LMFDB - Embedded newform 6864.2.a.bj.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.23607 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.23607 q^{5} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.23607 q^{15} -5.23607 q^{17} -2.47214 q^{19} -4.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +1.23607 q^{29} -3.23607 q^{31} +1.00000 q^{33} +0.472136 q^{37} -1.00000 q^{39} +12.4721 q^{41} -5.70820 q^{43} +1.23607 q^{45} -4.00000 q^{47} -7.00000 q^{49} -5.23607 q^{51} -2.00000 q^{53} +1.23607 q^{55} -2.47214 q^{57} -10.4721 q^{59} -4.47214 q^{61} -1.23607 q^{65} -3.23607 q^{67} -4.00000 q^{69} +2.47214 q^{71} +3.52786 q^{73} -3.47214 q^{75} +0.763932 q^{79} +1.00000 q^{81} -4.00000 q^{83} -6.47214 q^{85} +1.23607 q^{87} -11.7082 q^{89} -3.23607 q^{93} -3.05573 q^{95} +13.4164 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{19} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 2 q^{31} + 2 q^{33} - 8 q^{37} - 2 q^{39} + 16 q^{41} + 2 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 6 q^{51} - 4 q^{53} - 2 q^{55} + 4 q^{57} - 12 q^{59} + 2 q^{65} - 2 q^{67} - 8 q^{69} - 4 q^{71} + 16 q^{73} + 2 q^{75} + 6 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} - 2 q^{87} - 10 q^{89} - 2 q^{93} - 24 q^{95} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^17 + 4 * q^19 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 2 * q^31 + 2 * q^33 - 8 * q^37 - 2 * q^39 + 16 * q^41 + 2 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 6 * q^51 - 4 * q^53 - 2 * q^55 + 4 * q^57 - 12 * q^59 + 2 * q^65 - 2 * q^67 - 8 * q^69 - 4 * q^71 + 16 * q^73 + 2 * q^75 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 - 2 * q^87 - 10 * q^89 - 2 * q^93 - 24 * q^95 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	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
$15$	1.23607	0.319151
$16$	0	0
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.23607	0.229532	0.114766	−	0.993393i	$-0.463388\pi$
$29$	1.23607	0.229532	0.114766	+	0.993393i	$0.463388\pi$
$30$	0	0
$31$	−3.23607	−0.581215	−0.290607	−	0.956842i	$-0.593857\pi$
$31$	−3.23607	−0.581215	−0.290607	+	0.956842i	$0.593857\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	12.4721	1.94782	0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721	1.94782	0.973910	+	0.226934i	$0.0728701\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	1.23607	0.184262
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−5.23607	−0.733196
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	0	0
$57$	−2.47214	−0.327442
$58$	0	0
$59$	−10.4721	−1.36336	−0.681678	−	0.731652i	$-0.738749\pi$
$59$	−10.4721	−1.36336	−0.681678	+	0.731652i	$0.738749\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−3.23607	−0.395349	−0.197674	−	0.980268i	$-0.563339\pi$
$67$	−3.23607	−0.395349	−0.197674	+	0.980268i	$0.563339\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	3.52786	0.412905	0.206453	−	0.978457i	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.763932	0.0859491	0.0429745	−	0.999076i	$-0.486317\pi$
$79$	0.763932	0.0859491	0.0429745	+	0.999076i	$0.486317\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−6.47214	−0.702002
$86$	0	0
$87$	1.23607	0.132520
$88$	0	0
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.23607	−0.335565
$94$	0	0
$95$	−3.05573	−0.313511
$96$	0	0
$97$	13.4164	1.36223	0.681115	−	0.732177i	$-0.261495\pi$
$97$	13.4164	1.36223	0.681115	+	0.732177i	$0.261495\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−16.6525	−1.65698	−0.828492	−	0.560001i	$-0.810801\pi$
$101$	−16.6525	−1.65698	−0.828492	+	0.560001i	$0.810801\pi$
$102$	0	0
$103$	−6.47214	−0.637719	−0.318859	−	0.947802i	$-0.603300\pi$
$103$	−6.47214	−0.637719	−0.318859	+	0.947802i	$0.603300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.4721	−1.39907	−0.699537	−	0.714596i	$-0.746610\pi$
$107$	−14.4721	−1.39907	−0.699537	+	0.714596i	$0.746610\pi$
$108$	0	0
$109$	4.47214	0.428353	0.214176	−	0.976795i	$-0.431293\pi$
$109$	4.47214	0.428353	0.214176	+	0.976795i	$0.431293\pi$
$110$	0	0
$111$	0.472136	0.0448132
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−4.94427	−0.461056
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.4721	1.12457
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−13.7082	−1.21641	−0.608203	−	0.793781i	$-0.708109\pi$
$127$	−13.7082	−1.21641	−0.608203	+	0.793781i	$0.708109\pi$
$128$	0	0
$129$	−5.70820	−0.502579
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.23607	0.106384
$136$	0	0
$137$	14.1803	1.21151	0.605754	−	0.795652i	$-0.292872\pi$
$137$	14.1803	1.21151	0.605754	+	0.795652i	$0.292872\pi$
$138$	0	0
$139$	17.1246	1.45249	0.726245	−	0.687436i	$-0.241264\pi$
$139$	17.1246	1.45249	0.726245	+	0.687436i	$0.241264\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	1.52786	0.126882
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	11.8885	0.973947	0.486974	−	0.873417i	$-0.338101\pi$
$149$	11.8885	0.973947	0.486974	+	0.873417i	$0.338101\pi$
$150$	0	0
$151$	−14.4721	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$151$	−14.4721	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$152$	0	0
$153$	−5.23607	−0.423311
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	3.52786	0.281554	0.140777	−	0.990041i	$-0.455040\pi$
$157$	3.52786	0.281554	0.140777	+	0.990041i	$0.455040\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.1246	1.65461	0.827304	−	0.561755i	$-0.189874\pi$
$163$	21.1246	1.65461	0.827304	+	0.561755i	$0.189874\pi$
$164$	0	0
$165$	1.23607	0.0962278
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.47214	−0.189049
$172$	0	0
$173$	25.2361	1.91866	0.959331	−	0.282282i	$-0.0910915\pi$
$173$	25.2361	1.91866	0.959331	+	0.282282i	$0.0910915\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.4721	−0.787134
$178$	0	0
$179$	9.88854	0.739104	0.369552	−	0.929210i	$-0.379511\pi$
$179$	9.88854	0.739104	0.369552	+	0.929210i	$0.379511\pi$
$180$	0	0
$181$	0.472136	0.0350936	0.0175468	−	0.999846i	$-0.494414\pi$
$181$	0.472136	0.0350936	0.0175468	+	0.999846i	$0.494414\pi$
$182$	0	0
$183$	−4.47214	−0.330590
$184$	0	0
$185$	0.583592	0.0429065
$186$	0	0
$187$	−5.23607	−0.382899
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8885	1.29437	0.647185	−	0.762333i	$-0.275946\pi$
$191$	17.8885	1.29437	0.647185	+	0.762333i	$0.275946\pi$
$192$	0	0
$193$	0.472136	0.0339851	0.0169925	−	0.999856i	$-0.494591\pi$
$193$	0.472136	0.0339851	0.0169925	+	0.999856i	$0.494591\pi$
$194$	0	0
$195$	−1.23607	−0.0885167
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	11.4164	0.809288	0.404644	−	0.914474i	$-0.367395\pi$
$199$	11.4164	0.809288	0.404644	+	0.914474i	$0.367395\pi$
$200$	0	0
$201$	−3.23607	−0.228255
$202$	0	0
$203$	0	0
$204$	0	0
$205$	15.4164	1.07673
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−2.47214	−0.171001
$210$	0	0
$211$	−25.1246	−1.72965	−0.864825	−	0.502074i	$-0.832571\pi$
$211$	−25.1246	−1.72965	−0.864825	+	0.502074i	$0.832571\pi$
$212$	0	0
$213$	2.47214	0.169388
$214$	0	0
$215$	−7.05573	−0.481197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.52786	0.238391
$220$	0	0
$221$	5.23607	0.352216
$222$	0	0
$223$	−9.70820	−0.650109	−0.325055	−	0.945695i	$-0.605383\pi$
$223$	−9.70820	−0.650109	−0.325055	+	0.945695i	$0.605383\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	−12.9443	−0.859142	−0.429571	−	0.903033i	$-0.641335\pi$
$227$	−12.9443	−0.859142	−0.429571	+	0.903033i	$0.641335\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.6525	1.87709	0.938543	−	0.345162i	$-0.112176\pi$
$233$	28.6525	1.87709	0.938543	+	0.345162i	$0.112176\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	0	0
$237$	0.763932	0.0496227
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−8.65248	−0.552786
$246$	0	0
$247$	2.47214	0.157298
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−6.47214	−0.405301
$256$	0	0
$257$	26.3607	1.64433	0.822167	−	0.569246i	$-0.192765\pi$
$257$	26.3607	1.64433	0.822167	+	0.569246i	$0.192765\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.23607	0.0765107
$262$	0	0
$263$	−14.4721	−0.892390	−0.446195	−	0.894936i	$-0.647221\pi$
$263$	−14.4721	−0.892390	−0.446195	+	0.894936i	$0.647221\pi$
$264$	0	0
$265$	−2.47214	−0.151862
$266$	0	0
$267$	−11.7082	−0.716530
$268$	0	0
$269$	−8.47214	−0.516555	−0.258278	−	0.966071i	$-0.583155\pi$
$269$	−8.47214	−0.516555	−0.258278	+	0.966071i	$0.583155\pi$
$270$	0	0
$271$	−4.58359	−0.278433	−0.139217	−	0.990262i	$-0.544458\pi$
$271$	−4.58359	−0.278433	−0.139217	+	0.990262i	$0.544458\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.47214	−0.209378
$276$	0	0
$277$	−28.8328	−1.73240	−0.866198	−	0.499701i	$-0.833443\pi$
$277$	−28.8328	−1.73240	−0.866198	+	0.499701i	$0.833443\pi$
$278$	0	0
$279$	−3.23607	−0.193738
$280$	0	0
$281$	−6.58359	−0.392744	−0.196372	−	0.980529i	$-0.562916\pi$
$281$	−6.58359	−0.392744	−0.196372	+	0.980529i	$0.562916\pi$
$282$	0	0
$283$	10.6525	0.633224	0.316612	−	0.948555i	$-0.397455\pi$
$283$	10.6525	0.633224	0.316612	+	0.948555i	$0.397455\pi$
$284$	0	0
$285$	−3.05573	−0.181006
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	13.4164	0.786484
$292$	0	0
$293$	−18.9443	−1.10674	−0.553368	−	0.832937i	$-0.686658\pi$
$293$	−18.9443	−1.10674	−0.553368	+	0.832937i	$0.686658\pi$
$294$	0	0
$295$	−12.9443	−0.753645
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−16.6525	−0.956660
$304$	0	0
$305$	−5.52786	−0.316525
$306$	0	0
$307$	−2.47214	−0.141092	−0.0705461	−	0.997509i	$-0.522474\pi$
$307$	−2.47214	−0.141092	−0.0705461	+	0.997509i	$0.522474\pi$
$308$	0	0
$309$	−6.47214	−0.368187
$310$	0	0
$311$	−16.9443	−0.960822	−0.480411	−	0.877044i	$-0.659512\pi$
$311$	−16.9443	−0.960822	−0.480411	+	0.877044i	$0.659512\pi$
$312$	0	0
$313$	9.41641	0.532247	0.266123	−	0.963939i	$-0.414257\pi$
$313$	9.41641	0.532247	0.266123	+	0.963939i	$0.414257\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.1803	0.796447	0.398224	−	0.917288i	$-0.369627\pi$
$317$	14.1803	0.796447	0.398224	+	0.917288i	$0.369627\pi$
$318$	0	0
$319$	1.23607	0.0692065
$320$	0	0
$321$	−14.4721	−0.807756
$322$	0	0
$323$	12.9443	0.720239
$324$	0	0
$325$	3.47214	0.192599
$326$	0	0
$327$	4.47214	0.247310
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.1803	1.32907	0.664536	−	0.747256i	$-0.268629\pi$
$331$	24.1803	1.32907	0.664536	+	0.747256i	$0.268629\pi$
$332$	0	0
$333$	0.472136	0.0258729
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−22.3607	−1.21806	−0.609032	−	0.793146i	$-0.708442\pi$
$337$	−22.3607	−1.21806	−0.609032	+	0.793146i	$0.708442\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−3.23607	−0.175243
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.94427	−0.266191
$346$	0	0
$347$	3.05573	0.164040	0.0820200	−	0.996631i	$-0.473863\pi$
$347$	3.05573	0.164040	0.0820200	+	0.996631i	$0.473863\pi$
$348$	0	0
$349$	14.9443	0.799949	0.399974	−	0.916526i	$-0.369019\pi$
$349$	14.9443	0.799949	0.399974	+	0.916526i	$0.369019\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	23.7082	1.26186	0.630930	−	0.775840i	$-0.282673\pi$
$353$	23.7082	1.26186	0.630930	+	0.775840i	$0.282673\pi$
$354$	0	0
$355$	3.05573	0.162181
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.8885	−1.99968	−0.999840	−	0.0178638i	$-0.994313\pi$
$359$	−37.8885	−1.99968	−0.999840	+	0.0178638i	$0.994313\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	4.36068	0.228248
$366$	0	0
$367$	25.8885	1.35137	0.675685	−	0.737190i	$-0.263848\pi$
$367$	25.8885	1.35137	0.675685	+	0.737190i	$0.263848\pi$
$368$	0	0
$369$	12.4721	0.649273
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.472136	0.0244463	0.0122231	−	0.999925i	$-0.496109\pi$
$373$	0.472136	0.0244463	0.0122231	+	0.999925i	$0.496109\pi$
$374$	0	0
$375$	−10.4721	−0.540779
$376$	0	0
$377$	−1.23607	−0.0636607
$378$	0	0
$379$	−0.180340	−0.00926344	−0.00463172	−	0.999989i	$-0.501474\pi$
$379$	−0.180340	−0.00926344	−0.00463172	+	0.999989i	$0.501474\pi$
$380$	0	0
$381$	−13.7082	−0.702293
$382$	0	0
$383$	12.3607	0.631601	0.315801	−	0.948826i	$-0.397727\pi$
$383$	12.3607	0.631601	0.315801	+	0.948826i	$0.397727\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70820	−0.290164
$388$	0	0
$389$	−29.4164	−1.49147	−0.745736	−	0.666242i	$-0.767902\pi$
$389$	−29.4164	−1.49147	−0.745736	+	0.666242i	$0.767902\pi$
$390$	0	0
$391$	20.9443	1.05920
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	0.944272	0.0475115
$396$	0	0
$397$	−4.47214	−0.224450	−0.112225	−	0.993683i	$-0.535798\pi$
$397$	−4.47214	−0.224450	−0.112225	+	0.993683i	$0.535798\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.6525	1.03134	0.515668	−	0.856789i	$-0.327544\pi$
$401$	20.6525	1.03134	0.515668	+	0.856789i	$0.327544\pi$
$402$	0	0
$403$	3.23607	0.161200
$404$	0	0
$405$	1.23607	0.0614207
$406$	0	0
$407$	0.472136	0.0234029
$408$	0	0
$409$	−28.4721	−1.40786	−0.703928	−	0.710271i	$-0.748572\pi$
$409$	−28.4721	−1.40786	−0.703928	+	0.710271i	$0.748572\pi$
$410$	0	0
$411$	14.1803	0.699465
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.94427	−0.242705
$416$	0	0
$417$	17.1246	0.838596
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	18.1803	0.881876
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−12.9443	−0.623504	−0.311752	−	0.950164i	$-0.600916\pi$
$431$	−12.9443	−0.623504	−0.311752	+	0.950164i	$0.600916\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	1.52786	0.0732555
$436$	0	0
$437$	9.88854	0.473033
$438$	0	0
$439$	−21.7082	−1.03608	−0.518038	−	0.855358i	$-0.673337\pi$
$439$	−21.7082	−1.03608	−0.518038	+	0.855358i	$0.673337\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−16.9443	−0.805047	−0.402523	−	0.915410i	$-0.631867\pi$
$443$	−16.9443	−0.805047	−0.402523	+	0.915410i	$0.631867\pi$
$444$	0	0
$445$	−14.4721	−0.686045
$446$	0	0
$447$	11.8885	0.562309
$448$	0	0
$449$	−1.81966	−0.0858751	−0.0429375	−	0.999078i	$-0.513672\pi$
$449$	−1.81966	−0.0858751	−0.0429375	+	0.999078i	$0.513672\pi$
$450$	0	0
$451$	12.4721	0.587290
$452$	0	0
$453$	−14.4721	−0.679960
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.05573	0.0493849	0.0246924	−	0.999695i	$-0.492139\pi$
$457$	1.05573	0.0493849	0.0246924	+	0.999695i	$0.492139\pi$
$458$	0	0
$459$	−5.23607	−0.244399
$460$	0	0
$461$	−36.4721	−1.69868	−0.849338	−	0.527849i	$-0.822999\pi$
$461$	−36.4721	−1.69868	−0.849338	+	0.527849i	$0.822999\pi$
$462$	0	0
$463$	21.1246	0.981744	0.490872	−	0.871232i	$-0.336678\pi$
$463$	21.1246	0.981744	0.490872	+	0.871232i	$0.336678\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	17.8885	0.827783	0.413892	−	0.910326i	$-0.364169\pi$
$467$	17.8885	0.827783	0.413892	+	0.910326i	$0.364169\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.52786	0.162555
$472$	0	0
$473$	−5.70820	−0.262463
$474$	0	0
$475$	8.58359	0.393842
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−32.9443	−1.50526	−0.752631	−	0.658442i	$-0.771216\pi$
$479$	−32.9443	−1.50526	−0.752631	+	0.658442i	$0.771216\pi$
$480$	0	0
$481$	−0.472136	−0.0215275
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.5836	0.753022
$486$	0	0
$487$	15.8197	0.716857	0.358429	−	0.933557i	$-0.383313\pi$
$487$	15.8197	0.716857	0.358429	+	0.933557i	$0.383313\pi$
$488$	0	0
$489$	21.1246	0.955288
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−6.47214	−0.291490
$494$	0	0
$495$	1.23607	0.0555571
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.81966	0.350056	0.175028	−	0.984563i	$-0.443998\pi$
$499$	7.81966	0.350056	0.175028	+	0.984563i	$0.443998\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−19.4164	−0.865735	−0.432867	−	0.901458i	$-0.642498\pi$
$503$	−19.4164	−0.865735	−0.432867	+	0.901458i	$0.642498\pi$
$504$	0	0
$505$	−20.5836	−0.915958
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−23.1246	−1.02498	−0.512490	−	0.858693i	$-0.671277\pi$
$509$	−23.1246	−1.02498	−0.512490	+	0.858693i	$0.671277\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.47214	−0.109147
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	25.2361	1.10774
$520$	0	0
$521$	−1.05573	−0.0462523	−0.0231261	−	0.999733i	$-0.507362\pi$
$521$	−1.05573	−0.0462523	−0.0231261	+	0.999733i	$0.507362\pi$
$522$	0	0
$523$	21.7082	0.949233	0.474617	−	0.880193i	$-0.342587\pi$
$523$	21.7082	0.949233	0.474617	+	0.880193i	$0.342587\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.9443	0.738104
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−10.4721	−0.454452
$532$	0	0
$533$	−12.4721	−0.540228
$534$	0	0
$535$	−17.8885	−0.773389
$536$	0	0
$537$	9.88854	0.426722
$538$	0	0
$539$	−7.00000	−0.301511
$540$	0	0
$541$	11.8885	0.511128	0.255564	−	0.966792i	$-0.417739\pi$
$541$	11.8885	0.511128	0.255564	+	0.966792i	$0.417739\pi$
$542$	0	0
$543$	0.472136	0.0202613
$544$	0	0
$545$	5.52786	0.236788
$546$	0	0
$547$	5.70820	0.244065	0.122033	−	0.992526i	$-0.461059\pi$
$547$	5.70820	0.244065	0.122033	+	0.992526i	$0.461059\pi$
$548$	0	0
$549$	−4.47214	−0.190866
$550$	0	0
$551$	−3.05573	−0.130178
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.583592	0.0247721
$556$	0	0
$557$	−44.8328	−1.89963	−0.949814	−	0.312816i	$-0.898728\pi$
$557$	−44.8328	−1.89963	−0.949814	+	0.312816i	$0.898728\pi$
$558$	0	0
$559$	5.70820	0.241431
$560$	0	0
$561$	−5.23607	−0.221067
$562$	0	0
$563$	27.0557	1.14026	0.570131	−	0.821553i	$-0.306892\pi$
$563$	27.0557	1.14026	0.570131	+	0.821553i	$0.306892\pi$
$564$	0	0
$565$	2.47214	0.104004
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.70820	−0.155456	−0.0777280	−	0.996975i	$-0.524767\pi$
$569$	−3.70820	−0.155456	−0.0777280	+	0.996975i	$0.524767\pi$
$570$	0	0
$571$	29.7082	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$571$	29.7082	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$572$	0	0
$573$	17.8885	0.747305
$574$	0	0
$575$	13.8885	0.579192
$576$	0	0
$577$	13.0557	0.543517	0.271759	−	0.962365i	$-0.412395\pi$
$577$	13.0557	0.543517	0.271759	+	0.962365i	$0.412395\pi$
$578$	0	0
$579$	0.472136	0.0196213
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−1.23607	−0.0511051
$586$	0	0
$587$	−33.3050	−1.37464	−0.687321	−	0.726354i	$-0.741214\pi$
$587$	−33.3050	−1.37464	−0.687321	+	0.726354i	$0.741214\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−6.94427	−0.285167	−0.142584	−	0.989783i	$-0.545541\pi$
$593$	−6.94427	−0.285167	−0.142584	+	0.989783i	$0.545541\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.4164	0.467242
$598$	0	0
$599$	3.05573	0.124854	0.0624268	−	0.998050i	$-0.480116\pi$
$599$	3.05573	0.124854	0.0624268	+	0.998050i	$0.480116\pi$
$600$	0	0
$601$	13.0557	0.532554	0.266277	−	0.963897i	$-0.414206\pi$
$601$	13.0557	0.532554	0.266277	+	0.963897i	$0.414206\pi$
$602$	0	0
$603$	−3.23607	−0.131783
$604$	0	0
$605$	1.23607	0.0502533
$606$	0	0
$607$	7.23607	0.293703	0.146851	−	0.989159i	$-0.453086\pi$
$607$	7.23607	0.293703	0.146851	+	0.989159i	$0.453086\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−0.472136	−0.0190694	−0.00953470	−	0.999955i	$-0.503035\pi$
$613$	−0.472136	−0.0190694	−0.00953470	+	0.999955i	$0.503035\pi$
$614$	0	0
$615$	15.4164	0.621650
$616$	0	0
$617$	−19.7082	−0.793422	−0.396711	−	0.917943i	$-0.629849\pi$
$617$	−19.7082	−0.793422	−0.396711	+	0.917943i	$0.629849\pi$
$618$	0	0
$619$	22.6525	0.910480	0.455240	−	0.890369i	$-0.349553\pi$
$619$	22.6525	0.910480	0.455240	+	0.890369i	$0.349553\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−2.47214	−0.0987276
$628$	0	0
$629$	−2.47214	−0.0985705
$630$	0	0
$631$	34.0689	1.35626	0.678130	−	0.734942i	$-0.262790\pi$
$631$	34.0689	1.35626	0.678130	+	0.734942i	$0.262790\pi$
$632$	0	0
$633$	−25.1246	−0.998614
$634$	0	0
$635$	−16.9443	−0.672413
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	2.47214	0.0977962
$640$	0	0
$641$	−2.94427	−0.116292	−0.0581459	−	0.998308i	$-0.518519\pi$
$641$	−2.94427	−0.116292	−0.0581459	+	0.998308i	$0.518519\pi$
$642$	0	0
$643$	−3.59675	−0.141842	−0.0709209	−	0.997482i	$-0.522594\pi$
$643$	−3.59675	−0.141842	−0.0709209	+	0.997482i	$0.522594\pi$
$644$	0	0
$645$	−7.05573	−0.277819
$646$	0	0
$647$	14.8328	0.583138	0.291569	−	0.956550i	$-0.405823\pi$
$647$	14.8328	0.583138	0.291569	+	0.956550i	$0.405823\pi$
$648$	0	0
$649$	−10.4721	−0.411067
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.2492	−1.41854	−0.709271	−	0.704936i	$-0.750976\pi$
$653$	−36.2492	−1.41854	−0.709271	+	0.704936i	$0.750976\pi$
$654$	0	0
$655$	−9.88854	−0.386377
$656$	0	0
$657$	3.52786	0.137635
$658$	0	0
$659$	6.47214	0.252119	0.126059	−	0.992023i	$-0.459767\pi$
$659$	6.47214	0.252119	0.126059	+	0.992023i	$0.459767\pi$
$660$	0	0
$661$	10.3607	0.402984	0.201492	−	0.979490i	$-0.435421\pi$
$661$	10.3607	0.402984	0.201492	+	0.979490i	$0.435421\pi$
$662$	0	0
$663$	5.23607	0.203352
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.94427	−0.191443
$668$	0	0
$669$	−9.70820	−0.375341
$670$	0	0
$671$	−4.47214	−0.172645
$672$	0	0
$673$	8.11146	0.312674	0.156337	−	0.987704i	$-0.450031\pi$
$673$	8.11146	0.312674	0.156337	+	0.987704i	$0.450031\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	8.87539	0.341109	0.170554	−	0.985348i	$-0.445444\pi$
$677$	8.87539	0.341109	0.170554	+	0.985348i	$0.445444\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.9443	−0.496026
$682$	0	0
$683$	33.3050	1.27438	0.637189	−	0.770707i	$-0.280097\pi$
$683$	33.3050	1.27438	0.637189	+	0.770707i	$0.280097\pi$
$684$	0	0
$685$	17.5279	0.669705
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	16.1803	0.615529	0.307765	−	0.951463i	$-0.400419\pi$
$691$	16.1803	0.615529	0.307765	+	0.951463i	$0.400419\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.1672	0.802917
$696$	0	0
$697$	−65.3050	−2.47360
$698$	0	0
$699$	28.6525	1.08374
$700$	0	0
$701$	−6.76393	−0.255470	−0.127735	−	0.991808i	$-0.540771\pi$
$701$	−6.76393	−0.255470	−0.127735	+	0.991808i	$0.540771\pi$
$702$	0	0
$703$	−1.16718	−0.0440212
$704$	0	0
$705$	−4.94427	−0.186212
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−44.8328	−1.68373	−0.841866	−	0.539687i	$-0.818543\pi$
$709$	−44.8328	−1.68373	−0.841866	+	0.539687i	$0.818543\pi$
$710$	0	0
$711$	0.763932	0.0286497
$712$	0	0
$713$	12.9443	0.484767
$714$	0	0
$715$	−1.23607	−0.0462263
$716$	0	0
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−28.9443	−1.07944	−0.539720	−	0.841845i	$-0.681470\pi$
$719$	−28.9443	−1.07944	−0.539720	+	0.841845i	$0.681470\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	−4.29180	−0.159393
$726$	0	0
$727$	38.4721	1.42685	0.713426	−	0.700730i	$-0.247142\pi$
$727$	38.4721	1.42685	0.713426	+	0.700730i	$0.247142\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.8885	1.10547
$732$	0	0
$733$	25.4164	0.938776	0.469388	−	0.882992i	$-0.344474\pi$
$733$	25.4164	0.938776	0.469388	+	0.882992i	$0.344474\pi$
$734$	0	0
$735$	−8.65248	−0.319151
$736$	0	0
$737$	−3.23607	−0.119202
$738$	0	0
$739$	34.8328	1.28135	0.640673	−	0.767814i	$-0.278655\pi$
$739$	34.8328	1.28135	0.640673	+	0.767814i	$0.278655\pi$
$740$	0	0
$741$	2.47214	0.0908162
$742$	0	0
$743$	−30.8328	−1.13115	−0.565573	−	0.824698i	$-0.691345\pi$
$743$	−30.8328	−1.13115	−0.565573	+	0.824698i	$0.691345\pi$
$744$	0	0
$745$	14.6950	0.538385
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.7771	−1.01360	−0.506800	−	0.862064i	$-0.669172\pi$
$751$	−27.7771	−1.01360	−0.506800	+	0.862064i	$0.669172\pi$
$752$	0	0
$753$	−16.9443	−0.617484
$754$	0	0
$755$	−17.8885	−0.651031
$756$	0	0
$757$	39.3050	1.42856	0.714281	−	0.699859i	$-0.246754\pi$
$757$	39.3050	1.42856	0.714281	+	0.699859i	$0.246754\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.47214	−0.234001
$766$	0	0
$767$	10.4721	0.378127
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	26.3607	0.949357
$772$	0	0
$773$	20.2918	0.729845	0.364923	−	0.931038i	$-0.381095\pi$
$773$	20.2918	0.729845	0.364923	+	0.931038i	$0.381095\pi$
$774$	0	0
$775$	11.2361	0.403611
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.8328	−1.10470
$780$	0	0
$781$	2.47214	0.0884600
$782$	0	0
$783$	1.23607	0.0441735
$784$	0	0
$785$	4.36068	0.155639
$786$	0	0
$787$	−24.9443	−0.889167	−0.444584	−	0.895737i	$-0.646648\pi$
$787$	−24.9443	−0.889167	−0.444584	+	0.895737i	$0.646648\pi$
$788$	0	0
$789$	−14.4721	−0.515222
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.47214	0.158810
$794$	0	0
$795$	−2.47214	−0.0876776
$796$	0	0
$797$	19.3050	0.683816	0.341908	−	0.939733i	$-0.388927\pi$
$797$	19.3050	0.683816	0.341908	+	0.939733i	$0.388927\pi$
$798$	0	0
$799$	20.9443	0.740955
$800$	0	0
$801$	−11.7082	−0.413689
$802$	0	0
$803$	3.52786	0.124496
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.47214	−0.298233
$808$	0	0
$809$	28.2918	0.994687	0.497343	−	0.867554i	$-0.334309\pi$
$809$	28.2918	0.994687	0.497343	+	0.867554i	$0.334309\pi$
$810$	0	0
$811$	−26.4721	−0.929562	−0.464781	−	0.885426i	$-0.653867\pi$
$811$	−26.4721	−0.929562	−0.464781	+	0.885426i	$0.653867\pi$
$812$	0	0
$813$	−4.58359	−0.160754
$814$	0	0
$815$	26.1115	0.914644
$816$	0	0
$817$	14.1115	0.493697
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.47214	0.295680	0.147840	−	0.989011i	$-0.452768\pi$
$821$	8.47214	0.295680	0.147840	+	0.989011i	$0.452768\pi$
$822$	0	0
$823$	−9.52786	−0.332120	−0.166060	−	0.986116i	$-0.553105\pi$
$823$	−9.52786	−0.332120	−0.166060	+	0.986116i	$0.553105\pi$
$824$	0	0
$825$	−3.47214	−0.120884
$826$	0	0
$827$	28.9443	1.00649	0.503245	−	0.864144i	$-0.332139\pi$
$827$	28.9443	1.00649	0.503245	+	0.864144i	$0.332139\pi$
$828$	0	0
$829$	−43.8885	−1.52431	−0.762156	−	0.647393i	$-0.775859\pi$
$829$	−43.8885	−1.52431	−0.762156	+	0.647393i	$0.775859\pi$
$830$	0	0
$831$	−28.8328	−1.00020
$832$	0	0
$833$	36.6525	1.26993
$834$	0	0
$835$	−9.88854	−0.342207
$836$	0	0
$837$	−3.23607	−0.111855
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−27.4721	−0.947315
$842$	0	0
$843$	−6.58359	−0.226751
$844$	0	0
$845$	1.23607	0.0425220
$846$	0	0
$847$	0	0
$848$	0	0
$849$	10.6525	0.365592
$850$	0	0
$851$	−1.88854	−0.0647384
$852$	0	0
$853$	6.94427	0.237767	0.118884	−	0.992908i	$-0.462068\pi$
$853$	6.94427	0.237767	0.118884	+	0.992908i	$0.462068\pi$
$854$	0	0
$855$	−3.05573	−0.104504
$856$	0	0
$857$	−29.5967	−1.01101	−0.505503	−	0.862825i	$-0.668693\pi$
$857$	−29.5967	−1.01101	−0.505503	+	0.862825i	$0.668693\pi$
$858$	0	0
$859$	10.4721	0.357305	0.178652	−	0.983912i	$-0.442826\pi$
$859$	10.4721	0.357305	0.178652	+	0.983912i	$0.442826\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.5279	−0.460494	−0.230247	−	0.973132i	$-0.573953\pi$
$863$	−13.5279	−0.460494	−0.230247	+	0.973132i	$0.573953\pi$
$864$	0	0
$865$	31.1935	1.06061
$866$	0	0
$867$	10.4164	0.353760
$868$	0	0
$869$	0.763932	0.0259146
$870$	0	0
$871$	3.23607	0.109650
$872$	0	0
$873$	13.4164	0.454077
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.11146	−0.138834	−0.0694170	−	0.997588i	$-0.522114\pi$
$877$	−4.11146	−0.138834	−0.0694170	+	0.997588i	$0.522114\pi$
$878$	0	0
$879$	−18.9443	−0.638974
$880$	0	0
$881$	6.94427	0.233958	0.116979	−	0.993134i	$-0.462679\pi$
$881$	6.94427	0.233958	0.116979	+	0.993134i	$0.462679\pi$
$882$	0	0
$883$	−18.4721	−0.621637	−0.310818	−	0.950469i	$-0.600603\pi$
$883$	−18.4721	−0.621637	−0.310818	+	0.950469i	$0.600603\pi$
$884$	0	0
$885$	−12.9443	−0.435117
$886$	0	0
$887$	−24.3607	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$887$	−24.3607	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	9.88854	0.330908
$894$	0	0
$895$	12.2229	0.408567
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	10.4721	0.348877
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.583592	0.0193993
$906$	0	0
$907$	31.0557	1.03119	0.515594	−	0.856833i	$-0.327571\pi$
$907$	31.0557	1.03119	0.515594	+	0.856833i	$0.327571\pi$
$908$	0	0
$909$	−16.6525	−0.552328
$910$	0	0
$911$	6.11146	0.202482	0.101241	−	0.994862i	$-0.467719\pi$
$911$	6.11146	0.202482	0.101241	+	0.994862i	$0.467719\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	−5.52786	−0.182746
$916$	0	0
$917$	0	0
$918$	0	0
$919$	60.1803	1.98517	0.992583	−	0.121568i	$-0.0387922\pi$
$919$	60.1803	1.98517	0.992583	+	0.121568i	$0.0387922\pi$
$920$	0	0
$921$	−2.47214	−0.0814596
$922$	0	0
$923$	−2.47214	−0.0813713
$924$	0	0
$925$	−1.63932	−0.0539005
$926$	0	0
$927$	−6.47214	−0.212573
$928$	0	0
$929$	−51.7082	−1.69649	−0.848246	−	0.529603i	$-0.822341\pi$
$929$	−51.7082	−1.69649	−0.848246	+	0.529603i	$0.822341\pi$
$930$	0	0
$931$	17.3050	0.567147
$932$	0	0
$933$	−16.9443	−0.554731
$934$	0	0
$935$	−6.47214	−0.211661
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	9.41641	0.307293
$940$	0	0
$941$	3.52786	0.115005	0.0575025	−	0.998345i	$-0.481686\pi$
$941$	3.52786	0.115005	0.0575025	+	0.998345i	$0.481686\pi$
$942$	0	0
$943$	−49.8885	−1.62459
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−3.52786	−0.114519
$950$	0	0
$951$	14.1803	0.459829
$952$	0	0
$953$	−6.76393	−0.219105	−0.109553	−	0.993981i	$-0.534942\pi$
$953$	−6.76393	−0.219105	−0.109553	+	0.993981i	$0.534942\pi$
$954$	0	0
$955$	22.1115	0.715510
$956$	0	0
$957$	1.23607	0.0399564
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.5279	−0.662189
$962$	0	0
$963$	−14.4721	−0.466358
$964$	0	0
$965$	0.583592	0.0187865
$966$	0	0
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	0	0
$969$	12.9443	0.415830
$970$	0	0
$971$	35.7771	1.14814	0.574071	−	0.818806i	$-0.305363\pi$
$971$	35.7771	1.14814	0.574071	+	0.818806i	$0.305363\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.47214	0.111197
$976$	0	0
$977$	−44.0689	−1.40989	−0.704944	−	0.709263i	$-0.749028\pi$
$977$	−44.0689	−1.40989	−0.704944	+	0.709263i	$0.749028\pi$
$978$	0	0
$979$	−11.7082	−0.374196
$980$	0	0
$981$	4.47214	0.142784
$982$	0	0
$983$	−48.9443	−1.56108	−0.780540	−	0.625106i	$-0.785056\pi$
$983$	−48.9443	−1.56108	−0.780540	+	0.625106i	$0.785056\pi$
$984$	0	0
$985$	2.47214	0.0787688
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.8328	0.726041
$990$	0	0
$991$	−29.3050	−0.930902	−0.465451	−	0.885074i	$-0.654108\pi$
$991$	−29.3050	−0.930902	−0.465451	+	0.885074i	$0.654108\pi$
$992$	0	0
$993$	24.1803	0.767340
$994$	0	0
$995$	14.1115	0.447363
$996$	0	0
$997$	−27.3050	−0.864756	−0.432378	−	0.901692i	$-0.642325\pi$
$997$	−27.3050	−0.864756	−0.432378	+	0.901692i	$0.642325\pi$
$998$	0	0
$999$	0.472136	0.0149377

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.k.1.2	✓	2	4.3	odd	2
6864.2.a.bj.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.23607 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.23607 q^{5} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.23607 q^{15} -5.23607 q^{17} -2.47214 q^{19} -4.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +1.23607 q^{29} -3.23607 q^{31} +1.00000 q^{33} +0.472136 q^{37} -1.00000 q^{39} +12.4721 q^{41} -5.70820 q^{43} +1.23607 q^{45} -4.00000 q^{47} -7.00000 q^{49} -5.23607 q^{51} -2.00000 q^{53} +1.23607 q^{55} -2.47214 q^{57} -10.4721 q^{59} -4.47214 q^{61} -1.23607 q^{65} -3.23607 q^{67} -4.00000 q^{69} +2.47214 q^{71} +3.52786 q^{73} -3.47214 q^{75} +0.763932 q^{79} +1.00000 q^{81} -4.00000 q^{83} -6.47214 q^{85} +1.23607 q^{87} -11.7082 q^{89} -3.23607 q^{93} -3.05573 q^{95} +13.4164 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{19} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 2 q^{31} + 2 q^{33} - 8 q^{37} - 2 q^{39} + 16 q^{41} + 2 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 6 q^{51} - 4 q^{53} - 2 q^{55} + 4 q^{57} - 12 q^{59} + 2 q^{65} - 2 q^{67} - 8 q^{69} - 4 q^{71} + 16 q^{73} + 2 q^{75} + 6 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} - 2 q^{87} - 10 q^{89} - 2 q^{93} - 24 q^{95} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^17 + 4 * q^19 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 2 * q^31 + 2 * q^33 - 8 * q^37 - 2 * q^39 + 16 * q^41 + 2 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 6 * q^51 - 4 * q^53 - 2 * q^55 + 4 * q^57 - 12 * q^59 + 2 * q^65 - 2 * q^67 - 8 * q^69 - 4 * q^71 + 16 * q^73 + 2 * q^75 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 - 2 * q^87 - 10 * q^89 - 2 * q^93 - 24 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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