LMFDB - Embedded newform 207.4.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [207,4,Mod(1,207)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("207.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4] 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:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 26x^{3} + 10x^{2} + 144x + 56 $ x^5 - x^4 - 26x^3 + 10x^2 + 144*x + 56
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.28650$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.28650 q^{2} +2.80110 q^{4} +21.8905 q^{5} +23.4895 q^{7} -17.0862 q^{8} +71.9433 q^{10} -53.0888 q^{11} +18.8707 q^{13} +77.1983 q^{14} -78.5626 q^{16} +107.017 q^{17} -43.5150 q^{19} +61.3175 q^{20} -174.477 q^{22} -23.0000 q^{23} +354.195 q^{25} +62.0186 q^{26} +65.7964 q^{28} +101.638 q^{29} -224.819 q^{31} -121.507 q^{32} +351.710 q^{34} +514.198 q^{35} +70.5460 q^{37} -143.012 q^{38} -374.026 q^{40} +42.6933 q^{41} -297.280 q^{43} -148.707 q^{44} -75.5896 q^{46} +174.746 q^{47} +208.757 q^{49} +1164.06 q^{50} +52.8587 q^{52} -684.506 q^{53} -1162.14 q^{55} -401.346 q^{56} +334.032 q^{58} -226.519 q^{59} -782.305 q^{61} -738.869 q^{62} +229.169 q^{64} +413.090 q^{65} -637.736 q^{67} +299.764 q^{68} +1689.91 q^{70} +828.705 q^{71} -820.013 q^{73} +231.850 q^{74} -121.890 q^{76} -1247.03 q^{77} +649.539 q^{79} -1719.78 q^{80} +140.311 q^{82} -7.84073 q^{83} +2342.65 q^{85} -977.012 q^{86} +907.086 q^{88} +582.459 q^{89} +443.264 q^{91} -64.4253 q^{92} +574.302 q^{94} -952.567 q^{95} -389.191 q^{97} +686.079 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 16 q^{4} + 20 q^{5} - 10 q^{7} + 48 q^{8} + 50 q^{10} + 46 q^{11} + 54 q^{13} + 164 q^{14} - 60 q^{16} + 250 q^{17} - 28 q^{19} + 242 q^{20} - 10 q^{22} - 115 q^{23} + 239 q^{25} + 368 q^{26}+ \cdots - 2400 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 16 * q^4 + 20 * q^5 - 10 * q^7 + 48 * q^8 + 50 * q^10 + 46 * q^11 + 54 * q^13 + 164 * q^14 - 60 * q^16 + 250 * q^17 - 28 * q^19 + 242 * q^20 - 10 * q^22 - 115 * q^23 + 239 * q^25 + 368 * q^26 + 224 * q^28 + 460 * q^29 - 360 * q^31 + 268 * q^32 - 20 * q^34 + 964 * q^35 - 92 * q^37 + 266 * q^38 - 278 * q^40 + 788 * q^41 + 108 * q^43 + 354 * q^44 - 92 * q^46 + 724 * q^47 + 149 * q^49 + 128 * q^50 - 288 * q^52 + 580 * q^53 - 672 * q^55 - 1172 * q^56 - 48 * q^58 - 408 * q^59 - 620 * q^61 - 1088 * q^62 + 696 * q^64 + 184 * q^65 + 100 * q^67 - 1584 * q^68 + 728 * q^70 - 124 * q^71 - 1946 * q^73 - 2866 * q^74 + 1362 * q^76 - 396 * q^77 - 198 * q^79 - 2398 * q^80 + 76 * q^82 - 578 * q^83 + 376 * q^85 - 3910 * q^86 + 1366 * q^88 + 930 * q^89 + 1972 * q^91 - 368 * q^92 + 976 * q^94 - 908 * q^95 + 2618 * q^97 - 2400 * 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 $

$2$	3.28650	1.16195	0.580977	−	0.813920i	$-0.302671\pi$
$2$	3.28650	1.16195	0.580977	+	0.813920i	$0.302671\pi$
$3$	0	0
$3$	0	0
$4$	2.80110	0.350137
$5$	21.8905	1.95795	0.978974	−	0.203984i	$-0.0653891\pi$
$5$	21.8905	1.95795	0.978974	+	0.203984i	$0.0653891\pi$
$6$	0	0
$7$	23.4895	1.26831	0.634157	−	0.773204i	$-0.281347\pi$
$7$	23.4895	1.26831	0.634157	+	0.773204i	$0.281347\pi$
$8$	−17.0862	−0.755111
$9$	0	0
$10$	71.9433	2.27505
$11$	−53.0888	−1.45517	−0.727586	−	0.686017i	$-0.759358\pi$
$11$	−53.0888	−1.45517	−0.727586	+	0.686017i	$0.759358\pi$
$12$	0	0
$13$	18.8707	0.402600	0.201300	−	0.979530i	$-0.435483\pi$
$13$	18.8707	0.402600	0.201300	+	0.979530i	$0.435483\pi$
$14$	77.1983	1.47372
$15$	0	0
$16$	−78.5626	−1.22754
$17$	107.017	1.52679	0.763393	−	0.645935i	$-0.223532\pi$
$17$	107.017	1.52679	0.763393	+	0.645935i	$0.223532\pi$
$18$	0	0
$19$	−43.5150	−0.525423	−0.262711	−	0.964874i	$-0.584617\pi$
$19$	−43.5150	−0.525423	−0.262711	+	0.964874i	$0.584617\pi$
$20$	61.3175	0.685551
$21$	0	0
$22$	−174.477	−1.69084
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	354.195	2.83356
$26$	62.0186	0.467802
$27$	0	0
$28$	65.7964	0.444084
$29$	101.638	0.650815	0.325407	−	0.945574i	$-0.394499\pi$
$29$	101.638	0.650815	0.325407	+	0.945574i	$0.394499\pi$
$30$	0	0
$31$	−224.819	−1.30254	−0.651270	−	0.758846i	$-0.725763\pi$
$31$	−224.819	−1.30254	−0.651270	+	0.758846i	$0.725763\pi$
$32$	−121.507	−0.671236
$33$	0	0
$34$	351.710	1.77405
$35$	514.198	2.48329
$36$	0	0
$37$	70.5460	0.313451	0.156726	−	0.987642i	$-0.449906\pi$
$37$	70.5460	0.313451	0.156726	+	0.987642i	$0.449906\pi$
$38$	−143.012	−0.610517
$39$	0	0
$40$	−374.026	−1.47847
$41$	42.6933	0.162624	0.0813118	−	0.996689i	$-0.474089\pi$
$41$	42.6933	0.162624	0.0813118	+	0.996689i	$0.474089\pi$
$42$	0	0
$43$	−297.280	−1.05430	−0.527149	−	0.849773i	$-0.676739\pi$
$43$	−297.280	−1.05430	−0.527149	+	0.849773i	$0.676739\pi$
$44$	−148.707	−0.509510
$45$	0	0
$46$	−75.5896	−0.242284
$47$	174.746	0.542325	0.271162	−	0.962534i	$-0.412592\pi$
$47$	174.746	0.542325	0.271162	+	0.962534i	$0.412592\pi$
$48$	0	0
$49$	208.757	0.608620
$50$	1164.06	3.29247
$51$	0	0
$52$	52.8587	0.140965
$53$	−684.506	−1.77404	−0.887020	−	0.461731i	$-0.847228\pi$
$53$	−684.506	−1.77404	−0.887020	+	0.461731i	$0.847228\pi$
$54$	0	0
$55$	−1162.14	−2.84915
$56$	−401.346	−0.957717
$57$	0	0
$58$	334.032	0.756217
$59$	−226.519	−0.499835	−0.249917	−	0.968267i	$-0.580403\pi$
$59$	−226.519	−0.499835	−0.249917	+	0.968267i	$0.580403\pi$
$60$	0	0
$61$	−782.305	−1.64203	−0.821015	−	0.570906i	$-0.806592\pi$
$61$	−782.305	−1.64203	−0.821015	+	0.570906i	$0.806592\pi$
$62$	−738.869	−1.51349
$63$	0	0
$64$	229.169	0.447596
$65$	413.090	0.788269
$66$	0	0
$67$	−637.736	−1.16286	−0.581432	−	0.813595i	$-0.697507\pi$
$67$	−637.736	−1.16286	−0.581432	+	0.813595i	$0.697507\pi$
$68$	299.764	0.534584
$69$	0	0
$70$	1689.91	2.88547
$71$	828.705	1.38520	0.692600	−	0.721322i	$-0.256465\pi$
$71$	828.705	1.38520	0.692600	+	0.721322i	$0.256465\pi$
$72$	0	0
$73$	−820.013	−1.31473	−0.657365	−	0.753573i	$-0.728329\pi$
$73$	−820.013	−1.31473	−0.657365	+	0.753573i	$0.728329\pi$
$74$	231.850	0.364216
$75$	0	0
$76$	−121.890	−0.183970
$77$	−1247.03	−1.84561
$78$	0	0
$79$	649.539	0.925049	0.462525	−	0.886606i	$-0.346944\pi$
$79$	649.539	0.925049	0.462525	+	0.886606i	$0.346944\pi$
$80$	−1719.78	−2.40346
$81$	0	0
$82$	140.311	0.188961
$83$	−7.84073	−0.0103691	−0.00518453	−	0.999987i	$-0.501650\pi$
$83$	−7.84073	−0.0103691	−0.00518453	+	0.999987i	$0.501650\pi$
$84$	0	0
$85$	2342.65	2.98937
$86$	−977.012	−1.22505
$87$	0	0
$88$	907.086	1.09882
$89$	582.459	0.693713	0.346857	−	0.937918i	$-0.387249\pi$
$89$	582.459	0.693713	0.346857	+	0.937918i	$0.387249\pi$
$90$	0	0
$91$	443.264	0.510623
$92$	−64.4253	−0.0730087
$93$	0	0
$94$	574.302	0.630156
$95$	−952.567	−1.02875
$96$	0	0
$97$	−389.191	−0.407385	−0.203693	−	0.979035i	$-0.565294\pi$
$97$	−389.191	−0.407385	−0.203693	+	0.979035i	$0.565294\pi$
$98$	686.079	0.707188
$99$	0	0
$100$	992.136	0.992136
$101$	−271.413	−0.267392	−0.133696	−	0.991022i	$-0.542685\pi$
$101$	−271.413	−0.267392	−0.133696	+	0.991022i	$0.542685\pi$
$102$	0	0
$103$	58.3654	0.0558341	0.0279170	−	0.999610i	$-0.491113\pi$
$103$	58.3654	0.0558341	0.0279170	+	0.999610i	$0.491113\pi$
$104$	−322.429	−0.304007
$105$	0	0
$106$	−2249.63	−2.06135
$107$	559.848	0.505818	0.252909	−	0.967490i	$-0.418613\pi$
$107$	559.848	0.505818	0.252909	+	0.967490i	$0.418613\pi$
$108$	0	0
$109$	−96.8835	−0.0851353	−0.0425677	−	0.999094i	$-0.513554\pi$
$109$	−96.8835	−0.0851353	−0.0425677	+	0.999094i	$0.513554\pi$
$110$	−3819.38	−3.31058
$111$	0	0
$112$	−1845.40	−1.55691
$113$	675.286	0.562173	0.281086	−	0.959682i	$-0.409305\pi$
$113$	675.286	0.562173	0.281086	+	0.959682i	$0.409305\pi$
$114$	0	0
$115$	−503.482	−0.408260
$116$	284.697	0.227874
$117$	0	0
$118$	−744.455	−0.580785
$119$	2513.77	1.93644
$120$	0	0
$121$	1487.42	1.11752
$122$	−2571.05	−1.90796
$123$	0	0
$124$	−629.741	−0.456068
$125$	5017.20	3.59002
$126$	0	0
$127$	−1417.21	−0.990210	−0.495105	−	0.868833i	$-0.664870\pi$
$127$	−1417.21	−0.990210	−0.495105	+	0.868833i	$0.664870\pi$
$128$	1725.22	1.19132
$129$	0	0
$130$	1357.62	0.915933
$131$	−792.935	−0.528848	−0.264424	−	0.964407i	$-0.585182\pi$
$131$	−792.935	−0.528848	−0.264424	+	0.964407i	$0.585182\pi$
$132$	0	0
$133$	−1022.15	−0.666401
$134$	−2095.92	−1.35119
$135$	0	0
$136$	−1828.51	−1.15289
$137$	−841.176	−0.524573	−0.262287	−	0.964990i	$-0.584477\pi$
$137$	−841.176	−0.524573	−0.262287	+	0.964990i	$0.584477\pi$
$138$	0	0
$139$	−1418.00	−0.865276	−0.432638	−	0.901568i	$-0.642417\pi$
$139$	−1418.00	−0.865276	−0.432638	+	0.901568i	$0.642417\pi$
$140$	1440.32	0.869493
$141$	0	0
$142$	2723.54	1.60954
$143$	−1001.82	−0.585851
$144$	0	0
$145$	2224.90	1.27426
$146$	−2694.97	−1.52766
$147$	0	0
$148$	197.606	0.109751
$149$	326.139	0.179318	0.0896589	−	0.995973i	$-0.471422\pi$
$149$	326.139	0.179318	0.0896589	+	0.995973i	$0.471422\pi$
$150$	0	0
$151$	1362.97	0.734551	0.367275	−	0.930112i	$-0.380291\pi$
$151$	1362.97	0.734551	0.367275	+	0.930112i	$0.380291\pi$
$152$	743.507	0.396752
$153$	0	0
$154$	−4098.37	−2.14452
$155$	−4921.41	−2.55031
$156$	0	0
$157$	362.468	0.184256	0.0921278	−	0.995747i	$-0.470633\pi$
$157$	362.468	0.184256	0.0921278	+	0.995747i	$0.470633\pi$
$158$	2134.71	1.07486
$159$	0	0
$160$	−2659.85	−1.31425
$161$	−540.258	−0.264462
$162$	0	0
$163$	−1251.83	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$163$	−1251.83	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$164$	119.588	0.0569406
$165$	0	0
$166$	−25.7686	−0.0120484
$167$	3586.72	1.66197	0.830984	−	0.556296i	$-0.187778\pi$
$167$	3586.72	1.66197	0.830984	+	0.556296i	$0.187778\pi$
$168$	0	0
$169$	−1840.90	−0.837914
$170$	7699.13	3.47351
$171$	0	0
$172$	−832.711	−0.369149
$173$	−1898.77	−0.834455	−0.417227	−	0.908802i	$-0.636998\pi$
$173$	−1898.77	−0.834455	−0.417227	+	0.908802i	$0.636998\pi$
$174$	0	0
$175$	8319.87	3.59385
$176$	4170.80	1.78628
$177$	0	0
$178$	1914.25	0.806063
$179$	410.977	0.171608	0.0858041	−	0.996312i	$-0.472654\pi$
$179$	410.977	0.171608	0.0858041	+	0.996312i	$0.472654\pi$
$180$	0	0
$181$	1133.03	0.465291	0.232646	−	0.972562i	$-0.425262\pi$
$181$	1133.03	0.465291	0.232646	+	0.972562i	$0.425262\pi$
$182$	1456.79	0.593320
$183$	0	0
$184$	392.983	0.157451
$185$	1544.29	0.613721
$186$	0	0
$187$	−5681.39	−2.22173
$188$	489.480	0.189888
$189$	0	0
$190$	−3130.61	−1.19536
$191$	2929.96	1.10997	0.554986	−	0.831859i	$-0.312723\pi$
$191$	2929.96	1.10997	0.554986	+	0.831859i	$0.312723\pi$
$192$	0	0
$193$	3634.92	1.35569	0.677843	−	0.735207i	$-0.262915\pi$
$193$	3634.92	1.35569	0.677843	+	0.735207i	$0.262915\pi$
$194$	−1279.08	−0.473363
$195$	0	0
$196$	584.748	0.213100
$197$	3054.04	1.10452	0.552262	−	0.833671i	$-0.313765\pi$
$197$	3054.04	1.10452	0.552262	+	0.833671i	$0.313765\pi$
$198$	0	0
$199$	2953.78	1.05220	0.526101	−	0.850422i	$-0.323653\pi$
$199$	2953.78	1.05220	0.526101	+	0.850422i	$0.323653\pi$
$200$	−6051.85	−2.13965
$201$	0	0
$202$	−891.999	−0.310697
$203$	2387.42	0.825437
$204$	0	0
$205$	934.578	0.318409
$206$	191.818	0.0648766
$207$	0	0
$208$	−1482.53	−0.494208
$209$	2310.16	0.764580
$210$	0	0
$211$	753.365	0.245800	0.122900	−	0.992419i	$-0.460781\pi$
$211$	753.365	0.245800	0.122900	+	0.992419i	$0.460781\pi$
$212$	−1917.37	−0.621157
$213$	0	0
$214$	1839.94	0.587738
$215$	−6507.62	−2.06426
$216$	0	0
$217$	−5280.89	−1.65203
$218$	−318.408	−0.0989234
$219$	0	0
$220$	−3255.27	−0.997594
$221$	2019.48	0.614683
$222$	0	0
$223$	−2422.36	−0.727414	−0.363707	−	0.931513i	$-0.618489\pi$
$223$	−2422.36	−0.727414	−0.363707	+	0.931513i	$0.618489\pi$
$224$	−2854.13	−0.851338
$225$	0	0
$226$	2219.33	0.653219
$227$	−2527.79	−0.739097	−0.369548	−	0.929211i	$-0.620488\pi$
$227$	−2527.79	−0.739097	−0.369548	+	0.929211i	$0.620488\pi$
$228$	0	0
$229$	4478.46	1.29233	0.646167	−	0.763196i	$-0.276371\pi$
$229$	4478.46	1.29233	0.646167	+	0.763196i	$0.276371\pi$
$230$	−1654.70	−0.474380
$231$	0	0
$232$	−1736.60	−0.491437
$233$	2666.13	0.749631	0.374816	−	0.927099i	$-0.377706\pi$
$233$	2666.13	0.749631	0.374816	+	0.927099i	$0.377706\pi$
$234$	0	0
$235$	3825.27	1.06184
$236$	−634.502	−0.175011
$237$	0	0
$238$	8261.50	2.25006
$239$	−4039.81	−1.09336	−0.546681	−	0.837341i	$-0.684109\pi$
$239$	−4039.81	−1.09336	−0.546681	+	0.837341i	$0.684109\pi$
$240$	0	0
$241$	−1792.22	−0.479034	−0.239517	−	0.970892i	$-0.576989\pi$
$241$	−1792.22	−0.479034	−0.239517	+	0.970892i	$0.576989\pi$
$242$	4888.42	1.29851
$243$	0	0
$244$	−2191.31	−0.574936
$245$	4569.79	1.19165
$246$	0	0
$247$	−821.160	−0.211535
$248$	3841.31	0.983562
$249$	0	0
$250$	16489.1	4.17144
$251$	1328.35	0.334042	0.167021	−	0.985953i	$-0.446585\pi$
$251$	1328.35	0.334042	0.167021	+	0.985953i	$0.446585\pi$
$252$	0	0
$253$	1221.04	0.303424
$254$	−4657.65	−1.15058
$255$	0	0
$256$	3836.58	0.936665
$257$	6605.61	1.60330	0.801648	−	0.597797i	$-0.203957\pi$
$257$	6605.61	1.60330	0.801648	+	0.597797i	$0.203957\pi$
$258$	0	0
$259$	1657.09	0.397554
$260$	1157.11	0.276002
$261$	0	0
$262$	−2605.98	−0.614497
$263$	2912.35	0.682825	0.341413	−	0.939913i	$-0.389095\pi$
$263$	2912.35	0.682825	0.341413	+	0.939913i	$0.389095\pi$
$264$	0	0
$265$	−14984.2	−3.47348
$266$	−3359.29	−0.774327
$267$	0	0
$268$	−1786.36	−0.407162
$269$	−922.528	−0.209098	−0.104549	−	0.994520i	$-0.533340\pi$
$269$	−922.528	−0.209098	−0.104549	+	0.994520i	$0.533340\pi$
$270$	0	0
$271$	4350.19	0.975112	0.487556	−	0.873092i	$-0.337888\pi$
$271$	4350.19	0.975112	0.487556	+	0.873092i	$0.337888\pi$
$272$	−8407.51	−1.87419
$273$	0	0
$274$	−2764.53	−0.609530
$275$	−18803.8	−4.12332
$276$	0	0
$277$	1521.88	0.330112	0.165056	−	0.986284i	$-0.447219\pi$
$277$	1521.88	0.330112	0.165056	+	0.986284i	$0.447219\pi$
$278$	−4660.27	−1.00541
$279$	0	0
$280$	−8785.68	−1.87516
$281$	−5639.43	−1.19723	−0.598613	−	0.801039i	$-0.704281\pi$
$281$	−5639.43	−1.19723	−0.598613	+	0.801039i	$0.704281\pi$
$282$	0	0
$283$	−1098.07	−0.230648	−0.115324	−	0.993328i	$-0.536791\pi$
$283$	−1098.07	−0.230648	−0.115324	+	0.993328i	$0.536791\pi$
$284$	2321.28	0.485010
$285$	0	0
$286$	−3292.50	−0.680732
$287$	1002.84	0.206258
$288$	0	0
$289$	6539.56	1.33107
$290$	7312.14	1.48063
$291$	0	0
$292$	−2296.94	−0.460336
$293$	3403.18	0.678552	0.339276	−	0.940687i	$-0.389818\pi$
$293$	3403.18	0.678552	0.339276	+	0.940687i	$0.389818\pi$
$294$	0	0
$295$	−4958.62	−0.978651
$296$	−1205.36	−0.236690
$297$	0	0
$298$	1071.86	0.208359
$299$	−434.026	−0.0839478
$300$	0	0
$301$	−6982.96	−1.33718
$302$	4479.41	0.853514
$303$	0	0
$304$	3418.66	0.644978
$305$	−17125.1	−3.21501
$306$	0	0
$307$	−3712.90	−0.690250	−0.345125	−	0.938557i	$-0.612163\pi$
$307$	−3712.90	−0.690250	−0.345125	+	0.938557i	$0.612163\pi$
$308$	−3493.05	−0.646218
$309$	0	0
$310$	−16174.2	−2.96334
$311$	2638.89	0.481150	0.240575	−	0.970631i	$-0.422664\pi$
$311$	2638.89	0.481150	0.240575	+	0.970631i	$0.422664\pi$
$312$	0	0
$313$	−8072.97	−1.45786	−0.728932	−	0.684586i	$-0.759983\pi$
$313$	−8072.97	−1.45786	−0.728932	+	0.684586i	$0.759983\pi$
$314$	1191.25	0.214096
$315$	0	0
$316$	1819.42	0.323894
$317$	1682.83	0.298162	0.149081	−	0.988825i	$-0.452368\pi$
$317$	1682.83	0.298162	0.149081	+	0.988825i	$0.452368\pi$
$318$	0	0
$319$	−5395.82	−0.947046
$320$	5016.63	0.876370
$321$	0	0
$322$	−1775.56	−0.307292
$323$	−4656.83	−0.802208
$324$	0	0
$325$	6683.92	1.14079
$326$	−4114.13	−0.698959
$327$	0	0
$328$	−729.466	−0.122799
$329$	4104.69	0.687838
$330$	0	0
$331$	4251.71	0.706028	0.353014	−	0.935618i	$-0.385157\pi$
$331$	4251.71	0.706028	0.353014	+	0.935618i	$0.385157\pi$
$332$	−21.9627	−0.00363059
$333$	0	0
$334$	11787.8	1.93113
$335$	−13960.4	−2.27683
$336$	0	0
$337$	−10303.0	−1.66540	−0.832701	−	0.553723i	$-0.813207\pi$
$337$	−10303.0	−1.66540	−0.832701	+	0.553723i	$0.813207\pi$
$338$	−6050.11	−0.973617
$339$	0	0
$340$	6562.00	1.04669
$341$	11935.4	1.89542
$342$	0	0
$343$	−3153.31	−0.496393
$344$	5079.39	0.796111
$345$	0	0
$346$	−6240.30	−0.969598
$347$	4414.54	0.682954	0.341477	−	0.939890i	$-0.389073\pi$
$347$	4414.54	0.682954	0.341477	+	0.939890i	$0.389073\pi$
$348$	0	0
$349$	7388.50	1.13323	0.566615	−	0.823983i	$-0.308253\pi$
$349$	7388.50	1.13323	0.566615	+	0.823983i	$0.308253\pi$
$350$	27343.3	4.17588
$351$	0	0
$352$	6450.65	0.976763
$353$	−4899.89	−0.738795	−0.369398	−	0.929271i	$-0.620436\pi$
$353$	−4899.89	−0.738795	−0.369398	+	0.929271i	$0.620436\pi$
$354$	0	0
$355$	18140.8	2.71215
$356$	1631.52	0.242895
$357$	0	0
$358$	1350.68	0.199401
$359$	779.159	0.114547	0.0572736	−	0.998359i	$-0.481759\pi$
$359$	779.159	0.114547	0.0572736	+	0.998359i	$0.481759\pi$
$360$	0	0
$361$	−4965.44	−0.723931
$362$	3723.72	0.540647
$363$	0	0
$364$	1241.62	0.178788
$365$	−17950.5	−2.57417
$366$	0	0
$367$	13201.8	1.87773	0.938865	−	0.344286i	$-0.111879\pi$
$367$	13201.8	1.87773	0.938865	+	0.344286i	$0.111879\pi$
$368$	1806.94	0.255960
$369$	0	0
$370$	5075.31	0.713116
$371$	−16078.7	−2.25004
$372$	0	0
$373$	12365.3	1.71648	0.858242	−	0.513245i	$-0.171557\pi$
$373$	12365.3	1.71648	0.858242	+	0.513245i	$0.171557\pi$
$374$	−18671.9	−2.58155
$375$	0	0
$376$	−2985.74	−0.409515
$377$	1917.97	0.262018
$378$	0	0
$379$	11978.8	1.62351	0.811756	−	0.583997i	$-0.198512\pi$
$379$	11978.8	1.62351	0.811756	+	0.583997i	$0.198512\pi$
$380$	−2668.23	−0.360204
$381$	0	0
$382$	9629.34	1.28974
$383$	12441.8	1.65992	0.829958	−	0.557825i	$-0.188364\pi$
$383$	12441.8	1.65992	0.829958	+	0.557825i	$0.188364\pi$
$384$	0	0
$385$	−27298.1	−3.61362
$386$	11946.2	1.57524
$387$	0	0
$388$	−1090.16	−0.142641
$389$	−9254.95	−1.20628	−0.603142	−	0.797634i	$-0.706085\pi$
$389$	−9254.95	−1.20628	−0.603142	+	0.797634i	$0.706085\pi$
$390$	0	0
$391$	−2461.38	−0.318357
$392$	−3566.86	−0.459575
$393$	0	0
$394$	10037.1	1.28341
$395$	14218.8	1.81120
$396$	0	0
$397$	−2957.45	−0.373880	−0.186940	−	0.982371i	$-0.559857\pi$
$397$	−2957.45	−0.373880	−0.186940	+	0.982371i	$0.559857\pi$
$398$	9707.62	1.22261
$399$	0	0
$400$	−27826.5	−3.47831
$401$	8475.76	1.05551	0.527755	−	0.849397i	$-0.323034\pi$
$401$	8475.76	1.05551	0.527755	+	0.849397i	$0.323034\pi$
$402$	0	0
$403$	−4242.50	−0.524402
$404$	−760.254	−0.0936239
$405$	0	0
$406$	7846.25	0.959120
$407$	−3745.20	−0.456125
$408$	0	0
$409$	1851.69	0.223863	0.111932	−	0.993716i	$-0.464296\pi$
$409$	1851.69	0.223863	0.111932	+	0.993716i	$0.464296\pi$
$410$	3071.49	0.369976
$411$	0	0
$412$	163.487	0.0195496
$413$	−5320.82	−0.633947
$414$	0	0
$415$	−171.638	−0.0203021
$416$	−2292.92	−0.270239
$417$	0	0
$418$	7592.35	0.888407
$419$	4303.52	0.501768	0.250884	−	0.968017i	$-0.419279\pi$
$419$	4303.52	0.501768	0.250884	+	0.968017i	$0.419279\pi$
$420$	0	0
$421$	15653.6	1.81213	0.906067	−	0.423134i	$-0.139070\pi$
$421$	15653.6	1.81213	0.906067	+	0.423134i	$0.139070\pi$
$422$	2475.94	0.285608
$423$	0	0
$424$	11695.6	1.33960
$425$	37904.8	4.32624
$426$	0	0
$427$	−18376.0	−2.08261
$428$	1568.19	0.177106
$429$	0	0
$430$	−21387.3	−2.39858
$431$	685.365	0.0765960	0.0382980	−	0.999266i	$-0.487806\pi$
$431$	685.365	0.0765960	0.0382980	+	0.999266i	$0.487806\pi$
$432$	0	0
$433$	8771.01	0.973459	0.486729	−	0.873553i	$-0.338190\pi$
$433$	8771.01	0.973459	0.486729	+	0.873553i	$0.338190\pi$
$434$	−17355.7	−1.91958
$435$	0	0
$436$	−271.380	−0.0298091
$437$	1000.85	0.109558
$438$	0	0
$439$	−1983.55	−0.215649	−0.107824	−	0.994170i	$-0.534388\pi$
$439$	−1983.55	−0.215649	−0.107824	+	0.994170i	$0.534388\pi$
$440$	19856.6	2.15142
$441$	0	0
$442$	6637.03	0.714233
$443$	−9467.17	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$443$	−9467.17	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$444$	0	0
$445$	12750.3	1.35825
$446$	−7961.10	−0.845222
$447$	0	0
$448$	5383.07	0.567692
$449$	−14377.4	−1.51116	−0.755581	−	0.655055i	$-0.772645\pi$
$449$	−14377.4	−1.51116	−0.755581	+	0.655055i	$0.772645\pi$
$450$	0	0
$451$	−2266.53	−0.236645
$452$	1891.54	0.196838
$453$	0	0
$454$	−8307.57	−0.858797
$455$	9703.27	0.999772
$456$	0	0
$457$	−10508.5	−1.07564	−0.537819	−	0.843060i	$-0.680752\pi$
$457$	−10508.5	−1.07564	−0.537819	+	0.843060i	$0.680752\pi$
$458$	14718.5	1.50163
$459$	0	0
$460$	−1410.30	−0.142947
$461$	−1640.19	−0.165708	−0.0828539	−	0.996562i	$-0.526403\pi$
$461$	−1640.19	−0.165708	−0.0828539	+	0.996562i	$0.526403\pi$
$462$	0	0
$463$	−5863.97	−0.588600	−0.294300	−	0.955713i	$-0.595087\pi$
$463$	−5863.97	−0.588600	−0.294300	+	0.955713i	$0.595087\pi$
$464$	−7984.91	−0.798902
$465$	0	0
$466$	8762.24	0.871037
$467$	−5323.17	−0.527467	−0.263733	−	0.964596i	$-0.584954\pi$
$467$	−5323.17	−0.527467	−0.263733	+	0.964596i	$0.584954\pi$
$468$	0	0
$469$	−14980.1	−1.47488
$470$	12571.8	1.23381
$471$	0	0
$472$	3870.35	0.377431
$473$	15782.2	1.53418
$474$	0	0
$475$	−15412.8	−1.48882
$476$	7041.31	0.678021
$477$	0	0
$478$	−13276.9	−1.27044
$479$	−19819.2	−1.89053	−0.945263	−	0.326310i	$-0.894195\pi$
$479$	−19819.2	−1.89053	−0.945263	+	0.326310i	$0.894195\pi$
$480$	0	0
$481$	1331.25	0.126195
$482$	−5890.15	−0.556616
$483$	0	0
$484$	4166.42	0.391286
$485$	−8519.60	−0.797639
$486$	0	0
$487$	−15258.0	−1.41972	−0.709861	−	0.704342i	$-0.751242\pi$
$487$	−15258.0	−1.41972	−0.709861	+	0.704342i	$0.751242\pi$
$488$	13366.6	1.23991
$489$	0	0
$490$	15018.6	1.38464
$491$	11077.0	1.01813	0.509063	−	0.860729i	$-0.329992\pi$
$491$	11077.0	1.01813	0.509063	+	0.860729i	$0.329992\pi$
$492$	0	0
$493$	10876.9	0.993654
$494$	−2698.74	−0.245794
$495$	0	0
$496$	17662.4	1.59892
$497$	19465.9	1.75687
$498$	0	0
$499$	12511.3	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$499$	12511.3	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$500$	14053.7	1.25700
$501$	0	0
$502$	4365.61	0.388141
$503$	−9688.28	−0.858805	−0.429402	−	0.903113i	$-0.641276\pi$
$503$	−9688.28	−0.858805	−0.429402	+	0.903113i	$0.641276\pi$
$504$	0	0
$505$	−5941.37	−0.523540
$506$	4012.96	0.352565
$507$	0	0
$508$	−3969.73	−0.346710
$509$	−12217.5	−1.06391	−0.531956	−	0.846772i	$-0.678543\pi$
$509$	−12217.5	−1.06391	−0.531956	+	0.846772i	$0.678543\pi$
$510$	0	0
$511$	−19261.7	−1.66749
$512$	−1192.81	−0.102960
$513$	0	0
$514$	21709.4	1.86296
$515$	1277.65	0.109320
$516$	0	0
$517$	−9277.04	−0.789175
$518$	5446.03	0.461940
$519$	0	0
$520$	−7058.14	−0.595230
$521$	5006.39	0.420986	0.210493	−	0.977595i	$-0.432493\pi$
$521$	5006.39	0.420986	0.210493	+	0.977595i	$0.432493\pi$
$522$	0	0
$523$	3643.50	0.304626	0.152313	−	0.988332i	$-0.451328\pi$
$523$	3643.50	0.304626	0.152313	+	0.988332i	$0.451328\pi$
$524$	−2221.09	−0.185169
$525$	0	0
$526$	9571.44	0.793412
$527$	−24059.4	−1.98870
$528$	0	0
$529$	529.000	0.0434783
$530$	−49245.6	−4.03602
$531$	0	0
$532$	−2863.13	−0.233332
$533$	805.652	0.0654722
$534$	0	0
$535$	12255.4	0.990366
$536$	10896.5	0.878091
$537$	0	0
$538$	−3031.89	−0.242963
$539$	−11082.6	−0.885646
$540$	0	0
$541$	−5995.24	−0.476443	−0.238221	−	0.971211i	$-0.576564\pi$
$541$	−5995.24	−0.476443	−0.238221	+	0.971211i	$0.576564\pi$
$542$	14296.9	1.13304
$543$	0	0
$544$	−13003.2	−1.02483
$545$	−2120.83	−0.166691
$546$	0	0
$547$	6643.19	0.519273	0.259636	−	0.965706i	$-0.416397\pi$
$547$	6643.19	0.519273	0.259636	+	0.965706i	$0.416397\pi$
$548$	−2356.22	−0.183673
$549$	0	0
$550$	−61798.8	−4.79110
$551$	−4422.76	−0.341953
$552$	0	0
$553$	15257.4	1.17325
$554$	5001.67	0.383575
$555$	0	0
$556$	−3971.96	−0.302965
$557$	−2254.46	−0.171498	−0.0857492	−	0.996317i	$-0.527328\pi$
$557$	−2254.46	−0.171498	−0.0857492	+	0.996317i	$0.527328\pi$
$558$	0	0
$559$	−5609.89	−0.424460
$560$	−40396.7	−3.04834
$561$	0	0
$562$	−18534.0	−1.39112
$563$	−7591.34	−0.568272	−0.284136	−	0.958784i	$-0.591707\pi$
$563$	−7591.34	−0.568272	−0.284136	+	0.958784i	$0.591707\pi$
$564$	0	0
$565$	14782.4	1.10071
$566$	−3608.80	−0.268002
$567$	0	0
$568$	−14159.4	−1.04598
$569$	5189.45	0.382343	0.191171	−	0.981557i	$-0.438771\pi$
$569$	5189.45	0.382343	0.191171	+	0.981557i	$0.438771\pi$
$570$	0	0
$571$	−20887.1	−1.53082	−0.765409	−	0.643544i	$-0.777463\pi$
$571$	−20887.1	−1.53082	−0.765409	+	0.643544i	$0.777463\pi$
$572$	−2806.21	−0.205128
$573$	0	0
$574$	3295.85	0.239662
$575$	−8146.49	−0.590838
$576$	0	0
$577$	−17317.6	−1.24947	−0.624734	−	0.780838i	$-0.714793\pi$
$577$	−17317.6	−1.24947	−0.624734	+	0.780838i	$0.714793\pi$
$578$	21492.3	1.54665
$579$	0	0
$580$	6232.16	0.446166
$581$	−184.175	−0.0131512
$582$	0	0
$583$	36339.6	2.58153
$584$	14010.9	0.992766
$585$	0	0
$586$	11184.6	0.788447
$587$	−8520.93	−0.599142	−0.299571	−	0.954074i	$-0.596844\pi$
$587$	−8520.93	−0.599142	−0.299571	+	0.954074i	$0.596844\pi$
$588$	0	0
$589$	9783.02	0.684384
$590$	−16296.5	−1.13715
$591$	0	0
$592$	−5542.28	−0.384774
$593$	7785.03	0.539111	0.269555	−	0.962985i	$-0.413123\pi$
$593$	7785.03	0.539111	0.269555	+	0.962985i	$0.413123\pi$
$594$	0	0
$595$	55027.7	3.79145
$596$	913.548	0.0627859
$597$	0	0
$598$	−1426.43	−0.0975435
$599$	18699.4	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$599$	18699.4	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$600$	0	0
$601$	8297.05	0.563135	0.281567	−	0.959541i	$-0.409146\pi$
$601$	8297.05	0.563135	0.281567	+	0.959541i	$0.409146\pi$
$602$	−22949.5	−1.55374
$603$	0	0
$604$	3817.82	0.257194
$605$	32560.5	2.18805
$606$	0	0
$607$	−15542.6	−1.03930	−0.519650	−	0.854379i	$-0.673938\pi$
$607$	−15542.6	−1.03930	−0.519650	+	0.854379i	$0.673938\pi$
$608$	5287.37	0.352683
$609$	0	0
$610$	−56281.6	−3.73570
$611$	3297.57	0.218340
$612$	0	0
$613$	21102.0	1.39038	0.695190	−	0.718826i	$-0.255320\pi$
$613$	21102.0	1.39038	0.695190	+	0.718826i	$0.255320\pi$
$614$	−12202.5	−0.802038
$615$	0	0
$616$	21307.0	1.39364
$617$	5872.11	0.383148	0.191574	−	0.981478i	$-0.438641\pi$
$617$	5872.11	0.383148	0.191574	+	0.981478i	$0.438641\pi$
$618$	0	0
$619$	−22326.9	−1.44974	−0.724872	−	0.688883i	$-0.758101\pi$
$619$	−22326.9	−1.44974	−0.724872	+	0.688883i	$0.758101\pi$
$620$	−13785.4	−0.892957
$621$	0	0
$622$	8672.71	0.559074
$623$	13681.7	0.879846
$624$	0	0
$625$	65554.8	4.19551
$626$	−26531.9	−1.69397
$627$	0	0
$628$	1015.31	0.0645147
$629$	7549.59	0.478572
$630$	0	0
$631$	−5221.71	−0.329434	−0.164717	−	0.986341i	$-0.552671\pi$
$631$	−5221.71	−0.329434	−0.164717	+	0.986341i	$0.552671\pi$
$632$	−11098.2	−0.698514
$633$	0	0
$634$	5530.64	0.346451
$635$	−31023.4	−1.93878
$636$	0	0
$637$	3939.38	0.245030
$638$	−17733.4	−1.10042
$639$	0	0
$640$	37765.9	2.33255
$641$	2982.45	0.183775	0.0918875	−	0.995769i	$-0.470710\pi$
$641$	2982.45	0.183775	0.0918875	+	0.995769i	$0.470710\pi$
$642$	0	0
$643$	−7759.82	−0.475921	−0.237961	−	0.971275i	$-0.576479\pi$
$643$	−7759.82	−0.475921	−0.237961	+	0.971275i	$0.576479\pi$
$644$	−1513.32	−0.0925979
$645$	0	0
$646$	−15304.7	−0.932128
$647$	−10968.0	−0.666458	−0.333229	−	0.942846i	$-0.608138\pi$
$647$	−10968.0	−0.666458	−0.333229	+	0.942846i	$0.608138\pi$
$648$	0	0
$649$	12025.6	0.727345
$650$	21966.7	1.32555
$651$	0	0
$652$	−3506.49	−0.210621
$653$	14340.6	0.859407	0.429703	−	0.902970i	$-0.358618\pi$
$653$	14340.6	0.859407	0.429703	+	0.902970i	$0.358618\pi$
$654$	0	0
$655$	−17357.8	−1.03546
$656$	−3354.09	−0.199627
$657$	0	0
$658$	13490.1	0.799236
$659$	8567.91	0.506462	0.253231	−	0.967406i	$-0.418507\pi$
$659$	8567.91	0.506462	0.253231	+	0.967406i	$0.418507\pi$
$660$	0	0
$661$	33151.8	1.95076	0.975382	−	0.220523i	$-0.0707762\pi$
$661$	33151.8	1.95076	0.975382	+	0.220523i	$0.0707762\pi$
$662$	13973.3	0.820372
$663$	0	0
$664$	133.968	0.00782979
$665$	−22375.3	−1.30478
$666$	0	0
$667$	−2337.66	−0.135704
$668$	10046.8	0.581917
$669$	0	0
$670$	−45880.8	−2.64557
$671$	41531.6	2.38944
$672$	0	0
$673$	−13629.8	−0.780667	−0.390334	−	0.920674i	$-0.627640\pi$
$673$	−13629.8	−0.780667	−0.390334	+	0.920674i	$0.627640\pi$
$674$	−33860.8	−1.93512
$675$	0	0
$676$	−5156.53	−0.293385
$677$	−22637.0	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$677$	−22637.0	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$678$	0	0
$679$	−9141.90	−0.516692
$680$	−40027.0	−2.25730
$681$	0	0
$682$	39225.7	2.20239
$683$	−5161.78	−0.289180	−0.144590	−	0.989492i	$-0.546186\pi$
$683$	−5161.78	−0.289180	−0.144590	+	0.989492i	$0.546186\pi$
$684$	0	0
$685$	−18413.8	−1.02709
$686$	−10363.4	−0.576786
$687$	0	0
$688$	23355.1	1.29419
$689$	−12917.1	−0.714228
$690$	0	0
$691$	−13302.9	−0.732368	−0.366184	−	0.930543i	$-0.619336\pi$
$691$	−13302.9	−0.732368	−0.366184	+	0.930543i	$0.619336\pi$
$692$	−5318.64	−0.292174
$693$	0	0
$694$	14508.4	0.793561
$695$	−31040.8	−1.69417
$696$	0	0
$697$	4568.89	0.248291
$698$	24282.3	1.31676
$699$	0	0
$700$	23304.8	1.25834
$701$	26588.4	1.43257	0.716285	−	0.697808i	$-0.245841\pi$
$701$	26588.4	1.43257	0.716285	+	0.697808i	$0.245841\pi$
$702$	0	0
$703$	−3069.81	−0.164694
$704$	−12166.3	−0.651329
$705$	0	0
$706$	−16103.5	−0.858446
$707$	−6375.35	−0.339137
$708$	0	0
$709$	−15897.2	−0.842074	−0.421037	−	0.907043i	$-0.638334\pi$
$709$	−15897.2	−0.842074	−0.421037	+	0.907043i	$0.638334\pi$
$710$	59619.8	3.15139
$711$	0	0
$712$	−9952.01	−0.523830
$713$	5170.84	0.271598
$714$	0	0
$715$	−21930.5	−1.14707
$716$	1151.19	0.0600864
$717$	0	0
$718$	2560.71	0.133099
$719$	22399.4	1.16183	0.580916	−	0.813963i	$-0.302694\pi$
$719$	22399.4	1.16183	0.580916	+	0.813963i	$0.302694\pi$
$720$	0	0
$721$	1370.97	0.0708151
$722$	−16318.9	−0.841175
$723$	0	0
$724$	3173.74	0.162916
$725$	35999.5	1.84412
$726$	0	0
$727$	35251.9	1.79838	0.899189	−	0.437560i	$-0.144157\pi$
$727$	35251.9	1.79838	0.899189	+	0.437560i	$0.144157\pi$
$728$	−7573.69	−0.385576
$729$	0	0
$730$	−58994.4	−2.99107
$731$	−31813.9	−1.60969
$732$	0	0
$733$	8709.67	0.438880	0.219440	−	0.975626i	$-0.429577\pi$
$733$	8709.67	0.438880	0.219440	+	0.975626i	$0.429577\pi$
$734$	43387.6	2.18184
$735$	0	0
$736$	2794.65	0.139962
$737$	33856.7	1.69217
$738$	0	0
$739$	−14437.1	−0.718641	−0.359320	−	0.933214i	$-0.616991\pi$
$739$	−14437.1	−0.718641	−0.359320	+	0.933214i	$0.616991\pi$
$740$	4325.70	0.214887
$741$	0	0
$742$	−52842.7	−2.61444
$743$	−28347.2	−1.39967	−0.699836	−	0.714303i	$-0.746744\pi$
$743$	−28347.2	−1.39967	−0.699836	+	0.714303i	$0.746744\pi$
$744$	0	0
$745$	7139.36	0.351095
$746$	40638.4	1.99448
$747$	0	0
$748$	−15914.1	−0.777912
$749$	13150.6	0.641536
$750$	0	0
$751$	−23848.6	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$751$	−23848.6	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$752$	−13728.5	−0.665726
$753$	0	0
$754$	6303.42	0.304452
$755$	29836.2	1.43821
$756$	0	0
$757$	−19676.7	−0.944730	−0.472365	−	0.881403i	$-0.656599\pi$
$757$	−19676.7	−0.944730	−0.472365	+	0.881403i	$0.656599\pi$
$758$	39368.5	1.88645
$759$	0	0
$760$	16275.8	0.776820
$761$	28325.8	1.34929	0.674646	−	0.738142i	$-0.264296\pi$
$761$	28325.8	1.34929	0.674646	+	0.738142i	$0.264296\pi$
$762$	0	0
$763$	−2275.74	−0.107978
$764$	8207.12	0.388643
$765$	0	0
$766$	40890.1	1.92875
$767$	−4274.57	−0.201233
$768$	0	0
$769$	−9264.43	−0.434439	−0.217220	−	0.976123i	$-0.569699\pi$
$769$	−9264.43	−0.434439	−0.217220	+	0.976123i	$0.569699\pi$
$770$	−89715.4	−4.19886
$771$	0	0
$772$	10181.8	0.474676
$773$	−8356.74	−0.388837	−0.194418	−	0.980919i	$-0.562282\pi$
$773$	−8356.74	−0.388837	−0.194418	+	0.980919i	$0.562282\pi$
$774$	0	0
$775$	−79629.9	−3.69083
$776$	6649.80	0.307621
$777$	0	0
$778$	−30416.4	−1.40165
$779$	−1857.80	−0.0854461
$780$	0	0
$781$	−43995.0	−2.01570
$782$	−8089.34	−0.369916
$783$	0	0
$784$	−16400.5	−0.747106
$785$	7934.62	0.360763
$786$	0	0
$787$	24846.8	1.12540	0.562701	−	0.826660i	$-0.309762\pi$
$787$	24846.8	1.12540	0.562701	+	0.826660i	$0.309762\pi$
$788$	8554.65	0.386735
$789$	0	0
$790$	46730.0	2.10453
$791$	15862.1	0.713011
$792$	0	0
$793$	−14762.7	−0.661081
$794$	−9719.68	−0.434431
$795$	0	0
$796$	8273.84	0.368415
$797$	17079.6	0.759085	0.379542	−	0.925174i	$-0.376081\pi$
$797$	17079.6	0.759085	0.379542	+	0.925174i	$0.376081\pi$
$798$	0	0
$799$	18700.7	0.828013
$800$	−43037.1	−1.90199
$801$	0	0
$802$	27855.6	1.22645
$803$	43533.5	1.91316
$804$	0	0
$805$	−11826.5	−0.517802
$806$	−13943.0	−0.609331
$807$	0	0
$808$	4637.42	0.201911
$809$	−26016.1	−1.13063	−0.565314	−	0.824876i	$-0.691245\pi$
$809$	−26016.1	−1.13063	−0.565314	+	0.824876i	$0.691245\pi$
$810$	0	0
$811$	25743.2	1.11463	0.557315	−	0.830301i	$-0.311832\pi$
$811$	25743.2	1.11463	0.557315	+	0.830301i	$0.311832\pi$
$812$	6687.39	0.289016
$813$	0	0
$814$	−12308.6	−0.529996
$815$	−27403.2	−1.17778
$816$	0	0
$817$	12936.2	0.553952
$818$	6085.57	0.260119
$819$	0	0
$820$	2617.84	0.111487
$821$	27463.9	1.16747	0.583737	−	0.811943i	$-0.301590\pi$
$821$	27463.9	1.16747	0.583737	+	0.811943i	$0.301590\pi$
$822$	0	0
$823$	10385.9	0.439889	0.219945	−	0.975512i	$-0.429412\pi$
$823$	10385.9	0.439889	0.219945	+	0.975512i	$0.429412\pi$
$824$	−997.242	−0.0421609
$825$	0	0
$826$	−17486.9	−0.736618
$827$	9008.63	0.378792	0.189396	−	0.981901i	$-0.439347\pi$
$827$	9008.63	0.378792	0.189396	+	0.981901i	$0.439347\pi$
$828$	0	0
$829$	−36173.8	−1.51552	−0.757762	−	0.652531i	$-0.773707\pi$
$829$	−36173.8	−1.51552	−0.757762	+	0.652531i	$0.773707\pi$
$830$	−564.088	−0.0235901
$831$	0	0
$832$	4324.58	0.180202
$833$	22340.4	0.929231
$834$	0	0
$835$	78515.2	3.25405
$836$	6470.99	0.267708
$837$	0	0
$838$	14143.5	0.583031
$839$	−22289.2	−0.917173	−0.458586	−	0.888650i	$-0.651644\pi$
$839$	−22289.2	−0.917173	−0.458586	+	0.888650i	$0.651644\pi$
$840$	0	0
$841$	−14058.8	−0.576440
$842$	51445.5	2.10562
$843$	0	0
$844$	2110.25	0.0860637
$845$	−40298.2	−1.64059
$846$	0	0
$847$	34938.8	1.41737
$848$	53776.6	2.17771
$849$	0	0
$850$	124574.	5.02689
$851$	−1622.56	−0.0653591
$852$	0	0
$853$	10796.1	0.433353	0.216677	−	0.976243i	$-0.430478\pi$
$853$	10796.1	0.433353	0.216677	+	0.976243i	$0.430478\pi$
$854$	−60392.6	−2.41990
$855$	0	0
$856$	−9565.68	−0.381949
$857$	38628.3	1.53969	0.769847	−	0.638229i	$-0.220333\pi$
$857$	38628.3	1.53969	0.769847	+	0.638229i	$0.220333\pi$
$858$	0	0
$859$	−3141.74	−0.124790	−0.0623951	−	0.998052i	$-0.519874\pi$
$859$	−3141.74	−0.124790	−0.0623951	+	0.998052i	$0.519874\pi$
$860$	−18228.5	−0.722774
$861$	0	0
$862$	2252.45	0.0890010
$863$	15237.5	0.601034	0.300517	−	0.953776i	$-0.402841\pi$
$863$	15237.5	0.601034	0.300517	+	0.953776i	$0.402841\pi$
$864$	0	0
$865$	−41565.0	−1.63382
$866$	28825.9	1.13111
$867$	0	0
$868$	−14792.3	−0.578437
$869$	−34483.3	−1.34610
$870$	0	0
$871$	−12034.5	−0.468168
$872$	1655.37	0.0642866
$873$	0	0
$874$	3289.28	0.127302
$875$	117852.	4.55327
$876$	0	0
$877$	14291.5	0.550275	0.275137	−	0.961405i	$-0.411277\pi$
$877$	14291.5	0.550275	0.275137	+	0.961405i	$0.411277\pi$
$878$	−6518.95	−0.250574
$879$	0	0
$880$	91301.0	3.49745
$881$	−45521.5	−1.74081	−0.870407	−	0.492333i	$-0.836144\pi$
$881$	−45521.5	−1.74081	−0.870407	+	0.492333i	$0.836144\pi$
$882$	0	0
$883$	−28192.8	−1.07448	−0.537239	−	0.843430i	$-0.680533\pi$
$883$	−28192.8	−1.07448	−0.537239	+	0.843430i	$0.680533\pi$
$884$	5656.76	0.215223
$885$	0	0
$886$	−31113.9	−1.17979
$887$	4138.35	0.156654	0.0783271	−	0.996928i	$-0.475042\pi$
$887$	4138.35	0.156654	0.0783271	+	0.996928i	$0.475042\pi$
$888$	0	0
$889$	−33289.5	−1.25590
$890$	41904.0	1.57823
$891$	0	0
$892$	−6785.28	−0.254695
$893$	−7604.06	−0.284950
$894$	0	0
$895$	8996.50	0.336000
$896$	40524.5	1.51097
$897$	0	0
$898$	−47251.4	−1.75590
$899$	−22850.1	−0.847712
$900$	0	0
$901$	−73253.5	−2.70858
$902$	−7448.97	−0.274971
$903$	0	0
$904$	−11538.1	−0.424503
$905$	24802.7	0.911016
$906$	0	0
$907$	−23216.2	−0.849924	−0.424962	−	0.905211i	$-0.639713\pi$
$907$	−23216.2	−0.849924	−0.424962	+	0.905211i	$0.639713\pi$
$908$	−7080.57	−0.258785
$909$	0	0
$910$	31889.8	1.16169
$911$	−17961.0	−0.653211	−0.326606	−	0.945161i	$-0.605905\pi$
$911$	−17961.0	−0.653211	−0.326606	+	0.945161i	$0.605905\pi$
$912$	0	0
$913$	416.255	0.0150888
$914$	−34536.2	−1.24984
$915$	0	0
$916$	12544.6	0.452495
$917$	−18625.6	−0.670745
$918$	0	0
$919$	27253.0	0.978230	0.489115	−	0.872219i	$-0.337320\pi$
$919$	27253.0	0.978230	0.489115	+	0.872219i	$0.337320\pi$
$920$	8602.60	0.308282
$921$	0	0
$922$	−5390.49	−0.192545
$923$	15638.3	0.557681
$924$	0	0
$925$	24987.0	0.888183
$926$	−19272.0	−0.683927
$927$	0	0
$928$	−12349.6	−0.436850
$929$	−33578.2	−1.18586	−0.592931	−	0.805254i	$-0.702029\pi$
$929$	−33578.2	−1.18586	−0.592931	+	0.805254i	$0.702029\pi$
$930$	0	0
$931$	−9084.05	−0.319783
$932$	7468.09	0.262474
$933$	0	0
$934$	−17494.6	−0.612892
$935$	−124369.	−4.35004
$936$	0	0
$937$	−18381.8	−0.640883	−0.320442	−	0.947268i	$-0.603831\pi$
$937$	−18381.8	−0.640883	−0.320442	+	0.947268i	$0.603831\pi$
$938$	−49232.1	−1.71374
$939$	0	0
$940$	10715.0	0.371791
$941$	−50541.5	−1.75091	−0.875454	−	0.483301i	$-0.839438\pi$
$941$	−50541.5	−1.75091	−0.875454	+	0.483301i	$0.839438\pi$
$942$	0	0
$943$	−981.945	−0.0339094
$944$	17795.9	0.613568
$945$	0	0
$946$	51868.4	1.78265
$947$	32533.7	1.11637	0.558186	−	0.829716i	$-0.311498\pi$
$947$	32533.7	1.11637	0.558186	+	0.829716i	$0.311498\pi$
$948$	0	0
$949$	−15474.2	−0.529309
$950$	−50654.3	−1.72994
$951$	0	0
$952$	−42950.7	−1.46223
$953$	−27408.1	−0.931623	−0.465812	−	0.884884i	$-0.654238\pi$
$953$	−27408.1	−0.931623	−0.465812	+	0.884884i	$0.654238\pi$
$954$	0	0
$955$	64138.5	2.17327
$956$	−11315.9	−0.382827
$957$	0	0
$958$	−65135.8	−2.19670
$959$	−19758.8	−0.665324
$960$	0	0
$961$	20752.7	0.696610
$962$	4375.17	0.146633
$963$	0	0
$964$	−5020.20	−0.167728
$965$	79570.3	2.65436
$966$	0	0
$967$	52463.9	1.74470	0.872351	−	0.488881i	$-0.162595\pi$
$967$	52463.9	1.74470	0.872351	+	0.488881i	$0.162595\pi$
$968$	−25414.4	−0.843853
$969$	0	0
$970$	−27999.7	−0.926820
$971$	−52736.1	−1.74293	−0.871463	−	0.490461i	$-0.836828\pi$
$971$	−52736.1	−1.74293	−0.871463	+	0.490461i	$0.836828\pi$
$972$	0	0
$973$	−33308.2	−1.09744
$974$	−50145.3	−1.64965
$975$	0	0
$976$	61459.9	2.01566
$977$	38462.0	1.25948	0.629739	−	0.776807i	$-0.283162\pi$
$977$	38462.0	1.25948	0.629739	+	0.776807i	$0.283162\pi$
$978$	0	0
$979$	−30922.0	−1.00947
$980$	12800.4	0.417240
$981$	0	0
$982$	36404.7	1.18302
$983$	−11477.7	−0.372414	−0.186207	−	0.982511i	$-0.559620\pi$
$983$	−11477.7	−0.372414	−0.186207	+	0.982511i	$0.559620\pi$
$984$	0	0
$985$	66854.5	2.16260
$986$	35747.0	1.15458
$987$	0	0
$988$	−2300.15	−0.0740663
$989$	6837.44	0.219836
$990$	0	0
$991$	43500.8	1.39440	0.697199	−	0.716878i	$-0.254430\pi$
$991$	43500.8	1.39440	0.697199	+	0.716878i	$0.254430\pi$
$992$	27317.0	0.874312
$993$	0	0
$994$	63974.6	2.04140
$995$	64659.9	2.06016
$996$	0	0
$997$	10602.9	0.336806	0.168403	−	0.985718i	$-0.446139\pi$
$997$	10602.9	0.336806	0.168403	+	0.985718i	$0.446139\pi$
$998$	41118.5	1.30419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.h.1.4	yes	5
3.2	odd	2		207.4.a.g.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.a.g.1.2	✓	5	3.2	odd	2
207.4.a.h.1.4	yes	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+3.28650 q^{2} +2.80110 q^{4} +21.8905 q^{5} +23.4895 q^{7} -17.0862 q^{8} +71.9433 q^{10} -53.0888 q^{11} +18.8707 q^{13} +77.1983 q^{14} -78.5626 q^{16} +107.017 q^{17} -43.5150 q^{19} +61.3175 q^{20} -174.477 q^{22} -23.0000 q^{23} +354.195 q^{25} +62.0186 q^{26} +65.7964 q^{28} +101.638 q^{29} -224.819 q^{31} -121.507 q^{32} +351.710 q^{34} +514.198 q^{35} +70.5460 q^{37} -143.012 q^{38} -374.026 q^{40} +42.6933 q^{41} -297.280 q^{43} -148.707 q^{44} -75.5896 q^{46} +174.746 q^{47} +208.757 q^{49} +1164.06 q^{50} +52.8587 q^{52} -684.506 q^{53} -1162.14 q^{55} -401.346 q^{56} +334.032 q^{58} -226.519 q^{59} -782.305 q^{61} -738.869 q^{62} +229.169 q^{64} +413.090 q^{65} -637.736 q^{67} +299.764 q^{68} +1689.91 q^{70} +828.705 q^{71} -820.013 q^{73} +231.850 q^{74} -121.890 q^{76} -1247.03 q^{77} +649.539 q^{79} -1719.78 q^{80} +140.311 q^{82} -7.84073 q^{83} +2342.65 q^{85} -977.012 q^{86} +907.086 q^{88} +582.459 q^{89} +443.264 q^{91} -64.4253 q^{92} +574.302 q^{94} -952.567 q^{95} -389.191 q^{97} +686.079 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 16 q^{4} + 20 q^{5} - 10 q^{7} + 48 q^{8} + 50 q^{10} + 46 q^{11} + 54 q^{13} + 164 q^{14} - 60 q^{16} + 250 q^{17} - 28 q^{19} + 242 q^{20} - 10 q^{22} - 115 q^{23} + 239 q^{25} + 368 q^{26}+ \cdots - 2400 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 16 * q^4 + 20 * q^5 - 10 * q^7 + 48 * q^8 + 50 * q^10 + 46 * q^11 + 54 * q^13 + 164 * q^14 - 60 * q^16 + 250 * q^17 - 28 * q^19 + 242 * q^20 - 10 * q^22 - 115 * q^23 + 239 * q^25 + 368 * q^26 + 224 * q^28 + 460 * q^29 - 360 * q^31 + 268 * q^32 - 20 * q^34 + 964 * q^35 - 92 * q^37 + 266 * q^38 - 278 * q^40 + 788 * q^41 + 108 * q^43 + 354 * q^44 - 92 * q^46 + 724 * q^47 + 149 * q^49 + 128 * q^50 - 288 * q^52 + 580 * q^53 - 672 * q^55 - 1172 * q^56 - 48 * q^58 - 408 * q^59 - 620 * q^61 - 1088 * q^62 + 696 * q^64 + 184 * q^65 + 100 * q^67 - 1584 * q^68 + 728 * q^70 - 124 * q^71 - 1946 * q^73 - 2866 * q^74 + 1362 * q^76 - 396 * q^77 - 198 * q^79 - 2398 * q^80 + 76 * q^82 - 578 * q^83 + 376 * q^85 - 3910 * q^86 + 1366 * q^88 + 930 * q^89 + 1972 * q^91 - 368 * q^92 + 976 * q^94 - 908 * q^95 + 2618 * q^97 - 2400 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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