LMFDB - Embedded newform 2800.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +4.89898 q^{11} -4.44949 q^{13} -2.00000 q^{17} -1.55051 q^{19} +2.44949 q^{21} +2.89898 q^{23} +6.89898 q^{29} -8.89898 q^{31} -12.0000 q^{33} -2.00000 q^{37} +10.8990 q^{39} -1.10102 q^{41} -0.898979 q^{43} +8.89898 q^{47} +1.00000 q^{49} +4.89898 q^{51} +10.8990 q^{53} +3.79796 q^{57} +1.55051 q^{59} +3.55051 q^{61} -3.00000 q^{63} -8.00000 q^{67} -7.10102 q^{69} +1.10102 q^{71} -2.89898 q^{73} -4.89898 q^{77} -6.89898 q^{79} -9.00000 q^{81} -2.44949 q^{83} -16.8990 q^{87} -10.0000 q^{89} +4.44949 q^{91} +21.7980 q^{93} -15.7980 q^{97} +14.6969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{29} - 8 q^{31} - 24 q^{33} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{49} + 12 q^{53} - 12 q^{57} + 8 q^{59} + 12 q^{61} - 6 q^{63} - 16 q^{67} - 24 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{79} - 18 q^{81} - 24 q^{87} - 20 q^{89} + 4 q^{91} + 24 q^{93} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 - 8 * q^19 - 4 * q^23 + 4 * q^29 - 8 * q^31 - 24 * q^33 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^49 + 12 * q^53 - 12 * q^57 + 8 * q^59 + 12 * q^61 - 6 * q^63 - 16 * q^67 - 24 * q^69 + 12 * q^71 + 4 * q^73 - 4 * q^79 - 18 * q^81 - 24 * q^87 - 20 * q^89 + 4 * q^91 + 24 * q^93 - 12 * q^97

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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	0	0
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.55051	−0.355711	−0.177856	−	0.984057i	$-0.556916\pi$
$19$	−1.55051	−0.355711	−0.177856	+	0.984057i	$0.556916\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	0	0
$23$	2.89898	0.604479	0.302240	−	0.953232i	$-0.402266\pi$
$23$	2.89898	0.604479	0.302240	+	0.953232i	$0.402266\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.89898	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$30$	0	0
$31$	−8.89898	−1.59830	−0.799152	−	0.601129i	$-0.794718\pi$
$31$	−8.89898	−1.59830	−0.799152	+	0.601129i	$0.794718\pi$
$32$	0	0
$33$	−12.0000	−2.08893
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	10.8990	1.74523
$40$	0	0
$41$	−1.10102	−0.171951	−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102	−0.171951	−0.0859753	+	0.996297i	$0.527401\pi$
$42$	0	0
$43$	−0.898979	−0.137093	−0.0685465	−	0.997648i	$-0.521836\pi$
$43$	−0.898979	−0.137093	−0.0685465	+	0.997648i	$0.521836\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.89898	1.29805	0.649025	−	0.760767i	$-0.275177\pi$
$47$	8.89898	1.29805	0.649025	+	0.760767i	$0.275177\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.89898	0.685994
$52$	0	0
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.79796	0.503052
$58$	0	0
$59$	1.55051	0.201859	0.100930	−	0.994894i	$-0.467818\pi$
$59$	1.55051	0.201859	0.100930	+	0.994894i	$0.467818\pi$
$60$	0	0
$61$	3.55051	0.454596	0.227298	−	0.973825i	$-0.427011\pi$
$61$	3.55051	0.454596	0.227298	+	0.973825i	$0.427011\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−7.10102	−0.854862
$70$	0	0
$71$	1.10102	0.130667	0.0653335	−	0.997863i	$-0.479189\pi$
$71$	1.10102	0.130667	0.0653335	+	0.997863i	$0.479189\pi$
$72$	0	0
$73$	−2.89898	−0.339300	−0.169650	−	0.985504i	$-0.554264\pi$
$73$	−2.89898	−0.339300	−0.169650	+	0.985504i	$0.554264\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.89898	−0.558291
$78$	0	0
$79$	−6.89898	−0.776196	−0.388098	−	0.921618i	$-0.626868\pi$
$79$	−6.89898	−0.776196	−0.388098	+	0.921618i	$0.626868\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−2.44949	−0.268866	−0.134433	−	0.990923i	$-0.542921\pi$
$83$	−2.44949	−0.268866	−0.134433	+	0.990923i	$0.542921\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−16.8990	−1.81176
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	4.44949	0.466433
$92$	0	0
$93$	21.7980	2.26034
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.7980	−1.60404	−0.802020	−	0.597297i	$-0.796241\pi$
$97$	−15.7980	−1.60404	−0.802020	+	0.597297i	$0.796241\pi$
$98$	0	0
$99$	14.6969	1.47710
$100$	0	0
$101$	3.55051	0.353289	0.176644	−	0.984275i	$-0.443476\pi$
$101$	3.55051	0.353289	0.176644	+	0.984275i	$0.443476\pi$
$102$	0	0
$103$	12.8990	1.27097	0.635487	−	0.772111i	$-0.280799\pi$
$103$	12.8990	1.27097	0.635487	+	0.772111i	$0.280799\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	6.89898	0.660802	0.330401	−	0.943841i	$-0.392816\pi$
$109$	6.89898	0.660802	0.330401	+	0.943841i	$0.392816\pi$
$110$	0	0
$111$	4.89898	0.464991
$112$	0	0
$113$	−19.7980	−1.86244	−0.931218	−	0.364464i	$-0.881252\pi$
$113$	−19.7980	−1.86244	−0.931218	+	0.364464i	$0.881252\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.3485	−1.23407
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	13.0000	1.18182
$122$	0	0
$123$	2.69694	0.243175
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.8990	−1.32207	−0.661035	−	0.750355i	$-0.729883\pi$
$127$	−14.8990	−1.32207	−0.661035	+	0.750355i	$0.729883\pi$
$128$	0	0
$129$	2.20204	0.193879
$130$	0	0
$131$	6.44949	0.563495	0.281747	−	0.959489i	$-0.409086\pi$
$131$	6.44949	0.563495	0.281747	+	0.959489i	$0.409086\pi$
$132$	0	0
$133$	1.55051	0.134446
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.79796	0.153610	0.0768050	−	0.997046i	$-0.475528\pi$
$137$	1.79796	0.153610	0.0768050	+	0.997046i	$0.475528\pi$
$138$	0	0
$139$	−1.55051	−0.131513	−0.0657563	−	0.997836i	$-0.520946\pi$
$139$	−1.55051	−0.131513	−0.0657563	+	0.997836i	$0.520946\pi$
$140$	0	0
$141$	−21.7980	−1.83572
$142$	0	0
$143$	−21.7980	−1.82284
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.44949	−0.202031
$148$	0	0
$149$	−3.79796	−0.311141	−0.155570	−	0.987825i	$-0.549722\pi$
$149$	−3.79796	−0.311141	−0.155570	+	0.987825i	$0.549722\pi$
$150$	0	0
$151$	−19.5959	−1.59469	−0.797347	−	0.603522i	$-0.793764\pi$
$151$	−19.5959	−1.59469	−0.797347	+	0.603522i	$0.793764\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.55051	−0.283362	−0.141681	−	0.989912i	$-0.545251\pi$
$157$	−3.55051	−0.283362	−0.141681	+	0.989912i	$0.545251\pi$
$158$	0	0
$159$	−26.6969	−2.11720
$160$	0	0
$161$	−2.89898	−0.228472
$162$	0	0
$163$	−7.10102	−0.556195	−0.278097	−	0.960553i	$-0.589704\pi$
$163$	−7.10102	−0.556195	−0.278097	+	0.960553i	$0.589704\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	−4.65153	−0.355711
$172$	0	0
$173$	6.24745	0.474985	0.237492	−	0.971389i	$-0.423675\pi$
$173$	6.24745	0.474985	0.237492	+	0.971389i	$0.423675\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.79796	−0.285472
$178$	0	0
$179$	−13.7980	−1.03131	−0.515654	−	0.856797i	$-0.672451\pi$
$179$	−13.7980	−1.03131	−0.515654	+	0.856797i	$0.672451\pi$
$180$	0	0
$181$	−10.2474	−0.761687	−0.380843	−	0.924640i	$-0.624366\pi$
$181$	−10.2474	−0.761687	−0.380843	+	0.924640i	$0.624366\pi$
$182$	0	0
$183$	−8.69694	−0.642896
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.79796	−0.716498
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.6969	−0.918718	−0.459359	−	0.888251i	$-0.651921\pi$
$191$	−12.6969	−0.918718	−0.459359	+	0.888251i	$0.651921\pi$
$192$	0	0
$193$	21.5959	1.55451	0.777254	−	0.629187i	$-0.216612\pi$
$193$	21.5959	1.55451	0.777254	+	0.629187i	$0.216612\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.8990	−1.34650	−0.673248	−	0.739417i	$-0.735101\pi$
$197$	−18.8990	−1.34650	−0.673248	+	0.739417i	$0.735101\pi$
$198$	0	0
$199$	−16.8990	−1.19794	−0.598968	−	0.800773i	$-0.704423\pi$
$199$	−16.8990	−1.19794	−0.598968	+	0.800773i	$0.704423\pi$
$200$	0	0
$201$	19.5959	1.38219
$202$	0	0
$203$	−6.89898	−0.484213
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.69694	0.604479
$208$	0	0
$209$	−7.59592	−0.525421
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−2.69694	−0.184791
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.89898	0.604102
$218$	0	0
$219$	7.10102	0.479842
$220$	0	0
$221$	8.89898	0.598610
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.34847	0.487735	0.243868	−	0.969809i	$-0.421584\pi$
$227$	7.34847	0.487735	0.243868	+	0.969809i	$0.421584\pi$
$228$	0	0
$229$	−19.1464	−1.26523	−0.632616	−	0.774466i	$-0.718019\pi$
$229$	−19.1464	−1.26523	−0.632616	+	0.774466i	$0.718019\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	−29.7980	−1.95213	−0.976065	−	0.217481i	$-0.930216\pi$
$233$	−29.7980	−1.95213	−0.976065	+	0.217481i	$0.930216\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.8990	1.09771
$238$	0	0
$239$	6.20204	0.401177	0.200588	−	0.979676i	$-0.435715\pi$
$239$	6.20204	0.401177	0.200588	+	0.979676i	$0.435715\pi$
$240$	0	0
$241$	−8.69694	−0.560219	−0.280110	−	0.959968i	$-0.590371\pi$
$241$	−8.69694	−0.560219	−0.280110	+	0.959968i	$0.590371\pi$
$242$	0	0
$243$	22.0454	1.41421
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.89898	0.438972
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	6.44949	0.407088	0.203544	−	0.979066i	$-0.434754\pi$
$251$	6.44949	0.407088	0.203544	+	0.979066i	$0.434754\pi$
$252$	0	0
$253$	14.2020	0.892875
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.69694	0.542500	0.271250	−	0.962509i	$-0.412563\pi$
$257$	8.69694	0.542500	0.271250	+	0.962509i	$0.412563\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	20.6969	1.28111
$262$	0	0
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.4949	1.49906
$268$	0	0
$269$	19.1464	1.16738	0.583689	−	0.811977i	$-0.301609\pi$
$269$	19.1464	1.16738	0.583689	+	0.811977i	$0.301609\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−10.8990	−0.659636
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.8990	0.895193	0.447596	−	0.894236i	$-0.352280\pi$
$277$	14.8990	0.895193	0.447596	+	0.894236i	$0.352280\pi$
$278$	0	0
$279$	−26.6969	−1.59830
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	3.75255	0.223066	0.111533	−	0.993761i	$-0.464424\pi$
$283$	3.75255	0.223066	0.111533	+	0.993761i	$0.464424\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.10102	0.0649912
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	38.6969	2.26845
$292$	0	0
$293$	−18.2474	−1.06603	−0.533014	−	0.846107i	$-0.678941\pi$
$293$	−18.2474	−1.06603	−0.533014	+	0.846107i	$0.678941\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.8990	−0.745967
$300$	0	0
$301$	0.898979	0.0518163
$302$	0	0
$303$	−8.69694	−0.499626
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.2474	−1.15558	−0.577791	−	0.816184i	$-0.696085\pi$
$307$	−20.2474	−1.15558	−0.577791	+	0.816184i	$0.696085\pi$
$308$	0	0
$309$	−31.5959	−1.79743
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	21.5959	1.22067	0.610337	−	0.792142i	$-0.291034\pi$
$313$	21.5959	1.22067	0.610337	+	0.792142i	$0.291034\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.4949	1.26344	0.631720	−	0.775197i	$-0.282349\pi$
$317$	22.4949	1.26344	0.631720	+	0.775197i	$0.282349\pi$
$318$	0	0
$319$	33.7980	1.89232
$320$	0	0
$321$	19.5959	1.09374
$322$	0	0
$323$	3.10102	0.172545
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−16.8990	−0.934516
$328$	0	0
$329$	−8.89898	−0.490617
$330$	0	0
$331$	18.6969	1.02768	0.513838	−	0.857887i	$-0.328223\pi$
$331$	18.6969	1.02768	0.513838	+	0.857887i	$0.328223\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.59592	−0.522723	−0.261361	−	0.965241i	$-0.584171\pi$
$337$	−9.59592	−0.522723	−0.261361	+	0.965241i	$0.584171\pi$
$338$	0	0
$339$	48.4949	2.63388
$340$	0	0
$341$	−43.5959	−2.36085
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.8990	1.55138	0.775689	−	0.631115i	$-0.217402\pi$
$347$	28.8990	1.55138	0.775689	+	0.631115i	$0.217402\pi$
$348$	0	0
$349$	−8.44949	−0.452291	−0.226145	−	0.974094i	$-0.572612\pi$
$349$	−8.44949	−0.452291	−0.226145	+	0.974094i	$0.572612\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.8990	−1.21879	−0.609395	−	0.792867i	$-0.708588\pi$
$353$	−22.8990	−1.21879	−0.609395	+	0.792867i	$0.708588\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.89898	−0.259281
$358$	0	0
$359$	27.5959	1.45646	0.728228	−	0.685334i	$-0.240344\pi$
$359$	27.5959	1.45646	0.728228	+	0.685334i	$0.240344\pi$
$360$	0	0
$361$	−16.5959	−0.873469
$362$	0	0
$363$	−31.8434	−1.67134
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−3.30306	−0.171951
$370$	0	0
$371$	−10.8990	−0.565847
$372$	0	0
$373$	4.69694	0.243198	0.121599	−	0.992579i	$-0.461198\pi$
$373$	4.69694	0.243198	0.121599	+	0.992579i	$0.461198\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−30.6969	−1.58097
$378$	0	0
$379$	−30.6969	−1.57680	−0.788398	−	0.615166i	$-0.789089\pi$
$379$	−30.6969	−1.57680	−0.788398	+	0.615166i	$0.789089\pi$
$380$	0	0
$381$	36.4949	1.86969
$382$	0	0
$383$	−7.10102	−0.362845	−0.181423	−	0.983405i	$-0.558070\pi$
$383$	−7.10102	−0.362845	−0.181423	+	0.983405i	$0.558070\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.69694	−0.137093
$388$	0	0
$389$	13.1010	0.664248	0.332124	−	0.943236i	$-0.392235\pi$
$389$	13.1010	0.664248	0.332124	+	0.943236i	$0.392235\pi$
$390$	0	0
$391$	−5.79796	−0.293215
$392$	0	0
$393$	−15.7980	−0.796902
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.65153	0.133077	0.0665383	−	0.997784i	$-0.478805\pi$
$397$	2.65153	0.133077	0.0665383	+	0.997784i	$0.478805\pi$
$398$	0	0
$399$	−3.79796	−0.190136
$400$	0	0
$401$	−29.3939	−1.46786	−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939	−1.46786	−0.733930	+	0.679225i	$0.762316\pi$
$402$	0	0
$403$	39.5959	1.97241
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.79796	−0.485667
$408$	0	0
$409$	34.4949	1.70566	0.852831	−	0.522186i	$-0.174883\pi$
$409$	34.4949	1.70566	0.852831	+	0.522186i	$0.174883\pi$
$410$	0	0
$411$	−4.40408	−0.217237
$412$	0	0
$413$	−1.55051	−0.0762956
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.79796	0.185987
$418$	0	0
$419$	1.55051	0.0757474	0.0378737	−	0.999283i	$-0.487942\pi$
$419$	1.55051	0.0757474	0.0378737	+	0.999283i	$0.487942\pi$
$420$	0	0
$421$	−4.20204	−0.204795	−0.102397	−	0.994744i	$-0.532651\pi$
$421$	−4.20204	−0.204795	−0.102397	+	0.994744i	$0.532651\pi$
$422$	0	0
$423$	26.6969	1.29805
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.55051	−0.171821
$428$	0	0
$429$	53.3939	2.57788
$430$	0	0
$431$	1.79796	0.0866046	0.0433023	−	0.999062i	$-0.486212\pi$
$431$	1.79796	0.0866046	0.0433023	+	0.999062i	$0.486212\pi$
$432$	0	0
$433$	0.202041	0.00970947	0.00485474	−	0.999988i	$-0.498455\pi$
$433$	0.202041	0.00970947	0.00485474	+	0.999988i	$0.498455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.49490	−0.215020
$438$	0	0
$439$	21.3939	1.02107	0.510537	−	0.859856i	$-0.329447\pi$
$439$	21.3939	1.02107	0.510537	+	0.859856i	$0.329447\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	9.79796	0.465515	0.232758	−	0.972535i	$-0.425225\pi$
$443$	9.79796	0.465515	0.232758	+	0.972535i	$0.425225\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.30306	0.440020
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−5.39388	−0.253988
$452$	0	0
$453$	48.0000	2.25524
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.5959	−1.38444	−0.692219	−	0.721687i	$-0.743367\pi$
$457$	−29.5959	−1.38444	−0.692219	+	0.721687i	$0.743367\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.3485	0.807999	0.403999	−	0.914759i	$-0.367620\pi$
$461$	17.3485	0.807999	0.403999	+	0.914759i	$0.367620\pi$
$462$	0	0
$463$	3.59592	0.167116	0.0835582	−	0.996503i	$-0.473372\pi$
$463$	3.59592	0.167116	0.0835582	+	0.996503i	$0.473372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.4495	0.483545	0.241772	−	0.970333i	$-0.422271\pi$
$467$	10.4495	0.483545	0.241772	+	0.970333i	$0.422271\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	8.69694	0.400734
$472$	0	0
$473$	−4.40408	−0.202500
$474$	0	0
$475$	0	0
$476$	0	0
$477$	32.6969	1.49709
$478$	0	0
$479$	−9.30306	−0.425068	−0.212534	−	0.977154i	$-0.568172\pi$
$479$	−9.30306	−0.425068	−0.212534	+	0.977154i	$0.568172\pi$
$480$	0	0
$481$	8.89898	0.405759
$482$	0	0
$483$	7.10102	0.323108
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.30306	−0.330933	−0.165467	−	0.986215i	$-0.552913\pi$
$487$	−7.30306	−0.330933	−0.165467	+	0.986215i	$0.552913\pi$
$488$	0	0
$489$	17.3939	0.786578
$490$	0	0
$491$	−19.5959	−0.884351	−0.442176	−	0.896928i	$-0.645793\pi$
$491$	−19.5959	−0.884351	−0.442176	+	0.896928i	$0.645793\pi$
$492$	0	0
$493$	−13.7980	−0.621429
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.10102	−0.0493875
$498$	0	0
$499$	−6.20204	−0.277641	−0.138821	−	0.990318i	$-0.544331\pi$
$499$	−6.20204	−0.277641	−0.138821	+	0.990318i	$0.544331\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−16.6515	−0.739520
$508$	0	0
$509$	−31.5505	−1.39845	−0.699226	−	0.714901i	$-0.746472\pi$
$509$	−31.5505	−1.39845	−0.699226	+	0.714901i	$0.746472\pi$
$510$	0	0
$511$	2.89898	0.128243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	43.5959	1.91735
$518$	0	0
$519$	−15.3031	−0.671730
$520$	0	0
$521$	32.6969	1.43248	0.716239	−	0.697855i	$-0.245862\pi$
$521$	32.6969	1.43248	0.716239	+	0.697855i	$0.245862\pi$
$522$	0	0
$523$	−33.1464	−1.44939	−0.724696	−	0.689069i	$-0.758020\pi$
$523$	−33.1464	−1.44939	−0.724696	+	0.689069i	$0.758020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.7980	0.775291
$528$	0	0
$529$	−14.5959	−0.634605
$530$	0	0
$531$	4.65153	0.201859
$532$	0	0
$533$	4.89898	0.212198
$534$	0	0
$535$	0	0
$536$	0	0
$537$	33.7980	1.45849
$538$	0	0
$539$	4.89898	0.211014
$540$	0	0
$541$	9.59592	0.412561	0.206280	−	0.978493i	$-0.433864\pi$
$541$	9.59592	0.412561	0.206280	+	0.978493i	$0.433864\pi$
$542$	0	0
$543$	25.1010	1.07719
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.6969	−0.799423	−0.399712	−	0.916641i	$-0.630890\pi$
$547$	−18.6969	−0.799423	−0.399712	+	0.916641i	$0.630890\pi$
$548$	0	0
$549$	10.6515	0.454596
$550$	0	0
$551$	−10.6969	−0.455705
$552$	0	0
$553$	6.89898	0.293374
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.6969	−0.537987	−0.268993	−	0.963142i	$-0.586691\pi$
$557$	−12.6969	−0.537987	−0.268993	+	0.963142i	$0.586691\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	−30.0454	−1.26626	−0.633131	−	0.774044i	$-0.718231\pi$
$563$	−30.0454	−1.26626	−0.633131	+	0.774044i	$0.718231\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	33.7980	1.41688	0.708442	−	0.705769i	$-0.249398\pi$
$569$	33.7980	1.41688	0.708442	+	0.705769i	$0.249398\pi$
$570$	0	0
$571$	11.1010	0.464563	0.232282	−	0.972649i	$-0.425381\pi$
$571$	11.1010	0.464563	0.232282	+	0.972649i	$0.425381\pi$
$572$	0	0
$573$	31.1010	1.29926
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.49490	0.103864	0.0519320	−	0.998651i	$-0.483462\pi$
$577$	2.49490	0.103864	0.0519320	+	0.998651i	$0.483462\pi$
$578$	0	0
$579$	−52.8990	−2.19841
$580$	0	0
$581$	2.44949	0.101622
$582$	0	0
$583$	53.3939	2.21135
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.14643	0.0473182	0.0236591	−	0.999720i	$-0.492468\pi$
$587$	1.14643	0.0473182	0.0236591	+	0.999720i	$0.492468\pi$
$588$	0	0
$589$	13.7980	0.568535
$590$	0	0
$591$	46.2929	1.90423
$592$	0	0
$593$	10.8990	0.447567	0.223784	−	0.974639i	$-0.428159\pi$
$593$	10.8990	0.447567	0.223784	+	0.974639i	$0.428159\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.3939	1.69414
$598$	0	0
$599$	13.1010	0.535293	0.267647	−	0.963517i	$-0.413754\pi$
$599$	13.1010	0.535293	0.267647	+	0.963517i	$0.413754\pi$
$600$	0	0
$601$	−39.3939	−1.60691	−0.803455	−	0.595366i	$-0.797007\pi$
$601$	−39.3939	−1.60691	−0.803455	+	0.595366i	$0.797007\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.3939	1.35542	0.677708	−	0.735331i	$-0.262973\pi$
$607$	33.3939	1.35542	0.677708	+	0.735331i	$0.262973\pi$
$608$	0	0
$609$	16.8990	0.684781
$610$	0	0
$611$	−39.5959	−1.60188
$612$	0	0
$613$	27.7980	1.12275	0.561374	−	0.827562i	$-0.310273\pi$
$613$	27.7980	1.12275	0.561374	+	0.827562i	$0.310273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.5959	−1.19149	−0.595743	−	0.803175i	$-0.703142\pi$
$617$	−29.5959	−1.19149	−0.595743	+	0.803175i	$0.703142\pi$
$618$	0	0
$619$	41.5505	1.67006	0.835028	−	0.550207i	$-0.185451\pi$
$619$	41.5505	1.67006	0.835028	+	0.550207i	$0.185451\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.6061	0.743057
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	42.4949	1.69170	0.845848	−	0.533425i	$-0.179095\pi$
$631$	42.4949	1.69170	0.845848	+	0.533425i	$0.179095\pi$
$632$	0	0
$633$	29.3939	1.16830
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.44949	−0.176295
$638$	0	0
$639$	3.30306	0.130667
$640$	0	0
$641$	25.7980	1.01896	0.509479	−	0.860483i	$-0.329838\pi$
$641$	25.7980	1.01896	0.509479	+	0.860483i	$0.329838\pi$
$642$	0	0
$643$	25.1464	0.991678	0.495839	−	0.868414i	$-0.334861\pi$
$643$	25.1464	0.991678	0.495839	+	0.868414i	$0.334861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.2929	−1.81996	−0.909980	−	0.414652i	$-0.863903\pi$
$647$	−46.2929	−1.81996	−0.909980	+	0.414652i	$0.863903\pi$
$648$	0	0
$649$	7.59592	0.298166
$650$	0	0
$651$	−21.7980	−0.854329
$652$	0	0
$653$	20.2020	0.790567	0.395283	−	0.918559i	$-0.370646\pi$
$653$	20.2020	0.790567	0.395283	+	0.918559i	$0.370646\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.69694	−0.339300
$658$	0	0
$659$	−16.8990	−0.658291	−0.329145	−	0.944279i	$-0.606761\pi$
$659$	−16.8990	−0.658291	−0.329145	+	0.944279i	$0.606761\pi$
$660$	0	0
$661$	−40.9444	−1.59255	−0.796276	−	0.604933i	$-0.793200\pi$
$661$	−40.9444	−1.59255	−0.796276	+	0.604933i	$0.793200\pi$
$662$	0	0
$663$	−21.7980	−0.846563
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.0000	0.774403
$668$	0	0
$669$	9.79796	0.378811
$670$	0	0
$671$	17.3939	0.671483
$672$	0	0
$673$	17.7980	0.686061	0.343030	−	0.939324i	$-0.388547\pi$
$673$	17.7980	0.686061	0.343030	+	0.939324i	$0.388547\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.4495	1.40087	0.700434	−	0.713717i	$-0.252990\pi$
$677$	36.4495	1.40087	0.700434	+	0.713717i	$0.252990\pi$
$678$	0	0
$679$	15.7980	0.606270
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	3.59592	0.137594	0.0687970	−	0.997631i	$-0.478084\pi$
$683$	3.59592	0.137594	0.0687970	+	0.997631i	$0.478084\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	46.8990	1.78931
$688$	0	0
$689$	−48.4949	−1.84751
$690$	0	0
$691$	−21.1464	−0.804448	−0.402224	−	0.915541i	$-0.631763\pi$
$691$	−21.1464	−0.804448	−0.402224	+	0.915541i	$0.631763\pi$
$692$	0	0
$693$	−14.6969	−0.558291
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.20204	0.0834083
$698$	0	0
$699$	72.9898	2.76073
$700$	0	0
$701$	11.3031	0.426911	0.213455	−	0.976953i	$-0.431528\pi$
$701$	11.3031	0.426911	0.213455	+	0.976953i	$0.431528\pi$
$702$	0	0
$703$	3.10102	0.116957
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.55051	−0.133531
$708$	0	0
$709$	−28.2929	−1.06256	−0.531280	−	0.847196i	$-0.678289\pi$
$709$	−28.2929	−1.06256	−0.531280	+	0.847196i	$0.678289\pi$
$710$	0	0
$711$	−20.6969	−0.776196
$712$	0	0
$713$	−25.7980	−0.966141
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.1918	−0.567350
$718$	0	0
$719$	4.49490	0.167631	0.0838157	−	0.996481i	$-0.473289\pi$
$719$	4.49490	0.167631	0.0838157	+	0.996481i	$0.473289\pi$
$720$	0	0
$721$	−12.8990	−0.480383
$722$	0	0
$723$	21.3031	0.792269
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.6969	0.841783	0.420891	−	0.907111i	$-0.361717\pi$
$727$	22.6969	0.841783	0.420891	+	0.907111i	$0.361717\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	1.79796	0.0664999
$732$	0	0
$733$	−39.6413	−1.46419	−0.732093	−	0.681205i	$-0.761456\pi$
$733$	−39.6413	−1.46419	−0.732093	+	0.681205i	$0.761456\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.1918	−1.44365
$738$	0	0
$739$	−4.49490	−0.165347	−0.0826737	−	0.996577i	$-0.526346\pi$
$739$	−4.49490	−0.165347	−0.0826737	+	0.996577i	$0.526346\pi$
$740$	0	0
$741$	−16.8990	−0.620800
$742$	0	0
$743$	−44.6969	−1.63977	−0.819886	−	0.572527i	$-0.805963\pi$
$743$	−44.6969	−1.63977	−0.819886	+	0.572527i	$0.805963\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.34847	−0.268866
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	41.7980	1.52523	0.762615	−	0.646853i	$-0.223915\pi$
$751$	41.7980	1.52523	0.762615	+	0.646853i	$0.223915\pi$
$752$	0	0
$753$	−15.7980	−0.575710
$754$	0	0
$755$	0	0
$756$	0	0
$757$	51.7980	1.88263	0.941314	−	0.337531i	$-0.109592\pi$
$757$	51.7980	1.88263	0.941314	+	0.337531i	$0.109592\pi$
$758$	0	0
$759$	−34.7878	−1.26272
$760$	0	0
$761$	−21.1010	−0.764911	−0.382456	−	0.923974i	$-0.624922\pi$
$761$	−21.1010	−0.764911	−0.382456	+	0.923974i	$0.624922\pi$
$762$	0	0
$763$	−6.89898	−0.249760
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.89898	−0.249108
$768$	0	0
$769$	−40.6969	−1.46757	−0.733785	−	0.679382i	$-0.762248\pi$
$769$	−40.6969	−1.46757	−0.733785	+	0.679382i	$0.762248\pi$
$770$	0	0
$771$	−21.3031	−0.767211
$772$	0	0
$773$	−1.34847	−0.0485011	−0.0242505	−	0.999706i	$-0.507720\pi$
$773$	−1.34847	−0.0485011	−0.0242505	+	0.999706i	$0.507720\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.89898	−0.175750
$778$	0	0
$779$	1.70714	0.0611648
$780$	0	0
$781$	5.39388	0.193008
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.4495	1.79833	0.899165	−	0.437610i	$-0.144175\pi$
$787$	50.4495	1.79833	0.899165	+	0.437610i	$0.144175\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	19.7980	0.703934
$792$	0	0
$793$	−15.7980	−0.561002
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.944387	0.0334519	0.0167260	−	0.999860i	$-0.494676\pi$
$797$	0.944387	0.0334519	0.0167260	+	0.999860i	$0.494676\pi$
$798$	0	0
$799$	−17.7980	−0.629647
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	−14.2020	−0.501179
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−46.8990	−1.65092
$808$	0	0
$809$	47.5959	1.67338	0.836692	−	0.547674i	$-0.184487\pi$
$809$	47.5959	1.67338	0.836692	+	0.547674i	$0.184487\pi$
$810$	0	0
$811$	−14.9444	−0.524768	−0.262384	−	0.964963i	$-0.584509\pi$
$811$	−14.9444	−0.524768	−0.262384	+	0.964963i	$0.584509\pi$
$812$	0	0
$813$	29.3939	1.03089
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.39388	0.0487656
$818$	0	0
$819$	13.3485	0.466433
$820$	0	0
$821$	8.20204	0.286253	0.143127	−	0.989704i	$-0.454284\pi$
$821$	8.20204	0.286253	0.143127	+	0.989704i	$0.454284\pi$
$822$	0	0
$823$	−39.1918	−1.36614	−0.683071	−	0.730352i	$-0.739356\pi$
$823$	−39.1918	−1.36614	−0.683071	+	0.730352i	$0.739356\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.5959	−0.542323	−0.271162	−	0.962534i	$-0.587408\pi$
$827$	−15.5959	−0.542323	−0.271162	+	0.962534i	$0.587408\pi$
$828$	0	0
$829$	−43.6413	−1.51573	−0.757863	−	0.652414i	$-0.773756\pi$
$829$	−43.6413	−1.51573	−0.757863	+	0.652414i	$0.773756\pi$
$830$	0	0
$831$	−36.4949	−1.26599
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.8990	−1.27389	−0.636947	−	0.770907i	$-0.719803\pi$
$839$	−36.8990	−1.27389	−0.636947	+	0.770907i	$0.719803\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	0	0
$843$	44.0908	1.51857
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.0000	−0.446685
$848$	0	0
$849$	−9.19184	−0.315463
$850$	0	0
$851$	−5.79796	−0.198751
$852$	0	0
$853$	33.8434	1.15877	0.579387	−	0.815052i	$-0.303292\pi$
$853$	33.8434	1.15877	0.579387	+	0.815052i	$0.303292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	53.1918	1.81700	0.908499	−	0.417886i	$-0.137229\pi$
$857$	53.1918	1.81700	0.908499	+	0.417886i	$0.137229\pi$
$858$	0	0
$859$	−53.6413	−1.83022	−0.915109	−	0.403206i	$-0.867896\pi$
$859$	−53.6413	−1.83022	−0.915109	+	0.403206i	$0.867896\pi$
$860$	0	0
$861$	−2.69694	−0.0919114
$862$	0	0
$863$	−45.3939	−1.54523	−0.772613	−	0.634878i	$-0.781051\pi$
$863$	−45.3939	−1.54523	−0.772613	+	0.634878i	$0.781051\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	31.8434	1.08146
$868$	0	0
$869$	−33.7980	−1.14652
$870$	0	0
$871$	35.5959	1.20612
$872$	0	0
$873$	−47.3939	−1.60404
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.3939	1.33024	0.665118	−	0.746738i	$-0.268381\pi$
$877$	39.3939	1.33024	0.665118	+	0.746738i	$0.268381\pi$
$878$	0	0
$879$	44.6969	1.50759
$880$	0	0
$881$	8.20204	0.276334	0.138167	−	0.990409i	$-0.455879\pi$
$881$	8.20204	0.276334	0.138167	+	0.990409i	$0.455879\pi$
$882$	0	0
$883$	22.2020	0.747158	0.373579	−	0.927598i	$-0.378130\pi$
$883$	22.2020	0.747158	0.373579	+	0.927598i	$0.378130\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.69694	0.0905543	0.0452772	−	0.998974i	$-0.485583\pi$
$887$	2.69694	0.0905543	0.0452772	+	0.998974i	$0.485583\pi$
$888$	0	0
$889$	14.8990	0.499696
$890$	0	0
$891$	−44.0908	−1.47710
$892$	0	0
$893$	−13.7980	−0.461731
$894$	0	0
$895$	0	0
$896$	0	0
$897$	31.5959	1.05496
$898$	0	0
$899$	−61.3939	−2.04760
$900$	0	0
$901$	−21.7980	−0.726195
$902$	0	0
$903$	−2.20204	−0.0732793
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.7980	−1.38788	−0.693939	−	0.720034i	$-0.744126\pi$
$907$	−41.7980	−1.38788	−0.693939	+	0.720034i	$0.744126\pi$
$908$	0	0
$909$	10.6515	0.353289
$910$	0	0
$911$	35.5959	1.17935	0.589673	−	0.807642i	$-0.299257\pi$
$911$	35.5959	1.17935	0.589673	+	0.807642i	$0.299257\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.44949	−0.212981
$918$	0	0
$919$	26.8990	0.887315	0.443658	−	0.896196i	$-0.353681\pi$
$919$	26.8990	0.887315	0.443658	+	0.896196i	$0.353681\pi$
$920$	0	0
$921$	49.5959	1.63424
$922$	0	0
$923$	−4.89898	−0.161252
$924$	0	0
$925$	0	0
$926$	0	0
$927$	38.6969	1.27097
$928$	0	0
$929$	−28.2929	−0.928259	−0.464129	−	0.885767i	$-0.653633\pi$
$929$	−28.2929	−0.928259	−0.464129	+	0.885767i	$0.653633\pi$
$930$	0	0
$931$	−1.55051	−0.0508159
$932$	0	0
$933$	29.3939	0.962312
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.1010	1.34271	0.671356	−	0.741135i	$-0.265712\pi$
$937$	41.1010	1.34271	0.671356	+	0.741135i	$0.265712\pi$
$938$	0	0
$939$	−52.8990	−1.72629
$940$	0	0
$941$	−19.5505	−0.637328	−0.318664	−	0.947868i	$-0.603234\pi$
$941$	−19.5505	−0.637328	−0.318664	+	0.947868i	$0.603234\pi$
$942$	0	0
$943$	−3.19184	−0.103940
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.0908	1.43276	0.716379	−	0.697711i	$-0.245798\pi$
$947$	44.0908	1.43276	0.716379	+	0.697711i	$0.245798\pi$
$948$	0	0
$949$	12.8990	0.418719
$950$	0	0
$951$	−55.1010	−1.78677
$952$	0	0
$953$	−2.20204	−0.0713311	−0.0356656	−	0.999364i	$-0.511355\pi$
$953$	−2.20204	−0.0713311	−0.0356656	+	0.999364i	$0.511355\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−82.7878	−2.67615
$958$	0	0
$959$	−1.79796	−0.0580591
$960$	0	0
$961$	48.1918	1.55458
$962$	0	0
$963$	−24.0000	−0.773389
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.2929	−1.16710	−0.583550	−	0.812077i	$-0.698337\pi$
$967$	−36.2929	−1.16710	−0.583550	+	0.812077i	$0.698337\pi$
$968$	0	0
$969$	−7.59592	−0.244016
$970$	0	0
$971$	9.55051	0.306490	0.153245	−	0.988188i	$-0.451028\pi$
$971$	9.55051	0.306490	0.153245	+	0.988188i	$0.451028\pi$
$972$	0	0
$973$	1.55051	0.0497071
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.3939	0.940393	0.470197	−	0.882562i	$-0.344183\pi$
$977$	29.3939	0.940393	0.470197	+	0.882562i	$0.344183\pi$
$978$	0	0
$979$	−48.9898	−1.56572
$980$	0	0
$981$	20.6969	0.660802
$982$	0	0
$983$	−13.3031	−0.424302	−0.212151	−	0.977237i	$-0.568047\pi$
$983$	−13.3031	−0.424302	−0.212151	+	0.977237i	$0.568047\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.7980	0.693837
$988$	0	0
$989$	−2.60612	−0.0828699
$990$	0	0
$991$	−31.3031	−0.994375	−0.497187	−	0.867643i	$-0.665634\pi$
$991$	−31.3031	−0.994375	−0.497187	+	0.867643i	$0.665634\pi$
$992$	0	0
$993$	−45.7980	−1.45335
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−57.3485	−1.81624	−0.908122	−	0.418705i	$-0.862484\pi$
$997$	−57.3485	−1.81624	−0.908122	+	0.418705i	$0.862484\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bl.1.1		2
4.3	odd	2		350.2.a.h.1.2		2
5.2	odd	4		560.2.g.e.449.4		4
5.3	odd	4		560.2.g.e.449.2		4
5.4	even	2		2800.2.a.bm.1.2		2
12.11	even	2		3150.2.a.bs.1.2		2
15.2	even	4		5040.2.t.t.1009.1		4
15.8	even	4		5040.2.t.t.1009.2		4
20.3	even	4		70.2.c.a.29.2	✓	4
20.7	even	4		70.2.c.a.29.3	yes	4
20.19	odd	2		350.2.a.g.1.1		2
28.27	even	2		2450.2.a.bq.1.1		2
40.3	even	4		2240.2.g.j.449.1		4
40.13	odd	4		2240.2.g.i.449.3		4
40.27	even	4		2240.2.g.j.449.3		4
40.37	odd	4		2240.2.g.i.449.1		4
60.23	odd	4		630.2.g.g.379.3		4
60.47	odd	4		630.2.g.g.379.1		4
60.59	even	2		3150.2.a.bt.1.2		2
140.3	odd	12		490.2.i.f.79.3		8
140.23	even	12		490.2.i.c.459.1		8
140.27	odd	4		490.2.c.e.99.4		4
140.47	odd	12		490.2.i.f.459.3		8
140.67	even	12		490.2.i.c.79.1		8
140.83	odd	4		490.2.c.e.99.1		4
140.87	odd	12		490.2.i.f.79.2		8
140.103	odd	12		490.2.i.f.459.2		8
140.107	even	12		490.2.i.c.459.4		8
140.123	even	12		490.2.i.c.79.4		8
140.139	even	2		2450.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.2	✓	4	20.3	even	4
70.2.c.a.29.3	yes	4	20.7	even	4
350.2.a.g.1.1		2	20.19	odd	2
350.2.a.h.1.2		2	4.3	odd	2
490.2.c.e.99.1		4	140.83	odd	4
490.2.c.e.99.4		4	140.27	odd	4
490.2.i.c.79.1		8	140.67	even	12
490.2.i.c.79.4		8	140.123	even	12
490.2.i.c.459.1		8	140.23	even	12
490.2.i.c.459.4		8	140.107	even	12
490.2.i.f.79.2		8	140.87	odd	12
490.2.i.f.79.3		8	140.3	odd	12
490.2.i.f.459.2		8	140.103	odd	12
490.2.i.f.459.3		8	140.47	odd	12
560.2.g.e.449.2		4	5.3	odd	4
560.2.g.e.449.4		4	5.2	odd	4
630.2.g.g.379.1		4	60.47	odd	4
630.2.g.g.379.3		4	60.23	odd	4
2240.2.g.i.449.1		4	40.37	odd	4
2240.2.g.i.449.3		4	40.13	odd	4
2240.2.g.j.449.1		4	40.3	even	4
2240.2.g.j.449.3		4	40.27	even	4
2450.2.a.bl.1.2		2	140.139	even	2
2450.2.a.bq.1.1		2	28.27	even	2
2800.2.a.bl.1.1		2	1.1	even	1	trivial
2800.2.a.bm.1.2		2	5.4	even	2
3150.2.a.bs.1.2		2	12.11	even	2
3150.2.a.bt.1.2		2	60.59	even	2
5040.2.t.t.1009.1		4	15.2	even	4
5040.2.t.t.1009.2		4	15.8	even	4

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +4.89898 q^{11} -4.44949 q^{13} -2.00000 q^{17} -1.55051 q^{19} +2.44949 q^{21} +2.89898 q^{23} +6.89898 q^{29} -8.89898 q^{31} -12.0000 q^{33} -2.00000 q^{37} +10.8990 q^{39} -1.10102 q^{41} -0.898979 q^{43} +8.89898 q^{47} +1.00000 q^{49} +4.89898 q^{51} +10.8990 q^{53} +3.79796 q^{57} +1.55051 q^{59} +3.55051 q^{61} -3.00000 q^{63} -8.00000 q^{67} -7.10102 q^{69} +1.10102 q^{71} -2.89898 q^{73} -4.89898 q^{77} -6.89898 q^{79} -9.00000 q^{81} -2.44949 q^{83} -16.8990 q^{87} -10.0000 q^{89} +4.44949 q^{91} +21.7980 q^{93} -15.7980 q^{97} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{29} - 8 q^{31} - 24 q^{33} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{49} + 12 q^{53} - 12 q^{57} + 8 q^{59} + 12 q^{61} - 6 q^{63} - 16 q^{67} - 24 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{79} - 18 q^{81} - 24 q^{87} - 20 q^{89} + 4 q^{91} + 24 q^{93} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 - 8 * q^19 - 4 * q^23 + 4 * q^29 - 8 * q^31 - 24 * q^33 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^49 + 12 * q^53 - 12 * q^57 + 8 * q^59 + 12 * q^61 - 6 * q^63 - 16 * q^67 - 24 * q^69 + 12 * q^71 + 4 * q^73 - 4 * q^79 - 18 * q^81 - 24 * q^87 - 20 * q^89 + 4 * q^91 + 24 * q^93 - 12 * q^97

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