LMFDB - Embedded newform 1100.4.b.h.749.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,4,Mod(749,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1100.749"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1100.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-48,0,66,0,0,0,0,0,0,0,-342] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$64.9021010063$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1351885824.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 37x^{4} + 384x^{2} + 900 $ x^6 + 37x^4 + 384x^2 + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.5
Root			$3.67648i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.4.b.h.749.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.86946i q^{3} -15.2554i q^{7} -20.1894 q^{9} +11.0000 q^{11} +30.1909i q^{13} -74.4793i q^{17} -30.6126 q^{19} +104.796 q^{21} +111.339i q^{23} +46.7850i q^{27} -23.5336 q^{29} -272.083 q^{31} +75.5640i q^{33} -292.415i q^{37} -207.395 q^{39} -127.493 q^{41} -466.210i q^{43} -430.418i q^{47} +110.273 q^{49} +511.632 q^{51} +235.761i q^{53} -210.292i q^{57} -167.162 q^{59} -363.291 q^{61} +307.998i q^{63} -611.297i q^{67} -764.836 q^{69} -315.466 q^{71} -372.861i q^{73} -167.809i q^{77} +300.829 q^{79} -866.502 q^{81} -1218.18i q^{83} -161.663i q^{87} -73.6581 q^{89} +460.574 q^{91} -1869.06i q^{93} +1389.82i q^{97} -222.084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{9} + 66 q^{11} - 342 q^{19} + 518 q^{21} - 110 q^{29} + 362 q^{31} - 1108 q^{39} + 604 q^{41} + 536 q^{49} + 1666 q^{51} + 84 q^{59} + 698 q^{61} + 516 q^{69} + 1854 q^{71} + 2292 q^{79} - 3066 q^{81}+ \cdots - 528 q^{99}+O(q^{100}) $ 6 * q - 48 * q^9 + 66 * q^11 - 342 * q^19 + 518 * q^21 - 110 * q^29 + 362 * q^31 - 1108 * q^39 + 604 * q^41 + 536 * q^49 + 1666 * q^51 + 84 * q^59 + 698 * q^61 + 516 * q^69 + 1854 * q^71 + 2292 * q^79 - 3066 * q^81 + 1750 * q^89 + 3964 * q^91 - 528 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	6.86946i	1.32203i	0.750374	+	0.661014i	$0.229873\pi$
$3$	6.86946i	1.32203i	−0.750374	+	0.661014i	$0.770127\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 15.2554i	− 0.823715i	−0.911248	−	0.411857i	$-0.864880\pi$
$7$	− 15.2554i	− 0.823715i	0.911248	−	0.411857i	$-0.135120\pi$
$8$	0	0
$9$	−20.1894	−0.747756
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	30.1909i	0.644110i	0.946721	+	0.322055i	$0.104374\pi$
$13$	30.1909i	0.644110i	−0.946721	+	0.322055i	$0.895626\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 74.4793i	− 1.06258i	−0.847190	−	0.531290i	$-0.821707\pi$
$17$	− 74.4793i	− 1.06258i	0.847190	−	0.531290i	$-0.178293\pi$
$18$	0	0
$19$	−30.6126	−0.369632	−0.184816	−	0.982773i	$-0.559169\pi$
$19$	−30.6126	−0.369632	−0.184816	+	0.982773i	$0.559169\pi$
$20$	0	0
$21$	104.796	1.08897
$22$	0	0
$23$	111.339i	1.00938i	0.863301	+	0.504690i	$0.168393\pi$
$23$	111.339i	1.00938i	−0.863301	+	0.504690i	$0.831607\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	46.7850i	0.333473i
$28$	0	0
$29$	−23.5336	−0.150692	−0.0753462	−	0.997157i	$-0.524006\pi$
$29$	−23.5336	−0.150692	−0.0753462	+	0.997157i	$0.524006\pi$
$30$	0	0
$31$	−272.083	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$31$	−272.083	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$32$	0	0
$33$	75.5640i	0.398606i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 292.415i	− 1.29926i	−0.760249	−	0.649632i	$-0.774923\pi$
$37$	− 292.415i	− 1.29926i	0.760249	−	0.649632i	$-0.225077\pi$
$38$	0	0
$39$	−207.395	−0.851532
$40$	0	0
$41$	−127.493	−0.485634	−0.242817	−	0.970072i	$-0.578071\pi$
$41$	−127.493	−0.485634	−0.242817	+	0.970072i	$0.578071\pi$
$42$	0	0
$43$	− 466.210i	− 1.65341i	−0.562639	−	0.826703i	$-0.690214\pi$
$43$	− 466.210i	− 1.65341i	0.562639	−	0.826703i	$-0.309786\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 430.418i	− 1.33581i	−0.744248	−	0.667903i	$-0.767192\pi$
$47$	− 430.418i	− 1.33581i	0.744248	−	0.667903i	$-0.232808\pi$
$48$	0	0
$49$	110.273	0.321494
$50$	0	0
$51$	511.632	1.40476
$52$	0	0
$53$	235.761i	0.611024i	0.952188	+	0.305512i	$0.0988276\pi$
$53$	235.761i	0.611024i	−0.952188	+	0.305512i	$0.901172\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 210.292i	− 0.488664i
$58$	0	0
$59$	−167.162	−0.368859	−0.184429	−	0.982846i	$-0.559044\pi$
$59$	−167.162	−0.368859	−0.184429	+	0.982846i	$0.559044\pi$
$60$	0	0
$61$	−363.291	−0.762535	−0.381267	−	0.924465i	$-0.624512\pi$
$61$	−363.291	−0.762535	−0.381267	+	0.924465i	$0.624512\pi$
$62$	0	0
$63$	307.998i	0.615938i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 611.297i	− 1.11465i	−0.830293	−	0.557327i	$-0.811827\pi$
$67$	− 611.297i	− 1.11465i	0.830293	−	0.557327i	$-0.188173\pi$
$68$	0	0
$69$	−764.836	−1.33443
$70$	0	0
$71$	−315.466	−0.527309	−0.263654	−	0.964617i	$-0.584928\pi$
$71$	−315.466	−0.527309	−0.263654	+	0.964617i	$0.584928\pi$
$72$	0	0
$73$	− 372.861i	− 0.597810i	−0.954283	−	0.298905i	$-0.903379\pi$
$73$	− 372.861i	− 0.597810i	0.954283	−	0.298905i	$-0.0966214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 167.809i	− 0.248359i
$78$	0	0
$79$	300.829	0.428430	0.214215	−	0.976787i	$-0.431281\pi$
$79$	300.829	0.428430	0.214215	+	0.976787i	$0.431281\pi$
$80$	0	0
$81$	−866.502	−1.18862
$82$	0	0
$83$	− 1218.18i	− 1.61100i	−0.592599	−	0.805498i	$-0.701898\pi$
$83$	− 1218.18i	− 1.61100i	0.592599	−	0.805498i	$-0.298102\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 161.663i	− 0.199219i
$88$	0	0
$89$	−73.6581	−0.0877275	−0.0438637	−	0.999038i	$-0.513967\pi$
$89$	−73.6581	−0.0877275	−0.0438637	+	0.999038i	$0.513967\pi$
$90$	0	0
$91$	460.574	0.530563
$92$	0	0
$93$	− 1869.06i	− 2.08400i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1389.82i	1.45480i	0.686216	+	0.727398i	$0.259271\pi$
$97$	1389.82i	1.45480i	−0.686216	+	0.727398i	$0.740729\pi$
$98$	0	0
$99$	−222.084	−0.225457
$100$	0	0
$101$	−321.478	−0.316715	−0.158358	−	0.987382i	$-0.550620\pi$
$101$	−321.478	−0.316715	−0.158358	+	0.987382i	$0.550620\pi$
$102$	0	0
$103$	− 566.635i	− 0.542060i	−0.962571	−	0.271030i	$-0.912636\pi$
$103$	− 566.635i	− 0.542060i	0.962571	−	0.271030i	$-0.0873642\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	866.060i	0.782478i	0.920289	+	0.391239i	$0.127953\pi$
$107$	866.060i	0.782478i	−0.920289	+	0.391239i	$0.872047\pi$
$108$	0	0
$109$	1407.61	1.23693	0.618463	−	0.785814i	$-0.287756\pi$
$109$	1407.61	1.23693	0.618463	+	0.785814i	$0.287756\pi$
$110$	0	0
$111$	2008.73	1.71766
$112$	0	0
$113$	− 426.591i	− 0.355135i	−0.984109	−	0.177568i	$-0.943177\pi$
$113$	− 426.591i	− 0.355135i	0.984109	−	0.177568i	$-0.0568229\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 609.536i	− 0.481638i
$118$	0	0
$119$	−1136.21	−0.875263
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	− 875.805i	− 0.642022i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	398.684i	0.278563i	0.990253	+	0.139282i	$0.0444793\pi$
$127$	398.684i	0.278563i	−0.990253	+	0.139282i	$0.955521\pi$
$128$	0	0
$129$	3202.61	2.18585
$130$	0	0
$131$	1504.91	1.00370	0.501849	−	0.864955i	$-0.332653\pi$
$131$	1504.91	1.00370	0.501849	+	0.864955i	$0.332653\pi$
$132$	0	0
$133$	467.008i	0.304472i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	189.347i	0.118080i	0.998256	+	0.0590401i	$0.0188040\pi$
$137$	189.347i	0.118080i	−0.998256	+	0.0590401i	$0.981196\pi$
$138$	0	0
$139$	−1581.63	−0.965125	−0.482563	−	0.875861i	$-0.660294\pi$
$139$	−1581.63	−0.965125	−0.482563	+	0.875861i	$0.660294\pi$
$140$	0	0
$141$	2956.74	1.76597
$142$	0	0
$143$	332.099i	0.194207i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	757.512i	0.425024i
$148$	0	0
$149$	2015.57	1.10820	0.554101	−	0.832449i	$-0.313062\pi$
$149$	2015.57	1.10820	0.554101	+	0.832449i	$0.313062\pi$
$150$	0	0
$151$	2275.41	1.22629	0.613146	−	0.789969i	$-0.289904\pi$
$151$	2275.41	1.22629	0.613146	+	0.789969i	$0.289904\pi$
$152$	0	0
$153$	1503.69i	0.794551i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1028.52i	− 0.522833i	−0.965226	−	0.261416i	$-0.915810\pi$
$157$	− 1028.52i	− 0.522833i	0.965226	−	0.261416i	$-0.0841896\pi$
$158$	0	0
$159$	−1619.55	−0.807790
$160$	0	0
$161$	1698.52	0.831441
$162$	0	0
$163$	− 1518.33i	− 0.729602i	−0.931086	−	0.364801i	$-0.881137\pi$
$163$	− 1518.33i	− 0.729602i	0.931086	−	0.364801i	$-0.118863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1047.55i	− 0.485401i	−0.970101	−	0.242701i	$-0.921967\pi$
$167$	− 1047.55i	− 0.485401i	0.970101	−	0.242701i	$-0.0780332\pi$
$168$	0	0
$169$	1285.51	0.585122
$170$	0	0
$171$	618.051	0.276395
$172$	0	0
$173$	− 2607.51i	− 1.14593i	−0.819581	−	0.572963i	$-0.805794\pi$
$173$	− 2607.51i	− 1.14593i	0.819581	−	0.572963i	$-0.194206\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1148.31i	− 0.487641i
$178$	0	0
$179$	−3017.20	−1.25987	−0.629934	−	0.776649i	$-0.716918\pi$
$179$	−3017.20	−1.25987	−0.629934	+	0.776649i	$0.716918\pi$
$180$	0	0
$181$	441.344	0.181242	0.0906210	−	0.995885i	$-0.471115\pi$
$181$	441.344	0.181242	0.0906210	+	0.995885i	$0.471115\pi$
$182$	0	0
$183$	− 2495.61i	− 1.00809i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 819.272i	− 0.320380i
$188$	0	0
$189$	713.724	0.274687
$190$	0	0
$191$	4113.45	1.55832	0.779159	−	0.626826i	$-0.215646\pi$
$191$	4113.45	1.55832	0.779159	+	0.626826i	$0.215646\pi$
$192$	0	0
$193$	− 2255.66i	− 0.841275i	−0.907229	−	0.420638i	$-0.861806\pi$
$193$	− 2255.66i	− 0.841275i	0.907229	−	0.420638i	$-0.138194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3284.67i	− 1.18793i	−0.804490	−	0.593967i	$-0.797561\pi$
$197$	− 3284.67i	− 1.18793i	0.804490	−	0.593967i	$-0.202439\pi$
$198$	0	0
$199$	−2296.22	−0.817963	−0.408982	−	0.912543i	$-0.634116\pi$
$199$	−2296.22	−0.817963	−0.408982	+	0.912543i	$0.634116\pi$
$200$	0	0
$201$	4199.28	1.47360
$202$	0	0
$203$	359.015i	0.124128i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2247.86i	− 0.754770i
$208$	0	0
$209$	−336.739	−0.111448
$210$	0	0
$211$	−4649.74	−1.51707	−0.758533	−	0.651634i	$-0.774084\pi$
$211$	−4649.74	−1.51707	−0.758533	+	0.651634i	$0.774084\pi$
$212$	0	0
$213$	− 2167.08i	− 0.697117i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4150.73i	1.29848i
$218$	0	0
$219$	2561.35	0.790321
$220$	0	0
$221$	2248.59	0.684419
$222$	0	0
$223$	689.598i	0.207080i	0.994625	+	0.103540i	$0.0330170\pi$
$223$	689.598i	0.207080i	−0.994625	+	0.103540i	$0.966983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5832.01i	1.70522i	0.522551	+	0.852608i	$0.324981\pi$
$227$	5832.01i	1.70522i	−0.522551	+	0.852608i	$0.675019\pi$
$228$	0	0
$229$	−2877.35	−0.830309	−0.415154	−	0.909751i	$-0.636272\pi$
$229$	−2877.35	−0.830309	−0.415154	+	0.909751i	$0.636272\pi$
$230$	0	0
$231$	1152.76	0.328338
$232$	0	0
$233$	− 919.503i	− 0.258535i	−0.991610	−	0.129267i	$-0.958737\pi$
$233$	− 919.503i	− 0.258535i	0.991610	−	0.129267i	$-0.0412626\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2066.53i	0.566396i
$238$	0	0
$239$	−4917.54	−1.33092	−0.665459	−	0.746434i	$-0.731764\pi$
$239$	−4917.54	−1.33092	−0.665459	+	0.746434i	$0.731764\pi$
$240$	0	0
$241$	2908.06	0.777282	0.388641	−	0.921389i	$-0.372945\pi$
$241$	2908.06	0.777282	0.388641	+	0.921389i	$0.372945\pi$
$242$	0	0
$243$	− 4689.20i	− 1.23791i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 924.221i	− 0.238084i
$248$	0	0
$249$	8368.23	2.12978
$250$	0	0
$251$	−720.590	−0.181208	−0.0906041	−	0.995887i	$-0.528880\pi$
$251$	−720.590	−0.181208	−0.0906041	+	0.995887i	$0.528880\pi$
$252$	0	0
$253$	1224.73i	0.304339i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4113.91i	− 0.998516i	−0.866453	−	0.499258i	$-0.833606\pi$
$257$	− 4113.91i	− 0.998516i	0.866453	−	0.499258i	$-0.166394\pi$
$258$	0	0
$259$	−4460.91	−1.07022
$260$	0	0
$261$	475.130	0.112681
$262$	0	0
$263$	3564.90i	0.835822i	0.908488	+	0.417911i	$0.137238\pi$
$263$	3564.90i	0.835822i	−0.908488	+	0.417911i	$0.862762\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 505.991i	− 0.115978i
$268$	0	0
$269$	−4405.46	−0.998533	−0.499267	−	0.866448i	$-0.666397\pi$
$269$	−4405.46	−0.998533	−0.499267	+	0.866448i	$0.666397\pi$
$270$	0	0
$271$	−7032.65	−1.57639	−0.788197	−	0.615423i	$-0.788985\pi$
$271$	−7032.65	−1.57639	−0.788197	+	0.615423i	$0.788985\pi$
$272$	0	0
$273$	3163.89i	0.701419i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7403.56i	− 1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$277$	− 7403.56i	− 1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$278$	0	0
$279$	5493.19	1.17874
$280$	0	0
$281$	5709.48	1.21210	0.606048	−	0.795428i	$-0.292754\pi$
$281$	5709.48	1.21210	0.606048	+	0.795428i	$0.292754\pi$
$282$	0	0
$283$	− 3030.73i	− 0.636602i	−0.947990	−	0.318301i	$-0.896888\pi$
$283$	− 3030.73i	− 0.636602i	0.947990	−	0.318301i	$-0.103112\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1944.95i	0.400024i
$288$	0	0
$289$	−634.160	−0.129078
$290$	0	0
$291$	−9547.34	−1.92328
$292$	0	0
$293$	2559.81i	0.510395i	0.966889	+	0.255197i	$0.0821405\pi$
$293$	2559.81i	0.510395i	−0.966889	+	0.255197i	$0.917860\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	514.635i	0.100546i
$298$	0	0
$299$	−3361.41	−0.650152
$300$	0	0
$301$	−7112.23	−1.36193
$302$	0	0
$303$	− 2208.38i	− 0.418707i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4897.98i	− 0.910561i	−0.890348	−	0.455281i	$-0.849539\pi$
$307$	− 4897.98i	− 0.910561i	0.890348	−	0.455281i	$-0.150461\pi$
$308$	0	0
$309$	3892.47	0.716618
$310$	0	0
$311$	3220.87	0.587263	0.293631	−	0.955919i	$-0.405136\pi$
$311$	3220.87	0.587263	0.293631	+	0.955919i	$0.405136\pi$
$312$	0	0
$313$	− 2546.74i	− 0.459905i	−0.973202	−	0.229952i	$-0.926143\pi$
$313$	− 2546.74i	− 0.459905i	0.973202	−	0.229952i	$-0.0738570\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3054.11i	− 0.541122i	−0.962703	−	0.270561i	$-0.912791\pi$
$317$	− 3054.11i	− 0.541122i	0.962703	−	0.270561i	$-0.0872092\pi$
$318$	0	0
$319$	−258.870	−0.0454355
$320$	0	0
$321$	−5949.36	−1.03446
$322$	0	0
$323$	2280.00i	0.392764i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9669.53i	1.63525i
$328$	0	0
$329$	−6566.20	−1.10032
$330$	0	0
$331$	−8086.37	−1.34280	−0.671401	−	0.741095i	$-0.734307\pi$
$331$	−8086.37	−1.34280	−0.671401	+	0.741095i	$0.734307\pi$
$332$	0	0
$333$	5903.69i	0.971533i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4367.51i	0.705975i	0.935628	+	0.352987i	$0.114834\pi$
$337$	4367.51i	0.705975i	−0.935628	+	0.352987i	$0.885166\pi$
$338$	0	0
$339$	2930.45	0.469499
$340$	0	0
$341$	−2992.91	−0.475293
$342$	0	0
$343$	− 6914.86i	− 1.08853i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9426.38i	− 1.45831i	−0.684347	−	0.729157i	$-0.739912\pi$
$347$	− 9426.38i	− 1.45831i	0.684347	−	0.729157i	$-0.260088\pi$
$348$	0	0
$349$	8924.60	1.36883	0.684417	−	0.729091i	$-0.260057\pi$
$349$	8924.60	1.36883	0.684417	+	0.729091i	$0.260057\pi$
$350$	0	0
$351$	−1412.48	−0.214794
$352$	0	0
$353$	− 9858.79i	− 1.48649i	−0.669020	−	0.743244i	$-0.733286\pi$
$353$	− 9858.79i	− 1.48649i	0.669020	−	0.743244i	$-0.266714\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 7805.15i	− 1.15712i
$358$	0	0
$359$	12720.2	1.87005	0.935026	−	0.354580i	$-0.115376\pi$
$359$	12720.2	1.87005	0.935026	+	0.354580i	$0.115376\pi$
$360$	0	0
$361$	−5921.87	−0.863372
$362$	0	0
$363$	831.204i	0.120184i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6175.96i	0.878427i	0.898383	+	0.439213i	$0.144743\pi$
$367$	6175.96i	0.878427i	−0.898383	+	0.439213i	$0.855257\pi$
$368$	0	0
$369$	2574.00	0.363136
$370$	0	0
$371$	3596.63	0.503309
$372$	0	0
$373$	10635.6i	1.47638i	0.674591	+	0.738192i	$0.264320\pi$
$373$	10635.6i	1.47638i	−0.674591	+	0.738192i	$0.735680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 710.500i	− 0.0970626i
$378$	0	0
$379$	−12614.8	−1.70971	−0.854855	−	0.518868i	$-0.826354\pi$
$379$	−12614.8	−1.70971	−0.854855	+	0.518868i	$0.826354\pi$
$380$	0	0
$381$	−2738.75	−0.368268
$382$	0	0
$383$	11523.9i	1.53745i	0.639582	+	0.768723i	$0.279107\pi$
$383$	11523.9i	1.53745i	−0.639582	+	0.768723i	$0.720893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9412.52i	1.23634i
$388$	0	0
$389$	8729.61	1.13781	0.568906	−	0.822402i	$-0.307367\pi$
$389$	8729.61	1.13781	0.568906	+	0.822402i	$0.307367\pi$
$390$	0	0
$391$	8292.43	1.07255
$392$	0	0
$393$	10337.9i	1.32692i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 313.914i	− 0.0396848i	−0.999803	−	0.0198424i	$-0.993684\pi$
$397$	− 313.914i	− 0.0396848i	0.999803	−	0.0198424i	$-0.00631645\pi$
$398$	0	0
$399$	−3208.09	−0.402520
$400$	0	0
$401$	−15699.2	−1.95506	−0.977531	−	0.210792i	$-0.932396\pi$
$401$	−15699.2	−1.95506	−0.977531	+	0.210792i	$0.932396\pi$
$402$	0	0
$403$	− 8214.40i	− 1.01536i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3216.57i	− 0.391743i
$408$	0	0
$409$	334.332	0.0404197	0.0202098	−	0.999796i	$-0.493567\pi$
$409$	334.332	0.0404197	0.0202098	+	0.999796i	$0.493567\pi$
$410$	0	0
$411$	−1300.71	−0.156105
$412$	0	0
$413$	2550.13i	0.303834i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 10865.0i	− 1.27592i
$418$	0	0
$419$	−6745.52	−0.786493	−0.393246	−	0.919433i	$-0.628648\pi$
$419$	−6745.52	−0.786493	−0.393246	+	0.919433i	$0.628648\pi$
$420$	0	0
$421$	14779.7	1.71097	0.855484	−	0.517830i	$-0.173260\pi$
$421$	14779.7	1.71097	0.855484	+	0.517830i	$0.173260\pi$
$422$	0	0
$423$	8689.89i	0.998858i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5542.15i	0.628111i
$428$	0	0
$429$	−2281.34	−0.256746
$430$	0	0
$431$	−10394.2	−1.16165	−0.580823	−	0.814030i	$-0.697269\pi$
$431$	−10394.2	−1.16165	−0.580823	+	0.814030i	$0.697269\pi$
$432$	0	0
$433$	− 5818.62i	− 0.645785i	−0.946436	−	0.322893i	$-0.895345\pi$
$433$	− 5818.62i	− 0.645785i	0.946436	−	0.322893i	$-0.104655\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3408.37i	− 0.373099i
$438$	0	0
$439$	−7542.46	−0.820005	−0.410002	−	0.912084i	$-0.634472\pi$
$439$	−7542.46	−0.820005	−0.410002	+	0.912084i	$0.634472\pi$
$440$	0	0
$441$	−2226.34	−0.240399
$442$	0	0
$443$	6599.99i	0.707845i	0.935275	+	0.353922i	$0.115152\pi$
$443$	6599.99i	0.707845i	−0.935275	+	0.353922i	$0.884848\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13845.9i	1.46507i
$448$	0	0
$449$	−10081.9	−1.05968	−0.529840	−	0.848098i	$-0.677748\pi$
$449$	−10081.9	−1.05968	−0.529840	+	0.848098i	$0.677748\pi$
$450$	0	0
$451$	−1402.42	−0.146424
$452$	0	0
$453$	15630.8i	1.62119i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15159.8i	1.55175i	0.630890	+	0.775873i	$0.282690\pi$
$457$	15159.8i	1.55175i	−0.630890	+	0.775873i	$0.717310\pi$
$458$	0	0
$459$	3484.51	0.354342
$460$	0	0
$461$	11761.3	1.18824	0.594118	−	0.804378i	$-0.297501\pi$
$461$	11761.3	1.18824	0.594118	+	0.804378i	$0.297501\pi$
$462$	0	0
$463$	6693.19i	0.671833i	0.941892	+	0.335917i	$0.109046\pi$
$463$	6693.19i	0.671833i	−0.941892	+	0.335917i	$0.890954\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19784.8i	− 1.96046i	−0.197873	−	0.980228i	$-0.563403\pi$
$467$	− 19784.8i	− 1.96046i	0.197873	−	0.980228i	$-0.436597\pi$
$468$	0	0
$469$	−9325.58	−0.918156
$470$	0	0
$471$	7065.37	0.691199
$472$	0	0
$473$	− 5128.32i	− 0.498520i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4759.88i	− 0.456897i
$478$	0	0
$479$	−4432.99	−0.422857	−0.211428	−	0.977393i	$-0.567812\pi$
$479$	−4432.99	−0.422857	−0.211428	+	0.977393i	$0.567812\pi$
$480$	0	0
$481$	8828.26	0.836870
$482$	0	0
$483$	11667.9i	1.09919i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18994.9i	1.76743i	0.468023	+	0.883716i	$0.344966\pi$
$487$	18994.9i	1.76743i	−0.468023	+	0.883716i	$0.655034\pi$
$488$	0	0
$489$	10430.1	0.964553
$490$	0	0
$491$	4821.55	0.443164	0.221582	−	0.975142i	$-0.428878\pi$
$491$	4821.55	0.443164	0.221582	+	0.975142i	$0.428878\pi$
$492$	0	0
$493$	1752.77i	0.160123i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4812.56i	0.434352i
$498$	0	0
$499$	−17440.3	−1.56460	−0.782300	−	0.622902i	$-0.785953\pi$
$499$	−17440.3	−1.56460	−0.782300	+	0.622902i	$0.785953\pi$
$500$	0	0
$501$	7196.11	0.641714
$502$	0	0
$503$	− 9153.19i	− 0.811373i	−0.914012	−	0.405687i	$-0.867032\pi$
$503$	− 9153.19i	− 0.811373i	0.914012	−	0.405687i	$-0.132968\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8830.77i	0.773547i
$508$	0	0
$509$	1383.24	0.120454	0.0602268	−	0.998185i	$-0.480818\pi$
$509$	1383.24	0.120454	0.0602268	+	0.998185i	$0.480818\pi$
$510$	0	0
$511$	−5688.15	−0.492425
$512$	0	0
$513$	− 1432.21i	− 0.123262i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4734.60i	− 0.402761i
$518$	0	0
$519$	17912.2	1.51494
$520$	0	0
$521$	−22764.1	−1.91423	−0.957113	−	0.289713i	$-0.906440\pi$
$521$	−22764.1	−1.91423	−0.957113	+	0.289713i	$0.906440\pi$
$522$	0	0
$523$	12616.5i	1.05484i	0.849605	+	0.527420i	$0.176841\pi$
$523$	12616.5i	1.05484i	−0.849605	+	0.527420i	$0.823159\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20264.5i	1.67502i
$528$	0	0
$529$	−229.312	−0.0188470
$530$	0	0
$531$	3374.91	0.275816
$532$	0	0
$533$	− 3849.11i	− 0.312802i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20726.5i	− 1.66558i
$538$	0	0
$539$	1213.00	0.0969342
$540$	0	0
$541$	−21440.2	−1.70386	−0.851929	−	0.523657i	$-0.824567\pi$
$541$	−21440.2	−1.70386	−0.851929	+	0.523657i	$0.824567\pi$
$542$	0	0
$543$	3031.79i	0.239607i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22138.5i	1.73048i	0.501358	+	0.865240i	$0.332834\pi$
$547$	22138.5i	1.73048i	−0.501358	+	0.865240i	$0.667166\pi$
$548$	0	0
$549$	7334.63	0.570190
$550$	0	0
$551$	720.425	0.0557008
$552$	0	0
$553$	− 4589.27i	− 0.352904i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18160.8i	1.38151i	0.723090	+	0.690753i	$0.242721\pi$
$557$	18160.8i	1.38151i	−0.723090	+	0.690753i	$0.757279\pi$
$558$	0	0
$559$	14075.3	1.06498
$560$	0	0
$561$	5627.95	0.423551
$562$	0	0
$563$	11886.6i	0.889806i	0.895579	+	0.444903i	$0.146762\pi$
$563$	11886.6i	0.889806i	−0.895579	+	0.444903i	$0.853238\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	13218.8i	0.979081i
$568$	0	0
$569$	19547.3	1.44018	0.720092	−	0.693879i	$-0.244100\pi$
$569$	19547.3	1.44018	0.720092	+	0.693879i	$0.244100\pi$
$570$	0	0
$571$	−6953.38	−0.509614	−0.254807	−	0.966992i	$-0.582012\pi$
$571$	−6953.38	−0.509614	−0.254807	+	0.966992i	$0.582012\pi$
$572$	0	0
$573$	28257.2i	2.06014i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1267.04i	0.0914170i	0.998955	+	0.0457085i	$0.0145545\pi$
$577$	1267.04i	0.0914170i	−0.998955	+	0.0457085i	$0.985445\pi$
$578$	0	0
$579$	15495.2	1.11219
$580$	0	0
$581$	−18583.8	−1.32700
$582$	0	0
$583$	2593.37i	0.184231i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16816.0i	− 1.18241i	−0.806523	−	0.591203i	$-0.798653\pi$
$587$	− 16816.0i	− 1.18241i	0.806523	−	0.591203i	$-0.201347\pi$
$588$	0	0
$589$	8329.16	0.582677
$590$	0	0
$591$	22563.9	1.57048
$592$	0	0
$593$	− 1411.66i	− 0.0977573i	−0.998805	−	0.0488786i	$-0.984435\pi$
$593$	− 1411.66i	− 0.0977573i	0.998805	−	0.0488786i	$-0.0155648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15773.8i	− 1.08137i
$598$	0	0
$599$	4859.48	0.331474	0.165737	−	0.986170i	$-0.447000\pi$
$599$	4859.48	0.331474	0.165737	+	0.986170i	$0.447000\pi$
$600$	0	0
$601$	11695.5	0.793793	0.396897	−	0.917863i	$-0.370087\pi$
$601$	11695.5	0.793793	0.396897	+	0.917863i	$0.370087\pi$
$602$	0	0
$603$	12341.7i	0.833489i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.57260i	0 0.000506363i	−1.00000		0.000253181i	$-0.999919\pi$
$607$	− 7.57260i	0 0.000506363i	1.00000		0.000253181i	$-8.05902e-5\pi$
$608$	0	0
$609$	−2466.24	−0.164100
$610$	0	0
$611$	12994.7	0.860407
$612$	0	0
$613$	− 19891.5i	− 1.31062i	−0.755359	−	0.655311i	$-0.772537\pi$
$613$	− 19891.5i	− 1.31062i	0.755359	−	0.655311i	$-0.227463\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10779.1i	− 0.703320i	−0.936128	−	0.351660i	$-0.885617\pi$
$617$	− 10779.1i	− 0.703320i	0.936128	−	0.351660i	$-0.114383\pi$
$618$	0	0
$619$	6513.73	0.422955	0.211477	−	0.977383i	$-0.432173\pi$
$619$	6513.73	0.422955	0.211477	+	0.977383i	$0.432173\pi$
$620$	0	0
$621$	−5208.98	−0.336601
$622$	0	0
$623$	1123.68i	0.0722624i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2313.21i	− 0.147338i
$628$	0	0
$629$	−21778.9	−1.38057
$630$	0	0
$631$	10913.6	0.688533	0.344266	−	0.938872i	$-0.388128\pi$
$631$	10913.6	0.688533	0.344266	+	0.938872i	$0.388128\pi$
$632$	0	0
$633$	− 31941.2i	− 2.00560i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3329.22i	0.207078i
$638$	0	0
$639$	6369.08	0.394298
$640$	0	0
$641$	7378.43	0.454650	0.227325	−	0.973819i	$-0.427002\pi$
$641$	7378.43	0.454650	0.227325	+	0.973819i	$0.427002\pi$
$642$	0	0
$643$	297.203i	0.0182279i	0.999958	+	0.00911395i	$0.00290110\pi$
$643$	297.203i	0.0182279i	−0.999958	+	0.00911395i	$0.997099\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17665.5i	1.07342i	0.843768	+	0.536709i	$0.180333\pi$
$647$	17665.5i	1.07342i	−0.843768	+	0.536709i	$0.819667\pi$
$648$	0	0
$649$	−1838.78	−0.111215
$650$	0	0
$651$	−28513.3	−1.71662
$652$	0	0
$653$	− 28420.0i	− 1.70316i	−0.524228	−	0.851578i	$-0.675646\pi$
$653$	− 28420.0i	− 1.70316i	0.524228	−	0.851578i	$-0.324354\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7527.86i	0.447016i
$658$	0	0
$659$	−19552.3	−1.15576	−0.577882	−	0.816120i	$-0.696121\pi$
$659$	−19552.3	−1.15576	−0.577882	+	0.816120i	$0.696121\pi$
$660$	0	0
$661$	27600.3	1.62410	0.812049	−	0.583590i	$-0.198352\pi$
$661$	27600.3	1.62410	0.812049	+	0.583590i	$0.198352\pi$
$662$	0	0
$663$	15446.6i	0.904821i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2620.20i	− 0.152106i
$668$	0	0
$669$	−4737.16	−0.273766
$670$	0	0
$671$	−3996.20	−0.229913
$672$	0	0
$673$	24272.9i	1.39027i	0.718879	+	0.695135i	$0.244656\pi$
$673$	24272.9i	1.39027i	−0.718879	+	0.695135i	$0.755344\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18258.3i	1.03652i	0.855223	+	0.518261i	$0.173420\pi$
$677$	18258.3i	1.03652i	−0.855223	+	0.518261i	$0.826580\pi$
$678$	0	0
$679$	21202.3	1.19834
$680$	0	0
$681$	−40062.7	−2.25434
$682$	0	0
$683$	− 2171.51i	− 0.121655i	−0.998148	−	0.0608277i	$-0.980626\pi$
$683$	− 2171.51i	− 0.121655i	0.998148	−	0.0608277i	$-0.0193740\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 19765.8i	− 1.09769i
$688$	0	0
$689$	−7117.83	−0.393567
$690$	0	0
$691$	−19495.9	−1.07331	−0.536657	−	0.843801i	$-0.680313\pi$
$691$	−19495.9	−1.07331	−0.536657	+	0.843801i	$0.680313\pi$
$692$	0	0
$693$	3387.98i	0.185712i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9495.56i	0.516026i
$698$	0	0
$699$	6316.48	0.341790
$700$	0	0
$701$	−13625.9	−0.734158	−0.367079	−	0.930190i	$-0.619642\pi$
$701$	−13625.9	−0.734158	−0.367079	+	0.930190i	$0.619642\pi$
$702$	0	0
$703$	8951.59i	0.480250i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4904.28i	0.260883i
$708$	0	0
$709$	−6810.99	−0.360779	−0.180389	−	0.983595i	$-0.557736\pi$
$709$	−6810.99	−0.360779	−0.180389	+	0.983595i	$0.557736\pi$
$710$	0	0
$711$	−6073.57	−0.320361
$712$	0	0
$713$	− 30293.3i	− 1.59116i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 33780.8i	− 1.75951i
$718$	0	0
$719$	−13166.6	−0.682934	−0.341467	−	0.939894i	$-0.610924\pi$
$719$	−13166.6	−0.682934	−0.341467	+	0.939894i	$0.610924\pi$
$720$	0	0
$721$	−8644.24	−0.446503
$722$	0	0
$723$	19976.8i	1.02759i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8075.12i	0.411952i	0.978557	+	0.205976i	$0.0660370\pi$
$727$	8075.12i	0.411952i	−0.978557	+	0.205976i	$0.933963\pi$
$728$	0	0
$729$	8816.71	0.447935
$730$	0	0
$731$	−34723.0	−1.75688
$732$	0	0
$733$	26226.1i	1.32153i	0.750593	+	0.660765i	$0.229768\pi$
$733$	26226.1i	1.32153i	−0.750593	+	0.660765i	$0.770232\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6724.26i	− 0.336081i
$738$	0	0
$739$	−37686.2	−1.87592	−0.937962	−	0.346737i	$-0.887290\pi$
$739$	−37686.2	−1.87592	−0.937962	+	0.346737i	$0.887290\pi$
$740$	0	0
$741$	6348.89	0.314754
$742$	0	0
$743$	− 5461.18i	− 0.269652i	−0.990869	−	0.134826i	$-0.956952\pi$
$743$	− 5461.18i	− 0.269652i	0.990869	−	0.134826i	$-0.0430475\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24594.3i	1.20463i
$748$	0	0
$749$	13212.1	0.644539
$750$	0	0
$751$	16325.5	0.793242	0.396621	−	0.917983i	$-0.370183\pi$
$751$	16325.5	0.793242	0.396621	+	0.917983i	$0.370183\pi$
$752$	0	0
$753$	− 4950.06i	− 0.239562i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15157.7i	− 0.727763i	−0.931445	−	0.363881i	$-0.881451\pi$
$757$	− 15157.7i	− 0.727763i	0.931445	−	0.363881i	$-0.118549\pi$
$758$	0	0
$759$	−8413.20	−0.402345
$760$	0	0
$761$	−22985.6	−1.09491	−0.547456	−	0.836835i	$-0.684404\pi$
$761$	−22985.6	−1.09491	−0.547456	+	0.836835i	$0.684404\pi$
$762$	0	0
$763$	− 21473.7i	− 1.01887i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5046.77i	− 0.237586i
$768$	0	0
$769$	3715.50	0.174232	0.0871159	−	0.996198i	$-0.472235\pi$
$769$	3715.50	0.174232	0.0871159	+	0.996198i	$0.472235\pi$
$770$	0	0
$771$	28260.3	1.32007
$772$	0	0
$773$	− 11510.5i	− 0.535582i	−0.963477	−	0.267791i	$-0.913706\pi$
$773$	− 11510.5i	− 0.535582i	0.963477	−	0.267791i	$-0.0862936\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 30644.0i	− 1.41486i
$778$	0	0
$779$	3902.88	0.179506
$780$	0	0
$781$	−3470.13	−0.158990
$782$	0	0
$783$	− 1101.02i	− 0.0502519i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5418.55i	− 0.245426i	−0.992442	−	0.122713i	$-0.960841\pi$
$787$	− 5418.55i	− 0.245426i	0.992442	−	0.122713i	$-0.0391595\pi$
$788$	0	0
$789$	−24488.9	−1.10498
$790$	0	0
$791$	−6507.82	−0.292530
$792$	0	0
$793$	− 10968.1i	− 0.491157i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11751.3i	− 0.522275i	−0.965302	−	0.261138i	$-0.915902\pi$
$797$	− 11751.3i	− 0.522275i	0.965302	−	0.261138i	$-0.0840977\pi$
$798$	0	0
$799$	−32057.2	−1.41940
$800$	0	0
$801$	1487.11	0.0655988
$802$	0	0
$803$	− 4101.48i	− 0.180246i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 30263.1i	− 1.32009i
$808$	0	0
$809$	35385.4	1.53780	0.768902	−	0.639366i	$-0.220803\pi$
$809$	35385.4	1.53780	0.768902	+	0.639366i	$0.220803\pi$
$810$	0	0
$811$	11033.5	0.477731	0.238865	−	0.971053i	$-0.423225\pi$
$811$	11033.5	0.477731	0.238865	+	0.971053i	$0.423225\pi$
$812$	0	0
$813$	− 48310.4i	− 2.08404i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14271.9i	0.611152i
$818$	0	0
$819$	−9298.72	−0.396732
$820$	0	0
$821$	−449.705	−0.0191167	−0.00955834	−	0.999954i	$-0.503043\pi$
$821$	−449.705	−0.0191167	−0.00955834	+	0.999954i	$0.503043\pi$
$822$	0	0
$823$	23306.1i	0.987120i	0.869712	+	0.493560i	$0.164305\pi$
$823$	23306.1i	0.987120i	−0.869712	+	0.493560i	$0.835695\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33096.0i	− 1.39161i	−0.718232	−	0.695804i	$-0.755048\pi$
$827$	− 33096.0i	− 1.39161i	0.718232	−	0.695804i	$-0.244952\pi$
$828$	0	0
$829$	−42098.2	−1.76373	−0.881865	−	0.471502i	$-0.843712\pi$
$829$	−42098.2	−1.76373	−0.881865	+	0.471502i	$0.843712\pi$
$830$	0	0
$831$	50858.4	2.12306
$832$	0	0
$833$	− 8213.01i	− 0.341614i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 12729.4i	− 0.525677i
$838$	0	0
$839$	452.609	0.0186243	0.00931215	−	0.999957i	$-0.497036\pi$
$839$	452.609	0.0186243	0.00931215	+	0.999957i	$0.497036\pi$
$840$	0	0
$841$	−23835.2	−0.977292
$842$	0	0
$843$	39221.0i	1.60242i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1845.90i	− 0.0748831i
$848$	0	0
$849$	20819.5	0.841605
$850$	0	0
$851$	32557.1	1.31145
$852$	0	0
$853$	− 32079.3i	− 1.28766i	−0.765169	−	0.643830i	$-0.777344\pi$
$853$	− 32079.3i	− 1.28766i	0.765169	−	0.643830i	$-0.222656\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15049.5i	− 0.599862i	−0.953961	−	0.299931i	$-0.903036\pi$
$857$	− 15049.5i	− 0.599862i	0.953961	−	0.299931i	$-0.0969636\pi$
$858$	0	0
$859$	−15862.2	−0.630049	−0.315025	−	0.949083i	$-0.602013\pi$
$859$	−15862.2	−0.630049	−0.315025	+	0.949083i	$0.602013\pi$
$860$	0	0
$861$	−13360.8	−0.528843
$862$	0	0
$863$	26892.7i	1.06076i	0.847759	+	0.530381i	$0.177951\pi$
$863$	26892.7i	1.06076i	−0.847759	+	0.530381i	$0.822049\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4356.33i	− 0.170645i
$868$	0	0
$869$	3309.12	0.129176
$870$	0	0
$871$	18455.6	0.717960
$872$	0	0
$873$	− 28059.7i	− 1.08783i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29275.7i	1.12722i	0.826041	+	0.563609i	$0.190588\pi$
$877$	29275.7i	1.12722i	−0.826041	+	0.563609i	$0.809412\pi$
$878$	0	0
$879$	−17584.5	−0.674756
$880$	0	0
$881$	27090.9	1.03600	0.518000	−	0.855381i	$-0.326677\pi$
$881$	27090.9	1.03600	0.518000	+	0.855381i	$0.326677\pi$
$882$	0	0
$883$	21772.9i	0.829805i	0.909866	+	0.414902i	$0.136184\pi$
$883$	21772.9i	0.829805i	−0.909866	+	0.414902i	$0.863816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3101.20i	0.117393i	0.998276	+	0.0586967i	$0.0186945\pi$
$887$	3101.20i	0.117393i	−0.998276	+	0.0586967i	$0.981306\pi$
$888$	0	0
$889$	6082.09	0.229457
$890$	0	0
$891$	−9531.52	−0.358381
$892$	0	0
$893$	13176.2i	0.493757i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 23091.1i	− 0.859519i
$898$	0	0
$899$	6403.08	0.237547
$900$	0	0
$901$	17559.3	0.649262
$902$	0	0
$903$	− 48857.2i	− 1.80051i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 15615.2i	− 0.571659i	−0.958281	−	0.285829i	$-0.907731\pi$
$907$	− 15615.2i	− 0.571659i	0.958281	−	0.285829i	$-0.0922690\pi$
$908$	0	0
$909$	6490.46	0.236826
$910$	0	0
$911$	14189.7	0.516053	0.258027	−	0.966138i	$-0.416928\pi$
$911$	14189.7	0.516053	0.258027	+	0.966138i	$0.416928\pi$
$912$	0	0
$913$	− 13400.0i	− 0.485733i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 22958.0i	− 0.826760i
$918$	0	0
$919$	−38188.9	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$919$	−38188.9	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$920$	0	0
$921$	33646.4	1.20379
$922$	0	0
$923$	− 9524.19i	− 0.339645i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11440.0i	0.405329i
$928$	0	0
$929$	41832.9	1.47739	0.738693	−	0.674042i	$-0.235443\pi$
$929$	41832.9	1.47739	0.738693	+	0.674042i	$0.235443\pi$
$930$	0	0
$931$	−3375.73	−0.118835
$932$	0	0
$933$	22125.6i	0.776378i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6058.27i	− 0.211222i	−0.994408	−	0.105611i	$-0.966320\pi$
$937$	− 6058.27i	− 0.211222i	0.994408	−	0.105611i	$-0.0336798\pi$
$938$	0	0
$939$	17494.7	0.608007
$940$	0	0
$941$	32923.7	1.14058	0.570288	−	0.821445i	$-0.306832\pi$
$941$	32923.7	1.14058	0.570288	+	0.821445i	$0.306832\pi$
$942$	0	0
$943$	− 14194.9i	− 0.490189i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3942.99i	0.135301i	0.997709	+	0.0676505i	$0.0215503\pi$
$947$	3942.99i	0.135301i	−0.997709	+	0.0676505i	$0.978450\pi$
$948$	0	0
$949$	11257.0	0.385056
$950$	0	0
$951$	20980.1	0.715379
$952$	0	0
$953$	9152.99i	0.311117i	0.987827	+	0.155558i	$0.0497177\pi$
$953$	9152.99i	0.311117i	−0.987827	+	0.155558i	$0.950282\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1778.29i	− 0.0600669i
$958$	0	0
$959$	2888.56	0.0972644
$960$	0	0
$961$	44237.9	1.48494
$962$	0	0
$963$	− 17485.2i	− 0.585103i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4536.39i	− 0.150859i	−0.997151	−	0.0754294i	$-0.975967\pi$
$967$	− 4536.39i	− 0.150859i	0.997151	−	0.0754294i	$-0.0240328\pi$
$968$	0	0
$969$	−15662.4	−0.519245
$970$	0	0
$971$	−53683.7	−1.77424	−0.887122	−	0.461535i	$-0.847299\pi$
$971$	−53683.7	−1.77424	−0.887122	+	0.461535i	$0.847299\pi$
$972$	0	0
$973$	24128.5i	0.794988i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39392.4i	1.28994i	0.764206	+	0.644972i	$0.223131\pi$
$977$	39392.4i	1.28994i	−0.764206	+	0.644972i	$0.776869\pi$
$978$	0	0
$979$	−810.239	−0.0264508
$980$	0	0
$981$	−28418.9	−0.924919
$982$	0	0
$983$	− 33348.6i	− 1.08205i	−0.841006	−	0.541026i	$-0.818036\pi$
$983$	− 33348.6i	− 1.08205i	0.841006	−	0.541026i	$-0.181964\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 45106.2i	− 1.45466i
$988$	0	0
$989$	51907.3	1.66891
$990$	0	0
$991$	−696.027	−0.0223108	−0.0111554	−	0.999938i	$-0.503551\pi$
$991$	−696.027	−0.0223108	−0.0111554	+	0.999938i	$0.503551\pi$
$992$	0	0
$993$	− 55549.0i	− 1.77522i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	45135.4i	1.43375i	0.697200	+	0.716877i	$0.254429\pi$
$997$	45135.4i	1.43375i	−0.697200	+	0.716877i	$0.745571\pi$
$998$	0	0
$999$	13680.6	0.433270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.b.h.749.5		6
5.2	odd	4		220.4.a.f.1.3	✓	3
5.3	odd	4		1100.4.a.i.1.1		3
5.4	even	2	inner	1100.4.b.h.749.2		6
15.2	even	4		1980.4.a.l.1.3		3
20.7	even	4		880.4.a.w.1.1		3
55.32	even	4		2420.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.3	✓	3	5.2	odd	4
880.4.a.w.1.1		3	20.7	even	4
1100.4.a.i.1.1		3	5.3	odd	4
1100.4.b.h.749.2		6	5.4	even	2	inner
1100.4.b.h.749.5		6	1.1	even	1	trivial
1980.4.a.l.1.3		3	15.2	even	4
2420.4.a.i.1.3		3	55.32	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.86946i q^{3} -15.2554i q^{7} -20.1894 q^{9} +11.0000 q^{11} +30.1909i q^{13} -74.4793i q^{17} -30.6126 q^{19} +104.796 q^{21} +111.339i q^{23} +46.7850i q^{27} -23.5336 q^{29} -272.083 q^{31} +75.5640i q^{33} -292.415i q^{37} -207.395 q^{39} -127.493 q^{41} -466.210i q^{43} -430.418i q^{47} +110.273 q^{49} +511.632 q^{51} +235.761i q^{53} -210.292i q^{57} -167.162 q^{59} -363.291 q^{61} +307.998i q^{63} -611.297i q^{67} -764.836 q^{69} -315.466 q^{71} -372.861i q^{73} -167.809i q^{77} +300.829 q^{79} -866.502 q^{81} -1218.18i q^{83} -161.663i q^{87} -73.6581 q^{89} +460.574 q^{91} -1869.06i q^{93} +1389.82i q^{97} -222.084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{9} + 66 q^{11} - 342 q^{19} + 518 q^{21} - 110 q^{29} + 362 q^{31} - 1108 q^{39} + 604 q^{41} + 536 q^{49} + 1666 q^{51} + 84 q^{59} + 698 q^{61} + 516 q^{69} + 1854 q^{71} + 2292 q^{79} - 3066 q^{81}+ \cdots - 528 q^{99}+O(q^{100}) \) 6 * q - 48 * q^9 + 66 * q^11 - 342 * q^19 + 518 * q^21 - 110 * q^29 + 362 * q^31 - 1108 * q^39 + 604 * q^41 + 536 * q^49 + 1666 * q^51 + 84 * q^59 + 698 * q^61 + 516 * q^69 + 1854 * q^71 + 2292 * q^79 - 3066 * q^81 + 1750 * q^89 + 3964 * q^91 - 528 * q^99

Introduction

L-functions

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