LMFDB - Embedded newform 6864.2.a.bx.1.4

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			1.4
Root			$-0.386887$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.58473 q^{5} -2.19785 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.58473 q^{5} -2.19785 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.58473 q^{15} +3.38689 q^{17} +6.85032 q^{19} +2.19785 q^{21} -2.97162 q^{23} +7.85032 q^{25} -1.00000 q^{27} +1.38689 q^{29} +6.23721 q^{31} -1.00000 q^{33} -7.87870 q^{35} +1.22623 q^{37} +1.00000 q^{39} +8.19785 q^{41} -11.6613 q^{43} +3.58473 q^{45} -7.04817 q^{47} -2.16947 q^{49} -3.38689 q^{51} +0.121303 q^{53} +3.58473 q^{55} -6.85032 q^{57} +7.16947 q^{59} +14.5957 q^{61} -2.19785 q^{63} -3.58473 q^{65} -2.41527 q^{67} +2.97162 q^{69} +9.82194 q^{71} -8.01979 q^{73} -7.85032 q^{75} -2.19785 q^{77} -16.3783 q^{79} +1.00000 q^{81} -2.77377 q^{83} +12.1411 q^{85} -1.38689 q^{87} -7.46343 q^{89} +2.19785 q^{91} -6.23721 q^{93} +24.5566 q^{95} -10.2744 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.58473	1.60314	0.801571	−	0.597900i	$-0.203998\pi$
$5$	3.58473	1.60314	0.801571	+	0.597900i	$0.203998\pi$
$6$	0	0
$7$	−2.19785	−0.830708	−0.415354	−	0.909660i	$-0.636342\pi$
$7$	−2.19785	−0.830708	−0.415354	+	0.909660i	$0.636342\pi$
$8$	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
$15$	−3.58473	−0.925574
$16$	0	0
$17$	3.38689	0.821441	0.410720	−	0.911761i	$-0.365277\pi$
$17$	3.38689	0.821441	0.410720	+	0.911761i	$0.365277\pi$
$18$	0	0
$19$	6.85032	1.57157	0.785785	−	0.618499i	$-0.212259\pi$
$19$	6.85032	1.57157	0.785785	+	0.618499i	$0.212259\pi$
$20$	0	0
$21$	2.19785	0.479610
$22$	0	0
$23$	−2.97162	−0.619626	−0.309813	−	0.950798i	$-0.600266\pi$
$23$	−2.97162	−0.619626	−0.309813	+	0.950798i	$0.600266\pi$
$24$	0	0
$25$	7.85032	1.57006
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.38689	0.257538	0.128769	−	0.991675i	$-0.458897\pi$
$29$	1.38689	0.257538	0.128769	+	0.991675i	$0.458897\pi$
$30$	0	0
$31$	6.23721	1.12024	0.560118	−	0.828413i	$-0.310756\pi$
$31$	6.23721	1.12024	0.560118	+	0.828413i	$0.310756\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−7.87870	−1.33174
$36$	0	0
$37$	1.22623	0.201590	0.100795	−	0.994907i	$-0.467861\pi$
$37$	1.22623	0.201590	0.100795	+	0.994907i	$0.467861\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.19785	1.28029	0.640144	−	0.768255i	$-0.278875\pi$
$41$	8.19785	1.28029	0.640144	+	0.768255i	$0.278875\pi$
$42$	0	0
$43$	−11.6613	−1.77833	−0.889164	−	0.457588i	$-0.848713\pi$
$43$	−11.6613	−1.77833	−0.889164	+	0.457588i	$0.848713\pi$
$44$	0	0
$45$	3.58473	0.534381
$46$	0	0
$47$	−7.04817	−1.02808	−0.514040	−	0.857766i	$-0.671852\pi$
$47$	−7.04817	−1.02808	−0.514040	+	0.857766i	$0.671852\pi$
$48$	0	0
$49$	−2.16947	−0.309924
$50$	0	0
$51$	−3.38689	−0.474259
$52$	0	0
$53$	0.121303	0.0166622	0.00833110	−	0.999965i	$-0.497348\pi$
$53$	0.121303	0.0166622	0.00833110	+	0.999965i	$0.497348\pi$
$54$	0	0
$55$	3.58473	0.483365
$56$	0	0
$57$	−6.85032	−0.907347
$58$	0	0
$59$	7.16947	0.933385	0.466693	−	0.884420i	$-0.345445\pi$
$59$	7.16947	0.933385	0.466693	+	0.884420i	$0.345445\pi$
$60$	0	0
$61$	14.5957	1.86879	0.934395	−	0.356239i	$-0.115941\pi$
$61$	14.5957	1.86879	0.934395	+	0.356239i	$0.115941\pi$
$62$	0	0
$63$	−2.19785	−0.276903
$64$	0	0
$65$	−3.58473	−0.444632
$66$	0	0
$67$	−2.41527	−0.295072	−0.147536	−	0.989057i	$-0.547134\pi$
$67$	−2.41527	−0.295072	−0.147536	+	0.989057i	$0.547134\pi$
$68$	0	0
$69$	2.97162	0.357741
$70$	0	0
$71$	9.82194	1.16565	0.582825	−	0.812598i	$-0.301947\pi$
$71$	9.82194	1.16565	0.582825	+	0.812598i	$0.301947\pi$
$72$	0	0
$73$	−8.01979	−0.938645	−0.469323	−	0.883027i	$-0.655502\pi$
$73$	−8.01979	−0.938645	−0.469323	+	0.883027i	$0.655502\pi$
$74$	0	0
$75$	−7.85032	−0.906477
$76$	0	0
$77$	−2.19785	−0.250468
$78$	0	0
$79$	−16.3783	−1.84270	−0.921351	−	0.388732i	$-0.872913\pi$
$79$	−16.3783	−1.84270	−0.921351	+	0.388732i	$0.872913\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.77377	−0.304461	−0.152231	−	0.988345i	$-0.548646\pi$
$83$	−2.77377	−0.304461	−0.152231	+	0.988345i	$0.548646\pi$
$84$	0	0
$85$	12.1411	1.31689
$86$	0	0
$87$	−1.38689	−0.148690
$88$	0	0
$89$	−7.46343	−0.791122	−0.395561	−	0.918440i	$-0.629450\pi$
$89$	−7.46343	−0.791122	−0.395561	+	0.918440i	$0.629450\pi$
$90$	0	0
$91$	2.19785	0.230397
$92$	0	0
$93$	−6.23721	−0.646768
$94$	0	0
$95$	24.5566	2.51945
$96$	0	0
$97$	−10.2744	−1.04321	−0.521603	−	0.853188i	$-0.674666\pi$
$97$	−10.2744	−1.04321	−0.521603	+	0.853188i	$0.674666\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	18.0570	1.79674	0.898368	−	0.439244i	$-0.144754\pi$
$101$	18.0570	1.79674	0.898368	+	0.439244i	$0.144754\pi$
$102$	0	0
$103$	5.94324	0.585605	0.292803	−	0.956173i	$-0.405412\pi$
$103$	5.94324	0.585605	0.292803	+	0.956173i	$0.405412\pi$
$104$	0	0
$105$	7.87870	0.768882
$106$	0	0
$107$	4.33115	0.418708	0.209354	−	0.977840i	$-0.432864\pi$
$107$	4.33115	0.418708	0.209354	+	0.977840i	$0.432864\pi$
$108$	0	0
$109$	−3.36732	−0.322530	−0.161265	−	0.986911i	$-0.551557\pi$
$109$	−3.36732	−0.322530	−0.161265	+	0.986911i	$0.551557\pi$
$110$	0	0
$111$	−1.22623	−0.116388
$112$	0	0
$113$	4.77377	0.449079	0.224539	−	0.974465i	$-0.427912\pi$
$113$	4.77377	0.449079	0.224539	+	0.974465i	$0.427912\pi$
$114$	0	0
$115$	−10.6525	−0.993348
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−7.44386	−0.682378
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.19785	−0.739175
$124$	0	0
$125$	10.2176	0.913893
$126$	0	0
$127$	20.6527	1.83263	0.916315	−	0.400459i	$-0.131149\pi$
$127$	20.6527	1.83263	0.916315	+	0.400459i	$0.131149\pi$
$128$	0	0
$129$	11.6613	1.02672
$130$	0	0
$131$	−0.895077	−0.0782032	−0.0391016	−	0.999235i	$-0.512450\pi$
$131$	−0.895077	−0.0782032	−0.0391016	+	0.999235i	$0.512450\pi$
$132$	0	0
$133$	−15.0560	−1.30552
$134$	0	0
$135$	−3.58473	−0.308525
$136$	0	0
$137$	−13.9159	−1.18891	−0.594457	−	0.804127i	$-0.702633\pi$
$137$	−13.9159	−1.18891	−0.594457	+	0.804127i	$0.702633\pi$
$138$	0	0
$139$	10.9520	0.928941	0.464470	−	0.885589i	$-0.346245\pi$
$139$	10.9520	0.928941	0.464470	+	0.885589i	$0.346245\pi$
$140$	0	0
$141$	7.04817	0.593563
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	4.97162	0.412871
$146$	0	0
$147$	2.16947	0.178935
$148$	0	0
$149$	20.3509	1.66721	0.833607	−	0.552358i	$-0.186272\pi$
$149$	20.3509	1.66721	0.833607	+	0.552358i	$0.186272\pi$
$150$	0	0
$151$	−16.2942	−1.32600	−0.663001	−	0.748619i	$-0.730717\pi$
$151$	−16.2942	−1.32600	−0.663001	+	0.748619i	$0.730717\pi$
$152$	0	0
$153$	3.38689	0.273814
$154$	0	0
$155$	22.3587	1.79590
$156$	0	0
$157$	−13.1893	−1.05262	−0.526309	−	0.850294i	$-0.676424\pi$
$157$	−13.1893	−1.05262	−0.526309	+	0.850294i	$0.676424\pi$
$158$	0	0
$159$	−0.121303	−0.00961992
$160$	0	0
$161$	6.53117	0.514728
$162$	0	0
$163$	15.0110	1.17575	0.587875	−	0.808952i	$-0.299965\pi$
$163$	15.0110	1.17575	0.587875	+	0.808952i	$0.299965\pi$
$164$	0	0
$165$	−3.58473	−0.279071
$166$	0	0
$167$	18.7346	1.44973	0.724865	−	0.688891i	$-0.241902\pi$
$167$	18.7346	1.44973	0.724865	+	0.688891i	$0.241902\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.85032	0.523857
$172$	0	0
$173$	−18.7740	−1.42736	−0.713680	−	0.700472i	$-0.752973\pi$
$173$	−18.7740	−1.42736	−0.713680	+	0.700472i	$0.752973\pi$
$174$	0	0
$175$	−17.2538	−1.30426
$176$	0	0
$177$	−7.16947	−0.538890
$178$	0	0
$179$	16.7368	1.25097	0.625484	−	0.780237i	$-0.284902\pi$
$179$	16.7368	1.25097	0.625484	+	0.780237i	$0.284902\pi$
$180$	0	0
$181$	8.39787	0.624208	0.312104	−	0.950048i	$-0.398966\pi$
$181$	8.39787	0.624208	0.312104	+	0.950048i	$0.398966\pi$
$182$	0	0
$183$	−14.5957	−1.07895
$184$	0	0
$185$	4.39569	0.323178
$186$	0	0
$187$	3.38689	0.247674
$188$	0	0
$189$	2.19785	0.159870
$190$	0	0
$191$	26.2744	1.90115	0.950574	−	0.310498i	$-0.100496\pi$
$191$	26.2744	1.90115	0.950574	+	0.310498i	$0.100496\pi$
$192$	0	0
$193$	−2.07654	−0.149473	−0.0747365	−	0.997203i	$-0.523812\pi$
$193$	−2.07654	−0.149473	−0.0747365	+	0.997203i	$0.523812\pi$
$194$	0	0
$195$	3.58473	0.256708
$196$	0	0
$197$	13.3106	0.948338	0.474169	−	0.880434i	$-0.342749\pi$
$197$	13.3106	0.948338	0.474169	+	0.880434i	$0.342749\pi$
$198$	0	0
$199$	−7.49079	−0.531008	−0.265504	−	0.964110i	$-0.585538\pi$
$199$	−7.49079	−0.531008	−0.265504	+	0.964110i	$0.585538\pi$
$200$	0	0
$201$	2.41527	0.170360
$202$	0	0
$203$	−3.04817	−0.213939
$204$	0	0
$205$	29.3871	2.05248
$206$	0	0
$207$	−2.97162	−0.206542
$208$	0	0
$209$	6.85032	0.473846
$210$	0	0
$211$	5.45143	0.375292	0.187646	−	0.982237i	$-0.439914\pi$
$211$	5.45143	0.375292	0.187646	+	0.982237i	$0.439914\pi$
$212$	0	0
$213$	−9.82194	−0.672988
$214$	0	0
$215$	−41.8026	−2.85091
$216$	0	0
$217$	−13.7084	−0.930588
$218$	0	0
$219$	8.01979	0.541927
$220$	0	0
$221$	−3.38689	−0.227827
$222$	0	0
$223$	−4.68966	−0.314043	−0.157021	−	0.987595i	$-0.550189\pi$
$223$	−4.68966	−0.314043	−0.157021	+	0.987595i	$0.550189\pi$
$224$	0	0
$225$	7.85032	0.523355
$226$	0	0
$227$	4.65030	0.308651	0.154326	−	0.988020i	$-0.450680\pi$
$227$	4.65030	0.308651	0.154326	+	0.988020i	$0.450680\pi$
$228$	0	0
$229$	16.7914	1.10961	0.554803	−	0.831982i	$-0.312794\pi$
$229$	16.7914	1.10961	0.554803	+	0.831982i	$0.312794\pi$
$230$	0	0
$231$	2.19785	0.144608
$232$	0	0
$233$	8.29958	0.543723	0.271862	−	0.962336i	$-0.412361\pi$
$233$	8.29958	0.543723	0.271862	+	0.962336i	$0.412361\pi$
$234$	0	0
$235$	−25.2658	−1.64816
$236$	0	0
$237$	16.3783	1.06388
$238$	0	0
$239$	−15.5028	−1.00279	−0.501396	−	0.865218i	$-0.667180\pi$
$239$	−15.5028	−1.00279	−0.501396	+	0.865218i	$0.667180\pi$
$240$	0	0
$241$	−22.1041	−1.42385	−0.711926	−	0.702255i	$-0.752177\pi$
$241$	−22.1041	−1.42385	−0.711926	+	0.702255i	$0.752177\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.77697	−0.496852
$246$	0	0
$247$	−6.85032	−0.435875
$248$	0	0
$249$	2.77377	0.175781
$250$	0	0
$251$	−17.6615	−1.11478	−0.557392	−	0.830250i	$-0.688198\pi$
$251$	−17.6615	−1.11478	−0.557392	+	0.830250i	$0.688198\pi$
$252$	0	0
$253$	−2.97162	−0.186824
$254$	0	0
$255$	−12.1411	−0.760305
$256$	0	0
$257$	−1.09510	−0.0683102	−0.0341551	−	0.999417i	$-0.510874\pi$
$257$	−1.09510	−0.0683102	−0.0341551	+	0.999417i	$0.510874\pi$
$258$	0	0
$259$	−2.69506	−0.167463
$260$	0	0
$261$	1.38689	0.0858461
$262$	0	0
$263$	22.0220	1.35793	0.678966	−	0.734170i	$-0.262428\pi$
$263$	22.0220	1.35793	0.678966	+	0.734170i	$0.262428\pi$
$264$	0	0
$265$	0.434838	0.0267119
$266$	0	0
$267$	7.46343	0.456755
$268$	0	0
$269$	−21.8219	−1.33051	−0.665254	−	0.746617i	$-0.731677\pi$
$269$	−21.8219	−1.33051	−0.665254	+	0.746617i	$0.731677\pi$
$270$	0	0
$271$	16.2155	0.985019	0.492510	−	0.870307i	$-0.336080\pi$
$271$	16.2155	0.985019	0.492510	+	0.870307i	$0.336080\pi$
$272$	0	0
$273$	−2.19785	−0.133020
$274$	0	0
$275$	7.85032	0.473392
$276$	0	0
$277$	6.39569	0.384280	0.192140	−	0.981368i	$-0.438457\pi$
$277$	6.39569	0.384280	0.192140	+	0.981368i	$0.438457\pi$
$278$	0	0
$279$	6.23721	0.373412
$280$	0	0
$281$	1.66668	0.0994257	0.0497128	−	0.998764i	$-0.484169\pi$
$281$	1.66668	0.0994257	0.0497128	+	0.998764i	$0.484169\pi$
$282$	0	0
$283$	−24.8953	−1.47987	−0.739936	−	0.672678i	$-0.765144\pi$
$283$	−24.8953	−1.47987	−0.739936	+	0.672678i	$0.765144\pi$
$284$	0	0
$285$	−24.5566	−1.45461
$286$	0	0
$287$	−18.0176	−1.06355
$288$	0	0
$289$	−5.52900	−0.325235
$290$	0	0
$291$	10.2744	0.602295
$292$	0	0
$293$	−27.2538	−1.59218	−0.796092	−	0.605176i	$-0.793103\pi$
$293$	−27.2538	−1.59218	−0.796092	+	0.605176i	$0.793103\pi$
$294$	0	0
$295$	25.7006	1.49635
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.97162	0.171853
$300$	0	0
$301$	25.6297	1.47727
$302$	0	0
$303$	−18.0570	−1.03735
$304$	0	0
$305$	52.3218	2.99593
$306$	0	0
$307$	−21.9806	−1.25450	−0.627251	−	0.778817i	$-0.715820\pi$
$307$	−21.9806	−1.25450	−0.627251	+	0.778817i	$0.715820\pi$
$308$	0	0
$309$	−5.94324	−0.338099
$310$	0	0
$311$	−14.2198	−0.806331	−0.403166	−	0.915127i	$-0.632090\pi$
$311$	−14.2198	−0.806331	−0.403166	+	0.915127i	$0.632090\pi$
$312$	0	0
$313$	25.2680	1.42823	0.714115	−	0.700028i	$-0.246829\pi$
$313$	25.2680	1.42823	0.714115	+	0.700028i	$0.246829\pi$
$314$	0	0
$315$	−7.87870	−0.443914
$316$	0	0
$317$	−21.8024	−1.22454	−0.612271	−	0.790648i	$-0.709744\pi$
$317$	−21.8024	−1.22454	−0.612271	+	0.790648i	$0.709744\pi$
$318$	0	0
$319$	1.38689	0.0776508
$320$	0	0
$321$	−4.33115	−0.241741
$322$	0	0
$323$	23.2013	1.29095
$324$	0	0
$325$	−7.85032	−0.435457
$326$	0	0
$327$	3.36732	0.186213
$328$	0	0
$329$	15.4908	0.854035
$330$	0	0
$331$	−1.14211	−0.0627760	−0.0313880	−	0.999507i	$-0.509993\pi$
$331$	−1.14211	−0.0627760	−0.0313880	+	0.999507i	$0.509993\pi$
$332$	0	0
$333$	1.22623	0.0671968
$334$	0	0
$335$	−8.65809	−0.473042
$336$	0	0
$337$	−11.2658	−0.613687	−0.306844	−	0.951760i	$-0.599273\pi$
$337$	−11.2658	−0.613687	−0.306844	+	0.951760i	$0.599273\pi$
$338$	0	0
$339$	−4.77377	−0.259276
$340$	0	0
$341$	6.23721	0.337764
$342$	0	0
$343$	20.1531	1.08816
$344$	0	0
$345$	10.6525	0.573510
$346$	0	0
$347$	23.9041	1.28324	0.641620	−	0.767023i	$-0.278263\pi$
$347$	23.9041	1.28324	0.641620	+	0.767023i	$0.278263\pi$
$348$	0	0
$349$	3.62192	0.193877	0.0969385	−	0.995290i	$-0.469095\pi$
$349$	3.62192	0.193877	0.0969385	+	0.995290i	$0.469095\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	28.6079	1.52265	0.761323	−	0.648373i	$-0.224550\pi$
$353$	28.6079	1.52265	0.761323	+	0.648373i	$0.224550\pi$
$354$	0	0
$355$	35.2090	1.86870
$356$	0	0
$357$	7.44386	0.393971
$358$	0	0
$359$	24.2942	1.28220	0.641099	−	0.767458i	$-0.278479\pi$
$359$	24.2942	1.28220	0.641099	+	0.767458i	$0.278479\pi$
$360$	0	0
$361$	27.9269	1.46983
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−28.7488	−1.50478
$366$	0	0
$367$	−13.2658	−0.692469	−0.346235	−	0.938148i	$-0.612540\pi$
$367$	−13.2658	−0.692469	−0.346235	+	0.938148i	$0.612540\pi$
$368$	0	0
$369$	8.19785	0.426763
$370$	0	0
$371$	−0.266605	−0.0138414
$372$	0	0
$373$	14.7170	0.762018	0.381009	−	0.924571i	$-0.375577\pi$
$373$	14.7170	0.762018	0.381009	+	0.924571i	$0.375577\pi$
$374$	0	0
$375$	−10.2176	−0.527636
$376$	0	0
$377$	−1.38689	−0.0714283
$378$	0	0
$379$	7.19887	0.369781	0.184890	−	0.982759i	$-0.440807\pi$
$379$	7.19887	0.369781	0.184890	+	0.982759i	$0.440807\pi$
$380$	0	0
$381$	−20.6527	−1.05807
$382$	0	0
$383$	−17.6615	−0.902460	−0.451230	−	0.892408i	$-0.649015\pi$
$383$	−17.6615	−0.902460	−0.451230	+	0.892408i	$0.649015\pi$
$384$	0	0
$385$	−7.87870	−0.401536
$386$	0	0
$387$	−11.6613	−0.592776
$388$	0	0
$389$	−10.5738	−0.536111	−0.268055	−	0.963404i	$-0.586381\pi$
$389$	−10.5738	−0.536111	−0.268055	+	0.963404i	$0.586381\pi$
$390$	0	0
$391$	−10.0645	−0.508986
$392$	0	0
$393$	0.895077	0.0451506
$394$	0	0
$395$	−58.7118	−2.95411
$396$	0	0
$397$	10.5312	0.528544	0.264272	−	0.964448i	$-0.414868\pi$
$397$	10.5312	0.528544	0.264272	+	0.964448i	$0.414868\pi$
$398$	0	0
$399$	15.0560	0.753740
$400$	0	0
$401$	−7.43165	−0.371119	−0.185559	−	0.982633i	$-0.559410\pi$
$401$	−7.43165	−0.371119	−0.185559	+	0.982633i	$0.559410\pi$
$402$	0	0
$403$	−6.23721	−0.310697
$404$	0	0
$405$	3.58473	0.178127
$406$	0	0
$407$	1.22623	0.0607818
$408$	0	0
$409$	24.4155	1.20727	0.603634	−	0.797262i	$-0.293719\pi$
$409$	24.4155	1.20727	0.603634	+	0.797262i	$0.293719\pi$
$410$	0	0
$411$	13.9159	0.686420
$412$	0	0
$413$	−15.7574	−0.775371
$414$	0	0
$415$	−9.94324	−0.488095
$416$	0	0
$417$	−10.9520	−0.536324
$418$	0	0
$419$	−13.1325	−0.641564	−0.320782	−	0.947153i	$-0.603946\pi$
$419$	−13.1325	−0.641564	−0.320782	+	0.947153i	$0.603946\pi$
$420$	0	0
$421$	22.6525	1.10401	0.552007	−	0.833840i	$-0.313862\pi$
$421$	22.6525	1.10401	0.552007	+	0.833840i	$0.313862\pi$
$422$	0	0
$423$	−7.04817	−0.342694
$424$	0	0
$425$	26.5881	1.28971
$426$	0	0
$427$	−32.0791	−1.55242
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	22.0568	1.06244	0.531218	−	0.847235i	$-0.321734\pi$
$431$	22.0568	1.06244	0.531218	+	0.847235i	$0.321734\pi$
$432$	0	0
$433$	36.0963	1.73468	0.867339	−	0.497717i	$-0.165828\pi$
$433$	36.0963	1.73468	0.867339	+	0.497717i	$0.165828\pi$
$434$	0	0
$435$	−4.97162	−0.238371
$436$	0	0
$437$	−20.3566	−0.973786
$438$	0	0
$439$	14.4351	0.688947	0.344474	−	0.938796i	$-0.388057\pi$
$439$	14.4351	0.688947	0.344474	+	0.938796i	$0.388057\pi$
$440$	0	0
$441$	−2.16947	−0.103308
$442$	0	0
$443$	10.8525	0.515617	0.257809	−	0.966196i	$-0.417000\pi$
$443$	10.8525	0.515617	0.257809	+	0.966196i	$0.417000\pi$
$444$	0	0
$445$	−26.7544	−1.26828
$446$	0	0
$447$	−20.3509	−0.962566
$448$	0	0
$449$	−33.4888	−1.58044	−0.790218	−	0.612826i	$-0.790033\pi$
$449$	−33.4888	−1.58044	−0.790218	+	0.612826i	$0.790033\pi$
$450$	0	0
$451$	8.19785	0.386022
$452$	0	0
$453$	16.2942	0.765567
$454$	0	0
$455$	7.87870	0.369359
$456$	0	0
$457$	30.0254	1.40453	0.702265	−	0.711916i	$-0.252172\pi$
$457$	30.0254	1.40453	0.702265	+	0.711916i	$0.252172\pi$
$458$	0	0
$459$	−3.38689	−0.158086
$460$	0	0
$461$	−10.9325	−0.509176	−0.254588	−	0.967050i	$-0.581940\pi$
$461$	−10.9325	−0.509176	−0.254588	+	0.967050i	$0.581940\pi$
$462$	0	0
$463$	32.2122	1.49703	0.748515	−	0.663118i	$-0.230767\pi$
$463$	32.2122	1.49703	0.748515	+	0.663118i	$0.230767\pi$
$464$	0	0
$465$	−22.3587	−1.03686
$466$	0	0
$467$	3.11054	0.143939	0.0719693	−	0.997407i	$-0.477072\pi$
$467$	3.11054	0.143939	0.0719693	+	0.997407i	$0.477072\pi$
$468$	0	0
$469$	5.30839	0.245118
$470$	0	0
$471$	13.1893	0.607729
$472$	0	0
$473$	−11.6613	−0.536186
$474$	0	0
$475$	53.7772	2.46747
$476$	0	0
$477$	0.121303	0.00555407
$478$	0	0
$479$	−18.2374	−0.833289	−0.416645	−	0.909070i	$-0.636794\pi$
$479$	−18.2374	−0.833289	−0.416645	+	0.909070i	$0.636794\pi$
$480$	0	0
$481$	−1.22623	−0.0559111
$482$	0	0
$483$	−6.53117	−0.297178
$484$	0	0
$485$	−36.8310	−1.67241
$486$	0	0
$487$	26.7151	1.21057	0.605287	−	0.796007i	$-0.293058\pi$
$487$	26.7151	1.21057	0.605287	+	0.796007i	$0.293058\pi$
$488$	0	0
$489$	−15.0110	−0.678820
$490$	0	0
$491$	26.3609	1.18965	0.594825	−	0.803855i	$-0.297221\pi$
$491$	26.3609	1.18965	0.594825	+	0.803855i	$0.297221\pi$
$492$	0	0
$493$	4.69723	0.211553
$494$	0	0
$495$	3.58473	0.161122
$496$	0	0
$497$	−21.5871	−0.968315
$498$	0	0
$499$	−0.714629	−0.0319912	−0.0159956	−	0.999872i	$-0.505092\pi$
$499$	−0.714629	−0.0319912	−0.0159956	+	0.999872i	$0.505092\pi$
$500$	0	0
$501$	−18.7346	−0.837002
$502$	0	0
$503$	24.7170	1.10208	0.551039	−	0.834480i	$-0.314232\pi$
$503$	24.7170	1.10208	0.551039	+	0.834480i	$0.314232\pi$
$504$	0	0
$505$	64.7294	2.88042
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	12.2548	0.543185	0.271593	−	0.962412i	$-0.412450\pi$
$509$	12.2548	0.543185	0.271593	+	0.962412i	$0.412450\pi$
$510$	0	0
$511$	17.6263	0.779740
$512$	0	0
$513$	−6.85032	−0.302449
$514$	0	0
$515$	21.3049	0.938808
$516$	0	0
$517$	−7.04817	−0.309978
$518$	0	0
$519$	18.7740	0.824087
$520$	0	0
$521$	3.66106	0.160394	0.0801971	−	0.996779i	$-0.474445\pi$
$521$	3.66106	0.160394	0.0801971	+	0.996779i	$0.474445\pi$
$522$	0	0
$523$	−3.85351	−0.168502	−0.0842511	−	0.996445i	$-0.526850\pi$
$523$	−3.85351	−0.168502	−0.0842511	+	0.996445i	$0.526850\pi$
$524$	0	0
$525$	17.2538	0.753018
$526$	0	0
$527$	21.1247	0.920207
$528$	0	0
$529$	−14.1695	−0.616064
$530$	0	0
$531$	7.16947	0.311128
$532$	0	0
$533$	−8.19785	−0.355088
$534$	0	0
$535$	15.5260	0.671248
$536$	0	0
$537$	−16.7368	−0.722246
$538$	0	0
$539$	−2.16947	−0.0934456
$540$	0	0
$541$	24.3389	1.04641	0.523206	−	0.852206i	$-0.324736\pi$
$541$	24.3389	1.04641	0.523206	+	0.852206i	$0.324736\pi$
$542$	0	0
$543$	−8.39787	−0.360387
$544$	0	0
$545$	−12.0709	−0.517062
$546$	0	0
$547$	2.90168	0.124067	0.0620334	−	0.998074i	$-0.480241\pi$
$547$	2.90168	0.124067	0.0620334	+	0.998074i	$0.480241\pi$
$548$	0	0
$549$	14.5957	0.622930
$550$	0	0
$551$	9.50062	0.404740
$552$	0	0
$553$	35.9970	1.53075
$554$	0	0
$555$	−4.39569	−0.186587
$556$	0	0
$557$	−30.5368	−1.29389	−0.646943	−	0.762539i	$-0.723953\pi$
$557$	−30.5368	−1.29389	−0.646943	+	0.762539i	$0.723953\pi$
$558$	0	0
$559$	11.6613	0.493220
$560$	0	0
$561$	−3.38689	−0.142994
$562$	0	0
$563$	36.5742	1.54142	0.770709	−	0.637187i	$-0.219902\pi$
$563$	36.5742	1.54142	0.770709	+	0.637187i	$0.219902\pi$
$564$	0	0
$565$	17.1127	0.719937
$566$	0	0
$567$	−2.19785	−0.0923009
$568$	0	0
$569$	12.7422	0.534181	0.267090	−	0.963671i	$-0.413938\pi$
$569$	12.7422	0.534181	0.267090	+	0.963671i	$0.413938\pi$
$570$	0	0
$571$	−16.4918	−0.690161	−0.345080	−	0.938573i	$-0.612148\pi$
$571$	−16.4918	−0.690161	−0.345080	+	0.938573i	$0.612148\pi$
$572$	0	0
$573$	−26.2744	−1.09763
$574$	0	0
$575$	−23.3282	−0.972852
$576$	0	0
$577$	−23.4297	−0.975391	−0.487695	−	0.873014i	$-0.662162\pi$
$577$	−23.4297	−0.975391	−0.487695	+	0.873014i	$0.662162\pi$
$578$	0	0
$579$	2.07654	0.0862982
$580$	0	0
$581$	6.09633	0.252918
$582$	0	0
$583$	0.121303	0.00502384
$584$	0	0
$585$	−3.58473	−0.148211
$586$	0	0
$587$	19.2986	0.796537	0.398268	−	0.917269i	$-0.369611\pi$
$587$	19.2986	0.796537	0.398268	+	0.917269i	$0.369611\pi$
$588$	0	0
$589$	42.7268	1.76053
$590$	0	0
$591$	−13.3106	−0.547523
$592$	0	0
$593$	−9.46365	−0.388625	−0.194313	−	0.980940i	$-0.562248\pi$
$593$	−9.46365	−0.388625	−0.194313	+	0.980940i	$0.562248\pi$
$594$	0	0
$595$	−26.6843	−1.09395
$596$	0	0
$597$	7.49079	0.306578
$598$	0	0
$599$	−0.613328	−0.0250599	−0.0125299	−	0.999921i	$-0.503989\pi$
$599$	−0.613328	−0.0250599	−0.0125299	+	0.999921i	$0.503989\pi$
$600$	0	0
$601$	−23.8537	−0.973014	−0.486507	−	0.873677i	$-0.661729\pi$
$601$	−23.8537	−0.973014	−0.486507	+	0.873677i	$0.661729\pi$
$602$	0	0
$603$	−2.41527	−0.0983572
$604$	0	0
$605$	3.58473	0.145740
$606$	0	0
$607$	47.7478	1.93802	0.969011	−	0.247016i	$-0.0794501\pi$
$607$	47.7478	1.93802	0.969011	+	0.247016i	$0.0794501\pi$
$608$	0	0
$609$	3.04817	0.123518
$610$	0	0
$611$	7.04817	0.285138
$612$	0	0
$613$	15.3673	0.620680	0.310340	−	0.950626i	$-0.399557\pi$
$613$	15.3673	0.620680	0.310340	+	0.950626i	$0.399557\pi$
$614$	0	0
$615$	−29.3871	−1.18500
$616$	0	0
$617$	−6.90729	−0.278077	−0.139039	−	0.990287i	$-0.544401\pi$
$617$	−6.90729	−0.278077	−0.139039	+	0.990287i	$0.544401\pi$
$618$	0	0
$619$	−28.9895	−1.16518	−0.582592	−	0.812765i	$-0.697962\pi$
$619$	−28.9895	−1.16518	−0.582592	+	0.812765i	$0.697962\pi$
$620$	0	0
$621$	2.97162	0.119247
$622$	0	0
$623$	16.4035	0.657192
$624$	0	0
$625$	−2.62409	−0.104964
$626$	0	0
$627$	−6.85032	−0.273575
$628$	0	0
$629$	4.15309	0.165595
$630$	0	0
$631$	−13.0428	−0.519224	−0.259612	−	0.965713i	$-0.583595\pi$
$631$	−13.0428	−0.519224	−0.259612	+	0.965713i	$0.583595\pi$
$632$	0	0
$633$	−5.45143	−0.216675
$634$	0	0
$635$	74.0344	2.93797
$636$	0	0
$637$	2.16947	0.0859575
$638$	0	0
$639$	9.82194	0.388550
$640$	0	0
$641$	11.8142	0.466631	0.233315	−	0.972401i	$-0.425043\pi$
$641$	11.8142	0.466631	0.233315	+	0.972401i	$0.425043\pi$
$642$	0	0
$643$	−25.2286	−0.994919	−0.497460	−	0.867487i	$-0.665734\pi$
$643$	−25.2286	−0.994919	−0.497460	+	0.867487i	$0.665734\pi$
$644$	0	0
$645$	41.8026	1.64598
$646$	0	0
$647$	−22.6744	−0.891424	−0.445712	−	0.895176i	$-0.647049\pi$
$647$	−22.6744	−0.891424	−0.445712	+	0.895176i	$0.647049\pi$
$648$	0	0
$649$	7.16947	0.281426
$650$	0	0
$651$	13.7084	0.537275
$652$	0	0
$653$	−16.1957	−0.633786	−0.316893	−	0.948461i	$-0.602640\pi$
$653$	−16.1957	−0.633786	−0.316893	+	0.948461i	$0.602640\pi$
$654$	0	0
$655$	−3.20861	−0.125371
$656$	0	0
$657$	−8.01979	−0.312882
$658$	0	0
$659$	−24.7390	−0.963694	−0.481847	−	0.876255i	$-0.660034\pi$
$659$	−24.7390	−0.963694	−0.481847	+	0.876255i	$0.660034\pi$
$660$	0	0
$661$	−38.0146	−1.47860	−0.739298	−	0.673378i	$-0.764843\pi$
$661$	−38.0146	−1.47860	−0.739298	+	0.673378i	$0.764843\pi$
$662$	0	0
$663$	3.38689	0.131536
$664$	0	0
$665$	−53.9716	−2.09293
$666$	0	0
$667$	−4.12130	−0.159577
$668$	0	0
$669$	4.68966	0.181313
$670$	0	0
$671$	14.5957	0.563461
$672$	0	0
$673$	5.24040	0.202002	0.101001	−	0.994886i	$-0.467795\pi$
$673$	5.24040	0.202002	0.101001	+	0.994886i	$0.467795\pi$
$674$	0	0
$675$	−7.85032	−0.302159
$676$	0	0
$677$	7.01315	0.269537	0.134769	−	0.990877i	$-0.456971\pi$
$677$	7.01315	0.269537	0.134769	+	0.990877i	$0.456971\pi$
$678$	0	0
$679$	22.5815	0.866600
$680$	0	0
$681$	−4.65030	−0.178200
$682$	0	0
$683$	−28.9523	−1.10783	−0.553914	−	0.832574i	$-0.686866\pi$
$683$	−28.9523	−1.10783	−0.553914	+	0.832574i	$0.686866\pi$
$684$	0	0
$685$	−49.8847	−1.90600
$686$	0	0
$687$	−16.7914	−0.640631
$688$	0	0
$689$	−0.121303	−0.00462126
$690$	0	0
$691$	14.1017	0.536455	0.268228	−	0.963356i	$-0.413562\pi$
$691$	14.1017	0.536455	0.268228	+	0.963356i	$0.413562\pi$
$692$	0	0
$693$	−2.19785	−0.0834893
$694$	0	0
$695$	39.2602	1.48922
$696$	0	0
$697$	27.7652	1.05168
$698$	0	0
$699$	−8.29958	−0.313919
$700$	0	0
$701$	−8.28196	−0.312805	−0.156403	−	0.987693i	$-0.549990\pi$
$701$	−8.28196	−0.312805	−0.156403	+	0.987693i	$0.549990\pi$
$702$	0	0
$703$	8.40004	0.316813
$704$	0	0
$705$	25.2658	0.951565
$706$	0	0
$707$	−39.6865	−1.49256
$708$	0	0
$709$	−42.1007	−1.58112	−0.790562	−	0.612382i	$-0.790211\pi$
$709$	−42.1007	−1.58112	−0.790562	+	0.612382i	$0.790211\pi$
$710$	0	0
$711$	−16.3783	−0.614234
$712$	0	0
$713$	−18.5346	−0.694127
$714$	0	0
$715$	−3.58473	−0.134061
$716$	0	0
$717$	15.5028	0.578962
$718$	0	0
$719$	41.7260	1.55612	0.778059	−	0.628191i	$-0.216204\pi$
$719$	41.7260	1.55612	0.778059	+	0.628191i	$0.216204\pi$
$720$	0	0
$721$	−13.0623	−0.486467
$722$	0	0
$723$	22.1041	0.822061
$724$	0	0
$725$	10.8875	0.404352
$726$	0	0
$727$	−13.6263	−0.505370	−0.252685	−	0.967549i	$-0.581314\pi$
$727$	−13.6263	−0.505370	−0.252685	+	0.967549i	$0.581314\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.4954	−1.46079
$732$	0	0
$733$	44.9784	1.66132	0.830658	−	0.556783i	$-0.187964\pi$
$733$	44.9784	1.66132	0.830658	+	0.556783i	$0.187964\pi$
$734$	0	0
$735$	7.77697	0.286858
$736$	0	0
$737$	−2.41527	−0.0889675
$738$	0	0
$739$	22.0702	0.811864	0.405932	−	0.913903i	$-0.366947\pi$
$739$	22.0702	0.811864	0.405932	+	0.913903i	$0.366947\pi$
$740$	0	0
$741$	6.85032	0.251653
$742$	0	0
$743$	−11.7183	−0.429901	−0.214950	−	0.976625i	$-0.568959\pi$
$743$	−11.7183	−0.429901	−0.214950	+	0.976625i	$0.568959\pi$
$744$	0	0
$745$	72.9527	2.67278
$746$	0	0
$747$	−2.77377	−0.101487
$748$	0	0
$749$	−9.51920	−0.347824
$750$	0	0
$751$	−19.5695	−0.714101	−0.357051	−	0.934085i	$-0.616218\pi$
$751$	−19.5695	−0.714101	−0.357051	+	0.934085i	$0.616218\pi$
$752$	0	0
$753$	17.6615	0.643621
$754$	0	0
$755$	−58.4103	−2.12577
$756$	0	0
$757$	38.5338	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$757$	38.5338	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$758$	0	0
$759$	2.97162	0.107863
$760$	0	0
$761$	20.8013	0.754048	0.377024	−	0.926203i	$-0.376947\pi$
$761$	20.8013	0.754048	0.377024	+	0.926203i	$0.376947\pi$
$762$	0	0
$763$	7.40084	0.267929
$764$	0	0
$765$	12.1411	0.438962
$766$	0	0
$767$	−7.16947	−0.258874
$768$	0	0
$769$	−40.9958	−1.47834	−0.739172	−	0.673516i	$-0.764783\pi$
$769$	−40.9958	−1.47834	−0.739172	+	0.673516i	$0.764783\pi$
$770$	0	0
$771$	1.09510	0.0394389
$772$	0	0
$773$	17.8571	0.642275	0.321137	−	0.947033i	$-0.395935\pi$
$773$	17.8571	0.642275	0.321137	+	0.947033i	$0.395935\pi$
$774$	0	0
$775$	48.9640	1.75884
$776$	0	0
$777$	2.69506	0.0966846
$778$	0	0
$779$	56.1579	2.01206
$780$	0	0
$781$	9.82194	0.351457
$782$	0	0
$783$	−1.38689	−0.0495633
$784$	0	0
$785$	−47.2800	−1.68749
$786$	0	0
$787$	−1.26754	−0.0451830	−0.0225915	−	0.999745i	$-0.507192\pi$
$787$	−1.26754	−0.0451830	−0.0225915	+	0.999745i	$0.507192\pi$
$788$	0	0
$789$	−22.0220	−0.784002
$790$	0	0
$791$	−10.4920	−0.373053
$792$	0	0
$793$	−14.5957	−0.518309
$794$	0	0
$795$	−0.434838	−0.0154221
$796$	0	0
$797$	−9.59013	−0.339700	−0.169850	−	0.985470i	$-0.554328\pi$
$797$	−9.59013	−0.339700	−0.169850	+	0.985470i	$0.554328\pi$
$798$	0	0
$799$	−23.8713	−0.844507
$800$	0	0
$801$	−7.46343	−0.263707
$802$	0	0
$803$	−8.01979	−0.283012
$804$	0	0
$805$	23.4125	0.825182
$806$	0	0
$807$	21.8219	0.768169
$808$	0	0
$809$	−4.14305	−0.145662	−0.0728309	−	0.997344i	$-0.523203\pi$
$809$	−4.14305	−0.145662	−0.0728309	+	0.997344i	$0.523203\pi$
$810$	0	0
$811$	18.3763	0.645280	0.322640	−	0.946522i	$-0.395430\pi$
$811$	18.3763	0.645280	0.322640	+	0.946522i	$0.395430\pi$
$812$	0	0
$813$	−16.2155	−0.568701
$814$	0	0
$815$	53.8104	1.88490
$816$	0	0
$817$	−79.8835	−2.79477
$818$	0	0
$819$	2.19785	0.0767990
$820$	0	0
$821$	−42.2550	−1.47471	−0.737355	−	0.675505i	$-0.763926\pi$
$821$	−42.2550	−1.47471	−0.737355	+	0.675505i	$0.763926\pi$
$822$	0	0
$823$	44.8701	1.56407	0.782037	−	0.623232i	$-0.214181\pi$
$823$	44.8701	1.56407	0.782037	+	0.623232i	$0.214181\pi$
$824$	0	0
$825$	−7.85032	−0.273313
$826$	0	0
$827$	−16.6546	−0.579139	−0.289569	−	0.957157i	$-0.593512\pi$
$827$	−16.6546	−0.579139	−0.289569	+	0.957157i	$0.593512\pi$
$828$	0	0
$829$	−10.9488	−0.380268	−0.190134	−	0.981758i	$-0.560892\pi$
$829$	−10.9488	−0.380268	−0.190134	+	0.981758i	$0.560892\pi$
$830$	0	0
$831$	−6.39569	−0.221864
$832$	0	0
$833$	−7.34774	−0.254584
$834$	0	0
$835$	67.1587	2.32412
$836$	0	0
$837$	−6.23721	−0.215589
$838$	0	0
$839$	35.0056	1.20853	0.604263	−	0.796785i	$-0.293467\pi$
$839$	35.0056	1.20853	0.604263	+	0.796785i	$0.293467\pi$
$840$	0	0
$841$	−27.0765	−0.933674
$842$	0	0
$843$	−1.66668	−0.0574035
$844$	0	0
$845$	3.58473	0.123319
$846$	0	0
$847$	−2.19785	−0.0755189
$848$	0	0
$849$	24.8953	0.854404
$850$	0	0
$851$	−3.64388	−0.124911
$852$	0	0
$853$	19.9432	0.682844	0.341422	−	0.939910i	$-0.389092\pi$
$853$	19.9432	0.682844	0.341422	+	0.939910i	$0.389092\pi$
$854$	0	0
$855$	24.5566	0.839817
$856$	0	0
$857$	34.9838	1.19503	0.597513	−	0.801860i	$-0.296156\pi$
$857$	34.9838	1.19503	0.597513	+	0.801860i	$0.296156\pi$
$858$	0	0
$859$	−45.7906	−1.56235	−0.781177	−	0.624309i	$-0.785381\pi$
$859$	−45.7906	−1.56235	−0.781177	+	0.624309i	$0.785381\pi$
$860$	0	0
$861$	18.0176	0.614039
$862$	0	0
$863$	−17.4688	−0.594646	−0.297323	−	0.954777i	$-0.596094\pi$
$863$	−17.4688	−0.594646	−0.297323	+	0.954777i	$0.596094\pi$
$864$	0	0
$865$	−67.2998	−2.28826
$866$	0	0
$867$	5.52900	0.187775
$868$	0	0
$869$	−16.3783	−0.555596
$870$	0	0
$871$	2.41527	0.0818382
$872$	0	0
$873$	−10.2744	−0.347735
$874$	0	0
$875$	−22.4568	−0.759178
$876$	0	0
$877$	16.2842	0.549879	0.274940	−	0.961462i	$-0.411342\pi$
$877$	16.2842	0.549879	0.274940	+	0.961462i	$0.411342\pi$
$878$	0	0
$879$	27.2538	0.919248
$880$	0	0
$881$	31.8254	1.07222	0.536112	−	0.844147i	$-0.319892\pi$
$881$	31.8254	1.07222	0.536112	+	0.844147i	$0.319892\pi$
$882$	0	0
$883$	−29.1519	−0.981038	−0.490519	−	0.871430i	$-0.663193\pi$
$883$	−29.1519	−0.981038	−0.490519	+	0.871430i	$0.663193\pi$
$884$	0	0
$885$	−25.7006	−0.863917
$886$	0	0
$887$	23.8396	0.800454	0.400227	−	0.916416i	$-0.368931\pi$
$887$	23.8396	0.800454	0.400227	+	0.916416i	$0.368931\pi$
$888$	0	0
$889$	−45.3914	−1.52238
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−48.2822	−1.61570
$894$	0	0
$895$	59.9970	2.00548
$896$	0	0
$897$	−2.97162	−0.0992195
$898$	0	0
$899$	8.65030	0.288504
$900$	0	0
$901$	0.410838	0.0136870
$902$	0	0
$903$	−25.6297	−0.852903
$904$	0	0
$905$	30.1041	1.00069
$906$	0	0
$907$	57.7011	1.91593	0.957966	−	0.286881i	$-0.0926184\pi$
$907$	57.7011	1.91593	0.957966	+	0.286881i	$0.0926184\pi$
$908$	0	0
$909$	18.0570	0.598912
$910$	0	0
$911$	−18.6461	−0.617772	−0.308886	−	0.951099i	$-0.599956\pi$
$911$	−18.6461	−0.617772	−0.308886	+	0.951099i	$0.599956\pi$
$912$	0	0
$913$	−2.77377	−0.0917985
$914$	0	0
$915$	−52.3218	−1.72970
$916$	0	0
$917$	1.96724	0.0649640
$918$	0	0
$919$	−54.5851	−1.80060	−0.900299	−	0.435273i	$-0.856652\pi$
$919$	−54.5851	−1.80060	−0.900299	+	0.435273i	$0.856652\pi$
$920$	0	0
$921$	21.9806	0.724287
$922$	0	0
$923$	−9.82194	−0.323293
$924$	0	0
$925$	9.62626	0.316510
$926$	0	0
$927$	5.94324	0.195202
$928$	0	0
$929$	−50.9542	−1.67175	−0.835877	−	0.548917i	$-0.815040\pi$
$929$	−50.9542	−1.67175	−0.835877	+	0.548917i	$0.815040\pi$
$930$	0	0
$931$	−14.8615	−0.487068
$932$	0	0
$933$	14.2198	0.465536
$934$	0	0
$935$	12.1411	0.397056
$936$	0	0
$937$	−3.02621	−0.0988619	−0.0494309	−	0.998778i	$-0.515741\pi$
$937$	−3.02621	−0.0988619	−0.0494309	+	0.998778i	$0.515741\pi$
$938$	0	0
$939$	−25.2680	−0.824589
$940$	0	0
$941$	10.7638	0.350890	0.175445	−	0.984489i	$-0.443864\pi$
$941$	10.7638	0.350890	0.175445	+	0.984489i	$0.443864\pi$
$942$	0	0
$943$	−24.3609	−0.793300
$944$	0	0
$945$	7.87870	0.256294
$946$	0	0
$947$	−52.8779	−1.71830	−0.859150	−	0.511723i	$-0.829007\pi$
$947$	−52.8779	−1.71830	−0.859150	+	0.511723i	$0.829007\pi$
$948$	0	0
$949$	8.01979	0.260333
$950$	0	0
$951$	21.8024	0.706990
$952$	0	0
$953$	17.9258	0.580675	0.290338	−	0.956924i	$-0.406232\pi$
$953$	17.9258	0.580675	0.290338	+	0.956924i	$0.406232\pi$
$954$	0	0
$955$	94.1867	3.04781
$956$	0	0
$957$	−1.38689	−0.0448317
$958$	0	0
$959$	30.5850	0.987641
$960$	0	0
$961$	7.90273	0.254927
$962$	0	0
$963$	4.33115	0.139569
$964$	0	0
$965$	−7.44386	−0.239626
$966$	0	0
$967$	7.79577	0.250695	0.125347	−	0.992113i	$-0.459995\pi$
$967$	7.79577	0.250695	0.125347	+	0.992113i	$0.459995\pi$
$968$	0	0
$969$	−23.2013	−0.745332
$970$	0	0
$971$	−45.9630	−1.47502	−0.737512	−	0.675334i	$-0.763999\pi$
$971$	−45.9630	−1.47502	−0.737512	+	0.675334i	$0.763999\pi$
$972$	0	0
$973$	−24.0709	−0.771679
$974$	0	0
$975$	7.85032	0.251411
$976$	0	0
$977$	−6.98902	−0.223599	−0.111799	−	0.993731i	$-0.535661\pi$
$977$	−6.98902	−0.223599	−0.111799	+	0.993731i	$0.535661\pi$
$978$	0	0
$979$	−7.46343	−0.238532
$980$	0	0
$981$	−3.36732	−0.107510
$982$	0	0
$983$	7.88214	0.251401	0.125701	−	0.992068i	$-0.459882\pi$
$983$	7.88214	0.251401	0.125701	+	0.992068i	$0.459882\pi$
$984$	0	0
$985$	47.7148	1.52032
$986$	0	0
$987$	−15.4908	−0.493077
$988$	0	0
$989$	34.6529	1.10190
$990$	0	0
$991$	43.0912	1.36884	0.684418	−	0.729090i	$-0.260056\pi$
$991$	43.0912	1.36884	0.684418	+	0.729090i	$0.260056\pi$
$992$	0	0
$993$	1.14211	0.0362437
$994$	0	0
$995$	−26.8525	−0.851281
$996$	0	0
$997$	29.6047	0.937591	0.468796	−	0.883307i	$-0.344688\pi$
$997$	29.6047	0.937591	0.468796	+	0.883307i	$0.344688\pi$
$998$	0	0
$999$	−1.22623	−0.0387961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bx.1.4		4
4.3	odd	2		3432.2.a.v.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.v.1.4	✓	4	4.3	odd	2
6864.2.a.bx.1.4		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.58473 q^{5} -2.19785 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.58473 q^{5} -2.19785 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.58473 q^{15} +3.38689 q^{17} +6.85032 q^{19} +2.19785 q^{21} -2.97162 q^{23} +7.85032 q^{25} -1.00000 q^{27} +1.38689 q^{29} +6.23721 q^{31} -1.00000 q^{33} -7.87870 q^{35} +1.22623 q^{37} +1.00000 q^{39} +8.19785 q^{41} -11.6613 q^{43} +3.58473 q^{45} -7.04817 q^{47} -2.16947 q^{49} -3.38689 q^{51} +0.121303 q^{53} +3.58473 q^{55} -6.85032 q^{57} +7.16947 q^{59} +14.5957 q^{61} -2.19785 q^{63} -3.58473 q^{65} -2.41527 q^{67} +2.97162 q^{69} +9.82194 q^{71} -8.01979 q^{73} -7.85032 q^{75} -2.19785 q^{77} -16.3783 q^{79} +1.00000 q^{81} -2.77377 q^{83} +12.1411 q^{85} -1.38689 q^{87} -7.46343 q^{89} +2.19785 q^{91} -6.23721 q^{93} +24.5566 q^{95} -10.2744 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 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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