LMFDB - Embedded newform 930.2.d.i.559.5

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			559.5
Root			$0.630356 - 2.20530i$ of defining polynomial
Character	$\chi$	$=$	930.559
Dual form			930.2.d.i.559.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-0.369644 - 2.20530i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-0.369644 - 2.20530i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.20530 - 0.369644i) q^{10} +3.26071 q^{11} +1.00000i q^{12} -2.73929i q^{13} -2.00000 q^{14} +(-2.20530 + 0.369644i) q^{15} +1.00000 q^{16} -4.93203i q^{17} -1.00000i q^{18} -4.93203 q^{19} +(0.369644 + 2.20530i) q^{20} +2.00000 q^{21} +3.26071i q^{22} +0.521423i q^{23} -1.00000 q^{24} +(-4.72673 + 1.63036i) q^{25} +2.73929 q^{26} +1.00000i q^{27} -2.00000i q^{28} -0.521423 q^{29} +(-0.369644 - 2.20530i) q^{30} -1.00000 q^{31} +1.00000i q^{32} -3.26071i q^{33} +4.93203 q^{34} +(4.41061 - 0.739289i) q^{35} +1.00000 q^{36} -10.8212i q^{37} -4.93203i q^{38} -2.73929 q^{39} +(-2.20530 + 0.369644i) q^{40} +2.00000 q^{41} +2.00000i q^{42} -8.82121i q^{43} -3.26071 q^{44} +(0.369644 + 2.20530i) q^{45} -0.521423 q^{46} +4.93203i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-1.63036 - 4.72673i) q^{50} -4.93203 q^{51} +2.73929i q^{52} -13.3426i q^{53} -1.00000 q^{54} +(-1.20530 - 7.19086i) q^{55} +2.00000 q^{56} +4.93203i q^{57} -0.521423i q^{58} -9.34264 q^{59} +(2.20530 - 0.369644i) q^{60} -9.75324 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(-6.04096 + 1.01256i) q^{65} +3.26071 q^{66} -13.5605i q^{67} +4.93203i q^{68} +0.521423 q^{69} +(0.739289 + 4.41061i) q^{70} +5.56050 q^{71} +1.00000i q^{72} +3.04285i q^{73} +10.8212 q^{74} +(1.63036 + 4.72673i) q^{75} +4.93203 q^{76} +6.52142i q^{77} -2.73929i q^{78} +6.41061 q^{79} +(-0.369644 - 2.20530i) q^{80} +1.00000 q^{81} +2.00000i q^{82} +1.36776i q^{83} -2.00000 q^{84} +(-10.8766 + 1.82310i) q^{85} +8.82121 q^{86} +0.521423i q^{87} -3.26071i q^{88} +11.8641 q^{89} +(-2.20530 + 0.369644i) q^{90} +5.47858 q^{91} -0.521423i q^{92} +1.00000i q^{93} -4.93203 q^{94} +(1.82310 + 10.8766i) q^{95} +1.00000 q^{96} -7.26071i q^{97} +3.00000i q^{98} -3.26071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.369644	−	2.20530i	−0.165310	−	0.986242i
$5$	−0.369644	−	2.20530i	−0.165310	−	0.986242i
$6$	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
$10$	2.20530	−	0.369644i	0.697378	−	0.116892i
$11$	3.26071			0.983141			0.491571	−	0.870838i	$-0.336423\pi$
$11$	3.26071			0.983141			0.491571	+	0.870838i	$0.336423\pi$
$12$			1.00000i			0.288675i
$13$		−	2.73929i		−	0.759742i	−0.925039	−	0.379871i	$-0.875968\pi$
$13$		−	2.73929i		−	0.759742i	0.925039	−	0.379871i	$-0.124032\pi$
$14$	−2.00000			−0.534522
$15$	−2.20530	+	0.369644i	−0.569407	+	0.0954417i
$16$	1.00000			0.250000
$17$		−	4.93203i		−	1.19619i	−0.801424	−	0.598096i	$-0.795924\pi$
$17$		−	4.93203i		−	1.19619i	0.801424	−	0.598096i	$-0.204076\pi$
$18$		−	1.00000i		−	0.235702i
$19$	−4.93203			−1.13149			−0.565743	−	0.824582i	$-0.691410\pi$
$19$	−4.93203			−1.13149			−0.565743	+	0.824582i	$0.691410\pi$
$20$	0.369644	+	2.20530i	0.0826550	+	0.493121i
$21$	2.00000			0.436436
$22$			3.26071i			0.695186i
$23$			0.521423i			0.108724i	0.998521	+	0.0543621i	$0.0173125\pi$
$23$			0.521423i			0.108724i	−0.998521	+	0.0543621i	$0.982687\pi$
$24$	−1.00000			−0.204124
$25$	−4.72673	+	1.63036i	−0.945345	+	0.326071i
$26$	2.73929			0.537219
$27$			1.00000i			0.192450i
$28$		−	2.00000i		−	0.377964i
$29$	−0.521423			−0.0968258			−0.0484129	−	0.998827i	$-0.515416\pi$
$29$	−0.521423			−0.0968258			−0.0484129	+	0.998827i	$0.515416\pi$
$30$	−0.369644	−	2.20530i	−0.0674875	−	0.402631i
$31$	−1.00000			−0.179605
$31$	−1.00000			−0.179605
$32$			1.00000i			0.176777i
$33$		−	3.26071i		−	0.567617i
$34$	4.93203			0.845836
$35$	4.41061	−	0.739289i	0.745529	−	0.124963i
$36$	1.00000			0.166667
$37$		−	10.8212i		−	1.77900i	−0.456939	−	0.889498i	$-0.651054\pi$
$37$		−	10.8212i		−	1.77900i	0.456939	−	0.889498i	$-0.348946\pi$
$38$		−	4.93203i		−	0.800081i
$39$	−2.73929			−0.438637
$40$	−2.20530	+	0.369644i	−0.348689	+	0.0584459i
$41$	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
$43$		−	8.82121i		−	1.34522i	−0.739996	−	0.672611i	$-0.765173\pi$
$43$		−	8.82121i		−	1.34522i	0.739996	−	0.672611i	$-0.234827\pi$
$44$	−3.26071			−0.491571
$45$	0.369644	+	2.20530i	0.0551033	+	0.328747i
$46$	−0.521423			−0.0768796
$47$			4.93203i			0.719410i	0.933066	+	0.359705i	$0.117123\pi$
$47$			4.93203i			0.719410i	−0.933066	+	0.359705i	$0.882877\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	−1.63036	−	4.72673i	−0.230567	−	0.668460i
$51$	−4.93203			−0.690622
$52$			2.73929i			0.379871i
$53$		−	13.3426i		−	1.83275i	−0.400319	−	0.916376i	$-0.631101\pi$
$53$		−	13.3426i		−	1.83275i	0.400319	−	0.916376i	$-0.368899\pi$
$54$	−1.00000			−0.136083
$55$	−1.20530	−	7.19086i	−0.162523	−	0.969615i
$56$	2.00000			0.267261
$57$			4.93203i			0.653263i
$58$		−	0.521423i		−	0.0684662i
$59$	−9.34264			−1.21631			−0.608154	−	0.793819i	$-0.708090\pi$
$59$	−9.34264			−1.21631			−0.608154	+	0.793819i	$0.708090\pi$
$60$	2.20530	−	0.369644i	0.284703	−	0.0477209i
$61$	−9.75324			−1.24877			−0.624387	−	0.781115i	$-0.714651\pi$
$61$	−9.75324			−1.24877			−0.624387	+	0.781115i	$0.714651\pi$
$62$		−	1.00000i		−	0.127000i
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$	−6.04096	+	1.01256i	−0.749289	+	0.125593i
$66$	3.26071			0.401366
$67$		−	13.5605i		−	1.65668i	−0.560226	−	0.828340i	$-0.689286\pi$
$67$		−	13.5605i		−	1.65668i	0.560226	−	0.828340i	$-0.310714\pi$
$68$			4.93203i			0.598096i
$69$	0.521423			0.0627719
$70$	0.739289	+	4.41061i	0.0883619	+	0.527168i
$71$	5.56050			0.659910			0.329955	−	0.943997i	$-0.392966\pi$
$71$	5.56050			0.659910			0.329955	+	0.943997i	$0.392966\pi$
$72$			1.00000i			0.117851i
$73$			3.04285i			0.356138i	0.984018	+	0.178069i	$0.0569850\pi$
$73$			3.04285i			0.356138i	−0.984018	+	0.178069i	$0.943015\pi$
$74$	10.8212			1.25794
$75$	1.63036	+	4.72673i	0.188257	+	0.545795i
$76$	4.93203			0.565743
$77$			6.52142i			0.743185i
$78$		−	2.73929i		−	0.310163i
$79$	6.41061			0.721250			0.360625	−	0.932711i	$-0.382563\pi$
$79$	6.41061			0.721250			0.360625	+	0.932711i	$0.382563\pi$
$80$	−0.369644	−	2.20530i	−0.0413275	−	0.246560i
$81$	1.00000			0.111111
$82$			2.00000i			0.220863i
$83$			1.36776i			0.150131i	0.997179	+	0.0750657i	$0.0239166\pi$
$83$			1.36776i			0.150131i	−0.997179	+	0.0750657i	$0.976083\pi$
$84$	−2.00000			−0.218218
$85$	−10.8766	+	1.82310i	−1.17974	+	0.197743i
$86$	8.82121			0.951216
$87$			0.521423i			0.0559024i
$88$		−	3.26071i		−	0.347593i
$89$	11.8641			1.25759			0.628794	−	0.777572i	$-0.283549\pi$
$89$	11.8641			1.25759			0.628794	+	0.777572i	$0.283549\pi$
$90$	−2.20530	+	0.369644i	−0.232459	+	0.0389639i
$91$	5.47858			0.574311
$92$		−	0.521423i		−	0.0543621i
$93$			1.00000i			0.103695i
$94$	−4.93203			−0.508700
$95$	1.82310	+	10.8766i	0.187046	+	1.11592i
$96$	1.00000			0.102062
$97$		−	7.26071i		−	0.737214i	−0.929585	−	0.368607i	$-0.879835\pi$
$97$		−	7.26071i		−	0.737214i	0.929585	−	0.368607i	$-0.120165\pi$
$98$			3.00000i			0.303046i
$99$	−3.26071			−0.327714
$100$	4.72673	−	1.63036i	0.472673	−	0.163036i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		−	4.93203i		−	0.488344i
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−2.73929			−0.268609
$105$	−0.739289	−	4.41061i	−0.0721472	−	0.430431i
$106$	13.3426			1.29595
$107$			18.2998i			1.76911i	0.466438	+	0.884554i	$0.345537\pi$
$107$			18.2998i			1.76911i	−0.466438	+	0.884554i	$0.654463\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	−5.34264			−0.511732			−0.255866	−	0.966712i	$-0.582361\pi$
$109$	−5.34264			−0.511732			−0.255866	+	0.966712i	$0.582361\pi$
$110$	7.19086	−	1.20530i	0.685621	−	0.114921i
$111$	−10.8212			−1.02710
$112$			2.00000i			0.188982i
$113$			16.6853i			1.56962i	0.619737	+	0.784809i	$0.287239\pi$
$113$			16.6853i			1.56962i	−0.619737	+	0.784809i	$0.712761\pi$
$114$	−4.93203			−0.461927
$115$	1.14990	−	0.192741i	0.107228	−	0.0179732i
$116$	0.521423			0.0484129
$117$			2.73929i			0.253247i
$118$		−	9.34264i		−	0.860059i
$119$	9.86406			0.904237
$120$	0.369644	+	2.20530i	0.0337438	+	0.201316i
$121$	−0.367761			−0.0334328
$122$		−	9.75324i		−	0.883017i
$123$		−	2.00000i		−	0.180334i
$124$	1.00000			0.0898027
$125$	5.34264	+	9.82121i	0.477860	+	0.878436i
$126$	2.00000			0.178174
$127$		−	6.52142i		−	0.578683i	−0.957226	−	0.289341i	$-0.906564\pi$
$127$		−	6.52142i		−	0.578683i	0.957226	−	0.289341i	$-0.0934363\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−8.82121			−0.776665
$130$	−1.01256	−	6.04096i	−0.0888076	−	0.529827i
$131$	13.3426			1.16575			0.582876	−	0.812561i	$-0.301927\pi$
$131$	13.3426			1.16575			0.582876	+	0.812561i	$0.301927\pi$
$132$			3.26071i			0.283808i
$133$		−	9.86406i		−	0.855322i
$134$	13.5605			1.17145
$135$	2.20530	−	0.369644i	0.189802	−	0.0318139i
$136$	−4.93203			−0.422918
$137$			3.77837i			0.322808i	0.986888	+	0.161404i	$0.0516022\pi$
$137$			3.77837i			0.322808i	−0.986888	+	0.161404i	$0.948398\pi$
$138$			0.521423i			0.0443865i
$139$	−12.5214			−1.06205			−0.531027	−	0.847355i	$-0.678194\pi$
$139$	−12.5214			−1.06205			−0.531027	+	0.847355i	$0.678194\pi$
$140$	−4.41061	+	0.739289i	−0.372764	+	0.0624813i
$141$	4.93203			0.415352
$142$			5.56050i			0.466627i
$143$		−	8.93203i		−	0.746934i
$144$	−1.00000			−0.0833333
$145$	0.192741	+	1.14990i	0.0160063	+	0.0954936i
$146$	−3.04285			−0.251828
$147$		−	3.00000i		−	0.247436i
$148$			10.8212i			0.889498i
$149$	−2.21787			−0.181695			−0.0908473	−	0.995865i	$-0.528958\pi$
$149$	−2.21787			−0.181695			−0.0908473	+	0.995865i	$0.528958\pi$
$150$	−4.72673	+	1.63036i	−0.385936	+	0.133118i
$151$	18.1890			1.48020			0.740099	−	0.672498i	$-0.234779\pi$
$151$	18.1890			1.48020			0.740099	+	0.672498i	$0.234779\pi$
$152$			4.93203i			0.400040i
$153$			4.93203i			0.398731i
$154$	−6.52142			−0.525511
$155$	0.369644	+	2.20530i	0.0296905	+	0.177134i
$156$	2.73929			0.219319
$157$			19.2067i			1.53286i	0.642327	+	0.766431i	$0.277969\pi$
$157$			19.2067i			1.53286i	−0.642327	+	0.766431i	$0.722031\pi$
$158$			6.41061i			0.510000i
$159$	−13.3426			−1.05814
$160$	2.20530	−	0.369644i	0.174345	−	0.0292229i
$161$	−1.04285			−0.0821877
$162$			1.00000i			0.0785674i
$163$			9.56050i			0.748836i	0.927260	+	0.374418i	$0.122158\pi$
$163$			9.56050i			0.748836i	−0.927260	+	0.374418i	$0.877842\pi$
$164$	−2.00000			−0.156174
$165$	−7.19086	+	1.20530i	−0.559808	+	0.0938327i
$166$	−1.36776			−0.106159
$167$			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
$169$	5.49630			0.422792
$170$	−1.82310	−	10.8766i	−0.139825	−	0.834199i
$171$	4.93203			0.377162
$172$			8.82121i			0.672611i
$173$			1.45345i			0.110504i	0.998472	+	0.0552520i	$0.0175962\pi$
$173$			1.45345i			0.110504i	−0.998472	+	0.0552520i	$0.982404\pi$
$174$	−0.521423			−0.0395290
$175$	−3.26071	−	9.45345i	−0.246487	−	0.714614i
$176$	3.26071			0.245785
$177$			9.34264i			0.702236i
$178$			11.8641i			0.889249i
$179$	26.3817			1.97186			0.985931	−	0.167153i	$-0.0534573\pi$
$179$	26.3817			1.97186			0.985931	+	0.167153i	$0.0534573\pi$
$180$	−0.369644	−	2.20530i	−0.0275517	−	0.164374i
$181$	−7.77837			−0.578162			−0.289081	−	0.957305i	$-0.593350\pi$
$181$	−7.77837			−0.578162			−0.289081	+	0.957305i	$0.593350\pi$
$182$			5.47858i			0.406099i
$183$			9.75324i			0.720980i
$184$	0.521423			0.0384398
$185$	−23.8641	+	4.00000i	−1.75452	+	0.294086i
$186$	−1.00000			−0.0733236
$187$		−	16.0819i		−	1.17603i
$188$		−	4.93203i		−	0.359705i
$189$	−2.00000			−0.145479
$190$	−10.8766	+	1.82310i	−0.789073	+	0.132261i
$191$	7.56427			0.547331			0.273666	−	0.961825i	$-0.411764\pi$
$191$	7.56427			0.547331			0.273666	+	0.961825i	$0.411764\pi$
$192$			1.00000i			0.0721688i
$193$			22.3817i			1.61107i	0.592547	+	0.805536i	$0.298122\pi$
$193$			22.3817i			1.61107i	−0.592547	+	0.805536i	$0.701878\pi$
$194$	7.26071			0.521289
$195$	1.01256	+	6.04096i	0.0725111	+	0.432602i
$196$	−3.00000			−0.214286
$197$		−	16.6853i		−	1.18878i	−0.804178	−	0.594388i	$-0.797394\pi$
$197$		−	16.6853i		−	1.18878i	0.804178	−	0.594388i	$-0.202606\pi$
$198$		−	3.26071i		−	0.231729i
$199$	−15.4535			−1.09547			−0.547733	−	0.836653i	$-0.684509\pi$
$199$	−15.4535			−1.09547			−0.547733	+	0.836653i	$0.684509\pi$
$200$	1.63036	+	4.72673i	0.115284	+	0.334230i
$201$	−13.5605			−0.956484
$202$		0			0
$203$		−	1.04285i		−	0.0731934i
$204$	4.93203			0.345311
$205$	−0.739289	−	4.41061i	−0.0516342	−	0.308050i
$206$	−6.00000			−0.418040
$207$		−	0.521423i		−	0.0362414i
$208$		−	2.73929i		−	0.189935i
$209$	−16.0819			−1.11241
$210$	4.41061	−	0.739289i	0.304361	−	0.0510158i
$211$	17.0428			1.17328			0.586639	−	0.809849i	$-0.300451\pi$
$211$	17.0428			1.17328			0.586639	+	0.809849i	$0.300451\pi$
$212$			13.3426i			0.916376i
$213$		−	5.56050i		−	0.380999i
$214$	−18.2998			−1.25095
$215$	−19.4535	+	3.26071i	−1.32671	+	0.222379i
$216$	1.00000			0.0680414
$217$		−	2.00000i		−	0.135769i
$218$		−	5.34264i		−	0.361849i
$219$	3.04285			0.205616
$220$	1.20530	+	7.19086i	0.0812615	+	0.484808i
$221$	−13.5103			−0.908798
$222$		−	10.8212i		−	0.726272i
$223$			13.7821i			0.922920i	0.887161	+	0.461460i	$0.152674\pi$
$223$			13.7821i			0.922920i	−0.887161	+	0.461460i	$0.847326\pi$
$224$	−2.00000			−0.133631
$225$	4.72673	−	1.63036i	0.315115	−	0.108690i
$226$	−16.6853			−1.10989
$227$		−	2.52142i		−	0.167353i	−0.996493	−	0.0836764i	$-0.973334\pi$
$227$		−	2.52142i		−	0.167353i	0.996493	−	0.0836764i	$-0.0266662\pi$
$228$		−	4.93203i		−	0.326632i
$229$	−27.3957			−1.81036			−0.905178	−	0.425032i	$-0.860263\pi$
$229$	−27.3957			−1.81036			−0.905178	+	0.425032i	$0.860263\pi$
$230$	0.192741	+	1.14990i	0.0127090	+	0.0758219i
$231$	6.52142			0.429078
$232$			0.521423i			0.0342331i
$233$		−	22.8212i		−	1.49507i	−0.664224	−	0.747534i	$-0.731238\pi$
$233$		−	22.8212i		−	1.49507i	0.664224	−	0.747534i	$-0.268762\pi$
$234$	−2.73929			−0.179073
$235$	10.8766	−	1.82310i	0.709513	−	0.118926i
$236$	9.34264			0.608154
$237$		−	6.41061i		−	0.416414i
$238$			9.86406i			0.639392i
$239$	14.2998			0.924977			0.462488	−	0.886625i	$-0.346957\pi$
$239$	14.2998			0.924977			0.462488	+	0.886625i	$0.346957\pi$
$240$	−2.20530	+	0.369644i	−0.142352	+	0.0238604i
$241$	13.3426			0.859475			0.429737	−	0.902954i	$-0.358606\pi$
$241$	13.3426			0.859475			0.429737	+	0.902954i	$0.358606\pi$
$242$		−	0.367761i		−	0.0236406i
$243$		−	1.00000i		−	0.0641500i
$244$	9.75324			0.624387
$245$	−1.10893	−	6.61591i	−0.0708471	−	0.422675i
$246$	2.00000			0.127515
$247$			13.5103i			0.859637i
$248$			1.00000i			0.0635001i
$249$	1.36776			0.0866783
$250$	−9.82121	+	5.34264i	−0.621148	+	0.337898i
$251$	10.5214			0.664106			0.332053	−	0.943261i	$-0.392259\pi$
$251$	10.5214			0.664106			0.332053	+	0.943261i	$0.392259\pi$
$252$			2.00000i			0.125988i
$253$			1.70021i			0.106891i
$254$	6.52142			0.409190
$255$	1.82310	+	10.8766i	0.114167	+	0.681120i
$256$	1.00000			0.0625000
$257$			7.86406i			0.490547i	0.969454	+	0.245273i	$0.0788777\pi$
$257$			7.86406i			0.490547i	−0.969454	+	0.245273i	$0.921122\pi$
$258$		−	8.82121i		−	0.549185i
$259$	21.6424			1.34479
$260$	6.04096	−	1.01256i	0.374645	−	0.0627965i
$261$	0.521423			0.0322753
$262$			13.3426i			0.824311i
$263$		−	23.8641i		−	1.47152i	−0.677242	−	0.735760i	$-0.736825\pi$
$263$		−	23.8641i		−	1.47152i	0.677242	−	0.735760i	$-0.263175\pi$
$264$	−3.26071			−0.200683
$265$	−29.4246	+	4.93203i	−1.80754	+	0.302972i
$266$	9.86406			0.604804
$267$		−	11.8641i		−	0.726069i
$268$			13.5605i			0.828340i
$269$	8.95715			0.546127			0.273064	−	0.961996i	$-0.411963\pi$
$269$	8.95715			0.546127			0.273064	+	0.961996i	$0.411963\pi$
$270$	0.369644	+	2.20530i	0.0224958	+	0.134210i
$271$	−13.0959			−0.795518			−0.397759	−	0.917490i	$-0.630212\pi$
$271$	−13.0959			−0.795518			−0.397759	+	0.917490i	$0.630212\pi$
$272$		−	4.93203i		−	0.299048i
$273$		−	5.47858i		−	0.331579i
$274$	−3.77837			−0.228260
$275$	−15.4125	+	5.31612i	−0.929408	+	0.320574i
$276$	−0.521423			−0.0313860
$277$		−	3.78213i		−	0.227246i	−0.993524	−	0.113623i	$-0.963754\pi$
$277$		−	3.78213i		−	0.227246i	0.993524	−	0.113623i	$-0.0362457\pi$
$278$		−	12.5214i		−	0.750985i
$279$	1.00000			0.0598684
$280$	−0.739289	−	4.41061i	−0.0441809	−	0.263584i
$281$	3.04285			0.181521			0.0907605	−	0.995873i	$-0.471070\pi$
$281$	3.04285			0.181521			0.0907605	+	0.995873i	$0.471070\pi$
$282$			4.93203i			0.293698i
$283$		−	22.7672i		−	1.35337i	−0.736273	−	0.676685i	$-0.763416\pi$
$283$		−	22.7672i		−	1.35337i	0.736273	−	0.676685i	$-0.236584\pi$
$284$	−5.56050			−0.329955
$285$	10.8766	−	1.82310i	0.644275	−	0.107991i
$286$	8.93203			0.528162
$287$			4.00000i			0.236113i
$288$		−	1.00000i		−	0.0589256i
$289$	−7.32492			−0.430877
$290$	−1.14990	+	0.192741i	−0.0675242	+	0.0113181i
$291$	−7.26071			−0.425630
$292$		−	3.04285i		−	0.178069i
$293$			3.64243i			0.212793i	0.994324	+	0.106396i	$0.0339313\pi$
$293$			3.64243i			0.212793i	−0.994324	+	0.106396i	$0.966069\pi$
$294$	3.00000			0.174964
$295$	3.45345	+	20.6033i	0.201068	+	1.19957i
$296$	−10.8212			−0.628970
$297$			3.26071i			0.189206i
$298$		−	2.21787i		−	0.128478i
$299$	1.42833			0.0826023
$300$	−1.63036	−	4.72673i	−0.0941286	−	0.272898i
$301$	17.6424			1.01689
$302$			18.1890i			1.04666i
$303$		0			0
$304$	−4.93203			−0.282871
$305$	3.60523	+	21.5089i	0.206435	+	1.23159i
$306$	−4.93203			−0.281945
$307$		−	17.4786i		−	0.997555i	−0.866730	−	0.498778i	$-0.833782\pi$
$307$		−	17.4786i		−	0.997555i	0.866730	−	0.498778i	$-0.166218\pi$
$308$		−	6.52142i		−	0.371593i
$309$	6.00000			0.341328
$310$	−2.20530	+	0.369644i	−0.125253	+	0.0209944i
$311$	18.1676			1.03019			0.515095	−	0.857133i	$-0.327756\pi$
$311$	18.1676			1.03019			0.515095	+	0.857133i	$0.327756\pi$
$312$			2.73929i			0.155082i
$313$			33.7281i			1.90643i	0.302300	+	0.953213i	$0.402246\pi$
$313$			33.7281i			1.90643i	−0.302300	+	0.953213i	$0.597754\pi$
$314$	−19.2067			−1.08390
$315$	−4.41061	+	0.739289i	−0.248510	+	0.0416542i
$316$	−6.41061			−0.360625
$317$		−	2.54655i		−	0.143028i	−0.997440	−	0.0715142i	$-0.977217\pi$
$317$		−	2.54655i		−	0.143028i	0.997440	−	0.0715142i	$-0.0227831\pi$
$318$		−	13.3426i		−	0.748218i
$319$	−1.70021			−0.0951934
$320$	0.369644	+	2.20530i	0.0206637	+	0.123280i
$321$	18.2998			1.02139
$322$		−	1.04285i		−	0.0581155i
$323$			24.3249i			1.35347i
$324$	−1.00000			−0.0555556
$325$	4.46601	+	12.9479i	0.247730	+	0.718218i
$326$	−9.56050			−0.529507
$327$			5.34264i			0.295448i
$328$		−	2.00000i		−	0.110432i
$329$	−9.86406			−0.543823
$330$	−1.20530	−	7.19086i	−0.0663498	−	0.395844i
$331$	10.1638			0.558656			0.279328	−	0.960196i	$-0.409888\pi$
$331$	10.1638			0.558656			0.279328	+	0.960196i	$0.409888\pi$
$332$		−	1.36776i		−	0.0750657i
$333$			10.8212i			0.592999i
$334$	−2.00000			−0.109435
$335$	−29.9050	+	5.01256i	−1.63389	+	0.273866i
$336$	2.00000			0.109109
$337$		−	24.9069i		−	1.35677i	−0.734709	−	0.678383i	$-0.762681\pi$
$337$		−	24.9069i		−	1.35677i	0.734709	−	0.678383i	$-0.237319\pi$
$338$			5.49630i			0.298959i
$339$	16.6853			0.906220
$340$	10.8766	−	1.82310i	0.589868	−	0.0988713i
$341$	−3.26071			−0.176577
$342$			4.93203i			0.266694i
$343$			20.0000i			1.07990i
$344$	−8.82121			−0.475608
$345$	−0.192741	−	1.14990i	−0.0103768	−	0.0619083i
$346$	−1.45345			−0.0781381
$347$		−	7.67508i		−	0.412020i	−0.978550	−	0.206010i	$-0.933952\pi$
$347$		−	7.67508i		−	0.412020i	0.978550	−	0.206010i	$-0.0660480\pi$
$348$		−	0.521423i		−	0.0279512i
$349$	1.77837			0.0951939			0.0475969	−	0.998867i	$-0.484844\pi$
$349$	1.77837			0.0951939			0.0475969	+	0.998867i	$0.484844\pi$
$350$	9.45345	−	3.26071i	0.505308	−	0.174292i
$351$	2.73929			0.146212
$352$			3.26071i			0.173797i
$353$			0.932030i			0.0496069i	0.999692	+	0.0248035i	$0.00789600\pi$
$353$			0.932030i			0.0496069i	−0.999692	+	0.0248035i	$0.992104\pi$
$354$	−9.34264			−0.496556
$355$	−2.05541	−	12.2626i	−0.109090	−	0.650831i
$356$	−11.8641			−0.628794
$357$		−	9.86406i		−	0.522061i
$358$			26.3817i			1.39432i
$359$	−17.2886			−0.912458			−0.456229	−	0.889862i	$-0.650800\pi$
$359$	−17.2886			−0.912458			−0.456229	+	0.889862i	$0.650800\pi$
$360$	2.20530	−	0.369644i	0.116230	−	0.0194820i
$361$	5.32492			0.280259
$362$		−	7.77837i		−	0.408822i
$363$			0.367761i			0.0193025i
$364$	−5.47858			−0.287155
$365$	6.71040	−	1.12477i	0.351238	−	0.0588732i
$366$	−9.75324			−0.509810
$367$		−	6.82498i		−	0.356261i	−0.984007	−	0.178131i	$-0.942995\pi$
$367$		−	6.82498i		−	0.356261i	0.984007	−	0.178131i	$-0.0570049\pi$
$368$			0.521423i			0.0271810i
$369$	−2.00000			−0.104116
$370$	−4.00000	−	23.8641i	−0.207950	−	1.24063i
$371$	26.6853			1.38543
$372$		−	1.00000i		−	0.0518476i
$373$			0.135941i			0.00703875i	0.999994	+	0.00351938i	$0.00112025\pi$
$373$			0.135941i			0.00703875i	−0.999994	+	0.00351938i	$0.998880\pi$
$374$	16.0819			0.831577
$375$	9.82121	−	5.34264i	0.507165	−	0.275893i
$376$	4.93203			0.254350
$377$			1.42833i			0.0735626i
$378$		−	2.00000i		−	0.102869i
$379$	10.7961			0.554558			0.277279	−	0.960789i	$-0.410567\pi$
$379$	10.7961			0.554558			0.277279	+	0.960789i	$0.410567\pi$
$380$	−1.82310	−	10.8766i	−0.0935229	−	0.557959i
$381$	−6.52142			−0.334103
$382$			7.56427i			0.387022i
$383$			23.4283i			1.19713i	0.801074	+	0.598566i	$0.204262\pi$
$383$			23.4283i			1.19713i	−0.801074	+	0.598566i	$0.795738\pi$
$384$	−1.00000			−0.0510310
$385$	14.3817	−	2.41061i	0.732960	−	0.122856i
$386$	−22.3817			−1.13920
$387$			8.82121i			0.448407i
$388$			7.26071i			0.368607i
$389$	−10.3855			−0.526565			−0.263282	−	0.964719i	$-0.584805\pi$
$389$	−10.3855			−0.526565			−0.263282	+	0.964719i	$0.584805\pi$
$390$	−6.04096	+	1.01256i	−0.305896	+	0.0512731i
$391$	2.57167			0.130055
$392$		−	3.00000i		−	0.151523i
$393$		−	13.3426i		−	0.673047i
$394$	16.6853			0.840592
$395$	−2.36964	−	14.1373i	−0.119230	−	0.711326i
$396$	3.26071			0.163857
$397$		−	10.2141i		−	0.512631i	−0.966593	−	0.256315i	$-0.917491\pi$
$397$		−	10.2141i		−	0.512631i	0.966593	−	0.256315i	$-0.0825086\pi$
$398$		−	15.4535i		−	0.774612i
$399$	−9.86406			−0.493821
$400$	−4.72673	+	1.63036i	−0.236336	+	0.0815178i
$401$	6.30356			0.314785			0.157392	−	0.987536i	$-0.449691\pi$
$401$	6.30356			0.314785			0.157392	+	0.987536i	$0.449691\pi$
$402$		−	13.5605i		−	0.676336i
$403$			2.73929i			0.136454i
$404$		0			0
$405$	−0.369644	−	2.20530i	−0.0183678	−	0.109582i
$406$	1.04285			0.0517556
$407$		−	35.2849i		−	1.74901i
$408$			4.93203i			0.244172i
$409$	19.2569			0.952195			0.476097	−	0.879393i	$-0.342051\pi$
$409$	19.2569			0.952195			0.476097	+	0.879393i	$0.342051\pi$
$410$	4.41061	−	0.739289i	0.217824	−	0.0365109i
$411$	3.77837			0.186373
$412$		−	6.00000i		−	0.295599i
$413$		−	18.6853i		−	0.919442i
$414$	0.521423			0.0256265
$415$	3.01633	−	0.505585i	0.148066	−	0.0248182i
$416$	2.73929			0.134305
$417$			12.5214i			0.613177i
$418$		−	16.0819i		−	0.786593i
$419$	−3.47858			−0.169940			−0.0849698	−	0.996384i	$-0.527079\pi$
$419$	−3.47858			−0.169940			−0.0849698	+	0.996384i	$0.527079\pi$
$420$	0.739289	+	4.41061i	0.0360736	+	0.215216i
$421$	−4.35757			−0.212375			−0.106188	−	0.994346i	$-0.533864\pi$
$421$	−4.35757			−0.212375			−0.106188	+	0.994346i	$0.533864\pi$
$422$			17.0428i			0.829633i
$423$		−	4.93203i		−	0.239803i
$424$	−13.3426			−0.647976
$425$	8.04096	+	23.3124i	0.390044	+	1.13082i
$426$	5.56050			0.269407
$427$		−	19.5065i		−	0.943985i
$428$		−	18.2998i		−	0.884554i
$429$	−8.93203			−0.431242
$430$	−3.26071	−	19.4535i	−0.157245	−	0.938129i
$431$	−22.2495			−1.07172			−0.535861	−	0.844306i	$-0.680013\pi$
$431$	−22.2495			−1.07172			−0.535861	+	0.844306i	$0.680013\pi$
$432$			1.00000i			0.0481125i
$433$			18.8212i			0.904490i	0.891894	+	0.452245i	$0.149377\pi$
$433$			18.8212i			0.904490i	−0.891894	+	0.452245i	$0.850623\pi$
$434$	2.00000			0.0960031
$435$	1.14990	−	0.192741i	0.0551333	−	0.00924122i
$436$	5.34264			0.255866
$437$		−	2.57167i		−	0.123020i
$438$			3.04285i			0.145393i
$439$	9.25695			0.441810			0.220905	−	0.975295i	$-0.429099\pi$
$439$	9.25695			0.441810			0.220905	+	0.975295i	$0.429099\pi$
$440$	−7.19086	+	1.20530i	−0.342811	+	0.0574606i
$441$	−3.00000			−0.142857
$442$		−	13.5103i		−	0.642617i
$443$			8.82121i			0.419109i	0.977797	+	0.209554i	$0.0672013\pi$
$443$			8.82121i			0.419109i	−0.977797	+	0.209554i	$0.932799\pi$
$444$	10.8212			0.513552
$445$	−4.38548	−	26.1638i	−0.207892	−	1.24029i
$446$	−13.7821			−0.652603
$447$			2.21787i			0.104901i
$448$		−	2.00000i		−	0.0944911i
$449$	16.2179			0.765368			0.382684	−	0.923879i	$-0.375000\pi$
$449$	16.2179			0.765368			0.382684	+	0.923879i	$0.375000\pi$
$450$	1.63036	+	4.72673i	0.0768557	+	0.222820i
$451$	6.52142			0.307082
$452$		−	16.6853i		−	0.784809i
$453$		−	18.1890i		−	0.854593i
$454$	2.52142			0.118336
$455$	−2.02512	−	12.0819i	−0.0949393	−	0.566409i
$456$	4.93203			0.230963
$457$		−	19.4283i		−	0.908819i	−0.890793	−	0.454409i	$-0.849850\pi$
$457$		−	19.4283i		−	0.908819i	0.890793	−	0.454409i	$-0.150150\pi$
$458$		−	27.3957i		−	1.28012i
$459$	4.93203			0.230207
$460$	−1.14990	+	0.192741i	−0.0536142	+	0.00898659i
$461$	−2.38548			−0.111103			−0.0555515	−	0.998456i	$-0.517692\pi$
$461$	−2.38548			−0.111103			−0.0555515	+	0.998456i	$0.517692\pi$
$462$			6.52142i			0.303404i
$463$		−	4.90314i		−	0.227868i	−0.993488	−	0.113934i	$-0.963655\pi$
$463$		−	4.90314i		−	0.227868i	0.993488	−	0.113934i	$-0.0363453\pi$
$464$	−0.521423			−0.0242064
$465$	2.20530	−	0.369644i	0.102268	−	0.0171418i
$466$	22.8212			1.05717
$467$		−	22.0857i		−	1.02200i	−0.859580	−	0.511002i	$-0.829274\pi$
$467$		−	22.0857i		−	1.02200i	0.859580	−	0.511002i	$-0.170726\pi$
$468$		−	2.73929i		−	0.126624i
$469$	27.1210			1.25233
$470$	1.82310	+	10.8766i	0.0840932	+	0.501701i
$471$	19.2067			0.884998
$472$			9.34264i			0.430030i
$473$		−	28.7634i		−	1.32254i
$474$	6.41061			0.294449
$475$	23.3124	−	8.04096i	1.06964	−	0.368945i
$476$	−9.86406			−0.452118
$477$			13.3426i			0.610917i
$478$			14.2998i			0.654057i
$479$	10.1676			0.464570			0.232285	−	0.972648i	$-0.425380\pi$
$479$	10.1676			0.464570			0.232285	+	0.972648i	$0.425380\pi$
$480$	−0.369644	−	2.20530i	−0.0168719	−	0.100658i
$481$	−29.6424			−1.35158
$482$			13.3426i			0.607740i
$483$			1.04285i			0.0474511i
$484$	0.367761			0.0167164
$485$	−16.0121	+	2.68388i	−0.727071	+	0.121869i
$486$	1.00000			0.0453609
$487$		−	38.5531i		−	1.74701i	−0.486817	−	0.873504i	$-0.661842\pi$
$487$		−	38.5531i		−	1.74701i	0.486817	−	0.873504i	$-0.338158\pi$
$488$			9.75324i			0.441509i
$489$	9.56050			0.432341
$490$	6.61591	−	1.10893i	0.298876	−	0.0500965i
$491$	39.7206			1.79256			0.896282	−	0.443484i	$-0.146258\pi$
$491$	39.7206			1.79256			0.896282	+	0.443484i	$0.146258\pi$
$492$			2.00000i			0.0901670i
$493$			2.57167i			0.115822i
$494$	−13.5103			−0.607855
$495$	1.20530	+	7.19086i	0.0541744	+	0.323205i
$496$	−1.00000			−0.0449013
$497$			11.1210i			0.498845i
$498$			1.36776i			0.0612908i
$499$	−31.8641			−1.42643			−0.713216	−	0.700945i	$-0.752762\pi$
$499$	−31.8641			−1.42643			−0.713216	+	0.700945i	$0.752762\pi$
$500$	−5.34264	−	9.82121i	−0.238930	−	0.439218i
$501$	2.00000			0.0893534
$502$			10.5214i			0.469594i
$503$			26.7961i			1.19478i	0.801951	+	0.597389i	$0.203795\pi$
$503$			26.7961i			1.19478i	−0.801951	+	0.597389i	$0.796205\pi$
$504$	−2.00000			−0.0890871
$505$		0			0
$506$	−1.70021			−0.0755835
$507$		−	5.49630i		−	0.244099i
$508$			6.52142i			0.289341i
$509$	13.5065			0.598664			0.299332	−	0.954149i	$-0.403236\pi$
$509$	13.5065			0.598664			0.299332	+	0.954149i	$0.403236\pi$
$510$	−10.8766	+	1.82310i	−0.481625	+	0.0807281i
$511$	−6.08569			−0.269215
$512$			1.00000i			0.0441942i
$513$		−	4.93203i		−	0.217754i
$514$	−7.86406			−0.346869
$515$	13.2318	−	2.21787i	0.583064	−	0.0977308i
$516$	8.82121			0.388332
$517$			16.0819i			0.707282i
$518$			21.6424i			0.950914i
$519$	1.45345			0.0637995
$520$	1.01256	+	6.04096i	0.0444038	+	0.264914i
$521$	−2.16385			−0.0948000			−0.0474000	−	0.998876i	$-0.515094\pi$
$521$	−2.16385			−0.0948000			−0.0474000	+	0.998876i	$0.515094\pi$
$522$			0.521423i			0.0228221i
$523$			18.1359i			0.793029i	0.918028	+	0.396515i	$0.129780\pi$
$523$			18.1359i			0.793029i	−0.918028	+	0.396515i	$0.870220\pi$
$524$	−13.3426			−0.582876
$525$	−9.45345	+	3.26071i	−0.412582	+	0.142309i
$526$	23.8641			1.04052
$527$			4.93203i			0.214843i
$528$		−	3.26071i		−	0.141904i
$529$	22.7281			0.988179
$530$	−4.93203	−	29.4246i	−0.214234	−	1.27812i
$531$	9.34264			0.405436
$532$			9.86406i			0.427661i
$533$		−	5.47858i		−	0.237304i
$534$	11.8641			0.513408
$535$	40.3566	−	6.76441i	1.74477	−	0.292451i
$536$	−13.5605			−0.585724
$537$		−	26.3817i		−	1.13846i
$538$			8.95715i			0.386170i
$539$	9.78213			0.421346
$540$	−2.20530	+	0.369644i	−0.0949011	+	0.0159070i
$541$	7.64996			0.328897			0.164449	−	0.986386i	$-0.447415\pi$
$541$	7.64996			0.328897			0.164449	+	0.986386i	$0.447415\pi$
$542$		−	13.0959i		−	0.562516i
$543$			7.77837i			0.333802i
$544$	4.93203			0.211459
$545$	1.97488	+	11.7821i	0.0845944	+	0.504691i
$546$	5.47858			0.234461
$547$			15.1210i			0.646527i	0.946309	+	0.323264i	$0.104780\pi$
$547$			15.1210i			0.646527i	−0.946309	+	0.323264i	$0.895220\pi$
$548$		−	3.77837i		−	0.161404i
$549$	9.75324			0.416258
$550$	−5.31612	−	15.4125i	−0.226680	−	0.657191i
$551$	2.57167			0.109557
$552$		−	0.521423i		−	0.0221932i
$553$			12.8212i			0.545213i
$554$	3.78213			0.160687
$555$	4.00000	+	23.8641i	0.169791	+	1.01297i
$556$	12.5214			0.531027
$557$		−	21.5065i		−	0.911259i	−0.890169	−	0.455630i	$-0.849414\pi$
$557$		−	21.5065i		−	0.911259i	0.890169	−	0.455630i	$-0.150586\pi$
$558$			1.00000i			0.0423334i
$559$	−24.1638			−1.02202
$560$	4.41061	−	0.739289i	0.186382	−	0.0312406i
$561$	−16.0819			−0.678979
$562$			3.04285i			0.128355i
$563$			21.2569i			0.895873i	0.894065	+	0.447937i	$0.147841\pi$
$563$			21.2569i			0.895873i	−0.894065	+	0.447937i	$0.852159\pi$
$564$	−4.93203			−0.207676
$565$	36.7961	−	6.16762i	1.54802	−	0.259474i
$566$	22.7672			0.956977
$567$			2.00000i			0.0839921i
$568$		−	5.56050i		−	0.233313i
$569$	−19.4208			−0.814162			−0.407081	−	0.913392i	$-0.633453\pi$
$569$	−19.4208			−0.814162			−0.407081	+	0.913392i	$0.633453\pi$
$570$	1.82310	+	10.8766i	0.0763611	+	0.455572i
$571$	23.8641			0.998680			0.499340	−	0.866406i	$-0.333576\pi$
$571$	23.8641			0.998680			0.499340	+	0.866406i	$0.333576\pi$
$572$			8.93203i			0.373467i
$573$		−	7.56427i		−	0.316002i
$574$	−4.00000			−0.166957
$575$	−0.850105	−	2.46462i	−0.0354518	−	0.102782i
$576$	1.00000			0.0416667
$577$			1.78213i			0.0741912i	0.999312	+	0.0370956i	$0.0118106\pi$
$577$			1.78213i			0.0741912i	−0.999312	+	0.0370956i	$0.988189\pi$
$578$		−	7.32492i		−	0.304676i
$579$	22.3817			0.930152
$580$	−0.192741	−	1.14990i	−0.00800313	−	0.0477468i
$581$	−2.73552			−0.113489
$582$		−	7.26071i		−	0.300966i
$583$		−	43.5065i		−	1.80185i
$584$	3.04285			0.125914
$585$	6.04096	−	1.01256i	0.249763	−	0.0418643i
$586$	−3.64243			−0.150467
$587$			10.1387i			0.418470i	0.977865	+	0.209235i	$0.0670974\pi$
$587$			10.1387i			0.418470i	−0.977865	+	0.209235i	$0.932903\pi$
$588$			3.00000i			0.123718i
$589$	4.93203			0.203221
$590$	−20.6033	+	3.45345i	−0.848226	+	0.142176i
$591$	−16.6853			−0.686340
$592$		−	10.8212i		−	0.444749i
$593$		−	43.5846i		−	1.78981i	−0.446260	−	0.894903i	$-0.647244\pi$
$593$		−	43.5846i		−	1.78981i	0.446260	−	0.894903i	$-0.352756\pi$
$594$	−3.26071			−0.133789
$595$	−3.64619	−	21.7532i	−0.149479	−	0.891796i
$596$	2.21787			0.0908473
$597$			15.4535i			0.632468i
$598$			1.42833i			0.0584087i
$599$	16.6815			0.681588			0.340794	−	0.940138i	$-0.389304\pi$
$599$	16.6815			0.681588			0.340794	+	0.940138i	$0.389304\pi$
$600$	4.72673	−	1.63036i	0.192968	−	0.0665590i
$601$	−43.6424			−1.78021			−0.890106	−	0.455754i	$-0.849370\pi$
$601$	−43.6424			−1.78021			−0.890106	+	0.455754i	$0.849370\pi$
$602$			17.6424i			0.719052i
$603$			13.5605i			0.552226i
$604$	−18.1890			−0.740099
$605$	0.135941	+	0.811025i	0.00552678	+	0.0329729i
$606$		0			0
$607$		−	10.5996i		−	0.430224i	−0.976589	−	0.215112i	$-0.930988\pi$
$607$		−	10.5996i		−	0.430224i	0.976589	−	0.215112i	$-0.0690116\pi$
$608$		−	4.93203i		−	0.200020i
$609$	−1.04285			−0.0422582
$610$	−21.5089	+	3.60523i	−0.870868	+	0.145972i
$611$	13.5103			0.546566
$612$		−	4.93203i		−	0.199365i
$613$		−	44.5456i		−	1.79918i	−0.436737	−	0.899589i	$-0.643866\pi$
$613$		−	44.5456i		−	1.79918i	0.436737	−	0.899589i	$-0.356134\pi$
$614$	17.4786			0.705378
$615$	−4.41061	+	0.739289i	−0.177853	+	0.0298110i
$616$	6.52142			0.262756
$617$		−	40.6853i		−	1.63793i	−0.573845	−	0.818964i	$-0.694549\pi$
$617$		−	40.6853i		−	1.63793i	0.573845	−	0.818964i	$-0.305451\pi$
$618$			6.00000i			0.241355i
$619$	−42.8212			−1.72113			−0.860565	−	0.509341i	$-0.829889\pi$
$619$	−42.8212			−1.72113			−0.860565	+	0.509341i	$0.829889\pi$
$620$	−0.369644	−	2.20530i	−0.0148453	−	0.0885671i
$621$	−0.521423			−0.0209240
$622$			18.1676i			0.728455i
$623$			23.7281i			0.950647i
$624$	−2.73929			−0.109659
$625$	19.6839	−	15.4125i	0.787355	−	0.616500i
$626$	−33.7281			−1.34805
$627$			16.0819i			0.642250i
$628$		−	19.2067i		−	0.766431i
$629$	−53.3705			−2.12802
$630$	−0.739289	−	4.41061i	−0.0294540	−	0.175723i
$631$	−1.31473			−0.0523385			−0.0261692	−	0.999658i	$-0.508331\pi$
$631$	−1.31473			−0.0523385			−0.0261692	+	0.999658i	$0.508331\pi$
$632$		−	6.41061i		−	0.255000i
$633$		−	17.0428i		−	0.677392i
$634$	2.54655			0.101136
$635$	−14.3817	+	2.41061i	−0.570721	+	0.0956620i
$636$	13.3426			0.529070
$637$		−	8.21787i		−	0.325604i
$638$		−	1.70021i		−	0.0673119i
$639$	−5.56050			−0.219970
$640$	−2.20530	+	0.369644i	−0.0871723	+	0.0146115i
$641$	−25.4246			−1.00421			−0.502105	−	0.864807i	$-0.667441\pi$
$641$	−25.4246			−1.00421			−0.502105	+	0.864807i	$0.667441\pi$
$642$			18.2998i			0.722235i
$643$			44.3855i			1.75039i	0.483768	+	0.875196i	$0.339268\pi$
$643$			44.3855i			1.75039i	−0.483768	+	0.875196i	$0.660732\pi$
$644$	1.04285			0.0410939
$645$	3.26071	+	19.4535i	0.128390	+	0.765979i
$646$	−24.3249			−0.957051
$647$		−	17.5567i		−	0.690227i	−0.938561	−	0.345113i	$-0.887841\pi$
$647$		−	17.5567i		−	0.690227i	0.938561	−	0.345113i	$-0.112159\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−30.4636			−1.19580
$650$	−12.9479	+	4.46601i	−0.507857	+	0.175172i
$651$	−2.00000			−0.0783862
$652$		−	9.56050i		−	0.374418i
$653$			18.0530i			0.706470i	0.935535	+	0.353235i	$0.114918\pi$
$653$			18.0530i			0.706470i	−0.935535	+	0.353235i	$0.885082\pi$
$654$	−5.34264			−0.208914
$655$	−4.93203	−	29.4246i	−0.192710	−	1.14971i
$656$	2.00000			0.0780869
$657$		−	3.04285i		−	0.118713i
$658$		−	9.86406i		−	0.384541i
$659$	0.957154			0.0372854			0.0186427	−	0.999826i	$-0.494065\pi$
$659$	0.957154			0.0372854			0.0186427	+	0.999826i	$0.494065\pi$
$660$	7.19086	−	1.20530i	0.279904	−	0.0469164i
$661$	−9.56427			−0.372007			−0.186003	−	0.982549i	$-0.559554\pi$
$661$	−9.56427			−0.372007			−0.186003	+	0.982549i	$0.559554\pi$
$662$			10.1638i			0.395029i
$663$			13.5103i			0.524695i
$664$	1.36776			0.0530794
$665$	−21.7532	+	3.64619i	−0.843555	+	0.141393i
$666$	−10.8212			−0.419314
$667$		−	0.271882i		−	0.0105273i
$668$		−	2.00000i		−	0.0773823i
$669$	13.7821			0.532848
$670$	−5.01256	−	29.9050i	−0.193652	−	1.15533i
$671$	−31.8025			−1.22772
$672$			2.00000i			0.0771517i
$673$			41.7206i			1.60821i	0.594487	+	0.804105i	$0.297355\pi$
$673$			41.7206i			1.60821i	−0.594487	+	0.804105i	$0.702645\pi$
$674$	24.9069			0.959378
$675$	−1.63036	−	4.72673i	−0.0627524	−	0.181932i
$676$	−5.49630			−0.211396
$677$			21.7784i			0.837011i	0.908214	+	0.418505i	$0.137446\pi$
$677$			21.7784i			0.837011i	−0.908214	+	0.418505i	$0.862554\pi$
$678$			16.6853i			0.640794i
$679$	14.5214			0.557281
$680$	1.82310	+	10.8766i	0.0699126	+	0.417099i
$681$	−2.52142			−0.0966211
$682$		−	3.26071i		−	0.124859i
$683$			39.7281i			1.52015i	0.649833	+	0.760077i	$0.274839\pi$
$683$			39.7281i			1.52015i	−0.649833	+	0.760077i	$0.725161\pi$
$684$	−4.93203			−0.188581
$685$	8.33245	−	1.39665i	0.318366	−	0.0533633i
$686$	−20.0000			−0.763604
$687$			27.3957i			1.04521i
$688$		−	8.82121i		−	0.336306i
$689$	−36.5493			−1.39242
$690$	1.14990	−	0.192741i	0.0437758	−	0.00733752i
$691$	37.5316			1.42777			0.713885	−	0.700263i	$-0.246934\pi$
$691$	37.5316			1.42777			0.713885	+	0.700263i	$0.246934\pi$
$692$		−	1.45345i		−	0.0552520i
$693$		−	6.52142i		−	0.247728i
$694$	7.67508			0.291342
$695$	4.62847	+	27.6135i	0.175568	+	1.04744i
$696$	0.521423			0.0197645
$697$		−	9.86406i		−	0.373628i
$698$			1.77837i			0.0673122i
$699$	−22.8212			−0.863178
$700$	3.26071	+	9.45345i	0.123243	+	0.357307i
$701$	−16.7393			−0.632234			−0.316117	−	0.948720i	$-0.602379\pi$
$701$	−16.7393			−0.632234			−0.316117	+	0.948720i	$0.602379\pi$
$702$			2.73929i			0.103388i
$703$			53.3705i			2.01291i
$704$	−3.26071			−0.122893
$705$	−1.82310	−	10.8766i	−0.0686618	−	0.409637i
$706$	−0.932030			−0.0350774
$707$		0			0
$708$		−	9.34264i		−	0.351118i
$709$	30.8463			1.15846			0.579229	−	0.815165i	$-0.303354\pi$
$709$	30.8463			1.15846			0.579229	+	0.815165i	$0.303354\pi$
$710$	12.2626	−	2.05541i	0.460207	−	0.0771381i
$711$	−6.41061			−0.240417
$712$		−	11.8641i		−	0.444624i
$713$		−	0.521423i		−	0.0195274i
$714$	9.86406			0.369153
$715$	−19.6978	+	3.30167i	−0.736657	+	0.123476i
$716$	−26.3817			−0.985931
$717$		−	14.2998i		−	0.534035i
$718$		−	17.2886i		−	0.645206i
$719$	10.4712			0.390509			0.195254	−	0.980753i	$-0.437447\pi$
$719$	10.4712			0.390509			0.195254	+	0.980753i	$0.437447\pi$
$720$	0.369644	+	2.20530i	0.0137758	+	0.0821868i
$721$	−12.0000			−0.446903
$722$			5.32492i			0.198173i
$723$		−	13.3426i		−	0.496218i
$724$	7.77837			0.289081
$725$	2.46462	−	0.850105i	0.0915338	−	0.0315721i
$726$	−0.367761			−0.0136489
$727$			30.3779i			1.12666i	0.826233	+	0.563328i	$0.190479\pi$
$727$			30.3779i			1.12666i	−0.826233	+	0.563328i	$0.809521\pi$
$728$		−	5.47858i		−	0.203050i
$729$	−1.00000			−0.0370370
$730$	1.12477	+	6.71040i	0.0416296	+	0.248363i
$731$	−43.5065			−1.60915
$732$		−	9.75324i		−	0.360490i
$733$		−	34.3855i		−	1.27006i	−0.772489	−	0.635028i	$-0.780988\pi$
$733$		−	34.3855i		−	1.27006i	0.772489	−	0.635028i	$-0.219012\pi$
$734$	6.82498			0.251915
$735$	−6.61591	+	1.10893i	−0.244032	+	0.0409036i
$736$	−0.521423			−0.0192199
$737$		−	44.2169i		−	1.62875i
$738$		−	2.00000i		−	0.0736210i
$739$	17.1210			0.629806			0.314903	−	0.949124i	$-0.398028\pi$
$739$	17.1210			0.629806			0.314903	+	0.949124i	$0.398028\pi$
$740$	23.8641	−	4.00000i	0.877260	−	0.147043i
$741$	13.5103			0.496312
$742$			26.6853i			0.979647i
$743$		−	3.86406i		−	0.141759i	−0.997485	−	0.0708793i	$-0.977419\pi$
$743$		−	3.86406i		−	0.141759i	0.997485	−	0.0708793i	$-0.0225805\pi$
$744$	1.00000			0.0366618
$745$	0.819821	+	4.89107i	0.0300359	+	0.179195i
$746$	−0.135941			−0.00497715
$747$		−	1.36776i		−	0.0500438i
$748$			16.0819i			0.588013i
$749$	−36.5996			−1.33732
$750$	5.34264	+	9.82121i	0.195085	+	0.358620i
$751$	−22.1359			−0.807752			−0.403876	−	0.914814i	$-0.632337\pi$
$751$	−22.1359			−0.807752			−0.403876	+	0.914814i	$0.632337\pi$
$752$			4.93203i			0.179853i
$753$		−	10.5214i		−	0.383422i
$754$	−1.42833			−0.0520166
$755$	−6.72345	−	40.1122i	−0.244691	−	1.45983i
$756$	2.00000			0.0727393
$757$			43.5922i			1.58438i	0.610272	+	0.792192i	$0.291060\pi$
$757$			43.5922i			1.58438i	−0.610272	+	0.792192i	$0.708940\pi$
$758$			10.7961i			0.392132i
$759$	1.70021			0.0617137
$760$	10.8766	−	1.82310i	0.394537	−	0.0661307i
$761$	14.9107			0.540511			0.270256	−	0.962789i	$-0.412892\pi$
$761$	14.9107			0.540511			0.270256	+	0.962789i	$0.412892\pi$
$762$		−	6.52142i		−	0.236246i
$763$		−	10.6853i		−	0.386833i
$764$	−7.56427			−0.273666
$765$	10.8766	−	1.82310i	0.393245	−	0.0659142i
$766$	−23.4283			−0.846500
$767$			25.5922i			0.924080i
$768$		−	1.00000i		−	0.0360844i
$769$	−48.4134			−1.74583			−0.872916	−	0.487871i	$-0.837774\pi$
$769$	−48.4134			−1.74583			−0.872916	+	0.487871i	$0.837774\pi$
$770$	2.41061	+	14.3817i	0.0868722	+	0.518281i
$771$	7.86406			0.283217
$772$		−	22.3817i		−	0.805536i
$773$			6.54933i			0.235563i	0.993040	+	0.117782i	$0.0375783\pi$
$773$			6.54933i			0.235563i	−0.993040	+	0.117782i	$0.962422\pi$
$774$	−8.82121			−0.317072
$775$	4.72673	−	1.63036i	0.169789	−	0.0585641i
$776$	−7.26071			−0.260644
$777$		−	21.6424i		−	0.776418i
$778$		−	10.3855i		−	0.372338i
$779$	−9.86406			−0.353417
$780$	−1.01256	−	6.04096i	−0.0362556	−	0.216301i
$781$	18.1312			0.648785

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-0.369644 - 2.20530i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-0.369644 - 2.20530i) q^{5} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.20530 - 0.369644i) q^{10} +3.26071 q^{11} +1.00000i q^{12} -2.73929i q^{13} -2.00000 q^{14} +(-2.20530 + 0.369644i) q^{15} +1.00000 q^{16} -4.93203i q^{17} -1.00000i q^{18} -4.93203 q^{19} +(0.369644 + 2.20530i) q^{20} +2.00000 q^{21} +3.26071i q^{22} +0.521423i q^{23} -1.00000 q^{24} +(-4.72673 + 1.63036i) q^{25} +2.73929 q^{26} +1.00000i q^{27} -2.00000i q^{28} -0.521423 q^{29} +(-0.369644 - 2.20530i) q^{30} -1.00000 q^{31} +1.00000i q^{32} -3.26071i q^{33} +4.93203 q^{34} +(4.41061 - 0.739289i) q^{35} +1.00000 q^{36} -10.8212i q^{37} -4.93203i q^{38} -2.73929 q^{39} +(-2.20530 + 0.369644i) q^{40} +2.00000 q^{41} +2.00000i q^{42} -8.82121i q^{43} -3.26071 q^{44} +(0.369644 + 2.20530i) q^{45} -0.521423 q^{46} +4.93203i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-1.63036 - 4.72673i) q^{50} -4.93203 q^{51} +2.73929i q^{52} -13.3426i q^{53} -1.00000 q^{54} +(-1.20530 - 7.19086i) q^{55} +2.00000 q^{56} +4.93203i q^{57} -0.521423i q^{58} -9.34264 q^{59} +(2.20530 - 0.369644i) q^{60} -9.75324 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(-6.04096 + 1.01256i) q^{65} +3.26071 q^{66} -13.5605i q^{67} +4.93203i q^{68} +0.521423 q^{69} +(0.739289 + 4.41061i) q^{70} +5.56050 q^{71} +1.00000i q^{72} +3.04285i q^{73} +10.8212 q^{74} +(1.63036 + 4.72673i) q^{75} +4.93203 q^{76} +6.52142i q^{77} -2.73929i q^{78} +6.41061 q^{79} +(-0.369644 - 2.20530i) q^{80} +1.00000 q^{81} +2.00000i q^{82} +1.36776i q^{83} -2.00000 q^{84} +(-10.8766 + 1.82310i) q^{85} +8.82121 q^{86} +0.521423i q^{87} -3.26071i q^{88} +11.8641 q^{89} +(-2.20530 + 0.369644i) q^{90} +5.47858 q^{91} -0.521423i q^{92} +1.00000i q^{93} -4.93203 q^{94} +(1.82310 + 10.8766i) q^{95} +1.00000 q^{96} -7.26071i q^{97} +3.00000i q^{98} -3.26071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 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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