LMFDB - Embedded newform 640.3.e.i.319.16

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			319.16
Root			$-0.332843 - 1.47845i$ of defining polynomial
Character	$\chi$	$=$	640.319
Dual form			640.3.e.i.319.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.62258i q^{3} +(4.96837 + 0.561553i) q^{5} -6.45101 q^{7} -4.12311 q^{9} +O(q^{10})$	$q+3.62258i q^{3} +(4.96837 + 0.561553i) q^{5} -6.45101 q^{7} -4.12311 q^{9} +3.94566 q^{11} +15.5167 q^{13} +(-2.03427 + 17.9983i) q^{15} +25.4535i q^{17} -32.0510 q^{19} -23.3693i q^{21} +15.2845 q^{23} +(24.3693 + 5.58000i) q^{25} +17.6670i q^{27} +34.7386i q^{29} +20.2140i q^{31} +14.2935i q^{33} +(-32.0510 - 3.62258i) q^{35} +9.93673 q^{37} +56.2106i q^{39} -42.3542 q^{41} -52.9460i q^{43} +(-20.4851 - 2.31534i) q^{45} +21.6377 q^{47} -7.38447 q^{49} -92.2073 q^{51} -35.3902 q^{53} +(19.6035 + 2.21569i) q^{55} -116.107i q^{57} -16.2684 q^{59} -77.3390i q^{61} +26.5982 q^{63} +(77.0928 + 8.71346i) q^{65} +14.0444i q^{67} +55.3693i q^{69} +107.990i q^{71} -93.7873i q^{73} +(-20.2140 + 88.2799i) q^{75} -25.4535 q^{77} +92.2073i q^{79} -101.108 q^{81} +26.7509i q^{83} +(-14.2935 + 126.462i) q^{85} -125.844 q^{87} +27.7538 q^{89} -100.099 q^{91} -73.2269 q^{93} +(-159.241 - 17.9983i) q^{95} +133.534i q^{97} -16.2684 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			3.62258i			1.20753i	0.797163	+	0.603764i	$0.206333\pi$
$3$			3.62258i			1.20753i	−0.797163	+	0.603764i	$0.793667\pi$
$4$		0			0
$5$	4.96837	+	0.561553i	0.993673	+	0.112311i
$5$	4.96837	+	0.561553i	0.993673	+	0.112311i
$6$		0			0
$7$	−6.45101			−0.921573			−0.460786	−	0.887511i	$-0.652433\pi$
$7$	−6.45101			−0.921573			−0.460786	+	0.887511i	$0.652433\pi$
$8$		0			0
$9$	−4.12311			−0.458123
$10$		0			0
$11$	3.94566			0.358696			0.179348	−	0.983786i	$-0.442601\pi$
$11$	3.94566			0.358696			0.179348	+	0.983786i	$0.442601\pi$
$12$		0			0
$13$	15.5167			1.19359			0.596797	−	0.802392i	$-0.296440\pi$
$13$	15.5167			1.19359			0.596797	+	0.802392i	$0.296440\pi$
$14$		0			0
$15$	−2.03427	+	17.9983i	−0.135618	+	1.19989i
$16$		0			0
$17$			25.4535i			1.49726i	0.662987	+	0.748631i	$0.269289\pi$
$17$			25.4535i			1.49726i	−0.662987	+	0.748631i	$0.730711\pi$
$18$		0			0
$19$	−32.0510			−1.68689			−0.843447	−	0.537213i	$-0.819477\pi$
$19$	−32.0510			−1.68689			−0.843447	+	0.537213i	$0.819477\pi$
$20$		0			0
$21$		−	23.3693i		−	1.11282i
$22$		0			0
$23$	15.2845			0.664543			0.332271	−	0.943184i	$-0.392185\pi$
$23$	15.2845			0.664543			0.332271	+	0.943184i	$0.392185\pi$
$24$		0			0
$25$	24.3693	+	5.58000i	0.974773	+	0.223200i
$26$		0			0
$27$			17.6670i			0.654332i
$28$		0			0
$29$			34.7386i			1.19788i	0.800792	+	0.598942i	$0.204412\pi$
$29$			34.7386i			1.19788i	−0.800792	+	0.598942i	$0.795588\pi$
$30$		0			0
$31$			20.2140i			0.652065i	0.945359	+	0.326032i	$0.105712\pi$
$31$			20.2140i			0.652065i	−0.945359	+	0.326032i	$0.894288\pi$
$32$		0			0
$33$			14.2935i			0.433135i
$34$		0			0
$35$	−32.0510	−	3.62258i	−0.915742	−	0.103502i
$36$		0			0
$37$	9.93673			0.268560			0.134280	−	0.990943i	$-0.457128\pi$
$37$	9.93673			0.268560			0.134280	+	0.990943i	$0.457128\pi$
$38$		0			0
$39$			56.2106i			1.44130i
$40$		0			0
$41$	−42.3542			−1.03303			−0.516514	−	0.856279i	$-0.672771\pi$
$41$	−42.3542			−1.03303			−0.516514	+	0.856279i	$0.672771\pi$
$42$		0			0
$43$		−	52.9460i		−	1.23130i	−0.788019	−	0.615651i	$-0.788893\pi$
$43$		−	52.9460i		−	1.23130i	0.788019	−	0.615651i	$-0.211107\pi$
$44$		0			0
$45$	−20.4851	−	2.31534i	−0.455224	−	0.0514520i
$46$		0			0
$47$	21.6377			0.460377			0.230189	−	0.973146i	$-0.426066\pi$
$47$	21.6377			0.460377			0.230189	+	0.973146i	$0.426066\pi$
$48$		0			0
$49$	−7.38447			−0.150704
$50$		0			0
$51$	−92.2073			−1.80799
$52$		0			0
$53$	−35.3902			−0.667740			−0.333870	−	0.942619i	$-0.608355\pi$
$53$	−35.3902			−0.667740			−0.333870	+	0.942619i	$0.608355\pi$
$54$		0			0
$55$	19.6035	+	2.21569i	0.356427	+	0.0402853i
$56$		0			0
$57$		−	116.107i		−	2.03697i
$58$		0			0
$59$	−16.2684			−0.275735			−0.137867	−	0.990451i	$-0.544025\pi$
$59$	−16.2684			−0.275735			−0.137867	+	0.990451i	$0.544025\pi$
$60$		0			0
$61$		−	77.3390i		−	1.26785i	−0.773393	−	0.633926i	$-0.781442\pi$
$61$		−	77.3390i		−	1.26785i	0.773393	−	0.633926i	$-0.218558\pi$
$62$		0			0
$63$	26.5982			0.422194
$64$		0			0
$65$	77.0928	+	8.71346i	1.18604	+	0.134053i
$66$		0			0
$67$			14.0444i			0.209617i	0.994492	+	0.104809i	$0.0334230\pi$
$67$			14.0444i			0.209617i	−0.994492	+	0.104809i	$0.966577\pi$
$68$		0			0
$69$			55.3693i			0.802454i
$70$		0			0
$71$			107.990i			1.52098i	0.649347	+	0.760492i	$0.275042\pi$
$71$			107.990i			1.52098i	−0.649347	+	0.760492i	$0.724958\pi$
$72$		0			0
$73$		−	93.7873i		−	1.28476i	−0.766387	−	0.642379i	$-0.777948\pi$
$73$		−	93.7873i		−	1.28476i	0.766387	−	0.642379i	$-0.222052\pi$
$74$		0			0
$75$	−20.2140	+	88.2799i	−0.269520	+	1.17706i
$76$		0			0
$77$	−25.4535			−0.330564
$78$		0			0
$79$			92.2073i			1.16718i	0.812048	+	0.583590i	$0.198353\pi$
$79$			92.2073i			1.16718i	−0.812048	+	0.583590i	$0.801647\pi$
$80$		0			0
$81$	−101.108			−1.24825
$82$		0			0
$83$			26.7509i			0.322300i	0.986930	+	0.161150i	$0.0515202\pi$
$83$			26.7509i			0.322300i	−0.986930	+	0.161150i	$0.948480\pi$
$84$		0			0
$85$	−14.2935	+	126.462i	−0.168158	+	1.48779i
$86$		0			0
$87$	−125.844			−1.44648
$88$		0			0
$89$	27.7538			0.311840			0.155920	−	0.987770i	$-0.450166\pi$
$89$	27.7538			0.311840			0.155920	+	0.987770i	$0.450166\pi$
$90$		0			0
$91$	−100.099			−1.09998
$92$		0			0
$93$	−73.2269			−0.787386
$94$		0			0
$95$	−159.241	−	17.9983i	−1.67622	−	0.189456i
$96$		0			0
$97$			133.534i			1.37664i	0.725406	+	0.688321i	$0.241652\pi$
$97$			133.534i			1.37664i	−0.725406	+	0.688321i	$0.758348\pi$
$98$		0			0
$99$	−16.2684			−0.164327
$100$		0			0
$101$		−	148.708i		−	1.47236i	−0.676786	−	0.736180i	$-0.736628\pi$
$101$		−	148.708i		−	1.47236i	0.676786	−	0.736180i	$-0.263372\pi$
$102$		0			0
$103$	146.089			1.41834			0.709168	−	0.705040i	$-0.249071\pi$
$103$	146.089			1.41834			0.709168	+	0.705040i	$0.249071\pi$
$104$		0			0
$105$	13.1231	−	116.107i	0.124982	−	1.10578i
$106$		0			0
$107$			35.2790i			0.329710i	0.986318	+	0.164855i	$0.0527157\pi$
$107$			35.2790i			0.329710i	−0.986318	+	0.164855i	$0.947284\pi$
$108$		0			0
$109$		−	24.1383i		−	0.221452i	−0.993851	−	0.110726i	$-0.964682\pi$
$109$		−	24.1383i		−	0.221452i	0.993851	−	0.110726i	$-0.0353176\pi$
$110$		0			0
$111$			35.9966i			0.324294i
$112$		0			0
$113$			181.308i			1.60449i	0.596993	+	0.802246i	$0.296362\pi$
$113$			181.308i			1.60449i	−0.596993	+	0.802246i	$0.703638\pi$
$114$		0			0
$115$	75.9389	+	8.58305i	0.660338	+	0.0746352i
$116$		0			0
$117$	−63.9771			−0.546813
$118$		0			0
$119$		−	164.201i		−	1.37984i
$120$		0			0
$121$	−105.432			−0.871337
$122$		0			0
$123$		−	153.431i		−	1.24741i
$124$		0			0
$125$	117.942	+	41.4081i	0.943538	+	0.331265i
$126$		0			0
$127$	191.453			1.50751			0.753753	−	0.657158i	$-0.228242\pi$
$127$	191.453			1.50751			0.753753	+	0.657158i	$0.228242\pi$
$128$		0			0
$129$	191.801			1.48683
$130$		0			0
$131$	224.357			1.71265			0.856324	−	0.516439i	$-0.172743\pi$
$131$	224.357			1.71265			0.856324	+	0.516439i	$0.172743\pi$
$132$		0			0
$133$	206.761			1.55460
$134$		0			0
$135$	−9.92093	+	87.7759i	−0.0734884	+	0.650192i
$136$		0			0
$137$			55.8000i			0.407299i	0.979044	+	0.203650i	$0.0652803\pi$
$137$			55.8000i			0.407299i	−0.979044	+	0.203650i	$0.934720\pi$
$138$		0			0
$139$	−115.395			−0.830183			−0.415092	−	0.909780i	$-0.636250\pi$
$139$	−115.395			−0.830183			−0.415092	+	0.909780i	$0.636250\pi$
$140$		0			0
$141$			78.3845i			0.555918i
$142$		0			0
$143$	61.2237			0.428138
$144$		0			0
$145$	−19.5076	+	172.594i	−0.134535	+	1.19031i
$146$		0			0
$147$		−	26.7509i		−	0.181979i
$148$		0			0
$149$		−	52.6307i		−	0.353226i	−0.984280	−	0.176613i	$-0.943486\pi$
$149$		−	52.6307i		−	0.353226i	0.984280	−	0.176613i	$-0.0565141\pi$
$150$		0			0
$151$		−	231.762i		−	1.53485i	−0.641138	−	0.767425i	$-0.721538\pi$
$151$		−	231.762i		−	1.53485i	0.641138	−	0.767425i	$-0.278462\pi$
$152$		0			0
$153$		−	104.947i		−	0.685930i
$154$		0			0
$155$	−11.3512	+	100.431i	−0.0732338	+	0.647939i
$156$		0			0
$157$	162.658			1.03604			0.518018	−	0.855370i	$-0.326670\pi$
$157$	162.658			1.03604			0.518018	+	0.855370i	$0.326670\pi$
$158$		0			0
$159$		−	128.204i		−	0.806314i
$160$		0			0
$161$	−98.6004			−0.612425
$162$		0			0
$163$		−	293.319i		−	1.79951i	−0.436400	−	0.899753i	$-0.643747\pi$
$163$		−	293.319i		−	1.79951i	0.436400	−	0.899753i	$-0.356253\pi$
$164$		0			0
$165$	−8.02653	+	71.0152i	−0.0486457	+	0.430395i
$166$		0			0
$167$	−246.934			−1.47865			−0.739324	−	0.673350i	$-0.764855\pi$
$167$	−246.934			−1.47865			−0.739324	+	0.673350i	$0.764855\pi$
$168$		0			0
$169$	71.7689			0.424668
$170$		0			0
$171$	132.150			0.772804
$172$		0			0
$173$	76.8967			0.444490			0.222245	−	0.974991i	$-0.428662\pi$
$173$	76.8967			0.444490			0.222245	+	0.974991i	$0.428662\pi$
$174$		0			0
$175$	−157.207	−	35.9966i	−0.898324	−	0.205695i
$176$		0			0
$177$		−	58.9335i		−	0.332957i
$178$		0			0
$179$	284.999			1.59217			0.796086	−	0.605183i	$-0.206900\pi$
$179$	284.999			1.59217			0.796086	+	0.605183i	$0.206900\pi$
$180$		0			0
$181$		−	176.769i		−	0.976624i	−0.872669	−	0.488312i	$-0.837613\pi$
$181$		−	176.769i		−	0.976624i	0.872669	−	0.488312i	$-0.162387\pi$
$182$		0			0
$183$	280.167			1.53097
$184$		0			0
$185$	49.3693	+	5.58000i	0.266861	+	0.0301622i
$186$		0			0
$187$			100.431i			0.537062i
$188$		0			0
$189$		−	113.970i		−	0.603014i
$190$		0			0
$191$		−	265.271i		−	1.38885i	−0.719564	−	0.694426i	$-0.755659\pi$
$191$		−	265.271i		−	1.38885i	0.719564	−	0.694426i	$-0.244341\pi$
$192$		0			0
$193$			82.6273i			0.428121i	0.976820	+	0.214060i	$0.0686689\pi$
$193$			82.6273i			0.428121i	−0.976820	+	0.214060i	$0.931331\pi$
$194$		0			0
$195$	−31.5652	+	279.275i	−0.161873	+	1.43218i
$196$		0			0
$197$	180.771			0.917621			0.458811	−	0.888534i	$-0.348276\pi$
$197$	180.771			0.917621			0.458811	+	0.888534i	$0.348276\pi$
$198$		0			0
$199$			171.120i			0.859901i	0.902852	+	0.429951i	$0.141469\pi$
$199$			171.120i			0.859901i	−0.902852	+	0.429951i	$0.858531\pi$
$200$		0			0
$201$	−50.8769			−0.253119
$202$		0			0
$203$		−	224.099i		−	1.10394i
$204$		0			0
$205$	−210.431	−	23.7841i	−1.02649	−	0.116020i
$206$		0			0
$207$	−63.0196			−0.304442
$208$		0			0
$209$	−126.462			−0.605082
$210$		0			0
$211$	96.1529			0.455701			0.227851	−	0.973696i	$-0.426830\pi$
$211$	96.1529			0.455701			0.227851	+	0.973696i	$0.426830\pi$
$212$		0			0
$213$	−391.202			−1.83663
$214$		0			0
$215$	29.7320	−	263.055i	0.138288	−	1.22351i
$216$		0			0
$217$		−	130.401i		−	0.600925i
$218$		0			0
$219$	339.752			1.55138
$220$		0			0
$221$			394.955i			1.78712i
$222$		0			0
$223$	79.8803			0.358208			0.179104	−	0.983830i	$-0.442680\pi$
$223$	79.8803			0.358208			0.179104	+	0.983830i	$0.442680\pi$
$224$		0			0
$225$	−100.477	−	23.0069i	−0.446566	−	0.102253i
$226$		0			0
$227$			229.897i			1.01276i	0.862310	+	0.506381i	$0.169017\pi$
$227$			229.897i			1.01276i	−0.862310	+	0.506381i	$0.830983\pi$
$228$		0			0
$229$			132.277i			0.577627i	0.957385	+	0.288813i	$0.0932607\pi$
$229$			132.277i			0.577627i	−0.957385	+	0.288813i	$0.906739\pi$
$230$		0			0
$231$		−	92.2073i		−	0.399166i
$232$		0			0
$233$		−	81.2535i		−	0.348727i	−0.984681	−	0.174364i	$-0.944213\pi$
$233$		−	81.2535i		−	0.348727i	0.984681	−	0.174364i	$-0.0557868\pi$
$234$		0			0
$235$	107.504	+	12.1507i	0.457465	+	0.0517052i
$236$		0			0
$237$	−334.028			−1.40940
$238$		0			0
$239$		−	317.050i		−	1.32657i	−0.748368	−	0.663284i	$-0.769162\pi$
$239$		−	317.050i		−	1.32657i	0.748368	−	0.663284i	$-0.230838\pi$
$240$		0			0
$241$	446.725			1.85363			0.926816	−	0.375516i	$-0.122534\pi$
$241$	446.725			1.85363			0.926816	+	0.375516i	$0.122534\pi$
$242$		0			0
$243$		−	207.269i		−	0.852960i
$244$		0			0
$245$	−36.6888	−	4.14677i	−0.149750	−	0.0169256i
$246$		0			0
$247$	−497.326			−2.01347
$248$		0			0
$249$	−96.9072			−0.389186
$250$		0			0
$251$	253.434			1.00970			0.504848	−	0.863208i	$-0.331549\pi$
$251$	253.434			1.00970			0.504848	+	0.863208i	$0.331549\pi$
$252$		0			0
$253$	60.3073			0.238369
$254$		0			0
$255$	−458.119	−	51.7793i	−1.79655	−	0.203056i
$256$		0			0
$257$		−	85.7608i		−	0.333700i	−0.985982	−	0.166850i	$-0.946641\pi$
$257$		−	85.7608i		−	0.333700i	0.985982	−	0.166850i	$-0.0533595\pi$
$258$		0			0
$259$	−64.1020			−0.247498
$260$		0			0
$261$		−	143.231i		−	0.548778i
$262$		0			0
$263$	302.587			1.15052			0.575260	−	0.817971i	$-0.304901\pi$
$263$	302.587			1.15052			0.575260	+	0.817971i	$0.304901\pi$
$264$		0			0
$265$	−175.831	−	19.8735i	−0.663515	−	0.0749942i
$266$		0			0
$267$			100.540i			0.376556i
$268$		0			0
$269$		−	173.339i		−	0.644383i	−0.946675	−	0.322191i	$-0.895581\pi$
$269$		−	173.339i		−	0.644383i	0.946675	−	0.322191i	$-0.104419\pi$
$270$		0			0
$271$			195.766i			0.722383i	0.932492	+	0.361191i	$0.117630\pi$
$271$			195.766i			0.722383i	−0.932492	+	0.361191i	$0.882370\pi$
$272$		0			0
$273$		−	362.615i		−	1.32826i
$274$		0			0
$275$	96.1529	+	22.0168i	0.349647	+	0.0800609i
$276$		0			0
$277$	−365.213			−1.31846			−0.659228	−	0.751943i	$-0.729117\pi$
$277$	−365.213			−1.31846			−0.659228	+	0.751943i	$0.729117\pi$
$278$		0			0
$279$		−	83.3445i		−	0.298726i
$280$		0			0
$281$	268.756			0.956426			0.478213	−	0.878244i	$-0.341285\pi$
$281$	268.756			0.956426			0.478213	+	0.878244i	$0.341285\pi$
$282$		0			0
$283$			43.6357i			0.154190i	0.997024	+	0.0770949i	$0.0245644\pi$
$283$			43.6357i			0.154190i	−0.997024	+	0.0770949i	$0.975436\pi$
$284$		0			0
$285$	65.2004	−	576.864i	0.228773	−	2.02408i
$286$		0			0
$287$	273.227			0.952011
$288$		0			0
$289$	−358.879			−1.24180
$290$		0			0
$291$	−483.739			−1.66233
$292$		0			0
$293$	226.098			0.771667			0.385833	−	0.922569i	$-0.373914\pi$
$293$	226.098			0.771667			0.385833	+	0.922569i	$0.373914\pi$
$294$		0			0
$295$	−80.8271	−	9.13554i	−0.273990	−	0.0309679i
$296$		0			0
$297$			69.7077i			0.234706i
$298$		0			0
$299$	237.165			0.793195
$300$		0			0
$301$			341.555i			1.13473i
$302$		0			0
$303$	538.708			1.77791
$304$		0			0
$305$	43.4299	−	384.248i	0.142393	−	1.25983i
$306$		0			0
$307$			172.436i			0.561682i	0.959754	+	0.280841i	$0.0906133\pi$
$307$			172.436i			0.561682i	−0.959754	+	0.280841i	$0.909387\pi$
$308$		0			0
$309$			529.218i			1.71268i
$310$		0			0
$311$			78.9131i			0.253740i	0.991919	+	0.126870i	$0.0404931\pi$
$311$			78.9131i			0.253740i	−0.991919	+	0.126870i	$0.959507\pi$
$312$		0			0
$313$			338.922i			1.08282i	0.840760	+	0.541408i	$0.182109\pi$
$313$			338.922i			1.08282i	−0.840760	+	0.541408i	$0.817891\pi$
$314$		0			0
$315$	132.150	+	14.9363i	0.419522	+	0.0474168i
$316$		0			0
$317$	101.278			0.319487			0.159744	−	0.987159i	$-0.448933\pi$
$317$	101.278			0.319487			0.159744	+	0.987159i	$0.448933\pi$
$318$		0			0
$319$			137.067i			0.429676i
$320$		0			0
$321$	−127.801			−0.398134
$322$		0			0
$323$		−	815.808i		−	2.52572i
$324$		0			0
$325$	378.132	+	86.5834i	1.16348	+	0.266410i
$326$		0			0
$327$	87.4428			0.267409
$328$		0			0
$329$	−139.585			−0.424271
$330$		0			0
$331$	−153.335			−0.463248			−0.231624	−	0.972805i	$-0.574404\pi$
$331$	−153.335			−0.463248			−0.231624	+	0.972805i	$0.574404\pi$
$332$		0			0
$333$	−40.9702			−0.123034
$334$		0			0
$335$	−7.88666	+	69.7776i	−0.0235423	+	0.208291i
$336$		0			0
$337$			146.454i			0.434581i	0.976107	+	0.217291i	$0.0697219\pi$
$337$			146.454i			0.434581i	−0.976107	+	0.217291i	$0.930278\pi$
$338$		0			0
$339$	−656.802			−1.93747
$340$		0			0
$341$			79.7575i			0.233893i
$342$		0			0
$343$	363.737			1.06046
$344$		0			0
$345$	−31.0928	+	275.095i	−0.0901240	+	0.797377i
$346$		0			0
$347$		−	332.832i		−	0.959169i	−0.877496	−	0.479585i	$-0.840787\pi$
$347$		−	332.832i		−	0.959169i	0.877496	−	0.479585i	$-0.159213\pi$
$348$		0			0
$349$		−	307.170i		−	0.880145i	−0.897962	−	0.440072i	$-0.854953\pi$
$349$		−	307.170i		−	0.880145i	0.897962	−	0.440072i	$-0.145047\pi$
$350$		0			0
$351$			274.133i			0.781007i
$352$		0			0
$353$		−	177.788i		−	0.503650i	−0.967773	−	0.251825i	$-0.918969\pi$
$353$		−	177.788i		−	0.503650i	0.967773	−	0.251825i	$-0.0810308\pi$
$354$		0			0
$355$	−60.6420	+	536.533i	−0.170823	+	1.51136i
$356$		0			0
$357$	594.830			1.66619
$358$		0			0
$359$		−	447.742i		−	1.24719i	−0.781746	−	0.623596i	$-0.785671\pi$
$359$		−	447.742i		−	1.24719i	0.781746	−	0.623596i	$-0.214329\pi$
$360$		0			0
$361$	666.265			1.84561
$362$		0			0
$363$		−	381.935i		−	1.05216i
$364$		0			0
$365$	52.6665	−	465.970i	0.144292	−	1.27663i
$366$		0			0
$367$	−417.312			−1.13709			−0.568545	−	0.822652i	$-0.692494\pi$
$367$	−417.312			−1.13709			−0.568545	+	0.822652i	$0.692494\pi$
$368$		0			0
$369$	174.631			0.473254
$370$		0			0
$371$	228.302			0.615371
$372$		0			0
$373$	−206.225			−0.552882			−0.276441	−	0.961031i	$-0.589155\pi$
$373$	−206.225			−0.552882			−0.276441	+	0.961031i	$0.589155\pi$
$374$		0			0
$375$	−150.004	+	427.255i	−0.400012	+	1.13935i
$376$		0			0
$377$			539.030i			1.42979i
$378$		0			0
$379$	−573.943			−1.51436			−0.757181	−	0.653205i	$-0.773424\pi$
$379$	−573.943			−1.51436			−0.757181	+	0.653205i	$0.773424\pi$
$380$		0			0
$381$			693.555i			1.82035i
$382$		0			0
$383$	−144.219			−0.376551			−0.188275	−	0.982116i	$-0.560290\pi$
$383$	−144.219			−0.376551			−0.188275	+	0.982116i	$0.560290\pi$
$384$		0			0
$385$	−126.462	−	14.2935i	−0.328473	−	0.0371259i
$386$		0			0
$387$			218.302i			0.564087i
$388$		0			0
$389$		−	470.358i		−	1.20915i	−0.796550	−	0.604573i	$-0.793344\pi$
$389$		−	470.358i		−	1.20915i	0.796550	−	0.604573i	$-0.206656\pi$
$390$		0			0
$391$			389.043i			0.994995i
$392$		0			0
$393$			812.751i			2.06807i
$394$		0			0
$395$	−51.7793	+	458.119i	−0.131087	+	1.15980i
$396$		0			0
$397$	337.011			0.848895			0.424448	−	0.905453i	$-0.360468\pi$
$397$	337.011			0.848895			0.424448	+	0.905453i	$0.360468\pi$
$398$		0			0
$399$			749.009i			1.87722i
$400$		0			0
$401$	228.739			0.570421			0.285210	−	0.958465i	$-0.407937\pi$
$401$	228.739			0.570421			0.285210	+	0.958465i	$0.407937\pi$
$402$		0			0
$403$			313.655i			0.778301i
$404$		0			0
$405$	−502.341	−	56.7775i	−1.24035	−	0.140191i
$406$		0			0
$407$	39.2069			0.0963315
$408$		0			0
$409$	−576.263			−1.40896			−0.704478	−	0.709726i	$-0.748819\pi$
$409$	−576.263			−1.40896			−0.704478	+	0.709726i	$0.748819\pi$
$410$		0			0
$411$	−202.140			−0.491825
$412$		0			0
$413$	104.947			0.254110
$414$		0			0
$415$	−15.0220	+	132.908i	−0.0361976	+	0.320260i
$416$		0			0
$417$		−	418.030i		−	1.00247i
$418$		0			0
$419$	435.905			1.04035			0.520173	−	0.854061i	$-0.325867\pi$
$419$	435.905			1.04035			0.520173	+	0.854061i	$0.325867\pi$
$420$		0			0
$421$		−	429.616i		−	1.02046i	−0.860037	−	0.510232i	$-0.829559\pi$
$421$		−	429.616i		−	1.02046i	0.860037	−	0.510232i	$-0.170441\pi$
$422$		0			0
$423$	−89.2147			−0.210909
$424$		0			0
$425$	−142.030	+	620.283i	−0.334189	+	1.45949i
$426$		0			0
$427$			498.915i			1.16842i
$428$		0			0
$429$			221.788i			0.516988i
$430$		0			0
$431$		−	690.856i		−	1.60291i	−0.598053	−	0.801457i	$-0.704059\pi$
$431$		−	690.856i		−	1.60291i	0.598053	−	0.801457i	$-0.295941\pi$
$432$		0			0
$433$		−	70.0935i		−	0.161879i	−0.996719	−	0.0809393i	$-0.974208\pi$
$433$		−	70.0935i		−	0.161879i	0.996719	−	0.0809393i	$-0.0257920\pi$
$434$		0			0
$435$	−625.237	−	70.6678i	−1.43733	−	0.162455i
$436$		0			0
$437$	−489.883			−1.12101
$438$		0			0
$439$			816.571i			1.86007i	0.367469	+	0.930036i	$0.380224\pi$
$439$			816.571i			1.86007i	−0.367469	+	0.930036i	$0.619776\pi$
$440$		0			0
$441$	30.4470			0.0690407
$442$		0			0
$443$			199.242i			0.449756i	0.974387	+	0.224878i	$0.0721984\pi$
$443$			199.242i			0.449756i	−0.974387	+	0.224878i	$0.927802\pi$
$444$		0			0
$445$	137.891	+	15.5852i	0.309867	+	0.0350230i
$446$		0			0
$447$	190.659			0.426530
$448$		0			0
$449$	−3.00187			−0.00668567			−0.00334283	−	0.999994i	$-0.501064\pi$
$449$	−3.00187			−0.00668567			−0.00334283	+	0.999994i	$0.501064\pi$
$450$		0			0
$451$	−167.115			−0.370543
$452$		0			0
$453$	839.579			1.85337
$454$		0			0
$455$	−497.326	−	56.2106i	−1.09303	−	0.123540i
$456$		0			0
$457$		−	345.188i		−	0.755336i	−0.925941	−	0.377668i	$-0.876726\pi$
$457$		−	345.188i		−	0.755336i	0.925941	−	0.377668i	$-0.123274\pi$
$458$		0			0
$459$	−449.685			−0.979706
$460$		0			0
$461$			799.663i			1.73463i	0.497763	+	0.867313i	$0.334155\pi$
$461$			799.663i			1.73463i	−0.497763	+	0.867313i	$0.665845\pi$
$462$		0			0
$463$	245.737			0.530749			0.265375	−	0.964145i	$-0.414504\pi$
$463$	245.737			0.530749			0.265375	+	0.964145i	$0.414504\pi$
$464$		0			0
$465$	−363.818	−	41.1208i	−0.782405	−	0.0884318i
$466$		0			0
$467$		−	331.439i		−	0.709719i	−0.934920	−	0.354860i	$-0.884529\pi$
$467$		−	331.439i		−	0.709719i	0.934920	−	0.354860i	$-0.115471\pi$
$468$		0			0
$469$		−	90.6004i		−	0.193178i
$470$		0			0
$471$			589.240i			1.25104i
$472$		0			0
$473$		−	208.907i		−	0.441663i
$474$		0			0
$475$	−781.060	−	178.844i	−1.64434	−	0.376515i
$476$		0			0
$477$	145.918			0.305907
$478$		0			0
$479$			310.676i			0.648592i	0.945956	+	0.324296i	$0.105127\pi$
$479$			310.676i			0.648592i	−0.945956	+	0.324296i	$0.894873\pi$
$480$		0			0
$481$	154.186			0.320552
$482$		0			0
$483$		−	357.188i		−	0.739520i
$484$		0			0
$485$	−74.9865	+	663.447i	−0.154611	+	1.36793i
$486$		0			0
$487$	173.615			0.356498			0.178249	−	0.983985i	$-0.442957\pi$
$487$	173.615			0.356498			0.178249	+	0.983985i	$0.442957\pi$
$488$		0			0
$489$	1062.57			2.17295
$490$		0			0
$491$	−470.930			−0.959125			−0.479563	−	0.877508i	$-0.659205\pi$
$491$	−470.930			−0.959125			−0.479563	+	0.877508i	$0.659205\pi$
$492$		0			0
$493$	−884.219			−1.79355
$494$		0			0
$495$	−80.8271	−	9.13554i	−0.163287	−	0.0184556i
$496$		0			0
$497$		−	696.644i		−	1.40170i
$498$		0			0
$499$	114.424			0.229307			0.114653	−	0.993406i	$-0.463424\pi$
$499$	114.424			0.229307			0.114653	+	0.993406i	$0.463424\pi$
$500$		0			0
$501$		−	894.540i		−	1.78551i
$502$		0			0
$503$	279.067			0.554806			0.277403	−	0.960754i	$-0.410526\pi$
$503$	279.067			0.554806			0.277403	+	0.960754i	$0.410526\pi$
$504$		0			0
$505$	83.5076	−	738.837i	0.165362	−	1.46304i
$506$		0			0
$507$			259.989i			0.512799i
$508$		0			0
$509$		−	332.155i		−	0.652564i	−0.945272	−	0.326282i	$-0.894204\pi$
$509$		−	332.155i		−	0.652564i	0.945272	−	0.326282i	$-0.105796\pi$
$510$		0			0
$511$			605.023i			1.18400i
$512$		0			0
$513$		−	566.243i		−	1.10379i
$514$		0			0
$515$	725.821	+	82.0364i	1.40936	+	0.159294i
$516$		0			0
$517$	85.3750			0.165135
$518$		0			0
$519$			278.565i			0.536734i
$520$		0			0
$521$	−746.924			−1.43364			−0.716818	−	0.697260i	$-0.754402\pi$
$521$	−746.924			−1.43364			−0.716818	+	0.697260i	$0.754402\pi$
$522$		0			0
$523$		−	584.018i		−	1.11667i	−0.829616	−	0.558335i	$-0.811441\pi$
$523$		−	584.018i		−	1.11667i	0.829616	−	0.558335i	$-0.188559\pi$
$524$		0			0
$525$	130.401	−	569.494i	0.248382	−	1.08475i
$526$		0			0
$527$	−514.517			−0.976312
$528$		0			0
$529$	−295.384			−0.558383
$530$		0			0
$531$	67.0761			0.126320
$532$		0			0
$533$	−657.198			−1.23302
$534$		0			0
$535$	−19.8110	+	175.279i	−0.0370300	+	0.327624i
$536$		0			0
$537$			1032.43i			1.92259i
$538$		0			0
$539$	−29.1366			−0.0540567
$540$		0			0
$541$			271.905i			0.502598i	0.967910	+	0.251299i	$0.0808577\pi$
$541$			271.905i			0.502598i	−0.967910	+	0.251299i	$0.919142\pi$
$542$		0			0
$543$	640.360			1.17930
$544$		0			0
$545$	13.5549	−	119.928i	0.0248714	−	0.220051i
$546$		0			0
$547$			19.1146i			0.0349445i	0.999847	+	0.0174722i	$0.00556187\pi$
$547$			19.1146i			0.0349445i	−0.999847	+	0.0174722i	$0.994438\pi$
$548$		0			0
$549$			318.877i			0.580832i
$550$		0			0
$551$		−	1113.41i		−	2.02070i
$552$		0			0
$553$		−	594.830i		−	1.07564i
$554$		0			0
$555$	−20.2140	+	178.844i	−0.0364216	+	0.322242i
$556$		0			0
$557$	−868.015			−1.55837			−0.779187	−	0.626791i	$-0.784368\pi$
$557$	−868.015			−1.55837			−0.779187	+	0.626791i	$0.784368\pi$
$558$		0			0
$559$		−	821.548i		−	1.46967i
$560$		0			0
$561$	−363.818			−0.648517
$562$		0			0
$563$			912.726i			1.62118i	0.585612	+	0.810592i	$0.300854\pi$
$563$			912.726i			1.62118i	−0.585612	+	0.810592i	$0.699146\pi$
$564$		0			0
$565$	−101.814	+	900.803i	−0.180202	+	1.59434i
$566$		0			0
$567$	652.248			1.15035
$568$		0			0
$569$	115.434			0.202871			0.101436	−	0.994842i	$-0.467656\pi$
$569$	115.434			0.202871			0.101436	+	0.994842i	$0.467656\pi$
$570$		0			0
$571$	−347.158			−0.607982			−0.303991	−	0.952675i	$-0.598319\pi$
$571$	−347.158			−0.607982			−0.303991	+	0.952675i	$0.598319\pi$
$572$		0			0
$573$	960.965			1.67708
$574$		0			0
$575$	372.472	+	85.2874i	0.647778	+	0.148326i
$576$		0			0
$577$		−	70.4792i		−	0.122148i	−0.998133	−	0.0610738i	$-0.980547\pi$
$577$		−	70.4792i		−	0.122148i	0.998133	−	0.0610738i	$-0.0194525\pi$
$578$		0			0
$579$	−299.324			−0.516968
$580$		0			0
$581$		−	172.570i		−	0.297022i
$582$		0			0
$583$	−139.638			−0.239515
$584$		0			0
$585$	−317.862	−	35.9265i	−0.543353	−	0.0614129i
$586$		0			0
$587$		−	461.461i		−	0.786134i	−0.919510	−	0.393067i	$-0.871414\pi$
$587$		−	461.461i		−	0.786134i	0.919510	−	0.393067i	$-0.128586\pi$
$588$		0			0
$589$		−	647.879i		−	1.09996i
$590$		0			0
$591$			654.859i			1.10805i
$592$		0			0
$593$		−	222.428i		−	0.375090i	−0.982256	−	0.187545i	$-0.939947\pi$
$593$		−	222.428i		−	0.375090i	0.982256	−	0.187545i	$-0.0600531\pi$
$594$		0			0
$595$	92.2073	−	815.808i	0.154970	−	1.37111i
$596$		0			0
$597$	−619.898			−1.03835
$598$		0			0
$599$		−	206.571i		−	0.344861i	−0.985022	−	0.172430i	$-0.944838\pi$
$599$		−	206.571i		−	0.344861i	0.985022	−	0.172430i	$-0.0551620\pi$
$600$		0			0
$601$	−102.941			−0.171283			−0.0856416	−	0.996326i	$-0.527294\pi$
$601$	−102.941			−0.171283			−0.0856416	+	0.996326i	$0.527294\pi$
$602$		0			0
$603$		−	57.9064i		−	0.0960306i
$604$		0			0
$605$	−523.824	−	59.2055i	−0.865824	−	0.0978604i
$606$		0			0
$607$	−857.703			−1.41302			−0.706510	−	0.707703i	$-0.749731\pi$
$607$	−857.703			−1.41302			−0.706510	+	0.707703i	$0.749731\pi$
$608$		0			0
$609$	811.818			1.33303
$610$		0			0
$611$	335.747			0.549504
$612$		0			0
$613$	−299.711			−0.488925			−0.244462	−	0.969659i	$-0.578611\pi$
$613$	−299.711			−0.488925			−0.244462	+	0.969659i	$0.578611\pi$
$614$		0			0
$615$	86.1599	−	762.304i	0.140097	−	1.23952i
$616$		0			0
$617$			189.334i			0.306863i	0.988159	+	0.153431i	$0.0490324\pi$
$617$			189.334i			0.306863i	−0.988159	+	0.153431i	$0.950968\pi$
$618$		0			0
$619$	269.762			0.435803			0.217901	−	0.975971i	$-0.430079\pi$
$619$	269.762			0.435803			0.217901	+	0.975971i	$0.430079\pi$
$620$		0			0
$621$			270.030i			0.434831i
$622$		0			0
$623$	−179.040			−0.287384
$624$		0			0
$625$	562.727	+	271.962i	0.900364	+	0.435138i
$626$		0			0
$627$		−	458.119i		−	0.730653i
$628$		0			0
$629$			252.924i			0.402105i
$630$		0			0
$631$			569.026i			0.901785i	0.892578	+	0.450892i	$0.148894\pi$
$631$			569.026i			0.901785i	−0.892578	+	0.450892i	$0.851106\pi$
$632$		0			0
$633$			348.322i			0.550272i
$634$		0			0
$635$	951.209	+	107.511i	1.49797	+	0.169309i
$636$		0			0
$637$	−114.583			−0.179879
$638$		0			0
$639$		−	445.254i		−	0.696798i
$640$		0			0
$641$	−173.093			−0.270036			−0.135018	−	0.990843i	$-0.543109\pi$
$641$	−173.093			−0.270036			−0.135018	+	0.990843i	$0.543109\pi$
$642$		0			0
$643$		−	551.298i		−	0.857384i	−0.903451	−	0.428692i	$-0.858974\pi$
$643$		−	551.298i		−	0.857384i	0.903451	−	0.428692i	$-0.141026\pi$
$644$		0			0
$645$	952.938	+	107.706i	1.47742	+	0.166987i
$646$		0			0
$647$	518.940			0.802071			0.401036	−	0.916062i	$-0.368650\pi$
$647$	518.940			0.802071			0.401036	+	0.916062i	$0.368650\pi$
$648$		0			0
$649$	−64.1893			−0.0989050
$650$		0			0
$651$	472.388			0.725634
$652$		0			0
$653$	−150.509			−0.230489			−0.115245	−	0.993337i	$-0.536765\pi$
$653$	−150.509			−0.230489			−0.115245	+	0.993337i	$0.536765\pi$
$654$		0			0
$655$	1114.69	+	125.988i	1.70181	+	0.192348i
$656$		0			0
$657$			386.695i			0.588577i
$658$		0			0
$659$	397.420			0.603065			0.301533	−	0.953456i	$-0.402502\pi$
$659$	397.420			0.603065			0.301533	+	0.953456i	$0.402502\pi$
$660$		0			0
$661$		−	673.953i		−	1.01960i	−0.860294	−	0.509798i	$-0.829720\pi$
$661$		−	673.953i		−	1.01960i	0.860294	−	0.509798i	$-0.170280\pi$
$662$		0			0
$663$	−1430.76			−2.15800
$664$		0			0
$665$	1027.27	+	116.107i	1.54476	+	0.174597i
$666$		0			0
$667$			530.962i			0.796045i
$668$		0			0
$669$			289.373i			0.432546i
$670$		0			0
$671$		−	305.153i		−	0.454774i
$672$		0			0
$673$			1132.10i			1.68217i	0.540903	+	0.841085i	$0.318083\pi$
$673$			1132.10i			1.68217i	−0.540903	+	0.841085i	$0.681917\pi$
$674$		0			0
$675$	−98.5816	+	430.532i	−0.146047	+	0.637825i
$676$		0			0
$677$	1088.08			1.60721			0.803605	−	0.595163i	$-0.202912\pi$
$677$	1088.08			1.60721			0.803605	+	0.595163i	$0.202912\pi$
$678$		0			0
$679$		−	861.431i		−	1.26868i
$680$		0			0
$681$	−832.820			−1.22294
$682$		0			0
$683$		−	24.4045i		−	0.0357313i	−0.999840	−	0.0178657i	$-0.994313\pi$
$683$		−	24.4045i		−	0.0357313i	0.999840	−	0.0178657i	$-0.00568712\pi$
$684$		0			0
$685$	−31.3346	+	277.235i	−0.0457440	+	0.404722i
$686$		0			0
$687$	−479.183			−0.697500
$688$		0			0
$689$	−549.140			−0.797010
$690$		0			0
$691$	−349.646			−0.506001			−0.253000	−	0.967466i	$-0.581417\pi$
$691$	−349.646			−0.506001			−0.253000	+	0.967466i	$0.581417\pi$
$692$		0			0
$693$	104.947			0.151439
$694$		0			0
$695$	−573.327	−	64.8007i	−0.824931	−	0.0932384i
$696$		0			0
$697$		−	1078.06i		−	1.54671i
$698$		0			0
$699$	294.347			0.421098
$700$		0			0
$701$		−	819.714i		−	1.16935i	−0.811268	−	0.584675i	$-0.801222\pi$
$701$		−	819.714i		−	1.16935i	0.811268	−	0.584675i	$-0.198778\pi$
$702$		0			0
$703$	−318.482			−0.453033
$704$		0			0
$705$	−44.0170	+	389.443i	−0.0624355	+	0.552401i
$706$		0			0
$707$			959.319i			1.35689i
$708$		0			0
$709$			719.231i			1.01443i	0.861819	+	0.507215i	$0.169325\pi$
$709$			719.231i			1.01443i	−0.861819	+	0.507215i	$0.830675\pi$
$710$		0			0
$711$		−	380.180i		−	0.534712i
$712$		0			0
$713$			308.961i			0.433325i
$714$		0			0
$715$	304.182	+	34.3803i	0.425429	+	0.0480844i
$716$		0			0
$717$	1148.54			1.60187
$718$		0			0
$719$		−	600.046i		−	0.834556i	−0.908779	−	0.417278i	$-0.862984\pi$
$719$		−	600.046i		−	0.834556i	0.908779	−	0.417278i	$-0.137016\pi$
$720$		0			0
$721$	−942.419			−1.30710
$722$		0			0
$723$			1618.30i			2.23831i
$724$		0			0
$725$	−193.842	+	846.557i	−0.267368	+	1.16766i
$726$		0			0
$727$	452.279			0.622117			0.311059	−	0.950391i	$-0.399316\pi$
$727$	452.279			0.622117			0.311059	+	0.950391i	$0.399316\pi$
$728$		0			0
$729$	−159.121			−0.218273
$730$		0			0
$731$	1347.66			1.84358
$732$		0			0
$733$	773.541			1.05531			0.527654	−	0.849459i	$-0.323072\pi$
$733$	773.541			1.05531			0.527654	+	0.849459i	$0.323072\pi$
$734$		0			0
$735$	15.0220	−	132.908i	0.0204381	−	0.180827i
$736$		0			0
$737$			55.4142i			0.0751889i
$738$		0			0
$739$	−34.5394			−0.0467381			−0.0233690	−	0.999727i	$-0.507439\pi$
$739$	−34.5394			−0.0467381			−0.0233690	+	0.999727i	$0.507439\pi$
$740$		0			0
$741$		−	1801.61i		−	2.43132i
$742$		0			0
$743$	−396.078			−0.533079			−0.266539	−	0.963824i	$-0.585880\pi$
$743$	−396.078			−0.533079			−0.266539	+	0.963824i	$0.585880\pi$
$744$		0			0
$745$	29.5549	−	261.488i	0.0396710	−	0.350991i
$746$		0			0
$747$		−	110.297i		−	0.147653i
$748$		0			0
$749$		−	227.585i		−	0.303852i
$750$		0			0
$751$		−	850.079i		−	1.13193i	−0.824429	−	0.565965i	$-0.808504\pi$
$751$		−	850.079i		−	1.13193i	0.824429	−	0.565965i	$-0.191496\pi$
$752$		0			0
$753$			918.084i			1.21924i
$754$		0			0
$755$	130.147	−	1151.48i	0.172380	−	1.52514i
$756$		0			0
$757$	458.313			0.605433			0.302717	−	0.953081i	$-0.402106\pi$
$757$	458.313			0.605433			0.302717	+	0.953081i	$0.402106\pi$
$758$		0			0
$759$			218.468i			0.287837i
$760$		0			0
$761$	788.678			1.03637			0.518185	−	0.855268i	$-0.326608\pi$
$761$	788.678			1.03637			0.518185	+	0.855268i	$0.326608\pi$
$762$		0			0
$763$			155.716i			0.204084i
$764$		0			0
$765$	58.9335	−	521.417i	0.0770372	−	0.681590i
$766$		0			0
$767$	−252.432			−0.329116
$768$		0			0
$769$	980.098			1.27451			0.637255	−	0.770653i	$-0.280070\pi$
$769$	980.098			1.27451			0.637255	+	0.770653i	$0.280070\pi$
$770$		0			0
$771$	310.676			0.402951
$772$		0			0
$773$	193.305			0.250071			0.125036	−	0.992152i	$-0.460095\pi$
$773$	193.305			0.250071			0.125036	+	0.992152i	$0.460095\pi$
$774$		0			0
$775$	−112.794	+	492.602i	−0.145541	+	0.635615i
$776$		0			0
$777$		−	232.215i		−	0.298861i
$778$		0			0
$779$	1357.49			1.74261
$780$		0			0
$781$			426.091i			0.545571i
$782$		0			0
$783$	−613.726			−0.783813
$784$		0			0
$785$	808.142	+	91.3408i	1.02948	+	0.116358i
$786$		0			0
$787$		−	679.927i		−	0.863948i	−0.901886	−	0.431974i	$-0.857817\pi$
$787$		−	679.927i		−	0.863948i	0.901886	−	0.431974i	$-0.142183\pi$
$788$		0			0
$789$			1096.15i			1.38928i
$790$		0			0
$791$		−	1169.62i		−	1.47866i
$792$		0			0
$793$		−	1200.05i		−	1.51330i
$794$		0			0
$795$	71.9933	−	636.964i	0.0905576	−	0.801212i
$796$		0			0
$797$	324.477			0.407124			0.203562	−	0.979062i	$-0.434748\pi$
$797$	324.477			0.407124			0.203562	+	0.979062i	$0.434748\pi$
$798$		0			0
$799$			550.755i			0.689306i
$800$		0			0
$801$	−114.432			−0.142861
$802$		0			0
$803$		−	370.052i		−	0.460837i
$804$		0			0
$805$	−489.883	−	55.3693i	−0.608550	−	0.0687818i
$806$		0			0
$807$	627.935			0.778110
$808$		0			0
$809$	−392.712			−0.485429			−0.242714	−	0.970098i	$-0.578038\pi$
$809$	−392.712			−0.485429			−0.242714	+	0.970098i	$0.578038\pi$
$810$		0			0
$811$	606.480			0.747818			0.373909	−	0.927465i	$-0.378017\pi$
$811$	606.480			0.747818			0.373909	+	0.927465i	$0.378017\pi$
$812$		0			0
$813$	−709.178			−0.872297
$814$		0			0
$815$	164.714	−	1457.32i	0.202103	−	1.78812i
$816$		0			0
$817$			1696.97i			2.07707i
$818$		0			0
$819$	412.717			0.503928
$820$		0			0
$821$		−	750.634i		−	0.914293i	−0.889391	−	0.457146i	$-0.848872\pi$
$821$		−	750.634i		−	0.914293i	0.889391	−	0.457146i	$-0.151128\pi$
$822$		0			0
$823$	557.389			0.677265			0.338632	−	0.940919i	$-0.390036\pi$
$823$	557.389			0.677265			0.338632	+	0.940919i	$0.390036\pi$
$824$		0			0
$825$	−79.7575	+	348.322i	−0.0966758	+	0.422208i
$826$		0			0
$827$		−	1394.76i		−	1.68652i	−0.537502	−	0.843262i	$-0.680632\pi$
$827$		−	1394.76i		−	1.68652i	0.537502	−	0.843262i	$-0.319368\pi$
$828$		0			0
$829$			392.570i			0.473547i	0.971565	+	0.236773i	$0.0760899\pi$
$829$			392.570i			0.473547i	−0.971565	+	0.236773i	$0.923910\pi$
$830$		0			0
$831$		−	1323.01i		−	1.59207i
$832$		0			0
$833$		−	187.960i		−	0.225643i
$834$		0			0
$835$	−1226.86	−	138.667i	−1.46929	−	0.166068i
$836$		0			0
$837$	−357.120			−0.426667
$838$		0			0
$839$		−	10.8057i		−	0.0128793i	−0.999979	−	0.00643963i	$-0.997950\pi$
$839$		−	10.8057i		−	0.0128793i	0.999979	−	0.00643963i	$-0.00204981\pi$
$840$		0			0
$841$	−365.773			−0.434926
$842$		0			0
$843$			973.590i			1.15491i
$844$		0			0
$845$	356.574	+	40.3021i	0.421981	+	0.0476947i
$846$		0			0
$847$	680.142			0.803001
$848$		0			0
$849$	−158.074			−0.186188
$850$		0			0
$851$	151.878			0.178470
$852$		0			0
$853$	−222.965			−0.261389			−0.130695	−	0.991423i	$-0.541721\pi$
$853$	−222.965			−0.261389			−0.130695	+	0.991423i	$0.541721\pi$
$854$		0			0
$855$	656.567	+	74.2090i	0.767915	+	0.0867941i
$856$		0			0
$857$		−	1258.77i		−	1.46880i	−0.678714	−	0.734402i	$-0.737463\pi$
$857$		−	1258.77i		−	1.46880i	0.678714	−	0.734402i	$-0.262537\pi$
$858$		0			0
$859$	667.122			0.776626			0.388313	−	0.921527i	$-0.373058\pi$
$859$	667.122			0.776626			0.388313	+	0.921527i	$0.373058\pi$
$860$		0			0
$861$			989.788i			1.14958i
$862$		0			0
$863$	−235.010			−0.272317			−0.136159	−	0.990687i	$-0.543476\pi$
$863$	−235.010			−0.272317			−0.136159	+	0.990687i	$0.543476\pi$
$864$		0			0
$865$	382.051	+	43.1816i	0.441678	+	0.0499209i
$866$		0			0
$867$		−	1300.07i		−	1.49950i
$868$		0			0
$869$			363.818i			0.418663i
$870$		0			0
$871$			217.923i			0.250198i
$872$		0			0
$873$		−	550.576i		−	0.630671i
$874$		0			0
$875$	−760.846	−	267.124i	−0.869539	−	0.305285i
$876$		0			0
$877$	−1577.19			−1.79840			−0.899198	−	0.437543i	$-0.855849\pi$
$877$	−1577.19			−1.79840			−0.899198	+	0.437543i	$0.855849\pi$
$878$		0			0
$879$			819.060i			0.931809i
$880$		0			0
$881$	−220.324			−0.250084			−0.125042	−	0.992151i	$-0.539907\pi$
$881$	−220.324			−0.250084			−0.125042	+	0.992151i	$0.539907\pi$
$882$		0			0
$883$			1144.57i			1.29623i	0.761542	+	0.648115i	$0.224442\pi$
$883$			1144.57i			1.29623i	−0.761542	+	0.648115i	$0.775558\pi$
$884$		0			0
$885$	33.0943	−	292.803i	0.0373946	−	0.330851i
$886$		0			0
$887$	773.413			0.871942			0.435971	−	0.899961i	$-0.356405\pi$
$887$	773.413			0.871942			0.435971	+	0.899961i	$0.356405\pi$
$888$		0			0
$889$	−1235.07			−1.38928
$890$		0			0
$891$	−398.937			−0.447741
$892$		0			0
$893$	−693.510			−0.776607
$894$		0			0
$895$	1415.98	+	160.042i	1.58210	+	0.178818i
$896$		0			0
$897$			859.151i			0.957805i
$898$		0			0
$899$	−702.207			−0.781098
$900$		0			0
$901$		−	900.803i		−	0.999781i
$902$		0			0
$903$	−1237.31			−1.37022
$904$		0			0
$905$	99.2651	−	878.253i	0.109685	−	0.970445i
$906$		0			0
$907$			271.474i			0.299310i	0.988738	+	0.149655i	$0.0478163\pi$
$907$			271.474i			0.299310i	−0.988738	+	0.149655i	$0.952184\pi$
$908$		0			0
$909$			613.140i			0.674522i
$910$		0			0
$911$			663.176i			0.727965i	0.931406	+	0.363983i	$0.118583\pi$
$911$			663.176i			0.727965i	−0.931406	+	0.363983i	$0.881417\pi$
$912$		0			0
$913$			105.550i			0.115608i
$914$		0			0
$915$	1391.97	+	157.329i	1.52128	+	0.171944i
$916$		0			0
$917$	−1447.33			−1.57833
$918$		0			0
$919$		−	1017.31i		−	1.10698i	−0.832856	−	0.553490i	$-0.813296\pi$
$919$		−	1017.31i		−	1.10698i	0.832856	−	0.553490i	$-0.186704\pi$
$920$		0			0
$921$	−624.665			−0.678246
$922$		0			0
$923$			1675.65i			1.81544i
$924$		0			0
$925$	242.151	+	55.4470i	0.261785	+	0.0599427i
$926$		0			0
$927$	−602.338			−0.649772
$928$		0			0
$929$	−800.790			−0.861991			−0.430996	−	0.902354i	$-0.641838\pi$
$929$	−800.790			−0.861991			−0.430996	+	0.902354i	$0.641838\pi$
$930$		0			0
$931$	236.680			0.254221
$932$		0			0
$933$	−285.869			−0.306398
$934$		0			0
$935$	−56.3971	+	498.976i	−0.0603177	+	0.533664i
$936$		0			0
$937$		−	795.927i		−	0.849441i	−0.905324	−	0.424721i	$-0.860372\pi$
$937$		−	795.927i		−	0.849441i	0.905324	−	0.424721i	$-0.139628\pi$
$938$		0			0
$939$	−1227.77			−1.30753
$940$		0			0
$941$			159.663i			0.169674i	0.996395	+	0.0848368i	$0.0270369\pi$
$941$			159.663i			0.169674i	−0.996395	+	0.0848368i	$0.972963\pi$
$942$		0			0
$943$	−647.362			−0.686492
$944$		0			0
$945$	64.0000	−	566.243i	0.0677249	−	0.599199i
$946$		0			0
$947$		−	138.776i		−	0.146543i	−0.997312	−	0.0732716i	$-0.976656\pi$
$947$		−	138.776i		−	0.146543i	0.997312	−	0.0732716i	$-0.0233440\pi$
$948$		0			0
$949$		−	1455.27i		−	1.53348i
$950$		0			0
$951$			366.886i			0.385790i
$952$		0			0
$953$			141.561i			0.148542i	0.997238	+	0.0742711i	$0.0236630\pi$
$953$			141.561i			0.148542i	−0.997238	+	0.0742711i	$0.976337\pi$
$954$		0			0
$955$	148.963	−	1317.96i	0.155983	−	1.38006i
$956$		0			0
$957$	−496.535			−0.518846
$958$		0			0
$959$		−	359.966i		−	0.375356i
$960$		0			0
$961$	552.394			0.574811
$962$		0			0
$963$		−	145.459i		−	0.151048i
$964$		0			0
$965$	−46.3996	+	410.523i	−0.0480825	+	0.425412i
$966$		0			0
$967$	−278.872			−0.288389			−0.144194	−	0.989549i	$-0.546059\pi$
$967$	−278.872			−0.288389			−0.144194	+	0.989549i	$0.546059\pi$
$968$		0			0
$969$	2955.33			3.04988
$970$		0			0
$971$	−88.2616			−0.0908977			−0.0454488	−	0.998967i	$-0.514472\pi$
$971$	−88.2616			−0.0908977			−0.0454488	+	0.998967i	$0.514472\pi$
$972$		0			0
$973$	744.417			0.765074
$974$		0			0
$975$	−313.655	+	1369.81i	−0.321698	+	1.40494i
$976$		0			0
$977$		−	521.989i		−	0.534277i	−0.963658	−	0.267139i	$-0.913922\pi$
$977$		−	521.989i		−	0.534277i	0.963658	−	0.267139i	$-0.0860782\pi$
$978$		0			0
$979$	109.507			0.111856
$980$		0			0
$981$			99.5246i			0.101452i
$982$		0			0
$983$	−1819.43			−1.85090			−0.925448	−	0.378876i	$-0.876311\pi$
$983$	−1819.43			−1.85090			−0.925448	+	0.378876i	$0.876311\pi$
$984$		0			0
$985$	898.138	+	101.513i	0.911815	+	0.103059i
$986$		0			0
$987$		−	505.659i		−	0.512319i
$988$		0			0
$989$		−	809.252i		−	0.818253i
$990$		0			0
$991$			593.672i			0.599063i	0.954086	+	0.299532i	$0.0968304\pi$
$991$			593.672i			0.599063i	−0.954086	+	0.299532i	$0.903170\pi$
$992$		0			0
$993$		−	555.469i		−	0.559385i
$994$		0			0
$995$	−96.0931	+	850.189i	−0.0965760	+	0.854461i
$996$		0			0
$997$	−639.235			−0.641158			−0.320579	−	0.947222i	$-0.603878\pi$
$997$	−639.235			−0.641158			−0.320579	+	0.947222i	$0.603878\pi$
$998$		0			0
$999$			175.552i			0.175727i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.3.e.i.319.16	yes	16
4.3	odd	2	inner	640.3.e.i.319.4	yes	16
5.4	even	2	inner	640.3.e.i.319.2	yes	16
8.3	odd	2	inner	640.3.e.i.319.13	yes	16
8.5	even	2	inner	640.3.e.i.319.1	✓	16
16.3	odd	4		1280.3.h.h.1279.7		8
16.5	even	4		1280.3.h.l.1279.8		8
16.11	odd	4		1280.3.h.l.1279.2		8
16.13	even	4		1280.3.h.h.1279.1		8
20.19	odd	2	inner	640.3.e.i.319.14	yes	16
40.19	odd	2	inner	640.3.e.i.319.3	yes	16
40.29	even	2	inner	640.3.e.i.319.15	yes	16
80.19	odd	4		1280.3.h.h.1279.2		8
80.29	even	4		1280.3.h.h.1279.8		8
80.59	odd	4		1280.3.h.l.1279.7		8
80.69	even	4		1280.3.h.l.1279.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.e.i.319.1	✓	16	8.5	even	2	inner
640.3.e.i.319.2	yes	16	5.4	even	2	inner
640.3.e.i.319.3	yes	16	40.19	odd	2	inner
640.3.e.i.319.4	yes	16	4.3	odd	2	inner
640.3.e.i.319.13	yes	16	8.3	odd	2	inner
640.3.e.i.319.14	yes	16	20.19	odd	2	inner
640.3.e.i.319.15	yes	16	40.29	even	2	inner
640.3.e.i.319.16	yes	16	1.1	even	1	trivial
1280.3.h.h.1279.1		8	16.13	even	4
1280.3.h.h.1279.2		8	80.19	odd	4
1280.3.h.h.1279.7		8	16.3	odd	4
1280.3.h.h.1279.8		8	80.29	even	4
1280.3.h.l.1279.1		8	80.69	even	4
1280.3.h.l.1279.2		8	16.11	odd	4
1280.3.h.l.1279.7		8	80.59	odd	4
1280.3.h.l.1279.8		8	16.5	even	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)

\(f(q)\)	\(=\)	\(q+3.62258i q^{3} +(4.96837 + 0.561553i) q^{5} -6.45101 q^{7} -4.12311 q^{9} +O(q^{10})\)	\(q+3.62258i q^{3} +(4.96837 + 0.561553i) q^{5} -6.45101 q^{7} -4.12311 q^{9} +3.94566 q^{11} +15.5167 q^{13} +(-2.03427 + 17.9983i) q^{15} +25.4535i q^{17} -32.0510 q^{19} -23.3693i q^{21} +15.2845 q^{23} +(24.3693 + 5.58000i) q^{25} +17.6670i q^{27} +34.7386i q^{29} +20.2140i q^{31} +14.2935i q^{33} +(-32.0510 - 3.62258i) q^{35} +9.93673 q^{37} +56.2106i q^{39} -42.3542 q^{41} -52.9460i q^{43} +(-20.4851 - 2.31534i) q^{45} +21.6377 q^{47} -7.38447 q^{49} -92.2073 q^{51} -35.3902 q^{53} +(19.6035 + 2.21569i) q^{55} -116.107i q^{57} -16.2684 q^{59} -77.3390i q^{61} +26.5982 q^{63} +(77.0928 + 8.71346i) q^{65} +14.0444i q^{67} +55.3693i q^{69} +107.990i q^{71} -93.7873i q^{73} +(-20.2140 + 88.2799i) q^{75} -25.4535 q^{77} +92.2073i q^{79} -101.108 q^{81} +26.7509i q^{83} +(-14.2935 + 126.462i) q^{85} -125.844 q^{87} +27.7538 q^{89} -100.099 q^{91} -73.2269 q^{93} +(-159.241 - 17.9983i) q^{95} +133.534i q^{97} -16.2684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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