LMFDB - Embedded newform 6864.2.a.bo.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.31955$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.31955 q^{5} +2.25879 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.31955 q^{5} +2.25879 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.31955 q^{15} -4.93923 q^{17} -2.00000 q^{19} -2.25879 q^{21} +3.61968 q^{23} -3.25879 q^{25} -1.00000 q^{27} +9.19802 q^{29} -4.93923 q^{31} -1.00000 q^{33} -2.98058 q^{35} +6.00000 q^{37} -1.00000 q^{39} -1.61968 q^{41} -8.55892 q^{43} -1.31955 q^{45} -5.87847 q^{47} -1.89789 q^{49} +4.93923 q^{51} +2.00000 q^{53} -1.31955 q^{55} +2.00000 q^{57} -11.4982 q^{59} +0.380316 q^{61} +2.25879 q^{63} -1.31955 q^{65} -8.59775 q^{67} -3.61968 q^{69} +7.23937 q^{71} +15.4155 q^{73} +3.25879 q^{75} +2.25879 q^{77} -1.06077 q^{79} +1.00000 q^{81} +8.51757 q^{83} +6.51757 q^{85} -9.19802 q^{87} -0.300133 q^{89} +2.25879 q^{91} +4.93923 q^{93} +2.63910 q^{95} -15.0351 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.31955	−0.590121	−0.295061	−	0.955479i	$-0.595340\pi$
$5$	−1.31955	−0.590121	−0.295061	+	0.955479i	$0.595340\pi$
$6$	0	0
$7$	2.25879	0.853741	0.426870	−	0.904313i	$-0.359616\pi$
$7$	2.25879	0.853741	0.426870	+	0.904313i	$0.359616\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.31955	0.340707
$16$	0	0
$17$	−4.93923	−1.19794	−0.598970	−	0.800771i	$-0.704423\pi$
$17$	−4.93923	−1.19794	−0.598970	+	0.800771i	$0.704423\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−2.25879	−0.492907
$22$	0	0
$23$	3.61968	0.754756	0.377378	−	0.926059i	$-0.376826\pi$
$23$	3.61968	0.754756	0.377378	+	0.926059i	$0.376826\pi$
$24$	0	0
$25$	−3.25879	−0.651757
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.19802	1.70803	0.854015	−	0.520249i	$-0.174161\pi$
$29$	9.19802	1.70803	0.854015	+	0.520249i	$0.174161\pi$
$30$	0	0
$31$	−4.93923	−0.887113	−0.443556	−	0.896246i	$-0.646283\pi$
$31$	−4.93923	−0.887113	−0.443556	+	0.896246i	$0.646283\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.98058	−0.503810
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.61968	−0.252952	−0.126476	−	0.991970i	$-0.540367\pi$
$41$	−1.61968	−0.252952	−0.126476	+	0.991970i	$0.540367\pi$
$42$	0	0
$43$	−8.55892	−1.30522	−0.652611	−	0.757693i	$-0.726327\pi$
$43$	−8.55892	−1.30522	−0.652611	+	0.757693i	$0.726327\pi$
$44$	0	0
$45$	−1.31955	−0.196707
$46$	0	0
$47$	−5.87847	−0.857463	−0.428731	−	0.903432i	$-0.641039\pi$
$47$	−5.87847	−0.857463	−0.428731	+	0.903432i	$0.641039\pi$
$48$	0	0
$49$	−1.89789	−0.271127
$50$	0	0
$51$	4.93923	0.691631
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−1.31955	−0.177928
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−11.4982	−1.49693	−0.748466	−	0.663173i	$-0.769209\pi$
$59$	−11.4982	−1.49693	−0.748466	+	0.663173i	$0.769209\pi$
$60$	0	0
$61$	0.380316	0.0486945	0.0243472	−	0.999704i	$-0.492249\pi$
$61$	0.380316	0.0486945	0.0243472	+	0.999704i	$0.492249\pi$
$62$	0	0
$63$	2.25879	0.284580
$64$	0	0
$65$	−1.31955	−0.163670
$66$	0	0
$67$	−8.59775	−1.05038	−0.525191	−	0.850984i	$-0.676006\pi$
$67$	−8.59775	−1.05038	−0.525191	+	0.850984i	$0.676006\pi$
$68$	0	0
$69$	−3.61968	−0.435759
$70$	0	0
$71$	7.23937	0.859155	0.429577	−	0.903030i	$-0.358663\pi$
$71$	7.23937	0.859155	0.429577	+	0.903030i	$0.358663\pi$
$72$	0	0
$73$	15.4155	1.80424	0.902121	−	0.431482i	$-0.142009\pi$
$73$	15.4155	1.80424	0.902121	+	0.431482i	$0.142009\pi$
$74$	0	0
$75$	3.25879	0.376292
$76$	0	0
$77$	2.25879	0.257413
$78$	0	0
$79$	−1.06077	−0.119345	−0.0596727	−	0.998218i	$-0.519006\pi$
$79$	−1.06077	−0.119345	−0.0596727	+	0.998218i	$0.519006\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.51757	0.934925	0.467462	−	0.884013i	$-0.345168\pi$
$83$	8.51757	0.934925	0.467462	+	0.884013i	$0.345168\pi$
$84$	0	0
$85$	6.51757	0.706930
$86$	0	0
$87$	−9.19802	−0.986131
$88$	0	0
$89$	−0.300133	−0.0318141	−0.0159070	−	0.999873i	$-0.505064\pi$
$89$	−0.300133	−0.0318141	−0.0159070	+	0.999873i	$0.505064\pi$
$90$	0	0
$91$	2.25879	0.236785
$92$	0	0
$93$	4.93923	0.512175
$94$	0	0
$95$	2.63910	0.270766
$96$	0	0
$97$	−15.0351	−1.52659	−0.763294	−	0.646052i	$-0.776419\pi$
$97$	−15.0351	−1.52659	−0.763294	+	0.646052i	$0.776419\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	5.45681	0.542973	0.271486	−	0.962442i	$-0.412485\pi$
$101$	5.45681	0.542973	0.271486	+	0.962442i	$0.412485\pi$
$102$	0	0
$103$	3.49815	0.344683	0.172342	−	0.985037i	$-0.444867\pi$
$103$	3.49815	0.344683	0.172342	+	0.985037i	$0.444867\pi$
$104$	0	0
$105$	2.98058	0.290875
$106$	0	0
$107$	1.01942	0.0985508	0.0492754	−	0.998785i	$-0.484309\pi$
$107$	1.01942	0.0985508	0.0492754	+	0.998785i	$0.484309\pi$
$108$	0	0
$109$	−11.7569	−1.12611	−0.563055	−	0.826419i	$-0.690374\pi$
$109$	−11.7569	−1.12611	−0.563055	+	0.826419i	$0.690374\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−8.13726	−0.765489	−0.382744	−	0.923854i	$-0.625021\pi$
$113$	−8.13726	−0.765489	−0.382744	+	0.923854i	$0.625021\pi$
$114$	0	0
$115$	−4.77636	−0.445398
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−11.1567	−1.02273
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.61968	0.146042
$124$	0	0
$125$	10.8979	0.974737
$126$	0	0
$127$	−21.8528	−1.93913	−0.969563	−	0.244841i	$-0.921264\pi$
$127$	−21.8528	−1.93913	−0.969563	+	0.244841i	$0.921264\pi$
$128$	0	0
$129$	8.55892	0.753571
$130$	0	0
$131$	−5.61968	−0.490994	−0.245497	−	0.969397i	$-0.578951\pi$
$131$	−5.61968	−0.490994	−0.245497	+	0.969397i	$0.578951\pi$
$132$	0	0
$133$	−4.51757	−0.391723
$134$	0	0
$135$	1.31955	0.113569
$136$	0	0
$137$	15.9744	1.36478	0.682392	−	0.730987i	$-0.260940\pi$
$137$	15.9744	1.36478	0.682392	+	0.730987i	$0.260940\pi$
$138$	0	0
$139$	−9.57834	−0.812424	−0.406212	−	0.913779i	$-0.633151\pi$
$139$	−9.57834	−0.812424	−0.406212	+	0.913779i	$0.633151\pi$
$140$	0	0
$141$	5.87847	0.495056
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−12.1373	−1.00794
$146$	0	0
$147$	1.89789	0.156535
$148$	0	0
$149$	7.15667	0.586297	0.293149	−	0.956067i	$-0.405297\pi$
$149$	7.15667	0.586297	0.293149	+	0.956067i	$0.405297\pi$
$150$	0	0
$151$	−2.76063	−0.224657	−0.112329	−	0.993671i	$-0.535831\pi$
$151$	−2.76063	−0.224657	−0.112329	+	0.993671i	$0.535831\pi$
$152$	0	0
$153$	−4.93923	−0.399313
$154$	0	0
$155$	6.51757	0.523504
$156$	0	0
$157$	11.9173	0.951104	0.475552	−	0.879687i	$-0.342248\pi$
$157$	11.9173	0.951104	0.475552	+	0.879687i	$0.342248\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	8.17609	0.644366
$162$	0	0
$163$	−6.55892	−0.513734	−0.256867	−	0.966447i	$-0.582690\pi$
$163$	−6.55892	−0.513734	−0.256867	+	0.966447i	$0.582690\pi$
$164$	0	0
$165$	1.31955	0.102727
$166$	0	0
$167$	−5.10211	−0.394813	−0.197407	−	0.980322i	$-0.563252\pi$
$167$	−5.10211	−0.394813	−0.197407	+	0.980322i	$0.563252\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−5.19802	−0.395198	−0.197599	−	0.980283i	$-0.563314\pi$
$173$	−5.19802	−0.395198	−0.197599	+	0.980283i	$0.563314\pi$
$174$	0	0
$175$	−7.36090	−0.556432
$176$	0	0
$177$	11.4982	0.864254
$178$	0	0
$179$	22.5176	1.68304	0.841521	−	0.540224i	$-0.181661\pi$
$179$	22.5176	1.68304	0.841521	+	0.540224i	$0.181661\pi$
$180$	0	0
$181$	−1.60396	−0.119221	−0.0596107	−	0.998222i	$-0.518986\pi$
$181$	−1.60396	−0.119221	−0.0596107	+	0.998222i	$0.518986\pi$
$182$	0	0
$183$	−0.380316	−0.0281138
$184$	0	0
$185$	−7.91730	−0.582092
$186$	0	0
$187$	−4.93923	−0.361193
$188$	0	0
$189$	−2.25879	−0.164302
$190$	0	0
$191$	−6.38032	−0.461663	−0.230832	−	0.972994i	$-0.574145\pi$
$191$	−6.38032	−0.461663	−0.230832	+	0.972994i	$0.574145\pi$
$192$	0	0
$193$	15.2394	1.09695	0.548477	−	0.836166i	$-0.315208\pi$
$193$	15.2394	1.09695	0.548477	+	0.836166i	$0.315208\pi$
$194$	0	0
$195$	1.31955	0.0944950
$196$	0	0
$197$	18.9136	1.34754	0.673770	−	0.738942i	$-0.264674\pi$
$197$	18.9136	1.34754	0.673770	+	0.738942i	$0.264674\pi$
$198$	0	0
$199$	−15.7412	−1.11587	−0.557933	−	0.829886i	$-0.688405\pi$
$199$	−15.7412	−1.11587	−0.557933	+	0.829886i	$0.688405\pi$
$200$	0	0
$201$	8.59775	0.606439
$202$	0	0
$203$	20.7764	1.45821
$204$	0	0
$205$	2.13726	0.149272
$206$	0	0
$207$	3.61968	0.251585
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	27.2137	1.87347	0.936736	−	0.350037i	$-0.113831\pi$
$211$	27.2137	1.87347	0.936736	+	0.350037i	$0.113831\pi$
$212$	0	0
$213$	−7.23937	−0.496033
$214$	0	0
$215$	11.2939	0.770240
$216$	0	0
$217$	−11.1567	−0.757364
$218$	0	0
$219$	−15.4155	−1.04168
$220$	0	0
$221$	−4.93923	−0.332249
$222$	0	0
$223$	−21.3741	−1.43132	−0.715658	−	0.698451i	$-0.753873\pi$
$223$	−21.3741	−1.43132	−0.715658	+	0.698451i	$0.753873\pi$
$224$	0	0
$225$	−3.25879	−0.217252
$226$	0	0
$227$	−9.91730	−0.658235	−0.329117	−	0.944289i	$-0.606751\pi$
$227$	−9.91730	−0.658235	−0.329117	+	0.944289i	$0.606751\pi$
$228$	0	0
$229$	3.53699	0.233731	0.116865	−	0.993148i	$-0.462715\pi$
$229$	3.53699	0.233731	0.116865	+	0.993148i	$0.462715\pi$
$230$	0	0
$231$	−2.25879	−0.148617
$232$	0	0
$233$	−20.6135	−1.35043	−0.675217	−	0.737619i	$-0.735950\pi$
$233$	−20.6135	−1.35043	−0.675217	+	0.737619i	$0.735950\pi$
$234$	0	0
$235$	7.75694	0.506007
$236$	0	0
$237$	1.06077	0.0689041
$238$	0	0
$239$	5.49815	0.355646	0.177823	−	0.984062i	$-0.443095\pi$
$239$	5.49815	0.355646	0.177823	+	0.984062i	$0.443095\pi$
$240$	0	0
$241$	−28.3133	−1.82382	−0.911911	−	0.410387i	$-0.865394\pi$
$241$	−28.3133	−1.82382	−0.911911	+	0.410387i	$0.865394\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.50436	0.159998
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−8.51757	−0.539779
$250$	0	0
$251$	29.5527	1.86535	0.932675	−	0.360717i	$-0.117468\pi$
$251$	29.5527	1.86535	0.932675	+	0.360717i	$0.117468\pi$
$252$	0	0
$253$	3.61968	0.227568
$254$	0	0
$255$	−6.51757	−0.408146
$256$	0	0
$257$	−4.46301	−0.278395	−0.139197	−	0.990265i	$-0.544452\pi$
$257$	−4.46301	−0.278395	−0.139197	+	0.990265i	$0.544452\pi$
$258$	0	0
$259$	13.5527	0.842125
$260$	0	0
$261$	9.19802	0.569343
$262$	0	0
$263$	−22.9136	−1.41291	−0.706457	−	0.707756i	$-0.749708\pi$
$263$	−22.9136	−1.41291	−0.706457	+	0.707756i	$0.749708\pi$
$264$	0	0
$265$	−2.63910	−0.162119
$266$	0	0
$267$	0.300133	0.0183679
$268$	0	0
$269$	−26.9524	−1.64332	−0.821660	−	0.569978i	$-0.806952\pi$
$269$	−26.9524	−1.64332	−0.821660	+	0.569978i	$0.806952\pi$
$270$	0	0
$271$	−22.5176	−1.36785	−0.683923	−	0.729555i	$-0.739727\pi$
$271$	−22.5176	−1.36785	−0.683923	+	0.729555i	$0.739727\pi$
$272$	0	0
$273$	−2.25879	−0.136708
$274$	0	0
$275$	−3.25879	−0.196512
$276$	0	0
$277$	4.13726	0.248584	0.124292	−	0.992246i	$-0.460334\pi$
$277$	4.13726	0.248584	0.124292	+	0.992246i	$0.460334\pi$
$278$	0	0
$279$	−4.93923	−0.295704
$280$	0	0
$281$	−6.13726	−0.366118	−0.183059	−	0.983102i	$-0.558600\pi$
$281$	−6.13726	−0.366118	−0.183059	+	0.983102i	$0.558600\pi$
$282$	0	0
$283$	−23.0377	−1.36945	−0.684723	−	0.728803i	$-0.740077\pi$
$283$	−23.0377	−1.36945	−0.684723	+	0.728803i	$0.740077\pi$
$284$	0	0
$285$	−2.63910	−0.156327
$286$	0	0
$287$	−3.65852	−0.215956
$288$	0	0
$289$	7.39604	0.435061
$290$	0	0
$291$	15.0351	0.881376
$292$	0	0
$293$	15.6742	0.915699	0.457850	−	0.889030i	$-0.348620\pi$
$293$	15.6742	0.915699	0.457850	+	0.889030i	$0.348620\pi$
$294$	0	0
$295$	15.1724	0.883371
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	3.61968	0.209332
$300$	0	0
$301$	−19.3328	−1.11432
$302$	0	0
$303$	−5.45681	−0.313485
$304$	0	0
$305$	−0.501846	−0.0287356
$306$	0	0
$307$	−3.27820	−0.187097	−0.0935485	−	0.995615i	$-0.529821\pi$
$307$	−3.27820	−0.187097	−0.0935485	+	0.995615i	$0.529821\pi$
$308$	0	0
$309$	−3.49815	−0.199003
$310$	0	0
$311$	−21.2394	−1.20437	−0.602187	−	0.798355i	$-0.705704\pi$
$311$	−21.2394	−1.20437	−0.602187	+	0.798355i	$0.705704\pi$
$312$	0	0
$313$	8.77636	0.496069	0.248035	−	0.968751i	$-0.420215\pi$
$313$	8.77636	0.496069	0.248035	+	0.968751i	$0.420215\pi$
$314$	0	0
$315$	−2.98058	−0.167937
$316$	0	0
$317$	−24.9938	−1.40379	−0.701896	−	0.712280i	$-0.747663\pi$
$317$	−24.9938	−1.40379	−0.701896	+	0.712280i	$0.747663\pi$
$318$	0	0
$319$	9.19802	0.514990
$320$	0	0
$321$	−1.01942	−0.0568983
$322$	0	0
$323$	9.87847	0.549653
$324$	0	0
$325$	−3.25879	−0.180765
$326$	0	0
$327$	11.7569	0.650160
$328$	0	0
$329$	−13.2782	−0.732051
$330$	0	0
$331$	−32.1116	−1.76501	−0.882507	−	0.470298i	$-0.844146\pi$
$331$	−32.1116	−1.76501	−0.882507	+	0.470298i	$0.844146\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	11.3452	0.619853
$336$	0	0
$337$	2.60027	0.141646	0.0708228	−	0.997489i	$-0.477438\pi$
$337$	2.60027	0.141646	0.0708228	+	0.997489i	$0.477438\pi$
$338$	0	0
$339$	8.13726	0.441955
$340$	0	0
$341$	−4.93923	−0.267475
$342$	0	0
$343$	−20.0984	−1.08521
$344$	0	0
$345$	4.77636	0.257150
$346$	0	0
$347$	−17.6354	−0.946718	−0.473359	−	0.880870i	$-0.656959\pi$
$347$	−17.6354	−0.946718	−0.473359	+	0.880870i	$0.656959\pi$
$348$	0	0
$349$	−21.0740	−1.12806	−0.564032	−	0.825753i	$-0.690751\pi$
$349$	−21.0740	−1.12806	−0.564032	+	0.825753i	$0.690751\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−28.5746	−1.52087	−0.760437	−	0.649411i	$-0.775015\pi$
$353$	−28.5746	−1.52087	−0.760437	+	0.649411i	$0.775015\pi$
$354$	0	0
$355$	−9.55271	−0.507005
$356$	0	0
$357$	11.1567	0.590474
$358$	0	0
$359$	−1.74121	−0.0918978	−0.0459489	−	0.998944i	$-0.514631\pi$
$359$	−1.74121	−0.0918978	−0.0459489	+	0.998944i	$0.514631\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−20.3415	−1.06472
$366$	0	0
$367$	8.79208	0.458943	0.229471	−	0.973315i	$-0.426300\pi$
$367$	8.79208	0.458943	0.229471	+	0.973315i	$0.426300\pi$
$368$	0	0
$369$	−1.61968	−0.0843174
$370$	0	0
$371$	4.51757	0.234541
$372$	0	0
$373$	−0.297621	−0.0154102	−0.00770511	−	0.999970i	$-0.502453\pi$
$373$	−0.297621	−0.0154102	−0.00770511	+	0.999970i	$0.502453\pi$
$374$	0	0
$375$	−10.8979	−0.562764
$376$	0	0
$377$	9.19802	0.473722
$378$	0	0
$379$	−18.5746	−0.954115	−0.477058	−	0.878872i	$-0.658297\pi$
$379$	−18.5746	−0.954115	−0.477058	+	0.878872i	$0.658297\pi$
$380$	0	0
$381$	21.8528	1.11956
$382$	0	0
$383$	−2.63910	−0.134852	−0.0674259	−	0.997724i	$-0.521479\pi$
$383$	−2.63910	−0.134852	−0.0674259	+	0.997724i	$0.521479\pi$
$384$	0	0
$385$	−2.98058	−0.151905
$386$	0	0
$387$	−8.55892	−0.435074
$388$	0	0
$389$	−34.3572	−1.74198	−0.870990	−	0.491301i	$-0.836521\pi$
$389$	−34.3572	−1.74198	−0.870990	+	0.491301i	$0.836521\pi$
$390$	0	0
$391$	−17.8785	−0.904153
$392$	0	0
$393$	5.61968	0.283476
$394$	0	0
$395$	1.39973	0.0704282
$396$	0	0
$397$	22.4506	1.12676	0.563382	−	0.826197i	$-0.309500\pi$
$397$	22.4506	1.12676	0.563382	+	0.826197i	$0.309500\pi$
$398$	0	0
$399$	4.51757	0.226161
$400$	0	0
$401$	−27.8140	−1.38897	−0.694483	−	0.719509i	$-0.744367\pi$
$401$	−27.8140	−1.38897	−0.694483	+	0.719509i	$0.744367\pi$
$402$	0	0
$403$	−4.93923	−0.246041
$404$	0	0
$405$	−1.31955	−0.0655690
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	9.45429	0.467485	0.233742	−	0.972299i	$-0.424903\pi$
$409$	9.45429	0.467485	0.233742	+	0.972299i	$0.424903\pi$
$410$	0	0
$411$	−15.9744	−0.787958
$412$	0	0
$413$	−25.9719	−1.27799
$414$	0	0
$415$	−11.2394	−0.551719
$416$	0	0
$417$	9.57834	0.469053
$418$	0	0
$419$	20.9963	1.02574	0.512868	−	0.858467i	$-0.328583\pi$
$419$	20.9963	1.02574	0.512868	+	0.858467i	$0.328583\pi$
$420$	0	0
$421$	−4.89789	−0.238708	−0.119354	−	0.992852i	$-0.538082\pi$
$421$	−4.89789	−0.238708	−0.119354	+	0.992852i	$0.538082\pi$
$422$	0	0
$423$	−5.87847	−0.285821
$424$	0	0
$425$	16.0959	0.780766
$426$	0	0
$427$	0.859052	0.0415724
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−6.72180	−0.323778	−0.161889	−	0.986809i	$-0.551759\pi$
$431$	−6.72180	−0.323778	−0.161889	+	0.986809i	$0.551759\pi$
$432$	0	0
$433$	−17.6585	−0.848614	−0.424307	−	0.905518i	$-0.639482\pi$
$433$	−17.6585	−0.848614	−0.424307	+	0.905518i	$0.639482\pi$
$434$	0	0
$435$	12.1373	0.581937
$436$	0	0
$437$	−7.23937	−0.346306
$438$	0	0
$439$	13.3353	0.636458	0.318229	−	0.948014i	$-0.396912\pi$
$439$	13.3353	0.636458	0.318229	+	0.948014i	$0.396912\pi$
$440$	0	0
$441$	−1.89789	−0.0903756
$442$	0	0
$443$	22.5564	1.07169	0.535844	−	0.844317i	$-0.319994\pi$
$443$	22.5564	1.07169	0.535844	+	0.844317i	$0.319994\pi$
$444$	0	0
$445$	0.396041	0.0187741
$446$	0	0
$447$	−7.15667	−0.338499
$448$	0	0
$449$	−23.4568	−1.10700	−0.553498	−	0.832851i	$-0.686707\pi$
$449$	−23.4568	−1.10700	−0.553498	+	0.832851i	$0.686707\pi$
$450$	0	0
$451$	−1.61968	−0.0762679
$452$	0	0
$453$	2.76063	0.129706
$454$	0	0
$455$	−2.98058	−0.139732
$456$	0	0
$457$	27.8942	1.30484	0.652418	−	0.757860i	$-0.273755\pi$
$457$	27.8942	1.30484	0.652418	+	0.757860i	$0.273755\pi$
$458$	0	0
$459$	4.93923	0.230544
$460$	0	0
$461$	1.44359	0.0672349	0.0336174	−	0.999435i	$-0.489297\pi$
$461$	1.44359	0.0672349	0.0336174	+	0.999435i	$0.489297\pi$
$462$	0	0
$463$	−1.69987	−0.0789995	−0.0394998	−	0.999220i	$-0.512576\pi$
$463$	−1.69987	−0.0789995	−0.0394998	+	0.999220i	$0.512576\pi$
$464$	0	0
$465$	−6.51757	−0.302245
$466$	0	0
$467$	−34.0315	−1.57479	−0.787394	−	0.616450i	$-0.788570\pi$
$467$	−34.0315	−1.57479	−0.787394	+	0.616450i	$0.788570\pi$
$468$	0	0
$469$	−19.4205	−0.896755
$470$	0	0
$471$	−11.9173	−0.549120
$472$	0	0
$473$	−8.55892	−0.393540
$474$	0	0
$475$	6.51757	0.299047
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−8.21995	−0.375579	−0.187790	−	0.982209i	$-0.560132\pi$
$479$	−8.21995	−0.375579	−0.187790	+	0.982209i	$0.560132\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−8.17609	−0.372025
$484$	0	0
$485$	19.8396	0.900871
$486$	0	0
$487$	−15.5783	−0.705922	−0.352961	−	0.935638i	$-0.614825\pi$
$487$	−15.5783	−0.705922	−0.352961	+	0.935638i	$0.614825\pi$
$488$	0	0
$489$	6.55892	0.296605
$490$	0	0
$491$	−2.05456	−0.0927210	−0.0463605	−	0.998925i	$-0.514762\pi$
$491$	−2.05456	−0.0927210	−0.0463605	+	0.998925i	$0.514762\pi$
$492$	0	0
$493$	−45.4312	−2.04612
$494$	0	0
$495$	−1.31955	−0.0593094
$496$	0	0
$497$	16.3522	0.733496
$498$	0	0
$499$	15.7544	0.705265	0.352633	−	0.935762i	$-0.385287\pi$
$499$	15.7544	0.705265	0.352633	+	0.935762i	$0.385287\pi$
$500$	0	0
$501$	5.10211	0.227946
$502$	0	0
$503$	−0.434876	−0.0193902	−0.00969509	−	0.999953i	$-0.503086\pi$
$503$	−0.434876	−0.0193902	−0.00969509	+	0.999953i	$0.503086\pi$
$504$	0	0
$505$	−7.20053	−0.320420
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	35.1311	1.55716	0.778578	−	0.627548i	$-0.215941\pi$
$509$	35.1311	1.55716	0.778578	+	0.627548i	$0.215941\pi$
$510$	0	0
$511$	34.8202	1.54036
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−4.61599	−0.203405
$516$	0	0
$517$	−5.87847	−0.258535
$518$	0	0
$519$	5.19802	0.228168
$520$	0	0
$521$	−14.5845	−0.638960	−0.319480	−	0.947593i	$-0.603508\pi$
$521$	−14.5845	−0.638960	−0.319480	+	0.947593i	$0.603508\pi$
$522$	0	0
$523$	−3.02193	−0.132140	−0.0660699	−	0.997815i	$-0.521046\pi$
$523$	−3.02193	−0.132140	−0.0660699	+	0.997815i	$0.521046\pi$
$524$	0	0
$525$	7.36090	0.321256
$526$	0	0
$527$	24.3960	1.06271
$528$	0	0
$529$	−9.89789	−0.430343
$530$	0	0
$531$	−11.4982	−0.498977
$532$	0	0
$533$	−1.61968	−0.0701563
$534$	0	0
$535$	−1.34517	−0.0581569
$536$	0	0
$537$	−22.5176	−0.971705
$538$	0	0
$539$	−1.89789	−0.0817478
$540$	0	0
$541$	17.7569	0.763430	0.381715	−	0.924280i	$-0.375334\pi$
$541$	17.7569	0.763430	0.381715	+	0.924280i	$0.375334\pi$
$542$	0	0
$543$	1.60396	0.0688325
$544$	0	0
$545$	15.5139	0.664542
$546$	0	0
$547$	29.2683	1.25142	0.625711	−	0.780055i	$-0.284809\pi$
$547$	29.2683	1.25142	0.625711	+	0.780055i	$0.284809\pi$
$548$	0	0
$549$	0.380316	0.0162315
$550$	0	0
$551$	−18.3960	−0.783698
$552$	0	0
$553$	−2.39604	−0.101890
$554$	0	0
$555$	7.91730	0.336071
$556$	0	0
$557$	9.36090	0.396634	0.198317	−	0.980138i	$-0.436452\pi$
$557$	9.36090	0.396634	0.198317	+	0.980138i	$0.436452\pi$
$558$	0	0
$559$	−8.55892	−0.362004
$560$	0	0
$561$	4.93923	0.208535
$562$	0	0
$563$	20.9575	0.883252	0.441626	−	0.897199i	$-0.354402\pi$
$563$	20.9575	0.883252	0.441626	+	0.897199i	$0.354402\pi$
$564$	0	0
$565$	10.7375	0.451731
$566$	0	0
$567$	2.25879	0.0948601
$568$	0	0
$569$	42.2489	1.77117	0.885583	−	0.464482i	$-0.153759\pi$
$569$	42.2489	1.77117	0.885583	+	0.464482i	$0.153759\pi$
$570$	0	0
$571$	−19.7933	−0.828322	−0.414161	−	0.910204i	$-0.635925\pi$
$571$	−19.7933	−0.828322	−0.414161	+	0.910204i	$0.635925\pi$
$572$	0	0
$573$	6.38032	0.266542
$574$	0	0
$575$	−11.7958	−0.491918
$576$	0	0
$577$	−7.00369	−0.291568	−0.145784	−	0.989316i	$-0.546570\pi$
$577$	−7.00369	−0.291568	−0.145784	+	0.989316i	$0.546570\pi$
$578$	0	0
$579$	−15.2394	−0.633327
$580$	0	0
$581$	19.2394	0.798183
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	−1.31955	−0.0545567
$586$	0	0
$587$	−1.53699	−0.0634383	−0.0317192	−	0.999497i	$-0.510098\pi$
$587$	−1.53699	−0.0634383	−0.0317192	+	0.999497i	$0.510098\pi$
$588$	0	0
$589$	9.87847	0.407035
$590$	0	0
$591$	−18.9136	−0.778002
$592$	0	0
$593$	9.11784	0.374425	0.187212	−	0.982319i	$-0.440055\pi$
$593$	9.11784	0.374425	0.187212	+	0.982319i	$0.440055\pi$
$594$	0	0
$595$	14.7218	0.603535
$596$	0	0
$597$	15.7412	0.644245
$598$	0	0
$599$	4.85905	0.198535	0.0992677	−	0.995061i	$-0.468350\pi$
$599$	4.85905	0.198535	0.0992677	+	0.995061i	$0.468350\pi$
$600$	0	0
$601$	−32.8309	−1.33920	−0.669601	−	0.742721i	$-0.733535\pi$
$601$	−32.8309	−1.33920	−0.669601	+	0.742721i	$0.733535\pi$
$602$	0	0
$603$	−8.59775	−0.350128
$604$	0	0
$605$	−1.31955	−0.0536474
$606$	0	0
$607$	3.01691	0.122452	0.0612262	−	0.998124i	$-0.480499\pi$
$607$	3.01691	0.122452	0.0612262	+	0.998124i	$0.480499\pi$
$608$	0	0
$609$	−20.7764	−0.841900
$610$	0	0
$611$	−5.87847	−0.237817
$612$	0	0
$613$	−10.4787	−0.423232	−0.211616	−	0.977353i	$-0.567873\pi$
$613$	−10.4787	−0.423232	−0.211616	+	0.977353i	$0.567873\pi$
$614$	0	0
$615$	−2.13726	−0.0861825
$616$	0	0
$617$	−9.41797	−0.379153	−0.189577	−	0.981866i	$-0.560712\pi$
$617$	−9.41797	−0.379153	−0.189577	+	0.981866i	$0.560712\pi$
$618$	0	0
$619$	−41.9901	−1.68772	−0.843862	−	0.536560i	$-0.819724\pi$
$619$	−41.9901	−1.68772	−0.843862	+	0.536560i	$0.819724\pi$
$620$	0	0
$621$	−3.61968	−0.145253
$622$	0	0
$623$	−0.677937	−0.0271610
$624$	0	0
$625$	1.91361	0.0765445
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−29.6354	−1.18164
$630$	0	0
$631$	−22.9004	−0.911651	−0.455825	−	0.890069i	$-0.650656\pi$
$631$	−22.9004	−0.911651	−0.455825	+	0.890069i	$0.650656\pi$
$632$	0	0
$633$	−27.2137	−1.08165
$634$	0	0
$635$	28.8359	1.14432
$636$	0	0
$637$	−1.89789	−0.0751970
$638$	0	0
$639$	7.23937	0.286385
$640$	0	0
$641$	−29.6900	−1.17268	−0.586342	−	0.810064i	$-0.699432\pi$
$641$	−29.6900	−1.17268	−0.586342	+	0.810064i	$0.699432\pi$
$642$	0	0
$643$	39.2526	1.54797	0.773985	−	0.633204i	$-0.218260\pi$
$643$	39.2526	1.54797	0.773985	+	0.633204i	$0.218260\pi$
$644$	0	0
$645$	−11.2939	−0.444698
$646$	0	0
$647$	−10.0703	−0.395904	−0.197952	−	0.980212i	$-0.563429\pi$
$647$	−10.0703	−0.395904	−0.197952	+	0.980212i	$0.563429\pi$
$648$	0	0
$649$	−11.4982	−0.451342
$650$	0	0
$651$	11.1567	0.437264
$652$	0	0
$653$	31.4700	1.23152	0.615759	−	0.787935i	$-0.288850\pi$
$653$	31.4700	1.23152	0.615759	+	0.787935i	$0.288850\pi$
$654$	0	0
$655$	7.41546	0.289746
$656$	0	0
$657$	15.4155	0.601414
$658$	0	0
$659$	44.5490	1.73538	0.867692	−	0.497103i	$-0.165603\pi$
$659$	44.5490	1.73538	0.867692	+	0.497103i	$0.165603\pi$
$660$	0	0
$661$	28.9136	1.12461	0.562305	−	0.826930i	$-0.309915\pi$
$661$	28.9136	1.12461	0.562305	+	0.826930i	$0.309915\pi$
$662$	0	0
$663$	4.93923	0.191824
$664$	0	0
$665$	5.96116	0.231164
$666$	0	0
$667$	33.2939	1.28915
$668$	0	0
$669$	21.3741	0.826371
$670$	0	0
$671$	0.380316	0.0146819
$672$	0	0
$673$	0.639102	0.0246356	0.0123178	−	0.999924i	$-0.496079\pi$
$673$	0.639102	0.0246356	0.0123178	+	0.999924i	$0.496079\pi$
$674$	0	0
$675$	3.25879	0.125431
$676$	0	0
$677$	−25.8140	−0.992113	−0.496057	−	0.868290i	$-0.665219\pi$
$677$	−25.8140	−0.992113	−0.496057	+	0.868290i	$0.665219\pi$
$678$	0	0
$679$	−33.9612	−1.30331
$680$	0	0
$681$	9.91730	0.380032
$682$	0	0
$683$	−14.6548	−0.560751	−0.280376	−	0.959890i	$-0.590459\pi$
$683$	−14.6548	−0.560751	−0.280376	+	0.959890i	$0.590459\pi$
$684$	0	0
$685$	−21.0790	−0.805387
$686$	0	0
$687$	−3.53699	−0.134945
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	−35.2526	−1.34107	−0.670536	−	0.741877i	$-0.733936\pi$
$691$	−35.2526	−1.34107	−0.670536	+	0.741877i	$0.733936\pi$
$692$	0	0
$693$	2.25879	0.0858042
$694$	0	0
$695$	12.6391	0.479428
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	20.6135	0.779673
$700$	0	0
$701$	−13.9587	−0.527211	−0.263606	−	0.964631i	$-0.584912\pi$
$701$	−13.9587	−0.527211	−0.263606	+	0.964631i	$0.584912\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−7.75694	−0.292143
$706$	0	0
$707$	12.3258	0.463558
$708$	0	0
$709$	−36.8466	−1.38380	−0.691902	−	0.721991i	$-0.743227\pi$
$709$	−36.8466	−1.38380	−0.691902	+	0.721991i	$0.743227\pi$
$710$	0	0
$711$	−1.06077	−0.0397818
$712$	0	0
$713$	−17.8785	−0.669554
$714$	0	0
$715$	−1.31955	−0.0493484
$716$	0	0
$717$	−5.49815	−0.205332
$718$	0	0
$719$	10.9756	0.409319	0.204660	−	0.978833i	$-0.434391\pi$
$719$	10.9756	0.409319	0.204660	+	0.978833i	$0.434391\pi$
$720$	0	0
$721$	7.90158	0.294270
$722$	0	0
$723$	28.3133	1.05298
$724$	0	0
$725$	−29.9744	−1.11322
$726$	0	0
$727$	−10.1215	−0.375387	−0.187693	−	0.982228i	$-0.560101\pi$
$727$	−10.1215	−0.375387	−0.187693	+	0.982228i	$0.560101\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.2745	1.56358
$732$	0	0
$733$	−25.7958	−0.952789	−0.476394	−	0.879232i	$-0.658057\pi$
$733$	−25.7958	−0.952789	−0.476394	+	0.879232i	$0.658057\pi$
$734$	0	0
$735$	−2.50436	−0.0923747
$736$	0	0
$737$	−8.59775	−0.316702
$738$	0	0
$739$	7.43857	0.273632	0.136816	−	0.990596i	$-0.456313\pi$
$739$	7.43857	0.273632	0.136816	+	0.990596i	$0.456313\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−0.936722	−0.0343650	−0.0171825	−	0.999852i	$-0.505470\pi$
$743$	−0.936722	−0.0343650	−0.0171825	+	0.999852i	$0.505470\pi$
$744$	0	0
$745$	−9.44359	−0.345986
$746$	0	0
$747$	8.51757	0.311642
$748$	0	0
$749$	2.30265	0.0841368
$750$	0	0
$751$	49.2427	1.79689	0.898446	−	0.439085i	$-0.144697\pi$
$751$	49.2427	1.79689	0.898446	+	0.439085i	$0.144697\pi$
$752$	0	0
$753$	−29.5527	−1.07696
$754$	0	0
$755$	3.64279	0.132575
$756$	0	0
$757$	33.9099	1.23248	0.616239	−	0.787560i	$-0.288656\pi$
$757$	33.9099	1.23248	0.616239	+	0.787560i	$0.288656\pi$
$758$	0	0
$759$	−3.61968	−0.131386
$760$	0	0
$761$	34.8466	1.26319	0.631595	−	0.775299i	$-0.282401\pi$
$761$	34.8466	1.26319	0.631595	+	0.775299i	$0.282401\pi$
$762$	0	0
$763$	−26.5564	−0.961406
$764$	0	0
$765$	6.51757	0.235643
$766$	0	0
$767$	−11.4982	−0.415174
$768$	0	0
$769$	−5.41546	−0.195286	−0.0976432	−	0.995221i	$-0.531130\pi$
$769$	−5.41546	−0.195286	−0.0976432	+	0.995221i	$0.531130\pi$
$770$	0	0
$771$	4.46301	0.160731
$772$	0	0
$773$	9.90409	0.356225	0.178113	−	0.984010i	$-0.443001\pi$
$773$	9.90409	0.356225	0.178113	+	0.984010i	$0.443001\pi$
$774$	0	0
$775$	16.0959	0.578182
$776$	0	0
$777$	−13.5527	−0.486201
$778$	0	0
$779$	3.23937	0.116062
$780$	0	0
$781$	7.23937	0.259045
$782$	0	0
$783$	−9.19802	−0.328710
$784$	0	0
$785$	−15.7255	−0.561267
$786$	0	0
$787$	−36.1530	−1.28871	−0.644357	−	0.764725i	$-0.722875\pi$
$787$	−36.1530	−1.28871	−0.644357	+	0.764725i	$0.722875\pi$
$788$	0	0
$789$	22.9136	0.815746
$790$	0	0
$791$	−18.3803	−0.653529
$792$	0	0
$793$	0.380316	0.0135054
$794$	0	0
$795$	2.63910	0.0935993
$796$	0	0
$797$	−12.8822	−0.456310	−0.228155	−	0.973625i	$-0.573269\pi$
$797$	−12.8822	−0.456310	−0.228155	+	0.973625i	$0.573269\pi$
$798$	0	0
$799$	29.0351	1.02719
$800$	0	0
$801$	−0.300133	−0.0106047
$802$	0	0
$803$	15.4155	0.544000
$804$	0	0
$805$	−10.7888	−0.380254
$806$	0	0
$807$	26.9524	0.948771
$808$	0	0
$809$	38.4919	1.35330	0.676652	−	0.736303i	$-0.263430\pi$
$809$	38.4919	1.35330	0.676652	+	0.736303i	$0.263430\pi$
$810$	0	0
$811$	−19.7958	−0.695124	−0.347562	−	0.937657i	$-0.612990\pi$
$811$	−19.7958	−0.695124	−0.347562	+	0.937657i	$0.612990\pi$
$812$	0	0
$813$	22.5176	0.789726
$814$	0	0
$815$	8.65483	0.303165
$816$	0	0
$817$	17.1178	0.598877
$818$	0	0
$819$	2.25879	0.0789284
$820$	0	0
$821$	9.00369	0.314231	0.157116	−	0.987580i	$-0.449781\pi$
$821$	9.00369	0.314231	0.157116	+	0.987580i	$0.449781\pi$
$822$	0	0
$823$	5.94544	0.207245	0.103623	−	0.994617i	$-0.466957\pi$
$823$	5.94544	0.207245	0.103623	+	0.994617i	$0.466957\pi$
$824$	0	0
$825$	3.25879	0.113456
$826$	0	0
$827$	4.88216	0.169769	0.0848847	−	0.996391i	$-0.472948\pi$
$827$	4.88216	0.169769	0.0848847	+	0.996391i	$0.472948\pi$
$828$	0	0
$829$	−24.4787	−0.850182	−0.425091	−	0.905151i	$-0.639758\pi$
$829$	−24.4787	−0.850182	−0.425091	+	0.905151i	$0.639758\pi$
$830$	0	0
$831$	−4.13726	−0.143520
$832$	0	0
$833$	9.37411	0.324794
$834$	0	0
$835$	6.73250	0.232988
$836$	0	0
$837$	4.93923	0.170725
$838$	0	0
$839$	44.1091	1.52282	0.761408	−	0.648273i	$-0.224508\pi$
$839$	44.1091	1.52282	0.761408	+	0.648273i	$0.224508\pi$
$840$	0	0
$841$	55.6036	1.91736
$842$	0	0
$843$	6.13726	0.211378
$844$	0	0
$845$	−1.31955	−0.0453939
$846$	0	0
$847$	2.25879	0.0776128
$848$	0	0
$849$	23.0377	0.790650
$850$	0	0
$851$	21.7181	0.744487
$852$	0	0
$853$	14.5952	0.499732	0.249866	−	0.968280i	$-0.419614\pi$
$853$	14.5952	0.499732	0.249866	+	0.968280i	$0.419614\pi$
$854$	0	0
$855$	2.63910	0.0902554
$856$	0	0
$857$	−39.2526	−1.34084	−0.670421	−	0.741981i	$-0.733887\pi$
$857$	−39.2526	−1.34084	−0.670421	+	0.741981i	$0.733887\pi$
$858$	0	0
$859$	−29.4700	−1.00550	−0.502752	−	0.864431i	$-0.667679\pi$
$859$	−29.4700	−1.00550	−0.502752	+	0.864431i	$0.667679\pi$
$860$	0	0
$861$	3.65852	0.124682
$862$	0	0
$863$	31.3997	1.06886	0.534430	−	0.845213i	$-0.320526\pi$
$863$	31.3997	1.06886	0.534430	+	0.845213i	$0.320526\pi$
$864$	0	0
$865$	6.85905	0.233215
$866$	0	0
$867$	−7.39604	−0.251183
$868$	0	0
$869$	−1.06077	−0.0359840
$870$	0	0
$871$	−8.59775	−0.291324
$872$	0	0
$873$	−15.0351	−0.508862
$874$	0	0
$875$	24.6160	0.832172
$876$	0	0
$877$	−16.9963	−0.573925	−0.286962	−	0.957942i	$-0.592645\pi$
$877$	−16.9963	−0.573925	−0.286962	+	0.957942i	$0.592645\pi$
$878$	0	0
$879$	−15.6742	−0.528679
$880$	0	0
$881$	21.8554	0.736326	0.368163	−	0.929761i	$-0.379987\pi$
$881$	21.8554	0.736326	0.368163	+	0.929761i	$0.379987\pi$
$882$	0	0
$883$	0.0826952	0.00278291	0.00139146	−	0.999999i	$-0.499557\pi$
$883$	0.0826952	0.00278291	0.00139146	+	0.999999i	$0.499557\pi$
$884$	0	0
$885$	−15.1724	−0.510015
$886$	0	0
$887$	−9.52126	−0.319693	−0.159846	−	0.987142i	$-0.551100\pi$
$887$	−9.52126	−0.319693	−0.159846	+	0.987142i	$0.551100\pi$
$888$	0	0
$889$	−49.3609	−1.65551
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	11.7569	0.393431
$894$	0	0
$895$	−29.7131	−0.993199
$896$	0	0
$897$	−3.61968	−0.120858
$898$	0	0
$899$	−45.4312	−1.51521
$900$	0	0
$901$	−9.87847	−0.329100
$902$	0	0
$903$	19.3328	0.643354
$904$	0	0
$905$	2.11651	0.0703550
$906$	0	0
$907$	38.4787	1.27767	0.638833	−	0.769346i	$-0.279418\pi$
$907$	38.4787	1.27767	0.638833	+	0.769346i	$0.279418\pi$
$908$	0	0
$909$	5.45681	0.180991
$910$	0	0
$911$	−47.0278	−1.55810	−0.779050	−	0.626962i	$-0.784298\pi$
$911$	−47.0278	−1.55810	−0.779050	+	0.626962i	$0.784298\pi$
$912$	0	0
$913$	8.51757	0.281890
$914$	0	0
$915$	0.501846	0.0165905
$916$	0	0
$917$	−12.6937	−0.419182
$918$	0	0
$919$	−9.82140	−0.323978	−0.161989	−	0.986793i	$-0.551791\pi$
$919$	−9.82140	−0.323978	−0.161989	+	0.986793i	$0.551791\pi$
$920$	0	0
$921$	3.27820	0.108020
$922$	0	0
$923$	7.23937	0.238287
$924$	0	0
$925$	−19.5527	−0.642889
$926$	0	0
$927$	3.49815	0.114894
$928$	0	0
$929$	−12.3001	−0.403554	−0.201777	−	0.979431i	$-0.564672\pi$
$929$	−12.3001	−0.403554	−0.201777	+	0.979431i	$0.564672\pi$
$930$	0	0
$931$	3.79577	0.124401
$932$	0	0
$933$	21.2394	0.695346
$934$	0	0
$935$	6.51757	0.213147
$936$	0	0
$937$	−41.9173	−1.36938	−0.684689	−	0.728835i	$-0.740062\pi$
$937$	−41.9173	−1.36938	−0.684689	+	0.728835i	$0.740062\pi$
$938$	0	0
$939$	−8.77636	−0.286406
$940$	0	0
$941$	−17.2782	−0.563253	−0.281627	−	0.959524i	$-0.590874\pi$
$941$	−17.2782	−0.563253	−0.281627	+	0.959524i	$0.590874\pi$
$942$	0	0
$943$	−5.86274	−0.190917
$944$	0	0
$945$	2.98058	0.0969584
$946$	0	0
$947$	−37.3923	−1.21509	−0.607544	−	0.794286i	$-0.707845\pi$
$947$	−37.3923	−1.21509	−0.607544	+	0.794286i	$0.707845\pi$
$948$	0	0
$949$	15.4155	0.500407
$950$	0	0
$951$	24.9938	0.810479
$952$	0	0
$953$	21.5395	0.697733	0.348866	−	0.937172i	$-0.386567\pi$
$953$	21.5395	0.697733	0.348866	+	0.937172i	$0.386567\pi$
$954$	0	0
$955$	8.41915	0.272437
$956$	0	0
$957$	−9.19802	−0.297330
$958$	0	0
$959$	36.0827	1.16517
$960$	0	0
$961$	−6.60396	−0.213031
$962$	0	0
$963$	1.01942	0.0328503
$964$	0	0
$965$	−20.1091	−0.647335
$966$	0	0
$967$	21.3328	0.686015	0.343008	−	0.939333i	$-0.388554\pi$
$967$	21.3328	0.686015	0.343008	+	0.939333i	$0.388554\pi$
$968$	0	0
$969$	−9.87847	−0.317342
$970$	0	0
$971$	29.5915	0.949638	0.474819	−	0.880083i	$-0.342514\pi$
$971$	29.5915	0.949638	0.474819	+	0.880083i	$0.342514\pi$
$972$	0	0
$973$	−21.6354	−0.693599
$974$	0	0
$975$	3.25879	0.104365
$976$	0	0
$977$	3.02193	0.0966801	0.0483401	−	0.998831i	$-0.484607\pi$
$977$	3.02193	0.0966801	0.0483401	+	0.998831i	$0.484607\pi$
$978$	0	0
$979$	−0.300133	−0.00959230
$980$	0	0
$981$	−11.7569	−0.375370
$982$	0	0
$983$	40.1968	1.28208	0.641040	−	0.767507i	$-0.278503\pi$
$983$	40.1968	1.28208	0.641040	+	0.767507i	$0.278503\pi$
$984$	0	0
$985$	−24.9575	−0.795211
$986$	0	0
$987$	13.2782	0.422650
$988$	0	0
$989$	−30.9806	−0.985125
$990$	0	0
$991$	27.3328	0.868254	0.434127	−	0.900852i	$-0.357057\pi$
$991$	27.3328	0.868254	0.434127	+	0.900852i	$0.357057\pi$
$992$	0	0
$993$	32.1116	1.01903
$994$	0	0
$995$	20.7713	0.658495
$996$	0	0
$997$	34.3679	1.08844	0.544221	−	0.838942i	$-0.316825\pi$
$997$	34.3679	1.08844	0.544221	+	0.838942i	$0.316825\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bo.1.2		3
4.3	odd	2		3432.2.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.o.1.2	✓	3	4.3	odd	2
6864.2.a.bo.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.31955 q^{5} +2.25879 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.31955 q^{5} +2.25879 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.31955 q^{15} -4.93923 q^{17} -2.00000 q^{19} -2.25879 q^{21} +3.61968 q^{23} -3.25879 q^{25} -1.00000 q^{27} +9.19802 q^{29} -4.93923 q^{31} -1.00000 q^{33} -2.98058 q^{35} +6.00000 q^{37} -1.00000 q^{39} -1.61968 q^{41} -8.55892 q^{43} -1.31955 q^{45} -5.87847 q^{47} -1.89789 q^{49} +4.93923 q^{51} +2.00000 q^{53} -1.31955 q^{55} +2.00000 q^{57} -11.4982 q^{59} +0.380316 q^{61} +2.25879 q^{63} -1.31955 q^{65} -8.59775 q^{67} -3.61968 q^{69} +7.23937 q^{71} +15.4155 q^{73} +3.25879 q^{75} +2.25879 q^{77} -1.06077 q^{79} +1.00000 q^{81} +8.51757 q^{83} +6.51757 q^{85} -9.19802 q^{87} -0.300133 q^{89} +2.25879 q^{91} +4.93923 q^{93} +2.63910 q^{95} -15.0351 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

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