LMFDB - Embedded newform 1800.3.c.a.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,3,Mod(449,1800)] 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("1800.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,64] 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1800.449
Dual form			1800.3.c.a.449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{7} -5.65685i q^{11} -8.00000i q^{13} +9.89949 q^{17} +16.0000 q^{19} +39.5980 q^{23} +29.6985i q^{29} -4.00000 q^{31} -30.0000i q^{37} -21.2132i q^{41} -8.00000i q^{43} -16.9706 q^{47} -95.0000 q^{49} -49.4975 q^{53} -79.1960i q^{59} -14.0000 q^{61} +88.0000i q^{67} -28.2843i q^{71} -80.0000i q^{73} -67.8823 q^{77} -100.000 q^{79} +130.108 q^{83} -148.492i q^{89} -96.0000 q^{91} +112.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 64 q^{19} - 16 q^{31} - 380 q^{49} - 56 q^{61} - 400 q^{79} - 384 q^{91}+O(q^{100}) $ 4 * q + 64 * q^19 - 16 * q^31 - 380 * q^49 - 56 * q^61 - 400 * q^79 - 384 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 12.0000i	− 1.71429i	−0.515079	−	0.857143i	$-0.672237\pi$
$7$	− 12.0000i	− 1.71429i	0.515079	−	0.857143i	$-0.327763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.65685i	− 0.514259i	−0.966377	−	0.257130i	$-0.917223\pi$
$11$	− 5.65685i	− 0.514259i	0.966377	−	0.257130i	$-0.0827768\pi$
$12$	0	0
$13$	− 8.00000i	− 0.615385i	−0.951486	−	0.307692i	$-0.900443\pi$
$13$	− 8.00000i	− 0.615385i	0.951486	−	0.307692i	$-0.0995567\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.89949	0.582323	0.291162	−	0.956674i	$-0.405958\pi$
$17$	9.89949	0.582323	0.291162	+	0.956674i	$0.405958\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.5980	1.72165	0.860826	−	0.508900i	$-0.169948\pi$
$23$	39.5980	1.72165	0.860826	+	0.508900i	$0.169948\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.6985i	1.02409i	0.858960	+	0.512043i	$0.171111\pi$
$29$	29.6985i	1.02409i	−0.858960	+	0.512043i	$0.828889\pi$
$30$	0	0
$31$	−4.00000	−0.129032	−0.0645161	−	0.997917i	$-0.520550\pi$
$31$	−4.00000	−0.129032	−0.0645161	+	0.997917i	$0.520550\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 30.0000i	− 0.810811i	−0.914137	−	0.405405i	$-0.867130\pi$
$37$	− 30.0000i	− 0.810811i	0.914137	−	0.405405i	$-0.132870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 21.2132i	− 0.517395i	−0.965958	−	0.258698i	$-0.916707\pi$
$41$	− 21.2132i	− 0.517395i	0.965958	−	0.258698i	$-0.0832933\pi$
$42$	0	0
$43$	− 8.00000i	− 0.186047i	−0.995664	−	0.0930233i	$-0.970347\pi$
$43$	− 8.00000i	− 0.186047i	0.995664	−	0.0930233i	$-0.0296531\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−16.9706	−0.361076	−0.180538	−	0.983568i	$-0.557784\pi$
$47$	−16.9706	−0.361076	−0.180538	+	0.983568i	$0.557784\pi$
$48$	0	0
$49$	−95.0000	−1.93878
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−49.4975	−0.933915	−0.466957	−	0.884280i	$-0.654650\pi$
$53$	−49.4975	−0.933915	−0.466957	+	0.884280i	$0.654650\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 79.1960i	− 1.34230i	−0.741320	−	0.671152i	$-0.765800\pi$
$59$	− 79.1960i	− 1.34230i	0.741320	−	0.671152i	$-0.234200\pi$
$60$	0	0
$61$	−14.0000	−0.229508	−0.114754	−	0.993394i	$-0.536608\pi$
$61$	−14.0000	−0.229508	−0.114754	+	0.993394i	$0.536608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	88.0000i	1.31343i	0.754138	+	0.656716i	$0.228055\pi$
$67$	88.0000i	1.31343i	−0.754138	+	0.656716i	$0.771945\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 28.2843i	− 0.398370i	−0.979962	−	0.199185i	$-0.936171\pi$
$71$	− 28.2843i	− 0.398370i	0.979962	−	0.199185i	$-0.0638295\pi$
$72$	0	0
$73$	− 80.0000i	− 1.09589i	−0.836514	−	0.547945i	$-0.815410\pi$
$73$	− 80.0000i	− 1.09589i	0.836514	−	0.547945i	$-0.184590\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−67.8823	−0.881588
$78$	0	0
$79$	−100.000	−1.26582	−0.632911	−	0.774224i	$-0.718140\pi$
$79$	−100.000	−1.26582	−0.632911	+	0.774224i	$0.718140\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	130.108	1.56756	0.783781	−	0.621037i	$-0.213288\pi$
$83$	130.108	1.56756	0.783781	+	0.621037i	$0.213288\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 148.492i	− 1.66845i	−0.551421	−	0.834227i	$-0.685914\pi$
$89$	− 148.492i	− 1.66845i	0.551421	−	0.834227i	$-0.314086\pi$
$90$	0	0
$91$	−96.0000	−1.05495
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	112.000i	1.15464i	0.816518	+	0.577320i	$0.195901\pi$
$97$	112.000i	1.15464i	−0.816518	+	0.577320i	$0.804099\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	63.6396i	0.630095i	0.949076	+	0.315048i	$0.102020\pi$
$101$	63.6396i	0.630095i	−0.949076	+	0.315048i	$0.897980\pi$
$102$	0	0
$103$	− 44.0000i	− 0.427184i	−0.976923	−	0.213592i	$-0.931484\pi$
$103$	− 44.0000i	− 0.427184i	0.976923	−	0.213592i	$-0.0685164\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−136.000	−1.24771	−0.623853	−	0.781542i	$-0.714434\pi$
$109$	−136.000	−1.24771	−0.623853	+	0.781542i	$0.714434\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	21.2132	0.187727	0.0938637	−	0.995585i	$-0.470078\pi$
$113$	21.2132	0.187727	0.0938637	+	0.995585i	$0.470078\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 118.794i	− 0.998268i
$120$	0	0
$121$	89.0000	0.735537
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.0000i	0.157480i	0.996895	+	0.0787402i	$0.0250897\pi$
$127$	20.0000i	0.157480i	−0.996895	+	0.0787402i	$0.974910\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	56.5685i	0.431821i	0.976413	+	0.215910i	$0.0692719\pi$
$131$	56.5685i	0.431821i	−0.976413	+	0.215910i	$0.930728\pi$
$132$	0	0
$133$	− 192.000i	− 1.44361i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	89.0955	0.650332	0.325166	−	0.945657i	$-0.394580\pi$
$137$	89.0955	0.650332	0.325166	+	0.945657i	$0.394580\pi$
$138$	0	0
$139$	−120.000	−0.863309	−0.431655	−	0.902039i	$-0.642070\pi$
$139$	−120.000	−0.863309	−0.431655	+	0.902039i	$0.642070\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−45.2548	−0.316467
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 128.693i	− 0.863714i	−0.901942	−	0.431857i	$-0.857858\pi$
$149$	− 128.693i	− 0.863714i	0.901942	−	0.431857i	$-0.142142\pi$
$150$	0	0
$151$	−196.000	−1.29801	−0.649007	−	0.760783i	$-0.724815\pi$
$151$	−196.000	−1.29801	−0.649007	+	0.760783i	$0.724815\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 238.000i	− 1.51592i	−0.652299	−	0.757962i	$-0.726195\pi$
$157$	− 238.000i	− 1.51592i	0.652299	−	0.757962i	$-0.273805\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 475.176i	− 2.95140i
$162$	0	0
$163$	− 168.000i	− 1.03067i	−0.856987	−	0.515337i	$-0.827667\pi$
$163$	− 168.000i	− 1.03067i	0.856987	−	0.515337i	$-0.172333\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	169.706	1.01620	0.508101	−	0.861298i	$-0.330348\pi$
$167$	169.706	1.01620	0.508101	+	0.861298i	$0.330348\pi$
$168$	0	0
$169$	105.000	0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	207.889	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$173$	207.889	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.647i	1.13769i	0.822444	+	0.568846i	$0.192610\pi$
$179$	203.647i	1.13769i	−0.822444	+	0.568846i	$0.807390\pi$
$180$	0	0
$181$	−216.000	−1.19337	−0.596685	−	0.802475i	$-0.703516\pi$
$181$	−216.000	−1.19337	−0.596685	+	0.802475i	$0.703516\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 56.0000i	− 0.299465i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	373.352i	1.95472i	0.211573	+	0.977362i	$0.432141\pi$
$191$	373.352i	1.95472i	−0.211573	+	0.977362i	$0.567859\pi$
$192$	0	0
$193$	− 178.000i	− 0.922280i	−0.887327	−	0.461140i	$-0.847441\pi$
$193$	− 178.000i	− 0.922280i	0.887327	−	0.461140i	$-0.152559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−207.889	−1.05528	−0.527638	−	0.849469i	$-0.676922\pi$
$197$	−207.889	−1.05528	−0.527638	+	0.849469i	$0.676922\pi$
$198$	0	0
$199$	−196.000	−0.984925	−0.492462	−	0.870334i	$-0.663903\pi$
$199$	−196.000	−0.984925	−0.492462	+	0.870334i	$0.663903\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	356.382	1.75558
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 90.5097i	− 0.433061i
$210$	0	0
$211$	−56.0000	−0.265403	−0.132701	−	0.991156i	$-0.542365\pi$
$211$	−56.0000	−0.265403	−0.132701	+	0.991156i	$0.542365\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	48.0000i	0.221198i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 79.1960i	− 0.358353i
$222$	0	0
$223$	28.0000i	0.125561i	0.998027	+	0.0627803i	$0.0199967\pi$
$223$	28.0000i	0.125561i	−0.998027	+	0.0627803i	$0.980003\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	84.8528	0.373801	0.186900	−	0.982379i	$-0.440156\pi$
$227$	84.8528	0.373801	0.186900	+	0.982379i	$0.440156\pi$
$228$	0	0
$229$	−184.000	−0.803493	−0.401747	−	0.915751i	$-0.631597\pi$
$229$	−184.000	−0.803493	−0.401747	+	0.915751i	$0.631597\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−304.056	−1.30496	−0.652481	−	0.757806i	$-0.726272\pi$
$233$	−304.056	−1.30496	−0.652481	+	0.757806i	$0.726272\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 226.274i	− 0.946754i	−0.880860	−	0.473377i	$-0.843035\pi$
$239$	− 226.274i	− 0.946754i	0.880860	−	0.473377i	$-0.156965\pi$
$240$	0	0
$241$	384.000	1.59336	0.796680	−	0.604401i	$-0.206587\pi$
$241$	384.000	1.59336	0.796680	+	0.604401i	$0.206587\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 128.000i	− 0.518219i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 197.990i	− 0.788804i	−0.918938	−	0.394402i	$-0.870952\pi$
$251$	− 197.990i	− 0.788804i	0.918938	−	0.394402i	$-0.129048\pi$
$252$	0	0
$253$	− 224.000i	− 0.885375i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−91.9239	−0.357680	−0.178840	−	0.983878i	$-0.557235\pi$
$257$	−91.9239	−0.357680	−0.178840	+	0.983878i	$0.557235\pi$
$258$	0	0
$259$	−360.000	−1.38996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	395.980	1.50563	0.752813	−	0.658234i	$-0.228696\pi$
$263$	395.980	1.50563	0.752813	+	0.658234i	$0.228696\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	83.4386i	0.310181i	0.987900	+	0.155090i	$0.0495669\pi$
$269$	83.4386i	0.310181i	−0.987900	+	0.155090i	$0.950433\pi$
$270$	0	0
$271$	140.000	0.516605	0.258303	−	0.966064i	$-0.416837\pi$
$271$	140.000	0.516605	0.258303	+	0.966064i	$0.416837\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	248.000i	0.895307i	0.894207	+	0.447653i	$0.147740\pi$
$277$	248.000i	0.895307i	−0.894207	+	0.447653i	$0.852260\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 168.291i	− 0.598902i	−0.954112	−	0.299451i	$-0.903197\pi$
$281$	− 168.291i	− 0.598902i	0.954112	−	0.299451i	$-0.0968035\pi$
$282$	0	0
$283$	368.000i	1.30035i	0.759783	+	0.650177i	$0.225305\pi$
$283$	368.000i	1.30035i	−0.759783	+	0.650177i	$0.774695\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−254.558	−0.886963
$288$	0	0
$289$	−191.000	−0.660900
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−207.889	−0.709520	−0.354760	−	0.934957i	$-0.615437\pi$
$293$	−207.889	−0.709520	−0.354760	+	0.934957i	$0.615437\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 316.784i	− 1.05948i
$300$	0	0
$301$	−96.0000	−0.318937
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	552.000i	1.79805i	0.437902	+	0.899023i	$0.355722\pi$
$307$	552.000i	1.79805i	−0.437902	+	0.899023i	$0.644278\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.9411i	0.109135i	0.998510	+	0.0545677i	$0.0173781\pi$
$311$	33.9411i	0.109135i	−0.998510	+	0.0545677i	$0.982622\pi$
$312$	0	0
$313$	− 222.000i	− 0.709265i	−0.935006	−	0.354633i	$-0.884606\pi$
$313$	− 222.000i	− 0.709265i	0.935006	−	0.354633i	$-0.115394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−445.477	−1.40529	−0.702646	−	0.711540i	$-0.747998\pi$
$317$	−445.477	−1.40529	−0.702646	+	0.711540i	$0.747998\pi$
$318$	0	0
$319$	168.000	0.526646
$320$	0	0
$321$	0	0
$322$	0	0
$323$	158.392	0.490377
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	203.647i	0.618987i
$330$	0	0
$331$	−520.000	−1.57100	−0.785498	−	0.618864i	$-0.787593\pi$
$331$	−520.000	−1.57100	−0.785498	+	0.618864i	$0.787593\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 48.0000i	− 0.142433i	−0.997461	−	0.0712166i	$-0.977312\pi$
$337$	− 48.0000i	− 0.142433i	0.997461	−	0.0712166i	$-0.0226882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.6274i	0.0663561i
$342$	0	0
$343$	552.000i	1.60933i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	356.382	1.02704	0.513518	−	0.858079i	$-0.328342\pi$
$347$	356.382	1.02704	0.513518	+	0.858079i	$0.328342\pi$
$348$	0	0
$349$	−210.000	−0.601719	−0.300860	−	0.953668i	$-0.597274\pi$
$349$	−210.000	−0.601719	−0.300860	+	0.953668i	$0.597274\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−272.943	−0.773210	−0.386605	−	0.922245i	$-0.626352\pi$
$353$	−272.943	−0.773210	−0.386605	+	0.922245i	$0.626352\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 593.970i	− 1.65451i	−0.561825	−	0.827256i	$-0.689901\pi$
$359$	− 593.970i	− 1.65451i	0.561825	−	0.827256i	$-0.310099\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 172.000i	− 0.468665i	−0.972157	−	0.234332i	$-0.924710\pi$
$367$	− 172.000i	− 0.468665i	0.972157	−	0.234332i	$-0.0752904\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	593.970i	1.60100i
$372$	0	0
$373$	2.00000i	0.00536193i	0.999996	+	0.00268097i	$0.000853379\pi$
$373$	2.00000i	0.00536193i	−0.999996	+	0.00268097i	$0.999147\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	237.588	0.630207
$378$	0	0
$379$	736.000	1.94195	0.970976	−	0.239176i	$-0.0768773\pi$
$379$	736.000	1.94195	0.970976	+	0.239176i	$0.0768773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	459.619i	1.18154i	0.806840	+	0.590770i	$0.201176\pi$
$389$	459.619i	1.18154i	−0.806840	+	0.590770i	$0.798824\pi$
$390$	0	0
$391$	392.000	1.00256
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 338.000i	− 0.851385i	−0.904868	−	0.425693i	$-0.860030\pi$
$397$	− 338.000i	− 0.851385i	0.904868	−	0.425693i	$-0.139970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 148.492i	− 0.370305i	−0.982710	−	0.185153i	$-0.940722\pi$
$401$	− 148.492i	− 0.370305i	0.982710	−	0.185153i	$-0.0592780\pi$
$402$	0	0
$403$	32.0000i	0.0794045i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−169.706	−0.416967
$408$	0	0
$409$	−144.000	−0.352078	−0.176039	−	0.984383i	$-0.556329\pi$
$409$	−144.000	−0.352078	−0.176039	+	0.984383i	$0.556329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−950.352	−2.30109
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	763.675i	1.82261i	0.411727	+	0.911307i	$0.364926\pi$
$419$	763.675i	1.82261i	−0.411727	+	0.911307i	$0.635074\pi$
$420$	0	0
$421$	680.000	1.61520	0.807601	−	0.589729i	$-0.200766\pi$
$421$	680.000	1.61520	0.807601	+	0.589729i	$0.200766\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	168.000i	0.393443i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.65685i	0.0131250i	0.999978	+	0.00656248i	$0.00208892\pi$
$431$	5.65685i	0.0131250i	−0.999978	+	0.00656248i	$0.997911\pi$
$432$	0	0
$433$	− 482.000i	− 1.11316i	−0.830793	−	0.556582i	$-0.812113\pi$
$433$	− 482.000i	− 1.11316i	0.830793	−	0.556582i	$-0.187887\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	633.568	1.44981
$438$	0	0
$439$	756.000	1.72210	0.861048	−	0.508524i	$-0.169809\pi$
$439$	756.000	1.72210	0.861048	+	0.508524i	$0.169809\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	197.990	0.446930	0.223465	−	0.974712i	$-0.428263\pi$
$443$	197.990	0.446930	0.223465	+	0.974712i	$0.428263\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	77.7817i	0.173233i	0.996242	+	0.0866166i	$0.0276055\pi$
$449$	77.7817i	0.173233i	−0.996242	+	0.0866166i	$0.972394\pi$
$450$	0	0
$451$	−120.000	−0.266075
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 464.000i	− 1.01532i	−0.861558	−	0.507659i	$-0.830511\pi$
$457$	− 464.000i	− 1.01532i	0.861558	−	0.507659i	$-0.169489\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	49.4975i	0.107370i	0.998558	+	0.0536849i	$0.0170967\pi$
$461$	49.4975i	0.107370i	−0.998558	+	0.0536849i	$0.982903\pi$
$462$	0	0
$463$	20.0000i	0.0431965i	0.999767	+	0.0215983i	$0.00687548\pi$
$463$	20.0000i	0.0431965i	−0.999767	+	0.0215983i	$0.993125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	695.793	1.48992	0.744960	−	0.667109i	$-0.232468\pi$
$467$	695.793	1.48992	0.744960	+	0.667109i	$0.232468\pi$
$468$	0	0
$469$	1056.00	2.25160
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−45.2548	−0.0956762
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	673.166i	1.40536i	0.711508	+	0.702678i	$0.248013\pi$
$479$	673.166i	1.40536i	−0.711508	+	0.702678i	$0.751987\pi$
$480$	0	0
$481$	−240.000	−0.498960
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 388.000i	− 0.796715i	−0.917230	−	0.398357i	$-0.869580\pi$
$487$	− 388.000i	− 0.796715i	0.917230	−	0.398357i	$-0.130420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	294.000i	0.596349i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−339.411	−0.682920
$498$	0	0
$499$	−96.0000	−0.192385	−0.0961924	−	0.995363i	$-0.530666\pi$
$499$	−96.0000	−0.192385	−0.0961924	+	0.995363i	$0.530666\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	424.264	0.843467	0.421734	−	0.906720i	$-0.361422\pi$
$503$	424.264	0.843467	0.421734	+	0.906720i	$0.361422\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 309.713i	− 0.608473i	−0.952597	−	0.304237i	$-0.901599\pi$
$509$	− 309.713i	− 0.608473i	0.952597	−	0.304237i	$-0.0984013\pi$
$510$	0	0
$511$	−960.000	−1.87867
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	96.0000i	0.185687i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	722.663i	1.38707i	0.720423	+	0.693535i	$0.243948\pi$
$521$	722.663i	1.38707i	−0.720423	+	0.693535i	$0.756052\pi$
$522$	0	0
$523$	352.000i	0.673040i	0.941676	+	0.336520i	$0.109250\pi$
$523$	352.000i	0.673040i	−0.941676	+	0.336520i	$0.890750\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−39.5980	−0.0751385
$528$	0	0
$529$	1039.00	1.96408
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−169.706	−0.318397
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	537.401i	0.997034i
$540$	0	0
$541$	936.000	1.73013	0.865065	−	0.501660i	$-0.167277\pi$
$541$	936.000	1.73013	0.865065	+	0.501660i	$0.167277\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 56.0000i	− 0.102377i	−0.998689	−	0.0511883i	$-0.983699\pi$
$547$	− 56.0000i	− 0.102377i	0.998689	−	0.0511883i	$-0.0163009\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	475.176i	0.862388i
$552$	0	0
$553$	1200.00i	2.16998i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	321.026	0.576349	0.288175	−	0.957578i	$-0.406952\pi$
$557$	321.026	0.576349	0.288175	+	0.957578i	$0.406952\pi$
$558$	0	0
$559$	−64.0000	−0.114490
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−469.519	−0.833959	−0.416979	−	0.908916i	$-0.636911\pi$
$563$	−469.519	−0.833959	−0.416979	+	0.908916i	$0.636911\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	408.708i	0.718291i	0.933282	+	0.359146i	$0.116932\pi$
$569$	408.708i	0.718291i	−0.933282	+	0.359146i	$0.883068\pi$
$570$	0	0
$571$	816.000	1.42907	0.714536	−	0.699599i	$-0.246638\pi$
$571$	816.000	1.42907	0.714536	+	0.699599i	$0.246638\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 242.000i	− 0.419411i	−0.977765	−	0.209705i	$-0.932750\pi$
$577$	− 242.000i	− 0.419411i	0.977765	−	0.209705i	$-0.0672505\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1561.29i	− 2.68725i
$582$	0	0
$583$	280.000i	0.480274i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−395.980	−0.674582	−0.337291	−	0.941400i	$-0.609511\pi$
$587$	−395.980	−0.674582	−0.337291	+	0.941400i	$0.609511\pi$
$588$	0	0
$589$	−64.0000	−0.108659
$590$	0	0
$591$	0	0
$592$	0	0
$593$	205.061	0.345803	0.172901	−	0.984939i	$-0.444686\pi$
$593$	205.061	0.345803	0.172901	+	0.984939i	$0.444686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 593.970i	− 0.991602i	−0.868436	−	0.495801i	$-0.834875\pi$
$599$	− 593.970i	− 0.991602i	0.868436	−	0.495801i	$-0.165125\pi$
$600$	0	0
$601$	286.000	0.475874	0.237937	−	0.971281i	$-0.423529\pi$
$601$	286.000	0.475874	0.237937	+	0.971281i	$0.423529\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	100.000i	0.164745i	0.996602	+	0.0823723i	$0.0262497\pi$
$607$	100.000i	0.164745i	−0.996602	+	0.0823723i	$0.973750\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	135.765i	0.222200i
$612$	0	0
$613$	770.000i	1.25612i	0.778166	+	0.628059i	$0.216150\pi$
$613$	770.000i	1.25612i	−0.778166	+	0.628059i	$0.783850\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−869.741	−1.40963	−0.704815	−	0.709391i	$-0.748970\pi$
$617$	−869.741	−1.40963	−0.704815	+	0.709391i	$0.748970\pi$
$618$	0	0
$619$	−256.000	−0.413570	−0.206785	−	0.978386i	$-0.566300\pi$
$619$	−256.000	−0.413570	−0.206785	+	0.978386i	$0.566300\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1781.91	−2.86021
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 296.985i	− 0.472154i
$630$	0	0
$631$	196.000	0.310618	0.155309	−	0.987866i	$-0.450363\pi$
$631$	196.000	0.310618	0.155309	+	0.987866i	$0.450363\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	760.000i	1.19309i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	643.467i	1.00385i	0.864911	+	0.501924i	$0.167374\pi$
$641$	643.467i	1.00385i	−0.864911	+	0.501924i	$0.832626\pi$
$642$	0	0
$643$	120.000i	0.186625i	0.995637	+	0.0933126i	$0.0297456\pi$
$643$	120.000i	0.186625i	−0.995637	+	0.0933126i	$0.970254\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−707.107	−1.09290	−0.546450	−	0.837491i	$-0.684021\pi$
$647$	−707.107	−1.09290	−0.546450	+	0.837491i	$0.684021\pi$
$648$	0	0
$649$	−448.000	−0.690293
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−131.522	−0.201412	−0.100706	−	0.994916i	$-0.532110\pi$
$653$	−131.522	−0.201412	−0.100706	+	0.994916i	$0.532110\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1187.94i	1.80264i	0.433154	+	0.901320i	$0.357400\pi$
$659$	1187.94i	1.80264i	−0.433154	+	0.901320i	$0.642600\pi$
$660$	0	0
$661$	−130.000	−0.196672	−0.0983359	−	0.995153i	$-0.531352\pi$
$661$	−130.000	−0.196672	−0.0983359	+	0.995153i	$0.531352\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1176.00i	1.76312i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	79.1960i	0.118027i
$672$	0	0
$673$	610.000i	0.906389i	0.891412	+	0.453195i	$0.149716\pi$
$673$	610.000i	0.906389i	−0.891412	+	0.453195i	$0.850284\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−603.869	−0.891978	−0.445989	−	0.895038i	$-0.647148\pi$
$677$	−603.869	−0.891978	−0.445989	+	0.895038i	$0.647148\pi$
$678$	0	0
$679$	1344.00	1.97938
$680$	0	0
$681$	0	0
$682$	0	0
$683$	989.949	1.44941	0.724707	−	0.689057i	$-0.241975\pi$
$683$	989.949	1.44941	0.724707	+	0.689057i	$0.241975\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	395.980i	0.574717i
$690$	0	0
$691$	632.000	0.914616	0.457308	−	0.889308i	$-0.348814\pi$
$691$	632.000	0.914616	0.457308	+	0.889308i	$0.348814\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 210.000i	− 0.301291i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 29.6985i	− 0.0423659i	−0.999776	−	0.0211829i	$-0.993257\pi$
$701$	− 29.6985i	− 0.0423659i	0.999776	−	0.0211829i	$-0.00674324\pi$
$702$	0	0
$703$	− 480.000i	− 0.682788i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	763.675	1.08016
$708$	0	0
$709$	−56.0000	−0.0789845	−0.0394922	−	0.999220i	$-0.512574\pi$
$709$	−56.0000	−0.0789845	−0.0394922	+	0.999220i	$0.512574\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−158.392	−0.222149
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	616.597i	0.857576i	0.903405	+	0.428788i	$0.141059\pi$
$719$	616.597i	0.857576i	−0.903405	+	0.428788i	$0.858941\pi$
$720$	0	0
$721$	−528.000	−0.732316
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 372.000i	− 0.511692i	−0.966718	−	0.255846i	$-0.917646\pi$
$727$	− 372.000i	− 0.511692i	0.966718	−	0.255846i	$-0.0823540\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 79.1960i	− 0.108339i
$732$	0	0
$733$	− 520.000i	− 0.709413i	−0.934978	−	0.354707i	$-0.884581\pi$
$733$	− 520.000i	− 0.709413i	0.934978	−	0.354707i	$-0.115419\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	497.803	0.675445
$738$	0	0
$739$	−1040.00	−1.40731	−0.703654	−	0.710543i	$-0.748449\pi$
$739$	−1040.00	−1.40731	−0.703654	+	0.710543i	$0.748449\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−622.254	−0.837489	−0.418744	−	0.908104i	$-0.637530\pi$
$743$	−622.254	−0.837489	−0.418744	+	0.908104i	$0.637530\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−180.000	−0.239680	−0.119840	−	0.992793i	$-0.538238\pi$
$751$	−180.000	−0.239680	−0.119840	+	0.992793i	$0.538238\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 152.000i	− 0.200793i	−0.994948	−	0.100396i	$-0.967989\pi$
$757$	− 152.000i	− 0.200793i	0.994948	−	0.100396i	$-0.0320111\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	544.472i	0.715469i	0.933823	+	0.357735i	$0.116451\pi$
$761$	544.472i	0.715469i	−0.933823	+	0.357735i	$0.883549\pi$
$762$	0	0
$763$	1632.00i	2.13893i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−633.568	−0.826033
$768$	0	0
$769$	526.000	0.684005	0.342003	−	0.939699i	$-0.388895\pi$
$769$	526.000	0.684005	0.342003	+	0.939699i	$0.388895\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1421.28	1.83866	0.919330	−	0.393487i	$-0.128731\pi$
$773$	1421.28	1.83866	0.919330	+	0.393487i	$0.128731\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 339.411i	− 0.435701i
$780$	0	0
$781$	−160.000	−0.204866
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1072.00i	− 1.36213i	−0.732221	−	0.681067i	$-0.761516\pi$
$787$	− 1072.00i	− 1.36213i	0.732221	−	0.681067i	$-0.238484\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 254.558i	− 0.321819i
$792$	0	0
$793$	112.000i	0.141236i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1534.42	1.92525	0.962623	−	0.270843i	$-0.0873026\pi$
$797$	1534.42	1.92525	0.962623	+	0.270843i	$0.0873026\pi$
$798$	0	0
$799$	−168.000	−0.210263
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−452.548	−0.563572
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	134.350i	0.166070i	0.996547	+	0.0830348i	$0.0264613\pi$
$809$	134.350i	0.166070i	−0.996547	+	0.0830348i	$0.973539\pi$
$810$	0	0
$811$	544.000	0.670777	0.335388	−	0.942080i	$-0.391132\pi$
$811$	544.000	0.670777	0.335388	+	0.942080i	$0.391132\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 128.000i	− 0.156671i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	878.227i	1.06970i	0.844946	+	0.534852i	$0.179633\pi$
$821$	878.227i	1.06970i	−0.844946	+	0.534852i	$0.820367\pi$
$822$	0	0
$823$	− 868.000i	− 1.05468i	−0.849655	−	0.527339i	$-0.823190\pi$
$823$	− 868.000i	− 1.05468i	0.849655	−	0.527339i	$-0.176810\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1414.21	1.71005	0.855026	−	0.518585i	$-0.173541\pi$
$827$	1414.21	1.71005	0.855026	+	0.518585i	$0.173541\pi$
$828$	0	0
$829$	1208.00	1.45718	0.728589	−	0.684952i	$-0.240177\pi$
$829$	1208.00	1.45718	0.728589	+	0.684952i	$0.240177\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−940.452	−1.12899
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1069.15i	− 1.27431i	−0.770736	−	0.637155i	$-0.780111\pi$
$839$	− 1069.15i	− 1.27431i	0.770736	−	0.637155i	$-0.219889\pi$
$840$	0	0
$841$	−41.0000	−0.0487515
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1068.00i	− 1.26092i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1187.94i	− 1.39593i
$852$	0	0
$853$	− 1282.00i	− 1.50293i	−0.659772	−	0.751465i	$-0.729347\pi$
$853$	− 1282.00i	− 1.50293i	0.659772	−	0.751465i	$-0.270653\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1039.45	−1.21289	−0.606445	−	0.795125i	$-0.707405\pi$
$857$	−1039.45	−1.21289	−0.606445	+	0.795125i	$0.707405\pi$
$858$	0	0
$859$	1032.00	1.20140	0.600698	−	0.799476i	$-0.294889\pi$
$859$	1032.00	1.20140	0.600698	+	0.799476i	$0.294889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	56.5685	0.0655487	0.0327744	−	0.999463i	$-0.489566\pi$
$863$	56.5685	0.0655487	0.0327744	+	0.999463i	$0.489566\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	565.685i	0.650961i
$870$	0	0
$871$	704.000	0.808266
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	462.000i	0.526796i	0.964687	+	0.263398i	$0.0848432\pi$
$877$	462.000i	0.526796i	−0.964687	+	0.263398i	$0.915157\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	247.487i	0.280916i	0.990087	+	0.140458i	$0.0448576\pi$
$881$	247.487i	0.280916i	−0.990087	+	0.140458i	$0.955142\pi$
$882$	0	0
$883$	8.00000i	0.00906002i	0.999990	+	0.00453001i	$0.00144195\pi$
$883$	8.00000i	0.00906002i	−0.999990	+	0.00453001i	$0.998558\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−893.783	−1.00765	−0.503824	−	0.863807i	$-0.668074\pi$
$887$	−893.783	−1.00765	−0.503824	+	0.863807i	$0.668074\pi$
$888$	0	0
$889$	240.000	0.269966
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−271.529	−0.304064
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 118.794i	− 0.132140i
$900$	0	0
$901$	−490.000	−0.543840
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	200.000i	0.220507i	0.993903	+	0.110254i	$0.0351663\pi$
$907$	200.000i	0.220507i	−0.993903	+	0.110254i	$0.964834\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	79.1960i	0.0869330i	0.999055	+	0.0434665i	$0.0138402\pi$
$911$	79.1960i	0.0869330i	−0.999055	+	0.0434665i	$0.986160\pi$
$912$	0	0
$913$	− 736.000i	− 0.806134i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823	0.740264
$918$	0	0
$919$	884.000	0.961915	0.480958	−	0.876744i	$-0.340289\pi$
$919$	884.000	0.961915	0.480958	+	0.876744i	$0.340289\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−226.274	−0.245151
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1076.22i	1.15847i	0.815161	+	0.579234i	$0.196648\pi$
$929$	1076.22i	1.15847i	−0.815161	+	0.579234i	$0.803352\pi$
$930$	0	0
$931$	−1520.00	−1.63265
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 78.0000i	− 0.0832444i	−0.999133	−	0.0416222i	$-0.986747\pi$
$937$	− 78.0000i	− 0.0832444i	0.999133	−	0.0416222i	$-0.0132526\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 267.286i	− 0.284045i	−0.989863	−	0.142023i	$-0.954639\pi$
$941$	− 267.286i	− 0.284045i	0.989863	−	0.142023i	$-0.0453605\pi$
$942$	0	0
$943$	− 840.000i	− 0.890774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	576.999	0.609292	0.304646	−	0.952466i	$-0.401462\pi$
$947$	576.999	0.609292	0.304646	+	0.952466i	$0.401462\pi$
$948$	0	0
$949$	−640.000	−0.674394
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1336.43	1.40234	0.701171	−	0.712993i	$-0.252661\pi$
$953$	1336.43	1.40234	0.701171	+	0.712993i	$0.252661\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1069.15i	− 1.11485i
$960$	0	0
$961$	−945.000	−0.983351
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 20.0000i	− 0.0206825i	−0.999947	−	0.0103413i	$-0.996708\pi$
$967$	− 20.0000i	− 0.0206825i	0.999947	−	0.0103413i	$-0.00329178\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	118.794i	0.122342i	0.998127	+	0.0611709i	$0.0194835\pi$
$971$	118.794i	0.122342i	−0.998127	+	0.0611709i	$0.980517\pi$
$972$	0	0
$973$	1440.00i	1.47996i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	420.021	0.429909	0.214955	−	0.976624i	$-0.431040\pi$
$977$	420.021	0.429909	0.214955	+	0.976624i	$0.431040\pi$
$978$	0	0
$979$	−840.000	−0.858018
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1187.94	1.20848	0.604242	−	0.796801i	$-0.293476\pi$
$983$	1187.94	1.20848	0.604242	+	0.796801i	$0.293476\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 316.784i	− 0.320307i
$990$	0	0
$991$	−116.000	−0.117053	−0.0585267	−	0.998286i	$-0.518640\pi$
$991$	−116.000	−0.117053	−0.0585267	+	0.998286i	$0.518640\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 130.000i	− 0.130391i	−0.997873	−	0.0651956i	$-0.979233\pi$
$997$	− 130.000i	− 0.130391i	0.997873	−	0.0651956i	$-0.0207671\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.c.a.449.1		4
3.2	odd	2	inner	1800.3.c.a.449.2		4
4.3	odd	2		3600.3.c.c.449.4		4
5.2	odd	4		72.3.e.a.17.2	yes	2
5.3	odd	4		1800.3.l.a.1601.1		2
5.4	even	2	inner	1800.3.c.a.449.3		4
12.11	even	2		3600.3.c.c.449.3		4
15.2	even	4		72.3.e.a.17.1	✓	2
15.8	even	4		1800.3.l.a.1601.2		2
15.14	odd	2	inner	1800.3.c.a.449.4		4
20.3	even	4		3600.3.l.l.1601.2		2
20.7	even	4		144.3.e.a.17.2		2
20.19	odd	2		3600.3.c.c.449.2		4
35.27	even	4		3528.3.d.a.1961.1		2
40.27	even	4		576.3.e.a.449.1		2
40.37	odd	4		576.3.e.h.449.1		2
45.2	even	12		648.3.m.a.377.2		4
45.7	odd	12		648.3.m.a.377.1		4
45.22	odd	12		648.3.m.a.593.2		4
45.32	even	12		648.3.m.a.593.1		4
60.23	odd	4		3600.3.l.l.1601.1		2
60.47	odd	4		144.3.e.a.17.1		2
60.59	even	2		3600.3.c.c.449.1		4
80.27	even	4		2304.3.h.h.2177.2		4
80.37	odd	4		2304.3.h.a.2177.2		4
80.67	even	4		2304.3.h.h.2177.3		4
80.77	odd	4		2304.3.h.a.2177.3		4
105.62	odd	4		3528.3.d.a.1961.2		2
120.77	even	4		576.3.e.h.449.2		2
120.107	odd	4		576.3.e.a.449.2		2
180.7	even	12		1296.3.q.k.1025.1		4
180.47	odd	12		1296.3.q.k.1025.2		4
180.67	even	12		1296.3.q.k.593.2		4
180.167	odd	12		1296.3.q.k.593.1		4
240.77	even	4		2304.3.h.a.2177.1		4
240.107	odd	4		2304.3.h.h.2177.4		4
240.197	even	4		2304.3.h.a.2177.4		4
240.227	odd	4		2304.3.h.h.2177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.e.a.17.1	✓	2	15.2	even	4
72.3.e.a.17.2	yes	2	5.2	odd	4
144.3.e.a.17.1		2	60.47	odd	4
144.3.e.a.17.2		2	20.7	even	4
576.3.e.a.449.1		2	40.27	even	4
576.3.e.a.449.2		2	120.107	odd	4
576.3.e.h.449.1		2	40.37	odd	4
576.3.e.h.449.2		2	120.77	even	4
648.3.m.a.377.1		4	45.7	odd	12
648.3.m.a.377.2		4	45.2	even	12
648.3.m.a.593.1		4	45.32	even	12
648.3.m.a.593.2		4	45.22	odd	12
1296.3.q.k.593.1		4	180.167	odd	12
1296.3.q.k.593.2		4	180.67	even	12
1296.3.q.k.1025.1		4	180.7	even	12
1296.3.q.k.1025.2		4	180.47	odd	12
1800.3.c.a.449.1		4	1.1	even	1	trivial
1800.3.c.a.449.2		4	3.2	odd	2	inner
1800.3.c.a.449.3		4	5.4	even	2	inner
1800.3.c.a.449.4		4	15.14	odd	2	inner
1800.3.l.a.1601.1		2	5.3	odd	4
1800.3.l.a.1601.2		2	15.8	even	4
2304.3.h.a.2177.1		4	240.77	even	4
2304.3.h.a.2177.2		4	80.37	odd	4
2304.3.h.a.2177.3		4	80.77	odd	4
2304.3.h.a.2177.4		4	240.197	even	4
2304.3.h.h.2177.1		4	240.227	odd	4
2304.3.h.h.2177.2		4	80.27	even	4
2304.3.h.h.2177.3		4	80.67	even	4
2304.3.h.h.2177.4		4	240.107	odd	4
3528.3.d.a.1961.1		2	35.27	even	4
3528.3.d.a.1961.2		2	105.62	odd	4
3600.3.c.c.449.1		4	60.59	even	2
3600.3.c.c.449.2		4	20.19	odd	2
3600.3.c.c.449.3		4	12.11	even	2
3600.3.c.c.449.4		4	4.3	odd	2
3600.3.l.l.1601.1		2	60.23	odd	4
3600.3.l.l.1601.2		2	20.3	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.0000i q^{7} -5.65685i q^{11} -8.00000i q^{13} +9.89949 q^{17} +16.0000 q^{19} +39.5980 q^{23} +29.6985i q^{29} -4.00000 q^{31} -30.0000i q^{37} -21.2132i q^{41} -8.00000i q^{43} -16.9706 q^{47} -95.0000 q^{49} -49.4975 q^{53} -79.1960i q^{59} -14.0000 q^{61} +88.0000i q^{67} -28.2843i q^{71} -80.0000i q^{73} -67.8823 q^{77} -100.000 q^{79} +130.108 q^{83} -148.492i q^{89} -96.0000 q^{91} +112.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 64 q^{19} - 16 q^{31} - 380 q^{49} - 56 q^{61} - 400 q^{79} - 384 q^{91}+O(q^{100}) \) 4 * q + 64 * q^19 - 16 * q^31 - 380 * q^49 - 56 * q^61 - 400 * q^79 - 384 * q^91

Introduction

L-functions

Modular forms

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Representations

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Database

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