LMFDB - Embedded newform 6728.2.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.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:	$53.7233504799$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 18 x^{10} + 83 x^{9} + 83 x^{8} - 577 x^{7} + 121 x^{6} + 1416 x^{5} - 1289 x^{4} + \cdots + 64 $ x^12 - 4x^11 - 18x^10 + 83x^9 + 83x^8 - 577x^7 + 121x^6 + 1416x^5 - 1289x^4 - 465x^3 + 1056x^2 - 464*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.777781$ of defining polynomial
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.777781 q^{3} +0.314394 q^{5} -4.74923 q^{7} -2.39506 q^{9} +O(q^{10})$	$q-0.777781 q^{3} +0.314394 q^{5} -4.74923 q^{7} -2.39506 q^{9} -0.169220 q^{11} +0.141399 q^{13} -0.244529 q^{15} +0.833683 q^{17} +6.86919 q^{19} +3.69386 q^{21} +4.40308 q^{23} -4.90116 q^{25} +4.19617 q^{27} +2.81199 q^{31} +0.131616 q^{33} -1.49313 q^{35} -0.0436497 q^{37} -0.109977 q^{39} +7.26282 q^{41} -5.21816 q^{43} -0.752991 q^{45} -8.96047 q^{47} +15.5551 q^{49} -0.648423 q^{51} -6.12892 q^{53} -0.0532016 q^{55} -5.34272 q^{57} +13.7067 q^{59} -11.2647 q^{61} +11.3747 q^{63} +0.0444549 q^{65} +6.84051 q^{67} -3.42463 q^{69} +5.54279 q^{71} +9.86223 q^{73} +3.81203 q^{75} +0.803662 q^{77} -0.408500 q^{79} +3.92147 q^{81} +2.59146 q^{83} +0.262105 q^{85} -8.84300 q^{89} -0.671535 q^{91} -2.18711 q^{93} +2.15963 q^{95} +10.7025 q^{97} +0.405291 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * 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
$3$	−0.777781	−0.449052	−0.224526	−	0.974468i	$-0.572083\pi$
$3$	−0.777781	−0.449052	−0.224526	+	0.974468i	$0.572083\pi$
$4$	0	0
$5$	0.314394	0.140601	0.0703006	−	0.997526i	$-0.477604\pi$
$5$	0.314394	0.140601	0.0703006	+	0.997526i	$0.477604\pi$
$6$	0	0
$7$	−4.74923	−1.79504	−0.897519	−	0.440975i	$-0.854633\pi$
$7$	−4.74923	−1.79504	−0.897519	+	0.440975i	$0.854633\pi$
$8$	0	0
$9$	−2.39506	−0.798352
$10$	0	0
$11$	−0.169220	−0.0510216	−0.0255108	−	0.999675i	$-0.508121\pi$
$11$	−0.169220	−0.0510216	−0.0255108	+	0.999675i	$0.508121\pi$
$12$	0	0
$13$	0.141399	0.0392170	0.0196085	−	0.999808i	$-0.493758\pi$
$13$	0.141399	0.0392170	0.0196085	+	0.999808i	$0.493758\pi$
$14$	0	0
$15$	−0.244529	−0.0631372
$16$	0	0
$17$	0.833683	0.202198	0.101099	−	0.994876i	$-0.467764\pi$
$17$	0.833683	0.202198	0.101099	+	0.994876i	$0.467764\pi$
$18$	0	0
$19$	6.86919	1.57590	0.787950	−	0.615739i	$-0.211143\pi$
$19$	6.86919	1.57590	0.787950	+	0.615739i	$0.211143\pi$
$20$	0	0
$21$	3.69386	0.806066
$22$	0	0
$23$	4.40308	0.918106	0.459053	−	0.888409i	$-0.348189\pi$
$23$	4.40308	0.918106	0.459053	+	0.888409i	$0.348189\pi$
$24$	0	0
$25$	−4.90116	−0.980231
$26$	0	0
$27$	4.19617	0.807554
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	2.81199	0.505048	0.252524	−	0.967591i	$-0.418739\pi$
$31$	2.81199	0.505048	0.252524	+	0.967591i	$0.418739\pi$
$32$	0	0
$33$	0.131616	0.0229114
$34$	0	0
$35$	−1.49313	−0.252384
$36$	0	0
$37$	−0.0436497	−0.00717596	−0.00358798	−	0.999994i	$-0.501142\pi$
$37$	−0.0436497	−0.00717596	−0.00358798	+	0.999994i	$0.501142\pi$
$38$	0	0
$39$	−0.109977	−0.0176105
$40$	0	0
$41$	7.26282	1.13426	0.567131	−	0.823628i	$-0.308053\pi$
$41$	7.26282	1.13426	0.567131	+	0.823628i	$0.308053\pi$
$42$	0	0
$43$	−5.21816	−0.795761	−0.397881	−	0.917437i	$-0.630254\pi$
$43$	−5.21816	−0.795761	−0.397881	+	0.917437i	$0.630254\pi$
$44$	0	0
$45$	−0.752991	−0.112249
$46$	0	0
$47$	−8.96047	−1.30702	−0.653510	−	0.756918i	$-0.726704\pi$
$47$	−8.96047	−1.30702	−0.653510	+	0.756918i	$0.726704\pi$
$48$	0	0
$49$	15.5551	2.22216
$50$	0	0
$51$	−0.648423	−0.0907974
$52$	0	0
$53$	−6.12892	−0.841872	−0.420936	−	0.907090i	$-0.638298\pi$
$53$	−6.12892	−0.841872	−0.420936	+	0.907090i	$0.638298\pi$
$54$	0	0
$55$	−0.0532016	−0.00717370
$56$	0	0
$57$	−5.34272	−0.707661
$58$	0	0
$59$	13.7067	1.78445	0.892227	−	0.451587i	$-0.149142\pi$
$59$	13.7067	1.78445	0.892227	+	0.451587i	$0.149142\pi$
$60$	0	0
$61$	−11.2647	−1.44230	−0.721150	−	0.692779i	$-0.756386\pi$
$61$	−11.2647	−1.44230	−0.721150	+	0.692779i	$0.756386\pi$
$62$	0	0
$63$	11.3747	1.43307
$64$	0	0
$65$	0.0444549	0.00551395
$66$	0	0
$67$	6.84051	0.835702	0.417851	−	0.908516i	$-0.362783\pi$
$67$	6.84051	0.835702	0.417851	+	0.908516i	$0.362783\pi$
$68$	0	0
$69$	−3.42463	−0.412277
$70$	0	0
$71$	5.54279	0.657808	0.328904	−	0.944363i	$-0.393321\pi$
$71$	5.54279	0.657808	0.328904	+	0.944363i	$0.393321\pi$
$72$	0	0
$73$	9.86223	1.15429	0.577143	−	0.816643i	$-0.304167\pi$
$73$	9.86223	1.15429	0.577143	+	0.816643i	$0.304167\pi$
$74$	0	0
$75$	3.81203	0.440175
$76$	0	0
$77$	0.803662	0.0915858
$78$	0	0
$79$	−0.408500	−0.0459599	−0.0229799	−	0.999736i	$-0.507315\pi$
$79$	−0.408500	−0.0459599	−0.0229799	+	0.999736i	$0.507315\pi$
$80$	0	0
$81$	3.92147	0.435718
$82$	0	0
$83$	2.59146	0.284450	0.142225	−	0.989834i	$-0.454574\pi$
$83$	2.59146	0.284450	0.142225	+	0.989834i	$0.454574\pi$
$84$	0	0
$85$	0.262105	0.0284293
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.84300	−0.937356	−0.468678	−	0.883369i	$-0.655270\pi$
$89$	−8.84300	−0.937356	−0.468678	+	0.883369i	$0.655270\pi$
$90$	0	0
$91$	−0.671535	−0.0703960
$92$	0	0
$93$	−2.18711	−0.226793
$94$	0	0
$95$	2.15963	0.221573
$96$	0	0
$97$	10.7025	1.08667	0.543335	−	0.839516i	$-0.317161\pi$
$97$	10.7025	1.08667	0.543335	+	0.839516i	$0.317161\pi$
$98$	0	0
$99$	0.405291	0.0407332
$100$	0	0
$101$	−0.696499	−0.0693042	−0.0346521	−	0.999399i	$-0.511032\pi$
$101$	−0.696499	−0.0693042	−0.0346521	+	0.999399i	$0.511032\pi$
$102$	0	0
$103$	−6.94013	−0.683831	−0.341916	−	0.939731i	$-0.611076\pi$
$103$	−6.94013	−0.683831	−0.341916	+	0.939731i	$0.611076\pi$
$104$	0	0
$105$	1.16133	0.113334
$106$	0	0
$107$	0.319192	0.0308575	0.0154287	−	0.999881i	$-0.495089\pi$
$107$	0.319192	0.0308575	0.0154287	+	0.999881i	$0.495089\pi$
$108$	0	0
$109$	−3.21614	−0.308051	−0.154025	−	0.988067i	$-0.549224\pi$
$109$	−3.21614	−0.308051	−0.154025	+	0.988067i	$0.549224\pi$
$110$	0	0
$111$	0.0339499	0.00322238
$112$	0	0
$113$	−16.6161	−1.56311	−0.781554	−	0.623838i	$-0.785573\pi$
$113$	−16.6161	−1.56311	−0.781554	+	0.623838i	$0.785573\pi$
$114$	0	0
$115$	1.38430	0.129087
$116$	0	0
$117$	−0.338658	−0.0313090
$118$	0	0
$119$	−3.95935	−0.362953
$120$	0	0
$121$	−10.9714	−0.997397
$122$	0	0
$123$	−5.64888	−0.509343
$124$	0	0
$125$	−3.11286	−0.278423
$126$	0	0
$127$	11.1247	0.987159	0.493580	−	0.869701i	$-0.335688\pi$
$127$	11.1247	0.987159	0.493580	+	0.869701i	$0.335688\pi$
$128$	0	0
$129$	4.05858	0.357338
$130$	0	0
$131$	−14.1255	−1.23415	−0.617075	−	0.786904i	$-0.711683\pi$
$131$	−14.1255	−1.23415	−0.617075	+	0.786904i	$0.711683\pi$
$132$	0	0
$133$	−32.6233	−2.82880
$134$	0	0
$135$	1.31925	0.113543
$136$	0	0
$137$	−8.22509	−0.702717	−0.351359	−	0.936241i	$-0.614280\pi$
$137$	−8.22509	−0.702717	−0.351359	+	0.936241i	$0.614280\pi$
$138$	0	0
$139$	−12.4313	−1.05441	−0.527205	−	0.849738i	$-0.676760\pi$
$139$	−12.4313	−1.05441	−0.527205	+	0.849738i	$0.676760\pi$
$140$	0	0
$141$	6.96929	0.586920
$142$	0	0
$143$	−0.0239275	−0.00200092
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.0985	−0.997867
$148$	0	0
$149$	−21.0231	−1.72228	−0.861140	−	0.508369i	$-0.830249\pi$
$149$	−21.0231	−1.72228	−0.861140	+	0.508369i	$0.830249\pi$
$150$	0	0
$151$	15.1802	1.23535	0.617675	−	0.786434i	$-0.288075\pi$
$151$	15.1802	1.23535	0.617675	+	0.786434i	$0.288075\pi$
$152$	0	0
$153$	−1.99672	−0.161425
$154$	0	0
$155$	0.884071	0.0710103
$156$	0	0
$157$	−1.66764	−0.133092	−0.0665460	−	0.997783i	$-0.521198\pi$
$157$	−1.66764	−0.133092	−0.0665460	+	0.997783i	$0.521198\pi$
$158$	0	0
$159$	4.76696	0.378045
$160$	0	0
$161$	−20.9112	−1.64803
$162$	0	0
$163$	−3.02129	−0.236645	−0.118323	−	0.992975i	$-0.537752\pi$
$163$	−3.02129	−0.236645	−0.118323	+	0.992975i	$0.537752\pi$
$164$	0	0
$165$	0.0413792	0.00322137
$166$	0	0
$167$	15.0782	1.16678	0.583392	−	0.812191i	$-0.301725\pi$
$167$	15.0782	1.16678	0.583392	+	0.812191i	$0.301725\pi$
$168$	0	0
$169$	−12.9800	−0.998462
$170$	0	0
$171$	−16.4521	−1.25812
$172$	0	0
$173$	15.0190	1.14187	0.570937	−	0.820994i	$-0.306580\pi$
$173$	15.0190	1.14187	0.570937	+	0.820994i	$0.306580\pi$
$174$	0	0
$175$	23.2767	1.75955
$176$	0	0
$177$	−10.6608	−0.801313
$178$	0	0
$179$	15.8926	1.18787	0.593936	−	0.804513i	$-0.297573\pi$
$179$	15.8926	1.18787	0.593936	+	0.804513i	$0.297573\pi$
$180$	0	0
$181$	−12.0361	−0.894633	−0.447316	−	0.894376i	$-0.647620\pi$
$181$	−12.0361	−0.894633	−0.447316	+	0.894376i	$0.647620\pi$
$182$	0	0
$183$	8.76149	0.647668
$184$	0	0
$185$	−0.0137232	−0.00100895
$186$	0	0
$187$	−0.141076	−0.0103165
$188$	0	0
$189$	−19.9286	−1.44959
$190$	0	0
$191$	−20.5470	−1.48673	−0.743366	−	0.668885i	$-0.766772\pi$
$191$	−20.5470	−1.48673	−0.743366	+	0.668885i	$0.766772\pi$
$192$	0	0
$193$	−22.9871	−1.65464	−0.827322	−	0.561728i	$-0.810137\pi$
$193$	−22.9871	−1.65464	−0.827322	+	0.561728i	$0.810137\pi$
$194$	0	0
$195$	−0.0345762	−0.00247605
$196$	0	0
$197$	−24.0745	−1.71524	−0.857618	−	0.514287i	$-0.828057\pi$
$197$	−24.0745	−1.71524	−0.857618	+	0.514287i	$0.828057\pi$
$198$	0	0
$199$	−3.54964	−0.251628	−0.125814	−	0.992054i	$-0.540154\pi$
$199$	−3.54964	−0.251628	−0.125814	+	0.992054i	$0.540154\pi$
$200$	0	0
$201$	−5.32042	−0.375274
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.28339	0.159479
$206$	0	0
$207$	−10.5456	−0.732972
$208$	0	0
$209$	−1.16240	−0.0804050
$210$	0	0
$211$	5.06611	0.348765	0.174383	−	0.984678i	$-0.444207\pi$
$211$	5.06611	0.348765	0.174383	+	0.984678i	$0.444207\pi$
$212$	0	0
$213$	−4.31107	−0.295390
$214$	0	0
$215$	−1.64056	−0.111885
$216$	0	0
$217$	−13.3548	−0.906580
$218$	0	0
$219$	−7.67065	−0.518335
$220$	0	0
$221$	0.117882	0.00792960
$222$	0	0
$223$	−14.2325	−0.953076	−0.476538	−	0.879154i	$-0.658109\pi$
$223$	−14.2325	−0.953076	−0.476538	+	0.879154i	$0.658109\pi$
$224$	0	0
$225$	11.7385	0.782570
$226$	0	0
$227$	−8.54523	−0.567167	−0.283584	−	0.958948i	$-0.591523\pi$
$227$	−8.54523	−0.567167	−0.283584	+	0.958948i	$0.591523\pi$
$228$	0	0
$229$	24.3983	1.61229	0.806143	−	0.591721i	$-0.201551\pi$
$229$	24.3983	1.61229	0.806143	+	0.591721i	$0.201551\pi$
$230$	0	0
$231$	−0.625073	−0.0411268
$232$	0	0
$233$	−5.75011	−0.376702	−0.188351	−	0.982102i	$-0.560314\pi$
$233$	−5.75011	−0.376702	−0.188351	+	0.982102i	$0.560314\pi$
$234$	0	0
$235$	−2.81712	−0.183768
$236$	0	0
$237$	0.317724	0.0206384
$238$	0	0
$239$	19.8514	1.28408	0.642041	−	0.766670i	$-0.278088\pi$
$239$	19.8514	1.28408	0.642041	+	0.766670i	$0.278088\pi$
$240$	0	0
$241$	29.4347	1.89605	0.948027	−	0.318189i	$-0.103075\pi$
$241$	29.4347	1.89605	0.948027	+	0.318189i	$0.103075\pi$
$242$	0	0
$243$	−15.6386	−1.00321
$244$	0	0
$245$	4.89044	0.312439
$246$	0	0
$247$	0.971296	0.0618021
$248$	0	0
$249$	−2.01559	−0.127733
$250$	0	0
$251$	−2.66312	−0.168095	−0.0840473	−	0.996462i	$-0.526785\pi$
$251$	−2.66312	−0.168095	−0.0840473	+	0.996462i	$0.526785\pi$
$252$	0	0
$253$	−0.745088	−0.0468433
$254$	0	0
$255$	−0.203860	−0.0127662
$256$	0	0
$257$	8.76704	0.546873	0.273437	−	0.961890i	$-0.411840\pi$
$257$	8.76704	0.546873	0.273437	+	0.961890i	$0.411840\pi$
$258$	0	0
$259$	0.207302	0.0128811
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.3378	−1.13075	−0.565377	−	0.824832i	$-0.691269\pi$
$263$	−18.3378	−1.13075	−0.565377	+	0.824832i	$0.691269\pi$
$264$	0	0
$265$	−1.92690	−0.118368
$266$	0	0
$267$	6.87792	0.420922
$268$	0	0
$269$	−16.6056	−1.01246	−0.506231	−	0.862398i	$-0.668962\pi$
$269$	−16.6056	−1.01246	−0.506231	+	0.862398i	$0.668962\pi$
$270$	0	0
$271$	22.7698	1.38316	0.691582	−	0.722298i	$-0.256914\pi$
$271$	22.7698	1.38316	0.691582	+	0.722298i	$0.256914\pi$
$272$	0	0
$273$	0.522307	0.0316115
$274$	0	0
$275$	0.829372	0.0500130
$276$	0	0
$277$	−20.0776	−1.20635	−0.603175	−	0.797609i	$-0.706098\pi$
$277$	−20.0776	−1.20635	−0.603175	+	0.797609i	$0.706098\pi$
$278$	0	0
$279$	−6.73487	−0.403206
$280$	0	0
$281$	24.0322	1.43364	0.716820	−	0.697259i	$-0.245597\pi$
$281$	24.0322	1.43364	0.716820	+	0.697259i	$0.245597\pi$
$282$	0	0
$283$	−13.2595	−0.788194	−0.394097	−	0.919069i	$-0.628943\pi$
$283$	−13.2595	−0.788194	−0.394097	+	0.919069i	$0.628943\pi$
$284$	0	0
$285$	−1.67972	−0.0994980
$286$	0	0
$287$	−34.4928	−2.03604
$288$	0	0
$289$	−16.3050	−0.959116
$290$	0	0
$291$	−8.32417	−0.487971
$292$	0	0
$293$	−21.2734	−1.24281	−0.621403	−	0.783491i	$-0.713437\pi$
$293$	−21.2734	−1.24281	−0.621403	+	0.783491i	$0.713437\pi$
$294$	0	0
$295$	4.30929	0.250896
$296$	0	0
$297$	−0.710075	−0.0412027
$298$	0	0
$299$	0.622591	0.0360053
$300$	0	0
$301$	24.7822	1.42842
$302$	0	0
$303$	0.541724	0.0311212
$304$	0	0
$305$	−3.54156	−0.202789
$306$	0	0
$307$	−31.3569	−1.78963	−0.894817	−	0.446433i	$-0.852694\pi$
$307$	−31.3569	−1.78963	−0.894817	+	0.446433i	$0.852694\pi$
$308$	0	0
$309$	5.39790	0.307076
$310$	0	0
$311$	7.66248	0.434499	0.217250	−	0.976116i	$-0.430291\pi$
$311$	7.66248	0.434499	0.217250	+	0.976116i	$0.430291\pi$
$312$	0	0
$313$	1.02777	0.0580931	0.0290465	−	0.999578i	$-0.490753\pi$
$313$	1.02777	0.0580931	0.0290465	+	0.999578i	$0.490753\pi$
$314$	0	0
$315$	3.57612	0.201492
$316$	0	0
$317$	4.23443	0.237829	0.118915	−	0.992904i	$-0.462059\pi$
$317$	4.23443	0.237829	0.118915	+	0.992904i	$0.462059\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.248261	−0.0138566
$322$	0	0
$323$	5.72673	0.318644
$324$	0	0
$325$	−0.693018	−0.0384417
$326$	0	0
$327$	2.50146	0.138331
$328$	0	0
$329$	42.5553	2.34615
$330$	0	0
$331$	14.3575	0.789158	0.394579	−	0.918862i	$-0.370890\pi$
$331$	14.3575	0.789158	0.394579	+	0.918862i	$0.370890\pi$
$332$	0	0
$333$	0.104543	0.00572895
$334$	0	0
$335$	2.15061	0.117501
$336$	0	0
$337$	1.89981	0.103489	0.0517447	−	0.998660i	$-0.483522\pi$
$337$	1.89981	0.103489	0.0517447	+	0.998660i	$0.483522\pi$
$338$	0	0
$339$	12.9237	0.701917
$340$	0	0
$341$	−0.475844	−0.0257684
$342$	0	0
$343$	−40.6303	−2.19383
$344$	0	0
$345$	−1.07668	−0.0579667
$346$	0	0
$347$	−18.8015	−1.00932	−0.504659	−	0.863319i	$-0.668382\pi$
$347$	−18.8015	−1.00932	−0.504659	+	0.863319i	$0.668382\pi$
$348$	0	0
$349$	27.9360	1.49538	0.747689	−	0.664049i	$-0.231163\pi$
$349$	27.9360	1.49538	0.747689	+	0.664049i	$0.231163\pi$
$350$	0	0
$351$	0.593334	0.0316698
$352$	0	0
$353$	27.2393	1.44980	0.724901	−	0.688853i	$-0.241886\pi$
$353$	27.2393	1.44980	0.724901	+	0.688853i	$0.241886\pi$
$354$	0	0
$355$	1.74262	0.0924885
$356$	0	0
$357$	3.07951	0.162985
$358$	0	0
$359$	−23.5830	−1.24466	−0.622331	−	0.782754i	$-0.713814\pi$
$359$	−23.5830	−1.24466	−0.622331	+	0.782754i	$0.713814\pi$
$360$	0	0
$361$	28.1857	1.48346
$362$	0	0
$363$	8.53332	0.447883
$364$	0	0
$365$	3.10062	0.162294
$366$	0	0
$367$	25.6809	1.34053	0.670266	−	0.742121i	$-0.266180\pi$
$367$	25.6809	1.34053	0.670266	+	0.742121i	$0.266180\pi$
$368$	0	0
$369$	−17.3949	−0.905541
$370$	0	0
$371$	29.1076	1.51119
$372$	0	0
$373$	−32.3234	−1.67364	−0.836821	−	0.547477i	$-0.815589\pi$
$373$	−32.3234	−1.67364	−0.836821	+	0.547477i	$0.815589\pi$
$374$	0	0
$375$	2.42112	0.125026
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.3000	−1.45367	−0.726837	−	0.686810i	$-0.759010\pi$
$379$	−28.3000	−1.45367	−0.726837	+	0.686810i	$0.759010\pi$
$380$	0	0
$381$	−8.65259	−0.443286
$382$	0	0
$383$	−13.9660	−0.713629	−0.356814	−	0.934175i	$-0.616137\pi$
$383$	−13.9660	−0.713629	−0.356814	+	0.934175i	$0.616137\pi$
$384$	0	0
$385$	0.252666	0.0128771
$386$	0	0
$387$	12.4978	0.635298
$388$	0	0
$389$	−15.3730	−0.779440	−0.389720	−	0.920933i	$-0.627428\pi$
$389$	−15.3730	−0.779440	−0.389720	+	0.920933i	$0.627428\pi$
$390$	0	0
$391$	3.67077	0.185639
$392$	0	0
$393$	10.9865	0.554198
$394$	0	0
$395$	−0.128430	−0.00646201
$396$	0	0
$397$	5.04455	0.253179	0.126589	−	0.991955i	$-0.459597\pi$
$397$	5.04455	0.253179	0.126589	+	0.991955i	$0.459597\pi$
$398$	0	0
$399$	25.3738	1.27028
$400$	0	0
$401$	11.2476	0.561679	0.280839	−	0.959755i	$-0.409387\pi$
$401$	11.2476	0.561679	0.280839	+	0.959755i	$0.409387\pi$
$402$	0	0
$403$	0.397612	0.0198065
$404$	0	0
$405$	1.23288	0.0612625
$406$	0	0
$407$	0.00738638	0.000366129 0
$408$	0	0
$409$	−11.7530	−0.581149	−0.290574	−	0.956852i	$-0.593846\pi$
$409$	−11.7530	−0.581149	−0.290574	+	0.956852i	$0.593846\pi$
$410$	0	0
$411$	6.39732	0.315557
$412$	0	0
$413$	−65.0960	−3.20316
$414$	0	0
$415$	0.814738	0.0399939
$416$	0	0
$417$	9.66884	0.473485
$418$	0	0
$419$	10.2603	0.501251	0.250625	−	0.968084i	$-0.419364\pi$
$419$	10.2603	0.501251	0.250625	+	0.968084i	$0.419364\pi$
$420$	0	0
$421$	−12.6119	−0.614669	−0.307334	−	0.951602i	$-0.599437\pi$
$421$	−12.6119	−0.614669	−0.307334	+	0.951602i	$0.599437\pi$
$422$	0	0
$423$	21.4608	1.04346
$424$	0	0
$425$	−4.08601	−0.198201
$426$	0	0
$427$	53.4987	2.58898
$428$	0	0
$429$	0.0186103	0.000898515 0
$430$	0	0
$431$	−33.6589	−1.62129	−0.810645	−	0.585537i	$-0.800884\pi$
$431$	−33.6589	−1.62129	−0.810645	+	0.585537i	$0.800884\pi$
$432$	0	0
$433$	−18.7039	−0.898853	−0.449427	−	0.893317i	$-0.648372\pi$
$433$	−18.7039	−0.898853	−0.449427	+	0.893317i	$0.648372\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.2456	1.44684
$438$	0	0
$439$	13.5776	0.648023	0.324011	−	0.946053i	$-0.394968\pi$
$439$	13.5776	0.648023	0.324011	+	0.946053i	$0.394968\pi$
$440$	0	0
$441$	−37.2554	−1.77407
$442$	0	0
$443$	−2.40600	−0.114312	−0.0571561	−	0.998365i	$-0.518203\pi$
$443$	−2.40600	−0.114312	−0.0571561	+	0.998365i	$0.518203\pi$
$444$	0	0
$445$	−2.78018	−0.131793
$446$	0	0
$447$	16.3514	0.773393
$448$	0	0
$449$	−23.1475	−1.09240	−0.546199	−	0.837656i	$-0.683926\pi$
$449$	−23.1475	−1.09240	−0.546199	+	0.837656i	$0.683926\pi$
$450$	0	0
$451$	−1.22901	−0.0578719
$452$	0	0
$453$	−11.8069	−0.554736
$454$	0	0
$455$	−0.211126	−0.00989776
$456$	0	0
$457$	−23.8639	−1.11631	−0.558153	−	0.829738i	$-0.688490\pi$
$457$	−23.8639	−1.11631	−0.558153	+	0.829738i	$0.688490\pi$
$458$	0	0
$459$	3.49828	0.163286
$460$	0	0
$461$	26.8691	1.25142	0.625709	−	0.780056i	$-0.284810\pi$
$461$	26.8691	1.25142	0.625709	+	0.780056i	$0.284810\pi$
$462$	0	0
$463$	4.03771	0.187648	0.0938242	−	0.995589i	$-0.470091\pi$
$463$	4.03771	0.187648	0.0938242	+	0.995589i	$0.470091\pi$
$464$	0	0
$465$	−0.687614	−0.0318873
$466$	0	0
$467$	7.17273	0.331914	0.165957	−	0.986133i	$-0.446929\pi$
$467$	7.17273	0.331914	0.165957	+	0.986133i	$0.446929\pi$
$468$	0	0
$469$	−32.4871	−1.50012
$470$	0	0
$471$	1.29706	0.0597652
$472$	0	0
$473$	0.883014	0.0406010
$474$	0	0
$475$	−33.6670	−1.54475
$476$	0	0
$477$	14.6791	0.672111
$478$	0	0
$479$	−13.0287	−0.595295	−0.297648	−	0.954676i	$-0.596202\pi$
$479$	−13.0287	−0.595295	−0.297648	+	0.954676i	$0.596202\pi$
$480$	0	0
$481$	−0.00617202	−0.000281420 0
$482$	0	0
$483$	16.2643	0.740053
$484$	0	0
$485$	3.36478	0.152787
$486$	0	0
$487$	8.06465	0.365444	0.182722	−	0.983165i	$-0.441509\pi$
$487$	8.06465	0.365444	0.182722	+	0.983165i	$0.441509\pi$
$488$	0	0
$489$	2.34990	0.106266
$490$	0	0
$491$	42.1695	1.90308	0.951541	−	0.307521i	$-0.0994995\pi$
$491$	42.1695	1.90308	0.951541	+	0.307521i	$0.0994995\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.127421	0.00572714
$496$	0	0
$497$	−26.3239	−1.18079
$498$	0	0
$499$	35.0280	1.56807	0.784034	−	0.620718i	$-0.213159\pi$
$499$	35.0280	1.56807	0.784034	+	0.620718i	$0.213159\pi$
$500$	0	0
$501$	−11.7275	−0.523947
$502$	0	0
$503$	−18.8141	−0.838880	−0.419440	−	0.907783i	$-0.637773\pi$
$503$	−18.8141	−0.838880	−0.419440	+	0.907783i	$0.637773\pi$
$504$	0	0
$505$	−0.218975	−0.00974426
$506$	0	0
$507$	10.0956	0.448361
$508$	0	0
$509$	−26.6043	−1.17921	−0.589607	−	0.807690i	$-0.700717\pi$
$509$	−26.6043	−1.17921	−0.589607	+	0.807690i	$0.700717\pi$
$510$	0	0
$511$	−46.8379	−2.07199
$512$	0	0
$513$	28.8243	1.27262
$514$	0	0
$515$	−2.18193	−0.0961475
$516$	0	0
$517$	1.51629	0.0666863
$518$	0	0
$519$	−11.6815	−0.512761
$520$	0	0
$521$	−22.8061	−0.999154	−0.499577	−	0.866269i	$-0.666511\pi$
$521$	−22.8061	−0.999154	−0.499577	+	0.866269i	$0.666511\pi$
$522$	0	0
$523$	35.0763	1.53378	0.766890	−	0.641778i	$-0.221803\pi$
$523$	35.0763	1.53378	0.766890	+	0.641778i	$0.221803\pi$
$524$	0	0
$525$	−18.1042	−0.790131
$526$	0	0
$527$	2.34431	0.102120
$528$	0	0
$529$	−3.61289	−0.157082
$530$	0	0
$531$	−32.8282	−1.42462
$532$	0	0
$533$	1.02695	0.0444824
$534$	0	0
$535$	0.100352	0.00433859
$536$	0	0
$537$	−12.3610	−0.533416
$538$	0	0
$539$	−2.63224	−0.113378
$540$	0	0
$541$	32.5823	1.40082	0.700410	−	0.713740i	$-0.253000\pi$
$541$	32.5823	1.40082	0.700410	+	0.713740i	$0.253000\pi$
$542$	0	0
$543$	9.36142	0.401737
$544$	0	0
$545$	−1.01114	−0.0433123
$546$	0	0
$547$	−30.0637	−1.28543	−0.642716	−	0.766105i	$-0.722192\pi$
$547$	−30.0637	−1.28543	−0.642716	+	0.766105i	$0.722192\pi$
$548$	0	0
$549$	26.9797	1.15146
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.94006	0.0824997
$554$	0	0
$555$	0.0106736	0.000453070 0
$556$	0	0
$557$	−12.5344	−0.531098	−0.265549	−	0.964097i	$-0.585553\pi$
$557$	−12.5344	−0.531098	−0.265549	+	0.964097i	$0.585553\pi$
$558$	0	0
$559$	−0.737841	−0.0312074
$560$	0	0
$561$	0.109726	0.00463263
$562$	0	0
$563$	21.4944	0.905880	0.452940	−	0.891541i	$-0.350375\pi$
$563$	21.4944	0.905880	0.452940	+	0.891541i	$0.350375\pi$
$564$	0	0
$565$	−5.22399	−0.219775
$566$	0	0
$567$	−18.6239	−0.782131
$568$	0	0
$569$	−11.8577	−0.497099	−0.248550	−	0.968619i	$-0.579954\pi$
$569$	−11.8577	−0.497099	−0.248550	+	0.968619i	$0.579954\pi$
$570$	0	0
$571$	−1.50474	−0.0629714	−0.0314857	−	0.999504i	$-0.510024\pi$
$571$	−1.50474	−0.0629714	−0.0314857	+	0.999504i	$0.510024\pi$
$572$	0	0
$573$	15.9811	0.667620
$574$	0	0
$575$	−21.5802	−0.899956
$576$	0	0
$577$	4.49515	0.187135	0.0935677	−	0.995613i	$-0.470173\pi$
$577$	4.49515	0.187135	0.0935677	+	0.995613i	$0.470173\pi$
$578$	0	0
$579$	17.8789	0.743021
$580$	0	0
$581$	−12.3074	−0.510598
$582$	0	0
$583$	1.03713	0.0429537
$584$	0	0
$585$	−0.106472	−0.00440208
$586$	0	0
$587$	7.35457	0.303555	0.151778	−	0.988415i	$-0.451500\pi$
$587$	7.35457	0.303555	0.151778	+	0.988415i	$0.451500\pi$
$588$	0	0
$589$	19.3161	0.795905
$590$	0	0
$591$	18.7247	0.770231
$592$	0	0
$593$	14.3814	0.590573	0.295286	−	0.955409i	$-0.404585\pi$
$593$	14.3814	0.590573	0.295286	+	0.955409i	$0.404585\pi$
$594$	0	0
$595$	−1.24479	−0.0510316
$596$	0	0
$597$	2.76085	0.112994
$598$	0	0
$599$	1.43394	0.0585893	0.0292946	−	0.999571i	$-0.490674\pi$
$599$	1.43394	0.0585893	0.0292946	+	0.999571i	$0.490674\pi$
$600$	0	0
$601$	9.03472	0.368534	0.184267	−	0.982876i	$-0.441009\pi$
$601$	9.03472	0.368534	0.184267	+	0.982876i	$0.441009\pi$
$602$	0	0
$603$	−16.3834	−0.667184
$604$	0	0
$605$	−3.44933	−0.140235
$606$	0	0
$607$	−26.5041	−1.07577	−0.537885	−	0.843018i	$-0.680777\pi$
$607$	−26.5041	−1.07577	−0.537885	+	0.843018i	$0.680777\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.26700	−0.0512574
$612$	0	0
$613$	12.8279	0.518114	0.259057	−	0.965862i	$-0.416588\pi$
$613$	12.8279	0.518114	0.259057	+	0.965862i	$0.416588\pi$
$614$	0	0
$615$	−1.77597	−0.0716142
$616$	0	0
$617$	−23.1526	−0.932087	−0.466044	−	0.884762i	$-0.654321\pi$
$617$	−23.1526	−0.932087	−0.466044	+	0.884762i	$0.654321\pi$
$618$	0	0
$619$	−12.7337	−0.511811	−0.255906	−	0.966702i	$-0.582374\pi$
$619$	−12.7337	−0.511811	−0.255906	+	0.966702i	$0.582374\pi$
$620$	0	0
$621$	18.4761	0.741420
$622$	0	0
$623$	41.9974	1.68259
$624$	0	0
$625$	23.5271	0.941085
$626$	0	0
$627$	0.904094	0.0361060
$628$	0	0
$629$	−0.0363900	−0.00145096
$630$	0	0
$631$	−40.4571	−1.61057	−0.805285	−	0.592888i	$-0.797988\pi$
$631$	−40.4571	−1.61057	−0.805285	+	0.592888i	$0.797988\pi$
$632$	0	0
$633$	−3.94033	−0.156614
$634$	0	0
$635$	3.49754	0.138796
$636$	0	0
$637$	2.19948	0.0871466
$638$	0	0
$639$	−13.2753	−0.525162
$640$	0	0
$641$	−8.28216	−0.327126	−0.163563	−	0.986533i	$-0.552299\pi$
$641$	−8.28216	−0.327126	−0.163563	+	0.986533i	$0.552299\pi$
$642$	0	0
$643$	33.1662	1.30795	0.653973	−	0.756518i	$-0.273101\pi$
$643$	33.1662	1.30795	0.653973	+	0.756518i	$0.273101\pi$
$644$	0	0
$645$	1.27599	0.0502422
$646$	0	0
$647$	−2.68178	−0.105432	−0.0527158	−	0.998610i	$-0.516788\pi$
$647$	−2.68178	−0.105432	−0.0527158	+	0.998610i	$0.516788\pi$
$648$	0	0
$649$	−2.31943	−0.0910458
$650$	0	0
$651$	10.3871	0.407102
$652$	0	0
$653$	35.7432	1.39874	0.699370	−	0.714759i	$-0.253464\pi$
$653$	35.7432	1.39874	0.699370	+	0.714759i	$0.253464\pi$
$654$	0	0
$655$	−4.44097	−0.173523
$656$	0	0
$657$	−23.6206	−0.921527
$658$	0	0
$659$	−36.6152	−1.42633	−0.713163	−	0.700998i	$-0.752738\pi$
$659$	−36.6152	−1.42633	−0.713163	+	0.700998i	$0.752738\pi$
$660$	0	0
$661$	0.343812	0.0133727	0.00668636	−	0.999978i	$-0.497872\pi$
$661$	0.343812	0.0133727	0.00668636	+	0.999978i	$0.497872\pi$
$662$	0	0
$663$	−0.0916863	−0.00356080
$664$	0	0
$665$	−10.2566	−0.397733
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.0697	0.427981
$670$	0	0
$671$	1.90621	0.0735885
$672$	0	0
$673$	−23.5751	−0.908754	−0.454377	−	0.890809i	$-0.650138\pi$
$673$	−23.5751	−0.908754	−0.454377	+	0.890809i	$0.650138\pi$
$674$	0	0
$675$	−20.5661	−0.791590
$676$	0	0
$677$	29.9830	1.15234	0.576170	−	0.817330i	$-0.304547\pi$
$677$	29.9830	1.15234	0.576170	+	0.817330i	$0.304547\pi$
$678$	0	0
$679$	−50.8284	−1.95061
$680$	0	0
$681$	6.64632	0.254688
$682$	0	0
$683$	30.0457	1.14967	0.574833	−	0.818271i	$-0.305067\pi$
$683$	30.0457	1.14967	0.574833	+	0.818271i	$0.305067\pi$
$684$	0	0
$685$	−2.58592	−0.0988028
$686$	0	0
$687$	−18.9765	−0.724000
$688$	0	0
$689$	−0.866623	−0.0330157
$690$	0	0
$691$	23.5325	0.895220	0.447610	−	0.894229i	$-0.352275\pi$
$691$	23.5325	0.895220	0.447610	+	0.894229i	$0.352275\pi$
$692$	0	0
$693$	−1.92482	−0.0731177
$694$	0	0
$695$	−3.90833	−0.148251
$696$	0	0
$697$	6.05489	0.229345
$698$	0	0
$699$	4.47232	0.169159
$700$	0	0
$701$	−30.8235	−1.16419	−0.582094	−	0.813122i	$-0.697766\pi$
$701$	−30.8235	−1.16419	−0.582094	+	0.813122i	$0.697766\pi$
$702$	0	0
$703$	−0.299838	−0.0113086
$704$	0	0
$705$	2.19110	0.0825216
$706$	0	0
$707$	3.30783	0.124404
$708$	0	0
$709$	−15.0691	−0.565930	−0.282965	−	0.959130i	$-0.591318\pi$
$709$	−15.0691	−0.565930	−0.282965	+	0.959130i	$0.591318\pi$
$710$	0	0
$711$	0.978381	0.0366922
$712$	0	0
$713$	12.3814	0.463687
$714$	0	0
$715$	−0.00752265	−0.000281331 0
$716$	0	0
$717$	−15.4401	−0.576620
$718$	0	0
$719$	11.0174	0.410880	0.205440	−	0.978670i	$-0.434137\pi$
$719$	11.0174	0.410880	0.205440	+	0.978670i	$0.434137\pi$
$720$	0	0
$721$	32.9602	1.22750
$722$	0	0
$723$	−22.8937	−0.851427
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−30.7008	−1.13863	−0.569316	−	0.822119i	$-0.692792\pi$
$727$	−30.7008	−1.13863	−0.569316	+	0.822119i	$0.692792\pi$
$728$	0	0
$729$	0.398977	0.0147769
$730$	0	0
$731$	−4.35029	−0.160901
$732$	0	0
$733$	0.740314	0.0273441	0.0136721	−	0.999907i	$-0.495648\pi$
$733$	0.740314	0.0273441	0.0136721	+	0.999907i	$0.495648\pi$
$734$	0	0
$735$	−3.80369	−0.140301
$736$	0	0
$737$	−1.15755	−0.0426389
$738$	0	0
$739$	1.07803	0.0396560	0.0198280	−	0.999803i	$-0.493688\pi$
$739$	1.07803	0.0396560	0.0198280	+	0.999803i	$0.493688\pi$
$740$	0	0
$741$	−0.755455	−0.0277523
$742$	0	0
$743$	9.86478	0.361904	0.180952	−	0.983492i	$-0.442082\pi$
$743$	9.86478	0.361904	0.180952	+	0.983492i	$0.442082\pi$
$744$	0	0
$745$	−6.60953	−0.242154
$746$	0	0
$747$	−6.20669	−0.227091
$748$	0	0
$749$	−1.51591	−0.0553903
$750$	0	0
$751$	−13.9508	−0.509072	−0.254536	−	0.967063i	$-0.581923\pi$
$751$	−13.9508	−0.509072	−0.254536	+	0.967063i	$0.581923\pi$
$752$	0	0
$753$	2.07132	0.0754832
$754$	0	0
$755$	4.77257	0.173691
$756$	0	0
$757$	26.5113	0.963570	0.481785	−	0.876289i	$-0.339989\pi$
$757$	26.5113	0.963570	0.481785	+	0.876289i	$0.339989\pi$
$758$	0	0
$759$	0.579515	0.0210351
$760$	0	0
$761$	41.5289	1.50542	0.752710	−	0.658352i	$-0.228746\pi$
$761$	41.5289	1.50542	0.752710	+	0.658352i	$0.228746\pi$
$762$	0	0
$763$	15.2742	0.552963
$764$	0	0
$765$	−0.627756	−0.0226966
$766$	0	0
$767$	1.93811	0.0699809
$768$	0	0
$769$	11.2343	0.405118	0.202559	−	0.979270i	$-0.435074\pi$
$769$	11.2343	0.405118	0.202559	+	0.979270i	$0.435074\pi$
$770$	0	0
$771$	−6.81884	−0.245574
$772$	0	0
$773$	−53.6797	−1.93073	−0.965363	−	0.260911i	$-0.915977\pi$
$773$	−53.6797	−1.93073	−0.965363	+	0.260911i	$0.915977\pi$
$774$	0	0
$775$	−13.7820	−0.495064
$776$	0	0
$777$	−0.161236	−0.00578430
$778$	0	0
$779$	49.8897	1.78748
$780$	0	0
$781$	−0.937948	−0.0335624
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.524295	−0.0187129
$786$	0	0
$787$	14.3077	0.510014	0.255007	−	0.966939i	$-0.417922\pi$
$787$	14.3077	0.510014	0.255007	+	0.966939i	$0.417922\pi$
$788$	0	0
$789$	14.2628	0.507768
$790$	0	0
$791$	78.9134	2.80584
$792$	0	0
$793$	−1.59282	−0.0565627
$794$	0	0
$795$	1.49870	0.0531535
$796$	0	0
$797$	−17.4661	−0.618680	−0.309340	−	0.950952i	$-0.600108\pi$
$797$	−17.4661	−0.618680	−0.309340	+	0.950952i	$0.600108\pi$
$798$	0	0
$799$	−7.47020	−0.264277
$800$	0	0
$801$	21.1795	0.748341
$802$	0	0
$803$	−1.66888	−0.0588936
$804$	0	0
$805$	−6.57436	−0.231716
$806$	0	0
$807$	12.9155	0.454648
$808$	0	0
$809$	−33.9252	−1.19275	−0.596374	−	0.802707i	$-0.703392\pi$
$809$	−33.9252	−1.19275	−0.596374	+	0.802707i	$0.703392\pi$
$810$	0	0
$811$	−6.50175	−0.228307	−0.114154	−	0.993463i	$-0.536416\pi$
$811$	−6.50175	−0.228307	−0.114154	+	0.993463i	$0.536416\pi$
$812$	0	0
$813$	−17.7099	−0.621113
$814$	0	0
$815$	−0.949873	−0.0332726
$816$	0	0
$817$	−35.8445	−1.25404
$818$	0	0
$819$	1.60836	0.0562008
$820$	0	0
$821$	−13.4721	−0.470181	−0.235091	−	0.971973i	$-0.575539\pi$
$821$	−13.4721	−0.470181	−0.235091	+	0.971973i	$0.575539\pi$
$822$	0	0
$823$	−11.3371	−0.395186	−0.197593	−	0.980284i	$-0.563312\pi$
$823$	−11.3371	−0.395186	−0.197593	+	0.980284i	$0.563312\pi$
$824$	0	0
$825$	−0.645070	−0.0224584
$826$	0	0
$827$	−56.5747	−1.96729	−0.983647	−	0.180105i	$-0.942356\pi$
$827$	−56.5747	−1.96729	−0.983647	+	0.180105i	$0.942356\pi$
$828$	0	0
$829$	−42.4183	−1.47325	−0.736625	−	0.676302i	$-0.763582\pi$
$829$	−42.4183	−1.47325	−0.736625	+	0.676302i	$0.763582\pi$
$830$	0	0
$831$	15.6160	0.541714
$832$	0	0
$833$	12.9681	0.449317
$834$	0	0
$835$	4.74048	0.164051
$836$	0	0
$837$	11.7996	0.407853
$838$	0	0
$839$	9.39372	0.324307	0.162154	−	0.986766i	$-0.448156\pi$
$839$	9.39372	0.324307	0.162154	+	0.986766i	$0.448156\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−18.6918	−0.643779
$844$	0	0
$845$	−4.08083	−0.140385
$846$	0	0
$847$	52.1055	1.79037
$848$	0	0
$849$	10.3130	0.353940
$850$	0	0
$851$	−0.192193	−0.00658829
$852$	0	0
$853$	5.84208	0.200029	0.100014	−	0.994986i	$-0.468111\pi$
$853$	5.84208	0.200029	0.100014	+	0.994986i	$0.468111\pi$
$854$	0	0
$855$	−5.17243	−0.176894
$856$	0	0
$857$	−34.3911	−1.17478	−0.587389	−	0.809305i	$-0.699844\pi$
$857$	−34.3911	−1.17478	−0.587389	+	0.809305i	$0.699844\pi$
$858$	0	0
$859$	5.04070	0.171987	0.0859933	−	0.996296i	$-0.472594\pi$
$859$	5.04070	0.171987	0.0859933	+	0.996296i	$0.472594\pi$
$860$	0	0
$861$	26.8278	0.914290
$862$	0	0
$863$	50.8624	1.73138	0.865689	−	0.500583i	$-0.166881\pi$
$863$	50.8624	1.73138	0.865689	+	0.500583i	$0.166881\pi$
$864$	0	0
$865$	4.72188	0.160549
$866$	0	0
$867$	12.6817	0.430693
$868$	0	0
$869$	0.0691262	0.00234495
$870$	0	0
$871$	0.967241	0.0327737
$872$	0	0
$873$	−25.6330	−0.867545
$874$	0	0
$875$	14.7837	0.499780
$876$	0	0
$877$	40.5961	1.37083	0.685417	−	0.728151i	$-0.259620\pi$
$877$	40.5961	1.37083	0.685417	+	0.728151i	$0.259620\pi$
$878$	0	0
$879$	16.5461	0.558084
$880$	0	0
$881$	18.8820	0.636152	0.318076	−	0.948065i	$-0.396963\pi$
$881$	18.8820	0.636152	0.318076	+	0.948065i	$0.396963\pi$
$882$	0	0
$883$	−42.9589	−1.44568	−0.722841	−	0.691015i	$-0.757164\pi$
$883$	−42.9589	−1.44568	−0.722841	+	0.691015i	$0.757164\pi$
$884$	0	0
$885$	−3.35168	−0.112666
$886$	0	0
$887$	26.6470	0.894718	0.447359	−	0.894354i	$-0.352365\pi$
$887$	26.6470	0.894718	0.447359	+	0.894354i	$0.352365\pi$
$888$	0	0
$889$	−52.8338	−1.77199
$890$	0	0
$891$	−0.663589	−0.0222311
$892$	0	0
$893$	−61.5512	−2.05973
$894$	0	0
$895$	4.99654	0.167016
$896$	0	0
$897$	−0.484239	−0.0161683
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−5.10958	−0.170225
$902$	0	0
$903$	−19.2751	−0.641436
$904$	0	0
$905$	−3.78406	−0.125786
$906$	0	0
$907$	−14.2443	−0.472973	−0.236486	−	0.971635i	$-0.575996\pi$
$907$	−14.2443	−0.472973	−0.236486	+	0.971635i	$0.575996\pi$
$908$	0	0
$909$	1.66815	0.0553292
$910$	0	0
$911$	−31.0572	−1.02897	−0.514486	−	0.857499i	$-0.672017\pi$
$911$	−31.0572	−1.02897	−0.514486	+	0.857499i	$0.672017\pi$
$912$	0	0
$913$	−0.438526	−0.0145131
$914$	0	0
$915$	2.75456	0.0910629
$916$	0	0
$917$	67.0852	2.21535
$918$	0	0
$919$	−59.8452	−1.97411	−0.987056	−	0.160375i	$-0.948730\pi$
$919$	−59.8452	−1.97411	−0.987056	+	0.160375i	$0.948730\pi$
$920$	0	0
$921$	24.3888	0.803639
$922$	0	0
$923$	0.783744	0.0257972
$924$	0	0
$925$	0.213934	0.00703410
$926$	0	0
$927$	16.6220	0.545938
$928$	0	0
$929$	−36.2614	−1.18970	−0.594850	−	0.803837i	$-0.702788\pi$
$929$	−36.2614	−1.18970	−0.594850	+	0.803837i	$0.702788\pi$
$930$	0	0
$931$	106.851	3.50191
$932$	0	0
$933$	−5.95973	−0.195113
$934$	0	0
$935$	−0.0443533	−0.00145051
$936$	0	0
$937$	−15.1929	−0.496330	−0.248165	−	0.968718i	$-0.579828\pi$
$937$	−15.1929	−0.496330	−0.248165	+	0.968718i	$0.579828\pi$
$938$	0	0
$939$	−0.799381	−0.0260868
$940$	0	0
$941$	−26.1995	−0.854079	−0.427040	−	0.904233i	$-0.640443\pi$
$941$	−26.1995	−0.854079	−0.427040	+	0.904233i	$0.640443\pi$
$942$	0	0
$943$	31.9788	1.04137
$944$	0	0
$945$	−6.26542	−0.203814
$946$	0	0
$947$	−0.168582	−0.00547818	−0.00273909	−	0.999996i	$-0.500872\pi$
$947$	−0.168582	−0.00547818	−0.00273909	+	0.999996i	$0.500872\pi$
$948$	0	0
$949$	1.39451	0.0452676
$950$	0	0
$951$	−3.29346	−0.106798
$952$	0	0
$953$	−25.9451	−0.840443	−0.420222	−	0.907422i	$-0.638048\pi$
$953$	−25.9451	−0.840443	−0.420222	+	0.907422i	$0.638048\pi$
$954$	0	0
$955$	−6.45986	−0.209036
$956$	0	0
$957$	0	0
$958$	0	0
$959$	39.0628	1.26140
$960$	0	0
$961$	−23.0927	−0.744927
$962$	0	0
$963$	−0.764483	−0.0246351
$964$	0	0
$965$	−7.22698	−0.232645
$966$	0	0
$967$	−49.5428	−1.59319	−0.796595	−	0.604514i	$-0.793367\pi$
$967$	−49.5428	−1.59319	−0.796595	+	0.604514i	$0.793367\pi$
$968$	0	0
$969$	−4.45414	−0.143088
$970$	0	0
$971$	−41.6311	−1.33600	−0.668002	−	0.744159i	$-0.732850\pi$
$971$	−41.6311	−1.33600	−0.668002	+	0.744159i	$0.732850\pi$
$972$	0	0
$973$	59.0391	1.89271
$974$	0	0
$975$	0.539016	0.0172623
$976$	0	0
$977$	−30.0107	−0.960129	−0.480064	−	0.877233i	$-0.659387\pi$
$977$	−30.0107	−0.960129	−0.480064	+	0.877233i	$0.659387\pi$
$978$	0	0
$979$	1.49641	0.0478255
$980$	0	0
$981$	7.70285	0.245933
$982$	0	0
$983$	43.2684	1.38005	0.690024	−	0.723787i	$-0.257600\pi$
$983$	43.2684	1.38005	0.690024	+	0.723787i	$0.257600\pi$
$984$	0	0
$985$	−7.56887	−0.241164
$986$	0	0
$987$	−33.0987	−1.05354
$988$	0	0
$989$	−22.9760	−0.730593
$990$	0	0
$991$	6.46898	0.205494	0.102747	−	0.994708i	$-0.467237\pi$
$991$	6.46898	0.205494	0.102747	+	0.994708i	$0.467237\pi$
$992$	0	0
$993$	−11.1670	−0.354373
$994$	0	0
$995$	−1.11599	−0.0353791
$996$	0	0
$997$	14.6991	0.465525	0.232763	−	0.972534i	$-0.425223\pi$
$997$	14.6991	0.465525	0.232763	+	0.972534i	$0.425223\pi$
$998$	0	0
$999$	−0.183162	−0.00579498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.z.1.5		12
29.7	even	7		232.2.m.d.49.3	✓	24
29.25	even	7		232.2.m.d.161.3	yes	24
29.28	even	2		6728.2.a.bb.1.8		12
116.7	odd	14		464.2.u.i.49.2		24
116.83	odd	14		464.2.u.i.161.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.m.d.49.3	✓	24	29.7	even	7
232.2.m.d.161.3	yes	24	29.25	even	7
464.2.u.i.49.2		24	116.7	odd	14
464.2.u.i.161.2		24	116.83	odd	14
6728.2.a.z.1.5		12	1.1	even	1	trivial
6728.2.a.bb.1.8		12	29.28	even	2

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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q-0.777781 q^{3} +0.314394 q^{5} -4.74923 q^{7} -2.39506 q^{9} +O(q^{10})\)	\(q-0.777781 q^{3} +0.314394 q^{5} -4.74923 q^{7} -2.39506 q^{9} -0.169220 q^{11} +0.141399 q^{13} -0.244529 q^{15} +0.833683 q^{17} +6.86919 q^{19} +3.69386 q^{21} +4.40308 q^{23} -4.90116 q^{25} +4.19617 q^{27} +2.81199 q^{31} +0.131616 q^{33} -1.49313 q^{35} -0.0436497 q^{37} -0.109977 q^{39} +7.26282 q^{41} -5.21816 q^{43} -0.752991 q^{45} -8.96047 q^{47} +15.5551 q^{49} -0.648423 q^{51} -6.12892 q^{53} -0.0532016 q^{55} -5.34272 q^{57} +13.7067 q^{59} -11.2647 q^{61} +11.3747 q^{63} +0.0444549 q^{65} +6.84051 q^{67} -3.42463 q^{69} +5.54279 q^{71} +9.86223 q^{73} +3.81203 q^{75} +0.803662 q^{77} -0.408500 q^{79} +3.92147 q^{81} +2.59146 q^{83} +0.262105 q^{85} -8.84300 q^{89} -0.671535 q^{91} -2.18711 q^{93} +2.15963 q^{95} +10.7025 q^{97} +0.405291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * q^99

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