LMFDB - Embedded newform 99.8.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [99,8,Mod(1,99)] 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("99.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	99.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9261175229$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.74597 q^{2} -113.968 q^{4} +80.0807 q^{5} +21.1693 q^{7} +906.403 q^{8} -299.980 q^{10} -1331.00 q^{11} +4184.41 q^{13} -79.2994 q^{14} +11192.5 q^{16} +32630.7 q^{17} -40738.5 q^{19} -9126.61 q^{20} +4985.88 q^{22} -2487.20 q^{23} -71712.1 q^{25} -15674.7 q^{26} -2412.61 q^{28} -689.254 q^{29} +127963. q^{31} -157946. q^{32} -122234. q^{34} +1695.25 q^{35} -467881. q^{37} +152605. q^{38} +72585.4 q^{40} -391657. q^{41} +236581. q^{43} +151691. q^{44} +9316.96 q^{46} -713115. q^{47} -823095. q^{49} +268631. q^{50} -476888. q^{52} -1.32248e6 q^{53} -106587. q^{55} +19187.9 q^{56} +2581.92 q^{58} +2.36633e6 q^{59} -3.23345e6 q^{61} -479345. q^{62} -840980. q^{64} +335090. q^{65} -3.12834e6 q^{67} -3.71885e6 q^{68} -6350.35 q^{70} -2.93234e6 q^{71} -1.06930e6 q^{73} +1.75267e6 q^{74} +4.64287e6 q^{76} -28176.3 q^{77} +4.40962e6 q^{79} +896304. q^{80} +1.46713e6 q^{82} +2.49753e6 q^{83} +2.61309e6 q^{85} -886223. q^{86} -1.20642e6 q^{88} +5.17831e6 q^{89} +88580.9 q^{91} +283460. q^{92} +2.67130e6 q^{94} -3.26236e6 q^{95} +1.32101e6 q^{97} +3.08329e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8} + 4280 q^{10} - 2662 q^{11} + 344 q^{13} - 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 35280 q^{19} - 5240 q^{20} - 10648 q^{22} + 61486 q^{23}+ \cdots + 11738664 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 - 104 * q^4 + 470 * q^5 - 1228 * q^7 - 480 * q^8 + 4280 * q^10 - 2662 * q^11 + 344 * q^13 - 14752 * q^14 - 6368 * q^16 + 8468 * q^17 - 35280 * q^19 - 5240 * q^20 - 10648 * q^22 + 61486 * q^23 + 2200 * q^25 - 60784 * q^26 - 14864 * q^28 - 179040 * q^29 - 57166 * q^31 - 186752 * q^32 - 406048 * q^34 - 485380 * q^35 - 877698 * q^37 + 216720 * q^38 - 468000 * q^40 + 283616 * q^41 + 275484 * q^43 + 138424 * q^44 + 760744 * q^46 - 1662512 * q^47 - 86214 * q^49 + 1136800 * q^50 - 515168 * q^52 - 1616484 * q^53 - 625570 * q^55 + 1751040 * q^56 - 2092320 * q^58 + 2454130 * q^59 - 6019176 * q^61 - 2653864 * q^62 + 1068416 * q^64 - 1162360 * q^65 - 174698 * q^67 - 3959696 * q^68 - 5727520 * q^70 + 1151466 * q^71 + 885944 * q^73 - 3061032 * q^74 + 4697280 * q^76 + 1634468 * q^77 + 3801460 * q^79 - 5950880 * q^80 + 9398864 * q^82 + 2282916 * q^83 - 6808420 * q^85 - 429264 * q^86 + 638880 * q^88 + 13481970 * q^89 + 4885904 * q^91 + 921128 * q^92 - 8480288 * q^94 - 1134000 * q^95 - 68078 * q^97 + 11738664 * q^98

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 $

$2$	−3.74597	−0.331100	−0.165550	−	0.986201i	$-0.552940\pi$
$2$	−3.74597	−0.331100	−0.165550	+	0.986201i	$0.552940\pi$
$3$	0	0
$3$	0	0
$4$	−113.968	−0.890373
$5$	80.0807	0.286505	0.143253	−	0.989686i	$-0.454244\pi$
$5$	80.0807	0.286505	0.143253	+	0.989686i	$0.454244\pi$
$6$	0	0
$7$	21.1693	0.0233272	0.0116636	−	0.999932i	$-0.496287\pi$
$7$	21.1693	0.0233272	0.0116636	+	0.999932i	$0.496287\pi$
$8$	906.403	0.625902
$9$	0	0
$10$	−299.980	−0.0948618
$11$	−1331.00	−0.301511
$11$	−1331.00	−0.301511
$12$	0	0
$13$	4184.41	0.528242	0.264121	−	0.964490i	$-0.414918\pi$
$13$	4184.41	0.528242	0.264121	+	0.964490i	$0.414918\pi$
$14$	−79.2994	−0.00772363
$15$	0	0
$16$	11192.5	0.683137
$17$	32630.7	1.61085	0.805425	−	0.592697i	$-0.201937\pi$
$17$	32630.7	1.61085	0.805425	+	0.592697i	$0.201937\pi$
$18$	0	0
$19$	−40738.5	−1.36260	−0.681298	−	0.732006i	$-0.738584\pi$
$19$	−40738.5	−1.36260	−0.681298	+	0.732006i	$0.738584\pi$
$20$	−9126.61	−0.255097
$21$	0	0
$22$	4985.88	0.0998303
$23$	−2487.20	−0.0426248	−0.0213124	−	0.999773i	$-0.506784\pi$
$23$	−2487.20	−0.0426248	−0.0213124	+	0.999773i	$0.506784\pi$
$24$	0	0
$25$	−71712.1	−0.917915
$26$	−15674.7	−0.174901
$27$	0	0
$28$	−2412.61	−0.0207699
$29$	−689.254	−0.00524791	−0.00262395	−	0.999997i	$-0.500835\pi$
$29$	−689.254	−0.00524791	−0.00262395	+	0.999997i	$0.500835\pi$
$30$	0	0
$31$	127963.	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$31$	127963.	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$32$	−157946.	−0.852089
$33$	0	0
$34$	−122234.	−0.533352
$35$	1695.25	0.00668337
$36$	0	0
$37$	−467881.	−1.51855	−0.759275	−	0.650770i	$-0.774446\pi$
$37$	−467881.	−1.51855	−0.759275	+	0.650770i	$0.774446\pi$
$38$	152605.	0.451155
$39$	0	0
$40$	72585.4	0.179324
$41$	−391657.	−0.887487	−0.443744	−	0.896154i	$-0.646350\pi$
$41$	−391657.	−0.887487	−0.443744	+	0.896154i	$0.646350\pi$
$42$	0	0
$43$	236581.	0.453774	0.226887	−	0.973921i	$-0.427145\pi$
$43$	236581.	0.453774	0.226887	+	0.973921i	$0.427145\pi$
$44$	151691.	0.268458
$45$	0	0
$46$	9316.96	0.0141131
$47$	−713115.	−1.00188	−0.500941	−	0.865481i	$-0.667013\pi$
$47$	−713115.	−1.00188	−0.500941	+	0.865481i	$0.667013\pi$
$48$	0	0
$49$	−823095.	−0.999456
$50$	268631.	0.303921
$51$	0	0
$52$	−476888.	−0.470332
$53$	−1.32248e6	−1.22018	−0.610090	−	0.792332i	$-0.708867\pi$
$53$	−1.32248e6	−1.22018	−0.610090	+	0.792332i	$0.708867\pi$
$54$	0	0
$55$	−106587.	−0.0863846
$56$	19187.9	0.0146005
$57$	0	0
$58$	2581.92	0.00173758
$59$	2.36633e6	1.50000	0.750002	−	0.661435i	$-0.230052\pi$
$59$	2.36633e6	1.50000	0.750002	+	0.661435i	$0.230052\pi$
$60$	0	0
$61$	−3.23345e6	−1.82394	−0.911971	−	0.410254i	$-0.865440\pi$
$61$	−3.23345e6	−1.82394	−0.911971	+	0.410254i	$0.865440\pi$
$62$	−479345.	−0.255433
$63$	0	0
$64$	−840980.	−0.401010
$65$	335090.	0.151344
$66$	0	0
$67$	−3.12834e6	−1.27073	−0.635363	−	0.772214i	$-0.719149\pi$
$67$	−3.12834e6	−1.27073	−0.635363	+	0.772214i	$0.719149\pi$
$68$	−3.71885e6	−1.43426
$69$	0	0
$70$	−6350.35	−0.00221286
$71$	−2.93234e6	−0.972321	−0.486161	−	0.873869i	$-0.661603\pi$
$71$	−2.93234e6	−0.972321	−0.486161	+	0.873869i	$0.661603\pi$
$72$	0	0
$73$	−1.06930e6	−0.321715	−0.160857	−	0.986978i	$-0.551426\pi$
$73$	−1.06930e6	−0.321715	−0.160857	+	0.986978i	$0.551426\pi$
$74$	1.75267e6	0.502792
$75$	0	0
$76$	4.64287e6	1.21322
$77$	−28176.3	−0.00703342
$78$	0	0
$79$	4.40962e6	1.00625	0.503125	−	0.864214i	$-0.332183\pi$
$79$	4.40962e6	1.00625	0.503125	+	0.864214i	$0.332183\pi$
$80$	896304.	0.195722
$81$	0	0
$82$	1.46713e6	0.293847
$83$	2.49753e6	0.479443	0.239722	−	0.970842i	$-0.422944\pi$
$83$	2.49753e6	0.479443	0.239722	+	0.970842i	$0.422944\pi$
$84$	0	0
$85$	2.61309e6	0.461517
$86$	−886223.	−0.150244
$87$	0	0
$88$	−1.20642e6	−0.188717
$89$	5.17831e6	0.778616	0.389308	−	0.921108i	$-0.372714\pi$
$89$	5.17831e6	0.778616	0.389308	+	0.921108i	$0.372714\pi$
$90$	0	0
$91$	88580.9	0.0123224
$92$	283460.	0.0379520
$93$	0	0
$94$	2.67130e6	0.331723
$95$	−3.26236e6	−0.390391
$96$	0	0
$97$	1.32101e6	0.146962	0.0734810	−	0.997297i	$-0.476589\pi$
$97$	1.32101e6	0.146962	0.0734810	+	0.997297i	$0.476589\pi$
$98$	3.08329e6	0.330920
$99$	0	0
$100$	8.17286e6	0.817286
$101$	1.71975e7	1.66089	0.830447	−	0.557098i	$-0.188085\pi$
$101$	1.71975e7	1.66089	0.830447	+	0.557098i	$0.188085\pi$
$102$	0	0
$103$	4.02516e6	0.362955	0.181477	−	0.983395i	$-0.441912\pi$
$103$	4.02516e6	0.362955	0.181477	+	0.983395i	$0.441912\pi$
$104$	3.79276e6	0.330628
$105$	0	0
$106$	4.95397e6	0.404001
$107$	−2.17827e7	−1.71897	−0.859486	−	0.511160i	$-0.829216\pi$
$107$	−2.17827e7	−1.71897	−0.859486	+	0.511160i	$0.829216\pi$
$108$	0	0
$109$	−2.69177e6	−0.199088	−0.0995439	−	0.995033i	$-0.531738\pi$
$109$	−2.69177e6	−0.199088	−0.0995439	+	0.995033i	$0.531738\pi$
$110$	399273.	0.0286019
$111$	0	0
$112$	236937.	0.0159357
$113$	598776.	0.0390382	0.0195191	−	0.999809i	$-0.493786\pi$
$113$	598776.	0.0390382	0.0195191	+	0.999809i	$0.493786\pi$
$114$	0	0
$115$	−199176.	−0.0122122
$116$	78552.7	0.00467260
$117$	0	0
$118$	−8.86418e6	−0.496651
$119$	690768.	0.0375767
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	1.21124e7	0.603907
$123$	0	0
$124$	−1.45837e7	−0.686895
$125$	−1.19991e7	−0.549493
$126$	0	0
$127$	3.59136e7	1.55577	0.777886	−	0.628405i	$-0.216292\pi$
$127$	3.59136e7	1.55577	0.777886	+	0.628405i	$0.216292\pi$
$128$	2.33674e7	0.984863
$129$	0	0
$130$	−1.25524e6	−0.0501100
$131$	1.65324e7	0.642521	0.321261	−	0.946991i	$-0.395893\pi$
$131$	1.65324e7	0.642521	0.321261	+	0.946991i	$0.395893\pi$
$132$	0	0
$133$	−862404.	−0.0317856
$134$	1.17187e7	0.420737
$135$	0	0
$136$	2.95766e7	1.00823
$137$	−2.77432e7	−0.921797	−0.460899	−	0.887453i	$-0.652473\pi$
$137$	−2.77432e7	−0.921797	−0.460899	+	0.887453i	$0.652473\pi$
$138$	0	0
$139$	−5.04731e7	−1.59407	−0.797037	−	0.603930i	$-0.793601\pi$
$139$	−5.04731e7	−1.59407	−0.797037	+	0.603930i	$0.793601\pi$
$140$	−193204.	−0.00595069
$141$	0	0
$142$	1.09844e7	0.321935
$143$	−5.56945e6	−0.159271
$144$	0	0
$145$	−55195.9	−0.00150355
$146$	4.00558e6	0.106520
$147$	0	0
$148$	5.33233e7	1.35208
$149$	−3.09903e7	−0.767492	−0.383746	−	0.923439i	$-0.625366\pi$
$149$	−3.09903e7	−0.767492	−0.383746	+	0.923439i	$0.625366\pi$
$150$	0	0
$151$	−6.75757e6	−0.159724	−0.0798622	−	0.996806i	$-0.525448\pi$
$151$	−6.75757e6	−0.159724	−0.0798622	+	0.996806i	$0.525448\pi$
$152$	−3.69255e7	−0.852852
$153$	0	0
$154$	105547.	0.00232876
$155$	1.02474e7	0.221030
$156$	0	0
$157$	1.89989e7	0.391813	0.195907	−	0.980623i	$-0.437235\pi$
$157$	1.89989e7	0.391813	0.195907	+	0.980623i	$0.437235\pi$
$158$	−1.65183e7	−0.333169
$159$	0	0
$160$	−1.26485e7	−0.244128
$161$	−52652.1	−0.000994318 0
$162$	0	0
$163$	−6.97589e7	−1.26166	−0.630831	−	0.775921i	$-0.717286\pi$
$163$	−6.97589e7	−1.26166	−0.630831	+	0.775921i	$0.717286\pi$
$164$	4.46362e7	0.790195
$165$	0	0
$166$	−9.35566e6	−0.158744
$167$	2.42233e7	0.402463	0.201231	−	0.979544i	$-0.435506\pi$
$167$	2.42233e7	0.402463	0.201231	+	0.979544i	$0.435506\pi$
$168$	0	0
$169$	−4.52392e7	−0.720961
$170$	−9.78855e6	−0.152808
$171$	0	0
$172$	−2.69625e7	−0.404028
$173$	−6.00438e7	−0.881671	−0.440836	−	0.897588i	$-0.645318\pi$
$173$	−6.00438e7	−0.881671	−0.440836	+	0.897588i	$0.645318\pi$
$174$	0	0
$175$	−1.51809e6	−0.0214124
$176$	−1.48972e7	−0.205974
$177$	0	0
$178$	−1.93978e7	−0.257800
$179$	−1.82943e6	−0.0238413	−0.0119206	−	0.999929i	$-0.503795\pi$
$179$	−1.82943e6	−0.0238413	−0.0119206	+	0.999929i	$0.503795\pi$
$180$	0	0
$181$	−1.24476e8	−1.56030	−0.780152	−	0.625590i	$-0.784858\pi$
$181$	−1.24476e8	−1.56030	−0.780152	+	0.625590i	$0.784858\pi$
$182$	−331821.	−0.00407994
$183$	0	0
$184$	−2.25440e6	−0.0266790
$185$	−3.74682e7	−0.435073
$186$	0	0
$187$	−4.34315e7	−0.485690
$188$	8.12720e7	0.892049
$189$	0	0
$190$	1.22207e7	0.129258
$191$	7.76142e7	0.805980	0.402990	−	0.915204i	$-0.367971\pi$
$191$	7.76142e7	0.805980	0.402990	+	0.915204i	$0.367971\pi$
$192$	0	0
$193$	3.85250e7	0.385738	0.192869	−	0.981225i	$-0.438221\pi$
$193$	3.85250e7	0.385738	0.192869	+	0.981225i	$0.438221\pi$
$194$	−4.94846e6	−0.0486591
$195$	0	0
$196$	9.38063e7	0.889888
$197$	−1.49067e8	−1.38915	−0.694575	−	0.719421i	$-0.744407\pi$
$197$	−1.49067e8	−1.38915	−0.694575	+	0.719421i	$0.744407\pi$
$198$	0	0
$199$	−6.45185e7	−0.580362	−0.290181	−	0.956972i	$-0.593715\pi$
$199$	−6.45185e7	−0.580362	−0.290181	+	0.956972i	$0.593715\pi$
$200$	−6.50001e7	−0.574525
$201$	0	0
$202$	−6.44214e7	−0.549921
$203$	−14591.0	−0.000122419 0
$204$	0	0
$205$	−3.13641e7	−0.254270
$206$	−1.50781e7	−0.120174
$207$	0	0
$208$	4.68341e7	0.360861
$209$	5.42229e7	0.410838
$210$	0	0
$211$	3.55799e7	0.260745	0.130373	−	0.991465i	$-0.458383\pi$
$211$	3.55799e7	0.260745	0.130373	+	0.991465i	$0.458383\pi$
$212$	1.50720e8	1.08642
$213$	0	0
$214$	8.15973e7	0.569151
$215$	1.89455e7	0.130009
$216$	0	0
$217$	2.70888e6	0.0179962
$218$	1.00833e7	0.0659180
$219$	0	0
$220$	1.21475e7	0.0769145
$221$	1.36540e8	0.850918
$222$	0	0
$223$	−1.55599e8	−0.939590	−0.469795	−	0.882776i	$-0.655672\pi$
$223$	−1.55599e8	−0.939590	−0.469795	+	0.882776i	$0.655672\pi$
$224$	−3.34361e6	−0.0198768
$225$	0	0
$226$	−2.24300e6	−0.0129255
$227$	−1.39233e8	−0.790046	−0.395023	−	0.918671i	$-0.629263\pi$
$227$	−1.39233e8	−0.790046	−0.395023	+	0.918671i	$0.629263\pi$
$228$	0	0
$229$	2.52955e8	1.39193	0.695966	−	0.718074i	$-0.254976\pi$
$229$	2.52955e8	1.39193	0.695966	+	0.718074i	$0.254976\pi$
$230$	746108.	0.00404347
$231$	0	0
$232$	−624742.	−0.00328468
$233$	−1.00338e8	−0.519658	−0.259829	−	0.965655i	$-0.583666\pi$
$233$	−1.00338e8	−0.519658	−0.259829	+	0.965655i	$0.583666\pi$
$234$	0	0
$235$	−5.71067e7	−0.287045
$236$	−2.69685e8	−1.33556
$237$	0	0
$238$	−2.58760e6	−0.0124416
$239$	−7.53046e7	−0.356803	−0.178402	−	0.983958i	$-0.557093\pi$
$239$	−7.53046e7	−0.356803	−0.178402	+	0.983958i	$0.557093\pi$
$240$	0	0
$241$	1.25802e8	0.578934	0.289467	−	0.957188i	$-0.406522\pi$
$241$	1.25802e8	0.578934	0.289467	+	0.957188i	$0.406522\pi$
$242$	−6.63621e6	−0.0301000
$243$	0	0
$244$	3.68509e8	1.62399
$245$	−6.59140e7	−0.286349
$246$	0	0
$247$	−1.70467e8	−0.719780
$248$	1.15986e8	0.482864
$249$	0	0
$250$	4.49481e7	0.181937
$251$	1.64527e8	0.656718	0.328359	−	0.944553i	$-0.393504\pi$
$251$	1.64527e8	0.656718	0.328359	+	0.944553i	$0.393504\pi$
$252$	0	0
$253$	3.31046e6	0.0128519
$254$	−1.34531e8	−0.515116
$255$	0	0
$256$	2.01119e7	0.0749225
$257$	1.10435e8	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$257$	1.10435e8	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$258$	0	0
$259$	−9.90470e6	−0.0354235
$260$	−3.81895e7	−0.134753
$261$	0	0
$262$	−6.19300e7	−0.212739
$263$	3.98541e8	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$263$	3.98541e8	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$264$	0	0
$265$	−1.05905e8	−0.349588
$266$	3.23054e6	0.0105242
$267$	0	0
$268$	3.56530e8	1.13142
$269$	2.99250e8	0.937350	0.468675	−	0.883371i	$-0.344732\pi$
$269$	2.99250e8	0.937350	0.468675	+	0.883371i	$0.344732\pi$
$270$	0	0
$271$	−1.37158e7	−0.0418628	−0.0209314	−	0.999781i	$-0.506663\pi$
$271$	−1.37158e7	−0.0418628	−0.0209314	+	0.999781i	$0.506663\pi$
$272$	3.65220e8	1.10043
$273$	0	0
$274$	1.03925e8	0.305207
$275$	9.54488e7	0.276762
$276$	0	0
$277$	−6.73807e8	−1.90483	−0.952415	−	0.304804i	$-0.901409\pi$
$277$	−6.73807e8	−1.90483	−0.952415	+	0.304804i	$0.901409\pi$
$278$	1.89071e8	0.527798
$279$	0	0
$280$	1.53658e6	0.00418313
$281$	−5.66275e8	−1.52249	−0.761246	−	0.648463i	$-0.775412\pi$
$281$	−5.66275e8	−1.52249	−0.761246	+	0.648463i	$0.775412\pi$
$282$	0	0
$283$	6.29800e8	1.65177	0.825886	−	0.563837i	$-0.190675\pi$
$283$	6.29800e8	1.65177	0.825886	+	0.563837i	$0.190675\pi$
$284$	3.34192e8	0.865729
$285$	0	0
$286$	2.08630e7	0.0527345
$287$	−8.29109e6	−0.0207026
$288$	0	0
$289$	6.54425e8	1.59484
$290$	206762.	0.000497826 0
$291$	0	0
$292$	1.21866e8	0.286446
$293$	1.23830e6	0.00287600	0.00143800	−	0.999999i	$-0.499542\pi$
$293$	1.23830e6	0.00287600	0.00143800	+	0.999999i	$0.499542\pi$
$294$	0	0
$295$	1.89497e8	0.429759
$296$	−4.24089e8	−0.950464
$297$	0	0
$298$	1.16089e8	0.254116
$299$	−1.04075e7	−0.0225162
$300$	0	0
$301$	5.00824e6	0.0105853
$302$	2.53136e7	0.0528847
$303$	0	0
$304$	−4.55966e8	−0.930840
$305$	−2.58937e8	−0.522569
$306$	0	0
$307$	−4.88441e8	−0.963447	−0.481723	−	0.876323i	$-0.659989\pi$
$307$	−4.88441e8	−0.963447	−0.481723	+	0.876323i	$0.659989\pi$
$308$	3.21119e6	0.00626236
$309$	0	0
$310$	−3.83863e7	−0.0731830
$311$	4.81123e8	0.906974	0.453487	−	0.891263i	$-0.350180\pi$
$311$	4.81123e8	0.906974	0.453487	+	0.891263i	$0.350180\pi$
$312$	0	0
$313$	5.17645e7	0.0954172	0.0477086	−	0.998861i	$-0.484808\pi$
$313$	5.17645e7	0.0954172	0.0477086	+	0.998861i	$0.484808\pi$
$314$	−7.11691e7	−0.129729
$315$	0	0
$316$	−5.02554e8	−0.895938
$317$	4.71412e8	0.831177	0.415588	−	0.909553i	$-0.363576\pi$
$317$	4.71412e8	0.831177	0.415588	+	0.909553i	$0.363576\pi$
$318$	0	0
$319$	917397.	0.00158230
$320$	−6.73462e7	−0.114892
$321$	0	0
$322$	197233.	0.000329219 0
$323$	−1.32933e9	−2.19494
$324$	0	0
$325$	−3.00073e8	−0.484881
$326$	2.61314e8	0.417736
$327$	0	0
$328$	−3.54999e8	−0.555480
$329$	−1.50961e7	−0.0233711
$330$	0	0
$331$	5.97269e8	0.905258	0.452629	−	0.891699i	$-0.350486\pi$
$331$	5.97269e8	0.905258	0.452629	+	0.891699i	$0.350486\pi$
$332$	−2.84638e8	−0.426883
$333$	0	0
$334$	−9.07397e7	−0.133255
$335$	−2.50519e8	−0.364070
$336$	0	0
$337$	−8.11952e8	−1.15565	−0.577824	−	0.816161i	$-0.696098\pi$
$337$	−8.11952e8	−1.15565	−0.577824	+	0.816161i	$0.696098\pi$
$338$	1.69465e8	0.238710
$339$	0	0
$340$	−2.97808e8	−0.410923
$341$	−1.70319e8	−0.232607
$342$	0	0
$343$	−3.48581e7	−0.0466417
$344$	2.14437e8	0.284018
$345$	0	0
$346$	2.24922e8	0.291921
$347$	4.61385e8	0.592803	0.296401	−	0.955063i	$-0.404213\pi$
$347$	4.61385e8	0.592803	0.296401	+	0.955063i	$0.404213\pi$
$348$	0	0
$349$	−9.00068e8	−1.13341	−0.566704	−	0.823922i	$-0.691782\pi$
$349$	−9.00068e8	−1.13341	−0.566704	+	0.823922i	$0.691782\pi$
$350$	5.68672e6	0.00708964
$351$	0	0
$352$	2.10227e8	0.256914
$353$	−7.73849e8	−0.936364	−0.468182	−	0.883632i	$-0.655091\pi$
$353$	−7.73849e8	−0.936364	−0.468182	+	0.883632i	$0.655091\pi$
$354$	0	0
$355$	−2.34824e8	−0.278575
$356$	−5.90161e8	−0.693259
$357$	0	0
$358$	6.85298e6	0.00789384
$359$	−1.71828e9	−1.96003	−0.980016	−	0.198918i	$-0.936257\pi$
$359$	−1.71828e9	−1.96003	−0.980016	+	0.198918i	$0.936257\pi$
$360$	0	0
$361$	7.65751e8	0.856668
$362$	4.66281e8	0.516616
$363$	0	0
$364$	−1.00954e7	−0.0109715
$365$	−8.56306e7	−0.0921730
$366$	0	0
$367$	−3.00191e8	−0.317005	−0.158503	−	0.987359i	$-0.550667\pi$
$367$	−3.00191e8	−0.317005	−0.158503	+	0.987359i	$0.550667\pi$
$368$	−2.78380e7	−0.0291186
$369$	0	0
$370$	1.40355e8	0.144053
$371$	−2.79960e7	−0.0284634
$372$	0	0
$373$	−9.39634e7	−0.0937514	−0.0468757	−	0.998901i	$-0.514926\pi$
$373$	−9.39634e7	−0.0937514	−0.0468757	+	0.998901i	$0.514926\pi$
$374$	1.62693e8	0.160812
$375$	0	0
$376$	−6.46369e8	−0.627080
$377$	−2.88412e6	−0.00277216
$378$	0	0
$379$	−7.85315e8	−0.740980	−0.370490	−	0.928836i	$-0.620810\pi$
$379$	−7.85315e8	−0.740980	−0.370490	+	0.928836i	$0.620810\pi$
$380$	3.71804e8	0.347594
$381$	0	0
$382$	−2.90740e8	−0.266860
$383$	1.56792e9	1.42603	0.713014	−	0.701149i	$-0.247329\pi$
$383$	1.56792e9	1.42603	0.713014	+	0.701149i	$0.247329\pi$
$384$	0	0
$385$	−2.25638e6	−0.00201511
$386$	−1.44313e8	−0.127718
$387$	0	0
$388$	−1.50552e8	−0.130851
$389$	1.59542e9	1.37420	0.687101	−	0.726562i	$-0.258883\pi$
$389$	1.59542e9	1.37420	0.687101	+	0.726562i	$0.258883\pi$
$390$	0	0
$391$	−8.11590e7	−0.0686623
$392$	−7.46056e8	−0.625562
$393$	0	0
$394$	5.58399e8	0.459947
$395$	3.53125e8	0.288296
$396$	0	0
$397$	1.23380e8	0.0989645	0.0494822	−	0.998775i	$-0.484243\pi$
$397$	1.23380e8	0.0989645	0.0494822	+	0.998775i	$0.484243\pi$
$398$	2.41684e8	0.192158
$399$	0	0
$400$	−8.02639e8	−0.627061
$401$	7.80784e8	0.604680	0.302340	−	0.953200i	$-0.402232\pi$
$401$	7.80784e8	0.604680	0.302340	+	0.953200i	$0.402232\pi$
$402$	0	0
$403$	5.35450e8	0.407522
$404$	−1.95997e9	−1.47881
$405$	0	0
$406$	54657.4	4.05329e−5 0
$407$	6.22749e8	0.457860
$408$	0	0
$409$	1.78733e9	1.29174	0.645868	−	0.763449i	$-0.276495\pi$
$409$	1.78733e9	1.29174	0.645868	+	0.763449i	$0.276495\pi$
$410$	1.17489e8	0.0841887
$411$	0	0
$412$	−4.58738e8	−0.323165
$413$	5.00934e7	0.0349909
$414$	0	0
$415$	2.00004e8	0.137363
$416$	−6.60913e8	−0.450109
$417$	0	0
$418$	−2.03117e8	−0.136028
$419$	1.51640e9	1.00708	0.503542	−	0.863971i	$-0.332030\pi$
$419$	1.51640e9	1.00708	0.503542	+	0.863971i	$0.332030\pi$
$420$	0	0
$421$	2.24522e9	1.46646	0.733231	−	0.679980i	$-0.238011\pi$
$421$	2.24522e9	1.46646	0.733231	+	0.679980i	$0.238011\pi$
$422$	−1.33281e8	−0.0863326
$423$	0	0
$424$	−1.19870e9	−0.763713
$425$	−2.34002e9	−1.47862
$426$	0	0
$427$	−6.84497e7	−0.0425475
$428$	2.48253e9	1.53053
$429$	0	0
$430$	−7.09693e7	−0.0430458
$431$	−2.62807e9	−1.58113	−0.790564	−	0.612380i	$-0.790212\pi$
$431$	−2.62807e9	−1.58113	−0.790564	+	0.612380i	$0.790212\pi$
$432$	0	0
$433$	3.73140e8	0.220884	0.110442	−	0.993883i	$-0.464773\pi$
$433$	3.73140e8	0.220884	0.110442	+	0.993883i	$0.464773\pi$
$434$	−1.01474e7	−0.00595854
$435$	0	0
$436$	3.06775e8	0.177262
$437$	1.01325e8	0.0580804
$438$	0	0
$439$	2.04384e9	1.15298	0.576488	−	0.817105i	$-0.304423\pi$
$439$	2.04384e9	1.15298	0.576488	+	0.817105i	$0.304423\pi$
$440$	−9.66111e7	−0.0540683
$441$	0	0
$442$	−5.11475e8	−0.281739
$443$	3.44729e9	1.88393	0.941965	−	0.335712i	$-0.108977\pi$
$443$	3.44729e9	1.88393	0.941965	+	0.335712i	$0.108977\pi$
$444$	0	0
$445$	4.14683e8	0.223078
$446$	5.82867e8	0.311098
$447$	0	0
$448$	−1.78029e7	−0.00935445
$449$	−1.53897e9	−0.802357	−0.401178	−	0.916000i	$-0.631399\pi$
$449$	−1.53897e9	−0.802357	−0.401178	+	0.916000i	$0.631399\pi$
$450$	0	0
$451$	5.21295e8	0.267587
$452$	−6.82412e7	−0.0347586
$453$	0	0
$454$	5.21563e8	0.261584
$455$	7.09362e6	0.00353043
$456$	0	0
$457$	−1.48389e9	−0.727268	−0.363634	−	0.931542i	$-0.618464\pi$
$457$	−1.48389e9	−0.727268	−0.363634	+	0.931542i	$0.618464\pi$
$458$	−9.47559e8	−0.460869
$459$	0	0
$460$	2.26997e7	0.0108735
$461$	2.44680e9	1.16317	0.581587	−	0.813484i	$-0.302432\pi$
$461$	2.44680e9	1.16317	0.581587	+	0.813484i	$0.302432\pi$
$462$	0	0
$463$	1.45022e9	0.679049	0.339524	−	0.940597i	$-0.389734\pi$
$463$	1.45022e9	0.679049	0.339524	+	0.940597i	$0.389734\pi$
$464$	−7.71448e6	−0.00358504
$465$	0	0
$466$	3.75861e8	0.172059
$467$	−1.48083e9	−0.672814	−0.336407	−	0.941717i	$-0.609212\pi$
$467$	−1.48083e9	−0.672814	−0.336407	+	0.941717i	$0.609212\pi$
$468$	0	0
$469$	−6.62246e7	−0.0296425
$470$	2.13920e8	0.0950404
$471$	0	0
$472$	2.14485e9	0.938856
$473$	−3.14889e8	−0.136818
$474$	0	0
$475$	2.92144e9	1.25075
$476$	−7.87253e7	−0.0334572
$477$	0	0
$478$	2.82089e8	0.118138
$479$	3.76893e9	1.56691	0.783455	−	0.621449i	$-0.213456\pi$
$479$	3.76893e9	1.56691	0.783455	+	0.621449i	$0.213456\pi$
$480$	0	0
$481$	−1.95781e9	−0.802162
$482$	−4.71251e8	−0.191685
$483$	0	0
$484$	−2.01901e8	−0.0809430
$485$	1.05787e8	0.0421054
$486$	0	0
$487$	9.84963e8	0.386428	0.193214	−	0.981157i	$-0.438109\pi$
$487$	9.84963e8	0.386428	0.193214	+	0.981157i	$0.438109\pi$
$488$	−2.93081e9	−1.14161
$489$	0	0
$490$	2.46912e8	0.0948102
$491$	3.56853e8	0.136052	0.0680259	−	0.997684i	$-0.478330\pi$
$491$	3.56853e8	0.136052	0.0680259	+	0.997684i	$0.478330\pi$
$492$	0	0
$493$	−2.24908e7	−0.00845360
$494$	6.38562e8	0.238319
$495$	0	0
$496$	1.43223e9	0.527019
$497$	−6.20754e7	−0.0226815
$498$	0	0
$499$	−2.25684e9	−0.813111	−0.406555	−	0.913626i	$-0.633270\pi$
$499$	−2.25684e9	−0.813111	−0.406555	+	0.913626i	$0.633270\pi$
$500$	1.36750e9	0.489253
$501$	0	0
$502$	−6.16312e8	−0.217439
$503$	−1.74836e9	−0.612553	−0.306277	−	0.951943i	$-0.599083\pi$
$503$	−1.74836e9	−0.612553	−0.306277	+	0.951943i	$0.599083\pi$
$504$	0	0
$505$	1.37719e9	0.475855
$506$	−1.24009e7	−0.00425525
$507$	0	0
$508$	−4.09299e9	−1.38522
$509$	4.30470e9	1.44687	0.723437	−	0.690391i	$-0.242561\pi$
$509$	4.30470e9	1.44687	0.723437	+	0.690391i	$0.242561\pi$
$510$	0	0
$511$	−2.26364e7	−0.00750471
$512$	−3.06637e9	−1.00967
$513$	0	0
$514$	−4.13687e8	−0.134370
$515$	3.22337e8	0.103989
$516$	0	0
$517$	9.49155e8	0.302079
$518$	3.71027e7	0.0117287
$519$	0	0
$520$	3.03727e8	0.0947265
$521$	−3.38119e9	−1.04746	−0.523730	−	0.851885i	$-0.675460\pi$
$521$	−3.38119e9	−1.04746	−0.523730	+	0.851885i	$0.675460\pi$
$522$	0	0
$523$	4.20231e9	1.28449	0.642247	−	0.766497i	$-0.278002\pi$
$523$	4.20231e9	1.28449	0.642247	+	0.766497i	$0.278002\pi$
$524$	−1.88417e9	−0.572084
$525$	0	0
$526$	−1.49292e9	−0.447287
$527$	4.17552e9	1.24272
$528$	0	0
$529$	−3.39864e9	−0.998183
$530$	3.96717e8	0.115749
$531$	0	0
$532$	9.82862e7	0.0283010
$533$	−1.63885e9	−0.468808
$534$	0	0
$535$	−1.74437e9	−0.492495
$536$	−2.83554e9	−0.795350
$537$	0	0
$538$	−1.12098e9	−0.310356
$539$	1.09554e9	0.301347
$540$	0	0
$541$	−6.90757e9	−1.87558	−0.937788	−	0.347208i	$-0.887130\pi$
$541$	−6.90757e9	−1.87558	−0.937788	+	0.347208i	$0.887130\pi$
$542$	5.13789e7	0.0138608
$543$	0	0
$544$	−5.15390e9	−1.37259
$545$	−2.15559e8	−0.0570397
$546$	0	0
$547$	−1.87201e9	−0.489049	−0.244525	−	0.969643i	$-0.578632\pi$
$547$	−1.87201e9	−0.489049	−0.244525	+	0.969643i	$0.578632\pi$
$548$	3.16184e9	0.820743
$549$	0	0
$550$	−3.57548e8	−0.0916357
$551$	2.80792e7	0.00715078
$552$	0	0
$553$	9.33484e7	0.0234730
$554$	2.52406e9	0.630689
$555$	0	0
$556$	5.75231e9	1.41932
$557$	−2.57944e9	−0.632458	−0.316229	−	0.948683i	$-0.602417\pi$
$557$	−2.57944e9	−0.632458	−0.316229	+	0.948683i	$0.602417\pi$
$558$	0	0
$559$	9.89950e8	0.239702
$560$	1.89741e7	0.00456566
$561$	0	0
$562$	2.12125e9	0.504097
$563$	2.44826e7	0.00578199	0.00289100	−	0.999996i	$-0.499080\pi$
$563$	2.44826e7	0.00578199	0.00289100	+	0.999996i	$0.499080\pi$
$564$	0	0
$565$	4.79504e7	0.0111847
$566$	−2.35921e9	−0.546902
$567$	0	0
$568$	−2.65788e9	−0.608578
$569$	−6.24602e9	−1.42138	−0.710690	−	0.703505i	$-0.751617\pi$
$569$	−6.24602e9	−1.42138	−0.710690	+	0.703505i	$0.751617\pi$
$570$	0	0
$571$	2.67106e9	0.600423	0.300211	−	0.953873i	$-0.402943\pi$
$571$	2.67106e9	0.600423	0.300211	+	0.953873i	$0.402943\pi$
$572$	6.34738e8	0.141810
$573$	0	0
$574$	3.10581e7	0.00685463
$575$	1.78362e8	0.0391260
$576$	0	0
$577$	2.92211e9	0.633258	0.316629	−	0.948549i	$-0.397449\pi$
$577$	2.92211e9	0.633258	0.316629	+	0.948549i	$0.397449\pi$
$578$	−2.45145e9	−0.528051
$579$	0	0
$580$	6.29055e6	0.00133872
$581$	5.28709e7	0.0111841
$582$	0	0
$583$	1.76022e9	0.367898
$584$	−9.69220e8	−0.201362
$585$	0	0
$586$	−4.63863e6	−0.000952245 0
$587$	−8.18950e9	−1.67118	−0.835592	−	0.549351i	$-0.814875\pi$
$587$	−8.18950e9	−1.67118	−0.835592	+	0.549351i	$0.814875\pi$
$588$	0	0
$589$	−5.21302e9	−1.05120
$590$	−7.09849e8	−0.142293
$591$	0	0
$592$	−5.23676e9	−1.03738
$593$	9.08514e9	1.78912	0.894562	−	0.446944i	$-0.147488\pi$
$593$	9.08514e9	1.78912	0.894562	+	0.446944i	$0.147488\pi$
$594$	0	0
$595$	5.53172e7	0.0107659
$596$	3.53189e9	0.683354
$597$	0	0
$598$	3.89860e7	0.00745511
$599$	−3.46726e9	−0.659162	−0.329581	−	0.944127i	$-0.606907\pi$
$599$	−3.46726e9	−0.659162	−0.329581	+	0.944127i	$0.606907\pi$
$600$	0	0
$601$	−2.56544e9	−0.482060	−0.241030	−	0.970518i	$-0.577485\pi$
$601$	−2.56544e9	−0.482060	−0.241030	+	0.970518i	$0.577485\pi$
$602$	−1.87607e7	−0.00350478
$603$	0	0
$604$	7.70145e8	0.142214
$605$	1.41868e8	0.0260459
$606$	0	0
$607$	1.86107e9	0.337755	0.168877	−	0.985637i	$-0.445986\pi$
$607$	1.86107e9	0.337755	0.168877	+	0.985637i	$0.445986\pi$
$608$	6.43449e9	1.16105
$609$	0	0
$610$	9.69968e8	0.173023
$611$	−2.98396e9	−0.529236
$612$	0	0
$613$	−2.35285e9	−0.412556	−0.206278	−	0.978493i	$-0.566135\pi$
$613$	−2.35285e9	−0.412556	−0.206278	+	0.978493i	$0.566135\pi$
$614$	1.82968e9	0.318997
$615$	0	0
$616$	−2.55391e7	−0.00440223
$617$	5.81579e9	0.996806	0.498403	−	0.866946i	$-0.333920\pi$
$617$	5.81579e9	0.996806	0.498403	+	0.866946i	$0.333920\pi$
$618$	0	0
$619$	3.71473e9	0.629520	0.314760	−	0.949171i	$-0.398076\pi$
$619$	3.71473e9	0.629520	0.314760	+	0.949171i	$0.398076\pi$
$620$	−1.16787e9	−0.196799
$621$	0	0
$622$	−1.80227e9	−0.300299
$623$	1.09621e8	0.0181629
$624$	0	0
$625$	4.64161e9	0.760482
$626$	−1.93908e8	−0.0315926
$627$	0	0
$628$	−2.16526e9	−0.348860
$629$	−1.52673e10	−2.44616
$630$	0	0
$631$	−6.61199e9	−1.04768	−0.523841	−	0.851816i	$-0.675501\pi$
$631$	−6.61199e9	−1.04768	−0.523841	+	0.851816i	$0.675501\pi$
$632$	3.99689e9	0.629814
$633$	0	0
$634$	−1.76589e9	−0.275202
$635$	2.87599e9	0.445737
$636$	0	0
$637$	−3.44417e9	−0.527954
$638$	−3.43654e6	−0.000523901 0
$639$	0	0
$640$	1.87128e9	0.282168
$641$	1.31108e10	1.96619	0.983093	−	0.183106i	$-0.0586151\pi$
$641$	1.31108e10	1.96619	0.983093	+	0.183106i	$0.0586151\pi$
$642$	0	0
$643$	8.73321e9	1.29549	0.647747	−	0.761856i	$-0.275711\pi$
$643$	8.73321e9	1.29549	0.647747	+	0.761856i	$0.275711\pi$
$644$	6.00065e6	0.000885314 0
$645$	0	0
$646$	4.97961e9	0.726744
$647$	4.05488e9	0.588590	0.294295	−	0.955715i	$-0.404915\pi$
$647$	4.05488e9	0.588590	0.294295	+	0.955715i	$0.404915\pi$
$648$	0	0
$649$	−3.14958e9	−0.452268
$650$	1.12406e9	0.160544
$651$	0	0
$652$	7.95026e9	1.12335
$653$	−2.96057e9	−0.416082	−0.208041	−	0.978120i	$-0.566709\pi$
$653$	−2.96057e9	−0.416082	−0.208041	+	0.978120i	$0.566709\pi$
$654$	0	0
$655$	1.32393e9	0.184086
$656$	−4.38362e9	−0.606275
$657$	0	0
$658$	5.65495e7	0.00773817
$659$	−1.25563e10	−1.70908	−0.854540	−	0.519386i	$-0.826161\pi$
$659$	−1.25563e10	−1.70908	−0.854540	+	0.519386i	$0.826161\pi$
$660$	0	0
$661$	−8.54786e9	−1.15120	−0.575602	−	0.817730i	$-0.695232\pi$
$661$	−8.54786e9	−1.15120	−0.575602	+	0.817730i	$0.695232\pi$
$662$	−2.23735e9	−0.299731
$663$	0	0
$664$	2.26377e9	0.300085
$665$	−6.90619e7	−0.00910673
$666$	0	0
$667$	1.71431e6	0.000223691 0
$668$	−2.76068e9	−0.358342
$669$	0	0
$670$	9.38437e8	0.120543
$671$	4.30372e9	0.549939
$672$	0	0
$673$	−7.90222e9	−0.999301	−0.499650	−	0.866227i	$-0.666538\pi$
$673$	−7.90222e9	−0.999301	−0.499650	+	0.866227i	$0.666538\pi$
$674$	3.04154e9	0.382635
$675$	0	0
$676$	5.15581e9	0.641924
$677$	8.15033e9	1.00952	0.504760	−	0.863260i	$-0.331581\pi$
$677$	8.15033e9	1.00952	0.504760	+	0.863260i	$0.331581\pi$
$678$	0	0
$679$	2.79648e7	0.00342821
$680$	2.36851e9	0.288865
$681$	0	0
$682$	6.38008e8	0.0770160
$683$	6.93643e8	0.0833036	0.0416518	−	0.999132i	$-0.486738\pi$
$683$	6.93643e8	0.0833036	0.0416518	+	0.999132i	$0.486738\pi$
$684$	0	0
$685$	−2.22170e9	−0.264100
$686$	1.30577e8	0.0154431
$687$	0	0
$688$	2.64793e9	0.309990
$689$	−5.53380e9	−0.644550
$690$	0	0
$691$	3.17516e7	0.00366094	0.00183047	−	0.999998i	$-0.499417\pi$
$691$	3.17516e7	0.00366094	0.00183047	+	0.999998i	$0.499417\pi$
$692$	6.84305e9	0.785016
$693$	0	0
$694$	−1.72833e9	−0.196277
$695$	−4.04192e9	−0.456711
$696$	0	0
$697$	−1.27800e10	−1.42961
$698$	3.37162e9	0.375271
$699$	0	0
$700$	1.73014e8	0.0190650
$701$	−7.68876e9	−0.843030	−0.421515	−	0.906821i	$-0.638502\pi$
$701$	−7.68876e9	−0.843030	−0.421515	+	0.906821i	$0.638502\pi$
$702$	0	0
$703$	1.90608e10	2.06917
$704$	1.11934e9	0.120909
$705$	0	0
$706$	2.89881e9	0.310030
$707$	3.64059e8	0.0387440
$708$	0	0
$709$	−4.65053e9	−0.490051	−0.245025	−	0.969517i	$-0.578796\pi$
$709$	−4.65053e9	−0.490051	−0.245025	+	0.969517i	$0.578796\pi$
$710$	8.79641e8	0.0922362
$711$	0	0
$712$	4.69364e9	0.487337
$713$	−3.18269e8	−0.0328837
$714$	0	0
$715$	−4.46005e8	−0.0456319
$716$	2.08496e8	0.0212276
$717$	0	0
$718$	6.43662e9	0.648966
$719$	9.19022e9	0.922093	0.461047	−	0.887376i	$-0.347474\pi$
$719$	9.19022e9	0.922093	0.461047	+	0.887376i	$0.347474\pi$
$720$	0	0
$721$	8.52097e7	0.00846672
$722$	−2.86848e9	−0.283643
$723$	0	0
$724$	1.41862e10	1.38925
$725$	4.94278e7	0.00481713
$726$	0	0
$727$	9.36688e9	0.904117	0.452059	−	0.891988i	$-0.350690\pi$
$727$	9.36688e9	0.904117	0.452059	+	0.891988i	$0.350690\pi$
$728$	8.02900e7	0.00771262
$729$	0	0
$730$	3.20769e8	0.0305185
$731$	7.71979e9	0.730962
$732$	0	0
$733$	1.29034e10	1.21015	0.605076	−	0.796168i	$-0.293143\pi$
$733$	1.29034e10	1.21015	0.605076	+	0.796168i	$0.293143\pi$
$734$	1.12451e9	0.104960
$735$	0	0
$736$	3.92844e8	0.0363201
$737$	4.16382e9	0.383138
$738$	0	0
$739$	6.13118e9	0.558841	0.279421	−	0.960169i	$-0.409858\pi$
$739$	6.13118e9	0.558841	0.279421	+	0.960169i	$0.409858\pi$
$740$	4.27017e9	0.387377
$741$	0	0
$742$	1.04872e8	0.00942422
$743$	−9.12544e9	−0.816193	−0.408096	−	0.912939i	$-0.633807\pi$
$743$	−9.12544e9	−0.816193	−0.408096	+	0.912939i	$0.633807\pi$
$744$	0	0
$745$	−2.48172e9	−0.219891
$746$	3.51984e8	0.0310411
$747$	0	0
$748$	4.94979e9	0.432445
$749$	−4.61124e8	−0.0400988
$750$	0	0
$751$	−1.80702e9	−0.155676	−0.0778381	−	0.996966i	$-0.524802\pi$
$751$	−1.80702e9	−0.155676	−0.0778381	+	0.996966i	$0.524802\pi$
$752$	−7.98154e9	−0.684423
$753$	0	0
$754$	1.08038e7	0.000917863 0
$755$	−5.41151e8	−0.0457619
$756$	0	0
$757$	4.78438e9	0.400858	0.200429	−	0.979708i	$-0.435766\pi$
$757$	4.78438e9	0.400858	0.200429	+	0.979708i	$0.435766\pi$
$758$	2.94176e9	0.245338
$759$	0	0
$760$	−2.95702e9	−0.244347
$761$	−8.11100e9	−0.667157	−0.333578	−	0.942722i	$-0.608256\pi$
$761$	−8.11100e9	−0.667157	−0.333578	+	0.942722i	$0.608256\pi$
$762$	0	0
$763$	−5.69828e7	−0.00464416
$764$	−8.84551e9	−0.717623
$765$	0	0
$766$	−5.87338e9	−0.472158
$767$	9.90168e9	0.792365
$768$	0	0
$769$	−4.03304e9	−0.319809	−0.159904	−	0.987133i	$-0.551119\pi$
$769$	−4.03304e9	−0.319809	−0.159904	+	0.987133i	$0.551119\pi$
$770$	8.45231e6	0.000667203 0
$771$	0	0
$772$	−4.39061e9	−0.343451
$773$	−3.46346e9	−0.269701	−0.134850	−	0.990866i	$-0.543055\pi$
$773$	−3.46346e9	−0.269701	−0.134850	+	0.990866i	$0.543055\pi$
$774$	0	0
$775$	−9.17649e9	−0.708143
$776$	1.19737e9	0.0919838
$777$	0	0
$778$	−5.97638e9	−0.454998
$779$	1.59555e10	1.20929
$780$	0	0
$781$	3.90294e9	0.293166
$782$	3.04019e8	0.0227341
$783$	0	0
$784$	−9.21250e9	−0.682765
$785$	1.52144e9	0.112257
$786$	0	0
$787$	−7.81853e9	−0.571760	−0.285880	−	0.958265i	$-0.592286\pi$
$787$	−7.81853e9	−0.571760	−0.285880	+	0.958265i	$0.592286\pi$
$788$	1.69888e10	1.23686
$789$	0	0
$790$	−1.32279e9	−0.0954548
$791$	1.26757e7	0.000910652 0
$792$	0	0
$793$	−1.35301e10	−0.963482
$794$	−4.62179e8	−0.0327671
$795$	0	0
$796$	7.35303e9	0.516738
$797$	−1.19003e10	−0.832635	−0.416318	−	0.909219i	$-0.636680\pi$
$797$	−1.19003e10	−0.832635	−0.416318	+	0.909219i	$0.636680\pi$
$798$	0	0
$799$	−2.32694e10	−1.61388
$800$	1.13267e10	0.782145
$801$	0	0
$802$	−2.92479e9	−0.200209
$803$	1.42324e9	0.0970007
$804$	0	0
$805$	−4.21642e6	−0.000284878 0
$806$	−2.00578e9	−0.134930
$807$	0	0
$808$	1.55879e10	1.03956
$809$	−1.04001e10	−0.690584	−0.345292	−	0.938495i	$-0.612220\pi$
$809$	−1.04001e10	−0.690584	−0.345292	+	0.938495i	$0.612220\pi$
$810$	0	0
$811$	−1.00469e10	−0.661392	−0.330696	−	0.943737i	$-0.607283\pi$
$811$	−1.00469e10	−0.661392	−0.330696	+	0.943737i	$0.607283\pi$
$812$	1.66290e6	0.000108999 0
$813$	0	0
$814$	−2.33280e9	−0.151597
$815$	−5.58634e9	−0.361473
$816$	0	0
$817$	−9.63793e9	−0.618310
$818$	−6.69529e9	−0.427694
$819$	0	0
$820$	3.57450e9	0.226395
$821$	−2.01156e10	−1.26862	−0.634309	−	0.773079i	$-0.718715\pi$
$821$	−2.01156e10	−1.26862	−0.634309	+	0.773079i	$0.718715\pi$
$822$	0	0
$823$	−4.01271e9	−0.250922	−0.125461	−	0.992099i	$-0.540041\pi$
$823$	−4.01271e9	−0.250922	−0.125461	+	0.992099i	$0.540041\pi$
$824$	3.64842e9	0.227174
$825$	0	0
$826$	−1.87648e8	−0.0115855
$827$	1.18312e10	0.727378	0.363689	−	0.931520i	$-0.381517\pi$
$827$	1.18312e10	0.727378	0.363689	+	0.931520i	$0.381517\pi$
$828$	0	0
$829$	7.67662e9	0.467982	0.233991	−	0.972239i	$-0.424821\pi$
$829$	7.67662e9	0.467982	0.233991	+	0.972239i	$0.424821\pi$
$830$	−7.49207e8	−0.0454809
$831$	0	0
$832$	−3.51901e9	−0.211830
$833$	−2.68582e10	−1.60997
$834$	0	0
$835$	1.93982e9	0.115308
$836$	−6.17966e9	−0.365799
$837$	0	0
$838$	−5.68039e9	−0.333445
$839$	9.41603e9	0.550429	0.275214	−	0.961383i	$-0.411251\pi$
$839$	9.41603e9	0.550429	0.275214	+	0.961383i	$0.411251\pi$
$840$	0	0
$841$	−1.72494e10	−0.999972
$842$	−8.41050e9	−0.485545
$843$	0	0
$844$	−4.05496e9	−0.232160
$845$	−3.62279e9	−0.206559
$846$	0	0
$847$	3.75027e7	0.00212066
$848$	−1.48019e10	−0.833550
$849$	0	0
$850$	8.76562e9	0.489572
$851$	1.16371e9	0.0647280
$852$	0	0
$853$	−1.58189e9	−0.0872680	−0.0436340	−	0.999048i	$-0.513894\pi$
$853$	−1.58189e9	−0.0872680	−0.0436340	+	0.999048i	$0.513894\pi$
$854$	2.56410e8	0.0140875
$855$	0	0
$856$	−1.97439e10	−1.07591
$857$	1.73709e10	0.942734	0.471367	−	0.881937i	$-0.343761\pi$
$857$	1.73709e10	0.942734	0.471367	+	0.881937i	$0.343761\pi$
$858$	0	0
$859$	8.52987e9	0.459163	0.229581	−	0.973289i	$-0.426264\pi$
$859$	8.52987e9	0.459163	0.229581	+	0.973289i	$0.426264\pi$
$860$	−2.15918e9	−0.115756
$861$	0	0
$862$	9.84467e9	0.523511
$863$	2.50924e10	1.32894	0.664470	−	0.747315i	$-0.268657\pi$
$863$	2.50924e10	1.32894	0.664470	+	0.747315i	$0.268657\pi$
$864$	0	0
$865$	−4.80834e9	−0.252603
$866$	−1.39777e9	−0.0731346
$867$	0	0
$868$	−3.08725e8	−0.0160233
$869$	−5.86920e9	−0.303396
$870$	0	0
$871$	−1.30903e10	−0.671250
$872$	−2.43983e9	−0.124610
$873$	0	0
$874$	−3.79559e8	−0.0192304
$875$	−2.54011e8	−0.0128181
$876$	0	0
$877$	1.68456e10	0.843314	0.421657	−	0.906756i	$-0.361449\pi$
$877$	1.68456e10	0.843314	0.421657	+	0.906756i	$0.361449\pi$
$878$	−7.65615e9	−0.381750
$879$	0	0
$880$	−1.19298e9	−0.0590125
$881$	1.17288e10	0.577881	0.288941	−	0.957347i	$-0.406697\pi$
$881$	1.17288e10	0.577881	0.288941	+	0.957347i	$0.406697\pi$
$882$	0	0
$883$	3.11743e10	1.52382	0.761911	−	0.647682i	$-0.224261\pi$
$883$	3.11743e10	1.52382	0.761911	+	0.647682i	$0.224261\pi$
$884$	−1.55612e10	−0.757635
$885$	0	0
$886$	−1.29134e10	−0.623769
$887$	−2.88046e10	−1.38589	−0.692944	−	0.720991i	$-0.743687\pi$
$887$	−2.88046e10	−1.38589	−0.692944	+	0.720991i	$0.743687\pi$
$888$	0	0
$889$	7.60265e8	0.0362918
$890$	−1.55339e9	−0.0738610
$891$	0	0
$892$	1.77332e10	0.836586
$893$	2.90512e10	1.36516
$894$	0	0
$895$	−1.46502e8	−0.00683065
$896$	4.94671e8	0.0229741
$897$	0	0
$898$	5.76492e9	0.265660
$899$	−8.81990e7	−0.00404860
$900$	0	0
$901$	−4.31535e10	−1.96553
$902$	−1.95275e9	−0.0885982
$903$	0	0
$904$	5.42733e8	0.0244341
$905$	−9.96809e9	−0.447035
$906$	0	0
$907$	−1.47721e10	−0.657381	−0.328690	−	0.944438i	$-0.606607\pi$
$907$	−1.47721e10	−0.657381	−0.328690	+	0.944438i	$0.606607\pi$
$908$	1.58681e10	0.703435
$909$	0	0
$910$	−2.65725e7	−0.00116893
$911$	−1.23248e10	−0.540089	−0.270044	−	0.962848i	$-0.587038\pi$
$911$	−1.23248e10	−0.540089	−0.270044	+	0.962848i	$0.587038\pi$
$912$	0	0
$913$	−3.32421e9	−0.144558
$914$	5.55859e9	0.240798
$915$	0	0
$916$	−2.88287e10	−1.23934
$917$	3.49980e8	0.0149882
$918$	0	0
$919$	2.59401e10	1.10247	0.551235	−	0.834350i	$-0.314157\pi$
$919$	2.59401e10	1.10247	0.551235	+	0.834350i	$0.314157\pi$
$920$	−1.80534e8	−0.00764367
$921$	0	0
$922$	−9.16562e9	−0.385127
$923$	−1.22701e10	−0.513621
$924$	0	0
$925$	3.35527e10	1.39390
$926$	−5.43248e9	−0.224833
$927$	0	0
$928$	1.08865e8	0.00447168
$929$	−2.78900e10	−1.14128	−0.570642	−	0.821199i	$-0.693306\pi$
$929$	−2.78900e10	−1.14128	−0.570642	+	0.821199i	$0.693306\pi$
$930$	0	0
$931$	3.35316e10	1.36185
$932$	1.14352e10	0.462690
$933$	0	0
$934$	5.54713e9	0.222769
$935$	−3.47802e9	−0.139153
$936$	0	0
$937$	9.94837e9	0.395060	0.197530	−	0.980297i	$-0.436708\pi$
$937$	9.94837e9	0.395060	0.197530	+	0.980297i	$0.436708\pi$
$938$	2.48075e8	0.00981462
$939$	0	0
$940$	6.50832e9	0.255577
$941$	2.18049e10	0.853082	0.426541	−	0.904468i	$-0.359732\pi$
$941$	2.18049e10	0.853082	0.426541	+	0.904468i	$0.359732\pi$
$942$	0	0
$943$	9.74127e8	0.0378290
$944$	2.64851e10	1.02471
$945$	0	0
$946$	1.17956e9	0.0453004
$947$	−1.74659e10	−0.668290	−0.334145	−	0.942522i	$-0.608448\pi$
$947$	−1.74659e10	−0.668290	−0.334145	+	0.942522i	$0.608448\pi$
$948$	0	0
$949$	−4.47441e9	−0.169943
$950$	−1.09436e10	−0.414122
$951$	0	0
$952$	6.26115e8	0.0235193
$953$	4.46489e10	1.67104	0.835518	−	0.549464i	$-0.185168\pi$
$953$	4.46489e10	1.67104	0.835518	+	0.549464i	$0.185168\pi$
$954$	0	0
$955$	6.21539e9	0.230917
$956$	8.58230e9	0.317688
$957$	0	0
$958$	−1.41183e10	−0.518804
$959$	−5.87304e8	−0.0215030
$960$	0	0
$961$	−1.11381e10	−0.404836
$962$	7.33388e9	0.265596
$963$	0	0
$964$	−1.43374e10	−0.515467
$965$	3.08511e9	0.110516
$966$	0	0
$967$	2.46485e10	0.876594	0.438297	−	0.898830i	$-0.355582\pi$
$967$	2.46485e10	0.876594	0.438297	+	0.898830i	$0.355582\pi$
$968$	1.60575e9	0.0569002
$969$	0	0
$970$	−3.96276e8	−0.0139411
$971$	1.77832e10	0.623365	0.311682	−	0.950186i	$-0.399108\pi$
$971$	1.77832e10	0.623365	0.311682	+	0.950186i	$0.399108\pi$
$972$	0	0
$973$	−1.06848e9	−0.0371853
$974$	−3.68964e9	−0.127946
$975$	0	0
$976$	−3.61904e10	−1.24600
$977$	−4.72834e10	−1.62210	−0.811051	−	0.584976i	$-0.801104\pi$
$977$	−4.72834e10	−1.62210	−0.811051	+	0.584976i	$0.801104\pi$
$978$	0	0
$979$	−6.89234e9	−0.234762
$980$	7.51207e9	0.254958
$981$	0	0
$982$	−1.33676e9	−0.0450468
$983$	4.34353e10	1.45850	0.729249	−	0.684249i	$-0.239870\pi$
$983$	4.34353e10	1.45850	0.729249	+	0.684249i	$0.239870\pi$
$984$	0	0
$985$	−1.19374e10	−0.397999
$986$	8.42500e7	0.00279898
$987$	0	0
$988$	1.94277e10	0.640873
$989$	−5.88422e8	−0.0193420
$990$	0	0
$991$	3.94776e9	0.128853	0.0644263	−	0.997922i	$-0.479478\pi$
$991$	3.94776e9	0.128853	0.0644263	+	0.997922i	$0.479478\pi$
$992$	−2.02113e10	−0.657360
$993$	0	0
$994$	2.32533e8	0.00750985
$995$	−5.16669e9	−0.166277
$996$	0	0
$997$	1.90391e10	0.608434	0.304217	−	0.952603i	$-0.401605\pi$
$997$	1.90391e10	0.608434	0.304217	+	0.952603i	$0.401605\pi$
$998$	8.45406e9	0.269221
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.8.a.c.1.1		2
3.2	odd	2		11.8.a.a.1.2	✓	2
12.11	even	2		176.8.a.d.1.2		2
15.14	odd	2		275.8.a.a.1.1		2
21.20	even	2		539.8.a.a.1.2		2
33.32	even	2		121.8.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.8.a.a.1.2	✓	2	3.2	odd	2
99.8.a.c.1.1		2	1.1	even	1	trivial
121.8.a.b.1.1		2	33.32	even	2
176.8.a.d.1.2		2	12.11	even	2
275.8.a.a.1.1		2	15.14	odd	2
539.8.a.a.1.2		2	21.20	even	2

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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q-3.74597 q^{2} -113.968 q^{4} +80.0807 q^{5} +21.1693 q^{7} +906.403 q^{8} -299.980 q^{10} -1331.00 q^{11} +4184.41 q^{13} -79.2994 q^{14} +11192.5 q^{16} +32630.7 q^{17} -40738.5 q^{19} -9126.61 q^{20} +4985.88 q^{22} -2487.20 q^{23} -71712.1 q^{25} -15674.7 q^{26} -2412.61 q^{28} -689.254 q^{29} +127963. q^{31} -157946. q^{32} -122234. q^{34} +1695.25 q^{35} -467881. q^{37} +152605. q^{38} +72585.4 q^{40} -391657. q^{41} +236581. q^{43} +151691. q^{44} +9316.96 q^{46} -713115. q^{47} -823095. q^{49} +268631. q^{50} -476888. q^{52} -1.32248e6 q^{53} -106587. q^{55} +19187.9 q^{56} +2581.92 q^{58} +2.36633e6 q^{59} -3.23345e6 q^{61} -479345. q^{62} -840980. q^{64} +335090. q^{65} -3.12834e6 q^{67} -3.71885e6 q^{68} -6350.35 q^{70} -2.93234e6 q^{71} -1.06930e6 q^{73} +1.75267e6 q^{74} +4.64287e6 q^{76} -28176.3 q^{77} +4.40962e6 q^{79} +896304. q^{80} +1.46713e6 q^{82} +2.49753e6 q^{83} +2.61309e6 q^{85} -886223. q^{86} -1.20642e6 q^{88} +5.17831e6 q^{89} +88580.9 q^{91} +283460. q^{92} +2.67130e6 q^{94} -3.26236e6 q^{95} +1.32101e6 q^{97} +3.08329e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8} + 4280 q^{10} - 2662 q^{11} + 344 q^{13} - 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 35280 q^{19} - 5240 q^{20} - 10648 q^{22} + 61486 q^{23}+ \cdots + 11738664 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 - 104 * q^4 + 470 * q^5 - 1228 * q^7 - 480 * q^8 + 4280 * q^10 - 2662 * q^11 + 344 * q^13 - 14752 * q^14 - 6368 * q^16 + 8468 * q^17 - 35280 * q^19 - 5240 * q^20 - 10648 * q^22 + 61486 * q^23 + 2200 * q^25 - 60784 * q^26 - 14864 * q^28 - 179040 * q^29 - 57166 * q^31 - 186752 * q^32 - 406048 * q^34 - 485380 * q^35 - 877698 * q^37 + 216720 * q^38 - 468000 * q^40 + 283616 * q^41 + 275484 * q^43 + 138424 * q^44 + 760744 * q^46 - 1662512 * q^47 - 86214 * q^49 + 1136800 * q^50 - 515168 * q^52 - 1616484 * q^53 - 625570 * q^55 + 1751040 * q^56 - 2092320 * q^58 + 2454130 * q^59 - 6019176 * q^61 - 2653864 * q^62 + 1068416 * q^64 - 1162360 * q^65 - 174698 * q^67 - 3959696 * q^68 - 5727520 * q^70 + 1151466 * q^71 + 885944 * q^73 - 3061032 * q^74 + 4697280 * q^76 + 1634468 * q^77 + 3801460 * q^79 - 5950880 * q^80 + 9398864 * q^82 + 2282916 * q^83 - 6808420 * q^85 - 429264 * q^86 + 638880 * q^88 + 13481970 * q^89 + 4885904 * q^91 + 921128 * q^92 - 8480288 * q^94 - 1134000 * q^95 - 68078 * q^97 + 11738664 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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