LMFDB - Embedded newform 140.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,6,Mod(1,140)]

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, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("140.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	140.a (trivial)

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:	yes
Analytic conductor:	$22.4537347738$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3101016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 312x + 1740 $ x^3 - x^2 - 312*x + 1740
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$14.3069$ of defining polynomial
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.3069 q^{3} -25.0000 q^{5} -49.0000 q^{7} +56.5286 q^{9} +O(q^{10})$	$q-17.3069 q^{3} -25.0000 q^{5} -49.0000 q^{7} +56.5286 q^{9} -151.996 q^{11} -1053.36 q^{13} +432.672 q^{15} +1191.49 q^{17} -2189.99 q^{19} +848.038 q^{21} +4202.71 q^{23} +625.000 q^{25} +3227.24 q^{27} -657.888 q^{29} +1071.74 q^{31} +2630.57 q^{33} +1225.00 q^{35} +960.000 q^{37} +18230.3 q^{39} +9478.23 q^{41} +17449.1 q^{43} -1413.21 q^{45} +11168.5 q^{47} +2401.00 q^{49} -20621.0 q^{51} -32190.2 q^{53} +3799.89 q^{55} +37902.0 q^{57} -26557.6 q^{59} +18849.9 q^{61} -2769.90 q^{63} +26333.9 q^{65} +57924.9 q^{67} -72735.8 q^{69} -3735.92 q^{71} -68457.7 q^{73} -10816.8 q^{75} +7447.78 q^{77} +20067.7 q^{79} -69590.0 q^{81} +103082. q^{83} -29787.2 q^{85} +11386.0 q^{87} -52326.1 q^{89} +51614.4 q^{91} -18548.5 q^{93} +54749.8 q^{95} +49265.9 q^{97} -8592.09 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9}+O(q^{10}) $ 3 * q - 10 * q^3 - 75 * q^5 - 147 * q^7 - 71 * q^9	$ 3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9} + 58 q^{11} + 148 q^{13} + 250 q^{15} + 1360 q^{17} + 1692 q^{19} + 490 q^{21} + 4204 q^{23} + 1875 q^{25} + 3410 q^{27} + 5572 q^{29} + 2816 q^{31} + 11794 q^{33} + 3675 q^{35} + 15210 q^{37} + 14174 q^{39} + 7706 q^{41} + 24904 q^{43} + 1775 q^{45} + 21350 q^{47} + 7203 q^{49} + 31722 q^{51} + 342 q^{53} - 1450 q^{55} + 52548 q^{57} + 12888 q^{59} - 5614 q^{61} + 3479 q^{63} - 3700 q^{65} + 35196 q^{67} - 31036 q^{69} - 19520 q^{71} - 60506 q^{73} - 6250 q^{75} - 2842 q^{77} + 61122 q^{79} - 130853 q^{81} - 11664 q^{83} - 34000 q^{85} - 5686 q^{87} - 144878 q^{89} - 7252 q^{91} - 123544 q^{93} - 42300 q^{95} + 17304 q^{97} + 39368 q^{99}+O(q^{100}) $ 3 * q - 10 * q^3 - 75 * q^5 - 147 * q^7 - 71 * q^9 + 58 * q^11 + 148 * q^13 + 250 * q^15 + 1360 * q^17 + 1692 * q^19 + 490 * q^21 + 4204 * q^23 + 1875 * q^25 + 3410 * q^27 + 5572 * q^29 + 2816 * q^31 + 11794 * q^33 + 3675 * q^35 + 15210 * q^37 + 14174 * q^39 + 7706 * q^41 + 24904 * q^43 + 1775 * q^45 + 21350 * q^47 + 7203 * q^49 + 31722 * q^51 + 342 * q^53 - 1450 * q^55 + 52548 * q^57 + 12888 * q^59 - 5614 * q^61 + 3479 * q^63 - 3700 * q^65 + 35196 * q^67 - 31036 * q^69 - 19520 * q^71 - 60506 * q^73 - 6250 * q^75 - 2842 * q^77 + 61122 * q^79 - 130853 * q^81 - 11664 * q^83 - 34000 * q^85 - 5686 * q^87 - 144878 * q^89 - 7252 * q^91 - 123544 * q^93 - 42300 * q^95 + 17304 * q^97 + 39368 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−17.3069	−1.11024	−0.555119	−	0.831771i	$-0.687327\pi$
$3$	−17.3069	−1.11024	−0.555119	+	0.831771i	$0.687327\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	56.5286	0.232628
$10$	0	0
$11$	−151.996	−0.378747	−0.189373	−	0.981905i	$-0.560646\pi$
$11$	−151.996	−0.378747	−0.189373	+	0.981905i	$0.560646\pi$
$12$	0	0
$13$	−1053.36	−1.72869	−0.864344	−	0.502900i	$-0.832266\pi$
$13$	−1053.36	−1.72869	−0.864344	+	0.502900i	$0.832266\pi$
$14$	0	0
$15$	432.672	0.496513
$16$	0	0
$17$	1191.49	0.999925	0.499962	−	0.866047i	$-0.333347\pi$
$17$	1191.49	0.999925	0.499962	+	0.866047i	$0.333347\pi$
$18$	0	0
$19$	−2189.99	−1.39174	−0.695871	−	0.718167i	$-0.744982\pi$
$19$	−2189.99	−1.39174	−0.695871	+	0.718167i	$0.744982\pi$
$20$	0	0
$21$	848.038	0.419630
$22$	0	0
$23$	4202.71	1.65657	0.828284	−	0.560308i	$-0.189317\pi$
$23$	4202.71	1.65657	0.828284	+	0.560308i	$0.189317\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	3227.24	0.851966
$28$	0	0
$29$	−657.888	−0.145264	−0.0726318	−	0.997359i	$-0.523140\pi$
$29$	−657.888	−0.145264	−0.0726318	+	0.997359i	$0.523140\pi$
$30$	0	0
$31$	1071.74	0.200302	0.100151	−	0.994972i	$-0.468067\pi$
$31$	1071.74	0.200302	0.100151	+	0.994972i	$0.468067\pi$
$32$	0	0
$33$	2630.57	0.420499
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	960.000	0.115283	0.0576417	−	0.998337i	$-0.481642\pi$
$37$	960.000	0.115283	0.0576417	+	0.998337i	$0.481642\pi$
$38$	0	0
$39$	18230.3	1.91926
$40$	0	0
$41$	9478.23	0.880577	0.440289	−	0.897856i	$-0.354876\pi$
$41$	9478.23	0.880577	0.440289	+	0.897856i	$0.354876\pi$
$42$	0	0
$43$	17449.1	1.43914	0.719568	−	0.694422i	$-0.244340\pi$
$43$	17449.1	1.43914	0.719568	+	0.694422i	$0.244340\pi$
$44$	0	0
$45$	−1413.21	−0.104034
$46$	0	0
$47$	11168.5	0.737478	0.368739	−	0.929533i	$-0.379790\pi$
$47$	11168.5	0.737478	0.368739	+	0.929533i	$0.379790\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−20621.0	−1.11015
$52$	0	0
$53$	−32190.2	−1.57411	−0.787054	−	0.616884i	$-0.788395\pi$
$53$	−32190.2	−1.57411	−0.787054	+	0.616884i	$0.788395\pi$
$54$	0	0
$55$	3799.89	0.169381
$56$	0	0
$57$	37902.0	1.54516
$58$	0	0
$59$	−26557.6	−0.993251	−0.496626	−	0.867965i	$-0.665428\pi$
$59$	−26557.6	−0.993251	−0.496626	+	0.867965i	$0.665428\pi$
$60$	0	0
$61$	18849.9	0.648613	0.324306	−	0.945952i	$-0.394869\pi$
$61$	18849.9	0.648613	0.324306	+	0.945952i	$0.394869\pi$
$62$	0	0
$63$	−2769.90	−0.0879251
$64$	0	0
$65$	26333.9	0.773093
$66$	0	0
$67$	57924.9	1.57644	0.788222	−	0.615391i	$-0.211002\pi$
$67$	57924.9	1.57644	0.788222	+	0.615391i	$0.211002\pi$
$68$	0	0
$69$	−72735.8	−1.83919
$70$	0	0
$71$	−3735.92	−0.0879532	−0.0439766	−	0.999033i	$-0.514003\pi$
$71$	−3735.92	−0.0879532	−0.0439766	+	0.999033i	$0.514003\pi$
$72$	0	0
$73$	−68457.7	−1.50354	−0.751770	−	0.659426i	$-0.770800\pi$
$73$	−68457.7	−1.50354	−0.751770	+	0.659426i	$0.770800\pi$
$74$	0	0
$75$	−10816.8	−0.222048
$76$	0	0
$77$	7447.78	0.143153
$78$	0	0
$79$	20067.7	0.361767	0.180884	−	0.983505i	$-0.442104\pi$
$79$	20067.7	0.361767	0.180884	+	0.983505i	$0.442104\pi$
$80$	0	0
$81$	−69590.0	−1.17851
$82$	0	0
$83$	103082.	1.64243	0.821216	−	0.570617i	$-0.193296\pi$
$83$	103082.	1.64243	0.821216	+	0.570617i	$0.193296\pi$
$84$	0	0
$85$	−29787.2	−0.447180
$86$	0	0
$87$	11386.0	0.161277
$88$	0	0
$89$	−52326.1	−0.700234	−0.350117	−	0.936706i	$-0.613858\pi$
$89$	−52326.1	−0.700234	−0.350117	+	0.936706i	$0.613858\pi$
$90$	0	0
$91$	51614.4	0.653383
$92$	0	0
$93$	−18548.5	−0.222383
$94$	0	0
$95$	54749.8	0.622406
$96$	0	0
$97$	49265.9	0.531639	0.265820	−	0.964023i	$-0.414357\pi$
$97$	49265.9	0.531639	0.265820	+	0.964023i	$0.414357\pi$
$98$	0	0
$99$	−8592.09	−0.0881071
$100$	0	0
$101$	−164990.	−1.60936	−0.804681	−	0.593707i	$-0.797664\pi$
$101$	−164990.	−1.60936	−0.804681	+	0.593707i	$0.797664\pi$
$102$	0	0
$103$	−182419.	−1.69425	−0.847126	−	0.531393i	$-0.821669\pi$
$103$	−182419.	−1.69425	−0.847126	+	0.531393i	$0.821669\pi$
$104$	0	0
$105$	−21200.9	−0.187664
$106$	0	0
$107$	−44842.6	−0.378644	−0.189322	−	0.981915i	$-0.560629\pi$
$107$	−44842.6	−0.378644	−0.189322	+	0.981915i	$0.560629\pi$
$108$	0	0
$109$	188177.	1.51705	0.758525	−	0.651644i	$-0.225920\pi$
$109$	188177.	1.51705	0.758525	+	0.651644i	$0.225920\pi$
$110$	0	0
$111$	−16614.6	−0.127992
$112$	0	0
$113$	109084.	0.803643	0.401822	−	0.915718i	$-0.368377\pi$
$113$	109084.	0.803643	0.401822	+	0.915718i	$0.368377\pi$
$114$	0	0
$115$	−105068.	−0.740840
$116$	0	0
$117$	−59544.7	−0.402141
$118$	0	0
$119$	−58382.9	−0.377936
$120$	0	0
$121$	−137948.	−0.856551
$122$	0	0
$123$	−164039.	−0.977650
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	194260.	1.06875	0.534373	−	0.845249i	$-0.320548\pi$
$127$	194260.	1.06875	0.534373	+	0.845249i	$0.320548\pi$
$128$	0	0
$129$	−301990.	−1.59778
$130$	0	0
$131$	−150777.	−0.767637	−0.383818	−	0.923409i	$-0.625391\pi$
$131$	−150777.	−0.767637	−0.383818	+	0.923409i	$0.625391\pi$
$132$	0	0
$133$	107310.	0.526029
$134$	0	0
$135$	−80681.0	−0.381011
$136$	0	0
$137$	−72757.1	−0.331188	−0.165594	−	0.986194i	$-0.552954\pi$
$137$	−72757.1	−0.331188	−0.165594	+	0.986194i	$0.552954\pi$
$138$	0	0
$139$	59396.6	0.260750	0.130375	−	0.991465i	$-0.458382\pi$
$139$	59396.6	0.260750	0.130375	+	0.991465i	$0.458382\pi$
$140$	0	0
$141$	−193291.	−0.818776
$142$	0	0
$143$	160105.	0.654735
$144$	0	0
$145$	16447.2	0.0649639
$146$	0	0
$147$	−41553.9	−0.158605
$148$	0	0
$149$	143343.	0.528944	0.264472	−	0.964393i	$-0.414802\pi$
$149$	143343.	0.528944	0.264472	+	0.964393i	$0.414802\pi$
$150$	0	0
$151$	−94778.1	−0.338272	−0.169136	−	0.985593i	$-0.554098\pi$
$151$	−94778.1	−0.338272	−0.169136	+	0.985593i	$0.554098\pi$
$152$	0	0
$153$	67353.1	0.232610
$154$	0	0
$155$	−26793.5	−0.0895778
$156$	0	0
$157$	−67015.7	−0.216984	−0.108492	−	0.994097i	$-0.534602\pi$
$157$	−67015.7	−0.216984	−0.108492	+	0.994097i	$0.534602\pi$
$158$	0	0
$159$	557113.	1.74763
$160$	0	0
$161$	−205933.	−0.626124
$162$	0	0
$163$	438023.	1.29130	0.645651	−	0.763632i	$-0.276586\pi$
$163$	438023.	1.29130	0.645651	+	0.763632i	$0.276586\pi$
$164$	0	0
$165$	−65764.3	−0.188053
$166$	0	0
$167$	701642.	1.94681	0.973407	−	0.229081i	$-0.0735720\pi$
$167$	701642.	1.94681	0.973407	+	0.229081i	$0.0735720\pi$
$168$	0	0
$169$	738266.	1.98836
$170$	0	0
$171$	−123797.	−0.323758
$172$	0	0
$173$	60788.2	0.154420	0.0772101	−	0.997015i	$-0.475399\pi$
$173$	60788.2	0.154420	0.0772101	+	0.997015i	$0.475399\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	459630.	1.10275
$178$	0	0
$179$	171551.	0.400185	0.200093	−	0.979777i	$-0.435876\pi$
$179$	171551.	0.400185	0.200093	+	0.979777i	$0.435876\pi$
$180$	0	0
$181$	258526.	0.586554	0.293277	−	0.956028i	$-0.405254\pi$
$181$	258526.	0.586554	0.293277	+	0.956028i	$0.405254\pi$
$182$	0	0
$183$	−326234.	−0.720114
$184$	0	0
$185$	−24000.0	−0.0515563
$186$	0	0
$187$	−181101.	−0.378718
$188$	0	0
$189$	−158135.	−0.322013
$190$	0	0
$191$	694925.	1.37834	0.689168	−	0.724602i	$-0.257976\pi$
$191$	694925.	1.37834	0.689168	+	0.724602i	$0.257976\pi$
$192$	0	0
$193$	450549.	0.870659	0.435330	−	0.900271i	$-0.356632\pi$
$193$	450549.	0.870659	0.435330	+	0.900271i	$0.356632\pi$
$194$	0	0
$195$	−455758.	−0.858317
$196$	0	0
$197$	−435355.	−0.799242	−0.399621	−	0.916680i	$-0.630858\pi$
$197$	−435355.	−0.799242	−0.399621	+	0.916680i	$0.630858\pi$
$198$	0	0
$199$	642732.	1.15053	0.575264	−	0.817968i	$-0.304899\pi$
$199$	642732.	1.15053	0.575264	+	0.817968i	$0.304899\pi$
$200$	0	0
$201$	−1.00250e6	−1.75023
$202$	0	0
$203$	32236.5	0.0549045
$204$	0	0
$205$	−236956.	−0.393806
$206$	0	0
$207$	237573.	0.385364
$208$	0	0
$209$	332869.	0.527118
$210$	0	0
$211$	302080.	0.467107	0.233554	−	0.972344i	$-0.424965\pi$
$211$	302080.	0.467107	0.233554	+	0.972344i	$0.424965\pi$
$212$	0	0
$213$	64657.2	0.0976490
$214$	0	0
$215$	−436228.	−0.643602
$216$	0	0
$217$	−52515.3	−0.0757071
$218$	0	0
$219$	1.18479e6	1.66929
$220$	0	0
$221$	−1.25506e6	−1.72856
$222$	0	0
$223$	221742.	0.298597	0.149298	−	0.988792i	$-0.452299\pi$
$223$	221742.	0.298597	0.149298	+	0.988792i	$0.452299\pi$
$224$	0	0
$225$	35330.3	0.0465256
$226$	0	0
$227$	−40755.6	−0.0524955	−0.0262478	−	0.999655i	$-0.508356\pi$
$227$	−40755.6	−0.0524955	−0.0262478	+	0.999655i	$0.508356\pi$
$228$	0	0
$229$	−932910.	−1.17558	−0.587789	−	0.809015i	$-0.700001\pi$
$229$	−932910.	−1.17558	−0.587789	+	0.809015i	$0.700001\pi$
$230$	0	0
$231$	−128898.	−0.158934
$232$	0	0
$233$	362225.	0.437107	0.218554	−	0.975825i	$-0.429866\pi$
$233$	362225.	0.437107	0.218554	+	0.975825i	$0.429866\pi$
$234$	0	0
$235$	−279212.	−0.329810
$236$	0	0
$237$	−347309.	−0.401648
$238$	0	0
$239$	−447627.	−0.506900	−0.253450	−	0.967349i	$-0.581565\pi$
$239$	−447627.	−0.506900	−0.253450	+	0.967349i	$0.581565\pi$
$240$	0	0
$241$	−1.43410e6	−1.59051	−0.795255	−	0.606275i	$-0.792663\pi$
$241$	−1.43410e6	−1.59051	−0.795255	+	0.606275i	$0.792663\pi$
$242$	0	0
$243$	420166.	0.456463
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	2.30684e6	2.40589
$248$	0	0
$249$	−1.78403e6	−1.82349
$250$	0	0
$251$	−1.46572e6	−1.46848	−0.734240	−	0.678890i	$-0.762461\pi$
$251$	−1.46572e6	−1.46848	−0.734240	+	0.678890i	$0.762461\pi$
$252$	0	0
$253$	−638793.	−0.627420
$254$	0	0
$255$	515524.	0.496476
$256$	0	0
$257$	2.00020e6	1.88904	0.944518	−	0.328459i	$-0.106529\pi$
$257$	2.00020e6	1.88904	0.944518	+	0.328459i	$0.106529\pi$
$258$	0	0
$259$	−47040.0	−0.0435730
$260$	0	0
$261$	−37189.4	−0.0337924
$262$	0	0
$263$	−1.02544e6	−0.914161	−0.457081	−	0.889425i	$-0.651105\pi$
$263$	−1.02544e6	−0.914161	−0.457081	+	0.889425i	$0.651105\pi$
$264$	0	0
$265$	804756.	0.703962
$266$	0	0
$267$	905602.	0.777426
$268$	0	0
$269$	1.45144e6	1.22298	0.611488	−	0.791254i	$-0.290571\pi$
$269$	1.45144e6	1.22298	0.611488	+	0.791254i	$0.290571\pi$
$270$	0	0
$271$	573373.	0.474257	0.237128	−	0.971478i	$-0.423794\pi$
$271$	573373.	0.474257	0.237128	+	0.971478i	$0.423794\pi$
$272$	0	0
$273$	−893286.	−0.725410
$274$	0	0
$275$	−94997.2	−0.0757494
$276$	0	0
$277$	−681469.	−0.533638	−0.266819	−	0.963747i	$-0.585973\pi$
$277$	−681469.	−0.533638	−0.266819	+	0.963747i	$0.585973\pi$
$278$	0	0
$279$	60584.0	0.0465958
$280$	0	0
$281$	−1.40359e6	−1.06041	−0.530205	−	0.847870i	$-0.677885\pi$
$281$	−1.40359e6	−1.06041	−0.530205	+	0.847870i	$0.677885\pi$
$282$	0	0
$283$	−2.07792e6	−1.54227	−0.771137	−	0.636669i	$-0.780312\pi$
$283$	−2.07792e6	−1.54227	−0.771137	+	0.636669i	$0.780312\pi$
$284$	0	0
$285$	−947550.	−0.691019
$286$	0	0
$287$	−464433.	−0.332827
$288$	0	0
$289$	−213.069	−0.000150063 0
$290$	0	0
$291$	−852639.	−0.590246
$292$	0	0
$293$	166790.	0.113502	0.0567508	−	0.998388i	$-0.481926\pi$
$293$	166790.	0.113502	0.0567508	+	0.998388i	$0.481926\pi$
$294$	0	0
$295$	663941.	0.444195
$296$	0	0
$297$	−490526.	−0.322679
$298$	0	0
$299$	−4.42695e6	−2.86369
$300$	0	0
$301$	−855006.	−0.543943
$302$	0	0
$303$	2.85546e6	1.78677
$304$	0	0
$305$	−471248.	−0.290068
$306$	0	0
$307$	1.69175e6	1.02445	0.512225	−	0.858851i	$-0.328821\pi$
$307$	1.69175e6	1.02445	0.512225	+	0.858851i	$0.328821\pi$
$308$	0	0
$309$	3.15711e6	1.88102
$310$	0	0
$311$	318544.	0.186753	0.0933766	−	0.995631i	$-0.470234\pi$
$311$	318544.	0.186753	0.0933766	+	0.995631i	$0.470234\pi$
$312$	0	0
$313$	1.77015e6	1.02129	0.510646	−	0.859791i	$-0.329406\pi$
$313$	1.77015e6	1.02129	0.510646	+	0.859791i	$0.329406\pi$
$314$	0	0
$315$	69247.5	0.0393213
$316$	0	0
$317$	281588.	0.157386	0.0786929	−	0.996899i	$-0.474925\pi$
$317$	281588.	0.157386	0.0786929	+	0.996899i	$0.474925\pi$
$318$	0	0
$319$	99996.0	0.0550181
$320$	0	0
$321$	776086.	0.420385
$322$	0	0
$323$	−2.60935e6	−1.39164
$324$	0	0
$325$	−658347.	−0.345738
$326$	0	0
$327$	−3.25676e6	−1.68429
$328$	0	0
$329$	−547255.	−0.278740
$330$	0	0
$331$	462210.	0.231883	0.115942	−	0.993256i	$-0.463011\pi$
$331$	462210.	0.231883	0.115942	+	0.993256i	$0.463011\pi$
$332$	0	0
$333$	54267.4	0.0268181
$334$	0	0
$335$	−1.44812e6	−0.705007
$336$	0	0
$337$	2.03370e6	0.975464	0.487732	−	0.872993i	$-0.337824\pi$
$337$	2.03370e6	0.975464	0.487732	+	0.872993i	$0.337824\pi$
$338$	0	0
$339$	−1.88790e6	−0.892235
$340$	0	0
$341$	−162900.	−0.0758638
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.81840e6	0.822509
$346$	0	0
$347$	1.63046e6	0.726918	0.363459	−	0.931610i	$-0.381596\pi$
$347$	1.63046e6	0.726918	0.363459	+	0.931610i	$0.381596\pi$
$348$	0	0
$349$	144115.	0.0633354	0.0316677	−	0.999498i	$-0.489918\pi$
$349$	144115.	0.0633354	0.0316677	+	0.999498i	$0.489918\pi$
$350$	0	0
$351$	−3.39943e6	−1.47278
$352$	0	0
$353$	−3.59502e6	−1.53555	−0.767776	−	0.640718i	$-0.778637\pi$
$353$	−3.59502e6	−1.53555	−0.767776	+	0.640718i	$0.778637\pi$
$354$	0	0
$355$	93398.0	0.0393339
$356$	0	0
$357$	1.01043e6	0.419599
$358$	0	0
$359$	−278912.	−0.114217	−0.0571086	−	0.998368i	$-0.518188\pi$
$359$	−278912.	−0.114217	−0.0571086	+	0.998368i	$0.518188\pi$
$360$	0	0
$361$	2.31997e6	0.936947
$362$	0	0
$363$	2.38746e6	0.950975
$364$	0	0
$365$	1.71144e6	0.672403
$366$	0	0
$367$	1.42779e6	0.553347	0.276674	−	0.960964i	$-0.410768\pi$
$367$	1.42779e6	0.553347	0.276674	+	0.960964i	$0.410768\pi$
$368$	0	0
$369$	535791.	0.204847
$370$	0	0
$371$	1.57732e6	0.594957
$372$	0	0
$373$	3.49805e6	1.30183	0.650915	−	0.759150i	$-0.274385\pi$
$373$	3.49805e6	1.30183	0.650915	+	0.759150i	$0.274385\pi$
$374$	0	0
$375$	270420.	0.0993027
$376$	0	0
$377$	692990.	0.251116
$378$	0	0
$379$	2.29686e6	0.821367	0.410684	−	0.911778i	$-0.365290\pi$
$379$	2.29686e6	0.821367	0.410684	+	0.911778i	$0.365290\pi$
$380$	0	0
$381$	−3.36204e6	−1.18656
$382$	0	0
$383$	−3.43087e6	−1.19511	−0.597554	−	0.801829i	$-0.703861\pi$
$383$	−3.43087e6	−1.19511	−0.597554	+	0.801829i	$0.703861\pi$
$384$	0	0
$385$	−186195.	−0.0640199
$386$	0	0
$387$	986373.	0.334783
$388$	0	0
$389$	20353.4	0.00681966	0.00340983	−	0.999994i	$-0.498915\pi$
$389$	20353.4	0.00681966	0.00340983	+	0.999994i	$0.498915\pi$
$390$	0	0
$391$	5.00748e6	1.65644
$392$	0	0
$393$	2.60948e6	0.852260
$394$	0	0
$395$	−501692.	−0.161787
$396$	0	0
$397$	1.04727e6	0.333489	0.166745	−	0.986000i	$-0.446674\pi$
$397$	1.04727e6	0.333489	0.166745	+	0.986000i	$0.446674\pi$
$398$	0	0
$399$	−1.85720e6	−0.584017
$400$	0	0
$401$	4.79111e6	1.48790	0.743952	−	0.668233i	$-0.232949\pi$
$401$	4.79111e6	1.48790	0.743952	+	0.668233i	$0.232949\pi$
$402$	0	0
$403$	−1.12892e6	−0.346260
$404$	0	0
$405$	1.73975e6	0.527047
$406$	0	0
$407$	−145916.	−0.0436632
$408$	0	0
$409$	5.18232e6	1.53185	0.765924	−	0.642931i	$-0.222282\pi$
$409$	5.18232e6	1.53185	0.765924	+	0.642931i	$0.222282\pi$
$410$	0	0
$411$	1.25920e6	0.367697
$412$	0	0
$413$	1.30132e6	0.375414
$414$	0	0
$415$	−2.57705e6	−0.734518
$416$	0	0
$417$	−1.02797e6	−0.289495
$418$	0	0
$419$	−320345.	−0.0891422	−0.0445711	−	0.999006i	$-0.514192\pi$
$419$	−320345.	−0.0891422	−0.0445711	+	0.999006i	$0.514192\pi$
$420$	0	0
$421$	2.85464e6	0.784956	0.392478	−	0.919761i	$-0.371618\pi$
$421$	2.85464e6	0.784956	0.392478	+	0.919761i	$0.371618\pi$
$422$	0	0
$423$	631337.	0.171558
$424$	0	0
$425$	744680.	0.199985
$426$	0	0
$427$	−923647.	−0.245153
$428$	0	0
$429$	−2.77093e6	−0.726912
$430$	0	0
$431$	2.74009e6	0.710512	0.355256	−	0.934769i	$-0.384394\pi$
$431$	2.74009e6	0.710512	0.355256	+	0.934769i	$0.384394\pi$
$432$	0	0
$433$	5.80476e6	1.48787	0.743934	−	0.668253i	$-0.232957\pi$
$433$	5.80476e6	1.48787	0.743934	+	0.668253i	$0.232957\pi$
$434$	0	0
$435$	−284650.	−0.0721253
$436$	0	0
$437$	−9.20390e6	−2.30552
$438$	0	0
$439$	−58765.0	−0.0145532	−0.00727658	−	0.999974i	$-0.502316\pi$
$439$	−58765.0	−0.0145532	−0.00727658	+	0.999974i	$0.502316\pi$
$440$	0	0
$441$	135725.	0.0332325
$442$	0	0
$443$	−733652.	−0.177615	−0.0888077	−	0.996049i	$-0.528306\pi$
$443$	−733652.	−0.177615	−0.0888077	+	0.996049i	$0.528306\pi$
$444$	0	0
$445$	1.30815e6	0.313154
$446$	0	0
$447$	−2.48082e6	−0.587253
$448$	0	0
$449$	7.88727e6	1.84634	0.923168	−	0.384397i	$-0.125591\pi$
$449$	7.88727e6	1.84634	0.923168	+	0.384397i	$0.125591\pi$
$450$	0	0
$451$	−1.44065e6	−0.333516
$452$	0	0
$453$	1.64031e6	0.375562
$454$	0	0
$455$	−1.29036e6	−0.292202
$456$	0	0
$457$	−5.12896e6	−1.14879	−0.574393	−	0.818580i	$-0.694762\pi$
$457$	−5.12896e6	−1.14879	−0.574393	+	0.818580i	$0.694762\pi$
$458$	0	0
$459$	3.84522e6	0.851902
$460$	0	0
$461$	3.43821e6	0.753494	0.376747	−	0.926316i	$-0.377043\pi$
$461$	3.43821e6	0.753494	0.376747	+	0.926316i	$0.377043\pi$
$462$	0	0
$463$	−6.57405e6	−1.42522	−0.712608	−	0.701562i	$-0.752486\pi$
$463$	−6.57405e6	−1.42522	−0.712608	+	0.701562i	$0.752486\pi$
$464$	0	0
$465$	463713.	0.0994527
$466$	0	0
$467$	−6.74445e6	−1.43105	−0.715524	−	0.698588i	$-0.753812\pi$
$467$	−6.74445e6	−1.43105	−0.715524	+	0.698588i	$0.753812\pi$
$468$	0	0
$469$	−2.83832e6	−0.595840
$470$	0	0
$471$	1.15983e6	0.240904
$472$	0	0
$473$	−2.65219e6	−0.545069
$474$	0	0
$475$	−1.36875e6	−0.278348
$476$	0	0
$477$	−1.81967e6	−0.366181
$478$	0	0
$479$	−6.39635e6	−1.27378	−0.636888	−	0.770956i	$-0.719779\pi$
$479$	−6.39635e6	−1.27378	−0.636888	+	0.770956i	$0.719779\pi$
$480$	0	0
$481$	−1.01122e6	−0.199289
$482$	0	0
$483$	3.56405e6	0.695147
$484$	0	0
$485$	−1.23165e6	−0.237756
$486$	0	0
$487$	−5.36517e6	−1.02509	−0.512544	−	0.858661i	$-0.671297\pi$
$487$	−5.36517e6	−1.02509	−0.512544	+	0.858661i	$0.671297\pi$
$488$	0	0
$489$	−7.58082e6	−1.43365
$490$	0	0
$491$	3.08244e6	0.577021	0.288510	−	0.957477i	$-0.406840\pi$
$491$	3.08244e6	0.577021	0.288510	+	0.957477i	$0.406840\pi$
$492$	0	0
$493$	−783865.	−0.145253
$494$	0	0
$495$	214802.	0.0394027
$496$	0	0
$497$	183060.	0.0332432
$498$	0	0
$499$	2.94445e6	0.529362	0.264681	−	0.964336i	$-0.414733\pi$
$499$	2.94445e6	0.529362	0.264681	+	0.964336i	$0.414733\pi$
$500$	0	0
$501$	−1.21432e7	−2.16143
$502$	0	0
$503$	4.03407e6	0.710925	0.355462	−	0.934691i	$-0.384323\pi$
$503$	4.03407e6	0.710925	0.355462	+	0.934691i	$0.384323\pi$
$504$	0	0
$505$	4.12475e6	0.719729
$506$	0	0
$507$	−1.27771e7	−2.20756
$508$	0	0
$509$	−2.89070e6	−0.494549	−0.247274	−	0.968945i	$-0.579535\pi$
$509$	−2.89070e6	−0.494549	−0.247274	+	0.968945i	$0.579535\pi$
$510$	0	0
$511$	3.35443e6	0.568285
$512$	0	0
$513$	−7.06764e6	−1.18572
$514$	0	0
$515$	4.56048e6	0.757692
$516$	0	0
$517$	−1.69756e6	−0.279317
$518$	0	0
$519$	−1.05206e6	−0.171443
$520$	0	0
$521$	7.55318e6	1.21909	0.609544	−	0.792752i	$-0.291352\pi$
$521$	7.55318e6	1.21909	0.609544	+	0.792752i	$0.291352\pi$
$522$	0	0
$523$	7.13825e6	1.14114	0.570568	−	0.821250i	$-0.306723\pi$
$523$	7.13825e6	1.14114	0.570568	+	0.821250i	$0.306723\pi$
$524$	0	0
$525$	530024.	0.0839261
$526$	0	0
$527$	1.27697e6	0.200287
$528$	0	0
$529$	1.12264e7	1.74422
$530$	0	0
$531$	−1.50126e6	−0.231058
$532$	0	0
$533$	−9.98395e6	−1.52224
$534$	0	0
$535$	1.12106e6	0.169335
$536$	0	0
$537$	−2.96902e6	−0.444301
$538$	0	0
$539$	−364941.	−0.0541067
$540$	0	0
$541$	7.76233e6	1.14025	0.570124	−	0.821559i	$-0.306895\pi$
$541$	7.76233e6	1.14025	0.570124	+	0.821559i	$0.306895\pi$
$542$	0	0
$543$	−4.47428e6	−0.651214
$544$	0	0
$545$	−4.70442e6	−0.678445
$546$	0	0
$547$	−5.79171e6	−0.827635	−0.413817	−	0.910360i	$-0.635805\pi$
$547$	−5.79171e6	−0.827635	−0.413817	+	0.910360i	$0.635805\pi$
$548$	0	0
$549$	1.06556e6	0.150885
$550$	0	0
$551$	1.44077e6	0.202170
$552$	0	0
$553$	−983316.	−0.136735
$554$	0	0
$555$	415365.	0.0572397
$556$	0	0
$557$	5.97241e6	0.815665	0.407833	−	0.913057i	$-0.366285\pi$
$557$	5.97241e6	0.815665	0.407833	+	0.913057i	$0.366285\pi$
$558$	0	0
$559$	−1.83801e7	−2.48782
$560$	0	0
$561$	3.13429e6	0.420468
$562$	0	0
$563$	8.00301e6	1.06410	0.532050	−	0.846713i	$-0.321422\pi$
$563$	8.00301e6	1.06410	0.532050	+	0.846713i	$0.321422\pi$
$564$	0	0
$565$	−2.72709e6	−0.359400
$566$	0	0
$567$	3.40991e6	0.445436
$568$	0	0
$569$	−2.32831e6	−0.301481	−0.150741	−	0.988573i	$-0.548166\pi$
$569$	−2.32831e6	−0.301481	−0.150741	+	0.988573i	$0.548166\pi$
$570$	0	0
$571$	1.13657e7	1.45883	0.729417	−	0.684069i	$-0.239791\pi$
$571$	1.13657e7	1.45883	0.729417	+	0.684069i	$0.239791\pi$
$572$	0	0
$573$	−1.20270e7	−1.53028
$574$	0	0
$575$	2.62669e6	0.331314
$576$	0	0
$577$	1.56351e7	1.95506	0.977532	−	0.210785i	$-0.0676021\pi$
$577$	1.56351e7	1.95506	0.977532	+	0.210785i	$0.0676021\pi$
$578$	0	0
$579$	−7.79760e6	−0.966639
$580$	0	0
$581$	−5.05102e6	−0.620781
$582$	0	0
$583$	4.89277e6	0.596188
$584$	0	0
$585$	1.48862e6	0.179843
$586$	0	0
$587$	7.76965e6	0.930692	0.465346	−	0.885129i	$-0.345930\pi$
$587$	7.76965e6	0.930692	0.465346	+	0.885129i	$0.345930\pi$
$588$	0	0
$589$	−2.34711e6	−0.278769
$590$	0	0
$591$	7.53465e6	0.887348
$592$	0	0
$593$	6.10126e6	0.712497	0.356249	−	0.934391i	$-0.384056\pi$
$593$	6.10126e6	0.712497	0.356249	+	0.934391i	$0.384056\pi$
$594$	0	0
$595$	1.45957e6	0.169018
$596$	0	0
$597$	−1.11237e7	−1.27736
$598$	0	0
$599$	−1.30790e7	−1.48939	−0.744693	−	0.667408i	$-0.767404\pi$
$599$	−1.30790e7	−1.48939	−0.744693	+	0.667408i	$0.767404\pi$
$600$	0	0
$601$	8.63978e6	0.975701	0.487850	−	0.872927i	$-0.337781\pi$
$601$	8.63978e6	0.975701	0.487850	+	0.872927i	$0.337781\pi$
$602$	0	0
$603$	3.27441e6	0.366725
$604$	0	0
$605$	3.44871e6	0.383061
$606$	0	0
$607$	7.88899e6	0.869060	0.434530	−	0.900657i	$-0.356915\pi$
$607$	7.88899e6	0.869060	0.434530	+	0.900657i	$0.356915\pi$
$608$	0	0
$609$	−557914.	−0.0609570
$610$	0	0
$611$	−1.17644e7	−1.27487
$612$	0	0
$613$	−1.48033e7	−1.59114	−0.795568	−	0.605864i	$-0.792828\pi$
$613$	−1.48033e7	−1.59114	−0.795568	+	0.605864i	$0.792828\pi$
$614$	0	0
$615$	4.10097e6	0.437218
$616$	0	0
$617$	−8.32643e6	−0.880533	−0.440267	−	0.897867i	$-0.645116\pi$
$617$	−8.32643e6	−0.880533	−0.440267	+	0.897867i	$0.645116\pi$
$618$	0	0
$619$	1.87820e7	1.97022	0.985109	−	0.171932i	$-0.0550008\pi$
$619$	1.87820e7	1.97022	0.985109	+	0.171932i	$0.0550008\pi$
$620$	0	0
$621$	1.35631e7	1.41134
$622$	0	0
$623$	2.56398e6	0.264664
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−5.76093e6	−0.585226
$628$	0	0
$629$	1.14383e6	0.115275
$630$	0	0
$631$	−1.11975e7	−1.11956	−0.559782	−	0.828640i	$-0.689115\pi$
$631$	−1.11975e7	−1.11956	−0.559782	+	0.828640i	$0.689115\pi$
$632$	0	0
$633$	−5.22807e6	−0.518600
$634$	0	0
$635$	−4.85651e6	−0.477958
$636$	0	0
$637$	−2.52911e6	−0.246956
$638$	0	0
$639$	−211186.	−0.0204604
$640$	0	0
$641$	3.71579e6	0.357195	0.178598	−	0.983922i	$-0.442844\pi$
$641$	3.71579e6	0.357195	0.178598	+	0.983922i	$0.442844\pi$
$642$	0	0
$643$	2.06858e6	0.197308	0.0986539	−	0.995122i	$-0.468546\pi$
$643$	2.06858e6	0.197308	0.0986539	+	0.995122i	$0.468546\pi$
$644$	0	0
$645$	7.54975e6	0.714551
$646$	0	0
$647$	−7.37789e6	−0.692902	−0.346451	−	0.938068i	$-0.612613\pi$
$647$	−7.37789e6	−0.692902	−0.346451	+	0.938068i	$0.612613\pi$
$648$	0	0
$649$	4.03664e6	0.376191
$650$	0	0
$651$	908877.	0.0840529
$652$	0	0
$653$	−4.17493e6	−0.383147	−0.191574	−	0.981478i	$-0.561359\pi$
$653$	−4.17493e6	−0.383147	−0.191574	+	0.981478i	$0.561359\pi$
$654$	0	0
$655$	3.76942e6	0.343298
$656$	0	0
$657$	−3.86981e6	−0.349765
$658$	0	0
$659$	−5.07376e6	−0.455110	−0.227555	−	0.973765i	$-0.573073\pi$
$659$	−5.07376e6	−0.455110	−0.227555	+	0.973765i	$0.573073\pi$
$660$	0	0
$661$	−4.07717e6	−0.362957	−0.181479	−	0.983395i	$-0.558088\pi$
$661$	−4.07717e6	−0.362957	−0.181479	+	0.983395i	$0.558088\pi$
$662$	0	0
$663$	2.17212e7	1.91911
$664$	0	0
$665$	−2.68274e6	−0.235247
$666$	0	0
$667$	−2.76491e6	−0.240639
$668$	0	0
$669$	−3.83766e6	−0.331513
$670$	0	0
$671$	−2.86511e6	−0.245660
$672$	0	0
$673$	−1.71220e7	−1.45720	−0.728598	−	0.684942i	$-0.759828\pi$
$673$	−1.71220e7	−1.45720	−0.728598	+	0.684942i	$0.759828\pi$
$674$	0	0
$675$	2.01703e6	0.170393
$676$	0	0
$677$	1.23875e7	1.03875	0.519376	−	0.854546i	$-0.326165\pi$
$677$	1.23875e7	1.03875	0.519376	+	0.854546i	$0.326165\pi$
$678$	0	0
$679$	−2.41403e6	−0.200941
$680$	0	0
$681$	705352.	0.0582825
$682$	0	0
$683$	1.27206e7	1.04341	0.521706	−	0.853126i	$-0.325296\pi$
$683$	1.27206e7	1.04341	0.521706	+	0.853126i	$0.325296\pi$
$684$	0	0
$685$	1.81893e6	0.148112
$686$	0	0
$687$	1.61458e7	1.30517
$688$	0	0
$689$	3.39078e7	2.72114
$690$	0	0
$691$	1.82320e7	1.45257	0.726287	−	0.687392i	$-0.241244\pi$
$691$	1.82320e7	1.45257	0.726287	+	0.687392i	$0.241244\pi$
$692$	0	0
$693$	421012.	0.0333013
$694$	0	0
$695$	−1.48491e6	−0.116611
$696$	0	0
$697$	1.12932e7	0.880511
$698$	0	0
$699$	−6.26898e6	−0.485293
$700$	0	0
$701$	4.39555e6	0.337846	0.168923	−	0.985629i	$-0.445971\pi$
$701$	4.39555e6	0.337846	0.168923	+	0.985629i	$0.445971\pi$
$702$	0	0
$703$	−2.10239e6	−0.160445
$704$	0	0
$705$	4.83229e6	0.366168
$706$	0	0
$707$	8.08450e6	0.608282
$708$	0	0
$709$	−2.14394e7	−1.60176	−0.800879	−	0.598826i	$-0.795634\pi$
$709$	−2.14394e7	−1.60176	−0.800879	+	0.598826i	$0.795634\pi$
$710$	0	0
$711$	1.13440e6	0.0841571
$712$	0	0
$713$	4.50421e6	0.331814
$714$	0	0
$715$	−4.00263e6	−0.292807
$716$	0	0
$717$	7.74704e6	0.562779
$718$	0	0
$719$	−1.15783e7	−0.835262	−0.417631	−	0.908617i	$-0.637140\pi$
$719$	−1.15783e7	−0.835262	−0.417631	+	0.908617i	$0.637140\pi$
$720$	0	0
$721$	8.93855e6	0.640367
$722$	0	0
$723$	2.48198e7	1.76584
$724$	0	0
$725$	−411180.	−0.0290527
$726$	0	0
$727$	9.88538e6	0.693677	0.346839	−	0.937925i	$-0.387255\pi$
$727$	9.88538e6	0.693677	0.346839	+	0.937925i	$0.387255\pi$
$728$	0	0
$729$	9.63859e6	0.671730
$730$	0	0
$731$	2.07904e7	1.43903
$732$	0	0
$733$	−2.80117e7	−1.92566	−0.962829	−	0.270110i	$-0.912940\pi$
$733$	−2.80117e7	−1.92566	−0.962829	+	0.270110i	$0.912940\pi$
$734$	0	0
$735$	1.03885e6	0.0709305
$736$	0	0
$737$	−8.80433e6	−0.597073
$738$	0	0
$739$	−1.73017e7	−1.16540	−0.582702	−	0.812686i	$-0.698005\pi$
$739$	−1.73017e7	−1.16540	−0.582702	+	0.812686i	$0.698005\pi$
$740$	0	0
$741$	−3.99243e7	−2.67111
$742$	0	0
$743$	−1.92495e7	−1.27923	−0.639614	−	0.768697i	$-0.720906\pi$
$743$	−1.92495e7	−1.27923	−0.639614	+	0.768697i	$0.720906\pi$
$744$	0	0
$745$	−3.58357e6	−0.236551
$746$	0	0
$747$	5.82708e6	0.382076
$748$	0	0
$749$	2.19729e6	0.143114
$750$	0	0
$751$	−7.74910e6	−0.501362	−0.250681	−	0.968070i	$-0.580655\pi$
$751$	−7.74910e6	−0.501362	−0.250681	+	0.968070i	$0.580655\pi$
$752$	0	0
$753$	2.53671e7	1.63036
$754$	0	0
$755$	2.36945e6	0.151280
$756$	0	0
$757$	1.47840e7	0.937674	0.468837	−	0.883285i	$-0.344673\pi$
$757$	1.47840e7	0.937674	0.468837	+	0.883285i	$0.344673\pi$
$758$	0	0
$759$	1.10555e7	0.696586
$760$	0	0
$761$	−8.06279e6	−0.504689	−0.252344	−	0.967637i	$-0.581202\pi$
$761$	−8.06279e6	−0.504689	−0.252344	+	0.967637i	$0.581202\pi$
$762$	0	0
$763$	−9.22067e6	−0.573391
$764$	0	0
$765$	−1.68383e6	−0.104027
$766$	0	0
$767$	2.79746e7	1.71702
$768$	0	0
$769$	1.06105e6	0.0647021	0.0323510	−	0.999477i	$-0.489701\pi$
$769$	1.06105e6	0.0647021	0.0323510	+	0.999477i	$0.489701\pi$
$770$	0	0
$771$	−3.46172e7	−2.09728
$772$	0	0
$773$	1.32792e7	0.799322	0.399661	−	0.916663i	$-0.369128\pi$
$773$	1.32792e7	0.799322	0.399661	+	0.916663i	$0.369128\pi$
$774$	0	0
$775$	669838.	0.0400604
$776$	0	0
$777$	814116.	0.0483764
$778$	0	0
$779$	−2.07573e7	−1.22554
$780$	0	0
$781$	567843.	0.0333120
$782$	0	0
$783$	−2.12316e6	−0.123760
$784$	0	0
$785$	1.67539e6	0.0970382
$786$	0	0
$787$	−8.95012e6	−0.515101	−0.257550	−	0.966265i	$-0.582915\pi$
$787$	−8.95012e6	−0.515101	−0.257550	+	0.966265i	$0.582915\pi$
$788$	0	0
$789$	1.77473e7	1.01494
$790$	0	0
$791$	−5.34510e6	−0.303749
$792$	0	0
$793$	−1.98557e7	−1.12125
$794$	0	0
$795$	−1.39278e7	−0.781566
$796$	0	0
$797$	−2.14809e7	−1.19786	−0.598930	−	0.800802i	$-0.704407\pi$
$797$	−2.14809e7	−1.19786	−0.598930	+	0.800802i	$0.704407\pi$
$798$	0	0
$799$	1.33071e7	0.737422
$800$	0	0
$801$	−2.95792e6	−0.162894
$802$	0	0
$803$	1.04053e7	0.569461
$804$	0	0
$805$	5.14832e6	0.280011
$806$	0	0
$807$	−2.51199e7	−1.35779
$808$	0	0
$809$	−7.49384e6	−0.402563	−0.201281	−	0.979533i	$-0.564511\pi$
$809$	−7.49384e6	−0.402563	−0.201281	+	0.979533i	$0.564511\pi$
$810$	0	0
$811$	1.20726e7	0.644539	0.322269	−	0.946648i	$-0.395554\pi$
$811$	1.20726e7	0.644539	0.322269	+	0.946648i	$0.395554\pi$
$812$	0	0
$813$	−9.92330e6	−0.526538
$814$	0	0
$815$	−1.09506e7	−0.577488
$816$	0	0
$817$	−3.82134e7	−2.00291
$818$	0	0
$819$	2.91769e6	0.151995
$820$	0	0
$821$	−3.15534e7	−1.63376	−0.816879	−	0.576809i	$-0.804298\pi$
$821$	−3.15534e7	−1.63376	−0.816879	+	0.576809i	$0.804298\pi$
$822$	0	0
$823$	1.20898e7	0.622186	0.311093	−	0.950379i	$-0.399305\pi$
$823$	1.20898e7	0.622186	0.311093	+	0.950379i	$0.399305\pi$
$824$	0	0
$825$	1.64411e6	0.0840998
$826$	0	0
$827$	−2.65153e7	−1.34813	−0.674066	−	0.738671i	$-0.735454\pi$
$827$	−2.65153e7	−1.34813	−0.674066	+	0.738671i	$0.735454\pi$
$828$	0	0
$829$	412122.	0.0208276	0.0104138	−	0.999946i	$-0.496685\pi$
$829$	412122.	0.0208276	0.0104138	+	0.999946i	$0.496685\pi$
$830$	0	0
$831$	1.17941e7	0.592465
$832$	0	0
$833$	2.86076e6	0.142846
$834$	0	0
$835$	−1.75411e7	−0.870642
$836$	0	0
$837$	3.45877e6	0.170650
$838$	0	0
$839$	1.56082e7	0.765505	0.382753	−	0.923851i	$-0.374976\pi$
$839$	1.56082e7	0.765505	0.382753	+	0.923851i	$0.374976\pi$
$840$	0	0
$841$	−2.00783e7	−0.978898
$842$	0	0
$843$	2.42917e7	1.17731
$844$	0	0
$845$	−1.84566e7	−0.889224
$846$	0	0
$847$	6.75947e6	0.323746
$848$	0	0
$849$	3.59623e7	1.71229
$850$	0	0
$851$	4.03460e6	0.190975
$852$	0	0
$853$	−3.49328e7	−1.64385	−0.821923	−	0.569599i	$-0.807099\pi$
$853$	−3.49328e7	−1.64385	−0.821923	+	0.569599i	$0.807099\pi$
$854$	0	0
$855$	3.09493e6	0.144789
$856$	0	0
$857$	−3.12764e7	−1.45467	−0.727335	−	0.686282i	$-0.759241\pi$
$857$	−3.12764e7	−1.45467	−0.727335	+	0.686282i	$0.759241\pi$
$858$	0	0
$859$	3.18771e7	1.47400	0.736998	−	0.675895i	$-0.236243\pi$
$859$	3.18771e7	1.47400	0.736998	+	0.675895i	$0.236243\pi$
$860$	0	0
$861$	8.03789e6	0.369517
$862$	0	0
$863$	−8.54790e6	−0.390690	−0.195345	−	0.980735i	$-0.562583\pi$
$863$	−8.54790e6	−0.390690	−0.195345	+	0.980735i	$0.562583\pi$
$864$	0	0
$865$	−1.51971e6	−0.0690588
$866$	0	0
$867$	3687.56	0.000166606 0
$868$	0	0
$869$	−3.05020e6	−0.137018
$870$	0	0
$871$	−6.10156e7	−2.72518
$872$	0	0
$873$	2.78493e6	0.123674
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−1.08208e7	−0.475071	−0.237536	−	0.971379i	$-0.576340\pi$
$877$	−1.08208e7	−0.475071	−0.237536	+	0.971379i	$0.576340\pi$
$878$	0	0
$879$	−2.88662e6	−0.126014
$880$	0	0
$881$	1.93246e7	0.838824	0.419412	−	0.907796i	$-0.362236\pi$
$881$	1.93246e7	0.838824	0.419412	+	0.907796i	$0.362236\pi$
$882$	0	0
$883$	1.87836e7	0.810731	0.405365	−	0.914155i	$-0.367144\pi$
$883$	1.87836e7	0.810731	0.405365	+	0.914155i	$0.367144\pi$
$884$	0	0
$885$	−1.14907e7	−0.493163
$886$	0	0
$887$	3.48834e7	1.48871	0.744355	−	0.667784i	$-0.232757\pi$
$887$	3.48834e7	1.48871	0.744355	+	0.667784i	$0.232757\pi$
$888$	0	0
$889$	−9.51875e6	−0.403948
$890$	0	0
$891$	1.05774e7	0.446358
$892$	0	0
$893$	−2.44589e7	−1.02638
$894$	0	0
$895$	−4.28878e6	−0.178968
$896$	0	0
$897$	7.66167e7	3.17938
$898$	0	0
$899$	−705085.	−0.0290966
$900$	0	0
$901$	−3.83543e7	−1.57399
$902$	0	0
$903$	1.47975e7	0.603906
$904$	0	0
$905$	−6.46315e6	−0.262315
$906$	0	0
$907$	−485424.	−0.0195931	−0.00979655	−	0.999952i	$-0.503118\pi$
$907$	−485424.	−0.0195931	−0.00979655	+	0.999952i	$0.503118\pi$
$908$	0	0
$909$	−9.32664e6	−0.374382
$910$	0	0
$911$	3.48489e7	1.39121	0.695605	−	0.718425i	$-0.255136\pi$
$911$	3.48489e7	1.39121	0.695605	+	0.718425i	$0.255136\pi$
$912$	0	0
$913$	−1.56680e7	−0.622066
$914$	0	0
$915$	8.15585e6	0.322045
$916$	0	0
$917$	7.38806e6	0.290140
$918$	0	0
$919$	−3.02317e7	−1.18079	−0.590396	−	0.807114i	$-0.701028\pi$
$919$	−3.02317e7	−1.18079	−0.590396	+	0.807114i	$0.701028\pi$
$920$	0	0
$921$	−2.92790e7	−1.13738
$922$	0	0
$923$	3.93525e6	0.152044
$924$	0	0
$925$	600000.	0.0230567
$926$	0	0
$927$	−1.03119e7	−0.394130
$928$	0	0
$929$	−3.14486e6	−0.119554	−0.0597768	−	0.998212i	$-0.519039\pi$
$929$	−3.14486e6	−0.119554	−0.0597768	+	0.998212i	$0.519039\pi$
$930$	0	0
$931$	−5.25817e6	−0.198820
$932$	0	0
$933$	−5.51300e6	−0.207340
$934$	0	0
$935$	4.52752e6	0.169368
$936$	0	0
$937$	1.52536e7	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$937$	1.52536e7	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$938$	0	0
$939$	−3.06358e7	−1.13388
$940$	0	0
$941$	4.35966e7	1.60501	0.802507	−	0.596643i	$-0.203499\pi$
$941$	4.35966e7	1.60501	0.802507	+	0.596643i	$0.203499\pi$
$942$	0	0
$943$	3.98342e7	1.45874
$944$	0	0
$945$	3.95337e6	0.144008
$946$	0	0
$947$	9.50274e6	0.344329	0.172165	−	0.985068i	$-0.444924\pi$
$947$	9.50274e6	0.344329	0.172165	+	0.985068i	$0.444924\pi$
$948$	0	0
$949$	7.21103e7	2.59915
$950$	0	0
$951$	−4.87341e6	−0.174736
$952$	0	0
$953$	3.79636e7	1.35405	0.677026	−	0.735959i	$-0.263268\pi$
$953$	3.79636e7	1.35405	0.677026	+	0.735959i	$0.263268\pi$
$954$	0	0
$955$	−1.73731e7	−0.616410
$956$	0	0
$957$	−1.73062e6	−0.0610832
$958$	0	0
$959$	3.56510e6	0.125177
$960$	0	0
$961$	−2.74805e7	−0.959879
$962$	0	0
$963$	−2.53489e6	−0.0880831
$964$	0	0
$965$	−1.12637e7	−0.389371
$966$	0	0
$967$	9.58977e6	0.329794	0.164897	−	0.986311i	$-0.447271\pi$
$967$	9.58977e6	0.329794	0.164897	+	0.986311i	$0.447271\pi$
$968$	0	0
$969$	4.51598e7	1.54505
$970$	0	0
$971$	−1.48924e7	−0.506892	−0.253446	−	0.967350i	$-0.581564\pi$
$971$	−1.48924e7	−0.506892	−0.253446	+	0.967350i	$0.581564\pi$
$972$	0	0
$973$	−2.91043e6	−0.0985543
$974$	0	0
$975$	1.13939e7	0.383851
$976$	0	0
$977$	4.83055e7	1.61905	0.809525	−	0.587085i	$-0.199725\pi$
$977$	4.83055e7	1.61905	0.809525	+	0.587085i	$0.199725\pi$
$978$	0	0
$979$	7.95333e6	0.265211
$980$	0	0
$981$	1.06374e7	0.352908
$982$	0	0
$983$	1.03155e7	0.340492	0.170246	−	0.985402i	$-0.445544\pi$
$983$	1.03155e7	0.340492	0.170246	+	0.985402i	$0.445544\pi$
$984$	0	0
$985$	1.08839e7	0.357432
$986$	0	0
$987$	9.47128e6	0.309468
$988$	0	0
$989$	7.33335e7	2.38403
$990$	0	0
$991$	−2.25954e7	−0.730863	−0.365432	−	0.930838i	$-0.619079\pi$
$991$	−2.25954e7	−0.730863	−0.365432	+	0.930838i	$0.619079\pi$
$992$	0	0
$993$	−7.99942e6	−0.257446
$994$	0	0
$995$	−1.60683e7	−0.514532
$996$	0	0
$997$	−5.03579e7	−1.60446	−0.802231	−	0.597014i	$-0.796354\pi$
$997$	−5.03579e7	−1.60446	−0.802231	+	0.597014i	$0.796354\pi$
$998$	0	0
$999$	3.09815e6	0.0982175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.c.1.1	✓	3
4.3	odd	2		560.6.a.u.1.3		3
5.2	odd	4		700.6.e.h.449.6		6
5.3	odd	4		700.6.e.h.449.1		6
5.4	even	2		700.6.a.j.1.3		3
7.6	odd	2		980.6.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.c.1.1	✓	3	1.1	even	1	trivial
560.6.a.u.1.3		3	4.3	odd	2
700.6.a.j.1.3		3	5.4	even	2
700.6.e.h.449.1		6	5.3	odd	4
700.6.e.h.449.6		6	5.2	odd	4
980.6.a.i.1.3		3	7.6	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

\(f(q)\)	\(=\)	\(q-17.3069 q^{3} -25.0000 q^{5} -49.0000 q^{7} +56.5286 q^{9} +O(q^{10})\)	\(q-17.3069 q^{3} -25.0000 q^{5} -49.0000 q^{7} +56.5286 q^{9} -151.996 q^{11} -1053.36 q^{13} +432.672 q^{15} +1191.49 q^{17} -2189.99 q^{19} +848.038 q^{21} +4202.71 q^{23} +625.000 q^{25} +3227.24 q^{27} -657.888 q^{29} +1071.74 q^{31} +2630.57 q^{33} +1225.00 q^{35} +960.000 q^{37} +18230.3 q^{39} +9478.23 q^{41} +17449.1 q^{43} -1413.21 q^{45} +11168.5 q^{47} +2401.00 q^{49} -20621.0 q^{51} -32190.2 q^{53} +3799.89 q^{55} +37902.0 q^{57} -26557.6 q^{59} +18849.9 q^{61} -2769.90 q^{63} +26333.9 q^{65} +57924.9 q^{67} -72735.8 q^{69} -3735.92 q^{71} -68457.7 q^{73} -10816.8 q^{75} +7447.78 q^{77} +20067.7 q^{79} -69590.0 q^{81} +103082. q^{83} -29787.2 q^{85} +11386.0 q^{87} -52326.1 q^{89} +51614.4 q^{91} -18548.5 q^{93} +54749.8 q^{95} +49265.9 q^{97} -8592.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9}+O(q^{10}) \) 3 * q - 10 * q^3 - 75 * q^5 - 147 * q^7 - 71 * q^9	\( 3 q - 10 q^{3} - 75 q^{5} - 147 q^{7} - 71 q^{9} + 58 q^{11} + 148 q^{13} + 250 q^{15} + 1360 q^{17} + 1692 q^{19} + 490 q^{21} + 4204 q^{23} + 1875 q^{25} + 3410 q^{27} + 5572 q^{29} + 2816 q^{31} + 11794 q^{33} + 3675 q^{35} + 15210 q^{37} + 14174 q^{39} + 7706 q^{41} + 24904 q^{43} + 1775 q^{45} + 21350 q^{47} + 7203 q^{49} + 31722 q^{51} + 342 q^{53} - 1450 q^{55} + 52548 q^{57} + 12888 q^{59} - 5614 q^{61} + 3479 q^{63} - 3700 q^{65} + 35196 q^{67} - 31036 q^{69} - 19520 q^{71} - 60506 q^{73} - 6250 q^{75} - 2842 q^{77} + 61122 q^{79} - 130853 q^{81} - 11664 q^{83} - 34000 q^{85} - 5686 q^{87} - 144878 q^{89} - 7252 q^{91} - 123544 q^{93} - 42300 q^{95} + 17304 q^{97} + 39368 q^{99}+O(q^{100}) \) 3 * q - 10 * q^3 - 75 * q^5 - 147 * q^7 - 71 * q^9 + 58 * q^11 + 148 * q^13 + 250 * q^15 + 1360 * q^17 + 1692 * q^19 + 490 * q^21 + 4204 * q^23 + 1875 * q^25 + 3410 * q^27 + 5572 * q^29 + 2816 * q^31 + 11794 * q^33 + 3675 * q^35 + 15210 * q^37 + 14174 * q^39 + 7706 * q^41 + 24904 * q^43 + 1775 * q^45 + 21350 * q^47 + 7203 * q^49 + 31722 * q^51 + 342 * q^53 - 1450 * q^55 + 52548 * q^57 + 12888 * q^59 - 5614 * q^61 + 3479 * q^63 - 3700 * q^65 + 35196 * q^67 - 31036 * q^69 - 19520 * q^71 - 60506 * q^73 - 6250 * q^75 - 2842 * q^77 + 61122 * q^79 - 130853 * q^81 - 11664 * q^83 - 34000 * q^85 - 5686 * q^87 - 144878 * q^89 - 7252 * q^91 - 123544 * q^93 - 42300 * q^95 + 17304 * q^97 + 39368 * q^99

Introduction

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