LMFDB - Embedded newform 640.3.e.i.319.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,3,Mod(319,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("640.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	640.e (of order $2$, degree $1$, 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:	$17.4387369191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 16x^{14} + 104x^{12} - 208x^{10} - 352x^{8} + 2312x^{6} + 2497x^{4} - 9072x^{2} + 5184 $ x^16 - 16x^14 + 104x^12 - 208x^10 - 352x^8 + 2312x^6 + 2497x^4 - 9072*x^2 + 5184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.8
Root			$-0.973371 + 0.130814i$ of defining polynomial
Character	$\chi$	$=$	640.319
Dual form			640.3.e.i.319.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.20837i q^{3} +(3.50932 + 3.56155i) q^{5} +0.620058 q^{7} +4.12311 q^{9} +O(q^{10})$	$q-2.20837i q^{3} +(3.50932 + 3.56155i) q^{5} +0.620058 q^{7} +4.12311 q^{9} +17.6757 q^{11} -17.9786 q^{13} +(7.86522 - 7.74988i) q^{15} +10.9600i q^{17} +2.17598 q^{19} -1.36932i q^{21} +13.8703 q^{23} +(-0.369317 + 24.9973i) q^{25} -28.9807i q^{27} +14.7386i q^{29} +55.2032i q^{31} -39.0346i q^{33} +(2.17598 + 2.20837i) q^{35} +7.01864 q^{37} +39.7034i q^{39} +48.3542 q^{41} -58.7769i q^{43} +(14.4693 + 14.6847i) q^{45} +54.1646 q^{47} -48.6155 q^{49} +24.2037 q^{51} +3.94134 q^{53} +(62.0299 + 62.9531i) q^{55} -4.80537i q^{57} +72.8789 q^{59} -79.3390i q^{61} +2.55656 q^{63} +(-63.0928 - 64.0318i) q^{65} -26.7723i q^{67} -30.6307i q^{69} -46.4993i q^{71} +95.1837i q^{73} +(55.2032 + 0.815588i) q^{75} +10.9600 q^{77} +24.2037i q^{79} -26.8920 q^{81} -107.361i q^{83} +(-39.0346 + 38.4621i) q^{85} +32.5483 q^{87} +44.2462 q^{89} -11.1478 q^{91} +121.909 q^{93} +(7.63623 + 7.74988i) q^{95} -123.258i q^{97} +72.8789 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 192 q^{25} + 48 q^{41} - 448 q^{49} + 112 q^{65} - 1024 q^{81} + 576 q^{89}+O(q^{100}) $ 16 * q + 192 * q^25 + 48 * q^41 - 448 * q^49 + 112 * q^65 - 1024 * q^81 + 576 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	2.20837i		−	0.736123i	−0.929801	−	0.368062i	$-0.880022\pi$
$3$		−	2.20837i		−	0.736123i	0.929801	−	0.368062i	$-0.119978\pi$
$4$		0			0
$5$	3.50932	+	3.56155i	0.701864	+	0.712311i
$5$	3.50932	+	3.56155i	0.701864	+	0.712311i
$6$		0			0
$7$	0.620058			0.0885797			0.0442899	−	0.999019i	$-0.485897\pi$
$7$	0.620058			0.0885797			0.0442899	+	0.999019i	$0.485897\pi$
$8$		0			0
$9$	4.12311			0.458123
$10$		0			0
$11$	17.6757			1.60689			0.803443	−	0.595382i	$-0.202999\pi$
$11$	17.6757			1.60689			0.803443	+	0.595382i	$0.202999\pi$
$12$		0			0
$13$	−17.9786			−1.38297			−0.691486	−	0.722390i	$-0.743043\pi$
$13$	−17.9786			−1.38297			−0.691486	+	0.722390i	$0.743043\pi$
$14$		0			0
$15$	7.86522	−	7.74988i	0.524348	−	0.516659i
$16$		0			0
$17$			10.9600i			0.644705i	0.946620	+	0.322352i	$0.104474\pi$
$17$			10.9600i			0.644705i	−0.946620	+	0.322352i	$0.895526\pi$
$18$		0			0
$19$	2.17598			0.114525			0.0572627	−	0.998359i	$-0.481763\pi$
$19$	2.17598			0.114525			0.0572627	+	0.998359i	$0.481763\pi$
$20$		0			0
$21$		−	1.36932i		−	0.0652056i
$22$		0			0
$23$	13.8703			0.603055			0.301528	−	0.953457i	$-0.402503\pi$
$23$	13.8703			0.603055			0.301528	+	0.953457i	$0.402503\pi$
$24$		0			0
$25$	−0.369317	+	24.9973i	−0.0147727	+	0.999891i
$26$		0			0
$27$		−	28.9807i		−	1.07336i
$28$		0			0
$29$			14.7386i			0.508229i	0.967174	+	0.254114i	$0.0817840\pi$
$29$			14.7386i			0.508229i	−0.967174	+	0.254114i	$0.918216\pi$
$30$		0			0
$31$			55.2032i			1.78075i	0.455229	+	0.890374i	$0.349557\pi$
$31$			55.2032i			1.78075i	−0.455229	+	0.890374i	$0.650443\pi$
$32$		0			0
$33$		−	39.0346i		−	1.18287i
$34$		0			0
$35$	2.17598	+	2.20837i	0.0621709	+	0.0630963i
$36$		0			0
$37$	7.01864			0.189693			0.0948465	−	0.995492i	$-0.469764\pi$
$37$	7.01864			0.189693			0.0948465	+	0.995492i	$0.469764\pi$
$38$		0			0
$39$			39.7034i			1.01804i
$40$		0			0
$41$	48.3542			1.17937			0.589685	−	0.807633i	$-0.299252\pi$
$41$	48.3542			1.17937			0.589685	+	0.807633i	$0.299252\pi$
$42$		0			0
$43$		−	58.7769i		−	1.36690i	−0.729995	−	0.683452i	$-0.760478\pi$
$43$		−	58.7769i		−	1.36690i	0.729995	−	0.683452i	$-0.239522\pi$
$44$		0			0
$45$	14.4693	+	14.6847i	0.321540	+	0.326326i
$46$		0			0
$47$	54.1646			1.15244			0.576220	−	0.817295i	$-0.304527\pi$
$47$	54.1646			1.15244			0.576220	+	0.817295i	$0.304527\pi$
$48$		0			0
$49$	−48.6155			−0.992154
$50$		0			0
$51$	24.2037			0.474582
$52$		0			0
$53$	3.94134			0.0743649			0.0371824	−	0.999308i	$-0.488162\pi$
$53$	3.94134			0.0743649			0.0371824	+	0.999308i	$0.488162\pi$
$54$		0			0
$55$	62.0299	+	62.9531i	1.12782	+	1.14460i
$56$		0			0
$57$		−	4.80537i		−	0.0843048i
$58$		0			0
$59$	72.8789			1.23524			0.617618	−	0.786478i	$-0.288098\pi$
$59$	72.8789			1.23524			0.617618	+	0.786478i	$0.288098\pi$
$60$		0			0
$61$		−	79.3390i		−	1.30064i	−0.759661	−	0.650320i	$-0.774635\pi$
$61$		−	79.3390i		−	1.30064i	0.759661	−	0.650320i	$-0.225365\pi$
$62$		0			0
$63$	2.55656			0.0405804
$64$		0			0
$65$	−63.0928	−	64.0318i	−0.970658	−	0.985105i
$66$		0			0
$67$		−	26.7723i		−	0.399586i	−0.979838	−	0.199793i	$-0.935973\pi$
$67$		−	26.7723i		−	0.399586i	0.979838	−	0.199793i	$-0.0640270\pi$
$68$		0			0
$69$		−	30.6307i		−	0.443923i
$70$		0			0
$71$		−	46.4993i		−	0.654919i	−0.944865	−	0.327460i	$-0.893807\pi$
$71$		−	46.4993i		−	0.654919i	0.944865	−	0.327460i	$-0.106193\pi$
$72$		0			0
$73$			95.1837i			1.30389i	0.758268	+	0.651943i	$0.226046\pi$
$73$			95.1837i			1.30389i	−0.758268	+	0.651943i	$0.773954\pi$
$74$		0			0
$75$	55.2032	+	0.815588i	0.736043	+	0.0108745i
$76$		0			0
$77$	10.9600			0.142337
$78$		0			0
$79$			24.2037i			0.306376i	0.988197	+	0.153188i	$0.0489540\pi$
$79$			24.2037i			0.306376i	−0.988197	+	0.153188i	$0.951046\pi$
$80$		0			0
$81$	−26.8920			−0.332001
$82$		0			0
$83$		−	107.361i		−	1.29351i	−0.762699	−	0.646753i	$-0.776126\pi$
$83$		−	107.361i		−	1.29351i	0.762699	−	0.646753i	$-0.223874\pi$
$84$		0			0
$85$	−39.0346	+	38.4621i	−0.459230	+	0.452495i
$86$		0			0
$87$	32.5483			0.374119
$88$		0			0
$89$	44.2462			0.497148			0.248574	−	0.968613i	$-0.420038\pi$
$89$	44.2462			0.497148			0.248574	+	0.968613i	$0.420038\pi$
$90$		0			0
$91$	−11.1478			−0.122503
$92$		0			0
$93$	121.909			1.31085
$94$		0			0
$95$	7.63623	+	7.74988i	0.0803813	+	0.0815777i
$96$		0			0
$97$		−	123.258i		−	1.27070i	−0.772223	−	0.635352i	$-0.780855\pi$
$97$		−	123.258i		−	1.27070i	0.772223	−	0.635352i	$-0.219145\pi$
$98$		0			0
$99$	72.8789			0.736151
$100$		0			0
$101$		−	32.7083i		−	0.323845i	−0.986803	−	0.161922i	$-0.948231\pi$
$101$		−	32.7083i		−	0.323845i	0.986803	−	0.161922i	$-0.0517694\pi$
$102$		0			0
$103$	−70.2862			−0.682390			−0.341195	−	0.939993i	$-0.610832\pi$
$103$	−70.2862			−0.682390			−0.341195	+	0.939993i	$0.610832\pi$
$104$		0			0
$105$	4.87689	−	4.80537i	0.0464466	−	0.0457655i
$106$		0			0
$107$			87.7576i			0.820164i	0.912049	+	0.410082i	$0.134500\pi$
$107$			87.7576i			0.820164i	−0.912049	+	0.410082i	$0.865500\pi$
$108$		0			0
$109$			81.8617i			0.751025i	0.926817	+	0.375513i	$0.122533\pi$
$109$			81.8617i			0.751025i	−0.926817	+	0.375513i	$0.877467\pi$
$110$		0			0
$111$		−	15.4998i		−	0.139637i
$112$		0			0
$113$		−	12.3092i		−	0.108931i	−0.998516	−	0.0544656i	$-0.982654\pi$
$113$		−	12.3092i		−	0.108931i	0.998516	−	0.0544656i	$-0.0173455\pi$
$114$		0			0
$115$	48.6753	+	49.3997i	0.423263	+	0.429563i
$116$		0			0
$117$	−74.1278			−0.633571
$118$		0			0
$119$			6.79583i			0.0571078i
$120$		0			0
$121$	191.432			1.58208
$122$		0			0
$123$		−	106.784i		−	0.868161i
$124$		0			0
$125$	−90.3252	+	86.4081i	−0.722601	+	0.691265i
$126$		0			0
$127$	175.897			1.38501			0.692507	−	0.721411i	$-0.256506\pi$
$127$	175.897			1.38501			0.692507	+	0.721411i	$0.256506\pi$
$128$		0			0
$129$	−129.801			−1.00621
$130$		0			0
$131$	−15.2319			−0.116274			−0.0581370	−	0.998309i	$-0.518516\pi$
$131$	−15.2319			−0.116274			−0.0581370	+	0.998309i	$0.518516\pi$
$132$		0			0
$133$	1.34924			0.0101446
$134$		0			0
$135$	103.216	−	101.702i	0.764564	−	0.753352i
$136$		0			0
$137$			249.973i			1.82462i	0.409502	+	0.912309i	$0.365702\pi$
$137$			249.973i			1.82462i	−0.409502	+	0.912309i	$0.634298\pi$
$138$		0			0
$139$	−225.433			−1.62182			−0.810909	−	0.585172i	$-0.801027\pi$
$139$	−225.433			−1.62182			−0.810909	+	0.585172i	$0.801027\pi$
$140$		0			0
$141$		−	119.616i		−	0.848337i
$142$		0			0
$143$	−317.786			−2.22228
$144$		0			0
$145$	−52.4924	+	51.7226i	−0.362017	+	0.356708i
$146$		0			0
$147$			107.361i			0.730347i
$148$		0			0
$149$			77.3693i			0.519257i	0.965709	+	0.259629i	$0.0836001\pi$
$149$			77.3693i			0.519257i	−0.965709	+	0.259629i	$0.916400\pi$
$150$		0			0
$151$			163.702i			1.08412i	0.840341	+	0.542058i	$0.182355\pi$
$151$			163.702i			1.08412i	−0.840341	+	0.542058i	$0.817645\pi$
$152$		0			0
$153$			45.1892i			0.295354i
$154$		0			0
$155$	−196.609	+	193.726i	−1.26845	+	1.24984i
$156$		0			0
$157$	−58.7413			−0.374148			−0.187074	−	0.982346i	$-0.559900\pi$
$157$	−58.7413			−0.374148			−0.187074	+	0.982346i	$0.559900\pi$
$158$		0			0
$159$		−	8.70393i		−	0.0547417i
$160$		0			0
$161$	8.60037			0.0534185
$162$		0			0
$163$		−	112.560i		−	0.690551i	−0.938501	−	0.345276i	$-0.887785\pi$
$163$		−	112.560i		−	0.690551i	0.938501	−	0.345276i	$-0.112215\pi$
$164$		0			0
$165$	139.024	−	136.985i	0.842568	−	0.830211i
$166$		0			0
$167$	−237.035			−1.41937			−0.709685	−	0.704519i	$-0.751163\pi$
$167$	−237.035			−1.41937			−0.709685	+	0.704519i	$0.751163\pi$
$168$		0			0
$169$	154.231			0.912610
$170$		0			0
$171$	8.97181			0.0524667
$172$		0			0
$173$	−292.949			−1.69334			−0.846672	−	0.532115i	$-0.821398\pi$
$173$	−292.949			−1.69334			−0.846672	+	0.532115i	$0.821398\pi$
$174$		0			0
$175$	−0.228998	+	15.4998i	−0.00130856	+	0.0885700i
$176$		0			0
$177$		−	160.944i		−	0.909286i
$178$		0			0
$179$	−180.841			−1.01029			−0.505144	−	0.863035i	$-0.668561\pi$
$179$	−180.841			−1.01029			−0.505144	+	0.863035i	$0.668561\pi$
$180$		0			0
$181$			259.231i			1.43222i	0.697990	+	0.716108i	$0.254078\pi$
$181$			259.231i			1.43222i	−0.697990	+	0.716108i	$0.745922\pi$
$182$		0			0
$183$	−175.210			−0.957431
$184$		0			0
$185$	24.6307	+	24.9973i	0.133139	+	0.135120i
$186$		0			0
$187$			193.726i			1.03597i
$188$		0			0
$189$		−	17.9697i		−	0.0950777i
$190$		0			0
$191$		−	269.220i		−	1.40953i	−0.709441	−	0.704765i	$-0.751052\pi$
$191$		−	269.220i		−	1.40953i	0.709441	−	0.704765i	$-0.248948\pi$
$192$		0			0
$193$		−	145.178i		−	0.752219i	−0.926575	−	0.376109i	$-0.877262\pi$
$193$		−	145.178i		−	0.752219i	0.926575	−	0.376109i	$-0.122738\pi$
$194$		0			0
$195$	−141.406	+	139.332i	−0.725159	+	0.714524i
$196$		0			0
$197$	272.378			1.38263			0.691314	−	0.722554i	$-0.257032\pi$
$197$	272.378			1.38263			0.691314	+	0.722554i	$0.257032\pi$
$198$		0			0
$199$		−	329.311i		−	1.65483i	−0.561591	−	0.827415i	$-0.689811\pi$
$199$		−	329.311i		−	1.65483i	0.561591	−	0.827415i	$-0.310189\pi$
$200$		0			0
$201$	−59.1231			−0.294145
$202$		0			0
$203$			9.13881i			0.0450188i
$204$		0			0
$205$	169.690	+	172.216i	0.827758	+	0.840078i
$206$		0			0
$207$	57.1886			0.276273
$208$		0			0
$209$	38.4621			0.184029
$210$		0			0
$211$	−6.52795			−0.0309381			−0.0154691	−	0.999880i	$-0.504924\pi$
$211$	−6.52795			−0.0309381			−0.0154691	+	0.999880i	$0.504924\pi$
$212$		0			0
$213$	−102.688			−0.482101
$214$		0			0
$215$	209.337	−	206.267i	0.973661	−	0.959382i
$216$		0			0
$217$			34.2292i			0.157738i
$218$		0			0
$219$	210.201			0.959821
$220$		0			0
$221$		−	197.045i		−	0.891608i
$222$		0			0
$223$	−365.597			−1.63945			−0.819724	−	0.572759i	$-0.805873\pi$
$223$	−365.597			−1.63945			−0.819724	+	0.572759i	$0.805873\pi$
$224$		0			0
$225$	−1.52273	+	103.066i	−0.00676770	+	0.458073i
$226$		0			0
$227$			224.066i			0.987074i	0.869725	+	0.493537i	$0.164296\pi$
$227$			224.066i			0.987074i	−0.869725	+	0.493537i	$0.835704\pi$
$228$		0			0
$229$		−	247.723i		−	1.08176i	−0.841099	−	0.540881i	$-0.818091\pi$
$229$		−	247.723i		−	1.08176i	0.841099	−	0.540881i	$-0.181909\pi$
$230$		0			0
$231$		−	24.2037i		−	0.104778i
$232$		0			0
$233$		−	260.933i		−	1.11988i	−0.828532	−	0.559941i	$-0.810824\pi$
$233$		−	260.933i		−	1.11988i	0.828532	−	0.559941i	$-0.189176\pi$
$234$		0			0
$235$	190.081	+	192.910i	0.808856	+	0.820895i
$236$		0			0
$237$	53.4507			0.225530
$238$		0			0
$239$		−	183.017i		−	0.765764i	−0.923797	−	0.382882i	$-0.874932\pi$
$239$		−	183.017i		−	0.765764i	0.923797	−	0.382882i	$-0.125068\pi$
$240$		0			0
$241$	−204.725			−0.849483			−0.424741	−	0.905315i	$-0.639635\pi$
$241$	−204.725			−0.849483			−0.424741	+	0.905315i	$0.639635\pi$
$242$		0			0
$243$		−	201.438i		−	0.828965i
$244$		0			0
$245$	−170.608	−	173.147i	−0.696357	−	0.706722i
$246$		0			0
$247$	−39.1212			−0.158385
$248$		0			0
$249$	−237.093			−0.952180
$250$		0			0
$251$	−322.247			−1.28385			−0.641927	−	0.766766i	$-0.721865\pi$
$251$	−322.247			−1.28385			−0.641927	+	0.766766i	$0.721865\pi$
$252$		0			0
$253$	245.167			0.969041
$254$		0			0
$255$	84.9385	+	86.2027i	0.333092	+	0.338050i
$256$		0			0
$257$			234.207i			0.911313i	0.890156	+	0.455656i	$0.150595\pi$
$257$			234.207i			0.911313i	−0.890156	+	0.455656i	$0.849405\pi$
$258$		0			0
$259$	4.35197			0.0168030
$260$		0			0
$261$			60.7689i			0.232831i
$262$		0			0
$263$	−448.361			−1.70479			−0.852397	−	0.522896i	$-0.824852\pi$
$263$	−448.361			−1.70479			−0.852397	+	0.522896i	$0.824852\pi$
$264$		0			0
$265$	13.8314	+	14.0373i	0.0521941	+	0.0529709i
$266$		0			0
$267$		−	97.7120i		−	0.365962i
$268$		0			0
$269$			16.6610i			0.0619368i	0.999520	+	0.0309684i	$0.00985912\pi$
$269$			16.6610i			0.0619368i	−0.999520	+	0.0309684i	$0.990141\pi$
$270$		0			0
$271$		−	148.202i		−	0.546870i	−0.961890	−	0.273435i	$-0.911840\pi$
$271$		−	148.202i		−	0.546870i	0.961890	−	0.273435i	$-0.0881598\pi$
$272$		0			0
$273$			24.6184i			0.0901774i
$274$		0			0
$275$	−6.52795	+	441.845i	−0.0237380	+	1.60671i
$276$		0			0
$277$	−373.716			−1.34916			−0.674578	−	0.738204i	$-0.735674\pi$
$277$	−373.716			−1.34916			−0.674578	+	0.738204i	$0.735674\pi$
$278$		0			0
$279$			227.609i			0.815802i
$280$		0			0
$281$	−250.756			−0.892369			−0.446184	−	0.894941i	$-0.647217\pi$
$281$	−250.756			−0.892369			−0.446184	+	0.894941i	$0.647217\pi$
$282$		0			0
$283$		−	463.657i		−	1.63836i	−0.573533	−	0.819182i	$-0.694428\pi$
$283$		−	463.657i		−	1.63836i	0.573533	−	0.819182i	$-0.305572\pi$
$284$		0			0
$285$	17.1146	−	16.8636i	0.0600512	−	0.0591705i
$286$		0			0
$287$	29.9824			0.104468
$288$		0			0
$289$	168.879			0.584356
$290$		0			0
$291$	−272.200			−0.935395
$292$		0			0
$293$	275.455			0.940120			0.470060	−	0.882634i	$-0.344232\pi$
$293$	275.455			0.940120			0.470060	+	0.882634i	$0.344232\pi$
$294$		0			0
$295$	255.756	+	259.562i	0.866968	+	0.879872i
$296$		0			0
$297$		−	512.255i		−	1.72476i
$298$		0			0
$299$	−249.368			−0.834008
$300$		0			0
$301$		−	36.4451i		−	0.121080i
$302$		0			0
$303$	−72.2321			−0.238390
$304$		0			0
$305$	282.570	−	278.426i	0.926459	−	0.912873i
$306$		0			0
$307$			131.620i			0.428728i	0.976754	+	0.214364i	$0.0687679\pi$
$307$			131.620i			0.428728i	−0.976754	+	0.214364i	$0.931232\pi$
$308$		0			0
$309$			155.218i			0.502323i
$310$		0			0
$311$		−	353.515i		−	1.13670i	−0.822786	−	0.568352i	$-0.807581\pi$
$311$		−	353.515i		−	1.13670i	0.822786	−	0.568352i	$-0.192419\pi$
$312$		0			0
$313$			281.503i			0.899372i	0.893187	+	0.449686i	$0.148464\pi$
$313$			281.503i			0.899372i	−0.893187	+	0.449686i	$0.851536\pi$
$314$		0			0
$315$	8.97181	+	9.10534i	0.0284819	+	0.0289058i
$316$		0			0
$317$	216.229			0.682110			0.341055	−	0.940043i	$-0.389216\pi$
$317$	216.229			0.682110			0.341055	+	0.940043i	$0.389216\pi$
$318$		0			0
$319$			260.516i			0.816665i
$320$		0			0
$321$	193.801			0.603742
$322$		0			0
$323$			23.8487i			0.0738351i
$324$		0			0
$325$	6.63981	−	449.417i	0.0204302	−	1.38282i
$326$		0			0
$327$	180.781			0.552847
$328$		0			0
$329$	33.5852			0.102083
$330$		0			0
$331$	333.395			1.00724			0.503618	−	0.863926i	$-0.332002\pi$
$331$	333.395			1.00724			0.503618	+	0.863926i	$0.332002\pi$
$332$		0			0
$333$	28.9386			0.0869027
$334$		0			0
$335$	95.3509	−	93.9526i	0.284630	−	0.280456i
$336$		0			0
$337$			243.818i			0.723496i	0.932276	+	0.361748i	$0.117820\pi$
$337$			243.818i			0.723496i	−0.932276	+	0.361748i	$0.882180\pi$
$338$		0			0
$339$	−27.1833			−0.0801867
$340$		0			0
$341$			975.758i			2.86146i
$342$		0			0
$343$	−60.5273			−0.176464
$344$		0			0
$345$	109.093	−	107.493i	0.316211	−	0.311574i
$346$		0			0
$347$			221.109i			0.637201i	0.947889	+	0.318601i	$0.103213\pi$
$347$			221.109i			0.637201i	−0.947889	+	0.318601i	$0.896787\pi$
$348$		0			0
$349$		−	39.1704i		−	0.112236i	−0.998424	−	0.0561181i	$-0.982128\pi$
$349$		−	39.1704i		−	0.112236i	0.998424	−	0.0561181i	$-0.0178723\pi$
$350$		0			0
$351$			521.033i			1.48442i
$352$		0			0
$353$			646.473i			1.83137i	0.401899	+	0.915684i	$0.368350\pi$
$353$			646.473i			1.83137i	−0.401899	+	0.915684i	$0.631650\pi$
$354$		0			0
$355$	165.610	−	163.181i	0.466506	−	0.459665i
$356$		0			0
$357$	15.0077			0.0420383
$358$		0			0
$359$			256.700i			0.715042i	0.933905	+	0.357521i	$0.116378\pi$
$359$			256.700i			0.715042i	−0.933905	+	0.357521i	$0.883622\pi$
$360$		0			0
$361$	−356.265			−0.986884
$362$		0			0
$363$		−	422.752i		−	1.16461i
$364$		0			0
$365$	−339.002	+	334.030i	−0.928772	+	0.915152i
$366$		0			0
$367$	551.424			1.50252			0.751259	−	0.660008i	$-0.229447\pi$
$367$	551.424			1.50252			0.751259	+	0.660008i	$0.229447\pi$
$368$		0			0
$369$	199.369			0.540296
$370$		0			0
$371$	2.44386			0.00658722
$372$		0			0
$373$	−261.418			−0.700852			−0.350426	−	0.936590i	$-0.613963\pi$
$373$	−261.418			−0.700852			−0.350426	+	0.936590i	$0.613963\pi$
$374$		0			0
$375$	190.821	+	199.471i	0.508856	+	0.531923i
$376$		0			0
$377$		−	264.980i		−	0.702866i
$378$		0			0
$379$	344.007			0.907671			0.453835	−	0.891086i	$-0.350055\pi$
$379$	344.007			0.907671			0.453835	+	0.891086i	$0.350055\pi$
$380$		0			0
$381$		−	388.445i		−	1.01954i
$382$		0			0
$383$	−374.736			−0.978422			−0.489211	−	0.872165i	$-0.662715\pi$
$383$	−374.736			−0.978422			−0.489211	+	0.872165i	$0.662715\pi$
$384$		0			0
$385$	38.4621	+	39.0346i	0.0999016	+	0.101388i
$386$		0			0
$387$		−	242.343i		−	0.626210i
$388$		0			0
$389$		−	692.358i		−	1.77984i	−0.456116	−	0.889920i	$-0.650760\pi$
$389$		−	692.358i		−	1.77984i	0.456116	−	0.889920i	$-0.349240\pi$
$390$		0			0
$391$			152.018i			0.388793i
$392$		0			0
$393$			33.6376i			0.0855919i
$394$		0			0
$395$	−86.2027	+	84.9385i	−0.218235	+	0.215034i
$396$		0			0
$397$	−427.546			−1.07694			−0.538471	−	0.842644i	$-0.680998\pi$
$397$	−427.546			−1.07694			−0.538471	+	0.842644i	$0.680998\pi$
$398$		0			0
$399$		−	2.97961i		−	0.00746770i
$400$		0			0
$401$	179.261			0.447036			0.223518	−	0.974700i	$-0.428246\pi$
$401$	179.261			0.447036			0.223518	+	0.974700i	$0.428246\pi$
$402$		0			0
$403$		−	992.478i		−	2.46272i
$404$		0			0
$405$	−94.3729	−	95.7775i	−0.233019	−	0.236488i
$406$		0			0
$407$	124.060			0.304815
$408$		0			0
$409$	−89.7368			−0.219405			−0.109703	−	0.993964i	$-0.534990\pi$
$409$	−89.7368			−0.219405			−0.109703	+	0.993964i	$0.534990\pi$
$410$		0			0
$411$	552.032			1.34314
$412$		0			0
$413$	45.1892			0.109417
$414$		0			0
$415$	382.372	−	376.764i	0.921378	−	0.907866i
$416$		0			0
$417$			497.839i			1.19386i
$418$		0			0
$419$	203.673			0.486093			0.243046	−	0.970015i	$-0.421853\pi$
$419$	203.673			0.486093			0.243046	+	0.970015i	$0.421853\pi$
$420$		0			0
$421$			388.384i			0.922528i	0.887263	+	0.461264i	$0.152604\pi$
$421$			388.384i			0.922528i	−0.887263	+	0.461264i	$0.847396\pi$
$422$		0			0
$423$	223.327			0.527959
$424$		0			0
$425$	−273.970	−	4.04771i	−0.644635	−	0.00952402i
$426$		0			0
$427$		−	49.1948i		−	0.115210i
$428$		0			0
$429$			701.788i			1.63587i
$430$		0			0
$431$			617.011i			1.43158i	0.698316	+	0.715790i	$0.253933\pi$
$431$			617.011i			1.43158i	−0.698316	+	0.715790i	$0.746067\pi$
$432$		0			0
$433$		−	210.938i		−	0.487155i	−0.969881	−	0.243578i	$-0.921679\pi$
$433$		−	210.938i		−	0.487155i	0.969881	−	0.243578i	$-0.0783210\pi$
$434$		0			0
$435$	114.223	+	115.923i	0.262581	+	0.266489i
$436$		0			0
$437$	30.1815			0.0690652
$438$		0			0
$439$		−	159.885i		−	0.364203i	−0.983280	−	0.182102i	$-0.941710\pi$
$439$		−	159.885i		−	0.364203i	0.983280	−	0.182102i	$-0.0582900\pi$
$440$		0			0
$441$	−200.447			−0.454528
$442$		0			0
$443$		−	121.460i		−	0.274177i	−0.990559	−	0.137088i	$-0.956226\pi$
$443$		−	121.460i		−	0.274177i	0.990559	−	0.137088i	$-0.0437744\pi$
$444$		0			0
$445$	155.274	+	157.585i	0.348931	+	0.354124i
$446$		0			0
$447$	170.860			0.382237
$448$		0			0
$449$	533.002			1.18709			0.593543	−	0.804802i	$-0.297729\pi$
$449$	533.002			1.18709			0.593543	+	0.804802i	$0.297729\pi$
$450$		0			0
$451$	854.696			1.89511
$452$		0			0
$453$	361.513			0.798043
$454$		0			0
$455$	−39.1212	−	39.7034i	−0.0859806	−	0.0872603i
$456$		0			0
$457$		−	103.445i		−	0.226357i	−0.993575	−	0.113179i	$-0.963897\pi$
$457$		−	103.445i		−	0.226357i	0.993575	−	0.113179i	$-0.0361032\pi$
$458$		0			0
$459$	317.628			0.691999
$460$		0			0
$461$		−	420.337i		−	0.911794i	−0.890032	−	0.455897i	$-0.849318\pi$
$461$		−	420.337i		−	0.911794i	0.890032	−	0.455897i	$-0.150682\pi$
$462$		0			0
$463$	63.3035			0.136725			0.0683623	−	0.997661i	$-0.478223\pi$
$463$	63.3035			0.136725			0.0683623	+	0.997661i	$0.478223\pi$
$464$		0			0
$465$	427.818	+	434.186i	0.920039	+	0.933732i
$466$		0			0
$467$			129.206i			0.276673i	0.990385	+	0.138337i	$0.0441756\pi$
$467$			129.206i			0.276673i	−0.990385	+	0.138337i	$0.955824\pi$
$468$		0			0
$469$		−	16.6004i		−	0.0353953i
$470$		0			0
$471$			129.722i			0.275419i
$472$		0			0
$473$		−	1038.93i		−	2.19646i
$474$		0			0
$475$	−0.803627	+	54.3936i	−0.00169185	+	0.114513i
$476$		0			0
$477$	16.2506			0.0340683
$478$		0			0
$479$		−	517.216i		−	1.07978i	−0.841734	−	0.539892i	$-0.818465\pi$
$479$		−	517.216i		−	1.07978i	0.841734	−	0.539892i	$-0.181535\pi$
$480$		0			0
$481$	−126.186			−0.262340
$482$		0			0
$483$		−	18.9928i		−	0.0393226i
$484$		0			0
$485$	438.991	−	432.553i	0.905136	−	0.891862i
$486$		0			0
$487$	893.449			1.83460			0.917299	−	0.398199i	$-0.130365\pi$
$487$	893.449			1.83460			0.917299	+	0.398199i	$0.130365\pi$
$488$		0			0
$489$	−248.574			−0.508331
$490$		0			0
$491$	−506.336			−1.03124			−0.515618	−	0.856819i	$-0.672437\pi$
$491$	−506.336			−1.03124			−0.515618	+	0.856819i	$0.672437\pi$
$492$		0			0
$493$	−161.535			−0.327658
$494$		0			0
$495$	255.756	+	259.562i	0.516678	+	0.524368i
$496$		0			0
$497$		−	28.8322i		−	0.0580126i
$498$		0			0
$499$	512.596			1.02725			0.513624	−	0.858016i	$-0.328303\pi$
$499$	512.596			1.02725			0.513624	+	0.858016i	$0.328303\pi$
$500$		0			0
$501$			523.460i			1.04483i
$502$		0			0
$503$	−389.856			−0.775061			−0.387530	−	0.921857i	$-0.626672\pi$
$503$	−389.856			−0.775061			−0.387530	+	0.921857i	$0.626672\pi$
$504$		0			0
$505$	116.492	−	114.784i	0.230678	−	0.227295i
$506$		0			0
$507$		−	340.599i		−	0.671793i
$508$		0			0
$509$		−	80.1553i		−	0.157476i	−0.996895	−	0.0787380i	$-0.974911\pi$
$509$		−	80.1553i		−	0.157476i	0.996895	−	0.0787380i	$-0.0250891\pi$
$510$		0			0
$511$			59.0194i			0.115498i
$512$		0			0
$513$		−	63.0614i		−	0.122927i
$514$		0			0
$515$	−246.657	−	250.328i	−0.478945	−	0.486073i
$516$		0			0
$517$	957.400			1.85184
$518$		0			0
$519$			646.939i			1.24651i
$520$		0			0
$521$	−417.076			−0.800529			−0.400265	−	0.916400i	$-0.631082\pi$
$521$	−417.076			−0.800529			−0.400265	+	0.916400i	$0.631082\pi$
$522$		0			0
$523$			28.2320i			0.0539809i	0.999636	+	0.0269904i	$0.00859237\pi$
$523$			28.2320i			0.0539809i	−0.999636	+	0.0269904i	$0.991408\pi$
$524$		0			0
$525$	34.2292	+	0.505712i	0.0651985	+	0.000963261i
$526$		0			0
$527$	−605.026			−1.14806
$528$		0			0
$529$	−336.616			−0.636324
$530$		0			0
$531$	300.488			0.565890
$532$		0			0
$533$	−869.341			−1.63103
$534$		0			0
$535$	−312.553	+	307.970i	−0.584212	+	0.575644i
$536$		0			0
$537$			399.365i			0.743696i
$538$		0			0
$539$	−859.315			−1.59428
$540$		0			0
$541$		−	948.095i		−	1.75249i	−0.481870	−	0.876243i	$-0.660042\pi$
$541$		−	948.095i		−	1.75249i	0.481870	−	0.876243i	$-0.339958\pi$
$542$		0			0
$543$	572.478			1.05429
$544$		0			0
$545$	−291.555	+	287.279i	−0.534963	+	0.527118i
$546$		0			0
$547$		−	266.602i		−	0.487389i	−0.969852	−	0.243695i	$-0.921641\pi$
$547$		−	266.602i		−	0.487389i	0.969852	−	0.243695i	$-0.0783595\pi$
$548$		0			0
$549$		−	327.123i		−	0.595853i
$550$		0			0
$551$			32.0710i			0.0582051i
$552$		0			0
$553$			15.0077i			0.0271387i
$554$		0			0
$555$	55.2032	−	54.3936i	0.0994652	−	0.0980066i
$556$		0			0
$557$	23.5417			0.0422651			0.0211326	−	0.999777i	$-0.493273\pi$
$557$	23.5417			0.0422651			0.0211326	+	0.999777i	$0.493273\pi$
$558$		0			0
$559$			1056.73i			1.89039i
$560$		0			0
$561$	427.818			0.762599
$562$		0			0
$563$		−	119.352i		−	0.211993i	−0.994366	−	0.105997i	$-0.966197\pi$
$563$		−	119.352i		−	0.211993i	0.994366	−	0.105997i	$-0.0338033\pi$
$564$		0			0
$565$	43.8399	−	43.1970i	0.0775928	−	0.0764549i
$566$		0			0
$567$	−16.6746			−0.0294085
$568$		0			0
$569$	−717.434			−1.26087			−0.630434	−	0.776243i	$-0.717123\pi$
$569$	−717.434			−1.26087			−0.630434	+	0.776243i	$0.717123\pi$
$570$		0			0
$571$	−389.134			−0.681496			−0.340748	−	0.940155i	$-0.610680\pi$
$571$	−389.134			−0.681496			−0.340748	+	0.940155i	$0.610680\pi$
$572$		0			0
$573$	−594.538			−1.03759
$574$		0			0
$575$	−5.12253	+	346.719i	−0.00890874	+	0.602990i
$576$		0			0
$577$		−	934.131i		−	1.61894i	−0.587158	−	0.809472i	$-0.699753\pi$
$577$		−	934.131i		−	1.61894i	0.587158	−	0.809472i	$-0.300247\pi$
$578$		0			0
$579$	−320.607			−0.553726
$580$		0			0
$581$		−	66.5701i		−	0.114578i
$582$		0			0
$583$	69.6661			0.119496
$584$		0			0
$585$	−260.138	−	264.010i	−0.444681	−	0.451299i
$586$		0			0
$587$			372.365i			0.634353i	0.948366	+	0.317177i	$0.102735\pi$
$587$			372.365i			0.634353i	−0.948366	+	0.317177i	$0.897265\pi$
$588$		0			0
$589$			120.121i			0.203941i
$590$		0			0
$591$		−	601.511i		−	1.01778i
$592$		0			0
$593$			446.495i			0.752942i	0.926428	+	0.376471i	$0.122863\pi$
$593$			446.495i			0.752942i	−0.926428	+	0.376471i	$0.877137\pi$
$594$		0			0
$595$	−24.2037	+	23.8487i	−0.0406785	+	0.0400819i
$596$		0			0
$597$	−727.241			−1.21816
$598$		0			0
$599$		−	677.938i		−	1.13178i	−0.824480	−	0.565892i	$-0.808532\pi$
$599$		−	677.938i		−	1.13178i	0.824480	−	0.565892i	$-0.191468\pi$
$600$		0			0
$601$	696.941			1.15964			0.579818	−	0.814746i	$-0.303124\pi$
$601$	696.941			1.15964			0.579818	+	0.814746i	$0.303124\pi$
$602$		0			0
$603$		−	110.385i		−	0.183060i
$604$		0			0
$605$	671.796	+	681.794i	1.11041	+	1.12693i
$606$		0			0
$607$	−372.628			−0.613884			−0.306942	−	0.951728i	$-0.599306\pi$
$607$	−372.628			−0.613884			−0.306942	+	0.951728i	$0.599306\pi$
$608$		0			0
$609$	20.1819			0.0331393
$610$		0			0
$611$	−973.806			−1.59379
$612$		0			0
$613$	569.647			0.929277			0.464638	−	0.885501i	$-0.346184\pi$
$613$	569.647			0.929277			0.464638	+	0.885501i	$0.346184\pi$
$614$		0			0
$615$	380.316	−	374.739i	0.618400	−	0.609332i
$616$		0			0
$617$			126.714i			0.205372i	0.994714	+	0.102686i	$0.0327437\pi$
$617$			126.714i			0.205372i	−0.994714	+	0.102686i	$0.967256\pi$
$618$		0			0
$619$	771.205			1.24589			0.622944	−	0.782266i	$-0.285936\pi$
$619$	771.205			1.24589			0.622944	+	0.782266i	$0.285936\pi$
$620$		0			0
$621$		−	401.970i		−	0.647294i
$622$		0			0
$623$	27.4352			0.0440373
$624$		0			0
$625$	−624.727	−	18.4638i	−0.999564	−	0.0295421i
$626$		0			0
$627$		−	84.9385i		−	0.135468i
$628$		0			0
$629$			76.9242i			0.122296i
$630$		0			0
$631$			74.5192i			0.118097i	0.998255	+	0.0590485i	$0.0188067\pi$
$631$			74.5192i			0.118097i	−0.998255	+	0.0590485i	$0.981193\pi$
$632$		0			0
$633$			14.4161i			0.0227743i
$634$		0			0
$635$	617.279	+	626.466i	0.972092	+	0.986560i
$636$		0			0
$637$	874.040			1.37212
$638$		0			0
$639$		−	191.721i		−	0.300034i
$640$		0			0
$641$	−32.9072			−0.0513373			−0.0256686	−	0.999671i	$-0.508171\pi$
$641$	−32.9072			−0.0513373			−0.0256686	+	0.999671i	$0.508171\pi$
$642$		0			0
$643$			900.609i			1.40064i	0.713831	+	0.700318i	$0.246958\pi$
$643$			900.609i			1.40064i	−0.713831	+	0.700318i	$0.753042\pi$
$644$		0			0
$645$	−455.514	−	462.294i	−0.706223	−	0.716734i
$646$		0			0
$647$	−571.419			−0.883182			−0.441591	−	0.897217i	$-0.645586\pi$
$647$	−571.419			−0.883182			−0.441591	+	0.897217i	$0.645586\pi$
$648$		0			0
$649$	1288.19			1.98488
$650$		0			0
$651$	75.5907			0.116115
$652$		0			0
$653$	1138.05			1.74280			0.871402	−	0.490570i	$-0.163212\pi$
$653$	1138.05			1.74280			0.871402	+	0.490570i	$0.163212\pi$
$654$		0			0
$655$	−53.4536	−	54.2492i	−0.0816085	−	0.0828231i
$656$		0			0
$657$			392.453i			0.597340i
$658$		0			0
$659$	−260.248			−0.394914			−0.197457	−	0.980312i	$-0.563268\pi$
$659$	−260.248			−0.394914			−0.197457	+	0.980312i	$0.563268\pi$
$660$		0			0
$661$			1012.05i			1.53109i	0.643385	+	0.765543i	$0.277529\pi$
$661$			1012.05i			1.53109i	−0.643385	+	0.765543i	$0.722471\pi$
$662$		0			0
$663$	−435.149			−0.656334
$664$		0			0
$665$	4.73490	+	4.80537i	0.00712015	+	0.00722613i
$666$		0			0
$667$			204.429i			0.306490i
$668$		0			0
$669$			807.373i			1.20684i
$670$		0			0
$671$		−	1402.38i		−	2.08998i
$672$		0			0
$673$		−	597.070i		−	0.887177i	−0.896231	−	0.443588i	$-0.853705\pi$
$673$		−	597.070i		−	0.887177i	0.896231	−	0.443588i	$-0.146295\pi$
$674$		0			0
$675$	724.437	+	10.7030i	1.07324	+	0.0158564i
$676$		0			0
$677$	−1112.46			−1.64322			−0.821611	−	0.570048i	$-0.806925\pi$
$677$	−1112.46			−1.64322			−0.821611	+	0.570048i	$0.806925\pi$
$678$		0			0
$679$		−	76.4273i		−	0.112559i
$680$		0			0
$681$	494.820			0.726608
$682$		0			0
$683$			1089.31i			1.59489i	0.603394	+	0.797443i	$0.293815\pi$
$683$			1089.31i			1.59489i	−0.603394	+	0.797443i	$0.706185\pi$
$684$		0			0
$685$	−890.291	+	877.235i	−1.29969	+	1.28063i
$686$		0			0
$687$	−547.065			−0.796310
$688$		0			0
$689$	−70.8599			−0.102845
$690$		0			0
$691$	−837.556			−1.21209			−0.606046	−	0.795430i	$-0.707245\pi$
$691$	−837.556			−1.21209			−0.606046	+	0.795430i	$0.707245\pi$
$692$		0			0
$693$	45.1892			0.0652080
$694$		0			0
$695$	−791.116	−	802.890i	−1.13830	−	1.15524i
$696$		0			0
$697$			529.961i			0.760346i
$698$		0			0
$699$	−576.236			−0.824372
$700$		0			0
$701$		−	969.714i		−	1.38333i	−0.722219	−	0.691665i	$-0.756878\pi$
$701$		−	969.714i		−	1.38333i	0.722219	−	0.691665i	$-0.243122\pi$
$702$		0			0
$703$	15.2725			0.0217247
$704$		0			0
$705$	426.017	−	419.769i	0.604279	−	0.595418i
$706$		0			0
$707$		−	20.2811i		−	0.0286861i
$708$		0			0
$709$		−	636.769i		−	0.898123i	−0.893501	−	0.449061i	$-0.851758\pi$
$709$		−	636.769i		−	0.898123i	0.893501	−	0.449061i	$-0.148242\pi$
$710$		0			0
$711$			99.7944i			0.140358i
$712$		0			0
$713$			765.684i			1.07389i
$714$		0			0
$715$	−1115.21	−	1131.81i	−1.55974	−	1.58295i
$716$		0			0
$717$	−404.170			−0.563696
$718$		0			0
$719$		−	955.862i		−	1.32943i	−0.747096	−	0.664717i	$-0.768552\pi$
$719$		−	955.862i		−	1.32943i	0.747096	−	0.664717i	$-0.231448\pi$
$720$		0			0
$721$	−43.5815			−0.0604459
$722$		0			0
$723$			452.109i			0.625324i
$724$		0			0
$725$	−368.426	−	5.44323i	−0.508173	−	0.00750790i
$726$		0			0
$727$	334.899			0.460659			0.230330	−	0.973113i	$-0.426020\pi$
$727$	334.899			0.460659			0.230330	+	0.973113i	$0.426020\pi$
$728$		0			0
$729$	−686.879			−0.942221
$730$		0			0
$731$	644.194			0.881250
$732$		0			0
$733$	−321.781			−0.438992			−0.219496	−	0.975613i	$-0.570441\pi$
$733$	−321.781			−0.438992			−0.219496	+	0.975613i	$0.570441\pi$
$734$		0			0
$735$	−382.372	+	376.764i	−0.520234	+	0.512605i
$736$		0			0
$737$		−	473.220i		−	0.642090i
$738$		0			0
$739$	−446.245			−0.603850			−0.301925	−	0.953332i	$-0.597629\pi$
$739$	−446.245			−0.603850			−0.301925	+	0.953332i	$0.597629\pi$
$740$		0			0
$741$			86.3940i			0.116591i
$742$		0			0
$743$	436.894			0.588014			0.294007	−	0.955803i	$-0.405011\pi$
$743$	436.894			0.588014			0.294007	+	0.955803i	$0.405011\pi$
$744$		0			0
$745$	−275.555	+	271.514i	−0.369872	+	0.364448i
$746$		0			0
$747$		−	442.661i		−	0.592585i
$748$		0			0
$749$			54.4148i			0.0726499i
$750$		0			0
$751$		−	273.036i		−	0.363564i	−0.983339	−	0.181782i	$-0.941814\pi$
$751$		−	273.036i		−	0.363564i	0.983339	−	0.181782i	$-0.0581865\pi$
$752$		0			0
$753$			711.641i			0.945075i
$754$		0			0
$755$	−583.032	+	574.481i	−0.772227	+	0.760902i
$756$		0			0
$757$	265.844			0.351182			0.175591	−	0.984463i	$-0.443816\pi$
$757$	265.844			0.351182			0.175591	+	0.984463i	$0.443816\pi$
$758$		0			0
$759$		−	541.420i		−	0.713333i
$760$		0			0
$761$	475.322			0.624602			0.312301	−	0.949983i	$-0.398900\pi$
$761$	475.322			0.624602			0.312301	+	0.949983i	$0.398900\pi$
$762$		0			0
$763$			50.7590i			0.0665256i
$764$		0			0
$765$	−160.944	+	158.583i	−0.210384	+	0.207298i
$766$		0			0
$767$	−1310.26			−1.70830
$768$		0			0
$769$	−768.098			−0.998828			−0.499414	−	0.866364i	$-0.666451\pi$
$769$	−768.098			−0.998828			−0.499414	+	0.866364i	$0.666451\pi$
$770$		0			0
$771$	517.216			0.670838
$772$		0			0
$773$	628.494			0.813059			0.406529	−	0.913638i	$-0.366739\pi$
$773$	628.494			0.813059			0.406529	+	0.913638i	$0.366739\pi$
$774$		0			0
$775$	−1379.93	−	20.3875i	−1.78055	−	0.0263064i
$776$		0			0
$777$		−	9.61075i		−	0.0123690i
$778$		0			0
$779$	105.218			0.135068
$780$		0			0
$781$		−	821.909i		−	1.05238i
$782$		0			0
$783$	427.135			0.545511
$784$		0			0
$785$	−206.142	−	209.210i	−0.262601	−	0.266510i
$786$		0			0
$787$			1051.87i			1.33655i	0.743914	+	0.668275i	$0.232967\pi$
$787$			1051.87i			1.33655i	−0.743914	+	0.668275i	$0.767033\pi$
$788$		0			0
$789$			990.146i			1.25494i
$790$		0			0
$791$		−	7.63243i		−	0.00964909i
$792$		0			0
$793$			1426.41i			1.79875i
$794$		0			0
$795$	30.9995	−	30.5449i	0.0389931	−	0.0384213i
$796$		0			0
$797$	−783.662			−0.983265			−0.491632	−	0.870803i	$-0.663600\pi$
$797$	−783.662			−0.983265			−0.491632	+	0.870803i	$0.663600\pi$
$798$		0			0
$799$			593.644i			0.742983i
$800$		0			0
$801$	182.432			0.227755
$802$		0			0
$803$			1682.44i			2.09520i
$804$		0			0
$805$	30.1815	+	30.6307i	0.0374925	+	0.0380505i
$806$		0			0
$807$	36.7936			0.0455931
$808$		0			0
$809$	860.712			1.06392			0.531960	−	0.846769i	$-0.321455\pi$
$809$	860.712			1.06392			0.531960	+	0.846769i	$0.321455\pi$
$810$		0			0
$811$	−489.765			−0.603903			−0.301951	−	0.953323i	$-0.597638\pi$
$811$	−489.765			−0.603903			−0.301951	+	0.953323i	$0.597638\pi$
$812$		0			0
$813$	−327.284			−0.402564
$814$		0			0
$815$	400.888	−	395.009i	0.491887	−	0.484674i
$816$		0			0
$817$		−	127.898i		−	0.156545i
$818$		0			0
$819$	−45.9635			−0.0561215
$820$		0			0
$821$		−	296.634i		−	0.361309i	−0.983547	−	0.180654i	$-0.942178\pi$
$821$		−	296.634i		−	0.361309i	0.983547	−	0.180654i	$-0.0578215\pi$
$822$		0			0
$823$	544.661			0.661800			0.330900	−	0.943666i	$-0.392648\pi$
$823$	544.661			0.661800			0.330900	+	0.943666i	$0.392648\pi$
$824$		0			0
$825$	975.758	+	14.4161i	1.18274	+	0.0174741i
$826$		0			0
$827$		−	187.749i		−	0.227024i	−0.993537	−	0.113512i	$-0.963790\pi$
$827$		−	187.749i		−	0.227024i	0.993537	−	0.113512i	$-0.0362101\pi$
$828$		0			0
$829$		−	153.430i		−	0.185078i	−0.995709	−	0.0925392i	$-0.970502\pi$
$829$		−	153.430i		−	0.185078i	0.995709	−	0.0925392i	$-0.0294983\pi$
$830$		0			0
$831$			825.303i			0.993145i
$832$		0			0
$833$		−	532.825i		−	0.639646i
$834$		0			0
$835$	−831.831	−	844.212i	−0.996205	−	1.01103i
$836$		0			0
$837$	1599.83			1.91138
$838$		0			0
$839$		−	826.140i		−	0.984672i	−0.870405	−	0.492336i	$-0.836143\pi$
$839$		−	826.140i		−	0.984672i	0.870405	−	0.492336i	$-0.163857\pi$
$840$		0			0
$841$	623.773			0.741704
$842$		0			0
$843$			553.761i			0.656893i
$844$		0			0
$845$	541.246	+	549.302i	0.640528	+	0.650062i
$846$		0			0
$847$	118.699			0.140140
$848$		0			0
$849$	−1023.93			−1.20604
$850$		0			0
$851$	97.3505			0.114395
$852$		0			0
$853$	−186.426			−0.218553			−0.109277	−	0.994011i	$-0.534853\pi$
$853$	−186.426			−0.218553			−0.109277	+	0.994011i	$0.534853\pi$
$854$		0			0
$855$	31.4850	+	31.9536i	0.0368245	+	0.0373726i
$856$		0			0
$857$		−	1310.23i		−	1.52885i	−0.644711	−	0.764426i	$-0.723022\pi$
$857$		−	1310.23i		−	1.52885i	0.644711	−	0.764426i	$-0.276978\pi$
$858$		0			0
$859$	−655.375			−0.762951			−0.381475	−	0.924379i	$-0.624584\pi$
$859$	−655.375			−0.762951			−0.381475	+	0.924379i	$0.624584\pi$
$860$		0			0
$861$		−	66.2122i		−	0.0769015i
$862$		0			0
$863$	170.869			0.197995			0.0989973	−	0.995088i	$-0.468436\pi$
$863$	170.869			0.197995			0.0989973	+	0.995088i	$0.468436\pi$
$864$		0			0
$865$	−1028.05	−	1043.35i	−1.18850	−	1.20619i
$866$		0			0
$867$		−	372.947i		−	0.430158i
$868$		0			0
$869$			427.818i			0.492311i
$870$		0			0
$871$			481.329i			0.552617i
$872$		0			0
$873$		−	508.207i		−	0.582139i
$874$		0			0
$875$	−56.0068	+	53.5781i	−0.0640078	+	0.0612321i
$876$		0			0
$877$	−303.743			−0.346343			−0.173171	−	0.984892i	$-0.555401\pi$
$877$	−303.743			−0.346343			−0.173171	+	0.984892i	$0.555401\pi$
$878$		0			0
$879$		−	608.307i		−	0.692044i
$880$		0			0
$881$	2.32385			0.00263774			0.00131887	−	0.999999i	$-0.499580\pi$
$881$	2.32385			0.00263774			0.00131887	+	0.999999i	$0.499580\pi$
$882$		0			0
$883$		−	260.688i		−	0.295230i	−0.989045	−	0.147615i	$-0.952840\pi$
$883$		−	260.688i		−	0.295230i	0.989045	−	0.147615i	$-0.0471596\pi$
$884$		0			0
$885$	573.209	−	564.803i	0.647694	−	0.638195i
$886$		0			0
$887$	−452.710			−0.510384			−0.255192	−	0.966890i	$-0.582139\pi$
$887$	−452.710			−0.510384			−0.255192	+	0.966890i	$0.582139\pi$
$888$		0			0
$889$	109.066			0.122684
$890$		0			0
$891$	−475.337			−0.533487
$892$		0			0
$893$	117.861			0.131984
$894$		0			0
$895$	−634.631	−	644.077i	−0.709085	−	0.719639i
$896$		0			0
$897$			550.698i			0.613933i
$898$		0			0
$899$	−813.620			−0.905028
$900$		0			0
$901$			43.1970i			0.0479434i
$902$		0			0
$903$	−80.4842			−0.0891298
$904$		0			0
$905$	−923.265	+	909.725i	−1.02018	+	1.00522i
$906$		0			0
$907$			417.248i			0.460031i	0.973187	+	0.230015i	$0.0738777\pi$
$907$			417.248i			0.460031i	−0.973187	+	0.230015i	$0.926122\pi$
$908$		0			0
$909$		−	134.860i		−	0.148361i
$910$		0			0
$911$			673.050i			0.738804i	0.929270	+	0.369402i	$0.120437\pi$
$911$			673.050i			0.738804i	−0.929270	+	0.369402i	$0.879563\pi$
$912$		0			0
$913$		−	1897.69i		−	2.07852i
$914$		0			0
$915$	−614.868	−	624.019i	−0.671987	−	0.681988i
$916$		0			0
$917$	−9.44465			−0.0102995
$918$		0			0
$919$			1204.93i			1.31113i	0.755138	+	0.655566i	$0.227570\pi$
$919$			1204.93i			1.31113i	−0.755138	+	0.655566i	$0.772430\pi$
$920$		0			0
$921$	290.665			0.315597
$922$		0			0
$923$			835.993i			0.905735i
$924$		0			0
$925$	−2.59210	+	175.447i	−0.00280227	+	0.189672i
$926$		0			0
$927$	−289.797			−0.312618
$928$		0			0
$929$	658.790			0.709139			0.354569	−	0.935030i	$-0.384628\pi$
$929$	658.790			0.709139			0.354569	+	0.935030i	$0.384628\pi$
$930$		0			0
$931$	−105.787			−0.113627
$932$		0			0
$933$	−780.691			−0.836754
$934$		0			0
$935$	−689.965	+	679.846i	−0.737930	+	0.727108i
$936$		0			0
$937$			1690.80i			1.80448i	0.431237	+	0.902239i	$0.358077\pi$
$937$			1690.80i			1.80448i	−0.431237	+	0.902239i	$0.641923\pi$
$938$		0			0
$939$	621.664			0.662048
$940$		0			0
$941$			219.663i			0.233436i	0.993165	+	0.116718i	$0.0372373\pi$
$941$			219.663i			0.233436i	−0.993165	+	0.116718i	$0.962763\pi$
$942$		0			0
$943$	670.685			0.711225
$944$		0			0
$945$	64.0000	−	63.0614i	0.0677249	−	0.0667317i
$946$		0			0
$947$		−	552.774i		−	0.583711i	−0.956462	−	0.291855i	$-0.905727\pi$
$947$		−	552.774i		−	0.583711i	0.956462	−	0.291855i	$-0.0942726\pi$
$948$		0			0
$949$		−	1711.27i		−	1.80324i
$950$		0			0
$951$		−	477.513i		−	0.502117i
$952$		0			0
$953$			15.7654i			0.0165429i	0.999966	+	0.00827144i	$0.00263291\pi$
$953$			15.7654i			0.0165429i	−0.999966	+	0.00827144i	$0.997367\pi$
$954$		0			0
$955$	958.842	−	944.780i	1.00402	−	0.989299i
$956$		0			0
$957$	575.316			0.601166
$958$		0			0
$959$			154.998i			0.161624i
$960$		0			0
$961$	−2086.39			−2.17107
$962$		0			0
$963$			361.834i			0.375736i
$964$		0			0
$965$	517.060	−	509.477i	0.535814	−	0.527956i
$966$		0			0
$967$	308.027			0.318538			0.159269	−	0.987235i	$-0.449086\pi$
$967$	308.027			0.318538			0.159269	+	0.987235i	$0.449086\pi$
$968$		0			0
$969$	52.6668			0.0543517
$970$		0			0
$971$	41.8794			0.0431302			0.0215651	−	0.999767i	$-0.493135\pi$
$971$	41.8794			0.0431302			0.0215651	+	0.999767i	$0.493135\pi$
$972$		0			0
$973$	−139.781			−0.143660
$974$		0			0
$975$	−992.478	−	14.6632i	−1.01793	−	0.0150391i
$976$		0			0
$977$		−	586.276i		−	0.600078i	−0.953927	−	0.300039i	$-0.903000\pi$
$977$		−	586.276i		−	0.600078i	0.953927	−	0.300039i	$-0.0969997\pi$
$978$		0			0
$979$	782.085			0.798861
$980$		0			0
$981$			337.525i			0.344062i
$982$		0			0
$983$	1627.01			1.65515			0.827573	−	0.561358i	$-0.189721\pi$
$983$	1627.01			1.65515			0.827573	+	0.561358i	$0.189721\pi$
$984$		0			0
$985$	955.862	+	970.088i	0.970418	+	0.984861i
$986$		0			0
$987$		−	74.1686i		−	0.0751455i
$988$		0			0
$989$		−	815.252i		−	0.824319i
$990$		0			0
$991$			255.629i			0.257950i	0.991648	+	0.128975i	$0.0411687\pi$
$991$			255.629i			0.257950i	−0.991648	+	0.128975i	$0.958831\pi$
$992$		0			0
$993$		−	736.260i		−	0.741450i
$994$		0			0
$995$	1172.86	−	1155.66i	1.17875	−	1.16147i
$996$		0			0
$997$	−1001.35			−1.00436			−0.502180	−	0.864763i	$-0.667468\pi$
$997$	−1001.35			−1.00436			−0.502180	+	0.864763i	$0.667468\pi$
$998$		0			0
$999$		−	203.405i		−	0.203609i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.3.e.i.319.8	yes	16
4.3	odd	2	inner	640.3.e.i.319.12	yes	16
5.4	even	2	inner	640.3.e.i.319.10	yes	16
8.3	odd	2	inner	640.3.e.i.319.5	✓	16
8.5	even	2	inner	640.3.e.i.319.9	yes	16
16.3	odd	4		1280.3.h.l.1279.3		8
16.5	even	4		1280.3.h.h.1279.4		8
16.11	odd	4		1280.3.h.h.1279.6		8
16.13	even	4		1280.3.h.l.1279.5		8
20.19	odd	2	inner	640.3.e.i.319.6	yes	16
40.19	odd	2	inner	640.3.e.i.319.11	yes	16
40.29	even	2	inner	640.3.e.i.319.7	yes	16
80.19	odd	4		1280.3.h.l.1279.6		8
80.29	even	4		1280.3.h.l.1279.4		8
80.59	odd	4		1280.3.h.h.1279.3		8
80.69	even	4		1280.3.h.h.1279.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.e.i.319.5	✓	16	8.3	odd	2	inner
640.3.e.i.319.6	yes	16	20.19	odd	2	inner
640.3.e.i.319.7	yes	16	40.29	even	2	inner
640.3.e.i.319.8	yes	16	1.1	even	1	trivial
640.3.e.i.319.9	yes	16	8.5	even	2	inner
640.3.e.i.319.10	yes	16	5.4	even	2	inner
640.3.e.i.319.11	yes	16	40.19	odd	2	inner
640.3.e.i.319.12	yes	16	4.3	odd	2	inner
1280.3.h.h.1279.3		8	80.59	odd	4
1280.3.h.h.1279.4		8	16.5	even	4
1280.3.h.h.1279.5		8	80.69	even	4
1280.3.h.h.1279.6		8	16.11	odd	4
1280.3.h.l.1279.3		8	16.3	odd	4
1280.3.h.l.1279.4		8	80.29	even	4
1280.3.h.l.1279.5		8	16.13	even	4
1280.3.h.l.1279.6		8	80.19	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	640.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.20837i q^{3} +(3.50932 + 3.56155i) q^{5} +0.620058 q^{7} +4.12311 q^{9} +O(q^{10})\)	\(q-2.20837i q^{3} +(3.50932 + 3.56155i) q^{5} +0.620058 q^{7} +4.12311 q^{9} +17.6757 q^{11} -17.9786 q^{13} +(7.86522 - 7.74988i) q^{15} +10.9600i q^{17} +2.17598 q^{19} -1.36932i q^{21} +13.8703 q^{23} +(-0.369317 + 24.9973i) q^{25} -28.9807i q^{27} +14.7386i q^{29} +55.2032i q^{31} -39.0346i q^{33} +(2.17598 + 2.20837i) q^{35} +7.01864 q^{37} +39.7034i q^{39} +48.3542 q^{41} -58.7769i q^{43} +(14.4693 + 14.6847i) q^{45} +54.1646 q^{47} -48.6155 q^{49} +24.2037 q^{51} +3.94134 q^{53} +(62.0299 + 62.9531i) q^{55} -4.80537i q^{57} +72.8789 q^{59} -79.3390i q^{61} +2.55656 q^{63} +(-63.0928 - 64.0318i) q^{65} -26.7723i q^{67} -30.6307i q^{69} -46.4993i q^{71} +95.1837i q^{73} +(55.2032 + 0.815588i) q^{75} +10.9600 q^{77} +24.2037i q^{79} -26.8920 q^{81} -107.361i q^{83} +(-39.0346 + 38.4621i) q^{85} +32.5483 q^{87} +44.2462 q^{89} -11.1478 q^{91} +121.909 q^{93} +(7.63623 + 7.74988i) q^{95} -123.258i q^{97} +72.8789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 192 q^{25} + 48 q^{41} - 448 q^{49} + 112 q^{65} - 1024 q^{81} + 576 q^{89}+O(q^{100}) \) 16 * q + 192 * q^25 + 48 * q^41 - 448 * q^49 + 112 * q^65 - 1024 * q^81 + 576 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more