LMFDB - Embedded newform 112.10.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-170,0,544] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6840136504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-170.000 q^{3} +544.000 q^{5} +2401.00 q^{7} +9217.00 q^{9} -48824.0 q^{11} -15876.0 q^{13} -92480.0 q^{15} -21418.0 q^{17} +716410. q^{19} -408170. q^{21} +2.47000e6 q^{23} -1.65719e6 q^{25} +1.77922e6 q^{27} +5.55683e6 q^{29} -5.79935e6 q^{31} +8.30008e6 q^{33} +1.30614e6 q^{35} -3.89443e6 q^{37} +2.69892e6 q^{39} -6.36086e6 q^{41} +1.87013e7 q^{43} +5.01405e6 q^{45} -5.65391e7 q^{47} +5.76480e6 q^{49} +3.64106e6 q^{51} -5.98947e7 q^{53} -2.65603e7 q^{55} -1.21790e8 q^{57} -1.65630e8 q^{59} +5.14190e7 q^{61} +2.21300e7 q^{63} -8.63654e6 q^{65} -9.35465e7 q^{67} -4.19900e8 q^{69} +9.56335e7 q^{71} +3.06496e8 q^{73} +2.81722e8 q^{75} -1.17226e8 q^{77} -4.96474e8 q^{79} -4.83886e8 q^{81} +3.71487e8 q^{83} -1.16514e7 q^{85} -9.44660e8 q^{87} -1.65483e8 q^{89} -3.81183e7 q^{91} +9.85889e8 q^{93} +3.89727e8 q^{95} +7.58017e8 q^{97} -4.50011e8 q^{99} +O(q^{100})$

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
$3$	−170.000	−1.21172	−0.605861	−	0.795570i	$-0.707171\pi$
$3$	−170.000	−1.21172	−0.605861	+	0.795570i	$0.707171\pi$
$4$	0	0
$5$	544.000	0.389255	0.194627	−	0.980877i	$-0.437650\pi$
$5$	544.000	0.389255	0.194627	+	0.980877i	$0.437650\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	9217.00	0.468272
$10$	0	0
$11$	−48824.0	−1.00546	−0.502732	−	0.864442i	$-0.667672\pi$
$11$	−48824.0	−1.00546	−0.502732	+	0.864442i	$0.667672\pi$
$12$	0	0
$13$	−15876.0	−0.154169	−0.0770843	−	0.997025i	$-0.524561\pi$
$13$	−15876.0	−0.154169	−0.0770843	+	0.997025i	$0.524561\pi$
$14$	0	0
$15$	−92480.0	−0.471669
$16$	0	0
$17$	−21418.0	−0.0621955	−0.0310977	−	0.999516i	$-0.509900\pi$
$17$	−21418.0	−0.0621955	−0.0310977	+	0.999516i	$0.509900\pi$
$18$	0	0
$19$	716410.	1.26116	0.630580	−	0.776124i	$-0.282817\pi$
$19$	716410.	1.26116	0.630580	+	0.776124i	$0.282817\pi$
$20$	0	0
$21$	−408170.	−0.457988
$22$	0	0
$23$	2.47000e6	1.84044	0.920220	−	0.391401i	$-0.128010\pi$
$23$	2.47000e6	1.84044	0.920220	+	0.391401i	$0.128010\pi$
$24$	0	0
$25$	−1.65719e6	−0.848481
$26$	0	0
$27$	1.77922e6	0.644307
$28$	0	0
$29$	5.55683e6	1.45893	0.729467	−	0.684016i	$-0.239768\pi$
$29$	5.55683e6	1.45893	0.729467	+	0.684016i	$0.239768\pi$
$30$	0	0
$31$	−5.79935e6	−1.12785	−0.563925	−	0.825826i	$-0.690709\pi$
$31$	−5.79935e6	−1.12785	−0.563925	+	0.825826i	$0.690709\pi$
$32$	0	0
$33$	8.30008e6	1.21834
$34$	0	0
$35$	1.30614e6	0.147124
$36$	0	0
$37$	−3.89443e6	−0.341614	−0.170807	−	0.985304i	$-0.554638\pi$
$37$	−3.89443e6	−0.341614	−0.170807	+	0.985304i	$0.554638\pi$
$38$	0	0
$39$	2.69892e6	0.186810
$40$	0	0
$41$	−6.36086e6	−0.351551	−0.175776	−	0.984430i	$-0.556243\pi$
$41$	−6.36086e6	−0.351551	−0.175776	+	0.984430i	$0.556243\pi$
$42$	0	0
$43$	1.87013e7	0.834187	0.417094	−	0.908863i	$-0.363049\pi$
$43$	1.87013e7	0.834187	0.417094	+	0.908863i	$0.363049\pi$
$44$	0	0
$45$	5.01405e6	0.182277
$46$	0	0
$47$	−5.65391e7	−1.69008	−0.845042	−	0.534700i	$-0.820425\pi$
$47$	−5.65391e7	−1.69008	−0.845042	+	0.534700i	$0.820425\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	3.64106e6	0.0753637
$52$	0	0
$53$	−5.98947e7	−1.04267	−0.521335	−	0.853352i	$-0.674566\pi$
$53$	−5.98947e7	−1.04267	−0.521335	+	0.853352i	$0.674566\pi$
$54$	0	0
$55$	−2.65603e7	−0.391381
$56$	0	0
$57$	−1.21790e8	−1.52818
$58$	0	0
$59$	−1.65630e8	−1.77952	−0.889762	−	0.456424i	$-0.849130\pi$
$59$	−1.65630e8	−1.77952	−0.889762	+	0.456424i	$0.849130\pi$
$60$	0	0
$61$	5.14190e7	0.475488	0.237744	−	0.971328i	$-0.423592\pi$
$61$	5.14190e7	0.475488	0.237744	+	0.971328i	$0.423592\pi$
$62$	0	0
$63$	2.21300e7	0.176990
$64$	0	0
$65$	−8.63654e6	−0.0600109
$66$	0	0
$67$	−9.35465e7	−0.567141	−0.283570	−	0.958951i	$-0.591519\pi$
$67$	−9.35465e7	−0.567141	−0.283570	+	0.958951i	$0.591519\pi$
$68$	0	0
$69$	−4.19900e8	−2.23010
$70$	0	0
$71$	9.56335e7	0.446630	0.223315	−	0.974746i	$-0.428312\pi$
$71$	9.56335e7	0.446630	0.223315	+	0.974746i	$0.428312\pi$
$72$	0	0
$73$	3.06496e8	1.26320	0.631601	−	0.775294i	$-0.282398\pi$
$73$	3.06496e8	1.26320	0.631601	+	0.775294i	$0.282398\pi$
$74$	0	0
$75$	2.81722e8	1.02812
$76$	0	0
$77$	−1.17226e8	−0.380029
$78$	0	0
$79$	−4.96474e8	−1.43408	−0.717042	−	0.697030i	$-0.754505\pi$
$79$	−4.96474e8	−1.43408	−0.717042	+	0.697030i	$0.754505\pi$
$80$	0	0
$81$	−4.83886e8	−1.24899
$82$	0	0
$83$	3.71487e8	0.859196	0.429598	−	0.903020i	$-0.358655\pi$
$83$	3.71487e8	0.859196	0.429598	+	0.903020i	$0.358655\pi$
$84$	0	0
$85$	−1.16514e7	−0.0242099
$86$	0	0
$87$	−9.44660e8	−1.76782
$88$	0	0
$89$	−1.65483e8	−0.279574	−0.139787	−	0.990182i	$-0.544642\pi$
$89$	−1.65483e8	−0.279574	−0.139787	+	0.990182i	$0.544642\pi$
$90$	0	0
$91$	−3.81183e7	−0.0582703
$92$	0	0
$93$	9.85889e8	1.36664
$94$	0	0
$95$	3.89727e8	0.490913
$96$	0	0
$97$	7.58017e8	0.869373	0.434686	−	0.900582i	$-0.356859\pi$
$97$	7.58017e8	0.869373	0.434686	+	0.900582i	$0.356859\pi$
$98$	0	0
$99$	−4.50011e8	−0.470830
$100$	0	0
$101$	−9.04212e8	−0.864618	−0.432309	−	0.901726i	$-0.642301\pi$
$101$	−9.04212e8	−0.864618	−0.432309	+	0.901726i	$0.642301\pi$
$102$	0	0
$103$	−1.98157e9	−1.73477	−0.867384	−	0.497639i	$-0.834200\pi$
$103$	−1.98157e9	−1.73477	−0.867384	+	0.497639i	$0.834200\pi$
$104$	0	0
$105$	−2.22044e8	−0.178274
$106$	0	0
$107$	−4.16379e8	−0.307087	−0.153544	−	0.988142i	$-0.549069\pi$
$107$	−4.16379e8	−0.307087	−0.153544	+	0.988142i	$0.549069\pi$
$108$	0	0
$109$	−1.26921e9	−0.861220	−0.430610	−	0.902538i	$-0.641701\pi$
$109$	−1.26921e9	−0.861220	−0.430610	+	0.902538i	$0.641701\pi$
$110$	0	0
$111$	6.62053e8	0.413942
$112$	0	0
$113$	−2.83528e9	−1.63585	−0.817923	−	0.575328i	$-0.804874\pi$
$113$	−2.83528e9	−1.63585	−0.817923	+	0.575328i	$0.804874\pi$
$114$	0	0
$115$	1.34368e9	0.716400
$116$	0	0
$117$	−1.46329e8	−0.0721929
$118$	0	0
$119$	−5.14246e7	−0.0235077
$120$	0	0
$121$	2.58353e7	0.0109567
$122$	0	0
$123$	1.08135e9	0.425982
$124$	0	0
$125$	−1.96401e9	−0.719530
$126$	0	0
$127$	−5.44282e9	−1.85655	−0.928277	−	0.371889i	$-0.878710\pi$
$127$	−5.44282e9	−1.85655	−0.928277	+	0.371889i	$0.878710\pi$
$128$	0	0
$129$	−3.17922e9	−1.01080
$130$	0	0
$131$	6.44057e8	0.191075	0.0955374	−	0.995426i	$-0.469543\pi$
$131$	6.44057e8	0.191075	0.0955374	+	0.995426i	$0.469543\pi$
$132$	0	0
$133$	1.72010e9	0.476674
$134$	0	0
$135$	9.67896e8	0.250799
$136$	0	0
$137$	1.67376e9	0.405928	0.202964	−	0.979186i	$-0.434942\pi$
$137$	1.67376e9	0.405928	0.202964	+	0.979186i	$0.434942\pi$
$138$	0	0
$139$	4.17330e9	0.948229	0.474115	−	0.880463i	$-0.342768\pi$
$139$	4.17330e9	0.948229	0.474115	+	0.880463i	$0.342768\pi$
$140$	0	0
$141$	9.61164e9	2.04791
$142$	0	0
$143$	7.75130e8	0.155011
$144$	0	0
$145$	3.02291e9	0.567897
$146$	0	0
$147$	−9.80016e8	−0.173103
$148$	0	0
$149$	−4.64096e8	−0.0771382	−0.0385691	−	0.999256i	$-0.512280\pi$
$149$	−4.64096e8	−0.0771382	−0.0385691	+	0.999256i	$0.512280\pi$
$150$	0	0
$151$	−7.31929e9	−1.14571	−0.572853	−	0.819658i	$-0.694163\pi$
$151$	−7.31929e9	−1.14571	−0.572853	+	0.819658i	$0.694163\pi$
$152$	0	0
$153$	−1.97410e8	−0.0291244
$154$	0	0
$155$	−3.15485e9	−0.439021
$156$	0	0
$157$	−4.43050e9	−0.581975	−0.290987	−	0.956727i	$-0.593984\pi$
$157$	−4.43050e9	−0.581975	−0.290987	+	0.956727i	$0.593984\pi$
$158$	0	0
$159$	1.01821e10	1.26343
$160$	0	0
$161$	5.93047e9	0.695621
$162$	0	0
$163$	1.33645e10	1.48289	0.741446	−	0.671013i	$-0.234140\pi$
$163$	1.33645e10	1.48289	0.741446	+	0.671013i	$0.234140\pi$
$164$	0	0
$165$	4.51524e9	0.474246
$166$	0	0
$167$	1.24456e10	1.23821	0.619103	−	0.785310i	$-0.287496\pi$
$167$	1.24456e10	1.23821	0.619103	+	0.785310i	$0.287496\pi$
$168$	0	0
$169$	−1.03525e10	−0.976232
$170$	0	0
$171$	6.60315e9	0.590566
$172$	0	0
$173$	1.04544e10	0.887345	0.443672	−	0.896189i	$-0.353675\pi$
$173$	1.04544e10	0.887345	0.443672	+	0.896189i	$0.353675\pi$
$174$	0	0
$175$	−3.97891e9	−0.320696
$176$	0	0
$177$	2.81570e10	2.15629
$178$	0	0
$179$	4.04391e9	0.294417	0.147208	−	0.989105i	$-0.452971\pi$
$179$	4.04391e9	0.294417	0.147208	+	0.989105i	$0.452971\pi$
$180$	0	0
$181$	1.24735e10	0.863843	0.431922	−	0.901911i	$-0.357836\pi$
$181$	1.24735e10	0.863843	0.431922	+	0.901911i	$0.357836\pi$
$182$	0	0
$183$	−8.74123e9	−0.576160
$184$	0	0
$185$	−2.11857e9	−0.132975
$186$	0	0
$187$	1.04571e9	0.0625353
$188$	0	0
$189$	4.27191e9	0.243525
$190$	0	0
$191$	3.81947e9	0.207660	0.103830	−	0.994595i	$-0.466890\pi$
$191$	3.81947e9	0.207660	0.103830	+	0.994595i	$0.466890\pi$
$192$	0	0
$193$	−2.41193e10	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$193$	−2.41193e10	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$194$	0	0
$195$	1.46821e9	0.0727165
$196$	0	0
$197$	−1.24798e10	−0.590351	−0.295176	−	0.955443i	$-0.595378\pi$
$197$	−1.24798e10	−0.590351	−0.295176	+	0.955443i	$0.595378\pi$
$198$	0	0
$199$	2.93127e9	0.132500	0.0662502	−	0.997803i	$-0.478896\pi$
$199$	2.93127e9	0.132500	0.0662502	+	0.997803i	$0.478896\pi$
$200$	0	0
$201$	1.59029e10	0.687218
$202$	0	0
$203$	1.33419e10	0.551425
$204$	0	0
$205$	−3.46031e9	−0.136843
$206$	0	0
$207$	2.27660e10	0.861827
$208$	0	0
$209$	−3.49780e10	−1.26805
$210$	0	0
$211$	−3.36978e10	−1.17039	−0.585195	−	0.810892i	$-0.698982\pi$
$211$	−3.36978e10	−1.17039	−0.585195	+	0.810892i	$0.698982\pi$
$212$	0	0
$213$	−1.62577e10	−0.541191
$214$	0	0
$215$	1.01735e10	0.324711
$216$	0	0
$217$	−1.39242e10	−0.426287
$218$	0	0
$219$	−5.21044e10	−1.53065
$220$	0	0
$221$	3.40032e8	0.00958859
$222$	0	0
$223$	3.87208e10	1.04851	0.524255	−	0.851561i	$-0.324344\pi$
$223$	3.87208e10	1.04851	0.524255	+	0.851561i	$0.324344\pi$
$224$	0	0
$225$	−1.52743e10	−0.397320
$226$	0	0
$227$	−7.69011e10	−1.92228	−0.961139	−	0.276063i	$-0.910970\pi$
$227$	−7.69011e10	−1.92228	−0.961139	+	0.276063i	$0.910970\pi$
$228$	0	0
$229$	4.35114e10	1.04555	0.522773	−	0.852472i	$-0.324897\pi$
$229$	4.35114e10	1.04555	0.522773	+	0.852472i	$0.324897\pi$
$230$	0	0
$231$	1.99285e10	0.460490
$232$	0	0
$233$	−2.07043e10	−0.460213	−0.230107	−	0.973165i	$-0.573907\pi$
$233$	−2.07043e10	−0.460213	−0.230107	+	0.973165i	$0.573907\pi$
$234$	0	0
$235$	−3.07573e10	−0.657873
$236$	0	0
$237$	8.44006e10	1.73771
$238$	0	0
$239$	−2.16220e10	−0.428653	−0.214326	−	0.976762i	$-0.568756\pi$
$239$	−2.16220e10	−0.428653	−0.214326	+	0.976762i	$0.568756\pi$
$240$	0	0
$241$	6.77789e10	1.29425	0.647124	−	0.762385i	$-0.275971\pi$
$241$	6.77789e10	1.29425	0.647124	+	0.762385i	$0.275971\pi$
$242$	0	0
$243$	4.72402e10	0.869127
$244$	0	0
$245$	3.13605e9	0.0556078
$246$	0	0
$247$	−1.13737e10	−0.194431
$248$	0	0
$249$	−6.31528e10	−1.04111
$250$	0	0
$251$	−4.87895e9	−0.0775881	−0.0387940	−	0.999247i	$-0.512352\pi$
$251$	−4.87895e9	−0.0775881	−0.0387940	+	0.999247i	$0.512352\pi$
$252$	0	0
$253$	−1.20595e11	−1.85050
$254$	0	0
$255$	1.98074e9	0.0293357
$256$	0	0
$257$	−2.75029e10	−0.393259	−0.196630	−	0.980478i	$-0.563000\pi$
$257$	−2.75029e10	−0.393259	−0.196630	+	0.980478i	$0.563000\pi$
$258$	0	0
$259$	−9.35053e9	−0.129118
$260$	0	0
$261$	5.12173e10	0.683178
$262$	0	0
$263$	2.22595e10	0.286889	0.143445	−	0.989658i	$-0.454182\pi$
$263$	2.22595e10	0.286889	0.143445	+	0.989658i	$0.454182\pi$
$264$	0	0
$265$	−3.25827e10	−0.405864
$266$	0	0
$267$	2.81320e10	0.338766
$268$	0	0
$269$	1.73017e10	0.201466	0.100733	−	0.994913i	$-0.467881\pi$
$269$	1.73017e10	0.201466	0.100733	+	0.994913i	$0.467881\pi$
$270$	0	0
$271$	−4.81901e10	−0.542745	−0.271372	−	0.962474i	$-0.587477\pi$
$271$	−4.81901e10	−0.542745	−0.271372	+	0.962474i	$0.587477\pi$
$272$	0	0
$273$	6.48011e9	0.0706074
$274$	0	0
$275$	8.09106e10	0.853116
$276$	0	0
$277$	8.03834e10	0.820365	0.410183	−	0.912003i	$-0.365465\pi$
$277$	8.03834e10	0.820365	0.410183	+	0.912003i	$0.365465\pi$
$278$	0	0
$279$	−5.34526e10	−0.528141
$280$	0	0
$281$	−1.95595e11	−1.87146	−0.935729	−	0.352719i	$-0.885257\pi$
$281$	−1.95595e11	−1.87146	−0.935729	+	0.352719i	$0.885257\pi$
$282$	0	0
$283$	6.02802e10	0.558645	0.279322	−	0.960197i	$-0.409890\pi$
$283$	6.02802e10	0.558645	0.279322	+	0.960197i	$0.409890\pi$
$284$	0	0
$285$	−6.62536e10	−0.594850
$286$	0	0
$287$	−1.52724e10	−0.132874
$288$	0	0
$289$	−1.18129e11	−0.996132
$290$	0	0
$291$	−1.28863e11	−1.05344
$292$	0	0
$293$	−4.86743e10	−0.385830	−0.192915	−	0.981216i	$-0.561794\pi$
$293$	−4.86743e10	−0.385830	−0.192915	+	0.981216i	$0.561794\pi$
$294$	0	0
$295$	−9.01025e10	−0.692688
$296$	0	0
$297$	−8.68686e10	−0.647827
$298$	0	0
$299$	−3.92137e10	−0.283738
$300$	0	0
$301$	4.49018e10	0.315293
$302$	0	0
$303$	1.53716e11	1.04768
$304$	0	0
$305$	2.79719e10	0.185086
$306$	0	0
$307$	−2.75178e11	−1.76804	−0.884018	−	0.467453i	$-0.845172\pi$
$307$	−2.75178e11	−1.76804	−0.884018	+	0.467453i	$0.845172\pi$
$308$	0	0
$309$	3.36867e11	2.10206
$310$	0	0
$311$	−1.12322e11	−0.680835	−0.340418	−	0.940274i	$-0.610568\pi$
$311$	−1.12322e11	−0.680835	−0.340418	+	0.940274i	$0.610568\pi$
$312$	0	0
$313$	−1.06140e11	−0.625069	−0.312535	−	0.949906i	$-0.601178\pi$
$313$	−1.06140e11	−0.625069	−0.312535	+	0.949906i	$0.601178\pi$
$314$	0	0
$315$	1.20387e10	0.0688943
$316$	0	0
$317$	2.31358e9	0.0128682	0.00643409	−	0.999979i	$-0.497952\pi$
$317$	2.31358e9	0.0128682	0.00643409	+	0.999979i	$0.497952\pi$
$318$	0	0
$319$	−2.71306e11	−1.46691
$320$	0	0
$321$	7.07844e10	0.372105
$322$	0	0
$323$	−1.53441e10	−0.0784385
$324$	0	0
$325$	2.63095e10	0.130809
$326$	0	0
$327$	2.15766e11	1.04356
$328$	0	0
$329$	−1.35750e11	−0.638792
$330$	0	0
$331$	2.16185e11	0.989921	0.494960	−	0.868916i	$-0.335183\pi$
$331$	2.16185e11	0.989921	0.494960	+	0.868916i	$0.335183\pi$
$332$	0	0
$333$	−3.58950e10	−0.159968
$334$	0	0
$335$	−5.08893e10	−0.220762
$336$	0	0
$337$	5.00291e10	0.211294	0.105647	−	0.994404i	$-0.466309\pi$
$337$	5.00291e10	0.211294	0.105647	+	0.994404i	$0.466309\pi$
$338$	0	0
$339$	4.81997e11	1.98219
$340$	0	0
$341$	2.83147e11	1.13401
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	−2.28426e11	−0.868078
$346$	0	0
$347$	1.84606e11	0.683541	0.341770	−	0.939784i	$-0.388974\pi$
$347$	1.84606e11	0.683541	0.341770	+	0.939784i	$0.388974\pi$
$348$	0	0
$349$	2.74666e11	0.991039	0.495520	−	0.868597i	$-0.334978\pi$
$349$	2.74666e11	0.991039	0.495520	+	0.868597i	$0.334978\pi$
$350$	0	0
$351$	−2.82469e10	−0.0993319
$352$	0	0
$353$	−1.58053e11	−0.541774	−0.270887	−	0.962611i	$-0.587317\pi$
$353$	−1.58053e11	−0.541774	−0.270887	+	0.962611i	$0.587317\pi$
$354$	0	0
$355$	5.20246e10	0.173853
$356$	0	0
$357$	8.74219e9	0.0284848
$358$	0	0
$359$	−3.40759e11	−1.08274	−0.541368	−	0.840786i	$-0.682093\pi$
$359$	−3.40759e11	−1.08274	−0.541368	+	0.840786i	$0.682093\pi$
$360$	0	0
$361$	1.90556e11	0.590526
$362$	0	0
$363$	−4.39200e9	−0.0132765
$364$	0	0
$365$	1.66734e11	0.491707
$366$	0	0
$367$	6.10216e11	1.75584	0.877922	−	0.478803i	$-0.158929\pi$
$367$	6.10216e11	1.75584	0.877922	+	0.478803i	$0.158929\pi$
$368$	0	0
$369$	−5.86280e10	−0.164622
$370$	0	0
$371$	−1.43807e11	−0.394092
$372$	0	0
$373$	4.34930e11	1.16340	0.581701	−	0.813402i	$-0.302387\pi$
$373$	4.34930e11	1.16340	0.581701	+	0.813402i	$0.302387\pi$
$374$	0	0
$375$	3.33882e11	0.871871
$376$	0	0
$377$	−8.82202e10	−0.224922
$378$	0	0
$379$	7.30677e11	1.81907	0.909534	−	0.415630i	$-0.136439\pi$
$379$	7.30677e11	1.81907	0.909534	+	0.415630i	$0.136439\pi$
$380$	0	0
$381$	9.25280e11	2.24963
$382$	0	0
$383$	−2.11074e11	−0.501233	−0.250617	−	0.968086i	$-0.580633\pi$
$383$	−2.11074e11	−0.501233	−0.250617	+	0.968086i	$0.580633\pi$
$384$	0	0
$385$	−6.37712e10	−0.147928
$386$	0	0
$387$	1.72370e11	0.390627
$388$	0	0
$389$	7.21857e9	0.0159837	0.00799186	−	0.999968i	$-0.497456\pi$
$389$	7.21857e9	0.0159837	0.00799186	+	0.999968i	$0.497456\pi$
$390$	0	0
$391$	−5.29025e10	−0.114467
$392$	0	0
$393$	−1.09490e11	−0.231530
$394$	0	0
$395$	−2.70082e11	−0.558224
$396$	0	0
$397$	−6.99387e11	−1.41306	−0.706529	−	0.707684i	$-0.749740\pi$
$397$	−6.99387e11	−1.41306	−0.706529	+	0.707684i	$0.749740\pi$
$398$	0	0
$399$	−2.92417e11	−0.577597
$400$	0	0
$401$	6.40644e11	1.23728	0.618638	−	0.785676i	$-0.287685\pi$
$401$	6.40644e11	1.23728	0.618638	+	0.785676i	$0.287685\pi$
$402$	0	0
$403$	9.20704e10	0.173879
$404$	0	0
$405$	−2.63234e11	−0.486177
$406$	0	0
$407$	1.90142e11	0.343481
$408$	0	0
$409$	−1.31500e10	−0.0232365	−0.0116182	−	0.999933i	$-0.503698\pi$
$409$	−1.31500e10	−0.0232365	−0.0116182	+	0.999933i	$0.503698\pi$
$410$	0	0
$411$	−2.84538e11	−0.491873
$412$	0	0
$413$	−3.97677e11	−0.672597
$414$	0	0
$415$	2.02089e11	0.334446
$416$	0	0
$417$	−7.09461e11	−1.14899
$418$	0	0
$419$	5.79915e11	0.919182	0.459591	−	0.888131i	$-0.347996\pi$
$419$	5.79915e11	0.919182	0.459591	+	0.888131i	$0.347996\pi$
$420$	0	0
$421$	1.66175e11	0.257808	0.128904	−	0.991657i	$-0.458854\pi$
$421$	1.66175e11	0.257808	0.128904	+	0.991657i	$0.458854\pi$
$422$	0	0
$423$	−5.21121e11	−0.791419
$424$	0	0
$425$	3.54937e10	0.0527717
$426$	0	0
$427$	1.23457e11	0.179718
$428$	0	0
$429$	−1.31772e11	−0.187830
$430$	0	0
$431$	−7.57723e11	−1.05770	−0.528850	−	0.848715i	$-0.677377\pi$
$431$	−7.57723e11	−1.05770	−0.528850	+	0.848715i	$0.677377\pi$
$432$	0	0
$433$	−1.07485e12	−1.46944	−0.734719	−	0.678371i	$-0.762686\pi$
$433$	−1.07485e12	−1.46944	−0.734719	+	0.678371i	$0.762686\pi$
$434$	0	0
$435$	−5.13895e11	−0.688134
$436$	0	0
$437$	1.76953e12	2.32109
$438$	0	0
$439$	−1.70418e11	−0.218991	−0.109496	−	0.993987i	$-0.534924\pi$
$439$	−1.70418e11	−0.218991	−0.109496	+	0.993987i	$0.534924\pi$
$440$	0	0
$441$	5.31342e10	0.0668960
$442$	0	0
$443$	−1.22937e12	−1.51658	−0.758290	−	0.651918i	$-0.773965\pi$
$443$	−1.22937e12	−1.51658	−0.758290	+	0.651918i	$0.773965\pi$
$444$	0	0
$445$	−9.00225e10	−0.108826
$446$	0	0
$447$	7.88963e10	0.0934701
$448$	0	0
$449$	−7.25792e10	−0.0842759	−0.0421380	−	0.999112i	$-0.513417\pi$
$449$	−7.25792e10	−0.0842759	−0.0421380	+	0.999112i	$0.513417\pi$
$450$	0	0
$451$	3.10563e11	0.353472
$452$	0	0
$453$	1.24428e12	1.38828
$454$	0	0
$455$	−2.07363e10	−0.0226820
$456$	0	0
$457$	−6.64172e11	−0.712291	−0.356146	−	0.934431i	$-0.615909\pi$
$457$	−6.64172e11	−0.712291	−0.356146	+	0.934431i	$0.615909\pi$
$458$	0	0
$459$	−3.81073e10	−0.0400730
$460$	0	0
$461$	−1.21501e12	−1.25293	−0.626463	−	0.779451i	$-0.715498\pi$
$461$	−1.21501e12	−1.25293	−0.626463	+	0.779451i	$0.715498\pi$
$462$	0	0
$463$	−2.93878e11	−0.297202	−0.148601	−	0.988897i	$-0.547477\pi$
$463$	−2.93878e11	−0.297202	−0.148601	+	0.988897i	$0.547477\pi$
$464$	0	0
$465$	5.36324e11	0.531972
$466$	0	0
$467$	−4.73112e11	−0.460297	−0.230149	−	0.973155i	$-0.573921\pi$
$467$	−4.73112e11	−0.460297	−0.230149	+	0.973155i	$0.573921\pi$
$468$	0	0
$469$	−2.24605e11	−0.214359
$470$	0	0
$471$	7.53185e11	0.705192
$472$	0	0
$473$	−9.13072e11	−0.838745
$474$	0	0
$475$	−1.18723e12	−1.07007
$476$	0	0
$477$	−5.52049e11	−0.488253
$478$	0	0
$479$	2.05945e12	1.78748	0.893742	−	0.448582i	$-0.148071\pi$
$479$	2.05945e12	1.78748	0.893742	+	0.448582i	$0.148071\pi$
$480$	0	0
$481$	6.18280e10	0.0526662
$482$	0	0
$483$	−1.00818e12	−0.842900
$484$	0	0
$485$	4.12361e11	0.338407
$486$	0	0
$487$	1.22247e12	0.984821	0.492411	−	0.870363i	$-0.336116\pi$
$487$	1.22247e12	0.984821	0.492411	+	0.870363i	$0.336116\pi$
$488$	0	0
$489$	−2.27197e12	−1.79685
$490$	0	0
$491$	−1.98225e11	−0.153918	−0.0769592	−	0.997034i	$-0.524521\pi$
$491$	−1.98225e11	−0.153918	−0.0769592	+	0.997034i	$0.524521\pi$
$492$	0	0
$493$	−1.19016e11	−0.0907391
$494$	0	0
$495$	−2.44806e11	−0.183273
$496$	0	0
$497$	2.29616e11	0.168810
$498$	0	0
$499$	3.00745e11	0.217143	0.108571	−	0.994089i	$-0.465372\pi$
$499$	3.00745e11	0.217143	0.108571	+	0.994089i	$0.465372\pi$
$500$	0	0
$501$	−2.11576e12	−1.50036
$502$	0	0
$503$	−3.30194e11	−0.229993	−0.114996	−	0.993366i	$-0.536686\pi$
$503$	−3.30194e11	−0.229993	−0.114996	+	0.993366i	$0.536686\pi$
$504$	0	0
$505$	−4.91891e11	−0.336556
$506$	0	0
$507$	1.75992e12	1.18292
$508$	0	0
$509$	−6.32399e10	−0.0417600	−0.0208800	−	0.999782i	$-0.506647\pi$
$509$	−6.32399e10	−0.0417600	−0.0208800	+	0.999782i	$0.506647\pi$
$510$	0	0
$511$	7.35898e11	0.477445
$512$	0	0
$513$	1.27465e12	0.812574
$514$	0	0
$515$	−1.07797e12	−0.675267
$516$	0	0
$517$	2.76046e12	1.69932
$518$	0	0
$519$	−1.77725e12	−1.07522
$520$	0	0
$521$	−1.88994e12	−1.12377	−0.561886	−	0.827215i	$-0.689924\pi$
$521$	−1.88994e12	−1.12377	−0.561886	+	0.827215i	$0.689924\pi$
$522$	0	0
$523$	−8.95863e11	−0.523581	−0.261791	−	0.965125i	$-0.584313\pi$
$523$	−8.95863e11	−0.523581	−0.261791	+	0.965125i	$0.584313\pi$
$524$	0	0
$525$	6.76415e11	0.388594
$526$	0	0
$527$	1.24210e11	0.0701472
$528$	0	0
$529$	4.29975e12	2.38722
$530$	0	0
$531$	−1.52661e12	−0.833302
$532$	0	0
$533$	1.00985e11	0.0541981
$534$	0	0
$535$	−2.26510e11	−0.119535
$536$	0	0
$537$	−6.87464e11	−0.356752
$538$	0	0
$539$	−2.81461e11	−0.143638
$540$	0	0
$541$	1.00221e12	0.503005	0.251502	−	0.967857i	$-0.419075\pi$
$541$	1.00221e12	0.503005	0.251502	+	0.967857i	$0.419075\pi$
$542$	0	0
$543$	−2.12050e12	−1.04674
$544$	0	0
$545$	−6.90450e11	−0.335234
$546$	0	0
$547$	−2.73436e12	−1.30591	−0.652955	−	0.757396i	$-0.726471\pi$
$547$	−2.73436e12	−1.30591	−0.652955	+	0.757396i	$0.726471\pi$
$548$	0	0
$549$	4.73929e11	0.222658
$550$	0	0
$551$	3.98097e12	1.83995
$552$	0	0
$553$	−1.19203e12	−0.542033
$554$	0	0
$555$	3.60157e11	0.161129
$556$	0	0
$557$	−4.22359e12	−1.85923	−0.929614	−	0.368534i	$-0.879860\pi$
$557$	−4.22359e12	−1.85923	−0.929614	+	0.368534i	$0.879860\pi$
$558$	0	0
$559$	−2.96902e11	−0.128606
$560$	0	0
$561$	−1.77771e11	−0.0757754
$562$	0	0
$563$	3.08311e12	1.29330	0.646652	−	0.762785i	$-0.276169\pi$
$563$	3.08311e12	1.29330	0.646652	+	0.762785i	$0.276169\pi$
$564$	0	0
$565$	−1.54239e12	−0.636761
$566$	0	0
$567$	−1.16181e12	−0.472075
$568$	0	0
$569$	−2.74669e11	−0.109851	−0.0549256	−	0.998490i	$-0.517492\pi$
$569$	−2.74669e11	−0.109851	−0.0549256	+	0.998490i	$0.517492\pi$
$570$	0	0
$571$	−2.82499e12	−1.11213	−0.556064	−	0.831140i	$-0.687689\pi$
$571$	−2.82499e12	−1.11213	−0.556064	+	0.831140i	$0.687689\pi$
$572$	0	0
$573$	−6.49310e11	−0.251626
$574$	0	0
$575$	−4.09326e12	−1.56158
$576$	0	0
$577$	3.76585e12	1.41440	0.707199	−	0.707014i	$-0.249958\pi$
$577$	3.76585e12	1.41440	0.707199	+	0.707014i	$0.249958\pi$
$578$	0	0
$579$	4.10028e12	1.51621
$580$	0	0
$581$	8.91940e11	0.324745
$582$	0	0
$583$	2.92430e12	1.04837
$584$	0	0
$585$	−7.96030e10	−0.0281014
$586$	0	0
$587$	3.00831e12	1.04580	0.522902	−	0.852393i	$-0.324849\pi$
$587$	3.00831e12	1.04580	0.522902	+	0.852393i	$0.324849\pi$
$588$	0	0
$589$	−4.15471e12	−1.42240
$590$	0	0
$591$	2.12157e12	0.715342
$592$	0	0
$593$	3.64775e12	1.21138	0.605688	−	0.795703i	$-0.292898\pi$
$593$	3.64775e12	1.21138	0.605688	+	0.795703i	$0.292898\pi$
$594$	0	0
$595$	−2.79750e10	−0.00915047
$596$	0	0
$597$	−4.98316e11	−0.160554
$598$	0	0
$599$	−4.17778e12	−1.32594	−0.662972	−	0.748644i	$-0.730705\pi$
$599$	−4.17778e12	−1.32594	−0.662972	+	0.748644i	$0.730705\pi$
$600$	0	0
$601$	4.84445e12	1.51464	0.757321	−	0.653043i	$-0.226508\pi$
$601$	4.84445e12	1.51464	0.757321	+	0.653043i	$0.226508\pi$
$602$	0	0
$603$	−8.62218e11	−0.265576
$604$	0	0
$605$	1.40544e10	0.00426494
$606$	0	0
$607$	1.58444e12	0.473726	0.236863	−	0.971543i	$-0.423881\pi$
$607$	1.58444e12	0.473726	0.236863	+	0.971543i	$0.423881\pi$
$608$	0	0
$609$	−2.26813e12	−0.668175
$610$	0	0
$611$	8.97614e11	0.260558
$612$	0	0
$613$	−2.79203e11	−0.0798635	−0.0399318	−	0.999202i	$-0.512714\pi$
$613$	−2.79203e11	−0.0798635	−0.0399318	+	0.999202i	$0.512714\pi$
$614$	0	0
$615$	5.88252e11	0.165816
$616$	0	0
$617$	−5.55576e12	−1.54333	−0.771667	−	0.636027i	$-0.780577\pi$
$617$	−5.55576e12	−1.54333	−0.771667	+	0.636027i	$0.780577\pi$
$618$	0	0
$619$	2.70437e12	0.740385	0.370192	−	0.928955i	$-0.379292\pi$
$619$	2.70437e12	0.740385	0.370192	+	0.928955i	$0.379292\pi$
$620$	0	0
$621$	4.39467e12	1.18581
$622$	0	0
$623$	−3.97324e11	−0.105669
$624$	0	0
$625$	2.16828e12	0.568400
$626$	0	0
$627$	5.94626e12	1.53653
$628$	0	0
$629$	8.34109e10	0.0212469
$630$	0	0
$631$	4.73158e12	1.18816	0.594078	−	0.804407i	$-0.297517\pi$
$631$	4.73158e12	1.18816	0.594078	+	0.804407i	$0.297517\pi$
$632$	0	0
$633$	5.72863e12	1.41819
$634$	0	0
$635$	−2.96090e12	−0.722672
$636$	0	0
$637$	−9.15220e10	−0.0220241
$638$	0	0
$639$	8.81454e11	0.209144
$640$	0	0
$641$	1.38865e12	0.324887	0.162444	−	0.986718i	$-0.448062\pi$
$641$	1.38865e12	0.324887	0.162444	+	0.986718i	$0.448062\pi$
$642$	0	0
$643$	−3.09398e12	−0.713786	−0.356893	−	0.934145i	$-0.616164\pi$
$643$	−3.09398e12	−0.713786	−0.356893	+	0.934145i	$0.616164\pi$
$644$	0	0
$645$	−1.72950e12	−0.393460
$646$	0	0
$647$	−2.31453e12	−0.519270	−0.259635	−	0.965707i	$-0.583602\pi$
$647$	−2.31453e12	−0.519270	−0.259635	+	0.965707i	$0.583602\pi$
$648$	0	0
$649$	8.08670e12	1.78925
$650$	0	0
$651$	2.36712e12	0.516542
$652$	0	0
$653$	−8.10575e11	−0.174455	−0.0872276	−	0.996188i	$-0.527801\pi$
$653$	−8.10575e11	−0.174455	−0.0872276	+	0.996188i	$0.527801\pi$
$654$	0	0
$655$	3.50367e11	0.0743768
$656$	0	0
$657$	2.82498e12	0.591522
$658$	0	0
$659$	−4.57355e11	−0.0944645	−0.0472323	−	0.998884i	$-0.515040\pi$
$659$	−4.57355e11	−0.0944645	−0.0472323	+	0.998884i	$0.515040\pi$
$660$	0	0
$661$	7.94557e12	1.61889	0.809447	−	0.587193i	$-0.199767\pi$
$661$	7.94557e12	1.61889	0.809447	+	0.587193i	$0.199767\pi$
$662$	0	0
$663$	−5.78055e10	−0.0116187
$664$	0	0
$665$	9.35735e11	0.185548
$666$	0	0
$667$	1.37254e13	2.68508
$668$	0	0
$669$	−6.58254e12	−1.27050
$670$	0	0
$671$	−2.51048e12	−0.478086
$672$	0	0
$673$	−8.92805e12	−1.67760	−0.838801	−	0.544439i	$-0.816743\pi$
$673$	−8.92805e12	−1.67760	−0.838801	+	0.544439i	$0.816743\pi$
$674$	0	0
$675$	−2.94850e12	−0.546682
$676$	0	0
$677$	−8.01730e12	−1.46683	−0.733414	−	0.679782i	$-0.762074\pi$
$677$	−8.01730e12	−1.46683	−0.733414	+	0.679782i	$0.762074\pi$
$678$	0	0
$679$	1.82000e12	0.328592
$680$	0	0
$681$	1.30732e13	2.32927
$682$	0	0
$683$	−4.14724e12	−0.729233	−0.364617	−	0.931158i	$-0.618800\pi$
$683$	−4.14724e12	−0.729233	−0.364617	+	0.931158i	$0.618800\pi$
$684$	0	0
$685$	9.10523e11	0.158010
$686$	0	0
$687$	−7.39694e12	−1.26691
$688$	0	0
$689$	9.50888e11	0.160747
$690$	0	0
$691$	6.05580e12	1.01046	0.505231	−	0.862984i	$-0.331407\pi$
$691$	6.05580e12	1.01046	0.505231	+	0.862984i	$0.331407\pi$
$692$	0	0
$693$	−1.08048e12	−0.177957
$694$	0	0
$695$	2.27028e12	0.369103
$696$	0	0
$697$	1.36237e11	0.0218649
$698$	0	0
$699$	3.51973e12	0.557651
$700$	0	0
$701$	1.88599e12	0.294990	0.147495	−	0.989063i	$-0.452879\pi$
$701$	1.88599e12	0.294990	0.147495	+	0.989063i	$0.452879\pi$
$702$	0	0
$703$	−2.79001e12	−0.430831
$704$	0	0
$705$	5.22873e12	0.797160
$706$	0	0
$707$	−2.17101e12	−0.326795
$708$	0	0
$709$	−4.76210e12	−0.707767	−0.353884	−	0.935289i	$-0.615139\pi$
$709$	−4.76210e12	−0.707767	−0.353884	+	0.935289i	$0.615139\pi$
$710$	0	0
$711$	−4.57600e12	−0.671542
$712$	0	0
$713$	−1.43244e13	−2.07574
$714$	0	0
$715$	4.21671e11	0.0603387
$716$	0	0
$717$	3.67574e12	0.519408
$718$	0	0
$719$	−5.34893e12	−0.746427	−0.373213	−	0.927746i	$-0.621744\pi$
$719$	−5.34893e12	−0.746427	−0.373213	+	0.927746i	$0.621744\pi$
$720$	0	0
$721$	−4.75774e12	−0.655681
$722$	0	0
$723$	−1.15224e13	−1.56827
$724$	0	0
$725$	−9.20871e12	−1.23788
$726$	0	0
$727$	−9.08222e12	−1.20583	−0.602917	−	0.797804i	$-0.705995\pi$
$727$	−9.08222e12	−1.20583	−0.602917	+	0.797804i	$0.705995\pi$
$728$	0	0
$729$	1.49349e12	0.195852
$730$	0	0
$731$	−4.00544e11	−0.0518827
$732$	0	0
$733$	−1.20547e13	−1.54237	−0.771185	−	0.636611i	$-0.780336\pi$
$733$	−1.20547e13	−1.54237	−0.771185	+	0.636611i	$0.780336\pi$
$734$	0	0
$735$	−5.33129e11	−0.0673813
$736$	0	0
$737$	4.56731e12	0.570239
$738$	0	0
$739$	−1.08128e13	−1.33364	−0.666822	−	0.745217i	$-0.732346\pi$
$739$	−1.08128e13	−1.33364	−0.666822	+	0.745217i	$0.732346\pi$
$740$	0	0
$741$	1.93353e12	0.235597
$742$	0	0
$743$	−6.39433e12	−0.769742	−0.384871	−	0.922970i	$-0.625754\pi$
$743$	−6.39433e12	−0.769742	−0.384871	+	0.922970i	$0.625754\pi$
$744$	0	0
$745$	−2.52468e11	−0.0300264
$746$	0	0
$747$	3.42400e12	0.402337
$748$	0	0
$749$	−9.99726e11	−0.116068
$750$	0	0
$751$	2.30580e12	0.264510	0.132255	−	0.991216i	$-0.457778\pi$
$751$	2.30580e12	0.264510	0.132255	+	0.991216i	$0.457778\pi$
$752$	0	0
$753$	8.29422e11	0.0940152
$754$	0	0
$755$	−3.98170e12	−0.445971
$756$	0	0
$757$	−6.85316e12	−0.758507	−0.379253	−	0.925293i	$-0.623819\pi$
$757$	−6.85316e12	−0.758507	−0.379253	+	0.925293i	$0.623819\pi$
$758$	0	0
$759$	2.05012e13	2.24229
$760$	0	0
$761$	−1.55520e12	−0.168095	−0.0840474	−	0.996462i	$-0.526785\pi$
$761$	−1.55520e12	−0.168095	−0.0840474	+	0.996462i	$0.526785\pi$
$762$	0	0
$763$	−3.04737e12	−0.325510
$764$	0	0
$765$	−1.07391e11	−0.0113368
$766$	0	0
$767$	2.62954e12	0.274347
$768$	0	0
$769$	−1.31148e12	−0.135236	−0.0676179	−	0.997711i	$-0.521540\pi$
$769$	−1.31148e12	−0.135236	−0.0676179	+	0.997711i	$0.521540\pi$
$770$	0	0
$771$	4.67549e12	0.476521
$772$	0	0
$773$	−9.82010e12	−0.989255	−0.494627	−	0.869105i	$-0.664695\pi$
$773$	−9.82010e12	−0.989255	−0.494627	+	0.869105i	$0.664695\pi$
$774$	0	0
$775$	9.61062e12	0.956959
$776$	0	0
$777$	1.58959e12	0.156455
$778$	0	0
$779$	−4.55698e12	−0.443362
$780$	0	0
$781$	−4.66921e12	−0.449070
$782$	0	0
$783$	9.88682e12	0.940001
$784$	0	0
$785$	−2.41019e12	−0.226536
$786$	0	0
$787$	4.81658e12	0.447562	0.223781	−	0.974639i	$-0.428160\pi$
$787$	4.81658e12	0.447562	0.223781	+	0.974639i	$0.428160\pi$
$788$	0	0
$789$	−3.78411e12	−0.347630
$790$	0	0
$791$	−6.80750e12	−0.618292
$792$	0	0
$793$	−8.16328e11	−0.0733053
$794$	0	0
$795$	5.53906e12	0.491795
$796$	0	0
$797$	7.71344e12	0.677151	0.338575	−	0.940939i	$-0.390055\pi$
$797$	7.71344e12	0.677151	0.338575	+	0.940939i	$0.390055\pi$
$798$	0	0
$799$	1.21095e12	0.105116
$800$	0	0
$801$	−1.52525e12	−0.130917
$802$	0	0
$803$	−1.49644e13	−1.27010
$804$	0	0
$805$	3.22618e12	0.270774
$806$	0	0
$807$	−2.94128e12	−0.244121
$808$	0	0
$809$	2.12869e13	1.74721	0.873604	−	0.486637i	$-0.161777\pi$
$809$	2.12869e13	1.74721	0.873604	+	0.486637i	$0.161777\pi$
$810$	0	0
$811$	−2.45053e13	−1.98914	−0.994570	−	0.104067i	$-0.966814\pi$
$811$	−2.45053e13	−1.98914	−0.994570	+	0.104067i	$0.966814\pi$
$812$	0	0
$813$	8.19231e12	0.657656
$814$	0	0
$815$	7.27030e12	0.577223
$816$	0	0
$817$	1.33978e13	1.05204
$818$	0	0
$819$	−3.51336e11	−0.0272863
$820$	0	0
$821$	−9.72826e12	−0.747293	−0.373646	−	0.927571i	$-0.621893\pi$
$821$	−9.72826e12	−0.747293	−0.373646	+	0.927571i	$0.621893\pi$
$822$	0	0
$823$	8.28745e11	0.0629683	0.0314841	−	0.999504i	$-0.489977\pi$
$823$	8.28745e11	0.0629683	0.0314841	+	0.999504i	$0.489977\pi$
$824$	0	0
$825$	−1.37548e13	−1.03374
$826$	0	0
$827$	−2.42842e13	−1.80530	−0.902649	−	0.430376i	$-0.858381\pi$
$827$	−2.42842e13	−1.80530	−0.902649	+	0.430376i	$0.858381\pi$
$828$	0	0
$829$	1.95570e13	1.43816	0.719078	−	0.694929i	$-0.244564\pi$
$829$	1.95570e13	1.43816	0.719078	+	0.694929i	$0.244564\pi$
$830$	0	0
$831$	−1.36652e13	−0.994055
$832$	0	0
$833$	−1.23471e11	−0.00888507
$834$	0	0
$835$	6.77043e12	0.481978
$836$	0	0
$837$	−1.03183e13	−0.726682
$838$	0	0
$839$	7.69577e12	0.536196	0.268098	−	0.963392i	$-0.413605\pi$
$839$	7.69577e12	0.536196	0.268098	+	0.963392i	$0.413605\pi$
$840$	0	0
$841$	1.63712e13	1.12849
$842$	0	0
$843$	3.32512e13	2.26769
$844$	0	0
$845$	−5.63173e12	−0.380003
$846$	0	0
$847$	6.20305e10	0.00414124
$848$	0	0
$849$	−1.02476e13	−0.676923
$850$	0	0
$851$	−9.61924e12	−0.628721
$852$	0	0
$853$	−2.79895e12	−0.181019	−0.0905097	−	0.995896i	$-0.528850\pi$
$853$	−2.79895e12	−0.181019	−0.0905097	+	0.995896i	$0.528850\pi$
$854$	0	0
$855$	3.59211e12	0.229881
$856$	0	0
$857$	−5.56141e12	−0.352185	−0.176093	−	0.984374i	$-0.556346\pi$
$857$	−5.56141e12	−0.352185	−0.176093	+	0.984374i	$0.556346\pi$
$858$	0	0
$859$	−3.51052e12	−0.219989	−0.109995	−	0.993932i	$-0.535083\pi$
$859$	−3.51052e12	−0.219989	−0.109995	+	0.993932i	$0.535083\pi$
$860$	0	0
$861$	2.59631e12	0.161006
$862$	0	0
$863$	−1.43838e13	−0.882721	−0.441361	−	0.897330i	$-0.645504\pi$
$863$	−1.43838e13	−0.882721	−0.441361	+	0.897330i	$0.645504\pi$
$864$	0	0
$865$	5.68720e12	0.345403
$866$	0	0
$867$	2.00820e13	1.20704
$868$	0	0
$869$	2.42399e13	1.44192
$870$	0	0
$871$	1.48514e12	0.0874353
$872$	0	0
$873$	6.98664e12	0.407103
$874$	0	0
$875$	−4.71559e12	−0.271957
$876$	0	0
$877$	−6.34278e12	−0.362061	−0.181030	−	0.983477i	$-0.557943\pi$
$877$	−6.34278e12	−0.362061	−0.181030	+	0.983477i	$0.557943\pi$
$878$	0	0
$879$	8.27463e12	0.467519
$880$	0	0
$881$	−2.89282e13	−1.61782	−0.808910	−	0.587933i	$-0.799942\pi$
$881$	−2.89282e13	−1.61782	−0.808910	+	0.587933i	$0.799942\pi$
$882$	0	0
$883$	7.17154e12	0.396999	0.198500	−	0.980101i	$-0.436393\pi$
$883$	7.17154e12	0.396999	0.198500	+	0.980101i	$0.436393\pi$
$884$	0	0
$885$	1.53174e13	0.839346
$886$	0	0
$887$	1.68020e13	0.911389	0.455695	−	0.890136i	$-0.349391\pi$
$887$	1.68020e13	0.911389	0.455695	+	0.890136i	$0.349391\pi$
$888$	0	0
$889$	−1.30682e13	−0.701711
$890$	0	0
$891$	2.36252e13	1.25582
$892$	0	0
$893$	−4.05052e13	−2.13147
$894$	0	0
$895$	2.19989e12	0.114603
$896$	0	0
$897$	6.66633e12	0.343812
$898$	0	0
$899$	−3.22260e13	−1.64546
$900$	0	0
$901$	1.28282e12	0.0648493
$902$	0	0
$903$	−7.63331e12	−0.382048
$904$	0	0
$905$	6.78559e12	0.336255
$906$	0	0
$907$	1.87924e13	0.922038	0.461019	−	0.887390i	$-0.347484\pi$
$907$	1.87924e13	0.922038	0.461019	+	0.887390i	$0.347484\pi$
$908$	0	0
$909$	−8.33412e12	−0.404876
$910$	0	0
$911$	3.39496e13	1.63306	0.816529	−	0.577304i	$-0.195895\pi$
$911$	3.39496e13	1.63306	0.816529	+	0.577304i	$0.195895\pi$
$912$	0	0
$913$	−1.81375e13	−0.863890
$914$	0	0
$915$	−4.75523e12	−0.224273
$916$	0	0
$917$	1.54638e12	0.0722195
$918$	0	0
$919$	−1.03287e13	−0.477667	−0.238833	−	0.971061i	$-0.576765\pi$
$919$	−1.03287e13	−0.477667	−0.238833	+	0.971061i	$0.576765\pi$
$920$	0	0
$921$	4.67803e13	2.14237
$922$	0	0
$923$	−1.51828e12	−0.0688563
$924$	0	0
$925$	6.45381e12	0.289853
$926$	0	0
$927$	−1.82641e13	−0.812344
$928$	0	0
$929$	−2.79767e13	−1.23233	−0.616163	−	0.787619i	$-0.711314\pi$
$929$	−2.79767e13	−1.23233	−0.616163	+	0.787619i	$0.711314\pi$
$930$	0	0
$931$	4.12996e12	0.180166
$932$	0	0
$933$	1.90947e13	0.824984
$934$	0	0
$935$	5.68868e11	0.0243421
$936$	0	0
$937$	3.31649e13	1.40556	0.702782	−	0.711406i	$-0.251941\pi$
$937$	3.31649e13	1.40556	0.702782	+	0.711406i	$0.251941\pi$
$938$	0	0
$939$	1.80437e13	0.757411
$940$	0	0
$941$	3.19309e13	1.32757	0.663786	−	0.747923i	$-0.268949\pi$
$941$	3.19309e13	1.32757	0.663786	+	0.747923i	$0.268949\pi$
$942$	0	0
$943$	−1.57113e13	−0.647009
$944$	0	0
$945$	2.32392e12	0.0947933
$946$	0	0
$947$	−3.56921e13	−1.44210	−0.721052	−	0.692881i	$-0.756341\pi$
$947$	−3.56921e13	−1.44210	−0.721052	+	0.692881i	$0.756341\pi$
$948$	0	0
$949$	−4.86594e12	−0.194746
$950$	0	0
$951$	−3.93308e11	−0.0155927
$952$	0	0
$953$	−3.04923e13	−1.19749	−0.598745	−	0.800940i	$-0.704334\pi$
$953$	−3.04923e13	−1.19749	−0.598745	+	0.800940i	$0.704334\pi$
$954$	0	0
$955$	2.07779e12	0.0808327
$956$	0	0
$957$	4.61221e13	1.77748
$958$	0	0
$959$	4.01869e12	0.153427
$960$	0	0
$961$	7.19282e12	0.272047
$962$	0	0
$963$	−3.83776e12	−0.143800
$964$	0	0
$965$	−1.31209e13	−0.487069
$966$	0	0
$967$	3.45533e13	1.27078	0.635389	−	0.772192i	$-0.280840\pi$
$967$	3.45533e13	1.27078	0.635389	+	0.772192i	$0.280840\pi$
$968$	0	0
$969$	2.60849e12	0.0950457
$970$	0	0
$971$	2.06708e13	0.746225	0.373112	−	0.927786i	$-0.378291\pi$
$971$	2.06708e13	0.746225	0.373112	+	0.927786i	$0.378291\pi$
$972$	0	0
$973$	1.00201e13	0.358397
$974$	0	0
$975$	−4.47262e12	−0.158504
$976$	0	0
$977$	1.78789e11	0.00627791	0.00313896	−	0.999995i	$-0.499001\pi$
$977$	1.78789e11	0.00627791	0.00313896	+	0.999995i	$0.499001\pi$
$978$	0	0
$979$	8.07952e12	0.281102
$980$	0	0
$981$	−1.16983e13	−0.403285
$982$	0	0
$983$	1.42657e13	0.487305	0.243652	−	0.969863i	$-0.421654\pi$
$983$	1.42657e13	0.487305	0.243652	+	0.969863i	$0.421654\pi$
$984$	0	0
$985$	−6.78902e12	−0.229797
$986$	0	0
$987$	2.30776e13	0.774038
$988$	0	0
$989$	4.61922e13	1.53527
$990$	0	0
$991$	2.71296e13	0.893537	0.446768	−	0.894650i	$-0.352575\pi$
$991$	2.71296e13	0.893537	0.446768	+	0.894650i	$0.352575\pi$
$992$	0	0
$993$	−3.67515e13	−1.19951
$994$	0	0
$995$	1.59461e12	0.0515764
$996$	0	0
$997$	−3.86723e12	−0.123957	−0.0619787	−	0.998077i	$-0.519741\pi$
$997$	−3.86723e12	−0.123957	−0.0619787	+	0.998077i	$0.519741\pi$
$998$	0	0
$999$	−6.92905e12	−0.220104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.a.1.1		1
4.3	odd	2		14.10.a.b.1.1	✓	1
12.11	even	2		126.10.a.a.1.1		1
20.3	even	4		350.10.c.d.99.1		2
20.7	even	4		350.10.c.d.99.2		2
20.19	odd	2		350.10.a.a.1.1		1
28.3	even	6		98.10.c.d.79.1		2
28.11	odd	6		98.10.c.a.79.1		2
28.19	even	6		98.10.c.d.67.1		2
28.23	odd	6		98.10.c.a.67.1		2
28.27	even	2		98.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.a.b.1.1	✓	1	4.3	odd	2
98.10.a.b.1.1		1	28.27	even	2
98.10.c.a.67.1		2	28.23	odd	6
98.10.c.a.79.1		2	28.11	odd	6
98.10.c.d.67.1		2	28.19	even	6
98.10.c.d.79.1		2	28.3	even	6
112.10.a.a.1.1		1	1.1	even	1	trivial
126.10.a.a.1.1		1	12.11	even	2
350.10.a.a.1.1		1	20.19	odd	2
350.10.c.d.99.1		2	20.3	even	4
350.10.c.d.99.2		2	20.7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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