LMFDB - Embedded newform 6864.2.a.bw.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.70164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 10x - 2 $ x^4 - x^3 - 10x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$0.279954$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.36642 q^{5} -0.777601 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.36642 q^{5} -0.777601 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.36642 q^{15} -5.14402 q^{17} -0.559909 q^{19} +0.777601 q^{21} +0.777601 q^{23} +14.0656 q^{25} -1.00000 q^{27} -3.80651 q^{29} -10.5841 q^{31} +1.00000 q^{33} -3.39534 q^{35} -8.28805 q^{37} +1.00000 q^{39} +10.3906 q^{41} +4.48153 q^{43} +4.36642 q^{45} -8.00000 q^{47} -6.39534 q^{49} +5.14402 q^{51} +6.00000 q^{53} -4.36642 q^{55} +0.559909 q^{57} +0.217693 q^{59} +2.21769 q^{61} -0.777601 q^{63} -4.36642 q^{65} -11.0993 q^{67} -0.777601 q^{69} -12.7328 q^{71} +10.6209 q^{73} -14.0656 q^{75} +0.777601 q^{77} -0.855976 q^{79} +1.00000 q^{81} -6.67502 q^{83} -22.4610 q^{85} +3.80651 q^{87} -11.2591 q^{89} +0.777601 q^{91} +10.5841 q^{93} -2.44480 q^{95} -8.28805 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 - q^15 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * q^95 + 4 * 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
$5$	4.36642	1.95272	0.976362	−	0.216142i	$-0.0693475\pi$
$5$	4.36642	1.95272	0.976362	+	0.216142i	$0.0693475\pi$
$6$	0	0
$7$	−0.777601	−0.293906	−0.146953	−	0.989144i	$-0.546947\pi$
$7$	−0.777601	−0.293906	−0.146953	+	0.989144i	$0.546947\pi$
$8$	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
$15$	−4.36642	−1.12741
$16$	0	0
$17$	−5.14402	−1.24761	−0.623805	−	0.781580i	$-0.714414\pi$
$17$	−5.14402	−1.24761	−0.623805	+	0.781580i	$0.714414\pi$
$18$	0	0
$19$	−0.559909	−0.128452	−0.0642259	−	0.997935i	$-0.520458\pi$
$19$	−0.559909	−0.128452	−0.0642259	+	0.997935i	$0.520458\pi$
$20$	0	0
$21$	0.777601	0.169687
$22$	0	0
$23$	0.777601	0.162141	0.0810705	−	0.996708i	$-0.474166\pi$
$23$	0.777601	0.162141	0.0810705	+	0.996708i	$0.474166\pi$
$24$	0	0
$25$	14.0656	2.81313
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.80651	−0.706852	−0.353426	−	0.935462i	$-0.614983\pi$
$29$	−3.80651	−0.706852	−0.353426	+	0.935462i	$0.614983\pi$
$30$	0	0
$31$	−10.5841	−1.90096	−0.950482	−	0.310781i	$-0.899410\pi$
$31$	−10.5841	−1.90096	−0.950482	+	0.310781i	$0.899410\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.39534	−0.573917
$36$	0	0
$37$	−8.28805	−1.36255	−0.681273	−	0.732029i	$-0.738574\pi$
$37$	−8.28805	−1.36255	−0.681273	+	0.732029i	$0.738574\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	10.3906	1.62274	0.811372	−	0.584530i	$-0.198721\pi$
$41$	10.3906	1.62274	0.811372	+	0.584530i	$0.198721\pi$
$42$	0	0
$43$	4.48153	0.683428	0.341714	−	0.939804i	$-0.388993\pi$
$43$	4.48153	0.683428	0.341714	+	0.939804i	$0.388993\pi$
$44$	0	0
$45$	4.36642	0.650908
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−6.39534	−0.913619
$50$	0	0
$51$	5.14402	0.720307
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.36642	−0.588768
$56$	0	0
$57$	0.559909	0.0741617
$58$	0	0
$59$	0.217693	0.0283412	0.0141706	−	0.999900i	$-0.495489\pi$
$59$	0.217693	0.0283412	0.0141706	+	0.999900i	$0.495489\pi$
$60$	0	0
$61$	2.21769	0.283946	0.141973	−	0.989870i	$-0.454655\pi$
$61$	2.21769	0.283946	0.141973	+	0.989870i	$0.454655\pi$
$62$	0	0
$63$	−0.777601	−0.0979686
$64$	0	0
$65$	−4.36642	−0.541588
$66$	0	0
$67$	−11.0993	−1.35599	−0.677996	−	0.735066i	$-0.737151\pi$
$67$	−11.0993	−1.35599	−0.677996	+	0.735066i	$0.737151\pi$
$68$	0	0
$69$	−0.777601	−0.0936122
$70$	0	0
$71$	−12.7328	−1.51111	−0.755555	−	0.655085i	$-0.772633\pi$
$71$	−12.7328	−1.51111	−0.755555	+	0.655085i	$0.772633\pi$
$72$	0	0
$73$	10.6209	1.24308	0.621538	−	0.783384i	$-0.286508\pi$
$73$	10.6209	1.24308	0.621538	+	0.783384i	$0.286508\pi$
$74$	0	0
$75$	−14.0656	−1.62416
$76$	0	0
$77$	0.777601	0.0886159
$78$	0	0
$79$	−0.855976	−0.0963048	−0.0481524	−	0.998840i	$-0.515333\pi$
$79$	−0.855976	−0.0963048	−0.0481524	+	0.998840i	$0.515333\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.67502	−0.732679	−0.366339	−	0.930481i	$-0.619389\pi$
$83$	−6.67502	−0.732679	−0.366339	+	0.930481i	$0.619389\pi$
$84$	0	0
$85$	−22.4610	−2.43624
$86$	0	0
$87$	3.80651	0.408101
$88$	0	0
$89$	−11.2591	−1.19347	−0.596733	−	0.802440i	$-0.703535\pi$
$89$	−11.2591	−1.19347	−0.596733	+	0.802440i	$0.703535\pi$
$90$	0	0
$91$	0.777601	0.0815148
$92$	0	0
$93$	10.5841	1.09752
$94$	0	0
$95$	−2.44480	−0.250831
$96$	0	0
$97$	−8.28805	−0.841524	−0.420762	−	0.907171i	$-0.638237\pi$
$97$	−8.28805	−0.841524	−0.420762	+	0.907171i	$0.638237\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−8.75705	−0.871359	−0.435680	−	0.900102i	$-0.643492\pi$
$101$	−8.75705	−0.871359	−0.435680	+	0.900102i	$0.643492\pi$
$102$	0	0
$103$	16.7938	1.65474	0.827371	−	0.561656i	$-0.189836\pi$
$103$	16.7938	1.65474	0.827371	+	0.561656i	$0.189836\pi$
$104$	0	0
$105$	3.39534	0.331351
$106$	0	0
$107$	−14.8032	−1.43108	−0.715540	−	0.698572i	$-0.753819\pi$
$107$	−14.8032	−1.43108	−0.715540	+	0.698572i	$0.753819\pi$
$108$	0	0
$109$	−18.0162	−1.72564	−0.862819	−	0.505513i	$-0.831303\pi$
$109$	−18.0162	−1.72564	−0.862819	+	0.505513i	$0.831303\pi$
$110$	0	0
$111$	8.28805	0.786667
$112$	0	0
$113$	−6.95054	−0.653852	−0.326926	−	0.945050i	$-0.606013\pi$
$113$	−6.95054	−0.653852	−0.326926	+	0.945050i	$0.606013\pi$
$114$	0	0
$115$	3.39534	0.316617
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−10.3906	−0.936892
$124$	0	0
$125$	39.5845	3.54054
$126$	0	0
$127$	7.44148	0.660325	0.330162	−	0.943924i	$-0.392896\pi$
$127$	7.44148	0.660325	0.330162	+	0.943924i	$0.392896\pi$
$128$	0	0
$129$	−4.48153	−0.394577
$130$	0	0
$131$	20.7454	1.81253	0.906266	−	0.422708i	$-0.138920\pi$
$131$	20.7454	1.81253	0.906266	+	0.422708i	$0.138920\pi$
$132$	0	0
$133$	0.435386	0.0377527
$134$	0	0
$135$	−4.36642	−0.375802
$136$	0	0
$137$	−6.59353	−0.563323	−0.281662	−	0.959514i	$-0.590886\pi$
$137$	−6.59353	−0.563323	−0.281662	+	0.959514i	$0.590886\pi$
$138$	0	0
$139$	6.75705	0.573126	0.286563	−	0.958061i	$-0.407487\pi$
$139$	6.75705	0.573126	0.286563	+	0.958061i	$0.407487\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−16.6209	−1.38029
$146$	0	0
$147$	6.39534	0.527478
$148$	0	0
$149$	−14.6083	−1.19676	−0.598380	−	0.801212i	$-0.704189\pi$
$149$	−14.6083	−1.19676	−0.598380	+	0.801212i	$0.704189\pi$
$150$	0	0
$151$	−4.99529	−0.406511	−0.203256	−	0.979126i	$-0.565152\pi$
$151$	−4.99529	−0.406511	−0.203256	+	0.979126i	$0.565152\pi$
$152$	0	0
$153$	−5.14402	−0.415870
$154$	0	0
$155$	−46.2147	−3.71206
$156$	0	0
$157$	18.4704	1.47410	0.737049	−	0.675839i	$-0.236219\pi$
$157$	18.4704	1.47410	0.737049	+	0.675839i	$0.236219\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−0.604664	−0.0476542
$162$	0	0
$163$	−9.24661	−0.724250	−0.362125	−	0.932130i	$-0.617949\pi$
$163$	−9.24661	−0.724250	−0.362125	+	0.932130i	$0.617949\pi$
$164$	0	0
$165$	4.36642	0.339926
$166$	0	0
$167$	6.27552	0.485614	0.242807	−	0.970075i	$-0.421932\pi$
$167$	6.27552	0.485614	0.242807	+	0.970075i	$0.421932\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.559909	−0.0428173
$172$	0	0
$173$	−18.8274	−1.43142	−0.715711	−	0.698397i	$-0.753897\pi$
$173$	−18.8274	−1.43142	−0.715711	+	0.698397i	$0.753897\pi$
$174$	0	0
$175$	−10.9375	−0.826795
$176$	0	0
$177$	−0.217693	−0.0163628
$178$	0	0
$179$	10.1823	0.761065	0.380532	−	0.924768i	$-0.375741\pi$
$179$	10.1823	0.761065	0.380532	+	0.924768i	$0.375741\pi$
$180$	0	0
$181$	−9.58080	−0.712135	−0.356068	−	0.934460i	$-0.615883\pi$
$181$	−9.58080	−0.712135	−0.356068	+	0.934460i	$0.615883\pi$
$182$	0	0
$183$	−2.21769	−0.163937
$184$	0	0
$185$	−36.1891	−2.66068
$186$	0	0
$187$	5.14402	0.376168
$188$	0	0
$189$	0.777601	0.0565622
$190$	0	0
$191$	11.6256	0.841196	0.420598	−	0.907247i	$-0.361820\pi$
$191$	11.6256	0.841196	0.420598	+	0.907247i	$0.361820\pi$
$192$	0	0
$193$	19.5714	1.40878	0.704390	−	0.709813i	$-0.251221\pi$
$193$	19.5714	1.40878	0.704390	+	0.709813i	$0.251221\pi$
$194$	0	0
$195$	4.36642	0.312686
$196$	0	0
$197$	−4.61773	−0.329000	−0.164500	−	0.986377i	$-0.552601\pi$
$197$	−4.61773	−0.329000	−0.164500	+	0.986377i	$0.552601\pi$
$198$	0	0
$199$	−12.2177	−0.866089	−0.433045	−	0.901372i	$-0.642561\pi$
$199$	−12.2177	−0.866089	−0.433045	+	0.901372i	$0.642561\pi$
$200$	0	0
$201$	11.0993	0.782882
$202$	0	0
$203$	2.95995	0.207748
$204$	0	0
$205$	45.3699	3.16877
$206$	0	0
$207$	0.777601	0.0540470
$208$	0	0
$209$	0.559909	0.0387297
$210$	0	0
$211$	10.1649	0.699782	0.349891	−	0.936790i	$-0.386219\pi$
$211$	10.1649	0.699782	0.349891	+	0.936790i	$0.386219\pi$
$212$	0	0
$213$	12.7328	0.872440
$214$	0	0
$215$	19.5683	1.33455
$216$	0	0
$217$	8.23022	0.558704
$218$	0	0
$219$	−10.6209	−0.717691
$220$	0	0
$221$	5.14402	0.346025
$222$	0	0
$223$	−1.64611	−0.110231	−0.0551157	−	0.998480i	$-0.517553\pi$
$223$	−1.64611	−0.110231	−0.0551157	+	0.998480i	$0.517553\pi$
$224$	0	0
$225$	14.0656	0.937710
$226$	0	0
$227$	−24.5188	−1.62737	−0.813685	−	0.581306i	$-0.802542\pi$
$227$	−24.5188	−1.62737	−0.813685	+	0.581306i	$0.802542\pi$
$228$	0	0
$229$	−22.0609	−1.45783	−0.728914	−	0.684605i	$-0.759975\pi$
$229$	−22.0609	−1.45783	−0.728914	+	0.684605i	$0.759975\pi$
$230$	0	0
$231$	−0.777601	−0.0511624
$232$	0	0
$233$	1.53100	0.100299	0.0501494	−	0.998742i	$-0.484030\pi$
$233$	1.53100	0.100299	0.0501494	+	0.998742i	$0.484030\pi$
$234$	0	0
$235$	−34.9314	−2.27867
$236$	0	0
$237$	0.855976	0.0556016
$238$	0	0
$239$	23.3631	1.51123	0.755617	−	0.655014i	$-0.227337\pi$
$239$	23.3631	1.51123	0.755617	+	0.655014i	$0.227337\pi$
$240$	0	0
$241$	17.4563	1.12446	0.562229	−	0.826982i	$-0.309944\pi$
$241$	17.4563	1.12446	0.562229	+	0.826982i	$0.309944\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−27.9247	−1.78405
$246$	0	0
$247$	0.559909	0.0356261
$248$	0	0
$249$	6.67502	0.423012
$250$	0	0
$251$	−2.44480	−0.154314	−0.0771571	−	0.997019i	$-0.524584\pi$
$251$	−2.44480	−0.154314	−0.0771571	+	0.997019i	$0.524584\pi$
$252$	0	0
$253$	−0.777601	−0.0488874
$254$	0	0
$255$	22.4610	1.40656
$256$	0	0
$257$	−20.8032	−1.29767	−0.648834	−	0.760930i	$-0.724743\pi$
$257$	−20.8032	−1.29767	−0.648834	+	0.760930i	$0.724743\pi$
$258$	0	0
$259$	6.44480	0.400460
$260$	0	0
$261$	−3.80651	−0.235617
$262$	0	0
$263$	11.1776	0.689243	0.344622	−	0.938742i	$-0.388007\pi$
$263$	11.1776	0.689243	0.344622	+	0.938742i	$0.388007\pi$
$264$	0	0
$265$	26.1985	1.60936
$266$	0	0
$267$	11.2591	0.689048
$268$	0	0
$269$	−18.9631	−1.15620	−0.578099	−	0.815966i	$-0.696205\pi$
$269$	−18.9631	−1.15620	−0.578099	+	0.815966i	$0.696205\pi$
$270$	0	0
$271$	7.23493	0.439491	0.219745	−	0.975557i	$-0.429477\pi$
$271$	7.23493	0.439491	0.219745	+	0.975557i	$0.429477\pi$
$272$	0	0
$273$	−0.777601	−0.0470626
$274$	0	0
$275$	−14.0656	−0.848191
$276$	0	0
$277$	28.8516	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$277$	28.8516	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$278$	0	0
$279$	−10.5841	−0.633654
$280$	0	0
$281$	−20.4000	−1.21696	−0.608482	−	0.793567i	$-0.708221\pi$
$281$	−20.4000	−1.21696	−0.608482	+	0.793567i	$0.708221\pi$
$282$	0	0
$283$	26.2259	1.55897	0.779483	−	0.626424i	$-0.215482\pi$
$283$	26.2259	1.55897	0.779483	+	0.626424i	$0.215482\pi$
$284$	0	0
$285$	2.44480	0.144817
$286$	0	0
$287$	−8.07977	−0.476934
$288$	0	0
$289$	9.46099	0.556529
$290$	0	0
$291$	8.28805	0.485854
$292$	0	0
$293$	25.8594	1.51072	0.755362	−	0.655307i	$-0.227461\pi$
$293$	25.8594	1.51072	0.755362	+	0.655307i	$0.227461\pi$
$294$	0	0
$295$	0.950539	0.0553425
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−0.777601	−0.0449698
$300$	0	0
$301$	−3.48485	−0.200863
$302$	0	0
$303$	8.75705	0.503080
$304$	0	0
$305$	9.68338	0.554469
$306$	0	0
$307$	−11.1454	−0.636103	−0.318051	−	0.948074i	$-0.603028\pi$
$307$	−11.1454	−0.636103	−0.318051	+	0.948074i	$0.603028\pi$
$308$	0	0
$309$	−16.7938	−0.955365
$310$	0	0
$311$	26.7584	1.51733	0.758666	−	0.651480i	$-0.225851\pi$
$311$	26.7584	1.51733	0.758666	+	0.651480i	$0.225851\pi$
$312$	0	0
$313$	2.18547	0.123530	0.0617649	−	0.998091i	$-0.480327\pi$
$313$	2.18547	0.123530	0.0617649	+	0.998091i	$0.480327\pi$
$314$	0	0
$315$	−3.39534	−0.191306
$316$	0	0
$317$	32.2769	1.81285	0.906426	−	0.422365i	$-0.138800\pi$
$317$	32.2769	1.81285	0.906426	+	0.422365i	$0.138800\pi$
$318$	0	0
$319$	3.80651	0.213124
$320$	0	0
$321$	14.8032	0.826234
$322$	0	0
$323$	2.88018	0.160258
$324$	0	0
$325$	−14.0656	−0.780222
$326$	0	0
$327$	18.0162	0.996298
$328$	0	0
$329$	6.22081	0.342964
$330$	0	0
$331$	4.94915	0.272030	0.136015	−	0.990707i	$-0.456571\pi$
$331$	4.94915	0.272030	0.136015	+	0.990707i	$0.456571\pi$
$332$	0	0
$333$	−8.28805	−0.454182
$334$	0	0
$335$	−48.4641	−2.64788
$336$	0	0
$337$	−7.39845	−0.403019	−0.201510	−	0.979487i	$-0.564585\pi$
$337$	−7.39845	−0.403019	−0.201510	+	0.979487i	$0.564585\pi$
$338$	0	0
$339$	6.95054	0.377501
$340$	0	0
$341$	10.5841	0.573162
$342$	0	0
$343$	10.4162	0.562424
$344$	0	0
$345$	−3.39534	−0.182799
$346$	0	0
$347$	26.8157	1.43954	0.719772	−	0.694211i	$-0.244246\pi$
$347$	26.8157	1.43954	0.719772	+	0.694211i	$0.244246\pi$
$348$	0	0
$349$	−12.4448	−0.666155	−0.333077	−	0.942900i	$-0.608087\pi$
$349$	−12.4448	−0.666155	−0.333077	+	0.942900i	$0.608087\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−3.18566	−0.169556	−0.0847778	−	0.996400i	$-0.527018\pi$
$353$	−3.18566	−0.169556	−0.0847778	+	0.996400i	$0.527018\pi$
$354$	0	0
$355$	−55.5970	−2.95078
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−7.36311	−0.388610	−0.194305	−	0.980941i	$-0.562245\pi$
$359$	−7.36311	−0.388610	−0.194305	+	0.980941i	$0.562245\pi$
$360$	0	0
$361$	−18.6865	−0.983500
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	46.3751	2.42739
$366$	0	0
$367$	−28.5761	−1.49166	−0.745830	−	0.666136i	$-0.767947\pi$
$367$	−28.5761	−1.49166	−0.745830	+	0.666136i	$0.767947\pi$
$368$	0	0
$369$	10.3906	0.540915
$370$	0	0
$371$	−4.66561	−0.242226
$372$	0	0
$373$	−12.8927	−0.667559	−0.333780	−	0.942651i	$-0.608324\pi$
$373$	−12.8927	−0.667559	−0.333780	+	0.942651i	$0.608324\pi$
$374$	0	0
$375$	−39.5845	−2.04413
$376$	0	0
$377$	3.80651	0.196045
$378$	0	0
$379$	−28.9394	−1.48652	−0.743259	−	0.669004i	$-0.766721\pi$
$379$	−28.9394	−1.48652	−0.743259	+	0.669004i	$0.766721\pi$
$380$	0	0
$381$	−7.44148	−0.381239
$382$	0	0
$383$	−25.0787	−1.28146	−0.640731	−	0.767765i	$-0.721369\pi$
$383$	−25.0787	−1.28146	−0.640731	+	0.767765i	$0.721369\pi$
$384$	0	0
$385$	3.39534	0.173042
$386$	0	0
$387$	4.48153	0.227809
$388$	0	0
$389$	11.1682	0.566252	0.283126	−	0.959083i	$-0.408629\pi$
$389$	11.1682	0.566252	0.283126	+	0.959083i	$0.408629\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−20.7454	−1.04647
$394$	0	0
$395$	−3.73755	−0.188057
$396$	0	0
$397$	−11.5427	−0.579310	−0.289655	−	0.957131i	$-0.593541\pi$
$397$	−11.5427	−0.579310	−0.289655	+	0.957131i	$0.593541\pi$
$398$	0	0
$399$	−0.435386	−0.0217965
$400$	0	0
$401$	−15.0773	−0.752926	−0.376463	−	0.926432i	$-0.622860\pi$
$401$	−15.0773	−0.752926	−0.376463	+	0.926432i	$0.622860\pi$
$402$	0	0
$403$	10.5841	0.527232
$404$	0	0
$405$	4.36642	0.216969
$406$	0	0
$407$	8.28805	0.410823
$408$	0	0
$409$	33.4115	1.65209	0.826047	−	0.563602i	$-0.190585\pi$
$409$	33.4115	1.65209	0.826047	+	0.563602i	$0.190585\pi$
$410$	0	0
$411$	6.59353	0.325235
$412$	0	0
$413$	−0.169278	−0.00832964
$414$	0	0
$415$	−29.1460	−1.43072
$416$	0	0
$417$	−6.75705	−0.330894
$418$	0	0
$419$	27.0531	1.32163	0.660816	−	0.750548i	$-0.270210\pi$
$419$	27.0531	1.32163	0.660816	+	0.750548i	$0.270210\pi$
$420$	0	0
$421$	−10.0609	−0.490340	−0.245170	−	0.969480i	$-0.578844\pi$
$421$	−10.0609	−0.490340	−0.245170	+	0.969480i	$0.578844\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−72.3540	−3.50969
$426$	0	0
$427$	−1.72448	−0.0834535
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−20.9220	−1.00778	−0.503888	−	0.863769i	$-0.668098\pi$
$431$	−20.9220	−1.00778	−0.503888	+	0.863769i	$0.668098\pi$
$432$	0	0
$433$	−5.83072	−0.280207	−0.140103	−	0.990137i	$-0.544743\pi$
$433$	−5.83072	−0.280207	−0.140103	+	0.990137i	$0.544743\pi$
$434$	0	0
$435$	16.6209	0.796909
$436$	0	0
$437$	−0.435386	−0.0208273
$438$	0	0
$439$	−18.6898	−0.892016	−0.446008	−	0.895029i	$-0.647155\pi$
$439$	−18.6898	−0.892016	−0.446008	+	0.895029i	$0.647155\pi$
$440$	0	0
$441$	−6.39534	−0.304540
$442$	0	0
$443$	19.2261	0.913458	0.456729	−	0.889606i	$-0.349021\pi$
$443$	19.2261	0.913458	0.456729	+	0.889606i	$0.349021\pi$
$444$	0	0
$445$	−49.1621	−2.33051
$446$	0	0
$447$	14.6083	0.690950
$448$	0	0
$449$	−17.8446	−0.842141	−0.421070	−	0.907028i	$-0.638345\pi$
$449$	−17.8446	−0.842141	−0.421070	+	0.907028i	$0.638345\pi$
$450$	0	0
$451$	−10.3906	−0.489276
$452$	0	0
$453$	4.99529	0.234699
$454$	0	0
$455$	3.39534	0.159176
$456$	0	0
$457$	−16.7360	−0.782875	−0.391438	−	0.920205i	$-0.628022\pi$
$457$	−16.7360	−0.782875	−0.391438	+	0.920205i	$0.628022\pi$
$458$	0	0
$459$	5.14402	0.240102
$460$	0	0
$461$	6.24641	0.290924	0.145462	−	0.989364i	$-0.453533\pi$
$461$	6.24641	0.290924	0.145462	+	0.989364i	$0.453533\pi$
$462$	0	0
$463$	−32.5524	−1.51284	−0.756420	−	0.654086i	$-0.773053\pi$
$463$	−32.5524	−1.51284	−0.756420	+	0.654086i	$0.773053\pi$
$464$	0	0
$465$	46.2147	2.14316
$466$	0	0
$467$	−8.71666	−0.403359	−0.201679	−	0.979452i	$-0.564640\pi$
$467$	−8.71666	−0.403359	−0.201679	+	0.979452i	$0.564640\pi$
$468$	0	0
$469$	8.63081	0.398534
$470$	0	0
$471$	−18.4704	−0.851071
$472$	0	0
$473$	−4.48153	−0.206061
$474$	0	0
$475$	−7.87548	−0.361352
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−28.3495	−1.29532	−0.647662	−	0.761928i	$-0.724253\pi$
$479$	−28.3495	−1.29532	−0.647662	+	0.761928i	$0.724253\pi$
$480$	0	0
$481$	8.28805	0.377902
$482$	0	0
$483$	0.604664	0.0275132
$484$	0	0
$485$	−36.1891	−1.64326
$486$	0	0
$487$	−7.01950	−0.318084	−0.159042	−	0.987272i	$-0.550841\pi$
$487$	−7.01950	−0.318084	−0.159042	+	0.987272i	$0.550841\pi$
$488$	0	0
$489$	9.24661	0.418146
$490$	0	0
$491$	−19.4204	−0.876430	−0.438215	−	0.898870i	$-0.644389\pi$
$491$	−19.4204	−0.876430	−0.438215	+	0.898870i	$0.644389\pi$
$492$	0	0
$493$	19.5808	0.881875
$494$	0	0
$495$	−4.36642	−0.196256
$496$	0	0
$497$	9.90108	0.444124
$498$	0	0
$499$	−9.29502	−0.416102	−0.208051	−	0.978118i	$-0.566712\pi$
$499$	−9.29502	−0.416102	−0.208051	+	0.978118i	$0.566712\pi$
$500$	0	0
$501$	−6.27552	−0.280370
$502$	0	0
$503$	1.71195	0.0763321	0.0381661	−	0.999271i	$-0.487848\pi$
$503$	1.71195	0.0763321	0.0381661	+	0.999271i	$0.487848\pi$
$504$	0	0
$505$	−38.2370	−1.70152
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−18.2961	−0.810959	−0.405479	−	0.914104i	$-0.632895\pi$
$509$	−18.2961	−0.810959	−0.405479	+	0.914104i	$0.632895\pi$
$510$	0	0
$511$	−8.25879	−0.365347
$512$	0	0
$513$	0.559909	0.0247206
$514$	0	0
$515$	73.3288	3.23125
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	18.8274	0.826432
$520$	0	0
$521$	−0.0452970	−0.00198450	−0.000992248	−	1.00000i	$-0.500316\pi$
$521$	−0.0452970	−0.00198450	−0.000992248		1.00000i	$0.500316\pi$
$522$	0	0
$523$	30.3951	1.32909	0.664543	−	0.747250i	$-0.268626\pi$
$523$	30.3951	1.32909	0.664543	+	0.747250i	$0.268626\pi$
$524$	0	0
$525$	10.9375	0.477350
$526$	0	0
$527$	54.4449	2.37166
$528$	0	0
$529$	−22.3953	−0.973710
$530$	0	0
$531$	0.217693	0.00944706
$532$	0	0
$533$	−10.3906	−0.450068
$534$	0	0
$535$	−64.6370	−2.79450
$536$	0	0
$537$	−10.1823	−0.439401
$538$	0	0
$539$	6.39534	0.275467
$540$	0	0
$541$	−0.649961	−0.0279440	−0.0139720	−	0.999902i	$-0.504448\pi$
$541$	−0.649961	−0.0279440	−0.0139720	+	0.999902i	$0.504448\pi$
$542$	0	0
$543$	9.58080	0.411152
$544$	0	0
$545$	−78.6663	−3.36969
$546$	0	0
$547$	−22.1524	−0.947168	−0.473584	−	0.880749i	$-0.657040\pi$
$547$	−22.1524	−0.947168	−0.473584	+	0.880749i	$0.657040\pi$
$548$	0	0
$549$	2.21769	0.0946488
$550$	0	0
$551$	2.13130	0.0907964
$552$	0	0
$553$	0.665608	0.0283045
$554$	0	0
$555$	36.1891	1.53614
$556$	0	0
$557$	28.7584	1.21853	0.609267	−	0.792965i	$-0.291464\pi$
$557$	28.7584	1.21853	0.609267	+	0.792965i	$0.291464\pi$
$558$	0	0
$559$	−4.48153	−0.189549
$560$	0	0
$561$	−5.14402	−0.217181
$562$	0	0
$563$	24.8063	1.04546	0.522731	−	0.852498i	$-0.324913\pi$
$563$	24.8063	1.04546	0.522731	+	0.852498i	$0.324913\pi$
$564$	0	0
$565$	−30.3490	−1.27679
$566$	0	0
$567$	−0.777601	−0.0326562
$568$	0	0
$569$	12.1649	0.509980	0.254990	−	0.966944i	$-0.417928\pi$
$569$	12.1649	0.509980	0.254990	+	0.966944i	$0.417928\pi$
$570$	0	0
$571$	−37.8577	−1.58430	−0.792148	−	0.610329i	$-0.791037\pi$
$571$	−37.8577	−1.58430	−0.792148	+	0.610329i	$0.791037\pi$
$572$	0	0
$573$	−11.6256	−0.485665
$574$	0	0
$575$	10.9375	0.456124
$576$	0	0
$577$	−10.9147	−0.454383	−0.227191	−	0.973850i	$-0.572954\pi$
$577$	−10.9147	−0.454383	−0.227191	+	0.973850i	$0.572954\pi$
$578$	0	0
$579$	−19.5714	−0.813359
$580$	0	0
$581$	5.19050	0.215338
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−4.36642	−0.180529
$586$	0	0
$587$	32.5886	1.34508	0.672538	−	0.740062i	$-0.265204\pi$
$587$	32.5886	1.34508	0.672538	+	0.740062i	$0.265204\pi$
$588$	0	0
$589$	5.92614	0.244182
$590$	0	0
$591$	4.61773	0.189948
$592$	0	0
$593$	−9.05312	−0.371767	−0.185884	−	0.982572i	$-0.559515\pi$
$593$	−9.05312	−0.371767	−0.185884	+	0.982572i	$0.559515\pi$
$594$	0	0
$595$	17.4657	0.716024
$596$	0	0
$597$	12.2177	0.500037
$598$	0	0
$599$	22.9923	0.939441	0.469721	−	0.882815i	$-0.344355\pi$
$599$	22.9923	0.939441	0.469721	+	0.882815i	$0.344355\pi$
$600$	0	0
$601$	−37.9105	−1.54640	−0.773201	−	0.634162i	$-0.781345\pi$
$601$	−37.9105	−1.54640	−0.773201	+	0.634162i	$0.781345\pi$
$602$	0	0
$603$	−11.0993	−0.451997
$604$	0	0
$605$	4.36642	0.177520
$606$	0	0
$607$	−9.15344	−0.371527	−0.185763	−	0.982595i	$-0.559476\pi$
$607$	−9.15344	−0.371527	−0.185763	+	0.982595i	$0.559476\pi$
$608$	0	0
$609$	−2.95995	−0.119943
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−17.8048	−0.719129	−0.359564	−	0.933120i	$-0.617075\pi$
$613$	−17.8048	−0.719129	−0.359564	+	0.933120i	$0.617075\pi$
$614$	0	0
$615$	−45.3699	−1.82949
$616$	0	0
$617$	21.7523	0.875717	0.437858	−	0.899044i	$-0.355737\pi$
$617$	21.7523	0.875717	0.437858	+	0.899044i	$0.355737\pi$
$618$	0	0
$619$	33.5280	1.34761	0.673803	−	0.738911i	$-0.264660\pi$
$619$	33.5280	1.34761	0.673803	+	0.738911i	$0.264660\pi$
$620$	0	0
$621$	−0.777601	−0.0312041
$622$	0	0
$623$	8.75512	0.350766
$624$	0	0
$625$	102.514	4.10057
$626$	0	0
$627$	−0.559909	−0.0223606
$628$	0	0
$629$	42.6339	1.69993
$630$	0	0
$631$	42.8217	1.70470	0.852352	−	0.522969i	$-0.175176\pi$
$631$	42.8217	1.70470	0.852352	+	0.522969i	$0.175176\pi$
$632$	0	0
$633$	−10.1649	−0.404019
$634$	0	0
$635$	32.4927	1.28943
$636$	0	0
$637$	6.39534	0.253392
$638$	0	0
$639$	−12.7328	−0.503704
$640$	0	0
$641$	−10.3995	−0.410756	−0.205378	−	0.978683i	$-0.565842\pi$
$641$	−10.3995	−0.410756	−0.205378	+	0.978683i	$0.565842\pi$
$642$	0	0
$643$	−23.7523	−0.936701	−0.468351	−	0.883543i	$-0.655152\pi$
$643$	−23.7523	−0.936701	−0.468351	+	0.883543i	$0.655152\pi$
$644$	0	0
$645$	−19.5683	−0.770500
$646$	0	0
$647$	−12.2302	−0.480820	−0.240410	−	0.970671i	$-0.577282\pi$
$647$	−12.2302	−0.480820	−0.240410	+	0.970671i	$0.577282\pi$
$648$	0	0
$649$	−0.217693	−0.00854519
$650$	0	0
$651$	−8.23022	−0.322568
$652$	0	0
$653$	13.4079	0.524690	0.262345	−	0.964974i	$-0.415504\pi$
$653$	13.4079	0.524690	0.262345	+	0.964974i	$0.415504\pi$
$654$	0	0
$655$	90.5831	3.53937
$656$	0	0
$657$	10.6209	0.414359
$658$	0	0
$659$	23.8370	0.928558	0.464279	−	0.885689i	$-0.346313\pi$
$659$	23.8370	0.928558	0.464279	+	0.885689i	$0.346313\pi$
$660$	0	0
$661$	−11.4173	−0.444081	−0.222040	−	0.975037i	$-0.571272\pi$
$661$	−11.4173	−0.444081	−0.222040	+	0.975037i	$0.571272\pi$
$662$	0	0
$663$	−5.14402	−0.199777
$664$	0	0
$665$	1.90108	0.0737206
$666$	0	0
$667$	−2.95995	−0.114610
$668$	0	0
$669$	1.64611	0.0636422
$670$	0	0
$671$	−2.21769	−0.0856131
$672$	0	0
$673$	−0.656196	−0.0252945	−0.0126472	−	0.999920i	$-0.504026\pi$
$673$	−0.656196	−0.0252945	−0.0126472	+	0.999920i	$0.504026\pi$
$674$	0	0
$675$	−14.0656	−0.541387
$676$	0	0
$677$	18.4018	0.707237	0.353619	−	0.935390i	$-0.384951\pi$
$677$	18.4018	0.707237	0.353619	+	0.935390i	$0.384951\pi$
$678$	0	0
$679$	6.44480	0.247329
$680$	0	0
$681$	24.5188	0.939563
$682$	0	0
$683$	17.5677	0.672211	0.336105	−	0.941824i	$-0.390890\pi$
$683$	17.5677	0.672211	0.336105	+	0.941824i	$0.390890\pi$
$684$	0	0
$685$	−28.7901	−1.10001
$686$	0	0
$687$	22.0609	0.841678
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−38.6837	−1.47160	−0.735799	−	0.677200i	$-0.763193\pi$
$691$	−38.6837	−1.47160	−0.735799	+	0.677200i	$0.763193\pi$
$692$	0	0
$693$	0.777601	0.0295386
$694$	0	0
$695$	29.5042	1.11916
$696$	0	0
$697$	−53.4497	−2.02455
$698$	0	0
$699$	−1.53100	−0.0579076
$700$	0	0
$701$	−26.1430	−0.987407	−0.493703	−	0.869630i	$-0.664357\pi$
$701$	−26.1430	−0.987407	−0.493703	+	0.869630i	$0.664357\pi$
$702$	0	0
$703$	4.64055	0.175022
$704$	0	0
$705$	34.9314	1.31559
$706$	0	0
$707$	6.80950	0.256097
$708$	0	0
$709$	4.73596	0.177863	0.0889314	−	0.996038i	$-0.471655\pi$
$709$	4.73596	0.177863	0.0889314	+	0.996038i	$0.471655\pi$
$710$	0	0
$711$	−0.855976	−0.0321016
$712$	0	0
$713$	−8.23022	−0.308224
$714$	0	0
$715$	4.36642	0.163295
$716$	0	0
$717$	−23.3631	−0.872511
$718$	0	0
$719$	−32.1121	−1.19758	−0.598790	−	0.800906i	$-0.704352\pi$
$719$	−32.1121	−1.19758	−0.598790	+	0.800906i	$0.704352\pi$
$720$	0	0
$721$	−13.0589	−0.486338
$722$	0	0
$723$	−17.4563	−0.649206
$724$	0	0
$725$	−53.5411	−1.98847
$726$	0	0
$727$	9.71195	0.360196	0.180098	−	0.983649i	$-0.442358\pi$
$727$	9.71195	0.360196	0.180098	+	0.983649i	$0.442358\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.0531	−0.852650
$732$	0	0
$733$	−23.6870	−0.874901	−0.437450	−	0.899243i	$-0.644118\pi$
$733$	−23.6870	−0.874901	−0.437450	+	0.899243i	$0.644118\pi$
$734$	0	0
$735$	27.9247	1.03002
$736$	0	0
$737$	11.0993	0.408847
$738$	0	0
$739$	−11.5136	−0.423533	−0.211767	−	0.977320i	$-0.567922\pi$
$739$	−11.5136	−0.423533	−0.211767	+	0.977320i	$0.567922\pi$
$740$	0	0
$741$	−0.559909	−0.0205688
$742$	0	0
$743$	16.5152	0.605882	0.302941	−	0.953009i	$-0.402031\pi$
$743$	16.5152	0.605882	0.302941	+	0.953009i	$0.402031\pi$
$744$	0	0
$745$	−63.7861	−2.33694
$746$	0	0
$747$	−6.67502	−0.244226
$748$	0	0
$749$	11.5110	0.420602
$750$	0	0
$751$	−17.6350	−0.643509	−0.321755	−	0.946823i	$-0.604273\pi$
$751$	−17.6350	−0.643509	−0.321755	+	0.946823i	$0.604273\pi$
$752$	0	0
$753$	2.44480	0.0890934
$754$	0	0
$755$	−21.8116	−0.793804
$756$	0	0
$757$	41.5620	1.51060	0.755298	−	0.655382i	$-0.227492\pi$
$757$	41.5620	1.51060	0.755298	+	0.655382i	$0.227492\pi$
$758$	0	0
$759$	0.777601	0.0282251
$760$	0	0
$761$	−40.3840	−1.46392	−0.731960	−	0.681348i	$-0.761394\pi$
$761$	−40.3840	−1.46392	−0.731960	+	0.681348i	$0.761394\pi$
$762$	0	0
$763$	14.0094	0.507175
$764$	0	0
$765$	−22.4610	−0.812079
$766$	0	0
$767$	−0.217693	−0.00786043
$768$	0	0
$769$	−14.8673	−0.536127	−0.268064	−	0.963401i	$-0.586384\pi$
$769$	−14.8673	−0.536127	−0.268064	+	0.963401i	$0.586384\pi$
$770$	0	0
$771$	20.8032	0.749209
$772$	0	0
$773$	−24.9300	−0.896669	−0.448335	−	0.893866i	$-0.647983\pi$
$773$	−24.9300	−0.896669	−0.448335	+	0.893866i	$0.647983\pi$
$774$	0	0
$775$	−148.872	−5.34766
$776$	0	0
$777$	−6.44480	−0.231206
$778$	0	0
$779$	−5.81780	−0.208444
$780$	0	0
$781$	12.7328	0.455617
$782$	0	0
$783$	3.80651	0.136034
$784$	0	0
$785$	80.6496	2.87851
$786$	0	0
$787$	40.2886	1.43613	0.718067	−	0.695974i	$-0.245027\pi$
$787$	40.2886	1.43613	0.718067	+	0.695974i	$0.245027\pi$
$788$	0	0
$789$	−11.1776	−0.397935
$790$	0	0
$791$	5.40475	0.192171
$792$	0	0
$793$	−2.21769	−0.0787526
$794$	0	0
$795$	−26.1985	−0.929166
$796$	0	0
$797$	−35.5203	−1.25820	−0.629098	−	0.777326i	$-0.716575\pi$
$797$	−35.5203	−1.25820	−0.629098	+	0.777326i	$0.716575\pi$
$798$	0	0
$799$	41.1522	1.45586
$800$	0	0
$801$	−11.2591	−0.397822
$802$	0	0
$803$	−10.6209	−0.374802
$804$	0	0
$805$	−2.64022	−0.0930555
$806$	0	0
$807$	18.9631	0.667532
$808$	0	0
$809$	−11.8351	−0.416099	−0.208050	−	0.978118i	$-0.566712\pi$
$809$	−11.8351	−0.416099	−0.208050	+	0.978118i	$0.566712\pi$
$810$	0	0
$811$	28.7974	1.01122	0.505608	−	0.862764i	$-0.331268\pi$
$811$	28.7974	1.01122	0.505608	+	0.862764i	$0.331268\pi$
$812$	0	0
$813$	−7.23493	−0.253740
$814$	0	0
$815$	−40.3746	−1.41426
$816$	0	0
$817$	−2.50925	−0.0877875
$818$	0	0
$819$	0.777601	0.0271716
$820$	0	0
$821$	16.1475	0.563551	0.281776	−	0.959480i	$-0.409077\pi$
$821$	16.1475	0.563551	0.281776	+	0.959480i	$0.409077\pi$
$822$	0	0
$823$	−14.8677	−0.518254	−0.259127	−	0.965843i	$-0.583435\pi$
$823$	−14.8677	−0.518254	−0.259127	+	0.965843i	$0.583435\pi$
$824$	0	0
$825$	14.0656	0.489703
$826$	0	0
$827$	21.5230	0.748427	0.374214	−	0.927343i	$-0.377913\pi$
$827$	21.5230	0.748427	0.374214	+	0.927343i	$0.377913\pi$
$828$	0	0
$829$	17.1104	0.594269	0.297134	−	0.954836i	$-0.403969\pi$
$829$	17.1104	0.594269	0.297134	+	0.954836i	$0.403969\pi$
$830$	0	0
$831$	−28.8516	−1.00085
$832$	0	0
$833$	32.8978	1.13984
$834$	0	0
$835$	27.4016	0.948270
$836$	0	0
$837$	10.5841	0.365841
$838$	0	0
$839$	27.2511	0.940813	0.470407	−	0.882450i	$-0.344107\pi$
$839$	27.2511	0.940813	0.470407	+	0.882450i	$0.344107\pi$
$840$	0	0
$841$	−14.5104	−0.500360
$842$	0	0
$843$	20.4000	0.702615
$844$	0	0
$845$	4.36642	0.150210
$846$	0	0
$847$	−0.777601	−0.0267187
$848$	0	0
$849$	−26.2259	−0.900069
$850$	0	0
$851$	−6.44480	−0.220925
$852$	0	0
$853$	38.1731	1.30702	0.653511	−	0.756917i	$-0.273295\pi$
$853$	38.1731	1.30702	0.653511	+	0.756917i	$0.273295\pi$
$854$	0	0
$855$	−2.44480	−0.0836103
$856$	0	0
$857$	−54.2478	−1.85307	−0.926535	−	0.376209i	$-0.877227\pi$
$857$	−54.2478	−1.85307	−0.926535	+	0.376209i	$0.877227\pi$
$858$	0	0
$859$	−22.2692	−0.759816	−0.379908	−	0.925024i	$-0.624044\pi$
$859$	−22.2692	−0.759816	−0.379908	+	0.925024i	$0.624044\pi$
$860$	0	0
$861$	8.07977	0.275358
$862$	0	0
$863$	3.93276	0.133873	0.0669364	−	0.997757i	$-0.478678\pi$
$863$	3.93276	0.133873	0.0669364	+	0.997757i	$0.478678\pi$
$864$	0	0
$865$	−82.2084	−2.79517
$866$	0	0
$867$	−9.46099	−0.321312
$868$	0	0
$869$	0.855976	0.0290370
$870$	0	0
$871$	11.0993	0.376084
$872$	0	0
$873$	−8.28805	−0.280508
$874$	0	0
$875$	−30.7809	−1.04059
$876$	0	0
$877$	13.4312	0.453540	0.226770	−	0.973948i	$-0.427183\pi$
$877$	13.4312	0.453540	0.226770	+	0.973948i	$0.427183\pi$
$878$	0	0
$879$	−25.8594	−0.872217
$880$	0	0
$881$	20.5057	0.690856	0.345428	−	0.938445i	$-0.387734\pi$
$881$	20.5057	0.690856	0.345428	+	0.938445i	$0.387734\pi$
$882$	0	0
$883$	−28.6496	−0.964135	−0.482067	−	0.876134i	$-0.660114\pi$
$883$	−28.6496	−0.964135	−0.482067	+	0.876134i	$0.660114\pi$
$884$	0	0
$885$	−0.950539	−0.0319520
$886$	0	0
$887$	−48.8547	−1.64038	−0.820191	−	0.572090i	$-0.806133\pi$
$887$	−48.8547	−1.64038	−0.820191	+	0.572090i	$0.806133\pi$
$888$	0	0
$889$	−5.78651	−0.194073
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	4.47927	0.149893
$894$	0	0
$895$	44.4604	1.48615
$896$	0	0
$897$	0.777601	0.0259634
$898$	0	0
$899$	40.2886	1.34370
$900$	0	0
$901$	−30.8641	−1.02823
$902$	0	0
$903$	3.48485	0.115968
$904$	0	0
$905$	−41.8338	−1.39060
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−8.75705	−0.290453
$910$	0	0
$911$	12.3647	0.409661	0.204830	−	0.978797i	$-0.434336\pi$
$911$	12.3647	0.409661	0.204830	+	0.978797i	$0.434336\pi$
$912$	0	0
$913$	6.67502	0.220911
$914$	0	0
$915$	−9.68338	−0.320123
$916$	0	0
$917$	−16.1316	−0.532713
$918$	0	0
$919$	−25.5244	−0.841971	−0.420986	−	0.907067i	$-0.638316\pi$
$919$	−25.5244	−0.841971	−0.420986	+	0.907067i	$0.638316\pi$
$920$	0	0
$921$	11.1454	0.367254
$922$	0	0
$923$	12.7328	0.419107
$924$	0	0
$925$	−116.577	−3.83302
$926$	0	0
$927$	16.7938	0.551580
$928$	0	0
$929$	−14.7154	−0.482797	−0.241399	−	0.970426i	$-0.577606\pi$
$929$	−14.7154	−0.482797	−0.241399	+	0.970426i	$0.577606\pi$
$930$	0	0
$931$	3.58080	0.117356
$932$	0	0
$933$	−26.7584	−0.876032
$934$	0	0
$935$	22.4610	0.734553
$936$	0	0
$937$	26.4542	0.864221	0.432111	−	0.901821i	$-0.357769\pi$
$937$	26.4542	0.864221	0.432111	+	0.901821i	$0.357769\pi$
$938$	0	0
$939$	−2.18547	−0.0713200
$940$	0	0
$941$	7.56516	0.246617	0.123309	−	0.992368i	$-0.460650\pi$
$941$	7.56516	0.246617	0.123309	+	0.992368i	$0.460650\pi$
$942$	0	0
$943$	8.07977	0.263113
$944$	0	0
$945$	3.39534	0.110450
$946$	0	0
$947$	45.2355	1.46996	0.734978	−	0.678091i	$-0.237193\pi$
$947$	45.2355	1.46996	0.734978	+	0.678091i	$0.237193\pi$
$948$	0	0
$949$	−10.6209	−0.344767
$950$	0	0
$951$	−32.2769	−1.04665
$952$	0	0
$953$	23.4805	0.760608	0.380304	−	0.924862i	$-0.375819\pi$
$953$	23.4805	0.760608	0.380304	+	0.924862i	$0.375819\pi$
$954$	0	0
$955$	50.7621	1.64262
$956$	0	0
$957$	−3.80651	−0.123047
$958$	0	0
$959$	5.12714	0.165564
$960$	0	0
$961$	81.0235	2.61366
$962$	0	0
$963$	−14.8032	−0.477027
$964$	0	0
$965$	85.4570	2.75096
$966$	0	0
$967$	−9.55886	−0.307392	−0.153696	−	0.988118i	$-0.549118\pi$
$967$	−9.55886	−0.307392	−0.153696	+	0.988118i	$0.549118\pi$
$968$	0	0
$969$	−2.88018	−0.0925248
$970$	0	0
$971$	21.2677	0.682513	0.341256	−	0.939970i	$-0.389148\pi$
$971$	21.2677	0.682513	0.341256	+	0.939970i	$0.389148\pi$
$972$	0	0
$973$	−5.25429	−0.168445
$974$	0	0
$975$	14.0656	0.450461
$976$	0	0
$977$	−26.4180	−0.845185	−0.422593	−	0.906320i	$-0.638880\pi$
$977$	−26.4180	−0.845185	−0.422593	+	0.906320i	$0.638880\pi$
$978$	0	0
$979$	11.2591	0.359844
$980$	0	0
$981$	−18.0162	−0.575213
$982$	0	0
$983$	17.6064	0.561557	0.280779	−	0.959773i	$-0.409407\pi$
$983$	17.6064	0.561557	0.280779	+	0.959773i	$0.409407\pi$
$984$	0	0
$985$	−20.1630	−0.642446
$986$	0	0
$987$	−6.22081	−0.198011
$988$	0	0
$989$	3.48485	0.110812
$990$	0	0
$991$	1.31869	0.0418894	0.0209447	−	0.999781i	$-0.493333\pi$
$991$	1.31869	0.0418894	0.0209447	+	0.999781i	$0.493333\pi$
$992$	0	0
$993$	−4.94915	−0.157056
$994$	0	0
$995$	−53.3476	−1.69123
$996$	0	0
$997$	−49.3793	−1.56386	−0.781929	−	0.623367i	$-0.785764\pi$
$997$	−49.3793	−1.56386	−0.781929	+	0.623367i	$0.785764\pi$
$998$	0	0
$999$	8.28805	0.262222

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.36642 q^{5} -0.777601 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.36642 q^{5} -0.777601 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.36642 q^{15} -5.14402 q^{17} -0.559909 q^{19} +0.777601 q^{21} +0.777601 q^{23} +14.0656 q^{25} -1.00000 q^{27} -3.80651 q^{29} -10.5841 q^{31} +1.00000 q^{33} -3.39534 q^{35} -8.28805 q^{37} +1.00000 q^{39} +10.3906 q^{41} +4.48153 q^{43} +4.36642 q^{45} -8.00000 q^{47} -6.39534 q^{49} +5.14402 q^{51} +6.00000 q^{53} -4.36642 q^{55} +0.559909 q^{57} +0.217693 q^{59} +2.21769 q^{61} -0.777601 q^{63} -4.36642 q^{65} -11.0993 q^{67} -0.777601 q^{69} -12.7328 q^{71} +10.6209 q^{73} -14.0656 q^{75} +0.777601 q^{77} -0.855976 q^{79} +1.00000 q^{81} -6.67502 q^{83} -22.4610 q^{85} +3.80651 q^{87} -11.2591 q^{89} +0.777601 q^{91} +10.5841 q^{93} -2.44480 q^{95} -8.28805 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 - q^15 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * q^95 + 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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