LMFDB - Embedded newform 507.4.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,4,Mod(1,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,30,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 $ x^10 - 70x^8 + 1645x^6 - 14700x^4 + 44100x^2 - 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.36472$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.36472 q^{2} +3.00000 q^{3} +20.7803 q^{4} +2.69631 q^{5} -16.0942 q^{6} +15.2025 q^{7} -68.5626 q^{8} +9.00000 q^{9} -14.4650 q^{10} -66.8848 q^{11} +62.3408 q^{12} -81.5570 q^{14} +8.08894 q^{15} +201.577 q^{16} +4.16354 q^{17} -48.2825 q^{18} +26.0850 q^{19} +56.0301 q^{20} +45.6074 q^{21} +358.819 q^{22} +47.3242 q^{23} -205.688 q^{24} -117.730 q^{25} +27.0000 q^{27} +315.911 q^{28} +257.007 q^{29} -43.3949 q^{30} +206.242 q^{31} -532.906 q^{32} -200.655 q^{33} -22.3362 q^{34} +40.9906 q^{35} +187.022 q^{36} +175.686 q^{37} -139.939 q^{38} -184.866 q^{40} +156.463 q^{41} -244.671 q^{42} +51.9845 q^{43} -1389.88 q^{44} +24.2668 q^{45} -253.881 q^{46} -354.222 q^{47} +604.732 q^{48} -111.885 q^{49} +631.588 q^{50} +12.4906 q^{51} -10.4723 q^{53} -144.848 q^{54} -180.342 q^{55} -1042.32 q^{56} +78.2550 q^{57} -1378.77 q^{58} +445.114 q^{59} +168.090 q^{60} +119.696 q^{61} -1106.43 q^{62} +136.822 q^{63} +1246.28 q^{64} +1076.46 q^{66} -22.4078 q^{67} +86.5195 q^{68} +141.973 q^{69} -219.903 q^{70} +285.207 q^{71} -617.064 q^{72} +740.989 q^{73} -942.507 q^{74} -353.190 q^{75} +542.053 q^{76} -1016.81 q^{77} -547.679 q^{79} +543.516 q^{80} +81.0000 q^{81} -839.378 q^{82} +603.056 q^{83} +947.734 q^{84} +11.2262 q^{85} -278.882 q^{86} +771.021 q^{87} +4585.80 q^{88} +215.668 q^{89} -130.185 q^{90} +983.409 q^{92} +618.726 q^{93} +1900.31 q^{94} +70.3333 q^{95} -1598.72 q^{96} +1447.50 q^{97} +600.233 q^{98} -601.964 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36}+ \cdots - 2760 q^{95}+O(q^{100}) $ 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 + 80 * q^10 + 180 * q^12 - 60 * q^14 + 500 * q^16 + 210 * q^17 + 580 * q^22 - 120 * q^23 + 960 * q^25 + 270 * q^27 + 990 * q^29 + 240 * q^30 - 120 * q^35 + 540 * q^36 + 1380 * q^38 + 2000 * q^40 - 180 * q^42 - 740 * q^43 + 1500 * q^48 + 1550 * q^49 + 630 * q^51 + 330 * q^53 + 520 * q^55 - 5340 * q^56 + 2750 * q^61 - 1560 * q^62 + 3140 * q^64 + 1740 * q^66 + 1200 * q^68 - 360 * q^69 - 4380 * q^74 + 2880 * q^75 + 4320 * q^77 + 1100 * q^79 + 810 * q^81 - 4780 * q^82 + 2970 * q^87 + 6340 * q^88 + 720 * q^90 - 1740 * q^92 + 6460 * q^94 - 2760 * q^95

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 $

$2$	−5.36472	−1.89672	−0.948358	−	0.317201i	$-0.897257\pi$
$2$	−5.36472	−1.89672	−0.948358	+	0.317201i	$0.897257\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	20.7803	2.59753
$5$	2.69631	0.241165	0.120583	−	0.992703i	$-0.461524\pi$
$5$	2.69631	0.241165	0.120583	+	0.992703i	$0.461524\pi$
$6$	−16.0942	−1.09507
$7$	15.2025	0.820856	0.410428	−	0.911893i	$-0.365379\pi$
$7$	15.2025	0.820856	0.410428	+	0.911893i	$0.365379\pi$
$8$	−68.5626	−3.03007
$9$	9.00000	0.333333
$10$	−14.4650	−0.457423
$11$	−66.8848	−1.83332	−0.916661	−	0.399666i	$-0.869126\pi$
$11$	−66.8848	−1.83332	−0.916661	+	0.399666i	$0.869126\pi$
$12$	62.3408	1.49969
$13$	0	0
$13$	0	0
$14$	−81.5570	−1.55693
$15$	8.08894	0.139237
$16$	201.577	3.14965
$17$	4.16354	0.0594004	0.0297002	−	0.999559i	$-0.490545\pi$
$17$	4.16354	0.0594004	0.0297002	+	0.999559i	$0.490545\pi$
$18$	−48.2825	−0.632239
$19$	26.0850	0.314963	0.157482	−	0.987522i	$-0.449662\pi$
$19$	26.0850	0.314963	0.157482	+	0.987522i	$0.449662\pi$
$20$	56.0301	0.626436
$21$	45.6074	0.473921
$22$	358.819	3.47729
$23$	47.3242	0.429034	0.214517	−	0.976720i	$-0.431182\pi$
$23$	47.3242	0.429034	0.214517	+	0.976720i	$0.431182\pi$
$24$	−205.688	−1.74941
$25$	−117.730	−0.941839
$26$	0	0
$27$	27.0000	0.192450
$28$	315.911	2.13220
$29$	257.007	1.64569	0.822845	−	0.568266i	$-0.192386\pi$
$29$	257.007	1.64569	0.822845	+	0.568266i	$0.192386\pi$
$30$	−43.3949	−0.264093
$31$	206.242	1.19491	0.597455	−	0.801903i	$-0.296179\pi$
$31$	206.242	1.19491	0.597455	+	0.801903i	$0.296179\pi$
$32$	−532.906	−2.94392
$33$	−200.655	−1.05847
$34$	−22.3362	−0.112666
$35$	40.9906	0.197962
$36$	187.022	0.865845
$37$	175.686	0.780611	0.390305	−	0.920685i	$-0.372369\pi$
$37$	175.686	0.780611	0.390305	+	0.920685i	$0.372369\pi$
$38$	−139.939	−0.597396
$39$	0	0
$40$	−184.866	−0.730748
$41$	156.463	0.595984	0.297992	−	0.954568i	$-0.403683\pi$
$41$	156.463	0.595984	0.297992	+	0.954568i	$0.403683\pi$
$42$	−244.671	−0.898895
$43$	51.9845	0.184362	0.0921809	−	0.995742i	$-0.470616\pi$
$43$	51.9845	0.184362	0.0921809	+	0.995742i	$0.470616\pi$
$44$	−1389.88	−4.76211
$45$	24.2668	0.0803885
$46$	−253.881	−0.813755
$47$	−354.222	−1.09933	−0.549666	−	0.835384i	$-0.685245\pi$
$47$	−354.222	−1.09933	−0.549666	+	0.835384i	$0.685245\pi$
$48$	604.732	1.81845
$49$	−111.885	−0.326196
$50$	631.588	1.78640
$51$	12.4906	0.0342948
$52$	0	0
$53$	−10.4723	−0.0271412	−0.0135706	−	0.999908i	$-0.504320\pi$
$53$	−10.4723	−0.0271412	−0.0135706	+	0.999908i	$0.504320\pi$
$54$	−144.848	−0.365023
$55$	−180.342	−0.442134
$56$	−1042.32	−2.48725
$57$	78.2550	0.181844
$58$	−1378.77	−3.12141
$59$	445.114	0.982185	0.491092	−	0.871107i	$-0.336598\pi$
$59$	445.114	0.982185	0.491092	+	0.871107i	$0.336598\pi$
$60$	168.090	0.361673
$61$	119.696	0.251238	0.125619	−	0.992079i	$-0.459908\pi$
$61$	119.696	0.251238	0.125619	+	0.992079i	$0.459908\pi$
$62$	−1106.43	−2.26640
$63$	136.822	0.273619
$64$	1246.28	2.43413
$65$	0	0
$66$	1076.46	2.00761
$67$	−22.4078	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$67$	−22.4078	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$68$	86.5195	0.154295
$69$	141.973	0.247703
$70$	−219.903	−0.375478
$71$	285.207	0.476731	0.238365	−	0.971176i	$-0.423388\pi$
$71$	285.207	0.476731	0.238365	+	0.971176i	$0.423388\pi$
$72$	−617.064	−1.01002
$73$	740.989	1.18803	0.594015	−	0.804454i	$-0.297542\pi$
$73$	740.989	1.18803	0.594015	+	0.804454i	$0.297542\pi$
$74$	−942.507	−1.48060
$75$	−353.190	−0.543771
$76$	542.053	0.818128
$77$	−1016.81	−1.50489
$78$	0	0
$79$	−547.679	−0.779983	−0.389992	−	0.920818i	$-0.627522\pi$
$79$	−547.679	−0.779983	−0.389992	+	0.920818i	$0.627522\pi$
$80$	543.516	0.759586
$81$	81.0000	0.111111
$82$	−839.378	−1.13041
$83$	603.056	0.797518	0.398759	−	0.917056i	$-0.369441\pi$
$83$	603.056	0.797518	0.398759	+	0.917056i	$0.369441\pi$
$84$	947.734	1.23103
$85$	11.2262	0.0143253
$86$	−278.882	−0.349682
$87$	771.021	0.950139
$88$	4585.80	5.55509
$89$	215.668	0.256863	0.128431	−	0.991718i	$-0.459006\pi$
$89$	215.668	0.256863	0.128431	+	0.991718i	$0.459006\pi$
$90$	−130.185	−0.152474
$91$	0	0
$92$	983.409	1.11443
$93$	618.726	0.689881
$94$	1900.31	2.08512
$95$	70.3333	0.0759583
$96$	−1598.72	−1.69967
$97$	1447.50	1.51517	0.757586	−	0.652735i	$-0.226378\pi$
$97$	1447.50	1.51517	0.757586	+	0.652735i	$0.226378\pi$
$98$	600.233	0.618700
$99$	−601.964	−0.611107
$100$	−2446.46	−2.44646
$101$	−883.450	−0.870362	−0.435181	−	0.900343i	$-0.643316\pi$
$101$	−883.450	−0.870362	−0.435181	+	0.900343i	$0.643316\pi$
$102$	−67.0087	−0.0650476
$103$	1251.74	1.19745	0.598726	−	0.800954i	$-0.295674\pi$
$103$	1251.74	1.19745	0.598726	+	0.800954i	$0.295674\pi$
$104$	0	0
$105$	122.972	0.114293
$106$	56.1812	0.0514792
$107$	−341.614	−0.308645	−0.154323	−	0.988021i	$-0.549319\pi$
$107$	−341.614	−0.308645	−0.154323	+	0.988021i	$0.549319\pi$
$108$	561.067	0.499896
$109$	775.177	0.681179	0.340589	−	0.940212i	$-0.389373\pi$
$109$	775.177	0.681179	0.340589	+	0.940212i	$0.389373\pi$
$110$	967.487	0.838603
$111$	527.058	0.450686
$112$	3064.47	2.58541
$113$	1279.05	1.06480	0.532402	−	0.846492i	$-0.321290\pi$
$113$	1279.05	1.06480	0.532402	+	0.846492i	$0.321290\pi$
$114$	−419.816	−0.344907
$115$	127.601	0.103468
$116$	5340.67	4.27473
$117$	0	0
$118$	−2387.91	−1.86293
$119$	63.2961	0.0487592
$120$	−554.599	−0.421898
$121$	3142.58	2.36107
$122$	−642.137	−0.476528
$123$	469.388	0.344091
$124$	4285.77	3.10382
$125$	−654.476	−0.468305
$126$	−734.013	−0.518977
$127$	−1113.82	−0.778233	−0.389117	−	0.921188i	$-0.627220\pi$
$127$	−1113.82	−0.778233	−0.389117	+	0.921188i	$0.627220\pi$
$128$	−2422.68	−1.67294
$129$	155.953	0.106441
$130$	0	0
$131$	2100.12	1.40068	0.700339	−	0.713811i	$-0.253032\pi$
$131$	2100.12	1.40068	0.700339	+	0.713811i	$0.253032\pi$
$132$	−4169.65	−2.74941
$133$	396.556	0.258540
$134$	120.211	0.0774977
$135$	72.8004	0.0464123
$136$	−285.463	−0.179987
$137$	−1205.02	−0.751471	−0.375736	−	0.926727i	$-0.622610\pi$
$137$	−1205.02	−0.751471	−0.375736	+	0.926727i	$0.622610\pi$
$138$	−761.643	−0.469822
$139$	322.890	0.197030	0.0985149	−	0.995136i	$-0.468591\pi$
$139$	322.890	0.197030	0.0985149	+	0.995136i	$0.468591\pi$
$140$	851.796	0.514213
$141$	−1062.67	−0.634700
$142$	−1530.06	−0.904223
$143$	0	0
$144$	1814.20	1.04988
$145$	692.971	0.396884
$146$	−3975.20	−2.25336
$147$	−335.655	−0.188329
$148$	3650.80	2.02766
$149$	1128.86	0.620669	0.310335	−	0.950627i	$-0.399559\pi$
$149$	1128.86	0.620669	0.310335	+	0.950627i	$0.399559\pi$
$150$	1894.77	1.03138
$151$	−2940.44	−1.58470	−0.792350	−	0.610066i	$-0.791143\pi$
$151$	−2940.44	−1.58470	−0.792350	+	0.610066i	$0.791143\pi$
$152$	−1788.46	−0.954361
$153$	37.4719	0.0198001
$154$	5454.93	2.85436
$155$	556.093	0.288171
$156$	0	0
$157$	629.388	0.319940	0.159970	−	0.987122i	$-0.448860\pi$
$157$	629.388	0.319940	0.159970	+	0.987122i	$0.448860\pi$
$158$	2938.15	1.47941
$159$	−31.4170	−0.0156700
$160$	−1436.88	−0.709972
$161$	719.444	0.352175
$162$	−434.543	−0.210746
$163$	−394.912	−0.189766	−0.0948832	−	0.995488i	$-0.530248\pi$
$163$	−394.912	−0.189766	−0.0948832	+	0.995488i	$0.530248\pi$
$164$	3251.33	1.54809
$165$	−541.027	−0.255266
$166$	−3235.23	−1.51267
$167$	−151.860	−0.0703669	−0.0351834	−	0.999381i	$-0.511202\pi$
$167$	−151.860	−0.0703669	−0.0351834	+	0.999381i	$0.511202\pi$
$168$	−3126.96	−1.43601
$169$	0	0
$170$	−60.2255	−0.0271711
$171$	234.765	0.104988
$172$	1080.25	0.478886
$173$	538.813	0.236793	0.118397	−	0.992966i	$-0.462225\pi$
$173$	538.813	0.236793	0.118397	+	0.992966i	$0.462225\pi$
$174$	−4136.31	−1.80214
$175$	−1789.78	−0.773114
$176$	−13482.5	−5.77432
$177$	1335.34	0.567065
$178$	−1157.00	−0.487196
$179$	−2220.80	−0.927319	−0.463659	−	0.886014i	$-0.653464\pi$
$179$	−2220.80	−0.927319	−0.463659	+	0.886014i	$0.653464\pi$
$180$	504.271	0.208812
$181$	−3822.78	−1.56986	−0.784932	−	0.619582i	$-0.787302\pi$
$181$	−3822.78	−1.56986	−0.784932	+	0.619582i	$0.787302\pi$
$182$	0	0
$183$	359.089	0.145052
$184$	−3244.67	−1.30000
$185$	473.704	0.188256
$186$	−3319.30	−1.30851
$187$	−278.478	−0.108900
$188$	−7360.84	−2.85555
$189$	410.467	0.157974
$190$	−377.319	−0.144071
$191$	3464.19	1.31236	0.656178	−	0.754606i	$-0.272172\pi$
$191$	3464.19	1.31236	0.656178	+	0.754606i	$0.272172\pi$
$192$	3738.83	1.40535
$193$	4697.40	1.75195	0.875975	−	0.482357i	$-0.160219\pi$
$193$	4697.40	1.75195	0.875975	+	0.482357i	$0.160219\pi$
$194$	−7765.46	−2.87385
$195$	0	0
$196$	−2325.00	−0.847304
$197$	−2887.89	−1.04443	−0.522217	−	0.852813i	$-0.674895\pi$
$197$	−2887.89	−1.04443	−0.522217	+	0.852813i	$0.674895\pi$
$198$	3229.37	1.15910
$199$	63.0092	0.0224453	0.0112226	−	0.999937i	$-0.496428\pi$
$199$	63.0092	0.0224453	0.0112226	+	0.999937i	$0.496428\pi$
$200$	8071.87	2.85384
$201$	−67.2233	−0.0235899
$202$	4739.47	1.65083
$203$	3907.14	1.35087
$204$	259.558	0.0890820
$205$	421.872	0.143731
$206$	−6715.24	−2.27123
$207$	425.918	0.143011
$208$	0	0
$209$	−1744.69	−0.577429
$210$	−659.710	−0.216782
$211$	−1049.70	−0.342484	−0.171242	−	0.985229i	$-0.554778\pi$
$211$	−1049.70	−0.342484	−0.171242	+	0.985229i	$0.554778\pi$
$212$	−217.618	−0.0705002
$213$	855.622	0.275241
$214$	1832.66	0.585412
$215$	140.166	0.0444617
$216$	−1851.19	−0.583137
$217$	3135.39	0.980848
$218$	−4158.61	−1.29200
$219$	2222.97	0.685909
$220$	−3747.56	−1.14846
$221$	0	0
$222$	−2827.52	−0.854823
$223$	2313.49	0.694722	0.347361	−	0.937732i	$-0.387078\pi$
$223$	2313.49	0.694722	0.347361	+	0.937732i	$0.387078\pi$
$224$	−8101.49	−2.41653
$225$	−1059.57	−0.313946
$226$	−6861.74	−2.01963
$227$	3799.46	1.11092	0.555460	−	0.831543i	$-0.312542\pi$
$227$	3799.46	1.11092	0.555460	+	0.831543i	$0.312542\pi$
$228$	1626.16	0.472347
$229$	−4321.07	−1.24692	−0.623459	−	0.781856i	$-0.714273\pi$
$229$	−4321.07	−1.24692	−0.623459	+	0.781856i	$0.714273\pi$
$230$	−684.543	−0.196250
$231$	−3050.44	−0.868850
$232$	−17621.1	−4.98655
$233$	5279.77	1.48450	0.742251	−	0.670122i	$-0.233758\pi$
$233$	5279.77	1.48450	0.742251	+	0.670122i	$0.233758\pi$
$234$	0	0
$235$	−955.094	−0.265121
$236$	9249.59	2.55126
$237$	−1643.04	−0.450323
$238$	−339.566	−0.0924823
$239$	1547.92	0.418939	0.209469	−	0.977815i	$-0.432826\pi$
$239$	1547.92	0.418939	0.209469	+	0.977815i	$0.432826\pi$
$240$	1630.55	0.438547
$241$	−4918.01	−1.31451	−0.657255	−	0.753669i	$-0.728282\pi$
$241$	−4918.01	−1.31451	−0.657255	+	0.753669i	$0.728282\pi$
$242$	−16859.1	−4.47828
$243$	243.000	0.0641500
$244$	2487.32	0.652600
$245$	−301.677	−0.0786671
$246$	−2518.14	−0.652644
$247$	0	0
$248$	−14140.5	−3.62066
$249$	1809.17	0.460447
$250$	3511.08	0.888241
$251$	1155.78	0.290646	0.145323	−	0.989384i	$-0.453578\pi$
$251$	1155.78	0.290646	0.145323	+	0.989384i	$0.453578\pi$
$252$	2843.20	0.710734
$253$	−3165.27	−0.786556
$254$	5975.34	1.47609
$255$	33.6786	0.00827073
$256$	3026.80	0.738965
$257$	2351.95	0.570859	0.285429	−	0.958400i	$-0.407864\pi$
$257$	2351.95	0.570859	0.285429	+	0.958400i	$0.407864\pi$
$258$	−836.647	−0.201889
$259$	2670.86	0.640769
$260$	0	0
$261$	2313.06	0.548563
$262$	−11266.6	−2.65669
$263$	5521.88	1.29465	0.647326	−	0.762213i	$-0.275887\pi$
$263$	5521.88	1.29465	0.647326	+	0.762213i	$0.275887\pi$
$264$	13757.4	3.20723
$265$	−28.2367	−0.00654553
$266$	−2127.41	−0.490376
$267$	647.004	0.148300
$268$	−465.639	−0.106132
$269$	3916.95	0.887810	0.443905	−	0.896074i	$-0.353593\pi$
$269$	3916.95	0.887810	0.443905	+	0.896074i	$0.353593\pi$
$270$	−390.554	−0.0880310
$271$	−2777.53	−0.622593	−0.311297	−	0.950313i	$-0.600763\pi$
$271$	−2777.53	−0.622593	−0.311297	+	0.950313i	$0.600763\pi$
$272$	839.276	0.187090
$273$	0	0
$274$	6464.58	1.42533
$275$	7874.34	1.72669
$276$	2950.23	0.643416
$277$	6583.08	1.42794	0.713969	−	0.700177i	$-0.246896\pi$
$277$	6583.08	1.42794	0.713969	+	0.700177i	$0.246896\pi$
$278$	−1732.21	−0.373710
$279$	1856.18	0.398303
$280$	−2810.42	−0.599839
$281$	2871.66	0.609640	0.304820	−	0.952410i	$-0.401404\pi$
$281$	2871.66	0.609640	0.304820	+	0.952410i	$0.401404\pi$
$282$	5700.92	1.20385
$283$	7518.04	1.57916	0.789578	−	0.613651i	$-0.210300\pi$
$283$	7518.04	1.57916	0.789578	+	0.613651i	$0.210300\pi$
$284$	5926.69	1.23832
$285$	211.000	0.0438546
$286$	0	0
$287$	2378.62	0.489217
$288$	−4796.16	−0.981307
$289$	−4895.66	−0.996472
$290$	−3717.60	−0.752776
$291$	4342.51	0.874785
$292$	15397.9	3.08595
$293$	4506.57	0.898555	0.449278	−	0.893392i	$-0.351681\pi$
$293$	4506.57	0.898555	0.449278	+	0.893392i	$0.351681\pi$
$294$	1800.70	0.357207
$295$	1200.17	0.236869
$296$	−12045.5	−2.36530
$297$	−1805.89	−0.352823
$298$	−6056.02	−1.17723
$299$	0	0
$300$	−7339.38	−1.41246
$301$	790.292	0.151335
$302$	15774.7	3.00573
$303$	−2650.35	−0.502504
$304$	5258.15	0.992024
$305$	322.738	0.0605900
$306$	−201.026	−0.0375552
$307$	9538.89	1.77333	0.886667	−	0.462409i	$-0.153015\pi$
$307$	9538.89	1.77333	0.886667	+	0.462409i	$0.153015\pi$
$308$	−21129.7	−3.90901
$309$	3755.22	0.691350
$310$	−2983.29	−0.546578
$311$	−7466.28	−1.36133	−0.680666	−	0.732594i	$-0.738309\pi$
$311$	−7466.28	−1.36133	−0.680666	+	0.732594i	$0.738309\pi$
$312$	0	0
$313$	−1821.65	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$313$	−1821.65	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$314$	−3376.49	−0.606836
$315$	368.915	0.0659874
$316$	−11380.9	−2.02603
$317$	3125.14	0.553708	0.276854	−	0.960912i	$-0.410708\pi$
$317$	3125.14	0.553708	0.276854	+	0.960912i	$0.410708\pi$
$318$	168.543	0.0297215
$319$	−17189.9	−3.01708
$320$	3360.35	0.587029
$321$	−1024.84	−0.178196
$322$	−3859.62	−0.667976
$323$	108.606	0.0187090
$324$	1683.20	0.288615
$325$	0	0
$326$	2118.60	0.359933
$327$	2325.53	0.393279
$328$	−10727.5	−1.80587
$329$	−5385.05	−0.902394
$330$	2902.46	0.484167
$331$	−1553.67	−0.257999	−0.128999	−	0.991645i	$-0.541177\pi$
$331$	−1553.67	−0.257999	−0.128999	+	0.991645i	$0.541177\pi$
$332$	12531.7	2.07158
$333$	1581.17	0.260204
$334$	814.686	0.133466
$335$	−60.4183	−0.00985375
$336$	9193.42	1.49269
$337$	−3190.43	−0.515709	−0.257855	−	0.966184i	$-0.583016\pi$
$337$	−3190.43	−0.515709	−0.257855	+	0.966184i	$0.583016\pi$
$338$	0	0
$339$	3837.15	0.614764
$340$	233.283	0.0372105
$341$	−13794.5	−2.19065
$342$	−1259.45	−0.199132
$343$	−6915.37	−1.08862
$344$	−3564.19	−0.558629
$345$	382.802	0.0597373
$346$	−2890.59	−0.449130
$347$	−5718.68	−0.884712	−0.442356	−	0.896840i	$-0.645857\pi$
$347$	−5718.68	−0.884712	−0.442356	+	0.896840i	$0.645857\pi$
$348$	16022.0	2.46802
$349$	−3328.46	−0.510511	−0.255256	−	0.966874i	$-0.582160\pi$
$349$	−3328.46	−0.510511	−0.255256	+	0.966874i	$0.582160\pi$
$350$	9601.70	1.46638
$351$	0	0
$352$	35643.4	5.39715
$353$	−12306.5	−1.85555	−0.927774	−	0.373142i	$-0.878281\pi$
$353$	−12306.5	−1.85555	−0.927774	+	0.373142i	$0.878281\pi$
$354$	−7163.74	−1.07556
$355$	769.008	0.114971
$356$	4481.64	0.667210
$357$	189.888	0.0281511
$358$	11914.0	1.75886
$359$	8539.97	1.25549	0.627747	−	0.778418i	$-0.283977\pi$
$359$	8539.97	1.25549	0.627747	+	0.778418i	$0.283977\pi$
$360$	−1663.80	−0.243583
$361$	−6178.57	−0.900798
$362$	20508.2	2.97759
$363$	9427.74	1.36316
$364$	0	0
$365$	1997.94	0.286512
$366$	−1926.41	−0.275123
$367$	2496.65	0.355107	0.177553	−	0.984111i	$-0.443182\pi$
$367$	2496.65	0.355107	0.177553	+	0.984111i	$0.443182\pi$
$368$	9539.49	1.35130
$369$	1408.16	0.198661
$370$	−2541.29	−0.357069
$371$	−159.205	−0.0222790
$372$	12857.3	1.79199
$373$	−1142.91	−0.158653	−0.0793264	−	0.996849i	$-0.525277\pi$
$373$	−1142.91	−0.158653	−0.0793264	+	0.996849i	$0.525277\pi$
$374$	1493.96	0.206552
$375$	−1963.43	−0.270376
$376$	24286.4	3.33105
$377$	0	0
$378$	−2202.04	−0.299632
$379$	−12181.8	−1.65102	−0.825512	−	0.564384i	$-0.809114\pi$
$379$	−12181.8	−1.65102	−0.825512	+	0.564384i	$0.809114\pi$
$380$	1461.54	0.197304
$381$	−3341.46	−0.449313
$382$	−18584.4	−2.48917
$383$	−10180.6	−1.35824	−0.679118	−	0.734029i	$-0.737638\pi$
$383$	−10180.6	−1.35824	−0.679118	+	0.734029i	$0.737638\pi$
$384$	−7268.04	−0.965874
$385$	−2741.65	−0.362928
$386$	−25200.3	−3.32295
$387$	467.860	0.0614539
$388$	30079.5	3.93571
$389$	5845.83	0.761941	0.380971	−	0.924587i	$-0.375590\pi$
$389$	5845.83	0.761941	0.380971	+	0.924587i	$0.375590\pi$
$390$	0	0
$391$	197.036	0.0254848
$392$	7671.13	0.988395
$393$	6300.37	0.808681
$394$	15492.7	1.98100
$395$	−1476.71	−0.188105
$396$	−12509.0	−1.58737
$397$	−2500.92	−0.316166	−0.158083	−	0.987426i	$-0.550531\pi$
$397$	−2500.92	−0.316166	−0.158083	+	0.987426i	$0.550531\pi$
$398$	−338.027	−0.0425723
$399$	1189.67	0.149268
$400$	−23731.7	−2.96646
$401$	−9189.25	−1.14436	−0.572181	−	0.820127i	$-0.693902\pi$
$401$	−9189.25	−1.14436	−0.572181	+	0.820127i	$0.693902\pi$
$402$	360.634	0.0447433
$403$	0	0
$404$	−18358.3	−2.26080
$405$	218.401	0.0267962
$406$	−20960.7	−2.56223
$407$	−11750.7	−1.43111
$408$	−856.390	−0.103916
$409$	9214.38	1.11399	0.556995	−	0.830516i	$-0.311954\pi$
$409$	9214.38	1.11399	0.556995	+	0.830516i	$0.311954\pi$
$410$	−2263.23	−0.272617
$411$	−3615.05	−0.433862
$412$	26011.5	3.11042
$413$	6766.83	0.806232
$414$	−2284.93	−0.271252
$415$	1626.03	0.192334
$416$	0	0
$417$	968.669	0.113755
$418$	9359.78	1.09522
$419$	−6494.41	−0.757214	−0.378607	−	0.925558i	$-0.623597\pi$
$419$	−6494.41	−0.757214	−0.378607	+	0.925558i	$0.623597\pi$
$420$	2555.39	0.296881
$421$	3059.56	0.354190	0.177095	−	0.984194i	$-0.443330\pi$
$421$	3059.56	0.354190	0.177095	+	0.984194i	$0.443330\pi$
$422$	5631.33	0.649595
$423$	−3188.00	−0.366444
$424$	718.010	0.0822398
$425$	−490.173	−0.0559456
$426$	−4590.18	−0.522054
$427$	1819.68	0.206230
$428$	−7098.82	−0.801716
$429$	0	0
$430$	−751.954	−0.0843313
$431$	7937.05	0.887040	0.443520	−	0.896264i	$-0.353729\pi$
$431$	7937.05	0.887040	0.443520	+	0.896264i	$0.353729\pi$
$432$	5442.59	0.606150
$433$	7294.37	0.809573	0.404786	−	0.914411i	$-0.367346\pi$
$433$	7294.37	0.809573	0.404786	+	0.914411i	$0.367346\pi$
$434$	−16820.5	−1.86039
$435$	2078.91	0.229141
$436$	16108.4	1.76938
$437$	1234.45	0.135130
$438$	−11925.6	−1.30098
$439$	15214.7	1.65412	0.827059	−	0.562115i	$-0.190012\pi$
$439$	15214.7	1.65412	0.827059	+	0.562115i	$0.190012\pi$
$440$	12364.7	1.33970
$441$	−1006.97	−0.108732
$442$	0	0
$443$	1517.05	0.162703	0.0813515	−	0.996685i	$-0.474076\pi$
$443$	1517.05	0.162703	0.0813515	+	0.996685i	$0.474076\pi$
$444$	10952.4	1.17067
$445$	581.509	0.0619464
$446$	−12411.3	−1.31769
$447$	3386.58	0.358343
$448$	18946.5	1.99807
$449$	−705.247	−0.0741262	−0.0370631	−	0.999313i	$-0.511800\pi$
$449$	−705.247	−0.0741262	−0.0370631	+	0.999313i	$0.511800\pi$
$450$	5684.30	0.595467
$451$	−10465.0	−1.09263
$452$	26579.0	2.76586
$453$	−8821.33	−0.914927
$454$	−20383.1	−2.10710
$455$	0	0
$456$	−5365.37	−0.551001
$457$	−7277.73	−0.744940	−0.372470	−	0.928044i	$-0.621489\pi$
$457$	−7277.73	−0.744940	−0.372470	+	0.928044i	$0.621489\pi$
$458$	23181.4	2.36505
$459$	112.416	0.0114316
$460$	2651.58	0.268762
$461$	1961.88	0.198208	0.0991041	−	0.995077i	$-0.468402\pi$
$461$	1961.88	0.198208	0.0991041	+	0.995077i	$0.468402\pi$
$462$	16364.8	1.64796
$463$	−10374.1	−1.04131	−0.520653	−	0.853768i	$-0.674311\pi$
$463$	−10374.1	−1.04131	−0.520653	+	0.853768i	$0.674311\pi$
$464$	51806.8	5.18334
$465$	1668.28	0.166376
$466$	−28324.5	−2.81568
$467$	8788.92	0.870883	0.435442	−	0.900217i	$-0.356592\pi$
$467$	8788.92	0.870883	0.435442	+	0.900217i	$0.356592\pi$
$468$	0	0
$469$	−340.653	−0.0335392
$470$	5123.82	0.502860
$471$	1888.16	0.184718
$472$	−30518.2	−2.97609
$473$	−3476.97	−0.337995
$474$	8814.44	0.854136
$475$	−3070.98	−0.296645
$476$	1315.31	0.126654
$477$	−94.2510	−0.00904707
$478$	−8304.14	−0.794608
$479$	−11141.7	−1.06279	−0.531394	−	0.847125i	$-0.678332\pi$
$479$	−11141.7	−1.06279	−0.531394	+	0.847125i	$0.678332\pi$
$480$	−4310.65	−0.409903
$481$	0	0
$482$	26383.8	2.49325
$483$	2158.33	0.203328
$484$	65303.7	6.13295
$485$	3902.92	0.365407
$486$	−1303.63	−0.121674
$487$	−19640.5	−1.82750	−0.913752	−	0.406273i	$-0.866828\pi$
$487$	−19640.5	−1.82750	−0.913752	+	0.406273i	$0.866828\pi$
$488$	−8206.69	−0.761269
$489$	−1184.74	−0.109562
$490$	1618.41	0.149209
$491$	−3410.31	−0.313453	−0.156726	−	0.987642i	$-0.550094\pi$
$491$	−3410.31	−0.313453	−0.156726	+	0.987642i	$0.550094\pi$
$492$	9754.00	0.893789
$493$	1070.06	0.0977546
$494$	0	0
$495$	−1623.08	−0.147378
$496$	41573.8	3.76354
$497$	4335.86	0.391327
$498$	−9705.68	−0.873338
$499$	−5032.44	−0.451469	−0.225735	−	0.974189i	$-0.572478\pi$
$499$	−5032.44	−0.451469	−0.225735	+	0.974189i	$0.572478\pi$
$500$	−13600.2	−1.21644
$501$	−455.580	−0.0406263
$502$	−6200.44	−0.551274
$503$	17189.4	1.52373	0.761866	−	0.647735i	$-0.224284\pi$
$503$	17189.4	1.52373	0.761866	+	0.647735i	$0.224284\pi$
$504$	−9380.89	−0.829083
$505$	−2382.06	−0.209901
$506$	16980.8	1.49187
$507$	0	0
$508$	−23145.5	−2.02149
$509$	−930.560	−0.0810341	−0.0405170	−	0.999179i	$-0.512901\pi$
$509$	−930.560	−0.0810341	−0.0405170	+	0.999179i	$0.512901\pi$
$510$	−180.676	−0.0156872
$511$	11264.9	0.975201
$512$	3143.50	0.271337
$513$	704.295	0.0606148
$514$	−12617.6	−1.08276
$515$	3375.08	0.288784
$516$	3240.75	0.276485
$517$	23692.1	2.01543
$518$	−14328.4	−1.21536
$519$	1616.44	0.136713
$520$	0	0
$521$	−9869.60	−0.829933	−0.414966	−	0.909837i	$-0.636207\pi$
$521$	−9869.60	−0.829933	−0.414966	+	0.909837i	$0.636207\pi$
$522$	−12408.9	−1.04047
$523$	−21420.6	−1.79093	−0.895466	−	0.445129i	$-0.853158\pi$
$523$	−21420.6	−1.79093	−0.895466	+	0.445129i	$0.853158\pi$
$524$	43641.1	3.63831
$525$	−5369.35	−0.446358
$526$	−29623.4	−2.45559
$527$	858.697	0.0709781
$528$	−40447.4	−3.33380
$529$	−9927.42	−0.815930
$530$	151.482	0.0124150
$531$	4006.03	0.327395
$532$	8240.54	0.671565
$533$	0	0
$534$	−3471.00	−0.281283
$535$	−921.097	−0.0744345
$536$	1536.34	0.123805
$537$	−6662.39	−0.535388
$538$	−21013.4	−1.68392
$539$	7483.41	0.598021
$540$	1512.81	0.120558
$541$	7771.50	0.617602	0.308801	−	0.951127i	$-0.400072\pi$
$541$	7771.50	0.617602	0.308801	+	0.951127i	$0.400072\pi$
$542$	14900.7	1.18088
$543$	−11468.3	−0.906361
$544$	−2218.78	−0.174870
$545$	2090.12	0.164277
$546$	0	0
$547$	−15577.5	−1.21763	−0.608817	−	0.793310i	$-0.708356\pi$
$547$	−15577.5	−1.21763	−0.608817	+	0.793310i	$0.708356\pi$
$548$	−25040.6	−1.95197
$549$	1077.27	0.0837461
$550$	−42243.7	−3.27505
$551$	6704.02	0.518332
$552$	−9734.01	−0.750556
$553$	−8326.07	−0.640254
$554$	−35316.4	−2.70840
$555$	1421.11	0.108690
$556$	6709.74	0.511792
$557$	−22804.5	−1.73475	−0.867377	−	0.497652i	$-0.834196\pi$
$557$	−22804.5	−1.73475	−0.867377	+	0.497652i	$0.834196\pi$
$558$	−9957.89	−0.755468
$559$	0	0
$560$	8262.78	0.623511
$561$	−835.433	−0.0628734
$562$	−15405.7	−1.15631
$563$	−3517.41	−0.263306	−0.131653	−	0.991296i	$-0.542028\pi$
$563$	−3517.41	−0.263306	−0.131653	+	0.991296i	$0.542028\pi$
$564$	−22082.5	−1.64865
$565$	3448.71	0.256794
$566$	−40332.2	−2.99521
$567$	1231.40	0.0912062
$568$	−19554.6	−1.44453
$569$	−6093.44	−0.448946	−0.224473	−	0.974480i	$-0.572066\pi$
$569$	−6093.44	−0.448946	−0.224473	+	0.974480i	$0.572066\pi$
$570$	−1131.96	−0.0831797
$571$	10460.2	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$571$	10460.2	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$572$	0	0
$573$	10392.6	0.757689
$574$	−12760.6	−0.927906
$575$	−5571.47	−0.404081
$576$	11216.5	0.811378
$577$	−9648.19	−0.696117	−0.348058	−	0.937473i	$-0.613159\pi$
$577$	−9648.19	−0.696117	−0.348058	+	0.937473i	$0.613159\pi$
$578$	26263.9	1.89002
$579$	14092.2	1.01149
$580$	14400.1	1.03092
$581$	9167.93	0.654647
$582$	−23296.4	−1.65922
$583$	700.440	0.0497586
$584$	−50804.1	−3.59981
$585$	0	0
$586$	−24176.5	−1.70430
$587$	2170.73	0.152633	0.0763166	−	0.997084i	$-0.475684\pi$
$587$	2170.73	0.152633	0.0763166	+	0.997084i	$0.475684\pi$
$588$	−6975.01	−0.489191
$589$	5379.82	0.376353
$590$	−6438.56	−0.449274
$591$	−8663.66	−0.603004
$592$	35414.3	2.45865
$593$	22885.9	1.58484	0.792421	−	0.609975i	$-0.208820\pi$
$593$	22885.9	1.58484	0.792421	+	0.609975i	$0.208820\pi$
$594$	9688.11	0.669205
$595$	170.666	0.0117590
$596$	23458.0	1.61221
$597$	189.028	0.0129588
$598$	0	0
$599$	−23978.7	−1.63563	−0.817815	−	0.575482i	$-0.804815\pi$
$599$	−23978.7	−1.63563	−0.817815	+	0.575482i	$0.804815\pi$
$600$	24215.6	1.64766
$601$	12873.2	0.873728	0.436864	−	0.899528i	$-0.356089\pi$
$601$	12873.2	0.873728	0.436864	+	0.899528i	$0.356089\pi$
$602$	−4239.70	−0.287039
$603$	−201.670	−0.0136196
$604$	−61103.2	−4.11631
$605$	8473.38	0.569408
$606$	14218.4	0.953107
$607$	−7117.15	−0.475908	−0.237954	−	0.971276i	$-0.576477\pi$
$607$	−7117.15	−0.475908	−0.237954	+	0.971276i	$0.576477\pi$
$608$	−13900.9	−0.927227
$609$	11721.4	0.779927
$610$	−1731.40	−0.114922
$611$	0	0
$612$	778.675	0.0514315
$613$	173.297	0.0114183	0.00570913	−	0.999984i	$-0.498183\pi$
$613$	173.297	0.0114183	0.00570913	+	0.999984i	$0.498183\pi$
$614$	−51173.5	−3.36351
$615$	1265.62	0.0829830
$616$	69715.5	4.55993
$617$	−6102.75	−0.398197	−0.199099	−	0.979979i	$-0.563801\pi$
$617$	−6102.75	−0.398197	−0.199099	+	0.979979i	$0.563801\pi$
$618$	−20145.7	−1.31129
$619$	−14867.8	−0.965409	−0.482705	−	0.875783i	$-0.660346\pi$
$619$	−14867.8	−0.965409	−0.482705	+	0.875783i	$0.660346\pi$
$620$	11555.8	0.748534
$621$	1277.75	0.0825676
$622$	40054.6	2.58206
$623$	3278.69	0.210847
$624$	0	0
$625$	12951.6	0.828900
$626$	9772.64	0.623951
$627$	−5234.07	−0.333379
$628$	13078.9	0.831056
$629$	731.475	0.0463686
$630$	−1979.13	−0.125159
$631$	−17210.6	−1.08580	−0.542902	−	0.839796i	$-0.682675\pi$
$631$	−17210.6	−1.08580	−0.542902	+	0.839796i	$0.682675\pi$
$632$	37550.3	2.36340
$633$	−3149.09	−0.197733
$634$	−16765.5	−1.05023
$635$	−3003.21	−0.187683
$636$	−652.853	−0.0407033
$637$	0	0
$638$	92218.9	5.72254
$639$	2566.87	0.158910
$640$	−6532.30	−0.403456
$641$	−12636.0	−0.778616	−0.389308	−	0.921108i	$-0.627286\pi$
$641$	−12636.0	−0.778616	−0.389308	+	0.921108i	$0.627286\pi$
$642$	5497.99	0.337988
$643$	−9586.37	−0.587947	−0.293973	−	0.955814i	$-0.594978\pi$
$643$	−9586.37	−0.587947	−0.293973	+	0.955814i	$0.594978\pi$
$644$	14950.2	0.914786
$645$	420.499	0.0256700
$646$	−582.641	−0.0354856
$647$	5244.32	0.318664	0.159332	−	0.987225i	$-0.449066\pi$
$647$	5244.32	0.318664	0.159332	+	0.987225i	$0.449066\pi$
$648$	−5553.57	−0.336674
$649$	−29771.4	−1.80066
$650$	0	0
$651$	9406.17	0.566293
$652$	−8206.39	−0.492925
$653$	18869.0	1.13078	0.565392	−	0.824822i	$-0.308725\pi$
$653$	18869.0	1.13078	0.565392	+	0.824822i	$0.308725\pi$
$654$	−12475.8	−0.745938
$655$	5662.59	0.337795
$656$	31539.3	1.87714
$657$	6668.90	0.396010
$658$	28889.3	1.71159
$659$	−24299.8	−1.43639	−0.718197	−	0.695839i	$-0.755033\pi$
$659$	−24299.8	−1.43639	−0.718197	+	0.695839i	$0.755033\pi$
$660$	−11242.7	−0.663062
$661$	29915.0	1.76030	0.880151	−	0.474694i	$-0.157441\pi$
$661$	29915.0	1.76030	0.880151	+	0.474694i	$0.157441\pi$
$662$	8335.03	0.489350
$663$	0	0
$664$	−41347.1	−2.41653
$665$	1069.24	0.0623508
$666$	−8482.56	−0.493532
$667$	12162.6	0.706056
$668$	−3155.69	−0.182780
$669$	6940.48	0.401098
$670$	324.128	0.0186898
$671$	−8005.86	−0.460600
$672$	−24304.5	−1.39519
$673$	15493.7	0.887426	0.443713	−	0.896169i	$-0.353661\pi$
$673$	15493.7	0.887426	0.443713	+	0.896169i	$0.353661\pi$
$674$	17115.8	0.978154
$675$	−3178.71	−0.181257
$676$	0	0
$677$	11729.7	0.665891	0.332945	−	0.942946i	$-0.391958\pi$
$677$	11729.7	0.665891	0.332945	+	0.942946i	$0.391958\pi$
$678$	−20585.2	−1.16603
$679$	22005.6	1.24374
$680$	−769.698	−0.0434067
$681$	11398.4	0.641390
$682$	74003.5	4.15505
$683$	−1168.42	−0.0654587	−0.0327294	−	0.999464i	$-0.510420\pi$
$683$	−1168.42	−0.0654587	−0.0327294	+	0.999464i	$0.510420\pi$
$684$	4878.48	0.272709
$685$	−3249.10	−0.181229
$686$	37099.1	2.06480
$687$	−12963.2	−0.719909
$688$	10478.9	0.580675
$689$	0	0
$690$	−2053.63	−0.113305
$691$	32992.9	1.81637	0.908183	−	0.418574i	$-0.137470\pi$
$691$	32992.9	1.81637	0.908183	+	0.418574i	$0.137470\pi$
$692$	11196.7	0.615078
$693$	−9151.33	−0.501631
$694$	30679.2	1.67805
$695$	870.612	0.0475168
$696$	−52863.2	−2.87899
$697$	651.438	0.0354017
$698$	17856.3	0.968295
$699$	15839.3	0.857078
$700$	−37192.2	−2.00819
$701$	−14785.8	−0.796651	−0.398326	−	0.917244i	$-0.630409\pi$
$701$	−14785.8	−0.796651	−0.398326	+	0.917244i	$0.630409\pi$
$702$	0	0
$703$	4582.77	0.245864
$704$	−83357.0	−4.46255
$705$	−2865.28	−0.153068
$706$	66021.0	3.51945
$707$	−13430.6	−0.714442
$708$	27748.8	1.47297
$709$	12634.0	0.669225	0.334612	−	0.942356i	$-0.391395\pi$
$709$	12634.0	0.669225	0.334612	+	0.942356i	$0.391395\pi$
$710$	−4125.52	−0.218067
$711$	−4929.11	−0.259994
$712$	−14786.8	−0.778312
$713$	9760.24	0.512656
$714$	−1018.70	−0.0533947
$715$	0	0
$716$	−46148.7	−2.40874
$717$	4643.75	0.241874
$718$	−45814.6	−2.38132
$719$	−27296.5	−1.41584	−0.707920	−	0.706292i	$-0.750366\pi$
$719$	−27296.5	−1.41584	−0.707920	+	0.706292i	$0.750366\pi$
$720$	4891.64	0.253195
$721$	19029.5	0.982936
$722$	33146.3	1.70856
$723$	−14754.0	−0.758932
$724$	−79438.5	−4.07777
$725$	−30257.4	−1.54997
$726$	−50577.2	−2.58553
$727$	4658.21	0.237639	0.118819	−	0.992916i	$-0.462089\pi$
$727$	4658.21	0.237639	0.118819	+	0.992916i	$0.462089\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−10718.4	−0.543432
$731$	216.439	0.0109512
$732$	7461.96	0.376779
$733$	−166.474	−0.00838864	−0.00419432	−	0.999991i	$-0.501335\pi$
$733$	−166.474	−0.00838864	−0.00419432	+	0.999991i	$0.501335\pi$
$734$	−13393.9	−0.673537
$735$	−905.031	−0.0454185
$736$	−25219.4	−1.26304
$737$	1498.74	0.0749074
$738$	−7554.41	−0.376804
$739$	12738.1	0.634069	0.317035	−	0.948414i	$-0.397313\pi$
$739$	12738.1	0.634069	0.317035	+	0.948414i	$0.397313\pi$
$740$	9843.70	0.489002
$741$	0	0
$742$	854.092	0.0422570
$743$	−30724.5	−1.51705	−0.758527	−	0.651641i	$-0.774081\pi$
$743$	−30724.5	−1.51705	−0.758527	+	0.651641i	$0.774081\pi$
$744$	−42421.5	−2.09039
$745$	3043.76	0.149684
$746$	6131.38	0.300919
$747$	5427.50	0.265839
$748$	−5786.84	−0.282871
$749$	−5193.37	−0.253353
$750$	10533.2	0.512826
$751$	−39538.6	−1.92115	−0.960575	−	0.278021i	$-0.910322\pi$
$751$	−39538.6	−1.92115	−0.960575	+	0.278021i	$0.910322\pi$
$752$	−71403.2	−3.46251
$753$	3467.34	0.167805
$754$	0	0
$755$	−7928.35	−0.382175
$756$	8529.61	0.410342
$757$	23035.1	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$757$	23035.1	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$758$	65352.1	3.13153
$759$	−9495.81	−0.454119
$760$	−4822.23	−0.230159
$761$	32454.0	1.54594	0.772968	−	0.634445i	$-0.218771\pi$
$761$	32454.0	1.54594	0.772968	+	0.634445i	$0.218771\pi$
$762$	17926.0	0.852220
$763$	11784.6	0.559150
$764$	71986.8	3.40889
$765$	101.036	0.00477511
$766$	54616.1	2.57619
$767$	0	0
$768$	9080.40	0.426641
$769$	−32216.2	−1.51072	−0.755362	−	0.655307i	$-0.772539\pi$
$769$	−32216.2	−1.51072	−0.755362	+	0.655307i	$0.772539\pi$
$770$	14708.2	0.688372
$771$	7055.86	0.329586
$772$	97613.2	4.55075
$773$	2924.60	0.136081	0.0680404	−	0.997683i	$-0.478325\pi$
$773$	2924.60	0.136081	0.0680404	+	0.997683i	$0.478325\pi$
$774$	−2509.94	−0.116561
$775$	−24280.9	−1.12541
$776$	−99244.7	−4.59108
$777$	8012.58	0.369948
$778$	−31361.2	−1.44519
$779$	4081.32	0.187713
$780$	0	0
$781$	−19076.1	−0.874001
$782$	−1057.04	−0.0483374
$783$	6939.19	0.316713
$784$	−22553.5	−1.02740
$785$	1697.03	0.0771586
$786$	−33799.8	−1.53384
$787$	25507.1	1.15531	0.577656	−	0.816280i	$-0.303967\pi$
$787$	25507.1	1.15531	0.577656	+	0.816280i	$0.303967\pi$
$788$	−60011.1	−2.71295
$789$	16565.6	0.747468
$790$	7922.16	0.356782
$791$	19444.7	0.874050
$792$	41272.2	1.85170
$793$	0	0
$794$	13416.8	0.599677
$795$	−84.7100	−0.00377906
$796$	1309.35	0.0583023
$797$	5448.98	0.242174	0.121087	−	0.992642i	$-0.461362\pi$
$797$	5448.98	0.242174	0.121087	+	0.992642i	$0.461362\pi$
$798$	−6382.24	−0.283119
$799$	−1474.82	−0.0653008
$800$	62739.0	2.77270
$801$	1941.01	0.0856209
$802$	49297.8	2.17053
$803$	−49560.9	−2.17804
$804$	−1396.92	−0.0612755
$805$	1939.85	0.0849324
$806$	0	0
$807$	11750.9	0.512577
$808$	60571.7	2.63726
$809$	−2453.12	−0.106610	−0.0533048	−	0.998578i	$-0.516975\pi$
$809$	−2453.12	−0.106610	−0.0533048	+	0.998578i	$0.516975\pi$
$810$	−1171.66	−0.0508247
$811$	−5133.85	−0.222286	−0.111143	−	0.993804i	$-0.535451\pi$
$811$	−5133.85	−0.222286	−0.111143	+	0.993804i	$0.535451\pi$
$812$	81191.4	3.50894
$813$	−8332.58	−0.359454
$814$	63039.4	2.71441
$815$	−1064.81	−0.0457651
$816$	2517.83	0.108017
$817$	1356.01	0.0580673
$818$	−49432.6	−2.11292
$819$	0	0
$820$	8766.61	0.373345
$821$	−23854.0	−1.01402	−0.507009	−	0.861941i	$-0.669249\pi$
$821$	−23854.0	−1.01402	−0.507009	+	0.861941i	$0.669249\pi$
$822$	19393.8	0.822913
$823$	757.156	0.0320690	0.0160345	−	0.999871i	$-0.494896\pi$
$823$	757.156	0.0320690	0.0160345	+	0.999871i	$0.494896\pi$
$824$	−85822.6	−3.62836
$825$	23623.0	0.996907
$826$	−36302.2	−1.52919
$827$	28621.2	1.20345	0.601726	−	0.798702i	$-0.294480\pi$
$827$	28621.2	1.20345	0.601726	+	0.798702i	$0.294480\pi$
$828$	8850.68	0.371476
$829$	27429.2	1.14916	0.574582	−	0.818447i	$-0.305165\pi$
$829$	27429.2	1.14916	0.574582	+	0.818447i	$0.305165\pi$
$830$	−8723.19	−0.364803
$831$	19749.2	0.824421
$832$	0	0
$833$	−465.838	−0.0193761
$834$	−5196.64	−0.215761
$835$	−409.462	−0.0169701
$836$	−36255.1	−1.49989
$837$	5568.54	0.229960
$838$	34840.7	1.43622
$839$	−4633.62	−0.190668	−0.0953339	−	0.995445i	$-0.530392\pi$
$839$	−4633.62	−0.190668	−0.0953339	+	0.995445i	$0.530392\pi$
$840$	−8431.27	−0.346317
$841$	41663.6	1.70829
$842$	−16413.7	−0.671798
$843$	8614.98	0.351976
$844$	−21813.0	−0.889614
$845$	0	0
$846$	17102.7	0.695041
$847$	47775.0	1.93810
$848$	−2110.99	−0.0854853
$849$	22554.1	0.911726
$850$	2629.64	0.106113
$851$	8314.19	0.334908
$852$	17780.1	0.714947
$853$	−14854.6	−0.596261	−0.298131	−	0.954525i	$-0.596363\pi$
$853$	−14854.6	−0.596261	−0.298131	+	0.954525i	$0.596363\pi$
$854$	−9762.07	−0.391161
$855$	632.999	0.0253194
$856$	23421.9	0.935216
$857$	−42799.5	−1.70595	−0.852977	−	0.521948i	$-0.825206\pi$
$857$	−42799.5	−1.70595	−0.852977	+	0.521948i	$0.825206\pi$
$858$	0	0
$859$	−8246.47	−0.327551	−0.163775	−	0.986498i	$-0.552367\pi$
$859$	−8246.47	−0.327551	−0.163775	+	0.986498i	$0.552367\pi$
$860$	2912.70	0.115491
$861$	7135.85	0.282450
$862$	−42580.1	−1.68246
$863$	17695.4	0.697983	0.348991	−	0.937126i	$-0.386524\pi$
$863$	17695.4	0.697983	0.348991	+	0.937126i	$0.386524\pi$
$864$	−14388.5	−0.566558
$865$	1452.81	0.0571064
$866$	−39132.3	−1.53553
$867$	−14687.0	−0.575313
$868$	65154.2	2.54779
$869$	36631.4	1.42996
$870$	−11152.8	−0.434615
$871$	0	0
$872$	−53148.2	−2.06402
$873$	13027.5	0.505058
$874$	−6622.49	−0.256303
$875$	−9949.64	−0.384411
$876$	46193.8	1.78167
$877$	−14346.8	−0.552401	−0.276200	−	0.961100i	$-0.589075\pi$
$877$	−14346.8	−0.552401	−0.276200	+	0.961100i	$0.589075\pi$
$878$	−81622.7	−3.13739
$879$	13519.7	0.518781
$880$	−36353.0	−1.39257
$881$	−2063.51	−0.0789121	−0.0394560	−	0.999221i	$-0.512563\pi$
$881$	−2063.51	−0.0789121	−0.0394560	+	0.999221i	$0.512563\pi$
$882$	5402.09	0.206233
$883$	−34137.1	−1.30103	−0.650513	−	0.759495i	$-0.725446\pi$
$883$	−34137.1	−1.30103	−0.650513	+	0.759495i	$0.725446\pi$
$884$	0	0
$885$	3600.50	0.136756
$886$	−8138.58	−0.308602
$887$	−22238.5	−0.841821	−0.420910	−	0.907102i	$-0.638289\pi$
$887$	−22238.5	−0.841821	−0.420910	+	0.907102i	$0.638289\pi$
$888$	−36136.5	−1.36561
$889$	−16932.8	−0.638817
$890$	−3119.63	−0.117495
$891$	−5417.67	−0.203702
$892$	48075.0	1.80456
$893$	−9239.89	−0.346250
$894$	−18168.0	−0.679676
$895$	−5987.96	−0.223637
$896$	−36830.7	−1.37325
$897$	0	0
$898$	3783.45	0.140596
$899$	53005.7	1.96645
$900$	−22018.1	−0.815486
$901$	−43.6019	−0.00161220
$902$	56141.7	2.07241
$903$	2370.88	0.0873730
$904$	−87694.9	−3.22643
$905$	−10307.4	−0.378597
$906$	47324.0	1.73536
$907$	−44160.3	−1.61667	−0.808335	−	0.588723i	$-0.799631\pi$
$907$	−44160.3	−1.61667	−0.808335	+	0.588723i	$0.799631\pi$
$908$	78953.8	2.88565
$909$	−7951.05	−0.290121
$910$	0	0
$911$	11916.3	0.433376	0.216688	−	0.976241i	$-0.430475\pi$
$911$	11916.3	0.433376	0.216688	+	0.976241i	$0.430475\pi$
$912$	15774.4	0.572745
$913$	−40335.3	−1.46211
$914$	39043.0	1.41294
$915$	968.215	0.0349816
$916$	−89793.0	−3.23891
$917$	31927.1	1.14975
$918$	−603.078	−0.0216825
$919$	−20128.9	−0.722516	−0.361258	−	0.932466i	$-0.617653\pi$
$919$	−20128.9	−0.722516	−0.361258	+	0.932466i	$0.617653\pi$
$920$	−8748.64	−0.313515
$921$	28616.7	1.02383
$922$	−10525.0	−0.375945
$923$	0	0
$924$	−63389.0	−2.25687
$925$	−20683.5	−0.735210
$926$	55654.1	1.97506
$927$	11265.7	0.399151
$928$	−136961.	−4.84478
$929$	−31428.4	−1.10994	−0.554969	−	0.831871i	$-0.687270\pi$
$929$	−31428.4	−1.10994	−0.554969	+	0.831871i	$0.687270\pi$
$930$	−8949.86	−0.315567
$931$	−2918.52	−0.102740
$932$	109715.	3.85605
$933$	−22398.9	−0.785965
$934$	−47150.1	−1.65182
$935$	−750.863	−0.0262629
$936$	0	0
$937$	42473.2	1.48083	0.740416	−	0.672149i	$-0.234629\pi$
$937$	42473.2	1.48083	0.740416	+	0.672149i	$0.234629\pi$
$938$	1827.51	0.0636144
$939$	−5464.94	−0.189927
$940$	−19847.1	−0.688661
$941$	−42644.0	−1.47732	−0.738659	−	0.674079i	$-0.764541\pi$
$941$	−42644.0	−1.47732	−0.738659	+	0.674079i	$0.764541\pi$
$942$	−10129.5	−0.350357
$943$	7404.46	0.255697
$944$	89725.0	3.09354
$945$	1106.75	0.0380978
$946$	18653.0	0.641080
$947$	4282.95	0.146966	0.0734831	−	0.997296i	$-0.476588\pi$
$947$	4282.95	0.146966	0.0734831	+	0.997296i	$0.476588\pi$
$948$	−34142.7	−1.16973
$949$	0	0
$950$	16475.0	0.562651
$951$	9375.42	0.319683
$952$	−4339.74	−0.147744
$953$	40593.9	1.37982	0.689908	−	0.723897i	$-0.257651\pi$
$953$	40593.9	1.37982	0.689908	+	0.723897i	$0.257651\pi$
$954$	505.630	0.0171597
$955$	9340.54	0.316495
$956$	32166.1	1.08821
$957$	−51569.6	−1.74191
$958$	59771.9	2.01581
$959$	−18319.2	−0.616850
$960$	10081.1	0.338922
$961$	12744.8	0.427808
$962$	0	0
$963$	−3074.52	−0.102882
$964$	−102198.	−3.41448
$965$	12665.7	0.422510
$966$	−11578.9	−0.385656
$967$	17709.4	0.588931	0.294466	−	0.955662i	$-0.404858\pi$
$967$	17709.4	0.588931	0.294466	+	0.955662i	$0.404858\pi$
$968$	−215464.	−7.15420
$969$	325.818	0.0108016
$970$	−20938.1	−0.693074
$971$	4038.97	0.133488	0.0667440	−	0.997770i	$-0.478739\pi$
$971$	4038.97	0.133488	0.0667440	+	0.997770i	$0.478739\pi$
$972$	5049.61	0.166632
$973$	4908.72	0.161733
$974$	105366.	3.46626
$975$	0	0
$976$	24128.1	0.791312
$977$	−2764.30	−0.0905197	−0.0452598	−	0.998975i	$-0.514412\pi$
$977$	−2764.30	−0.0905197	−0.0452598	+	0.998975i	$0.514412\pi$
$978$	6355.79	0.207807
$979$	−14424.9	−0.470912
$980$	−6268.93	−0.204340
$981$	6976.59	0.227060
$982$	18295.4	0.594531
$983$	−17804.5	−0.577695	−0.288848	−	0.957375i	$-0.593272\pi$
$983$	−17804.5	−0.577695	−0.288848	+	0.957375i	$0.593272\pi$
$984$	−32182.4	−1.04262
$985$	−7786.65	−0.251881
$986$	−5740.57	−0.185413
$987$	−16155.2	−0.520997
$988$	0	0
$989$	2460.12	0.0790974
$990$	8707.39	0.279534
$991$	−8129.69	−0.260593	−0.130297	−	0.991475i	$-0.541593\pi$
$991$	−8129.69	−0.260593	−0.130297	+	0.991475i	$0.541593\pi$
$992$	−109908.	−3.51772
$993$	−4661.02	−0.148956
$994$	−23260.7	−0.742237
$995$	169.893	0.00541302
$996$	37595.0	1.19603
$997$	−29452.9	−0.935589	−0.467795	−	0.883837i	$-0.654951\pi$
$997$	−29452.9	−0.935589	−0.467795	+	0.883837i	$0.654951\pi$
$998$	26997.7	0.856309
$999$	4743.52	0.150229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.r.1.1		10
3.2	odd	2		1521.4.a.bk.1.10		10
13.2	odd	12		39.4.j.c.4.1	✓	10
13.5	odd	4		507.4.b.i.337.10		10
13.7	odd	12		39.4.j.c.10.1	yes	10
13.8	odd	4		507.4.b.i.337.1		10
13.12	even	2	inner	507.4.a.r.1.10		10
39.2	even	12		117.4.q.e.82.5		10
39.20	even	12		117.4.q.e.10.5		10
39.38	odd	2		1521.4.a.bk.1.1		10
52.7	even	12		624.4.bv.h.49.3		10
52.15	even	12		624.4.bv.h.433.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.1	✓	10	13.2	odd	12
39.4.j.c.10.1	yes	10	13.7	odd	12
117.4.q.e.10.5		10	39.20	even	12
117.4.q.e.82.5		10	39.2	even	12
507.4.a.r.1.1		10	1.1	even	1	trivial
507.4.a.r.1.10		10	13.12	even	2	inner
507.4.b.i.337.1		10	13.8	odd	4
507.4.b.i.337.10		10	13.5	odd	4
624.4.bv.h.49.3		10	52.7	even	12
624.4.bv.h.433.3		10	52.15	even	12
1521.4.a.bk.1.1		10	39.38	odd	2
1521.4.a.bk.1.10		10	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-5.36472 q^{2} +3.00000 q^{3} +20.7803 q^{4} +2.69631 q^{5} -16.0942 q^{6} +15.2025 q^{7} -68.5626 q^{8} +9.00000 q^{9} -14.4650 q^{10} -66.8848 q^{11} +62.3408 q^{12} -81.5570 q^{14} +8.08894 q^{15} +201.577 q^{16} +4.16354 q^{17} -48.2825 q^{18} +26.0850 q^{19} +56.0301 q^{20} +45.6074 q^{21} +358.819 q^{22} +47.3242 q^{23} -205.688 q^{24} -117.730 q^{25} +27.0000 q^{27} +315.911 q^{28} +257.007 q^{29} -43.3949 q^{30} +206.242 q^{31} -532.906 q^{32} -200.655 q^{33} -22.3362 q^{34} +40.9906 q^{35} +187.022 q^{36} +175.686 q^{37} -139.939 q^{38} -184.866 q^{40} +156.463 q^{41} -244.671 q^{42} +51.9845 q^{43} -1389.88 q^{44} +24.2668 q^{45} -253.881 q^{46} -354.222 q^{47} +604.732 q^{48} -111.885 q^{49} +631.588 q^{50} +12.4906 q^{51} -10.4723 q^{53} -144.848 q^{54} -180.342 q^{55} -1042.32 q^{56} +78.2550 q^{57} -1378.77 q^{58} +445.114 q^{59} +168.090 q^{60} +119.696 q^{61} -1106.43 q^{62} +136.822 q^{63} +1246.28 q^{64} +1076.46 q^{66} -22.4078 q^{67} +86.5195 q^{68} +141.973 q^{69} -219.903 q^{70} +285.207 q^{71} -617.064 q^{72} +740.989 q^{73} -942.507 q^{74} -353.190 q^{75} +542.053 q^{76} -1016.81 q^{77} -547.679 q^{79} +543.516 q^{80} +81.0000 q^{81} -839.378 q^{82} +603.056 q^{83} +947.734 q^{84} +11.2262 q^{85} -278.882 q^{86} +771.021 q^{87} +4585.80 q^{88} +215.668 q^{89} -130.185 q^{90} +983.409 q^{92} +618.726 q^{93} +1900.31 q^{94} +70.3333 q^{95} -1598.72 q^{96} +1447.50 q^{97} +600.233 q^{98} -601.964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36}+ \cdots - 2760 q^{95}+O(q^{100}) \) 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 + 80 * q^10 + 180 * q^12 - 60 * q^14 + 500 * q^16 + 210 * q^17 + 580 * q^22 - 120 * q^23 + 960 * q^25 + 270 * q^27 + 990 * q^29 + 240 * q^30 - 120 * q^35 + 540 * q^36 + 1380 * q^38 + 2000 * q^40 - 180 * q^42 - 740 * q^43 + 1500 * q^48 + 1550 * q^49 + 630 * q^51 + 330 * q^53 + 520 * q^55 - 5340 * q^56 + 2750 * q^61 - 1560 * q^62 + 3140 * q^64 + 1740 * q^66 + 1200 * q^68 - 360 * q^69 - 4380 * q^74 + 2880 * q^75 + 4320 * q^77 + 1100 * q^79 + 810 * q^81 - 4780 * q^82 + 2970 * q^87 + 6340 * q^88 + 720 * q^90 - 1740 * q^92 + 6460 * q^94 - 2760 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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