LMFDB - Embedded newform 675.2.e.c.451.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(226,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("675.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.e (of order $3$, degree $2$, not 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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1223810289.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 $ x^8 - 2x^7 + 8x^6 - 2x^5 + 23x^4 - 8x^3 + 37x^2 + 15*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.4
Root			$-0.816862 + 1.41485i$ of defining polynomial
Character	$\chi$	$=$	675.451
Dual form			675.2.e.c.226.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.816862 - 1.41485i) q^{2} +(-0.334526 - 0.579416i) q^{4} +(-0.252674 + 0.437645i) q^{7} +2.17440 q^{8} +O(q^{10})$	\(q+(0.816862 - 1.41485i) q^{2} +(-0.334526 - 0.579416i) q^{4} +(-0.252674 + 0.437645i) q^{7} +2.17440 q^{8} +(1.55010 - 2.68485i) q^{11} +(3.11964 + 5.40337i) q^{13} +(0.412800 + 0.714990i) q^{14} +(2.44524 - 4.23527i) q^{16} +6.10020 q^{17} -5.57022 q^{19} +(-2.53244 - 4.38631i) q^{22} +(-1.91280 - 3.31307i) q^{23} +10.1932 q^{26} +0.338104 q^{28} +(1.22966 - 2.12984i) q^{29} +(-2.11429 - 3.66206i) q^{31} +(-1.82044 - 3.15309i) q^{32} +(4.98302 - 8.63085i) q^{34} +6.72677 q^{37} +(-4.55010 + 7.88101i) q^{38} +(-2.72092 - 4.71278i) q^{41} +(-0.663704 + 1.14957i) q^{43} -2.07420 q^{44} -6.24997 q^{46} +(-1.85396 + 3.21115i) q^{47} +(3.37231 + 5.84101i) q^{49} +(2.08720 - 3.61514i) q^{52} -2.54205 q^{53} +(-0.549415 + 0.951614i) q^{56} +(-2.00893 - 3.47956i) q^{58} +(1.44116 + 2.49616i) q^{59} +(1.42173 - 2.46250i) q^{61} -6.90833 q^{62} +3.83276 q^{64} +(-1.20326 - 2.08411i) q^{67} +(-2.04068 - 3.53456i) q^{68} -5.54205 q^{71} -11.7988 q^{73} +(5.49484 - 9.51734i) q^{74} +(1.86338 + 3.22748i) q^{76} +(0.783341 + 1.35679i) q^{77} +(-1.70149 + 2.94707i) q^{79} -8.89047 q^{82} +(-6.95059 + 12.0388i) q^{83} +(1.08431 + 1.87808i) q^{86} +(3.37054 - 5.83795i) q^{88} -3.38513 q^{89} -3.15301 q^{91} +(-1.27976 + 2.21661i) q^{92} +(3.02886 + 5.24614i) q^{94} +(-5.53779 + 9.59173i) q^{97} +11.0188 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	0.816862	−	1.41485i	0.577608	−	1.00045i	−0.418144	−	0.908381i	$-0.637319\pi$
$2$	0.816862	−	1.41485i	0.577608	−	1.00045i	0.995753	−	0.0920666i	$-0.0293473\pi$
$3$		0			0
$3$		0			0
$4$	−0.334526	−	0.579416i	−0.167263	−	0.289708i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.252674	+	0.437645i	−0.0955019	+	0.165414i	−0.909818	−	0.415008i	$-0.863779\pi$
$7$	−0.252674	+	0.437645i	−0.0955019	+	0.165414i	0.814316	+	0.580422i	$0.197112\pi$
$8$	2.17440			0.768767
$9$		0			0
$10$		0			0
$11$	1.55010	−	2.68485i	0.467373	−	0.809514i	−0.531932	−	0.846787i	$-0.678534\pi$
$11$	1.55010	−	2.68485i	0.467373	−	0.809514i	0.999305	+	0.0372730i	$0.0118671\pi$
$12$		0			0
$13$	3.11964	+	5.40337i	0.865232	+	1.49863i	0.866817	+	0.498627i	$0.166162\pi$
$13$	3.11964	+	5.40337i	0.865232	+	1.49863i	−0.00158518	+	0.999999i	$0.500505\pi$
$14$	0.412800	+	0.714990i	0.110325	+	0.191089i
$15$		0			0
$16$	2.44524	−	4.23527i	0.611309	−	1.05882i
$17$	6.10020			1.47952			0.739758	−	0.672873i	$-0.234940\pi$
$17$	6.10020			1.47952			0.739758	+	0.672873i	$0.234940\pi$
$18$		0			0
$19$	−5.57022			−1.27790			−0.638948	−	0.769250i	$-0.720630\pi$
$19$	−5.57022			−1.27790			−0.638948	+	0.769250i	$0.720630\pi$
$20$		0			0
$21$		0			0
$22$	−2.53244	−	4.38631i	−0.539917	−	0.935164i
$23$	−1.91280	−	3.31307i	−0.398846	−	0.690822i	0.594738	−	0.803920i	$-0.297256\pi$
$23$	−1.91280	−	3.31307i	−0.398846	−	0.690822i	−0.993584	+	0.113098i	$0.963923\pi$
$24$		0			0
$25$		0			0
$26$	10.1932			1.99906
$27$		0			0
$28$	0.338104			0.0638957
$29$	1.22966	−	2.12984i	0.228342	−	0.395501i	−0.728975	−	0.684541i	$-0.760003\pi$
$29$	1.22966	−	2.12984i	0.228342	−	0.395501i	0.957317	+	0.289040i	$0.0933361\pi$
$30$		0			0
$31$	−2.11429	−	3.66206i	−0.379738	−	0.657725i	0.611286	−	0.791409i	$-0.290652\pi$
$31$	−2.11429	−	3.66206i	−0.379738	−	0.657725i	−0.991024	+	0.133685i	$0.957319\pi$
$32$	−1.82044	−	3.15309i	−0.321811	−	0.557394i
$33$		0			0
$34$	4.98302	−	8.63085i	0.854581	−	1.48018i
$35$		0			0
$36$		0			0
$37$	6.72677			1.10587			0.552937	−	0.833223i	$-0.313507\pi$
$37$	6.72677			1.10587			0.552937	+	0.833223i	$0.313507\pi$
$38$	−4.55010	+	7.88101i	−0.738124	+	1.27847i
$39$		0			0
$40$		0			0
$41$	−2.72092	−	4.71278i	−0.424937	−	0.736012i	0.571478	−	0.820618i	$-0.306370\pi$
$41$	−2.72092	−	4.71278i	−0.424937	−	0.736012i	−0.996415	+	0.0846053i	$0.973037\pi$
$42$		0			0
$43$	−0.663704	+	1.14957i	−0.101214	+	0.175308i	−0.912185	−	0.409779i	$-0.865606\pi$
$43$	−0.663704	+	1.14957i	−0.101214	+	0.175308i	0.810971	+	0.585086i	$0.198939\pi$
$44$	−2.07420			−0.312697
$45$		0			0
$46$	−6.24997			−0.921508
$47$	−1.85396	+	3.21115i	−0.270428	+	0.468395i	−0.968971	−	0.247173i	$-0.920499\pi$
$47$	−1.85396	+	3.21115i	−0.270428	+	0.468395i	0.698544	+	0.715568i	$0.253832\pi$
$48$		0			0
$49$	3.37231	+	5.84101i	0.481759	+	0.834431i
$50$		0			0
$51$		0			0
$52$	2.08720	−	3.61514i	0.289443	−	0.501329i
$53$	−2.54205			−0.349177			−0.174589	−	0.984641i	$-0.555860\pi$
$53$	−2.54205			−0.349177			−0.174589	+	0.984641i	$0.555860\pi$
$54$		0			0
$55$		0			0
$56$	−0.549415	+	0.951614i	−0.0734187	+	0.127165i
$57$		0			0
$58$	−2.00893	−	3.47956i	−0.263785	−	0.456889i
$59$	1.44116	+	2.49616i	0.187623	+	0.324973i	0.944457	−	0.328634i	$-0.106588\pi$
$59$	1.44116	+	2.49616i	0.187623	+	0.324973i	−0.756834	+	0.653607i	$0.773255\pi$
$60$		0			0
$61$	1.42173	−	2.46250i	0.182033	−	0.315291i	−0.760539	−	0.649292i	$-0.775065\pi$
$61$	1.42173	−	2.46250i	0.182033	−	0.315291i	0.942573	+	0.334001i	$0.108399\pi$
$62$	−6.90833			−0.877358
$63$		0			0
$64$	3.83276			0.479095
$65$		0			0
$66$		0			0
$67$	−1.20326	−	2.08411i	−0.147002	−	0.254614i	0.783116	−	0.621875i	$-0.213629\pi$
$67$	−1.20326	−	2.08411i	−0.147002	−	0.254614i	−0.930118	+	0.367261i	$0.880296\pi$
$68$	−2.04068	−	3.53456i	−0.247468	−	0.428628i
$69$		0			0
$70$		0			0
$71$	−5.54205			−0.657720			−0.328860	−	0.944379i	$-0.606664\pi$
$71$	−5.54205			−0.657720			−0.328860	+	0.944379i	$0.606664\pi$
$72$		0			0
$73$	−11.7988			−1.38095			−0.690473	−	0.723359i	$-0.742597\pi$
$73$	−11.7988			−1.38095			−0.690473	+	0.723359i	$0.742597\pi$
$74$	5.49484	−	9.51734i	0.638762	−	1.10637i
$75$		0			0
$76$	1.86338	+	3.22748i	0.213745	+	0.370217i
$77$	0.783341	+	1.35679i	0.0892700	+	0.154620i
$78$		0			0
$79$	−1.70149	+	2.94707i	−0.191433	+	0.331571i	−0.945725	−	0.324967i	$-0.894647\pi$
$79$	−1.70149	+	2.94707i	−0.191433	+	0.331571i	0.754293	+	0.656538i	$0.227980\pi$
$80$		0			0
$81$		0			0
$82$	−8.89047			−0.981789
$83$	−6.95059	+	12.0388i	−0.762926	+	1.32143i	0.178410	+	0.983956i	$0.442905\pi$
$83$	−6.95059	+	12.0388i	−0.762926	+	1.32143i	−0.941336	+	0.337470i	$0.890429\pi$
$84$		0			0
$85$		0			0
$86$	1.08431	+	1.87808i	0.116924	+	0.202518i
$87$		0			0
$88$	3.37054	−	5.83795i	0.359301	−	0.622327i
$89$	−3.38513			−0.358823			−0.179411	−	0.983774i	$-0.557419\pi$
$89$	−3.38513			−0.358823			−0.179411	+	0.983774i	$0.557419\pi$
$90$		0			0
$91$	−3.15301			−0.330525
$92$	−1.27976	+	2.21661i	−0.133425	+	0.231098i
$93$		0			0
$94$	3.02886	+	5.24614i	0.312403	+	0.541098i
$95$		0			0
$96$		0			0
$97$	−5.53779	+	9.59173i	−0.562277	+	0.973892i	0.435020	+	0.900421i	$0.356741\pi$
$97$	−5.53779	+	9.59173i	−0.562277	+	0.973892i	−0.997297	+	0.0734716i	$0.976592\pi$
$98$	11.0188			1.11307
$99$		0			0
$100$		0			0
$101$	−8.68451	+	15.0420i	−0.864141	+	1.49674i	0.00375621	+	0.999993i	$0.498804\pi$
$101$	−8.68451	+	15.0420i	−0.864141	+	1.49674i	−0.867897	+	0.496744i	$0.834529\pi$
$102$		0			0
$103$	−0.416378	−	0.721188i	−0.0410269	−	0.0710608i	0.844783	−	0.535109i	$-0.179730\pi$
$103$	−0.416378	−	0.721188i	−0.0410269	−	0.0710608i	−0.885810	+	0.464049i	$0.846396\pi$
$104$	6.78334	+	11.7491i	0.665161	+	1.15209i
$105$		0			0
$106$	−2.07650	+	3.59661i	−0.201688	+	0.349334i
$107$	11.0684			1.07002			0.535012	−	0.844844i	$-0.320307\pi$
$107$	11.0684			1.07002			0.535012	+	0.844844i	$0.320307\pi$
$108$		0			0
$109$	−4.65836			−0.446190			−0.223095	−	0.974797i	$-0.571616\pi$
$109$	−4.65836			−0.446190			−0.223095	+	0.974797i	$0.571616\pi$
$110$		0			0
$111$		0			0
$112$	1.23570	+	2.14029i	0.116762	+	0.202238i
$113$	−5.99711	−	10.3873i	−0.564160	−	0.977155i	−0.997127	−	0.0757447i	$-0.975867\pi$
$113$	−5.99711	−	10.3873i	−0.564160	−	0.977155i	0.432967	−	0.901410i	$-0.357467\pi$
$114$		0			0
$115$		0			0
$116$	−1.64542			−0.152773
$117$		0			0
$118$	4.70892			0.433491
$119$	−1.54136	+	2.66972i	−0.141297	+	0.244733i
$120$		0			0
$121$	0.694371	+	1.20269i	0.0631246	+	0.109335i
$122$	−2.32271	−	4.02305i	−0.210288	−	0.364230i
$123$		0			0
$124$	−1.41457	+	2.45011i	−0.127032	+	0.220026i
$125$		0			0
$126$		0			0
$127$	3.22858			0.286490			0.143245	−	0.989687i	$-0.454246\pi$
$127$	3.22858			0.286490			0.143245	+	0.989687i	$0.454246\pi$
$128$	6.77171	−	11.7289i	0.598540	−	1.03670i
$129$		0			0
$130$		0			0
$131$	−4.69256	−	8.12776i	−0.409991	−	0.710125i	0.584897	−	0.811107i	$-0.301135\pi$
$131$	−4.69256	−	8.12776i	−0.409991	−	0.710125i	−0.994888	+	0.100982i	$0.967802\pi$
$132$		0			0
$133$	1.40745	−	2.43778i	0.122042	−	0.211382i
$134$	−3.93159			−0.339637
$135$		0			0
$136$	13.2643			1.13740
$137$	−1.15478	+	2.00013i	−0.0986593	+	0.170883i	−0.911130	−	0.412119i	$-0.864789\pi$
$137$	−1.15478	+	2.00013i	−0.0986593	+	0.170883i	0.812471	+	0.583002i	$0.198122\pi$
$138$		0			0
$139$	−5.44701	−	9.43449i	−0.462009	−	0.800223i	0.537052	−	0.843549i	$-0.319538\pi$
$139$	−5.44701	−	9.43449i	−0.462009	−	0.800223i	−0.999061	+	0.0433260i	$0.986205\pi$
$140$		0			0
$141$		0			0
$142$	−4.52709	+	7.84115i	−0.379905	+	0.658014i
$143$	19.3430			1.61754
$144$		0			0
$145$		0			0
$146$	−9.63799	+	16.6935i	−0.797646	+	1.38156i
$147$		0			0
$148$	−2.25028	−	3.89760i	−0.184972	−	0.320381i
$149$	−8.17151	−	14.1535i	−0.669436	−	1.15950i	−0.978062	−	0.208314i	$-0.933202\pi$
$149$	−8.17151	−	14.1535i	−0.669436	−	1.15950i	0.308626	−	0.951183i	$-0.400131\pi$
$150$		0			0
$151$	−11.3913	+	19.7304i	−0.927015	+	1.60564i	−0.138727	+	0.990331i	$0.544301\pi$
$151$	−11.3913	+	19.7304i	−0.927015	+	1.60564i	−0.788288	+	0.615306i	$0.789032\pi$
$152$	−12.1119			−0.982404
$153$		0			0
$154$	2.55953			0.206252
$155$		0			0
$156$		0			0
$157$	6.23035	+	10.7913i	0.497236	+	0.861238i	0.999995	−	0.00318877i	$-0.00101502\pi$
$157$	6.23035	+	10.7913i	0.497236	+	0.861238i	−0.502759	+	0.864427i	$0.667682\pi$
$158$	2.77976	+	4.81469i	0.221146	+	0.383036i
$159$		0			0
$160$		0			0
$161$	1.93326			0.152362
$162$		0			0
$163$	7.57384			0.593229			0.296614	−	0.954997i	$-0.404142\pi$
$163$	7.57384			0.593229			0.296614	+	0.954997i	$0.404142\pi$
$164$	−1.82044	+	3.15309i	−0.142152	+	0.246215i
$165$		0			0
$166$	11.3553	+	19.6680i	0.881345	+	1.52653i
$167$	1.48837	+	2.57793i	0.115174	+	0.199486i	0.917849	−	0.396929i	$-0.129924\pi$
$167$	1.48837	+	2.57793i	0.115174	+	0.199486i	−0.802676	+	0.596416i	$0.796591\pi$
$168$		0			0
$169$	−12.9643	+	22.4548i	−0.997252	+	1.72729i
$170$		0			0
$171$		0			0
$172$	0.888105			0.0677174
$173$	7.92649	−	13.7291i	0.602640	−	1.04380i	−0.389780	−	0.920908i	$-0.627449\pi$
$173$	7.92649	−	13.7291i	0.602640	−	1.04380i	0.992420	−	0.122895i	$-0.0392177\pi$
$174$		0			0
$175$		0			0
$176$	−7.58073	−	13.1302i	−0.571419	−	0.989727i
$177$		0			0
$178$	−2.76518	+	4.78943i	−0.207259	+	0.358983i
$179$	−17.0841			−1.27693			−0.638463	−	0.769653i	$-0.720429\pi$
$179$	−17.0841			−1.27693			−0.638463	+	0.769653i	$0.720429\pi$
$180$		0			0
$181$	13.3690			0.993712			0.496856	−	0.867833i	$-0.334488\pi$
$181$	13.3690			0.993712			0.496856	+	0.867833i	$0.334488\pi$
$182$	−2.57557	+	4.46102i	−0.190914	+	0.330673i
$183$		0			0
$184$	−4.15919	−	7.20393i	−0.306620	−	0.531081i
$185$		0			0
$186$		0			0
$187$	9.45593	−	16.3782i	0.691486	−	1.19769i
$188$	2.48079			0.180930
$189$		0			0
$190$		0			0
$191$	12.6686	−	21.9427i	0.916669	−	1.58772i	0.112230	−	0.993682i	$-0.464201\pi$
$191$	12.6686	−	21.9427i	0.916669	−	1.58772i	0.804439	−	0.594035i	$-0.202466\pi$
$192$		0			0
$193$	−4.77976	−	8.27879i	−0.344055	−	0.595921i	0.641127	−	0.767435i	$-0.278467\pi$
$193$	−4.77976	−	8.27879i	−0.344055	−	0.595921i	−0.985182	+	0.171515i	$0.945134\pi$
$194$	9.04721	+	15.6702i	0.649552	+	1.12506i
$195$		0			0
$196$	2.25625	−	3.90794i	0.161161	−	0.279139i
$197$	−2.06841			−0.147368			−0.0736842	−	0.997282i	$-0.523476\pi$
$197$	−2.06841			−0.147368			−0.0736842	+	0.997282i	$0.523476\pi$
$198$		0			0
$199$	13.0970			0.928419			0.464210	−	0.885725i	$-0.346338\pi$
$199$	13.0970			0.928419			0.464210	+	0.885725i	$0.346338\pi$
$200$		0			0
$201$		0			0
$202$	14.1881	+	24.5745i	0.998271	+	1.72906i
$203$	0.621407	+	1.07631i	0.0436142	+	0.0755421i
$204$		0			0
$205$		0			0
$206$	−1.36049			−0.0947900
$207$		0			0
$208$	30.5130			2.11570
$209$	−8.63441	+	14.9552i	−0.597255	+	1.03448i
$210$		0			0
$211$	−5.55595	−	9.62318i	−0.382487	−	0.662487i	0.608930	−	0.793224i	$-0.291599\pi$
$211$	−5.55595	−	9.62318i	−0.382487	−	0.662487i	−0.991417	+	0.130737i	$0.958266\pi$
$212$	0.850382	+	1.47291i	0.0584045	+	0.101160i
$213$		0			0
$214$	9.04136	−	15.6601i	0.618055	−	1.07050i
$215$		0			0
$216$		0			0
$217$	2.13690			0.145063
$218$	−3.80523	+	6.59086i	−0.257723	+	0.446389i
$219$		0			0
$220$		0			0
$221$	19.0304	+	32.9617i	1.28012	+	2.21724i
$222$		0			0
$223$	−1.94701	+	3.37231i	−0.130381	+	0.225827i	−0.923824	−	0.382819i	$-0.874953\pi$
$223$	−1.94701	+	3.37231i	−0.130381	+	0.225827i	0.793442	+	0.608645i	$0.208287\pi$
$224$	1.83991			0.122934
$225$		0			0
$226$	−19.5952			−1.30346
$227$	−6.40406	+	11.0922i	−0.425053	+	0.736213i	−0.996425	−	0.0844781i	$-0.973078\pi$
$227$	−6.40406	+	11.0922i	−0.425053	+	0.736213i	0.571373	+	0.820691i	$0.306411\pi$
$228$		0			0
$229$	−3.32647	−	5.76162i	−0.219820	−	0.380739i	0.734933	−	0.678140i	$-0.237214\pi$
$229$	−3.32647	−	5.76162i	−0.219820	−	0.380739i	−0.954753	+	0.297401i	$0.903880\pi$
$230$		0			0
$231$		0			0
$232$	2.67378	−	4.63112i	0.175542	−	0.304048i
$233$	−3.65836			−0.239667			−0.119833	−	0.992794i	$-0.538236\pi$
$233$	−3.65836			−0.239667			−0.119833	+	0.992794i	$0.538236\pi$
$234$		0			0
$235$		0			0
$236$	0.964212	−	1.67006i	0.0627648	−	0.108712i
$237$		0			0
$238$	2.51816	+	4.36158i	0.163228	+	0.282720i
$239$	−7.84576	−	13.5893i	−0.507500	−	0.879016i	−0.999962	−	0.00868195i	$-0.997236\pi$
$239$	−7.84576	−	13.5893i	−0.507500	−	0.879016i	0.492462	−	0.870334i	$-0.336097\pi$
$240$		0			0
$241$	−5.61248	+	9.72110i	−0.361532	+	0.626191i	−0.988213	−	0.153084i	$-0.951079\pi$
$241$	−5.61248	+	9.72110i	−0.361532	+	0.626191i	0.626681	+	0.779276i	$0.284413\pi$
$242$	2.26882			0.145845
$243$		0			0
$244$	−1.90242			−0.121790
$245$		0			0
$246$		0			0
$247$	−17.3771	−	30.0980i	−1.10568	−	1.91509i
$248$	−4.59731	−	7.96278i	−0.291930	−	0.505637i
$249$		0			0
$250$		0			0
$251$	−6.94042			−0.438075			−0.219038	−	0.975716i	$-0.570292\pi$
$251$	−6.94042			−0.438075			−0.219038	+	0.975716i	$0.570292\pi$
$252$		0			0
$253$	−11.8601			−0.745640
$254$	2.63730	−	4.56794i	0.165479	−	0.286618i
$255$		0			0
$256$	−7.23035	−	12.5233i	−0.451897	−	0.782708i
$257$	−9.16635	−	15.8766i	−0.571781	−	0.990354i	−0.996383	−	0.0849739i	$-0.972919\pi$
$257$	−9.16635	−	15.8766i	−0.571781	−	0.990354i	0.424602	−	0.905380i	$-0.360414\pi$
$258$		0			0
$259$	−1.69968	+	2.94393i	−0.105613	+	0.182927i
$260$		0			0
$261$		0			0
$262$	−15.3327			−0.947257
$263$	−8.03832	+	13.9228i	−0.495664	+	0.858515i	−0.999988	−	0.00499942i	$-0.998409\pi$
$263$	−8.03832	+	13.9228i	−0.495664	+	0.858515i	0.504323	+	0.863515i	$0.331742\pi$
$264$		0			0
$265$		0			0
$266$	−2.29939	−	3.98265i	−0.140984	−	0.244192i
$267$		0			0
$268$	−0.805043	+	1.39438i	−0.0491758	+	0.0851751i
$269$	18.2004			1.10970			0.554849	−	0.831951i	$-0.312776\pi$
$269$	18.2004			1.10970			0.554849	+	0.831951i	$0.312776\pi$
$270$		0			0
$271$	−2.48571			−0.150996			−0.0754979	−	0.997146i	$-0.524055\pi$
$271$	−2.48571			−0.150996			−0.0754979	+	0.997146i	$0.524055\pi$
$272$	14.9164	−	25.8360i	0.904442	−	1.56654i
$273$		0			0
$274$	1.88659	+	3.26766i	0.113973	+	0.197407i
$275$		0			0
$276$		0			0
$277$	3.83363	−	6.64004i	0.230341	−	0.398962i	−0.727568	−	0.686036i	$-0.759349\pi$
$277$	3.83363	−	6.64004i	0.230341	−	0.398962i	0.957908	+	0.287074i	$0.0926826\pi$
$278$	−17.7978			−1.06744
$279$		0			0
$280$		0			0
$281$	0.136615	−	0.236624i	0.00814978	−	0.0141158i	−0.861922	−	0.507041i	$-0.830739\pi$
$281$	0.136615	−	0.236624i	0.00814978	−	0.0141158i	0.870072	+	0.492925i	$0.164072\pi$
$282$		0			0
$283$	1.68544	+	2.91928i	0.100189	+	0.173533i	0.911763	−	0.410718i	$-0.134722\pi$
$283$	1.68544	+	2.91928i	0.100189	+	0.173533i	−0.811573	+	0.584251i	$0.801389\pi$
$284$	1.85396	+	3.21115i	0.110012	+	0.190547i
$285$		0			0
$286$	15.8006	−	27.3674i	0.934307	−	1.61827i
$287$	2.75003			0.162329
$288$		0			0
$289$	20.2125			1.18897
$290$		0			0
$291$		0			0
$292$	3.94701	+	6.83642i	0.230981	+	0.400071i
$293$	2.82202	+	4.88788i	0.164864	+	0.285553i	0.936607	−	0.350382i	$-0.113948\pi$
$293$	2.82202	+	4.88788i	0.164864	+	0.285553i	−0.771743	+	0.635935i	$0.780615\pi$
$294$		0			0
$295$		0			0
$296$	14.6267			0.850159
$297$		0			0
$298$	−26.7000			−1.54669
$299$	11.9345	−	20.6711i	0.690189	−	1.19544i
$300$		0			0
$301$	−0.335402	−	0.580933i	−0.0193322	−	0.0334844i
$302$	18.6103	+	32.2340i	1.07090	+	1.85486i
$303$		0			0
$304$	−13.6205	+	23.5914i	−0.781190	+	1.35306i
$305$		0			0
$306$		0			0
$307$	−5.44105			−0.310537			−0.155269	−	0.987872i	$-0.549624\pi$
$307$	−5.44105			−0.310537			−0.155269	+	0.987872i	$0.549624\pi$
$308$	0.524096	−	0.907761i	0.0298632	−	0.0517245i
$309$		0			0
$310$		0			0
$311$	−9.53985	−	16.5235i	−0.540955	−	0.936962i	−0.998849	−	0.0479550i	$-0.984730\pi$
$311$	−9.53985	−	16.5235i	−0.540955	−	0.936962i	0.457895	−	0.889007i	$-0.348604\pi$
$312$		0			0
$313$	−4.57116	+	7.91747i	−0.258377	+	0.447522i	−0.965807	−	0.259261i	$-0.916521\pi$
$313$	−4.57116	+	7.91747i	−0.258377	+	0.447522i	0.707430	+	0.706783i	$0.249854\pi$
$314$	20.3573			1.14883
$315$		0			0
$316$	2.27677			0.128078
$317$	−7.11836	+	12.3294i	−0.399807	+	0.692486i	−0.993702	−	0.112056i	$-0.964256\pi$
$317$	−7.11836	+	12.3294i	−0.399807	+	0.692486i	0.593895	+	0.804543i	$0.297590\pi$
$318$		0			0
$319$	−3.81220	−	6.60292i	−0.213442	−	0.369693i
$320$		0			0
$321$		0			0
$322$	1.57921	−	2.73527i	0.0880057	−	0.152430i
$323$	−33.9795			−1.89067
$324$		0			0
$325$		0			0
$326$	6.18678	−	10.7158i	0.342654	−	0.593494i
$327$		0			0
$328$	−5.91638	−	10.2475i	−0.326677	−	0.565822i
$329$	−0.936896	−	1.62275i	−0.0516527	−	0.0894652i
$330$		0			0
$331$	6.10001	−	10.5655i	0.335287	−	0.580734i	−0.648253	−	0.761425i	$-0.724500\pi$
$331$	6.10001	−	10.5655i	0.335287	−	0.580734i	0.983540	+	0.180691i	$0.0578334\pi$
$332$	9.30061			0.510437
$333$		0			0
$334$	4.86317			0.266101
$335$		0			0
$336$		0			0
$337$	2.29493	+	3.97494i	0.125013	+	0.216529i	0.921738	−	0.387813i	$-0.126769\pi$
$337$	2.29493	+	3.97494i	0.125013	+	0.216529i	−0.796725	+	0.604342i	$0.793436\pi$
$338$	21.1800	+	36.6849i	1.15204	+	1.99540i
$339$		0			0
$340$		0			0
$341$	−13.1094			−0.709917
$342$		0			0
$343$	−6.94582			−0.375039
$344$	−1.44316	+	2.49962i	−0.0778099	+	0.134771i
$345$		0			0
$346$	−12.9497	−	22.4295i	−0.696180	−	1.20582i
$347$	−16.7301	−	28.9775i	−0.898121	−	1.55559i	−0.829894	−	0.557921i	$-0.811599\pi$
$347$	−16.7301	−	28.9775i	−0.898121	−	1.55559i	−0.0682272	−	0.997670i	$-0.521734\pi$
$348$		0			0
$349$	14.0408	−	24.3193i	0.751586	−	1.30178i	−0.195468	−	0.980710i	$-0.562623\pi$
$349$	14.0408	−	24.3193i	0.751586	−	1.30178i	0.947054	−	0.321074i	$-0.104044\pi$
$350$		0			0
$351$		0			0
$352$	−11.2875			−0.601624
$353$	0.920851	−	1.59496i	0.0490119	−	0.0848912i	−0.840479	−	0.541845i	$-0.817726\pi$
$353$	0.920851	−	1.59496i	0.0490119	−	0.0848912i	0.889491	+	0.456954i	$0.151059\pi$
$354$		0			0
$355$		0			0
$356$	1.13241	+	1.96140i	0.0600178	+	0.103954i
$357$		0			0
$358$	−13.9553	+	24.1714i	−0.737563	+	1.27750i
$359$	12.1119			0.639241			0.319621	−	0.947546i	$-0.396445\pi$
$359$	12.1119			0.639241			0.319621	+	0.947546i	$0.396445\pi$
$360$		0			0
$361$	12.0274			0.633020
$362$	10.9206	−	18.9151i	0.573976	−	0.994156i
$363$		0			0
$364$	1.05476	+	1.82690i	0.0552846	+	0.0957558i
$365$		0			0
$366$		0			0
$367$	−7.28688	+	12.6212i	−0.380372	+	0.658824i	−0.991115	−	0.133005i	$-0.957537\pi$
$367$	−7.28688	+	12.6212i	−0.380372	+	0.658824i	0.610743	+	0.791829i	$0.290871\pi$
$368$	−18.7090			−0.975274
$369$		0			0
$370$		0			0
$371$	0.642310	−	1.11251i	0.0333471	−	0.0577589i
$372$		0			0
$373$	−4.72323	−	8.18087i	−0.244560	−	0.423590i	0.717448	−	0.696612i	$-0.245310\pi$
$373$	−4.72323	−	8.18087i	−0.244560	−	0.423590i	−0.962008	+	0.273022i	$0.911977\pi$
$374$	−15.4484	−	26.7574i	−0.798817	−	1.38359i
$375$		0			0
$376$	−4.03125	+	6.98233i	−0.207896	+	0.360086i
$377$	15.3444			0.790276
$378$		0			0
$379$	−28.5541			−1.46673			−0.733363	−	0.679837i	$-0.762051\pi$
$379$	−28.5541			−1.46673			−0.733363	+	0.679837i	$0.762051\pi$
$380$		0			0
$381$		0			0
$382$	−20.6970	−	35.8483i	−1.05895	−	1.83416i
$383$	−0.732704	−	1.26908i	−0.0374394	−	0.0648470i	0.846699	−	0.532073i	$-0.178587\pi$
$383$	−0.732704	−	1.26908i	−0.0374394	−	0.0648470i	−0.884138	+	0.467226i	$0.845253\pi$
$384$		0			0
$385$		0			0
$386$	−15.6176			−0.794916
$387$		0			0
$388$	7.41014			0.376193
$389$	−6.45506	+	11.1805i	−0.327284	+	0.566873i	−0.981972	−	0.189026i	$-0.939467\pi$
$389$	−6.45506	+	11.1805i	−0.327284	+	0.566873i	0.654688	+	0.755900i	$0.272800\pi$
$390$		0			0
$391$	−11.6685	−	20.2104i	−0.590100	−	1.02208i
$392$	7.33276	+	12.7007i	0.370360	+	0.641483i
$393$		0			0
$394$	−1.68961	+	2.92649i	−0.0851212	+	0.147434i
$395$		0			0
$396$		0			0
$397$	0.868386			0.0435831			0.0217915	−	0.999763i	$-0.493063\pi$
$397$	0.868386			0.0435831			0.0217915	+	0.999763i	$0.493063\pi$
$398$	10.6984	−	18.5302i	0.536263	−	0.928834i
$399$		0			0
$400$		0			0
$401$	16.7063	+	28.9361i	0.834270	+	1.44500i	0.894623	+	0.446822i	$0.147444\pi$
$401$	16.7063	+	28.9361i	0.834270	+	1.44500i	−0.0603527	+	0.998177i	$0.519223\pi$
$402$		0			0
$403$	13.1916	−	22.8486i	0.657122	−	1.13817i
$404$	11.6208			0.578156
$405$		0			0
$406$	2.03042			0.100768
$407$	10.4272	−	18.0604i	0.516856	−	0.895221i
$408$		0			0
$409$	−2.52767	−	4.37806i	−0.124985	−	0.216481i	0.796742	−	0.604320i	$-0.206555\pi$
$409$	−2.52767	−	4.37806i	−0.124985	−	0.216481i	−0.921727	+	0.387839i	$0.873222\pi$
$410$		0			0
$411$		0			0
$412$	−0.278579	+	0.482512i	−0.0137246	+	0.0237717i
$413$	−1.45658			−0.0716734
$414$		0			0
$415$		0			0
$416$	11.3582	−	19.6730i	0.556883	−	0.964549i
$417$		0			0
$418$	14.1062	+	24.4327i	0.689959	+	1.19504i
$419$	5.47880	+	9.48955i	0.267657	+	0.463595i	0.968256	−	0.249960i	$-0.0804174\pi$
$419$	5.47880	+	9.48955i	0.267657	+	0.463595i	−0.700600	+	0.713555i	$0.747084\pi$
$420$		0			0
$421$	5.31932	−	9.21333i	0.259248	−	0.449030i	−0.706793	−	0.707421i	$-0.749859\pi$
$421$	5.31932	−	9.21333i	0.259248	−	0.449030i	0.966041	+	0.258390i	$0.0831921\pi$
$422$	−18.1538			−0.883711
$423$		0			0
$424$	−5.52744			−0.268436
$425$		0			0
$426$		0			0
$427$	0.718467	+	1.24442i	0.0347691	+	0.0602218i
$428$	−3.70267	−	6.41322i	−0.178975	−	0.309995i
$429$		0			0
$430$		0			0
$431$	37.3529			1.79923			0.899613	−	0.436687i	$-0.143848\pi$
$431$	37.3529			1.79923			0.899613	+	0.436687i	$0.143848\pi$
$432$		0			0
$433$	17.2125			0.827179			0.413589	−	0.910464i	$-0.364275\pi$
$433$	17.2125			0.827179			0.413589	+	0.910464i	$0.364275\pi$
$434$	1.74556	−	3.02339i	0.0837894	−	0.145127i
$435$		0			0
$436$	1.55834	+	2.69913i	0.0746310	+	0.129265i
$437$	10.6547	+	18.4545i	0.509684	+	0.882799i
$438$		0			0
$439$	15.8744	−	27.4952i	0.757642	−	1.31228i	−0.186408	−	0.982473i	$-0.559684\pi$
$439$	15.8744	−	27.4952i	0.757642	−	1.31228i	0.944050	−	0.329803i	$-0.106982\pi$
$440$		0			0
$441$		0			0
$442$	62.1809			2.95764
$443$	0.177979	−	0.308268i	0.00845603	−	0.0146463i	−0.861766	−	0.507305i	$-0.830642\pi$
$443$	0.177979	−	0.308268i	0.00845603	−	0.0146463i	0.870222	+	0.492659i	$0.163975\pi$
$444$		0			0
$445$		0			0
$446$	3.18087	+	5.50943i	0.150619	+	0.260879i
$447$		0			0
$448$	−0.968438	+	1.67738i	−0.0457544	+	0.0792490i
$449$	7.85632			0.370762			0.185381	−	0.982667i	$-0.440648\pi$
$449$	7.85632			0.370762			0.185381	+	0.982667i	$0.440648\pi$
$450$		0			0
$451$	−16.8708			−0.794416
$452$	−4.01238	+	6.94964i	−0.188726	+	0.326884i
$453$		0			0
$454$	10.4625	+	18.1215i	0.491028	+	0.850485i
$455$		0			0
$456$		0			0
$457$	10.7455	−	18.6118i	0.502654	−	0.870622i	−0.497341	−	0.867555i	$-0.665690\pi$
$457$	10.7455	−	18.6118i	0.502654	−	0.870622i	0.999995	−	0.00306742i	$-0.000976391\pi$
$458$	−10.8691			−0.507879
$459$		0			0
$460$		0			0
$461$	20.4964	−	35.5007i	0.954611	−	1.65343i	0.219355	−	0.975645i	$-0.429605\pi$
$461$	20.4964	−	35.5007i	0.954611	−	1.65343i	0.735256	−	0.677789i	$-0.237062\pi$
$462$		0			0
$463$	21.0669	+	36.4890i	0.979063	+	1.69579i	0.665816	+	0.746116i	$0.268084\pi$
$463$	21.0669	+	36.4890i	0.979063	+	1.69579i	0.313248	+	0.949671i	$0.398583\pi$
$464$	−6.01363	−	10.4159i	−0.279176	−	0.483546i
$465$		0			0
$466$	−2.98837	+	5.17601i	−0.138434	+	0.239774i
$467$	−22.5376			−1.04292			−0.521459	−	0.853276i	$-0.674612\pi$
$467$	−22.5376			−1.04292			−0.521459	+	0.853276i	$0.674612\pi$
$468$		0			0
$469$	1.21613			0.0561557
$470$		0			0
$471$		0			0
$472$	3.13366	+	5.42766i	0.144238	+	0.249828i
$473$	2.05762	+	3.56390i	0.0946093	+	0.163868i
$474$		0			0
$475$		0			0
$476$	2.06251			0.0945348
$477$		0			0
$478$	−25.6356			−1.17255
$479$	−16.6440	+	28.8282i	−0.760483	+	1.31720i	0.182119	+	0.983277i	$0.441704\pi$
$479$	−16.6440	+	28.8282i	−0.760483	+	1.31720i	−0.942602	+	0.333919i	$0.891629\pi$
$480$		0			0
$481$	20.9851	+	36.3472i	0.956837	+	1.65729i
$482$	9.16924	+	15.8816i	0.417647	+	0.723387i
$483$		0			0
$484$	0.464570	−	0.804660i	0.0211168	−	0.0365754i
$485$		0			0
$486$		0			0
$487$	−23.7703			−1.07713			−0.538566	−	0.842583i	$-0.681034\pi$
$487$	−23.7703			−1.07713			−0.538566	+	0.842583i	$0.681034\pi$
$488$	3.09140	−	5.35447i	0.139941	−	0.242385i
$489$		0			0
$490$		0			0
$491$	−2.30281	−	3.98859i	−0.103925	−	0.180003i	0.809374	−	0.587294i	$-0.199807\pi$
$491$	−2.30281	−	3.98859i	−0.103925	−	0.180003i	−0.913298	+	0.407291i	$0.866473\pi$
$492$		0			0
$493$	7.50118	−	12.9924i	0.337836	−	0.585150i
$494$	−56.7787			−2.55459
$495$		0			0
$496$	−20.6797			−0.928548
$497$	1.40033	−	2.42545i	0.0628135	−	0.108796i
$498$		0			0
$499$	9.44878	+	16.3658i	0.422985	+	0.732632i	0.996230	−	0.0867522i	$-0.0276488\pi$
$499$	9.44878	+	16.3658i	0.422985	+	0.732632i	−0.573245	+	0.819384i	$0.694316\pi$
$500$		0			0
$501$		0			0
$502$	−5.66936	+	9.81962i	−0.253036	+	0.438271i
$503$	35.7581			1.59438			0.797188	−	0.603731i	$-0.206320\pi$
$503$	35.7581			1.59438			0.797188	+	0.603731i	$0.206320\pi$
$504$		0			0
$505$		0			0
$506$	−9.68809	+	16.7803i	−0.430688	+	0.745974i
$507$		0			0
$508$	−1.08004	−	1.87069i	−0.0479192	−	0.0829985i
$509$	12.2034	+	21.1368i	0.540904	+	0.936874i	0.998852	+	0.0478949i	$0.0152513\pi$
$509$	12.2034	+	21.1368i	0.540904	+	0.936874i	−0.457948	+	0.888979i	$0.651415\pi$
$510$		0			0
$511$	2.98125	−	5.16368i	0.131883	−	0.228428i
$512$	3.46207			0.153003
$513$		0			0
$514$	−29.9506			−1.32106
$515$		0			0
$516$		0			0
$517$	5.74765	+	9.95523i	0.252782	+	0.437830i
$518$	2.77681	+	4.80957i	0.122006	+	0.211321i
$519$		0			0
$520$		0			0
$521$	−33.3968			−1.46314			−0.731571	−	0.681766i	$-0.761212\pi$
$521$	−33.3968			−1.46314			−0.731571	+	0.681766i	$0.761212\pi$
$522$		0			0
$523$	37.3654			1.63388			0.816938	−	0.576726i	$-0.195670\pi$
$523$	37.3654			1.63388			0.816938	+	0.576726i	$0.195670\pi$
$524$	−3.13957	+	5.43789i	−0.137153	+	0.237555i
$525$		0			0
$526$	13.1324	+	22.7460i	0.572600	+	0.991772i
$527$	−12.8976	−	22.3393i	−0.561828	−	0.973115i
$528$		0			0
$529$	4.18239	−	7.24412i	0.181843	−	0.314962i
$530$		0			0
$531$		0			0
$532$	−1.88332			−0.0816521
$533$	16.9766	−	29.4043i	0.735338	−	1.27364i
$534$		0			0
$535$		0			0
$536$	−2.61637	−	4.53168i	−0.113010	−	0.195739i
$537$		0			0
$538$	14.8672	−	25.7508i	0.640971	−	1.11019i
$539$	20.9097			0.900645
$540$		0			0
$541$	28.2560			1.21482			0.607409	−	0.794389i	$-0.292209\pi$
$541$	28.2560			1.21482			0.607409	+	0.794389i	$0.292209\pi$
$542$	−2.03048	+	3.51689i	−0.0872165	+	0.151063i
$543$		0			0
$544$	−11.1051	−	19.2345i	−0.476125	−	0.824673i
$545$		0			0
$546$		0			0
$547$	−19.2726	+	33.3811i	−0.824036	+	1.42727i	0.0786172	+	0.996905i	$0.474950\pi$
$547$	−19.2726	+	33.3811i	−0.824036	+	1.42727i	−0.902654	+	0.430368i	$0.858384\pi$
$548$	1.54521			0.0660082
$549$		0			0
$550$		0			0
$551$	−6.84949	+	11.8637i	−0.291798	+	0.505409i
$552$		0			0
$553$	−0.859845	−	1.48929i	−0.0365643	−	0.0633313i
$554$	−6.26309	−	10.8480i	−0.266093	−	0.460887i
$555$		0			0
$556$	−3.64433	+	6.31217i	−0.154554	+	0.267696i
$557$	−27.4125			−1.16151			−0.580753	−	0.814080i	$-0.697242\pi$
$557$	−27.4125			−1.16151			−0.580753	+	0.814080i	$0.697242\pi$
$558$		0			0
$559$	−8.28206			−0.350294
$560$		0			0
$561$		0			0
$562$	−0.223191	−	0.386579i	−0.00941476	−	0.0163068i
$563$	13.8196	+	23.9363i	0.582427	+	1.00879i	0.995191	+	0.0979551i	$0.0312302\pi$
$563$	13.8196	+	23.9363i	0.582427	+	1.00879i	−0.412764	+	0.910838i	$0.635437\pi$
$564$		0			0
$565$		0			0
$566$	5.50710			0.231481
$567$		0			0
$568$	−12.0506			−0.505634
$569$	−7.35807	+	12.7446i	−0.308467	+	0.534280i	−0.978027	−	0.208478i	$-0.933149\pi$
$569$	−7.35807	+	12.7446i	−0.308467	+	0.534280i	0.669561	+	0.742757i	$0.266482\pi$
$570$		0			0
$571$	14.1503	+	24.5090i	0.592172	+	1.02567i	0.993939	+	0.109930i	$0.0350627\pi$
$571$	14.1503	+	24.5090i	0.592172	+	1.02567i	−0.401768	+	0.915742i	$0.631604\pi$
$572$	−6.47074	−	11.2077i	−0.270555	−	0.468616i
$573$		0			0
$574$	2.24639	−	3.89087i	0.0937626	−	0.162402i
$575$		0			0
$576$		0			0
$577$	40.7976			1.69843			0.849214	−	0.528049i	$-0.177076\pi$
$577$	40.7976			1.69843			0.849214	+	0.528049i	$0.177076\pi$
$578$	16.5108	−	28.5975i	0.686759	−	1.18950i
$579$		0			0
$580$		0			0
$581$	−3.51247	−	6.08377i	−0.145722	−	0.252397i
$582$		0			0
$583$	−3.94044	+	6.82504i	−0.163196	+	0.282664i
$584$	−25.6553			−1.06162
$585$		0			0
$586$	9.22080			0.380908
$587$	−1.39016	+	2.40784i	−0.0573782	+	0.0993820i	−0.893288	−	0.449485i	$-0.851607\pi$
$587$	−1.39016	+	2.40784i	−0.0573782	+	0.0993820i	0.835910	+	0.548867i	$0.184941\pi$
$588$		0			0
$589$	11.7771	+	20.3985i	0.485265	+	0.840504i
$590$		0			0
$591$		0			0
$592$	16.4485	−	28.4897i	0.676031	−	1.17092i
$593$	−14.8084			−0.608109			−0.304055	−	0.952655i	$-0.598341\pi$
$593$	−14.8084			−0.608109			−0.304055	+	0.952655i	$0.598341\pi$
$594$		0			0
$595$		0			0
$596$	−5.46717	+	9.46941i	−0.223944	+	0.387882i
$597$		0			0
$598$	−19.4976	−	33.7709i	−0.797318	−	1.38100i
$599$	−8.17151	−	14.1535i	−0.333879	−	0.578295i	0.649390	−	0.760456i	$-0.275024\pi$
$599$	−8.17151	−	14.1535i	−0.333879	−	0.578295i	−0.983269	+	0.182160i	$0.941691\pi$
$600$		0			0
$601$	3.31185	−	5.73630i	0.135093	−	0.233988i	−0.790540	−	0.612411i	$-0.790200\pi$
$601$	3.31185	−	5.73630i	0.135093	−	0.233988i	0.925633	+	0.378422i	$0.123533\pi$
$602$	−1.09591			−0.0446658
$603$		0			0
$604$	15.2428			0.620221
$605$		0			0
$606$		0			0
$607$	15.1547	+	26.2487i	0.615110	+	1.06540i	0.990365	+	0.138480i	$0.0442216\pi$
$607$	15.1547	+	26.2487i	0.615110	+	1.06540i	−0.375256	+	0.926921i	$0.622445\pi$
$608$	10.1403	+	17.5634i	0.411242	+	0.712291i
$609$		0			0
$610$		0			0
$611$	−23.1347			−0.935931
$612$		0			0
$613$	−14.7803			−0.596969			−0.298484	−	0.954415i	$-0.596481\pi$
$613$	−14.7803			−0.596969			−0.298484	+	0.954415i	$0.596481\pi$
$614$	−4.44459	+	7.69825i	−0.179369	+	0.310676i
$615$		0			0
$616$	1.70330	+	2.95020i	0.0686278	+	0.118867i
$617$	16.9256	+	29.3160i	0.681399	+	1.18022i	0.974554	+	0.224152i	$0.0719614\pi$
$617$	16.9256	+	29.3160i	0.681399	+	1.18022i	−0.293155	+	0.956065i	$0.594705\pi$
$618$		0			0
$619$	−5.84433	+	10.1227i	−0.234903	+	0.406865i	−0.959245	−	0.282577i	$-0.908811\pi$
$619$	−5.84433	+	10.1227i	−0.234903	+	0.406865i	0.724341	+	0.689442i	$0.242144\pi$
$620$		0			0
$621$		0			0
$622$	−31.1709			−1.24984
$623$	0.855334	−	1.48148i	0.0342682	−	0.0593543i
$624$		0			0
$625$		0			0
$626$	7.46800	+	12.9350i	0.298481	+	0.516985i
$627$		0			0
$628$	4.16843	−	7.21993i	0.166338	−	0.288107i
$629$	41.0347			1.63616
$630$		0			0
$631$	−38.1357			−1.51816			−0.759078	−	0.650999i	$-0.774350\pi$
$631$	−38.1357			−1.51816			−0.759078	+	0.650999i	$0.774350\pi$
$632$	−3.69972	+	6.40810i	−0.147167	+	0.254901i
$633$		0			0
$634$	11.6294	+	20.1428i	0.461864	+	0.799972i
$635$		0			0
$636$		0			0
$637$	−21.0408	+	36.4437i	−0.833666	+	1.44395i
$638$	−12.4562			−0.493144
$639$		0			0
$640$		0			0
$641$	−17.3827	+	30.1077i	−0.686576	+	1.18918i	0.286363	+	0.958121i	$0.407554\pi$
$641$	−17.3827	+	30.1077i	−0.686576	+	1.18918i	−0.972939	+	0.231063i	$0.925780\pi$
$642$		0			0
$643$	1.34258	+	2.32541i	0.0529461	+	0.0917053i	0.891284	−	0.453446i	$-0.149806\pi$
$643$	1.34258	+	2.32541i	0.0529461	+	0.0917053i	−0.838338	+	0.545151i	$0.816472\pi$
$644$	−0.646726	−	1.12016i	−0.0254846	−	0.0441406i
$645$		0			0
$646$	−27.7565	+	48.0757i	−1.09207	+	1.89151i
$647$	40.5103			1.59262			0.796311	−	0.604887i	$-0.206782\pi$
$647$	40.5103			1.59262			0.796311	+	0.604887i	$0.206782\pi$
$648$		0			0
$649$	8.93578			0.350760
$650$		0			0
$651$		0			0
$652$	−2.53365	−	4.38841i	−0.0992253	−	0.171863i
$653$	−6.66772	−	11.5488i	−0.260928	−	0.451941i	0.705561	−	0.708650i	$-0.250695\pi$
$653$	−6.66772	−	11.5488i	−0.260928	−	0.451941i	−0.966489	+	0.256709i	$0.917362\pi$
$654$		0			0
$655$		0			0
$656$	−26.6132			−1.03907
$657$		0			0
$658$	−3.06126			−0.119340
$659$	15.5772	−	26.9804i	0.606800	−	1.05101i	−0.384965	−	0.922931i	$-0.625786\pi$
$659$	15.5772	−	26.9804i	0.606800	−	1.05101i	0.991764	−	0.128077i	$-0.0408803\pi$
$660$		0			0
$661$	−3.15894	−	5.47145i	−0.122869	−	0.212815i	0.798029	−	0.602619i	$-0.205876\pi$
$661$	−3.15894	−	5.47145i	−0.122869	−	0.212815i	−0.920898	+	0.389804i	$0.872543\pi$
$662$	−9.96574	−	17.2612i	−0.387329	−	0.670874i
$663$		0			0
$664$	−15.1134	+	26.1771i	−0.586512	+	1.01587i
$665$		0			0
$666$		0			0
$667$	−9.40838			−0.364294
$668$	0.995798	−	1.72477i	0.0385286	−	0.0667334i
$669$		0			0
$670$		0			0
$671$	−4.40764	−	7.63426i	−0.170155	−	0.294717i
$672$		0			0
$673$	−3.29610	+	5.70901i	−0.127055	+	0.220066i	−0.922534	−	0.385915i	$-0.873886\pi$
$673$	−3.29610	+	5.70901i	−0.127055	+	0.220066i	0.795479	+	0.605981i	$0.207219\pi$
$674$	7.49857			0.288834
$675$		0			0
$676$	17.3476			0.667214
$677$	−17.4473	+	30.2197i	−0.670556	+	1.16144i	0.307191	+	0.951648i	$0.400611\pi$
$677$	−17.4473	+	30.2197i	−0.670556	+	1.16144i	−0.977747	+	0.209788i	$0.932722\pi$
$678$		0			0
$679$	−2.79851	−	4.84716i	−0.107397	−	0.186017i
$680$		0			0
$681$		0			0
$682$	−10.7086	+	18.5479i	−0.410054	+	0.710234i
$683$	26.0958			0.998528			0.499264	−	0.866450i	$-0.333604\pi$
$683$	26.0958			0.998528			0.499264	+	0.866450i	$0.333604\pi$
$684$		0			0
$685$		0			0
$686$	−5.67378	+	9.82727i	−0.216626	+	0.375207i
$687$		0			0
$688$	3.24583	+	5.62194i	0.123746	+	0.214334i
$689$	−7.93028	−	13.7356i	−0.302119	−	0.523286i
$690$		0			0
$691$	14.6529	−	25.3796i	0.557423	−	0.965485i	−0.440288	−	0.897857i	$-0.645124\pi$
$691$	14.6529	−	25.3796i	0.557423	−	0.965485i	0.997711	−	0.0676282i	$-0.0215432\pi$
$692$	−10.6065			−0.403197
$693$		0			0
$694$	−54.6648			−2.07505
$695$		0			0
$696$		0			0
$697$	−16.5982	−	28.7489i	−0.628701	−	1.08894i
$698$	−22.9387	−	39.7311i	−0.868244	−	1.50384i
$699$		0			0
$700$		0			0
$701$	−15.3891			−0.581239			−0.290620	−	0.956839i	$-0.593861\pi$
$701$	−15.3891			−0.581239			−0.290620	+	0.956839i	$0.593861\pi$
$702$		0			0
$703$	−37.4696			−1.41319
$704$	5.94116	−	10.2904i	0.223916	−	0.387834i
$705$		0			0
$706$	−1.50442	−	2.60572i	−0.0566194	−	0.0980677i
$707$	−4.38870	−	7.60146i	−0.165054	−	0.285882i
$708$		0			0
$709$	3.86996	−	6.70296i	0.145339	−	0.251735i	−0.784160	−	0.620558i	$-0.786906\pi$
$709$	3.86996	−	6.70296i	0.145339	−	0.251735i	0.929499	+	0.368823i	$0.120239\pi$
$710$		0			0
$711$		0			0
$712$	−7.36062			−0.275851
$713$	−8.08842	+	14.0096i	−0.302914	+	0.524662i
$714$		0			0
$715$		0			0
$716$	5.71508	+	9.89880i	0.213582	+	0.369936i
$717$		0			0
$718$	9.89374	−	17.1365i	0.369231	−	0.639527i
$719$	15.1316			0.564313			0.282156	−	0.959368i	$-0.408950\pi$
$719$	15.1316			0.564313			0.282156	+	0.959368i	$0.408950\pi$
$720$		0			0
$721$	0.420832			0.0156726
$722$	9.82470	−	17.0169i	0.365638	−	0.633303i
$723$		0			0
$724$	−4.47229	−	7.74623i	−0.166211	−	0.287886i
$725$		0			0
$726$		0			0
$727$	0.0809381	−	0.140189i	0.00300183	−	0.00519932i	−0.864521	−	0.502597i	$-0.832378\pi$
$727$	0.0809381	−	0.140189i	0.00300183	−	0.00519932i	0.867522	+	0.497398i	$0.165711\pi$
$728$	−6.85590			−0.254097
$729$		0			0
$730$		0			0
$731$	−4.04873	+	7.01260i	−0.149748	+	0.259370i
$732$		0			0
$733$	−25.0166	−	43.3300i	−0.924009	−	1.60043i	−0.793148	−	0.609029i	$-0.791559\pi$
$733$	−25.0166	−	43.3300i	−0.924009	−	1.60043i	−0.130861	−	0.991401i	$-0.541774\pi$
$734$	11.9047	+	20.6196i	0.439412	+	0.761084i
$735$		0			0
$736$	−6.96427	+	12.0625i	−0.256707	+	0.444629i
$737$	−7.46070			−0.274818
$738$		0			0
$739$	30.5505			1.12382			0.561909	−	0.827199i	$-0.310067\pi$
$739$	30.5505			1.12382			0.561909	+	0.827199i	$0.310067\pi$
$740$		0			0
$741$		0			0
$742$	−1.04936	−	1.81754i	−0.0385231	−	0.0667240i
$743$	−2.98342	−	5.16743i	−0.109451	−	0.189575i	0.806097	−	0.591783i	$-0.201576\pi$
$743$	−2.98342	−	5.16743i	−0.109451	−	0.189575i	−0.915548	+	0.402209i	$0.868243\pi$
$744$		0			0
$745$		0			0
$746$	−15.4329			−0.565039
$747$		0			0
$748$	−12.6530			−0.462640
$749$	−2.79670	+	4.84403i	−0.102189	+	0.176997i
$750$		0			0
$751$	−17.1988	−	29.7892i	−0.627593	−	1.08702i	−0.988033	−	0.154240i	$-0.950707\pi$
$751$	−17.1988	−	29.7892i	−0.627593	−	1.08702i	0.360441	−	0.932782i	$-0.382626\pi$
$752$	9.06674	+	15.7041i	0.330630	+	0.572668i
$753$		0			0
$754$	12.5342	−	21.7099i	0.456470	−	0.790630i
$755$		0			0
$756$		0			0
$757$	−40.6873			−1.47881			−0.739403	−	0.673263i	$-0.764892\pi$
$757$	−40.6873			−1.47881			−0.739403	+	0.673263i	$0.764892\pi$
$758$	−23.3248	+	40.3997i	−0.847194	+	1.46738i
$759$		0			0
$760$		0			0
$761$	14.1298	+	24.4735i	0.512204	+	0.887164i	0.999900	+	0.0141502i	$0.00450429\pi$
$761$	14.1298	+	24.4735i	0.512204	+	0.887164i	−0.487696	+	0.873014i	$0.662162\pi$
$762$		0			0
$763$	1.17705	−	2.03870i	0.0426119	−	0.0738060i
$764$	−16.9519			−0.613299
$765$		0			0
$766$	−2.39407			−0.0865013
$767$	−8.99180	+	15.5743i	−0.324675	+	0.562354i
$768$		0			0
$769$	23.4518	+	40.6197i	0.845694	+	1.46478i	0.885017	+	0.465558i	$0.154146\pi$
$769$	23.4518	+	40.6197i	0.845694	+	1.46478i	−0.0393235	+	0.999227i	$0.512520\pi$
$770$		0			0
$771$		0			0
$772$	−3.19791	+	5.53894i	−0.115095	+	0.199351i
$773$	9.19641			0.330772			0.165386	−	0.986229i	$-0.447113\pi$
$773$	9.19641			0.330772			0.165386	+	0.986229i	$0.447113\pi$
$774$		0			0
$775$		0			0
$776$	−12.0414	+	20.8563i	−0.432260	+	0.748696i
$777$		0			0
$778$	10.5458	+	18.2658i	0.378085	+	0.654862i
$779$	15.1562	+	26.2512i	0.543025	+	0.940548i
$780$		0			0
$781$	−8.59074	+	14.8796i	−0.307401	+	0.532434i
$782$	−38.1261			−1.36339
$783$		0			0
$784$	32.9844			1.17801
$785$		0			0
$786$		0			0
$787$	2.87319	+	4.97651i	0.102418	+	0.177393i	0.912680	−	0.408674i	$-0.134009\pi$
$787$	2.87319	+	4.97651i	0.102418	+	0.177393i	−0.810262	+	0.586067i	$0.800675\pi$
$788$	0.691939	+	1.19847i	0.0246493	+	0.0426938i
$789$		0			0
$790$

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.816862 - 1.41485i) q^{2} +(-0.334526 - 0.579416i) q^{4} +(-0.252674 + 0.437645i) q^{7} +2.17440 q^{8} +O(q^{10})\)	\(q+(0.816862 - 1.41485i) q^{2} +(-0.334526 - 0.579416i) q^{4} +(-0.252674 + 0.437645i) q^{7} +2.17440 q^{8} +(1.55010 - 2.68485i) q^{11} +(3.11964 + 5.40337i) q^{13} +(0.412800 + 0.714990i) q^{14} +(2.44524 - 4.23527i) q^{16} +6.10020 q^{17} -5.57022 q^{19} +(-2.53244 - 4.38631i) q^{22} +(-1.91280 - 3.31307i) q^{23} +10.1932 q^{26} +0.338104 q^{28} +(1.22966 - 2.12984i) q^{29} +(-2.11429 - 3.66206i) q^{31} +(-1.82044 - 3.15309i) q^{32} +(4.98302 - 8.63085i) q^{34} +6.72677 q^{37} +(-4.55010 + 7.88101i) q^{38} +(-2.72092 - 4.71278i) q^{41} +(-0.663704 + 1.14957i) q^{43} -2.07420 q^{44} -6.24997 q^{46} +(-1.85396 + 3.21115i) q^{47} +(3.37231 + 5.84101i) q^{49} +(2.08720 - 3.61514i) q^{52} -2.54205 q^{53} +(-0.549415 + 0.951614i) q^{56} +(-2.00893 - 3.47956i) q^{58} +(1.44116 + 2.49616i) q^{59} +(1.42173 - 2.46250i) q^{61} -6.90833 q^{62} +3.83276 q^{64} +(-1.20326 - 2.08411i) q^{67} +(-2.04068 - 3.53456i) q^{68} -5.54205 q^{71} -11.7988 q^{73} +(5.49484 - 9.51734i) q^{74} +(1.86338 + 3.22748i) q^{76} +(0.783341 + 1.35679i) q^{77} +(-1.70149 + 2.94707i) q^{79} -8.89047 q^{82} +(-6.95059 + 12.0388i) q^{83} +(1.08431 + 1.87808i) q^{86} +(3.37054 - 5.83795i) q^{88} -3.38513 q^{89} -3.15301 q^{91} +(-1.27976 + 2.21661i) q^{92} +(3.02886 + 5.24614i) q^{94} +(-5.53779 + 9.59173i) q^{97} +11.0188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

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