LMFDB - Embedded newform 8400.2.a.dl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8400,2,Mod(1,8400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8400, 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("8400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.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:	$67.0743376979$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	8400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -6.31265 q^{11} +6.96239 q^{13} +6.57452 q^{17} +3.73813 q^{19} +1.00000 q^{21} -5.73813 q^{23} +1.00000 q^{27} -2.00000 q^{29} +1.03761 q^{31} -6.31265 q^{33} +10.7005 q^{37} +6.96239 q^{39} -6.96239 q^{41} +5.92478 q^{43} +1.00000 q^{49} +6.57452 q^{51} +1.03761 q^{53} +3.73813 q^{57} +3.22425 q^{59} -13.8496 q^{61} +1.00000 q^{63} +4.77575 q^{67} -5.73813 q^{69} -8.23743 q^{71} +4.26187 q^{73} -6.31265 q^{77} +5.92478 q^{79} +1.00000 q^{81} -3.22425 q^{83} -2.00000 q^{87} +2.18664 q^{89} +6.96239 q^{91} +1.03761 q^{93} +3.73813 q^{97} -6.31265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 10 q^{13} + 8 q^{17} + 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} + 14 q^{31} + 2 q^{33} + 12 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} + 8 q^{51} + 14 q^{53} + 2 q^{57} + 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} + 18 q^{71} + 22 q^{73} + 2 q^{77} - 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} + 10 q^{91} + 14 q^{93} + 2 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 10 * q^13 + 8 * q^17 + 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 + 14 * q^31 + 2 * q^33 + 12 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 + 8 * q^51 + 14 * q^53 + 2 * q^57 + 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 + 18 * q^71 + 22 * q^73 + 2 * q^77 - 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 + 10 * q^91 + 14 * q^93 + 2 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.31265	−1.90334	−0.951668	−	0.307129i	$-0.900632\pi$
$11$	−6.31265	−1.90334	−0.951668	+	0.307129i	$0.900632\pi$
$12$	0	0
$13$	6.96239	1.93102	0.965510	−	0.260368i	$-0.0838437\pi$
$13$	6.96239	1.93102	0.965510	+	0.260368i	$0.0838437\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.57452	1.59455	0.797277	−	0.603613i	$-0.206273\pi$
$17$	6.57452	1.59455	0.797277	+	0.603613i	$0.206273\pi$
$18$	0	0
$19$	3.73813	0.857587	0.428793	−	0.903403i	$-0.358939\pi$
$19$	3.73813	0.857587	0.428793	+	0.903403i	$0.358939\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.73813	−1.19648	−0.598242	−	0.801316i	$-0.704134\pi$
$23$	−5.73813	−1.19648	−0.598242	+	0.801316i	$0.704134\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	1.03761	0.186361	0.0931803	−	0.995649i	$-0.470297\pi$
$31$	1.03761	0.186361	0.0931803	+	0.995649i	$0.470297\pi$
$32$	0	0
$33$	−6.31265	−1.09889
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.7005	1.75916	0.879578	−	0.475755i	$-0.157825\pi$
$37$	10.7005	1.75916	0.879578	+	0.475755i	$0.157825\pi$
$38$	0	0
$39$	6.96239	1.11487
$40$	0	0
$41$	−6.96239	−1.08734	−0.543671	−	0.839298i	$-0.682966\pi$
$41$	−6.96239	−1.08734	−0.543671	+	0.839298i	$0.682966\pi$
$42$	0	0
$43$	5.92478	0.903520	0.451760	−	0.892139i	$-0.350796\pi$
$43$	5.92478	0.903520	0.451760	+	0.892139i	$0.350796\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.57452	0.920616
$52$	0	0
$53$	1.03761	0.142527	0.0712634	−	0.997458i	$-0.477297\pi$
$53$	1.03761	0.142527	0.0712634	+	0.997458i	$0.477297\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.73813	0.495128
$58$	0	0
$59$	3.22425	0.419762	0.209881	−	0.977727i	$-0.432692\pi$
$59$	3.22425	0.419762	0.209881	+	0.977727i	$0.432692\pi$
$60$	0	0
$61$	−13.8496	−1.77325	−0.886627	−	0.462485i	$-0.846958\pi$
$61$	−13.8496	−1.77325	−0.886627	+	0.462485i	$0.846958\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.77575	0.583450	0.291725	−	0.956502i	$-0.405771\pi$
$67$	4.77575	0.583450	0.291725	+	0.956502i	$0.405771\pi$
$68$	0	0
$69$	−5.73813	−0.690790
$70$	0	0
$71$	−8.23743	−0.977603	−0.488801	−	0.872395i	$-0.662566\pi$
$71$	−8.23743	−0.977603	−0.488801	+	0.872395i	$0.662566\pi$
$72$	0	0
$73$	4.26187	0.498814	0.249407	−	0.968399i	$-0.419764\pi$
$73$	4.26187	0.498814	0.249407	+	0.968399i	$0.419764\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.31265	−0.719393
$78$	0	0
$79$	5.92478	0.666590	0.333295	−	0.942823i	$-0.391840\pi$
$79$	5.92478	0.666590	0.333295	+	0.942823i	$0.391840\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.22425	−0.353908	−0.176954	−	0.984219i	$-0.556624\pi$
$83$	−3.22425	−0.353908	−0.176954	+	0.984219i	$0.556624\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	2.18664	0.231784	0.115892	−	0.993262i	$-0.463027\pi$
$89$	2.18664	0.231784	0.115892	+	0.993262i	$0.463027\pi$
$90$	0	0
$91$	6.96239	0.729857
$92$	0	0
$93$	1.03761	0.107595
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.73813	0.379550	0.189775	−	0.981828i	$-0.439224\pi$
$97$	3.73813	0.379550	0.189775	+	0.981828i	$0.439224\pi$
$98$	0	0
$99$	−6.31265	−0.634445
$100$	0	0
$101$	9.66291	0.961496	0.480748	−	0.876859i	$-0.340365\pi$
$101$	9.66291	0.961496	0.480748	+	0.876859i	$0.340365\pi$
$102$	0	0
$103$	−1.29948	−0.128041	−0.0640206	−	0.997949i	$-0.520392\pi$
$103$	−1.29948	−0.128041	−0.0640206	+	0.997949i	$0.520392\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.66291	0.354107	0.177054	−	0.984201i	$-0.443343\pi$
$107$	3.66291	0.354107	0.177054	+	0.984201i	$0.443343\pi$
$108$	0	0
$109$	6.77575	0.648999	0.324499	−	0.945886i	$-0.394804\pi$
$109$	6.77575	0.648999	0.324499	+	0.945886i	$0.394804\pi$
$110$	0	0
$111$	10.7005	1.01565
$112$	0	0
$113$	−8.88717	−0.836034	−0.418017	−	0.908439i	$-0.637275\pi$
$113$	−8.88717	−0.836034	−0.418017	+	0.908439i	$0.637275\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.96239	0.643673
$118$	0	0
$119$	6.57452	0.602685
$120$	0	0
$121$	28.8496	2.62269
$122$	0	0
$123$	−6.96239	−0.627777
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.70052	0.239633	0.119816	−	0.992796i	$-0.461769\pi$
$127$	2.70052	0.239633	0.119816	+	0.992796i	$0.461769\pi$
$128$	0	0
$129$	5.92478	0.521648
$130$	0	0
$131$	9.40105	0.821373	0.410687	−	0.911777i	$-0.365289\pi$
$131$	9.40105	0.821373	0.410687	+	0.911777i	$0.365289\pi$
$132$	0	0
$133$	3.73813	0.324137
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.66291	−0.825558	−0.412779	−	0.910831i	$-0.635442\pi$
$137$	−9.66291	−0.825558	−0.412779	+	0.910831i	$0.635442\pi$
$138$	0	0
$139$	−20.8872	−1.77163	−0.885813	−	0.464042i	$-0.846399\pi$
$139$	−20.8872	−1.77163	−0.885813	+	0.464042i	$0.846399\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−43.9511	−3.67538
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	0.0752228	0.00616249	0.00308125	−	0.999995i	$-0.499019\pi$
$149$	0.0752228	0.00616249	0.00308125	+	0.999995i	$0.499019\pi$
$150$	0	0
$151$	−6.70052	−0.545281	−0.272640	−	0.962116i	$-0.587897\pi$
$151$	−6.70052	−0.545281	−0.272640	+	0.962116i	$0.587897\pi$
$152$	0	0
$153$	6.57452	0.531518
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.66291	−0.451950	−0.225975	−	0.974133i	$-0.572557\pi$
$157$	−5.66291	−0.451950	−0.225975	+	0.974133i	$0.572557\pi$
$158$	0	0
$159$	1.03761	0.0822879
$160$	0	0
$161$	−5.73813	−0.452228
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.6253	1.59603	0.798017	−	0.602635i	$-0.205883\pi$
$167$	20.6253	1.59603	0.798017	+	0.602635i	$0.205883\pi$
$168$	0	0
$169$	35.4749	2.72884
$170$	0	0
$171$	3.73813	0.285862
$172$	0	0
$173$	10.5745	0.803966	0.401983	−	0.915647i	$-0.368321\pi$
$173$	10.5745	0.803966	0.401983	+	0.915647i	$0.368321\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.22425	0.242350
$178$	0	0
$179$	−0.911603	−0.0681364	−0.0340682	−	0.999420i	$-0.510846\pi$
$179$	−0.911603	−0.0681364	−0.0340682	+	0.999420i	$0.510846\pi$
$180$	0	0
$181$	−13.3258	−0.990501	−0.495250	−	0.868750i	$-0.664924\pi$
$181$	−13.3258	−0.990501	−0.495250	+	0.868750i	$0.664924\pi$
$182$	0	0
$183$	−13.8496	−1.02379
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−41.5026	−3.03497
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	25.7889	1.86602	0.933010	−	0.359849i	$-0.117172\pi$
$191$	25.7889	1.86602	0.933010	+	0.359849i	$0.117172\pi$
$192$	0	0
$193$	15.3258	1.10318	0.551588	−	0.834116i	$-0.314022\pi$
$193$	15.3258	1.10318	0.551588	+	0.834116i	$0.314022\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.513881	0.0366125	0.0183063	−	0.999832i	$-0.494173\pi$
$197$	0.513881	0.0366125	0.0183063	+	0.999832i	$0.494173\pi$
$198$	0	0
$199$	−3.73813	−0.264989	−0.132495	−	0.991184i	$-0.542299\pi$
$199$	−3.73813	−0.264989	−0.132495	+	0.991184i	$0.542299\pi$
$200$	0	0
$201$	4.77575	0.336855
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.73813	−0.398828
$208$	0	0
$209$	−23.5975	−1.63228
$210$	0	0
$211$	3.22425	0.221967	0.110983	−	0.993822i	$-0.464600\pi$
$211$	3.22425	0.221967	0.110983	+	0.993822i	$0.464600\pi$
$212$	0	0
$213$	−8.23743	−0.564419
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.03761	0.0704377
$218$	0	0
$219$	4.26187	0.287990
$220$	0	0
$221$	45.7743	3.07911
$222$	0	0
$223$	11.8496	0.793505	0.396752	−	0.917926i	$-0.370137\pi$
$223$	11.8496	0.793505	0.396752	+	0.917926i	$0.370137\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.7005	−0.975708	−0.487854	−	0.872925i	$-0.662220\pi$
$227$	−14.7005	−0.975708	−0.487854	+	0.872925i	$0.662220\pi$
$228$	0	0
$229$	9.47627	0.626210	0.313105	−	0.949719i	$-0.398631\pi$
$229$	9.47627	0.626210	0.313105	+	0.949719i	$0.398631\pi$
$230$	0	0
$231$	−6.31265	−0.415342
$232$	0	0
$233$	29.5125	1.93343	0.966713	−	0.255863i	$-0.0823597\pi$
$233$	29.5125	1.93343	0.966713	+	0.255863i	$0.0823597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.92478	0.384856
$238$	0	0
$239$	25.0132	1.61797	0.808984	−	0.587831i	$-0.200018\pi$
$239$	25.0132	1.61797	0.808984	+	0.587831i	$0.200018\pi$
$240$	0	0
$241$	7.92478	0.510480	0.255240	−	0.966878i	$-0.417846\pi$
$241$	7.92478	0.510480	0.255240	+	0.966878i	$0.417846\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.0263	1.65602
$248$	0	0
$249$	−3.22425	−0.204329
$250$	0	0
$251$	15.0738	0.951450	0.475725	−	0.879594i	$-0.342186\pi$
$251$	15.0738	0.951450	0.475725	+	0.879594i	$0.342186\pi$
$252$	0	0
$253$	36.2228	2.27731
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.0508	1.37549	0.687745	−	0.725952i	$-0.258601\pi$
$257$	22.0508	1.37549	0.687745	+	0.725952i	$0.258601\pi$
$258$	0	0
$259$	10.7005	0.664898
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−11.8134	−0.728443	−0.364221	−	0.931312i	$-0.618665\pi$
$263$	−11.8134	−0.728443	−0.364221	+	0.931312i	$0.618665\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.18664	0.133820
$268$	0	0
$269$	−27.0640	−1.65012	−0.825059	−	0.565046i	$-0.808858\pi$
$269$	−27.0640	−1.65012	−0.825059	+	0.565046i	$0.808858\pi$
$270$	0	0
$271$	10.8119	0.656779	0.328389	−	0.944542i	$-0.393494\pi$
$271$	10.8119	0.656779	0.328389	+	0.944542i	$0.393494\pi$
$272$	0	0
$273$	6.96239	0.421383
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.5501	−1.35490	−0.677451	−	0.735568i	$-0.736916\pi$
$277$	−22.5501	−1.35490	−0.677451	+	0.735568i	$0.736916\pi$
$278$	0	0
$279$	1.03761	0.0621202
$280$	0	0
$281$	−10.3733	−0.618818	−0.309409	−	0.950929i	$-0.600131\pi$
$281$	−10.3733	−0.618818	−0.309409	+	0.950929i	$0.600131\pi$
$282$	0	0
$283$	10.7005	0.636080	0.318040	−	0.948077i	$-0.396975\pi$
$283$	10.7005	0.636080	0.318040	+	0.948077i	$0.396975\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.96239	−0.410977
$288$	0	0
$289$	26.2243	1.54260
$290$	0	0
$291$	3.73813	0.219133
$292$	0	0
$293$	−10.5745	−0.617770	−0.308885	−	0.951099i	$-0.599956\pi$
$293$	−10.5745	−0.617770	−0.308885	+	0.951099i	$0.599956\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.31265	−0.366297
$298$	0	0
$299$	−39.9511	−2.31043
$300$	0	0
$301$	5.92478	0.341498
$302$	0	0
$303$	9.66291	0.555120
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.55008	−0.373833	−0.186916	−	0.982376i	$-0.559849\pi$
$307$	−6.55008	−0.373833	−0.186916	+	0.982376i	$0.559849\pi$
$308$	0	0
$309$	−1.29948	−0.0739246
$310$	0	0
$311$	17.2995	0.980963	0.490482	−	0.871452i	$-0.336821\pi$
$311$	17.2995	0.980963	0.490482	+	0.871452i	$0.336821\pi$
$312$	0	0
$313$	−20.2130	−1.14251	−0.571253	−	0.820774i	$-0.693542\pi$
$313$	−20.2130	−1.14251	−0.571253	+	0.820774i	$0.693542\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3357	1.08600	0.543000	−	0.839733i	$-0.317288\pi$
$317$	19.3357	1.08600	0.543000	+	0.839733i	$0.317288\pi$
$318$	0	0
$319$	12.6253	0.706881
$320$	0	0
$321$	3.66291	0.204444
$322$	0	0
$323$	24.5764	1.36747
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.77575	0.374700
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.6253	−1.35353	−0.676764	−	0.736200i	$-0.736618\pi$
$331$	−24.6253	−1.35353	−0.676764	+	0.736200i	$0.736618\pi$
$332$	0	0
$333$	10.7005	0.586385
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.44851	0.351273	0.175636	−	0.984455i	$-0.443802\pi$
$337$	6.44851	0.351273	0.175636	+	0.984455i	$0.443802\pi$
$338$	0	0
$339$	−8.88717	−0.482685
$340$	0	0
$341$	−6.55008	−0.354707
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.6629	0.626098	0.313049	−	0.949737i	$-0.398650\pi$
$347$	11.6629	0.626098	0.313049	+	0.949737i	$0.398650\pi$
$348$	0	0
$349$	12.4485	0.666353	0.333177	−	0.942864i	$-0.391879\pi$
$349$	12.4485	0.666353	0.333177	+	0.942864i	$0.391879\pi$
$350$	0	0
$351$	6.96239	0.371625
$352$	0	0
$353$	−13.0230	−0.693146	−0.346573	−	0.938023i	$-0.612655\pi$
$353$	−13.0230	−0.693146	−0.346573	+	0.938023i	$0.612655\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.57452	0.347960
$358$	0	0
$359$	20.3879	1.07603	0.538015	−	0.842935i	$-0.319174\pi$
$359$	20.3879	1.07603	0.538015	+	0.842935i	$0.319174\pi$
$360$	0	0
$361$	−5.02635	−0.264545
$362$	0	0
$363$	28.8496	1.51421
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.8496	1.03614	0.518069	−	0.855339i	$-0.326651\pi$
$367$	19.8496	1.03614	0.518069	+	0.855339i	$0.326651\pi$
$368$	0	0
$369$	−6.96239	−0.362447
$370$	0	0
$371$	1.03761	0.0538701
$372$	0	0
$373$	2.85097	0.147618	0.0738088	−	0.997272i	$-0.476485\pi$
$373$	2.85097	0.147618	0.0738088	+	0.997272i	$0.476485\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.9248	−0.717163
$378$	0	0
$379$	−5.29948	−0.272216	−0.136108	−	0.990694i	$-0.543459\pi$
$379$	−5.29948	−0.272216	−0.136108	+	0.990694i	$0.543459\pi$
$380$	0	0
$381$	2.70052	0.138352
$382$	0	0
$383$	12.2520	0.626049	0.313024	−	0.949745i	$-0.398658\pi$
$383$	12.2520	0.626049	0.313024	+	0.949745i	$0.398658\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.92478	0.301173
$388$	0	0
$389$	16.4485	0.833972	0.416986	−	0.908913i	$-0.363086\pi$
$389$	16.4485	0.833972	0.416986	+	0.908913i	$0.363086\pi$
$390$	0	0
$391$	−37.7255	−1.90786
$392$	0	0
$393$	9.40105	0.474220
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.58769	−0.180061	−0.0900305	−	0.995939i	$-0.528696\pi$
$397$	−3.58769	−0.180061	−0.0900305	+	0.995939i	$0.528696\pi$
$398$	0	0
$399$	3.73813	0.187141
$400$	0	0
$401$	19.9248	0.994996	0.497498	−	0.867465i	$-0.334252\pi$
$401$	19.9248	0.994996	0.497498	+	0.867465i	$0.334252\pi$
$402$	0	0
$403$	7.22425	0.359866
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−67.5487	−3.34826
$408$	0	0
$409$	−25.1754	−1.24484	−0.622421	−	0.782682i	$-0.713851\pi$
$409$	−25.1754	−1.24484	−0.622421	+	0.782682i	$0.713851\pi$
$410$	0	0
$411$	−9.66291	−0.476636
$412$	0	0
$413$	3.22425	0.158655
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−20.8872	−1.02285
$418$	0	0
$419$	−1.14903	−0.0561338	−0.0280669	−	0.999606i	$-0.508935\pi$
$419$	−1.14903	−0.0561338	−0.0280669	+	0.999606i	$0.508935\pi$
$420$	0	0
$421$	−0.176793	−0.00861637	−0.00430819	−	0.999991i	$-0.501371\pi$
$421$	−0.176793	−0.00861637	−0.00430819	+	0.999991i	$0.501371\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.8496	−0.670227
$428$	0	0
$429$	−43.9511	−2.12198
$430$	0	0
$431$	−12.3879	−0.596703	−0.298351	−	0.954456i	$-0.596437\pi$
$431$	−12.3879	−0.596703	−0.298351	+	0.954456i	$0.596437\pi$
$432$	0	0
$433$	−1.41090	−0.0678033	−0.0339017	−	0.999425i	$-0.510793\pi$
$433$	−1.41090	−0.0678033	−0.0339017	+	0.999425i	$0.510793\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.4499	−1.02609
$438$	0	0
$439$	−31.5877	−1.50760	−0.753799	−	0.657105i	$-0.771781\pi$
$439$	−31.5877	−1.50760	−0.753799	+	0.657105i	$0.771781\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.0362	−1.33204	−0.666020	−	0.745934i	$-0.732003\pi$
$443$	−28.0362	−1.33204	−0.666020	+	0.745934i	$0.732003\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.0752228	0.00355792
$448$	0	0
$449$	−19.4010	−0.915592	−0.457796	−	0.889057i	$-0.651361\pi$
$449$	−19.4010	−0.915592	−0.457796	+	0.889057i	$0.651361\pi$
$450$	0	0
$451$	43.9511	2.06958
$452$	0	0
$453$	−6.70052	−0.314818
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.3258	0.904024	0.452012	−	0.892012i	$-0.350706\pi$
$457$	19.3258	0.904024	0.452012	+	0.892012i	$0.350706\pi$
$458$	0	0
$459$	6.57452	0.306872
$460$	0	0
$461$	2.81194	0.130965	0.0654826	−	0.997854i	$-0.479141\pi$
$461$	2.81194	0.130965	0.0654826	+	0.997854i	$0.479141\pi$
$462$	0	0
$463$	−24.1016	−1.12009	−0.560047	−	0.828461i	$-0.689217\pi$
$463$	−24.1016	−1.12009	−0.560047	+	0.828461i	$0.689217\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.1490	0.978660	0.489330	−	0.872099i	$-0.337241\pi$
$467$	21.1490	0.978660	0.489330	+	0.872099i	$0.337241\pi$
$468$	0	0
$469$	4.77575	0.220523
$470$	0	0
$471$	−5.66291	−0.260933
$472$	0	0
$473$	−37.4010	−1.71970
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.03761	0.0475090
$478$	0	0
$479$	17.2995	0.790433	0.395217	−	0.918588i	$-0.370670\pi$
$479$	17.2995	0.790433	0.395217	+	0.918588i	$0.370670\pi$
$480$	0	0
$481$	74.5012	3.39696
$482$	0	0
$483$	−5.73813	−0.261094
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−40.4749	−1.83409	−0.917045	−	0.398783i	$-0.869433\pi$
$487$	−40.4749	−1.83409	−0.917045	+	0.398783i	$0.869433\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	19.9854	0.901929	0.450964	−	0.892542i	$-0.351080\pi$
$491$	19.9854	0.901929	0.450964	+	0.892542i	$0.351080\pi$
$492$	0	0
$493$	−13.1490	−0.592203
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.23743	−0.369499
$498$	0	0
$499$	−14.8510	−0.664821	−0.332410	−	0.943135i	$-0.607862\pi$
$499$	−14.8510	−0.664821	−0.332410	+	0.943135i	$0.607862\pi$
$500$	0	0
$501$	20.6253	0.921470
$502$	0	0
$503$	22.5501	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$503$	22.5501	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	35.4749	1.57549
$508$	0	0
$509$	17.5125	0.776226	0.388113	−	0.921612i	$-0.373127\pi$
$509$	17.5125	0.776226	0.388113	+	0.921612i	$0.373127\pi$
$510$	0	0
$511$	4.26187	0.188534
$512$	0	0
$513$	3.73813	0.165043
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	10.5745	0.464170
$520$	0	0
$521$	26.4387	1.15830	0.579149	−	0.815221i	$-0.303385\pi$
$521$	26.4387	1.15830	0.579149	+	0.815221i	$0.303385\pi$
$522$	0	0
$523$	−31.8496	−1.39268	−0.696342	−	0.717710i	$-0.745190\pi$
$523$	−31.8496	−1.39268	−0.696342	+	0.717710i	$0.745190\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.82179	0.297162
$528$	0	0
$529$	9.92619	0.431574
$530$	0	0
$531$	3.22425	0.139921
$532$	0	0
$533$	−48.4749	−2.09968
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.911603	−0.0393386
$538$	0	0
$539$	−6.31265	−0.271905
$540$	0	0
$541$	24.1768	1.03944	0.519721	−	0.854336i	$-0.326036\pi$
$541$	24.1768	1.03944	0.519721	+	0.854336i	$0.326036\pi$
$542$	0	0
$543$	−13.3258	−0.571866
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.1768	−1.46129	−0.730647	−	0.682755i	$-0.760781\pi$
$547$	−34.1768	−1.46129	−0.730647	+	0.682755i	$0.760781\pi$
$548$	0	0
$549$	−13.8496	−0.591085
$550$	0	0
$551$	−7.47627	−0.318500
$552$	0	0
$553$	5.92478	0.251947
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.7367	1.04813	0.524064	−	0.851679i	$-0.324415\pi$
$557$	24.7367	1.04813	0.524064	+	0.851679i	$0.324415\pi$
$558$	0	0
$559$	41.2506	1.74471
$560$	0	0
$561$	−41.5026	−1.75224
$562$	0	0
$563$	7.47627	0.315087	0.157544	−	0.987512i	$-0.449643\pi$
$563$	7.47627	0.315087	0.157544	+	0.987512i	$0.449643\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	17.3258	0.726336	0.363168	−	0.931724i	$-0.381695\pi$
$569$	17.3258	0.726336	0.363168	+	0.931724i	$0.381695\pi$
$570$	0	0
$571$	−31.8496	−1.33286	−0.666431	−	0.745567i	$-0.732179\pi$
$571$	−31.8496	−1.33286	−0.666431	+	0.745567i	$0.732179\pi$
$572$	0	0
$573$	25.7889	1.07735
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.5139	−0.854004	−0.427002	−	0.904251i	$-0.640430\pi$
$577$	−20.5139	−0.854004	−0.427002	+	0.904251i	$0.640430\pi$
$578$	0	0
$579$	15.3258	0.636920
$580$	0	0
$581$	−3.22425	−0.133765
$582$	0	0
$583$	−6.55008	−0.271277
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.8773	−0.861699	−0.430850	−	0.902424i	$-0.641786\pi$
$587$	−20.8773	−0.861699	−0.430850	+	0.902424i	$0.641786\pi$
$588$	0	0
$589$	3.87873	0.159820
$590$	0	0
$591$	0.513881	0.0211382
$592$	0	0
$593$	−5.57593	−0.228976	−0.114488	−	0.993425i	$-0.536523\pi$
$593$	−5.57593	−0.228976	−0.114488	+	0.993425i	$0.536523\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.73813	−0.152992
$598$	0	0
$599$	−1.01317	−0.0413972	−0.0206986	−	0.999786i	$-0.506589\pi$
$599$	−1.01317	−0.0413972	−0.0206986	+	0.999786i	$0.506589\pi$
$600$	0	0
$601$	−20.2981	−0.827975	−0.413988	−	0.910283i	$-0.635864\pi$
$601$	−20.2981	−0.827975	−0.413988	+	0.910283i	$0.635864\pi$
$602$	0	0
$603$	4.77575	0.194483
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.9525	0.606905	0.303452	−	0.952847i	$-0.401861\pi$
$607$	14.9525	0.606905	0.303452	+	0.952847i	$0.401861\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.0738	0.770384	0.385192	−	0.922836i	$-0.374135\pi$
$613$	19.0738	0.770384	0.385192	+	0.922836i	$0.374135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.8886	−0.639650	−0.319825	−	0.947477i	$-0.603624\pi$
$617$	−15.8886	−0.639650	−0.319825	+	0.947477i	$0.603624\pi$
$618$	0	0
$619$	−32.7367	−1.31580	−0.657900	−	0.753105i	$-0.728555\pi$
$619$	−32.7367	−1.31580	−0.657900	+	0.753105i	$0.728555\pi$
$620$	0	0
$621$	−5.73813	−0.230263
$622$	0	0
$623$	2.18664	0.0876060
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−23.5975	−0.942395
$628$	0	0
$629$	70.3508	2.80507
$630$	0	0
$631$	40.7269	1.62131	0.810656	−	0.585523i	$-0.199111\pi$
$631$	40.7269	1.62131	0.810656	+	0.585523i	$0.199111\pi$
$632$	0	0
$633$	3.22425	0.128153
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.96239	0.275860
$638$	0	0
$639$	−8.23743	−0.325868
$640$	0	0
$641$	−37.1754	−1.46834	−0.734170	−	0.678966i	$-0.762428\pi$
$641$	−37.1754	−1.46834	−0.734170	+	0.678966i	$0.762428\pi$
$642$	0	0
$643$	−18.3996	−0.725611	−0.362805	−	0.931865i	$-0.618181\pi$
$643$	−18.3996	−0.725611	−0.362805	+	0.931865i	$0.618181\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.2243	−0.598527	−0.299264	−	0.954170i	$-0.596741\pi$
$647$	−15.2243	−0.598527	−0.299264	+	0.954170i	$0.596741\pi$
$648$	0	0
$649$	−20.3536	−0.798948
$650$	0	0
$651$	1.03761	0.0406672
$652$	0	0
$653$	−0.785595	−0.0307427	−0.0153714	−	0.999882i	$-0.504893\pi$
$653$	−0.785595	−0.0307427	−0.0153714	+	0.999882i	$0.504893\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.26187	0.166271
$658$	0	0
$659$	21.2652	0.828374	0.414187	−	0.910192i	$-0.364066\pi$
$659$	21.2652	0.828374	0.414187	+	0.910192i	$0.364066\pi$
$660$	0	0
$661$	−32.4485	−1.26210	−0.631050	−	0.775742i	$-0.717376\pi$
$661$	−32.4485	−1.26210	−0.631050	+	0.775742i	$0.717376\pi$
$662$	0	0
$663$	45.7743	1.77773
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.4763	0.444363
$668$	0	0
$669$	11.8496	0.458130
$670$	0	0
$671$	87.4274	3.37510
$672$	0	0
$673$	−5.55149	−0.213994	−0.106997	−	0.994259i	$-0.534124\pi$
$673$	−5.55149	−0.213994	−0.106997	+	0.994259i	$0.534124\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.72355	−0.296840	−0.148420	−	0.988924i	$-0.547419\pi$
$677$	−7.72355	−0.296840	−0.148420	+	0.988924i	$0.547419\pi$
$678$	0	0
$679$	3.73813	0.143456
$680$	0	0
$681$	−14.7005	−0.563325
$682$	0	0
$683$	15.5125	0.593568	0.296784	−	0.954945i	$-0.404086\pi$
$683$	15.5125	0.593568	0.296784	+	0.954945i	$0.404086\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.47627	0.361542
$688$	0	0
$689$	7.22425	0.275222
$690$	0	0
$691$	24.7367	0.941029	0.470515	−	0.882392i	$-0.344068\pi$
$691$	24.7367	0.941029	0.470515	+	0.882392i	$0.344068\pi$
$692$	0	0
$693$	−6.31265	−0.239798
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−45.7743	−1.73383
$698$	0	0
$699$	29.5125	1.11626
$700$	0	0
$701$	−3.55149	−0.134138	−0.0670690	−	0.997748i	$-0.521365\pi$
$701$	−3.55149	−0.134138	−0.0670690	+	0.997748i	$0.521365\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.66291	0.363411
$708$	0	0
$709$	17.6991	0.664704	0.332352	−	0.943155i	$-0.392158\pi$
$709$	17.6991	0.664704	0.332352	+	0.943155i	$0.392158\pi$
$710$	0	0
$711$	5.92478	0.222197
$712$	0	0
$713$	−5.95395	−0.222977
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.0132	0.934134
$718$	0	0
$719$	12.3733	0.461446	0.230723	−	0.973020i	$-0.425891\pi$
$719$	12.3733	0.461446	0.230723	+	0.973020i	$0.425891\pi$
$720$	0	0
$721$	−1.29948	−0.0483950
$722$	0	0
$723$	7.92478	0.294726
$724$	0	0
$725$	0	0
$726$	0	0
$727$	47.9511	1.77841	0.889204	−	0.457510i	$-0.151259\pi$
$727$	47.9511	1.77841	0.889204	+	0.457510i	$0.151259\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.9525	1.44071
$732$	0	0
$733$	37.9149	1.40042	0.700210	−	0.713937i	$-0.253090\pi$
$733$	37.9149	1.40042	0.700210	+	0.713937i	$0.253090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30.1476	−1.11050
$738$	0	0
$739$	−41.4010	−1.52296	−0.761481	−	0.648187i	$-0.775527\pi$
$739$	−41.4010	−1.52296	−0.761481	+	0.648187i	$0.775527\pi$
$740$	0	0
$741$	26.0263	0.956102
$742$	0	0
$743$	−7.28963	−0.267430	−0.133715	−	0.991020i	$-0.542691\pi$
$743$	−7.28963	−0.267430	−0.133715	+	0.991020i	$0.542691\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.22425	−0.117969
$748$	0	0
$749$	3.66291	0.133840
$750$	0	0
$751$	−4.12127	−0.150387	−0.0751936	−	0.997169i	$-0.523957\pi$
$751$	−4.12127	−0.150387	−0.0751936	+	0.997169i	$0.523957\pi$
$752$	0	0
$753$	15.0738	0.549320
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.4471	−0.852199	−0.426100	−	0.904676i	$-0.640113\pi$
$757$	−23.4471	−0.852199	−0.426100	+	0.904676i	$0.640113\pi$
$758$	0	0
$759$	36.2228	1.31481
$760$	0	0
$761$	10.7104	0.388251	0.194125	−	0.980977i	$-0.437813\pi$
$761$	10.7104	0.388251	0.194125	+	0.980977i	$0.437813\pi$
$762$	0	0
$763$	6.77575	0.245298
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.4485	0.810569
$768$	0	0
$769$	−21.8496	−0.787915	−0.393958	−	0.919129i	$-0.628894\pi$
$769$	−21.8496	−0.787915	−0.393958	+	0.919129i	$0.628894\pi$
$770$	0	0
$771$	22.0508	0.794140
$772$	0	0
$773$	−31.9756	−1.15008	−0.575041	−	0.818125i	$-0.695014\pi$
$773$	−31.9756	−1.15008	−0.575041	+	0.818125i	$0.695014\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.7005	0.383879
$778$	0	0
$779$	−26.0263	−0.932491
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.2506	1.04267	0.521336	−	0.853352i	$-0.325434\pi$
$787$	29.2506	1.04267	0.521336	+	0.853352i	$0.325434\pi$
$788$	0	0
$789$	−11.8134	−0.420567
$790$	0	0
$791$	−8.88717	−0.315991
$792$	0	0
$793$	−96.4260	−3.42419
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.1246	−1.03165	−0.515823	−	0.856695i	$-0.672514\pi$
$797$	−29.1246	−1.03165	−0.515823	+	0.856695i	$0.672514\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2.18664	0.0772612
$802$	0	0
$803$	−26.9037	−0.949410
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−27.0640	−0.952696
$808$	0	0
$809$	4.95254	0.174122	0.0870610	−	0.996203i	$-0.472252\pi$
$809$	4.95254	0.174122	0.0870610	+	0.996203i	$0.472252\pi$
$810$	0	0
$811$	−31.6893	−1.11276	−0.556380	−	0.830928i	$-0.687810\pi$
$811$	−31.6893	−1.11276	−0.556380	+	0.830928i	$0.687810\pi$
$812$	0	0
$813$	10.8119	0.379191
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.1476	0.774847
$818$	0	0
$819$	6.96239	0.243286
$820$	0	0
$821$	−16.8218	−0.587085	−0.293542	−	0.955946i	$-0.594834\pi$
$821$	−16.8218	−0.587085	−0.293542	+	0.955946i	$0.594834\pi$
$822$	0	0
$823$	26.7005	0.930722	0.465361	−	0.885121i	$-0.345925\pi$
$823$	26.7005	0.930722	0.465361	+	0.885121i	$0.345925\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.1866	−1.11924	−0.559620	−	0.828750i	$-0.689053\pi$
$827$	−32.1866	−1.11924	−0.559620	+	0.828750i	$0.689053\pi$
$828$	0	0
$829$	46.7269	1.62289	0.811446	−	0.584428i	$-0.198681\pi$
$829$	46.7269	1.62289	0.811446	+	0.584428i	$0.198681\pi$
$830$	0	0
$831$	−22.5501	−0.782254
$832$	0	0
$833$	6.57452	0.227793
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.03761	0.0358651
$838$	0	0
$839$	44.6253	1.54064	0.770318	−	0.637660i	$-0.220097\pi$
$839$	44.6253	1.54064	0.770318	+	0.637660i	$0.220097\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−10.3733	−0.357275
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.8496	0.991282
$848$	0	0
$849$	10.7005	0.367241
$850$	0	0
$851$	−61.4010	−2.10480
$852$	0	0
$853$	−24.2422	−0.830036	−0.415018	−	0.909813i	$-0.636225\pi$
$853$	−24.2422	−0.830036	−0.415018	+	0.909813i	$0.636225\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.6023	0.532964	0.266482	−	0.963840i	$-0.414139\pi$
$857$	15.6023	0.532964	0.266482	+	0.963840i	$0.414139\pi$
$858$	0	0
$859$	−38.3371	−1.30804	−0.654022	−	0.756475i	$-0.726920\pi$
$859$	−38.3371	−1.30804	−0.654022	+	0.756475i	$0.726920\pi$
$860$	0	0
$861$	−6.96239	−0.237278
$862$	0	0
$863$	29.0640	0.989349	0.494674	−	0.869078i	$-0.335287\pi$
$863$	29.0640	0.989349	0.494674	+	0.869078i	$0.335287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.2243	0.890622
$868$	0	0
$869$	−37.4010	−1.26874
$870$	0	0
$871$	33.2506	1.12665
$872$	0	0
$873$	3.73813	0.126517
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.7757	0.566477	0.283238	−	0.959050i	$-0.408591\pi$
$877$	16.7757	0.566477	0.283238	+	0.959050i	$0.408591\pi$
$878$	0	0
$879$	−10.5745	−0.356670
$880$	0	0
$881$	−30.6907	−1.03400	−0.516998	−	0.855987i	$-0.672950\pi$
$881$	−30.6907	−1.03400	−0.516998	+	0.855987i	$0.672950\pi$
$882$	0	0
$883$	32.9986	1.11049	0.555245	−	0.831687i	$-0.312624\pi$
$883$	32.9986	1.11049	0.555245	+	0.831687i	$0.312624\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.1002	1.51432	0.757158	−	0.653232i	$-0.226588\pi$
$887$	45.1002	1.51432	0.757158	+	0.653232i	$0.226588\pi$
$888$	0	0
$889$	2.70052	0.0905727
$890$	0	0
$891$	−6.31265	−0.211482
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−39.9511	−1.33393
$898$	0	0
$899$	−2.07522	−0.0692126
$900$	0	0
$901$	6.82179	0.227267
$902$	0	0
$903$	5.92478	0.197164
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.9986	1.09570	0.547850	−	0.836577i	$-0.315446\pi$
$907$	32.9986	1.09570	0.547850	+	0.836577i	$0.315446\pi$
$908$	0	0
$909$	9.66291	0.320499
$910$	0	0
$911$	−35.8153	−1.18661	−0.593306	−	0.804977i	$-0.702178\pi$
$911$	−35.8153	−1.18661	−0.593306	+	0.804977i	$0.702178\pi$
$912$	0	0
$913$	20.3536	0.673605
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.40105	0.310450
$918$	0	0
$919$	27.0738	0.893083	0.446541	−	0.894763i	$-0.352656\pi$
$919$	27.0738	0.893083	0.446541	+	0.894763i	$0.352656\pi$
$920$	0	0
$921$	−6.55008	−0.215832
$922$	0	0
$923$	−57.3522	−1.88777
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.29948	−0.0426804
$928$	0	0
$929$	−23.5877	−0.773887	−0.386943	−	0.922103i	$-0.626469\pi$
$929$	−23.5877	−0.773887	−0.386943	+	0.922103i	$0.626469\pi$
$930$	0	0
$931$	3.73813	0.122512
$932$	0	0
$933$	17.2995	0.566359
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.0625	1.57013	0.785067	−	0.619410i	$-0.212628\pi$
$937$	48.0625	1.57013	0.785067	+	0.619410i	$0.212628\pi$
$938$	0	0
$939$	−20.2130	−0.659626
$940$	0	0
$941$	−17.4109	−0.567579	−0.283789	−	0.958887i	$-0.591592\pi$
$941$	−17.4109	−0.567579	−0.283789	+	0.958887i	$0.591592\pi$
$942$	0	0
$943$	39.9511	1.30099
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.9610	−0.453671	−0.226835	−	0.973933i	$-0.572838\pi$
$947$	−13.9610	−0.453671	−0.226835	+	0.973933i	$0.572838\pi$
$948$	0	0
$949$	29.6728	0.963219
$950$	0	0
$951$	19.3357	0.627002
$952$	0	0
$953$	−27.4861	−0.890363	−0.445181	−	0.895440i	$-0.646861\pi$
$953$	−27.4861	−0.890363	−0.445181	+	0.895440i	$0.646861\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.6253	0.408118
$958$	0	0
$959$	−9.66291	−0.312032
$960$	0	0
$961$	−29.9234	−0.965270
$962$	0	0
$963$	3.66291	0.118036
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.6713	0.857693	0.428846	−	0.903377i	$-0.358920\pi$
$967$	26.6713	0.857693	0.428846	+	0.903377i	$0.358920\pi$
$968$	0	0
$969$	24.5764	0.789509
$970$	0	0
$971$	−49.6239	−1.59251	−0.796253	−	0.604964i	$-0.793188\pi$
$971$	−49.6239	−1.59251	−0.796253	+	0.604964i	$0.793188\pi$
$972$	0	0
$973$	−20.8872	−0.669612
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.5125	1.20013	0.600065	−	0.799951i	$-0.295141\pi$
$977$	37.5125	1.20013	0.600065	+	0.799951i	$0.295141\pi$
$978$	0	0
$979$	−13.8035	−0.441162
$980$	0	0
$981$	6.77575	0.216333
$982$	0	0
$983$	−52.2228	−1.66565	−0.832825	−	0.553536i	$-0.813278\pi$
$983$	−52.2228	−1.66565	−0.832825	+	0.553536i	$0.813278\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−33.9972	−1.08105
$990$	0	0
$991$	−26.3272	−0.836312	−0.418156	−	0.908375i	$-0.637324\pi$
$991$	−26.3272	−0.836312	−0.418156	+	0.908375i	$0.637324\pi$
$992$	0	0
$993$	−24.6253	−0.781460
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.3620	0.929905	0.464952	−	0.885336i	$-0.346071\pi$
$997$	29.3620	0.929905	0.464952	+	0.885336i	$0.346071\pi$
$998$	0	0
$999$	10.7005	0.338550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.dl.1.1		3
4.3	odd	2		4200.2.a.bn.1.3		3
5.2	odd	4		1680.2.t.j.1009.3		6
5.3	odd	4		1680.2.t.j.1009.6		6
5.4	even	2		8400.2.a.di.1.1		3
15.2	even	4		5040.2.t.z.1009.1		6
15.8	even	4		5040.2.t.z.1009.2		6
20.3	even	4		840.2.t.d.169.3	✓	6
20.7	even	4		840.2.t.d.169.6	yes	6
20.19	odd	2		4200.2.a.bp.1.3		3
60.23	odd	4		2520.2.t.k.1009.2		6
60.47	odd	4		2520.2.t.k.1009.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.d.169.3	✓	6	20.3	even	4
840.2.t.d.169.6	yes	6	20.7	even	4
1680.2.t.j.1009.3		6	5.2	odd	4
1680.2.t.j.1009.6		6	5.3	odd	4
2520.2.t.k.1009.1		6	60.47	odd	4
2520.2.t.k.1009.2		6	60.23	odd	4
4200.2.a.bn.1.3		3	4.3	odd	2
4200.2.a.bp.1.3		3	20.19	odd	2
5040.2.t.z.1009.1		6	15.2	even	4
5040.2.t.z.1009.2		6	15.8	even	4
8400.2.a.di.1.1		3	5.4	even	2
8400.2.a.dl.1.1		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:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -6.31265 q^{11} +6.96239 q^{13} +6.57452 q^{17} +3.73813 q^{19} +1.00000 q^{21} -5.73813 q^{23} +1.00000 q^{27} -2.00000 q^{29} +1.03761 q^{31} -6.31265 q^{33} +10.7005 q^{37} +6.96239 q^{39} -6.96239 q^{41} +5.92478 q^{43} +1.00000 q^{49} +6.57452 q^{51} +1.03761 q^{53} +3.73813 q^{57} +3.22425 q^{59} -13.8496 q^{61} +1.00000 q^{63} +4.77575 q^{67} -5.73813 q^{69} -8.23743 q^{71} +4.26187 q^{73} -6.31265 q^{77} +5.92478 q^{79} +1.00000 q^{81} -3.22425 q^{83} -2.00000 q^{87} +2.18664 q^{89} +6.96239 q^{91} +1.03761 q^{93} +3.73813 q^{97} -6.31265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 10 q^{13} + 8 q^{17} + 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} + 14 q^{31} + 2 q^{33} + 12 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} + 8 q^{51} + 14 q^{53} + 2 q^{57} + 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} + 18 q^{71} + 22 q^{73} + 2 q^{77} - 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} + 10 q^{91} + 14 q^{93} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 10 * q^13 + 8 * q^17 + 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 + 14 * q^31 + 2 * q^33 + 12 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 + 8 * q^51 + 14 * q^53 + 2 * q^57 + 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 + 18 * q^71 + 22 * q^73 + 2 * q^77 - 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 + 10 * q^91 + 14 * q^93 + 2 * q^97 + 2 * q^99

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