LMFDB - Embedded newform 675.2.e.c.451.1

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			451.1
Root			$1.31686 - 2.28087i$ of defining polynomial
Character	$\chi$	$=$	675.451
Dual form			675.2.e.c.226.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.31686 + 2.28087i) q^{2} +(-2.46825 - 4.27513i) q^{4} +(-0.898714 + 1.55662i) q^{7} +7.73393 q^{8} +O(q^{10})$	\(q+(-1.31686 + 2.28087i) q^{2} +(-2.46825 - 4.27513i) q^{4} +(-0.898714 + 1.55662i) q^{7} +7.73393 q^{8} +(0.904062 - 1.56588i) q^{11} +(0.985914 + 1.70765i) q^{13} +(-2.36696 - 4.09970i) q^{14} +(-5.24801 + 9.08982i) q^{16} +4.80812 q^{17} +2.96467 q^{19} +(2.38105 + 4.12410i) q^{22} +(0.866963 + 1.50162i) q^{23} -5.19325 q^{26} +8.87300 q^{28} +(-3.68382 + 6.38057i) q^{29} +(1.31151 + 2.27161i) q^{31} +(-6.08789 - 10.5445i) q^{32} +(-6.33163 + 10.9667i) q^{34} -11.6351 q^{37} +(-3.90406 + 6.76203i) q^{38} +(-1.23324 - 2.13603i) q^{41} +(-3.63907 + 6.30306i) q^{43} -8.92580 q^{44} -4.56668 q^{46} +(-3.14604 + 5.44910i) q^{47} +(1.88463 + 3.26427i) q^{49} +(4.86696 - 8.42983i) q^{52} +1.72540 q^{53} +(-6.95059 + 12.0388i) q^{56} +(-9.70218 - 16.8047i) q^{58} +(5.51300 + 9.54880i) q^{59} +(6.33521 - 10.9729i) q^{61} -6.90833 q^{62} +11.0756 q^{64} +(4.55187 + 7.88407i) q^{67} +(-11.8676 - 20.5554i) q^{68} -1.27460 q^{71} +3.58770 q^{73} +(15.3218 - 26.5382i) q^{74} +(-7.31755 - 12.6744i) q^{76} +(1.62499 + 2.81456i) q^{77} +(-1.05545 + 1.82809i) q^{79} +6.49602 q^{82} +(-0.549415 + 0.951614i) q^{83} +(-9.58431 - 16.6005i) q^{86} +(6.99195 - 12.1104i) q^{88} +13.2935 q^{89} -3.54422 q^{91} +(4.27976 - 7.41277i) q^{92} +(-8.28580 - 14.3514i) q^{94} +(-1.91638 + 3.31926i) q^{97} -9.92718 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.31686	+	2.28087i	−0.931162	+	1.61282i	−0.149823	+	0.988713i	$0.547870\pi$
$2$	−1.31686	+	2.28087i	−0.931162	+	1.61282i	−0.781339	+	0.624107i	$0.785463\pi$
$3$		0			0
$3$		0			0
$4$	−2.46825	−	4.27513i	−1.23412	−	2.13757i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.898714	+	1.55662i	−0.339682	+	0.588346i	−0.984373	−	0.176097i	$-0.943653\pi$
$7$	−0.898714	+	1.55662i	−0.339682	+	0.588346i	0.644691	+	0.764443i	$0.276986\pi$
$8$	7.73393			2.73436
$9$		0			0
$10$		0			0
$11$	0.904062	−	1.56588i	0.272585	−	0.472131i	−0.696938	−	0.717131i	$-0.745455\pi$
$11$	0.904062	−	1.56588i	0.272585	−	0.472131i	0.969523	+	0.245000i	$0.0787881\pi$
$12$		0			0
$13$	0.985914	+	1.70765i	0.273443	+	0.473618i	0.969741	−	0.244135i	$-0.0785041\pi$
$13$	0.985914	+	1.70765i	0.273443	+	0.473618i	−0.696298	+	0.717753i	$0.745171\pi$
$14$	−2.36696	−	4.09970i	−0.632598	−	1.09569i
$15$		0			0
$16$	−5.24801	+	9.08982i	−1.31200	+	2.27246i
$17$	4.80812			1.16614			0.583071	−	0.812421i	$-0.301851\pi$
$17$	4.80812			1.16614			0.583071	+	0.812421i	$0.301851\pi$
$18$		0			0
$19$	2.96467			0.680142			0.340071	−	0.940400i	$-0.389549\pi$
$19$	2.96467			0.680142			0.340071	+	0.940400i	$0.389549\pi$
$20$		0			0
$21$		0			0
$22$	2.38105	+	4.12410i	0.507641	+	0.879261i
$23$	0.866963	+	1.50162i	0.180774	+	0.313110i	0.942144	−	0.335207i	$-0.108806\pi$
$23$	0.866963	+	1.50162i	0.180774	+	0.313110i	−0.761370	+	0.648317i	$0.775473\pi$
$24$		0			0
$25$		0			0
$26$	−5.19325			−1.01848
$27$		0			0
$28$	8.87300			1.67684
$29$	−3.68382	+	6.38057i	−0.684069	+	1.18484i	0.289659	+	0.957130i	$0.406458\pi$
$29$	−3.68382	+	6.38057i	−0.684069	+	1.18484i	−0.973728	+	0.227713i	$0.926875\pi$
$30$		0			0
$31$	1.31151	+	2.27161i	0.235555	+	0.407993i	0.959434	−	0.281934i	$-0.0909760\pi$
$31$	1.31151	+	2.27161i	0.235555	+	0.407993i	−0.723879	+	0.689927i	$0.757643\pi$
$32$	−6.08789	−	10.5445i	−1.07620	−	1.86403i
$33$		0			0
$34$	−6.33163	+	10.9667i	−1.08587	+	1.88078i
$35$		0			0
$36$		0			0
$37$	−11.6351			−1.91280			−0.956399	−	0.292063i	$-0.905658\pi$
$37$	−11.6351			−1.91280			−0.956399	+	0.292063i	$0.905658\pi$
$38$	−3.90406	+	6.76203i	−0.633322	+	1.09695i
$39$		0			0
$40$		0			0
$41$	−1.23324	−	2.13603i	−0.192600	−	0.333592i	0.753511	−	0.657435i	$-0.228359\pi$
$41$	−1.23324	−	2.13603i	−0.192600	−	0.333592i	−0.946111	+	0.323842i	$0.895025\pi$
$42$		0			0
$43$	−3.63907	+	6.30306i	−0.554953	+	0.961207i	0.442954	+	0.896544i	$0.353931\pi$
$43$	−3.63907	+	6.30306i	−0.554953	+	0.961207i	−0.997907	+	0.0646628i	$0.979403\pi$
$44$	−8.92580			−1.34562
$45$		0			0
$46$	−4.56668			−0.673321
$47$	−3.14604	+	5.44910i	−0.458897	+	0.794833i	−0.998903	−	0.0468283i	$-0.985089\pi$
$47$	−3.14604	+	5.44910i	−0.458897	+	0.794833i	0.540006	+	0.841661i	$0.318422\pi$
$48$		0			0
$49$	1.88463	+	3.26427i	0.269233	+	0.466324i
$50$		0			0
$51$		0			0
$52$	4.86696	−	8.42983i	0.674926	−	1.16901i
$53$	1.72540			0.237001			0.118501	−	0.992954i	$-0.462191\pi$
$53$	1.72540			0.237001			0.118501	+	0.992954i	$0.462191\pi$
$54$		0			0
$55$		0			0
$56$	−6.95059	+	12.0388i	−0.928811	+	1.60875i
$57$		0			0
$58$	−9.70218	−	16.8047i	−1.27396	−	2.20656i
$59$	5.51300	+	9.54880i	0.717732	+	1.24315i	0.961896	+	0.273414i	$0.0881529\pi$
$59$	5.51300	+	9.54880i	0.717732	+	1.24315i	−0.244165	+	0.969734i	$0.578514\pi$
$60$		0			0
$61$	6.33521	−	10.9729i	0.811141	−	1.40494i	−0.100925	−	0.994894i	$-0.532180\pi$
$61$	6.33521	−	10.9729i	0.811141	−	1.40494i	0.912066	−	0.410043i	$-0.134486\pi$
$62$	−6.90833			−0.877358
$63$		0			0
$64$	11.0756			1.38445
$65$		0			0
$66$		0			0
$67$	4.55187	+	7.88407i	0.556100	+	0.963193i	0.997817	+	0.0660386i	$0.0210360\pi$
$67$	4.55187	+	7.88407i	0.556100	+	0.963193i	−0.441717	+	0.897154i	$0.645631\pi$
$68$	−11.8676	−	20.5554i	−1.43916	−	2.49271i
$69$		0			0
$70$		0			0
$71$	−1.27460			−0.151268			−0.0756338	−	0.997136i	$-0.524098\pi$
$71$	−1.27460			−0.151268			−0.0756338	+	0.997136i	$0.524098\pi$
$72$		0			0
$73$	3.58770			0.419908			0.209954	−	0.977711i	$-0.432669\pi$
$73$	3.58770			0.419908			0.209954	+	0.977711i	$0.432669\pi$
$74$	15.3218	−	26.5382i	1.78112	−	3.08500i
$75$		0			0
$76$	−7.31755	−	12.6744i	−0.839380	−	1.45385i
$77$	1.62499	+	2.81456i	0.185184	+	0.320749i
$78$		0			0
$79$	−1.05545	+	1.82809i	−0.118747	+	0.205676i	−0.919272	−	0.393624i	$-0.871221\pi$
$79$	−1.05545	+	1.82809i	−0.118747	+	0.205676i	0.800524	+	0.599300i	$0.204555\pi$
$80$		0			0
$81$		0			0
$82$	6.49602			0.717366
$83$	−0.549415	+	0.951614i	−0.0603061	+	0.104453i	−0.894602	−	0.446863i	$-0.852541\pi$
$83$	−0.549415	+	0.951614i	−0.0603061	+	0.104453i	0.834296	+	0.551317i	$0.185874\pi$
$84$		0			0
$85$		0			0
$86$	−9.58431	−	16.6005i	−1.03350	−	1.79008i
$87$		0			0
$88$	6.99195	−	12.1104i	0.745344	−	1.29097i
$89$	13.2935			1.40910			0.704552	−	0.709653i	$-0.251148\pi$
$89$	13.2935			1.40910			0.704552	+	0.709653i	$0.251148\pi$
$90$		0			0
$91$	−3.54422			−0.371535
$92$	4.27976	−	7.41277i	0.446196	−	0.772834i
$93$		0			0
$94$	−8.28580	−	14.3514i	−0.854615	−	1.48024i
$95$		0			0
$96$		0			0
$97$	−1.91638	+	3.31926i	−0.194579	+	0.337020i	−0.946762	−	0.321933i	$-0.895667\pi$
$97$	−1.91638	+	3.31926i	−0.194579	+	0.337020i	0.752184	+	0.658954i	$0.229001\pi$
$98$	−9.92718			−1.00280
$99$		0			0
$100$		0			0
$101$	3.27618	−	5.67452i	0.325993	−	0.564636i	−0.655720	−	0.755004i	$-0.727635\pi$
$101$	3.27618	−	5.67452i	0.325993	−	0.564636i	0.981713	+	0.190368i	$0.0609682\pi$
$102$		0			0
$103$	−4.03779	−	6.99365i	−0.397855	−	0.689105i	0.595606	−	0.803277i	$-0.296912\pi$
$103$	−4.03779	−	6.99365i	−0.397855	−	0.689105i	−0.993461	+	0.114172i	$0.963579\pi$
$104$	7.62499	+	13.2069i	0.747691	+	1.29504i
$105$		0			0
$106$	−2.27211	+	3.93541i	−0.220687	+	0.382241i
$107$	−8.97674			−0.867814			−0.433907	−	0.900958i	$-0.642865\pi$
$107$	−8.97674			−0.867814			−0.433907	+	0.900958i	$0.642865\pi$
$108$		0			0
$109$	−6.34164			−0.607419			−0.303710	−	0.952765i	$-0.598225\pi$
$109$	−6.34164			−0.607419			−0.303710	+	0.952765i	$0.598225\pi$
$110$		0			0
$111$		0			0
$112$	−9.43292	−	16.3383i	−0.891327	−	1.54382i
$113$	7.45127	+	12.9060i	0.700957	+	1.21409i	0.968131	+	0.250444i	$0.0805767\pi$
$113$	7.45127	+	12.9060i	0.700957	+	1.21409i	−0.267174	+	0.963648i	$0.586090\pi$
$114$		0			0
$115$		0			0
$116$	36.3704			3.37691
$117$		0			0
$118$	−29.0394			−2.67330
$119$	−4.32113	+	7.48441i	−0.396117	+	0.686095i
$120$		0			0
$121$	3.86534	+	6.69497i	0.351395	+	0.608634i
$122$	16.6852	+	28.8996i	1.51061	+	2.61645i
$123$		0			0
$124$	6.47428	−	11.2138i	0.581408	−	1.00703i
$125$		0			0
$126$		0			0
$127$	−3.62303			−0.321492			−0.160746	−	0.986996i	$-0.551390\pi$
$127$	−3.62303			−0.321492			−0.160746	+	0.986996i	$0.551390\pi$
$128$	−2.40922	+	4.17289i	−0.212947	+	0.368835i
$129$		0			0
$130$		0			0
$131$	3.64673	+	6.31631i	0.318616	+	0.551859i	0.980200	−	0.198012i	$-0.0634486\pi$
$131$	3.64673	+	6.31631i	0.318616	+	0.551859i	−0.661584	+	0.749871i	$0.730115\pi$
$132$		0			0
$133$	−2.66439	+	4.61486i	−0.231032	+	0.400159i
$134$	−23.9767			−2.07127
$135$		0			0
$136$	37.1857			3.18865
$137$	3.56310	−	6.17148i	0.304417	−	0.527265i	−0.672715	−	0.739902i	$-0.734872\pi$
$137$	3.56310	−	6.17148i	0.304417	−	0.527265i	0.977131	+	0.212637i	$0.0682052\pi$
$138$		0			0
$139$	7.35533	+	12.7398i	0.623871	+	1.08058i	0.988758	+	0.149525i	$0.0477744\pi$
$139$	7.35533	+	12.7398i	0.623871	+	1.08058i	−0.364887	+	0.931052i	$0.618892\pi$
$140$		0			0
$141$		0			0
$142$	1.67848	−	2.90721i	0.140855	−	0.243967i
$143$	3.56531			0.298146
$144$		0			0
$145$		0			0
$146$	−4.72450	+	8.18308i	−0.391003	+	0.677236i
$147$		0			0
$148$	28.7183	+	49.7416i	2.36063	+	4.08873i
$149$	−0.282655	−	0.489572i	−0.0231560	−	0.0401073i	0.854215	−	0.519920i	$-0.174038\pi$
$149$	−0.282655	−	0.489572i	−0.0231560	−	0.0401073i	−0.877371	+	0.479812i	$0.840705\pi$
$150$		0			0
$151$	−0.0766925	+	0.132835i	−0.00624115	+	0.0108100i	−0.869129	−	0.494585i	$-0.835320\pi$
$151$	−0.0766925	+	0.132835i	−0.00624115	+	0.0108100i	0.862888	+	0.505395i	$0.168653\pi$
$152$	22.9285			1.85975
$153$		0			0
$154$	−8.55953			−0.689746
$155$		0			0
$156$		0			0
$157$	−5.73035	−	9.92525i	−0.457332	−	0.792121i	0.541487	−	0.840709i	$-0.317861\pi$
$157$	−5.73035	−	9.92525i	−0.457332	−	0.792121i	−0.998819	+	0.0485874i	$0.984528\pi$
$158$	−2.77976	−	4.81469i	−0.221146	−	0.383036i
$159$		0			0
$160$		0			0
$161$	−3.11661			−0.245623
$162$		0			0
$163$	22.0595			1.72783			0.863915	−	0.503637i	$-0.168005\pi$
$163$	22.0595			1.72783			0.863915	+	0.503637i	$0.168005\pi$
$164$	−6.08789	+	10.5445i	−0.475384	+	0.823389i
$165$		0			0
$166$	−1.44701	−	2.50629i	−0.112310	−	0.194526i
$167$	−8.53421	−	14.7817i	−0.660397	−	1.14384i	−0.980511	−	0.196462i	$-0.937055\pi$
$167$	−8.53421	−	14.7817i	−0.660397	−	1.14384i	0.320115	−	0.947379i	$-0.396279\pi$
$168$		0			0
$169$	4.55595	−	7.89113i	0.350457	−	0.607010i
$170$		0			0
$171$		0			0
$172$	35.9285			2.73953
$173$	−5.97233	+	10.3444i	−0.454067	+	0.786468i	−0.998634	−	0.0522497i	$-0.983361\pi$
$173$	−5.97233	+	10.3444i	−0.454067	+	0.786468i	0.544567	+	0.838718i	$0.316694\pi$
$174$		0			0
$175$		0			0
$176$	9.48906	+	16.4355i	0.715265	+	1.23887i
$177$		0			0
$178$	−17.5056	+	30.3207i	−1.31210	+	2.27263i
$179$	−8.54921			−0.638998			−0.319499	−	0.947587i	$-0.603515\pi$
$179$	−8.54921			−0.638998			−0.319499	+	0.947587i	$0.603515\pi$
$180$		0			0
$181$	−10.5524			−0.784351			−0.392176	−	0.919890i	$-0.628277\pi$
$181$	−10.5524			−0.784351			−0.392176	+	0.919890i	$0.628277\pi$
$182$	4.66724	−	8.08390i	0.345959	−	0.599219i
$183$		0			0
$184$	6.70503	+	11.6135i	0.494301	+	0.856155i
$185$		0			0
$186$		0			0
$187$	4.34684	−	7.52895i	0.317873	−	0.550571i
$188$	31.0608			2.26534
$189$		0			0
$190$		0			0
$191$	−8.66862	+	15.0145i	−0.627239	+	1.08641i	0.360864	+	0.932618i	$0.382482\pi$
$191$	−8.66862	+	15.0145i	−0.627239	+	1.08641i	−0.988103	+	0.153792i	$0.950852\pi$
$192$		0			0
$193$	0.779763	+	1.35059i	0.0561286	+	0.0972175i	0.892724	−	0.450603i	$-0.148791\pi$
$193$	0.779763	+	1.35059i	0.0561286	+	0.0972175i	−0.836596	+	0.547821i	$0.815458\pi$
$194$	−5.04721	−	8.74202i	−0.362369	−	0.627641i
$195$		0			0
$196$	9.30346	−	16.1141i	0.664533	−	1.15100i
$197$	17.9767			1.28079			0.640395	−	0.768046i	$-0.278771\pi$
$197$	17.9767			1.28079			0.640395	+	0.768046i	$0.278771\pi$
$198$		0			0
$199$	11.0225			0.781362			0.390681	−	0.920526i	$-0.372240\pi$
$199$	11.0225			0.781362			0.390681	+	0.920526i	$0.372240\pi$
$200$		0			0
$201$		0			0
$202$	8.62856	+	14.9451i	0.607104	+	1.05153i
$203$	−6.62141	−	11.4686i	−0.464732	−	0.804939i
$204$		0			0
$205$		0			0
$206$	21.2688			1.48187
$207$		0			0
$208$	−20.6964			−1.43503
$209$	2.68025	−	4.64232i	0.185397	−	0.321116i
$210$		0			0
$211$	11.9643	+	20.7227i	0.823655	+	1.42661i	0.902943	+	0.429760i	$0.141402\pi$
$211$	11.9643	+	20.7227i	0.823655	+	1.42661i	−0.0792886	+	0.996852i	$0.525265\pi$
$212$	−4.25871	−	7.37630i	−0.292489	−	0.506606i
$213$		0			0
$214$	11.8211	−	20.4748i	0.808076	−	1.39963i
$215$		0			0
$216$		0			0
$217$	−4.71470			−0.320055
$218$	8.35107	−	14.4645i	0.565606	−	0.979658i
$219$		0			0
$220$		0			0
$221$	4.74040	+	8.21061i	0.318874	+	0.552305i
$222$		0			0
$223$	10.8553	−	18.8020i	0.726927	−	1.25907i	−0.231249	−	0.972895i	$-0.574281\pi$
$223$	10.8553	−	18.8020i	0.726927	−	1.25907i	0.958176	−	0.286180i	$-0.0923855\pi$
$224$	21.8851			1.46226
$225$		0			0
$226$	−39.2492			−2.61082
$227$	−7.05010	+	12.2111i	−0.467932	+	0.810481i	−0.999328	−	0.0366416i	$-0.988334\pi$
$227$	−7.05010	+	12.2111i	−0.467932	+	0.810481i	0.531397	+	0.847123i	$0.321667\pi$
$228$		0			0
$229$	−1.83879	−	3.18488i	−0.121511	−	0.210463i	0.798853	−	0.601526i	$-0.205441\pi$
$229$	−1.83879	−	3.18488i	−0.121511	−	0.210463i	−0.920364	+	0.391064i	$0.872107\pi$
$230$		0			0
$231$		0			0
$232$	−28.4904	+	49.3469i	−1.87049	+	3.23978i
$233$	−5.34164			−0.349943			−0.174971	−	0.984574i	$-0.555983\pi$
$233$	−5.34164			−0.349943			−0.174971	+	0.984574i	$0.555983\pi$
$234$		0			0
$235$		0			0
$236$	27.2149	−	47.1376i	1.77154	−	3.06840i
$237$		0			0
$238$	−11.3807	−	19.7119i	−0.737698	−	1.27773i
$239$	−11.0167	−	19.0815i	−0.712613	−	1.23428i	−0.963873	−	0.266362i	$-0.914178\pi$
$239$	−11.0167	−	19.0815i	−0.712613	−	1.23428i	0.251260	−	0.967920i	$-0.419155\pi$
$240$		0			0
$241$	9.32358	−	16.1489i	0.600585	−	1.04024i	−0.392148	−	0.919902i	$-0.628268\pi$
$241$	9.32358	−	16.1489i	0.600585	−	1.04024i	0.992733	−	0.120341i	$-0.0383988\pi$
$242$	−20.3605			−1.30882
$243$		0			0
$244$	−62.5475			−4.00420
$245$		0			0
$246$		0			0
$247$	2.92291	+	5.06263i	0.185980	+	0.322127i
$248$	10.1431	+	17.5684i	0.644091	+	1.11560i
$249$		0			0
$250$		0			0
$251$	−14.6929			−0.927407			−0.463704	−	0.885990i	$-0.653480\pi$
$251$	−14.6929			−0.927407			−0.463704	+	0.885990i	$0.653480\pi$
$252$		0			0
$253$	3.13515			0.197105
$254$	4.77103	−	8.26366i	0.299361	−	0.518508i
$255$		0			0
$256$	4.73035	+	8.19320i	0.295647	+	0.512075i
$257$	−11.1045	−	19.2335i	−0.692678	−	1.19975i	−0.970957	−	0.239253i	$-0.923098\pi$
$257$	−11.1045	−	19.2335i	−0.692678	−	1.19975i	0.278280	−	0.960500i	$-0.410236\pi$
$258$		0			0
$259$	10.4566	−	18.1114i	0.649743	−	1.12539i
$260$		0			0
$261$		0			0
$262$	−19.2089			−1.18673
$263$	−2.87001	+	4.97100i	−0.176972	+	0.306525i	−0.940842	−	0.338846i	$-0.889964\pi$
$263$	−2.87001	+	4.97100i	−0.176972	+	0.306525i	0.763870	+	0.645370i	$0.223297\pi$
$264$		0			0
$265$		0			0
$266$	−7.01727	−	12.1543i	−0.430256	−	0.745226i
$267$		0			0
$268$	22.4703	−	38.9197i	1.37259	−	2.37740i
$269$	15.6162			0.952139			0.476070	−	0.879408i	$-0.342061\pi$
$269$	15.6162			0.952139			0.476070	+	0.879408i	$0.342061\pi$
$270$		0			0
$271$	−6.75315			−0.410225			−0.205112	−	0.978738i	$-0.565756\pi$
$271$	−6.75315			−0.410225			−0.205112	+	0.978738i	$0.565756\pi$
$272$	−25.2331	+	43.7050i	−1.52998	+	2.65000i
$273$		0			0
$274$	9.38423	+	16.2540i	0.566922	+	0.981938i
$275$		0			0
$276$		0			0
$277$	15.1483	−	26.2376i	0.910172	−	1.57646i	0.0963529	−	0.995347i	$-0.469282\pi$
$277$	15.1483	−	26.2376i	0.910172	−	1.57646i	0.813820	−	0.581118i	$-0.197384\pi$
$278$	−38.7438			−2.32370
$279$		0			0
$280$		0			0
$281$	9.31755	−	16.1385i	0.555838	−	0.962740i	−0.441999	−	0.897015i	$-0.645731\pi$
$281$	9.31755	−	16.1385i	0.555838	−	0.962740i	0.997838	−	0.0657249i	$-0.0209360\pi$
$282$		0			0
$283$	−2.83683	−	4.91354i	−0.168632	−	0.292079i	0.769307	−	0.638879i	$-0.220602\pi$
$283$	−2.83683	−	4.91354i	−0.168632	−	0.292079i	−0.937939	+	0.346800i	$0.887268\pi$
$284$	3.14604	+	5.44910i	0.186683	+	0.323345i
$285$		0			0
$286$	−4.69502	+	8.13201i	−0.277622	+	0.480856i
$287$	4.43332			0.261690
$288$		0			0
$289$	6.11806			0.359886
$290$		0			0
$291$		0			0
$292$	−8.85533	−	15.3379i	−0.518219	−	0.897582i
$293$	−9.13867	−	15.8286i	−0.533887	−	0.924720i	−0.999216	−	0.0395819i	$-0.987397\pi$
$293$	−9.13867	−	15.8286i	−0.533887	−	0.924720i	0.465329	−	0.885138i	$-0.345936\pi$
$294$		0			0
$295$		0			0
$296$	−89.9850			−5.23027
$297$		0			0
$298$	1.48887			0.0862478
$299$	−1.70950	+	2.96094i	−0.0988631	+	0.171236i
$300$		0			0
$301$	−6.54097	−	11.3293i	−0.377015	−	0.653009i
$302$	−0.201987	−	0.349852i	−0.0116230	−	0.0201317i
$303$		0			0
$304$	−15.5586	+	26.9483i	−0.892349	+	1.54559i
$305$		0			0
$306$		0			0
$307$	15.5050			0.884915			0.442458	−	0.896789i	$-0.354107\pi$
$307$	15.5050			0.884915			0.442458	+	0.896789i	$0.354107\pi$
$308$	8.02174	−	13.8941i	0.457081	−	0.791688i
$309$		0			0
$310$		0			0
$311$	15.2232	+	26.3673i	0.863228	+	1.49515i	0.868796	+	0.495170i	$0.164894\pi$
$311$	15.2232	+	26.3673i	0.863228	+	1.49515i	−0.00556798	+	0.999984i	$0.501772\pi$
$312$		0			0
$313$	−3.47468	+	6.01832i	−0.196401	+	0.340176i	−0.947359	−	0.320174i	$-0.896259\pi$
$313$	−3.47468	+	6.01832i	−0.196401	+	0.340176i	0.750958	+	0.660350i	$0.229592\pi$
$314$	30.1843			1.70340
$315$		0			0
$316$	10.4205			0.586196
$317$	8.07253	−	13.9820i	0.453398	−	0.785309i	−0.545196	−	0.838308i	$-0.683545\pi$
$317$	8.07253	−	13.9820i	0.453398	−	0.785309i	0.998595	+	0.0529995i	$0.0168782\pi$
$318$		0			0
$319$	6.66081	+	11.5369i	0.372934	+	0.645940i
$320$		0			0
$321$		0			0
$322$	4.10414	−	7.10858i	0.228715	−	0.396146i
$323$	14.2545			0.793142
$324$		0			0
$325$		0			0
$326$	−29.0493	+	50.3148i	−1.60889	+	2.78668i
$327$		0			0
$328$	−9.53779	−	16.5199i	−0.526636	−	0.912160i
$329$	−5.65478	−	9.79436i	−0.311758	−	0.539981i
$330$		0			0
$331$	−6.31112	+	10.9312i	−0.346890	+	0.600832i	−0.985695	−	0.168537i	$-0.946096\pi$
$331$	−6.31112	+	10.9312i	−0.346890	+	0.600832i	0.638805	+	0.769369i	$0.279429\pi$
$332$	5.42437			0.297701
$333$		0			0
$334$	44.9535			2.45975
$335$		0			0
$336$		0			0
$337$	−3.46020	−	5.99324i	−0.188489	−	0.326473i	0.756258	−	0.654274i	$-0.227026\pi$
$337$	−3.46020	−	5.99324i	−0.188489	−	0.326473i	−0.944747	+	0.327801i	$0.893692\pi$
$338$	11.9991	+	20.7831i	0.652665	+	1.13045i
$339$		0			0
$340$		0			0
$341$	4.74276			0.256835
$342$		0			0
$343$	−19.3570			−1.04518
$344$	−28.1443	+	48.7474i	−1.51744	+	2.62828i
$345$		0			0
$346$	−15.7295	−	27.2442i	−0.845621	−	1.46466i
$347$	−6.90317	−	11.9566i	−0.370581	−	0.641866i	0.619074	−	0.785333i	$-0.287508\pi$
$347$	−6.90317	−	11.9566i	−0.370581	−	0.641866i	−0.989655	+	0.143467i	$0.954175\pi$
$348$		0			0
$349$	−3.28384	+	5.68778i	−0.175780	+	0.304460i	−0.940431	−	0.339985i	$-0.889578\pi$
$349$	−3.28384	+	5.68778i	−0.175780	+	0.304460i	0.764651	+	0.644445i	$0.222911\pi$
$350$		0			0
$351$		0			0
$352$	−22.0153			−1.17342
$353$	1.76250	−	3.05273i	0.0938082	−	0.162481i	−0.815302	−	0.579035i	$-0.803429\pi$
$353$	1.76250	−	3.05273i	0.0938082	−	0.162481i	0.909111	+	0.416555i	$0.136763\pi$
$354$		0			0
$355$		0			0
$356$	−32.8116	−	56.8313i	−1.73901	−	3.01205i
$357$		0			0
$358$	11.2581	−	19.4996i	0.595010	−	1.03059i
$359$	−22.9285			−1.21012			−0.605061	−	0.796179i	$-0.706851\pi$
$359$	−22.9285			−1.21012			−0.605061	+	0.796179i	$0.706851\pi$
$360$		0			0
$361$	−10.2107			−0.537407
$362$	13.8960	−	24.0686i	0.730358	−	1.26502i
$363$		0			0
$364$	8.74801	+	15.1520i	0.458520	+	0.794181i
$365$		0			0
$366$		0			0
$367$	2.08966	−	3.61939i	0.109079	−	0.188931i	−0.806318	−	0.591482i	$-0.798543\pi$
$367$	2.08966	−	3.61939i	0.109079	−	0.188931i	0.915397	+	0.402551i	$0.131876\pi$
$368$	−18.1993			−0.948706
$369$		0			0
$370$		0			0
$371$	−1.55064	+	2.68578i	−0.0805051	+	0.139439i
$372$		0			0
$373$	3.42045	+	5.92440i	0.177104	+	0.306754i	0.940888	−	0.338719i	$-0.109994\pi$
$373$	3.42045	+	5.92440i	0.177104	+	0.306754i	−0.763783	+	0.645473i	$0.776660\pi$
$374$	11.4484	+	19.8292i	0.591982	+	1.02534i
$375$		0			0
$376$	−24.3312	+	42.1429i	−1.25479	+	2.17336i
$377$	−14.5277			−0.748217
$378$		0			0
$379$	−12.7764			−0.656280			−0.328140	−	0.944629i	$-0.606422\pi$
$379$	−12.7764			−0.656280			−0.328140	+	0.944629i	$0.606422\pi$
$380$		0			0
$381$		0			0
$382$	−22.8307	−	39.5440i	−1.16812	−	2.02325i
$383$	−3.76730	−	6.52515i	−0.192500	−	0.333420i	0.753578	−	0.657358i	$-0.228326\pi$
$383$	−3.76730	−	6.52515i	−0.192500	−	0.333420i	−0.946078	+	0.323939i	$0.894993\pi$
$384$		0			0
$385$		0			0
$386$	−4.10736			−0.209059
$387$		0			0
$388$	18.9204			0.960538
$389$	2.72588	−	4.72135i	0.138207	−	0.239382i	−0.788611	−	0.614893i	$-0.789199\pi$
$389$	2.72588	−	4.72135i	0.138207	−	0.239382i	0.926818	+	0.375511i	$0.122533\pi$
$390$		0			0
$391$	4.16847	+	7.22000i	0.210808	+	0.365131i
$392$	14.5756	+	25.2456i	0.736177	+	1.27510i
$393$		0			0
$394$	−23.6729	+	41.0026i	−1.19262	+	2.06568i
$395$		0			0
$396$		0			0
$397$	5.64549			0.283339			0.141670	−	0.989914i	$-0.454753\pi$
$397$	5.64549			0.283339			0.141670	+	0.989914i	$0.454753\pi$
$398$	−14.5151	+	25.1408i	−0.727574	+	1.26020i
$399$		0			0
$400$		0			0
$401$	−2.75209	−	4.76676i	−0.137433	−	0.238040i	0.789091	−	0.614276i	$-0.210552\pi$
$401$	−2.75209	−	4.76676i	−0.137433	−	0.238040i	−0.926524	+	0.376235i	$0.877218\pi$
$402$		0			0
$403$	−2.58608	+	4.47922i	−0.128822	+	0.223126i
$404$	−32.3458			−1.60926
$405$		0			0
$406$	34.8779			1.73096
$407$	−10.5188	+	18.2192i	−0.521400	+	0.903091i
$408$		0			0
$409$	−16.4265	−	28.4515i	−0.812238	−	1.40684i	−0.911295	−	0.411755i	$-0.864916\pi$
$409$	−16.4265	−	28.4515i	−0.812238	−	1.40684i	0.0990570	−	0.995082i	$-0.468417\pi$
$410$		0			0
$411$		0			0
$412$	−19.9325	+	34.5241i	−0.982005	+	1.70088i
$413$	−19.8184			−0.975202
$414$		0			0
$415$		0			0
$416$	12.0043	−	20.7920i	0.588558	−	1.01941i
$417$		0			0
$418$	7.05903	+	12.2266i	0.345268	+	0.598022i
$419$	11.4295	+	19.7965i	0.558369	+	0.967124i	0.997633	+	0.0687656i	$0.0219060\pi$
$419$	11.4295	+	19.7965i	0.558369	+	0.967124i	−0.439264	+	0.898358i	$0.644761\pi$
$420$		0			0
$421$	−8.97071	+	15.5377i	−0.437205	+	0.757262i	−0.997473	−	0.0710498i	$-0.977365\pi$
$421$	−8.97071	+	15.5377i	−0.437205	+	0.757262i	0.560267	+	0.828312i	$0.310698\pi$
$422$	−63.0212			−3.06782
$423$		0			0
$424$	13.3441			0.648046
$425$		0			0
$426$		0			0
$427$	11.3871	+	19.7230i	0.551060	+	0.954463i
$428$	22.1568	+	38.3768i	1.07099	+	1.85501i
$429$		0			0
$430$		0			0
$431$	6.18871			0.298100			0.149050	−	0.988830i	$-0.452378\pi$
$431$	6.18871			0.298100			0.149050	+	0.988830i	$0.452378\pi$
$432$		0			0
$433$	3.11806			0.149844			0.0749221	−	0.997189i	$-0.476129\pi$
$433$	3.11806			0.149844			0.0749221	+	0.997189i	$0.476129\pi$
$434$	6.20861	−	10.7536i	0.298023	−	0.516190i
$435$		0			0
$436$	15.6528	+	27.1114i	0.749631	+	1.29840i
$437$	2.57026	+	4.45182i	0.122952	+	0.212960i
$438$		0			0
$439$	−6.75494	+	11.6999i	−0.322396	+	0.558406i	−0.980982	−	0.194100i	$-0.937822\pi$
$439$	−6.75494	+	11.6999i	−0.322396	+	0.558406i	0.658586	+	0.752505i	$0.271155\pi$
$440$		0			0
$441$		0			0
$442$	−24.9698			−1.18769
$443$	12.1387	−	21.0248i	0.576726	−	0.998918i	−0.419126	−	0.907928i	$-0.637663\pi$
$443$	12.1387	−	21.0248i	0.576726	−	0.998918i	0.995852	−	0.0909904i	$-0.0290032\pi$
$444$		0			0
$445$		0			0
$446$	28.5899	+	49.5192i	1.35377	+	2.34480i
$447$		0			0
$448$	−9.95377	+	17.2404i	−0.470271	+	0.814534i
$449$	24.1437			1.13941			0.569705	−	0.821849i	$-0.307057\pi$
$449$	24.1437			1.13941			0.569705	+	0.821849i	$0.307057\pi$
$450$		0			0
$451$	−4.45970			−0.209999
$452$	36.7832	−	63.7104i	1.73014	−	2.99668i
$453$		0			0
$454$	−18.5680	−	32.1607i	−0.871440	−	1.50938i
$455$		0			0
$456$		0			0
$457$	−1.41078	+	2.44355i	−0.0659937	+	0.114304i	−0.897134	−	0.441758i	$-0.854355\pi$
$457$	−1.41078	+	2.44355i	−0.0659937	+	0.114304i	0.831141	+	0.556062i	$0.187688\pi$
$458$	9.68573			0.452585
$459$		0			0
$460$		0			0
$461$	10.7286	−	18.5825i	0.499681	−	0.865474i	−0.500318	−	0.865841i	$-0.666784\pi$
$461$	10.7286	−	18.5825i	0.499681	−	0.865474i	1.00000		0.000367761i	$0.000117062\pi$
$462$		0			0
$463$	−9.90167	−	17.1502i	−0.460170	−	0.797037i	0.538799	−	0.842434i	$-0.318878\pi$
$463$	−9.90167	−	17.1502i	−0.460170	−	0.797037i	−0.998969	+	0.0453970i	$0.985545\pi$
$464$	−38.6655	−	66.9706i	−1.79500	−	3.10903i
$465$		0			0
$466$	7.03421	−	12.1836i	0.325853	−	0.564395i
$467$	22.7210			1.05140			0.525701	−	0.850669i	$-0.323803\pi$
$467$	22.7210			1.05140			0.525701	+	0.850669i	$0.323803\pi$
$468$		0			0
$469$	−16.3633			−0.755588
$470$		0			0
$471$		0			0
$472$	42.6372	+	73.8497i	1.96253	+	3.39921i
$473$	6.57989	+	11.3967i	0.302544	+	0.524021i
$474$		0			0
$475$		0			0
$476$	42.6625			1.95543
$477$		0			0
$478$	58.0300			2.65423
$479$	10.6440	−	18.4359i	0.486336	−	0.842359i	−0.513541	−	0.858065i	$-0.671666\pi$
$479$	10.6440	−	18.4359i	0.486336	−	0.842359i	0.999877	+	0.0157065i	$0.00499974\pi$
$480$		0			0
$481$	−11.4712	−	19.8687i	−0.523042	−	0.905935i
$482$	24.5557	+	42.5318i	1.11848	+	1.93727i
$483$		0			0
$484$	19.0813	−	33.0497i	0.867330	−	1.50226i
$485$		0			0
$486$		0			0
$487$	9.58690			0.434424			0.217212	−	0.976124i	$-0.430304\pi$
$487$	9.58690			0.434424			0.217212	+	0.976124i	$0.430304\pi$
$488$	48.9961	−	84.8637i	2.21795	−	3.84160i
$489$		0			0
$490$		0			0
$491$	−18.9222	−	32.7742i	−0.853945	−	1.47908i	−0.877620	−	0.479357i	$-0.840870\pi$
$491$	−18.9222	−	32.7742i	−0.853945	−	1.47908i	0.0236745	−	0.999720i	$-0.492463\pi$
$492$		0			0
$493$	−17.7123	+	30.6786i	−0.797721	+	1.38169i
$494$	−15.3963			−0.692711
$495$		0			0
$496$	−27.5314			−1.23619
$497$	1.14550	−	1.98407i	0.0513829	−	0.0889977i
$498$		0			0
$499$	−8.46266	−	14.6577i	−0.378840	−	0.656171i	0.612053	−	0.790816i	$-0.290344\pi$
$499$	−8.46266	−	14.6577i	−0.378840	−	0.656171i	−0.990894	+	0.134646i	$0.957010\pi$
$500$		0			0
$501$		0			0
$502$	19.3485	−	33.5126i	0.863566	−	1.49574i
$503$	40.4168			1.80210			0.901048	−	0.433719i	$-0.142799\pi$
$503$	40.4168			1.80210			0.901048	+	0.433719i	$0.142799\pi$
$504$		0			0
$505$		0			0
$506$	−4.12856	+	7.15088i	−0.183537	+	0.317896i
$507$		0			0
$508$	8.94253	+	15.4889i	0.396761	+	0.687210i
$509$	−20.7034	−	35.8593i	−0.917660	−	1.58943i	−0.802959	−	0.596034i	$-0.796742\pi$
$509$	−20.7034	−	35.8593i	−0.917660	−	1.58943i	−0.114701	−	0.993400i	$-0.536591\pi$
$510$		0			0
$511$	−3.22431	+	5.58467i	−0.142635	+	0.247051i
$512$	−34.5537			−1.52707
$513$		0			0
$514$	58.4922			2.57998
$515$		0			0
$516$		0			0
$517$	5.68843	+	9.85265i	0.250177	+	0.433319i
$518$	27.5398	+	47.7004i	1.21003	+	2.09584i
$519$		0			0
$520$		0			0
$521$	17.0301			0.746103			0.373052	−	0.927811i	$-0.378311\pi$
$521$	17.0301			0.746103			0.373052	+	0.927811i	$0.378311\pi$
$522$		0			0
$523$	−9.57651			−0.418751			−0.209376	−	0.977835i	$-0.567143\pi$
$523$	−9.57651			−0.418751			−0.209376	+	0.977835i	$0.567143\pi$
$524$	18.0021	−	31.1805i	0.786424	−	1.36213i
$525$		0			0
$526$	−7.55880	−	13.0922i	−0.329579	−	0.570848i
$527$	6.30592	+	10.9222i	0.274690	+	0.475777i
$528$		0			0
$529$	9.99675	−	17.3149i	0.434641	−	0.752821i
$530$		0			0
$531$		0			0
$532$	26.3055			1.14049
$533$	2.43174	−	4.21189i	0.105330	−	0.182437i
$534$		0			0
$535$		0			0
$536$	35.2038	+	60.9748i	1.52057	+	2.63371i
$537$		0			0
$538$	−20.5644	+	35.6187i	−0.886596	+	1.53563i
$539$	6.81528			0.293555
$540$		0			0
$541$	−0.833751			−0.0358458			−0.0179229	−	0.999839i	$-0.505705\pi$
$541$	−0.833751			−0.0358458			−0.0179229	+	0.999839i	$0.505705\pi$
$542$	8.89297	−	15.4031i	0.381986	−	0.661619i
$543$		0			0
$544$	−29.2713	−	50.6994i	−1.25500	−	2.17372i
$545$		0			0
$546$		0			0
$547$	−14.1635	+	24.5319i	−0.605587	+	1.04891i	0.386371	+	0.922343i	$0.373728\pi$
$547$	−14.1635	+	24.5319i	−0.605587	+	1.04891i	−0.991958	+	0.126565i	$0.959605\pi$
$548$	−35.1785			−1.50275
$549$		0			0
$550$		0			0
$551$	−10.9213	+	18.9163i	−0.465264	+	0.805861i
$552$		0			0
$553$	−1.89709	−	3.28586i	−0.0806727	−	0.139729i
$554$	39.8964	+	69.1026i	1.69504	+	2.93589i
$555$		0			0
$556$	36.3096	−	62.8901i	1.53987	−	2.66713i
$557$	11.5042			0.487448			0.243724	−	0.969845i	$-0.421631\pi$
$557$	11.5042			0.487448			0.243724	+	0.969845i	$0.421631\pi$
$558$		0			0
$559$	−14.3512			−0.606993
$560$		0			0
$561$		0			0
$562$	24.5398	+	42.5043i	1.03515	+	1.79293i
$563$	−16.5030	−	28.5840i	−0.695517	−	1.20467i	−0.970006	−	0.243080i	$-0.921842\pi$
$563$	−16.5030	−	28.5840i	−0.695517	−	1.20467i	0.274490	−	0.961590i	$-0.411491\pi$
$564$		0			0
$565$		0			0
$566$	14.9429			0.628095
$567$		0			0
$568$	−9.85769			−0.413619
$569$	−13.5044	+	23.3903i	−0.566135	+	0.980574i	0.430809	+	0.902443i	$0.358228\pi$
$569$	−13.5044	+	23.3903i	−0.566135	+	0.980574i	−0.996943	+	0.0781305i	$0.975105\pi$
$570$		0			0
$571$	12.2122	+	21.1521i	0.511064	+	0.885189i	0.999918	+	0.0128232i	$0.00408185\pi$
$571$	12.2122	+	21.1521i	0.511064	+	0.885189i	−0.488854	+	0.872366i	$0.662585\pi$
$572$	−8.80007	−	15.2422i	−0.367950	−	0.637307i
$573$		0			0
$574$	−5.83807	+	10.1118i	−0.243676	+	0.422060i
$575$		0			0
$576$		0			0
$577$	−14.7976			−0.616033			−0.308017	−	0.951381i	$-0.599665\pi$
$577$	−14.7976			−0.616033			−0.308017	+	0.951381i	$0.599665\pi$
$578$	−8.05663	+	13.9545i	−0.335112	+	0.580431i
$579$		0			0
$580$		0			0
$581$	−0.987533	−	1.71046i	−0.0409698	−	0.0709617i
$582$		0			0
$583$	1.55987	−	2.70177i	0.0646030	−	0.111896i
$584$	27.7470			1.14818
$585$		0			0
$586$	48.1375			1.98854
$587$	−15.2890	+	26.4813i	−0.631044	+	1.09300i	0.356295	+	0.934374i	$0.384040\pi$
$587$	−15.2890	+	26.4813i	−0.631044	+	1.09300i	−0.987339	+	0.158626i	$0.949294\pi$
$588$		0			0
$589$	3.88821	+	6.73457i	0.160211	+	0.277493i
$590$		0			0
$591$		0			0
$592$	61.0611	−	105.761i	2.50960	−	4.34675i
$593$	−5.09990			−0.209428			−0.104714	−	0.994502i	$-0.533393\pi$
$593$	−5.09990			−0.209428			−0.104714	+	0.994502i	$0.533393\pi$
$594$		0			0
$595$		0			0
$596$	−1.39532	+	2.41677i	−0.0571547	+	0.0989949i
$597$		0			0
$598$	−4.50236	−	7.79831i	−0.184115	−	0.318897i
$599$	−0.282655	−	0.489572i	−0.0115490	−	0.0200034i	0.860193	−	0.509968i	$-0.170343\pi$
$599$	−0.282655	−	0.489572i	−0.0115490	−	0.0200034i	−0.871742	+	0.489965i	$0.837010\pi$
$600$		0			0
$601$	5.50480	−	9.53459i	0.224546	−	0.388924i	−0.731637	−	0.681694i	$-0.761244\pi$
$601$	5.50480	−	9.53459i	0.224546	−	0.388924i	0.956183	+	0.292770i	$0.0945769\pi$
$602$	34.4542			1.40425
$603$		0			0
$604$	0.757185			0.0308094
$605$		0			0
$606$		0			0
$607$	−9.54913	−	16.5396i	−0.387587	−	0.671321i	0.604537	−	0.796577i	$-0.293358\pi$
$607$	−9.54913	−	16.5396i	−0.387587	−	0.671321i	−0.992124	+	0.125256i	$0.960025\pi$
$608$	−18.0486	−	31.2611i	−0.731967	−	1.26780i
$609$		0			0
$610$		0			0
$611$	−12.4069			−0.501929
$612$		0			0
$613$	−9.33918			−0.377206			−0.188603	−	0.982053i	$-0.560396\pi$
$613$	−9.33918			−0.377206			−0.188603	+	0.982053i	$0.560396\pi$
$614$	−20.4179	+	35.3648i	−0.823999	+	1.42721i
$615$		0			0
$616$	12.5675	+	21.7676i	0.506360	+	0.877041i
$617$	12.2077	+	21.1444i	0.491464	+	0.851241i	0.999952	−	0.00982861i	$-0.00312859\pi$
$617$	12.2077	+	21.1444i	0.491464	+	0.851241i	−0.508488	+	0.861069i	$0.669795\pi$
$618$		0			0
$619$	−19.7431	+	34.1961i	−0.793544	+	1.37446i	0.130216	+	0.991486i	$0.458433\pi$
$619$	−19.7431	+	34.1961i	−0.793544	+	1.37446i	−0.923760	+	0.382973i	$0.874900\pi$
$620$		0			0
$621$		0			0
$622$	−80.1874			−3.21522
$623$	−11.9470	+	20.6928i	−0.478647	+	0.829040i
$624$		0			0
$625$		0			0
$626$	−9.15135	−	15.8506i	−0.365761	−	0.633517i
$627$		0			0
$628$	−28.2879	+	48.9960i	−1.12881	+	1.95515i
$629$	−55.9430			−2.23059
$630$		0			0
$631$	42.1634			1.67850			0.839249	−	0.543747i	$-0.182995\pi$
$631$	42.1634			1.67850			0.839249	+	0.543747i	$0.182995\pi$
$632$	−8.16277	+	14.1383i	−0.324698	+	0.562393i
$633$		0			0
$634$	21.2608	+	36.8248i	0.844375	+	1.46250i
$635$		0			0
$636$		0			0
$637$	−3.71616	+	6.43658i	−0.147240	+	0.255027i
$638$	−35.0855			−1.38905
$639$		0			0
$640$		0			0
$641$	17.6577	−	30.5841i	0.697438	−	1.20800i	−0.271913	−	0.962322i	$-0.587656\pi$
$641$	17.6577	−	30.5841i	0.697438	−	1.20800i	0.969352	−	0.245677i	$-0.0790103\pi$
$642$		0			0
$643$	7.09771	+	12.2936i	0.279906	+	0.484812i	0.971361	−	0.237608i	$-0.0763633\pi$
$643$	7.09771	+	12.2936i	0.279906	+	0.484812i	−0.691455	+	0.722420i	$0.743030\pi$
$644$	7.69256	+	13.3239i	0.303129	+	0.525036i
$645$		0			0
$646$	−18.7712	+	32.5127i	−0.738544	+	1.27919i
$647$	17.4897			0.687593			0.343796	−	0.939044i	$-0.388287\pi$
$647$	17.4897			0.687593			0.343796	+	0.939044i	$0.388287\pi$
$648$		0			0
$649$	19.9364			0.782572
$650$		0			0
$651$		0			0
$652$	−54.4483	−	94.3072i	−2.13236	−	3.69335i
$653$	5.48858	+	9.50650i	0.214785	+	0.372018i	0.953206	−	0.302322i	$-0.0977617\pi$
$653$	5.48858	+	9.50650i	0.214785	+	0.372018i	−0.738421	+	0.674340i	$0.764428\pi$
$654$		0			0
$655$		0			0
$656$	25.8882			1.01077
$657$		0			0
$658$	29.7862			1.16119
$659$	−7.89381	+	13.6725i	−0.307499	+	0.532604i	−0.977815	−	0.209472i	$-0.932825\pi$
$659$	−7.89381	+	13.6725i	−0.307499	+	0.532604i	0.670316	+	0.742076i	$0.266159\pi$
$660$		0			0
$661$	−24.9466	−	43.2088i	−0.970311	−	1.68063i	−0.694614	−	0.719383i	$-0.744425\pi$
$661$	−24.9466	−	43.2088i	−0.970311	−	1.68063i	−0.275697	−	0.961245i	$-0.588909\pi$
$662$	−16.6217	−	28.7897i	−0.646022	−	1.11894i
$663$		0			0
$664$	−4.24913	+	7.35972i	−0.164898	+	0.285612i
$665$		0			0
$666$		0			0
$667$	−12.7750			−0.494649
$668$	−42.1291	+	72.9698i	−1.63002	+	2.82328i
$669$		0			0
$670$		0			0
$671$	−11.4549	−	19.8404i	−0.442210	−	0.765929i
$672$		0			0
$673$	14.4197	−	24.9757i	0.555840	−	0.962743i	−0.441998	−	0.897016i	$-0.645730\pi$
$673$	14.4197	−	24.9757i	0.555840	−	0.962743i	0.997838	−	0.0657266i	$-0.0209365\pi$
$674$	18.2264			0.702055
$675$		0			0
$676$	−44.9809			−1.73003
$677$	−5.23181	+	9.06176i	−0.201075	+	0.348272i	−0.948875	−	0.315652i	$-0.897777\pi$
$677$	−5.23181	+	9.06176i	−0.201075	+	0.348272i	0.747800	+	0.663924i	$0.231110\pi$
$678$		0			0
$679$	−3.44455	−	5.96614i	−0.132190	−	0.228959i
$680$		0			0
$681$		0			0
$682$	−6.24556	+	10.8176i	−0.239155	+	0.414228i
$683$	−16.1875			−0.619396			−0.309698	−	0.950835i	$-0.600228\pi$
$683$	−16.1875			−0.619396			−0.309698	+	0.950835i	$0.600228\pi$
$684$		0			0
$685$		0			0
$686$	25.4904	−	44.1507i	0.973229	−	1.68568i
$687$		0			0
$688$	−38.1958	−	66.1570i	−1.45620	−	2.52221i
$689$	1.70109	+	2.94638i	0.0648065	+	0.112248i
$690$		0			0
$691$	−4.94181	+	8.55946i	−0.187995	+	0.325617i	−0.944582	−	0.328276i	$-0.893532\pi$
$691$	−4.94181	+	8.55946i	−0.187995	+	0.325617i	0.756586	+	0.653894i	$0.226866\pi$
$692$	58.9648			2.24150
$693$		0			0
$694$	36.3621			1.38029
$695$		0			0
$696$		0			0
$697$	−5.92957	−	10.2703i	−0.224598	−	0.389016i
$698$	−8.64872	−	14.9800i	−0.327359	−	0.567002i
$699$		0			0
$700$		0			0
$701$	−43.9692			−1.66069			−0.830346	−	0.557248i	$-0.811857\pi$
$701$	−43.9692			−1.66069			−0.830346	+	0.557248i	$0.811857\pi$
$702$		0			0
$703$	−34.4942			−1.30097
$704$	10.0130	−	17.3430i	0.377379	−	0.653640i
$705$		0			0
$706$	4.64193	+	8.04005i	0.174701	+	0.302591i
$707$	5.88870	+	10.1995i	0.221467	+	0.383593i
$708$		0			0
$709$	−12.6130	+	21.8464i	−0.473692	+	0.820458i	−0.999546	−	0.0301162i	$-0.990412\pi$
$709$	−12.6130	+	21.8464i	−0.473692	+	0.820458i	0.525855	+	0.850574i	$0.323746\pi$
$710$		0			0
$711$		0			0
$712$	102.811			3.85299
$713$	−2.27407	+	3.93880i	−0.0851645	+	0.147509i
$714$		0			0
$715$		0			0
$716$	21.1016	+	36.5490i	0.788603	+	1.36590i
$717$		0			0
$718$	30.1937	−	52.2971i	1.12682	−	1.95171i
$719$	36.8600			1.37465			0.687323	−	0.726352i	$-0.258786\pi$
$719$	36.8600			1.37465			0.687323	+	0.726352i	$0.258786\pi$
$720$		0			0
$721$	14.5153			0.540576
$722$	13.4461	−	23.2893i	0.500412	−	0.866740i
$723$		0			0
$724$	26.0459	+	45.1128i	0.967988	+	1.67660i
$725$		0			0
$726$		0			0
$727$	−19.1226	+	33.1213i	−0.709217	+	1.22840i	0.255931	+	0.966695i	$0.417618\pi$
$727$	−19.1226	+	33.1213i	−0.709217	+	1.22840i	−0.965148	+	0.261705i	$0.915715\pi$
$728$	−27.4107			−1.01591
$729$		0			0
$730$		0			0
$731$	−17.4971	+	30.3059i	−0.647154	+	1.12090i
$732$		0			0
$733$	19.7916	+	34.2801i	0.731020	+	1.26616i	0.956448	+	0.291903i	$0.0942886\pi$
$733$	19.7916	+	34.2801i	0.731020	+	1.26616i	−0.225428	+	0.974260i	$0.572378\pi$
$734$	5.50358	+	9.53248i	0.203141	+	0.351850i
$735$		0			0
$736$	10.5559	−	18.2834i	0.389097	−	0.673936i
$737$	16.4607			0.606338
$738$		0			0
$739$	−8.24773			−0.303398			−0.151699	−	0.988427i	$-0.548474\pi$
$739$	−8.24773			−0.303398			−0.151699	+	0.988427i	$0.548474\pi$
$740$		0			0
$741$		0			0
$742$	−4.08395	−	7.07361i	−0.149927	−	0.259680i
$743$	−0.654091	−	1.13292i	−0.0239963	−	0.0415627i	0.853778	−	0.520637i	$-0.174306\pi$
$743$	−0.654091	−	1.13292i	−0.0239963	−	0.0415627i	−0.877774	+	0.479075i	$0.840972\pi$
$744$		0			0
$745$		0			0
$746$	−18.0171			−0.659651
$747$		0			0
$748$	−42.9164			−1.56918
$749$	8.06752	−	13.9734i	0.294781	−	0.510575i
$750$		0			0
$751$	−14.2234	−	24.6357i	−0.519020	−	0.898969i	−0.999756	−	0.0221034i	$-0.992964\pi$
$751$	−14.2234	−	24.6357i	−0.519020	−	0.898969i	0.480736	−	0.876866i	$-0.340370\pi$
$752$	−33.0209	−	57.1939i	−1.20415	−	2.08565i
$753$		0			0
$754$	19.1310	−	33.1359i	0.696711	−	1.20674i
$755$		0			0
$756$		0			0
$757$	38.2012			1.38845			0.694223	−	0.719760i	$-0.255748\pi$
$757$	38.2012			1.38845			0.694223	+	0.719760i	$0.255748\pi$
$758$	16.8248	−	29.1414i	0.611103	−	1.05846i
$759$		0			0
$760$		0			0
$761$	11.0952	+	19.2174i	0.402200	+	0.696632i	0.993991	−	0.109460i	$-0.0349121\pi$
$761$	11.0952	+	19.2174i	0.402200	+	0.696632i	−0.591791	+	0.806092i	$0.701579\pi$
$762$		0			0
$763$	5.69932	−	9.87152i	0.206329	−	0.357373i
$764$	85.5852			3.09637
$765$		0			0
$766$	19.8440			0.716994
$767$	−10.8707	+	18.8286i	−0.392518	+	0.679861i
$768$		0			0
$769$	8.45652	+	14.6471i	0.304950	+	0.528189i	0.977250	−	0.212090i	$-0.0680270\pi$
$769$	8.45652	+	14.6471i	0.304950	+	0.528189i	−0.672300	+	0.740279i	$0.734694\pi$
$770$		0			0
$771$		0			0
$772$	3.84930	−	6.66718i	0.138539	−	0.239957i
$773$	−38.6464			−1.39001			−0.695007	−	0.719003i	$-0.744599\pi$
$773$	−38.6464			−1.39001			−0.695007	+	0.719003i	$0.744599\pi$
$774$		0			0
$775$		0			0
$776$	−14.8211	+	25.6709i	−0.532047	+	0.921533i
$777$		0			0
$778$	7.17920	+	12.4347i	0.257387	+	0.445807i
$779$	−3.65615	−	6.33264i	−0.130995	−	0.226890i
$780$		0			0
$781$	−1.15232	+	1.99588i	−0.0412333	+	0.0714181i
$782$	−21.9572			−0.785187
$783$		0			0
$784$	−39.5622			−1.41294
$785$		0			0
$786$		0			0
$787$	5.45734	+	9.45240i	0.194533	+	0.336942i	0.946747	−	0.321977i	$-0.104347\pi$
$787$	5.45734	+	9.45240i	0.194533	+	0.336942i	−0.752214	+	0.658919i	$0.771014\pi$
$788$	−44.3711	−	76.8530i	−1.58065	−	2.73777i
$789$		0			0
$790$		0

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)

\(f(q)\)	\(=\)	\(q+(-1.31686 + 2.28087i) q^{2} +(-2.46825 - 4.27513i) q^{4} +(-0.898714 + 1.55662i) q^{7} +7.73393 q^{8} +O(q^{10})\)	\(q+(-1.31686 + 2.28087i) q^{2} +(-2.46825 - 4.27513i) q^{4} +(-0.898714 + 1.55662i) q^{7} +7.73393 q^{8} +(0.904062 - 1.56588i) q^{11} +(0.985914 + 1.70765i) q^{13} +(-2.36696 - 4.09970i) q^{14} +(-5.24801 + 9.08982i) q^{16} +4.80812 q^{17} +2.96467 q^{19} +(2.38105 + 4.12410i) q^{22} +(0.866963 + 1.50162i) q^{23} -5.19325 q^{26} +8.87300 q^{28} +(-3.68382 + 6.38057i) q^{29} +(1.31151 + 2.27161i) q^{31} +(-6.08789 - 10.5445i) q^{32} +(-6.33163 + 10.9667i) q^{34} -11.6351 q^{37} +(-3.90406 + 6.76203i) q^{38} +(-1.23324 - 2.13603i) q^{41} +(-3.63907 + 6.30306i) q^{43} -8.92580 q^{44} -4.56668 q^{46} +(-3.14604 + 5.44910i) q^{47} +(1.88463 + 3.26427i) q^{49} +(4.86696 - 8.42983i) q^{52} +1.72540 q^{53} +(-6.95059 + 12.0388i) q^{56} +(-9.70218 - 16.8047i) q^{58} +(5.51300 + 9.54880i) q^{59} +(6.33521 - 10.9729i) q^{61} -6.90833 q^{62} +11.0756 q^{64} +(4.55187 + 7.88407i) q^{67} +(-11.8676 - 20.5554i) q^{68} -1.27460 q^{71} +3.58770 q^{73} +(15.3218 - 26.5382i) q^{74} +(-7.31755 - 12.6744i) q^{76} +(1.62499 + 2.81456i) q^{77} +(-1.05545 + 1.82809i) q^{79} +6.49602 q^{82} +(-0.549415 + 0.951614i) q^{83} +(-9.58431 - 16.6005i) q^{86} +(6.99195 - 12.1104i) q^{88} +13.2935 q^{89} -3.54422 q^{91} +(4.27976 - 7.41277i) q^{92} +(-8.28580 - 14.3514i) q^{94} +(-1.91638 + 3.31926i) q^{97} -9.92718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more