LMFDB - Embedded newform 930.2.d.i.559.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(559,930)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("930.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.11669056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58 $ x^6 - 2x^5 + 7x^4 + 8x^3 - x^2 + 54x + 58
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.4
Root			$-1.23545 + 0.0526623i$ of defining polynomial
Character	$\chi$	$=$	930.559
Dual form			930.2.d.i.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-2.23545 + 0.0526623i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-2.23545 + 0.0526623i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(-0.0526623 - 2.23545i) q^{10} -0.470896 q^{11} +1.00000i q^{12} -6.47090i q^{13} -2.00000 q^{14} +(0.0526623 + 2.23545i) q^{15} +1.00000 q^{16} +7.04712i q^{17} -1.00000i q^{18} +7.04712 q^{19} +(2.23545 - 0.0526623i) q^{20} +2.00000 q^{21} -0.470896i q^{22} -6.94179i q^{23} -1.00000 q^{24} +(4.99445 - 0.235448i) q^{25} +6.47090 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.94179 q^{29} +(-2.23545 + 0.0526623i) q^{30} -1.00000 q^{31} +1.00000i q^{32} +0.470896i q^{33} -7.04712 q^{34} +(-0.105325 - 4.47090i) q^{35} +1.00000 q^{36} -1.78935i q^{37} +7.04712i q^{38} -6.47090 q^{39} +(0.0526623 + 2.23545i) q^{40} +2.00000 q^{41} +2.00000i q^{42} +0.210649i q^{43} +0.470896 q^{44} +(2.23545 - 0.0526623i) q^{45} +6.94179 q^{46} -7.04712i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(0.235448 + 4.99445i) q^{50} +7.04712 q^{51} +6.47090i q^{52} +3.15244i q^{53} -1.00000 q^{54} +(1.05266 - 0.0247985i) q^{55} +2.00000 q^{56} -7.04712i q^{57} +6.94179i q^{58} +7.15244 q^{59} +(-0.0526623 - 2.23545i) q^{60} +11.2578 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(0.340772 + 14.4653i) q^{65} -0.470896 q^{66} -8.26025i q^{67} -7.04712i q^{68} -6.94179 q^{69} +(4.47090 - 0.105325i) q^{70} +0.260246 q^{71} +1.00000i q^{72} -11.8836i q^{73} +1.78935 q^{74} +(-0.235448 - 4.99445i) q^{75} -7.04712 q^{76} -0.941791i q^{77} -6.47090i q^{78} +1.89468 q^{79} +(-2.23545 + 0.0526623i) q^{80} +1.00000 q^{81} +2.00000i q^{82} +11.7783i q^{83} -2.00000 q^{84} +(-0.371118 - 15.7535i) q^{85} -0.210649 q^{86} -6.94179i q^{87} +0.470896i q^{88} -12.0942 q^{89} +(0.0526623 + 2.23545i) q^{90} +12.9418 q^{91} +6.94179i q^{92} +1.00000i q^{93} +7.04712 q^{94} +(-15.7535 + 0.371118i) q^{95} +1.00000 q^{96} -3.52910i q^{97} +3.00000i q^{98} +0.470896 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9	$ 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 + 16 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 + 4 * q^20 + 12 * q^21 - 6 * q^24 - 8 * q^25 + 20 * q^26 + 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^34 + 6 * q^36 - 20 * q^39 + 12 * q^41 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 4 * q^51 - 6 * q^54 + 6 * q^55 + 12 * q^56 + 4 * q^59 + 28 * q^61 - 6 * q^64 - 8 * q^65 + 16 * q^66 - 4 * q^69 + 8 * q^70 - 16 * q^71 + 12 * q^74 + 8 * q^75 - 4 * q^76 + 12 * q^79 - 4 * q^80 + 6 * q^81 - 12 * q^84 - 22 * q^85 + 4 * q^89 + 40 * q^91 + 4 * q^94 - 28 * q^95 + 6 * q^96 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	−2.23545	+	0.0526623i	−0.999723	+	0.0235513i
$5$	−2.23545	+	0.0526623i	−0.999723	+	0.0235513i
$6$	1.00000			0.408248
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	−0.0526623	−	2.23545i	−0.0166533	−	0.706911i
$11$	−0.470896			−0.141980			−0.0709902	−	0.997477i	$-0.522616\pi$
$11$	−0.470896			−0.141980			−0.0709902	+	0.997477i	$0.522616\pi$
$12$			1.00000i			0.288675i
$13$		−	6.47090i		−	1.79470i	−0.441316	−	0.897352i	$-0.645488\pi$
$13$		−	6.47090i		−	1.79470i	0.441316	−	0.897352i	$-0.354512\pi$
$14$	−2.00000			−0.534522
$15$	0.0526623	+	2.23545i	0.0135974	+	0.577190i
$16$	1.00000			0.250000
$17$			7.04712i			1.70918i	0.519306	+	0.854588i	$0.326190\pi$
$17$			7.04712i			1.70918i	−0.519306	+	0.854588i	$0.673810\pi$
$18$		−	1.00000i		−	0.235702i
$19$	7.04712			1.61672			0.808360	−	0.588689i	$-0.200356\pi$
$19$	7.04712			1.61672			0.808360	+	0.588689i	$0.200356\pi$
$20$	2.23545	−	0.0526623i	0.499861	−	0.0117757i
$21$	2.00000			0.436436
$22$		−	0.470896i		−	0.100395i
$23$		−	6.94179i		−	1.44746i	−0.690081	−	0.723732i	$-0.742425\pi$
$23$		−	6.94179i		−	1.44746i	0.690081	−	0.723732i	$-0.257575\pi$
$24$	−1.00000			−0.204124
$25$	4.99445	−	0.235448i	0.998891	−	0.0470896i
$26$	6.47090			1.26905
$27$			1.00000i			0.192450i
$28$		−	2.00000i		−	0.377964i
$29$	6.94179			1.28906			0.644529	−	0.764580i	$-0.277053\pi$
$29$	6.94179			1.28906			0.644529	+	0.764580i	$0.277053\pi$
$30$	−2.23545	+	0.0526623i	−0.408135	+	0.00961478i
$31$	−1.00000			−0.179605
$31$	−1.00000			−0.179605
$32$			1.00000i			0.176777i
$33$			0.470896i			0.0819724i
$34$	−7.04712			−1.20857
$35$	−0.105325	−	4.47090i	−0.0178031	−	0.755719i
$36$	1.00000			0.166667
$37$		−	1.78935i		−	0.294167i	−0.989124	−	0.147084i	$-0.953011\pi$
$37$		−	1.78935i		−	0.294167i	0.989124	−	0.147084i	$-0.0469887\pi$
$38$			7.04712i			1.14319i
$39$	−6.47090			−1.03617
$40$	0.0526623	+	2.23545i	0.00832664	+	0.353455i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$			2.00000i			0.308607i
$43$			0.210649i			0.0321237i	0.999871	+	0.0160619i	$0.00511287\pi$
$43$			0.210649i			0.0321237i	−0.999871	+	0.0160619i	$0.994887\pi$
$44$	0.470896			0.0709902
$45$	2.23545	−	0.0526623i	0.333241	−	0.00785044i
$46$	6.94179			1.02351
$47$		−	7.04712i		−	1.02793i	−0.857812	−	0.513964i	$-0.828177\pi$
$47$		−	7.04712i		−	1.02793i	0.857812	−	0.513964i	$-0.171823\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	0.235448	+	4.99445i	0.0332973	+	0.706322i
$51$	7.04712			0.986794
$52$			6.47090i			0.897352i
$53$			3.15244i			0.433021i	0.976280	+	0.216510i	$0.0694676\pi$
$53$			3.15244i			0.433021i	−0.976280	+	0.216510i	$0.930532\pi$
$54$	−1.00000			−0.136083
$55$	1.05266	−	0.0247985i	0.141941	−	0.00334382i
$56$	2.00000			0.267261
$57$		−	7.04712i		−	0.933413i
$58$			6.94179i			0.911502i
$59$	7.15244			0.931168			0.465584	−	0.885004i	$-0.345844\pi$
$59$	7.15244			0.931168			0.465584	+	0.885004i	$0.345844\pi$
$60$	−0.0526623	−	2.23545i	−0.00679868	−	0.288595i
$61$	11.2578			1.44141			0.720705	−	0.693242i	$-0.243818\pi$
$61$	11.2578			1.44141			0.720705	+	0.693242i	$0.243818\pi$
$62$		−	1.00000i		−	0.127000i
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$	0.340772	+	14.4653i	0.0422676	+	1.79421i
$66$	−0.470896			−0.0579632
$67$		−	8.26025i		−	1.00915i	−0.863368	−	0.504575i	$-0.831649\pi$
$67$		−	8.26025i		−	1.00915i	0.863368	−	0.504575i	$-0.168351\pi$
$68$		−	7.04712i		−	0.854588i
$69$	−6.94179			−0.835693
$70$	4.47090	−	0.105325i	0.534374	−	0.0125887i
$71$	0.260246			0.0308855			0.0154428	−	0.999881i	$-0.495084\pi$
$71$	0.260246			0.0308855			0.0154428	+	0.999881i	$0.495084\pi$
$72$			1.00000i			0.117851i
$73$		−	11.8836i		−	1.39087i	−0.718590	−	0.695434i	$-0.755212\pi$
$73$		−	11.8836i		−	1.39087i	0.718590	−	0.695434i	$-0.244788\pi$
$74$	1.78935			0.208008
$75$	−0.235448	−	4.99445i	−0.0271872	−	0.576710i
$76$	−7.04712			−0.808360
$77$		−	0.941791i		−	0.107327i
$78$		−	6.47090i		−	0.732685i
$79$	1.89468			0.213168			0.106584	−	0.994304i	$-0.466009\pi$
$79$	1.89468			0.213168			0.106584	+	0.994304i	$0.466009\pi$
$80$	−2.23545	+	0.0526623i	−0.249931	+	0.00588783i
$81$	1.00000			0.111111
$82$			2.00000i			0.220863i
$83$			11.7783i			1.29283i	0.762985	+	0.646416i	$0.223733\pi$
$83$			11.7783i			1.29283i	−0.762985	+	0.646416i	$0.776267\pi$
$84$	−2.00000			−0.218218
$85$	−0.371118	−	15.7535i	−0.0402533	−	1.70870i
$86$	−0.210649			−0.0227149
$87$		−	6.94179i		−	0.744238i
$88$			0.470896i			0.0501976i
$89$	−12.0942			−1.28199			−0.640993	−	0.767547i	$-0.721477\pi$
$89$	−12.0942			−1.28199			−0.640993	+	0.767547i	$0.721477\pi$
$90$	0.0526623	+	2.23545i	0.00555110	+	0.235637i
$91$	12.9418			1.35667
$92$			6.94179i			0.723732i
$93$			1.00000i			0.103695i
$94$	7.04712			0.726854
$95$	−15.7535	+	0.371118i	−1.61627	+	0.0380759i
$96$	1.00000			0.102062
$97$		−	3.52910i		−	0.358326i	−0.983819	−	0.179163i	$-0.942661\pi$
$97$		−	3.52910i		−	0.358326i	0.983819	−	0.179163i	$-0.0573390\pi$
$98$			3.00000i			0.303046i
$99$	0.470896			0.0473268
$100$	−4.99445	+	0.235448i	−0.499445	+	0.0235448i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$			7.04712i			0.697768i
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−6.47090			−0.634524
$105$	−4.47090	+	0.105325i	−0.436315	+	0.0102786i
$106$	−3.15244			−0.306192
$107$			16.7311i			1.61746i	0.588180	+	0.808730i	$0.299845\pi$
$107$			16.7311i			1.61746i	−0.588180	+	0.808730i	$0.700155\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	11.1524			1.06821			0.534105	−	0.845418i	$-0.320649\pi$
$109$	11.1524			1.06821			0.534105	+	0.845418i	$0.320649\pi$
$110$	0.0247985	+	1.05266i	0.00236444	+	0.100367i
$111$	−1.78935			−0.169838
$112$			2.00000i			0.188982i
$113$		−	16.3049i		−	1.53383i	−0.641746	−	0.766917i	$-0.721790\pi$
$113$		−	16.3049i		−	1.53383i	0.641746	−	0.766917i	$-0.278210\pi$
$114$	7.04712			0.660023
$115$	0.365571	+	15.5180i	0.0340897	+	1.44706i
$116$	−6.94179			−0.644529
$117$			6.47090i			0.598235i
$118$			7.15244i			0.658436i
$119$	−14.0942			−1.29202
$120$	2.23545	−	0.0526623i	0.204068	−	0.00480739i
$121$	−10.7783			−0.979842
$122$			11.2578i			1.01923i
$123$		−	2.00000i		−	0.180334i
$124$	1.00000			0.0898027
$125$	−11.1524	+	0.789351i	−0.997505	+	0.0706017i
$126$	2.00000			0.178174
$127$			0.941791i			0.0835704i	0.999127	+	0.0417852i	$0.0133045\pi$
$127$			0.941791i			0.0835704i	−0.999127	+	0.0417852i	$0.986695\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	0.210649			0.0185466
$130$	−14.4653	+	0.340772i	−1.26870	+	0.0298877i
$131$	−3.15244			−0.275430			−0.137715	−	0.990472i	$-0.543976\pi$
$131$	−3.15244			−0.275430			−0.137715	+	0.990472i	$0.543976\pi$
$132$		−	0.470896i		−	0.0409862i
$133$			14.0942i			1.22212i
$134$	8.26025			0.713577
$135$	−0.0526623	−	2.23545i	−0.00453245	−	0.192397i
$136$	7.04712			0.604285
$137$			9.67293i			0.826414i	0.910637	+	0.413207i	$0.135591\pi$
$137$			9.67293i			0.826414i	−0.910637	+	0.413207i	$0.864409\pi$
$138$		−	6.94179i		−	0.590924i
$139$	−5.05821			−0.429032			−0.214516	−	0.976721i	$-0.568817\pi$
$139$	−5.05821			−0.429032			−0.214516	+	0.976721i	$0.568817\pi$
$140$	0.105325	+	4.47090i	0.00890156	+	0.377860i
$141$	−7.04712			−0.593474
$142$			0.260246i			0.0218394i
$143$			3.04712i			0.254813i
$144$	−1.00000			−0.0833333
$145$	−15.5180	+	0.365571i	−1.28870	+	0.0303590i
$146$	11.8836			0.983492
$147$		−	3.00000i		−	0.247436i
$148$			1.78935i			0.147084i
$149$	−13.4127			−1.09881			−0.549405	−	0.835556i	$-0.685146\pi$
$149$	−13.4127			−1.09881			−0.549405	+	0.835556i	$0.685146\pi$
$150$	4.99445	−	0.235448i	0.407795	−	0.0192242i
$151$	19.5676			1.59239			0.796195	−	0.605041i	$-0.206843\pi$
$151$	19.5676			1.59239			0.796195	+	0.605041i	$0.206843\pi$
$152$		−	7.04712i		−	0.571597i
$153$		−	7.04712i		−	0.569726i
$154$	0.941791			0.0758917
$155$	2.23545	−	0.0526623i	0.179555	−	0.00422994i
$156$	6.47090			0.518086
$157$		−	21.2467i		−	1.69567i	−0.530261	−	0.847835i	$-0.677906\pi$
$157$		−	21.2467i		−	1.69567i	0.530261	−	0.847835i	$-0.322094\pi$
$158$			1.89468i			0.150732i
$159$	3.15244			0.250005
$160$	−0.0526623	−	2.23545i	−0.00416332	−	0.176728i
$161$	13.8836			1.09418
$162$			1.00000i			0.0785674i
$163$			4.26025i			0.333688i	0.985983	+	0.166844i	$0.0533577\pi$
$163$			4.26025i			0.333688i	−0.985983	+	0.166844i	$0.946642\pi$
$164$	−2.00000			−0.156174
$165$	−0.0247985	−	1.05266i	−0.00193056	−	0.0819497i
$166$	−11.7783			−0.914170
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		−	2.00000i		−	0.154303i
$169$	−28.8725			−2.22096
$170$	15.7535	−	0.371118i	1.20824	−	0.0284634i
$171$	−7.04712			−0.538906
$172$		−	0.210649i		−	0.0160619i
$173$		−	17.9889i		−	1.36767i	−0.729636	−	0.683836i	$-0.760311\pi$
$173$		−	17.9889i		−	1.36767i	0.729636	−	0.683836i	$-0.239689\pi$
$174$	6.94179			0.526256
$175$	0.470896	+	9.98891i	0.0355964	+	0.755090i
$176$	−0.470896			−0.0354951
$177$		−	7.15244i		−	0.537610i
$178$		−	12.0942i		−	0.906501i
$179$	12.0496			0.900629			0.450315	−	0.892870i	$-0.351312\pi$
$179$	12.0496			0.900629			0.450315	+	0.892870i	$0.351312\pi$
$180$	−2.23545	+	0.0526623i	−0.166620	+	0.00392522i
$181$	−13.6729			−1.01630			−0.508151	−	0.861268i	$-0.669671\pi$
$181$	−13.6729			−1.01630			−0.508151	+	0.861268i	$0.669671\pi$
$182$			12.9418i			0.959309i
$183$		−	11.2578i		−	0.832198i
$184$	−6.94179			−0.511756
$185$	0.0942314	+	4.00000i	0.00692803	+	0.294086i
$186$	−1.00000			−0.0733236
$187$		−	3.31846i		−	0.242669i
$188$			7.04712i			0.513964i
$189$	−2.00000			−0.145479
$190$	−0.371118	−	15.7535i	−0.0269237	−	1.14288i
$191$	−14.8254			−1.07273			−0.536363	−	0.843987i	$-0.680202\pi$
$191$	−14.8254			−1.07273			−0.536363	+	0.843987i	$0.680202\pi$
$192$			1.00000i			0.0721688i
$193$			8.04960i			0.579423i	0.957114	+	0.289711i	$0.0935593\pi$
$193$			8.04960i			0.579423i	−0.957114	+	0.289711i	$0.906441\pi$
$194$	3.52910			0.253375
$195$	14.4653	−	0.340772i	1.03589	−	0.0244032i
$196$	−3.00000			−0.214286
$197$			16.3049i			1.16167i	0.814020	+	0.580837i	$0.197275\pi$
$197$			16.3049i			1.16167i	−0.814020	+	0.580837i	$0.802725\pi$
$198$			0.470896i			0.0334651i
$199$	3.98891			0.282766			0.141383	−	0.989955i	$-0.454845\pi$
$199$	3.98891			0.282766			0.141383	+	0.989955i	$0.454845\pi$
$200$	−0.235448	−	4.99445i	−0.0166487	−	0.353161i
$201$	−8.26025			−0.582633
$202$		0			0
$203$			13.8836i			0.974436i
$204$	−7.04712			−0.493397
$205$	−4.47090	+	0.105325i	−0.312261	+	0.00735619i
$206$	−6.00000			−0.418040
$207$			6.94179i			0.482488i
$208$		−	6.47090i		−	0.448676i
$209$	−3.31846			−0.229542
$210$	−0.105325	−	4.47090i	−0.00726809	−	0.308521i
$211$	2.11642			0.145700			0.0728501	−	0.997343i	$-0.476791\pi$
$211$	2.11642			0.145700			0.0728501	+	0.997343i	$0.476791\pi$
$212$		−	3.15244i		−	0.216510i
$213$		−	0.260246i		−	0.0178318i
$214$	−16.7311			−1.14372
$215$	−0.0110933	−	0.470896i	−0.000756556	−	0.0321148i
$216$	1.00000			0.0680414
$217$		−	2.00000i		−	0.135769i
$218$			11.1524i			0.755339i
$219$	−11.8836			−0.803018
$220$	−1.05266	+	0.0247985i	−0.0709705	+	0.00167191i
$221$	45.6011			3.06747
$222$		−	1.78935i		−	0.120093i
$223$			2.58731i			0.173259i	0.996241	+	0.0866297i	$0.0276097\pi$
$223$			2.58731i			0.173259i	−0.996241	+	0.0866297i	$0.972390\pi$
$224$	−2.00000			−0.133631
$225$	−4.99445	+	0.235448i	−0.332964	+	0.0156965i
$226$	16.3049			1.08458
$227$			4.94179i			0.327998i	0.986460	+	0.163999i	$0.0524394\pi$
$227$			4.94179i			0.327998i	−0.986460	+	0.163999i	$0.947561\pi$
$228$			7.04712i			0.466707i
$229$	11.6791			0.771774			0.385887	−	0.922546i	$-0.373895\pi$
$229$	11.6791			0.771774			0.385887	+	0.922546i	$0.373895\pi$
$230$	−15.5180	+	0.365571i	−1.02323	+	0.0241050i
$231$	−0.941791			−0.0619653
$232$		−	6.94179i		−	0.455751i
$233$		−	13.7894i		−	0.903370i	−0.892177	−	0.451685i	$-0.850823\pi$
$233$		−	13.7894i		−	0.903370i	0.892177	−	0.451685i	$-0.149177\pi$
$234$	−6.47090			−0.423016
$235$	0.371118	+	15.7535i	0.0242090	+	1.02764i
$236$	−7.15244			−0.465584
$237$		−	1.89468i		−	0.123072i
$238$		−	14.0942i		−	0.913593i
$239$	12.7311			0.823509			0.411755	−	0.911295i	$-0.364916\pi$
$239$	12.7311			0.823509			0.411755	+	0.911295i	$0.364916\pi$
$240$	0.0526623	+	2.23545i	0.00339934	+	0.144298i
$241$	−3.15244			−0.203067			−0.101533	−	0.994832i	$-0.532375\pi$
$241$	−3.15244			−0.203067			−0.101533	+	0.994832i	$0.532375\pi$
$242$		−	10.7783i		−	0.692853i
$243$		−	1.00000i		−	0.0641500i
$244$	−11.2578			−0.720705
$245$	−6.70634	+	0.157987i	−0.428453	+	0.0100934i
$246$	2.00000			0.127515
$247$		−	45.6011i		−	2.90153i
$248$			1.00000i			0.0635001i
$249$	11.7783			0.746417
$250$	−0.789351	−	11.1524i	−0.0499229	−	0.705342i
$251$	3.05821			0.193032			0.0965162	−	0.995331i	$-0.469230\pi$
$251$	3.05821			0.193032			0.0965162	+	0.995331i	$0.469230\pi$
$252$			2.00000i			0.125988i
$253$			3.26886i			0.205511i
$254$	−0.941791			−0.0590932
$255$	−15.7535	+	0.371118i	−0.986520	+	0.0232403i
$256$	1.00000			0.0625000
$257$		−	16.0942i		−	1.00393i	−0.864888	−	0.501965i	$-0.832611\pi$
$257$		−	16.0942i		−	1.00393i	0.864888	−	0.501965i	$-0.167389\pi$
$258$			0.210649i			0.0131145i
$259$	3.57870			0.222370
$260$	−0.340772	−	14.4653i	−0.0211338	−	0.897103i
$261$	−6.94179			−0.429686
$262$		−	3.15244i		−	0.194758i
$263$			0.0942314i			0.00581056i	0.999996	+	0.00290528i	$0.000924780\pi$
$263$			0.0942314i			0.00581056i	−0.999996	+	0.00290528i	$0.999075\pi$
$264$	0.470896			0.0289816
$265$	−0.166015	−	7.04712i	−0.0101982	−	0.432901i
$266$	−14.0942			−0.864173
$267$			12.0942i			0.740155i
$268$			8.26025i			0.504575i
$269$	23.8836			1.45621			0.728104	−	0.685467i	$-0.240402\pi$
$269$	23.8836			1.45621			0.728104	+	0.685467i	$0.240402\pi$
$270$	2.23545	−	0.0526623i	0.136045	−	0.00320493i
$271$	24.4102			1.48281			0.741407	−	0.671055i	$-0.234159\pi$
$271$	24.4102			1.48281			0.741407	+	0.671055i	$0.234159\pi$
$272$			7.04712i			0.427294i
$273$		−	12.9418i		−	0.783273i
$274$	−9.67293			−0.584363
$275$	−2.35187	+	0.110871i	−0.141823	+	0.00668579i
$276$	6.94179			0.417847
$277$			7.41269i			0.445385i	0.974889	+	0.222693i	$0.0714846\pi$
$277$			7.41269i			0.445385i	−0.974889	+	0.222693i	$0.928515\pi$
$278$		−	5.05821i		−	0.303371i
$279$	1.00000			0.0598684
$280$	−4.47090	+	0.105325i	−0.267187	+	0.00629435i
$281$	−11.8836			−0.708915			−0.354458	−	0.935072i	$-0.615334\pi$
$281$	−11.8836			−0.708915			−0.354458	+	0.935072i	$0.615334\pi$
$282$		−	7.04712i		−	0.419650i
$283$			22.9864i			1.36640i	0.730231	+	0.683201i	$0.239413\pi$
$283$			22.9864i			1.36640i	−0.730231	+	0.683201i	$0.760587\pi$
$284$	−0.260246			−0.0154428
$285$	0.371118	+	15.7535i	0.0219831	+	0.933154i
$286$	−3.04712			−0.180180
$287$			4.00000i			0.236113i
$288$		−	1.00000i		−	0.0589256i
$289$	−32.6618			−1.92128
$290$	−0.365571	−	15.5180i	−0.0214671	−	0.911249i
$291$	−3.52910			−0.206880
$292$			11.8836i			0.695434i
$293$		−	14.4213i		−	0.842501i	−0.906944	−	0.421251i	$-0.861591\pi$
$293$		−	14.4213i		−	0.842501i	0.906944	−	0.421251i	$-0.138409\pi$
$294$	3.00000			0.174964
$295$	−15.9889	+	0.376664i	−0.930910	+	0.0219302i
$296$	−1.78935			−0.104004
$297$		−	0.470896i		−	0.0273241i
$298$		−	13.4127i		−	0.776976i
$299$	−44.9196			−2.59777
$300$	0.235448	+	4.99445i	0.0135936	+	0.288355i
$301$	−0.421299			−0.0242832
$302$			19.5676i			1.12599i
$303$		0			0
$304$	7.04712			0.404180
$305$	−25.1661	+	0.592860i	−1.44101	+	0.0339471i
$306$	7.04712			0.402857
$307$		−	24.9418i		−	1.42350i	−0.702431	−	0.711752i	$-0.747902\pi$
$307$		−	24.9418i		−	1.42350i	0.702431	−	0.711752i	$-0.252098\pi$
$308$			0.941791i			0.0536635i
$309$	6.00000			0.341328
$310$	0.0526623	+	2.23545i	0.00299102	+	0.126965i
$311$	−24.4487			−1.38636			−0.693180	−	0.720765i	$-0.743791\pi$
$311$	−24.4487			−1.38636			−0.693180	+	0.720765i	$0.743791\pi$
$312$			6.47090i			0.366342i
$313$		−	14.1885i		−	0.801979i	−0.916083	−	0.400990i	$-0.868666\pi$
$313$		−	14.1885i		−	0.801979i	0.916083	−	0.400990i	$-0.131334\pi$
$314$	21.2467			1.19902
$315$	0.105325	+	4.47090i	0.00593437	+	0.251906i
$316$	−1.89468			−0.106584
$317$		−	21.9889i		−	1.23502i	−0.786563	−	0.617510i	$-0.788141\pi$
$317$		−	21.9889i		−	1.23502i	0.786563	−	0.617510i	$-0.211859\pi$
$318$			3.15244i			0.176780i
$319$	−3.26886			−0.183021
$320$	2.23545	−	0.0526623i	0.124965	−	0.00294391i
$321$	16.7311			0.933841
$322$			13.8836i			0.773702i
$323$			49.6618i			2.76326i
$324$	−1.00000			−0.0555556
$325$	−1.52356	−	32.3186i	−0.0845118	−	1.79271i
$326$	−4.26025			−0.235953
$327$		−	11.1524i		−	0.616731i
$328$		−	2.00000i		−	0.110432i
$329$	14.0942			0.777040
$330$	1.05266	−	0.0247985i	0.0579472	−	0.00136511i
$331$	−15.3631			−0.844432			−0.422216	−	0.906495i	$-0.638748\pi$
$331$	−15.3631			−0.844432			−0.422216	+	0.906495i	$0.638748\pi$
$332$		−	11.7783i		−	0.646416i
$333$			1.78935i			0.0980558i
$334$	−2.00000			−0.109435
$335$	0.435004	+	18.4653i	0.0237668	+	1.00887i
$336$	2.00000			0.109109
$337$			13.9778i			0.761420i	0.924694	+	0.380710i	$0.124320\pi$
$337$			13.9778i			0.761420i	−0.924694	+	0.380710i	$0.875680\pi$
$338$		−	28.8725i		−	1.57046i
$339$	−16.3049			−0.885560
$340$	0.371118	+	15.7535i	0.0201267	+	0.854351i
$341$	0.470896			0.0255004
$342$		−	7.04712i		−	0.381064i
$343$			20.0000i			1.07990i
$344$	0.210649			0.0113574
$345$	15.5180	−	0.365571i	0.835462	−	0.0196817i
$346$	17.9889			0.967090
$347$			17.6618i			0.948137i	0.880488	+	0.474069i	$0.157215\pi$
$347$			17.6618i			0.948137i	−0.880488	+	0.474069i	$0.842785\pi$
$348$			6.94179i			0.372119i
$349$	7.67293			0.410723			0.205361	−	0.978686i	$-0.434163\pi$
$349$	7.67293			0.410723			0.205361	+	0.978686i	$0.434163\pi$
$350$	−9.98891	+	0.470896i	−0.533930	+	0.0251704i
$351$	6.47090			0.345391
$352$		−	0.470896i		−	0.0250988i
$353$		−	11.0471i		−	0.587979i	−0.955809	−	0.293989i	$-0.905017\pi$
$353$		−	11.0471i		−	0.587979i	0.955809	−	0.293989i	$-0.0949830\pi$
$354$	7.15244			0.380148
$355$	−0.581767	+	0.0137052i	−0.0308770	+	0.000727395i
$356$	12.0942			0.640993
$357$			14.0942i			0.745946i
$358$			12.0496i			0.636841i
$359$	35.9282			1.89622			0.948109	−	0.317944i	$-0.102993\pi$
$359$	35.9282			1.89622			0.948109	+	0.317944i	$0.102993\pi$
$360$	−0.0526623	−	2.23545i	−0.00277555	−	0.117818i
$361$	30.6618			1.61378
$362$		−	13.6729i		−	0.718633i
$363$			10.7783i			0.565712i
$364$	−12.9418			−0.678334
$365$	0.625817	+	26.5651i	0.0327568	+	1.39048i
$366$	11.2578			0.588453
$367$			19.2963i			1.00726i	0.863920	+	0.503629i	$0.168002\pi$
$367$			19.2963i			1.00726i	−0.863920	+	0.503629i	$0.831998\pi$
$368$		−	6.94179i		−	0.361866i
$369$	−2.00000			−0.104116
$370$	−4.00000	+	0.0942314i	−0.207950	+	0.00489886i
$371$	−6.30488			−0.327333
$372$		−	1.00000i		−	0.0518476i
$373$			24.0942i			1.24755i	0.781603	+	0.623776i	$0.214402\pi$
$373$			24.0942i			1.24755i	−0.781603	+	0.623776i	$0.785598\pi$
$374$	3.31846			0.171593
$375$	0.789351	+	11.1524i	0.0407619	+	0.575910i
$376$	−7.04712			−0.363427
$377$		−	44.9196i		−	2.31348i
$378$		−	2.00000i		−	0.102869i
$379$	−25.1413			−1.29142			−0.645712	−	0.763581i	$-0.723439\pi$
$379$	−25.1413			−1.29142			−0.645712	+	0.763581i	$0.723439\pi$
$380$	15.7535	−	0.371118i	0.808135	−	0.0190379i
$381$	0.941791			0.0482494
$382$		−	14.8254i		−	0.758532i
$383$		−	22.9196i		−	1.17114i	−0.810623	−	0.585569i	$-0.800871\pi$
$383$		−	22.9196i		−	1.17114i	0.810623	−	0.585569i	$-0.199129\pi$
$384$	−1.00000			−0.0510310
$385$	0.0495969	+	2.10532i	0.00252769	+	0.107297i
$386$	−8.04960			−0.409714
$387$		−	0.210649i		−	0.0107079i
$388$			3.52910i			0.179163i
$389$	21.0360			1.06657			0.533284	−	0.845936i	$-0.320958\pi$
$389$	21.0360			1.06657			0.533284	+	0.845936i	$0.320958\pi$
$390$	0.340772	+	14.4653i	0.0172557	+	0.732481i
$391$	48.9196			2.47397
$392$		−	3.00000i		−	0.151523i
$393$			3.15244i			0.159020i
$394$	−16.3049			−0.821428
$395$	−4.23545	+	0.0997780i	−0.213109	+	0.00502038i
$396$	−0.470896			−0.0236634
$397$		−	38.4983i		−	1.93217i	−0.258216	−	0.966087i	$-0.583135\pi$
$397$		−	38.4983i		−	1.93217i	0.258216	−	0.966087i	$-0.416865\pi$
$398$			3.98891i			0.199946i
$399$	14.0942			0.705594
$400$	4.99445	−	0.235448i	0.249723	−	0.0117724i
$401$	−12.3545			−0.616953			−0.308477	−	0.951232i	$-0.599819\pi$
$401$	−12.3545			−0.616953			−0.308477	+	0.951232i	$0.599819\pi$
$402$		−	8.26025i		−	0.411984i
$403$			6.47090i			0.322338i
$404$		0			0
$405$	−2.23545	+	0.0526623i	−0.111080	+	0.00261681i
$406$	−13.8836			−0.689031
$407$			0.842597i			0.0417660i
$408$		−	7.04712i		−	0.348884i
$409$	32.6147			1.61269			0.806347	−	0.591443i	$-0.201441\pi$
$409$	32.6147			1.61269			0.806347	+	0.591443i	$0.201441\pi$
$410$	−0.105325	−	4.47090i	−0.00520161	−	0.220802i
$411$	9.67293			0.477131
$412$		−	6.00000i		−	0.295599i
$413$			14.3049i			0.703897i
$414$	−6.94179			−0.341170
$415$	−0.620270	−	26.3297i	−0.0304479	−	1.29247i
$416$	6.47090			0.317262
$417$			5.05821i			0.247702i
$418$		−	3.31846i		−	0.162311i
$419$	−10.9418			−0.534541			−0.267271	−	0.963621i	$-0.586122\pi$
$419$	−10.9418			−0.534541			−0.267271	+	0.963621i	$0.586122\pi$
$420$	4.47090	−	0.105325i	0.218157	−	0.00513932i
$421$	−22.4213			−1.09275			−0.546374	−	0.837542i	$-0.683992\pi$
$421$	−22.4213			−1.09275			−0.546374	+	0.837542i	$0.683992\pi$
$422$			2.11642i			0.103026i
$423$			7.04712i			0.342642i
$424$	3.15244			0.153096
$425$	1.65923	+	35.1965i	0.0804844	+	1.70728i
$426$	0.260246			0.0126090
$427$			22.5155i			1.08960i
$428$		−	16.7311i		−	0.808730i
$429$	3.04712			0.147116
$430$	0.470896	−	0.0110933i	0.0227086	−	0.000534966i
$431$	33.1303			1.59583			0.797914	−	0.602771i	$-0.205937\pi$
$431$	33.1303			1.59583			0.797914	+	0.602771i	$0.205937\pi$
$432$			1.00000i			0.0481125i
$433$			9.78935i			0.470446i	0.971941	+	0.235223i	$0.0755821\pi$
$433$			9.78935i			0.470446i	−0.971941	+	0.235223i	$0.924418\pi$
$434$	2.00000			0.0960031
$435$	0.365571	+	15.5180i	0.0175278	+	0.744032i
$436$	−11.1524			−0.534105
$437$		−	48.9196i		−	2.34014i
$438$		−	11.8836i		−	0.567820i
$439$	22.6147			1.07934			0.539671	−	0.841876i	$-0.318549\pi$
$439$	22.6147			1.07934			0.539671	+	0.841876i	$0.318549\pi$
$440$	−0.0247985	−	1.05266i	−0.00118222	−	0.0501837i
$441$	−3.00000			−0.142857
$442$			45.6011i			2.16903i
$443$		−	0.210649i		−	0.0100083i	−0.999987	−	0.00500413i	$-0.998407\pi$
$443$		−	0.210649i		−	0.0100083i	0.999987	−	0.00500413i	$-0.00159287\pi$
$444$	1.78935			0.0849188
$445$	27.0360	−	0.636910i	1.28163	−	0.0301924i
$446$	−2.58731			−0.122513
$447$			13.4127i			0.634398i
$448$		−	2.00000i		−	0.0944911i
$449$	27.4127			1.29368			0.646842	−	0.762624i	$-0.276089\pi$
$449$	27.4127			1.29368			0.646842	+	0.762624i	$0.276089\pi$
$450$	−0.235448	−	4.99445i	−0.0110991	−	0.235441i
$451$	−0.941791			−0.0443472
$452$			16.3049i			0.766917i
$453$		−	19.5676i		−	0.919366i
$454$	−4.94179			−0.231930
$455$	−28.9307	+	0.681545i	−1.35629	+	0.0319513i
$456$	−7.04712			−0.330011
$457$			26.9196i			1.25925i	0.776901	+	0.629623i	$0.216791\pi$
$457$			26.9196i			1.25925i	−0.776901	+	0.629623i	$0.783209\pi$
$458$			11.6791i			0.545727i
$459$	−7.04712			−0.328931
$460$	−0.365571	−	15.5180i	−0.0170448	−	0.723531i
$461$	29.0360			1.35234			0.676171	−	0.736745i	$-0.263638\pi$
$461$	29.0360			1.35234			0.676171	+	0.736745i	$0.263638\pi$
$462$		−	0.941791i		−	0.0438161i
$463$			16.8922i			0.785047i	0.919742	+	0.392523i	$0.128398\pi$
$463$			16.8922i			0.785047i	−0.919742	+	0.392523i	$0.871602\pi$
$464$	6.94179			0.322265
$465$	−0.0526623	−	2.23545i	−0.00244216	−	0.103666i
$466$	13.7894			0.638779
$467$			7.76716i			0.359421i	0.983719	+	0.179711i	$0.0575162\pi$
$467$			7.76716i			0.359421i	−0.983719	+	0.179711i	$0.942484\pi$
$468$		−	6.47090i		−	0.299117i
$469$	16.5205			0.762845
$470$	−15.7535	+	0.371118i	−0.726653	+	0.0171184i
$471$	−21.2467			−0.978995
$472$		−	7.15244i		−	0.329218i
$473$		−	0.0991938i		−	0.00456094i
$474$	1.89468			0.0870253
$475$	35.1965	−	1.65923i	1.61493	−	0.0761306i
$476$	14.0942			0.646008
$477$		−	3.15244i		−	0.144340i
$478$			12.7311i			0.582309i
$479$	−32.4487			−1.48262			−0.741310	−	0.671163i	$-0.765795\pi$
$479$	−32.4487			−1.48262			−0.741310	+	0.671163i	$0.765795\pi$
$480$	−2.23545	+	0.0526623i	−0.102034	+	0.00240370i
$481$	−11.5787			−0.527943
$482$		−	3.15244i		−	0.143590i
$483$		−	13.8836i		−	0.631725i
$484$	10.7783			0.489921
$485$	0.185851	+	7.88913i	0.00843905	+	0.358227i
$486$	1.00000			0.0453609
$487$			35.4847i			1.60797i	0.594652	+	0.803983i	$0.297290\pi$
$487$			35.4847i			1.60797i	−0.594652	+	0.803983i	$0.702710\pi$
$488$		−	11.2578i		−	0.509615i
$489$	4.26025			0.192655
$490$	−0.157987	−	6.70634i	−0.00713712	−	0.302962i
$491$	25.9828			1.17259			0.586293	−	0.810099i	$-0.300587\pi$
$491$	25.9828			1.17259			0.586293	+	0.810099i	$0.300587\pi$
$492$			2.00000i			0.0901670i
$493$			48.9196i			2.20323i
$494$	45.6011			2.05169
$495$	−1.05266	+	0.0247985i	−0.0473137	+	0.00111461i
$496$	−1.00000			−0.0449013
$497$			0.520492i			0.0233473i
$498$			11.7783i			0.527796i
$499$	−7.90577			−0.353911			−0.176955	−	0.984219i	$-0.556625\pi$
$499$	−7.90577			−0.353911			−0.176955	+	0.984219i	$0.556625\pi$
$500$	11.1524	−	0.789351i	0.498752	−	0.0353008i
$501$	2.00000			0.0893534
$502$			3.05821i			0.136495i
$503$		−	9.14135i		−	0.407593i	−0.979013	−	0.203796i	$-0.934672\pi$
$503$		−	9.14135i		−	0.407593i	0.979013	−	0.203796i	$-0.0653280\pi$
$504$	−2.00000			−0.0890871
$505$		0			0
$506$	−3.26886			−0.145318
$507$			28.8725i			1.28227i
$508$		−	0.941791i		−	0.0417852i
$509$	−28.5155			−1.26393			−0.631964	−	0.774997i	$-0.717751\pi$
$509$	−28.5155			−1.26393			−0.631964	+	0.774997i	$0.717751\pi$
$510$	−0.371118	−	15.7535i	−0.0164334	−	0.697575i
$511$	23.7672			1.05140
$512$			1.00000i			0.0441942i
$513$			7.04712i			0.311138i
$514$	16.0942			0.709886
$515$	−0.315974	−	13.4127i	−0.0139235	−	0.591034i
$516$	−0.210649			−0.00927332
$517$			3.31846i			0.145945i
$518$			3.57870i			0.157239i
$519$	−17.9889			−0.789625
$520$	14.4653	−	0.340772i	0.634348	−	0.0149439i
$521$	23.3631			1.02356			0.511778	−	0.859118i	$-0.328987\pi$
$521$	23.3631			1.02356			0.511778	+	0.859118i	$0.328987\pi$
$522$		−	6.94179i		−	0.303834i
$523$			42.0942i			1.84065i	0.391152	+	0.920326i	$0.372077\pi$
$523$			42.0942i			1.84065i	−0.391152	+	0.920326i	$0.627923\pi$
$524$	3.15244			0.137715
$525$	9.98891	−	0.470896i	0.435952	−	0.0205516i
$526$	−0.0942314			−0.00410868
$527$		−	7.04712i		−	0.306977i
$528$			0.470896i			0.0204931i
$529$	−25.1885			−1.09515
$530$	7.04712	−	0.166015i	0.306107	−	0.00721122i
$531$	−7.15244			−0.310389
$532$		−	14.0942i		−	0.611062i
$533$		−	12.9418i		−	0.560571i
$534$	−12.0942			−0.523369
$535$	−0.881101	−	37.4016i	−0.0380933	−	1.61701i
$536$	−8.26025			−0.356788
$537$		−	12.0496i		−	0.519978i
$538$			23.8836i			1.02969i
$539$	−1.41269			−0.0608487
$540$	0.0526623	+	2.23545i	0.00226623	+	0.0961984i
$541$	−44.5925			−1.91718			−0.958591	−	0.284785i	$-0.908078\pi$
$541$	−44.5925			−1.91718			−0.958591	+	0.284785i	$0.908078\pi$
$542$			24.4102i			1.04851i
$543$			13.6729i			0.586762i
$544$	−7.04712			−0.302143
$545$	−24.9307	+	0.587313i	−1.06791	+	0.0251577i
$546$	12.9418			0.553858
$547$			4.52049i			0.193282i	0.995319	+	0.0966411i	$0.0308099\pi$
$547$			4.52049i			0.193282i	−0.995319	+	0.0966411i	$0.969190\pi$
$548$		−	9.67293i		−	0.413207i
$549$	−11.2578			−0.480470
$550$	−0.110871	−	2.35187i	−0.00472757	−	0.100284i
$551$	48.9196			2.08405
$552$			6.94179i			0.295462i
$553$			3.78935i			0.161140i
$554$	−7.41269			−0.314935
$555$	4.00000	−	0.0942314i	0.169791	−	0.00399990i
$556$	5.05821			0.214516
$557$			20.5155i			0.869271i	0.900606	+	0.434635i	$0.143123\pi$
$557$			20.5155i			0.869271i	−0.900606	+	0.434635i	$0.856877\pi$
$558$			1.00000i			0.0423334i
$559$	1.36309			0.0576525
$560$	−0.105325	−	4.47090i	−0.00445078	−	0.188930i
$561$	−3.31846			−0.140105
$562$		−	11.8836i		−	0.501279i
$563$			34.6147i			1.45884i	0.684068	+	0.729418i	$0.260209\pi$
$563$			34.6147i			1.45884i	−0.684068	+	0.729418i	$0.739791\pi$
$564$	7.04712			0.296737
$565$	0.858653	+	36.4487i	0.0361238	+	1.53341i
$566$	−22.9864			−0.966192
$567$			2.00000i			0.0839921i
$568$		−	0.260246i		−	0.0109197i
$569$	−7.25163			−0.304004			−0.152002	−	0.988380i	$-0.548572\pi$
$569$	−7.25163			−0.304004			−0.152002	+	0.988380i	$0.548572\pi$
$570$	−15.7535	+	0.371118i	−0.659840	+	0.0155444i
$571$	−0.0942314			−0.00394346			−0.00197173	−	0.999998i	$-0.500628\pi$
$571$	−0.0942314			−0.00394346			−0.00197173	+	0.999998i	$0.500628\pi$
$572$		−	3.04712i		−	0.127406i
$573$			14.8254i			0.619339i
$574$	−4.00000			−0.166957
$575$	−1.63443	−	34.6705i	−0.0681604	−	1.44586i
$576$	1.00000			0.0416667
$577$		−	9.41269i		−	0.391855i	−0.980618	−	0.195928i	$-0.937228\pi$
$577$		−	9.41269i		−	0.391855i	0.980618	−	0.195928i	$-0.0627718\pi$
$578$		−	32.6618i		−	1.35855i
$579$	8.04960			0.334530
$580$	15.5180	−	0.365571i	0.644350	−	0.0151795i
$581$	−23.5565			−0.977289
$582$		−	3.52910i		−	0.146286i
$583$		−	1.48447i		−	0.0614805i
$584$	−11.8836			−0.491746
$585$	−0.340772	−	14.4653i	−0.0140892	−	0.598069i
$586$	14.4213			0.595738
$587$		−	42.2938i		−	1.74565i	−0.488032	−	0.872826i	$-0.662285\pi$
$587$		−	42.2938i		−	1.74565i	0.488032	−	0.872826i	$-0.337715\pi$
$588$			3.00000i			0.123718i
$589$	−7.04712			−0.290371
$590$	−0.376664	−	15.9889i	−0.0155070	−	0.658253i
$591$	16.3049			0.670693
$592$		−	1.78935i		−	0.0735419i
$593$		−	5.88854i		−	0.241814i	−0.992664	−	0.120907i	$-0.961420\pi$
$593$		−	5.88854i		−	0.241814i	0.992664	−	0.120907i	$-0.0385802\pi$
$594$	0.470896			0.0193211
$595$	31.5069	−	0.742235i	1.29166	−	0.0304287i
$596$	13.4127			0.549405
$597$		−	3.98891i		−	0.163255i
$598$		−	44.9196i		−	1.83690i
$599$	0.780739			0.0319001			0.0159501	−	0.999873i	$-0.494923\pi$
$599$	0.780739			0.0319001			0.0159501	+	0.999873i	$0.494923\pi$
$600$	−4.99445	+	0.235448i	−0.203898	+	0.00961211i
$601$	−25.5787			−1.04338			−0.521688	−	0.853136i	$-0.674698\pi$
$601$	−25.5787			−1.04338			−0.521688	+	0.853136i	$0.674698\pi$
$602$		−	0.421299i		−	0.0171708i
$603$			8.26025i			0.336383i
$604$	−19.5676			−0.796195
$605$	24.0942	−	0.567608i	0.979570	−	0.0230766i
$606$		0			0
$607$		−	7.46228i		−	0.302885i	−0.988466	−	0.151442i	$-0.951608\pi$
$607$		−	7.46228i		−	0.302885i	0.988466	−	0.151442i	$-0.0483918\pi$
$608$			7.04712i			0.285798i
$609$	13.8836			0.562591
$610$	−0.592860	−	25.1661i	−0.0240042	−	1.01895i
$611$	−45.6011			−1.84483
$612$			7.04712i			0.284863i
$613$		−	4.68651i		−	0.189286i	−0.995511	−	0.0946431i	$-0.969829\pi$
$613$		−	4.68651i		−	0.189286i	0.995511	−	0.0946431i	$-0.0301710\pi$
$614$	24.9418			1.00657
$615$	0.105325	+	4.47090i	0.00424710	+	0.180284i
$616$	−0.941791			−0.0379458
$617$		−	7.69512i		−	0.309794i	−0.987931	−	0.154897i	$-0.950495\pi$
$617$		−	7.69512i		−	0.309794i	0.987931	−	0.154897i	$-0.0495045\pi$
$618$			6.00000i			0.241355i
$619$	−33.7894			−1.35811			−0.679054	−	0.734088i	$-0.737610\pi$
$619$	−33.7894			−1.35811			−0.679054	+	0.734088i	$0.737610\pi$
$620$	−2.23545	+	0.0526623i	−0.0897777	+	0.00211497i
$621$	6.94179			0.278564
$622$		−	24.4487i		−	0.980304i
$623$		−	24.1885i		−	0.969090i
$624$	−6.47090			−0.259043
$625$	24.8891	−	2.35187i	0.995565	−	0.0940746i
$626$	14.1885			0.567085
$627$			3.31846i			0.132526i
$628$			21.2467i			0.847835i
$629$	12.6098			0.502784
$630$	−4.47090	+	0.105325i	−0.178125	+	0.00419623i
$631$	−34.3049			−1.36566			−0.682828	−	0.730579i	$-0.739250\pi$
$631$	−34.3049			−1.36566			−0.682828	+	0.730579i	$0.739250\pi$
$632$		−	1.89468i		−	0.0753661i
$633$		−	2.11642i		−	0.0841201i
$634$	21.9889			0.873291
$635$	−0.0495969	−	2.10532i	−0.00196819	−	0.0835473i
$636$	−3.15244			−0.125002
$637$		−	19.4127i		−	0.769159i
$638$		−	3.26886i		−	0.129415i
$639$	−0.260246			−0.0102952
$640$	0.0526623	+	2.23545i	0.00208166	+	0.0883638i
$641$	3.83399			0.151433			0.0757167	−	0.997129i	$-0.475876\pi$
$641$	3.83399			0.151433			0.0757167	+	0.997129i	$0.475876\pi$
$642$			16.7311i			0.660325i
$643$			12.9640i			0.511249i	0.966776	+	0.255625i	$0.0822811\pi$
$643$			12.9640i			0.511249i	−0.966776	+	0.255625i	$0.917719\pi$
$644$	−13.8836			−0.547090
$645$	−0.470896	+	0.0110933i	−0.0185415	+	0.000436798i
$646$	−49.6618			−1.95392
$647$		−	29.3459i		−	1.15371i	−0.816848	−	0.576853i	$-0.804281\pi$
$647$		−	29.3459i		−	1.15371i	0.816848	−	0.576853i	$-0.195719\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−3.36805			−0.132208
$650$	32.3186	−	1.52356i	1.26764	−	0.0597589i
$651$	−2.00000			−0.0783862
$652$		−	4.26025i		−	0.166844i
$653$		−	4.52662i		−	0.177140i	−0.996070	−	0.0885702i	$-0.971770\pi$
$653$		−	4.52662i		−	0.177140i	0.996070	−	0.0885702i	$-0.0282298\pi$
$654$	11.1524			0.436095
$655$	7.04712	−	0.166015i	0.275354	−	0.00648674i
$656$	2.00000			0.0780869
$657$			11.8836i			0.463623i
$658$			14.0942i			0.549450i
$659$	15.8836			0.618737			0.309368	−	0.950942i	$-0.399882\pi$
$659$	15.8836			0.618737			0.309368	+	0.950942i	$0.399882\pi$
$660$	0.0247985	+	1.05266i	0.000965278	+	0.0409748i
$661$	12.8254			0.498849			0.249425	−	0.968394i	$-0.419759\pi$
$661$	12.8254			0.498849			0.249425	+	0.968394i	$0.419759\pi$
$662$		−	15.3631i		−	0.597103i
$663$		−	45.6011i		−	1.77100i
$664$	11.7783			0.457085
$665$	−0.742235	−	31.5069i	−0.0287826	−	1.22179i
$666$	−1.78935			−0.0693359
$667$		−	48.1885i		−	1.86586i
$668$		−	2.00000i		−	0.0773823i
$669$	2.58731			0.100031
$670$	−18.4653	+	0.435004i	−0.713379	+	0.0168057i
$671$	−5.30123			−0.204652
$672$			2.00000i			0.0771517i
$673$			27.9828i			1.07866i	0.842096	+	0.539328i	$0.181322\pi$
$673$			27.9828i			1.07866i	−0.842096	+	0.539328i	$0.818678\pi$
$674$	−13.9778			−0.538405
$675$	0.235448	+	4.99445i	0.00906239	+	0.192237i
$676$	28.8725			1.11048
$677$			27.6729i			1.06356i	0.846883	+	0.531779i	$0.178476\pi$
$677$			27.6729i			1.06356i	−0.846883	+	0.531779i	$0.821524\pi$
$678$		−	16.3049i		−	0.626185i
$679$	7.05821			0.270869
$680$	−15.7535	+	0.371118i	−0.604118	+	0.0142317i
$681$	4.94179			0.189370
$682$			0.470896i			0.0180315i
$683$		−	8.18846i		−	0.313323i	−0.987652	−	0.156661i	$-0.949927\pi$
$683$		−	8.18846i		−	0.313323i	0.987652	−	0.156661i	$-0.0500731\pi$
$684$	7.04712			0.269453
$685$	−0.509399	−	21.6233i	−0.0194631	−	0.826185i
$686$	−20.0000			−0.763604
$687$		−	11.6791i		−	0.445584i
$688$			0.210649i			0.00803093i
$689$	20.3991			0.777144
$690$	0.365571	+	15.5180i	0.0139170	+	0.590761i
$691$	22.4152			0.852713			0.426357	−	0.904555i	$-0.359797\pi$
$691$	22.4152			0.852713			0.426357	+	0.904555i	$0.359797\pi$
$692$			17.9889i			0.683836i
$693$			0.941791i			0.0357757i
$694$	−17.6618			−0.670434
$695$	11.3074	−	0.266377i	0.428913	−	0.0101043i
$696$	−6.94179			−0.263128
$697$			14.0942i			0.533857i
$698$			7.67293i			0.290425i
$699$	−13.7894			−0.521561
$700$	−0.470896	−	9.98891i	−0.0177982	−	0.377545i
$701$	−20.4709			−0.773175			−0.386588	−	0.922253i	$-0.626346\pi$
$701$	−20.4709			−0.773175			−0.386588	+	0.922253i	$0.626346\pi$
$702$			6.47090i			0.244228i
$703$		−	12.6098i		−	0.475586i
$704$	0.470896			0.0177475
$705$	15.7535	−	0.371118i	0.593310	−	0.0139771i
$706$	11.0471			0.415764
$707$		0			0
$708$			7.15244i			0.268805i
$709$	48.7200			1.82972			0.914860	−	0.403771i	$-0.132301\pi$
$709$	48.7200			1.82972			0.914860	+	0.403771i	$0.132301\pi$
$710$	−0.0137052	−	0.581767i	−0.000514346	−	0.0218333i
$711$	−1.89468			−0.0710559
$712$			12.0942i			0.453250i
$713$			6.94179i			0.259972i
$714$	−14.0942			−0.527463
$715$	−0.160468	−	6.81167i	−0.00600117	−	0.254742i
$716$	−12.0496			−0.450315
$717$		−	12.7311i		−	0.475453i
$718$			35.9282i			1.34083i
$719$	−50.8032			−1.89464			−0.947320	−	0.320290i	$-0.896220\pi$
$719$	−50.8032			−1.89464			−0.947320	+	0.320290i	$0.896220\pi$
$720$	2.23545	−	0.0526623i	0.0833102	−	0.00196261i
$721$	−12.0000			−0.446903
$722$			30.6618i			1.14112i
$723$			3.15244i			0.117241i
$724$	13.6729			0.508151
$725$	34.6705	−	1.63443i	1.28763	−	0.0607012i
$726$	−10.7783			−0.400019
$727$			33.1352i			1.22892i	0.788949	+	0.614459i	$0.210625\pi$
$727$			33.1352i			1.22892i	−0.788949	+	0.614459i	$0.789375\pi$
$728$		−	12.9418i		−	0.479655i
$729$	−1.00000			−0.0370370
$730$	−26.5651	+	0.625817i	−0.983219	+	0.0231625i
$731$	−1.48447			−0.0549051
$732$			11.2578i			0.416099i
$733$		−	2.96398i		−	0.109477i	−0.998501	−	0.0547385i	$-0.982567\pi$
$733$		−	2.96398i		−	0.109477i	0.998501	−	0.0547385i	$-0.0174325\pi$
$734$	−19.2963			−0.712238
$735$	0.157987	+	6.70634i	0.00582744	+	0.247367i
$736$	6.94179			0.255878
$737$			3.88971i			0.143279i
$738$		−	2.00000i		−	0.0736210i
$739$	6.52049			0.239860			0.119930	−	0.992782i	$-0.461733\pi$
$739$	6.52049			0.239860			0.119930	+	0.992782i	$0.461733\pi$
$740$	−0.0942314	−	4.00000i	−0.00346401	−	0.147043i
$741$	−45.6011			−1.67520
$742$		−	6.30488i		−	0.231459i
$743$			20.0942i			0.737186i	0.929591	+	0.368593i	$0.120160\pi$
$743$			20.0942i			0.737186i	−0.929591	+	0.368593i	$0.879840\pi$
$744$	1.00000			0.0366618
$745$	29.9834	−	0.706343i	1.09851	−	0.0258784i
$746$	−24.0942			−0.882152
$747$		−	11.7783i		−	0.430944i
$748$			3.31846i			0.121335i
$749$	−33.4623			−1.22269
$750$	−11.1524	+	0.789351i	−0.407230	+	0.0288230i
$751$	−46.0942			−1.68200			−0.841001	−	0.541033i	$-0.818033\pi$
$751$	−46.0942			−1.68200			−0.841001	+	0.541033i	$0.818033\pi$
$752$		−	7.04712i		−	0.256982i
$753$		−	3.05821i		−	0.111447i
$754$	44.9196			1.63588
$755$	−43.7424	+	1.03048i	−1.59195	+	0.0375029i
$756$	2.00000			0.0727393
$757$		−	28.2827i		−	1.02795i	−0.857805	−	0.513976i	$-0.828172\pi$
$757$		−	28.2827i		−	1.02795i	0.857805	−	0.513976i	$-0.171828\pi$
$758$		−	25.1413i		−	0.913175i
$759$	3.26886			0.118652
$760$	0.371118	+	15.7535i	0.0134618	+	0.571438i
$761$	−41.0634			−1.48855			−0.744274	−	0.667874i	$-0.767204\pi$
$761$	−41.0634			−1.48855			−0.744274	+	0.667874i	$0.767204\pi$
$762$			0.941791i			0.0341175i
$763$			22.3049i			0.807491i
$764$	14.8254			0.536363
$765$	0.371118	+	15.7535i	0.0134178	+	0.569568i
$766$	22.9196			0.828119
$767$		−	46.2827i		−	1.67117i
$768$		−	1.00000i		−	0.0360844i
$769$	32.4933			1.17174			0.585870	−	0.810405i	$-0.300753\pi$
$769$	32.4933			1.17174			0.585870	+	0.810405i	$0.300753\pi$
$770$	−2.10532	+	0.0495969i	−0.0758706	+	0.00178735i
$771$	−16.0942			−0.579620
$772$		−	8.04960i		−	0.289711i
$773$		−	50.3991i		−	1.81273i	−0.422495	−	0.906365i	$-0.638846\pi$
$773$		−	50.3991i		−	1.81273i	0.422495	−	0.906365i	$-0.361154\pi$
$774$	0.210649			0.00757163
$775$	−4.99445	+	0.235448i	−0.179406	+	0.00845753i
$776$	−3.52910			−0.126687
$777$		−	3.57870i		−	0.128385i
$778$			21.0360i			0.754178i
$779$	14.0942			0.504978
$780$	−14.4653	+	0.340772i	−0.517943	+	0.0122016i
$781$	−0.122549

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-2.23545 + 0.0526623i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-2.23545 + 0.0526623i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(-0.0526623 - 2.23545i) q^{10} -0.470896 q^{11} +1.00000i q^{12} -6.47090i q^{13} -2.00000 q^{14} +(0.0526623 + 2.23545i) q^{15} +1.00000 q^{16} +7.04712i q^{17} -1.00000i q^{18} +7.04712 q^{19} +(2.23545 - 0.0526623i) q^{20} +2.00000 q^{21} -0.470896i q^{22} -6.94179i q^{23} -1.00000 q^{24} +(4.99445 - 0.235448i) q^{25} +6.47090 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.94179 q^{29} +(-2.23545 + 0.0526623i) q^{30} -1.00000 q^{31} +1.00000i q^{32} +0.470896i q^{33} -7.04712 q^{34} +(-0.105325 - 4.47090i) q^{35} +1.00000 q^{36} -1.78935i q^{37} +7.04712i q^{38} -6.47090 q^{39} +(0.0526623 + 2.23545i) q^{40} +2.00000 q^{41} +2.00000i q^{42} +0.210649i q^{43} +0.470896 q^{44} +(2.23545 - 0.0526623i) q^{45} +6.94179 q^{46} -7.04712i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(0.235448 + 4.99445i) q^{50} +7.04712 q^{51} +6.47090i q^{52} +3.15244i q^{53} -1.00000 q^{54} +(1.05266 - 0.0247985i) q^{55} +2.00000 q^{56} -7.04712i q^{57} +6.94179i q^{58} +7.15244 q^{59} +(-0.0526623 - 2.23545i) q^{60} +11.2578 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(0.340772 + 14.4653i) q^{65} -0.470896 q^{66} -8.26025i q^{67} -7.04712i q^{68} -6.94179 q^{69} +(4.47090 - 0.105325i) q^{70} +0.260246 q^{71} +1.00000i q^{72} -11.8836i q^{73} +1.78935 q^{74} +(-0.235448 - 4.99445i) q^{75} -7.04712 q^{76} -0.941791i q^{77} -6.47090i q^{78} +1.89468 q^{79} +(-2.23545 + 0.0526623i) q^{80} +1.00000 q^{81} +2.00000i q^{82} +11.7783i q^{83} -2.00000 q^{84} +(-0.371118 - 15.7535i) q^{85} -0.210649 q^{86} -6.94179i q^{87} +0.470896i q^{88} -12.0942 q^{89} +(0.0526623 + 2.23545i) q^{90} +12.9418 q^{91} +6.94179i q^{92} +1.00000i q^{93} +7.04712 q^{94} +(-15.7535 + 0.371118i) q^{95} +1.00000 q^{96} -3.52910i q^{97} +3.00000i q^{98} +0.470896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9	\( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 + 16 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 + 4 * q^20 + 12 * q^21 - 6 * q^24 - 8 * q^25 + 20 * q^26 + 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^34 + 6 * q^36 - 20 * q^39 + 12 * q^41 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 4 * q^51 - 6 * q^54 + 6 * q^55 + 12 * q^56 + 4 * q^59 + 28 * q^61 - 6 * q^64 - 8 * q^65 + 16 * q^66 - 4 * q^69 + 8 * q^70 - 16 * q^71 + 12 * q^74 + 8 * q^75 - 4 * q^76 + 12 * q^79 - 4 * q^80 + 6 * q^81 - 12 * q^84 - 22 * q^85 + 4 * q^89 + 40 * q^91 + 4 * q^94 - 28 * q^95 + 6 * q^96 - 16 * q^99

Introduction

L-functions

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