LMFDB - Embedded newform 140.4.e.c.29.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,4,Mod(29,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("140.29");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	140.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:	$8.26026740080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			29.3
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	140.29
Dual form			140.4.e.c.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} +(-7.57321 - 8.22474i) q^{5} +7.00000i q^{7} +21.0000 q^{9} +O(q^{10})$	$q+2.44949i q^{3} +(-7.57321 - 8.22474i) q^{5} +7.00000i q^{7} +21.0000 q^{9} +58.2929 q^{11} +70.3383i q^{13} +(20.1464 - 18.5505i) q^{15} +44.7878i q^{17} +26.8536 q^{19} -17.1464 q^{21} -18.8990i q^{23} +(-10.2929 + 124.576i) q^{25} +117.576i q^{27} +104.293 q^{29} +17.7071 q^{31} +142.788i q^{33} +(57.5732 - 53.0125i) q^{35} -299.576i q^{37} -172.293 q^{39} +467.221 q^{41} -34.2520i q^{43} +(-159.037 - 172.720i) q^{45} -199.464i q^{47} -49.0000 q^{49} -109.707 q^{51} +584.474i q^{53} +(-441.464 - 479.444i) q^{55} +65.7775i q^{57} -678.368 q^{59} -628.368 q^{61} +147.000i q^{63} +(578.514 - 532.687i) q^{65} -817.453i q^{67} +46.2929 q^{69} -520.293 q^{71} +417.950i q^{73} +(-305.146 - 25.2122i) q^{75} +408.050i q^{77} +375.321 q^{79} +279.000 q^{81} -367.004i q^{83} +(368.368 - 339.187i) q^{85} +255.464i q^{87} +414.486 q^{89} -492.368 q^{91} +43.3735i q^{93} +(-203.368 - 220.864i) q^{95} -762.990i q^{97} +1224.15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 84 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 84 * q^9	$ 4 q + 4 q^{5} + 84 q^{9} + 96 q^{11} + 12 q^{15} + 176 q^{19} + 96 q^{25} + 280 q^{29} + 208 q^{31} + 196 q^{35} - 552 q^{39} + 360 q^{41} + 84 q^{45} - 196 q^{49} - 576 q^{51} - 1080 q^{55} - 1136 q^{59} - 936 q^{61} + 668 q^{65} + 48 q^{69} - 1944 q^{71} - 1152 q^{75} - 1928 q^{79} + 1116 q^{81} - 104 q^{85} + 3304 q^{89} - 392 q^{91} + 764 q^{95} + 2016 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 84 * q^9 + 96 * q^11 + 12 * q^15 + 176 * q^19 + 96 * q^25 + 280 * q^29 + 208 * q^31 + 196 * q^35 - 552 * q^39 + 360 * q^41 + 84 * q^45 - 196 * q^49 - 576 * q^51 - 1080 * q^55 - 1136 * q^59 - 936 * q^61 + 668 * q^65 + 48 * q^69 - 1944 * q^71 - 1152 * q^75 - 1928 * q^79 + 1116 * q^81 - 104 * q^85 + 3304 * q^89 - 392 * q^91 + 764 * q^95 + 2016 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.44949i			0.471405i	0.971825	+	0.235702i	$0.0757390\pi$
$3$			2.44949i			0.471405i	−0.971825	+	0.235702i	$0.924261\pi$
$4$		0			0
$5$	−7.57321	−	8.22474i	−0.677369	−	0.735644i
$5$	−7.57321	−	8.22474i	−0.677369	−	0.735644i
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		0			0
$9$	21.0000			0.777778
$10$		0			0
$11$	58.2929			1.59781			0.798907	−	0.601454i	$-0.205412\pi$
$11$	58.2929			1.59781			0.798907	+	0.601454i	$0.205412\pi$
$12$		0			0
$13$			70.3383i			1.50064i	0.661075	+	0.750320i	$0.270101\pi$
$13$			70.3383i			1.50064i	−0.661075	+	0.750320i	$0.729899\pi$
$14$		0			0
$15$	20.1464	−	18.5505i	0.346786	−	0.319315i
$16$		0			0
$17$			44.7878i			0.638978i	0.947590	+	0.319489i	$0.103511\pi$
$17$			44.7878i			0.638978i	−0.947590	+	0.319489i	$0.896489\pi$
$18$		0			0
$19$	26.8536			0.324244			0.162122	−	0.986771i	$-0.448166\pi$
$19$	26.8536			0.324244			0.162122	+	0.986771i	$0.448166\pi$
$20$		0			0
$21$	−17.1464			−0.178174
$22$		0			0
$23$		−	18.8990i		−	0.171335i	−0.996324	−	0.0856676i	$-0.972698\pi$
$23$		−	18.8990i		−	0.171335i	0.996324	−	0.0856676i	$-0.0273023\pi$
$24$		0			0
$25$	−10.2929	+	124.576i	−0.0823429	+	0.996604i
$26$		0			0
$27$			117.576i			0.838052i
$28$		0			0
$29$	104.293			0.667817			0.333909	−	0.942605i	$-0.391632\pi$
$29$	104.293			0.667817			0.333909	+	0.942605i	$0.391632\pi$
$30$		0			0
$31$	17.7071			0.102590			0.0512951	−	0.998684i	$-0.483665\pi$
$31$	17.7071			0.102590			0.0512951	+	0.998684i	$0.483665\pi$
$32$		0			0
$33$			142.788i			0.753217i
$34$		0			0
$35$	57.5732	−	53.0125i	0.278047	−	0.256021i
$36$		0			0
$37$		−	299.576i		−	1.33108i	−0.746363	−	0.665539i	$-0.768202\pi$
$37$		−	299.576i		−	1.33108i	0.746363	−	0.665539i	$-0.231798\pi$
$38$		0			0
$39$	−172.293			−0.707409
$40$		0			0
$41$	467.221			1.77970			0.889850	−	0.456253i	$-0.150809\pi$
$41$	467.221			1.77970			0.889850	+	0.456253i	$0.150809\pi$
$42$		0			0
$43$		−	34.2520i		−	0.121474i	−0.998154	−	0.0607371i	$-0.980655\pi$
$43$		−	34.2520i		−	0.121474i	0.998154	−	0.0607371i	$-0.0193451\pi$
$44$		0			0
$45$	−159.037	−	172.720i	−0.526842	−	0.572167i
$46$		0			0
$47$		−	199.464i		−	0.619039i	−0.950893	−	0.309520i	$-0.899832\pi$
$47$		−	199.464i		−	0.619039i	0.950893	−	0.309520i	$-0.100168\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−109.707			−0.301217
$52$		0			0
$53$			584.474i			1.51479i	0.652958	+	0.757394i	$0.273528\pi$
$53$			584.474i			1.51479i	−0.652958	+	0.757394i	$0.726472\pi$
$54$		0			0
$55$	−441.464	−	479.444i	−1.08231	−	1.17542i
$56$		0			0
$57$			65.7775i			0.152850i
$58$		0			0
$59$	−678.368			−1.49688			−0.748440	−	0.663202i	$-0.769197\pi$
$59$	−678.368			−1.49688			−0.748440	+	0.663202i	$0.769197\pi$
$60$		0			0
$61$	−628.368			−1.31892			−0.659461	−	0.751739i	$-0.729215\pi$
$61$	−628.368			−1.31892			−0.659461	+	0.751739i	$0.729215\pi$
$62$		0			0
$63$			147.000i			0.293972i
$64$		0			0
$65$	578.514	−	532.687i	1.10394	−	1.01649i
$66$		0			0
$67$		−	817.453i		−	1.49056i	−0.666749	−	0.745282i	$-0.732315\pi$
$67$		−	817.453i		−	1.49056i	0.666749	−	0.745282i	$-0.267685\pi$
$68$		0			0
$69$	46.2929			0.0807682
$70$		0			0
$71$	−520.293			−0.869682			−0.434841	−	0.900507i	$-0.643195\pi$
$71$	−520.293			−0.869682			−0.434841	+	0.900507i	$0.643195\pi$
$72$		0			0
$73$			417.950i			0.670101i	0.942200	+	0.335050i	$0.108753\pi$
$73$			417.950i			0.670101i	−0.942200	+	0.335050i	$0.891247\pi$
$74$		0			0
$75$	−305.146	−	25.2122i	−0.469804	−	0.0388168i
$76$		0			0
$77$			408.050i			0.603917i
$78$		0			0
$79$	375.321			0.534518			0.267259	−	0.963625i	$-0.413882\pi$
$79$	375.321			0.534518			0.267259	+	0.963625i	$0.413882\pi$
$80$		0			0
$81$	279.000			0.382716
$82$		0			0
$83$		−	367.004i		−	0.485348i	−0.970108	−	0.242674i	$-0.921975\pi$
$83$		−	367.004i		−	0.485348i	0.970108	−	0.242674i	$-0.0780245\pi$
$84$		0			0
$85$	368.368	−	339.187i	0.470060	−	0.432824i
$86$		0			0
$87$			255.464i			0.314812i
$88$		0			0
$89$	414.486			0.493656			0.246828	−	0.969059i	$-0.420612\pi$
$89$	414.486			0.493656			0.246828	+	0.969059i	$0.420612\pi$
$90$		0			0
$91$	−492.368			−0.567189
$92$		0			0
$93$			43.3735i			0.0483615i
$94$		0			0
$95$	−203.368	−	220.864i	−0.219633	−	0.238528i
$96$		0			0
$97$		−	762.990i		−	0.798659i	−0.916808	−	0.399329i	$-0.869243\pi$
$97$		−	762.990i		−	0.798659i	0.916808	−	0.399329i	$-0.130757\pi$
$98$		0			0
$99$	1224.15			1.24274
$100$		0			0
$101$	−223.882			−0.220565			−0.110283	−	0.993900i	$-0.535176\pi$
$101$	−223.882			−0.220565			−0.110283	+	0.993900i	$0.535176\pi$
$102$		0			0
$103$			1966.25i			1.88098i	0.339826	+	0.940488i	$0.389632\pi$
$103$			1966.25i			1.88098i	−0.339826	+	0.940488i	$0.610368\pi$
$104$		0			0
$105$	129.854	+	141.025i	0.120690	+	0.131073i
$106$		0			0
$107$		−	985.698i		−	0.890570i	−0.895389	−	0.445285i	$-0.853102\pi$
$107$		−	985.698i		−	0.890570i	0.895389	−	0.445285i	$-0.146898\pi$
$108$		0			0
$109$	−958.736			−0.842479			−0.421240	−	0.906949i	$-0.638405\pi$
$109$	−958.736			−0.842479			−0.421240	+	0.906949i	$0.638405\pi$
$110$		0			0
$111$	733.807			0.627476
$112$		0			0
$113$			1689.53i			1.40653i	0.710928	+	0.703265i	$0.248275\pi$
$113$			1689.53i			1.40653i	−0.710928	+	0.703265i	$0.751725\pi$
$114$		0			0
$115$	−155.439	+	143.126i	−0.126042	+	0.116057i
$116$		0			0
$117$			1477.10i			1.16716i
$118$		0			0
$119$	−313.514			−0.241511
$120$		0			0
$121$	2067.06			1.55301
$122$		0			0
$123$			1144.45i			0.838959i
$124$		0			0
$125$	1102.55	−	858.781i	0.788922	−	0.614494i
$126$		0			0
$127$		−	2572.23i		−	1.79723i	−0.438736	−	0.898616i	$-0.644574\pi$
$127$		−	2572.23i		−	1.79723i	0.438736	−	0.898616i	$-0.355426\pi$
$128$		0			0
$129$	83.9000			0.0572634
$130$		0			0
$131$	−569.882			−0.380083			−0.190041	−	0.981776i	$-0.560862\pi$
$131$	−569.882			−0.380083			−0.190041	+	0.981776i	$0.560862\pi$
$132$		0			0
$133$			187.975i			0.122553i
$134$		0			0
$135$	967.029	−	890.424i	0.616508	−	0.567671i
$136$		0			0
$137$			2604.99i			1.62452i	0.583297	+	0.812259i	$0.301762\pi$
$137$			2604.99i			1.62452i	−0.583297	+	0.812259i	$0.698238\pi$
$138$		0			0
$139$	1165.40			0.711134			0.355567	−	0.934651i	$-0.384288\pi$
$139$	1165.40			0.711134			0.355567	+	0.934651i	$0.384288\pi$
$140$		0			0
$141$	488.586			0.291818
$142$		0			0
$143$			4100.22i			2.39774i
$144$		0			0
$145$	−789.832	−	857.782i	−0.452359	−	0.491275i
$146$		0			0
$147$		−	120.025i		−	0.0673435i
$148$		0			0
$149$	2034.50			1.11861			0.559304	−	0.828962i	$-0.311068\pi$
$149$	2034.50			1.11861			0.559304	+	0.828962i	$0.311068\pi$
$150$		0			0
$151$	−2551.23			−1.37494			−0.687470	−	0.726212i	$-0.741279\pi$
$151$	−2551.23			−1.37494			−0.687470	+	0.726212i	$0.741279\pi$
$152$		0			0
$153$			940.543i			0.496983i
$154$		0			0
$155$	−134.100	−	145.637i	−0.0694914	−	0.0754698i
$156$		0			0
$157$			149.130i			0.0758082i	0.999281	+	0.0379041i	$0.0120681\pi$
$157$			149.130i			0.0758082i	−0.999281	+	0.0379041i	$0.987932\pi$
$158$		0			0
$159$	−1431.66			−0.714078
$160$		0			0
$161$	132.293			0.0647586
$162$		0			0
$163$		−	3162.83i		−	1.51983i	−0.650023	−	0.759915i	$-0.725241\pi$
$163$		−	3162.83i		−	1.51983i	0.650023	−	0.759915i	$-0.274759\pi$
$164$		0			0
$165$	1174.39	−	1081.36i	0.554099	−	0.510206i
$166$		0			0
$167$		−	979.446i		−	0.453843i	−0.973913	−	0.226922i	$-0.927134\pi$
$167$		−	979.446i		−	0.453843i	0.973913	−	0.226922i	$-0.0728661\pi$
$168$		0			0
$169$	−2750.47			−1.25192
$170$		0			0
$171$	563.925			0.252190
$172$		0			0
$173$		−	2591.20i		−	1.13876i	−0.822075	−	0.569380i	$-0.807183\pi$
$173$		−	2591.20i		−	1.13876i	0.822075	−	0.569380i	$-0.192817\pi$
$174$		0			0
$175$	−872.029	−	72.0500i	−0.376681	−	0.0311227i
$176$		0			0
$177$		−	1661.66i		−	0.705636i
$178$		0			0
$179$	1144.59			0.477935			0.238967	−	0.971028i	$-0.423191\pi$
$179$	1144.59			0.477935			0.238967	+	0.971028i	$0.423191\pi$
$180$		0			0
$181$	−3224.27			−1.32408			−0.662039	−	0.749470i	$-0.730308\pi$
$181$	−3224.27			−1.32408			−0.662039	+	0.749470i	$0.730308\pi$
$182$		0			0
$183$		−	1539.18i		−	0.621746i
$184$		0			0
$185$	−2463.93	+	2268.75i	−0.979199	+	0.901631i
$186$		0			0
$187$			2610.81i			1.02097i
$188$		0			0
$189$	−823.029			−0.316754
$190$		0			0
$191$	3859.24			1.46201			0.731007	−	0.682370i	$-0.239051\pi$
$191$	3859.24			1.46201			0.731007	+	0.682370i	$0.239051\pi$
$192$		0			0
$193$		−	1186.89i		−	0.442663i	−0.975199	−	0.221331i	$-0.928960\pi$
$193$		−	1186.89i		−	0.442663i	0.975199	−	0.221331i	$-0.0710402\pi$
$194$		0			0
$195$	1304.81	+	1417.06i	0.479177	+	0.520401i
$196$		0			0
$197$		−	2545.22i		−	0.920503i	−0.887789	−	0.460251i	$-0.847759\pi$
$197$		−	2545.22i		−	0.920503i	0.887789	−	0.460251i	$-0.152241\pi$
$198$		0			0
$199$	−2652.25			−0.944789			−0.472395	−	0.881387i	$-0.656610\pi$
$199$	−2652.25			−0.944789			−0.472395	+	0.881387i	$0.656610\pi$
$200$		0			0
$201$	2002.34			0.702659
$202$		0			0
$203$			730.050i			0.252411i
$204$		0			0
$205$	−3538.37	−	3842.78i	−1.20551	−	1.30923i
$206$		0			0
$207$		−	396.879i		−	0.133261i
$208$		0			0
$209$	1565.37			0.518081
$210$		0			0
$211$	1360.11			0.443764			0.221882	−	0.975074i	$-0.428780\pi$
$211$	1360.11			0.443764			0.221882	+	0.975074i	$0.428780\pi$
$212$		0			0
$213$		−	1274.45i		−	0.409972i
$214$		0			0
$215$	−281.714	+	259.398i	−0.0893616	+	0.0822828i
$216$		0			0
$217$			123.950i			0.0387755i
$218$		0			0
$219$	−1023.76			−0.315888
$220$		0			0
$221$	−3150.29			−0.958876
$222$		0			0
$223$		−	2152.97i		−	0.646517i	−0.946311	−	0.323259i	$-0.895222\pi$
$223$		−	2152.97i		−	0.646517i	0.946311	−	0.323259i	$-0.104778\pi$
$224$		0			0
$225$	−216.150	+	2616.09i	−0.0640444	+	0.775136i
$226$		0			0
$227$		−	2814.95i		−	0.823062i	−0.911396	−	0.411531i	$-0.864994\pi$
$227$		−	2814.95i		−	0.823062i	0.911396	−	0.411531i	$-0.135006\pi$
$228$		0			0
$229$	−5309.05			−1.53202			−0.766009	−	0.642830i	$-0.777760\pi$
$229$	−5309.05			−1.53202			−0.766009	+	0.642830i	$0.777760\pi$
$230$		0			0
$231$	−999.514			−0.284689
$232$		0			0
$233$		−	441.439i		−	0.124119i	−0.998072	−	0.0620593i	$-0.980233\pi$
$233$		−	441.439i		−	0.124119i	0.998072	−	0.0620593i	$-0.0197668\pi$
$234$		0			0
$235$	−1640.54	+	1510.59i	−0.455392	+	0.419318i
$236$		0			0
$237$			919.346i			0.251974i
$238$		0			0
$239$	4768.70			1.29063			0.645317	−	0.763915i	$-0.276725\pi$
$239$	4768.70			1.29063			0.645317	+	0.763915i	$0.276725\pi$
$240$		0			0
$241$	−488.207			−0.130490			−0.0652452	−	0.997869i	$-0.520783\pi$
$241$	−488.207			−0.130490			−0.0652452	+	0.997869i	$0.520783\pi$
$242$		0			0
$243$			3857.95i			1.01847i
$244$		0			0
$245$	371.087	+	403.012i	0.0967670	+	0.105092i
$246$		0			0
$247$			1888.83i			0.486573i
$248$		0			0
$249$	898.971			0.228795
$250$		0			0
$251$	−2147.45			−0.540024			−0.270012	−	0.962857i	$-0.587028\pi$
$251$	−2147.45			−0.540024			−0.270012	+	0.962857i	$0.587028\pi$
$252$		0			0
$253$		−	1101.68i		−	0.273762i
$254$		0			0
$255$	830.836	+	902.313i	0.204035	+	0.221588i
$256$		0			0
$257$			724.270i			0.175793i	0.996130	+	0.0878964i	$0.0280144\pi$
$257$			724.270i			0.175793i	−0.996130	+	0.0878964i	$0.971986\pi$
$258$		0			0
$259$	2097.03			0.503100
$260$		0			0
$261$	2190.15			0.519413
$262$		0			0
$263$			1383.89i			0.324465i	0.986753	+	0.162233i	$0.0518695\pi$
$263$			1383.89i			0.324465i	−0.986753	+	0.162233i	$0.948130\pi$
$264$		0			0
$265$	4807.15	−	4426.35i	1.11434	−	1.02607i
$266$		0			0
$267$			1015.28i			0.232712i
$268$		0			0
$269$	721.539			0.163543			0.0817714	−	0.996651i	$-0.473942\pi$
$269$	721.539			0.163543			0.0817714	+	0.996651i	$0.473942\pi$
$270$		0			0
$271$	2220.00			0.497621			0.248811	−	0.968552i	$-0.419960\pi$
$271$	2220.00			0.497621			0.248811	+	0.968552i	$0.419960\pi$
$272$		0			0
$273$		−	1206.05i		−	0.267375i
$274$		0			0
$275$	−600.000	+	7261.86i	−0.131569	+	1.59239i
$276$		0			0
$277$			502.685i			0.109038i	0.998513	+	0.0545188i	$0.0173625\pi$
$277$			502.685i			0.109038i	−0.998513	+	0.0545188i	$0.982638\pi$
$278$		0			0
$279$	371.850			0.0797924
$280$		0			0
$281$	−2631.09			−0.558567			−0.279284	−	0.960209i	$-0.590097\pi$
$281$	−2631.09			−0.558567			−0.279284	+	0.960209i	$0.590097\pi$
$282$		0			0
$283$		−	7511.02i		−	1.57768i	−0.614598	−	0.788841i	$-0.710682\pi$
$283$		−	7511.02i		−	1.57768i	0.614598	−	0.788841i	$-0.289318\pi$
$284$		0			0
$285$	541.004	−	498.147i	0.112443	−	0.103536i
$286$		0			0
$287$			3270.55i			0.672664i
$288$		0			0
$289$	2907.06			0.591707
$290$		0			0
$291$	1868.94			0.376491
$292$		0			0
$293$		−	6353.76i		−	1.26686i	−0.773799	−	0.633431i	$-0.781646\pi$
$293$		−	6353.76i		−	1.26686i	0.773799	−	0.633431i	$-0.218354\pi$
$294$		0			0
$295$	5137.42	+	5579.40i	1.01394	+	1.10117i
$296$		0			0
$297$			6853.81i			1.33905i
$298$		0			0
$299$	1329.32			0.257113
$300$		0			0
$301$	239.764			0.0459129
$302$		0			0
$303$		−	548.397i		−	0.103976i
$304$		0			0
$305$	4758.76	+	5168.17i	0.893397	+	0.970257i
$306$		0			0
$307$			5284.47i			0.982413i	0.871043	+	0.491206i	$0.163444\pi$
$307$			5284.47i			0.982413i	−0.871043	+	0.491206i	$0.836556\pi$
$308$		0			0
$309$	−4816.31			−0.886701
$310$		0			0
$311$	10469.6			1.90892			0.954461	−	0.298335i	$-0.0964312\pi$
$311$	10469.6			1.90892			0.954461	+	0.298335i	$0.0964312\pi$
$312$		0			0
$313$		−	5757.07i		−	1.03965i	−0.854274	−	0.519823i	$-0.825998\pi$
$313$		−	5757.07i		−	1.03965i	0.854274	−	0.519823i	$-0.174002\pi$
$314$		0			0
$315$	1209.04	−	1113.26i	0.216259	−	0.199128i
$316$		0			0
$317$			4547.68i			0.805752i	0.915255	+	0.402876i	$0.131989\pi$
$317$			4547.68i			0.805752i	−0.915255	+	0.402876i	$0.868011\pi$
$318$		0			0
$319$	6079.53			1.06705
$320$		0			0
$321$	2414.46			0.419819
$322$		0			0
$323$			1202.71i			0.207185i
$324$		0			0
$325$	−8762.42	−	723.982i	−1.49554	−	0.123567i
$326$		0			0
$327$		−	2348.41i		−	0.397148i
$328$		0			0
$329$	1396.25			0.233975
$330$		0			0
$331$	−7106.44			−1.18008			−0.590038	−	0.807376i	$-0.700887\pi$
$331$	−7106.44			−1.18008			−0.590038	+	0.807376i	$0.700887\pi$
$332$		0			0
$333$		−	6291.09i		−	1.03528i
$334$		0			0
$335$	−6723.34	+	6190.75i	−1.09652	+	1.00966i
$336$		0			0
$337$			8724.75i			1.41029i	0.709064	+	0.705144i	$0.249118\pi$
$337$			8724.75i			1.41029i	−0.709064	+	0.705144i	$0.750882\pi$
$338$		0			0
$339$	−4138.49			−0.663044
$340$		0			0
$341$	1032.20			0.163920
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		0			0
$345$	−350.586	−	380.747i	−0.0547099	−	0.0594166i
$346$		0			0
$347$		−	11267.1i		−	1.74308i	−0.490323	−	0.871541i	$-0.663121\pi$
$347$		−	11267.1i		−	1.74308i	0.490323	−	0.871541i	$-0.336879\pi$
$348$		0			0
$349$	−2836.72			−0.435089			−0.217544	−	0.976050i	$-0.569805\pi$
$349$	−2836.72			−0.435089			−0.217544	+	0.976050i	$0.569805\pi$
$350$		0			0
$351$	−8270.06			−1.25762
$352$		0			0
$353$		−	190.544i		−	0.0287298i	−0.999897	−	0.0143649i	$-0.995427\pi$
$353$		−	190.544i		−	0.0287298i	0.999897	−	0.0143649i	$-0.00457265\pi$
$354$		0			0
$355$	3940.29	+	4279.28i	0.589095	+	0.639776i
$356$		0			0
$357$		−	767.950i		−	0.113849i
$358$		0			0
$359$	−9442.37			−1.38816			−0.694080	−	0.719898i	$-0.744189\pi$
$359$	−9442.37			−1.38816			−0.694080	+	0.719898i	$0.744189\pi$
$360$		0			0
$361$	−6137.89			−0.894866
$362$		0			0
$363$			5063.24i			0.732096i
$364$		0			0
$365$	3437.53	−	3165.22i	0.492955	−	0.453905i
$366$		0			0
$367$		−	8614.90i		−	1.22532i	−0.790345	−	0.612662i	$-0.790099\pi$
$367$		−	8614.90i		−	1.22532i	0.790345	−	0.612662i	$-0.209901\pi$
$368$		0			0
$369$	9811.65			1.38421
$370$		0			0
$371$	−4091.32			−0.572536
$372$		0			0
$373$			1059.49i			0.147073i	0.997293	+	0.0735365i	$0.0234286\pi$
$373$			1059.49i			0.147073i	−0.997293	+	0.0735365i	$0.976571\pi$
$374$		0			0
$375$	2103.57	+	2700.69i	0.289675	+	0.371901i
$376$		0			0
$377$			7335.78i			1.00215i
$378$		0			0
$379$	7641.05			1.03561			0.517803	−	0.855500i	$-0.326750\pi$
$379$	7641.05			1.03561			0.517803	+	0.855500i	$0.326750\pi$
$380$		0			0
$381$	6300.65			0.847223
$382$		0			0
$383$		−	10343.3i		−	1.37994i	−0.723837	−	0.689971i	$-0.757623\pi$
$383$		−	10343.3i		−	1.37994i	0.723837	−	0.689971i	$-0.242377\pi$
$384$		0			0
$385$	3356.11	−	3090.25i	0.444268	−	0.409075i
$386$		0			0
$387$		−	719.293i		−	0.0944799i
$388$		0			0
$389$	−2483.32			−0.323675			−0.161837	−	0.986817i	$-0.551742\pi$
$389$	−2483.32			−0.323675			−0.161837	+	0.986817i	$0.551742\pi$
$390$		0			0
$391$	846.443			0.109479
$392$		0			0
$393$		−	1395.92i		−	0.179173i
$394$		0			0
$395$	−2842.39	−	3086.92i	−0.362066	−	0.393215i
$396$		0			0
$397$		−	669.676i		−	0.0846601i	−0.999104	−	0.0423301i	$-0.986522\pi$
$397$		−	669.676i		−	0.0846601i	0.999104	−	0.0423301i	$-0.0134781\pi$
$398$		0			0
$399$	−460.443			−0.0577719
$400$		0			0
$401$	−2600.53			−0.323851			−0.161925	−	0.986803i	$-0.551770\pi$
$401$	−2600.53			−0.323851			−0.161925	+	0.986803i	$0.551770\pi$
$402$		0			0
$403$			1245.49i			0.153951i
$404$		0			0
$405$	−2112.93	−	2294.70i	−0.259240	−	0.281543i
$406$		0			0
$407$		−	17463.1i		−	2.12682i
$408$		0			0
$409$	7226.29			0.873636			0.436818	−	0.899550i	$-0.356105\pi$
$409$	7226.29			0.873636			0.436818	+	0.899550i	$0.356105\pi$
$410$		0			0
$411$	−6380.89			−0.765805
$412$		0			0
$413$		−	4748.57i		−	0.565768i
$414$		0			0
$415$	−3018.51	+	2779.40i	−0.357043	+	0.328760i
$416$		0			0
$417$			2854.63i			0.335232i
$418$		0			0
$419$	−8346.11			−0.973113			−0.486556	−	0.873649i	$-0.661747\pi$
$419$	−8346.11			−0.973113			−0.486556	+	0.873649i	$0.661747\pi$
$420$		0			0
$421$	−15721.8			−1.82004			−0.910019	−	0.414567i	$-0.863933\pi$
$421$	−15721.8			−1.82004			−0.910019	+	0.414567i	$0.863933\pi$
$422$		0			0
$423$		−	4188.75i		−	0.481475i
$424$		0			0
$425$	−5579.46	−	460.994i	−0.636808	−	0.0526153i
$426$		0			0
$427$		−	4398.57i		−	0.498506i
$428$		0			0
$429$	−10043.4			−1.13031
$430$		0			0
$431$	4199.96			0.469385			0.234692	−	0.972070i	$-0.424592\pi$
$431$	4199.96			0.469385			0.234692	+	0.972070i	$0.424592\pi$
$432$		0			0
$433$		−	88.8960i		−	0.00986621i	−0.999988	−	0.00493310i	$-0.998430\pi$
$433$		−	88.8960i		−	0.00986621i	0.999988	−	0.00493310i	$-0.00157026\pi$
$434$		0			0
$435$	2101.13	−	1934.69i	0.231589	−	0.213244i
$436$		0			0
$437$		−	507.505i		−	0.0555544i
$438$		0			0
$439$	−6352.53			−0.690637			−0.345318	−	0.938486i	$-0.612229\pi$
$439$	−6352.53			−0.690637			−0.345318	+	0.938486i	$0.612229\pi$
$440$		0			0
$441$	−1029.00			−0.111111
$442$		0			0
$443$			7517.73i			0.806271i	0.915140	+	0.403136i	$0.132080\pi$
$443$			7517.73i			0.806271i	−0.915140	+	0.403136i	$0.867920\pi$
$444$		0			0
$445$	−3138.99	−	3409.04i	−0.334387	−	0.363155i
$446$		0			0
$447$			4983.49i			0.527317i
$448$		0			0
$449$	−444.286			−0.0466974			−0.0233487	−	0.999727i	$-0.507433\pi$
$449$	−444.286			−0.0466974			−0.0233487	+	0.999727i	$0.507433\pi$
$450$		0			0
$451$	27235.7			2.84363
$452$		0			0
$453$		−	6249.21i		−	0.648153i
$454$		0			0
$455$	3728.81	+	4049.60i	0.384196	+	0.417249i
$456$		0			0
$457$			16815.5i			1.72122i	0.509266	+	0.860609i	$0.329917\pi$
$457$			16815.5i			1.72122i	−0.509266	+	0.860609i	$0.670083\pi$
$458$		0			0
$459$	−5265.94			−0.535497
$460$		0			0
$461$	−6995.01			−0.706703			−0.353352	−	0.935491i	$-0.614958\pi$
$461$	−6995.01			−0.706703			−0.353352	+	0.935491i	$0.614958\pi$
$462$		0			0
$463$		−	6615.96i		−	0.664081i	−0.943265	−	0.332041i	$-0.892263\pi$
$463$		−	6615.96i		−	0.664081i	0.943265	−	0.332041i	$-0.107737\pi$
$464$		0			0
$465$	356.736	−	328.477i	0.0355768	−	0.0327586i
$466$		0			0
$467$		−	6538.89i		−	0.647931i	−0.946069	−	0.323965i	$-0.894984\pi$
$467$		−	6538.89i		−	0.647931i	0.946069	−	0.323965i	$-0.105016\pi$
$468$		0			0
$469$	5722.17			0.563380
$470$		0			0
$471$	−365.293			−0.0357363
$472$		0			0
$473$		−	1996.65i		−	0.194093i
$474$		0			0
$475$	−276.400	+	3345.30i	−0.0266992	+	0.323143i
$476$		0			0
$477$			12274.0i			1.17817i
$478$		0			0
$479$	−10843.7			−1.03437			−0.517185	−	0.855874i	$-0.673020\pi$
$479$	−10843.7			−1.03437			−0.517185	+	0.855874i	$0.673020\pi$
$480$		0			0
$481$	21071.6			1.99747
$482$		0			0
$483$			324.050i			0.0305275i
$484$		0			0
$485$	−6275.40	+	5778.29i	−0.587528	+	0.540986i
$486$		0			0
$487$			7911.46i			0.736145i	0.929797	+	0.368073i	$0.119982\pi$
$487$			7911.46i			0.736145i	−0.929797	+	0.368073i	$0.880018\pi$
$488$		0			0
$489$	7747.33			0.716455
$490$		0			0
$491$	−7998.03			−0.735124			−0.367562	−	0.929999i	$-0.619807\pi$
$491$	−7998.03			−0.735124			−0.367562	+	0.929999i	$0.619807\pi$
$492$		0			0
$493$			4671.04i			0.426720i
$494$		0			0
$495$	−9270.75	−	10068.3i	−0.841796	−	0.914217i
$496$		0			0
$497$		−	3642.05i		−	0.328709i
$498$		0			0
$499$	−4573.79			−0.410322			−0.205161	−	0.978728i	$-0.565772\pi$
$499$	−4573.79			−0.410322			−0.205161	+	0.978728i	$0.565772\pi$
$500$		0			0
$501$	2399.14			0.213944
$502$		0			0
$503$			7902.19i			0.700479i	0.936660	+	0.350240i	$0.113900\pi$
$503$			7902.19i			0.700479i	−0.936660	+	0.350240i	$0.886100\pi$
$504$		0			0
$505$	1695.51	+	1841.37i	0.149404	+	0.162258i
$506$		0			0
$507$		−	6737.25i		−	0.590161i
$508$		0			0
$509$	19035.2			1.65761			0.828803	−	0.559540i	$-0.189022\pi$
$509$	19035.2			1.65761			0.828803	+	0.559540i	$0.189022\pi$
$510$		0			0
$511$	−2925.65			−0.253274
$512$		0			0
$513$			3157.32i			0.271733i
$514$		0			0
$515$	16171.9	−	14890.8i	1.38373	−	1.27411i
$516$		0			0
$517$		−	11627.3i		−	0.989110i
$518$		0			0
$519$	6347.12			0.536816
$520$		0			0
$521$	20280.6			1.70539			0.852695	−	0.522409i	$-0.174967\pi$
$521$	20280.6			1.70539			0.852695	+	0.522409i	$0.174967\pi$
$522$		0			0
$523$			22422.8i			1.87473i	0.348355	+	0.937363i	$0.386740\pi$
$523$			22422.8i			1.87473i	−0.348355	+	0.937363i	$0.613260\pi$
$524$		0			0
$525$	176.486	−	2136.02i	0.0146714	−	0.177569i
$526$		0			0
$527$			793.063i			0.0655529i
$528$		0			0
$529$	11809.8			0.970644
$530$		0			0
$531$	−14245.7			−1.16424
$532$		0			0
$533$			32863.5i			2.67069i
$534$		0			0
$535$	−8107.11	+	7464.90i	−0.655142	+	0.603245i
$536$		0			0
$537$			2803.65i			0.225301i
$538$		0			0
$539$	−2856.35			−0.228259
$540$		0			0
$541$	17136.9			1.36188			0.680938	−	0.732341i	$-0.261572\pi$
$541$	17136.9			1.36188			0.680938	+	0.732341i	$0.261572\pi$
$542$		0			0
$543$		−	7897.81i		−	0.624176i
$544$		0			0
$545$	7260.71	+	7885.36i	0.570669	+	0.619764i
$546$		0			0
$547$			531.580i			0.0415516i	0.999784	+	0.0207758i	$0.00661362\pi$
$547$			531.580i			0.0415516i	−0.999784	+	0.0207758i	$0.993386\pi$
$548$		0			0
$549$	−13195.7			−1.02583
$550$		0			0
$551$	2800.64			0.216536
$552$		0			0
$553$			2627.25i			0.202029i
$554$		0			0
$555$	−5557.28	−	6035.38i	−0.425033	−	0.461599i
$556$		0			0
$557$		−	3819.56i		−	0.290556i	−0.989391	−	0.145278i	$-0.953592\pi$
$557$		−	3819.56i		−	0.290556i	0.989391	−	0.145278i	$-0.0464077\pi$
$558$		0			0
$559$	2409.23			0.182289
$560$		0			0
$561$	−6395.14			−0.481289
$562$		0			0
$563$			1781.33i			0.133346i	0.997775	+	0.0666732i	$0.0212385\pi$
$563$			1781.33i			0.133346i	−0.997775	+	0.0666732i	$0.978762\pi$
$564$		0			0
$565$	13896.0	−	12795.2i	1.03470	−	0.952739i
$566$		0			0
$567$			1953.00i			0.144653i
$568$		0			0
$569$	−4639.53			−0.341826			−0.170913	−	0.985286i	$-0.554672\pi$
$569$	−4639.53			−0.341826			−0.170913	+	0.985286i	$0.554672\pi$
$570$		0			0
$571$	−25750.9			−1.88729			−0.943643	−	0.330966i	$-0.892626\pi$
$571$	−25750.9			−1.88729			−0.943643	+	0.330966i	$0.892626\pi$
$572$		0			0
$573$			9453.16i			0.689200i
$574$		0			0
$575$	2354.35	+	194.524i	0.170753	+	0.0141082i
$576$		0			0
$577$		−	640.189i		−	0.0461896i	−0.999733	−	0.0230948i	$-0.992648\pi$
$577$		−	640.189i		−	0.0461896i	0.999733	−	0.0230948i	$-0.00735196\pi$
$578$		0			0
$579$	2907.26			0.208673
$580$		0			0
$581$	2569.02			0.183444
$582$		0			0
$583$			34070.7i			2.42035i
$584$		0			0
$585$	12148.8	−	11186.4i	0.858617	−	0.790601i
$586$		0			0
$587$		−	27125.5i		−	1.90731i	−0.300909	−	0.953653i	$-0.597290\pi$
$587$		−	27125.5i		−	1.90731i	0.300909	−	0.953653i	$-0.402710\pi$
$588$		0			0
$589$	475.500			0.0332642
$590$		0			0
$591$	6234.48			0.433929
$592$		0			0
$593$			1298.29i			0.0899063i	0.998989	+	0.0449531i	$0.0143138\pi$
$593$			1298.29i			0.0899063i	−0.998989	+	0.0449531i	$0.985686\pi$
$594$		0			0
$595$	2374.31	+	2578.57i	0.163592	+	0.177666i
$596$		0			0
$597$		−	6496.66i		−	0.445378i
$598$		0			0
$599$	7020.52			0.478883			0.239441	−	0.970911i	$-0.423036\pi$
$599$	7020.52			0.478883			0.239441	+	0.970911i	$0.423036\pi$
$600$		0			0
$601$	−8642.67			−0.586592			−0.293296	−	0.956022i	$-0.594752\pi$
$601$	−8642.67			−0.586592			−0.293296	+	0.956022i	$0.594752\pi$
$602$		0			0
$603$		−	17166.5i		−	1.15933i
$604$		0			0
$605$	−15654.3	−	17001.0i	−1.05196	−	1.14246i
$606$		0			0
$607$			3806.85i			0.254556i	0.991867	+	0.127278i	$0.0406240\pi$
$607$			3806.85i			0.254556i	−0.991867	+	0.127278i	$0.959376\pi$
$608$		0			0
$609$	−1788.25			−0.118988
$610$		0			0
$611$	14030.0			0.928956
$612$		0			0
$613$			13895.2i			0.915533i	0.889073	+	0.457766i	$0.151350\pi$
$613$			13895.2i			0.915533i	−0.889073	+	0.457766i	$0.848650\pi$
$614$		0			0
$615$	9412.84	−	8667.20i	0.617175	−	0.568285i
$616$		0			0
$617$			9322.18i			0.608261i	0.952630	+	0.304130i	$0.0983659\pi$
$617$			9322.18i			0.608261i	−0.952630	+	0.304130i	$0.901634\pi$
$618$		0			0
$619$	21665.3			1.40679			0.703396	−	0.710799i	$-0.251666\pi$
$619$	21665.3			1.40679			0.703396	+	0.710799i	$0.251666\pi$
$620$		0			0
$621$	2222.06			0.143588
$622$		0			0
$623$			2901.40i			0.186584i
$624$		0			0
$625$	−15413.1	−	2564.48i	−0.986439	−	0.164126i
$626$		0			0
$627$			3834.36i			0.244226i
$628$		0			0
$629$	13417.3			0.850530
$630$		0			0
$631$	13570.1			0.856132			0.428066	−	0.903748i	$-0.359195\pi$
$631$	13570.1			0.856132			0.428066	+	0.903748i	$0.359195\pi$
$632$		0			0
$633$			3331.59i			0.209192i
$634$		0			0
$635$	−21155.9	+	19480.0i	−1.32212	+	1.21739i
$636$		0			0
$637$		−	3446.57i		−	0.214377i
$638$		0			0
$639$	−10926.1			−0.676419
$640$		0			0
$641$	−27598.8			−1.70060			−0.850301	−	0.526297i	$-0.823580\pi$
$641$	−27598.8			−1.70060			−0.850301	+	0.526297i	$0.823580\pi$
$642$		0			0
$643$			4977.02i			0.305248i	0.988284	+	0.152624i	$0.0487724\pi$
$643$			4977.02i			0.305248i	−0.988284	+	0.152624i	$0.951228\pi$
$644$		0			0
$645$	−635.393	−	690.056i	−0.0387885	−	0.0421255i
$646$		0			0
$647$		−	9309.09i		−	0.565654i	−0.959171	−	0.282827i	$-0.908728\pi$
$647$		−	9309.09i		−	0.565654i	0.959171	−	0.282827i	$-0.0912722\pi$
$648$		0			0
$649$	−39544.0			−2.39174
$650$		0			0
$651$	−303.614			−0.0182789
$652$		0			0
$653$		−	20913.3i		−	1.25330i	−0.779303	−	0.626648i	$-0.784427\pi$
$653$		−	20913.3i		−	1.25330i	0.779303	−	0.626648i	$-0.215573\pi$
$654$		0			0
$655$	4315.84	+	4687.14i	0.257456	+	0.279605i
$656$		0			0
$657$			8776.95i			0.521189i
$658$		0			0
$659$	3284.58			0.194156			0.0970782	−	0.995277i	$-0.469050\pi$
$659$	3284.58			0.194156			0.0970782	+	0.995277i	$0.469050\pi$
$660$		0			0
$661$	13530.3			0.796167			0.398084	−	0.917349i	$-0.369675\pi$
$661$	13530.3			0.796167			0.398084	+	0.917349i	$0.369675\pi$
$662$		0			0
$663$		−	7716.61i		−	0.452019i
$664$		0			0
$665$	1546.05	−	1423.57i	0.0901551	−	0.0830133i
$666$		0			0
$667$		−	1971.03i		−	0.114421i
$668$		0			0
$669$	5273.67			0.304771
$670$		0			0
$671$	−36629.4			−2.10739
$672$		0			0
$673$		−	13550.2i		−	0.776108i	−0.921637	−	0.388054i	$-0.873147\pi$
$673$		−	13550.2i		−	0.776108i	0.921637	−	0.388054i	$-0.126853\pi$
$674$		0			0
$675$	−14647.0	−	1210.19i	−0.835207	−	0.0690076i
$676$		0			0
$677$		−	20897.0i		−	1.18632i	−0.805085	−	0.593160i	$-0.797880\pi$
$677$		−	20897.0i		−	1.18632i	0.805085	−	0.593160i	$-0.202120\pi$
$678$		0			0
$679$	5340.93			0.301865
$680$		0			0
$681$	6895.20			0.387995
$682$		0			0
$683$			9344.53i			0.523512i	0.965134	+	0.261756i	$0.0843016\pi$
$683$			9344.53i			0.523512i	−0.965134	+	0.261756i	$0.915698\pi$
$684$		0			0
$685$	21425.3	−	19728.1i	1.19507	−	1.10040i
$686$		0			0
$687$		−	13004.5i		−	0.722200i
$688$		0			0
$689$	−41110.9			−2.27315
$690$		0			0
$691$	−1456.60			−0.0801903			−0.0400952	−	0.999196i	$-0.512766\pi$
$691$	−1456.60			−0.0801903			−0.0400952	+	0.999196i	$0.512766\pi$
$692$		0			0
$693$			8569.05i			0.469713i
$694$		0			0
$695$	−8825.80	−	9585.09i	−0.481700	−	0.523141i
$696$		0			0
$697$			20925.8i			1.13719i
$698$		0			0
$699$	1081.30			0.0585100
$700$		0			0
$701$	−31323.5			−1.68769			−0.843845	−	0.536587i	$-0.819713\pi$
$701$	−31323.5			−1.68769			−0.843845	+	0.536587i	$0.819713\pi$
$702$		0			0
$703$		−	8044.67i		−	0.431594i
$704$		0			0
$705$	−3700.16	−	4018.49i	−0.197668	−	0.214674i
$706$		0			0
$707$		−	1567.17i		−	0.0833659i
$708$		0			0
$709$	9483.38			0.502335			0.251168	−	0.967944i	$-0.419185\pi$
$709$	9483.38			0.502335			0.251168	+	0.967944i	$0.419185\pi$
$710$		0			0
$711$	7881.75			0.415737
$712$		0			0
$713$		−	334.647i		−	0.0175773i
$714$		0			0
$715$	33723.2	−	31051.8i	1.76389	−	1.62416i
$716$		0			0
$717$			11680.9i			0.608411i
$718$		0			0
$719$	1461.62			0.0758126			0.0379063	−	0.999281i	$-0.487931\pi$
$719$	1461.62			0.0758126			0.0379063	+	0.999281i	$0.487931\pi$
$720$		0			0
$721$	−13763.8			−0.710942
$722$		0			0
$723$		−	1195.86i		−	0.0615138i
$724$		0			0
$725$	−1073.47	+	12992.3i	−0.0549900	+	0.665549i
$726$		0			0
$727$			14807.2i			0.755388i	0.925931	+	0.377694i	$0.123283\pi$
$727$			14807.2i			0.755388i	−0.925931	+	0.377694i	$0.876717\pi$
$728$		0			0
$729$	−1917.00			−0.0973937
$730$		0			0
$731$	1534.07			0.0776193
$732$		0			0
$733$			17273.9i			0.870433i	0.900326	+	0.435217i	$0.143328\pi$
$733$			17273.9i			0.870433i	−0.900326	+	0.435217i	$0.856672\pi$
$734$		0			0
$735$	−987.175	+	908.975i	−0.0495408	+	0.0456164i
$736$		0			0
$737$		−	47651.7i		−	2.38164i
$738$		0			0
$739$	−1989.11			−0.0990128			−0.0495064	−	0.998774i	$-0.515765\pi$
$739$	−1989.11			−0.0990128			−0.0495064	+	0.998774i	$0.515765\pi$
$740$		0			0
$741$	−4626.68			−0.229373
$742$		0			0
$743$			14502.6i			0.716083i	0.933706	+	0.358041i	$0.116555\pi$
$743$			14502.6i			0.716083i	−0.933706	+	0.358041i	$0.883445\pi$
$744$		0			0
$745$	−15407.7	−	16733.2i	−0.757711	−	0.822897i
$746$		0			0
$747$		−	7707.07i		−	0.377493i
$748$		0			0
$749$	6899.89			0.336604
$750$		0			0
$751$	19929.7			0.968368			0.484184	−	0.874966i	$-0.339117\pi$
$751$	19929.7			0.968368			0.484184	+	0.874966i	$0.339117\pi$
$752$		0			0
$753$		−	5260.17i		−	0.254570i
$754$		0			0
$755$	19321.0	+	20983.2i	0.931342	+	1.01147i
$756$		0			0
$757$			20963.0i			1.00649i	0.864144	+	0.503244i	$0.167860\pi$
$757$			20963.0i			1.00649i	−0.864144	+	0.503244i	$0.832140\pi$
$758$		0			0
$759$	2698.54			0.129053
$760$		0			0
$761$	20252.5			0.964723			0.482361	−	0.875972i	$-0.339779\pi$
$761$	20252.5			0.964723			0.482361	+	0.875972i	$0.339779\pi$
$762$		0			0
$763$		−	6711.15i		−	0.318427i
$764$		0			0
$765$	7735.72	−	7122.93i	0.365602	−	0.336641i
$766$		0			0
$767$		−	47715.2i		−	2.24628i
$768$		0			0
$769$	9918.95			0.465132			0.232566	−	0.972581i	$-0.425288\pi$
$769$	9918.95			0.465132			0.232566	+	0.972581i	$0.425288\pi$
$770$		0			0
$771$	−1774.09			−0.0828695
$772$		0			0
$773$			32691.2i			1.52111i	0.649271	+	0.760557i	$0.275074\pi$
$773$			32691.2i			1.52111i	−0.649271	+	0.760557i	$0.724926\pi$
$774$		0			0
$775$	−182.257	+	2205.88i	−0.00844757	+	0.102242i
$776$		0			0
$777$			5136.65i			0.237164i
$778$		0			0
$779$	12546.6			0.577057
$780$		0			0
$781$	−30329.4			−1.38959
$782$		0			0
$783$			12262.3i			0.559666i
$784$		0			0
$785$	1226.56	−	1129.39i	0.0557678	−	0.0513501i
$786$		0			0
$787$			37914.4i			1.71728i	0.512576	+	0.858642i	$0.328691\pi$
$787$			37914.4i			1.71728i	−0.512576	+	0.858642i	$0.671309\pi$
$788$		0			0
$789$	−3389.83			−0.152954
$790$		0			0
$791$	−11826.7			−0.531618
$792$		0			0
$793$		−	44198.3i		−	1.97923i
$794$		0			0
$795$	10842.3	+	11775.1i	0.483694	+	0.525307i
$796$		0			0
$797$		−	5059.77i		−	0.224876i	−0.993659	−	0.112438i	$-0.964134\pi$
$797$		−	5059.77i		−	0.224876i	0.993659	−	0.112438i	$-0.0358660\pi$
$798$		0			0
$799$	8933.56			0.395553
$800$		0			0
$801$	8704.20			0.383955
$802$		0			0
$803$			24363.5i			1.07070i
$804$		0			0
$805$	−1001.88	−	1088.07i	−0.0438655	−	0.0476393i
$806$		0			0
$807$			1767.40i			0.0770948i
$808$		0			0
$809$	16998.1			0.738718			0.369359	−	0.929287i	$-0.379577\pi$
$809$	16998.1			0.738718			0.369359	+	0.929287i	$0.379577\pi$
$810$		0			0
$811$	33804.0			1.46365			0.731825	−	0.681492i	$-0.238669\pi$
$811$	33804.0			1.46365			0.731825	+	0.681492i	$0.238669\pi$
$812$		0			0
$813$			5437.87i			0.234581i
$814$		0			0
$815$	−26013.5	+	23952.8i	−1.11805	+	1.02949i
$816$		0			0
$817$		−	919.790i		−	0.0393872i
$818$		0			0
$819$	−10339.7			−0.441147
$820$		0			0
$821$	37069.6			1.57581			0.787903	−	0.615800i	$-0.211167\pi$
$821$	37069.6			1.57581			0.787903	+	0.615800i	$0.211167\pi$
$822$		0			0
$823$			31129.8i			1.31849i	0.751929	+	0.659245i	$0.229124\pi$
$823$			31129.8i			1.31849i	−0.751929	+	0.659245i	$0.770876\pi$
$824$		0			0
$825$	−17787.9	−	1469.69i	−0.750659	−	0.0620220i
$826$		0			0
$827$			13873.8i			0.583361i	0.956516	+	0.291680i	$0.0942143\pi$
$827$			13873.8i			0.583361i	−0.956516	+	0.291680i	$0.905786\pi$
$828$		0			0
$829$	−30471.6			−1.27663			−0.638313	−	0.769777i	$-0.720368\pi$
$829$	−30471.6			−1.27663			−0.638313	+	0.769777i	$0.720368\pi$
$830$		0			0
$831$	−1231.32			−0.0514008
$832$		0			0
$833$		−	2194.60i		−	0.0912826i
$834$		0			0
$835$	−8055.69	+	7417.55i	−0.333867	+	0.307419i
$836$		0			0
$837$			2081.93i			0.0859760i
$838$		0			0
$839$	−1714.65			−0.0705558			−0.0352779	−	0.999378i	$-0.511232\pi$
$839$	−1714.65			−0.0705558			−0.0352779	+	0.999378i	$0.511232\pi$
$840$		0			0
$841$	−13512.0			−0.554020
$842$		0			0
$843$		−	6444.82i		−	0.263311i
$844$		0			0
$845$	20829.9	+	22621.9i	0.848013	+	0.920968i
$846$		0			0
$847$			14469.4i			0.586983i
$848$		0			0
$849$	18398.2			0.743726
$850$		0			0
$851$	−5661.67			−0.228061
$852$		0			0
$853$		−	37242.7i		−	1.49492i	−0.664306	−	0.747460i	$-0.731273\pi$
$853$		−	37242.7i		−	1.49492i	0.664306	−	0.747460i	$-0.268727\pi$
$854$		0			0
$855$	−4270.72	−	4638.14i	−0.170825	−	0.185522i
$856$		0			0
$857$			5318.70i			0.211999i	0.994366	+	0.106000i	$0.0338042\pi$
$857$			5318.70i			0.211999i	−0.994366	+	0.106000i	$0.966196\pi$
$858$		0			0
$859$	−22196.9			−0.881664			−0.440832	−	0.897590i	$-0.645317\pi$
$859$	−22196.9			−0.881664			−0.440832	+	0.897590i	$0.645317\pi$
$860$		0			0
$861$	−8011.18			−0.317097
$862$		0			0
$863$		−	7729.42i		−	0.304881i	−0.988313	−	0.152441i	$-0.951287\pi$
$863$		−	7729.42i		−	0.304881i	0.988313	−	0.152441i	$-0.0487133\pi$
$864$		0			0
$865$	−21312.0	+	19623.7i	−0.837721	+	0.771360i
$866$		0			0
$867$			7120.81i			0.278933i
$868$		0			0
$869$	21878.6			0.854061
$870$		0			0
$871$	57498.2			2.23680
$872$		0			0
$873$		−	16022.8i		−	0.621179i
$874$		0			0
$875$	6011.47	+	7717.86i	0.232257	+	0.298184i
$876$		0			0
$877$		−	346.308i		−	0.0133341i	−0.999978	−	0.00666705i	$-0.997878\pi$
$877$		−	346.308i		−	0.0133341i	0.999978	−	0.00666705i	$-0.00212220\pi$
$878$		0			0
$879$	15563.5			0.597204
$880$		0			0
$881$	35242.6			1.34774			0.673868	−	0.738852i	$-0.264632\pi$
$881$	35242.6			1.34774			0.673868	+	0.738852i	$0.264632\pi$
$882$		0			0
$883$		−	37128.4i		−	1.41503i	−0.706700	−	0.707514i	$-0.749817\pi$
$883$		−	37128.4i		−	1.41503i	0.706700	−	0.707514i	$-0.250183\pi$
$884$		0			0
$885$	−13666.7	+	12584.1i	−0.519097	+	0.477976i
$886$		0			0
$887$			27830.8i			1.05351i	0.850016	+	0.526756i	$0.176592\pi$
$887$			27830.8i			1.05351i	−0.850016	+	0.526756i	$0.823408\pi$
$888$		0			0
$889$	18005.6			0.679290
$890$		0			0
$891$	16263.7			0.611509
$892$		0			0
$893$		−	5356.33i		−	0.200720i
$894$		0			0
$895$	−8668.19	−	9413.93i	−0.323738	−	0.351590i
$896$		0			0
$897$			3256.16i			0.121204i
$898$		0			0
$899$	1846.73			0.0685115
$900$		0			0
$901$	−26177.3			−0.967916
$902$		0			0
$903$			587.300i			0.0216435i
$904$		0			0
$905$	24418.1	+	26518.8i	0.896889	+	0.974049i
$906$		0			0
$907$			49505.1i			1.81234i	0.422916	+	0.906169i	$0.361006\pi$
$907$			49505.1i			1.81234i	−0.422916	+	0.906169i	$0.638994\pi$
$908$		0			0
$909$	−4701.52			−0.171551
$910$		0			0
$911$	−19701.6			−0.716514			−0.358257	−	0.933623i	$-0.616629\pi$
$911$	−19701.6			−0.716514			−0.358257	+	0.933623i	$0.616629\pi$
$912$		0			0
$913$		−	21393.7i		−	0.775496i
$914$		0			0
$915$	−12659.4	+	11656.5i	−0.457383	+	0.421151i
$916$		0			0
$917$		−	3989.17i		−	0.143658i
$918$		0			0
$919$	21044.6			0.755384			0.377692	−	0.925931i	$-0.376718\pi$
$919$	21044.6			0.755384			0.377692	+	0.925931i	$0.376718\pi$
$920$		0			0
$921$	−12944.3			−0.463114
$922$		0			0
$923$		−	36596.5i		−	1.30508i
$924$		0			0
$925$	37319.8	+	3083.49i	1.32656	+	0.109605i
$926$		0			0
$927$			41291.3i			1.46298i
$928$		0			0
$929$	−22745.2			−0.803278			−0.401639	−	0.915798i	$-0.631559\pi$
$929$	−22745.2			−0.803278			−0.401639	+	0.915798i	$0.631559\pi$
$930$		0			0
$931$	−1315.83			−0.0463205
$932$		0			0
$933$			25645.1i			0.899875i
$934$		0			0
$935$	21473.2	−	19772.2i	0.751069	−	0.691572i
$936$		0			0
$937$		−	48685.9i		−	1.69744i	−0.528846	−	0.848718i	$-0.677375\pi$
$937$		−	48685.9i		−	1.69744i	0.528846	−	0.848718i	$-0.322625\pi$
$938$		0			0
$939$	14101.9			0.490094
$940$		0			0
$941$	46364.6			1.60621			0.803104	−	0.595838i	$-0.203180\pi$
$941$	46364.6			1.60621			0.803104	+	0.595838i	$0.203180\pi$
$942$		0			0
$943$		−	8830.01i		−	0.304925i
$944$		0			0
$945$	6232.97	+	6769.20i	0.214559	+	0.233018i
$946$		0			0
$947$		−	32478.8i		−	1.11449i	−0.830349	−	0.557243i	$-0.811859\pi$
$947$		−	32478.8i		−	1.11449i	0.830349	−	0.557243i	$-0.188141\pi$
$948$		0			0
$949$	−29397.9			−1.00558
$950$		0			0
$951$	−11139.5			−0.379835
$952$		0			0
$953$		−	19614.1i		−	0.666698i	−0.942804	−	0.333349i	$-0.891821\pi$
$953$		−	19614.1i		−	0.666698i	0.942804	−	0.333349i	$-0.108179\pi$
$954$		0			0
$955$	−29226.8	−	31741.2i	−0.990322	−	1.07552i
$956$		0			0
$957$			14891.7i			0.503011i
$958$		0			0
$959$	−18234.9			−0.614010
$960$		0			0
$961$	−29477.5			−0.989475
$962$		0			0
$963$		−	20699.7i		−	0.692666i
$964$		0			0
$965$	−9761.83	+	8988.54i	−0.325642	+	0.299846i
$966$		0			0
$967$		−	11525.5i		−	0.383283i	−0.981465	−	0.191641i	$-0.938619\pi$
$967$		−	11525.5i		−	0.383283i	0.981465	−	0.191641i	$-0.0613810\pi$
$968$		0			0
$969$	−2946.03			−0.0976678
$970$		0			0
$971$	−7668.77			−0.253453			−0.126726	−	0.991938i	$-0.540447\pi$
$971$	−7668.77			−0.253453			−0.126726	+	0.991938i	$0.540447\pi$
$972$		0			0
$973$			8157.77i			0.268783i
$974$		0			0
$975$	1773.39	−	21463.5i	0.0582500	−	0.705006i
$976$		0			0
$977$		−	14981.6i		−	0.490589i	−0.969449	−	0.245294i	$-0.921115\pi$
$977$		−	14981.6i		−	0.490589i	0.969449	−	0.245294i	$-0.0788846\pi$
$978$		0			0
$979$	24161.6			0.788771
$980$		0			0
$981$	−20133.4			−0.655262
$982$		0			0
$983$			364.373i			0.0118227i	0.999983	+	0.00591134i	$0.00188165\pi$
$983$			364.373i			0.0118227i	−0.999983	+	0.00591134i	$0.998118\pi$
$984$		0			0
$985$	−20933.7	+	19275.5i	−0.677162	+	0.623520i
$986$		0			0
$987$			3420.10i			0.110297i
$988$		0			0
$989$	−647.329			−0.0208128
$990$		0			0
$991$	−25813.3			−0.827433			−0.413717	−	0.910406i	$-0.635770\pi$
$991$	−25813.3			−0.827433			−0.413717	+	0.910406i	$0.635770\pi$
$992$		0			0
$993$		−	17407.1i		−	0.556293i
$994$		0			0
$995$	20086.1	+	21814.1i	0.639971	+	0.695028i
$996$		0			0
$997$		−	19996.9i		−	0.635215i	−0.948222	−	0.317607i	$-0.897121\pi$
$997$		−	19996.9i		−	0.635215i	0.948222	−	0.317607i	$-0.102879\pi$
$998$		0			0
$999$	35222.7			1.11551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.4.e.c.29.3	yes	4
3.2	odd	2		1260.4.k.d.1009.4		4
4.3	odd	2		560.4.g.d.449.1		4
5.2	odd	4		700.4.a.p.1.2		2
5.3	odd	4		700.4.a.q.1.1		2
5.4	even	2	inner	140.4.e.c.29.1	✓	4
7.6	odd	2		980.4.e.d.589.2		4
15.14	odd	2		1260.4.k.d.1009.3		4
20.19	odd	2		560.4.g.d.449.3		4
35.34	odd	2		980.4.e.d.589.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.e.c.29.1	✓	4	5.4	even	2	inner
140.4.e.c.29.3	yes	4	1.1	even	1	trivial
560.4.g.d.449.1		4	4.3	odd	2
560.4.g.d.449.3		4	20.19	odd	2
700.4.a.p.1.2		2	5.2	odd	4
700.4.a.q.1.1		2	5.3	odd	4
980.4.e.d.589.2		4	7.6	odd	2
980.4.e.d.589.4		4	35.34	odd	2
1260.4.k.d.1009.3		4	15.14	odd	2
1260.4.k.d.1009.4		4	3.2	odd	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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	140.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} +(-7.57321 - 8.22474i) q^{5} +7.00000i q^{7} +21.0000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{3} +(-7.57321 - 8.22474i) q^{5} +7.00000i q^{7} +21.0000 q^{9} +58.2929 q^{11} +70.3383i q^{13} +(20.1464 - 18.5505i) q^{15} +44.7878i q^{17} +26.8536 q^{19} -17.1464 q^{21} -18.8990i q^{23} +(-10.2929 + 124.576i) q^{25} +117.576i q^{27} +104.293 q^{29} +17.7071 q^{31} +142.788i q^{33} +(57.5732 - 53.0125i) q^{35} -299.576i q^{37} -172.293 q^{39} +467.221 q^{41} -34.2520i q^{43} +(-159.037 - 172.720i) q^{45} -199.464i q^{47} -49.0000 q^{49} -109.707 q^{51} +584.474i q^{53} +(-441.464 - 479.444i) q^{55} +65.7775i q^{57} -678.368 q^{59} -628.368 q^{61} +147.000i q^{63} +(578.514 - 532.687i) q^{65} -817.453i q^{67} +46.2929 q^{69} -520.293 q^{71} +417.950i q^{73} +(-305.146 - 25.2122i) q^{75} +408.050i q^{77} +375.321 q^{79} +279.000 q^{81} -367.004i q^{83} +(368.368 - 339.187i) q^{85} +255.464i q^{87} +414.486 q^{89} -492.368 q^{91} +43.3735i q^{93} +(-203.368 - 220.864i) q^{95} -762.990i q^{97} +1224.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 84 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 84 * q^9	\( 4 q + 4 q^{5} + 84 q^{9} + 96 q^{11} + 12 q^{15} + 176 q^{19} + 96 q^{25} + 280 q^{29} + 208 q^{31} + 196 q^{35} - 552 q^{39} + 360 q^{41} + 84 q^{45} - 196 q^{49} - 576 q^{51} - 1080 q^{55} - 1136 q^{59} - 936 q^{61} + 668 q^{65} + 48 q^{69} - 1944 q^{71} - 1152 q^{75} - 1928 q^{79} + 1116 q^{81} - 104 q^{85} + 3304 q^{89} - 392 q^{91} + 764 q^{95} + 2016 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 84 * q^9 + 96 * q^11 + 12 * q^15 + 176 * q^19 + 96 * q^25 + 280 * q^29 + 208 * q^31 + 196 * q^35 - 552 * q^39 + 360 * q^41 + 84 * q^45 - 196 * q^49 - 576 * q^51 - 1080 * q^55 - 1136 * q^59 - 936 * q^61 + 668 * q^65 + 48 * q^69 - 1944 * q^71 - 1152 * q^75 - 1928 * q^79 + 1116 * q^81 - 104 * q^85 + 3304 * q^89 - 392 * q^91 + 764 * q^95 + 2016 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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