LMFDB - Embedded newform 6223.2.a.k.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,-2,4,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.33134$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.33134 q^{2} +1.35454 q^{3} -0.227522 q^{4} +2.92411 q^{5} -1.80336 q^{6} +2.96560 q^{8} -1.16521 q^{9} -3.89299 q^{10} +6.16901 q^{11} -0.308188 q^{12} +0.620036 q^{13} +3.96083 q^{15} -3.49319 q^{16} -0.644462 q^{17} +1.55130 q^{18} -2.75517 q^{19} -0.665298 q^{20} -8.21308 q^{22} +4.71184 q^{23} +4.01703 q^{24} +3.55041 q^{25} -0.825482 q^{26} -5.64196 q^{27} -8.85381 q^{29} -5.27323 q^{30} -2.41753 q^{31} -1.28056 q^{32} +8.35618 q^{33} +0.858001 q^{34} +0.265112 q^{36} -6.07476 q^{37} +3.66808 q^{38} +0.839865 q^{39} +8.67173 q^{40} +2.70008 q^{41} +10.3043 q^{43} -1.40358 q^{44} -3.40721 q^{45} -6.27308 q^{46} +10.7764 q^{47} -4.73167 q^{48} -4.72681 q^{50} -0.872951 q^{51} -0.141072 q^{52} +6.17224 q^{53} +7.51139 q^{54} +18.0388 q^{55} -3.73200 q^{57} +11.7875 q^{58} +11.6075 q^{59} -0.901175 q^{60} +12.3316 q^{61} +3.21857 q^{62} +8.69124 q^{64} +1.81305 q^{65} -11.1250 q^{66} +12.3886 q^{67} +0.146629 q^{68} +6.38238 q^{69} -10.0499 q^{71} -3.45556 q^{72} -8.20886 q^{73} +8.08760 q^{74} +4.80918 q^{75} +0.626862 q^{76} -1.11815 q^{78} +2.30230 q^{79} -10.2145 q^{80} -4.14663 q^{81} -3.59473 q^{82} +1.22797 q^{83} -1.88448 q^{85} -13.7186 q^{86} -11.9929 q^{87} +18.2948 q^{88} -11.7296 q^{89} +4.53618 q^{90} -1.07205 q^{92} -3.27465 q^{93} -14.3471 q^{94} -8.05642 q^{95} -1.73457 q^{96} -1.19035 q^{97} -7.18822 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) $ 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * 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 $

$2$	−1.33134	−0.941403	−0.470701	−	0.882293i	$-0.655999\pi$
$2$	−1.33134	−0.941403	−0.470701	+	0.882293i	$0.655999\pi$
$3$	1.35454	0.782045	0.391023	−	0.920381i	$-0.372121\pi$
$3$	1.35454	0.782045	0.391023	+	0.920381i	$0.372121\pi$
$4$	−0.227522	−0.113761
$5$	2.92411	1.30770	0.653850	−	0.756624i	$-0.273153\pi$
$5$	2.92411	1.30770	0.653850	+	0.756624i	$0.273153\pi$
$6$	−1.80336	−0.736220
$7$	0	0
$7$	0	0
$8$	2.96560	1.04850
$9$	−1.16521	−0.388405
$10$	−3.89299	−1.23107
$11$	6.16901	1.86003	0.930013	−	0.367526i	$-0.119795\pi$
$11$	6.16901	1.86003	0.930013	+	0.367526i	$0.119795\pi$
$12$	−0.308188	−0.0889662
$13$	0.620036	0.171967	0.0859836	−	0.996297i	$-0.472597\pi$
$13$	0.620036	0.171967	0.0859836	+	0.996297i	$0.472597\pi$
$14$	0	0
$15$	3.96083	1.02268
$16$	−3.49319	−0.873298
$17$	−0.644462	−0.156305	−0.0781525	−	0.996941i	$-0.524902\pi$
$17$	−0.644462	−0.156305	−0.0781525	+	0.996941i	$0.524902\pi$
$18$	1.55130	0.365645
$19$	−2.75517	−0.632080	−0.316040	−	0.948746i	$-0.602353\pi$
$19$	−2.75517	−0.632080	−0.316040	+	0.948746i	$0.602353\pi$
$20$	−0.665298	−0.148765
$21$	0	0
$22$	−8.21308	−1.75103
$23$	4.71184	0.982486	0.491243	−	0.871023i	$-0.336543\pi$
$23$	4.71184	0.982486	0.491243	+	0.871023i	$0.336543\pi$
$24$	4.01703	0.819973
$25$	3.55041	0.710081
$26$	−0.825482	−0.161890
$27$	−5.64196	−1.08580
$28$	0	0
$29$	−8.85381	−1.64411	−0.822056	−	0.569407i	$-0.807173\pi$
$29$	−8.85381	−1.64411	−0.822056	+	0.569407i	$0.807173\pi$
$30$	−5.27323	−0.962755
$31$	−2.41753	−0.434202	−0.217101	−	0.976149i	$-0.569660\pi$
$31$	−2.41753	−0.434202	−0.217101	+	0.976149i	$0.569660\pi$
$32$	−1.28056	−0.226373
$33$	8.35618	1.45463
$34$	0.858001	0.147146
$35$	0	0
$36$	0.265112	0.0441853
$37$	−6.07476	−0.998685	−0.499343	−	0.866405i	$-0.666425\pi$
$37$	−6.07476	−0.998685	−0.499343	+	0.866405i	$0.666425\pi$
$38$	3.66808	0.595042
$39$	0.839865	0.134486
$40$	8.67173	1.37112
$41$	2.70008	0.421681	0.210840	−	0.977520i	$-0.432380\pi$
$41$	2.70008	0.421681	0.210840	+	0.977520i	$0.432380\pi$
$42$	0	0
$43$	10.3043	1.57140	0.785698	−	0.618611i	$-0.212304\pi$
$43$	10.3043	1.57140	0.785698	+	0.618611i	$0.212304\pi$
$44$	−1.40358	−0.211598
$45$	−3.40721	−0.507917
$46$	−6.27308	−0.924915
$47$	10.7764	1.57190	0.785950	−	0.618290i	$-0.212174\pi$
$47$	10.7764	1.57190	0.785950	+	0.618290i	$0.212174\pi$
$48$	−4.73167	−0.682958
$49$	0	0
$50$	−4.72681	−0.668472
$51$	−0.872951	−0.122238
$52$	−0.141072	−0.0195631
$53$	6.17224	0.847822	0.423911	−	0.905704i	$-0.360657\pi$
$53$	6.17224	0.847822	0.423911	+	0.905704i	$0.360657\pi$
$54$	7.51139	1.02217
$55$	18.0388	2.43236
$56$	0	0
$57$	−3.73200	−0.494315
$58$	11.7875	1.54777
$59$	11.6075	1.51116	0.755581	−	0.655055i	$-0.227355\pi$
$59$	11.6075	1.51116	0.755581	+	0.655055i	$0.227355\pi$
$60$	−0.901175	−0.116341
$61$	12.3316	1.57890	0.789448	−	0.613818i	$-0.210367\pi$
$61$	12.3316	1.57890	0.789448	+	0.613818i	$0.210367\pi$
$62$	3.21857	0.408759
$63$	0	0
$64$	8.69124	1.08641
$65$	1.81305	0.224882
$66$	−11.1250	−1.36939
$67$	12.3886	1.51351	0.756753	−	0.653701i	$-0.226785\pi$
$67$	12.3886	1.51351	0.756753	+	0.653701i	$0.226785\pi$
$68$	0.146629	0.0177814
$69$	6.38238	0.768348
$70$	0	0
$71$	−10.0499	−1.19270	−0.596351	−	0.802724i	$-0.703383\pi$
$71$	−10.0499	−1.19270	−0.596351	+	0.802724i	$0.703383\pi$
$72$	−3.45556	−0.407242
$73$	−8.20886	−0.960775	−0.480387	−	0.877056i	$-0.659504\pi$
$73$	−8.20886	−0.960775	−0.480387	+	0.877056i	$0.659504\pi$
$74$	8.08760	0.940165
$75$	4.80918	0.555316
$76$	0.626862	0.0719060
$77$	0	0
$78$	−1.11815	−0.126606
$79$	2.30230	0.259029	0.129515	−	0.991578i	$-0.458658\pi$
$79$	2.30230	0.259029	0.129515	+	0.991578i	$0.458658\pi$
$80$	−10.2145	−1.14201
$81$	−4.14663	−0.460737
$82$	−3.59473	−0.396972
$83$	1.22797	0.134787	0.0673936	−	0.997726i	$-0.478532\pi$
$83$	1.22797	0.134787	0.0673936	+	0.997726i	$0.478532\pi$
$84$	0	0
$85$	−1.88448	−0.204400
$86$	−13.7186	−1.47932
$87$	−11.9929	−1.28577
$88$	18.2948	1.95023
$89$	−11.7296	−1.24334	−0.621668	−	0.783281i	$-0.713544\pi$
$89$	−11.7296	−1.24334	−0.621668	+	0.783281i	$0.713544\pi$
$90$	4.53618	0.478155
$91$	0	0
$92$	−1.07205	−0.111768
$93$	−3.27465	−0.339565
$94$	−14.3471	−1.47979
$95$	−8.05642	−0.826571
$96$	−1.73457	−0.177034
$97$	−1.19035	−0.120862	−0.0604309	−	0.998172i	$-0.519247\pi$
$97$	−1.19035	−0.120862	−0.0604309	+	0.998172i	$0.519247\pi$
$98$	0	0
$99$	−7.18822	−0.722443
$100$	−0.807795	−0.0807795
$101$	19.3595	1.92635	0.963173	−	0.268882i	$-0.0866542\pi$
$101$	19.3595	1.92635	0.963173	+	0.268882i	$0.0866542\pi$
$102$	1.16220	0.115075
$103$	10.2002	1.00506	0.502528	−	0.864561i	$-0.332403\pi$
$103$	10.2002	1.00506	0.502528	+	0.864561i	$0.332403\pi$
$104$	1.83878	0.180307
$105$	0	0
$106$	−8.21738	−0.798142
$107$	−15.3876	−1.48758	−0.743788	−	0.668415i	$-0.766973\pi$
$107$	−15.3876	−1.48758	−0.743788	+	0.668415i	$0.766973\pi$
$108$	1.28367	0.123521
$109$	−2.40780	−0.230626	−0.115313	−	0.993329i	$-0.536787\pi$
$109$	−2.40780	−0.230626	−0.115313	+	0.993329i	$0.536787\pi$
$110$	−24.0159	−2.28983
$111$	−8.22853	−0.781017
$112$	0	0
$113$	−2.71096	−0.255026	−0.127513	−	0.991837i	$-0.540699\pi$
$113$	−2.71096	−0.255026	−0.127513	+	0.991837i	$0.540699\pi$
$114$	4.96858	0.465350
$115$	13.7779	1.28480
$116$	2.01443	0.187036
$117$	−0.722475	−0.0667929
$118$	−15.4535	−1.42261
$119$	0	0
$120$	11.7462	1.07228
$121$	27.0567	2.45970
$122$	−16.4176	−1.48638
$123$	3.65737	0.329774
$124$	0.550041	0.0493952
$125$	−4.23877	−0.379127
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−9.00992	−0.796372
$129$	13.9576	1.22890
$130$	−2.41380	−0.211704
$131$	12.0134	1.04961	0.524807	−	0.851222i	$-0.324138\pi$
$131$	12.0134	1.04961	0.524807	+	0.851222i	$0.324138\pi$
$132$	−1.90121	−0.165479
$133$	0	0
$134$	−16.4935	−1.42482
$135$	−16.4977	−1.41990
$136$	−1.91122	−0.163885
$137$	−12.6214	−1.07832	−0.539161	−	0.842203i	$-0.681258\pi$
$137$	−12.6214	−1.07832	−0.539161	+	0.842203i	$0.681258\pi$
$138$	−8.49715	−0.723325
$139$	17.1553	1.45510	0.727548	−	0.686057i	$-0.240660\pi$
$139$	17.1553	1.45510	0.727548	+	0.686057i	$0.240660\pi$
$140$	0	0
$141$	14.5971	1.22930
$142$	13.3799	1.12281
$143$	3.82501	0.319863
$144$	4.07032	0.339193
$145$	−25.8895	−2.15001
$146$	10.9288	0.904476
$147$	0	0
$148$	1.38214	0.113611
$149$	5.64770	0.462677	0.231339	−	0.972873i	$-0.425689\pi$
$149$	5.64770	0.462677	0.231339	+	0.972873i	$0.425689\pi$
$150$	−6.40267	−0.522776
$151$	13.0571	1.06257	0.531287	−	0.847192i	$-0.321709\pi$
$151$	13.0571	1.06257	0.531287	+	0.847192i	$0.321709\pi$
$152$	−8.17074	−0.662734
$153$	0.750937	0.0607096
$154$	0	0
$155$	−7.06913	−0.567806
$156$	−0.191088	−0.0152993
$157$	6.53317	0.521404	0.260702	−	0.965419i	$-0.416046\pi$
$157$	6.53317	0.521404	0.260702	+	0.965419i	$0.416046\pi$
$158$	−3.06516	−0.243851
$159$	8.36056	0.663035
$160$	−3.74449	−0.296028
$161$	0	0
$162$	5.52059	0.433739
$163$	13.2598	1.03858	0.519292	−	0.854597i	$-0.326196\pi$
$163$	13.2598	1.03858	0.519292	+	0.854597i	$0.326196\pi$
$164$	−0.614326	−0.0479708
$165$	24.4344	1.90221
$166$	−1.63485	−0.126889
$167$	2.01803	0.156160	0.0780801	−	0.996947i	$-0.475121\pi$
$167$	2.01803	0.156160	0.0780801	+	0.996947i	$0.475121\pi$
$168$	0	0
$169$	−12.6156	−0.970427
$170$	2.50889	0.192423
$171$	3.21037	0.245503
$172$	−2.34446	−0.178763
$173$	−1.84088	−0.139960	−0.0699799	−	0.997548i	$-0.522294\pi$
$173$	−1.84088	−0.139960	−0.0699799	+	0.997548i	$0.522294\pi$
$174$	15.9666	1.21043
$175$	0	0
$176$	−21.5495	−1.62436
$177$	15.7228	1.18180
$178$	15.6161	1.17048
$179$	18.7622	1.40235	0.701177	−	0.712987i	$-0.252658\pi$
$179$	18.7622	1.40235	0.701177	+	0.712987i	$0.252658\pi$
$180$	0.775215	0.0577811
$181$	−23.1779	−1.72280	−0.861399	−	0.507929i	$-0.830411\pi$
$181$	−23.1779	−1.72280	−0.861399	+	0.507929i	$0.830411\pi$
$182$	0	0
$183$	16.7036	1.23477
$184$	13.9734	1.03013
$185$	−17.7633	−1.30598
$186$	4.35969	0.319668
$187$	−3.97569	−0.290731
$188$	−2.45187	−0.178821
$189$	0	0
$190$	10.7259	0.778137
$191$	−0.0673136	−0.00487064	−0.00243532	−	0.999997i	$-0.500775\pi$
$191$	−0.0673136	−0.00487064	−0.00243532	+	0.999997i	$0.500775\pi$
$192$	11.7727	0.849619
$193$	−22.7803	−1.63976	−0.819879	−	0.572536i	$-0.805960\pi$
$193$	−22.7803	−1.63976	−0.819879	+	0.572536i	$0.805960\pi$
$194$	1.58477	0.113780
$195$	2.45586	0.175868
$196$	0	0
$197$	−20.6532	−1.47148	−0.735741	−	0.677263i	$-0.763166\pi$
$197$	−20.6532	−1.47148	−0.735741	+	0.677263i	$0.763166\pi$
$198$	9.57000	0.680110
$199$	−0.546917	−0.0387699	−0.0193850	−	0.999812i	$-0.506171\pi$
$199$	−0.546917	−0.0387699	−0.0193850	+	0.999812i	$0.506171\pi$
$200$	10.5291	0.744518
$201$	16.7809	1.18363
$202$	−25.7742	−1.81347
$203$	0	0
$204$	0.198615	0.0139059
$205$	7.89531	0.551432
$206$	−13.5800	−0.946162
$207$	−5.49030	−0.381602
$208$	−2.16590	−0.150178
$209$	−16.9967	−1.17569
$210$	0	0
$211$	−7.86912	−0.541733	−0.270866	−	0.962617i	$-0.587310\pi$
$211$	−7.86912	−0.541733	−0.270866	+	0.962617i	$0.587310\pi$
$212$	−1.40432	−0.0964490
$213$	−13.6130	−0.932747
$214$	20.4862	1.40041
$215$	30.1310	2.05491
$216$	−16.7318	−1.13845
$217$	0	0
$218$	3.20561	0.217112
$219$	−11.1193	−0.751369
$220$	−4.10423	−0.276707
$221$	−0.399590	−0.0268793
$222$	10.9550	0.735252
$223$	−0.641278	−0.0429432	−0.0214716	−	0.999769i	$-0.506835\pi$
$223$	−0.641278	−0.0429432	−0.0214716	+	0.999769i	$0.506835\pi$
$224$	0	0
$225$	−4.13699	−0.275799
$226$	3.60923	0.240082
$227$	−25.4329	−1.68804	−0.844021	−	0.536310i	$-0.819818\pi$
$227$	−25.4329	−1.68804	−0.844021	+	0.536310i	$0.819818\pi$
$228$	0.849111	0.0562337
$229$	17.2560	1.14031	0.570156	−	0.821537i	$-0.306883\pi$
$229$	17.2560	1.14031	0.570156	+	0.821537i	$0.306883\pi$
$230$	−18.3432	−1.20951
$231$	0	0
$232$	−26.2568	−1.72385
$233$	17.2249	1.12844	0.564219	−	0.825625i	$-0.309177\pi$
$233$	17.2249	1.12844	0.564219	+	0.825625i	$0.309177\pi$
$234$	0.961864	0.0628790
$235$	31.5114	2.05557
$236$	−2.64095	−0.171911
$237$	3.11856	0.202573
$238$	0	0
$239$	−5.48952	−0.355088	−0.177544	−	0.984113i	$-0.556815\pi$
$239$	−5.48952	−0.355088	−0.177544	+	0.984113i	$0.556815\pi$
$240$	−13.8359	−0.893105
$241$	26.9366	1.73514	0.867571	−	0.497314i	$-0.165680\pi$
$241$	26.9366	1.73514	0.867571	+	0.497314i	$0.165680\pi$
$242$	−36.0218	−2.31557
$243$	11.3091	0.725479
$244$	−2.80570	−0.179617
$245$	0	0
$246$	−4.86921	−0.310450
$247$	−1.70831	−0.108697
$248$	−7.16943	−0.455259
$249$	1.66334	0.105410
$250$	5.64326	0.356911
$251$	−0.302222	−0.0190761	−0.00953804	−	0.999955i	$-0.503036\pi$
$251$	−0.302222	−0.0190761	−0.00953804	+	0.999955i	$0.503036\pi$
$252$	0	0
$253$	29.0674	1.82745
$254$	1.33134	0.0835360
$255$	−2.55260	−0.159850
$256$	−5.38718	−0.336698
$257$	18.7785	1.17137	0.585685	−	0.810539i	$-0.300826\pi$
$257$	18.7785	1.17137	0.585685	+	0.810539i	$0.300826\pi$
$258$	−18.5824	−1.15689
$259$	0	0
$260$	−0.412509	−0.0255827
$261$	10.3166	0.638581
$262$	−15.9939	−0.988109
$263$	−4.41926	−0.272503	−0.136251	−	0.990674i	$-0.543506\pi$
$263$	−4.41926	−0.272503	−0.136251	+	0.990674i	$0.543506\pi$
$264$	24.7811	1.52517
$265$	18.0483	1.10870
$266$	0	0
$267$	−15.8882	−0.972344
$268$	−2.81867	−0.172178
$269$	2.13768	0.130337	0.0651683	−	0.997874i	$-0.479242\pi$
$269$	2.13768	0.130337	0.0651683	+	0.997874i	$0.479242\pi$
$270$	21.9641	1.33669
$271$	3.17538	0.192891	0.0964454	−	0.995338i	$-0.469253\pi$
$271$	3.17538	0.192891	0.0964454	+	0.995338i	$0.469253\pi$
$272$	2.25123	0.136501
$273$	0	0
$274$	16.8035	1.01513
$275$	21.9025	1.32077
$276$	−1.45213	−0.0874080
$277$	20.5642	1.23559	0.617793	−	0.786341i	$-0.288027\pi$
$277$	20.5642	1.23559	0.617793	+	0.786341i	$0.288027\pi$
$278$	−22.8396	−1.36983
$279$	2.81695	0.168646
$280$	0	0
$281$	−18.9073	−1.12791	−0.563956	−	0.825805i	$-0.690721\pi$
$281$	−18.9073	−1.12791	−0.563956	+	0.825805i	$0.690721\pi$
$282$	−19.4338	−1.15726
$283$	15.0480	0.894513	0.447256	−	0.894406i	$-0.352401\pi$
$283$	15.0480	0.894513	0.447256	+	0.894406i	$0.352401\pi$
$284$	2.28657	0.135683
$285$	−10.9128	−0.646416
$286$	−5.09241	−0.301120
$287$	0	0
$288$	1.49213	0.0879243
$289$	−16.5847	−0.975569
$290$	34.4678	2.02402
$291$	−1.61238	−0.0945195
$292$	1.86770	0.109299
$293$	−12.4842	−0.729334	−0.364667	−	0.931138i	$-0.618817\pi$
$293$	−12.4842	−0.729334	−0.364667	+	0.931138i	$0.618817\pi$
$294$	0	0
$295$	33.9414	1.97615
$296$	−18.0153	−1.04712
$297$	−34.8053	−2.01961
$298$	−7.51903	−0.435566
$299$	2.92151	0.168955
$300$	−1.09419	−0.0631732
$301$	0	0
$302$	−17.3835	−1.00031
$303$	26.2233	1.50649
$304$	9.62434	0.551994
$305$	36.0588	2.06472
$306$	−0.999756	−0.0571522
$307$	−27.4111	−1.56443	−0.782216	−	0.623007i	$-0.785911\pi$
$307$	−27.4111	−1.56443	−0.782216	+	0.623007i	$0.785911\pi$
$308$	0	0
$309$	13.8166	0.785999
$310$	9.41144	0.534534
$311$	25.1878	1.42827	0.714135	−	0.700008i	$-0.246820\pi$
$311$	25.1878	1.42827	0.714135	+	0.700008i	$0.246820\pi$
$312$	2.49070	0.141008
$313$	11.7164	0.662251	0.331125	−	0.943587i	$-0.392572\pi$
$313$	11.7164	0.662251	0.331125	+	0.943587i	$0.392572\pi$
$314$	−8.69790	−0.490851
$315$	0	0
$316$	−0.523824	−0.0294674
$317$	18.0033	1.01117	0.505584	−	0.862777i	$-0.331277\pi$
$317$	18.0033	1.01117	0.505584	+	0.862777i	$0.331277\pi$
$318$	−11.1308	−0.624183
$319$	−54.6192	−3.05809
$320$	25.4141	1.42069
$321$	−20.8432	−1.16335
$322$	0	0
$323$	1.77560	0.0987973
$324$	0.943449	0.0524138
$325$	2.20138	0.122111
$326$	−17.6533	−0.977726
$327$	−3.26147	−0.180360
$328$	8.00734	0.442131
$329$	0	0
$330$	−32.5306	−1.79075
$331$	−14.0863	−0.774254	−0.387127	−	0.922026i	$-0.626533\pi$
$331$	−14.0863	−0.774254	−0.387127	+	0.922026i	$0.626533\pi$
$332$	−0.279390	−0.0153335
$333$	7.07841	0.387894
$334$	−2.68670	−0.147010
$335$	36.2255	1.97921
$336$	0	0
$337$	6.85642	0.373493	0.186747	−	0.982408i	$-0.440206\pi$
$337$	6.85642	0.373493	0.186747	+	0.982408i	$0.440206\pi$
$338$	16.7957	0.913563
$339$	−3.67212	−0.199442
$340$	0.428759	0.0232527
$341$	−14.9138	−0.807627
$342$	−4.27411	−0.231117
$343$	0	0
$344$	30.5585	1.64760
$345$	18.6628	1.00477
$346$	2.45085	0.131759
$347$	15.2537	0.818860	0.409430	−	0.912342i	$-0.365728\pi$
$347$	15.2537	0.818860	0.409430	+	0.912342i	$0.365728\pi$
$348$	2.72864	0.146270
$349$	−7.20325	−0.385581	−0.192791	−	0.981240i	$-0.561754\pi$
$349$	−7.20325	−0.385581	−0.192791	+	0.981240i	$0.561754\pi$
$350$	0	0
$351$	−3.49822	−0.186721
$352$	−7.89977	−0.421059
$353$	4.26996	0.227267	0.113633	−	0.993523i	$-0.463751\pi$
$353$	4.26996	0.227267	0.113633	+	0.993523i	$0.463751\pi$
$354$	−20.9324	−1.11255
$355$	−29.3869	−1.55970
$356$	2.66874	0.141443
$357$	0	0
$358$	−24.9790	−1.32018
$359$	−31.0817	−1.64043	−0.820214	−	0.572057i	$-0.806146\pi$
$359$	−31.0817	−1.64043	−0.820214	+	0.572057i	$0.806146\pi$
$360$	−10.1044	−0.532550
$361$	−11.4090	−0.600475
$362$	30.8577	1.62185
$363$	36.6494	1.92360
$364$	0	0
$365$	−24.0036	−1.25641
$366$	−22.2383	−1.16241
$367$	18.3623	0.958503	0.479251	−	0.877678i	$-0.340908\pi$
$367$	18.3623	0.958503	0.479251	+	0.877678i	$0.340908\pi$
$368$	−16.4593	−0.858002
$369$	−3.14617	−0.163783
$370$	23.6490	1.22945
$371$	0	0
$372$	0.745054	0.0386293
$373$	−20.8539	−1.07977	−0.539886	−	0.841738i	$-0.681533\pi$
$373$	−20.8539	−1.07977	−0.539886	+	0.841738i	$0.681533\pi$
$374$	5.29302	0.273695
$375$	−5.74159	−0.296495
$376$	31.9585	1.64813
$377$	−5.48968	−0.282733
$378$	0	0
$379$	6.97802	0.358437	0.179218	−	0.983809i	$-0.442643\pi$
$379$	6.97802	0.358437	0.179218	+	0.983809i	$0.442643\pi$
$380$	1.83301	0.0940315
$381$	−1.35454	−0.0693953
$382$	0.0896176	0.00458524
$383$	0.0360913	0.00184418	0.000922089	−	1.00000i	$-0.499706\pi$
$383$	0.0360913	0.00184418	0.000922089		1.00000i	$0.499706\pi$
$384$	−12.2043	−0.622799
$385$	0	0
$386$	30.3284	1.54367
$387$	−12.0068	−0.610338
$388$	0.270831	0.0137494
$389$	−19.6007	−0.993793	−0.496897	−	0.867810i	$-0.665527\pi$
$389$	−19.6007	−0.993793	−0.496897	+	0.867810i	$0.665527\pi$
$390$	−3.26959	−0.165562
$391$	−3.03660	−0.153567
$392$	0	0
$393$	16.2726	0.820845
$394$	27.4966	1.38526
$395$	6.73218	0.338733
$396$	1.63548	0.0821858
$397$	−20.6792	−1.03786	−0.518930	−	0.854817i	$-0.673669\pi$
$397$	−20.6792	−1.03786	−0.518930	+	0.854817i	$0.673669\pi$
$398$	0.728135	0.0364981
$399$	0	0
$400$	−12.4022	−0.620112
$401$	−24.7687	−1.23689	−0.618444	−	0.785829i	$-0.712237\pi$
$401$	−24.7687	−1.23689	−0.618444	+	0.785829i	$0.712237\pi$
$402$	−22.3411	−1.11427
$403$	−1.49896	−0.0746684
$404$	−4.40472	−0.219143
$405$	−12.1252	−0.602506
$406$	0	0
$407$	−37.4753	−1.85758
$408$	−2.58882	−0.128166
$409$	37.2721	1.84299	0.921494	−	0.388392i	$-0.126969\pi$
$409$	37.2721	1.84299	0.921494	+	0.388392i	$0.126969\pi$
$410$	−10.5114	−0.519120
$411$	−17.0963	−0.843296
$412$	−2.32077	−0.114336
$413$	0	0
$414$	7.30948	0.359241
$415$	3.59071	0.176261
$416$	−0.793992	−0.0389287
$417$	23.2376	1.13795
$418$	22.6284	1.10679
$419$	25.0016	1.22141	0.610705	−	0.791858i	$-0.290886\pi$
$419$	25.0016	1.22141	0.610705	+	0.791858i	$0.290886\pi$
$420$	0	0
$421$	10.6102	0.517108	0.258554	−	0.965997i	$-0.416754\pi$
$421$	10.6102	0.517108	0.258554	+	0.965997i	$0.416754\pi$
$422$	10.4765	0.509989
$423$	−12.5568	−0.610534
$424$	18.3044	0.888940
$425$	−2.28810	−0.110989
$426$	18.1236	0.878091
$427$	0	0
$428$	3.50102	0.169228
$429$	5.18114	0.250148
$430$	−40.1147	−1.93450
$431$	−33.4302	−1.61028	−0.805138	−	0.593088i	$-0.797909\pi$
$431$	−33.4302	−1.61028	−0.805138	+	0.593088i	$0.797909\pi$
$432$	19.7084	0.948223
$433$	−4.82890	−0.232062	−0.116031	−	0.993246i	$-0.537017\pi$
$433$	−4.82890	−0.232062	−0.116031	+	0.993246i	$0.537017\pi$
$434$	0	0
$435$	−35.0684	−1.68140
$436$	0.547827	0.0262362
$437$	−12.9819	−0.621009
$438$	14.8036	0.707341
$439$	−5.44232	−0.259748	−0.129874	−	0.991531i	$-0.541457\pi$
$439$	−5.44232	−0.259748	−0.129874	+	0.991531i	$0.541457\pi$
$440$	53.4960	2.55032
$441$	0	0
$442$	0.531992	0.0253043
$443$	−23.6814	−1.12514	−0.562568	−	0.826751i	$-0.690187\pi$
$443$	−23.6814	−1.12514	−0.562568	+	0.826751i	$0.690187\pi$
$444$	1.87217	0.0888492
$445$	−34.2986	−1.62591
$446$	0.853762	0.0404268
$447$	7.65004	0.361835
$448$	0	0
$449$	−7.30490	−0.344740	−0.172370	−	0.985032i	$-0.555142\pi$
$449$	−7.30490	−0.344740	−0.172370	+	0.985032i	$0.555142\pi$
$450$	5.50775	0.259638
$451$	16.6568	0.784338
$452$	0.616803	0.0290120
$453$	17.6864	0.830981
$454$	33.8600	1.58913
$455$	0	0
$456$	−11.0676	−0.518288
$457$	4.66805	0.218362	0.109181	−	0.994022i	$-0.465177\pi$
$457$	4.66805	0.218362	0.109181	+	0.994022i	$0.465177\pi$
$458$	−22.9737	−1.07349
$459$	3.63603	0.169715
$460$	−3.13478	−0.146160
$461$	−26.9450	−1.25495	−0.627477	−	0.778635i	$-0.715912\pi$
$461$	−26.9450	−1.25495	−0.627477	+	0.778635i	$0.715912\pi$
$462$	0	0
$463$	9.98472	0.464029	0.232015	−	0.972712i	$-0.425468\pi$
$463$	9.98472	0.464029	0.232015	+	0.972712i	$0.425468\pi$
$464$	30.9280	1.43580
$465$	−9.57543	−0.444050
$466$	−22.9322	−1.06231
$467$	37.0776	1.71575	0.857873	−	0.513861i	$-0.171785\pi$
$467$	37.0776	1.71575	0.857873	+	0.513861i	$0.171785\pi$
$468$	0.164379	0.00759842
$469$	0	0
$470$	−41.9525	−1.93512
$471$	8.84946	0.407761
$472$	34.4231	1.58445
$473$	63.5675	2.92284
$474$	−4.15188	−0.190702
$475$	−9.78198	−0.448828
$476$	0	0
$477$	−7.19198	−0.329298
$478$	7.30845	0.334281
$479$	18.8165	0.859748	0.429874	−	0.902889i	$-0.358558\pi$
$479$	18.8165	0.859748	0.429874	+	0.902889i	$0.358558\pi$
$480$	−5.07207	−0.231507
$481$	−3.76657	−0.171741
$482$	−35.8619	−1.63347
$483$	0	0
$484$	−6.15598	−0.279817
$485$	−3.48072	−0.158051
$486$	−15.0563	−0.682968
$487$	−26.1973	−1.18711	−0.593556	−	0.804793i	$-0.702276\pi$
$487$	−26.1973	−1.18711	−0.593556	+	0.804793i	$0.702276\pi$
$488$	36.5705	1.65547
$489$	17.9609	0.812220
$490$	0	0
$491$	18.9523	0.855306	0.427653	−	0.903943i	$-0.359340\pi$
$491$	18.9523	0.855306	0.427653	+	0.903943i	$0.359340\pi$
$492$	−0.832131	−0.0375153
$493$	5.70594	0.256983
$494$	2.27434	0.102328
$495$	−21.0191	−0.944740
$496$	8.44490	0.379187
$497$	0	0
$498$	−2.21447	−0.0992329
$499$	23.2998	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$499$	23.2998	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$500$	0.964412	0.0431298
$501$	2.73351	0.122124
$502$	0.402361	0.0179583
$503$	−6.11772	−0.272775	−0.136388	−	0.990656i	$-0.543549\pi$
$503$	−6.11772	−0.272775	−0.136388	+	0.990656i	$0.543549\pi$
$504$	0	0
$505$	56.6094	2.51908
$506$	−38.6987	−1.72037
$507$	−17.0883	−0.758918
$508$	0.227522	0.0100946
$509$	3.73713	0.165646	0.0828228	−	0.996564i	$-0.473606\pi$
$509$	3.73713	0.165646	0.0828228	+	0.996564i	$0.473606\pi$
$510$	3.39839	0.150483
$511$	0	0
$512$	25.1920	1.11334
$513$	15.5446	0.686310
$514$	−25.0006	−1.10273
$515$	29.8265	1.31431
$516$	−3.17567	−0.139801
$517$	66.4797	2.92378
$518$	0	0
$519$	−2.49356	−0.109455
$520$	5.37679	0.235788
$521$	21.0747	0.923299	0.461649	−	0.887062i	$-0.347258\pi$
$521$	21.0747	0.923299	0.461649	+	0.887062i	$0.347258\pi$
$522$	−13.7349	−0.601162
$523$	7.26739	0.317781	0.158890	−	0.987296i	$-0.449208\pi$
$523$	7.26739	0.317781	0.158890	+	0.987296i	$0.449208\pi$
$524$	−2.73330	−0.119405
$525$	0	0
$526$	5.88355	0.256535
$527$	1.55801	0.0678679
$528$	−29.1897	−1.27032
$529$	−0.798603	−0.0347219
$530$	−24.0285	−1.04373
$531$	−13.5252	−0.586943
$532$	0	0
$533$	1.67414	0.0725152
$534$	21.1527	0.915368
$535$	−44.9950	−1.94531
$536$	36.7396	1.58691
$537$	25.4142	1.09671
$538$	−2.84599	−0.122699
$539$	0	0
$540$	3.75359	0.161529
$541$	9.77573	0.420291	0.210146	−	0.977670i	$-0.432606\pi$
$541$	9.77573	0.420291	0.210146	+	0.977670i	$0.432606\pi$
$542$	−4.22753	−0.181588
$543$	−31.3954	−1.34731
$544$	0.825271	0.0353832
$545$	−7.04067	−0.301589
$546$	0	0
$547$	19.6478	0.840078	0.420039	−	0.907506i	$-0.362016\pi$
$547$	19.6478	0.840078	0.420039	+	0.907506i	$0.362016\pi$
$548$	2.87165	0.122671
$549$	−14.3689	−0.613251
$550$	−29.1598	−1.24338
$551$	24.3938	1.03921
$552$	18.9276	0.805611
$553$	0	0
$554$	−27.3781	−1.16318
$555$	−24.0611	−1.02134
$556$	−3.90321	−0.165533
$557$	23.3748	0.990423	0.495211	−	0.868773i	$-0.335091\pi$
$557$	23.3748	0.990423	0.495211	+	0.868773i	$0.335091\pi$
$558$	−3.75032	−0.158764
$559$	6.38906	0.270228
$560$	0	0
$561$	−5.38524	−0.227365
$562$	25.1721	1.06182
$563$	2.16432	0.0912152	0.0456076	−	0.998959i	$-0.485478\pi$
$563$	2.16432	0.0912152	0.0456076	+	0.998959i	$0.485478\pi$
$564$	−3.32116	−0.139846
$565$	−7.92715	−0.333498
$566$	−20.0341	−0.842097
$567$	0	0
$568$	−29.8039	−1.25054
$569$	4.40419	0.184633	0.0923165	−	0.995730i	$-0.470573\pi$
$569$	4.40419	0.184633	0.0923165	+	0.995730i	$0.470573\pi$
$570$	14.5286	0.608538
$571$	−2.17029	−0.0908239	−0.0454120	−	0.998968i	$-0.514460\pi$
$571$	−2.17029	−0.0908239	−0.0454120	+	0.998968i	$0.514460\pi$
$572$	−0.870273	−0.0363879
$573$	−0.0911791	−0.00380906
$574$	0	0
$575$	16.7289	0.697645
$576$	−10.1272	−0.421965
$577$	18.3420	0.763587	0.381794	−	0.924248i	$-0.375306\pi$
$577$	18.3420	0.763587	0.381794	+	0.924248i	$0.375306\pi$
$578$	22.0799	0.918403
$579$	−30.8568	−1.28237
$580$	5.89042	0.244586
$581$	0	0
$582$	2.14664	0.0889809
$583$	38.0766	1.57697
$584$	−24.3442	−1.00737
$585$	−2.11260	−0.0873451
$586$	16.6208	0.686597
$587$	4.64632	0.191774	0.0958871	−	0.995392i	$-0.469431\pi$
$587$	4.64632	0.191774	0.0958871	+	0.995392i	$0.469431\pi$
$588$	0	0
$589$	6.66072	0.274450
$590$	−45.1878	−1.86035
$591$	−27.9757	−1.15077
$592$	21.2203	0.872149
$593$	−35.2665	−1.44822	−0.724110	−	0.689684i	$-0.757749\pi$
$593$	−35.2665	−1.44822	−0.724110	+	0.689684i	$0.757749\pi$
$594$	46.3379	1.90127
$595$	0	0
$596$	−1.28497	−0.0526346
$597$	−0.740823	−0.0303199
$598$	−3.88953	−0.159055
$599$	−11.1046	−0.453720	−0.226860	−	0.973927i	$-0.572846\pi$
$599$	−11.1046	−0.453720	−0.226860	+	0.973927i	$0.572846\pi$
$600$	14.2621	0.582247
$601$	−27.6727	−1.12879	−0.564397	−	0.825504i	$-0.690891\pi$
$601$	−27.6727	−1.12879	−0.564397	+	0.825504i	$0.690891\pi$
$602$	0	0
$603$	−14.4354	−0.587853
$604$	−2.97078	−0.120879
$605$	79.1166	3.21655
$606$	−34.9123	−1.41821
$607$	−39.8271	−1.61653	−0.808266	−	0.588818i	$-0.799594\pi$
$607$	−39.8271	−1.61653	−0.808266	+	0.588818i	$0.799594\pi$
$608$	3.52816	0.143086
$609$	0	0
$610$	−48.0067	−1.94374
$611$	6.68176	0.270315
$612$	−0.170854	−0.00690638
$613$	−11.1909	−0.451997	−0.225999	−	0.974128i	$-0.572564\pi$
$613$	−11.1909	−0.451997	−0.225999	+	0.974128i	$0.572564\pi$
$614$	36.4936	1.47276
$615$	10.6945	0.431245
$616$	0	0
$617$	−21.1289	−0.850617	−0.425308	−	0.905049i	$-0.639834\pi$
$617$	−21.1289	−0.850617	−0.425308	+	0.905049i	$0.639834\pi$
$618$	−18.3947	−0.739942
$619$	−6.30389	−0.253375	−0.126687	−	0.991943i	$-0.540434\pi$
$619$	−6.30389	−0.253375	−0.126687	+	0.991943i	$0.540434\pi$
$620$	1.60838	0.0645941
$621$	−26.5840	−1.06678
$622$	−33.5337	−1.34458
$623$	0	0
$624$	−2.93381	−0.117446
$625$	−30.1466	−1.20587
$626$	−15.5986	−0.623445
$627$	−23.0227	−0.919439
$628$	−1.48644	−0.0593153
$629$	3.91496	0.156099
$630$	0	0
$631$	36.4561	1.45130	0.725648	−	0.688066i	$-0.241540\pi$
$631$	36.4561	1.45130	0.725648	+	0.688066i	$0.241540\pi$
$632$	6.82770	0.271591
$633$	−10.6591	−0.423660
$634$	−23.9687	−0.951917
$635$	−2.92411	−0.116040
$636$	−1.90221	−0.0754275
$637$	0	0
$638$	72.7170	2.87889
$639$	11.7103	0.463251
$640$	−26.3460	−1.04142
$641$	23.4463	0.926073	0.463037	−	0.886339i	$-0.346760\pi$
$641$	23.4463	0.926073	0.463037	+	0.886339i	$0.346760\pi$
$642$	27.7494	1.09518
$643$	12.6228	0.497796	0.248898	−	0.968530i	$-0.419932\pi$
$643$	12.6228	0.497796	0.248898	+	0.968530i	$0.419932\pi$
$644$	0	0
$645$	40.8137	1.60704
$646$	−2.36394	−0.0930080
$647$	−25.2402	−0.992294	−0.496147	−	0.868238i	$-0.665252\pi$
$647$	−25.2402	−0.992294	−0.496147	+	0.868238i	$0.665252\pi$
$648$	−12.2972	−0.483081
$649$	71.6065	2.81080
$650$	−2.93080	−0.114955
$651$	0	0
$652$	−3.01688	−0.118150
$653$	−3.66022	−0.143236	−0.0716178	−	0.997432i	$-0.522816\pi$
$653$	−3.66022	−0.143236	−0.0716178	+	0.997432i	$0.522816\pi$
$654$	4.34214	0.169791
$655$	35.1284	1.37258
$656$	−9.43188	−0.368253
$657$	9.56509	0.373170
$658$	0	0
$659$	9.14308	0.356164	0.178082	−	0.984016i	$-0.443011\pi$
$659$	9.14308	0.356164	0.178082	+	0.984016i	$0.443011\pi$
$660$	−5.55935	−0.216398
$661$	−16.9706	−0.660078	−0.330039	−	0.943967i	$-0.607062\pi$
$661$	−16.9706	−0.660078	−0.330039	+	0.943967i	$0.607062\pi$
$662$	18.7537	0.728885
$663$	−0.541261	−0.0210209
$664$	3.64166	0.141324
$665$	0	0
$666$	−9.42380	−0.365165
$667$	−41.7177	−1.61532
$668$	−0.459147	−0.0177649
$669$	−0.868639	−0.0335835
$670$	−48.2287	−1.86324
$671$	76.0736	2.93679
$672$	0	0
$673$	2.01062	0.0775038	0.0387519	−	0.999249i	$-0.487662\pi$
$673$	2.01062	0.0775038	0.0387519	+	0.999249i	$0.487662\pi$
$674$	−9.12826	−0.351607
$675$	−20.0312	−0.771003
$676$	2.87031	0.110397
$677$	−7.34704	−0.282370	−0.141185	−	0.989983i	$-0.545091\pi$
$677$	−7.34704	−0.282370	−0.141185	+	0.989983i	$0.545091\pi$
$678$	4.88885	0.187755
$679$	0	0
$680$	−5.58860	−0.214313
$681$	−34.4500	−1.32013
$682$	19.8554	0.760302
$683$	−30.7252	−1.17567	−0.587833	−	0.808982i	$-0.700019\pi$
$683$	−30.7252	−1.17567	−0.587833	+	0.808982i	$0.700019\pi$
$684$	−0.730429	−0.0279286
$685$	−36.9064	−1.41012
$686$	0	0
$687$	23.3740	0.891775
$688$	−35.9950	−1.37230
$689$	3.82701	0.145798
$690$	−24.8466	−0.945893
$691$	−41.4079	−1.57523	−0.787616	−	0.616166i	$-0.788685\pi$
$691$	−41.4079	−1.57523	−0.787616	+	0.616166i	$0.788685\pi$
$692$	0.418841	0.0159220
$693$	0	0
$694$	−20.3079	−0.770877
$695$	50.1640	1.90283
$696$	−35.5660	−1.34813
$697$	−1.74010	−0.0659108
$698$	9.59001	0.362987
$699$	23.3318	0.882490
$700$	0	0
$701$	5.45833	0.206158	0.103079	−	0.994673i	$-0.467131\pi$
$701$	5.45833	0.206158	0.103079	+	0.994673i	$0.467131\pi$
$702$	4.65734	0.175780
$703$	16.7370	0.631249
$704$	53.6164	2.02074
$705$	42.6835	1.60755
$706$	−5.68479	−0.213950
$707$	0	0
$708$	−3.57728	−0.134442
$709$	4.69396	0.176285	0.0881427	−	0.996108i	$-0.471907\pi$
$709$	4.69396	0.176285	0.0881427	+	0.996108i	$0.471907\pi$
$710$	39.1241	1.46830
$711$	−2.68268	−0.100608
$712$	−34.7853	−1.30363
$713$	−11.3910	−0.426597
$714$	0	0
$715$	11.1847	0.418286
$716$	−4.26882	−0.159533
$717$	−7.43579	−0.277695
$718$	41.3804	1.54430
$719$	32.7760	1.22234	0.611170	−	0.791499i	$-0.290699\pi$
$719$	32.7760	1.22234	0.611170	+	0.791499i	$0.290699\pi$
$720$	11.9020	0.443563
$721$	0	0
$722$	15.1893	0.565289
$723$	36.4868	1.35696
$724$	5.27347	0.195987
$725$	−31.4346	−1.16745
$726$	−48.7930	−1.81088
$727$	16.3785	0.607445	0.303723	−	0.952760i	$-0.401770\pi$
$727$	16.3785	0.607445	0.303723	+	0.952760i	$0.401770\pi$
$728$	0	0
$729$	27.7585	1.02809
$730$	31.9571	1.18278
$731$	−6.64075	−0.245617
$732$	−3.80044	−0.140468
$733$	−10.7267	−0.396199	−0.198100	−	0.980182i	$-0.563477\pi$
$733$	−10.7267	−0.396199	−0.198100	+	0.980182i	$0.563477\pi$
$734$	−24.4465	−0.902337
$735$	0	0
$736$	−6.03378	−0.222408
$737$	76.4253	2.81516
$738$	4.18863	0.154186
$739$	11.9344	0.439014	0.219507	−	0.975611i	$-0.429555\pi$
$739$	11.9344	0.439014	0.219507	+	0.975611i	$0.429555\pi$
$740$	4.04153	0.148570
$741$	−2.31397	−0.0850060
$742$	0	0
$743$	3.45152	0.126624	0.0633119	−	0.997994i	$-0.479834\pi$
$743$	3.45152	0.126624	0.0633119	+	0.997994i	$0.479834\pi$
$744$	−9.71130	−0.356034
$745$	16.5145	0.605043
$746$	27.7637	1.01650
$747$	−1.43085	−0.0523520
$748$	0.904557	0.0330739
$749$	0	0
$750$	7.64404	0.279121
$751$	−0.508332	−0.0185493	−0.00927465	−	0.999957i	$-0.502952\pi$
$751$	−0.508332	−0.0185493	−0.00927465	+	0.999957i	$0.502952\pi$
$752$	−37.6440	−1.37274
$753$	−0.409372	−0.0149184
$754$	7.30866	0.266166
$755$	38.1804	1.38953
$756$	0	0
$757$	11.2937	0.410477	0.205238	−	0.978712i	$-0.434203\pi$
$757$	11.2937	0.410477	0.205238	+	0.978712i	$0.434203\pi$
$758$	−9.29014	−0.337433
$759$	39.3730	1.42915
$760$	−23.8921	−0.866658
$761$	−27.6722	−1.00312	−0.501558	−	0.865124i	$-0.667240\pi$
$761$	−27.6722	−1.00312	−0.501558	+	0.865124i	$0.667240\pi$
$762$	1.80336	0.0653289
$763$	0	0
$764$	0.0153153	0.000554089 0
$765$	2.19582	0.0793900
$766$	−0.0480499	−0.00173611
$767$	7.19704	0.259870
$768$	−7.29716	−0.263313
$769$	−29.1720	−1.05197	−0.525984	−	0.850495i	$-0.676303\pi$
$769$	−29.1720	−1.05197	−0.525984	+	0.850495i	$0.676303\pi$
$770$	0	0
$771$	25.4362	0.916064
$772$	5.18300	0.186540
$773$	19.2965	0.694046	0.347023	−	0.937857i	$-0.387193\pi$
$773$	19.2965	0.694046	0.347023	+	0.937857i	$0.387193\pi$
$774$	15.9851	0.574574
$775$	−8.58322	−0.308318
$776$	−3.53010	−0.126723
$777$	0	0
$778$	26.0952	0.935560
$779$	−7.43917	−0.266536
$780$	−0.558761	−0.0200069
$781$	−61.9978	−2.21846
$782$	4.04276	0.144569
$783$	49.9528	1.78517
$784$	0	0
$785$	19.1037	0.681840
$786$	−21.6645	−0.772746
$787$	18.6400	0.664446	0.332223	−	0.943201i	$-0.392201\pi$
$787$	18.6400	0.664446	0.332223	+	0.943201i	$0.392201\pi$
$788$	4.69906	0.167397
$789$	−5.98607	−0.213110
$790$	−8.96285	−0.318884
$791$	0	0
$792$	−21.3174	−0.757480
$793$	7.64602	0.271518
$794$	27.5311	0.977044
$795$	24.4472	0.867052
$796$	0.124436	0.00441050
$797$	12.4612	0.441399	0.220699	−	0.975342i	$-0.429166\pi$
$797$	12.4612	0.441399	0.220699	+	0.975342i	$0.429166\pi$
$798$	0	0
$799$	−6.94498	−0.245696
$800$	−4.54650	−0.160743
$801$	13.6675	0.482917
$802$	32.9756	1.16441
$803$	−50.6405	−1.78707
$804$	−3.81801	−0.134651
$805$	0	0
$806$	1.99563	0.0702930
$807$	2.89558	0.101929
$808$	57.4126	2.01977
$809$	−27.8109	−0.977779	−0.488890	−	0.872346i	$-0.662598\pi$
$809$	−27.8109	−0.977779	−0.488890	+	0.872346i	$0.662598\pi$
$810$	16.1428	0.567200
$811$	27.0417	0.949561	0.474781	−	0.880104i	$-0.342527\pi$
$811$	27.0417	0.949561	0.474781	+	0.880104i	$0.342527\pi$
$812$	0	0
$813$	4.30119	0.150849
$814$	49.8925	1.74873
$815$	38.7730	1.35816
$816$	3.04938	0.106750
$817$	−28.3902	−0.993247
$818$	−49.6221	−1.73499
$819$	0	0
$820$	−1.79636	−0.0627314
$821$	−0.578331	−0.0201839	−0.0100920	−	0.999949i	$-0.503212\pi$
$821$	−0.578331	−0.0201839	−0.0100920	+	0.999949i	$0.503212\pi$
$822$	22.7610	0.793881
$823$	9.69777	0.338043	0.169022	−	0.985612i	$-0.445939\pi$
$823$	9.69777	0.338043	0.169022	+	0.985612i	$0.445939\pi$
$824$	30.2497	1.05380
$825$	29.6678	1.03290
$826$	0	0
$827$	27.9922	0.973384	0.486692	−	0.873574i	$-0.338203\pi$
$827$	27.9922	0.973384	0.486692	+	0.873574i	$0.338203\pi$
$828$	1.24916	0.0434114
$829$	−26.5981	−0.923791	−0.461895	−	0.886934i	$-0.652830\pi$
$829$	−26.5981	−0.923791	−0.461895	+	0.886934i	$0.652830\pi$
$830$	−4.78048	−0.165933
$831$	27.8551	0.966284
$832$	5.38889	0.186826
$833$	0	0
$834$	−30.9373	−1.07127
$835$	5.90095	0.204211
$836$	3.86712	0.133747
$837$	13.6396	0.471454
$838$	−33.2858	−1.14984
$839$	−4.90573	−0.169365	−0.0846823	−	0.996408i	$-0.526988\pi$
$839$	−4.90573	−0.169365	−0.0846823	+	0.996408i	$0.526988\pi$
$840$	0	0
$841$	49.3899	1.70310
$842$	−14.1258	−0.486807
$843$	−25.6107	−0.882079
$844$	1.79040	0.0616280
$845$	−36.8892	−1.26903
$846$	16.7175	0.574758
$847$	0	0
$848$	−21.5608	−0.740401
$849$	20.3832	0.699550
$850$	3.04625	0.104486
$851$	−28.6233	−0.981194
$852$	3.09725	0.106110
$853$	−44.0800	−1.50927	−0.754635	−	0.656145i	$-0.772186\pi$
$853$	−44.0800	−1.50927	−0.754635	+	0.656145i	$0.772186\pi$
$854$	0	0
$855$	9.38746	0.321044
$856$	−45.6335	−1.55972
$857$	−10.7388	−0.366829	−0.183415	−	0.983036i	$-0.558715\pi$
$857$	−10.7388	−0.366829	−0.183415	+	0.983036i	$0.558715\pi$
$858$	−6.89788	−0.235490
$859$	−51.5763	−1.75976	−0.879880	−	0.475197i	$-0.842377\pi$
$859$	−51.5763	−1.75976	−0.879880	+	0.475197i	$0.842377\pi$
$860$	−6.85545	−0.233769
$861$	0	0
$862$	44.5071	1.51592
$863$	−33.0290	−1.12432	−0.562160	−	0.827029i	$-0.690029\pi$
$863$	−33.0290	−1.12432	−0.562160	+	0.827029i	$0.690029\pi$
$864$	7.22486	0.245795
$865$	−5.38294	−0.183026
$866$	6.42894	0.218464
$867$	−22.4646	−0.762939
$868$	0	0
$869$	14.2029	0.481801
$870$	46.6881	1.58288
$871$	7.68137	0.260273
$872$	−7.14058	−0.241810
$873$	1.38702	0.0469434
$874$	17.2834	0.584620
$875$	0	0
$876$	2.52987	0.0854765
$877$	30.4104	1.02689	0.513443	−	0.858124i	$-0.328370\pi$
$877$	30.4104	1.02689	0.513443	+	0.858124i	$0.328370\pi$
$878$	7.24560	0.244527
$879$	−16.9104	−0.570372
$880$	−63.0131	−2.12417
$881$	2.70216	0.0910381	0.0455190	−	0.998963i	$-0.485506\pi$
$881$	2.70216	0.0910381	0.0455190	+	0.998963i	$0.485506\pi$
$882$	0	0
$883$	57.0235	1.91899	0.959497	−	0.281719i	$-0.0909045\pi$
$883$	57.0235	1.91899	0.959497	+	0.281719i	$0.0909045\pi$
$884$	0.0909154	0.00305782
$885$	45.9751	1.54544
$886$	31.5281	1.05921
$887$	−19.5634	−0.656876	−0.328438	−	0.944525i	$-0.606522\pi$
$887$	−19.5634	−0.656876	−0.328438	+	0.944525i	$0.606522\pi$
$888$	−24.4025	−0.818895
$889$	0	0
$890$	45.6633	1.53064
$891$	−25.5806	−0.856982
$892$	0.145905	0.00488525
$893$	−29.6909	−0.993566
$894$	−10.1848	−0.340632
$895$	54.8628	1.83386
$896$	0	0
$897$	3.95731	0.132131
$898$	9.72534	0.324539
$899$	21.4044	0.713876
$900$	0.941254	0.0313751
$901$	−3.97777	−0.132519
$902$	−22.1759	−0.738378
$903$	0	0
$904$	−8.03963	−0.267394
$905$	−67.7746	−2.25290
$906$	−23.5467	−0.782288
$907$	−38.9478	−1.29324	−0.646620	−	0.762812i	$-0.723818\pi$
$907$	−38.9478	−1.29324	−0.646620	+	0.762812i	$0.723818\pi$
$908$	5.78654	0.192033
$909$	−22.5580	−0.748202
$910$	0	0
$911$	49.6820	1.64604	0.823020	−	0.568013i	$-0.192288\pi$
$911$	49.6820	1.64604	0.823020	+	0.568013i	$0.192288\pi$
$912$	13.0366	0.431684
$913$	7.57535	0.250708
$914$	−6.21479	−0.205567
$915$	48.8432	1.61471
$916$	−3.92613	−0.129723
$917$	0	0
$918$	−4.84081	−0.159770
$919$	−17.4080	−0.574238	−0.287119	−	0.957895i	$-0.592698\pi$
$919$	−17.4080	−0.574238	−0.287119	+	0.957895i	$0.592698\pi$
$920$	40.8598	1.34711
$921$	−37.1295	−1.22346
$922$	35.8731	1.18142
$923$	−6.23129	−0.205105
$924$	0	0
$925$	−21.5679	−0.709148
$926$	−13.2931	−0.436838
$927$	−11.8854	−0.390369
$928$	11.3378	0.372182
$929$	13.5172	0.443486	0.221743	−	0.975105i	$-0.428825\pi$
$929$	13.5172	0.443486	0.221743	+	0.975105i	$0.428825\pi$
$930$	12.7482	0.418030
$931$	0	0
$932$	−3.91903	−0.128372
$933$	34.1180	1.11697
$934$	−49.3631	−1.61521
$935$	−11.6254	−0.380190
$936$	−2.14257	−0.0700322
$937$	−19.8645	−0.648944	−0.324472	−	0.945895i	$-0.605187\pi$
$937$	−19.8645	−0.648944	−0.324472	+	0.945895i	$0.605187\pi$
$938$	0	0
$939$	15.8704	0.517910
$940$	−7.16952	−0.233844
$941$	−2.52477	−0.0823051	−0.0411525	−	0.999153i	$-0.513103\pi$
$941$	−2.52477	−0.0823051	−0.0411525	+	0.999153i	$0.513103\pi$
$942$	−11.7817	−0.383868
$943$	12.7223	0.414295
$944$	−40.5470	−1.31969
$945$	0	0
$946$	−84.6302	−2.75157
$947$	−13.3429	−0.433586	−0.216793	−	0.976218i	$-0.569560\pi$
$947$	−13.3429	−0.433586	−0.216793	+	0.976218i	$0.569560\pi$
$948$	−0.709541	−0.0230448
$949$	−5.08979	−0.165222
$950$	13.0232	0.422528
$951$	24.3863	0.790780
$952$	0	0
$953$	−19.1650	−0.620816	−0.310408	−	0.950603i	$-0.600466\pi$
$953$	−19.1650	−0.620816	−0.310408	+	0.950603i	$0.600466\pi$
$954$	9.57501	0.310002
$955$	−0.196832	−0.00636934
$956$	1.24899	0.0403951
$957$	−73.9841	−2.39157
$958$	−25.0513	−0.809369
$959$	0	0
$960$	34.4245	1.11105
$961$	−25.1555	−0.811469
$962$	5.01461	0.161677
$963$	17.9299	0.577782
$964$	−6.12867	−0.197391
$965$	−66.6119	−2.14431
$966$	0	0
$967$	−41.8133	−1.34463	−0.672313	−	0.740267i	$-0.734699\pi$
$967$	−41.8133	−1.34463	−0.672313	+	0.740267i	$0.734699\pi$
$968$	80.2392	2.57899
$969$	2.40513	0.0772639
$970$	4.63403	0.148790
$971$	−4.23069	−0.135769	−0.0678846	−	0.997693i	$-0.521625\pi$
$971$	−4.23069	−0.135769	−0.0678846	+	0.997693i	$0.521625\pi$
$972$	−2.57307	−0.0825311
$973$	0	0
$974$	34.8776	1.11755
$975$	2.98186	0.0954960
$976$	−43.0765	−1.37885
$977$	−2.61078	−0.0835263	−0.0417631	−	0.999128i	$-0.513297\pi$
$977$	−2.61078	−0.0835263	−0.0417631	+	0.999128i	$0.513297\pi$
$978$	−23.9122	−0.764626
$979$	−72.3600	−2.31264
$980$	0	0
$981$	2.80561	0.0895761
$982$	−25.2321	−0.805188
$983$	−41.6466	−1.32832	−0.664159	−	0.747591i	$-0.731210\pi$
$983$	−41.6466	−1.32832	−0.664159	+	0.747591i	$0.731210\pi$
$984$	10.8463	0.345767
$985$	−60.3923	−1.92426
$986$	−7.59658	−0.241924
$987$	0	0
$988$	0.388677	0.0123655
$989$	48.5523	1.54387
$990$	27.9837	0.889381
$991$	0.195803	0.00621990	0.00310995	−	0.999995i	$-0.499010\pi$
$991$	0.195803	0.00621990	0.00310995	+	0.999995i	$0.499010\pi$
$992$	3.09579	0.0982915
$993$	−19.0805	−0.605502
$994$	0	0
$995$	−1.59925	−0.0506995
$996$	−0.378445	−0.0119915
$997$	0.840968	0.0266337	0.0133169	−	0.999911i	$-0.495761\pi$
$997$	0.840968	0.0266337	0.0133169	+	0.999911i	$0.495761\pi$
$998$	−31.0200	−0.981921
$999$	34.2736	1.08437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.5		16
7.6	odd	2		889.2.a.c.1.5	✓	16
21.20	even	2		8001.2.a.t.1.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.5	✓	16	7.6	odd	2
6223.2.a.k.1.5		16	1.1	even	1	trivial
8001.2.a.t.1.12		16	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-1.33134 q^{2} +1.35454 q^{3} -0.227522 q^{4} +2.92411 q^{5} -1.80336 q^{6} +2.96560 q^{8} -1.16521 q^{9} -3.89299 q^{10} +6.16901 q^{11} -0.308188 q^{12} +0.620036 q^{13} +3.96083 q^{15} -3.49319 q^{16} -0.644462 q^{17} +1.55130 q^{18} -2.75517 q^{19} -0.665298 q^{20} -8.21308 q^{22} +4.71184 q^{23} +4.01703 q^{24} +3.55041 q^{25} -0.825482 q^{26} -5.64196 q^{27} -8.85381 q^{29} -5.27323 q^{30} -2.41753 q^{31} -1.28056 q^{32} +8.35618 q^{33} +0.858001 q^{34} +0.265112 q^{36} -6.07476 q^{37} +3.66808 q^{38} +0.839865 q^{39} +8.67173 q^{40} +2.70008 q^{41} +10.3043 q^{43} -1.40358 q^{44} -3.40721 q^{45} -6.27308 q^{46} +10.7764 q^{47} -4.73167 q^{48} -4.72681 q^{50} -0.872951 q^{51} -0.141072 q^{52} +6.17224 q^{53} +7.51139 q^{54} +18.0388 q^{55} -3.73200 q^{57} +11.7875 q^{58} +11.6075 q^{59} -0.901175 q^{60} +12.3316 q^{61} +3.21857 q^{62} +8.69124 q^{64} +1.81305 q^{65} -11.1250 q^{66} +12.3886 q^{67} +0.146629 q^{68} +6.38238 q^{69} -10.0499 q^{71} -3.45556 q^{72} -8.20886 q^{73} +8.08760 q^{74} +4.80918 q^{75} +0.626862 q^{76} -1.11815 q^{78} +2.30230 q^{79} -10.2145 q^{80} -4.14663 q^{81} -3.59473 q^{82} +1.22797 q^{83} -1.88448 q^{85} -13.7186 q^{86} -11.9929 q^{87} +18.2948 q^{88} -11.7296 q^{89} +4.53618 q^{90} -1.07205 q^{92} -3.27465 q^{93} -14.3471 q^{94} -8.05642 q^{95} -1.73457 q^{96} -1.19035 q^{97} -7.18822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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