LMFDB - Embedded newform 588.6.i.e.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.i (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:	$94.3056860500$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.6.i.e.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 7.79423i) q^{3} +(-17.0000 - 29.4449i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 - 7.79423i) q^{3} +(-17.0000 - 29.4449i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(166.000 - 287.520i) q^{11} +1026.00 q^{13} -306.000 q^{15} +(461.000 - 798.475i) q^{17} +(226.000 + 391.443i) q^{19} +(1888.00 + 3270.11i) q^{23} +(984.500 - 1705.20i) q^{25} -729.000 q^{27} +1166.00 q^{29} +(-4896.00 + 8480.12i) q^{31} +(-1494.00 - 2587.68i) q^{33} +(-1195.00 - 2069.80i) q^{37} +(4617.00 - 7996.88i) q^{39} +7230.00 q^{41} +4652.00 q^{43} +(-1377.00 + 2385.03i) q^{45} +(12336.0 + 21366.6i) q^{47} +(-4149.00 - 7186.28i) q^{51} +(-555.000 + 961.288i) q^{53} -11288.0 q^{55} +4068.00 q^{57} +(23446.0 - 40609.7i) q^{59} +(-4881.00 - 8454.14i) q^{61} +(-17442.0 - 30210.4i) q^{65} +(13126.0 - 22734.9i) q^{67} +33984.0 q^{69} +65440.0 q^{71} +(-2803.00 + 4854.94i) q^{73} +(-8860.50 - 15346.8i) q^{75} +(4920.00 + 8521.69i) q^{79} +(-3280.50 + 5681.99i) q^{81} -61108.0 q^{83} -31348.0 q^{85} +(5247.00 - 9088.07i) q^{87} +(-31479.0 - 54523.2i) q^{89} +(44064.0 + 76321.1i) q^{93} +(7684.00 - 13309.1i) q^{95} +37838.0 q^{97} -26892.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{3} - 34 q^{5} - 81 q^{9}+O(q^{10}) $ 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9	$ 2 q + 9 q^{3} - 34 q^{5} - 81 q^{9} + 332 q^{11} + 2052 q^{13} - 612 q^{15} + 922 q^{17} + 452 q^{19} + 3776 q^{23} + 1969 q^{25} - 1458 q^{27} + 2332 q^{29} - 9792 q^{31} - 2988 q^{33} - 2390 q^{37} + 9234 q^{39} + 14460 q^{41} + 9304 q^{43} - 2754 q^{45} + 24672 q^{47} - 8298 q^{51} - 1110 q^{53} - 22576 q^{55} + 8136 q^{57} + 46892 q^{59} - 9762 q^{61} - 34884 q^{65} + 26252 q^{67} + 67968 q^{69} + 130880 q^{71} - 5606 q^{73} - 17721 q^{75} + 9840 q^{79} - 6561 q^{81} - 122216 q^{83} - 62696 q^{85} + 10494 q^{87} - 62958 q^{89} + 88128 q^{93} + 15368 q^{95} + 75676 q^{97} - 53784 q^{99}+O(q^{100}) $ 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9 + 332 * q^11 + 2052 * q^13 - 612 * q^15 + 922 * q^17 + 452 * q^19 + 3776 * q^23 + 1969 * q^25 - 1458 * q^27 + 2332 * q^29 - 9792 * q^31 - 2988 * q^33 - 2390 * q^37 + 9234 * q^39 + 14460 * q^41 + 9304 * q^43 - 2754 * q^45 + 24672 * q^47 - 8298 * q^51 - 1110 * q^53 - 22576 * q^55 + 8136 * q^57 + 46892 * q^59 - 9762 * q^61 - 34884 * q^65 + 26252 * q^67 + 67968 * q^69 + 130880 * q^71 - 5606 * q^73 - 17721 * q^75 + 9840 * q^79 - 6561 * q^81 - 122216 * q^83 - 62696 * q^85 + 10494 * q^87 - 62958 * q^89 + 88128 * q^93 + 15368 * q^95 + 75676 * q^97 - 53784 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$2$		0			0
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$4$		0			0
$5$	−17.0000	−	29.4449i	−0.304105	−	0.526726i	0.672956	−	0.739682i	$-0.265024\pi$
$5$	−17.0000	−	29.4449i	−0.304105	−	0.526726i	−0.977062	+	0.212956i	$0.931691\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	166.000	−	287.520i	0.413644	−	0.716452i	−0.581641	−	0.813445i	$-0.697589\pi$
$11$	166.000	−	287.520i	0.413644	−	0.716452i	0.995285	+	0.0969935i	$0.0309226\pi$
$12$		0			0
$13$	1026.00			1.68379			0.841897	−	0.539638i	$-0.181439\pi$
$13$	1026.00			1.68379			0.841897	+	0.539638i	$0.181439\pi$
$14$		0			0
$15$	−306.000			−0.351150
$16$		0			0
$17$	461.000	−	798.475i	0.386882	−	0.670099i	−0.605146	−	0.796114i	$-0.706885\pi$
$17$	461.000	−	798.475i	0.386882	−	0.670099i	0.992028	+	0.126015i	$0.0402187\pi$
$18$		0			0
$19$	226.000	+	391.443i	0.143623	+	0.248763i	0.928858	−	0.370435i	$-0.120791\pi$
$19$	226.000	+	391.443i	0.143623	+	0.248763i	−0.785235	+	0.619198i	$0.787458\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1888.00	+	3270.11i	0.744188	+	1.28897i	0.950573	+	0.310500i	$0.100496\pi$
$23$	1888.00	+	3270.11i	0.744188	+	1.28897i	−0.206386	+	0.978471i	$0.566170\pi$
$24$		0			0
$25$	984.500	−	1705.20i	0.315040	−	0.545665i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	1166.00			0.257456			0.128728	−	0.991680i	$-0.458911\pi$
$29$	1166.00			0.257456			0.128728	+	0.991680i	$0.458911\pi$
$30$		0			0
$31$	−4896.00	+	8480.12i	−0.915034	+	1.58489i	−0.108182	+	0.994131i	$0.534503\pi$
$31$	−4896.00	+	8480.12i	−0.915034	+	1.58489i	−0.806852	+	0.590754i	$0.798830\pi$
$32$		0			0
$33$	−1494.00	−	2587.68i	−0.238817	−	0.413644i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1195.00	−	2069.80i	−0.143504	−	0.248556i	0.785310	−	0.619103i	$-0.212504\pi$
$37$	−1195.00	−	2069.80i	−0.143504	−	0.248556i	−0.928814	+	0.370547i	$0.879170\pi$
$38$		0			0
$39$	4617.00	−	7996.88i	0.486069	−	0.841897i
$40$		0			0
$41$	7230.00			0.671705			0.335853	−	0.941915i	$-0.390976\pi$
$41$	7230.00			0.671705			0.335853	+	0.941915i	$0.390976\pi$
$42$		0			0
$43$	4652.00			0.383679			0.191840	−	0.981426i	$-0.438555\pi$
$43$	4652.00			0.383679			0.191840	+	0.981426i	$0.438555\pi$
$44$		0			0
$45$	−1377.00	+	2385.03i	−0.101368	+	0.175575i
$46$		0			0
$47$	12336.0	+	21366.6i	0.814572	+	1.41088i	0.909635	+	0.415410i	$0.136362\pi$
$47$	12336.0	+	21366.6i	0.814572	+	1.41088i	−0.0950621	+	0.995471i	$0.530305\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−4149.00	−	7186.28i	−0.223366	−	0.386882i
$52$		0			0
$53$	−555.000	+	961.288i	−0.0271396	+	0.0470072i	−0.879276	−	0.476312i	$-0.841973\pi$
$53$	−555.000	+	961.288i	−0.0271396	+	0.0470072i	0.852137	+	0.523319i	$0.175307\pi$
$54$		0			0
$55$	−11288.0			−0.503165
$56$		0			0
$57$	4068.00			0.165842
$58$		0			0
$59$	23446.0	−	40609.7i	0.876877	−	1.51880i	0.0221275	−	0.999755i	$-0.492956\pi$
$59$	23446.0	−	40609.7i	0.876877	−	1.51880i	0.854750	−	0.519041i	$-0.173711\pi$
$60$		0			0
$61$	−4881.00	−	8454.14i	−0.167952	−	0.290901i	0.769748	−	0.638348i	$-0.220382\pi$
$61$	−4881.00	−	8454.14i	−0.167952	−	0.290901i	−0.937700	+	0.347447i	$0.887049\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17442.0	−	30210.4i	−0.512051	−	0.886898i
$66$		0			0
$67$	13126.0	−	22734.9i	0.357228	−	0.618737i	−0.630269	−	0.776377i	$-0.717055\pi$
$67$	13126.0	−	22734.9i	0.357228	−	0.618737i	0.987497	+	0.157640i	$0.0503886\pi$
$68$		0			0
$69$	33984.0			0.859314
$70$		0			0
$71$	65440.0			1.54063			0.770313	−	0.637666i	$-0.220100\pi$
$71$	65440.0			1.54063			0.770313	+	0.637666i	$0.220100\pi$
$72$		0			0
$73$	−2803.00	+	4854.94i	−0.0615625	+	0.106629i	−0.895164	−	0.445737i	$-0.852942\pi$
$73$	−2803.00	+	4854.94i	−0.0615625	+	0.106629i	0.833602	+	0.552366i	$0.186275\pi$
$74$		0			0
$75$	−8860.50	−	15346.8i	−0.181888	−	0.315040i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4920.00	+	8521.69i	0.0886946	+	0.153624i	0.906960	−	0.421218i	$-0.138397\pi$
$79$	4920.00	+	8521.69i	0.0886946	+	0.153624i	−0.818265	+	0.574841i	$0.805064\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−61108.0			−0.973650			−0.486825	−	0.873500i	$-0.661845\pi$
$83$	−61108.0			−0.973650			−0.486825	+	0.873500i	$0.661845\pi$
$84$		0			0
$85$	−31348.0			−0.470611
$86$		0			0
$87$	5247.00	−	9088.07i	0.0743212	−	0.128728i
$88$		0			0
$89$	−31479.0	−	54523.2i	−0.421256	−	0.729636i	0.574807	−	0.818289i	$-0.305077\pi$
$89$	−31479.0	−	54523.2i	−0.421256	−	0.729636i	−0.996063	+	0.0886527i	$0.971744\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	44064.0	+	76321.1i	0.528295	+	0.915034i
$94$		0			0
$95$	7684.00	−	13309.1i	0.0873531	−	0.151300i
$96$		0			0
$97$	37838.0			0.408318			0.204159	−	0.978938i	$-0.434554\pi$
$97$	37838.0			0.408318			0.204159	+	0.978938i	$0.434554\pi$
$98$		0			0
$99$	−26892.0			−0.275762
$100$		0			0
$101$	−28073.0	+	48623.9i	−0.273833	+	0.474292i	−0.969840	−	0.243743i	$-0.921625\pi$
$101$	−28073.0	+	48623.9i	−0.273833	+	0.474292i	0.696007	+	0.718035i	$0.254958\pi$
$102$		0			0
$103$	−13196.0	−	22856.1i	−0.122560	−	0.212280i	0.798216	−	0.602371i	$-0.205777\pi$
$103$	−13196.0	−	22856.1i	−0.122560	−	0.212280i	−0.920777	+	0.390090i	$0.872444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−23562.0	−	40810.6i	−0.198954	−	0.344598i	0.749236	−	0.662304i	$-0.230421\pi$
$107$	−23562.0	−	40810.6i	−0.198954	−	0.344598i	−0.948190	+	0.317705i	$0.897088\pi$
$108$		0			0
$109$	110737.	−	191802.i	0.892743	−	1.54628i	0.0561699	−	0.998421i	$-0.482111\pi$
$109$	110737.	−	191802.i	0.892743	−	1.54628i	0.836573	−	0.547855i	$-0.184556\pi$
$110$		0			0
$111$	−21510.0			−0.165704
$112$		0			0
$113$	54194.0			0.399259			0.199630	−	0.979871i	$-0.436026\pi$
$113$	54194.0			0.399259			0.199630	+	0.979871i	$0.436026\pi$
$114$		0			0
$115$	64192.0	−	111184.i	0.452623	−	0.783965i
$116$		0			0
$117$	−41553.0	−	71971.9i	−0.280632	−	0.486069i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	25413.5	+	44017.5i	0.157798	+	0.273314i
$122$		0			0
$123$	32535.0	−	56352.3i	0.193905	−	0.335853i
$124$		0			0
$125$	−173196.			−0.991432
$126$		0			0
$127$	−245760.			−1.35208			−0.676039	−	0.736866i	$-0.736305\pi$
$127$	−245760.			−1.35208			−0.676039	+	0.736866i	$0.736305\pi$
$128$		0			0
$129$	20934.0	−	36258.8i	0.110759	−	0.191840i
$130$		0			0
$131$	−75134.0	−	130136.i	−0.382524	−	0.662550i	0.608899	−	0.793248i	$-0.291612\pi$
$131$	−75134.0	−	130136.i	−0.382524	−	0.662550i	−0.991422	+	0.130698i	$0.958278\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	12393.0	+	21465.3i	0.0585251	+	0.101368i
$136$		0			0
$137$	200819.	−	347829.i	0.914120	−	1.58330i	0.105936	−	0.994373i	$-0.466216\pi$
$137$	200819.	−	347829.i	0.914120	−	1.58330i	0.808184	−	0.588930i	$-0.200451\pi$
$138$		0			0
$139$	−374092.			−1.64226			−0.821129	−	0.570743i	$-0.806655\pi$
$139$	−374092.			−1.64226			−0.821129	+	0.570743i	$0.806655\pi$
$140$		0			0
$141$	222048.			0.940587
$142$		0			0
$143$	170316.	−	294996.i	0.696491	−	1.20636i
$144$		0			0
$145$	−19822.0	−	34332.7i	−0.0782938	−	0.135609i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	228021.	+	394944.i	0.841413	+	1.45737i	0.888700	+	0.458489i	$0.151609\pi$
$149$	228021.	+	394944.i	0.841413	+	1.45737i	−0.0472874	+	0.998881i	$0.515058\pi$
$150$		0			0
$151$	4012.00	−	6948.99i	0.0143192	−	0.0248016i	−0.858777	−	0.512350i	$-0.828775\pi$
$151$	4012.00	−	6948.99i	0.0143192	−	0.0248016i	0.873096	+	0.487548i	$0.162109\pi$
$152$		0			0
$153$	−74682.0			−0.257921
$154$		0			0
$155$	332928.			1.11307
$156$		0			0
$157$	55039.0	−	95330.3i	0.178206	−	0.308661i	−0.763060	−	0.646327i	$-0.776304\pi$
$157$	55039.0	−	95330.3i	0.178206	−	0.308661i	0.941266	+	0.337666i	$0.109637\pi$
$158$		0			0
$159$	4995.00	+	8651.59i	0.0156691	+	0.0271396i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1814.00	+	3141.94i	0.00534772	+	0.00926251i	0.868687	−	0.495362i	$-0.164964\pi$
$163$	1814.00	+	3141.94i	0.00534772	+	0.00926251i	−0.863339	+	0.504624i	$0.831631\pi$
$164$		0			0
$165$	−50796.0	+	87981.3i	−0.145251	+	0.251582i
$166$		0			0
$167$	−192824.			−0.535020			−0.267510	−	0.963555i	$-0.586201\pi$
$167$	−192824.			−0.535020			−0.267510	+	0.963555i	$0.586201\pi$
$168$		0			0
$169$	681383.			1.83516
$170$		0			0
$171$	18306.0	−	31706.9i	0.0478744	−	0.0829209i
$172$		0			0
$173$	78571.0	+	136089.i	0.199594	+	0.345707i	0.948397	−	0.317086i	$-0.102704\pi$
$173$	78571.0	+	136089.i	0.199594	+	0.345707i	−0.748803	+	0.662793i	$0.769371\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−211014.	−	365487.i	−0.506265	−	0.876877i
$178$		0			0
$179$	223434.	−	386999.i	0.521215	−	0.902770i	−0.478481	−	0.878098i	$-0.658812\pi$
$179$	223434.	−	386999.i	0.521215	−	0.902770i	0.999696	−	0.0246724i	$-0.00785427\pi$
$180$		0			0
$181$	−805638.			−1.82786			−0.913931	−	0.405869i	$-0.866969\pi$
$181$	−805638.			−1.82786			−0.913931	+	0.405869i	$0.866969\pi$
$182$		0			0
$183$	−87858.0			−0.193934
$184$		0			0
$185$	−40630.0	+	70373.2i	−0.0872806	+	0.151174i
$186$		0			0
$187$	−153052.	−	265094.i	−0.320063	−	0.554365i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	373956.	+	647711.i	0.741715	+	1.28469i	0.951714	+	0.306987i	$0.0993206\pi$
$191$	373956.	+	647711.i	0.741715	+	1.28469i	−0.209999	+	0.977702i	$0.567346\pi$
$192$		0			0
$193$	288767.	−	500159.i	0.558026	−	0.966529i	−0.439635	−	0.898176i	$-0.644892\pi$
$193$	288767.	−	500159.i	0.558026	−	0.966529i	0.997661	−	0.0683527i	$-0.0217743\pi$
$194$		0			0
$195$	−313956.			−0.591265
$196$		0			0
$197$	−771098.			−1.41561			−0.707806	−	0.706407i	$-0.750315\pi$
$197$	−771098.			−1.41561			−0.707806	+	0.706407i	$0.750315\pi$
$198$		0			0
$199$	278620.	−	482584.i	0.498746	−	0.863854i	−0.501253	−	0.865301i	$-0.667127\pi$
$199$	278620.	−	482584.i	0.498746	−	0.863854i	0.999999	+	0.00144716i	$0.000460646\pi$
$200$		0			0
$201$	−118134.	−	204614.i	−0.206246	−	0.357228i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−122910.	−	212886.i	−0.204269	−	0.353804i
$206$		0			0
$207$	152928.	−	264879.i	0.248063	−	0.429657i
$208$		0			0
$209$	150064.			0.237635
$210$		0			0
$211$	−19660.0			−0.0304003			−0.0152001	−	0.999884i	$-0.504839\pi$
$211$	−19660.0			−0.0304003			−0.0152001	+	0.999884i	$0.504839\pi$
$212$		0			0
$213$	294480.	−	510054.i	0.444741	−	0.770313i
$214$		0			0
$215$	−79084.0	−	136978.i	−0.116679	−	0.202094i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	25227.0	+	43694.4i	0.0355431	+	0.0615625i
$220$		0			0
$221$	472986.	−	819236.i	0.651430	−	1.12831i
$222$		0			0
$223$	896848.			1.20769			0.603847	−	0.797100i	$-0.293634\pi$
$223$	896848.			1.20769			0.603847	+	0.797100i	$0.293634\pi$
$224$		0			0
$225$	−159489.			−0.210027
$226$		0			0
$227$	117114.	−	202847.i	0.150850	−	0.261279i	−0.780690	−	0.624918i	$-0.785132\pi$
$227$	117114.	−	202847.i	0.150850	−	0.261279i	0.931540	+	0.363639i	$0.118466\pi$
$228$		0			0
$229$	−517813.	−	896878.i	−0.652506	−	1.13017i	−0.982513	−	0.186194i	$-0.940385\pi$
$229$	−517813.	−	896878.i	−0.652506	−	1.13017i	0.330007	−	0.943978i	$-0.392949\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−228557.	−	395872.i	−0.275807	−	0.477711i	0.694532	−	0.719462i	$-0.255612\pi$
$233$	−228557.	−	395872.i	−0.275807	−	0.477711i	−0.970338	+	0.241751i	$0.922278\pi$
$234$		0			0
$235$	419424.	−	726464.i	0.495432	−	0.858113i
$236$		0			0
$237$	88560.0			0.102416
$238$		0			0
$239$	676344.			0.765901			0.382951	−	0.923769i	$-0.374908\pi$
$239$	676344.			0.765901			0.382951	+	0.923769i	$0.374908\pi$
$240$		0			0
$241$	−48335.0	+	83718.7i	−0.0536067	+	0.0928495i	−0.891584	−	0.452856i	$-0.850405\pi$
$241$	−48335.0	+	83718.7i	−0.0536067	+	0.0928495i	0.837977	+	0.545706i	$0.183738\pi$
$242$		0			0
$243$	29524.5	+	51137.9i	0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	231876.	+	401621.i	0.241832	+	0.418865i
$248$		0			0
$249$	−274986.	+	476290.i	−0.281069	+	0.486825i
$250$		0			0
$251$	−288876.			−0.289419			−0.144710	−	0.989474i	$-0.546225\pi$
$251$	−288876.			−0.289419			−0.144710	+	0.989474i	$0.546225\pi$
$252$		0			0
$253$	1.25363e6			1.23131
$254$		0			0
$255$	−141066.	+	244333.i	−0.135854	+	0.235306i
$256$		0			0
$257$	−355923.	−	616477.i	−0.336142	−	0.582216i	0.647561	−	0.762013i	$-0.275789\pi$
$257$	−355923.	−	616477.i	−0.336142	−	0.582216i	−0.983704	+	0.179798i	$0.942456\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−47223.0	−	81792.6i	−0.0429094	−	0.0743212i
$262$		0			0
$263$	936840.	−	1.62265e6i	0.835172	−	1.44656i	−0.0587179	−	0.998275i	$-0.518701\pi$
$263$	936840.	−	1.62265e6i	0.835172	−	1.44656i	0.893890	−	0.448286i	$-0.147965\pi$
$264$		0			0
$265$	37740.0			0.0330132
$266$		0			0
$267$	−566622.			−0.486424
$268$		0			0
$269$	−688301.	+	1.19217e6i	−0.579960	+	1.00452i	0.415524	+	0.909582i	$0.363598\pi$
$269$	−688301.	+	1.19217e6i	−0.579960	+	1.00452i	−0.995483	+	0.0949371i	$0.969735\pi$
$270$		0			0
$271$	−390888.	−	677038.i	−0.323317	−	0.560002i	0.657853	−	0.753146i	$-0.271465\pi$
$271$	−390888.	−	677038.i	−0.323317	−	0.560002i	−0.981170	+	0.193144i	$0.938131\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−326854.	−	566128.i	−0.260629	−	0.451422i
$276$		0			0
$277$	−1.03466e6	+	1.79208e6i	−0.810210	+	1.40333i	0.102506	+	0.994732i	$0.467314\pi$
$277$	−1.03466e6	+	1.79208e6i	−0.810210	+	1.40333i	−0.912717	+	0.408593i	$0.866019\pi$
$278$		0			0
$279$	793152.			0.610023
$280$		0			0
$281$	1.87911e6			1.41967			0.709835	−	0.704368i	$-0.248770\pi$
$281$	1.87911e6			1.41967			0.709835	+	0.704368i	$0.248770\pi$
$282$		0			0
$283$	335078.	−	580372.i	0.248702	−	0.430765i	−0.714464	−	0.699673i	$-0.753329\pi$
$283$	335078.	−	580372.i	0.248702	−	0.430765i	0.963166	+	0.268907i	$0.0866626\pi$
$284$		0			0
$285$	−69156.0	−	119782.i	−0.0504333	−	0.0873531i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	284886.	+	493438.i	0.200645	+	0.347526i
$290$		0			0
$291$	170271.	−	294918.i	0.117871	−	0.204159i
$292$		0			0
$293$	−1.69611e6			−1.15421			−0.577105	−	0.816670i	$-0.695818\pi$
$293$	−1.69611e6			−1.15421			−0.577105	+	0.816670i	$0.695818\pi$
$294$		0			0
$295$	−1.59433e6			−1.06665
$296$		0			0
$297$	−121014.	+	209602.i	−0.0796058	+	0.137881i
$298$		0			0
$299$	1.93709e6	+	3.35513e6i	1.25306	+	2.17036i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	252657.	+	437615.i	0.158097	+	0.273833i
$304$		0			0
$305$	−165954.	+	287441.i	−0.102150	+	0.176929i
$306$		0			0
$307$	1.09459e6			0.662834			0.331417	−	0.943484i	$-0.392473\pi$
$307$	1.09459e6			0.662834			0.331417	+	0.943484i	$0.392473\pi$
$308$		0			0
$309$	−237528.			−0.141520
$310$		0			0
$311$	606244.	−	1.05005e6i	0.355424	−	0.615612i	−0.631767	−	0.775159i	$-0.717670\pi$
$311$	606244.	−	1.05005e6i	0.355424	−	0.615612i	0.987190	+	0.159547i	$0.0510032\pi$
$312$		0			0
$313$	847181.	+	1.46736e6i	0.488782	+	0.846596i	0.999917	−	0.0129051i	$-0.00410795\pi$
$313$	847181.	+	1.46736e6i	0.488782	+	0.846596i	−0.511135	+	0.859501i	$0.670775\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−166671.	−	288683.i	−0.0931562	−	0.161351i	0.815681	−	0.578501i	$-0.196362\pi$
$317$	−166671.	−	288683.i	−0.0931562	−	0.161351i	−0.908838	+	0.417150i	$0.863029\pi$
$318$		0			0
$319$	193556.	−	335249.i	0.106495	−	0.184455i
$320$		0			0
$321$	−424116.			−0.229732
$322$		0			0
$323$	416744.			0.222261
$324$		0			0
$325$	1.01010e6	−	1.74954e6i	0.530462	−	0.918788i
$326$		0			0
$327$	−996633.	−	1.72622e6i	−0.515425	−	0.892743i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−918070.	−	1.59014e6i	−0.460581	−	0.797749i	0.538409	−	0.842684i	$-0.319026\pi$
$331$	−918070.	−	1.59014e6i	−0.460581	−	0.797749i	−0.998990	+	0.0449342i	$0.985692\pi$
$332$		0			0
$333$	−96795.0	+	167654.i	−0.0478346	+	0.0828520i
$334$		0			0
$335$	−892568.			−0.434540
$336$		0			0
$337$	−973518.			−0.466949			−0.233474	−	0.972363i	$-0.575009\pi$
$337$	−973518.			−0.466949			−0.233474	+	0.972363i	$0.575009\pi$
$338$		0			0
$339$	243873.	−	422400.i	0.115256	−	0.199630i
$340$		0			0
$341$	1.62547e6	+	2.81540e6i	0.756996	+	1.31116i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−577728.	−	1.00065e6i	−0.261322	−	0.452623i
$346$		0			0
$347$	−1.69905e6	+	2.94284e6i	−0.757500	+	1.31203i	0.186622	+	0.982432i	$0.440246\pi$
$347$	−1.69905e6	+	2.94284e6i	−0.757500	+	1.31203i	−0.944122	+	0.329596i	$0.893087\pi$
$348$		0			0
$349$	34370.0			0.0151048			0.00755242	−	0.999971i	$-0.497596\pi$
$349$	34370.0			0.0151048			0.00755242	+	0.999971i	$0.497596\pi$
$350$		0			0
$351$	−747954.			−0.324046
$352$		0			0
$353$	−1.25120e6	+	2.16713e6i	−0.534427	+	0.925654i	0.464764	+	0.885435i	$0.346139\pi$
$353$	−1.25120e6	+	2.16713e6i	−0.534427	+	0.925654i	−0.999191	+	0.0402198i	$0.987194\pi$
$354$		0			0
$355$	−1.11248e6	−	1.92687e6i	−0.468513	−	0.811488i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.51400e6	+	2.62232e6i	0.619997	+	1.07387i	0.989486	+	0.144631i	$0.0461996\pi$
$359$	1.51400e6	+	2.62232e6i	0.619997	+	1.07387i	−0.369488	+	0.929235i	$0.620467\pi$
$360$		0			0
$361$	1.13590e6	−	1.96743e6i	0.458745	−	0.794569i
$362$		0			0
$363$	457443.			0.182209
$364$		0			0
$365$	190604.			0.0748859
$366$		0			0
$367$	−1.60472e6	+	2.77946e6i	−0.621919	+	1.07720i	0.367209	+	0.930139i	$0.380314\pi$
$367$	−1.60472e6	+	2.77946e6i	−0.621919	+	1.07720i	−0.989128	+	0.147057i	$0.953020\pi$
$368$		0			0
$369$	−292815.	−	507170.i	−0.111951	−	0.193905i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−757523.	−	1.31207e6i	−0.281919	−	0.488297i	0.689939	−	0.723868i	$-0.257637\pi$
$373$	−757523.	−	1.31207e6i	−0.281919	−	0.488297i	−0.971857	+	0.235570i	$0.924304\pi$
$374$		0			0
$375$	−779382.	+	1.34993e6i	−0.286202	+	0.495716i
$376$		0			0
$377$	1.19632e6			0.433503
$378$		0			0
$379$	643516.			0.230124			0.115062	−	0.993358i	$-0.463293\pi$
$379$	643516.			0.230124			0.115062	+	0.993358i	$0.463293\pi$
$380$		0			0
$381$	−1.10592e6	+	1.91551e6i	−0.390311	+	0.676039i
$382$		0			0
$383$	2.37541e6	+	4.11433e6i	0.827449	+	1.43318i	0.900033	+	0.435822i	$0.143542\pi$
$383$	2.37541e6	+	4.11433e6i	0.827449	+	1.43318i	−0.0725840	+	0.997362i	$0.523125\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−188406.	−	326329.i	−0.0639466	−	0.110759i
$388$		0			0
$389$	−189787.	+	328721.i	−0.0635905	+	0.110142i	−0.896068	−	0.443917i	$-0.853588\pi$
$389$	−189787.	+	328721.i	−0.0635905	+	0.110142i	0.832477	+	0.554059i	$0.186922\pi$
$390$		0			0
$391$	3.48147e6			1.15165
$392$		0			0
$393$	−1.35241e6			−0.441700
$394$		0			0
$395$	167280.	−	289737.i	0.0539450	−	0.0934355i
$396$		0			0
$397$	−2.71066e6	−	4.69501e6i	−0.863176	−	1.49507i	−0.868847	−	0.495082i	$-0.835138\pi$
$397$	−2.71066e6	−	4.69501e6i	−0.863176	−	1.49507i	0.00567007	−	0.999984i	$-0.498195\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.10322e6	+	5.37493e6i	0.963720	+	1.66921i	0.713015	+	0.701149i	$0.247329\pi$
$401$	3.10322e6	+	5.37493e6i	0.963720	+	1.66921i	0.250705	+	0.968064i	$0.419338\pi$
$402$		0			0
$403$	−5.02330e6	+	8.70060e6i	−1.54073	+	2.66862i
$404$		0			0
$405$	223074.			0.0675789
$406$		0			0
$407$	−793480.			−0.237438
$408$		0			0
$409$	−2.12699e6	+	3.68405e6i	−0.628719	+	1.08897i	0.359090	+	0.933303i	$0.383087\pi$
$409$	−2.12699e6	+	3.68405e6i	−0.628719	+	1.08897i	−0.987809	+	0.155670i	$0.950246\pi$
$410$		0			0
$411$	−1.80737e6	−	3.13046e6i	−0.527768	−	0.914120i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.03884e6	+	1.79932e6i	0.296092	+	0.512846i
$416$		0			0
$417$	−1.68341e6	+	2.91576e6i	−0.474079	+	0.821129i
$418$		0			0
$419$	725484.			0.201880			0.100940	−	0.994893i	$-0.467815\pi$
$419$	725484.			0.201880			0.100940	+	0.994893i	$0.467815\pi$
$420$		0			0
$421$	−6.49867e6			−1.78698			−0.893489	−	0.449086i	$-0.851750\pi$
$421$	−6.49867e6			−1.78698			−0.893489	+	0.449086i	$0.851750\pi$
$422$		0			0
$423$	999216.	−	1.73069e6i	0.271524	−	0.470294i
$424$		0			0
$425$	−907709.	−	1.57220e6i	−0.243767	−	0.422216i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−1.53284e6	−	2.65496e6i	−0.402119	−	0.696491i
$430$		0			0
$431$	982620.	−	1.70195e6i	0.254796	−	0.441319i	−0.710044	−	0.704157i	$-0.751325\pi$
$431$	982620.	−	1.70195e6i	0.254796	−	0.441319i	0.964840	+	0.262838i	$0.0846584\pi$
$432$		0			0
$433$	−4.33531e6			−1.11122			−0.555611	−	0.831442i	$-0.687516\pi$
$433$	−4.33531e6			−1.11122			−0.555611	+	0.831442i	$0.687516\pi$
$434$		0			0
$435$	−356796.			−0.0904059
$436$		0			0
$437$	−853376.	+	1.47809e6i	−0.213765	+	0.370252i
$438$		0			0
$439$	3.23874e6	+	5.60966e6i	0.802075	+	1.38923i	0.918248	+	0.396005i	$0.129604\pi$
$439$	3.23874e6	+	5.60966e6i	0.802075	+	1.38923i	−0.116174	+	0.993229i	$0.537063\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.16348e6	−	3.74726e6i	−0.523774	−	0.907203i	−0.999617	−	0.0276730i	$-0.991190\pi$
$443$	−2.16348e6	−	3.74726e6i	−0.523774	−	0.907203i	0.475843	−	0.879530i	$-0.342143\pi$
$444$		0			0
$445$	−1.07029e6	+	1.85379e6i	−0.256212	+	0.443773i
$446$		0			0
$447$	4.10438e6			0.971580
$448$		0			0
$449$	482210.			0.112881			0.0564404	−	0.998406i	$-0.482025\pi$
$449$	482210.			0.112881			0.0564404	+	0.998406i	$0.482025\pi$
$450$		0			0
$451$	1.20018e6	−	2.07877e6i	0.277847	−	0.481244i
$452$		0			0
$453$	−36108.0	−	62540.9i	−0.00826719	−	0.0143192i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.26082e6	−	7.37996e6i	−0.954339	−	1.65296i	−0.735873	−	0.677120i	$-0.763228\pi$
$457$	−4.26082e6	−	7.37996e6i	−0.954339	−	1.65296i	−0.218466	−	0.975845i	$-0.570105\pi$
$458$		0			0
$459$	−336069.	+	582089.i	−0.0744555	+	0.128961i
$460$		0			0
$461$	5.99857e6			1.31461			0.657303	−	0.753627i	$-0.271697\pi$
$461$	5.99857e6			1.31461			0.657303	+	0.753627i	$0.271697\pi$
$462$		0			0
$463$	−4.59483e6			−0.996133			−0.498066	−	0.867139i	$-0.665956\pi$
$463$	−4.59483e6			−0.996133			−0.498066	+	0.867139i	$0.665956\pi$
$464$		0			0
$465$	1.49818e6	−	2.59492e6i	0.321315	−	0.556533i
$466$		0			0
$467$	4.42165e6	+	7.65852e6i	0.938193	+	1.62500i	0.768839	+	0.639443i	$0.220835\pi$
$467$	4.42165e6	+	7.65852e6i	0.938193	+	1.62500i	0.169354	+	0.985555i	$0.445832\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−495351.	−	857973.i	−0.102887	−	0.178206i
$472$		0			0
$473$	772232.	−	1.33755e6i	0.158707	−	0.274888i
$474$		0			0
$475$	889988.			0.180988
$476$		0			0
$477$	89910.0			0.0180931
$478$		0			0
$479$	−3.28031e6	+	5.68167e6i	−0.653245	+	1.13145i	0.329085	+	0.944300i	$0.393260\pi$
$479$	−3.28031e6	+	5.68167e6i	−0.653245	+	1.13145i	−0.982331	+	0.187154i	$0.940074\pi$
$480$		0			0
$481$	−1.22607e6	−	2.12362e6i	−0.241631	−	0.418517i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−643246.	−	1.11413e6i	−0.124172	−	0.215072i
$486$		0			0
$487$	−3.93886e6	+	6.82231e6i	−0.752572	+	1.30349i	0.194000	+	0.981002i	$0.437854\pi$
$487$	−3.93886e6	+	6.82231e6i	−0.752572	+	1.30349i	−0.946572	+	0.322492i	$0.895480\pi$
$488$		0			0
$489$	32652.0			0.00617501
$490$		0			0
$491$	637860.			0.119405			0.0597024	−	0.998216i	$-0.480985\pi$
$491$	637860.			0.119405			0.0597024	+	0.998216i	$0.480985\pi$
$492$		0			0
$493$	537526.	−	931022.i	0.0996052	−	0.172521i
$494$		0			0
$495$	457164.	+	791831.i	0.0838608	+	0.145251i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.46823e6	+	4.27510e6i	0.443746	+	0.768590i	0.997964	−	0.0637812i	$-0.0203160\pi$
$499$	2.46823e6	+	4.27510e6i	0.443746	+	0.768590i	−0.554218	+	0.832371i	$0.686983\pi$
$500$		0			0
$501$	−867708.	+	1.50291e6i	−0.154447	+	0.267510i
$502$		0			0
$503$	−226872.			−0.0399817			−0.0199908	−	0.999800i	$-0.506364\pi$
$503$	−226872.			−0.0399817			−0.0199908	+	0.999800i	$0.506364\pi$
$504$		0			0
$505$	1.90896e6			0.333096
$506$		0			0
$507$	3.06622e6	−	5.31085e6i	0.529766	−	0.917581i
$508$		0			0
$509$	2.68702e6	+	4.65405e6i	0.459702	+	0.796227i	0.998945	−	0.0459231i	$-0.0146229\pi$
$509$	2.68702e6	+	4.65405e6i	0.459702	+	0.796227i	−0.539243	+	0.842150i	$0.681290\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−164754.	−	285362.i	−0.0276403	−	0.0478744i
$514$		0			0
$515$	−448664.	+	777109.i	−0.0745424	+	0.129111i
$516$		0			0
$517$	8.19110e6			1.34777
$518$		0			0
$519$	1.41428e6			0.230471
$520$		0			0
$521$	−4.80710e6	+	8.32613e6i	−0.775869	+	1.34384i	0.158436	+	0.987369i	$0.449355\pi$
$521$	−4.80710e6	+	8.32613e6i	−0.775869	+	1.34384i	−0.934305	+	0.356475i	$0.883978\pi$
$522$		0			0
$523$	2.48215e6	+	4.29921e6i	0.396802	+	0.687281i	0.993329	−	0.115312i	$-0.0367866\pi$
$523$	2.48215e6	+	4.29921e6i	0.396802	+	0.687281i	−0.596527	+	0.802593i	$0.703453\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.51411e6	+	7.81867e6i	0.708020	+	1.22633i
$528$		0			0
$529$	−3.91092e6	+	6.77391e6i	−0.607630	+	1.05245i
$530$		0			0
$531$	−3.79825e6			−0.584585
$532$		0			0
$533$	7.41798e6			1.13101
$534$		0			0
$535$	−801108.	+	1.38756e6i	−0.121006	+	0.209588i
$536$		0			0
$537$	−2.01091e6	−	3.48299e6i	−0.300923	−	0.521215i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−6.02246e6	−	1.04312e7i	−0.884668	−	1.53229i	−0.846094	−	0.533034i	$-0.821052\pi$
$541$	−6.02246e6	−	1.04312e7i	−0.884668	−	1.53229i	−0.0385747	−	0.999256i	$-0.512282\pi$
$542$		0			0
$543$	−3.62537e6	+	6.27933e6i	−0.527658	+	0.913931i
$544$		0			0
$545$	−7.53012e6			−1.08595
$546$		0			0
$547$	4.23695e6			0.605459			0.302730	−	0.953077i	$-0.402102\pi$
$547$	4.23695e6			0.605459			0.302730	+	0.953077i	$0.402102\pi$
$548$		0			0
$549$	−395361.	+	684785.i	−0.0559839	+	0.0969669i
$550$		0			0
$551$	263516.	+	456423.i	0.0369767	+	0.0640455i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	365670.	+	633359.i	0.0503915	+	0.0872806i
$556$		0			0
$557$	5.12874e6	−	8.88325e6i	0.700444	−	1.21320i	−0.267867	−	0.963456i	$-0.586319\pi$
$557$	5.12874e6	−	8.88325e6i	0.700444	−	1.21320i	0.968311	−	0.249748i	$-0.0803478\pi$
$558$		0			0
$559$	4.77295e6			0.646037
$560$		0			0
$561$	−2.75494e6			−0.369577
$562$		0			0
$563$	−2.70389e6	+	4.68327e6i	−0.359515	+	0.622699i	−0.987880	−	0.155220i	$-0.950391\pi$
$563$	−2.70389e6	+	4.68327e6i	−0.359515	+	0.622699i	0.628365	+	0.777919i	$0.283725\pi$
$564$		0			0
$565$	−921298.	−	1.59573e6i	−0.121417	−	0.210300i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.69484e6	−	4.66759e6i	−0.348941	−	0.604383i	0.637121	−	0.770764i	$-0.280125\pi$
$569$	−2.69484e6	−	4.66759e6i	−0.348941	−	0.604383i	−0.986062	+	0.166381i	$0.946792\pi$
$570$		0			0
$571$	4.12276e6	−	7.14083e6i	0.529173	−	0.916555i	−0.470248	−	0.882534i	$-0.655835\pi$
$571$	4.12276e6	−	7.14083e6i	0.529173	−	0.916555i	0.999421	−	0.0340207i	$-0.0108312\pi$
$572$		0			0
$573$	6.73121e6			0.856459
$574$		0			0
$575$	7.43494e6			0.937795
$576$		0			0
$577$	−577039.	+	999461.i	−0.0721549	+	0.124976i	−0.899845	−	0.436209i	$-0.856321\pi$
$577$	−577039.	+	999461.i	−0.0721549	+	0.124976i	0.827691	+	0.561185i	$0.189654\pi$
$578$		0			0
$579$	−2.59890e6	−	4.50143e6i	−0.322176	−	0.558026i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	184260.	+	319148.i	0.0224522	+	0.0388884i
$584$		0			0
$585$	−1.41280e6	+	2.44704e6i	−0.170684	+	0.295633i
$586$		0			0
$587$	7.16464e6			0.858221			0.429111	−	0.903252i	$-0.358827\pi$
$587$	7.16464e6			0.858221			0.429111	+	0.903252i	$0.358827\pi$
$588$		0			0
$589$	−4.42598e6			−0.525680
$590$		0			0
$591$	−3.46994e6	+	6.01011e6i	−0.408652	+	0.707806i
$592$		0			0
$593$	7.27670e6	+	1.26036e7i	0.849763	+	1.47183i	0.881420	+	0.472334i	$0.156588\pi$
$593$	7.27670e6	+	1.26036e7i	0.849763	+	1.47183i	−0.0316568	+	0.999499i	$0.510078\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.50758e6	−	4.34326e6i	−0.287951	−	0.498746i
$598$		0			0
$599$	5.21599e6	−	9.03436e6i	0.593977	−	1.02880i	−0.399713	−	0.916640i	$-0.630890\pi$
$599$	5.21599e6	−	9.03436e6i	0.593977	−	1.02880i	0.993690	−	0.112159i	$-0.0357765\pi$
$600$		0			0
$601$	−416858.			−0.0470763			−0.0235381	−	0.999723i	$-0.507493\pi$
$601$	−416858.			−0.0470763			−0.0235381	+	0.999723i	$0.507493\pi$
$602$		0			0
$603$	−2.12641e6			−0.238152
$604$		0			0
$605$	864059.	−	1.49659e6i	0.0959743	−	0.166232i
$606$		0			0
$607$	−3.95417e6	−	6.84882e6i	−0.435596	−	0.754474i	0.561749	−	0.827308i	$-0.310129\pi$
$607$	−3.95417e6	−	6.84882e6i	−0.435596	−	0.754474i	−0.997344	+	0.0728345i	$0.976796\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.26567e7	+	2.19221e7i	1.37157	+	2.37563i
$612$		0			0
$613$	5.68807e6	−	9.85202e6i	0.611383	−	1.05895i	−0.379624	−	0.925141i	$-0.623947\pi$
$613$	5.68807e6	−	9.85202e6i	0.611383	−	1.05895i	0.991007	−	0.133806i	$-0.0427200\pi$
$614$		0			0
$615$	−2.21238e6			−0.235870
$616$		0			0
$617$	−8.77271e6			−0.927728			−0.463864	−	0.885906i	$-0.653537\pi$
$617$	−8.77271e6			−0.927728			−0.463864	+	0.885906i	$0.653537\pi$
$618$		0			0
$619$	7.20551e6	−	1.24803e7i	0.755854	−	1.30918i	−0.189094	−	0.981959i	$-0.560555\pi$
$619$	7.20551e6	−	1.24803e7i	0.755854	−	1.30918i	0.944948	−	0.327219i	$-0.106112\pi$
$620$		0			0
$621$	−1.37635e6	−	2.38391e6i	−0.143219	−	0.248063i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−132230.	−	229030.i	−0.0135404	−	0.0234527i
$626$		0			0
$627$	675288.	−	1.16963e6i	0.0685994	−	0.118818i
$628$		0			0
$629$	−2.20358e6			−0.222076
$630$		0			0
$631$	−1.29466e7			−1.29444			−0.647221	−	0.762303i	$-0.724069\pi$
$631$	−1.29466e7			−1.29444			−0.647221	+	0.762303i	$0.724069\pi$
$632$		0			0
$633$	−88470.0	+	153235.i	−0.00877580	+	0.0152001i
$634$		0			0
$635$	4.17792e6	+	7.23637e6i	0.411174	+	0.712175i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.65032e6	−	4.59049e6i	−0.256771	−	0.444741i
$640$		0			0
$641$	2.44517e6	−	4.23517e6i	0.235052	−	0.407123i	−0.724235	−	0.689553i	$-0.757807\pi$
$641$	2.44517e6	−	4.23517e6i	0.235052	−	0.407123i	0.959288	+	0.282430i	$0.0911405\pi$
$642$		0			0
$643$	−1.22604e6			−0.116943			−0.0584717	−	0.998289i	$-0.518623\pi$
$643$	−1.22604e6			−0.116943			−0.0584717	+	0.998289i	$0.518623\pi$
$644$		0			0
$645$	−1.42351e6			−0.134729
$646$		0			0
$647$	6.05492e6	−	1.04874e7i	0.568654	−	0.984937i	−0.428046	−	0.903757i	$-0.640798\pi$
$647$	6.05492e6	−	1.04874e7i	0.568654	−	0.984937i	0.996699	−	0.0811802i	$-0.0258689\pi$
$648$		0			0
$649$	−7.78407e6	−	1.34824e7i	−0.725429	−	1.25648i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.64349e6	−	8.04276e6i	−0.426149	−	0.738111i	0.570378	−	0.821382i	$-0.306797\pi$
$653$	−4.64349e6	−	8.04276e6i	−0.426149	−	0.738111i	−0.996527	+	0.0832708i	$0.973463\pi$
$654$		0			0
$655$	−2.55456e6	+	4.42462e6i	−0.232655	+	0.402970i
$656$		0			0
$657$	454086.			0.0410416
$658$		0			0
$659$	451612.			0.0405090			0.0202545	−	0.999795i	$-0.493552\pi$
$659$	451612.			0.0405090			0.0202545	+	0.999795i	$0.493552\pi$
$660$		0			0
$661$	−927541.	+	1.60655e6i	−0.0825714	+	0.143018i	−0.904354	−	0.426784i	$-0.859647\pi$
$661$	−927541.	+	1.60655e6i	−0.0825714	+	0.143018i	0.821782	+	0.569801i	$0.192980\pi$
$662$		0			0
$663$	−4.25687e6	−	7.37312e6i	−0.376103	−	0.651430i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.20141e6	+	3.81295e6i	0.191596	+	0.331854i
$668$		0			0
$669$	4.03582e6	−	6.99024e6i	0.348631	−	0.603847i
$670$		0			0
$671$	−3.24098e6			−0.277889
$672$		0			0
$673$	2.14534e7			1.82582			0.912911	−	0.408158i	$-0.133829\pi$
$673$	2.14534e7			1.82582			0.912911	+	0.408158i	$0.133829\pi$
$674$		0			0
$675$	−717700.	+	1.24309e6i	−0.0606295	+	0.105013i
$676$		0			0
$677$	7.84934e6	+	1.35954e7i	0.658205	+	1.14004i	0.981080	+	0.193603i	$0.0620173\pi$
$677$	7.84934e6	+	1.35954e7i	0.658205	+	1.14004i	−0.322875	+	0.946442i	$0.604649\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.05403e6	−	1.82563e6i	−0.0870931	−	0.150850i
$682$		0			0
$683$	−7.01248e6	+	1.21460e7i	−0.575201	+	0.996278i	0.420818	+	0.907145i	$0.361743\pi$
$683$	−7.01248e6	+	1.21460e7i	−0.575201	+	0.996278i	−0.996020	+	0.0891332i	$0.971590\pi$
$684$		0			0
$685$	−1.36557e7			−1.11196
$686$		0			0
$687$	−9.32063e6			−0.753449
$688$		0			0
$689$	−569430.	+	986282.i	−0.0456975	+	0.0791504i
$690$		0			0
$691$	−9.49093e6	−	1.64388e7i	−0.756160	−	1.30971i	−0.944795	−	0.327661i	$-0.893740\pi$
$691$	−9.49093e6	−	1.64388e7i	−0.756160	−	1.30971i	0.188635	−	0.982047i	$-0.439594\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.35956e6	+	1.10151e7i	0.499419	+	0.865020i
$696$		0			0
$697$	3.33303e6	−	5.77298e6i	0.259871	−	0.450109i
$698$		0			0
$699$	−4.11403e6			−0.318474
$700$		0			0
$701$	2.22806e7			1.71250			0.856251	−	0.516560i	$-0.172788\pi$
$701$	2.22806e7			1.71250			0.856251	+	0.516560i	$0.172788\pi$
$702$		0			0
$703$	540140.	−	935550.i	0.0412210	−	0.0713968i
$704$		0			0
$705$	−3.77482e6	−	6.53817e6i	−0.286038	−	0.495432i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	238133.	+	412458.i	0.0177911	+	0.0308152i	0.874784	−	0.484513i	$-0.161003\pi$
$709$	238133.	+	412458.i	0.0177911	+	0.0308152i	−0.856993	+	0.515328i	$0.827670\pi$
$710$		0			0
$711$	398520.	−	690257.i	0.0295649	−	0.0512079i
$712$		0			0
$713$	−3.69746e7			−2.72383
$714$		0			0
$715$	−1.15815e7			−0.847226
$716$		0			0
$717$	3.04355e6	−	5.27158e6i	0.221097	−	0.382951i
$718$		0			0
$719$	−131784.	−	228257.i	−0.00950693	−	0.0164665i	0.861233	−	0.508211i	$-0.169693\pi$
$719$	−131784.	−	228257.i	−0.00950693	−	0.0164665i	−0.870740	+	0.491744i	$0.836360\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	435015.	+	753468.i	0.0309498	+	0.0536067i
$724$		0			0
$725$	1.14793e6	−	1.98827e6i	0.0811090	−	0.140485i
$726$		0			0
$727$	−9.28319e6			−0.651420			−0.325710	−	0.945470i	$-0.605603\pi$
$727$	−9.28319e6			−0.651420			−0.325710	+	0.945470i	$0.605603\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	2.14457e6	−	3.71451e6i	0.148439	−	0.257103i
$732$		0			0
$733$	−9.47737e6	−	1.64153e7i	−0.651520	−	1.12847i	−0.982754	−	0.184917i	$-0.940798\pi$
$733$	−9.47737e6	−	1.64153e7i	−0.651520	−	1.12847i	0.331234	−	0.943549i	$-0.392535\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.35783e6	−	7.54799e6i	−0.295530	−	0.511873i
$738$		0			0
$739$	−9.77270e6	+	1.69268e7i	−0.658269	+	1.14016i	0.322795	+	0.946469i	$0.395378\pi$
$739$	−9.77270e6	+	1.69268e7i	−0.658269	+	1.14016i	−0.981064	+	0.193686i	$0.937956\pi$
$740$		0			0
$741$	4.17377e6			0.279243
$742$		0			0
$743$	1.54683e7			1.02795			0.513973	−	0.857806i	$-0.328173\pi$
$743$	1.54683e7			1.02795			0.513973	+	0.857806i	$0.328173\pi$
$744$		0			0
$745$	7.75271e6	−	1.34281e7i	0.511756	−	0.886388i
$746$		0			0
$747$	2.47487e6	+	4.28661e6i	0.162275	+	0.281069i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	7.25938e6	+	1.25736e7i	0.469677	+	0.813505i	0.999399	−	0.0346668i	$-0.0110370\pi$
$751$	7.25938e6	+	1.25736e7i	0.469677	+	0.813505i	−0.529722	+	0.848171i	$0.677704\pi$
$752$		0			0
$753$	−1.29994e6	+	2.25157e6i	−0.0835481	+	0.144710i
$754$		0			0
$755$	−272816.			−0.0174182
$756$		0			0
$757$	8.54477e6			0.541952			0.270976	−	0.962586i	$-0.412654\pi$
$757$	8.54477e6			0.541952			0.270976	+	0.962586i	$0.412654\pi$
$758$		0			0
$759$	5.64134e6	−	9.77109e6i	0.355450	−	0.615657i
$760$		0			0
$761$	−4.25199e6	−	7.36466e6i	−0.266153	−	0.460990i	0.701712	−	0.712460i	$-0.252419\pi$
$761$	−4.25199e6	−	7.36466e6i	−0.266153	−	0.460990i	−0.967865	+	0.251471i	$0.919086\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.26959e6	+	2.19900e6i	0.0784352	+	0.135854i
$766$		0			0
$767$	2.40556e7	−	4.16655e7i	1.47648	−	2.55734i
$768$		0			0
$769$	1.66581e7			1.01580			0.507901	−	0.861415i	$-0.330422\pi$
$769$	1.66581e7			1.01580			0.507901	+	0.861415i	$0.330422\pi$
$770$		0			0
$771$	−6.40661e6			−0.388144
$772$		0			0
$773$	1.08663e7	−	1.88210e7i	0.654083	−	1.13290i	−0.328040	−	0.944664i	$-0.606388\pi$
$773$	1.08663e7	−	1.88210e7i	0.654083	−	1.13290i	0.982123	−	0.188241i	$-0.0602786\pi$
$774$		0			0
$775$	9.64022e6	+	1.66974e7i	0.576545	+	0.998604i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.63398e6	+	2.83014e6i	0.0964724	+	0.167095i
$780$		0			0
$781$	1.08630e7	−	1.88153e7i	0.637270	−	1.10378i
$782$		0			0
$783$	−850014.			−0.0495475
$784$		0			0
$785$	−3.74265e6			−0.216773
$786$		0			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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 7.79423i) q^{3} +(-17.0000 - 29.4449i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 7.79423i) q^{3} +(-17.0000 - 29.4449i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(166.000 - 287.520i) q^{11} +1026.00 q^{13} -306.000 q^{15} +(461.000 - 798.475i) q^{17} +(226.000 + 391.443i) q^{19} +(1888.00 + 3270.11i) q^{23} +(984.500 - 1705.20i) q^{25} -729.000 q^{27} +1166.00 q^{29} +(-4896.00 + 8480.12i) q^{31} +(-1494.00 - 2587.68i) q^{33} +(-1195.00 - 2069.80i) q^{37} +(4617.00 - 7996.88i) q^{39} +7230.00 q^{41} +4652.00 q^{43} +(-1377.00 + 2385.03i) q^{45} +(12336.0 + 21366.6i) q^{47} +(-4149.00 - 7186.28i) q^{51} +(-555.000 + 961.288i) q^{53} -11288.0 q^{55} +4068.00 q^{57} +(23446.0 - 40609.7i) q^{59} +(-4881.00 - 8454.14i) q^{61} +(-17442.0 - 30210.4i) q^{65} +(13126.0 - 22734.9i) q^{67} +33984.0 q^{69} +65440.0 q^{71} +(-2803.00 + 4854.94i) q^{73} +(-8860.50 - 15346.8i) q^{75} +(4920.00 + 8521.69i) q^{79} +(-3280.50 + 5681.99i) q^{81} -61108.0 q^{83} -31348.0 q^{85} +(5247.00 - 9088.07i) q^{87} +(-31479.0 - 54523.2i) q^{89} +(44064.0 + 76321.1i) q^{93} +(7684.00 - 13309.1i) q^{95} +37838.0 q^{97} -26892.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{3} - 34 q^{5} - 81 q^{9}+O(q^{10}) \) 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9	\( 2 q + 9 q^{3} - 34 q^{5} - 81 q^{9} + 332 q^{11} + 2052 q^{13} - 612 q^{15} + 922 q^{17} + 452 q^{19} + 3776 q^{23} + 1969 q^{25} - 1458 q^{27} + 2332 q^{29} - 9792 q^{31} - 2988 q^{33} - 2390 q^{37} + 9234 q^{39} + 14460 q^{41} + 9304 q^{43} - 2754 q^{45} + 24672 q^{47} - 8298 q^{51} - 1110 q^{53} - 22576 q^{55} + 8136 q^{57} + 46892 q^{59} - 9762 q^{61} - 34884 q^{65} + 26252 q^{67} + 67968 q^{69} + 130880 q^{71} - 5606 q^{73} - 17721 q^{75} + 9840 q^{79} - 6561 q^{81} - 122216 q^{83} - 62696 q^{85} + 10494 q^{87} - 62958 q^{89} + 88128 q^{93} + 15368 q^{95} + 75676 q^{97} - 53784 q^{99}+O(q^{100}) \) 2 * q + 9 * q^3 - 34 * q^5 - 81 * q^9 + 332 * q^11 + 2052 * q^13 - 612 * q^15 + 922 * q^17 + 452 * q^19 + 3776 * q^23 + 1969 * q^25 - 1458 * q^27 + 2332 * q^29 - 9792 * q^31 - 2988 * q^33 - 2390 * q^37 + 9234 * q^39 + 14460 * q^41 + 9304 * q^43 - 2754 * q^45 + 24672 * q^47 - 8298 * q^51 - 1110 * q^53 - 22576 * q^55 + 8136 * q^57 + 46892 * q^59 - 9762 * q^61 - 34884 * q^65 + 26252 * q^67 + 67968 * q^69 + 130880 * q^71 - 5606 * q^73 - 17721 * q^75 + 9840 * q^79 - 6561 * q^81 - 122216 * q^83 - 62696 * q^85 + 10494 * q^87 - 62958 * q^89 + 88128 * q^93 + 15368 * q^95 + 75676 * q^97 - 53784 * q^99

Introduction

L-functions

Modular forms

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