LMFDB - Embedded newform 6864.2.a.bx.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.83476.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} - 2x + 1 $ x^4 - 12x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.36409$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.29726 q^{5} +3.06684 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.29726 q^{5} +3.06684 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.29726 q^{15} +6.36409 q^{17} -4.31712 q^{19} -3.06684 q^{21} -3.66135 q^{23} -3.31712 q^{25} -1.00000 q^{27} +4.36409 q^{29} -1.95303 q^{31} -1.00000 q^{33} +3.97847 q^{35} -4.72819 q^{37} +1.00000 q^{39} +2.93316 q^{41} +7.74805 q^{43} +1.29726 q^{45} +9.38396 q^{47} +2.40549 q^{49} -6.36409 q^{51} +11.9785 q^{53} +1.29726 q^{55} +4.31712 q^{57} +2.59451 q^{59} +10.0724 q^{61} +3.06684 q^{63} -1.29726 q^{65} -4.70274 q^{67} +3.66135 q^{69} -0.655774 q^{71} +7.72261 q^{73} +3.31712 q^{75} +3.06684 q^{77} -4.30283 q^{79} +1.00000 q^{81} -8.72819 q^{83} +8.25586 q^{85} -4.36409 q^{87} +6.68122 q^{89} -3.06684 q^{91} +1.95303 q^{93} -5.60042 q^{95} +12.1121 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{15} + 12 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} - 4 q^{33} - 22 q^{35} + 8 q^{37} + 4 q^{39} + 22 q^{41} - 14 q^{43} + 2 q^{45} + 6 q^{47} + 16 q^{49} - 12 q^{51} + 10 q^{53} + 2 q^{55} - 4 q^{57} + 4 q^{59} + 18 q^{61} + 2 q^{63} - 2 q^{65} - 22 q^{67} - 2 q^{69} + 2 q^{71} + 16 q^{73} - 8 q^{75} + 2 q^{77} - 2 q^{79} + 4 q^{81} - 8 q^{83} + 10 q^{85} - 4 q^{87} - 8 q^{89} - 2 q^{91} - 10 q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 2 * q^15 + 12 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 - 4 * q^33 - 22 * q^35 + 8 * q^37 + 4 * q^39 + 22 * q^41 - 14 * q^43 + 2 * q^45 + 6 * q^47 + 16 * q^49 - 12 * q^51 + 10 * q^53 + 2 * q^55 - 4 * q^57 + 4 * q^59 + 18 * q^61 + 2 * q^63 - 2 * q^65 - 22 * q^67 - 2 * q^69 + 2 * q^71 + 16 * q^73 - 8 * q^75 + 2 * q^77 - 2 * q^79 + 4 * q^81 - 8 * q^83 + 10 * q^85 - 4 * q^87 - 8 * q^89 - 2 * q^91 - 10 * 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
$5$	1.29726	0.580151	0.290075	−	0.957004i	$-0.406320\pi$
$5$	1.29726	0.580151	0.290075	+	0.957004i	$0.406320\pi$
$6$	0	0
$7$	3.06684	1.15916	0.579578	−	0.814917i	$-0.303218\pi$
$7$	3.06684	1.15916	0.579578	+	0.814917i	$0.303218\pi$
$8$	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
$15$	−1.29726	−0.334950
$16$	0	0
$17$	6.36409	1.54352	0.771760	−	0.635914i	$-0.219377\pi$
$17$	6.36409	1.54352	0.771760	+	0.635914i	$0.219377\pi$
$18$	0	0
$19$	−4.31712	−0.990416	−0.495208	−	0.868774i	$-0.664908\pi$
$19$	−4.31712	−0.990416	−0.495208	+	0.868774i	$0.664908\pi$
$20$	0	0
$21$	−3.06684	−0.669239
$22$	0	0
$23$	−3.66135	−0.763444	−0.381722	−	0.924277i	$-0.624669\pi$
$23$	−3.66135	−0.763444	−0.381722	+	0.924277i	$0.624669\pi$
$24$	0	0
$25$	−3.31712	−0.663425
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.36409	0.810392	0.405196	−	0.914230i	$-0.367203\pi$
$29$	4.36409	0.810392	0.405196	+	0.914230i	$0.367203\pi$
$30$	0	0
$31$	−1.95303	−0.350775	−0.175387	−	0.984500i	$-0.556118\pi$
$31$	−1.95303	−0.350775	−0.175387	+	0.984500i	$0.556118\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.97847	0.672485
$36$	0	0
$37$	−4.72819	−0.777309	−0.388655	−	0.921384i	$-0.627060\pi$
$37$	−4.72819	−0.777309	−0.388655	+	0.921384i	$0.627060\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.93316	0.458083	0.229042	−	0.973417i	$-0.426441\pi$
$41$	2.93316	0.458083	0.229042	+	0.973417i	$0.426441\pi$
$42$	0	0
$43$	7.74805	1.18157	0.590784	−	0.806830i	$-0.298819\pi$
$43$	7.74805	1.18157	0.590784	+	0.806830i	$0.298819\pi$
$44$	0	0
$45$	1.29726	0.193384
$46$	0	0
$47$	9.38396	1.36879	0.684396	−	0.729111i	$-0.260066\pi$
$47$	9.38396	1.36879	0.684396	+	0.729111i	$0.260066\pi$
$48$	0	0
$49$	2.40549	0.343641
$50$	0	0
$51$	−6.36409	−0.891151
$52$	0	0
$53$	11.9785	1.64537	0.822685	−	0.568497i	$-0.192475\pi$
$53$	11.9785	1.64537	0.822685	+	0.568497i	$0.192475\pi$
$54$	0	0
$55$	1.29726	0.174922
$56$	0	0
$57$	4.31712	0.571817
$58$	0	0
$59$	2.59451	0.337777	0.168888	−	0.985635i	$-0.445982\pi$
$59$	2.59451	0.337777	0.168888	+	0.985635i	$0.445982\pi$
$60$	0	0
$61$	10.0724	1.28964	0.644820	−	0.764334i	$-0.276932\pi$
$61$	10.0724	1.28964	0.644820	+	0.764334i	$0.276932\pi$
$62$	0	0
$63$	3.06684	0.386385
$64$	0	0
$65$	−1.29726	−0.160905
$66$	0	0
$67$	−4.70274	−0.574532	−0.287266	−	0.957851i	$-0.592746\pi$
$67$	−4.70274	−0.574532	−0.287266	+	0.957851i	$0.592746\pi$
$68$	0	0
$69$	3.66135	0.440775
$70$	0	0
$71$	−0.655774	−0.0778261	−0.0389130	−	0.999243i	$-0.512390\pi$
$71$	−0.655774	−0.0778261	−0.0389130	+	0.999243i	$0.512390\pi$
$72$	0	0
$73$	7.72261	0.903863	0.451932	−	0.892053i	$-0.350735\pi$
$73$	7.72261	0.903863	0.451932	+	0.892053i	$0.350735\pi$
$74$	0	0
$75$	3.31712	0.383029
$76$	0	0
$77$	3.06684	0.349498
$78$	0	0
$79$	−4.30283	−0.484107	−0.242053	−	0.970263i	$-0.577821\pi$
$79$	−4.30283	−0.484107	−0.242053	+	0.970263i	$0.577821\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.72819	−0.958043	−0.479021	−	0.877803i	$-0.659008\pi$
$83$	−8.72819	−0.958043	−0.479021	+	0.877803i	$0.659008\pi$
$84$	0	0
$85$	8.25586	0.895474
$86$	0	0
$87$	−4.36409	−0.467880
$88$	0	0
$89$	6.68122	0.708208	0.354104	−	0.935206i	$-0.384786\pi$
$89$	6.68122	0.708208	0.354104	+	0.935206i	$0.384786\pi$
$90$	0	0
$91$	−3.06684	−0.321492
$92$	0	0
$93$	1.95303	0.202520
$94$	0	0
$95$	−5.60042	−0.574591
$96$	0	0
$97$	12.1121	1.22980	0.614901	−	0.788604i	$-0.289196\pi$
$97$	12.1121	1.22980	0.614901	+	0.788604i	$0.289196\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−11.8817	−1.18228	−0.591138	−	0.806570i	$-0.701321\pi$
$101$	−11.8817	−1.18228	−0.591138	+	0.806570i	$0.701321\pi$
$102$	0	0
$103$	7.32270	0.721527	0.360764	−	0.932657i	$-0.382516\pi$
$103$	7.32270	0.721527	0.360764	+	0.932657i	$0.382516\pi$
$104$	0	0
$105$	−3.97847	−0.388259
$106$	0	0
$107$	−19.4348	−1.87884	−0.939419	−	0.342771i	$-0.888634\pi$
$107$	−19.4348	−1.87884	−0.939419	+	0.342771i	$0.888634\pi$
$108$	0	0
$109$	6.47232	0.619936	0.309968	−	0.950747i	$-0.399682\pi$
$109$	6.47232	0.619936	0.309968	+	0.950747i	$0.399682\pi$
$110$	0	0
$111$	4.72819	0.448780
$112$	0	0
$113$	10.7282	1.00922	0.504611	−	0.863347i	$-0.331636\pi$
$113$	10.7282	1.00922	0.504611	+	0.863347i	$0.331636\pi$
$114$	0	0
$115$	−4.74971	−0.442913
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	19.5176	1.78918
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.93316	−0.264474
$124$	0	0
$125$	−10.7894	−0.965037
$126$	0	0
$127$	−13.8093	−1.22538	−0.612689	−	0.790324i	$-0.709912\pi$
$127$	−13.8093	−1.22538	−0.612689	+	0.790324i	$0.709912\pi$
$128$	0	0
$129$	−7.74805	−0.682178
$130$	0	0
$131$	−18.7067	−1.63441	−0.817204	−	0.576348i	$-0.804477\pi$
$131$	−18.7067	−1.63441	−0.817204	+	0.576348i	$0.804477\pi$
$132$	0	0
$133$	−13.2399	−1.14805
$134$	0	0
$135$	−1.29726	−0.111650
$136$	0	0
$137$	12.1376	1.03698	0.518492	−	0.855082i	$-0.326494\pi$
$137$	12.1376	1.03698	0.518492	+	0.855082i	$0.326494\pi$
$138$	0	0
$139$	−1.17507	−0.0996678	−0.0498339	−	0.998758i	$-0.515869\pi$
$139$	−1.17507	−0.0996678	−0.0498339	+	0.998758i	$0.515869\pi$
$140$	0	0
$141$	−9.38396	−0.790272
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	5.66135	0.470150
$146$	0	0
$147$	−2.40549	−0.198401
$148$	0	0
$149$	−19.1575	−1.56944	−0.784720	−	0.619850i	$-0.787193\pi$
$149$	−19.1575	−1.56944	−0.784720	+	0.619850i	$0.787193\pi$
$150$	0	0
$151$	21.8348	1.77689	0.888444	−	0.458986i	$-0.151787\pi$
$151$	21.8348	1.77689	0.888444	+	0.458986i	$0.151787\pi$
$152$	0	0
$153$	6.36409	0.514506
$154$	0	0
$155$	−2.53358	−0.203502
$156$	0	0
$157$	7.12810	0.568884	0.284442	−	0.958693i	$-0.408192\pi$
$157$	7.12810	0.568884	0.284442	+	0.958693i	$0.408192\pi$
$158$	0	0
$159$	−11.9785	−0.949955
$160$	0	0
$161$	−11.2288	−0.884950
$162$	0	0
$163$	12.7752	1.00063	0.500314	−	0.865844i	$-0.333218\pi$
$163$	12.7752	1.00063	0.500314	+	0.865844i	$0.333218\pi$
$164$	0	0
$165$	−1.29726	−0.100991
$166$	0	0
$167$	−0.944645	−0.0730989	−0.0365494	−	0.999332i	$-0.511637\pi$
$167$	−0.944645	−0.0730989	−0.0365494	+	0.999332i	$0.511637\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.31712	−0.330139
$172$	0	0
$173$	3.83084	0.291253	0.145627	−	0.989340i	$-0.453480\pi$
$173$	3.83084	0.291253	0.145627	+	0.989340i	$0.453480\pi$
$174$	0	0
$175$	−10.1731	−0.769012
$176$	0	0
$177$	−2.59451	−0.195016
$178$	0	0
$179$	8.32828	0.622485	0.311242	−	0.950331i	$-0.399255\pi$
$179$	8.32828	0.622485	0.311242	+	0.950331i	$0.399255\pi$
$180$	0	0
$181$	9.13925	0.679315	0.339658	−	0.940549i	$-0.389689\pi$
$181$	9.13925	0.679315	0.339658	+	0.940549i	$0.389689\pi$
$182$	0	0
$183$	−10.0724	−0.744574
$184$	0	0
$185$	−6.13367	−0.450957
$186$	0	0
$187$	6.36409	0.465389
$188$	0	0
$189$	−3.06684	−0.223080
$190$	0	0
$191$	3.88785	0.281315	0.140658	−	0.990058i	$-0.455078\pi$
$191$	3.88785	0.281315	0.140658	+	0.990058i	$0.455078\pi$
$192$	0	0
$193$	15.0453	1.08299	0.541493	−	0.840706i	$-0.317859\pi$
$193$	15.0453	1.08299	0.541493	+	0.840706i	$0.317859\pi$
$194$	0	0
$195$	1.29726	0.0928985
$196$	0	0
$197$	4.85038	0.345575	0.172788	−	0.984959i	$-0.444723\pi$
$197$	4.85038	0.345575	0.172788	+	0.984959i	$0.444723\pi$
$198$	0	0
$199$	−20.7791	−1.47299	−0.736495	−	0.676443i	$-0.763520\pi$
$199$	−20.7791	−1.47299	−0.736495	+	0.676443i	$0.763520\pi$
$200$	0	0
$201$	4.70274	0.331706
$202$	0	0
$203$	13.3840	0.939370
$204$	0	0
$205$	3.80507	0.265757
$206$	0	0
$207$	−3.66135	−0.254481
$208$	0	0
$209$	−4.31712	−0.298622
$210$	0	0
$211$	21.6653	1.49150	0.745749	−	0.666227i	$-0.232092\pi$
$211$	21.6653	1.49150	0.745749	+	0.666227i	$0.232092\pi$
$212$	0	0
$213$	0.655774	0.0449329
$214$	0	0
$215$	10.0512	0.685487
$216$	0	0
$217$	−5.98963	−0.406602
$218$	0	0
$219$	−7.72261	−0.521846
$220$	0	0
$221$	−6.36409	−0.428095
$222$	0	0
$223$	15.4094	1.03189	0.515945	−	0.856622i	$-0.327441\pi$
$223$	15.4094	1.03189	0.515945	+	0.856622i	$0.327441\pi$
$224$	0	0
$225$	−3.31712	−0.221142
$226$	0	0
$227$	−12.5232	−0.831195	−0.415597	−	0.909549i	$-0.636427\pi$
$227$	−12.5232	−0.831195	−0.415597	+	0.909549i	$0.636427\pi$
$228$	0	0
$229$	−4.26735	−0.281994	−0.140997	−	0.990010i	$-0.545031\pi$
$229$	−4.26735	−0.281994	−0.140997	+	0.990010i	$0.545031\pi$
$230$	0	0
$231$	−3.06684	−0.201783
$232$	0	0
$233$	2.07522	0.135952	0.0679762	−	0.997687i	$-0.478346\pi$
$233$	2.07522	0.135952	0.0679762	+	0.997687i	$0.478346\pi$
$234$	0	0
$235$	12.1734	0.794106
$236$	0	0
$237$	4.30283	0.279499
$238$	0	0
$239$	1.56741	0.101387	0.0506937	−	0.998714i	$-0.483857\pi$
$239$	1.56741	0.101387	0.0506937	+	0.998714i	$0.483857\pi$
$240$	0	0
$241$	−3.85595	−0.248384	−0.124192	−	0.992258i	$-0.539634\pi$
$241$	−3.85595	−0.248384	−0.124192	+	0.992258i	$0.539634\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.12053	0.199364
$246$	0	0
$247$	4.31712	0.274692
$248$	0	0
$249$	8.72819	0.553126
$250$	0	0
$251$	30.3071	1.91297	0.956483	−	0.291788i	$-0.0942502\pi$
$251$	30.3071	1.91297	0.956483	+	0.291788i	$0.0942502\pi$
$252$	0	0
$253$	−3.66135	−0.230187
$254$	0	0
$255$	−8.25586	−0.517002
$256$	0	0
$257$	−24.9127	−1.55401	−0.777007	−	0.629492i	$-0.783263\pi$
$257$	−24.9127	−1.55401	−0.777007	+	0.629492i	$0.783263\pi$
$258$	0	0
$259$	−14.5006	−0.901022
$260$	0	0
$261$	4.36409	0.270131
$262$	0	0
$263$	17.5503	1.08220	0.541099	−	0.840959i	$-0.318008\pi$
$263$	17.5503	1.08220	0.541099	+	0.840959i	$0.318008\pi$
$264$	0	0
$265$	15.5392	0.954563
$266$	0	0
$267$	−6.68122	−0.408884
$268$	0	0
$269$	−11.3442	−0.691670	−0.345835	−	0.938295i	$-0.612404\pi$
$269$	−11.3442	−0.691670	−0.345835	+	0.938295i	$0.612404\pi$
$270$	0	0
$271$	−16.0624	−0.975719	−0.487860	−	0.872922i	$-0.662222\pi$
$271$	−16.0624	−0.975719	−0.487860	+	0.872922i	$0.662222\pi$
$272$	0	0
$273$	3.06684	0.185613
$274$	0	0
$275$	−3.31712	−0.200030
$276$	0	0
$277$	−4.13367	−0.248368	−0.124184	−	0.992259i	$-0.539631\pi$
$277$	−4.13367	−0.248368	−0.124184	+	0.992259i	$0.539631\pi$
$278$	0	0
$279$	−1.95303	−0.116925
$280$	0	0
$281$	14.1619	0.844830	0.422415	−	0.906403i	$-0.361183\pi$
$281$	14.1619	0.844830	0.422415	+	0.906403i	$0.361183\pi$
$282$	0	0
$283$	−14.1476	−0.840990	−0.420495	−	0.907295i	$-0.638144\pi$
$283$	−14.1476	−0.840990	−0.420495	+	0.907295i	$0.638144\pi$
$284$	0	0
$285$	5.60042	0.331740
$286$	0	0
$287$	8.99553	0.530990
$288$	0	0
$289$	23.5017	1.38245
$290$	0	0
$291$	−12.1121	−0.710027
$292$	0	0
$293$	−20.1731	−1.17852	−0.589262	−	0.807942i	$-0.700581\pi$
$293$	−20.1731	−1.17852	−0.589262	+	0.807942i	$0.700581\pi$
$294$	0	0
$295$	3.36575	0.195962
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	3.66135	0.211741
$300$	0	0
$301$	23.7620	1.36962
$302$	0	0
$303$	11.8817	0.682587
$304$	0	0
$305$	13.0665	0.748186
$306$	0	0
$307$	19.3954	1.10696	0.553478	−	0.832864i	$-0.313300\pi$
$307$	19.3954	1.10696	0.553478	+	0.832864i	$0.313300\pi$
$308$	0	0
$309$	−7.32270	−0.416574
$310$	0	0
$311$	−4.48347	−0.254235	−0.127117	−	0.991888i	$-0.540572\pi$
$311$	−4.48347	−0.254235	−0.127117	+	0.991888i	$0.540572\pi$
$312$	0	0
$313$	−0.900486	−0.0508985	−0.0254492	−	0.999676i	$-0.508102\pi$
$313$	−0.900486	−0.0508985	−0.0254492	+	0.999676i	$0.508102\pi$
$314$	0	0
$315$	3.97847	0.224162
$316$	0	0
$317$	1.49219	0.0838098	0.0419049	−	0.999122i	$-0.486657\pi$
$317$	1.49219	0.0838098	0.0419049	+	0.999122i	$0.486657\pi$
$318$	0	0
$319$	4.36409	0.244342
$320$	0	0
$321$	19.4348	1.08475
$322$	0	0
$323$	−27.4746	−1.52873
$324$	0	0
$325$	3.31712	0.184001
$326$	0	0
$327$	−6.47232	−0.357920
$328$	0	0
$329$	28.7791	1.58664
$330$	0	0
$331$	30.8658	1.69654	0.848268	−	0.529567i	$-0.177645\pi$
$331$	30.8658	1.69654	0.848268	+	0.529567i	$0.177645\pi$
$332$	0	0
$333$	−4.72819	−0.259103
$334$	0	0
$335$	−6.10067	−0.333315
$336$	0	0
$337$	26.1734	1.42576	0.712878	−	0.701288i	$-0.247391\pi$
$337$	26.1734	1.42576	0.712878	+	0.701288i	$0.247391\pi$
$338$	0	0
$339$	−10.7282	−0.582675
$340$	0	0
$341$	−1.95303	−0.105763
$342$	0	0
$343$	−14.0906	−0.760822
$344$	0	0
$345$	4.74971	0.255716
$346$	0	0
$347$	−0.350132	−0.0187960	−0.00939802	−	0.999956i	$-0.502992\pi$
$347$	−0.350132	−0.0187960	−0.00939802	+	0.999956i	$0.502992\pi$
$348$	0	0
$349$	−12.8619	−0.688480	−0.344240	−	0.938882i	$-0.611863\pi$
$349$	−12.8619	−0.688480	−0.344240	+	0.938882i	$0.611863\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−34.8331	−1.85398	−0.926989	−	0.375087i	$-0.877613\pi$
$353$	−34.8331	−1.85398	−0.926989	+	0.375087i	$0.877613\pi$
$354$	0	0
$355$	−0.850708	−0.0451509
$356$	0	0
$357$	−19.5176	−1.03298
$358$	0	0
$359$	−13.8348	−0.730171	−0.365085	−	0.930974i	$-0.618960\pi$
$359$	−13.8348	−0.730171	−0.365085	+	0.930974i	$0.618960\pi$
$360$	0	0
$361$	−0.362436	−0.0190756
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	10.0182	0.524377
$366$	0	0
$367$	24.1734	1.26184	0.630921	−	0.775847i	$-0.282677\pi$
$367$	24.1734	1.26184	0.630921	+	0.775847i	$0.282677\pi$
$368$	0	0
$369$	2.93316	0.152694
$370$	0	0
$371$	36.7360	1.90724
$372$	0	0
$373$	22.0509	1.14175	0.570876	−	0.821036i	$-0.306604\pi$
$373$	22.0509	1.14175	0.570876	+	0.821036i	$0.306604\pi$
$374$	0	0
$375$	10.7894	0.557165
$376$	0	0
$377$	−4.36409	−0.224762
$378$	0	0
$379$	−26.1885	−1.34521	−0.672606	−	0.740001i	$-0.734825\pi$
$379$	−26.1885	−1.34521	−0.672606	+	0.740001i	$0.734825\pi$
$380$	0	0
$381$	13.8093	0.707473
$382$	0	0
$383$	30.3071	1.54862	0.774310	−	0.632807i	$-0.218097\pi$
$383$	30.3071	1.54862	0.774310	+	0.632807i	$0.218097\pi$
$384$	0	0
$385$	3.97847	0.202762
$386$	0	0
$387$	7.74805	0.393856
$388$	0	0
$389$	−10.5221	−0.533492	−0.266746	−	0.963767i	$-0.585948\pi$
$389$	−10.5221	−0.533492	−0.266746	+	0.963767i	$0.585948\pi$
$390$	0	0
$391$	−23.3012	−1.17839
$392$	0	0
$393$	18.7067	0.943626
$394$	0	0
$395$	−5.58188	−0.280855
$396$	0	0
$397$	−7.22876	−0.362801	−0.181401	−	0.983409i	$-0.558063\pi$
$397$	−7.22876	−0.362801	−0.181401	+	0.983409i	$0.558063\pi$
$398$	0	0
$399$	13.2399	0.662825
$400$	0	0
$401$	−39.3879	−1.96694	−0.983468	−	0.181080i	$-0.942041\pi$
$401$	−39.3879	−1.96694	−0.983468	+	0.181080i	$0.942041\pi$
$402$	0	0
$403$	1.95303	0.0972874
$404$	0	0
$405$	1.29726	0.0644612
$406$	0	0
$407$	−4.72819	−0.234368
$408$	0	0
$409$	−1.85628	−0.0917873	−0.0458937	−	0.998946i	$-0.514614\pi$
$409$	−1.85628	−0.0917873	−0.0458937	+	0.998946i	$0.514614\pi$
$410$	0	0
$411$	−12.1376	−0.598703
$412$	0	0
$413$	7.95695	0.391536
$414$	0	0
$415$	−11.3227	−0.555810
$416$	0	0
$417$	1.17507	0.0575432
$418$	0	0
$419$	5.80540	0.283612	0.141806	−	0.989894i	$-0.454709\pi$
$419$	5.80540	0.283612	0.141806	+	0.989894i	$0.454709\pi$
$420$	0	0
$421$	16.7497	0.816331	0.408166	−	0.912908i	$-0.366169\pi$
$421$	16.7497	0.816331	0.408166	+	0.912908i	$0.366169\pi$
$422$	0	0
$423$	9.38396	0.456264
$424$	0	0
$425$	−21.1105	−1.02401
$426$	0	0
$427$	30.8904	1.49489
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	20.6773	0.995990	0.497995	−	0.867180i	$-0.334070\pi$
$431$	20.6773	0.995990	0.497995	+	0.867180i	$0.334070\pi$
$432$	0	0
$433$	3.23208	0.155324	0.0776619	−	0.996980i	$-0.475255\pi$
$433$	3.23208	0.155324	0.0776619	+	0.996980i	$0.475255\pi$
$434$	0	0
$435$	−5.66135	−0.271441
$436$	0	0
$437$	15.8065	0.756128
$438$	0	0
$439$	0.980132	0.0467792	0.0233896	−	0.999726i	$-0.492554\pi$
$439$	0.980132	0.0467792	0.0233896	+	0.999726i	$0.492554\pi$
$440$	0	0
$441$	2.40549	0.114547
$442$	0	0
$443$	10.9558	0.520526	0.260263	−	0.965538i	$-0.416191\pi$
$443$	10.9558	0.520526	0.260263	+	0.965538i	$0.416191\pi$
$444$	0	0
$445$	8.66726	0.410867
$446$	0	0
$447$	19.1575	0.906117
$448$	0	0
$449$	−6.94713	−0.327855	−0.163928	−	0.986472i	$-0.552416\pi$
$449$	−6.94713	−0.327855	−0.163928	+	0.986472i	$0.552416\pi$
$450$	0	0
$451$	2.93316	0.138117
$452$	0	0
$453$	−21.8348	−1.02589
$454$	0	0
$455$	−3.97847	−0.186514
$456$	0	0
$457$	17.6283	0.824619	0.412310	−	0.911044i	$-0.364722\pi$
$457$	17.6283	0.824619	0.412310	+	0.911044i	$0.364722\pi$
$458$	0	0
$459$	−6.36409	−0.297050
$460$	0	0
$461$	14.0115	0.652580	0.326290	−	0.945270i	$-0.394201\pi$
$461$	14.0115	0.652580	0.326290	+	0.945270i	$0.394201\pi$
$462$	0	0
$463$	−20.6994	−0.961984	−0.480992	−	0.876725i	$-0.659723\pi$
$463$	−20.6994	−0.961984	−0.480992	+	0.876725i	$0.659723\pi$
$464$	0	0
$465$	2.53358	0.117492
$466$	0	0
$467$	−11.3557	−0.525479	−0.262740	−	0.964867i	$-0.584626\pi$
$467$	−11.3557	−0.525479	−0.262740	+	0.964867i	$0.584626\pi$
$468$	0	0
$469$	−14.4225	−0.665971
$470$	0	0
$471$	−7.12810	−0.328445
$472$	0	0
$473$	7.74805	0.356256
$474$	0	0
$475$	14.3204	0.657067
$476$	0	0
$477$	11.9785	0.548457
$478$	0	0
$479$	18.5121	0.845838	0.422919	−	0.906168i	$-0.361006\pi$
$479$	18.5121	0.845838	0.422919	+	0.906168i	$0.361006\pi$
$480$	0	0
$481$	4.72819	0.215587
$482$	0	0
$483$	11.2288	0.510926
$484$	0	0
$485$	15.7126	0.713471
$486$	0	0
$487$	−5.78106	−0.261965	−0.130982	−	0.991385i	$-0.541813\pi$
$487$	−5.78106	−0.261965	−0.130982	+	0.991385i	$0.541813\pi$
$488$	0	0
$489$	−12.7752	−0.577713
$490$	0	0
$491$	12.7393	0.574918	0.287459	−	0.957793i	$-0.407189\pi$
$491$	12.7393	0.574918	0.287459	+	0.957793i	$0.407189\pi$
$492$	0	0
$493$	27.7735	1.25086
$494$	0	0
$495$	1.29726	0.0583074
$496$	0	0
$497$	−2.01115	−0.0902125
$498$	0	0
$499$	−25.3370	−1.13424	−0.567120	−	0.823635i	$-0.691942\pi$
$499$	−25.3370	−1.13424	−0.567120	+	0.823635i	$0.691942\pi$
$500$	0	0
$501$	0.944645	0.0422036
$502$	0	0
$503$	32.0509	1.42908	0.714539	−	0.699595i	$-0.246636\pi$
$503$	32.0509	1.42908	0.714539	+	0.699595i	$0.246636\pi$
$504$	0	0
$505$	−15.4137	−0.685899
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−22.9486	−1.01718	−0.508589	−	0.861010i	$-0.669833\pi$
$509$	−22.9486	−1.01718	−0.508589	+	0.861010i	$0.669833\pi$
$510$	0	0
$511$	23.6840	1.04772
$512$	0	0
$513$	4.31712	0.190606
$514$	0	0
$515$	9.49942	0.418595
$516$	0	0
$517$	9.38396	0.412706
$518$	0	0
$519$	−3.83084	−0.168155
$520$	0	0
$521$	12.8110	0.561259	0.280629	−	0.959816i	$-0.409457\pi$
$521$	12.8110	0.561259	0.280629	+	0.959816i	$0.409457\pi$
$522$	0	0
$523$	24.1658	1.05670	0.528349	−	0.849027i	$-0.322811\pi$
$523$	24.1658	1.05670	0.528349	+	0.849027i	$0.322811\pi$
$524$	0	0
$525$	10.1731	0.443990
$526$	0	0
$527$	−12.4293	−0.541428
$528$	0	0
$529$	−9.59451	−0.417153
$530$	0	0
$531$	2.59451	0.112592
$532$	0	0
$533$	−2.93316	−0.127049
$534$	0	0
$535$	−25.2120	−1.09001
$536$	0	0
$537$	−8.32828	−0.359392
$538$	0	0
$539$	2.40549	0.103612
$540$	0	0
$541$	15.1890	0.653027	0.326514	−	0.945192i	$-0.394126\pi$
$541$	15.1890	0.653027	0.326514	+	0.945192i	$0.394126\pi$
$542$	0	0
$543$	−9.13925	−0.392203
$544$	0	0
$545$	8.39627	0.359656
$546$	0	0
$547$	−41.5498	−1.77654	−0.888271	−	0.459320i	$-0.848093\pi$
$547$	−41.5498	−1.77654	−0.888271	+	0.459320i	$0.848093\pi$
$548$	0	0
$549$	10.0724	0.429880
$550$	0	0
$551$	−18.8403	−0.802625
$552$	0	0
$553$	−13.1961	−0.561155
$554$	0	0
$555$	6.13367	0.260360
$556$	0	0
$557$	−16.1222	−0.683119	−0.341560	−	0.939860i	$-0.610955\pi$
$557$	−16.1222	−0.683119	−0.341560	+	0.939860i	$0.610955\pi$
$558$	0	0
$559$	−7.74805	−0.327708
$560$	0	0
$561$	−6.36409	−0.268692
$562$	0	0
$563$	−20.5960	−0.868016	−0.434008	−	0.900909i	$-0.642901\pi$
$563$	−20.5960	−0.868016	−0.434008	+	0.900909i	$0.642901\pi$
$564$	0	0
$565$	13.9172	0.585502
$566$	0	0
$567$	3.06684	0.128795
$568$	0	0
$569$	36.2383	1.51919	0.759593	−	0.650398i	$-0.225398\pi$
$569$	36.2383	1.51919	0.759593	+	0.650398i	$0.225398\pi$
$570$	0	0
$571$	−1.65743	−0.0693614	−0.0346807	−	0.999398i	$-0.511041\pi$
$571$	−1.65743	−0.0693614	−0.0346807	+	0.999398i	$0.511041\pi$
$572$	0	0
$573$	−3.88785	−0.162417
$574$	0	0
$575$	12.1452	0.506488
$576$	0	0
$577$	−15.5559	−0.647602	−0.323801	−	0.946125i	$-0.604961\pi$
$577$	−15.5559	−0.647602	−0.323801	+	0.946125i	$0.604961\pi$
$578$	0	0
$579$	−15.0453	−0.625262
$580$	0	0
$581$	−26.7679	−1.11052
$582$	0	0
$583$	11.9785	0.496098
$584$	0	0
$585$	−1.29726	−0.0536350
$586$	0	0
$587$	41.1969	1.70038	0.850188	−	0.526479i	$-0.176488\pi$
$587$	41.1969	1.70038	0.850188	+	0.526479i	$0.176488\pi$
$588$	0	0
$589$	8.43148	0.347413
$590$	0	0
$591$	−4.85038	−0.199518
$592$	0	0
$593$	33.2402	1.36501	0.682507	−	0.730879i	$-0.260890\pi$
$593$	33.2402	1.36501	0.682507	+	0.730879i	$0.260890\pi$
$594$	0	0
$595$	25.3194	1.03799
$596$	0	0
$597$	20.7791	0.850431
$598$	0	0
$599$	30.9231	1.26348	0.631742	−	0.775179i	$-0.282340\pi$
$599$	30.9231	1.26348	0.631742	+	0.775179i	$0.282340\pi$
$600$	0	0
$601$	32.7249	1.33488	0.667438	−	0.744666i	$-0.267391\pi$
$601$	32.7249	1.33488	0.667438	+	0.744666i	$0.267391\pi$
$602$	0	0
$603$	−4.70274	−0.191511
$604$	0	0
$605$	1.29726	0.0527410
$606$	0	0
$607$	37.1034	1.50598	0.752991	−	0.658031i	$-0.228610\pi$
$607$	37.1034	1.50598	0.752991	+	0.658031i	$0.228610\pi$
$608$	0	0
$609$	−13.3840	−0.542345
$610$	0	0
$611$	−9.38396	−0.379634
$612$	0	0
$613$	5.52768	0.223261	0.111630	−	0.993750i	$-0.464393\pi$
$613$	5.52768	0.223261	0.111630	+	0.993750i	$0.464393\pi$
$614$	0	0
$615$	−3.80507	−0.153435
$616$	0	0
$617$	34.1989	1.37679	0.688397	−	0.725334i	$-0.258315\pi$
$617$	34.1989	1.37679	0.688397	+	0.725334i	$0.258315\pi$
$618$	0	0
$619$	25.8932	1.04074	0.520368	−	0.853942i	$-0.325795\pi$
$619$	25.8932	1.04074	0.520368	+	0.853942i	$0.325795\pi$
$620$	0	0
$621$	3.66135	0.146925
$622$	0	0
$623$	20.4902	0.820923
$624$	0	0
$625$	2.58894	0.103558
$626$	0	0
$627$	4.31712	0.172409
$628$	0	0
$629$	−30.0906	−1.19979
$630$	0	0
$631$	35.2939	1.40503	0.702515	−	0.711669i	$-0.252060\pi$
$631$	35.2939	1.40503	0.702515	+	0.711669i	$0.252060\pi$
$632$	0	0
$633$	−21.6653	−0.861117
$634$	0	0
$635$	−17.9142	−0.710904
$636$	0	0
$637$	−2.40549	−0.0953088
$638$	0	0
$639$	−0.655774	−0.0259420
$640$	0	0
$641$	−13.2797	−0.524515	−0.262257	−	0.964998i	$-0.584467\pi$
$641$	−13.2797	−0.524515	−0.262257	+	0.964998i	$0.584467\pi$
$642$	0	0
$643$	−1.98571	−0.0783087	−0.0391543	−	0.999233i	$-0.512466\pi$
$643$	−1.98571	−0.0783087	−0.0391543	+	0.999233i	$0.512466\pi$
$644$	0	0
$645$	−10.0512	−0.395766
$646$	0	0
$647$	−12.3000	−0.483564	−0.241782	−	0.970331i	$-0.577732\pi$
$647$	−12.3000	−0.483564	−0.241782	+	0.970331i	$0.577732\pi$
$648$	0	0
$649$	2.59451	0.101844
$650$	0	0
$651$	5.98963	0.234752
$652$	0	0
$653$	0.339759	0.0132958	0.00664789	−	0.999978i	$-0.497884\pi$
$653$	0.339759	0.0132958	0.00664789	+	0.999978i	$0.497884\pi$
$654$	0	0
$655$	−24.2673	−0.948204
$656$	0	0
$657$	7.72261	0.301288
$658$	0	0
$659$	−27.6012	−1.07519	−0.537595	−	0.843203i	$-0.680667\pi$
$659$	−27.6012	−1.07519	−0.537595	+	0.843203i	$0.680667\pi$
$660$	0	0
$661$	38.1916	1.48548	0.742741	−	0.669579i	$-0.233525\pi$
$661$	38.1916	1.48548	0.742741	+	0.669579i	$0.233525\pi$
$662$	0	0
$663$	6.36409	0.247161
$664$	0	0
$665$	−17.1756	−0.666040
$666$	0	0
$667$	−15.9785	−0.618689
$668$	0	0
$669$	−15.4094	−0.595762
$670$	0	0
$671$	10.0724	0.388841
$672$	0	0
$673$	−19.8018	−0.763301	−0.381651	−	0.924307i	$-0.624644\pi$
$673$	−19.8018	−0.763301	−0.381651	+	0.924307i	$0.624644\pi$
$674$	0	0
$675$	3.31712	0.127676
$676$	0	0
$677$	16.0481	0.616778	0.308389	−	0.951260i	$-0.400210\pi$
$677$	16.0481	0.616778	0.308389	+	0.951260i	$0.400210\pi$
$678$	0	0
$679$	37.1460	1.42553
$680$	0	0
$681$	12.5232	0.479890
$682$	0	0
$683$	11.7341	0.448993	0.224496	−	0.974475i	$-0.427926\pi$
$683$	11.7341	0.448993	0.224496	+	0.974475i	$0.427926\pi$
$684$	0	0
$685$	15.7456	0.601607
$686$	0	0
$687$	4.26735	0.162809
$688$	0	0
$689$	−11.9785	−0.456344
$690$	0	0
$691$	13.1421	0.499947	0.249974	−	0.968253i	$-0.419578\pi$
$691$	13.1421	0.499947	0.249974	+	0.968253i	$0.419578\pi$
$692$	0	0
$693$	3.06684	0.116499
$694$	0	0
$695$	−1.52436	−0.0578224
$696$	0	0
$697$	18.6669	0.707060
$698$	0	0
$699$	−2.07522	−0.0784921
$700$	0	0
$701$	−29.0708	−1.09799	−0.548994	−	0.835827i	$-0.684989\pi$
$701$	−29.0708	−1.09799	−0.548994	+	0.835827i	$0.684989\pi$
$702$	0	0
$703$	20.4122	0.769860
$704$	0	0
$705$	−12.1734	−0.458477
$706$	0	0
$707$	−36.4393	−1.37044
$708$	0	0
$709$	−31.7779	−1.19344	−0.596722	−	0.802448i	$-0.703531\pi$
$709$	−31.7779	−1.19344	−0.596722	+	0.802448i	$0.703531\pi$
$710$	0	0
$711$	−4.30283	−0.161369
$712$	0	0
$713$	7.15073	0.267797
$714$	0	0
$715$	−1.29726	−0.0485147
$716$	0	0
$717$	−1.56741	−0.0585361
$718$	0	0
$719$	6.99409	0.260836	0.130418	−	0.991459i	$-0.458368\pi$
$719$	6.99409	0.260836	0.130418	+	0.991459i	$0.458368\pi$
$720$	0	0
$721$	22.4575	0.836362
$722$	0	0
$723$	3.85595	0.143404
$724$	0	0
$725$	−14.4762	−0.537634
$726$	0	0
$727$	−19.6840	−0.730039	−0.365019	−	0.931000i	$-0.618938\pi$
$727$	−19.6840	−0.730039	−0.365019	+	0.931000i	$0.618938\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	49.3093	1.82377
$732$	0	0
$733$	−45.1541	−1.66781	−0.833903	−	0.551911i	$-0.813899\pi$
$733$	−45.1541	−1.66781	−0.833903	+	0.551911i	$0.813899\pi$
$734$	0	0
$735$	−3.12053	−0.115103
$736$	0	0
$737$	−4.70274	−0.173228
$738$	0	0
$739$	38.6521	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$739$	38.6521	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$740$	0	0
$741$	−4.31712	−0.158594
$742$	0	0
$743$	37.6298	1.38050	0.690251	−	0.723570i	$-0.257500\pi$
$743$	37.6298	1.38050	0.690251	+	0.723570i	$0.257500\pi$
$744$	0	0
$745$	−24.8521	−0.910512
$746$	0	0
$747$	−8.72819	−0.319348
$748$	0	0
$749$	−59.6035	−2.17786
$750$	0	0
$751$	−27.0067	−0.985488	−0.492744	−	0.870174i	$-0.664006\pi$
$751$	−27.0067	−0.985488	−0.492744	+	0.870174i	$0.664006\pi$
$752$	0	0
$753$	−30.3071	−1.10445
$754$	0	0
$755$	28.3253	1.03086
$756$	0	0
$757$	−25.0739	−0.911326	−0.455663	−	0.890152i	$-0.650598\pi$
$757$	−25.0739	−0.911326	−0.455663	+	0.890152i	$0.650598\pi$
$758$	0	0
$759$	3.66135	0.132899
$760$	0	0
$761$	25.6295	0.929067	0.464533	−	0.885556i	$-0.346222\pi$
$761$	25.6295	0.929067	0.464533	+	0.885556i	$0.346222\pi$
$762$	0	0
$763$	19.8496	0.718602
$764$	0	0
$765$	8.25586	0.298491
$766$	0	0
$767$	−2.59451	−0.0936825
$768$	0	0
$769$	−48.4846	−1.74840	−0.874199	−	0.485567i	$-0.838613\pi$
$769$	−48.4846	−1.74840	−0.874199	+	0.485567i	$0.838613\pi$
$770$	0	0
$771$	24.9127	0.897211
$772$	0	0
$773$	49.4284	1.77782	0.888908	−	0.458086i	$-0.151465\pi$
$773$	49.4284	1.77782	0.888908	+	0.458086i	$0.151465\pi$
$774$	0	0
$775$	6.47845	0.232713
$776$	0	0
$777$	14.5006	0.520205
$778$	0	0
$779$	−12.6628	−0.453693
$780$	0	0
$781$	−0.655774	−0.0234654
$782$	0	0
$783$	−4.36409	−0.155960
$784$	0	0
$785$	9.24697	0.330039
$786$	0	0
$787$	−32.2176	−1.14843	−0.574216	−	0.818704i	$-0.694693\pi$
$787$	−32.2176	−1.14843	−0.574216	+	0.818704i	$0.694693\pi$
$788$	0	0
$789$	−17.5503	−0.624808
$790$	0	0
$791$	32.9016	1.16985
$792$	0	0
$793$	−10.0724	−0.357682
$794$	0	0
$795$	−15.5392	−0.551117
$796$	0	0
$797$	−39.2072	−1.38879	−0.694396	−	0.719593i	$-0.744328\pi$
$797$	−39.2072	−1.38879	−0.694396	+	0.719593i	$0.744328\pi$
$798$	0	0
$799$	59.7204	2.11276
$800$	0	0
$801$	6.68122	0.236069
$802$	0	0
$803$	7.72261	0.272525
$804$	0	0
$805$	−14.5666	−0.513405
$806$	0	0
$807$	11.3442	0.399336
$808$	0	0
$809$	−40.0878	−1.40941	−0.704706	−	0.709499i	$-0.748921\pi$
$809$	−40.0878	−1.40941	−0.704706	+	0.709499i	$0.748921\pi$
$810$	0	0
$811$	−33.5291	−1.17737	−0.588683	−	0.808364i	$-0.700353\pi$
$811$	−33.5291	−1.17737	−0.588683	+	0.808364i	$0.700353\pi$
$812$	0	0
$813$	16.0624	0.563332
$814$	0	0
$815$	16.5727	0.580515
$816$	0	0
$817$	−33.4493	−1.17024
$818$	0	0
$819$	−3.06684	−0.107164
$820$	0	0
$821$	21.5076	0.750620	0.375310	−	0.926899i	$-0.377536\pi$
$821$	21.5076	0.750620	0.375310	+	0.926899i	$0.377536\pi$
$822$	0	0
$823$	17.9603	0.626055	0.313028	−	0.949744i	$-0.398657\pi$
$823$	17.9603	0.626055	0.313028	+	0.949744i	$0.398657\pi$
$824$	0	0
$825$	3.31712	0.115487
$826$	0	0
$827$	−22.0226	−0.765802	−0.382901	−	0.923789i	$-0.625075\pi$
$827$	−22.0226	−0.765802	−0.382901	+	0.923789i	$0.625075\pi$
$828$	0	0
$829$	21.8121	0.757566	0.378783	−	0.925485i	$-0.376343\pi$
$829$	21.8121	0.757566	0.378783	+	0.925485i	$0.376343\pi$
$830$	0	0
$831$	4.13367	0.143396
$832$	0	0
$833$	15.3087	0.530416
$834$	0	0
$835$	−1.22545	−0.0424084
$836$	0	0
$837$	1.95303	0.0675066
$838$	0	0
$839$	0.865175	0.0298692	0.0149346	−	0.999888i	$-0.495246\pi$
$839$	0.865175	0.0298692	0.0149346	+	0.999888i	$0.495246\pi$
$840$	0	0
$841$	−9.95469	−0.343265
$842$	0	0
$843$	−14.1619	−0.487763
$844$	0	0
$845$	1.29726	0.0446270
$846$	0	0
$847$	3.06684	0.105378
$848$	0	0
$849$	14.1476	0.485546
$850$	0	0
$851$	17.3115	0.593432
$852$	0	0
$853$	21.3227	0.730075	0.365038	−	0.930993i	$-0.381056\pi$
$853$	21.3227	0.730075	0.365038	+	0.930993i	$0.381056\pi$
$854$	0	0
$855$	−5.60042	−0.191530
$856$	0	0
$857$	−23.2442	−0.794005	−0.397003	−	0.917817i	$-0.629950\pi$
$857$	−23.2442	−0.794005	−0.397003	+	0.917817i	$0.629950\pi$
$858$	0	0
$859$	−24.2953	−0.828944	−0.414472	−	0.910062i	$-0.636034\pi$
$859$	−24.2953	−0.828944	−0.414472	+	0.910062i	$0.636034\pi$
$860$	0	0
$861$	−8.99553	−0.306567
$862$	0	0
$863$	−35.2288	−1.19920	−0.599601	−	0.800299i	$-0.704674\pi$
$863$	−35.2288	−1.19920	−0.599601	+	0.800299i	$0.704674\pi$
$864$	0	0
$865$	4.96958	0.168971
$866$	0	0
$867$	−23.5017	−0.798159
$868$	0	0
$869$	−4.30283	−0.145964
$870$	0	0
$871$	4.70274	0.159346
$872$	0	0
$873$	12.1121	0.409934
$874$	0	0
$875$	−33.0895	−1.11863
$876$	0	0
$877$	−47.7316	−1.61178	−0.805890	−	0.592065i	$-0.798313\pi$
$877$	−47.7316	−1.61178	−0.805890	+	0.592065i	$0.798313\pi$
$878$	0	0
$879$	20.1731	0.680421
$880$	0	0
$881$	13.4223	0.452207	0.226104	−	0.974103i	$-0.427401\pi$
$881$	13.4223	0.452207	0.226104	+	0.974103i	$0.427401\pi$
$882$	0	0
$883$	−51.5900	−1.73614	−0.868072	−	0.496439i	$-0.834641\pi$
$883$	−51.5900	−1.73614	−0.868072	+	0.496439i	$0.834641\pi$
$884$	0	0
$885$	−3.36575	−0.113138
$886$	0	0
$887$	−13.6513	−0.458366	−0.229183	−	0.973383i	$-0.573605\pi$
$887$	−13.6513	−0.458366	−0.229183	+	0.973383i	$0.573605\pi$
$888$	0	0
$889$	−42.3509	−1.42040
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−40.5117	−1.35567
$894$	0	0
$895$	10.8039	0.361135
$896$	0	0
$897$	−3.66135	−0.122249
$898$	0	0
$899$	−8.52321	−0.284265
$900$	0	0
$901$	76.2321	2.53966
$902$	0	0
$903$	−23.7620	−0.790750
$904$	0	0
$905$	11.8560	0.394105
$906$	0	0
$907$	−21.7523	−0.722273	−0.361137	−	0.932513i	$-0.617611\pi$
$907$	−21.7523	−0.722273	−0.361137	+	0.932513i	$0.617611\pi$
$908$	0	0
$909$	−11.8817	−0.394092
$910$	0	0
$911$	−46.4472	−1.53886	−0.769431	−	0.638729i	$-0.779460\pi$
$911$	−46.4472	−1.53886	−0.769431	+	0.638729i	$0.779460\pi$
$912$	0	0
$913$	−8.72819	−0.288861
$914$	0	0
$915$	−13.0665	−0.431965
$916$	0	0
$917$	−57.3703	−1.89453
$918$	0	0
$919$	42.3066	1.39557	0.697783	−	0.716310i	$-0.254170\pi$
$919$	42.3066	1.39557	0.697783	+	0.716310i	$0.254170\pi$
$920$	0	0
$921$	−19.3954	−0.639102
$922$	0	0
$923$	0.655774	0.0215851
$924$	0	0
$925$	15.6840	0.515686
$926$	0	0
$927$	7.32270	0.240509
$928$	0	0
$929$	−50.0979	−1.64366	−0.821829	−	0.569735i	$-0.807046\pi$
$929$	−50.0979	−1.64366	−0.821829	+	0.569735i	$0.807046\pi$
$930$	0	0
$931$	−10.3848	−0.340348
$932$	0	0
$933$	4.48347	0.146782
$934$	0	0
$935$	8.25586	0.269996
$936$	0	0
$937$	8.93427	0.291870	0.145935	−	0.989294i	$-0.453381\pi$
$937$	8.93427	0.291870	0.145935	+	0.989294i	$0.453381\pi$
$938$	0	0
$939$	0.900486	0.0293863
$940$	0	0
$941$	−9.16861	−0.298888	−0.149444	−	0.988770i	$-0.547748\pi$
$941$	−9.16861	−0.298888	−0.149444	+	0.988770i	$0.547748\pi$
$942$	0	0
$943$	−10.7393	−0.349721
$944$	0	0
$945$	−3.97847	−0.129420
$946$	0	0
$947$	−40.5841	−1.31881	−0.659404	−	0.751789i	$-0.729191\pi$
$947$	−40.5841	−1.31881	−0.659404	+	0.751789i	$0.729191\pi$
$948$	0	0
$949$	−7.72261	−0.250687
$950$	0	0
$951$	−1.49219	−0.0483876
$952$	0	0
$953$	17.7592	0.575277	0.287639	−	0.957739i	$-0.407130\pi$
$953$	17.7592	0.575277	0.287639	+	0.957739i	$0.407130\pi$
$954$	0	0
$955$	5.04354	0.163205
$956$	0	0
$957$	−4.36409	−0.141071
$958$	0	0
$959$	37.2240	1.20203
$960$	0	0
$961$	−27.1857	−0.876957
$962$	0	0
$963$	−19.4348	−0.626279
$964$	0	0
$965$	19.5176	0.628295
$966$	0	0
$967$	46.7643	1.50384	0.751919	−	0.659256i	$-0.229128\pi$
$967$	46.7643	1.50384	0.751919	+	0.659256i	$0.229128\pi$
$968$	0	0
$969$	27.4746	0.882611
$970$	0	0
$971$	−31.6001	−1.01410	−0.507048	−	0.861918i	$-0.669263\pi$
$971$	−31.6001	−1.01410	−0.507048	+	0.861918i	$0.669263\pi$
$972$	0	0
$973$	−3.60373	−0.115530
$974$	0	0
$975$	−3.31712	−0.106233
$976$	0	0
$977$	−9.22484	−0.295129	−0.147564	−	0.989052i	$-0.547143\pi$
$977$	−9.22484	−0.295129	−0.147564	+	0.989052i	$0.547143\pi$
$978$	0	0
$979$	6.68122	0.213533
$980$	0	0
$981$	6.47232	0.206645
$982$	0	0
$983$	−11.9004	−0.379565	−0.189783	−	0.981826i	$-0.560778\pi$
$983$	−11.9004	−0.379565	−0.189783	+	0.981826i	$0.560778\pi$
$984$	0	0
$985$	6.29219	0.200486
$986$	0	0
$987$	−28.7791	−0.916048
$988$	0	0
$989$	−28.3683	−0.902061
$990$	0	0
$991$	−12.7512	−0.405054	−0.202527	−	0.979277i	$-0.564915\pi$
$991$	−12.7512	−0.405054	−0.202527	+	0.979277i	$0.564915\pi$
$992$	0	0
$993$	−30.8658	−0.979496
$994$	0	0
$995$	−26.9558	−0.854556
$996$	0	0
$997$	−16.9844	−0.537901	−0.268950	−	0.963154i	$-0.586677\pi$
$997$	−16.9844	−0.537901	−0.268950	+	0.963154i	$0.586677\pi$
$998$	0	0
$999$	4.72819	0.149593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bx.1.3		4
4.3	odd	2		3432.2.a.v.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.v.1.3	✓	4	4.3	odd	2
6864.2.a.bx.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.29726 q^{5} +3.06684 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.29726 q^{5} +3.06684 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.29726 q^{15} +6.36409 q^{17} -4.31712 q^{19} -3.06684 q^{21} -3.66135 q^{23} -3.31712 q^{25} -1.00000 q^{27} +4.36409 q^{29} -1.95303 q^{31} -1.00000 q^{33} +3.97847 q^{35} -4.72819 q^{37} +1.00000 q^{39} +2.93316 q^{41} +7.74805 q^{43} +1.29726 q^{45} +9.38396 q^{47} +2.40549 q^{49} -6.36409 q^{51} +11.9785 q^{53} +1.29726 q^{55} +4.31712 q^{57} +2.59451 q^{59} +10.0724 q^{61} +3.06684 q^{63} -1.29726 q^{65} -4.70274 q^{67} +3.66135 q^{69} -0.655774 q^{71} +7.72261 q^{73} +3.31712 q^{75} +3.06684 q^{77} -4.30283 q^{79} +1.00000 q^{81} -8.72819 q^{83} +8.25586 q^{85} -4.36409 q^{87} +6.68122 q^{89} -3.06684 q^{91} +1.95303 q^{93} -5.60042 q^{95} +12.1121 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{15} + 12 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} - 4 q^{33} - 22 q^{35} + 8 q^{37} + 4 q^{39} + 22 q^{41} - 14 q^{43} + 2 q^{45} + 6 q^{47} + 16 q^{49} - 12 q^{51} + 10 q^{53} + 2 q^{55} - 4 q^{57} + 4 q^{59} + 18 q^{61} + 2 q^{63} - 2 q^{65} - 22 q^{67} - 2 q^{69} + 2 q^{71} + 16 q^{73} - 8 q^{75} + 2 q^{77} - 2 q^{79} + 4 q^{81} - 8 q^{83} + 10 q^{85} - 4 q^{87} - 8 q^{89} - 2 q^{91} - 10 q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 2 * q^15 + 12 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 - 4 * q^33 - 22 * q^35 + 8 * q^37 + 4 * q^39 + 22 * q^41 - 14 * q^43 + 2 * q^45 + 6 * q^47 + 16 * q^49 - 12 * q^51 + 10 * q^53 + 2 * q^55 - 4 * q^57 + 4 * q^59 + 18 * q^61 + 2 * q^63 - 2 * q^65 - 22 * q^67 - 2 * q^69 + 2 * q^71 + 16 * q^73 - 8 * q^75 + 2 * q^77 - 2 * q^79 + 4 * q^81 - 8 * q^83 + 10 * q^85 - 4 * q^87 - 8 * q^89 - 2 * q^91 - 10 * q^95 - 10 * q^97 + 4 * q^99

Introduction

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