LMFDB - Embedded newform 230.3.d.a.91.12

Newspace parameters

Level:	$ N $	$=$	$ 230 = 2 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	230.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.26704608029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.12
Root			$3.68124i$ of defining polynomial
Character	$\chi$	$=$	230.91
Dual form			230.3.d.a.91.11

$q$-expansion

$f(q)$	$=$	$q+1.41421 q^{2} -3.36596 q^{3} +2.00000 q^{4} +2.23607i q^{5} -4.76019 q^{6} +1.16919i q^{7} +2.82843 q^{8} +2.32968 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -3.36596 q^{3} +2.00000 q^{4} +2.23607i q^{5} -4.76019 q^{6} +1.16919i q^{7} +2.82843 q^{8} +2.32968 q^{9} +3.16228i q^{10} +10.6148i q^{11} -6.73192 q^{12} -15.0913 q^{13} +1.65348i q^{14} -7.52651i q^{15} +4.00000 q^{16} +20.0887i q^{17} +3.29467 q^{18} +22.5221i q^{19} +4.47214i q^{20} -3.93543i q^{21} +15.0116i q^{22} +(-20.9683 + 9.45142i) q^{23} -9.52037 q^{24} -5.00000 q^{25} -21.3424 q^{26} +22.4520 q^{27} +2.33837i q^{28} +32.5993 q^{29} -10.6441i q^{30} -27.0975 q^{31} +5.65685 q^{32} -35.7289i q^{33} +28.4097i q^{34} -2.61438 q^{35} +4.65936 q^{36} -53.0568i q^{37} +31.8510i q^{38} +50.7968 q^{39} +6.32456i q^{40} +9.43720 q^{41} -5.56554i q^{42} +36.4382i q^{43} +21.2296i q^{44} +5.20933i q^{45} +(-29.6537 + 13.3663i) q^{46} -49.1365 q^{47} -13.4638 q^{48} +47.6330 q^{49} -7.07107 q^{50} -67.6176i q^{51} -30.1827 q^{52} -104.253i q^{53} +31.7520 q^{54} -23.7354 q^{55} +3.30696i q^{56} -75.8083i q^{57} +46.1023 q^{58} +53.5457 q^{59} -15.0530i q^{60} -23.5166i q^{61} -38.3217 q^{62} +2.72383i q^{63} +8.00000 q^{64} -33.7453i q^{65} -50.5284i q^{66} +59.4754i q^{67} +40.1773i q^{68} +(70.5785 - 31.8131i) q^{69} -3.69729 q^{70} +55.2130 q^{71} +6.58933 q^{72} -8.77305 q^{73} -75.0337i q^{74} +16.8298 q^{75} +45.0441i q^{76} -12.4107 q^{77} +71.8376 q^{78} +57.0848i q^{79} +8.94427i q^{80} -96.5397 q^{81} +13.3462 q^{82} -55.1788i q^{83} -7.87086i q^{84} -44.9196 q^{85} +51.5314i q^{86} -109.728 q^{87} +30.0232i q^{88} +139.825i q^{89} +7.36710i q^{90} -17.6446i q^{91} +(-41.9366 + 18.9028i) q^{92} +91.2091 q^{93} -69.4894 q^{94} -50.3608 q^{95} -19.0407 q^{96} -19.8635i q^{97} +67.3632 q^{98} +24.7291i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9	$ 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 + 24 * q^13 + 64 * q^16 - 32 * q^18 + 4 * q^23 - 16 * q^24 - 80 * q^25 + 96 * q^26 - 96 * q^27 - 108 * q^29 - 116 * q^31 + 60 * q^35 + 128 * q^36 + 248 * q^39 - 156 * q^41 - 124 * q^46 - 128 * q^47 - 28 * q^49 + 48 * q^52 + 224 * q^54 + 160 * q^58 + 204 * q^59 + 64 * q^62 + 128 * q^64 - 268 * q^69 - 120 * q^70 + 236 * q^71 - 64 * q^72 - 112 * q^73 - 936 * q^77 - 432 * q^78 - 136 * q^81 - 64 * q^82 + 60 * q^85 - 152 * q^87 + 8 * q^92 + 856 * q^93 - 216 * q^94 - 160 * q^95 - 32 * q^96 + 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$	−3.36596			−1.12199			−0.560993	−	0.827820i	$-0.689581\pi$
$3$	−3.36596			−1.12199			−0.560993	+	0.827820i	$0.689581\pi$
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$	−4.76019			−0.793364
$7$			1.16919i			0.167026i	0.996507	+	0.0835132i	$0.0266141\pi$
$7$			1.16919i			0.167026i	−0.996507	+	0.0835132i	$0.973386\pi$
$8$	2.82843			0.353553
$9$	2.32968			0.258853
$10$			3.16228i			0.316228i
$11$			10.6148i			0.964981i	0.875901	+	0.482490i	$0.160268\pi$
$11$			10.6148i			0.964981i	−0.875901	+	0.482490i	$0.839732\pi$
$12$	−6.73192			−0.560993
$13$	−15.0913			−1.16087			−0.580436	−	0.814306i	$-0.697118\pi$
$13$	−15.0913			−1.16087			−0.580436	+	0.814306i	$0.697118\pi$
$14$			1.65348i			0.118106i
$15$		−	7.52651i		−	0.501768i
$16$	4.00000			0.250000
$17$			20.0887i			1.18169i	0.806787	+	0.590843i	$0.201205\pi$
$17$			20.0887i			1.18169i	−0.806787	+	0.590843i	$0.798795\pi$
$18$	3.29467			0.183037
$19$			22.5221i			1.18537i	0.805434	+	0.592686i	$0.201932\pi$
$19$			22.5221i			1.18537i	−0.805434	+	0.592686i	$0.798068\pi$
$20$			4.47214i			0.223607i
$21$		−	3.93543i		−	0.187401i
$22$			15.0116i			0.682344i
$23$	−20.9683	+	9.45142i	−0.911666	+	0.410931i
$23$	−20.9683	+	9.45142i	−0.911666	+	0.410931i
$24$	−9.52037			−0.396682
$25$	−5.00000			−0.200000
$26$	−21.3424			−0.820860
$27$	22.4520			0.831556
$28$			2.33837i			0.0835132i
$29$	32.5993			1.12411			0.562056	−	0.827099i	$-0.310010\pi$
$29$	32.5993			1.12411			0.562056	+	0.827099i	$0.310010\pi$
$30$		−	10.6441i		−	0.354803i
$31$	−27.0975			−0.874113			−0.437056	−	0.899434i	$-0.643979\pi$
$31$	−27.0975			−0.874113			−0.437056	+	0.899434i	$0.643979\pi$
$32$	5.65685			0.176777
$33$		−	35.7289i		−	1.08270i
$34$			28.4097i			0.835578i
$35$	−2.61438			−0.0746965
$36$	4.65936			0.129427
$37$		−	53.0568i		−	1.43397i	−0.697089	−	0.716984i	$-0.745522\pi$
$37$		−	53.0568i		−	1.43397i	0.697089	−	0.716984i	$-0.254478\pi$
$38$			31.8510i			0.838184i
$39$	50.7968			1.30248
$40$			6.32456i			0.158114i
$41$	9.43720			0.230176			0.115088	−	0.993355i	$-0.463285\pi$
$41$	9.43720			0.230176			0.115088	+	0.993355i	$0.463285\pi$
$42$		−	5.56554i		−	0.132513i
$43$			36.4382i			0.847400i	0.905802	+	0.423700i	$0.139269\pi$
$43$			36.4382i			0.847400i	−0.905802	+	0.423700i	$0.860731\pi$
$44$			21.2296i			0.482490i
$45$			5.20933i			0.115763i
$46$	−29.6537	+	13.3663i	−0.644645	+	0.290572i
$47$	−49.1365			−1.04546			−0.522728	−	0.852499i	$-0.675086\pi$
$47$	−49.1365			−1.04546			−0.522728	+	0.852499i	$0.675086\pi$
$48$	−13.4638			−0.280497
$49$	47.6330			0.972102
$50$	−7.07107			−0.141421
$51$		−	67.6176i		−	1.32584i
$52$	−30.1827			−0.580436
$53$		−	104.253i		−	1.96703i	−0.180815	−	0.983517i	$-0.557874\pi$
$53$		−	104.253i		−	1.96703i	0.180815	−	0.983517i	$-0.442126\pi$
$54$	31.7520			0.587999
$55$	−23.7354			−0.431552
$56$			3.30696i			0.0590528i
$57$		−	75.8083i		−	1.32997i
$58$	46.1023			0.794868
$59$	53.5457			0.907554			0.453777	−	0.891115i	$-0.350076\pi$
$59$	53.5457			0.907554			0.453777	+	0.891115i	$0.350076\pi$
$60$		−	15.0530i		−	0.250884i
$61$		−	23.5166i		−	0.385518i	−0.981246	−	0.192759i	$-0.938257\pi$
$61$		−	23.5166i		−	0.385518i	0.981246	−	0.192759i	$-0.0617435\pi$
$62$	−38.3217			−0.618091
$63$			2.72383i			0.0432354i
$64$	8.00000			0.125000
$65$		−	33.7453i		−	0.519158i
$66$		−	50.5284i		−	0.765581i
$67$			59.4754i			0.887692i	0.896103	+	0.443846i	$0.146386\pi$
$67$			59.4754i			0.887692i	−0.896103	+	0.443846i	$0.853614\pi$
$68$			40.1773i			0.590843i
$69$	70.5785	−	31.8131i	1.02288	−	0.461060i
$70$	−3.69729			−0.0528184
$71$	55.2130			0.777648			0.388824	−	0.921312i	$-0.372881\pi$
$71$	55.2130			0.777648			0.388824	+	0.921312i	$0.372881\pi$
$72$	6.58933			0.0915185
$73$	−8.77305			−0.120179			−0.0600894	−	0.998193i	$-0.519139\pi$
$73$	−8.77305			−0.120179			−0.0600894	+	0.998193i	$0.519139\pi$
$74$		−	75.0337i		−	1.01397i
$75$	16.8298			0.224397
$76$			45.0441i			0.592686i
$77$	−12.4107			−0.161177
$78$	71.8376			0.920994
$79$			57.0848i			0.722592i	0.932451	+	0.361296i	$0.117666\pi$
$79$			57.0848i			0.722592i	−0.932451	+	0.361296i	$0.882334\pi$
$80$			8.94427i			0.111803i
$81$	−96.5397			−1.19185
$82$	13.3462			0.162759
$83$		−	55.1788i		−	0.664805i	−0.943138	−	0.332403i	$-0.892141\pi$
$83$		−	55.1788i		−	0.664805i	0.943138	−	0.332403i	$-0.107859\pi$
$84$		−	7.87086i		−	0.0937007i
$85$	−44.9196			−0.528466
$86$			51.5314i			0.599203i
$87$	−109.728			−1.26124
$88$			30.0232i			0.341172i
$89$			139.825i			1.57107i	0.618815	+	0.785536i	$0.287613\pi$
$89$			139.825i			1.57107i	−0.618815	+	0.785536i	$0.712387\pi$
$90$			7.36710i			0.0818567i
$91$		−	17.6446i		−	0.193896i
$92$	−41.9366	+	18.9028i	−0.455833	+	0.205466i
$93$	91.2091			0.980743
$94$	−69.4894			−0.739249
$95$	−50.3608			−0.530114
$96$	−19.0407			−0.198341
$97$		−	19.8635i		−	0.204778i	−0.994744	−	0.102389i	$-0.967351\pi$
$97$		−	19.8635i		−	0.204778i	0.994744	−	0.102389i	$-0.0326487\pi$
$98$	67.3632			0.687380
$99$			24.7291i			0.249789i
$100$	−10.0000			−0.100000
$101$	86.5639			0.857068			0.428534	−	0.903526i	$-0.359030\pi$
$101$	86.5639			0.857068			0.428534	+	0.903526i	$0.359030\pi$
$102$		−	95.6258i		−	0.937507i
$103$			144.118i			1.39920i	0.714535	+	0.699600i	$0.246639\pi$
$103$			144.118i			1.39920i	−0.714535	+	0.699600i	$0.753361\pi$
$104$	−42.6847			−0.410430
$105$	8.79989			0.0838085
$106$		−	147.436i		−	1.39090i
$107$		−	10.4544i		−	0.0977050i	−0.998806	−	0.0488525i	$-0.984444\pi$
$107$		−	10.4544i		−	0.0977050i	0.998806	−	0.0488525i	$-0.0155564\pi$
$108$	44.9040			0.415778
$109$		−	69.1221i		−	0.634148i	−0.948401	−	0.317074i	$-0.897300\pi$
$109$		−	69.1221i		−	0.634148i	0.948401	−	0.317074i	$-0.102700\pi$
$110$	−33.5669			−0.305154
$111$			178.587i			1.60889i
$112$			4.67674i			0.0417566i
$113$			137.264i			1.21473i	0.794423	+	0.607364i	$0.207773\pi$
$113$			137.264i			1.21473i	−0.794423	+	0.607364i	$0.792227\pi$
$114$		−	107.209i		−	0.940431i
$115$	−21.1340	−	46.8866i	−0.183774	−	0.407710i
$116$	65.1985			0.562056
$117$	−35.1580			−0.300496
$118$	75.7250			0.641738
$119$	−23.4874			−0.197373
$120$		−	21.2882i		−	0.177402i
$121$	8.32629			0.0688123
$122$		−	33.2575i		−	0.272602i
$123$	−31.7652			−0.258254
$124$	−54.1950			−0.437056
$125$		−	11.1803i		−	0.0894427i
$126$			3.85208i			0.0305720i
$127$	57.9697			0.456455			0.228227	−	0.973608i	$-0.426707\pi$
$127$	57.9697			0.456455			0.228227	+	0.973608i	$0.426707\pi$
$128$	11.3137			0.0883883
$129$		−	122.650i		−	0.950772i
$130$		−	47.7230i		−	0.367100i
$131$	−95.0810			−0.725809			−0.362905	−	0.931826i	$-0.618215\pi$
$131$	−95.0810			−0.725809			−0.362905	+	0.931826i	$0.618215\pi$
$132$		−	71.4579i		−	0.541348i
$133$	−26.3325			−0.197988
$134$			84.1109i			0.627693i
$135$			50.2042i			0.371883i
$136$			56.8193i			0.417789i
$137$		−	198.711i		−	1.45045i	−0.688514	−	0.725223i	$-0.741737\pi$
$137$		−	198.711i		−	1.45045i	0.688514	−	0.725223i	$-0.258263\pi$
$138$	99.8131	−	44.9905i	0.723283	−	0.326018i
$139$	−0.129480			−0.000931508			−0.000465754	−	1.00000i	$-0.500148\pi$
$139$	−0.129480			−0.000931508			−0.000465754		1.00000i	$0.500148\pi$
$140$	−5.22876			−0.0373483
$141$	165.391			1.17299
$142$	78.0830			0.549880
$143$		−	160.191i		−	1.12022i
$144$	9.31873			0.0647134
$145$			72.8942i			0.502718i
$146$	−12.4070			−0.0849792
$147$	−160.331			−1.09069
$148$		−	106.114i		−	0.716984i
$149$		−	63.8745i		−	0.428688i	−0.976758	−	0.214344i	$-0.931239\pi$
$149$		−	63.8745i		−	0.428688i	0.976758	−	0.214344i	$-0.0687613\pi$
$150$	23.8009			0.158673
$151$	−35.2673			−0.233558			−0.116779	−	0.993158i	$-0.537257\pi$
$151$	−35.2673			−0.233558			−0.116779	+	0.993158i	$0.537257\pi$
$152$			63.7020i			0.419092i
$153$			46.8002i			0.305884i
$154$	−17.5513			−0.113970
$155$		−	60.5919i		−	0.390915i
$156$	101.594			0.651241
$157$			289.136i			1.84163i	0.390002	+	0.920814i	$0.372474\pi$
$157$			289.136i			1.84163i	−0.390002	+	0.920814i	$0.627526\pi$
$158$			80.7300i			0.510950i
$159$			350.911i			2.20699i
$160$			12.6491i			0.0790569i
$161$	−11.0505	−	24.5159i	−0.0686364	−	0.152272i
$162$	−136.528			−0.842764
$163$	−46.2536			−0.283764			−0.141882	−	0.989884i	$-0.545315\pi$
$163$	−46.2536			−0.283764			−0.141882	+	0.989884i	$0.545315\pi$
$164$	18.8744			0.115088
$165$	79.8923			0.484196
$166$		−	78.0346i		−	0.470088i
$167$	−90.8735			−0.544153			−0.272076	−	0.962276i	$-0.587710\pi$
$167$	−90.8735			−0.544153			−0.272076	+	0.962276i	$0.587710\pi$
$168$		−	11.1311i		−	0.0662564i
$169$	58.7484			0.347624
$170$	−63.5259			−0.373682
$171$			52.4692i			0.306837i
$172$			72.8764i			0.423700i
$173$	−90.7893			−0.524794			−0.262397	−	0.964960i	$-0.584513\pi$
$173$	−90.7893			−0.524794			−0.262397	+	0.964960i	$0.584513\pi$
$174$	−155.179			−0.891831
$175$		−	5.84593i		−	0.0334053i
$176$			42.4591i			0.241245i
$177$	−180.233			−1.01826
$178$			197.743i			1.11092i
$179$	301.636			1.68511			0.842557	−	0.538607i	$-0.181049\pi$
$179$	301.636			1.68511			0.842557	+	0.538607i	$0.181049\pi$
$180$			10.4187i			0.0578814i
$181$			248.547i			1.37319i	0.727040	+	0.686595i	$0.240895\pi$
$181$			248.547i			1.37319i	−0.727040	+	0.686595i	$0.759105\pi$
$182$		−	24.9532i		−	0.137105i
$183$			79.1558i			0.432546i
$184$	−59.3074	+	26.7327i	−0.322323	+	0.145286i
$185$	118.639			0.641290
$186$	128.989			0.693490
$187$	−213.237			−1.14030
$188$	−98.2729			−0.522728
$189$			26.2506i			0.138892i
$190$	−71.2210			−0.374847
$191$			146.382i			0.766397i	0.923666	+	0.383199i	$0.125178\pi$
$191$			146.382i			0.766397i	−0.923666	+	0.383199i	$0.874822\pi$
$192$	−26.9277			−0.140248
$193$	−120.648			−0.625120			−0.312560	−	0.949898i	$-0.601187\pi$
$193$	−120.648			−0.625120			−0.312560	+	0.949898i	$0.601187\pi$
$194$		−	28.0912i		−	0.144800i
$195$			113.585i			0.582488i
$196$	95.2660			0.486051
$197$	31.6502			0.160661			0.0803305	−	0.996768i	$-0.474402\pi$
$197$	31.6502			0.160661			0.0803305	+	0.996768i	$0.474402\pi$
$198$			34.9722i			0.176627i
$199$			37.4496i			0.188189i	0.995563	+	0.0940944i	$0.0299955\pi$
$199$			37.4496i			0.188189i	−0.995563	+	0.0940944i	$0.970004\pi$
$200$	−14.1421			−0.0707107
$201$		−	200.192i		−	0.995979i
$202$	122.420			0.606039
$203$			38.1146i			0.187757i
$204$		−	135.235i		−	0.662918i
$205$			21.1022i			0.102938i
$206$			203.813i			0.989384i
$207$	−48.8495	+	22.0188i	−0.235988	+	0.106371i
$208$	−60.3653			−0.290218
$209$	−239.067			−1.14386
$210$	12.4449			0.0592615
$211$	−43.6572			−0.206906			−0.103453	−	0.994634i	$-0.532989\pi$
$211$	−43.6572			−0.206906			−0.103453	+	0.994634i	$0.532989\pi$
$212$		−	208.506i		−	0.983517i
$213$	−185.845			−0.872510
$214$		−	14.7848i		−	0.0690879i
$215$	−81.4783			−0.378969
$216$	63.5039			0.294000
$217$		−	31.6820i		−	0.146000i
$218$		−	97.7535i		−	0.448410i
$219$	29.5297			0.134839
$220$	−47.4708			−0.215776
$221$		−	303.165i		−	1.37179i
$222$			252.560i			1.13766i
$223$	348.623			1.56333			0.781666	−	0.623697i	$-0.214370\pi$
$223$	348.623			1.56333			0.781666	+	0.623697i	$0.214370\pi$
$224$			6.61391i			0.0295264i
$225$	−11.6484			−0.0517707
$226$			194.121i			0.858943i
$227$			175.793i			0.774421i	0.921991	+	0.387210i	$0.126561\pi$
$227$			175.793i			0.774421i	−0.921991	+	0.387210i	$0.873439\pi$
$228$		−	151.617i		−	0.664985i
$229$			1.44148i			0.00629466i	0.999995	+	0.00314733i	$0.00100183\pi$
$229$			1.44148i			0.00629466i	−0.999995	+	0.00314733i	$0.998998\pi$
$230$	−29.8880	−	66.3077i	−0.129948	−	0.288294i
$231$	41.7738			0.180839
$232$	92.2047			0.397434
$233$	−113.416			−0.486763			−0.243381	−	0.969931i	$-0.578257\pi$
$233$	−113.416			−0.486763			−0.243381	+	0.969931i	$0.578257\pi$
$234$	−49.7209			−0.212483
$235$		−	109.872i		−	0.467542i
$236$	107.091			0.453777
$237$		−	192.145i		−	0.810738i
$238$	−33.2162			−0.139564
$239$	367.260			1.53665			0.768326	−	0.640059i	$-0.221090\pi$
$239$	367.260			1.53665			0.768326	+	0.640059i	$0.221090\pi$
$240$		−	30.1061i		−	0.125442i
$241$			220.365i			0.914378i	0.889370	+	0.457189i	$0.151144\pi$
$241$			220.365i			0.914378i	−0.889370	+	0.457189i	$0.848856\pi$
$242$	11.7752			0.0486577
$243$	122.881			0.505681
$244$		−	47.0332i		−	0.192759i
$245$			106.511i			0.434737i
$246$	−44.9228			−0.182613
$247$		−	339.888i		−	1.37606i
$248$	−76.6433			−0.309046
$249$			185.730i			0.745902i
$250$		−	15.8114i		−	0.0632456i
$251$		−	440.210i		−	1.75382i	−0.480650	−	0.876912i	$-0.659599\pi$
$251$		−	440.210i		−	1.75382i	0.480650	−	0.876912i	$-0.340401\pi$
$252$			5.44766i			0.0216177i
$253$	−100.325	−	222.574i	−0.396541	−	0.879740i
$254$	81.9816			0.322762
$255$	151.198			0.592932
$256$	16.0000			0.0625000
$257$	491.519			1.91252			0.956262	−	0.292512i	$-0.0944913\pi$
$257$	491.519			1.91252			0.956262	+	0.292512i	$0.0944913\pi$
$258$		−	173.453i		−	0.672297i
$259$	62.0333			0.239511
$260$		−	67.4905i		−	0.259579i
$261$	75.9459			0.290980
$262$	−134.465			−0.513225
$263$		−	267.833i		−	1.01838i	−0.860655	−	0.509189i	$-0.829946\pi$
$263$		−	267.833i		−	1.01838i	0.860655	−	0.509189i	$-0.170054\pi$
$264$		−	101.057i		−	0.382791i
$265$	233.116			0.879684
$266$	−37.2397			−0.139999
$267$		−	470.647i		−	1.76272i
$268$			118.951i			0.443846i
$269$	95.6985			0.355756			0.177878	−	0.984053i	$-0.443077\pi$
$269$	95.6985			0.355756			0.177878	+	0.984053i	$0.443077\pi$
$270$			70.9995i			0.262961i
$271$	−452.255			−1.66884			−0.834419	−	0.551131i	$-0.814197\pi$
$271$	−452.255			−1.66884			−0.834419	+	0.551131i	$0.814197\pi$
$272$			80.3547i			0.295422i
$273$			59.3909i			0.217549i
$274$		−	281.020i		−	1.02562i
$275$		−	53.0739i		−	0.192996i
$276$	141.157	−	63.6262i	0.511439	−	0.230530i
$277$	−87.4783			−0.315806			−0.157903	−	0.987455i	$-0.550473\pi$
$277$	−87.4783			−0.315806			−0.157903	+	0.987455i	$0.550473\pi$
$278$	−0.183112			−0.000658676
$279$	−63.1285			−0.226267
$280$	−7.39458			−0.0264092
$281$		−	232.157i		−	0.826183i	−0.910690	−	0.413091i	$-0.864449\pi$
$281$		−	232.157i		−	0.826183i	0.910690	−	0.413091i	$-0.135551\pi$
$282$	233.899			0.829428
$283$			239.367i			0.845820i	0.906172	+	0.422910i	$0.138991\pi$
$283$			239.367i			0.845820i	−0.906172	+	0.422910i	$0.861009\pi$
$284$	110.426			0.388824
$285$	169.513			0.594781
$286$		−	226.545i		−	0.792114i
$287$			11.0338i			0.0384454i
$288$	13.1787			0.0457593
$289$	−114.554			−0.396382
$290$			103.088i			0.355476i
$291$			66.8597i			0.229758i
$292$	−17.5461			−0.0600894
$293$		−	581.830i		−	1.98577i	−0.119092	−	0.992883i	$-0.537998\pi$
$293$		−	581.830i		−	1.98577i	0.119092	−	0.992883i	$-0.462002\pi$
$294$	−226.742			−0.771231
$295$			119.732i			0.405871i
$296$		−	150.067i		−	0.506984i
$297$			238.323i			0.802436i
$298$		−	90.3322i		−	0.303128i
$299$	316.440	−	142.635i	1.05833	−	0.477039i
$300$	33.6596			0.112199
$301$	−42.6030			−0.141538
$302$	−49.8755			−0.165151
$303$	−291.370			−0.961619
$304$			90.0882i			0.296343i
$305$	52.5847			0.172409
$306$			66.1855i			0.216292i
$307$	110.134			0.358742			0.179371	−	0.983782i	$-0.442594\pi$
$307$	110.134			0.358742			0.179371	+	0.983782i	$0.442594\pi$
$308$	−24.8213			−0.0805887
$309$		−	485.094i		−	1.56988i
$310$		−	85.6898i		−	0.276419i
$311$	−298.100			−0.958520			−0.479260	−	0.877673i	$-0.659095\pi$
$311$	−298.100			−0.958520			−0.479260	+	0.877673i	$0.659095\pi$
$312$	143.675			0.460497
$313$		−	109.474i		−	0.349758i	−0.984590	−	0.174879i	$-0.944047\pi$
$313$		−	109.474i		−	0.349758i	0.984590	−	0.174879i	$-0.0559534\pi$
$314$			408.900i			1.30223i
$315$	−6.09067			−0.0193355
$316$			114.170i			0.361296i
$317$	474.434			1.49664			0.748318	−	0.663340i	$-0.230862\pi$
$317$	474.434			1.49664			0.748318	+	0.663340i	$0.230862\pi$
$318$			496.263i			1.56057i
$319$			346.034i			1.08475i
$320$			17.8885i			0.0559017i
$321$			35.1892i			0.109624i
$322$	−15.6277	−	34.6707i	−0.0485333	−	0.107673i
$323$	−452.438			−1.40074
$324$	−193.079			−0.595924
$325$	75.4567			0.232174
$326$	−65.4124			−0.200652
$327$			232.662i			0.711506i
$328$	26.6924			0.0813794
$329$		−	57.4496i		−	0.174619i
$330$	112.985			0.342378
$331$	255.367			0.771503			0.385751	−	0.922603i	$-0.373942\pi$
$331$	255.367			0.771503			0.385751	+	0.922603i	$0.373942\pi$
$332$		−	110.358i		−	0.332403i
$333$		−	123.606i		−	0.371188i
$334$	−128.515			−0.384774
$335$	−132.991			−0.396988
$336$		−	15.7417i		−	0.0468504i
$337$			458.486i			1.36049i	0.732983	+	0.680247i	$0.238127\pi$
$337$			458.486i			1.36049i	−0.732983	+	0.680247i	$0.761873\pi$
$338$	83.0828			0.245807
$339$		−	462.026i		−	1.36291i
$340$	−89.8392			−0.264233
$341$		−	287.634i		−	0.843502i
$342$			74.2027i			0.216967i
$343$			112.982i			0.329393i
$344$			103.063i			0.299601i
$345$	71.1363	+	157.818i	0.206192	+	0.457445i
$346$	−128.395			−0.371085
$347$	510.676			1.47169			0.735845	−	0.677151i	$-0.236785\pi$
$347$	510.676			1.47169			0.735845	+	0.677151i	$0.236785\pi$
$348$	−219.456			−0.630620
$349$	−591.870			−1.69590			−0.847951	−	0.530075i	$-0.822164\pi$
$349$	−591.870			−1.69590			−0.847951	+	0.530075i	$0.822164\pi$
$350$		−	8.26739i		−	0.0236211i
$351$	−338.831			−0.965330
$352$			60.0463i			0.170586i
$353$	168.334			0.476866			0.238433	−	0.971159i	$-0.423366\pi$
$353$	168.334			0.476866			0.238433	+	0.971159i	$0.423366\pi$
$354$	−254.887			−0.720021
$355$			123.460i			0.347775i
$356$			279.651i			0.785536i
$357$	79.0575			0.221450
$358$	426.577			1.19156
$359$			37.5719i			0.104657i	0.998630	+	0.0523285i	$0.0166643\pi$
$359$			37.5719i			0.104657i	−0.998630	+	0.0523285i	$0.983336\pi$
$360$			14.7342i			0.0409283i
$361$	−146.243			−0.405105
$362$			351.499i			0.970992i
$363$	−28.0260			−0.0772065
$364$		−	35.2891i		−	0.0969482i
$365$		−	19.6171i		−	0.0537456i
$366$			111.943i			0.305856i
$367$			427.036i			1.16359i	0.813337	+	0.581793i	$0.197649\pi$
$367$			427.036i			1.16359i	−0.813337	+	0.581793i	$0.802351\pi$
$368$	−83.8733	+	37.8057i	−0.227917	+	0.102733i
$369$	21.9857			0.0595818
$370$	167.780			0.453461
$371$	121.891			0.328547
$372$	182.418			0.490371
$373$			369.980i			0.991904i	0.868350	+	0.495952i	$0.165181\pi$
$373$			369.980i			0.991904i	−0.868350	+	0.495952i	$0.834819\pi$
$374$	−301.563			−0.806317
$375$			37.6326i			0.100354i
$376$	−138.979			−0.369625
$377$	−491.966			−1.30495
$378$			37.1239i			0.0982114i
$379$			78.8009i			0.207918i	0.994582	+	0.103959i	$0.0331511\pi$
$379$			78.8009i			0.207918i	−0.994582	+	0.103959i	$0.966849\pi$
$380$	−100.722			−0.265057
$381$	−195.124			−0.512136
$382$			207.015i			0.541925i
$383$			254.083i			0.663403i	0.943384	+	0.331702i	$0.107623\pi$
$383$			254.083i			0.663403i	−0.943384	+	0.331702i	$0.892377\pi$
$384$	−38.0815			−0.0991705
$385$		−	27.7511i		−	0.0720807i
$386$	−170.622			−0.442027
$387$			84.8894i			0.219353i
$388$		−	39.7270i		−	0.102389i
$389$		−	322.363i		−	0.828698i	−0.910118	−	0.414349i	$-0.864009\pi$
$389$		−	322.363i		−	0.828698i	0.910118	−	0.414349i	$-0.135991\pi$
$390$			160.634i			0.411881i
$391$	−189.866	−	421.226i	−0.485592	−	1.07730i
$392$	134.726			0.343690
$393$	320.039			0.814348
$394$	44.7602			0.113605
$395$	−127.645			−0.323153
$396$			49.4581i			0.124894i
$397$	715.832			1.80310			0.901552	−	0.432671i	$-0.142429\pi$
$397$	715.832			1.80310			0.901552	+	0.432671i	$0.142429\pi$
$398$			52.9617i			0.133070i
$399$	88.6340			0.222140
$400$	−20.0000			−0.0500000
$401$			441.435i			1.10083i	0.834890	+	0.550417i	$0.185531\pi$
$401$			441.435i			1.10083i	−0.834890	+	0.550417i	$0.814469\pi$
$402$		−	283.114i		−	0.704263i
$403$	408.937			1.01473
$404$	173.128			0.428534
$405$		−	215.869i		−	0.533011i
$406$			53.9022i			0.132764i
$407$	563.187			1.38375
$408$		−	191.252i		−	0.468754i
$409$	−608.727			−1.48833			−0.744164	−	0.667996i	$-0.767152\pi$
$409$	−608.727			−1.48833			−0.744164	+	0.667996i	$0.767152\pi$
$410$			29.8430i			0.0727879i
$411$			668.853i			1.62738i
$412$			288.235i			0.699600i
$413$			62.6048i			0.151586i
$414$	−69.0836	+	31.1393i	−0.166869	+	0.0752157i
$415$	123.384			0.297310
$416$	−85.3695			−0.205215
$417$	0.435823			0.00104514
$418$	−338.091			−0.808831
$419$		−	287.454i		−	0.686047i	−0.939327	−	0.343023i	$-0.888549\pi$
$419$		−	287.454i		−	0.686047i	0.939327	−	0.343023i	$-0.111451\pi$
$420$	17.5998			0.0419042
$421$		−	130.848i		−	0.310803i	−0.987851	−	0.155402i	$-0.950333\pi$
$421$		−	130.848i		−	0.310803i	0.987851	−	0.155402i	$-0.0496672\pi$
$422$	−61.7406			−0.146305
$423$	−114.472			−0.270620
$424$		−	294.871i		−	0.695452i
$425$		−	100.443i		−	0.236337i
$426$	−262.824			−0.616958
$427$	27.4952			0.0643917
$428$		−	20.9089i		−	0.0488525i
$429$			539.197i			1.25687i
$430$	−115.228			−0.267972
$431$		−	239.816i		−	0.556417i	−0.960521	−	0.278209i	$-0.910259\pi$
$431$		−	239.816i		−	0.556417i	0.960521	−	0.278209i	$-0.0897406\pi$
$432$	89.8081			0.207889
$433$		−	305.542i		−	0.705640i	−0.935691	−	0.352820i	$-0.885223\pi$
$433$		−	305.542i		−	0.705640i	0.935691	−	0.352820i	$-0.114777\pi$
$434$		−	44.8051i		−	0.103238i
$435$		−	245.359i		−	0.564043i
$436$		−	138.244i		−	0.317074i
$437$	−212.865	−	472.250i	−0.487106	−	1.08066i
$438$	41.7613			0.0953455
$439$	−579.018			−1.31895			−0.659474	−	0.751727i	$-0.729221\pi$
$439$	−579.018			−1.31895			−0.659474	+	0.751727i	$0.729221\pi$
$440$	−67.1338			−0.152577
$441$	110.970			0.251632
$442$		−	428.740i		−	0.969999i
$443$	−606.274			−1.36856			−0.684282	−	0.729217i	$-0.739884\pi$
$443$	−606.274			−1.36856			−0.684282	+	0.729217i	$0.739884\pi$
$444$			357.174i			0.804446i
$445$	−312.659			−0.702605
$446$	493.028			1.10544
$447$			214.999i			0.480982i
$448$			9.35348i			0.0208783i
$449$	−202.580			−0.451181			−0.225590	−	0.974222i	$-0.572431\pi$
$449$	−202.580			−0.451181			−0.225590	+	0.974222i	$0.572431\pi$
$450$	−16.4733			−0.0366074
$451$			100.174i			0.222115i
$452$			274.529i			0.607364i
$453$	118.708			0.262049
$454$			248.609i			0.547598i
$455$	39.4545			0.0867131
$456$		−	214.418i		−	0.470215i
$457$			373.760i			0.817857i	0.912567	+	0.408928i	$0.134097\pi$
$457$			373.760i			0.817857i	−0.912567	+	0.408928i	$0.865903\pi$
$458$			2.03856i			0.00445100i
$459$			451.031i			0.982639i
$460$	−42.2681	−	93.7732i	−0.0918871	−	0.203855i
$461$	709.377			1.53878			0.769390	−	0.638780i	$-0.220561\pi$
$461$	709.377			1.53878			0.769390	+	0.638780i	$0.220561\pi$
$462$	59.0770			0.127872
$463$	266.488			0.575569			0.287785	−	0.957695i	$-0.407081\pi$
$463$	266.488			0.575569			0.287785	+	0.957695i	$0.407081\pi$
$464$	130.397			0.281028
$465$			203.950i			0.438602i
$466$	−160.394			−0.344193
$467$			686.135i			1.46924i	0.678479	+	0.734620i	$0.262639\pi$
$467$			686.135i			1.46924i	−0.678479	+	0.734620i	$0.737361\pi$
$468$	−70.3160			−0.150248
$469$	−69.5378			−0.148268
$470$		−	155.383i		−	0.330602i
$471$		−	973.219i		−	2.06628i
$472$	151.450			0.320869
$473$	−386.784			−0.817725
$474$		−	271.734i		−	0.573279i
$475$		−	112.610i		−	0.237074i
$476$	−46.9747			−0.0986864
$477$		−	242.876i		−	0.509174i
$478$	519.384			1.08658
$479$		−	176.517i		−	0.368511i	−0.982878	−	0.184256i	$-0.941013\pi$
$479$		−	176.517i		−	0.368511i	0.982878	−	0.184256i	$-0.0589874\pi$
$480$		−	42.5764i		−	0.0887008i
$481$			800.698i			1.66465i
$482$			311.643i			0.646563i
$483$	37.1954	+	82.5194i	0.0770091	+	0.170848i
$484$	16.6526			0.0344062
$485$	44.4161			0.0915796
$486$	173.779			0.357571
$487$	524.802			1.07762			0.538811	−	0.842427i	$-0.318874\pi$
$487$	524.802			1.07762			0.538811	+	0.842427i	$0.318874\pi$
$488$		−	66.5149i		−	0.136301i
$489$	155.688			0.318380
$490$			150.629i			0.307406i
$491$	−551.047			−1.12229			−0.561147	−	0.827716i	$-0.689640\pi$
$491$	−551.047			−1.12229			−0.561147	+	0.827716i	$0.689640\pi$
$492$	−63.5304			−0.129127
$493$			654.876i			1.32835i
$494$		−	480.674i		−	0.973024i
$495$	−55.2959			−0.111709
$496$	−108.390			−0.218528
$497$			64.5542i			0.129888i
$498$			262.661i			0.527433i
$499$	−736.739			−1.47643			−0.738216	−	0.674565i	$-0.764331\pi$
$499$	−736.739			−1.47643			−0.738216	+	0.674565i	$0.764331\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	305.876			0.610532
$502$		−	622.551i		−	1.24014i
$503$			699.308i			1.39027i	0.718877	+	0.695137i	$0.244656\pi$
$503$			699.308i			1.39027i	−0.718877	+	0.695137i	$0.755344\pi$
$504$			7.70415i			0.0152860i
$505$			193.563i			0.383292i
$506$	−141.881	−	314.768i	−0.280397	−	0.622070i
$507$	−197.745			−0.390029
$508$	115.939			0.228227
$509$	−316.428			−0.621666			−0.310833	−	0.950464i	$-0.600608\pi$
$509$	−316.428			−0.621666			−0.310833	+	0.950464i	$0.600608\pi$
$510$	213.826			0.419266
$511$		−	10.2573i		−	0.0200730i
$512$	22.6274			0.0441942
$513$			505.666i			0.985703i
$514$	695.112			1.35236
$515$	−322.257			−0.625741
$516$		−	245.299i		−	0.475386i
$517$		−	521.573i		−	1.00885i
$518$	87.7283			0.169360
$519$	305.593			0.588811
$520$		−	95.4460i		−	0.183550i
$521$			798.875i			1.53335i	0.642035	+	0.766675i	$0.278090\pi$
$521$			798.875i			1.53335i	−0.642035	+	0.766675i	$0.721910\pi$
$522$	107.404			0.205754
$523$		−	819.091i		−	1.56614i	−0.621934	−	0.783070i	$-0.713653\pi$
$523$		−	819.091i		−	1.56614i	0.621934	−	0.783070i	$-0.286347\pi$
$524$	−190.162			−0.362905
$525$			19.6772i			0.0374803i
$526$		−	378.773i		−	0.720101i
$527$		−	544.353i		−	1.03293i
$528$		−	142.916i		−	0.270674i
$529$	350.341	−	396.361i	0.662271	−	0.749265i
$530$	329.676			0.622031
$531$	124.744			0.234924
$532$	−52.6649			−0.0989942
$533$	−142.420			−0.267204
$534$		−	665.595i		−	1.24643i
$535$	23.3768			0.0436950
$536$			168.222i			0.313847i
$537$	−1015.29			−1.89068
$538$	135.338			0.251558
$539$			505.614i			0.938060i
$540$			100.408i			0.185942i
$541$	289.616			0.535335			0.267668	−	0.963511i	$-0.413747\pi$
$541$	289.616			0.535335			0.267668	+	0.963511i	$0.413747\pi$
$542$	−639.585			−1.18005
$543$		−	836.600i		−	1.54070i
$544$			113.639i			0.208895i
$545$	154.562			0.283600
$546$			83.9914i			0.153830i
$547$	996.117			1.82106			0.910528	−	0.413448i	$-0.135676\pi$
$547$	996.117			1.82106			0.910528	+	0.413448i	$0.135676\pi$
$548$		−	397.422i		−	0.725223i
$549$		−	54.7861i		−	0.0997926i
$550$		−	75.0579i		−	0.136469i
$551$			734.202i			1.33249i
$552$	199.626	−	89.9811i	0.361642	−	0.163009i
$553$	−66.7427			−0.120692
$554$	−123.713			−0.223309
$555$	−399.333			−0.719519
$556$	−0.258959			−0.000465754
$557$			937.756i			1.68358i	0.539802	+	0.841792i	$0.318499\pi$
$557$			937.756i			1.68358i	−0.539802	+	0.841792i	$0.681501\pi$
$558$	−89.2772			−0.159995
$559$		−	549.901i		−	0.983723i
$560$	−10.4575			−0.0186741
$561$	717.747			1.27941
$562$		−	328.320i		−	0.584200i
$563$		−	785.252i		−	1.39476i	−0.716700	−	0.697382i	$-0.754348\pi$
$563$		−	785.252i		−	1.39476i	0.716700	−	0.697382i	$-0.245652\pi$
$564$	330.783			0.586494
$565$	−306.932			−0.543243
$566$			338.516i			0.598085i
$567$		−	112.873i		−	0.199070i
$568$	156.166			0.274940
$569$			403.851i			0.709755i	0.934913	+	0.354877i	$0.115477\pi$
$569$			403.851i			0.709755i	−0.934913	+	0.354877i	$0.884523\pi$
$570$	239.727			0.420574
$571$		−	335.474i		−	0.587519i	−0.955879	−	0.293760i	$-0.905093\pi$
$571$		−	335.474i		−	0.587519i	0.955879	−	0.293760i	$-0.0949065\pi$
$572$		−	320.383i		−	0.560110i
$573$		−	492.715i		−	0.859887i
$574$			15.6042i			0.0271850i
$575$	104.842	−	47.2571i	0.182333	−	0.0821863i
$576$	18.6375			0.0323567
$577$	−152.703			−0.264649			−0.132325	−	0.991206i	$-0.542244\pi$
$577$	−152.703			−0.264649			−0.132325	+	0.991206i	$0.542244\pi$
$578$	−162.004			−0.280284
$579$	406.097			0.701376
$580$			145.788i			0.251359i
$581$	64.5143			0.111040
$582$			94.5538i			0.162464i
$583$	1106.62			1.89815
$584$	−24.8139			−0.0424896
$585$		−	78.6157i		−	0.134386i
$586$		−	822.831i		−	1.40415i
$587$	129.262			0.220207			0.110104	−	0.993920i	$-0.464882\pi$
$587$	129.262			0.220207			0.110104	+	0.993920i	$0.464882\pi$
$588$	−320.662			−0.545343
$589$		−	610.291i		−	1.03615i
$590$			169.326i			0.286994i
$591$	−106.533			−0.180260
$592$		−	212.227i		−	0.358492i
$593$	−879.853			−1.48373			−0.741866	−	0.670548i	$-0.766059\pi$
$593$	−879.853			−1.48373			−0.741866	+	0.670548i	$0.766059\pi$
$594$			337.040i			0.567408i
$595$		−	52.5194i		−	0.0882678i
$596$		−	127.749i		−	0.214344i
$597$		−	126.054i		−	0.211145i
$598$	447.514	−	201.716i	0.748351	−	0.337317i
$599$	−709.708			−1.18482			−0.592411	−	0.805636i	$-0.701824\pi$
$599$	−709.708			−1.18482			−0.592411	+	0.805636i	$0.701824\pi$
$600$	47.6019			0.0793364
$601$	238.861			0.397439			0.198719	−	0.980056i	$-0.436322\pi$
$601$	238.861			0.397439			0.198719	+	0.980056i	$0.436322\pi$
$602$	−60.2498			−0.100083
$603$			138.559i			0.229782i
$604$	−70.5346			−0.116779
$605$			18.6182i			0.0307738i
$606$	−412.060			−0.679967
$607$	−1082.53			−1.78341			−0.891703	−	0.452621i	$-0.850489\pi$
$607$	−1082.53			−1.78341			−0.891703	+	0.452621i	$0.850489\pi$
$608$			127.404i			0.209546i
$609$		−	128.292i		−	0.210660i
$610$	74.3659			0.121911
$611$	741.535			1.21364
$612$			93.6004i			0.152942i
$613$			54.0902i			0.0882384i	0.999026	+	0.0441192i	$0.0140481\pi$
$613$			54.0902i			0.0882384i	−0.999026	+	0.0441192i	$0.985952\pi$
$614$	155.753			0.253669
$615$		−	71.0292i		−	0.115495i
$616$	−35.1026			−0.0569848
$617$		−	454.342i		−	0.736373i	−0.929752	−	0.368186i	$-0.879979\pi$
$617$		−	454.342i		−	0.736373i	0.929752	−	0.368186i	$-0.120021\pi$
$618$		−	686.027i		−	1.11008i
$619$		−	153.676i		−	0.248264i	−0.992266	−	0.124132i	$-0.960385\pi$
$619$		−	153.676i		−	0.248264i	0.992266	−	0.124132i	$-0.0396147\pi$
$620$		−	121.184i		−	0.195458i
$621$	−470.781	+	212.204i	−0.758102	+	0.341713i
$622$	−421.577			−0.677776
$623$	−163.482			−0.262411
$624$	203.187			0.325621
$625$	25.0000			0.0400000
$626$		−	154.820i		−	0.247316i
$627$	804.689			1.28340
$628$			578.271i			0.920814i
$629$	1065.84			1.69450
$630$	−8.61350			−0.0136722
$631$		−	1241.15i		−	1.96696i	−0.181027	−	0.983478i	$-0.557942\pi$
$631$		−	1241.15i		−	1.96696i	0.181027	−	0.983478i	$-0.442058\pi$
$632$			161.460i			0.255475i
$633$	146.948			0.232146
$634$	670.950			1.05828
$635$			129.624i			0.204133i
$636$			701.821i			1.10349i
$637$	−718.846			−1.12849
$638$			489.366i			0.767032i
$639$	128.629			0.201297
$640$			25.2982i			0.0395285i
$641$			545.978i			0.851759i	0.904780	+	0.425880i	$0.140035\pi$
$641$			545.978i			0.851759i	−0.904780	+	0.425880i	$0.859965\pi$
$642$			49.7651i			0.0775157i
$643$			539.118i			0.838441i	0.907884	+	0.419221i	$0.137697\pi$
$643$			539.118i			0.838441i	−0.907884	+	0.419221i	$0.862303\pi$
$644$	−22.1009	−	49.0317i	−0.0343182	−	0.0761362i
$645$	274.253			0.425198
$646$	−639.844			−0.990470
$647$	652.715			1.00883			0.504417	−	0.863460i	$-0.331708\pi$
$647$	652.715			1.00883			0.504417	+	0.863460i	$0.331708\pi$
$648$	−273.056			−0.421382
$649$			568.376i			0.875772i
$650$	106.712			0.164172
$651$			106.640i			0.163810i
$652$	−92.5071			−0.141882
$653$	374.815			0.573989			0.286995	−	0.957932i	$-0.407344\pi$
$653$	374.815			0.573989			0.286995	+	0.957932i	$0.407344\pi$
$654$			329.034i			0.503110i
$655$		−	212.608i		−	0.324592i
$656$	37.7488			0.0575439
$657$	−20.4384			−0.0311087
$658$		−	81.2460i		−	0.123474i
$659$		−	440.090i		−	0.667815i	−0.942606	−	0.333907i	$-0.891633\pi$
$659$		−	440.090i		−	0.667815i	0.942606	−	0.333907i	$-0.108367\pi$
$660$	159.785			0.242098
$661$		−	72.6725i		−	0.109943i	−0.998488	−	0.0549716i	$-0.982493\pi$
$661$		−	72.6725i		−	0.109943i	0.998488	−	0.0549716i	$-0.0175068\pi$
$662$	361.144			0.545535
$663$			1020.44i			1.53913i
$664$		−	156.069i		−	0.235044i
$665$		−	58.8812i		−	0.0885431i
$666$		−	174.805i		−	0.262469i
$667$	−683.552	+	308.110i	−1.02482	+	0.461933i
$668$	−181.747			−0.272076
$669$	−1173.45			−1.75404
$670$	−188.078			−0.280713
$671$	249.623			0.372017
$672$		−	22.2622i		−	0.0331282i
$673$	−1052.88			−1.56445			−0.782227	−	0.622993i	$-0.785916\pi$
$673$	−1052.88			−1.56445			−0.782227	+	0.622993i	$0.785916\pi$
$674$			648.398i			0.962014i
$675$	−112.260			−0.166311
$676$	117.497			0.173812
$677$		−	393.587i		−	0.581369i	−0.956819	−	0.290684i	$-0.906117\pi$
$677$		−	393.587i		−	0.581369i	0.956819	−	0.290684i	$-0.0938830\pi$
$678$		−	653.404i		−	0.963722i
$679$	23.2241			0.0342034
$680$	−127.052			−0.186841
$681$		−	591.714i		−	0.868889i
$682$		−	406.776i		−	0.596446i
$683$	−645.240			−0.944714			−0.472357	−	0.881407i	$-0.656597\pi$
$683$	−645.240			−0.944714			−0.472357	+	0.881407i	$0.656597\pi$
$684$			104.938i			0.153419i
$685$	444.331			0.648659
$686$			159.781i			0.232916i
$687$		−	4.85195i		−	0.00706252i
$688$			145.753i			0.211850i
$689$			1573.31i			2.28347i
$690$	100.602	+	223.189i	0.145800	+	0.323462i
$691$	548.216			0.793366			0.396683	−	0.917956i	$-0.370161\pi$
$691$	548.216			0.793366			0.396683	+	0.917956i	$0.370161\pi$
$692$	−181.579			−0.262397
$693$	−28.9129			−0.0417213
$694$	722.205			1.04064
$695$		−	0.289525i		−	0.000416583i
$696$	−310.357			−0.445915
$697$			189.581i			0.271995i
$698$	−837.030			−1.19918
$699$	381.753			0.546141
$700$		−	11.6919i		−	0.0167026i
$701$		−	92.2064i		−	0.131536i	−0.997835	−	0.0657678i	$-0.979050\pi$
$701$		−	92.2064i		−	0.131536i	0.997835	−	0.0657678i	$-0.0209496\pi$
$702$	−479.179			−0.682592
$703$	1194.95			1.69978
$704$			84.9183i			0.120623i
$705$			369.826i			0.524576i
$706$	238.060			0.337195
$707$			101.209i			0.143153i
$708$	−360.465			−0.509132
$709$		−	922.612i		−	1.30129i	−0.759384	−	0.650643i	$-0.774499\pi$
$709$		−	922.612i		−	1.30129i	0.759384	−	0.650643i	$-0.225501\pi$
$710$			174.599i			0.245914i
$711$			132.989i			0.187045i
$712$			395.486i			0.555458i
$713$	568.189	−	256.110i	0.796899	−	0.359201i
$714$	111.804			0.156589
$715$	358.199			0.500977
$716$	603.271			0.842557
$717$	−1236.18			−1.72410
$718$			53.1347i			0.0740037i
$719$	606.887			0.844071			0.422035	−	0.906579i	$-0.361316\pi$
$719$	606.887			0.844071			0.422035	+	0.906579i	$0.361316\pi$
$720$			20.8373i			0.0289407i
$721$	−168.500			−0.233704
$722$	−206.819			−0.286452
$723$		−	741.740i		−	1.02592i
$724$			497.095i			0.686595i
$725$	−162.996			−0.224823
$726$	−39.6347			−0.0545932
$727$		−	787.544i		−	1.08328i	−0.840611	−	0.541640i	$-0.817804\pi$
$727$		−	787.544i		−	1.08328i	0.840611	−	0.541640i	$-0.182196\pi$
$728$		−	49.9064i		−	0.0685527i
$729$	455.246			0.624481
$730$		−	27.7428i		−	0.0380038i
$731$	−731.995			−1.00136
$732$			158.312i			0.216273i
$733$			415.085i			0.566282i	0.959078	+	0.283141i	$0.0913765\pi$
$733$			415.085i			0.566282i	−0.959078	+	0.283141i	$0.908624\pi$
$734$			603.921i			0.822780i
$735$		−	358.510i		−	0.487769i
$736$	−118.615	+	53.4653i	−0.161161	+	0.0726431i
$737$	−631.319			−0.856606
$738$	31.0924			0.0421307
$739$	639.964			0.865987			0.432993	−	0.901397i	$-0.357457\pi$
$739$	639.964			0.865987			0.432993	+	0.901397i	$0.357457\pi$
$740$	237.277			0.320645
$741$			1144.05i			1.54393i
$742$	172.380			0.232318
$743$		−	743.739i		−	1.00099i	−0.865738	−	0.500497i	$-0.833151\pi$
$743$		−	743.739i		−	1.00099i	0.865738	−	0.500497i	$-0.166849\pi$
$744$	257.978			0.346745
$745$	142.828			0.191715
$746$			523.231i			0.701382i
$747$		−	128.549i		−	0.172087i
$748$	−426.474			−0.570152
$749$	12.2232			0.0163193
$750$			53.2205i			0.0709607i
$751$		−	517.047i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$751$		−	517.047i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$752$	−196.546			−0.261364
$753$			1481.73i			1.96777i
$754$	−695.746			−0.922740
$755$		−	78.8601i		−	0.104450i
$756$			52.5011i			0.0694460i
$757$		−	911.823i		−	1.20452i	−0.798299	−	0.602261i	$-0.794267\pi$
$757$		−	911.823i		−	1.20452i	0.798299	−	0.602261i	$-0.205733\pi$
$758$			111.441i			0.147020i
$759$	337.689	+	749.176i	0.444914	+	0.987057i
$760$	−142.442			−0.187424
$761$	954.073			1.25371			0.626855	−	0.779136i	$-0.284342\pi$
$761$	954.073			1.25371			0.626855	+	0.779136i	$0.284342\pi$
$762$	−275.947			−0.362135
$763$	80.8166			0.105920
$764$			292.764i			0.383199i
$765$	−104.648			−0.136795
$766$			359.328i			0.469097i
$767$	−808.076			−1.05355
$768$	−53.8553			−0.0701242
$769$		−	276.277i		−	0.359268i	−0.983734	−	0.179634i	$-0.942509\pi$
$769$		−	276.277i		−	0.359268i	0.983734	−	0.179634i	$-0.0574913\pi$
$770$		−	39.2459i		−	0.0509687i
$771$	−1654.43			−2.14583
$772$	−241.296			−0.312560
$773$		−	40.8440i		−	0.0528383i	−0.999651	−	0.0264192i	$-0.991590\pi$
$773$		−	40.8440i		−	0.0528383i	0.999651	−	0.0264192i	$-0.00841046\pi$
$774$			120.052i			0.155106i
$775$	135.488			0.174823
$776$		−	56.1824i		−	0.0724000i
$777$	−208.801			−0.268728
$778$		−	455.891i		−	0.585978i
$779$			212.545i			0.272843i
$780$			227.170i			0.291244i
$781$			586.074i			0.750415i
$782$	−268.512	−	595.703i	−0.343365	−	0.761768i
$783$	731.919			0.934763
$784$	190.532			0.243026
$785$	−646.527			−0.823601
$786$	452.603			0.575831
$787$		−	1364.69i		−	1.73404i	−0.498274	−	0.867019i	$-0.666033\pi$
$787$		−	1364.69i		−	1.73404i	0.498274

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -3.36596 q^{3} +2.00000 q^{4} +2.23607i q^{5} -4.76019 q^{6} +1.16919i q^{7} +2.82843 q^{8} +2.32968 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -3.36596 q^{3} +2.00000 q^{4} +2.23607i q^{5} -4.76019 q^{6} +1.16919i q^{7} +2.82843 q^{8} +2.32968 q^{9} +3.16228i q^{10} +10.6148i q^{11} -6.73192 q^{12} -15.0913 q^{13} +1.65348i q^{14} -7.52651i q^{15} +4.00000 q^{16} +20.0887i q^{17} +3.29467 q^{18} +22.5221i q^{19} +4.47214i q^{20} -3.93543i q^{21} +15.0116i q^{22} +(-20.9683 + 9.45142i) q^{23} -9.52037 q^{24} -5.00000 q^{25} -21.3424 q^{26} +22.4520 q^{27} +2.33837i q^{28} +32.5993 q^{29} -10.6441i q^{30} -27.0975 q^{31} +5.65685 q^{32} -35.7289i q^{33} +28.4097i q^{34} -2.61438 q^{35} +4.65936 q^{36} -53.0568i q^{37} +31.8510i q^{38} +50.7968 q^{39} +6.32456i q^{40} +9.43720 q^{41} -5.56554i q^{42} +36.4382i q^{43} +21.2296i q^{44} +5.20933i q^{45} +(-29.6537 + 13.3663i) q^{46} -49.1365 q^{47} -13.4638 q^{48} +47.6330 q^{49} -7.07107 q^{50} -67.6176i q^{51} -30.1827 q^{52} -104.253i q^{53} +31.7520 q^{54} -23.7354 q^{55} +3.30696i q^{56} -75.8083i q^{57} +46.1023 q^{58} +53.5457 q^{59} -15.0530i q^{60} -23.5166i q^{61} -38.3217 q^{62} +2.72383i q^{63} +8.00000 q^{64} -33.7453i q^{65} -50.5284i q^{66} +59.4754i q^{67} +40.1773i q^{68} +(70.5785 - 31.8131i) q^{69} -3.69729 q^{70} +55.2130 q^{71} +6.58933 q^{72} -8.77305 q^{73} -75.0337i q^{74} +16.8298 q^{75} +45.0441i q^{76} -12.4107 q^{77} +71.8376 q^{78} +57.0848i q^{79} +8.94427i q^{80} -96.5397 q^{81} +13.3462 q^{82} -55.1788i q^{83} -7.87086i q^{84} -44.9196 q^{85} +51.5314i q^{86} -109.728 q^{87} +30.0232i q^{88} +139.825i q^{89} +7.36710i q^{90} -17.6446i q^{91} +(-41.9366 + 18.9028i) q^{92} +91.2091 q^{93} -69.4894 q^{94} -50.3608 q^{95} -19.0407 q^{96} -19.8635i q^{97} +67.3632 q^{98} +24.7291i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9	\( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 + 24 * q^13 + 64 * q^16 - 32 * q^18 + 4 * q^23 - 16 * q^24 - 80 * q^25 + 96 * q^26 - 96 * q^27 - 108 * q^29 - 116 * q^31 + 60 * q^35 + 128 * q^36 + 248 * q^39 - 156 * q^41 - 124 * q^46 - 128 * q^47 - 28 * q^49 + 48 * q^52 + 224 * q^54 + 160 * q^58 + 204 * q^59 + 64 * q^62 + 128 * q^64 - 268 * q^69 - 120 * q^70 + 236 * q^71 - 64 * q^72 - 112 * q^73 - 936 * q^77 - 432 * q^78 - 136 * q^81 - 64 * q^82 + 60 * q^85 - 152 * q^87 + 8 * q^92 + 856 * q^93 - 216 * q^94 - 160 * q^95 - 32 * q^96 + 256 * q^98

Introduction

L-functions

Modular forms

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