LMFDB - Embedded newform 1100.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,4,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, 4, names="a")

//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);

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

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

Self dual:	yes
Analytic conductor:	$64.9021010063$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 93x^{3} + 71x^{2} + 1873x - 2715 $ x^5 - x^4 - 93x^3 + 71x^2 + 1873*x - 2715
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.57551$ 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)

$f(q)$	$=$	$q-5.57551 q^{3} +20.7640 q^{7} +4.08629 q^{9} +11.0000 q^{11} -37.3507 q^{13} -32.4508 q^{17} +71.7593 q^{19} -115.770 q^{21} -86.2767 q^{23} +127.756 q^{27} -19.4826 q^{29} -113.280 q^{31} -61.3306 q^{33} +36.6594 q^{37} +208.249 q^{39} +152.019 q^{41} -75.7880 q^{43} +216.190 q^{47} +88.1454 q^{49} +180.930 q^{51} -291.842 q^{53} -400.095 q^{57} +673.318 q^{59} +267.522 q^{61} +84.8479 q^{63} -682.646 q^{67} +481.036 q^{69} +410.937 q^{71} +137.968 q^{73} +228.404 q^{77} +918.618 q^{79} -822.632 q^{81} -959.791 q^{83} +108.626 q^{87} +622.193 q^{89} -775.552 q^{91} +631.594 q^{93} -1390.64 q^{97} +44.9492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{3} - 50 q^{7} + 59 q^{9} + 55 q^{11} - 45 q^{13} - 62 q^{17} + 3 q^{19} + 158 q^{21} - 73 q^{23} - 312 q^{27} + 157 q^{29} + 57 q^{31} - 66 q^{33} + 72 q^{37} + 518 q^{39} + 74 q^{41} + 213 q^{43}+ \cdots + 649 q^{99}+O(q^{100}) $ 5 * q - 6 * q^3 - 50 * q^7 + 59 * q^9 + 55 * q^11 - 45 * q^13 - 62 * q^17 + 3 * q^19 + 158 * q^21 - 73 * q^23 - 312 * q^27 + 157 * q^29 + 57 * q^31 - 66 * q^33 + 72 * q^37 + 518 * q^39 + 74 * q^41 + 213 * q^43 - 440 * q^47 + 527 * q^49 - 1146 * q^51 - 336 * q^53 - 44 * q^57 + 692 * q^59 - 998 * q^61 - 1070 * q^63 - 674 * q^67 + 1246 * q^69 - 1645 * q^71 - 1028 * q^73 - 550 * q^77 + 1320 * q^79 - 775 * q^81 - 811 * q^83 - 3608 * q^87 + 1241 * q^89 - 1216 * q^91 - 2400 * q^93 - 2587 * q^97 + 649 * 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
$3$	−5.57551	−1.07301	−0.536504	−	0.843898i	$-0.680255\pi$
$3$	−5.57551	−1.07301	−0.536504	+	0.843898i	$0.680255\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.7640	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$7$	20.7640	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$8$	0	0
$9$	4.08629	0.151344
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−37.3507	−0.796864	−0.398432	−	0.917198i	$-0.630446\pi$
$13$	−37.3507	−0.796864	−0.398432	+	0.917198i	$0.630446\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−32.4508	−0.462969	−0.231485	−	0.972839i	$-0.574358\pi$
$17$	−32.4508	−0.462969	−0.231485	+	0.972839i	$0.574358\pi$
$18$	0	0
$19$	71.7593	0.866459	0.433229	−	0.901284i	$-0.357374\pi$
$19$	71.7593	0.866459	0.433229	+	0.901284i	$0.357374\pi$
$20$	0	0
$21$	−115.770	−1.20300
$22$	0	0
$23$	−86.2767	−0.782171	−0.391086	−	0.920354i	$-0.627900\pi$
$23$	−86.2767	−0.782171	−0.391086	+	0.920354i	$0.627900\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	127.756	0.910614
$28$	0	0
$29$	−19.4826	−0.124753	−0.0623764	−	0.998053i	$-0.519868\pi$
$29$	−19.4826	−0.124753	−0.0623764	+	0.998053i	$0.519868\pi$
$30$	0	0
$31$	−113.280	−0.656313	−0.328156	−	0.944623i	$-0.606427\pi$
$31$	−113.280	−0.656313	−0.328156	+	0.944623i	$0.606427\pi$
$32$	0	0
$33$	−61.3306	−0.323524
$34$	0	0
$35$	0	0
$36$	0	0
$37$	36.6594	0.162886	0.0814429	−	0.996678i	$-0.474047\pi$
$37$	36.6594	0.162886	0.0814429	+	0.996678i	$0.474047\pi$
$38$	0	0
$39$	208.249	0.855041
$40$	0	0
$41$	152.019	0.579058	0.289529	−	0.957169i	$-0.406501\pi$
$41$	152.019	0.579058	0.289529	+	0.957169i	$0.406501\pi$
$42$	0	0
$43$	−75.7880	−0.268781	−0.134390	−	0.990928i	$-0.542908\pi$
$43$	−75.7880	−0.268781	−0.134390	+	0.990928i	$0.542908\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	216.190	0.670947	0.335474	−	0.942050i	$-0.391104\pi$
$47$	216.190	0.670947	0.335474	+	0.942050i	$0.391104\pi$
$48$	0	0
$49$	88.1454	0.256984
$50$	0	0
$51$	180.930	0.496769
$52$	0	0
$53$	−291.842	−0.756370	−0.378185	−	0.925730i	$-0.623452\pi$
$53$	−291.842	−0.756370	−0.378185	+	0.925730i	$0.623452\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−400.095	−0.929716
$58$	0	0
$59$	673.318	1.48574	0.742869	−	0.669437i	$-0.233465\pi$
$59$	673.318	1.48574	0.742869	+	0.669437i	$0.233465\pi$
$60$	0	0
$61$	267.522	0.561520	0.280760	−	0.959778i	$-0.409413\pi$
$61$	267.522	0.561520	0.280760	+	0.959778i	$0.409413\pi$
$62$	0	0
$63$	84.8479	0.169680
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−682.646	−1.24475	−0.622377	−	0.782718i	$-0.713833\pi$
$67$	−682.646	−1.24475	−0.622377	+	0.782718i	$0.713833\pi$
$68$	0	0
$69$	481.036	0.839275
$70$	0	0
$71$	410.937	0.686890	0.343445	−	0.939173i	$-0.388406\pi$
$71$	410.937	0.686890	0.343445	+	0.939173i	$0.388406\pi$
$72$	0	0
$73$	137.968	0.221205	0.110603	−	0.993865i	$-0.464722\pi$
$73$	137.968	0.221205	0.110603	+	0.993865i	$0.464722\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	228.404	0.338040
$78$	0	0
$79$	918.618	1.30826	0.654130	−	0.756382i	$-0.273035\pi$
$79$	918.618	1.30826	0.654130	+	0.756382i	$0.273035\pi$
$80$	0	0
$81$	−822.632	−1.12844
$82$	0	0
$83$	−959.791	−1.26929	−0.634643	−	0.772805i	$-0.718853\pi$
$83$	−959.791	−1.26929	−0.634643	+	0.772805i	$0.718853\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	108.626	0.133861
$88$	0	0
$89$	622.193	0.741038	0.370519	−	0.928825i	$-0.379180\pi$
$89$	622.193	0.741038	0.370519	+	0.928825i	$0.379180\pi$
$90$	0	0
$91$	−775.552	−0.893406
$92$	0	0
$93$	631.594	0.704228
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1390.64	−1.45565	−0.727823	−	0.685765i	$-0.759468\pi$
$97$	−1390.64	−1.45565	−0.727823	+	0.685765i	$0.759468\pi$
$98$	0	0
$99$	44.9492	0.0456320
$100$	0	0
$101$	94.0599	0.0926664	0.0463332	−	0.998926i	$-0.485246\pi$
$101$	94.0599	0.0926664	0.0463332	+	0.998926i	$0.485246\pi$
$102$	0	0
$103$	−1032.75	−0.987963	−0.493982	−	0.869472i	$-0.664459\pi$
$103$	−1032.75	−0.987963	−0.493982	+	0.869472i	$0.664459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−743.781	−0.672000	−0.336000	−	0.941862i	$-0.609074\pi$
$107$	−743.781	−0.672000	−0.336000	+	0.941862i	$0.609074\pi$
$108$	0	0
$109$	−668.924	−0.587811	−0.293905	−	0.955835i	$-0.594955\pi$
$109$	−668.924	−0.587811	−0.293905	+	0.955835i	$0.594955\pi$
$110$	0	0
$111$	−204.395	−0.174778
$112$	0	0
$113$	−80.1193	−0.0666990	−0.0333495	−	0.999444i	$-0.510617\pi$
$113$	−80.1193	−0.0666990	−0.0333495	+	0.999444i	$0.510617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−152.626	−0.120601
$118$	0	0
$119$	−673.810	−0.519059
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−847.583	−0.621333
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1126.07	−0.786788	−0.393394	−	0.919370i	$-0.628699\pi$
$127$	−1126.07	−0.786788	−0.393394	+	0.919370i	$0.628699\pi$
$128$	0	0
$129$	422.557	0.288403
$130$	0	0
$131$	−2360.03	−1.57402	−0.787011	−	0.616940i	$-0.788372\pi$
$131$	−2360.03	−1.57402	−0.787011	+	0.616940i	$0.788372\pi$
$132$	0	0
$133$	1490.01	0.971433
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1823.51	−1.13718	−0.568588	−	0.822622i	$-0.692510\pi$
$137$	−1823.51	−1.13718	−0.568588	+	0.822622i	$0.692510\pi$
$138$	0	0
$139$	−1826.28	−1.11441	−0.557206	−	0.830374i	$-0.688127\pi$
$139$	−1826.28	−1.11441	−0.557206	+	0.830374i	$0.688127\pi$
$140$	0	0
$141$	−1205.37	−0.719931
$142$	0	0
$143$	−410.858	−0.240264
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−491.455	−0.275745
$148$	0	0
$149$	287.973	0.158333	0.0791666	−	0.996861i	$-0.474774\pi$
$149$	287.973	0.158333	0.0791666	+	0.996861i	$0.474774\pi$
$150$	0	0
$151$	−2881.09	−1.55272	−0.776358	−	0.630292i	$-0.782935\pi$
$151$	−2881.09	−1.55272	−0.776358	+	0.630292i	$0.782935\pi$
$152$	0	0
$153$	−132.603	−0.0700677
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2405.81	−1.22296	−0.611480	−	0.791260i	$-0.709425\pi$
$157$	−2405.81	−1.22296	−0.611480	+	0.791260i	$0.709425\pi$
$158$	0	0
$159$	1627.17	0.811590
$160$	0	0
$161$	−1791.45	−0.876933
$162$	0	0
$163$	−1160.53	−0.557667	−0.278834	−	0.960339i	$-0.589948\pi$
$163$	−1160.53	−0.557667	−0.278834	+	0.960339i	$0.589948\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	468.123	0.216913	0.108456	−	0.994101i	$-0.465409\pi$
$167$	468.123	0.216913	0.108456	+	0.994101i	$0.465409\pi$
$168$	0	0
$169$	−801.922	−0.365008
$170$	0	0
$171$	293.229	0.131133
$172$	0	0
$173$	56.1596	0.0246806	0.0123403	−	0.999924i	$-0.496072\pi$
$173$	56.1596	0.0246806	0.0123403	+	0.999924i	$0.496072\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3754.09	−1.59421
$178$	0	0
$179$	1508.71	0.629978	0.314989	−	0.949095i	$-0.397999\pi$
$179$	1508.71	0.629978	0.314989	+	0.949095i	$0.397999\pi$
$180$	0	0
$181$	438.185	0.179945	0.0899725	−	0.995944i	$-0.471322\pi$
$181$	438.185	0.179945	0.0899725	+	0.995944i	$0.471322\pi$
$182$	0	0
$183$	−1491.57	−0.602515
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−356.959	−0.139591
$188$	0	0
$189$	2652.72	1.02094
$190$	0	0
$191$	−3870.97	−1.46646	−0.733229	−	0.679982i	$-0.761988\pi$
$191$	−3870.97	−1.46646	−0.733229	+	0.679982i	$0.761988\pi$
$192$	0	0
$193$	4160.69	1.55178	0.775888	−	0.630871i	$-0.217302\pi$
$193$	4160.69	1.55178	0.775888	+	0.630871i	$0.217302\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−419.528	−0.151727	−0.0758633	−	0.997118i	$-0.524171\pi$
$197$	−419.528	−0.151727	−0.0758633	+	0.997118i	$0.524171\pi$
$198$	0	0
$199$	−2278.61	−0.811689	−0.405845	−	0.913942i	$-0.633023\pi$
$199$	−2278.61	−0.811689	−0.405845	+	0.913942i	$0.633023\pi$
$200$	0	0
$201$	3806.10	1.33563
$202$	0	0
$203$	−404.538	−0.139867
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−352.552	−0.118377
$208$	0	0
$209$	789.352	0.261247
$210$	0	0
$211$	757.812	0.247251	0.123625	−	0.992329i	$-0.460548\pi$
$211$	757.812	0.247251	0.123625	+	0.992329i	$0.460548\pi$
$212$	0	0
$213$	−2291.18	−0.737038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2352.15	−0.735827
$218$	0	0
$219$	−769.243	−0.237355
$220$	0	0
$221$	1212.06	0.368924
$222$	0	0
$223$	2132.39	0.640337	0.320169	−	0.947361i	$-0.396260\pi$
$223$	2132.39	0.640337	0.320169	+	0.947361i	$0.396260\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3797.04	−1.11021	−0.555107	−	0.831779i	$-0.687323\pi$
$227$	−3797.04	−1.11021	−0.555107	+	0.831779i	$0.687323\pi$
$228$	0	0
$229$	1965.66	0.567226	0.283613	−	0.958939i	$-0.408467\pi$
$229$	1965.66	0.567226	0.283613	+	0.958939i	$0.408467\pi$
$230$	0	0
$231$	−1273.47	−0.362720
$232$	0	0
$233$	−638.658	−0.179570	−0.0897851	−	0.995961i	$-0.528618\pi$
$233$	−638.658	−0.179570	−0.0897851	+	0.995961i	$0.528618\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5121.76	−1.40377
$238$	0	0
$239$	−4736.52	−1.28193	−0.640963	−	0.767572i	$-0.721465\pi$
$239$	−4736.52	−1.28193	−0.640963	+	0.767572i	$0.721465\pi$
$240$	0	0
$241$	−2834.52	−0.757625	−0.378812	−	0.925473i	$-0.623667\pi$
$241$	−2834.52	−0.757625	−0.378812	+	0.925473i	$0.623667\pi$
$242$	0	0
$243$	1137.19	0.300209
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2680.26	−0.690450
$248$	0	0
$249$	5351.32	1.36195
$250$	0	0
$251$	3290.48	0.827464	0.413732	−	0.910399i	$-0.364225\pi$
$251$	3290.48	0.827464	0.413732	+	0.910399i	$0.364225\pi$
$252$	0	0
$253$	−949.044	−0.235833
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2904.96	0.705084	0.352542	−	0.935796i	$-0.385317\pi$
$257$	2904.96	0.705084	0.352542	+	0.935796i	$0.385317\pi$
$258$	0	0
$259$	761.198	0.182620
$260$	0	0
$261$	−79.6117	−0.0188806
$262$	0	0
$263$	−4947.48	−1.15998	−0.579989	−	0.814624i	$-0.696943\pi$
$263$	−4947.48	−1.15998	−0.579989	+	0.814624i	$0.696943\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3469.04	−0.795139
$268$	0	0
$269$	6891.83	1.56209	0.781046	−	0.624474i	$-0.214687\pi$
$269$	6891.83	1.56209	0.781046	+	0.624474i	$0.214687\pi$
$270$	0	0
$271$	−3875.28	−0.868660	−0.434330	−	0.900754i	$-0.643015\pi$
$271$	−3875.28	−0.868660	−0.434330	+	0.900754i	$0.643015\pi$
$272$	0	0
$273$	4324.10	0.958631
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6658.20	1.44423	0.722117	−	0.691771i	$-0.243169\pi$
$277$	6658.20	1.44423	0.722117	+	0.691771i	$0.243169\pi$
$278$	0	0
$279$	−462.895	−0.0993291
$280$	0	0
$281$	−2564.20	−0.544369	−0.272184	−	0.962245i	$-0.587746\pi$
$281$	−2564.20	−0.544369	−0.272184	+	0.962245i	$0.587746\pi$
$282$	0	0
$283$	−5367.22	−1.12738	−0.563689	−	0.825987i	$-0.690618\pi$
$283$	−5367.22	−1.12738	−0.563689	+	0.825987i	$0.690618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3156.53	0.649212
$288$	0	0
$289$	−3859.94	−0.785659
$290$	0	0
$291$	7753.50	1.56192
$292$	0	0
$293$	3189.28	0.635903	0.317952	−	0.948107i	$-0.397005\pi$
$293$	3189.28	0.635903	0.317952	+	0.948107i	$0.397005\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1405.31	0.274560
$298$	0	0
$299$	3222.50	0.623284
$300$	0	0
$301$	−1573.67	−0.301344
$302$	0	0
$303$	−524.431	−0.0994317
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5223.47	0.971072	0.485536	−	0.874217i	$-0.338624\pi$
$307$	5223.47	0.971072	0.485536	+	0.874217i	$0.338624\pi$
$308$	0	0
$309$	5758.13	1.06009
$310$	0	0
$311$	−778.172	−0.141885	−0.0709423	−	0.997480i	$-0.522601\pi$
$311$	−778.172	−0.141885	−0.0709423	+	0.997480i	$0.522601\pi$
$312$	0	0
$313$	−5903.88	−1.06616	−0.533078	−	0.846066i	$-0.678965\pi$
$313$	−5903.88	−1.06616	−0.533078	+	0.846066i	$0.678965\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8015.21	1.42012	0.710062	−	0.704140i	$-0.248667\pi$
$317$	8015.21	1.42012	0.710062	+	0.704140i	$0.248667\pi$
$318$	0	0
$319$	−214.309	−0.0376144
$320$	0	0
$321$	4146.96	0.721061
$322$	0	0
$323$	−2328.65	−0.401144
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3729.59	0.630725
$328$	0	0
$329$	4488.97	0.752234
$330$	0	0
$331$	1945.09	0.322996	0.161498	−	0.986873i	$-0.448367\pi$
$331$	1945.09	0.322996	0.161498	+	0.986873i	$0.448367\pi$
$332$	0	0
$333$	149.801	0.0246518
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3833.73	0.619693	0.309847	−	0.950787i	$-0.399722\pi$
$337$	3833.73	0.619693	0.309847	+	0.950787i	$0.399722\pi$
$338$	0	0
$339$	446.706	0.0715685
$340$	0	0
$341$	−1246.08	−0.197886
$342$	0	0
$343$	−5291.81	−0.833035
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3031.17	−0.468938	−0.234469	−	0.972124i	$-0.575335\pi$
$347$	−3031.17	−0.468938	−0.234469	+	0.972124i	$0.575335\pi$
$348$	0	0
$349$	−9140.37	−1.40193	−0.700964	−	0.713197i	$-0.747246\pi$
$349$	−9140.37	−1.40193	−0.700964	+	0.713197i	$0.747246\pi$
$350$	0	0
$351$	−4771.77	−0.725635
$352$	0	0
$353$	−352.940	−0.0532156	−0.0266078	−	0.999646i	$-0.508471\pi$
$353$	−352.940	−0.0532156	−0.0266078	+	0.999646i	$0.508471\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3756.83	0.556954
$358$	0	0
$359$	−9308.76	−1.36852	−0.684258	−	0.729240i	$-0.739874\pi$
$359$	−9308.76	−1.36852	−0.684258	+	0.729240i	$0.739874\pi$
$360$	0	0
$361$	−1709.60	−0.249249
$362$	0	0
$363$	−674.636	−0.0975461
$364$	0	0
$365$	0	0
$366$	0	0
$367$	730.769	0.103940	0.0519698	−	0.998649i	$-0.483450\pi$
$367$	730.769	0.103940	0.0519698	+	0.998649i	$0.483450\pi$
$368$	0	0
$369$	621.193	0.0876370
$370$	0	0
$371$	−6059.82	−0.848006
$372$	0	0
$373$	−3886.41	−0.539493	−0.269746	−	0.962931i	$-0.586940\pi$
$373$	−3886.41	−0.539493	−0.269746	+	0.962931i	$0.586940\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	727.690	0.0994111
$378$	0	0
$379$	9925.81	1.34526	0.672631	−	0.739978i	$-0.265164\pi$
$379$	9925.81	1.34526	0.672631	+	0.739978i	$0.265164\pi$
$380$	0	0
$381$	6278.38	0.844229
$382$	0	0
$383$	−4658.20	−0.621469	−0.310735	−	0.950497i	$-0.600575\pi$
$383$	−4658.20	−0.621469	−0.310735	+	0.950497i	$0.600575\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−309.692	−0.0406784
$388$	0	0
$389$	9334.00	1.21659	0.608294	−	0.793712i	$-0.291854\pi$
$389$	9334.00	1.21659	0.608294	+	0.793712i	$0.291854\pi$
$390$	0	0
$391$	2799.75	0.362121
$392$	0	0
$393$	13158.4	1.68894
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3045.11	−0.384961	−0.192481	−	0.981301i	$-0.561653\pi$
$397$	−3045.11	−0.384961	−0.192481	+	0.981301i	$0.561653\pi$
$398$	0	0
$399$	−8307.58	−1.04235
$400$	0	0
$401$	8566.65	1.06683	0.533414	−	0.845854i	$-0.320909\pi$
$401$	8566.65	1.06683	0.533414	+	0.845854i	$0.320909\pi$
$402$	0	0
$403$	4231.09	0.522992
$404$	0	0
$405$	0	0
$406$	0	0
$407$	403.254	0.0491119
$408$	0	0
$409$	−2673.21	−0.323183	−0.161592	−	0.986858i	$-0.551663\pi$
$409$	−2673.21	−0.323183	−0.161592	+	0.986858i	$0.551663\pi$
$410$	0	0
$411$	10167.0	1.22020
$412$	0	0
$413$	13980.8	1.66574
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10182.5	1.19577
$418$	0	0
$419$	585.643	0.0682829	0.0341415	−	0.999417i	$-0.489130\pi$
$419$	585.643	0.0682829	0.0341415	+	0.999417i	$0.489130\pi$
$420$	0	0
$421$	−2506.29	−0.290141	−0.145070	−	0.989421i	$-0.546341\pi$
$421$	−2506.29	−0.290141	−0.145070	+	0.989421i	$0.546341\pi$
$422$	0	0
$423$	883.414	0.101544
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5554.85	0.629550
$428$	0	0
$429$	2290.74	0.257804
$430$	0	0
$431$	4307.54	0.481408	0.240704	−	0.970599i	$-0.422622\pi$
$431$	4307.54	0.481408	0.240704	+	0.970599i	$0.422622\pi$
$432$	0	0
$433$	−7599.55	−0.843443	−0.421722	−	0.906725i	$-0.638574\pi$
$433$	−7599.55	−0.843443	−0.421722	+	0.906725i	$0.638574\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6191.16	−0.677719
$438$	0	0
$439$	−14291.1	−1.55371	−0.776855	−	0.629680i	$-0.783186\pi$
$439$	−14291.1	−1.55371	−0.776855	+	0.629680i	$0.783186\pi$
$440$	0	0
$441$	360.188	0.0388930
$442$	0	0
$443$	8334.33	0.893851	0.446925	−	0.894571i	$-0.352519\pi$
$443$	8334.33	0.893851	0.446925	+	0.894571i	$0.352519\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1605.59	−0.169893
$448$	0	0
$449$	−3682.55	−0.387061	−0.193530	−	0.981094i	$-0.561994\pi$
$449$	−3682.55	−0.387061	−0.193530	+	0.981094i	$0.561994\pi$
$450$	0	0
$451$	1672.21	0.174592
$452$	0	0
$453$	16063.6	1.66607
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1348.62	0.138044	0.0690218	−	0.997615i	$-0.478012\pi$
$457$	1348.62	0.138044	0.0690218	+	0.997615i	$0.478012\pi$
$458$	0	0
$459$	−4145.77	−0.421586
$460$	0	0
$461$	2058.01	0.207920	0.103960	−	0.994582i	$-0.466849\pi$
$461$	2058.01	0.207920	0.103960	+	0.994582i	$0.466849\pi$
$462$	0	0
$463$	−8850.65	−0.888390	−0.444195	−	0.895930i	$-0.646510\pi$
$463$	−8850.65	−0.888390	−0.444195	+	0.895930i	$0.646510\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19339.3	1.91631	0.958154	−	0.286252i	$-0.0924094\pi$
$467$	19339.3	1.91631	0.958154	+	0.286252i	$0.0924094\pi$
$468$	0	0
$469$	−14174.5	−1.39556
$470$	0	0
$471$	13413.6	1.31224
$472$	0	0
$473$	−833.668	−0.0810404
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1192.55	−0.114472
$478$	0	0
$479$	−19045.4	−1.81671	−0.908356	−	0.418198i	$-0.862662\pi$
$479$	−19045.4	−1.81671	−0.908356	+	0.418198i	$0.862662\pi$
$480$	0	0
$481$	−1369.26	−0.129798
$482$	0	0
$483$	9988.26	0.940956
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7083.95	−0.659147	−0.329573	−	0.944130i	$-0.606905\pi$
$487$	−7083.95	−0.659147	−0.329573	+	0.944130i	$0.606905\pi$
$488$	0	0
$489$	6470.55	0.598381
$490$	0	0
$491$	11531.5	1.05990	0.529949	−	0.848030i	$-0.322211\pi$
$491$	11531.5	1.05990	0.529949	+	0.848030i	$0.322211\pi$
$492$	0	0
$493$	632.227	0.0577567
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8532.71	0.770109
$498$	0	0
$499$	8817.35	0.791019	0.395510	−	0.918462i	$-0.370568\pi$
$499$	8817.35	0.791019	0.395510	+	0.918462i	$0.370568\pi$
$500$	0	0
$501$	−2610.03	−0.232749
$502$	0	0
$503$	10596.4	0.939308	0.469654	−	0.882851i	$-0.344379\pi$
$503$	10596.4	0.939308	0.469654	+	0.882851i	$0.344379\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4471.12	0.391656
$508$	0	0
$509$	−18347.7	−1.59773	−0.798867	−	0.601508i	$-0.794567\pi$
$509$	−18347.7	−1.59773	−0.798867	+	0.601508i	$0.794567\pi$
$510$	0	0
$511$	2864.78	0.248005
$512$	0	0
$513$	9167.65	0.789009
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2378.09	0.202298
$518$	0	0
$519$	−313.119	−0.0264824
$520$	0	0
$521$	−18494.2	−1.55518	−0.777588	−	0.628775i	$-0.783557\pi$
$521$	−18494.2	−1.55518	−0.777588	+	0.628775i	$0.783557\pi$
$522$	0	0
$523$	−15387.2	−1.28649	−0.643244	−	0.765661i	$-0.722412\pi$
$523$	−15387.2	−1.28649	−0.643244	+	0.765661i	$0.722412\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3676.03	0.303853
$528$	0	0
$529$	−4723.33	−0.388208
$530$	0	0
$531$	2751.37	0.224858
$532$	0	0
$533$	−5678.02	−0.461430
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8411.81	−0.675971
$538$	0	0
$539$	969.599	0.0774835
$540$	0	0
$541$	5424.06	0.431051	0.215525	−	0.976498i	$-0.430854\pi$
$541$	5424.06	0.431051	0.215525	+	0.976498i	$0.430854\pi$
$542$	0	0
$543$	−2443.10	−0.193082
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−10544.7	−0.824241	−0.412120	−	0.911129i	$-0.635212\pi$
$547$	−10544.7	−0.824241	−0.412120	+	0.911129i	$0.635212\pi$
$548$	0	0
$549$	1093.17	0.0849828
$550$	0	0
$551$	−1398.06	−0.108093
$552$	0	0
$553$	19074.2	1.46676
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15977.6	1.21542	0.607712	−	0.794158i	$-0.292088\pi$
$557$	15977.6	1.21542	0.607712	+	0.794158i	$0.292088\pi$
$558$	0	0
$559$	2830.74	0.214182
$560$	0	0
$561$	1990.23	0.149782
$562$	0	0
$563$	−17264.6	−1.29239	−0.646196	−	0.763172i	$-0.723641\pi$
$563$	−17264.6	−1.29239	−0.646196	+	0.763172i	$0.723641\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17081.2	−1.26515
$568$	0	0
$569$	−11626.6	−0.856613	−0.428306	−	0.903634i	$-0.640890\pi$
$569$	−11626.6	−0.856613	−0.428306	+	0.903634i	$0.640890\pi$
$570$	0	0
$571$	26947.0	1.97495	0.987474	−	0.157781i	$-0.0504339\pi$
$571$	26947.0	1.97495	0.987474	+	0.157781i	$0.0504339\pi$
$572$	0	0
$573$	21582.6	1.57352
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23399.2	1.68825	0.844127	−	0.536143i	$-0.180119\pi$
$577$	23399.2	1.68825	0.844127	+	0.536143i	$0.180119\pi$
$578$	0	0
$579$	−23197.9	−1.66507
$580$	0	0
$581$	−19929.1	−1.42306
$582$	0	0
$583$	−3210.26	−0.228054
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4653.92	−0.327236	−0.163618	−	0.986524i	$-0.552316\pi$
$587$	−4653.92	−0.327236	−0.163618	+	0.986524i	$0.552316\pi$
$588$	0	0
$589$	−8128.90	−0.568668
$590$	0	0
$591$	2339.08	0.162804
$592$	0	0
$593$	8192.48	0.567327	0.283663	−	0.958924i	$-0.408450\pi$
$593$	8192.48	0.567327	0.283663	+	0.958924i	$0.408450\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12704.4	0.870948
$598$	0	0
$599$	6761.31	0.461201	0.230601	−	0.973048i	$-0.425931\pi$
$599$	6761.31	0.461201	0.230601	+	0.973048i	$0.425931\pi$
$600$	0	0
$601$	−15266.1	−1.03613	−0.518067	−	0.855340i	$-0.673348\pi$
$601$	−15266.1	−1.03613	−0.518067	+	0.855340i	$0.673348\pi$
$602$	0	0
$603$	−2789.49	−0.188386
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25104.0	1.67865	0.839326	−	0.543629i	$-0.182950\pi$
$607$	25104.0	1.67865	0.839326	+	0.543629i	$0.182950\pi$
$608$	0	0
$609$	2255.50	0.150078
$610$	0	0
$611$	−8074.85	−0.534654
$612$	0	0
$613$	8848.85	0.583037	0.291519	−	0.956565i	$-0.405840\pi$
$613$	8848.85	0.583037	0.291519	+	0.956565i	$0.405840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3042.72	−0.198534	−0.0992668	−	0.995061i	$-0.531650\pi$
$617$	−3042.72	−0.198534	−0.0992668	+	0.995061i	$0.531650\pi$
$618$	0	0
$619$	−3391.98	−0.220251	−0.110125	−	0.993918i	$-0.535125\pi$
$619$	−3391.98	−0.220251	−0.110125	+	0.993918i	$0.535125\pi$
$620$	0	0
$621$	−11022.3	−0.712256
$622$	0	0
$623$	12919.2	0.830816
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4401.04	−0.280320
$628$	0	0
$629$	−1189.63	−0.0754111
$630$	0	0
$631$	5791.13	0.365358	0.182679	−	0.983173i	$-0.441523\pi$
$631$	5791.13	0.365358	0.182679	+	0.983173i	$0.441523\pi$
$632$	0	0
$633$	−4225.19	−0.265302
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3292.30	−0.204781
$638$	0	0
$639$	1679.21	0.103957
$640$	0	0
$641$	−25616.3	−1.57844	−0.789221	−	0.614110i	$-0.789515\pi$
$641$	−25616.3	−1.57844	−0.789221	+	0.614110i	$0.789515\pi$
$642$	0	0
$643$	−16568.5	−1.01617	−0.508086	−	0.861306i	$-0.669647\pi$
$643$	−16568.5	−1.01617	−0.508086	+	0.861306i	$0.669647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8340.07	0.506773	0.253386	−	0.967365i	$-0.418456\pi$
$647$	8340.07	0.506773	0.253386	+	0.967365i	$0.418456\pi$
$648$	0	0
$649$	7406.50	0.447967
$650$	0	0
$651$	13114.4	0.789547
$652$	0	0
$653$	8570.19	0.513595	0.256798	−	0.966465i	$-0.417333\pi$
$653$	8570.19	0.513595	0.256798	+	0.966465i	$0.417333\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	563.779	0.0334781
$658$	0	0
$659$	−3184.90	−0.188264	−0.0941320	−	0.995560i	$-0.530008\pi$
$659$	−3184.90	−0.188264	−0.0941320	+	0.995560i	$0.530008\pi$
$660$	0	0
$661$	−5043.56	−0.296780	−0.148390	−	0.988929i	$-0.547409\pi$
$661$	−5043.56	−0.296780	−0.148390	+	0.988929i	$0.547409\pi$
$662$	0	0
$663$	−6757.86	−0.395858
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1680.90	0.0975781
$668$	0	0
$669$	−11889.1	−0.687087
$670$	0	0
$671$	2942.75	0.169305
$672$	0	0
$673$	−17554.6	−1.00547	−0.502735	−	0.864440i	$-0.667673\pi$
$673$	−17554.6	−1.00547	−0.502735	+	0.864440i	$0.667673\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2667.67	−0.151443	−0.0757214	−	0.997129i	$-0.524126\pi$
$677$	−2667.67	−0.151443	−0.0757214	+	0.997129i	$0.524126\pi$
$678$	0	0
$679$	−28875.2	−1.63200
$680$	0	0
$681$	21170.4	1.19127
$682$	0	0
$683$	−23516.3	−1.31746	−0.658732	−	0.752378i	$-0.728907\pi$
$683$	−23516.3	−1.31746	−0.658732	+	0.752378i	$0.728907\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10959.6	−0.608637
$688$	0	0
$689$	10900.5	0.602724
$690$	0	0
$691$	32061.4	1.76508	0.882542	−	0.470234i	$-0.155831\pi$
$691$	32061.4	1.76508	0.882542	+	0.470234i	$0.155831\pi$
$692$	0	0
$693$	933.327	0.0511604
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4933.14	−0.268086
$698$	0	0
$699$	3560.84	0.192680
$700$	0	0
$701$	−4590.65	−0.247341	−0.123671	−	0.992323i	$-0.539467\pi$
$701$	−4590.65	−0.247341	−0.123671	+	0.992323i	$0.539467\pi$
$702$	0	0
$703$	2630.66	0.141134
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1953.06	0.103893
$708$	0	0
$709$	24836.3	1.31558	0.657791	−	0.753201i	$-0.271491\pi$
$709$	24836.3	1.31558	0.657791	+	0.753201i	$0.271491\pi$
$710$	0	0
$711$	3753.74	0.197998
$712$	0	0
$713$	9773.43	0.513349
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26408.5	1.37552
$718$	0	0
$719$	33950.9	1.76099	0.880497	−	0.474051i	$-0.157209\pi$
$719$	33950.9	1.76099	0.880497	+	0.474051i	$0.157209\pi$
$720$	0	0
$721$	−21444.1	−1.10766
$722$	0	0
$723$	15803.9	0.812937
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16954.1	−0.864913	−0.432457	−	0.901655i	$-0.642353\pi$
$727$	−16954.1	−0.864913	−0.432457	+	0.901655i	$0.642353\pi$
$728$	0	0
$729$	15870.6	0.806312
$730$	0	0
$731$	2459.38	0.124437
$732$	0	0
$733$	−23218.8	−1.16999	−0.584996	−	0.811036i	$-0.698904\pi$
$733$	−23218.8	−1.16999	−0.584996	+	0.811036i	$0.698904\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7509.11	−0.375307
$738$	0	0
$739$	33365.8	1.66086	0.830432	−	0.557119i	$-0.188093\pi$
$739$	33365.8	1.66086	0.830432	+	0.557119i	$0.188093\pi$
$740$	0	0
$741$	14943.8	0.740857
$742$	0	0
$743$	13612.4	0.672128	0.336064	−	0.941839i	$-0.390904\pi$
$743$	13612.4	0.672128	0.336064	+	0.941839i	$0.390904\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3921.99	−0.192099
$748$	0	0
$749$	−15443.9	−0.753415
$750$	0	0
$751$	18882.2	0.917470	0.458735	−	0.888573i	$-0.348303\pi$
$751$	18882.2	0.917470	0.458735	+	0.888573i	$0.348303\pi$
$752$	0	0
$753$	−18346.1	−0.887875
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31967.8	1.53486	0.767429	−	0.641134i	$-0.221536\pi$
$757$	31967.8	1.53486	0.767429	+	0.641134i	$0.221536\pi$
$758$	0	0
$759$	5291.40	0.253051
$760$	0	0
$761$	20438.6	0.973588	0.486794	−	0.873517i	$-0.338166\pi$
$761$	20438.6	0.973588	0.486794	+	0.873517i	$0.338166\pi$
$762$	0	0
$763$	−13889.6	−0.659025
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25148.9	−1.18393
$768$	0	0
$769$	26825.0	1.25791	0.628956	−	0.777441i	$-0.283483\pi$
$769$	26825.0	1.25791	0.628956	+	0.777441i	$0.283483\pi$
$770$	0	0
$771$	−16196.6	−0.756560
$772$	0	0
$773$	−1020.11	−0.0474655	−0.0237327	−	0.999718i	$-0.507555\pi$
$773$	−1020.11	−0.0474655	−0.0237327	+	0.999718i	$0.507555\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4244.07	−0.195952
$778$	0	0
$779$	10908.8	0.501729
$780$	0	0
$781$	4520.30	0.207105
$782$	0	0
$783$	−2489.01	−0.113602
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−23236.5	−1.05247	−0.526235	−	0.850339i	$-0.676397\pi$
$787$	−23236.5	−1.05247	−0.526235	+	0.850339i	$0.676397\pi$
$788$	0	0
$789$	27584.7	1.24467
$790$	0	0
$791$	−1663.60	−0.0747798
$792$	0	0
$793$	−9992.16	−0.447455
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20354.2	0.904619	0.452310	−	0.891861i	$-0.350600\pi$
$797$	20354.2	0.904619	0.452310	+	0.891861i	$0.350600\pi$
$798$	0	0
$799$	−7015.53	−0.310628
$800$	0	0
$801$	2542.46	0.112152
$802$	0	0
$803$	1517.65	0.0666958
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−38425.5	−1.67613
$808$	0	0
$809$	14015.2	0.609084	0.304542	−	0.952499i	$-0.401497\pi$
$809$	14015.2	0.609084	0.304542	+	0.952499i	$0.401497\pi$
$810$	0	0
$811$	18874.0	0.817207	0.408604	−	0.912712i	$-0.366016\pi$
$811$	18874.0	0.817207	0.408604	+	0.912712i	$0.366016\pi$
$812$	0	0
$813$	21606.7	0.932078
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5438.50	−0.232887
$818$	0	0
$819$	−3169.13	−0.135212
$820$	0	0
$821$	2299.94	0.0977690	0.0488845	−	0.998804i	$-0.484433\pi$
$821$	2299.94	0.0977690	0.0488845	+	0.998804i	$0.484433\pi$
$822$	0	0
$823$	11685.4	0.494931	0.247466	−	0.968897i	$-0.420402\pi$
$823$	11685.4	0.494931	0.247466	+	0.968897i	$0.420402\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38022.6	1.59876	0.799380	−	0.600825i	$-0.205161\pi$
$827$	38022.6	1.59876	0.799380	+	0.600825i	$0.205161\pi$
$828$	0	0
$829$	2077.11	0.0870219	0.0435110	−	0.999053i	$-0.486146\pi$
$829$	2077.11	0.0870219	0.0435110	+	0.999053i	$0.486146\pi$
$830$	0	0
$831$	−37122.9	−1.54967
$832$	0	0
$833$	−2860.39	−0.118976
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−14472.2	−0.597647
$838$	0	0
$839$	−26534.0	−1.09184	−0.545921	−	0.837837i	$-0.683820\pi$
$839$	−26534.0	−1.09184	−0.545921	+	0.837837i	$0.683820\pi$
$840$	0	0
$841$	−24009.4	−0.984437
$842$	0	0
$843$	14296.7	0.584111
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2512.45	0.101923
$848$	0	0
$849$	29925.0	1.20968
$850$	0	0
$851$	−3162.86	−0.127405
$852$	0	0
$853$	−24845.2	−0.997283	−0.498641	−	0.866808i	$-0.666168\pi$
$853$	−24845.2	−0.997283	−0.498641	+	0.866808i	$0.666168\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18710.0	−0.745767	−0.372883	−	0.927878i	$-0.621631\pi$
$857$	−18710.0	−0.745767	−0.372883	+	0.927878i	$0.621631\pi$
$858$	0	0
$859$	−4277.39	−0.169898	−0.0849492	−	0.996385i	$-0.527073\pi$
$859$	−4277.39	−0.169898	−0.0849492	+	0.996385i	$0.527073\pi$
$860$	0	0
$861$	−17599.2	−0.696609
$862$	0	0
$863$	−17711.3	−0.698611	−0.349306	−	0.937009i	$-0.613582\pi$
$863$	−17711.3	−0.698611	−0.349306	+	0.937009i	$0.613582\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21521.2	0.843018
$868$	0	0
$869$	10104.8	0.394455
$870$	0	0
$871$	25497.3	0.991899
$872$	0	0
$873$	−5682.54	−0.220303
$874$	0	0
$875$	0	0
$876$	0	0
$877$	789.730	0.0304074	0.0152037	−	0.999884i	$-0.495160\pi$
$877$	789.730	0.0304074	0.0152037	+	0.999884i	$0.495160\pi$
$878$	0	0
$879$	−17781.8	−0.682329
$880$	0	0
$881$	−34340.8	−1.31325	−0.656624	−	0.754218i	$-0.728016\pi$
$881$	−34340.8	−1.31325	−0.656624	+	0.754218i	$0.728016\pi$
$882$	0	0
$883$	−43311.3	−1.65067	−0.825335	−	0.564644i	$-0.809014\pi$
$883$	−43311.3	−1.65067	−0.825335	+	0.564644i	$0.809014\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39802.8	−1.50670	−0.753352	−	0.657617i	$-0.771565\pi$
$887$	−39802.8	−1.50670	−0.753352	+	0.657617i	$0.771565\pi$
$888$	0	0
$889$	−23381.7	−0.882110
$890$	0	0
$891$	−9048.95	−0.340237
$892$	0	0
$893$	15513.6	0.581348
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−17967.1	−0.668788
$898$	0	0
$899$	2206.99	0.0818769
$900$	0	0
$901$	9470.52	0.350176
$902$	0	0
$903$	8773.99	0.323344
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28916.2	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$907$	−28916.2	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$908$	0	0
$909$	384.356	0.0140245
$910$	0	0
$911$	−3182.17	−0.115730	−0.0578651	−	0.998324i	$-0.518429\pi$
$911$	−3182.17	−0.115730	−0.0578651	+	0.998324i	$0.518429\pi$
$912$	0	0
$913$	−10557.7	−0.382704
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−49003.8	−1.76472
$918$	0	0
$919$	6946.10	0.249326	0.124663	−	0.992199i	$-0.460215\pi$
$919$	6946.10	0.249326	0.124663	+	0.992199i	$0.460215\pi$
$920$	0	0
$921$	−29123.5	−1.04197
$922$	0	0
$923$	−15348.8	−0.547358
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4220.13	−0.149522
$928$	0	0
$929$	44519.8	1.57228	0.786139	−	0.618050i	$-0.212077\pi$
$929$	44519.8	1.57228	0.786139	+	0.618050i	$0.212077\pi$
$930$	0	0
$931$	6325.25	0.222666
$932$	0	0
$933$	4338.70	0.152243
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7802.11	−0.272021	−0.136011	−	0.990707i	$-0.543428\pi$
$937$	−7802.11	−0.272021	−0.136011	+	0.990707i	$0.543428\pi$
$938$	0	0
$939$	32917.1	1.14399
$940$	0	0
$941$	−24835.3	−0.860369	−0.430184	−	0.902741i	$-0.641551\pi$
$941$	−24835.3	−0.860369	−0.430184	+	0.902741i	$0.641551\pi$
$942$	0	0
$943$	−13115.7	−0.452922
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26433.4	0.907045	0.453522	−	0.891245i	$-0.350167\pi$
$947$	26433.4	0.907045	0.453522	+	0.891245i	$0.350167\pi$
$948$	0	0
$949$	−5153.22	−0.176270
$950$	0	0
$951$	−44688.9	−1.52380
$952$	0	0
$953$	40389.2	1.37286	0.686429	−	0.727197i	$-0.259177\pi$
$953$	40389.2	1.37286	0.686429	+	0.727197i	$0.259177\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1194.88	0.0403605
$958$	0	0
$959$	−37863.5	−1.27495
$960$	0	0
$961$	−16958.6	−0.569254
$962$	0	0
$963$	−3039.31	−0.101703
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20108.7	−0.668719	−0.334359	−	0.942446i	$-0.608520\pi$
$967$	−20108.7	−0.668719	−0.334359	+	0.942446i	$0.608520\pi$
$968$	0	0
$969$	12983.4	0.430430
$970$	0	0
$971$	−6054.35	−0.200096	−0.100048	−	0.994983i	$-0.531900\pi$
$971$	−6054.35	−0.200096	−0.100048	+	0.994983i	$0.531900\pi$
$972$	0	0
$973$	−37921.0	−1.24943
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4573.59	0.149767	0.0748833	−	0.997192i	$-0.476142\pi$
$977$	4573.59	0.149767	0.0748833	+	0.997192i	$0.476142\pi$
$978$	0	0
$979$	6844.12	0.223431
$980$	0	0
$981$	−2733.42	−0.0889617
$982$	0	0
$983$	−27498.3	−0.892226	−0.446113	−	0.894977i	$-0.647192\pi$
$983$	−27498.3	−0.892226	−0.446113	+	0.894977i	$0.647192\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−25028.3	−0.807153
$988$	0	0
$989$	6538.74	0.210232
$990$	0	0
$991$	−22415.6	−0.718521	−0.359261	−	0.933237i	$-0.616971\pi$
$991$	−22415.6	−0.718521	−0.359261	+	0.933237i	$0.616971\pi$
$992$	0	0
$993$	−10844.8	−0.346577
$994$	0	0
$995$	0	0
$996$	0	0
$997$	27064.0	0.859706	0.429853	−	0.902899i	$-0.358565\pi$
$997$	27064.0	0.859706	0.429853	+	0.902899i	$0.358565\pi$
$998$	0	0
$999$	4683.45	0.148326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.a.j.1.2	✓	5
5.2	odd	4		1100.4.b.j.749.8		10
5.3	odd	4		1100.4.b.j.749.3		10
5.4	even	2		1100.4.a.m.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.4.a.j.1.2	✓	5	1.1	even	1	trivial
1100.4.a.m.1.4	yes	5	5.4	even	2
1100.4.b.j.749.3		10	5.3	odd	4
1100.4.b.j.749.8		10	5.2	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q-5.57551 q^{3} +20.7640 q^{7} +4.08629 q^{9} +11.0000 q^{11} -37.3507 q^{13} -32.4508 q^{17} +71.7593 q^{19} -115.770 q^{21} -86.2767 q^{23} +127.756 q^{27} -19.4826 q^{29} -113.280 q^{31} -61.3306 q^{33} +36.6594 q^{37} +208.249 q^{39} +152.019 q^{41} -75.7880 q^{43} +216.190 q^{47} +88.1454 q^{49} +180.930 q^{51} -291.842 q^{53} -400.095 q^{57} +673.318 q^{59} +267.522 q^{61} +84.8479 q^{63} -682.646 q^{67} +481.036 q^{69} +410.937 q^{71} +137.968 q^{73} +228.404 q^{77} +918.618 q^{79} -822.632 q^{81} -959.791 q^{83} +108.626 q^{87} +622.193 q^{89} -775.552 q^{91} +631.594 q^{93} -1390.64 q^{97} +44.9492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{3} - 50 q^{7} + 59 q^{9} + 55 q^{11} - 45 q^{13} - 62 q^{17} + 3 q^{19} + 158 q^{21} - 73 q^{23} - 312 q^{27} + 157 q^{29} + 57 q^{31} - 66 q^{33} + 72 q^{37} + 518 q^{39} + 74 q^{41} + 213 q^{43}+ \cdots + 649 q^{99}+O(q^{100}) \) 5 * q - 6 * q^3 - 50 * q^7 + 59 * q^9 + 55 * q^11 - 45 * q^13 - 62 * q^17 + 3 * q^19 + 158 * q^21 - 73 * q^23 - 312 * q^27 + 157 * q^29 + 57 * q^31 - 66 * q^33 + 72 * q^37 + 518 * q^39 + 74 * q^41 + 213 * q^43 - 440 * q^47 + 527 * q^49 - 1146 * q^51 - 336 * q^53 - 44 * q^57 + 692 * q^59 - 998 * q^61 - 1070 * q^63 - 674 * q^67 + 1246 * q^69 - 1645 * q^71 - 1028 * q^73 - 550 * q^77 + 1320 * q^79 - 775 * q^81 - 811 * q^83 - 3608 * q^87 + 1241 * q^89 - 1216 * q^91 - 2400 * q^93 - 2587 * q^97 + 649 * q^99

Introduction

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