LMFDB - Embedded newform 1296.4.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$76.4664753674$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.72153.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 12 $ x^4 - x^3 - 8x^2 + 3x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.26386$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.99447 q^{5} -15.5776 q^{7} +O(q^{10})$	$q+5.99447 q^{5} -15.5776 q^{7} +34.8844 q^{11} -80.6284 q^{13} +70.1103 q^{17} +4.25374 q^{19} +118.709 q^{23} -89.0663 q^{25} -123.414 q^{29} +185.740 q^{31} -93.3796 q^{35} +151.468 q^{37} -212.444 q^{41} -290.035 q^{43} -212.585 q^{47} -100.337 q^{49} +556.480 q^{53} +209.114 q^{55} -853.068 q^{59} -688.999 q^{61} -483.324 q^{65} +915.750 q^{67} +786.367 q^{71} -993.028 q^{73} -543.416 q^{77} +568.375 q^{79} -747.091 q^{83} +420.274 q^{85} -1013.91 q^{89} +1256.00 q^{91} +25.4989 q^{95} -1219.51 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{5} + 3 q^{7}+O(q^{10}) $ 4 * q - 5 * q^5 + 3 * q^7	$ 4 q - 5 q^{5} + 3 q^{7} - 16 q^{11} - 29 q^{13} + 17 q^{17} + 109 q^{19} - 37 q^{23} - 97 q^{25} - 3 q^{29} + 331 q^{31} - 171 q^{35} - 366 q^{37} - 378 q^{41} + 506 q^{43} - 171 q^{47} - 829 q^{49} - 410 q^{53} + 1163 q^{55} - 616 q^{59} - 1331 q^{61} - 815 q^{65} + 1162 q^{67} - 344 q^{71} - 1307 q^{73} - 741 q^{77} + 1853 q^{79} - 1421 q^{83} - 2074 q^{85} - 816 q^{89} + 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) $ 4 * q - 5 * q^5 + 3 * q^7 - 16 * q^11 - 29 * q^13 + 17 * q^17 + 109 * q^19 - 37 * q^23 - 97 * q^25 - 3 * q^29 + 331 * q^31 - 171 * q^35 - 366 * q^37 - 378 * q^41 + 506 * q^43 - 171 * q^47 - 829 * q^49 - 410 * q^53 + 1163 * q^55 - 616 * q^59 - 1331 * q^61 - 815 * q^65 + 1162 * q^67 - 344 * q^71 - 1307 * q^73 - 741 * q^77 + 1853 * q^79 - 1421 * q^83 - 2074 * q^85 - 816 * q^89 + 1995 * q^91 - 1292 * q^95 - 2506 * q^97

Display 100 coefficients 10 coefficients

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
$5$	5.99447	0.536162	0.268081	−	0.963396i	$-0.413611\pi$
$5$	5.99447	0.536162	0.268081	+	0.963396i	$0.413611\pi$
$6$	0	0
$7$	−15.5776	−0.841113	−0.420557	−	0.907266i	$-0.638165\pi$
$7$	−15.5776	−0.841113	−0.420557	+	0.907266i	$0.638165\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	34.8844	0.956186	0.478093	−	0.878309i	$-0.341328\pi$
$11$	34.8844	0.956186	0.478093	+	0.878309i	$0.341328\pi$
$12$	0	0
$13$	−80.6284	−1.72018	−0.860088	−	0.510146i	$-0.829591\pi$
$13$	−80.6284	−1.72018	−0.860088	+	0.510146i	$0.829591\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	70.1103	1.00025	0.500125	−	0.865953i	$-0.333287\pi$
$17$	70.1103	1.00025	0.500125	+	0.865953i	$0.333287\pi$
$18$	0	0
$19$	4.25374	0.0513619	0.0256809	−	0.999670i	$-0.491825\pi$
$19$	4.25374	0.0513619	0.0256809	+	0.999670i	$0.491825\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	118.709	1.07620	0.538099	−	0.842882i	$-0.319143\pi$
$23$	118.709	1.07620	0.538099	+	0.842882i	$0.319143\pi$
$24$	0	0
$25$	−89.0663	−0.712531
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−123.414	−0.790258	−0.395129	−	0.918626i	$-0.629300\pi$
$29$	−123.414	−0.790258	−0.395129	+	0.918626i	$0.629300\pi$
$30$	0	0
$31$	185.740	1.07613	0.538063	−	0.842904i	$-0.319156\pi$
$31$	185.740	1.07613	0.538063	+	0.842904i	$0.319156\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−93.3796	−0.450973
$36$	0	0
$37$	151.468	0.673005	0.336503	−	0.941683i	$-0.390756\pi$
$37$	151.468	0.673005	0.336503	+	0.941683i	$0.390756\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−212.444	−0.809223	−0.404612	−	0.914489i	$-0.632593\pi$
$41$	−212.444	−0.809223	−0.404612	+	0.914489i	$0.632593\pi$
$42$	0	0
$43$	−290.035	−1.02860	−0.514301	−	0.857610i	$-0.671949\pi$
$43$	−290.035	−1.02860	−0.514301	+	0.857610i	$0.671949\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−212.585	−0.659759	−0.329879	−	0.944023i	$-0.607008\pi$
$47$	−212.585	−0.659759	−0.329879	+	0.944023i	$0.607008\pi$
$48$	0	0
$49$	−100.337	−0.292529
$50$	0	0
$51$	0	0
$52$	0	0
$53$	556.480	1.44223	0.721117	−	0.692813i	$-0.243629\pi$
$53$	556.480	1.44223	0.721117	+	0.692813i	$0.243629\pi$
$54$	0	0
$55$	209.114	0.512670
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−853.068	−1.88237	−0.941187	−	0.337887i	$-0.890288\pi$
$59$	−853.068	−1.88237	−0.941187	+	0.337887i	$0.890288\pi$
$60$	0	0
$61$	−688.999	−1.44618	−0.723092	−	0.690752i	$-0.757280\pi$
$61$	−688.999	−1.44618	−0.723092	+	0.690752i	$0.757280\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−483.324	−0.922292
$66$	0	0
$67$	915.750	1.66980	0.834900	−	0.550401i	$-0.185525\pi$
$67$	915.750	1.66980	0.834900	+	0.550401i	$0.185525\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	786.367	1.31443	0.657216	−	0.753702i	$-0.271734\pi$
$71$	786.367	1.31443	0.657216	+	0.753702i	$0.271734\pi$
$72$	0	0
$73$	−993.028	−1.59213	−0.796063	−	0.605214i	$-0.793088\pi$
$73$	−993.028	−1.59213	−0.796063	+	0.605214i	$0.793088\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−543.416	−0.804260
$78$	0	0
$79$	568.375	0.809458	0.404729	−	0.914437i	$-0.367366\pi$
$79$	568.375	0.809458	0.404729	+	0.914437i	$0.367366\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−747.091	−0.987999	−0.494000	−	0.869462i	$-0.664466\pi$
$83$	−747.091	−0.987999	−0.494000	+	0.869462i	$0.664466\pi$
$84$	0	0
$85$	420.274	0.536296
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1013.91	−1.20757	−0.603787	−	0.797145i	$-0.706342\pi$
$89$	−1013.91	−1.20757	−0.603787	+	0.797145i	$0.706342\pi$
$90$	0	0
$91$	1256.00	1.44686
$92$	0	0
$93$	0	0
$94$	0	0
$95$	25.4989	0.0275383
$96$	0	0
$97$	−1219.51	−1.27652	−0.638260	−	0.769821i	$-0.720345\pi$
$97$	−1219.51	−1.27652	−0.638260	+	0.769821i	$0.720345\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	452.505	0.445802	0.222901	−	0.974841i	$-0.428447\pi$
$101$	452.505	0.445802	0.222901	+	0.974841i	$0.428447\pi$
$102$	0	0
$103$	−1155.50	−1.10538	−0.552692	−	0.833386i	$-0.686399\pi$
$103$	−1155.50	−1.10538	−0.552692	+	0.833386i	$0.686399\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	82.5186	0.0745549	0.0372774	−	0.999305i	$-0.488131\pi$
$107$	82.5186	0.0745549	0.0372774	+	0.999305i	$0.488131\pi$
$108$	0	0
$109$	326.121	0.286576	0.143288	−	0.989681i	$-0.454233\pi$
$109$	326.121	0.286576	0.143288	+	0.989681i	$0.454233\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	383.526	0.319284	0.159642	−	0.987175i	$-0.448966\pi$
$113$	383.526	0.319284	0.159642	+	0.987175i	$0.448966\pi$
$114$	0	0
$115$	711.597	0.577016
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1092.15	−0.841323
$120$	0	0
$121$	−114.078	−0.0857083
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1283.21	−0.918193
$126$	0	0
$127$	835.221	0.583574	0.291787	−	0.956483i	$-0.405750\pi$
$127$	835.221	0.583574	0.291787	+	0.956483i	$0.405750\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−613.637	−0.409265	−0.204632	−	0.978839i	$-0.565600\pi$
$131$	−613.637	−0.409265	−0.204632	+	0.978839i	$0.565600\pi$
$132$	0	0
$133$	−66.2632	−0.0432011
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2016.84	−1.25774	−0.628869	−	0.777511i	$-0.716482\pi$
$137$	−2016.84	−1.25774	−0.628869	+	0.777511i	$0.716482\pi$
$138$	0	0
$139$	1156.12	0.705477	0.352738	−	0.935722i	$-0.385251\pi$
$139$	1156.12	0.705477	0.352738	+	0.935722i	$0.385251\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2812.67	−1.64481
$144$	0	0
$145$	−739.804	−0.423706
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1283.88	−0.705902	−0.352951	−	0.935642i	$-0.614822\pi$
$149$	−1283.88	−0.705902	−0.352951	+	0.935642i	$0.614822\pi$
$150$	0	0
$151$	270.153	0.145594	0.0727972	−	0.997347i	$-0.476807\pi$
$151$	270.153	0.145594	0.0727972	+	0.997347i	$0.476807\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1113.41	0.576978
$156$	0	0
$157$	−3244.83	−1.64946	−0.824731	−	0.565525i	$-0.808673\pi$
$157$	−3244.83	−1.64946	−0.824731	+	0.565525i	$0.808673\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1849.20	−0.905203
$162$	0	0
$163$	−689.397	−0.331275	−0.165637	−	0.986187i	$-0.552968\pi$
$163$	−689.397	−0.331275	−0.165637	+	0.986187i	$0.552968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−535.907	−0.248322	−0.124161	−	0.992262i	$-0.539624\pi$
$167$	−535.907	−0.248322	−0.124161	+	0.992262i	$0.539624\pi$
$168$	0	0
$169$	4303.93	1.95900
$170$	0	0
$171$	0	0
$172$	0	0
$173$	596.031	0.261939	0.130969	−	0.991386i	$-0.458191\pi$
$173$	596.031	0.261939	0.130969	+	0.991386i	$0.458191\pi$
$174$	0	0
$175$	1387.44	0.599319
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2853.20	−1.19139	−0.595693	−	0.803212i	$-0.703122\pi$
$179$	−2853.20	−1.19139	−0.595693	+	0.803212i	$0.703122\pi$
$180$	0	0
$181$	941.824	0.386769	0.193385	−	0.981123i	$-0.438053\pi$
$181$	941.824	0.386769	0.193385	+	0.981123i	$0.438053\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	907.971	0.360840
$186$	0	0
$187$	2445.76	0.956425
$188$	0	0
$189$	0	0
$190$	0	0
$191$	745.675	0.282488	0.141244	−	0.989975i	$-0.454890\pi$
$191$	745.675	0.282488	0.141244	+	0.989975i	$0.454890\pi$
$192$	0	0
$193$	995.122	0.371142	0.185571	−	0.982631i	$-0.440586\pi$
$193$	995.122	0.371142	0.185571	+	0.982631i	$0.440586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3537.26	1.27929	0.639643	−	0.768672i	$-0.279082\pi$
$197$	3537.26	1.27929	0.639643	+	0.768672i	$0.279082\pi$
$198$	0	0
$199$	−4567.63	−1.62709	−0.813545	−	0.581502i	$-0.802465\pi$
$199$	−4567.63	−1.62709	−0.813545	+	0.581502i	$0.802465\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1922.50	0.664696
$204$	0	0
$205$	−1273.49	−0.433875
$206$	0	0
$207$	0	0
$208$	0	0
$209$	148.389	0.0491115
$210$	0	0
$211$	−2127.31	−0.694077	−0.347039	−	0.937851i	$-0.612813\pi$
$211$	−2127.31	−0.694077	−0.347039	+	0.937851i	$0.612813\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1738.60	−0.551497
$216$	0	0
$217$	−2893.39	−0.905144
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5652.88	−1.72061
$222$	0	0
$223$	−1961.78	−0.589105	−0.294553	−	0.955635i	$-0.595171\pi$
$223$	−1961.78	−0.589105	−0.294553	+	0.955635i	$0.595171\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1926.50	−0.563289	−0.281645	−	0.959519i	$-0.590880\pi$
$227$	−1926.50	−0.563289	−0.281645	+	0.959519i	$0.590880\pi$
$228$	0	0
$229$	−1524.52	−0.439925	−0.219962	−	0.975508i	$-0.570593\pi$
$229$	−1524.52	−0.439925	−0.219962	+	0.975508i	$0.570593\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3761.71	−1.05767	−0.528837	−	0.848723i	$-0.677372\pi$
$233$	−3761.71	−1.05767	−0.528837	+	0.848723i	$0.677372\pi$
$234$	0	0
$235$	−1274.33	−0.353737
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3731.99	−1.01005	−0.505025	−	0.863105i	$-0.668517\pi$
$239$	−3731.99	−1.01005	−0.505025	+	0.863105i	$0.668517\pi$
$240$	0	0
$241$	2978.20	0.796029	0.398014	−	0.917379i	$-0.369699\pi$
$241$	2978.20	0.796029	0.398014	+	0.917379i	$0.369699\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−601.470	−0.156843
$246$	0	0
$247$	−342.972	−0.0883514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3088.56	−0.776685	−0.388342	−	0.921515i	$-0.626952\pi$
$251$	−3088.56	−0.776685	−0.388342	+	0.921515i	$0.626952\pi$
$252$	0	0
$253$	4141.09	1.02904
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3921.72	0.951868	0.475934	−	0.879481i	$-0.342110\pi$
$257$	3921.72	0.951868	0.475934	+	0.879481i	$0.342110\pi$
$258$	0	0
$259$	−2359.51	−0.566073
$260$	0	0
$261$	0	0
$262$	0	0
$263$	959.305	0.224918	0.112459	−	0.993656i	$-0.464127\pi$
$263$	959.305	0.224918	0.112459	+	0.993656i	$0.464127\pi$
$264$	0	0
$265$	3335.80	0.773271
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7499.82	−1.69990	−0.849948	−	0.526866i	$-0.823367\pi$
$269$	−7499.82	−1.69990	−0.849948	+	0.526866i	$0.823367\pi$
$270$	0	0
$271$	723.420	0.162157	0.0810787	−	0.996708i	$-0.474163\pi$
$271$	723.420	0.162157	0.0810787	+	0.996708i	$0.474163\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3107.03	−0.681312
$276$	0	0
$277$	636.610	0.138087	0.0690436	−	0.997614i	$-0.478005\pi$
$277$	636.610	0.138087	0.0690436	+	0.997614i	$0.478005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6890.27	−1.46277	−0.731386	−	0.681964i	$-0.761126\pi$
$281$	−6890.27	−1.46277	−0.731386	+	0.681964i	$0.761126\pi$
$282$	0	0
$283$	−8831.63	−1.85507	−0.927537	−	0.373730i	$-0.878079\pi$
$283$	−8831.63	−1.85507	−0.927537	+	0.373730i	$0.878079\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3309.37	0.680648
$288$	0	0
$289$	2.45352	0.000499394 0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2757.43	−0.549797	−0.274899	−	0.961473i	$-0.588644\pi$
$293$	−2757.43	−0.549797	−0.274899	+	0.961473i	$0.588644\pi$
$294$	0	0
$295$	−5113.69	−1.00926
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9571.31	−1.85125
$300$	0	0
$301$	4518.05	0.865171
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4130.18	−0.775389
$306$	0	0
$307$	−1922.19	−0.357346	−0.178673	−	0.983908i	$-0.557180\pi$
$307$	−1922.19	−0.357346	−0.178673	+	0.983908i	$0.557180\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4394.67	0.801283	0.400641	−	0.916235i	$-0.368787\pi$
$311$	4394.67	0.801283	0.400641	+	0.916235i	$0.368787\pi$
$312$	0	0
$313$	2230.64	0.402821	0.201411	−	0.979507i	$-0.435447\pi$
$313$	2230.64	0.402821	0.201411	+	0.979507i	$0.435447\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1201.46	0.212872	0.106436	−	0.994320i	$-0.466056\pi$
$317$	1201.46	0.212872	0.106436	+	0.994320i	$0.466056\pi$
$318$	0	0
$319$	−4305.24	−0.755633
$320$	0	0
$321$	0	0
$322$	0	0
$323$	298.231	0.0513747
$324$	0	0
$325$	7181.27	1.22568
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3311.57	0.554932
$330$	0	0
$331$	2909.82	0.483196	0.241598	−	0.970376i	$-0.422328\pi$
$331$	2909.82	0.483196	0.241598	+	0.970376i	$0.422328\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5489.43	0.895283
$336$	0	0
$337$	6653.21	1.07544	0.537720	−	0.843123i	$-0.319286\pi$
$337$	6653.21	1.07544	0.537720	+	0.843123i	$0.319286\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6479.44	1.02898
$342$	0	0
$343$	6906.15	1.08716
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8086.63	−1.25105	−0.625523	−	0.780206i	$-0.715114\pi$
$347$	−8086.63	−1.25105	−0.625523	+	0.780206i	$0.715114\pi$
$348$	0	0
$349$	−7608.80	−1.16702	−0.583509	−	0.812106i	$-0.698321\pi$
$349$	−7608.80	−1.16702	−0.583509	+	0.812106i	$0.698321\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11710.7	1.76571	0.882856	−	0.469644i	$-0.155618\pi$
$353$	11710.7	1.76571	0.882856	+	0.469644i	$0.155618\pi$
$354$	0	0
$355$	4713.86	0.704748
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11013.5	1.61914	0.809571	−	0.587022i	$-0.199700\pi$
$359$	11013.5	1.61914	0.809571	+	0.587022i	$0.199700\pi$
$360$	0	0
$361$	−6840.91	−0.997362
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5952.68	−0.853637
$366$	0	0
$367$	3979.29	0.565988	0.282994	−	0.959122i	$-0.408672\pi$
$367$	3979.29	0.565988	0.282994	+	0.959122i	$0.408672\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8668.64	−1.21308
$372$	0	0
$373$	3583.15	0.497395	0.248697	−	0.968581i	$-0.419998\pi$
$373$	3583.15	0.497395	0.248697	+	0.968581i	$0.419998\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9950.70	1.35938
$378$	0	0
$379$	2337.28	0.316776	0.158388	−	0.987377i	$-0.449370\pi$
$379$	2337.28	0.316776	0.158388	+	0.987377i	$0.449370\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3387.03	0.451877	0.225939	−	0.974142i	$-0.427455\pi$
$383$	3387.03	0.451877	0.225939	+	0.974142i	$0.427455\pi$
$384$	0	0
$385$	−3257.49	−0.431214
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7251.79	−0.945193	−0.472597	−	0.881279i	$-0.656683\pi$
$389$	−7251.79	−0.945193	−0.472597	+	0.881279i	$0.656683\pi$
$390$	0	0
$391$	8322.72	1.07647
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3407.11	0.434000
$396$	0	0
$397$	11636.7	1.47110	0.735550	−	0.677470i	$-0.236924\pi$
$397$	11636.7	1.47110	0.735550	+	0.677470i	$0.236924\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6032.03	0.751184	0.375592	−	0.926785i	$-0.377439\pi$
$401$	6032.03	0.751184	0.375592	+	0.926785i	$0.377439\pi$
$402$	0	0
$403$	−14975.9	−1.85113
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5283.87	0.643518
$408$	0	0
$409$	−282.404	−0.0341418	−0.0170709	−	0.999854i	$-0.505434\pi$
$409$	−282.404	−0.0341418	−0.0170709	+	0.999854i	$0.505434\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13288.8	1.58329
$414$	0	0
$415$	−4478.42	−0.529727
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1214.87	0.141647	0.0708237	−	0.997489i	$-0.477437\pi$
$419$	1214.87	0.141647	0.0708237	+	0.997489i	$0.477437\pi$
$420$	0	0
$421$	9428.14	1.09145	0.545724	−	0.837965i	$-0.316255\pi$
$421$	9428.14	1.09145	0.545724	+	0.837965i	$0.316255\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6244.47	−0.712709
$426$	0	0
$427$	10733.0	1.21640
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2125.36	0.237529	0.118765	−	0.992922i	$-0.462107\pi$
$431$	2125.36	0.237529	0.118765	+	0.992922i	$0.462107\pi$
$432$	0	0
$433$	1802.72	0.200077	0.100038	−	0.994984i	$-0.468103\pi$
$433$	1802.72	0.200077	0.100038	+	0.994984i	$0.468103\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	504.957	0.0552755
$438$	0	0
$439$	−6231.22	−0.677448	−0.338724	−	0.940886i	$-0.609995\pi$
$439$	−6231.22	−0.677448	−0.338724	+	0.940886i	$0.609995\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16831.6	1.80518	0.902590	−	0.430501i	$-0.141663\pi$
$443$	16831.6	1.80518	0.902590	+	0.430501i	$0.141663\pi$
$444$	0	0
$445$	−6077.85	−0.647455
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4338.18	0.455972	0.227986	−	0.973664i	$-0.426786\pi$
$449$	4338.18	0.455972	0.227986	+	0.973664i	$0.426786\pi$
$450$	0	0
$451$	−7410.98	−0.773768
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7529.05	0.775752
$456$	0	0
$457$	−6361.40	−0.651146	−0.325573	−	0.945517i	$-0.605557\pi$
$457$	−6361.40	−0.651146	−0.325573	+	0.945517i	$0.605557\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4837.83	0.488764	0.244382	−	0.969679i	$-0.421415\pi$
$461$	4837.83	0.488764	0.244382	+	0.969679i	$0.421415\pi$
$462$	0	0
$463$	−2461.45	−0.247070	−0.123535	−	0.992340i	$-0.539423\pi$
$463$	−2461.45	−0.247070	−0.123535	+	0.992340i	$0.539423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12271.6	−1.21598	−0.607988	−	0.793946i	$-0.708023\pi$
$467$	−12271.6	−1.21598	−0.607988	+	0.793946i	$0.708023\pi$
$468$	0	0
$469$	−14265.2	−1.40449
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10117.7	−0.983535
$474$	0	0
$475$	−378.865	−0.0365969
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3033.45	−0.289357	−0.144679	−	0.989479i	$-0.546215\pi$
$479$	−3033.45	−0.289357	−0.144679	+	0.989479i	$0.546215\pi$
$480$	0	0
$481$	−12212.6	−1.15769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7310.31	−0.684421
$486$	0	0
$487$	−8610.41	−0.801181	−0.400590	−	0.916257i	$-0.631195\pi$
$487$	−8610.41	−0.801181	−0.400590	+	0.916257i	$0.631195\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4582.57	−0.421199	−0.210599	−	0.977572i	$-0.567542\pi$
$491$	−4582.57	−0.421199	−0.210599	+	0.977572i	$0.567542\pi$
$492$	0	0
$493$	−8652.62	−0.790455
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12249.7	−1.10559
$498$	0	0
$499$	3837.52	0.344270	0.172135	−	0.985073i	$-0.444933\pi$
$499$	3837.52	0.344270	0.172135	+	0.985073i	$0.444933\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7189.58	0.637311	0.318656	−	0.947871i	$-0.396769\pi$
$503$	7189.58	0.637311	0.318656	+	0.947871i	$0.396769\pi$
$504$	0	0
$505$	2712.53	0.239022
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6787.74	0.591083	0.295542	−	0.955330i	$-0.404500\pi$
$509$	6787.74	0.591083	0.295542	+	0.955330i	$0.404500\pi$
$510$	0	0
$511$	15469.0	1.33916
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6926.59	−0.592664
$516$	0	0
$517$	−7415.89	−0.630852
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8153.95	0.685664	0.342832	−	0.939397i	$-0.388614\pi$
$521$	8153.95	0.685664	0.342832	+	0.939397i	$0.388614\pi$
$522$	0	0
$523$	18260.0	1.52668	0.763339	−	0.645998i	$-0.223559\pi$
$523$	18260.0	1.52668	0.763339	+	0.645998i	$0.223559\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13022.3	1.07640
$528$	0	0
$529$	1924.82	0.158200
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17129.0	1.39201
$534$	0	0
$535$	494.655	0.0399735
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3500.21	−0.279712
$540$	0	0
$541$	−13335.5	−1.05978	−0.529888	−	0.848068i	$-0.677766\pi$
$541$	−13335.5	−1.05978	−0.529888	+	0.848068i	$0.677766\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1954.92	0.153651
$546$	0	0
$547$	14802.1	1.15703	0.578513	−	0.815673i	$-0.303633\pi$
$547$	14802.1	1.15703	0.578513	+	0.815673i	$0.303633\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−524.973	−0.0405891
$552$	0	0
$553$	−8853.94	−0.680846
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10454.8	−0.795302	−0.397651	−	0.917537i	$-0.630175\pi$
$557$	−10454.8	−0.795302	−0.397651	+	0.917537i	$0.630175\pi$
$558$	0	0
$559$	23385.0	1.76938
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5533.26	−0.414208	−0.207104	−	0.978319i	$-0.566404\pi$
$563$	−5533.26	−0.414208	−0.207104	+	0.978319i	$0.566404\pi$
$564$	0	0
$565$	2299.03	0.171188
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21509.6	1.58476	0.792380	−	0.610028i	$-0.208842\pi$
$569$	21509.6	1.58476	0.792380	+	0.610028i	$0.208842\pi$
$570$	0	0
$571$	−11031.0	−0.808467	−0.404234	−	0.914656i	$-0.632462\pi$
$571$	−11031.0	−0.808467	−0.404234	+	0.914656i	$0.632462\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10573.0	−0.766823
$576$	0	0
$577$	12890.5	0.930047	0.465024	−	0.885298i	$-0.346046\pi$
$577$	12890.5	0.930047	0.465024	+	0.885298i	$0.346046\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11637.9	0.831019
$582$	0	0
$583$	19412.5	1.37904
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24230.6	1.70376	0.851879	−	0.523739i	$-0.175463\pi$
$587$	24230.6	1.70376	0.851879	+	0.523739i	$0.175463\pi$
$588$	0	0
$589$	790.091	0.0552719
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23815.8	−1.64924	−0.824619	−	0.565688i	$-0.808611\pi$
$593$	−23815.8	−1.64924	−0.824619	+	0.565688i	$0.808611\pi$
$594$	0	0
$595$	−6546.87	−0.451085
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26285.2	1.79296	0.896482	−	0.443081i	$-0.146114\pi$
$599$	26285.2	1.79296	0.896482	+	0.443081i	$0.146114\pi$
$600$	0	0
$601$	−5626.41	−0.381874	−0.190937	−	0.981602i	$-0.561153\pi$
$601$	−5626.41	−0.381874	−0.190937	+	0.981602i	$0.561153\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−683.836	−0.0459535
$606$	0	0
$607$	−7493.17	−0.501052	−0.250526	−	0.968110i	$-0.580603\pi$
$607$	−7493.17	−0.501052	−0.250526	+	0.968110i	$0.580603\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17140.4	1.13490
$612$	0	0
$613$	−14061.2	−0.926470	−0.463235	−	0.886235i	$-0.653311\pi$
$613$	−14061.2	−0.926470	−0.463235	+	0.886235i	$0.653311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27610.3	1.80154	0.900768	−	0.434301i	$-0.143005\pi$
$617$	27610.3	1.80154	0.900768	+	0.434301i	$0.143005\pi$
$618$	0	0
$619$	18.6759	0.00121268	0.000606340	−	1.00000i	$-0.499807\pi$
$619$	18.6759	0.00121268	0.000606340		1.00000i	$0.499807\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15794.3	1.01571
$624$	0	0
$625$	3441.10	0.220231
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10619.5	0.673173
$630$	0	0
$631$	−739.432	−0.0466503	−0.0233251	−	0.999728i	$-0.507425\pi$
$631$	−739.432	−0.0466503	−0.0233251	+	0.999728i	$0.507425\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5006.71	0.312890
$636$	0	0
$637$	8090.04	0.503201
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10981.3	0.676656	0.338328	−	0.941028i	$-0.390139\pi$
$641$	10981.3	0.676656	0.338328	+	0.941028i	$0.390139\pi$
$642$	0	0
$643$	83.7747	0.00513803	0.00256902	−	0.999997i	$-0.499182\pi$
$643$	83.7747	0.00513803	0.00256902	+	0.999997i	$0.499182\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23261.3	1.41344	0.706719	−	0.707494i	$-0.250175\pi$
$647$	23261.3	1.41344	0.706719	+	0.707494i	$0.250175\pi$
$648$	0	0
$649$	−29758.8	−1.79990
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32640.5	1.95608	0.978042	−	0.208410i	$-0.0668288\pi$
$653$	32640.5	1.95608	0.978042	+	0.208410i	$0.0668288\pi$
$654$	0	0
$655$	−3678.43	−0.219432
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22481.2	−1.32890	−0.664448	−	0.747335i	$-0.731333\pi$
$659$	−22481.2	−1.32890	−0.664448	+	0.747335i	$0.731333\pi$
$660$	0	0
$661$	10759.5	0.633125	0.316562	−	0.948572i	$-0.397471\pi$
$661$	10759.5	0.633125	0.316562	+	0.948572i	$0.397471\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−397.213	−0.0231628
$666$	0	0
$667$	−14650.4	−0.850473
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24035.3	−1.38282
$672$	0	0
$673$	3652.61	0.209209	0.104605	−	0.994514i	$-0.466642\pi$
$673$	3652.61	0.209209	0.104605	+	0.994514i	$0.466642\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	828.979	0.0470609	0.0235305	−	0.999723i	$-0.492509\pi$
$677$	828.979	0.0470609	0.0235305	+	0.999723i	$0.492509\pi$
$678$	0	0
$679$	18997.1	1.07370
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26257.6	−1.47104	−0.735520	−	0.677503i	$-0.763062\pi$
$683$	−26257.6	−1.47104	−0.735520	+	0.677503i	$0.763062\pi$
$684$	0	0
$685$	−12089.9	−0.674351
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−44868.1	−2.48090
$690$	0	0
$691$	7945.55	0.437428	0.218714	−	0.975789i	$-0.429814\pi$
$691$	7945.55	0.437428	0.218714	+	0.975789i	$0.429814\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6930.36	0.378249
$696$	0	0
$697$	−14894.5	−0.809425
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2445.89	−0.131783	−0.0658917	−	0.997827i	$-0.520989\pi$
$701$	−2445.89	−0.131783	−0.0658917	+	0.997827i	$0.520989\pi$
$702$	0	0
$703$	644.306	0.0345668
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7048.96	−0.374970
$708$	0	0
$709$	−8726.31	−0.462233	−0.231117	−	0.972926i	$-0.574238\pi$
$709$	−8726.31	−0.462233	−0.231117	+	0.972926i	$0.574238\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22049.0	1.15812
$714$	0	0
$715$	−16860.5	−0.881883
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20119.7	1.04359	0.521793	−	0.853072i	$-0.325263\pi$
$719$	20119.7	1.04359	0.521793	+	0.853072i	$0.325263\pi$
$720$	0	0
$721$	17999.9	0.929753
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10992.1	0.563083
$726$	0	0
$727$	7108.88	0.362660	0.181330	−	0.983422i	$-0.441960\pi$
$727$	7108.88	0.362660	0.181330	+	0.983422i	$0.441960\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20334.4	−1.02886
$732$	0	0
$733$	919.161	0.0463165	0.0231582	−	0.999732i	$-0.492628\pi$
$733$	919.161	0.0463165	0.0231582	+	0.999732i	$0.492628\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31945.4	1.59664
$738$	0	0
$739$	12028.3	0.598737	0.299369	−	0.954138i	$-0.403224\pi$
$739$	12028.3	0.598737	0.299369	+	0.954138i	$0.403224\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27701.3	1.36778	0.683890	−	0.729585i	$-0.260287\pi$
$743$	27701.3	1.36778	0.683890	+	0.729585i	$0.260287\pi$
$744$	0	0
$745$	−7696.17	−0.378478
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1285.44	−0.0627091
$750$	0	0
$751$	−29075.0	−1.41273	−0.706367	−	0.707846i	$-0.749667\pi$
$751$	−29075.0	−1.41273	−0.706367	+	0.707846i	$0.749667\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1619.42	0.0780621
$756$	0	0
$757$	5123.69	0.246002	0.123001	−	0.992407i	$-0.460748\pi$
$757$	5123.69	0.246002	0.123001	+	0.992407i	$0.460748\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25490.2	−1.21422	−0.607109	−	0.794619i	$-0.707671\pi$
$761$	−25490.2	−1.21422	−0.607109	+	0.794619i	$0.707671\pi$
$762$	0	0
$763$	−5080.20	−0.241043
$764$	0	0
$765$	0	0
$766$	0	0
$767$	68781.5	3.23801
$768$	0	0
$769$	28307.5	1.32743	0.663715	−	0.747986i	$-0.268979\pi$
$769$	28307.5	1.32743	0.663715	+	0.747986i	$0.268979\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22838.2	−1.06265	−0.531327	−	0.847167i	$-0.678306\pi$
$773$	−22838.2	−1.06265	−0.531327	+	0.847167i	$0.678306\pi$
$774$	0	0
$775$	−16543.2	−0.766773
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−903.681	−0.0415632
$780$	0	0
$781$	27432.0	1.25684
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19451.0	−0.884378
$786$	0	0
$787$	40758.7	1.84611	0.923056	−	0.384665i	$-0.125683\pi$
$787$	40758.7	1.84611	0.923056	+	0.384665i	$0.125683\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5974.42	−0.268554
$792$	0	0
$793$	55552.8	2.48769
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1949.41	0.0866395	0.0433198	−	0.999061i	$-0.486207\pi$
$797$	1949.41	0.0866395	0.0433198	+	0.999061i	$0.486207\pi$
$798$	0	0
$799$	−14904.4	−0.659923
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34641.2	−1.52237
$804$	0	0
$805$	−11085.0	−0.485335
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23875.1	1.03758	0.518792	−	0.854901i	$-0.326382\pi$
$809$	23875.1	1.03758	0.518792	+	0.854901i	$0.326382\pi$
$810$	0	0
$811$	−7453.77	−0.322734	−0.161367	−	0.986894i	$-0.551590\pi$
$811$	−7453.77	−0.322734	−0.161367	+	0.986894i	$0.551590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4132.57	−0.177617
$816$	0	0
$817$	−1233.73	−0.0528309
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33235.3	−1.41281	−0.706407	−	0.707806i	$-0.749685\pi$
$821$	−33235.3	−1.41281	−0.706407	+	0.707806i	$0.749685\pi$
$822$	0	0
$823$	43362.0	1.83658	0.918289	−	0.395910i	$-0.129571\pi$
$823$	43362.0	1.83658	0.918289	+	0.395910i	$0.129571\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20087.0	−0.844609	−0.422305	−	0.906454i	$-0.638779\pi$
$827$	−20087.0	−0.844609	−0.422305	+	0.906454i	$0.638779\pi$
$828$	0	0
$829$	−23725.2	−0.993981	−0.496990	−	0.867756i	$-0.665562\pi$
$829$	−23725.2	−0.993981	−0.496990	+	0.867756i	$0.665562\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7034.69	−0.292602
$834$	0	0
$835$	−3212.48	−0.133140
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20910.2	−0.860430	−0.430215	−	0.902726i	$-0.641562\pi$
$839$	−20910.2	−0.860430	−0.430215	+	0.902726i	$0.641562\pi$
$840$	0	0
$841$	−9157.89	−0.375493
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25799.8	1.05034
$846$	0	0
$847$	1777.06	0.0720904
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17980.6	0.724286
$852$	0	0
$853$	17522.3	0.703343	0.351672	−	0.936123i	$-0.385613\pi$
$853$	17522.3	0.703343	0.351672	+	0.936123i	$0.385613\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4314.33	−0.171966	−0.0859830	−	0.996297i	$-0.527403\pi$
$857$	−4314.33	−0.171966	−0.0859830	+	0.996297i	$0.527403\pi$
$858$	0	0
$859$	−19004.2	−0.754849	−0.377424	−	0.926040i	$-0.623190\pi$
$859$	−19004.2	−0.754849	−0.377424	+	0.926040i	$0.623190\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17542.6	0.691955	0.345978	−	0.938243i	$-0.387547\pi$
$863$	17542.6	0.691955	0.345978	+	0.938243i	$0.387547\pi$
$864$	0	0
$865$	3572.89	0.140441
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19827.4	0.773993
$870$	0	0
$871$	−73835.4	−2.87235
$872$	0	0
$873$	0	0
$874$	0	0
$875$	19989.4	0.772304
$876$	0	0
$877$	−6870.70	−0.264546	−0.132273	−	0.991213i	$-0.542228\pi$
$877$	−6870.70	−0.264546	−0.132273	+	0.991213i	$0.542228\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9708.01	−0.371250	−0.185625	−	0.982621i	$-0.559431\pi$
$881$	−9708.01	−0.371250	−0.185625	+	0.982621i	$0.559431\pi$
$882$	0	0
$883$	20123.9	0.766958	0.383479	−	0.923550i	$-0.374726\pi$
$883$	20123.9	0.766958	0.383479	+	0.923550i	$0.374726\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2479.59	0.0938630	0.0469315	−	0.998898i	$-0.485056\pi$
$887$	2479.59	0.0938630	0.0469315	+	0.998898i	$0.485056\pi$
$888$	0	0
$889$	−13010.8	−0.490852
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−904.280	−0.0338864
$894$	0	0
$895$	−17103.4	−0.638775
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−22923.0	−0.850418
$900$	0	0
$901$	39015.0	1.44259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5645.73	0.207371
$906$	0	0
$907$	23849.5	0.873109	0.436555	−	0.899678i	$-0.356199\pi$
$907$	23849.5	0.873109	0.436555	+	0.899678i	$0.356199\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3403.51	0.123780	0.0618899	−	0.998083i	$-0.480287\pi$
$911$	3403.51	0.123780	0.0618899	+	0.998083i	$0.480287\pi$
$912$	0	0
$913$	−26061.8	−0.944711
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9559.00	0.344238
$918$	0	0
$919$	−26965.8	−0.967922	−0.483961	−	0.875090i	$-0.660802\pi$
$919$	−26965.8	−0.967922	−0.483961	+	0.875090i	$0.660802\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−63403.5	−2.26105
$924$	0	0
$925$	−13490.7	−0.479537
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25920.6	−0.915421	−0.457710	−	0.889101i	$-0.651330\pi$
$929$	−25920.6	−0.915421	−0.457710	+	0.889101i	$0.651330\pi$
$930$	0	0
$931$	−426.809	−0.0150248
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14661.0	0.512798
$936$	0	0
$937$	43039.2	1.50056	0.750282	−	0.661118i	$-0.229918\pi$
$937$	43039.2	1.50056	0.750282	+	0.661118i	$0.229918\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15360.7	−0.532141	−0.266070	−	0.963954i	$-0.585725\pi$
$941$	−15360.7	−0.532141	−0.266070	+	0.963954i	$0.585725\pi$
$942$	0	0
$943$	−25219.0	−0.870884
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48082.6	−1.64992	−0.824960	−	0.565192i	$-0.808802\pi$
$947$	−48082.6	−1.64992	−0.824960	+	0.565192i	$0.808802\pi$
$948$	0	0
$949$	80066.3	2.73874
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2729.77	−0.0927869	−0.0463934	−	0.998923i	$-0.514773\pi$
$953$	−2729.77	−0.0927869	−0.0463934	+	0.998923i	$0.514773\pi$
$954$	0	0
$955$	4469.93	0.151459
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31417.6	1.05790
$960$	0	0
$961$	4708.44	0.158049
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5965.23	0.198992
$966$	0	0
$967$	−19583.3	−0.651249	−0.325625	−	0.945499i	$-0.605575\pi$
$967$	−19583.3	−0.651249	−0.325625	+	0.945499i	$0.605575\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7284.17	−0.240742	−0.120371	−	0.992729i	$-0.538408\pi$
$971$	−7284.17	−0.240742	−0.120371	+	0.992729i	$0.538408\pi$
$972$	0	0
$973$	−18009.7	−0.593385
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19756.9	−0.646961	−0.323480	−	0.946235i	$-0.604853\pi$
$977$	−19756.9	−0.646961	−0.323480	+	0.946235i	$0.604853\pi$
$978$	0	0
$979$	−35369.6	−1.15467
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14242.5	−0.462123	−0.231061	−	0.972939i	$-0.574220\pi$
$983$	−14242.5	−0.462123	−0.231061	+	0.972939i	$0.574220\pi$
$984$	0	0
$985$	21204.0	0.685905
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34429.7	−1.10698
$990$	0	0
$991$	−1791.55	−0.0574273	−0.0287136	−	0.999588i	$-0.509141\pi$
$991$	−1791.55	−0.0574273	−0.0287136	+	0.999588i	$0.509141\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27380.5	−0.872383
$996$	0	0
$997$	−31561.2	−1.00256	−0.501280	−	0.865285i	$-0.667137\pi$
$997$	−31561.2	−1.00256	−0.501280	+	0.865285i	$0.667137\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.y.1.3		4
3.2	odd	2		1296.4.a.ba.1.2		4
4.3	odd	2		648.4.a.h.1.3		4
9.2	odd	6		144.4.i.e.49.4		8
9.4	even	3		432.4.i.e.289.2		8
9.5	odd	6		144.4.i.e.97.4		8
9.7	even	3		432.4.i.e.145.2		8
12.11	even	2		648.4.a.i.1.2		4
36.7	odd	6		216.4.i.a.145.2		8
36.11	even	6		72.4.i.a.49.1	yes	8
36.23	even	6		72.4.i.a.25.1	✓	8
36.31	odd	6		216.4.i.a.73.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.a.25.1	✓	8	36.23	even	6
72.4.i.a.49.1	yes	8	36.11	even	6
144.4.i.e.49.4		8	9.2	odd	6
144.4.i.e.97.4		8	9.5	odd	6
216.4.i.a.73.2		8	36.31	odd	6
216.4.i.a.145.2		8	36.7	odd	6
432.4.i.e.145.2		8	9.7	even	3
432.4.i.e.289.2		8	9.4	even	3
648.4.a.h.1.3		4	4.3	odd	2
648.4.a.i.1.2		4	12.11	even	2
1296.4.a.y.1.3		4	1.1	even	1	trivial
1296.4.a.ba.1.2		4	3.2	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q+5.99447 q^{5} -15.5776 q^{7} +O(q^{10})\)	\(q+5.99447 q^{5} -15.5776 q^{7} +34.8844 q^{11} -80.6284 q^{13} +70.1103 q^{17} +4.25374 q^{19} +118.709 q^{23} -89.0663 q^{25} -123.414 q^{29} +185.740 q^{31} -93.3796 q^{35} +151.468 q^{37} -212.444 q^{41} -290.035 q^{43} -212.585 q^{47} -100.337 q^{49} +556.480 q^{53} +209.114 q^{55} -853.068 q^{59} -688.999 q^{61} -483.324 q^{65} +915.750 q^{67} +786.367 q^{71} -993.028 q^{73} -543.416 q^{77} +568.375 q^{79} -747.091 q^{83} +420.274 q^{85} -1013.91 q^{89} +1256.00 q^{91} +25.4989 q^{95} -1219.51 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{5} + 3 q^{7}+O(q^{10}) \) 4 * q - 5 * q^5 + 3 * q^7	\( 4 q - 5 q^{5} + 3 q^{7} - 16 q^{11} - 29 q^{13} + 17 q^{17} + 109 q^{19} - 37 q^{23} - 97 q^{25} - 3 q^{29} + 331 q^{31} - 171 q^{35} - 366 q^{37} - 378 q^{41} + 506 q^{43} - 171 q^{47} - 829 q^{49} - 410 q^{53} + 1163 q^{55} - 616 q^{59} - 1331 q^{61} - 815 q^{65} + 1162 q^{67} - 344 q^{71} - 1307 q^{73} - 741 q^{77} + 1853 q^{79} - 1421 q^{83} - 2074 q^{85} - 816 q^{89} + 1995 q^{91} - 1292 q^{95} - 2506 q^{97}+O(q^{100}) \) 4 * q - 5 * q^5 + 3 * q^7 - 16 * q^11 - 29 * q^13 + 17 * q^17 + 109 * q^19 - 37 * q^23 - 97 * q^25 - 3 * q^29 + 331 * q^31 - 171 * q^35 - 366 * q^37 - 378 * q^41 + 506 * q^43 - 171 * q^47 - 829 * q^49 - 410 * q^53 + 1163 * q^55 - 616 * q^59 - 1331 * q^61 - 815 * q^65 + 1162 * q^67 - 344 * q^71 - 1307 * q^73 - 741 * q^77 + 1853 * q^79 - 1421 * q^83 - 2074 * q^85 - 816 * q^89 + 1995 * q^91 - 1292 * q^95 - 2506 * q^97

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