LMFDB - Embedded newform 648.4.i.v.433.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.3
Root			$2.07341 - 1.19709i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.v.217.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.16966 - 5.49001i) q^{5} +(-9.59728 - 16.6230i) q^{7} +O(q^{10})$	$q+(3.16966 - 5.49001i) q^{5} +(-9.59728 - 16.6230i) q^{7} +(20.6230 + 35.7201i) q^{11} +(-19.6328 + 34.0049i) q^{13} -59.9385 q^{17} -41.2583 q^{19} +(-26.9040 + 46.5990i) q^{23} +(42.4065 + 73.4502i) q^{25} +(-32.6297 - 56.5164i) q^{29} +(-32.3747 + 56.0745i) q^{31} -121.681 q^{35} +293.515 q^{37} +(103.636 - 179.503i) q^{41} +(188.999 + 327.355i) q^{43} +(161.891 + 280.404i) q^{47} +(-12.7157 + 22.0243i) q^{49} +340.588 q^{53} +261.471 q^{55} +(203.481 - 352.439i) q^{59} +(378.617 + 655.784i) q^{61} +(124.458 + 215.568i) q^{65} +(-422.172 + 731.223i) q^{67} -859.728 q^{71} -789.079 q^{73} +(395.849 - 685.631i) q^{77} +(108.955 + 188.715i) q^{79} +(50.5889 + 87.6225i) q^{83} +(-189.985 + 329.063i) q^{85} -1247.68 q^{89} +753.685 q^{91} +(-130.775 + 226.508i) q^{95} +(873.504 + 1512.95i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 8 q^{11} - 4 q^{13} - 32 q^{17} + 160 q^{19} + 200 q^{23} - 8 q^{25} + 216 q^{29} - 80 q^{31} - 816 q^{35} - 552 q^{37} + 384 q^{41} - 160 q^{43} + 768 q^{47} + 268 q^{49} - 1888 q^{53} + 608 q^{55} + 992 q^{59} + 548 q^{61} + 1328 q^{65} - 464 q^{67} - 3440 q^{71} - 1528 q^{73} + 1728 q^{77} - 688 q^{79} + 2128 q^{83} + 1324 q^{85} - 4224 q^{89} + 3552 q^{91} + 2056 q^{95} + 1816 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 8 * q^11 - 4 * q^13 - 32 * q^17 + 160 * q^19 + 200 * q^23 - 8 * q^25 + 216 * q^29 - 80 * q^31 - 816 * q^35 - 552 * q^37 + 384 * q^41 - 160 * q^43 + 768 * q^47 + 268 * q^49 - 1888 * q^53 + 608 * q^55 + 992 * q^59 + 548 * q^61 + 1328 * q^65 - 464 * q^67 - 3440 * q^71 - 1528 * q^73 + 1728 * q^77 - 688 * q^79 + 2128 * q^83 + 1324 * q^85 - 4224 * q^89 + 3552 * q^91 + 2056 * q^95 + 1816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.16966	−	5.49001i	0.283503	−	0.491042i	−0.688742	−	0.725007i	$-0.741837\pi$
$5$	3.16966	−	5.49001i	0.283503	−	0.491042i	0.972245	+	0.233965i	$0.0751701\pi$
$6$		0			0
$7$	−9.59728	−	16.6230i	−0.518205	−	0.897557i	−0.999776	−	0.0211500i	$-0.993267\pi$
$7$	−9.59728	−	16.6230i	−0.518205	−	0.897557i	0.481572	−	0.876407i	$-0.340066\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.6230	+	35.7201i	0.565279	+	0.979091i	0.997024	+	0.0770955i	$0.0245646\pi$
$11$	20.6230	+	35.7201i	0.565279	+	0.979091i	−0.431745	+	0.901996i	$0.642102\pi$
$12$		0			0
$13$	−19.6328	+	34.0049i	−0.418857	+	0.725482i	−0.995825	−	0.0912848i	$-0.970903\pi$
$13$	−19.6328	+	34.0049i	−0.418857	+	0.725482i	0.576967	+	0.816767i	$0.304236\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−59.9385			−0.855131			−0.427565	−	0.903984i	$-0.640629\pi$
$17$	−59.9385			−0.855131			−0.427565	+	0.903984i	$0.640629\pi$
$18$		0			0
$19$	−41.2583			−0.498173			−0.249087	−	0.968481i	$-0.580130\pi$
$19$	−41.2583			−0.498173			−0.249087	+	0.968481i	$0.580130\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−26.9040	+	46.5990i	−0.243907	+	0.422460i	−0.961824	−	0.273669i	$-0.911763\pi$
$23$	−26.9040	+	46.5990i	−0.243907	+	0.422460i	0.717917	+	0.696129i	$0.245096\pi$
$24$		0			0
$25$	42.4065	+	73.4502i	0.339252	+	0.587602i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−32.6297	−	56.5164i	−0.208938	−	0.361891i	0.742443	−	0.669910i	$-0.233667\pi$
$29$	−32.6297	−	56.5164i	−0.208938	−	0.361891i	−0.951380	+	0.308019i	$0.900334\pi$
$30$		0			0
$31$	−32.3747	+	56.0745i	−0.187570	+	0.324880i	−0.944439	−	0.328686i	$-0.893394\pi$
$31$	−32.3747	+	56.0745i	−0.187570	+	0.324880i	0.756870	+	0.653566i	$0.226728\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−121.681			−0.587650
$36$		0			0
$37$	293.515			1.30415			0.652076	−	0.758154i	$-0.273898\pi$
$37$	293.515			1.30415			0.652076	+	0.758154i	$0.273898\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	103.636	−	179.503i	0.394762	−	0.683748i	−0.598309	−	0.801266i	$-0.704160\pi$
$41$	103.636	−	179.503i	0.394762	−	0.683748i	0.993071	+	0.117518i	$0.0374937\pi$
$42$		0			0
$43$	188.999	+	327.355i	0.670280	+	1.16096i	0.977825	+	0.209425i	$0.0671592\pi$
$43$	188.999	+	327.355i	0.670280	+	1.16096i	−0.307545	+	0.951534i	$0.599507\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	161.891	+	280.404i	0.502432	+	0.870238i	0.999996	+	0.00281042i	$0.000894586\pi$
$47$	161.891	+	280.404i	0.502432	+	0.870238i	−0.497564	+	0.867427i	$0.665772\pi$
$48$		0			0
$49$	−12.7157	+	22.0243i	−0.0370721	+	0.0642107i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	340.588			0.882706			0.441353	−	0.897334i	$-0.354499\pi$
$53$	340.588			0.882706			0.441353	+	0.897334i	$0.354499\pi$
$54$		0			0
$55$	261.471			0.641033
$56$		0			0
$57$		0			0
$58$		0			0
$59$	203.481	−	352.439i	0.448999	−	0.777690i	−0.549322	−	0.835611i	$-0.685114\pi$
$59$	203.481	−	352.439i	0.448999	−	0.777690i	0.998321	+	0.0579213i	$0.0184472\pi$
$60$		0			0
$61$	378.617	+	655.784i	0.794705	+	1.37647i	0.923027	+	0.384736i	$0.125708\pi$
$61$	378.617	+	655.784i	0.794705	+	1.37647i	−0.128322	+	0.991733i	$0.540959\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	124.458	+	215.568i	0.237495	+	0.411353i
$66$		0			0
$67$	−422.172	+	731.223i	−0.769799	+	1.33333i	0.167873	+	0.985809i	$0.446310\pi$
$67$	−422.172	+	731.223i	−0.769799	+	1.33333i	−0.937672	+	0.347522i	$0.887023\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−859.728			−1.43706			−0.718528	−	0.695499i	$-0.755184\pi$
$71$	−859.728			−1.43706			−0.718528	+	0.695499i	$0.755184\pi$
$72$		0			0
$73$	−789.079			−1.26513			−0.632566	−	0.774506i	$-0.717998\pi$
$73$	−789.079			−1.26513			−0.632566	+	0.774506i	$0.717998\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	395.849	−	685.631i	0.585860	−	1.01474i
$78$		0			0
$79$	108.955	+	188.715i	0.155169	+	0.268761i	0.933121	−	0.359564i	$-0.117074\pi$
$79$	108.955	+	188.715i	0.155169	+	0.268761i	−0.777952	+	0.628324i	$0.783741\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	50.5889	+	87.6225i	0.0669018	+	0.115877i	0.897536	−	0.440941i	$-0.145355\pi$
$83$	50.5889	+	87.6225i	0.0669018	+	0.115877i	−0.830634	+	0.556818i	$0.812022\pi$
$84$		0			0
$85$	−189.985	+	329.063i	−0.242432	+	0.419905i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1247.68			−1.48600			−0.742998	−	0.669294i	$-0.766597\pi$
$89$	−1247.68			−1.48600			−0.742998	+	0.669294i	$0.766597\pi$
$90$		0			0
$91$	753.685			0.868216
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−130.775	+	226.508i	−0.141234	+	0.244624i
$96$		0			0
$97$	873.504	+	1512.95i	0.914340	+	1.58368i	0.807865	+	0.589367i	$0.200623\pi$
$97$	873.504	+	1512.95i	0.914340	+	1.58368i	0.106474	+	0.994315i	$0.466044\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	705.269	+	1221.56i	0.694821	+	1.20346i	0.970241	+	0.242141i	$0.0778497\pi$
$101$	705.269	+	1221.56i	0.694821	+	1.20346i	−0.275420	+	0.961324i	$0.588817\pi$
$102$		0			0
$103$	−439.941	+	761.999i	−0.420861	+	0.728952i	−0.996024	−	0.0890867i	$-0.971605\pi$
$103$	−439.941	+	761.999i	−0.420861	+	0.728952i	0.575163	+	0.818039i	$0.304938\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−504.967			−0.456234			−0.228117	−	0.973634i	$-0.573257\pi$
$107$	−504.967			−0.456234			−0.228117	+	0.973634i	$0.573257\pi$
$108$		0			0
$109$	726.560			0.638457			0.319229	−	0.947678i	$-0.396576\pi$
$109$	726.560			0.638457			0.319229	+	0.947678i	$0.396576\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	759.382	−	1315.29i	0.632182	−	1.09497i	−0.354922	−	0.934896i	$-0.615493\pi$
$113$	759.382	−	1315.29i	0.632182	−	1.09497i	0.987105	−	0.160076i	$-0.0511740\pi$
$114$		0			0
$115$	170.553	+	295.406i	0.138297	+	0.239537i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	575.247	+	996.357i	0.443133	+	0.767528i
$120$		0			0
$121$	−185.115	+	320.628i	−0.139080	+	0.240893i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1330.07			0.951722
$126$		0			0
$127$	2119.86			1.48116			0.740578	−	0.671970i	$-0.234552\pi$
$127$	2119.86			1.48116			0.740578	+	0.671970i	$0.234552\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1249.87	+	2164.84i	−0.833599	+	1.44384i	0.0615662	+	0.998103i	$0.480390\pi$
$131$	−1249.87	+	2164.84i	−0.833599	+	1.44384i	−0.895166	+	0.445734i	$0.852943\pi$
$132$		0			0
$133$	395.967	+	685.836i	0.258156	+	0.447139i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−686.556	−	1189.15i	−0.428149	−	0.741576i	0.568559	−	0.822642i	$-0.307501\pi$
$137$	−686.556	−	1189.15i	−0.428149	−	0.741576i	−0.996709	+	0.0810659i	$0.974168\pi$
$138$		0			0
$139$	−799.067	+	1384.02i	−0.487597	+	0.844542i	−0.999898	−	0.0142633i	$-0.995460\pi$
$139$	−799.067	+	1384.02i	−0.487597	+	0.844542i	0.512302	+	0.858806i	$0.328793\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1619.54			−0.947085
$144$		0			0
$145$	−413.701			−0.236938
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1247.98	−	2161.57i	0.686167	−	1.18848i	−0.286902	−	0.957960i	$-0.592625\pi$
$149$	1247.98	−	2161.57i	0.686167	−	1.18848i	0.973069	−	0.230516i	$-0.0740413\pi$
$150$		0			0
$151$	988.977	+	1712.96i	0.532992	+	0.923169i	0.999258	+	0.0385243i	$0.0122657\pi$
$151$	988.977	+	1712.96i	0.532992	+	0.923169i	−0.466266	+	0.884645i	$0.654401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	205.233	+	355.475i	0.106353	+	0.184209i
$156$		0			0
$157$	−76.6389	+	132.742i	−0.0389583	+	0.0674777i	−0.884847	−	0.465882i	$-0.845737\pi$
$157$	−76.6389	+	132.742i	−0.0389583	+	0.0674777i	0.845889	+	0.533359i	$0.179071\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1032.82			0.505575
$162$		0			0
$163$	55.9069			0.0268648			0.0134324	−	0.999910i	$-0.495724\pi$
$163$	55.9069			0.0268648			0.0134324	+	0.999910i	$0.495724\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−201.617	+	349.211i	−0.0934226	+	0.161813i	−0.908949	−	0.416907i	$-0.863114\pi$
$167$	−201.617	+	349.211i	−0.0934226	+	0.161813i	0.815527	+	0.578720i	$0.196447\pi$
$168$		0			0
$169$	327.610	+	567.437i	0.149117	+	0.258278i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−932.870	−	1615.78i	−0.409970	−	0.710089i	0.584916	−	0.811094i	$-0.301128\pi$
$173$	−932.870	−	1615.78i	−0.409970	−	0.710089i	−0.994886	+	0.101005i	$0.967794\pi$
$174$		0			0
$175$	813.974	−	1409.85i	0.351604	−	0.608996i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2223.69			−0.928527			−0.464263	−	0.885697i	$-0.653681\pi$
$179$	−2223.69			−0.928527			−0.464263	+	0.885697i	$0.653681\pi$
$180$		0			0
$181$	−3287.28			−1.34995			−0.674976	−	0.737840i	$-0.735846\pi$
$181$	−3287.28			−1.34995			−0.674976	+	0.737840i	$0.735846\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	930.344	−	1611.40i	0.369731	−	0.640393i
$186$		0			0
$187$	−1236.11	−	2141.01i	−0.483387	−	0.837251i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2050.07	+	3550.83i	0.776638	+	1.34518i	0.933869	+	0.357615i	$0.116410\pi$
$191$	2050.07	+	3550.83i	0.776638	+	1.34518i	−0.157231	+	0.987562i	$0.550257\pi$
$192$		0			0
$193$	1970.77	−	3413.47i	0.735020	−	1.27309i	−0.219695	−	0.975569i	$-0.570506\pi$
$193$	1970.77	−	3413.47i	0.735020	−	1.27309i	0.954715	−	0.297523i	$-0.0961605\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	466.195			0.168604			0.0843020	−	0.996440i	$-0.473134\pi$
$197$	466.195			0.168604			0.0843020	+	0.996440i	$0.473134\pi$
$198$		0			0
$199$	−2136.33			−0.761007			−0.380503	−	0.924780i	$-0.624249\pi$
$199$	−2136.33			−0.761007			−0.380503	+	0.924780i	$0.624249\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−626.314	+	1084.81i	−0.216545	+	0.375067i
$204$		0			0
$205$	−656.982	−	1137.93i	−0.223832	−	0.387689i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−850.869	−	1473.75i	−0.281607	−	0.487757i
$210$		0			0
$211$	−2072.67	+	3589.97i	−0.676249	+	1.17130i	0.299853	+	0.953985i	$0.403062\pi$
$211$	−2072.67	+	3589.97i	−0.676249	+	1.17130i	−0.976102	+	0.217312i	$0.930271\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2396.25			0.760106
$216$		0			0
$217$	1242.83			0.388798
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1176.76	−	2038.20i	0.358178	−	0.620382i
$222$		0			0
$223$	−3255.78	−	5639.18i	−0.977682	−	1.69340i	−0.670783	−	0.741654i	$-0.734042\pi$
$223$	−3255.78	−	5639.18i	−0.977682	−	1.69340i	−0.306899	−	0.951742i	$-0.599291\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1786.96	−	3095.11i	−0.522488	−	0.904975i	−0.999658	−	0.0261642i	$-0.991671\pi$
$227$	−1786.96	−	3095.11i	−0.522488	−	0.904975i	0.477170	−	0.878811i	$-0.341663\pi$
$228$		0			0
$229$	924.632	−	1601.51i	0.266818	−	0.462143i	−0.701220	−	0.712945i	$-0.747361\pi$
$229$	924.632	−	1601.51i	0.266818	−	0.462143i	0.968038	+	0.250802i	$0.0806943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5359.31			1.50687			0.753434	−	0.657524i	$-0.228396\pi$
$233$	5359.31			1.50687			0.753434	+	0.657524i	$0.228396\pi$
$234$		0			0
$235$	2052.56			0.569764
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2667.24	+	4619.80i	−0.721881	+	1.25033i	0.238364	+	0.971176i	$0.423389\pi$
$239$	−2667.24	+	4619.80i	−0.721881	+	1.25033i	−0.960245	+	0.279158i	$0.909945\pi$
$240$		0			0
$241$	1163.69	+	2015.58i	0.311038	+	0.538733i	0.978587	−	0.205832i	$-0.0659902\pi$
$241$	1163.69	+	2015.58i	0.311038	+	0.538733i	−0.667550	+	0.744565i	$0.732657\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	80.6091	+	139.619i	0.0210201	+	0.0364079i
$246$		0			0
$247$	810.014	−	1402.98i	0.208664	−	0.361416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−938.870			−0.236099			−0.118050	−	0.993008i	$-0.537664\pi$
$251$	−938.870			−0.236099			−0.118050	+	0.993008i	$0.537664\pi$
$252$		0			0
$253$	−2219.36			−0.551502
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2191.72	+	3796.17i	−0.531967	+	0.921395i	0.467336	+	0.884080i	$0.345214\pi$
$257$	−2191.72	+	3796.17i	−0.531967	+	0.921395i	−0.999304	+	0.0373149i	$0.988120\pi$
$258$		0			0
$259$	−2816.95	−	4879.10i	−0.675817	−	1.17055i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2189.66	−	3792.60i	−0.513384	−	0.889208i	−0.999879	−	0.0155246i	$-0.995058\pi$
$263$	−2189.66	−	3792.60i	−0.513384	−	0.889208i	0.486495	−	0.873683i	$-0.338275\pi$
$264$		0			0
$265$	1079.55	−	1869.83i	0.250250	−	0.433445i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1981.97			0.449231			0.224615	−	0.974447i	$-0.427887\pi$
$269$	1981.97			0.449231			0.224615	+	0.974447i	$0.427887\pi$
$270$		0			0
$271$	806.801			0.180847			0.0904237	−	0.995903i	$-0.471178\pi$
$271$	806.801			0.180847			0.0904237	+	0.995903i	$0.471178\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1749.10	+	3029.53i	−0.383544	+	0.664317i
$276$		0			0
$277$	−1.27779	−	2.21320i	−0.000277166	−	0.000480066i	0.865887	−	0.500240i	$-0.166755\pi$
$277$	−1.27779	−	2.21320i	−0.000277166	−	0.000480066i	−0.866164	+	0.499760i	$0.833422\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−41.3095	−	71.5501i	−0.00876981	−	0.0151897i	0.861607	−	0.507576i	$-0.169458\pi$
$281$	−41.3095	−	71.5501i	−0.00876981	−	0.0151897i	−0.870377	+	0.492386i	$0.836125\pi$
$282$		0			0
$283$	2224.13	−	3852.30i	0.467176	−	0.809172i	−0.532121	−	0.846668i	$-0.678605\pi$
$283$	2224.13	−	3852.30i	0.467176	−	0.809172i	0.999297	+	0.0374962i	$0.0119382\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3978.50			−0.818270
$288$		0			0
$289$	−1320.38			−0.268752
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1706.78	+	2956.23i	−0.340311	+	0.589437i	−0.984490	−	0.175438i	$-0.943866\pi$
$293$	−1706.78	+	2956.23i	−0.340311	+	0.589437i	0.644179	+	0.764875i	$0.277199\pi$
$294$		0			0
$295$	−1289.93	−	2234.23i	−0.254585	−	0.440955i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1056.40	−	1829.74i	−0.204325	−	0.353901i
$300$		0			0
$301$	3627.75	−	6283.44i	0.694684	−	1.20323i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4800.35			0.901205
$306$		0			0
$307$	2102.23			0.390817			0.195409	−	0.980722i	$-0.437397\pi$
$307$	2102.23			0.390817			0.195409	+	0.980722i	$0.437397\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3546.10	+	6142.02i	−0.646562	+	1.11988i	0.337377	+	0.941370i	$0.390460\pi$
$311$	−3546.10	+	6142.02i	−0.646562	+	1.11988i	−0.983938	+	0.178508i	$0.942873\pi$
$312$		0			0
$313$	−5242.71	−	9080.64i	−0.946759	−	1.63983i	−0.752191	−	0.658945i	$-0.771003\pi$
$313$	−5242.71	−	9080.64i	−0.946759	−	1.63983i	−0.194567	−	0.980889i	$-0.562330\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3758.74	−	6510.34i	−0.665969	−	1.15349i	−0.979022	−	0.203756i	$-0.934685\pi$
$317$	−3758.74	−	6510.34i	−0.665969	−	1.15349i	0.313053	−	0.949736i	$-0.398648\pi$
$318$		0			0
$319$	1345.85	−	2331.07i	0.236216	−	0.409138i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2472.96			0.426003
$324$		0			0
$325$	−3330.23			−0.568393
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3107.44	−	5382.24i	0.520725	−	0.901922i
$330$		0			0
$331$	−1113.57	−	1928.76i	−0.184916	−	0.320284i	0.758632	−	0.651519i	$-0.225868\pi$
$331$	−1113.57	−	1928.76i	−0.184916	−	0.320284i	−0.943548	+	0.331235i	$0.892535\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2676.28	+	4635.46i	0.436480	+	0.756006i
$336$		0			0
$337$	−1961.76	+	3397.87i	−0.317104	+	0.549240i	−0.979882	−	0.199575i	$-0.936044\pi$
$337$	−1961.76	+	3397.87i	−0.317104	+	0.549240i	0.662779	+	0.748815i	$0.269377\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2670.65			−0.424116
$342$		0			0
$343$	−6095.59			−0.959566
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6054.23	−	10486.2i	0.936623	−	1.62228i	0.164908	−	0.986309i	$-0.447267\pi$
$347$	6054.23	−	10486.2i	0.936623	−	1.62228i	0.771715	−	0.635969i	$-0.219399\pi$
$348$		0			0
$349$	−2793.89	−	4839.16i	−0.428520	−	0.742219i	0.568222	−	0.822875i	$-0.307632\pi$
$349$	−2793.89	−	4839.16i	−0.428520	−	0.742219i	−0.996742	+	0.0806569i	$0.974298\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1957.08	+	3389.77i	0.295085	+	0.511102i	0.975005	−	0.222184i	$-0.0713187\pi$
$353$	1957.08	+	3389.77i	0.295085	+	0.511102i	−0.679920	+	0.733287i	$0.737985\pi$
$354$		0			0
$355$	−2725.05	+	4719.92i	−0.407410	+	0.705654i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9037.04			1.32857			0.664285	−	0.747479i	$-0.268736\pi$
$359$	9037.04			1.32857			0.664285	+	0.747479i	$0.268736\pi$
$360$		0			0
$361$	−5156.76			−0.751823
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2501.11	+	4332.05i	−0.358669	+	0.621233i
$366$		0			0
$367$	2141.37	+	3708.95i	0.304573	+	0.527536i	0.977166	−	0.212476i	$-0.0681529\pi$
$367$	2141.37	+	3708.95i	0.304573	+	0.527536i	−0.672593	+	0.740013i	$0.734820\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3268.72	−	5661.59i	−0.457422	−	0.792278i
$372$		0			0
$373$	1453.56	−	2517.63i	0.201775	−	0.349485i	−0.747325	−	0.664459i	$-0.768662\pi$
$373$	1453.56	−	2517.63i	0.201775	−	0.349485i	0.949101	+	0.314973i	$0.101996\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2562.45			0.350060
$378$		0			0
$379$	−2096.88			−0.284194			−0.142097	−	0.989853i	$-0.545385\pi$
$379$	−2096.88			−0.284194			−0.142097	+	0.989853i	$0.545385\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3587.46	+	6213.66i	−0.478618	+	0.828990i	−0.999699	−	0.0245165i	$-0.992195\pi$
$383$	−3587.46	+	6213.66i	−0.478618	+	0.828990i	0.521082	+	0.853507i	$0.325529\pi$
$384$		0			0
$385$	−2509.42	−	4346.44i	−0.332186	−	0.575363i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1368.85	−	2370.92i	−0.178415	−	0.309025i	0.762923	−	0.646490i	$-0.223764\pi$
$389$	−1368.85	−	2370.92i	−0.178415	−	0.309025i	−0.941338	+	0.337465i	$0.890430\pi$
$390$		0			0
$391$	1612.58	−	2793.08i	0.208573	−	0.361258i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1381.40			0.175964
$396$		0			0
$397$	1561.87			0.197451			0.0987256	−	0.995115i	$-0.468523\pi$
$397$	1561.87			0.197451			0.0987256	+	0.995115i	$0.468523\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2426.93	−	4203.57i	0.302232	−	0.523482i	−0.674409	−	0.738358i	$-0.735601\pi$
$401$	2426.93	−	4203.57i	0.302232	−	0.523482i	0.976641	+	0.214876i	$0.0689348\pi$
$402$		0			0
$403$	−1271.21	−	2201.80i	−0.157130	−	0.272157i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6053.16	+	10484.4i	0.737209	+	1.27688i
$408$		0			0
$409$	−571.868	+	990.504i	−0.0691370	+	0.119749i	−0.898522	−	0.438929i	$-0.855358\pi$
$409$	−571.868	+	990.504i	−0.0691370	+	0.119749i	0.829385	+	0.558678i	$0.188691\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7811.46			−0.930694
$414$		0			0
$415$	641.398			0.0758675
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−660.371	+	1143.80i	−0.0769958	+	0.133361i	−0.901952	−	0.431835i	$-0.857866\pi$
$419$	−660.371	+	1143.80i	−0.0769958	+	0.133361i	0.824957	+	0.565196i	$0.191199\pi$
$420$		0			0
$421$	5641.34	+	9771.08i	0.653069	+	1.13115i	0.982374	+	0.186925i	$0.0598521\pi$
$421$	5641.34	+	9771.08i	0.653069	+	1.13115i	−0.329305	+	0.944223i	$0.606815\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2541.78	−	4402.50i	−0.290105	−	0.502476i
$426$		0			0
$427$	7267.40	−	12587.5i	0.823639	−	1.42658i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5443.17			0.608325			0.304163	−	0.952620i	$-0.401623\pi$
$431$	5443.17			0.608325			0.304163	+	0.952620i	$0.401623\pi$
$432$		0			0
$433$	4235.84			0.470119			0.235060	−	0.971981i	$-0.424471\pi$
$433$	4235.84			0.470119			0.235060	+	0.971981i	$0.424471\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1110.01	−	1922.60i	0.121508	−	0.210458i
$438$		0			0
$439$	7080.35	+	12263.5i	0.769765	+	1.33327i	0.937690	+	0.347472i	$0.112960\pi$
$439$	7080.35	+	12263.5i	0.769765	+	1.33327i	−0.167926	+	0.985800i	$0.553707\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7835.50	−	13571.5i	−0.840352	−	1.45553i	−0.889598	−	0.456745i	$-0.849015\pi$
$443$	−7835.50	−	13571.5i	−0.840352	−	1.45553i	0.0492461	−	0.998787i	$-0.484318\pi$
$444$		0			0
$445$	−3954.72	+	6849.77i	−0.421284	+	0.729686i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14045.2			−1.47624			−0.738122	−	0.674667i	$-0.764287\pi$
$449$	−14045.2			−1.47624			−0.738122	+	0.674667i	$0.764287\pi$
$450$		0			0
$451$	8549.14			0.892602
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2388.92	−	4137.74i	0.246142	−	0.426330i
$456$		0			0
$457$	445.435	+	771.517i	0.0455943	+	0.0789716i	0.887922	−	0.459994i	$-0.152149\pi$
$457$	445.435	+	771.517i	0.0455943	+	0.0789716i	−0.842328	+	0.538966i	$0.818815\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3990.93	+	6912.50i	0.403202	+	0.698367i	0.994110	−	0.108372i	$-0.0345638\pi$
$461$	3990.93	+	6912.50i	0.403202	+	0.698367i	−0.590908	+	0.806739i	$0.701230\pi$
$462$		0			0
$463$	−1650.38	+	2858.54i	−0.165658	+	0.286928i	−0.936889	−	0.349628i	$-0.886308\pi$
$463$	−1650.38	+	2858.54i	−0.165658	+	0.286928i	0.771231	+	0.636555i	$0.219641\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11181.4			−1.10795			−0.553977	−	0.832532i	$-0.686891\pi$
$467$	−11181.4			−1.10795			−0.553977	+	0.832532i	$0.686891\pi$
$468$		0			0
$469$	16206.8			1.59565
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7795.43	+	13502.1i	−0.757789	+	1.31253i
$474$		0			0
$475$	−1749.62	−	3030.43i	−0.169006	−	0.292728i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1444.77	+	2502.42i	0.137815	+	0.238703i	0.926669	−	0.375878i	$-0.122659\pi$
$479$	1444.77	+	2502.42i	0.137815	+	0.238703i	−0.788854	+	0.614580i	$0.789325\pi$
$480$		0			0
$481$	−5762.51	+	9980.97i	−0.546254	+	0.946139i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11074.9			1.03687
$486$		0			0
$487$	−13486.6			−1.25490			−0.627448	−	0.778658i	$-0.715901\pi$
$487$	−13486.6			−1.25490			−0.627448	+	0.778658i	$0.715901\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10544.0	−	18262.8i	0.969134	−	1.67859i	0.271061	−	0.962562i	$-0.412625\pi$
$491$	10544.0	−	18262.8i	0.969134	−	1.67859i	0.698073	−	0.716027i	$-0.254041\pi$
$492$		0			0
$493$	1955.78	+	3387.51i	0.178669	+	0.309464i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8251.05	+	14291.2i	0.744689	+	1.28984i
$498$		0			0
$499$	−4935.08	+	8547.80i	−0.442734	+	0.766838i	−0.997891	−	0.0649073i	$-0.979325\pi$
$499$	−4935.08	+	8547.80i	−0.442734	+	0.766838i	0.555157	+	0.831746i	$0.312658\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3945.75			0.349766			0.174883	−	0.984589i	$-0.444045\pi$
$503$	3945.75			0.349766			0.174883	+	0.984589i	$0.444045\pi$
$504$		0			0
$505$	8941.86			0.787935
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6802.07	+	11781.5i	−0.592331	+	1.02595i	0.401587	+	0.915821i	$0.368459\pi$
$509$	−6802.07	+	11781.5i	−0.592331	+	1.02595i	−0.993918	+	0.110126i	$0.964875\pi$
$510$		0			0
$511$	7573.01	+	13116.8i	0.655597	+	1.13553i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2788.92	+	4830.56i	0.238631	+	0.413320i
$516$		0			0
$517$	−6677.37	+	11565.5i	−0.568028	+	0.983853i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8044.57			−0.676467			−0.338233	−	0.941062i	$-0.609829\pi$
$521$	−8044.57			−0.676467			−0.338233	+	0.941062i	$0.609829\pi$
$522$		0			0
$523$	7785.42			0.650923			0.325462	−	0.945555i	$-0.394480\pi$
$523$	7785.42			0.650923			0.325462	+	0.945555i	$0.394480\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1940.49	−	3361.02i	0.160397	−	0.277815i
$528$		0			0
$529$	4635.85	+	8029.53i	0.381019	+	0.659944i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4069.32	+	7048.28i	0.330698	+	0.572786i
$534$		0			0
$535$	−1600.57	+	2772.28i	−0.129344	+	0.224030i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1048.94			−0.0838242
$540$		0			0
$541$	−2520.37			−0.200294			−0.100147	−	0.994973i	$-0.531931\pi$
$541$	−2520.37			−0.200294			−0.100147	+	0.994973i	$0.531931\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2302.95	−	3988.82i	0.181005	−	0.313509i
$546$		0			0
$547$	621.636	+	1076.71i	0.0485910	+	0.0841620i	0.889298	−	0.457328i	$-0.151194\pi$
$547$	621.636	+	1076.71i	0.0485910	+	0.0841620i	−0.840707	+	0.541490i	$0.817860\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1346.25	+	2331.77i	0.104087	+	0.180284i
$552$		0			0
$553$	2091.34	−	3622.30i	0.160819	−	0.278546i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14215.4			−1.08137			−0.540687	−	0.841224i	$-0.681836\pi$
$557$	−14215.4			−1.08137			−0.540687	+	0.841224i	$0.681836\pi$
$558$		0			0
$559$	−14842.3			−1.12301
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10806.2	−	18716.9i	0.808928	−	1.40110i	−0.104679	−	0.994506i	$-0.533381\pi$
$563$	10806.2	−	18716.9i	0.808928	−	1.40110i	0.913607	−	0.406599i	$-0.133285\pi$
$564$		0			0
$565$	−4813.96	−	8338.03i	−0.358451	−	0.620856i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5692.80	+	9860.22i	0.419428	+	0.726471i	0.995882	−	0.0906586i	$-0.0288972\pi$
$569$	5692.80	+	9860.22i	0.419428	+	0.726471i	−0.576454	+	0.817130i	$0.695564\pi$
$570$		0			0
$571$	−11632.7	+	20148.4i	−0.852560	+	1.47668i	0.0263309	+	0.999653i	$0.491618\pi$
$571$	−11632.7	+	20148.4i	−0.852560	+	1.47668i	−0.878891	+	0.477023i	$0.841716\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4563.61			−0.330984
$576$		0			0
$577$	−5871.82			−0.423652			−0.211826	−	0.977307i	$-0.567941\pi$
$577$	−5871.82			−0.423652			−0.211826	+	0.977307i	$0.567941\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	971.032	−	1681.88i	0.0693377	−	0.120096i
$582$		0			0
$583$	7023.95	+	12165.8i	0.498975	+	0.864249i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4871.80	−	8438.21i	−0.342557	−	0.593325i	0.642350	−	0.766411i	$-0.277960\pi$
$587$	−4871.80	−	8438.21i	−0.342557	−	0.593325i	−0.984907	+	0.173086i	$0.944626\pi$
$588$		0			0
$589$	1335.72	−	2313.54i	0.0934422	−	0.161847i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11058.0			−0.765764			−0.382882	−	0.923797i	$-0.625068\pi$
$593$	−11058.0			−0.765764			−0.382882	+	0.923797i	$0.625068\pi$
$594$		0			0
$595$	7293.35			0.502518
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13174.1	−	22818.2i	0.898629	−	1.55647i	0.0693819	−	0.997590i	$-0.477897\pi$
$599$	13174.1	−	22818.2i	0.898629	−	1.55647i	0.829247	−	0.558882i	$-0.188769\pi$
$600$		0			0
$601$	7555.17	+	13085.9i	0.512782	+	0.888164i	0.999890	+	0.0148226i	$0.00471836\pi$
$601$	7555.17	+	13085.9i	0.512782	+	0.888164i	−0.487108	+	0.873342i	$0.661948\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1173.50	+	2032.57i	0.0788590	+	0.136588i
$606$		0			0
$607$	6741.37	−	11676.4i	0.450780	−	0.780774i	−0.547654	−	0.836705i	$-0.684479\pi$
$607$	6741.37	−	11676.4i	0.450780	−	0.780774i	0.998435	+	0.0559303i	$0.0178125\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12713.5			−0.841789
$612$		0			0
$613$	12821.6			0.844798			0.422399	−	0.906410i	$-0.361188\pi$
$613$	12821.6			0.844798			0.422399	+	0.906410i	$0.361188\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9062.50	+	15696.7i	−0.591317	+	1.02419i	0.402738	+	0.915315i	$0.368058\pi$
$617$	−9062.50	+	15696.7i	−0.591317	+	1.02419i	−0.994055	+	0.108876i	$0.965275\pi$
$618$		0			0
$619$	11749.9	+	20351.4i	0.762954	+	1.32148i	0.941322	+	0.337511i	$0.109585\pi$
$619$	11749.9	+	20351.4i	0.762954	+	1.32148i	−0.178368	+	0.983964i	$0.557082\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11974.3	+	20740.1i	0.770050	+	1.33377i
$624$		0			0
$625$	−1084.94	+	1879.16i	−0.0694359	+	0.120266i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−17592.9			−1.11522
$630$		0			0
$631$	−4893.60			−0.308734			−0.154367	−	0.988014i	$-0.549334\pi$
$631$	−4893.60			−0.308734			−0.154367	+	0.988014i	$0.549334\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6719.23	−	11638.0i	0.419912	−	0.727310i
$636$		0			0
$637$	−499.289	−	864.795i	−0.0310558	−	0.0537903i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14392.0	−	24927.7i	−0.886816	−	1.53601i	−0.843618	−	0.536944i	$-0.819578\pi$
$641$	−14392.0	−	24927.7i	−0.886816	−	1.53601i	−0.0431987	−	0.999067i	$-0.513755\pi$
$642$		0			0
$643$	448.147	−	776.214i	0.0274855	−	0.0476064i	−0.851955	−	0.523614i	$-0.824583\pi$
$643$	448.147	−	776.214i	0.0274855	−	0.0476064i	0.879441	+	0.476008i	$0.157917\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30780.9			1.87036			0.935179	−	0.354174i	$-0.115238\pi$
$647$	30780.9			1.87036			0.935179	+	0.354174i	$0.115238\pi$
$648$		0			0
$649$	16785.5			1.01524
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6867.06	−	11894.1i	0.411529	−	0.712790i	−0.583528	−	0.812093i	$-0.698328\pi$
$653$	6867.06	−	11894.1i	0.411529	−	0.712790i	0.995057	+	0.0993032i	$0.0316614\pi$
$654$		0			0
$655$	7923.32	+	13723.6i	0.472656	+	0.818664i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8293.60	−	14364.9i	−0.490247	−	0.849133i	0.509690	−	0.860358i	$-0.329760\pi$
$659$	−8293.60	−	14364.9i	−0.490247	−	0.849133i	−0.999937	+	0.0112252i	$0.996427\pi$
$660$		0			0
$661$	−14123.0	+	24461.8i	−0.831047	+	1.43942i	0.0661617	+	0.997809i	$0.478925\pi$
$661$	−14123.0	+	24461.8i	−0.831047	+	1.43942i	−0.897209	+	0.441607i	$0.854409\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5020.33			0.292752
$666$		0			0
$667$	3511.48			0.203846
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15616.4	+	27048.5i	−0.898459	+	1.55618i
$672$		0			0
$673$	6281.36	+	10879.6i	0.359775	+	0.623149i	0.987923	−	0.154945i	$-0.0495200\pi$
$673$	6281.36	+	10879.6i	0.359775	+	0.623149i	−0.628148	+	0.778094i	$0.716187\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13266.2	+	22977.7i	0.753119	+	1.30444i	0.946304	+	0.323277i	$0.104785\pi$
$677$	13266.2	+	22977.7i	0.753119	+	1.30444i	−0.193186	+	0.981162i	$0.561882\pi$
$678$		0			0
$679$	16766.5	−	29040.5i	0.947630	−	1.64134i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2528.33			0.141645			0.0708226	−	0.997489i	$-0.477438\pi$
$683$	2528.33			0.141645			0.0708226	+	0.997489i	$0.477438\pi$
$684$		0			0
$685$	−8704.60			−0.485527
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6686.69	+	11581.7i	−0.369728	+	0.640388i
$690$		0			0
$691$	−9418.09	−	16312.6i	−0.518496	−	0.898062i	−0.999769	−	0.0214911i	$-0.993159\pi$
$691$	−9418.09	−	16312.6i	−0.518496	−	0.898062i	0.481273	−	0.876571i	$-0.340175\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5065.54	+	8773.77i	0.276470	+	0.478861i
$696$		0			0
$697$	−6211.79	+	10759.1i	−0.337573	+	0.584694i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26573.0			1.43174			0.715869	−	0.698234i	$-0.246031\pi$
$701$	26573.0			1.43174			0.715869	+	0.698234i	$0.246031\pi$
$702$		0			0
$703$	−12109.9			−0.649694
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13537.3	−	23447.4i	0.720119	−	1.24728i
$708$		0			0
$709$	−3110.41	−	5387.39i	−0.164759	−	0.285370i	0.771811	−	0.635852i	$-0.219351\pi$
$709$	−3110.41	−	5387.39i	−0.164759	−	0.285370i	−0.936570	+	0.350482i	$0.886018\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1742.01	−	3017.26i	−0.0914992	−	0.158481i
$714$		0			0
$715$	−5133.41	+	8891.32i	−0.268501	+	0.465058i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13175.7			0.683407			0.341704	−	0.939808i	$-0.388996\pi$
$719$	13175.7			0.683407			0.341704	+	0.939808i	$0.388996\pi$
$720$		0			0
$721$	16888.9			0.872368
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2767.43	−	4793.32i	0.141765	−	0.245544i
$726$		0			0
$727$	10337.5	+	17905.1i	0.527370	+	0.913431i	0.999491	+	0.0318974i	$0.0101550\pi$
$727$	10337.5	+	17905.1i	0.527370	+	0.913431i	−0.472122	+	0.881533i	$0.656512\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11328.3	−	19621.2i	−0.573177	−	0.992771i
$732$		0			0
$733$	−2804.51	+	4857.55i	−0.141319	+	0.244772i	−0.927994	−	0.372596i	$-0.878468\pi$
$733$	−2804.51	+	4857.55i	−0.141319	+	0.244772i	0.786675	+	0.617368i	$0.211801\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−34825.8			−1.74060
$738$		0			0
$739$	14434.8			0.718526			0.359263	−	0.933236i	$-0.383028\pi$
$739$	14434.8			0.718526			0.359263	+	0.933236i	$0.383028\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12384.7	+	21450.9i	−0.611506	+	1.05916i	0.379480	+	0.925200i	$0.376103\pi$
$743$	−12384.7	+	21450.9i	−0.611506	+	1.05916i	−0.990987	+	0.133960i	$0.957231\pi$
$744$		0			0
$745$	−7911.37	−	13702.9i	−0.389061	−	0.673873i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4846.31	+	8394.06i	0.236423	+	0.409496i
$750$		0			0
$751$	−1190.73	+	2062.40i	−0.0578566	+	0.100211i	−0.893503	−	0.449057i	$-0.851760\pi$
$751$	−1190.73	+	2062.40i	−0.0578566	+	0.100211i	0.835646	+	0.549268i	$0.185093\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12538.9			0.604419
$756$		0			0
$757$	14556.5			0.698897			0.349449	−	0.936956i	$-0.386369\pi$
$757$	14556.5			0.698897			0.349449	+	0.936956i	$0.386369\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5052.73	−	8751.59i	0.240685	−	0.416879i	−0.720225	−	0.693741i	$-0.755961\pi$
$761$	5052.73	−	8751.59i	0.240685	−	0.416879i	0.960910	+	0.276862i	$0.0892946\pi$
$762$		0			0
$763$	−6973.00	−	12077.6i	−0.330851	−	0.573051i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7989.78	+	13838.7i	0.376133	+	0.651482i
$768$		0			0
$769$	5815.57	−	10072.9i	0.272711	−	0.472349i	−0.696844	−	0.717223i	$-0.745413\pi$
$769$	5815.57	−	10072.9i	0.272711	−	0.472349i	0.969555	+	0.244873i	$0.0787464\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16482.4			−0.766922			−0.383461	−	0.923557i	$-0.625268\pi$
$773$	−16482.4			−0.766922			−0.383461	+	0.923557i	$0.625268\pi$
$774$		0			0
$775$	−5491.58			−0.254534
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4275.85	+	7405.98i	−0.196660	+	0.340625i
$780$		0			0
$781$	−17730.2	−	30709.5i	−0.812336	−	1.40701i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	485.838	+	841.497i	0.0220896	+	0.0382603i
$786$		0			0
$787$	7300.35	−	12644.6i	0.330660	−	0.572720i	−0.651981	−	0.758235i	$-0.726062\pi$
$787$	7300.35	−	12644.6i	0.330660	−	0.572720i	0.982641	+	0.185515i	$0.0593953\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−29152.0			−1.31040
$792$		0			0
$793$	−29733.2			−1.33147
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10915.1	+	18905.5i	−0.485111	+	0.840237i	−0.999854	−	0.0171080i	$-0.994554\pi$
$797$	−10915.1	+	18905.5i	−0.485111	+	0.840237i	0.514743	+	0.857345i	$0.327887\pi$
$798$		0			0
$799$	−9703.53	−	16807.0i	−0.429645	−	0.744167i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16273.2	−	28185.9i	−0.715152	−	1.23868i
$804$		0			0
$805$	3273.69	−	5670.20i	0.143332	−	0.248259i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14798.7			0.643133			0.321567	−	0.946887i	$-0.395791\pi$
$809$	14798.7			0.643133			0.321567	+	0.946887i	$0.395791\pi$
$810$		0			0
$811$	−23276.2			−1.00782			−0.503908	−	0.863757i	$-0.668105\pi$
$811$	−23276.2			−1.00782			−0.503908	+	0.863757i	$0.668105\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	177.206	−	306.929i	0.00761626	−	0.0131917i
$816$		0			0
$817$	−7797.76	−	13506.1i	−0.333916	−	0.578359i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13670.5	−	23678.0i	−0.581124	−	1.00654i	−0.995346	−	0.0963606i	$-0.969280\pi$
$821$	−13670.5	−	23678.0i	−0.581124	−	1.00654i	0.414223	−	0.910176i	$-0.364054\pi$
$822$		0			0
$823$	7465.04	−	12929.8i	0.316179	−	0.547637i	−0.663509	−	0.748169i	$-0.730933\pi$
$823$	7465.04	−	12929.8i	0.316179	−	0.547637i	0.979687	+	0.200531i	$0.0642668\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10707.3			−0.450219			−0.225109	−	0.974334i	$-0.572274\pi$
$827$	−10707.3			−0.450219			−0.225109	+	0.974334i	$0.572274\pi$
$828$		0			0
$829$	−28515.9			−1.19469			−0.597345	−	0.801984i	$-0.703778\pi$
$829$	−28515.9			−1.19469			−0.597345	+	0.801984i	$0.703778\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	762.161	−	1320.10i	0.0317015	−	0.0549086i
$834$		0			0
$835$	1278.11	+	2213.76i	0.0529712	+	0.0917488i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19211.0	+	33274.5i	0.790510	+	1.36920i	0.925651	+	0.378377i	$0.123518\pi$
$839$	19211.0	+	33274.5i	0.790510	+	1.36920i	−0.135141	+	0.990826i	$0.543149\pi$
$840$		0			0
$841$	10065.1	−	17433.3i	0.412690	−	0.714800i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4153.65			0.169100
$846$		0			0
$847$	7106.40			0.288287
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7896.72	+	13677.5i	−0.318092	+	0.550951i
$852$		0			0
$853$	8348.31	+	14459.7i	0.335100	+	0.580411i	0.983504	−	0.180886i	$-0.0578964\pi$
$853$	8348.31	+	14459.7i	0.335100	+	0.580411i	−0.648404	+	0.761297i	$0.724563\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4668.07	+	8085.33i	0.186066	+	0.322275i	0.943935	−	0.330131i	$-0.107093\pi$
$857$	4668.07	+	8085.33i	0.186066	+	0.322275i	−0.757869	+	0.652406i	$0.773760\pi$
$858$		0			0
$859$	−11451.7	+	19834.8i	−0.454861	+	0.787842i	−0.998680	−	0.0513604i	$-0.983644\pi$
$859$	−11451.7	+	19834.8i	−0.454861	+	0.787842i	0.543819	+	0.839202i	$0.316978\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31426.0			1.23958			0.619789	−	0.784769i	$-0.287218\pi$
$863$	31426.0			1.23958			0.619789	+	0.784769i	$0.287218\pi$
$864$		0			0
$865$	−11827.5			−0.464911
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4493.94	+	7783.73i	−0.175427	+	0.303849i
$870$		0			0
$871$	−16576.8	−	28711.9i	−0.644872	−	1.11695i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12765.1	−	22109.8i	−0.493187	−	0.854225i
$876$		0			0
$877$	19590.0	−	33930.9i	0.754286	−	1.30646i	−0.191443	−	0.981504i	$-0.561317\pi$
$877$	19590.0	−	33930.9i	0.754286	−	1.30646i	0.945729	−	0.324958i	$-0.105350\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23946.2			0.915743			0.457871	−	0.889018i	$-0.348612\pi$
$881$	23946.2			0.915743			0.457871	+	0.889018i	$0.348612\pi$
$882$		0			0
$883$	3969.15			0.151271			0.0756357	−	0.997136i	$-0.475901\pi$
$883$	3969.15			0.151271			0.0756357	+	0.997136i	$0.475901\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4725.15	+	8184.20i	−0.178867	+	0.309807i	−0.941493	−	0.337033i	$-0.890576\pi$
$887$	−4725.15	+	8184.20i	−0.178867	+	0.309807i	0.762626	+	0.646840i	$0.223910\pi$
$888$		0			0
$889$	−20344.9	−	35238.3i	−0.767542	−	1.32942i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6679.36	−	11569.0i	−0.250298	−	0.433529i
$894$		0			0
$895$	−7048.34	+	12208.1i	−0.263240	+	0.455945i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4225.51			0.156761
$900$		0			0
$901$	−20414.4			−0.754829
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10419.5	+	18047.2i	−0.382715	+	0.662883i
$906$		0			0
$907$	−20185.7	−	34962.6i	−0.738980	−	1.27995i	−0.952955	−	0.303111i	$-0.901975\pi$
$907$	−20185.7	−	34962.6i	−0.738980	−	1.27995i	0.213975	−	0.976839i	$-0.431359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8140.33	−	14099.5i	−0.296050	−	0.512773i	0.679179	−	0.733973i	$-0.262336\pi$
$911$	−8140.33	−	14099.5i	−0.296050	−	0.512773i	−0.975229	+	0.221200i	$0.929003\pi$
$912$		0			0
$913$	−2086.59	+	3614.08i	−0.0756363	+	0.131006i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	47981.4			1.72790
$918$		0			0
$919$	17967.5			0.644934			0.322467	−	0.946581i	$-0.395488\pi$
$919$	17967.5			0.644934			0.322467	+	0.946581i	$0.395488\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16878.8	−	29235.0i	0.601921	−	1.04256i
$924$		0			0
$925$	12447.0	+	21558.8i	0.442436	+	0.766322i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8708.28	−	15083.2i	−0.307545	−	0.532684i	0.670280	−	0.742109i	$-0.266174\pi$
$929$	−8708.28	−	15083.2i	−0.307545	−	0.532684i	−0.977825	+	0.209425i	$0.932841\pi$
$930$		0			0
$931$	524.629	−	908.684i	0.0184683	−	0.0319881i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15672.2			−0.548167
$936$		0			0
$937$	12121.1			0.422604			0.211302	−	0.977421i	$-0.432230\pi$
$937$	12121.1			0.422604			0.211302	+	0.977421i	$0.432230\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7701.73	+	13339.8i	−0.266811	+	0.462130i	−0.968037	−	0.250809i	$-0.919303\pi$
$941$	−7701.73	+	13339.8i	−0.266811	+	0.462130i	0.701226	+	0.712940i	$0.252637\pi$
$942$		0			0
$943$	5576.44	+	9658.69i	0.192571	+	0.333542i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15943.3	−	27614.5i	−0.547082	−	0.947573i	−0.998473	−	0.0552471i	$-0.982405\pi$
$947$	−15943.3	−	27614.5i	−0.547082	−	0.947573i	0.451391	−	0.892326i	$-0.350928\pi$
$948$		0			0
$949$	15491.8	−	26832.6i	0.529910	−	0.917831i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38339.2			1.30318			0.651589	−	0.758572i	$-0.274103\pi$
$953$	38339.2			1.30318			0.651589	+	0.758572i	$0.274103\pi$
$954$		0			0
$955$	25992.1			0.880717
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13178.1	+	22825.2i	−0.443738	+	0.768577i
$960$		0			0
$961$	12799.3	+	22169.0i	0.429635	+	0.744150i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12493.3	−	21639.1i	−0.416761	−	0.721851i
$966$		0			0
$967$	28153.5	−	48763.3i	0.936253	−	1.62164i	0.163868	−	0.986482i	$-0.447603\pi$
$967$	28153.5	−	48763.3i	0.936253	−	1.62164i	0.772385	−	0.635155i	$-0.219064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35994.8			1.18963			0.594814	−	0.803864i	$-0.297226\pi$
$971$	35994.8			1.18963			0.594814	+	0.803864i	$0.297226\pi$
$972$		0			0
$973$	30675.5			1.01070
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15072.1	−	26105.6i	0.493549	−	0.854852i	−0.506423	−	0.862285i	$-0.669033\pi$
$977$	15072.1	−	26105.6i	0.493549	−	0.854852i	0.999972	+	0.00743267i	$0.00236592\pi$
$978$		0			0
$979$	−25730.8	−	44567.1i	−0.840002	−	1.45493i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19432.9	+	33658.7i	0.630532	+	1.09211i	0.987443	+	0.157975i	$0.0504965\pi$
$983$	19432.9	+	33658.7i	0.630532	+	1.09211i	−0.356911	+	0.934138i	$0.616170\pi$
$984$		0			0
$985$	1477.68	−	2559.42i	0.0477998	−	0.0827916i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20339.3			−0.653944
$990$		0			0
$991$	−43210.3			−1.38509			−0.692543	−	0.721377i	$-0.743510\pi$
$991$	−43210.3			−1.38509			−0.692543	+	0.721377i	$0.743510\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6771.44	+	11728.5i	−0.215748	+	0.373686i
$996$		0			0
$997$	−15907.3	−	27552.3i	−0.505306	−	0.875215i	−0.999981	−	0.00613735i	$-0.998046\pi$
$997$	−15907.3	−	27552.3i	−0.505306	−	0.875215i	0.494675	−	0.869078i	$-0.335287\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.v.433.3		8
3.2	odd	2		648.4.i.u.433.2		8
9.2	odd	6		648.4.i.u.217.2		8
9.4	even	3		648.4.a.g.1.2	✓	4
9.5	odd	6		648.4.a.j.1.3	yes	4
9.7	even	3	inner	648.4.i.v.217.3		8
36.23	even	6		1296.4.a.bb.1.3		4
36.31	odd	6		1296.4.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.2	✓	4	9.4	even	3
648.4.a.j.1.3	yes	4	9.5	odd	6
648.4.i.u.217.2		8	9.2	odd	6
648.4.i.u.433.2		8	3.2	odd	2
648.4.i.v.217.3		8	9.7	even	3	inner
648.4.i.v.433.3		8	1.1	even	1	trivial
1296.4.a.x.1.2		4	36.31	odd	6
1296.4.a.bb.1.3		4	36.23	even	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.16966 - 5.49001i) q^{5} +(-9.59728 - 16.6230i) q^{7} +O(q^{10})\)	\(q+(3.16966 - 5.49001i) q^{5} +(-9.59728 - 16.6230i) q^{7} +(20.6230 + 35.7201i) q^{11} +(-19.6328 + 34.0049i) q^{13} -59.9385 q^{17} -41.2583 q^{19} +(-26.9040 + 46.5990i) q^{23} +(42.4065 + 73.4502i) q^{25} +(-32.6297 - 56.5164i) q^{29} +(-32.3747 + 56.0745i) q^{31} -121.681 q^{35} +293.515 q^{37} +(103.636 - 179.503i) q^{41} +(188.999 + 327.355i) q^{43} +(161.891 + 280.404i) q^{47} +(-12.7157 + 22.0243i) q^{49} +340.588 q^{53} +261.471 q^{55} +(203.481 - 352.439i) q^{59} +(378.617 + 655.784i) q^{61} +(124.458 + 215.568i) q^{65} +(-422.172 + 731.223i) q^{67} -859.728 q^{71} -789.079 q^{73} +(395.849 - 685.631i) q^{77} +(108.955 + 188.715i) q^{79} +(50.5889 + 87.6225i) q^{83} +(-189.985 + 329.063i) q^{85} -1247.68 q^{89} +753.685 q^{91} +(-130.775 + 226.508i) q^{95} +(873.504 + 1512.95i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 8 q^{11} - 4 q^{13} - 32 q^{17} + 160 q^{19} + 200 q^{23} - 8 q^{25} + 216 q^{29} - 80 q^{31} - 816 q^{35} - 552 q^{37} + 384 q^{41} - 160 q^{43} + 768 q^{47} + 268 q^{49} - 1888 q^{53} + 608 q^{55} + 992 q^{59} + 548 q^{61} + 1328 q^{65} - 464 q^{67} - 3440 q^{71} - 1528 q^{73} + 1728 q^{77} - 688 q^{79} + 2128 q^{83} + 1324 q^{85} - 4224 q^{89} + 3552 q^{91} + 2056 q^{95} + 1816 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 8 * q^11 - 4 * q^13 - 32 * q^17 + 160 * q^19 + 200 * q^23 - 8 * q^25 + 216 * q^29 - 80 * q^31 - 816 * q^35 - 552 * q^37 + 384 * q^41 - 160 * q^43 + 768 * q^47 + 268 * q^49 - 1888 * q^53 + 608 * q^55 + 992 * q^59 + 548 * q^61 + 1328 * q^65 - 464 * q^67 - 3440 * q^71 - 1528 * q^73 + 1728 * q^77 - 688 * q^79 + 2128 * q^83 + 1324 * q^85 - 4224 * q^89 + 3552 * q^91 + 2056 * q^95 + 1816 * q^97

Introduction

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