LMFDB - Embedded newform 370.2.b.d.149.6

Newspace parameters

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 $ x^10 - 2x^9 + 2x^8 - 4x^7 + 51x^6 - 124x^5 + 154x^4 - 46x^3 + x^2 + 4x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.6
Root			$-0.282359 - 0.282359i$ of defining polynomial
Character	$\chi$	$=$	370.149
Dual form			370.2.b.d.149.5

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} -2.62545i q^{3} -1.00000 q^{4} +(1.70518 + 1.44650i) q^{5} +2.62545 q^{6} -1.83227i q^{7} -1.00000i q^{8} -3.89300 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.62545i q^{3} -1.00000 q^{4} +(1.70518 + 1.44650i) q^{5} +2.62545 q^{6} -1.83227i q^{7} -1.00000i q^{8} -3.89300 q^{9} +(-1.44650 + 1.70518i) q^{10} +4.19017 q^{11} +2.62545i q^{12} +0.369454i q^{13} +1.83227 q^{14} +(3.79772 - 4.47687i) q^{15} +1.00000 q^{16} -5.08317i q^{17} -3.89300i q^{18} -3.55963 q^{19} +(-1.70518 - 1.44650i) q^{20} -4.81053 q^{21} +4.19017i q^{22} -5.62036i q^{23} -2.62545 q^{24} +(0.815273 + 4.93309i) q^{25} -0.369454 q^{26} +2.34453i q^{27} +1.83227i q^{28} -1.20681 q^{29} +(4.47687 + 3.79772i) q^{30} +10.1030 q^{31} +1.00000i q^{32} -11.0011i q^{33} +5.08317 q^{34} +(2.65038 - 3.12434i) q^{35} +3.89300 q^{36} +1.00000i q^{37} -3.55963i q^{38} +0.969984 q^{39} +(1.44650 - 1.70518i) q^{40} -8.01447 q^{41} -4.81053i q^{42} +2.27264i q^{43} -4.19017 q^{44} +(-6.63826 - 5.63123i) q^{45} +5.62036 q^{46} +10.9154i q^{47} -2.62545i q^{48} +3.64280 q^{49} +(-4.93309 + 0.815273i) q^{50} -13.3456 q^{51} -0.369454i q^{52} +9.94355i q^{53} -2.34453 q^{54} +(7.14499 + 6.06108i) q^{55} -1.83227 q^{56} +9.34563i q^{57} -1.20681i q^{58} +5.34563 q^{59} +(-3.79772 + 4.47687i) q^{60} -9.79428 q^{61} +10.1030i q^{62} +7.13302i q^{63} -1.00000 q^{64} +(-0.534415 + 0.629985i) q^{65} +11.0011 q^{66} +1.85073i q^{67} +5.08317i q^{68} -14.7560 q^{69} +(3.12434 + 2.65038i) q^{70} +2.86038 q^{71} +3.89300i q^{72} +8.09942i q^{73} -1.00000 q^{74} +(12.9516 - 2.14046i) q^{75} +3.55963 q^{76} -7.67751i q^{77} +0.969984i q^{78} -6.06361 q^{79} +(1.70518 + 1.44650i) q^{80} -5.52355 q^{81} -8.01447i q^{82} +8.93200i q^{83} +4.81053 q^{84} +(7.35281 - 8.66772i) q^{85} -2.27264 q^{86} +3.16843i q^{87} -4.19017i q^{88} -11.4773 q^{89} +(5.63123 - 6.63826i) q^{90} +0.676938 q^{91} +5.62036i q^{92} -26.5249i q^{93} -10.9154 q^{94} +(-6.06980 - 5.14900i) q^{95} +2.62545 q^{96} +6.05864i q^{97} +3.64280i q^{98} -16.3123 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9}+O(q^{10}) $ 10 * q - 10 * q^4 + 6 * q^5 - 6 * q^9	$ 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9} + 2 q^{10} + 6 q^{11} + 2 q^{14} + 10 q^{16} - 8 q^{19} - 6 q^{20} + 32 q^{21} + 4 q^{25} - 12 q^{26} - 22 q^{29} + 20 q^{30} + 46 q^{31} - 18 q^{34} + 32 q^{35} + 6 q^{36} - 40 q^{39} - 2 q^{40} - 14 q^{41} - 6 q^{44} + 2 q^{45} + 12 q^{46} - 60 q^{49} + 8 q^{50} - 40 q^{51} + 42 q^{55} - 2 q^{56} - 40 q^{59} - 18 q^{61} - 10 q^{64} + 4 q^{65} + 40 q^{66} - 32 q^{69} - 6 q^{70} + 12 q^{71} - 10 q^{74} + 50 q^{75} + 8 q^{76} - 40 q^{79} + 6 q^{80} - 14 q^{81} - 32 q^{84} + 36 q^{85} - 34 q^{86} - 24 q^{89} + 44 q^{90} + 32 q^{91} - 24 q^{94} + 12 q^{95} + 22 q^{99}+O(q^{100}) $ 10 * q - 10 * q^4 + 6 * q^5 - 6 * q^9 + 2 * q^10 + 6 * q^11 + 2 * q^14 + 10 * q^16 - 8 * q^19 - 6 * q^20 + 32 * q^21 + 4 * q^25 - 12 * q^26 - 22 * q^29 + 20 * q^30 + 46 * q^31 - 18 * q^34 + 32 * q^35 + 6 * q^36 - 40 * q^39 - 2 * q^40 - 14 * q^41 - 6 * q^44 + 2 * q^45 + 12 * q^46 - 60 * q^49 + 8 * q^50 - 40 * q^51 + 42 * q^55 - 2 * q^56 - 40 * q^59 - 18 * q^61 - 10 * q^64 + 4 * q^65 + 40 * q^66 - 32 * q^69 - 6 * q^70 + 12 * q^71 - 10 * q^74 + 50 * q^75 + 8 * q^76 - 40 * q^79 + 6 * q^80 - 14 * q^81 - 32 * q^84 + 36 * q^85 - 34 * q^86 - 24 * q^89 + 44 * q^90 + 32 * q^91 - 24 * q^94 + 12 * q^95 + 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$.

$n$	$261$	$297$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	2.62545i		−	1.51581i	−0.652367	−	0.757903i	$-0.726224\pi$
$3$		−	2.62545i		−	1.51581i	0.652367	−	0.757903i	$-0.273776\pi$
$4$	−1.00000			−0.500000
$5$	1.70518	+	1.44650i	0.762579	+	0.646895i
$5$	1.70518	+	1.44650i	0.762579	+	0.646895i
$6$	2.62545			1.07184
$7$		−	1.83227i		−	0.692532i	−0.938136	−	0.346266i	$-0.887449\pi$
$7$		−	1.83227i		−	0.692532i	0.938136	−	0.346266i	$-0.112551\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−3.89300			−1.29767
$10$	−1.44650	+	1.70518i	−0.457424	+	0.539225i
$11$	4.19017			1.26338			0.631692	−	0.775219i	$-0.282361\pi$
$11$	4.19017			1.26338			0.631692	+	0.775219i	$0.282361\pi$
$12$			2.62545i			0.757903i
$13$			0.369454i			0.102468i	0.998687	+	0.0512340i	$0.0163154\pi$
$13$			0.369454i			0.102468i	−0.998687	+	0.0512340i	$0.983685\pi$
$14$	1.83227			0.489694
$15$	3.79772	−	4.47687i	0.980567	−	1.15592i
$16$	1.00000			0.250000
$17$		−	5.08317i		−	1.23285i	−0.787414	−	0.616425i	$-0.788580\pi$
$17$		−	5.08317i		−	1.23285i	0.787414	−	0.616425i	$-0.211420\pi$
$18$		−	3.89300i		−	0.917589i
$19$	−3.55963			−0.816634			−0.408317	−	0.912840i	$-0.633884\pi$
$19$	−3.55963			−0.816634			−0.408317	+	0.912840i	$0.633884\pi$
$20$	−1.70518	−	1.44650i	−0.381290	−	0.323447i
$21$	−4.81053			−1.04974
$22$			4.19017i			0.893348i
$23$		−	5.62036i		−	1.17193i	−0.810338	−	0.585963i	$-0.800716\pi$
$23$		−	5.62036i		−	1.17193i	0.810338	−	0.585963i	$-0.199284\pi$
$24$	−2.62545			−0.535918
$25$	0.815273	+	4.93309i	0.163055	+	0.986617i
$26$	−0.369454			−0.0724559
$27$			2.34453i			0.451205i
$28$			1.83227i			0.346266i
$29$	−1.20681			−0.224100			−0.112050	−	0.993703i	$-0.535742\pi$
$29$	−1.20681			−0.224100			−0.112050	+	0.993703i	$0.535742\pi$
$30$	4.47687	+	3.79772i	0.817360	+	0.693365i
$31$	10.1030			1.81455			0.907276	−	0.420535i	$-0.138158\pi$
$31$	10.1030			1.81455			0.907276	+	0.420535i	$0.138158\pi$
$32$			1.00000i			0.176777i
$33$		−	11.0011i		−	1.91504i
$34$	5.08317			0.871757
$35$	2.65038	−	3.12434i	0.447995	−	0.528111i
$36$	3.89300			0.648833
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		−	3.55963i		−	0.577447i
$39$	0.969984			0.155322
$40$	1.44650	−	1.70518i	0.228712	−	0.269613i
$41$	−8.01447			−1.25165			−0.625825	−	0.779964i	$-0.715238\pi$
$41$	−8.01447			−1.25165			−0.625825	+	0.779964i	$0.715238\pi$
$42$		−	4.81053i		−	0.742281i
$43$			2.27264i			0.346575i	0.984871	+	0.173287i	$0.0554389\pi$
$43$			2.27264i			0.346575i	−0.984871	+	0.173287i	$0.944561\pi$
$44$	−4.19017			−0.631692
$45$	−6.63826	−	5.63123i	−0.989574	−	0.839454i
$46$	5.62036			0.828677
$47$			10.9154i			1.59218i	0.605178	+	0.796090i	$0.293102\pi$
$47$			10.9154i			1.59218i	−0.605178	+	0.796090i	$0.706898\pi$
$48$		−	2.62545i		−	0.378951i
$49$	3.64280			0.520400
$50$	−4.93309	+	0.815273i	−0.697644	+	0.115297i
$51$	−13.3456			−1.86876
$52$		−	0.369454i		−	0.0512340i
$53$			9.94355i			1.36585i	0.730488	+	0.682926i	$0.239293\pi$
$53$			9.94355i			1.36585i	−0.730488	+	0.682926i	$0.760707\pi$
$54$	−2.34453			−0.319050
$55$	7.14499	+	6.06108i	0.963431	+	0.817276i
$56$	−1.83227			−0.244847
$57$			9.34563i			1.23786i
$58$		−	1.20681i		−	0.158463i
$59$	5.34563			0.695941			0.347971	−	0.937505i	$-0.386871\pi$
$59$	5.34563			0.695941			0.347971	+	0.937505i	$0.386871\pi$
$60$	−3.79772	+	4.47687i	−0.490283	+	0.577961i
$61$	−9.79428			−1.25403			−0.627015	−	0.779008i	$-0.715723\pi$
$61$	−9.79428			−1.25403			−0.627015	+	0.779008i	$0.715723\pi$
$62$			10.1030i			1.28308i
$63$			7.13302i			0.898676i
$64$	−1.00000			−0.125000
$65$	−0.534415	+	0.629985i	−0.0662861	+	0.0781401i
$66$	11.0011			1.35414
$67$			1.85073i			0.226103i	0.993589	+	0.113052i	$0.0360625\pi$
$67$			1.85073i			0.226103i	−0.993589	+	0.113052i	$0.963937\pi$
$68$			5.08317i			0.616425i
$69$	−14.7560			−1.77641
$70$	3.12434	+	2.65038i	0.373431	+	0.316780i
$71$	2.86038			0.339464			0.169732	−	0.985490i	$-0.445710\pi$
$71$	2.86038			0.339464			0.169732	+	0.985490i	$0.445710\pi$
$72$			3.89300i			0.458795i
$73$			8.09942i			0.947966i	0.880534	+	0.473983i	$0.157184\pi$
$73$			8.09942i			0.947966i	−0.880534	+	0.473983i	$0.842816\pi$
$74$	−1.00000			−0.116248
$75$	12.9516	−	2.14046i	1.49552	−	0.247159i
$76$	3.55963			0.408317
$77$		−	7.67751i		−	0.874934i
$78$			0.969984i			0.109829i
$79$	−6.06361			−0.682209			−0.341105	−	0.940025i	$-0.610801\pi$
$79$	−6.06361			−0.682209			−0.341105	+	0.940025i	$0.610801\pi$
$80$	1.70518	+	1.44650i	0.190645	+	0.161724i
$81$	−5.52355			−0.613727
$82$		−	8.01447i		−	0.885050i
$83$			8.93200i			0.980414i	0.871606	+	0.490207i	$0.163079\pi$
$83$			8.93200i			0.980414i	−0.871606	+	0.490207i	$0.836921\pi$
$84$	4.81053			0.524872
$85$	7.35281	−	8.66772i	0.797524	−	0.940146i
$86$	−2.27264			−0.245065
$87$			3.16843i			0.339692i
$88$		−	4.19017i		−	0.446674i
$89$	−11.4773			−1.21659			−0.608295	−	0.793711i	$-0.708146\pi$
$89$	−11.4773			−1.21659			−0.608295	+	0.793711i	$0.708146\pi$
$90$	5.63123	−	6.63826i	0.593583	−	0.699735i
$91$	0.676938			0.0709624
$92$			5.62036i			0.585963i
$93$		−	26.5249i		−	2.75051i
$94$	−10.9154			−1.12584
$95$	−6.06980	−	5.14900i	−0.622748	−	0.528276i
$96$	2.62545			0.267959
$97$			6.05864i			0.615162i	0.951522	+	0.307581i	$0.0995195\pi$
$97$			6.05864i			0.615162i	−0.951522	+	0.307581i	$0.900480\pi$
$98$			3.64280i			0.367978i
$99$	−16.3123			−1.63945
$100$	−0.815273	−	4.93309i	−0.0815273	−	0.493309i
$101$	4.96528			0.494064			0.247032	−	0.969007i	$-0.420545\pi$
$101$	4.96528			0.494064			0.247032	+	0.969007i	$0.420545\pi$
$102$		−	13.3456i		−	1.32141i
$103$			7.33338i			0.722579i	0.932454	+	0.361289i	$0.117663\pi$
$103$			7.33338i			0.722579i	−0.932454	+	0.361289i	$0.882337\pi$
$104$	0.369454			0.0362279
$105$	−8.20282	−	6.95843i	−0.800513	−	0.679074i
$106$	−9.94355			−0.965803
$107$		−	8.21182i		−	0.793867i	−0.917847	−	0.396933i	$-0.870074\pi$
$107$		−	8.21182i		−	0.793867i	0.917847	−	0.396933i	$-0.129926\pi$
$108$		−	2.34453i		−	0.225603i
$109$	20.5503			1.96836			0.984179	−	0.177175i	$-0.0566960\pi$
$109$	20.5503			1.96836			0.984179	+	0.177175i	$0.0566960\pi$
$110$	−6.06108	+	7.14499i	−0.577902	+	0.681248i
$111$	2.62545			0.249197
$112$		−	1.83227i		−	0.173133i
$113$			0.652984i			0.0614276i	0.999528	+	0.0307138i	$0.00977804\pi$
$113$			0.652984i			0.0614276i	−0.999528	+	0.0307138i	$0.990222\pi$
$114$	−9.34563			−0.875298
$115$	8.12985	−	9.58372i	0.758113	−	0.893686i
$116$	1.20681			0.112050
$117$		−	1.43828i		−	0.132969i
$118$			5.34563i			0.492105i
$119$	−9.31373			−0.853788
$120$	−4.47687	−	3.79772i	−0.408680	−	0.346683i
$121$	6.55754			0.596140
$122$		−	9.79428i		−	0.886733i
$123$			21.0416i			1.89726i
$124$	−10.1030			−0.907276
$125$	−5.74552	+	9.59109i	−0.513895	+	0.857853i
$126$	−7.13302			−0.635460
$127$		−	19.2856i		−	1.71132i	−0.517539	−	0.855660i	$-0.673152\pi$
$127$		−	19.2856i		−	1.71132i	0.517539	−	0.855660i	$-0.326848\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	5.96671			0.525340
$130$	−0.629985	−	0.534415i	−0.0552534	−	0.0468713i
$131$	−8.82072			−0.770670			−0.385335	−	0.922777i	$-0.625914\pi$
$131$	−8.82072			−0.770670			−0.385335	+	0.922777i	$0.625914\pi$
$132$			11.0011i			0.957522i
$133$			6.52218i			0.565545i
$134$	−1.85073			−0.159879
$135$	−3.39137	+	3.99785i	−0.291882	+	0.344080i
$136$	−5.08317			−0.435878
$137$			4.66126i			0.398239i	0.979975	+	0.199119i	$0.0638081\pi$
$137$			4.66126i			0.398239i	−0.979975	+	0.199119i	$0.936192\pi$
$138$		−	14.7560i		−	1.25611i
$139$	−16.4050			−1.39145			−0.695727	−	0.718306i	$-0.744918\pi$
$139$	−16.4050			−1.39145			−0.695727	+	0.718306i	$0.744918\pi$
$140$	−2.65038	+	3.12434i	−0.223998	+	0.264055i
$141$	28.6580			2.41344
$142$			2.86038i			0.240037i
$143$			1.54808i			0.129457i
$144$	−3.89300			−0.324417
$145$	−2.05784	−	1.74566i	−0.170894	−	0.144969i
$146$	−8.09942			−0.670313
$147$		−	9.56399i		−	0.788825i
$148$		−	1.00000i		−	0.0821995i
$149$	5.86259			0.480282			0.240141	−	0.970738i	$-0.422806\pi$
$149$	5.86259			0.480282			0.240141	+	0.970738i	$0.422806\pi$
$150$	2.14046	+	12.9516i	0.174768	+	1.05749i
$151$	13.1460			1.06981			0.534903	−	0.844913i	$-0.320348\pi$
$151$	13.1460			1.06981			0.534903	+	0.844913i	$0.320348\pi$
$152$			3.55963i			0.288724i
$153$			19.7888i			1.59983i
$154$	7.67751			0.618672
$155$	17.2274	+	14.6140i	1.38374	+	1.17382i
$156$	−0.969984			−0.0776609
$157$		−	7.33629i		−	0.585500i	−0.956189	−	0.292750i	$-0.905430\pi$
$157$		−	7.33629i		−	0.585500i	0.956189	−	0.292750i	$-0.0945704\pi$
$158$		−	6.06361i		−	0.482395i
$159$	26.1063			2.07037
$160$	−1.44650	+	1.70518i	−0.114356	+	0.134806i
$161$	−10.2980			−0.811596
$162$		−	5.52355i		−	0.433971i
$163$			18.7968i			1.47228i	0.676831	+	0.736138i	$0.263353\pi$
$163$			18.7968i			1.47228i	−0.676831	+	0.736138i	$0.736647\pi$
$164$	8.01447			0.625825
$165$	15.9131	−	18.7588i	1.23883	−	1.46037i
$166$	−8.93200			−0.693257
$167$		−	5.92772i		−	0.458700i	−0.973344	−	0.229350i	$-0.926340\pi$
$167$		−	5.92772i		−	0.458700i	0.973344	−	0.229350i	$-0.0736601\pi$
$168$			4.81053i			0.371140i
$169$	12.8635			0.989500
$170$	8.66772	+	7.35281i	0.664784	+	0.563935i
$171$	13.8576			1.05972
$172$		−	2.27264i		−	0.173287i
$173$		−	5.78242i		−	0.439629i	−0.975542	−	0.219815i	$-0.929455\pi$
$173$		−	5.78242i		−	0.439629i	0.975542	−	0.219815i	$-0.0705453\pi$
$174$	−3.16843			−0.240198
$175$	9.03873	−	1.49380i	0.683264	−	0.112921i
$176$	4.19017			0.315846
$177$		−	14.0347i		−	1.05491i
$178$		−	11.4773i		−	0.860259i
$179$	−7.13581			−0.533355			−0.266678	−	0.963786i	$-0.585926\pi$
$179$	−7.13581			−0.533355			−0.266678	+	0.963786i	$0.585926\pi$
$180$	6.63826	+	5.63123i	0.494787	+	0.419727i
$181$	11.5365			0.857503			0.428752	−	0.903422i	$-0.358954\pi$
$181$	11.5365			0.857503			0.428752	+	0.903422i	$0.358954\pi$
$182$			0.676938i			0.0501780i
$183$			25.7144i			1.90086i
$184$	−5.62036			−0.414338
$185$	−1.44650	+	1.70518i	−0.106349	+	0.125367i
$186$	26.5249			1.94490
$187$		−	21.2994i		−	1.55756i
$188$		−	10.9154i		−	0.796090i
$189$	4.29581			0.312474
$190$	5.14900	−	6.06980i	0.373548	−	0.440350i
$191$	−20.6048			−1.49091			−0.745456	−	0.666555i	$-0.767768\pi$
$191$	−20.6048			−1.49091			−0.745456	+	0.666555i	$0.767768\pi$
$192$			2.62545i			0.189476i
$193$			26.1663i			1.88349i	0.336321	+	0.941747i	$0.390817\pi$
$193$			26.1663i			1.88349i	−0.336321	+	0.941747i	$0.609183\pi$
$194$	−6.05864			−0.434985
$195$	1.65400	+	1.40308i	0.118445	+	0.100477i
$196$	−3.64280			−0.260200
$197$			14.6067i			1.04069i	0.853957	+	0.520343i	$0.174196\pi$
$197$			14.6067i			1.04069i	−0.853957	+	0.520343i	$0.825804\pi$
$198$		−	16.3123i		−	1.15927i
$199$	−5.81275			−0.412055			−0.206027	−	0.978546i	$-0.566054\pi$
$199$	−5.81275			−0.412055			−0.206027	+	0.978546i	$0.566054\pi$
$200$	4.93309	−	0.815273i	0.348822	−	0.0576485i
$201$	4.85901			0.342728
$202$			4.96528i			0.349356i
$203$			2.21121i			0.155196i
$204$	13.3456			0.934381
$205$	−13.6661	−	11.5929i	−0.954482	−	0.809685i
$206$	−7.33338			−0.510940
$207$			21.8801i			1.52077i
$208$			0.369454i			0.0256170i
$209$	−14.9154			−1.03172
$210$	6.95843	−	8.20282i	0.480178	−	0.566048i
$211$	16.0976			1.10821			0.554104	−	0.832448i	$-0.313061\pi$
$211$	16.0976			1.10821			0.554104	+	0.832448i	$0.313061\pi$
$212$		−	9.94355i		−	0.682926i
$213$		−	7.50978i		−	0.514562i
$214$	8.21182			0.561349
$215$	−3.28738	+	3.87526i	−0.224197	+	0.264291i
$216$	2.34453			0.159525
$217$		−	18.5114i		−	1.25664i
$218$			20.5503i			1.39184i
$219$	21.2646			1.43693
$220$	−7.14499	−	6.06108i	−0.481715	−	0.408638i
$221$	1.87800			0.126328
$222$			2.62545i			0.176209i
$223$		−	25.3608i		−	1.69828i	−0.528164	−	0.849142i	$-0.677119\pi$
$223$		−	25.3608i		−	1.69828i	0.528164	−	0.849142i	$-0.322881\pi$
$224$	1.83227			0.122423
$225$	−3.17386	−	19.2045i	−0.211591	−	1.28030i
$226$	−0.652984			−0.0434359
$227$		−	5.57339i		−	0.369919i	−0.982746	−	0.184960i	$-0.940785\pi$
$227$		−	5.57339i		−	0.369919i	0.982746	−	0.184960i	$-0.0592154\pi$
$228$		−	9.34563i		−	0.618929i
$229$	−9.02035			−0.596081			−0.298041	−	0.954553i	$-0.596333\pi$
$229$	−9.02035			−0.596081			−0.298041	+	0.954553i	$0.596333\pi$
$230$	9.58372	+	8.12985i	0.631932	+	0.536067i
$231$	−20.1569			−1.32623
$232$			1.20681i			0.0792313i
$233$		−	11.8100i		−	0.773696i	−0.922144	−	0.386848i	$-0.873564\pi$
$233$		−	11.8100i		−	0.773696i	0.922144	−	0.386848i	$-0.126436\pi$
$234$	1.43828			0.0940236
$235$	−15.7892	+	18.6128i	−1.02997	+	1.21416i
$236$	−5.34563			−0.347971
$237$			15.9197i			1.03410i
$238$		−	9.31373i		−	0.603719i
$239$	−0.907160			−0.0586793			−0.0293396	−	0.999569i	$-0.509340\pi$
$239$	−0.907160			−0.0586793			−0.0293396	+	0.999569i	$0.509340\pi$
$240$	3.79772	−	4.47687i	0.245142	−	0.288981i
$241$	−4.15616			−0.267722			−0.133861	−	0.991000i	$-0.542738\pi$
$241$	−4.15616			−0.267722			−0.133861	+	0.991000i	$0.542738\pi$
$242$			6.55754i			0.421534i
$243$			21.5354i			1.38150i
$244$	9.79428			0.627015
$245$	6.21162	+	5.26931i	0.396846	+	0.336644i
$246$	−21.0416			−1.34156
$247$		−	1.31512i		−	0.0836789i
$248$		−	10.1030i		−	0.641541i
$249$	23.4505			1.48612
$250$	−9.59109	−	5.74552i	−0.606594	−	0.363379i
$251$	9.31888			0.588203			0.294101	−	0.955774i	$-0.404980\pi$
$251$	9.31888			0.588203			0.294101	+	0.955774i	$0.404980\pi$
$252$		−	7.13302i		−	0.449338i
$253$		−	23.5503i		−	1.48059i
$254$	19.2856			1.21009
$255$	−22.7567	−	19.3045i	−1.42508	−	1.20889i
$256$	1.00000			0.0625000
$257$		−	13.6312i		−	0.850294i	−0.905124	−	0.425147i	$-0.860222\pi$
$257$		−	13.6312i		−	0.850294i	0.905124	−	0.425147i	$-0.139778\pi$
$258$			5.96671i			0.371471i
$259$	1.83227			0.113852
$260$	0.534415	−	0.629985i	0.0331430	−	0.0390700i
$261$	4.69813			0.290807
$262$		−	8.82072i		−	0.544946i
$263$		−	27.2170i		−	1.67827i	−0.543921	−	0.839137i	$-0.683061\pi$
$263$		−	27.2170i		−	1.67827i	0.543921	−	0.839137i	$-0.316939\pi$
$264$	−11.0011			−0.677071
$265$	−14.3833	+	16.9555i	−0.883562	+	1.04157i
$266$	−6.52218			−0.399901
$267$			30.1331i			1.84411i
$268$		−	1.85073i		−	0.113052i
$269$	−23.0550			−1.40569			−0.702845	−	0.711343i	$-0.748087\pi$
$269$	−23.0550			−1.40569			−0.702845	+	0.711343i	$0.748087\pi$
$270$	−3.99785	−	3.39137i	−0.243301	−	0.206392i
$271$	22.9320			1.39302			0.696510	−	0.717547i	$-0.254735\pi$
$271$	22.9320			1.39302			0.696510	+	0.717547i	$0.254735\pi$
$272$		−	5.08317i		−	0.308213i
$273$		−	1.77727i		−	0.107565i
$274$	−4.66126			−0.281597
$275$	3.41613	+	20.6705i	0.206001	+	1.24648i
$276$	14.7560			0.888206
$277$		−	11.7534i		−	0.706192i	−0.935587	−	0.353096i	$-0.885129\pi$
$277$		−	11.7534i		−	0.706192i	0.935587	−	0.353096i	$-0.114871\pi$
$278$		−	16.4050i		−	0.983906i
$279$	−39.3310			−2.35468
$280$	−3.12434	−	2.65038i	−0.186715	−	0.158390i
$281$	−29.0962			−1.73573			−0.867865	−	0.496799i	$-0.834508\pi$
$281$	−29.0962			−1.73573			−0.867865	+	0.496799i	$0.834508\pi$
$282$			28.6580i			1.70656i
$283$			10.1829i			0.605311i	0.953100	+	0.302655i	$0.0978731\pi$
$283$			10.1829i			0.605311i	−0.953100	+	0.302655i	$0.902127\pi$
$284$	−2.86038			−0.169732
$285$	−13.5185	+	15.9360i	−0.800764	+	0.943965i
$286$	−1.54808			−0.0915396
$287$			14.6846i			0.866807i
$288$		−	3.89300i		−	0.229397i
$289$	−8.83864			−0.519920
$290$	1.74566	−	2.05784i	0.102509	−	0.120840i
$291$	15.9067			0.932466
$292$		−	8.09942i		−	0.473983i
$293$		−	18.1243i		−	1.05883i	−0.848363	−	0.529415i	$-0.822411\pi$
$293$		−	18.1243i		−	1.05883i	0.848363	−	0.529415i	$-0.177589\pi$
$294$	9.56399			0.557783
$295$	9.11525	+	7.73245i	0.530711	+	0.450201i
$296$	1.00000			0.0581238
$297$			9.82399i			0.570046i
$298$			5.86259i			0.339611i
$299$	2.07646			0.120085
$300$	−12.9516	+	2.14046i	−0.747760	+	0.123580i
$301$	4.16409			0.240014
$302$			13.1460i			0.756467i
$303$		−	13.0361i		−	0.748905i
$304$	−3.55963			−0.204159
$305$	−16.7010	−	14.1674i	−0.956297	−	0.811225i
$306$	−19.7888			−1.13125
$307$		−	6.15561i		−	0.351319i	−0.984451	−	0.175660i	$-0.943794\pi$
$307$		−	6.15561i		−	0.351319i	0.984451	−	0.175660i	$-0.0562058\pi$
$308$			7.67751i			0.437467i
$309$	19.2534			1.09529
$310$	−14.6140	+	17.2274i	−0.830019	+	0.978452i
$311$	−29.4512			−1.67002			−0.835011	−	0.550234i	$-0.814539\pi$
$311$	−29.4512			−1.67002			−0.835011	+	0.550234i	$0.814539\pi$
$312$		−	0.969984i		−	0.0549145i
$313$		−	13.0102i		−	0.735378i	−0.929949	−	0.367689i	$-0.880149\pi$
$313$		−	13.0102i		−	0.735378i	0.929949	−	0.367689i	$-0.119851\pi$
$314$	7.33629			0.414011
$315$	−10.3179	+	12.1631i	−0.581349	+	0.685312i
$316$	6.06361			0.341105
$317$		−	7.39826i		−	0.415528i	−0.978179	−	0.207764i	$-0.933381\pi$
$317$		−	7.39826i		−	0.415528i	0.978179	−	0.207764i	$-0.0666186\pi$
$318$			26.1063i			1.46397i
$319$	−5.05676			−0.283124
$320$	−1.70518	−	1.44650i	−0.0953224	−	0.0808618i
$321$	−21.5598			−1.20335
$322$		−	10.2980i		−	0.573885i
$323$			18.0942i			1.00679i
$324$	5.52355			0.306864
$325$	−1.82255	+	0.301206i	−0.101097	+	0.0167079i
$326$	−18.7968			−1.04106
$327$		−	53.9537i		−	2.98365i
$328$			8.01447i			0.442525i
$329$	20.0000			1.10264
$330$	18.7588	+	15.9131i	1.03264	+	0.875987i
$331$	33.2545			1.82783			0.913916	−	0.405903i	$-0.133043\pi$
$331$	33.2545			1.82783			0.913916	+	0.405903i	$0.133043\pi$
$332$		−	8.93200i		−	0.490207i
$333$		−	3.89300i		−	0.213335i
$334$	5.92772			0.324350
$335$	−2.67709	+	3.15583i	−0.146265	+	0.172422i
$336$	−4.81053			−0.262436
$337$			22.6288i			1.23267i	0.787485	+	0.616334i	$0.211383\pi$
$337$			22.6288i			1.23267i	−0.787485	+	0.616334i	$0.788617\pi$
$338$			12.8635i			0.699682i
$339$	1.71438			0.0931123
$340$	−7.35281	+	8.66772i	−0.398762	+	0.470073i
$341$	42.3333			2.29248
$342$			13.8576i			0.749334i
$343$		−	19.5004i		−	1.05293i
$344$	2.27264			0.122533
$345$	−25.1616	−	21.3445i	−1.35466	−	1.14915i
$346$	5.78242			0.310865
$347$		−	24.1396i		−	1.29588i	−0.761691	−	0.647941i	$-0.775630\pi$
$347$		−	24.1396i		−	1.29588i	0.761691	−	0.647941i	$-0.224370\pi$
$348$		−	3.16843i		−	0.169846i
$349$	10.3702			0.555102			0.277551	−	0.960711i	$-0.410477\pi$
$349$	10.3702			0.555102			0.277551	+	0.960711i	$0.410477\pi$
$350$	1.49380	+	9.03873i	0.0798469	+	0.483140i
$351$	−0.866196			−0.0462341
$352$			4.19017i			0.223337i
$353$			3.92699i			0.209013i	0.994524	+	0.104506i	$0.0333262\pi$
$353$			3.92699i			0.209013i	−0.994524	+	0.104506i	$0.966674\pi$
$354$	14.0347			0.745935
$355$	4.87745	+	4.13753i	0.258868	+	0.219598i
$356$	11.4773			0.608295
$357$			24.4528i			1.29418i
$358$		−	7.13581i		−	0.377139i
$359$	9.85179			0.519957			0.259979	−	0.965614i	$-0.416284\pi$
$359$	9.85179			0.519957			0.259979	+	0.965614i	$0.416284\pi$
$360$	−5.63123	+	6.63826i	−0.296792	+	0.349867i
$361$	−6.32907			−0.333109
$362$			11.5365i			0.606346i
$363$		−	17.2165i		−	0.903632i
$364$	−0.676938			−0.0354812
$365$	−11.7158	+	13.8110i	−0.613234	+	0.722899i
$366$	−25.7144			−1.34411
$367$			33.2559i			1.73594i	0.496614	+	0.867972i	$0.334577\pi$
$367$			33.2559i			1.73594i	−0.496614	+	0.867972i	$0.665423\pi$
$368$		−	5.62036i		−	0.292981i
$369$	31.2003			1.62422
$370$	−1.70518	−	1.44650i	−0.0886481	−	0.0752000i
$371$	18.2192			0.945896
$372$			26.5249i			1.37525i
$373$			28.5387i			1.47768i	0.673882	+	0.738839i	$0.264625\pi$
$373$			28.5387i			1.47768i	−0.673882	+	0.738839i	$0.735375\pi$
$374$	21.2994			1.10136
$375$	25.1809	+	15.0846i	1.30034	+	0.778965i
$376$	10.9154			0.562921
$377$		−	0.445862i		−	0.0229631i
$378$			4.29581i			0.220953i
$379$	−0.476554			−0.0244789			−0.0122395	−	0.999925i	$-0.503896\pi$
$379$	−0.476554			−0.0244789			−0.0122395	+	0.999925i	$0.503896\pi$
$380$	6.06980	+	5.14900i	0.311374	+	0.264138i
$381$	−50.6334			−2.59403
$382$		−	20.6048i		−	1.05423i
$383$		−	7.13962i		−	0.364818i	−0.983223	−	0.182409i	$-0.941611\pi$
$383$		−	7.13962i		−	0.364818i	0.983223	−	0.182409i	$-0.0583895\pi$
$384$	−2.62545			−0.133980
$385$	11.1055	−	13.0915i	0.565990	−	0.667206i
$386$	−26.1663			−1.33183
$387$		−	8.84740i		−	0.449738i
$388$		−	6.05864i		−	0.307581i
$389$	24.2060			1.22729			0.613646	−	0.789581i	$-0.289702\pi$
$389$	24.2060			1.22729			0.613646	+	0.789581i	$0.289702\pi$
$390$	−1.40308	+	1.65400i	−0.0710478	+	0.0837534i
$391$	−28.5693			−1.44481
$392$		−	3.64280i		−	0.183989i
$393$			23.1584i			1.16819i
$394$	−14.6067			−0.735876
$395$	−10.3395	−	8.77101i	−0.520239	−	0.441318i
$396$	16.3123			0.819726
$397$			20.7491i			1.04137i	0.853750	+	0.520684i	$0.174323\pi$
$397$			20.7491i			1.04137i	−0.853750	+	0.520684i	$0.825677\pi$
$398$		−	5.81275i		−	0.291367i
$399$	17.1237			0.857256
$400$	0.815273	+	4.93309i	0.0407637	+	0.246654i
$401$	−34.8410			−1.73988			−0.869939	−	0.493159i	$-0.835842\pi$
$401$	−34.8410			−1.73988			−0.869939	+	0.493159i	$0.835842\pi$
$402$			4.85901i			0.242346i
$403$			3.73259i			0.185934i
$404$	−4.96528			−0.247032
$405$	−9.41864	−	7.98981i	−0.468016	−	0.397017i
$406$	−2.21121			−0.109740
$407$			4.19017i			0.207699i
$408$			13.3456i			0.660707i
$409$	−25.2132			−1.24671			−0.623356	−	0.781938i	$-0.714231\pi$
$409$	−25.2132			−1.24671			−0.623356	+	0.781938i	$0.714231\pi$
$410$	11.5929	−	13.6661i	0.572534	−	0.674921i
$411$	12.2379			0.603652
$412$		−	7.33338i		−	0.361289i
$413$		−	9.79462i		−	0.481962i
$414$	−21.8801			−1.07535
$415$	−12.9201	+	15.2307i	−0.634225	+	0.747644i
$416$	−0.369454			−0.0181140
$417$			43.0705i			2.10917i
$418$		−	14.9154i		−	0.729538i
$419$	−5.69695			−0.278314			−0.139157	−	0.990270i	$-0.544439\pi$
$419$	−5.69695			−0.278314			−0.139157	+	0.990270i	$0.544439\pi$
$420$	8.20282	+	6.95843i	0.400256	+	0.339537i
$421$	3.38092			0.164776			0.0823880	−	0.996600i	$-0.473745\pi$
$421$	3.38092			0.164776			0.0823880	+	0.996600i	$0.473745\pi$
$422$			16.0976i			0.783621i
$423$		−	42.4938i		−	2.06612i
$424$	9.94355			0.482901
$425$	25.0757	−	4.14417i	1.21635	−	0.201022i
$426$	7.50978			0.363850
$427$			17.9457i			0.868455i
$428$			8.21182i			0.396933i
$429$	4.06440			0.196231
$430$	−3.87526	−	3.28738i	−0.186882	−	0.158531i
$431$	−0.381731			−0.0183873			−0.00919367	−	0.999958i	$-0.502926\pi$
$431$	−0.381731			−0.0183873			−0.00919367	+	0.999958i	$0.502926\pi$
$432$			2.34453i			0.112801i
$433$			21.6393i			1.03992i	0.854192	+	0.519958i	$0.174052\pi$
$433$			21.6393i			1.03992i	−0.854192	+	0.519958i	$0.825948\pi$
$434$	18.5114			0.888575
$435$	−4.58314	+	5.40275i	−0.219745	+	0.259042i
$436$	−20.5503			−0.984179
$437$			20.0064i			0.957035i
$438$			21.2646i			1.01606i
$439$	−26.4663			−1.26317			−0.631583	−	0.775308i	$-0.717595\pi$
$439$	−26.4663			−1.26317			−0.631583	+	0.775308i	$0.717595\pi$
$440$	6.06108	−	7.14499i	0.288951	−	0.340624i
$441$	−14.1814			−0.675305
$442$			1.87800i			0.0893273i
$443$			4.65489i			0.221161i	0.993867	+	0.110580i	$0.0352709\pi$
$443$			4.65489i			0.221161i	−0.993867	+	0.110580i	$0.964729\pi$
$444$	−2.62545			−0.124598
$445$	−19.5708	−	16.6019i	−0.927746	−	0.787005i
$446$	25.3608			1.20087
$447$		−	15.3920i		−	0.728015i
$448$			1.83227i			0.0865665i
$449$	26.5531			1.25312			0.626558	−	0.779375i	$-0.284463\pi$
$449$	26.5531			1.25312			0.626558	+	0.779375i	$0.284463\pi$
$450$	19.2045	−	3.17386i	0.905309	−	0.149617i
$451$	−33.5820			−1.58131
$452$		−	0.652984i		−	0.0307138i
$453$		−	34.5142i		−	1.62162i
$454$	5.57339			0.261572
$455$	1.15430	+	0.979192i	0.0541145	+	0.0459052i
$456$	9.34563			0.437649
$457$			4.27264i			0.199866i	0.994994	+	0.0999329i	$0.0318628\pi$
$457$			4.27264i			0.199866i	−0.994994	+	0.0999329i	$0.968137\pi$
$458$		−	9.02035i		−	0.421493i
$459$	11.9177			0.556269
$460$	−8.12985	+	9.58372i	−0.379056	+	0.446843i
$461$	32.1873			1.49911			0.749555	−	0.661942i	$-0.230267\pi$
$461$	32.1873			1.49911			0.749555	+	0.661942i	$0.230267\pi$
$462$		−	20.1569i		−	0.937786i
$463$		−	18.4014i		−	0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$463$		−	18.4014i		−	0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$464$	−1.20681			−0.0560250
$465$	38.3684	−	45.2298i	1.77929	−	2.09748i
$466$	11.8100			0.547086
$467$		−	3.06777i		−	0.141959i	−0.997478	−	0.0709797i	$-0.977387\pi$
$467$		−	3.06777i		−	0.141959i	0.997478	−	0.0709797i	$-0.0226125\pi$
$468$			1.43828i			0.0664847i
$469$	3.39104			0.156584
$470$	−18.6128	−	15.7892i	−0.858544	−	0.728301i
$471$	−19.2611			−0.887504
$472$		−	5.34563i		−	0.246052i
$473$			9.52276i			0.437857i
$474$	−15.9197			−0.731217
$475$	−2.90207	−	17.5599i	−0.133156	−	0.805705i
$476$	9.31373			0.426894
$477$		−	38.7102i		−	1.77242i
$478$		−	0.907160i		−	0.0414925i
$479$	−8.78798			−0.401533			−0.200767	−	0.979639i	$-0.564343\pi$
$479$	−8.78798			−0.401533			−0.200767	+	0.979639i	$0.564343\pi$
$480$	4.47687	+	3.79772i	0.204340	+	0.173341i
$481$	−0.369454			−0.0168457
$482$		−	4.15616i		−	0.189308i
$483$			27.0369i			1.23022i
$484$	−6.55754			−0.298070
$485$	−8.76383	+	10.3311i	−0.397945	+	0.469110i
$486$	−21.5354			−0.976866
$487$		−	22.4814i		−	1.01873i	−0.860550	−	0.509366i	$-0.829880\pi$
$487$		−	22.4814i		−	1.01873i	0.860550	−	0.509366i	$-0.170120\pi$
$488$			9.79428i			0.443366i
$489$	49.3500			2.23168
$490$	−5.26931	+	6.21162i	−0.238043	+	0.280613i
$491$	−0.0441752			−0.00199360			−0.000996799	−	1.00000i	$-0.500317\pi$
$491$	−0.0441752			−0.00199360			−0.000996799		1.00000i	$0.500317\pi$
$492$		−	21.0416i		−	0.948629i
$493$			6.13445i			0.276282i
$494$	1.31512			0.0591699
$495$	−27.8155	−	23.5958i	−1.25021	−	1.06055i
$496$	10.1030			0.453638
$497$		−	5.24097i		−	0.235090i
$498$			23.4505i			1.05084i
$499$	−28.2445			−1.26440			−0.632199	−	0.774806i	$-0.717847\pi$
$499$	−28.2445			−1.26440			−0.632199	+	0.774806i	$0.717847\pi$
$500$	5.74552	−	9.59109i	0.256948	−	0.428926i
$501$	−15.5629			−0.695301
$502$			9.31888i			0.415922i
$503$			19.7713i			0.881560i	0.897615	+	0.440780i	$0.145298\pi$
$503$			19.7713i			0.881560i	−0.897615	+	0.440780i	$0.854702\pi$
$504$	7.13302			0.317730
$505$	8.46670	+	7.18229i	0.376763	+	0.319608i
$506$	23.5503			1.04694
$507$		−	33.7725i		−	1.49989i
$508$			19.2856i			0.855660i
$509$	−43.4809			−1.92726			−0.963628	−	0.267247i	$-0.913886\pi$
$509$	−43.4809			−1.92726			−0.963628	+	0.267247i	$0.913886\pi$
$510$	19.3045	−	22.7567i	0.854816	−	1.00768i
$511$	14.8403			0.656496
$512$			1.00000i			0.0441942i
$513$		−	8.34565i		−	0.368470i
$514$	13.6312			0.601249
$515$	−10.6077	+	12.5047i	−0.467432	+	0.551024i
$516$	−5.96671			−0.262670
$517$			45.7376i			2.01154i
$518$			1.83227i			0.0805052i
$519$	−15.1815			−0.666393
$520$	0.629985	+	0.534415i	0.0276267	+	0.0234357i
$521$	−18.0325			−0.790019			−0.395009	−	0.918677i	$-0.629259\pi$
$521$	−18.0325			−0.790019			−0.395009	+	0.918677i	$0.629259\pi$
$522$			4.69813i			0.205632i
$523$		−	31.6788i		−	1.38522i	−0.721313	−	0.692610i	$-0.756461\pi$
$523$		−	31.6788i		−	1.38522i	0.721313	−	0.692610i	$-0.243539\pi$
$524$	8.82072			0.385335
$525$	−3.92190	−	23.7308i	−0.171166	−	1.03570i
$526$	27.2170			1.18672
$527$		−	51.3553i		−	2.23707i
$528$		−	11.0011i		−	0.478761i
$529$	−8.58844			−0.373410
$530$	−16.9555	−	14.3833i	−0.736501	−	0.624773i
$531$	−20.8105			−0.903100
$532$		−	6.52218i		−	0.282773i
$533$		−	2.96098i		−	0.128254i
$534$	−30.1331			−1.30398
$535$	11.8784	−	14.0026i	0.513548	−	0.605387i
$536$	1.85073			0.0799395
$537$			18.7347i			0.808463i
$538$		−	23.0550i		−	0.993973i
$539$	15.2639			0.657465
$540$	3.39137	−	3.99785i	0.145941	−	0.172040i
$541$	9.39442			0.403898			0.201949	−	0.979396i	$-0.435273\pi$
$541$	9.39442			0.403898			0.201949	+	0.979396i	$0.435273\pi$
$542$			22.9320i			0.985014i
$543$		−	30.2886i		−	1.29981i
$544$	5.08317			0.217939
$545$	35.0419	+	29.7260i	1.50103	+	1.27332i
$546$	1.77727			0.0760601
$547$			24.7824i			1.05962i	0.848117	+	0.529809i	$0.177737\pi$
$547$			24.7824i			1.05962i	−0.848117	+	0.529809i	$0.822263\pi$
$548$		−	4.66126i		−	0.199119i
$549$	38.1291			1.62731
$550$	−20.6705	+	3.41613i	−0.881392	+	0.145664i
$551$	4.29581			0.183008
$552$			14.7560i			0.628056i
$553$			11.1102i			0.472452i
$554$	11.7534			0.499353
$555$	4.47687	+	3.79772i	0.190032	+	0.161204i
$556$	16.4050			0.695727
$557$		−	22.1627i		−	0.939062i	−0.882916	−	0.469531i	$-0.844423\pi$
$557$		−	22.1627i		−	0.939062i	0.882916	−	0.469531i	$-0.155577\pi$
$558$		−	39.3310i		−	1.66501i
$559$	−0.839637			−0.0355128
$560$	2.65038	−	3.12434i	0.111999	−	0.132028i
$561$	−55.9205			−2.36096
$562$		−	29.0962i		−	1.22735i
$563$		−	6.46846i		−	0.272613i	−0.990667	−	0.136306i	$-0.956477\pi$
$563$		−	6.46846i		−	0.272613i	0.990667	−	0.136306i	$-0.0435232\pi$
$564$	−28.6580			−1.20672
$565$	−0.944542	+	1.11346i	−0.0397372	+	0.0468434i
$566$	−10.1829			−0.428019
$567$			10.1206i			0.425026i
$568$		−	2.86038i		−	0.120019i
$569$	−8.26550			−0.346508			−0.173254	−	0.984877i	$-0.555428\pi$
$569$	−8.26550			−0.346508			−0.173254	+	0.984877i	$0.555428\pi$
$570$	−15.9360	−	13.5185i	−0.667484	−	0.566226i
$571$	34.5179			1.44453			0.722264	−	0.691618i	$-0.243102\pi$
$571$	34.5179			1.44453			0.722264	+	0.691618i	$0.243102\pi$
$572$		−	1.54808i		−	0.0647283i
$573$			54.0970i			2.25993i
$574$	−14.6846			−0.612925
$575$	27.7257	−	4.58213i	1.15624	−	0.191088i
$576$	3.89300			0.162208
$577$		−	3.05347i		−	0.127117i	−0.997978	−	0.0635587i	$-0.979755\pi$
$577$		−	3.05347i		−	0.127117i	0.997978	−	0.0635587i	$-0.0202450\pi$
$578$		−	8.83864i		−	0.367639i
$579$	68.6985			2.85501
$580$	2.05784	+	1.74566i	0.0854470	+	0.0724845i
$581$	16.3658			0.678968
$582$			15.9067i			0.659353i
$583$			41.6652i			1.72559i
$584$	8.09942			0.335156
$585$	2.08048	−	2.45253i	0.0860172	−	0.101400i
$586$	18.1243			0.748706
$587$		−	35.1344i		−	1.45015i	−0.688668	−	0.725076i	$-0.741804\pi$
$587$		−	35.1344i		−	1.45015i	0.688668	−	0.725076i	$-0.258196\pi$
$588$			9.56399i			0.394412i
$589$	−35.9629			−1.48183
$590$	−7.73245	+	9.11525i	−0.318340	+	0.375269i
$591$	38.3492			1.57748
$592$			1.00000i			0.0410997i
$593$			12.0400i			0.494424i	0.968961	+	0.247212i	$0.0795144\pi$
$593$			12.0400i			0.494424i	−0.968961	+	0.247212i	$0.920486\pi$
$594$	−9.82399			−0.403083
$595$	−15.8816	−	13.4723i	−0.651081	−	0.552311i
$596$	−5.86259			−0.240141
$597$			15.2611i			0.624595i
$598$			2.07646i			0.0849129i
$599$	17.4701			0.713810			0.356905	−	0.934141i	$-0.383832\pi$
$599$	17.4701			0.713810			0.356905	+	0.934141i	$0.383832\pi$
$600$	−2.14046	−	12.9516i	−0.0873839	−	0.528746i
$601$	38.7044			1.57878			0.789392	−	0.613890i	$-0.210396\pi$
$601$	38.7044			1.57878			0.789392	+	0.613890i	$0.210396\pi$
$602$			4.16409i			0.169716i
$603$		−	7.20491i		−	0.293406i
$604$	−13.1460			−0.534903
$605$	11.1818	+	9.48548i	0.454604	+	0.385639i
$606$	13.0361			0.529556
$607$		−	28.7903i		−	1.16856i	−0.811551	−	0.584282i	$-0.801376\pi$
$607$		−	28.7903i		−	1.16856i	0.811551	−	0.584282i	$-0.198624\pi$
$608$		−	3.55963i		−	0.144362i
$609$	5.80542			0.235247
$610$	14.1674	−	16.7010i	0.573623	−	0.676204i
$611$	−4.03275			−0.163148
$612$		−	19.7888i		−	0.799915i
$613$			7.66539i			0.309602i	0.987946	+	0.154801i	$0.0494737\pi$
$613$			7.66539i			0.309602i	−0.987946	+	0.154801i	$0.950526\pi$
$614$	6.15561			0.248420
$615$	−30.4367	+	35.8797i	−1.22733	+	1.44681i
$616$	−7.67751			−0.309336
$617$			36.5388i			1.47100i	0.677527	+	0.735498i	$0.263052\pi$
$617$			36.5388i			1.47100i	−0.677527	+	0.735498i	$0.736948\pi$
$618$			19.2534i			0.774487i
$619$	45.8085			1.84120			0.920599	−	0.390510i	$-0.127701\pi$
$619$	45.8085			1.84120			0.920599	+	0.390510i	$0.127701\pi$
$620$	−17.2274	−	14.6140i	−0.691870	−	0.586912i
$621$	13.1771			0.528779
$622$		−	29.4512i		−	1.18088i
$623$			21.0294i			0.842527i
$624$	0.969984			0.0388304
$625$	−23.6707	+	8.04362i	−0.946826	+	0.321745i
$626$	13.0102			0.519991
$627$			39.1598i			1.56389i
$628$			7.33629i			0.292750i
$629$	5.08317			0.202679
$630$	−12.1631	−	10.3179i	−0.484588	−	0.411075i
$631$	3.42682			0.136420			0.0682098	−	0.997671i	$-0.478271\pi$
$631$	3.42682			0.136420			0.0682098	+	0.997671i	$0.478271\pi$
$632$			6.06361i			0.241197i
$633$		−	42.2636i		−	1.67983i
$634$	7.39826			0.293823
$635$	27.8966	−	32.8854i	1.10704	−	1.30502i
$636$	−26.1063			−1.03518
$637$			1.34585i			0.0533244i
$638$		−	5.05676i		−	0.200199i
$639$	−11.1354			−0.440511
$640$	1.44650	−	1.70518i	0.0571779	−	0.0674031i
$641$	38.3241			1.51371			0.756855	−	0.653583i	$-0.226735\pi$
$641$	38.3241			1.51371			0.756855	+	0.653583i	$0.226735\pi$
$642$		−	21.5598i		−	0.850896i
$643$		−	1.36660i		−	0.0538935i	−0.999637	−	0.0269468i	$-0.991422\pi$
$643$		−	1.36660i		−	0.0538935i	0.999637	−	0.0269468i	$-0.00857846\pi$
$644$	10.2980			0.405798
$645$	10.1743	+	8.63085i	0.400613	+	0.339840i
$646$	−18.0942			−0.711906
$647$			11.2690i			0.443028i	0.975157	+	0.221514i	$0.0710999\pi$
$647$			11.2690i			0.443028i	−0.975157	+	0.221514i	$0.928900\pi$
$648$			5.52355i			0.216985i
$649$	22.3991			0.879241
$650$	−0.301206	−	1.82255i	−0.0118143	−	0.0714862i
$651$	−48.6008			−1.90482
$652$		−	18.7968i		−	0.736138i
$653$			29.5439i			1.15614i	0.815987	+	0.578070i	$0.196194\pi$
$653$			29.5439i			1.15614i	−0.815987	+	0.578070i	$0.803806\pi$
$654$	53.9537			2.10976
$655$	−15.0409	−	12.7592i	−0.587697	−	0.498542i
$656$	−8.01447			−0.312912
$657$		−	31.5311i		−	1.23014i
$658$			20.0000i			0.779681i
$659$	49.9126			1.94432			0.972159	−	0.234324i	$-0.0752877\pi$
$659$	49.9126			1.94432			0.972159	+	0.234324i	$0.0752877\pi$
$660$	−15.9131	+	18.7588i	−0.619416	+	0.730187i
$661$	25.9950			1.01109			0.505545	−	0.862800i	$-0.331292\pi$
$661$	25.9950			1.01109			0.505545	+	0.862800i	$0.331292\pi$
$662$			33.2545i			1.29247i
$663$		−	4.93059i		−	0.191488i
$664$	8.93200			0.346629
$665$	−9.43434	+	11.1215i	−0.365848	+	0.431273i
$666$	3.89300			0.150851
$667$			6.78273i			0.262628i
$668$			5.92772i			0.229350i
$669$	−66.5836			−2.57427
$670$	−3.15583	−	2.67709i	−0.121920	−	0.103425i
$671$	−41.0397			−1.58432
$672$		−	4.81053i		−	0.185570i
$673$		−	6.91054i		−	0.266382i	−0.991090	−	0.133191i	$-0.957478\pi$
$673$		−	6.91054i		−	0.266382i	0.991090	−	0.133191i	$-0.0425223\pi$
$674$	−22.6288			−0.871627
$675$	−11.5658	+	1.91143i	−0.445167	+	0.0735711i
$676$	−12.8635			−0.494750
$677$		−	15.6913i		−	0.603065i	−0.953456	−	0.301532i	$-0.902502\pi$
$677$		−	15.6913i		−	0.603065i	0.953456	−	0.301532i	$-0.0974982\pi$
$678$			1.71438i			0.0658403i
$679$	11.1011			0.426019
$680$	−8.66772	−	7.35281i	−0.332392	−	0.281967i
$681$	−14.6327			−0.560725
$682$			42.3333i			1.62103i
$683$			18.2002i			0.696412i	0.937418	+	0.348206i	$0.113209\pi$
$683$			18.2002i			0.696412i	−0.937418	+	0.348206i	$0.886791\pi$
$684$	−13.8576			−0.529859
$685$	−6.74252	+	7.94829i	−0.257618	+	0.303689i
$686$	19.5004			0.744531
$687$			23.6825i			0.903544i
$688$			2.27264i			0.0866437i
$689$	−3.67368			−0.139956
$690$	21.3445	−	25.1616i	0.812573	−	0.957886i
$691$	−2.58658			−0.0983982			−0.0491991	−	0.998789i	$-0.515667\pi$
$691$	−2.58658			−0.0983982			−0.0491991	+	0.998789i	$0.515667\pi$
$692$			5.78242i			0.219815i
$693$			29.8886i			1.13537i
$694$	24.1396			0.916327
$695$	−27.9735	−	23.7298i	−1.06109	−	0.900124i
$696$	3.16843			0.120099
$697$			40.7389i			1.54310i
$698$			10.3702i			0.392516i
$699$	−31.0065			−1.17277
$700$	−9.03873	+	1.49380i	−0.341632	+	0.0564603i
$701$	−26.5092			−1.00124			−0.500620	−	0.865667i	$-0.666895\pi$
$701$	−26.5092			−1.00124			−0.500620	+	0.865667i	$0.666895\pi$
$702$		−	0.866196i		−	0.0326925i
$703$		−	3.55963i		−	0.134254i
$704$	−4.19017			−0.157923
$705$	48.8670	+	41.4538i	1.84044	+	1.56124i
$706$	−3.92699			−0.147794
$707$		−	9.09773i		−	0.342155i
$708$			14.0347i			0.527456i
$709$	25.0474			0.940674			0.470337	−	0.882487i	$-0.344132\pi$
$709$	25.0474			0.940674			0.470337	+	0.882487i	$0.344132\pi$
$710$	−4.13753	+	4.87745i	−0.155279	+	0.183048i
$711$	23.6056			0.885281
$712$			11.4773i			0.430129i
$713$		−	56.7825i		−	2.12652i
$714$	−24.4528			−0.915121
$715$	−2.23929	+	2.63975i	−0.0837448	+	0.0987209i
$716$	7.13581			0.266678
$717$			2.38171i			0.0889464i
$718$			9.85179i			0.367665i
$719$	−24.9551			−0.930668			−0.465334	−	0.885135i	$-0.654066\pi$
$719$	−24.9551			−0.930668			−0.465334	+	0.885135i	$0.654066\pi$
$720$	−6.63826	−	5.63123i	−0.247394	−	0.209863i
$721$	13.4367			0.500409
$722$		−	6.32907i		−	0.235544i
$723$			10.9118i			0.405814i
$724$	−11.5365			−0.428752
$725$	−0.983883	−	5.95332i	−0.0365405	−	0.221101i
$726$	17.2165			0.638964
$727$		−	21.6774i		−	0.803970i	−0.915646	−	0.401985i	$-0.868320\pi$
$727$		−	21.6774i		−	0.803970i	0.915646	−	0.401985i	$-0.131680\pi$
$728$		−	0.676938i		−	0.0250890i
$729$	39.9695			1.48035
$730$	−13.8110	−	11.7158i	−0.511167	−	0.433622i
$731$	11.5522			0.427275
$732$		−	25.7144i		−	0.950432i
$733$		−	6.33267i		−	0.233903i	−0.993138	−	0.116951i	$-0.962688\pi$
$733$		−	6.33267i		−	0.233903i	0.993138	−	0.116951i	$-0.0373122\pi$
$734$	−33.2559			−1.22750
$735$	13.8343	−	16.3083i	0.510286	−	0.601541i
$736$	5.62036			0.207169
$737$			7.75489i			0.285655i
$738$			31.2003i			1.14850i
$739$	−12.8076			−0.471136			−0.235568	−	0.971858i	$-0.575695\pi$
$739$	−12.8076			−0.471136			−0.235568	+	0.971858i	$0.575695\pi$
$740$	1.44650	−	1.70518i	0.0531744	−	0.0626836i
$741$	−3.45278			−0.126841
$742$			18.2192i			0.668849i
$743$		−	11.0694i		−	0.406097i	−0.979169	−	0.203048i	$-0.934915\pi$
$743$		−	11.0694i		−	0.406097i	0.979169	−	0.203048i	$-0.0650848\pi$
$744$	−26.5249			−0.972452
$745$	9.99677	+	8.48024i	0.366253	+	0.310692i
$746$	−28.5387			−1.04488
$747$		−	34.7723i		−	1.27225i
$748$			21.2994i			0.778782i
$749$	−15.0463			−0.549778
$750$	−15.0846	+	25.1809i	−0.550812	+	0.919478i
$751$	−46.5316			−1.69796			−0.848981	−	0.528423i	$-0.822784\pi$
$751$	−46.5316			−1.69796			−0.848981	+	0.528423i	$0.822784\pi$
$752$			10.9154i			0.398045i
$753$		−	24.4663i		−	0.891601i
$754$	0.445862			0.0162374
$755$	22.4163	+	19.0157i	0.815812	+	0.692052i
$756$	−4.29581			−0.156237
$757$			32.9872i			1.19894i	0.800397	+	0.599470i	$0.204622\pi$
$757$			32.9872i			1.19894i	−0.800397	+	0.599470i	$0.795378\pi$
$758$		−	0.476554i		−	0.0173092i
$759$	−61.8301			−2.24429
$760$	−5.14900	+	6.06980i	−0.186774	+	0.220175i
$761$	28.7913			1.04369			0.521843	−	0.853042i	$-0.325245\pi$
$761$	28.7913			1.04369			0.521843	+	0.853042i	$0.325245\pi$
$762$		−	50.6334i		−	1.83426i
$763$		−	37.6536i		−	1.36315i
$764$	20.6048			0.745456
$765$	−28.6245	+	33.7434i	−1.03492	+	1.22000i
$766$	7.13962			0.257965
$767$			1.97496i			0.0713118i
$768$		−	2.62545i		−	0.0947379i
$769$	−41.9025			−1.51104			−0.755521	−	0.655124i	$-0.772616\pi$
$769$	−41.9025			−1.51104			−0.755521	+	0.655124i	$0.772616\pi$
$770$	13.0915	+	11.1055i	0.471786	+	0.400215i
$771$	−35.7882			−1.28888
$772$		−	26.1663i		−	0.941747i
$773$		−	31.0931i		−	1.11834i	−0.829052	−	0.559171i	$-0.811119\pi$
$773$		−	31.0931i		−	1.11834i	0.829052	−	0.559171i	$-0.188881\pi$
$774$	8.84740			0.318013
$775$	8.23670	+	49.8390i	0.295871	+	1.79027i
$776$	6.05864			0.217493
$777$		−	4.81053i		−	0.172577i
$778$			24.2060i			0.867827i
$779$	28.5285			1.02214
$780$	−1.65400	−	1.40308i	−0.0592226	−	0.0502384i
$781$	11.9855			0.428874
$782$		−	28.5693i		−	1.02163i
$783$		−	2.82941i		−	0.101115i
$784$	3.64280			0.130100
$785$	10.6120	−	12.5097i	0.378757	−	0.446490i
$786$	−23.1584			−0.826032

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:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.62545i q^{3} -1.00000 q^{4} +(1.70518 + 1.44650i) q^{5} +2.62545 q^{6} -1.83227i q^{7} -1.00000i q^{8} -3.89300 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.62545i q^{3} -1.00000 q^{4} +(1.70518 + 1.44650i) q^{5} +2.62545 q^{6} -1.83227i q^{7} -1.00000i q^{8} -3.89300 q^{9} +(-1.44650 + 1.70518i) q^{10} +4.19017 q^{11} +2.62545i q^{12} +0.369454i q^{13} +1.83227 q^{14} +(3.79772 - 4.47687i) q^{15} +1.00000 q^{16} -5.08317i q^{17} -3.89300i q^{18} -3.55963 q^{19} +(-1.70518 - 1.44650i) q^{20} -4.81053 q^{21} +4.19017i q^{22} -5.62036i q^{23} -2.62545 q^{24} +(0.815273 + 4.93309i) q^{25} -0.369454 q^{26} +2.34453i q^{27} +1.83227i q^{28} -1.20681 q^{29} +(4.47687 + 3.79772i) q^{30} +10.1030 q^{31} +1.00000i q^{32} -11.0011i q^{33} +5.08317 q^{34} +(2.65038 - 3.12434i) q^{35} +3.89300 q^{36} +1.00000i q^{37} -3.55963i q^{38} +0.969984 q^{39} +(1.44650 - 1.70518i) q^{40} -8.01447 q^{41} -4.81053i q^{42} +2.27264i q^{43} -4.19017 q^{44} +(-6.63826 - 5.63123i) q^{45} +5.62036 q^{46} +10.9154i q^{47} -2.62545i q^{48} +3.64280 q^{49} +(-4.93309 + 0.815273i) q^{50} -13.3456 q^{51} -0.369454i q^{52} +9.94355i q^{53} -2.34453 q^{54} +(7.14499 + 6.06108i) q^{55} -1.83227 q^{56} +9.34563i q^{57} -1.20681i q^{58} +5.34563 q^{59} +(-3.79772 + 4.47687i) q^{60} -9.79428 q^{61} +10.1030i q^{62} +7.13302i q^{63} -1.00000 q^{64} +(-0.534415 + 0.629985i) q^{65} +11.0011 q^{66} +1.85073i q^{67} +5.08317i q^{68} -14.7560 q^{69} +(3.12434 + 2.65038i) q^{70} +2.86038 q^{71} +3.89300i q^{72} +8.09942i q^{73} -1.00000 q^{74} +(12.9516 - 2.14046i) q^{75} +3.55963 q^{76} -7.67751i q^{77} +0.969984i q^{78} -6.06361 q^{79} +(1.70518 + 1.44650i) q^{80} -5.52355 q^{81} -8.01447i q^{82} +8.93200i q^{83} +4.81053 q^{84} +(7.35281 - 8.66772i) q^{85} -2.27264 q^{86} +3.16843i q^{87} -4.19017i q^{88} -11.4773 q^{89} +(5.63123 - 6.63826i) q^{90} +0.676938 q^{91} +5.62036i q^{92} -26.5249i q^{93} -10.9154 q^{94} +(-6.06980 - 5.14900i) q^{95} +2.62545 q^{96} +6.05864i q^{97} +3.64280i q^{98} -16.3123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9}+O(q^{10}) \) 10 * q - 10 * q^4 + 6 * q^5 - 6 * q^9	\( 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9} + 2 q^{10} + 6 q^{11} + 2 q^{14} + 10 q^{16} - 8 q^{19} - 6 q^{20} + 32 q^{21} + 4 q^{25} - 12 q^{26} - 22 q^{29} + 20 q^{30} + 46 q^{31} - 18 q^{34} + 32 q^{35} + 6 q^{36} - 40 q^{39} - 2 q^{40} - 14 q^{41} - 6 q^{44} + 2 q^{45} + 12 q^{46} - 60 q^{49} + 8 q^{50} - 40 q^{51} + 42 q^{55} - 2 q^{56} - 40 q^{59} - 18 q^{61} - 10 q^{64} + 4 q^{65} + 40 q^{66} - 32 q^{69} - 6 q^{70} + 12 q^{71} - 10 q^{74} + 50 q^{75} + 8 q^{76} - 40 q^{79} + 6 q^{80} - 14 q^{81} - 32 q^{84} + 36 q^{85} - 34 q^{86} - 24 q^{89} + 44 q^{90} + 32 q^{91} - 24 q^{94} + 12 q^{95} + 22 q^{99}+O(q^{100}) \) 10 * q - 10 * q^4 + 6 * q^5 - 6 * q^9 + 2 * q^10 + 6 * q^11 + 2 * q^14 + 10 * q^16 - 8 * q^19 - 6 * q^20 + 32 * q^21 + 4 * q^25 - 12 * q^26 - 22 * q^29 + 20 * q^30 + 46 * q^31 - 18 * q^34 + 32 * q^35 + 6 * q^36 - 40 * q^39 - 2 * q^40 - 14 * q^41 - 6 * q^44 + 2 * q^45 + 12 * q^46 - 60 * q^49 + 8 * q^50 - 40 * q^51 + 42 * q^55 - 2 * q^56 - 40 * q^59 - 18 * q^61 - 10 * q^64 + 4 * q^65 + 40 * q^66 - 32 * q^69 - 6 * q^70 + 12 * q^71 - 10 * q^74 + 50 * q^75 + 8 * q^76 - 40 * q^79 + 6 * q^80 - 14 * q^81 - 32 * q^84 + 36 * q^85 - 34 * q^86 - 24 * q^89 + 44 * q^90 + 32 * q^91 - 24 * q^94 + 12 * q^95 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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