LMFDB - Embedded newform 1728.4.d.e.865.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2261390379264.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122 $ x^8 - 4x^7 - 8x^6 + 38x^5 - 38x^4 + 8x^3 + 325x^2 - 322*x + 2122
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.8
Root			$3.75792 + 0.460416i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.e.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.7850i q^{5} +22.3100 q^{7} +O(q^{10})$	$q+16.7850i q^{5} +22.3100 q^{7} -51.9124i q^{11} +28.0451i q^{13} -60.9124 q^{17} -162.737i q^{19} -83.2950 q^{23} -156.737 q^{25} +217.575i q^{29} +174.005 q^{31} +374.474i q^{35} +407.946i q^{37} +196.562 q^{41} +345.474i q^{43} -335.701 q^{47} +154.737 q^{49} +234.360i q^{53} +871.351 q^{55} -321.124i q^{59} +739.381i q^{61} -470.737 q^{65} -20.0000i q^{67} -418.996 q^{71} +397.212 q^{73} -1158.17i q^{77} +95.6053 q^{79} +348.263i q^{83} -1022.42i q^{85} -1492.42 q^{89} +625.686i q^{91} +2731.55 q^{95} +881.949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100}) $ 8 * q - 168 * q^17 - 296 * q^25 - 24 * q^41 + 280 * q^49 - 2808 * q^65 + 304 * q^73 - 6192 * q^89 + 3224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	16.7850i	1.50130i	0.660701	+	0.750649i	$0.270259\pi$
$5$	16.7850i	1.50130i	−0.660701	+	0.750649i	$0.729741\pi$
$6$	0	0
$7$	22.3100	1.20463	0.602314	−	0.798259i	$-0.294245\pi$
$7$	22.3100	1.20463	0.602314	+	0.798259i	$0.294245\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 51.9124i	− 1.42293i	−0.702724	−	0.711463i	$-0.748033\pi$
$11$	− 51.9124i	− 1.42293i	0.702724	−	0.711463i	$-0.251967\pi$
$12$	0	0
$13$	28.0451i	0.598331i	0.954201	+	0.299165i	$0.0967082\pi$
$13$	28.0451i	0.598331i	−0.954201	+	0.299165i	$0.903292\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−60.9124	−0.869025	−0.434513	−	0.900666i	$-0.643079\pi$
$17$	−60.9124	−0.869025	−0.434513	+	0.900666i	$0.643079\pi$
$18$	0	0
$19$	− 162.737i	− 1.96497i	−0.186335	−	0.982486i	$-0.559661\pi$
$19$	− 162.737i	− 1.96497i	0.186335	−	0.982486i	$-0.440339\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−83.2950	−0.755140	−0.377570	−	0.925981i	$-0.623240\pi$
$23$	−83.2950	−0.755140	−0.377570	+	0.925981i	$0.623240\pi$
$24$	0	0
$25$	−156.737	−1.25390
$26$	0	0
$27$	0	0
$28$	0	0
$29$	217.575i	1.39320i	0.717461	+	0.696598i	$0.245304\pi$
$29$	217.575i	1.39320i	−0.717461	+	0.696598i	$0.754696\pi$
$30$	0	0
$31$	174.005	1.00814	0.504069	−	0.863663i	$-0.331836\pi$
$31$	174.005	1.00814	0.504069	+	0.863663i	$0.331836\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	374.474i	1.80851i
$36$	0	0
$37$	407.946i	1.81259i	0.422645	+	0.906295i	$0.361102\pi$
$37$	407.946i	1.81259i	−0.422645	+	0.906295i	$0.638898\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	196.562	0.748728	0.374364	−	0.927282i	$-0.377861\pi$
$41$	196.562	0.748728	0.374364	+	0.927282i	$0.377861\pi$
$42$	0	0
$43$	345.474i	1.22522i	0.790386	+	0.612609i	$0.209880\pi$
$43$	345.474i	1.22522i	−0.790386	+	0.612609i	$0.790120\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−335.701	−1.04185	−0.520925	−	0.853602i	$-0.674413\pi$
$47$	−335.701	−1.04185	−0.520925	+	0.853602i	$0.674413\pi$
$48$	0	0
$49$	154.737	0.451129
$50$	0	0
$51$	0	0
$52$	0	0
$53$	234.360i	0.607394i	0.952769	+	0.303697i	$0.0982210\pi$
$53$	234.360i	0.607394i	−0.952769	+	0.303697i	$0.901779\pi$
$54$	0	0
$55$	871.351	2.13624
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 321.124i	− 0.708590i	−0.935134	−	0.354295i	$-0.884721\pi$
$59$	− 321.124i	− 0.708590i	0.935134	−	0.354295i	$-0.115279\pi$
$60$	0	0
$61$	739.381i	1.55194i	0.630772	+	0.775968i	$0.282738\pi$
$61$	739.381i	1.55194i	−0.630772	+	0.775968i	$0.717262\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−470.737	−0.898273
$66$	0	0
$67$	− 20.0000i	− 0.0364685i	−0.999834	−	0.0182342i	$-0.994196\pi$
$67$	− 20.0000i	− 0.0364685i	0.999834	−	0.0182342i	$-0.00580446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−418.996	−0.700361	−0.350180	−	0.936682i	$-0.613880\pi$
$71$	−418.996	−0.700361	−0.350180	+	0.936682i	$0.613880\pi$
$72$	0	0
$73$	397.212	0.636851	0.318425	−	0.947948i	$-0.396846\pi$
$73$	397.212	0.636851	0.318425	+	0.947948i	$0.396846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1158.17i	− 1.71410i
$78$	0	0
$79$	95.6053	0.136157	0.0680787	−	0.997680i	$-0.478313\pi$
$79$	95.6053	0.136157	0.0680787	+	0.997680i	$0.478313\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	348.263i	0.460564i	0.973124	+	0.230282i	$0.0739649\pi$
$83$	348.263i	0.460564i	−0.973124	+	0.230282i	$0.926035\pi$
$84$	0	0
$85$	− 1022.42i	− 1.30467i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1492.42	−1.77749	−0.888745	−	0.458403i	$-0.848422\pi$
$89$	−1492.42	−1.77749	−0.888745	+	0.458403i	$0.848422\pi$
$90$	0	0
$91$	625.686i	0.720766i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2731.55	2.95001
$96$	0	0
$97$	881.949	0.923179	0.461589	−	0.887094i	$-0.347279\pi$
$97$	881.949	0.923179	0.461589	+	0.887094i	$0.347279\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	642.242i	0.632727i	0.948638	+	0.316364i	$0.102462\pi$
$101$	642.242i	0.632727i	−0.948638	+	0.316364i	$0.897538\pi$
$102$	0	0
$103$	−1322.03	−1.26469	−0.632345	−	0.774687i	$-0.717908\pi$
$103$	−1322.03	−1.26469	−0.632345	+	0.774687i	$0.717908\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1671.77i	1.51043i	0.655475	+	0.755217i	$0.272469\pi$
$107$	1671.77i	1.51043i	−0.655475	+	0.755217i	$0.727531\pi$
$108$	0	0
$109$	1667.52i	1.46531i	0.680598	+	0.732657i	$0.261720\pi$
$109$	1667.52i	1.46531i	−0.680598	+	0.732657i	$0.738280\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−418.314	−0.348245	−0.174122	−	0.984724i	$-0.555709\pi$
$113$	−418.314	−0.348245	−0.174122	+	0.984724i	$0.555709\pi$
$114$	0	0
$115$	− 1398.11i	− 1.13369i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1358.96	−1.04685
$120$	0	0
$121$	−1363.90	−1.02472
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 532.710i	− 0.381176i
$126$	0	0
$127$	106.509	0.0744185	0.0372093	−	0.999307i	$-0.488153\pi$
$127$	106.509	0.0744185	0.0372093	+	0.999307i	$0.488153\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1814.47i	1.21016i	0.796163	+	0.605082i	$0.206860\pi$
$131$	1814.47i	1.21016i	−0.796163	+	0.605082i	$0.793140\pi$
$132$	0	0
$133$	− 3630.67i	− 2.36706i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	401.197	0.250194	0.125097	−	0.992145i	$-0.460076\pi$
$137$	401.197	0.250194	0.125097	+	0.992145i	$0.460076\pi$
$138$	0	0
$139$	− 381.474i	− 0.232779i	−0.993204	−	0.116389i	$-0.962868\pi$
$139$	− 381.474i	− 0.232779i	0.993204	−	0.116389i	$-0.0371320\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1455.89	0.851380
$144$	0	0
$145$	−3652.01	−2.09160
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3021.94i	− 1.66152i	−0.556630	−	0.830760i	$-0.687906\pi$
$149$	− 3021.94i	− 1.66152i	0.556630	−	0.830760i	$-0.312094\pi$
$150$	0	0
$151$	−2165.27	−1.16693	−0.583467	−	0.812137i	$-0.698304\pi$
$151$	−2165.27	−1.16693	−0.583467	+	0.812137i	$0.698304\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2920.69i	1.51352i
$156$	0	0
$157$	− 1478.89i	− 0.751773i	−0.926666	−	0.375886i	$-0.877338\pi$
$157$	− 1478.89i	− 0.751773i	0.926666	−	0.375886i	$-0.122662\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1858.31	−0.909662
$162$	0	0
$163$	− 1140.21i	− 0.547903i	−0.961743	−	0.273952i	$-0.911669\pi$
$163$	− 1140.21i	− 0.547903i	0.961743	−	0.273952i	$-0.0883309\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	646.653	0.299638	0.149819	−	0.988713i	$-0.452131\pi$
$167$	646.653	0.299638	0.149819	+	0.988713i	$0.452131\pi$
$168$	0	0
$169$	1410.47	0.642000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4036.37i	1.77387i	0.461894	+	0.886935i	$0.347170\pi$
$173$	4036.37i	1.77387i	−0.461894	+	0.886935i	$0.652830\pi$
$174$	0	0
$175$	−3496.81	−1.51048
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1592.17i	− 0.664828i	−0.943134	−	0.332414i	$-0.892137\pi$
$179$	− 1592.17i	− 0.664828i	0.943134	−	0.332414i	$-0.107863\pi$
$180$	0	0
$181$	− 3849.93i	− 1.58101i	−0.612456	−	0.790505i	$-0.709818\pi$
$181$	− 3849.93i	− 1.58101i	0.612456	−	0.790505i	$-0.290182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6847.38	−2.72124
$186$	0	0
$187$	3162.11i	1.23656i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4049.60	−1.53413	−0.767065	−	0.641569i	$-0.778284\pi$
$191$	−4049.60	−1.53413	−0.767065	+	0.641569i	$0.778284\pi$
$192$	0	0
$193$	3421.96	1.27626	0.638129	−	0.769930i	$-0.279709\pi$
$193$	3421.96	1.27626	0.638129	+	0.769930i	$0.279709\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1955.03i	0.707055i	0.935424	+	0.353528i	$0.115018\pi$
$197$	1955.03i	0.707055i	−0.935424	+	0.353528i	$0.884982\pi$
$198$	0	0
$199$	−109.030	−0.0388387	−0.0194194	−	0.999811i	$-0.506182\pi$
$199$	−109.030	−0.0388387	−0.0194194	+	0.999811i	$0.506182\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4854.11i	1.67828i
$204$	0	0
$205$	3299.30i	1.12406i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8448.08	−2.79601
$210$	0	0
$211$	1491.58i	0.486658i	0.969944	+	0.243329i	$0.0782395\pi$
$211$	1491.58i	0.486658i	−0.969944	+	0.243329i	$0.921761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5798.80	−1.83942
$216$	0	0
$217$	3882.07	1.21443
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1708.29i	− 0.519965i
$222$	0	0
$223$	2183.67	0.655737	0.327869	−	0.944723i	$-0.393670\pi$
$223$	2183.67	0.655737	0.327869	+	0.944723i	$0.393670\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4868.07i	1.42337i	0.702498	+	0.711686i	$0.252068\pi$
$227$	4868.07i	1.42337i	−0.702498	+	0.711686i	$0.747932\pi$
$228$	0	0
$229$	1619.18i	0.467242i	0.972328	+	0.233621i	$0.0750575\pi$
$229$	1619.18i	0.467242i	−0.972328	+	0.233621i	$0.924943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4354.61	1.22438	0.612189	−	0.790711i	$-0.290289\pi$
$233$	4354.61	1.22438	0.612189	+	0.790711i	$0.290289\pi$
$234$	0	0
$235$	− 5634.74i	− 1.56413i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2102.54	0.569046	0.284523	−	0.958669i	$-0.408165\pi$
$239$	2102.54	0.569046	0.284523	+	0.958669i	$0.408165\pi$
$240$	0	0
$241$	−664.744	−0.177676	−0.0888381	−	0.996046i	$-0.528315\pi$
$241$	−664.744	−0.177676	−0.0888381	+	0.996046i	$0.528315\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2597.27i	0.677279i
$246$	0	0
$247$	4563.98	1.17570
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4513.82i	1.13510i	0.823339	+	0.567550i	$0.192109\pi$
$251$	4513.82i	1.13510i	−0.823339	+	0.567550i	$0.807891\pi$
$252$	0	0
$253$	4324.04i	1.07451i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−942.446	−0.228748	−0.114374	−	0.993438i	$-0.536486\pi$
$257$	−942.446	−0.228748	−0.114374	+	0.993438i	$0.536486\pi$
$258$	0	0
$259$	9101.28i	2.18350i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1570.46	0.368208	0.184104	−	0.982907i	$-0.441062\pi$
$263$	1570.46	0.368208	0.184104	+	0.982907i	$0.441062\pi$
$264$	0	0
$265$	−3933.74	−0.911879
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3268.67i	0.740871i	0.928858	+	0.370436i	$0.120792\pi$
$269$	3268.67i	0.740871i	−0.928858	+	0.370436i	$0.879208\pi$
$270$	0	0
$271$	−1081.79	−0.242486	−0.121243	−	0.992623i	$-0.538688\pi$
$271$	−1081.79	−0.242486	−0.121243	+	0.992623i	$0.538688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8136.61i	1.78420i
$276$	0	0
$277$	− 1402.51i	− 0.304219i	−0.988364	−	0.152109i	$-0.951393\pi$
$277$	− 1402.51i	− 0.304219i	0.988364	−	0.152109i	$-0.0486066\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−432.416	−0.0918000	−0.0459000	−	0.998946i	$-0.514616\pi$
$281$	−432.416	−0.0918000	−0.0459000	+	0.998946i	$0.514616\pi$
$282$	0	0
$283$	− 3529.48i	− 0.741364i	−0.928760	−	0.370682i	$-0.879124\pi$
$283$	− 3529.48i	− 0.741364i	0.928760	−	0.370682i	$-0.120876\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4385.30	0.901938
$288$	0	0
$289$	−1202.68	−0.244795
$290$	0	0
$291$	0	0
$292$	0	0
$293$	985.906i	0.196578i	0.995158	+	0.0982888i	$0.0313369\pi$
$293$	985.906i	0.196578i	−0.995158	+	0.0982888i	$0.968663\pi$
$294$	0	0
$295$	5390.08	1.06380
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2336.01i	− 0.451823i
$300$	0	0
$301$	7707.54i	1.47593i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12410.5	−2.32992
$306$	0	0
$307$	1061.90i	0.197414i	0.995117	+	0.0987070i	$0.0314707\pi$
$307$	1061.90i	0.197414i	−0.995117	+	0.0987070i	$0.968529\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2470.09	0.450373	0.225187	−	0.974316i	$-0.427701\pi$
$311$	2470.09	0.450373	0.225187	+	0.974316i	$0.427701\pi$
$312$	0	0
$313$	3270.89	0.590676	0.295338	−	0.955393i	$-0.404568\pi$
$313$	3270.89	0.590676	0.295338	+	0.955393i	$0.404568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7925.23i	− 1.40418i	−0.712088	−	0.702090i	$-0.752250\pi$
$317$	− 7925.23i	− 1.40418i	0.712088	−	0.702090i	$-0.247750\pi$
$318$	0	0
$319$	11294.9	1.98242
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9912.71i	1.70761i
$324$	0	0
$325$	− 4395.71i	− 0.750246i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7489.49	−1.25504
$330$	0	0
$331$	− 5577.08i	− 0.926115i	−0.886328	−	0.463057i	$-0.846752\pi$
$331$	− 5577.08i	− 0.926115i	0.886328	−	0.463057i	$-0.153248\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	335.701	0.0547501
$336$	0	0
$337$	−5957.59	−0.962999	−0.481499	−	0.876446i	$-0.659908\pi$
$337$	−5957.59	−0.962999	−0.481499	+	0.876446i	$0.659908\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9033.04i	− 1.43451i
$342$	0	0
$343$	−4200.15	−0.661186
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3873.30i	− 0.599221i	−0.954062	−	0.299610i	$-0.903143\pi$
$347$	− 3873.30i	− 0.599221i	0.954062	−	0.299610i	$-0.0968567\pi$
$348$	0	0
$349$	11789.8i	1.80830i	0.427218	+	0.904148i	$0.359494\pi$
$349$	11789.8i	1.80830i	−0.427218	+	0.904148i	$0.640506\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6449.61	−0.972460	−0.486230	−	0.873831i	$-0.661628\pi$
$353$	−6449.61	−0.972460	−0.486230	+	0.873831i	$0.661628\pi$
$354$	0	0
$355$	− 7032.85i	− 1.05145i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4333.86	0.637138	0.318569	−	0.947900i	$-0.396798\pi$
$359$	4333.86	0.637138	0.318569	+	0.947900i	$0.396798\pi$
$360$	0	0
$361$	−19624.4	−2.86112
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6667.21i	0.956103i
$366$	0	0
$367$	−1773.77	−0.252289	−0.126144	−	0.992012i	$-0.540260\pi$
$367$	−1773.77	−0.252289	−0.126144	+	0.992012i	$0.540260\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5228.58i	0.731683i
$372$	0	0
$373$	8187.21i	1.13651i	0.822853	+	0.568254i	$0.192381\pi$
$373$	8187.21i	1.13651i	−0.822853	+	0.568254i	$0.807619\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6101.91	−0.833593
$378$	0	0
$379$	− 4548.20i	− 0.616426i	−0.951317	−	0.308213i	$-0.900269\pi$
$379$	− 4548.20i	− 0.616426i	0.951317	−	0.308213i	$-0.0997310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14304.4	−1.90841	−0.954204	−	0.299155i	$-0.903295\pi$
$383$	−14304.4	−1.90841	−0.954204	+	0.299155i	$0.903295\pi$
$384$	0	0
$385$	19439.9	2.57337
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9949.97i	− 1.29687i	−0.761269	−	0.648436i	$-0.775423\pi$
$389$	− 9949.97i	− 1.29687i	0.761269	−	0.648436i	$-0.224577\pi$
$390$	0	0
$391$	5073.70	0.656235
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1604.74i	0.204413i
$396$	0	0
$397$	− 8804.08i	− 1.11301i	−0.830845	−	0.556504i	$-0.812143\pi$
$397$	− 8804.08i	− 1.11301i	0.830845	−	0.556504i	$-0.187857\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5447.33	0.678370	0.339185	−	0.940720i	$-0.389849\pi$
$401$	5447.33	0.678370	0.339185	+	0.940720i	$0.389849\pi$
$402$	0	0
$403$	4879.99i	0.603200i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21177.4	2.57918
$408$	0	0
$409$	11192.0	1.35307	0.676537	−	0.736408i	$-0.263480\pi$
$409$	11192.0	1.35307	0.676537	+	0.736408i	$0.263480\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7164.29i	− 0.853587i
$414$	0	0
$415$	−5845.60	−0.691444
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 8012.23i	− 0.934184i	−0.884209	−	0.467092i	$-0.845302\pi$
$419$	− 8012.23i	− 0.934184i	0.884209	−	0.467092i	$-0.154698\pi$
$420$	0	0
$421$	− 5895.02i	− 0.682436i	−0.939984	−	0.341218i	$-0.889161\pi$
$421$	− 5895.02i	− 0.682436i	0.939984	−	0.341218i	$-0.110839\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9547.24	1.08967
$426$	0	0
$427$	16495.6i	1.86951i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3869.38	0.432439	0.216220	−	0.976345i	$-0.430627\pi$
$431$	3869.38	0.432439	0.216220	+	0.976345i	$0.430627\pi$
$432$	0	0
$433$	−2257.93	−0.250599	−0.125300	−	0.992119i	$-0.539989\pi$
$433$	−2257.93	−0.250599	−0.125300	+	0.992119i	$0.539989\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13555.2i	1.48383i
$438$	0	0
$439$	−13979.2	−1.51979	−0.759897	−	0.650043i	$-0.774751\pi$
$439$	−13979.2	−1.51979	−0.759897	+	0.650043i	$0.774751\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16551.5i	1.77513i	0.460679	+	0.887567i	$0.347606\pi$
$443$	16551.5i	1.77513i	−0.460679	+	0.887567i	$0.652394\pi$
$444$	0	0
$445$	− 25050.4i	− 2.66854i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4494.66	−0.472419	−0.236210	−	0.971702i	$-0.575905\pi$
$449$	−4494.66	−0.472419	−0.236210	+	0.971702i	$0.575905\pi$
$450$	0	0
$451$	− 10204.0i	− 1.06538i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10502.2	−1.08209
$456$	0	0
$457$	16034.1	1.64123	0.820615	−	0.571482i	$-0.193631\pi$
$457$	16034.1	1.64123	0.820615	+	0.571482i	$0.193631\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1432.00i	− 0.144674i	−0.997380	−	0.0723370i	$-0.976954\pi$
$461$	− 1432.00i	− 0.144674i	0.997380	−	0.0723370i	$-0.0230457\pi$
$462$	0	0
$463$	−12090.6	−1.21360	−0.606800	−	0.794855i	$-0.707547\pi$
$463$	−12090.6	−1.21360	−0.606800	+	0.794855i	$0.707547\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2767.50i	0.274229i	0.990555	+	0.137114i	$0.0437828\pi$
$467$	2767.50i	0.274229i	−0.990555	+	0.137114i	$0.956217\pi$
$468$	0	0
$469$	− 446.200i	− 0.0439310i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17934.4	1.74339
$474$	0	0
$475$	25507.0i	2.46387i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14484.6	−1.38167	−0.690834	−	0.723013i	$-0.742757\pi$
$479$	−14484.6	−1.38167	−0.690834	+	0.723013i	$0.742757\pi$
$480$	0	0
$481$	−11440.9	−1.08453
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14803.5i	1.38597i
$486$	0	0
$487$	3300.69	0.307122	0.153561	−	0.988139i	$-0.450926\pi$
$487$	3300.69	0.307122	0.153561	+	0.988139i	$0.450926\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8400.25i	0.772094i	0.922479	+	0.386047i	$0.126160\pi$
$491$	8400.25i	0.772094i	−0.922479	+	0.386047i	$0.873840\pi$
$492$	0	0
$493$	− 13253.0i	− 1.21072i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9347.80	−0.843675
$498$	0	0
$499$	− 5884.85i	− 0.527940i	−0.964531	−	0.263970i	$-0.914968\pi$
$499$	− 5884.85i	− 0.527940i	0.964531	−	0.263970i	$-0.0850319\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15361.0	−1.36166	−0.680828	−	0.732443i	$-0.738380\pi$
$503$	−15361.0	−1.36166	−0.680828	+	0.732443i	$0.738380\pi$
$504$	0	0
$505$	−10780.1	−0.949913
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4734.24i	0.412262i	0.978524	+	0.206131i	$0.0660873\pi$
$509$	4734.24i	0.412262i	−0.978524	+	0.206131i	$0.933913\pi$
$510$	0	0
$511$	8861.80	0.767168
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 22190.3i	− 1.89868i
$516$	0	0
$517$	17427.0i	1.48248i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1802.84	−0.151600	−0.0758002	−	0.997123i	$-0.524151\pi$
$521$	−1802.84	−0.151600	−0.0758002	+	0.997123i	$0.524151\pi$
$522$	0	0
$523$	− 14334.3i	− 1.19846i	−0.800576	−	0.599231i	$-0.795473\pi$
$523$	− 14334.3i	− 1.19846i	0.800576	−	0.599231i	$-0.204527\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10599.1	−0.876098
$528$	0	0
$529$	−5228.94	−0.429764
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5512.59i	0.447987i
$534$	0	0
$535$	−28060.8	−2.26761
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 8032.78i	− 0.641923i
$540$	0	0
$541$	− 80.7842i	− 0.00641993i	−0.999995	−	0.00320997i	$-0.998978\pi$
$541$	− 80.7842i	− 0.00641993i	0.999995	−	0.00320997i	$-0.00102177\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−27989.3	−2.19987
$546$	0	0
$547$	16156.2i	1.26287i	0.775429	+	0.631435i	$0.217534\pi$
$547$	16156.2i	1.26287i	−0.775429	+	0.631435i	$0.782466\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35407.6	2.73759
$552$	0	0
$553$	2132.96	0.164019
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11647.7i	0.886050i	0.896509	+	0.443025i	$0.146095\pi$
$557$	11647.7i	0.886050i	−0.896509	+	0.443025i	$0.853905\pi$
$558$	0	0
$559$	−9688.85	−0.733085
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 15422.5i	− 1.15450i	−0.816568	−	0.577249i	$-0.804126\pi$
$563$	− 15422.5i	− 1.15450i	0.816568	−	0.577249i	$-0.195874\pi$
$564$	0	0
$565$	− 7021.41i	− 0.522819i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22141.8	1.63134	0.815669	−	0.578519i	$-0.196369\pi$
$569$	22141.8	1.63134	0.815669	+	0.578519i	$0.196369\pi$
$570$	0	0
$571$	− 22320.5i	− 1.63587i	−0.575309	−	0.817936i	$-0.695118\pi$
$571$	− 22320.5i	− 1.63587i	0.575309	−	0.817936i	$-0.304882\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13055.4	0.946868
$576$	0	0
$577$	1412.77	0.101932	0.0509658	−	0.998700i	$-0.483770\pi$
$577$	1412.77	0.101932	0.0509658	+	0.998700i	$0.483770\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7769.75i	0.554808i
$582$	0	0
$583$	12166.2	0.864276
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20519.0i	− 1.44278i	−0.692531	−	0.721388i	$-0.743505\pi$
$587$	− 20519.0i	− 1.44278i	0.692531	−	0.721388i	$-0.256495\pi$
$588$	0	0
$589$	− 28317.2i	− 1.98096i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15003.2	1.03897	0.519483	−	0.854480i	$-0.326124\pi$
$593$	15003.2	1.03897	0.519483	+	0.854480i	$0.326124\pi$
$594$	0	0
$595$	− 22810.1i	− 1.57164i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15673.9	−1.06915	−0.534573	−	0.845123i	$-0.679527\pi$
$599$	−15673.9	−1.06915	−0.534573	+	0.845123i	$0.679527\pi$
$600$	0	0
$601$	8395.61	0.569824	0.284912	−	0.958554i	$-0.408036\pi$
$601$	8395.61	0.569824	0.284912	+	0.958554i	$0.408036\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 22893.1i	− 1.53841i
$606$	0	0
$607$	5507.73	0.368290	0.184145	−	0.982899i	$-0.441048\pi$
$607$	5507.73	0.368290	0.184145	+	0.982899i	$0.441048\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9414.74i	− 0.623371i
$612$	0	0
$613$	22607.4i	1.48956i	0.667308	+	0.744782i	$0.267446\pi$
$613$	22607.4i	1.48956i	−0.667308	+	0.744782i	$0.732554\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28342.0	1.84928	0.924639	−	0.380844i	$-0.124366\pi$
$617$	28342.0	1.84928	0.924639	+	0.380844i	$0.124366\pi$
$618$	0	0
$619$	22779.4i	1.47913i	0.673086	+	0.739564i	$0.264968\pi$
$619$	22779.4i	1.47913i	−0.673086	+	0.739564i	$0.735032\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−33296.0	−2.14121
$624$	0	0
$625$	−10650.6	−0.681638
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 24848.9i	− 1.57519i
$630$	0	0
$631$	19354.6	1.22107	0.610536	−	0.791989i	$-0.290954\pi$
$631$	19354.6	1.22107	0.610536	+	0.791989i	$0.290954\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1787.76i	0.111724i
$636$	0	0
$637$	4339.61i	0.269924i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1458.61	0.0898775	0.0449387	−	0.998990i	$-0.485691\pi$
$641$	1458.61	0.0898775	0.0449387	+	0.998990i	$0.485691\pi$
$642$	0	0
$643$	− 14874.5i	− 0.912278i	−0.889909	−	0.456139i	$-0.849232\pi$
$643$	− 14874.5i	− 0.912278i	0.889909	−	0.456139i	$-0.150768\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13724.8	−0.833966	−0.416983	−	0.908914i	$-0.636913\pi$
$647$	−13724.8	−0.833966	−0.416983	+	0.908914i	$0.636913\pi$
$648$	0	0
$649$	−16670.3	−1.00827
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14706.2i	0.881311i	0.897676	+	0.440656i	$0.145254\pi$
$653$	14706.2i	0.881311i	−0.897676	+	0.440656i	$0.854746\pi$
$654$	0	0
$655$	−30456.0	−1.81682
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14302.2i	0.845426i	0.906264	+	0.422713i	$0.138922\pi$
$659$	14302.2i	0.845426i	−0.906264	+	0.422713i	$0.861078\pi$
$660$	0	0
$661$	18402.5i	1.08287i	0.840743	+	0.541434i	$0.182118\pi$
$661$	18402.5i	1.08287i	−0.840743	+	0.541434i	$0.817882\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	60940.9	3.55367
$666$	0	0
$667$	− 18122.9i	− 1.05206i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38383.1	2.20829
$672$	0	0
$673$	20057.4	1.14882	0.574411	−	0.818567i	$-0.305231\pi$
$673$	20057.4	1.14882	0.574411	+	0.818567i	$0.305231\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3332.38i	− 0.189178i	−0.995516	−	0.0945891i	$-0.969846\pi$
$677$	− 3332.38i	− 0.189178i	0.995516	−	0.0945891i	$-0.0301537\pi$
$678$	0	0
$679$	19676.3	1.11209
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 25093.2i	− 1.40581i	−0.711286	−	0.702903i	$-0.751887\pi$
$683$	− 25093.2i	− 1.40581i	0.711286	−	0.702903i	$-0.248113\pi$
$684$	0	0
$685$	6734.10i	0.375616i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6572.65	−0.363422
$690$	0	0
$691$	5265.23i	0.289868i	0.989441	+	0.144934i	$0.0462970\pi$
$691$	5265.23i	0.289868i	−0.989441	+	0.144934i	$0.953703\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6403.06	0.349470
$696$	0	0
$697$	−11973.1	−0.650663
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 20984.1i	− 1.13061i	−0.824882	−	0.565306i	$-0.808758\pi$
$701$	− 20984.1i	− 1.13061i	0.824882	−	0.565306i	$-0.191242\pi$
$702$	0	0
$703$	66387.9	3.56169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14328.4i	0.762201i
$708$	0	0
$709$	12441.7i	0.659036i	0.944149	+	0.329518i	$0.106886\pi$
$709$	12441.7i	0.659036i	−0.944149	+	0.329518i	$0.893114\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14493.8	−0.761285
$714$	0	0
$715$	24437.1i	1.27818i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34573.7	1.79330	0.896650	−	0.442740i	$-0.145994\pi$
$719$	34573.7	1.79330	0.896650	+	0.442740i	$0.145994\pi$
$720$	0	0
$721$	−29494.4	−1.52348
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 34102.1i	− 1.74693i
$726$	0	0
$727$	20704.6	1.05624	0.528122	−	0.849168i	$-0.322896\pi$
$727$	20704.6	1.05624	0.528122	+	0.849168i	$0.322896\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 21043.7i	− 1.06474i
$732$	0	0
$733$	11999.1i	0.604633i	0.953208	+	0.302317i	$0.0977600\pi$
$733$	11999.1i	0.604633i	−0.953208	+	0.302317i	$0.902240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1038.25	−0.0518919
$738$	0	0
$739$	4576.46i	0.227805i	0.993492	+	0.113902i	$0.0363351\pi$
$739$	4576.46i	0.227805i	−0.993492	+	0.113902i	$0.963665\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	457.260	0.0225777	0.0112889	−	0.999936i	$-0.496407\pi$
$743$	457.260	0.0225777	0.0112889	+	0.999936i	$0.496407\pi$
$744$	0	0
$745$	50723.3	2.49444
$746$	0	0
$747$	0	0
$748$	0	0
$749$	37297.3i	1.81951i
$750$	0	0
$751$	−15889.2	−0.772045	−0.386022	−	0.922489i	$-0.626151\pi$
$751$	−15889.2	−0.772045	−0.386022	+	0.922489i	$0.626151\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 36344.1i	− 1.75192i
$756$	0	0
$757$	34433.6i	1.65325i	0.562754	+	0.826624i	$0.309742\pi$
$757$	34433.6i	1.65325i	−0.562754	+	0.826624i	$0.690258\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21391.1	1.01896	0.509480	−	0.860483i	$-0.329838\pi$
$761$	21391.1	1.01896	0.509480	+	0.860483i	$0.329838\pi$
$762$	0	0
$763$	37202.3i	1.76516i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9005.94	0.423971
$768$	0	0
$769$	23915.1	1.12146	0.560728	−	0.828000i	$-0.310521\pi$
$769$	23915.1	1.12146	0.560728	+	0.828000i	$0.310521\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19446.0i	0.904818i	0.891811	+	0.452409i	$0.149435\pi$
$773$	19446.0i	0.904818i	−0.891811	+	0.452409i	$0.850565\pi$
$774$	0	0
$775$	−27273.1	−1.26410
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 31988.0i	− 1.47123i
$780$	0	0
$781$	21751.1i	0.996561i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24823.2	1.12864
$786$	0	0
$787$	29340.7i	1.32895i	0.747310	+	0.664475i	$0.231345\pi$
$787$	29340.7i	1.32895i	−0.747310	+	0.664475i	$0.768655\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9332.59	−0.419506
$792$	0	0
$793$	−20736.0	−0.928571
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 33505.7i	− 1.48913i	−0.667552	−	0.744563i	$-0.732658\pi$
$797$	− 33505.7i	− 1.48913i	0.667552	−	0.744563i	$-0.267342\pi$
$798$	0	0
$799$	20448.3	0.905394
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 20620.2i	− 0.906191i
$804$	0	0
$805$	− 31191.9i	− 1.36568i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19418.0	0.843882	0.421941	−	0.906623i	$-0.361349\pi$
$809$	19418.0	0.843882	0.421941	+	0.906623i	$0.361349\pi$
$810$	0	0
$811$	− 24228.8i	− 1.04906i	−0.851391	−	0.524531i	$-0.824241\pi$
$811$	− 24228.8i	− 1.04906i	0.851391	−	0.524531i	$-0.175759\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19138.5	0.822567
$816$	0	0
$817$	56221.5	2.40752
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 16355.5i	− 0.695263i	−0.937631	−	0.347631i	$-0.886986\pi$
$821$	− 16355.5i	− 0.695263i	0.937631	−	0.347631i	$-0.113014\pi$
$822$	0	0
$823$	3280.51	0.138945	0.0694724	−	0.997584i	$-0.477868\pi$
$823$	3280.51	0.138945	0.0694724	+	0.997584i	$0.477868\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23480.5i	− 0.987298i	−0.869661	−	0.493649i	$-0.835663\pi$
$827$	− 23480.5i	− 0.987298i	0.869661	−	0.493649i	$-0.164337\pi$
$828$	0	0
$829$	− 16306.2i	− 0.683160i	−0.939853	−	0.341580i	$-0.889038\pi$
$829$	− 16306.2i	− 0.683160i	0.939853	−	0.341580i	$-0.110962\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9425.42	−0.392042
$834$	0	0
$835$	10854.1i	0.449846i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29259.0	−1.20397	−0.601986	−	0.798506i	$-0.705624\pi$
$839$	−29259.0	−1.20397	−0.601986	+	0.798506i	$0.705624\pi$
$840$	0	0
$841$	−22950.0	−0.940998
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23674.9i	0.963834i
$846$	0	0
$847$	−30428.6	−1.23440
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 33979.8i	− 1.36876i
$852$	0	0
$853$	14076.0i	0.565008i	0.959266	+	0.282504i	$0.0911651\pi$
$853$	14076.0i	0.565008i	−0.959266	+	0.282504i	$0.908835\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43553.3	1.73600	0.868000	−	0.496565i	$-0.165405\pi$
$857$	43553.3	1.73600	0.868000	+	0.496565i	$0.165405\pi$
$858$	0	0
$859$	− 43728.9i	− 1.73692i	−0.495761	−	0.868459i	$-0.665111\pi$
$859$	− 43728.9i	− 1.73692i	0.495761	−	0.868459i	$-0.334889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21807.9	−0.860198	−0.430099	−	0.902782i	$-0.641521\pi$
$863$	−21807.9	−0.860198	−0.430099	+	0.902782i	$0.641521\pi$
$864$	0	0
$865$	−67750.6	−2.66311
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4963.10i	− 0.193742i
$870$	0	0
$871$	560.901	0.0218202
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11884.8i	− 0.459176i
$876$	0	0
$877$	− 6228.27i	− 0.239810i	−0.992785	−	0.119905i	$-0.961741\pi$
$877$	− 6228.27i	− 0.239810i	0.992785	−	0.119905i	$-0.0382591\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40608.1	−1.55292	−0.776459	−	0.630168i	$-0.782986\pi$
$881$	−40608.1	−1.55292	−0.776459	+	0.630168i	$0.782986\pi$
$882$	0	0
$883$	78.9699i	0.00300968i	0.999999	+	0.00150484i	$0.000479006\pi$
$883$	78.9699i	0.00300968i	−0.999999	+	0.00150484i	$0.999521\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33039.9	1.25070	0.625351	−	0.780344i	$-0.284956\pi$
$887$	33039.9	1.25070	0.625351	+	0.780344i	$0.284956\pi$
$888$	0	0
$889$	2376.22	0.0896467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	54631.0i	2.04721i
$894$	0	0
$895$	26724.6	0.998105
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37859.3i	1.40454i
$900$	0	0
$901$	− 14275.4i	− 0.527840i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	64621.1	2.37357
$906$	0	0
$907$	− 23459.7i	− 0.858838i	−0.903105	−	0.429419i	$-0.858718\pi$
$907$	− 23459.7i	− 0.858838i	0.903105	−	0.429419i	$-0.141282\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36386.3	−1.32331	−0.661653	−	0.749811i	$-0.730145\pi$
$911$	−36386.3	−1.32331	−0.661653	+	0.749811i	$0.730145\pi$
$912$	0	0
$913$	18079.2	0.655348
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40481.0i	1.45780i
$918$	0	0
$919$	−19058.6	−0.684097	−0.342049	−	0.939682i	$-0.611121\pi$
$919$	−19058.6	−0.684097	−0.342049	+	0.939682i	$0.611121\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 11750.8i	− 0.419048i
$924$	0	0
$925$	− 63940.3i	− 2.27280i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27020.7	0.954274	0.477137	−	0.878829i	$-0.341674\pi$
$929$	27020.7	0.954274	0.477137	+	0.878829i	$0.341674\pi$
$930$	0	0
$931$	− 25181.5i	− 0.886456i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−53076.1	−1.85644
$936$	0	0
$937$	11862.3	0.413581	0.206791	−	0.978385i	$-0.433698\pi$
$937$	11862.3	0.413581	0.206791	+	0.978385i	$0.433698\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 53659.9i	− 1.85894i	−0.368897	−	0.929470i	$-0.620265\pi$
$941$	− 53659.9i	− 1.85894i	0.368897	−	0.929470i	$-0.379735\pi$
$942$	0	0
$943$	−16372.6	−0.565394
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1063.85i	− 0.0365052i	−0.999833	−	0.0182526i	$-0.994190\pi$
$947$	− 1063.85i	− 0.0365052i	0.999833	−	0.0182526i	$-0.00581030\pi$
$948$	0	0
$949$	11139.8i	0.381047i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5715.17	0.194263	0.0971314	−	0.995272i	$-0.469033\pi$
$953$	5715.17	0.194263	0.0971314	+	0.995272i	$0.469033\pi$
$954$	0	0
$955$	− 67972.7i	− 2.30319i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8950.71	0.301391
$960$	0	0
$961$	486.886	0.0163434
$962$	0	0
$963$	0	0
$964$	0	0
$965$	57437.6i	1.91604i
$966$	0	0
$967$	−124.600	−0.00414360	−0.00207180	−	0.999998i	$-0.500659\pi$
$967$	−124.600	−0.00414360	−0.00207180	+	0.999998i	$0.500659\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40767.0i	1.34735i	0.739028	+	0.673674i	$0.235285\pi$
$971$	40767.0i	1.34735i	−0.739028	+	0.673674i	$0.764715\pi$
$972$	0	0
$973$	− 8510.70i	− 0.280412i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11600.1	−0.379856	−0.189928	−	0.981798i	$-0.560825\pi$
$977$	−11600.1	−0.379856	−0.189928	+	0.981798i	$0.560825\pi$
$978$	0	0
$979$	77475.3i	2.52923i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16979.6	−0.550930	−0.275465	−	0.961311i	$-0.588832\pi$
$983$	−16979.6	−0.550930	−0.275465	+	0.961311i	$0.588832\pi$
$984$	0	0
$985$	−32815.2	−1.06150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 28776.3i	− 0.925210i
$990$	0	0
$991$	−15762.5	−0.505259	−0.252630	−	0.967563i	$-0.581295\pi$
$991$	−15762.5	−0.505259	−0.252630	+	0.967563i	$0.581295\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1830.07i	− 0.0583085i
$996$	0	0
$997$	8861.70i	0.281497i	0.990045	+	0.140749i	$0.0449509\pi$
$997$	8861.70i	0.281497i	−0.990045	+	0.140749i	$0.955049\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.e.865.8	yes	8
3.2	odd	2		1728.4.d.h.865.2	yes	8
4.3	odd	2	inner	1728.4.d.e.865.7	yes	8
8.3	odd	2	inner	1728.4.d.e.865.1	✓	8
8.5	even	2	inner	1728.4.d.e.865.2	yes	8
12.11	even	2		1728.4.d.h.865.1	yes	8
24.5	odd	2		1728.4.d.h.865.8	yes	8
24.11	even	2		1728.4.d.h.865.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.e.865.1	✓	8	8.3	odd	2	inner
1728.4.d.e.865.2	yes	8	8.5	even	2	inner
1728.4.d.e.865.7	yes	8	4.3	odd	2	inner
1728.4.d.e.865.8	yes	8	1.1	even	1	trivial
1728.4.d.h.865.1	yes	8	12.11	even	2
1728.4.d.h.865.2	yes	8	3.2	odd	2
1728.4.d.h.865.7	yes	8	24.11	even	2
1728.4.d.h.865.8	yes	8	24.5	odd	2

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.7850i q^{5} +22.3100 q^{7} +O(q^{10})\)	\(q+16.7850i q^{5} +22.3100 q^{7} -51.9124i q^{11} +28.0451i q^{13} -60.9124 q^{17} -162.737i q^{19} -83.2950 q^{23} -156.737 q^{25} +217.575i q^{29} +174.005 q^{31} +374.474i q^{35} +407.946i q^{37} +196.562 q^{41} +345.474i q^{43} -335.701 q^{47} +154.737 q^{49} +234.360i q^{53} +871.351 q^{55} -321.124i q^{59} +739.381i q^{61} -470.737 q^{65} -20.0000i q^{67} -418.996 q^{71} +397.212 q^{73} -1158.17i q^{77} +95.6053 q^{79} +348.263i q^{83} -1022.42i q^{85} -1492.42 q^{89} +625.686i q^{91} +2731.55 q^{95} +881.949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100}) \) 8 * q - 168 * q^17 - 296 * q^25 - 24 * q^41 + 280 * q^49 - 2808 * q^65 + 304 * q^73 - 6192 * q^89 + 3224 * q^97

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