LMFDB - Embedded newform 6864.2.a.bl.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.37228 q^{15} -4.00000 q^{17} +6.74456 q^{19} +3.37228 q^{21} +3.37228 q^{23} +6.37228 q^{25} +1.00000 q^{27} -3.37228 q^{29} -8.74456 q^{31} +1.00000 q^{33} +11.3723 q^{35} -2.00000 q^{37} -1.00000 q^{39} +5.37228 q^{41} +2.62772 q^{43} +3.37228 q^{45} +4.00000 q^{47} +4.37228 q^{49} -4.00000 q^{51} -10.0000 q^{53} +3.37228 q^{55} +6.74456 q^{57} +11.3723 q^{59} +8.11684 q^{61} +3.37228 q^{63} -3.37228 q^{65} -12.1168 q^{67} +3.37228 q^{69} +14.7446 q^{71} -8.11684 q^{73} +6.37228 q^{75} +3.37228 q^{77} +10.0000 q^{79} +1.00000 q^{81} -2.74456 q^{83} -13.4891 q^{85} -3.37228 q^{87} +8.00000 q^{89} -3.37228 q^{91} -8.74456 q^{93} +22.7446 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} - 8 q^{17} + 2 q^{19} + q^{21} + q^{23} + 7 q^{25} + 2 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 17 q^{35} - 4 q^{37} - 2 q^{39} + 5 q^{41} + 11 q^{43} + q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 20 q^{53} + q^{55} + 2 q^{57} + 17 q^{59} - q^{61} + q^{63} - q^{65} - 7 q^{67} + q^{69} + 18 q^{71} + q^{73} + 7 q^{75} + q^{77} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 4 q^{85} - q^{87} + 16 q^{89} - q^{91} - 6 q^{93} + 34 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 - 8 * q^17 + 2 * q^19 + q^21 + q^23 + 7 * q^25 + 2 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 17 * q^35 - 4 * q^37 - 2 * q^39 + 5 * q^41 + 11 * q^43 + q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 20 * q^53 + q^55 + 2 * q^57 + 17 * q^59 - q^61 + q^63 - q^65 - 7 * q^67 + q^69 + 18 * q^71 + q^73 + 7 * q^75 + q^77 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 4 * q^85 - q^87 + 16 * q^89 - q^91 - 6 * q^93 + 34 * q^95 - 20 * q^97 + 2 * 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
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	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
$15$	3.37228	0.870719
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.74456	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$19$	6.74456	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	0	0
$23$	3.37228	0.703169	0.351585	−	0.936156i	$-0.385643\pi$
$23$	3.37228	0.703169	0.351585	+	0.936156i	$0.385643\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.37228	−0.626217	−0.313108	−	0.949717i	$-0.601370\pi$
$29$	−3.37228	−0.626217	−0.313108	+	0.949717i	$0.601370\pi$
$30$	0	0
$31$	−8.74456	−1.57057	−0.785285	−	0.619135i	$-0.787484\pi$
$31$	−8.74456	−1.57057	−0.785285	+	0.619135i	$0.787484\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	11.3723	1.92227
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.37228	0.839009	0.419505	−	0.907753i	$-0.362204\pi$
$41$	5.37228	0.839009	0.419505	+	0.907753i	$0.362204\pi$
$42$	0	0
$43$	2.62772	0.400723	0.200362	−	0.979722i	$-0.435788\pi$
$43$	2.62772	0.400723	0.200362	+	0.979722i	$0.435788\pi$
$44$	0	0
$45$	3.37228	0.502710
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	3.37228	0.454718
$56$	0	0
$57$	6.74456	0.893339
$58$	0	0
$59$	11.3723	1.48054	0.740272	−	0.672307i	$-0.234696\pi$
$59$	11.3723	1.48054	0.740272	+	0.672307i	$0.234696\pi$
$60$	0	0
$61$	8.11684	1.03926	0.519628	−	0.854393i	$-0.326071\pi$
$61$	8.11684	1.03926	0.519628	+	0.854393i	$0.326071\pi$
$62$	0	0
$63$	3.37228	0.424868
$64$	0	0
$65$	−3.37228	−0.418280
$66$	0	0
$67$	−12.1168	−1.48031	−0.740154	−	0.672437i	$-0.765247\pi$
$67$	−12.1168	−1.48031	−0.740154	+	0.672437i	$0.765247\pi$
$68$	0	0
$69$	3.37228	0.405975
$70$	0	0
$71$	14.7446	1.74986	0.874929	−	0.484252i	$-0.160908\pi$
$71$	14.7446	1.74986	0.874929	+	0.484252i	$0.160908\pi$
$72$	0	0
$73$	−8.11684	−0.950005	−0.475002	−	0.879985i	$-0.657553\pi$
$73$	−8.11684	−0.950005	−0.475002	+	0.879985i	$0.657553\pi$
$74$	0	0
$75$	6.37228	0.735808
$76$	0	0
$77$	3.37228	0.384307
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	−13.4891	−1.46310
$86$	0	0
$87$	−3.37228	−0.361547
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−3.37228	−0.353511
$92$	0	0
$93$	−8.74456	−0.906769
$94$	0	0
$95$	22.7446	2.33354
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−14.7446	−1.46714	−0.733569	−	0.679615i	$-0.762147\pi$
$101$	−14.7446	−1.46714	−0.733569	+	0.679615i	$0.762147\pi$
$102$	0	0
$103$	14.1168	1.39097	0.695487	−	0.718539i	$-0.255189\pi$
$103$	14.1168	1.39097	0.695487	+	0.718539i	$0.255189\pi$
$104$	0	0
$105$	11.3723	1.10982
$106$	0	0
$107$	−0.627719	−0.0606839	−0.0303419	−	0.999540i	$-0.509660\pi$
$107$	−0.627719	−0.0606839	−0.0303419	+	0.999540i	$0.509660\pi$
$108$	0	0
$109$	12.7446	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$109$	12.7446	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−9.37228	−0.881670	−0.440835	−	0.897588i	$-0.645318\pi$
$113$	−9.37228	−0.881670	−0.440835	+	0.897588i	$0.645318\pi$
$114$	0	0
$115$	11.3723	1.06047
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−13.4891	−1.23655
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.37228	0.484402
$124$	0	0
$125$	4.62772	0.413916
$126$	0	0
$127$	−15.4891	−1.37444	−0.687219	−	0.726450i	$-0.741169\pi$
$127$	−15.4891	−1.37444	−0.687219	+	0.726450i	$0.741169\pi$
$128$	0	0
$129$	2.62772	0.231358
$130$	0	0
$131$	4.62772	0.404326	0.202163	−	0.979352i	$-0.435203\pi$
$131$	4.62772	0.404326	0.202163	+	0.979352i	$0.435203\pi$
$132$	0	0
$133$	22.7446	1.97220
$134$	0	0
$135$	3.37228	0.290240
$136$	0	0
$137$	1.25544	0.107259	0.0536296	−	0.998561i	$-0.482921\pi$
$137$	1.25544	0.107259	0.0536296	+	0.998561i	$0.482921\pi$
$138$	0	0
$139$	−14.2337	−1.20729	−0.603643	−	0.797255i	$-0.706285\pi$
$139$	−14.2337	−1.20729	−0.603643	+	0.797255i	$0.706285\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−11.3723	−0.944417
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	16.7446	1.37177	0.685884	−	0.727711i	$-0.259416\pi$
$149$	16.7446	1.37177	0.685884	+	0.727711i	$0.259416\pi$
$150$	0	0
$151$	1.48913	0.121183	0.0605916	−	0.998163i	$-0.480701\pi$
$151$	1.48913	0.121183	0.0605916	+	0.998163i	$0.480701\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−29.4891	−2.36862
$156$	0	0
$157$	7.48913	0.597697	0.298849	−	0.954301i	$-0.403397\pi$
$157$	7.48913	0.597697	0.298849	+	0.954301i	$0.403397\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	11.3723	0.896261
$162$	0	0
$163$	−9.37228	−0.734094	−0.367047	−	0.930202i	$-0.619631\pi$
$163$	−9.37228	−0.734094	−0.367047	+	0.930202i	$0.619631\pi$
$164$	0	0
$165$	3.37228	0.262532
$166$	0	0
$167$	−2.11684	−0.163806	−0.0819032	−	0.996640i	$-0.526100\pi$
$167$	−2.11684	−0.163806	−0.0819032	+	0.996640i	$0.526100\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.74456	0.515770
$172$	0	0
$173$	16.8614	1.28195	0.640975	−	0.767562i	$-0.278530\pi$
$173$	16.8614	1.28195	0.640975	+	0.767562i	$0.278530\pi$
$174$	0	0
$175$	21.4891	1.62443
$176$	0	0
$177$	11.3723	0.854793
$178$	0	0
$179$	17.4891	1.30720	0.653599	−	0.756841i	$-0.273258\pi$
$179$	17.4891	1.30720	0.653599	+	0.756841i	$0.273258\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	8.11684	0.600014
$184$	0	0
$185$	−6.74456	−0.495870
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	3.37228	0.245297
$190$	0	0
$191$	−19.3723	−1.40173	−0.700865	−	0.713294i	$-0.747202\pi$
$191$	−19.3723	−1.40173	−0.700865	+	0.713294i	$0.747202\pi$
$192$	0	0
$193$	−11.4891	−0.827005	−0.413503	−	0.910503i	$-0.635695\pi$
$193$	−11.4891	−0.827005	−0.413503	+	0.910503i	$0.635695\pi$
$194$	0	0
$195$	−3.37228	−0.241494
$196$	0	0
$197$	−12.7446	−0.908012	−0.454006	−	0.890999i	$-0.650006\pi$
$197$	−12.7446	−0.908012	−0.454006	+	0.890999i	$0.650006\pi$
$198$	0	0
$199$	4.86141	0.344616	0.172308	−	0.985043i	$-0.444878\pi$
$199$	4.86141	0.344616	0.172308	+	0.985043i	$0.444878\pi$
$200$	0	0
$201$	−12.1168	−0.854656
$202$	0	0
$203$	−11.3723	−0.798178
$204$	0	0
$205$	18.1168	1.26534
$206$	0	0
$207$	3.37228	0.234390
$208$	0	0
$209$	6.74456	0.466531
$210$	0	0
$211$	8.74456	0.602001	0.301000	−	0.953624i	$-0.402679\pi$
$211$	8.74456	0.602001	0.301000	+	0.953624i	$0.402679\pi$
$212$	0	0
$213$	14.7446	1.01028
$214$	0	0
$215$	8.86141	0.604343
$216$	0	0
$217$	−29.4891	−2.00185
$218$	0	0
$219$	−8.11684	−0.548485
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	0.744563	0.0498596	0.0249298	−	0.999689i	$-0.492064\pi$
$223$	0.744563	0.0498596	0.0249298	+	0.999689i	$0.492064\pi$
$224$	0	0
$225$	6.37228	0.424819
$226$	0	0
$227$	−22.9783	−1.52512	−0.762560	−	0.646917i	$-0.776058\pi$
$227$	−22.9783	−1.52512	−0.762560	+	0.646917i	$0.776058\pi$
$228$	0	0
$229$	−21.6060	−1.42776	−0.713881	−	0.700267i	$-0.753064\pi$
$229$	−21.6060	−1.42776	−0.713881	+	0.700267i	$0.753064\pi$
$230$	0	0
$231$	3.37228	0.221880
$232$	0	0
$233$	5.25544	0.344295	0.172148	−	0.985071i	$-0.444929\pi$
$233$	5.25544	0.344295	0.172148	+	0.985071i	$0.444929\pi$
$234$	0	0
$235$	13.4891	0.879934
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	−24.8614	−1.60815	−0.804075	−	0.594527i	$-0.797339\pi$
$239$	−24.8614	−1.60815	−0.804075	+	0.594527i	$0.797339\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.7446	0.941996
$246$	0	0
$247$	−6.74456	−0.429146
$248$	0	0
$249$	−2.74456	−0.173930
$250$	0	0
$251$	−8.23369	−0.519706	−0.259853	−	0.965648i	$-0.583674\pi$
$251$	−8.23369	−0.519706	−0.259853	+	0.965648i	$0.583674\pi$
$252$	0	0
$253$	3.37228	0.212014
$254$	0	0
$255$	−13.4891	−0.844722
$256$	0	0
$257$	−4.11684	−0.256802	−0.128401	−	0.991722i	$-0.540984\pi$
$257$	−4.11684	−0.256802	−0.128401	+	0.991722i	$0.540984\pi$
$258$	0	0
$259$	−6.74456	−0.419087
$260$	0	0
$261$	−3.37228	−0.208739
$262$	0	0
$263$	17.4891	1.07843	0.539213	−	0.842170i	$-0.318722\pi$
$263$	17.4891	1.07843	0.539213	+	0.842170i	$0.318722\pi$
$264$	0	0
$265$	−33.7228	−2.07158
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	−26.2337	−1.59950	−0.799748	−	0.600336i	$-0.795034\pi$
$269$	−26.2337	−1.59950	−0.799748	+	0.600336i	$0.795034\pi$
$270$	0	0
$271$	−9.48913	−0.576423	−0.288212	−	0.957567i	$-0.593061\pi$
$271$	−9.48913	−0.576423	−0.288212	+	0.957567i	$0.593061\pi$
$272$	0	0
$273$	−3.37228	−0.204100
$274$	0	0
$275$	6.37228	0.384263
$276$	0	0
$277$	−17.3723	−1.04380	−0.521900	−	0.853007i	$-0.674776\pi$
$277$	−17.3723	−1.04380	−0.521900	+	0.853007i	$0.674776\pi$
$278$	0	0
$279$	−8.74456	−0.523523
$280$	0	0
$281$	22.6277	1.34986	0.674928	−	0.737883i	$-0.264175\pi$
$281$	22.6277	1.34986	0.674928	+	0.737883i	$0.264175\pi$
$282$	0	0
$283$	22.6277	1.34508	0.672539	−	0.740062i	$-0.265204\pi$
$283$	22.6277	1.34508	0.672539	+	0.740062i	$0.265204\pi$
$284$	0	0
$285$	22.7446	1.34727
$286$	0	0
$287$	18.1168	1.06940
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−27.4891	−1.60593	−0.802966	−	0.596025i	$-0.796746\pi$
$293$	−27.4891	−1.60593	−0.802966	+	0.596025i	$0.796746\pi$
$294$	0	0
$295$	38.3505	2.23285
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.37228	−0.195024
$300$	0	0
$301$	8.86141	0.510763
$302$	0	0
$303$	−14.7446	−0.847053
$304$	0	0
$305$	27.3723	1.56733
$306$	0	0
$307$	−25.7228	−1.46808	−0.734039	−	0.679107i	$-0.762367\pi$
$307$	−25.7228	−1.46808	−0.734039	+	0.679107i	$0.762367\pi$
$308$	0	0
$309$	14.1168	0.803079
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−28.1168	−1.58926	−0.794629	−	0.607095i	$-0.792335\pi$
$313$	−28.1168	−1.58926	−0.794629	+	0.607095i	$0.792335\pi$
$314$	0	0
$315$	11.3723	0.640755
$316$	0	0
$317$	−16.8614	−0.947031	−0.473515	−	0.880785i	$-0.657015\pi$
$317$	−16.8614	−0.947031	−0.473515	+	0.880785i	$0.657015\pi$
$318$	0	0
$319$	−3.37228	−0.188812
$320$	0	0
$321$	−0.627719	−0.0350358
$322$	0	0
$323$	−26.9783	−1.50111
$324$	0	0
$325$	−6.37228	−0.353471
$326$	0	0
$327$	12.7446	0.704776
$328$	0	0
$329$	13.4891	0.743680
$330$	0	0
$331$	−26.6277	−1.46359	−0.731796	−	0.681524i	$-0.761318\pi$
$331$	−26.6277	−1.46359	−0.731796	+	0.681524i	$0.761318\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−40.8614	−2.23250
$336$	0	0
$337$	32.9783	1.79644	0.898220	−	0.439546i	$-0.144861\pi$
$337$	32.9783	1.79644	0.898220	+	0.439546i	$0.144861\pi$
$338$	0	0
$339$	−9.37228	−0.509032
$340$	0	0
$341$	−8.74456	−0.473545
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	11.3723	0.612263
$346$	0	0
$347$	29.4891	1.58306	0.791530	−	0.611131i	$-0.209285\pi$
$347$	29.4891	1.58306	0.791530	+	0.611131i	$0.209285\pi$
$348$	0	0
$349$	0.744563	0.0398555	0.0199278	−	0.999801i	$-0.493656\pi$
$349$	0.744563	0.0398555	0.0199278	+	0.999801i	$0.493656\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	2.74456	0.146078	0.0730392	−	0.997329i	$-0.476730\pi$
$353$	2.74456	0.146078	0.0730392	+	0.997329i	$0.476730\pi$
$354$	0	0
$355$	49.7228	2.63901
$356$	0	0
$357$	−13.4891	−0.713920
$358$	0	0
$359$	−2.11684	−0.111723	−0.0558614	−	0.998439i	$-0.517790\pi$
$359$	−2.11684	−0.111723	−0.0558614	+	0.998439i	$0.517790\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−27.3723	−1.43273
$366$	0	0
$367$	−34.9783	−1.82585	−0.912925	−	0.408128i	$-0.866182\pi$
$367$	−34.9783	−1.82585	−0.912925	+	0.408128i	$0.866182\pi$
$368$	0	0
$369$	5.37228	0.279670
$370$	0	0
$371$	−33.7228	−1.75080
$372$	0	0
$373$	0.116844	0.00604995	0.00302498	−	0.999995i	$-0.499037\pi$
$373$	0.116844	0.00604995	0.00302498	+	0.999995i	$0.499037\pi$
$374$	0	0
$375$	4.62772	0.238974
$376$	0	0
$377$	3.37228	0.173681
$378$	0	0
$379$	−6.23369	−0.320203	−0.160102	−	0.987101i	$-0.551182\pi$
$379$	−6.23369	−0.320203	−0.160102	+	0.987101i	$0.551182\pi$
$380$	0	0
$381$	−15.4891	−0.793532
$382$	0	0
$383$	−33.7228	−1.72316	−0.861578	−	0.507626i	$-0.830523\pi$
$383$	−33.7228	−1.72316	−0.861578	+	0.507626i	$0.830523\pi$
$384$	0	0
$385$	11.3723	0.579585
$386$	0	0
$387$	2.62772	0.133574
$388$	0	0
$389$	−16.9783	−0.860831	−0.430416	−	0.902631i	$-0.641633\pi$
$389$	−16.9783	−0.860831	−0.430416	+	0.902631i	$0.641633\pi$
$390$	0	0
$391$	−13.4891	−0.682174
$392$	0	0
$393$	4.62772	0.233438
$394$	0	0
$395$	33.7228	1.69678
$396$	0	0
$397$	25.3723	1.27340	0.636699	−	0.771112i	$-0.280299\pi$
$397$	25.3723	1.27340	0.636699	+	0.771112i	$0.280299\pi$
$398$	0	0
$399$	22.7446	1.13865
$400$	0	0
$401$	6.51087	0.325138	0.162569	−	0.986697i	$-0.448022\pi$
$401$	6.51087	0.325138	0.162569	+	0.986697i	$0.448022\pi$
$402$	0	0
$403$	8.74456	0.435598
$404$	0	0
$405$	3.37228	0.167570
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	9.60597	0.474985	0.237492	−	0.971389i	$-0.423675\pi$
$409$	9.60597	0.474985	0.237492	+	0.971389i	$0.423675\pi$
$410$	0	0
$411$	1.25544	0.0619262
$412$	0	0
$413$	38.3505	1.88711
$414$	0	0
$415$	−9.25544	−0.454332
$416$	0	0
$417$	−14.2337	−0.697027
$418$	0	0
$419$	33.4891	1.63605	0.818025	−	0.575182i	$-0.195069\pi$
$419$	33.4891	1.63605	0.818025	+	0.575182i	$0.195069\pi$
$420$	0	0
$421$	−32.1168	−1.56528	−0.782640	−	0.622475i	$-0.786127\pi$
$421$	−32.1168	−1.56528	−0.782640	+	0.622475i	$0.786127\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−25.4891	−1.23640
$426$	0	0
$427$	27.3723	1.32464
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	26.9783	1.29950	0.649748	−	0.760149i	$-0.274874\pi$
$431$	26.9783	1.29950	0.649748	+	0.760149i	$0.274874\pi$
$432$	0	0
$433$	4.11684	0.197843	0.0989214	−	0.995095i	$-0.468461\pi$
$433$	4.11684	0.197843	0.0989214	+	0.995095i	$0.468461\pi$
$434$	0	0
$435$	−11.3723	−0.545259
$436$	0	0
$437$	22.7446	1.08802
$438$	0	0
$439$	15.2554	0.728102	0.364051	−	0.931379i	$-0.381393\pi$
$439$	15.2554	0.728102	0.364051	+	0.931379i	$0.381393\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	26.9783	1.27889
$446$	0	0
$447$	16.7446	0.791991
$448$	0	0
$449$	10.9783	0.518096	0.259048	−	0.965864i	$-0.416591\pi$
$449$	10.9783	0.518096	0.259048	+	0.965864i	$0.416591\pi$
$450$	0	0
$451$	5.37228	0.252971
$452$	0	0
$453$	1.48913	0.0699652
$454$	0	0
$455$	−11.3723	−0.533141
$456$	0	0
$457$	−5.37228	−0.251305	−0.125652	−	0.992074i	$-0.540102\pi$
$457$	−5.37228	−0.251305	−0.125652	+	0.992074i	$0.540102\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	1.76631	0.0822654	0.0411327	−	0.999154i	$-0.486903\pi$
$461$	1.76631	0.0822654	0.0411327	+	0.999154i	$0.486903\pi$
$462$	0	0
$463$	−35.4891	−1.64932	−0.824660	−	0.565629i	$-0.808633\pi$
$463$	−35.4891	−1.64932	−0.824660	+	0.565629i	$0.808633\pi$
$464$	0	0
$465$	−29.4891	−1.36753
$466$	0	0
$467$	6.97825	0.322915	0.161457	−	0.986880i	$-0.448381\pi$
$467$	6.97825	0.322915	0.161457	+	0.986880i	$0.448381\pi$
$468$	0	0
$469$	−40.8614	−1.88680
$470$	0	0
$471$	7.48913	0.345081
$472$	0	0
$473$	2.62772	0.120823
$474$	0	0
$475$	42.9783	1.97198
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	30.3505	1.38675	0.693376	−	0.720576i	$-0.256123\pi$
$479$	30.3505	1.38675	0.693376	+	0.720576i	$0.256123\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	11.3723	0.517457
$484$	0	0
$485$	−33.7228	−1.53127
$486$	0	0
$487$	30.4674	1.38061	0.690304	−	0.723519i	$-0.257477\pi$
$487$	30.4674	1.38061	0.690304	+	0.723519i	$0.257477\pi$
$488$	0	0
$489$	−9.37228	−0.423829
$490$	0	0
$491$	28.6277	1.29195	0.645975	−	0.763358i	$-0.276451\pi$
$491$	28.6277	1.29195	0.645975	+	0.763358i	$0.276451\pi$
$492$	0	0
$493$	13.4891	0.607520
$494$	0	0
$495$	3.37228	0.151573
$496$	0	0
$497$	49.7228	2.23037
$498$	0	0
$499$	−34.8614	−1.56061	−0.780305	−	0.625399i	$-0.784936\pi$
$499$	−34.8614	−1.56061	−0.780305	+	0.625399i	$0.784936\pi$
$500$	0	0
$501$	−2.11684	−0.0945736
$502$	0	0
$503$	17.4891	0.779802	0.389901	−	0.920857i	$-0.372509\pi$
$503$	17.4891	0.779802	0.389901	+	0.920857i	$0.372509\pi$
$504$	0	0
$505$	−49.7228	−2.21264
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	26.7446	1.18543	0.592716	−	0.805412i	$-0.298056\pi$
$509$	26.7446	1.18543	0.592716	+	0.805412i	$0.298056\pi$
$510$	0	0
$511$	−27.3723	−1.21088
$512$	0	0
$513$	6.74456	0.297780
$514$	0	0
$515$	47.6060	2.09777
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	16.8614	0.740134
$520$	0	0
$521$	15.8832	0.695854	0.347927	−	0.937522i	$-0.386886\pi$
$521$	15.8832	0.695854	0.347927	+	0.937522i	$0.386886\pi$
$522$	0	0
$523$	−7.25544	−0.317258	−0.158629	−	0.987338i	$-0.550707\pi$
$523$	−7.25544	−0.317258	−0.158629	+	0.987338i	$0.550707\pi$
$524$	0	0
$525$	21.4891	0.937862
$526$	0	0
$527$	34.9783	1.52368
$528$	0	0
$529$	−11.6277	−0.505553
$530$	0	0
$531$	11.3723	0.493515
$532$	0	0
$533$	−5.37228	−0.232699
$534$	0	0
$535$	−2.11684	−0.0915191
$536$	0	0
$537$	17.4891	0.754711
$538$	0	0
$539$	4.37228	0.188327
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	42.9783	1.84099
$546$	0	0
$547$	20.1168	0.860134	0.430067	−	0.902797i	$-0.358490\pi$
$547$	20.1168	0.860134	0.430067	+	0.902797i	$0.358490\pi$
$548$	0	0
$549$	8.11684	0.346418
$550$	0	0
$551$	−22.7446	−0.968951
$552$	0	0
$553$	33.7228	1.43404
$554$	0	0
$555$	−6.74456	−0.286291
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−2.62772	−0.111141
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	10.9783	0.462678	0.231339	−	0.972873i	$-0.425689\pi$
$563$	10.9783	0.462678	0.231339	+	0.972873i	$0.425689\pi$
$564$	0	0
$565$	−31.6060	−1.32967
$566$	0	0
$567$	3.37228	0.141623
$568$	0	0
$569$	−4.23369	−0.177485	−0.0887427	−	0.996055i	$-0.528285\pi$
$569$	−4.23369	−0.177485	−0.0887427	+	0.996055i	$0.528285\pi$
$570$	0	0
$571$	12.1168	0.507074	0.253537	−	0.967326i	$-0.418406\pi$
$571$	12.1168	0.507074	0.253537	+	0.967326i	$0.418406\pi$
$572$	0	0
$573$	−19.3723	−0.809289
$574$	0	0
$575$	21.4891	0.896158
$576$	0	0
$577$	−34.2337	−1.42517	−0.712584	−	0.701587i	$-0.752475\pi$
$577$	−34.2337	−1.42517	−0.712584	+	0.701587i	$0.752475\pi$
$578$	0	0
$579$	−11.4891	−0.477472
$580$	0	0
$581$	−9.25544	−0.383980
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−3.37228	−0.139427
$586$	0	0
$587$	4.62772	0.191006	0.0955032	−	0.995429i	$-0.469554\pi$
$587$	4.62772	0.191006	0.0955032	+	0.995429i	$0.469554\pi$
$588$	0	0
$589$	−58.9783	−2.43016
$590$	0	0
$591$	−12.7446	−0.524241
$592$	0	0
$593$	16.9783	0.697213	0.348607	−	0.937269i	$-0.386655\pi$
$593$	16.9783	0.697213	0.348607	+	0.937269i	$0.386655\pi$
$594$	0	0
$595$	−45.4891	−1.86487
$596$	0	0
$597$	4.86141	0.198964
$598$	0	0
$599$	13.8832	0.567250	0.283625	−	0.958935i	$-0.408463\pi$
$599$	13.8832	0.567250	0.283625	+	0.958935i	$0.408463\pi$
$600$	0	0
$601$	−12.7446	−0.519862	−0.259931	−	0.965627i	$-0.583700\pi$
$601$	−12.7446	−0.519862	−0.259931	+	0.965627i	$0.583700\pi$
$602$	0	0
$603$	−12.1168	−0.493436
$604$	0	0
$605$	3.37228	0.137103
$606$	0	0
$607$	−12.9783	−0.526771	−0.263385	−	0.964691i	$-0.584839\pi$
$607$	−12.9783	−0.526771	−0.263385	+	0.964691i	$0.584839\pi$
$608$	0	0
$609$	−11.3723	−0.460828
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	23.7228	0.958155	0.479078	−	0.877772i	$-0.340971\pi$
$613$	23.7228	0.958155	0.479078	+	0.877772i	$0.340971\pi$
$614$	0	0
$615$	18.1168	0.730542
$616$	0	0
$617$	−14.7446	−0.593594	−0.296797	−	0.954941i	$-0.595918\pi$
$617$	−14.7446	−0.593594	−0.296797	+	0.954941i	$0.595918\pi$
$618$	0	0
$619$	−12.1168	−0.487017	−0.243509	−	0.969899i	$-0.578298\pi$
$619$	−12.1168	−0.487017	−0.243509	+	0.969899i	$0.578298\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	0	0
$623$	26.9783	1.08086
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	6.74456	0.269352
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	8.74456	0.347565
$634$	0	0
$635$	−52.2337	−2.07283
$636$	0	0
$637$	−4.37228	−0.173236
$638$	0	0
$639$	14.7446	0.583286
$640$	0	0
$641$	29.8397	1.17860	0.589298	−	0.807916i	$-0.299404\pi$
$641$	29.8397	1.17860	0.589298	+	0.807916i	$0.299404\pi$
$642$	0	0
$643$	24.7446	0.975830	0.487915	−	0.872891i	$-0.337758\pi$
$643$	24.7446	0.975830	0.487915	+	0.872891i	$0.337758\pi$
$644$	0	0
$645$	8.86141	0.348918
$646$	0	0
$647$	−34.9783	−1.37514	−0.687568	−	0.726120i	$-0.741322\pi$
$647$	−34.9783	−1.37514	−0.687568	+	0.726120i	$0.741322\pi$
$648$	0	0
$649$	11.3723	0.446401
$650$	0	0
$651$	−29.4891	−1.15577
$652$	0	0
$653$	7.48913	0.293072	0.146536	−	0.989205i	$-0.453188\pi$
$653$	7.48913	0.293072	0.146536	+	0.989205i	$0.453188\pi$
$654$	0	0
$655$	15.6060	0.609776
$656$	0	0
$657$	−8.11684	−0.316668
$658$	0	0
$659$	−9.48913	−0.369644	−0.184822	−	0.982772i	$-0.559171\pi$
$659$	−9.48913	−0.369644	−0.184822	+	0.982772i	$0.559171\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	76.7011	2.97434
$666$	0	0
$667$	−11.3723	−0.440336
$668$	0	0
$669$	0.744563	0.0287865
$670$	0	0
$671$	8.11684	0.313347
$672$	0	0
$673$	−10.2337	−0.394480	−0.197240	−	0.980355i	$-0.563198\pi$
$673$	−10.2337	−0.394480	−0.197240	+	0.980355i	$0.563198\pi$
$674$	0	0
$675$	6.37228	0.245269
$676$	0	0
$677$	32.2337	1.23884	0.619421	−	0.785059i	$-0.287368\pi$
$677$	32.2337	1.23884	0.619421	+	0.785059i	$0.287368\pi$
$678$	0	0
$679$	−33.7228	−1.29416
$680$	0	0
$681$	−22.9783	−0.880528
$682$	0	0
$683$	−34.5842	−1.32333	−0.661664	−	0.749800i	$-0.730150\pi$
$683$	−34.5842	−1.32333	−0.661664	+	0.749800i	$0.730150\pi$
$684$	0	0
$685$	4.23369	0.161761
$686$	0	0
$687$	−21.6060	−0.824319
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	49.2119	1.87211	0.936055	−	0.351853i	$-0.114448\pi$
$691$	49.2119	1.87211	0.936055	+	0.351853i	$0.114448\pi$
$692$	0	0
$693$	3.37228	0.128102
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	−21.4891	−0.813959
$698$	0	0
$699$	5.25544	0.198779
$700$	0	0
$701$	−27.8397	−1.05149	−0.525745	−	0.850642i	$-0.676213\pi$
$701$	−27.8397	−1.05149	−0.525745	+	0.850642i	$0.676213\pi$
$702$	0	0
$703$	−13.4891	−0.508752
$704$	0	0
$705$	13.4891	0.508030
$706$	0	0
$707$	−49.7228	−1.87002
$708$	0	0
$709$	29.3723	1.10310	0.551550	−	0.834142i	$-0.314037\pi$
$709$	29.3723	1.10310	0.551550	+	0.834142i	$0.314037\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−29.4891	−1.10438
$714$	0	0
$715$	−3.37228	−0.126116
$716$	0	0
$717$	−24.8614	−0.928466
$718$	0	0
$719$	47.6060	1.77540	0.887702	−	0.460419i	$-0.152301\pi$
$719$	47.6060	1.77540	0.887702	+	0.460419i	$0.152301\pi$
$720$	0	0
$721$	47.6060	1.77294
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	−21.4891	−0.798086
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.5109	−0.388759
$732$	0	0
$733$	−44.9783	−1.66131	−0.830655	−	0.556788i	$-0.812034\pi$
$733$	−44.9783	−1.66131	−0.830655	+	0.556788i	$0.812034\pi$
$734$	0	0
$735$	14.7446	0.543861
$736$	0	0
$737$	−12.1168	−0.446330
$738$	0	0
$739$	36.4674	1.34147	0.670737	−	0.741695i	$-0.265978\pi$
$739$	36.4674	1.34147	0.670737	+	0.741695i	$0.265978\pi$
$740$	0	0
$741$	−6.74456	−0.247768
$742$	0	0
$743$	35.8397	1.31483	0.657415	−	0.753529i	$-0.271650\pi$
$743$	35.8397	1.31483	0.657415	+	0.753529i	$0.271650\pi$
$744$	0	0
$745$	56.4674	2.06880
$746$	0	0
$747$	−2.74456	−0.100418
$748$	0	0
$749$	−2.11684	−0.0773478
$750$	0	0
$751$	−28.6277	−1.04464	−0.522320	−	0.852749i	$-0.674933\pi$
$751$	−28.6277	−1.04464	−0.522320	+	0.852749i	$0.674933\pi$
$752$	0	0
$753$	−8.23369	−0.300052
$754$	0	0
$755$	5.02175	0.182760
$756$	0	0
$757$	−16.9783	−0.617085	−0.308543	−	0.951211i	$-0.599841\pi$
$757$	−16.9783	−0.617085	−0.308543	+	0.951211i	$0.599841\pi$
$758$	0	0
$759$	3.37228	0.122406
$760$	0	0
$761$	−23.8832	−0.865764	−0.432882	−	0.901451i	$-0.642503\pi$
$761$	−23.8832	−0.865764	−0.432882	+	0.901451i	$0.642503\pi$
$762$	0	0
$763$	42.9783	1.55592
$764$	0	0
$765$	−13.4891	−0.487700
$766$	0	0
$767$	−11.3723	−0.410629
$768$	0	0
$769$	−22.6277	−0.815976	−0.407988	−	0.912987i	$-0.633770\pi$
$769$	−22.6277	−0.815976	−0.407988	+	0.912987i	$0.633770\pi$
$770$	0	0
$771$	−4.11684	−0.148265
$772$	0	0
$773$	3.76631	0.135465	0.0677324	−	0.997704i	$-0.478424\pi$
$773$	3.76631	0.135465	0.0677324	+	0.997704i	$0.478424\pi$
$774$	0	0
$775$	−55.7228	−2.00162
$776$	0	0
$777$	−6.74456	−0.241960
$778$	0	0
$779$	36.2337	1.29821
$780$	0	0
$781$	14.7446	0.527602
$782$	0	0
$783$	−3.37228	−0.120516
$784$	0	0
$785$	25.2554	0.901405
$786$	0	0
$787$	−1.02175	−0.0364214	−0.0182107	−	0.999834i	$-0.505797\pi$
$787$	−1.02175	−0.0364214	−0.0182107	+	0.999834i	$0.505797\pi$
$788$	0	0
$789$	17.4891	0.622629
$790$	0	0
$791$	−31.6060	−1.12378
$792$	0	0
$793$	−8.11684	−0.288238
$794$	0	0
$795$	−33.7228	−1.19602
$796$	0	0
$797$	27.7228	0.981992	0.490996	−	0.871162i	$-0.336633\pi$
$797$	27.7228	0.981992	0.490996	+	0.871162i	$0.336633\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	−8.11684	−0.286437
$804$	0	0
$805$	38.3505	1.35168
$806$	0	0
$807$	−26.2337	−0.923470
$808$	0	0
$809$	−38.7446	−1.36219	−0.681093	−	0.732197i	$-0.738495\pi$
$809$	−38.7446	−1.36219	−0.681093	+	0.732197i	$0.738495\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−9.48913	−0.332798
$814$	0	0
$815$	−31.6060	−1.10711
$816$	0	0
$817$	17.7228	0.620043
$818$	0	0
$819$	−3.37228	−0.117837
$820$	0	0
$821$	−4.97825	−0.173742	−0.0868711	−	0.996220i	$-0.527687\pi$
$821$	−4.97825	−0.173742	−0.0868711	+	0.996220i	$0.527687\pi$
$822$	0	0
$823$	−43.6060	−1.52001	−0.760004	−	0.649918i	$-0.774803\pi$
$823$	−43.6060	−1.52001	−0.760004	+	0.649918i	$0.774803\pi$
$824$	0	0
$825$	6.37228	0.221854
$826$	0	0
$827$	44.4674	1.54628	0.773141	−	0.634234i	$-0.218684\pi$
$827$	44.4674	1.54628	0.773141	+	0.634234i	$0.218684\pi$
$828$	0	0
$829$	3.02175	0.104950	0.0524748	−	0.998622i	$-0.483289\pi$
$829$	3.02175	0.104950	0.0524748	+	0.998622i	$0.483289\pi$
$830$	0	0
$831$	−17.3723	−0.602638
$832$	0	0
$833$	−17.4891	−0.605962
$834$	0	0
$835$	−7.13859	−0.247041
$836$	0	0
$837$	−8.74456	−0.302256
$838$	0	0
$839$	13.2554	0.457629	0.228814	−	0.973470i	$-0.426515\pi$
$839$	13.2554	0.457629	0.228814	+	0.973470i	$0.426515\pi$
$840$	0	0
$841$	−17.6277	−0.607852
$842$	0	0
$843$	22.6277	0.779340
$844$	0	0
$845$	3.37228	0.116010
$846$	0	0
$847$	3.37228	0.115873
$848$	0	0
$849$	22.6277	0.776581
$850$	0	0
$851$	−6.74456	−0.231201
$852$	0	0
$853$	−34.2337	−1.17214	−0.586070	−	0.810261i	$-0.699325\pi$
$853$	−34.2337	−1.17214	−0.586070	+	0.810261i	$0.699325\pi$
$854$	0	0
$855$	22.7446	0.777848
$856$	0	0
$857$	50.9783	1.74138	0.870692	−	0.491829i	$-0.163671\pi$
$857$	50.9783	1.74138	0.870692	+	0.491829i	$0.163671\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	18.1168	0.617420
$862$	0	0
$863$	−29.4891	−1.00382	−0.501911	−	0.864919i	$-0.667369\pi$
$863$	−29.4891	−1.00382	−0.501911	+	0.864919i	$0.667369\pi$
$864$	0	0
$865$	56.8614	1.93335
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	12.1168	0.410564
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	15.6060	0.527578
$876$	0	0
$877$	51.7228	1.74656	0.873278	−	0.487223i	$-0.161990\pi$
$877$	51.7228	1.74656	0.873278	+	0.487223i	$0.161990\pi$
$878$	0	0
$879$	−27.4891	−0.927185
$880$	0	0
$881$	−30.8614	−1.03975	−0.519874	−	0.854243i	$-0.674021\pi$
$881$	−30.8614	−1.03975	−0.519874	+	0.854243i	$0.674021\pi$
$882$	0	0
$883$	39.2119	1.31959	0.659793	−	0.751447i	$-0.270644\pi$
$883$	39.2119	1.31959	0.659793	+	0.751447i	$0.270644\pi$
$884$	0	0
$885$	38.3505	1.28914
$886$	0	0
$887$	22.9783	0.771534	0.385767	−	0.922596i	$-0.373937\pi$
$887$	22.9783	0.771534	0.385767	+	0.922596i	$0.373937\pi$
$888$	0	0
$889$	−52.2337	−1.75186
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	26.9783	0.902793
$894$	0	0
$895$	58.9783	1.97143
$896$	0	0
$897$	−3.37228	−0.112597
$898$	0	0
$899$	29.4891	0.983517
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	8.86141	0.294889
$904$	0	0
$905$	33.7228	1.12098
$906$	0	0
$907$	36.4674	1.21088	0.605440	−	0.795891i	$-0.292997\pi$
$907$	36.4674	1.21088	0.605440	+	0.795891i	$0.292997\pi$
$908$	0	0
$909$	−14.7446	−0.489046
$910$	0	0
$911$	42.9783	1.42393	0.711966	−	0.702213i	$-0.247805\pi$
$911$	42.9783	1.42393	0.711966	+	0.702213i	$0.247805\pi$
$912$	0	0
$913$	−2.74456	−0.0908318
$914$	0	0
$915$	27.3723	0.904900
$916$	0	0
$917$	15.6060	0.515355
$918$	0	0
$919$	−46.7011	−1.54053	−0.770263	−	0.637726i	$-0.779875\pi$
$919$	−46.7011	−1.54053	−0.770263	+	0.637726i	$0.779875\pi$
$920$	0	0
$921$	−25.7228	−0.847596
$922$	0	0
$923$	−14.7446	−0.485323
$924$	0	0
$925$	−12.7446	−0.419039
$926$	0	0
$927$	14.1168	0.463658
$928$	0	0
$929$	−40.4674	−1.32769	−0.663846	−	0.747870i	$-0.731077\pi$
$929$	−40.4674	−1.32769	−0.663846	+	0.747870i	$0.731077\pi$
$930$	0	0
$931$	29.4891	0.966467
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−13.4891	−0.441142
$936$	0	0
$937$	−37.2119	−1.21566	−0.607831	−	0.794067i	$-0.707960\pi$
$937$	−37.2119	−1.21566	−0.607831	+	0.794067i	$0.707960\pi$
$938$	0	0
$939$	−28.1168	−0.917559
$940$	0	0
$941$	−32.7446	−1.06744	−0.533721	−	0.845661i	$-0.679207\pi$
$941$	−32.7446	−1.06744	−0.533721	+	0.845661i	$0.679207\pi$
$942$	0	0
$943$	18.1168	0.589966
$944$	0	0
$945$	11.3723	0.369940
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	8.11684	0.263484
$950$	0	0
$951$	−16.8614	−0.546769
$952$	0	0
$953$	10.9783	0.355620	0.177810	−	0.984065i	$-0.443099\pi$
$953$	10.9783	0.355620	0.177810	+	0.984065i	$0.443099\pi$
$954$	0	0
$955$	−65.3288	−2.11399
$956$	0	0
$957$	−3.37228	−0.109010
$958$	0	0
$959$	4.23369	0.136713
$960$	0	0
$961$	45.4674	1.46669
$962$	0	0
$963$	−0.627719	−0.0202280
$964$	0	0
$965$	−38.7446	−1.24723
$966$	0	0
$967$	36.6277	1.17787	0.588934	−	0.808181i	$-0.299548\pi$
$967$	36.6277	1.17787	0.588934	+	0.808181i	$0.299548\pi$
$968$	0	0
$969$	−26.9783	−0.866666
$970$	0	0
$971$	18.7446	0.601542	0.300771	−	0.953696i	$-0.402756\pi$
$971$	18.7446	0.601542	0.300771	+	0.953696i	$0.402756\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	0	0
$975$	−6.37228	−0.204076
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	12.7446	0.406903
$982$	0	0
$983$	18.7446	0.597859	0.298929	−	0.954275i	$-0.403371\pi$
$983$	18.7446	0.597859	0.298929	+	0.954275i	$0.403371\pi$
$984$	0	0
$985$	−42.9783	−1.36940
$986$	0	0
$987$	13.4891	0.429364
$988$	0	0
$989$	8.86141	0.281776
$990$	0	0
$991$	−39.3723	−1.25070	−0.625351	−	0.780344i	$-0.715044\pi$
$991$	−39.3723	−1.25070	−0.625351	+	0.780344i	$0.715044\pi$
$992$	0	0
$993$	−26.6277	−0.845005
$994$	0	0
$995$	16.3940	0.519726
$996$	0	0
$997$	−33.6060	−1.06431	−0.532156	−	0.846646i	$-0.678618\pi$
$997$	−33.6060	−1.06431	−0.532156	+	0.846646i	$0.678618\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bl.1.2		2
4.3	odd	2		858.2.a.o.1.2	✓	2
12.11	even	2		2574.2.a.bc.1.1		2
44.43	even	2		9438.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.o.1.2	✓	2	4.3	odd	2
2574.2.a.bc.1.1		2	12.11	even	2
6864.2.a.bl.1.2		2	1.1	even	1	trivial
9438.2.a.cb.1.2		2	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.37228 q^{5} +3.37228 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.37228 q^{15} -4.00000 q^{17} +6.74456 q^{19} +3.37228 q^{21} +3.37228 q^{23} +6.37228 q^{25} +1.00000 q^{27} -3.37228 q^{29} -8.74456 q^{31} +1.00000 q^{33} +11.3723 q^{35} -2.00000 q^{37} -1.00000 q^{39} +5.37228 q^{41} +2.62772 q^{43} +3.37228 q^{45} +4.00000 q^{47} +4.37228 q^{49} -4.00000 q^{51} -10.0000 q^{53} +3.37228 q^{55} +6.74456 q^{57} +11.3723 q^{59} +8.11684 q^{61} +3.37228 q^{63} -3.37228 q^{65} -12.1168 q^{67} +3.37228 q^{69} +14.7446 q^{71} -8.11684 q^{73} +6.37228 q^{75} +3.37228 q^{77} +10.0000 q^{79} +1.00000 q^{81} -2.74456 q^{83} -13.4891 q^{85} -3.37228 q^{87} +8.00000 q^{89} -3.37228 q^{91} -8.74456 q^{93} +22.7446 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} - 8 q^{17} + 2 q^{19} + q^{21} + q^{23} + 7 q^{25} + 2 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 17 q^{35} - 4 q^{37} - 2 q^{39} + 5 q^{41} + 11 q^{43} + q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 20 q^{53} + q^{55} + 2 q^{57} + 17 q^{59} - q^{61} + q^{63} - q^{65} - 7 q^{67} + q^{69} + 18 q^{71} + q^{73} + 7 q^{75} + q^{77} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 4 q^{85} - q^{87} + 16 q^{89} - q^{91} - 6 q^{93} + 34 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 - 8 * q^17 + 2 * q^19 + q^21 + q^23 + 7 * q^25 + 2 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 17 * q^35 - 4 * q^37 - 2 * q^39 + 5 * q^41 + 11 * q^43 + q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 20 * q^53 + q^55 + 2 * q^57 + 17 * q^59 - q^61 + q^63 - q^65 - 7 * q^67 + q^69 + 18 * q^71 + q^73 + 7 * q^75 + q^77 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 4 * q^85 - q^87 + 16 * q^89 - q^91 - 6 * q^93 + 34 * q^95 - 20 * q^97 + 2 * q^99

Introduction

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