LMFDB - Embedded newform 1764.3.z.l.901.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.4
Root			$-1.60021 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1764.901
Dual form			1764.3.z.l.325.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.04718 + 2.91399i) q^{5} +O(q^{10})$	$q+(5.04718 + 2.91399i) q^{5} +(-3.43068 - 5.94212i) q^{11} -3.62063i q^{13} +(-8.41362 + 4.85761i) q^{17} +(-26.2322 - 15.1451i) q^{19} +(-9.07789 + 15.7234i) q^{23} +(4.48269 + 7.76425i) q^{25} +40.4570 q^{29} +(-47.8153 + 27.6062i) q^{31} +(-27.4873 + 47.6093i) q^{37} +56.3322i q^{41} -66.0512 q^{43} +(42.8003 + 24.7108i) q^{47} +(-40.5081 - 70.1621i) q^{53} -39.9879i q^{55} +(-30.1302 + 17.3957i) q^{59} +(0.0331519 + 0.0191403i) q^{61} +(10.5505 - 18.2740i) q^{65} +(32.0449 + 55.5034i) q^{67} -50.2730 q^{71} +(18.4865 - 10.6732i) q^{73} +(23.7522 - 41.1400i) q^{79} +33.6039i q^{83} -56.6201 q^{85} +(135.180 + 78.0459i) q^{89} +(-88.2656 - 152.881i) q^{95} -43.7452i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{17} - 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} + 264 q^{47} - 72 q^{53} - 168 q^{59} - 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} + 336 q^{73} + 216 q^{79} - 96 q^{85} - 96 q^{89} - 136 q^{95}+O(q^{100}) $ 8 * q - 48 * q^17 - 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 + 48 * q^31 - 64 * q^37 - 112 * q^43 + 264 * q^47 - 72 * q^53 - 168 * q^59 - 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 + 336 * q^73 + 216 * q^79 - 96 * q^85 - 96 * q^89 - 136 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.04718	+	2.91399i	1.00944	+	0.582798i	0.911027	−	0.412348i	$-0.135291\pi$
$5$	5.04718	+	2.91399i	1.00944	+	0.582798i	0.0984097	+	0.995146i	$0.468624\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.43068	−	5.94212i	−0.311880	−	0.540193i	0.666889	−	0.745157i	$-0.267626\pi$
$11$	−3.43068	−	5.94212i	−0.311880	−	0.540193i	−0.978769	+	0.204964i	$0.934292\pi$
$12$		0			0
$13$		−	3.62063i		−	0.278510i	−0.990257	−	0.139255i	$-0.955529\pi$
$13$		−	3.62063i		−	0.278510i	0.990257	−	0.139255i	$-0.0444708\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.41362	+	4.85761i	−0.494919	+	0.285742i	−0.726613	−	0.687047i	$-0.758907\pi$
$17$	−8.41362	+	4.85761i	−0.494919	+	0.285742i	0.231694	+	0.972789i	$0.425573\pi$
$18$		0			0
$19$	−26.2322	−	15.1451i	−1.38064	−	0.797113i	−0.388405	−	0.921489i	$-0.626974\pi$
$19$	−26.2322	−	15.1451i	−1.38064	−	0.797113i	−0.992235	+	0.124376i	$0.960307\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.07789	+	15.7234i	−0.394691	+	0.683625i	−0.993062	−	0.117595i	$-0.962482\pi$
$23$	−9.07789	+	15.7234i	−0.394691	+	0.683625i	0.598371	+	0.801219i	$0.295815\pi$
$24$		0			0
$25$	4.48269	+	7.76425i	0.179308	+	0.310570i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	40.4570			1.39507			0.697534	−	0.716552i	$-0.254281\pi$
$29$	40.4570			1.39507			0.697534	+	0.716552i	$0.254281\pi$
$30$		0			0
$31$	−47.8153	+	27.6062i	−1.54243	+	0.890522i	−0.543744	+	0.839251i	$0.682994\pi$
$31$	−47.8153	+	27.6062i	−1.54243	+	0.890522i	−0.998685	+	0.0512709i	$0.983673\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−27.4873	+	47.6093i	−0.742899	+	1.28674i	0.208271	+	0.978071i	$0.433216\pi$
$37$	−27.4873	+	47.6093i	−0.742899	+	1.28674i	−0.951170	+	0.308668i	$0.900117\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			56.3322i			1.37396i	0.726678	+	0.686978i	$0.241063\pi$
$41$			56.3322i			1.37396i	−0.726678	+	0.686978i	$0.758937\pi$
$42$		0			0
$43$	−66.0512			−1.53608			−0.768038	−	0.640405i	$-0.778767\pi$
$43$	−66.0512			−1.53608			−0.768038	+	0.640405i	$0.778767\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.8003	+	24.7108i	0.910645	+	0.525761i	0.880639	−	0.473789i	$-0.157114\pi$
$47$	42.8003	+	24.7108i	0.910645	+	0.525761i	0.0300061	+	0.999550i	$0.490447\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−40.5081	−	70.1621i	−0.764303	−	1.32381i	−0.940614	−	0.339478i	$-0.889750\pi$
$53$	−40.5081	−	70.1621i	−0.764303	−	1.32381i	0.176311	−	0.984335i	$-0.443584\pi$
$54$		0			0
$55$		−	39.9879i		−	0.727054i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−30.1302	+	17.3957i	−0.510682	+	0.294842i	−0.733114	−	0.680106i	$-0.761934\pi$
$59$	−30.1302	+	17.3957i	−0.510682	+	0.294842i	0.222432	+	0.974948i	$0.428600\pi$
$60$		0			0
$61$	0.0331519	+	0.0191403i	0.000543474	+	0.000313775i	0.500272	−	0.865868i	$-0.333233\pi$
$61$	0.0331519	+	0.0191403i	0.000543474	+	0.000313775i	−0.499728	+	0.866182i	$0.666567\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.5505	−	18.2740i	0.162315	−	0.281138i
$66$		0			0
$67$	32.0449	+	55.5034i	0.478282	+	0.828409i	0.999690	−	0.0248985i	$-0.00792626\pi$
$67$	32.0449	+	55.5034i	0.478282	+	0.828409i	−0.521408	+	0.853308i	$0.674593\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−50.2730			−0.708070			−0.354035	−	0.935232i	$-0.615191\pi$
$71$	−50.2730			−0.708070			−0.354035	+	0.935232i	$0.615191\pi$
$72$		0			0
$73$	18.4865	−	10.6732i	0.253240	−	0.146208i	−0.368007	−	0.929823i	$-0.619960\pi$
$73$	18.4865	−	10.6732i	0.253240	−	0.146208i	0.621247	+	0.783615i	$0.286626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	23.7522	−	41.1400i	0.300660	−	0.520759i	−0.675625	−	0.737245i	$-0.736126\pi$
$79$	23.7522	−	41.1400i	0.300660	−	0.520759i	0.976286	+	0.216486i	$0.0694596\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			33.6039i			0.404866i	0.979296	+	0.202433i	$0.0648849\pi$
$83$			33.6039i			0.404866i	−0.979296	+	0.202433i	$0.935115\pi$
$84$		0			0
$85$	−56.6201			−0.666119
$86$		0			0
$87$		0			0
$88$		0			0
$89$	135.180	+	78.0459i	1.51887	+	0.876920i	0.999753	+	0.0222177i	$0.00707269\pi$
$89$	135.180	+	78.0459i	1.51887	+	0.876920i	0.519118	+	0.854703i	$0.326261\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−88.2656	−	152.881i	−0.929112	−	1.60927i
$96$		0			0
$97$		−	43.7452i		−	0.450981i	−0.974245	−	0.225491i	$-0.927601\pi$
$97$		−	43.7452i		−	0.450981i	0.974245	−	0.225491i	$-0.0723985\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−147.005	+	84.8732i	−1.45549	+	0.840329i	−0.998785	−	0.0492878i	$-0.984305\pi$
$101$	−147.005	+	84.8732i	−1.45549	+	0.840329i	−0.456708	+	0.889617i	$0.650972\pi$
$102$		0			0
$103$	51.7204	+	29.8608i	0.502140	+	0.289911i	0.729597	−	0.683877i	$-0.239708\pi$
$103$	51.7204	+	29.8608i	0.502140	+	0.289911i	−0.227457	+	0.973788i	$0.573041\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−28.8839	+	50.0284i	−0.269943	+	0.467555i	−0.968847	−	0.247661i	$-0.920338\pi$
$107$	−28.8839	+	50.0284i	−0.269943	+	0.467555i	0.698904	+	0.715216i	$0.253671\pi$
$108$		0			0
$109$	−89.9228	−	155.751i	−0.824980	−	1.42891i	−0.901935	−	0.431872i	$-0.857853\pi$
$109$	−89.9228	−	155.751i	−0.824980	−	1.42891i	0.0769549	−	0.997035i	$-0.475480\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−96.6900			−0.855664			−0.427832	−	0.903858i	$-0.640722\pi$
$113$	−96.6900			−0.855664			−0.427832	+	0.903858i	$0.640722\pi$
$114$		0			0
$115$	−91.6355	+	52.9058i	−0.796831	+	0.460050i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	36.9608	−	64.0180i	0.305461	−	0.529074i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	93.4495i		−	0.747596i
$126$		0			0
$127$	−136.454			−1.07444			−0.537221	−	0.843441i	$-0.680526\pi$
$127$	−136.454			−1.07444			−0.537221	+	0.843441i	$0.680526\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−22.9812	−	13.2682i	−0.175429	−	0.101284i	0.409714	−	0.912214i	$-0.365628\pi$
$131$	−22.9812	−	13.2682i	−0.175429	−	0.101284i	−0.585143	+	0.810930i	$0.698962\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	105.433	+	182.615i	0.769583	+	1.33296i	0.937789	+	0.347206i	$0.112869\pi$
$137$	105.433	+	182.615i	0.769583	+	1.33296i	−0.168206	+	0.985752i	$0.553797\pi$
$138$		0			0
$139$			83.7490i			0.602511i	0.953543	+	0.301256i	$0.0974057\pi$
$139$			83.7490i			0.602511i	−0.953543	+	0.301256i	$0.902594\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−21.5142	+	12.4212i	−0.150449	+	0.0868619i
$144$		0			0
$145$	204.194	+	117.891i	1.40823	+	0.813043i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.99740	−	6.92371i	0.0268282	−	0.0464678i	−0.852300	−	0.523054i	$-0.824793\pi$
$149$	3.99740	−	6.92371i	0.0268282	−	0.0464678i	0.879128	+	0.476586i	$0.158126\pi$
$150$		0			0
$151$	−30.4415	−	52.7263i	−0.201600	−	0.349181i	0.747444	−	0.664324i	$-0.231281\pi$
$151$	−30.4415	−	52.7263i	−0.201600	−	0.349181i	−0.949044	+	0.315144i	$0.897947\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−321.777			−2.07598
$156$		0			0
$157$	25.2670	−	14.5879i	0.160936	−	0.0929166i	−0.417369	−	0.908737i	$-0.637048\pi$
$157$	25.2670	−	14.5879i	0.160936	−	0.0929166i	0.578305	+	0.815821i	$0.303714\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	31.2278	−	54.0881i	0.191582	−	0.331829i	−0.754193	−	0.656653i	$-0.771972\pi$
$163$	31.2278	−	54.0881i	0.191582	−	0.331829i	0.945775	+	0.324824i	$0.105305\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	141.404i		−	0.846730i	−0.905959	−	0.423365i	$-0.860849\pi$
$167$		−	141.404i		−	0.846730i	0.905959	−	0.423365i	$-0.139151\pi$
$168$		0			0
$169$	155.891			0.922432
$170$		0			0
$171$		0			0
$172$		0			0
$173$	211.178	+	121.924i	1.22068	+	0.704763i	0.965064	−	0.262015i	$-0.0843868\pi$
$173$	211.178	+	121.924i	1.22068	+	0.704763i	0.255621	+	0.966777i	$0.417720\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−143.321	−	248.239i	−0.800676	−	1.38681i	−0.919172	−	0.393856i	$-0.871141\pi$
$179$	−143.321	−	248.239i	−0.800676	−	1.38681i	0.118496	−	0.992954i	$-0.462193\pi$
$180$		0			0
$181$			214.838i			1.18695i	0.804852	+	0.593475i	$0.202245\pi$
$181$			214.838i			1.18695i	−0.804852	+	0.593475i	$0.797755\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−277.466	+	160.195i	−1.49982	+	0.865921i
$186$		0			0
$187$	57.7290	+	33.3298i	0.308711	+	0.178234i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.6624	+	39.2525i	−0.118652	+	0.205511i	−0.919234	−	0.393713i	$-0.871190\pi$
$191$	−22.6624	+	39.2525i	−0.118652	+	0.205511i	0.800582	+	0.599223i	$0.204524\pi$
$192$		0			0
$193$	−157.015	−	271.958i	−0.813551	−	1.40911i	−0.910364	−	0.413809i	$-0.864198\pi$
$193$	−157.015	−	271.958i	−0.813551	−	1.40911i	0.0968130	−	0.995303i	$-0.469135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−259.231			−1.31590			−0.657948	−	0.753063i	$-0.728575\pi$
$197$	−259.231			−1.31590			−0.657948	+	0.753063i	$0.728575\pi$
$198$		0			0
$199$	−63.6749	+	36.7627i	−0.319975	+	0.184737i	−0.651381	−	0.758751i	$-0.725810\pi$
$199$	−63.6749	+	36.7627i	−0.319975	+	0.184737i	0.331407	+	0.943488i	$0.392477\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−164.151	+	284.319i	−0.800739	+	1.38692i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			207.833i			0.994415i
$210$		0			0
$211$	263.537			1.24899			0.624496	−	0.781028i	$-0.285304\pi$
$211$	263.537			1.24899			0.624496	+	0.781028i	$0.285304\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−333.372	−	192.473i	−1.55057	−	0.895222i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.5876	+	30.4626i	0.0795819	+	0.137840i
$222$		0			0
$223$			191.042i			0.856689i	0.903616	+	0.428344i	$0.140903\pi$
$223$			191.042i			0.856689i	−0.903616	+	0.428344i	$0.859097\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−373.608	+	215.702i	−1.64585	+	0.950231i	−0.667151	+	0.744923i	$0.732486\pi$
$227$	−373.608	+	215.702i	−1.64585	+	0.950231i	−0.978697	+	0.205308i	$0.934180\pi$
$228$		0			0
$229$	30.1973	+	17.4344i	0.131866	+	0.0761329i	0.564482	−	0.825445i	$-0.309076\pi$
$229$	30.1973	+	17.4344i	0.131866	+	0.0761329i	−0.432616	+	0.901578i	$0.642409\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−175.386	+	303.777i	−0.752728	+	1.30376i	0.193768	+	0.981047i	$0.437929\pi$
$233$	−175.386	+	303.777i	−0.752728	+	1.30376i	−0.946496	+	0.322715i	$0.895404\pi$
$234$		0			0
$235$	144.014	+	249.439i	0.612825	+	1.06144i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−167.400			−0.700419			−0.350209	−	0.936671i	$-0.613890\pi$
$239$	−167.400			−0.700419			−0.350209	+	0.936671i	$0.613890\pi$
$240$		0			0
$241$	−71.0711	+	41.0329i	−0.294901	+	0.170261i	−0.640150	−	0.768250i	$-0.721128\pi$
$241$	−71.0711	+	41.0329i	−0.294901	+	0.170261i	0.345249	+	0.938511i	$0.387794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−54.8350	+	94.9770i	−0.222004	+	0.384522i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	232.918i		−	0.927959i	−0.885846	−	0.463979i	$-0.846421\pi$
$251$		−	232.918i		−	0.927959i	0.885846	−	0.463979i	$-0.153579\pi$
$252$		0			0
$253$	124.574			0.492385
$254$		0			0
$255$		0			0
$256$		0			0
$257$	32.3538	+	18.6795i	0.125890	+	0.0726827i	0.561623	−	0.827394i	$-0.310177\pi$
$257$	32.3538	+	18.6795i	0.125890	+	0.0726827i	−0.435732	+	0.900076i	$0.643511\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.68631	+	2.92077i	0.00641181	+	0.0111056i	0.869214	−	0.494437i	$-0.164626\pi$
$263$	1.68631	+	2.92077i	0.00641181	+	0.0111056i	−0.862802	+	0.505543i	$0.831292\pi$
$264$		0			0
$265$		−	472.161i		−	1.78174i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	75.5768	−	43.6343i	0.280955	−	0.162209i	−0.352901	−	0.935661i	$-0.614805\pi$
$269$	75.5768	−	43.6343i	0.280955	−	0.162209i	0.633856	+	0.773451i	$0.281471\pi$
$270$		0			0
$271$	−79.6808	−	46.0037i	−0.294025	−	0.169756i	0.345731	−	0.938334i	$-0.387631\pi$
$271$	−79.6808	−	46.0037i	−0.294025	−	0.169756i	−0.639756	+	0.768578i	$0.720964\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	30.7574	−	53.2734i	0.111845	−	0.193721i
$276$		0			0
$277$	−76.2406	−	132.053i	−0.275237	−	0.476724i	0.694958	−	0.719050i	$-0.255423\pi$
$277$	−76.2406	−	132.053i	−0.275237	−	0.476724i	−0.970195	+	0.242326i	$0.922090\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	219.880			0.782491			0.391245	−	0.920286i	$-0.372044\pi$
$281$	219.880			0.782491			0.391245	+	0.920286i	$0.372044\pi$
$282$		0			0
$283$	−335.489	+	193.695i	−1.18547	+	0.684433i	−0.957274	−	0.289181i	$-0.906617\pi$
$283$	−335.489	+	193.695i	−1.18547	+	0.684433i	−0.228199	+	0.973615i	$0.573284\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−97.3073	+	168.541i	−0.336704	+	0.583188i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	14.8794i		−	0.0507828i	−0.999678	−	0.0253914i	$-0.991917\pi$
$293$		−	14.8794i		−	0.0507828i	0.999678	−	0.0253914i	$-0.00808320\pi$
$294$		0			0
$295$	−202.764			−0.687335
$296$		0			0
$297$		0			0
$298$		0			0
$299$	56.9285	+	32.8677i	0.190396	+	0.109925i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.111549	+	0.193209i	0.000365735	+	0.000633471i
$306$		0			0
$307$			453.211i			1.47626i	0.674660	+	0.738128i	$0.264290\pi$
$307$			453.211i			1.47626i	−0.674660	+	0.738128i	$0.735710\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	376.934	−	217.623i	1.21201	−	0.699752i	0.248810	−	0.968552i	$-0.419961\pi$
$311$	376.934	−	217.623i	1.21201	−	0.699752i	0.963196	+	0.268801i	$0.0866273\pi$
$312$		0			0
$313$	47.5799	+	27.4703i	0.152012	+	0.0877644i	0.574077	−	0.818801i	$-0.305361\pi$
$313$	47.5799	+	27.4703i	0.152012	+	0.0877644i	−0.422064	+	0.906566i	$0.638694\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	216.846	−	375.587i	0.684055	−	1.18482i	−0.289677	−	0.957124i	$-0.593548\pi$
$317$	216.846	−	375.587i	0.684055	−	1.18482i	0.973733	−	0.227694i	$-0.0731186\pi$
$318$		0			0
$319$	−138.795	−	240.400i	−0.435095	−	0.753606i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	294.277			0.911073
$324$		0			0
$325$	28.1115	−	16.2302i	0.0864968	−	0.0499390i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	288.274	−	499.305i	0.870919	−	1.50848i	0.00987084	−	0.999951i	$-0.496858\pi$
$331$	288.274	−	499.305i	0.870919	−	1.50848i	0.861048	−	0.508524i	$-0.169809\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			373.514i			1.11497i
$336$		0			0
$337$	−301.108			−0.893495			−0.446747	−	0.894660i	$-0.647418\pi$
$337$	−301.108			−0.893495			−0.446747	+	0.894660i	$0.647418\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	328.078	+	189.416i	0.962107	+	0.555473i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−126.056	−	218.335i	−0.363273	−	0.629208i	0.625224	−	0.780445i	$-0.285008\pi$
$347$	−126.056	−	218.335i	−0.363273	−	0.629208i	−0.988497	+	0.151237i	$0.951674\pi$
$348$		0			0
$349$		−	406.452i		−	1.16462i	−0.812967	−	0.582309i	$-0.802149\pi$
$349$		−	406.452i		−	1.16462i	0.812967	−	0.582309i	$-0.197851\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−116.419	+	67.2146i	−0.329799	+	0.190410i	−0.655752	−	0.754976i	$-0.727648\pi$
$353$	−116.419	+	67.2146i	−0.329799	+	0.190410i	0.325953	+	0.945386i	$0.394315\pi$
$354$		0			0
$355$	−253.737	−	146.495i	−0.714752	−	0.412662i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−58.3380	+	101.044i	−0.162501	+	0.281461i	−0.935765	−	0.352624i	$-0.885290\pi$
$359$	−58.3380	+	101.044i	−0.162501	+	0.281461i	0.773264	+	0.634084i	$0.218623\pi$
$360$		0			0
$361$	278.251	+	481.944i	0.770777	+	1.33503i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	124.406			0.340839
$366$		0			0
$367$	362.993	−	209.574i	0.989081	−	0.571046i	0.0840817	−	0.996459i	$-0.473204\pi$
$367$	362.993	−	209.574i	0.989081	−	0.571046i	0.905000	+	0.425413i	$0.139871\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	63.4962	−	109.979i	0.170231	−	0.294849i	−0.768270	−	0.640127i	$-0.778882\pi$
$373$	63.4962	−	109.979i	0.170231	−	0.294849i	0.938501	+	0.345278i	$0.112215\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	146.480i		−	0.388541i
$378$		0			0
$379$	−366.675			−0.967479			−0.483740	−	0.875212i	$-0.660722\pi$
$379$	−366.675			−0.967479			−0.483740	+	0.875212i	$0.660722\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	223.750	+	129.182i	0.584203	+	0.337290i	0.762802	−	0.646632i	$-0.223823\pi$
$383$	223.750	+	129.182i	0.584203	+	0.337290i	−0.178599	+	0.983922i	$0.557157\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	271.198	+	469.728i	0.697166	+	1.20753i	0.969445	+	0.245309i	$0.0788893\pi$
$389$	271.198	+	469.728i	0.697166	+	1.20753i	−0.272279	+	0.962218i	$0.587777\pi$
$390$		0			0
$391$		−	176.387i		−	0.451118i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	239.763	−	138.427i	0.606995	−	0.350449i
$396$		0			0
$397$	−350.536	−	202.382i	−0.882962	−	0.509778i	−0.0113280	−	0.999936i	$-0.503606\pi$
$397$	−350.536	−	202.382i	−0.882962	−	0.509778i	−0.871634	+	0.490158i	$0.836939\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−236.718	+	410.008i	−0.590319	+	1.02246i	0.403870	+	0.914816i	$0.367665\pi$
$401$	−236.718	+	410.008i	−0.590319	+	1.02246i	−0.994189	+	0.107646i	$0.965669\pi$
$402$		0			0
$403$	99.9518	+	173.122i	0.248019	+	0.429582i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	377.201			0.926783
$408$		0			0
$409$	349.578	−	201.829i	0.854714	−	0.493469i	−0.00752476	−	0.999972i	$-0.502395\pi$
$409$	349.578	−	201.829i	0.854714	−	0.493469i	0.862239	+	0.506502i	$0.169062\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−97.9215	+	169.605i	−0.235955	+	0.408687i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			454.684i			1.08517i	0.840003	+	0.542583i	$0.182553\pi$
$419$			454.684i			1.08517i	−0.840003	+	0.542583i	$0.817447\pi$
$420$		0			0
$421$	−180.928			−0.429758			−0.214879	−	0.976641i	$-0.568936\pi$
$421$	−180.928			−0.429758			−0.214879	+	0.976641i	$0.568936\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−75.4313	−	43.5503i	−0.177485	−	0.102471i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−403.946	−	699.655i	−0.937229	−	1.62333i	−0.770610	−	0.637307i	$-0.780048\pi$
$431$	−403.946	−	699.655i	−0.937229	−	1.62333i	−0.166619	−	0.986021i	$-0.553285\pi$
$432$		0			0
$433$		−	166.000i		−	0.383372i	−0.981456	−	0.191686i	$-0.938604\pi$
$433$		−	166.000i		−	0.383372i	0.981456	−	0.191686i	$-0.0613955\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	476.265	−	274.972i	1.08985	−	0.629226i
$438$		0			0
$439$	−25.3654	−	14.6447i	−0.0577800	−	0.0333593i	0.470832	−	0.882223i	$-0.343954\pi$
$439$	−25.3654	−	14.6447i	−0.0577800	−	0.0333593i	−0.528612	+	0.848864i	$0.677287\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	411.376	−	712.524i	0.928613	−	1.60841i	0.142969	−	0.989727i	$-0.454335\pi$
$443$	411.376	−	712.524i	0.928613	−	1.60841i	0.785645	−	0.618678i	$-0.212331\pi$
$444$		0			0
$445$	454.850	+	787.824i	1.02214	+	1.77039i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	428.131			0.953522			0.476761	−	0.879033i	$-0.341811\pi$
$449$	428.131			0.953522			0.476761	+	0.879033i	$0.341811\pi$
$450$		0			0
$451$	334.733	−	193.258i	0.742201	−	0.428510i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	93.6555	−	162.216i	0.204935	−	0.354959i	−0.745177	−	0.666867i	$-0.767635\pi$
$457$	93.6555	−	162.216i	0.204935	−	0.354959i	0.950112	+	0.311908i	$0.100968\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	182.821i		−	0.396576i	−0.980144	−	0.198288i	$-0.936462\pi$
$461$		−	182.821i		−	0.396576i	0.980144	−	0.198288i	$-0.0635381\pi$
$462$		0			0
$463$	232.389			0.501920			0.250960	−	0.967997i	$-0.419254\pi$
$463$	232.389			0.501920			0.250960	+	0.967997i	$0.419254\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−388.782	−	224.463i	−0.832509	−	0.480650i	0.0222017	−	0.999754i	$-0.492932\pi$
$467$	−388.782	−	224.463i	−0.832509	−	0.480650i	−0.854711	+	0.519104i	$0.826266\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	226.601	+	392.484i	0.479072	+	0.829777i
$474$		0			0
$475$		−	271.564i		−	0.571713i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	547.020	−	315.822i	1.14200	−	0.659336i	0.195078	−	0.980788i	$-0.437504\pi$
$479$	547.020	−	315.822i	1.14200	−	0.659336i	0.946926	+	0.321452i	$0.104171\pi$
$480$		0			0
$481$	172.376	+	99.5213i	0.358370	+	0.206905i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	127.473	−	220.790i	0.262831	−	0.455237i
$486$		0			0
$487$	−350.767	−	607.547i	−0.720262	−	1.24753i	−0.960895	−	0.276914i	$-0.910688\pi$
$487$	−350.767	−	607.547i	−0.720262	−	1.24753i	0.240633	−	0.970616i	$-0.422645\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	414.789			0.844785			0.422392	−	0.906413i	$-0.361190\pi$
$491$	414.789			0.844785			0.422392	+	0.906413i	$0.361190\pi$
$492$		0			0
$493$	−340.390	+	196.524i	−0.690446	+	0.398629i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	306.170	−	530.302i	0.613567	−	1.06273i	−0.377067	−	0.926186i	$-0.623067\pi$
$499$	306.170	−	530.302i	0.613567	−	1.06273i	0.990634	−	0.136543i	$-0.0435992\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			467.449i			0.929323i	0.885488	+	0.464661i	$0.153824\pi$
$503$			467.449i			0.929323i	−0.885488	+	0.464661i	$0.846176\pi$
$504$		0			0
$505$	−989.279			−1.95897
$506$		0			0
$507$		0			0
$508$		0			0
$509$	54.1843	+	31.2833i	0.106452	+	0.0614603i	0.552281	−	0.833658i	$-0.313758\pi$
$509$	54.1843	+	31.2833i	0.106452	+	0.0614603i	−0.445829	+	0.895118i	$0.647091\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	174.028	+	301.426i	0.337919	+	0.585293i
$516$		0			0
$517$		−	339.099i		−	0.655898i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	617.108	−	356.288i	1.18447	−	0.683853i	0.227425	−	0.973796i	$-0.426969\pi$
$521$	617.108	−	356.288i	1.18447	−	0.683853i	0.957044	+	0.289942i	$0.0936362\pi$
$522$		0			0
$523$	242.586	+	140.057i	0.463835	+	0.267795i	0.713655	−	0.700497i	$-0.247038\pi$
$523$	242.586	+	140.057i	0.463835	+	0.267795i	−0.249821	+	0.968292i	$0.580372\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	268.200	−	464.536i	0.508918	−	0.881472i
$528$		0			0
$529$	99.6838	+	172.657i	0.188438	+	0.326385i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	203.958			0.382660
$534$		0			0
$535$	−291.564	+	168.335i	−0.544980	+	0.314645i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−316.501	+	548.196i	−0.585030	+	1.01330i	0.409842	+	0.912157i	$0.365584\pi$
$541$	−316.501	+	548.196i	−0.585030	+	1.01330i	−0.994872	+	0.101145i	$0.967749\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1048.14i		−	1.92319i
$546$		0			0
$547$	1047.16			1.91438			0.957189	−	0.289465i	$-0.0934773\pi$
$547$	1047.16			1.91438			0.957189	+	0.289465i	$0.0934773\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1061.27	−	612.727i	−1.92609	−	1.11203i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	340.866	+	590.397i	0.611967	+	1.05996i	0.990908	+	0.134538i	$0.0429550\pi$
$557$	340.866	+	590.397i	0.611967	+	1.05996i	−0.378941	+	0.925421i	$0.623712\pi$
$558$		0			0
$559$			239.147i			0.427812i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−174.581	+	100.794i	−0.310090	+	0.179031i	−0.646967	−	0.762518i	$-0.723963\pi$
$563$	−174.581	+	100.794i	−0.310090	+	0.179031i	0.336877	+	0.941549i	$0.390629\pi$
$564$		0			0
$565$	−488.012	−	281.754i	−0.863738	−	0.498679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−342.400	+	593.054i	−0.601758	+	1.04227i	0.390797	+	0.920477i	$0.372199\pi$
$569$	−342.400	+	593.054i	−0.601758	+	1.04227i	−0.992555	+	0.121798i	$0.961134\pi$
$570$		0			0
$571$	12.5301	+	21.7028i	0.0219442	+	0.0380084i	0.876789	−	0.480875i	$-0.159681\pi$
$571$	12.5301	+	21.7028i	0.0219442	+	0.0380084i	−0.854845	+	0.518884i	$0.826348\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−162.773			−0.283084
$576$		0			0
$577$	883.312	−	509.981i	1.53087	−	0.883849i	0.531549	−	0.847028i	$-0.321610\pi$
$577$	883.312	−	509.981i	1.53087	−	0.883849i	0.999322	−	0.0368210i	$-0.0117231\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−277.941	+	481.408i	−0.476743	+	0.825742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			581.810i			0.991158i	0.868563	+	0.495579i	$0.165044\pi$
$587$			581.810i			0.991158i	−0.868563	+	0.495579i	$0.834956\pi$
$588$		0			0
$589$	1672.40			2.83939
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−124.589	−	71.9314i	−0.210099	−	0.121301i	0.391258	−	0.920281i	$-0.372040\pi$
$593$	−124.589	−	71.9314i	−0.210099	−	0.121301i	−0.601358	+	0.798980i	$0.705373\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	355.368	+	615.515i	0.593269	+	1.02757i	0.993789	+	0.111283i	$0.0354961\pi$
$599$	355.368	+	615.515i	0.593269	+	1.02757i	−0.400520	+	0.916288i	$0.631171\pi$
$600$		0			0
$601$			914.930i			1.52235i	0.648549	+	0.761173i	$0.275376\pi$
$601$			914.930i			1.52235i	−0.648549	+	0.761173i	$0.724624\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	373.096	−	215.407i	0.616687	−	0.356044i
$606$		0			0
$607$	−539.084	−	311.240i	−0.888112	−	0.512752i	−0.0147875	−	0.999891i	$-0.504707\pi$
$607$	−539.084	−	311.240i	−0.888112	−	0.512752i	−0.873324	+	0.487139i	$0.838041\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	89.4686	−	154.964i	0.146430	−	0.253624i
$612$		0			0
$613$	172.510	+	298.795i	0.281419	+	0.487431i	0.971734	−	0.236077i	$-0.0758618\pi$
$613$	172.510	+	298.795i	0.281419	+	0.487431i	−0.690316	+	0.723508i	$0.742528\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	854.311			1.38462			0.692310	−	0.721600i	$-0.256593\pi$
$617$	854.311			1.38462			0.692310	+	0.721600i	$0.256593\pi$
$618$		0			0
$619$	42.6240	−	24.6090i	0.0688595	−	0.0397561i	−0.465175	−	0.885219i	$-0.654009\pi$
$619$	42.6240	−	24.6090i	0.0688595	−	0.0397561i	0.534034	+	0.845463i	$0.320675\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	384.378	−	665.763i	0.615005	−	1.06522i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	534.089i		−	0.849109i
$630$		0			0
$631$	457.699			0.725355			0.362677	−	0.931915i	$-0.381863\pi$
$631$	457.699			0.725355			0.362677	+	0.931915i	$0.381863\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−688.709	−	397.626i	−1.08458	−	0.626183i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−161.966	−	280.533i	−0.252677	−	0.437649i	0.711585	−	0.702600i	$-0.247978\pi$
$641$	−161.966	−	280.533i	−0.252677	−	0.437649i	−0.964262	+	0.264951i	$0.914644\pi$
$642$		0			0
$643$		−	117.018i		−	0.181987i	−0.995851	−	0.0909935i	$-0.970996\pi$
$643$		−	117.018i		−	0.181987i	0.995851	−	0.0909935i	$-0.0290043\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−467.002	+	269.624i	−0.721796	+	0.416729i	−0.815413	−	0.578879i	$-0.803490\pi$
$647$	−467.002	+	269.624i	−0.721796	+	0.416729i	0.0936176	+	0.995608i	$0.470157\pi$
$648$		0			0
$649$	206.735	+	119.358i	0.318543	+	0.183911i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−398.953	+	691.007i	−0.610954	+	1.05820i	0.380126	+	0.924935i	$0.375881\pi$
$653$	−398.953	+	691.007i	−0.610954	+	1.05820i	−0.991080	+	0.133268i	$0.957453\pi$
$654$		0			0
$655$	−77.3268	−	133.934i	−0.118056	−	0.204479i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−295.186			−0.447929			−0.223965	−	0.974597i	$-0.571900\pi$
$659$	−295.186			−0.447929			−0.223965	+	0.974597i	$0.571900\pi$
$660$		0			0
$661$	−893.947	+	516.120i	−1.35242	+	0.780817i	−0.988587	−	0.150649i	$-0.951864\pi$
$661$	−893.947	+	516.120i	−1.35242	+	0.780817i	−0.363828	+	0.931466i	$0.618530\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−367.264	+	636.120i	−0.550621	+	0.953703i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.262657i		−	0.000391441i
$672$		0			0
$673$	418.188			0.621379			0.310689	−	0.950512i	$-0.399440\pi$
$673$	418.188			0.621379			0.310689	+	0.950512i	$0.399440\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	403.100	+	232.730i	0.595422	+	0.343767i	0.767238	−	0.641362i	$-0.221630\pi$
$677$	403.100	+	232.730i	0.595422	+	0.343767i	−0.171817	+	0.985129i	$0.554964\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	629.705	+	1090.68i	0.921969	+	1.59690i	0.796364	+	0.604817i	$0.206754\pi$
$683$	629.705	+	1090.68i	0.921969	+	1.59690i	0.125605	+	0.992080i	$0.459913\pi$
$684$		0			0
$685$			1228.92i			1.79405i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−254.031	+	146.665i	−0.368695	+	0.212866i
$690$		0			0
$691$	−397.397	−	229.437i	−0.575104	−	0.332036i	0.184081	−	0.982911i	$-0.441069\pi$
$691$	−397.397	−	229.437i	−0.575104	−	0.332036i	−0.759185	+	0.650875i	$0.774402\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−244.044	+	422.697i	−0.351142	+	0.608196i
$696$		0			0
$697$	−273.640	−	473.958i	−0.392596	−	0.679997i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−822.907			−1.17390			−0.586952	−	0.809622i	$-0.699672\pi$
$701$	−822.907			−1.17390			−0.586952	+	0.809622i	$0.699672\pi$
$702$		0			0
$703$	1442.10	−	832.597i	2.05135	−	1.18435i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−142.954	+	247.603i	−0.201627	+	0.349229i	−0.949053	−	0.315117i	$-0.897956\pi$
$709$	−142.954	+	247.603i	−0.201627	+	0.349229i	0.747426	+	0.664346i	$0.231290\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1002.42i		−	1.40592i
$714$		0			0
$715$	−144.782			−0.202492
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−605.561	−	349.621i	−0.842227	−	0.486260i	0.0157934	−	0.999875i	$-0.494973\pi$
$719$	−605.561	−	349.621i	−0.842227	−	0.486260i	−0.858021	+	0.513615i	$0.828306\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	181.356	+	314.118i	0.250146	+	0.433266i
$726$		0			0
$727$		−	216.138i		−	0.297302i	−0.988890	−	0.148651i	$-0.952507\pi$
$727$		−	216.138i		−	0.297302i	0.988890	−	0.148651i	$-0.0474930\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	555.730	−	320.851i	0.760233	−	0.438920i
$732$		0			0
$733$	629.763	+	363.594i	0.859159	+	0.496035i	0.863730	−	0.503954i	$-0.168122\pi$
$733$	629.763	+	363.594i	0.859159	+	0.496035i	−0.00457181	+	0.999990i	$0.501455\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	219.872	−	380.829i	0.298334	−	0.516729i
$738$		0			0
$739$	−198.206	−	343.303i	−0.268209	−	0.464551i	0.700191	−	0.713956i	$-0.253098\pi$
$739$	−198.206	−	343.303i	−0.268209	−	0.464551i	−0.968399	+	0.249405i	$0.919765\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−739.481			−0.995263			−0.497632	−	0.867388i	$-0.665797\pi$
$743$	−739.481			−0.995263			−0.497632	+	0.867388i	$0.665797\pi$
$744$		0			0
$745$	40.3512	−	23.2968i	0.0541627	−	0.0312709i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	374.166	−	648.075i	0.498224	−	0.862949i	−0.501774	−	0.864999i	$-0.667319\pi$
$751$	374.166	−	648.075i	0.498224	−	0.862949i	0.999998	+	0.00204963i	$0.000652418\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	354.825i		−	0.469967i
$756$		0			0
$757$	199.145			0.263071			0.131536	−	0.991311i	$-0.458009\pi$
$757$	199.145			0.263071			0.131536	+	0.991311i	$0.458009\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−849.966	−	490.728i	−1.11691	−	0.644846i	−0.176297	−	0.984337i	$-0.556412\pi$
$761$	−849.966	−	490.728i	−1.11691	−	0.644846i	−0.940609	+	0.339491i	$0.889745\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	62.9834	+	109.090i	0.0821166	+	0.142230i
$768$		0			0
$769$		−	724.214i		−	0.941760i	−0.882197	−	0.470880i	$-0.843936\pi$
$769$		−	724.214i		−	0.941760i	0.882197	−	0.470880i	$-0.156064\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1216.68	−	702.449i	1.57397	−	0.908731i	0.578293	−	0.815829i	$-0.303719\pi$
$773$	1216.68	−	702.449i	1.57397	−	0.908731i	0.995675	−	0.0929015i	$-0.0296142\pi$
$774$		0			0
$775$	−428.682	−	247.500i	−0.553139	−	0.319355i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	853.159	−	1477.71i	1.09520	−	1.89694i
$780$		0			0
$781$	172.471	+	298.728i	0.220833	+	0.382495i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	170.036			0.216607
$786$		0			0
$787$	−221.541	+	127.907i	−0.281501	+	0.162525i	−0.634103	−	0.773249i	$-0.718630\pi$
$787$	−221.541	+	127.907i	−0.281501	+	0.162525i	0.352602	+	0.935773i	$0.385297\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0692998	−	0.120031i	8.73894e−5	−	0.000151363i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			474.641i			0.595534i	0.954639	+	0.297767i	$0.0962418\pi$
$797$			474.641i			0.595534i	−0.954639	+	0.297767i	$0.903758\pi$
$798$		0			0
$799$	−480.141			−0.600927
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−126.843	−	73.2327i	−0.157961	−	0.0911989i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	233.864	+	405.064i	0.289077	+	0.500697i	0.973590	−	0.228304i	$-0.0733182\pi$
$809$	233.864	+	405.064i	0.289077	+	0.500697i	−0.684512	+	0.729001i	$0.739985\pi$
$810$		0			0
$811$			1567.70i			1.93305i	0.256571	+	0.966525i	$0.417407\pi$
$811$			1567.70i			1.93305i	−0.256571	+	0.966525i	$0.582593\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	315.225	−	181.995i	0.386779	−	0.223307i
$816$		0			0
$817$	1732.67	+	1000.36i	2.12077	+	1.22442i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−219.125	+	379.536i	−0.266900	+	0.462285i	−0.968060	−	0.250720i	$-0.919333\pi$
$821$	−219.125	+	379.536i	−0.266900	+	0.462285i	0.701159	+	0.713005i	$0.252666\pi$
$822$		0			0
$823$	−468.606	−	811.650i	−0.569388	−	0.986209i	−0.996627	−	0.0820702i	$-0.973847\pi$
$823$	−468.606	−	811.650i	−0.569388	−	0.986209i	0.427238	−	0.904139i	$-0.359487\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1160.31			1.40303			0.701516	−	0.712654i	$-0.252507\pi$
$827$	1160.31			1.40303			0.701516	+	0.712654i	$0.252507\pi$
$828$		0			0
$829$	−220.489	+	127.300i	−0.265970	+	0.153558i	−0.627055	−	0.778975i	$-0.715740\pi$
$829$	−220.489	+	127.300i	−0.265970	+	0.153558i	0.361085	+	0.932533i	$0.382406\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	412.050	−	713.691i	0.493473	−	0.854720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			129.902i			0.154829i	0.996999	+	0.0774146i	$0.0246665\pi$
$839$			129.902i			0.154829i	−0.996999	+	0.0774146i	$0.975333\pi$
$840$		0			0
$841$	795.767			0.946215
$842$		0			0
$843$		0			0
$844$		0			0
$845$	786.810	+	454.265i	0.931136	+	0.537592i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−499.053	−	864.385i	−0.586431	−	1.01573i
$852$		0			0
$853$		−	1229.45i		−	1.44133i	−0.693284	−	0.720665i	$-0.743837\pi$
$853$		−	1229.45i		−	1.44133i	0.693284	−	0.720665i	$-0.256163\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−578.670	+	334.095i	−0.675227	+	0.389843i	−0.798054	−	0.602585i	$-0.794137\pi$
$857$	−578.670	+	334.095i	−0.675227	+	0.389843i	0.122827	+	0.992428i	$0.460804\pi$
$858$		0			0
$859$	16.7929	+	9.69537i	0.0195493	+	0.0112868i	0.509743	−	0.860327i	$-0.329741\pi$
$859$	16.7929	+	9.69537i	0.0195493	+	0.0112868i	−0.490193	+	0.871614i	$0.663074\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	46.4856	−	80.5154i	0.0538651	−	0.0932971i	−0.837836	−	0.545923i	$-0.816179\pi$
$863$	46.4856	−	80.5154i	0.0538651	−	0.0932971i	0.891701	+	0.452626i	$0.149513\pi$
$864$		0			0
$865$	710.571	+	1230.74i	0.821469	+	1.42283i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−325.945			−0.375080
$870$		0			0
$871$	200.957	−	116.023i	0.230720	−	0.133206i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−156.154	+	270.467i	−0.178055	+	0.308400i	−0.941214	−	0.337810i	$-0.890314\pi$
$877$	−156.154	+	270.467i	−0.178055	+	0.308400i	0.763159	+	0.646210i	$0.223647\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			772.018i			0.876298i	0.898902	+	0.438149i	$0.144366\pi$
$881$			772.018i			0.876298i	−0.898902	+	0.438149i	$0.855634\pi$
$882$		0			0
$883$	−959.154			−1.08624			−0.543122	−	0.839654i	$-0.682758\pi$
$883$	−959.154			−1.08624			−0.543122	+	0.839654i	$0.682758\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	480.613	+	277.482i	0.541841	+	0.312832i	0.745825	−	0.666142i	$-0.232056\pi$
$887$	480.613	+	277.482i	0.541841	+	0.312832i	−0.203984	+	0.978974i	$0.565389\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−748.496	−	1296.43i	−0.838181	−	1.45177i
$894$		0			0
$895$		−	1670.54i		−	1.86653i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1934.46	+	1116.86i	−2.15179	+	1.24234i
$900$		0			0
$901$	681.639	+	393.545i	0.756536	+	0.436787i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−626.036	+	1084.33i	−0.691753	+	1.19815i
$906$		0			0
$907$	−286.558	−	496.333i	−0.315940	−	0.547225i	0.663697	−	0.748002i	$-0.268987\pi$
$907$	−286.558	−	496.333i	−0.315940	−	0.547225i	−0.979637	+	0.200777i	$0.935653\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−726.088			−0.797023			−0.398511	−	0.917163i	$-0.630473\pi$
$911$	−726.088			−0.797023			−0.398511	+	0.917163i	$0.630473\pi$
$912$		0			0
$913$	199.679	−	115.284i	0.218706	−	0.126270i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−392.266	+	679.425i	−0.426840	+	0.739309i	−0.996590	−	0.0825089i	$-0.973707\pi$
$919$	−392.266	+	679.425i	−0.426840	+	0.739309i	0.569750	+	0.821818i	$0.307040\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			182.020i			0.197205i
$924$		0			0
$925$	−492.868			−0.532830
$926$		0			0
$927$		0			0
$928$		0			0
$929$	137.793	+	79.5547i	0.148324	+	0.0856347i	0.572325	−	0.820027i	$-0.306042\pi$
$929$	137.793	+	79.5547i	0.148324	+	0.0856347i	−0.424001	+	0.905662i	$0.639375\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	194.246	+	336.443i	0.207749	+	0.359833i
$936$		0			0
$937$			728.876i			0.777883i	0.921262	+	0.388941i	$0.127159\pi$
$937$			728.876i			0.777883i	−0.921262	+	0.388941i	$0.872841\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	279.939	−	161.623i	0.297491	−	0.171756i	−0.343824	−	0.939034i	$-0.611722\pi$
$941$	279.939	−	161.623i	0.297491	−	0.171756i	0.641315	+	0.767278i	$0.278389\pi$
$942$		0			0
$943$	−885.732	−	511.377i	−0.939270	−	0.542288i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	336.525	−	582.878i	0.355359	−	0.615499i	−0.631821	−	0.775115i	$-0.717692\pi$
$947$	336.525	−	582.878i	0.355359	−	0.615499i	0.987179	+	0.159615i	$0.0510254\pi$
$948$		0			0
$949$	−38.6437	−	66.9328i	−0.0407204	−	0.0705298i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	195.034			0.204653			0.102326	−	0.994751i	$-0.467371\pi$
$953$	195.034			0.204653			0.102326	+	0.994751i	$0.467371\pi$
$954$		0			0
$955$	−228.763	+	132.076i	−0.239542	+	0.138300i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1043.70	−	1807.75i	1.08606	−	1.88111i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1830.16i		−	1.89654i
$966$		0			0
$967$	−731.573			−0.756538			−0.378269	−	0.925696i	$-0.623481\pi$
$967$	−731.573			−0.756538			−0.378269	+	0.925696i	$0.623481\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	431.992	+	249.411i	0.444894	+	0.256860i	0.705672	−	0.708539i	$-0.250645\pi$
$971$	431.992	+	249.411i	0.444894	+	0.256860i	−0.260777	+	0.965399i	$0.583979\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−90.3523	−	156.495i	−0.0924794	−	0.160179i	0.816074	−	0.577947i	$-0.196146\pi$
$977$	−90.3523	−	156.495i	−0.0924794	−	0.160179i	−0.908554	+	0.417768i	$0.862813\pi$
$978$		0			0
$979$		−	1071.00i		−	1.09398i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	282.894	−	163.329i	0.287786	−	0.166153i	−0.349157	−	0.937064i	$-0.613532\pi$
$983$	282.894	−	163.329i	0.287786	−	0.166153i	0.636943	+	0.770911i	$0.280199\pi$
$984$		0			0
$985$	−1308.39	−	755.398i	−1.32831	−	0.766902i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	599.606	−	1038.55i	0.606275	−	1.05010i
$990$		0			0
$991$	823.376	+	1426.13i	0.830854	+	1.43908i	0.897362	+	0.441295i	$0.145481\pi$
$991$	823.376	+	1426.13i	0.830854	+	1.43908i	−0.0665084	+	0.997786i	$0.521186\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−428.505			−0.430659
$996$		0			0
$997$	121.381	−	70.0792i	0.121746	−	0.0702900i	−0.437890	−	0.899028i	$-0.644274\pi$
$997$	121.381	−	70.0792i	0.121746	−	0.0702900i	0.559636	+	0.828738i	$0.310941\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.l.901.4		8
3.2	odd	2		588.3.m.f.313.1		8
7.2	even	3		1764.3.d.h.685.1		8
7.3	odd	6	inner	1764.3.z.l.325.4		8
7.4	even	3		1764.3.z.m.325.1		8
7.5	odd	6		1764.3.d.h.685.8		8
7.6	odd	2		1764.3.z.m.901.1		8
21.2	odd	6		588.3.d.c.97.8	yes	8
21.5	even	6		588.3.d.c.97.1	✓	8
21.11	odd	6		588.3.m.e.325.4		8
21.17	even	6		588.3.m.f.325.1		8
21.20	even	2		588.3.m.e.313.4		8
84.23	even	6		2352.3.f.j.97.4		8
84.47	odd	6		2352.3.f.j.97.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.1	✓	8	21.5	even	6
588.3.d.c.97.8	yes	8	21.2	odd	6
588.3.m.e.313.4		8	21.20	even	2
588.3.m.e.325.4		8	21.11	odd	6
588.3.m.f.313.1		8	3.2	odd	2
588.3.m.f.325.1		8	21.17	even	6
1764.3.d.h.685.1		8	7.2	even	3
1764.3.d.h.685.8		8	7.5	odd	6
1764.3.z.l.325.4		8	7.3	odd	6	inner
1764.3.z.l.901.4		8	1.1	even	1	trivial
1764.3.z.m.325.1		8	7.4	even	3
1764.3.z.m.901.1		8	7.6	odd	2
2352.3.f.j.97.4		8	84.23	even	6
2352.3.f.j.97.5		8	84.47	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.04718 + 2.91399i) q^{5} +O(q^{10})\)	\(q+(5.04718 + 2.91399i) q^{5} +(-3.43068 - 5.94212i) q^{11} -3.62063i q^{13} +(-8.41362 + 4.85761i) q^{17} +(-26.2322 - 15.1451i) q^{19} +(-9.07789 + 15.7234i) q^{23} +(4.48269 + 7.76425i) q^{25} +40.4570 q^{29} +(-47.8153 + 27.6062i) q^{31} +(-27.4873 + 47.6093i) q^{37} +56.3322i q^{41} -66.0512 q^{43} +(42.8003 + 24.7108i) q^{47} +(-40.5081 - 70.1621i) q^{53} -39.9879i q^{55} +(-30.1302 + 17.3957i) q^{59} +(0.0331519 + 0.0191403i) q^{61} +(10.5505 - 18.2740i) q^{65} +(32.0449 + 55.5034i) q^{67} -50.2730 q^{71} +(18.4865 - 10.6732i) q^{73} +(23.7522 - 41.1400i) q^{79} +33.6039i q^{83} -56.6201 q^{85} +(135.180 + 78.0459i) q^{89} +(-88.2656 - 152.881i) q^{95} -43.7452i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{17} - 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} + 264 q^{47} - 72 q^{53} - 168 q^{59} - 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} + 336 q^{73} + 216 q^{79} - 96 q^{85} - 96 q^{89} - 136 q^{95}+O(q^{100}) \) 8 * q - 48 * q^17 - 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 + 48 * q^31 - 64 * q^37 - 112 * q^43 + 264 * q^47 - 72 * q^53 - 168 * q^59 - 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 + 336 * q^73 + 216 * q^79 - 96 * q^85 - 96 * q^89 - 136 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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