LMFDB - Embedded newform 6864.2.a.bx.1.1

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			1.1
Root			$0.217439$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.59899 q^{5} +4.38155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.59899 q^{5} +4.38155 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.59899 q^{15} +2.78256 q^{17} +6.95272 q^{19} -4.38155 q^{21} +4.81643 q^{23} +7.95272 q^{25} -1.00000 q^{27} +0.782561 q^{29} +5.73528 q^{31} -1.00000 q^{33} -15.7691 q^{35} +2.43488 q^{37} +1.00000 q^{39} +1.61845 q^{41} -5.78861 q^{43} -3.59899 q^{45} -0.571171 q^{47} +12.1980 q^{49} -2.78256 q^{51} -7.76915 q^{53} -3.59899 q^{55} -6.95272 q^{57} -7.19798 q^{59} +5.70141 q^{61} +4.38155 q^{63} +3.59899 q^{65} -9.59899 q^{67} -4.81643 q^{69} +2.13629 q^{71} +6.24526 q^{73} -7.95272 q^{75} +4.38155 q^{77} +6.27912 q^{79} +1.00000 q^{81} -1.56512 q^{83} -10.0144 q^{85} -0.782561 q^{87} -8.17016 q^{89} -4.38155 q^{91} -5.73528 q^{93} -25.0228 q^{95} -5.00605 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.59899	−1.60952	−0.804758	−	0.593603i	$-0.797705\pi$
$5$	−3.59899	−1.60952	−0.804758	+	0.593603i	$0.797705\pi$
$6$	0	0
$7$	4.38155	1.65607	0.828035	−	0.560676i	$-0.189459\pi$
$7$	4.38155	1.65607	0.828035	+	0.560676i	$0.189459\pi$
$8$	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
$15$	3.59899	0.929255
$16$	0	0
$17$	2.78256	0.674870	0.337435	−	0.941349i	$-0.390441\pi$
$17$	2.78256	0.674870	0.337435	+	0.941349i	$0.390441\pi$
$18$	0	0
$19$	6.95272	1.59506	0.797532	−	0.603277i	$-0.206139\pi$
$19$	6.95272	1.59506	0.797532	+	0.603277i	$0.206139\pi$
$20$	0	0
$21$	−4.38155	−0.956133
$22$	0	0
$23$	4.81643	1.00429	0.502147	−	0.864782i	$-0.332544\pi$
$23$	4.81643	1.00429	0.502147	+	0.864782i	$0.332544\pi$
$24$	0	0
$25$	7.95272	1.59054
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.782561	0.145318	0.0726590	−	0.997357i	$-0.476852\pi$
$29$	0.782561	0.145318	0.0726590	+	0.997357i	$0.476852\pi$
$30$	0	0
$31$	5.73528	1.03009	0.515043	−	0.857164i	$-0.327776\pi$
$31$	5.73528	1.03009	0.515043	+	0.857164i	$0.327776\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−15.7691	−2.66547
$36$	0	0
$37$	2.43488	0.400292	0.200146	−	0.979766i	$-0.435858\pi$
$37$	2.43488	0.400292	0.200146	+	0.979766i	$0.435858\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.61845	0.252759	0.126380	−	0.991982i	$-0.459664\pi$
$41$	1.61845	0.252759	0.126380	+	0.991982i	$0.459664\pi$
$42$	0	0
$43$	−5.78861	−0.882755	−0.441377	−	0.897322i	$-0.645510\pi$
$43$	−5.78861	−0.882755	−0.441377	+	0.897322i	$0.645510\pi$
$44$	0	0
$45$	−3.59899	−0.536506
$46$	0	0
$47$	−0.571171	−0.0833138	−0.0416569	−	0.999132i	$-0.513264\pi$
$47$	−0.571171	−0.0833138	−0.0416569	+	0.999132i	$0.513264\pi$
$48$	0	0
$49$	12.1980	1.74257
$50$	0	0
$51$	−2.78256	−0.389636
$52$	0	0
$53$	−7.76915	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$53$	−7.76915	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$54$	0	0
$55$	−3.59899	−0.485288
$56$	0	0
$57$	−6.95272	−0.920910
$58$	0	0
$59$	−7.19798	−0.937097	−0.468548	−	0.883438i	$-0.655223\pi$
$59$	−7.19798	−0.937097	−0.468548	+	0.883438i	$0.655223\pi$
$60$	0	0
$61$	5.70141	0.729991	0.364996	−	0.931009i	$-0.381070\pi$
$61$	5.70141	0.729991	0.364996	+	0.931009i	$0.381070\pi$
$62$	0	0
$63$	4.38155	0.552023
$64$	0	0
$65$	3.59899	0.446400
$66$	0	0
$67$	−9.59899	−1.17270	−0.586352	−	0.810057i	$-0.699436\pi$
$67$	−9.59899	−1.17270	−0.586352	+	0.810057i	$0.699436\pi$
$68$	0	0
$69$	−4.81643	−0.579830
$70$	0	0
$71$	2.13629	0.253531	0.126766	−	0.991933i	$-0.459540\pi$
$71$	2.13629	0.253531	0.126766	+	0.991933i	$0.459540\pi$
$72$	0	0
$73$	6.24526	0.730952	0.365476	−	0.930821i	$-0.380906\pi$
$73$	6.24526	0.730952	0.365476	+	0.930821i	$0.380906\pi$
$74$	0	0
$75$	−7.95272	−0.918301
$76$	0	0
$77$	4.38155	0.499324
$78$	0	0
$79$	6.27912	0.706457	0.353228	−	0.935537i	$-0.385084\pi$
$79$	6.27912	0.706457	0.353228	+	0.935537i	$0.385084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.56512	−0.171794	−0.0858972	−	0.996304i	$-0.527376\pi$
$83$	−1.56512	−0.171794	−0.0858972	+	0.996304i	$0.527376\pi$
$84$	0	0
$85$	−10.0144	−1.08621
$86$	0	0
$87$	−0.782561	−0.0838993
$88$	0	0
$89$	−8.17016	−0.866035	−0.433018	−	0.901385i	$-0.642551\pi$
$89$	−8.17016	−0.866035	−0.433018	+	0.901385i	$0.642551\pi$
$90$	0	0
$91$	−4.38155	−0.459311
$92$	0	0
$93$	−5.73528	−0.594721
$94$	0	0
$95$	−25.0228	−2.56728
$96$	0	0
$97$	−5.00605	−0.508287	−0.254144	−	0.967166i	$-0.581794\pi$
$97$	−5.00605	−0.508287	−0.254144	+	0.967166i	$0.581794\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−0.974490	−0.0969653	−0.0484827	−	0.998824i	$-0.515439\pi$
$101$	−0.974490	−0.0969653	−0.0484827	+	0.998824i	$0.515439\pi$
$102$	0	0
$103$	−9.63286	−0.949153	−0.474577	−	0.880214i	$-0.657399\pi$
$103$	−9.63286	−0.949153	−0.474577	+	0.880214i	$0.657399\pi$
$104$	0	0
$105$	15.7691	1.53891
$106$	0	0
$107$	14.6389	1.41520	0.707598	−	0.706615i	$-0.249779\pi$
$107$	14.6389	1.41520	0.707598	+	0.706615i	$0.249779\pi$
$108$	0	0
$109$	17.5795	1.68381	0.841907	−	0.539623i	$-0.181433\pi$
$109$	17.5795	1.68381	0.841907	+	0.539623i	$0.181433\pi$
$110$	0	0
$111$	−2.43488	−0.231108
$112$	0	0
$113$	3.56512	0.335378	0.167689	−	0.985840i	$-0.446369\pi$
$113$	3.56512	0.335378	0.167689	+	0.985840i	$0.446369\pi$
$114$	0	0
$115$	−17.3343	−1.61643
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	12.1919	1.11763
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.61845	−0.145931
$124$	0	0
$125$	−10.6268	−0.950491
$126$	0	0
$127$	−7.27308	−0.645381	−0.322691	−	0.946505i	$-0.604587\pi$
$127$	−7.27308	−0.645381	−0.322691	+	0.946505i	$0.604587\pi$
$128$	0	0
$129$	5.78861	0.509659
$130$	0	0
$131$	8.20403	0.716789	0.358395	−	0.933570i	$-0.383324\pi$
$131$	8.20403	0.716789	0.358395	+	0.933570i	$0.383324\pi$
$132$	0	0
$133$	30.4637	2.64154
$134$	0	0
$135$	3.59899	0.309752
$136$	0	0
$137$	−17.0399	−1.45582	−0.727909	−	0.685674i	$-0.759508\pi$
$137$	−17.0399	−1.45582	−0.727909	+	0.685674i	$0.759508\pi$
$138$	0	0
$139$	−17.1785	−1.45706	−0.728531	−	0.685012i	$-0.759797\pi$
$139$	−17.1785	−1.45706	−0.728531	+	0.685012i	$0.759797\pi$
$140$	0	0
$141$	0.571171	0.0481012
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.81643	−0.233892
$146$	0	0
$147$	−12.1980	−1.00607
$148$	0	0
$149$	16.3936	1.34302	0.671510	−	0.740996i	$-0.265646\pi$
$149$	16.3936	1.34302	0.671510	+	0.740996i	$0.265646\pi$
$150$	0	0
$151$	3.23921	0.263603	0.131802	−	0.991276i	$-0.457924\pi$
$151$	3.23921	0.263603	0.131802	+	0.991276i	$0.457924\pi$
$152$	0	0
$153$	2.78256	0.224957
$154$	0	0
$155$	−20.6412	−1.65794
$156$	0	0
$157$	15.4432	1.23250	0.616252	−	0.787549i	$-0.288650\pi$
$157$	15.4432	1.23250	0.616252	+	0.787549i	$0.288650\pi$
$158$	0	0
$159$	7.76915	0.616134
$160$	0	0
$161$	21.1034	1.66318
$162$	0	0
$163$	13.3004	1.04177	0.520884	−	0.853627i	$-0.325602\pi$
$163$	13.3004	1.04177	0.520884	+	0.853627i	$0.325602\pi$
$164$	0	0
$165$	3.59899	0.280181
$166$	0	0
$167$	−23.1591	−1.79210	−0.896051	−	0.443952i	$-0.853576\pi$
$167$	−23.1591	−1.79210	−0.896051	+	0.443952i	$0.853576\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.95272	0.531688
$172$	0	0
$173$	17.0422	1.29570	0.647848	−	0.761770i	$-0.275669\pi$
$173$	17.0422	1.29570	0.647848	+	0.761770i	$0.275669\pi$
$174$	0	0
$175$	34.8452	2.63405
$176$	0	0
$177$	7.19798	0.541033
$178$	0	0
$179$	−14.3130	−1.06980	−0.534902	−	0.844914i	$-0.679651\pi$
$179$	−14.3130	−1.06980	−0.534902	+	0.844914i	$0.679651\pi$
$180$	0	0
$181$	6.08296	0.452143	0.226072	−	0.974111i	$-0.427412\pi$
$181$	6.08296	0.452143	0.226072	+	0.974111i	$0.427412\pi$
$182$	0	0
$183$	−5.70141	−0.421461
$184$	0	0
$185$	−8.76310	−0.644276
$186$	0	0
$187$	2.78256	0.203481
$188$	0	0
$189$	−4.38155	−0.318711
$190$	0	0
$191$	21.0060	1.51994	0.759972	−	0.649956i	$-0.225213\pi$
$191$	21.0060	1.51994	0.759972	+	0.649956i	$0.225213\pi$
$192$	0	0
$193$	−3.38760	−0.243845	−0.121922	−	0.992540i	$-0.538906\pi$
$193$	−3.38760	−0.243845	−0.121922	+	0.992540i	$0.538906\pi$
$194$	0	0
$195$	−3.59899	−0.257729
$196$	0	0
$197$	−23.2124	−1.65381	−0.826907	−	0.562339i	$-0.809902\pi$
$197$	−23.2124	−1.65381	−0.826907	+	0.562339i	$0.809902\pi$
$198$	0	0
$199$	10.5026	0.744510	0.372255	−	0.928130i	$-0.378585\pi$
$199$	10.5026	0.744510	0.372255	+	0.928130i	$0.378585\pi$
$200$	0	0
$201$	9.59899	0.677060
$202$	0	0
$203$	3.42883	0.240657
$204$	0	0
$205$	−5.82478	−0.406821
$206$	0	0
$207$	4.81643	0.334765
$208$	0	0
$209$	6.95272	0.480930
$210$	0	0
$211$	−18.6194	−1.28182	−0.640908	−	0.767618i	$-0.721442\pi$
$211$	−18.6194	−1.28182	−0.640908	+	0.767618i	$0.721442\pi$
$212$	0	0
$213$	−2.13629	−0.146376
$214$	0	0
$215$	20.8331	1.42081
$216$	0	0
$217$	25.1294	1.70590
$218$	0	0
$219$	−6.24526	−0.422015
$220$	0	0
$221$	−2.78256	−0.187175
$222$	0	0
$223$	−6.60504	−0.442306	−0.221153	−	0.975239i	$-0.570982\pi$
$223$	−6.60504	−0.442306	−0.221153	+	0.975239i	$0.570982\pi$
$224$	0	0
$225$	7.95272	0.530181
$226$	0	0
$227$	0.488207	0.0324034	0.0162017	−	0.999869i	$-0.494843\pi$
$227$	0.488207	0.0324034	0.0162017	+	0.999869i	$0.494843\pi$
$228$	0	0
$229$	−9.52620	−0.629509	−0.314754	−	0.949173i	$-0.601922\pi$
$229$	−9.52620	−0.629509	−0.314754	+	0.949173i	$0.601922\pi$
$230$	0	0
$231$	−4.38155	−0.288285
$232$	0	0
$233$	−26.5128	−1.73691	−0.868455	−	0.495768i	$-0.834887\pi$
$233$	−26.5128	−1.73691	−0.868455	+	0.495768i	$0.834887\pi$
$234$	0	0
$235$	2.05564	0.134095
$236$	0	0
$237$	−6.27912	−0.407873
$238$	0	0
$239$	−22.2870	−1.44163	−0.720813	−	0.693130i	$-0.756231\pi$
$239$	−22.2870	−1.44163	−0.720813	+	0.693130i	$0.756231\pi$
$240$	0	0
$241$	29.8925	1.92555	0.962773	−	0.270311i	$-0.0871264\pi$
$241$	29.8925	1.92555	0.962773	+	0.270311i	$0.0871264\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−43.9004	−2.80469
$246$	0	0
$247$	−6.95272	−0.442391
$248$	0	0
$249$	1.56512	0.0991856
$250$	0	0
$251$	22.8187	1.44031	0.720153	−	0.693815i	$-0.244072\pi$
$251$	22.8187	1.44031	0.720153	+	0.693815i	$0.244072\pi$
$252$	0	0
$253$	4.81643	0.302806
$254$	0	0
$255$	10.0144	0.627126
$256$	0	0
$257$	3.73951	0.233264	0.116632	−	0.993175i	$-0.462790\pi$
$257$	3.73951	0.233264	0.116632	+	0.993175i	$0.462790\pi$
$258$	0	0
$259$	10.6685	0.662911
$260$	0	0
$261$	0.782561	0.0484393
$262$	0	0
$263$	18.6008	1.14697	0.573487	−	0.819214i	$-0.305590\pi$
$263$	18.6008	1.14697	0.573487	+	0.819214i	$0.305590\pi$
$264$	0	0
$265$	27.9611	1.71764
$266$	0	0
$267$	8.17016	0.500006
$268$	0	0
$269$	−14.1363	−0.861905	−0.430952	−	0.902375i	$-0.641822\pi$
$269$	−14.1363	−0.861905	−0.430952	+	0.902375i	$0.641822\pi$
$270$	0	0
$271$	−15.4729	−0.939910	−0.469955	−	0.882690i	$-0.655730\pi$
$271$	−15.4729	−0.939910	−0.469955	+	0.882690i	$0.655730\pi$
$272$	0	0
$273$	4.38155	0.265183
$274$	0	0
$275$	7.95272	0.479567
$276$	0	0
$277$	−6.76310	−0.406355	−0.203178	−	0.979142i	$-0.565127\pi$
$277$	−6.76310	−0.406355	−0.203178	+	0.979142i	$0.565127\pi$
$278$	0	0
$279$	5.73528	0.343362
$280$	0	0
$281$	−19.4850	−1.16238	−0.581188	−	0.813769i	$-0.697412\pi$
$281$	−19.4850	−1.16238	−0.581188	+	0.813769i	$0.697412\pi$
$282$	0	0
$283$	18.8114	1.11822	0.559110	−	0.829093i	$-0.311143\pi$
$283$	18.8114	1.11822	0.559110	+	0.829093i	$0.311143\pi$
$284$	0	0
$285$	25.0228	1.48222
$286$	0	0
$287$	7.09132	0.418587
$288$	0	0
$289$	−9.25735	−0.544550
$290$	0	0
$291$	5.00605	0.293460
$292$	0	0
$293$	24.8452	1.45147	0.725737	−	0.687972i	$-0.241499\pi$
$293$	24.8452	1.45147	0.725737	+	0.687972i	$0.241499\pi$
$294$	0	0
$295$	25.9054	1.50827
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−4.81643	−0.278541
$300$	0	0
$301$	−25.3631	−1.46190
$302$	0	0
$303$	0.974490	0.0559830
$304$	0	0
$305$	−20.5193	−1.17493
$306$	0	0
$307$	32.9694	1.88166	0.940832	−	0.338872i	$-0.110045\pi$
$307$	32.9694	1.88166	0.940832	+	0.338872i	$0.110045\pi$
$308$	0	0
$309$	9.63286	0.547994
$310$	0	0
$311$	−4.21926	−0.239252	−0.119626	−	0.992819i	$-0.538170\pi$
$311$	−4.21926	−0.239252	−0.119626	+	0.992819i	$0.538170\pi$
$312$	0	0
$313$	8.79043	0.496864	0.248432	−	0.968649i	$-0.420085\pi$
$313$	8.79043	0.496864	0.248432	+	0.968649i	$0.420085\pi$
$314$	0	0
$315$	−15.7691	−0.888491
$316$	0	0
$317$	6.22580	0.349676	0.174838	−	0.984597i	$-0.444060\pi$
$317$	6.22580	0.349676	0.174838	+	0.984597i	$0.444060\pi$
$318$	0	0
$319$	0.782561	0.0438150
$320$	0	0
$321$	−14.6389	−0.817064
$322$	0	0
$323$	19.3464	1.07646
$324$	0	0
$325$	−7.95272	−0.441138
$326$	0	0
$327$	−17.5795	−0.972150
$328$	0	0
$329$	−2.50261	−0.137973
$330$	0	0
$331$	−5.47479	−0.300922	−0.150461	−	0.988616i	$-0.548076\pi$
$331$	−5.47479	−0.300922	−0.150461	+	0.988616i	$0.548076\pi$
$332$	0	0
$333$	2.43488	0.133431
$334$	0	0
$335$	34.5467	1.88749
$336$	0	0
$337$	16.0556	0.874606	0.437303	−	0.899314i	$-0.355934\pi$
$337$	16.0556	0.874606	0.437303	+	0.899314i	$0.355934\pi$
$338$	0	0
$339$	−3.56512	−0.193631
$340$	0	0
$341$	5.73528	0.310583
$342$	0	0
$343$	22.7752	1.22975
$344$	0	0
$345$	17.3343	0.933246
$346$	0	0
$347$	−32.3570	−1.73702	−0.868508	−	0.495675i	$-0.834921\pi$
$347$	−32.3570	−1.73702	−0.868508	+	0.495675i	$0.834921\pi$
$348$	0	0
$349$	−8.32822	−0.445799	−0.222900	−	0.974841i	$-0.571552\pi$
$349$	−8.32822	−0.445799	−0.222900	+	0.974841i	$0.571552\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	9.88367	0.526055	0.263027	−	0.964788i	$-0.415279\pi$
$353$	9.88367	0.526055	0.263027	+	0.964788i	$0.415279\pi$
$354$	0	0
$355$	−7.68849	−0.408063
$356$	0	0
$357$	−12.1919	−0.645265
$358$	0	0
$359$	4.76079	0.251265	0.125632	−	0.992077i	$-0.459904\pi$
$359$	4.76079	0.251265	0.125632	+	0.992077i	$0.459904\pi$
$360$	0	0
$361$	29.3403	1.54423
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−22.4766	−1.17648
$366$	0	0
$367$	14.0556	0.733698	0.366849	−	0.930280i	$-0.380437\pi$
$367$	14.0556	0.733698	0.366849	+	0.930280i	$0.380437\pi$
$368$	0	0
$369$	1.61845	0.0842532
$370$	0	0
$371$	−34.0409	−1.76732
$372$	0	0
$373$	−2.06773	−0.107063	−0.0535316	−	0.998566i	$-0.517048\pi$
$373$	−2.06773	−0.107063	−0.0535316	+	0.998566i	$0.517048\pi$
$374$	0	0
$375$	10.6268	0.548766
$376$	0	0
$377$	−0.782561	−0.0403039
$378$	0	0
$379$	27.1076	1.39243	0.696213	−	0.717835i	$-0.254867\pi$
$379$	27.1076	1.39243	0.696213	+	0.717835i	$0.254867\pi$
$380$	0	0
$381$	7.27308	0.372611
$382$	0	0
$383$	22.8187	1.16598	0.582991	−	0.812478i	$-0.301882\pi$
$383$	22.8187	1.16598	0.582991	+	0.812478i	$0.301882\pi$
$384$	0	0
$385$	−15.7691	−0.803670
$386$	0	0
$387$	−5.78861	−0.294252
$388$	0	0
$389$	−5.10061	−0.258611	−0.129306	−	0.991605i	$-0.541275\pi$
$389$	−5.10061	−0.258611	−0.129306	+	0.991605i	$0.541275\pi$
$390$	0	0
$391$	13.4020	0.677768
$392$	0	0
$393$	−8.20403	−0.413838
$394$	0	0
$395$	−22.5985	−1.13705
$396$	0	0
$397$	25.1034	1.25990	0.629952	−	0.776634i	$-0.283074\pi$
$397$	25.1034	1.25990	0.629952	+	0.776634i	$0.283074\pi$
$398$	0	0
$399$	−30.4637	−1.52509
$400$	0	0
$401$	2.37419	0.118561	0.0592806	−	0.998241i	$-0.481119\pi$
$401$	2.37419	0.118561	0.0592806	+	0.998241i	$0.481119\pi$
$402$	0	0
$403$	−5.73528	−0.285695
$404$	0	0
$405$	−3.59899	−0.178835
$406$	0	0
$407$	2.43488	0.120692
$408$	0	0
$409$	−3.00836	−0.148754	−0.0743768	−	0.997230i	$-0.523697\pi$
$409$	−3.00836	−0.148754	−0.0743768	+	0.997230i	$0.523697\pi$
$410$	0	0
$411$	17.0399	0.840517
$412$	0	0
$413$	−31.5383	−1.55190
$414$	0	0
$415$	5.63286	0.276506
$416$	0	0
$417$	17.1785	0.841236
$418$	0	0
$419$	31.0761	1.51817	0.759083	−	0.650994i	$-0.225648\pi$
$419$	31.0761	1.51817	0.759083	+	0.650994i	$0.225648\pi$
$420$	0	0
$421$	29.3343	1.42966	0.714832	−	0.699296i	$-0.246503\pi$
$421$	29.3343	1.42966	0.714832	+	0.699296i	$0.246503\pi$
$422$	0	0
$423$	−0.571171	−0.0277713
$424$	0	0
$425$	22.1289	1.07341
$426$	0	0
$427$	24.9810	1.20892
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	37.6329	1.81271	0.906355	−	0.422516i	$-0.138853\pi$
$431$	37.6329	1.81271	0.906355	+	0.422516i	$0.138853\pi$
$432$	0	0
$433$	23.1423	1.11215	0.556075	−	0.831132i	$-0.312307\pi$
$433$	23.1423	1.11215	0.556075	+	0.831132i	$0.312307\pi$
$434$	0	0
$435$	2.81643	0.135037
$436$	0	0
$437$	33.4873	1.60191
$438$	0	0
$439$	7.35373	0.350974	0.175487	−	0.984482i	$-0.443850\pi$
$439$	7.35373	0.350974	0.175487	+	0.984482i	$0.443850\pi$
$440$	0	0
$441$	12.1980	0.580856
$442$	0	0
$443$	21.7988	1.03569	0.517846	−	0.855474i	$-0.326734\pi$
$443$	21.7988	1.03569	0.517846	+	0.855474i	$0.326734\pi$
$444$	0	0
$445$	29.4043	1.39390
$446$	0	0
$447$	−16.3936	−0.775393
$448$	0	0
$449$	29.9560	1.41371	0.706856	−	0.707357i	$-0.250113\pi$
$449$	29.9560	1.41371	0.706856	+	0.707357i	$0.250113\pi$
$450$	0	0
$451$	1.61845	0.0762098
$452$	0	0
$453$	−3.23921	−0.152191
$454$	0	0
$455$	15.7691	0.739269
$456$	0	0
$457$	−34.1262	−1.59636	−0.798178	−	0.602422i	$-0.794202\pi$
$457$	−34.1262	−1.59636	−0.798178	+	0.602422i	$0.794202\pi$
$458$	0	0
$459$	−2.78256	−0.129879
$460$	0	0
$461$	37.5406	1.74844	0.874220	−	0.485530i	$-0.161373\pi$
$461$	37.5406	1.74844	0.874220	+	0.485530i	$0.161373\pi$
$462$	0	0
$463$	26.6468	1.23838	0.619190	−	0.785241i	$-0.287461\pi$
$463$	26.6468	1.23838	0.619190	+	0.785241i	$0.287461\pi$
$464$	0	0
$465$	20.6412	0.957213
$466$	0	0
$467$	−37.6769	−1.74348	−0.871739	−	0.489970i	$-0.837008\pi$
$467$	−37.6769	−1.74348	−0.871739	+	0.489970i	$0.837008\pi$
$468$	0	0
$469$	−42.0584	−1.94208
$470$	0	0
$471$	−15.4432	−0.711587
$472$	0	0
$473$	−5.78861	−0.266161
$474$	0	0
$475$	55.2930	2.53702
$476$	0	0
$477$	−7.76915	−0.355725
$478$	0	0
$479$	16.8721	0.770904	0.385452	−	0.922728i	$-0.374045\pi$
$479$	16.8721	0.770904	0.385452	+	0.922728i	$0.374045\pi$
$480$	0	0
$481$	−2.43488	−0.111021
$482$	0	0
$483$	−21.1034	−0.960239
$484$	0	0
$485$	18.0167	0.818097
$486$	0	0
$487$	−35.5211	−1.60962	−0.804808	−	0.593535i	$-0.797732\pi$
$487$	−35.5211	−1.60962	−0.804808	+	0.593535i	$0.797732\pi$
$488$	0	0
$489$	−13.3004	−0.601465
$490$	0	0
$491$	−5.79515	−0.261531	−0.130766	−	0.991413i	$-0.541744\pi$
$491$	−5.79515	−0.261531	−0.130766	+	0.991413i	$0.541744\pi$
$492$	0	0
$493$	2.17752	0.0980707
$494$	0	0
$495$	−3.59899	−0.161763
$496$	0	0
$497$	9.36027	0.419866
$498$	0	0
$499$	−7.69355	−0.344411	−0.172205	−	0.985061i	$-0.555089\pi$
$499$	−7.69355	−0.344411	−0.172205	+	0.985061i	$0.555089\pi$
$500$	0	0
$501$	23.1591	1.03467
$502$	0	0
$503$	7.93227	0.353682	0.176841	−	0.984239i	$-0.443412\pi$
$503$	7.93227	0.353682	0.176841	+	0.984239i	$0.443412\pi$
$504$	0	0
$505$	3.50718	0.156067
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−13.3560	−0.591996	−0.295998	−	0.955189i	$-0.595652\pi$
$509$	−13.3560	−0.591996	−0.295998	+	0.955189i	$0.595652\pi$
$510$	0	0
$511$	27.3639	1.21051
$512$	0	0
$513$	−6.95272	−0.306970
$514$	0	0
$515$	34.6685	1.52768
$516$	0	0
$517$	−0.571171	−0.0251201
$518$	0	0
$519$	−17.0422	−0.748071
$520$	0	0
$521$	32.3960	1.41929	0.709646	−	0.704558i	$-0.248855\pi$
$521$	32.3960	1.41929	0.709646	+	0.704558i	$0.248855\pi$
$522$	0	0
$523$	−41.2880	−1.80540	−0.902699	−	0.430273i	$-0.858417\pi$
$523$	−41.2880	−1.80540	−0.902699	+	0.430273i	$0.858417\pi$
$524$	0	0
$525$	−34.8452	−1.52077
$526$	0	0
$527$	15.9588	0.695175
$528$	0	0
$529$	0.197978	0.00860772
$530$	0	0
$531$	−7.19798	−0.312366
$532$	0	0
$533$	−1.61845	−0.0701029
$534$	0	0
$535$	−52.6853	−2.27778
$536$	0	0
$537$	14.3130	0.617651
$538$	0	0
$539$	12.1980	0.525404
$540$	0	0
$541$	−4.39596	−0.188997	−0.0944984	−	0.995525i	$-0.530125\pi$
$541$	−4.39596	−0.188997	−0.0944984	+	0.995525i	$0.530125\pi$
$542$	0	0
$543$	−6.08296	−0.261045
$544$	0	0
$545$	−63.2685	−2.71013
$546$	0	0
$547$	33.8592	1.44771	0.723856	−	0.689951i	$-0.242368\pi$
$547$	33.8592	1.44771	0.723856	+	0.689951i	$0.242368\pi$
$548$	0	0
$549$	5.70141	0.243330
$550$	0	0
$551$	5.44093	0.231791
$552$	0	0
$553$	27.5123	1.16994
$554$	0	0
$555$	8.76310	0.371973
$556$	0	0
$557$	4.77750	0.202429	0.101215	−	0.994865i	$-0.467727\pi$
$557$	4.77750	0.202429	0.101215	+	0.994865i	$0.467727\pi$
$558$	0	0
$559$	5.78861	0.244832
$560$	0	0
$561$	−2.78256	−0.117480
$562$	0	0
$563$	−38.1141	−1.60632	−0.803159	−	0.595765i	$-0.796849\pi$
$563$	−38.1141	−1.60632	−0.803159	+	0.595765i	$0.796849\pi$
$564$	0	0
$565$	−12.8308	−0.539797
$566$	0	0
$567$	4.38155	0.184008
$568$	0	0
$569$	−33.5866	−1.40802	−0.704011	−	0.710189i	$-0.748609\pi$
$569$	−33.5866	−1.40802	−0.704011	+	0.710189i	$0.748609\pi$
$570$	0	0
$571$	−24.9866	−1.04566	−0.522828	−	0.852438i	$-0.675123\pi$
$571$	−24.9866	−1.04566	−0.522828	+	0.852438i	$0.675123\pi$
$572$	0	0
$573$	−21.0060	−0.877540
$574$	0	0
$575$	38.3037	1.59737
$576$	0	0
$577$	31.8276	1.32500	0.662500	−	0.749062i	$-0.269495\pi$
$577$	31.8276	1.32500	0.662500	+	0.749062i	$0.269495\pi$
$578$	0	0
$579$	3.38760	0.140784
$580$	0	0
$581$	−6.85766	−0.284504
$582$	0	0
$583$	−7.76915	−0.321765
$584$	0	0
$585$	3.59899	0.148800
$586$	0	0
$587$	−42.0020	−1.73361	−0.866804	−	0.498649i	$-0.833830\pi$
$587$	−42.0020	−1.73361	−0.866804	+	0.498649i	$0.833830\pi$
$588$	0	0
$589$	39.8758	1.64305
$590$	0	0
$591$	23.2124	0.954830
$592$	0	0
$593$	24.4372	1.00351	0.501757	−	0.865008i	$-0.332687\pi$
$593$	24.4372	1.00351	0.501757	+	0.865008i	$0.332687\pi$
$594$	0	0
$595$	−43.8786	−1.79885
$596$	0	0
$597$	−10.5026	−0.429843
$598$	0	0
$599$	33.3899	1.36427	0.682137	−	0.731224i	$-0.261051\pi$
$599$	33.3899	1.36427	0.682137	+	0.731224i	$0.261051\pi$
$600$	0	0
$601$	−26.6806	−1.08833	−0.544163	−	0.838980i	$-0.683153\pi$
$601$	−26.6806	−1.08833	−0.544163	+	0.838980i	$0.683153\pi$
$602$	0	0
$603$	−9.59899	−0.390901
$604$	0	0
$605$	−3.59899	−0.146320
$606$	0	0
$607$	14.9874	0.608320	0.304160	−	0.952621i	$-0.401624\pi$
$607$	14.9874	0.608320	0.304160	+	0.952621i	$0.401624\pi$
$608$	0	0
$609$	−3.42883	−0.138943
$610$	0	0
$611$	0.571171	0.0231071
$612$	0	0
$613$	−5.57953	−0.225355	−0.112677	−	0.993632i	$-0.535943\pi$
$613$	−5.57953	−0.225355	−0.112677	+	0.993632i	$0.535943\pi$
$614$	0	0
$615$	5.82478	0.234878
$616$	0	0
$617$	12.0218	0.483978	0.241989	−	0.970279i	$-0.422200\pi$
$617$	12.0218	0.483978	0.241989	+	0.970279i	$0.422200\pi$
$618$	0	0
$619$	38.5151	1.54805	0.774026	−	0.633154i	$-0.218240\pi$
$619$	38.5151	1.54805	0.774026	+	0.633154i	$0.218240\pi$
$620$	0	0
$621$	−4.81643	−0.193277
$622$	0	0
$623$	−35.7980	−1.43421
$624$	0	0
$625$	−1.51784	−0.0607137
$626$	0	0
$627$	−6.95272	−0.277665
$628$	0	0
$629$	6.77520	0.270145
$630$	0	0
$631$	−21.8447	−0.869626	−0.434813	−	0.900521i	$-0.643185\pi$
$631$	−21.8447	−0.869626	−0.434813	+	0.900521i	$0.643185\pi$
$632$	0	0
$633$	18.6194	0.740056
$634$	0	0
$635$	26.1757	1.03875
$636$	0	0
$637$	−12.1980	−0.483301
$638$	0	0
$639$	2.13629	0.0845104
$640$	0	0
$641$	43.1712	1.70516	0.852579	−	0.522598i	$-0.175037\pi$
$641$	43.1712	1.70516	0.852579	+	0.522598i	$0.175037\pi$
$642$	0	0
$643$	−2.67360	−0.105436	−0.0527182	−	0.998609i	$-0.516788\pi$
$643$	−2.67360	−0.105436	−0.0527182	+	0.998609i	$0.516788\pi$
$644$	0	0
$645$	−20.8331	−0.820304
$646$	0	0
$647$	−25.9351	−1.01961	−0.509807	−	0.860289i	$-0.670283\pi$
$647$	−25.9351	−1.01961	−0.509807	+	0.860289i	$0.670283\pi$
$648$	0	0
$649$	−7.19798	−0.282545
$650$	0	0
$651$	−25.1294	−0.984900
$652$	0	0
$653$	1.22761	0.0480402	0.0240201	−	0.999711i	$-0.492353\pi$
$653$	1.22761	0.0480402	0.0240201	+	0.999711i	$0.492353\pi$
$654$	0	0
$655$	−29.5262	−1.15368
$656$	0	0
$657$	6.24526	0.243651
$658$	0	0
$659$	−4.53307	−0.176583	−0.0882917	−	0.996095i	$-0.528141\pi$
$659$	−4.53307	−0.176583	−0.0882917	+	0.996095i	$0.528141\pi$
$660$	0	0
$661$	−4.42097	−0.171956	−0.0859780	−	0.996297i	$-0.527401\pi$
$661$	−4.42097	−0.171956	−0.0859780	+	0.996297i	$0.527401\pi$
$662$	0	0
$663$	2.78256	0.108066
$664$	0	0
$665$	−109.638	−4.25160
$666$	0	0
$667$	3.76915	0.145942
$668$	0	0
$669$	6.60504	0.255366
$670$	0	0
$671$	5.70141	0.220101
$672$	0	0
$673$	42.0705	1.62170	0.810850	−	0.585254i	$-0.199005\pi$
$673$	42.0705	1.62170	0.810850	+	0.585254i	$0.199005\pi$
$674$	0	0
$675$	−7.95272	−0.306100
$676$	0	0
$677$	16.1465	0.620559	0.310280	−	0.950645i	$-0.399577\pi$
$677$	16.1465	0.620559	0.310280	+	0.950645i	$0.399577\pi$
$678$	0	0
$679$	−21.9343	−0.841759
$680$	0	0
$681$	−0.488207	−0.0187081
$682$	0	0
$683$	33.7859	1.29278	0.646390	−	0.763007i	$-0.276278\pi$
$683$	33.7859	1.29278	0.646390	+	0.763007i	$0.276278\pi$
$684$	0	0
$685$	61.3265	2.34316
$686$	0	0
$687$	9.52620	0.363447
$688$	0	0
$689$	7.76915	0.295981
$690$	0	0
$691$	−14.1312	−0.537578	−0.268789	−	0.963199i	$-0.586623\pi$
$691$	−14.1312	−0.537578	−0.268789	+	0.963199i	$0.586623\pi$
$692$	0	0
$693$	4.38155	0.166441
$694$	0	0
$695$	61.8253	2.34517
$696$	0	0
$697$	4.50344	0.170580
$698$	0	0
$699$	26.5128	1.00281
$700$	0	0
$701$	1.42147	0.0536880	0.0268440	−	0.999640i	$-0.491454\pi$
$701$	1.42147	0.0536880	0.0268440	+	0.999640i	$0.491454\pi$
$702$	0	0
$703$	16.9290	0.638490
$704$	0	0
$705$	−2.05564	−0.0774197
$706$	0	0
$707$	−4.26977	−0.160581
$708$	0	0
$709$	−50.8345	−1.90913	−0.954564	−	0.298005i	$-0.903679\pi$
$709$	−50.8345	−1.90913	−0.954564	+	0.298005i	$0.903679\pi$
$710$	0	0
$711$	6.27912	0.235486
$712$	0	0
$713$	27.6236	1.03451
$714$	0	0
$715$	3.59899	0.134595
$716$	0	0
$717$	22.2870	0.832323
$718$	0	0
$719$	−22.2207	−0.828694	−0.414347	−	0.910119i	$-0.635990\pi$
$719$	−22.2207	−0.828694	−0.414347	+	0.910119i	$0.635990\pi$
$720$	0	0
$721$	−42.2068	−1.57186
$722$	0	0
$723$	−29.8925	−1.11171
$724$	0	0
$725$	6.22349	0.231135
$726$	0	0
$727$	−23.3639	−0.866519	−0.433260	−	0.901269i	$-0.642637\pi$
$727$	−23.3639	−0.866519	−0.433260	+	0.901269i	$0.642637\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.1072	−0.595745
$732$	0	0
$733$	42.6394	1.57492	0.787461	−	0.616364i	$-0.211395\pi$
$733$	42.6394	1.57492	0.787461	+	0.616364i	$0.211395\pi$
$734$	0	0
$735$	43.9004	1.61929
$736$	0	0
$737$	−9.59899	−0.353583
$738$	0	0
$739$	−51.2829	−1.88647	−0.943236	−	0.332122i	$-0.892235\pi$
$739$	−51.2829	−1.88647	−0.943236	+	0.332122i	$0.892235\pi$
$740$	0	0
$741$	6.95272	0.255415
$742$	0	0
$743$	13.1859	0.483743	0.241872	−	0.970308i	$-0.422239\pi$
$743$	13.1859	0.483743	0.241872	+	0.970308i	$0.422239\pi$
$744$	0	0
$745$	−59.0006	−2.16161
$746$	0	0
$747$	−1.56512	−0.0572648
$748$	0	0
$749$	64.1411	2.34366
$750$	0	0
$751$	−13.7310	−0.501053	−0.250527	−	0.968110i	$-0.580604\pi$
$751$	−13.7310	−0.501053	−0.250527	+	0.968110i	$0.580604\pi$
$752$	0	0
$753$	−22.8187	−0.831561
$754$	0	0
$755$	−11.6579	−0.424274
$756$	0	0
$757$	−5.26521	−0.191367	−0.0956837	−	0.995412i	$-0.530504\pi$
$757$	−5.26521	−0.191367	−0.0956837	+	0.995412i	$0.530504\pi$
$758$	0	0
$759$	−4.81643	−0.174825
$760$	0	0
$761$	−33.7150	−1.22217	−0.611084	−	0.791566i	$-0.709266\pi$
$761$	−33.7150	−1.22217	−0.611084	+	0.791566i	$0.709266\pi$
$762$	0	0
$763$	77.0256	2.78851
$764$	0	0
$765$	−10.0144	−0.362072
$766$	0	0
$767$	7.19798	0.259904
$768$	0	0
$769$	−40.6304	−1.46517	−0.732586	−	0.680675i	$-0.761687\pi$
$769$	−40.6304	−1.46517	−0.732586	+	0.680675i	$0.761687\pi$
$770$	0	0
$771$	−3.73951	−0.134675
$772$	0	0
$773$	−47.5713	−1.71102	−0.855511	−	0.517785i	$-0.826757\pi$
$773$	−47.5713	−1.71102	−0.855511	+	0.517785i	$0.826757\pi$
$774$	0	0
$775$	45.6111	1.63840
$776$	0	0
$777$	−10.6685	−0.382732
$778$	0	0
$779$	11.2526	0.403167
$780$	0	0
$781$	2.13629	0.0764426
$782$	0	0
$783$	−0.782561	−0.0279664
$784$	0	0
$785$	−55.5800	−1.98374
$786$	0	0
$787$	−54.0051	−1.92507	−0.962537	−	0.271149i	$-0.912596\pi$
$787$	−54.0051	−1.92507	−0.962537	+	0.271149i	$0.912596\pi$
$788$	0	0
$789$	−18.6008	−0.662206
$790$	0	0
$791$	15.6208	0.555410
$792$	0	0
$793$	−5.70141	−0.202463
$794$	0	0
$795$	−27.9611	−0.991677
$796$	0	0
$797$	12.8726	0.455970	0.227985	−	0.973665i	$-0.426786\pi$
$797$	12.8726	0.455970	0.227985	+	0.973665i	$0.426786\pi$
$798$	0	0
$799$	−1.58932	−0.0562260
$800$	0	0
$801$	−8.17016	−0.288678
$802$	0	0
$803$	6.24526	0.220390
$804$	0	0
$805$	−75.9510	−2.67692
$806$	0	0
$807$	14.1363	0.497621
$808$	0	0
$809$	−27.4390	−0.964704	−0.482352	−	0.875977i	$-0.660217\pi$
$809$	−27.4390	−0.964704	−0.482352	+	0.875977i	$0.660217\pi$
$810$	0	0
$811$	−49.7325	−1.74635	−0.873173	−	0.487411i	$-0.837941\pi$
$811$	−49.7325	−1.74635	−0.873173	+	0.487411i	$0.837941\pi$
$812$	0	0
$813$	15.4729	0.542657
$814$	0	0
$815$	−47.8680	−1.67674
$816$	0	0
$817$	−40.2466	−1.40805
$818$	0	0
$819$	−4.38155	−0.153104
$820$	0	0
$821$	17.9634	0.626926	0.313463	−	0.949600i	$-0.398511\pi$
$821$	17.9634	0.626926	0.313463	+	0.949600i	$0.398511\pi$
$822$	0	0
$823$	30.7075	1.07039	0.535197	−	0.844727i	$-0.320237\pi$
$823$	30.7075	1.07039	0.535197	+	0.844727i	$0.320237\pi$
$824$	0	0
$825$	−7.95272	−0.276878
$826$	0	0
$827$	−34.1803	−1.18857	−0.594283	−	0.804256i	$-0.702564\pi$
$827$	−34.1803	−1.18857	−0.594283	+	0.804256i	$0.702564\pi$
$828$	0	0
$829$	−8.94112	−0.310538	−0.155269	−	0.987872i	$-0.549624\pi$
$829$	−8.94112	−0.310538	−0.155269	+	0.987872i	$0.549624\pi$
$830$	0	0
$831$	6.76310	0.234609
$832$	0	0
$833$	33.9416	1.17601
$834$	0	0
$835$	83.3492	2.88442
$836$	0	0
$837$	−5.73528	−0.198240
$838$	0	0
$839$	48.5740	1.67696	0.838480	−	0.544932i	$-0.183445\pi$
$839$	48.5740	1.67696	0.838480	+	0.544932i	$0.183445\pi$
$840$	0	0
$841$	−28.3876	−0.978883
$842$	0	0
$843$	19.4850	0.671098
$844$	0	0
$845$	−3.59899	−0.123809
$846$	0	0
$847$	4.38155	0.150552
$848$	0	0
$849$	−18.8114	−0.645605
$850$	0	0
$851$	11.7274	0.402011
$852$	0	0
$853$	4.36714	0.149528	0.0747641	−	0.997201i	$-0.476180\pi$
$853$	4.36714	0.149528	0.0747641	+	0.997201i	$0.476180\pi$
$854$	0	0
$855$	−25.0228	−0.855760
$856$	0	0
$857$	17.3658	0.593205	0.296603	−	0.955001i	$-0.404146\pi$
$857$	17.3658	0.593205	0.296603	+	0.955001i	$0.404146\pi$
$858$	0	0
$859$	41.6227	1.42015	0.710075	−	0.704126i	$-0.248661\pi$
$859$	41.6227	1.42015	0.710075	+	0.704126i	$0.248661\pi$
$860$	0	0
$861$	−7.09132	−0.241672
$862$	0	0
$863$	−2.89658	−0.0986008	−0.0493004	−	0.998784i	$-0.515699\pi$
$863$	−2.89658	−0.0986008	−0.0493004	+	0.998784i	$0.515699\pi$
$864$	0	0
$865$	−61.3348	−2.08544
$866$	0	0
$867$	9.25735	0.314396
$868$	0	0
$869$	6.27912	0.213005
$870$	0	0
$871$	9.59899	0.325249
$872$	0	0
$873$	−5.00605	−0.169429
$874$	0	0
$875$	−46.5619	−1.57408
$876$	0	0
$877$	24.9496	0.842488	0.421244	−	0.906947i	$-0.361594\pi$
$877$	24.9496	0.842488	0.421244	+	0.906947i	$0.361594\pi$
$878$	0	0
$879$	−24.8452	−0.838009
$880$	0	0
$881$	−36.5907	−1.23277	−0.616386	−	0.787444i	$-0.711404\pi$
$881$	−36.5907	−1.23277	−0.616386	+	0.787444i	$0.711404\pi$
$882$	0	0
$883$	−39.8933	−1.34252	−0.671259	−	0.741223i	$-0.734246\pi$
$883$	−39.8933	−1.34252	−0.671259	+	0.741223i	$0.734246\pi$
$884$	0	0
$885$	−25.9054	−0.870802
$886$	0	0
$887$	−8.95503	−0.300680	−0.150340	−	0.988634i	$-0.548037\pi$
$887$	−8.95503	−0.300680	−0.150340	+	0.988634i	$0.548037\pi$
$888$	0	0
$889$	−31.8673	−1.06880
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−3.97119	−0.132891
$894$	0	0
$895$	51.5123	1.72187
$896$	0	0
$897$	4.81643	0.160816
$898$	0	0
$899$	4.48821	0.149690
$900$	0	0
$901$	−21.6181	−0.720204
$902$	0	0
$903$	25.3631	0.844030
$904$	0	0
$905$	−21.8925	−0.727732
$906$	0	0
$907$	−11.3093	−0.375517	−0.187759	−	0.982215i	$-0.560122\pi$
$907$	−11.3093	−0.375517	−0.187759	+	0.982215i	$0.560122\pi$
$908$	0	0
$909$	−0.974490	−0.0323218
$910$	0	0
$911$	49.3363	1.63458	0.817292	−	0.576224i	$-0.195475\pi$
$911$	49.3363	1.63458	0.817292	+	0.576224i	$0.195475\pi$
$912$	0	0
$913$	−1.56512	−0.0517980
$914$	0	0
$915$	20.5193	0.678348
$916$	0	0
$917$	35.9463	1.18705
$918$	0	0
$919$	−41.6412	−1.37362	−0.686809	−	0.726838i	$-0.740989\pi$
$919$	−41.6412	−1.37362	−0.686809	+	0.726838i	$0.740989\pi$
$920$	0	0
$921$	−32.9694	−1.08638
$922$	0	0
$923$	−2.13629	−0.0703169
$924$	0	0
$925$	19.3639	0.636681
$926$	0	0
$927$	−9.63286	−0.316384
$928$	0	0
$929$	−33.6675	−1.10460	−0.552298	−	0.833647i	$-0.686249\pi$
$929$	−33.6675	−1.10460	−0.552298	+	0.833647i	$0.686249\pi$
$930$	0	0
$931$	84.8091	2.77951
$932$	0	0
$933$	4.21926	0.138132
$934$	0	0
$935$	−10.0144	−0.327506
$936$	0	0
$937$	0.0296353	0.000968143 0	0.000484071	−	1.00000i	$-0.499846\pi$
$937$	0.0296353	0.000968143 0	0.000484071		1.00000i	$0.499846\pi$
$938$	0	0
$939$	−8.79043	−0.286865
$940$	0	0
$941$	37.7539	1.23074	0.615371	−	0.788237i	$-0.289006\pi$
$941$	37.7539	1.23074	0.615371	+	0.788237i	$0.289006\pi$
$942$	0	0
$943$	7.79515	0.253845
$944$	0	0
$945$	15.7691	0.512970
$946$	0	0
$947$	0.327397	0.0106390	0.00531948	−	0.999986i	$-0.498307\pi$
$947$	0.327397	0.0106390	0.00531948	+	0.999986i	$0.498307\pi$
$948$	0	0
$949$	−6.24526	−0.202730
$950$	0	0
$951$	−6.22580	−0.201885
$952$	0	0
$953$	−7.14888	−0.231575	−0.115787	−	0.993274i	$-0.536939\pi$
$953$	−7.14888	−0.231575	−0.115787	+	0.993274i	$0.536939\pi$
$954$	0	0
$955$	−75.6005	−2.44638
$956$	0	0
$957$	−0.782561	−0.0252966
$958$	0	0
$959$	−74.6612	−2.41094
$960$	0	0
$961$	1.89345	0.0610790
$962$	0	0
$963$	14.6389	0.471732
$964$	0	0
$965$	12.1919	0.392472
$966$	0	0
$967$	−60.2890	−1.93876	−0.969381	−	0.245560i	$-0.921028\pi$
$967$	−60.2890	−1.93876	−0.969381	+	0.245560i	$0.921028\pi$
$968$	0	0
$969$	−19.3464	−0.621495
$970$	0	0
$971$	−16.1219	−0.517376	−0.258688	−	0.965961i	$-0.583290\pi$
$971$	−16.1219	−0.517376	−0.258688	+	0.965961i	$0.583290\pi$
$972$	0	0
$973$	−75.2685	−2.41300
$974$	0	0
$975$	7.95272	0.254691
$976$	0	0
$977$	−8.69960	−0.278325	−0.139162	−	0.990270i	$-0.544441\pi$
$977$	−8.69960	−0.278325	−0.139162	+	0.990270i	$0.544441\pi$
$978$	0	0
$979$	−8.17016	−0.261119
$980$	0	0
$981$	17.5795	0.561271
$982$	0	0
$983$	−44.9578	−1.43393	−0.716966	−	0.697108i	$-0.754470\pi$
$983$	−44.9578	−1.43393	−0.716966	+	0.697108i	$0.754470\pi$
$984$	0	0
$985$	83.5411	2.66184
$986$	0	0
$987$	2.50261	0.0796590
$988$	0	0
$989$	−27.8804	−0.886546
$990$	0	0
$991$	−52.6463	−1.67237	−0.836183	−	0.548451i	$-0.815218\pi$
$991$	−52.6463	−1.67237	−0.836183	+	0.548451i	$0.815218\pi$
$992$	0	0
$993$	5.47479	0.173737
$994$	0	0
$995$	−37.7988	−1.19830
$996$	0	0
$997$	−26.4516	−0.837730	−0.418865	−	0.908048i	$-0.637572\pi$
$997$	−26.4516	−0.837730	−0.418865	+	0.908048i	$0.637572\pi$
$998$	0	0
$999$	−2.43488	−0.0770361

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.v.1.1	✓	4	4.3	odd	2
6864.2.a.bx.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.59899 q^{5} +4.38155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.59899 q^{5} +4.38155 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.59899 q^{15} +2.78256 q^{17} +6.95272 q^{19} -4.38155 q^{21} +4.81643 q^{23} +7.95272 q^{25} -1.00000 q^{27} +0.782561 q^{29} +5.73528 q^{31} -1.00000 q^{33} -15.7691 q^{35} +2.43488 q^{37} +1.00000 q^{39} +1.61845 q^{41} -5.78861 q^{43} -3.59899 q^{45} -0.571171 q^{47} +12.1980 q^{49} -2.78256 q^{51} -7.76915 q^{53} -3.59899 q^{55} -6.95272 q^{57} -7.19798 q^{59} +5.70141 q^{61} +4.38155 q^{63} +3.59899 q^{65} -9.59899 q^{67} -4.81643 q^{69} +2.13629 q^{71} +6.24526 q^{73} -7.95272 q^{75} +4.38155 q^{77} +6.27912 q^{79} +1.00000 q^{81} -1.56512 q^{83} -10.0144 q^{85} -0.782561 q^{87} -8.17016 q^{89} -4.38155 q^{91} -5.73528 q^{93} -25.0228 q^{95} -5.00605 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 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

L-functions

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