LMFDB - Embedded newform 165.2.c.a.34.3

Newspace parameters

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	165.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.31753163335$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			34.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	165.34
Dual form			165.2.c.a.34.4

$q$-expansion

$f(q)$	$=$	$q-1.21432i q^{2} +1.00000i q^{3} +0.525428 q^{4} +(0.311108 + 2.21432i) q^{5} +1.21432 q^{6} +4.90321i q^{7} -3.06668i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.21432i q^{2} +1.00000i q^{3} +0.525428 q^{4} +(0.311108 + 2.21432i) q^{5} +1.21432 q^{6} +4.90321i q^{7} -3.06668i q^{8} -1.00000 q^{9} +(2.68889 - 0.377784i) q^{10} -1.00000 q^{11} +0.525428i q^{12} -4.14764i q^{13} +5.95407 q^{14} +(-2.21432 + 0.311108i) q^{15} -2.67307 q^{16} -5.33185i q^{17} +1.21432i q^{18} +5.18421 q^{19} +(0.163465 + 1.16346i) q^{20} -4.90321 q^{21} +1.21432i q^{22} -4.00000i q^{23} +3.06668 q^{24} +(-4.80642 + 1.37778i) q^{25} -5.03657 q^{26} -1.00000i q^{27} +2.57628i q^{28} -1.80642 q^{29} +(0.377784 + 2.68889i) q^{30} +2.62222 q^{31} -2.88739i q^{32} -1.00000i q^{33} -6.47457 q^{34} +(-10.8573 + 1.52543i) q^{35} -0.525428 q^{36} +5.80642i q^{37} -6.29529i q^{38} +4.14764 q^{39} +(6.79060 - 0.954067i) q^{40} +1.80642 q^{41} +5.95407i q^{42} -4.90321i q^{43} -0.525428 q^{44} +(-0.311108 - 2.21432i) q^{45} -4.85728 q^{46} +7.05086i q^{47} -2.67307i q^{48} -17.0415 q^{49} +(1.67307 + 5.83654i) q^{50} +5.33185 q^{51} -2.17929i q^{52} +7.18421i q^{53} -1.21432 q^{54} +(-0.311108 - 2.21432i) q^{55} +15.0366 q^{56} +5.18421i q^{57} +2.19358i q^{58} -1.67307 q^{59} +(-1.16346 + 0.163465i) q^{60} +0.755569 q^{61} -3.18421i q^{62} -4.90321i q^{63} -8.85236 q^{64} +(9.18421 - 1.29036i) q^{65} -1.21432 q^{66} -4.85728i q^{67} -2.80150i q^{68} +4.00000 q^{69} +(1.85236 + 13.1842i) q^{70} +0.428639 q^{71} +3.06668i q^{72} -12.7096i q^{73} +7.05086 q^{74} +(-1.37778 - 4.80642i) q^{75} +2.72393 q^{76} -4.90321i q^{77} -5.03657i q^{78} +6.42864 q^{79} +(-0.831613 - 5.91903i) q^{80} +1.00000 q^{81} -2.19358i q^{82} +2.90321i q^{83} -2.57628 q^{84} +(11.8064 - 1.65878i) q^{85} -5.95407 q^{86} -1.80642i q^{87} +3.06668i q^{88} -0.622216 q^{89} +(-2.68889 + 0.377784i) q^{90} +20.3368 q^{91} -2.10171i q^{92} +2.62222i q^{93} +8.56199 q^{94} +(1.61285 + 11.4795i) q^{95} +2.88739 q^{96} -2.75557i q^{97} +20.6938i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9	$ 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} + 16 q^{10} - 6 q^{11} - 4 q^{14} + 10 q^{16} + 4 q^{19} + 14 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 16 q^{26} + 16 q^{29} + 2 q^{30} + 16 q^{31} - 52 q^{34} - 12 q^{35} + 10 q^{36} + 12 q^{39} - 12 q^{40} - 16 q^{41} + 10 q^{44} - 2 q^{45} + 24 q^{46} - 22 q^{49} - 16 q^{50} - 8 q^{51} + 6 q^{54} - 2 q^{55} + 76 q^{56} + 16 q^{59} - 20 q^{60} + 4 q^{61} - 66 q^{64} + 28 q^{65} + 6 q^{66} + 24 q^{69} + 24 q^{70} - 24 q^{71} + 16 q^{74} - 8 q^{75} - 36 q^{76} + 12 q^{79} - 58 q^{80} + 6 q^{81} + 24 q^{84} + 44 q^{85} + 4 q^{86} - 4 q^{89} - 16 q^{90} + 16 q^{91} + 24 q^{94} - 44 q^{95} - 22 q^{96} + 6 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 + 16 * q^10 - 6 * q^11 - 4 * q^14 + 10 * q^16 + 4 * q^19 + 14 * q^20 - 16 * q^21 + 18 * q^24 - 2 * q^25 - 16 * q^26 + 16 * q^29 + 2 * q^30 + 16 * q^31 - 52 * q^34 - 12 * q^35 + 10 * q^36 + 12 * q^39 - 12 * q^40 - 16 * q^41 + 10 * q^44 - 2 * q^45 + 24 * q^46 - 22 * q^49 - 16 * q^50 - 8 * q^51 + 6 * q^54 - 2 * q^55 + 76 * q^56 + 16 * q^59 - 20 * q^60 + 4 * q^61 - 66 * q^64 + 28 * q^65 + 6 * q^66 + 24 * q^69 + 24 * q^70 - 24 * q^71 + 16 * q^74 - 8 * q^75 - 36 * q^76 + 12 * q^79 - 58 * q^80 + 6 * q^81 + 24 * q^84 + 44 * q^85 + 4 * q^86 - 4 * q^89 - 16 * q^90 + 16 * q^91 + 24 * q^94 - 44 * q^95 - 22 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$67$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	1.21432i		−	0.858654i	−0.903149	−	0.429327i	$-0.858751\pi$
$2$		−	1.21432i		−	0.858654i	0.903149	−	0.429327i	$-0.141249\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	0.525428			0.262714
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$6$	1.21432			0.495744
$7$			4.90321i			1.85324i	0.375999	+	0.926620i	$0.377300\pi$
$7$			4.90321i			1.85324i	−0.375999	+	0.926620i	$0.622700\pi$
$8$		−	3.06668i		−	1.08423i
$9$	−1.00000			−0.333333
$10$	2.68889	−	0.377784i	0.850302	−	0.119466i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			0.525428i			0.151678i
$13$		−	4.14764i		−	1.15035i	−0.818031	−	0.575175i	$-0.804934\pi$
$13$		−	4.14764i		−	1.15035i	0.818031	−	0.575175i	$-0.195066\pi$
$14$	5.95407			1.59129
$15$	−2.21432	+	0.311108i	−0.571735	+	0.0803277i
$16$	−2.67307			−0.668268
$17$		−	5.33185i		−	1.29316i	−0.762845	−	0.646582i	$-0.776198\pi$
$17$		−	5.33185i		−	1.29316i	0.762845	−	0.646582i	$-0.223802\pi$
$18$			1.21432i			0.286218i
$19$	5.18421			1.18934			0.594669	−	0.803970i	$-0.297283\pi$
$19$	5.18421			1.18934			0.594669	+	0.803970i	$0.297283\pi$
$20$	0.163465	+	1.16346i	0.0365518	+	0.260159i
$21$	−4.90321			−1.06997
$22$			1.21432i			0.258894i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	3.06668			0.625983
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$	−5.03657			−0.987752
$27$		−	1.00000i		−	0.192450i
$28$			2.57628i			0.486872i
$29$	−1.80642			−0.335444			−0.167722	−	0.985834i	$-0.553641\pi$
$29$	−1.80642			−0.335444			−0.167722	+	0.985834i	$0.553641\pi$
$30$	0.377784	+	2.68889i	0.0689737	+	0.490922i
$31$	2.62222			0.470964			0.235482	−	0.971879i	$-0.424333\pi$
$31$	2.62222			0.470964			0.235482	+	0.971879i	$0.424333\pi$
$32$		−	2.88739i		−	0.510423i
$33$		−	1.00000i		−	0.174078i
$34$	−6.47457			−1.11038
$35$	−10.8573	+	1.52543i	−1.83522	+	0.257844i
$36$	−0.525428			−0.0875713
$37$			5.80642i			0.954570i	0.878749	+	0.477285i	$0.158379\pi$
$37$			5.80642i			0.954570i	−0.878749	+	0.477285i	$0.841621\pi$
$38$		−	6.29529i		−	1.02123i
$39$	4.14764			0.664154
$40$	6.79060	−	0.954067i	1.07369	−	0.150851i
$41$	1.80642			0.282116			0.141058	−	0.990001i	$-0.454950\pi$
$41$	1.80642			0.282116			0.141058	+	0.990001i	$0.454950\pi$
$42$			5.95407i			0.918732i
$43$		−	4.90321i		−	0.747733i	−0.927483	−	0.373866i	$-0.878032\pi$
$43$		−	4.90321i		−	0.747733i	0.927483	−	0.373866i	$-0.121968\pi$
$44$	−0.525428			−0.0792112
$45$	−0.311108	−	2.21432i	−0.0463772	−	0.330091i
$46$	−4.85728			−0.716167
$47$			7.05086i			1.02847i	0.857648	+	0.514236i	$0.171925\pi$
$47$			7.05086i			1.02847i	−0.857648	+	0.514236i	$0.828075\pi$
$48$		−	2.67307i		−	0.385825i
$49$	−17.0415			−2.43450
$50$	1.67307	+	5.83654i	0.236608	+	0.825411i
$51$	5.33185			0.746609
$52$		−	2.17929i		−	0.302213i
$53$			7.18421i			0.986827i	0.869795	+	0.493413i	$0.164251\pi$
$53$			7.18421i			0.986827i	−0.869795	+	0.493413i	$0.835749\pi$
$54$	−1.21432			−0.165248
$55$	−0.311108	−	2.21432i	−0.0419498	−	0.298579i
$56$	15.0366			2.00935
$57$			5.18421i			0.686665i
$58$			2.19358i			0.288031i
$59$	−1.67307			−0.217815			−0.108908	−	0.994052i	$-0.534735\pi$
$59$	−1.67307			−0.217815			−0.108908	+	0.994052i	$0.534735\pi$
$60$	−1.16346	+	0.163465i	−0.150203	+	0.0211032i
$61$	0.755569			0.0967407			0.0483703	−	0.998829i	$-0.484597\pi$
$61$	0.755569			0.0967407			0.0483703	+	0.998829i	$0.484597\pi$
$62$		−	3.18421i		−	0.404395i
$63$		−	4.90321i		−	0.617747i
$64$	−8.85236			−1.10654
$65$	9.18421	−	1.29036i	1.13916	−	0.160050i
$66$	−1.21432			−0.149472
$67$		−	4.85728i		−	0.593411i	−0.954969	−	0.296706i	$-0.904112\pi$
$67$		−	4.85728i		−	0.593411i	0.954969	−	0.296706i	$-0.0958880\pi$
$68$		−	2.80150i		−	0.339732i
$69$	4.00000			0.481543
$70$	1.85236	+	13.1842i	0.221399	+	1.57581i
$71$	0.428639			0.0508701			0.0254351	−	0.999676i	$-0.491903\pi$
$71$	0.428639			0.0508701			0.0254351	+	0.999676i	$0.491903\pi$
$72$			3.06668i			0.361411i
$73$		−	12.7096i		−	1.48755i	−0.668430	−	0.743775i	$-0.733033\pi$
$73$		−	12.7096i		−	1.48755i	0.668430	−	0.743775i	$-0.266967\pi$
$74$	7.05086			0.819645
$75$	−1.37778	−	4.80642i	−0.159093	−	0.554998i
$76$	2.72393			0.312456
$77$		−	4.90321i		−	0.558773i
$78$		−	5.03657i		−	0.570279i
$79$	6.42864			0.723278			0.361639	−	0.932318i	$-0.382217\pi$
$79$	6.42864			0.723278			0.361639	+	0.932318i	$0.382217\pi$
$80$	−0.831613	−	5.91903i	−0.0929772	−	0.661768i
$81$	1.00000			0.111111
$82$		−	2.19358i		−	0.242240i
$83$			2.90321i			0.318669i	0.987225	+	0.159334i	$0.0509348\pi$
$83$			2.90321i			0.318669i	−0.987225	+	0.159334i	$0.949065\pi$
$84$	−2.57628			−0.281095
$85$	11.8064	−	1.65878i	1.28059	−	0.179920i
$86$	−5.95407			−0.642044
$87$		−	1.80642i		−	0.193669i
$88$			3.06668i			0.326909i
$89$	−0.622216			−0.0659547			−0.0329774	−	0.999456i	$-0.510499\pi$
$89$	−0.622216			−0.0659547			−0.0329774	+	0.999456i	$0.510499\pi$
$90$	−2.68889	+	0.377784i	−0.283434	+	0.0398220i
$91$	20.3368			2.13187
$92$		−	2.10171i		−	0.219118i
$93$			2.62222i			0.271911i
$94$	8.56199			0.883102
$95$	1.61285	+	11.4795i	0.165475	+	1.17777i
$96$	2.88739			0.294693
$97$		−	2.75557i		−	0.279786i	−0.990167	−	0.139893i	$-0.955324\pi$
$97$		−	2.75557i		−	0.279786i	0.990167	−	0.139893i	$-0.0446758\pi$
$98$			20.6938i			2.09039i
$99$	1.00000			0.100504
$100$	−2.52543	+	0.723926i	−0.252543	+	0.0723926i
$101$	−17.8064			−1.77181			−0.885903	−	0.463871i	$-0.846460\pi$
$101$	−17.8064			−1.77181			−0.885903	+	0.463871i	$0.846460\pi$
$102$		−	6.47457i		−	0.641078i
$103$		−	4.94914i		−	0.487654i	−0.969819	−	0.243827i	$-0.921597\pi$
$103$		−	4.94914i		−	0.487654i	0.969819	−	0.243827i	$-0.0784029\pi$
$104$	−12.7195			−1.24725
$105$	−1.52543	−	10.8573i	−0.148866	−	1.05956i
$106$	8.72393			0.847343
$107$			11.1985i			1.08260i	0.840830	+	0.541300i	$0.182068\pi$
$107$			11.1985i			1.08260i	−0.840830	+	0.541300i	$0.817932\pi$
$108$		−	0.525428i		−	0.0505593i
$109$	−15.7146			−1.50518			−0.752591	−	0.658488i	$-0.771196\pi$
$109$	−15.7146			−1.50518			−0.752591	+	0.658488i	$0.771196\pi$
$110$	−2.68889	+	0.377784i	−0.256376	+	0.0360203i
$111$	−5.80642			−0.551121
$112$		−	13.1066i		−	1.23846i
$113$			1.76494i			0.166031i	0.996548	+	0.0830156i	$0.0264551\pi$
$113$			1.76494i			0.166031i	−0.996548	+	0.0830156i	$0.973545\pi$
$114$	6.29529			0.589608
$115$	8.85728	−	1.24443i	0.825946	−	0.116044i
$116$	−0.949145			−0.0881259
$117$			4.14764i			0.383450i
$118$			2.03164i			0.187028i
$119$	26.1432			2.39654
$120$	0.954067	+	6.79060i	0.0870940	+	0.619894i
$121$	1.00000			0.0909091
$122$		−	0.917502i		−	0.0830667i
$123$			1.80642i			0.162880i
$124$	1.37778			0.123729
$125$	−4.54617	−	10.2143i	−0.406622	−	0.913597i
$126$	−5.95407			−0.530430
$127$			18.7096i			1.66021i	0.557606	+	0.830106i	$0.311720\pi$
$127$			18.7096i			1.66021i	−0.557606	+	0.830106i	$0.688280\pi$
$128$			4.97481i			0.439715i
$129$	4.90321			0.431704
$130$	−1.56691	−	11.1526i	−0.137428	−	0.978145i
$131$	−1.24443			−0.108726			−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443			−0.108726			−0.0543632	+	0.998521i	$0.517313\pi$
$132$		−	0.525428i		−	0.0457326i
$133$			25.4193i			2.20413i
$134$	−5.89829			−0.509535
$135$	2.21432	−	0.311108i	0.190578	−	0.0267759i
$136$	−16.3511			−1.40209
$137$			18.7971i			1.60594i	0.596019	+	0.802970i	$0.296748\pi$
$137$			18.7971i			1.60594i	−0.596019	+	0.802970i	$0.703252\pi$
$138$		−	4.85728i		−	0.413479i
$139$	−14.0415			−1.19098			−0.595492	−	0.803361i	$-0.703043\pi$
$139$	−14.0415			−1.19098			−0.595492	+	0.803361i	$0.703043\pi$
$140$	−5.70471	+	0.801502i	−0.482136	+	0.0677393i
$141$	−7.05086			−0.593789
$142$		−	0.520505i		−	0.0436798i
$143$			4.14764i			0.346843i
$144$	2.67307			0.222756
$145$	−0.561993	−	4.00000i	−0.0466709	−	0.332182i
$146$	−15.4336			−1.27729
$147$		−	17.0415i		−	1.40556i
$148$			3.05086i			0.250779i
$149$	3.05086			0.249936			0.124968	−	0.992161i	$-0.460117\pi$
$149$	3.05086			0.249936			0.124968	+	0.992161i	$0.460117\pi$
$150$	−5.83654	+	1.67307i	−0.476551	+	0.136606i
$151$	−0.326929			−0.0266051			−0.0133026	−	0.999912i	$-0.504234\pi$
$151$	−0.326929			−0.0266051			−0.0133026	+	0.999912i	$0.504234\pi$
$152$		−	15.8983i		−	1.28952i
$153$			5.33185i			0.431055i
$154$	−5.95407			−0.479792
$155$	0.815792	+	5.80642i	0.0655260	+	0.466383i
$156$	2.17929			0.174483
$157$		−	19.9081i		−	1.58884i	−0.607367	−	0.794421i	$-0.707774\pi$
$157$		−	19.9081i		−	1.58884i	0.607367	−	0.794421i	$-0.292226\pi$
$158$		−	7.80642i		−	0.621046i
$159$	−7.18421			−0.569745
$160$	6.39361	−	0.898290i	0.505459	−	0.0710160i
$161$	19.6128			1.54571
$162$		−	1.21432i		−	0.0954060i
$163$		−	12.1748i		−	0.953607i	−0.879010	−	0.476804i	$-0.841795\pi$
$163$		−	12.1748i		−	0.953607i	0.879010	−	0.476804i	$-0.158205\pi$
$164$	0.949145			0.0741158
$165$	2.21432	−	0.311108i	0.172385	−	0.0242197i
$166$	3.52543			0.273626
$167$			13.0049i			1.00635i	0.864184	+	0.503176i	$0.167835\pi$
$167$			13.0049i			1.00635i	−0.864184	+	0.503176i	$0.832165\pi$
$168$			15.0366i			1.16010i
$169$	−4.20294			−0.323303
$170$	−2.01429	−	14.3368i	−0.154489	−	1.09958i
$171$	−5.18421			−0.396446
$172$		−	2.57628i		−	0.196440i
$173$		−	13.8938i		−	1.05633i	−0.849142	−	0.528165i	$-0.822880\pi$
$173$		−	13.8938i		−	1.05633i	0.849142	−	0.528165i	$-0.177120\pi$
$174$	−2.19358			−0.166295
$175$	−6.75557	−	23.5669i	−0.510673	−	1.78149i
$176$	2.67307			0.201490
$177$		−	1.67307i		−	0.125756i
$178$			0.755569i			0.0566323i
$179$	−12.8573			−0.960998			−0.480499	−	0.876995i	$-0.659544\pi$
$179$	−12.8573			−0.960998			−0.480499	+	0.876995i	$0.659544\pi$
$180$	−0.163465	−	1.16346i	−0.0121839	−	0.0867195i
$181$	0.917502			0.0681974			0.0340987	−	0.999418i	$-0.489144\pi$
$181$	0.917502			0.0681974			0.0340987	+	0.999418i	$0.489144\pi$
$182$		−	24.6953i		−	1.83054i
$183$			0.755569i			0.0558532i
$184$	−12.2667			−0.904314
$185$	−12.8573	+	1.80642i	−0.945286	+	0.132811i
$186$	3.18421			0.233477
$187$			5.33185i			0.389904i
$188$			3.70471i			0.270194i
$189$	4.90321			0.356656
$190$	13.9398	−	1.95851i	1.01130	−	0.142085i
$191$	14.3684			1.03966			0.519831	−	0.854269i	$-0.325995\pi$
$191$	14.3684			1.03966			0.519831	+	0.854269i	$0.325995\pi$
$192$		−	8.85236i		−	0.638864i
$193$			11.7605i			0.846539i	0.906004	+	0.423269i	$0.139118\pi$
$193$			11.7605i			0.846539i	−0.906004	+	0.423269i	$0.860882\pi$
$194$	−3.34614			−0.240239
$195$	1.29036	+	9.18421i	0.0924049	+	0.657695i
$196$	−8.95407			−0.639576
$197$		−	3.82071i		−	0.272215i	−0.990694	−	0.136107i	$-0.956541\pi$
$197$		−	3.82071i		−	0.272215i	0.990694	−	0.136107i	$-0.0434592\pi$
$198$		−	1.21432i		−	0.0862979i
$199$	13.7146			0.972199			0.486100	−	0.873903i	$-0.338419\pi$
$199$	13.7146			0.972199			0.486100	+	0.873903i	$0.338419\pi$
$200$	4.22522	+	14.7397i	0.298768	+	1.04226i
$201$	4.85728			0.342606
$202$			21.6227i			1.52137i
$203$		−	8.85728i		−	0.621659i
$204$	2.80150			0.196144
$205$	0.561993	+	4.00000i	0.0392513	+	0.279372i
$206$	−6.00984			−0.418726
$207$			4.00000i			0.278019i
$208$			11.0869i			0.768741i
$209$	−5.18421			−0.358599
$210$	−13.1842	+	1.85236i	−0.909797	+	0.127825i
$211$	1.95851			0.134830			0.0674148	−	0.997725i	$-0.478525\pi$
$211$	1.95851			0.134830			0.0674148	+	0.997725i	$0.478525\pi$
$212$			3.77478i			0.259253i
$213$			0.428639i			0.0293699i
$214$	13.5986			0.929578
$215$	10.8573	−	1.52543i	0.740460	−	0.104033i
$216$	−3.06668			−0.208661
$217$			12.8573i			0.872809i
$218$			19.0825i			1.29243i
$219$	12.7096			0.858838
$220$	−0.163465	−	1.16346i	−0.0110208	−	0.0784408i
$221$	−22.1146			−1.48759
$222$			7.05086i			0.473222i
$223$			26.0098i			1.74175i	0.491506	+	0.870874i	$0.336446\pi$
$223$			26.0098i			1.74175i	−0.491506	+	0.870874i	$0.663554\pi$
$224$	14.1575			0.945937
$225$	4.80642	−	1.37778i	0.320428	−	0.0918523i
$226$	2.14320			0.142563
$227$		−	6.34122i		−	0.420882i	−0.977607	−	0.210441i	$-0.932510\pi$
$227$		−	6.34122i		−	0.420882i	0.977607	−	0.210441i	$-0.0674899\pi$
$228$			2.72393i			0.180396i
$229$	23.3274			1.54152			0.770759	−	0.637127i	$-0.219877\pi$
$229$	23.3274			1.54152			0.770759	+	0.637127i	$0.219877\pi$
$230$	−1.51114	−	10.7556i	−0.0996415	−	0.709201i
$231$	4.90321			0.322608
$232$			5.53972i			0.363700i
$233$			1.42372i			0.0932708i	0.998912	+	0.0466354i	$0.0148499\pi$
$233$			1.42372i			0.0932708i	−0.998912	+	0.0466354i	$0.985150\pi$
$234$	5.03657			0.329251
$235$	−15.6128	+	2.19358i	−1.01847	+	0.143093i
$236$	−0.879077			−0.0572231
$237$			6.42864i			0.417585i
$238$		−	31.7462i		−	2.05780i
$239$	18.9590			1.22636			0.613178	−	0.789945i	$-0.289891\pi$
$239$	18.9590			1.22636			0.613178	+	0.789945i	$0.289891\pi$
$240$	5.91903	−	0.831613i	0.382072	−	0.0536804i
$241$	−1.34614			−0.0867126			−0.0433563	−	0.999060i	$-0.513805\pi$
$241$	−1.34614			−0.0867126			−0.0433563	+	0.999060i	$0.513805\pi$
$242$		−	1.21432i		−	0.0780594i
$243$			1.00000i			0.0641500i
$244$	0.396997			0.0254151
$245$	−5.30174	−	37.7353i	−0.338716	−	2.41082i
$246$	2.19358			0.139857
$247$		−	21.5022i		−	1.36816i
$248$		−	8.04149i		−	0.510635i
$249$	−2.90321			−0.183984
$250$	−12.4035	+	5.52051i	−0.784463	+	0.349147i
$251$	1.08250			0.0683267			0.0341633	−	0.999416i	$-0.489123\pi$
$251$	1.08250			0.0683267			0.0341633	+	0.999416i	$0.489123\pi$
$252$		−	2.57628i		−	0.162291i
$253$			4.00000i			0.251478i
$254$	22.7195			1.42555
$255$	1.65878	+	11.8064i	0.103877	+	0.739347i
$256$	−11.6637			−0.728981
$257$		−	0.133353i		−	0.00831834i	−0.999991	−	0.00415917i	$-0.998676\pi$
$257$		−	0.133353i		−	0.00831834i	0.999991	−	0.00415917i	$-0.00132391\pi$
$258$		−	5.95407i		−	0.370684i
$259$	−28.4701			−1.76905
$260$	4.82564	−	0.677993i	0.299273	−	0.0420473i
$261$	1.80642			0.111815
$262$			1.51114i			0.0933584i
$263$			0.147643i			0.00910407i	0.999990	+	0.00455203i	$0.00144896\pi$
$263$			0.147643i			0.00910407i	−0.999990	+	0.00455203i	$0.998551\pi$
$264$	−3.06668			−0.188741
$265$	−15.9081	+	2.23506i	−0.977229	+	0.137299i
$266$	30.8671			1.89258
$267$		−	0.622216i		−	0.0380790i
$268$		−	2.55215i		−	0.155897i
$269$	26.8573			1.63752			0.818759	−	0.574138i	$-0.194663\pi$
$269$	26.8573			1.63752			0.818759	+	0.574138i	$0.194663\pi$
$270$	−0.377784	−	2.68889i	−0.0229912	−	0.163641i
$271$	3.08250			0.187248			0.0936242	−	0.995608i	$-0.470155\pi$
$271$	3.08250			0.187248			0.0936242	+	0.995608i	$0.470155\pi$
$272$			14.2524i			0.864180i
$273$			20.3368i			1.23084i
$274$	22.8256			1.37895
$275$	4.80642	−	1.37778i	0.289838	−	0.0830835i
$276$	2.10171			0.126508
$277$		−	8.70964i		−	0.523311i	−0.965161	−	0.261656i	$-0.915732\pi$
$277$		−	8.70964i		−	0.523311i	0.965161	−	0.261656i	$-0.0842685\pi$
$278$			17.0509i			1.02264i
$279$	−2.62222			−0.156988
$280$	4.67799	+	33.2958i	0.279564	+	1.98980i
$281$	−20.3783			−1.21567			−0.607833	−	0.794065i	$-0.707961\pi$
$281$	−20.3783			−1.21567			−0.607833	+	0.794065i	$0.707961\pi$
$282$			8.56199i			0.509859i
$283$			6.32248i			0.375833i	0.982185	+	0.187916i	$0.0601734\pi$
$283$			6.32248i			0.375833i	−0.982185	+	0.187916i	$0.939827\pi$
$284$	0.225219			0.0133643
$285$	−11.4795	+	1.61285i	−0.679987	+	0.0955369i
$286$	5.03657			0.297818
$287$			8.85728i			0.522829i
$288$			2.88739i			0.170141i
$289$	−11.4286			−0.672273
$290$	−4.85728	+	0.682439i	−0.285229	+	0.0400742i
$291$	2.75557			0.161534
$292$		−	6.67799i		−	0.390800i
$293$		−	16.6780i		−	0.974339i	−0.873308	−	0.487169i	$-0.838029\pi$
$293$		−	16.6780i		−	0.974339i	0.873308	−	0.487169i	$-0.161971\pi$
$294$	−20.6938			−1.20689
$295$	−0.520505	−	3.70471i	−0.0303050	−	0.215697i
$296$	17.8064			1.03498
$297$			1.00000i			0.0580259i
$298$		−	3.70471i		−	0.214608i
$299$	−16.5906			−0.959458
$300$	−0.723926	−	2.52543i	−0.0417959	−	0.145806i
$301$	24.0415			1.38573
$302$			0.396997i			0.0228446i
$303$		−	17.8064i		−	1.02295i
$304$	−13.8578			−0.794797
$305$	0.235063	+	1.67307i	0.0134597	+	0.0957998i
$306$	6.47457			0.370127
$307$		−	9.58565i		−	0.547082i	−0.961860	−	0.273541i	$-0.911805\pi$
$307$		−	9.58565i		−	0.547082i	0.961860	−	0.273541i	$-0.0881949\pi$
$308$		−	2.57628i		−	0.146797i
$309$	4.94914			0.281547
$310$	7.05086	−	0.990632i	0.400462	−	0.0562641i
$311$	14.5303			0.823941			0.411970	−	0.911197i	$-0.364841\pi$
$311$	14.5303			0.823941			0.411970	+	0.911197i	$0.364841\pi$
$312$		−	12.7195i		−	0.720099i
$313$			21.0321i			1.18881i	0.804167	+	0.594403i	$0.202612\pi$
$313$			21.0321i			1.18881i	−0.804167	+	0.594403i	$0.797388\pi$
$314$	−24.1748			−1.36427
$315$	10.8573	−	1.52543i	0.611738	−	0.0859481i
$316$	3.37778			0.190015
$317$		−	0.990632i		−	0.0556394i	−0.999613	−	0.0278197i	$-0.991144\pi$
$317$		−	0.990632i		−	0.0556394i	0.999613	−	0.0278197i	$-0.00885643\pi$
$318$			8.72393i			0.489213i
$319$	1.80642			0.101140
$320$	−2.75404	−	19.6019i	−0.153955	−	1.09578i
$321$	−11.1985			−0.625039
$322$		−	23.8163i		−	1.32723i
$323$		−	27.6414i		−	1.53801i
$324$	0.525428			0.0291904
$325$	5.71456	+	19.9353i	0.316987	+	1.10581i
$326$	−14.7841			−0.818818
$327$		−	15.7146i		−	0.869017i
$328$		−	5.53972i		−	0.305880i
$329$	−34.5718			−1.90601
$330$	−0.377784	−	2.68889i	−0.0207963	−	0.148019i
$331$	−17.5812			−0.966350			−0.483175	−	0.875524i	$-0.660517\pi$
$331$	−17.5812			−0.966350			−0.483175	+	0.875524i	$0.660517\pi$
$332$			1.52543i			0.0837187i
$333$		−	5.80642i		−	0.318190i
$334$	15.7921			0.864107
$335$	10.7556	−	1.51114i	0.587639	−	0.0825623i
$336$	13.1066			0.715025
$337$		−	3.16992i		−	0.172676i	−0.996266	−	0.0863382i	$-0.972483\pi$
$337$		−	3.16992i		−	0.172676i	0.996266	−	0.0863382i	$-0.0275166\pi$
$338$			5.10372i			0.277606i
$339$	−1.76494			−0.0958582
$340$	6.20342	−	0.871569i	0.336428	−	0.0472675i
$341$	−2.62222			−0.142001
$342$			6.29529i			0.340410i
$343$		−	49.2355i		−	2.65847i
$344$	−15.0366			−0.810717
$345$	1.24443	+	8.85728i	0.0669979	+	0.476860i
$346$	−16.8716			−0.907021
$347$			4.97634i			0.267144i	0.991039	+	0.133572i	$0.0426447\pi$
$347$			4.97634i			0.267144i	−0.991039	+	0.133572i	$0.957355\pi$
$348$		−	0.949145i		−	0.0508795i
$349$	−18.2034			−0.974407			−0.487203	−	0.873289i	$-0.661983\pi$
$349$	−18.2034			−0.974407			−0.487203	+	0.873289i	$0.661983\pi$
$350$	−28.6178	+	8.20342i	−1.52968	+	0.438491i
$351$	−4.14764			−0.221385
$352$			2.88739i			0.153898i
$353$		−	22.4099i		−	1.19276i	−0.802703	−	0.596379i	$-0.796605\pi$
$353$		−	22.4099i		−	1.19276i	0.802703	−	0.596379i	$-0.203395\pi$
$354$	−2.03164			−0.107981
$355$	0.133353	+	0.949145i	0.00707765	+	0.0503754i
$356$	−0.326929			−0.0173272
$357$			26.1432i			1.38364i
$358$			15.6128i			0.825165i
$359$	−21.3274			−1.12562			−0.562809	−	0.826587i	$-0.690279\pi$
$359$	−21.3274			−1.12562			−0.562809	+	0.826587i	$0.690279\pi$
$360$	−6.79060	+	0.954067i	−0.357896	+	0.0502837i
$361$	7.87601			0.414527
$362$		−	1.11414i		−	0.0585579i
$363$			1.00000i			0.0524864i
$364$	10.6855			0.560072
$365$	28.1432	−	3.95407i	1.47308	−	0.206965i
$366$	0.917502			0.0479586
$367$			35.1338i			1.83397i	0.398921	+	0.916985i	$0.369385\pi$
$367$			35.1338i			1.83397i	−0.398921	+	0.916985i	$0.630615\pi$
$368$			10.6923i			0.557374i
$369$	−1.80642			−0.0940387
$370$	2.19358	+	15.6128i	0.114039	+	0.811673i
$371$	−35.2257			−1.82883
$372$			1.37778i			0.0714348i
$373$			17.0049i			0.880481i	0.897880	+	0.440241i	$0.145107\pi$
$373$			17.0049i			0.880481i	−0.897880	+	0.440241i	$0.854893\pi$
$374$	6.47457			0.334792
$375$	10.2143	−	4.54617i	0.527465	−	0.234763i
$376$	21.6227			1.11511
$377$			7.49240i			0.385878i
$378$		−	5.95407i		−	0.306244i
$379$	−2.36842			−0.121657			−0.0608287	−	0.998148i	$-0.519374\pi$
$379$	−2.36842			−0.121657			−0.0608287	+	0.998148i	$0.519374\pi$
$380$	0.847435	+	6.03164i	0.0434725	+	0.309417i
$381$	−18.7096			−0.958524
$382$		−	17.4479i		−	0.892710i
$383$		−	1.21585i		−	0.0621271i	−0.999517	−	0.0310635i	$-0.990111\pi$
$383$		−	1.21585i		−	0.0621271i	0.999517	−	0.0310635i	$-0.00988942\pi$
$384$	−4.97481			−0.253870
$385$	10.8573	−	1.52543i	0.553338	−	0.0777430i
$386$	14.2810			0.726884
$387$			4.90321i			0.249244i
$388$		−	1.44785i		−	0.0735035i
$389$	−2.26671			−0.114927			−0.0574633	−	0.998348i	$-0.518301\pi$
$389$	−2.26671			−0.114927			−0.0574633	+	0.998348i	$0.518301\pi$
$390$	11.1526	−	1.56691i	0.564732	−	0.0793438i
$391$	−21.3274			−1.07857
$392$			52.2607i			2.63957i
$393$		−	1.24443i		−	0.0627733i
$394$	−4.63957			−0.233738
$395$	2.00000	+	14.2351i	0.100631	+	0.716244i
$396$	0.525428			0.0264037
$397$		−	18.4889i		−	0.927929i	−0.885854	−	0.463965i	$-0.846426\pi$
$397$		−	18.4889i		−	0.927929i	0.885854	−	0.463965i	$-0.153574\pi$
$398$		−	16.6539i		−	0.834782i
$399$	−25.4193			−1.27256
$400$	12.8479	−	3.68292i	0.642396	−	0.184146i
$401$	17.5625			0.877028			0.438514	−	0.898724i	$-0.355505\pi$
$401$	17.5625			0.877028			0.438514	+	0.898724i	$0.355505\pi$
$402$		−	5.89829i		−	0.294180i
$403$		−	10.8760i		−	0.541773i
$404$	−9.35599			−0.465478
$405$	0.311108	+	2.21432i	0.0154591	+	0.110030i
$406$	−10.7556			−0.533790
$407$		−	5.80642i		−	0.287814i
$408$		−	16.3511i		−	0.809498i
$409$	21.3461			1.05550			0.527749	−	0.849400i	$-0.323036\pi$
$409$	21.3461			1.05550			0.527749	+	0.849400i	$0.323036\pi$
$410$	4.85728	−	0.682439i	0.239884	−	0.0337032i
$411$	−18.7971			−0.927190
$412$		−	2.60042i		−	0.128113i
$413$		−	8.20342i		−	0.403664i
$414$	4.85728			0.238722
$415$	−6.42864	+	0.903212i	−0.315570	+	0.0443369i
$416$	−11.9759			−0.587165
$417$		−	14.0415i		−	0.687615i
$418$			6.29529i			0.307913i
$419$	−28.8573			−1.40977			−0.704885	−	0.709321i	$-0.749001\pi$
$419$	−28.8573			−1.40977			−0.704885	+	0.709321i	$0.749001\pi$
$420$	−0.801502	−	5.70471i	−0.0391093	−	0.278362i
$421$	−35.4893			−1.72964			−0.864822	−	0.502078i	$-0.832569\pi$
$421$	−35.4893			−1.72964			−0.864822	+	0.502078i	$0.832569\pi$
$422$		−	2.37826i		−	0.115772i
$423$		−	7.05086i		−	0.342824i
$424$	22.0316			1.06995
$425$	7.34614	+	25.6271i	0.356340	+	1.24310i
$426$	0.520505			0.0252186
$427$			3.70471i			0.179284i
$428$			5.88400i			0.284414i
$429$	−4.14764			−0.200250
$430$	−1.85236	−	13.1842i	−0.0893286	−	0.635799i
$431$	9.24443			0.445289			0.222644	−	0.974900i	$-0.428531\pi$
$431$	9.24443			0.445289			0.222644	+	0.974900i	$0.428531\pi$
$432$			2.67307i			0.128608i
$433$			6.28544i			0.302059i	0.988529	+	0.151030i	$0.0482589\pi$
$433$			6.28544i			0.302059i	−0.988529	+	0.151030i	$0.951741\pi$
$434$	15.6128			0.749441
$435$	4.00000	−	0.561993i	0.191785	−	0.0269455i
$436$	−8.25686			−0.395432
$437$		−	20.7368i		−	0.991977i
$438$		−	15.4336i		−	0.737444i
$439$	36.5303			1.74350			0.871749	−	0.489952i	$-0.162986\pi$
$439$	36.5303			1.74350			0.871749	+	0.489952i	$0.162986\pi$
$440$	−6.79060	+	0.954067i	−0.323729	+	0.0454834i
$441$	17.0415			0.811499
$442$			26.8542i			1.27732i
$443$			38.2766i			1.81857i	0.416170	+	0.909287i	$0.363372\pi$
$443$			38.2766i			1.81857i	−0.416170	+	0.909287i	$0.636628\pi$
$444$	−3.05086			−0.144787
$445$	−0.193576	−	1.37778i	−0.00917639	−	0.0653132i
$446$	31.5843			1.49556
$447$			3.05086i			0.144300i
$448$		−	43.4050i		−	2.05069i
$449$	31.8479			1.50300			0.751498	−	0.659735i	$-0.229332\pi$
$449$	31.8479			1.50300			0.751498	+	0.659735i	$0.229332\pi$
$450$	−1.67307	−	5.83654i	−0.0788693	−	0.275137i
$451$	−1.80642			−0.0850612
$452$			0.927346i			0.0436187i
$453$		−	0.326929i		−	0.0153605i
$454$	−7.70027			−0.361391
$455$	6.32693	+	45.0321i	0.296611	+	2.11114i
$456$	15.8983			0.744506
$457$			1.39207i			0.0651185i	0.999470	+	0.0325592i	$0.0103658\pi$
$457$			1.39207i			0.0651185i	−0.999470	+	0.0325592i	$0.989634\pi$
$458$		−	28.3269i		−	1.32363i
$459$	−5.33185			−0.248870
$460$	4.65386	−	0.653858i	0.216987	−	0.0304863i
$461$	−7.70471			−0.358844			−0.179422	−	0.983772i	$-0.557423\pi$
$461$	−7.70471			−0.358844			−0.179422	+	0.983772i	$0.557423\pi$
$462$		−	5.95407i		−	0.277008i
$463$		−	4.68244i		−	0.217611i	−0.994063	−	0.108806i	$-0.965297\pi$
$463$		−	4.68244i		−	0.217611i	0.994063	−	0.108806i	$-0.0347026\pi$
$464$	4.82870			0.224167
$465$	−5.80642	+	0.815792i	−0.269266	+	0.0378314i
$466$	1.72885			0.0800873
$467$			12.8573i			0.594964i	0.954727	+	0.297482i	$0.0961468\pi$
$467$			12.8573i			0.594964i	−0.954727	+	0.297482i	$0.903853\pi$
$468$			2.17929i			0.100738i
$469$	23.8163			1.09973
$470$	2.66370	+	18.9590i	0.122867	+	0.874513i
$471$	19.9081			0.917318
$472$			5.13077i			0.236163i
$473$			4.90321i			0.225450i
$474$	7.80642			0.358561
$475$	−24.9175	+	7.14272i	−1.14329	+	0.327731i
$476$	13.7364			0.629605
$477$		−	7.18421i		−	0.328942i
$478$		−	23.0223i		−	1.05301i
$479$	−8.38715			−0.383219			−0.191609	−	0.981471i	$-0.561371\pi$
$479$	−8.38715			−0.383219			−0.191609	+	0.981471i	$0.561371\pi$
$480$	0.898290	+	6.39361i	0.0410011	+	0.291827i
$481$	24.0830			1.09809
$482$			1.63465i			0.0744561i
$483$			19.6128i			0.892415i
$484$	0.525428			0.0238831
$485$	6.10171	−	0.857279i	0.277064	−	0.0389270i
$486$	1.21432			0.0550827
$487$			9.83500i			0.445667i	0.974857	+	0.222833i	$0.0715306\pi$
$487$			9.83500i			0.445667i	−0.974857	+	0.222833i	$0.928469\pi$
$488$		−	2.31708i		−	0.104890i
$489$	12.1748			0.550565
$490$	−45.8227	+	6.43801i	−2.07006	+	0.290840i
$491$	−32.9403			−1.48657			−0.743286	−	0.668973i	$-0.766734\pi$
$491$	−32.9403			−1.48657			−0.743286	+	0.668973i	$0.766734\pi$
$492$			0.949145i			0.0427908i
$493$			9.63158i			0.433785i
$494$	−26.1106			−1.17477
$495$	0.311108	+	2.21432i	0.0139833	+	0.0995263i
$496$	−7.00937			−0.314730
$497$			2.10171i			0.0942746i
$498$			3.52543i			0.157978i
$499$	1.63158			0.0730397			0.0365199	−	0.999333i	$-0.488373\pi$
$499$	1.63158			0.0730397			0.0365199	+	0.999333i	$0.488373\pi$
$500$	−2.38868	−	5.36689i	−0.106825	−	0.240014i
$501$	−13.0049			−0.581017
$502$		−	1.31450i		−	0.0586689i
$503$		−	41.8622i		−	1.86654i	−0.359171	−	0.933272i	$-0.616941\pi$
$503$		−	41.8622i		−	1.86654i	0.359171	−	0.933272i	$-0.383059\pi$
$504$	−15.0366			−0.669782
$505$	−5.53972	−	39.4291i	−0.246514	−	1.75457i
$506$	4.85728			0.215932
$507$		−	4.20294i		−	0.186659i
$508$			9.83056i			0.436160i
$509$	38.8573			1.72232			0.861159	−	0.508335i	$-0.169739\pi$
$509$	38.8573			1.72232			0.861159	+	0.508335i	$0.169739\pi$
$510$	14.3368	−	2.01429i	0.634843	−	0.0891943i
$511$	62.3180			2.75679
$512$			24.1131i			1.06566i
$513$		−	5.18421i		−	0.228888i
$514$	−0.161933			−0.00714257
$515$	10.9590	−	1.53972i	0.482911	−	0.0678481i
$516$	2.57628			0.113415
$517$		−	7.05086i		−	0.310096i
$518$			34.5718i			1.51900i
$519$	13.8938			0.609872
$520$	−3.95713	−	28.1650i	−0.173532	−	1.23512i
$521$	11.1111			0.486785			0.243393	−	0.969928i	$-0.421740\pi$
$521$	11.1111			0.486785			0.243393	+	0.969928i	$0.421740\pi$
$522$		−	2.19358i		−	0.0960102i
$523$		−	27.3002i		−	1.19375i	−0.802332	−	0.596877i	$-0.796408\pi$
$523$		−	27.3002i		−	1.19375i	0.802332	−	0.596877i	$-0.203592\pi$
$524$	−0.653858			−0.0285639
$525$	23.5669	−	6.75557i	1.02854	−	0.294837i
$526$	0.179286			0.00781724
$527$		−	13.9813i		−	0.609033i
$528$			2.67307i			0.116330i
$529$	7.00000			0.304348
$530$	2.71408	+	19.3176i	0.117892	+	0.839101i
$531$	1.67307			0.0726051
$532$			13.3560i			0.579055i
$533$		−	7.49240i		−	0.324532i
$534$	−0.755569			−0.0326967
$535$	−24.7971	+	3.48394i	−1.07207	+	0.150624i
$536$	−14.8957			−0.643396
$537$		−	12.8573i		−	0.554833i
$538$		−	32.6133i		−	1.40606i
$539$	17.0415			0.734029
$540$	1.16346	−	0.163465i	0.0500675	−	0.00703440i
$541$	−16.1017			−0.692267			−0.346133	−	0.938185i	$-0.612506\pi$
$541$	−16.1017			−0.692267			−0.346133	+	0.938185i	$0.612506\pi$
$542$		−	3.74314i		−	0.160782i
$543$			0.917502i			0.0393738i
$544$	−15.3951			−0.660061
$545$	−4.88892	−	34.7971i	−0.209418	−	1.49054i
$546$	24.6953			1.05686
$547$			40.0370i			1.71186i	0.517091	+	0.855930i	$0.327015\pi$
$547$			40.0370i			1.71186i	−0.517091	+	0.855930i	$0.672985\pi$
$548$			9.87649i			0.421903i
$549$	−0.755569			−0.0322469
$550$	−1.67307	−	5.83654i	−0.0713400	−	0.248871i
$551$	−9.36488			−0.398957
$552$		−	12.2667i		−	0.522106i
$553$			31.5210i			1.34041i
$554$	−10.5763			−0.449343
$555$	−1.80642	−	12.8573i	−0.0766784	−	0.545761i
$556$	−7.37778			−0.312888
$557$		−	28.2908i		−	1.19872i	−0.800479	−	0.599361i	$-0.795422\pi$
$557$		−	28.2908i		−	1.19872i	0.800479	−	0.599361i	$-0.204578\pi$
$558$			3.18421i			0.134798i
$559$	−20.3368			−0.860154
$560$	29.0223	−	4.07758i	1.22641	−	0.172309i
$561$	−5.33185			−0.225111
$562$			24.7457i			1.04384i
$563$			32.7926i			1.38204i	0.722834	+	0.691022i	$0.242839\pi$
$563$			32.7926i			1.38204i	−0.722834	+	0.691022i	$0.757161\pi$
$564$	−3.70471			−0.155997
$565$	−3.90813	+	0.549086i	−0.164416	+	0.0231002i
$566$	7.67752			0.322710
$567$			4.90321i			0.205916i
$568$		−	1.31450i		−	0.0551551i
$569$	8.88586			0.372515			0.186257	−	0.982501i	$-0.440364\pi$
$569$	8.88586			0.372515			0.186257	+	0.982501i	$0.440364\pi$
$570$	1.95851	+	13.9398i	0.0820331	+	0.583873i
$571$	−10.6953			−0.447586			−0.223793	−	0.974637i	$-0.571844\pi$
$571$	−10.6953			−0.447586			−0.223793	+	0.974637i	$0.571844\pi$
$572$			2.17929i			0.0911205i
$573$			14.3684i			0.600249i
$574$	10.7556			0.448929
$575$	5.51114	+	19.2257i	0.229830	+	0.801767i
$576$	8.85236			0.368848
$577$		−	27.1338i		−	1.12960i	−0.825229	−	0.564798i	$-0.808954\pi$
$577$		−	27.1338i		−	1.12960i	0.825229	−	0.564798i	$-0.191046\pi$
$578$			13.8780i			0.577250i
$579$	−11.7605			−0.488749
$580$	−0.295286	−	2.10171i	−0.0122611	−	0.0872688i
$581$	−14.2351			−0.590570
$582$		−	3.34614i		−	0.138702i
$583$		−	7.18421i		−	0.297540i
$584$	−38.9763			−1.61285
$585$	−9.18421	+	1.29036i	−0.379720	+	0.0533500i
$586$	−20.2524			−0.836620
$587$		−	10.9590i		−	0.452326i	−0.974089	−	0.226163i	$-0.927382\pi$
$587$		−	10.9590i		−	0.452326i	0.974089	−	0.226163i	$-0.0726182\pi$
$588$		−	8.95407i		−	0.369260i
$589$	13.5941			0.560136
$590$	−4.49871	+	0.632060i	−0.185209	+	0.0260215i
$591$	3.82071			0.157163
$592$		−	15.5210i		−	0.637908i
$593$			23.7003i			0.973253i	0.873610	+	0.486627i	$0.161773\pi$
$593$			23.7003i			0.973253i	−0.873610	+	0.486627i	$0.838227\pi$
$594$	1.21432			0.0498241
$595$	8.13335	+	57.8894i	0.333435	+	2.37323i
$596$	1.60300			0.0656616
$597$			13.7146i			0.561299i
$598$			20.1463i			0.823842i
$599$	−41.7146			−1.70441			−0.852205	−	0.523208i	$-0.824735\pi$
$599$	−41.7146			−1.70441			−0.852205	+	0.523208i	$0.824735\pi$
$600$	−14.7397	+	4.22522i	−0.601748	+	0.172494i
$601$	14.5906			0.595162			0.297581	−	0.954697i	$-0.403820\pi$
$601$	14.5906			0.595162			0.297581	+	0.954697i	$0.403820\pi$
$602$		−	29.1941i		−	1.18986i
$603$			4.85728i			0.197804i
$604$	−0.171778			−0.00698953
$605$	0.311108	+	2.21432i	0.0126483	+	0.0900249i
$606$	−21.6227			−0.878362
$607$		−	19.9826i		−	0.811071i	−0.914079	−	0.405535i	$-0.867085\pi$
$607$		−	19.9826i		−	0.811071i	0.914079	−	0.405535i	$-0.132915\pi$
$608$		−	14.9688i		−	0.607066i
$609$	8.85728			0.358915
$610$	2.03164	−	0.285442i	0.0822588	−	0.0115572i
$611$	29.2444			1.18310
$612$			2.80150i			0.113244i
$613$		−	19.0781i		−	0.770555i	−0.922801	−	0.385278i	$-0.874106\pi$
$613$		−	19.0781i		−	0.770555i	0.922801	−	0.385278i	$-0.125894\pi$
$614$	−11.6400			−0.469754
$615$	−4.00000	+	0.561993i	−0.161296	+	0.0226617i
$616$	−15.0366			−0.605840
$617$		−	39.3590i		−	1.58454i	−0.610174	−	0.792268i	$-0.708900\pi$
$617$		−	39.3590i		−	1.58454i	0.610174	−	0.792268i	$-0.291100\pi$
$618$		−	6.00984i		−	0.241751i
$619$	23.0923			0.928160			0.464080	−	0.885793i	$-0.346385\pi$
$619$	23.0923			0.928160			0.464080	+	0.885793i	$0.346385\pi$
$620$	0.428639	+	3.05086i	0.0172146	+	0.122525i
$621$	−4.00000			−0.160514
$622$		−	17.6445i		−	0.707480i
$623$		−	3.05086i		−	0.122230i
$624$	−11.0869			−0.443833
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$	25.5397			1.02077
$627$		−	5.18421i		−	0.207037i
$628$		−	10.4603i		−	0.417411i
$629$	30.9590			1.23442
$630$	−1.85236	−	13.1842i	−0.0737997	−	0.525271i
$631$	−25.5111			−1.01558			−0.507791	−	0.861480i	$-0.669538\pi$
$631$	−25.5111			−1.01558			−0.507791	+	0.861480i	$0.669538\pi$
$632$		−	19.7146i		−	0.784203i
$633$			1.95851i			0.0778439i
$634$	−1.20294			−0.0477750
$635$	−41.4291	+	5.82071i	−1.64406	+	0.230988i
$636$	−3.77478			−0.149680
$637$			70.6820i			2.80052i
$638$		−	2.19358i		−	0.0868445i
$639$	−0.428639			−0.0169567
$640$	−11.0158	+	1.54770i	−0.435439	+	0.0611783i
$641$	−6.25380			−0.247010			−0.123505	−	0.992344i	$-0.539414\pi$
$641$	−6.25380			−0.247010			−0.123505	+	0.992344i	$0.539414\pi$
$642$			13.5986i			0.536692i
$643$		−	6.84743i		−	0.270036i	−0.990843	−	0.135018i	$-0.956891\pi$
$643$		−	6.84743i		−	0.270036i	0.990843	−	0.135018i	$-0.0431093\pi$
$644$	10.3051			0.406079
$645$	1.52543	+	10.8573i	0.0600637	+	0.427505i
$646$	−33.5655			−1.32062
$647$			20.2953i			0.797890i	0.916975	+	0.398945i	$0.130624\pi$
$647$			20.2953i			0.797890i	−0.916975	+	0.398945i	$0.869376\pi$
$648$		−	3.06668i		−	0.120470i
$649$	1.67307			0.0656738
$650$	24.2079	−	6.93930i	0.949511	−	0.272182i
$651$	−12.8573			−0.503916
$652$		−	6.39700i		−	0.250526i
$653$			10.6222i			0.415679i	0.978163	+	0.207840i	$0.0666432\pi$
$653$			10.6222i			0.415679i	−0.978163	+	0.207840i	$0.933357\pi$
$654$	−19.0825			−0.746185
$655$	−0.387152	−	2.75557i	−0.0151273	−	0.107669i
$656$	−4.82870			−0.188529
$657$			12.7096i			0.495850i
$658$			41.9813i			1.63660i
$659$	−10.1017			−0.393507			−0.196753	−	0.980453i	$-0.563040\pi$
$659$	−10.1017			−0.393507			−0.196753	+	0.980453i	$0.563040\pi$
$660$	1.16346	−	0.163465i	0.0452878	−	0.00636285i
$661$	21.6128			0.840642			0.420321	−	0.907375i	$-0.361917\pi$
$661$	21.6128			0.840642			0.420321	+	0.907375i	$0.361917\pi$
$662$			21.3492i			0.829760i
$663$		−	22.1146i		−	0.858861i
$664$	8.90321			0.345512
$665$	−56.2864	+	7.90813i	−2.18269	+	0.306664i
$666$	−7.05086			−0.273215
$667$			7.22570i			0.279780i
$668$			6.83314i			0.264382i
$669$	−26.0098			−1.00560
$670$	−1.83500	−	13.0607i	−0.0708924	−	0.504579i
$671$	−0.755569			−0.0291684
$672$			14.1575i			0.546137i
$673$		−	10.2208i		−	0.393982i	−0.980405	−	0.196991i	$-0.936883\pi$
$673$		−	10.2208i		−	0.393982i	0.980405	−	0.196991i	$-0.0631170\pi$
$674$	−3.84929			−0.148269
$675$	1.37778	+	4.80642i	0.0530309	+	0.184999i
$676$	−2.20834			−0.0849363
$677$			13.9224i			0.535082i	0.963546	+	0.267541i	$0.0862111\pi$
$677$			13.9224i			0.535082i	−0.963546	+	0.267541i	$0.913789\pi$
$678$			2.14320i			0.0823090i
$679$	13.5111			0.518510
$680$	−5.08694	−	36.2065i	−0.195075	−	1.38846i
$681$	6.34122			0.242996
$682$			3.18421i			0.121930i
$683$			10.3970i			0.397830i	0.980017	+	0.198915i	$0.0637418\pi$
$683$			10.3970i			0.397830i	−0.980017	+	0.198915i	$0.936258\pi$
$684$	−2.72393			−0.104152
$685$	−41.6227	+	5.84791i	−1.59032	+	0.223437i
$686$	−59.7877			−2.28270
$687$			23.3274i			0.889996i
$688$			13.1066i			0.499686i
$689$	29.7975			1.13520
$690$	10.7556	−	1.51114i	0.409458	−	0.0575280i
$691$	−0.977725			−0.0371944			−0.0185972	−	0.999827i	$-0.505920\pi$
$691$	−0.977725			−0.0371944			−0.0185972	+	0.999827i	$0.505920\pi$
$692$		−	7.30021i		−	0.277512i
$693$			4.90321i			0.186258i
$694$	6.04287			0.229384
$695$	−4.36842	−	31.0923i	−0.165703	−	1.17940i
$696$	−5.53972			−0.209982
$697$		−	9.63158i		−	0.364822i
$698$			22.1048i			0.836678i
$699$	−1.42372			−0.0538499
$700$	−3.54956	−	12.3827i	−0.134161	−	0.468022i
$701$	48.9688			1.84953			0.924764	−	0.380542i	$-0.124263\pi$
$701$	48.9688			1.84953			0.924764	+	0.380542i	$0.124263\pi$
$702$			5.03657i			0.190093i
$703$			30.1017i			1.13531i
$704$	8.85236			0.333636
$705$	−2.19358	−	15.6128i	−0.0826149	−	0.588014i
$706$	−27.2128			−1.02417
$707$		−	87.3087i		−	3.28358i
$708$		−	0.879077i		−	0.0330378i
$709$	37.2672			1.39960			0.699799	−	0.714340i	$-0.253273\pi$
$709$	37.2672			1.39960			0.699799	+	0.714340i	$0.253273\pi$
$710$	1.15257	−	0.161933i	0.0432550	−	0.00607725i
$711$	−6.42864			−0.241093
$712$			1.90813i			0.0715103i
$713$		−	10.4889i		−	0.392811i
$714$	31.7462			1.18807
$715$	−9.18421	+	1.29036i	−0.343470	+	0.0482569i
$716$	−6.75557			−0.252467
$717$			18.9590i			0.708036i
$718$			25.8983i			0.966516i
$719$	−5.83500			−0.217609			−0.108804	−	0.994063i	$-0.534702\pi$
$719$	−5.83500			−0.217609			−0.108804	+	0.994063i	$0.534702\pi$
$720$	0.831613	+	5.91903i	0.0309924	+	0.220589i
$721$	24.2667			0.903739
$722$		−	9.56400i		−	0.355935i
$723$		−	1.34614i		−	0.0500635i
$724$	0.482081			0.0179164
$725$	8.68244	−	2.48886i	0.322458	−	0.0924340i
$726$	1.21432			0.0450676
$727$		−	46.8385i		−	1.73715i	−0.495562	−	0.868573i	$-0.665038\pi$
$727$		−	46.8385i		−	1.73715i	0.495562	−	0.868573i	$-0.334962\pi$
$728$		−	62.3663i		−	2.31145i
$729$	−1.00000			−0.0370370
$730$	−4.80150	−	34.1748i	−0.177712	−	1.26487i
$731$	−26.1432			−0.966941
$732$			0.396997i			0.0146734i
$733$			45.2083i			1.66981i	0.550395	+	0.834904i	$0.314477\pi$
$733$			45.2083i			1.66981i	−0.550395	+	0.834904i	$0.685523\pi$
$734$	42.6637			1.57475
$735$	37.7353	−	5.30174i	1.39189	−	0.195558i
$736$	−11.5496			−0.425723
$737$			4.85728i			0.178920i
$738$			2.19358i			0.0807467i
$739$	−5.65433			−0.207998			−0.103999	−	0.994577i	$-0.533164\pi$
$739$	−5.65433			−0.207998			−0.103999	+	0.994577i	$0.533164\pi$
$740$	−6.75557	+	0.949145i	−0.248340	+	0.0348913i
$741$	21.5022			0.789905
$742$			42.7753i			1.57033i
$743$		−	4.50622i		−	0.165317i	−0.996578	−	0.0826585i	$-0.973659\pi$
$743$		−	4.50622i		−	0.165317i	0.996578	−	0.0826585i	$-0.0263411\pi$
$744$	8.04149			0.294815
$745$	0.949145	+	6.75557i	0.0347740	+	0.247505i
$746$	20.6494			0.756029
$747$		−	2.90321i		−	0.106223i
$748$			2.80150i			0.102433i
$749$	−54.9086			−2.00632
$750$	−5.52051	−	12.4035i	−0.201580	−	0.452910i
$751$	47.5121			1.73374			0.866870	−	0.498534i	$-0.166128\pi$
$751$	47.5121			1.73374			0.866870	+	0.498534i	$0.166128\pi$
$752$		−	18.8474i		−	0.687295i
$753$			1.08250i			0.0394484i
$754$	9.09817			0.331336
$755$	−0.101710	−	0.723926i	−0.00370161	−	0.0263464i
$756$	2.57628			0.0936985
$757$		−	46.6637i		−	1.69602i	−0.529979	−	0.848011i	$-0.677800\pi$
$757$		−	46.6637i		−	1.69602i	0.529979	−	0.848011i	$-0.322200\pi$
$758$			2.87601i			0.104462i
$759$	−4.00000			−0.145191
$760$	35.2039	−	4.94608i	1.27698	−	0.179413i
$761$	14.9304			0.541227			0.270613	−	0.962688i	$-0.412773\pi$
$761$	14.9304			0.541227			0.270613	+	0.962688i	$0.412773\pi$
$762$			22.7195i			0.823040i
$763$		−	77.0518i		−	2.78946i
$764$	7.54956			0.273134
$765$	−11.8064	+	1.65878i	−0.426862	+	0.0599733i
$766$	−1.47643			−0.0533457
$767$			6.93930i			0.250564i
$768$		−	11.6637i		−	0.420878i
$769$	−38.8573			−1.40123			−0.700615	−	0.713540i	$-0.747091\pi$
$769$	−38.8573			−1.40123			−0.700615	+	0.713540i	$0.747091\pi$
$770$	−1.85236	−	13.1842i	−0.0667543	−	0.475126i
$771$	0.133353			0.00480259
$772$			6.17929i			0.222397i
$773$			36.3368i			1.30694i	0.756951	+	0.653471i	$0.226688\pi$
$773$			36.3368i			1.30694i	−0.756951	+	0.653471i	$0.773312\pi$
$774$	5.95407			0.214015
$775$	−12.6035	+	3.61285i	−0.452730	+	0.129777i
$776$	−8.45044			−0.303353
$777$		−	28.4701i		−	1.02136i
$778$			2.75251i			0.0986821i
$779$	9.36488			0.335532
$780$	0.677993	+	4.82564i	0.0242760	+	0.172785i
$781$	−0.428639			−0.0153379
$782$			25.8983i			0.926121i
$783$			1.80642i			0.0645563i
$784$	45.5531			1.62690
$785$	44.0830	−	6.19358i	1.57339	−	0.221058i

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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.21432i q^{2} +1.00000i q^{3} +0.525428 q^{4} +(0.311108 + 2.21432i) q^{5} +1.21432 q^{6} +4.90321i q^{7} -3.06668i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.21432i q^{2} +1.00000i q^{3} +0.525428 q^{4} +(0.311108 + 2.21432i) q^{5} +1.21432 q^{6} +4.90321i q^{7} -3.06668i q^{8} -1.00000 q^{9} +(2.68889 - 0.377784i) q^{10} -1.00000 q^{11} +0.525428i q^{12} -4.14764i q^{13} +5.95407 q^{14} +(-2.21432 + 0.311108i) q^{15} -2.67307 q^{16} -5.33185i q^{17} +1.21432i q^{18} +5.18421 q^{19} +(0.163465 + 1.16346i) q^{20} -4.90321 q^{21} +1.21432i q^{22} -4.00000i q^{23} +3.06668 q^{24} +(-4.80642 + 1.37778i) q^{25} -5.03657 q^{26} -1.00000i q^{27} +2.57628i q^{28} -1.80642 q^{29} +(0.377784 + 2.68889i) q^{30} +2.62222 q^{31} -2.88739i q^{32} -1.00000i q^{33} -6.47457 q^{34} +(-10.8573 + 1.52543i) q^{35} -0.525428 q^{36} +5.80642i q^{37} -6.29529i q^{38} +4.14764 q^{39} +(6.79060 - 0.954067i) q^{40} +1.80642 q^{41} +5.95407i q^{42} -4.90321i q^{43} -0.525428 q^{44} +(-0.311108 - 2.21432i) q^{45} -4.85728 q^{46} +7.05086i q^{47} -2.67307i q^{48} -17.0415 q^{49} +(1.67307 + 5.83654i) q^{50} +5.33185 q^{51} -2.17929i q^{52} +7.18421i q^{53} -1.21432 q^{54} +(-0.311108 - 2.21432i) q^{55} +15.0366 q^{56} +5.18421i q^{57} +2.19358i q^{58} -1.67307 q^{59} +(-1.16346 + 0.163465i) q^{60} +0.755569 q^{61} -3.18421i q^{62} -4.90321i q^{63} -8.85236 q^{64} +(9.18421 - 1.29036i) q^{65} -1.21432 q^{66} -4.85728i q^{67} -2.80150i q^{68} +4.00000 q^{69} +(1.85236 + 13.1842i) q^{70} +0.428639 q^{71} +3.06668i q^{72} -12.7096i q^{73} +7.05086 q^{74} +(-1.37778 - 4.80642i) q^{75} +2.72393 q^{76} -4.90321i q^{77} -5.03657i q^{78} +6.42864 q^{79} +(-0.831613 - 5.91903i) q^{80} +1.00000 q^{81} -2.19358i q^{82} +2.90321i q^{83} -2.57628 q^{84} +(11.8064 - 1.65878i) q^{85} -5.95407 q^{86} -1.80642i q^{87} +3.06668i q^{88} -0.622216 q^{89} +(-2.68889 + 0.377784i) q^{90} +20.3368 q^{91} -2.10171i q^{92} +2.62222i q^{93} +8.56199 q^{94} +(1.61285 + 11.4795i) q^{95} +2.88739 q^{96} -2.75557i q^{97} +20.6938i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9	\( 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} + 16 q^{10} - 6 q^{11} - 4 q^{14} + 10 q^{16} + 4 q^{19} + 14 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 16 q^{26} + 16 q^{29} + 2 q^{30} + 16 q^{31} - 52 q^{34} - 12 q^{35} + 10 q^{36} + 12 q^{39} - 12 q^{40} - 16 q^{41} + 10 q^{44} - 2 q^{45} + 24 q^{46} - 22 q^{49} - 16 q^{50} - 8 q^{51} + 6 q^{54} - 2 q^{55} + 76 q^{56} + 16 q^{59} - 20 q^{60} + 4 q^{61} - 66 q^{64} + 28 q^{65} + 6 q^{66} + 24 q^{69} + 24 q^{70} - 24 q^{71} + 16 q^{74} - 8 q^{75} - 36 q^{76} + 12 q^{79} - 58 q^{80} + 6 q^{81} + 24 q^{84} + 44 q^{85} + 4 q^{86} - 4 q^{89} - 16 q^{90} + 16 q^{91} + 24 q^{94} - 44 q^{95} - 22 q^{96} + 6 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 + 16 * q^10 - 6 * q^11 - 4 * q^14 + 10 * q^16 + 4 * q^19 + 14 * q^20 - 16 * q^21 + 18 * q^24 - 2 * q^25 - 16 * q^26 + 16 * q^29 + 2 * q^30 + 16 * q^31 - 52 * q^34 - 12 * q^35 + 10 * q^36 + 12 * q^39 - 12 * q^40 - 16 * q^41 + 10 * q^44 - 2 * q^45 + 24 * q^46 - 22 * q^49 - 16 * q^50 - 8 * q^51 + 6 * q^54 - 2 * q^55 + 76 * q^56 + 16 * q^59 - 20 * q^60 + 4 * q^61 - 66 * q^64 + 28 * q^65 + 6 * q^66 + 24 * q^69 + 24 * q^70 - 24 * q^71 + 16 * q^74 - 8 * q^75 - 36 * q^76 + 12 * q^79 - 58 * q^80 + 6 * q^81 + 24 * q^84 + 44 * q^85 + 4 * q^86 - 4 * q^89 - 16 * q^90 + 16 * q^91 + 24 * q^94 - 44 * q^95 - 22 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

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