LMFDB - Embedded newform 98.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,4,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	98.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:	$5.78218718056$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} -39.5980 q^{10} -14.0000 q^{11} +28.2843 q^{12} -50.9117 q^{13} +140.000 q^{15} +16.0000 q^{16} -1.41421 q^{17} -46.0000 q^{18} +1.41421 q^{19} +79.1960 q^{20} +28.0000 q^{22} +140.000 q^{23} -56.5685 q^{24} +267.000 q^{25} +101.823 q^{26} -28.2843 q^{27} -286.000 q^{29} -280.000 q^{30} +93.3381 q^{31} -32.0000 q^{32} -98.9949 q^{33} +2.82843 q^{34} +92.0000 q^{36} -38.0000 q^{37} -2.82843 q^{38} -360.000 q^{39} -158.392 q^{40} +125.865 q^{41} -34.0000 q^{43} -56.0000 q^{44} +455.377 q^{45} -280.000 q^{46} -523.259 q^{47} +113.137 q^{48} -534.000 q^{50} -10.0000 q^{51} -203.647 q^{52} -74.0000 q^{53} +56.5685 q^{54} -277.186 q^{55} +10.0000 q^{57} +572.000 q^{58} -434.164 q^{59} +560.000 q^{60} -14.1421 q^{61} -186.676 q^{62} +64.0000 q^{64} -1008.00 q^{65} +197.990 q^{66} +684.000 q^{67} -5.65685 q^{68} +989.949 q^{69} +588.000 q^{71} -184.000 q^{72} +270.115 q^{73} +76.0000 q^{74} +1887.98 q^{75} +5.65685 q^{76} +720.000 q^{78} +1220.00 q^{79} +316.784 q^{80} -821.000 q^{81} -251.730 q^{82} -422.850 q^{83} -28.0000 q^{85} +68.0000 q^{86} -2022.33 q^{87} +112.000 q^{88} -618.011 q^{89} -910.754 q^{90} +560.000 q^{92} +660.000 q^{93} +1046.52 q^{94} +28.0000 q^{95} -226.274 q^{96} -1483.51 q^{97} -322.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9} - 28 q^{11} + 280 q^{15} + 32 q^{16} - 92 q^{18} + 56 q^{22} + 280 q^{23} + 534 q^{25} - 572 q^{29} - 560 q^{30} - 64 q^{32} + 184 q^{36} - 76 q^{37} - 720 q^{39}+ \cdots - 644 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 46 * q^9 - 28 * q^11 + 280 * q^15 + 32 * q^16 - 92 * q^18 + 56 * q^22 + 280 * q^23 + 534 * q^25 - 572 * q^29 - 560 * q^30 - 64 * q^32 + 184 * q^36 - 76 * q^37 - 720 * q^39 - 68 * q^43 - 112 * q^44 - 560 * q^46 - 1068 * q^50 - 20 * q^51 - 148 * q^53 + 20 * q^57 + 1144 * q^58 + 1120 * q^60 + 128 * q^64 - 2016 * q^65 + 1368 * q^67 + 1176 * q^71 - 368 * q^72 + 152 * q^74 + 1440 * q^78 + 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 136 * q^86 + 224 * q^88 + 1120 * q^92 + 1320 * q^93 + 56 * q^95 - 644 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	7.07107	1.36083	0.680414	−	0.732828i	$-0.261800\pi$
$3$	7.07107	1.36083	0.680414	+	0.732828i	$0.261800\pi$
$4$	4.00000	0.500000
$5$	19.7990	1.77088	0.885438	−	0.464758i	$-0.153859\pi$
$5$	19.7990	1.77088	0.885438	+	0.464758i	$0.153859\pi$
$6$	−14.1421	−0.962250
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	23.0000	0.851852
$10$	−39.5980	−1.25220
$11$	−14.0000	−0.383742	−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000	−0.383742	−0.191871	+	0.981420i	$0.561455\pi$
$12$	28.2843	0.680414
$13$	−50.9117	−1.08618	−0.543091	−	0.839674i	$-0.682746\pi$
$13$	−50.9117	−1.08618	−0.543091	+	0.839674i	$0.682746\pi$
$14$	0	0
$15$	140.000	2.40986
$16$	16.0000	0.250000
$17$	−1.41421	−0.0201763	−0.0100882	−	0.999949i	$-0.503211\pi$
$17$	−1.41421	−0.0201763	−0.0100882	+	0.999949i	$0.503211\pi$
$18$	−46.0000	−0.602350
$19$	1.41421	0.0170759	0.00853797	−	0.999964i	$-0.497282\pi$
$19$	1.41421	0.0170759	0.00853797	+	0.999964i	$0.497282\pi$
$20$	79.1960	0.885438
$21$	0	0
$22$	28.0000	0.271346
$23$	140.000	1.26922	0.634609	−	0.772833i	$-0.281161\pi$
$23$	140.000	1.26922	0.634609	+	0.772833i	$0.281161\pi$
$24$	−56.5685	−0.481125
$25$	267.000	2.13600
$26$	101.823	0.768046
$27$	−28.2843	−0.201604
$28$	0	0
$29$	−286.000	−1.83134	−0.915670	−	0.401931i	$-0.868339\pi$
$29$	−286.000	−1.83134	−0.915670	+	0.401931i	$0.868339\pi$
$30$	−280.000	−1.70403
$31$	93.3381	0.540775	0.270387	−	0.962752i	$-0.412848\pi$
$31$	93.3381	0.540775	0.270387	+	0.962752i	$0.412848\pi$
$32$	−32.0000	−0.176777
$33$	−98.9949	−0.522206
$34$	2.82843	0.0142668
$35$	0	0
$36$	92.0000	0.425926
$37$	−38.0000	−0.168842	−0.0844211	−	0.996430i	$-0.526904\pi$
$37$	−38.0000	−0.168842	−0.0844211	+	0.996430i	$0.526904\pi$
$38$	−2.82843	−0.0120745
$39$	−360.000	−1.47811
$40$	−158.392	−0.626099
$41$	125.865	0.479434	0.239717	−	0.970843i	$-0.422945\pi$
$41$	125.865	0.479434	0.239717	+	0.970843i	$0.422945\pi$
$42$	0	0
$43$	−34.0000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$43$	−34.0000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$44$	−56.0000	−0.191871
$45$	455.377	1.50852
$46$	−280.000	−0.897473
$47$	−523.259	−1.62394	−0.811970	−	0.583699i	$-0.801605\pi$
$47$	−523.259	−1.62394	−0.811970	+	0.583699i	$0.801605\pi$
$48$	113.137	0.340207
$49$	0	0
$50$	−534.000	−1.51038
$51$	−10.0000	−0.0274565
$52$	−203.647	−0.543091
$53$	−74.0000	−0.191786	−0.0958932	−	0.995392i	$-0.530571\pi$
$53$	−74.0000	−0.191786	−0.0958932	+	0.995392i	$0.530571\pi$
$54$	56.5685	0.142556
$55$	−277.186	−0.679559
$56$	0	0
$57$	10.0000	0.0232374
$58$	572.000	1.29495
$59$	−434.164	−0.958022	−0.479011	−	0.877809i	$-0.659005\pi$
$59$	−434.164	−0.958022	−0.479011	+	0.877809i	$0.659005\pi$
$60$	560.000	1.20493
$61$	−14.1421	−0.0296839	−0.0148419	−	0.999890i	$-0.504725\pi$
$61$	−14.1421	−0.0296839	−0.0148419	+	0.999890i	$0.504725\pi$
$62$	−186.676	−0.382385
$63$	0	0
$64$	64.0000	0.125000
$65$	−1008.00	−1.92349
$66$	197.990	0.369256
$67$	684.000	1.24722	0.623611	−	0.781735i	$-0.285665\pi$
$67$	684.000	1.24722	0.623611	+	0.781735i	$0.285665\pi$
$68$	−5.65685	−0.0100882
$69$	989.949	1.72719
$70$	0	0
$71$	588.000	0.982856	0.491428	−	0.870918i	$-0.336475\pi$
$71$	588.000	0.982856	0.491428	+	0.870918i	$0.336475\pi$
$72$	−184.000	−0.301175
$73$	270.115	0.433076	0.216538	−	0.976274i	$-0.430523\pi$
$73$	270.115	0.433076	0.216538	+	0.976274i	$0.430523\pi$
$74$	76.0000	0.119389
$75$	1887.98	2.90673
$76$	5.65685	0.00853797
$77$	0	0
$78$	720.000	1.04518
$79$	1220.00	1.73748	0.868739	−	0.495271i	$-0.164931\pi$
$79$	1220.00	1.73748	0.868739	+	0.495271i	$0.164931\pi$
$80$	316.784	0.442719
$81$	−821.000	−1.12620
$82$	−251.730	−0.339011
$83$	−422.850	−0.559202	−0.279601	−	0.960116i	$-0.590202\pi$
$83$	−422.850	−0.559202	−0.279601	+	0.960116i	$0.590202\pi$
$84$	0	0
$85$	−28.0000	−0.0357297
$86$	68.0000	0.0852631
$87$	−2022.33	−2.49214
$88$	112.000	0.135673
$89$	−618.011	−0.736057	−0.368028	−	0.929815i	$-0.619967\pi$
$89$	−618.011	−0.736057	−0.368028	+	0.929815i	$0.619967\pi$
$90$	−910.754	−1.06669
$91$	0	0
$92$	560.000	0.634609
$93$	660.000	0.735901
$94$	1046.52	1.14830
$95$	28.0000	0.0302394
$96$	−226.274	−0.240563
$97$	−1483.51	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$97$	−1483.51	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$98$	0	0
$99$	−322.000	−0.326891
$100$	1068.00	1.06800
$101$	−1128.54	−1.11182	−0.555912	−	0.831241i	$-0.687631\pi$
$101$	−1128.54	−1.11182	−0.555912	+	0.831241i	$0.687631\pi$
$102$	20.0000	0.0194147
$103$	868.327	0.830668	0.415334	−	0.909669i	$-0.363665\pi$
$103$	868.327	0.830668	0.415334	+	0.909669i	$0.363665\pi$
$104$	407.294	0.384023
$105$	0	0
$106$	148.000	0.135613
$107$	−1684.00	−1.52148	−0.760740	−	0.649056i	$-0.775164\pi$
$107$	−1684.00	−1.52148	−0.760740	+	0.649056i	$0.775164\pi$
$108$	−113.137	−0.100802
$109$	−818.000	−0.718809	−0.359405	−	0.933182i	$-0.617020\pi$
$109$	−818.000	−0.718809	−0.359405	+	0.933182i	$0.617020\pi$
$110$	554.372	0.480521
$111$	−268.701	−0.229765
$112$	0	0
$113$	−540.000	−0.449548	−0.224774	−	0.974411i	$-0.572164\pi$
$113$	−540.000	−0.449548	−0.224774	+	0.974411i	$0.572164\pi$
$114$	−20.0000	−0.0164313
$115$	2771.86	2.24763
$116$	−1144.00	−0.915670
$117$	−1170.97	−0.925266
$118$	868.327	0.677424
$119$	0	0
$120$	−1120.00	−0.852013
$121$	−1135.00	−0.852742
$122$	28.2843	0.0209897
$123$	890.000	0.652428
$124$	373.352	0.270387
$125$	2811.46	2.01171
$126$	0	0
$127$	1720.00	1.20177	0.600887	−	0.799334i	$-0.294814\pi$
$127$	1720.00	1.20177	0.600887	+	0.799334i	$0.294814\pi$
$128$	−128.000	−0.0883883
$129$	−240.416	−0.164089
$130$	2016.00	1.36011
$131$	1735.24	1.15732	0.578659	−	0.815570i	$-0.303576\pi$
$131$	1735.24	1.15732	0.578659	+	0.815570i	$0.303576\pi$
$132$	−395.980	−0.261103
$133$	0	0
$134$	−1368.00	−0.881919
$135$	−560.000	−0.357016
$136$	11.3137	0.00713340
$137$	828.000	0.516356	0.258178	−	0.966097i	$-0.416878\pi$
$137$	828.000	0.516356	0.258178	+	0.966097i	$0.416878\pi$
$138$	−1979.90	−1.22131
$139$	425.678	0.259752	0.129876	−	0.991530i	$-0.458542\pi$
$139$	425.678	0.259752	0.129876	+	0.991530i	$0.458542\pi$
$140$	0	0
$141$	−3700.00	−2.20990
$142$	−1176.00	−0.694984
$143$	712.764	0.416813
$144$	368.000	0.212963
$145$	−5662.51	−3.24308
$146$	−540.230	−0.306231
$147$	0	0
$148$	−152.000	−0.0844211
$149$	2050.00	1.12713	0.563566	−	0.826071i	$-0.309429\pi$
$149$	2050.00	1.12713	0.563566	+	0.826071i	$0.309429\pi$
$150$	−3775.95	−2.05537
$151$	−472.000	−0.254376	−0.127188	−	0.991879i	$-0.540595\pi$
$151$	−472.000	−0.254376	−0.127188	+	0.991879i	$0.540595\pi$
$152$	−11.3137	−0.00603726
$153$	−32.5269	−0.0171872
$154$	0	0
$155$	1848.00	0.957645
$156$	−1440.00	−0.739053
$157$	2211.83	1.12435	0.562176	−	0.827018i	$-0.309964\pi$
$157$	2211.83	1.12435	0.562176	+	0.827018i	$0.309964\pi$
$158$	−2440.00	−1.22858
$159$	−523.259	−0.260988
$160$	−633.568	−0.313050
$161$	0	0
$162$	1642.00	0.796344
$163$	3286.00	1.57901	0.789507	−	0.613741i	$-0.210336\pi$
$163$	3286.00	1.57901	0.789507	+	0.613741i	$0.210336\pi$
$164$	503.460	0.239717
$165$	−1960.00	−0.924762
$166$	845.700	0.395416
$167$	1490.58	0.690686	0.345343	−	0.938476i	$-0.387763\pi$
$167$	1490.58	0.690686	0.345343	+	0.938476i	$0.387763\pi$
$168$	0	0
$169$	395.000	0.179791
$170$	56.0000	0.0252647
$171$	32.5269	0.0145462
$172$	−136.000	−0.0602901
$173$	2070.41	0.909886	0.454943	−	0.890521i	$-0.349660\pi$
$173$	2070.41	0.909886	0.454943	+	0.890521i	$0.349660\pi$
$174$	4044.65	1.76221
$175$	0	0
$176$	−224.000	−0.0959354
$177$	−3070.00	−1.30370
$178$	1236.02	0.520471
$179$	540.000	0.225483	0.112742	−	0.993624i	$-0.464037\pi$
$179$	540.000	0.225483	0.112742	+	0.993624i	$0.464037\pi$
$180$	1821.51	0.754262
$181$	−3784.44	−1.55412	−0.777058	−	0.629429i	$-0.783289\pi$
$181$	−3784.44	−1.55412	−0.777058	+	0.629429i	$0.783289\pi$
$182$	0	0
$183$	−100.000	−0.0403946
$184$	−1120.00	−0.448736
$185$	−752.362	−0.298999
$186$	−1320.00	−0.520361
$187$	19.7990	0.00774249
$188$	−2093.04	−0.811970
$189$	0	0
$190$	−56.0000	−0.0213825
$191$	1028.00	0.389442	0.194721	−	0.980859i	$-0.437620\pi$
$191$	1028.00	0.389442	0.194721	+	0.980859i	$0.437620\pi$
$192$	452.548	0.170103
$193$	4592.00	1.71264	0.856320	−	0.516446i	$-0.172745\pi$
$193$	4592.00	1.71264	0.856320	+	0.516446i	$0.172745\pi$
$194$	2967.02	1.09804
$195$	−7127.64	−2.61754
$196$	0	0
$197$	794.000	0.287158	0.143579	−	0.989639i	$-0.454139\pi$
$197$	794.000	0.287158	0.143579	+	0.989639i	$0.454139\pi$
$198$	644.000	0.231147
$199$	−2486.19	−0.885634	−0.442817	−	0.896612i	$-0.646021\pi$
$199$	−2486.19	−0.885634	−0.442817	+	0.896612i	$0.646021\pi$
$200$	−2136.00	−0.755190
$201$	4836.61	1.69725
$202$	2257.08	0.786178
$203$	0	0
$204$	−40.0000	−0.0137282
$205$	2492.00	0.849019
$206$	−1736.65	−0.587371
$207$	3220.00	1.08119
$208$	−814.587	−0.271545
$209$	−19.7990	−0.00655275
$210$	0	0
$211$	−2748.00	−0.896588	−0.448294	−	0.893886i	$-0.647968\pi$
$211$	−2748.00	−0.896588	−0.448294	+	0.893886i	$0.647968\pi$
$212$	−296.000	−0.0958932
$213$	4157.79	1.33750
$214$	3368.00	1.07585
$215$	−673.166	−0.213533
$216$	226.274	0.0712778
$217$	0	0
$218$	1636.00	0.508275
$219$	1910.00	0.589342
$220$	−1108.74	−0.339779
$221$	72.0000	0.0219151
$222$	537.401	0.162468
$223$	3428.05	1.02941	0.514707	−	0.857366i	$-0.327901\pi$
$223$	3428.05	1.02941	0.514707	+	0.857366i	$0.327901\pi$
$224$	0	0
$225$	6141.00	1.81956
$226$	1080.00	0.317878
$227$	−5290.57	−1.54691	−0.773453	−	0.633854i	$-0.781472\pi$
$227$	−5290.57	−1.54691	−0.773453	+	0.633854i	$0.781472\pi$
$228$	40.0000	0.0116187
$229$	2749.23	0.793338	0.396669	−	0.917962i	$-0.370166\pi$
$229$	2749.23	0.793338	0.396669	+	0.917962i	$0.370166\pi$
$230$	−5543.72	−1.58931
$231$	0	0
$232$	2288.00	0.647477
$233$	72.0000	0.0202441	0.0101221	−	0.999949i	$-0.496778\pi$
$233$	72.0000	0.0202441	0.0101221	+	0.999949i	$0.496778\pi$
$234$	2341.94	0.654262
$235$	−10360.0	−2.87580
$236$	−1736.65	−0.479011
$237$	8626.70	2.36441
$238$	0	0
$239$	4308.00	1.16595	0.582974	−	0.812491i	$-0.301889\pi$
$239$	4308.00	1.16595	0.582974	+	0.812491i	$0.301889\pi$
$240$	2240.00	0.602464
$241$	1540.08	0.411640	0.205820	−	0.978590i	$-0.434014\pi$
$241$	1540.08	0.411640	0.205820	+	0.978590i	$0.434014\pi$
$242$	2270.00	0.602980
$243$	−5041.67	−1.33096
$244$	−56.5685	−0.0148419
$245$	0	0
$246$	−1780.00	−0.461336
$247$	−72.0000	−0.0185476
$248$	−746.705	−0.191193
$249$	−2990.00	−0.760978
$250$	−5622.91	−1.42250
$251$	−931.967	−0.234363	−0.117182	−	0.993110i	$-0.537386\pi$
$251$	−931.967	−0.234363	−0.117182	+	0.993110i	$0.537386\pi$
$252$	0	0
$253$	−1960.00	−0.487052
$254$	−3440.00	−0.849783
$255$	−197.990	−0.0486220
$256$	256.000	0.0625000
$257$	−937.624	−0.227577	−0.113789	−	0.993505i	$-0.536299\pi$
$257$	−937.624	−0.227577	−0.113789	+	0.993505i	$0.536299\pi$
$258$	480.833	0.116028
$259$	0	0
$260$	−4032.00	−0.961746
$261$	−6578.00	−1.56003
$262$	−3470.48	−0.818347
$263$	7140.00	1.67404	0.837018	−	0.547176i	$-0.184297\pi$
$263$	7140.00	1.67404	0.837018	+	0.547176i	$0.184297\pi$
$264$	791.960	0.184628
$265$	−1465.13	−0.339630
$266$	0	0
$267$	−4370.00	−1.00165
$268$	2736.00	0.623611
$269$	−4610.34	−1.04497	−0.522485	−	0.852648i	$-0.674995\pi$
$269$	−4610.34	−1.04497	−0.522485	+	0.852648i	$0.674995\pi$
$270$	1120.00	0.252448
$271$	2364.57	0.530026	0.265013	−	0.964245i	$-0.414624\pi$
$271$	2364.57	0.530026	0.265013	+	0.964245i	$0.414624\pi$
$272$	−22.6274	−0.00504408
$273$	0	0
$274$	−1656.00	−0.365119
$275$	−3738.00	−0.819672
$276$	3959.80	0.863594
$277$	4006.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$277$	4006.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$278$	−851.357	−0.183673
$279$	2146.78	0.460660
$280$	0	0
$281$	−5984.00	−1.27038	−0.635188	−	0.772358i	$-0.719077\pi$
$281$	−5984.00	−1.27038	−0.635188	+	0.772358i	$0.719077\pi$
$282$	7400.00	1.56264
$283$	4928.53	1.03523	0.517617	−	0.855613i	$-0.326819\pi$
$283$	4928.53	1.03523	0.517617	+	0.855613i	$0.326819\pi$
$284$	2352.00	0.491428
$285$	197.990	0.0411506
$286$	−1425.53	−0.294731
$287$	0	0
$288$	−736.000	−0.150588
$289$	−4911.00	−0.999593
$290$	11325.0	2.29320
$291$	−10490.0	−2.11318
$292$	1080.46	0.216538
$293$	−1971.41	−0.393076	−0.196538	−	0.980496i	$-0.562970\pi$
$293$	−1971.41	−0.393076	−0.196538	+	0.980496i	$0.562970\pi$
$294$	0	0
$295$	−8596.00	−1.69654
$296$	304.000	0.0596947
$297$	395.980	0.0773639
$298$	−4100.00	−0.797002
$299$	−7127.64	−1.37860
$300$	7551.90	1.45336
$301$	0	0
$302$	944.000	0.179871
$303$	−7980.00	−1.51300
$304$	22.6274	0.00426898
$305$	−280.000	−0.0525664
$306$	65.0538	0.0121532
$307$	4767.31	0.886270	0.443135	−	0.896455i	$-0.353866\pi$
$307$	4767.31	0.886270	0.443135	+	0.896455i	$0.353866\pi$
$308$	0	0
$309$	6140.00	1.13040
$310$	−3696.00	−0.677157
$311$	6776.91	1.23564	0.617819	−	0.786320i	$-0.288016\pi$
$311$	6776.91	1.23564	0.617819	+	0.786320i	$0.288016\pi$
$312$	2880.00	0.522589
$313$	6190.01	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$313$	6190.01	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$314$	−4423.66	−0.795037
$315$	0	0
$316$	4880.00	0.868739
$317$	9826.00	1.74096	0.870478	−	0.492207i	$-0.163810\pi$
$317$	9826.00	1.74096	0.870478	+	0.492207i	$0.163810\pi$
$318$	1046.52	0.184547
$319$	4004.00	0.702762
$320$	1267.14	0.221359
$321$	−11907.7	−2.07047
$322$	0	0
$323$	−2.00000	−0.000344529 0
$324$	−3284.00	−0.563100
$325$	−13593.4	−2.32008
$326$	−6572.00	−1.11653
$327$	−5784.13	−0.978175
$328$	−1006.92	−0.169506
$329$	0	0
$330$	3920.00	0.653906
$331$	−5738.00	−0.952837	−0.476418	−	0.879219i	$-0.658065\pi$
$331$	−5738.00	−0.952837	−0.476418	+	0.879219i	$0.658065\pi$
$332$	−1691.40	−0.279601
$333$	−874.000	−0.143829
$334$	−2981.16	−0.488389
$335$	13542.5	2.20868
$336$	0	0
$337$	−2254.00	−0.364342	−0.182171	−	0.983267i	$-0.558312\pi$
$337$	−2254.00	−0.364342	−0.182171	+	0.983267i	$0.558312\pi$
$338$	−790.000	−0.127131
$339$	−3818.38	−0.611757
$340$	−112.000	−0.0178649
$341$	−1306.73	−0.207518
$342$	−65.0538	−0.0102857
$343$	0	0
$344$	272.000	0.0426316
$345$	19600.0	3.05863
$346$	−4140.82	−0.643386
$347$	−1986.00	−0.307245	−0.153623	−	0.988130i	$-0.549094\pi$
$347$	−1986.00	−0.307245	−0.153623	+	0.988130i	$0.549094\pi$
$348$	−8089.30	−1.24607
$349$	−6771.25	−1.03856	−0.519279	−	0.854605i	$-0.673800\pi$
$349$	−6771.25	−1.03856	−0.519279	+	0.854605i	$0.673800\pi$
$350$	0	0
$351$	1440.00	0.218979
$352$	448.000	0.0678366
$353$	−6993.29	−1.05443	−0.527217	−	0.849731i	$-0.676764\pi$
$353$	−6993.29	−1.05443	−0.527217	+	0.849731i	$0.676764\pi$
$354$	6140.00	0.921857
$355$	11641.8	1.74052
$356$	−2472.05	−0.368028
$357$	0	0
$358$	−1080.00	−0.159441
$359$	5944.00	0.873850	0.436925	−	0.899498i	$-0.356067\pi$
$359$	5944.00	0.873850	0.436925	+	0.899498i	$0.356067\pi$
$360$	−3643.01	−0.533344
$361$	−6857.00	−0.999708
$362$	7568.87	1.09893
$363$	−8025.66	−1.16044
$364$	0	0
$365$	5348.00	0.766924
$366$	200.000	0.0285633
$367$	842.871	0.119884	0.0599421	−	0.998202i	$-0.480908\pi$
$367$	842.871	0.119884	0.0599421	+	0.998202i	$0.480908\pi$
$368$	2240.00	0.317305
$369$	2894.90	0.408407
$370$	1504.72	0.211424
$371$	0	0
$372$	2640.00	0.367951
$373$	−5726.00	−0.794855	−0.397428	−	0.917634i	$-0.630097\pi$
$373$	−5726.00	−0.794855	−0.397428	+	0.917634i	$0.630097\pi$
$374$	−39.5980	−0.00547477
$375$	19880.0	2.73760
$376$	4186.07	0.574149
$377$	14560.7	1.98917
$378$	0	0
$379$	10330.0	1.40004	0.700022	−	0.714122i	$-0.253174\pi$
$379$	10330.0	1.40004	0.700022	+	0.714122i	$0.253174\pi$
$380$	112.000	0.0151197
$381$	12162.2	1.63541
$382$	−2056.00	−0.275377
$383$	−1004.09	−0.133960	−0.0669800	−	0.997754i	$-0.521336\pi$
$383$	−1004.09	−0.133960	−0.0669800	+	0.997754i	$0.521336\pi$
$384$	−905.097	−0.120281
$385$	0	0
$386$	−9184.00	−1.21102
$387$	−782.000	−0.102717
$388$	−5934.04	−0.776431
$389$	5210.00	0.679068	0.339534	−	0.940594i	$-0.389731\pi$
$389$	5210.00	0.679068	0.339534	+	0.940594i	$0.389731\pi$
$390$	14255.3	1.85088
$391$	−197.990	−0.0256081
$392$	0	0
$393$	12270.0	1.57491
$394$	−1588.00	−0.203051
$395$	24154.8	3.07686
$396$	−1288.00	−0.163446
$397$	73.5391	0.00929678	0.00464839	−	0.999989i	$-0.498520\pi$
$397$	73.5391	0.00929678	0.00464839	+	0.999989i	$0.498520\pi$
$398$	4972.37	0.626238
$399$	0	0
$400$	4272.00	0.534000
$401$	−498.000	−0.0620173	−0.0310086	−	0.999519i	$-0.509872\pi$
$401$	−498.000	−0.0620173	−0.0310086	+	0.999519i	$0.509872\pi$
$402$	−9673.22	−1.20014
$403$	−4752.00	−0.587380
$404$	−4514.17	−0.555912
$405$	−16255.0	−1.99436
$406$	0	0
$407$	532.000	0.0647918
$408$	80.0000	0.00970733
$409$	−3355.93	−0.405721	−0.202861	−	0.979208i	$-0.565024\pi$
$409$	−3355.93	−0.405721	−0.202861	+	0.979208i	$0.565024\pi$
$410$	−4984.00	−0.600347
$411$	5854.84	0.702672
$412$	3473.31	0.415334
$413$	0	0
$414$	−6440.00	−0.764514
$415$	−8372.00	−0.990278
$416$	1629.17	0.192012
$417$	3010.00	0.353478
$418$	39.5980	0.00463349
$419$	−14545.2	−1.69589	−0.847946	−	0.530082i	$-0.822161\pi$
$419$	−14545.2	−1.69589	−0.847946	+	0.530082i	$0.822161\pi$
$420$	0	0
$421$	10854.0	1.25651	0.628256	−	0.778007i	$-0.283769\pi$
$421$	10854.0	1.25651	0.628256	+	0.778007i	$0.283769\pi$
$422$	5496.00	0.633984
$423$	−12035.0	−1.38336
$424$	592.000	0.0678067
$425$	−377.595	−0.0430966
$426$	−8315.58	−0.945753
$427$	0	0
$428$	−6736.00	−0.760740
$429$	5040.00	0.567211
$430$	1346.33	0.150990
$431$	−5364.00	−0.599477	−0.299739	−	0.954021i	$-0.596900\pi$
$431$	−5364.00	−0.599477	−0.299739	+	0.954021i	$0.596900\pi$
$432$	−452.548	−0.0504010
$433$	6487.00	0.719966	0.359983	−	0.932959i	$-0.382783\pi$
$433$	6487.00	0.719966	0.359983	+	0.932959i	$0.382783\pi$
$434$	0	0
$435$	−40040.0	−4.41327
$436$	−3272.00	−0.359405
$437$	197.990	0.0216731
$438$	−3820.00	−0.416728
$439$	−13932.8	−1.51476	−0.757378	−	0.652977i	$-0.773520\pi$
$439$	−13932.8	−1.51476	−0.757378	+	0.652977i	$0.773520\pi$
$440$	2217.49	0.240260
$441$	0	0
$442$	−144.000	−0.0154963
$443$	5996.00	0.643067	0.321533	−	0.946898i	$-0.395802\pi$
$443$	5996.00	0.643067	0.321533	+	0.946898i	$0.395802\pi$
$444$	−1074.80	−0.114883
$445$	−12236.0	−1.30347
$446$	−6856.11	−0.727906
$447$	14495.7	1.53383
$448$	0	0
$449$	2622.00	0.275590	0.137795	−	0.990461i	$-0.455999\pi$
$449$	2622.00	0.275590	0.137795	+	0.990461i	$0.455999\pi$
$450$	−12282.0	−1.28662
$451$	−1762.11	−0.183979
$452$	−2160.00	−0.224774
$453$	−3337.54	−0.346162
$454$	10581.1	1.09383
$455$	0	0
$456$	−80.0000	−0.00821567
$457$	11208.0	1.14724	0.573619	−	0.819122i	$-0.305539\pi$
$457$	11208.0	1.14724	0.573619	+	0.819122i	$0.305539\pi$
$458$	−5498.46	−0.560974
$459$	40.0000	0.00406763
$460$	11087.4	1.12381
$461$	−9786.36	−0.988712	−0.494356	−	0.869260i	$-0.664596\pi$
$461$	−9786.36	−0.988712	−0.494356	+	0.869260i	$0.664596\pi$
$462$	0	0
$463$	3952.00	0.396685	0.198342	−	0.980133i	$-0.436444\pi$
$463$	3952.00	0.396685	0.198342	+	0.980133i	$0.436444\pi$
$464$	−4576.00	−0.457835
$465$	13067.3	1.30319
$466$	−144.000	−0.0143147
$467$	−17506.5	−1.73470	−0.867352	−	0.497696i	$-0.834180\pi$
$467$	−17506.5	−1.73470	−0.867352	+	0.497696i	$0.834180\pi$
$468$	−4683.88	−0.462633
$469$	0	0
$470$	20720.0	2.03349
$471$	15640.0	1.53005
$472$	3473.31	0.338712
$473$	476.000	0.0462717
$474$	−17253.4	−1.67189
$475$	377.595	0.0364742
$476$	0	0
$477$	−1702.00	−0.163374
$478$	−8616.00	−0.824449
$479$	−2288.20	−0.218268	−0.109134	−	0.994027i	$-0.534808\pi$
$479$	−2288.20	−0.218268	−0.109134	+	0.994027i	$0.534808\pi$
$480$	−4480.00	−0.426006
$481$	1934.64	0.183393
$482$	−3080.16	−0.291073
$483$	0	0
$484$	−4540.00	−0.426371
$485$	−29372.0	−2.74993
$486$	10083.3	0.941131
$487$	972.000	0.0904426	0.0452213	−	0.998977i	$-0.485601\pi$
$487$	972.000	0.0904426	0.0452213	+	0.998977i	$0.485601\pi$
$488$	113.137	0.0104948
$489$	23235.5	2.14877
$490$	0	0
$491$	−7404.00	−0.680525	−0.340263	−	0.940330i	$-0.610516\pi$
$491$	−7404.00	−0.680525	−0.340263	+	0.940330i	$0.610516\pi$
$492$	3560.00	0.326214
$493$	404.465	0.0369497
$494$	144.000	0.0131151
$495$	−6375.27	−0.578883
$496$	1493.41	0.135194
$497$	0	0
$498$	5980.00	0.538093
$499$	−12244.0	−1.09843	−0.549215	−	0.835681i	$-0.685073\pi$
$499$	−12244.0	−1.09843	−0.549215	+	0.835681i	$0.685073\pi$
$500$	11245.8	1.00586
$501$	10540.0	0.939905
$502$	1863.93	0.165720
$503$	2415.48	0.214117	0.107058	−	0.994253i	$-0.465857\pi$
$503$	2415.48	0.214117	0.107058	+	0.994253i	$0.465857\pi$
$504$	0	0
$505$	−22344.0	−1.96890
$506$	3920.00	0.344398
$507$	2793.07	0.244664
$508$	6880.00	0.600887
$509$	5707.77	0.497038	0.248519	−	0.968627i	$-0.420056\pi$
$509$	5707.77	0.497038	0.248519	+	0.968627i	$0.420056\pi$
$510$	395.980	0.0343809
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−40.0000	−0.00344258
$514$	1875.25	0.160921
$515$	17192.0	1.47101
$516$	−961.665	−0.0820445
$517$	7325.63	0.623173
$518$	0	0
$519$	14640.0	1.23820
$520$	8064.00	0.680057
$521$	−1.41421	−0.000118921 0	−5.94605e−5	−	1.00000i	$-0.500019\pi$
$521$	−1.41421	−0.000118921 0	−5.94605e−5		1.00000i	$0.500019\pi$
$522$	13156.0	1.10311
$523$	12257.0	1.02478	0.512391	−	0.858752i	$-0.328760\pi$
$523$	12257.0	1.02478	0.512391	+	0.858752i	$0.328760\pi$
$524$	6940.96	0.578659
$525$	0	0
$526$	−14280.0	−1.18372
$527$	−132.000	−0.0109108
$528$	−1583.92	−0.130552
$529$	7433.00	0.610915
$530$	2930.25	0.240155
$531$	−9985.76	−0.816093
$532$	0	0
$533$	−6408.00	−0.520753
$534$	8740.00	0.708271
$535$	−33341.5	−2.69435
$536$	−5472.00	−0.440960
$537$	3818.38	0.306844
$538$	9220.67	0.738906
$539$	0	0
$540$	−2240.00	−0.178508
$541$	2050.00	0.162914	0.0814569	−	0.996677i	$-0.474043\pi$
$541$	2050.00	0.162914	0.0814569	+	0.996677i	$0.474043\pi$
$542$	−4729.13	−0.374785
$543$	−26760.0	−2.11488
$544$	45.2548	0.00356670
$545$	−16195.6	−1.27292
$546$	0	0
$547$	14554.0	1.13763	0.568815	−	0.822465i	$-0.307402\pi$
$547$	14554.0	1.13763	0.568815	+	0.822465i	$0.307402\pi$
$548$	3312.00	0.258178
$549$	−325.269	−0.0252862
$550$	7476.00	0.579596
$551$	−404.465	−0.0312719
$552$	−7919.60	−0.610653
$553$	0	0
$554$	−8012.00	−0.614435
$555$	−5320.00	−0.406885
$556$	1702.71	0.129876
$557$	6954.00	0.528995	0.264498	−	0.964386i	$-0.414794\pi$
$557$	6954.00	0.528995	0.264498	+	0.964386i	$0.414794\pi$
$558$	−4293.55	−0.325736
$559$	1731.00	0.130972
$560$	0	0
$561$	140.000	0.0105362
$562$	11968.0	0.898291
$563$	1636.25	0.122486	0.0612429	−	0.998123i	$-0.480494\pi$
$563$	1636.25	0.122486	0.0612429	+	0.998123i	$0.480494\pi$
$564$	−14800.0	−1.10495
$565$	−10691.5	−0.796094
$566$	−9857.07	−0.732020
$567$	0	0
$568$	−4704.00	−0.347492
$569$	−7142.00	−0.526201	−0.263100	−	0.964768i	$-0.584745\pi$
$569$	−7142.00	−0.526201	−0.263100	+	0.964768i	$0.584745\pi$
$570$	−395.980	−0.0290978
$571$	−20606.0	−1.51022	−0.755109	−	0.655599i	$-0.772416\pi$
$571$	−20606.0	−1.51022	−0.755109	+	0.655599i	$0.772416\pi$
$572$	2851.05	0.208407
$573$	7269.06	0.529964
$574$	0	0
$575$	37380.0	2.71105
$576$	1472.00	0.106481
$577$	8803.48	0.635171	0.317585	−	0.948230i	$-0.397128\pi$
$577$	8803.48	0.635171	0.317585	+	0.948230i	$0.397128\pi$
$578$	9822.00	0.706819
$579$	32470.3	2.33061
$580$	−22650.0	−1.62154
$581$	0	0
$582$	20980.0	1.49424
$583$	1036.00	0.0735965
$584$	−2160.92	−0.153115
$585$	−23184.0	−1.63853
$586$	3942.83	0.277947
$587$	−6503.97	−0.457321	−0.228661	−	0.973506i	$-0.573435\pi$
$587$	−6503.97	−0.457321	−0.228661	+	0.973506i	$0.573435\pi$
$588$	0	0
$589$	132.000	0.00923424
$590$	17192.0	1.19963
$591$	5614.43	0.390773
$592$	−608.000	−0.0422106
$593$	23140.8	1.60249	0.801246	−	0.598335i	$-0.204171\pi$
$593$	23140.8	1.60249	0.801246	+	0.598335i	$0.204171\pi$
$594$	−791.960	−0.0547045
$595$	0	0
$596$	8200.00	0.563566
$597$	−17580.0	−1.20520
$598$	14255.3	0.974818
$599$	−11296.0	−0.770521	−0.385260	−	0.922808i	$-0.625888\pi$
$599$	−11296.0	−0.770521	−0.385260	+	0.922808i	$0.625888\pi$
$600$	−15103.8	−1.02768
$601$	−8727.11	−0.592323	−0.296162	−	0.955138i	$-0.595707\pi$
$601$	−8727.11	−0.592323	−0.296162	+	0.955138i	$0.595707\pi$
$602$	0	0
$603$	15732.0	1.06245
$604$	−1888.00	−0.127188
$605$	−22471.9	−1.51010
$606$	15960.0	1.06985
$607$	−19736.8	−1.31975	−0.659877	−	0.751374i	$-0.729392\pi$
$607$	−19736.8	−1.31975	−0.659877	+	0.751374i	$0.729392\pi$
$608$	−45.2548	−0.00301863
$609$	0	0
$610$	560.000	0.0371701
$611$	26640.0	1.76389
$612$	−130.108	−0.00859361
$613$	16962.0	1.11760	0.558800	−	0.829302i	$-0.311262\pi$
$613$	16962.0	1.11760	0.558800	+	0.829302i	$0.311262\pi$
$614$	−9534.63	−0.626688
$615$	17621.1	1.15537
$616$	0	0
$617$	−19034.0	−1.24194	−0.620972	−	0.783832i	$-0.713262\pi$
$617$	−19034.0	−1.24194	−0.620972	+	0.783832i	$0.713262\pi$
$618$	−12280.0	−0.799311
$619$	18677.5	1.21278	0.606392	−	0.795166i	$-0.292616\pi$
$619$	18677.5	1.21278	0.606392	+	0.795166i	$0.292616\pi$
$620$	7392.00	0.478822
$621$	−3959.80	−0.255880
$622$	−13553.8	−0.873728
$623$	0	0
$624$	−5760.00	−0.369527
$625$	22289.0	1.42650
$626$	−12380.0	−0.790424
$627$	−140.000	−0.00891716
$628$	8847.32	0.562176
$629$	53.7401	0.00340661
$630$	0	0
$631$	−14716.0	−0.928423	−0.464211	−	0.885724i	$-0.653662\pi$
$631$	−14716.0	−0.928423	−0.464211	+	0.885724i	$0.653662\pi$
$632$	−9760.00	−0.614291
$633$	−19431.3	−1.22010
$634$	−19652.0	−1.23104
$635$	34054.3	2.12819
$636$	−2093.04	−0.130494
$637$	0	0
$638$	−8008.00	−0.496928
$639$	13524.0	0.837248
$640$	−2534.27	−0.156525
$641$	4730.00	0.291457	0.145728	−	0.989325i	$-0.453447\pi$
$641$	4730.00	0.291457	0.145728	+	0.989325i	$0.453447\pi$
$642$	23815.4	1.46405
$643$	−19056.5	−1.16877	−0.584383	−	0.811478i	$-0.698663\pi$
$643$	−19056.5	−1.16877	−0.584383	+	0.811478i	$0.698663\pi$
$644$	0	0
$645$	−4760.00	−0.290581
$646$	4.00000	0.000243619 0
$647$	−9342.29	−0.567672	−0.283836	−	0.958873i	$-0.591607\pi$
$647$	−9342.29	−0.567672	−0.283836	+	0.958873i	$0.591607\pi$
$648$	6568.00	0.398172
$649$	6078.29	0.367633
$650$	27186.8	1.64055
$651$	0	0
$652$	13144.0	0.789507
$653$	3774.00	0.226169	0.113084	−	0.993585i	$-0.463927\pi$
$653$	3774.00	0.226169	0.113084	+	0.993585i	$0.463927\pi$
$654$	11568.3	0.691674
$655$	34356.0	2.04947
$656$	2013.84	0.119859
$657$	6212.64	0.368917
$658$	0	0
$659$	−21150.0	−1.25021	−0.625104	−	0.780541i	$-0.714943\pi$
$659$	−21150.0	−1.25021	−0.625104	+	0.780541i	$0.714943\pi$
$660$	−7840.00	−0.462381
$661$	10377.5	0.610647	0.305324	−	0.952249i	$-0.401235\pi$
$661$	10377.5	0.610647	0.305324	+	0.952249i	$0.401235\pi$
$662$	11476.0	0.673757
$663$	509.117	0.0298227
$664$	3382.80	0.197708
$665$	0	0
$666$	1748.00	0.101702
$667$	−40040.0	−2.32437
$668$	5962.32	0.345343
$669$	24240.0	1.40086
$670$	−27085.0	−1.56177
$671$	197.990	0.0113909
$672$	0	0
$673$	−1164.00	−0.0666700	−0.0333350	−	0.999444i	$-0.510613\pi$
$673$	−1164.00	−0.0666700	−0.0333350	+	0.999444i	$0.510613\pi$
$674$	4508.00	0.257629
$675$	−7551.90	−0.430626
$676$	1580.00	0.0898953
$677$	−27152.9	−1.54146	−0.770732	−	0.637160i	$-0.780109\pi$
$677$	−27152.9	−1.54146	−0.770732	+	0.637160i	$0.780109\pi$
$678$	7636.75	0.432578
$679$	0	0
$680$	224.000	0.0126324
$681$	−37410.0	−2.10507
$682$	2613.47	0.146737
$683$	−16596.0	−0.929763	−0.464882	−	0.885373i	$-0.653903\pi$
$683$	−16596.0	−0.929763	−0.464882	+	0.885373i	$0.653903\pi$
$684$	130.108	0.00727309
$685$	16393.6	0.914403
$686$	0	0
$687$	19440.0	1.07960
$688$	−544.000	−0.0301451
$689$	3767.46	0.208315
$690$	−39200.0	−2.16278
$691$	−11298.2	−0.622000	−0.311000	−	0.950410i	$-0.600664\pi$
$691$	−11298.2	−0.622000	−0.311000	+	0.950410i	$0.600664\pi$
$692$	8281.63	0.454943
$693$	0	0
$694$	3972.00	0.217255
$695$	8428.00	0.459989
$696$	16178.6	0.881104
$697$	−178.000	−0.00967321
$698$	13542.5	0.734372
$699$	509.117	0.0275487
$700$	0	0
$701$	−2754.00	−0.148384	−0.0741920	−	0.997244i	$-0.523638\pi$
$701$	−2754.00	−0.148384	−0.0741920	+	0.997244i	$0.523638\pi$
$702$	−2880.00	−0.154841
$703$	−53.7401	−0.00288314
$704$	−896.000	−0.0479677
$705$	−73256.3	−3.91346
$706$	13986.6	0.745597
$707$	0	0
$708$	−12280.0	−0.651851
$709$	29434.0	1.55912	0.779561	−	0.626327i	$-0.215442\pi$
$709$	29434.0	1.55912	0.779561	+	0.626327i	$0.215442\pi$
$710$	−23283.6	−1.23073
$711$	28060.0	1.48007
$712$	4944.09	0.260235
$713$	13067.3	0.686361
$714$	0	0
$715$	14112.0	0.738124
$716$	2160.00	0.112742
$717$	30462.2	1.58665
$718$	−11888.0	−0.617906
$719$	17669.2	0.916480	0.458240	−	0.888828i	$-0.348480\pi$
$719$	17669.2	0.916480	0.458240	+	0.888828i	$0.348480\pi$
$720$	7286.03	0.377131
$721$	0	0
$722$	13714.0	0.706901
$723$	10890.0	0.560171
$724$	−15137.7	−0.777058
$725$	−76362.0	−3.91174
$726$	16051.3	0.820552
$727$	−28445.5	−1.45115	−0.725574	−	0.688144i	$-0.758426\pi$
$727$	−28445.5	−1.45115	−0.725574	+	0.688144i	$0.758426\pi$
$728$	0	0
$729$	−13483.0	−0.685007
$730$	−10696.0	−0.542297
$731$	48.0833	0.00243286
$732$	−400.000	−0.0201973
$733$	−22341.7	−1.12580	−0.562900	−	0.826525i	$-0.690314\pi$
$733$	−22341.7	−1.12580	−0.562900	+	0.826525i	$0.690314\pi$
$734$	−1685.74	−0.0847710
$735$	0	0
$736$	−4480.00	−0.224368
$737$	−9576.00	−0.478611
$738$	−5789.79	−0.288787
$739$	20670.0	1.02890	0.514451	−	0.857520i	$-0.327996\pi$
$739$	20670.0	1.02890	0.514451	+	0.857520i	$0.327996\pi$
$740$	−3009.45	−0.149499
$741$	−509.117	−0.0252400
$742$	0	0
$743$	−25400.0	−1.25415	−0.627076	−	0.778958i	$-0.715749\pi$
$743$	−25400.0	−1.25415	−0.627076	+	0.778958i	$0.715749\pi$
$744$	−5280.00	−0.260180
$745$	40587.9	1.99601
$746$	11452.0	0.562048
$747$	−9725.55	−0.476358
$748$	79.1960	0.00387124
$749$	0	0
$750$	−39760.0	−1.93577
$751$	29180.0	1.41783	0.708917	−	0.705292i	$-0.249184\pi$
$751$	29180.0	1.41783	0.708917	+	0.705292i	$0.249184\pi$
$752$	−8372.14	−0.405985
$753$	−6590.00	−0.318928
$754$	−29121.5	−1.40655
$755$	−9345.12	−0.450469
$756$	0	0
$757$	−26206.0	−1.25822	−0.629110	−	0.777316i	$-0.716581\pi$
$757$	−26206.0	−1.25822	−0.629110	+	0.777316i	$0.716581\pi$
$758$	−20660.0	−0.989980
$759$	−13859.3	−0.662794
$760$	−224.000	−0.0106912
$761$	−6863.18	−0.326925	−0.163463	−	0.986550i	$-0.552266\pi$
$761$	−6863.18	−0.326925	−0.163463	+	0.986550i	$0.552266\pi$
$762$	−24324.5	−1.15641
$763$	0	0
$764$	4112.00	0.194721
$765$	−644.000	−0.0304364
$766$	2008.18	0.0947240
$767$	22104.0	1.04059
$768$	1810.19	0.0850517
$769$	9058.04	0.424761	0.212380	−	0.977187i	$-0.431878\pi$
$769$	9058.04	0.424761	0.212380	+	0.977187i	$0.431878\pi$
$770$	0	0
$771$	−6630.00	−0.309693
$772$	18368.0	0.856320
$773$	132.936	0.00618548	0.00309274	−	0.999995i	$-0.499016\pi$
$773$	132.936	0.00618548	0.00309274	+	0.999995i	$0.499016\pi$
$774$	1564.00	0.0726315
$775$	24921.3	1.15509
$776$	11868.1	0.549020
$777$	0	0
$778$	−10420.0	−0.480174
$779$	178.000	0.00818679
$780$	−28510.5	−1.30877
$781$	−8232.00	−0.377163
$782$	395.980	0.0181077
$783$	8089.30	0.369206
$784$	0	0
$785$	43792.0	1.99109
$786$	−24540.0	−1.11363
$787$	8729.94	0.395411	0.197706	−	0.980261i	$-0.436651\pi$
$787$	8729.94	0.395411	0.197706	+	0.980261i	$0.436651\pi$
$788$	3176.00	0.143579
$789$	50487.4	2.27807
$790$	−48309.5	−2.17567
$791$	0	0
$792$	2576.00	0.115573
$793$	720.000	0.0322421
$794$	−147.078	−0.00657382
$795$	−10360.0	−0.462178
$796$	−9944.75	−0.442817
$797$	−7517.96	−0.334128	−0.167064	−	0.985946i	$-0.553429\pi$
$797$	−7517.96	−0.334128	−0.167064	+	0.985946i	$0.553429\pi$
$798$	0	0
$799$	740.000	0.0327651
$800$	−8544.00	−0.377595
$801$	−14214.3	−0.627011
$802$	996.000	0.0438528
$803$	−3781.61	−0.166189
$804$	19346.4	0.848627
$805$	0	0
$806$	9504.00	0.415340
$807$	−32600.0	−1.42203
$808$	9028.34	0.393089
$809$	3776.00	0.164100	0.0820501	−	0.996628i	$-0.473853\pi$
$809$	3776.00	0.164100	0.0820501	+	0.996628i	$0.473853\pi$
$810$	32509.9	1.41023
$811$	36227.9	1.56860	0.784300	−	0.620382i	$-0.213023\pi$
$811$	36227.9	1.56860	0.784300	+	0.620382i	$0.213023\pi$
$812$	0	0
$813$	16720.0	0.721274
$814$	−1064.00	−0.0458147
$815$	65059.5	2.79624
$816$	−160.000	−0.00686412
$817$	−48.0833	−0.00205902
$818$	6711.86	0.286888
$819$	0	0
$820$	9968.00	0.424509
$821$	−16410.0	−0.697580	−0.348790	−	0.937201i	$-0.613407\pi$
$821$	−16410.0	−0.697580	−0.348790	+	0.937201i	$0.613407\pi$
$822$	−11709.7	−0.496864
$823$	22072.0	0.934850	0.467425	−	0.884033i	$-0.345182\pi$
$823$	22072.0	0.934850	0.467425	+	0.884033i	$0.345182\pi$
$824$	−6946.62	−0.293686
$825$	−26431.7	−1.11543
$826$	0	0
$827$	−11628.0	−0.488930	−0.244465	−	0.969658i	$-0.578612\pi$
$827$	−11628.0	−0.488930	−0.244465	+	0.969658i	$0.578612\pi$
$828$	12880.0	0.540593
$829$	30906.2	1.29483	0.647417	−	0.762136i	$-0.275849\pi$
$829$	30906.2	1.29483	0.647417	+	0.762136i	$0.275849\pi$
$830$	16744.0	0.700232
$831$	28326.7	1.18248
$832$	−3258.35	−0.135773
$833$	0	0
$834$	−6020.00	−0.249947
$835$	29512.0	1.22312
$836$	−79.1960	−0.00327638
$837$	−2640.00	−0.109022
$838$	29090.4	1.19918
$839$	−17884.1	−0.735911	−0.367955	−	0.929843i	$-0.619942\pi$
$839$	−17884.1	−0.735911	−0.367955	+	0.929843i	$0.619942\pi$
$840$	0	0
$841$	57407.0	2.35381
$842$	−21708.0	−0.888488
$843$	−42313.3	−1.72876
$844$	−10992.0	−0.448294
$845$	7820.60	0.318387
$846$	24069.9	0.978181
$847$	0	0
$848$	−1184.00	−0.0479466
$849$	34850.0	1.40877
$850$	755.190	0.0304739
$851$	−5320.00	−0.214298
$852$	16631.2	0.668749
$853$	20755.0	0.833104	0.416552	−	0.909112i	$-0.363238\pi$
$853$	20755.0	0.833104	0.416552	+	0.909112i	$0.363238\pi$
$854$	0	0
$855$	644.000	0.0257595
$856$	13472.0	0.537925
$857$	−44919.7	−1.79046	−0.895231	−	0.445602i	$-0.852990\pi$
$857$	−44919.7	−1.79046	−0.895231	+	0.445602i	$0.852990\pi$
$858$	−10080.0	−0.401079
$859$	−69.2965	−0.00275246	−0.00137623	−	0.999999i	$-0.500438\pi$
$859$	−69.2965	−0.00275246	−0.00137623	+	0.999999i	$0.500438\pi$
$860$	−2692.66	−0.106766
$861$	0	0
$862$	10728.0	0.423895
$863$	−5452.00	−0.215050	−0.107525	−	0.994202i	$-0.534293\pi$
$863$	−5452.00	−0.215050	−0.107525	+	0.994202i	$0.534293\pi$
$864$	905.097	0.0356389
$865$	40992.0	1.61129
$866$	−12974.0	−0.509093
$867$	−34726.0	−1.36027
$868$	0	0
$869$	−17080.0	−0.666743
$870$	80080.0	3.12065
$871$	−34823.6	−1.35471
$872$	6544.00	0.254137
$873$	−34120.7	−1.32281
$874$	−395.980	−0.0153252
$875$	0	0
$876$	7640.00	0.294671
$877$	31106.0	1.19769	0.598845	−	0.800865i	$-0.295626\pi$
$877$	31106.0	1.19769	0.598845	+	0.800865i	$0.295626\pi$
$878$	27865.7	1.07109
$879$	−13940.0	−0.534908
$880$	−4434.97	−0.169890
$881$	−5943.94	−0.227306	−0.113653	−	0.993521i	$-0.536255\pi$
$881$	−5943.94	−0.227306	−0.113653	+	0.993521i	$0.536255\pi$
$882$	0	0
$883$	34796.0	1.32614	0.663068	−	0.748559i	$-0.269254\pi$
$883$	34796.0	1.32614	0.663068	+	0.748559i	$0.269254\pi$
$884$	288.000	0.0109576
$885$	−60782.9	−2.30869
$886$	−11992.0	−0.454717
$887$	−9964.55	−0.377200	−0.188600	−	0.982054i	$-0.560395\pi$
$887$	−9964.55	−0.377200	−0.188600	+	0.982054i	$0.560395\pi$
$888$	2149.60	0.0812342
$889$	0	0
$890$	24472.0	0.921689
$891$	11494.0	0.432170
$892$	13712.2	0.514707
$893$	−740.000	−0.0277303
$894$	−28991.4	−1.08458
$895$	10691.5	0.399303
$896$	0	0
$897$	−50400.0	−1.87604
$898$	−5244.00	−0.194871
$899$	−26694.7	−0.990343
$900$	24564.0	0.909778
$901$	104.652	0.00386954
$902$	3524.22	0.130093
$903$	0	0
$904$	4320.00	0.158939
$905$	−74928.0	−2.75214
$906$	6675.09	0.244774
$907$	−29756.0	−1.08934	−0.544670	−	0.838650i	$-0.683345\pi$
$907$	−29756.0	−1.08934	−0.544670	+	0.838650i	$0.683345\pi$
$908$	−21162.3	−0.773453
$909$	−25956.5	−0.947109
$910$	0	0
$911$	21440.0	0.779735	0.389868	−	0.920871i	$-0.372521\pi$
$911$	21440.0	0.779735	0.389868	+	0.920871i	$0.372521\pi$
$912$	160.000	0.00580935
$913$	5919.90	0.214589
$914$	−22416.0	−0.811220
$915$	−1979.90	−0.0715338
$916$	10996.9	0.396669
$917$	0	0
$918$	−80.0000	−0.00287625
$919$	−8288.00	−0.297493	−0.148746	−	0.988875i	$-0.547524\pi$
$919$	−8288.00	−0.297493	−0.148746	+	0.988875i	$0.547524\pi$
$920$	−22174.9	−0.794656
$921$	33710.0	1.20606
$922$	19572.7	0.699125
$923$	−29936.1	−1.06756
$924$	0	0
$925$	−10146.0	−0.360647
$926$	−7904.00	−0.280498
$927$	19971.5	0.707606
$928$	9152.00	0.323738
$929$	45581.5	1.60978	0.804888	−	0.593427i	$-0.202226\pi$
$929$	45581.5	1.60978	0.804888	+	0.593427i	$0.202226\pi$
$930$	−26134.7	−0.921494
$931$	0	0
$932$	288.000	0.0101221
$933$	47920.0	1.68149
$934$	35013.1	1.22662
$935$	392.000	0.0137110
$936$	9367.75	0.327131
$937$	−11665.8	−0.406731	−0.203365	−	0.979103i	$-0.565188\pi$
$937$	−11665.8	−0.406731	−0.203365	+	0.979103i	$0.565188\pi$
$938$	0	0
$939$	43770.0	1.52117
$940$	−41440.0	−1.43790
$941$	−14.1421	−0.000489926 0	−0.000244963	−	1.00000i	$-0.500078\pi$
$941$	−14.1421	−0.000489926 0	−0.000244963		1.00000i	$0.500078\pi$
$942$	−31280.0	−1.08191
$943$	17621.1	0.608507
$944$	−6946.62	−0.239505
$945$	0	0
$946$	−952.000	−0.0327190
$947$	14034.0	0.481567	0.240783	−	0.970579i	$-0.422596\pi$
$947$	14034.0	0.481567	0.240783	+	0.970579i	$0.422596\pi$
$948$	34506.8	1.18220
$949$	−13752.0	−0.470399
$950$	−755.190	−0.0257912
$951$	69480.3	2.36914
$952$	0	0
$953$	−42698.0	−1.45134	−0.725668	−	0.688045i	$-0.758469\pi$
$953$	−42698.0	−1.45134	−0.725668	+	0.688045i	$0.758469\pi$
$954$	3404.00	0.115523
$955$	20353.4	0.689654
$956$	17232.0	0.582974
$957$	28312.6	0.956337
$958$	4576.40	0.154339
$959$	0	0
$960$	8960.00	0.301232
$961$	−21079.0	−0.707563
$962$	−3869.29	−0.129679
$963$	−38732.0	−1.29608
$964$	6160.31	0.205820
$965$	90917.0	3.03287
$966$	0	0
$967$	−48492.0	−1.61261	−0.806307	−	0.591497i	$-0.798537\pi$
$967$	−48492.0	−1.61261	−0.806307	+	0.591497i	$0.798537\pi$
$968$	9080.00	0.301490
$969$	−14.1421	−0.000468845 0
$970$	58744.0	1.94449
$971$	−52669.6	−1.74073	−0.870364	−	0.492409i	$-0.836116\pi$
$971$	−52669.6	−1.74073	−0.870364	+	0.492409i	$0.836116\pi$
$972$	−20166.7	−0.665480
$973$	0	0
$974$	−1944.00	−0.0639525
$975$	−96120.0	−3.15723
$976$	−226.274	−0.00742096
$977$	−55380.0	−1.81347	−0.906737	−	0.421698i	$-0.861434\pi$
$977$	−55380.0	−1.81347	−0.906737	+	0.421698i	$0.861434\pi$
$978$	−46471.1	−1.51941
$979$	8652.16	0.282456
$980$	0	0
$981$	−18814.0	−0.612319
$982$	14808.0	0.481204
$983$	50535.5	1.63971	0.819854	−	0.572573i	$-0.194055\pi$
$983$	50535.5	1.63971	0.819854	+	0.572573i	$0.194055\pi$
$984$	−7120.00	−0.230668
$985$	15720.4	0.508521
$986$	−808.930	−0.0261274
$987$	0	0
$988$	−288.000	−0.00927379
$989$	−4760.00	−0.153043
$990$	12750.5	0.409332
$991$	−39712.0	−1.27295	−0.636475	−	0.771297i	$-0.719608\pi$
$991$	−39712.0	−1.27295	−0.636475	+	0.771297i	$0.719608\pi$
$992$	−2986.82	−0.0955964
$993$	−40573.8	−1.29665
$994$	0	0
$995$	−49224.0	−1.56835
$996$	−11960.0	−0.380489
$997$	−2186.37	−0.0694515	−0.0347258	−	0.999397i	$-0.511056\pi$
$997$	−2186.37	−0.0694515	−0.0347258	+	0.999397i	$0.511056\pi$
$998$	24488.0	0.776708
$999$	1074.80	0.0340393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.a.g.1.2	yes	2
3.2	odd	2		882.4.a.bg.1.1		2
4.3	odd	2		784.4.a.y.1.1		2
5.4	even	2		2450.4.a.bx.1.1		2
7.2	even	3		98.4.c.h.67.1		4
7.3	odd	6		98.4.c.h.79.2		4
7.4	even	3		98.4.c.h.79.1		4
7.5	odd	6		98.4.c.h.67.2		4
7.6	odd	2	inner	98.4.a.g.1.1	✓	2
21.2	odd	6		882.4.g.ba.361.2		4
21.5	even	6		882.4.g.ba.361.1		4
21.11	odd	6		882.4.g.ba.667.2		4
21.17	even	6		882.4.g.ba.667.1		4
21.20	even	2		882.4.a.bg.1.2		2
28.27	even	2		784.4.a.y.1.2		2
35.34	odd	2		2450.4.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.a.g.1.1	✓	2	7.6	odd	2	inner
98.4.a.g.1.2	yes	2	1.1	even	1	trivial
98.4.c.h.67.1		4	7.2	even	3
98.4.c.h.67.2		4	7.5	odd	6
98.4.c.h.79.1		4	7.4	even	3
98.4.c.h.79.2		4	7.3	odd	6
784.4.a.y.1.1		2	4.3	odd	2
784.4.a.y.1.2		2	28.27	even	2
882.4.a.bg.1.1		2	3.2	odd	2
882.4.a.bg.1.2		2	21.20	even	2
882.4.g.ba.361.1		4	21.5	even	6
882.4.g.ba.361.2		4	21.2	odd	6
882.4.g.ba.667.1		4	21.17	even	6
882.4.g.ba.667.2		4	21.11	odd	6
2450.4.a.bx.1.1		2	5.4	even	2
2450.4.a.bx.1.2		2	35.34	odd	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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +7.07107 q^{3} +4.00000 q^{4} +19.7990 q^{5} -14.1421 q^{6} -8.00000 q^{8} +23.0000 q^{9} -39.5980 q^{10} -14.0000 q^{11} +28.2843 q^{12} -50.9117 q^{13} +140.000 q^{15} +16.0000 q^{16} -1.41421 q^{17} -46.0000 q^{18} +1.41421 q^{19} +79.1960 q^{20} +28.0000 q^{22} +140.000 q^{23} -56.5685 q^{24} +267.000 q^{25} +101.823 q^{26} -28.2843 q^{27} -286.000 q^{29} -280.000 q^{30} +93.3381 q^{31} -32.0000 q^{32} -98.9949 q^{33} +2.82843 q^{34} +92.0000 q^{36} -38.0000 q^{37} -2.82843 q^{38} -360.000 q^{39} -158.392 q^{40} +125.865 q^{41} -34.0000 q^{43} -56.0000 q^{44} +455.377 q^{45} -280.000 q^{46} -523.259 q^{47} +113.137 q^{48} -534.000 q^{50} -10.0000 q^{51} -203.647 q^{52} -74.0000 q^{53} +56.5685 q^{54} -277.186 q^{55} +10.0000 q^{57} +572.000 q^{58} -434.164 q^{59} +560.000 q^{60} -14.1421 q^{61} -186.676 q^{62} +64.0000 q^{64} -1008.00 q^{65} +197.990 q^{66} +684.000 q^{67} -5.65685 q^{68} +989.949 q^{69} +588.000 q^{71} -184.000 q^{72} +270.115 q^{73} +76.0000 q^{74} +1887.98 q^{75} +5.65685 q^{76} +720.000 q^{78} +1220.00 q^{79} +316.784 q^{80} -821.000 q^{81} -251.730 q^{82} -422.850 q^{83} -28.0000 q^{85} +68.0000 q^{86} -2022.33 q^{87} +112.000 q^{88} -618.011 q^{89} -910.754 q^{90} +560.000 q^{92} +660.000 q^{93} +1046.52 q^{94} +28.0000 q^{95} -226.274 q^{96} -1483.51 q^{97} -322.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 46 q^{9} - 28 q^{11} + 280 q^{15} + 32 q^{16} - 92 q^{18} + 56 q^{22} + 280 q^{23} + 534 q^{25} - 572 q^{29} - 560 q^{30} - 64 q^{32} + 184 q^{36} - 76 q^{37} - 720 q^{39}+ \cdots - 644 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 46 * q^9 - 28 * q^11 + 280 * q^15 + 32 * q^16 - 92 * q^18 + 56 * q^22 + 280 * q^23 + 534 * q^25 - 572 * q^29 - 560 * q^30 - 64 * q^32 + 184 * q^36 - 76 * q^37 - 720 * q^39 - 68 * q^43 - 112 * q^44 - 560 * q^46 - 1068 * q^50 - 20 * q^51 - 148 * q^53 + 20 * q^57 + 1144 * q^58 + 1120 * q^60 + 128 * q^64 - 2016 * q^65 + 1368 * q^67 + 1176 * q^71 - 368 * q^72 + 152 * q^74 + 1440 * q^78 + 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 136 * q^86 + 224 * q^88 + 1120 * q^92 + 1320 * q^93 + 56 * q^95 - 644 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more