LMFDB - Embedded newform 9800.2.a.co.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.546295$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.546295 q^{3} -2.70156 q^{9} +O(q^{10})$	$q-0.546295 q^{3} -2.70156 q^{9} -1.70156 q^{11} +0.546295 q^{13} -3.11473 q^{17} +4.75362 q^{19} +5.40312 q^{23} +3.11473 q^{27} -0.298438 q^{29} -7.32206 q^{31} +0.929554 q^{33} +5.40312 q^{37} -0.298438 q^{39} +7.32206 q^{41} -4.00000 q^{43} -8.25161 q^{47} +1.70156 q^{51} +1.40312 q^{53} -2.59688 q^{57} +7.70532 q^{59} -9.89049 q^{61} -8.00000 q^{67} -2.95170 q^{69} -5.40312 q^{71} +11.3663 q^{73} +11.1047 q^{79} +6.40312 q^{81} -9.89049 q^{83} +0.163035 q^{87} +16.8293 q^{89} +4.00000 q^{93} -5.29991 q^{97} +4.59688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} + 6 q^{11} - 4 q^{23} - 14 q^{29} - 4 q^{37} - 14 q^{39} - 16 q^{43} - 6 q^{51} - 20 q^{53} - 36 q^{57} - 32 q^{67} + 4 q^{71} + 6 q^{79} + 16 q^{93} + 44 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 6 * q^11 - 4 * q^23 - 14 * q^29 - 4 * q^37 - 14 * q^39 - 16 * q^43 - 6 * q^51 - 20 * q^53 - 36 * q^57 - 32 * q^67 + 4 * q^71 + 6 * q^79 + 16 * q^93 + 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.546295	−0.315403	−0.157702	−	0.987487i	$-0.550408\pi$
$3$	−0.546295	−0.315403	−0.157702	+	0.987487i	$0.550408\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.70156	−0.900521
$10$	0	0
$11$	−1.70156	−0.513040	−0.256520	−	0.966539i	$-0.582576\pi$
$11$	−1.70156	−0.513040	−0.256520	+	0.966539i	$0.582576\pi$
$12$	0	0
$13$	0.546295	0.151515	0.0757574	−	0.997126i	$-0.475863\pi$
$13$	0.546295	0.151515	0.0757574	+	0.997126i	$0.475863\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.11473	−0.755434	−0.377717	−	0.925921i	$-0.623291\pi$
$17$	−3.11473	−0.755434	−0.377717	+	0.925921i	$0.623291\pi$
$18$	0	0
$19$	4.75362	1.09055	0.545277	−	0.838256i	$-0.316424\pi$
$19$	4.75362	1.09055	0.545277	+	0.838256i	$0.316424\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.40312	1.12663	0.563315	−	0.826242i	$-0.309526\pi$
$23$	5.40312	1.12663	0.563315	+	0.826242i	$0.309526\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.11473	0.599431
$28$	0	0
$29$	−0.298438	−0.0554185	−0.0277093	−	0.999616i	$-0.508821\pi$
$29$	−0.298438	−0.0554185	−0.0277093	+	0.999616i	$0.508821\pi$
$30$	0	0
$31$	−7.32206	−1.31508	−0.657540	−	0.753420i	$-0.728403\pi$
$31$	−7.32206	−1.31508	−0.657540	+	0.753420i	$0.728403\pi$
$32$	0	0
$33$	0.929554	0.161815
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.40312	0.888268	0.444134	−	0.895960i	$-0.353511\pi$
$37$	5.40312	0.888268	0.444134	+	0.895960i	$0.353511\pi$
$38$	0	0
$39$	−0.298438	−0.0477883
$40$	0	0
$41$	7.32206	1.14351	0.571756	−	0.820423i	$-0.306262\pi$
$41$	7.32206	1.14351	0.571756	+	0.820423i	$0.306262\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.25161	−1.20362	−0.601811	−	0.798639i	$-0.705554\pi$
$47$	−8.25161	−1.20362	−0.601811	+	0.798639i	$0.705554\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.70156	0.238266
$52$	0	0
$53$	1.40312	0.192734	0.0963670	−	0.995346i	$-0.469278\pi$
$53$	1.40312	0.192734	0.0963670	+	0.995346i	$0.469278\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.59688	−0.343965
$58$	0	0
$59$	7.70532	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$59$	7.70532	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$60$	0	0
$61$	−9.89049	−1.26635	−0.633174	−	0.774009i	$-0.718248\pi$
$61$	−9.89049	−1.26635	−0.633174	+	0.774009i	$0.718248\pi$
$62$	0	0
$63$	0	0
$64$	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
$69$	−2.95170	−0.355343
$70$	0	0
$71$	−5.40312	−0.641233	−0.320616	−	0.947209i	$-0.603890\pi$
$71$	−5.40312	−0.641233	−0.320616	+	0.947209i	$0.603890\pi$
$72$	0	0
$73$	11.3663	1.33033	0.665165	−	0.746696i	$-0.268361\pi$
$73$	11.3663	1.33033	0.665165	+	0.746696i	$0.268361\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.1047	1.24937	0.624687	−	0.780875i	$-0.285226\pi$
$79$	11.1047	1.24937	0.624687	+	0.780875i	$0.285226\pi$
$80$	0	0
$81$	6.40312	0.711458
$82$	0	0
$83$	−9.89049	−1.08562	−0.542811	−	0.839855i	$-0.682640\pi$
$83$	−9.89049	−1.08562	−0.542811	+	0.839855i	$0.682640\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.163035	0.0174792
$88$	0	0
$89$	16.8293	1.78390	0.891951	−	0.452133i	$-0.149337\pi$
$89$	16.8293	1.78390	0.891951	+	0.452133i	$0.149337\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.29991	−0.538125	−0.269062	−	0.963123i	$-0.586714\pi$
$97$	−5.29991	−0.538125	−0.269062	+	0.963123i	$0.586714\pi$
$98$	0	0
$99$	4.59688	0.462003
$100$	0	0
$101$	0.383260	0.0381358	0.0190679	−	0.999818i	$-0.493930\pi$
$101$	0.383260	0.0381358	0.0190679	+	0.999818i	$0.493930\pi$
$102$	0	0
$103$	18.8514	1.85749	0.928743	−	0.370723i	$-0.120890\pi$
$103$	18.8514	1.85749	0.928743	+	0.370723i	$0.120890\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	4.29844	0.411716	0.205858	−	0.978582i	$-0.434002\pi$
$109$	4.29844	0.411716	0.205858	+	0.978582i	$0.434002\pi$
$110$	0	0
$111$	−2.95170	−0.280163
$112$	0	0
$113$	−5.40312	−0.508283	−0.254142	−	0.967167i	$-0.581793\pi$
$113$	−5.40312	−0.508283	−0.254142	+	0.967167i	$0.581793\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.47585	−0.136442
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.10469	−0.736790
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.4031	−1.54428	−0.772139	−	0.635454i	$-0.780813\pi$
$127$	−17.4031	−1.54428	−0.772139	+	0.635454i	$0.780813\pi$
$128$	0	0
$129$	2.18518	0.192394
$130$	0	0
$131$	−9.89049	−0.864136	−0.432068	−	0.901841i	$-0.642216\pi$
$131$	−9.89049	−0.864136	−0.432068	+	0.901841i	$0.642216\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.0000	−1.70872	−0.854358	−	0.519685i	$-0.826049\pi$
$137$	−20.0000	−1.70872	−0.854358	+	0.519685i	$0.826049\pi$
$138$	0	0
$139$	6.93880	0.588541	0.294270	−	0.955722i	$-0.404923\pi$
$139$	6.93880	0.588541	0.294270	+	0.955722i	$0.404923\pi$
$140$	0	0
$141$	4.50781	0.379626
$142$	0	0
$143$	−0.929554	−0.0777332
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−6.29844	−0.512560	−0.256280	−	0.966603i	$-0.582497\pi$
$151$	−6.29844	−0.512560	−0.256280	+	0.966603i	$0.582497\pi$
$152$	0	0
$153$	8.41464	0.680284
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.3051	−1.46091	−0.730455	−	0.682961i	$-0.760692\pi$
$157$	−18.3051	−1.46091	−0.730455	+	0.682961i	$0.760692\pi$
$158$	0	0
$159$	−0.766519	−0.0607889
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.8062	−0.846411	−0.423205	−	0.906034i	$-0.639095\pi$
$163$	−10.8062	−0.846411	−0.423205	+	0.906034i	$0.639095\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.6663	1.28967	0.644837	−	0.764320i	$-0.276925\pi$
$167$	16.6663	1.28967	0.644837	+	0.764320i	$0.276925\pi$
$168$	0	0
$169$	−12.7016	−0.977043
$170$	0	0
$171$	−12.8422	−0.982067
$172$	0	0
$173$	−4.91665	−0.373806	−0.186903	−	0.982378i	$-0.559845\pi$
$173$	−4.91665	−0.373806	−0.186903	+	0.982378i	$0.559845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.20937	−0.316396
$178$	0	0
$179$	−10.8062	−0.807697	−0.403848	−	0.914826i	$-0.632328\pi$
$179$	−10.8062	−0.807697	−0.403848	+	0.914826i	$0.632328\pi$
$180$	0	0
$181$	2.56844	0.190911	0.0954553	−	0.995434i	$-0.469569\pi$
$181$	2.56844	0.190911	0.0954553	+	0.995434i	$0.469569\pi$
$182$	0	0
$183$	5.40312	0.399410
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.29991	0.387568
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7016	0.846695	0.423348	−	0.905967i	$-0.360855\pi$
$191$	11.7016	0.846695	0.423348	+	0.905967i	$0.360855\pi$
$192$	0	0
$193$	−16.2094	−1.16678	−0.583388	−	0.812194i	$-0.698273\pi$
$193$	−16.2094	−1.16678	−0.583388	+	0.812194i	$0.698273\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.2094	−1.72485	−0.862423	−	0.506188i	$-0.831054\pi$
$197$	−24.2094	−1.72485	−0.862423	+	0.506188i	$0.831054\pi$
$198$	0	0
$199$	5.13688	0.364144	0.182072	−	0.983285i	$-0.441720\pi$
$199$	5.13688	0.364144	0.182072	+	0.983285i	$0.441720\pi$
$200$	0	0
$201$	4.37036	0.308261
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−14.5969	−1.01455
$208$	0	0
$209$	−8.08857	−0.559498
$210$	0	0
$211$	13.7016	0.943254	0.471627	−	0.881798i	$-0.343667\pi$
$211$	13.7016	0.943254	0.471627	+	0.881798i	$0.343667\pi$
$212$	0	0
$213$	2.95170	0.202247
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.20937	−0.419591
$220$	0	0
$221$	−1.70156	−0.114459
$222$	0	0
$223$	−7.48509	−0.501239	−0.250619	−	0.968086i	$-0.580634\pi$
$223$	−7.48509	−0.501239	−0.250619	+	0.968086i	$0.580634\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.5567	1.76263	0.881317	−	0.472526i	$-0.156658\pi$
$227$	26.5567	1.76263	0.881317	+	0.472526i	$0.156658\pi$
$228$	0	0
$229$	−4.75362	−0.314128	−0.157064	−	0.987588i	$-0.550203\pi$
$229$	−4.75362	−0.314128	−0.157064	+	0.987588i	$0.550203\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.06643	−0.394057
$238$	0	0
$239$	9.70156	0.627542	0.313771	−	0.949499i	$-0.398408\pi$
$239$	9.70156	0.627542	0.313771	+	0.949499i	$0.398408\pi$
$240$	0	0
$241$	2.95170	0.190136	0.0950678	−	0.995471i	$-0.469693\pi$
$241$	2.95170	0.190136	0.0950678	+	0.995471i	$0.469693\pi$
$242$	0	0
$243$	−12.8422	−0.823827
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.59688	0.165235
$248$	0	0
$249$	5.40312	0.342409
$250$	0	0
$251$	6.93880	0.437973	0.218986	−	0.975728i	$-0.429725\pi$
$251$	6.93880	0.437973	0.218986	+	0.975728i	$0.429725\pi$
$252$	0	0
$253$	−9.19375	−0.578006
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.46295	−0.340769	−0.170385	−	0.985378i	$-0.554501\pi$
$257$	−5.46295	−0.340769	−0.170385	+	0.985378i	$0.554501\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.806248	0.0499055
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.19375	−0.562648
$268$	0	0
$269$	−26.7198	−1.62913	−0.814567	−	0.580070i	$-0.803025\pi$
$269$	−26.7198	−1.62913	−0.814567	+	0.580070i	$0.803025\pi$
$270$	0	0
$271$	22.7327	1.38091	0.690456	−	0.723374i	$-0.257410\pi$
$271$	22.7327	1.38091	0.690456	+	0.723374i	$0.257410\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.4031	−1.52633	−0.763163	−	0.646206i	$-0.776355\pi$
$277$	−25.4031	−1.52633	−0.763163	+	0.646206i	$0.776355\pi$
$278$	0	0
$279$	19.7810	1.18426
$280$	0	0
$281$	−17.9109	−1.06848	−0.534238	−	0.845334i	$-0.679402\pi$
$281$	−17.9109	−1.06848	−0.534238	+	0.845334i	$0.679402\pi$
$282$	0	0
$283$	−19.2347	−1.14338	−0.571692	−	0.820468i	$-0.693713\pi$
$283$	−19.2347	−1.14338	−0.571692	+	0.820468i	$0.693713\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.29844	−0.429320
$290$	0	0
$291$	2.89531	0.169726
$292$	0	0
$293$	13.3313	0.778823	0.389411	−	0.921064i	$-0.372678\pi$
$293$	13.3313	0.778823	0.389411	+	0.921064i	$0.372678\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.29991	−0.307532
$298$	0	0
$299$	2.95170	0.170701
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.209373	−0.0120281
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.4199	−1.22250	−0.611248	−	0.791439i	$-0.709332\pi$
$307$	−21.4199	−1.22250	−0.611248	+	0.791439i	$0.709332\pi$
$308$	0	0
$309$	−10.2984	−0.585858
$310$	0	0
$311$	27.1030	1.53687	0.768436	−	0.639926i	$-0.221035\pi$
$311$	27.1030	1.53687	0.768436	+	0.639926i	$0.221035\pi$
$312$	0	0
$313$	−26.1735	−1.47941	−0.739707	−	0.672930i	$-0.765036\pi$
$313$	−26.1735	−1.47941	−0.739707	+	0.672930i	$0.765036\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.4031	0.977457	0.488728	−	0.872436i	$-0.337461\pi$
$317$	17.4031	0.977457	0.488728	+	0.872436i	$0.337461\pi$
$318$	0	0
$319$	0.507811	0.0284319
$320$	0	0
$321$	−4.37036	−0.243930
$322$	0	0
$323$	−14.8062	−0.823842
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.34821	−0.129857
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10.8062	−0.593965	−0.296983	−	0.954883i	$-0.595980\pi$
$331$	−10.8062	−0.593965	−0.296983	+	0.954883i	$0.595980\pi$
$332$	0	0
$333$	−14.5969	−0.799904
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.5969	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$337$	−14.5969	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$338$	0	0
$339$	2.95170	0.160314
$340$	0	0
$341$	12.4589	0.674689
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.8062	−1.65377	−0.826883	−	0.562374i	$-0.809888\pi$
$347$	−30.8062	−1.65377	−0.826883	+	0.562374i	$0.809888\pi$
$348$	0	0
$349$	15.0274	0.804397	0.402198	−	0.915553i	$-0.368246\pi$
$349$	15.0274	0.804397	0.402198	+	0.915553i	$0.368246\pi$
$350$	0	0
$351$	1.70156	0.0908227
$352$	0	0
$353$	−20.7105	−1.10231	−0.551155	−	0.834403i	$-0.685813\pi$
$353$	−20.7105	−1.10231	−0.551155	+	0.834403i	$0.685813\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.8062	1.62589	0.812946	−	0.582339i	$-0.197862\pi$
$359$	30.8062	1.62589	0.812946	+	0.582339i	$0.197862\pi$
$360$	0	0
$361$	3.59688	0.189309
$362$	0	0
$363$	4.42755	0.232386
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.29991	−0.276653	−0.138327	−	0.990387i	$-0.544172\pi$
$367$	−5.29991	−0.276653	−0.138327	+	0.990387i	$0.544172\pi$
$368$	0	0
$369$	−19.7810	−1.02976
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.40312	−0.279763	−0.139882	−	0.990168i	$-0.544672\pi$
$373$	−5.40312	−0.279763	−0.139882	+	0.990168i	$0.544672\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.163035	−0.00839673
$378$	0	0
$379$	37.6125	1.93202	0.966012	−	0.258497i	$-0.0832272\pi$
$379$	37.6125	1.93202	0.966012	+	0.258497i	$0.0832272\pi$
$380$	0	0
$381$	9.50723	0.487070
$382$	0	0
$383$	13.8776	0.709112	0.354556	−	0.935035i	$-0.384632\pi$
$383$	13.8776	0.709112	0.354556	+	0.935035i	$0.384632\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.8062	0.549312
$388$	0	0
$389$	11.1047	0.563030	0.281515	−	0.959557i	$-0.409163\pi$
$389$	11.1047	0.563030	0.281515	+	0.959557i	$0.409163\pi$
$390$	0	0
$391$	−16.8293	−0.851094
$392$	0	0
$393$	5.40312	0.272552
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.5567	1.33285	0.666423	−	0.745574i	$-0.267825\pi$
$397$	26.5567	1.33285	0.666423	+	0.745574i	$0.267825\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.29844	−0.314529	−0.157264	−	0.987557i	$-0.550268\pi$
$401$	−6.29844	−0.314529	−0.157264	+	0.987557i	$0.550268\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.19375	−0.455717
$408$	0	0
$409$	−5.13688	−0.254002	−0.127001	−	0.991903i	$-0.540535\pi$
$409$	−5.13688	−0.254002	−0.127001	+	0.991903i	$0.540535\pi$
$410$	0	0
$411$	10.9259	0.538935
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.79063	−0.185628
$418$	0	0
$419$	−21.5829	−1.05439	−0.527197	−	0.849743i	$-0.676757\pi$
$419$	−21.5829	−1.05439	−0.527197	+	0.849743i	$0.676757\pi$
$420$	0	0
$421$	−22.5078	−1.09696	−0.548482	−	0.836163i	$-0.684794\pi$
$421$	−22.5078	−1.09696	−0.548482	+	0.836163i	$0.684794\pi$
$422$	0	0
$423$	22.2922	1.08389
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.507811	0.0245173
$430$	0	0
$431$	−2.89531	−0.139462	−0.0697312	−	0.997566i	$-0.522214\pi$
$431$	−2.89531	−0.139462	−0.0697312	+	0.997566i	$0.522214\pi$
$432$	0	0
$433$	1.85911	0.0893431	0.0446715	−	0.999002i	$-0.485776\pi$
$433$	1.85911	0.0893431	0.0446715	+	0.999002i	$0.485776\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.6844	1.22865
$438$	0	0
$439$	−39.5620	−1.88819	−0.944095	−	0.329673i	$-0.893061\pi$
$439$	−39.5620	−1.88819	−0.944095	+	0.329673i	$0.893061\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.4187	1.73031	0.865153	−	0.501507i	$-0.167221\pi$
$443$	36.4187	1.73031	0.865153	+	0.501507i	$0.167221\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.46295	0.258389
$448$	0	0
$449$	−31.1047	−1.46792	−0.733960	−	0.679192i	$-0.762330\pi$
$449$	−31.1047	−1.46792	−0.733960	+	0.679192i	$0.762330\pi$
$450$	0	0
$451$	−12.4589	−0.586668
$452$	0	0
$453$	3.44080	0.161663
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.2094	1.13247	0.566233	−	0.824245i	$-0.308400\pi$
$457$	24.2094	1.13247	0.566233	+	0.824245i	$0.308400\pi$
$458$	0	0
$459$	−9.70156	−0.452830
$460$	0	0
$461$	6.93880	0.323172	0.161586	−	0.986859i	$-0.448339\pi$
$461$	6.93880	0.323172	0.161586	+	0.986859i	$0.448339\pi$
$462$	0	0
$463$	−30.8062	−1.43169	−0.715844	−	0.698260i	$-0.753958\pi$
$463$	−30.8062	−1.43169	−0.715844	+	0.698260i	$0.753958\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.0197	1.48169	0.740847	−	0.671673i	$-0.234424\pi$
$467$	32.0197	1.48169	0.740847	+	0.671673i	$0.234424\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	6.80625	0.312952
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.79063	−0.173561
$478$	0	0
$479$	−42.5137	−1.94250	−0.971250	−	0.238063i	$-0.923487\pi$
$479$	−42.5137	−1.94250	−0.971250	+	0.238063i	$0.923487\pi$
$480$	0	0
$481$	2.95170	0.134586
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.2094	−1.64080	−0.820402	−	0.571788i	$-0.806250\pi$
$487$	−36.2094	−1.64080	−0.820402	+	0.571788i	$0.806250\pi$
$488$	0	0
$489$	5.90340	0.266961
$490$	0	0
$491$	32.5078	1.46706	0.733528	−	0.679659i	$-0.237872\pi$
$491$	32.5078	1.46706	0.733528	+	0.679659i	$0.237872\pi$
$492$	0	0
$493$	0.929554	0.0418650
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.507811	−0.0227327	−0.0113664	−	0.999935i	$-0.503618\pi$
$499$	−0.507811	−0.0227327	−0.0113664	+	0.999935i	$0.503618\pi$
$500$	0	0
$501$	−9.10469	−0.406767
$502$	0	0
$503$	−9.34420	−0.416637	−0.208319	−	0.978061i	$-0.566799\pi$
$503$	−9.34420	−0.416637	−0.208319	+	0.978061i	$0.566799\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.93880	0.308163
$508$	0	0
$509$	−6.93880	−0.307557	−0.153778	−	0.988105i	$-0.549144\pi$
$509$	−6.93880	−0.307557	−0.153778	+	0.988105i	$0.549144\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.8062	0.653712
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.0406	0.617506
$518$	0	0
$519$	2.68594	0.117900
$520$	0	0
$521$	26.3365	1.15382	0.576912	−	0.816806i	$-0.304258\pi$
$521$	26.3365	1.15382	0.576912	+	0.816806i	$0.304258\pi$
$522$	0	0
$523$	−25.3011	−1.10634	−0.553170	−	0.833068i	$-0.686582\pi$
$523$	−25.3011	−1.10634	−0.553170	+	0.833068i	$0.686582\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.8062	0.993456
$528$	0	0
$529$	6.19375	0.269294
$530$	0	0
$531$	−20.8164	−0.903354
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.90340	0.254750
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.5078	0.623739	0.311870	−	0.950125i	$-0.399045\pi$
$541$	14.5078	0.623739	0.311870	+	0.950125i	$0.399045\pi$
$542$	0	0
$543$	−1.40312	−0.0602138
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−41.6125	−1.77922	−0.889611	−	0.456719i	$-0.849024\pi$
$547$	−41.6125	−1.77922	−0.889611	+	0.456719i	$0.849024\pi$
$548$	0	0
$549$	26.7198	1.14037
$550$	0	0
$551$	−1.41866	−0.0604369
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.4031	−1.07636	−0.538182	−	0.842828i	$-0.680889\pi$
$557$	−25.4031	−1.07636	−0.538182	+	0.842828i	$0.680889\pi$
$558$	0	0
$559$	−2.18518	−0.0924232
$560$	0	0
$561$	−2.89531	−0.122240
$562$	0	0
$563$	−15.7939	−0.665633	−0.332817	−	0.942992i	$-0.607999\pi$
$563$	−15.7939	−0.665633	−0.332817	+	0.942992i	$0.607999\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.4031	−1.48418	−0.742088	−	0.670302i	$-0.766164\pi$
$569$	−35.4031	−1.48418	−0.742088	+	0.670302i	$0.766164\pi$
$570$	0	0
$571$	21.6125	0.904455	0.452227	−	0.891903i	$-0.350630\pi$
$571$	21.6125	0.904455	0.452227	+	0.891903i	$0.350630\pi$
$572$	0	0
$573$	−6.39250	−0.267051
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.9923	0.707400	0.353700	−	0.935359i	$-0.384923\pi$
$577$	16.9923	0.707400	0.353700	+	0.935359i	$0.384923\pi$
$578$	0	0
$579$	8.85509	0.368005
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.38750	−0.0988803
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.98710	−0.164565	−0.0822826	−	0.996609i	$-0.526221\pi$
$587$	−3.98710	−0.164565	−0.0822826	+	0.996609i	$0.526221\pi$
$588$	0	0
$589$	−34.8062	−1.43417
$590$	0	0
$591$	13.2255	0.544022
$592$	0	0
$593$	20.7105	0.850480	0.425240	−	0.905081i	$-0.360190\pi$
$593$	20.7105	0.850480	0.425240	+	0.905081i	$0.360190\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.80625	−0.114852
$598$	0	0
$599$	−11.1047	−0.453725	−0.226863	−	0.973927i	$-0.572847\pi$
$599$	−11.1047	−0.453725	−0.226863	+	0.973927i	$0.572847\pi$
$600$	0	0
$601$	−35.1916	−1.43550	−0.717748	−	0.696303i	$-0.754827\pi$
$601$	−35.1916	−1.43550	−0.717748	+	0.696303i	$0.754827\pi$
$602$	0	0
$603$	21.6125	0.880129
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.3587	−1.15104	−0.575521	−	0.817787i	$-0.695201\pi$
$607$	−28.3587	−1.15104	−0.575521	+	0.817787i	$0.695201\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.50781	−0.182366
$612$	0	0
$613$	9.40312	0.379789	0.189894	−	0.981805i	$-0.439185\pi$
$613$	9.40312	0.379789	0.189894	+	0.981805i	$0.439185\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.4031	−1.02269	−0.511346	−	0.859375i	$-0.670853\pi$
$617$	−25.4031	−1.02269	−0.511346	+	0.859375i	$0.670853\pi$
$618$	0	0
$619$	9.89049	0.397533	0.198766	−	0.980047i	$-0.436307\pi$
$619$	9.89049	0.397533	0.198766	+	0.980047i	$0.436307\pi$
$620$	0	0
$621$	16.8293	0.675336
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.41875	0.176468
$628$	0	0
$629$	−16.8293	−0.671028
$630$	0	0
$631$	2.50781	0.0998344	0.0499172	−	0.998753i	$-0.484104\pi$
$631$	2.50781	0.0998344	0.0499172	+	0.998753i	$0.484104\pi$
$632$	0	0
$633$	−7.48509	−0.297506
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.5969	0.577443
$640$	0	0
$641$	−4.59688	−0.181566	−0.0907828	−	0.995871i	$-0.528937\pi$
$641$	−4.59688	−0.181566	−0.0907828	+	0.995871i	$0.528937\pi$
$642$	0	0
$643$	45.5712	1.79715	0.898577	−	0.438817i	$-0.144602\pi$
$643$	45.5712	1.79715	0.898577	+	0.438817i	$0.144602\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.5217	−1.12130	−0.560652	−	0.828052i	$-0.689449\pi$
$647$	−28.5217	−1.12130	−0.560652	+	0.828052i	$0.689449\pi$
$648$	0	0
$649$	−13.1111	−0.514655
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.4031	0.837569	0.418784	−	0.908086i	$-0.362456\pi$
$653$	21.4031	0.837569	0.418784	+	0.908086i	$0.362456\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.7069	−1.19799
$658$	0	0
$659$	41.1047	1.60121	0.800606	−	0.599192i	$-0.204511\pi$
$659$	41.1047	1.60121	0.800606	+	0.599192i	$0.204511\pi$
$660$	0	0
$661$	−12.8422	−0.499503	−0.249752	−	0.968310i	$-0.580349\pi$
$661$	−12.8422	−0.499503	−0.249752	+	0.968310i	$0.580349\pi$
$662$	0	0
$663$	0.929554	0.0361009
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.61250	−0.0624361
$668$	0	0
$669$	4.08907	0.158092
$670$	0	0
$671$	16.8293	0.649688
$672$	0	0
$673$	−41.6125	−1.60404	−0.802022	−	0.597295i	$-0.796242\pi$
$673$	−41.6125	−1.60404	−0.802022	+	0.597295i	$0.796242\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.3233	−1.05012	−0.525059	−	0.851066i	$-0.675957\pi$
$677$	−27.3233	−1.05012	−0.525059	+	0.851066i	$0.675957\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.5078	−0.555941
$682$	0	0
$683$	−9.19375	−0.351789	−0.175895	−	0.984409i	$-0.556282\pi$
$683$	−9.19375	−0.351789	−0.175895	+	0.984409i	$0.556282\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.59688	0.0990770
$688$	0	0
$689$	0.766519	0.0292021
$690$	0	0
$691$	−26.7198	−1.01647	−0.508234	−	0.861219i	$-0.669702\pi$
$691$	−26.7198	−1.01647	−0.508234	+	0.861219i	$0.669702\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.8062	−0.863848
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.3141	0.502865	0.251433	−	0.967875i	$-0.419098\pi$
$701$	13.3141	0.502865	0.251433	+	0.967875i	$0.419098\pi$
$702$	0	0
$703$	25.6844	0.968705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35.1047	−1.31838	−0.659192	−	0.751975i	$-0.729102\pi$
$709$	−35.1047	−1.31838	−0.659192	+	0.751975i	$0.729102\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	0	0
$713$	−39.5620	−1.48161
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.29991	−0.197929
$718$	0	0
$719$	2.95170	0.110080	0.0550399	−	0.998484i	$-0.482471\pi$
$719$	2.95170	0.110080	0.0550399	+	0.998484i	$0.482471\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.61250	−0.0599694
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.7810	−0.733636	−0.366818	−	0.930293i	$-0.619553\pi$
$727$	−19.7810	−0.733636	−0.366818	+	0.930293i	$0.619553\pi$
$728$	0	0
$729$	−12.1938	−0.451620
$730$	0	0
$731$	12.4589	0.460810
$732$	0	0
$733$	−16.2830	−0.601426	−0.300713	−	0.953715i	$-0.597225\pi$
$733$	−16.2830	−0.601426	−0.300713	+	0.953715i	$0.597225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.6125	0.501423
$738$	0	0
$739$	−20.5078	−0.754392	−0.377196	−	0.926133i	$-0.623112\pi$
$739$	−20.5078	−0.754392	−0.377196	+	0.926133i	$0.623112\pi$
$740$	0	0
$741$	−1.41866	−0.0521158
$742$	0	0
$743$	9.40312	0.344967	0.172484	−	0.985012i	$-0.444821\pi$
$743$	9.40312	0.344967	0.172484	+	0.985012i	$0.444821\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	26.7198	0.977626
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.5078	0.894303	0.447151	−	0.894458i	$-0.352439\pi$
$751$	24.5078	0.894303	0.447151	+	0.894458i	$0.352439\pi$
$752$	0	0
$753$	−3.79063	−0.138138
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−14.5969	−0.530532	−0.265266	−	0.964175i	$-0.585460\pi$
$757$	−14.5969	−0.530532	−0.265266	+	0.964175i	$0.585460\pi$
$758$	0	0
$759$	5.02250	0.182305
$760$	0	0
$761$	−30.7069	−1.11312	−0.556562	−	0.830806i	$-0.687880\pi$
$761$	−30.7069	−1.11312	−0.556562	+	0.830806i	$0.687880\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.20937	0.151992
$768$	0	0
$769$	11.0403	0.398122	0.199061	−	0.979987i	$-0.436211\pi$
$769$	11.0403	0.398122	0.199061	+	0.979987i	$0.436211\pi$
$770$	0	0
$771$	2.98438	0.107480
$772$	0	0
$773$	−31.6936	−1.13994	−0.569970	−	0.821665i	$-0.693045\pi$
$773$	−31.6936	−1.13994	−0.569970	+	0.821665i	$0.693045\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.8062	1.24706
$780$	0	0
$781$	9.19375	0.328978
$782$	0	0
$783$	−0.929554	−0.0332196
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.9569	−0.568803	−0.284401	−	0.958705i	$-0.591795\pi$
$787$	−15.9569	−0.568803	−0.284401	+	0.958705i	$0.591795\pi$
$788$	0	0
$789$	8.74071	0.311178
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.40312	−0.191871
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.4343	1.43226	0.716129	−	0.697968i	$-0.245912\pi$
$797$	40.4343	1.43226	0.716129	+	0.697968i	$0.245912\pi$
$798$	0	0
$799$	25.7016	0.909256
$800$	0	0
$801$	−45.4654	−1.60644
$802$	0	0
$803$	−19.3405	−0.682513
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.5969	0.513834
$808$	0	0
$809$	16.5078	0.580384	0.290192	−	0.956968i	$-0.406281\pi$
$809$	16.5078	0.580384	0.290192	+	0.956968i	$0.406281\pi$
$810$	0	0
$811$	52.4042	1.84016	0.920080	−	0.391731i	$-0.128124\pi$
$811$	52.4042	1.84016	0.920080	+	0.391731i	$0.128124\pi$
$812$	0	0
$813$	−12.4187	−0.435544
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19.0145	−0.665232
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.3141	−1.58147	−0.790736	−	0.612158i	$-0.790302\pi$
$821$	−45.3141	−1.58147	−0.790736	+	0.612158i	$0.790302\pi$
$822$	0	0
$823$	14.8062	0.516113	0.258057	−	0.966130i	$-0.416918\pi$
$823$	14.8062	0.516113	0.258057	+	0.966130i	$0.416918\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.8062	−0.375770	−0.187885	−	0.982191i	$-0.560163\pi$
$827$	−10.8062	−0.375770	−0.187885	+	0.982191i	$0.560163\pi$
$828$	0	0
$829$	38.4122	1.33411	0.667055	−	0.745008i	$-0.267554\pi$
$829$	38.4122	1.33411	0.667055	+	0.745008i	$0.267554\pi$
$830$	0	0
$831$	13.8776	0.481408
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−22.8062	−0.788299
$838$	0	0
$839$	−30.7069	−1.06012	−0.530060	−	0.847960i	$-0.677830\pi$
$839$	−30.7069	−1.06012	−0.530060	+	0.847960i	$0.677830\pi$
$840$	0	0
$841$	−28.9109	−0.996929
$842$	0	0
$843$	9.78465	0.337001
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	10.5078	0.360627
$850$	0	0
$851$	29.1938	1.00075
$852$	0	0
$853$	−1.47585	−0.0505321	−0.0252661	−	0.999681i	$-0.508043\pi$
$853$	−1.47585	−0.0505321	−0.0252661	+	0.999681i	$0.508043\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.2439	0.862316	0.431158	−	0.902276i	$-0.358105\pi$
$857$	25.2439	0.862316	0.431158	+	0.902276i	$0.358105\pi$
$858$	0	0
$859$	32.6232	1.11309	0.556544	−	0.830818i	$-0.312127\pi$
$859$	32.6232	1.11309	0.556544	+	0.830818i	$0.312127\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.4187	−0.967385	−0.483693	−	0.875238i	$-0.660705\pi$
$863$	−28.4187	−0.967385	−0.483693	+	0.875238i	$0.660705\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.98710	0.135409
$868$	0	0
$869$	−18.8953	−0.640980
$870$	0	0
$871$	−4.37036	−0.148084
$872$	0	0
$873$	14.3180	0.484592
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.0156	−1.18239	−0.591197	−	0.806527i	$-0.701344\pi$
$877$	−35.0156	−1.18239	−0.591197	+	0.806527i	$0.701344\pi$
$878$	0	0
$879$	−7.28282	−0.245643
$880$	0	0
$881$	39.5620	1.33288	0.666438	−	0.745560i	$-0.267818\pi$
$881$	39.5620	1.33288	0.666438	+	0.745560i	$0.267818\pi$
$882$	0	0
$883$	−9.19375	−0.309394	−0.154697	−	0.987962i	$-0.549440\pi$
$883$	−9.19375	−0.309394	−0.154697	+	0.987962i	$0.549440\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.6844	−0.862397	−0.431199	−	0.902257i	$-0.641909\pi$
$887$	−25.6844	−0.862397	−0.431199	+	0.902257i	$0.641909\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10.8953	−0.365007
$892$	0	0
$893$	−39.2250	−1.31261
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.61250	−0.0538397
$898$	0	0
$899$	2.18518	0.0728798
$900$	0	0
$901$	−4.37036	−0.145598
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.19375	0.305274	0.152637	−	0.988282i	$-0.451224\pi$
$907$	9.19375	0.305274	0.152637	+	0.988282i	$0.451224\pi$
$908$	0	0
$909$	−1.03540	−0.0343420
$910$	0	0
$911$	−30.8062	−1.02066	−0.510328	−	0.859980i	$-0.670476\pi$
$911$	−30.8062	−1.02066	−0.510328	+	0.859980i	$0.670476\pi$
$912$	0	0
$913$	16.8293	0.556968
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41.9109	1.38251	0.691257	−	0.722609i	$-0.257057\pi$
$919$	41.9109	1.38251	0.691257	+	0.722609i	$0.257057\pi$
$920$	0	0
$921$	11.7016	0.385580
$922$	0	0
$923$	−2.95170	−0.0971563
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−50.9283	−1.67271
$928$	0	0
$929$	−5.13688	−0.168535	−0.0842677	−	0.996443i	$-0.526855\pi$
$929$	−5.13688	−0.168535	−0.0842677	+	0.996443i	$0.526855\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.8062	−0.484735
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54.0431	1.76551	0.882755	−	0.469834i	$-0.155686\pi$
$937$	54.0431	1.76551	0.882755	+	0.469834i	$0.155686\pi$
$938$	0	0
$939$	14.2984	0.466612
$940$	0	0
$941$	−23.7681	−0.774817	−0.387409	−	0.921908i	$-0.626630\pi$
$941$	−23.7681	−0.774817	−0.387409	+	0.921908i	$0.626630\pi$
$942$	0	0
$943$	39.5620	1.28832
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.8062	−0.741103	−0.370552	−	0.928812i	$-0.620831\pi$
$947$	−22.8062	−0.741103	−0.370552	+	0.928812i	$0.620831\pi$
$948$	0	0
$949$	6.20937	0.201565
$950$	0	0
$951$	−9.50723	−0.308293
$952$	0	0
$953$	25.6125	0.829670	0.414835	−	0.909897i	$-0.363839\pi$
$953$	25.6125	0.829670	0.414835	+	0.909897i	$0.363839\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.277414	−0.00896753
$958$	0	0
$959$	0	0
$960$	0	0
$961$	22.6125	0.729435
$962$	0	0
$963$	−21.6125	−0.696453
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.20937	0.135364	0.0676822	−	0.997707i	$-0.478440\pi$
$967$	4.20937	0.135364	0.0676822	+	0.997707i	$0.478440\pi$
$968$	0	0
$969$	8.08857	0.259842
$970$	0	0
$971$	−58.9597	−1.89211	−0.946053	−	0.324011i	$-0.894969\pi$
$971$	−58.9597	−1.89211	−0.946053	+	0.324011i	$0.894969\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−28.6361	−0.915213
$980$	0	0
$981$	−11.6125	−0.370758
$982$	0	0
$983$	37.5398	1.19733	0.598667	−	0.800998i	$-0.295697\pi$
$983$	37.5398	1.19733	0.598667	+	0.800998i	$0.295697\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.6125	−0.687237
$990$	0	0
$991$	−48.2094	−1.53142	−0.765711	−	0.643185i	$-0.777613\pi$
$991$	−48.2094	−1.53142	−0.765711	+	0.643185i	$0.777613\pi$
$992$	0	0
$993$	5.90340	0.187339
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.07933	−0.0658531	−0.0329266	−	0.999458i	$-0.510483\pi$
$997$	−2.07933	−0.0658531	−0.0329266	+	0.999458i	$0.510483\pi$
$998$	0	0
$999$	16.8293	0.532455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.co.1.2		4
5.2	odd	4		1960.2.g.d.1569.5	yes	8
5.3	odd	4		1960.2.g.d.1569.3	✓	8
5.4	even	2		9800.2.a.cp.1.3		4
7.6	odd	2	inner	9800.2.a.co.1.3		4
35.13	even	4		1960.2.g.d.1569.6	yes	8
35.27	even	4		1960.2.g.d.1569.4	yes	8
35.34	odd	2		9800.2.a.cp.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.g.d.1569.3	✓	8	5.3	odd	4
1960.2.g.d.1569.4	yes	8	35.27	even	4
1960.2.g.d.1569.5	yes	8	5.2	odd	4
1960.2.g.d.1569.6	yes	8	35.13	even	4
9800.2.a.co.1.2		4	1.1	even	1	trivial
9800.2.a.co.1.3		4	7.6	odd	2	inner
9800.2.a.cp.1.2		4	35.34	odd	2
9800.2.a.cp.1.3		4	5.4	even	2

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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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q-0.546295 q^{3} -2.70156 q^{9} +O(q^{10})\)	\(q-0.546295 q^{3} -2.70156 q^{9} -1.70156 q^{11} +0.546295 q^{13} -3.11473 q^{17} +4.75362 q^{19} +5.40312 q^{23} +3.11473 q^{27} -0.298438 q^{29} -7.32206 q^{31} +0.929554 q^{33} +5.40312 q^{37} -0.298438 q^{39} +7.32206 q^{41} -4.00000 q^{43} -8.25161 q^{47} +1.70156 q^{51} +1.40312 q^{53} -2.59688 q^{57} +7.70532 q^{59} -9.89049 q^{61} -8.00000 q^{67} -2.95170 q^{69} -5.40312 q^{71} +11.3663 q^{73} +11.1047 q^{79} +6.40312 q^{81} -9.89049 q^{83} +0.163035 q^{87} +16.8293 q^{89} +4.00000 q^{93} -5.29991 q^{97} +4.59688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} + 6 q^{11} - 4 q^{23} - 14 q^{29} - 4 q^{37} - 14 q^{39} - 16 q^{43} - 6 q^{51} - 20 q^{53} - 36 q^{57} - 32 q^{67} + 4 q^{71} + 6 q^{79} + 16 q^{93} + 44 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 6 * q^11 - 4 * q^23 - 14 * q^29 - 4 * q^37 - 14 * q^39 - 16 * q^43 - 6 * q^51 - 20 * q^53 - 36 * q^57 - 32 * q^67 + 4 * q^71 + 6 * q^79 + 16 * q^93 + 44 * q^99

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