51712	0.758558	−	0.651606i	$-0.225904\pi$
$757$	2.39198e7	1.51712	0.758558	+	0.651606i	$0.225904\pi$
$758$	0	0
$759$	−3.38254e6	−0.213127
$760$	0	0
$761$	−2.78047e7	−1.74043	−0.870215	−	0.492672i	$-0.836020\pi$
$761$	−2.78047e7	−1.74043	−0.870215	+	0.492672i	$0.836020\pi$
$762$	0	0
$763$	3.80228e6	0.236446
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.69623e6	−0.411000
$768$	0	0
$769$	1.25298e7	0.764063	0.382031	−	0.924149i	$-0.375225\pi$
$769$	1.25298e7	0.764063	0.382031	+	0.924149i	$0.375225\pi$
$770$	0	0
$771$	6.34507e6	0.384415
$772$	0	0
$773$	2.17300e7	1.30801	0.654006	−	0.756490i	$-0.273087\pi$
$773$	2.17300e7	1.30801	0.654006	+	0.756490i	$0.273087\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.55812e6	−0.330275
$778$	0	0
$779$	−4.35686e7	−2.57235
$780$	0	0
$781$	1.35759e6	0.0796419
$782$	0	0
$783$	−2.13893e6	−0.124679
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.47679e6	−0.430307	−0.215153	−	0.976580i	$-0.569025\pi$
$787$	−7.47679e6	−0.430307	−0.215153	+	0.976580i	$0.569025\pi$
$788$	0	0
$789$	−1.28646e7	−0.735706
$790$	0	0
$791$	6.43109e6	0.365463
$792$	0	0
$793$	1.45007e7	0.818852
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.58463e6	−0.478714	−0.239357	−	0.970932i	$-0.576937\pi$
$797$	−8.58463e6	−0.478714	−0.239357	+	0.970932i	$0.576937\pi$
$798$	0	0
$799$	−6.47642e6	−0.358895
$800$	0	0
$801$	−3.73920e7	−2.05920
$802$	0	0
$803$	−6.19386e6	−0.338979
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.52309e7	−1.36380
$808$	0	0
$809$	2.00739e7	1.07835	0.539176	−	0.842193i	$-0.318736\pi$
$809$	2.00739e7	1.07835	0.539176	+	0.842193i	$0.318736\pi$
$810$	0	0
$811$	345528.	0.0184472	0.00922362	−	0.999957i	$-0.497064\pi$
$811$	345528.	0.0184472	0.00922362	+	0.999957i	$0.497064\pi$
$812$	0	0
$813$	−3.82039e7	−2.02713
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.46152e7	2.33845
$818$	0	0
$819$	8.01601e6	0.417589
$820$	0	0
$821$	−3.55582e7	−1.84112	−0.920560	−	0.390601i	$-0.872267\pi$
$821$	−3.55582e7	−1.84112	−0.920560	+	0.390601i	$0.872267\pi$
$822$	0	0
$823$	2.18881e7	1.12644	0.563220	−	0.826307i	$-0.309562\pi$
$823$	2.18881e7	1.12644	0.563220	+	0.826307i	$0.309562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.08664e7	−1.56936	−0.784680	−	0.619901i	$-0.787173\pi$
$827$	−3.08664e7	−1.56936	−0.784680	+	0.619901i	$0.787173\pi$
$828$	0	0
$829$	−1.02344e7	−0.517219	−0.258610	−	0.965982i	$-0.583264\pi$
$829$	−1.02344e7	−0.517219	−0.258610	+	0.965982i	$0.583264\pi$
$830$	0	0
$831$	−1.82571e7	−0.917125
$832$	0	0
$833$	1.30538e7	0.651816
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.03585e7	−1.00446
$838$	0	0
$839$	−1.42644e7	−0.699596	−0.349798	−	0.936825i	$-0.613750\pi$
$839$	−1.42644e7	−0.699596	−0.349798	+	0.936825i	$0.613750\pi$
$840$	0	0
$841$	−1.99746e7	−0.973842
$842$	0	0
$843$	3.62199e7	1.75541
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.00462e6	−0.0481163
$848$	0	0
$849$	3.10700e7	1.47936
$850$	0	0
$851$	−3.74441e6	−0.177239
$852$	0	0
$853$	−6.85288e6	−0.322478	−0.161239	−	0.986915i	$-0.551549\pi$
$853$	−6.85288e6	−0.322478	−0.161239	+	0.986915i	$0.551549\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.51342e7	1.63410	0.817050	−	0.576567i	$-0.195608\pi$
$857$	3.51342e7	1.63410	0.817050	+	0.576567i	$0.195608\pi$
$858$	0	0
$859$	8.48281e6	0.392244	0.196122	−	0.980579i	$-0.437165\pi$
$859$	8.48281e6	0.392244	0.196122	+	0.980579i	$0.437165\pi$
$860$	0	0
$861$	2.53387e7	1.16487
$862$	0	0
$863$	−3.84664e7	−1.75815	−0.879073	−	0.476687i	$-0.841838\pi$
$863$	−3.84664e7	−1.75815	−0.879073	+	0.476687i	$0.841838\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.28924e6	0.284152
$868$	0	0
$869$	−6.01122e6	−0.270031
$870$	0	0
$871$	2.19320e7	0.979563
$872$	0	0
$873$	195733.	0.00869218
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.87813e7	0.824567	0.412283	−	0.911056i	$-0.364731\pi$
$877$	1.87813e7	0.824567	0.412283	+	0.911056i	$0.364731\pi$
$878$	0	0
$879$	−3.70479e7	−1.61730
$880$	0	0
$881$	4.01661e7	1.74349	0.871745	−	0.489960i	$-0.162989\pi$
$881$	4.01661e7	1.74349	0.871745	+	0.489960i	$0.162989\pi$
$882$	0	0
$883$	5.55332e6	0.239690	0.119845	−	0.992793i	$-0.461760\pi$
$883$	5.55332e6	0.239690	0.119845	+	0.992793i	$0.461760\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.28130e7	−0.973584	−0.486792	−	0.873518i	$-0.661833\pi$
$887$	−2.28130e7	−0.973584	−0.486792	+	0.873518i	$0.661833\pi$
$888$	0	0
$889$	−2.02216e6	−0.0858147
$890$	0	0
$891$	−1.94735e6	−0.0821767
$892$	0	0
$893$	−1.74158e7	−0.730828
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.02785e6	0.374630
$898$	0	0
$899$	5.10671e6	0.210738
$900$	0	0
$901$	3.81347e7	1.56498
$902$	0	0
$903$	−2.59474e7	−1.05895
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.90037e7	−1.57430	−0.787150	−	0.616761i	$-0.788444\pi$
$907$	−3.90037e7	−1.57430	−0.787150	+	0.616761i	$0.788444\pi$
$908$	0	0
$909$	2.31671e6	0.0929953
$910$	0	0
$911$	−3.68587e7	−1.47145	−0.735723	−	0.677283i	$-0.763157\pi$
$911$	−3.68587e7	−1.47145	−0.735723	+	0.677283i	$0.763157\pi$
$912$	0	0
$913$	−187774.	−0.00745517
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.69278e6	0.105749
$918$	0	0
$919$	1.03588e7	0.404597	0.202299	−	0.979324i	$-0.435159\pi$
$919$	1.03588e7	0.404597	0.202299	+	0.979324i	$0.435159\pi$
$920$	0	0
$921$	−5.33083e7	−2.07084
$922$	0	0
$923$	−3.62335e6	−0.139993
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.65465e7	0.632422
$928$	0	0
$929$	9.13177e6	0.347149	0.173574	−	0.984821i	$-0.444468\pi$
$929$	9.13177e6	0.347149	0.173574	+	0.984821i	$0.444468\pi$
$930$	0	0
$931$	3.51032e7	1.32731
$932$	0	0
$933$	−4.48093e7	−1.68525
$934$	0	0
$935$	0	0
$936$	0	0
$937$	219135.	0.00815384	0.00407692	−	0.999992i	$-0.498702\pi$
$937$	219135.	0.00815384	0.00407692	+	0.999992i	$0.498702\pi$
$938$	0	0
$939$	1.17022e7	0.433116
$940$	0	0
$941$	1.48284e7	0.545907	0.272954	−	0.962027i	$-0.411999\pi$
$941$	1.48284e7	0.545907	0.272954	+	0.962027i	$0.411999\pi$
$942$	0	0
$943$	1.70702e7	0.625116
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.18145e7	−1.87748	−0.938742	−	0.344621i	$-0.888007\pi$
$947$	−5.18145e7	−1.87748	−0.938742	+	0.344621i	$0.888007\pi$
$948$	0	0
$949$	1.65311e7	0.595850
$950$	0	0
$951$	−1.87783e7	−0.673294
$952$	0	0
$953$	3.57043e7	1.27347	0.636735	−	0.771083i	$-0.280285\pi$
$953$	3.57043e7	1.27347	0.636735	+	0.771083i	$0.280285\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.17955e6	−0.0769286
$958$	0	0
$959$	−2.31915e7	−0.814295
$960$	0	0
$961$	1.99769e7	0.697781
$962$	0	0
$963$	−3.44350e7	−1.19656
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.61627e6	0.330705	0.165352	−	0.986235i	$-0.447124\pi$
$967$	9.61627e6	0.330705	0.165352	+	0.986235i	$0.447124\pi$
$968$	0	0
$969$	−7.69819e7	−2.63378
$970$	0	0
$971$	2.39248e7	0.814330	0.407165	−	0.913355i	$-0.366517\pi$
$971$	2.39248e7	0.814330	0.407165	+	0.913355i	$0.366517\pi$
$972$	0	0
$973$	−1.25887e6	−0.0426285
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.46267e6	0.116058	0.0580290	−	0.998315i	$-0.481518\pi$
$977$	3.46267e6	0.116058	0.0580290	+	0.998315i	$0.481518\pi$
$978$	0	0
$979$	−1.25073e7	−0.417067
$980$	0	0
$981$	−2.00455e7	−0.665033
$982$	0	0
$983$	−3.74695e7	−1.23678	−0.618392	−	0.785870i	$-0.712216\pi$
$983$	−3.74695e7	−1.23678	−0.618392	+	0.785870i	$0.712216\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.01287e7	0.330949
$988$	0	0
$989$	−1.74803e7	−0.568275
$990$	0	0
$991$	2.73653e7	0.885150	0.442575	−	0.896732i	$-0.354065\pi$
$991$	2.73653e7	0.885150	0.442575	+	0.896732i	$0.354065\pi$
$992$	0	0
$993$	−2.79112e7	−0.898267
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.37185e7	0.437087	0.218543	−	0.975827i	$-0.429869\pi$
$997$	1.37185e7	0.437087	0.218543	+	0.975827i	$0.429869\pi$
$998$	0	0
$999$	9.61861e6	0.304929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.m.1.2		14
5.2	odd	4		220.6.b.b.89.13	yes	14
5.3	odd	4		220.6.b.b.89.2	✓	14
5.4	even	2	inner	1100.6.a.m.1.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.b.b.89.2	✓	14	5.3	odd	4
220.6.b.b.89.13	yes	14	5.2	odd	4
1100.6.a.m.1.2		14	1.1	even	1	trivial
1100.6.a.m.1.13		14	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-24.5916 q^{3} -68.6167 q^{7} +361.745 q^{9} +121.000 q^{11} -322.943 q^{13} -1078.94 q^{17} -2901.39 q^{19} +1687.39 q^{21} +1136.77 q^{23} -2920.12 q^{27} +732.481 q^{29} +6971.80 q^{31} -2975.58 q^{33} -3293.91 q^{37} +7941.68 q^{39} +15016.5 q^{41} -15377.2 q^{43} +6002.58 q^{47} -12098.7 q^{49} +26532.8 q^{51} -35344.7 q^{53} +71349.6 q^{57} +20735.0 q^{59} -44901.6 q^{61} -24821.7 q^{63} -67912.8 q^{67} -27954.9 q^{69} +11219.8 q^{71} -51188.9 q^{73} -8302.62 q^{77} -49679.5 q^{79} -16093.8 q^{81} -1551.85 q^{83} -18012.8 q^{87} -103366. q^{89} +22159.3 q^{91} -171447. q^{93} +541.081 q^{97} +43771.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1142 q^{9} + 1694 q^{11} + 4540 q^{19} + 3824 q^{21} + 9972 q^{29} + 19076 q^{31} + 13616 q^{39} + 15052 q^{41} + 55346 q^{49} - 13380 q^{51} + 2108 q^{59} + 11660 q^{61} - 1620 q^{69} - 89284 q^{71}+ \cdots + 138182 q^{99}+O(q^{100}) \) 14 * q + 1142 * q^9 + 1694 * q^11 + 4540 * q^19 + 3824 * q^21 + 9972 * q^29 + 19076 * q^31 + 13616 * q^39 + 15052 * q^41 + 55346 * q^49 - 13380 * q^51 + 2108 * q^59 + 11660 * q^61 - 1620 * q^69 - 89284 * q^71 - 19432 * q^79 - 94714 * q^81 + 38836 * q^89 + 39632 * q^91 + 138182 * q^99

Introduction

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