LMFDB - Embedded newform 640.4.c.a.129.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,4,Mod(129,640)]

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([0, 0, 1]))

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

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

chi := DirichletCharacter("640.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	640.c (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:	$37.7612224037$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} + 18 x^{16} + 54 x^{15} + 2104 x^{14} - 10372 x^{13} + 25818 x^{12} + 35384 x^{11} + \cdots + 44745800 $ x^18 - 6x^17 + 18x^16 + 54x^15 + 2104x^14 - 10372x^13 + 25818x^12 + 35384x^11 + 1142136x^10 - 4885024x^9 + 10130552x^8 - 4037264x^7 + 156562660x^6 - 640608248x^5 + 1298643352x^4 - 1377944104x^3 + 862714384x^2 - 277859120*x + 44745800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{54}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.9
Root			$0.272660 - 0.272660i$ of defining polynomial
Character	$\chi$	$=$	640.129
Dual form			640.4.c.a.129.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.545319i q^{3} +(-10.0777 + 4.84141i) q^{5} -11.8531i q^{7} +26.7026 q^{9} -41.0997 q^{11} +55.5377i q^{13} +(2.64012 + 5.49558i) q^{15} -31.1506i q^{17} +39.7575 q^{19} -6.46373 q^{21} +62.7363i q^{23} +(78.1215 - 97.5809i) q^{25} -29.2851i q^{27} +1.87745 q^{29} -99.5774 q^{31} +22.4124i q^{33} +(57.3858 + 119.453i) q^{35} -101.358i q^{37} +30.2858 q^{39} -154.785 q^{41} -348.961i q^{43} +(-269.102 + 129.278i) q^{45} -604.903i q^{47} +202.504 q^{49} -16.9870 q^{51} +488.561i q^{53} +(414.191 - 198.980i) q^{55} -21.6805i q^{57} +696.407 q^{59} +161.860 q^{61} -316.509i q^{63} +(-268.881 - 559.694i) q^{65} -987.849i q^{67} +34.2113 q^{69} +768.530 q^{71} -474.591i q^{73} +(-53.2128 - 42.6011i) q^{75} +487.159i q^{77} +8.84939 q^{79} +705.001 q^{81} -597.884i q^{83} +(150.813 + 313.928i) q^{85} -1.02381i q^{87} +651.808 q^{89} +658.295 q^{91} +54.3015i q^{93} +(-400.665 + 192.482i) q^{95} -822.543i q^{97} -1097.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{5} - 162 q^{9} - 28 q^{11} + 4 q^{15} - 68 q^{19} - 136 q^{21} - 22 q^{25} - 340 q^{29} + 336 q^{31} - 236 q^{35} + 1000 q^{39} + 236 q^{41} + 90 q^{45} - 882 q^{49} - 48 q^{51} - 1532 q^{55}+ \cdots + 3628 q^{99}+O(q^{100}) $ 18 * q - 2 * q^5 - 162 * q^9 - 28 * q^11 + 4 * q^15 - 68 * q^19 - 136 * q^21 - 22 * q^25 - 340 * q^29 + 336 * q^31 - 236 * q^35 + 1000 * q^39 + 236 * q^41 + 90 * q^45 - 882 * q^49 - 48 * q^51 - 1532 * q^55 - 1756 * q^59 + 780 * q^61 + 104 * q^65 - 1016 * q^69 + 72 * q^71 + 3496 * q^75 - 1952 * q^79 + 2074 * q^81 + 664 * q^85 - 292 * q^89 - 520 * q^91 + 1612 * q^95 + 3628 * q^99

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
$3$		−	0.545319i		−	0.104947i	−0.998622	−	0.0524734i	$-0.983290\pi$
$3$		−	0.545319i		−	0.104947i	0.998622	−	0.0524734i	$-0.0167105\pi$
$4$		0			0
$5$	−10.0777	+	4.84141i	−0.901380	+	0.433029i
$5$	−10.0777	+	4.84141i	−0.901380	+	0.433029i
$6$		0			0
$7$		−	11.8531i		−	0.640008i	−0.947416	−	0.320004i	$-0.896316\pi$
$7$		−	11.8531i		−	0.640008i	0.947416	−	0.320004i	$-0.103684\pi$
$8$		0			0
$9$	26.7026			0.988986
$10$		0			0
$11$	−41.0997			−1.12655			−0.563273	−	0.826271i	$-0.690458\pi$
$11$	−41.0997			−1.12655			−0.563273	+	0.826271i	$0.690458\pi$
$12$		0			0
$13$			55.5377i			1.18488i	0.805616	+	0.592438i	$0.201835\pi$
$13$			55.5377i			1.18488i	−0.805616	+	0.592438i	$0.798165\pi$
$14$		0			0
$15$	2.64012	+	5.49558i	0.0454450	+	0.0945969i
$16$		0			0
$17$		−	31.1506i		−	0.444420i	−0.974999	−	0.222210i	$-0.928673\pi$
$17$		−	31.1506i		−	0.444420i	0.974999	−	0.222210i	$-0.0713271\pi$
$18$		0			0
$19$	39.7575			0.480052			0.240026	−	0.970766i	$-0.422844\pi$
$19$	39.7575			0.480052			0.240026	+	0.970766i	$0.422844\pi$
$20$		0			0
$21$	−6.46373			−0.0671668
$22$		0			0
$23$			62.7363i			0.568757i	0.958712	+	0.284379i	$0.0917873\pi$
$23$			62.7363i			0.568757i	−0.958712	+	0.284379i	$0.908213\pi$
$24$		0			0
$25$	78.1215	−	97.5809i	0.624972	−	0.780647i
$26$		0			0
$27$		−	29.2851i		−	0.208738i
$28$		0			0
$29$	1.87745			0.0120219			0.00601094	−	0.999982i	$-0.498087\pi$
$29$	1.87745			0.0120219			0.00601094	+	0.999982i	$0.498087\pi$
$30$		0			0
$31$	−99.5774			−0.576924			−0.288462	−	0.957491i	$-0.593144\pi$
$31$	−99.5774			−0.576924			−0.288462	+	0.957491i	$0.593144\pi$
$32$		0			0
$33$			22.4124i			0.118227i
$34$		0			0
$35$	57.3858	+	119.453i	0.277142	+	0.576890i
$36$		0			0
$37$		−	101.358i		−	0.450357i	−0.974318	−	0.225178i	$-0.927704\pi$
$37$		−	101.358i		−	0.450357i	0.974318	−	0.225178i	$-0.0722965\pi$
$38$		0			0
$39$	30.2858			0.124349
$40$		0			0
$41$	−154.785			−0.589594			−0.294797	−	0.955560i	$-0.595252\pi$
$41$	−154.785			−0.589594			−0.294797	+	0.955560i	$0.595252\pi$
$42$		0			0
$43$		−	348.961i		−	1.23758i	−0.785556	−	0.618791i	$-0.787623\pi$
$43$		−	348.961i		−	1.23758i	0.785556	−	0.618791i	$-0.212377\pi$
$44$		0			0
$45$	−269.102	+	129.278i	−0.891452	+	0.428260i
$46$		0			0
$47$		−	604.903i		−	1.87732i	−0.344842	−	0.938661i	$-0.612067\pi$
$47$		−	604.903i		−	1.87732i	0.344842	−	0.938661i	$-0.387933\pi$
$48$		0			0
$49$	202.504			0.590390
$50$		0			0
$51$	−16.9870			−0.0466404
$52$		0			0
$53$			488.561i			1.26621i	0.774067	+	0.633104i	$0.218219\pi$
$53$			488.561i			1.26621i	−0.774067	+	0.633104i	$0.781781\pi$
$54$		0			0
$55$	414.191	−	198.980i	1.01545	−	0.487827i
$56$		0			0
$57$		−	21.6805i		−	0.0503799i
$58$		0			0
$59$	696.407			1.53669			0.768343	−	0.640038i	$-0.221081\pi$
$59$	696.407			1.53669			0.768343	+	0.640038i	$0.221081\pi$
$60$		0			0
$61$	161.860			0.339739			0.169869	−	0.985467i	$-0.445665\pi$
$61$	161.860			0.339739			0.169869	+	0.985467i	$0.445665\pi$
$62$		0			0
$63$		−	316.509i		−	0.632959i
$64$		0			0
$65$	−268.881	−	559.694i	−0.513086	−	1.06802i
$66$		0			0
$67$		−	987.849i		−	1.80127i	−0.434578	−	0.900634i	$-0.643103\pi$
$67$		−	987.849i		−	1.80127i	0.434578	−	0.900634i	$-0.356897\pi$
$68$		0			0
$69$	34.2113			0.0596892
$70$		0			0
$71$	768.530			1.28462			0.642308	−	0.766447i	$-0.277977\pi$
$71$	768.530			1.28462			0.642308	+	0.766447i	$0.277977\pi$
$72$		0			0
$73$		−	474.591i		−	0.760914i	−0.924799	−	0.380457i	$-0.875767\pi$
$73$		−	474.591i		−	0.760914i	0.924799	−	0.380457i	$-0.124233\pi$
$74$		0			0
$75$	−53.2128	−	42.6011i	−0.0819264	−	0.0655888i
$76$		0			0
$77$			487.159i			0.720999i
$78$		0			0
$79$	8.84939			0.0126030			0.00630148	−	0.999980i	$-0.497994\pi$
$79$	8.84939			0.0126030			0.00630148	+	0.999980i	$0.497994\pi$
$80$		0			0
$81$	705.001			0.967080
$82$		0			0
$83$		−	597.884i		−	0.790678i	−0.918535	−	0.395339i	$-0.870627\pi$
$83$		−	597.884i		−	0.790678i	0.918535	−	0.395339i	$-0.129373\pi$
$84$		0			0
$85$	150.813	+	313.928i	0.192447	+	0.400591i
$86$		0			0
$87$		−	1.02381i		−	0.00126166i
$88$		0			0
$89$	651.808			0.776309			0.388155	−	0.921594i	$-0.373113\pi$
$89$	651.808			0.776309			0.388155	+	0.921594i	$0.373113\pi$
$90$		0			0
$91$	658.295			0.758330
$92$		0			0
$93$			54.3015i			0.0605463i
$94$		0			0
$95$	−400.665	+	192.482i	−0.432709	+	0.207876i
$96$		0			0
$97$		−	822.543i		−	0.860996i	−0.902592	−	0.430498i	$-0.858338\pi$
$97$		−	822.543i		−	0.860996i	0.902592	−	0.430498i	$-0.141662\pi$
$98$		0			0
$99$	−1097.47			−1.11414
$100$		0			0
$101$	−1559.82			−1.53671			−0.768357	−	0.640021i	$-0.778925\pi$
$101$	−1559.82			−1.53671			−0.768357	+	0.640021i	$0.778925\pi$
$102$		0			0
$103$		−	1251.12i		−	1.19686i	−0.801174	−	0.598431i	$-0.795791\pi$
$103$		−	1251.12i		−	1.19686i	0.801174	−	0.598431i	$-0.204209\pi$
$104$		0			0
$105$	65.1398	−	31.2936i	0.0605428	−	0.0290852i
$106$		0			0
$107$			354.176i			0.319995i	0.987117	+	0.159997i	$0.0511486\pi$
$107$			354.176i			0.319995i	−0.987117	+	0.159997i	$0.948851\pi$
$108$		0			0
$109$	−513.960			−0.451637			−0.225818	−	0.974169i	$-0.572506\pi$
$109$	−513.960			−0.451637			−0.225818	+	0.974169i	$0.572506\pi$
$110$		0			0
$111$	−55.2726			−0.0472635
$112$		0			0
$113$		−	530.511i		−	0.441648i	−0.975314	−	0.220824i	$-0.929125\pi$
$113$		−	530.511i		−	0.441648i	0.975314	−	0.220824i	$-0.0708747\pi$
$114$		0			0
$115$	−303.732	−	632.239i	−0.246288	−	0.512666i
$116$		0			0
$117$			1483.00i			1.17183i
$118$		0			0
$119$	−369.232			−0.284432
$120$		0			0
$121$	358.182			0.269107
$122$		0			0
$123$			84.4072i			0.0618760i
$124$		0			0
$125$	−314.858	+	1361.61i	−0.225294	+	0.974291i
$126$		0			0
$127$		−	253.892i		−	0.177396i	−0.996059	−	0.0886981i	$-0.971729\pi$
$127$		−	253.892i		−	0.177396i	0.996059	−	0.0886981i	$-0.0282706\pi$
$128$		0			0
$129$	−190.295			−0.129880
$130$		0			0
$131$	−271.914			−0.181353			−0.0906764	−	0.995880i	$-0.528903\pi$
$131$	−271.914			−0.181353			−0.0906764	+	0.995880i	$0.528903\pi$
$132$		0			0
$133$		−	471.250i		−	0.307237i
$134$		0			0
$135$	141.781	+	295.127i	0.0903895	+	0.188152i
$136$		0			0
$137$		−	1596.26i		−	0.995457i	−0.867333	−	0.497729i	$-0.834168\pi$
$137$		−	1596.26i		−	0.995457i	0.867333	−	0.497729i	$-0.165832\pi$
$138$		0			0
$139$	2497.29			1.52386			0.761932	−	0.647657i	$-0.224251\pi$
$139$	2497.29			1.52386			0.761932	+	0.647657i	$0.224251\pi$
$140$		0			0
$141$	−329.865			−0.197019
$142$		0			0
$143$		−	2282.58i		−	1.33482i
$144$		0			0
$145$	−18.9205	+	9.08953i	−0.0108363	+	0.00520582i
$146$		0			0
$147$		−	110.429i		−	0.0619595i
$148$		0			0
$149$	163.816			0.0900694			0.0450347	−	0.998985i	$-0.485660\pi$
$149$	163.816			0.0900694			0.0450347	+	0.998985i	$0.485660\pi$
$150$		0			0
$151$	2478.52			1.33576			0.667879	−	0.744270i	$-0.267203\pi$
$151$	2478.52			1.33576			0.667879	+	0.744270i	$0.267203\pi$
$152$		0			0
$153$		−	831.804i		−	0.439525i
$154$		0			0
$155$	1003.52	−	482.095i	0.520028	−	0.249825i
$156$		0			0
$157$			1602.74i			0.814730i	0.913265	+	0.407365i	$0.133552\pi$
$157$			1602.74i			0.814730i	−0.913265	+	0.407365i	$0.866448\pi$
$158$		0			0
$159$	266.422			0.132884
$160$		0			0
$161$	743.620			0.364009
$162$		0			0
$163$		−	2762.68i		−	1.32754i	−0.747935	−	0.663772i	$-0.768955\pi$
$163$		−	2762.68i		−	1.32754i	0.747935	−	0.663772i	$-0.231045\pi$
$164$		0			0
$165$	−108.508	−	225.867i	−0.0511959	−	0.106568i
$166$		0			0
$167$			1344.88i			0.623173i	0.950218	+	0.311587i	$0.100860\pi$
$167$			1344.88i			0.623173i	−0.950218	+	0.311587i	$0.899140\pi$
$168$		0			0
$169$	−887.437			−0.403931
$170$		0			0
$171$	1061.63			0.474765
$172$		0			0
$173$		−	3017.80i		−	1.32624i	−0.748514	−	0.663119i	$-0.769232\pi$
$173$		−	3017.80i		−	1.32624i	0.748514	−	0.663119i	$-0.230768\pi$
$174$		0			0
$175$	−1156.64	−	925.983i	−0.499621	−	0.399987i
$176$		0			0
$177$		−	379.764i		−	0.161270i
$178$		0			0
$179$	−2377.40			−0.992711			−0.496356	−	0.868119i	$-0.665329\pi$
$179$	−2377.40			−0.992711			−0.496356	+	0.868119i	$0.665329\pi$
$180$		0			0
$181$	3367.21			1.38278			0.691390	−	0.722482i	$-0.256999\pi$
$181$	3367.21			1.38278			0.691390	+	0.722482i	$0.256999\pi$
$182$		0			0
$183$		−	88.2655i		−	0.0356545i
$184$		0			0
$185$	490.717	+	1021.46i	0.195018	+	0.405943i
$186$		0			0
$187$			1280.28i			0.500660i
$188$		0			0
$189$	−347.119			−0.133594
$190$		0			0
$191$	−4993.98			−1.89189			−0.945946	−	0.324324i	$-0.894863\pi$
$191$	−4993.98			−1.89189			−0.945946	+	0.324324i	$0.894863\pi$
$192$		0			0
$193$			5119.21i			1.90927i	0.297783	+	0.954634i	$0.403753\pi$
$193$			5119.21i			1.90927i	−0.297783	+	0.954634i	$0.596247\pi$
$194$		0			0
$195$	−305.212	+	146.626i	−0.112086	+	0.0538467i
$196$		0			0
$197$		−	1521.99i		−	0.550442i	−0.961381	−	0.275221i	$-0.911249\pi$
$197$		−	1521.99i		−	0.550442i	0.961381	−	0.275221i	$-0.0887510\pi$
$198$		0			0
$199$	4990.52			1.77773			0.888866	−	0.458167i	$-0.151494\pi$
$199$	4990.52			1.77773			0.888866	+	0.458167i	$0.151494\pi$
$200$		0			0
$201$	−538.693			−0.189037
$202$		0			0
$203$		−	22.2537i		−	0.00769410i
$204$		0			0
$205$	1559.88	−	749.378i	0.531448	−	0.255311i
$206$		0			0
$207$			1675.22i			0.562493i
$208$		0			0
$209$	−1634.02			−0.540801
$210$		0			0
$211$	−2328.32			−0.759661			−0.379830	−	0.925056i	$-0.624018\pi$
$211$	−2328.32			−0.759661			−0.379830	+	0.925056i	$0.624018\pi$
$212$		0			0
$213$		−	419.094i		−	0.134816i
$214$		0			0
$215$	1689.46	+	3516.73i	0.535909	+	1.11553i
$216$		0			0
$217$			1180.30i			0.369236i
$218$		0			0
$219$	−258.804			−0.0798555
$220$		0			0
$221$	1730.04			0.526583
$222$		0			0
$223$		−	1868.69i		−	0.561150i	−0.959832	−	0.280575i	$-0.909475\pi$
$223$		−	1868.69i		−	0.561150i	0.959832	−	0.280575i	$-0.0905252\pi$
$224$		0			0
$225$	2086.05	−	2605.67i	0.618088	−	0.772049i
$226$		0			0
$227$			4388.31i			1.28309i	0.767084	+	0.641547i	$0.221707\pi$
$227$			4388.31i			1.28309i	−0.767084	+	0.641547i	$0.778293\pi$
$228$		0			0
$229$	−1841.81			−0.531487			−0.265744	−	0.964044i	$-0.585617\pi$
$229$	−1841.81			−0.531487			−0.265744	+	0.964044i	$0.585617\pi$
$230$		0			0
$231$	265.657			0.0756665
$232$		0			0
$233$			5763.09i			1.62040i	0.586156	+	0.810198i	$0.300641\pi$
$233$			5763.09i			1.62040i	−0.586156	+	0.810198i	$0.699359\pi$
$234$		0			0
$235$	2928.58	+	6096.05i	0.812935	+	1.69218i
$236$		0			0
$237$		−	4.82574i		−	0.00132264i
$238$		0			0
$239$	−5346.35			−1.44697			−0.723486	−	0.690339i	$-0.757462\pi$
$239$	−5346.35			−1.44697			−0.723486	+	0.690339i	$0.757462\pi$
$240$		0			0
$241$	−4869.08			−1.30143			−0.650716	−	0.759321i	$-0.725531\pi$
$241$	−4869.08			−1.30143			−0.650716	+	0.759321i	$0.725531\pi$
$242$		0			0
$243$		−	1175.15i		−	0.310230i
$244$		0			0
$245$	−2040.78	+	980.404i	−0.532165	+	0.255656i
$246$		0			0
$247$			2208.04i			0.568802i
$248$		0			0
$249$	−326.038			−0.0829791
$250$		0			0
$251$	−1651.90			−0.415405			−0.207703	−	0.978192i	$-0.566599\pi$
$251$	−1651.90			−0.415405			−0.207703	+	0.978192i	$0.566599\pi$
$252$		0			0
$253$		−	2578.44i		−	0.640732i
$254$		0			0
$255$	171.191	−	82.2413i	0.0420408	−	0.0201967i
$256$		0			0
$257$			556.334i			0.135032i	0.997718	+	0.0675159i	$0.0215073\pi$
$257$			556.334i			0.135032i	−0.997718	+	0.0675159i	$0.978493\pi$
$258$		0			0
$259$	−1201.41			−0.288232
$260$		0			0
$261$	50.1330			0.0118895
$262$		0			0
$263$			4709.05i			1.10408i	0.833818	+	0.552039i	$0.186150\pi$
$263$			4709.05i			1.10408i	−0.833818	+	0.552039i	$0.813850\pi$
$264$		0			0
$265$	−2365.32	−	4923.59i	−0.548305	−	1.14133i
$266$		0			0
$267$		−	355.444i		−	0.0814711i
$268$		0			0
$269$	1863.29			0.422330			0.211165	−	0.977450i	$-0.432274\pi$
$269$	1863.29			0.422330			0.211165	+	0.977450i	$0.432274\pi$
$270$		0			0
$271$	−2272.69			−0.509433			−0.254717	−	0.967016i	$-0.581982\pi$
$271$	−2272.69			−0.509433			−0.254717	+	0.967016i	$0.581982\pi$
$272$		0			0
$273$		−	358.981i		−	0.0795843i
$274$		0			0
$275$	−3210.77	+	4010.54i	−0.704060	+	0.879436i
$276$		0			0
$277$		−	1528.11i		−	0.331462i	−0.986171	−	0.165731i	$-0.947002\pi$
$277$		−	1528.11i		−	0.331462i	0.986171	−	0.165731i	$-0.0529983\pi$
$278$		0			0
$279$	−2658.98			−0.570570
$280$		0			0
$281$	3981.09			0.845167			0.422584	−	0.906324i	$-0.361123\pi$
$281$	3981.09			0.845167			0.422584	+	0.906324i	$0.361123\pi$
$282$		0			0
$283$		−	2191.09i		−	0.460237i	−0.973163	−	0.230118i	$-0.926089\pi$
$283$		−	2191.09i		−	0.460237i	0.973163	−	0.230118i	$-0.0739113\pi$
$284$		0			0
$285$	104.964	+	218.490i	0.0218160	+	0.0454114i
$286$		0			0
$287$			1834.68i			0.377345i
$288$		0			0
$289$	3942.64			0.802491
$290$		0			0
$291$	−448.549			−0.0903587
$292$		0			0
$293$		−	5987.75i		−	1.19388i	−0.802284	−	0.596942i	$-0.796382\pi$
$293$		−	5987.75i		−	1.19388i	0.802284	−	0.596942i	$-0.203618\pi$
$294$		0			0
$295$	−7018.21	+	3371.60i	−1.38514	+	0.665430i
$296$		0			0
$297$			1203.61i			0.235153i
$298$		0			0
$299$	−3484.23			−0.673907
$300$		0			0
$301$	−4136.27			−0.792063
$302$		0			0
$303$			850.601i			0.161273i
$304$		0			0
$305$	−1631.18	+	783.632i	−0.306234	+	0.147117i
$306$		0			0
$307$			1592.84i			0.296117i	0.988979	+	0.148059i	$0.0473025\pi$
$307$			1592.84i			0.296117i	−0.988979	+	0.148059i	$0.952698\pi$
$308$		0			0
$309$	−682.262			−0.125607
$310$		0			0
$311$	3115.01			0.567961			0.283981	−	0.958830i	$-0.408345\pi$
$311$	3115.01			0.567961			0.283981	+	0.958830i	$0.408345\pi$
$312$		0			0
$313$			7516.74i			1.35742i	0.734408	+	0.678708i	$0.237460\pi$
$313$			7516.74i			1.35742i	−0.734408	+	0.678708i	$0.762540\pi$
$314$		0			0
$315$	1532.35	+	3189.70i	0.274090	+	0.570537i
$316$		0			0
$317$		−	6955.62i		−	1.23239i	−0.787595	−	0.616194i	$-0.788674\pi$
$317$		−	6955.62i		−	1.23239i	0.787595	−	0.616194i	$-0.211326\pi$
$318$		0			0
$319$	−77.1627			−0.0135432
$320$		0			0
$321$	193.139			0.0335824
$322$		0			0
$323$		−	1238.47i		−	0.213345i
$324$		0			0
$325$	5419.42	+	4338.69i	0.924970	+	0.740514i
$326$		0			0
$327$			280.272i			0.0473978i
$328$		0			0
$329$	−7169.98			−1.20150
$330$		0			0
$331$	−1132.02			−0.187980			−0.0939898	−	0.995573i	$-0.529962\pi$
$331$	−1132.02			−0.187980			−0.0939898	+	0.995573i	$0.529962\pi$
$332$		0			0
$333$		−	2706.53i		−	0.445397i
$334$		0			0
$335$	4782.58	+	9955.28i	0.780001	+	1.62363i
$336$		0			0
$337$			3993.78i			0.645564i	0.946473	+	0.322782i	$0.104618\pi$
$337$			3993.78i			0.645564i	−0.946473	+	0.322782i	$0.895382\pi$
$338$		0			0
$339$	−289.298			−0.0463496
$340$		0			0
$341$	4092.60			0.649932
$342$		0			0
$343$		−	6465.92i		−	1.01786i
$344$		0			0
$345$	−344.772	+	165.631i	−0.0538027	+	0.0258472i
$346$		0			0
$347$			9909.90i			1.53312i	0.642175	+	0.766558i	$0.278032\pi$
$347$			9909.90i			1.53312i	−0.642175	+	0.766558i	$0.721968\pi$
$348$		0			0
$349$	3602.90			0.552604			0.276302	−	0.961071i	$-0.410891\pi$
$349$	3602.90			0.552604			0.276302	+	0.961071i	$0.410891\pi$
$350$		0			0
$351$	1626.43			0.247328
$352$		0			0
$353$			11700.1i			1.76412i	0.471136	+	0.882061i	$0.343844\pi$
$353$			11700.1i			1.76412i	−0.471136	+	0.882061i	$0.656156\pi$
$354$		0			0
$355$	−7745.04	+	3720.77i	−1.15793	+	0.556276i
$356$		0			0
$357$			201.349i			0.0298503i
$358$		0			0
$359$	1461.06			0.214796			0.107398	−	0.994216i	$-0.465748\pi$
$359$	1461.06			0.214796			0.107398	+	0.994216i	$0.465748\pi$
$360$		0			0
$361$	−5278.35			−0.769550
$362$		0			0
$363$		−	195.324i		−	0.0282419i
$364$		0			0
$365$	2297.69	+	4782.81i	0.329498	+	0.685873i
$366$		0			0
$367$			2290.70i			0.325814i	0.986641	+	0.162907i	$0.0520870\pi$
$367$			2290.70i			0.325814i	−0.986641	+	0.162907i	$0.947913\pi$
$368$		0			0
$369$	−4133.16			−0.583100
$370$		0			0
$371$	5790.97			0.810383
$372$		0			0
$373$		−	12561.0i		−	1.74366i	−0.489809	−	0.871830i	$-0.662933\pi$
$373$		−	12561.0i		−	1.74366i	0.489809	−	0.871830i	$-0.337067\pi$
$374$		0			0
$375$	742.514	+	171.698i	0.102249	+	0.0236439i
$376$		0			0
$377$			104.270i			0.0142444i
$378$		0			0
$379$	5349.81			0.725069			0.362535	−	0.931970i	$-0.381911\pi$
$379$	5349.81			0.725069			0.362535	+	0.931970i	$0.381911\pi$
$380$		0			0
$381$	−138.452			−0.0186172
$382$		0			0
$383$		−	6001.35i		−	0.800665i	−0.916370	−	0.400332i	$-0.868895\pi$
$383$		−	6001.35i		−	0.800665i	0.916370	−	0.400332i	$-0.131105\pi$
$384$		0			0
$385$	−2358.54	−	4909.46i	−0.312213	−	0.649894i
$386$		0			0
$387$		−	9318.17i		−	1.22395i
$388$		0			0
$389$	11671.8			1.52130			0.760649	−	0.649163i	$-0.224881\pi$
$389$	11671.8			1.52130			0.760649	+	0.649163i	$0.224881\pi$
$390$		0			0
$391$	1954.28			0.252767
$392$		0			0
$393$			148.280i			0.0190324i
$394$		0			0
$395$	−89.1818	+	42.8436i	−0.0113601	+	0.00545745i
$396$		0			0
$397$			7746.29i			0.979282i	0.871924	+	0.489641i	$0.162872\pi$
$397$			7746.29i			0.979282i	−0.871924	+	0.489641i	$0.837128\pi$
$398$		0			0
$399$	−256.982			−0.0322435
$400$		0			0
$401$	1756.61			0.218755			0.109377	−	0.994000i	$-0.465114\pi$
$401$	1756.61			0.218755			0.109377	+	0.994000i	$0.465114\pi$
$402$		0			0
$403$		−	5530.30i		−	0.683583i
$404$		0			0
$405$	−7104.81	+	3413.20i	−0.871706	+	0.418774i
$406$		0			0
$407$			4165.79i			0.507348i
$408$		0			0
$409$	12636.3			1.52769			0.763846	−	0.645398i	$-0.223308\pi$
$409$	12636.3			1.52769			0.763846	+	0.645398i	$0.223308\pi$
$410$		0			0
$411$	−870.471			−0.104470
$412$		0			0
$413$		−	8254.60i		−	0.983492i
$414$		0			0
$415$	2894.60	+	6025.32i	0.342387	+	0.712702i
$416$		0			0
$417$		−	1361.82i		−	0.159925i
$418$		0			0
$419$	4006.19			0.467101			0.233550	−	0.972345i	$-0.424966\pi$
$419$	4006.19			0.467101			0.233550	+	0.972345i	$0.424966\pi$
$420$		0			0
$421$	−10349.4			−1.19810			−0.599048	−	0.800713i	$-0.704454\pi$
$421$	−10349.4			−1.19810			−0.599048	+	0.800713i	$0.704454\pi$
$422$		0			0
$423$		−	16152.5i		−	1.85665i
$424$		0			0
$425$	−3039.71	−	2433.53i	−0.346935	−	0.277750i
$426$		0			0
$427$		−	1918.55i		−	0.217436i
$428$		0			0
$429$	−1244.74			−0.140085
$430$		0			0
$431$	−3700.85			−0.413605			−0.206802	−	0.978383i	$-0.566306\pi$
$431$	−3700.85			−0.413605			−0.206802	+	0.978383i	$0.566306\pi$
$432$		0			0
$433$		−	9058.98i		−	1.00542i	−0.864455	−	0.502710i	$-0.832336\pi$
$433$		−	9058.98i		−	1.00542i	0.864455	−	0.502710i	$-0.167664\pi$
$434$		0			0
$435$	4.95670	+	10.3177i	0.000546334	+	0.00113723i
$436$		0			0
$437$			2494.23i			0.273033i
$438$		0			0
$439$	−10811.6			−1.17542			−0.587711	−	0.809071i	$-0.699971\pi$
$439$	−10811.6			−1.17542			−0.587711	+	0.809071i	$0.699971\pi$
$440$		0			0
$441$	5407.38			0.583887
$442$		0			0
$443$		−	11501.8i		−	1.23356i	−0.787134	−	0.616782i	$-0.788436\pi$
$443$		−	11501.8i		−	1.23356i	0.787134	−	0.616782i	$-0.211564\pi$
$444$		0			0
$445$	−6568.75	+	3155.67i	−0.699750	+	0.336164i
$446$		0			0
$447$		−	89.3321i		−	0.00945249i
$448$		0			0
$449$	−266.343			−0.0279944			−0.0139972	−	0.999902i	$-0.504456\pi$
$449$	−266.343			−0.0279944			−0.0139972	+	0.999902i	$0.504456\pi$
$450$		0			0
$451$	6361.61			0.664205
$452$		0			0
$453$		−	1351.59i		−	0.140183i
$454$		0			0
$455$	−6634.12	+	3187.08i	−0.683544	+	0.328379i
$456$		0			0
$457$		−	17225.0i		−	1.76313i	−0.472059	−	0.881567i	$-0.656489\pi$
$457$		−	17225.0i		−	1.76313i	0.472059	−	0.881567i	$-0.343511\pi$
$458$		0			0
$459$	−912.249			−0.0927672
$460$		0			0
$461$	5506.51			0.556321			0.278160	−	0.960535i	$-0.410275\pi$
$461$	5506.51			0.556321			0.278160	+	0.960535i	$0.410275\pi$
$462$		0			0
$463$		−	10758.9i		−	1.07993i	−0.841687	−	0.539965i	$-0.818437\pi$
$463$		−	10758.9i		−	1.07993i	0.841687	−	0.539965i	$-0.181563\pi$
$464$		0			0
$465$	−262.896	−	547.236i	−0.0262183	−	0.0545752i
$466$		0			0
$467$			13714.8i			1.35899i	0.733681	+	0.679494i	$0.237801\pi$
$467$			13714.8i			1.35899i	−0.733681	+	0.679494i	$0.762199\pi$
$468$		0			0
$469$	−11709.1			−1.15283
$470$		0			0
$471$	874.005			0.0855033
$472$		0			0
$473$			14342.2i			1.39419i
$474$		0			0
$475$	3105.91	−	3879.57i	0.300019	−	0.374751i
$476$		0			0
$477$			13045.9i			1.25226i
$478$		0			0
$479$	10741.8			1.02465			0.512323	−	0.858793i	$-0.328785\pi$
$479$	10741.8			1.02465			0.512323	+	0.858793i	$0.328785\pi$
$480$		0			0
$481$	5629.21			0.533617
$482$		0			0
$483$		−	405.511i		−	0.0382016i
$484$		0			0
$485$	3982.27	+	8289.37i	0.372836	+	0.776085i
$486$		0			0
$487$			8440.40i			0.785361i	0.919675	+	0.392681i	$0.128452\pi$
$487$			8440.40i			0.785361i	−0.919675	+	0.392681i	$0.871548\pi$
$488$		0			0
$489$	−1506.54			−0.139321
$490$		0			0
$491$	−20132.1			−1.85041			−0.925204	−	0.379471i	$-0.876106\pi$
$491$	−20132.1			−1.85041			−0.925204	+	0.379471i	$0.876106\pi$
$492$		0			0
$493$		−	58.4839i		−	0.00534276i
$494$		0			0
$495$	11060.0	−	5313.30i	1.00426	−	0.482455i
$496$		0			0
$497$		−	9109.47i		−	0.822165i
$498$		0			0
$499$	4481.50			0.402043			0.201021	−	0.979587i	$-0.435574\pi$
$499$	4481.50			0.402043			0.201021	+	0.979587i	$0.435574\pi$
$500$		0			0
$501$	733.389			0.0654000
$502$		0			0
$503$		−	6946.42i		−	0.615757i	−0.951426	−	0.307878i	$-0.900381\pi$
$503$		−	6946.42i		−	0.615757i	0.951426	−	0.307878i	$-0.0996189\pi$
$504$		0			0
$505$	15719.5	−	7551.74i	1.38516	−	0.665442i
$506$		0			0
$507$			483.937i			0.0423913i
$508$		0			0
$509$	−8349.21			−0.727058			−0.363529	−	0.931583i	$-0.618428\pi$
$509$	−8349.21			−0.727058			−0.363529	+	0.931583i	$0.618428\pi$
$510$		0			0
$511$	−5625.39			−0.486991
$512$		0			0
$513$		−	1164.30i		−	0.100205i
$514$		0			0
$515$	6057.20	+	12608.5i	0.518276	+	1.07883i
$516$		0			0
$517$			24861.3i			2.11489i
$518$		0			0
$519$	−1645.66			−0.139184
$520$		0			0
$521$	−23197.5			−1.95067			−0.975337	−	0.220722i	$-0.929159\pi$
$521$	−23197.5			−1.95067			−0.975337	+	0.220722i	$0.929159\pi$
$522$		0			0
$523$		−	18221.1i		−	1.52343i	−0.647914	−	0.761713i	$-0.724359\pi$
$523$		−	18221.1i		−	1.52343i	0.647914	−	0.761713i	$-0.275641\pi$
$524$		0			0
$525$	−504.956	+	630.737i	−0.0419773	+	0.0524336i
$526$		0			0
$527$			3101.90i			0.256396i
$528$		0			0
$529$	8231.16			0.676515
$530$		0			0
$531$	18595.9			1.51976
$532$		0			0
$533$		−	8596.40i		−	0.698596i
$534$		0			0
$535$	−1714.71	−	3569.29i	−0.138567	−	0.288437i
$536$		0			0
$537$			1296.44i			0.104182i
$538$		0			0
$539$	−8322.83			−0.665101
$540$		0			0
$541$	1421.71			0.112984			0.0564919	−	0.998403i	$-0.482008\pi$
$541$	1421.71			0.112984			0.0564919	+	0.998403i	$0.482008\pi$
$542$		0			0
$543$		−	1836.21i		−	0.145118i
$544$		0			0
$545$	5179.55	−	2488.29i	0.407096	−	0.195572i
$546$		0			0
$547$			1467.65i			0.114720i	0.998354	+	0.0573601i	$0.0182683\pi$
$547$			1467.65i			0.114720i	−0.998354	+	0.0573601i	$0.981732\pi$
$548$		0			0
$549$	4322.09			0.335997
$550$		0			0
$551$	74.6428			0.00577113
$552$		0			0
$553$		−	104.893i		−	0.00806600i
$554$		0			0
$555$	557.023	−	267.598i	0.0426024	−	0.0204665i
$556$		0			0
$557$		−	4430.55i		−	0.337035i	−0.985699	−	0.168518i	$-0.946102\pi$
$557$		−	4430.55i		−	0.337035i	0.985699	−	0.168518i	$-0.0538980\pi$
$558$		0			0
$559$	19380.5			1.46638
$560$		0			0
$561$	698.162			0.0525426
$562$		0			0
$563$		−	16432.8i		−	1.23012i	−0.788478	−	0.615062i	$-0.789131\pi$
$563$		−	16432.8i		−	1.23012i	0.788478	−	0.615062i	$-0.210869\pi$
$564$		0			0
$565$	2568.42	+	5346.35i	0.191247	+	0.398093i
$566$		0			0
$567$		−	8356.46i		−	0.618939i
$568$		0			0
$569$	13114.1			0.966205			0.483103	−	0.875564i	$-0.339510\pi$
$569$	13114.1			0.966205			0.483103	+	0.875564i	$0.339510\pi$
$570$		0			0
$571$	−23508.3			−1.72293			−0.861463	−	0.507821i	$-0.830451\pi$
$571$	−23508.3			−1.72293			−0.861463	+	0.507821i	$0.830451\pi$
$572$		0			0
$573$			2723.31i			0.198548i
$574$		0			0
$575$	6121.86	+	4901.05i	0.443999	+	0.355457i
$576$		0			0
$577$		−	16108.6i		−	1.16223i	−0.813820	−	0.581117i	$-0.802616\pi$
$577$		−	16108.6i		−	1.16223i	0.813820	−	0.581117i	$-0.197384\pi$
$578$		0			0
$579$	2791.60			0.200371
$580$		0			0
$581$	−7086.79			−0.506041
$582$		0			0
$583$		−	20079.7i		−	1.42644i
$584$		0			0
$585$	−7179.83	−	14945.3i	−0.507435	−	1.05626i
$586$		0			0
$587$		−	13302.5i		−	0.935356i	−0.883899	−	0.467678i	$-0.845091\pi$
$587$		−	13302.5i		−	0.935356i	0.883899	−	0.467678i	$-0.154909\pi$
$588$		0			0
$589$	−3958.95			−0.276953
$590$		0			0
$591$	−829.969			−0.0577671
$592$		0			0
$593$			23711.4i			1.64201i	0.570923	+	0.821003i	$0.306585\pi$
$593$			23711.4i			1.64201i	−0.570923	+	0.821003i	$0.693415\pi$
$594$		0			0
$595$	3721.02	−	1787.61i	0.256382	−	0.123167i
$596$		0			0
$597$		−	2721.43i		−	0.186567i
$598$		0			0
$599$	−16146.5			−1.10138			−0.550691	−	0.834709i	$-0.685636\pi$
$599$	−16146.5			−1.10138			−0.550691	+	0.834709i	$0.685636\pi$
$600$		0			0
$601$	19397.1			1.31652			0.658258	−	0.752793i	$-0.271294\pi$
$601$	19397.1			1.31652			0.658258	+	0.752793i	$0.271294\pi$
$602$		0			0
$603$		−	26378.2i		−	1.78143i
$604$		0			0
$605$	−3609.66	+	1734.11i	−0.242568	+	0.116531i
$606$		0			0
$607$		−	2169.94i		−	0.145099i	−0.997365	−	0.0725494i	$-0.976886\pi$
$607$		−	2169.94i		−	0.145099i	0.997365	−	0.0725494i	$-0.0231135\pi$
$608$		0			0
$609$	−12.1354			−0.000807471
$610$		0			0
$611$	33594.9			2.22439
$612$		0			0
$613$			9626.49i			0.634274i	0.948380	+	0.317137i	$0.102722\pi$
$613$			9626.49i			0.634274i	−0.948380	+	0.317137i	$0.897278\pi$
$614$		0			0
$615$	−408.650	−	850.634i	−0.0267941	−	0.0557738i
$616$		0			0
$617$		−	10198.8i		−	0.665462i	−0.943022	−	0.332731i	$-0.892030\pi$
$617$		−	10198.8i		−	0.665462i	0.943022	−	0.332731i	$-0.107970\pi$
$618$		0			0
$619$	14305.6			0.928905			0.464452	−	0.885598i	$-0.346251\pi$
$619$	14305.6			0.928905			0.464452	+	0.885598i	$0.346251\pi$
$620$		0			0
$621$	1837.24			0.118721
$622$		0			0
$623$		−	7725.96i		−	0.496844i
$624$		0			0
$625$	−3419.07	−	15246.3i	−0.218821	−	0.975765i
$626$		0			0
$627$			891.061i			0.0567553i
$628$		0			0
$629$	−3157.38			−0.200148
$630$		0			0
$631$	25116.4			1.58457			0.792287	−	0.610148i	$-0.208890\pi$
$631$	25116.4			1.58457			0.792287	+	0.610148i	$0.208890\pi$
$632$		0			0
$633$			1269.68i			0.0797240i
$634$		0			0
$635$	1229.20	+	2558.66i	0.0768177	+	0.159901i
$636$		0			0
$637$			11246.6i			0.699539i
$638$		0			0
$639$	20521.8			1.27047
$640$		0			0
$641$	7003.81			0.431566			0.215783	−	0.976441i	$-0.430770\pi$
$641$	7003.81			0.431566			0.215783	+	0.976441i	$0.430770\pi$
$642$		0			0
$643$		−	17474.1i		−	1.07171i	−0.844310	−	0.535856i	$-0.819989\pi$
$643$		−	17474.1i		−	1.07171i	0.844310	−	0.535856i	$-0.180011\pi$
$644$		0			0
$645$	1917.74	−	921.297i	0.117071	−	0.0562419i
$646$		0			0
$647$		−	19277.1i		−	1.17134i	−0.810548	−	0.585672i	$-0.800831\pi$
$647$		−	19277.1i		−	1.17134i	0.810548	−	0.585672i	$-0.199169\pi$
$648$		0			0
$649$	−28622.1			−1.73115
$650$		0			0
$651$	643.642			0.0387501
$652$		0			0
$653$		−	8622.09i		−	0.516705i	−0.966051	−	0.258353i	$-0.916820\pi$
$653$		−	8622.09i		−	0.516705i	0.966051	−	0.258353i	$-0.0831797\pi$
$654$		0			0
$655$	2740.27	−	1316.45i	0.163468	−	0.0785310i
$656$		0			0
$657$		−	12672.8i		−	0.752533i
$658$		0			0
$659$	−12333.0			−0.729022			−0.364511	−	0.931199i	$-0.618764\pi$
$659$	−12333.0			−0.729022			−0.364511	+	0.931199i	$0.618764\pi$
$660$		0			0
$661$	50.0662			0.00294606			0.00147303	−	0.999999i	$-0.499531\pi$
$661$	50.0662			0.00294606			0.00147303	+	0.999999i	$0.499531\pi$
$662$		0			0
$663$		−	943.422i		−	0.0552631i
$664$		0			0
$665$	2281.51	+	4749.13i	0.133043	+	0.276937i
$666$		0			0
$667$			117.784i			0.00683753i
$668$		0			0
$669$	−1019.03			−0.0588909
$670$		0			0
$671$	−6652.40			−0.382732
$672$		0			0
$673$			4922.49i			0.281944i	0.990014	+	0.140972i	$0.0450227\pi$
$673$			4922.49i			0.281944i	−0.990014	+	0.140972i	$0.954977\pi$
$674$		0			0
$675$	−2857.67	−	2287.79i	−0.162951	−	0.130455i
$676$		0			0
$677$		−	20087.4i		−	1.14036i	−0.821521	−	0.570178i	$-0.806874\pi$
$677$		−	20087.4i		−	1.14036i	0.821521	−	0.570178i	$-0.193126\pi$
$678$		0			0
$679$	−9749.70			−0.551044
$680$		0			0
$681$	2393.03			0.134657
$682$		0			0
$683$			15728.0i			0.881133i	0.897720	+	0.440567i	$0.145222\pi$
$683$			15728.0i			0.881133i	−0.897720	+	0.440567i	$0.854778\pi$
$684$		0			0
$685$	7728.15	+	16086.7i	0.431062	+	0.897285i
$686$		0			0
$687$			1004.38i			0.0557779i
$688$		0			0
$689$	−27133.6			−1.50030
$690$		0			0
$691$	14146.4			0.778808			0.389404	−	0.921067i	$-0.372681\pi$
$691$	14146.4			0.778808			0.389404	+	0.921067i	$0.372681\pi$
$692$		0			0
$693$			13008.4i			0.713058i
$694$		0			0
$695$	−25167.0	+	12090.4i	−1.37358	+	0.659878i
$696$		0			0
$697$			4821.65i			0.262027i
$698$		0			0
$699$	3142.72			0.170055
$700$		0			0
$701$	−982.768			−0.0529510			−0.0264755	−	0.999649i	$-0.508428\pi$
$701$	−982.768			−0.0529510			−0.0264755	+	0.999649i	$0.508428\pi$
$702$		0			0
$703$		−	4029.75i		−	0.216195i
$704$		0			0
$705$	3324.29	−	1597.01i	0.177589	−	0.0853149i
$706$		0			0
$707$			18488.8i			0.983509i
$708$		0			0
$709$	−9464.68			−0.501345			−0.250673	−	0.968072i	$-0.580652\pi$
$709$	−9464.68			−0.501345			−0.250673	+	0.968072i	$0.580652\pi$
$710$		0			0
$711$	236.302			0.0124642
$712$		0			0
$713$		−	6247.12i		−	0.328130i
$714$		0			0
$715$	11050.9	+	23003.2i	0.578015	+	1.20318i
$716$		0			0
$717$			2915.47i			0.151855i
$718$		0			0
$719$	18281.5			0.948239			0.474120	−	0.880460i	$-0.342766\pi$
$719$	18281.5			0.948239			0.474120	+	0.880460i	$0.342766\pi$
$720$		0			0
$721$	−14829.7			−0.766002
$722$		0			0
$723$			2655.21i			0.136581i
$724$		0			0
$725$	146.669	−	183.204i	0.00751333	−	0.00938485i
$726$		0			0
$727$		−	8275.38i		−	0.422169i	−0.977468	−	0.211084i	$-0.932301\pi$
$727$		−	8275.38i		−	0.422169i	0.977468	−	0.211084i	$-0.0676995\pi$
$728$		0			0
$729$	18394.2			0.934522
$730$		0			0
$731$	−10870.4			−0.550006
$732$		0			0
$733$		−	12103.8i		−	0.609912i	−0.952366	−	0.304956i	$-0.901358\pi$
$733$		−	12103.8i		−	0.609912i	0.952366	−	0.304956i	$-0.0986418\pi$
$734$		0			0
$735$	534.633	+	1112.88i	0.0268303	+	0.0558490i
$736$		0			0
$737$			40600.3i			2.02921i
$738$		0			0
$739$	−25912.5			−1.28986			−0.644929	−	0.764242i	$-0.723113\pi$
$739$	−25912.5			−1.28986			−0.644929	+	0.764242i	$0.723113\pi$
$740$		0			0
$741$	1204.09			0.0596939
$742$		0			0
$743$			18108.6i			0.894133i	0.894501	+	0.447067i	$0.147531\pi$
$743$			18108.6i			0.894133i	−0.894501	+	0.447067i	$0.852469\pi$
$744$		0			0
$745$	−1650.90	+	793.102i	−0.0811868	+	0.0390027i
$746$		0			0
$747$		−	15965.1i		−	0.781970i
$748$		0			0
$749$	4198.09			0.204799
$750$		0			0
$751$	−16532.6			−0.803306			−0.401653	−	0.915792i	$-0.631564\pi$
$751$	−16532.6			−0.803306			−0.401653	+	0.915792i	$0.631564\pi$
$752$		0			0
$753$			900.811i			0.0435954i
$754$		0			0
$755$	−24977.9	+	11999.6i	−1.20402	+	0.578422i
$756$		0			0
$757$			7668.83i			0.368201i	0.982907	+	0.184101i	$0.0589372\pi$
$757$			7668.83i			0.368201i	−0.982907	+	0.184101i	$0.941063\pi$
$758$		0			0
$759$	−1406.07			−0.0672427
$760$		0			0
$761$	23891.6			1.13807			0.569035	−	0.822313i	$-0.307317\pi$
$761$	23891.6			1.13807			0.569035	+	0.822313i	$0.307317\pi$
$762$		0			0
$763$			6092.02i			0.289051i
$764$		0			0
$765$	4027.11	+	8382.70i	0.190327	+	0.396179i
$766$		0			0
$767$			38676.9i			1.82078i
$768$		0			0
$769$	31394.7			1.47220			0.736099	−	0.676873i	$-0.236666\pi$
$769$	31394.7			1.47220			0.736099	+	0.676873i	$0.236666\pi$
$770$		0			0
$771$	303.380			0.0141712
$772$		0			0
$773$		−	34865.4i		−	1.62228i	−0.584853	−	0.811140i	$-0.698848\pi$
$773$		−	34865.4i		−	1.62228i	0.584853	−	0.811140i	$-0.301152\pi$
$774$		0			0
$775$	−7779.14	+	9716.86i	−0.360561	+	0.450374i
$776$		0			0
$777$			655.153i			0.0302490i
$778$		0			0
$779$	−6153.86			−0.283036
$780$		0			0
$781$	−31586.3			−1.44718
$782$		0			0
$783$		−	54.9814i		−	0.00250942i
$784$		0			0
$785$	−7759.52	−	16152.0i	−0.352802	−	0.734381i
$786$		0			0
$787$		−	14825.6i		−	0.671505i	−0.941950	−	0.335753i	$-0.891009\pi$
$787$		−	14825.6i		−	0.671505i	0.941950	−	0.335753i	$-0.108991\pi$
$788$		0			0
$789$	2567.94			0.115869
$790$		0			0
$791$	−6288.21			−0.282658
$792$		0			0
$793$			8989.34i			0.402549i
$794$		0			0
$795$	−2684.93	+	1289.86i	−0.119779	+	0.0575428i
$796$		0			0
$797$		−	16971.4i		−	0.754277i	−0.926157	−	0.377139i	$-0.876908\pi$
$797$		−	16971.4i		−	0.754277i	0.926157	−	0.377139i	$-0.123092\pi$
$798$		0			0
$799$	−18843.1			−0.834319
$800$		0			0
$801$	17405.0			0.767759
$802$		0			0
$803$			19505.5i			0.857205i
$804$		0			0
$805$	−7494.01	+	3600.17i	−0.328111	+	0.157627i
$806$		0			0
$807$		−	1016.09i		−	0.0443222i
$808$		0			0
$809$	−17505.4			−0.760764			−0.380382	−	0.924829i	$-0.624208\pi$
$809$	−17505.4			−0.760764			−0.380382	+	0.924829i	$0.624208\pi$
$810$		0			0
$811$	−21749.0			−0.941690			−0.470845	−	0.882216i	$-0.656051\pi$
$811$	−21749.0			−0.941690			−0.470845	+	0.882216i	$0.656051\pi$
$812$		0			0
$813$			1239.34i			0.0534633i
$814$		0			0
$815$	13375.3	+	27841.5i	0.574865	+	1.19662i
$816$		0			0
$817$		−	13873.8i		−	0.594104i
$818$		0			0
$819$	17578.2			0.749978
$820$		0			0
$821$	27359.8			1.16305			0.581525	−	0.813529i	$-0.302456\pi$
$821$	27359.8			1.16305			0.581525	+	0.813529i	$0.302456\pi$
$822$		0			0
$823$		−	19012.1i		−	0.805249i	−0.915365	−	0.402625i	$-0.868098\pi$
$823$		−	19012.1i		−	0.805249i	0.915365	−	0.402625i	$-0.131902\pi$
$824$		0			0
$825$	2187.03	+	1750.89i	0.0922939	+	0.0738888i
$826$		0			0
$827$			20243.8i			0.851204i	0.904911	+	0.425602i	$0.139938\pi$
$827$			20243.8i			0.851204i	−0.904911	+	0.425602i	$0.860062\pi$
$828$		0			0
$829$	36927.4			1.54710			0.773548	−	0.633738i	$-0.218480\pi$
$829$	36927.4			1.54710			0.773548	+	0.633738i	$0.218480\pi$
$830$		0			0
$831$	−833.305			−0.0347859
$832$		0			0
$833$		−	6308.12i		−	0.262381i
$834$		0			0
$835$	−6511.12	−	13553.3i	−0.269852	−	0.561716i
$836$		0			0
$837$			2916.13i			0.120426i
$838$		0			0
$839$	−37067.9			−1.52530			−0.762650	−	0.646812i	$-0.776102\pi$
$839$	−37067.9			−1.52530			−0.762650	+	0.646812i	$0.776102\pi$
$840$		0			0
$841$	−24385.5			−0.999855
$842$		0			0
$843$		−	2170.97i		−	0.0886975i
$844$		0			0
$845$	8943.36	−	4296.45i	0.364096	−	0.174914i
$846$		0			0
$847$		−	4245.57i		−	0.172231i
$848$		0			0
$849$	−1194.85			−0.0483003
$850$		0			0
$851$	6358.84			0.256144
$852$		0			0
$853$			3194.55i			0.128229i	0.997943	+	0.0641145i	$0.0204223\pi$
$853$			3194.55i			0.128229i	−0.997943	+	0.0641145i	$0.979578\pi$
$854$		0			0
$855$	−10698.8	+	5139.78i	−0.427943	+	0.205587i
$856$		0			0
$857$			32759.1i			1.30575i	0.757466	+	0.652875i	$0.226437\pi$
$857$			32759.1i			1.30575i	−0.757466	+	0.652875i	$0.773563\pi$
$858$		0			0
$859$	11475.2			0.455795			0.227898	−	0.973685i	$-0.426815\pi$
$859$	11475.2			0.455795			0.227898	+	0.973685i	$0.426815\pi$
$860$		0			0
$861$	1000.49			0.0396011
$862$		0			0
$863$			16763.0i			0.661204i	0.943770	+	0.330602i	$0.107252\pi$
$863$			16763.0i			0.661204i	−0.943770	+	0.330602i	$0.892748\pi$
$864$		0			0
$865$	14610.4	+	30412.6i	0.574299	+	1.19544i
$866$		0			0
$867$		−	2150.00i		−	0.0842188i
$868$		0			0
$869$	−363.707			−0.0141978
$870$		0			0
$871$	54862.9			2.13428
$872$		0			0
$873$		−	21964.1i		−	0.851513i
$874$		0			0
$875$	16139.4	+	3732.05i	0.623554	+	0.144190i
$876$		0			0
$877$		−	14645.2i		−	0.563893i	−0.959430	−	0.281947i	$-0.909020\pi$
$877$		−	14645.2i		−	0.563893i	0.959430	−	0.281947i	$-0.0909801\pi$
$878$		0			0
$879$	−3265.24			−0.125294
$880$		0			0
$881$	−34249.7			−1.30976			−0.654882	−	0.755731i	$-0.727282\pi$
$881$	−34249.7			−1.30976			−0.654882	+	0.755731i	$0.727282\pi$
$882$		0			0
$883$			19054.4i			0.726197i	0.931751	+	0.363099i	$0.118281\pi$
$883$			19054.4i			0.726197i	−0.931751	+	0.363099i	$0.881719\pi$
$884$		0			0
$885$	1838.60	+	3827.17i	0.0698347	+	0.145366i
$886$		0			0
$887$			32595.7i			1.23389i	0.787008	+	0.616943i	$0.211629\pi$
$887$			32595.7i			1.23389i	−0.787008	+	0.616943i	$0.788371\pi$
$888$		0			0
$889$	−3009.42			−0.113535
$890$		0			0
$891$	−28975.3			−1.08946
$892$		0			0
$893$		−	24049.4i		−	0.901212i
$894$		0			0
$895$	23958.8	−	11510.0i	0.894810	−	0.429873i
$896$		0			0
$897$			1900.02i			0.0707243i
$898$		0			0
$899$	−186.952			−0.00693571
$900$		0			0
$901$	15219.0			0.562728
$902$		0			0
$903$			2255.59i			0.0831244i
$904$		0			0
$905$	−33933.9	+	16302.1i	−1.24641	+	0.598784i
$906$		0			0
$907$			18673.5i			0.683621i	0.939769	+	0.341811i	$0.111040\pi$
$907$			18673.5i			0.683621i	−0.939769	+	0.341811i	$0.888960\pi$
$908$		0			0
$909$	−41651.4			−1.51979
$910$		0			0
$911$	−45247.4			−1.64557			−0.822785	−	0.568353i	$-0.807581\pi$
$911$	−45247.4			−1.64557			−0.822785	+	0.568353i	$0.807581\pi$
$912$		0			0
$913$			24572.8i			0.890736i
$914$		0			0
$915$	427.330	+	889.516i	0.0154394	+	0.0321383i
$916$		0			0
$917$			3223.02i			0.116067i
$918$		0			0
$919$	37660.2			1.35179			0.675896	−	0.736997i	$-0.263757\pi$
$919$	37660.2			1.35179			0.675896	+	0.736997i	$0.263757\pi$
$920$		0			0
$921$	868.605			0.0310766
$922$		0			0
$923$			42682.4i			1.52211i
$924$		0			0
$925$	−9890.63	−	7918.26i	−0.351570	−	0.281460i
$926$		0			0
$927$		−	33408.3i		−	1.18368i
$928$		0			0
$929$	−5793.93			−0.204621			−0.102310	−	0.994753i	$-0.532624\pi$
$929$	−5793.93			−0.204621			−0.102310	+	0.994753i	$0.532624\pi$
$930$		0			0
$931$	8051.03			0.283418
$932$		0			0
$933$		−	1698.67i		−	0.0596057i
$934$		0			0
$935$	−6198.37	−	12902.3i	−0.216800	−	0.451285i
$936$		0			0
$937$			16162.2i			0.563496i	0.959488	+	0.281748i	$0.0909142\pi$
$937$			16162.2i			0.563496i	−0.959488	+	0.281748i	$0.909086\pi$
$938$		0			0
$939$	4099.02			0.142456
$940$		0			0
$941$	50202.2			1.73915			0.869577	−	0.493797i	$-0.164391\pi$
$941$	50202.2			1.73915			0.869577	+	0.493797i	$0.164391\pi$
$942$		0			0
$943$		−	9710.63i		−	0.335336i
$944$		0			0
$945$	3498.18	−	1680.55i	0.120419	−	0.0578500i
$946$		0			0
$947$			13964.1i			0.479167i	0.970876	+	0.239583i	$0.0770108\pi$
$947$			13964.1i			0.479167i	−0.970876	+	0.239583i	$0.922989\pi$
$948$		0			0
$949$	26357.7			0.901589
$950$		0			0
$951$	−3793.04			−0.129335
$952$		0			0
$953$			46121.1i			1.56769i	0.620956	+	0.783846i	$0.286745\pi$
$953$			46121.1i			1.56769i	−0.620956	+	0.783846i	$0.713255\pi$
$954$		0			0
$955$	50328.0	−	24177.9i	1.70531	−	0.819244i
$956$		0			0
$957$			42.0783i			0.00142132i
$958$		0			0
$959$	−18920.7			−0.637101
$960$		0			0
$961$	−19875.3			−0.667159
$962$		0			0
$963$			9457.42i			0.316471i
$964$		0			0
$965$	−24784.2	−	51590.0i	−0.826768	−	1.72098i
$966$		0			0
$967$			44011.4i			1.46361i	0.681514	+	0.731805i	$0.261322\pi$
$967$			44011.4i			1.46361i	−0.681514	+	0.731805i	$0.738678\pi$
$968$		0			0
$969$	−675.362			−0.0223898
$970$		0			0
$971$	−31284.0			−1.03393			−0.516967	−	0.856005i	$-0.672939\pi$
$971$	−31284.0			−1.03393			−0.516967	+	0.856005i	$0.672939\pi$
$972$		0			0
$973$		−	29600.6i		−	0.975286i
$974$		0			0
$975$	2365.97	−	2955.32i	0.0777146	−	0.0970727i
$976$		0			0
$977$			13753.6i			0.450377i	0.974315	+	0.225188i	$0.0722997\pi$
$977$			13753.6i			0.450377i	−0.974315	+	0.225188i	$0.927700\pi$
$978$		0			0
$979$	−26789.1			−0.874548
$980$		0			0
$981$	−13724.1			−0.446663
$982$		0			0
$983$			45210.8i			1.46694i	0.679722	+	0.733470i	$0.262100\pi$
$983$			45210.8i			1.46694i	−0.679722	+	0.733470i	$0.737900\pi$
$984$		0			0
$985$	7368.56	+	15338.2i	0.238357	+	0.496157i
$986$		0			0
$987$			3909.93i			0.126094i
$988$		0			0
$989$	21892.5			0.703884
$990$		0			0
$991$	−58560.9			−1.87714			−0.938572	−	0.345084i	$-0.887850\pi$
$991$	−58560.9			−1.87714			−0.938572	+	0.345084i	$0.887850\pi$
$992$		0			0
$993$			617.311i			0.0197279i
$994$		0			0
$995$	−50293.1	+	24161.2i	−1.60241	+	0.769810i
$996$		0			0
$997$		−	42394.2i		−	1.34668i	−0.739334	−	0.673339i	$-0.764859\pi$
$997$		−	42394.2i		−	1.34668i	0.739334	−	0.673339i	$-0.235141\pi$
$998$		0			0
$999$	−2968.29			−0.0940064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.4.c.a.129.9	✓	18
4.3	odd	2		640.4.c.b.129.10	yes	18
5.4	even	2	inner	640.4.c.a.129.10	yes	18
8.3	odd	2		640.4.c.c.129.9	yes	18
8.5	even	2		640.4.c.d.129.10	yes	18
20.19	odd	2		640.4.c.b.129.9	yes	18
40.19	odd	2		640.4.c.c.129.10	yes	18
40.29	even	2		640.4.c.d.129.9	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.4.c.a.129.9	✓	18	1.1	even	1	trivial
640.4.c.a.129.10	yes	18	5.4	even	2	inner
640.4.c.b.129.9	yes	18	20.19	odd	2
640.4.c.b.129.10	yes	18	4.3	odd	2
640.4.c.c.129.9	yes	18	8.3	odd	2
640.4.c.c.129.10	yes	18	40.19	odd	2
640.4.c.d.129.9	yes	18	40.29	even	2
640.4.c.d.129.10	yes	18	8.5	even	2

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	640.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.545319i q^{3} +(-10.0777 + 4.84141i) q^{5} -11.8531i q^{7} +26.7026 q^{9} -41.0997 q^{11} +55.5377i q^{13} +(2.64012 + 5.49558i) q^{15} -31.1506i q^{17} +39.7575 q^{19} -6.46373 q^{21} +62.7363i q^{23} +(78.1215 - 97.5809i) q^{25} -29.2851i q^{27} +1.87745 q^{29} -99.5774 q^{31} +22.4124i q^{33} +(57.3858 + 119.453i) q^{35} -101.358i q^{37} +30.2858 q^{39} -154.785 q^{41} -348.961i q^{43} +(-269.102 + 129.278i) q^{45} -604.903i q^{47} +202.504 q^{49} -16.9870 q^{51} +488.561i q^{53} +(414.191 - 198.980i) q^{55} -21.6805i q^{57} +696.407 q^{59} +161.860 q^{61} -316.509i q^{63} +(-268.881 - 559.694i) q^{65} -987.849i q^{67} +34.2113 q^{69} +768.530 q^{71} -474.591i q^{73} +(-53.2128 - 42.6011i) q^{75} +487.159i q^{77} +8.84939 q^{79} +705.001 q^{81} -597.884i q^{83} +(150.813 + 313.928i) q^{85} -1.02381i q^{87} +651.808 q^{89} +658.295 q^{91} +54.3015i q^{93} +(-400.665 + 192.482i) q^{95} -822.543i q^{97} -1097.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{5} - 162 q^{9} - 28 q^{11} + 4 q^{15} - 68 q^{19} - 136 q^{21} - 22 q^{25} - 340 q^{29} + 336 q^{31} - 236 q^{35} + 1000 q^{39} + 236 q^{41} + 90 q^{45} - 882 q^{49} - 48 q^{51} - 1532 q^{55}+ \cdots + 3628 q^{99}+O(q^{100}) \) 18 * q - 2 * q^5 - 162 * q^9 - 28 * q^11 + 4 * q^15 - 68 * q^19 - 136 * q^21 - 22 * q^25 - 340 * q^29 + 336 * q^31 - 236 * q^35 + 1000 * q^39 + 236 * q^41 + 90 * q^45 - 882 * q^49 - 48 * q^51 - 1532 * q^55 - 1756 * q^59 + 780 * q^61 + 104 * q^65 - 1016 * q^69 + 72 * q^71 + 3496 * q^75 - 1952 * q^79 + 2074 * q^81 + 664 * q^85 - 292 * q^89 - 520 * q^91 + 1612 * q^95 + 3628 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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