LMFDB - Embedded newform 140.6.a.b.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1009}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$16.3824$ 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)

$f(q)$	$=$	$q-24.3824 q^{3} +25.0000 q^{5} -49.0000 q^{7} +351.500 q^{9} +O(q^{10})$	$q-24.3824 q^{3} +25.0000 q^{5} -49.0000 q^{7} +351.500 q^{9} +59.4414 q^{11} +873.678 q^{13} -609.560 q^{15} +414.795 q^{17} -1397.59 q^{19} +1194.74 q^{21} +839.708 q^{23} +625.000 q^{25} -2645.50 q^{27} -7593.04 q^{29} -7018.83 q^{31} -1449.32 q^{33} -1225.00 q^{35} +2763.65 q^{37} -21302.3 q^{39} -14373.8 q^{41} -22368.9 q^{43} +8787.51 q^{45} +12046.6 q^{47} +2401.00 q^{49} -10113.7 q^{51} -14422.9 q^{53} +1486.04 q^{55} +34076.6 q^{57} +16743.3 q^{59} +33828.2 q^{61} -17223.5 q^{63} +21841.9 q^{65} +12135.2 q^{67} -20474.1 q^{69} -43847.6 q^{71} +33432.7 q^{73} -15239.0 q^{75} -2912.63 q^{77} -36877.6 q^{79} -20911.0 q^{81} -114590. q^{83} +10369.9 q^{85} +185136. q^{87} -58796.4 q^{89} -42810.2 q^{91} +171136. q^{93} -34939.7 q^{95} +34474.9 q^{97} +20893.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9}+O(q^{10}) $ 2 * q - 17 * q^3 + 50 * q^5 - 98 * q^7 + 163 * q^9	$ 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} - 167 q^{11} + 445 q^{13} - 425 q^{15} + 99 q^{17} - 1334 q^{19} + 833 q^{21} - 290 q^{23} + 1250 q^{25} - 5831 q^{27} - 6959 q^{29} - 10480 q^{31} - 3121 q^{33} - 2450 q^{35} - 1588 q^{37} - 24467 q^{39} - 24110 q^{41} - 20406 q^{43} + 4075 q^{45} + 619 q^{47} + 4802 q^{49} - 12445 q^{51} - 21794 q^{53} - 4175 q^{55} + 34546 q^{57} + 1976 q^{59} + 22614 q^{61} - 7987 q^{63} + 11125 q^{65} + 4068 q^{67} - 28814 q^{69} - 46528 q^{71} + 27096 q^{73} - 10625 q^{75} + 8183 q^{77} - 12481 q^{79} + 1378 q^{81} - 100088 q^{83} + 2475 q^{85} + 189817 q^{87} + 33798 q^{89} - 21805 q^{91} + 145584 q^{93} - 33350 q^{95} + 138991 q^{97} + 63578 q^{99}+O(q^{100}) $ 2 * q - 17 * q^3 + 50 * q^5 - 98 * q^7 + 163 * q^9 - 167 * q^11 + 445 * q^13 - 425 * q^15 + 99 * q^17 - 1334 * q^19 + 833 * q^21 - 290 * q^23 + 1250 * q^25 - 5831 * q^27 - 6959 * q^29 - 10480 * q^31 - 3121 * q^33 - 2450 * q^35 - 1588 * q^37 - 24467 * q^39 - 24110 * q^41 - 20406 * q^43 + 4075 * q^45 + 619 * q^47 + 4802 * q^49 - 12445 * q^51 - 21794 * q^53 - 4175 * q^55 + 34546 * q^57 + 1976 * q^59 + 22614 * q^61 - 7987 * q^63 + 11125 * q^65 + 4068 * q^67 - 28814 * q^69 - 46528 * q^71 + 27096 * q^73 - 10625 * q^75 + 8183 * q^77 - 12481 * q^79 + 1378 * q^81 - 100088 * q^83 + 2475 * q^85 + 189817 * q^87 + 33798 * q^89 - 21805 * q^91 + 145584 * q^93 - 33350 * q^95 + 138991 * q^97 + 63578 * 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
$3$	−24.3824	−1.56413	−0.782065	−	0.623197i	$-0.785834\pi$
$3$	−24.3824	−1.56413	−0.782065	+	0.623197i	$0.785834\pi$
$4$	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
$9$	351.500	1.44650
$10$	0	0
$11$	59.4414	0.148118	0.0740589	−	0.997254i	$-0.476405\pi$
$11$	59.4414	0.148118	0.0740589	+	0.997254i	$0.476405\pi$
$12$	0	0
$13$	873.678	1.43381	0.716907	−	0.697169i	$-0.245557\pi$
$13$	873.678	1.43381	0.716907	+	0.697169i	$0.245557\pi$
$14$	0	0
$15$	−609.560	−0.699500
$16$	0	0
$17$	414.795	0.348106	0.174053	−	0.984736i	$-0.444314\pi$
$17$	414.795	0.348106	0.174053	+	0.984736i	$0.444314\pi$
$18$	0	0
$19$	−1397.59	−0.888169	−0.444085	−	0.895985i	$-0.646471\pi$
$19$	−1397.59	−0.888169	−0.444085	+	0.895985i	$0.646471\pi$
$20$	0	0
$21$	1194.74	0.591186
$22$	0	0
$23$	839.708	0.330985	0.165493	−	0.986211i	$-0.447079\pi$
$23$	839.708	0.330985	0.165493	+	0.986211i	$0.447079\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2645.50	−0.698390
$28$	0	0
$29$	−7593.04	−1.67657	−0.838283	−	0.545236i	$-0.816440\pi$
$29$	−7593.04	−1.67657	−0.838283	+	0.545236i	$0.816440\pi$
$30$	0	0
$31$	−7018.83	−1.31178	−0.655889	−	0.754857i	$-0.727706\pi$
$31$	−7018.83	−1.31178	−0.655889	+	0.754857i	$0.727706\pi$
$32$	0	0
$33$	−1449.32	−0.231676
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	2763.65	0.331879	0.165939	−	0.986136i	$-0.446934\pi$
$37$	2763.65	0.331879	0.165939	+	0.986136i	$0.446934\pi$
$38$	0	0
$39$	−21302.3	−2.24267
$40$	0	0
$41$	−14373.8	−1.33540	−0.667702	−	0.744429i	$-0.732722\pi$
$41$	−14373.8	−1.33540	−0.667702	+	0.744429i	$0.732722\pi$
$42$	0	0
$43$	−22368.9	−1.84490	−0.922452	−	0.386113i	$-0.873818\pi$
$43$	−22368.9	−1.84490	−0.922452	+	0.386113i	$0.873818\pi$
$44$	0	0
$45$	8787.51	0.646896
$46$	0	0
$47$	12046.6	0.795461	0.397731	−	0.917502i	$-0.369798\pi$
$47$	12046.6	0.795461	0.397731	+	0.917502i	$0.369798\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−10113.7	−0.544482
$52$	0	0
$53$	−14422.9	−0.705282	−0.352641	−	0.935759i	$-0.614716\pi$
$53$	−14422.9	−0.705282	−0.352641	+	0.935759i	$0.614716\pi$
$54$	0	0
$55$	1486.04	0.0662403
$56$	0	0
$57$	34076.6	1.38921
$58$	0	0
$59$	16743.3	0.626198	0.313099	−	0.949721i	$-0.398633\pi$
$59$	16743.3	0.626198	0.313099	+	0.949721i	$0.398633\pi$
$60$	0	0
$61$	33828.2	1.16400	0.582002	−	0.813187i	$-0.302270\pi$
$61$	33828.2	1.16400	0.582002	+	0.813187i	$0.302270\pi$
$62$	0	0
$63$	−17223.5	−0.546727
$64$	0	0
$65$	21841.9	0.641221
$66$	0	0
$67$	12135.2	0.330263	0.165131	−	0.986272i	$-0.447195\pi$
$67$	12135.2	0.330263	0.165131	+	0.986272i	$0.447195\pi$
$68$	0	0
$69$	−20474.1	−0.517704
$70$	0	0
$71$	−43847.6	−1.03228	−0.516142	−	0.856503i	$-0.672633\pi$
$71$	−43847.6	−1.03228	−0.516142	+	0.856503i	$0.672633\pi$
$72$	0	0
$73$	33432.7	0.734285	0.367143	−	0.930165i	$-0.380336\pi$
$73$	33432.7	0.734285	0.367143	+	0.930165i	$0.380336\pi$
$74$	0	0
$75$	−15239.0	−0.312826
$76$	0	0
$77$	−2912.63	−0.0559833
$78$	0	0
$79$	−36877.6	−0.664806	−0.332403	−	0.943137i	$-0.607859\pi$
$79$	−36877.6	−0.664806	−0.332403	+	0.943137i	$0.607859\pi$
$80$	0	0
$81$	−20911.0	−0.354130
$82$	0	0
$83$	−114590.	−1.82579	−0.912896	−	0.408191i	$-0.866160\pi$
$83$	−114590.	−1.82579	−0.912896	+	0.408191i	$0.866160\pi$
$84$	0	0
$85$	10369.9	0.155678
$86$	0	0
$87$	185136.	2.62237
$88$	0	0
$89$	−58796.4	−0.786821	−0.393410	−	0.919363i	$-0.628705\pi$
$89$	−58796.4	−0.786821	−0.393410	+	0.919363i	$0.628705\pi$
$90$	0	0
$91$	−42810.2	−0.541931
$92$	0	0
$93$	171136.	2.05179
$94$	0	0
$95$	−34939.7	−0.397201
$96$	0	0
$97$	34474.9	0.372026	0.186013	−	0.982547i	$-0.440443\pi$
$97$	34474.9	0.372026	0.186013	+	0.982547i	$0.440443\pi$
$98$	0	0
$99$	20893.7	0.214253
$100$	0	0
$101$	101322.	0.988330	0.494165	−	0.869368i	$-0.335474\pi$
$101$	101322.	0.988330	0.494165	+	0.869368i	$0.335474\pi$
$102$	0	0
$103$	−28647.8	−0.266071	−0.133036	−	0.991111i	$-0.542472\pi$
$103$	−28647.8	−0.266071	−0.133036	+	0.991111i	$0.542472\pi$
$104$	0	0
$105$	29868.4	0.264386
$106$	0	0
$107$	89387.2	0.754772	0.377386	−	0.926056i	$-0.376823\pi$
$107$	89387.2	0.754772	0.377386	+	0.926056i	$0.376823\pi$
$108$	0	0
$109$	−143199.	−1.15444	−0.577221	−	0.816588i	$-0.695863\pi$
$109$	−143199.	−1.15444	−0.577221	+	0.816588i	$0.695863\pi$
$110$	0	0
$111$	−67384.4	−0.519101
$112$	0	0
$113$	−130142.	−0.958784	−0.479392	−	0.877601i	$-0.659143\pi$
$113$	−130142.	−0.958784	−0.479392	+	0.877601i	$0.659143\pi$
$114$	0	0
$115$	20992.7	0.148021
$116$	0	0
$117$	307098.	2.07402
$118$	0	0
$119$	−20324.9	−0.131572
$120$	0	0
$121$	−157518.	−0.978061
$122$	0	0
$123$	350468.	2.08875
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−163710.	−0.900671	−0.450336	−	0.892859i	$-0.648696\pi$
$127$	−163710.	−0.900671	−0.450336	+	0.892859i	$0.648696\pi$
$128$	0	0
$129$	545407.	2.88567
$130$	0	0
$131$	306287.	1.55937	0.779686	−	0.626171i	$-0.215379\pi$
$131$	306287.	1.55937	0.779686	+	0.626171i	$0.215379\pi$
$132$	0	0
$133$	68481.9	0.335696
$134$	0	0
$135$	−66137.5	−0.312330
$136$	0	0
$137$	−27852.7	−0.126784	−0.0633922	−	0.997989i	$-0.520192\pi$
$137$	−27852.7	−0.126784	−0.0633922	+	0.997989i	$0.520192\pi$
$138$	0	0
$139$	176535.	0.774986	0.387493	−	0.921873i	$-0.373341\pi$
$139$	176535.	0.774986	0.387493	+	0.921873i	$0.373341\pi$
$140$	0	0
$141$	−293724.	−1.24421
$142$	0	0
$143$	51932.6	0.212373
$144$	0	0
$145$	−189826.	−0.749783
$146$	0	0
$147$	−58542.1	−0.223447
$148$	0	0
$149$	−47427.0	−0.175009	−0.0875045	−	0.996164i	$-0.527889\pi$
$149$	−47427.0	−0.175009	−0.0875045	+	0.996164i	$0.527889\pi$
$150$	0	0
$151$	−509583.	−1.81875	−0.909374	−	0.415979i	$-0.863439\pi$
$151$	−509583.	−1.81875	−0.909374	+	0.415979i	$0.863439\pi$
$152$	0	0
$153$	145801.	0.503536
$154$	0	0
$155$	−175471.	−0.586645
$156$	0	0
$157$	−285482.	−0.924334	−0.462167	−	0.886793i	$-0.652928\pi$
$157$	−285482.	−0.924334	−0.462167	+	0.886793i	$0.652928\pi$
$158$	0	0
$159$	351664.	1.10315
$160$	0	0
$161$	−41145.7	−0.125101
$162$	0	0
$163$	−312305.	−0.920682	−0.460341	−	0.887742i	$-0.652273\pi$
$163$	−312305.	−0.920682	−0.460341	+	0.887742i	$0.652273\pi$
$164$	0	0
$165$	−36233.1	−0.103609
$166$	0	0
$167$	−247746.	−0.687408	−0.343704	−	0.939078i	$-0.611682\pi$
$167$	−247746.	−0.687408	−0.343704	+	0.939078i	$0.611682\pi$
$168$	0	0
$169$	392020.	1.05582
$170$	0	0
$171$	−491253.	−1.28474
$172$	0	0
$173$	−301061.	−0.764784	−0.382392	−	0.924000i	$-0.624900\pi$
$173$	−301061.	−0.764784	−0.382392	+	0.924000i	$0.624900\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−408242.	−0.979455
$178$	0	0
$179$	134306.	0.313301	0.156651	−	0.987654i	$-0.449930\pi$
$179$	134306.	0.313301	0.156651	+	0.987654i	$0.449930\pi$
$180$	0	0
$181$	−384828.	−0.873113	−0.436556	−	0.899677i	$-0.643802\pi$
$181$	−384828.	−0.873113	−0.436556	+	0.899677i	$0.643802\pi$
$182$	0	0
$183$	−824812.	−1.82065
$184$	0	0
$185$	69091.3	0.148421
$186$	0	0
$187$	24656.0	0.0515607
$188$	0	0
$189$	129629.	0.263967
$190$	0	0
$191$	−689905.	−1.36838	−0.684189	−	0.729305i	$-0.739843\pi$
$191$	−689905.	−1.36838	−0.684189	+	0.729305i	$0.739843\pi$
$192$	0	0
$193$	702104.	1.35678	0.678388	−	0.734704i	$-0.262679\pi$
$193$	702104.	1.35678	0.678388	+	0.734704i	$0.262679\pi$
$194$	0	0
$195$	−532558.	−1.00295
$196$	0	0
$197$	544207.	0.999077	0.499538	−	0.866292i	$-0.333503\pi$
$197$	544207.	0.999077	0.499538	+	0.866292i	$0.333503\pi$
$198$	0	0
$199$	−312684.	−0.559723	−0.279862	−	0.960040i	$-0.590289\pi$
$199$	−312684.	−0.559723	−0.279862	+	0.960040i	$0.590289\pi$
$200$	0	0
$201$	−295885.	−0.516574
$202$	0	0
$203$	372059.	0.633682
$204$	0	0
$205$	−359346.	−0.597211
$206$	0	0
$207$	295158.	0.478771
$208$	0	0
$209$	−83074.7	−0.131554
$210$	0	0
$211$	552813.	0.854815	0.427407	−	0.904059i	$-0.359427\pi$
$211$	552813.	0.854815	0.427407	+	0.904059i	$0.359427\pi$
$212$	0	0
$213$	1.06911e6	1.61463
$214$	0	0
$215$	−559223.	−0.825066
$216$	0	0
$217$	343923.	0.495805
$218$	0	0
$219$	−815170.	−1.14852
$220$	0	0
$221$	362397.	0.499119
$222$	0	0
$223$	−717458.	−0.966127	−0.483064	−	0.875585i	$-0.660476\pi$
$223$	−717458.	−0.966127	−0.483064	+	0.875585i	$0.660476\pi$
$224$	0	0
$225$	219688.	0.289301
$226$	0	0
$227$	680024.	0.875911	0.437955	−	0.898997i	$-0.355703\pi$
$227$	680024.	0.875911	0.437955	+	0.898997i	$0.355703\pi$
$228$	0	0
$229$	−238359.	−0.300361	−0.150180	−	0.988659i	$-0.547985\pi$
$229$	−238359.	−0.300361	−0.150180	+	0.988659i	$0.547985\pi$
$230$	0	0
$231$	71016.8	0.0875652
$232$	0	0
$233$	442042.	0.533425	0.266712	−	0.963776i	$-0.414063\pi$
$233$	442042.	0.533425	0.266712	+	0.963776i	$0.414063\pi$
$234$	0	0
$235$	301164.	0.355741
$236$	0	0
$237$	899164.	1.03984
$238$	0	0
$239$	1.09492e6	1.23990	0.619952	−	0.784639i	$-0.287152\pi$
$239$	1.09492e6	1.23990	0.619952	+	0.784639i	$0.287152\pi$
$240$	0	0
$241$	−1.47128e6	−1.63175	−0.815875	−	0.578228i	$-0.803744\pi$
$241$	−1.47128e6	−1.63175	−0.815875	+	0.578228i	$0.803744\pi$
$242$	0	0
$243$	1.15272e6	1.25230
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−1.22104e6	−1.27347
$248$	0	0
$249$	2.79398e6	2.85578
$250$	0	0
$251$	579697.	0.580787	0.290394	−	0.956907i	$-0.406214\pi$
$251$	579697.	0.580787	0.290394	+	0.956907i	$0.406214\pi$
$252$	0	0
$253$	49913.4	0.0490248
$254$	0	0
$255$	−252842.	−0.243500
$256$	0	0
$257$	1.05839e6	0.999573	0.499786	−	0.866149i	$-0.333412\pi$
$257$	1.05839e6	0.999573	0.499786	+	0.866149i	$0.333412\pi$
$258$	0	0
$259$	−135419.	−0.125438
$260$	0	0
$261$	−2.66896e6	−2.42516
$262$	0	0
$263$	−1.76047e6	−1.56942	−0.784710	−	0.619863i	$-0.787188\pi$
$263$	−1.76047e6	−1.56942	−0.784710	+	0.619863i	$0.787188\pi$
$264$	0	0
$265$	−360572.	−0.315412
$266$	0	0
$267$	1.43360e6	1.23069
$268$	0	0
$269$	385475.	0.324799	0.162400	−	0.986725i	$-0.448077\pi$
$269$	385475.	0.324799	0.162400	+	0.986725i	$0.448077\pi$
$270$	0	0
$271$	1.02368e6	0.846723	0.423362	−	0.905961i	$-0.360850\pi$
$271$	1.02368e6	0.846723	0.423362	+	0.905961i	$0.360850\pi$
$272$	0	0
$273$	1.04381e6	0.847650
$274$	0	0
$275$	37150.9	0.0296236
$276$	0	0
$277$	1.29828e6	1.01664	0.508322	−	0.861167i	$-0.330266\pi$
$277$	1.29828e6	1.01664	0.508322	+	0.861167i	$0.330266\pi$
$278$	0	0
$279$	−2.46712e6	−1.89749
$280$	0	0
$281$	−1.98354e6	−1.49856	−0.749282	−	0.662251i	$-0.769601\pi$
$281$	−1.98354e6	−1.49856	−0.749282	+	0.662251i	$0.769601\pi$
$282$	0	0
$283$	−1.84574e6	−1.36995	−0.684975	−	0.728566i	$-0.740187\pi$
$283$	−1.84574e6	−1.36995	−0.684975	+	0.728566i	$0.740187\pi$
$284$	0	0
$285$	851914.	0.621275
$286$	0	0
$287$	704318.	0.504735
$288$	0	0
$289$	−1.24780e6	−0.878823
$290$	0	0
$291$	−840579.	−0.581897
$292$	0	0
$293$	2.01175e6	1.36901	0.684504	−	0.729010i	$-0.260019\pi$
$293$	2.01175e6	1.36901	0.684504	+	0.729010i	$0.260019\pi$
$294$	0	0
$295$	418583.	0.280044
$296$	0	0
$297$	−157252.	−0.103444
$298$	0	0
$299$	733634.	0.474571
$300$	0	0
$301$	1.09608e6	0.697308
$302$	0	0
$303$	−2.47048e6	−1.54588
$304$	0	0
$305$	845705.	0.520558
$306$	0	0
$307$	2.09488e6	1.26857	0.634284	−	0.773100i	$-0.281295\pi$
$307$	2.09488e6	1.26857	0.634284	+	0.773100i	$0.281295\pi$
$308$	0	0
$309$	698501.	0.416170
$310$	0	0
$311$	2.02927e6	1.18970	0.594852	−	0.803835i	$-0.297211\pi$
$311$	2.02927e6	1.18970	0.594852	+	0.803835i	$0.297211\pi$
$312$	0	0
$313$	233443.	0.134685	0.0673427	−	0.997730i	$-0.478548\pi$
$313$	233443.	0.134685	0.0673427	+	0.997730i	$0.478548\pi$
$314$	0	0
$315$	−430588.	−0.244504
$316$	0	0
$317$	2.88793e6	1.61413	0.807065	−	0.590462i	$-0.201055\pi$
$317$	2.88793e6	1.61413	0.807065	+	0.590462i	$0.201055\pi$
$318$	0	0
$319$	−451341.	−0.248329
$320$	0	0
$321$	−2.17947e6	−1.18056
$322$	0	0
$323$	−579713.	−0.309177
$324$	0	0
$325$	546048.	0.286763
$326$	0	0
$327$	3.49152e6	1.80570
$328$	0	0
$329$	−590282.	−0.300656
$330$	0	0
$331$	1.33037e6	0.667423	0.333712	−	0.942675i	$-0.391699\pi$
$331$	1.33037e6	0.667423	0.333712	+	0.942675i	$0.391699\pi$
$332$	0	0
$333$	971425.	0.480064
$334$	0	0
$335$	303380.	0.147698
$336$	0	0
$337$	−3.47180e6	−1.66525	−0.832626	−	0.553835i	$-0.813164\pi$
$337$	−3.47180e6	−1.66525	−0.832626	+	0.553835i	$0.813164\pi$
$338$	0	0
$339$	3.17317e6	1.49966
$340$	0	0
$341$	−417209.	−0.194298
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−511852.	−0.231524
$346$	0	0
$347$	3.02779e6	1.34990	0.674951	−	0.737862i	$-0.264165\pi$
$347$	3.02779e6	1.34990	0.674951	+	0.737862i	$0.264165\pi$
$348$	0	0
$349$	−3.98788e6	−1.75258	−0.876292	−	0.481781i	$-0.839990\pi$
$349$	−3.98788e6	−1.75258	−0.876292	+	0.481781i	$0.839990\pi$
$350$	0	0
$351$	−2.31131e6	−1.00136
$352$	0	0
$353$	−1.53257e6	−0.654611	−0.327305	−	0.944919i	$-0.606141\pi$
$353$	−1.53257e6	−0.654611	−0.327305	+	0.944919i	$0.606141\pi$
$354$	0	0
$355$	−1.09619e6	−0.461652
$356$	0	0
$357$	495570.	0.205795
$358$	0	0
$359$	−3.34312e6	−1.36904	−0.684519	−	0.728995i	$-0.739988\pi$
$359$	−3.34312e6	−1.36904	−0.684519	+	0.728995i	$0.739988\pi$
$360$	0	0
$361$	−522843.	−0.211156
$362$	0	0
$363$	3.84066e6	1.52982
$364$	0	0
$365$	835818.	0.328382
$366$	0	0
$367$	3.80285e6	1.47382	0.736910	−	0.675991i	$-0.236284\pi$
$367$	3.80285e6	1.47382	0.736910	+	0.675991i	$0.236284\pi$
$368$	0	0
$369$	−5.05241e6	−1.93167
$370$	0	0
$371$	706722.	0.266571
$372$	0	0
$373$	−1.26664e6	−0.471391	−0.235695	−	0.971827i	$-0.575737\pi$
$373$	−1.26664e6	−0.471391	−0.235695	+	0.971827i	$0.575737\pi$
$374$	0	0
$375$	−380975.	−0.139900
$376$	0	0
$377$	−6.63387e6	−2.40388
$378$	0	0
$379$	3.00151e6	1.07335	0.536675	−	0.843789i	$-0.319680\pi$
$379$	3.00151e6	1.07335	0.536675	+	0.843789i	$0.319680\pi$
$380$	0	0
$381$	3.99164e6	1.40877
$382$	0	0
$383$	5.55564e6	1.93525	0.967625	−	0.252393i	$-0.0812176\pi$
$383$	5.55564e6	1.93525	0.967625	+	0.252393i	$0.0812176\pi$
$384$	0	0
$385$	−72815.7	−0.0250365
$386$	0	0
$387$	−7.86268e6	−2.66866
$388$	0	0
$389$	1.20865e6	0.404972	0.202486	−	0.979285i	$-0.435098\pi$
$389$	1.20865e6	0.404972	0.202486	+	0.979285i	$0.435098\pi$
$390$	0	0
$391$	348306.	0.115218
$392$	0	0
$393$	−7.46800e6	−2.43906
$394$	0	0
$395$	−921940.	−0.297310
$396$	0	0
$397$	−5.78510e6	−1.84219	−0.921095	−	0.389338i	$-0.872704\pi$
$397$	−5.78510e6	−1.84219	−0.921095	+	0.389338i	$0.872704\pi$
$398$	0	0
$399$	−1.66975e6	−0.525073
$400$	0	0
$401$	202477.	0.0628802	0.0314401	−	0.999506i	$-0.489991\pi$
$401$	202477.	0.0628802	0.0314401	+	0.999506i	$0.489991\pi$
$402$	0	0
$403$	−6.13219e6	−1.88085
$404$	0	0
$405$	−522776.	−0.158372
$406$	0	0
$407$	164275.	0.0491572
$408$	0	0
$409$	5.20097e6	1.53736	0.768681	−	0.639633i	$-0.220914\pi$
$409$	5.20097e6	1.53736	0.768681	+	0.639633i	$0.220914\pi$
$410$	0	0
$411$	679115.	0.198307
$412$	0	0
$413$	−820423.	−0.236681
$414$	0	0
$415$	−2.86475e6	−0.816519
$416$	0	0
$417$	−4.30434e6	−1.21218
$418$	0	0
$419$	−3.77891e6	−1.05155	−0.525777	−	0.850622i	$-0.676225\pi$
$419$	−3.77891e6	−1.05155	−0.525777	+	0.850622i	$0.676225\pi$
$420$	0	0
$421$	−5.87618e6	−1.61581	−0.807904	−	0.589314i	$-0.799398\pi$
$421$	−5.87618e6	−1.61581	−0.807904	+	0.589314i	$0.799398\pi$
$422$	0	0
$423$	4.23438e6	1.15064
$424$	0	0
$425$	259247.	0.0696211
$426$	0	0
$427$	−1.65758e6	−0.439952
$428$	0	0
$429$	−1.26624e6	−0.332180
$430$	0	0
$431$	4.43938e6	1.15114	0.575571	−	0.817751i	$-0.304780\pi$
$431$	4.43938e6	1.15114	0.575571	+	0.817751i	$0.304780\pi$
$432$	0	0
$433$	−5.99703e6	−1.53715	−0.768576	−	0.639759i	$-0.779034\pi$
$433$	−5.99703e6	−1.53715	−0.768576	+	0.639759i	$0.779034\pi$
$434$	0	0
$435$	4.62841e6	1.17276
$436$	0	0
$437$	−1.17357e6	−0.293971
$438$	0	0
$439$	2.22003e6	0.549791	0.274895	−	0.961474i	$-0.411357\pi$
$439$	2.22003e6	0.549791	0.274895	+	0.961474i	$0.411357\pi$
$440$	0	0
$441$	843953.	0.206643
$442$	0	0
$443$	4.93550e6	1.19487	0.597437	−	0.801916i	$-0.296186\pi$
$443$	4.93550e6	1.19487	0.597437	+	0.801916i	$0.296186\pi$
$444$	0	0
$445$	−1.46991e6	−0.351877
$446$	0	0
$447$	1.15638e6	0.273737
$448$	0	0
$449$	6.01484e6	1.40802	0.704009	−	0.710191i	$-0.251391\pi$
$449$	6.01484e6	1.40802	0.704009	+	0.710191i	$0.251391\pi$
$450$	0	0
$451$	−854401.	−0.197797
$452$	0	0
$453$	1.24248e7	2.84476
$454$	0	0
$455$	−1.07026e6	−0.242359
$456$	0	0
$457$	2.74872e6	0.615660	0.307830	−	0.951441i	$-0.400397\pi$
$457$	2.74872e6	0.615660	0.307830	+	0.951441i	$0.400397\pi$
$458$	0	0
$459$	−1.09734e6	−0.243114
$460$	0	0
$461$	3.53770e6	0.775297	0.387649	−	0.921807i	$-0.373287\pi$
$461$	3.53770e6	0.775297	0.387649	+	0.921807i	$0.373287\pi$
$462$	0	0
$463$	−4.30850e6	−0.934058	−0.467029	−	0.884242i	$-0.654676\pi$
$463$	−4.30850e6	−0.934058	−0.467029	+	0.884242i	$0.654676\pi$
$464$	0	0
$465$	4.27839e6	0.917589
$466$	0	0
$467$	−2.69910e6	−0.572700	−0.286350	−	0.958125i	$-0.592442\pi$
$467$	−2.69910e6	−0.572700	−0.286350	+	0.958125i	$0.592442\pi$
$468$	0	0
$469$	−594624.	−0.124828
$470$	0	0
$471$	6.96072e6	1.44578
$472$	0	0
$473$	−1.32964e6	−0.273263
$474$	0	0
$475$	−873493.	−0.177634
$476$	0	0
$477$	−5.06965e6	−1.02019
$478$	0	0
$479$	7.97101e6	1.58736	0.793678	−	0.608338i	$-0.208164\pi$
$479$	7.97101e6	1.58736	0.793678	+	0.608338i	$0.208164\pi$
$480$	0	0
$481$	2.41454e6	0.475852
$482$	0	0
$483$	1.00323e6	0.195674
$484$	0	0
$485$	861871.	0.166375
$486$	0	0
$487$	−3.99688e6	−0.763658	−0.381829	−	0.924233i	$-0.624706\pi$
$487$	−3.99688e6	−0.763658	−0.381829	+	0.924233i	$0.624706\pi$
$488$	0	0
$489$	7.61473e6	1.44007
$490$	0	0
$491$	789283.	0.147751	0.0738753	−	0.997267i	$-0.476463\pi$
$491$	789283.	0.147751	0.0738753	+	0.997267i	$0.476463\pi$
$492$	0	0
$493$	−3.14955e6	−0.583622
$494$	0	0
$495$	522342.	0.0958169
$496$	0	0
$497$	2.14853e6	0.390167
$498$	0	0
$499$	171664.	0.0308622	0.0154311	−	0.999881i	$-0.495088\pi$
$499$	171664.	0.0308622	0.0154311	+	0.999881i	$0.495088\pi$
$500$	0	0
$501$	6.04063e6	1.07520
$502$	0	0
$503$	1.12380e6	0.198047	0.0990235	−	0.995085i	$-0.468428\pi$
$503$	1.12380e6	0.198047	0.0990235	+	0.995085i	$0.468428\pi$
$504$	0	0
$505$	2.53306e6	0.441995
$506$	0	0
$507$	−9.55837e6	−1.65144
$508$	0	0
$509$	−8.64586e6	−1.47915	−0.739577	−	0.673072i	$-0.764975\pi$
$509$	−8.64586e6	−1.47915	−0.739577	+	0.673072i	$0.764975\pi$
$510$	0	0
$511$	−1.63820e6	−0.277534
$512$	0	0
$513$	3.69732e6	0.620289
$514$	0	0
$515$	−716194.	−0.118991
$516$	0	0
$517$	716066.	0.117822
$518$	0	0
$519$	7.34058e6	1.19622
$520$	0	0
$521$	2.82649e6	0.456198	0.228099	−	0.973638i	$-0.426749\pi$
$521$	2.82649e6	0.456198	0.228099	+	0.973638i	$0.426749\pi$
$522$	0	0
$523$	9.25594e6	1.47967	0.739837	−	0.672786i	$-0.234903\pi$
$523$	9.25594e6	1.47967	0.739837	+	0.672786i	$0.234903\pi$
$524$	0	0
$525$	746710.	0.118237
$526$	0	0
$527$	−2.91137e6	−0.456637
$528$	0	0
$529$	−5.73123e6	−0.890449
$530$	0	0
$531$	5.88529e6	0.905798
$532$	0	0
$533$	−1.25581e7	−1.91472
$534$	0	0
$535$	2.23468e6	0.337544
$536$	0	0
$537$	−3.27470e6	−0.490044
$538$	0	0
$539$	142719.	0.0211597
$540$	0	0
$541$	6.96253e6	1.02276	0.511380	−	0.859355i	$-0.329134\pi$
$541$	6.96253e6	1.02276	0.511380	+	0.859355i	$0.329134\pi$
$542$	0	0
$543$	9.38303e6	1.36566
$544$	0	0
$545$	−3.57996e6	−0.516282
$546$	0	0
$547$	−4.71143e6	−0.673263	−0.336631	−	0.941637i	$-0.609288\pi$
$547$	−4.71143e6	−0.673263	−0.336631	+	0.941637i	$0.609288\pi$
$548$	0	0
$549$	1.18906e7	1.68374
$550$	0	0
$551$	1.06119e7	1.48907
$552$	0	0
$553$	1.80700e6	0.251273
$554$	0	0
$555$	−1.68461e6	−0.232149
$556$	0	0
$557$	4.69349e6	0.641000	0.320500	−	0.947249i	$-0.396149\pi$
$557$	4.69349e6	0.641000	0.320500	+	0.947249i	$0.396149\pi$
$558$	0	0
$559$	−1.95432e7	−2.64525
$560$	0	0
$561$	−601172.	−0.0806476
$562$	0	0
$563$	−9.21973e6	−1.22588	−0.612939	−	0.790130i	$-0.710013\pi$
$563$	−9.21973e6	−1.22588	−0.612939	+	0.790130i	$0.710013\pi$
$564$	0	0
$565$	−3.25354e6	−0.428781
$566$	0	0
$567$	1.02464e6	0.133849
$568$	0	0
$569$	−162896.	−0.0210926	−0.0105463	−	0.999944i	$-0.503357\pi$
$569$	−162896.	−0.0210926	−0.0105463	+	0.999944i	$0.503357\pi$
$570$	0	0
$571$	−733222.	−0.0941120	−0.0470560	−	0.998892i	$-0.514984\pi$
$571$	−733222.	−0.0941120	−0.0470560	+	0.998892i	$0.514984\pi$
$572$	0	0
$573$	1.68215e7	2.14032
$574$	0	0
$575$	524817.	0.0661970
$576$	0	0
$577$	−1.25596e7	−1.57049	−0.785245	−	0.619185i	$-0.787463\pi$
$577$	−1.25596e7	−1.57049	−0.785245	+	0.619185i	$0.787463\pi$
$578$	0	0
$579$	−1.71190e7	−2.12217
$580$	0	0
$581$	5.61491e6	0.690085
$582$	0	0
$583$	−857317.	−0.104465
$584$	0	0
$585$	7.67745e6	0.927529
$586$	0	0
$587$	−1.99237e6	−0.238658	−0.119329	−	0.992855i	$-0.538074\pi$
$587$	−1.99237e6	−0.238658	−0.119329	+	0.992855i	$0.538074\pi$
$588$	0	0
$589$	9.80944e6	1.16508
$590$	0	0
$591$	−1.32691e7	−1.56269
$592$	0	0
$593$	−1.60192e7	−1.87070	−0.935350	−	0.353725i	$-0.884915\pi$
$593$	−1.60192e7	−1.87070	−0.935350	+	0.353725i	$0.884915\pi$
$594$	0	0
$595$	−508124.	−0.0588406
$596$	0	0
$597$	7.62399e6	0.875480
$598$	0	0
$599$	−6.72762e6	−0.766116	−0.383058	−	0.923724i	$-0.625129\pi$
$599$	−6.72762e6	−0.766116	−0.383058	+	0.923724i	$0.625129\pi$
$600$	0	0
$601$	1.53688e6	0.173561	0.0867806	−	0.996227i	$-0.472342\pi$
$601$	1.53688e6	0.173561	0.0867806	+	0.996227i	$0.472342\pi$
$602$	0	0
$603$	4.26553e6	0.477727
$604$	0	0
$605$	−3.93794e6	−0.437402
$606$	0	0
$607$	8.43829e6	0.929571	0.464786	−	0.885423i	$-0.346131\pi$
$607$	8.43829e6	0.929571	0.464786	+	0.885423i	$0.346131\pi$
$608$	0	0
$609$	−9.07168e6	−0.991162
$610$	0	0
$611$	1.05248e7	1.14054
$612$	0	0
$613$	−9.55026e6	−1.02651	−0.513256	−	0.858236i	$-0.671561\pi$
$613$	−9.55026e6	−1.02651	−0.513256	+	0.858236i	$0.671561\pi$
$614$	0	0
$615$	8.76170e6	0.934116
$616$	0	0
$617$	1.46544e7	1.54972	0.774862	−	0.632131i	$-0.217819\pi$
$617$	1.46544e7	1.54972	0.774862	+	0.632131i	$0.217819\pi$
$618$	0	0
$619$	−1.28694e7	−1.34999	−0.674997	−	0.737820i	$-0.735855\pi$
$619$	−1.28694e7	−1.34999	−0.674997	+	0.737820i	$0.735855\pi$
$620$	0	0
$621$	−2.22145e6	−0.231157
$622$	0	0
$623$	2.88102e6	0.297390
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	2.02556e6	0.205767
$628$	0	0
$629$	1.14635e6	0.115529
$630$	0	0
$631$	1.95977e7	1.95944	0.979719	−	0.200378i	$-0.0642171\pi$
$631$	1.95977e7	1.95944	0.979719	+	0.200378i	$0.0642171\pi$
$632$	0	0
$633$	−1.34789e7	−1.33704
$634$	0	0
$635$	−4.09275e6	−0.402792
$636$	0	0
$637$	2.09770e6	0.204831
$638$	0	0
$639$	−1.54124e7	−1.49320
$640$	0	0
$641$	−1.23136e7	−1.18369	−0.591847	−	0.806050i	$-0.701601\pi$
$641$	−1.23136e7	−1.18369	−0.591847	+	0.806050i	$0.701601\pi$
$642$	0	0
$643$	423026.	0.0403496	0.0201748	−	0.999796i	$-0.493578\pi$
$643$	423026.	0.0403496	0.0201748	+	0.999796i	$0.493578\pi$
$644$	0	0
$645$	1.36352e7	1.29051
$646$	0	0
$647$	−3.53941e6	−0.332407	−0.166203	−	0.986091i	$-0.553151\pi$
$647$	−3.53941e6	−0.332407	−0.166203	+	0.986091i	$0.553151\pi$
$648$	0	0
$649$	995247.	0.0927511
$650$	0	0
$651$	−8.38565e6	−0.775504
$652$	0	0
$653$	544725.	0.0499913	0.0249956	−	0.999688i	$-0.492043\pi$
$653$	544725.	0.0499913	0.0249956	+	0.999688i	$0.492043\pi$
$654$	0	0
$655$	7.65716e6	0.697372
$656$	0	0
$657$	1.17516e7	1.06215
$658$	0	0
$659$	1.50778e7	1.35246	0.676229	−	0.736691i	$-0.263613\pi$
$659$	1.50778e7	1.35246	0.676229	+	0.736691i	$0.263613\pi$
$660$	0	0
$661$	2.87160e6	0.255635	0.127817	−	0.991798i	$-0.459203\pi$
$661$	2.87160e6	0.255635	0.127817	+	0.991798i	$0.459203\pi$
$662$	0	0
$663$	−8.83610e6	−0.780687
$664$	0	0
$665$	1.71205e6	0.150128
$666$	0	0
$667$	−6.37593e6	−0.554918
$668$	0	0
$669$	1.74933e7	1.51115
$670$	0	0
$671$	2.01080e6	0.172410
$672$	0	0
$673$	−2.11706e7	−1.80175	−0.900875	−	0.434078i	$-0.857074\pi$
$673$	−2.11706e7	−1.80175	−0.900875	+	0.434078i	$0.857074\pi$
$674$	0	0
$675$	−1.65344e6	−0.139678
$676$	0	0
$677$	5.86653e6	0.491938	0.245969	−	0.969278i	$-0.420894\pi$
$677$	5.86653e6	0.491938	0.245969	+	0.969278i	$0.420894\pi$
$678$	0	0
$679$	−1.68927e6	−0.140613
$680$	0	0
$681$	−1.65806e7	−1.37004
$682$	0	0
$683$	441447.	0.0362099	0.0181049	−	0.999836i	$-0.494237\pi$
$683$	441447.	0.0362099	0.0181049	+	0.999836i	$0.494237\pi$
$684$	0	0
$685$	−696317.	−0.0566997
$686$	0	0
$687$	5.81176e6	0.469803
$688$	0	0
$689$	−1.26010e7	−1.01124
$690$	0	0
$691$	−5.89504e6	−0.469669	−0.234834	−	0.972035i	$-0.575455\pi$
$691$	−5.89504e6	−0.469669	−0.234834	+	0.972035i	$0.575455\pi$
$692$	0	0
$693$	−1.02379e6	−0.0809801
$694$	0	0
$695$	4.41337e6	0.346584
$696$	0	0
$697$	−5.96219e6	−0.464862
$698$	0	0
$699$	−1.07780e7	−0.834346
$700$	0	0
$701$	−1.25764e7	−0.966635	−0.483317	−	0.875445i	$-0.660568\pi$
$701$	−1.25764e7	−0.966635	−0.483317	+	0.875445i	$0.660568\pi$
$702$	0	0
$703$	−3.86245e6	−0.294764
$704$	0	0
$705$	−7.34311e6	−0.556426
$706$	0	0
$707$	−4.96480e6	−0.373554
$708$	0	0
$709$	1.41839e7	1.05969	0.529847	−	0.848093i	$-0.322249\pi$
$709$	1.41839e7	1.05969	0.529847	+	0.848093i	$0.322249\pi$
$710$	0	0
$711$	−1.29625e7	−0.961645
$712$	0	0
$713$	−5.89376e6	−0.434179
$714$	0	0
$715$	1.29832e6	0.0949763
$716$	0	0
$717$	−2.66968e7	−1.93937
$718$	0	0
$719$	−2.16964e7	−1.56518	−0.782591	−	0.622537i	$-0.786102\pi$
$719$	−2.16964e7	−1.56518	−0.782591	+	0.622537i	$0.786102\pi$
$720$	0	0
$721$	1.40374e6	0.100565
$722$	0	0
$723$	3.58734e7	2.55227
$724$	0	0
$725$	−4.74565e6	−0.335313
$726$	0	0
$727$	2.51847e7	1.76726	0.883629	−	0.468187i	$-0.155093\pi$
$727$	2.51847e7	1.76726	0.883629	+	0.468187i	$0.155093\pi$
$728$	0	0
$729$	−2.30246e7	−1.60462
$730$	0	0
$731$	−9.27850e6	−0.642221
$732$	0	0
$733$	−1.92772e7	−1.32521	−0.662603	−	0.748971i	$-0.730548\pi$
$733$	−1.92772e7	−1.32521	−0.662603	+	0.748971i	$0.730548\pi$
$734$	0	0
$735$	−1.46355e6	−0.0999286
$736$	0	0
$737$	721333.	0.0489178
$738$	0	0
$739$	4.95610e6	0.333833	0.166916	−	0.985971i	$-0.446619\pi$
$739$	4.95610e6	0.333833	0.166916	+	0.985971i	$0.446619\pi$
$740$	0	0
$741$	2.97719e7	1.99187
$742$	0	0
$743$	1.03298e7	0.686465	0.343233	−	0.939250i	$-0.388478\pi$
$743$	1.03298e7	0.686465	0.343233	+	0.939250i	$0.388478\pi$
$744$	0	0
$745$	−1.18568e6	−0.0782664
$746$	0	0
$747$	−4.02784e7	−2.64102
$748$	0	0
$749$	−4.37997e6	−0.285277
$750$	0	0
$751$	−1.23752e7	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$751$	−1.23752e7	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$752$	0	0
$753$	−1.41344e7	−0.908427
$754$	0	0
$755$	−1.27396e7	−0.813369
$756$	0	0
$757$	2.53583e7	1.60835	0.804175	−	0.594392i	$-0.202607\pi$
$757$	2.53583e7	1.60835	0.804175	+	0.594392i	$0.202607\pi$
$758$	0	0
$759$	−1.21701e6	−0.0766812
$760$	0	0
$761$	2.94609e7	1.84410	0.922048	−	0.387075i	$-0.126514\pi$
$761$	2.94609e7	1.84410	0.922048	+	0.387075i	$0.126514\pi$
$762$	0	0
$763$	7.01673e6	0.436338
$764$	0	0
$765$	3.64501e6	0.225188
$766$	0	0
$767$	1.46283e7	0.897851
$768$	0	0
$769$	1.05271e7	0.641936	0.320968	−	0.947090i	$-0.395992\pi$
$769$	1.05271e7	0.641936	0.320968	+	0.947090i	$0.395992\pi$
$770$	0	0
$771$	−2.58062e7	−1.56346
$772$	0	0
$773$	1.11836e7	0.673180	0.336590	−	0.941651i	$-0.390726\pi$
$773$	1.11836e7	0.673180	0.336590	+	0.941651i	$0.390726\pi$
$774$	0	0
$775$	−4.38677e6	−0.262356
$776$	0	0
$777$	3.30184e6	0.196202
$778$	0	0
$779$	2.00887e7	1.18606
$780$	0	0
$781$	−2.60636e6	−0.152900
$782$	0	0
$783$	2.00874e7	1.17090
$784$	0	0
$785$	−7.13704e6	−0.413375
$786$	0	0
$787$	3.14414e6	0.180953	0.0904764	−	0.995899i	$-0.471161\pi$
$787$	3.14414e6	0.180953	0.0904764	+	0.995899i	$0.471161\pi$
$788$	0	0
$789$	4.29244e7	2.45478
$790$	0	0
$791$	6.37695e6	0.362386
$792$	0	0
$793$	2.95550e7	1.66897
$794$	0	0
$795$	8.79161e6	0.493345
$796$	0	0
$797$	−1.10953e7	−0.618720	−0.309360	−	0.950945i	$-0.600115\pi$
$797$	−1.10953e7	−0.618720	−0.309360	+	0.950945i	$0.600115\pi$
$798$	0	0
$799$	4.99686e6	0.276905
$800$	0	0
$801$	−2.06670e7	−1.13814
$802$	0	0
$803$	1.98729e6	0.108761
$804$	0	0
$805$	−1.02864e6	−0.0559467
$806$	0	0
$807$	−9.39879e6	−0.508029
$808$	0	0
$809$	−3.05083e6	−0.163888	−0.0819439	−	0.996637i	$-0.526113\pi$
$809$	−3.05083e6	−0.163888	−0.0819439	+	0.996637i	$0.526113\pi$
$810$	0	0
$811$	−5.20426e6	−0.277848	−0.138924	−	0.990303i	$-0.544364\pi$
$811$	−5.20426e6	−0.277848	−0.138924	+	0.990303i	$0.544364\pi$
$812$	0	0
$813$	−2.49598e7	−1.32439
$814$	0	0
$815$	−7.80762e6	−0.411741
$816$	0	0
$817$	3.12625e7	1.63859
$818$	0	0
$819$	−1.50478e7	−0.783905
$820$	0	0
$821$	2.92071e7	1.51227	0.756137	−	0.654413i	$-0.227084\pi$
$821$	2.92071e7	1.51227	0.756137	+	0.654413i	$0.227084\pi$
$822$	0	0
$823$	−5.08765e6	−0.261829	−0.130915	−	0.991394i	$-0.541791\pi$
$823$	−5.08765e6	−0.261829	−0.130915	+	0.991394i	$0.541791\pi$
$824$	0	0
$825$	−905827.	−0.0463351
$826$	0	0
$827$	6.43585e6	0.327222	0.163611	−	0.986525i	$-0.447686\pi$
$827$	6.43585e6	0.327222	0.163611	+	0.986525i	$0.447686\pi$
$828$	0	0
$829$	3.80871e7	1.92483	0.962414	−	0.271586i	$-0.0875482\pi$
$829$	3.80871e7	1.92483	0.962414	+	0.271586i	$0.0875482\pi$
$830$	0	0
$831$	−3.16552e7	−1.59016
$832$	0	0
$833$	995922.	0.0497294
$834$	0	0
$835$	−6.19364e6	−0.307418
$836$	0	0
$837$	1.85683e7	0.916133
$838$	0	0
$839$	−1.84812e7	−0.906412	−0.453206	−	0.891406i	$-0.649720\pi$
$839$	−1.84812e7	−0.906412	−0.453206	+	0.891406i	$0.649720\pi$
$840$	0	0
$841$	3.71431e7	1.81087
$842$	0	0
$843$	4.83635e7	2.34395
$844$	0	0
$845$	9.80049e6	0.472178
$846$	0	0
$847$	7.71837e6	0.369672
$848$	0	0
$849$	4.50036e7	2.14278
$850$	0	0
$851$	2.32066e6	0.109847
$852$	0	0
$853$	1.45787e7	0.686033	0.343017	−	0.939329i	$-0.388551\pi$
$853$	1.45787e7	0.686033	0.343017	+	0.939329i	$0.388551\pi$
$854$	0	0
$855$	−1.22813e7	−0.574553
$856$	0	0
$857$	−1.71760e7	−0.798858	−0.399429	−	0.916764i	$-0.630792\pi$
$857$	−1.71760e7	−0.798858	−0.399429	+	0.916764i	$0.630792\pi$
$858$	0	0
$859$	1.06813e7	0.493904	0.246952	−	0.969028i	$-0.420571\pi$
$859$	1.06813e7	0.493904	0.246952	+	0.969028i	$0.420571\pi$
$860$	0	0
$861$	−1.71729e7	−0.789472
$862$	0	0
$863$	7.01227e6	0.320503	0.160251	−	0.987076i	$-0.448770\pi$
$863$	7.01227e6	0.320503	0.160251	+	0.987076i	$0.448770\pi$
$864$	0	0
$865$	−7.52652e6	−0.342022
$866$	0	0
$867$	3.04244e7	1.37459
$868$	0	0
$869$	−2.19206e6	−0.0984697
$870$	0	0
$871$	1.06022e7	0.473535
$872$	0	0
$873$	1.21179e7	0.538137
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	2.38378e7	1.04657	0.523284	−	0.852158i	$-0.324707\pi$
$877$	2.38378e7	1.04657	0.523284	+	0.852158i	$0.324707\pi$
$878$	0	0
$879$	−4.90513e7	−2.14131
$880$	0	0
$881$	−1.43085e7	−0.621088	−0.310544	−	0.950559i	$-0.600511\pi$
$881$	−1.43085e7	−0.621088	−0.310544	+	0.950559i	$0.600511\pi$
$882$	0	0
$883$	1.61976e7	0.699114	0.349557	−	0.936915i	$-0.386332\pi$
$883$	1.61976e7	0.699114	0.349557	+	0.936915i	$0.386332\pi$
$884$	0	0
$885$	−1.02061e7	−0.438026
$886$	0	0
$887$	−1.76151e7	−0.751755	−0.375878	−	0.926669i	$-0.622659\pi$
$887$	−1.76151e7	−0.751755	−0.375878	+	0.926669i	$0.622659\pi$
$888$	0	0
$889$	8.02180e6	0.340422
$890$	0	0
$891$	−1.24298e6	−0.0524530
$892$	0	0
$893$	−1.68362e7	−0.706504
$894$	0	0
$895$	3.35765e6	0.140113
$896$	0	0
$897$	−1.78877e7	−0.742291
$898$	0	0
$899$	5.32942e7	2.19928
$900$	0	0
$901$	−5.98254e6	−0.245512
$902$	0	0
$903$	−2.67249e7	−1.09068
$904$	0	0
$905$	−9.62070e6	−0.390468
$906$	0	0
$907$	−3.08532e7	−1.24532	−0.622662	−	0.782491i	$-0.713949\pi$
$907$	−3.08532e7	−1.24532	−0.622662	+	0.782491i	$0.713949\pi$
$908$	0	0
$909$	3.56149e7	1.42962
$910$	0	0
$911$	−1.09518e7	−0.437210	−0.218605	−	0.975813i	$-0.570151\pi$
$911$	−1.09518e7	−0.437210	−0.218605	+	0.975813i	$0.570151\pi$
$912$	0	0
$913$	−6.81139e6	−0.270433
$914$	0	0
$915$	−2.06203e7	−0.814221
$916$	0	0
$917$	−1.50080e7	−0.589387
$918$	0	0
$919$	1.71813e7	0.671067	0.335534	−	0.942028i	$-0.391083\pi$
$919$	1.71813e7	0.671067	0.335534	+	0.942028i	$0.391083\pi$
$920$	0	0
$921$	−5.10782e7	−1.98421
$922$	0	0
$923$	−3.83086e7	−1.48010
$924$	0	0
$925$	1.72728e6	0.0663757
$926$	0	0
$927$	−1.00697e7	−0.384873
$928$	0	0
$929$	4.19050e6	0.159304	0.0796520	−	0.996823i	$-0.474619\pi$
$929$	4.19050e6	0.159304	0.0796520	+	0.996823i	$0.474619\pi$
$930$	0	0
$931$	−3.35561e6	−0.126881
$932$	0	0
$933$	−4.94784e7	−1.86085
$934$	0	0
$935$	616400.	0.0230586
$936$	0	0
$937$	−3.45468e7	−1.28546	−0.642731	−	0.766092i	$-0.722199\pi$
$937$	−3.45468e7	−1.28546	−0.642731	+	0.766092i	$0.722199\pi$
$938$	0	0
$939$	−5.69190e6	−0.210666
$940$	0	0
$941$	−2.31574e7	−0.852542	−0.426271	−	0.904596i	$-0.640173\pi$
$941$	−2.31574e7	−0.852542	−0.426271	+	0.904596i	$0.640173\pi$
$942$	0	0
$943$	−1.20698e7	−0.441999
$944$	0	0
$945$	3.24074e6	0.118050
$946$	0	0
$947$	1.71790e7	0.622477	0.311238	−	0.950332i	$-0.399256\pi$
$947$	1.71790e7	0.622477	0.311238	+	0.950332i	$0.399256\pi$
$948$	0	0
$949$	2.92094e7	1.05283
$950$	0	0
$951$	−7.04146e7	−2.52471
$952$	0	0
$953$	2.26025e7	0.806164	0.403082	−	0.915164i	$-0.367939\pi$
$953$	2.26025e7	0.806164	0.403082	+	0.915164i	$0.367939\pi$
$954$	0	0
$955$	−1.72476e7	−0.611957
$956$	0	0
$957$	1.10048e7	0.388419
$958$	0	0
$959$	1.36478e6	0.0479200
$960$	0	0
$961$	2.06348e7	0.720761
$962$	0	0
$963$	3.14196e7	1.09178
$964$	0	0
$965$	1.75526e7	0.606768
$966$	0	0
$967$	3.56482e7	1.22595	0.612974	−	0.790103i	$-0.289973\pi$
$967$	3.56482e7	1.22595	0.612974	+	0.790103i	$0.289973\pi$
$968$	0	0
$969$	1.41348e7	0.483592
$970$	0	0
$971$	−8.70185e6	−0.296185	−0.148093	−	0.988973i	$-0.547313\pi$
$971$	−8.70185e6	−0.296185	−0.148093	+	0.988973i	$0.547313\pi$
$972$	0	0
$973$	−8.65021e6	−0.292917
$974$	0	0
$975$	−1.33140e7	−0.448534
$976$	0	0
$977$	−703138.	−0.0235670	−0.0117835	−	0.999931i	$-0.503751\pi$
$977$	−703138.	−0.0235670	−0.0117835	+	0.999931i	$0.503751\pi$
$978$	0	0
$979$	−3.49494e6	−0.116542
$980$	0	0
$981$	−5.03344e7	−1.66991
$982$	0	0
$983$	3.74622e7	1.23654	0.618272	−	0.785964i	$-0.287833\pi$
$983$	3.74622e7	1.23654	0.618272	+	0.785964i	$0.287833\pi$
$984$	0	0
$985$	1.36052e7	0.446801
$986$	0	0
$987$	1.43925e7	0.470265
$988$	0	0
$989$	−1.87833e7	−0.610635
$990$	0	0
$991$	2.39813e7	0.775689	0.387845	−	0.921725i	$-0.373220\pi$
$991$	2.39813e7	0.775689	0.387845	+	0.921725i	$0.373220\pi$
$992$	0	0
$993$	−3.24375e7	−1.04394
$994$	0	0
$995$	−7.81711e6	−0.250316
$996$	0	0
$997$	−8.35171e6	−0.266096	−0.133048	−	0.991110i	$-0.542476\pi$
$997$	−8.35171e6	−0.266096	−0.133048	+	0.991110i	$0.542476\pi$
$998$	0	0
$999$	−7.31124e6	−0.231781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.b.1.1	✓	2
4.3	odd	2		560.6.a.o.1.2		2
5.2	odd	4		700.6.e.f.449.4		4
5.3	odd	4		700.6.e.f.449.1		4
5.4	even	2		700.6.a.g.1.2		2
7.6	odd	2		980.6.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.b.1.1	✓	2	1.1	even	1	trivial
560.6.a.o.1.2		2	4.3	odd	2
700.6.a.g.1.2		2	5.4	even	2
700.6.e.f.449.1		4	5.3	odd	4
700.6.e.f.449.4		4	5.2	odd	4
980.6.a.f.1.2		2	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)

\(f(q)\)	\(=\)	\(q-24.3824 q^{3} +25.0000 q^{5} -49.0000 q^{7} +351.500 q^{9} +O(q^{10})\)	\(q-24.3824 q^{3} +25.0000 q^{5} -49.0000 q^{7} +351.500 q^{9} +59.4414 q^{11} +873.678 q^{13} -609.560 q^{15} +414.795 q^{17} -1397.59 q^{19} +1194.74 q^{21} +839.708 q^{23} +625.000 q^{25} -2645.50 q^{27} -7593.04 q^{29} -7018.83 q^{31} -1449.32 q^{33} -1225.00 q^{35} +2763.65 q^{37} -21302.3 q^{39} -14373.8 q^{41} -22368.9 q^{43} +8787.51 q^{45} +12046.6 q^{47} +2401.00 q^{49} -10113.7 q^{51} -14422.9 q^{53} +1486.04 q^{55} +34076.6 q^{57} +16743.3 q^{59} +33828.2 q^{61} -17223.5 q^{63} +21841.9 q^{65} +12135.2 q^{67} -20474.1 q^{69} -43847.6 q^{71} +33432.7 q^{73} -15239.0 q^{75} -2912.63 q^{77} -36877.6 q^{79} -20911.0 q^{81} -114590. q^{83} +10369.9 q^{85} +185136. q^{87} -58796.4 q^{89} -42810.2 q^{91} +171136. q^{93} -34939.7 q^{95} +34474.9 q^{97} +20893.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9}+O(q^{10}) \) 2 * q - 17 * q^3 + 50 * q^5 - 98 * q^7 + 163 * q^9	\( 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} - 167 q^{11} + 445 q^{13} - 425 q^{15} + 99 q^{17} - 1334 q^{19} + 833 q^{21} - 290 q^{23} + 1250 q^{25} - 5831 q^{27} - 6959 q^{29} - 10480 q^{31} - 3121 q^{33} - 2450 q^{35} - 1588 q^{37} - 24467 q^{39} - 24110 q^{41} - 20406 q^{43} + 4075 q^{45} + 619 q^{47} + 4802 q^{49} - 12445 q^{51} - 21794 q^{53} - 4175 q^{55} + 34546 q^{57} + 1976 q^{59} + 22614 q^{61} - 7987 q^{63} + 11125 q^{65} + 4068 q^{67} - 28814 q^{69} - 46528 q^{71} + 27096 q^{73} - 10625 q^{75} + 8183 q^{77} - 12481 q^{79} + 1378 q^{81} - 100088 q^{83} + 2475 q^{85} + 189817 q^{87} + 33798 q^{89} - 21805 q^{91} + 145584 q^{93} - 33350 q^{95} + 138991 q^{97} + 63578 q^{99}+O(q^{100}) \) 2 * q - 17 * q^3 + 50 * q^5 - 98 * q^7 + 163 * q^9 - 167 * q^11 + 445 * q^13 - 425 * q^15 + 99 * q^17 - 1334 * q^19 + 833 * q^21 - 290 * q^23 + 1250 * q^25 - 5831 * q^27 - 6959 * q^29 - 10480 * q^31 - 3121 * q^33 - 2450 * q^35 - 1588 * q^37 - 24467 * q^39 - 24110 * q^41 - 20406 * q^43 + 4075 * q^45 + 619 * q^47 + 4802 * q^49 - 12445 * q^51 - 21794 * q^53 - 4175 * q^55 + 34546 * q^57 + 1976 * q^59 + 22614 * q^61 - 7987 * q^63 + 11125 * q^65 + 4068 * q^67 - 28814 * q^69 - 46528 * q^71 + 27096 * q^73 - 10625 * q^75 + 8183 * q^77 - 12481 * q^79 + 1378 * q^81 - 100088 * q^83 + 2475 * q^85 + 189817 * q^87 + 33798 * q^89 - 21805 * q^91 + 145584 * q^93 - 33350 * q^95 + 138991 * q^97 + 63578 * q^99

Introduction

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