LMFDB - Embedded newform 6864.2.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.27841$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.477194 q^{5} -0.435561 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.477194 q^{5} -0.435561 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.477194 q^{15} +3.29509 q^{17} -4.43556 q^{19} -0.435561 q^{21} -3.16688 q^{23} -4.77229 q^{25} +1.00000 q^{27} +6.02641 q^{29} -10.6365 q^{31} +1.00000 q^{33} +0.207847 q^{35} +3.42795 q^{37} +1.00000 q^{39} -7.54922 q^{41} -12.4240 q^{43} -0.477194 q^{45} +3.25346 q^{47} -6.81029 q^{49} +3.29509 q^{51} -9.81029 q^{53} -0.477194 q^{55} -4.43556 q^{57} +12.9239 q^{59} +10.3671 q^{61} -0.435561 q^{63} -0.477194 q^{65} -7.50758 q^{67} -3.16688 q^{69} -14.5416 q^{71} +0.344337 q^{73} -4.77229 q^{75} -0.435561 q^{77} +7.55285 q^{79} +1.00000 q^{81} +9.03039 q^{83} -1.57240 q^{85} +6.02641 q^{87} +5.90514 q^{89} -0.435561 q^{91} -10.6365 q^{93} +2.11662 q^{95} +12.8906 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * q^95 + 10 * q^97 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.477194	−0.213408	−0.106704	−	0.994291i	$-0.534030\pi$
$5$	−0.477194	−0.213408	−0.106704	+	0.994291i	$0.534030\pi$
$6$	0	0
$7$	−0.435561	−0.164627	−0.0823133	−	0.996607i	$-0.526231\pi$
$7$	−0.435561	−0.164627	−0.0823133	+	0.996607i	$0.526231\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.477194	−0.123211
$16$	0	0
$17$	3.29509	0.799177	0.399589	−	0.916695i	$-0.369153\pi$
$17$	3.29509	0.799177	0.399589	+	0.916695i	$0.369153\pi$
$18$	0	0
$19$	−4.43556	−1.01759	−0.508794	−	0.860888i	$-0.669908\pi$
$19$	−4.43556	−1.01759	−0.508794	+	0.860888i	$0.669908\pi$
$20$	0	0
$21$	−0.435561	−0.0950472
$22$	0	0
$23$	−3.16688	−0.660340	−0.330170	−	0.943922i	$-0.607106\pi$
$23$	−3.16688	−0.660340	−0.330170	+	0.943922i	$0.607106\pi$
$24$	0	0
$25$	−4.77229	−0.954457
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.02641	1.11908	0.559538	−	0.828805i	$-0.310979\pi$
$29$	6.02641	1.11908	0.559538	+	0.828805i	$0.310979\pi$
$30$	0	0
$31$	−10.6365	−1.91036	−0.955182	−	0.296018i	$-0.904341\pi$
$31$	−10.6365	−1.91036	−0.955182	+	0.296018i	$0.904341\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0.207847	0.0351326
$36$	0	0
$37$	3.42795	0.563551	0.281776	−	0.959480i	$-0.409077\pi$
$37$	3.42795	0.563551	0.281776	+	0.959480i	$0.409077\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−7.54922	−1.17899	−0.589495	−	0.807772i	$-0.700673\pi$
$41$	−7.54922	−1.17899	−0.589495	+	0.807772i	$0.700673\pi$
$42$	0	0
$43$	−12.4240	−1.89464	−0.947319	−	0.320292i	$-0.896219\pi$
$43$	−12.4240	−1.89464	−0.947319	+	0.320292i	$0.896219\pi$
$44$	0	0
$45$	−0.477194	−0.0711359
$46$	0	0
$47$	3.25346	0.474566	0.237283	−	0.971441i	$-0.423743\pi$
$47$	3.25346	0.474566	0.237283	+	0.971441i	$0.423743\pi$
$48$	0	0
$49$	−6.81029	−0.972898
$50$	0	0
$51$	3.29509	0.461405
$52$	0	0
$53$	−9.81029	−1.34755	−0.673773	−	0.738938i	$-0.735328\pi$
$53$	−9.81029	−1.34755	−0.673773	+	0.738938i	$0.735328\pi$
$54$	0	0
$55$	−0.477194	−0.0643448
$56$	0	0
$57$	−4.43556	−0.587504
$58$	0	0
$59$	12.9239	1.68255	0.841277	−	0.540604i	$-0.181804\pi$
$59$	12.9239	1.68255	0.841277	+	0.540604i	$0.181804\pi$
$60$	0	0
$61$	10.3671	1.32737	0.663686	−	0.748011i	$-0.268991\pi$
$61$	10.3671	1.32737	0.663686	+	0.748011i	$0.268991\pi$
$62$	0	0
$63$	−0.435561	−0.0548755
$64$	0	0
$65$	−0.477194	−0.0591886
$66$	0	0
$67$	−7.50758	−0.917197	−0.458599	−	0.888644i	$-0.651648\pi$
$67$	−7.50758	−0.917197	−0.458599	+	0.888644i	$0.651648\pi$
$68$	0	0
$69$	−3.16688	−0.381247
$70$	0	0
$71$	−14.5416	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$71$	−14.5416	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$72$	0	0
$73$	0.344337	0.0403016	0.0201508	−	0.999797i	$-0.493585\pi$
$73$	0.344337	0.0403016	0.0201508	+	0.999797i	$0.493585\pi$
$74$	0	0
$75$	−4.77229	−0.551056
$76$	0	0
$77$	−0.435561	−0.0496368
$78$	0	0
$79$	7.55285	0.849762	0.424881	−	0.905249i	$-0.360316\pi$
$79$	7.55285	0.849762	0.424881	+	0.905249i	$0.360316\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.03039	0.991214	0.495607	−	0.868547i	$-0.334946\pi$
$83$	9.03039	0.991214	0.495607	+	0.868547i	$0.334946\pi$
$84$	0	0
$85$	−1.57240	−0.170550
$86$	0	0
$87$	6.02641	0.646099
$88$	0	0
$89$	5.90514	0.625944	0.312972	−	0.949762i	$-0.398675\pi$
$89$	5.90514	0.625944	0.312972	+	0.949762i	$0.398675\pi$
$90$	0	0
$91$	−0.435561	−0.0456592
$92$	0	0
$93$	−10.6365	−1.10295
$94$	0	0
$95$	2.11662	0.217161
$96$	0	0
$97$	12.8906	1.30884	0.654420	−	0.756131i	$-0.272913\pi$
$97$	12.8906	1.30884	0.654420	+	0.756131i	$0.272913\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.07499	−0.604484	−0.302242	−	0.953231i	$-0.597735\pi$
$101$	−6.07499	−0.604484	−0.302242	+	0.953231i	$0.597735\pi$
$102$	0	0
$103$	1.98478	0.195566	0.0977829	−	0.995208i	$-0.468825\pi$
$103$	1.98478	0.195566	0.0977829	+	0.995208i	$0.468825\pi$
$104$	0	0
$105$	0.207847	0.0202838
$106$	0	0
$107$	−16.5750	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$107$	−16.5750	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$108$	0	0
$109$	5.21546	0.499550	0.249775	−	0.968304i	$-0.419643\pi$
$109$	5.21546	0.499550	0.249775	+	0.968304i	$0.419643\pi$
$110$	0	0
$111$	3.42795	0.325367
$112$	0	0
$113$	−17.2881	−1.62633	−0.813166	−	0.582032i	$-0.802258\pi$
$113$	−17.2881	−1.62633	−0.813166	+	0.582032i	$0.802258\pi$
$114$	0	0
$115$	1.51121	0.140922
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.43521	−0.131566
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.54922	−0.680690
$124$	0	0
$125$	4.66328	0.417096
$126$	0	0
$127$	4.37539	0.388253	0.194127	−	0.980977i	$-0.437813\pi$
$127$	4.37539	0.388253	0.194127	+	0.980977i	$0.437813\pi$
$128$	0	0
$129$	−12.4240	−1.09387
$130$	0	0
$131$	−15.8936	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$131$	−15.8936	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$132$	0	0
$133$	1.93196	0.167522
$134$	0	0
$135$	−0.477194	−0.0410703
$136$	0	0
$137$	7.01880	0.599656	0.299828	−	0.953993i	$-0.403071\pi$
$137$	7.01880	0.599656	0.299828	+	0.953993i	$0.403071\pi$
$138$	0	0
$139$	−7.86714	−0.667282	−0.333641	−	0.942700i	$-0.608277\pi$
$139$	−7.86714	−0.667282	−0.333641	+	0.942700i	$0.608277\pi$
$140$	0	0
$141$	3.25346	0.273991
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.87577	−0.238819
$146$	0	0
$147$	−6.81029	−0.561703
$148$	0	0
$149$	−10.6781	−0.874783	−0.437392	−	0.899271i	$-0.644098\pi$
$149$	−10.6781	−0.874783	−0.437392	+	0.899271i	$0.644098\pi$
$150$	0	0
$151$	12.3291	1.00333	0.501665	−	0.865062i	$-0.332721\pi$
$151$	12.3291	1.00333	0.501665	+	0.865062i	$0.332721\pi$
$152$	0	0
$153$	3.29509	0.266392
$154$	0	0
$155$	5.07565	0.407686
$156$	0	0
$157$	2.56741	0.204901	0.102451	−	0.994738i	$-0.467332\pi$
$157$	2.56741	0.204901	0.102451	+	0.994738i	$0.467332\pi$
$158$	0	0
$159$	−9.81029	−0.778006
$160$	0	0
$161$	1.37937	0.108710
$162$	0	0
$163$	1.47423	0.115470	0.0577351	−	0.998332i	$-0.481612\pi$
$163$	1.47423	0.115470	0.0577351	+	0.998332i	$0.481612\pi$
$164$	0	0
$165$	−0.477194	−0.0371495
$166$	0	0
$167$	19.0984	1.47788	0.738940	−	0.673771i	$-0.235326\pi$
$167$	19.0984	1.47788	0.738940	+	0.673771i	$0.235326\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.43556	−0.339196
$172$	0	0
$173$	8.67446	0.659507	0.329754	−	0.944067i	$-0.393034\pi$
$173$	8.67446	0.659507	0.329754	+	0.944067i	$0.393034\pi$
$174$	0	0
$175$	2.07862	0.157129
$176$	0	0
$177$	12.9239	0.971423
$178$	0	0
$179$	−18.4550	−1.37939	−0.689697	−	0.724098i	$-0.742256\pi$
$179$	−18.4550	−1.37939	−0.689697	+	0.724098i	$0.742256\pi$
$180$	0	0
$181$	6.99239	0.519740	0.259870	−	0.965644i	$-0.416320\pi$
$181$	6.99239	0.519740	0.259870	+	0.965644i	$0.416320\pi$
$182$	0	0
$183$	10.3671	0.766359
$184$	0	0
$185$	−1.63580	−0.120266
$186$	0	0
$187$	3.29509	0.240961
$188$	0	0
$189$	−0.435561	−0.0316824
$190$	0	0
$191$	−21.4461	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$191$	−21.4461	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$192$	0	0
$193$	−21.6100	−1.55552	−0.777761	−	0.628561i	$-0.783644\pi$
$193$	−21.6100	−1.55552	−0.777761	+	0.628561i	$0.783644\pi$
$194$	0	0
$195$	−0.477194	−0.0341726
$196$	0	0
$197$	−8.42760	−0.600442	−0.300221	−	0.953870i	$-0.597060\pi$
$197$	−8.42760	−0.600442	−0.300221	+	0.953870i	$0.597060\pi$
$198$	0	0
$199$	−13.0072	−0.922056	−0.461028	−	0.887385i	$-0.652519\pi$
$199$	−13.0072	−0.922056	−0.461028	+	0.887385i	$0.652519\pi$
$200$	0	0
$201$	−7.50758	−0.529544
$202$	0	0
$203$	−2.62487	−0.184230
$204$	0	0
$205$	3.60244	0.251605
$206$	0	0
$207$	−3.16688	−0.220113
$208$	0	0
$209$	−4.43556	−0.306814
$210$	0	0
$211$	−7.39358	−0.508995	−0.254498	−	0.967073i	$-0.581910\pi$
$211$	−7.39358	−0.508995	−0.254498	+	0.967073i	$0.581910\pi$
$212$	0	0
$213$	−14.5416	−0.996374
$214$	0	0
$215$	5.92864	0.404330
$216$	0	0
$217$	4.63283	0.314497
$218$	0	0
$219$	0.344337	0.0232681
$220$	0	0
$221$	3.29509	0.221652
$222$	0	0
$223$	15.0826	1.01001	0.505003	−	0.863118i	$-0.331491\pi$
$223$	15.0826	1.01001	0.505003	+	0.863118i	$0.331491\pi$
$224$	0	0
$225$	−4.77229	−0.318152
$226$	0	0
$227$	−21.3899	−1.41970	−0.709850	−	0.704352i	$-0.751237\pi$
$227$	−21.3899	−1.41970	−0.709850	+	0.704352i	$0.751237\pi$
$228$	0	0
$229$	3.69366	0.244084	0.122042	−	0.992525i	$-0.461056\pi$
$229$	3.69366	0.244084	0.122042	+	0.992525i	$0.461056\pi$
$230$	0	0
$231$	−0.435561	−0.0286578
$232$	0	0
$233$	−24.6398	−1.61421	−0.807103	−	0.590411i	$-0.798966\pi$
$233$	−24.6398	−1.61421	−0.807103	+	0.590411i	$0.798966\pi$
$234$	0	0
$235$	−1.55253	−0.101276
$236$	0	0
$237$	7.55285	0.490610
$238$	0	0
$239$	15.2683	0.987623	0.493811	−	0.869569i	$-0.335603\pi$
$239$	15.2683	0.987623	0.493811	+	0.869569i	$0.335603\pi$
$240$	0	0
$241$	2.50825	0.161570	0.0807852	−	0.996732i	$-0.474257\pi$
$241$	2.50825	0.161570	0.0807852	+	0.996732i	$0.474257\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.24983	0.207624
$246$	0	0
$247$	−4.43556	−0.282228
$248$	0	0
$249$	9.03039	0.572278
$250$	0	0
$251$	−11.7343	−0.740662	−0.370331	−	0.928900i	$-0.620756\pi$
$251$	−11.7343	−0.740662	−0.370331	+	0.928900i	$0.620756\pi$
$252$	0	0
$253$	−3.16688	−0.199100
$254$	0	0
$255$	−1.57240	−0.0984674
$256$	0	0
$257$	12.2273	0.762719	0.381359	−	0.924427i	$-0.375456\pi$
$257$	12.2273	0.762719	0.381359	+	0.924427i	$0.375456\pi$
$258$	0	0
$259$	−1.49308	−0.0927756
$260$	0	0
$261$	6.02641	0.373025
$262$	0	0
$263$	−17.7191	−1.09260	−0.546302	−	0.837588i	$-0.683965\pi$
$263$	−17.7191	−1.09260	−0.546302	+	0.837588i	$0.683965\pi$
$264$	0	0
$265$	4.68141	0.287577
$266$	0	0
$267$	5.90514	0.361389
$268$	0	0
$269$	17.7038	1.07942	0.539711	−	0.841850i	$-0.318533\pi$
$269$	17.7038	1.07942	0.539711	+	0.841850i	$0.318533\pi$
$270$	0	0
$271$	−9.45666	−0.574451	−0.287226	−	0.957863i	$-0.592733\pi$
$271$	−9.45666	−0.574451	−0.287226	+	0.957863i	$0.592733\pi$
$272$	0	0
$273$	−0.435561	−0.0263614
$274$	0	0
$275$	−4.77229	−0.287780
$276$	0	0
$277$	−18.4917	−1.11106	−0.555529	−	0.831497i	$-0.687484\pi$
$277$	−18.4917	−1.11106	−0.555529	+	0.831497i	$0.687484\pi$
$278$	0	0
$279$	−10.6365	−0.636788
$280$	0	0
$281$	−15.0198	−0.896007	−0.448003	−	0.894032i	$-0.647865\pi$
$281$	−15.0198	−0.896007	−0.448003	+	0.894032i	$0.647865\pi$
$282$	0	0
$283$	12.2921	0.730691	0.365345	−	0.930872i	$-0.380951\pi$
$283$	12.2921	0.730691	0.365345	+	0.930872i	$0.380951\pi$
$284$	0	0
$285$	2.11662	0.125378
$286$	0	0
$287$	3.28814	0.194093
$288$	0	0
$289$	−6.14237	−0.361316
$290$	0	0
$291$	12.8906	0.755659
$292$	0	0
$293$	−30.4579	−1.77937	−0.889686	−	0.456573i	$-0.849077\pi$
$293$	−30.4579	−1.77937	−0.889686	+	0.456573i	$0.849077\pi$
$294$	0	0
$295$	−6.16723	−0.359070
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.16688	−0.183145
$300$	0	0
$301$	5.41140	0.311908
$302$	0	0
$303$	−6.07499	−0.348999
$304$	0	0
$305$	−4.94712	−0.283271
$306$	0	0
$307$	1.82883	0.104377	0.0521883	−	0.998637i	$-0.483380\pi$
$307$	1.82883	0.104377	0.0521883	+	0.998637i	$0.483380\pi$
$308$	0	0
$309$	1.98478	0.112910
$310$	0	0
$311$	0.144448	0.00819092	0.00409546	−	0.999992i	$-0.498696\pi$
$311$	0.144448	0.00819092	0.00409546	+	0.999992i	$0.498696\pi$
$312$	0	0
$313$	6.69623	0.378493	0.189247	−	0.981930i	$-0.439395\pi$
$313$	6.69623	0.378493	0.189247	+	0.981930i	$0.439395\pi$
$314$	0	0
$315$	0.207847	0.0117109
$316$	0	0
$317$	−21.0175	−1.18046	−0.590229	−	0.807236i	$-0.700963\pi$
$317$	−21.0175	−1.18046	−0.590229	+	0.807236i	$0.700963\pi$
$318$	0	0
$319$	6.02641	0.337414
$320$	0	0
$321$	−16.5750	−0.925124
$322$	0	0
$323$	−14.6156	−0.813233
$324$	0	0
$325$	−4.77229	−0.264719
$326$	0	0
$327$	5.21546	0.288416
$328$	0	0
$329$	−1.41708	−0.0781262
$330$	0	0
$331$	−2.00363	−0.110130	−0.0550648	−	0.998483i	$-0.517537\pi$
$331$	−2.00363	−0.110130	−0.0550648	+	0.998483i	$0.517537\pi$
$332$	0	0
$333$	3.42795	0.187850
$334$	0	0
$335$	3.58257	0.195737
$336$	0	0
$337$	−18.5750	−1.01184	−0.505921	−	0.862580i	$-0.668847\pi$
$337$	−18.5750	−1.01184	−0.505921	+	0.862580i	$0.668847\pi$
$338$	0	0
$339$	−17.2881	−0.938963
$340$	0	0
$341$	−10.6365	−0.575997
$342$	0	0
$343$	6.01522	0.324792
$344$	0	0
$345$	1.51121	0.0813611
$346$	0	0
$347$	31.3063	1.68061	0.840305	−	0.542115i	$-0.182376\pi$
$347$	31.3063	1.68061	0.840305	+	0.542115i	$0.182376\pi$
$348$	0	0
$349$	−28.9847	−1.55152	−0.775758	−	0.631030i	$-0.782632\pi$
$349$	−28.9847	−1.55152	−0.775758	+	0.631030i	$0.782632\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−1.11305	−0.0592416	−0.0296208	−	0.999561i	$-0.509430\pi$
$353$	−1.11305	−0.0592416	−0.0296208	+	0.999561i	$0.509430\pi$
$354$	0	0
$355$	6.93916	0.368293
$356$	0	0
$357$	−1.43521	−0.0759596
$358$	0	0
$359$	−14.0575	−0.741924	−0.370962	−	0.928648i	$-0.620972\pi$
$359$	−14.0575	−0.741924	−0.370962	+	0.928648i	$0.620972\pi$
$360$	0	0
$361$	0.674202	0.0354843
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−0.164315	−0.00860066
$366$	0	0
$367$	−11.0456	−0.576576	−0.288288	−	0.957544i	$-0.593086\pi$
$367$	−11.0456	−0.576576	−0.288288	+	0.957544i	$0.593086\pi$
$368$	0	0
$369$	−7.54922	−0.392996
$370$	0	0
$371$	4.27298	0.221842
$372$	0	0
$373$	8.41702	0.435817	0.217908	−	0.975969i	$-0.430077\pi$
$373$	8.41702	0.435817	0.217908	+	0.975969i	$0.430077\pi$
$374$	0	0
$375$	4.66328	0.240810
$376$	0	0
$377$	6.02641	0.310376
$378$	0	0
$379$	18.4953	0.950041	0.475021	−	0.879975i	$-0.342441\pi$
$379$	18.4953	0.950041	0.475021	+	0.879975i	$0.342441\pi$
$380$	0	0
$381$	4.37539	0.224158
$382$	0	0
$383$	6.93916	0.354575	0.177287	−	0.984159i	$-0.443268\pi$
$383$	6.93916	0.354575	0.177287	+	0.984159i	$0.443268\pi$
$384$	0	0
$385$	0.207847	0.0105929
$386$	0	0
$387$	−12.4240	−0.631546
$388$	0	0
$389$	7.13684	0.361852	0.180926	−	0.983497i	$-0.442091\pi$
$389$	7.13684	0.361852	0.180926	+	0.983497i	$0.442091\pi$
$390$	0	0
$391$	−10.4352	−0.527729
$392$	0	0
$393$	−15.8936	−0.801724
$394$	0	0
$395$	−3.60417	−0.181346
$396$	0	0
$397$	−37.2091	−1.86747	−0.933736	−	0.357962i	$-0.883472\pi$
$397$	−37.2091	−1.86747	−0.933736	+	0.357962i	$0.883472\pi$
$398$	0	0
$399$	1.93196	0.0967189
$400$	0	0
$401$	−6.97022	−0.348076	−0.174038	−	0.984739i	$-0.555682\pi$
$401$	−6.97022	−0.348076	−0.174038	+	0.984739i	$0.555682\pi$
$402$	0	0
$403$	−10.6365	−0.529840
$404$	0	0
$405$	−0.477194	−0.0237120
$406$	0	0
$407$	3.42795	0.169917
$408$	0	0
$409$	8.66287	0.428351	0.214176	−	0.976795i	$-0.431293\pi$
$409$	8.66287	0.428351	0.214176	+	0.976795i	$0.431293\pi$
$410$	0	0
$411$	7.01880	0.346212
$412$	0	0
$413$	−5.62917	−0.276993
$414$	0	0
$415$	−4.30925	−0.211533
$416$	0	0
$417$	−7.86714	−0.385256
$418$	0	0
$419$	13.3869	0.653994	0.326997	−	0.945025i	$-0.393963\pi$
$419$	13.3869	0.653994	0.326997	+	0.945025i	$0.393963\pi$
$420$	0	0
$421$	23.6053	1.15045	0.575227	−	0.817994i	$-0.304914\pi$
$421$	23.6053	1.15045	0.575227	+	0.817994i	$0.304914\pi$
$422$	0	0
$423$	3.25346	0.158189
$424$	0	0
$425$	−15.7251	−0.762780
$426$	0	0
$427$	−4.51551	−0.218521
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	34.4005	1.65701	0.828506	−	0.559980i	$-0.189191\pi$
$431$	34.4005	1.65701	0.828506	+	0.559980i	$0.189191\pi$
$432$	0	0
$433$	7.55980	0.363301	0.181650	−	0.983363i	$-0.441856\pi$
$433$	7.55980	0.363301	0.181650	+	0.983363i	$0.441856\pi$
$434$	0	0
$435$	−2.87577	−0.137882
$436$	0	0
$437$	14.0469	0.671954
$438$	0	0
$439$	25.4239	1.21342	0.606709	−	0.794924i	$-0.292490\pi$
$439$	25.4239	1.21342	0.606709	+	0.794924i	$0.292490\pi$
$440$	0	0
$441$	−6.81029	−0.324299
$442$	0	0
$443$	−5.04428	−0.239661	−0.119831	−	0.992794i	$-0.538235\pi$
$443$	−5.04428	−0.239661	−0.119831	+	0.992794i	$0.538235\pi$
$444$	0	0
$445$	−2.81790	−0.133581
$446$	0	0
$447$	−10.6781	−0.505056
$448$	0	0
$449$	4.97615	0.234839	0.117420	−	0.993082i	$-0.462538\pi$
$449$	4.97615	0.234839	0.117420	+	0.993082i	$0.462538\pi$
$450$	0	0
$451$	−7.54922	−0.355479
$452$	0	0
$453$	12.3291	0.579272
$454$	0	0
$455$	0.207847	0.00974402
$456$	0	0
$457$	12.6827	0.593273	0.296637	−	0.954990i	$-0.404135\pi$
$457$	12.6827	0.593273	0.296637	+	0.954990i	$0.404135\pi$
$458$	0	0
$459$	3.29509	0.153802
$460$	0	0
$461$	10.8605	0.505826	0.252913	−	0.967489i	$-0.418611\pi$
$461$	10.8605	0.505826	0.252913	+	0.967489i	$0.418611\pi$
$462$	0	0
$463$	36.3208	1.68797	0.843986	−	0.536365i	$-0.180203\pi$
$463$	36.3208	1.68797	0.843986	+	0.536365i	$0.180203\pi$
$464$	0	0
$465$	5.07565	0.235378
$466$	0	0
$467$	6.54492	0.302863	0.151431	−	0.988468i	$-0.451612\pi$
$467$	6.54492	0.302863	0.151431	+	0.988468i	$0.451612\pi$
$468$	0	0
$469$	3.27001	0.150995
$470$	0	0
$471$	2.56741	0.118300
$472$	0	0
$473$	−12.4240	−0.571255
$474$	0	0
$475$	21.1678	0.971244
$476$	0	0
$477$	−9.81029	−0.449182
$478$	0	0
$479$	2.42760	0.110920	0.0554600	−	0.998461i	$-0.482337\pi$
$479$	2.42760	0.110920	0.0554600	+	0.998461i	$0.482337\pi$
$480$	0	0
$481$	3.42795	0.156301
$482$	0	0
$483$	1.37937	0.0627635
$484$	0	0
$485$	−6.15131	−0.279316
$486$	0	0
$487$	27.1628	1.23087	0.615433	−	0.788189i	$-0.288981\pi$
$487$	27.1628	1.23087	0.615433	+	0.788189i	$0.288981\pi$
$488$	0	0
$489$	1.47423	0.0666668
$490$	0	0
$491$	−23.0637	−1.04085	−0.520426	−	0.853907i	$-0.674227\pi$
$491$	−23.0637	−1.04085	−0.520426	+	0.853907i	$0.674227\pi$
$492$	0	0
$493$	19.8576	0.894340
$494$	0	0
$495$	−0.477194	−0.0214483
$496$	0	0
$497$	6.33376	0.284108
$498$	0	0
$499$	10.0978	0.452038	0.226019	−	0.974123i	$-0.427429\pi$
$499$	10.0978	0.452038	0.226019	+	0.974123i	$0.427429\pi$
$500$	0	0
$501$	19.0984	0.853255
$502$	0	0
$503$	17.3034	0.771519	0.385760	−	0.922599i	$-0.373939\pi$
$503$	17.3034	0.771519	0.385760	+	0.922599i	$0.373939\pi$
$504$	0	0
$505$	2.89895	0.129001
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	29.3664	1.30164	0.650822	−	0.759230i	$-0.274424\pi$
$509$	29.3664	1.30164	0.650822	+	0.759230i	$0.274424\pi$
$510$	0	0
$511$	−0.149980	−0.00663471
$512$	0	0
$513$	−4.43556	−0.195835
$514$	0	0
$515$	−0.947123	−0.0417352
$516$	0	0
$517$	3.25346	0.143087
$518$	0	0
$519$	8.67446	0.380767
$520$	0	0
$521$	−26.2881	−1.15170	−0.575851	−	0.817555i	$-0.695329\pi$
$521$	−26.2881	−1.15170	−0.575851	+	0.817555i	$0.695329\pi$
$522$	0	0
$523$	−41.7729	−1.82660	−0.913301	−	0.407286i	$-0.866475\pi$
$523$	−41.7729	−1.82660	−0.913301	+	0.407286i	$0.866475\pi$
$524$	0	0
$525$	2.07862	0.0907185
$526$	0	0
$527$	−35.0481	−1.52672
$528$	0	0
$529$	−12.9709	−0.563951
$530$	0	0
$531$	12.9239	0.560851
$532$	0	0
$533$	−7.54922	−0.326993
$534$	0	0
$535$	7.90947	0.341956
$536$	0	0
$537$	−18.4550	−0.796393
$538$	0	0
$539$	−6.81029	−0.293340
$540$	0	0
$541$	−1.62057	−0.0696739	−0.0348369	−	0.999393i	$-0.511091\pi$
$541$	−1.62057	−0.0696739	−0.0348369	+	0.999393i	$0.511091\pi$
$542$	0	0
$543$	6.99239	0.300072
$544$	0	0
$545$	−2.48879	−0.106608
$546$	0	0
$547$	5.74547	0.245659	0.122829	−	0.992428i	$-0.460803\pi$
$547$	5.74547	0.245659	0.122829	+	0.992428i	$0.460803\pi$
$548$	0	0
$549$	10.3671	0.442458
$550$	0	0
$551$	−26.7305	−1.13876
$552$	0	0
$553$	−3.28973	−0.139893
$554$	0	0
$555$	−1.63580	−0.0694357
$556$	0	0
$557$	−15.5399	−0.658448	−0.329224	−	0.944252i	$-0.606787\pi$
$557$	−15.5399	−0.658448	−0.329224	+	0.944252i	$0.606787\pi$
$558$	0	0
$559$	−12.4240	−0.525478
$560$	0	0
$561$	3.29509	0.139119
$562$	0	0
$563$	29.4533	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$563$	29.4533	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$564$	0	0
$565$	8.24980	0.347072
$566$	0	0
$567$	−0.435561	−0.0182918
$568$	0	0
$569$	17.1206	0.717733	0.358866	−	0.933389i	$-0.383163\pi$
$569$	17.1206	0.717733	0.358866	+	0.933389i	$0.383163\pi$
$570$	0	0
$571$	−20.3036	−0.849680	−0.424840	−	0.905268i	$-0.639670\pi$
$571$	−20.3036	−0.849680	−0.424840	+	0.905268i	$0.639670\pi$
$572$	0	0
$573$	−21.4461	−0.895923
$574$	0	0
$575$	15.1132	0.630266
$576$	0	0
$577$	−9.09843	−0.378773	−0.189386	−	0.981903i	$-0.560650\pi$
$577$	−9.09843	−0.378773	−0.189386	+	0.981903i	$0.560650\pi$
$578$	0	0
$579$	−21.6100	−0.898081
$580$	0	0
$581$	−3.93329	−0.163180
$582$	0	0
$583$	−9.81029	−0.406301
$584$	0	0
$585$	−0.477194	−0.0197295
$586$	0	0
$587$	−36.1288	−1.49120	−0.745598	−	0.666396i	$-0.767836\pi$
$587$	−36.1288	−1.49120	−0.745598	+	0.666396i	$0.767836\pi$
$588$	0	0
$589$	47.1787	1.94396
$590$	0	0
$591$	−8.42760	−0.346665
$592$	0	0
$593$	−7.60203	−0.312178	−0.156089	−	0.987743i	$-0.549889\pi$
$593$	−7.60203	−0.312178	−0.156089	+	0.987743i	$0.549889\pi$
$594$	0	0
$595$	0.684875	0.0280771
$596$	0	0
$597$	−13.0072	−0.532350
$598$	0	0
$599$	−27.5373	−1.12514	−0.562572	−	0.826748i	$-0.690188\pi$
$599$	−27.5373	−1.12514	−0.562572	+	0.826748i	$0.690188\pi$
$600$	0	0
$601$	34.2028	1.39516	0.697581	−	0.716506i	$-0.254260\pi$
$601$	34.2028	1.39516	0.697581	+	0.716506i	$0.254260\pi$
$602$	0	0
$603$	−7.50758	−0.305732
$604$	0	0
$605$	−0.477194	−0.0194007
$606$	0	0
$607$	−26.5194	−1.07639	−0.538195	−	0.842820i	$-0.680894\pi$
$607$	−26.5194	−1.07639	−0.538195	+	0.842820i	$0.680894\pi$
$608$	0	0
$609$	−2.62487	−0.106365
$610$	0	0
$611$	3.25346	0.131621
$612$	0	0
$613$	−23.0938	−0.932749	−0.466375	−	0.884587i	$-0.654440\pi$
$613$	−23.0938	−0.932749	−0.466375	+	0.884587i	$0.654440\pi$
$614$	0	0
$615$	3.60244	0.145264
$616$	0	0
$617$	−6.14344	−0.247325	−0.123663	−	0.992324i	$-0.539464\pi$
$617$	−6.14344	−0.247325	−0.123663	+	0.992324i	$0.539464\pi$
$618$	0	0
$619$	−11.1767	−0.449231	−0.224615	−	0.974447i	$-0.572113\pi$
$619$	−11.1767	−0.449231	−0.224615	+	0.974447i	$0.572113\pi$
$620$	0	0
$621$	−3.16688	−0.127082
$622$	0	0
$623$	−2.57205	−0.103047
$624$	0	0
$625$	21.6361	0.865446
$626$	0	0
$627$	−4.43556	−0.177139
$628$	0	0
$629$	11.2954	0.450377
$630$	0	0
$631$	22.1325	0.881079	0.440540	−	0.897733i	$-0.354787\pi$
$631$	22.1325	0.881079	0.440540	+	0.897733i	$0.354787\pi$
$632$	0	0
$633$	−7.39358	−0.293868
$634$	0	0
$635$	−2.08791	−0.0828561
$636$	0	0
$637$	−6.81029	−0.269833
$638$	0	0
$639$	−14.5416	−0.575257
$640$	0	0
$641$	11.3562	0.448542	0.224271	−	0.974527i	$-0.428000\pi$
$641$	11.3562	0.448542	0.224271	+	0.974527i	$0.428000\pi$
$642$	0	0
$643$	−19.8334	−0.782152	−0.391076	−	0.920358i	$-0.627897\pi$
$643$	−19.8334	−0.782152	−0.391076	+	0.920358i	$0.627897\pi$
$644$	0	0
$645$	5.92864	0.233440
$646$	0	0
$647$	−8.58425	−0.337482	−0.168741	−	0.985660i	$-0.553970\pi$
$647$	−8.58425	−0.337482	−0.168741	+	0.985660i	$0.553970\pi$
$648$	0	0
$649$	12.9239	0.507309
$650$	0	0
$651$	4.63283	0.181575
$652$	0	0
$653$	19.8783	0.777899	0.388950	−	0.921259i	$-0.372838\pi$
$653$	19.8783	0.777899	0.388950	+	0.921259i	$0.372838\pi$
$654$	0	0
$655$	7.58431	0.296343
$656$	0	0
$657$	0.344337	0.0134339
$658$	0	0
$659$	29.9818	1.16792	0.583962	−	0.811781i	$-0.301502\pi$
$659$	29.9818	1.16792	0.583962	+	0.811781i	$0.301502\pi$
$660$	0	0
$661$	37.9919	1.47771	0.738857	−	0.673862i	$-0.235366\pi$
$661$	37.9919	1.47771	0.738857	+	0.673862i	$0.235366\pi$
$662$	0	0
$663$	3.29509	0.127971
$664$	0	0
$665$	−0.921918	−0.0357505
$666$	0	0
$667$	−19.0849	−0.738970
$668$	0	0
$669$	15.0826	0.583127
$670$	0	0
$671$	10.3671	0.400218
$672$	0	0
$673$	25.5859	0.986263	0.493132	−	0.869955i	$-0.335852\pi$
$673$	25.5859	0.986263	0.493132	+	0.869955i	$0.335852\pi$
$674$	0	0
$675$	−4.77229	−0.183685
$676$	0	0
$677$	−24.4362	−0.939158	−0.469579	−	0.882890i	$-0.655594\pi$
$677$	−24.4362	−0.939158	−0.469579	+	0.882890i	$0.655594\pi$
$678$	0	0
$679$	−5.61464	−0.215470
$680$	0	0
$681$	−21.3899	−0.819665
$682$	0	0
$683$	39.0934	1.49587	0.747933	−	0.663774i	$-0.231046\pi$
$683$	39.0934	1.49587	0.747933	+	0.663774i	$0.231046\pi$
$684$	0	0
$685$	−3.34933	−0.127971
$686$	0	0
$687$	3.69366	0.140922
$688$	0	0
$689$	−9.81029	−0.373742
$690$	0	0
$691$	40.6394	1.54599	0.772997	−	0.634409i	$-0.218757\pi$
$691$	40.6394	1.54599	0.772997	+	0.634409i	$0.218757\pi$
$692$	0	0
$693$	−0.435561	−0.0165456
$694$	0	0
$695$	3.75415	0.142403
$696$	0	0
$697$	−24.8754	−0.942221
$698$	0	0
$699$	−24.6398	−0.931962
$700$	0	0
$701$	−4.06905	−0.153686	−0.0768430	−	0.997043i	$-0.524484\pi$
$701$	−4.06905	−0.153686	−0.0768430	+	0.997043i	$0.524484\pi$
$702$	0	0
$703$	−15.2049	−0.573463
$704$	0	0
$705$	−1.55253	−0.0584717
$706$	0	0
$707$	2.64603	0.0995142
$708$	0	0
$709$	30.7457	1.15468	0.577340	−	0.816504i	$-0.304091\pi$
$709$	30.7457	1.15468	0.577340	+	0.816504i	$0.304091\pi$
$710$	0	0
$711$	7.55285	0.283254
$712$	0	0
$713$	33.6844	1.26149
$714$	0	0
$715$	−0.477194	−0.0178460
$716$	0	0
$717$	15.2683	0.570204
$718$	0	0
$719$	−25.7422	−0.960024	−0.480012	−	0.877262i	$-0.659368\pi$
$719$	−25.7422	−0.960024	−0.480012	+	0.877262i	$0.659368\pi$
$720$	0	0
$721$	−0.864491	−0.0321953
$722$	0	0
$723$	2.50825	0.0932827
$724$	0	0
$725$	−28.7597	−1.06811
$726$	0	0
$727$	−3.21877	−0.119378	−0.0596889	−	0.998217i	$-0.519011\pi$
$727$	−3.21877	−0.119378	−0.0596889	+	0.998217i	$0.519011\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.9381	−1.51415
$732$	0	0
$733$	38.8208	1.43388	0.716940	−	0.697135i	$-0.245542\pi$
$733$	38.8208	1.43388	0.716940	+	0.697135i	$0.245542\pi$
$734$	0	0
$735$	3.24983	0.119872
$736$	0	0
$737$	−7.50758	−0.276545
$738$	0	0
$739$	−14.7389	−0.542178	−0.271089	−	0.962554i	$-0.587384\pi$
$739$	−14.7389	−0.542178	−0.271089	+	0.962554i	$0.587384\pi$
$740$	0	0
$741$	−4.43556	−0.162944
$742$	0	0
$743$	21.1064	0.774318	0.387159	−	0.922013i	$-0.373456\pi$
$743$	21.1064	0.774318	0.387159	+	0.922013i	$0.373456\pi$
$744$	0	0
$745$	5.09552	0.186685
$746$	0	0
$747$	9.03039	0.330405
$748$	0	0
$749$	7.21941	0.263791
$750$	0	0
$751$	13.3701	0.487881	0.243941	−	0.969790i	$-0.421560\pi$
$751$	13.3701	0.487881	0.243941	+	0.969790i	$0.421560\pi$
$752$	0	0
$753$	−11.7343	−0.427621
$754$	0	0
$755$	−5.88338	−0.214118
$756$	0	0
$757$	2.09082	0.0759921	0.0379961	−	0.999278i	$-0.487903\pi$
$757$	2.09082	0.0759921	0.0379961	+	0.999278i	$0.487903\pi$
$758$	0	0
$759$	−3.16688	−0.114950
$760$	0	0
$761$	30.7680	1.11534	0.557669	−	0.830063i	$-0.311696\pi$
$761$	30.7680	1.11534	0.557669	+	0.830063i	$0.311696\pi$
$762$	0	0
$763$	−2.27165	−0.0822393
$764$	0	0
$765$	−1.57240	−0.0568502
$766$	0	0
$767$	12.9239	0.466656
$768$	0	0
$769$	−37.4540	−1.35063	−0.675314	−	0.737531i	$-0.735992\pi$
$769$	−37.4540	−1.35063	−0.675314	+	0.737531i	$0.735992\pi$
$770$	0	0
$771$	12.2273	0.440356
$772$	0	0
$773$	40.3657	1.45185	0.725927	−	0.687772i	$-0.241411\pi$
$773$	40.3657	1.45185	0.725927	+	0.687772i	$0.241411\pi$
$774$	0	0
$775$	50.7602	1.82336
$776$	0	0
$777$	−1.49308	−0.0535640
$778$	0	0
$779$	33.4850	1.19972
$780$	0	0
$781$	−14.5416	−0.520339
$782$	0	0
$783$	6.02641	0.215366
$784$	0	0
$785$	−1.22515	−0.0437275
$786$	0	0
$787$	−14.7389	−0.525384	−0.262692	−	0.964880i	$-0.584610\pi$
$787$	−14.7389	−0.525384	−0.262692	+	0.964880i	$0.584610\pi$
$788$	0	0
$789$	−17.7191	−0.630815
$790$	0	0
$791$	7.53004	0.267738
$792$	0	0
$793$	10.3671	0.368147
$794$	0	0
$795$	4.68141	0.166032
$796$	0	0
$797$	−43.3925	−1.53704	−0.768520	−	0.639826i	$-0.779006\pi$
$797$	−43.3925	−1.53704	−0.768520	+	0.639826i	$0.779006\pi$
$798$	0	0
$799$	10.7204	0.379262
$800$	0	0
$801$	5.90514	0.208648
$802$	0	0
$803$	0.344337	0.0121514
$804$	0	0
$805$	−0.658226	−0.0231994
$806$	0	0
$807$	17.7038	0.623205
$808$	0	0
$809$	28.0821	0.987315	0.493658	−	0.869656i	$-0.335660\pi$
$809$	28.0821	0.987315	0.493658	+	0.869656i	$0.335660\pi$
$810$	0	0
$811$	−42.2927	−1.48510	−0.742549	−	0.669791i	$-0.766384\pi$
$811$	−42.2927	−1.48510	−0.742549	+	0.669791i	$0.766384\pi$
$812$	0	0
$813$	−9.45666	−0.331660
$814$	0	0
$815$	−0.703491	−0.0246422
$816$	0	0
$817$	55.1073	1.92796
$818$	0	0
$819$	−0.435561	−0.0152197
$820$	0	0
$821$	−47.6687	−1.66365	−0.831825	−	0.555037i	$-0.812704\pi$
$821$	−47.6687	−1.66365	−0.831825	+	0.555037i	$0.812704\pi$
$822$	0	0
$823$	29.8618	1.04092	0.520458	−	0.853887i	$-0.325761\pi$
$823$	29.8618	1.04092	0.520458	+	0.853887i	$0.325761\pi$
$824$	0	0
$825$	−4.77229	−0.166150
$826$	0	0
$827$	−53.2133	−1.85041	−0.925204	−	0.379470i	$-0.876107\pi$
$827$	−53.2133	−1.85041	−0.925204	+	0.379470i	$0.876107\pi$
$828$	0	0
$829$	−14.2881	−0.496246	−0.248123	−	0.968729i	$-0.579814\pi$
$829$	−14.2881	−0.496246	−0.248123	+	0.968729i	$0.579814\pi$
$830$	0	0
$831$	−18.4917	−0.641470
$832$	0	0
$833$	−22.4405	−0.777518
$834$	0	0
$835$	−9.11365	−0.315391
$836$	0	0
$837$	−10.6365	−0.367650
$838$	0	0
$839$	−31.0680	−1.07259	−0.536293	−	0.844032i	$-0.680176\pi$
$839$	−31.0680	−1.07259	−0.536293	+	0.844032i	$0.680176\pi$
$840$	0	0
$841$	7.31761	0.252331
$842$	0	0
$843$	−15.0198	−0.517310
$844$	0	0
$845$	−0.477194	−0.0164160
$846$	0	0
$847$	−0.435561	−0.0149661
$848$	0	0
$849$	12.2921	0.421865
$850$	0	0
$851$	−10.8559	−0.372135
$852$	0	0
$853$	−16.0515	−0.549593	−0.274796	−	0.961502i	$-0.588610\pi$
$853$	−16.0515	−0.549593	−0.274796	+	0.961502i	$0.588610\pi$
$854$	0	0
$855$	2.11662	0.0723870
$856$	0	0
$857$	−11.4725	−0.391893	−0.195946	−	0.980615i	$-0.562778\pi$
$857$	−11.4725	−0.391893	−0.195946	+	0.980615i	$0.562778\pi$
$858$	0	0
$859$	26.7956	0.914255	0.457128	−	0.889401i	$-0.348878\pi$
$859$	26.7956	0.914255	0.457128	+	0.889401i	$0.348878\pi$
$860$	0	0
$861$	3.28814	0.112060
$862$	0	0
$863$	43.5095	1.48108	0.740541	−	0.672012i	$-0.234570\pi$
$863$	43.5095	1.48108	0.740541	+	0.672012i	$0.234570\pi$
$864$	0	0
$865$	−4.13940	−0.140744
$866$	0	0
$867$	−6.14237	−0.208606
$868$	0	0
$869$	7.55285	0.256213
$870$	0	0
$871$	−7.50758	−0.254385
$872$	0	0
$873$	12.8906	0.436280
$874$	0	0
$875$	−2.03114	−0.0686651
$876$	0	0
$877$	−11.9320	−0.402914	−0.201457	−	0.979497i	$-0.564568\pi$
$877$	−11.9320	−0.402914	−0.201457	+	0.979497i	$0.564568\pi$
$878$	0	0
$879$	−30.4579	−1.02732
$880$	0	0
$881$	13.8176	0.465525	0.232763	−	0.972534i	$-0.425223\pi$
$881$	13.8176	0.465525	0.232763	+	0.972534i	$0.425223\pi$
$882$	0	0
$883$	−29.8413	−1.00424	−0.502119	−	0.864799i	$-0.667446\pi$
$883$	−29.8413	−1.00424	−0.502119	+	0.864799i	$0.667446\pi$
$884$	0	0
$885$	−6.16723	−0.207309
$886$	0	0
$887$	53.2070	1.78652	0.893258	−	0.449545i	$-0.148414\pi$
$887$	53.2070	1.78652	0.893258	+	0.449545i	$0.148414\pi$
$888$	0	0
$889$	−1.90575	−0.0639168
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−14.4309	−0.482912
$894$	0	0
$895$	8.80662	0.294373
$896$	0	0
$897$	−3.16688	−0.105739
$898$	0	0
$899$	−64.0997	−2.13784
$900$	0	0
$901$	−32.3258	−1.07693
$902$	0	0
$903$	5.41140	0.180080
$904$	0	0
$905$	−3.33672	−0.110916
$906$	0	0
$907$	45.8017	1.52082	0.760410	−	0.649443i	$-0.224998\pi$
$907$	45.8017	1.52082	0.760410	+	0.649443i	$0.224998\pi$
$908$	0	0
$909$	−6.07499	−0.201495
$910$	0	0
$911$	10.9696	0.363439	0.181720	−	0.983350i	$-0.441834\pi$
$911$	10.9696	0.363439	0.181720	+	0.983350i	$0.441834\pi$
$912$	0	0
$913$	9.03039	0.298862
$914$	0	0
$915$	−4.94712	−0.163547
$916$	0	0
$917$	6.92261	0.228605
$918$	0	0
$919$	19.2008	0.633377	0.316689	−	0.948530i	$-0.397429\pi$
$919$	19.2008	0.633377	0.316689	+	0.948530i	$0.397429\pi$
$920$	0	0
$921$	1.82883	0.0602618
$922$	0	0
$923$	−14.5416	−0.478643
$924$	0	0
$925$	−16.3592	−0.537886
$926$	0	0
$927$	1.98478	0.0651886
$928$	0	0
$929$	−3.91310	−0.128385	−0.0641924	−	0.997938i	$-0.520447\pi$
$929$	−3.91310	−0.128385	−0.0641924	+	0.997938i	$0.520447\pi$
$930$	0	0
$931$	30.2074	0.990009
$932$	0	0
$933$	0.144448	0.00472903
$934$	0	0
$935$	−1.57240	−0.0514229
$936$	0	0
$937$	49.4490	1.61543	0.807714	−	0.589574i	$-0.200704\pi$
$937$	49.4490	1.61543	0.807714	+	0.589574i	$0.200704\pi$
$938$	0	0
$939$	6.69623	0.218523
$940$	0	0
$941$	−27.1698	−0.885710	−0.442855	−	0.896593i	$-0.646034\pi$
$941$	−27.1698	−0.885710	−0.442855	+	0.896593i	$0.646034\pi$
$942$	0	0
$943$	23.9074	0.778534
$944$	0	0
$945$	0.207847	0.00676127
$946$	0	0
$947$	−3.67844	−0.119533	−0.0597666	−	0.998212i	$-0.519036\pi$
$947$	−3.67844	−0.119533	−0.0597666	+	0.998212i	$0.519036\pi$
$948$	0	0
$949$	0.344337	0.0111776
$950$	0	0
$951$	−21.0175	−0.681538
$952$	0	0
$953$	−14.3681	−0.465429	−0.232715	−	0.972545i	$-0.574761\pi$
$953$	−14.3681	−0.465429	−0.232715	+	0.972545i	$0.574761\pi$
$954$	0	0
$955$	10.2339	0.331163
$956$	0	0
$957$	6.02641	0.194806
$958$	0	0
$959$	−3.05712	−0.0987194
$960$	0	0
$961$	82.1343	2.64949
$962$	0	0
$963$	−16.5750	−0.534121
$964$	0	0
$965$	10.3122	0.331960
$966$	0	0
$967$	15.7904	0.507786	0.253893	−	0.967232i	$-0.418289\pi$
$967$	15.7904	0.507786	0.253893	+	0.967232i	$0.418289\pi$
$968$	0	0
$969$	−14.6156	−0.469520
$970$	0	0
$971$	−34.6447	−1.11180	−0.555901	−	0.831248i	$-0.687627\pi$
$971$	−34.6447	−1.11180	−0.555901	+	0.831248i	$0.687627\pi$
$972$	0	0
$973$	3.42662	0.109852
$974$	0	0
$975$	−4.77229	−0.152835
$976$	0	0
$977$	1.18666	0.0379645	0.0189823	−	0.999820i	$-0.493957\pi$
$977$	1.18666	0.0379645	0.0189823	+	0.999820i	$0.493957\pi$
$978$	0	0
$979$	5.90514	0.188729
$980$	0	0
$981$	5.21546	0.166517
$982$	0	0
$983$	−0.173162	−0.00552301	−0.00276150	−	0.999996i	$-0.500879\pi$
$983$	−0.173162	−0.00552301	−0.00276150	+	0.999996i	$0.500879\pi$
$984$	0	0
$985$	4.02160	0.128139
$986$	0	0
$987$	−1.41708	−0.0451062
$988$	0	0
$989$	39.3452	1.25110
$990$	0	0
$991$	60.8935	1.93435	0.967173	−	0.254120i	$-0.0817857\pi$
$991$	60.8935	1.93435	0.967173	+	0.254120i	$0.0817857\pi$
$992$	0	0
$993$	−2.00363	−0.0635833
$994$	0	0
$995$	6.20696	0.196774
$996$	0	0
$997$	−50.8776	−1.61131	−0.805655	−	0.592386i	$-0.798186\pi$
$997$	−50.8776	−1.61131	−0.805655	+	0.592386i	$0.798186\pi$
$998$	0	0
$999$	3.42795	0.108456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bz.1.3		4
4.3	odd	2		429.2.a.h.1.3	✓	4
12.11	even	2		1287.2.a.m.1.2		4
44.43	even	2		4719.2.a.z.1.2		4
52.51	odd	2		5577.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.3	✓	4	4.3	odd	2
1287.2.a.m.1.2		4	12.11	even	2
4719.2.a.z.1.2		4	44.43	even	2
5577.2.a.m.1.2		4	52.51	odd	2
6864.2.a.bz.1.3		4	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.477194 q^{5} -0.435561 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.477194 q^{5} -0.435561 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.477194 q^{15} +3.29509 q^{17} -4.43556 q^{19} -0.435561 q^{21} -3.16688 q^{23} -4.77229 q^{25} +1.00000 q^{27} +6.02641 q^{29} -10.6365 q^{31} +1.00000 q^{33} +0.207847 q^{35} +3.42795 q^{37} +1.00000 q^{39} -7.54922 q^{41} -12.4240 q^{43} -0.477194 q^{45} +3.25346 q^{47} -6.81029 q^{49} +3.29509 q^{51} -9.81029 q^{53} -0.477194 q^{55} -4.43556 q^{57} +12.9239 q^{59} +10.3671 q^{61} -0.435561 q^{63} -0.477194 q^{65} -7.50758 q^{67} -3.16688 q^{69} -14.5416 q^{71} +0.344337 q^{73} -4.77229 q^{75} -0.435561 q^{77} +7.55285 q^{79} +1.00000 q^{81} +9.03039 q^{83} -1.57240 q^{85} +6.02641 q^{87} +5.90514 q^{89} -0.435561 q^{91} -10.6365 q^{93} +2.11662 q^{95} +12.8906 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * q^95 + 10 * q^97 + 4 * q^99

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