LMFDB - Embedded newform 140.6.a.d.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:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 499x - 210 $ x^3 - 499*x - 210
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			$-22.1248$ 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)

$f(q)$	$=$	$q-20.1248 q^{3} +25.0000 q^{5} +49.0000 q^{7} +162.009 q^{9} +O(q^{10})$	$q-20.1248 q^{3} +25.0000 q^{5} +49.0000 q^{7} +162.009 q^{9} -427.006 q^{11} -646.889 q^{13} -503.121 q^{15} -1029.14 q^{17} +2253.95 q^{19} -986.117 q^{21} +2595.11 q^{23} +625.000 q^{25} +1629.93 q^{27} +869.860 q^{29} +4407.19 q^{31} +8593.44 q^{33} +1225.00 q^{35} +2555.64 q^{37} +13018.5 q^{39} +6222.90 q^{41} +7757.57 q^{43} +4050.23 q^{45} +1887.76 q^{47} +2401.00 q^{49} +20711.2 q^{51} -11718.0 q^{53} -10675.2 q^{55} -45360.4 q^{57} +33385.2 q^{59} -1212.23 q^{61} +7938.44 q^{63} -16172.2 q^{65} -50421.1 q^{67} -52226.1 q^{69} +57251.5 q^{71} +84023.5 q^{73} -12578.0 q^{75} -20923.3 q^{77} +98400.4 q^{79} -72170.3 q^{81} -32157.5 q^{83} -25728.4 q^{85} -17505.8 q^{87} +54323.7 q^{89} -31697.6 q^{91} -88694.1 q^{93} +56348.8 q^{95} -21700.3 q^{97} -69178.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9}+O(q^{10}) $ 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9	$ 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33} + 3675 q^{35} + 11378 q^{37} + 13702 q^{39} + 11450 q^{41} + 18544 q^{43} + 7025 q^{45} + 21754 q^{47} + 7203 q^{49} + 31762 q^{51} + 7494 q^{53} - 350 q^{55} + 7332 q^{57} + 12388 q^{59} + 27182 q^{61} + 13769 q^{63} - 100 q^{65} - 7676 q^{67} - 62140 q^{69} + 81992 q^{71} + 230 q^{73} + 3750 q^{75} - 686 q^{77} - 15926 q^{79} - 32101 q^{81} - 86100 q^{83} + 1100 q^{85} - 85098 q^{87} - 95710 q^{89} - 196 q^{91} - 229472 q^{93} + 58200 q^{95} - 176188 q^{97} - 114368 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * 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
$3$	−20.1248	−1.29101	−0.645504	−	0.763757i	$-0.723353\pi$
$3$	−20.1248	−1.29101	−0.645504	+	0.763757i	$0.723353\pi$
$4$	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
$9$	162.009	0.666704
$10$	0	0
$11$	−427.006	−1.06403	−0.532014	−	0.846736i	$-0.678564\pi$
$11$	−427.006	−1.06403	−0.532014	+	0.846736i	$0.678564\pi$
$12$	0	0
$13$	−646.889	−1.06163	−0.530813	−	0.847489i	$-0.678113\pi$
$13$	−646.889	−1.06163	−0.530813	+	0.847489i	$0.678113\pi$
$14$	0	0
$15$	−503.121	−0.577357
$16$	0	0
$17$	−1029.14	−0.863676	−0.431838	−	0.901951i	$-0.642135\pi$
$17$	−1029.14	−0.863676	−0.431838	+	0.901951i	$0.642135\pi$
$18$	0	0
$19$	2253.95	1.43239	0.716193	−	0.697902i	$-0.245883\pi$
$19$	2253.95	1.43239	0.716193	+	0.697902i	$0.245883\pi$
$20$	0	0
$21$	−986.117	−0.487955
$22$	0	0
$23$	2595.11	1.02291	0.511453	−	0.859311i	$-0.329108\pi$
$23$	2595.11	1.02291	0.511453	+	0.859311i	$0.329108\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	1629.93	0.430288
$28$	0	0
$29$	869.860	0.192068	0.0960339	−	0.995378i	$-0.469384\pi$
$29$	869.860	0.192068	0.0960339	+	0.995378i	$0.469384\pi$
$30$	0	0
$31$	4407.19	0.823679	0.411839	−	0.911256i	$-0.364886\pi$
$31$	4407.19	0.823679	0.411839	+	0.911256i	$0.364886\pi$
$32$	0	0
$33$	8593.44	1.37367
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	2555.64	0.306899	0.153450	−	0.988156i	$-0.450962\pi$
$37$	2555.64	0.306899	0.153450	+	0.988156i	$0.450962\pi$
$38$	0	0
$39$	13018.5	1.37057
$40$	0	0
$41$	6222.90	0.578141	0.289070	−	0.957308i	$-0.406654\pi$
$41$	6222.90	0.578141	0.289070	+	0.957308i	$0.406654\pi$
$42$	0	0
$43$	7757.57	0.639815	0.319907	−	0.947449i	$-0.396348\pi$
$43$	7757.57	0.639815	0.319907	+	0.947449i	$0.396348\pi$
$44$	0	0
$45$	4050.23	0.298159
$46$	0	0
$47$	1887.76	0.124653	0.0623265	−	0.998056i	$-0.480148\pi$
$47$	1887.76	0.124653	0.0623265	+	0.998056i	$0.480148\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	20711.2	1.11501
$52$	0	0
$53$	−11718.0	−0.573010	−0.286505	−	0.958079i	$-0.592493\pi$
$53$	−11718.0	−0.573010	−0.286505	+	0.958079i	$0.592493\pi$
$54$	0	0
$55$	−10675.2	−0.475847
$56$	0	0
$57$	−45360.4	−1.84922
$58$	0	0
$59$	33385.2	1.24860	0.624300	−	0.781184i	$-0.285384\pi$
$59$	33385.2	1.24860	0.624300	+	0.781184i	$0.285384\pi$
$60$	0	0
$61$	−1212.23	−0.0417119	−0.0208559	−	0.999782i	$-0.506639\pi$
$61$	−1212.23	−0.0417119	−0.0208559	+	0.999782i	$0.506639\pi$
$62$	0	0
$63$	7938.44	0.251990
$64$	0	0
$65$	−16172.2	−0.474774
$66$	0	0
$67$	−50421.1	−1.37222	−0.686112	−	0.727496i	$-0.740684\pi$
$67$	−50421.1	−1.37222	−0.686112	+	0.727496i	$0.740684\pi$
$68$	0	0
$69$	−52226.1	−1.32058
$70$	0	0
$71$	57251.5	1.34785	0.673925	−	0.738800i	$-0.264607\pi$
$71$	57251.5	1.34785	0.673925	+	0.738800i	$0.264607\pi$
$72$	0	0
$73$	84023.5	1.84541	0.922707	−	0.385502i	$-0.125972\pi$
$73$	84023.5	1.84541	0.922707	+	0.385502i	$0.125972\pi$
$74$	0	0
$75$	−12578.0	−0.258202
$76$	0	0
$77$	−20923.3	−0.402164
$78$	0	0
$79$	98400.4	1.77390	0.886950	−	0.461866i	$-0.152820\pi$
$79$	98400.4	1.77390	0.886950	+	0.461866i	$0.152820\pi$
$80$	0	0
$81$	−72170.3	−1.22221
$82$	0	0
$83$	−32157.5	−0.512375	−0.256187	−	0.966627i	$-0.582466\pi$
$83$	−32157.5	−0.512375	−0.256187	+	0.966627i	$0.582466\pi$
$84$	0	0
$85$	−25728.4	−0.386247
$86$	0	0
$87$	−17505.8	−0.247961
$88$	0	0
$89$	54323.7	0.726966	0.363483	−	0.931601i	$-0.381587\pi$
$89$	54323.7	0.726966	0.363483	+	0.931601i	$0.381587\pi$
$90$	0	0
$91$	−31697.6	−0.401257
$92$	0	0
$93$	−88694.1	−1.06338
$94$	0	0
$95$	56348.8	0.640583
$96$	0	0
$97$	−21700.3	−0.234173	−0.117086	−	0.993122i	$-0.537355\pi$
$97$	−21700.3	−0.234173	−0.117086	+	0.993122i	$0.537355\pi$
$98$	0	0
$99$	−69178.9	−0.709391
$100$	0	0
$101$	127360.	1.24231	0.621154	−	0.783689i	$-0.286664\pi$
$101$	127360.	1.24231	0.621154	+	0.783689i	$0.286664\pi$
$102$	0	0
$103$	44694.5	0.415108	0.207554	−	0.978224i	$-0.433450\pi$
$103$	44694.5	0.415108	0.207554	+	0.978224i	$0.433450\pi$
$104$	0	0
$105$	−24652.9	−0.218220
$106$	0	0
$107$	−125360.	−1.05852	−0.529258	−	0.848461i	$-0.677530\pi$
$107$	−125360.	−1.05852	−0.529258	+	0.848461i	$0.677530\pi$
$108$	0	0
$109$	−22940.1	−0.184939	−0.0924696	−	0.995716i	$-0.529476\pi$
$109$	−22940.1	−0.184939	−0.0924696	+	0.995716i	$0.529476\pi$
$110$	0	0
$111$	−51431.9	−0.396210
$112$	0	0
$113$	−191652.	−1.41194	−0.705971	−	0.708240i	$-0.749489\pi$
$113$	−191652.	−1.41194	−0.705971	+	0.708240i	$0.749489\pi$
$114$	0	0
$115$	64877.6	0.457457
$116$	0	0
$117$	−104802.	−0.707790
$118$	0	0
$119$	−50427.7	−0.326439
$120$	0	0
$121$	21283.5	0.132154
$122$	0	0
$123$	−125235.	−0.746385
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−280121.	−1.54112	−0.770559	−	0.637368i	$-0.780023\pi$
$127$	−280121.	−1.54112	−0.770559	+	0.637368i	$0.780023\pi$
$128$	0	0
$129$	−156120.	−0.826007
$130$	0	0
$131$	245780.	1.25132	0.625660	−	0.780096i	$-0.284830\pi$
$131$	245780.	1.25132	0.625660	+	0.780096i	$0.284830\pi$
$132$	0	0
$133$	110444.	0.541391
$134$	0	0
$135$	40748.2	0.192431
$136$	0	0
$137$	−13417.8	−0.0610773	−0.0305386	−	0.999534i	$-0.509722\pi$
$137$	−13417.8	−0.0610773	−0.0305386	+	0.999534i	$0.509722\pi$
$138$	0	0
$139$	−363369.	−1.59519	−0.797593	−	0.603196i	$-0.793894\pi$
$139$	−363369.	−1.59519	−0.797593	+	0.603196i	$0.793894\pi$
$140$	0	0
$141$	−37990.9	−0.160928
$142$	0	0
$143$	276226.	1.12960
$144$	0	0
$145$	21746.5	0.0858953
$146$	0	0
$147$	−48319.7	−0.184430
$148$	0	0
$149$	−260049.	−0.959597	−0.479798	−	0.877379i	$-0.659290\pi$
$149$	−260049.	−0.959597	−0.479798	+	0.877379i	$0.659290\pi$
$150$	0	0
$151$	219894.	0.784821	0.392410	−	0.919790i	$-0.371641\pi$
$151$	219894.	0.784821	0.392410	+	0.919790i	$0.371641\pi$
$152$	0	0
$153$	−166729.	−0.575816
$154$	0	0
$155$	110180.	0.368360
$156$	0	0
$157$	430845.	1.39499	0.697496	−	0.716589i	$-0.254298\pi$
$157$	430845.	1.39499	0.697496	+	0.716589i	$0.254298\pi$
$158$	0	0
$159$	235822.	0.739761
$160$	0	0
$161$	127160.	0.386622
$162$	0	0
$163$	462115.	1.36233	0.681163	−	0.732132i	$-0.261474\pi$
$163$	462115.	1.36233	0.681163	+	0.732132i	$0.261474\pi$
$164$	0	0
$165$	214836.	0.614323
$166$	0	0
$167$	493069.	1.36809	0.684047	−	0.729438i	$-0.260218\pi$
$167$	493069.	1.36809	0.684047	+	0.729438i	$0.260218\pi$
$168$	0	0
$169$	47172.8	0.127050
$170$	0	0
$171$	365160.	0.954978
$172$	0	0
$173$	−614223.	−1.56031	−0.780155	−	0.625586i	$-0.784860\pi$
$173$	−614223.	−1.56031	−0.780155	+	0.625586i	$0.784860\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−671871.	−1.61195
$178$	0	0
$179$	−92331.1	−0.215385	−0.107692	−	0.994184i	$-0.534346\pi$
$179$	−92331.1	−0.215385	−0.107692	+	0.994184i	$0.534346\pi$
$180$	0	0
$181$	694842.	1.57649	0.788243	−	0.615365i	$-0.210991\pi$
$181$	694842.	1.57649	0.788243	+	0.615365i	$0.210991\pi$
$182$	0	0
$183$	24395.9	0.0538504
$184$	0	0
$185$	63891.1	0.137250
$186$	0	0
$187$	439448.	0.918974
$188$	0	0
$189$	79866.5	0.162634
$190$	0	0
$191$	−459214.	−0.910818	−0.455409	−	0.890282i	$-0.650507\pi$
$191$	−459214.	−0.910818	−0.455409	+	0.890282i	$0.650507\pi$
$192$	0	0
$193$	182792.	0.353236	0.176618	−	0.984279i	$-0.443484\pi$
$193$	182792.	0.353236	0.176618	+	0.984279i	$0.443484\pi$
$194$	0	0
$195$	325464.	0.612937
$196$	0	0
$197$	345493.	0.634270	0.317135	−	0.948380i	$-0.397279\pi$
$197$	345493.	0.634270	0.317135	+	0.948380i	$0.397279\pi$
$198$	0	0
$199$	455123.	0.814697	0.407348	−	0.913273i	$-0.366453\pi$
$199$	455123.	0.814697	0.407348	+	0.913273i	$0.366453\pi$
$200$	0	0
$201$	1.01472e6	1.77155
$202$	0	0
$203$	42623.2	0.0725948
$204$	0	0
$205$	155573.	0.258552
$206$	0	0
$207$	420431.	0.681975
$208$	0	0
$209$	−962451.	−1.52410
$210$	0	0
$211$	364003.	0.562858	0.281429	−	0.959582i	$-0.409192\pi$
$211$	364003.	0.562858	0.281429	+	0.959582i	$0.409192\pi$
$212$	0	0
$213$	−1.15218e6	−1.74009
$214$	0	0
$215$	193939.	0.286134
$216$	0	0
$217$	215953.	0.311321
$218$	0	0
$219$	−1.69096e6	−2.38245
$220$	0	0
$221$	665737.	0.916901
$222$	0	0
$223$	275681.	0.371232	0.185616	−	0.982622i	$-0.440572\pi$
$223$	275681.	0.371232	0.185616	+	0.982622i	$0.440572\pi$
$224$	0	0
$225$	101256.	0.133341
$226$	0	0
$227$	−1.14571e6	−1.47574	−0.737869	−	0.674943i	$-0.764168\pi$
$227$	−1.14571e6	−1.47574	−0.737869	+	0.674943i	$0.764168\pi$
$228$	0	0
$229$	−908996.	−1.14544	−0.572721	−	0.819750i	$-0.694112\pi$
$229$	−908996.	−1.14544	−0.572721	+	0.819750i	$0.694112\pi$
$230$	0	0
$231$	421078.	0.519198
$232$	0	0
$233$	−788658.	−0.951698	−0.475849	−	0.879527i	$-0.657859\pi$
$233$	−788658.	−0.951698	−0.475849	+	0.879527i	$0.657859\pi$
$234$	0	0
$235$	47194.0	0.0557465
$236$	0	0
$237$	−1.98029e6	−2.29012
$238$	0	0
$239$	1.50376e6	1.70288	0.851441	−	0.524451i	$-0.175729\pi$
$239$	1.50376e6	1.70288	0.851441	+	0.524451i	$0.175729\pi$
$240$	0	0
$241$	114870.	0.127399	0.0636993	−	0.997969i	$-0.479710\pi$
$241$	114870.	0.127399	0.0636993	+	0.997969i	$0.479710\pi$
$242$	0	0
$243$	1.05634e6	1.14760
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−1.45806e6	−1.52066
$248$	0	0
$249$	647165.	0.661480
$250$	0	0
$251$	−893691.	−0.895371	−0.447686	−	0.894191i	$-0.647752\pi$
$251$	−893691.	−0.895371	−0.447686	+	0.894191i	$0.647752\pi$
$252$	0	0
$253$	−1.10813e6	−1.08840
$254$	0	0
$255$	517780.	0.498649
$256$	0	0
$257$	−839193.	−0.792555	−0.396277	−	0.918131i	$-0.629698\pi$
$257$	−839193.	−0.792555	−0.396277	+	0.918131i	$0.629698\pi$
$258$	0	0
$259$	125227.	0.115997
$260$	0	0
$261$	140925.	0.128052
$262$	0	0
$263$	1.19862e6	1.06855	0.534273	−	0.845312i	$-0.320585\pi$
$263$	1.19862e6	1.06855	0.534273	+	0.845312i	$0.320585\pi$
$264$	0	0
$265$	−292949.	−0.256258
$266$	0	0
$267$	−1.09326e6	−0.938520
$268$	0	0
$269$	333286.	0.280825	0.140413	−	0.990093i	$-0.455157\pi$
$269$	333286.	0.280825	0.140413	+	0.990093i	$0.455157\pi$
$270$	0	0
$271$	1.99242e6	1.64800	0.824000	−	0.566589i	$-0.191737\pi$
$271$	1.99242e6	1.64800	0.824000	+	0.566589i	$0.191737\pi$
$272$	0	0
$273$	637909.	0.518026
$274$	0	0
$275$	−266879.	−0.212805
$276$	0	0
$277$	−201374.	−0.157690	−0.0788449	−	0.996887i	$-0.525123\pi$
$277$	−201374.	−0.157690	−0.0788449	+	0.996887i	$0.525123\pi$
$278$	0	0
$279$	714005.	0.549150
$280$	0	0
$281$	−456782.	−0.345099	−0.172549	−	0.985001i	$-0.555200\pi$
$281$	−456782.	−0.345099	−0.172549	+	0.985001i	$0.555200\pi$
$282$	0	0
$283$	590693.	0.438425	0.219213	−	0.975677i	$-0.429651\pi$
$283$	590693.	0.438425	0.219213	+	0.975677i	$0.429651\pi$
$284$	0	0
$285$	−1.13401e6	−0.826998
$286$	0	0
$287$	304922.	0.218517
$288$	0	0
$289$	−360735.	−0.254064
$290$	0	0
$291$	436715.	0.302319
$292$	0	0
$293$	−416363.	−0.283337	−0.141668	−	0.989914i	$-0.545247\pi$
$293$	−416363.	−0.283337	−0.141668	+	0.989914i	$0.545247\pi$
$294$	0	0
$295$	834629.	0.558391
$296$	0	0
$297$	−695990.	−0.457838
$298$	0	0
$299$	−1.67875e6	−1.08594
$300$	0	0
$301$	380121.	0.241827
$302$	0	0
$303$	−2.56310e6	−1.60383
$304$	0	0
$305$	−30305.7	−0.0186541
$306$	0	0
$307$	−1.25575e6	−0.760427	−0.380213	−	0.924899i	$-0.624149\pi$
$307$	−1.25575e6	−0.760427	−0.380213	+	0.924899i	$0.624149\pi$
$308$	0	0
$309$	−899470.	−0.535909
$310$	0	0
$311$	−1.16270e6	−0.681659	−0.340830	−	0.940125i	$-0.610708\pi$
$311$	−1.16270e6	−0.681659	−0.340830	+	0.940125i	$0.610708\pi$
$312$	0	0
$313$	3.45256e6	1.99196	0.995980	−	0.0895747i	$-0.0285508\pi$
$313$	3.45256e6	1.99196	0.995980	+	0.0895747i	$0.0285508\pi$
$314$	0	0
$315$	198461.	0.112694
$316$	0	0
$317$	−1.06034e6	−0.592651	−0.296325	−	0.955087i	$-0.595761\pi$
$317$	−1.06034e6	−0.592651	−0.296325	+	0.955087i	$0.595761\pi$
$318$	0	0
$319$	−371436.	−0.204365
$320$	0	0
$321$	2.52284e6	1.36655
$322$	0	0
$323$	−2.31962e6	−1.23712
$324$	0	0
$325$	−404306.	−0.212325
$326$	0	0
$327$	461666.	0.238758
$328$	0	0
$329$	92500.3	0.0471144
$330$	0	0
$331$	−1.80273e6	−0.904402	−0.452201	−	0.891916i	$-0.649361\pi$
$331$	−1.80273e6	−0.904402	−0.452201	+	0.891916i	$0.649361\pi$
$332$	0	0
$333$	414037.	0.204611
$334$	0	0
$335$	−1.26053e6	−0.613677
$336$	0	0
$337$	390541.	0.187323	0.0936616	−	0.995604i	$-0.470143\pi$
$337$	390541.	0.187323	0.0936616	+	0.995604i	$0.470143\pi$
$338$	0	0
$339$	3.85696e6	1.82283
$340$	0	0
$341$	−1.88190e6	−0.876417
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.30565e6	−0.590581
$346$	0	0
$347$	−3.38976e6	−1.51128	−0.755640	−	0.654988i	$-0.772674\pi$
$347$	−3.38976e6	−1.51128	−0.755640	+	0.654988i	$0.772674\pi$
$348$	0	0
$349$	312893.	0.137510	0.0687548	−	0.997634i	$-0.478097\pi$
$349$	312893.	0.137510	0.0687548	+	0.997634i	$0.478097\pi$
$350$	0	0
$351$	−1.05438e6	−0.456805
$352$	0	0
$353$	1.24260e6	0.530756	0.265378	−	0.964144i	$-0.414503\pi$
$353$	1.24260e6	0.530756	0.265378	+	0.964144i	$0.414503\pi$
$354$	0	0
$355$	1.43129e6	0.602776
$356$	0	0
$357$	1.01485e6	0.421435
$358$	0	0
$359$	−2.84568e6	−1.16533	−0.582666	−	0.812712i	$-0.697990\pi$
$359$	−2.84568e6	−1.16533	−0.582666	+	0.812712i	$0.697990\pi$
$360$	0	0
$361$	2.60419e6	1.05173
$362$	0	0
$363$	−428327.	−0.170612
$364$	0	0
$365$	2.10059e6	0.825294
$366$	0	0
$367$	384128.	0.148871	0.0744357	−	0.997226i	$-0.476284\pi$
$367$	384128.	0.148871	0.0744357	+	0.997226i	$0.476284\pi$
$368$	0	0
$369$	1.00817e6	0.385449
$370$	0	0
$371$	−574180.	−0.216577
$372$	0	0
$373$	−76006.4	−0.0282864	−0.0141432	−	0.999900i	$-0.504502\pi$
$373$	−76006.4	−0.0282864	−0.0141432	+	0.999900i	$0.504502\pi$
$374$	0	0
$375$	−314451.	−0.115471
$376$	0	0
$377$	−562703.	−0.203904
$378$	0	0
$379$	5.54214e6	1.98189	0.990944	−	0.134273i	$-0.0428698\pi$
$379$	5.54214e6	1.98189	0.990944	+	0.134273i	$0.0428698\pi$
$380$	0	0
$381$	5.63739e6	1.98960
$382$	0	0
$383$	4.85688e6	1.69185	0.845923	−	0.533305i	$-0.179050\pi$
$383$	4.85688e6	1.69185	0.845923	+	0.533305i	$0.179050\pi$
$384$	0	0
$385$	−523083.	−0.179853
$386$	0	0
$387$	1.25680e6	0.426567
$388$	0	0
$389$	3.98689e6	1.33586	0.667928	−	0.744226i	$-0.267181\pi$
$389$	3.98689e6	1.33586	0.667928	+	0.744226i	$0.267181\pi$
$390$	0	0
$391$	−2.67072e6	−0.883458
$392$	0	0
$393$	−4.94629e6	−1.61547
$394$	0	0
$395$	2.46001e6	0.793312
$396$	0	0
$397$	3.96757e6	1.26342	0.631710	−	0.775205i	$-0.282353\pi$
$397$	3.96757e6	1.26342	0.631710	+	0.775205i	$0.282353\pi$
$398$	0	0
$399$	−2.22266e6	−0.698941
$400$	0	0
$401$	4.93573e6	1.53282	0.766409	−	0.642353i	$-0.222042\pi$
$401$	4.93573e6	1.53282	0.766409	+	0.642353i	$0.222042\pi$
$402$	0	0
$403$	−2.85097e6	−0.874439
$404$	0	0
$405$	−1.80426e6	−0.546589
$406$	0	0
$407$	−1.09128e6	−0.326549
$408$	0	0
$409$	6.35318e6	1.87794	0.938972	−	0.343994i	$-0.111780\pi$
$409$	6.35318e6	1.87794	0.938972	+	0.343994i	$0.111780\pi$
$410$	0	0
$411$	270031.	0.0788513
$412$	0	0
$413$	1.63587e6	0.471927
$414$	0	0
$415$	−803939.	−0.229141
$416$	0	0
$417$	7.31275e6	2.05940
$418$	0	0
$419$	−5.86656e6	−1.63248	−0.816241	−	0.577711i	$-0.803946\pi$
$419$	−5.86656e6	−1.63248	−0.816241	+	0.577711i	$0.803946\pi$
$420$	0	0
$421$	−4.13207e6	−1.13622	−0.568109	−	0.822953i	$-0.692325\pi$
$421$	−4.13207e6	−1.13622	−0.568109	+	0.822953i	$0.692325\pi$
$422$	0	0
$423$	305834.	0.0831066
$424$	0	0
$425$	−643210.	−0.172735
$426$	0	0
$427$	−59399.2	−0.0157656
$428$	0	0
$429$	−5.55900e6	−1.45832
$430$	0	0
$431$	−3.28044e6	−0.850627	−0.425314	−	0.905046i	$-0.639836\pi$
$431$	−3.28044e6	−0.850627	−0.425314	+	0.905046i	$0.639836\pi$
$432$	0	0
$433$	7.35755e6	1.88588	0.942938	−	0.332968i	$-0.108050\pi$
$433$	7.35755e6	1.88588	0.942938	+	0.332968i	$0.108050\pi$
$434$	0	0
$435$	−437645.	−0.110892
$436$	0	0
$437$	5.84924e6	1.46520
$438$	0	0
$439$	−296949.	−0.0735395	−0.0367698	−	0.999324i	$-0.511707\pi$
$439$	−296949.	−0.0735395	−0.0367698	+	0.999324i	$0.511707\pi$
$440$	0	0
$441$	388984.	0.0952434
$442$	0	0
$443$	3.07626e6	0.744756	0.372378	−	0.928081i	$-0.378542\pi$
$443$	3.07626e6	0.744756	0.372378	+	0.928081i	$0.378542\pi$
$444$	0	0
$445$	1.35809e6	0.325109
$446$	0	0
$447$	5.23343e6	1.23885
$448$	0	0
$449$	−2.95674e6	−0.692145	−0.346073	−	0.938208i	$-0.612485\pi$
$449$	−2.95674e6	−0.692145	−0.346073	+	0.938208i	$0.612485\pi$
$450$	0	0
$451$	−2.65722e6	−0.615157
$452$	0	0
$453$	−4.42532e6	−1.01321
$454$	0	0
$455$	−792439.	−0.179448
$456$	0	0
$457$	5.68155e6	1.27255	0.636277	−	0.771461i	$-0.280474\pi$
$457$	5.68155e6	1.27255	0.636277	+	0.771461i	$0.280474\pi$
$458$	0	0
$459$	−1.67742e6	−0.371629
$460$	0	0
$461$	−5.75693e6	−1.26165	−0.630825	−	0.775925i	$-0.717283\pi$
$461$	−5.75693e6	−1.26165	−0.630825	+	0.775925i	$0.717283\pi$
$462$	0	0
$463$	−2.82533e6	−0.612514	−0.306257	−	0.951949i	$-0.599077\pi$
$463$	−2.82533e6	−0.612514	−0.306257	+	0.951949i	$0.599077\pi$
$464$	0	0
$465$	−2.21735e6	−0.475557
$466$	0	0
$467$	1.29610e6	0.275008	0.137504	−	0.990501i	$-0.456092\pi$
$467$	1.29610e6	0.275008	0.137504	+	0.990501i	$0.456092\pi$
$468$	0	0
$469$	−2.47063e6	−0.518652
$470$	0	0
$471$	−8.67068e6	−1.80095
$472$	0	0
$473$	−3.31253e6	−0.680781
$474$	0	0
$475$	1.40872e6	0.286477
$476$	0	0
$477$	−1.89841e6	−0.382028
$478$	0	0
$479$	854121.	0.170091	0.0850453	−	0.996377i	$-0.472896\pi$
$479$	854121.	0.170091	0.0850453	+	0.996377i	$0.472896\pi$
$480$	0	0
$481$	−1.65322e6	−0.325812
$482$	0	0
$483$	−2.55908e6	−0.499132
$484$	0	0
$485$	−542508.	−0.104725
$486$	0	0
$487$	2.05453e6	0.392545	0.196273	−	0.980549i	$-0.437116\pi$
$487$	2.05453e6	0.392545	0.196273	+	0.980549i	$0.437116\pi$
$488$	0	0
$489$	−9.29999e6	−1.75877
$490$	0	0
$491$	−7.21316e6	−1.35027	−0.675137	−	0.737692i	$-0.735916\pi$
$491$	−7.21316e6	−1.35027	−0.675137	+	0.737692i	$0.735916\pi$
$492$	0	0
$493$	−895205.	−0.165884
$494$	0	0
$495$	−1.72947e6	−0.317249
$496$	0	0
$497$	2.80533e6	0.509439
$498$	0	0
$499$	4.85347e6	0.872571	0.436286	−	0.899808i	$-0.356294\pi$
$499$	4.85347e6	0.872571	0.436286	+	0.899808i	$0.356294\pi$
$500$	0	0
$501$	−9.92292e6	−1.76622
$502$	0	0
$503$	−2.60704e6	−0.459439	−0.229720	−	0.973257i	$-0.573781\pi$
$503$	−2.60704e6	−0.459439	−0.229720	+	0.973257i	$0.573781\pi$
$504$	0	0
$505$	3.18400e6	0.555577
$506$	0	0
$507$	−949346.	−0.164023
$508$	0	0
$509$	−4.56269e6	−0.780596	−0.390298	−	0.920689i	$-0.627628\pi$
$509$	−4.56269e6	−0.780596	−0.390298	+	0.920689i	$0.627628\pi$
$510$	0	0
$511$	4.11715e6	0.697501
$512$	0	0
$513$	3.67378e6	0.616339
$514$	0	0
$515$	1.11736e6	0.185642
$516$	0	0
$517$	−806086.	−0.132634
$518$	0	0
$519$	1.23611e7	2.01437
$520$	0	0
$521$	−3.14831e6	−0.508140	−0.254070	−	0.967186i	$-0.581769\pi$
$521$	−3.14831e6	−0.508140	−0.254070	+	0.967186i	$0.581769\pi$
$522$	0	0
$523$	7.28996e6	1.16539	0.582694	−	0.812692i	$-0.301999\pi$
$523$	7.28996e6	1.16539	0.582694	+	0.812692i	$0.301999\pi$
$524$	0	0
$525$	−616323.	−0.0975911
$526$	0	0
$527$	−4.53560e6	−0.711391
$528$	0	0
$529$	298229.	0.0463351
$530$	0	0
$531$	5.40870e6	0.832447
$532$	0	0
$533$	−4.02553e6	−0.613769
$534$	0	0
$535$	−3.13399e6	−0.473383
$536$	0	0
$537$	1.85815e6	0.278064
$538$	0	0
$539$	−1.02524e6	−0.152004
$540$	0	0
$541$	−4131.68	−0.000606923 0	−0.000303462	−	1.00000i	$-0.500097\pi$
$541$	−4131.68	−0.000606923 0	−0.000303462		1.00000i	$0.500097\pi$
$542$	0	0
$543$	−1.39836e7	−2.03526
$544$	0	0
$545$	−573503.	−0.0827074
$546$	0	0
$547$	1.13419e7	1.62075	0.810376	−	0.585910i	$-0.199263\pi$
$547$	1.13419e7	1.62075	0.810376	+	0.585910i	$0.199263\pi$
$548$	0	0
$549$	−196392.	−0.0278095
$550$	0	0
$551$	1.96062e6	0.275115
$552$	0	0
$553$	4.82162e6	0.670471
$554$	0	0
$555$	−1.28580e6	−0.177190
$556$	0	0
$557$	−3.06648e6	−0.418796	−0.209398	−	0.977831i	$-0.567150\pi$
$557$	−3.06648e6	−0.418796	−0.209398	+	0.977831i	$0.567150\pi$
$558$	0	0
$559$	−5.01829e6	−0.679244
$560$	0	0
$561$	−8.84382e6	−1.18640
$562$	0	0
$563$	−8.73658e6	−1.16164	−0.580819	−	0.814033i	$-0.697268\pi$
$563$	−8.73658e6	−1.16164	−0.580819	+	0.814033i	$0.697268\pi$
$564$	0	0
$565$	−4.79130e6	−0.631440
$566$	0	0
$567$	−3.53634e6	−0.461952
$568$	0	0
$569$	8.05550e6	1.04307	0.521533	−	0.853231i	$-0.325360\pi$
$569$	8.05550e6	1.04307	0.521533	+	0.853231i	$0.325360\pi$
$570$	0	0
$571$	2.25192e6	0.289043	0.144522	−	0.989502i	$-0.453836\pi$
$571$	2.25192e6	0.289043	0.144522	+	0.989502i	$0.453836\pi$
$572$	0	0
$573$	9.24161e6	1.17587
$574$	0	0
$575$	1.62194e6	0.204581
$576$	0	0
$577$	−1.56246e6	−0.195375	−0.0976874	−	0.995217i	$-0.531145\pi$
$577$	−1.56246e6	−0.195375	−0.0976874	+	0.995217i	$0.531145\pi$
$578$	0	0
$579$	−3.67867e6	−0.456031
$580$	0	0
$581$	−1.57572e6	−0.193659
$582$	0	0
$583$	5.00364e6	0.609698
$584$	0	0
$585$	−2.62005e6	−0.316533
$586$	0	0
$587$	1.31305e7	1.57284	0.786421	−	0.617690i	$-0.211931\pi$
$587$	1.31305e7	1.57284	0.786421	+	0.617690i	$0.211931\pi$
$588$	0	0
$589$	9.93360e6	1.17983
$590$	0	0
$591$	−6.95299e6	−0.818848
$592$	0	0
$593$	8.96601e6	1.04704	0.523519	−	0.852014i	$-0.324619\pi$
$593$	8.96601e6	1.04704	0.523519	+	0.852014i	$0.324619\pi$
$594$	0	0
$595$	−1.26069e6	−0.145988
$596$	0	0
$597$	−9.15927e6	−1.05178
$598$	0	0
$599$	1.29511e7	1.47483	0.737414	−	0.675441i	$-0.236047\pi$
$599$	1.29511e7	1.47483	0.737414	+	0.675441i	$0.236047\pi$
$600$	0	0
$601$	1.96489e6	0.221897	0.110949	−	0.993826i	$-0.464611\pi$
$601$	1.96489e6	0.221897	0.110949	+	0.993826i	$0.464611\pi$
$602$	0	0
$603$	−8.16867e6	−0.914867
$604$	0	0
$605$	532088.	0.0591010
$606$	0	0
$607$	−1.13138e7	−1.24634	−0.623172	−	0.782085i	$-0.714156\pi$
$607$	−1.13138e7	−1.24634	−0.623172	+	0.782085i	$0.714156\pi$
$608$	0	0
$609$	−857784.	−0.0937205
$610$	0	0
$611$	−1.22117e6	−0.132335
$612$	0	0
$613$	7.29872e6	0.784505	0.392252	−	0.919858i	$-0.371696\pi$
$613$	7.29872e6	0.784505	0.392252	+	0.919858i	$0.371696\pi$
$614$	0	0
$615$	−3.13087e6	−0.333793
$616$	0	0
$617$	19580.9	0.00207071	0.00103536	−	0.999999i	$-0.499670\pi$
$617$	19580.9	0.00207071	0.00103536	+	0.999999i	$0.499670\pi$
$618$	0	0
$619$	6.49314e6	0.681127	0.340563	−	0.940222i	$-0.389382\pi$
$619$	6.49314e6	0.681127	0.340563	+	0.940222i	$0.389382\pi$
$620$	0	0
$621$	4.22984e6	0.440144
$622$	0	0
$623$	2.66186e6	0.274767
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.93692e7	1.96762
$628$	0	0
$629$	−2.63011e6	−0.265061
$630$	0	0
$631$	4.09998e6	0.409929	0.204964	−	0.978769i	$-0.434292\pi$
$631$	4.09998e6	0.409929	0.204964	+	0.978769i	$0.434292\pi$
$632$	0	0
$633$	−7.32550e6	−0.726655
$634$	0	0
$635$	−7.00302e6	−0.689209
$636$	0	0
$637$	−1.55318e6	−0.151661
$638$	0	0
$639$	9.27527e6	0.898616
$640$	0	0
$641$	−8.51977e6	−0.818998	−0.409499	−	0.912311i	$-0.634296\pi$
$641$	−8.51977e6	−0.818998	−0.409499	+	0.912311i	$0.634296\pi$
$642$	0	0
$643$	−4.17027e6	−0.397774	−0.198887	−	0.980022i	$-0.563733\pi$
$643$	−4.17027e6	−0.397774	−0.198887	+	0.980022i	$0.563733\pi$
$644$	0	0
$645$	−3.90299e6	−0.369401
$646$	0	0
$647$	−2.00875e7	−1.88654	−0.943269	−	0.332029i	$-0.892267\pi$
$647$	−2.00875e7	−1.88654	−0.943269	+	0.332029i	$0.892267\pi$
$648$	0	0
$649$	−1.42557e7	−1.32855
$650$	0	0
$651$	−4.34601e6	−0.401919
$652$	0	0
$653$	1.88656e7	1.73136	0.865682	−	0.500595i	$-0.166885\pi$
$653$	1.88656e7	1.73136	0.865682	+	0.500595i	$0.166885\pi$
$654$	0	0
$655$	6.14450e6	0.559608
$656$	0	0
$657$	1.36126e7	1.23034
$658$	0	0
$659$	1.99030e7	1.78527	0.892636	−	0.450777i	$-0.148853\pi$
$659$	1.99030e7	1.78527	0.892636	+	0.450777i	$0.148853\pi$
$660$	0	0
$661$	−2.07323e7	−1.84563	−0.922815	−	0.385244i	$-0.874117\pi$
$661$	−2.07323e7	−1.84563	−0.922815	+	0.385244i	$0.874117\pi$
$662$	0	0
$663$	−1.33979e7	−1.18373
$664$	0	0
$665$	2.76109e6	0.242118
$666$	0	0
$667$	2.25738e6	0.196467
$668$	0	0
$669$	−5.54805e6	−0.479264
$670$	0	0
$671$	517629.	0.0443826
$672$	0	0
$673$	1.43855e7	1.22430	0.612151	−	0.790741i	$-0.290304\pi$
$673$	1.43855e7	1.22430	0.612151	+	0.790741i	$0.290304\pi$
$674$	0	0
$675$	1.01871e6	0.0860576
$676$	0	0
$677$	−7.15539e6	−0.600014	−0.300007	−	0.953937i	$-0.596989\pi$
$677$	−7.15539e6	−0.600014	−0.300007	+	0.953937i	$0.596989\pi$
$678$	0	0
$679$	−1.06332e6	−0.0885091
$680$	0	0
$681$	2.30572e7	1.90519
$682$	0	0
$683$	9.42223e6	0.772862	0.386431	−	0.922318i	$-0.373708\pi$
$683$	9.42223e6	0.772862	0.386431	+	0.922318i	$0.373708\pi$
$684$	0	0
$685$	−335445.	−0.0273146
$686$	0	0
$687$	1.82934e7	1.47878
$688$	0	0
$689$	7.58022e6	0.608322
$690$	0	0
$691$	−7.56428e6	−0.602660	−0.301330	−	0.953520i	$-0.597431\pi$
$691$	−7.56428e6	−0.602660	−0.301330	+	0.953520i	$0.597431\pi$
$692$	0	0
$693$	−3.38977e6	−0.268125
$694$	0	0
$695$	−9.08424e6	−0.713389
$696$	0	0
$697$	−6.40422e6	−0.499326
$698$	0	0
$699$	1.58716e7	1.22865
$700$	0	0
$701$	7.75589e6	0.596124	0.298062	−	0.954547i	$-0.403660\pi$
$701$	7.75589e6	0.596124	0.298062	+	0.954547i	$0.403660\pi$
$702$	0	0
$703$	5.76029e6	0.439599
$704$	0	0
$705$	−949772.	−0.0719692
$706$	0	0
$707$	6.24064e6	0.469548
$708$	0	0
$709$	−1.08957e7	−0.814030	−0.407015	−	0.913422i	$-0.633430\pi$
$709$	−1.08957e7	−0.814030	−0.407015	+	0.913422i	$0.633430\pi$
$710$	0	0
$711$	1.59418e7	1.18267
$712$	0	0
$713$	1.14371e7	0.842546
$714$	0	0
$715$	6.90565e6	0.505172
$716$	0	0
$717$	−3.02630e7	−2.19843
$718$	0	0
$719$	1.83953e7	1.32704	0.663519	−	0.748159i	$-0.269062\pi$
$719$	1.83953e7	1.32704	0.663519	+	0.748159i	$0.269062\pi$
$720$	0	0
$721$	2.19003e6	0.156896
$722$	0	0
$723$	−2.31174e6	−0.164473
$724$	0	0
$725$	543663.	0.0384136
$726$	0	0
$727$	−3.74257e6	−0.262623	−0.131312	−	0.991341i	$-0.541919\pi$
$727$	−3.74257e6	−0.262623	−0.131312	+	0.991341i	$0.541919\pi$
$728$	0	0
$729$	−3.72133e6	−0.259346
$730$	0	0
$731$	−7.98359e6	−0.552593
$732$	0	0
$733$	1.63705e7	1.12539	0.562695	−	0.826665i	$-0.309765\pi$
$733$	1.63705e7	1.12539	0.562695	+	0.826665i	$0.309765\pi$
$734$	0	0
$735$	−1.20799e6	−0.0824795
$736$	0	0
$737$	2.15301e7	1.46008
$738$	0	0
$739$	−2.72667e7	−1.83663	−0.918315	−	0.395851i	$-0.870450\pi$
$739$	−2.72667e7	−1.83663	−0.918315	+	0.395851i	$0.870450\pi$
$740$	0	0
$741$	2.93431e7	1.96318
$742$	0	0
$743$	−2.19134e7	−1.45625	−0.728127	−	0.685443i	$-0.759609\pi$
$743$	−2.19134e7	−1.45625	−0.728127	+	0.685443i	$0.759609\pi$
$744$	0	0
$745$	−6.50121e6	−0.429145
$746$	0	0
$747$	−5.20981e6	−0.341602
$748$	0	0
$749$	−6.14262e6	−0.400082
$750$	0	0
$751$	1.19293e7	0.771820	0.385910	−	0.922536i	$-0.373887\pi$
$751$	1.19293e7	0.771820	0.385910	+	0.922536i	$0.373887\pi$
$752$	0	0
$753$	1.79854e7	1.15593
$754$	0	0
$755$	5.49734e6	0.350982
$756$	0	0
$757$	−2.46216e7	−1.56162	−0.780811	−	0.624767i	$-0.785194\pi$
$757$	−2.46216e7	−1.56162	−0.780811	+	0.624767i	$0.785194\pi$
$758$	0	0
$759$	2.23009e7	1.40513
$760$	0	0
$761$	−1.11065e6	−0.0695208	−0.0347604	−	0.999396i	$-0.511067\pi$
$761$	−1.11065e6	−0.0695208	−0.0347604	+	0.999396i	$0.511067\pi$
$762$	0	0
$763$	−1.12407e6	−0.0699005
$764$	0	0
$765$	−4.16824e6	−0.257513
$766$	0	0
$767$	−2.15965e7	−1.32555
$768$	0	0
$769$	−516268.	−0.0314818	−0.0157409	−	0.999876i	$-0.505011\pi$
$769$	−516268.	−0.0314818	−0.0157409	+	0.999876i	$0.505011\pi$
$770$	0	0
$771$	1.68886e7	1.02319
$772$	0	0
$773$	−5.63544e6	−0.339218	−0.169609	−	0.985511i	$-0.554251\pi$
$773$	−5.63544e6	−0.339218	−0.169609	+	0.985511i	$0.554251\pi$
$774$	0	0
$775$	2.75450e6	0.164736
$776$	0	0
$777$	−2.52016e6	−0.149753
$778$	0	0
$779$	1.40261e7	0.828121
$780$	0	0
$781$	−2.44468e7	−1.43415
$782$	0	0
$783$	1.41781e6	0.0826445
$784$	0	0
$785$	1.07711e7	0.623859
$786$	0	0
$787$	−1.77616e6	−0.102222	−0.0511110	−	0.998693i	$-0.516276\pi$
$787$	−1.77616e6	−0.102222	−0.0511110	+	0.998693i	$0.516276\pi$
$788$	0	0
$789$	−2.41221e7	−1.37950
$790$	0	0
$791$	−9.39094e6	−0.533664
$792$	0	0
$793$	784177.	0.0442824
$794$	0	0
$795$	5.89555e6	0.330831
$796$	0	0
$797$	4.79357e6	0.267309	0.133654	−	0.991028i	$-0.457329\pi$
$797$	4.79357e6	0.267309	0.133654	+	0.991028i	$0.457329\pi$
$798$	0	0
$799$	−1.94276e6	−0.107660
$800$	0	0
$801$	8.80093e6	0.484671
$802$	0	0
$803$	−3.58786e7	−1.96357
$804$	0	0
$805$	3.17900e6	0.172903
$806$	0	0
$807$	−6.70733e6	−0.362548
$808$	0	0
$809$	−2.32430e7	−1.24860	−0.624298	−	0.781187i	$-0.714615\pi$
$809$	−2.32430e7	−1.24860	−0.624298	+	0.781187i	$0.714615\pi$
$810$	0	0
$811$	−1.55958e7	−0.832637	−0.416318	−	0.909219i	$-0.636680\pi$
$811$	−1.55958e7	−0.832637	−0.416318	+	0.909219i	$0.636680\pi$
$812$	0	0
$813$	−4.00971e7	−2.12758
$814$	0	0
$815$	1.15529e7	0.609251
$816$	0	0
$817$	1.74852e7	0.916463
$818$	0	0
$819$	−5.13529e6	−0.267520
$820$	0	0
$821$	−7.07206e6	−0.366174	−0.183087	−	0.983097i	$-0.558609\pi$
$821$	−7.07206e6	−0.366174	−0.183087	+	0.983097i	$0.558609\pi$
$822$	0	0
$823$	−3.51052e7	−1.80664	−0.903321	−	0.428965i	$-0.858878\pi$
$823$	−3.51052e7	−1.80664	−0.903321	+	0.428965i	$0.858878\pi$
$824$	0	0
$825$	5.37090e6	0.274734
$826$	0	0
$827$	1.26303e7	0.642168	0.321084	−	0.947051i	$-0.395953\pi$
$827$	1.26303e7	0.642168	0.321084	+	0.947051i	$0.395953\pi$
$828$	0	0
$829$	−2.71570e7	−1.37245	−0.686223	−	0.727391i	$-0.740733\pi$
$829$	−2.71570e7	−1.37245	−0.686223	+	0.727391i	$0.740733\pi$
$830$	0	0
$831$	4.05262e6	0.203579
$832$	0	0
$833$	−2.47096e6	−0.123382
$834$	0	0
$835$	1.23267e7	0.611830
$836$	0	0
$837$	7.18342e6	0.354419
$838$	0	0
$839$	−1.26528e7	−0.620555	−0.310278	−	0.950646i	$-0.600422\pi$
$839$	−1.26528e7	−0.620555	−0.310278	+	0.950646i	$0.600422\pi$
$840$	0	0
$841$	−1.97545e7	−0.963110
$842$	0	0
$843$	9.19267e6	0.445526
$844$	0	0
$845$	1.17932e6	0.0568186
$846$	0	0
$847$	1.04289e6	0.0499495
$848$	0	0
$849$	−1.18876e7	−0.566011
$850$	0	0
$851$	6.63216e6	0.313929
$852$	0	0
$853$	−1.52399e7	−0.717151	−0.358576	−	0.933501i	$-0.616737\pi$
$853$	−1.52399e7	−0.717151	−0.358576	+	0.933501i	$0.616737\pi$
$854$	0	0
$855$	9.12901e6	0.427079
$856$	0	0
$857$	−3.63637e7	−1.69128	−0.845641	−	0.533753i	$-0.820781\pi$
$857$	−3.63637e7	−1.69128	−0.845641	+	0.533753i	$0.820781\pi$
$858$	0	0
$859$	−2.51710e7	−1.16391	−0.581953	−	0.813222i	$-0.697711\pi$
$859$	−2.51710e7	−1.16391	−0.581953	+	0.813222i	$0.697711\pi$
$860$	0	0
$861$	−6.13651e6	−0.282107
$862$	0	0
$863$	2.21176e7	1.01091	0.505454	−	0.862853i	$-0.331325\pi$
$863$	2.21176e7	1.01091	0.505454	+	0.862853i	$0.331325\pi$
$864$	0	0
$865$	−1.53556e7	−0.697792
$866$	0	0
$867$	7.25974e6	0.327999
$868$	0	0
$869$	−4.20176e7	−1.88748
$870$	0	0
$871$	3.26169e7	1.45679
$872$	0	0
$873$	−3.51565e6	−0.156124
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	4.77311e6	0.209557	0.104779	−	0.994496i	$-0.466587\pi$
$877$	4.77311e6	0.209557	0.104779	+	0.994496i	$0.466587\pi$
$878$	0	0
$879$	8.37923e6	0.365790
$880$	0	0
$881$	−1.79025e7	−0.777095	−0.388548	−	0.921429i	$-0.627023\pi$
$881$	−1.79025e7	−0.777095	−0.388548	+	0.921429i	$0.627023\pi$
$882$	0	0
$883$	1.23636e7	0.533634	0.266817	−	0.963747i	$-0.414028\pi$
$883$	1.23636e7	0.533634	0.266817	+	0.963747i	$0.414028\pi$
$884$	0	0
$885$	−1.67968e7	−0.720888
$886$	0	0
$887$	−2.33951e7	−0.998427	−0.499214	−	0.866479i	$-0.666378\pi$
$887$	−2.33951e7	−0.998427	−0.499214	+	0.866479i	$0.666378\pi$
$888$	0	0
$889$	−1.37259e7	−0.582488
$890$	0	0
$891$	3.08172e7	1.30046
$892$	0	0
$893$	4.25492e6	0.178551
$894$	0	0
$895$	−2.30828e6	−0.0963231
$896$	0	0
$897$	3.37845e7	1.40196
$898$	0	0
$899$	3.83364e6	0.158202
$900$	0	0
$901$	1.20594e7	0.494895
$902$	0	0
$903$	−7.64987e6	−0.312201
$904$	0	0
$905$	1.73711e7	0.705026
$906$	0	0
$907$	1.76517e7	0.712474	0.356237	−	0.934396i	$-0.384060\pi$
$907$	1.76517e7	0.712474	0.356237	+	0.934396i	$0.384060\pi$
$908$	0	0
$909$	2.06335e7	0.828252
$910$	0	0
$911$	2.50785e7	1.00117	0.500583	−	0.865688i	$-0.333119\pi$
$911$	2.50785e7	1.00117	0.500583	+	0.865688i	$0.333119\pi$
$912$	0	0
$913$	1.37315e7	0.545181
$914$	0	0
$915$	609897.	0.0240826
$916$	0	0
$917$	1.20432e7	0.472955
$918$	0	0
$919$	4.36173e7	1.70361	0.851805	−	0.523859i	$-0.175508\pi$
$919$	4.36173e7	1.70361	0.851805	+	0.523859i	$0.175508\pi$
$920$	0	0
$921$	2.52718e7	0.981718
$922$	0	0
$923$	−3.70354e7	−1.43091
$924$	0	0
$925$	1.59728e6	0.0613799
$926$	0	0
$927$	7.24092e6	0.276754
$928$	0	0
$929$	4.60105e7	1.74911	0.874556	−	0.484924i	$-0.161153\pi$
$929$	4.60105e7	1.74911	0.874556	+	0.484924i	$0.161153\pi$
$930$	0	0
$931$	5.41173e6	0.204627
$932$	0	0
$933$	2.33992e7	0.880028
$934$	0	0
$935$	1.09862e7	0.410978
$936$	0	0
$937$	−4.54956e7	−1.69286	−0.846428	−	0.532503i	$-0.821252\pi$
$937$	−4.54956e7	−1.69286	−0.846428	+	0.532503i	$0.821252\pi$
$938$	0	0
$939$	−6.94822e7	−2.57164
$940$	0	0
$941$	5.13895e7	1.89191	0.945955	−	0.324299i	$-0.105128\pi$
$941$	5.13895e7	1.89191	0.945955	+	0.324299i	$0.105128\pi$
$942$	0	0
$943$	1.61491e7	0.591383
$944$	0	0
$945$	1.99666e6	0.0727320
$946$	0	0
$947$	−8.35685e6	−0.302808	−0.151404	−	0.988472i	$-0.548379\pi$
$947$	−8.35685e6	−0.302808	−0.151404	+	0.988472i	$0.548379\pi$
$948$	0	0
$949$	−5.43539e7	−1.95914
$950$	0	0
$951$	2.13393e7	0.765118
$952$	0	0
$953$	3.47459e7	1.23929	0.619643	−	0.784884i	$-0.287277\pi$
$953$	3.47459e7	1.23929	0.619643	+	0.784884i	$0.287277\pi$
$954$	0	0
$955$	−1.14804e7	−0.407330
$956$	0	0
$957$	7.47509e6	0.263837
$958$	0	0
$959$	−657472.	−0.0230850
$960$	0	0
$961$	−9.20579e6	−0.321553
$962$	0	0
$963$	−2.03094e7	−0.705717
$964$	0	0
$965$	4.56981e6	0.157972
$966$	0	0
$967$	1.15760e7	0.398101	0.199051	−	0.979989i	$-0.436214\pi$
$967$	1.15760e7	0.398101	0.199051	+	0.979989i	$0.436214\pi$
$968$	0	0
$969$	4.66820e7	1.59713
$970$	0	0
$971$	−1.65286e7	−0.562583	−0.281292	−	0.959622i	$-0.590763\pi$
$971$	−1.65286e7	−0.562583	−0.281292	+	0.959622i	$0.590763\pi$
$972$	0	0
$973$	−1.78051e7	−0.602924
$974$	0	0
$975$	8.13659e6	0.274114
$976$	0	0
$977$	−4.85257e6	−0.162643	−0.0813215	−	0.996688i	$-0.525914\pi$
$977$	−4.85257e6	−0.162643	−0.0813215	+	0.996688i	$0.525914\pi$
$978$	0	0
$979$	−2.31966e7	−0.773512
$980$	0	0
$981$	−3.71650e6	−0.123300
$982$	0	0
$983$	8.52706e6	0.281459	0.140730	−	0.990048i	$-0.455055\pi$
$983$	8.52706e6	0.281459	0.140730	+	0.990048i	$0.455055\pi$
$984$	0	0
$985$	8.63733e6	0.283654
$986$	0	0
$987$	−1.86155e6	−0.0608251
$988$	0	0
$989$	2.01317e7	0.654470
$990$	0	0
$991$	6271.13	0.000202844 0	0.000101422	−	1.00000i	$-0.499968\pi$
$991$	6271.13	0.000202844 0	0.000101422		1.00000i	$0.499968\pi$
$992$	0	0
$993$	3.62797e7	1.16759
$994$	0	0
$995$	1.13781e7	0.364343
$996$	0	0
$997$	−3.44729e7	−1.09835	−0.549174	−	0.835708i	$-0.685058\pi$
$997$	−3.44729e7	−1.09835	−0.549174	+	0.835708i	$0.685058\pi$
$998$	0	0
$999$	4.16552e6	0.132055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.d.1.1	✓	3
4.3	odd	2		560.6.a.t.1.3		3
5.2	odd	4		700.6.e.g.449.5		6
5.3	odd	4		700.6.e.g.449.2		6
5.4	even	2		700.6.a.i.1.3		3
7.6	odd	2		980.6.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.d.1.1	✓	3	1.1	even	1	trivial
560.6.a.t.1.3		3	4.3	odd	2
700.6.a.i.1.3		3	5.4	even	2
700.6.e.g.449.2		6	5.3	odd	4
700.6.e.g.449.5		6	5.2	odd	4
980.6.a.h.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)

\(f(q)\)	\(=\)	\(q-20.1248 q^{3} +25.0000 q^{5} +49.0000 q^{7} +162.009 q^{9} +O(q^{10})\)	\(q-20.1248 q^{3} +25.0000 q^{5} +49.0000 q^{7} +162.009 q^{9} -427.006 q^{11} -646.889 q^{13} -503.121 q^{15} -1029.14 q^{17} +2253.95 q^{19} -986.117 q^{21} +2595.11 q^{23} +625.000 q^{25} +1629.93 q^{27} +869.860 q^{29} +4407.19 q^{31} +8593.44 q^{33} +1225.00 q^{35} +2555.64 q^{37} +13018.5 q^{39} +6222.90 q^{41} +7757.57 q^{43} +4050.23 q^{45} +1887.76 q^{47} +2401.00 q^{49} +20711.2 q^{51} -11718.0 q^{53} -10675.2 q^{55} -45360.4 q^{57} +33385.2 q^{59} -1212.23 q^{61} +7938.44 q^{63} -16172.2 q^{65} -50421.1 q^{67} -52226.1 q^{69} +57251.5 q^{71} +84023.5 q^{73} -12578.0 q^{75} -20923.3 q^{77} +98400.4 q^{79} -72170.3 q^{81} -32157.5 q^{83} -25728.4 q^{85} -17505.8 q^{87} +54323.7 q^{89} -31697.6 q^{91} -88694.1 q^{93} +56348.8 q^{95} -21700.3 q^{97} -69178.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9}+O(q^{10}) \) 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9	\( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33} + 3675 q^{35} + 11378 q^{37} + 13702 q^{39} + 11450 q^{41} + 18544 q^{43} + 7025 q^{45} + 21754 q^{47} + 7203 q^{49} + 31762 q^{51} + 7494 q^{53} - 350 q^{55} + 7332 q^{57} + 12388 q^{59} + 27182 q^{61} + 13769 q^{63} - 100 q^{65} - 7676 q^{67} - 62140 q^{69} + 81992 q^{71} + 230 q^{73} + 3750 q^{75} - 686 q^{77} - 15926 q^{79} - 32101 q^{81} - 86100 q^{83} + 1100 q^{85} - 85098 q^{87} - 95710 q^{89} - 196 q^{91} - 229472 q^{93} + 58200 q^{95} - 176188 q^{97} - 114368 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * q^99

Introduction

L-functions

Modular forms

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Database

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