LMFDB - Embedded newform 2700.3.p.e.1601.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(1601,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2700.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.p (of order $6$, 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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} + \cdots + 43046721 $ x^16 - 2x^15 + 10x^14 - 12x^13 + 6x^12 - 27x^11 - 585x^10 + 3618x^9 - 12150x^8 + 32562x^7 - 47385x^6 - 19683x^5 + 39366x^4 - 708588x^3 + 5314410x^2 - 9565938*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 3^{14} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.6
Root			$-1.82249 + 2.38297i$ of defining polynomial
Character	$\chi$	$=$	2700.1601
Dual form			2700.3.p.e.2501.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.32731 - 4.03103i) q^{7} +O(q^{10})$	$q+(2.32731 - 4.03103i) q^{7} +(12.4900 + 7.21110i) q^{11} +(12.2909 + 21.2885i) q^{13} -6.56655i q^{17} -33.0870 q^{19} +(-31.9638 + 18.4543i) q^{23} +(17.4825 + 10.0935i) q^{29} +(6.48775 + 11.2371i) q^{31} -19.9373 q^{37} +(-44.9196 + 25.9344i) q^{41} +(-3.70519 + 6.41757i) q^{43} +(-23.7134 - 13.6910i) q^{47} +(13.6672 + 23.6723i) q^{49} -79.3730i q^{53} +(-63.3853 + 36.5955i) q^{59} +(-12.3357 + 21.3660i) q^{61} +(20.4754 + 35.4645i) q^{67} -117.149i q^{71} +40.1824 q^{73} +(58.1363 - 33.5650i) q^{77} +(-11.1389 + 19.2931i) q^{79} +(-116.582 - 67.3085i) q^{83} -4.69877i q^{89} +114.419 q^{91} +(69.4664 - 120.319i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{7}+O(q^{10}) $ 16 * q + q^7	$ 16 q + q^{7} + 10 q^{13} + 2 q^{19} + 27 q^{23} - 9 q^{29} + 8 q^{31} + 22 q^{37} - 54 q^{41} - 44 q^{43} - 108 q^{47} - 45 q^{49} - 9 q^{59} - 55 q^{61} + 28 q^{67} - 86 q^{73} + 342 q^{77} + 11 q^{79} - 306 q^{83} - 134 q^{91} - 41 q^{97}+O(q^{100}) $ 16 * q + q^7 + 10 * q^13 + 2 * q^19 + 27 * q^23 - 9 * q^29 + 8 * q^31 + 22 * q^37 - 54 * q^41 - 44 * q^43 - 108 * q^47 - 45 * q^49 - 9 * q^59 - 55 * q^61 + 28 * q^67 - 86 * q^73 + 342 * q^77 + 11 * q^79 - 306 * q^83 - 134 * q^91 - 41 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.32731	−	4.03103i	0.332474	−	0.575861i	−0.650523	−	0.759487i	$-0.725450\pi$
$7$	2.32731	−	4.03103i	0.332474	−	0.575861i	0.982996	+	0.183626i	$0.0587835\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.4900	+	7.21110i	1.13545	+	0.655554i	0.945301	−	0.326200i	$-0.105768\pi$
$11$	12.4900	+	7.21110i	1.13545	+	0.655554i	0.190153	+	0.981755i	$0.439102\pi$
$12$		0			0
$13$	12.2909	+	21.2885i	0.945456	+	1.63758i	0.754835	+	0.655915i	$0.227717\pi$
$13$	12.2909	+	21.2885i	0.945456	+	1.63758i	0.190621	+	0.981664i	$0.438950\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.56655i		−	0.386268i	−0.981172	−	0.193134i	$-0.938135\pi$
$17$		−	6.56655i		−	0.386268i	0.981172	−	0.193134i	$-0.0618652\pi$
$18$		0			0
$19$	−33.0870			−1.74142			−0.870711	−	0.491795i	$-0.836341\pi$
$19$	−33.0870			−1.74142			−0.870711	+	0.491795i	$0.836341\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−31.9638	+	18.4543i	−1.38973	+	0.802361i	−0.993285	−	0.115697i	$-0.963090\pi$
$23$	−31.9638	+	18.4543i	−1.38973	+	0.802361i	−0.396446	+	0.918058i	$0.629757\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	17.4825	+	10.0935i	0.602844	+	0.348052i	0.770160	−	0.637851i	$-0.220176\pi$
$29$	17.4825	+	10.0935i	0.602844	+	0.348052i	−0.167315	+	0.985903i	$0.553510\pi$
$30$		0			0
$31$	6.48775	+	11.2371i	0.209282	+	0.362487i	0.951489	−	0.307684i	$-0.0995540\pi$
$31$	6.48775	+	11.2371i	0.209282	+	0.362487i	−0.742206	+	0.670171i	$0.766221\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−19.9373			−0.538845			−0.269423	−	0.963022i	$-0.586833\pi$
$37$	−19.9373			−0.538845			−0.269423	+	0.963022i	$0.586833\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−44.9196	+	25.9344i	−1.09560	+	0.632546i	−0.935062	−	0.354484i	$-0.884657\pi$
$41$	−44.9196	+	25.9344i	−1.09560	+	0.632546i	−0.160539	+	0.987029i	$0.551323\pi$
$42$		0			0
$43$	−3.70519	+	6.41757i	−0.0861672	+	0.149246i	−0.905888	−	0.423517i	$-0.860795\pi$
$43$	−3.70519	+	6.41757i	−0.0861672	+	0.149246i	0.819721	+	0.572763i	$0.194129\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−23.7134	−	13.6910i	−0.504541	−	0.291297i	0.226046	−	0.974117i	$-0.427420\pi$
$47$	−23.7134	−	13.6910i	−0.504541	−	0.291297i	−0.730587	+	0.682820i	$0.760753\pi$
$48$		0			0
$49$	13.6672	+	23.6723i	0.278923	+	0.483108i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	79.3730i		−	1.49760i	−0.662794	−	0.748802i	$-0.730630\pi$
$53$		−	79.3730i		−	1.49760i	0.662794	−	0.748802i	$-0.269370\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−63.3853	+	36.5955i	−1.07433	+	0.620263i	−0.929360	−	0.369174i	$-0.879641\pi$
$59$	−63.3853	+	36.5955i	−1.07433	+	0.620263i	−0.144966	+	0.989437i	$0.546307\pi$
$60$		0			0
$61$	−12.3357	+	21.3660i	−0.202224	+	0.350263i	−0.949245	−	0.314538i	$-0.898150\pi$
$61$	−12.3357	+	21.3660i	−0.202224	+	0.350263i	0.747020	+	0.664801i	$0.231484\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	20.4754	+	35.4645i	0.305603	+	0.529321i	0.977396	−	0.211419i	$-0.0678085\pi$
$67$	20.4754	+	35.4645i	0.305603	+	0.529321i	−0.671792	+	0.740740i	$0.734475\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	117.149i		−	1.64998i	−0.565145	−	0.824992i	$-0.691180\pi$
$71$		−	117.149i		−	1.64998i	0.565145	−	0.824992i	$-0.308820\pi$
$72$		0			0
$73$	40.1824			0.550444			0.275222	−	0.961381i	$-0.411249\pi$
$73$	40.1824			0.550444			0.275222	+	0.961381i	$0.411249\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	58.1363	−	33.5650i	0.755016	−	0.435909i
$78$		0			0
$79$	−11.1389	+	19.2931i	−0.140998	+	0.244216i	−0.927873	−	0.372897i	$-0.878365\pi$
$79$	−11.1389	+	19.2931i	−0.140998	+	0.244216i	0.786874	+	0.617113i	$0.211698\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−116.582	−	67.3085i	−1.40460	−	0.810945i	−0.409738	−	0.912203i	$-0.634380\pi$
$83$	−116.582	−	67.3085i	−1.40460	−	0.810945i	−0.994860	+	0.101258i	$0.967713\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.69877i		−	0.0527951i	−0.999652	−	0.0263976i	$-0.991596\pi$
$89$		−	4.69877i		−	0.0527951i	0.999652	−	0.0263976i	$-0.00840358\pi$
$90$		0			0
$91$	114.419			1.25736
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	69.4664	−	120.319i	0.716148	−	1.24040i	−0.246367	−	0.969177i	$-0.579237\pi$
$97$	69.4664	−	120.319i	0.716148	−	1.24040i	0.962515	−	0.271228i	$-0.0874298\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−83.0415	−	47.9440i	−0.822193	−	0.474693i	0.0289792	−	0.999580i	$-0.490774\pi$
$101$	−83.0415	−	47.9440i	−0.822193	−	0.474693i	−0.851172	+	0.524887i	$0.824108\pi$
$102$		0			0
$103$	64.4175	+	111.574i	0.625413	+	1.08325i	0.988461	+	0.151476i	$0.0484027\pi$
$103$	64.4175	+	111.574i	0.625413	+	1.08325i	−0.363048	+	0.931770i	$0.618264\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.4712i			0.181974i	0.995852	+	0.0909870i	$0.0290022\pi$
$107$			19.4712i			0.181974i	−0.995852	+	0.0909870i	$0.970998\pi$
$108$		0			0
$109$	−27.4421			−0.251762			−0.125881	−	0.992045i	$-0.540176\pi$
$109$	−27.4421			−0.251762			−0.125881	+	0.992045i	$0.540176\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−47.5616	+	27.4597i	−0.420899	+	0.243006i	−0.695462	−	0.718563i	$-0.744800\pi$
$113$	−47.5616	+	27.4597i	−0.420899	+	0.243006i	0.274563	+	0.961569i	$0.411467\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−26.4699	−	15.2824i	−0.222437	−	0.128424i
$120$		0			0
$121$	43.4999	+	75.3440i	0.359503	+	0.622677i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	93.6328			0.737266			0.368633	−	0.929575i	$-0.379826\pi$
$127$	93.6328			0.737266			0.368633	+	0.929575i	$0.379826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−161.160	+	93.0460i	−1.23023	+	0.710274i	−0.967078	−	0.254480i	$-0.918096\pi$
$131$	−161.160	+	93.0460i	−1.23023	+	0.710274i	−0.263153	+	0.964754i	$0.584762\pi$
$132$		0			0
$133$	−77.0039	+	133.375i	−0.578977	+	1.00282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	61.0773	+	35.2630i	0.445820	+	0.257394i	0.706063	−	0.708149i	$-0.250469\pi$
$137$	61.0773	+	35.2630i	0.445820	+	0.257394i	−0.260243	+	0.965543i	$0.583803\pi$
$138$		0			0
$139$	24.3745	+	42.2179i	0.175356	+	0.303726i	0.940285	−	0.340389i	$-0.110559\pi$
$139$	24.3745	+	42.2179i	0.175356	+	0.303726i	−0.764928	+	0.644116i	$0.777226\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			354.524i			2.47919i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−125.457	+	72.4324i	−0.841990	+	0.486123i	−0.857940	−	0.513750i	$-0.828256\pi$
$149$	−125.457	+	72.4324i	−0.841990	+	0.486123i	0.0159500	+	0.999873i	$0.494923\pi$
$150$		0			0
$151$	−116.217	+	201.293i	−0.769647	+	1.33307i	0.168108	+	0.985769i	$0.446234\pi$
$151$	−116.217	+	201.293i	−0.769647	+	1.33307i	−0.937755	+	0.347299i	$0.887099\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−30.8620	−	53.4545i	−0.196573	−	0.340474i	0.750842	−	0.660482i	$-0.229648\pi$
$157$	−30.8620	−	53.4545i	−0.196573	−	0.340474i	−0.947415	+	0.320007i	$0.896315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			171.796i			1.06706i
$162$		0			0
$163$	53.3693			0.327419			0.163710	−	0.986509i	$-0.447654\pi$
$163$	53.3693			0.327419			0.163710	+	0.986509i	$0.447654\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	116.376	−	67.1898i	0.696863	−	0.402334i	−0.109315	−	0.994007i	$-0.534866\pi$
$167$	116.376	−	67.1898i	0.696863	−	0.402334i	0.806178	+	0.591673i	$0.201532\pi$
$168$		0			0
$169$	−217.634	+	376.953i	−1.28778	+	2.23049i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−204.285	−	117.944i	−1.18084	−	0.681758i	−0.224630	−	0.974444i	$-0.572117\pi$
$173$	−204.285	−	117.944i	−1.18084	−	0.681758i	−0.956208	+	0.292686i	$0.905451\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			324.935i			1.81528i	0.419752	+	0.907639i	$0.362117\pi$
$179$			324.935i			1.81528i	−0.419752	+	0.907639i	$0.637883\pi$
$180$		0			0
$181$	−205.548			−1.13563			−0.567813	−	0.823158i	$-0.692210\pi$
$181$	−205.548			−1.13563			−0.567813	+	0.823158i	$0.692210\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	47.3520	−	82.0161i	0.253219	−	0.438589i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	55.6781	+	32.1458i	0.291508	+	0.168302i	0.638622	−	0.769521i	$-0.279505\pi$
$191$	55.6781	+	32.1458i	0.291508	+	0.168302i	−0.347114	+	0.937823i	$0.612838\pi$
$192$		0			0
$193$	65.9311	+	114.196i	0.341612	+	0.591689i	0.984732	−	0.174076i	$-0.0556939\pi$
$193$	65.9311	+	114.196i	0.341612	+	0.591689i	−0.643120	+	0.765765i	$0.722361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	97.7058i		−	0.495968i	−0.968764	−	0.247984i	$-0.920232\pi$
$197$		−	97.7058i		−	0.495968i	0.968764	−	0.247984i	$-0.0797681\pi$
$198$		0			0
$199$	−67.9123			−0.341268			−0.170634	−	0.985334i	$-0.554582\pi$
$199$	−67.9123			−0.341268			−0.170634	+	0.985334i	$0.554582\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	81.3745	−	46.9816i	0.400859	−	0.231436i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−413.257	−	238.594i	−1.97730	−	1.14160i
$210$		0			0
$211$	4.53364	+	7.85250i	0.0214865	+	0.0372156i	0.876569	−	0.481277i	$-0.159827\pi$
$211$	4.53364	+	7.85250i	0.0214865	+	0.0372156i	−0.855082	+	0.518492i	$0.826493\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	60.3961			0.278323
$218$		0			0
$219$		0			0
$220$		0			0
$221$	139.792	−	80.7090i	0.632544	−	0.365199i
$222$		0			0
$223$	−64.1704	+	111.146i	−0.287760	+	0.498415i	−0.973275	−	0.229644i	$-0.926244\pi$
$223$	−64.1704	+	111.146i	−0.287760	+	0.498415i	0.685515	+	0.728059i	$0.259577\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	179.936	+	103.886i	0.792671	+	0.457649i	0.840902	−	0.541187i	$-0.182025\pi$
$227$	179.936	+	103.886i	0.792671	+	0.457649i	−0.0482311	+	0.998836i	$0.515358\pi$
$228$		0			0
$229$	195.885	+	339.283i	0.855394	+	1.48159i	0.876279	+	0.481803i	$0.160018\pi$
$229$	195.885	+	339.283i	0.855394	+	1.48159i	−0.0208858	+	0.999782i	$0.506649\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	152.713i		−	0.655419i	−0.944779	−	0.327709i	$-0.893723\pi$
$233$		−	152.713i		−	0.655419i	0.944779	−	0.327709i	$-0.106277\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−63.4669	+	36.6426i	−0.265552	+	0.153316i	−0.626865	−	0.779128i	$-0.715662\pi$
$239$	−63.4669	+	36.6426i	−0.265552	+	0.153316i	0.361313	+	0.932445i	$0.382329\pi$
$240$		0			0
$241$	214.984	−	372.364i	0.892051	−	1.54508i	0.0546375	−	0.998506i	$-0.482600\pi$
$241$	214.984	−	372.364i	0.892051	−	1.54508i	0.837413	−	0.546571i	$-0.184067\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−406.670	−	704.374i	−1.64644	−	2.85172i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	165.235i		−	0.658309i	−0.944276	−	0.329154i	$-0.893236\pi$
$251$		−	165.235i		−	0.658309i	0.944276	−	0.329154i	$-0.106764\pi$
$252$		0			0
$253$	−532.303			−2.10397
$254$		0			0
$255$		0			0
$256$		0			0
$257$	405.971	−	234.388i	1.57965	−	0.912014i	0.584747	−	0.811216i	$-0.301194\pi$
$257$	405.971	−	234.388i	1.57965	−	0.912014i	0.994907	−	0.100798i	$-0.0321397\pi$
$258$		0			0
$259$	−46.4003	+	80.3677i	−0.179152	+	0.310300i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	30.6821	+	17.7143i	0.116662	+	0.0673549i	0.557196	−	0.830381i	$-0.311877\pi$
$263$	30.6821	+	17.7143i	0.116662	+	0.0673549i	−0.440533	+	0.897736i	$0.645211\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			141.461i			0.525876i	0.964813	+	0.262938i	$0.0846914\pi$
$269$			141.461i			0.525876i	−0.964813	+	0.262938i	$0.915309\pi$
$270$		0			0
$271$	−372.478			−1.37446			−0.687229	−	0.726441i	$-0.741173\pi$
$271$	−372.478			−1.37446			−0.687229	+	0.726441i	$0.741173\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−129.107	+	223.619i	−0.466089	+	0.807289i	−0.999250	−	0.0387244i	$-0.987671\pi$
$277$	−129.107	+	223.619i	−0.466089	+	0.807289i	0.533161	+	0.846014i	$0.321004\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	283.705	+	163.797i	1.00963	+	0.582908i	0.911082	−	0.412224i	$-0.135248\pi$
$281$	283.705	+	163.797i	1.00963	+	0.582908i	0.0985444	+	0.995133i	$0.468581\pi$
$282$		0			0
$283$	−207.142	−	358.781i	−0.731952	−	1.26778i	−0.956048	−	0.293210i	$-0.905276\pi$
$283$	−207.142	−	358.781i	−0.731952	−	1.26778i	0.224096	−	0.974567i	$-0.428057\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			241.430i			0.841219i
$288$		0			0
$289$	245.880			0.850797
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−359.962	+	207.824i	−1.22854	+	0.709298i	−0.966724	−	0.255820i	$-0.917655\pi$
$293$	−359.962	+	207.824i	−1.22854	+	0.709298i	−0.261816	+	0.965118i	$0.584321\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−785.730	−	453.641i	−2.62786	−	1.51720i
$300$		0			0
$301$	17.2463	+	29.8714i	0.0572966	+	0.0992406i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−91.4750			−0.297964			−0.148982	−	0.988840i	$-0.547600\pi$
$307$	−91.4750			−0.297964			−0.148982	+	0.988840i	$0.547600\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−385.023	+	222.293i	−1.23802	+	0.714769i	−0.968688	−	0.248280i	$-0.920135\pi$
$311$	−385.023	+	222.293i	−1.23802	+	0.714769i	−0.269328	+	0.963049i	$0.586801\pi$
$312$		0			0
$313$	−67.6156	+	117.114i	−0.216024	+	0.374165i	−0.953589	−	0.301111i	$-0.902642\pi$
$313$	−67.6156	+	117.114i	−0.216024	+	0.374165i	0.737565	+	0.675277i	$0.235976\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−297.699	−	171.876i	−0.939113	−	0.542197i	−0.0494307	−	0.998778i	$-0.515741\pi$
$317$	−297.699	−	171.876i	−0.939113	−	0.542197i	−0.889682	+	0.456580i	$0.849074\pi$
$318$		0			0
$319$	145.571	+	252.136i	0.456334	+	0.790394i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			217.268i			0.672655i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−110.377	+	63.7263i	−0.335493	+	0.193697i
$330$		0			0
$331$	−269.871	+	467.430i	−0.815319	+	1.41217i	0.0937793	+	0.995593i	$0.470105\pi$
$331$	−269.871	+	467.430i	−0.815319	+	1.41217i	−0.909098	+	0.416581i	$0.863228\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	274.336	+	475.164i	0.814054	+	1.40998i	0.910005	+	0.414597i	$0.136077\pi$
$337$	274.336	+	475.164i	0.814054	+	1.40998i	−0.0959512	+	0.995386i	$0.530589\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			187.135i			0.548783i
$342$		0			0
$343$	355.308			1.03588
$344$		0			0
$345$		0			0
$346$		0			0
$347$	100.553	−	58.0540i	0.289777	−	0.167303i	−0.348064	−	0.937471i	$-0.613161\pi$
$347$	100.553	−	58.0540i	0.289777	−	0.167303i	0.637841	+	0.770168i	$0.279828\pi$
$348$		0			0
$349$	−119.155	+	206.383i	−0.341419	+	0.591355i	−0.984696	−	0.174278i	$-0.944241\pi$
$349$	−119.155	+	206.383i	−0.341419	+	0.591355i	0.643278	+	0.765633i	$0.277574\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	481.520	+	278.006i	1.36408	+	0.787551i	0.990164	−	0.139912i	$-0.0446819\pi$
$353$	481.520	+	278.006i	1.36408	+	0.787551i	0.373915	+	0.927463i	$0.378015\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	119.443i		−	0.332710i	−0.986066	−	0.166355i	$-0.946800\pi$
$359$		−	119.443i		−	0.332710i	0.986066	−	0.166355i	$-0.0531998\pi$
$360$		0			0
$361$	733.751			2.03255
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	287.355	−	497.713i	0.782983	−	1.35617i	−0.147214	−	0.989105i	$-0.547030\pi$
$367$	287.355	−	497.713i	0.782983	−	1.35617i	0.930197	−	0.367062i	$-0.119636\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−319.955	−	184.726i	−0.862411	−	0.497913i
$372$		0			0
$373$	108.136	+	187.298i	0.289910	+	0.502139i	0.973788	−	0.227458i	$-0.0730413\pi$
$373$	108.136	+	187.298i	0.289910	+	0.502139i	−0.683878	+	0.729596i	$0.739708\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			496.235i			1.31627i
$378$		0			0
$379$	−252.187			−0.665400			−0.332700	−	0.943033i	$-0.607960\pi$
$379$	−252.187			−0.665400			−0.332700	+	0.943033i	$0.607960\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	111.402	−	64.3180i	0.290867	−	0.167932i	−0.347466	−	0.937693i	$-0.612958\pi$
$383$	111.402	−	64.3180i	0.290867	−	0.167932i	0.638333	+	0.769761i	$0.279624\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	214.436	+	123.805i	0.551250	+	0.318264i	0.749626	−	0.661862i	$-0.230233\pi$
$389$	214.436	+	123.805i	0.551250	+	0.318264i	−0.198376	+	0.980126i	$0.563567\pi$
$390$		0			0
$391$	121.181	+	209.892i	0.309926	+	0.536808i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	123.266			0.310493			0.155246	−	0.987876i	$-0.450383\pi$
$397$	123.266			0.310493			0.155246	+	0.987876i	$0.450383\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−397.425	+	229.454i	−0.991085	+	0.572203i	−0.905599	−	0.424136i	$-0.860578\pi$
$401$	−397.425	+	229.454i	−0.991085	+	0.572203i	−0.0854868	+	0.996339i	$0.527245\pi$
$402$		0			0
$403$	−159.481	+	276.229i	−0.395734	+	0.685432i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−249.016	−	143.770i	−0.611833	−	0.353242i
$408$		0			0
$409$	−38.8984	−	67.3740i	−0.0951061	−	0.164729i	0.814547	−	0.580098i	$-0.196986\pi$
$409$	−38.8984	−	67.3740i	−0.0951061	−	0.164729i	−0.909653	+	0.415369i	$0.863652\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			340.677i			0.824884i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−363.032	+	209.597i	−0.866425	+	0.500230i	−0.866158	−	0.499770i	$-0.833418\pi$
$419$	−363.032	+	209.597i	−0.866425	+	0.500230i	−0.000266138		1.00000i	$0.500085\pi$
$420$		0			0
$421$	−41.7402	+	72.2962i	−0.0991454	+	0.171725i	−0.911331	−	0.411674i	$-0.864944\pi$
$421$	−41.7402	+	72.2962i	−0.0991454	+	0.171725i	0.812186	+	0.583399i	$0.198278\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	57.4181	+	99.4510i	0.134469	+	0.232906i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	464.864i		−	1.07857i	−0.842123	−	0.539285i	$-0.818694\pi$
$431$		−	464.864i		−	1.07857i	0.842123	−	0.539285i	$-0.181306\pi$
$432$		0			0
$433$	−201.140			−0.464525			−0.232263	−	0.972653i	$-0.574613\pi$
$433$	−201.140			−0.464525			−0.232263	+	0.972653i	$0.574613\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1057.59	−	610.598i	2.42011	−	1.39725i
$438$		0			0
$439$	−252.879	+	437.999i	−0.576033	+	0.997719i	0.419895	+	0.907573i	$0.362067\pi$
$439$	−252.879	+	437.999i	−0.576033	+	0.997719i	−0.995929	+	0.0901465i	$0.971266\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−197.026	−	113.753i	−0.444755	−	0.256779i	0.260858	−	0.965377i	$-0.415995\pi$
$443$	−197.026	−	113.753i	−0.444755	−	0.256779i	−0.705612	+	0.708598i	$0.749328\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			224.825i			0.500724i	0.968152	+	0.250362i	$0.0805496\pi$
$449$			224.825i			0.500724i	−0.968152	+	0.250362i	$0.919450\pi$
$450$		0			0
$451$	−748.061			−1.65867
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	308.178	−	533.780i	0.674350	−	1.16801i	−0.302309	−	0.953210i	$-0.597757\pi$
$457$	308.178	−	533.780i	0.674350	−	1.16801i	0.976659	−	0.214798i	$-0.0689093\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	616.599	+	355.994i	1.33753	+	0.772221i	0.986440	−	0.164122i	$-0.0524791\pi$
$461$	616.599	+	355.994i	1.33753	+	0.772221i	0.351086	+	0.936343i	$0.385812\pi$
$462$		0			0
$463$	126.318	+	218.788i	0.272824	+	0.472545i	0.969584	−	0.244759i	$-0.0787090\pi$
$463$	126.318	+	218.788i	0.272824	+	0.472545i	−0.696760	+	0.717304i	$0.745376\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			508.789i			1.08948i	0.838604	+	0.544742i	$0.183372\pi$
$467$			508.789i			1.08948i	−0.838604	+	0.544742i	$0.816628\pi$
$468$		0			0
$469$	190.611			0.406420
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−92.5555	+	53.4370i	−0.195678	+	0.112975i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	683.840	+	394.815i	1.42764	+	0.824249i	0.996934	−	0.0782437i	$-0.0249312\pi$
$479$	683.840	+	394.815i	1.42764	+	0.824249i	0.430706	+	0.902492i	$0.358265\pi$
$480$		0			0
$481$	−245.048	−	424.435i	−0.509454	−	0.882401i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−317.579			−0.652112			−0.326056	−	0.945350i	$-0.605720\pi$
$487$	−317.579			−0.652112			−0.326056	+	0.945350i	$0.605720\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	597.835	−	345.160i	1.21759	−	0.702973i	0.253185	−	0.967418i	$-0.418522\pi$
$491$	597.835	−	345.160i	1.21759	−	0.702973i	0.964401	+	0.264444i	$0.0851886\pi$
$492$		0			0
$493$	66.2796	−	114.800i	0.134441	−	0.232859i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−472.230	−	272.642i	−0.950161	−	0.548576i
$498$		0			0
$499$	25.6761	+	44.4723i	0.0514550	+	0.0891228i	0.890606	−	0.454776i	$-0.150281\pi$
$499$	25.6761	+	44.4723i	0.0514550	+	0.0891228i	−0.839151	+	0.543899i	$0.816947\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	239.928i		−	0.476994i	−0.971143	−	0.238497i	$-0.923345\pi$
$503$		−	239.928i		−	0.476994i	0.971143	−	0.238497i	$-0.0766547\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	440.034	−	254.054i	0.864507	−	0.499123i	−0.00101197	−	0.999999i	$-0.500322\pi$
$509$	440.034	−	254.054i	0.864507	−	0.499123i	0.865519	+	0.500876i	$0.166989\pi$
$510$		0			0
$511$	93.5171	−	161.976i	0.183008	−	0.316979i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−197.454	−	342.000i	−0.381922	−	0.661508i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			66.1586i			0.126984i	0.997982	+	0.0634920i	$0.0202237\pi$
$521$			66.1586i			0.126984i	−0.997982	+	0.0634920i	$0.979776\pi$
$522$		0			0
$523$	−133.503			−0.255264			−0.127632	−	0.991822i	$-0.540738\pi$
$523$	−133.503			−0.255264			−0.127632	+	0.991822i	$0.540738\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	73.7890	−	42.6021i	0.140017	−	0.0808389i
$528$		0			0
$529$	416.623	−	721.613i	0.787568	−	1.36411i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1104.21	−	637.515i	−2.07169	−	1.19609i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			394.222i			0.731396i
$540$		0			0
$541$	263.866			0.487737			0.243868	−	0.969808i	$-0.421584\pi$
$541$	263.866			0.487737			0.243868	+	0.969808i	$0.421584\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−302.353	+	523.690i	−0.552747	+	0.957386i	0.445328	+	0.895368i	$0.353087\pi$
$547$	−302.353	+	523.690i	−0.552747	+	0.957386i	−0.998075	+	0.0620188i	$0.980246\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−578.443	−	333.964i	−1.04981	−	0.606106i
$552$		0			0
$553$	51.8473	+	89.8022i	0.0937565	+	0.162391i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			510.522i			0.916556i	0.888809	+	0.458278i	$0.151534\pi$
$557$			510.522i			0.916556i	−0.888809	+	0.458278i	$0.848466\pi$
$558$		0			0
$559$	−182.161			−0.325869
$560$		0			0
$561$		0			0
$562$		0			0
$563$	333.774	−	192.705i	0.592849	−	0.342282i	−0.173374	−	0.984856i	$-0.555467\pi$
$563$	333.774	−	192.705i	0.592849	−	0.342282i	0.766223	+	0.642574i	$0.222134\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−192.215	−	110.975i	−0.337812	−	0.195036i	0.321492	−	0.946912i	$-0.395816\pi$
$569$	−192.215	−	110.975i	−0.337812	−	0.195036i	−0.659304	+	0.751877i	$0.729149\pi$
$570$		0			0
$571$	−187.067	−	324.010i	−0.327613	−	0.567443i	0.654425	−	0.756127i	$-0.272911\pi$
$571$	−187.067	−	324.010i	−0.327613	−	0.567443i	−0.982038	+	0.188685i	$0.939578\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	237.121			0.410955			0.205478	−	0.978662i	$-0.434125\pi$
$577$	237.121			0.410955			0.205478	+	0.978662i	$0.434125\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−542.645	+	313.296i	−0.933984	+	0.539236i
$582$		0			0
$583$	572.366	−	991.367i	0.981760	−	1.70046i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−820.431	−	473.676i	−1.39767	−	0.806944i	−0.403519	−	0.914971i	$-0.632213\pi$
$587$	−820.431	−	473.676i	−1.39767	−	0.806944i	−0.994148	+	0.108028i	$0.965547\pi$
$588$		0			0
$589$	−214.660	−	371.802i	−0.364449	−	0.631243i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	744.224i		−	1.25502i	−0.778610	−	0.627508i	$-0.784075\pi$
$593$		−	744.224i		−	1.25502i	0.778610	−	0.627508i	$-0.215925\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	532.457	−	307.414i	0.888910	−	0.513212i	0.0153242	−	0.999883i	$-0.495122\pi$
$599$	532.457	−	307.414i	0.888910	−	0.513212i	0.873586	+	0.486670i	$0.161789\pi$
$600$		0			0
$601$	−156.574	+	271.195i	−0.260523	+	0.451239i	−0.966381	−	0.257114i	$-0.917228\pi$
$601$	−156.574	+	271.195i	−0.260523	+	0.451239i	0.705858	+	0.708353i	$0.250562\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	485.085	+	840.192i	0.799152	+	1.38417i	0.920169	+	0.391521i	$0.128051\pi$
$607$	485.085	+	840.192i	0.799152	+	1.38417i	−0.121018	+	0.992650i	$0.538616\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	673.099i		−	1.10163i
$612$		0			0
$613$	758.446			1.23727			0.618635	−	0.785679i	$-0.287686\pi$
$613$	758.446			1.23727			0.618635	+	0.785679i	$0.287686\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−270.760	+	156.323i	−0.438833	+	0.253360i	−0.703102	−	0.711089i	$-0.748202\pi$
$617$	−270.760	+	156.323i	−0.438833	+	0.253360i	0.264270	+	0.964449i	$0.414869\pi$
$618$		0			0
$619$	488.656	−	846.377i	0.789428	−	1.36733i	−0.136889	−	0.990586i	$-0.543710\pi$
$619$	488.656	−	846.377i	0.789428	−	1.36733i	0.926318	−	0.376744i	$-0.122956\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.9409	−	10.9355i	−0.0304027	−	0.0175530i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			130.919i			0.208138i
$630$		0			0
$631$	904.406			1.43329			0.716645	−	0.697438i	$-0.245677\pi$
$631$	904.406			1.43329			0.716645	+	0.697438i	$0.245677\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−335.966	+	581.910i	−0.527419	+	0.913516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	764.357	+	441.302i	1.19245	+	0.688459i	0.958860	−	0.283878i	$-0.0916210\pi$
$641$	764.357	+	441.302i	1.19245	+	0.688459i	0.233585	+	0.972336i	$0.424954\pi$
$642$		0			0
$643$	285.894	+	495.184i	0.444626	+	0.770114i	0.998026	−	0.0628010i	$-0.0200033\pi$
$643$	285.894	+	495.184i	0.444626	+	0.770114i	−0.553400	+	0.832915i	$0.686670\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	113.496i		−	0.175419i	−0.996146	−	0.0877096i	$-0.972045\pi$
$647$		−	113.496i		−	0.175419i	0.996146	−	0.0877096i	$-0.0279547\pi$
$648$		0			0
$649$	−1055.58			−1.62646
$650$		0			0
$651$		0			0
$652$		0			0
$653$	880.891	−	508.582i	1.34899	−	0.778840i	0.360884	−	0.932611i	$-0.382475\pi$
$653$	880.891	−	508.582i	1.34899	−	0.778840i	0.988106	+	0.153771i	$0.0491418\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−697.672	−	402.801i	−1.05868	−	0.611231i	−0.133615	−	0.991033i	$-0.542659\pi$
$659$	−697.672	−	402.801i	−1.05868	−	0.611231i	−0.925068	+	0.379802i	$0.875992\pi$
$660$		0			0
$661$	491.116	+	850.638i	0.742989	+	1.28690i	0.951128	+	0.308796i	$0.0999261\pi$
$661$	491.116	+	850.638i	0.742989	+	1.28690i	−0.208139	+	0.978099i	$0.566741\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−745.076			−1.11705
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−308.145	+	177.908i	−0.459233	+	0.265138i
$672$		0			0
$673$	−142.183	+	246.268i	−0.211267	+	0.365926i	−0.952111	−	0.305751i	$-0.901092\pi$
$673$	−142.183	+	246.268i	−0.211267	+	0.365926i	0.740844	+	0.671677i	$0.234426\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	222.141	+	128.253i	0.328125	+	0.189443i	0.655009	−	0.755621i	$-0.272665\pi$
$677$	222.141	+	128.253i	0.328125	+	0.189443i	−0.326883	+	0.945065i	$0.605998\pi$
$678$		0			0
$679$	−323.340	−	560.042i	−0.476200	−	0.824803i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1126.46i			1.64928i	0.565660	+	0.824639i	$0.308622\pi$
$683$			1126.46i			1.64928i	−0.565660	+	0.824639i	$0.691378\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1689.73	−	975.568i	2.45244	−	1.41592i
$690$		0			0
$691$	144.035	−	249.476i	0.208444	−	0.361036i	−0.742780	−	0.669535i	$-0.766493\pi$
$691$	144.035	−	249.476i	0.208444	−	0.361036i	0.951225	+	0.308499i	$0.0998267\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	170.299	+	294.967i	0.244332	+	0.423195i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	854.593i		−	1.21911i	−0.792745	−	0.609553i	$-0.791349\pi$
$701$		−	854.593i		−	1.21911i	0.792745	−	0.609553i	$-0.208651\pi$
$702$		0			0
$703$	659.665			0.938357
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−386.527	+	223.162i	−0.546715	+	0.315646i
$708$		0			0
$709$	183.387	−	317.636i	0.258656	−	0.448006i	−0.707226	−	0.706988i	$-0.750054\pi$
$709$	183.387	−	317.636i	0.258656	−	0.448006i	0.965882	+	0.258982i	$0.0833870\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−414.746	−	239.454i	−0.581692	−	0.335840i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			167.078i			0.232376i	0.993227	+	0.116188i	$0.0370675\pi$
$719$			167.078i			0.232376i	−0.993227	+	0.116188i	$0.962932\pi$
$720$		0			0
$721$	599.679			0.831733
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−330.352	+	572.186i	−0.454404	+	0.787051i	−0.998654	−	0.0518723i	$-0.983481\pi$
$727$	−330.352	+	572.186i	−0.454404	+	0.787051i	0.544250	+	0.838923i	$0.316814\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	42.1413	+	24.3303i	0.0576489	+	0.0332836i
$732$		0			0
$733$	−13.6300	−	23.6078i	−0.0185947	−	0.0322071i	0.856578	−	0.516017i	$-0.172586\pi$
$733$	−13.6300	−	23.6078i	−0.0185947	−	0.0322071i	−0.875173	+	0.483810i	$0.839253\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			590.601i			0.801359i
$738$		0			0
$739$	549.148			0.743096			0.371548	−	0.928414i	$-0.378827\pi$
$739$	549.148			0.743096			0.371548	+	0.928414i	$0.378827\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−152.537	+	88.0673i	−0.205299	+	0.118529i	−0.599125	−	0.800656i	$-0.704485\pi$
$743$	−152.537	+	88.0673i	−0.205299	+	0.118529i	0.393826	+	0.919185i	$0.371151\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	78.4890	+	45.3157i	0.104792	+	0.0605016i
$750$		0			0
$751$	108.231	+	187.462i	0.144116	+	0.249616i	0.929043	−	0.369972i	$-0.120633\pi$
$751$	108.231	+	187.462i	0.144116	+	0.249616i	−0.784927	+	0.619589i	$0.787299\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1104.91			1.45959			0.729793	−	0.683668i	$-0.239616\pi$
$757$	1104.91			1.45959			0.729793	+	0.683668i	$0.239616\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−774.245	+	447.011i	−1.01741	+	0.587399i	−0.913351	−	0.407173i	$-0.866515\pi$
$761$	−774.245	+	447.011i	−1.01741	+	0.587399i	−0.104054	+	0.994572i	$0.533181\pi$
$762$		0			0
$763$	−63.8663	+	110.620i	−0.0837042	+	0.144980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1558.13	−	899.586i	−2.03146	−	1.17286i
$768$		0			0
$769$	−287.360	−	497.722i	−0.373680	−	0.647233i	0.616448	−	0.787395i	$-0.288571\pi$
$769$	−287.360	−	497.722i	−0.373680	−	0.647233i	−0.990129	+	0.140162i	$0.955238\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			71.2870i			0.0922212i	0.998936	+	0.0461106i	$0.0146827\pi$
$773$			71.2870i			0.0922212i	−0.998936	+	0.0461106i	$0.985317\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1486.26	−	858.091i	1.90790	−	1.10153i
$780$		0			0
$781$	844.772	−	1463.19i	1.08165	−	1.87348i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−17.4363	−	30.2005i	−0.0221554	−	0.0383742i	0.854735	−	0.519064i	$-0.173720\pi$
$787$	−17.4363	−	30.2005i	−0.0221554	−	0.0383742i	−0.876891	+	0.480690i	$0.840386\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			255.629i			0.323172i
$792$		0			0
$793$	−606.469			−0.764777
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−782.440	+	451.742i	−0.981731	+	0.566803i	−0.902792	−	0.430077i	$-0.858487\pi$
$797$	−782.440	+	451.742i	−0.981731	+	0.566803i	−0.0789390	+	0.996879i	$0.525153\pi$
$798$		0			0
$799$	−89.9024	+	155.716i	−0.112519	+	0.194888i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	501.878	+	289.759i	0.625004	+	0.360846i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1019.43i			1.26011i	0.776551	+	0.630054i	$0.216967\pi$
$809$			1019.43i			1.26011i	−0.776551	+	0.630054i	$0.783033\pi$
$810$		0			0
$811$	−1022.62			−1.26093			−0.630467	−	0.776216i	$-0.717136\pi$
$811$	−1022.62			−1.26093			−0.630467	+	0.776216i	$0.717136\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	122.594	−	212.338i	0.150053	−	0.259900i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1352.85	+	781.066i	1.64780	+	0.951359i	0.977943	+	0.208871i	$0.0669789\pi$
$821$	1352.85	+	781.066i	1.64780	+	0.951359i	0.669859	+	0.742488i	$0.266354\pi$
$822$		0			0
$823$	−707.151	−	1224.82i	−0.859236	−	1.48824i	−0.872659	−	0.488331i	$-0.837606\pi$
$823$	−707.151	−	1224.82i	−0.859236	−	1.48824i	0.0134224	−	0.999910i	$-0.495727\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	949.074i		−	1.14761i	−0.818992	−	0.573805i	$-0.805467\pi$
$827$		−	949.074i		−	1.14761i	0.818992	−	0.573805i	$-0.194533\pi$
$828$		0			0
$829$	1144.99			1.38118			0.690588	−	0.723249i	$-0.257352\pi$
$829$	1144.99			1.38118			0.690588	+	0.723249i	$0.257352\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	155.445	−	89.7465i	0.186609	−	0.107739i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	218.826	+	126.340i	0.260818	+	0.150583i	0.624708	−	0.780859i	$-0.285218\pi$
$839$	218.826	+	126.340i	0.260818	+	0.150583i	−0.363890	+	0.931442i	$0.618551\pi$
$840$		0			0
$841$	−216.742	−	375.408i	−0.257719	−	0.446383i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	404.951			0.478101
$848$		0			0
$849$		0			0
$850$		0			0
$851$	637.271	−	367.929i	0.748849	−	0.432348i
$852$		0			0
$853$	−136.549	+	236.510i	−0.160081	+	0.277269i	−0.934898	−	0.354917i	$-0.884509\pi$
$853$	−136.549	+	236.510i	−0.160081	+	0.277269i	0.774816	+	0.632186i	$0.217842\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	358.085	+	206.740i	0.417835	+	0.241237i	0.694151	−	0.719830i	$-0.255780\pi$
$857$	358.085	+	206.740i	0.417835	+	0.241237i	−0.276315	+	0.961067i	$0.589113\pi$
$858$		0			0
$859$	−664.377	−	1150.73i	−0.773431	−	1.33962i	−0.935672	−	0.352870i	$-0.885206\pi$
$859$	−664.377	−	1150.73i	−0.773431	−	1.33962i	0.162242	−	0.986751i	$-0.448128\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			283.552i			0.328565i	0.986413	+	0.164283i	$0.0525309\pi$
$863$			283.552i			0.328565i	−0.986413	+	0.164283i	$0.947469\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−278.249	+	160.647i	−0.320194	+	0.184864i
$870$		0			0
$871$	−503.324	+	871.783i	−0.577870	+	1.00090i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−229.117	−	396.843i	−0.261251	−	0.452500i	0.705323	−	0.708886i	$-0.250802\pi$
$877$	−229.117	−	396.843i	−0.261251	−	0.452500i	−0.966575	+	0.256385i	$0.917468\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1408.20i		−	1.59841i	−0.601056	−	0.799207i	$-0.705253\pi$
$881$		−	1408.20i		−	1.59841i	0.601056	−	0.799207i	$-0.294747\pi$
$882$		0			0
$883$	−200.279			−0.226817			−0.113408	−	0.993548i	$-0.536177\pi$
$883$	−200.279			−0.226817			−0.113408	+	0.993548i	$0.536177\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1026.67	−	592.748i	1.15746	−	0.668262i	0.206769	−	0.978390i	$-0.433705\pi$
$887$	1026.67	−	592.748i	1.15746	−	0.668262i	0.950695	+	0.310128i	$0.100372\pi$
$888$		0			0
$889$	217.913	−	377.436i	0.245121	−	0.424563i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	784.607	+	452.993i	0.878620	+	0.507271i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			261.937i			0.291364i
$900$		0			0
$901$	−521.207			−0.578476
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−665.240	+	1152.23i	−0.733451	+	1.27037i	0.221949	+	0.975058i	$0.428758\pi$
$907$	−665.240	+	1152.23i	−0.733451	+	1.27037i	−0.955400	+	0.295316i	$0.904575\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−311.564	−	179.882i	−0.342003	−	0.197455i	0.319155	−	0.947703i	$-0.396601\pi$
$911$	−311.564	−	179.882i	−0.342003	−	0.197455i	−0.661157	+	0.750247i	$0.729934\pi$
$912$		0			0
$913$	−970.736	−	1681.36i	−1.06324	−	1.84158i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			866.189i			0.944590i
$918$		0			0
$919$	114.308			0.124383			0.0621913	−	0.998064i	$-0.480191\pi$
$919$	114.308			0.124383			0.0621913	+	0.998064i	$0.480191\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2493.93	−	1439.87i	2.70198	−	1.55999i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	692.105	+	399.587i	0.745000	+	0.430126i	0.823885	−	0.566758i	$-0.191802\pi$
$929$	692.105	+	399.587i	0.745000	+	0.430126i	−0.0788843	+	0.996884i	$0.525136\pi$
$930$		0			0
$931$	−452.207	−	783.246i	−0.485722	−	0.841296i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−937.387			−1.00041			−0.500206	−	0.865906i	$-0.666743\pi$
$937$	−937.387			−1.00041			−0.500206	+	0.865906i	$0.666743\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	490.911	−	283.428i	0.521691	−	0.301198i	−0.215935	−	0.976408i	$-0.569280\pi$
$941$	490.911	−	283.428i	0.521691	−	0.301198i	0.737626	+	0.675209i	$0.235947\pi$
$942$		0			0
$943$	957.202	−	1657.92i	1.01506	−	1.75814i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−619.197	−	357.494i	−0.653851	−	0.377501i	0.136079	−	0.990698i	$-0.456550\pi$
$947$	−619.197	−	357.494i	−0.653851	−	0.377501i	−0.789930	+	0.613197i	$0.789883\pi$
$948$		0			0
$949$	493.880	+	855.424i	0.520421	+	0.901396i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	918.880i		−	0.964197i	−0.876117	−	0.482099i	$-0.839875\pi$
$953$		−	918.880i		−	0.964197i	0.876117	−	0.482099i	$-0.160125\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	284.292	−	164.136i	0.296447	−	0.171154i
$960$		0			0
$961$	396.318	−	686.443i	0.412402	−	0.714301i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−9.16023	−	15.8660i	−0.00947283	−	0.0164074i	0.861250	−	0.508181i	$-0.169682\pi$
$967$	−9.16023	−	15.8660i	−0.00947283	−	0.0164074i	−0.870723	+	0.491774i	$0.836349\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	35.9529i		−	0.0370266i	−0.999829	−	0.0185133i	$-0.994107\pi$
$971$		−	35.9529i		−	0.0370266i	0.999829	−	0.0185133i	$-0.00589331\pi$
$972$		0			0
$973$	226.909			0.233205
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−486.908	+	281.116i	−0.498370	+	0.287734i	−0.728040	−	0.685534i	$-0.759569\pi$
$977$	−486.908	+	281.116i	−0.498370	+	0.287734i	0.229670	+	0.973269i	$0.426235\pi$
$978$		0			0
$979$	33.8833	−	58.6875i	0.0346101	−	0.0599464i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	418.957	+	241.885i	0.426202	+	0.246068i	0.697727	−	0.716363i	$-0.254195\pi$
$983$	418.957	+	241.885i	0.426202	+	0.246068i	−0.271525	+	0.962431i	$0.587528\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	273.507i		−	0.276549i
$990$		0			0
$991$	−1071.66			−1.08139			−0.540695	−	0.841219i	$-0.681839\pi$
$991$	−1071.66			−1.08139			−0.540695	+	0.841219i	$0.681839\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	155.387	−	269.139i	0.155855	−	0.269949i	−0.777515	−	0.628864i	$-0.783520\pi$
$997$	155.387	−	269.139i	0.155855	−	0.269949i	0.933370	+	0.358916i	$0.116853\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.p.e.1601.6		16
3.2	odd	2		900.3.p.d.101.1	✓	16
5.2	odd	4		2700.3.u.d.2249.11		32
5.3	odd	4		2700.3.u.d.2249.6		32
5.4	even	2		2700.3.p.d.1601.3		16
9.4	even	3		900.3.p.d.401.1	yes	16
9.5	odd	6	inner	2700.3.p.e.2501.6		16
15.2	even	4		900.3.u.d.749.8		32
15.8	even	4		900.3.u.d.749.9		32
15.14	odd	2		900.3.p.e.101.8	yes	16
45.4	even	6		900.3.p.e.401.8	yes	16
45.13	odd	12		900.3.u.d.149.8		32
45.14	odd	6		2700.3.p.d.2501.3		16
45.22	odd	12		900.3.u.d.149.9		32
45.23	even	12		2700.3.u.d.449.11		32
45.32	even	12		2700.3.u.d.449.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.p.d.101.1	✓	16	3.2	odd	2
900.3.p.d.401.1	yes	16	9.4	even	3
900.3.p.e.101.8	yes	16	15.14	odd	2
900.3.p.e.401.8	yes	16	45.4	even	6
900.3.u.d.149.8		32	45.13	odd	12
900.3.u.d.149.9		32	45.22	odd	12
900.3.u.d.749.8		32	15.2	even	4
900.3.u.d.749.9		32	15.8	even	4
2700.3.p.d.1601.3		16	5.4	even	2
2700.3.p.d.2501.3		16	45.14	odd	6
2700.3.p.e.1601.6		16	1.1	even	1	trivial
2700.3.p.e.2501.6		16	9.5	odd	6	inner
2700.3.u.d.449.6		32	45.32	even	12
2700.3.u.d.449.11		32	45.23	even	12
2700.3.u.d.2249.6		32	5.3	odd	4
2700.3.u.d.2249.11		32	5.2	odd	4

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.32731 - 4.03103i) q^{7} +O(q^{10})\)	\(q+(2.32731 - 4.03103i) q^{7} +(12.4900 + 7.21110i) q^{11} +(12.2909 + 21.2885i) q^{13} -6.56655i q^{17} -33.0870 q^{19} +(-31.9638 + 18.4543i) q^{23} +(17.4825 + 10.0935i) q^{29} +(6.48775 + 11.2371i) q^{31} -19.9373 q^{37} +(-44.9196 + 25.9344i) q^{41} +(-3.70519 + 6.41757i) q^{43} +(-23.7134 - 13.6910i) q^{47} +(13.6672 + 23.6723i) q^{49} -79.3730i q^{53} +(-63.3853 + 36.5955i) q^{59} +(-12.3357 + 21.3660i) q^{61} +(20.4754 + 35.4645i) q^{67} -117.149i q^{71} +40.1824 q^{73} +(58.1363 - 33.5650i) q^{77} +(-11.1389 + 19.2931i) q^{79} +(-116.582 - 67.3085i) q^{83} -4.69877i q^{89} +114.419 q^{91} +(69.4664 - 120.319i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{7}+O(q^{10}) \) 16 * q + q^7	\( 16 q + q^{7} + 10 q^{13} + 2 q^{19} + 27 q^{23} - 9 q^{29} + 8 q^{31} + 22 q^{37} - 54 q^{41} - 44 q^{43} - 108 q^{47} - 45 q^{49} - 9 q^{59} - 55 q^{61} + 28 q^{67} - 86 q^{73} + 342 q^{77} + 11 q^{79} - 306 q^{83} - 134 q^{91} - 41 q^{97}+O(q^{100}) \) 16 * q + q^7 + 10 * q^13 + 2 * q^19 + 27 * q^23 - 9 * q^29 + 8 * q^31 + 22 * q^37 - 54 * q^41 - 44 * q^43 - 108 * q^47 - 45 * q^49 - 9 * q^59 - 55 * q^61 + 28 * q^67 - 86 * q^73 + 342 * q^77 + 11 * q^79 - 306 * q^83 - 134 * q^91 - 41 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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