LMFDB - Embedded newform 1296.4.a.s.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 162)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+0.803848 q^{5} +2.39230 q^{7} -59.5692 q^{11} +77.7461 q^{13} +2.84232 q^{17} +118.315 q^{19} -110.785 q^{23} -124.354 q^{25} +125.627 q^{29} -63.0615 q^{31} +1.92305 q^{35} -227.392 q^{37} +324.631 q^{41} +272.823 q^{43} +4.93851 q^{47} -337.277 q^{49} -598.908 q^{53} -47.8846 q^{55} +670.277 q^{59} +464.531 q^{61} +62.4960 q^{65} +769.069 q^{67} +611.569 q^{71} +923.831 q^{73} -142.508 q^{77} -39.8076 q^{79} -443.769 q^{83} +2.28479 q^{85} -78.4576 q^{89} +185.992 q^{91} +95.1075 q^{95} -91.0155 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} - 16 q^{7} - 36 q^{11} + 10 q^{13} + 120 q^{17} + 8 q^{19} - 180 q^{23} - 124 q^{25} + 324 q^{29} + 248 q^{31} - 204 q^{35} - 434 q^{37} + 192 q^{41} + 608 q^{43} + 384 q^{47} - 342 q^{49}+ \cdots - 764 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 - 16 * q^7 - 36 * q^11 + 10 * q^13 + 120 * q^17 + 8 * q^19 - 180 * q^23 - 124 * q^25 + 324 * q^29 + 248 * q^31 - 204 * q^35 - 434 * q^37 + 192 * q^41 + 608 * q^43 + 384 * q^47 - 342 * q^49 - 408 * q^53 + 216 * q^55 + 1008 * q^59 + 742 * q^61 - 696 * q^65 + 104 * q^67 + 1140 * q^71 + 850 * q^73 - 576 * q^77 + 440 * q^79 - 264 * q^83 + 1314 * q^85 + 768 * q^89 + 1432 * q^91 - 1140 * q^95 - 764 * q^97

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
$5$	0.803848	0.0718983	0.0359492	−	0.999354i	$-0.488555\pi$
$5$	0.803848	0.0718983	0.0359492	+	0.999354i	$0.488555\pi$
$6$	0	0
$7$	2.39230	0.129172	0.0645862	−	0.997912i	$-0.479427\pi$
$7$	2.39230	0.129172	0.0645862	+	0.997912i	$0.479427\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−59.5692	−1.63280	−0.816400	−	0.577487i	$-0.804033\pi$
$11$	−59.5692	−1.63280	−0.816400	+	0.577487i	$0.804033\pi$
$12$	0	0
$13$	77.7461	1.65868	0.829342	−	0.558741i	$-0.188715\pi$
$13$	77.7461	1.65868	0.829342	+	0.558741i	$0.188715\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.84232	0.0405509	0.0202754	−	0.999794i	$-0.493546\pi$
$17$	2.84232	0.0405509	0.0202754	+	0.999794i	$0.493546\pi$
$18$	0	0
$19$	118.315	1.42860	0.714300	−	0.699840i	$-0.246745\pi$
$19$	118.315	1.42860	0.714300	+	0.699840i	$0.246745\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−110.785	−1.00436	−0.502178	−	0.864764i	$-0.667468\pi$
$23$	−110.785	−1.00436	−0.502178	+	0.864764i	$0.667468\pi$
$24$	0	0
$25$	−124.354	−0.994831
$26$	0	0
$27$	0	0
$28$	0	0
$29$	125.627	0.804425	0.402213	−	0.915546i	$-0.368241\pi$
$29$	125.627	0.804425	0.402213	+	0.915546i	$0.368241\pi$
$30$	0	0
$31$	−63.0615	−0.365361	−0.182680	−	0.983172i	$-0.558477\pi$
$31$	−63.0615	−0.365361	−0.182680	+	0.983172i	$0.558477\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.92305	0.00928727
$36$	0	0
$37$	−227.392	−1.01035	−0.505177	−	0.863016i	$-0.668573\pi$
$37$	−227.392	−1.01035	−0.505177	+	0.863016i	$0.668573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	324.631	1.23656	0.618278	−	0.785959i	$-0.287831\pi$
$41$	324.631	1.23656	0.618278	+	0.785959i	$0.287831\pi$
$42$	0	0
$43$	272.823	0.967561	0.483781	−	0.875189i	$-0.339263\pi$
$43$	272.823	0.967561	0.483781	+	0.875189i	$0.339263\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.93851	0.0153267	0.00766336	−	0.999971i	$-0.497561\pi$
$47$	4.93851	0.0153267	0.00766336	+	0.999971i	$0.497561\pi$
$48$	0	0
$49$	−337.277	−0.983315
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−598.908	−1.55219	−0.776097	−	0.630614i	$-0.782803\pi$
$53$	−598.908	−1.55219	−0.776097	+	0.630614i	$0.782803\pi$
$54$	0	0
$55$	−47.8846	−0.117396
$56$	0	0
$57$	0	0
$58$	0	0
$59$	670.277	1.47903	0.739514	−	0.673142i	$-0.235056\pi$
$59$	670.277	1.47903	0.739514	+	0.673142i	$0.235056\pi$
$60$	0	0
$61$	464.531	0.975034	0.487517	−	0.873114i	$-0.337903\pi$
$61$	464.531	0.975034	0.487517	+	0.873114i	$0.337903\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	62.4960	0.119257
$66$	0	0
$67$	769.069	1.40234	0.701170	−	0.712994i	$-0.252662\pi$
$67$	769.069	1.40234	0.701170	+	0.712994i	$0.252662\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	611.569	1.02225	0.511126	−	0.859506i	$-0.329228\pi$
$71$	611.569	1.02225	0.511126	+	0.859506i	$0.329228\pi$
$72$	0	0
$73$	923.831	1.48118	0.740590	−	0.671957i	$-0.234546\pi$
$73$	923.831	1.48118	0.740590	+	0.671957i	$0.234546\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−142.508	−0.210913
$78$	0	0
$79$	−39.8076	−0.0566925	−0.0283462	−	0.999598i	$-0.509024\pi$
$79$	−39.8076	−0.0566925	−0.0283462	+	0.999598i	$0.509024\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−443.769	−0.586867	−0.293434	−	0.955979i	$-0.594798\pi$
$83$	−443.769	−0.586867	−0.293434	+	0.955979i	$0.594798\pi$
$84$	0	0
$85$	2.28479	0.00291554
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−78.4576	−0.0934437	−0.0467218	−	0.998908i	$-0.514877\pi$
$89$	−78.4576	−0.0934437	−0.0467218	+	0.998908i	$0.514877\pi$
$90$	0	0
$91$	185.992	0.214256
$92$	0	0
$93$	0	0
$94$	0	0
$95$	95.1075	0.102714
$96$	0	0
$97$	−91.0155	−0.0952703	−0.0476352	−	0.998865i	$-0.515168\pi$
$97$	−91.0155	−0.0952703	−0.0476352	+	0.998865i	$0.515168\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1622.52	−1.59849	−0.799243	−	0.601008i	$-0.794766\pi$
$101$	−1622.52	−1.59849	−0.799243	+	0.601008i	$0.794766\pi$
$102$	0	0
$103$	748.231	0.715780	0.357890	−	0.933764i	$-0.383496\pi$
$103$	748.231	0.715780	0.357890	+	0.933764i	$0.383496\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1435.00	−1.29651	−0.648255	−	0.761423i	$-0.724501\pi$
$107$	−1435.00	−1.29651	−0.648255	+	0.761423i	$0.724501\pi$
$108$	0	0
$109$	−83.1305	−0.0730501	−0.0365250	−	0.999333i	$-0.511629\pi$
$109$	−83.1305	−0.0730501	−0.0365250	+	0.999333i	$0.511629\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	972.196	0.809349	0.404675	−	0.914461i	$-0.367385\pi$
$113$	972.196	0.809349	0.404675	+	0.914461i	$0.367385\pi$
$114$	0	0
$115$	−89.0539	−0.0722115
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.79970	0.00523805
$120$	0	0
$121$	2217.49	1.66603
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−200.442	−0.143425
$126$	0	0
$127$	1316.13	0.919588	0.459794	−	0.888026i	$-0.347923\pi$
$127$	1316.13	0.919588	0.459794	+	0.888026i	$0.347923\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1115.28	0.743833	0.371917	−	0.928266i	$-0.378701\pi$
$131$	1115.28	0.743833	0.371917	+	0.928266i	$0.378701\pi$
$132$	0	0
$133$	283.046	0.184536
$134$	0	0
$135$	0	0
$136$	0	0
$137$	617.143	0.384862	0.192431	−	0.981311i	$-0.438363\pi$
$137$	617.143	0.384862	0.192431	+	0.981311i	$0.438363\pi$
$138$	0	0
$139$	−2156.57	−1.31596	−0.657978	−	0.753037i	$-0.728588\pi$
$139$	−2156.57	−1.31596	−0.657978	+	0.753037i	$0.728588\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4631.28	−2.70830
$144$	0	0
$145$	100.985	0.0578368
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3571.69	1.96379	0.981893	−	0.189437i	$-0.0606662\pi$
$149$	3571.69	1.96379	0.981893	+	0.189437i	$0.0606662\pi$
$150$	0	0
$151$	2261.43	1.21876	0.609379	−	0.792879i	$-0.291419\pi$
$151$	2261.43	1.21876	0.609379	+	0.792879i	$0.291419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−50.6918	−0.0262688
$156$	0	0
$157$	1829.25	0.929875	0.464937	−	0.885344i	$-0.346077\pi$
$157$	1829.25	0.929875	0.464937	+	0.885344i	$0.346077\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−265.031	−0.129735
$162$	0	0
$163$	−268.554	−0.129048	−0.0645238	−	0.997916i	$-0.520553\pi$
$163$	−268.554	−0.129048	−0.0645238	+	0.997916i	$0.520553\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4054.52	1.87873	0.939366	−	0.342915i	$-0.111414\pi$
$167$	4054.52	1.87873	0.939366	+	0.342915i	$0.111414\pi$
$168$	0	0
$169$	3847.46	1.75123
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1747.35	0.767911	0.383955	−	0.923352i	$-0.374562\pi$
$173$	1747.35	0.767911	0.383955	+	0.923352i	$0.374562\pi$
$174$	0	0
$175$	−297.492	−0.128505
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2037.71	0.850868	0.425434	−	0.904989i	$-0.360121\pi$
$179$	2037.71	0.850868	0.425434	+	0.904989i	$0.360121\pi$
$180$	0	0
$181$	2820.68	1.15834	0.579169	−	0.815207i	$-0.303377\pi$
$181$	2820.68	1.15834	0.579169	+	0.815207i	$0.303377\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−182.789	−0.0726427
$186$	0	0
$187$	−169.315	−0.0662114
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−439.723	−0.166582	−0.0832912	−	0.996525i	$-0.526543\pi$
$191$	−439.723	−0.166582	−0.0832912	+	0.996525i	$0.526543\pi$
$192$	0	0
$193$	−462.676	−0.172560	−0.0862802	−	0.996271i	$-0.527498\pi$
$193$	−462.676	−0.172560	−0.0862802	+	0.996271i	$0.527498\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1036.60	0.374896	0.187448	−	0.982275i	$-0.439978\pi$
$197$	1036.60	0.374896	0.187448	+	0.982275i	$0.439978\pi$
$198$	0	0
$199$	1190.22	0.423980	0.211990	−	0.977272i	$-0.432006\pi$
$199$	1190.22	0.423980	0.211990	+	0.977272i	$0.432006\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	300.538	0.103909
$204$	0	0
$205$	260.954	0.0889063
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7047.95	−2.33262
$210$	0	0
$211$	4928.01	1.60786	0.803929	−	0.594725i	$-0.202739\pi$
$211$	4928.01	1.60786	0.803929	+	0.594725i	$0.202739\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	219.308	0.0695660
$216$	0	0
$217$	−150.862	−0.0471945
$218$	0	0
$219$	0	0
$220$	0	0
$221$	220.980	0.0672611
$222$	0	0
$223$	1470.47	0.441569	0.220785	−	0.975323i	$-0.429138\pi$
$223$	1470.47	0.441569	0.220785	+	0.975323i	$0.429138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2102.26	−0.614678	−0.307339	−	0.951600i	$-0.599439\pi$
$227$	−2102.26	−0.614678	−0.307339	+	0.951600i	$0.599439\pi$
$228$	0	0
$229$	−1001.04	−0.288867	−0.144433	−	0.989515i	$-0.546136\pi$
$229$	−1001.04	−0.288867	−0.144433	+	0.989515i	$0.546136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6280.78	−1.76596	−0.882978	−	0.469415i	$-0.844465\pi$
$233$	−6280.78	−1.76596	−0.882978	+	0.469415i	$0.844465\pi$
$234$	0	0
$235$	3.96981	0.00110197
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1017.37	0.275348	0.137674	−	0.990478i	$-0.456037\pi$
$239$	1017.37	0.275348	0.137674	+	0.990478i	$0.456037\pi$
$240$	0	0
$241$	−921.292	−0.246247	−0.123124	−	0.992391i	$-0.539291\pi$
$241$	−921.292	−0.246247	−0.123124	+	0.992391i	$0.539291\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−271.119	−0.0706987
$246$	0	0
$247$	9198.56	2.36960
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2262.40	−0.568930	−0.284465	−	0.958686i	$-0.591816\pi$
$251$	−2262.40	−0.568930	−0.284465	+	0.958686i	$0.591816\pi$
$252$	0	0
$253$	6599.35	1.63991
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4786.65	1.16180	0.580901	−	0.813974i	$-0.302700\pi$
$257$	4786.65	1.16180	0.580901	+	0.813974i	$0.302700\pi$
$258$	0	0
$259$	−543.992	−0.130510
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3007.92	0.705234	0.352617	−	0.935768i	$-0.385292\pi$
$263$	3007.92	0.705234	0.352617	+	0.935768i	$0.385292\pi$
$264$	0	0
$265$	−481.430	−0.111600
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6559.73	−1.48682	−0.743409	−	0.668837i	$-0.766792\pi$
$269$	−6559.73	−1.48682	−0.743409	+	0.668837i	$0.766792\pi$
$270$	0	0
$271$	5147.64	1.15386	0.576931	−	0.816793i	$-0.304250\pi$
$271$	5147.64	1.15386	0.576931	+	0.816793i	$0.304250\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7407.66	1.62436
$276$	0	0
$277$	−1067.66	−0.231587	−0.115793	−	0.993273i	$-0.536941\pi$
$277$	−1067.66	−0.231587	−0.115793	+	0.993273i	$0.536941\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4186.47	0.888769	0.444384	−	0.895836i	$-0.353422\pi$
$281$	4186.47	0.888769	0.444384	+	0.895836i	$0.353422\pi$
$282$	0	0
$283$	−5356.95	−1.12522	−0.562611	−	0.826722i	$-0.690203\pi$
$283$	−5356.95	−1.12522	−0.562611	+	0.826722i	$0.690203\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	776.616	0.159729
$288$	0	0
$289$	−4904.92	−0.998356
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2895.79	0.577385	0.288692	−	0.957422i	$-0.406780\pi$
$293$	2895.79	0.577385	0.288692	+	0.957422i	$0.406780\pi$
$294$	0	0
$295$	538.800	0.106340
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8613.08	−1.66591
$300$	0	0
$301$	652.676	0.124982
$302$	0	0
$303$	0	0
$304$	0	0
$305$	373.412	0.0701033
$306$	0	0
$307$	−1855.49	−0.344947	−0.172473	−	0.985014i	$-0.555176\pi$
$307$	−1855.49	−0.344947	−0.172473	+	0.985014i	$0.555176\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7171.69	−1.30762	−0.653809	−	0.756659i	$-0.726830\pi$
$311$	−7171.69	−1.30762	−0.653809	+	0.756659i	$0.726830\pi$
$312$	0	0
$313$	−1310.83	−0.236717	−0.118359	−	0.992971i	$-0.537763\pi$
$313$	−1310.83	−0.236717	−0.118359	+	0.992971i	$0.537763\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3689.26	0.653658	0.326829	−	0.945083i	$-0.394020\pi$
$317$	3689.26	0.653658	0.326829	+	0.945083i	$0.394020\pi$
$318$	0	0
$319$	−7483.50	−1.31347
$320$	0	0
$321$	0	0
$322$	0	0
$323$	336.290	0.0579310
$324$	0	0
$325$	−9668.03	−1.65011
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.8144	0.00197979
$330$	0	0
$331$	3244.96	0.538849	0.269425	−	0.963021i	$-0.413166\pi$
$331$	3244.96	0.538849	0.269425	+	0.963021i	$0.413166\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	618.214	0.100826
$336$	0	0
$337$	4301.66	0.695331	0.347665	−	0.937619i	$-0.386975\pi$
$337$	4301.66	0.695331	0.347665	+	0.937619i	$0.386975\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3756.52	0.596561
$342$	0	0
$343$	−1627.43	−0.256189
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3304.20	0.511178	0.255589	−	0.966786i	$-0.417731\pi$
$347$	3304.20	0.511178	0.255589	+	0.966786i	$0.417731\pi$
$348$	0	0
$349$	4508.71	0.691535	0.345767	−	0.938320i	$-0.387619\pi$
$349$	4508.71	0.691535	0.345767	+	0.938320i	$0.387619\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6886.74	1.03837	0.519184	−	0.854662i	$-0.326236\pi$
$353$	6886.74	1.03837	0.519184	+	0.854662i	$0.326236\pi$
$354$	0	0
$355$	491.608	0.0734982
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1529.31	0.224829	0.112415	−	0.993661i	$-0.464141\pi$
$359$	1529.31	0.224829	0.112415	+	0.993661i	$0.464141\pi$
$360$	0	0
$361$	7139.52	1.04090
$362$	0	0
$363$	0	0
$364$	0	0
$365$	742.619	0.106494
$366$	0	0
$367$	10231.4	1.45524	0.727620	−	0.685980i	$-0.240626\pi$
$367$	10231.4	1.45524	0.727620	+	0.685980i	$0.240626\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1432.77	−0.200501
$372$	0	0
$373$	946.430	0.131379	0.0656894	−	0.997840i	$-0.479075\pi$
$373$	946.430	0.131379	0.0656894	+	0.997840i	$0.479075\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9767.01	1.33429
$378$	0	0
$379$	8717.46	1.18149	0.590746	−	0.806857i	$-0.298833\pi$
$379$	8717.46	1.18149	0.590746	+	0.806857i	$0.298833\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12261.5	−1.63586	−0.817931	−	0.575316i	$-0.804879\pi$
$383$	−12261.5	−1.63586	−0.817931	+	0.575316i	$0.804879\pi$
$384$	0	0
$385$	−114.554	−0.0151643
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13211.1	−1.72192	−0.860961	−	0.508672i	$-0.830137\pi$
$389$	−13211.1	−1.72192	−0.860961	+	0.508672i	$0.830137\pi$
$390$	0	0
$391$	−314.886	−0.0407275
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−31.9993	−0.00407609
$396$	0	0
$397$	−5211.93	−0.658890	−0.329445	−	0.944175i	$-0.606862\pi$
$397$	−5211.93	−0.658890	−0.329445	+	0.944175i	$0.606862\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	985.634	0.122744	0.0613719	−	0.998115i	$-0.480452\pi$
$401$	985.634	0.122744	0.0613719	+	0.998115i	$0.480452\pi$
$402$	0	0
$403$	−4902.79	−0.606018
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13545.6	1.64970
$408$	0	0
$409$	−674.647	−0.0815627	−0.0407813	−	0.999168i	$-0.512985\pi$
$409$	−674.647	−0.0815627	−0.0407813	+	0.999168i	$0.512985\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1603.51	0.191049
$414$	0	0
$415$	−356.723	−0.0421948
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7505.63	0.875117	0.437559	−	0.899190i	$-0.355843\pi$
$419$	7505.63	0.875117	0.437559	+	0.899190i	$0.355843\pi$
$420$	0	0
$421$	−1352.62	−0.156586	−0.0782932	−	0.996930i	$-0.524947\pi$
$421$	−1352.62	−0.156586	−0.0782932	+	0.996930i	$0.524947\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−353.454	−0.0403412
$426$	0	0
$427$	1111.30	0.125947
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4003.21	−0.447397	−0.223698	−	0.974658i	$-0.571813\pi$
$431$	−4003.21	−0.447397	−0.223698	+	0.974658i	$0.571813\pi$
$432$	0	0
$433$	9975.38	1.10713	0.553564	−	0.832807i	$-0.313267\pi$
$433$	9975.38	1.10713	0.553564	+	0.832807i	$0.313267\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13107.5	−1.43482
$438$	0	0
$439$	−17360.9	−1.88745	−0.943726	−	0.330728i	$-0.892706\pi$
$439$	−17360.9	−1.88745	−0.943726	+	0.330728i	$0.892706\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8575.43	−0.919709	−0.459854	−	0.887994i	$-0.652098\pi$
$443$	−8575.43	−0.919709	−0.459854	+	0.887994i	$0.652098\pi$
$444$	0	0
$445$	−63.0679	−0.00671844
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4703.77	−0.494398	−0.247199	−	0.968965i	$-0.579510\pi$
$449$	−4703.77	−0.494398	−0.247199	+	0.968965i	$0.579510\pi$
$450$	0	0
$451$	−19338.0	−2.01905
$452$	0	0
$453$	0	0
$454$	0	0
$455$	149.510	0.0154047
$456$	0	0
$457$	−7046.80	−0.721303	−0.360651	−	0.932701i	$-0.617446\pi$
$457$	−7046.80	−0.721303	−0.360651	+	0.932701i	$0.617446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5770.26	−0.582967	−0.291484	−	0.956576i	$-0.594149\pi$
$461$	−5770.26	−0.582967	−0.291484	+	0.956576i	$0.594149\pi$
$462$	0	0
$463$	13599.1	1.36502	0.682511	−	0.730875i	$-0.260888\pi$
$463$	13599.1	1.36502	0.682511	+	0.730875i	$0.260888\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5730.52	0.567831	0.283915	−	0.958849i	$-0.408367\pi$
$467$	5730.52	0.567831	0.283915	+	0.958849i	$0.408367\pi$
$468$	0	0
$469$	1839.85	0.181143
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16251.9	−1.57983
$474$	0	0
$475$	−14713.0	−1.42122
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14568.3	1.38965	0.694824	−	0.719180i	$-0.255482\pi$
$479$	14568.3	1.38965	0.694824	+	0.719180i	$0.255482\pi$
$480$	0	0
$481$	−17678.9	−1.67586
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−73.1626	−0.00684977
$486$	0	0
$487$	−7164.51	−0.666642	−0.333321	−	0.942813i	$-0.608169\pi$
$487$	−7164.51	−0.666642	−0.333321	+	0.942813i	$0.608169\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9510.06	−0.874100	−0.437050	−	0.899437i	$-0.643977\pi$
$491$	−9510.06	−0.874100	−0.437050	+	0.899437i	$0.643977\pi$
$492$	0	0
$493$	357.072	0.0326201
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1463.06	0.132047
$498$	0	0
$499$	−21485.6	−1.92751	−0.963756	−	0.266785i	$-0.914039\pi$
$499$	−21485.6	−1.92751	−0.963756	+	0.266785i	$0.914039\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11603.2	−1.02855	−0.514274	−	0.857626i	$-0.671939\pi$
$503$	−11603.2	−1.02855	−0.514274	+	0.857626i	$0.671939\pi$
$504$	0	0
$505$	−1304.26	−0.114928
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8939.74	−0.778481	−0.389241	−	0.921136i	$-0.627262\pi$
$509$	−8939.74	−0.778481	−0.389241	+	0.921136i	$0.627262\pi$
$510$	0	0
$511$	2210.08	0.191328
$512$	0	0
$513$	0	0
$514$	0	0
$515$	601.464	0.0514634
$516$	0	0
$517$	−294.183	−0.0250255
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7410.10	0.623114	0.311557	−	0.950227i	$-0.399149\pi$
$521$	7410.10	0.623114	0.311557	+	0.950227i	$0.399149\pi$
$522$	0	0
$523$	−18970.8	−1.58611	−0.793054	−	0.609151i	$-0.791510\pi$
$523$	−18970.8	−1.58611	−0.793054	+	0.609151i	$0.791510\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−179.241	−0.0148157
$528$	0	0
$529$	106.230	0.00873097
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25238.8	2.05106
$534$	0	0
$535$	−1153.52	−0.0932169
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20091.3	1.60556
$540$	0	0
$541$	−11440.4	−0.909169	−0.454584	−	0.890704i	$-0.650212\pi$
$541$	−11440.4	−0.909169	−0.454584	+	0.890704i	$0.650212\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−66.8243	−0.00525218
$546$	0	0
$547$	−10253.0	−0.801434	−0.400717	−	0.916202i	$-0.631239\pi$
$547$	−10253.0	−0.801434	−0.400717	+	0.916202i	$0.631239\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14863.6	1.14920
$552$	0	0
$553$	−95.2320	−0.00732310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15336.5	1.16666	0.583328	−	0.812236i	$-0.301750\pi$
$557$	15336.5	1.16666	0.583328	+	0.812236i	$0.301750\pi$
$558$	0	0
$559$	21210.9	1.60488
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7701.37	−0.576508	−0.288254	−	0.957554i	$-0.593075\pi$
$563$	−7701.37	−0.576508	−0.288254	+	0.957554i	$0.593075\pi$
$564$	0	0
$565$	781.497	0.0581909
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17739.3	−1.30697	−0.653487	−	0.756938i	$-0.726695\pi$
$569$	−17739.3	−1.30697	−0.653487	+	0.756938i	$0.726695\pi$
$570$	0	0
$571$	−13823.1	−1.01310	−0.506549	−	0.862211i	$-0.669079\pi$
$571$	−13823.1	−1.01310	−0.506549	+	0.862211i	$0.669079\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13776.5	0.999164
$576$	0	0
$577$	−15339.1	−1.10672	−0.553359	−	0.832943i	$-0.686654\pi$
$577$	−15339.1	−1.10672	−0.553359	+	0.832943i	$0.686654\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1061.63	−0.0758070
$582$	0	0
$583$	35676.5	2.53442
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14766.5	−1.03830	−0.519148	−	0.854684i	$-0.673751\pi$
$587$	−14766.5	−1.03830	−0.519148	+	0.854684i	$0.673751\pi$
$588$	0	0
$589$	−7461.14	−0.521954
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17603.6	1.21904	0.609522	−	0.792769i	$-0.291361\pi$
$593$	17603.6	1.21904	0.609522	+	0.792769i	$0.291361\pi$
$594$	0	0
$595$	5.46593	0.000376607 0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22801.1	−1.55531	−0.777653	−	0.628694i	$-0.783590\pi$
$599$	−22801.1	−1.55531	−0.777653	+	0.628694i	$0.783590\pi$
$600$	0	0
$601$	1905.25	0.129312	0.0646562	−	0.997908i	$-0.479405\pi$
$601$	1905.25	0.129312	0.0646562	+	0.997908i	$0.479405\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1782.53	0.119785
$606$	0	0
$607$	2589.22	0.173136	0.0865678	−	0.996246i	$-0.472410\pi$
$607$	2589.22	0.173136	0.0865678	+	0.996246i	$0.472410\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	383.950	0.0254222
$612$	0	0
$613$	16930.4	1.11552	0.557759	−	0.830003i	$-0.311661\pi$
$613$	16930.4	1.11552	0.557759	+	0.830003i	$0.311661\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12813.4	0.836061	0.418030	−	0.908433i	$-0.362721\pi$
$617$	12813.4	0.836061	0.418030	+	0.908433i	$0.362721\pi$
$618$	0	0
$619$	−10267.6	−0.666703	−0.333351	−	0.942803i	$-0.608180\pi$
$619$	−10267.6	−0.666703	−0.333351	+	0.942803i	$0.608180\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−187.694	−0.0120703
$624$	0	0
$625$	15383.1	0.984519
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−646.322	−0.0409707
$630$	0	0
$631$	21887.5	1.38087	0.690433	−	0.723397i	$-0.257420\pi$
$631$	21887.5	1.38087	0.690433	+	0.723397i	$0.257420\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1057.97	0.0661168
$636$	0	0
$637$	−26222.0	−1.63101
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17203.7	1.06007	0.530035	−	0.847976i	$-0.322179\pi$
$641$	17203.7	1.06007	0.530035	+	0.847976i	$0.322179\pi$
$642$	0	0
$643$	−12565.6	−0.770666	−0.385333	−	0.922778i	$-0.625913\pi$
$643$	−12565.6	−0.770666	−0.385333	+	0.922778i	$0.625913\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3552.55	−0.215866	−0.107933	−	0.994158i	$-0.534423\pi$
$647$	−3552.55	−0.215866	−0.107933	+	0.994158i	$0.534423\pi$
$648$	0	0
$649$	−39927.9	−2.41496
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5771.00	0.345845	0.172922	−	0.984935i	$-0.444679\pi$
$653$	5771.00	0.345845	0.172922	+	0.984935i	$0.444679\pi$
$654$	0	0
$655$	896.512	0.0534804
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16478.4	−0.974063	−0.487032	−	0.873384i	$-0.661920\pi$
$659$	−16478.4	−0.974063	−0.487032	+	0.873384i	$0.661920\pi$
$660$	0	0
$661$	−4563.24	−0.268516	−0.134258	−	0.990946i	$-0.542865\pi$
$661$	−4563.24	−0.268516	−0.134258	+	0.990946i	$0.542865\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	227.526	0.0132678
$666$	0	0
$667$	−13917.5	−0.807929
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27671.7	−1.59203
$672$	0	0
$673$	−13152.3	−0.753318	−0.376659	−	0.926352i	$-0.622927\pi$
$673$	−13152.3	−0.753318	−0.376659	+	0.926352i	$0.622927\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14876.1	0.844515	0.422257	−	0.906476i	$-0.361238\pi$
$677$	14876.1	0.844515	0.422257	+	0.906476i	$0.361238\pi$
$678$	0	0
$679$	−217.737	−0.0123063
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24082.4	1.34918	0.674588	−	0.738195i	$-0.264321\pi$
$683$	24082.4	1.34918	0.674588	+	0.738195i	$0.264321\pi$
$684$	0	0
$685$	496.089	0.0276709
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−46562.7	−2.57460
$690$	0	0
$691$	−12890.2	−0.709648	−0.354824	−	0.934933i	$-0.615459\pi$
$691$	−12890.2	−0.709648	−0.354824	+	0.934933i	$0.615459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1733.55	−0.0946150
$696$	0	0
$697$	922.705	0.0501434
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11887.0	−0.640467	−0.320233	−	0.947339i	$-0.603761\pi$
$701$	−11887.0	−0.640467	−0.320233	+	0.947339i	$0.603761\pi$
$702$	0	0
$703$	−26904.0	−1.44339
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3881.57	−0.206480
$708$	0	0
$709$	5814.43	0.307991	0.153996	−	0.988072i	$-0.450786\pi$
$709$	5814.43	0.307991	0.153996	+	0.988072i	$0.450786\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6986.24	0.366952
$714$	0	0
$715$	−3722.84	−0.194722
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4488.23	−0.232800	−0.116400	−	0.993202i	$-0.537135\pi$
$719$	−4488.23	−0.232800	−0.116400	+	0.993202i	$0.537135\pi$
$720$	0	0
$721$	1790.00	0.0924590
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15622.2	−0.800267
$726$	0	0
$727$	−23184.0	−1.18273	−0.591366	−	0.806403i	$-0.701411\pi$
$727$	−23184.0	−1.18273	−0.591366	+	0.806403i	$0.701411\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	775.451	0.0392354
$732$	0	0
$733$	−20677.4	−1.04193	−0.520967	−	0.853577i	$-0.674429\pi$
$733$	−20677.4	−1.04193	−0.520967	+	0.853577i	$0.674429\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−45812.8	−2.28974
$738$	0	0
$739$	13001.7	0.647191	0.323595	−	0.946196i	$-0.395108\pi$
$739$	13001.7	0.647191	0.323595	+	0.946196i	$0.395108\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9267.09	−0.457573	−0.228786	−	0.973477i	$-0.573476\pi$
$743$	−9267.09	−0.457573	−0.228786	+	0.973477i	$0.573476\pi$
$744$	0	0
$745$	2871.09	0.141193
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3432.96	−0.167473
$750$	0	0
$751$	30722.3	1.49277	0.746385	−	0.665514i	$-0.231788\pi$
$751$	30722.3	1.49277	0.746385	+	0.665514i	$0.231788\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1817.85	0.0876267
$756$	0	0
$757$	5356.86	0.257197	0.128599	−	0.991697i	$-0.458952\pi$
$757$	5356.86	0.257197	0.128599	+	0.991697i	$0.458952\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10019.0	0.477252	0.238626	−	0.971111i	$-0.423303\pi$
$761$	10019.0	0.477252	0.238626	+	0.971111i	$0.423303\pi$
$762$	0	0
$763$	−198.874	−0.00943605
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52111.4	2.45324
$768$	0	0
$769$	−17551.1	−0.823029	−0.411514	−	0.911403i	$-0.635000\pi$
$769$	−17551.1	−0.823029	−0.411514	+	0.911403i	$0.635000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17144.5	0.797731	0.398866	−	0.917009i	$-0.369404\pi$
$773$	17144.5	0.797731	0.398866	+	0.917009i	$0.369404\pi$
$774$	0	0
$775$	7841.94	0.363472
$776$	0	0
$777$	0	0
$778$	0	0
$779$	38408.8	1.76654
$780$	0	0
$781$	−36430.7	−1.66913
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1470.44	0.0668564
$786$	0	0
$787$	8595.46	0.389320	0.194660	−	0.980871i	$-0.437640\pi$
$787$	8595.46	0.389320	0.194660	+	0.980871i	$0.437640\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2325.79	0.104546
$792$	0	0
$793$	36115.5	1.61727
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20837.8	0.926115	0.463057	−	0.886328i	$-0.346752\pi$
$797$	20837.8	0.926115	0.463057	+	0.886328i	$0.346752\pi$
$798$	0	0
$799$	14.0369	0.000621512 0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−55031.9	−2.41847
$804$	0	0
$805$	−213.044	−0.00932773
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6529.43	0.283761	0.141880	−	0.989884i	$-0.454685\pi$
$809$	6529.43	0.283761	0.141880	+	0.989884i	$0.454685\pi$
$810$	0	0
$811$	−29764.0	−1.28873	−0.644363	−	0.764720i	$-0.722877\pi$
$811$	−29764.0	−1.28873	−0.644363	+	0.764720i	$0.722877\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−215.876	−0.00927830
$816$	0	0
$817$	32279.2	1.38226
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29571.2	−1.25705	−0.628527	−	0.777788i	$-0.716342\pi$
$821$	−29571.2	−1.25705	−0.628527	+	0.777788i	$0.716342\pi$
$822$	0	0
$823$	−6112.43	−0.258889	−0.129445	−	0.991587i	$-0.541319\pi$
$823$	−6112.43	−0.258889	−0.129445	+	0.991587i	$0.541319\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3265.87	0.137322	0.0686612	−	0.997640i	$-0.478127\pi$
$827$	3265.87	0.137322	0.0686612	+	0.997640i	$0.478127\pi$
$828$	0	0
$829$	−19327.8	−0.809747	−0.404874	−	0.914373i	$-0.632685\pi$
$829$	−19327.8	−0.809747	−0.404874	+	0.914373i	$0.632685\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−958.650	−0.0398743
$834$	0	0
$835$	3259.22	0.135078
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37638.1	1.54876	0.774382	−	0.632719i	$-0.218061\pi$
$839$	37638.1	1.54876	0.774382	+	0.632719i	$0.218061\pi$
$840$	0	0
$841$	−8606.87	−0.352900
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3092.77	0.125911
$846$	0	0
$847$	5304.92	0.215206
$848$	0	0
$849$	0	0
$850$	0	0
$851$	25191.6	1.01475
$852$	0	0
$853$	41869.6	1.68064	0.840322	−	0.542088i	$-0.182366\pi$
$853$	41869.6	1.68064	0.840322	+	0.542088i	$0.182366\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33072.9	−1.31826	−0.659131	−	0.752029i	$-0.729076\pi$
$857$	−33072.9	−1.31826	−0.659131	+	0.752029i	$0.729076\pi$
$858$	0	0
$859$	14609.7	0.580298	0.290149	−	0.956981i	$-0.406295\pi$
$859$	14609.7	0.580298	0.290149	+	0.956981i	$0.406295\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	118.786	0.00468543	0.00234271	−	0.999997i	$-0.499254\pi$
$863$	118.786	0.00468543	0.00234271	+	0.999997i	$0.499254\pi$
$864$	0	0
$865$	1404.60	0.0552115
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2371.31	0.0925675
$870$	0	0
$871$	59792.1	2.32604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−479.519	−0.0185265
$876$	0	0
$877$	−6360.18	−0.244889	−0.122445	−	0.992475i	$-0.539073\pi$
$877$	−6360.18	−0.244889	−0.122445	+	0.992475i	$0.539073\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32601.5	1.24673	0.623367	−	0.781930i	$-0.285764\pi$
$881$	32601.5	1.24673	0.623367	+	0.781930i	$0.285764\pi$
$882$	0	0
$883$	−1478.26	−0.0563392	−0.0281696	−	0.999603i	$-0.508968\pi$
$883$	−1478.26	−0.0563392	−0.0281696	+	0.999603i	$0.508968\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45267.1	−1.71355	−0.856776	−	0.515688i	$-0.827536\pi$
$887$	−45267.1	−1.71355	−0.856776	+	0.515688i	$0.827536\pi$
$888$	0	0
$889$	3148.59	0.118785
$890$	0	0
$891$	0	0
$892$	0	0
$893$	584.302	0.0218958
$894$	0	0
$895$	1638.01	0.0611760
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7922.22	−0.293905
$900$	0	0
$901$	−1702.29	−0.0629428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2267.39	0.0832826
$906$	0	0
$907$	−7261.52	−0.265838	−0.132919	−	0.991127i	$-0.542435\pi$
$907$	−7261.52	−0.265838	−0.132919	+	0.991127i	$0.542435\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19434.4	0.706795	0.353398	−	0.935473i	$-0.385026\pi$
$911$	19434.4	0.706795	0.353398	+	0.935473i	$0.385026\pi$
$912$	0	0
$913$	26435.0	0.958237
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2668.08	0.0960827
$918$	0	0
$919$	24010.2	0.861830	0.430915	−	0.902392i	$-0.358191\pi$
$919$	24010.2	0.861830	0.430915	+	0.902392i	$0.358191\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	47547.1	1.69559
$924$	0	0
$925$	28277.1	1.00513
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41554.6	1.46756	0.733780	−	0.679387i	$-0.237754\pi$
$929$	41554.6	1.46756	0.733780	+	0.679387i	$0.237754\pi$
$930$	0	0
$931$	−39905.0	−1.40476
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−136.103	−0.00476049
$936$	0	0
$937$	16811.0	0.586116	0.293058	−	0.956095i	$-0.405327\pi$
$937$	16811.0	0.586116	0.293058	+	0.956095i	$0.405327\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10732.9	−0.371819	−0.185909	−	0.982567i	$-0.559523\pi$
$941$	−10732.9	−0.371819	−0.185909	+	0.982567i	$0.559523\pi$
$942$	0	0
$943$	−35964.1	−1.24194
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31549.3	−1.08259	−0.541297	−	0.840832i	$-0.682066\pi$
$947$	−31549.3	−1.08259	−0.541297	+	0.840832i	$0.682066\pi$
$948$	0	0
$949$	71824.3	2.45681
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31955.2	1.08618	0.543090	−	0.839675i	$-0.317254\pi$
$953$	31955.2	1.08618	0.543090	+	0.839675i	$0.317254\pi$
$954$	0	0
$955$	−353.470	−0.0119770
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1476.39	0.0497135
$960$	0	0
$961$	−25814.2	−0.866512
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−371.921	−0.0124068
$966$	0	0
$967$	−38372.2	−1.27608	−0.638038	−	0.770005i	$-0.720254\pi$
$967$	−38372.2	−1.27608	−0.638038	+	0.770005i	$0.720254\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46445.9	−1.53503	−0.767517	−	0.641028i	$-0.778508\pi$
$971$	−46445.9	−1.53503	−0.767517	+	0.641028i	$0.778508\pi$
$972$	0	0
$973$	−5159.17	−0.169985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47908.3	1.56880	0.784402	−	0.620253i	$-0.212970\pi$
$977$	47908.3	1.56880	0.784402	+	0.620253i	$0.212970\pi$
$978$	0	0
$979$	4673.66	0.152575
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40547.7	−1.31564	−0.657818	−	0.753177i	$-0.728520\pi$
$983$	−40547.7	−1.31564	−0.657818	+	0.753177i	$0.728520\pi$
$984$	0	0
$985$	833.265	0.0269544
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30224.6	−0.971776
$990$	0	0
$991$	10651.5	0.341428	0.170714	−	0.985321i	$-0.445393\pi$
$991$	10651.5	0.341428	0.170714	+	0.985321i	$0.445393\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	956.751	0.0304835
$996$	0	0
$997$	−18758.4	−0.595871	−0.297935	−	0.954586i	$-0.596298\pi$
$997$	−18758.4	−0.595871	−0.297935	+	0.954586i	$0.596298\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.s.1.1		2
3.2	odd	2		1296.4.a.j.1.2		2
4.3	odd	2		162.4.a.h.1.1	yes	2
12.11	even	2		162.4.a.e.1.2	✓	2
36.7	odd	6		162.4.c.i.109.2		4
36.11	even	6		162.4.c.j.109.1		4
36.23	even	6		162.4.c.j.55.1		4
36.31	odd	6		162.4.c.i.55.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.2	✓	2	12.11	even	2
162.4.a.h.1.1	yes	2	4.3	odd	2
162.4.c.i.55.2		4	36.31	odd	6
162.4.c.i.109.2		4	36.7	odd	6
162.4.c.j.55.1		4	36.23	even	6
162.4.c.j.109.1		4	36.11	even	6
1296.4.a.j.1.2		2	3.2	odd	2
1296.4.a.s.1.1		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.803848 q^{5} +2.39230 q^{7} -59.5692 q^{11} +77.7461 q^{13} +2.84232 q^{17} +118.315 q^{19} -110.785 q^{23} -124.354 q^{25} +125.627 q^{29} -63.0615 q^{31} +1.92305 q^{35} -227.392 q^{37} +324.631 q^{41} +272.823 q^{43} +4.93851 q^{47} -337.277 q^{49} -598.908 q^{53} -47.8846 q^{55} +670.277 q^{59} +464.531 q^{61} +62.4960 q^{65} +769.069 q^{67} +611.569 q^{71} +923.831 q^{73} -142.508 q^{77} -39.8076 q^{79} -443.769 q^{83} +2.28479 q^{85} -78.4576 q^{89} +185.992 q^{91} +95.1075 q^{95} -91.0155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} - 16 q^{7} - 36 q^{11} + 10 q^{13} + 120 q^{17} + 8 q^{19} - 180 q^{23} - 124 q^{25} + 324 q^{29} + 248 q^{31} - 204 q^{35} - 434 q^{37} + 192 q^{41} + 608 q^{43} + 384 q^{47} - 342 q^{49}+ \cdots - 764 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 - 16 * q^7 - 36 * q^11 + 10 * q^13 + 120 * q^17 + 8 * q^19 - 180 * q^23 - 124 * q^25 + 324 * q^29 + 248 * q^31 - 204 * q^35 - 434 * q^37 + 192 * q^41 + 608 * q^43 + 384 * q^47 - 342 * q^49 - 408 * q^53 + 216 * q^55 + 1008 * q^59 + 742 * q^61 - 696 * q^65 + 104 * q^67 + 1140 * q^71 + 850 * q^73 - 576 * q^77 + 440 * q^79 - 264 * q^83 + 1314 * q^85 + 768 * q^89 + 1432 * q^91 - 1140 * q^95 - 764 * q^97

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