LMFDB - Embedded newform 99.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,4,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	99.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:	$5.84118909057$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-2.73205 q^{2} -0.535898 q^{4} -14.8564 q^{5} +3.07180 q^{7} +23.3205 q^{8} +O(q^{10})$	\(q-2.73205 q^{2} -0.535898 q^{4} -14.8564 q^{5} +3.07180 q^{7} +23.3205 q^{8} +40.5885 q^{10} +11.0000 q^{11} +5.35898 q^{13} -8.39230 q^{14} -59.4256 q^{16} +41.2154 q^{17} +139.923 q^{19} +7.96152 q^{20} -30.0526 q^{22} +111.354 q^{23} +95.7128 q^{25} -14.6410 q^{26} -1.64617 q^{28} +24.9948 q^{29} +31.4974 q^{31} -24.2102 q^{32} -112.603 q^{34} -45.6359 q^{35} +13.1436 q^{37} -382.277 q^{38} -346.459 q^{40} -261.072 q^{41} -57.7128 q^{43} -5.89488 q^{44} -304.224 q^{46} +343.846 q^{47} -333.564 q^{49} -261.492 q^{50} -2.87187 q^{52} +342.995 q^{53} -163.420 q^{55} +71.6359 q^{56} -68.2872 q^{58} -88.3693 q^{59} +738.697 q^{61} -86.0526 q^{62} +541.549 q^{64} -79.6152 q^{65} +342.359 q^{67} -22.0873 q^{68} +124.679 q^{70} +207.364 q^{71} -1010.60 q^{73} -35.9090 q^{74} -74.9845 q^{76} +33.7898 q^{77} +1294.23 q^{79} +882.851 q^{80} +713.261 q^{82} -441.846 q^{83} -612.313 q^{85} +157.674 q^{86} +256.526 q^{88} +1489.11 q^{89} +16.4617 q^{91} -59.6743 q^{92} -939.405 q^{94} -2078.75 q^{95} +1346.42 q^{97} +911.314 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8	$ 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 + 50 * q^10 + 22 * q^11 + 80 * q^13 + 4 * q^14 - 8 * q^16 + 124 * q^17 + 72 * q^19 - 88 * q^20 - 22 * q^22 + 98 * q^23 + 136 * q^25 + 40 * q^26 - 128 * q^28 - 144 * q^29 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 + 54 * q^37 - 432 * q^38 - 492 * q^40 - 536 * q^41 - 60 * q^43 - 88 * q^44 - 314 * q^46 + 272 * q^47 - 390 * q^49 - 232 * q^50 - 560 * q^52 + 492 * q^53 - 22 * q^55 - 120 * q^56 - 192 * q^58 - 634 * q^59 + 840 * q^61 - 134 * q^62 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 284 * q^70 + 678 * q^71 - 400 * q^73 - 6 * q^74 + 432 * q^76 + 220 * q^77 + 316 * q^79 + 1544 * q^80 + 512 * q^82 - 468 * q^83 + 452 * q^85 + 156 * q^86 + 132 * q^88 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 - 992 * q^94 - 2952 * q^95 + 2194 * q^97 + 870 * q^98

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 $

$2$	−2.73205	−0.965926	−0.482963	−	0.875641i	$-0.660439\pi$
$2$	−2.73205	−0.965926	−0.482963	+	0.875641i	$0.660439\pi$
$3$	0	0
$3$	0	0
$4$	−0.535898	−0.0669873
$5$	−14.8564	−1.32880	−0.664399	−	0.747378i	$-0.731312\pi$
$5$	−14.8564	−1.32880	−0.664399	+	0.747378i	$0.731312\pi$
$6$	0	0
$7$	3.07180	0.165861	0.0829307	−	0.996555i	$-0.473572\pi$
$7$	3.07180	0.165861	0.0829307	+	0.996555i	$0.473572\pi$
$8$	23.3205	1.03063
$9$	0	0
$10$	40.5885	1.28352
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	5.35898	0.114332	0.0571659	−	0.998365i	$-0.481794\pi$
$13$	5.35898	0.114332	0.0571659	+	0.998365i	$0.481794\pi$
$14$	−8.39230	−0.160210
$15$	0	0
$16$	−59.4256	−0.928525
$17$	41.2154	0.588012	0.294006	−	0.955804i	$-0.405011\pi$
$17$	41.2154	0.588012	0.294006	+	0.955804i	$0.405011\pi$
$18$	0	0
$19$	139.923	1.68950	0.844751	−	0.535159i	$-0.179748\pi$
$19$	139.923	1.68950	0.844751	+	0.535159i	$0.179748\pi$
$20$	7.96152	0.0890125
$21$	0	0
$22$	−30.0526	−0.291238
$23$	111.354	1.00952	0.504758	−	0.863261i	$-0.331582\pi$
$23$	111.354	1.00952	0.504758	+	0.863261i	$0.331582\pi$
$24$	0	0
$25$	95.7128	0.765703
$26$	−14.6410	−0.110436
$27$	0	0
$28$	−1.64617	−0.0111106
$29$	24.9948	0.160049	0.0800246	−	0.996793i	$-0.474500\pi$
$29$	24.9948	0.160049	0.0800246	+	0.996793i	$0.474500\pi$
$30$	0	0
$31$	31.4974	0.182487	0.0912436	−	0.995829i	$-0.470916\pi$
$31$	31.4974	0.182487	0.0912436	+	0.995829i	$0.470916\pi$
$32$	−24.2102	−0.133744
$33$	0	0
$34$	−112.603	−0.567976
$35$	−45.6359	−0.220396
$36$	0	0
$37$	13.1436	0.0583998	0.0291999	−	0.999574i	$-0.490704\pi$
$37$	13.1436	0.0583998	0.0291999	+	0.999574i	$0.490704\pi$
$38$	−382.277	−1.63193
$39$	0	0
$40$	−346.459	−1.36950
$41$	−261.072	−0.994453	−0.497226	−	0.867621i	$-0.665648\pi$
$41$	−261.072	−0.994453	−0.497226	+	0.867621i	$0.665648\pi$
$42$	0	0
$43$	−57.7128	−0.204677	−0.102339	−	0.994750i	$-0.532633\pi$
$43$	−57.7128	−0.204677	−0.102339	+	0.994750i	$0.532633\pi$
$44$	−5.89488	−0.0201974
$45$	0	0
$46$	−304.224	−0.975118
$47$	343.846	1.06713	0.533565	−	0.845759i	$-0.320852\pi$
$47$	343.846	1.06713	0.533565	+	0.845759i	$0.320852\pi$
$48$	0	0
$49$	−333.564	−0.972490
$50$	−261.492	−0.739612
$51$	0	0
$52$	−2.87187	−0.00765879
$53$	342.995	0.888943	0.444471	−	0.895793i	$-0.353392\pi$
$53$	342.995	0.888943	0.444471	+	0.895793i	$0.353392\pi$
$54$	0	0
$55$	−163.420	−0.400647
$56$	71.6359	0.170942
$57$	0	0
$58$	−68.2872	−0.154596
$59$	−88.3693	−0.194995	−0.0974975	−	0.995236i	$-0.531084\pi$
$59$	−88.3693	−0.194995	−0.0974975	+	0.995236i	$0.531084\pi$
$60$	0	0
$61$	738.697	1.55050	0.775250	−	0.631654i	$-0.217624\pi$
$61$	738.697	1.55050	0.775250	+	0.631654i	$0.217624\pi$
$62$	−86.0526	−0.176269
$63$	0	0
$64$	541.549	1.05771
$65$	−79.6152	−0.151924
$66$	0	0
$67$	342.359	0.624266	0.312133	−	0.950038i	$-0.398957\pi$
$67$	342.359	0.624266	0.312133	+	0.950038i	$0.398957\pi$
$68$	−22.0873	−0.0393893
$69$	0	0
$70$	124.679	0.212886
$71$	207.364	0.346614	0.173307	−	0.984868i	$-0.444555\pi$
$71$	207.364	0.346614	0.173307	+	0.984868i	$0.444555\pi$
$72$	0	0
$73$	−1010.60	−1.62030	−0.810149	−	0.586224i	$-0.800614\pi$
$73$	−1010.60	−1.62030	−0.810149	+	0.586224i	$0.800614\pi$
$74$	−35.9090	−0.0564099
$75$	0	0
$76$	−74.9845	−0.113175
$77$	33.7898	0.0500091
$78$	0	0
$79$	1294.23	1.84319	0.921593	−	0.388157i	$-0.126888\pi$
$79$	1294.23	1.84319	0.921593	+	0.388157i	$0.126888\pi$
$80$	882.851	1.23382
$81$	0	0
$82$	713.261	0.960568
$83$	−441.846	−0.584324	−0.292162	−	0.956369i	$-0.594375\pi$
$83$	−441.846	−0.584324	−0.292162	+	0.956369i	$0.594375\pi$
$84$	0	0
$85$	−612.313	−0.781349
$86$	157.674	0.197703
$87$	0	0
$88$	256.526	0.310747
$89$	1489.11	1.77355	0.886773	−	0.462205i	$-0.152942\pi$
$89$	1489.11	1.77355	0.886773	+	0.462205i	$0.152942\pi$
$90$	0	0
$91$	16.4617	0.0189633
$92$	−59.6743	−0.0676248
$93$	0	0
$94$	−939.405	−1.03077
$95$	−2078.75	−2.24501
$96$	0	0
$97$	1346.42	1.40936	0.704679	−	0.709526i	$-0.251091\pi$
$97$	1346.42	1.40936	0.704679	+	0.709526i	$0.251091\pi$
$98$	911.314	0.939353
$99$	0	0
$100$	−51.2923	−0.0512923
$101$	161.461	0.159069	0.0795347	−	0.996832i	$-0.474657\pi$
$101$	161.461	0.159069	0.0795347	+	0.996832i	$0.474657\pi$
$102$	0	0
$103$	−34.7592	−0.0332517	−0.0166259	−	0.999862i	$-0.505292\pi$
$103$	−34.7592	−0.0332517	−0.0166259	+	0.999862i	$0.505292\pi$
$104$	124.974	0.117834
$105$	0	0
$106$	−937.079	−0.858653
$107$	−832.179	−0.751867	−0.375934	−	0.926647i	$-0.622678\pi$
$107$	−832.179	−0.751867	−0.375934	+	0.926647i	$0.622678\pi$
$108$	0	0
$109$	1044.26	0.917629	0.458815	−	0.888532i	$-0.348274\pi$
$109$	1044.26	0.917629	0.458815	+	0.888532i	$0.348274\pi$
$110$	446.473	0.386996
$111$	0	0
$112$	−182.543	−0.154007
$113$	−295.082	−0.245654	−0.122827	−	0.992428i	$-0.539196\pi$
$113$	−295.082	−0.245654	−0.122827	+	0.992428i	$0.539196\pi$
$114$	0	0
$115$	−1654.32	−1.34144
$116$	−13.3947	−0.0107213
$117$	0	0
$118$	241.429	0.188351
$119$	126.605	0.0975285
$120$	0	0
$121$	121.000	0.0909091
$122$	−2018.16	−1.49767
$123$	0	0
$124$	−16.8794	−0.0122243
$125$	435.102	0.311334
$126$	0	0
$127$	−1317.60	−0.920618	−0.460309	−	0.887759i	$-0.652261\pi$
$127$	−1317.60	−0.920618	−0.460309	+	0.887759i	$0.652261\pi$
$128$	−1285.86	−0.887928
$129$	0	0
$130$	217.513	0.146747
$131$	1600.71	1.06759	0.533797	−	0.845612i	$-0.320765\pi$
$131$	1600.71	1.06759	0.533797	+	0.845612i	$0.320765\pi$
$132$	0	0
$133$	429.815	0.280223
$134$	−935.342	−0.602994
$135$	0	0
$136$	961.164	0.606023
$137$	−1611.68	−1.00507	−0.502536	−	0.864556i	$-0.667600\pi$
$137$	−1611.68	−1.00507	−0.502536	+	0.864556i	$0.667600\pi$
$138$	0	0
$139$	−31.8619	−0.0194424	−0.00972120	−	0.999953i	$-0.503094\pi$
$139$	−31.8619	−0.0194424	−0.00972120	+	0.999953i	$0.503094\pi$
$140$	24.4562	0.0147637
$141$	0	0
$142$	−566.529	−0.334803
$143$	58.9488	0.0344724
$144$	0	0
$145$	−371.334	−0.212673
$146$	2761.01	1.56509
$147$	0	0
$148$	−7.04363	−0.00391205
$149$	2428.34	1.33515	0.667576	−	0.744542i	$-0.267332\pi$
$149$	2428.34	1.33515	0.667576	+	0.744542i	$0.267332\pi$
$150$	0	0
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$152$	3263.08	1.74125
$153$	0	0
$154$	−92.3154	−0.0483051
$155$	−467.939	−0.242489
$156$	0	0
$157$	2475.94	1.25861	0.629305	−	0.777158i	$-0.283340\pi$
$157$	2475.94	1.25861	0.629305	+	0.777158i	$0.283340\pi$
$158$	−3535.89	−1.78038
$159$	0	0
$160$	359.677	0.177719
$161$	342.056	0.167440
$162$	0	0
$163$	−2725.11	−1.30949	−0.654745	−	0.755850i	$-0.727224\pi$
$163$	−2725.11	−1.30949	−0.654745	+	0.755850i	$0.727224\pi$
$164$	139.908	0.0666157
$165$	0	0
$166$	1207.15	0.564414
$167$	−2737.30	−1.26837	−0.634187	−	0.773180i	$-0.718665\pi$
$167$	−2737.30	−1.26837	−0.634187	+	0.773180i	$0.718665\pi$
$168$	0	0
$169$	−2168.28	−0.986928
$170$	1672.87	0.754725
$171$	0	0
$172$	30.9282	0.0137108
$173$	−2307.42	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$173$	−2307.42	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$174$	0	0
$175$	294.010	0.127001
$176$	−653.682	−0.279961
$177$	0	0
$178$	−4068.33	−1.71311
$179$	1312.15	0.547905	0.273953	−	0.961743i	$-0.411669\pi$
$179$	1312.15	0.547905	0.273953	+	0.961743i	$0.411669\pi$
$180$	0	0
$181$	−803.174	−0.329831	−0.164916	−	0.986308i	$-0.552735\pi$
$181$	−803.174	−0.329831	−0.164916	+	0.986308i	$0.552735\pi$
$182$	−44.9742	−0.0183171
$183$	0	0
$184$	2596.83	1.04044
$185$	−195.267	−0.0776015
$186$	0	0
$187$	453.369	0.177292
$188$	−184.267	−0.0714842
$189$	0	0
$190$	5679.26	2.16851
$191$	−1718.25	−0.650932	−0.325466	−	0.945554i	$-0.605521\pi$
$191$	−1718.25	−0.650932	−0.325466	+	0.945554i	$0.605521\pi$
$192$	0	0
$193$	1340.18	0.499837	0.249919	−	0.968267i	$-0.419596\pi$
$193$	1340.18	0.499837	0.249919	+	0.968267i	$0.419596\pi$
$194$	−3678.48	−1.36134
$195$	0	0
$196$	178.756	0.0651445
$197$	3518.33	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$197$	3518.33	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$198$	0	0
$199$	823.692	0.293417	0.146709	−	0.989180i	$-0.453132\pi$
$199$	823.692	0.293417	0.146709	+	0.989180i	$0.453132\pi$
$200$	2232.07	0.789156
$201$	0	0
$202$	−441.121	−0.153649
$203$	76.7791	0.0265460
$204$	0	0
$205$	3878.59	1.32143
$206$	94.9639	0.0321187
$207$	0	0
$208$	−318.461	−0.106160
$209$	1539.15	0.509404
$210$	0	0
$211$	−107.343	−0.0350228	−0.0175114	−	0.999847i	$-0.505574\pi$
$211$	−107.343	−0.0350228	−0.0175114	+	0.999847i	$0.505574\pi$
$212$	−183.810	−0.0595479
$213$	0	0
$214$	2273.56	0.726248
$215$	857.405	0.271975
$216$	0	0
$217$	96.7537	0.0302676
$218$	−2852.96	−0.886362
$219$	0	0
$220$	87.5768	0.0268383
$221$	220.873	0.0672285
$222$	0	0
$223$	−3933.68	−1.18125	−0.590625	−	0.806946i	$-0.701119\pi$
$223$	−3933.68	−1.18125	−0.590625	+	0.806946i	$0.701119\pi$
$224$	−74.3689	−0.0221830
$225$	0	0
$226$	806.178	0.237284
$227$	1771.90	0.518085	0.259042	−	0.965866i	$-0.416593\pi$
$227$	1771.90	0.518085	0.259042	+	0.965866i	$0.416593\pi$
$228$	0	0
$229$	1915.37	0.552713	0.276356	−	0.961055i	$-0.410873\pi$
$229$	1915.37	0.552713	0.276356	+	0.961055i	$0.410873\pi$
$230$	4519.68	1.29573
$231$	0	0
$232$	582.892	0.164952
$233$	−4396.32	−1.23610	−0.618052	−	0.786137i	$-0.712078\pi$
$233$	−4396.32	−1.23610	−0.618052	+	0.786137i	$0.712078\pi$
$234$	0	0
$235$	−5108.32	−1.41800
$236$	47.3570	0.0130622
$237$	0	0
$238$	−345.892	−0.0942053
$239$	4084.49	1.10546	0.552728	−	0.833362i	$-0.313587\pi$
$239$	4084.49	1.10546	0.552728	+	0.833362i	$0.313587\pi$
$240$	0	0
$241$	3908.58	1.04471	0.522353	−	0.852730i	$-0.325054\pi$
$241$	3908.58	1.04471	0.522353	+	0.852730i	$0.325054\pi$
$242$	−330.578	−0.0878114
$243$	0	0
$244$	−395.867	−0.103864
$245$	4955.56	1.29224
$246$	0	0
$247$	749.845	0.193164
$248$	734.536	0.188077
$249$	0	0
$250$	−1188.72	−0.300725
$251$	−1094.89	−0.275335	−0.137667	−	0.990479i	$-0.543960\pi$
$251$	−1094.89	−0.275335	−0.137667	+	0.990479i	$0.543960\pi$
$252$	0	0
$253$	1224.89	0.304381
$254$	3599.76	0.889249
$255$	0	0
$256$	−819.364	−0.200040
$257$	−783.179	−0.190091	−0.0950454	−	0.995473i	$-0.530300\pi$
$257$	−783.179	−0.190091	−0.0950454	+	0.995473i	$0.530300\pi$
$258$	0	0
$259$	40.3744	0.00968628
$260$	42.6657	0.0101770
$261$	0	0
$262$	−4373.23	−1.03122
$263$	−6180.06	−1.44897	−0.724484	−	0.689292i	$-0.757922\pi$
$263$	−6180.06	−1.44897	−0.724484	+	0.689292i	$0.757922\pi$
$264$	0	0
$265$	−5095.67	−1.18122
$266$	−1174.28	−0.270675
$267$	0	0
$268$	−183.470	−0.0418179
$269$	−986.965	−0.223704	−0.111852	−	0.993725i	$-0.535678\pi$
$269$	−986.965	−0.223704	−0.111852	+	0.993725i	$0.535678\pi$
$270$	0	0
$271$	4576.99	1.02595	0.512975	−	0.858404i	$-0.328543\pi$
$271$	4576.99	1.02595	0.512975	+	0.858404i	$0.328543\pi$
$272$	−2449.25	−0.545984
$273$	0	0
$274$	4403.18	0.970825
$275$	1052.84	0.230868
$276$	0	0
$277$	567.836	0.123169	0.0615847	−	0.998102i	$-0.480385\pi$
$277$	567.836	0.123169	0.0615847	+	0.998102i	$0.480385\pi$
$278$	87.0484	0.0187799
$279$	0	0
$280$	−1064.25	−0.227147
$281$	−5311.01	−1.12750	−0.563752	−	0.825944i	$-0.690643\pi$
$281$	−5311.01	−1.12750	−0.563752	+	0.825944i	$0.690643\pi$
$282$	0	0
$283$	−4728.44	−0.993204	−0.496602	−	0.867978i	$-0.665419\pi$
$283$	−4728.44	−0.993204	−0.496602	+	0.867978i	$0.665419\pi$
$284$	−111.126	−0.0232187
$285$	0	0
$286$	−161.051	−0.0332977
$287$	−801.960	−0.164941
$288$	0	0
$289$	−3214.29	−0.654242
$290$	1014.50	0.205426
$291$	0	0
$292$	541.579	0.108539
$293$	−2328.92	−0.464358	−0.232179	−	0.972673i	$-0.574585\pi$
$293$	−2328.92	−0.464358	−0.232179	+	0.972673i	$0.574585\pi$
$294$	0	0
$295$	1312.85	0.259109
$296$	306.515	0.0601886
$297$	0	0
$298$	−6634.36	−1.28966
$299$	596.743	0.115420
$300$	0	0
$301$	−177.282	−0.0339481
$302$	7039.61	1.34134
$303$	0	0
$304$	−8315.01	−1.56875
$305$	−10974.4	−2.06030
$306$	0	0
$307$	−1678.07	−0.311962	−0.155981	−	0.987760i	$-0.549854\pi$
$307$	−1678.07	−0.311962	−0.155981	+	0.987760i	$0.549854\pi$
$308$	−18.1079	−0.00334997
$309$	0	0
$310$	1278.43	0.234226
$311$	−3572.71	−0.651413	−0.325707	−	0.945471i	$-0.605602\pi$
$311$	−3572.71	−0.651413	−0.325707	+	0.945471i	$0.605602\pi$
$312$	0	0
$313$	7184.36	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$313$	7184.36	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$314$	−6764.40	−1.21572
$315$	0	0
$316$	−693.573	−0.123470
$317$	15.7077	0.00278306	0.00139153	−	0.999999i	$-0.499557\pi$
$317$	15.7077	0.00278306	0.00139153	+	0.999999i	$0.499557\pi$
$318$	0	0
$319$	274.943	0.0482566
$320$	−8045.47	−1.40549
$321$	0	0
$322$	−934.515	−0.161734
$323$	5766.98	0.993447
$324$	0	0
$325$	512.923	0.0875442
$326$	7445.13	1.26487
$327$	0	0
$328$	−6088.33	−1.02491
$329$	1056.23	0.176996
$330$	0	0
$331$	−1318.95	−0.219022	−0.109511	−	0.993986i	$-0.534928\pi$
$331$	−1318.95	−0.219022	−0.109511	+	0.993986i	$0.534928\pi$
$332$	236.785	0.0391423
$333$	0	0
$334$	7478.43	1.22515
$335$	−5086.22	−0.829523
$336$	0	0
$337$	−239.183	−0.0386621	−0.0193310	−	0.999813i	$-0.506154\pi$
$337$	−239.183	−0.0386621	−0.0193310	+	0.999813i	$0.506154\pi$
$338$	5923.85	0.953299
$339$	0	0
$340$	328.137	0.0523404
$341$	346.472	0.0550220
$342$	0	0
$343$	−2078.27	−0.327160
$344$	−1345.89	−0.210947
$345$	0	0
$346$	6303.98	0.979491
$347$	5862.79	0.907006	0.453503	−	0.891255i	$-0.350174\pi$
$347$	5862.79	0.907006	0.453503	+	0.891255i	$0.350174\pi$
$348$	0	0
$349$	3491.73	0.535553	0.267776	−	0.963481i	$-0.413711\pi$
$349$	3491.73	0.535553	0.267776	+	0.963481i	$0.413711\pi$
$350$	−803.251	−0.122673
$351$	0	0
$352$	−266.313	−0.0403253
$353$	10916.7	1.64600	0.822999	−	0.568043i	$-0.192299\pi$
$353$	10916.7	1.64600	0.822999	+	0.568043i	$0.192299\pi$
$354$	0	0
$355$	−3080.69	−0.460580
$356$	−798.013	−0.118805
$357$	0	0
$358$	−3584.87	−0.529236
$359$	11500.7	1.69077	0.845384	−	0.534160i	$-0.179372\pi$
$359$	11500.7	1.69077	0.845384	+	0.534160i	$0.179372\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	2194.31	0.318592
$363$	0	0
$364$	−8.82180	−0.00127030
$365$	15013.9	2.15305
$366$	0	0
$367$	6767.01	0.962493	0.481246	−	0.876585i	$-0.340184\pi$
$367$	6767.01	0.962493	0.481246	+	0.876585i	$0.340184\pi$
$368$	−6617.27	−0.937362
$369$	0	0
$370$	533.478	0.0749573
$371$	1053.61	0.147441
$372$	0	0
$373$	−5310.22	−0.737139	−0.368569	−	0.929600i	$-0.620152\pi$
$373$	−5310.22	−0.737139	−0.368569	+	0.929600i	$0.620152\pi$
$374$	−1238.63	−0.171251
$375$	0	0
$376$	8018.67	1.09982
$377$	133.947	0.0182987
$378$	0	0
$379$	−838.267	−0.113612	−0.0568059	−	0.998385i	$-0.518092\pi$
$379$	−838.267	−0.113612	−0.0568059	+	0.998385i	$0.518092\pi$
$380$	1114.00	0.150387
$381$	0	0
$382$	4694.34	0.628752
$383$	2832.16	0.377851	0.188925	−	0.981991i	$-0.439500\pi$
$383$	2832.16	0.377851	0.188925	+	0.981991i	$0.439500\pi$
$384$	0	0
$385$	−501.994	−0.0664520
$386$	−3661.45	−0.482806
$387$	0	0
$388$	−721.542	−0.0944091
$389$	−3111.25	−0.405519	−0.202759	−	0.979229i	$-0.564991\pi$
$389$	−3111.25	−0.405519	−0.202759	+	0.979229i	$0.564991\pi$
$390$	0	0
$391$	4589.49	0.593608
$392$	−7778.88	−1.00228
$393$	0	0
$394$	−9612.25	−1.22908
$395$	−19227.5	−2.44922
$396$	0	0
$397$	14208.7	1.79626	0.898131	−	0.439728i	$-0.144925\pi$
$397$	14208.7	1.79626	0.898131	+	0.439728i	$0.144925\pi$
$398$	−2250.37	−0.283419
$399$	0	0
$400$	−5687.79	−0.710974
$401$	6261.68	0.779784	0.389892	−	0.920861i	$-0.372512\pi$
$401$	6261.68	0.779784	0.389892	+	0.920861i	$0.372512\pi$
$402$	0	0
$403$	168.794	0.0208641
$404$	−86.5269	−0.0106556
$405$	0	0
$406$	−209.764	−0.0256415
$407$	144.580	0.0176082
$408$	0	0
$409$	−4192.50	−0.506860	−0.253430	−	0.967354i	$-0.581559\pi$
$409$	−4192.50	−0.506860	−0.253430	+	0.967354i	$0.581559\pi$
$410$	−10596.5	−1.27640
$411$	0	0
$412$	18.6274	0.00222744
$413$	−271.453	−0.0323421
$414$	0	0
$415$	6564.25	0.776448
$416$	−129.742	−0.0152912
$417$	0	0
$418$	−4205.05	−0.492047
$419$	9287.15	1.08283	0.541416	−	0.840755i	$-0.317888\pi$
$419$	9287.15	1.08283	0.541416	+	0.840755i	$0.317888\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	0.760923	+	0.648842i	$0.224746\pi$
$422$	293.267	0.0338294
$423$	0	0
$424$	7998.81	0.916172
$425$	3944.84	0.450242
$426$	0	0
$427$	2269.13	0.257168
$428$	445.964	0.0503656
$429$	0	0
$430$	−2342.47	−0.262707
$431$	−4909.67	−0.548701	−0.274351	−	0.961630i	$-0.588463\pi$
$431$	−4909.67	−0.548701	−0.274351	+	0.961630i	$0.588463\pi$
$432$	0	0
$433$	−11743.3	−1.30334	−0.651671	−	0.758502i	$-0.725932\pi$
$433$	−11743.3	−1.30334	−0.651671	+	0.758502i	$0.725932\pi$
$434$	−264.336	−0.0292363
$435$	0	0
$436$	−559.615	−0.0614695
$437$	15581.0	1.70558
$438$	0	0
$439$	−11824.2	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$439$	−11824.2	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$440$	−3811.05	−0.412920
$441$	0	0
$442$	−603.435	−0.0649377
$443$	−10102.1	−1.08344	−0.541722	−	0.840558i	$-0.682228\pi$
$443$	−10102.1	−1.08344	−0.541722	+	0.840558i	$0.682228\pi$
$444$	0	0
$445$	−22122.9	−2.35668
$446$	10747.0	1.14100
$447$	0	0
$448$	1663.53	0.175434
$449$	345.254	0.0362885	0.0181443	−	0.999835i	$-0.494224\pi$
$449$	345.254	0.0362885	0.0181443	+	0.999835i	$0.494224\pi$
$450$	0	0
$451$	−2871.79	−0.299839
$452$	158.134	0.0164557
$453$	0	0
$454$	−4840.93	−0.500431
$455$	−244.562	−0.0251983
$456$	0	0
$457$	−10567.1	−1.08164	−0.540821	−	0.841138i	$-0.681886\pi$
$457$	−10567.1	−1.08164	−0.540821	+	0.841138i	$0.681886\pi$
$458$	−5232.89	−0.533879
$459$	0	0
$460$	886.546	0.0898596
$461$	−4733.96	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$461$	−4733.96	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$462$	0	0
$463$	3431.20	0.344409	0.172204	−	0.985061i	$-0.444911\pi$
$463$	3431.20	0.344409	0.172204	+	0.985061i	$0.444911\pi$
$464$	−1485.33	−0.148610
$465$	0	0
$466$	12011.0	1.19399
$467$	−5116.96	−0.507034	−0.253517	−	0.967331i	$-0.581587\pi$
$467$	−5116.96	−0.507034	−0.253517	+	0.967331i	$0.581587\pi$
$468$	0	0
$469$	1051.66	0.103542
$470$	13956.2	1.36968
$471$	0	0
$472$	−2060.82	−0.200968
$473$	−634.841	−0.0617125
$474$	0	0
$475$	13392.4	1.29366
$476$	−67.8476	−0.00653317
$477$	0	0
$478$	−11159.0	−1.06779
$479$	−11566.9	−1.10335	−0.551675	−	0.834059i	$-0.686011\pi$
$479$	−11566.9	−1.10335	−0.551675	+	0.834059i	$0.686011\pi$
$480$	0	0
$481$	70.4363	0.00667696
$482$	−10678.4	−1.00911
$483$	0	0
$484$	−64.8437	−0.00608975
$485$	−20002.9	−1.87275
$486$	0	0
$487$	−18326.5	−1.70525	−0.852623	−	0.522527i	$-0.824990\pi$
$487$	−18326.5	−1.70525	−0.852623	+	0.522527i	$0.824990\pi$
$488$	17226.8	1.59799
$489$	0	0
$490$	−13538.9	−1.24821
$491$	7617.58	0.700156	0.350078	−	0.936721i	$-0.386155\pi$
$491$	7617.58	0.700156	0.350078	+	0.936721i	$0.386155\pi$
$492$	0	0
$493$	1030.17	0.0941108
$494$	−2048.62	−0.186582
$495$	0	0
$496$	−1871.75	−0.169444
$497$	636.980	0.0574899
$498$	0	0
$499$	12909.1	1.15810	0.579050	−	0.815292i	$-0.303424\pi$
$499$	12909.1	1.15810	0.579050	+	0.815292i	$0.303424\pi$
$500$	−233.171	−0.0208554
$501$	0	0
$502$	2991.30	0.265953
$503$	−10165.7	−0.901121	−0.450561	−	0.892746i	$-0.648776\pi$
$503$	−10165.7	−0.901121	−0.450561	+	0.892746i	$0.648776\pi$
$504$	0	0
$505$	−2398.74	−0.211371
$506$	−3346.47	−0.294009
$507$	0	0
$508$	706.102	0.0616697
$509$	−6449.93	−0.561666	−0.280833	−	0.959757i	$-0.590611\pi$
$509$	−6449.93	−0.561666	−0.280833	+	0.959757i	$0.590611\pi$
$510$	0	0
$511$	−3104.36	−0.268745
$512$	12525.4	1.08115
$513$	0	0
$514$	2139.68	0.183614
$515$	516.397	0.0441848
$516$	0	0
$517$	3782.31	0.321752
$518$	−110.305	−0.00935623
$519$	0	0
$520$	−1856.67	−0.156577
$521$	19327.4	1.62524	0.812620	−	0.582794i	$-0.198041\pi$
$521$	19327.4	1.62524	0.812620	+	0.582794i	$0.198041\pi$
$522$	0	0
$523$	6259.09	0.523310	0.261655	−	0.965161i	$-0.415732\pi$
$523$	6259.09	0.523310	0.261655	+	0.965161i	$0.415732\pi$
$524$	−857.819	−0.0715153
$525$	0	0
$526$	16884.2	1.39960
$527$	1298.18	0.107305
$528$	0	0
$529$	232.675	0.0191235
$530$	13921.6	1.14098
$531$	0	0
$532$	−230.337	−0.0187714
$533$	−1399.08	−0.113698
$534$	0	0
$535$	12363.2	0.999079
$536$	7983.99	0.643387
$537$	0	0
$538$	2696.44	0.216081
$539$	−3669.20	−0.293217
$540$	0	0
$541$	−14008.2	−1.11323	−0.556616	−	0.830770i	$-0.687900\pi$
$541$	−14008.2	−1.11323	−0.556616	+	0.830770i	$0.687900\pi$
$542$	−12504.6	−0.990991
$543$	0	0
$544$	−997.834	−0.0786430
$545$	−15513.9	−1.21934
$546$	0	0
$547$	−4949.45	−0.386879	−0.193440	−	0.981112i	$-0.561964\pi$
$547$	−4949.45	−0.386879	−0.193440	+	0.981112i	$0.561964\pi$
$548$	863.695	0.0673270
$549$	0	0
$550$	−2876.41	−0.223001
$551$	3497.35	0.270404
$552$	0	0
$553$	3975.60	0.305714
$554$	−1551.36	−0.118973
$555$	0	0
$556$	17.0748	0.00130239
$557$	3801.58	0.289188	0.144594	−	0.989491i	$-0.453812\pi$
$557$	3801.58	0.289188	0.144594	+	0.989491i	$0.453812\pi$
$558$	0	0
$559$	−309.282	−0.0234011
$560$	2711.94	0.204644
$561$	0	0
$562$	14510.0	1.08908
$563$	9900.11	0.741101	0.370551	−	0.928812i	$-0.379169\pi$
$563$	9900.11	0.741101	0.370551	+	0.928812i	$0.379169\pi$
$564$	0	0
$565$	4383.85	0.326425
$566$	12918.3	0.959361
$567$	0	0
$568$	4835.84	0.357231
$569$	−5329.16	−0.392636	−0.196318	−	0.980540i	$-0.562898\pi$
$569$	−5329.16	−0.392636	−0.196318	+	0.980540i	$0.562898\pi$
$570$	0	0
$571$	−16962.6	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$571$	−16962.6	−1.24319	−0.621597	+	0.783337i	$0.713516\pi$
$572$	−31.5906	−0.00230921
$573$	0	0
$574$	2190.99	0.159321
$575$	10658.0	0.772989
$576$	0	0
$577$	−15487.0	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$577$	−15487.0	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$578$	8781.61	0.631949
$579$	0	0
$580$	198.997	0.0142464
$581$	−1357.26	−0.0969169
$582$	0	0
$583$	3772.94	0.268026
$584$	−23567.7	−1.66993
$585$	0	0
$586$	6362.72	0.448535
$587$	−11084.2	−0.779373	−0.389686	−	0.920948i	$-0.627417\pi$
$587$	−11084.2	−0.779373	−0.389686	+	0.920948i	$0.627417\pi$
$588$	0	0
$589$	4407.22	0.308313
$590$	−3586.77	−0.250280
$591$	0	0
$592$	−781.066	−0.0542257
$593$	−4349.68	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$593$	−4349.68	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$594$	0	0
$595$	−1880.90	−0.129596
$596$	−1301.34	−0.0894382
$597$	0	0
$598$	−1630.33	−0.111487
$599$	−13183.9	−0.899299	−0.449650	−	0.893205i	$-0.648451\pi$
$599$	−13183.9	−0.899299	−0.449650	+	0.893205i	$0.648451\pi$
$600$	0	0
$601$	−18765.0	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$601$	−18765.0	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$602$	484.344	0.0327913
$603$	0	0
$604$	1380.84	0.0930223
$605$	−1797.63	−0.120800
$606$	0	0
$607$	21871.4	1.46249	0.731244	−	0.682116i	$-0.238940\pi$
$607$	21871.4	1.46249	0.731244	+	0.682116i	$0.238940\pi$
$608$	−3387.57	−0.225961
$609$	0	0
$610$	29982.6	1.99010
$611$	1842.67	0.122007
$612$	0	0
$613$	−3527.85	−0.232445	−0.116222	−	0.993223i	$-0.537079\pi$
$613$	−3527.85	−0.232445	−0.116222	+	0.993223i	$0.537079\pi$
$614$	4584.56	0.301332
$615$	0	0
$616$	787.994	0.0515409
$617$	22728.1	1.48298	0.741490	−	0.670963i	$-0.234119\pi$
$617$	22728.1	1.48298	0.741490	+	0.670963i	$0.234119\pi$
$618$	0	0
$619$	−21443.3	−1.39237	−0.696187	−	0.717861i	$-0.745121\pi$
$619$	−21443.3	−1.39237	−0.696187	+	0.717861i	$0.745121\pi$
$620$	250.767	0.0162437
$621$	0	0
$622$	9760.81	0.629217
$623$	4574.25	0.294163
$624$	0	0
$625$	−18428.2	−1.17940
$626$	−19628.0	−1.25319
$627$	0	0
$628$	−1326.85	−0.0843109
$629$	541.718	0.0343398
$630$	0	0
$631$	21532.0	1.35844	0.679219	−	0.733936i	$-0.262319\pi$
$631$	21532.0	1.35844	0.679219	+	0.733936i	$0.262319\pi$
$632$	30182.0	1.89964
$633$	0	0
$634$	−42.9141	−0.00268823
$635$	19574.9	1.22332
$636$	0	0
$637$	−1787.56	−0.111187
$638$	−751.159	−0.0466123
$639$	0	0
$640$	19103.2	1.17988
$641$	−20148.3	−1.24151	−0.620756	−	0.784004i	$-0.713174\pi$
$641$	−20148.3	−1.24151	−0.620756	+	0.784004i	$0.713174\pi$
$642$	0	0
$643$	28869.7	1.77062	0.885310	−	0.465000i	$-0.153946\pi$
$643$	28869.7	1.77062	0.885310	+	0.465000i	$0.153946\pi$
$644$	−183.307	−0.0112163
$645$	0	0
$646$	−15755.7	−0.959597
$647$	1590.02	0.0966155	0.0483077	−	0.998833i	$-0.484617\pi$
$647$	1590.02	0.0966155	0.0483077	+	0.998833i	$0.484617\pi$
$648$	0	0
$649$	−972.062	−0.0587932
$650$	−1401.33	−0.0845612
$651$	0	0
$652$	1460.38	0.0877192
$653$	−20028.1	−1.20024	−0.600122	−	0.799909i	$-0.704881\pi$
$653$	−20028.1	−1.20024	−0.600122	+	0.799909i	$0.704881\pi$
$654$	0	0
$655$	−23780.8	−1.41862
$656$	15514.4	0.923375
$657$	0	0
$658$	−2885.66	−0.170965
$659$	10520.7	0.621897	0.310948	−	0.950427i	$-0.399353\pi$
$659$	10520.7	0.621897	0.310948	+	0.950427i	$0.399353\pi$
$660$	0	0
$661$	3295.83	0.193938	0.0969690	−	0.995287i	$-0.469085\pi$
$661$	3295.83	0.193938	0.0969690	+	0.995287i	$0.469085\pi$
$662$	3603.45	0.211559
$663$	0	0
$664$	−10304.1	−0.602222
$665$	−6385.51	−0.372360
$666$	0	0
$667$	2783.27	0.161572
$668$	1466.91	0.0849649
$669$	0	0
$670$	13895.8	0.801257
$671$	8125.67	0.467493
$672$	0	0
$673$	−1187.64	−0.0680239	−0.0340119	−	0.999421i	$-0.510828\pi$
$673$	−1187.64	−0.0680239	−0.0340119	+	0.999421i	$0.510828\pi$
$674$	653.460	0.0373447
$675$	0	0
$676$	1161.98	0.0661117
$677$	−13221.4	−0.750574	−0.375287	−	0.926909i	$-0.622456\pi$
$677$	−13221.4	−0.750574	−0.375287	+	0.926909i	$0.622456\pi$
$678$	0	0
$679$	4135.91	0.233758
$680$	−14279.4	−0.805282
$681$	0	0
$682$	−946.578	−0.0531471
$683$	13831.4	0.774882	0.387441	−	0.921894i	$-0.373359\pi$
$683$	13831.4	0.774882	0.387441	+	0.921894i	$0.373359\pi$
$684$	0	0
$685$	23943.7	1.33554
$686$	5677.93	0.316012
$687$	0	0
$688$	3429.62	0.190048
$689$	1838.10	0.101635
$690$	0	0
$691$	−9817.07	−0.540462	−0.270231	−	0.962796i	$-0.587100\pi$
$691$	−9817.07	−0.540462	−0.270231	+	0.962796i	$0.587100\pi$
$692$	1236.54	0.0679280
$693$	0	0
$694$	−16017.4	−0.876101
$695$	473.354	0.0258350
$696$	0	0
$697$	−10760.2	−0.584750
$698$	−9539.58	−0.517304
$699$	0	0
$700$	−157.560	−0.00850742
$701$	−29949.8	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$701$	−29949.8	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$702$	0	0
$703$	1839.09	0.0986667
$704$	5957.03	0.318912
$705$	0	0
$706$	−29825.0	−1.58991
$707$	495.976	0.0263835
$708$	0	0
$709$	11307.5	0.598959	0.299479	−	0.954103i	$-0.403187\pi$
$709$	11307.5	0.598959	0.299479	+	0.954103i	$0.403187\pi$
$710$	8416.59	0.444886
$711$	0	0
$712$	34726.9	1.82787
$713$	3507.36	0.184224
$714$	0	0
$715$	−875.768	−0.0458068
$716$	−703.181	−0.0367027
$717$	0	0
$718$	−31420.6	−1.63316
$719$	32623.4	1.69214	0.846070	−	0.533071i	$-0.178962\pi$
$719$	32623.4	1.69214	0.846070	+	0.533071i	$0.178962\pi$
$720$	0	0
$721$	−106.773	−0.00551518
$722$	−34750.2	−1.79123
$723$	0	0
$724$	430.420	0.0220945
$725$	2392.33	0.122550
$726$	0	0
$727$	−502.545	−0.0256373	−0.0128187	−	0.999918i	$-0.504080\pi$
$727$	−502.545	−0.0256373	−0.0128187	+	0.999918i	$0.504080\pi$
$728$	383.895	0.0195441
$729$	0	0
$730$	−41018.7	−2.07968
$731$	−2378.66	−0.120353
$732$	0	0
$733$	8631.37	0.434935	0.217467	−	0.976068i	$-0.430220\pi$
$733$	8631.37	0.434935	0.217467	+	0.976068i	$0.430220\pi$
$734$	−18487.8	−0.929697
$735$	0	0
$736$	−2695.90	−0.135017
$737$	3765.95	0.188223
$738$	0	0
$739$	−18357.5	−0.913792	−0.456896	−	0.889520i	$-0.651039\pi$
$739$	−18357.5	−0.913792	−0.456896	+	0.889520i	$0.651039\pi$
$740$	104.643	0.00519832
$741$	0	0
$742$	−2878.52	−0.142417
$743$	−11182.6	−0.552155	−0.276078	−	0.961135i	$-0.589035\pi$
$743$	−11182.6	−0.552155	−0.276078	+	0.961135i	$0.589035\pi$
$744$	0	0
$745$	−36076.4	−1.77415
$746$	14507.8	0.712021
$747$	0	0
$748$	−242.960	−0.0118763
$749$	−2556.29	−0.124706
$750$	0	0
$751$	16733.4	0.813063	0.406531	−	0.913637i	$-0.366738\pi$
$751$	16733.4	0.813063	0.406531	+	0.913637i	$0.366738\pi$
$752$	−20433.3	−0.990857
$753$	0	0
$754$	−365.950	−0.0176752
$755$	38280.2	1.84524
$756$	0	0
$757$	−24402.4	−1.17163	−0.585813	−	0.810446i	$-0.699225\pi$
$757$	−24402.4	−1.17163	−0.585813	+	0.810446i	$0.699225\pi$
$758$	2290.19	0.109741
$759$	0	0
$760$	−48477.6	−2.31377
$761$	−8469.33	−0.403434	−0.201717	−	0.979444i	$-0.564652\pi$
$761$	−8469.33	−0.403434	−0.201717	+	0.979444i	$0.564652\pi$
$762$	0	0
$763$	3207.74	0.152199
$764$	920.805	0.0436042
$765$	0	0
$766$	−7737.62	−0.364976
$767$	−473.570	−0.0222941
$768$	0	0
$769$	32834.7	1.53973	0.769864	−	0.638208i	$-0.220324\pi$
$769$	32834.7	1.53973	0.769864	+	0.638208i	$0.220324\pi$
$770$	1371.47	0.0641877
$771$	0	0
$772$	−718.202	−0.0334827
$773$	35571.4	1.65513	0.827564	−	0.561371i	$-0.189726\pi$
$773$	35571.4	1.65513	0.827564	+	0.561371i	$0.189726\pi$
$774$	0	0
$775$	3014.71	0.139731
$776$	31399.1	1.45253
$777$	0	0
$778$	8500.11	0.391701
$779$	−36530.0	−1.68013
$780$	0	0
$781$	2281.01	0.104508
$782$	−12538.7	−0.573381
$783$	0	0
$784$	19822.3	0.902982
$785$	−36783.6	−1.67244
$786$	0	0
$787$	15729.6	0.712452	0.356226	−	0.934400i	$-0.384063\pi$
$787$	15729.6	0.712452	0.356226	+	0.934400i	$0.384063\pi$
$788$	−1885.47	−0.0852373
$789$	0	0
$790$	52530.6	2.36577
$791$	−906.431	−0.0407446
$792$	0	0
$793$	3958.67	0.177272
$794$	−38819.0	−1.73506
$795$	0	0
$796$	−441.415	−0.0196552
$797$	−7888.07	−0.350577	−0.175288	−	0.984517i	$-0.556086\pi$
$797$	−7888.07	−0.350577	−0.175288	+	0.984517i	$0.556086\pi$
$798$	0	0
$799$	14171.8	0.627485
$800$	−2317.23	−0.102408
$801$	0	0
$802$	−17107.2	−0.753214
$803$	−11116.6	−0.488538
$804$	0	0
$805$	−5081.73	−0.222494
$806$	−461.154	−0.0201532
$807$	0	0
$808$	3765.36	0.163942
$809$	−5896.97	−0.256275	−0.128138	−	0.991756i	$-0.540900\pi$
$809$	−5896.97	−0.256275	−0.128138	+	0.991756i	$0.540900\pi$
$810$	0	0
$811$	14197.9	0.614744	0.307372	−	0.951589i	$-0.400550\pi$
$811$	14197.9	0.614744	0.307372	+	0.951589i	$0.400550\pi$
$812$	−41.1458	−0.00177824
$813$	0	0
$814$	−394.999	−0.0170082
$815$	40485.3	1.74005
$816$	0	0
$817$	−8075.35	−0.345803
$818$	11454.1	0.489589
$819$	0	0
$820$	−2078.53	−0.0885188
$821$	19841.7	0.843459	0.421729	−	0.906722i	$-0.361423\pi$
$821$	19841.7	0.843459	0.421729	+	0.906722i	$0.361423\pi$
$822$	0	0
$823$	−28202.2	−1.19449	−0.597246	−	0.802058i	$-0.703738\pi$
$823$	−28202.2	−1.19449	−0.597246	+	0.802058i	$0.703738\pi$
$824$	−810.602	−0.0342702
$825$	0	0
$826$	741.622	0.0312401
$827$	−34031.0	−1.43092	−0.715462	−	0.698651i	$-0.753784\pi$
$827$	−34031.0	−1.43092	−0.715462	+	0.698651i	$0.753784\pi$
$828$	0	0
$829$	4931.55	0.206610	0.103305	−	0.994650i	$-0.467058\pi$
$829$	4931.55	0.206610	0.103305	+	0.994650i	$0.467058\pi$
$830$	−17933.9	−0.749992
$831$	0	0
$832$	2902.15	0.120930
$833$	−13748.0	−0.571836
$834$	0	0
$835$	40666.4	1.68541
$836$	−824.830	−0.0341236
$837$	0	0
$838$	−25373.0	−1.04594
$839$	38189.8	1.57146	0.785731	−	0.618568i	$-0.212287\pi$
$839$	38189.8	1.57146	0.785731	+	0.618568i	$0.212287\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	−35915.6	−1.46999
$843$	0	0
$844$	57.5250	0.00234608
$845$	32212.9	1.31143
$846$	0	0
$847$	371.687	0.0150783
$848$	−20382.7	−0.825406
$849$	0	0
$850$	−10777.5	−0.434900
$851$	1463.59	0.0589556
$852$	0	0
$853$	42966.8	1.72469	0.862343	−	0.506325i	$-0.168997\pi$
$853$	42966.8	1.72469	0.862343	+	0.506325i	$0.168997\pi$
$854$	−6199.37	−0.248405
$855$	0	0
$856$	−19406.8	−0.774898
$857$	17281.5	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$857$	17281.5	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$858$	0	0
$859$	9316.75	0.370062	0.185031	−	0.982733i	$-0.440761\pi$
$859$	9316.75	0.370062	0.185031	+	0.982733i	$0.440761\pi$
$860$	−459.482	−0.0182188
$861$	0	0
$862$	13413.5	0.530005
$863$	9647.65	0.380544	0.190272	−	0.981731i	$-0.439063\pi$
$863$	9647.65	0.380544	0.190272	+	0.981731i	$0.439063\pi$
$864$	0	0
$865$	34279.9	1.34746
$866$	32083.3	1.25893
$867$	0	0
$868$	−51.8501	−0.00202754
$869$	14236.5	0.555742
$870$	0	0
$871$	1834.70	0.0713735
$872$	24352.6	0.945737
$873$	0	0
$874$	−42568.0	−1.64746
$875$	1336.55	0.0516383
$876$	0	0
$877$	19728.7	0.759624	0.379812	−	0.925064i	$-0.375989\pi$
$877$	19728.7	0.759624	0.379812	+	0.925064i	$0.375989\pi$
$878$	32304.3	1.24171
$879$	0	0
$880$	9711.36	0.372011
$881$	−19473.9	−0.744712	−0.372356	−	0.928090i	$-0.621450\pi$
$881$	−19473.9	−0.744712	−0.372356	+	0.928090i	$0.621450\pi$
$882$	0	0
$883$	49092.4	1.87100	0.935499	−	0.353329i	$-0.114950\pi$
$883$	49092.4	1.87100	0.935499	+	0.353329i	$0.114950\pi$
$884$	−118.365	−0.00450346
$885$	0	0
$886$	27599.5	1.04653
$887$	−9292.86	−0.351774	−0.175887	−	0.984410i	$-0.556279\pi$
$887$	−9292.86	−0.351774	−0.175887	+	0.984410i	$0.556279\pi$
$888$	0	0
$889$	−4047.41	−0.152695
$890$	60440.8	2.27638
$891$	0	0
$892$	2108.05	0.0791288
$893$	48112.0	1.80292
$894$	0	0
$895$	−19493.9	−0.728055
$896$	−3949.89	−0.147273
$897$	0	0
$898$	−943.252	−0.0350520
$899$	787.273	0.0292069
$900$	0	0
$901$	14136.7	0.522709
$902$	7845.88	0.289622
$903$	0	0
$904$	−6881.46	−0.253179
$905$	11932.3	0.438279
$906$	0	0
$907$	37688.7	1.37975	0.689875	−	0.723928i	$-0.257665\pi$
$907$	37688.7	1.37975	0.689875	+	0.723928i	$0.257665\pi$
$908$	−949.559	−0.0347051
$909$	0	0
$910$	668.155	0.0243397
$911$	−33049.6	−1.20196	−0.600979	−	0.799265i	$-0.705222\pi$
$911$	−33049.6	−1.20196	−0.600979	+	0.799265i	$0.705222\pi$
$912$	0	0
$913$	−4860.31	−0.176180
$914$	28870.0	1.04479
$915$	0	0
$916$	−1026.44	−0.0370247
$917$	4917.06	0.177073
$918$	0	0
$919$	−23148.0	−0.830883	−0.415442	−	0.909620i	$-0.636373\pi$
$919$	−23148.0	−0.830883	−0.415442	+	0.909620i	$0.636373\pi$
$920$	−38579.5	−1.38253
$921$	0	0
$922$	12933.4	0.461973
$923$	1111.26	0.0396290
$924$	0	0
$925$	1258.01	0.0447169
$926$	−9374.21	−0.332673
$927$	0	0
$928$	−605.131	−0.0214056
$929$	23177.9	0.818561	0.409280	−	0.912409i	$-0.365780\pi$
$929$	23177.9	0.818561	0.409280	+	0.912409i	$0.365780\pi$
$930$	0	0
$931$	−46673.3	−1.64302
$932$	2355.98	0.0828033
$933$	0	0
$934$	13979.8	0.489757
$935$	−6735.44	−0.235585
$936$	0	0
$937$	−34574.7	−1.20545	−0.602724	−	0.797950i	$-0.705918\pi$
$937$	−34574.7	−1.20545	−0.602724	+	0.797950i	$0.705918\pi$
$938$	−2873.18	−0.100014
$939$	0	0
$940$	2737.54	0.0949880
$941$	−41831.2	−1.44916	−0.724578	−	0.689192i	$-0.757966\pi$
$941$	−41831.2	−1.44916	−0.724578	+	0.689192i	$0.757966\pi$
$942$	0	0
$943$	−29071.3	−1.00392
$944$	5251.40	0.181058
$945$	0	0
$946$	1734.42	0.0596097
$947$	−27231.2	−0.934419	−0.467209	−	0.884147i	$-0.654741\pi$
$947$	−27231.2	−0.934419	−0.467209	+	0.884147i	$0.654741\pi$
$948$	0	0
$949$	−5415.79	−0.185252
$950$	−36588.8	−1.24958
$951$	0	0
$952$	2952.50	0.100516
$953$	−40939.4	−1.39156	−0.695781	−	0.718254i	$-0.744942\pi$
$953$	−40939.4	−1.39156	−0.695781	+	0.718254i	$0.744942\pi$
$954$	0	0
$955$	25527.0	0.864956
$956$	−2188.87	−0.0740515
$957$	0	0
$958$	31601.3	1.06575
$959$	−4950.74	−0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	−192.436	−0.00644945
$963$	0	0
$964$	−2094.60	−0.0699820
$965$	−19910.3	−0.664182
$966$	0	0
$967$	−46173.1	−1.53550	−0.767750	−	0.640750i	$-0.778624\pi$
$967$	−46173.1	−1.53550	−0.767750	+	0.640750i	$0.778624\pi$
$968$	2821.78	0.0936937
$969$	0	0
$970$	54648.9	1.80894
$971$	5153.91	0.170337	0.0851683	−	0.996367i	$-0.472857\pi$
$971$	5153.91	0.170337	0.0851683	+	0.996367i	$0.472857\pi$
$972$	0	0
$973$	−97.8734	−0.00322474
$974$	50069.0	1.64714
$975$	0	0
$976$	−43897.6	−1.43968
$977$	−9692.13	−0.317378	−0.158689	−	0.987329i	$-0.550727\pi$
$977$	−9692.13	−0.317378	−0.158689	+	0.987329i	$0.550727\pi$
$978$	0	0
$979$	16380.2	0.534744
$980$	−2655.68	−0.0865638
$981$	0	0
$982$	−20811.6	−0.676299
$983$	−32915.7	−1.06800	−0.534002	−	0.845483i	$-0.679313\pi$
$983$	−32915.7	−1.06800	−0.534002	+	0.845483i	$0.679313\pi$
$984$	0	0
$985$	−52269.7	−1.69081
$986$	−2814.48	−0.0909041
$987$	0	0
$988$	−401.841	−0.0129395
$989$	−6426.54	−0.206625
$990$	0	0
$991$	29477.9	0.944901	0.472451	−	0.881357i	$-0.343370\pi$
$991$	29477.9	0.944901	0.472451	+	0.881357i	$0.343370\pi$
$992$	−762.560	−0.0244066
$993$	0	0
$994$	−1740.26	−0.0555310
$995$	−12237.1	−0.389892
$996$	0	0
$997$	−31944.4	−1.01473	−0.507366	−	0.861731i	$-0.669381\pi$
$997$	−31944.4	−1.01473	−0.507366	+	0.861731i	$0.669381\pi$
$998$	−35268.4	−1.11864
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.a.c.1.1		2
3.2	odd	2		11.4.a.a.1.2	✓	2
4.3	odd	2		1584.4.a.bc.1.1		2
5.4	even	2		2475.4.a.q.1.2		2
11.10	odd	2		1089.4.a.v.1.2		2
12.11	even	2		176.4.a.i.1.2		2
15.2	even	4		275.4.b.c.199.4		4
15.8	even	4		275.4.b.c.199.1		4
15.14	odd	2		275.4.a.b.1.1		2
21.20	even	2		539.4.a.e.1.2		2
24.5	odd	2		704.4.a.p.1.2		2
24.11	even	2		704.4.a.n.1.1		2
33.2	even	10		121.4.c.f.81.2		8
33.5	odd	10		121.4.c.c.3.1		8
33.8	even	10		121.4.c.f.9.1		8
33.14	odd	10		121.4.c.c.9.2		8
33.17	even	10		121.4.c.f.3.2		8
33.20	odd	10		121.4.c.c.81.1		8
33.26	odd	10		121.4.c.c.27.2		8
33.29	even	10		121.4.c.f.27.1		8
33.32	even	2		121.4.a.c.1.1		2
39.38	odd	2		1859.4.a.a.1.1		2
132.131	odd	2		1936.4.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	3.2	odd	2
99.4.a.c.1.1		2	1.1	even	1	trivial
121.4.a.c.1.1		2	33.32	even	2
121.4.c.c.3.1		8	33.5	odd	10
121.4.c.c.9.2		8	33.14	odd	10
121.4.c.c.27.2		8	33.26	odd	10
121.4.c.c.81.1		8	33.20	odd	10
121.4.c.f.3.2		8	33.17	even	10
121.4.c.f.9.1		8	33.8	even	10
121.4.c.f.27.1		8	33.29	even	10
121.4.c.f.81.2		8	33.2	even	10
176.4.a.i.1.2		2	12.11	even	2
275.4.a.b.1.1		2	15.14	odd	2
275.4.b.c.199.1		4	15.8	even	4
275.4.b.c.199.4		4	15.2	even	4
539.4.a.e.1.2		2	21.20	even	2
704.4.a.n.1.1		2	24.11	even	2
704.4.a.p.1.2		2	24.5	odd	2
1089.4.a.v.1.2		2	11.10	odd	2
1584.4.a.bc.1.1		2	4.3	odd	2
1859.4.a.a.1.1		2	39.38	odd	2
1936.4.a.w.1.2		2	132.131	odd	2
2475.4.a.q.1.2		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} -0.535898 q^{4} -14.8564 q^{5} +3.07180 q^{7} +23.3205 q^{8} +O(q^{10})\)	\(q-2.73205 q^{2} -0.535898 q^{4} -14.8564 q^{5} +3.07180 q^{7} +23.3205 q^{8} +40.5885 q^{10} +11.0000 q^{11} +5.35898 q^{13} -8.39230 q^{14} -59.4256 q^{16} +41.2154 q^{17} +139.923 q^{19} +7.96152 q^{20} -30.0526 q^{22} +111.354 q^{23} +95.7128 q^{25} -14.6410 q^{26} -1.64617 q^{28} +24.9948 q^{29} +31.4974 q^{31} -24.2102 q^{32} -112.603 q^{34} -45.6359 q^{35} +13.1436 q^{37} -382.277 q^{38} -346.459 q^{40} -261.072 q^{41} -57.7128 q^{43} -5.89488 q^{44} -304.224 q^{46} +343.846 q^{47} -333.564 q^{49} -261.492 q^{50} -2.87187 q^{52} +342.995 q^{53} -163.420 q^{55} +71.6359 q^{56} -68.2872 q^{58} -88.3693 q^{59} +738.697 q^{61} -86.0526 q^{62} +541.549 q^{64} -79.6152 q^{65} +342.359 q^{67} -22.0873 q^{68} +124.679 q^{70} +207.364 q^{71} -1010.60 q^{73} -35.9090 q^{74} -74.9845 q^{76} +33.7898 q^{77} +1294.23 q^{79} +882.851 q^{80} +713.261 q^{82} -441.846 q^{83} -612.313 q^{85} +157.674 q^{86} +256.526 q^{88} +1489.11 q^{89} +16.4617 q^{91} -59.6743 q^{92} -939.405 q^{94} -2078.75 q^{95} +1346.42 q^{97} +911.314 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8	\( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 + 50 * q^10 + 22 * q^11 + 80 * q^13 + 4 * q^14 - 8 * q^16 + 124 * q^17 + 72 * q^19 - 88 * q^20 - 22 * q^22 + 98 * q^23 + 136 * q^25 + 40 * q^26 - 128 * q^28 - 144 * q^29 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 + 54 * q^37 - 432 * q^38 - 492 * q^40 - 536 * q^41 - 60 * q^43 - 88 * q^44 - 314 * q^46 + 272 * q^47 - 390 * q^49 - 232 * q^50 - 560 * q^52 + 492 * q^53 - 22 * q^55 - 120 * q^56 - 192 * q^58 - 634 * q^59 + 840 * q^61 - 134 * q^62 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 284 * q^70 + 678 * q^71 - 400 * q^73 - 6 * q^74 + 432 * q^76 + 220 * q^77 + 316 * q^79 + 1544 * q^80 + 512 * q^82 - 468 * q^83 + 452 * q^85 + 156 * q^86 + 132 * q^88 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 - 992 * q^94 - 2952 * q^95 + 2194 * q^97 + 870 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more