LMFDB - Embedded newform 6800.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6800,2,Mod(1,6800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.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:	$54.2982733745$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53407 q^{3} +2.53407 q^{7} +3.42151 q^{9} +O(q^{10})$	$q+2.53407 q^{3} +2.53407 q^{7} +3.42151 q^{9} +2.95558 q^{11} +5.48965 q^{13} -1.00000 q^{17} +5.06814 q^{19} +6.42151 q^{21} +7.37709 q^{23} +1.06814 q^{27} -7.91116 q^{29} -1.88744 q^{31} +7.48965 q^{33} -3.06814 q^{37} +13.9112 q^{39} +11.0681 q^{41} -9.48965 q^{43} +0.421512 q^{47} -0.578488 q^{49} -2.53407 q^{51} -6.84302 q^{53} +12.8430 q^{57} -1.91116 q^{59} +8.13628 q^{61} +8.67035 q^{63} -7.57849 q^{67} +18.6941 q^{69} -1.04442 q^{71} +6.22512 q^{73} +7.48965 q^{77} +1.04442 q^{79} -7.55779 q^{81} -14.3327 q^{83} -20.0474 q^{87} -2.55779 q^{89} +13.9112 q^{91} -4.78291 q^{93} -13.8223 q^{97} +10.1126 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{9}+O(q^{10}) $ 3 * q + 7 * q^9	$ 3 q + 7 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{17} + 16 q^{21} + 8 q^{23} - 12 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 6 q^{37} + 20 q^{39} + 18 q^{41} - 10 q^{43} - 2 q^{47} - 5 q^{49} - 14 q^{53} + 32 q^{57} + 16 q^{59} - 6 q^{61} - 12 q^{63} - 26 q^{67} - 8 q^{69} - 14 q^{71} + 10 q^{73} + 4 q^{77} + 14 q^{79} + 11 q^{81} - 18 q^{83} - 8 q^{87} + 26 q^{89} + 20 q^{91} + 28 q^{93} + 2 q^{97} + 26 q^{99}+O(q^{100}) $ 3 * q + 7 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^17 + 16 * q^21 + 8 * q^23 - 12 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 6 * q^37 + 20 * q^39 + 18 * q^41 - 10 * q^43 - 2 * q^47 - 5 * q^49 - 14 * q^53 + 32 * q^57 + 16 * q^59 - 6 * q^61 - 12 * q^63 - 26 * q^67 - 8 * q^69 - 14 * q^71 + 10 * q^73 + 4 * q^77 + 14 * q^79 + 11 * q^81 - 18 * q^83 - 8 * q^87 + 26 * q^89 + 20 * q^91 + 28 * q^93 + 2 * q^97 + 26 * 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
$3$	2.53407	1.46305	0.731523	−	0.681817i	$-0.238810\pi$
$3$	2.53407	1.46305	0.731523	+	0.681817i	$0.238810\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.53407	0.957789	0.478894	−	0.877873i	$-0.341038\pi$
$7$	2.53407	0.957789	0.478894	+	0.877873i	$0.341038\pi$
$8$	0	0
$9$	3.42151	1.14050
$10$	0	0
$11$	2.95558	0.891141	0.445571	−	0.895247i	$-0.353001\pi$
$11$	2.95558	0.891141	0.445571	+	0.895247i	$0.353001\pi$
$12$	0	0
$13$	5.48965	1.52256	0.761278	−	0.648426i	$-0.224572\pi$
$13$	5.48965	1.52256	0.761278	+	0.648426i	$0.224572\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.06814	1.16271	0.581356	−	0.813650i	$-0.302523\pi$
$19$	5.06814	1.16271	0.581356	+	0.813650i	$0.302523\pi$
$20$	0	0
$21$	6.42151	1.40129
$22$	0	0
$23$	7.37709	1.53823	0.769115	−	0.639110i	$-0.220697\pi$
$23$	7.37709	1.53823	0.769115	+	0.639110i	$0.220697\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.06814	0.205564
$28$	0	0
$29$	−7.91116	−1.46907	−0.734533	−	0.678573i	$-0.762599\pi$
$29$	−7.91116	−1.46907	−0.734533	+	0.678573i	$0.762599\pi$
$30$	0	0
$31$	−1.88744	−0.338995	−0.169497	−	0.985531i	$-0.554214\pi$
$31$	−1.88744	−0.338995	−0.169497	+	0.985531i	$0.554214\pi$
$32$	0	0
$33$	7.48965	1.30378
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.06814	−0.504399	−0.252200	−	0.967675i	$-0.581154\pi$
$37$	−3.06814	−0.504399	−0.252200	+	0.967675i	$0.581154\pi$
$38$	0	0
$39$	13.9112	2.22757
$40$	0	0
$41$	11.0681	1.72855	0.864277	−	0.503017i	$-0.167777\pi$
$41$	11.0681	1.72855	0.864277	+	0.503017i	$0.167777\pi$
$42$	0	0
$43$	−9.48965	−1.44716	−0.723579	−	0.690241i	$-0.757504\pi$
$43$	−9.48965	−1.44716	−0.723579	+	0.690241i	$0.757504\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.421512	0.0614838	0.0307419	−	0.999527i	$-0.490213\pi$
$47$	0.421512	0.0614838	0.0307419	+	0.999527i	$0.490213\pi$
$48$	0	0
$49$	−0.578488	−0.0826412
$50$	0	0
$51$	−2.53407	−0.354841
$52$	0	0
$53$	−6.84302	−0.939962	−0.469981	−	0.882677i	$-0.655739\pi$
$53$	−6.84302	−0.939962	−0.469981	+	0.882677i	$0.655739\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.8430	1.70110
$58$	0	0
$59$	−1.91116	−0.248812	−0.124406	−	0.992231i	$-0.539703\pi$
$59$	−1.91116	−0.248812	−0.124406	+	0.992231i	$0.539703\pi$
$60$	0	0
$61$	8.13628	1.04174	0.520872	−	0.853635i	$-0.325607\pi$
$61$	8.13628	1.04174	0.520872	+	0.853635i	$0.325607\pi$
$62$	0	0
$63$	8.67035	1.09236
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.57849	−0.925860	−0.462930	−	0.886395i	$-0.653202\pi$
$67$	−7.57849	−0.925860	−0.462930	+	0.886395i	$0.653202\pi$
$68$	0	0
$69$	18.6941	2.25050
$70$	0	0
$71$	−1.04442	−0.123950	−0.0619748	−	0.998078i	$-0.519740\pi$
$71$	−1.04442	−0.123950	−0.0619748	+	0.998078i	$0.519740\pi$
$72$	0	0
$73$	6.22512	0.728595	0.364297	−	0.931283i	$-0.381309\pi$
$73$	6.22512	0.728595	0.364297	+	0.931283i	$0.381309\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.48965	0.853525
$78$	0	0
$79$	1.04442	0.117506	0.0587531	−	0.998273i	$-0.481288\pi$
$79$	1.04442	0.117506	0.0587531	+	0.998273i	$0.481288\pi$
$80$	0	0
$81$	−7.55779	−0.839755
$82$	0	0
$83$	−14.3327	−1.57322	−0.786608	−	0.617453i	$-0.788165\pi$
$83$	−14.3327	−1.57322	−0.786608	+	0.617453i	$0.788165\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−20.0474	−2.14931
$88$	0	0
$89$	−2.55779	−0.271125	−0.135563	−	0.990769i	$-0.543284\pi$
$89$	−2.55779	−0.271125	−0.135563	+	0.990769i	$0.543284\pi$
$90$	0	0
$91$	13.9112	1.45829
$92$	0	0
$93$	−4.78291	−0.495965
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.8223	−1.40344	−0.701722	−	0.712450i	$-0.747585\pi$
$97$	−13.8223	−1.40344	−0.701722	+	0.712450i	$0.747585\pi$
$98$	0	0
$99$	10.1126	1.01635
$100$	0	0
$101$	8.42151	0.837972	0.418986	−	0.907993i	$-0.362386\pi$
$101$	8.42151	0.837972	0.418986	+	0.907993i	$0.362386\pi$
$102$	0	0
$103$	−10.5578	−1.04029	−0.520145	−	0.854078i	$-0.674122\pi$
$103$	−10.5578	−1.04029	−0.520145	+	0.854078i	$0.674122\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.75919	−0.653435	−0.326718	−	0.945122i	$-0.605943\pi$
$107$	−6.75919	−0.653435	−0.326718	+	0.945122i	$0.605943\pi$
$108$	0	0
$109$	16.1363	1.54558	0.772788	−	0.634665i	$-0.218862\pi$
$109$	16.1363	1.54558	0.772788	+	0.634665i	$0.218862\pi$
$110$	0	0
$111$	−7.77488	−0.737959
$112$	0	0
$113$	3.06814	0.288626	0.144313	−	0.989532i	$-0.453903\pi$
$113$	3.06814	0.288626	0.144313	+	0.989532i	$0.453903\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	18.7829	1.73648
$118$	0	0
$119$	−2.53407	−0.232298
$120$	0	0
$121$	−2.26454	−0.205867
$122$	0	0
$123$	28.0474	2.52895
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.3327	−1.62676	−0.813381	−	0.581731i	$-0.802375\pi$
$127$	−18.3327	−1.62676	−0.813381	+	0.581731i	$0.802375\pi$
$128$	0	0
$129$	−24.0474	−2.11726
$130$	0	0
$131$	−11.7986	−1.03085	−0.515424	−	0.856935i	$-0.672366\pi$
$131$	−11.7986	−1.03085	−0.515424	+	0.856935i	$0.672366\pi$
$132$	0	0
$133$	12.8430	1.11363
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.6466	−1.08047	−0.540237	−	0.841513i	$-0.681665\pi$
$137$	−12.6466	−1.08047	−0.540237	+	0.841513i	$0.681665\pi$
$138$	0	0
$139$	8.02372	0.680563	0.340282	−	0.940324i	$-0.389478\pi$
$139$	8.02372	0.680563	0.340282	+	0.940324i	$0.389478\pi$
$140$	0	0
$141$	1.06814	0.0899536
$142$	0	0
$143$	16.2251	1.35681
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.46593	−0.120908
$148$	0	0
$149$	20.1363	1.64963	0.824814	−	0.565404i	$-0.191280\pi$
$149$	20.1363	1.64963	0.824814	+	0.565404i	$0.191280\pi$
$150$	0	0
$151$	−9.91116	−0.806559	−0.403280	−	0.915077i	$-0.632130\pi$
$151$	−9.91116	−0.806559	−0.403280	+	0.915077i	$0.632130\pi$
$152$	0	0
$153$	−3.42151	−0.276613
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.97930	0.397392	0.198696	−	0.980061i	$-0.436329\pi$
$157$	4.97930	0.397392	0.198696	+	0.980061i	$0.436329\pi$
$158$	0	0
$159$	−17.3407	−1.37521
$160$	0	0
$161$	18.6941	1.47330
$162$	0	0
$163$	14.5815	1.14211	0.571056	−	0.820911i	$-0.306534\pi$
$163$	14.5815	1.14211	0.571056	+	0.820911i	$0.306534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.62291	0.357731	0.178866	−	0.983874i	$-0.442757\pi$
$167$	4.62291	0.357731	0.178866	+	0.983874i	$0.442757\pi$
$168$	0	0
$169$	17.1363	1.31818
$170$	0	0
$171$	17.3407	1.32608
$172$	0	0
$173$	−15.9112	−1.20970	−0.604852	−	0.796338i	$-0.706768\pi$
$173$	−15.9112	−1.20970	−0.604852	+	0.796338i	$0.706768\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.84302	−0.364024
$178$	0	0
$179$	−0.225117	−0.0168260	−0.00841301	−	0.999965i	$-0.502678\pi$
$179$	−0.225117	−0.0168260	−0.00841301	+	0.999965i	$0.502678\pi$
$180$	0	0
$181$	−0.931860	−0.0692646	−0.0346323	−	0.999400i	$-0.511026\pi$
$181$	−0.931860	−0.0692646	−0.0346323	+	0.999400i	$0.511026\pi$
$182$	0	0
$183$	20.6179	1.52412
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.95558	−0.216134
$188$	0	0
$189$	2.70674	0.196887
$190$	0	0
$191$	−17.2933	−1.25130	−0.625648	−	0.780105i	$-0.715165\pi$
$191$	−17.2933	−1.25130	−0.625648	+	0.780105i	$0.715165\pi$
$192$	0	0
$193$	24.3614	1.75357	0.876786	−	0.480881i	$-0.159683\pi$
$193$	24.3614	1.75357	0.876786	+	0.480881i	$0.159683\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.91116	−0.563647	−0.281824	−	0.959466i	$-0.590939\pi$
$197$	−7.91116	−0.563647	−0.281824	+	0.959466i	$0.590939\pi$
$198$	0	0
$199$	9.09186	0.644505	0.322253	−	0.946654i	$-0.395560\pi$
$199$	9.09186	0.644505	0.322253	+	0.946654i	$0.395560\pi$
$200$	0	0
$201$	−19.2044	−1.35458
$202$	0	0
$203$	−20.0474	−1.40705
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.2408	1.75436
$208$	0	0
$209$	14.9793	1.03614
$210$	0	0
$211$	−26.1600	−1.80093	−0.900464	−	0.434930i	$-0.856773\pi$
$211$	−26.1600	−1.80093	−0.900464	+	0.434930i	$0.856773\pi$
$212$	0	0
$213$	−2.64663	−0.181344
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.78291	−0.324685
$218$	0	0
$219$	15.7749	1.06597
$220$	0	0
$221$	−5.48965	−0.369274
$222$	0	0
$223$	16.6466	1.11474	0.557370	−	0.830264i	$-0.311810\pi$
$223$	16.6466	1.11474	0.557370	+	0.830264i	$0.311810\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.39779	−0.291892	−0.145946	−	0.989293i	$-0.546623\pi$
$227$	−4.39779	−0.291892	−0.145946	+	0.989293i	$0.546623\pi$
$228$	0	0
$229$	−16.6941	−1.10318	−0.551588	−	0.834117i	$-0.685978\pi$
$229$	−16.6941	−1.10318	−0.551588	+	0.834117i	$0.685978\pi$
$230$	0	0
$231$	18.9793	1.24875
$232$	0	0
$233$	17.8223	1.16758	0.583790	−	0.811905i	$-0.301569\pi$
$233$	17.8223	1.16758	0.583790	+	0.811905i	$0.301569\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.64663	0.171917
$238$	0	0
$239$	−17.1156	−1.10712	−0.553558	−	0.832811i	$-0.686730\pi$
$239$	−17.1156	−1.10712	−0.553558	+	0.832811i	$0.686730\pi$
$240$	0	0
$241$	−4.93186	−0.317689	−0.158845	−	0.987304i	$-0.550777\pi$
$241$	−4.93186	−0.317689	−0.158845	+	0.987304i	$0.550777\pi$
$242$	0	0
$243$	−22.3564	−1.43416
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.8223	1.77029
$248$	0	0
$249$	−36.3200	−2.30169
$250$	0	0
$251$	11.8223	0.746219	0.373109	−	0.927787i	$-0.378292\pi$
$251$	11.8223	0.746219	0.373109	+	0.927787i	$0.378292\pi$
$252$	0	0
$253$	21.8036	1.37078
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.7622	−1.85652	−0.928258	−	0.371937i	$-0.878694\pi$
$257$	−29.7622	−1.85652	−0.928258	+	0.371937i	$0.878694\pi$
$258$	0	0
$259$	−7.77488	−0.483108
$260$	0	0
$261$	−27.0681	−1.67548
$262$	0	0
$263$	8.19639	0.505411	0.252706	−	0.967543i	$-0.418680\pi$
$263$	8.19639	0.505411	0.252706	+	0.967543i	$0.418680\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.48163	−0.396669
$268$	0	0
$269$	7.68605	0.468627	0.234313	−	0.972161i	$-0.424716\pi$
$269$	7.68605	0.468627	0.234313	+	0.972161i	$0.424716\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	35.2519	2.13354
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.6654	1.60217	0.801083	−	0.598553i	$-0.204258\pi$
$277$	26.6654	1.60217	0.801083	+	0.598553i	$0.204258\pi$
$278$	0	0
$279$	−6.45790	−0.386625
$280$	0	0
$281$	−0.313953	−0.0187289	−0.00936443	−	0.999956i	$-0.502981\pi$
$281$	−0.313953	−0.0187289	−0.00936443	+	0.999956i	$0.502981\pi$
$282$	0	0
$283$	24.7178	1.46932	0.734660	−	0.678435i	$-0.237342\pi$
$283$	24.7178	1.46932	0.734660	+	0.678435i	$0.237342\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.0474	1.65559
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−35.0267	−2.05330
$292$	0	0
$293$	4.97930	0.290894	0.145447	−	0.989366i	$-0.453538\pi$
$293$	4.97930	0.290894	0.145447	+	0.989366i	$0.453538\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.15698	0.183186
$298$	0	0
$299$	40.4977	2.34204
$300$	0	0
$301$	−24.0474	−1.38607
$302$	0	0
$303$	21.3407	1.22599
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.4215	0.937225	0.468613	−	0.883404i	$-0.344754\pi$
$307$	16.4215	0.937225	0.468613	+	0.883404i	$0.344754\pi$
$308$	0	0
$309$	−26.7542	−1.52199
$310$	0	0
$311$	14.9556	0.848053	0.424027	−	0.905650i	$-0.360616\pi$
$311$	14.9556	0.848053	0.424027	+	0.905650i	$0.360616\pi$
$312$	0	0
$313$	15.9112	0.899352	0.449676	−	0.893192i	$-0.351539\pi$
$313$	15.9112	0.899352	0.449676	+	0.893192i	$0.351539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.15698	−0.0649823	−0.0324911	−	0.999472i	$-0.510344\pi$
$317$	−1.15698	−0.0649823	−0.0324911	+	0.999472i	$0.510344\pi$
$318$	0	0
$319$	−23.3821	−1.30915
$320$	0	0
$321$	−17.1283	−0.956006
$322$	0	0
$323$	−5.06814	−0.281999
$324$	0	0
$325$	0	0
$326$	0	0
$327$	40.8905	2.26125
$328$	0	0
$329$	1.06814	0.0588885
$330$	0	0
$331$	27.2044	1.49529	0.747645	−	0.664098i	$-0.231184\pi$
$331$	27.2044	1.49529	0.747645	+	0.664098i	$0.231184\pi$
$332$	0	0
$333$	−10.4977	−0.575269
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.1156	−1.04129	−0.520646	−	0.853773i	$-0.674309\pi$
$337$	−19.1156	−1.04129	−0.520646	+	0.853773i	$0.674309\pi$
$338$	0	0
$339$	7.77488	0.422274
$340$	0	0
$341$	−5.57849	−0.302092
$342$	0	0
$343$	−19.2044	−1.03694
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.4927	−1.52957	−0.764784	−	0.644287i	$-0.777154\pi$
$347$	−28.4927	−1.52957	−0.764784	+	0.644287i	$0.777154\pi$
$348$	0	0
$349$	34.2726	1.83457	0.917284	−	0.398233i	$-0.130377\pi$
$349$	34.2726	1.83457	0.917284	+	0.398233i	$0.130377\pi$
$350$	0	0
$351$	5.86372	0.312982
$352$	0	0
$353$	−12.1363	−0.645949	−0.322975	−	0.946408i	$-0.604683\pi$
$353$	−12.1363	−0.645949	−0.322975	+	0.946408i	$0.604683\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.42151	−0.339862
$358$	0	0
$359$	28.8905	1.52478	0.762390	−	0.647117i	$-0.224026\pi$
$359$	28.8905	1.52478	0.762390	+	0.647117i	$0.224026\pi$
$360$	0	0
$361$	6.68605	0.351897
$362$	0	0
$363$	−5.73849	−0.301193
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.28826	−0.0672465	−0.0336233	−	0.999435i	$-0.510705\pi$
$367$	−1.28826	−0.0672465	−0.0336233	+	0.999435i	$0.510705\pi$
$368$	0	0
$369$	37.8698	1.97142
$370$	0	0
$371$	−17.3407	−0.900284
$372$	0	0
$373$	−16.4690	−0.852730	−0.426365	−	0.904551i	$-0.640206\pi$
$373$	−16.4690	−0.852730	−0.426365	+	0.904551i	$0.640206\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−43.4295	−2.23674
$378$	0	0
$379$	−24.9142	−1.27976	−0.639878	−	0.768477i	$-0.721015\pi$
$379$	−24.9142	−1.27976	−0.639878	+	0.768477i	$0.721015\pi$
$380$	0	0
$381$	−46.4563	−2.38003
$382$	0	0
$383$	1.21709	0.0621904	0.0310952	−	0.999516i	$-0.490100\pi$
$383$	1.21709	0.0621904	0.0310952	+	0.999516i	$0.490100\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−32.4690	−1.65049
$388$	0	0
$389$	−26.5578	−1.34653	−0.673267	−	0.739400i	$-0.735109\pi$
$389$	−26.5578	−1.34653	−0.673267	+	0.739400i	$0.735109\pi$
$390$	0	0
$391$	−7.37709	−0.373076
$392$	0	0
$393$	−29.8985	−1.50818
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.5865	1.43472	0.717358	−	0.696705i	$-0.245351\pi$
$397$	28.5865	1.43472	0.717358	+	0.696705i	$0.245351\pi$
$398$	0	0
$399$	32.5451	1.62929
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	−10.3614	−0.516138
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.06814	−0.449491
$408$	0	0
$409$	24.5865	1.21572	0.607862	−	0.794042i	$-0.292027\pi$
$409$	24.5865	1.21572	0.607862	+	0.794042i	$0.292027\pi$
$410$	0	0
$411$	−32.0474	−1.58078
$412$	0	0
$413$	−4.84302	−0.238310
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.3327	0.995695
$418$	0	0
$419$	−1.31698	−0.0643387	−0.0321693	−	0.999482i	$-0.510242\pi$
$419$	−1.31698	−0.0643387	−0.0321693	+	0.999482i	$0.510242\pi$
$420$	0	0
$421$	36.0662	1.75776	0.878879	−	0.477045i	$-0.158292\pi$
$421$	36.0662	1.75776	0.878879	+	0.477045i	$0.158292\pi$
$422$	0	0
$423$	1.44221	0.0701225
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.6179	0.997770
$428$	0	0
$429$	41.1156	1.98508
$430$	0	0
$431$	24.2963	1.17031	0.585155	−	0.810921i	$-0.301034\pi$
$431$	24.2963	1.17031	0.585155	+	0.810921i	$0.301034\pi$
$432$	0	0
$433$	−11.6259	−0.558707	−0.279353	−	0.960188i	$-0.590120\pi$
$433$	−11.6259	−0.558707	−0.279353	+	0.960188i	$0.590120\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	37.3881	1.78852
$438$	0	0
$439$	37.7572	1.80205	0.901027	−	0.433764i	$-0.142815\pi$
$439$	37.7572	1.80205	0.901027	+	0.433764i	$0.142815\pi$
$440$	0	0
$441$	−1.97930	−0.0942526
$442$	0	0
$443$	−12.0287	−0.571502	−0.285751	−	0.958304i	$-0.592243\pi$
$443$	−12.0287	−0.571502	−0.285751	+	0.958304i	$0.592243\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	51.0267	2.41348
$448$	0	0
$449$	−36.0061	−1.69923	−0.849615	−	0.527403i	$-0.823166\pi$
$449$	−36.0061	−1.69923	−0.849615	+	0.527403i	$0.823166\pi$
$450$	0	0
$451$	32.7128	1.54039
$452$	0	0
$453$	−25.1156	−1.18003
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.1550	0.662143	0.331072	−	0.943606i	$-0.392590\pi$
$457$	14.1550	0.662143	0.331072	+	0.943606i	$0.392590\pi$
$458$	0	0
$459$	−1.06814	−0.0498565
$460$	0	0
$461$	−3.68605	−0.171676	−0.0858382	−	0.996309i	$-0.527357\pi$
$461$	−3.68605	−0.171676	−0.0858382	+	0.996309i	$0.527357\pi$
$462$	0	0
$463$	−32.0662	−1.49024	−0.745121	−	0.666930i	$-0.767608\pi$
$463$	−32.0662	−1.49024	−0.745121	+	0.666930i	$0.767608\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.6673	−0.817546	−0.408773	−	0.912636i	$-0.634043\pi$
$467$	−17.6673	−0.817546	−0.408773	+	0.912636i	$0.634043\pi$
$468$	0	0
$469$	−19.2044	−0.886778
$470$	0	0
$471$	12.6179	0.581402
$472$	0	0
$473$	−28.0474	−1.28962
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−23.4135	−1.07203
$478$	0	0
$479$	−6.55279	−0.299405	−0.149702	−	0.988731i	$-0.547832\pi$
$479$	−6.55279	−0.299405	−0.149702	+	0.988731i	$0.547832\pi$
$480$	0	0
$481$	−16.8430	−0.767976
$482$	0	0
$483$	47.3721	2.15550
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.4927	1.47238	0.736192	−	0.676773i	$-0.236622\pi$
$487$	32.4927	1.47238	0.736192	+	0.676773i	$0.236622\pi$
$488$	0	0
$489$	36.9506	1.67096
$490$	0	0
$491$	−12.8905	−0.581739	−0.290869	−	0.956763i	$-0.593944\pi$
$491$	−12.8905	−0.581739	−0.290869	+	0.956763i	$0.593944\pi$
$492$	0	0
$493$	7.91116	0.356301
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.64663	−0.118718
$498$	0	0
$499$	−41.3170	−1.84960	−0.924801	−	0.380451i	$-0.875769\pi$
$499$	−41.3170	−1.84960	−0.924801	+	0.380451i	$0.875769\pi$
$500$	0	0
$501$	11.7148	0.523377
$502$	0	0
$503$	1.69105	0.0754000	0.0377000	−	0.999289i	$-0.487997\pi$
$503$	1.69105	0.0754000	0.0377000	+	0.999289i	$0.487997\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	43.4245	1.92855
$508$	0	0
$509$	42.2726	1.87370	0.936849	−	0.349734i	$-0.113728\pi$
$509$	42.2726	1.87370	0.936849	+	0.349734i	$0.113728\pi$
$510$	0	0
$511$	15.7749	0.697840
$512$	0	0
$513$	5.41349	0.239011
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.24581	0.0547908
$518$	0	0
$519$	−40.3200	−1.76985
$520$	0	0
$521$	27.6860	1.21295	0.606474	−	0.795103i	$-0.292583\pi$
$521$	27.6860	1.21295	0.606474	+	0.795103i	$0.292583\pi$
$522$	0	0
$523$	−12.2438	−0.535386	−0.267693	−	0.963504i	$-0.586261\pi$
$523$	−12.2438	−0.535386	−0.267693	+	0.963504i	$0.586261\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.88744	0.0822182
$528$	0	0
$529$	31.4215	1.36615
$530$	0	0
$531$	−6.53907	−0.283771
$532$	0	0
$533$	60.7602	2.63182
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.570462	−0.0246173
$538$	0	0
$539$	−1.70977	−0.0736450
$540$	0	0
$541$	9.20442	0.395729	0.197864	−	0.980229i	$-0.436599\pi$
$541$	9.20442	0.395729	0.197864	+	0.980229i	$0.436599\pi$
$542$	0	0
$543$	−2.36140	−0.101337
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−22.4038	−0.957919	−0.478960	−	0.877837i	$-0.658986\pi$
$547$	−22.4038	−0.957919	−0.478960	+	0.877837i	$0.658986\pi$
$548$	0	0
$549$	27.8384	1.18811
$550$	0	0
$551$	−40.0949	−1.70810
$552$	0	0
$553$	2.64663	0.112546
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.66732	0.0706468	0.0353234	−	0.999376i	$-0.488754\pi$
$557$	1.66732	0.0706468	0.0353234	+	0.999376i	$0.488754\pi$
$558$	0	0
$559$	−52.0949	−2.20338
$560$	0	0
$561$	−7.48965	−0.316213
$562$	0	0
$563$	5.93989	0.250336	0.125168	−	0.992136i	$-0.460053\pi$
$563$	5.93989	0.250336	0.125168	+	0.992136i	$0.460053\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−19.1520	−0.804307
$568$	0	0
$569$	−15.5084	−0.650145	−0.325072	−	0.945689i	$-0.605389\pi$
$569$	−15.5084	−0.650145	−0.325072	+	0.945689i	$0.605389\pi$
$570$	0	0
$571$	−33.7098	−1.41071	−0.705355	−	0.708854i	$-0.749212\pi$
$571$	−33.7098	−1.41071	−0.705355	+	0.708854i	$0.749212\pi$
$572$	0	0
$573$	−43.8223	−1.83070
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.1757	1.29786	0.648931	−	0.760848i	$-0.275217\pi$
$577$	31.1757	1.29786	0.648931	+	0.760848i	$0.275217\pi$
$578$	0	0
$579$	61.7335	2.56556
$580$	0	0
$581$	−36.3200	−1.50681
$582$	0	0
$583$	−20.2251	−0.837639
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.3327	−0.756671	−0.378335	−	0.925669i	$-0.623503\pi$
$587$	−18.3327	−0.756671	−0.378335	+	0.925669i	$0.623503\pi$
$588$	0	0
$589$	−9.56582	−0.394153
$590$	0	0
$591$	−20.0474	−0.824642
$592$	0	0
$593$	−7.95861	−0.326821	−0.163410	−	0.986558i	$-0.552249\pi$
$593$	−7.95861	−0.326821	−0.163410	+	0.986558i	$0.552249\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.0394	0.942941
$598$	0	0
$599$	22.9793	0.938909	0.469454	−	0.882957i	$-0.344451\pi$
$599$	22.9793	0.938909	0.469454	+	0.882957i	$0.344451\pi$
$600$	0	0
$601$	20.7542	0.846581	0.423290	−	0.905994i	$-0.360875\pi$
$601$	20.7542	0.846581	0.423290	+	0.905994i	$0.360875\pi$
$602$	0	0
$603$	−25.9299	−1.05595
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11.6497	−0.472845	−0.236422	−	0.971650i	$-0.575975\pi$
$607$	−11.6497	−0.472845	−0.236422	+	0.971650i	$0.575975\pi$
$608$	0	0
$609$	−50.8016	−2.05859
$610$	0	0
$611$	2.31395	0.0936125
$612$	0	0
$613$	1.15698	0.0467298	0.0233649	−	0.999727i	$-0.492562\pi$
$613$	1.15698	0.0467298	0.0233649	+	0.999727i	$0.492562\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.2044	0.531590	0.265795	−	0.964030i	$-0.414366\pi$
$617$	13.2044	0.531590	0.265795	+	0.964030i	$0.414366\pi$
$618$	0	0
$619$	−21.8874	−0.879731	−0.439865	−	0.898064i	$-0.644974\pi$
$619$	−21.8874	−0.879731	−0.439865	+	0.898064i	$0.644974\pi$
$620$	0	0
$621$	7.87977	0.316204
$622$	0	0
$623$	−6.48163	−0.259681
$624$	0	0
$625$	0	0
$626$	0	0
$627$	37.9586	1.51592
$628$	0	0
$629$	3.06814	0.122335
$630$	0	0
$631$	−19.0267	−0.757443	−0.378721	−	0.925511i	$-0.623636\pi$
$631$	−19.0267	−0.757443	−0.378721	+	0.925511i	$0.623636\pi$
$632$	0	0
$633$	−66.2913	−2.63484
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.17570	−0.125826
$638$	0	0
$639$	−3.57349	−0.141365
$640$	0	0
$641$	18.0474	0.712831	0.356416	−	0.934328i	$-0.383999\pi$
$641$	18.0474	0.712831	0.356416	+	0.934328i	$0.383999\pi$
$642$	0	0
$643$	6.35640	0.250672	0.125336	−	0.992114i	$-0.459999\pi$
$643$	6.35640	0.250672	0.125336	+	0.992114i	$0.459999\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.0869	−1.14352	−0.571761	−	0.820420i	$-0.693740\pi$
$647$	−29.0869	−1.14352	−0.571761	+	0.820420i	$0.693740\pi$
$648$	0	0
$649$	−5.64860	−0.221727
$650$	0	0
$651$	−12.1202	−0.475029
$652$	0	0
$653$	−32.0061	−1.25249	−0.626247	−	0.779625i	$-0.715410\pi$
$653$	−32.0061	−1.25249	−0.626247	+	0.779625i	$0.715410\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	21.2993	0.830965
$658$	0	0
$659$	−23.2519	−0.905764	−0.452882	−	0.891570i	$-0.649604\pi$
$659$	−23.2519	−0.905764	−0.452882	+	0.891570i	$0.649604\pi$
$660$	0	0
$661$	−22.0949	−0.859392	−0.429696	−	0.902974i	$-0.641379\pi$
$661$	−22.0949	−0.859392	−0.429696	+	0.902974i	$0.641379\pi$
$662$	0	0
$663$	−13.9112	−0.540265
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−58.3614	−2.25976
$668$	0	0
$669$	42.1837	1.63092
$670$	0	0
$671$	24.0474	0.928341
$672$	0	0
$673$	21.2044	0.817370	0.408685	−	0.912675i	$-0.365987\pi$
$673$	21.2044	0.817370	0.408685	+	0.912675i	$0.365987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.1156	0.427207	0.213603	−	0.976920i	$-0.431480\pi$
$677$	11.1156	0.427207	0.213603	+	0.976920i	$0.431480\pi$
$678$	0	0
$679$	−35.0267	−1.34420
$680$	0	0
$681$	−11.1443	−0.427051
$682$	0	0
$683$	−5.46593	−0.209148	−0.104574	−	0.994517i	$-0.533348\pi$
$683$	−5.46593	−0.209148	−0.104574	+	0.994517i	$0.533348\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−42.3040	−1.61400
$688$	0	0
$689$	−37.5658	−1.43114
$690$	0	0
$691$	12.6416	0.480910	0.240455	−	0.970660i	$-0.422703\pi$
$691$	12.6416	0.480910	0.240455	+	0.970660i	$0.422703\pi$
$692$	0	0
$693$	25.6259	0.973449
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.0681	−0.419236
$698$	0	0
$699$	45.1630	1.70822
$700$	0	0
$701$	−45.8097	−1.73021	−0.865103	−	0.501593i	$-0.832747\pi$
$701$	−45.8097	−1.73021	−0.865103	+	0.501593i	$0.832747\pi$
$702$	0	0
$703$	−15.5498	−0.586471
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.3407	0.802600
$708$	0	0
$709$	−12.1837	−0.457569	−0.228785	−	0.973477i	$-0.573475\pi$
$709$	−12.1837	−0.457569	−0.228785	+	0.973477i	$0.573475\pi$
$710$	0	0
$711$	3.57349	0.134016
$712$	0	0
$713$	−13.9238	−0.521452
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.3721	−1.61976
$718$	0	0
$719$	43.6209	1.62679	0.813393	−	0.581714i	$-0.197618\pi$
$719$	43.6209	1.62679	0.813393	+	0.581714i	$0.197618\pi$
$720$	0	0
$721$	−26.7542	−0.996378
$722$	0	0
$723$	−12.4977	−0.464794
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−12.8717	−0.477387	−0.238693	−	0.971095i	$-0.576719\pi$
$727$	−12.8717	−0.477387	−0.238693	+	0.971095i	$0.576719\pi$
$728$	0	0
$729$	−33.9793	−1.25849
$730$	0	0
$731$	9.48965	0.350987
$732$	0	0
$733$	−5.15698	−0.190477	−0.0952386	−	0.995454i	$-0.530361\pi$
$733$	−5.15698	−0.190477	−0.0952386	+	0.995454i	$0.530361\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.3988	−0.825072
$738$	0	0
$739$	−13.0681	−0.480719	−0.240360	−	0.970684i	$-0.577265\pi$
$739$	−13.0681	−0.480719	−0.240360	+	0.970684i	$0.577265\pi$
$740$	0	0
$741$	70.5037	2.59002
$742$	0	0
$743$	−10.7117	−0.392976	−0.196488	−	0.980506i	$-0.562954\pi$
$743$	−10.7117	−0.392976	−0.196488	+	0.980506i	$0.562954\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−49.0394	−1.79426
$748$	0	0
$749$	−17.1283	−0.625853
$750$	0	0
$751$	21.2695	0.776136	0.388068	−	0.921631i	$-0.373143\pi$
$751$	21.2695	0.776136	0.388068	+	0.921631i	$0.373143\pi$
$752$	0	0
$753$	29.9586	1.09175
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.5845	−0.639121	−0.319561	−	0.947566i	$-0.603535\pi$
$757$	−17.5845	−0.639121	−0.319561	+	0.947566i	$0.603535\pi$
$758$	0	0
$759$	55.2519	2.00552
$760$	0	0
$761$	−22.3801	−0.811279	−0.405639	−	0.914033i	$-0.632951\pi$
$761$	−22.3801	−0.811279	−0.405639	+	0.914033i	$0.632951\pi$
$762$	0	0
$763$	40.8905	1.48033
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.4916	−0.378831
$768$	0	0
$769$	−20.2438	−0.730012	−0.365006	−	0.931005i	$-0.618933\pi$
$769$	−20.2438	−0.730012	−0.365006	+	0.931005i	$0.618933\pi$
$770$	0	0
$771$	−75.4195	−2.71617
$772$	0	0
$773$	−33.9399	−1.22073	−0.610366	−	0.792119i	$-0.708978\pi$
$773$	−33.9399	−1.22073	−0.610366	+	0.792119i	$0.708978\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−19.7021	−0.706809
$778$	0	0
$779$	56.0949	2.00981
$780$	0	0
$781$	−3.08686	−0.110457
$782$	0	0
$783$	−8.45023	−0.301987
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.3771	0.833303	0.416652	−	0.909066i	$-0.363203\pi$
$787$	23.3771	0.833303	0.416652	+	0.909066i	$0.363203\pi$
$788$	0	0
$789$	20.7702	0.739440
$790$	0	0
$791$	7.77488	0.276443
$792$	0	0
$793$	44.6654	1.58611
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.97930	0.176376	0.0881880	−	0.996104i	$-0.471892\pi$
$797$	4.97930	0.176376	0.0881880	+	0.996104i	$0.471892\pi$
$798$	0	0
$799$	−0.421512	−0.0149120
$800$	0	0
$801$	−8.75152	−0.309220
$802$	0	0
$803$	18.3988	0.649281
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.4770	0.685622
$808$	0	0
$809$	−37.4295	−1.31595	−0.657976	−	0.753039i	$-0.728587\pi$
$809$	−37.4295	−1.31595	−0.657976	+	0.753039i	$0.728587\pi$
$810$	0	0
$811$	4.68907	0.164656	0.0823278	−	0.996605i	$-0.473765\pi$
$811$	4.68907	0.164656	0.0823278	+	0.996605i	$0.473765\pi$
$812$	0	0
$813$	50.6814	1.77747
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−48.0949	−1.68263
$818$	0	0
$819$	47.5972	1.66318
$820$	0	0
$821$	10.8430	0.378424	0.189212	−	0.981936i	$-0.439407\pi$
$821$	10.8430	0.378424	0.189212	+	0.981936i	$0.439407\pi$
$822$	0	0
$823$	−6.13128	−0.213723	−0.106862	−	0.994274i	$-0.534080\pi$
$823$	−6.13128	−0.213723	−0.106862	+	0.994274i	$0.534080\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.5608	1.02793	0.513965	−	0.857811i	$-0.328176\pi$
$827$	29.5608	1.02793	0.513965	+	0.857811i	$0.328176\pi$
$828$	0	0
$829$	−21.6447	−0.751750	−0.375875	−	0.926670i	$-0.622658\pi$
$829$	−21.6447	−0.751750	−0.375875	+	0.926670i	$0.622658\pi$
$830$	0	0
$831$	67.5719	2.34404
$832$	0	0
$833$	0.578488	0.0200434
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.01605	−0.0696850
$838$	0	0
$839$	−29.6623	−1.02406	−0.512029	−	0.858968i	$-0.671106\pi$
$839$	−29.6623	−1.02406	−0.512029	+	0.858968i	$0.671106\pi$
$840$	0	0
$841$	33.5865	1.15816
$842$	0	0
$843$	−0.795579	−0.0274012
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.73849	−0.197177
$848$	0	0
$849$	62.6366	2.14968
$850$	0	0
$851$	−22.6340	−0.775882
$852$	0	0
$853$	48.7542	1.66931	0.834656	−	0.550772i	$-0.185667\pi$
$853$	48.7542	1.66931	0.834656	+	0.550772i	$0.185667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.0061	−0.683394	−0.341697	−	0.939810i	$-0.611002\pi$
$857$	−20.0061	−0.683394	−0.341697	+	0.939810i	$0.611002\pi$
$858$	0	0
$859$	−37.0681	−1.26475	−0.632374	−	0.774663i	$-0.717920\pi$
$859$	−37.0681	−1.26475	−0.632374	+	0.774663i	$0.717920\pi$
$860$	0	0
$861$	71.0742	2.42220
$862$	0	0
$863$	34.8304	1.18564	0.592820	−	0.805335i	$-0.298015\pi$
$863$	34.8304	1.18564	0.592820	+	0.805335i	$0.298015\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.53407	0.0860615
$868$	0	0
$869$	3.08686	0.104715
$870$	0	0
$871$	−41.6033	−1.40967
$872$	0	0
$873$	−47.2933	−1.60063
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.50837	0.253540	0.126770	−	0.991932i	$-0.459539\pi$
$877$	7.50837	0.253540	0.126770	+	0.991932i	$0.459539\pi$
$878$	0	0
$879$	12.6179	0.425591
$880$	0	0
$881$	29.2044	0.983922	0.491961	−	0.870617i	$-0.336280\pi$
$881$	29.2044	0.983922	0.491961	+	0.870617i	$0.336280\pi$
$882$	0	0
$883$	−17.4422	−0.586977	−0.293489	−	0.955963i	$-0.594816\pi$
$883$	−17.4422	−0.586977	−0.293489	+	0.955963i	$0.594816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.80058	0.161188	0.0805938	−	0.996747i	$-0.474318\pi$
$887$	4.80058	0.161188	0.0805938	+	0.996747i	$0.474318\pi$
$888$	0	0
$889$	−46.4563	−1.55809
$890$	0	0
$891$	−22.3377	−0.748340
$892$	0	0
$893$	2.13628	0.0714879
$894$	0	0
$895$	0	0
$896$	0	0
$897$	102.624	3.42651
$898$	0	0
$899$	14.9319	0.498005
$900$	0	0
$901$	6.84302	0.227974
$902$	0	0
$903$	−60.9379	−2.02789
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−18.3564	−0.609514	−0.304757	−	0.952430i	$-0.598575\pi$
$907$	−18.3564	−0.609514	−0.304757	+	0.952430i	$0.598575\pi$
$908$	0	0
$909$	28.8143	0.955710
$910$	0	0
$911$	16.6891	0.552934	0.276467	−	0.961023i	$-0.410836\pi$
$911$	16.6891	0.552934	0.276467	+	0.961023i	$0.410836\pi$
$912$	0	0
$913$	−42.3614	−1.40196
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29.8985	−0.987335
$918$	0	0
$919$	−33.3307	−1.09948	−0.549739	−	0.835336i	$-0.685273\pi$
$919$	−33.3307	−1.09948	−0.549739	+	0.835336i	$0.685273\pi$
$920$	0	0
$921$	41.6133	1.37120
$922$	0	0
$923$	−5.73349	−0.188720
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−36.1236	−1.18646
$928$	0	0
$929$	44.8491	1.47145	0.735725	−	0.677280i	$-0.236841\pi$
$929$	44.8491	1.47145	0.735725	+	0.677280i	$0.236841\pi$
$930$	0	0
$931$	−2.93186	−0.0960878
$932$	0	0
$933$	37.8985	1.24074
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.8223	0.582230	0.291115	−	0.956688i	$-0.405974\pi$
$937$	17.8223	0.582230	0.291115	+	0.956688i	$0.405974\pi$
$938$	0	0
$939$	40.3200	1.31579
$940$	0	0
$941$	28.3614	0.924555	0.462278	−	0.886735i	$-0.347032\pi$
$941$	28.3614	0.924555	0.462278	+	0.886735i	$0.347032\pi$
$942$	0	0
$943$	81.6507	2.65891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−54.5815	−1.77366	−0.886830	−	0.462095i	$-0.847098\pi$
$947$	−54.5815	−1.77366	−0.886830	+	0.462095i	$0.847098\pi$
$948$	0	0
$949$	34.1737	1.10933
$950$	0	0
$951$	−2.93186	−0.0950721
$952$	0	0
$953$	51.4483	1.66657	0.833286	−	0.552842i	$-0.186457\pi$
$953$	51.4483	1.66657	0.833286	+	0.552842i	$0.186457\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−59.2519	−1.91534
$958$	0	0
$959$	−32.0474	−1.03487
$960$	0	0
$961$	−27.4376	−0.885083
$962$	0	0
$963$	−23.1266	−0.745245
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.0969	1.19295	0.596477	−	0.802630i	$-0.296567\pi$
$967$	37.0969	1.19295	0.596477	+	0.802630i	$0.296567\pi$
$968$	0	0
$969$	−12.8430	−0.412577
$970$	0	0
$971$	40.1423	1.28823	0.644114	−	0.764929i	$-0.277226\pi$
$971$	40.1423	1.28823	0.644114	+	0.764929i	$0.277226\pi$
$972$	0	0
$973$	20.3327	0.651836
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.95861	0.126647	0.0633236	−	0.997993i	$-0.479830\pi$
$977$	3.95861	0.126647	0.0633236	+	0.997993i	$0.479830\pi$
$978$	0	0
$979$	−7.55977	−0.241611
$980$	0	0
$981$	55.2105	1.76273
$982$	0	0
$983$	13.4659	0.429496	0.214748	−	0.976669i	$-0.431107\pi$
$983$	13.4659	0.429496	0.214748	+	0.976669i	$0.431107\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.70674	0.0861566
$988$	0	0
$989$	−70.0061	−2.22606
$990$	0	0
$991$	34.7779	1.10476	0.552378	−	0.833594i	$-0.313720\pi$
$991$	34.7779	1.10476	0.552378	+	0.833594i	$0.313720\pi$
$992$	0	0
$993$	68.9379	2.18768
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18.0474	−0.571568	−0.285784	−	0.958294i	$-0.592254\pi$
$997$	−18.0474	−0.571568	−0.285784	+	0.958294i	$0.592254\pi$
$998$	0	0
$999$	−3.27720	−0.103686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bm.1.3		3
4.3	odd	2		1700.2.a.e.1.1		3
5.4	even	2		1360.2.a.s.1.1		3
20.3	even	4		1700.2.e.d.749.2		6
20.7	even	4		1700.2.e.d.749.5		6
20.19	odd	2		340.2.a.b.1.3	✓	3
40.19	odd	2		5440.2.a.bq.1.1		3
40.29	even	2		5440.2.a.br.1.3		3
60.59	even	2		3060.2.a.s.1.3		3
340.259	odd	4		5780.2.c.f.5201.2		6
340.319	odd	4		5780.2.c.f.5201.5		6
340.339	odd	2		5780.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.a.b.1.3	✓	3	20.19	odd	2
1360.2.a.s.1.1		3	5.4	even	2
1700.2.a.e.1.1		3	4.3	odd	2
1700.2.e.d.749.2		6	20.3	even	4
1700.2.e.d.749.5		6	20.7	even	4
3060.2.a.s.1.3		3	60.59	even	2
5440.2.a.bq.1.1		3	40.19	odd	2
5440.2.a.br.1.3		3	40.29	even	2
5780.2.a.j.1.1		3	340.339	odd	2
5780.2.c.f.5201.2		6	340.259	odd	4
5780.2.c.f.5201.5		6	340.319	odd	4
6800.2.a.bm.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53407 q^{3} +2.53407 q^{7} +3.42151 q^{9} +O(q^{10})\)	\(q+2.53407 q^{3} +2.53407 q^{7} +3.42151 q^{9} +2.95558 q^{11} +5.48965 q^{13} -1.00000 q^{17} +5.06814 q^{19} +6.42151 q^{21} +7.37709 q^{23} +1.06814 q^{27} -7.91116 q^{29} -1.88744 q^{31} +7.48965 q^{33} -3.06814 q^{37} +13.9112 q^{39} +11.0681 q^{41} -9.48965 q^{43} +0.421512 q^{47} -0.578488 q^{49} -2.53407 q^{51} -6.84302 q^{53} +12.8430 q^{57} -1.91116 q^{59} +8.13628 q^{61} +8.67035 q^{63} -7.57849 q^{67} +18.6941 q^{69} -1.04442 q^{71} +6.22512 q^{73} +7.48965 q^{77} +1.04442 q^{79} -7.55779 q^{81} -14.3327 q^{83} -20.0474 q^{87} -2.55779 q^{89} +13.9112 q^{91} -4.78291 q^{93} -13.8223 q^{97} +10.1126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{9}+O(q^{10}) \) 3 * q + 7 * q^9	\( 3 q + 7 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{17} + 16 q^{21} + 8 q^{23} - 12 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 6 q^{37} + 20 q^{39} + 18 q^{41} - 10 q^{43} - 2 q^{47} - 5 q^{49} - 14 q^{53} + 32 q^{57} + 16 q^{59} - 6 q^{61} - 12 q^{63} - 26 q^{67} - 8 q^{69} - 14 q^{71} + 10 q^{73} + 4 q^{77} + 14 q^{79} + 11 q^{81} - 18 q^{83} - 8 q^{87} + 26 q^{89} + 20 q^{91} + 28 q^{93} + 2 q^{97} + 26 q^{99}+O(q^{100}) \) 3 * q + 7 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^17 + 16 * q^21 + 8 * q^23 - 12 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 6 * q^37 + 20 * q^39 + 18 * q^41 - 10 * q^43 - 2 * q^47 - 5 * q^49 - 14 * q^53 + 32 * q^57 + 16 * q^59 - 6 * q^61 - 12 * q^63 - 26 * q^67 - 8 * q^69 - 14 * q^71 + 10 * q^73 + 4 * q^77 + 14 * q^79 + 11 * q^81 - 18 * q^83 - 8 * q^87 + 26 * q^89 + 20 * q^91 + 28 * q^93 + 2 * q^97 + 26 * q^99

Introduction

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