LMFDB - Embedded newform 1350.2.e.i.451.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(451,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1350.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1350.451
Dual form			1350.2.e.i.901.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +(-1.00000 + 1.73205i) q^{11} +(3.00000 + 5.19615i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{22} +(0.500000 + 0.866025i) q^{23} +6.00000 q^{26} -1.00000 q^{28} +(4.50000 - 7.79423i) q^{29} +(1.00000 + 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.00000 - 1.73205i) q^{34} +2.00000 q^{37} +(3.00000 - 5.19615i) q^{38} +(-5.50000 - 9.52628i) q^{41} +(2.00000 - 3.46410i) q^{43} +2.00000 q^{44} +1.00000 q^{46} +(-3.50000 + 6.06218i) q^{47} +(3.00000 + 5.19615i) q^{49} +(3.00000 - 5.19615i) q^{52} +(-0.500000 + 0.866025i) q^{56} +(-4.50000 - 7.79423i) q^{58} +(-2.00000 - 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(5.50000 + 9.52628i) q^{67} +(-1.00000 - 1.73205i) q^{68} +6.00000 q^{71} -4.00000 q^{73} +(1.00000 - 1.73205i) q^{74} +(-3.00000 - 5.19615i) q^{76} +(1.00000 + 1.73205i) q^{77} +(6.00000 - 10.3923i) q^{79} -11.0000 q^{82} +(-5.50000 + 9.52628i) q^{83} +(-2.00000 - 3.46410i) q^{86} +(1.00000 - 1.73205i) q^{88} -1.00000 q^{89} +6.00000 q^{91} +(0.500000 - 0.866025i) q^{92} +(3.50000 + 6.06218i) q^{94} +(4.00000 - 6.92820i) q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 + q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} + q^{7} - 2 q^{8} - 2 q^{11} + 6 q^{13} - q^{14} - q^{16} + 4 q^{17} + 12 q^{19} + 2 q^{22} + q^{23} + 12 q^{26} - 2 q^{28} + 9 q^{29} + 2 q^{31} + q^{32} + 2 q^{34} + 4 q^{37} + 6 q^{38} - 11 q^{41} + 4 q^{43} + 4 q^{44} + 2 q^{46} - 7 q^{47} + 6 q^{49} + 6 q^{52} - q^{56} - 9 q^{58} - 4 q^{59} + 7 q^{61} + 4 q^{62} + 2 q^{64} + 11 q^{67} - 2 q^{68} + 12 q^{71} - 8 q^{73} + 2 q^{74} - 6 q^{76} + 2 q^{77} + 12 q^{79} - 22 q^{82} - 11 q^{83} - 4 q^{86} + 2 q^{88} - 2 q^{89} + 12 q^{91} + q^{92} + 7 q^{94} + 8 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + q^7 - 2 * q^8 - 2 * q^11 + 6 * q^13 - q^14 - q^16 + 4 * q^17 + 12 * q^19 + 2 * q^22 + q^23 + 12 * q^26 - 2 * q^28 + 9 * q^29 + 2 * q^31 + q^32 + 2 * q^34 + 4 * q^37 + 6 * q^38 - 11 * q^41 + 4 * q^43 + 4 * q^44 + 2 * q^46 - 7 * q^47 + 6 * q^49 + 6 * q^52 - q^56 - 9 * q^58 - 4 * q^59 + 7 * q^61 + 4 * q^62 + 2 * q^64 + 11 * q^67 - 2 * q^68 + 12 * q^71 - 8 * q^73 + 2 * q^74 - 6 * q^76 + 2 * q^77 + 12 * q^79 - 22 * q^82 - 11 * q^83 - 4 * q^86 + 2 * q^88 - 2 * q^89 + 12 * q^91 + q^92 + 7 * q^94 + 8 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	0.896410	+	0.443227i	$0.146166\pi$
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	−0.0643593	+	0.997927i	$0.520500\pi$
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$	1.00000	+	1.73205i	0.213201	+	0.369274i
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	0.913434	−	0.406986i	$-0.133420\pi$
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	−0.809177	+	0.587565i	$0.800087\pi$
$24$		0			0
$25$		0			0
$26$	6.00000			1.17670
$27$		0			0
$28$	−1.00000			−0.188982
$29$	4.50000	−	7.79423i	0.835629	−	1.44735i	−0.0578882	−	0.998323i	$-0.518437\pi$
$29$	4.50000	−	7.79423i	0.835629	−	1.44735i	0.893517	−	0.449029i	$-0.148230\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	1.00000	−	1.73205i	0.171499	−	0.297044i
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	3.00000	−	5.19615i	0.486664	−	0.842927i
$39$		0			0
$40$		0			0
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	−0.872926	−	0.487852i	$-0.837780\pi$
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	0.0139704	−	0.999902i	$-0.495553\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	1.00000			0.147442
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	0.489398	+	0.872060i	$0.337217\pi$
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	−0.999926	+	0.0121990i	$0.996117\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$	3.00000	−	5.19615i	0.416025	−	0.720577i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	−4.50000	−	7.79423i	−0.590879	−	1.02343i
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$	2.00000			0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	−0.305424	+	0.952217i	$0.598798\pi$
$68$	−1.00000	−	1.73205i	−0.121268	−	0.210042i
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$	1.00000	−	1.73205i	0.116248	−	0.201347i
$75$		0			0
$76$	−3.00000	−	5.19615i	−0.344124	−	0.596040i
$77$	1.00000	+	1.73205i	0.113961	+	0.197386i
$78$		0			0
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	−0.301401	−	0.953498i	$-0.597454\pi$
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	0.976453	−	0.215728i	$-0.0692125\pi$
$80$		0			0
$81$		0			0
$82$	−11.0000			−1.21475
$83$	−5.50000	+	9.52628i	−0.603703	+	1.04565i	0.388552	+	0.921427i	$0.372976\pi$
$83$	−5.50000	+	9.52628i	−0.603703	+	1.04565i	−0.992255	+	0.124218i	$0.960358\pi$
$84$		0			0
$85$		0			0
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$	1.00000	−	1.73205i	0.106600	−	0.184637i
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	0.500000	−	0.866025i	0.0521286	−	0.0902894i
$93$		0			0
$94$	3.50000	+	6.06218i	0.360997	+	0.625266i
$95$		0			0
$96$		0			0
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	−0.588315	−	0.808632i	$-0.700208\pi$
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	0.994453	+	0.105180i	$0.0335417\pi$
$98$	6.00000			0.606092
$99$		0			0
$100$		0			0
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$	−3.00000	−	5.19615i	−0.294174	−	0.509525i
$105$		0			0
$106$		0			0
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	7.00000			0.670478			0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000			0.670478			0.335239	+	0.942133i	$0.391183\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$	6.00000	+	10.3923i	0.564433	+	0.977626i	0.997102	+	0.0760733i	$0.0242383\pi$
$113$	6.00000	+	10.3923i	0.564433	+	0.977626i	−0.432670	+	0.901553i	$0.642428\pi$
$114$		0			0
$115$		0			0
$116$	−9.00000			−0.835629
$117$		0			0
$118$	−4.00000			−0.368230
$119$	1.00000	−	1.73205i	0.0916698	−	0.158777i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	−3.50000	−	6.06218i	−0.316875	−	0.548844i
$123$		0			0
$124$	1.00000	−	1.73205i	0.0898027	−	0.155543i
$125$		0			0
$126$		0			0
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	3.00000	−	5.19615i	0.260133	−	0.450564i
$134$	11.0000			0.950255
$135$		0			0
$136$	−2.00000			−0.171499
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		0			0
$142$	3.00000	−	5.19615i	0.251754	−	0.436051i
$143$	−12.0000			−1.00349
$144$		0			0
$145$		0			0
$146$	−2.00000	+	3.46410i	−0.165521	+	0.286691i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	−0.500000	−	0.866025i	−0.0409616	−	0.0709476i	0.844818	−	0.535054i	$-0.179709\pi$
$149$	−0.500000	−	0.866025i	−0.0409616	−	0.0709476i	−0.885779	+	0.464107i	$0.846375\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$	−6.00000			−0.486664
$153$		0			0
$154$	2.00000			0.161165
$155$		0			0
$156$		0			0
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	0.775113	−	0.631822i	$-0.217693\pi$
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	−0.934731	+	0.355357i	$0.884359\pi$
$158$	−6.00000	−	10.3923i	−0.477334	−	0.826767i
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	−5.50000	+	9.52628i	−0.429478	+	0.743877i
$165$		0			0
$166$	5.50000	+	9.52628i	0.426883	+	0.739383i
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	0.802135	−	0.597143i	$-0.203697\pi$
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	−0.918208	+	0.396098i	$0.870364\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	−0.931984	−	0.362500i	$-0.881923\pi$
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	0.779926	+	0.625871i	$0.215256\pi$
$174$		0			0
$175$		0			0
$176$	−1.00000	−	1.73205i	−0.0753778	−	0.130558i
$177$		0			0
$178$	−0.500000	+	0.866025i	−0.0374766	+	0.0649113i
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$	3.00000	−	5.19615i	0.222375	−	0.385164i
$183$		0			0
$184$	−0.500000	−	0.866025i	−0.0368605	−	0.0638442i
$185$		0			0
$186$		0			0
$187$	−2.00000	+	3.46410i	−0.146254	+	0.253320i
$188$	7.00000			0.510527
$189$		0			0
$190$		0			0
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$		0			0
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	0.987945	−	0.154805i	$-0.0494748\pi$
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	−0.628037	+	0.778183i	$0.716141\pi$
$194$	−4.00000	−	6.92820i	−0.287183	−	0.497416i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		0			0
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$		0			0
$201$		0			0
$202$	−1.00000	−	1.73205i	−0.0703598	−	0.121867i
$203$	−4.50000	−	7.79423i	−0.315838	−	0.547048i
$204$		0			0
$205$		0			0
$206$	8.00000			0.557386
$207$		0			0
$208$	−6.00000			−0.416025
$209$	−6.00000	+	10.3923i	−0.415029	+	0.718851i
$210$		0			0
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	−0.989561	−	0.144112i	$-0.953967\pi$
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	0.369976	−	0.929041i	$-0.379366\pi$
$212$		0			0
$213$		0			0
$214$	−1.50000	+	2.59808i	−0.102538	+	0.177601i
$215$		0			0
$216$		0			0
$217$	2.00000			0.135769
$218$	3.50000	−	6.06218i	0.237050	−	0.410582i
$219$		0			0
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	−11.5000	+	19.9186i	−0.770097	+	1.33385i	0.167412	+	0.985887i	$0.446459\pi$
$223$	−11.5000	+	19.9186i	−0.770097	+	1.33385i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	12.0000			0.798228
$227$	−4.00000	+	6.92820i	−0.265489	+	0.459841i	−0.967692	−	0.252136i	$-0.918867\pi$
$227$	−4.00000	+	6.92820i	−0.265489	+	0.459841i	0.702202	+	0.711977i	$0.252200\pi$
$228$		0			0
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$	−4.50000	+	7.79423i	−0.295439	+	0.511716i
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		0			0
$235$		0			0
$236$	−2.00000	+	3.46410i	−0.130189	+	0.225494i
$237$		0			0
$238$	−1.00000	−	1.73205i	−0.0648204	−	0.112272i
$239$	14.0000	+	24.2487i	0.905585	+	1.56852i	0.820130	+	0.572177i	$0.193901\pi$
$239$	14.0000	+	24.2487i	0.905585	+	1.56852i	0.0854543	+	0.996342i	$0.472766\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$	7.00000			0.449977
$243$		0			0
$244$	−7.00000			−0.448129
$245$		0			0
$246$		0			0
$247$	18.0000	+	31.1769i	1.14531	+	1.98374i
$248$	−1.00000	−	1.73205i	−0.0635001	−	0.109985i
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$	−9.50000	+	16.4545i	−0.596083	+	1.03245i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	0.0621370	−	0.107624i
$260$		0			0
$261$		0			0
$262$	12.0000			0.741362
$263$	8.00000	−	13.8564i	0.493301	−	0.854423i	−0.506669	−	0.862141i	$-0.669123\pi$
$263$	8.00000	−	13.8564i	0.493301	−	0.854423i	0.999970	+	0.00771799i	$0.00245674\pi$
$264$		0			0
$265$		0			0
$266$	−3.00000	−	5.19615i	−0.183942	−	0.318597i
$267$		0			0
$268$	5.50000	−	9.52628i	0.335966	−	0.581910i
$269$	−3.00000			−0.182913			−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000			−0.182913			−0.0914566	+	0.995809i	$0.529152\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$	−1.00000	+	1.73205i	−0.0606339	+	0.105021i
$273$		0			0
$274$	−6.00000	−	10.3923i	−0.362473	−	0.627822i
$275$		0			0
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$	−16.0000			−0.959616
$279$		0			0
$280$		0			0
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	−0.907293	−	0.420500i	$-0.861855\pi$
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	0.817810	+	0.575488i	$0.195188\pi$
$282$		0			0
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	0.850782	−	0.525519i	$-0.176129\pi$
$283$	−0.500000	−	0.866025i	−0.0297219	−	0.0514799i	−0.880504	+	0.474039i	$0.842796\pi$
$284$	−3.00000	−	5.19615i	−0.178017	−	0.308335i
$285$		0			0
$286$	−6.00000	+	10.3923i	−0.354787	+	0.614510i
$287$	−11.0000			−0.649309
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$	2.00000	+	3.46410i	0.117041	+	0.202721i
$293$	−9.00000	−	15.5885i	−0.525786	−	0.910687i	−0.999549	−	0.0300351i	$-0.990438\pi$
$293$	−9.00000	−	15.5885i	−0.525786	−	0.910687i	0.473763	−	0.880652i	$-0.342895\pi$
$294$		0			0
$295$		0			0
$296$	−2.00000			−0.116248
$297$		0			0
$298$	−1.00000			−0.0579284
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$	5.00000	+	8.66025i	0.287718	+	0.498342i
$303$		0			0
$304$	−3.00000	+	5.19615i	−0.172062	+	0.298020i
$305$		0			0
$306$		0			0
$307$	−9.00000			−0.513657			−0.256829	−	0.966457i	$-0.582678\pi$
$307$	−9.00000			−0.513657			−0.256829	+	0.966457i	$0.582678\pi$
$308$	1.00000	−	1.73205i	0.0569803	−	0.0986928i
$309$		0			0
$310$		0			0
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	0.938460	−	0.345389i	$-0.112253\pi$
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	−0.768345	+	0.640036i	$0.778920\pi$
$312$		0			0
$313$	11.0000	−	19.0526i	0.621757	−	1.07691i	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	11.0000	−	19.0526i	0.621757	−	1.07691i	0.989158	−	0.146852i	$-0.0469141\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	−12.0000			−0.675053
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	−0.892741	−	0.450570i	$-0.851221\pi$
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	0.836576	+	0.547852i	$0.184554\pi$
$318$		0			0
$319$	9.00000	+	15.5885i	0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$	0.500000	−	0.866025i	0.0278639	−	0.0482617i
$323$	12.0000			0.667698
$324$		0			0
$325$		0			0
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$		0			0
$328$	5.50000	+	9.52628i	0.303687	+	0.526001i
$329$	3.50000	+	6.06218i	0.192961	+	0.334219i
$330$		0			0
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	−0.734905	−	0.678170i	$-0.762773\pi$
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	0.954765	+	0.297361i	$0.0961066\pi$
$332$	11.0000			0.603703
$333$		0			0
$334$	−3.00000			−0.164153
$335$		0			0
$336$		0			0
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	0.736270	−	0.676688i	$-0.236585\pi$
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	−0.954164	+	0.299285i	$0.903252\pi$
$338$	11.5000	+	19.9186i	0.625518	+	1.08343i
$339$		0			0
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$	13.0000			0.701934
$344$	−2.00000	+	3.46410i	−0.107833	+	0.186772i
$345$		0			0
$346$	2.00000	+	3.46410i	0.107521	+	0.186231i
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	5.50000	−	9.52628i	0.294408	−	0.509930i	−0.680439	−	0.732805i	$-0.738211\pi$
$349$	5.50000	−	9.52628i	0.294408	−	0.509930i	0.974847	+	0.222875i	$0.0715441\pi$
$350$		0			0
$351$		0			0
$352$	−2.00000			−0.106600
$353$	−8.00000	+	13.8564i	−0.425797	+	0.737502i	−0.996495	−	0.0836583i	$-0.973340\pi$
$353$	−8.00000	+	13.8564i	−0.425797	+	0.737502i	0.570697	+	0.821160i	$0.306673\pi$
$354$		0			0
$355$		0			0
$356$	0.500000	+	0.866025i	0.0264999	+	0.0458993i
$357$		0			0
$358$	−1.00000	+	1.73205i	−0.0528516	+	0.0915417i
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	−6.50000	+	11.2583i	−0.341632	+	0.591725i
$363$		0			0
$364$	−3.00000	−	5.19615i	−0.157243	−	0.272352i
$365$		0			0
$366$		0			0
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	−0.578101	−	0.815966i	$-0.696206\pi$
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	0.995697	+	0.0926670i	$0.0295392\pi$
$368$	−1.00000			−0.0521286
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.00000	−	10.3923i	−0.310668	−	0.538093i	0.667839	−	0.744306i	$-0.267219\pi$
$373$	−6.00000	−	10.3923i	−0.310668	−	0.538093i	−0.978507	+	0.206213i	$0.933886\pi$
$374$	2.00000	+	3.46410i	0.103418	+	0.179124i
$375$		0			0
$376$	3.50000	−	6.06218i	0.180499	−	0.312633i
$377$	54.0000			2.78114
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$	−3.00000	−	5.19615i	−0.153493	−	0.265858i
$383$	16.0000	+	27.7128i	0.817562	+	1.41606i	0.907474	+	0.420109i	$0.138008\pi$
$383$	16.0000	+	27.7128i	0.817562	+	1.41606i	−0.0899119	+	0.995950i	$0.528659\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$	−8.00000			−0.406138
$389$	9.50000	−	16.4545i	0.481669	−	0.834275i	−0.518110	−	0.855314i	$-0.673364\pi$
$389$	9.50000	−	16.4545i	0.481669	−	0.834275i	0.999779	+	0.0210389i	$0.00669738\pi$
$390$		0			0
$391$	1.00000	+	1.73205i	0.0505722	+	0.0875936i
$392$	−3.00000	−	5.19615i	−0.151523	−	0.262445i
$393$		0			0
$394$	−4.00000	+	6.92820i	−0.201517	+	0.349038i
$395$		0			0
$396$		0			0
$397$	−4.00000			−0.200754			−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000			−0.200754			−0.100377	+	0.994949i	$0.532005\pi$
$398$	−9.00000	+	15.5885i	−0.451129	+	0.781379i
$399$		0			0
$400$		0			0
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	0.713751	−	0.700399i	$-0.246995\pi$
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	−0.963439	+	0.267927i	$0.913661\pi$
$402$		0			0
$403$	−6.00000	+	10.3923i	−0.298881	+	0.517678i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	−9.00000			−0.446663
$407$	−2.00000	+	3.46410i	−0.0991363	+	0.171709i
$408$		0			0
$409$	−19.0000	−	32.9090i	−0.939490	−	1.62724i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−19.0000	−	32.9090i	−0.939490	−	1.62724i	−0.173064	−	0.984911i	$-0.555367\pi$
$410$		0			0
$411$		0			0
$412$	4.00000	−	6.92820i	0.197066	−	0.341328i
$413$	−4.00000			−0.196827
$414$		0			0
$415$		0			0
$416$	−3.00000	+	5.19615i	−0.147087	+	0.254762i
$417$		0			0
$418$	6.00000	+	10.3923i	0.293470	+	0.508304i
$419$	17.0000	+	29.4449i	0.830504	+	1.43848i	0.897639	+	0.440732i	$0.145281\pi$
$419$	17.0000	+	29.4449i	0.830504	+	1.43848i	−0.0671345	+	0.997744i	$0.521386\pi$
$420$		0			0
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	0.463002	+	0.886357i	$0.346772\pi$
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	−0.999109	+	0.0422075i	$0.986561\pi$
$422$	−18.0000			−0.876226
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.50000	−	6.06218i	−0.169377	−	0.293369i
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$	1.00000	−	1.73205i	0.0480015	−	0.0831411i
$435$		0			0
$436$	−3.50000	−	6.06218i	−0.167620	−	0.290326i
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	0.423556	+	0.905870i	$0.360782\pi$
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	−0.996284	+	0.0861252i	$0.972552\pi$
$440$		0			0
$441$		0			0
$442$	12.0000			0.570782
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	−0.952901	−	0.303281i	$-0.901918\pi$
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	0.739100	+	0.673596i	$0.235251\pi$
$444$		0			0
$445$		0			0
$446$	11.5000	+	19.9186i	0.544541	+	0.943172i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	22.0000			1.03594
$452$	6.00000	−	10.3923i	0.282216	−	0.488813i
$453$		0			0
$454$	4.00000	+	6.92820i	0.187729	+	0.325157i
$455$		0			0
$456$		0			0
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	0.958950	+	0.283577i	$0.0915211\pi$
$458$	7.00000			0.327089
$459$		0			0
$460$		0			0
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	−0.999920	−	0.0126168i	$-0.995984\pi$
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	0.510887	+	0.859648i	$0.329317\pi$
$462$		0			0
$463$	−18.0000	−	31.1769i	−0.836531	−	1.44891i	−0.892778	−	0.450497i	$-0.851247\pi$
$463$	−18.0000	−	31.1769i	−0.836531	−	1.44891i	0.0562469	−	0.998417i	$-0.482087\pi$
$464$	4.50000	+	7.79423i	0.208907	+	0.361838i
$465$		0			0
$466$	5.00000	−	8.66025i	0.231621	−	0.401179i
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	11.0000			0.507933
$470$		0			0
$471$		0			0
$472$	2.00000	+	3.46410i	0.0920575	+	0.159448i
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$		0			0
$476$	−2.00000			−0.0916698
$477$		0			0
$478$	28.0000			1.28069
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	−0.345827	−	0.938298i	$-0.612402\pi$
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	0.985504	−	0.169654i	$-0.0542649\pi$
$480$		0			0
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$	0.500000	+	0.866025i	0.0227744	+	0.0394464i
$483$		0			0
$484$	3.50000	−	6.06218i	0.159091	−	0.275554i
$485$		0			0
$486$		0			0
$487$	−12.0000			−0.543772			−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000			−0.543772			−0.271886	+	0.962329i	$0.587647\pi$
$488$	−3.50000	+	6.06218i	−0.158438	+	0.274422i
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$	36.0000			1.61972
$495$		0			0
$496$	−2.00000			−0.0898027
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	12.0000	+	20.7846i	0.537194	+	0.930447i	0.999054	+	0.0434940i	$0.0138489\pi$
$499$	12.0000	+	20.7846i	0.537194	+	0.930447i	−0.461860	+	0.886953i	$0.652818\pi$
$500$		0			0
$501$		0			0
$502$	−9.00000	+	15.5885i	−0.401690	+	0.695747i
$503$	−27.0000			−1.20387			−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000			−1.20387			−0.601935	+	0.798545i	$0.705603\pi$
$504$		0			0
$505$		0			0
$506$	−1.00000	+	1.73205i	−0.0444554	+	0.0769991i
$507$		0			0
$508$	9.50000	+	16.4545i	0.421494	+	0.730050i
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	0.650556	−	0.759458i	$-0.274536\pi$
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	−0.982988	+	0.183669i	$0.941202\pi$
$510$		0			0
$511$	−2.00000	+	3.46410i	−0.0884748	+	0.153243i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	12.0000			0.529297
$515$		0			0
$516$		0			0
$517$	−7.00000	−	12.1244i	−0.307860	−	0.533229i
$518$	−1.00000	−	1.73205i	−0.0439375	−	0.0761019i
$519$		0			0
$520$		0			0
$521$	37.0000			1.62100			0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000			1.62100			0.810500	+	0.585739i	$0.199196\pi$
$522$		0			0
$523$	−29.0000			−1.26808			−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000			−1.26808			−0.634041	+	0.773300i	$0.718605\pi$
$524$	6.00000	−	10.3923i	0.262111	−	0.453990i
$525$		0			0
$526$	−8.00000	−	13.8564i	−0.348817	−	0.604168i
$527$	2.00000	+	3.46410i	0.0871214	+	0.150899i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$		0			0
$531$		0			0
$532$	−6.00000			−0.260133
$533$	33.0000	−	57.1577i	1.42939	−	2.47577i
$534$		0			0
$535$		0			0
$536$	−5.50000	−	9.52628i	−0.237564	−	0.411473i
$537$		0			0
$538$	−1.50000	+	2.59808i	−0.0646696	+	0.112011i
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−17.0000			−0.730887			−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000			−0.730887			−0.365444	+	0.930834i	$0.619083\pi$
$542$	−7.00000	+	12.1244i	−0.300676	+	0.520786i
$543$		0			0
$544$	1.00000	+	1.73205i	0.0428746	+	0.0742611i
$545$		0			0
$546$		0			0
$547$	−17.5000	+	30.3109i	−0.748246	+	1.29600i	0.200417	+	0.979711i	$0.435770\pi$
$547$	−17.5000	+	30.3109i	−0.748246	+	1.29600i	−0.948663	+	0.316289i	$0.897563\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$		0			0
$551$	27.0000	−	46.7654i	1.15024	−	1.99227i
$552$		0			0
$553$	−6.00000	−	10.3923i	−0.255146	−	0.441926i
$554$	−11.0000	−	19.0526i	−0.467345	−	0.809466i
$555$		0			0
$556$	−8.00000	+	13.8564i	−0.339276	+	0.587643i
$557$	24.0000			1.01691			0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000			1.01691			0.508456	+	0.861088i	$0.330216\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$	1.50000	+	2.59808i	0.0632737	+	0.109593i
$563$	−18.5000	−	32.0429i	−0.779682	−	1.35045i	−0.932125	−	0.362137i	$-0.882047\pi$
$563$	−18.5000	−	32.0429i	−0.779682	−	1.35045i	0.152443	−	0.988312i	$-0.451286\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000			−0.0420331
$567$		0			0
$568$	−6.00000			−0.251754
$569$	−1.00000	+	1.73205i	−0.0419222	+	0.0726113i	−0.886225	−	0.463255i	$-0.846681\pi$
$569$	−1.00000	+	1.73205i	−0.0419222	+	0.0726113i	0.844303	+	0.535866i	$0.180015\pi$
$570$		0			0
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	0.995788	−	0.0916910i	$-0.0292272\pi$
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	−0.577301	+	0.816532i	$0.695894\pi$
$572$	6.00000	+	10.3923i	0.250873	+	0.434524i
$573$		0			0
$574$	−5.50000	+	9.52628i	−0.229566	+	0.397619i
$575$		0			0
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$	−6.50000	+	11.2583i	−0.270364	+	0.468285i
$579$		0			0
$580$		0			0
$581$	5.50000	+	9.52628i	0.228178	+	0.395217i
$582$		0			0
$583$		0			0
$584$	4.00000			0.165521
$585$		0			0
$586$	−18.0000			−0.743573
$587$	1.50000	−	2.59808i	0.0619116	−	0.107234i	−0.833408	−	0.552658i	$-0.813614\pi$
$587$	1.50000	−	2.59808i	0.0619116	−	0.107234i	0.895320	+	0.445424i	$0.146947\pi$
$588$		0			0
$589$	6.00000	+	10.3923i	0.247226	+	0.428207i
$590$		0			0
$591$		0			0
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		0			0
$596$	−0.500000	+	0.866025i	−0.0204808	+	0.0354738i
$597$		0			0
$598$	3.00000	+	5.19615i	0.122679	+	0.212486i
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	−0.549602	−	0.835426i	$-0.685221\pi$
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	0.998302	+	0.0582563i	$0.0185541\pi$
$602$	−4.00000			−0.163028
$603$		0			0
$604$	10.0000			0.406894
$605$		0			0
$606$		0			0
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	0.855700	−	0.517472i	$-0.173127\pi$
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	−0.875994	+	0.482322i	$0.839794\pi$
$608$	3.00000	+	5.19615i	0.121666	+	0.210732i
$609$		0			0
$610$		0			0
$611$	−42.0000			−1.69914
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	−4.50000	+	7.79423i	−0.181605	+	0.314549i
$615$		0			0
$616$	−1.00000	−	1.73205i	−0.0402911	−	0.0697863i
$617$	−16.0000	−	27.7128i	−0.644136	−	1.11568i	−0.984500	−	0.175382i	$-0.943884\pi$
$617$	−16.0000	−	27.7128i	−0.644136	−	1.11568i	0.340365	−	0.940294i	$-0.389449\pi$
$618$		0			0
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	−0.747873	−	0.663842i	$-0.768925\pi$
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	0.948840	+	0.315757i	$0.102258\pi$
$620$		0			0
$621$		0			0
$622$	6.00000			0.240578
$623$	−0.500000	+	0.866025i	−0.0200321	+	0.0346966i
$624$		0			0
$625$		0			0
$626$	−11.0000	−	19.0526i	−0.439648	−	0.761493i
$627$		0			0
$628$	−2.00000	+	3.46410i	−0.0798087	+	0.138233i
$629$	4.00000			0.159490
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	−6.00000	+	10.3923i	−0.238667	+	0.413384i
$633$		0			0
$634$	1.00000	+	1.73205i	0.0397151	+	0.0687885i
$635$		0			0
$636$		0			0
$637$	−18.0000	+	31.1769i	−0.713186	+	1.23527i
$638$	18.0000			0.712627
$639$		0			0
$640$		0			0
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	−0.708631	−	0.705580i	$-0.750687\pi$
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	0.965365	+	0.260902i	$0.0840201\pi$
$642$		0			0
$643$	16.5000	+	28.5788i	0.650696	+	1.12704i	0.982954	+	0.183851i	$0.0588563\pi$
$643$	16.5000	+	28.5788i	0.650696	+	1.12704i	−0.332258	+	0.943189i	$0.607810\pi$
$644$	−0.500000	−	0.866025i	−0.0197028	−	0.0341262i
$645$		0			0
$646$	6.00000	−	10.3923i	0.236067	−	0.408880i
$647$	−33.0000			−1.29736			−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000			−1.29736			−0.648682	+	0.761060i	$0.724679\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$	−2.00000	−	3.46410i	−0.0783260	−	0.135665i
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	−0.999949	−	0.0101092i	$-0.996782\pi$
$653$	−13.0000	−	22.5167i	−0.508729	−	0.881145i	0.491220	−	0.871036i	$-0.336551\pi$
$654$		0			0
$655$		0			0
$656$	11.0000			0.429478
$657$		0			0
$658$	7.00000			0.272888
$659$	10.0000	−	17.3205i	0.389545	−	0.674711i	−0.602844	−	0.797859i	$-0.705966\pi$
$659$	10.0000	−	17.3205i	0.389545	−	0.674711i	0.992388	+	0.123148i	$0.0392990\pi$
$660$		0			0
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	0.946729	−	0.322031i	$-0.104366\pi$
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	−0.752252	+	0.658876i	$0.771032\pi$
$662$	−4.00000	−	6.92820i	−0.155464	−	0.269272i
$663$		0			0
$664$	5.50000	−	9.52628i	0.213441	−	0.369691i
$665$		0			0
$666$		0			0
$667$	9.00000			0.348481
$668$	−1.50000	+	2.59808i	−0.0580367	+	0.100523i
$669$		0			0
$670$		0			0
$671$	7.00000	+	12.1244i	0.270232	+	0.468056i
$672$		0			0
$673$	−3.00000	+	5.19615i	−0.115642	+	0.200297i	−0.918036	−	0.396497i	$-0.870226\pi$
$673$	−3.00000	+	5.19615i	−0.115642	+	0.200297i	0.802395	+	0.596794i	$0.203559\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	23.0000			0.884615
$677$	11.0000	−	19.0526i	0.422764	−	0.732249i	−0.573444	−	0.819244i	$-0.694393\pi$
$677$	11.0000	−	19.0526i	0.422764	−	0.732249i	0.996209	+	0.0869952i	$0.0277265\pi$
$678$		0			0
$679$	−4.00000	−	6.92820i	−0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$	−2.00000	+	3.46410i	−0.0765840	+	0.132647i
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		0			0
$685$		0			0
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	2.00000	+	3.46410i	0.0762493	+	0.132068i
$689$		0			0
$690$		0			0
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	−0.0555386	−	0.998457i	$-0.517688\pi$
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	0.892458	−	0.451130i	$-0.148979\pi$
$692$	4.00000			0.152057
$693$		0			0
$694$	−12.0000			−0.455514
$695$		0			0
$696$		0			0
$697$	−11.0000	−	19.0526i	−0.416655	−	0.721667i
$698$	−5.50000	−	9.52628i	−0.208178	−	0.360575i
$699$		0			0
$700$		0			0
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$	12.0000			0.452589
$704$	−1.00000	+	1.73205i	−0.0376889	+	0.0652791i
$705$		0			0
$706$	8.00000	+	13.8564i	0.301084	+	0.521493i
$707$	−1.00000	−	1.73205i	−0.0376089	−	0.0651405i
$708$		0			0
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	−0.492964	−	0.870050i	$-0.664087\pi$
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	0.999967	−	0.00810550i	$-0.00258009\pi$
$710$		0			0
$711$		0			0
$712$	1.00000			0.0374766
$713$	−1.00000	+	1.73205i	−0.0374503	+	0.0648658i
$714$		0			0
$715$		0			0
$716$	1.00000	+	1.73205i	0.0373718	+	0.0647298i
$717$		0			0
$718$	−15.0000	+	25.9808i	−0.559795	+	0.969593i
$719$	44.0000			1.64092			0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000			1.64092			0.820462	+	0.571702i	$0.193717\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$	8.50000	−	14.7224i	0.316337	−	0.547912i
$723$		0			0
$724$	6.50000	+	11.2583i	0.241571	+	0.418413i
$725$		0			0
$726$		0			0
$727$	−10.5000	+	18.1865i	−0.389423	+	0.674501i	−0.992372	−	0.123279i	$-0.960659\pi$
$727$	−10.5000	+	18.1865i	−0.389423	+	0.674501i	0.602949	+	0.797780i	$0.293992\pi$
$728$	−6.00000			−0.222375
$729$		0			0
$730$		0			0
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	0.826723	−	0.562609i	$-0.190202\pi$
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	−0.900595	+	0.434659i	$0.856869\pi$
$734$	−8.00000	−	13.8564i	−0.295285	−	0.511449i
$735$		0			0
$736$	−0.500000	+	0.866025i	−0.0184302	+	0.0319221i
$737$	−22.0000			−0.810380
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.50000	−	12.9904i	−0.275148	−	0.476571i	0.695024	−	0.718986i	$-0.255394\pi$
$743$	−7.50000	−	12.9904i	−0.275148	−	0.476571i	−0.970173	+	0.242415i	$0.922060\pi$
$744$		0			0
$745$		0			0
$746$	−12.0000			−0.439351
$747$		0			0
$748$	4.00000			0.146254
$749$	−1.50000	+	2.59808i	−0.0548088	+	0.0949316i
$750$		0			0
$751$	−13.0000	−	22.5167i	−0.474377	−	0.821645i	0.525193	−	0.850983i	$-0.323993\pi$
$751$	−13.0000	−	22.5167i	−0.474377	−	0.821645i	−0.999570	+	0.0293387i	$0.990660\pi$
$752$	−3.50000	−	6.06218i	−0.127632	−	0.221065i
$753$		0			0
$754$	27.0000	−	46.7654i	0.983282	−	1.70309i
$755$		0			0
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$	−8.00000	+	13.8564i	−0.290573	+	0.503287i
$759$		0			0
$760$		0			0
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	0.772863	−	0.634573i	$-0.218824\pi$
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	−0.935988	+	0.352032i	$0.885491\pi$
$762$		0			0
$763$	3.50000	−	6.06218i	0.126709	−	0.219466i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	32.0000			1.15621
$767$	12.0000	−	20.7846i	0.433295	−	0.750489i
$768$		0			0
$769$	7.50000	+	12.9904i	0.270457	+	0.468445i	0.968979	−	0.247143i	$-0.0794919\pi$
$769$	7.50000	+	12.9904i	0.270457	+	0.468445i	−0.698522	+	0.715589i	$0.746159\pi$
$770$		0			0
$771$		0			0
$772$	5.00000	−	8.66025i	0.179954	−	0.311689i
$773$	−12.0000			−0.431610			−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000			−0.431610			−0.215805	+	0.976436i	$0.569238\pi$
$774$		0			0
$775$		0			0
$776$	−4.00000	+	6.92820i	−0.143592	+	0.248708i
$777$		0			0
$778$	−9.50000	−	16.4545i	−0.340592	−	0.589922i
$779$	−33.0000	−	57.1577i	−1.18235	−	2.04789i
$780$		0			0
$781$	−6.00000	+	10.3923i	−0.214697	+	0.371866i
$782$	2.00000			0.0715199
$783$		0			0
$784$	−6.00000			−0.214286
$785$		0			0
$786$		0			0
$787$	−22.0000	−	38.1051i	−0.784215	−	1.35830i	−0.929467	−	0.368906i	$-0.879732\pi$
$787$	−22.0000	−	38.1051i	−0.784215	−	1.35830i	0.145251	−	0.989395i	$-0.453601\pi$
$788$	4.00000	+	6.92820i	0.142494	+	0.246807i
$789$		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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +(-1.00000 + 1.73205i) q^{11} +(3.00000 + 5.19615i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{22} +(0.500000 + 0.866025i) q^{23} +6.00000 q^{26} -1.00000 q^{28} +(4.50000 - 7.79423i) q^{29} +(1.00000 + 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.00000 - 1.73205i) q^{34} +2.00000 q^{37} +(3.00000 - 5.19615i) q^{38} +(-5.50000 - 9.52628i) q^{41} +(2.00000 - 3.46410i) q^{43} +2.00000 q^{44} +1.00000 q^{46} +(-3.50000 + 6.06218i) q^{47} +(3.00000 + 5.19615i) q^{49} +(3.00000 - 5.19615i) q^{52} +(-0.500000 + 0.866025i) q^{56} +(-4.50000 - 7.79423i) q^{58} +(-2.00000 - 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} +2.00000 q^{62} +1.00000 q^{64} +(5.50000 + 9.52628i) q^{67} +(-1.00000 - 1.73205i) q^{68} +6.00000 q^{71} -4.00000 q^{73} +(1.00000 - 1.73205i) q^{74} +(-3.00000 - 5.19615i) q^{76} +(1.00000 + 1.73205i) q^{77} +(6.00000 - 10.3923i) q^{79} -11.0000 q^{82} +(-5.50000 + 9.52628i) q^{83} +(-2.00000 - 3.46410i) q^{86} +(1.00000 - 1.73205i) q^{88} -1.00000 q^{89} +6.00000 q^{91} +(0.500000 - 0.866025i) q^{92} +(3.50000 + 6.06218i) q^{94} +(4.00000 - 6.92820i) q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 + q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} + q^{7} - 2 q^{8} - 2 q^{11} + 6 q^{13} - q^{14} - q^{16} + 4 q^{17} + 12 q^{19} + 2 q^{22} + q^{23} + 12 q^{26} - 2 q^{28} + 9 q^{29} + 2 q^{31} + q^{32} + 2 q^{34} + 4 q^{37} + 6 q^{38} - 11 q^{41} + 4 q^{43} + 4 q^{44} + 2 q^{46} - 7 q^{47} + 6 q^{49} + 6 q^{52} - q^{56} - 9 q^{58} - 4 q^{59} + 7 q^{61} + 4 q^{62} + 2 q^{64} + 11 q^{67} - 2 q^{68} + 12 q^{71} - 8 q^{73} + 2 q^{74} - 6 q^{76} + 2 q^{77} + 12 q^{79} - 22 q^{82} - 11 q^{83} - 4 q^{86} + 2 q^{88} - 2 q^{89} + 12 q^{91} + q^{92} + 7 q^{94} + 8 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + q^7 - 2 * q^8 - 2 * q^11 + 6 * q^13 - q^14 - q^16 + 4 * q^17 + 12 * q^19 + 2 * q^22 + q^23 + 12 * q^26 - 2 * q^28 + 9 * q^29 + 2 * q^31 + q^32 + 2 * q^34 + 4 * q^37 + 6 * q^38 - 11 * q^41 + 4 * q^43 + 4 * q^44 + 2 * q^46 - 7 * q^47 + 6 * q^49 + 6 * q^52 - q^56 - 9 * q^58 - 4 * q^59 + 7 * q^61 + 4 * q^62 + 2 * q^64 + 11 * q^67 - 2 * q^68 + 12 * q^71 - 8 * q^73 + 2 * q^74 - 6 * q^76 + 2 * q^77 + 12 * q^79 - 22 * q^82 - 11 * q^83 - 4 * q^86 + 2 * q^88 - 2 * q^89 + 12 * q^91 + q^92 + 7 * q^94 + 8 * q^97 + 12 * q^98

Introduction

L-functions

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